diff --git a/lean/OAI/Probability/SLE/Mass/RandomRadonLimit.lean b/lean/OAI/Probability/SLE/Mass/RandomRadonLimit.lean
index b14dbbb531753747280b524da71a6baf8c452079..db21f73744c8e95155c10a389d19434128b6153c 100644
--- a/lean/OAI/Probability/SLE/Mass/RandomRadonLimit.lean
+++ b/lean/OAI/Probability/SLE/Mass/RandomRadonLimit.lean
@@ -1,116 +1,116 @@
-import OAI.Probability.SLE.Mass.RadonTestMeasurability
-
-namespace OAI
-
-/-! Measurable selection of the pathwise Radon limit. Only a measurable
-full-measure set is retained; the limit is set to zero on its complement. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology CompactlySupported
-
-namespace SLEExactGauge
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-/-- An almost-sure regular vague limit of random positive measures has a
-measurable regular version. The approximants need only have measurable
-compact-test integrals. -/
-theorem exists_measurable_random_radon_limit
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ : ℕ → Ω → Measure X)
-    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω))
-    (hex : ∀ᵐ ω ∂P, ∃ ν : Measure X, ν.Regular ∧
-      ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν))) :
-    ∃ ν : Ω → Measure X, Measurable ν ∧
-      (∀ ω, (ν ω).Regular ∧ IsFiniteMeasureOnCompacts (ν ω)) ∧
-      ∀ᵐ ω ∂P, ∀ f : C_c(X,ℝ),
-        Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν ω)) := by
-  let good (ω : Ω) : Prop := ∃ ν : Measure X, ν.Regular ∧
-    ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν))
-  have hnull : P {ω | ¬good ω}=0 := ae_iff.mp hex
-  obtain ⟨N,hN,hNm,hN0⟩ := exists_measurable_superset_of_null hnull
-  have hgood (ω : Ω) (hω : ω ∉ N) : good ω := by
-    by_contra h
-    exact hω (hN h)
-  have hall (ω : Ω) : ∃ ν : Measure X, ν.Regular ∧
-      ∀ f : C_c(X,ℝ), Tendsto (fun n => if ω ∈ N then (0:ℝ) else ∫ x, f x ∂μ n ω)
-        atTop (𝓝 (∫ x, f x ∂ν)) := by
-    by_cases hω : ω ∈ N
-    · refine ⟨0,inferInstance,fun f => ?_⟩
-      simpa only [ite_eq_left hω,integral_zero_measure] using
-        (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
-    · obtain ⟨ν,hν,hlim⟩ := hgood ω hω
-      exact ⟨ν,hν,fun f => by simpa only [ite_eq_right hω] using hlim f⟩
-  choose ν hν hlim using hall
-  have hνm : Measurable ν := by
-    apply measurable_radon_of_tests ν hν
-    intro f
-    apply measurable_of_tendsto_metrizable
-      (f := fun n ω => if ω ∈ N then (0:ℝ) else ∫ x, f x ∂μ n ω)
-    · intro n
-      exact measurable_const.ite hNm (hmeas n f)
-    · exact tendsto_pi_nhds.mpr (fun ω => hlim ω f)
-  refine ⟨ν,hνm,fun ω => ?_,?_⟩
-  · let : (ν ω).Regular := hν ω
-    exact ⟨hν ω,inferInstance⟩
-  · filter_upwards [compl_mem_ae_iff.mpr hN0] with ω hω
-    intro f
-    simpa only [ite_eq_right hω] using hlim ω f
-
-/-- Countably many times retain one common event of convergence, and each
-selected time slice is a measurable locally finite regular random measure. -/
-theorem exists_measurable_countable_radon_limits
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ : ℕ → ℕ → Ω → Measure X)
-    (hmeas : ∀ n j (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n j ω))
-    (hex : ∀ᵐ ω ∂P, ∃ ν : ℕ → Measure X, (∀ j, (ν j).Regular) ∧
-      ∀ j (f : C_c(X,ℝ)), Tendsto (fun n => ∫ x, f x ∂μ n j ω) atTop (𝓝 (∫ x, f x ∂ν j))) :
-    ∃ ν : ℕ → Ω → Measure X, (∀ j, Measurable (ν j)) ∧
-      (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧
-      ∀ᵐ ω ∂P, ∀ j (f : C_c(X,ℝ)),
-        Tendsto (fun n => ∫ x, f x ∂μ n j ω) atTop (𝓝 (∫ x, f x ∂ν j ω)) := by
-  have hj (j : ℕ) : ∀ᵐ ω ∂P, ∃ ν : Measure X, ν.Regular ∧
-      ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n j ω) atTop (𝓝 (∫ x, f x ∂ν)) := by
-    filter_upwards [hex] with ω hω
-    obtain ⟨ν,hν,hlim⟩ := hω
-    exact ⟨ν j,hν j,hlim j⟩
-  choose ν hνm hνr hlim using fun j =>
-    exists_measurable_random_radon_limit P (fun n => μ n j) (fun n => hmeas n j) (hj j)
-  exact ⟨ν,hνm,hνr,ae_all_iff.mpr hlim⟩
-
-omit [SecondCountableTopology X] in
-/-- Regular vague limits of the same sequence are unique. -/
-theorem radon_test_limit_unique (μ : ℕ → Measure X) (ν η : Measure X)
-    [ν.Regular] [η.Regular]
-    (hν : ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)))
-    (hη : ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂η))) :
-    ν=η :=
-  Measure.ext_of_integral_eq_on_compactlySupported fun f => tendsto_nhds_unique (hν f) (hη f)
-
-/-- Any pathwise version of the same limit is almost everywhere
-measurable, useful when the approximants are measurable in a smaller
-stopped or deterministic-time sigma algebra. -/
-theorem aemeasurable_radon_of_test_limits_ae
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ : ℕ → Ω → Measure X)
-    (ν : Ω → Measure X)
-    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω))
-    (hν : ∀ᵐ ω ∂P, (ν ω).Regular)
-    (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(X,ℝ),
-      Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν ω))) :
-    AEMeasurable ν P := by
-  have hex : ∀ᵐ ω ∂P, ∃ η : Measure X, η.Regular ∧
-      ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂η)) := by
-    filter_upwards [hν,hlim] with ω hr hl
-    exact ⟨ν ω,hr,hl⟩
-  obtain ⟨η,hηm,hηr,hηlim⟩ := exists_measurable_random_radon_limit P μ hmeas hex
-  apply hηm.aemeasurable.congr
-  filter_upwards [hν,hlim,hηlim] with ω hr hl hηl
-  let : (ν ω).Regular := hr
-  let : (η ω).Regular := (hηr ω).1
-  exact (radon_test_limit_unique (fun n => μ n ω) (ν ω) (η ω) hl hηl).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RadonTestMeasurability
+
+namespace OAI
+
+/-! Measurable selection of the pathwise Radon limit. Only a measurable
+full-measure set is retained; the limit is set to zero on its complement. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology CompactlySupported
+
+namespace SLEExactGauge
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+/-- An almost-sure regular vague limit of random positive measures has a
+measurable regular version. The approximants need only have measurable
+compact-test integrals. -/
+theorem exists_measurable_random_radon_limit
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ : ℕ → Ω → Measure X)
+    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω))
+    (hex : ∀ᵐ ω ∂P, ∃ ν : Measure X, ν.Regular ∧
+      ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν))) :
+    ∃ ν : Ω → Measure X, Measurable ν ∧
+      (∀ ω, (ν ω).Regular ∧ IsFiniteMeasureOnCompacts (ν ω)) ∧
+      ∀ᵐ ω ∂P, ∀ f : C_c(X,ℝ),
+        Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν ω)) := by
+  let good (ω : Ω) : Prop := ∃ ν : Measure X, ν.Regular ∧
+    ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν))
+  have hnull : P {ω | ¬good ω}=0 := ae_iff.mp hex
+  obtain ⟨N,hN,hNm,hN0⟩ := exists_measurable_superset_of_null hnull
+  have hgood (ω : Ω) (hω : ω ∉ N) : good ω := by
+    by_contra h
+    exact hω (hN h)
+  have hall (ω : Ω) : ∃ ν : Measure X, ν.Regular ∧
+      ∀ f : C_c(X,ℝ), Tendsto (fun n => if ω ∈ N then (0:ℝ) else ∫ x, f x ∂μ n ω)
+        atTop (𝓝 (∫ x, f x ∂ν)) := by
+    by_cases hω : ω ∈ N
+    · refine ⟨0,inferInstance,fun f => ?_⟩
+      simpa only [ite_eq_left hω,integral_zero_measure] using
+        (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
+    · obtain ⟨ν,hν,hlim⟩ := hgood ω hω
+      exact ⟨ν,hν,fun f => by simpa only [ite_eq_right hω] using hlim f⟩
+  choose ν hν hlim using hall
+  have hνm : Measurable ν := by
+    apply measurable_radon_of_tests ν hν
+    intro f
+    apply measurable_of_tendsto_metrizable
+      (f := fun n ω => if ω ∈ N then (0:ℝ) else ∫ x, f x ∂μ n ω)
+    · intro n
+      exact measurable_const.ite hNm (hmeas n f)
+    · exact tendsto_pi_nhds.mpr (fun ω => hlim ω f)
+  refine ⟨ν,hνm,fun ω => ?_,?_⟩
+  · let : (ν ω).Regular := hν ω
+    exact ⟨hν ω,inferInstance⟩
+  · filter_upwards [compl_mem_ae_iff.mpr hN0] with ω hω
+    intro f
+    simpa only [ite_eq_right hω] using hlim ω f
+
+/-- Countably many times retain one common event of convergence, and each
+selected time slice is a measurable locally finite regular random measure. -/
+theorem exists_measurable_countable_radon_limits
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ : ℕ → ℕ → Ω → Measure X)
+    (hmeas : ∀ n j (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n j ω))
+    (hex : ∀ᵐ ω ∂P, ∃ ν : ℕ → Measure X, (∀ j, (ν j).Regular) ∧
+      ∀ j (f : C_c(X,ℝ)), Tendsto (fun n => ∫ x, f x ∂μ n j ω) atTop (𝓝 (∫ x, f x ∂ν j))) :
+    ∃ ν : ℕ → Ω → Measure X, (∀ j, Measurable (ν j)) ∧
+      (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧
+      ∀ᵐ ω ∂P, ∀ j (f : C_c(X,ℝ)),
+        Tendsto (fun n => ∫ x, f x ∂μ n j ω) atTop (𝓝 (∫ x, f x ∂ν j ω)) := by
+  have hj (j : ℕ) : ∀ᵐ ω ∂P, ∃ ν : Measure X, ν.Regular ∧
+      ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n j ω) atTop (𝓝 (∫ x, f x ∂ν)) := by
+    filter_upwards [hex] with ω hω
+    obtain ⟨ν,hν,hlim⟩ := hω
+    exact ⟨ν j,hν j,hlim j⟩
+  choose ν hνm hνr hlim using fun j =>
+    exists_measurable_random_radon_limit P (fun n => μ n j) (fun n => hmeas n j) (hj j)
+  exact ⟨ν,hνm,hνr,ae_all_iff.mpr hlim⟩
+
+omit [SecondCountableTopology X] in
+/-- Regular vague limits of the same sequence are unique. -/
+theorem radon_test_limit_unique (μ : ℕ → Measure X) (ν η : Measure X)
+    [ν.Regular] [η.Regular]
+    (hν : ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)))
+    (hη : ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂η))) :
+    ν=η :=
+  Measure.ext_of_integral_eq_on_compactlySupported fun f => tendsto_nhds_unique (hν f) (hη f)
+
+/-- Any pathwise version of the same limit is almost everywhere
+measurable, useful when the approximants are measurable in a smaller
+stopped or deterministic-time sigma algebra. -/
+theorem aemeasurable_radon_of_test_limits_ae
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ : ℕ → Ω → Measure X)
+    (ν : Ω → Measure X)
+    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω))
+    (hν : ∀ᵐ ω ∂P, (ν ω).Regular)
+    (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(X,ℝ),
+      Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν ω))) :
+    AEMeasurable ν P := by
+  have hex : ∀ᵐ ω ∂P, ∃ η : Measure X, η.Regular ∧
+      ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂η)) := by
+    filter_upwards [hν,hlim] with ω hr hl
+    exact ⟨ν ω,hr,hl⟩
+  obtain ⟨η,hηm,hηr,hηlim⟩ := exists_measurable_random_radon_limit P μ hmeas hex
+  apply hηm.aemeasurable.congr
+  filter_upwards [hν,hlim,hηlim] with ω hr hl hηl
+  let : (ν ω).Regular := hr
+  let : (η ω).Regular := (hηr ω).1
+  exact (radon_test_limit_unique (fun n => μ n ω) (ν ω) (η ω) hl hηl).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalCanonicalMass.lean b/lean/OAI/Probability/SLE/Mass/RationalCanonicalMass.lean
index 0d9c3678ebeb546c1799cbda60745d95b9f0e77f..7ea70b864b814673071a3e00d35993e93e5e562b 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalCanonicalMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalCanonicalMass.lean
@@ -1,190 +1,190 @@
-import OAI.Probability.SLE.Mass.LevelRemainderSupport
-import OAI.Probability.SLE.Mass.RationalMassFromSupports
-import OAI.Probability.SLE.Mass.CountableLevelAdapted
-import OAI.Probability.SLE.Mass.LevelRadonNonatomic
-
-namespace OAI
-
-/-! The actual common convex-tail construction at rational times, with
-adapted versions, positive trace increments and its conditional Green kernel. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped NNReal ENNReal Topology CompactlySupported
-
-namespace SLEExactGauge.Multipoint
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-def rationalMassTime (n : ℕ) : ℝ≥0 :=
-  (((Encodable.decode (α := ℚ) n).getD 0 : ℚ) : ℝ).toNNReal
-
-@[simp] lemma rationalMassTime_encode (q : ℚ) :
-    rationalMassTime (Encodable.encode q) = (q : ℝ).toNNReal := by
-  simp only [rationalMassTime,Encodable.encodek,Option.getD_some]
-
-structure RationalCanonicalMass
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (hBm : Measurable (Function.uncurry B))
-    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) where
-  terminal : Ω → Measure upperHalfPlane
-  mass : ℚ → Ω → Measure upperHalfPlane
-  terminal_measurable : Measurable terminal
-  mass_measurable : ∀ q : ℚ, Measurable[Filtration.natural B (fun r =>
-    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)
-      (q : ℝ).toNNReal] (mass q)
-  terminal_radon : ∀ ω, (terminal ω).Regular ∧ IsFiniteMeasureOnCompacts (terminal ω)
-  mass_radon : ∀ q ω, (mass q ω).Regular ∧ IsFiniteMeasureOnCompacts (mass q ω)
-  supports : ℕ → Finset ℕ
-  weights : ℕ → ℕ → ℝ
-  weights_nonneg : ∀ n i, i ∈ supports n → 0 ≤ weights n i
-  weights_sum : ∀ n, ∑ i ∈ supports n, weights n i = 1
-  terminal_limit : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
-    Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
-      supports weights n ∞ ω) atTop (𝓝 (∫ z, f z ∂terminal ω))
-  mass_limit : ∀ᵐ ω ∂P, ∀ (q : ℚ) (f : C_c(upperHalfPlane,ℝ)),
-    Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
-      supports weights n (q : ℝ).toNNReal ω) atTop (𝓝 (∫ z, f z ∂mass q ω))
-  zero : ∀ᵐ ω ∂P, mass 0 ω = 0
-  increments : ∀ᵐ ω ∂P,
-    HalfPlaneTraceIncrements (fun q => mass q ω) (terminal ω) (fun r => γ ω r.toNNReal)
-  nonatomic : ∀ᵐ ω ∂P, NullSingletonClass (terminal ω)
-  moments : ∀ K : Set upperHalfPlane, IsCompact K → ∀ m : ℕ,
-    (∫⁻ ω, terminal ω K ^ m ∂P) < ∞
-  mean : P.bind terminal = initialGreenMeasure κ
-  conditional : ∀ q : ℚ, ∀ E : Set Ω,
-    MeasurableSet[Filtration.natural B (fun r =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)
-        (q : ℝ).toNNReal] E →
-    (P.restrict E).bind terminal = (P.restrict E).bind (mass q) +
-      (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G (q : ℝ).toNNReal)
-
-theorem exists_rationalCanonicalMass
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) :
-    Nonempty (RationalCanonicalMass P κ B hBm γ G) := by
-  obtain ⟨s,w,ν,η,hw₀,hw₁,hνm,hν,hlim,hηm,hη,hηsum,hmean,hkernel⟩ :=
-    exists_countable_radon_remainder_kernels hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
-      rationalMassTime
-  have hfiniteLimit : ∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)),
-      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
-        s w n (rationalMassTime j) ω) atTop (𝓝 (∫ z, f z ∂ν (j+1) ω)) := by
-    filter_upwards [hlim] with ω hω
-    intro j f
-    simpa only [levelMassTimes_succ] using hω (j+1) f
-  obtain ⟨M,hMm,hMr,hMeq⟩ := countable_level_limits_have_past_versions P
-    (fun r => hBm.comp (measurable_const.prodMk measurable_id)) hBc hκ hκ8 γ G F hw
-    (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) s w hw₀ rationalMassTime
-    (fun j => ν (j+1)) (Eventually.of_forall (fun ω j => (hν (j+1) ω).1)) hfiniteLimit
-  let μ (q : ℚ) := M (Encodable.encode q)
-  have hμeq : ∀ᵐ ω ∂P, ∀ q, μ q ω = ν (Encodable.encode q+1) ω := by
-    filter_upwards [hMeq] with ω hω
-    exact fun q => hω _
-  have hμlim : ∀ᵐ ω ∂P, ∀ (q : ℚ) (f : C_c(upperHalfPlane,ℝ)),
-      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
-        s w n (q:ℝ).toNNReal ω) atTop (𝓝 (∫ z, f z ∂μ q ω)) := by
-    filter_upwards [hfiniteLimit,hμeq] with ω hl he
-    intro q f
-    rw [he q]
-    simpa only [rationalMassTime_encode] using hl (Encodable.encode q) f
-  have htermLimit : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
-      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
-        s w n ∞ ω) atTop (𝓝 (∫ z, f z ∂ν 0 ω)) := by
-    filter_upwards [hlim] with ω hω
-    exact hω 0
-  have hfuture (j : ℕ) := ae_remainder_carried_by_surviving hB hBm hBc hB0 hκ hκ8 γ G F hw
-    (rationalMassTime j) (η j) (hηm j) (fun ω => (hη j ω).2) (hkernel j)
-  have hinc : ∀ᵐ ω ∂P,
-      HalfPlaneTraceIncrements (fun q => μ q ω) (ν 0 ω) (fun r => γ ω r.toNNReal) := by
-    filter_upwards [hμlim,htermLimit,hμeq,hηsum,ae_all_iff.mpr hfuture] with ω hl ht he hs hf
-    let : (ν 0 ω).Regular := (hν 0 ω).1
-    let (q : ℚ) : (μ q ω).Regular := (hMr (Encodable.encode q) ω).1
-    let (q : ℚ) : IsFiniteMeasureOnCompacts (μ q ω) := (hMr (Encodable.encode q) ω).2
-    let (n : ℕ) (v : ℝ≥0∞) : IsFiniteMeasureOnCompacts
-        (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n v ω) :=
-      convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ s w n v
-    have hdom (q : ℚ) : μ q ω ≤ ν 0 ω := by
-      apply radon_limits_mono _ _ _ _ (hl q) ht
-      intro n
-      exact convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) s w n ω le_top
-    have hmono : Monotone (fun q => μ q ω) := by
-      intro p q hpq
-      let (n : ℕ) : IsFiniteMeasureOnCompacts
-          (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n (p:ℝ).toNNReal ω) :=
-        convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ s w n _
-      apply radon_limits_mono _ _ _ _ (hl p) (hl q)
-      intro n
-      exact convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) s w n ω
-        (ENNReal.coe_le_coe.mpr (Real.toNNReal_mono (Rat.cast_le.mpr hpq)))
-    have hzero (q : ℚ) (hq : q ≤ 0) : μ q ω = 0 := by
-      apply convex_level_radon_limit_zero_at_zero ω (hBc ω) (hw ω) (fun n => (n:ℝ))
-        tendsto_natCast_atTop_atTop s w (μ q ω)
-      simpa only [Real.toNNReal_eq_zero.mpr (Rat.cast_nonpos.mpr hq),ENNReal.coe_zero] using hl q
-    apply halfPlaneTraceIncrements_of_past_and_future (fun q => μ q ω) (ν 0 ω) (γ ω)
-      (fun q => (hMr (Encodable.encode q) ω).1) hdom hmono hzero
-    · intro q _
-      exact convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) _ (fun n => (n:ℝ))
-        tendsto_natCast_atTop_atTop s w (μ q ω) (hl q)
-    · intro q _
-      have heη : η (Encodable.encode q) ω = ν 0 ω-μ q ω := by
-        apply (Measure.add_left_inj (μ q ω) _ _).mp
-        rw [sigmaFinite_sub_add_cancel_of_le (hdom q),he q]
-        exact hs _
-      rw [← heη]
-      simpa only [rationalMassTime_encode] using hf (Encodable.encode q)
-  refine ⟨{
-    terminal := ν 0
-    mass := μ
-    terminal_measurable := hνm 0
-    mass_measurable := ?_
-    terminal_radon := hν 0
-    mass_radon := fun q => hMr (Encodable.encode q)
-    supports := s
-    weights := w
-    weights_nonneg := hw₀
-    weights_sum := hw₁
-    terminal_limit := htermLimit
-    mass_limit := hμlim
-    zero := ?_
-    increments := hinc
-    nonatomic := ?_
-    moments := ?_
-    mean := hmean
-    conditional := ?_ }⟩
-  · intro q
-    simpa +instances only [rationalMassTime_encode,Function.comp_def,Function.uncurry] using hMm (Encodable.encode q)
-  · filter_upwards [hμlim] with ω hω
-    let : (μ 0 ω).Regular := (hMr (Encodable.encode (0:ℚ)) ω).1
-    apply convex_level_radon_limit_zero_at_zero ω (hBc ω) (hw ω) (fun n => (n:ℝ))
-      tendsto_natCast_atTop_atTop s w (μ 0 ω)
-    simpa only [Rat.cast_zero,Real.toNNReal_zero,ENNReal.coe_zero] using hω 0
-  · exact level_radon_limit_nonatomic hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
-      (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) s w hw₀ hw₁ ∞ (ν 0) (hνm 0)
-      (fun ω => (hν 0 ω).2) htermLimit
-  · intro K hK m
-    exact level_radon_limit_compact_moments hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
-      (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) s w hw₀ hw₁ ∞ (ν 0)
-      (fun ω => (hν 0 ω).2) htermLimit K hK m
-  · intro q E hE
-    have hμm : Measurable (μ q) := (hMm (Encodable.encode q)).mono (Filtration.le _ _) le_rfl
-    have hs : ∀ᵐ ω ∂P, η (Encodable.encode q) ω + μ q ω = ν 0 ω := by
-      filter_upwards [hμeq,hηsum] with ω he hs
-      rw [he q]
-      exact hs _
-    have hc := bind_add_of_ae (P.restrict E) (μ q) (ν 0) (η (Encodable.encode q))
-      hμm (hνm 0) (hηm _) (ae_restrict_of_ae hs)
-    rw [← hc,add_comm]
-    congr 1
-    simpa only [rationalMassTime_encode] using hkernel (Encodable.encode q) E (by simpa +instances only [rationalMassTime_encode] using hE)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.LevelRemainderSupport
+import OAI.Probability.SLE.Mass.RationalMassFromSupports
+import OAI.Probability.SLE.Mass.CountableLevelAdapted
+import OAI.Probability.SLE.Mass.LevelRadonNonatomic
+
+namespace OAI
+
+/-! The actual common convex-tail construction at rational times, with
+adapted versions, positive trace increments and its conditional Green kernel. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped NNReal ENNReal Topology CompactlySupported
+
+namespace SLEExactGauge.Multipoint
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+def rationalMassTime (n : ℕ) : ℝ≥0 :=
+  (((Encodable.decode (α := ℚ) n).getD 0 : ℚ) : ℝ).toNNReal
+
+@[simp] lemma rationalMassTime_encode (q : ℚ) :
+    rationalMassTime (Encodable.encode q) = (q : ℝ).toNNReal := by
+  simp only [rationalMassTime,Encodable.encodek,Option.getD_some]
+
+structure RationalCanonicalMass
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (hBm : Measurable (Function.uncurry B))
+    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) where
+  terminal : Ω → Measure upperHalfPlane
+  mass : ℚ → Ω → Measure upperHalfPlane
+  terminal_measurable : Measurable terminal
+  mass_measurable : ∀ q : ℚ, Measurable[Filtration.natural B (fun r =>
+    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)
+      (q : ℝ).toNNReal] (mass q)
+  terminal_radon : ∀ ω, (terminal ω).Regular ∧ IsFiniteMeasureOnCompacts (terminal ω)
+  mass_radon : ∀ q ω, (mass q ω).Regular ∧ IsFiniteMeasureOnCompacts (mass q ω)
+  supports : ℕ → Finset ℕ
+  weights : ℕ → ℕ → ℝ
+  weights_nonneg : ∀ n i, i ∈ supports n → 0 ≤ weights n i
+  weights_sum : ∀ n, ∑ i ∈ supports n, weights n i = 1
+  terminal_limit : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
+    Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
+      supports weights n ∞ ω) atTop (𝓝 (∫ z, f z ∂terminal ω))
+  mass_limit : ∀ᵐ ω ∂P, ∀ (q : ℚ) (f : C_c(upperHalfPlane,ℝ)),
+    Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
+      supports weights n (q : ℝ).toNNReal ω) atTop (𝓝 (∫ z, f z ∂mass q ω))
+  zero : ∀ᵐ ω ∂P, mass 0 ω = 0
+  increments : ∀ᵐ ω ∂P,
+    HalfPlaneTraceIncrements (fun q => mass q ω) (terminal ω) (fun r => γ ω r.toNNReal)
+  nonatomic : ∀ᵐ ω ∂P, NullSingletonClass (terminal ω)
+  moments : ∀ K : Set upperHalfPlane, IsCompact K → ∀ m : ℕ,
+    (∫⁻ ω, terminal ω K ^ m ∂P) < ∞
+  mean : P.bind terminal = initialGreenMeasure κ
+  conditional : ∀ q : ℚ, ∀ E : Set Ω,
+    MeasurableSet[Filtration.natural B (fun r =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)
+        (q : ℝ).toNNReal] E →
+    (P.restrict E).bind terminal = (P.restrict E).bind (mass q) +
+      (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G (q : ℝ).toNNReal)
+
+theorem exists_rationalCanonicalMass
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) :
+    Nonempty (RationalCanonicalMass P κ B hBm γ G) := by
+  obtain ⟨s,w,ν,η,hw₀,hw₁,hνm,hν,hlim,hηm,hη,hηsum,hmean,hkernel⟩ :=
+    exists_countable_radon_remainder_kernels hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
+      rationalMassTime
+  have hfiniteLimit : ∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)),
+      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
+        s w n (rationalMassTime j) ω) atTop (𝓝 (∫ z, f z ∂ν (j+1) ω)) := by
+    filter_upwards [hlim] with ω hω
+    intro j f
+    simpa only [levelMassTimes_succ] using hω (j+1) f
+  obtain ⟨M,hMm,hMr,hMeq⟩ := countable_level_limits_have_past_versions P
+    (fun r => hBm.comp (measurable_const.prodMk measurable_id)) hBc hκ hκ8 γ G F hw
+    (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) s w hw₀ rationalMassTime
+    (fun j => ν (j+1)) (Eventually.of_forall (fun ω j => (hν (j+1) ω).1)) hfiniteLimit
+  let μ (q : ℚ) := M (Encodable.encode q)
+  have hμeq : ∀ᵐ ω ∂P, ∀ q, μ q ω = ν (Encodable.encode q+1) ω := by
+    filter_upwards [hMeq] with ω hω
+    exact fun q => hω _
+  have hμlim : ∀ᵐ ω ∂P, ∀ (q : ℚ) (f : C_c(upperHalfPlane,ℝ)),
+      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
+        s w n (q:ℝ).toNNReal ω) atTop (𝓝 (∫ z, f z ∂μ q ω)) := by
+    filter_upwards [hfiniteLimit,hμeq] with ω hl he
+    intro q f
+    rw [he q]
+    simpa only [rationalMassTime_encode] using hl (Encodable.encode q) f
+  have htermLimit : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
+      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ))
+        s w n ∞ ω) atTop (𝓝 (∫ z, f z ∂ν 0 ω)) := by
+    filter_upwards [hlim] with ω hω
+    exact hω 0
+  have hfuture (j : ℕ) := ae_remainder_carried_by_surviving hB hBm hBc hB0 hκ hκ8 γ G F hw
+    (rationalMassTime j) (η j) (hηm j) (fun ω => (hη j ω).2) (hkernel j)
+  have hinc : ∀ᵐ ω ∂P,
+      HalfPlaneTraceIncrements (fun q => μ q ω) (ν 0 ω) (fun r => γ ω r.toNNReal) := by
+    filter_upwards [hμlim,htermLimit,hμeq,hηsum,ae_all_iff.mpr hfuture] with ω hl ht he hs hf
+    let : (ν 0 ω).Regular := (hν 0 ω).1
+    let (q : ℚ) : (μ q ω).Regular := (hMr (Encodable.encode q) ω).1
+    let (q : ℚ) : IsFiniteMeasureOnCompacts (μ q ω) := (hMr (Encodable.encode q) ω).2
+    let (n : ℕ) (v : ℝ≥0∞) : IsFiniteMeasureOnCompacts
+        (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n v ω) :=
+      convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ s w n v
+    have hdom (q : ℚ) : μ q ω ≤ ν 0 ω := by
+      apply radon_limits_mono _ _ _ _ (hl q) ht
+      intro n
+      exact convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) s w n ω le_top
+    have hmono : Monotone (fun q => μ q ω) := by
+      intro p q hpq
+      let (n : ℕ) : IsFiniteMeasureOnCompacts
+          (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n (p:ℝ).toNNReal ω) :=
+        convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ s w n _
+      apply radon_limits_mono _ _ _ _ (hl p) (hl q)
+      intro n
+      exact convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) s w n ω
+        (ENNReal.coe_le_coe.mpr (Real.toNNReal_mono (Rat.cast_le.mpr hpq)))
+    have hzero (q : ℚ) (hq : q ≤ 0) : μ q ω = 0 := by
+      apply convex_level_radon_limit_zero_at_zero ω (hBc ω) (hw ω) (fun n => (n:ℝ))
+        tendsto_natCast_atTop_atTop s w (μ q ω)
+      simpa only [Real.toNNReal_eq_zero.mpr (Rat.cast_nonpos.mpr hq),ENNReal.coe_zero] using hl q
+    apply halfPlaneTraceIncrements_of_past_and_future (fun q => μ q ω) (ν 0 ω) (γ ω)
+      (fun q => (hMr (Encodable.encode q) ω).1) hdom hmono hzero
+    · intro q _
+      exact convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) _ (fun n => (n:ℝ))
+        tendsto_natCast_atTop_atTop s w (μ q ω) (hl q)
+    · intro q _
+      have heη : η (Encodable.encode q) ω = ν 0 ω-μ q ω := by
+        apply (Measure.add_left_inj (μ q ω) _ _).mp
+        rw [sigmaFinite_sub_add_cancel_of_le (hdom q),he q]
+        exact hs _
+      rw [← heη]
+      simpa only [rationalMassTime_encode] using hf (Encodable.encode q)
+  refine ⟨{
+    terminal := ν 0
+    mass := μ
+    terminal_measurable := hνm 0
+    mass_measurable := ?_
+    terminal_radon := hν 0
+    mass_radon := fun q => hMr (Encodable.encode q)
+    supports := s
+    weights := w
+    weights_nonneg := hw₀
+    weights_sum := hw₁
+    terminal_limit := htermLimit
+    mass_limit := hμlim
+    zero := ?_
+    increments := hinc
+    nonatomic := ?_
+    moments := ?_
+    mean := hmean
+    conditional := ?_ }⟩
+  · intro q
+    simpa +instances only [rationalMassTime_encode,Function.comp_def,Function.uncurry] using hMm (Encodable.encode q)
+  · filter_upwards [hμlim] with ω hω
+    let : (μ 0 ω).Regular := (hMr (Encodable.encode (0:ℚ)) ω).1
+    apply convex_level_radon_limit_zero_at_zero ω (hBc ω) (hw ω) (fun n => (n:ℝ))
+      tendsto_natCast_atTop_atTop s w (μ 0 ω)
+    simpa only [Rat.cast_zero,Real.toNNReal_zero,ENNReal.coe_zero] using hω 0
+  · exact level_radon_limit_nonatomic hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
+      (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) s w hw₀ hw₁ ∞ (ν 0) (hνm 0)
+      (fun ω => (hν 0 ω).2) htermLimit
+  · intro K hK m
+    exact level_radon_limit_compact_moments hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
+      (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) s w hw₀ hw₁ ∞ (ν 0)
+      (fun ω => (hν 0 ω).2) htermLimit K hK m
+  · intro q E hE
+    have hμm : Measurable (μ q) := (hMm (Encodable.encode q)).mono (Filtration.le _ _) le_rfl
+    have hs : ∀ᵐ ω ∂P, η (Encodable.encode q) ω + μ q ω = ν 0 ω := by
+      filter_upwards [hμeq,hηsum] with ω he hs
+      rw [he q]
+      exact hs _
+    have hc := bind_add_of_ae (P.restrict E) (μ q) (ν 0) (η (Encodable.encode q))
+      hμm (hνm 0) (hηm _) (ae_restrict_of_ae hs)
+    rw [← hc,add_comm]
+    congr 1
+    simpa only [rationalMassTime_encode] using hkernel (Encodable.encode q) E (by simpa +instances only [rationalMassTime_encode] using hE)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalMassFromSupports.lean b/lean/OAI/Probability/SLE/Mass/RationalMassFromSupports.lean
index 63a2acadb0ba77f93bb55f66a1196e6a4d69effa..3249852dbd511867999bfbbfab6109d8d3cf155d 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalMassFromSupports.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalMassFromSupports.lean
@@ -1,85 +1,85 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassLocal
-import OAI.Probability.SLE.Mass.LevelRadonSupport
-import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract
-
-namespace OAI
-
-/-! Past support and the remaining-mass support give exactly the rational
-increment hypothesis used by the locally finite continuous extension. -/
-
-noncomputable section
-open Set MeasureTheory Classical
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-lemma trace_segment_toNNReal_image (γ : ℝ≥0 → ℂ) {p q : ℚ} (hp : 0 ≤ p) (hpq : p ≤ q) :
-    γ '' Icc (p : ℝ).toNNReal (q : ℝ).toNNReal ⊆
-      (fun r : ℝ => γ r.toNNReal) '' Icc (p:ℝ) q := by
-  rintro _ ⟨r,hr,rfl⟩
-  refine ⟨(r:ℝ),⟨?_,?_⟩,?_⟩
-  · have hr' : ((p:ℝ).toNNReal:ℝ) ≤ (r:ℝ) := by exact_mod_cast hr.1
-    simpa only [Real.coe_toNNReal _ (Rat.cast_nonneg.mpr hp)] using hr'
-  · have hq : (0:ℝ) ≤ q := Rat.cast_nonneg.mpr (hp.trans hpq)
-    have hr' : (r:ℝ) ≤ ((q:ℝ).toNNReal:ℝ) := by exact_mod_cast hr.2
-    simpa only [Real.coe_toNNReal _ hq] using hr'
-  · change γ (r:ℝ).toNNReal = γ r
-    rw [Real.toNNReal_coe]
-
-theorem halfPlaneTraceIncrements_of_past_and_future
-    (μ : ℚ → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular]
-    (γ : ℝ≥0 → ℂ) (hμr : ∀ q, (μ q).Regular)
-    (hdom : ∀ q, μ q ≤ ν) (hmono : Monotone μ)
-    (hzero : ∀ q, q ≤ 0 → μ q = 0)
-    (hpast : ∀ q, 0 ≤ q → μ q {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 (q : ℝ).toNNReal} = 0)
-    (hfuture : ∀ q, 0 ≤ q → (ν-μ q)
-      {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ (q : ℝ).toNNReal} = 0) :
-    HalfPlaneTraceIncrements μ ν (fun r : ℝ => γ r.toNNReal) := by
-  let (q : ℚ) : (μ q).Regular := hμr q
-  refine ⟨hμr,hdom,?_⟩
-  intro p q hpq
-  let η := μ q-μ p
-  have hηsum : μ q = μ p+η := by
-    rw [add_comm]
-    exact (sigmaFinite_sub_add_cancel_of_le (hmono hpq)).symm
-  refine ⟨η,hηsum,?_⟩
-  by_cases hq : q ≤ 0
-  · have hp : p ≤ 0 := hpq.trans hq
-    simp only [η,hzero p hp,hzero q hq,Measure.sub_self]
-    exact bot_le
-  by_cases hp : p < 0
-  · have heη : η = μ q := by simp only [η,hzero p hp.le,Measure.sub_zero]
-    rw [heη]
-    let A := {z : upperHalfPlane | (z : ℂ) ∈ (fun r : ℝ => γ r.toNNReal) '' Icc (p:ℝ) q}
-    have hqpos : 0 < (q:ℝ) := Rat.cast_pos.mpr (lt_of_not_ge hq)
-    have hsubset : γ '' Icc 0 (q:ℝ).toNNReal ⊆
-        (fun r : ℝ => γ r.toNNReal) '' Icc (p:ℝ) q := by
-      rintro _ ⟨r,hr,rfl⟩
-      refine ⟨(r:ℝ),⟨(Rat.cast_nonpos.mpr hp.le).trans r.coe_nonneg,?_⟩,?_⟩
-      · have hr' : (r:ℝ) ≤ ((q:ℝ).toNNReal:ℝ) := by exact_mod_cast hr.2
-        simpa only [Real.coe_toNNReal _ hqpos.le] using hr'
-      · change γ (r:ℝ).toNNReal = γ r
-        rw [Real.toNNReal_coe]
-    have ha : ∀ᵐ z : upperHalfPlane ∂μ q, z ∈ A := by
-      have hh := measure_eq_zero_iff_ae_notMem.mp (hpast q (lt_of_not_ge hq).le)
-      filter_upwards [hh] with z hz
-      exact hsubset (not_not.mp hz)
-    calc
-      μ q = (μ q).restrict A := (Measure.restrict_eq_self_of_ae_mem ha).symm
-      _ ≤ ν.restrict A := Measure.restrict_mono le_rfl (hdom q)
-  · have hp' : 0 ≤ p := le_of_not_gt hp
-    have hq' : 0 ≤ q := hp'.trans hpq
-    have hh := Multipoint.increment_le_restrict_trace_segment γ
-      (p:ℝ).toNNReal (q:ℝ).toNNReal η (μ q) (ν-μ p) ν
-      Measure.sub_le (sub_le_sub_right_of_le (hdom p) (hdom q))
-      (Measure.sub_le.trans (hdom q)) (hpast q hq') (hfuture p hp')
-    exact hh.trans (Measure.restrict_mono
-      (fun (z : upperHalfPlane) hz => trace_segment_toNNReal_image γ hp' hpq hz) le_rfl)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassLocal
+import OAI.Probability.SLE.Mass.LevelRadonSupport
+import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract
+
+namespace OAI
+
+/-! Past support and the remaining-mass support give exactly the rational
+increment hypothesis used by the locally finite continuous extension. -/
+
+noncomputable section
+open Set MeasureTheory Classical
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+lemma trace_segment_toNNReal_image (γ : ℝ≥0 → ℂ) {p q : ℚ} (hp : 0 ≤ p) (hpq : p ≤ q) :
+    γ '' Icc (p : ℝ).toNNReal (q : ℝ).toNNReal ⊆
+      (fun r : ℝ => γ r.toNNReal) '' Icc (p:ℝ) q := by
+  rintro _ ⟨r,hr,rfl⟩
+  refine ⟨(r:ℝ),⟨?_,?_⟩,?_⟩
+  · have hr' : ((p:ℝ).toNNReal:ℝ) ≤ (r:ℝ) := by exact_mod_cast hr.1
+    simpa only [Real.coe_toNNReal _ (Rat.cast_nonneg.mpr hp)] using hr'
+  · have hq : (0:ℝ) ≤ q := Rat.cast_nonneg.mpr (hp.trans hpq)
+    have hr' : (r:ℝ) ≤ ((q:ℝ).toNNReal:ℝ) := by exact_mod_cast hr.2
+    simpa only [Real.coe_toNNReal _ hq] using hr'
+  · change γ (r:ℝ).toNNReal = γ r
+    rw [Real.toNNReal_coe]
+
+theorem halfPlaneTraceIncrements_of_past_and_future
+    (μ : ℚ → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular]
+    (γ : ℝ≥0 → ℂ) (hμr : ∀ q, (μ q).Regular)
+    (hdom : ∀ q, μ q ≤ ν) (hmono : Monotone μ)
+    (hzero : ∀ q, q ≤ 0 → μ q = 0)
+    (hpast : ∀ q, 0 ≤ q → μ q {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 (q : ℝ).toNNReal} = 0)
+    (hfuture : ∀ q, 0 ≤ q → (ν-μ q)
+      {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ (q : ℝ).toNNReal} = 0) :
+    HalfPlaneTraceIncrements μ ν (fun r : ℝ => γ r.toNNReal) := by
+  let (q : ℚ) : (μ q).Regular := hμr q
+  refine ⟨hμr,hdom,?_⟩
+  intro p q hpq
+  let η := μ q-μ p
+  have hηsum : μ q = μ p+η := by
+    rw [add_comm]
+    exact (sigmaFinite_sub_add_cancel_of_le (hmono hpq)).symm
+  refine ⟨η,hηsum,?_⟩
+  by_cases hq : q ≤ 0
+  · have hp : p ≤ 0 := hpq.trans hq
+    simp only [η,hzero p hp,hzero q hq,Measure.sub_self]
+    exact bot_le
+  by_cases hp : p < 0
+  · have heη : η = μ q := by simp only [η,hzero p hp.le,Measure.sub_zero]
+    rw [heη]
+    let A := {z : upperHalfPlane | (z : ℂ) ∈ (fun r : ℝ => γ r.toNNReal) '' Icc (p:ℝ) q}
+    have hqpos : 0 < (q:ℝ) := Rat.cast_pos.mpr (lt_of_not_ge hq)
+    have hsubset : γ '' Icc 0 (q:ℝ).toNNReal ⊆
+        (fun r : ℝ => γ r.toNNReal) '' Icc (p:ℝ) q := by
+      rintro _ ⟨r,hr,rfl⟩
+      refine ⟨(r:ℝ),⟨(Rat.cast_nonpos.mpr hp.le).trans r.coe_nonneg,?_⟩,?_⟩
+      · have hr' : (r:ℝ) ≤ ((q:ℝ).toNNReal:ℝ) := by exact_mod_cast hr.2
+        simpa only [Real.coe_toNNReal _ hqpos.le] using hr'
+      · change γ (r:ℝ).toNNReal = γ r
+        rw [Real.toNNReal_coe]
+    have ha : ∀ᵐ z : upperHalfPlane ∂μ q, z ∈ A := by
+      have hh := measure_eq_zero_iff_ae_notMem.mp (hpast q (lt_of_not_ge hq).le)
+      filter_upwards [hh] with z hz
+      exact hsubset (not_not.mp hz)
+    calc
+      μ q = (μ q).restrict A := (Measure.restrict_eq_self_of_ae_mem ha).symm
+      _ ≤ ν.restrict A := Measure.restrict_mono le_rfl (hdom q)
+  · have hp' : 0 ≤ p := le_of_not_gt hp
+    have hq' : 0 ≤ q := hp'.trans hpq
+    have hh := Multipoint.increment_le_restrict_trace_segment γ
+      (p:ℝ).toNNReal (q:ℝ).toNNReal η (μ q) (ν-μ p) ν
+      Measure.sub_le (sub_le_sub_right_of_le (hdom p) (hdom q))
+      (Measure.sub_le.trans (hdom q)) (hpast q hq') (hfuture p hp')
+    exact hh.trans (Measure.restrict_mono
+      (fun (z : upperHalfPlane) hz => trace_segment_toNNReal_image γ hp' hpq hz) le_rfl)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMass.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMass.lean
index 13c02a5356481650491be2d7c2a549558f3c66ff..8cf38363a1d66e9f64a93e3015b21673e7991633 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMass.lean
@@ -1,137 +1,137 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassControl
-import Mathlib.Topology.Sequences
-
-namespace OAI
-
-/-! Continuous extension of a rational measure family whose positive
-increments are supported on the corresponding trace intervals. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology CompactlySupported ENNReal
-
-namespace SLEExactGauge
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-/-- This is the positive-increment and interval-support assertion used in
-the pathwise extension. Local applications restrict all measures to one
-compact set, making the terminal measure finite. -/
-structure RationalTraceIncrements (μ : ℚ → Measure X) (ν : Measure X) (γ : ℝ → X) : Prop where
-  regular : ∀ q, (μ q).Regular
-  dominated : ∀ q, μ q ≤ ν
-  increment : ∀ p q, p ≤ q → ∃ η : Measure X,
-    μ q=μ p+η ∧ η ≤ ν.restrict (γ '' Icc (p:ℝ) q)
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X] in
-lemma RationalTraceIncrements.monotone {μ : ℚ → Measure X} {ν : Measure X} {γ : ℝ → X}
-    (H : RationalTraceIncrements μ ν γ) : Monotone μ := by
-  intro p q hpq
-  obtain ⟨η,hη,_⟩ := H.increment p q hpq
-  rw [hη]
-  exact Measure.le_add_right le_rfl
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
-lemma RationalTraceIncrements.integral_dist_le {μ : ℚ → Measure X} {ν : Measure X}
-    [IsFiniteMeasure ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
-    (f : C_c(X,ℝ)) (p q : ℚ) :
-    dist (∫ x, f x ∂μ p) (∫ x, f x ∂μ q) ≤ ‖f.toBoundedContinuousFunction‖ *
-      (ν (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal := by
-  have ordered (a b : ℚ) (hab : a ≤ b) :
-      dist (∫ x, f x ∂μ a) (∫ x, f x ∂μ b) ≤ ‖f.toBoundedContinuousFunction‖ *
-        (ν (γ '' Icc (a:ℝ) b)).toReal := by
-    obtain ⟨η,hη,hην⟩ := H.increment a b hab
-    have hηle : η ≤ ν := hην.trans Measure.restrict_le_self
-    let : IsFiniteMeasure η := isFiniteMeasure_of_le ν hηle
-    let : (μ a).Regular := H.regular a
-    rw [hη,integral_add_measure f.integrable f.integrable,dist_eq_norm]
-    have hsub : (∫ x, f x ∂μ a)-((∫ x, f x ∂μ a)+(∫ x, f x ∂η))= -(∫ x, f x ∂η) := by ring
-    rw [hsub,norm_neg]
-    have hb := norm_integral_le_of_norm_le_const (μ := η)
-      (ae_of_all η (fun x => f.toBoundedContinuousFunction.norm_coe_le_norm x))
-    apply hb.trans
-    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
-    apply ENNReal.toReal_mono (measure_ne_top ν _)
-    exact (hην univ).trans_eq (by simp)
-  rcases le_total p q with h | h
-  · simpa only [min_eq_left (Rat.cast_le.mpr h),max_eq_right (Rat.cast_le.mpr h)] using ordered p q h
-  · rw [dist_comm]
-    simpa only [min_eq_right (Rat.cast_le.mpr h),max_eq_left (Rat.cast_le.mpr h)] using ordered q p h
-
-lemma exists_rational_sequence_tendsto (t : ℝ) :
-    ∃ q : ℕ → ℚ, Tendsto (fun n => (q n : ℝ)) atTop (𝓝 t) := by
-  obtain ⟨x,hx,ht⟩ := mem_closure_iff_seq_limit.mp (Rat.denseRange_cast t)
-  choose q hq using hx
-  exact ⟨q,by simpa only [hq] using ht⟩
-
-omit [SecondCountableTopology X] in
-/-- The rational family has an all-real-time regular extension, continuous
-against every compact continuous test. It is dominated by the same finite
-terminal measure, and agrees exactly at every rational time. -/
-theorem RationalTraceIncrements.exists_continuous_extension
-    {μ : ℚ → Measure X} {ν : Measure X} [ν.Regular] [IsFiniteMeasure ν]
-    [NullSingletonClass ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
-    (hγ : Continuous γ) :
-    ∃ M : ℝ → Measure X, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
-      (∀ q : ℚ, M q=μ q) ∧
-      (∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂M t)) ∧ Monotone M := by
-  have hf (f : C_c(X,ℝ)) := continuous_rational_extension_of_trace_control
-    ν γ hγ (fun q => ∫ x, f x ∂μ q) ‖f.toBoundedContinuousFunction‖ (norm_nonneg _)
-    (H.integral_dist_le f)
-  choose F hFc hFq _ using hf
-  have hex (t : ℝ) : ∃ η : Measure X, η.Regular ∧ IsFiniteMeasureOnCompacts η ∧
-      ∀ f : C_c(X,ℝ), (∫ x, f x ∂η)=F f t := by
-    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-    let (n : ℕ) : (μ (q n)).Regular := H.regular (q n)
-    have hlim (f : C_c(X,ℝ)) : Tendsto (fun n => ∫ x, f x ∂μ (q n)) atTop (𝓝 (F f t)) := by
-      simpa only [Function.comp_def,hFq] using (hFc f).continuousAt.tendsto.comp hq
-    obtain ⟨η,hηr,hηf,hηlim⟩ := radon_limit_of_all_test_limits (fun n => μ (q n))
-      (fun f => ⟨F f t,hlim f⟩)
-    exact ⟨η,hηr,hηf,fun f => tendsto_nhds_unique (hηlim f) (hlim f)⟩
-  choose M hMr hMf hMI using hex
-  have hcont (f : C_c(X,ℝ)) : Continuous (fun t => ∫ x, f x ∂M t) := by
-    simpa only [hMI] using hFc f
-  have hrat (q : ℚ) : M q=μ q := by
-    let : (M q).Regular := hMr q
-    let : (μ q).Regular := H.regular q
-    apply Measure.ext_of_integral_eq_on_compactlySupported
-    intro f
-    rw [hMI,hFq]
-  have hdom (t : ℝ) : M t ≤ ν := by
-    let : (M t).Regular := hMr t
-    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-    apply radon_measure_le_of_nonneg_tests
-    intro f hfn
-    have hlim : Tendsto (fun n => ∫ x, f x ∂μ (q n)) atTop (𝓝 (∫ x, f x ∂M t)) := by
-      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
-    apply le_of_tendsto_of_tendsto' hlim tendsto_const_nhds
-    intro n
-    exact integral_mono_measure (H.dominated (q n)) (ae_of_all ν hfn) f.integrable
-  refine ⟨M,hMr,hdom,hrat,hcont,?_⟩
-  intro s t hst
-  rcases eq_or_lt_of_le hst with rfl | hst
-  · exact le_rfl
-  let : (M s).Regular := hMr s
-  let : (M t).Regular := hMr t
-  obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
-  obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-  have hp' : ∀ᶠ n in atTop, (p n : ℝ)<(s+t)/2 := hp.eventually (gt_mem_nhds (by linarith))
-  have hq' : ∀ᶠ n in atTop, (s+t)/2<(q n : ℝ) := hq.eventually (lt_mem_nhds (by linarith))
-  apply radon_measure_le_of_nonneg_tests
-  intro f hfn
-  have hplim : Tendsto (fun n => ∫ x, f x ∂μ (p n)) atTop (𝓝 (∫ x, f x ∂M s)) := by
-    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
-  have hqlim : Tendsto (fun n => ∫ x, f x ∂μ (q n)) atTop (𝓝 (∫ x, f x ∂M t)) := by
-    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
-  apply le_of_tendsto_of_tendsto hplim hqlim
-  filter_upwards [hp',hq'] with n hpn hqn
-  let : (μ (q n)).Regular := H.regular (q n)
-  exact integral_mono_measure (H.monotone (Rat.cast_le.mp (le_of_lt (hpn.trans hqn))))
-    (ae_of_all _ hfn) f.integrable
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassControl
+import Mathlib.Topology.Sequences
+
+namespace OAI
+
+/-! Continuous extension of a rational measure family whose positive
+increments are supported on the corresponding trace intervals. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology CompactlySupported ENNReal
+
+namespace SLEExactGauge
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+/-- This is the positive-increment and interval-support assertion used in
+the pathwise extension. Local applications restrict all measures to one
+compact set, making the terminal measure finite. -/
+structure RationalTraceIncrements (μ : ℚ → Measure X) (ν : Measure X) (γ : ℝ → X) : Prop where
+  regular : ∀ q, (μ q).Regular
+  dominated : ∀ q, μ q ≤ ν
+  increment : ∀ p q, p ≤ q → ∃ η : Measure X,
+    μ q=μ p+η ∧ η ≤ ν.restrict (γ '' Icc (p:ℝ) q)
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X] in
+lemma RationalTraceIncrements.monotone {μ : ℚ → Measure X} {ν : Measure X} {γ : ℝ → X}
+    (H : RationalTraceIncrements μ ν γ) : Monotone μ := by
+  intro p q hpq
+  obtain ⟨η,hη,_⟩ := H.increment p q hpq
+  rw [hη]
+  exact Measure.le_add_right le_rfl
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
+lemma RationalTraceIncrements.integral_dist_le {μ : ℚ → Measure X} {ν : Measure X}
+    [IsFiniteMeasure ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
+    (f : C_c(X,ℝ)) (p q : ℚ) :
+    dist (∫ x, f x ∂μ p) (∫ x, f x ∂μ q) ≤ ‖f.toBoundedContinuousFunction‖ *
+      (ν (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal := by
+  have ordered (a b : ℚ) (hab : a ≤ b) :
+      dist (∫ x, f x ∂μ a) (∫ x, f x ∂μ b) ≤ ‖f.toBoundedContinuousFunction‖ *
+        (ν (γ '' Icc (a:ℝ) b)).toReal := by
+    obtain ⟨η,hη,hην⟩ := H.increment a b hab
+    have hηle : η ≤ ν := hην.trans Measure.restrict_le_self
+    let : IsFiniteMeasure η := isFiniteMeasure_of_le ν hηle
+    let : (μ a).Regular := H.regular a
+    rw [hη,integral_add_measure f.integrable f.integrable,dist_eq_norm]
+    have hsub : (∫ x, f x ∂μ a)-((∫ x, f x ∂μ a)+(∫ x, f x ∂η))= -(∫ x, f x ∂η) := by ring
+    rw [hsub,norm_neg]
+    have hb := norm_integral_le_of_norm_le_const (μ := η)
+      (ae_of_all η (fun x => f.toBoundedContinuousFunction.norm_coe_le_norm x))
+    apply hb.trans
+    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+    apply ENNReal.toReal_mono (measure_ne_top ν _)
+    exact (hην univ).trans_eq (by simp)
+  rcases le_total p q with h | h
+  · simpa only [min_eq_left (Rat.cast_le.mpr h),max_eq_right (Rat.cast_le.mpr h)] using ordered p q h
+  · rw [dist_comm]
+    simpa only [min_eq_right (Rat.cast_le.mpr h),max_eq_left (Rat.cast_le.mpr h)] using ordered q p h
+
+lemma exists_rational_sequence_tendsto (t : ℝ) :
+    ∃ q : ℕ → ℚ, Tendsto (fun n => (q n : ℝ)) atTop (𝓝 t) := by
+  obtain ⟨x,hx,ht⟩ := mem_closure_iff_seq_limit.mp (Rat.denseRange_cast t)
+  choose q hq using hx
+  exact ⟨q,by simpa only [hq] using ht⟩
+
+omit [SecondCountableTopology X] in
+/-- The rational family has an all-real-time regular extension, continuous
+against every compact continuous test. It is dominated by the same finite
+terminal measure, and agrees exactly at every rational time. -/
+theorem RationalTraceIncrements.exists_continuous_extension
+    {μ : ℚ → Measure X} {ν : Measure X} [ν.Regular] [IsFiniteMeasure ν]
+    [NullSingletonClass ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
+    (hγ : Continuous γ) :
+    ∃ M : ℝ → Measure X, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
+      (∀ q : ℚ, M q=μ q) ∧
+      (∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂M t)) ∧ Monotone M := by
+  have hf (f : C_c(X,ℝ)) := continuous_rational_extension_of_trace_control
+    ν γ hγ (fun q => ∫ x, f x ∂μ q) ‖f.toBoundedContinuousFunction‖ (norm_nonneg _)
+    (H.integral_dist_le f)
+  choose F hFc hFq _ using hf
+  have hex (t : ℝ) : ∃ η : Measure X, η.Regular ∧ IsFiniteMeasureOnCompacts η ∧
+      ∀ f : C_c(X,ℝ), (∫ x, f x ∂η)=F f t := by
+    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+    let (n : ℕ) : (μ (q n)).Regular := H.regular (q n)
+    have hlim (f : C_c(X,ℝ)) : Tendsto (fun n => ∫ x, f x ∂μ (q n)) atTop (𝓝 (F f t)) := by
+      simpa only [Function.comp_def,hFq] using (hFc f).continuousAt.tendsto.comp hq
+    obtain ⟨η,hηr,hηf,hηlim⟩ := radon_limit_of_all_test_limits (fun n => μ (q n))
+      (fun f => ⟨F f t,hlim f⟩)
+    exact ⟨η,hηr,hηf,fun f => tendsto_nhds_unique (hηlim f) (hlim f)⟩
+  choose M hMr hMf hMI using hex
+  have hcont (f : C_c(X,ℝ)) : Continuous (fun t => ∫ x, f x ∂M t) := by
+    simpa only [hMI] using hFc f
+  have hrat (q : ℚ) : M q=μ q := by
+    let : (M q).Regular := hMr q
+    let : (μ q).Regular := H.regular q
+    apply Measure.ext_of_integral_eq_on_compactlySupported
+    intro f
+    rw [hMI,hFq]
+  have hdom (t : ℝ) : M t ≤ ν := by
+    let : (M t).Regular := hMr t
+    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+    apply radon_measure_le_of_nonneg_tests
+    intro f hfn
+    have hlim : Tendsto (fun n => ∫ x, f x ∂μ (q n)) atTop (𝓝 (∫ x, f x ∂M t)) := by
+      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
+    apply le_of_tendsto_of_tendsto' hlim tendsto_const_nhds
+    intro n
+    exact integral_mono_measure (H.dominated (q n)) (ae_of_all ν hfn) f.integrable
+  refine ⟨M,hMr,hdom,hrat,hcont,?_⟩
+  intro s t hst
+  rcases eq_or_lt_of_le hst with rfl | hst
+  · exact le_rfl
+  let : (M s).Regular := hMr s
+  let : (M t).Regular := hMr t
+  obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
+  obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+  have hp' : ∀ᶠ n in atTop, (p n : ℝ)<(s+t)/2 := hp.eventually (gt_mem_nhds (by linarith))
+  have hq' : ∀ᶠ n in atTop, (s+t)/2<(q n : ℝ) := hq.eventually (lt_mem_nhds (by linarith))
+  apply radon_measure_le_of_nonneg_tests
+  intro f hfn
+  have hplim : Tendsto (fun n => ∫ x, f x ∂μ (p n)) atTop (𝓝 (∫ x, f x ∂M s)) := by
+    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
+  have hqlim : Tendsto (fun n => ∫ x, f x ∂μ (q n)) atTop (𝓝 (∫ x, f x ∂M t)) := by
+    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
+  apply le_of_tendsto_of_tendsto hplim hqlim
+  filter_upwards [hp',hq'] with n hpn hqn
+  let : (μ (q n)).Regular := H.regular (q n)
+  exact integral_mono_measure (H.monotone (Rat.cast_le.mp (le_of_lt (hpn.trans hqn))))
+    (ae_of_all _ hfn) f.integrable
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdapted.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdapted.lean
index ca8d460383a1e4f0bce3c9f166c21cf37427f57f..3f0e44dbaf0d01709498fbe6dac4c03fd716473e 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdapted.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdapted.lean
@@ -1,111 +1,111 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassLocalVariation
-import OAI.Probability.SLE.Mass.RadonTestMeasurability
-
-namespace OAI
-
-/-! Adapted and jointly measurable all-time Radon mass, obtained from
-rational left limits in the original filtration. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology ENNReal NNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-lemma measurable_halfPlane_compact_test_integral
-    {Ω : Type*} [MeasurableSpace Ω] (μ : Ω → Measure upperHalfPlane)
-    (hm : Measurable μ) (hr : ∀ ω, (μ ω).Regular) (f : C_c(upperHalfPlane,ℝ)) :
-    Measurable (fun ω => ∫ z, f z ∂μ ω) := by
-  have hp := (Measure.measurable_lintegral f.continuous.measurable.ennreal_ofReal).comp hm
-  have hn := (Measure.measurable_lintegral f.continuous.measurable.neg.ennreal_ofReal).comp hm
-  have he (ω : Ω) : (∫ z, f z ∂μ ω) =
-      (∫⁻ z, ENNReal.ofReal (f z) ∂μ ω).toReal -
-        (∫⁻ z, ENNReal.ofReal (-f z) ∂μ ω).toReal := by
-    let := hr ω
-    exact integral_eq_lintegral_pos_part_sub_lintegral_neg_part f.integrable
-  simp_rw [he]
-  exact hp.ennreal_toReal.sub hn.ennreal_toReal
-
-theorem halfPlane_extension_adapted
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
-    (hMr : ∀ t ω, (M t ω).Regular)
-    (hcont : ∀ ω (f : C_c(upperHalfPlane,ℝ)), Continuous (fun t => ∫ z, f z ∂M t ω))
-    (hrat : ∀ (q : ℚ) ω, M q ω = μ q ω)
-    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
-    Adapted ℱ (fun t : ℝ≥0 => M t) := by
-  intro t
-  by_cases ht : t = 0
-  · subst t
-    have he : M (0:ℝ) = μ (0:ℚ) := by
-      funext ω
-      simpa only [Rat.cast_zero] using hrat 0 ω
-    change Measurable[ℱ 0] (M (0:ℝ))
-    rw [he]
-    convert hμm (0:ℚ) le_rfl using 1
-    norm_num
-  · have htpos : 0 < (t:ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr ht)
-    obtain ⟨q,_,hq,hqt⟩ := Rat.denseRange_cast.exists_seq_strictMono_tendsto_of_lt
-      (fun a b hab => Rat.cast_le.mpr hab) htpos
-    let : MeasurableSpace Ω := ℱ t
-    apply measurable_radon_of_tests (M t) (hMr t)
-    intro f
-    have hqm (n : ℕ) : Measurable[ℱ t] (μ (q n)) := by
-      have hqn : 0 ≤ q n := Rat.cast_nonneg.mp (hq n).1.le
-      have htime : (q n : ℝ).toNNReal ≤ t := by
-        simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal (hq n).2.le
-      exact (hμm (q n) hqn).mono (ℱ.mono htime) le_rfl
-    apply measurable_of_tendsto_metrizable
-      (fun n => measurable_halfPlane_compact_test_integral (μ (q n)) (hqm n)
-        (fun ω => hrat (q n) ω ▸ hMr (q n) ω) f)
-    apply tendsto_pi_nhds.mpr
-    intro ω
-    simpa only [Function.comp_def,hrat] using
-      (hcont ω f).continuousAt.tendsto.comp hqt
-
-/-- Joint measurability follows through real compact-test integrals;
-no topology on the space of Radon measures is required. -/
-theorem halfPlane_extension_joint_measurable
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
-    (hMr : ∀ t ω, (M t ω).Regular)
-    (hcont : ∀ ω (f : C_c(upperHalfPlane,ℝ)), Continuous (fun t => ∫ z, f z ∂M t ω))
-    (hrat : ∀ (q : ℚ) ω, M q ω = μ q ω)
-    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
-    Measurable (fun q : ℝ≥0 × Ω => M q.1 q.2) := by
-  have had := halfPlane_extension_adapted ℱ μ M hMr hcont hrat hμm
-  apply measurable_radon_of_tests (fun q : ℝ≥0 × Ω => M q.1 q.2)
-    (fun q => hMr q.1 q.2)
-  intro f
-  apply measurable_uncurry_of_continuous_of_measurable
-    (fun ω => (hcont ω f).comp NNReal.continuous_coe)
-  intro t
-  exact measurable_halfPlane_compact_test_integral (M t) (had.measurable (i := t)) (hMr t) f
-
-theorem halfPlane_extension_progressive
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
-    (hMr : ∀ t ω, (M t ω).Regular)
-    (hcont : ∀ ω (f : C_c(upperHalfPlane,ℝ)), Continuous (fun t => ∫ z, f z ∂M t ω))
-    (hrat : ∀ (q : ℚ) ω, M q ω = μ q ω)
-    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
-    IsProgressive ℱ (fun t : ℝ≥0 => M t) := by
-  have had := halfPlane_extension_adapted ℱ μ M hMr hcont hrat hμm
-  intro T
-  let : MeasurableSpace Ω := ℱ T
-  apply measurable_radon_of_tests (fun q : Iic T × Ω => M q.1.1 q.2)
-    (fun q => hMr q.1.1 q.2)
-  intro f
-  apply measurable_uncurry_of_continuous_of_measurable
-    (fun ω => (hcont ω f).comp (NNReal.continuous_coe.comp continuous_subtype_val))
-  intro t
-  have ht : Measurable[ℱ T] (M (t.1 : ℝ)) := had.measurable_le t.2
-  exact measurable_halfPlane_compact_test_integral (M (t.1 : ℝ)) ht (hMr t.1) f
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassLocalVariation
+import OAI.Probability.SLE.Mass.RadonTestMeasurability
+
+namespace OAI
+
+/-! Adapted and jointly measurable all-time Radon mass, obtained from
+rational left limits in the original filtration. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology ENNReal NNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+lemma measurable_halfPlane_compact_test_integral
+    {Ω : Type*} [MeasurableSpace Ω] (μ : Ω → Measure upperHalfPlane)
+    (hm : Measurable μ) (hr : ∀ ω, (μ ω).Regular) (f : C_c(upperHalfPlane,ℝ)) :
+    Measurable (fun ω => ∫ z, f z ∂μ ω) := by
+  have hp := (Measure.measurable_lintegral f.continuous.measurable.ennreal_ofReal).comp hm
+  have hn := (Measure.measurable_lintegral f.continuous.measurable.neg.ennreal_ofReal).comp hm
+  have he (ω : Ω) : (∫ z, f z ∂μ ω) =
+      (∫⁻ z, ENNReal.ofReal (f z) ∂μ ω).toReal -
+        (∫⁻ z, ENNReal.ofReal (-f z) ∂μ ω).toReal := by
+    let := hr ω
+    exact integral_eq_lintegral_pos_part_sub_lintegral_neg_part f.integrable
+  simp_rw [he]
+  exact hp.ennreal_toReal.sub hn.ennreal_toReal
+
+theorem halfPlane_extension_adapted
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
+    (hMr : ∀ t ω, (M t ω).Regular)
+    (hcont : ∀ ω (f : C_c(upperHalfPlane,ℝ)), Continuous (fun t => ∫ z, f z ∂M t ω))
+    (hrat : ∀ (q : ℚ) ω, M q ω = μ q ω)
+    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
+    Adapted ℱ (fun t : ℝ≥0 => M t) := by
+  intro t
+  by_cases ht : t = 0
+  · subst t
+    have he : M (0:ℝ) = μ (0:ℚ) := by
+      funext ω
+      simpa only [Rat.cast_zero] using hrat 0 ω
+    change Measurable[ℱ 0] (M (0:ℝ))
+    rw [he]
+    convert hμm (0:ℚ) le_rfl using 1
+    norm_num
+  · have htpos : 0 < (t:ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr ht)
+    obtain ⟨q,_,hq,hqt⟩ := Rat.denseRange_cast.exists_seq_strictMono_tendsto_of_lt
+      (fun a b hab => Rat.cast_le.mpr hab) htpos
+    let : MeasurableSpace Ω := ℱ t
+    apply measurable_radon_of_tests (M t) (hMr t)
+    intro f
+    have hqm (n : ℕ) : Measurable[ℱ t] (μ (q n)) := by
+      have hqn : 0 ≤ q n := Rat.cast_nonneg.mp (hq n).1.le
+      have htime : (q n : ℝ).toNNReal ≤ t := by
+        simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal (hq n).2.le
+      exact (hμm (q n) hqn).mono (ℱ.mono htime) le_rfl
+    apply measurable_of_tendsto_metrizable
+      (fun n => measurable_halfPlane_compact_test_integral (μ (q n)) (hqm n)
+        (fun ω => hrat (q n) ω ▸ hMr (q n) ω) f)
+    apply tendsto_pi_nhds.mpr
+    intro ω
+    simpa only [Function.comp_def,hrat] using
+      (hcont ω f).continuousAt.tendsto.comp hqt
+
+/-- Joint measurability follows through real compact-test integrals;
+no topology on the space of Radon measures is required. -/
+theorem halfPlane_extension_joint_measurable
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
+    (hMr : ∀ t ω, (M t ω).Regular)
+    (hcont : ∀ ω (f : C_c(upperHalfPlane,ℝ)), Continuous (fun t => ∫ z, f z ∂M t ω))
+    (hrat : ∀ (q : ℚ) ω, M q ω = μ q ω)
+    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
+    Measurable (fun q : ℝ≥0 × Ω => M q.1 q.2) := by
+  have had := halfPlane_extension_adapted ℱ μ M hMr hcont hrat hμm
+  apply measurable_radon_of_tests (fun q : ℝ≥0 × Ω => M q.1 q.2)
+    (fun q => hMr q.1 q.2)
+  intro f
+  apply measurable_uncurry_of_continuous_of_measurable
+    (fun ω => (hcont ω f).comp NNReal.continuous_coe)
+  intro t
+  exact measurable_halfPlane_compact_test_integral (M t) (had.measurable (i := t)) (hMr t) f
+
+theorem halfPlane_extension_progressive
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
+    (hMr : ∀ t ω, (M t ω).Regular)
+    (hcont : ∀ ω (f : C_c(upperHalfPlane,ℝ)), Continuous (fun t => ∫ z, f z ∂M t ω))
+    (hrat : ∀ (q : ℚ) ω, M q ω = μ q ω)
+    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
+    IsProgressive ℱ (fun t : ℝ≥0 => M t) := by
+  have had := halfPlane_extension_adapted ℱ μ M hMr hcont hrat hμm
+  intro T
+  let : MeasurableSpace Ω := ℱ T
+  apply measurable_radon_of_tests (fun q : Iic T × Ω => M q.1.1 q.2)
+    (fun q => hMr q.1.1 q.2)
+  intro f
+  apply measurable_uncurry_of_continuous_of_measurable
+    (fun ω => (hcont ω f).comp (NNReal.continuous_coe.comp continuous_subtype_val))
+  intro t
+  have ht : Measurable[ℱ T] (M (t.1 : ℝ)) := had.measurable_le t.2
+  exact measurable_halfPlane_compact_test_integral (M (t.1 : ℝ)) ht (hMr t.1) f
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdaptedVersion.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdaptedVersion.lean
index 298b51aa673dbe35f1ceccfd3c748765d2c26506..95ceb9d65e5568ae1eba79373995879b3633ba2b 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdaptedVersion.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassAdaptedVersion.lean
@@ -1,76 +1,76 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassAdapted
-import OAI.Probability.SLE.Mass.SubSigmaRadonLimit
-
-namespace OAI
-
-/-! Rational adapted versions determine one adapted all-time version on the
-same probability-one set on which the limiting compact tests are continuous. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology NNReal ENNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-theorem halfPlane_extension_adapted_common_version
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ)
-    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
-    (hμr : ∀ q ω, (μ q ω).Regular) (hMr : ∀ t ω, (M t ω).Regular)
-    (hcont : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
-      Continuous (fun t => ∫ z, f z ∂M t ω))
-    (hrat : ∀ᵐ ω ∂P, ∀ q : ℚ, M q ω = μ q ω)
-    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
-    ∃ N : ℝ≥0 → Ω → Measure upperHalfPlane, Adapted ℱ N ∧
-      (∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) ∧
-      ∀ᵐ ω ∂P, ∀ t : ℝ≥0, N t ω = M t ω := by
-  let Good (ω : Ω) : Prop :=
-    (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t ω)) ∧
-      ∀ q : ℚ, M q ω = μ q ω
-  have hex (t : ℝ≥0) : ∃ N : Ω → Measure upperHalfPlane,
-      Measurable[ℱ t] N ∧ (∀ ω, (N ω).Regular ∧ IsFiniteMeasureOnCompacts (N ω)) ∧
-      ∀ ω, Good ω → N ω = M t ω := by
-    by_cases ht : t = 0
-    · subst t
-      refine ⟨μ 0,?_,?_,?_⟩
-      · convert hμm (0:ℚ) le_rfl using 1
-        norm_num
-      · intro ω
-        let := hμr 0 ω
-        exact ⟨hμr 0 ω,inferInstance⟩
-      · intro ω hω
-        simpa only [Rat.cast_zero,NNReal.coe_zero] using (hω.2 0).symm
-    · have htpos : 0 < (t:ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr ht)
-      obtain ⟨q,_,hq,hqt⟩ := Rat.denseRange_cast.exists_seq_strictMono_tendsto_of_lt
-        (fun a b hab => Rat.cast_le.mpr hab) htpos
-      have hqm (n : ℕ) : Measurable[ℱ t] (μ (q n)) := by
-        have hqn : 0 ≤ q n := Rat.cast_nonneg.mp (hq n).1.le
-        have htime : (q n : ℝ).toNNReal ≤ t := by
-          simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal (hq n).2.le
-        exact (hμm (q n) hqn).mono (ℱ.mono htime) le_rfl
-      have hfinite (n : ℕ) (ω : Ω) : IsFiniteMeasureOnCompacts (μ (q n) ω) := by
-        let := hμr (q n) ω
-        infer_instance
-      let : MeasurableSpace Ω := ℱ t
-      have hm (n : ℕ) (f : C_c(upperHalfPlane,ℝ)) :
-          Measurable[ℱ t] (fun ω => ∫ z, f z ∂μ (q n) ω) :=
-        measurable_halfPlane_compact_test_integral (μ (q n)) (hqm n) (hμr (q n)) f
-      obtain ⟨N,hNm,hNr,hNe⟩ := exists_subsigma_radon_version_identified
-        (ℱ t) (fun n => μ (q n)) hfinite hm
-      refine ⟨N,hNm,hNr,?_⟩
-      intro ω hω
-      apply hNe ω (M t ω) (hMr t ω)
-      intro f
-      simpa only [Function.comp_def,hω.2] using
-        (hω.1 f).continuousAt.tendsto.comp hqt
-  choose N hNm hNr hNe using hex
-  refine ⟨N,hNm,hNr,?_⟩
-  filter_upwards [hcont,hrat] with ω hc hr
-  exact fun t => hNe t ω ⟨hc,hr⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassAdapted
+import OAI.Probability.SLE.Mass.SubSigmaRadonLimit
+
+namespace OAI
+
+/-! Rational adapted versions determine one adapted all-time version on the
+same probability-one set on which the limiting compact tests are continuous. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology NNReal ENNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+theorem halfPlane_extension_adapted_common_version
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) (ℱ : Filtration ℝ≥0 mΩ)
+    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
+    (hμr : ∀ q ω, (μ q ω).Regular) (hMr : ∀ t ω, (M t ω).Regular)
+    (hcont : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
+      Continuous (fun t => ∫ z, f z ∂M t ω))
+    (hrat : ∀ᵐ ω ∂P, ∀ q : ℚ, M q ω = μ q ω)
+    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
+    ∃ N : ℝ≥0 → Ω → Measure upperHalfPlane, Adapted ℱ N ∧
+      (∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) ∧
+      ∀ᵐ ω ∂P, ∀ t : ℝ≥0, N t ω = M t ω := by
+  let Good (ω : Ω) : Prop :=
+    (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t ω)) ∧
+      ∀ q : ℚ, M q ω = μ q ω
+  have hex (t : ℝ≥0) : ∃ N : Ω → Measure upperHalfPlane,
+      Measurable[ℱ t] N ∧ (∀ ω, (N ω).Regular ∧ IsFiniteMeasureOnCompacts (N ω)) ∧
+      ∀ ω, Good ω → N ω = M t ω := by
+    by_cases ht : t = 0
+    · subst t
+      refine ⟨μ 0,?_,?_,?_⟩
+      · convert hμm (0:ℚ) le_rfl using 1
+        norm_num
+      · intro ω
+        let := hμr 0 ω
+        exact ⟨hμr 0 ω,inferInstance⟩
+      · intro ω hω
+        simpa only [Rat.cast_zero,NNReal.coe_zero] using (hω.2 0).symm
+    · have htpos : 0 < (t:ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr ht)
+      obtain ⟨q,_,hq,hqt⟩ := Rat.denseRange_cast.exists_seq_strictMono_tendsto_of_lt
+        (fun a b hab => Rat.cast_le.mpr hab) htpos
+      have hqm (n : ℕ) : Measurable[ℱ t] (μ (q n)) := by
+        have hqn : 0 ≤ q n := Rat.cast_nonneg.mp (hq n).1.le
+        have htime : (q n : ℝ).toNNReal ≤ t := by
+          simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal (hq n).2.le
+        exact (hμm (q n) hqn).mono (ℱ.mono htime) le_rfl
+      have hfinite (n : ℕ) (ω : Ω) : IsFiniteMeasureOnCompacts (μ (q n) ω) := by
+        let := hμr (q n) ω
+        infer_instance
+      let : MeasurableSpace Ω := ℱ t
+      have hm (n : ℕ) (f : C_c(upperHalfPlane,ℝ)) :
+          Measurable[ℱ t] (fun ω => ∫ z, f z ∂μ (q n) ω) :=
+        measurable_halfPlane_compact_test_integral (μ (q n)) (hqm n) (hμr (q n)) f
+      obtain ⟨N,hNm,hNr,hNe⟩ := exists_subsigma_radon_version_identified
+        (ℱ t) (fun n => μ (q n)) hfinite hm
+      refine ⟨N,hNm,hNr,?_⟩
+      intro ω hω
+      apply hNe ω (M t ω) (hMr t ω)
+      intro f
+      simpa only [Function.comp_def,hω.2] using
+        (hω.1 f).continuousAt.tendsto.comp hqt
+  choose N hNm hNr hNe using hex
+  refine ⟨N,hNm,hNr,?_⟩
+  filter_upwards [hcont,hrat] with ω hc hr
+  exact fun t => hNe t ω ⟨hc,hr⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassControl.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassControl.lean
index 6ea64c02ebcfb442691292deeac7fc129e583ec0..ab3ce0d1f67ecac0542adbead223296b7d8998e0 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassControl.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassControl.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Mass.RadonTestLimitProperties
-import Mathlib.Topology.Instances.Rat
-import Mathlib.Topology.DenseEmbedding
-
-namespace OAI
-
-/-! Shrinking trace intervals have vanishing nonatomic mass. This gives
-local oscillation control and continuous extension from rational times. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal
-
-namespace SLEExactGauge
-
-variable {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
-
-omit [BorelSpace X] in
-/-- Continuity and null singleton mass control every sufficiently short
-trace interval around any fixed time. It applies to a finite compact
-restriction of the terminal measure. -/
-theorem trace_interval_mass_small (ν : Measure X) [ν.OuterRegular]
-    [NullSingletonClass ν] (γ : ℝ → X) (hγ : Continuous γ)
-    (t : ℝ) (ε : ℝ≥0∞) (hε : 0 < ε) :
-    ∃ δ : ℝ, 0 < δ ∧ ∀ a b : ℝ, dist a t < δ → dist b t < δ →
-      ν (γ '' Icc a b) < ε := by
-  obtain ⟨U,hUsub,hU,hνU⟩ := ({γ t} : Set X).exists_isOpen_lt_of_lt (μ := ν) ε (by simpa using hε)
-  have htU : γ t ∈ U := hUsub (mem_singleton _)
-  have hpre : γ ⁻¹' U ∈ 𝓝 t := hγ.continuousAt.preimage_mem_nhds (hU.mem_nhds htU)
-  obtain ⟨δ,hδ,hδsub⟩ := Metric.mem_nhds_iff.mp hpre
-  refine ⟨δ,hδ,fun a b ha hb => (measure_mono ?_).trans_lt hνU⟩
-  rintro _ ⟨x,hx,rfl⟩
-  apply hδsub
-  rw [Metric.mem_ball,Real.dist_eq]
-  rw [Real.dist_eq,abs_lt] at ha hb
-  exact abs_lt.mpr ⟨lt_of_lt_of_le ha.1 (sub_le_sub_right hx.1 t),
-    lt_of_le_of_lt (sub_le_sub_right hx.2 t) hb.2⟩
-
-/-- A rational-time real observable with arbitrarily small local
-oscillation has a unique continuous extension to all real times. -/
-theorem exists_continuous_rational_extension
-    (f : ℚ → ℝ)
-    (hosc : ∀ t : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
-      ∀ p q : ℚ, dist (p:ℝ) t < δ → dist (q:ℝ) t < δ → dist (f p) (f q) < ε) :
-    ∃ F : ℝ → ℝ, Continuous F ∧ (∀ q : ℚ, F q=f q) ∧
-      ∀ G : ℝ → ℝ, Continuous G → (∀ q : ℚ, G q=f q) → G=F := by
-  let D := Rat.isDenseEmbedding_coe_real.isDenseInducing
-  have hlim (t : ℝ) : ∃ c : ℝ, Tendsto f (comap ((↑) : ℚ → ℝ) (𝓝 t)) (𝓝 c) := by
-    let : NeBot (comap ((↑) : ℚ → ℝ) (𝓝 t))  := D.comap_nhds_neBot t
-    apply cauchy_map_iff_exists_tendsto.mp
-    apply Metric.cauchy_iff.mpr
-    refine ⟨inferInstance,fun ε hε => ?_⟩
-    obtain ⟨δ,hδ,hcontrol⟩ := hosc t ε hε
-    let A : Set ℚ := ((↑) : ℚ → ℝ) ⁻¹' Metric.ball t δ
-    have hA : A ∈ comap ((↑) : ℚ → ℝ) (𝓝 t) :=
-      preimage_mem_comap (Metric.ball_mem_nhds t hδ)
-    refine ⟨f '' A,image_mem_map hA,?_⟩
-    rintro _ ⟨p,hp,rfl⟩ _ ⟨q,hq,rfl⟩
-    exact hcontrol p q hp hq
-  refine ⟨D.extend f,D.continuous_extend hlim,D.extend_eq' hlim,?_⟩
-  intro G hG heq
-  exact (D.extend_unique heq hG).symm
-
-omit [BorelSpace X] in
-/-- The trace-support estimate for increments supplies the local
-oscillation hypothesis of the rational extension theorem. -/
-theorem continuous_rational_extension_of_trace_control
-    (ν : Measure X) [ν.OuterRegular] [IsFiniteMeasure ν] [NullSingletonClass ν]
-    (γ : ℝ → X) (hγ : Continuous γ) (f : ℚ → ℝ) (C : ℝ) (hC : 0 ≤ C)
-    (hcontrol : ∀ p q : ℚ, dist (f p) (f q) ≤
-      C*(ν (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal) :
-    ∃ F : ℝ → ℝ, Continuous F ∧ (∀ q : ℚ, F q=f q) ∧
-      ∀ G : ℝ → ℝ, Continuous G → (∀ q : ℚ, G q=f q) → G=F := by
-  apply exists_continuous_rational_extension f
-  intro t ε hε
-  have he : 0 < ε/(C+1) := div_pos hε (by linarith)
-  obtain ⟨δ,hδ,hsmall⟩ := trace_interval_mass_small ν γ hγ t
-    (ENNReal.ofReal (ε/(C+1))) (ENNReal.ofReal_pos.mpr he)
-  refine ⟨δ,hδ,fun p q hp hq => ?_⟩
-  have hm : dist (min (p:ℝ) q) t < δ := by
-    rcases le_total (p:ℝ) q with h | h
-    · simpa only [min_eq_left h] using hp
-    · simpa only [min_eq_right h] using hq
-  have hM : dist (max (p:ℝ) q) t < δ := by
-    rcases le_total (p:ℝ) q with h | h
-    · simpa only [max_eq_right h] using hq
-    · simpa only [max_eq_left h] using hp
-  have hv : (ν (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal < ε/(C+1) := by
-    have hh := (ENNReal.toReal_lt_toReal (measure_ne_top ν _) ENNReal.ofReal_ne_top).mpr (hsmall _ _ hm hM)
-    simpa only [ENNReal.toReal_ofReal he.le] using hh
-  have hbound := hcontrol p q
-  have hratio : C*(ε/(C+1)) < ε := by
-    rw [← mul_div_assoc,div_lt_iff₀ (by linarith : 0<C+1)]
-    nlinarith
-  exact hbound.trans_lt ((mul_le_mul_of_nonneg_left hv.le hC).trans_lt hratio)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RadonTestLimitProperties
+import Mathlib.Topology.Instances.Rat
+import Mathlib.Topology.DenseEmbedding
+
+namespace OAI
+
+/-! Shrinking trace intervals have vanishing nonatomic mass. This gives
+local oscillation control and continuous extension from rational times. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+
+namespace SLEExactGauge
+
+variable {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
+
+omit [BorelSpace X] in
+/-- Continuity and null singleton mass control every sufficiently short
+trace interval around any fixed time. It applies to a finite compact
+restriction of the terminal measure. -/
+theorem trace_interval_mass_small (ν : Measure X) [ν.OuterRegular]
+    [NullSingletonClass ν] (γ : ℝ → X) (hγ : Continuous γ)
+    (t : ℝ) (ε : ℝ≥0∞) (hε : 0 < ε) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ a b : ℝ, dist a t < δ → dist b t < δ →
+      ν (γ '' Icc a b) < ε := by
+  obtain ⟨U,hUsub,hU,hνU⟩ := ({γ t} : Set X).exists_isOpen_lt_of_lt (μ := ν) ε (by simpa using hε)
+  have htU : γ t ∈ U := hUsub (mem_singleton _)
+  have hpre : γ ⁻¹' U ∈ 𝓝 t := hγ.continuousAt.preimage_mem_nhds (hU.mem_nhds htU)
+  obtain ⟨δ,hδ,hδsub⟩ := Metric.mem_nhds_iff.mp hpre
+  refine ⟨δ,hδ,fun a b ha hb => (measure_mono ?_).trans_lt hνU⟩
+  rintro _ ⟨x,hx,rfl⟩
+  apply hδsub
+  rw [Metric.mem_ball,Real.dist_eq]
+  rw [Real.dist_eq,abs_lt] at ha hb
+  exact abs_lt.mpr ⟨lt_of_lt_of_le ha.1 (sub_le_sub_right hx.1 t),
+    lt_of_le_of_lt (sub_le_sub_right hx.2 t) hb.2⟩
+
+/-- A rational-time real observable with arbitrarily small local
+oscillation has a unique continuous extension to all real times. -/
+theorem exists_continuous_rational_extension
+    (f : ℚ → ℝ)
+    (hosc : ∀ t : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
+      ∀ p q : ℚ, dist (p:ℝ) t < δ → dist (q:ℝ) t < δ → dist (f p) (f q) < ε) :
+    ∃ F : ℝ → ℝ, Continuous F ∧ (∀ q : ℚ, F q=f q) ∧
+      ∀ G : ℝ → ℝ, Continuous G → (∀ q : ℚ, G q=f q) → G=F := by
+  let D := Rat.isDenseEmbedding_coe_real.isDenseInducing
+  have hlim (t : ℝ) : ∃ c : ℝ, Tendsto f (comap ((↑) : ℚ → ℝ) (𝓝 t)) (𝓝 c) := by
+    let : NeBot (comap ((↑) : ℚ → ℝ) (𝓝 t))  := D.comap_nhds_neBot t
+    apply cauchy_map_iff_exists_tendsto.mp
+    apply Metric.cauchy_iff.mpr
+    refine ⟨inferInstance,fun ε hε => ?_⟩
+    obtain ⟨δ,hδ,hcontrol⟩ := hosc t ε hε
+    let A : Set ℚ := ((↑) : ℚ → ℝ) ⁻¹' Metric.ball t δ
+    have hA : A ∈ comap ((↑) : ℚ → ℝ) (𝓝 t) :=
+      preimage_mem_comap (Metric.ball_mem_nhds t hδ)
+    refine ⟨f '' A,image_mem_map hA,?_⟩
+    rintro _ ⟨p,hp,rfl⟩ _ ⟨q,hq,rfl⟩
+    exact hcontrol p q hp hq
+  refine ⟨D.extend f,D.continuous_extend hlim,D.extend_eq' hlim,?_⟩
+  intro G hG heq
+  exact (D.extend_unique heq hG).symm
+
+omit [BorelSpace X] in
+/-- The trace-support estimate for increments supplies the local
+oscillation hypothesis of the rational extension theorem. -/
+theorem continuous_rational_extension_of_trace_control
+    (ν : Measure X) [ν.OuterRegular] [IsFiniteMeasure ν] [NullSingletonClass ν]
+    (γ : ℝ → X) (hγ : Continuous γ) (f : ℚ → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hcontrol : ∀ p q : ℚ, dist (f p) (f q) ≤
+      C*(ν (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal) :
+    ∃ F : ℝ → ℝ, Continuous F ∧ (∀ q : ℚ, F q=f q) ∧
+      ∀ G : ℝ → ℝ, Continuous G → (∀ q : ℚ, G q=f q) → G=F := by
+  apply exists_continuous_rational_extension f
+  intro t ε hε
+  have he : 0 < ε/(C+1) := div_pos hε (by linarith)
+  obtain ⟨δ,hδ,hsmall⟩ := trace_interval_mass_small ν γ hγ t
+    (ENNReal.ofReal (ε/(C+1))) (ENNReal.ofReal_pos.mpr he)
+  refine ⟨δ,hδ,fun p q hp hq => ?_⟩
+  have hm : dist (min (p:ℝ) q) t < δ := by
+    rcases le_total (p:ℝ) q with h | h
+    · simpa only [min_eq_left h] using hp
+    · simpa only [min_eq_right h] using hq
+  have hM : dist (max (p:ℝ) q) t < δ := by
+    rcases le_total (p:ℝ) q with h | h
+    · simpa only [max_eq_right h] using hq
+    · simpa only [max_eq_left h] using hp
+  have hv : (ν (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal < ε/(C+1) := by
+    have hh := (ENNReal.toReal_lt_toReal (measure_ne_top ν _) ENNReal.ofReal_ne_top).mpr (hsmall _ _ hm hM)
+    simpa only [ENNReal.toReal_ofReal he.le] using hh
+  have hbound := hcontrol p q
+  have hratio : C*(ε/(C+1)) < ε := by
+    rw [← mul_div_assoc,div_lt_iff₀ (by linarith : 0<C+1)]
+    nlinarith
+  exact hbound.trans_lt ((mul_le_mul_of_nonneg_left hv.le hC).trans_lt hratio)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocal.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocal.lean
index b7a710d85f9c05217a4e7b8db72dec1e7f9ea5cc..3d39028179eefcc093421691b8a7f13f2f392237 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocal.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocal.lean
@@ -1,197 +1,197 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassVariation
-import OAI.Probability.SLE.LoewnerCharts
-
-namespace OAI
-
-/-! Rational-time reconstruction on the open half-plane. The terminal
-measure is only locally finite; the trace is allowed to touch the real
-line. Compact test supports supply the finite ambient control measures. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology ENNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-structure HalfPlaneTraceIncrements (μ : ℚ → Measure upperHalfPlane)
-    (ν : Measure upperHalfPlane) (γ : ℝ → ℂ) : Prop where
-  regular : ∀ q, (μ q).Regular
-  dominated : ∀ q, μ q ≤ ν
-  increment : ∀ p q, p ≤ q → ∃ η : Measure upperHalfPlane,
-    μ q = μ p + η ∧ η ≤ ν.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc (p:ℝ) q}
-
-lemma HalfPlaneTraceIncrements.monotone
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} {γ : ℝ → ℂ}
-    (H : HalfPlaneTraceIncrements μ ν γ) : Monotone μ := by
-  intro p q hpq
-  obtain ⟨η,hη,_⟩ := H.increment p q hpq
-  rw [hη]
-  exact Measure.le_add_right le_rfl
-
-def ambientCompactRestriction (ν : Measure upperHalfPlane) (K : Set upperHalfPlane) : Measure ℂ :=
-  Measure.map (Subtype.val : upperHalfPlane → ℂ) (ν.restrict K)
-
-lemma ambientCompactRestriction_finite (ν : Measure upperHalfPlane)
-    [IsFiniteMeasureOnCompacts ν] (K : Set upperHalfPlane) (hK : IsCompact K) :
-    IsFiniteMeasure (ambientCompactRestriction ν K) := by
-  have : Fact (ν K < ⊤) := ⟨hK.measure_lt_top⟩
-  exact inferInstanceAs (IsFiniteMeasure ((ν.restrict K).map (Subtype.val : upperHalfPlane → ℂ)))
-
-lemma ambientCompactRestriction_nullSingletons (ν : Measure upperHalfPlane)
-    [NullSingletonClass ν] (K : Set upperHalfPlane) :
-    NullSingletonClass (ambientCompactRestriction ν K) := by
-  constructor
-  intro z
-  rw [ambientCompactRestriction,Measure.map_apply measurable_subtype_coe (measurableSet_singleton z)]
-  apply Set.Subsingleton.measure_zero
-  intro a ha b hb
-  apply Subtype.ext
-  exact (mem_singleton_iff.mp ha).trans (mem_singleton_iff.mp hb).symm
-
-lemma HalfPlaneTraceIncrements.integral_dist_le
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
-    {γ : ℝ → ℂ} (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ)
-    (f : C_c(upperHalfPlane,ℝ)) (p q : ℚ) :
-    dist (∫ z, f z ∂μ p) (∫ z, f z ∂μ q) ≤ ‖f.toBoundedContinuousFunction‖ *
-      (ambientCompactRestriction ν (tsupport f)
-        (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal := by
-  have ordered (a b : ℚ) (hab : a ≤ b) :
-      dist (∫ z, f z ∂μ a) (∫ z, f z ∂μ b) ≤ ‖f.toBoundedContinuousFunction‖ *
-        (ambientCompactRestriction ν (tsupport f) (γ '' Icc (a:ℝ) b)).toReal := by
-    let K := tsupport f
-    let A : Set upperHalfPlane := {z | (z : ℂ) ∈ γ '' Icc (a:ℝ) b}
-    have hAm : MeasurableSet A :=
-      (isCompact_Icc.image hγ).measurableSet.preimage measurable_subtype_coe
-    obtain ⟨η,hη,hην⟩ := H.increment a b hab
-    have hηle : η ≤ ν := hην.trans Measure.restrict_le_self
-    let : IsFiniteMeasureOnCompacts η := ⟨fun L hL => (hηle L).trans_lt hL.measure_lt_top⟩
-    let : (μ a).Regular := H.regular a
-    have hηK : η K < ⊤ := f.hasCompactSupport.measure_lt_top
-    have : Fact (η K < ⊤) := ⟨hηK⟩
-    let : IsFiniteMeasure (η.restrict K) := inferInstance
-    let : IsFiniteMeasure (ambientCompactRestriction ν K) :=
-      ambientCompactRestriction_finite ν K f.hasCompactSupport
-    have hbound : η K ≤ ambientCompactRestriction ν K (γ '' Icc (a:ℝ) b) := by
-      calc
-        η K ≤ ν.restrict A K := hην K
-        _ = ν (K ∩ A) := Measure.restrict_apply f.hasCompactSupport.measurableSet
-        _ = ambientCompactRestriction ν K (γ '' Icc (a:ℝ) b) := by
-          rw [ambientCompactRestriction,Measure.map_apply measurable_subtype_coe
-            (isCompact_Icc.image hγ).measurableSet]
-          change ν (K ∩ A) = ν.restrict K A
-          rw [Measure.restrict_apply hAm,inter_comm]
-    rw [hη,integral_add_measure f.integrable f.integrable,dist_eq_norm]
-    have hsub : (∫ z, f z ∂μ a)-((∫ z, f z ∂μ a)+(∫ z, f z ∂η)) = -(∫ z, f z ∂η) := by ring
-    rw [hsub,norm_neg,← setIntegral_eq_integral_of_forall_compl_eq_zero
-      (fun z hz => image_eq_zero_of_notMem_tsupport hz)]
-    have hb := norm_integral_le_of_norm_le_const (μ := η.restrict K)
-      (ae_of_all _ (fun z => f.toBoundedContinuousFunction.norm_coe_le_norm z))
-    apply hb.trans
-    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
-    apply ENNReal.toReal_mono (measure_ne_top (ambientCompactRestriction ν K) _)
-    simpa only [Measure.restrict_apply_univ] using hbound
-  rcases le_total p q with h | h
-  · simpa only [min_eq_left (Rat.cast_le.mpr h),max_eq_right (Rat.cast_le.mpr h)] using ordered p q h
-  · rw [dist_comm]
-    simpa only [min_eq_right (Rat.cast_le.mpr h),max_eq_left (Rat.cast_le.mpr h)] using ordered q p h
-
-lemma HalfPlaneTraceIncrements.continuous_test_extension
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
-    [NullSingletonClass ν] {γ : ℝ → ℂ}
-    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ) (f : C_c(upperHalfPlane,ℝ)) :
-    ∃ F : ℝ → ℝ, Continuous F ∧ (∀ q : ℚ, F q = ∫ z, f z ∂μ q) ∧
-      ∀ G : ℝ → ℝ, Continuous G → (∀ q : ℚ, G q = ∫ z, f z ∂μ q) → G = F := by
-  let ρ := ambientCompactRestriction ν (tsupport f)
-  let : IsFiniteMeasure ρ := ambientCompactRestriction_finite ν _ f.hasCompactSupport
-  let : NullSingletonClass ρ := ambientCompactRestriction_nullSingletons ν _
-  exact continuous_rational_extension_of_trace_control ρ γ hγ
-    (fun q => ∫ z, f z ∂μ q) ‖f.toBoundedContinuousFunction‖ (norm_nonneg _)
-    (H.integral_dist_le hγ f)
-
-/-- One locally finite Radon family extends all rational times. Every
-compact test is continuous, even when the physical trace meets the real line. -/
-theorem HalfPlaneTraceIncrements.exists_continuous_extension
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
-    [NullSingletonClass ν] {γ : ℝ → ℂ}
-    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ) :
-    ∃ M : ℝ → Measure upperHalfPlane, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
-      (∀ q : ℚ, M q = μ q) ∧
-      (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t)) ∧ Monotone M := by
-  choose F hFc hFq _ using H.continuous_test_extension hγ
-  have hex (t : ℝ) : ∃ η : Measure upperHalfPlane, η.Regular ∧ IsFiniteMeasureOnCompacts η ∧
-      ∀ f : C_c(upperHalfPlane,ℝ), (∫ z, f z ∂η) = F f t := by
-    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-    let (n : ℕ) : (μ (q n)).Regular := H.regular (q n)
-    have hlim (f : C_c(upperHalfPlane,ℝ)) :
-        Tendsto (fun n => ∫ z, f z ∂μ (q n)) atTop (𝓝 (F f t)) := by
-      simpa only [Function.comp_def,hFq] using (hFc f).continuousAt.tendsto.comp hq
-    obtain ⟨η,hηr,hηf,hηlim⟩ := radon_limit_of_all_test_limits (fun n => μ (q n))
-      (fun f => ⟨F f t,hlim f⟩)
-    exact ⟨η,hηr,hηf,fun f => tendsto_nhds_unique (hηlim f) (hlim f)⟩
-  choose M hMr hMf hMI using hex
-  have hcont (f : C_c(upperHalfPlane,ℝ)) : Continuous (fun t => ∫ z, f z ∂M t) := by
-    simpa only [hMI] using hFc f
-  have hrat (q : ℚ) : M q = μ q := by
-    let : (M q).Regular := hMr q
-    let : (μ q).Regular := H.regular q
-    apply Measure.ext_of_integral_eq_on_compactlySupported
-    intro f
-    rw [hMI,hFq]
-  have hdom (t : ℝ) : M t ≤ ν := by
-    let : (M t).Regular := hMr t
-    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-    apply radon_measure_le_of_nonneg_tests
-    intro f hfn
-    have hlim : Tendsto (fun n => ∫ z, f z ∂μ (q n)) atTop (𝓝 (∫ z, f z ∂M t)) := by
-      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
-    apply le_of_tendsto_of_tendsto' hlim tendsto_const_nhds
-    intro n
-    exact integral_mono_measure (H.dominated (q n)) (ae_of_all ν hfn) f.integrable
-  refine ⟨M,hMr,hdom,hrat,hcont,?_⟩
-  intro s t hst
-  rcases eq_or_lt_of_le hst with rfl | hst
-  · exact le_rfl
-  let : (M s).Regular := hMr s
-  let : (M t).Regular := hMr t
-  obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
-  obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-  have hp' : ∀ᶠ n in atTop, (p n : ℝ) < (s+t)/2 := hp.eventually (gt_mem_nhds (by linarith))
-  have hq' : ∀ᶠ n in atTop, (s+t)/2 < (q n : ℝ) := hq.eventually (lt_mem_nhds (by linarith))
-  apply radon_measure_le_of_nonneg_tests
-  intro f hfn
-  have hplim : Tendsto (fun n => ∫ z, f z ∂μ (p n)) atTop (𝓝 (∫ z, f z ∂M s)) := by
-    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
-  have hqlim : Tendsto (fun n => ∫ z, f z ∂μ (q n)) atTop (𝓝 (∫ z, f z ∂M t)) := by
-    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
-  apply le_of_tendsto_of_tendsto hplim hqlim
-  filter_upwards [hp',hq'] with n hpn hqn
-  let : (μ (q n)).Regular := H.regular (q n)
-  exact integral_mono_measure (H.monotone (Rat.cast_le.mp (le_of_lt (hpn.trans hqn))))
-    (ae_of_all _ hfn) f.integrable
-
-/-- Agreement at rational times identifies continuous Radon families.
-Thus the reconstruction is independent of compact exhaustions. -/
-theorem halfPlane_continuous_measure_extension_unique
-    (M N : ℝ → Measure upperHalfPlane)
-    (hMr : ∀ t, (M t).Regular) (hNr : ∀ t, (N t).Regular)
-    (hMc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t))
-    (hNc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t))
-    (heq : ∀ q : ℚ, M q = N q) : M = N := by
-  funext t
-  let : (M t).Regular := hMr t
-  let : (N t).Regular := hNr t
-  apply Measure.ext_of_integral_eq_on_compactlySupported
-  intro f
-  obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
-  have hm := (hMc f).continuousAt.tendsto.comp hq
-  have hn := (hNc f).continuousAt.tendsto.comp hq
-  simp only [Function.comp_def,heq] at hm hn
-  exact tendsto_nhds_unique hm hn
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassVariation
+import OAI.Probability.SLE.LoewnerCharts
+
+namespace OAI
+
+/-! Rational-time reconstruction on the open half-plane. The terminal
+measure is only locally finite; the trace is allowed to touch the real
+line. Compact test supports supply the finite ambient control measures. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology ENNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+structure HalfPlaneTraceIncrements (μ : ℚ → Measure upperHalfPlane)
+    (ν : Measure upperHalfPlane) (γ : ℝ → ℂ) : Prop where
+  regular : ∀ q, (μ q).Regular
+  dominated : ∀ q, μ q ≤ ν
+  increment : ∀ p q, p ≤ q → ∃ η : Measure upperHalfPlane,
+    μ q = μ p + η ∧ η ≤ ν.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc (p:ℝ) q}
+
+lemma HalfPlaneTraceIncrements.monotone
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} {γ : ℝ → ℂ}
+    (H : HalfPlaneTraceIncrements μ ν γ) : Monotone μ := by
+  intro p q hpq
+  obtain ⟨η,hη,_⟩ := H.increment p q hpq
+  rw [hη]
+  exact Measure.le_add_right le_rfl
+
+def ambientCompactRestriction (ν : Measure upperHalfPlane) (K : Set upperHalfPlane) : Measure ℂ :=
+  Measure.map (Subtype.val : upperHalfPlane → ℂ) (ν.restrict K)
+
+lemma ambientCompactRestriction_finite (ν : Measure upperHalfPlane)
+    [IsFiniteMeasureOnCompacts ν] (K : Set upperHalfPlane) (hK : IsCompact K) :
+    IsFiniteMeasure (ambientCompactRestriction ν K) := by
+  have : Fact (ν K < ⊤) := ⟨hK.measure_lt_top⟩
+  exact inferInstanceAs (IsFiniteMeasure ((ν.restrict K).map (Subtype.val : upperHalfPlane → ℂ)))
+
+lemma ambientCompactRestriction_nullSingletons (ν : Measure upperHalfPlane)
+    [NullSingletonClass ν] (K : Set upperHalfPlane) :
+    NullSingletonClass (ambientCompactRestriction ν K) := by
+  constructor
+  intro z
+  rw [ambientCompactRestriction,Measure.map_apply measurable_subtype_coe (measurableSet_singleton z)]
+  apply Set.Subsingleton.measure_zero
+  intro a ha b hb
+  apply Subtype.ext
+  exact (mem_singleton_iff.mp ha).trans (mem_singleton_iff.mp hb).symm
+
+lemma HalfPlaneTraceIncrements.integral_dist_le
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
+    {γ : ℝ → ℂ} (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ)
+    (f : C_c(upperHalfPlane,ℝ)) (p q : ℚ) :
+    dist (∫ z, f z ∂μ p) (∫ z, f z ∂μ q) ≤ ‖f.toBoundedContinuousFunction‖ *
+      (ambientCompactRestriction ν (tsupport f)
+        (γ '' Icc (min (p:ℝ) q) (max (p:ℝ) q))).toReal := by
+  have ordered (a b : ℚ) (hab : a ≤ b) :
+      dist (∫ z, f z ∂μ a) (∫ z, f z ∂μ b) ≤ ‖f.toBoundedContinuousFunction‖ *
+        (ambientCompactRestriction ν (tsupport f) (γ '' Icc (a:ℝ) b)).toReal := by
+    let K := tsupport f
+    let A : Set upperHalfPlane := {z | (z : ℂ) ∈ γ '' Icc (a:ℝ) b}
+    have hAm : MeasurableSet A :=
+      (isCompact_Icc.image hγ).measurableSet.preimage measurable_subtype_coe
+    obtain ⟨η,hη,hην⟩ := H.increment a b hab
+    have hηle : η ≤ ν := hην.trans Measure.restrict_le_self
+    let : IsFiniteMeasureOnCompacts η := ⟨fun L hL => (hηle L).trans_lt hL.measure_lt_top⟩
+    let : (μ a).Regular := H.regular a
+    have hηK : η K < ⊤ := f.hasCompactSupport.measure_lt_top
+    have : Fact (η K < ⊤) := ⟨hηK⟩
+    let : IsFiniteMeasure (η.restrict K) := inferInstance
+    let : IsFiniteMeasure (ambientCompactRestriction ν K) :=
+      ambientCompactRestriction_finite ν K f.hasCompactSupport
+    have hbound : η K ≤ ambientCompactRestriction ν K (γ '' Icc (a:ℝ) b) := by
+      calc
+        η K ≤ ν.restrict A K := hην K
+        _ = ν (K ∩ A) := Measure.restrict_apply f.hasCompactSupport.measurableSet
+        _ = ambientCompactRestriction ν K (γ '' Icc (a:ℝ) b) := by
+          rw [ambientCompactRestriction,Measure.map_apply measurable_subtype_coe
+            (isCompact_Icc.image hγ).measurableSet]
+          change ν (K ∩ A) = ν.restrict K A
+          rw [Measure.restrict_apply hAm,inter_comm]
+    rw [hη,integral_add_measure f.integrable f.integrable,dist_eq_norm]
+    have hsub : (∫ z, f z ∂μ a)-((∫ z, f z ∂μ a)+(∫ z, f z ∂η)) = -(∫ z, f z ∂η) := by ring
+    rw [hsub,norm_neg,← setIntegral_eq_integral_of_forall_compl_eq_zero
+      (fun z hz => image_eq_zero_of_notMem_tsupport hz)]
+    have hb := norm_integral_le_of_norm_le_const (μ := η.restrict K)
+      (ae_of_all _ (fun z => f.toBoundedContinuousFunction.norm_coe_le_norm z))
+    apply hb.trans
+    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+    apply ENNReal.toReal_mono (measure_ne_top (ambientCompactRestriction ν K) _)
+    simpa only [Measure.restrict_apply_univ] using hbound
+  rcases le_total p q with h | h
+  · simpa only [min_eq_left (Rat.cast_le.mpr h),max_eq_right (Rat.cast_le.mpr h)] using ordered p q h
+  · rw [dist_comm]
+    simpa only [min_eq_right (Rat.cast_le.mpr h),max_eq_left (Rat.cast_le.mpr h)] using ordered q p h
+
+lemma HalfPlaneTraceIncrements.continuous_test_extension
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
+    [NullSingletonClass ν] {γ : ℝ → ℂ}
+    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ) (f : C_c(upperHalfPlane,ℝ)) :
+    ∃ F : ℝ → ℝ, Continuous F ∧ (∀ q : ℚ, F q = ∫ z, f z ∂μ q) ∧
+      ∀ G : ℝ → ℝ, Continuous G → (∀ q : ℚ, G q = ∫ z, f z ∂μ q) → G = F := by
+  let ρ := ambientCompactRestriction ν (tsupport f)
+  let : IsFiniteMeasure ρ := ambientCompactRestriction_finite ν _ f.hasCompactSupport
+  let : NullSingletonClass ρ := ambientCompactRestriction_nullSingletons ν _
+  exact continuous_rational_extension_of_trace_control ρ γ hγ
+    (fun q => ∫ z, f z ∂μ q) ‖f.toBoundedContinuousFunction‖ (norm_nonneg _)
+    (H.integral_dist_le hγ f)
+
+/-- One locally finite Radon family extends all rational times. Every
+compact test is continuous, even when the physical trace meets the real line. -/
+theorem HalfPlaneTraceIncrements.exists_continuous_extension
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
+    [NullSingletonClass ν] {γ : ℝ → ℂ}
+    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ) :
+    ∃ M : ℝ → Measure upperHalfPlane, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
+      (∀ q : ℚ, M q = μ q) ∧
+      (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t)) ∧ Monotone M := by
+  choose F hFc hFq _ using H.continuous_test_extension hγ
+  have hex (t : ℝ) : ∃ η : Measure upperHalfPlane, η.Regular ∧ IsFiniteMeasureOnCompacts η ∧
+      ∀ f : C_c(upperHalfPlane,ℝ), (∫ z, f z ∂η) = F f t := by
+    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+    let (n : ℕ) : (μ (q n)).Regular := H.regular (q n)
+    have hlim (f : C_c(upperHalfPlane,ℝ)) :
+        Tendsto (fun n => ∫ z, f z ∂μ (q n)) atTop (𝓝 (F f t)) := by
+      simpa only [Function.comp_def,hFq] using (hFc f).continuousAt.tendsto.comp hq
+    obtain ⟨η,hηr,hηf,hηlim⟩ := radon_limit_of_all_test_limits (fun n => μ (q n))
+      (fun f => ⟨F f t,hlim f⟩)
+    exact ⟨η,hηr,hηf,fun f => tendsto_nhds_unique (hηlim f) (hlim f)⟩
+  choose M hMr hMf hMI using hex
+  have hcont (f : C_c(upperHalfPlane,ℝ)) : Continuous (fun t => ∫ z, f z ∂M t) := by
+    simpa only [hMI] using hFc f
+  have hrat (q : ℚ) : M q = μ q := by
+    let : (M q).Regular := hMr q
+    let : (μ q).Regular := H.regular q
+    apply Measure.ext_of_integral_eq_on_compactlySupported
+    intro f
+    rw [hMI,hFq]
+  have hdom (t : ℝ) : M t ≤ ν := by
+    let : (M t).Regular := hMr t
+    obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+    apply radon_measure_le_of_nonneg_tests
+    intro f hfn
+    have hlim : Tendsto (fun n => ∫ z, f z ∂μ (q n)) atTop (𝓝 (∫ z, f z ∂M t)) := by
+      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
+    apply le_of_tendsto_of_tendsto' hlim tendsto_const_nhds
+    intro n
+    exact integral_mono_measure (H.dominated (q n)) (ae_of_all ν hfn) f.integrable
+  refine ⟨M,hMr,hdom,hrat,hcont,?_⟩
+  intro s t hst
+  rcases eq_or_lt_of_le hst with rfl | hst
+  · exact le_rfl
+  let : (M s).Regular := hMr s
+  let : (M t).Regular := hMr t
+  obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
+  obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+  have hp' : ∀ᶠ n in atTop, (p n : ℝ) < (s+t)/2 := hp.eventually (gt_mem_nhds (by linarith))
+  have hq' : ∀ᶠ n in atTop, (s+t)/2 < (q n : ℝ) := hq.eventually (lt_mem_nhds (by linarith))
+  apply radon_measure_le_of_nonneg_tests
+  intro f hfn
+  have hplim : Tendsto (fun n => ∫ z, f z ∂μ (p n)) atTop (𝓝 (∫ z, f z ∂M s)) := by
+    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
+  have hqlim : Tendsto (fun n => ∫ z, f z ∂μ (q n)) atTop (𝓝 (∫ z, f z ∂M t)) := by
+    simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hq
+  apply le_of_tendsto_of_tendsto hplim hqlim
+  filter_upwards [hp',hq'] with n hpn hqn
+  let : (μ (q n)).Regular := H.regular (q n)
+  exact integral_mono_measure (H.monotone (Rat.cast_le.mp (le_of_lt (hpn.trans hqn))))
+    (ae_of_all _ hfn) f.integrable
+
+/-- Agreement at rational times identifies continuous Radon families.
+Thus the reconstruction is independent of compact exhaustions. -/
+theorem halfPlane_continuous_measure_extension_unique
+    (M N : ℝ → Measure upperHalfPlane)
+    (hMr : ∀ t, (M t).Regular) (hNr : ∀ t, (N t).Regular)
+    (hMc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t))
+    (hNc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t))
+    (heq : ∀ q : ℚ, M q = N q) : M = N := by
+  funext t
+  let : (M t).Regular := hMr t
+  let : (N t).Regular := hNr t
+  apply Measure.ext_of_integral_eq_on_compactlySupported
+  intro f
+  obtain ⟨q,hq⟩ := exists_rational_sequence_tendsto t
+  have hm := (hMc f).continuousAt.tendsto.comp hq
+  have hn := (hNc f).continuousAt.tendsto.comp hq
+  simp only [Function.comp_def,heq] at hm hn
+  exact tendsto_nhds_unique hm hn
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocalVariation.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocalVariation.lean
index e599d17633d7dabeae5dcc331658285a030c2189..3058d583022fc4cbbae1a38c07f2f27ff6446338 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocalVariation.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassLocalVariation.lean
@@ -1,139 +1,139 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassLocal
-
-namespace OAI
-
-/-! Local total-variation continuity of the half-plane reconstruction.
-Only the mass of a fixed compact is used; the terminal mass may be infinite. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology ENNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-private lemma halfPlane_regular_add (μ ν : Measure upperHalfPlane) [μ.Regular] [ν.Regular] :
-    (μ + ν).Regular := by
-  let : IsFiniteMeasureOnCompacts (μ + ν) := ⟨fun K hK => by
-    rw [Measure.add_apply]
-    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩⟩
-  infer_instance
-
-theorem HalfPlaneTraceIncrements.local_variation_continuity
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
-    [NullSingletonClass ν] {γ : ℝ → ℂ}
-    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ)
-    (M : ℝ → Measure upperHalfPlane) (hMr : ∀ t, (M t).Regular)
-    (hdom : ∀ t, M t ≤ ν) (hrat : ∀ q : ℚ, M q = μ q)
-    (hcont : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t))
-    (K : Set upperHalfPlane) (hK : IsCompact K) :
-    ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
-      ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
-        ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K →
-          |(M s A).toReal-(M t A).toReal| < ε := by
-  intro t₀ ε hε
-  let ρ := ambientCompactRestriction ν K
-  let : IsFiniteMeasure ρ := ambientCompactRestriction_finite ν K hK
-  let : NullSingletonClass ρ := ambientCompactRestriction_nullSingletons ν K
-  obtain ⟨U,hUsub,hU,hρU⟩ := ({γ t₀} : Set ℂ).exists_isOpen_lt_of_lt (μ := ρ)
-    (ENNReal.ofReal ε) (by simpa using ENNReal.ofReal_pos.mpr hε)
-  let V : Set upperHalfPlane := (Subtype.val : upperHalfPlane → ℂ) ⁻¹' U
-  have hV : IsOpen V := hU.preimage continuous_subtype_val
-  have htU : γ t₀ ∈ U := hUsub (mem_singleton _)
-  obtain ⟨δ,hδ,hδsub⟩ := Metric.mem_nhds_iff.mp
-    (hγ.continuousAt.preimage_mem_nhds (hU.mem_nhds htU))
-  have hI (a b : ℝ) (ha : dist a t₀ < δ) (hb : dist b t₀ < δ) : γ '' Icc a b ⊆ U := by
-    rintro _ ⟨x,hx,rfl⟩
-    apply hδsub
-    rw [Metric.mem_ball,Real.dist_eq]
-    rw [Real.dist_eq,abs_lt] at ha hb
-    exact abs_lt.mpr ⟨lt_of_lt_of_le ha.1 (sub_le_sub_right hx.1 t₀),
-      lt_of_le_of_lt (sub_le_sub_right hx.2 t₀) hb.2⟩
-  have hq (p q : ℚ) (hp : dist (p:ℝ) t₀ < δ) (hq : dist (q:ℝ) t₀ < δ) :
-      μ p ≤ μ q+ν.restrict V := by
-    rcases le_total p q with hpq | hqp
-    · exact (H.monotone hpq).trans (Measure.le_add_right le_rfl)
-    · obtain ⟨η,hη,hην⟩ := H.increment q p hqp
-      rw [hη]
-      apply add_le_add le_rfl
-      exact hην.trans (Measure.restrict_mono
-        (show {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc (q:ℝ) p} ⊆ V
-          from fun z hz => hI q p hq hp hz) le_rfl)
-  have hreal (s t : ℝ) (hs : dist s t₀ < δ) (ht : dist t t₀ < δ) :
-      M s ≤ M t+ν.restrict V := by
-    let : (M s).Regular := hMr s
-    let : (M t).Regular := hMr t
-    let : (ν.restrict V).Regular := inferInstance
-    let : (M t+ν.restrict V).Regular := halfPlane_regular_add _ _
-    obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
-    obtain ⟨q,hqt⟩ := exists_rational_sequence_tendsto t
-    have hpU : ∀ᶠ n in atTop, dist (p n:ℝ) t₀ < δ :=
-      hp.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds hs)
-    have hqU : ∀ᶠ n in atTop, dist (q n:ℝ) t₀ < δ :=
-      hqt.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds ht)
-    apply radon_measure_le_of_nonneg_tests
-    intro f hfn
-    have hplim : Tendsto (fun n => ∫ z, f z ∂μ (p n)) atTop (𝓝 (∫ z, f z ∂M s)) := by
-      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
-    have hqlim : Tendsto (fun n => ∫ z, f z ∂(μ (q n)+ν.restrict V)) atTop
-        (𝓝 (∫ z, f z ∂(M t+ν.restrict V))) := by
-      have hi := ((hcont f).continuousAt.tendsto.comp hqt).add
-        (tendsto_const_nhds (x := ∫ z, f z ∂ν.restrict V))
-      simp only [Function.comp_def,hrat] at hi
-      have hqadd (n : ℕ) : (∫ z, f z ∂(μ (q n)+ν.restrict V)) =
-          (∫ z, f z ∂μ (q n))+(∫ z, f z ∂ν.restrict V) := by
-        let : (μ (q n)).Regular := H.regular (q n)
-        exact integral_add_measure f.integrable f.integrable
-      simpa only [hqadd,integral_add_measure f.integrable f.integrable] using hi
-    apply le_of_tendsto_of_tendsto hplim hqlim
-    filter_upwards [hpU,hqU] with n hpn hqn
-    let : (μ (q n)).Regular := H.regular (q n)
-    let : (μ (q n)+ν.restrict V).Regular := halfPlane_regular_add _ _
-    exact integral_mono_measure (hq (p n) (q n) hpn hqn) (ae_of_all _ hfn) f.integrable
-  refine ⟨δ,hδ,fun s t hs ht A hA hAK => ?_⟩
-  have hst := hreal s t hs ht A
-  have hts := hreal t s ht hs A
-  have hsn : M s A ≠ ⊤ := ((hdom s A).trans ((measure_mono hAK))).trans_lt hK.measure_lt_top |>.ne
-  have htn : M t A ≠ ⊤ := ((hdom t A).trans ((measure_mono hAK))).trans_lt hK.measure_lt_top |>.ne
-  have hUn : ρ U ≠ ⊤ := measure_ne_top ρ U
-  rw [Measure.add_apply,Measure.restrict_apply hA,inter_comm] at hst hts
-  have hUA : ν (V ∩ A) ≤ ρ U := by
-    change ν (V ∩ A) ≤ ambientCompactRestriction ν K U
-    rw [ambientCompactRestriction,Measure.map_apply measurable_subtype_coe hU.measurableSet]
-    change ν (V ∩ A) ≤ ν.restrict K V
-    rw [Measure.restrict_apply hV.measurableSet]
-    exact measure_mono (fun z hz => ⟨hz.1,hAK hz.2⟩)
-  have hcan : ν (V ∩ A) ≠ ⊤ := (hUA.trans_lt hUn.lt_top).ne
-  have hst' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨htn,hcan⟩) hst
-  have hts' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hsn,hcan⟩) hts
-  rw [ENNReal.toReal_add htn hcan] at hst'
-  rw [ENNReal.toReal_add hsn hcan] at hts'
-  have hless : (ν (V ∩ A)).toReal < ε := by
-    have hUl : (ρ U).toReal < ε := by
-      simpa only [ENNReal.toReal_ofReal hε.le] using
-        (ENNReal.toReal_lt_toReal hUn ENNReal.ofReal_ne_top).mpr hρU
-    exact (ENNReal.toReal_mono hUn hUA).trans_lt hUl
-  exact abs_lt.mpr ⟨by linarith,by linarith⟩
-
-theorem HalfPlaneTraceIncrements.exists_locally_tv_continuous_extension
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
-    [NullSingletonClass ν] {γ : ℝ → ℂ}
-    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ) :
-    ∃ M : ℝ → Measure upperHalfPlane, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
-      (∀ q : ℚ, M q = μ q) ∧ Monotone M ∧
-      (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t)) ∧
-      ∀ K : Set upperHalfPlane, IsCompact K →
-        ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
-          ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
-            ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K →
-              |(M s A).toReal-(M t A).toReal| < ε := by
-  obtain ⟨M,hMr,hdom,hrat,hcont,hmono⟩ := H.exists_continuous_extension hγ
-  exact ⟨M,hMr,hdom,hrat,hmono,hcont,
-    fun K hK => H.local_variation_continuity hγ M hMr hdom hrat hcont K hK⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassLocal
+
+namespace OAI
+
+/-! Local total-variation continuity of the half-plane reconstruction.
+Only the mass of a fixed compact is used; the terminal mass may be infinite. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology ENNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+private lemma halfPlane_regular_add (μ ν : Measure upperHalfPlane) [μ.Regular] [ν.Regular] :
+    (μ + ν).Regular := by
+  let : IsFiniteMeasureOnCompacts (μ + ν) := ⟨fun K hK => by
+    rw [Measure.add_apply]
+    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩⟩
+  infer_instance
+
+theorem HalfPlaneTraceIncrements.local_variation_continuity
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
+    [NullSingletonClass ν] {γ : ℝ → ℂ}
+    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ)
+    (M : ℝ → Measure upperHalfPlane) (hMr : ∀ t, (M t).Regular)
+    (hdom : ∀ t, M t ≤ ν) (hrat : ∀ q : ℚ, M q = μ q)
+    (hcont : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t))
+    (K : Set upperHalfPlane) (hK : IsCompact K) :
+    ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
+      ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
+        ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K →
+          |(M s A).toReal-(M t A).toReal| < ε := by
+  intro t₀ ε hε
+  let ρ := ambientCompactRestriction ν K
+  let : IsFiniteMeasure ρ := ambientCompactRestriction_finite ν K hK
+  let : NullSingletonClass ρ := ambientCompactRestriction_nullSingletons ν K
+  obtain ⟨U,hUsub,hU,hρU⟩ := ({γ t₀} : Set ℂ).exists_isOpen_lt_of_lt (μ := ρ)
+    (ENNReal.ofReal ε) (by simpa using ENNReal.ofReal_pos.mpr hε)
+  let V : Set upperHalfPlane := (Subtype.val : upperHalfPlane → ℂ) ⁻¹' U
+  have hV : IsOpen V := hU.preimage continuous_subtype_val
+  have htU : γ t₀ ∈ U := hUsub (mem_singleton _)
+  obtain ⟨δ,hδ,hδsub⟩ := Metric.mem_nhds_iff.mp
+    (hγ.continuousAt.preimage_mem_nhds (hU.mem_nhds htU))
+  have hI (a b : ℝ) (ha : dist a t₀ < δ) (hb : dist b t₀ < δ) : γ '' Icc a b ⊆ U := by
+    rintro _ ⟨x,hx,rfl⟩
+    apply hδsub
+    rw [Metric.mem_ball,Real.dist_eq]
+    rw [Real.dist_eq,abs_lt] at ha hb
+    exact abs_lt.mpr ⟨lt_of_lt_of_le ha.1 (sub_le_sub_right hx.1 t₀),
+      lt_of_le_of_lt (sub_le_sub_right hx.2 t₀) hb.2⟩
+  have hq (p q : ℚ) (hp : dist (p:ℝ) t₀ < δ) (hq : dist (q:ℝ) t₀ < δ) :
+      μ p ≤ μ q+ν.restrict V := by
+    rcases le_total p q with hpq | hqp
+    · exact (H.monotone hpq).trans (Measure.le_add_right le_rfl)
+    · obtain ⟨η,hη,hην⟩ := H.increment q p hqp
+      rw [hη]
+      apply add_le_add le_rfl
+      exact hην.trans (Measure.restrict_mono
+        (show {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc (q:ℝ) p} ⊆ V
+          from fun z hz => hI q p hq hp hz) le_rfl)
+  have hreal (s t : ℝ) (hs : dist s t₀ < δ) (ht : dist t t₀ < δ) :
+      M s ≤ M t+ν.restrict V := by
+    let : (M s).Regular := hMr s
+    let : (M t).Regular := hMr t
+    let : (ν.restrict V).Regular := inferInstance
+    let : (M t+ν.restrict V).Regular := halfPlane_regular_add _ _
+    obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
+    obtain ⟨q,hqt⟩ := exists_rational_sequence_tendsto t
+    have hpU : ∀ᶠ n in atTop, dist (p n:ℝ) t₀ < δ :=
+      hp.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds hs)
+    have hqU : ∀ᶠ n in atTop, dist (q n:ℝ) t₀ < δ :=
+      hqt.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds ht)
+    apply radon_measure_le_of_nonneg_tests
+    intro f hfn
+    have hplim : Tendsto (fun n => ∫ z, f z ∂μ (p n)) atTop (𝓝 (∫ z, f z ∂M s)) := by
+      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
+    have hqlim : Tendsto (fun n => ∫ z, f z ∂(μ (q n)+ν.restrict V)) atTop
+        (𝓝 (∫ z, f z ∂(M t+ν.restrict V))) := by
+      have hi := ((hcont f).continuousAt.tendsto.comp hqt).add
+        (tendsto_const_nhds (x := ∫ z, f z ∂ν.restrict V))
+      simp only [Function.comp_def,hrat] at hi
+      have hqadd (n : ℕ) : (∫ z, f z ∂(μ (q n)+ν.restrict V)) =
+          (∫ z, f z ∂μ (q n))+(∫ z, f z ∂ν.restrict V) := by
+        let : (μ (q n)).Regular := H.regular (q n)
+        exact integral_add_measure f.integrable f.integrable
+      simpa only [hqadd,integral_add_measure f.integrable f.integrable] using hi
+    apply le_of_tendsto_of_tendsto hplim hqlim
+    filter_upwards [hpU,hqU] with n hpn hqn
+    let : (μ (q n)).Regular := H.regular (q n)
+    let : (μ (q n)+ν.restrict V).Regular := halfPlane_regular_add _ _
+    exact integral_mono_measure (hq (p n) (q n) hpn hqn) (ae_of_all _ hfn) f.integrable
+  refine ⟨δ,hδ,fun s t hs ht A hA hAK => ?_⟩
+  have hst := hreal s t hs ht A
+  have hts := hreal t s ht hs A
+  have hsn : M s A ≠ ⊤ := ((hdom s A).trans ((measure_mono hAK))).trans_lt hK.measure_lt_top |>.ne
+  have htn : M t A ≠ ⊤ := ((hdom t A).trans ((measure_mono hAK))).trans_lt hK.measure_lt_top |>.ne
+  have hUn : ρ U ≠ ⊤ := measure_ne_top ρ U
+  rw [Measure.add_apply,Measure.restrict_apply hA,inter_comm] at hst hts
+  have hUA : ν (V ∩ A) ≤ ρ U := by
+    change ν (V ∩ A) ≤ ambientCompactRestriction ν K U
+    rw [ambientCompactRestriction,Measure.map_apply measurable_subtype_coe hU.measurableSet]
+    change ν (V ∩ A) ≤ ν.restrict K V
+    rw [Measure.restrict_apply hV.measurableSet]
+    exact measure_mono (fun z hz => ⟨hz.1,hAK hz.2⟩)
+  have hcan : ν (V ∩ A) ≠ ⊤ := (hUA.trans_lt hUn.lt_top).ne
+  have hst' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨htn,hcan⟩) hst
+  have hts' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hsn,hcan⟩) hts
+  rw [ENNReal.toReal_add htn hcan] at hst'
+  rw [ENNReal.toReal_add hsn hcan] at hts'
+  have hless : (ν (V ∩ A)).toReal < ε := by
+    have hUl : (ρ U).toReal < ε := by
+      simpa only [ENNReal.toReal_ofReal hε.le] using
+        (ENNReal.toReal_lt_toReal hUn ENNReal.ofReal_ne_top).mpr hρU
+    exact (ENNReal.toReal_mono hUn hUA).trans_lt hUl
+  exact abs_lt.mpr ⟨by linarith,by linarith⟩
+
+theorem HalfPlaneTraceIncrements.exists_locally_tv_continuous_extension
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
+    [NullSingletonClass ν] {γ : ℝ → ℂ}
+    (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ) :
+    ∃ M : ℝ → Measure upperHalfPlane, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
+      (∀ q : ℚ, M q = μ q) ∧ Monotone M ∧
+      (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t)) ∧
+      ∀ K : Set upperHalfPlane, IsCompact K →
+        ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
+          ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
+            ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K →
+              |(M s A).toReal-(M t A).toReal| < ε := by
+  obtain ⟨M,hMr,hdom,hrat,hcont,hmono⟩ := H.exists_continuous_extension hγ
+  exact ⟨M,hMr,hdom,hrat,hmono,hcont,
+    fun K hK => H.local_variation_continuity hγ M hMr hdom hrat hcont K hK⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassProgressiveVersion.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassProgressiveVersion.lean
index 98a573938e36c64dfca943ade7a255c8f3ac344e..66a6faa73b29ccf2eb32e293776c1381f00dd1a0 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassProgressiveVersion.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassProgressiveVersion.lean
@@ -1,111 +1,111 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassAdaptedVersion
-import OAI.Probability.SLE.Mass.CanonicalRadonLimit
-import Mathlib.MeasureTheory.Function.Floor
-import Mathlib.Analysis.SpecificLimits.Basic
-
-namespace OAI
-
-/-! A single reconstruction from rational left samples is progressive in
-the uncompleted filtration and agrees at every time on the common good set. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology NNReal ENNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-def leftRationalSample (n : ℕ) (t : ℝ≥0) : ℚ :=
-  (⌊(t:ℝ)*((n:ℝ)+1)⌋₊ : ℚ)/((n:ℚ)+1)
-
-lemma leftRationalSample_nonneg (n : ℕ) (t : ℝ≥0) : 0 ≤ leftRationalSample n t := by
-  unfold leftRationalSample
-  positivity
-
-lemma leftRationalSample_le (n : ℕ) (t : ℝ≥0) : (leftRationalSample n t : ℝ) ≤ t := by
-  unfold leftRationalSample
-  push_cast
-  apply (div_le_iff₀ (by positivity : 0 < (n:ℝ)+1)).mpr
-  exact Nat.floor_le (mul_nonneg t.coe_nonneg (by positivity))
-
-lemma leftRationalSample_tendsto (t : ℝ≥0) :
-    Tendsto (fun n => (leftRationalSample n t : ℝ)) atTop (𝓝 (t:ℝ)) := by
-  have h := (tendsto_nat_floor_mul_div_atTop t.coe_nonneg).comp
-    (tendsto_atTop_add_const_right atTop (1:ℝ) (tendsto_natCast_atTop_atTop (R := ℝ)))
-  simpa only [leftRationalSample,Rat.cast_div,Rat.cast_natCast,Rat.cast_add,Rat.cast_one,
-    Function.comp_def] using h
-
-lemma leftRationalSample_measurable (n : ℕ) : Measurable (leftRationalSample n) := by
-  exact (measurable_of_countable (fun k : ℕ => (k:ℚ)/((n:ℚ)+1))).comp
-    ((NNReal.continuous_coe.measurable.mul_const ((n:ℝ)+1)).nat_floor)
-
-def rationalRadonExtension {Ω : Type*} (μ : ℚ → Ω → Measure upperHalfPlane)
-    (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane :=
-  canonicalRadonLimit (fun n => μ (leftRationalSample n t) ω)
-
-theorem rationalRadonExtension_regular {Ω : Type*}
-    (μ : ℚ → Ω → Measure upperHalfPlane) (t : ℝ≥0) (ω : Ω) :
-    (rationalRadonExtension μ t ω).Regular ∧
-      IsFiniteMeasureOnCompacts (rationalRadonExtension μ t ω) := by
-  have h : (rationalRadonExtension μ t ω).Regular :=
-    canonicalRadonLimit_regular (fun n => μ (leftRationalSample n t) ω)
-  let := h
-  exact ⟨h,inferInstance⟩
-
-theorem rationalRadonExtension_progressive
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    (μ : ℚ → Ω → Measure upperHalfPlane)
-    (hμr : ∀ q ω, (μ q ω).Regular)
-    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
-    IsProgressive ℱ (rationalRadonExtension μ) := by
-  classical
-  intro T
-  let : MeasurableSpace Ω := ℱ T
-  have hm (n : ℕ) : Measurable (fun p : Iic T × Ω => μ (leftRationalSample n p.1) p.2) := by
-    let ν (q : ℚ) (ω : Ω) := if 0 ≤ q ∧ (q:ℝ) ≤ T then μ q ω else 0
-    have hνm (q : ℚ) : Measurable (ν q) := by
-      by_cases hq : 0 ≤ q ∧ (q:ℝ) ≤ T
-      · have hqt : (q:ℝ).toNNReal ≤ T := by
-          simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hq.2
-        simpa only [ν,ite_eq_left hq] using (hμm q hq.1).mono (ℱ.mono hqt) le_rfl
-      · simpa only [ν,ite_eq_right hq] using (measurable_const : Measurable (fun _ : Ω => (0 : Measure upperHalfPlane)))
-    have hv : Measurable (fun p : ℚ × Ω => ν p.1 p.2) :=
-      measurable_from_prod_countable_right hνm
-    have hq : Measurable (fun p : Iic T × Ω => (leftRationalSample n p.1,p.2)) :=
-      ((leftRationalSample_measurable n).comp
-        (measurable_subtype_coe.comp measurable_fst)).prodMk measurable_snd
-    have h := hv.comp hq
-    convert h using 1
-    funext p
-    dsimp only [Function.comp_def,ν]
-    rw [ite_eq_left ⟨leftRationalSample_nonneg n p.1,
-      (leftRationalSample_le n p.1).trans (NNReal.coe_le_coe.mpr p.1.property)⟩]
-  apply canonicalRadonLimit_measurable (fun (n : ℕ) (p : Iic T × Ω) => μ (leftRationalSample n p.1) p.2)
-  · intro n p
-    let := hμr (leftRationalSample n p.1) p.2
-    infer_instance
-  · intro n f
-    exact measurable_halfPlane_compact_test_integral _ (hm n)
-      (fun p => hμr (leftRationalSample n p.1) p.2) f
-
-theorem rationalRadonExtension_common_version
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
-    (hMr : ∀ t ω, (M t ω).Regular)
-    (hcont : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t ω))
-    (hrat : ∀ᵐ ω ∂P, ∀ q : ℚ, M q ω = μ q ω) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, rationalRadonExtension μ t ω = M t ω := by
-  filter_upwards [hcont,hrat] with ω hc hr
-  intro t
-  apply canonicalRadonLimit_identified _ (M t ω)
-  refine ⟨hMr t ω,?_⟩
-  intro f
-  simpa only [Function.comp_def,hr] using
-    (hc f).continuousAt.tendsto.comp (leftRationalSample_tendsto t)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassAdaptedVersion
+import OAI.Probability.SLE.Mass.CanonicalRadonLimit
+import Mathlib.MeasureTheory.Function.Floor
+import Mathlib.Analysis.SpecificLimits.Basic
+
+namespace OAI
+
+/-! A single reconstruction from rational left samples is progressive in
+the uncompleted filtration and agrees at every time on the common good set. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology NNReal ENNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+def leftRationalSample (n : ℕ) (t : ℝ≥0) : ℚ :=
+  (⌊(t:ℝ)*((n:ℝ)+1)⌋₊ : ℚ)/((n:ℚ)+1)
+
+lemma leftRationalSample_nonneg (n : ℕ) (t : ℝ≥0) : 0 ≤ leftRationalSample n t := by
+  unfold leftRationalSample
+  positivity
+
+lemma leftRationalSample_le (n : ℕ) (t : ℝ≥0) : (leftRationalSample n t : ℝ) ≤ t := by
+  unfold leftRationalSample
+  push_cast
+  apply (div_le_iff₀ (by positivity : 0 < (n:ℝ)+1)).mpr
+  exact Nat.floor_le (mul_nonneg t.coe_nonneg (by positivity))
+
+lemma leftRationalSample_tendsto (t : ℝ≥0) :
+    Tendsto (fun n => (leftRationalSample n t : ℝ)) atTop (𝓝 (t:ℝ)) := by
+  have h := (tendsto_nat_floor_mul_div_atTop t.coe_nonneg).comp
+    (tendsto_atTop_add_const_right atTop (1:ℝ) (tendsto_natCast_atTop_atTop (R := ℝ)))
+  simpa only [leftRationalSample,Rat.cast_div,Rat.cast_natCast,Rat.cast_add,Rat.cast_one,
+    Function.comp_def] using h
+
+lemma leftRationalSample_measurable (n : ℕ) : Measurable (leftRationalSample n) := by
+  exact (measurable_of_countable (fun k : ℕ => (k:ℚ)/((n:ℚ)+1))).comp
+    ((NNReal.continuous_coe.measurable.mul_const ((n:ℝ)+1)).nat_floor)
+
+def rationalRadonExtension {Ω : Type*} (μ : ℚ → Ω → Measure upperHalfPlane)
+    (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane :=
+  canonicalRadonLimit (fun n => μ (leftRationalSample n t) ω)
+
+theorem rationalRadonExtension_regular {Ω : Type*}
+    (μ : ℚ → Ω → Measure upperHalfPlane) (t : ℝ≥0) (ω : Ω) :
+    (rationalRadonExtension μ t ω).Regular ∧
+      IsFiniteMeasureOnCompacts (rationalRadonExtension μ t ω) := by
+  have h : (rationalRadonExtension μ t ω).Regular :=
+    canonicalRadonLimit_regular (fun n => μ (leftRationalSample n t) ω)
+  let := h
+  exact ⟨h,inferInstance⟩
+
+theorem rationalRadonExtension_progressive
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    (μ : ℚ → Ω → Measure upperHalfPlane)
+    (hμr : ∀ q ω, (μ q ω).Regular)
+    (hμm : ∀ q : ℚ, 0 ≤ q → Measurable[ℱ (q : ℝ).toNNReal] (μ q)) :
+    IsProgressive ℱ (rationalRadonExtension μ) := by
+  classical
+  intro T
+  let : MeasurableSpace Ω := ℱ T
+  have hm (n : ℕ) : Measurable (fun p : Iic T × Ω => μ (leftRationalSample n p.1) p.2) := by
+    let ν (q : ℚ) (ω : Ω) := if 0 ≤ q ∧ (q:ℝ) ≤ T then μ q ω else 0
+    have hνm (q : ℚ) : Measurable (ν q) := by
+      by_cases hq : 0 ≤ q ∧ (q:ℝ) ≤ T
+      · have hqt : (q:ℝ).toNNReal ≤ T := by
+          simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hq.2
+        simpa only [ν,ite_eq_left hq] using (hμm q hq.1).mono (ℱ.mono hqt) le_rfl
+      · simpa only [ν,ite_eq_right hq] using (measurable_const : Measurable (fun _ : Ω => (0 : Measure upperHalfPlane)))
+    have hv : Measurable (fun p : ℚ × Ω => ν p.1 p.2) :=
+      measurable_from_prod_countable_right hνm
+    have hq : Measurable (fun p : Iic T × Ω => (leftRationalSample n p.1,p.2)) :=
+      ((leftRationalSample_measurable n).comp
+        (measurable_subtype_coe.comp measurable_fst)).prodMk measurable_snd
+    have h := hv.comp hq
+    convert h using 1
+    funext p
+    dsimp only [Function.comp_def,ν]
+    rw [ite_eq_left ⟨leftRationalSample_nonneg n p.1,
+      (leftRationalSample_le n p.1).trans (NNReal.coe_le_coe.mpr p.1.property)⟩]
+  apply canonicalRadonLimit_measurable (fun (n : ℕ) (p : Iic T × Ω) => μ (leftRationalSample n p.1) p.2)
+  · intro n p
+    let := hμr (leftRationalSample n p.1) p.2
+    infer_instance
+  · intro n f
+    exact measurable_halfPlane_compact_test_integral _ (hm n)
+      (fun p => hμr (leftRationalSample n p.1) p.2) f
+
+theorem rationalRadonExtension_common_version
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (μ : ℚ → Ω → Measure upperHalfPlane) (M : ℝ → Ω → Measure upperHalfPlane)
+    (hMr : ∀ t ω, (M t ω).Regular)
+    (hcont : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t ω))
+    (hrat : ∀ᵐ ω ∂P, ∀ q : ℚ, M q ω = μ q ω) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, rationalRadonExtension μ t ω = M t ω := by
+  filter_upwards [hcont,hrat] with ω hc hr
+  intro t
+  apply canonicalRadonLimit_identified _ (M t ω)
+  refine ⟨hMr t ω,?_⟩
+  intro f
+  simpa only [Function.comp_def,hr] using
+    (hc f).continuousAt.tendsto.comp (leftRationalSample_tendsto t)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassSupport.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassSupport.lean
index 8ff6656964525fffb5fe76e870b33498340a7c49..99eba419e9d2523fe42067da83ce0b6fa7d0f71c 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassSupport.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassSupport.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Mass.RationalTraceMassLocal
-import OAI.Probability.SLE.Mass.RadonTestLimitProperties
-import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract
-
-namespace OAI
-
-/-! The trace support of rational increments passes simultaneously to every
-real interval, using only monotonicity and continuity of the trace. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology ENNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-lemma exists_rational_trace_enclosure_avoiding
-    (γ : ℝ → ℂ) (hγ : Continuous γ) {s t : ℝ} (hst : s ≤ t)
-    (K : Set ℂ) (hK : IsClosed K) (hdis : Disjoint K (γ '' Icc s t)) :
-    ∃ p q : ℚ, (p:ℝ) < s ∧ t < (q:ℝ) ∧ Disjoint K (γ '' Icc (p:ℝ) q) := by
-  let U := γ ⁻¹' Kᶜ
-  have hU : IsOpen U := hK.isOpen_compl.preimage hγ
-  have hs : s ∈ U := by
-    intro hs
-    exact Set.disjoint_left.mp hdis hs ⟨s,⟨le_rfl,hst⟩,rfl⟩
-  have ht : t ∈ U := by
-    intro ht
-    exact Set.disjoint_left.mp hdis ht ⟨t,⟨hst,le_rfl⟩,rfl⟩
-  obtain ⟨ε,hε,hballε⟩ := Metric.isOpen_iff.mp hU s hs
-  obtain ⟨δ,hδ,hballδ⟩ := Metric.isOpen_iff.mp hU t ht
-  obtain ⟨p,hp,hps⟩ := exists_rat_btwn (show s-ε < s by linarith)
-  obtain ⟨q,htq,hq⟩ := exists_rat_btwn (show t < t+δ by linarith)
-  refine ⟨p,q,hps,htq,Set.disjoint_left.mpr ?_⟩
-  rintro z hz ⟨r,hr,rfl⟩
-  by_cases hrs : r < s
-  · have hb : r ∈ Metric.ball s ε := by
-      rw [Metric.mem_ball,Real.dist_eq,abs_of_neg (sub_neg.mpr hrs)]
-      linarith [hr.1]
-    exact hballε hb hz
-  by_cases htr : t < r
-  · have hb : r ∈ Metric.ball t δ := by
-      rw [Metric.mem_ball,Real.dist_eq,abs_of_pos (sub_pos.mpr htr)]
-      linarith [hr.2]
-    exact hballδ hb hz
-  exact Set.disjoint_left.mp hdis hz ⟨r,⟨le_of_not_gt hrs,le_of_not_gt htr⟩,rfl⟩
-
-theorem HalfPlaneTraceIncrements.extension_increment_carried
-    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
-    {γ : ℝ → ℂ} (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ)
-    (M : ℝ → Measure upperHalfPlane) (hMr : ∀ t, (M t).Regular)
-    (hrat : ∀ q : ℚ, M q = μ q) (hmono : Monotone M)
-    {s t : ℝ} (hst : s ≤ t) :
-    (M t-M s) {z : upperHalfPlane | (z:ℂ) ∉ γ '' Icc s t} = 0 := by
-  let η := M t-M s
-  let : (M t).Regular := hMr t
-  let : (M s).Regular := hMr s
-  let : IsFiniteMeasureOnCompacts η :=
-    ⟨fun K hK => lt_of_le_of_lt (Measure.sub_le K) hK.measure_lt_top⟩
-  let : η.Regular := inferInstance
-  let S : Set upperHalfPlane := {z | (z:ℂ) ∈ γ '' Icc s t}
-  apply radon_limit_carried_by_closed (fun _ => η) η (fun _ => tendsto_const_nhds) S
-    (((isCompact_Icc.image hγ).isClosed).preimage continuous_subtype_val)
-  intro f hf
-  let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f
-  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
-  have hdis : Disjoint K (γ '' Icc s t) := by
-    apply Set.disjoint_left.mpr
-    rintro _ ⟨z,hz,rfl⟩ hs
-    exact hf hz hs
-  obtain ⟨p,q,hps,htq,havoid⟩ := exists_rational_trace_enclosure_avoiding γ hγ hst K hK.isClosed hdis
-  have hpq : p ≤ q := Rat.cast_le.mp (hps.le.trans (hst.trans htq.le))
-  obtain ⟨ξ,hξ,hξle⟩ := H.increment p q hpq
-  have hηξ : η ≤ ξ := by
-    apply Measure.sub_le_of_le_add
-    calc
-      M t ≤ M q := hmono htq.le
-      _ = μ q := hrat q
-      _ = μ p+ξ := hξ
-      _ = ξ+M p := by rw [hrat p,add_comm]
-      _ ≤ ξ+M s := add_le_add le_rfl (hmono hps.le)
-  have hzero : η (tsupport f) = 0 := by
-    apply le_zero_iff.mp
-    apply ((hηξ _).trans (hξle _)).trans_eq
-    rw [Measure.restrict_apply f.hasCompactSupport.measurableSet]
-    have hempty : tsupport f ∩ {z : upperHalfPlane | (z:ℂ) ∈ γ '' Icc (p:ℝ) q} = ∅ := by
-      apply Set.eq_empty_iff_forall_notMem.mpr
-      intro z hz
-      exact Set.disjoint_left.mp havoid ⟨z,hz.1,rfl⟩ hz.2
-    rw [hempty,measure_empty]
-  apply Eventually.of_forall
-  intro _
-  apply integral_eq_zero_of_ae
-  filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with z hz
-  exact image_eq_zero_of_notMem_tsupport hz
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMassLocal
+import OAI.Probability.SLE.Mass.RadonTestLimitProperties
+import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract
+
+namespace OAI
+
+/-! The trace support of rational increments passes simultaneously to every
+real interval, using only monotonicity and continuity of the trace. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology ENNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+lemma exists_rational_trace_enclosure_avoiding
+    (γ : ℝ → ℂ) (hγ : Continuous γ) {s t : ℝ} (hst : s ≤ t)
+    (K : Set ℂ) (hK : IsClosed K) (hdis : Disjoint K (γ '' Icc s t)) :
+    ∃ p q : ℚ, (p:ℝ) < s ∧ t < (q:ℝ) ∧ Disjoint K (γ '' Icc (p:ℝ) q) := by
+  let U := γ ⁻¹' Kᶜ
+  have hU : IsOpen U := hK.isOpen_compl.preimage hγ
+  have hs : s ∈ U := by
+    intro hs
+    exact Set.disjoint_left.mp hdis hs ⟨s,⟨le_rfl,hst⟩,rfl⟩
+  have ht : t ∈ U := by
+    intro ht
+    exact Set.disjoint_left.mp hdis ht ⟨t,⟨hst,le_rfl⟩,rfl⟩
+  obtain ⟨ε,hε,hballε⟩ := Metric.isOpen_iff.mp hU s hs
+  obtain ⟨δ,hδ,hballδ⟩ := Metric.isOpen_iff.mp hU t ht
+  obtain ⟨p,hp,hps⟩ := exists_rat_btwn (show s-ε < s by linarith)
+  obtain ⟨q,htq,hq⟩ := exists_rat_btwn (show t < t+δ by linarith)
+  refine ⟨p,q,hps,htq,Set.disjoint_left.mpr ?_⟩
+  rintro z hz ⟨r,hr,rfl⟩
+  by_cases hrs : r < s
+  · have hb : r ∈ Metric.ball s ε := by
+      rw [Metric.mem_ball,Real.dist_eq,abs_of_neg (sub_neg.mpr hrs)]
+      linarith [hr.1]
+    exact hballε hb hz
+  by_cases htr : t < r
+  · have hb : r ∈ Metric.ball t δ := by
+      rw [Metric.mem_ball,Real.dist_eq,abs_of_pos (sub_pos.mpr htr)]
+      linarith [hr.2]
+    exact hballδ hb hz
+  exact Set.disjoint_left.mp hdis hz ⟨r,⟨le_of_not_gt hrs,le_of_not_gt htr⟩,rfl⟩
+
+theorem HalfPlaneTraceIncrements.extension_increment_carried
+    {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} [ν.Regular]
+    {γ : ℝ → ℂ} (H : HalfPlaneTraceIncrements μ ν γ) (hγ : Continuous γ)
+    (M : ℝ → Measure upperHalfPlane) (hMr : ∀ t, (M t).Regular)
+    (hrat : ∀ q : ℚ, M q = μ q) (hmono : Monotone M)
+    {s t : ℝ} (hst : s ≤ t) :
+    (M t-M s) {z : upperHalfPlane | (z:ℂ) ∉ γ '' Icc s t} = 0 := by
+  let η := M t-M s
+  let : (M t).Regular := hMr t
+  let : (M s).Regular := hMr s
+  let : IsFiniteMeasureOnCompacts η :=
+    ⟨fun K hK => lt_of_le_of_lt (Measure.sub_le K) hK.measure_lt_top⟩
+  let : η.Regular := inferInstance
+  let S : Set upperHalfPlane := {z | (z:ℂ) ∈ γ '' Icc s t}
+  apply radon_limit_carried_by_closed (fun _ => η) η (fun _ => tendsto_const_nhds) S
+    (((isCompact_Icc.image hγ).isClosed).preimage continuous_subtype_val)
+  intro f hf
+  let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f
+  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
+  have hdis : Disjoint K (γ '' Icc s t) := by
+    apply Set.disjoint_left.mpr
+    rintro _ ⟨z,hz,rfl⟩ hs
+    exact hf hz hs
+  obtain ⟨p,q,hps,htq,havoid⟩ := exists_rational_trace_enclosure_avoiding γ hγ hst K hK.isClosed hdis
+  have hpq : p ≤ q := Rat.cast_le.mp (hps.le.trans (hst.trans htq.le))
+  obtain ⟨ξ,hξ,hξle⟩ := H.increment p q hpq
+  have hηξ : η ≤ ξ := by
+    apply Measure.sub_le_of_le_add
+    calc
+      M t ≤ M q := hmono htq.le
+      _ = μ q := hrat q
+      _ = μ p+ξ := hξ
+      _ = ξ+M p := by rw [hrat p,add_comm]
+      _ ≤ ξ+M s := add_le_add le_rfl (hmono hps.le)
+  have hzero : η (tsupport f) = 0 := by
+    apply le_zero_iff.mp
+    apply ((hηξ _).trans (hξle _)).trans_eq
+    rw [Measure.restrict_apply f.hasCompactSupport.measurableSet]
+    have hempty : tsupport f ∩ {z : upperHalfPlane | (z:ℂ) ∈ γ '' Icc (p:ℝ) q} = ∅ := by
+      apply Set.eq_empty_iff_forall_notMem.mpr
+      intro z hz
+      exact Set.disjoint_left.mp havoid ⟨z,hz.1,rfl⟩ hz.2
+    rw [hempty,measure_empty]
+  apply Eventually.of_forall
+  intro _
+  apply integral_eq_zero_of_ae
+  filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with z hz
+  exact image_eq_zero_of_notMem_tsupport hz
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RationalTraceMassVariation.lean b/lean/OAI/Probability/SLE/Mass/RationalTraceMassVariation.lean
index b5a166f47fef635bdacafdbe338fc4eee6ddaa36..77e6c6824307d963abac5655e9a1c20b15213d1e 100644
--- a/lean/OAI/Probability/SLE/Mass/RationalTraceMassVariation.lean
+++ b/lean/OAI/Probability/SLE/Mass/RationalTraceMassVariation.lean
@@ -1,118 +1,118 @@
-import OAI.Probability.SLE.Mass.RationalTraceMass
-
-namespace OAI
-
-/-! Uniform local total-variation control for the all-time extension. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal CompactlySupported
-
-namespace SLEExactGauge
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-/-- Local total variation is expressed uniformly on measurable sets. The
-finite-terminal form applies to each compact restriction in the source. -/
-theorem RationalTraceIncrements.local_variation_continuity
-    {μ : ℚ → Measure X} {ν : Measure X} [ν.Regular] [IsFiniteMeasure ν]
-    [NullSingletonClass ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
-    (hγ : Continuous γ) (M : ℝ → Measure X)
-    (hMr : ∀ t, (M t).Regular) (hdom : ∀ t, M t ≤ ν)
-    (hrat : ∀ q : ℚ, M q=μ q)
-    (hcont : ∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂M t)) :
-    ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
-      ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
-        ∀ A : Set X, MeasurableSet A → |(M s A).toReal-(M t A).toReal| < ε := by
-  intro t₀ ε hε
-  obtain ⟨U,hUsub,hU,hνU⟩ := ({γ t₀} : Set X).exists_isOpen_lt_of_lt (μ := ν)
-    (ENNReal.ofReal ε) (by simpa using ENNReal.ofReal_pos.mpr hε)
-  have htU : γ t₀ ∈ U := hUsub (mem_singleton _)
-  obtain ⟨δ,hδ,hδsub⟩ := Metric.mem_nhds_iff.mp
-    (hγ.continuousAt.preimage_mem_nhds (hU.mem_nhds htU))
-  have hI (a b : ℝ) (ha : dist a t₀ < δ) (hb : dist b t₀ < δ) : γ '' Icc a b ⊆ U := by
-    rintro _ ⟨x,hx,rfl⟩
-    apply hδsub
-    rw [Metric.mem_ball,Real.dist_eq]
-    rw [Real.dist_eq,abs_lt] at ha hb
-    exact abs_lt.mpr ⟨lt_of_lt_of_le ha.1 (sub_le_sub_right hx.1 t₀),
-      lt_of_le_of_lt (sub_le_sub_right hx.2 t₀) hb.2⟩
-  have hq (p q : ℚ) (hp : dist (p:ℝ) t₀ < δ) (hq : dist (q:ℝ) t₀ < δ) :
-      μ p ≤ μ q+ν.restrict U := by
-    rcases le_total p q with hpq | hqp
-    · exact (H.monotone hpq).trans (Measure.le_add_right le_rfl)
-    · obtain ⟨η,hη,hην⟩ := H.increment q p hqp
-      rw [hη]
-      exact add_le_add le_rfl (hην.trans (Measure.restrict_mono (hI q p hq hp) le_rfl))
-  have hreal (s t : ℝ) (hs : dist s t₀ < δ) (ht : dist t t₀ < δ) :
-      M s ≤ M t+ν.restrict U := by
-    let : (M s).Regular := hMr s
-    let : (M t).Regular := hMr t
-    let : IsFiniteMeasure (M t) := isFiniteMeasure_of_le ν (hdom t)
-    let : (ν.restrict U).Regular := Measure.Regular.restrict_of_measure_ne_top (measure_ne_top ν U)
-    obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
-    obtain ⟨q,hqt⟩ := exists_rational_sequence_tendsto t
-    have hpU : ∀ᶠ n in atTop, dist (p n:ℝ) t₀ < δ :=
-      hp.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds hs)
-    have hqU : ∀ᶠ n in atTop, dist (q n:ℝ) t₀ < δ :=
-      hqt.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds ht)
-    apply radon_measure_le_of_nonneg_tests
-    intro f hfn
-    have hplim : Tendsto (fun n => ∫ x, f x ∂μ (p n)) atTop (𝓝 (∫ x, f x ∂M s)) := by
-      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
-    have hqlim : Tendsto (fun n => ∫ x, f x ∂(μ (q n)+ν.restrict U)) atTop
-        (𝓝 (∫ x, f x ∂(M t+ν.restrict U))) := by
-      have hi := ((hcont f).continuousAt.tendsto.comp hqt).add
-        (tendsto_const_nhds (x := ∫ x, f x ∂ν.restrict U))
-      simp only [Function.comp_def,hrat] at hi
-      have hqadd (n : ℕ) : (∫ x, f x ∂(μ (q n)+ν.restrict U))=
-          (∫ x, f x ∂μ (q n))+(∫ x, f x ∂ν.restrict U) := by
-        let : (μ (q n)).Regular := H.regular (q n)
-        exact integral_add_measure f.integrable f.integrable
-      simpa only [hqadd,integral_add_measure f.integrable f.integrable] using hi
-    apply le_of_tendsto_of_tendsto hplim hqlim
-    filter_upwards [hpU,hqU] with n hpn hqn
-    let : (μ (q n)).Regular := H.regular (q n)
-    let : IsFiniteMeasure (μ (q n)) := isFiniteMeasure_of_le ν (H.dominated (q n))
-    exact integral_mono_measure (hq (p n) (q n) hpn hqn) (ae_of_all _ hfn) f.integrable
-  refine ⟨δ,hδ,fun s t hs ht A hA => ?_⟩
-  have hst := hreal s t hs ht A
-  have hts := hreal t s ht hs A
-  have hsn : M s A ≠ ⊤ := ((hdom s A).trans_lt (measure_lt_top ν A)).ne
-  have htn : M t A ≠ ⊤ := ((hdom t A).trans_lt (measure_lt_top ν A)).ne
-  have hUn : ν U ≠ ⊤ := measure_ne_top ν U
-  rw [Measure.add_apply,Measure.restrict_apply hA,inter_comm] at hst hts
-  have hUA : ν (U ∩ A) ≤ ν U := measure_mono inter_subset_left
-  have hcan : ν (U ∩ A) ≠ ⊤ := (hUA.trans_lt hUn.lt_top).ne
-  have hst' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨htn,hcan⟩) hst
-  have hts' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hsn,hcan⟩) hts
-  rw [ENNReal.toReal_add htn hcan] at hst'
-  rw [ENNReal.toReal_add hsn hcan] at hts'
-  have hless : (ν (U ∩ A)).toReal < ε := by
-    have hUl : (ν U).toReal < ε := by
-      simpa only [ENNReal.toReal_ofReal hε.le] using
-        (ENNReal.toReal_lt_toReal hUn ENNReal.ofReal_ne_top).mpr hνU
-    exact (ENNReal.toReal_mono hUn hUA).trans_lt hUl
-  exact abs_lt.mpr ⟨by linarith,by linarith⟩
-
-/-- The pathwise rational squeeze gives one increasing extension with
-local total-variation continuity. -/
-theorem RationalTraceIncrements.exists_locally_tv_continuous_extension
-    {μ : ℚ → Measure X} {ν : Measure X} [ν.Regular] [IsFiniteMeasure ν]
-    [NullSingletonClass ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
-    (hγ : Continuous γ) :
-    ∃ M : ℝ → Measure X, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
-      (∀ q : ℚ, M q=μ q) ∧ Monotone M ∧
-      (∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂M t)) ∧
-      ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
-        ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
-          ∀ A : Set X, MeasurableSet A → |(M s A).toReal-(M t A).toReal| < ε := by
-  obtain ⟨M,hMr,hdom,hrat,hcont,hmono⟩ := H.exists_continuous_extension hγ
-  exact ⟨M,hMr,hdom,hrat,hmono,hcont,H.local_variation_continuity hγ M hMr hdom hrat hcont⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RationalTraceMass
+
+namespace OAI
+
+/-! Uniform local total-variation control for the all-time extension. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal CompactlySupported
+
+namespace SLEExactGauge
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+/-- Local total variation is expressed uniformly on measurable sets. The
+finite-terminal form applies to each compact restriction in the source. -/
+theorem RationalTraceIncrements.local_variation_continuity
+    {μ : ℚ → Measure X} {ν : Measure X} [ν.Regular] [IsFiniteMeasure ν]
+    [NullSingletonClass ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
+    (hγ : Continuous γ) (M : ℝ → Measure X)
+    (hMr : ∀ t, (M t).Regular) (hdom : ∀ t, M t ≤ ν)
+    (hrat : ∀ q : ℚ, M q=μ q)
+    (hcont : ∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂M t)) :
+    ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
+      ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
+        ∀ A : Set X, MeasurableSet A → |(M s A).toReal-(M t A).toReal| < ε := by
+  intro t₀ ε hε
+  obtain ⟨U,hUsub,hU,hνU⟩ := ({γ t₀} : Set X).exists_isOpen_lt_of_lt (μ := ν)
+    (ENNReal.ofReal ε) (by simpa using ENNReal.ofReal_pos.mpr hε)
+  have htU : γ t₀ ∈ U := hUsub (mem_singleton _)
+  obtain ⟨δ,hδ,hδsub⟩ := Metric.mem_nhds_iff.mp
+    (hγ.continuousAt.preimage_mem_nhds (hU.mem_nhds htU))
+  have hI (a b : ℝ) (ha : dist a t₀ < δ) (hb : dist b t₀ < δ) : γ '' Icc a b ⊆ U := by
+    rintro _ ⟨x,hx,rfl⟩
+    apply hδsub
+    rw [Metric.mem_ball,Real.dist_eq]
+    rw [Real.dist_eq,abs_lt] at ha hb
+    exact abs_lt.mpr ⟨lt_of_lt_of_le ha.1 (sub_le_sub_right hx.1 t₀),
+      lt_of_le_of_lt (sub_le_sub_right hx.2 t₀) hb.2⟩
+  have hq (p q : ℚ) (hp : dist (p:ℝ) t₀ < δ) (hq : dist (q:ℝ) t₀ < δ) :
+      μ p ≤ μ q+ν.restrict U := by
+    rcases le_total p q with hpq | hqp
+    · exact (H.monotone hpq).trans (Measure.le_add_right le_rfl)
+    · obtain ⟨η,hη,hην⟩ := H.increment q p hqp
+      rw [hη]
+      exact add_le_add le_rfl (hην.trans (Measure.restrict_mono (hI q p hq hp) le_rfl))
+  have hreal (s t : ℝ) (hs : dist s t₀ < δ) (ht : dist t t₀ < δ) :
+      M s ≤ M t+ν.restrict U := by
+    let : (M s).Regular := hMr s
+    let : (M t).Regular := hMr t
+    let : IsFiniteMeasure (M t) := isFiniteMeasure_of_le ν (hdom t)
+    let : (ν.restrict U).Regular := Measure.Regular.restrict_of_measure_ne_top (measure_ne_top ν U)
+    obtain ⟨p,hp⟩ := exists_rational_sequence_tendsto s
+    obtain ⟨q,hqt⟩ := exists_rational_sequence_tendsto t
+    have hpU : ∀ᶠ n in atTop, dist (p n:ℝ) t₀ < δ :=
+      hp.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds hs)
+    have hqU : ∀ᶠ n in atTop, dist (q n:ℝ) t₀ < δ :=
+      hqt.eventually (isOpen_lt (continuous_id.dist continuous_const) continuous_const |>.mem_nhds ht)
+    apply radon_measure_le_of_nonneg_tests
+    intro f hfn
+    have hplim : Tendsto (fun n => ∫ x, f x ∂μ (p n)) atTop (𝓝 (∫ x, f x ∂M s)) := by
+      simpa only [Function.comp_def,hrat] using (hcont f).continuousAt.tendsto.comp hp
+    have hqlim : Tendsto (fun n => ∫ x, f x ∂(μ (q n)+ν.restrict U)) atTop
+        (𝓝 (∫ x, f x ∂(M t+ν.restrict U))) := by
+      have hi := ((hcont f).continuousAt.tendsto.comp hqt).add
+        (tendsto_const_nhds (x := ∫ x, f x ∂ν.restrict U))
+      simp only [Function.comp_def,hrat] at hi
+      have hqadd (n : ℕ) : (∫ x, f x ∂(μ (q n)+ν.restrict U))=
+          (∫ x, f x ∂μ (q n))+(∫ x, f x ∂ν.restrict U) := by
+        let : (μ (q n)).Regular := H.regular (q n)
+        exact integral_add_measure f.integrable f.integrable
+      simpa only [hqadd,integral_add_measure f.integrable f.integrable] using hi
+    apply le_of_tendsto_of_tendsto hplim hqlim
+    filter_upwards [hpU,hqU] with n hpn hqn
+    let : (μ (q n)).Regular := H.regular (q n)
+    let : IsFiniteMeasure (μ (q n)) := isFiniteMeasure_of_le ν (H.dominated (q n))
+    exact integral_mono_measure (hq (p n) (q n) hpn hqn) (ae_of_all _ hfn) f.integrable
+  refine ⟨δ,hδ,fun s t hs ht A hA => ?_⟩
+  have hst := hreal s t hs ht A
+  have hts := hreal t s ht hs A
+  have hsn : M s A ≠ ⊤ := ((hdom s A).trans_lt (measure_lt_top ν A)).ne
+  have htn : M t A ≠ ⊤ := ((hdom t A).trans_lt (measure_lt_top ν A)).ne
+  have hUn : ν U ≠ ⊤ := measure_ne_top ν U
+  rw [Measure.add_apply,Measure.restrict_apply hA,inter_comm] at hst hts
+  have hUA : ν (U ∩ A) ≤ ν U := measure_mono inter_subset_left
+  have hcan : ν (U ∩ A) ≠ ⊤ := (hUA.trans_lt hUn.lt_top).ne
+  have hst' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨htn,hcan⟩) hst
+  have hts' := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hsn,hcan⟩) hts
+  rw [ENNReal.toReal_add htn hcan] at hst'
+  rw [ENNReal.toReal_add hsn hcan] at hts'
+  have hless : (ν (U ∩ A)).toReal < ε := by
+    have hUl : (ν U).toReal < ε := by
+      simpa only [ENNReal.toReal_ofReal hε.le] using
+        (ENNReal.toReal_lt_toReal hUn ENNReal.ofReal_ne_top).mpr hνU
+    exact (ENNReal.toReal_mono hUn hUA).trans_lt hUl
+  exact abs_lt.mpr ⟨by linarith,by linarith⟩
+
+/-- The pathwise rational squeeze gives one increasing extension with
+local total-variation continuity. -/
+theorem RationalTraceIncrements.exists_locally_tv_continuous_extension
+    {μ : ℚ → Measure X} {ν : Measure X} [ν.Regular] [IsFiniteMeasure ν]
+    [NullSingletonClass ν] {γ : ℝ → X} (H : RationalTraceIncrements μ ν γ)
+    (hγ : Continuous γ) :
+    ∃ M : ℝ → Measure X, (∀ t, (M t).Regular) ∧ (∀ t, M t ≤ ν) ∧
+      (∀ q : ℚ, M q=μ q) ∧ Monotone M ∧
+      (∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂M t)) ∧
+      ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
+        ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ →
+          ∀ A : Set X, MeasurableSet A → |(M s A).toReal-(M t A).toReal| < ε := by
+  obtain ⟨M,hMr,hdom,hrat,hcont,hmono⟩ := H.exists_continuous_extension hγ
+  exact ⟨M,hMr,hdom,hrat,hmono,hcont,H.local_variation_continuity hγ M hMr hdom hrat hcont⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RawMassFunctional.lean b/lean/OAI/Probability/SLE/Mass/RawMassFunctional.lean
index b1532af672bdf4d8e81d35ace4078dadd062acb4..b9161a6d7f39a5e48d5d4721397db2b345232f63 100644
--- a/lean/OAI/Probability/SLE/Mass/RawMassFunctional.lean
+++ b/lean/OAI/Probability/SLE/Mass/RawMassFunctional.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Stochastic.DriverPastFactorization
-import OAI.Probability.SLE.Mass.RationalTraceMassProgressiveVersion
-
-namespace OAI
-
-/-! A fixed raw, nonanticipating mass functional on continuous driver paths.
-Each rational slice is first factored through the stopped driver, and the
-single rational reconstruction is progressive on the full path space. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance rawMassDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance rawMassDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-lemma raw_rational_reconstruction_depends_on_past
-    (F : ℚ → C(ℝ≥0,ℝ) → Measure upperHalfPlane)
-    (t : ℝ≥0) (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ t, p s=q s) :
-    rationalRadonExtension (fun r p => F r (driverPathStopped (r:ℝ).toNNReal p)) t p =
-      rationalRadonExtension (fun r p => F r (driverPathStopped (r:ℝ).toNNReal p)) t q := by
-  unfold rationalRadonExtension
-  apply congrArg canonicalRadonLimit
-  funext n
-  apply congrArg (F (leftRationalSample n t))
-  exact driverPathStopped_eq_of_agree t _ (by
-    simpa only [Real.toNNReal_coe] using
-      Real.toNNReal_le_toNNReal (leftRationalSample_le n t)) p q hpq
-
-theorem exists_raw_progressive_radon_factor
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (N : ℝ≥0 → Ω → Measure upperHalfPlane)
-    (hNm : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) N)
-    (hNr : ∀ t ω, (N t ω).Regular)
-    (hNc : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t ω)) :
-    ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane,
-      IsProgressive driverPathFiltration R ∧
-      (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧
-      (∀ t p q, (∀ s ≤ t, p s=q s) → R t p=R t q) ∧
-      ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hc ω) = N t ω := by
-  choose F hFm hFr hFe using fun q : ℚ => exists_stopped_driver_radon_factor B hm hc
-    (q:ℝ).toNNReal (N (q:ℝ).toNNReal) (hNr _) (hNm _)
-  let μ (q : ℚ) (p : C(ℝ≥0,ℝ)) := F q (driverPathStopped (q:ℝ).toNNReal p)
-  have hμr (q : ℚ) (p : C(ℝ≥0,ℝ)) : (μ q p).Regular := hFr q _
-  have hμm (q : ℚ) (_hq : 0 ≤ q) :
-      Measurable[driverPathFiltration (q:ℝ).toNNReal] (μ q) :=
-    (hFm q).comp (driverPathStopped_measurable_past (q:ℝ).toNNReal)
-  refine ⟨rationalRadonExtension μ,rationalRadonExtension_progressive
-    driverPathFiltration μ hμr hμm,rationalRadonExtension_regular μ,
-      raw_rational_reconstruction_depends_on_past F,?_⟩
-  let μ' (q : ℚ) (ω : Ω) := μ q (sourceDriverPath B hc ω)
-  have hcont : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
-      Continuous (fun t : ℝ => ∫ z, f z ∂N t.toNNReal ω) := by
-    filter_upwards [hNc] with ω hω
-    intro f
-    exact (hω f).comp continuous_real_toNNReal
-  have heq : ∀ᵐ ω ∂P, ∀ q : ℚ, N (q:ℝ).toNNReal ω = μ' q ω :=
-    ae_of_all _ (fun ω q => (hFe q ω).symm)
-  have he := rationalRadonExtension_common_version P μ' (fun t ω => N t.toNNReal ω)
-    (fun t ω => hNr t.toNNReal ω) hcont heq
-  filter_upwards [he] with ω hω
-  intro t
-  simpa only [rationalRadonExtension,μ',Real.toNNReal_coe] using hω t
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.DriverPastFactorization
+import OAI.Probability.SLE.Mass.RationalTraceMassProgressiveVersion
+
+namespace OAI
+
+/-! A fixed raw, nonanticipating mass functional on continuous driver paths.
+Each rational slice is first factored through the stopped driver, and the
+single rational reconstruction is progressive on the full path space. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance rawMassDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance rawMassDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+lemma raw_rational_reconstruction_depends_on_past
+    (F : ℚ → C(ℝ≥0,ℝ) → Measure upperHalfPlane)
+    (t : ℝ≥0) (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ t, p s=q s) :
+    rationalRadonExtension (fun r p => F r (driverPathStopped (r:ℝ).toNNReal p)) t p =
+      rationalRadonExtension (fun r p => F r (driverPathStopped (r:ℝ).toNNReal p)) t q := by
+  unfold rationalRadonExtension
+  apply congrArg canonicalRadonLimit
+  funext n
+  apply congrArg (F (leftRationalSample n t))
+  exact driverPathStopped_eq_of_agree t _ (by
+    simpa only [Real.toNNReal_coe] using
+      Real.toNNReal_le_toNNReal (leftRationalSample_le n t)) p q hpq
+
+theorem exists_raw_progressive_radon_factor
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (N : ℝ≥0 → Ω → Measure upperHalfPlane)
+    (hNm : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) N)
+    (hNr : ∀ t ω, (N t ω).Regular)
+    (hNc : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t ω)) :
+    ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane,
+      IsProgressive driverPathFiltration R ∧
+      (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧
+      (∀ t p q, (∀ s ≤ t, p s=q s) → R t p=R t q) ∧
+      ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hc ω) = N t ω := by
+  choose F hFm hFr hFe using fun q : ℚ => exists_stopped_driver_radon_factor B hm hc
+    (q:ℝ).toNNReal (N (q:ℝ).toNNReal) (hNr _) (hNm _)
+  let μ (q : ℚ) (p : C(ℝ≥0,ℝ)) := F q (driverPathStopped (q:ℝ).toNNReal p)
+  have hμr (q : ℚ) (p : C(ℝ≥0,ℝ)) : (μ q p).Regular := hFr q _
+  have hμm (q : ℚ) (_hq : 0 ≤ q) :
+      Measurable[driverPathFiltration (q:ℝ).toNNReal] (μ q) :=
+    (hFm q).comp (driverPathStopped_measurable_past (q:ℝ).toNNReal)
+  refine ⟨rationalRadonExtension μ,rationalRadonExtension_progressive
+    driverPathFiltration μ hμr hμm,rationalRadonExtension_regular μ,
+      raw_rational_reconstruction_depends_on_past F,?_⟩
+  let μ' (q : ℚ) (ω : Ω) := μ q (sourceDriverPath B hc ω)
+  have hcont : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
+      Continuous (fun t : ℝ => ∫ z, f z ∂N t.toNNReal ω) := by
+    filter_upwards [hNc] with ω hω
+    intro f
+    exact (hω f).comp continuous_real_toNNReal
+  have heq : ∀ᵐ ω ∂P, ∀ q : ℚ, N (q:ℝ).toNNReal ω = μ' q ω :=
+    ae_of_all _ (fun ω q => (hFe q ω).symm)
+  have he := rationalRadonExtension_common_version P μ' (fun t ω => N t.toNNReal ω)
+    (fun t ω => hNr t.toNNReal ω) hcont heq
+  filter_upwards [he] with ω hω
+  intro t
+  simpa only [rationalRadonExtension,μ',Real.toNNReal_coe] using hω t
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RawMassPathLaw.lean b/lean/OAI/Probability/SLE/Mass/RawMassPathLaw.lean
index 66add794f542fd9cc80ff83a10d08875ce879fe8..649f6003260249b839ed5ce9f1659d3cfab4a8e2 100644
--- a/lean/OAI/Probability/SLE/Mass/RawMassPathLaw.lean
+++ b/lean/OAI/Probability/SLE/Mass/RawMassPathLaw.lean
@@ -1,133 +1,133 @@
-import OAI.Probability.SLE.Mass.CanonicalRawMass
-import OAI.Probability.SLE.Mass.RawRadonGoodPaths
-
-namespace OAI
-
-/-! The raw mass has one measurable full-law event carrying all path
-properties simultaneously. Equal driver laws therefore preserve these
-properties without assuming measurability of an uncountable predicate. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped Topology CompactlySupported NNReal
-
-namespace SLEExactGauge
-
-local instance rawPathLawDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance rawPathLawDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-theorem exists_raw_rational_radon_factor
-    {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (N : ℝ≥0 → Ω → Measure upperHalfPlane)
-    (hNm : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) N)
-    (hNr : ∀ t ω, (N t ω).Regular) :
-    ∃ μ : ℚ → C(ℝ≥0,ℝ) → Measure upperHalfPlane,
-      (∀ q p, (μ q p).Regular) ∧
-      (∀ q : ℚ, Measurable[driverPathFiltration (q:ℝ).toNNReal] (μ q)) ∧
-      (∀ (q : ℚ) p r, (∀ s ≤ (q:ℝ).toNNReal, p s=r s) → μ q p=μ q r) ∧
-      ∀ ω q, μ q (sourceDriverPath B hc ω)=N (q:ℝ).toNNReal ω := by
-  choose F hFm hFr hFe using fun q : ℚ => exists_stopped_driver_radon_factor B hm hc
-    (q:ℝ).toNNReal (N (q:ℝ).toNNReal) (hNr _) (hNm _)
-  refine ⟨fun q p => F q (driverPathStopped (q:ℝ).toNNReal p),?_,?_,?_,?_⟩
-  · exact fun q p => hFr q _
-  · exact fun q => (hFm q).comp (driverPathStopped_measurable_past (q:ℝ).toNNReal)
-  · intro q p r hpr
-    exact congrArg (F q) (driverPathStopped_eq_of_agree _ _ le_rfl p r hpr)
-  · exact fun ω q => hFe q ω
-
-lemma rational_reconstruction_nonanticipating
-    (μ : ℚ → C(ℝ≥0,ℝ) → Measure upperHalfPlane)
-    (hμ : ∀ (q : ℚ) p r, (∀ s ≤ (q:ℝ).toNNReal, p s=r s) → μ q p=μ q r)
-    (t : ℝ≥0) (p r : C(ℝ≥0,ℝ)) (hpr : ∀ s ≤ t, p s=r s) :
-    rationalRadonExtension μ t p=rationalRadonExtension μ t r := by
-  unfold rationalRadonExtension
-  apply congrArg canonicalRadonLimit
-  funext n
-  apply hμ
-  intro s hs
-  apply hpr s
-  apply hs.trans
-  simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal (leftRationalSample_le n t)
-
-theorem raw_radon_path_properties_of_same_law
-    {Ω Ξ Y : Type*} [MeasurableSpace Ω] [MeasurableSpace Ξ] [MeasurableSpace Y]
-    (P : Measure Ω) (Q : Measure Ξ) (A : Ω → Y) (B : Ξ → Y)
-    (hA : Measurable A) (hB : Measurable B) (hlaw : Q.map B=P.map A)
-    (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Y → Measure upperHalfPlane)
-    (hm : ∀ q, Measurable (μ q)) (hr : ∀ q y, (μ q y).Regular)
-    (hg : ∀ᵐ ω ∂P, RawRadonGood D (fun q => μ q (A ω))) :
-    ∀ᵐ ξ ∂Q,
-      rationalRadonExtension μ 0 (B ξ)=0 ∧
-      Monotone (fun t => rationalRadonExtension μ t (B ξ)) ∧
-      ∀ f : C_c(upperHalfPlane,ℝ),
-        Continuous (fun t => ∫ z, f z ∂rationalRadonExtension μ t (B ξ)) := by
-  have hE := measurableSet_rawRadonGood D μ hm hr
-  have hpush : ∀ᵐ y ∂P.map A, RawRadonGood D (fun q => μ q y) :=
-    (ae_map_iff hA.aemeasurable hE).mpr hg
-  rw [← hlaw] at hpush
-  have hgood := (ae_map_iff hB.aemeasurable hE).mp hpush
-  filter_upwards [hgood] with ξ hξ
-  exact RawRadonGood.reconstructed_paths D (fun q => μ q (B ξ)) (fun q => hr q _) hξ
-
-namespace Multipoint
-
-theorem CanonicalMass.exists_raw_nonanticipating_with_good_set
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (C : CanonicalMass P κ B hBm γ G)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) :
-    ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane,
-      ∃ E : Set C(ℝ≥0,ℝ),
-      IsProgressive driverPathFiltration R ∧
-      (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧
-      (∀ t p r, (∀ s ≤ t, p s=r s) → R t p=R t r) ∧
-      MeasurableSet E ∧ (∀ᵐ p ∂P.map (sourceDriverPath B hBc), p ∈ E) ∧
-      (∀ p ∈ E, R 0 p=0 ∧ Monotone (fun t => R t p) ∧
-        ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂(R t p))) ∧
-      ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω)=C.process t ω := by
-  have hm (t : ℝ≥0) : Measurable (B t) :=
-    hBm.comp (measurable_const.prodMk measurable_id)
-  obtain ⟨μ,hμr,hμm,hμn,hμe⟩ := exists_raw_rational_radon_factor B hm hBc C.process
-    C.progressive.adapted (fun t ω => (C.radon t ω).1)
-  let R := rationalRadonExtension μ
-  let D := localTestFamily upperHalfPlane
-  let E : Set C(ℝ≥0,ℝ) := {p | RawRadonGood D (fun q => μ q p)}
-  have hμma (q : ℚ) : Measurable (μ q) :=
-    (hμm q).mono (driverPathFiltration.le _) le_rfl
-  have hEm : MeasurableSet E := measurableSet_rawRadonGood D μ hμma hμr
-  have hgood : ∀ᵐ ω ∂P, sourceDriverPath B hBc ω ∈ E := by
-    filter_upwards [C.paths] with ω hω
-    dsimp only [E,mem_ofPred_eq]
-    simp_rw [hμe]
-    exact rawRadonGood_of_continuous_path D (fun t => C.process t ω)
-      (fun t => (C.radon t ω).1) hω.zero hω.monotone hω.continuous_tests
-  have hbp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm
-  refine ⟨R,E,rationalRadonExtension_progressive driverPathFiltration μ hμr (fun q _ => hμm q),
-    rationalRadonExtension_regular μ,rational_reconstruction_nonanticipating μ hμn,
-    hEm,(ae_map_iff hbp.aemeasurable hEm).mpr hgood,?_,?_⟩
-  · intro p hp
-    exact RawRadonGood.reconstructed_paths D (fun q => μ q p) (fun q => hμr q p) hp
-  · let μ' (q : ℚ) (ω : Ω) := μ q (sourceDriverPath B hBc ω)
-    have hc : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
-        Continuous (fun t : ℝ => ∫ z, f z ∂C.process t.toNNReal ω) := by
-      filter_upwards [C.paths] with ω hω
-      intro f
-      exact (hω.continuous_tests f).comp continuous_real_toNNReal
-    have he : ∀ᵐ ω ∂P, ∀ q : ℚ, C.process (q:ℝ).toNNReal ω=μ' q ω :=
-      ae_of_all _ (fun ω q => (hμe ω q).symm)
-    have h := rationalRadonExtension_common_version P μ' (fun t ω => C.process t.toNNReal ω)
-      (fun t ω => (C.radon t.toNNReal ω).1) hc he
-    filter_upwards [h] with ω hω
-    intro t
-    simpa only [R,rationalRadonExtension,μ',Real.toNNReal_coe] using hω t
-
-end Multipoint
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CanonicalRawMass
+import OAI.Probability.SLE.Mass.RawRadonGoodPaths
+
+namespace OAI
+
+/-! The raw mass has one measurable full-law event carrying all path
+properties simultaneously. Equal driver laws therefore preserve these
+properties without assuming measurability of an uncountable predicate. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped Topology CompactlySupported NNReal
+
+namespace SLEExactGauge
+
+local instance rawPathLawDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance rawPathLawDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+theorem exists_raw_rational_radon_factor
+    {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (N : ℝ≥0 → Ω → Measure upperHalfPlane)
+    (hNm : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) N)
+    (hNr : ∀ t ω, (N t ω).Regular) :
+    ∃ μ : ℚ → C(ℝ≥0,ℝ) → Measure upperHalfPlane,
+      (∀ q p, (μ q p).Regular) ∧
+      (∀ q : ℚ, Measurable[driverPathFiltration (q:ℝ).toNNReal] (μ q)) ∧
+      (∀ (q : ℚ) p r, (∀ s ≤ (q:ℝ).toNNReal, p s=r s) → μ q p=μ q r) ∧
+      ∀ ω q, μ q (sourceDriverPath B hc ω)=N (q:ℝ).toNNReal ω := by
+  choose F hFm hFr hFe using fun q : ℚ => exists_stopped_driver_radon_factor B hm hc
+    (q:ℝ).toNNReal (N (q:ℝ).toNNReal) (hNr _) (hNm _)
+  refine ⟨fun q p => F q (driverPathStopped (q:ℝ).toNNReal p),?_,?_,?_,?_⟩
+  · exact fun q p => hFr q _
+  · exact fun q => (hFm q).comp (driverPathStopped_measurable_past (q:ℝ).toNNReal)
+  · intro q p r hpr
+    exact congrArg (F q) (driverPathStopped_eq_of_agree _ _ le_rfl p r hpr)
+  · exact fun ω q => hFe q ω
+
+lemma rational_reconstruction_nonanticipating
+    (μ : ℚ → C(ℝ≥0,ℝ) → Measure upperHalfPlane)
+    (hμ : ∀ (q : ℚ) p r, (∀ s ≤ (q:ℝ).toNNReal, p s=r s) → μ q p=μ q r)
+    (t : ℝ≥0) (p r : C(ℝ≥0,ℝ)) (hpr : ∀ s ≤ t, p s=r s) :
+    rationalRadonExtension μ t p=rationalRadonExtension μ t r := by
+  unfold rationalRadonExtension
+  apply congrArg canonicalRadonLimit
+  funext n
+  apply hμ
+  intro s hs
+  apply hpr s
+  apply hs.trans
+  simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal (leftRationalSample_le n t)
+
+theorem raw_radon_path_properties_of_same_law
+    {Ω Ξ Y : Type*} [MeasurableSpace Ω] [MeasurableSpace Ξ] [MeasurableSpace Y]
+    (P : Measure Ω) (Q : Measure Ξ) (A : Ω → Y) (B : Ξ → Y)
+    (hA : Measurable A) (hB : Measurable B) (hlaw : Q.map B=P.map A)
+    (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Y → Measure upperHalfPlane)
+    (hm : ∀ q, Measurable (μ q)) (hr : ∀ q y, (μ q y).Regular)
+    (hg : ∀ᵐ ω ∂P, RawRadonGood D (fun q => μ q (A ω))) :
+    ∀ᵐ ξ ∂Q,
+      rationalRadonExtension μ 0 (B ξ)=0 ∧
+      Monotone (fun t => rationalRadonExtension μ t (B ξ)) ∧
+      ∀ f : C_c(upperHalfPlane,ℝ),
+        Continuous (fun t => ∫ z, f z ∂rationalRadonExtension μ t (B ξ)) := by
+  have hE := measurableSet_rawRadonGood D μ hm hr
+  have hpush : ∀ᵐ y ∂P.map A, RawRadonGood D (fun q => μ q y) :=
+    (ae_map_iff hA.aemeasurable hE).mpr hg
+  rw [← hlaw] at hpush
+  have hgood := (ae_map_iff hB.aemeasurable hE).mp hpush
+  filter_upwards [hgood] with ξ hξ
+  exact RawRadonGood.reconstructed_paths D (fun q => μ q (B ξ)) (fun q => hr q _) hξ
+
+namespace Multipoint
+
+theorem CanonicalMass.exists_raw_nonanticipating_with_good_set
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (C : CanonicalMass P κ B hBm γ G)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) :
+    ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane,
+      ∃ E : Set C(ℝ≥0,ℝ),
+      IsProgressive driverPathFiltration R ∧
+      (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧
+      (∀ t p r, (∀ s ≤ t, p s=r s) → R t p=R t r) ∧
+      MeasurableSet E ∧ (∀ᵐ p ∂P.map (sourceDriverPath B hBc), p ∈ E) ∧
+      (∀ p ∈ E, R 0 p=0 ∧ Monotone (fun t => R t p) ∧
+        ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂(R t p))) ∧
+      ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω)=C.process t ω := by
+  have hm (t : ℝ≥0) : Measurable (B t) :=
+    hBm.comp (measurable_const.prodMk measurable_id)
+  obtain ⟨μ,hμr,hμm,hμn,hμe⟩ := exists_raw_rational_radon_factor B hm hBc C.process
+    C.progressive.adapted (fun t ω => (C.radon t ω).1)
+  let R := rationalRadonExtension μ
+  let D := localTestFamily upperHalfPlane
+  let E : Set C(ℝ≥0,ℝ) := {p | RawRadonGood D (fun q => μ q p)}
+  have hμma (q : ℚ) : Measurable (μ q) :=
+    (hμm q).mono (driverPathFiltration.le _) le_rfl
+  have hEm : MeasurableSet E := measurableSet_rawRadonGood D μ hμma hμr
+  have hgood : ∀ᵐ ω ∂P, sourceDriverPath B hBc ω ∈ E := by
+    filter_upwards [C.paths] with ω hω
+    dsimp only [E,mem_ofPred_eq]
+    simp_rw [hμe]
+    exact rawRadonGood_of_continuous_path D (fun t => C.process t ω)
+      (fun t => (C.radon t ω).1) hω.zero hω.monotone hω.continuous_tests
+  have hbp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm
+  refine ⟨R,E,rationalRadonExtension_progressive driverPathFiltration μ hμr (fun q _ => hμm q),
+    rationalRadonExtension_regular μ,rational_reconstruction_nonanticipating μ hμn,
+    hEm,(ae_map_iff hbp.aemeasurable hEm).mpr hgood,?_,?_⟩
+  · intro p hp
+    exact RawRadonGood.reconstructed_paths D (fun q => μ q p) (fun q => hμr q p) hp
+  · let μ' (q : ℚ) (ω : Ω) := μ q (sourceDriverPath B hBc ω)
+    have hc : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
+        Continuous (fun t : ℝ => ∫ z, f z ∂C.process t.toNNReal ω) := by
+      filter_upwards [C.paths] with ω hω
+      intro f
+      exact (hω.continuous_tests f).comp continuous_real_toNNReal
+    have he : ∀ᵐ ω ∂P, ∀ q : ℚ, C.process (q:ℝ).toNNReal ω=μ' q ω :=
+      ae_of_all _ (fun ω q => (hμe ω q).symm)
+    have h := rationalRadonExtension_common_version P μ' (fun t ω => C.process t.toNNReal ω)
+      (fun t ω => (C.radon t.toNNReal ω).1) hc he
+    filter_upwards [h] with ω hω
+    intro t
+    simpa only [R,rationalRadonExtension,μ',Real.toNNReal_coe] using hω t
+
+end Multipoint
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RawRadonGoodPaths.lean b/lean/OAI/Probability/SLE/Mass/RawRadonGoodPaths.lean
index 19151e9e70268c1b4b37599c7f2b9bc322029693..aaab5be7f2e2aeb2e810b41a859f1b60f07c296f 100644
--- a/lean/OAI/Probability/SLE/Mass/RawRadonGoodPaths.lean
+++ b/lean/OAI/Probability/SLE/Mass/RawRadonGoodPaths.lean
@@ -1,109 +1,109 @@
-import OAI.Probability.SLE.Geometry.RationalTestPath
-
-namespace OAI
-
-/-! A countably measurable event guaranteeing zero initial mass, monotone
-growth, and continuous compact-test paths for the raw rational reconstruction. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology CompactlySupported NNReal
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-def RawRadonGood (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Measure upperHalfPlane) : Prop :=
-  HasContinuousRationalTests D μ ∧
-  (∀ p q : ℚ, 0 ≤ p → p ≤ q → ∀ j,
-    (∫ z, (D.test j ⊔ 0) z ∂μ p) ≤ ∫ z, (D.test j ⊔ 0) z ∂μ q) ∧
-  ∀ j, (∫ z, D.test j z ∂μ 0)=0
-
-lemma measurableSet_rawRadonGood {Y : Type*} [MeasurableSpace Y]
-    (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Y → Measure upperHalfPlane)
-    (hm : ∀ q, Measurable (μ q)) (hr : ∀ q y, (μ q y).Regular) :
-    MeasurableSet {y | RawRadonGood D (fun q => μ q y)} := by
-  have htest (q : ℚ) (f : C_c(upperHalfPlane,ℝ)) :=
-    measurable_halfPlane_compact_test_integral (μ q) (hm q) (hr q) f
-  have hc := measurableSet_hasContinuousRationalTests D μ (fun q j => htest q (D.test j))
-  have ho : MeasurableSet {y | ∀ p q : ℚ, 0 ≤ p → p ≤ q → ∀ j,
-      (∫ z, (D.test j ⊔ 0) z ∂μ p y) ≤ ∫ z, (D.test j ⊔ 0) z ∂μ q y} := by
-    simp only [Set.ofPred_forall]
-    exact MeasurableSet.iInter fun p => MeasurableSet.iInter fun q =>
-      MeasurableSet.iInter fun _ => MeasurableSet.iInter fun _ =>
-        MeasurableSet.iInter fun j => measurableSet_le (htest p _) (htest q _)
-  have hz : MeasurableSet {y | ∀ j, (∫ z, D.test j z ∂μ 0 y)=0} := by
-    simp only [Set.ofPred_forall]
-    exact MeasurableSet.iInter fun j => measurableSet_eq_fun (htest 0 (D.test j)) measurable_const
-  simpa only [RawRadonGood,Set.ofPred_and] using hc.inter (ho.inter hz)
-
-theorem RawRadonGood.reconstructed_paths
-    (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Measure upperHalfPlane)
-    (hr : ∀ q, (μ q).Regular) (hg : RawRadonGood D μ) :
-    let N := fun t => rationalRadonExtension (fun q (_ : Unit) => μ q) t ()
-    N 0=0 ∧ Monotone N ∧ ∀ f : C_c(upperHalfPlane,ℝ),
-      Continuous (fun t => ∫ z, f z ∂N t) := by
-  dsimp only
-  let N := fun t => rationalRadonExtension (fun q (_ : Unit) => μ q) t ()
-  obtain ⟨hq,hc⟩ := HasContinuousRationalTests.continuous_extension D μ hr hg.1
-  have hzero : μ 0=0 := by
-    let := hr 0
-    apply D.measure_eq_of_test_integrals
-    intro j
-    simpa only [integral_zero_measure] using hg.2.2 j
-  have hz : N 0=0 := by
-    have h := hq 0 le_rfl
-    simpa only [Rat.cast_zero,Real.toNNReal_zero,hzero] using h
-  refine ⟨hz,?_,hc⟩
-  intro s t hst
-  rcases eq_or_lt_of_le hst with rfl | hst
-  · exact le_rfl
-  let := (rationalRadonExtension_regular (fun q (_ : Unit) => μ q) s ()).1
-  let := (rationalRadonExtension_regular (fun q (_ : Unit) => μ q) t ()).1
-  apply radon_measure_le_of_nonneg_tests
-  intro f hf
-  have hlim (r : ℝ≥0) : Tendsto (fun n => ∫ z, f z ∂μ (leftRationalSample n r)) atTop
-      (𝓝 (∫ z, f z ∂N r)) := by
-    have he (n : ℕ) := hq (leftRationalSample n r) (leftRationalSample_nonneg n r)
-    have ht := (continuous_real_toNNReal.tendsto (r:ℝ)).comp (leftRationalSample_tendsto r)
-    simpa only [Real.toNNReal_coe,Function.comp_def,he] using
-      (hc f).continuousAt.tendsto.comp ht
-  have hlt : ∀ᶠ n in atTop, leftRationalSample n s ≤ leftRationalSample n t := by
-    have hs := (leftRationalSample_tendsto s).eventually
-      (gt_mem_nhds (show (s:ℝ)<((s:ℝ)+t)/2 by exact (by have := NNReal.coe_lt_coe.mpr hst; linarith)))
-    have ht := (leftRationalSample_tendsto t).eventually
-      (lt_mem_nhds (show ((s:ℝ)+t)/2<(t:ℝ) by exact (by have := NNReal.coe_lt_coe.mpr hst; linarith)))
-    filter_upwards [hs,ht] with n hn hm
-    exact Rat.cast_le.mp (hn.trans hm).le
-  apply le_of_tendsto_of_tendsto (hlim s) (hlim t)
-  filter_upwards [hlt] with n hn
-  have hle : μ (leftRationalSample n s) ≤ μ (leftRationalSample n t) := by
-    let := hr (leftRationalSample n s)
-    let := hr (leftRationalSample n t)
-    exact D.measure_le_of_positive_tests _ _
-      (hg.2.1 _ _ (leftRationalSample_nonneg n s) hn)
-  let := hr (leftRationalSample n t)
-  exact integral_mono_measure hle (ae_of_all _ hf) f.integrable
-
-theorem rawRadonGood_of_continuous_path
-    (D : LocalTestFamily upperHalfPlane) (N : ℝ≥0 → Measure upperHalfPlane)
-    (hr : ∀ t, (N t).Regular) (hz : N 0=0) (hm : Monotone N)
-    (hc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) :
-    RawRadonGood D (fun q : ℚ => N (q:ℝ).toNNReal) := by
-  refine ⟨?_,?_,?_⟩
-  · intro j
-    refine ⟨⟨fun t => ∫ z, D.test j z ∂N t.toNNReal,
-      (hc (D.test j)).comp continuous_real_toNNReal⟩,?_⟩
-    rfl
-  · intro p q _hp hpq j
-    let := hr (q:ℝ).toNNReal
-    exact integral_mono_measure (hm (Real.toNNReal_le_toNNReal (Rat.cast_le.mpr hpq)))
-      (ae_of_all _ (fun _ => le_max_right _ _)) (D.test j ⊔ 0).integrable
-  · intro j
-    simp only [Rat.cast_zero,Real.toNNReal_zero,hz,integral_zero_measure]
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.RationalTestPath
+
+namespace OAI
+
+/-! A countably measurable event guaranteeing zero initial mass, monotone
+growth, and continuous compact-test paths for the raw rational reconstruction. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology CompactlySupported NNReal
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+def RawRadonGood (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Measure upperHalfPlane) : Prop :=
+  HasContinuousRationalTests D μ ∧
+  (∀ p q : ℚ, 0 ≤ p → p ≤ q → ∀ j,
+    (∫ z, (D.test j ⊔ 0) z ∂μ p) ≤ ∫ z, (D.test j ⊔ 0) z ∂μ q) ∧
+  ∀ j, (∫ z, D.test j z ∂μ 0)=0
+
+lemma measurableSet_rawRadonGood {Y : Type*} [MeasurableSpace Y]
+    (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Y → Measure upperHalfPlane)
+    (hm : ∀ q, Measurable (μ q)) (hr : ∀ q y, (μ q y).Regular) :
+    MeasurableSet {y | RawRadonGood D (fun q => μ q y)} := by
+  have htest (q : ℚ) (f : C_c(upperHalfPlane,ℝ)) :=
+    measurable_halfPlane_compact_test_integral (μ q) (hm q) (hr q) f
+  have hc := measurableSet_hasContinuousRationalTests D μ (fun q j => htest q (D.test j))
+  have ho : MeasurableSet {y | ∀ p q : ℚ, 0 ≤ p → p ≤ q → ∀ j,
+      (∫ z, (D.test j ⊔ 0) z ∂μ p y) ≤ ∫ z, (D.test j ⊔ 0) z ∂μ q y} := by
+    simp only [Set.ofPred_forall]
+    exact MeasurableSet.iInter fun p => MeasurableSet.iInter fun q =>
+      MeasurableSet.iInter fun _ => MeasurableSet.iInter fun _ =>
+        MeasurableSet.iInter fun j => measurableSet_le (htest p _) (htest q _)
+  have hz : MeasurableSet {y | ∀ j, (∫ z, D.test j z ∂μ 0 y)=0} := by
+    simp only [Set.ofPred_forall]
+    exact MeasurableSet.iInter fun j => measurableSet_eq_fun (htest 0 (D.test j)) measurable_const
+  simpa only [RawRadonGood,Set.ofPred_and] using hc.inter (ho.inter hz)
+
+theorem RawRadonGood.reconstructed_paths
+    (D : LocalTestFamily upperHalfPlane) (μ : ℚ → Measure upperHalfPlane)
+    (hr : ∀ q, (μ q).Regular) (hg : RawRadonGood D μ) :
+    let N := fun t => rationalRadonExtension (fun q (_ : Unit) => μ q) t ()
+    N 0=0 ∧ Monotone N ∧ ∀ f : C_c(upperHalfPlane,ℝ),
+      Continuous (fun t => ∫ z, f z ∂N t) := by
+  dsimp only
+  let N := fun t => rationalRadonExtension (fun q (_ : Unit) => μ q) t ()
+  obtain ⟨hq,hc⟩ := HasContinuousRationalTests.continuous_extension D μ hr hg.1
+  have hzero : μ 0=0 := by
+    let := hr 0
+    apply D.measure_eq_of_test_integrals
+    intro j
+    simpa only [integral_zero_measure] using hg.2.2 j
+  have hz : N 0=0 := by
+    have h := hq 0 le_rfl
+    simpa only [Rat.cast_zero,Real.toNNReal_zero,hzero] using h
+  refine ⟨hz,?_,hc⟩
+  intro s t hst
+  rcases eq_or_lt_of_le hst with rfl | hst
+  · exact le_rfl
+  let := (rationalRadonExtension_regular (fun q (_ : Unit) => μ q) s ()).1
+  let := (rationalRadonExtension_regular (fun q (_ : Unit) => μ q) t ()).1
+  apply radon_measure_le_of_nonneg_tests
+  intro f hf
+  have hlim (r : ℝ≥0) : Tendsto (fun n => ∫ z, f z ∂μ (leftRationalSample n r)) atTop
+      (𝓝 (∫ z, f z ∂N r)) := by
+    have he (n : ℕ) := hq (leftRationalSample n r) (leftRationalSample_nonneg n r)
+    have ht := (continuous_real_toNNReal.tendsto (r:ℝ)).comp (leftRationalSample_tendsto r)
+    simpa only [Real.toNNReal_coe,Function.comp_def,he] using
+      (hc f).continuousAt.tendsto.comp ht
+  have hlt : ∀ᶠ n in atTop, leftRationalSample n s ≤ leftRationalSample n t := by
+    have hs := (leftRationalSample_tendsto s).eventually
+      (gt_mem_nhds (show (s:ℝ)<((s:ℝ)+t)/2 by exact (by have := NNReal.coe_lt_coe.mpr hst; linarith)))
+    have ht := (leftRationalSample_tendsto t).eventually
+      (lt_mem_nhds (show ((s:ℝ)+t)/2<(t:ℝ) by exact (by have := NNReal.coe_lt_coe.mpr hst; linarith)))
+    filter_upwards [hs,ht] with n hn hm
+    exact Rat.cast_le.mp (hn.trans hm).le
+  apply le_of_tendsto_of_tendsto (hlim s) (hlim t)
+  filter_upwards [hlt] with n hn
+  have hle : μ (leftRationalSample n s) ≤ μ (leftRationalSample n t) := by
+    let := hr (leftRationalSample n s)
+    let := hr (leftRationalSample n t)
+    exact D.measure_le_of_positive_tests _ _
+      (hg.2.1 _ _ (leftRationalSample_nonneg n s) hn)
+  let := hr (leftRationalSample n t)
+  exact integral_mono_measure hle (ae_of_all _ hf) f.integrable
+
+theorem rawRadonGood_of_continuous_path
+    (D : LocalTestFamily upperHalfPlane) (N : ℝ≥0 → Measure upperHalfPlane)
+    (hr : ∀ t, (N t).Regular) (hz : N 0=0) (hm : Monotone N)
+    (hc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) :
+    RawRadonGood D (fun q : ℚ => N (q:ℝ).toNNReal) := by
+  refine ⟨?_,?_,?_⟩
+  · intro j
+    refine ⟨⟨fun t => ∫ z, D.test j z ∂N t.toNNReal,
+      (hc (D.test j)).comp continuous_real_toNNReal⟩,?_⟩
+    rfl
+  · intro p q _hp hpq j
+    let := hr (q:ℝ).toNNReal
+    exact integral_mono_measure (hm (Real.toNNReal_le_toNNReal (Rat.cast_le.mpr hpq)))
+      (ae_of_all _ (fun _ => le_max_right _ _)) (D.test j ⊔ 0).integrable
+  · intro j
+    simp only [Rat.cast_zero,Real.toNNReal_zero,hz,integral_zero_measure]
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RawRadonLimit.lean b/lean/OAI/Probability/SLE/Mass/RawRadonLimit.lean
index 19e1b30b65aa58eb1833ad46c219390e0d037cd0..32b0018ed0744df7b840b1ad532607e51087c822 100644
--- a/lean/OAI/Probability/SLE/Mass/RawRadonLimit.lean
+++ b/lean/OAI/Probability/SLE/Mass/RawRadonLimit.lean
@@ -1,44 +1,44 @@
-import OAI.Probability.SLE.Mass.CanonicalRadonLimit
-
-namespace OAI
-
-/-! Raw Radon reconstruction from countable test-Cauchy data. The existing
-canonical reconstruction tests Cauchy convergence in the data space itself;
-its cutoff coordinates supply the local mass bounds. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology CompactlySupported
-
-namespace SLEExactGauge
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-lemma LocalTestFamily.exists_raw_radon_limit
-    (D : LocalTestFamily X) {Y : Type*} [MeasurableSpace Y]
-    (μ : ℕ → Y → Measure X) (hμ : ∀ n y, (μ n y).Regular)
-    (hm : ∀ n (f : C_c(X,ℝ)), Measurable (fun y => ∫ x, f x ∂μ n y)) :
-    ∃ ν : Y → Measure X, Measurable ν ∧ (∀ y, (ν y).Regular) ∧
-      ∀ y, (∀ j, CauchySeq (fun n => ∫ x, D.test j x ∂μ n y)) →
-        ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n y) atTop (𝓝 (∫ x, f x ∂ν y)) := by
-  let ν (y : Y) := canonicalRadonLimit (fun n => μ n y)
-  refine ⟨ν,canonicalRadonLimit_measurable μ ?_ hm,
-    fun y => canonicalRadonLimit_regular _,?_⟩
-  · intro n y
-    let := hμ n y
-    infer_instance
-  · intro y hy
-    let (n : ℕ) : IsFiniteMeasureOnCompacts (μ n y) := by
-      let := hμ n y
-      infer_instance
-    obtain ⟨η,hη,_,hl⟩ := D.exists_radon_limit (fun n => μ n y)
-      (fun j => cauchySeq_tendsto_of_complete (hy j))
-    have he : ν y=η := canonicalRadonLimit_identified _ η ⟨hη,hl⟩
-    simpa only [he] using hl
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CanonicalRadonLimit
+
+namespace OAI
+
+/-! Raw Radon reconstruction from countable test-Cauchy data. The existing
+canonical reconstruction tests Cauchy convergence in the data space itself;
+its cutoff coordinates supply the local mass bounds. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology CompactlySupported
+
+namespace SLEExactGauge
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+lemma LocalTestFamily.exists_raw_radon_limit
+    (D : LocalTestFamily X) {Y : Type*} [MeasurableSpace Y]
+    (μ : ℕ → Y → Measure X) (hμ : ∀ n y, (μ n y).Regular)
+    (hm : ∀ n (f : C_c(X,ℝ)), Measurable (fun y => ∫ x, f x ∂μ n y)) :
+    ∃ ν : Y → Measure X, Measurable ν ∧ (∀ y, (ν y).Regular) ∧
+      ∀ y, (∀ j, CauchySeq (fun n => ∫ x, D.test j x ∂μ n y)) →
+        ∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n y) atTop (𝓝 (∫ x, f x ∂ν y)) := by
+  let ν (y : Y) := canonicalRadonLimit (fun n => μ n y)
+  refine ⟨ν,canonicalRadonLimit_measurable μ ?_ hm,
+    fun y => canonicalRadonLimit_regular _,?_⟩
+  · intro n y
+    let := hμ n y
+    infer_instance
+  · intro y hy
+    let (n : ℕ) : IsFiniteMeasureOnCompacts (μ n y) := by
+      let := hμ n y
+      infer_instance
+    obtain ⟨η,hη,_,hl⟩ := D.exists_radon_limit (fun n => μ n y)
+      (fun j => cauchySeq_tendsto_of_complete (hy j))
+    have he : ν y=η := canonicalRadonLimit_identified _ η ⟨hη,hl⟩
+    simpa only [he] using hl
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RawScaledMassMeans.lean b/lean/OAI/Probability/SLE/Mass/RawScaledMassMeans.lean
index 7b71165c133c6ef4f192308b08d0c151fbc5bfd3..0c7a823bb0544859e5706199ca9505e002492bc1 100644
--- a/lean/OAI/Probability/SLE/Mass/RawScaledMassMeans.lean
+++ b/lean/OAI/Probability/SLE/Mass/RawScaledMassMeans.lean
@@ -1,135 +1,135 @@
-import OAI.Probability.SLE.Mass.RawScaledMassPaths
-import OAI.Probability.SLE.Mass.ScaledGreenMeasure
-import OAI.Probability.SLE.Mass.GreenMassProcessUniqueness
-
-namespace OAI
-
-/-! Mean bounds and equal Green remainders for the physical dilation
-and the canonical functional of the scaled Brownian driver. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-local instance rawScaledMeansMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance rawScaledMeansBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-lemma bind_comp_eq_of_path_law {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (X Y : Ω → C(ℝ≥0,ℝ)) (hX : Measurable X) (hY : Measurable Y)
-    (hXY : P.map X=P.map Y) (N : C(ℝ≥0,ℝ) → Measure upperHalfPlane) (hN : Measurable N) :
-    P.bind (fun ω => N (X ω))=P.bind (fun ω => N (Y ω)) := by
-  calc
-    _ = (P.map X).bind N := by
-      unfold Measure.bind
-      rw [Measure.map_map hN hX]
-      rfl
-    _ = (P.map Y).bind N := by rw [hXY]
-    _ = _ := by
-      unfold Measure.bind
-      rw [Measure.map_map hN hY]
-      rfl
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
-  (hc : ∀ ω, Continuous (fun t => B t ω))
-  (R : RawCanonicalMass)
-  (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-    R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
-  (c : ℝ≥0) (hcp : 0 < c)
-
-include C hB hc R he c hcp in
-omit [IsProbabilityMeasure P] in
-lemma CanonicalMass.scaledRawMass_mean_le (t : ℝ≥0) :
-    P.bind (scaledRawMass R c B hc t) ≤ initialGreenMeasure κ := by
-  have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id)
-  have hp := source_path_measurable (sourceDriverPath B hc) hm
-  have hy := source_path_measurable
-    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc))
-    (scaledBrownianDriver_measurable c B hm)
-  have h := bind_comp_eq_of_path_law P _ _ hy hp
-    (scaledBrownianDriver_path_law c hcp B hB hm hc) (R.process (c^2*t))
-    (R.progressive.adapted.measurable (i:=c^2*t))
-  change P.bind (scaledRawMass R c B hc t)=_ at h
-  rw [h,Measure.bind_congr_right (he.mono (fun _ h => h.1 (c^2*t)))]
-  simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le (c^2*t) univ
-
-include C hB hc R he c hcp in
-omit [IsProbabilityMeasure P] in
-lemma CanonicalMass.scaledRawTerminal_mean :
-    P.bind (scaledRawTerminal R c B hc)=initialGreenMeasure κ := by
-  have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id)
-  have hp := source_path_measurable (sourceDriverPath B hc) hm
-  have hy := source_path_measurable
-    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc))
-    (scaledBrownianDriver_measurable c B hm)
-  have h := bind_comp_eq_of_path_law P _ _ hy hp
-    (scaledBrownianDriver_path_law c hcp B hB hm hc) R.terminal R.terminal_measurable
-  change P.bind (scaledRawTerminal R c B hc)=_ at h
-  rw [h,Measure.bind_congr_right (he.mono (fun _ h => h.2)),C.rational.mean]
-
-include C in
-omit [IsProbabilityMeasure P] in
-lemma CanonicalMass.deterministic_green_kernel (t : ℝ≥0) (E : Set Ω)
-    (hE : MeasurableSet[(Filtration.natural B (fun s =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) :
-    (P.restrict E).bind C.rational.terminal=(P.restrict E).bind (C.process t)+
-      (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) := by
-  let ℱ := Filtration.natural B (fun s =>
-    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)
-  have hs := isStoppingTime_const ℱ t
-  have hEs : MeasurableSet[hs.measurableSpace] E := by
-    change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E
-    rw [IsStoppingTime.measurableSpace_const]
-    exact hE
-  have h := C.kernel (fun _ => (t:WithTop ℝ≥0)) hs E hEs
-  change (P.restrict E).bind C.rational.terminal=(P.restrict E).bind (C.process t)+
-    (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) at h
-  exact h
-
-include C hB hc R he c hcp in
-lemma CanonicalMass.scaledRawMass_cross_kernel
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (t : ℝ≥0) (E : Set Ω)
-    (hE : MeasurableSet[(Filtration.natural B (fun s =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) :
-    (P.restrict E).bind (fun ω => dilationMass κ c hcp (C.rational.terminal ω))+
-      (P.restrict E).bind (scaledRawMass R c B hc t)=
-    (P.restrict E).bind (scaledRawTerminal R c B hc)+
-      (P.restrict E).bind (fun ω => dilationMass κ c hcp (C.process t ω)) := by
-  have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id)
-  have hCm := C.progressive.adapted.measurable (i:=t)
-  have hGm := halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t
-  have hdm : Measurable (dilationMass κ c hcp) := conformalMass_measurable _ _
-  have hNm := (scaledRawMass_adapted R c hcp B hm hc).measurable (i:=t)
-  have hd := congrArg (dilationMass κ c hcp) (C.deterministic_green_kernel t E hE)
-  change dilationMass κ c hcp ((P.restrict E).bind C.rational.terminal)=
-    conformalMass (halfPlaneDilation c hcp) _
-      ((P.restrict E).bind (C.process t)+(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) at hd
-  rw [conformalMass_add] at hd
-  change dilationMass κ c hcp ((P.restrict E).bind C.rational.terminal)=
-    dilationMass κ c hcp ((P.restrict E).bind (C.process t))+
-      dilationMass κ c hcp ((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) at hd
-  rw [← dilationMass_bind _ _ _ _ _ C.rational.terminal_measurable,
-    ← dilationMass_bind _ _ _ _ _ hCm,← dilationMass_bind _ _ _ _ _ hGm] at hd
-  have hr := C.raw_scaled_green_kernel hB hc F hw R he c hcp t E hE
-  dsimp only at hr
-  simp_rw [canonicalGreenMass_scaled κ B hc γ G F hw c hcp t] at hr
-  change (P.restrict E).bind (scaledRawTerminal R c B hc)=
-    (P.restrict E).bind (fun ω => scaledRawMass R c B hc t ω+
-      dilationMass κ c hcp (halfPlaneGreenMeasure κ B γ G t ω)) at hr
-  have hD : Measurable (fun ω => dilationMass κ c hcp (halfPlaneGreenMeasure κ B γ G t ω)) :=
-    hdm.comp hGm
-  rw [bind_kernel_add _ _ _ hNm hD] at hr
-  rw [hd,hr]
-  ac_rfl
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RawScaledMassPaths
+import OAI.Probability.SLE.Mass.ScaledGreenMeasure
+import OAI.Probability.SLE.Mass.GreenMassProcessUniqueness
+
+namespace OAI
+
+/-! Mean bounds and equal Green remainders for the physical dilation
+and the canonical functional of the scaled Brownian driver. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+local instance rawScaledMeansMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance rawScaledMeansBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+lemma bind_comp_eq_of_path_law {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (X Y : Ω → C(ℝ≥0,ℝ)) (hX : Measurable X) (hY : Measurable Y)
+    (hXY : P.map X=P.map Y) (N : C(ℝ≥0,ℝ) → Measure upperHalfPlane) (hN : Measurable N) :
+    P.bind (fun ω => N (X ω))=P.bind (fun ω => N (Y ω)) := by
+  calc
+    _ = (P.map X).bind N := by
+      unfold Measure.bind
+      rw [Measure.map_map hN hX]
+      rfl
+    _ = (P.map Y).bind N := by rw [hXY]
+    _ = _ := by
+      unfold Measure.bind
+      rw [Measure.map_map hN hY]
+      rfl
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
+  (hc : ∀ ω, Continuous (fun t => B t ω))
+  (R : RawCanonicalMass)
+  (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+    R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω)
+  (c : ℝ≥0) (hcp : 0 < c)
+
+include C hB hc R he c hcp in
+omit [IsProbabilityMeasure P] in
+lemma CanonicalMass.scaledRawMass_mean_le (t : ℝ≥0) :
+    P.bind (scaledRawMass R c B hc t) ≤ initialGreenMeasure κ := by
+  have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id)
+  have hp := source_path_measurable (sourceDriverPath B hc) hm
+  have hy := source_path_measurable
+    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc))
+    (scaledBrownianDriver_measurable c B hm)
+  have h := bind_comp_eq_of_path_law P _ _ hy hp
+    (scaledBrownianDriver_path_law c hcp B hB hm hc) (R.process (c^2*t))
+    (R.progressive.adapted.measurable (i:=c^2*t))
+  change P.bind (scaledRawMass R c B hc t)=_ at h
+  rw [h,Measure.bind_congr_right (he.mono (fun _ h => h.1 (c^2*t)))]
+  simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le (c^2*t) univ
+
+include C hB hc R he c hcp in
+omit [IsProbabilityMeasure P] in
+lemma CanonicalMass.scaledRawTerminal_mean :
+    P.bind (scaledRawTerminal R c B hc)=initialGreenMeasure κ := by
+  have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id)
+  have hp := source_path_measurable (sourceDriverPath B hc) hm
+  have hy := source_path_measurable
+    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc))
+    (scaledBrownianDriver_measurable c B hm)
+  have h := bind_comp_eq_of_path_law P _ _ hy hp
+    (scaledBrownianDriver_path_law c hcp B hB hm hc) R.terminal R.terminal_measurable
+  change P.bind (scaledRawTerminal R c B hc)=_ at h
+  rw [h,Measure.bind_congr_right (he.mono (fun _ h => h.2)),C.rational.mean]
+
+include C in
+omit [IsProbabilityMeasure P] in
+lemma CanonicalMass.deterministic_green_kernel (t : ℝ≥0) (E : Set Ω)
+    (hE : MeasurableSet[(Filtration.natural B (fun s =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) :
+    (P.restrict E).bind C.rational.terminal=(P.restrict E).bind (C.process t)+
+      (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) := by
+  let ℱ := Filtration.natural B (fun s =>
+    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)
+  have hs := isStoppingTime_const ℱ t
+  have hEs : MeasurableSet[hs.measurableSpace] E := by
+    change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E
+    rw [IsStoppingTime.measurableSpace_const]
+    exact hE
+  have h := C.kernel (fun _ => (t:WithTop ℝ≥0)) hs E hEs
+  change (P.restrict E).bind C.rational.terminal=(P.restrict E).bind (C.process t)+
+    (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) at h
+  exact h
+
+include C hB hc R he c hcp in
+lemma CanonicalMass.scaledRawMass_cross_kernel
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (t : ℝ≥0) (E : Set Ω)
+    (hE : MeasurableSet[(Filtration.natural B (fun s =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) :
+    (P.restrict E).bind (fun ω => dilationMass κ c hcp (C.rational.terminal ω))+
+      (P.restrict E).bind (scaledRawMass R c B hc t)=
+    (P.restrict E).bind (scaledRawTerminal R c B hc)+
+      (P.restrict E).bind (fun ω => dilationMass κ c hcp (C.process t ω)) := by
+  have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id)
+  have hCm := C.progressive.adapted.measurable (i:=t)
+  have hGm := halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t
+  have hdm : Measurable (dilationMass κ c hcp) := conformalMass_measurable _ _
+  have hNm := (scaledRawMass_adapted R c hcp B hm hc).measurable (i:=t)
+  have hd := congrArg (dilationMass κ c hcp) (C.deterministic_green_kernel t E hE)
+  change dilationMass κ c hcp ((P.restrict E).bind C.rational.terminal)=
+    conformalMass (halfPlaneDilation c hcp) _
+      ((P.restrict E).bind (C.process t)+(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) at hd
+  rw [conformalMass_add] at hd
+  change dilationMass κ c hcp ((P.restrict E).bind C.rational.terminal)=
+    dilationMass κ c hcp ((P.restrict E).bind (C.process t))+
+      dilationMass κ c hcp ((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) at hd
+  rw [← dilationMass_bind _ _ _ _ _ C.rational.terminal_measurable,
+    ← dilationMass_bind _ _ _ _ _ hCm,← dilationMass_bind _ _ _ _ _ hGm] at hd
+  have hr := C.raw_scaled_green_kernel hB hc F hw R he c hcp t E hE
+  dsimp only at hr
+  simp_rw [canonicalGreenMass_scaled κ B hc γ G F hw c hcp t] at hr
+  change (P.restrict E).bind (scaledRawTerminal R c B hc)=
+    (P.restrict E).bind (fun ω => scaledRawMass R c B hc t ω+
+      dilationMass κ c hcp (halfPlaneGreenMeasure κ B γ G t ω)) at hr
+  have hD : Measurable (fun ω => dilationMass κ c hcp (halfPlaneGreenMeasure κ B γ G t ω)) :=
+    hdm.comp hGm
+  rw [bind_kernel_add _ _ _ hNm hD] at hr
+  rw [hd,hr]
+  ac_rfl
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RawScaledMassPaths.lean b/lean/OAI/Probability/SLE/Mass/RawScaledMassPaths.lean
index 40c34628c5a68fdc06d5dd1698093dddea0ad465..880dbbcc4a3121d3c4e90eba8a6f4f1243d88de2 100644
--- a/lean/OAI/Probability/SLE/Mass/RawScaledMassPaths.lean
+++ b/lean/OAI/Probability/SLE/Mass/RawScaledMassPaths.lean
@@ -1,109 +1,109 @@
-import OAI.Probability.SLE.Loewner.RawScaledGreenKernel
-import OAI.Probability.SLE.Loewner.RawTerminalExhaustion
-import OAI.Probability.SLE.Loewner.StoppedInversePaths
-
-namespace OAI
-
-/-! One raw canonical functional has a measurable full-law certificate
-for continuous positive mass paths. Brownian scaling preserves it. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal CompactlySupported
-
-namespace SLEExactGauge.Multipoint
-
-local instance rawScaledMassMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance rawScaledMassBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance rawScaledMassLocallyCompact : LocallyCompactSpace upperHalfPlane :=
-  isOpen_upperHalfPlane.locallyCompactSpace
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-
-omit [IsProbabilityMeasure P] in
-lemma CanonicalMass.exists_raw_positive_paths (C : CanonicalMass P κ B hBm γ G)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) :
-    ∃ R : RawCanonicalMass,
-      (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
-        R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧
-      ∃ E : Set C(ℝ≥0,ℝ), MeasurableSet E ∧
-        (∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E) ∧
-        ∀ p ∈ E, PositiveMassPaths (fun t => R.process t p) (R.terminal p) := by
-  obtain ⟨N,E,hNp,hNr,hNn,hEm,hEf,hEp,he⟩ := C.exists_raw_nonanticipating_with_good_set hc
-  let R : RawCanonicalMass := ⟨N,rawTerminalMass N,hNp,hNr,
-    rawTerminalMass_measurable N hNp hNr,rawTerminalMass_radon N,hNn⟩
-  obtain ⟨hTm,hTf,hTp⟩ := C.raw_terminal_good_set hc N hNp hNr he
-  refine ⟨R,?_,E ∩ {p | RawTerminalGood (localTestFamily upperHalfPlane)
-    (fun t => N t p) (rawTerminalMass N p)},hEm.inter hTm,?_,?_⟩
-  · filter_upwards [he,C.rawTerminal_realizes hc N he] with ω hω ht
-    exact ⟨hω,ht⟩
-  · exact hEf.and hTf
-  · intro p hp
-    obtain ⟨hz,hm,hcont⟩ := hEp p hp.1
-    exact ⟨hz,hm,(hTp p hp.2 hm).1,(fun t => hNr t p),fun f _ => hcont f⟩
-
-def scaledRawMass (R : RawCanonicalMass) (c : ℝ≥0)
-    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane :=
-  R.process (c^2*t) (sourceDriverPath (scaledBrownianDriver c B)
-    (scaledBrownianDriver_continuous c B hc) ω)
-
-def scaledRawTerminal (R : RawCanonicalMass) (c : ℝ≥0)
-    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (ω : Ω) : Measure upperHalfPlane :=
-  R.terminal (sourceDriverPath (scaledBrownianDriver c B)
-    (scaledBrownianDriver_continuous c B hc) ω)
-
-lemma scaledRawMass_adapted (R : RawCanonicalMass) (c : ℝ≥0) (hcp : 0 < c)
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) :
-    Adapted (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) (scaledRawMass R c B hc) := by
-  let ℱc := Filtration.natural (scaledBrownianDriver c B)
-    (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable)
-  let Y := sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc)
-  have hY : Adapted ℱc (fun t ω => Y ω t) := fun t =>
-    (Filtration.stronglyAdapted_natural
-      (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable) t).measurable
-  have hp := (raw_process_comp_progressive ℱc Y hY R.process R.progressive).adapted
-  intro t
-  have h := hp (c^2*t)
-  change Measurable[ℱc (c^2*t)] (scaledRawMass R c B hc t) at h
-  rwa [show ℱc (c^2*t)=(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t from
-    scaledBrownianDriver_filtration c hcp B hm t] at h
-
-omit [IsProbabilityMeasure P] in
-lemma scaledRawMass_positive_paths
-    (R : RawCanonicalMass) (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E)
-    (hEp : ∀ p ∈ E, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
-    (c : ℝ≥0) (hcp : 0 < c) :
-    ∀ᵐ ω ∂P, PositiveMassPaths (fun t => scaledRawMass R c B hc t ω)
-      (scaledRawTerminal R c B hc ω) := by
-  have hY := source_path_measurable
-    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc))
-    (scaledBrownianDriver_measurable c B hm)
-  have hf : ∀ᵐ p ∂P.map (sourceDriverPath (scaledBrownianDriver c B)
-      (scaledBrownianDriver_continuous c B hc)), p ∈ E := by
-    rwa [scaledBrownianDriver_path_law c hcp B hB hm hc]
-  filter_upwards [(ae_map_iff hY.aemeasurable hE).mp hf] with ω hω
-  have hp := hEp _ hω
-  refine ⟨?_,?_,?_,?_,?_⟩
-  · simpa only [scaledRawMass,mul_zero] using hp.zero
-  · intro s t hst
-    exact hp.monotone (mul_le_mul_of_nonneg_left hst zero_le)
-  · intro t
-    exact hp.dominated (c^2*t)
-  · intro t
-    exact hp.radon (c^2*t)
-  · intro f hfp
-    exact (hp.continuous_tests f hfp).comp (continuous_const.mul continuous_id)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.RawScaledGreenKernel
+import OAI.Probability.SLE.Loewner.RawTerminalExhaustion
+import OAI.Probability.SLE.Loewner.StoppedInversePaths
+
+namespace OAI
+
+/-! One raw canonical functional has a measurable full-law certificate
+for continuous positive mass paths. Brownian scaling preserves it. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal CompactlySupported
+
+namespace SLEExactGauge.Multipoint
+
+local instance rawScaledMassMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance rawScaledMassBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance rawScaledMassLocallyCompact : LocallyCompactSpace upperHalfPlane :=
+  isOpen_upperHalfPlane.locallyCompactSpace
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+
+omit [IsProbabilityMeasure P] in
+lemma CanonicalMass.exists_raw_positive_paths (C : CanonicalMass P κ B hBm γ G)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) :
+    ∃ R : RawCanonicalMass,
+      (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧
+        R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧
+      ∃ E : Set C(ℝ≥0,ℝ), MeasurableSet E ∧
+        (∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E) ∧
+        ∀ p ∈ E, PositiveMassPaths (fun t => R.process t p) (R.terminal p) := by
+  obtain ⟨N,E,hNp,hNr,hNn,hEm,hEf,hEp,he⟩ := C.exists_raw_nonanticipating_with_good_set hc
+  let R : RawCanonicalMass := ⟨N,rawTerminalMass N,hNp,hNr,
+    rawTerminalMass_measurable N hNp hNr,rawTerminalMass_radon N,hNn⟩
+  obtain ⟨hTm,hTf,hTp⟩ := C.raw_terminal_good_set hc N hNp hNr he
+  refine ⟨R,?_,E ∩ {p | RawTerminalGood (localTestFamily upperHalfPlane)
+    (fun t => N t p) (rawTerminalMass N p)},hEm.inter hTm,?_,?_⟩
+  · filter_upwards [he,C.rawTerminal_realizes hc N he] with ω hω ht
+    exact ⟨hω,ht⟩
+  · exact hEf.and hTf
+  · intro p hp
+    obtain ⟨hz,hm,hcont⟩ := hEp p hp.1
+    exact ⟨hz,hm,(hTp p hp.2 hm).1,(fun t => hNr t p),fun f _ => hcont f⟩
+
+def scaledRawMass (R : RawCanonicalMass) (c : ℝ≥0)
+    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane :=
+  R.process (c^2*t) (sourceDriverPath (scaledBrownianDriver c B)
+    (scaledBrownianDriver_continuous c B hc) ω)
+
+def scaledRawTerminal (R : RawCanonicalMass) (c : ℝ≥0)
+    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (ω : Ω) : Measure upperHalfPlane :=
+  R.terminal (sourceDriverPath (scaledBrownianDriver c B)
+    (scaledBrownianDriver_continuous c B hc) ω)
+
+lemma scaledRawMass_adapted (R : RawCanonicalMass) (c : ℝ≥0) (hcp : 0 < c)
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) :
+    Adapted (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) (scaledRawMass R c B hc) := by
+  let ℱc := Filtration.natural (scaledBrownianDriver c B)
+    (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable)
+  let Y := sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc)
+  have hY : Adapted ℱc (fun t ω => Y ω t) := fun t =>
+    (Filtration.stronglyAdapted_natural
+      (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable) t).measurable
+  have hp := (raw_process_comp_progressive ℱc Y hY R.process R.progressive).adapted
+  intro t
+  have h := hp (c^2*t)
+  change Measurable[ℱc (c^2*t)] (scaledRawMass R c B hc t) at h
+  rwa [show ℱc (c^2*t)=(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t from
+    scaledBrownianDriver_filtration c hcp B hm t] at h
+
+omit [IsProbabilityMeasure P] in
+lemma scaledRawMass_positive_paths
+    (R : RawCanonicalMass) (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E)
+    (hEp : ∀ p ∈ E, PositiveMassPaths (fun t => R.process t p) (R.terminal p))
+    (c : ℝ≥0) (hcp : 0 < c) :
+    ∀ᵐ ω ∂P, PositiveMassPaths (fun t => scaledRawMass R c B hc t ω)
+      (scaledRawTerminal R c B hc ω) := by
+  have hY := source_path_measurable
+    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc))
+    (scaledBrownianDriver_measurable c B hm)
+  have hf : ∀ᵐ p ∂P.map (sourceDriverPath (scaledBrownianDriver c B)
+      (scaledBrownianDriver_continuous c B hc)), p ∈ E := by
+    rwa [scaledBrownianDriver_path_law c hcp B hB hm hc]
+  filter_upwards [(ae_map_iff hY.aemeasurable hE).mp hf] with ω hω
+  have hp := hEp _ hω
+  refine ⟨?_,?_,?_,?_,?_⟩
+  · simpa only [scaledRawMass,mul_zero] using hp.zero
+  · intro s t hst
+    exact hp.monotone (mul_le_mul_of_nonneg_left hst zero_le)
+  · intro t
+    exact hp.dominated (c^2*t)
+  · intro t
+    exact hp.radon (c^2*t)
+  · intro f hfp
+    exact (hp.continuous_tests f hfp).comp (continuous_const.mul continuous_id)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RestartMassCoordinates.lean b/lean/OAI/Probability/SLE/Mass/RestartMassCoordinates.lean
index 3a1eddd74327e56991cd952737bb20531379e0f1..d4404372e7f09cf5e8b9895ac64b9439a7a8b995 100644
--- a/lean/OAI/Probability/SLE/Mass/RestartMassCoordinates.lean
+++ b/lean/OAI/Probability/SLE/Mass/RestartMassCoordinates.lean
@@ -1,87 +1,87 @@
-import OAI.Probability.SLE.Mass.ConformalMassTransport
-import OAI.Probability.SLE.Geometry.RestartConformalRadius
-
-namespace OAI
-
-/-! The actual centered inverse Loewner map and its mass weight.
-These are defined only on interior points; no boundary extension is used. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Classical
-open scoped NNReal ENNReal Topology CompactlySupported
-
-namespace SLEExactGauge
-
-def IsCapacityTwoWitness.restartHomeomorph
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) :
-    upperHalfPlane ≃ₜ survivingDomain γ T := by
-  rcases hw with ⟨hc,_,_,hG,hF,hGM,hFM,hFG,hGF,_,_,_,_⟩
-  have hshift (z : upperHalfPlane) : (z:ℂ)+(U T:ℂ) ∈ upperHalfPlane := by
-    simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero]
-      using z.property
-  have hback (z : survivingDomain γ T) : G T z-(U T:ℂ) ∈ upperHalfPlane := by
-    simpa only [upperHalfPlane,mem_ofPred_eq,Complex.sub_im,Complex.ofReal_im,sub_zero]
-      using hGM T z.property
-  refine {
-    toFun := fun z => ⟨F T ((z:ℂ)+(U T:ℂ)),hFM T (hshift z)⟩
-    invFun := fun z => ⟨G T z-(U T:ℂ),hback z⟩
-    left_inv := ?_
-    right_inv := ?_
-    continuous_toFun := ?_
-    continuous_invFun := ?_ }
-  · intro z
-    apply Subtype.ext
-    exact (congrArg (fun v : ℂ => v-(U T:ℂ)) (hGF T _ (hshift z))).trans
-      (add_sub_cancel_right _ _)
-  · intro z
-    apply Subtype.ext
-    change F T ((G T z-(U T:ℂ))+(U T:ℂ)) = z
-    rw [sub_add_cancel,hFG T _ z.property]
-  · exact ((hF T).continuousOn.comp_continuous
-      (continuous_subtype_val.add continuous_const) hshift).subtype_mk _
-  · exact (((hG T).continuousOn.comp_continuous continuous_subtype_val
-      (fun z => z.property)).sub continuous_const).subtype_mk _
-
-@[simp] lemma IsCapacityTwoWitness.restartHomeomorph_apply
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : upperHalfPlane) :
-    ((hw.restartHomeomorph T z : survivingDomain γ T):ℂ) = F T ((z:ℂ)+(U T:ℂ)) := by
-  rcases hw with ⟨hc,h0,hi,hG,hF,hGM,hFM,hFG,hGF,hG0,hode,hn,ht⟩
-  rfl
-
-def IsCapacityTwoWitness.restartMassWeight
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (κ : ℝ) (hκ : 0 < κ) :
-    C(upperHalfPlane, ℝ≥0) where
-  toFun z := ⟨‖deriv (F T) ((z:ℂ)+(U T:ℂ))‖ ^ (1+κ/8),Real.rpow_nonneg (norm_nonneg _) _⟩
-  continuous_toFun := by
-    have hshift (z : upperHalfPlane) : (z:ℂ)+(U T:ℂ) ∈ upperHalfPlane := by
-      simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero]
-        using z.property
-    have hd := (hw.2.2.2.2.1 T).deriv isOpen_upperHalfPlane
-    have hc := hd.continuousOn.comp_continuous
-      (continuous_subtype_val.add continuous_const) hshift
-    exact ((Real.continuous_rpow_const (by linarith : 0 ≤ 1+κ/8)).comp hc.norm).subtype_mk _
-
-def IsCapacityTwoWitness.restartMass
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (κ : ℝ) (hκ : 0 < κ)
-    (μ : Measure upperHalfPlane) : Measure (survivingDomain γ T) :=
-  Multipoint.conformalMass (hw.restartHomeomorph T) (hw.restartMassWeight T κ hκ) μ
-
-lemma IsCapacityTwoWitness.restartMass_integral
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (κ : ℝ) (hκ : 0 < κ)
-    (μ : Measure upperHalfPlane) (f : C_c(survivingDomain γ T, ℝ)) :
-    (∫ z, f z ∂hw.restartMass T κ hκ μ) =
-      ∫ z : upperHalfPlane, ‖deriv (F T) ((z:ℂ)+(U T:ℂ))‖ ^ (1+κ/8) *
-        f (hw.restartHomeomorph T z) ∂μ :=
-  Multipoint.conformalMass_integral (hw.restartHomeomorph T)
-    (hw.restartMassWeight T κ hκ) μ f
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.ConformalMassTransport
+import OAI.Probability.SLE.Geometry.RestartConformalRadius
+
+namespace OAI
+
+/-! The actual centered inverse Loewner map and its mass weight.
+These are defined only on interior points; no boundary extension is used. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Classical
+open scoped NNReal ENNReal Topology CompactlySupported
+
+namespace SLEExactGauge
+
+def IsCapacityTwoWitness.restartHomeomorph
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) :
+    upperHalfPlane ≃ₜ survivingDomain γ T := by
+  rcases hw with ⟨hc,_,_,hG,hF,hGM,hFM,hFG,hGF,_,_,_,_⟩
+  have hshift (z : upperHalfPlane) : (z:ℂ)+(U T:ℂ) ∈ upperHalfPlane := by
+    simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero]
+      using z.property
+  have hback (z : survivingDomain γ T) : G T z-(U T:ℂ) ∈ upperHalfPlane := by
+    simpa only [upperHalfPlane,mem_ofPred_eq,Complex.sub_im,Complex.ofReal_im,sub_zero]
+      using hGM T z.property
+  refine {
+    toFun := fun z => ⟨F T ((z:ℂ)+(U T:ℂ)),hFM T (hshift z)⟩
+    invFun := fun z => ⟨G T z-(U T:ℂ),hback z⟩
+    left_inv := ?_
+    right_inv := ?_
+    continuous_toFun := ?_
+    continuous_invFun := ?_ }
+  · intro z
+    apply Subtype.ext
+    exact (congrArg (fun v : ℂ => v-(U T:ℂ)) (hGF T _ (hshift z))).trans
+      (add_sub_cancel_right _ _)
+  · intro z
+    apply Subtype.ext
+    change F T ((G T z-(U T:ℂ))+(U T:ℂ)) = z
+    rw [sub_add_cancel,hFG T _ z.property]
+  · exact ((hF T).continuousOn.comp_continuous
+      (continuous_subtype_val.add continuous_const) hshift).subtype_mk _
+  · exact (((hG T).continuousOn.comp_continuous continuous_subtype_val
+      (fun z => z.property)).sub continuous_const).subtype_mk _
+
+@[simp] lemma IsCapacityTwoWitness.restartHomeomorph_apply
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : upperHalfPlane) :
+    ((hw.restartHomeomorph T z : survivingDomain γ T):ℂ) = F T ((z:ℂ)+(U T:ℂ)) := by
+  rcases hw with ⟨hc,h0,hi,hG,hF,hGM,hFM,hFG,hGF,hG0,hode,hn,ht⟩
+  rfl
+
+def IsCapacityTwoWitness.restartMassWeight
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (κ : ℝ) (hκ : 0 < κ) :
+    C(upperHalfPlane, ℝ≥0) where
+  toFun z := ⟨‖deriv (F T) ((z:ℂ)+(U T:ℂ))‖ ^ (1+κ/8),Real.rpow_nonneg (norm_nonneg _) _⟩
+  continuous_toFun := by
+    have hshift (z : upperHalfPlane) : (z:ℂ)+(U T:ℂ) ∈ upperHalfPlane := by
+      simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero]
+        using z.property
+    have hd := (hw.2.2.2.2.1 T).deriv isOpen_upperHalfPlane
+    have hc := hd.continuousOn.comp_continuous
+      (continuous_subtype_val.add continuous_const) hshift
+    exact ((Real.continuous_rpow_const (by linarith : 0 ≤ 1+κ/8)).comp hc.norm).subtype_mk _
+
+def IsCapacityTwoWitness.restartMass
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (κ : ℝ) (hκ : 0 < κ)
+    (μ : Measure upperHalfPlane) : Measure (survivingDomain γ T) :=
+  Multipoint.conformalMass (hw.restartHomeomorph T) (hw.restartMassWeight T κ hκ) μ
+
+lemma IsCapacityTwoWitness.restartMass_integral
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (κ : ℝ) (hκ : 0 < κ)
+    (μ : Measure upperHalfPlane) (f : C_c(survivingDomain γ T, ℝ)) :
+    (∫ z, f z ∂hw.restartMass T κ hκ μ) =
+      ∫ z : upperHalfPlane, ‖deriv (F T) ((z:ℂ)+(U T:ℂ))‖ ^ (1+κ/8) *
+        f (hw.restartHomeomorph T z) ∂μ :=
+  Multipoint.conformalMass_integral (hw.restartHomeomorph T)
+    (hw.restartMassWeight T κ hκ) μ f
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RestartRandomMassTesting.lean b/lean/OAI/Probability/SLE/Mass/RestartRandomMassTesting.lean
index 9d38e1c332a2cd2add5de498ef22dad5a1e93c20..8b2c9a36ed1530ea2275d2cea2f116305f6bba27 100644
--- a/lean/OAI/Probability/SLE/Mass/RestartRandomMassTesting.lean
+++ b/lean/OAI/Probability/SLE/Mass/RestartRandomMassTesting.lean
@@ -1,55 +1,55 @@
-import OAI.Probability.SLE.Mass.RadonRandomTesting
-import OAI.Probability.SLE.Geometry.RandomRestartEvents
-
-namespace OAI
-
-/-! The fresh-driver law may be tested against a random spatial weight
-measurable in the stopped past. This is the conditional-Tonelli step for
-the restart candidate, before invoking uniqueness. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal
-
-namespace SLEExactGauge
-
-lemma fresh_mass_test_joint_measurable
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (M : X → Measure upperHalfPlane) (hM : Measurable M)
-    (hMf : ∀ x, IsFiniteMeasureOnCompacts (M x))
-    (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable f) :
-    Measurable (fun p : Ω × X => ∫⁻ z, f (p.1,z) ∂M p.2) := by
-  exact halfPlane_random_test_measurable (fun p : Ω × X => M p.2)
-    (hM.comp measurable_snd) (fun p => hMf p.2)
-    (fun p : (Ω × X) × upperHalfPlane => f (p.1.1,p.2))
-    (hf.comp (measurable_fst.fst.prodMk measurable_snd))
-
-theorem restart_random_mass_test
-    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
-    (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (Y : Ω → X) (hY : Measurable[mΩ] Y)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
-      @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν)
-    (M : X → Measure upperHalfPlane) (hM : Measurable M)
-    (hMf : ∀ x, IsFiniteMeasureOnCompacts (M x))
-    (f : Ω × upperHalfPlane → ℝ≥0∞)
-    (hf : Measurable[m.prod inferInstance] f) :
-    (∫⁻ ω, ∫⁻ z, f (ω,z) ∂M (Y ω) ∂P) =
-      ∫⁻ ω, ∫⁻ z, f (ω,z) ∂ν.bind M ∂P := by
-  let : MeasurableSpace Ω := mΩ
-  have hg : Measurable[m.prod mX]
-      (fun p : Ω × X => ∫⁻ z, f (p.1,z) ∂M p.2) :=
-    @fresh_mass_test_joint_measurable Ω X m mX M hM hMf f hf
-  rw [@restart_lintegral_random_test Ω X mΩ mX P inferInstance ν inferInstance
-    m hm Y hY hlaw _ hg]
-  apply lintegral_congr
-  intro ω
-  exact (Measure.lintegral_bind hM.aemeasurable
-    (hf.comp (measurable_const.prodMk measurable_id)).aemeasurable).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RadonRandomTesting
+import OAI.Probability.SLE.Geometry.RandomRestartEvents
+
+namespace OAI
+
+/-! The fresh-driver law may be tested against a random spatial weight
+measurable in the stopped past. This is the conditional-Tonelli step for
+the restart candidate, before invoking uniqueness. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal
+
+namespace SLEExactGauge
+
+lemma fresh_mass_test_joint_measurable
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (M : X → Measure upperHalfPlane) (hM : Measurable M)
+    (hMf : ∀ x, IsFiniteMeasureOnCompacts (M x))
+    (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable f) :
+    Measurable (fun p : Ω × X => ∫⁻ z, f (p.1,z) ∂M p.2) := by
+  exact halfPlane_random_test_measurable (fun p : Ω × X => M p.2)
+    (hM.comp measurable_snd) (fun p => hMf p.2)
+    (fun p : (Ω × X) × upperHalfPlane => f (p.1.1,p.2))
+    (hf.comp (measurable_fst.fst.prodMk measurable_snd))
+
+theorem restart_random_mass_test
+    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
+    (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (Y : Ω → X) (hY : Measurable[mΩ] Y)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
+      @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν)
+    (M : X → Measure upperHalfPlane) (hM : Measurable M)
+    (hMf : ∀ x, IsFiniteMeasureOnCompacts (M x))
+    (f : Ω × upperHalfPlane → ℝ≥0∞)
+    (hf : Measurable[m.prod inferInstance] f) :
+    (∫⁻ ω, ∫⁻ z, f (ω,z) ∂M (Y ω) ∂P) =
+      ∫⁻ ω, ∫⁻ z, f (ω,z) ∂ν.bind M ∂P := by
+  let : MeasurableSpace Ω := mΩ
+  have hg : Measurable[m.prod mX]
+      (fun p : Ω × X => ∫⁻ z, f (p.1,z) ∂M p.2) :=
+    @fresh_mass_test_joint_measurable Ω X m mX M hM hMf f hf
+  rw [@restart_lintegral_random_test Ω X mΩ mX P inferInstance ν inferInstance
+    m hm Y hY hlaw _ hg]
+  apply lintegral_congr
+  intro ω
+  exact (Measure.lintegral_bind hM.aemeasurable
+    (hf.comp (measurable_const.prodMk measurable_id)).aemeasurable).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/RestartStoppedMass.lean b/lean/OAI/Probability/SLE/Mass/RestartStoppedMass.lean
index 7cf2e5cbc8d51193a629e3db6218aca6f7694277..8d3ba999c90fb3792c4ea5c968e284bbf0204064 100644
--- a/lean/OAI/Probability/SLE/Mass/RestartStoppedMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/RestartStoppedMass.lean
@@ -1,93 +1,93 @@
-import OAI.Probability.SLE.Geometry.NormalizedRestartKernel
-import OAI.Probability.SLE.Gauge.ConformalDenseExperiment
-
-namespace OAI
-
-/-! A dense mass generated by the normalized fresh trace is present in the
-original stopped trace at the corresponding physical radius level. -/
-
-noncomputable section
-open Set Filter Metric MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open FrontSuccessMeasurability
-
-lemma segmentSausageCharge_clock_le
-    (κ : ℝ) (γ η : C(ℝ≥0, ℂ)) (T c σ : ℝ≥0)
-    (hpath : ∀ t, η t = γ (T + c * t)) (v : ℝ) (D : Set ℂ) :
-    segmentSausageCharge κ η 0 σ v D ≤ segmentSausageCharge κ γ 0 (T + c * σ) v D := by
-  apply mul_le_mul' le_rfl
-  apply measure_mono
-  rintro z ⟨hz, t, ht, hdist⟩
-  refine ⟨hz, T + c * t, ⟨bot_le,
-    add_le_add le_rfl (mul_le_mul_of_nonneg_left ht.2 (show 0 ≤ c from bot_le))⟩, ?_⟩
-  simpa only [hpath t] using hdist
-
-lemma stoppedWholeSausageCharge_clock_le
-    (κ : ℝ) (γ η : C(ℝ≥0, ℂ)) (T c σ : ℝ≥0) (ρ : WithTop ℝ≥0)
-    (hpath : ∀ t, η t = γ (T + c * t))
-    (hstop : ((T + c * σ : ℝ≥0) : WithTop ℝ≥0) ≤ ρ) (v : ℝ) (D : Set ℂ) :
-    stoppedWholeSausageCharge κ η (σ : WithTop ℝ≥0) v D ≤
-      stoppedWholeSausageCharge κ γ ρ v D := by
-  rw [stoppedWholeSausageCharge_finite]
-  exact (segmentSausageCharge_clock_le κ γ η T c σ hpath v D).trans
-    (segmentSausageCharge_le_stoppedWhole κ γ _ ρ hstop v D)
-
-lemma IsCapacityTwoWitness.normalized_restart_level_of_live_source
-    {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a)
-    (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) β G' F')
-    (w : ℂ) (u v : ℝ)
-    (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) =
-      (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u))
-    (t : ℝ≥0) (ht : w ∈ survivingDomain β t)
-    (hR : fullRadius G' t w = (2 * w.im) * Real.exp (-v)) :
-    fullRadius G (T + a ^ 2 * t) (F T ((a : ℂ) * w + (U T : ℂ))) =
-        (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-(u + v)) ∧
-      radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + v) =
-        (T + a ^ 2 * t : ℝ≥0) := by
-  have hbup := normalized_restart_witness T a ha hb
-  have haC : (a : ℂ) ≠ 0 := by exact_mod_cast ha.ne'
-  have htu : (a : ℂ) * w ∈ survivingDomain (sourceScaleTrace a β) (a ^ 2 * t) := by
-    rw [source_scale_domain_iff a _ ha β, source_scale_time_mul_sq a t ha,
-      mul_div_cancel_left₀ w haC]
-    exact ht
-  have hRu : fullRadius (sourceScaleChart a G') (a ^ 2 * t) ((a : ℂ) * w) =
-      (2 * ((a : ℂ) * w).im) * Real.exp (-v) := by
-    rw [hb.scaled_radius a ha t w ht, hR]
-    simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero]
-    ring
-  exact hw.restart_level_of_live_source hU T hbup ((a : ℂ) * w) u v hbase (a ^ 2 * t) htu hRu
-
-/-- At every finite source radius level, the stopped fresh physical mass
-is contained in the stopped original mass at the corresponding actual level. -/
-theorem IsCapacityTwoWitness.normalized_restart_stopped_mass_le
-    {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a)
-    (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) β G' F')
-    (w : ℂ) (u q : ℝ)
-    (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) =
-      (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u))
-    (η : C(ℝ≥0, ℂ)) (hpath : ∀ t, η t = γ (T + a ^ 2 * t))
-    (hfinite : radiusLevelTime β G' w q < ∞) (κ v : ℝ) (D : Set ℂ) :
-    stoppedWholeSausageCharge κ η (radiusLevelTime β G' w q) v D ≤
-      stoppedWholeSausageCharge κ ⟨γ, hw.1⟩
-        (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q)) v D := by
-  have hV : Continuous (normalizedRestartDriver U T a) := by
-    unfold normalizedRestartDriver
-    fun_prop
-  obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top β G' w q hfinite
-  have hs := radiusLevelTime_eq_of_live_level hV hb w q t ht hR
-  have hp := (hw.normalized_restart_level_of_live_source hU T a ha hb w u q hbase t ht hR).2
-  have hs' : (radiusLevelTime β G' w q : WithTop ℝ≥0) = (t : WithTop ℝ≥0) := by exact_mod_cast hs
-  have hp' : (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q) : WithTop ℝ≥0) =
-      ((T + a ^ 2 * t : ℝ≥0) : WithTop ℝ≥0) := by exact_mod_cast hp
-  rw [hs', hp']
-  exact stoppedWholeSausageCharge_clock_le κ ⟨γ, hw.1⟩ η T (a ^ 2) t _ hpath le_rfl v D
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.NormalizedRestartKernel
+import OAI.Probability.SLE.Gauge.ConformalDenseExperiment
+
+namespace OAI
+
+/-! A dense mass generated by the normalized fresh trace is present in the
+original stopped trace at the corresponding physical radius level. -/
+
+noncomputable section
+open Set Filter Metric MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open FrontSuccessMeasurability
+
+lemma segmentSausageCharge_clock_le
+    (κ : ℝ) (γ η : C(ℝ≥0, ℂ)) (T c σ : ℝ≥0)
+    (hpath : ∀ t, η t = γ (T + c * t)) (v : ℝ) (D : Set ℂ) :
+    segmentSausageCharge κ η 0 σ v D ≤ segmentSausageCharge κ γ 0 (T + c * σ) v D := by
+  apply mul_le_mul' le_rfl
+  apply measure_mono
+  rintro z ⟨hz, t, ht, hdist⟩
+  refine ⟨hz, T + c * t, ⟨bot_le,
+    add_le_add le_rfl (mul_le_mul_of_nonneg_left ht.2 (show 0 ≤ c from bot_le))⟩, ?_⟩
+  simpa only [hpath t] using hdist
+
+lemma stoppedWholeSausageCharge_clock_le
+    (κ : ℝ) (γ η : C(ℝ≥0, ℂ)) (T c σ : ℝ≥0) (ρ : WithTop ℝ≥0)
+    (hpath : ∀ t, η t = γ (T + c * t))
+    (hstop : ((T + c * σ : ℝ≥0) : WithTop ℝ≥0) ≤ ρ) (v : ℝ) (D : Set ℂ) :
+    stoppedWholeSausageCharge κ η (σ : WithTop ℝ≥0) v D ≤
+      stoppedWholeSausageCharge κ γ ρ v D := by
+  rw [stoppedWholeSausageCharge_finite]
+  exact (segmentSausageCharge_clock_le κ γ η T c σ hpath v D).trans
+    (segmentSausageCharge_le_stoppedWhole κ γ _ ρ hstop v D)
+
+lemma IsCapacityTwoWitness.normalized_restart_level_of_live_source
+    {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a)
+    (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) β G' F')
+    (w : ℂ) (u v : ℝ)
+    (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) =
+      (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u))
+    (t : ℝ≥0) (ht : w ∈ survivingDomain β t)
+    (hR : fullRadius G' t w = (2 * w.im) * Real.exp (-v)) :
+    fullRadius G (T + a ^ 2 * t) (F T ((a : ℂ) * w + (U T : ℂ))) =
+        (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-(u + v)) ∧
+      radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + v) =
+        (T + a ^ 2 * t : ℝ≥0) := by
+  have hbup := normalized_restart_witness T a ha hb
+  have haC : (a : ℂ) ≠ 0 := by exact_mod_cast ha.ne'
+  have htu : (a : ℂ) * w ∈ survivingDomain (sourceScaleTrace a β) (a ^ 2 * t) := by
+    rw [source_scale_domain_iff a _ ha β, source_scale_time_mul_sq a t ha,
+      mul_div_cancel_left₀ w haC]
+    exact ht
+  have hRu : fullRadius (sourceScaleChart a G') (a ^ 2 * t) ((a : ℂ) * w) =
+      (2 * ((a : ℂ) * w).im) * Real.exp (-v) := by
+    rw [hb.scaled_radius a ha t w ht, hR]
+    simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, zero_mul, add_zero]
+    ring
+  exact hw.restart_level_of_live_source hU T hbup ((a : ℂ) * w) u v hbase (a ^ 2 * t) htu hRu
+
+/-- At every finite source radius level, the stopped fresh physical mass
+is contained in the stopped original mass at the corresponding actual level. -/
+theorem IsCapacityTwoWitness.normalized_restart_stopped_mass_le
+    {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T a : ℝ≥0) (ha : 0 < a)
+    (hb : IsCapacityTwoWitness (normalizedRestartDriver U T a) β G' F')
+    (w : ℂ) (u q : ℝ)
+    (hbase : fullRadius G T (F T ((a : ℂ) * w + (U T : ℂ))) =
+      (2 * (F T ((a : ℂ) * w + (U T : ℂ))).im) * Real.exp (-u))
+    (η : C(ℝ≥0, ℂ)) (hpath : ∀ t, η t = γ (T + a ^ 2 * t))
+    (hfinite : radiusLevelTime β G' w q < ∞) (κ v : ℝ) (D : Set ℂ) :
+    stoppedWholeSausageCharge κ η (radiusLevelTime β G' w q) v D ≤
+      stoppedWholeSausageCharge κ ⟨γ, hw.1⟩
+        (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q)) v D := by
+  have hV : Continuous (normalizedRestartDriver U T a) := by
+    unfold normalizedRestartDriver
+    fun_prop
+  obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top β G' w q hfinite
+  have hs := radiusLevelTime_eq_of_live_level hV hb w q t ht hR
+  have hp := (hw.normalized_restart_level_of_live_source hU T a ha hb w u q hbase t ht hR).2
+  have hs' : (radiusLevelTime β G' w q : WithTop ℝ≥0) = (t : WithTop ℝ≥0) := by exact_mod_cast hs
+  have hp' : (radiusLevelTime γ G (F T ((a : ℂ) * w + (U T : ℂ))) (u + q) : WithTop ℝ≥0) =
+      ((T + a ^ 2 * t : ℝ≥0) : WithTop ℝ≥0) := by exact_mod_cast hp
+  rw [hs', hp']
+  exact stoppedWholeSausageCharge_clock_le κ ⟨γ, hw.1⟩ η T (a ^ 2) t _ hpath le_rfl v D
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/ScaledGreenMeasure.lean b/lean/OAI/Probability/SLE/Mass/ScaledGreenMeasure.lean
index 283aac110d0be191d59aa55e5e4bbf5db90369db..9000a9b9700a6e2784fb08fffaa0c44c562d3d9d 100644
--- a/lean/OAI/Probability/SLE/Mass/ScaledGreenMeasure.lean
+++ b/lean/OAI/Probability/SLE/Mass/ScaledGreenMeasure.lean
@@ -1,61 +1,61 @@
-import OAI.Probability.SLE.Mass.DilationMass
-import OAI.Probability.SLE.Loewner.RawScaledGreenKernel
-import OAI.Probability.SLE.Geometry.ScaledMarkedKernel
-
-namespace OAI
-
-/-! Killed Green measures scale by the same dimension-weighted spatial
-pushforward as the canonical mass, including points no longer surviving. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-local instance scaledGreenMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance scaledGreenBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-lemma halfPlaneGreenMeasure_scaled {Ω : Type*} (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (c : ℝ≥0) (hc : 0 < c) (t : ℝ≥0) (ω : Ω) :
-    halfPlaneGreenMeasure κ (scaledBrownianDriver c B)
-      (fun ω => sourceScaleTrace c (γ ω)) (fun ω => sourceScaleChart c (G ω)) (c^2*t) ω=
-      dilationMass κ c hc (halfPlaneGreenMeasure κ B γ G t ω) := by
-  unfold halfPlaneGreenMeasure
-  apply dilationMass_realDensity
-  intro z _hz
-  have h := source_scale_greenDensity κ c hc (fun t => Real.sqrt κ*B t ω)
-    (γ ω) (G ω) (F ω) (hw ω) (c^2*t) z
-  rw [source_scale_time_mul_sq c t hc] at h
-  simpa only [scaledBrownianDriver,mul_left_comm (Real.sqrt κ) (c:ℝ)] using h
-
-lemma canonicalGreenMass_scaled {Ω : Type*} (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (c : ℝ≥0) (hcp : 0 < c) (t : ℝ≥0) (ω : Ω) :
-    canonicalGreenMass κ (c^2*t)
-      (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) ω)=
-      dilationMass κ c hcp (halfPlaneGreenMeasure κ B γ G t ω) := by
-  have hwc : IsCapacityTwoWitness (fun t => Real.sqrt κ*scaledBrownianDriver c B t ω)
-      (sourceScaleTrace c (γ ω)) (sourceScaleChart c (G ω)) (sourceScaleChart c (F ω)) := by
-    have h := source_scale_witness c hcp (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) (F ω) (hw ω)
-    simpa only [scaledBrownianDriver,mul_left_comm (Real.sqrt κ) (c:ℝ)] using h
-  have he := canonicalGreenMass_eq κ
-    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) ω)
-    hwc (c^2*t)
-  change canonicalGreenMass κ (c^2*t)
-    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) ω)=
-      halfPlaneGreenMeasure κ (scaledBrownianDriver c B)
-        (fun ω => sourceScaleTrace c (γ ω)) (fun ω => sourceScaleChart c (G ω)) (c^2*t) ω at he
-  rw [he]
-  exact halfPlaneGreenMeasure_scaled κ B γ G F hw c hcp t ω
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.DilationMass
+import OAI.Probability.SLE.Loewner.RawScaledGreenKernel
+import OAI.Probability.SLE.Geometry.ScaledMarkedKernel
+
+namespace OAI
+
+/-! Killed Green measures scale by the same dimension-weighted spatial
+pushforward as the canonical mass, including points no longer surviving. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+local instance scaledGreenMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance scaledGreenBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+lemma halfPlaneGreenMeasure_scaled {Ω : Type*} (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (c : ℝ≥0) (hc : 0 < c) (t : ℝ≥0) (ω : Ω) :
+    halfPlaneGreenMeasure κ (scaledBrownianDriver c B)
+      (fun ω => sourceScaleTrace c (γ ω)) (fun ω => sourceScaleChart c (G ω)) (c^2*t) ω=
+      dilationMass κ c hc (halfPlaneGreenMeasure κ B γ G t ω) := by
+  unfold halfPlaneGreenMeasure
+  apply dilationMass_realDensity
+  intro z _hz
+  have h := source_scale_greenDensity κ c hc (fun t => Real.sqrt κ*B t ω)
+    (γ ω) (G ω) (F ω) (hw ω) (c^2*t) z
+  rw [source_scale_time_mul_sq c t hc] at h
+  simpa only [scaledBrownianDriver,mul_left_comm (Real.sqrt κ) (c:ℝ)] using h
+
+lemma canonicalGreenMass_scaled {Ω : Type*} (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (c : ℝ≥0) (hcp : 0 < c) (t : ℝ≥0) (ω : Ω) :
+    canonicalGreenMass κ (c^2*t)
+      (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) ω)=
+      dilationMass κ c hcp (halfPlaneGreenMeasure κ B γ G t ω) := by
+  have hwc : IsCapacityTwoWitness (fun t => Real.sqrt κ*scaledBrownianDriver c B t ω)
+      (sourceScaleTrace c (γ ω)) (sourceScaleChart c (G ω)) (sourceScaleChart c (F ω)) := by
+    have h := source_scale_witness c hcp (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) (F ω) (hw ω)
+    simpa only [scaledBrownianDriver,mul_left_comm (Real.sqrt κ) (c:ℝ)] using h
+  have he := canonicalGreenMass_eq κ
+    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) ω)
+    hwc (c^2*t)
+  change canonicalGreenMass κ (c^2*t)
+    (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) ω)=
+      halfPlaneGreenMeasure κ (scaledBrownianDriver c B)
+        (fun ω => sourceScaleTrace c (γ ω)) (fun ω => sourceScaleChart c (G ω)) (c^2*t) ω at he
+  rw [he]
+  exact halfPlaneGreenMeasure_scaled κ B γ G F hw c hcp t ω
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/SignedLevelTests.lean b/lean/OAI/Probability/SLE/Mass/SignedLevelTests.lean
index 9b1a110823853d6fe4d4aceb0a0ee254330a23d5..bd5c1051c8a8149bf1e59ae601434276bf198087 100644
--- a/lean/OAI/Probability/SLE/Mass/SignedLevelTests.lean
+++ b/lean/OAI/Probability/SLE/Mass/SignedLevelTests.lean
@@ -1,142 +1,142 @@
-import OAI.Probability.SLE.Mass.LevelApproximationL2
-
-namespace OAI
-
-/-! Real compact tests of the actual level measures. Both signs are
-controlled by the same compact moment estimate. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Multipoint
-
-def levelApproximationRealTest {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ) (ω : Ω) : ℝ :=
-  ∫ z, f z ∂levelApproximation κ B γ G u t ω
-
-lemma integrable_of_bounded_support (μ : Measure ℂ) (K : Set ℂ)
-    (hK : MeasurableSet K) (hμK : μ K < ⊤) (f : ℂ → ℝ)
-    (hfm : Measurable f) (C : ℝ) (hf : ∀ z ∈ K, ‖f z‖ ≤ C)
-    (hz : ∀ z ∉ K, f z = 0) : Integrable f μ := by
-  have hi : Integrable (K.indicator (fun _ : ℂ => C)) μ :=
-    (integrableOn_const hμK.ne).integrable_indicator hK
-  apply hi.mono' hfm.aestronglyMeasurable
-  filter_upwards [] with z
-  by_cases h : z ∈ K
-  · simpa only [indicator_of_mem h] using hf z h
-  · simp only [hz z h,norm_zero,indicator_of_notMem h,le_refl]
-
-lemma levelApproximationRealTest_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ u : ℝ) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω))
-    (t : ℝ≥0∞) (f : ℂ → ℝ) (hfm : Measurable f) :
-    Measurable (levelApproximationRealTest κ B γ G u t f) := by
-  have hd := levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t
-  have he (ω : Ω) : levelApproximationRealTest κ B γ G u t f ω =
-      ∫ z, (levelApproximationDensity κ B γ G u t ω z).toReal * f z := by
-    apply integral_withDensity_eq_integral_toReal_smul hd.of_uncurry_left
-    filter_upwards [] with z
-    unfold levelApproximationDensity
-    split_ifs <;> simp
-  simp_rw [show levelApproximationRealTest κ B γ G u t f =
-    (fun ω => ∫ z, (levelApproximationDensity κ B γ G u t ω z).toReal * f z)
-    from funext he]
-  exact (hd.ennreal_toReal.mul (hfm.comp measurable_snd)).stronglyMeasurable.integral_prod_right'.measurable
-
-lemma levelApproximationRealTest_eq_parts
-    {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ) (ω : Ω)
-    (K : Set ℂ) (hK : MeasurableSet K) (hM : levelMass κ B γ G u K ω < ⊤)
-    (hfm : Measurable f) (C : ℝ) (hf : ∀ z ∈ K, ‖f z‖ ≤ C)
-    (hz : ∀ z ∉ K, f z = 0) :
-    levelApproximationRealTest κ B γ G u t f ω =
-      (levelApproximationTest κ B γ G u t (fun z => ENNReal.ofReal (f z)) ω).toReal -
-      (levelApproximationTest κ B γ G u t (fun z => ENNReal.ofReal (-f z)) ω).toReal := by
-  exact integral_eq_lintegral_pos_part_sub_lintegral_neg_part
-    (integrable_of_bounded_support _ K hK
-      ((levelApproximation_apply_le κ B γ G u t ω hK).trans_lt hM) f hfm C hf hz)
-
-theorem levelMass_ae_finite
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
-    (u : ℝ) (hu : 0 ≤ u) : ∀ᵐ ω ∂P, levelMass κ B γ G u K ω < ⊤ := by
-  obtain ⟨M,hM,hbound⟩ := uniform_levelMass_compact_moments hB hm hBm hBc hB0 hκ hκ8
-    γ G F hw hgrowth K hK hKH 1
-  apply ae_lt_top (levelMass_measurable hm hBc κ u hu γ G F hw K)
-  exact (lt_of_le_of_lt (by simpa only [pow_one] using hbound u hu) hM).ne
-
-theorem uniform_signed_levelTest_L2
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
-    (f : ℂ → ℝ) (hfm : Measurable f) (C : ℝ) (hf : ∀ z ∈ K, ‖f z‖ ≤ C)
-    (hz : ∀ z ∉ K, f z = 0) :
-    (∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, MemLp (levelApproximationRealTest κ B γ G u t f) 2 P) ∧
-      ∃ D : ℝ, ∀ u : ℝ, ∀ _hu : 0 ≤ u, ∀ t : ℝ≥0∞,
-        ∀ hp : MemLp (levelApproximationRealTest κ B γ G u t f) 2 P,
-          ‖hp.toLp (levelApproximationRealTest κ B γ G u t f)‖ ≤ D := by
-  let fp : ℂ → ℝ≥0∞ := fun z => ENNReal.ofReal (f z)
-  let fn : ℂ → ℝ≥0∞ := fun z => ENNReal.ofReal (-f z)
-  have hfp (z : ℂ) (hzK : z ∈ K) : fp z ≤ ENNReal.ofReal C :=
-    ENNReal.ofReal_le_ofReal ((le_abs_self _).trans (hf z hzK))
-  have hfn (z : ℂ) (hzK : z ∈ K) : fn z ≤ ENNReal.ofReal C :=
-    ENNReal.ofReal_le_ofReal ((neg_le_abs _).trans (hf z hzK))
-  obtain ⟨Mp,hMp,hpb⟩ := uniform_levelApproximationTest_moments hB hm hBm hBc hB0 hκ hκ8
-    γ G F hw hgrowth K hK hKH fp (ENNReal.ofReal C) ENNReal.ofReal_lt_top hfp
-    (fun z hzK => by simp only [fp,hz z hzK,ENNReal.ofReal_zero]) 2
-  obtain ⟨Mn,hMn,hnb⟩ := uniform_levelApproximationTest_moments hB hm hBm hBc hB0 hκ hκ8
-    γ G F hw hgrowth K hK hKH fn (ENNReal.ofReal C) ENNReal.ofReal_lt_top hfn
-    (fun z hzK => by simp only [fn,hz z hzK,neg_zero,ENNReal.ofReal_zero]) 2
-  have hpm (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
-    levelApproximationTest_measurable hm hBc κ u hu γ G F hw t fp hfm.ennreal_ofReal
-  have hnm (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
-    levelApproximationTest_measurable hm hBc κ u hu γ G F hw t fn hfm.neg.ennreal_ofReal
-  have hpLp (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
-    memLp_toReal_two P _ (hpm u hu t) Mp hMp (hpb u hu t)
-  have hnLp (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
-    memLp_toReal_two P _ (hnm u hu t) Mn hMn (hnb u hu t)
-  have he (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) : levelApproximationRealTest κ B γ G u t f =ᵐ[P]
-      fun ω => (levelApproximationTest κ B γ G u t fp ω).toReal -
-        (levelApproximationTest κ B γ G u t fn ω).toReal := by
-    filter_upwards [levelMass_ae_finite hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth K hK hKH u hu]
-      with ω hω
-    exact levelApproximationRealTest_eq_parts κ B γ G u t f ω K hK.measurableSet hω hfm C hf hz
-  refine ⟨fun u hu t => MemLp.ae_eq (he u hu t).symm ((hpLp u hu t).sub (hnLp u hu t)),
-    ⟨(Mp ^ (1/2 : ℝ)).toReal + (Mn ^ (1/2 : ℝ)).toReal, ?_⟩⟩
-  intro u hu t hp
-  have hclass : hp.toLp (levelApproximationRealTest κ B γ G u t f) =
-      (hpLp u hu t).toLp _ - (hnLp u hu t).toLp _ := by
-    apply Lp.ext
-    filter_upwards [hp.coeFn_toLp, (hpLp u hu t).coeFn_toLp,
-      (hnLp u hu t).coeFn_toLp,Lp.coeFn_sub ((hpLp u hu t).toLp _) ((hnLp u hu t).toLp _),
-      he u hu t] with ω h₁ h₂ h₃ h₄ h₅
-    rw [h₁,h₄,Pi.sub_apply,h₂,h₃,h₅]
-  rw [hclass]
-  exact (norm_sub_le _ _).trans (add_le_add
-    (norm_toLp_toReal_two_le P _ (hpm u hu t) Mp hMp (hpb u hu t) (hpLp u hu t))
-    (norm_toLp_toReal_two_le P _ (hnm u hu t) Mn hMn (hnb u hu t) (hnLp u hu t)))
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.LevelApproximationL2
+
+namespace OAI
+
+/-! Real compact tests of the actual level measures. Both signs are
+controlled by the same compact moment estimate. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Multipoint
+
+def levelApproximationRealTest {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ) (ω : Ω) : ℝ :=
+  ∫ z, f z ∂levelApproximation κ B γ G u t ω
+
+lemma integrable_of_bounded_support (μ : Measure ℂ) (K : Set ℂ)
+    (hK : MeasurableSet K) (hμK : μ K < ⊤) (f : ℂ → ℝ)
+    (hfm : Measurable f) (C : ℝ) (hf : ∀ z ∈ K, ‖f z‖ ≤ C)
+    (hz : ∀ z ∉ K, f z = 0) : Integrable f μ := by
+  have hi : Integrable (K.indicator (fun _ : ℂ => C)) μ :=
+    (integrableOn_const hμK.ne).integrable_indicator hK
+  apply hi.mono' hfm.aestronglyMeasurable
+  filter_upwards [] with z
+  by_cases h : z ∈ K
+  · simpa only [indicator_of_mem h] using hf z h
+  · simp only [hz z h,norm_zero,indicator_of_notMem h,le_refl]
+
+lemma levelApproximationRealTest_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ u : ℝ) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω))
+    (t : ℝ≥0∞) (f : ℂ → ℝ) (hfm : Measurable f) :
+    Measurable (levelApproximationRealTest κ B γ G u t f) := by
+  have hd := levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t
+  have he (ω : Ω) : levelApproximationRealTest κ B γ G u t f ω =
+      ∫ z, (levelApproximationDensity κ B γ G u t ω z).toReal * f z := by
+    apply integral_withDensity_eq_integral_toReal_smul hd.of_uncurry_left
+    filter_upwards [] with z
+    unfold levelApproximationDensity
+    split_ifs <;> simp
+  simp_rw [show levelApproximationRealTest κ B γ G u t f =
+    (fun ω => ∫ z, (levelApproximationDensity κ B γ G u t ω z).toReal * f z)
+    from funext he]
+  exact (hd.ennreal_toReal.mul (hfm.comp measurable_snd)).stronglyMeasurable.integral_prod_right'.measurable
+
+lemma levelApproximationRealTest_eq_parts
+    {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ) (ω : Ω)
+    (K : Set ℂ) (hK : MeasurableSet K) (hM : levelMass κ B γ G u K ω < ⊤)
+    (hfm : Measurable f) (C : ℝ) (hf : ∀ z ∈ K, ‖f z‖ ≤ C)
+    (hz : ∀ z ∉ K, f z = 0) :
+    levelApproximationRealTest κ B γ G u t f ω =
+      (levelApproximationTest κ B γ G u t (fun z => ENNReal.ofReal (f z)) ω).toReal -
+      (levelApproximationTest κ B γ G u t (fun z => ENNReal.ofReal (-f z)) ω).toReal := by
+  exact integral_eq_lintegral_pos_part_sub_lintegral_neg_part
+    (integrable_of_bounded_support _ K hK
+      ((levelApproximation_apply_le κ B γ G u t ω hK).trans_lt hM) f hfm C hf hz)
+
+theorem levelMass_ae_finite
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
+    (u : ℝ) (hu : 0 ≤ u) : ∀ᵐ ω ∂P, levelMass κ B γ G u K ω < ⊤ := by
+  obtain ⟨M,hM,hbound⟩ := uniform_levelMass_compact_moments hB hm hBm hBc hB0 hκ hκ8
+    γ G F hw hgrowth K hK hKH 1
+  apply ae_lt_top (levelMass_measurable hm hBc κ u hu γ G F hw K)
+  exact (lt_of_le_of_lt (by simpa only [pow_one] using hbound u hu) hM).ne
+
+theorem uniform_signed_levelTest_L2
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
+    (f : ℂ → ℝ) (hfm : Measurable f) (C : ℝ) (hf : ∀ z ∈ K, ‖f z‖ ≤ C)
+    (hz : ∀ z ∉ K, f z = 0) :
+    (∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, MemLp (levelApproximationRealTest κ B γ G u t f) 2 P) ∧
+      ∃ D : ℝ, ∀ u : ℝ, ∀ _hu : 0 ≤ u, ∀ t : ℝ≥0∞,
+        ∀ hp : MemLp (levelApproximationRealTest κ B γ G u t f) 2 P,
+          ‖hp.toLp (levelApproximationRealTest κ B γ G u t f)‖ ≤ D := by
+  let fp : ℂ → ℝ≥0∞ := fun z => ENNReal.ofReal (f z)
+  let fn : ℂ → ℝ≥0∞ := fun z => ENNReal.ofReal (-f z)
+  have hfp (z : ℂ) (hzK : z ∈ K) : fp z ≤ ENNReal.ofReal C :=
+    ENNReal.ofReal_le_ofReal ((le_abs_self _).trans (hf z hzK))
+  have hfn (z : ℂ) (hzK : z ∈ K) : fn z ≤ ENNReal.ofReal C :=
+    ENNReal.ofReal_le_ofReal ((neg_le_abs _).trans (hf z hzK))
+  obtain ⟨Mp,hMp,hpb⟩ := uniform_levelApproximationTest_moments hB hm hBm hBc hB0 hκ hκ8
+    γ G F hw hgrowth K hK hKH fp (ENNReal.ofReal C) ENNReal.ofReal_lt_top hfp
+    (fun z hzK => by simp only [fp,hz z hzK,ENNReal.ofReal_zero]) 2
+  obtain ⟨Mn,hMn,hnb⟩ := uniform_levelApproximationTest_moments hB hm hBm hBc hB0 hκ hκ8
+    γ G F hw hgrowth K hK hKH fn (ENNReal.ofReal C) ENNReal.ofReal_lt_top hfn
+    (fun z hzK => by simp only [fn,hz z hzK,neg_zero,ENNReal.ofReal_zero]) 2
+  have hpm (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
+    levelApproximationTest_measurable hm hBc κ u hu γ G F hw t fp hfm.ennreal_ofReal
+  have hnm (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
+    levelApproximationTest_measurable hm hBc κ u hu γ G F hw t fn hfm.neg.ennreal_ofReal
+  have hpLp (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
+    memLp_toReal_two P _ (hpm u hu t) Mp hMp (hpb u hu t)
+  have hnLp (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :=
+    memLp_toReal_two P _ (hnm u hu t) Mn hMn (hnb u hu t)
+  have he (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) : levelApproximationRealTest κ B γ G u t f =ᵐ[P]
+      fun ω => (levelApproximationTest κ B γ G u t fp ω).toReal -
+        (levelApproximationTest κ B γ G u t fn ω).toReal := by
+    filter_upwards [levelMass_ae_finite hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth K hK hKH u hu]
+      with ω hω
+    exact levelApproximationRealTest_eq_parts κ B γ G u t f ω K hK.measurableSet hω hfm C hf hz
+  refine ⟨fun u hu t => MemLp.ae_eq (he u hu t).symm ((hpLp u hu t).sub (hnLp u hu t)),
+    ⟨(Mp ^ (1/2 : ℝ)).toReal + (Mn ^ (1/2 : ℝ)).toReal, ?_⟩⟩
+  intro u hu t hp
+  have hclass : hp.toLp (levelApproximationRealTest κ B γ G u t f) =
+      (hpLp u hu t).toLp _ - (hnLp u hu t).toLp _ := by
+    apply Lp.ext
+    filter_upwards [hp.coeFn_toLp, (hpLp u hu t).coeFn_toLp,
+      (hnLp u hu t).coeFn_toLp,Lp.coeFn_sub ((hpLp u hu t).toLp _) ((hnLp u hu t).toLp _),
+      he u hu t] with ω h₁ h₂ h₃ h₄ h₅
+    rw [h₁,h₄,Pi.sub_apply,h₂,h₃,h₅]
+  rw [hclass]
+  exact (norm_sub_le _ _).trans (add_le_add
+    (norm_toLp_toReal_two_le P _ (hpm u hu t) Mp hMp (hpb u hu t) (hpLp u hu t))
+    (norm_toLp_toReal_two_le P _ (hnm u hu t) Mn hMn (hnb u hu t) (hnLp u hu t)))
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/SignedMeasureConvergence.lean b/lean/OAI/Probability/SLE/Mass/SignedMeasureConvergence.lean
index c99ae9bfae7abd71879534d865e75ac94cc4d8b7..1149c99c9f552fefa78de0bef0d6ca1ace8273c7 100644
--- a/lean/OAI/Probability/SLE/Mass/SignedMeasureConvergence.lean
+++ b/lean/OAI/Probability/SLE/Mass/SignedMeasureConvergence.lean
@@ -1,102 +1,102 @@
-import OAI.Probability.SLE.Mass.RadonKernelIntegral
-import Mathlib.MeasureTheory.Integral.Bochner.Set
-import Mathlib.MeasureTheory.Function.LpSpace.Indicator
-import Mathlib.Analysis.Normed.Group.FunctionSeries
-
-namespace OAI
-
-/-! Small analytic passages used for the conditional mass kernel. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal Topology
-
-namespace SLEExactGauge
-
-theorem integral_tendsto_of_lintegral_pos_neg
-    {X : Type*} [MeasurableSpace X] (μ : ℕ → Measure X) (ν : Measure X)
-    (f : X → ℝ) (hμ : ∀ n, Integrable f (μ n)) (hν : Integrable f ν)
-    (hpos : Tendsto (fun n => ∫⁻ x, ENNReal.ofReal (f x) ∂μ n) atTop
-      (𝓝 (∫⁻ x, ENNReal.ofReal (f x) ∂ν)))
-    (hneg : Tendsto (fun n => ∫⁻ x, ENNReal.ofReal (-f x) ∂μ n) atTop
-      (𝓝 (∫⁻ x, ENNReal.ofReal (-f x) ∂ν))) :
-    Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) := by
-  have hp := (ENNReal.tendsto_toReal (lintegral_ofReal_lt_top_of_integrable hν).ne).comp hpos
-  have hn := (ENNReal.tendsto_toReal (lintegral_ofReal_lt_top_of_integrable hν.neg).ne).comp hneg
-  have hh := hp.sub hn
-  dsimp only [Function.comp_def,Pi.neg_apply] at hh
-  rw [← integral_eq_lintegral_pos_part_sub_lintegral_neg_part hν] at hh
-  exact hh.congr' (Eventually.of_forall (fun n =>
-    (integral_eq_lintegral_pos_part_sub_lintegral_neg_part (hμ n)).symm))
-
-theorem real_convex_tail_tendsto (f : ℕ → ℝ) {a : ℝ}
-    (hf : Tendsto f atTop (𝓝 a)) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ)
-    (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) :
-    Tendsto (fun n => ∑ i ∈ s n, w n i * f (n+i)) atTop (𝓝 a) := by
-  apply Metric.tendsto_atTop.mpr
-  intro ε hε
-  obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp hf (ε/2) (by positivity)
-  refine ⟨N,fun n hn => ?_⟩
-  have he : (∑ i ∈ s n, w n i * f (n+i)) - a =
-      ∑ i ∈ s n, w n i * (f (n+i)-a) := by
-    simp only [mul_sub,Finset.sum_sub_distrib,← Finset.sum_mul,hw₁ n,one_mul]
-  calc
-    dist (∑ i ∈ s n, w n i * f (n+i)) a =
-        ‖∑ i ∈ s n, w n i * (f (n+i)-a)‖ := by rw [dist_eq_norm,he]
-    _ ≤ ∑ i ∈ s n, ‖w n i * (f (n+i)-a)‖ := norm_sum_le _ _
-    _ ≤ ∑ i ∈ s n, w n i * (ε/2) := by
-      apply Finset.sum_le_sum
-      intro i hi
-      rw [norm_mul,Real.norm_of_nonneg (hw₀ n i hi)]
-      exact mul_le_mul_of_nonneg_left (le_of_lt (hN (n+i) (by omega))) (hw₀ n i hi)
-    _ = ε/2 := by rw [← Finset.sum_mul,hw₁ n,one_mul]
-    _ < ε := by linarith
-
-
-lemma integral_indicator_one_mul
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (E : Set Ω)
-    (hE : MeasurableSet E) (f : Ω → ℝ) :
-    (∫ ω, E.indicator (fun _ => (1 : ℝ)) ω * f ω ∂P) = ∫ ω in E, f ω ∂P := by
-  rw [← integral_indicator hE]
-  congr 1
-  funext ω
-  by_cases hω : ω ∈ E
-  · simp only [indicator_of_mem hω,one_mul]
-  · simp only [indicator_of_notMem hω,zero_mul]
-
-theorem setIntegral_tendsto_of_L2_pairings
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
-    (F : ℕ → Ω → ℝ) (f : Ω → ℝ)
-    (hp : ∀ a : Ω → ℝ, MemLp a 2 P →
-      Tendsto (fun n => ∫ ω, a ω * F n ω ∂P) atTop (𝓝 (∫ ω, a ω * f ω ∂P)))
-    (E : Set Ω) (hE : MeasurableSet E) :
-    Tendsto (fun n => ∫ ω in E, F n ω ∂P) atTop (𝓝 (∫ ω in E, f ω ∂P)) := by
-  have ha : MemLp (E.indicator (fun _ => (1 : ℝ))) 2 P := (memLp_const 1).indicator hE
-  have hh := hp (E.indicator (fun _ => 1)) ha
-  simpa only [integral_indicator_one_mul P E hE] using hh
-
-
-theorem mean_test_tendsto_of_L2_pairings
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (P : Measure Ω) [IsFiniteMeasure P] (μ : ℕ → Ω → Measure X) (ν : Ω → Measure X)
-    (hμ : ∀ n, Measurable (μ n)) (hν : Measurable ν) (f : X → ℝ) (hf : Measurable f)
-    (hμi : ∀ n ω, Integrable f (μ n ω)) (hνi : ∀ ω, Integrable f (ν ω))
-    (E : Set Ω) (hE : MeasurableSet E)
-    (hμP : ∀ n, Integrable f ((P.restrict E).bind (μ n)))
-    (hνP : Integrable f ((P.restrict E).bind ν))
-    (hp : ∀ a : Ω → ℝ, MemLp a 2 P →
-      Tendsto (fun n => ∫ ω, a ω * (∫ x, f x ∂μ n ω) ∂P) atTop
-        (𝓝 (∫ ω, a ω * (∫ x, f x ∂ν ω) ∂P))) :
-    Tendsto (fun n => ∫ x, f x ∂(P.restrict E).bind (μ n)) atTop
-      (𝓝 (∫ x, f x ∂(P.restrict E).bind ν)) := by
-  have he (n : ℕ) := integral_bind_of_integrable_sections (P.restrict E) (μ n) (hμ n) f hf
-    (hμi n) (hμP n)
-  have hev := integral_bind_of_integrable_sections (P.restrict E) ν hν f hf hνi hνP
-  simpa only [he,hev] using setIntegral_tendsto_of_L2_pairings P
-    (fun n ω => ∫ x, f x ∂μ n ω) (fun ω => ∫ x, f x ∂ν ω) hp E hE
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RadonKernelIntegral
+import Mathlib.MeasureTheory.Integral.Bochner.Set
+import Mathlib.MeasureTheory.Function.LpSpace.Indicator
+import Mathlib.Analysis.Normed.Group.FunctionSeries
+
+namespace OAI
+
+/-! Small analytic passages used for the conditional mass kernel. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal Topology
+
+namespace SLEExactGauge
+
+theorem integral_tendsto_of_lintegral_pos_neg
+    {X : Type*} [MeasurableSpace X] (μ : ℕ → Measure X) (ν : Measure X)
+    (f : X → ℝ) (hμ : ∀ n, Integrable f (μ n)) (hν : Integrable f ν)
+    (hpos : Tendsto (fun n => ∫⁻ x, ENNReal.ofReal (f x) ∂μ n) atTop
+      (𝓝 (∫⁻ x, ENNReal.ofReal (f x) ∂ν)))
+    (hneg : Tendsto (fun n => ∫⁻ x, ENNReal.ofReal (-f x) ∂μ n) atTop
+      (𝓝 (∫⁻ x, ENNReal.ofReal (-f x) ∂ν))) :
+    Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) := by
+  have hp := (ENNReal.tendsto_toReal (lintegral_ofReal_lt_top_of_integrable hν).ne).comp hpos
+  have hn := (ENNReal.tendsto_toReal (lintegral_ofReal_lt_top_of_integrable hν.neg).ne).comp hneg
+  have hh := hp.sub hn
+  dsimp only [Function.comp_def,Pi.neg_apply] at hh
+  rw [← integral_eq_lintegral_pos_part_sub_lintegral_neg_part hν] at hh
+  exact hh.congr' (Eventually.of_forall (fun n =>
+    (integral_eq_lintegral_pos_part_sub_lintegral_neg_part (hμ n)).symm))
+
+theorem real_convex_tail_tendsto (f : ℕ → ℝ) {a : ℝ}
+    (hf : Tendsto f atTop (𝓝 a)) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ)
+    (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) :
+    Tendsto (fun n => ∑ i ∈ s n, w n i * f (n+i)) atTop (𝓝 a) := by
+  apply Metric.tendsto_atTop.mpr
+  intro ε hε
+  obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp hf (ε/2) (by positivity)
+  refine ⟨N,fun n hn => ?_⟩
+  have he : (∑ i ∈ s n, w n i * f (n+i)) - a =
+      ∑ i ∈ s n, w n i * (f (n+i)-a) := by
+    simp only [mul_sub,Finset.sum_sub_distrib,← Finset.sum_mul,hw₁ n,one_mul]
+  calc
+    dist (∑ i ∈ s n, w n i * f (n+i)) a =
+        ‖∑ i ∈ s n, w n i * (f (n+i)-a)‖ := by rw [dist_eq_norm,he]
+    _ ≤ ∑ i ∈ s n, ‖w n i * (f (n+i)-a)‖ := norm_sum_le _ _
+    _ ≤ ∑ i ∈ s n, w n i * (ε/2) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      rw [norm_mul,Real.norm_of_nonneg (hw₀ n i hi)]
+      exact mul_le_mul_of_nonneg_left (le_of_lt (hN (n+i) (by omega))) (hw₀ n i hi)
+    _ = ε/2 := by rw [← Finset.sum_mul,hw₁ n,one_mul]
+    _ < ε := by linarith
+
+
+lemma integral_indicator_one_mul
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (E : Set Ω)
+    (hE : MeasurableSet E) (f : Ω → ℝ) :
+    (∫ ω, E.indicator (fun _ => (1 : ℝ)) ω * f ω ∂P) = ∫ ω in E, f ω ∂P := by
+  rw [← integral_indicator hE]
+  congr 1
+  funext ω
+  by_cases hω : ω ∈ E
+  · simp only [indicator_of_mem hω,one_mul]
+  · simp only [indicator_of_notMem hω,zero_mul]
+
+theorem setIntegral_tendsto_of_L2_pairings
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
+    (F : ℕ → Ω → ℝ) (f : Ω → ℝ)
+    (hp : ∀ a : Ω → ℝ, MemLp a 2 P →
+      Tendsto (fun n => ∫ ω, a ω * F n ω ∂P) atTop (𝓝 (∫ ω, a ω * f ω ∂P)))
+    (E : Set Ω) (hE : MeasurableSet E) :
+    Tendsto (fun n => ∫ ω in E, F n ω ∂P) atTop (𝓝 (∫ ω in E, f ω ∂P)) := by
+  have ha : MemLp (E.indicator (fun _ => (1 : ℝ))) 2 P := (memLp_const 1).indicator hE
+  have hh := hp (E.indicator (fun _ => 1)) ha
+  simpa only [integral_indicator_one_mul P E hE] using hh
+
+
+theorem mean_test_tendsto_of_L2_pairings
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (P : Measure Ω) [IsFiniteMeasure P] (μ : ℕ → Ω → Measure X) (ν : Ω → Measure X)
+    (hμ : ∀ n, Measurable (μ n)) (hν : Measurable ν) (f : X → ℝ) (hf : Measurable f)
+    (hμi : ∀ n ω, Integrable f (μ n ω)) (hνi : ∀ ω, Integrable f (ν ω))
+    (E : Set Ω) (hE : MeasurableSet E)
+    (hμP : ∀ n, Integrable f ((P.restrict E).bind (μ n)))
+    (hνP : Integrable f ((P.restrict E).bind ν))
+    (hp : ∀ a : Ω → ℝ, MemLp a 2 P →
+      Tendsto (fun n => ∫ ω, a ω * (∫ x, f x ∂μ n ω) ∂P) atTop
+        (𝓝 (∫ ω, a ω * (∫ x, f x ∂ν ω) ∂P))) :
+    Tendsto (fun n => ∫ x, f x ∂(P.restrict E).bind (μ n)) atTop
+      (𝓝 (∫ x, f x ∂(P.restrict E).bind ν)) := by
+  have he (n : ℕ) := integral_bind_of_integrable_sections (P.restrict E) (μ n) (hμ n) f hf
+    (hμi n) (hμP n)
+  have hev := integral_bind_of_integrable_sections (P.restrict E) ν hν f hf hνi hνP
+  simpa only [he,hev] using setIntegral_tendsto_of_L2_pairings P
+    (fun n ω => ∫ x, f x ∂μ n ω) (fun ω => ∫ x, f x ∂ν ω) hp E hE
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/SimultaneousConvexTails.lean b/lean/OAI/Probability/SLE/Mass/SimultaneousConvexTails.lean
index 8246ea22d5325a0b44b70e0a1187d83289a2e7b1..9abfaaf4c0596e7d3a920ec6eff8092902bc36c4 100644
--- a/lean/OAI/Probability/SLE/Mass/SimultaneousConvexTails.lean
+++ b/lean/OAI/Probability/SLE/Mass/SimultaneousConvexTails.lean
@@ -1,82 +1,82 @@
-import OAI.Probability.SLE.Mass.HilbertConvexTails
-import Mathlib.Analysis.InnerProductSpace.l2Space
-
-namespace OAI
-
-/-! The countable deterministic extraction in `mass:construction`.
-A weighted Hilbert sum ensures the same finite convex coefficients work
-for every member of the countable family of bounded test sequences. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology ENNReal
-
-namespace SLEExactGauge
-
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
-
-lemma geometric_half_mem_l2 : Memℓp (fun j : ℕ => (1/2 : ℝ)^j) 2 := by
-  apply memℓp_gen
-  have h := summable_geometric_of_abs_lt_one (show |(1/4 : ℝ)| < 1 by norm_num)
-  convert h using 1
-  funext j
-  simp only [ENNReal.toReal_ofNat,Real.rpow_two,Real.norm_eq_abs,
-    abs_of_nonneg (pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) j)]
-  rw [pow_two,← mul_pow]
-  norm_num
-
-theorem exists_simultaneous_convex_tail_combinations (u : ℕ → ℕ → E)
-    (hbound : ∀ j, ∃ B : ℝ, ∀ n, ‖u n j‖ ≤ B) :
-    ∃ x : ℕ → E, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ,
-      (∀ n i, i ∈ s n → 0 ≤ w n i) ∧
-      (∀ n, ∑ i ∈ s n, w n i = 1) ∧
-      (∀ j, Summable (fun n => ‖(∑ i ∈ s n, w n i • u (n+i) j) - x j‖)) ∧
-      (∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i • u (n+i) j) atTop (𝓝 (x j))) := by
-  choose B hB using hbound
-  let c : ℕ → ℝ := fun j => (1/2:ℝ)^j / max 1 (B j)
-  have hc (j : ℕ) : 0 < c j := by
-    apply div_pos (pow_pos (by norm_num) _)
-    exact lt_of_lt_of_le zero_lt_one (le_max_left _ _)
-  have hsmall (n j : ℕ) : ‖c j • u n j‖ ≤ (1/2:ℝ)^j := by
-    rw [norm_smul,Real.norm_eq_abs,abs_of_pos (hc j)]
-    dsimp only [c]
-    have hd : 0 < max 1 (B j) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
-    have hh : ‖u n j‖ ≤ max 1 (B j) := (hB j n).trans (le_max_right _ _)
-    calc
-      (1/2:ℝ)^j / max 1 (B j) * ‖u n j‖ =
-          (1/2:ℝ)^j * ‖u n j‖ / max 1 (B j) := by ring
-      _ ≤ (1/2:ℝ)^j := (div_le_iff₀ hd).mpr (by gcongr)
-  let g : lp (fun _ : ℕ => ℝ) 2 := ⟨_,geometric_half_mem_l2⟩
-  let v : ℕ → lp (fun _ : ℕ => E) 2 := fun n =>
-    ⟨fun j => c j • u n j,geometric_half_mem_l2.mono (hsmall n)⟩
-  have hv (n : ℕ) : ‖v n‖ ≤ ‖g‖ := by
-    apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
-    intro j
-    simpa only [g,Real.norm_eq_abs,
-      abs_of_nonneg (pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) j)] using hsmall n j
-  obtain ⟨x,s,w,hw,hs,herror,hconv⟩ := exists_convex_tail_combinations v ‖g‖ hv
-  let L (j : ℕ) : lp (fun _ : ℕ => E) 2 →L[ℝ] E :=
-    (c j)⁻¹ • lp.evalCLM ℝ (fun _ : ℕ => E) 2 j
-  have hL (j n : ℕ) : L j (v n) = u n j := by
-    change (c j)⁻¹ • (c j • u n j) = _
-    rw [smul_smul,inv_mul_cancel₀ (hc j).ne',one_smul]
-  refine ⟨fun j => L j x,s,w,hw,hs,?_,?_⟩
-  · intro j
-    have hgeo : Summable (fun n : ℕ => ‖L j‖ * (1/2:ℝ)^n) :=
-      (summable_geometric_of_abs_lt_one (show |(1/2:ℝ)| < 1 by norm_num)).mul_left _
-    apply Summable.of_nonneg_of_le (fun _ => norm_nonneg _) _ hgeo
-    intro n
-    have hnorm := (L j).le_opNorm ((∑ i ∈ s n, w n i • v (n+i))-x)
-    simp only [map_sub,map_sum,map_smul,hL] at hnorm
-    have hh : ‖(∑ i ∈ s n, w n i • v (n+i))-x‖ ≤ (1/2:ℝ)^n := by
-      simpa only [dist_eq_norm] using (herror n).le
-    exact hnorm.trans (mul_le_mul_of_nonneg_left hh (norm_nonneg _))
-  · intro j
-    have hh := ((L j).continuous.tendsto x).comp hconv
-    simpa only [Function.comp_def,map_sum,map_smul,hL] using hh
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.HilbertConvexTails
+import Mathlib.Analysis.InnerProductSpace.l2Space
+
+namespace OAI
+
+/-! The countable deterministic extraction in `mass:construction`.
+A weighted Hilbert sum ensures the same finite convex coefficients work
+for every member of the countable family of bounded test sequences. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology ENNReal
+
+namespace SLEExactGauge
+
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+
+lemma geometric_half_mem_l2 : Memℓp (fun j : ℕ => (1/2 : ℝ)^j) 2 := by
+  apply memℓp_gen
+  have h := summable_geometric_of_abs_lt_one (show |(1/4 : ℝ)| < 1 by norm_num)
+  convert h using 1
+  funext j
+  simp only [ENNReal.toReal_ofNat,Real.rpow_two,Real.norm_eq_abs,
+    abs_of_nonneg (pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) j)]
+  rw [pow_two,← mul_pow]
+  norm_num
+
+theorem exists_simultaneous_convex_tail_combinations (u : ℕ → ℕ → E)
+    (hbound : ∀ j, ∃ B : ℝ, ∀ n, ‖u n j‖ ≤ B) :
+    ∃ x : ℕ → E, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ,
+      (∀ n i, i ∈ s n → 0 ≤ w n i) ∧
+      (∀ n, ∑ i ∈ s n, w n i = 1) ∧
+      (∀ j, Summable (fun n => ‖(∑ i ∈ s n, w n i • u (n+i) j) - x j‖)) ∧
+      (∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i • u (n+i) j) atTop (𝓝 (x j))) := by
+  choose B hB using hbound
+  let c : ℕ → ℝ := fun j => (1/2:ℝ)^j / max 1 (B j)
+  have hc (j : ℕ) : 0 < c j := by
+    apply div_pos (pow_pos (by norm_num) _)
+    exact lt_of_lt_of_le zero_lt_one (le_max_left _ _)
+  have hsmall (n j : ℕ) : ‖c j • u n j‖ ≤ (1/2:ℝ)^j := by
+    rw [norm_smul,Real.norm_eq_abs,abs_of_pos (hc j)]
+    dsimp only [c]
+    have hd : 0 < max 1 (B j) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
+    have hh : ‖u n j‖ ≤ max 1 (B j) := (hB j n).trans (le_max_right _ _)
+    calc
+      (1/2:ℝ)^j / max 1 (B j) * ‖u n j‖ =
+          (1/2:ℝ)^j * ‖u n j‖ / max 1 (B j) := by ring
+      _ ≤ (1/2:ℝ)^j := (div_le_iff₀ hd).mpr (by gcongr)
+  let g : lp (fun _ : ℕ => ℝ) 2 := ⟨_,geometric_half_mem_l2⟩
+  let v : ℕ → lp (fun _ : ℕ => E) 2 := fun n =>
+    ⟨fun j => c j • u n j,geometric_half_mem_l2.mono (hsmall n)⟩
+  have hv (n : ℕ) : ‖v n‖ ≤ ‖g‖ := by
+    apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
+    intro j
+    simpa only [g,Real.norm_eq_abs,
+      abs_of_nonneg (pow_nonneg (by norm_num : (0:ℝ) ≤ 1/2) j)] using hsmall n j
+  obtain ⟨x,s,w,hw,hs,herror,hconv⟩ := exists_convex_tail_combinations v ‖g‖ hv
+  let L (j : ℕ) : lp (fun _ : ℕ => E) 2 →L[ℝ] E :=
+    (c j)⁻¹ • lp.evalCLM ℝ (fun _ : ℕ => E) 2 j
+  have hL (j n : ℕ) : L j (v n) = u n j := by
+    change (c j)⁻¹ • (c j • u n j) = _
+    rw [smul_smul,inv_mul_cancel₀ (hc j).ne',one_smul]
+  refine ⟨fun j => L j x,s,w,hw,hs,?_,?_⟩
+  · intro j
+    have hgeo : Summable (fun n : ℕ => ‖L j‖ * (1/2:ℝ)^n) :=
+      (summable_geometric_of_abs_lt_one (show |(1/2:ℝ)| < 1 by norm_num)).mul_left _
+    apply Summable.of_nonneg_of_le (fun _ => norm_nonneg _) _ hgeo
+    intro n
+    have hnorm := (L j).le_opNorm ((∑ i ∈ s n, w n i • v (n+i))-x)
+    simp only [map_sub,map_sum,map_smul,hL] at hnorm
+    have hh : ‖(∑ i ∈ s n, w n i • v (n+i))-x‖ ≤ (1/2:ℝ)^n := by
+      simpa only [dist_eq_norm] using (herror n).le
+    exact hnorm.trans (mul_le_mul_of_nonneg_left hh (norm_nonneg _))
+  · intro j
+    have hh := ((L j).continuous.tendsto x).comp hconv
+    simpa only [Function.comp_def,map_sum,map_smul,hL] using hh
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/StoppedConformalPassageMass.lean b/lean/OAI/Probability/SLE/Mass/StoppedConformalPassageMass.lean
index 3b244fdeb09f8cf03c08d7fa419891606bd67f05..e30821b517c1407aa830bd2883c8c99ae44f9d14 100644
--- a/lean/OAI/Probability/SLE/Mass/StoppedConformalPassageMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/StoppedConformalPassageMass.lean
@@ -1,218 +1,218 @@
-import OAI.Probability.SLE.Mass.CompactConformalPassageMass
-
-namespace OAI
-
-/-! Retained sausage mass before the actual passage stop. The physical
-curve may include boundary visits; only its retained interior interval is
-identified with the conformal image. All mass is charged to the prescribed
-stopped segment, rather than to later portions of the whole trace. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-
-namespace SLEExactGauge
-
-def segmentSausageCharge (κ : ℝ) (η : ℝ≥0 → ℂ) (s t : ℝ≥0)
-    (v : ℝ) (R : Set ℂ) : ℝ≥0∞ :=
-  ENNReal.ofReal (v ^ (-codimension κ)) * volume (R ∩ sourceSausageSet η s t v)
-
-theorem sourceSausageSet_mono_interval
-    (η : ℝ≥0 → ℂ) {s t a b : ℝ≥0} (hsa : s ≤ a) (hbt : b ≤ t) (v : ℝ) :
-    sourceSausageSet η a b v ⊆ sourceSausageSet η s t v := by
-  rintro z ⟨u, hu, hd⟩
-  exact ⟨u, ⟨hsa.trans hu.1, hu.2.trans hbt⟩, hd⟩
-
-theorem sourceSausageSet_congr_interval
-    {η ξ : ℝ≥0 → ℂ} {s t : ℝ≥0} (heq : EqOn η ξ (Icc s t)) (v : ℝ) :
-    sourceSausageSet η s t v = sourceSausageSet ξ s t v := by
-  ext z
-  constructor <;> rintro ⟨u, hu, hd⟩
-  · exact ⟨u, hu, by simpa only [heq hu] using hd⟩
-  · exact ⟨u, hu, by simpa only [heq hu] using hd⟩
-
-theorem segmentSausageCharge_mono
-    (κ : ℝ) (η : ℝ≥0 → ℂ) {s t a b : ℝ≥0}
-    (hsa : s ≤ a) (hbt : b ≤ t) (v : ℝ) {R S : Set ℂ} (hRS : R ⊆ S) :
-    segmentSausageCharge κ η a b v R ≤ segmentSausageCharge κ η s t v S :=
-  mul_le_mul' le_rfl (measure_mono (inter_subset_inter hRS
-    (sourceSausageSet_mono_interval η hsa hbt v)))
-
-/-- The finite-list conformal transfer charges only the actual stopped
-physical passage. Its values at parameter-boundary times can be supplied by
-the boundary-safe restart theorem. -/
-theorem compact_conformal_stopped_passage_mass
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
-    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R)
-    (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ)
-    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n))
-    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
-    (hretain : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n →
-      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2)) :
-    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
-      ENNReal.ofReal (q / 2) ≤ P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤
-        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by
-  obtain ⟨c, ell, v₀, hc, hell, hv₀, htransfer⟩ := eventually_compact_sausage_transfer
-    κ Fₙ F hFn hinj hF hne hconv K hK hKn hKH R hR himage
-  have hcoeff : 0 < c ^ 2 * ell ^ codimension κ :=
-    mul_pos (sq_pos_of_pos hc) (Real.rpow_pos_of_pos hell _)
-  have hsmall : ∀ᶠ n in atTop, v n < v₀ := hlim.eventually (gt_mem_nhds hv₀)
-  have hlim' : Tendsto (fun n => ell * v n) atTop (𝓝 0) := by
-    simpa only [mul_zero] using tendsto_const_nhds.mul hlim
-  have htrans : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → True ∧
-      ∃ J ∈ I, ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) *
-        sourceSausageMass κ (γ ω) J.1 J.2 (ell * v n) ≤
-        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n) := by
-    filter_upwards [htransfer, hretain, hsmall, hregion] with n hn hr hs hRsub
-    intro ω hω
-    obtain ⟨J, hJI, hJK, hJs, hJt, heq⟩ := hr ω hω
-    refine ⟨trivial, J, hJI, (hn (γ ω) J.1 J.2 hJK (v n) (hv n) hs).trans ?_⟩
-    change segmentSausageCharge κ (fun u => Fₙ n (γ ω u)) J.1 J.2 (v n) R ≤ _
-    have hsausage := sourceSausageSet_congr_interval heq (v n)
-    unfold segmentSausageCharge
-    rw [← hsausage]
-    exact segmentSausageCharge_mono κ (η n ω) hJs hJt (v n) hRsub
-  obtain ⟨a, ha, hmass⟩ := ribbon_passage_mass_of_eventually_finite_intervals
-    P κ hκ hκ8 γ hSLE I hI (fun n => ell * v n)
-    (fun n => mul_pos hell (hv n)) hlim' G q hq hG
-    (fun n ω => segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n))
-    (fun _ _ => True) (c ^ 2 * ell ^ codimension κ) hcoeff htrans
-  exact ⟨a, ha, by simpa only [true_and] using hmass⟩
-
-/-- Almost-sure chart identities suffice; exceptional restart paths are
-removed without changing the probabilities of the retained events. -/
-theorem compact_conformal_stopped_passage_mass_ae
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
-    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R)
-    (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ)
-    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n))
-    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
-    (hretain : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n →
-      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2)) :
-    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
-      ENNReal.ofReal (q / 2) ≤ P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤
-        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by
-  let H (n : ℕ) : Set Ω := {ω | ω ∈ G n ∧
-      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2)}
-  have hH : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (H n) := by
-    filter_upwards [hG, hretain] with n hn hr
-    have he : H n =ᵐ[P] G n := by
-      filter_upwards [hr] with ω hω
-      exact propext ⟨fun h => h.1, fun h => ⟨h, hω h⟩⟩
-    simpa only [measure_congr he] using hn
-  have hr : ∀ᶠ n in atTop, ∀ ω, ω ∈ H n →
-      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2) :=
-    Eventually.of_forall fun _ _ hω => hω.2
-  obtain ⟨a, ha, hm⟩ := compact_conformal_stopped_passage_mass P κ hκ hκ8 γ hSLE
-    Fₙ F hFn hinj hF hne hconv K hK hKn hKH R hR himage I hI v hv hlim
-    η s t Rₙ H q hq hH hregion hr
-  refine ⟨a, ha, ?_⟩
-  filter_upwards [hm] with n hn
-  exact hn.trans (measure_mono fun _ hω => ⟨hω.1.1, hω.2⟩)
-
-/-- Countable compact/time certificates are extracted before choosing the
-common mesh. The selected intervals lie before the actual passage stop. -/
-theorem countable_compact_conformal_stopped_passage_mass
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane)
-    (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R)
-    (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2)
-    (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i))
-    (hcover : P (E \ ⋃ i, D i) = 0)
-    (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ)
-    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
-    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
-    (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E))
-    (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ ω, ω ∈ G n ∩ E ∩ D i →
-      s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) :
-    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
-      ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤
-        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by
-  classical
-  obtain ⟨N, hN⟩ := varying_events_after_finite_retention P E D G hE hD hcover q hq hG
-  let A := ⋃ i ∈ Finset.range N, D i
-  let Q := ⋃ i ∈ Finset.range N, K i
-  let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A
-  have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i)
-  have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i)
-  have hQR : F '' Q ⊆ R := by
-    rintro w ⟨z, hz, rfl⟩
-    obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz
-    exact himage i (mem_image_of_mem F hi)
-  have hQn : Q.Nonempty := by
-    have hHpos : 0 < P (H 0) := (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le (hN 0)
-    obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne'
-    obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2
-    exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi,
-      hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩
-  let I := (Finset.range N).image J
-  have hI : ∀ L ∈ I, L.1 < L.2 := by
-    intro L hL
-    obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL
-    exact hJ i
-  have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ ω, ω ∈ G n ∩ E ∩ D i →
-      s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) :=
-    (Finset.eventually_all _).mpr (fun i _ => hwindow i)
-  have hr : ∀ᶠ n in atTop, ∀ ω, ω ∈ H n →
-      ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧
-        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by
-    filter_upwards [hwin] with n hn
-    intro ω hω
-    obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2
-    exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩,
-      (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl),
-      hn i hi ω ⟨hω.1, hDi⟩⟩
-  obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass P κ hκ hκ8 γ hSLE
-    Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI v hv hlim
-    η s t Rₙ H (q / 2) (by positivity) (Eventually.of_forall hN) hregion hr
-  refine ⟨a, ha, ?_⟩
-  filter_upwards [hmass] with n hn
-  have hqq : q / 2 / 2 = q / 4 := by ring
-  rw [hqq] at hn
-  exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩))
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CompactConformalPassageMass
+
+namespace OAI
+
+/-! Retained sausage mass before the actual passage stop. The physical
+curve may include boundary visits; only its retained interior interval is
+identified with the conformal image. All mass is charged to the prescribed
+stopped segment, rather than to later portions of the whole trace. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+
+namespace SLEExactGauge
+
+def segmentSausageCharge (κ : ℝ) (η : ℝ≥0 → ℂ) (s t : ℝ≥0)
+    (v : ℝ) (R : Set ℂ) : ℝ≥0∞ :=
+  ENNReal.ofReal (v ^ (-codimension κ)) * volume (R ∩ sourceSausageSet η s t v)
+
+theorem sourceSausageSet_mono_interval
+    (η : ℝ≥0 → ℂ) {s t a b : ℝ≥0} (hsa : s ≤ a) (hbt : b ≤ t) (v : ℝ) :
+    sourceSausageSet η a b v ⊆ sourceSausageSet η s t v := by
+  rintro z ⟨u, hu, hd⟩
+  exact ⟨u, ⟨hsa.trans hu.1, hu.2.trans hbt⟩, hd⟩
+
+theorem sourceSausageSet_congr_interval
+    {η ξ : ℝ≥0 → ℂ} {s t : ℝ≥0} (heq : EqOn η ξ (Icc s t)) (v : ℝ) :
+    sourceSausageSet η s t v = sourceSausageSet ξ s t v := by
+  ext z
+  constructor <;> rintro ⟨u, hu, hd⟩
+  · exact ⟨u, hu, by simpa only [heq hu] using hd⟩
+  · exact ⟨u, hu, by simpa only [heq hu] using hd⟩
+
+theorem segmentSausageCharge_mono
+    (κ : ℝ) (η : ℝ≥0 → ℂ) {s t a b : ℝ≥0}
+    (hsa : s ≤ a) (hbt : b ≤ t) (v : ℝ) {R S : Set ℂ} (hRS : R ⊆ S) :
+    segmentSausageCharge κ η a b v R ≤ segmentSausageCharge κ η s t v S :=
+  mul_le_mul' le_rfl (measure_mono (inter_subset_inter hRS
+    (sourceSausageSet_mono_interval η hsa hbt v)))
+
+/-- The finite-list conformal transfer charges only the actual stopped
+physical passage. Its values at parameter-boundary times can be supplied by
+the boundary-safe restart theorem. -/
+theorem compact_conformal_stopped_passage_mass
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
+    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R)
+    (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ)
+    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n))
+    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
+    (hretain : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n →
+      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2)) :
+    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
+      ENNReal.ofReal (q / 2) ≤ P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤
+        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by
+  obtain ⟨c, ell, v₀, hc, hell, hv₀, htransfer⟩ := eventually_compact_sausage_transfer
+    κ Fₙ F hFn hinj hF hne hconv K hK hKn hKH R hR himage
+  have hcoeff : 0 < c ^ 2 * ell ^ codimension κ :=
+    mul_pos (sq_pos_of_pos hc) (Real.rpow_pos_of_pos hell _)
+  have hsmall : ∀ᶠ n in atTop, v n < v₀ := hlim.eventually (gt_mem_nhds hv₀)
+  have hlim' : Tendsto (fun n => ell * v n) atTop (𝓝 0) := by
+    simpa only [mul_zero] using tendsto_const_nhds.mul hlim
+  have htrans : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → True ∧
+      ∃ J ∈ I, ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) *
+        sourceSausageMass κ (γ ω) J.1 J.2 (ell * v n) ≤
+        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n) := by
+    filter_upwards [htransfer, hretain, hsmall, hregion] with n hn hr hs hRsub
+    intro ω hω
+    obtain ⟨J, hJI, hJK, hJs, hJt, heq⟩ := hr ω hω
+    refine ⟨trivial, J, hJI, (hn (γ ω) J.1 J.2 hJK (v n) (hv n) hs).trans ?_⟩
+    change segmentSausageCharge κ (fun u => Fₙ n (γ ω u)) J.1 J.2 (v n) R ≤ _
+    have hsausage := sourceSausageSet_congr_interval heq (v n)
+    unfold segmentSausageCharge
+    rw [← hsausage]
+    exact segmentSausageCharge_mono κ (η n ω) hJs hJt (v n) hRsub
+  obtain ⟨a, ha, hmass⟩ := ribbon_passage_mass_of_eventually_finite_intervals
+    P κ hκ hκ8 γ hSLE I hI (fun n => ell * v n)
+    (fun n => mul_pos hell (hv n)) hlim' G q hq hG
+    (fun n ω => segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n))
+    (fun _ _ => True) (c ^ 2 * ell ^ codimension κ) hcoeff htrans
+  exact ⟨a, ha, by simpa only [true_and] using hmass⟩
+
+/-- Almost-sure chart identities suffice; exceptional restart paths are
+removed without changing the probabilities of the retained events. -/
+theorem compact_conformal_stopped_passage_mass_ae
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane)
+    (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R)
+    (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ)
+    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n))
+    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
+    (hretain : ∀ᶠ n in atTop, ∀ᵐ ω ∂P, ω ∈ G n →
+      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2)) :
+    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
+      ENNReal.ofReal (q / 2) ≤ P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤
+        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by
+  let H (n : ℕ) : Set Ω := {ω | ω ∈ G n ∧
+      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2)}
+  have hH : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (H n) := by
+    filter_upwards [hG, hretain] with n hn hr
+    have he : H n =ᵐ[P] G n := by
+      filter_upwards [hr] with ω hω
+      exact propext ⟨fun h => h.1, fun h => ⟨h, hω h⟩⟩
+    simpa only [measure_congr he] using hn
+  have hr : ∀ᶠ n in atTop, ∀ ω, ω ∈ H n →
+      ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K ∧ s n ω ≤ J.1 ∧ J.2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc J.1 J.2) :=
+    Eventually.of_forall fun _ _ hω => hω.2
+  obtain ⟨a, ha, hm⟩ := compact_conformal_stopped_passage_mass P κ hκ hκ8 γ hSLE
+    Fₙ F hFn hinj hF hne hconv K hK hKn hKH R hR himage I hI v hv hlim
+    η s t Rₙ H q hq hH hregion hr
+  refine ⟨a, ha, ?_⟩
+  filter_upwards [hm] with n hn
+  exact hn.trans (measure_mono fun _ hω => ⟨hω.1.1, hω.2⟩)
+
+/-- Countable compact/time certificates are extracted before choosing the
+common mesh. The selected intervals lie before the actual passage stop. -/
+theorem countable_compact_conformal_stopped_passage_mass
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) (hKH : ∀ i, K i ⊆ upperHalfPlane)
+    (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R)
+    (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2)
+    (E : Set Ω) (D : ℕ → Set Ω) (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i))
+    (hcover : P (E \ ⋃ i, D i) = 0)
+    (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (s t : ℕ → Ω → ℝ≥0) (Rₙ : ℕ → Set ℂ)
+    (hregion : ∀ᶠ n in atTop, R ⊆ Rₙ n)
+    (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q)
+    (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E))
+    (hwindow : ∀ i, ∀ᶠ n in atTop, ∀ ω, ω ∈ G n ∩ E ∩ D i →
+      s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2)) :
+    ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop,
+      ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤
+        segmentSausageCharge κ (η n ω) (s n ω) (t n ω) (v n) (Rₙ n)} := by
+  classical
+  obtain ⟨N, hN⟩ := varying_events_after_finite_retention P E D G hE hD hcover q hq hG
+  let A := ⋃ i ∈ Finset.range N, D i
+  let Q := ⋃ i ∈ Finset.range N, K i
+  let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A
+  have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i)
+  have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i)
+  have hQR : F '' Q ⊆ R := by
+    rintro w ⟨z, hz, rfl⟩
+    obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz
+    exact himage i (mem_image_of_mem F hi)
+  have hQn : Q.Nonempty := by
+    have hHpos : 0 < P (H 0) := (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le (hN 0)
+    obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne'
+    obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2
+    exact ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi,
+      hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩)⟩⟩
+  let I := (Finset.range N).image J
+  have hI : ∀ L ∈ I, L.1 < L.2 := by
+    intro L hL
+    obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL
+    exact hJ i
+  have hwin : ∀ᶠ n in atTop, ∀ i ∈ Finset.range N, ∀ ω, ω ∈ G n ∩ E ∩ D i →
+      s n ω ≤ (J i).1 ∧ (J i).2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc (J i).1 (J i).2) :=
+    (Finset.eventually_all _).mpr (fun i _ => hwindow i)
+  have hr : ∀ᶠ n in atTop, ∀ ω, ω ∈ H n →
+      ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q ∧ s n ω ≤ L.1 ∧ L.2 ≤ t n ω ∧
+        EqOn (η n ω) (fun u => Fₙ n (γ ω u)) (Icc L.1 L.2) := by
+    filter_upwards [hwin] with n hn
+    intro ω hω
+    obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2
+    exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩,
+      (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl),
+      hn i hi ω ⟨hω.1, hDi⟩⟩
+  obtain ⟨a, ha, hmass⟩ := compact_conformal_stopped_passage_mass P κ hκ hκ8 γ hSLE
+    Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI v hv hlim
+    η s t Rₙ H (q / 2) (by positivity) (Eventually.of_forall hN) hregion hr
+  refine ⟨a, ha, ?_⟩
+  filter_upwards [hmass] with n hn
+  have hqq : q / 2 / 2 = q / 4 := by ring
+  rw [hqq] at hn
+  exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩))
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/StoppedInverseMass.lean b/lean/OAI/Probability/SLE/Mass/StoppedInverseMass.lean
index 7f4ae26ea92a6f5046202a96c1cb225e8ec5375c..dcb7fef5b2481ec4e08f19ef7f7335a263b93e81 100644
--- a/lean/OAI/Probability/SLE/Mass/StoppedInverseMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/StoppedInverseMass.lean
@@ -1,105 +1,105 @@
-import OAI.Probability.SLE.Geometry.RestartDerivativeMeasurability
-import OAI.Probability.SLE.Geometry.CanonicalRawMean
-import OAI.Probability.SLE.Mass.WeightedImageRandomMass
-
-namespace OAI
-
-/-! Actual stopped inverse-Loewner transport of the fresh raw terminal
-mass, with measurable charts, derivative weights and exact spatial tests. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Multipoint
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  (B : ℝ≥0 → Ω → ℝ) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
-  (G F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-  (τ : Ω → ℝ≥0)
-
-def stoppedMassPoint (ω : Ω) (z : upperHalfPlane) : upperHalfPlane :=
-  ⟨F ω (τ ω) ((z:ℂ)+(Real.sqrt κ*B (τ ω) ω:ℝ)),
-    ((hw ω).2.2.2.2.2.2.1 (τ ω) (by
-      simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero]
-        using z.property)).1.1⟩
-
-def stoppedMassWeight (ω : Ω) (z : upperHalfPlane) : ℝ≥0∞ :=
-  ENNReal.ofReal (‖deriv (F ω (τ ω)) ((z:ℂ)+(Real.sqrt κ*B (τ ω) ω:ℝ))‖^dimension κ)
-
-variable (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-  (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun ω => (τ ω:WithTop ℝ≥0)))
-
-include hm hc hτ γ G hw
-
-theorem stoppedMassPoint_measurable : Measurable[hτ.measurableSpace.prod inferInstance]
-    (fun p : Ω × upperHalfPlane => stoppedMassPoint B κ γ G F hw τ p.1 p.2) := by
-  unfold stoppedMassPoint
-  apply Measurable.subtype_mk
-  exact continuous_eval.measurable.comp
-    (((source_stopped_inversePath_measurable B hm hc κ γ G F hw τ hτ).comp
-      measurable_fst).prodMk measurable_snd)
-
-theorem stoppedMassWeight_measurable (hκ : 0 < κ) :
-    Measurable[hτ.measurableSpace.prod inferInstance]
-      (fun p : Ω × upperHalfPlane => stoppedMassWeight B κ F τ p.1 p.2) := by
-  let : MeasurableSpace (Ω × upperHalfPlane) := hτ.measurableSpace.prod inferInstance
-  have hd := source_stopped_inverse_derivative_measurable B hm hc κ γ G F hw τ hτ
-  have he : 0 ≤ dimension κ := by unfold dimension; linarith
-  exact ((Real.continuous_rpow_const he).measurable.comp hd.norm).ennreal_ofReal
-
-omit hm hτ in
-def stoppedFreshTerminalMass (R : RawCanonicalMass) (ω : Ω) : Measure upperHalfPlane :=
-  weightedImageMass (stoppedMassPoint B κ γ G F hw τ ω) (stoppedMassWeight B κ F τ ω)
-    (R.terminal (sourceRestartDriverPath B hc τ ω))
-
-theorem stoppedFreshTerminalMass_measurable (hκ : 0 < κ) (R : RawCanonicalMass) :
-    Measurable (stoppedFreshTerminalMass B κ γ G F hw τ hc R) := by
-  have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _
-    (fun t => (measurable_pi_apply t).comp
-      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
-  have hM := @weightedImageMass_measurable_of_subsigma Ω upperHalfPlane mΩ inferInstance
-    hτ.measurableSpace hτ.measurableSpace_le
-    (fun ω => R.terminal (sourceRestartDriverPath B hc τ ω))
-    (R.terminal_measurable.comp hY) (fun ω => (R.terminal_radon _).2)
-    (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2)
-    (stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ)
-    (fun p => stoppedMassWeight B κ F τ p.1 p.2)
-    (stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ)
-  exact hM
-
-theorem CanonicalMass.stoppedFreshTerminalMass_test
-    {P : Measure Ω} [IsProbabilityMeasure P] {hBm : Measurable (Function.uncurry B)}
-    (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) (hκ : 0 < κ)
-    (R : RawCanonicalMass)
-    (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω) = C.rational.terminal ω)
-    (f : Ω × upperHalfPlane → ℝ≥0∞)
-    (hf : Measurable[hτ.measurableSpace.prod inferInstance] f) :
-    (∫⁻ ω, ∫⁻ x, f (ω,x) ∂(stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) ∂P) =
-      ∫⁻ ω, ∫⁻ z, stoppedMassWeight B κ F τ ω z *
-        f (ω,stoppedMassPoint B κ γ G F hw τ ω z) ∂initialGreenMeasure κ ∂P := by
-  have hp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
-  let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) :=
-    (Measure.isProbabilityMeasure_map_iff hp.aemeasurable).mpr inferInstance
-  have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _
-    (fun t => (measurable_pi_apply t).comp
-      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
-  have h := restart_weighted_mass_test P (P.map (sourceDriverPath B hc))
-    hτ.measurableSpace hτ.measurableSpace_le (sourceRestartDriverPath B hc τ) hY
-    (fun A hA => source_restart_continuous_driver_law hB hm hc τ hτ hA)
-    R.terminal R.terminal_measurable (fun p => (R.terminal_radon p).2) _
-    (stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ) _
-    (stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ) f hf
-  rwa [C.raw_terminal_mean hc R he] at h
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.RestartDerivativeMeasurability
+import OAI.Probability.SLE.Geometry.CanonicalRawMean
+import OAI.Probability.SLE.Mass.WeightedImageRandomMass
+
+namespace OAI
+
+/-! Actual stopped inverse-Loewner transport of the fresh raw terminal
+mass, with measurable charts, derivative weights and exact spatial tests. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Multipoint
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  (B : ℝ≥0 → Ω → ℝ) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
+  (G F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+  (τ : Ω → ℝ≥0)
+
+def stoppedMassPoint (ω : Ω) (z : upperHalfPlane) : upperHalfPlane :=
+  ⟨F ω (τ ω) ((z:ℂ)+(Real.sqrt κ*B (τ ω) ω:ℝ)),
+    ((hw ω).2.2.2.2.2.2.1 (τ ω) (by
+      simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero]
+        using z.property)).1.1⟩
+
+def stoppedMassWeight (ω : Ω) (z : upperHalfPlane) : ℝ≥0∞ :=
+  ENNReal.ofReal (‖deriv (F ω (τ ω)) ((z:ℂ)+(Real.sqrt κ*B (τ ω) ω:ℝ))‖^dimension κ)
+
+variable (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+  (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun ω => (τ ω:WithTop ℝ≥0)))
+
+include hm hc hτ γ G hw
+
+theorem stoppedMassPoint_measurable : Measurable[hτ.measurableSpace.prod inferInstance]
+    (fun p : Ω × upperHalfPlane => stoppedMassPoint B κ γ G F hw τ p.1 p.2) := by
+  unfold stoppedMassPoint
+  apply Measurable.subtype_mk
+  exact continuous_eval.measurable.comp
+    (((source_stopped_inversePath_measurable B hm hc κ γ G F hw τ hτ).comp
+      measurable_fst).prodMk measurable_snd)
+
+theorem stoppedMassWeight_measurable (hκ : 0 < κ) :
+    Measurable[hτ.measurableSpace.prod inferInstance]
+      (fun p : Ω × upperHalfPlane => stoppedMassWeight B κ F τ p.1 p.2) := by
+  let : MeasurableSpace (Ω × upperHalfPlane) := hτ.measurableSpace.prod inferInstance
+  have hd := source_stopped_inverse_derivative_measurable B hm hc κ γ G F hw τ hτ
+  have he : 0 ≤ dimension κ := by unfold dimension; linarith
+  exact ((Real.continuous_rpow_const he).measurable.comp hd.norm).ennreal_ofReal
+
+omit hm hτ in
+def stoppedFreshTerminalMass (R : RawCanonicalMass) (ω : Ω) : Measure upperHalfPlane :=
+  weightedImageMass (stoppedMassPoint B κ γ G F hw τ ω) (stoppedMassWeight B κ F τ ω)
+    (R.terminal (sourceRestartDriverPath B hc τ ω))
+
+theorem stoppedFreshTerminalMass_measurable (hκ : 0 < κ) (R : RawCanonicalMass) :
+    Measurable (stoppedFreshTerminalMass B κ γ G F hw τ hc R) := by
+  have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _
+    (fun t => (measurable_pi_apply t).comp
+      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
+  have hM := @weightedImageMass_measurable_of_subsigma Ω upperHalfPlane mΩ inferInstance
+    hτ.measurableSpace hτ.measurableSpace_le
+    (fun ω => R.terminal (sourceRestartDriverPath B hc τ ω))
+    (R.terminal_measurable.comp hY) (fun ω => (R.terminal_radon _).2)
+    (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2)
+    (stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ)
+    (fun p => stoppedMassWeight B κ F τ p.1 p.2)
+    (stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ)
+  exact hM
+
+theorem CanonicalMass.stoppedFreshTerminalMass_test
+    {P : Measure Ω} [IsProbabilityMeasure P] {hBm : Measurable (Function.uncurry B)}
+    (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) (hκ : 0 < κ)
+    (R : RawCanonicalMass)
+    (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω) = C.rational.terminal ω)
+    (f : Ω × upperHalfPlane → ℝ≥0∞)
+    (hf : Measurable[hτ.measurableSpace.prod inferInstance] f) :
+    (∫⁻ ω, ∫⁻ x, f (ω,x) ∂(stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) ∂P) =
+      ∫⁻ ω, ∫⁻ z, stoppedMassWeight B κ F τ ω z *
+        f (ω,stoppedMassPoint B κ γ G F hw τ ω z) ∂initialGreenMeasure κ ∂P := by
+  have hp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
+  let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) :=
+    (Measure.isProbabilityMeasure_map_iff hp.aemeasurable).mpr inferInstance
+  have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _
+    (fun t => (measurable_pi_apply t).comp
+      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
+  have h := restart_weighted_mass_test P (P.map (sourceDriverPath B hc))
+    hτ.measurableSpace hτ.measurableSpace_le (sourceRestartDriverPath B hc τ) hY
+    (fun A hA => source_restart_continuous_driver_law hB hm hc τ hτ hA)
+    R.terminal R.terminal_measurable (fun p => (R.terminal_radon p).2) _
+    (stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ) _
+    (stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ) f hf
+  rwa [C.raw_terminal_mean hc R he] at h
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/StrictPastMass.lean b/lean/OAI/Probability/SLE/Mass/StrictPastMass.lean
index 078893f86791f8da06e1a612b96efb3cb9498354..1f2a0ffaf4ca0cc9315df6edb3a9ea614d3f512a 100644
--- a/lean/OAI/Probability/SLE/Mass/StrictPastMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/StrictPastMass.lean
@@ -1,76 +1,76 @@
-import OAI.Probability.SLE.Mass.CanonicalMassFirstVisit
-
-namespace OAI
-
-/-! The expected terminal mass arriving strictly after a deterministic past
-is its current Green mass, including arbitrary events in that past. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-
-lemma strict_past_mass_kernel
-    (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (t : ℝ≥0) (E : Set Ω)
-    (hE : MeasurableSet[(Filtration.natural B (fun s =>
-      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E)
-    (K : Set upperHalfPlane) (hK : IsCompact K) :
-    (∫⁻ ω in E, C.rational.terminal ω
-      (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) ∂P)=
-      ∫⁻ ω in E, halfPlaneGreenMeasure κ B γ G t ω K ∂P := by
-  let ℱ := Filtration.natural B (fun s =>
-    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)
-  have hNm : Measurable (C.process t) := by
-    simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe]
-      using C.atTime_measurable (fun _ => (t:ℝ≥0∞)) measurable_const
-  have hNG : Measurable (fun ω => C.process t ω K) :=
-    (Measure.measurable_coe hK.measurableSet).comp hNm
-  have hGm := halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t
-  have hp : ∀ᵐ ω ∂P, C.process t ω=(C.rational.terminal ω).restrict
-      {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t} := by
-    simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe]
-      using C.atTime_eq_restrict_past hB hBm hc h0 hκ hκ8 F hw t
-  have hdecomp : (∫⁻ ω in E, C.rational.terminal ω K ∂P)=
-      (∫⁻ ω in E, C.process t ω K ∂P)+
-      ∫⁻ ω in E, C.rational.terminal ω
-        (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) ∂P := by
-    rw [← lintegral_add_left hNG]
-    apply lintegral_congr_ae
-    filter_upwards [ae_restrict_of_ae hp] with ω hω
-    rw [hω,Measure.restrict_apply hK.measurableSet]
-    exact (measure_inter_add_sdiff K
-      (((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet)).symm
-  have hS : IsStoppingTime ℱ (fun _ : Ω => (t:WithTop ℝ≥0)) := isStoppingTime_const ℱ t
-  have hEs : MeasurableSet[hS.measurableSpace] E := by
-    change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E
-    rw [IsStoppingTime.measurableSpace_const]
-    exact hE
-  have hk := congrArg (fun μ : Measure upperHalfPlane => μ K) (C.kernel _ hS E hEs)
-  change ((P.restrict E).bind C.rational.terminal) K=
-    (((P.restrict E).bind (C.process t))+((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) K at hk
-  rw [Measure.add_apply,Measure.bind_apply hK.measurableSet C.rational.terminal_measurable.aemeasurable,
-    Measure.bind_apply hK.measurableSet hNm.aemeasurable,
-    Measure.bind_apply hK.measurableSet hGm.aemeasurable] at hk
-  have hfin : (∫⁻ ω in E, C.process t ω K ∂P)≠∞ := by
-    let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ
-    have hb := C.atTime_mean_le (fun _ => (t:ℝ≥0∞)) measurable_const E K
-    change ((P.restrict E).bind (C.process t)) K ≤ initialGreenMeasure κ K at hb
-    rw [Measure.bind_apply hK.measurableSet hNm.aemeasurable] at hb
-    exact (hb.trans_lt hK.measure_lt_top).ne
-  apply (ENNReal.add_right_inj hfin).mp
-  simpa only [add_comm] using hdecomp.symm.trans hk
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CanonicalMassFirstVisit
+
+namespace OAI
+
+/-! The expected terminal mass arriving strictly after a deterministic past
+is its current Green mass, including arbitrary events in that past. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+
+lemma strict_past_mass_kernel
+    (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (t : ℝ≥0) (E : Set Ω)
+    (hE : MeasurableSet[(Filtration.natural B (fun s =>
+      (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E)
+    (K : Set upperHalfPlane) (hK : IsCompact K) :
+    (∫⁻ ω in E, C.rational.terminal ω
+      (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) ∂P)=
+      ∫⁻ ω in E, halfPlaneGreenMeasure κ B γ G t ω K ∂P := by
+  let ℱ := Filtration.natural B (fun s =>
+    (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)
+  have hNm : Measurable (C.process t) := by
+    simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe]
+      using C.atTime_measurable (fun _ => (t:ℝ≥0∞)) measurable_const
+  have hNG : Measurable (fun ω => C.process t ω K) :=
+    (Measure.measurable_coe hK.measurableSet).comp hNm
+  have hGm := halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t
+  have hp : ∀ᵐ ω ∂P, C.process t ω=(C.rational.terminal ω).restrict
+      {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t} := by
+    simpa only [atTime,extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe]
+      using C.atTime_eq_restrict_past hB hBm hc h0 hκ hκ8 F hw t
+  have hdecomp : (∫⁻ ω in E, C.rational.terminal ω K ∂P)=
+      (∫⁻ ω in E, C.process t ω K ∂P)+
+      ∫⁻ ω in E, C.rational.terminal ω
+        (K \ {z : upperHalfPlane | (z:ℂ)∈γ ω '' Icc 0 t}) ∂P := by
+    rw [← lintegral_add_left hNG]
+    apply lintegral_congr_ae
+    filter_upwards [ae_restrict_of_ae hp] with ω hω
+    rw [hω,Measure.restrict_apply hK.measurableSet]
+    exact (measure_inter_add_sdiff K
+      (((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet)).symm
+  have hS : IsStoppingTime ℱ (fun _ : Ω => (t:WithTop ℝ≥0)) := isStoppingTime_const ℱ t
+  have hEs : MeasurableSet[hS.measurableSpace] E := by
+    change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E
+    rw [IsStoppingTime.measurableSpace_const]
+    exact hE
+  have hk := congrArg (fun μ : Measure upperHalfPlane => μ K) (C.kernel _ hS E hEs)
+  change ((P.restrict E).bind C.rational.terminal) K=
+    (((P.restrict E).bind (C.process t))+((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) K at hk
+  rw [Measure.add_apply,Measure.bind_apply hK.measurableSet C.rational.terminal_measurable.aemeasurable,
+    Measure.bind_apply hK.measurableSet hNm.aemeasurable,
+    Measure.bind_apply hK.measurableSet hGm.aemeasurable] at hk
+  have hfin : (∫⁻ ω in E, C.process t ω K ∂P)≠∞ := by
+    let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ
+    have hb := C.atTime_mean_le (fun _ => (t:ℝ≥0∞)) measurable_const E K
+    change ((P.restrict E).bind (C.process t)) K ≤ initialGreenMeasure κ K at hb
+    rw [Measure.bind_apply hK.measurableSet hNm.aemeasurable] at hb
+    exact (hb.trans_lt hK.measure_lt_top).ne
+  apply (ENNReal.add_right_inj hfin).mp
+  simpa only [add_comm] using hdecomp.symm.trans hk
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/SubSigmaRadonLimit.lean b/lean/OAI/Probability/SLE/Mass/SubSigmaRadonLimit.lean
index b3c9b85ae68edcf5c41d6d2b083485293c71de2a..1d4d6b77a00129238d4e452c94b62f2695dbc3c9 100644
--- a/lean/OAI/Probability/SLE/Mass/SubSigmaRadonLimit.lean
+++ b/lean/OAI/Probability/SLE/Mass/SubSigmaRadonLimit.lean
@@ -1,88 +1,88 @@
-import OAI.Probability.SLE.Mass.RandomRadonLimit
-import OAI.Probability.SLE.Mass.RadonTestFamily
-
-namespace OAI
-
-/-! A vague Radon limit has a version measurable in the same sigma-algebra
-as its approximants. The probability measure may use a larger sigma-algebra;
-the exceptional convergence set is detected by countably many tests. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped Topology CompactlySupported
-
-namespace SLEExactGauge
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-theorem exists_subsigma_radon_version_identified
-    {Ω : Type*} (m : MeasurableSpace Ω) (μ : ℕ → Ω → Measure X)
-    (hfinite : ∀ n ω, IsFiniteMeasureOnCompacts (μ n ω))
-    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable[m] (fun ω => ∫ x, f x ∂μ n ω)) :
-    ∃ η : Ω → Measure X, Measurable[m] η ∧
-      (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧
-      ∀ ω (ν : Measure X), ν.Regular →
-        (∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν))) →
-        η ω = ν := by
-  let : MeasurableSpace Ω := m
-  let D := localTestFamily X
-  let G : Set Ω := {ω | ∀ j, ∃ a : ℝ,
-    Tendsto (fun n => ∫ x, D.test j x ∂μ n ω) atTop (𝓝 a)}
-  have hG : MeasurableSet[m] G := by
-    let : MeasurableSpace Ω := m
-    simpa only [G,Set.ofPred_forall] using MeasurableSet.iInter
-      (fun j => measurableSet_exists_tendsto (fun n => hmeas n (D.test j)))
-  have hex (ω : Ω) : ∃ η : Measure X, η.Regular ∧
-      ∀ f : C_c(X,ℝ), Tendsto
-        (fun n => if ω ∈ G then ∫ x, f x ∂μ n ω else 0) atTop (𝓝 (∫ x, f x ∂η)) := by
-    by_cases hω : ω ∈ G
-    · let (n : ℕ) : IsFiniteMeasureOnCompacts (μ n ω) := hfinite n ω
-      obtain ⟨η,hη,_,hl⟩ := D.exists_radon_limit (fun n => μ n ω) hω
-      exact ⟨η,hη,fun f => by simpa only [ite_eq_left hω] using hl f⟩
-    · refine ⟨0,inferInstance,fun f => ?_⟩
-      simpa only [ite_eq_right hω,integral_zero_measure] using
-        (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
-  choose η hηr hηlim using hex
-  have hηm : Measurable[m] η := by
-    let : MeasurableSpace Ω := m
-    apply measurable_radon_of_tests η hηr
-    intro f
-    apply measurable_of_tendsto_metrizable
-      (f := fun n ω => if ω ∈ G then ∫ x, f x ∂μ n ω else 0)
-    · intro n
-      exact (hmeas n f).ite hG measurable_const
-    · exact tendsto_pi_nhds.mpr (fun ω => hηlim ω f)
-  refine ⟨η,hηm,fun ω => ?_,?_⟩
-  · let := hηr ω
-    exact ⟨hηr ω,inferInstance⟩
-  · intro ω ν hr hl
-    have hg : ω ∈ G := fun j => ⟨∫ x, D.test j x ∂ν,hl (D.test j)⟩
-    let := hηr ω
-    let := hr
-    apply radon_test_limit_unique (fun n => μ n ω) (η ω) ν
-    · intro f
-      simpa only [ite_eq_left hg] using hηlim ω f
-    · exact hl
-
-theorem exists_subsigma_radon_version
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (m : MeasurableSpace Ω) (P : @Measure Ω mΩ)
-    (μ : ℕ → Ω → Measure X) (ν : Ω → Measure X)
-    (hfinite : ∀ n ω, IsFiniteMeasureOnCompacts (μ n ω))
-    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable[m] (fun ω => ∫ x, f x ∂μ n ω))
-    (hν : ∀ᵐ ω ∂P, (ν ω).Regular)
-    (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(X,ℝ),
-      Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν ω))) :
-    ∃ η : Ω → Measure X, Measurable[m] η ∧
-      (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ η =ᵐ[P] ν := by
-  obtain ⟨η,hm,hr,he⟩ := exists_subsigma_radon_version_identified m μ hfinite hmeas
-  refine ⟨η,hm,hr,?_⟩
-  filter_upwards [hν,hlim] with ω hν hlim
-  exact he ω (ν ω) hν hlim
-
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RandomRadonLimit
+import OAI.Probability.SLE.Mass.RadonTestFamily
+
+namespace OAI
+
+/-! A vague Radon limit has a version measurable in the same sigma-algebra
+as its approximants. The probability measure may use a larger sigma-algebra;
+the exceptional convergence set is detected by countably many tests. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped Topology CompactlySupported
+
+namespace SLEExactGauge
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+  [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+theorem exists_subsigma_radon_version_identified
+    {Ω : Type*} (m : MeasurableSpace Ω) (μ : ℕ → Ω → Measure X)
+    (hfinite : ∀ n ω, IsFiniteMeasureOnCompacts (μ n ω))
+    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable[m] (fun ω => ∫ x, f x ∂μ n ω)) :
+    ∃ η : Ω → Measure X, Measurable[m] η ∧
+      (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧
+      ∀ ω (ν : Measure X), ν.Regular →
+        (∀ f : C_c(X,ℝ), Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν))) →
+        η ω = ν := by
+  let : MeasurableSpace Ω := m
+  let D := localTestFamily X
+  let G : Set Ω := {ω | ∀ j, ∃ a : ℝ,
+    Tendsto (fun n => ∫ x, D.test j x ∂μ n ω) atTop (𝓝 a)}
+  have hG : MeasurableSet[m] G := by
+    let : MeasurableSpace Ω := m
+    simpa only [G,Set.ofPred_forall] using MeasurableSet.iInter
+      (fun j => measurableSet_exists_tendsto (fun n => hmeas n (D.test j)))
+  have hex (ω : Ω) : ∃ η : Measure X, η.Regular ∧
+      ∀ f : C_c(X,ℝ), Tendsto
+        (fun n => if ω ∈ G then ∫ x, f x ∂μ n ω else 0) atTop (𝓝 (∫ x, f x ∂η)) := by
+    by_cases hω : ω ∈ G
+    · let (n : ℕ) : IsFiniteMeasureOnCompacts (μ n ω) := hfinite n ω
+      obtain ⟨η,hη,_,hl⟩ := D.exists_radon_limit (fun n => μ n ω) hω
+      exact ⟨η,hη,fun f => by simpa only [ite_eq_left hω] using hl f⟩
+    · refine ⟨0,inferInstance,fun f => ?_⟩
+      simpa only [ite_eq_right hω,integral_zero_measure] using
+        (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
+  choose η hηr hηlim using hex
+  have hηm : Measurable[m] η := by
+    let : MeasurableSpace Ω := m
+    apply measurable_radon_of_tests η hηr
+    intro f
+    apply measurable_of_tendsto_metrizable
+      (f := fun n ω => if ω ∈ G then ∫ x, f x ∂μ n ω else 0)
+    · intro n
+      exact (hmeas n f).ite hG measurable_const
+    · exact tendsto_pi_nhds.mpr (fun ω => hηlim ω f)
+  refine ⟨η,hηm,fun ω => ?_,?_⟩
+  · let := hηr ω
+    exact ⟨hηr ω,inferInstance⟩
+  · intro ω ν hr hl
+    have hg : ω ∈ G := fun j => ⟨∫ x, D.test j x ∂ν,hl (D.test j)⟩
+    let := hηr ω
+    let := hr
+    apply radon_test_limit_unique (fun n => μ n ω) (η ω) ν
+    · intro f
+      simpa only [ite_eq_left hg] using hηlim ω f
+    · exact hl
+
+theorem exists_subsigma_radon_version
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (m : MeasurableSpace Ω) (P : @Measure Ω mΩ)
+    (μ : ℕ → Ω → Measure X) (ν : Ω → Measure X)
+    (hfinite : ∀ n ω, IsFiniteMeasureOnCompacts (μ n ω))
+    (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable[m] (fun ω => ∫ x, f x ∂μ n ω))
+    (hν : ∀ᵐ ω ∂P, (ν ω).Regular)
+    (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(X,ℝ),
+      Tendsto (fun n => ∫ x, f x ∂μ n ω) atTop (𝓝 (∫ x, f x ∂ν ω))) :
+    ∃ η : Ω → Measure X, Measurable[m] η ∧
+      (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ η =ᵐ[P] ν := by
+  obtain ⟨η,hm,hr,he⟩ := exists_subsigma_radon_version_identified m μ hfinite hmeas
+  refine ⟨η,hm,hr,?_⟩
+  filter_upwards [hν,hlim] with ω hν hlim
+  exact he ω (ν ω) hν hlim
+
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/TargetDiscreteMassRecovery.lean b/lean/OAI/Probability/SLE/Mass/TargetDiscreteMassRecovery.lean
index 052ebf993a04278ea850ca62555bca7af12d7379..ff6199ba0d6d6d52371107d0bc9d562362efe211 100644
--- a/lean/OAI/Probability/SLE/Mass/TargetDiscreteMassRecovery.lean
+++ b/lean/OAI/Probability/SLE/Mass/TargetDiscreteMassRecovery.lean
@@ -1,71 +1,71 @@
-import OAI.Probability.SLE.Mass.MassCompensatorRecovery
-import OAI.Probability.SLE.Gauge.MomentCoefficientConvergence
-
-namespace OAI
-
-/-! The exact finite sum in the prescribed coefficient formula agrees
-with the stopped predictable compensator once its potential increments
-have been identified. The endpoint is allowed to be infinite. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma compensator_before_extended_sum (n : ℕ) (T : ℝ≥0∞) (q : ℕ → ℝ) :
-    (∑ i ∈ Finset.range (n*2^n),
-      if (compensatorMeshTime n (i+1):ℝ≥0∞) ≤ T then q i else 0) =
-      ∑ i ∈ Finset.range (compensatorStopIndex n T), q i := by
-  by_cases hT : T=∞
-  · simp [hT,compensatorStopIndex]
-  · have he : T=(T.toNNReal:ℝ≥0∞) := (ENNReal.coe_toNNReal hT).symm
-    conv_lhs => rw [he]
-    simp only [ENNReal.coe_le_coe]
-    rw [compensator_before_sum,compensatorStopIndex,ite_eq_right hT]
-
-lemma target_mesh_time (n i : ℕ) :
-    ENNReal.ofReal (((i:ℝ)+1)*Target.mesh n) =
-      (compensatorMeshTime n (i+1):ℝ≥0∞) := by
-  have he : (compensatorMeshTime n (i+1):ℝ)=((i:ℝ)+1)*Target.mesh n := by
-    simp [compensatorMeshTime,Target.mesh]
-  rw [← he,ENNReal.ofReal_coe_nnreal]
-
-lemma target_discreteMass_eq_sum (κ θ : ℝ) (W : Target.WienerLaw)
-    (b : Target.Driver) (ℓ : ℝ) (n : ℕ) :
-    Target.discreteMass κ θ W b ℓ n =
-      ∑ i ∈ Finset.range (compensatorStopIndex n (Target.levelTime κ θ b ℓ)),
-        Target.potentialDifference κ θ W b n i := by
-  unfold Target.discreteMass
-  simp_rw [target_mesh_time]
-  exact compensator_before_extended_sum n _ _
-
-theorem target_discreteMass_tendstoInMeasure
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ θ : ℝ) (W : Target.WienerLaw) (b : Ω → Target.Driver) (ℓ : ℝ)
-    (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ)
-    (hA : ∀ t, Measurable (A t))
-    (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i)))
-    (hL : MemLp L 2 P)
-    (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧
-      Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧
-      ∀ t, A t ω ≤ L ω)
-    (hT : Measurable (fun ω => Target.levelTime κ θ (b ω) ℓ))
-    (hq : ∀ n i, (fun ω => Target.potentialDifference κ θ W (b ω) n i) =ᵐ[P]
-      condExp (ℱ n i) P (meshMassIncrement A n i)) :
-    TendstoInMeasure P (fun n ω => Target.discreteMass κ θ W (b ω) ℓ n) atTop
-      (fun ω => massAtExtendedTime (fun t => A t ω) (L ω)
-        (Target.levelTime κ θ (b ω) ℓ)) := by
-  have h := meshMassCompensator_stopped_tendstoInMeasure P ℱ A L hA hgrid hL hp
-    (fun ω => Target.levelTime κ θ (b ω) ℓ) hT
-  apply h.congr _ (Filter.EventuallyEq.refl _ _)
-  intro n
-  filter_upwards [ae_all_iff.mpr (hq n)] with ω hω
-  rw [target_discreteMass_eq_sum]
-  exact Finset.sum_congr rfl (fun i _ => (hω i).symm)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.MassCompensatorRecovery
+import OAI.Probability.SLE.Gauge.MomentCoefficientConvergence
+
+namespace OAI
+
+/-! The exact finite sum in the prescribed coefficient formula agrees
+with the stopped predictable compensator once its potential increments
+have been identified. The endpoint is allowed to be infinite. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma compensator_before_extended_sum (n : ℕ) (T : ℝ≥0∞) (q : ℕ → ℝ) :
+    (∑ i ∈ Finset.range (n*2^n),
+      if (compensatorMeshTime n (i+1):ℝ≥0∞) ≤ T then q i else 0) =
+      ∑ i ∈ Finset.range (compensatorStopIndex n T), q i := by
+  by_cases hT : T=∞
+  · simp [hT,compensatorStopIndex]
+  · have he : T=(T.toNNReal:ℝ≥0∞) := (ENNReal.coe_toNNReal hT).symm
+    conv_lhs => rw [he]
+    simp only [ENNReal.coe_le_coe]
+    rw [compensator_before_sum,compensatorStopIndex,ite_eq_right hT]
+
+lemma target_mesh_time (n i : ℕ) :
+    ENNReal.ofReal (((i:ℝ)+1)*Target.mesh n) =
+      (compensatorMeshTime n (i+1):ℝ≥0∞) := by
+  have he : (compensatorMeshTime n (i+1):ℝ)=((i:ℝ)+1)*Target.mesh n := by
+    simp [compensatorMeshTime,Target.mesh]
+  rw [← he,ENNReal.ofReal_coe_nnreal]
+
+lemma target_discreteMass_eq_sum (κ θ : ℝ) (W : Target.WienerLaw)
+    (b : Target.Driver) (ℓ : ℝ) (n : ℕ) :
+    Target.discreteMass κ θ W b ℓ n =
+      ∑ i ∈ Finset.range (compensatorStopIndex n (Target.levelTime κ θ b ℓ)),
+        Target.potentialDifference κ θ W b n i := by
+  unfold Target.discreteMass
+  simp_rw [target_mesh_time]
+  exact compensator_before_extended_sum n _ _
+
+theorem target_discreteMass_tendstoInMeasure
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ θ : ℝ) (W : Target.WienerLaw) (b : Ω → Target.Driver) (ℓ : ℝ)
+    (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ)
+    (hA : ∀ t, Measurable (A t))
+    (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i)))
+    (hL : MemLp L 2 P)
+    (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧
+      Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧
+      ∀ t, A t ω ≤ L ω)
+    (hT : Measurable (fun ω => Target.levelTime κ θ (b ω) ℓ))
+    (hq : ∀ n i, (fun ω => Target.potentialDifference κ θ W (b ω) n i) =ᵐ[P]
+      condExp (ℱ n i) P (meshMassIncrement A n i)) :
+    TendstoInMeasure P (fun n ω => Target.discreteMass κ θ W (b ω) ℓ n) atTop
+      (fun ω => massAtExtendedTime (fun t => A t ω) (L ω)
+        (Target.levelTime κ θ (b ω) ℓ)) := by
+  have h := meshMassCompensator_stopped_tendstoInMeasure P ℱ A L hA hgrid hL hp
+    (fun ω => Target.levelTime κ θ (b ω) ℓ) hT
+  apply h.congr _ (Filter.EventuallyEq.refl _ _)
+  intro n
+  filter_upwards [ae_all_iff.mpr (hq n)] with ω hω
+  rw [target_discreteMass_eq_sum]
+  exact Finset.sum_congr rfl (fun i _ => (hω i).symm)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/TerminalMassTestLimit.lean b/lean/OAI/Probability/SLE/Mass/TerminalMassTestLimit.lean
index 52cf03d05104dac38dfab235162341e0b579ae3b..b601410f3aae9fb4c0684bfe2fd6a5c9fbe44c88 100644
--- a/lean/OAI/Probability/SLE/Mass/TerminalMassTestLimit.lean
+++ b/lean/OAI/Probability/SLE/Mass/TerminalMassTestLimit.lean
@@ -1,57 +1,57 @@
-import OAI.Probability.SLE.Mass.CanonicalMassExhaustion
-
-namespace OAI
-
-/-! Local exhaustion implies convergence of every compact continuous test.
-This provides the terminal value of the fixed raw driver functional. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped NNReal ENNReal Topology CompactlySupported
-
-namespace SLEExactGauge
-
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-lemma compact_test_norm_le_mass (μ : Measure upperHalfPlane)
-    [IsFiniteMeasureOnCompacts μ] (f : C_c(upperHalfPlane,ℝ)) :
-    ‖∫ z, f z ∂μ‖ ≤ ‖f.toBoundedContinuousFunction‖ * (μ (tsupport f)).toReal := by
-  let : IsFiniteMeasure (μ.restrict (tsupport f)) :=
-    ⟨by simpa only [Measure.restrict_apply_univ] using f.hasCompactSupport.measure_lt_top (μ := μ)⟩
-  have he := setIntegral_eq_integral_of_forall_compl_eq_zero (μ := μ)
-    (s := tsupport f) (fun z hz => image_eq_zero_of_notMem_tsupport hz)
-  rw [← he]
-  have hb : ∀ᵐ z ∂μ.restrict (tsupport f), ‖f z‖ ≤ ‖f.toBoundedContinuousFunction‖ :=
-    ae_of_all _ (fun z => f.toBoundedContinuousFunction.norm_coe_le_norm z)
-  have h := norm_integral_le_of_norm_le_const hb
-  simpa only [Measure.real,Measure.restrict_apply_univ] using h
-
-theorem terminal_mass_test_tendsto
-    (ν : Measure upperHalfPlane) [ν.Regular]
-    (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular)
-    (hdom : ∀ t, N t ≤ ν)
-    (hlim : ∀ K : Set upperHalfPlane, IsCompact K →
-      Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0))
-    (f : C_c(upperHalfPlane,ℝ)) :
-    Tendsto (fun t : ℝ≥0 => ∫ z, f z ∂N t) atTop (𝓝 (∫ z, f z ∂ν)) := by
-  let (t : ℝ≥0) : IsFiniteMeasureOnCompacts (ν-N t) :=
-    ⟨fun K hK => lt_of_le_of_lt (Measure.sub_le K) hK.measure_lt_top⟩
-  have hrem : Tendsto (fun t : ℝ≥0 => ∫ z, f z ∂(ν-N t)) atTop (𝓝 0) := by
-    apply squeeze_zero_norm (fun t => compact_test_norm_le_mass (ν-N t) f)
-    have hh := (ENNReal.continuousAt_toReal ENNReal.zero_ne_top).tendsto.comp
-      (hlim (tsupport f) f.hasCompactSupport)
-    simpa only [Function.comp_def,ENNReal.toReal_zero,mul_zero] using
-      (tendsto_const_nhds (x := ‖f.toBoundedContinuousFunction‖)).mul hh
-  have he (t : ℝ≥0) : (∫ z, f z ∂ν)-(∫ z, f z ∂(ν-N t)) = ∫ z, f z ∂N t := by
-    let := hNr t
-    have : (ν-N t).Regular := inferInstance
-    have hsum := integral_add_measure (μ := ν-N t) (ν := N t) f.integrable f.integrable
-    rw [sigmaFinite_sub_add_cancel_of_le (hdom t)] at hsum
-    linarith
-  simpa only [he,sub_zero] using (tendsto_const_nhds (x := ∫ z, f z ∂ν)).sub hrem
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CanonicalMassExhaustion
+
+namespace OAI
+
+/-! Local exhaustion implies convergence of every compact continuous test.
+This provides the terminal value of the fixed raw driver functional. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped NNReal ENNReal Topology CompactlySupported
+
+namespace SLEExactGauge
+
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+lemma compact_test_norm_le_mass (μ : Measure upperHalfPlane)
+    [IsFiniteMeasureOnCompacts μ] (f : C_c(upperHalfPlane,ℝ)) :
+    ‖∫ z, f z ∂μ‖ ≤ ‖f.toBoundedContinuousFunction‖ * (μ (tsupport f)).toReal := by
+  let : IsFiniteMeasure (μ.restrict (tsupport f)) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using f.hasCompactSupport.measure_lt_top (μ := μ)⟩
+  have he := setIntegral_eq_integral_of_forall_compl_eq_zero (μ := μ)
+    (s := tsupport f) (fun z hz => image_eq_zero_of_notMem_tsupport hz)
+  rw [← he]
+  have hb : ∀ᵐ z ∂μ.restrict (tsupport f), ‖f z‖ ≤ ‖f.toBoundedContinuousFunction‖ :=
+    ae_of_all _ (fun z => f.toBoundedContinuousFunction.norm_coe_le_norm z)
+  have h := norm_integral_le_of_norm_le_const hb
+  simpa only [Measure.real,Measure.restrict_apply_univ] using h
+
+theorem terminal_mass_test_tendsto
+    (ν : Measure upperHalfPlane) [ν.Regular]
+    (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular)
+    (hdom : ∀ t, N t ≤ ν)
+    (hlim : ∀ K : Set upperHalfPlane, IsCompact K →
+      Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0))
+    (f : C_c(upperHalfPlane,ℝ)) :
+    Tendsto (fun t : ℝ≥0 => ∫ z, f z ∂N t) atTop (𝓝 (∫ z, f z ∂ν)) := by
+  let (t : ℝ≥0) : IsFiniteMeasureOnCompacts (ν-N t) :=
+    ⟨fun K hK => lt_of_le_of_lt (Measure.sub_le K) hK.measure_lt_top⟩
+  have hrem : Tendsto (fun t : ℝ≥0 => ∫ z, f z ∂(ν-N t)) atTop (𝓝 0) := by
+    apply squeeze_zero_norm (fun t => compact_test_norm_le_mass (ν-N t) f)
+    have hh := (ENNReal.continuousAt_toReal ENNReal.zero_ne_top).tendsto.comp
+      (hlim (tsupport f) f.hasCompactSupport)
+    simpa only [Function.comp_def,ENNReal.toReal_zero,mul_zero] using
+      (tendsto_const_nhds (x := ‖f.toBoundedContinuousFunction‖)).mul hh
+  have he (t : ℝ≥0) : (∫ z, f z ∂ν)-(∫ z, f z ∂(ν-N t)) = ∫ z, f z ∂N t := by
+    let := hNr t
+    have : (ν-N t).Regular := inferInstance
+    have hsum := integral_add_measure (μ := ν-N t) (ν := N t) f.integrable f.integrable
+    rw [sigmaFinite_sub_add_cancel_of_le (hdom t)] at hsum
+    linarith
+  simpa only [he,sub_zero] using (tendsto_const_nhds (x := ∫ z, f z ∂ν)).sub hrem
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/UniformCoverLevels.lean b/lean/OAI/Probability/SLE/Mass/UniformCoverLevels.lean
index dd7799176916c5b29c9a73d75e68f2d7144ff8f0..0e64ab80a1d4a99d87fe9e87ddf36be70f0d07d5 100644
--- a/lean/OAI/Probability/SLE/Mass/UniformCoverLevels.lean
+++ b/lean/OAI/Probability/SLE/Mass/UniformCoverLevels.lean
@@ -1,118 +1,118 @@
-import OAI.Probability.SLE.Gauge.CoverRadiusLevels
-import OAI.Probability.SLE.Gauge.CoverGrid
-
-namespace OAI
-
-/-! Uniform clock geometry over the moving grid centers retained above a
-fixed positive height. The terminal visit level is separated from every
-test endpoint by a full clock unit. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def coverVisitLevel (z : ℂ) (T : ℝ) (n : ℕ) : ℝ :=
-  Real.log (2 * z.im) + T * (n : ℝ) - Real.log (512 * Real.pi)
-
-/-- One deterministic cutoff works for every point above the retained height
-and every test index, including the endpoint after the last test. -/
-theorem eventually_uniform_cover_levels {T η : ℝ} (hT : 0 < T) (hη : 0 < η) :
-    ∀ᶠ n : ℕ in atTop, ∀ z : ℂ, η / 2 ≤ z.im →
-      (∀ j : ℕ, n / 3 ≤ j →
-        coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) ∧
-      coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n ∧
-      4 * coverRadius T n ≤ z.im := by
-  obtain ⟨N, hN⟩ := exists_nat_gt (-Real.log η / T)
-  have hNT : -Real.log η < T * (N : ℝ) := by
-    simpa only [mul_comm] using (div_lt_iff₀ hT).mp hN
-  have hlarge : ∀ᶠ n : ℕ in atTop,
-      3 * (Real.log (512 * Real.pi) + 1 + T) / T ≤ (n : ℝ) :=
-    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
-  have hlim : Tendsto (fun n => 4 * coverRadius T n) atTop (𝓝 0) := by
-    simpa only [mul_zero] using (coverRadius_tendsto_zero hT).const_mul 4
-  have hsmall : ∀ᶠ n : ℕ in atTop, 4 * coverRadius T n < η / 2 :=
-    hlim.eventually (gt_mem_nhds (half_pos hη))
-  filter_upwards [eventually_ge_atTop (3 * N), hlarge, hsmall] with n hn hnlarge hnsmall
-  intro z hz
-  have hlog : Real.log η ≤ Real.log (2 * z.im) :=
-    Real.log_le_log hη (by linarith)
-  have hlevels : ∀ j : ℕ, n / 3 ≤ j →
-      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ) := by
-    intro j hj
-    have hjN : N ≤ j := by omega
-    have hNj : T * (N : ℝ) ≤ T * (j : ℝ) :=
-      mul_le_mul_of_nonneg_left (by exact_mod_cast hjN) hT.le
-    exact max_eq_right (by linarith)
-  refine ⟨hlevels, ?_, hnsmall.le.trans hz⟩
-  rw [hlevels (2 * n / 3 + 1) (by omega)]
-  have hlast : 3 * (2 * n / 3 + 1) ≤ 2 * n + 3 := by omega
-  have hlastR : (3 : ℝ) * (2 * n / 3 + 1 : ℕ) ≤ 2 * n + 3 := by exact_mod_cast hlast
-  have hnT := (div_le_iff₀ hT).mp hnlarge
-  unfold coverVisitLevel
-  nlinarith
-
-lemma coverVisitLevel_nonneg_of_gap (z : ℂ) (T : ℝ) (n : ℕ)
-    (hgap : coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n) :
-    0 ≤ coverVisitLevel z T n := by
-  have h0 := coverLogLevel_nonneg z T (2 * n / 3 + 1)
-  linarith
-
-/-- The late visit level is the fixed multiple `512π r_n` of the fine mesh,
-independently of the varying center's height. -/
-lemma coverVisitLevel_radius (z : ℂ) (hz : 0 < z.im) (T : ℝ) (n : ℕ) :
-    (2 * z.im) * Real.exp (-coverVisitLevel z T n) =
-      (512 * Real.pi) * coverRadius T n := by
-  have hz2 : 0 < 2 * z.im := by positivity
-  have hM : 0 < 512 * Real.pi := by positivity
-  have he : -coverVisitLevel z T n = Real.log (512 * Real.pi) +
-      (-Real.log (2 * z.im) + -T * (n : ℝ)) := by
-    unfold coverVisitLevel
-    ring
-  unfold coverRadius
-  rw [he, Real.exp_add, Real.exp_add,
-    Real.exp_log hM, Real.exp_neg, Real.exp_log hz2]
-  field_simp [hz2.ne']
-
-/-- The exact test window inherits the common late-grid cutoff. -/
-theorem eventually_uniform_cover_window_levels {T η : ℝ} (hT : 0 < T) (hη : 0 < η) :
-    ∀ᶠ n : ℕ in atTop, ∀ z : ℂ, η / 2 ≤ z.im →
-      (∀ j ∈ coverTestWindow n,
-        coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) ∧
-      coverLogLevel z T (2 * n / 3 + 1) =
-        Real.log (2 * z.im) + T * ((2 * n / 3 + 1 : ℕ) : ℝ) ∧
-      coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n ∧
-      4 * coverRadius T n ≤ z.im := by
-  filter_upwards [eventually_uniform_cover_levels hT hη] with n hn
-  intro z hz
-  obtain ⟨hlevel, hgap, hsmall⟩ := hn z hz
-  exact ⟨fun j hj => hlevel j (Finset.mem_Icc.mp hj).1,
-    hlevel _ (by omega), hgap, hsmall⟩
-
-/-- In particular the changing retained grid centers satisfy all clock and
-visit geometry simultaneously; no center-dependent cutoff is required. -/
-theorem eventually_retained_grid_cover_levels {R T η : ℝ} {K : Set ℂ}
-    (hT : 0 < T) (hη : 0 < η) (hK : K ⊆ Metric.closedBall 0 R)
-    (hheight : ∀ z ∈ K, η ≤ z.im) :
-    ∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
-      let z := coverGridCenter (coverRadius T n) ij
-      z ∈ upperHalfPlane ∧
-      (∀ j ∈ coverTestWindow n,
-        coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) ∧
-      coverLogLevel z T (2 * n / 3 + 1) =
-        Real.log (2 * z.im) + T * ((2 * n / 3 + 1 : ℕ) : ℝ) ∧
-      coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n ∧
-      4 * coverRadius T n ≤ z.im := by
-  have hsmall := (coverRadius_tendsto_zero hT).eventually (gt_mem_nhds (show 0 < η / 4 by positivity))
-  filter_upwards [eventually_uniform_cover_window_levels hT hη, hsmall] with n hn hnr
-  intro ij hij
-  have hcenter := (retainedCoverGrid_center_bounds hK hheight hij).2
-  have hz : η / 2 ≤ (coverGridCenter (coverRadius T n) ij).im := by linarith
-  exact ⟨lt_of_lt_of_le (half_pos hη) hz, hn _ hz⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.CoverRadiusLevels
+import OAI.Probability.SLE.Gauge.CoverGrid
+
+namespace OAI
+
+/-! Uniform clock geometry over the moving grid centers retained above a
+fixed positive height. The terminal visit level is separated from every
+test endpoint by a full clock unit. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def coverVisitLevel (z : ℂ) (T : ℝ) (n : ℕ) : ℝ :=
+  Real.log (2 * z.im) + T * (n : ℝ) - Real.log (512 * Real.pi)
+
+/-- One deterministic cutoff works for every point above the retained height
+and every test index, including the endpoint after the last test. -/
+theorem eventually_uniform_cover_levels {T η : ℝ} (hT : 0 < T) (hη : 0 < η) :
+    ∀ᶠ n : ℕ in atTop, ∀ z : ℂ, η / 2 ≤ z.im →
+      (∀ j : ℕ, n / 3 ≤ j →
+        coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) ∧
+      coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n ∧
+      4 * coverRadius T n ≤ z.im := by
+  obtain ⟨N, hN⟩ := exists_nat_gt (-Real.log η / T)
+  have hNT : -Real.log η < T * (N : ℝ) := by
+    simpa only [mul_comm] using (div_lt_iff₀ hT).mp hN
+  have hlarge : ∀ᶠ n : ℕ in atTop,
+      3 * (Real.log (512 * Real.pi) + 1 + T) / T ≤ (n : ℝ) :=
+    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
+  have hlim : Tendsto (fun n => 4 * coverRadius T n) atTop (𝓝 0) := by
+    simpa only [mul_zero] using (coverRadius_tendsto_zero hT).const_mul 4
+  have hsmall : ∀ᶠ n : ℕ in atTop, 4 * coverRadius T n < η / 2 :=
+    hlim.eventually (gt_mem_nhds (half_pos hη))
+  filter_upwards [eventually_ge_atTop (3 * N), hlarge, hsmall] with n hn hnlarge hnsmall
+  intro z hz
+  have hlog : Real.log η ≤ Real.log (2 * z.im) :=
+    Real.log_le_log hη (by linarith)
+  have hlevels : ∀ j : ℕ, n / 3 ≤ j →
+      coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ) := by
+    intro j hj
+    have hjN : N ≤ j := by omega
+    have hNj : T * (N : ℝ) ≤ T * (j : ℝ) :=
+      mul_le_mul_of_nonneg_left (by exact_mod_cast hjN) hT.le
+    exact max_eq_right (by linarith)
+  refine ⟨hlevels, ?_, hnsmall.le.trans hz⟩
+  rw [hlevels (2 * n / 3 + 1) (by omega)]
+  have hlast : 3 * (2 * n / 3 + 1) ≤ 2 * n + 3 := by omega
+  have hlastR : (3 : ℝ) * (2 * n / 3 + 1 : ℕ) ≤ 2 * n + 3 := by exact_mod_cast hlast
+  have hnT := (div_le_iff₀ hT).mp hnlarge
+  unfold coverVisitLevel
+  nlinarith
+
+lemma coverVisitLevel_nonneg_of_gap (z : ℂ) (T : ℝ) (n : ℕ)
+    (hgap : coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n) :
+    0 ≤ coverVisitLevel z T n := by
+  have h0 := coverLogLevel_nonneg z T (2 * n / 3 + 1)
+  linarith
+
+/-- The late visit level is the fixed multiple `512π r_n` of the fine mesh,
+independently of the varying center's height. -/
+lemma coverVisitLevel_radius (z : ℂ) (hz : 0 < z.im) (T : ℝ) (n : ℕ) :
+    (2 * z.im) * Real.exp (-coverVisitLevel z T n) =
+      (512 * Real.pi) * coverRadius T n := by
+  have hz2 : 0 < 2 * z.im := by positivity
+  have hM : 0 < 512 * Real.pi := by positivity
+  have he : -coverVisitLevel z T n = Real.log (512 * Real.pi) +
+      (-Real.log (2 * z.im) + -T * (n : ℝ)) := by
+    unfold coverVisitLevel
+    ring
+  unfold coverRadius
+  rw [he, Real.exp_add, Real.exp_add,
+    Real.exp_log hM, Real.exp_neg, Real.exp_log hz2]
+  field_simp [hz2.ne']
+
+/-- The exact test window inherits the common late-grid cutoff. -/
+theorem eventually_uniform_cover_window_levels {T η : ℝ} (hT : 0 < T) (hη : 0 < η) :
+    ∀ᶠ n : ℕ in atTop, ∀ z : ℂ, η / 2 ≤ z.im →
+      (∀ j ∈ coverTestWindow n,
+        coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) ∧
+      coverLogLevel z T (2 * n / 3 + 1) =
+        Real.log (2 * z.im) + T * ((2 * n / 3 + 1 : ℕ) : ℝ) ∧
+      coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n ∧
+      4 * coverRadius T n ≤ z.im := by
+  filter_upwards [eventually_uniform_cover_levels hT hη] with n hn
+  intro z hz
+  obtain ⟨hlevel, hgap, hsmall⟩ := hn z hz
+  exact ⟨fun j hj => hlevel j (Finset.mem_Icc.mp hj).1,
+    hlevel _ (by omega), hgap, hsmall⟩
+
+/-- In particular the changing retained grid centers satisfy all clock and
+visit geometry simultaneously; no center-dependent cutoff is required. -/
+theorem eventually_retained_grid_cover_levels {R T η : ℝ} {K : Set ℂ}
+    (hT : 0 < T) (hη : 0 < η) (hK : K ⊆ Metric.closedBall 0 R)
+    (hheight : ∀ z ∈ K, η ≤ z.im) :
+    ∀ᶠ n : ℕ in atTop, ∀ ij ∈ retainedCoverGridIndex R (coverRadius T n) K,
+      let z := coverGridCenter (coverRadius T n) ij
+      z ∈ upperHalfPlane ∧
+      (∀ j ∈ coverTestWindow n,
+        coverLogLevel z T j = Real.log (2 * z.im) + T * (j : ℝ)) ∧
+      coverLogLevel z T (2 * n / 3 + 1) =
+        Real.log (2 * z.im) + T * ((2 * n / 3 + 1 : ℕ) : ℝ) ∧
+      coverLogLevel z T (2 * n / 3 + 1) + 1 ≤ coverVisitLevel z T n ∧
+      4 * coverRadius T n ≤ z.im := by
+  have hsmall := (coverRadius_tendsto_zero hT).eventually (gt_mem_nhds (show 0 < η / 4 by positivity))
+  filter_upwards [eventually_uniform_cover_window_levels hT hη, hsmall] with n hn hnr
+  intro ij hij
+  have hcenter := (retainedCoverGrid_center_bounds hK hheight hij).2
+  have hz : η / 2 ≤ (coverGridCenter (coverRadius T n) ij).im := by linarith
+  exact ⟨lt_of_lt_of_le (half_pos hη) hz, hn _ hz⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/VerticalDisplacementMass.lean b/lean/OAI/Probability/SLE/Mass/VerticalDisplacementMass.lean
index fe80f0e18f8a637fdde5f7652086f4290bf4de9d..afbecccc05b7516142e10a2c1eb6d085152f6307 100644
--- a/lean/OAI/Probability/SLE/Mass/VerticalDisplacementMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/VerticalDisplacementMass.lean
@@ -1,31 +1,31 @@
-import OAI.Probability.SLE.Loewner.HalfPlaneDisplacementBound
-
-namespace OAI
-
-/-! Reading the hydrodynamic mass from a vertical displacement bound. -/
-noncomputable section
-open Set Filter Complex
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma hydrodynamic_mass_le_of_displacement {f : ℂ → ℂ} {a b : ℝ}
-    (hcoef : Tendsto (fun z : ℂ => z*f z)
-      (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-a:ℂ)))
-    (hbound : ∀ z∈upperHalfPlane, ‖f z‖≤b/z.im) : a≤b := by
-  apply le_of_tendsto (hydrodynamic_vertical_limits hcoef).2
-  apply Eventually.of_forall
-  intro n
-  have hi := (Complex.im_le_norm (f (verticalInfinity n))).trans
-    (hbound _ (verticalInfinity_mem n))
-  have hy : 0<(n:ℝ)+1 := by positivity
-  have hb := (le_div_iff₀ hy).mp (show (f (verticalInfinity n)).im≤b/((n:ℝ)+1) from by
-    simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero]
-      using hi)
-  nlinarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.HalfPlaneDisplacementBound
+
+namespace OAI
+
+/-! Reading the hydrodynamic mass from a vertical displacement bound. -/
+noncomputable section
+open Set Filter Complex
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma hydrodynamic_mass_le_of_displacement {f : ℂ → ℂ} {a b : ℝ}
+    (hcoef : Tendsto (fun z : ℂ => z*f z)
+      (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-a:ℂ)))
+    (hbound : ∀ z∈upperHalfPlane, ‖f z‖≤b/z.im) : a≤b := by
+  apply le_of_tendsto (hydrodynamic_vertical_limits hcoef).2
+  apply Eventually.of_forall
+  intro n
+  have hi := (Complex.im_le_norm (f (verticalInfinity n))).trans
+    (hbound _ (verticalInfinity_mem n))
+  have hy : 0<(n:ℝ)+1 := by positivity
+  have hb := (le_div_iff₀ hy).mp (show (f (verticalInfinity n)).im≤b/((n:ℝ)+1) from by
+    simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero]
+      using hi)
+  nlinarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/WaypointHarmonicMeasure.lean b/lean/OAI/Probability/SLE/Mass/WaypointHarmonicMeasure.lean
index 0bc71e1ac8920510b3f916b833eea89941bf536a..03891a46b78f9f19010a5b26c0d94c9bb7c2df4a 100644
--- a/lean/OAI/Probability/SLE/Mass/WaypointHarmonicMeasure.lean
+++ b/lean/OAI/Probability/SLE/Mass/WaypointHarmonicMeasure.lean
@@ -1,135 +1,135 @@
-import OAI.Probability.SLE.Geometry.WaypointTubeGeometry
-import OAI.Probability.SLE.Geometry.CrosscutExitSides
-import OAI.Probability.SLE.Geometry.WaypointAngleNumerics
-
-namespace OAI
-
-/-! The actual two-sided physical route events bound the two half-plane
-exit arcs from below.  Uniform route support is proved elsewhere; only the
-generic published conformal Brownian exit law is parameterized here. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.RibbonCoordinates
-open Target CrosscutSides
-
-theorem finite_exit_event_measure_le_arc
-    (μ : Measure (Driver × Driver)) (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ)
-    (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ))
-    (hg : ContinuousOn g D) (E : Set (Driver × Driver)) {dir : ℝ}
-    (hdir : dir = -1 ∨ dir = 1)
-    (hexit : ∀ᵐ b ∂μ, ∀ τ : ℝ, UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ →
-      ∃ x : ℝ, x ≠ 0 ∧ Tendsto (fun t => g (physicalUnitBrownianPath w b t))
-        (𝓝[<] τ) (𝓝 (x : ℂ)))
-    (hgeom : ∀ b ∈ E, ∃ τ : ℝ,
-      UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ (1 / 4 : ℝ) < τ ∧
-      ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0)
-    (hstart : ∀ b ∈ E, 0 < dir * (h (g (physicalUnitBrownianPath w b (1 / 4)))).re) :
-    μ E ≤ μ (unitWindowExitArc D g w dir) := by
-  apply measure_mono_ae
-  filter_upwards [hexit] with b hb
-  intro he
-  obtain ⟨τ, hτ, haτ, hn⟩ := hgeom b he
-  obtain ⟨x, hx, hlim⟩ := hb τ hτ
-  refine ⟨τ, hτ, x, ?_, hlim⟩
-  have hs := hstart b he
-  rcases hdir with rfl | rfl
-  · have hxneg := negative_first_exit_belongs_to_arc D g h hreal hg
-      (physicalUnitBrownianPath w b) (physicalUnitBrownianPath_continuous w b)
-      hτ haτ hn (by nlinarith) x hx hlim
-    nlinarith
-  · have hxpos := positive_first_exit_belongs_to_arc D g h hreal hg
-      (physicalUnitBrownianPath w b) (physicalUnitBrownianPath_continuous w b)
-      hτ haτ hn (by simpa using hs) x hx hlim
-    simpa using hxpos
-
-theorem ribbon_tube_harmonic_bounds
-    (hopen : PublishedPlanarDiskOpenness) (hexitLaw : PublishedHalfPlaneConformalExitLaw)
-    (W : WienerLaw) (hW : IsWiener W) (K : ℝ≥0) (hK : 0 < (K : ℝ))
-    (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
-    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
-    {k ε x : ℝ} (hk : 1 ≤ k) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8)
-    (hx : |x| ≤ 1 / 16)
-    (hlower : ∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k,
-      dist u v ≤ K * dist (P u) (P v))
-    (γ xp : ℝ≥0 → ℂ) (hγc : Continuous γ) (T : ℝ≥0) (hT : 0 < T)
-    (hxp : ContinuousOn xp (Icc 0 T)) (hγ : EqOn γ (P ∘ xp) (Icc 0 T))
-    (hxp0 : (xp 0).re = 0) (hxpT : (xp T).re = k)
-    (hcoords : MapsTo xp (Icc 0 T) closedRibbon)
-    (hcore : ∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε)
-    (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k))
-    (g F : ℂ → ℂ)
-    (hg : DifferentiableOn ℂ g (survivingDomain γ T))
-    (hF : DifferentiableOn ℂ F upperHalfPlane)
-    (hgH : MapsTo g (survivingDomain γ T) upperHalfPlane)
-    (hFD : MapsTo F upperHalfPlane (survivingDomain γ T))
-    (hFg : ∀ z ∈ survivingDomain γ T, F (g z) = z)
-    (hgF : ∀ z ∈ upperHalfPlane, g (F z) = z)
-    (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ))
-    (hcut : ∀ z ∈ survivingDomain γ T, (h (g z)).re = 0 →
-      ∃ u : ℝ, 0 ≤ u ∧ P (guideCoordinate k x u) = z)
-    (hloc : ContinuousOn (fun z => h (g (P z)))
-      (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)))
-    (hloci : InjOn (fun z => h (g (P z)))
-      (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)))
-    (hlocz : ∀ z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4),
-      (h (g (P z))).re = 0 ↔ z.im = 0)
-    (q : ℝ≥0∞)
-    (hprob : ∀ side : ℝ, (side = -1 ∨ side = 1) →
-      q ≤ (W.prod W) {b | ∀ t ∈ Icc (0 : ℝ) 3,
-        dist (physicalUnitBrownianPath (P ((k + 1 / 2 : ℝ) : ℂ)) b t)
-          (P (waypointRoute k side t)) < (1 / 32 : ℝ) / K}) :
-    q ≤ ENNReal.ofReal ((g (P ((k + 1 / 2 : ℝ) : ℂ))).arg / Real.pi) ∧
-      q ≤ ENNReal.ofReal (1 - (g (P ((k + 1 / 2 : ℝ) : ℂ))).arg / Real.pi) := by
-  let D := survivingDomain γ T
-  let w := P ((k + 1 / 2 : ℝ) : ℂ)
-  let E := fun side : ℝ => {b | ∀ t ∈ Icc (0 : ℝ) 3,
-    dist (physicalUnitBrownianPath w b t) (P (waypointRoute k side t)) < (1 / 32 : ℝ) / K}
-  have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by
-    rintro z ⟨s, hs, rfl⟩
-    exact ⟨xp s, ⟨hcoords hs, (hcore s hs).1⟩, (hγ hs).symm⟩
-  have hwD : w ∈ D := by
-    apply futureRibbon_survives P hP hi (by linarith) hH γ T hpast hunbounded
-    exact ⟨((k + 1 / 2 : ℝ) : ℂ), ⟨by simp, by simp⟩, rfl⟩
-  obtain ⟨hae, hnegBound, hposBound⟩ := hexitLaw W hW D g F
-    (isOpen_survivingDomain hγc T) (fun z hz => hz.1.1) hg hF hgH hFD hFg hgF w hwD
-  obtain ⟨side, hside, hpos, hneg⟩ := physical_waypoint_tubes_opposite_sides hopen P hP hi
-    K hK hk hlower (fun z => h (g (P z))) (fun z => (h (g z)).re)
-    hloc hloci hlocz (fun _ _ => rfl)
-  have hgeom (side' : ℝ) (hside' : side' = -1 ∨ side' = 1) (b : Driver × Driver)
-      (hb : b ∈ E side') : ∃ τ : ℝ,
-      UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ (1 / 4 : ℝ) < τ ∧
-      ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0 := by
-    let ζ := physicalUnitBrownianPath w b
-    have hζ : Continuous ζ := physicalUnitBrownianPath_continuous w b
-    obtain ⟨hearly, havoid, hhit⟩ := physical_waypoint_tube_geometry hopen P hP hi hH K hK
-      hk hε hεsmall hx hside' hlower γ xp T hT hxp hγ hxp0 hxpT hcoords hcore hunbounded
-      ζ hζ hb
-    obtain ⟨τ, hτ, hout, hin⟩ := waypoint_first_exit_after_early_prefix γ hγc T ζ
-      hζ.continuousOn hearly hhit
-    refine ⟨τ, ⟨⟨by linarith [hτ.1], hτ.2⟩, hout, hin⟩, by linarith [hτ.1], ?_⟩
-    intro t ht he
-    obtain ⟨u, hu, heq⟩ := hcut (ζ t) (hin ⟨by linarith [ht.1], ht.2⟩) he
-    exact havoid t ⟨ht.1, ht.2.le.trans hτ.2⟩ u hu heq.symm
-  have hposProb : q ≤ (W.prod W) (unitWindowExitArc D g w 1) := by
-    refine (hprob side hside).trans (finite_exit_event_measure_le_arc (W.prod W) D g w h hreal
-      hg.continuousOn (E side) (Or.inr rfl) hae (hgeom side hside) ?_)
-    intro b hb
-    simpa only [one_mul] using hpos _ (hb _ (by norm_num))
-  have hsideNeg : -side = -1 ∨ -side = 1 := by
-    rcases hside with rfl | rfl <;> norm_num
-  have hnegProb : q ≤ (W.prod W) (unitWindowExitArc D g w (-1)) := by
-    refine (hprob (-side) hsideNeg).trans (finite_exit_event_measure_le_arc (W.prod W) D g w h hreal
-      hg.continuousOn (E (-side)) (Or.inl rfl) hae (hgeom (-side) hsideNeg) ?_)
-    intro b hb
-    have hh := hneg _ (hb _ (by norm_num))
-    nlinarith
-  exact ⟨hnegProb.trans hnegBound, hposProb.trans hposBound⟩
-
-end SLEExactGauge.RibbonCoordinates
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.WaypointTubeGeometry
+import OAI.Probability.SLE.Geometry.CrosscutExitSides
+import OAI.Probability.SLE.Geometry.WaypointAngleNumerics
+
+namespace OAI
+
+/-! The actual two-sided physical route events bound the two half-plane
+exit arcs from below.  Uniform route support is proved elsewhere; only the
+generic published conformal Brownian exit law is parameterized here. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.RibbonCoordinates
+open Target CrosscutSides
+
+theorem finite_exit_event_measure_le_arc
+    (μ : Measure (Driver × Driver)) (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ)
+    (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ))
+    (hg : ContinuousOn g D) (E : Set (Driver × Driver)) {dir : ℝ}
+    (hdir : dir = -1 ∨ dir = 1)
+    (hexit : ∀ᵐ b ∂μ, ∀ τ : ℝ, UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ →
+      ∃ x : ℝ, x ≠ 0 ∧ Tendsto (fun t => g (physicalUnitBrownianPath w b t))
+        (𝓝[<] τ) (𝓝 (x : ℂ)))
+    (hgeom : ∀ b ∈ E, ∃ τ : ℝ,
+      UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ (1 / 4 : ℝ) < τ ∧
+      ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0)
+    (hstart : ∀ b ∈ E, 0 < dir * (h (g (physicalUnitBrownianPath w b (1 / 4)))).re) :
+    μ E ≤ μ (unitWindowExitArc D g w dir) := by
+  apply measure_mono_ae
+  filter_upwards [hexit] with b hb
+  intro he
+  obtain ⟨τ, hτ, haτ, hn⟩ := hgeom b he
+  obtain ⟨x, hx, hlim⟩ := hb τ hτ
+  refine ⟨τ, hτ, x, ?_, hlim⟩
+  have hs := hstart b he
+  rcases hdir with rfl | rfl
+  · have hxneg := negative_first_exit_belongs_to_arc D g h hreal hg
+      (physicalUnitBrownianPath w b) (physicalUnitBrownianPath_continuous w b)
+      hτ haτ hn (by nlinarith) x hx hlim
+    nlinarith
+  · have hxpos := positive_first_exit_belongs_to_arc D g h hreal hg
+      (physicalUnitBrownianPath w b) (physicalUnitBrownianPath_continuous w b)
+      hτ haτ hn (by simpa using hs) x hx hlim
+    simpa using hxpos
+
+theorem ribbon_tube_harmonic_bounds
+    (hopen : PublishedPlanarDiskOpenness) (hexitLaw : PublishedHalfPlaneConformalExitLaw)
+    (W : WienerLaw) (hW : IsWiener W) (K : ℝ≥0) (hK : 0 < (K : ℝ))
+    (P : ℂ → ℂ) (hP : ContinuousOn P closedRibbon) (hi : InjOn P closedRibbon)
+    (hH : ∀ z ∈ closedRibbon, 0 < z.re → P z ∈ upperHalfPlane)
+    {k ε x : ℝ} (hk : 1 ≤ k) (hε : 0 < ε) (hεsmall : ε ≤ 1 / 8)
+    (hx : |x| ≤ 1 / 16)
+    (hlower : ∀ u ∈ crossingPatch k, ∀ v ∈ crossingPatch k,
+      dist u v ≤ K * dist (P u) (P v))
+    (γ xp : ℝ≥0 → ℂ) (hγc : Continuous γ) (T : ℝ≥0) (hT : 0 < T)
+    (hxp : ContinuousOn xp (Icc 0 T)) (hγ : EqOn γ (P ∘ xp) (Icc 0 T))
+    (hxp0 : (xp 0).re = 0) (hxpT : (xp T).re = k)
+    (hcoords : MapsTo xp (Icc 0 T) closedRibbon)
+    (hcore : ∀ t ∈ Icc 0 T, (xp t).re ≤ k ∧ |(xp t).im| ≤ ε)
+    (hunbounded : ¬ Bornology.IsBounded (P '' futureRibbon k))
+    (g F : ℂ → ℂ)
+    (hg : DifferentiableOn ℂ g (survivingDomain γ T))
+    (hF : DifferentiableOn ℂ F upperHalfPlane)
+    (hgH : MapsTo g (survivingDomain γ T) upperHalfPlane)
+    (hFD : MapsTo F upperHalfPlane (survivingDomain γ T))
+    (hFg : ∀ z ∈ survivingDomain γ T, F (g z) = z)
+    (hgF : ∀ z ∈ upperHalfPlane, g (F z) = z)
+    (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ))
+    (hcut : ∀ z ∈ survivingDomain γ T, (h (g z)).re = 0 →
+      ∃ u : ℝ, 0 ≤ u ∧ P (guideCoordinate k x u) = z)
+    (hloc : ContinuousOn (fun z => h (g (P z)))
+      (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)))
+    (hloci : InjOn (fun z => h (g (P z)))
+      (ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4)))
+    (hlocz : ∀ z ∈ ball ((k + 1 / 2 : ℝ) : ℂ) (1 / 4),
+      (h (g (P z))).re = 0 ↔ z.im = 0)
+    (q : ℝ≥0∞)
+    (hprob : ∀ side : ℝ, (side = -1 ∨ side = 1) →
+      q ≤ (W.prod W) {b | ∀ t ∈ Icc (0 : ℝ) 3,
+        dist (physicalUnitBrownianPath (P ((k + 1 / 2 : ℝ) : ℂ)) b t)
+          (P (waypointRoute k side t)) < (1 / 32 : ℝ) / K}) :
+    q ≤ ENNReal.ofReal ((g (P ((k + 1 / 2 : ℝ) : ℂ))).arg / Real.pi) ∧
+      q ≤ ENNReal.ofReal (1 - (g (P ((k + 1 / 2 : ℝ) : ℂ))).arg / Real.pi) := by
+  let D := survivingDomain γ T
+  let w := P ((k + 1 / 2 : ℝ) : ℂ)
+  let E := fun side : ℝ => {b | ∀ t ∈ Icc (0 : ℝ) 3,
+    dist (physicalUnitBrownianPath w b t) (P (waypointRoute k side t)) < (1 / 32 : ℝ) / K}
+  have hpast : γ '' Icc 0 T ⊆ P '' {z ∈ closedRibbon | z.re ≤ k} := by
+    rintro z ⟨s, hs, rfl⟩
+    exact ⟨xp s, ⟨hcoords hs, (hcore s hs).1⟩, (hγ hs).symm⟩
+  have hwD : w ∈ D := by
+    apply futureRibbon_survives P hP hi (by linarith) hH γ T hpast hunbounded
+    exact ⟨((k + 1 / 2 : ℝ) : ℂ), ⟨by simp, by simp⟩, rfl⟩
+  obtain ⟨hae, hnegBound, hposBound⟩ := hexitLaw W hW D g F
+    (isOpen_survivingDomain hγc T) (fun z hz => hz.1.1) hg hF hgH hFD hFg hgF w hwD
+  obtain ⟨side, hside, hpos, hneg⟩ := physical_waypoint_tubes_opposite_sides hopen P hP hi
+    K hK hk hlower (fun z => h (g (P z))) (fun z => (h (g z)).re)
+    hloc hloci hlocz (fun _ _ => rfl)
+  have hgeom (side' : ℝ) (hside' : side' = -1 ∨ side' = 1) (b : Driver × Driver)
+      (hb : b ∈ E side') : ∃ τ : ℝ,
+      UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ (1 / 4 : ℝ) < τ ∧
+      ∀ t ∈ Ico (1 / 4 : ℝ) τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0 := by
+    let ζ := physicalUnitBrownianPath w b
+    have hζ : Continuous ζ := physicalUnitBrownianPath_continuous w b
+    obtain ⟨hearly, havoid, hhit⟩ := physical_waypoint_tube_geometry hopen P hP hi hH K hK
+      hk hε hεsmall hx hside' hlower γ xp T hT hxp hγ hxp0 hxpT hcoords hcore hunbounded
+      ζ hζ hb
+    obtain ⟨τ, hτ, hout, hin⟩ := waypoint_first_exit_after_early_prefix γ hγc T ζ
+      hζ.continuousOn hearly hhit
+    refine ⟨τ, ⟨⟨by linarith [hτ.1], hτ.2⟩, hout, hin⟩, by linarith [hτ.1], ?_⟩
+    intro t ht he
+    obtain ⟨u, hu, heq⟩ := hcut (ζ t) (hin ⟨by linarith [ht.1], ht.2⟩) he
+    exact havoid t ⟨ht.1, ht.2.le.trans hτ.2⟩ u hu heq.symm
+  have hposProb : q ≤ (W.prod W) (unitWindowExitArc D g w 1) := by
+    refine (hprob side hside).trans (finite_exit_event_measure_le_arc (W.prod W) D g w h hreal
+      hg.continuousOn (E side) (Or.inr rfl) hae (hgeom side hside) ?_)
+    intro b hb
+    simpa only [one_mul] using hpos _ (hb _ (by norm_num))
+  have hsideNeg : -side = -1 ∨ -side = 1 := by
+    rcases hside with rfl | rfl <;> norm_num
+  have hnegProb : q ≤ (W.prod W) (unitWindowExitArc D g w (-1)) := by
+    refine (hprob (-side) hsideNeg).trans (finite_exit_event_measure_le_arc (W.prod W) D g w h hreal
+      hg.continuousOn (E (-side)) (Or.inl rfl) hae (hgeom (-side) hsideNeg) ?_)
+    intro b hb
+    have hh := hneg _ (hb _ (by norm_num))
+    nlinarith
+  exact ⟨hnegProb.trans hnegBound, hposProb.trans hposBound⟩
+
+end SLEExactGauge.RibbonCoordinates
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/WeightedCompactMass.lean b/lean/OAI/Probability/SLE/Mass/WeightedCompactMass.lean
index b7cee9250a2beebe09f710d088efd5fe9128e2bb..9eac64a08474e27ce03dc0f1de69323549d3ad23 100644
--- a/lean/OAI/Probability/SLE/Mass/WeightedCompactMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/WeightedCompactMass.lean
@@ -1,108 +1,108 @@
-import OAI.Probability.SLE.Mass.CanonicalMassContinuity
-import OAI.Probability.SLE.Geometry.ConformalRestartTransport
-
-namespace OAI
-
-/-! Bounded compact weights preserve the scalar continuity estimates
-used for the canonical disk mass. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma integral_weight_difference_le
-    {X : Type*} [MeasurableSpace X] (μ ν : Measure X)
-    [IsFiniteMeasure μ] [IsFiniteMeasure ν] (hμν : μ ≤ ν)
-    (f : X → ℝ) (hf : Integrable f ν) (M : ℝ)
-    (h0 : 0 ≤ᵐ[ν] f) (hM : ∀ᵐ x ∂ν, f x ≤ M) :
-    0 ≤ (∫ x, f x ∂ν)-(∫ x, f x ∂μ) ∧
-      (∫ x, f x ∂ν)-(∫ x, f x ∂μ) ≤ M*((ν univ).toReal-(μ univ).toReal) := by
-  let ξ := ν-μ
-  have hξν : ξ ≤ ν := Measure.sub_le
-  have hξ : Integrable f ξ := hf.mono_measure hξν
-  have hμ : Integrable f μ := hf.mono_measure hμν
-  have he : (∫ x, f x ∂ν) = (∫ x, f x ∂ξ)+(∫ x, f x ∂μ) := by
-    rw [← sigmaFinite_sub_add_cancel_of_le hμν]
-    exact integral_add_measure hξ hμ
-  have hn : 0 ≤ ∫ x, f x ∂ξ := integral_nonneg_of_ae (ae_mono hξν h0)
-  have hi : (∫ x, f x ∂ξ) ≤ ∫ _ : X, M ∂ξ :=
-    integral_mono_ae hξ (integrable_const M) (ae_mono hξν hM)
-  have hv : (ξ univ).toReal = (ν univ).toReal-(μ univ).toReal := by
-    rw [Measure.sub_apply MeasurableSet.univ hμν,
-      ENNReal.toReal_sub_of_le (hμν univ) (measure_ne_top ν univ)]
-  simp only [integral_const,smul_eq_mul,measureReal_def,hv] at hi
-  constructor
-  · linarith
-  · rw [mul_comm] at hi
-    linarith
-
-namespace Multipoint
-
-lemma compact_weight_bounded (w : C(upperHalfPlane,ℝ≥0))
-    (K : Set upperHalfPlane) (hK : IsCompact K) :
-    ∃ M : ℝ≥0, ∀ z ∈ K, w z ≤ M := by
-  obtain ⟨M,hM⟩ := hK.bddAbove_image w.continuous.continuousOn
-  exact ⟨M,fun z hz => hM (mem_image_of_mem _ hz)⟩
-
-lemma compact_weight_integrable (w : C(upperHalfPlane,ℝ≥0))
-    (K : Set upperHalfPlane) (hK : IsCompact K)
-    (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] :
-    IntegrableOn (fun z => (w z:ℝ)) K μ :=
-  (continuous_subtype_val.comp w.continuous).continuousOn.integrableOn_compact hK
-
-lemma compact_weight_difference_le
-    (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K)
-    (M : ℝ≥0) (hM : ∀ z ∈ K, w z ≤ M)
-    (μ ν : Measure upperHalfPlane)
-    [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] (hμν : μ ≤ ν) :
-    |(∫ z in K, (w z:ℝ) ∂ν)-(∫ z in K, (w z:ℝ) ∂μ)| ≤
-      (M:ℝ)*|(ν K).toReal-(μ K).toReal| := by
-  let : IsFiniteMeasure (μ.restrict K) := ⟨by simpa using hK.measure_lt_top (μ := μ)⟩
-  let : IsFiniteMeasure (ν.restrict K) := ⟨by simpa using hK.measure_lt_top (μ := ν)⟩
-  have h := integral_weight_difference_le (μ.restrict K) (ν.restrict K)
-    (Measure.restrict_mono le_rfl hμν) (fun z => (w z:ℝ))
-    (compact_weight_integrable w K hK ν) M
-    (ae_of_all _ (fun z => (w z).coe_nonneg))
-    ((ae_restrict_mem hK.measurableSet).mono (fun z hz => NNReal.coe_le_coe.mpr (hM z hz)))
-  have hv : (μ K).toReal ≤ (ν K).toReal :=
-    ENNReal.toReal_mono hK.measure_ne_top (hμν K)
-  simpa only [Measure.restrict_apply_univ,abs_of_nonneg h.1,
-    abs_of_nonneg (sub_nonneg.mpr hv)] using h.2
-
-theorem ContinuousMassPath.weighted_compact_continuous
-    {γ : ℝ≥0 → ℂ} {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane}
-    {N : ℝ≥0 → Measure upperHalfPlane} (hp : ContinuousMassPath γ μ ν N)
-    (hN : ∀ t, IsFiniteMeasureOnCompacts (N t))
-    (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) :
-    Continuous (fun t => ∫ z in K, (w z:ℝ) ∂N t) := by
-  obtain ⟨M,hM⟩ := compact_weight_bounded w K hK
-  apply continuous_iff_continuousAt.mpr
-  intro t₀
-  rw [Metric.continuousAt_iff]
-  intro ε hε
-  have hpos : 0 < ε/((M:ℝ)+1) := div_pos hε (by positivity)
-  obtain ⟨δ,hδ,hvar⟩ := hp.local_variation K hK t₀ _ hpos
-  refine ⟨δ,hδ,?_⟩
-  intro t ht
-  let := hN t
-  let := hN t₀
-  have ht' : dist (t:ℝ) (t₀:ℝ) < δ := ht
-  have hsmall := hvar t t₀ ht' (by simpa using hδ) K hK.measurableSet Subset.rfl
-  have hb : |(∫ z in K, (w z:ℝ) ∂N t)-(∫ z in K, (w z:ℝ) ∂N t₀)| ≤
-      (M:ℝ)*|((N t) K).toReal-((N t₀) K).toReal| := by
-    rcases le_total t t₀ with htt | htt
-    · have h := compact_weight_difference_le w K hK M hM (N t) (N t₀) (hp.monotone htt)
-      simpa only [abs_sub_comm] using h
-    · exact compact_weight_difference_le w K hK M hM (N t₀) (N t) (hp.monotone htt)
-  rw [Real.dist_eq]
-  have he : ε/((M:ℝ)+1)*((M:ℝ)+1)=ε := div_mul_cancel₀ _ (by positivity)
-  nlinarith [abs_nonneg (((N t) K).toReal-((N t₀) K).toReal),M.coe_nonneg]
-
-end Multipoint
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.CanonicalMassContinuity
+import OAI.Probability.SLE.Geometry.ConformalRestartTransport
+
+namespace OAI
+
+/-! Bounded compact weights preserve the scalar continuity estimates
+used for the canonical disk mass. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma integral_weight_difference_le
+    {X : Type*} [MeasurableSpace X] (μ ν : Measure X)
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν] (hμν : μ ≤ ν)
+    (f : X → ℝ) (hf : Integrable f ν) (M : ℝ)
+    (h0 : 0 ≤ᵐ[ν] f) (hM : ∀ᵐ x ∂ν, f x ≤ M) :
+    0 ≤ (∫ x, f x ∂ν)-(∫ x, f x ∂μ) ∧
+      (∫ x, f x ∂ν)-(∫ x, f x ∂μ) ≤ M*((ν univ).toReal-(μ univ).toReal) := by
+  let ξ := ν-μ
+  have hξν : ξ ≤ ν := Measure.sub_le
+  have hξ : Integrable f ξ := hf.mono_measure hξν
+  have hμ : Integrable f μ := hf.mono_measure hμν
+  have he : (∫ x, f x ∂ν) = (∫ x, f x ∂ξ)+(∫ x, f x ∂μ) := by
+    rw [← sigmaFinite_sub_add_cancel_of_le hμν]
+    exact integral_add_measure hξ hμ
+  have hn : 0 ≤ ∫ x, f x ∂ξ := integral_nonneg_of_ae (ae_mono hξν h0)
+  have hi : (∫ x, f x ∂ξ) ≤ ∫ _ : X, M ∂ξ :=
+    integral_mono_ae hξ (integrable_const M) (ae_mono hξν hM)
+  have hv : (ξ univ).toReal = (ν univ).toReal-(μ univ).toReal := by
+    rw [Measure.sub_apply MeasurableSet.univ hμν,
+      ENNReal.toReal_sub_of_le (hμν univ) (measure_ne_top ν univ)]
+  simp only [integral_const,smul_eq_mul,measureReal_def,hv] at hi
+  constructor
+  · linarith
+  · rw [mul_comm] at hi
+    linarith
+
+namespace Multipoint
+
+lemma compact_weight_bounded (w : C(upperHalfPlane,ℝ≥0))
+    (K : Set upperHalfPlane) (hK : IsCompact K) :
+    ∃ M : ℝ≥0, ∀ z ∈ K, w z ≤ M := by
+  obtain ⟨M,hM⟩ := hK.bddAbove_image w.continuous.continuousOn
+  exact ⟨M,fun z hz => hM (mem_image_of_mem _ hz)⟩
+
+lemma compact_weight_integrable (w : C(upperHalfPlane,ℝ≥0))
+    (K : Set upperHalfPlane) (hK : IsCompact K)
+    (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] :
+    IntegrableOn (fun z => (w z:ℝ)) K μ :=
+  (continuous_subtype_val.comp w.continuous).continuousOn.integrableOn_compact hK
+
+lemma compact_weight_difference_le
+    (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K)
+    (M : ℝ≥0) (hM : ∀ z ∈ K, w z ≤ M)
+    (μ ν : Measure upperHalfPlane)
+    [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] (hμν : μ ≤ ν) :
+    |(∫ z in K, (w z:ℝ) ∂ν)-(∫ z in K, (w z:ℝ) ∂μ)| ≤
+      (M:ℝ)*|(ν K).toReal-(μ K).toReal| := by
+  let : IsFiniteMeasure (μ.restrict K) := ⟨by simpa using hK.measure_lt_top (μ := μ)⟩
+  let : IsFiniteMeasure (ν.restrict K) := ⟨by simpa using hK.measure_lt_top (μ := ν)⟩
+  have h := integral_weight_difference_le (μ.restrict K) (ν.restrict K)
+    (Measure.restrict_mono le_rfl hμν) (fun z => (w z:ℝ))
+    (compact_weight_integrable w K hK ν) M
+    (ae_of_all _ (fun z => (w z).coe_nonneg))
+    ((ae_restrict_mem hK.measurableSet).mono (fun z hz => NNReal.coe_le_coe.mpr (hM z hz)))
+  have hv : (μ K).toReal ≤ (ν K).toReal :=
+    ENNReal.toReal_mono hK.measure_ne_top (hμν K)
+  simpa only [Measure.restrict_apply_univ,abs_of_nonneg h.1,
+    abs_of_nonneg (sub_nonneg.mpr hv)] using h.2
+
+theorem ContinuousMassPath.weighted_compact_continuous
+    {γ : ℝ≥0 → ℂ} {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane}
+    {N : ℝ≥0 → Measure upperHalfPlane} (hp : ContinuousMassPath γ μ ν N)
+    (hN : ∀ t, IsFiniteMeasureOnCompacts (N t))
+    (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) :
+    Continuous (fun t => ∫ z in K, (w z:ℝ) ∂N t) := by
+  obtain ⟨M,hM⟩ := compact_weight_bounded w K hK
+  apply continuous_iff_continuousAt.mpr
+  intro t₀
+  rw [Metric.continuousAt_iff]
+  intro ε hε
+  have hpos : 0 < ε/((M:ℝ)+1) := div_pos hε (by positivity)
+  obtain ⟨δ,hδ,hvar⟩ := hp.local_variation K hK t₀ _ hpos
+  refine ⟨δ,hδ,?_⟩
+  intro t ht
+  let := hN t
+  let := hN t₀
+  have ht' : dist (t:ℝ) (t₀:ℝ) < δ := ht
+  have hsmall := hvar t t₀ ht' (by simpa using hδ) K hK.measurableSet Subset.rfl
+  have hb : |(∫ z in K, (w z:ℝ) ∂N t)-(∫ z in K, (w z:ℝ) ∂N t₀)| ≤
+      (M:ℝ)*|((N t) K).toReal-((N t₀) K).toReal| := by
+    rcases le_total t t₀ with htt | htt
+    · have h := compact_weight_difference_le w K hK M hM (N t) (N t₀) (hp.monotone htt)
+      simpa only [abs_sub_comm] using h
+    · exact compact_weight_difference_le w K hK M hM (N t₀) (N t) (hp.monotone htt)
+  rw [Real.dist_eq]
+  have he : ε/((M:ℝ)+1)*((M:ℝ)+1)=ε := div_mul_cancel₀ _ (by positivity)
+  nlinarith [abs_nonneg (((N t) K).toReal-((N t₀) K).toReal),M.coe_nonneg]
+
+end Multipoint
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/WeightedImageRandomMass.lean b/lean/OAI/Probability/SLE/Mass/WeightedImageRandomMass.lean
index dd36f7cf7a8124e3d8c66e16144219635798f32f..7d644705b18429b3f2baa2d2160c272e044975b2 100644
--- a/lean/OAI/Probability/SLE/Mass/WeightedImageRandomMass.lean
+++ b/lean/OAI/Probability/SLE/Mass/WeightedImageRandomMass.lean
@@ -1,97 +1,97 @@
-import OAI.Probability.SLE.Mass.RestartRandomMassTesting
-
-namespace OAI
-
-/-! Measurable weighted images of locally finite random half-plane measures.
-The weight and image map can be chosen from the stopped past. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped ENNReal
-
-namespace SLEExactGauge
-
-def weightedImageMass {X : Type*} [MeasurableSpace X]
-    (J : upperHalfPlane → X) (w : upperHalfPlane → ℝ≥0∞)
-    (μ : Measure upperHalfPlane) : Measure X :=
-  (μ.withDensity w).map J
-
-lemma weightedImageMass_apply {X : Type*} [MeasurableSpace X]
-    (J : upperHalfPlane → X) (hJ : Measurable J)
-    (w : upperHalfPlane → ℝ≥0∞) (μ : Measure upperHalfPlane)
-    (A : Set X) (hA : MeasurableSet A) :
-    weightedImageMass J w μ A = ∫⁻ z, (J ⁻¹' A).indicator w z ∂μ := by
-  rw [weightedImageMass,Measure.map_apply hJ hA,withDensity_apply _ (hJ hA),
-    lintegral_indicator (hJ hA)]
-
-theorem weightedImageMass_measurable
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ)
-    (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω))
-    (J : Ω × upperHalfPlane → X) (hJ : Measurable J)
-    (w : Ω × upperHalfPlane → ℝ≥0∞) (hw : Measurable w) :
-    Measurable (fun ω => weightedImageMass (fun z => J (ω,z))
-      (fun z => w (ω,z)) (μ ω)) := by
-  apply Measure.measurable_of_measurable_coe
-  intro A hA
-  have he (ω : Ω) := weightedImageMass_apply (fun z => J (ω,z))
-    (hJ.comp (measurable_const.prodMk measurable_id)) (fun z => w (ω,z)) (μ ω) A hA
-  simp_rw [he]
-  exact halfPlane_random_test_measurable μ hμ hfin
-    ((J ⁻¹' A).indicator w) (hw.indicator (hJ hA))
-
-lemma weightedImageMass_lintegral {X : Type*} [MeasurableSpace X]
-    (J : upperHalfPlane → X) (hJ : Measurable J)
-    (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w)
-    (μ : Measure upperHalfPlane) (f : X → ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻ x, f x ∂weightedImageMass J w μ) = ∫⁻ z, w z*f (J z) ∂μ := by
-  rw [weightedImageMass,lintegral_map hf hJ]
-  exact lintegral_withDensity_eq_lintegral_mul μ hw
-    (show Measurable (fun z => f (J z)) from hf.comp hJ)
-
-theorem weightedImageMass_measurable_of_subsigma
-    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (μ : Ω → Measure upperHalfPlane) (hμ : Measurable[mΩ] μ)
-    (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω))
-    (J : Ω × upperHalfPlane → X) (hJ : Measurable[m.prod inferInstance] J)
-    (w : Ω × upperHalfPlane → ℝ≥0∞) (hw : Measurable[m.prod inferInstance] w) :
-    Measurable[mΩ] (fun ω => weightedImageMass (fun z => J (ω,z))
-      (fun z => w (ω,z)) (μ ω)) := by
-  let : MeasurableSpace Ω := mΩ
-  have hle : m.prod (inferInstance : MeasurableSpace upperHalfPlane) ≤ mΩ.prod inferInstance :=
-    sup_le_sup (MeasurableSpace.comap_mono hm) le_rfl
-  exact weightedImageMass_measurable μ hμ hfin J (hJ.mono hle le_rfl) w (hw.mono hle le_rfl)
-
-theorem restart_weighted_mass_test
-    {Ω X Y : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
-    [mY : MeasurableSpace Y]
-    (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (V : Ω → X) (hV : Measurable[mΩ] V)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
-      @Measure.map Ω X mΩ mX V (P.restrict A) = P A • ν)
-    (M : X → Measure upperHalfPlane) (hM : Measurable M)
-    (hMf : ∀ x, IsFiniteMeasureOnCompacts (M x))
-    (J : Ω × upperHalfPlane → Y) (hJ : Measurable[m.prod inferInstance] J)
-    (w : Ω × upperHalfPlane → ℝ≥0∞) (hw : Measurable[m.prod inferInstance] w)
-    (f : Ω × Y → ℝ≥0∞) (hf : Measurable[m.prod mY] f) :
-    (∫⁻ ω, ∫⁻ y, f (ω,y) ∂weightedImageMass (fun z => J (ω,z))
-        (fun z => w (ω,z)) (M (V ω)) ∂P) =
-      ∫⁻ ω, ∫⁻ z, w (ω,z)*f (ω,J (ω,z)) ∂ν.bind M ∂P := by
-  let : MeasurableSpace Ω := mΩ
-  have htest : Measurable[m.prod inferInstance]
-      (fun p : Ω × upperHalfPlane => w p*f (p.1,J p)) :=
-    hw.mul (hf.comp (measurable_fst.prodMk hJ))
-  have he (ω : Ω) := weightedImageMass_lintegral (fun z => J (ω,z))
-    (hJ.comp (measurable_const.prodMk measurable_id)) (fun z => w (ω,z))
-    (hw.comp (measurable_const.prodMk measurable_id)) (M (V ω)) (fun y => f (ω,y))
-    (hf.comp (measurable_const.prodMk measurable_id))
-  simp_rw [he]
-  exact restart_random_mass_test P ν m hm V hV hlaw M hM hMf _ htest
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RestartRandomMassTesting
+
+namespace OAI
+
+/-! Measurable weighted images of locally finite random half-plane measures.
+The weight and image map can be chosen from the stopped past. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped ENNReal
+
+namespace SLEExactGauge
+
+def weightedImageMass {X : Type*} [MeasurableSpace X]
+    (J : upperHalfPlane → X) (w : upperHalfPlane → ℝ≥0∞)
+    (μ : Measure upperHalfPlane) : Measure X :=
+  (μ.withDensity w).map J
+
+lemma weightedImageMass_apply {X : Type*} [MeasurableSpace X]
+    (J : upperHalfPlane → X) (hJ : Measurable J)
+    (w : upperHalfPlane → ℝ≥0∞) (μ : Measure upperHalfPlane)
+    (A : Set X) (hA : MeasurableSet A) :
+    weightedImageMass J w μ A = ∫⁻ z, (J ⁻¹' A).indicator w z ∂μ := by
+  rw [weightedImageMass,Measure.map_apply hJ hA,withDensity_apply _ (hJ hA),
+    lintegral_indicator (hJ hA)]
+
+theorem weightedImageMass_measurable
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ)
+    (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω))
+    (J : Ω × upperHalfPlane → X) (hJ : Measurable J)
+    (w : Ω × upperHalfPlane → ℝ≥0∞) (hw : Measurable w) :
+    Measurable (fun ω => weightedImageMass (fun z => J (ω,z))
+      (fun z => w (ω,z)) (μ ω)) := by
+  apply Measure.measurable_of_measurable_coe
+  intro A hA
+  have he (ω : Ω) := weightedImageMass_apply (fun z => J (ω,z))
+    (hJ.comp (measurable_const.prodMk measurable_id)) (fun z => w (ω,z)) (μ ω) A hA
+  simp_rw [he]
+  exact halfPlane_random_test_measurable μ hμ hfin
+    ((J ⁻¹' A).indicator w) (hw.indicator (hJ hA))
+
+lemma weightedImageMass_lintegral {X : Type*} [MeasurableSpace X]
+    (J : upperHalfPlane → X) (hJ : Measurable J)
+    (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w)
+    (μ : Measure upperHalfPlane) (f : X → ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻ x, f x ∂weightedImageMass J w μ) = ∫⁻ z, w z*f (J z) ∂μ := by
+  rw [weightedImageMass,lintegral_map hf hJ]
+  exact lintegral_withDensity_eq_lintegral_mul μ hw
+    (show Measurable (fun z => f (J z)) from hf.comp hJ)
+
+theorem weightedImageMass_measurable_of_subsigma
+    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (μ : Ω → Measure upperHalfPlane) (hμ : Measurable[mΩ] μ)
+    (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω))
+    (J : Ω × upperHalfPlane → X) (hJ : Measurable[m.prod inferInstance] J)
+    (w : Ω × upperHalfPlane → ℝ≥0∞) (hw : Measurable[m.prod inferInstance] w) :
+    Measurable[mΩ] (fun ω => weightedImageMass (fun z => J (ω,z))
+      (fun z => w (ω,z)) (μ ω)) := by
+  let : MeasurableSpace Ω := mΩ
+  have hle : m.prod (inferInstance : MeasurableSpace upperHalfPlane) ≤ mΩ.prod inferInstance :=
+    sup_le_sup (MeasurableSpace.comap_mono hm) le_rfl
+  exact weightedImageMass_measurable μ hμ hfin J (hJ.mono hle le_rfl) w (hw.mono hle le_rfl)
+
+theorem restart_weighted_mass_test
+    {Ω X Y : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
+    [mY : MeasurableSpace Y]
+    (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (V : Ω → X) (hV : Measurable[mΩ] V)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A →
+      @Measure.map Ω X mΩ mX V (P.restrict A) = P A • ν)
+    (M : X → Measure upperHalfPlane) (hM : Measurable M)
+    (hMf : ∀ x, IsFiniteMeasureOnCompacts (M x))
+    (J : Ω × upperHalfPlane → Y) (hJ : Measurable[m.prod inferInstance] J)
+    (w : Ω × upperHalfPlane → ℝ≥0∞) (hw : Measurable[m.prod inferInstance] w)
+    (f : Ω × Y → ℝ≥0∞) (hf : Measurable[m.prod mY] f) :
+    (∫⁻ ω, ∫⁻ y, f (ω,y) ∂weightedImageMass (fun z => J (ω,z))
+        (fun z => w (ω,z)) (M (V ω)) ∂P) =
+      ∫⁻ ω, ∫⁻ z, w (ω,z)*f (ω,J (ω,z)) ∂ν.bind M ∂P := by
+  let : MeasurableSpace Ω := mΩ
+  have htest : Measurable[m.prod inferInstance]
+      (fun p : Ω × upperHalfPlane => w p*f (p.1,J p)) :=
+    hw.mul (hf.comp (measurable_fst.prodMk hJ))
+  have he (ω : Ω) := weightedImageMass_lintegral (fun z => J (ω,z))
+    (hJ.comp (measurable_const.prodMk measurable_id)) (fun z => w (ω,z))
+    (hw.comp (measurable_const.prodMk measurable_id)) (M (V ω)) (fun y => f (ω,y))
+    (hf.comp (measurable_const.prodMk measurable_id))
+  simp_rw [he]
+  exact restart_random_mass_test P ν m hm V hV hlaw M hM hMf _ htest
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Mass/WeightedLevelMoments.lean b/lean/OAI/Probability/SLE/Mass/WeightedLevelMoments.lean
index 463bb1f0161a1db3c46551b039fbc97c07383f72..a540717dcc70e25b0cb2158ab738a579f7533344 100644
--- a/lean/OAI/Probability/SLE/Mass/WeightedLevelMoments.lean
+++ b/lean/OAI/Probability/SLE/Mass/WeightedLevelMoments.lean
@@ -1,188 +1,188 @@
-import OAI.Probability.SLE.Mass.LevelRadonMoments
-import OAI.Probability.SLE.Gauge.WeightedMomentIntegration
-
-namespace OAI
-
-/-! The actual multipoint estimate with spatial weights retained. This
-retains the near-diagonal gain needed for nonatomicity. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped NNReal ENNReal Topology CompactlySupported
-
-namespace SLEExactGauge.Multipoint
-
-def weightedLevelMass {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (u : ℝ) (K : Set ℂ) (f : ℂ → ℝ≥0∞) (ω : Ω) : ℝ≥0∞ :=
-  ∫⁻ z in K, f z * ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-    (γ ω) (G ω) z u)
-
-theorem weightedLevelMass_moment_identity
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [SigmaFinite P]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (K : Set ℂ) (f : ℂ → ℝ≥0∞) (hf : Measurable f) (m : ℕ) :
-    (∫⁻ ω, weightedLevelMass κ B γ G u K f ω ^ m ∂P) =
-      ∫⁻ z : Fin m → ℂ, (∏ i, f (z i)) *
-        (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-          (γ ω) (G ω) (z i) u) ∂P) ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
-  have hW := (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal
-  unfold weightedLevelMass
-  rw [lintegral_weighted_mass_pow_eq P (volume.restrict K)
-    (fun ω z => ENNReal.ofReal (stoppedGreen κ
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u)) hW f hf m]
-  apply lintegral_congr
-  intro z
-  apply congrArg (fun v : ℝ≥0∞ => (∏ i, f (z i)) * v)
-  apply lintegral_congr
-  intro ω
-  exact (ENNReal.ofReal_prod_of_nonneg (fun i _ => stoppedGreen_nonneg (hw ω) κ _ _)).symm
-
-theorem actual_product_le_cappedKernel
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    {m : ℕ} (z : Fin m → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hzi : Function.Injective z)
-    (b : ℝ) (hb : 0 < b) (hbR : ∀ i, b ≤ 2*(z i).im) (u : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (z i) u) ∂P) ≤
-      ENNReal.ofReal (multipointConstant κ m) * cappedDistanceKernel (codimension κ) b z := by
-  have hc := stoppedGreen_common_level_product_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw
-    hgrowth z hz hzi (cappedDistance b z) (fun i => cap_pos hb hzi i)
-    (fun i => (cap_le_fixed b z i).trans (hbR i))
-    (fun i j hij => cap_le_dist b z i j hij) u
-  apply hc.trans_eq
-  rw [ENNReal.ofReal_mul (by positivity)]
-  congr 1
-  · simp only [multipointConstant,Fintype.card_fin]
-  · exact ENNReal.ofReal_prod_of_nonneg
-      (fun i _ => Real.rpow_nonneg (cap_pos hb hzi i).le _)
-
-theorem weightedLevelMass_moment_le_cappedKernel
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane)
-    (f : ℂ → ℝ≥0∞) (hf : Measurable f)
-    (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ)
-    (u : ℝ) (hu : 0 ≤ u) :
-    (∫⁻ ω, weightedLevelMass κ B γ G u K f ω ^ m ∂P) ≤
-      ENNReal.ofReal (multipointConstant κ m) *
-        ∫⁻ z : Fin m → ℂ, (∏ i, f (z i)) * cappedDistanceKernel (codimension κ) b z
-          ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  rw [weightedLevelMass_moment_identity P hm hBc κ u hu γ G F hw K f hf m]
-  have hk : Measurable (fun z : Fin m → ℂ =>
-      (∏ i, f (z i)) * cappedDistanceKernel (codimension κ) b z) :=
-    (Finset.measurable_prod _ (fun i _ => hf.comp (measurable_pi_apply i))).mul
-      (measurable_cappedDistanceKernel _ _)
-  rw [← lintegral_const_mul _ hk]
-  apply lintegral_mono_ae
-  filter_upwards [ae_tuple_mem (ι := Fin m) K hK,ae_tuple_injective (ι := Fin m) K] with z hz hzi
-  have he := actual_product_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
-    z (fun i => hKH (hz i)) hzi b hb (fun i => hbR (z i) (hz i)) u
-  exact (mul_le_mul' le_rfl he).trans_eq (by ac_rfl)
-
-lemma levelApproximationTest_le_weightedLevelMass
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ u : ℝ) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (K : Set ℂ) (hK : MeasurableSet K) (f : ℂ → ℝ≥0∞) (hf : Measurable f)
-    (hz : ∀ z ∉ K, f z = 0) (t : ℝ≥0∞) (ω : Ω) :
-    levelApproximationTest κ B γ G u t f ω ≤ weightedLevelMass κ B γ G u K f ω := by
-  have hd := levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t
-  change (∫⁻ z, f z ∂volume.withDensity _) ≤ _
-  rw [lintegral_withDensity_eq_lintegral_mul volume hd.of_uncurry_left hf]
-  have he : (fun z => levelApproximationDensity κ B γ G u t ω z * f z) =
-      K.indicator (fun z => levelApproximationDensity κ B γ G u t ω z * f z) := by
-    funext z
-    by_cases hzK : z ∈ K
-    · simp only [indicator_of_mem hzK]
-    · simp only [indicator_of_notMem hzK,hz z hzK,mul_zero]
-  change (∫⁻ z, levelApproximationDensity κ B γ G u t ω z * f z) ≤ _
-  rw [he,lintegral_indicator hK]
-  apply lintegral_mono
-  intro z
-  unfold levelApproximationDensity
-  dsimp only
-  split_ifs
-  · exact (mul_comm _ _).le
-  · simp only [zero_mul,zero_le]
-
-theorem levelApproximationTest_moment_le_cappedKernel
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane)
-    (f : ℂ → ℝ≥0∞) (hf : Measurable f) (hz : ∀ z ∉ K, f z = 0)
-    (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ)
-    (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :
-    (∫⁻ ω, levelApproximationTest κ B γ G u t f ω ^ m ∂P) ≤
-      ENNReal.ofReal (multipointConstant κ m) *
-        ∫⁻ z : Fin m → ℂ, (∏ i, f (z i)) * cappedDistanceKernel (codimension κ) b z
-          ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
-  apply (lintegral_mono (fun ω => pow_le_pow_left'
-    (levelApproximationTest_le_weightedLevelMass hm hBc κ u hu γ G F hw K hK f hf hz t ω) m)).trans
-  exact weightedLevelMass_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
-    K hK hKH f hf b hb hbR m u hu
-
-theorem level_radon_limit_test_moment_le
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ)
-    (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1)
-    (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane)
-    (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
-      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω)
-        atTop (𝓝 (∫ z, f z ∂ν ω)))
-    (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) (M : ℝ≥0∞)
-    (hbound : ∀ n, (∫⁻ ω, ENNReal.ofReal
-      (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u n) t ω)^m ∂P) ≤ M) :
-    (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂ν ω)^m ∂P) ≤ M := by
-  apply lintegral_le_of_ae_tendsto_bound P
-    (fun n ω => ENNReal.ofReal (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω)^m)
-    (fun ω => ENNReal.ofReal (∫ z, f z ∂ν ω)^m)
-    (fun n => ((convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n t f).ennreal_ofReal.pow_const m).aemeasurable)
-  · filter_upwards [hlim] with ω hω
-    exact (ENNReal.continuous_pow m).continuousAt.tendsto.comp
-      ((ENNReal.continuous_ofReal.tendsto _).comp (hω f))
-  · intro n
-    have hb := lintegral_pow_convex_sum_le P (s n) (w n)
-      (fun i ω => ∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω)
-      (hw₀ n) (hw₁ n) (fun i _ ω => integral_nonneg hf)
-      (fun i _ => by
-        simp_rw [integral_halfPlaneLevelMeasure]
-        exact levelApproximationRealTest_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t
-          _ (extendHalfPlaneTest_measurable f f.continuous.measurable))
-      m M (fun i _ => hbound (n+i))
-    simpa only [integral_convexLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) u s w n t (hw₀ n) f,
-      integral_halfPlaneLevelMeasure] using hb
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.LevelRadonMoments
+import OAI.Probability.SLE.Gauge.WeightedMomentIntegration
+
+namespace OAI
+
+/-! The actual multipoint estimate with spatial weights retained. This
+retains the near-diagonal gain needed for nonatomicity. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped NNReal ENNReal Topology CompactlySupported
+
+namespace SLEExactGauge.Multipoint
+
+def weightedLevelMass {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (u : ℝ) (K : Set ℂ) (f : ℂ → ℝ≥0∞) (ω : Ω) : ℝ≥0∞ :=
+  ∫⁻ z in K, f z * ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+    (γ ω) (G ω) z u)
+
+theorem weightedLevelMass_moment_identity
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [SigmaFinite P]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (K : Set ℂ) (f : ℂ → ℝ≥0∞) (hf : Measurable f) (m : ℕ) :
+    (∫⁻ ω, weightedLevelMass κ B γ G u K f ω ^ m ∂P) =
+      ∫⁻ z : Fin m → ℂ, (∏ i, f (z i)) *
+        (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+          (γ ω) (G ω) (z i) u) ∂P) ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
+  have hW := (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal
+  unfold weightedLevelMass
+  rw [lintegral_weighted_mass_pow_eq P (volume.restrict K)
+    (fun ω z => ENNReal.ofReal (stoppedGreen κ
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u)) hW f hf m]
+  apply lintegral_congr
+  intro z
+  apply congrArg (fun v : ℝ≥0∞ => (∏ i, f (z i)) * v)
+  apply lintegral_congr
+  intro ω
+  exact (ENNReal.ofReal_prod_of_nonneg (fun i _ => stoppedGreen_nonneg (hw ω) κ _ _)).symm
+
+theorem actual_product_le_cappedKernel
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    {m : ℕ} (z : Fin m → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hzi : Function.Injective z)
+    (b : ℝ) (hb : 0 < b) (hbR : ∀ i, b ≤ 2*(z i).im) (u : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (z i) u) ∂P) ≤
+      ENNReal.ofReal (multipointConstant κ m) * cappedDistanceKernel (codimension κ) b z := by
+  have hc := stoppedGreen_common_level_product_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw
+    hgrowth z hz hzi (cappedDistance b z) (fun i => cap_pos hb hzi i)
+    (fun i => (cap_le_fixed b z i).trans (hbR i))
+    (fun i j hij => cap_le_dist b z i j hij) u
+  apply hc.trans_eq
+  rw [ENNReal.ofReal_mul (by positivity)]
+  congr 1
+  · simp only [multipointConstant,Fintype.card_fin]
+  · exact ENNReal.ofReal_prod_of_nonneg
+      (fun i _ => Real.rpow_nonneg (cap_pos hb hzi i).le _)
+
+theorem weightedLevelMass_moment_le_cappedKernel
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane)
+    (f : ℂ → ℝ≥0∞) (hf : Measurable f)
+    (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ)
+    (u : ℝ) (hu : 0 ≤ u) :
+    (∫⁻ ω, weightedLevelMass κ B γ G u K f ω ^ m ∂P) ≤
+      ENNReal.ofReal (multipointConstant κ m) *
+        ∫⁻ z : Fin m → ℂ, (∏ i, f (z i)) * cappedDistanceKernel (codimension κ) b z
+          ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  rw [weightedLevelMass_moment_identity P hm hBc κ u hu γ G F hw K f hf m]
+  have hk : Measurable (fun z : Fin m → ℂ =>
+      (∏ i, f (z i)) * cappedDistanceKernel (codimension κ) b z) :=
+    (Finset.measurable_prod _ (fun i _ => hf.comp (measurable_pi_apply i))).mul
+      (measurable_cappedDistanceKernel _ _)
+  rw [← lintegral_const_mul _ hk]
+  apply lintegral_mono_ae
+  filter_upwards [ae_tuple_mem (ι := Fin m) K hK,ae_tuple_injective (ι := Fin m) K] with z hz hzi
+  have he := actual_product_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
+    z (fun i => hKH (hz i)) hzi b hb (fun i => hbR (z i) (hz i)) u
+  exact (mul_le_mul' le_rfl he).trans_eq (by ac_rfl)
+
+lemma levelApproximationTest_le_weightedLevelMass
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ u : ℝ) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (K : Set ℂ) (hK : MeasurableSet K) (f : ℂ → ℝ≥0∞) (hf : Measurable f)
+    (hz : ∀ z ∉ K, f z = 0) (t : ℝ≥0∞) (ω : Ω) :
+    levelApproximationTest κ B γ G u t f ω ≤ weightedLevelMass κ B γ G u K f ω := by
+  have hd := levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t
+  change (∫⁻ z, f z ∂volume.withDensity _) ≤ _
+  rw [lintegral_withDensity_eq_lintegral_mul volume hd.of_uncurry_left hf]
+  have he : (fun z => levelApproximationDensity κ B γ G u t ω z * f z) =
+      K.indicator (fun z => levelApproximationDensity κ B γ G u t ω z * f z) := by
+    funext z
+    by_cases hzK : z ∈ K
+    · simp only [indicator_of_mem hzK]
+    · simp only [indicator_of_notMem hzK,hz z hzK,mul_zero]
+  change (∫⁻ z, levelApproximationDensity κ B γ G u t ω z * f z) ≤ _
+  rw [he,lintegral_indicator hK]
+  apply lintegral_mono
+  intro z
+  unfold levelApproximationDensity
+  dsimp only
+  split_ifs
+  · exact (mul_comm _ _).le
+  · simp only [zero_mul,zero_le]
+
+theorem levelApproximationTest_moment_le_cappedKernel
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane)
+    (f : ℂ → ℝ≥0∞) (hf : Measurable f) (hz : ∀ z ∉ K, f z = 0)
+    (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ)
+    (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0∞) :
+    (∫⁻ ω, levelApproximationTest κ B γ G u t f ω ^ m ∂P) ≤
+      ENNReal.ofReal (multipointConstant κ m) *
+        ∫⁻ z : Fin m → ℂ, (∏ i, f (z i)) * cappedDistanceKernel (codimension κ) b z
+          ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
+  apply (lintegral_mono (fun ω => pow_le_pow_left'
+    (levelApproximationTest_le_weightedLevelMass hm hBc κ u hu γ G F hw K hK f hf hz t ω) m)).trans
+  exact weightedLevelMass_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth
+    K hK hKH f hf b hb hbR m u hu
+
+theorem level_radon_limit_test_moment_le
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ)
+    (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1)
+    (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane)
+    (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ),
+      Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω)
+        atTop (𝓝 (∫ z, f z ∂ν ω)))
+    (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) (M : ℝ≥0∞)
+    (hbound : ∀ n, (∫⁻ ω, ENNReal.ofReal
+      (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u n) t ω)^m ∂P) ≤ M) :
+    (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂ν ω)^m ∂P) ≤ M := by
+  apply lintegral_le_of_ae_tendsto_bound P
+    (fun n ω => ENNReal.ofReal (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω)^m)
+    (fun ω => ENNReal.ofReal (∫ z, f z ∂ν ω)^m)
+    (fun n => ((convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n t f).ennreal_ofReal.pow_const m).aemeasurable)
+  · filter_upwards [hlim] with ω hω
+    exact (ENNReal.continuous_pow m).continuousAt.tendsto.comp
+      ((ENNReal.continuous_ofReal.tendsto _).comp (hω f))
+  · intro n
+    have hb := lintegral_pow_convex_sum_le P (s n) (w n)
+      (fun i ω => ∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω)
+      (hw₀ n) (hw₁ n) (fun i _ ω => integral_nonneg hf)
+      (fun i _ => by
+        simp_rw [integral_halfPlaneLevelMeasure]
+        exact levelApproximationRealTest_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t
+          _ (extendHalfPlaneTest_measurable f f.continuous.measurable))
+      m M (fun i _ => hbound (n+i))
+    simpa only [integral_convexLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) u s w n t (hw₀ n) f,
+      integral_halfPlaneLevelMeasure] using hb
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/MeasureSupport.lean b/lean/OAI/Probability/SLE/MeasureSupport.lean
index afe6dd79645edaf3c3b82061453f32427e6b91dd..ccd42fa24b03bac48df12c647a81a0d3a093d5e9 100644
--- a/lean/OAI/Probability/SLE/MeasureSupport.lean
+++ b/lean/OAI/Probability/SLE/MeasureSupport.lean
@@ -1,698 +1,698 @@
-import Mathlib
-import OAI.Probability.SLE.InverseMoments
-import OAI.Probability.SLE.SausagePointBound
-import OAI.Probability.SLE.SpatialMeans
-
-namespace OAI
-
-/-! Measure Support for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable def spatialGreenMass (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
-    (G : ℝ≥0 → ℂ → ℂ) (r : ℝ) (A : Set ℂ) : ℝ≥0∞ :=
-  ∫⁻ z in A, ENNReal.ofReal (spatialGreenWeight κ U γ G z r)
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
-lemma spatialGreenMass_measurable (r : ℝ) (A : Set ℂ) :
-    Measurable (fun ω : a.samples => spatialGreenMass κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A) :=
-  (a.spatialGreenWeight_measurable r).ennreal_ofReal.lintegral_prod_left'
-
- 
-
-lemma spatialGreenMass_expectation [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, r ≤ z.im) :
-    (∫⁻ ω : a.samples, spatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
-      (γ ω) (a.G ω) r A ∂(P.comap Subtype.val)) =
-      ∫⁻ z in A, ENNReal.ofReal (initialGreen κ z) := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  simp only [spatialGreenMass]
-  rw [← lintegral_lintegral_swap (a.spatialGreenWeight_measurable r).ennreal_ofReal.aemeasurable]
-  apply setLIntegral_congr_fun hA
-  intro z hz
-  obtain ⟨hi, he⟩ := a.spatialGreen_expectation hκ hκ8 z r hr (hheight z hz)
-  change (∫⁻ ω : a.samples, ENNReal.ofReal (spatialGreenWeight κ
-    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) ∂μ) = ENNReal.ofReal (initialGreen κ z)
-  rw [← he]
-  exact (ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall fun ω => by
-    dsimp only [Pi.zero_apply, spatialGreenWeight]
-    split_ifs
-    · exact le_rfl
-    · exact greenDensity_nonneg (a.witness ω) κ _ z)).symm
-
- 
-
-lemma spatialGreenMass_expectation_le [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, r ≤ z.im) :
-    (∫⁻ ω : a.samples, spatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
-      (γ ω) (a.G ω) r A ∂(P.comap Subtype.val)) ≤
-      ENNReal.ofReal ((2 : ℝ) ^ (-codimension κ) * r ^ (-codimension κ)) * volume A := by
-  rw [a.spatialGreenMass_expectation hκ hκ8 r hr A hA hheight]
-  calc
-    _ ≤ ∫⁻ _z in A, ENNReal.ofReal ((2 : ℝ) ^ (-codimension κ) * r ^ (-codimension κ)) := by
-      apply setLIntegral_mono' hA
-      intro z hz
-      apply ENNReal.ofReal_le_ofReal
-      calc
-        _ ≤ (2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ) :=
-          initialGreen_le_height κ hκ hκ8 z (hr.trans_le (hheight z hz))
-        _ ≤ _ := mul_le_mul_of_nonneg_left
-          (Real.rpow_le_rpow_of_nonpos hr (hheight z hz) (neg_nonpos.mpr (codimension_pos κ hκ8).le))
-          (Real.rpow_nonneg (by norm_num) _)
-    _ = _ := by rw [lintegral_const, Measure.restrict_apply_univ]
-
-end RegularSLEPresentation
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma finite_density_crossing_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P Q : Measure Ω)
-    (F : Filtration ℕ mΩ) (B : ℕ → Set Ω)
-    (hBm : ∀ n, MeasurableSet[F n] (B n)) (c : ℝ≥0∞)
-    (hQ : ∀ n A, MeasurableSet[F n] A → A ⊆ B n → Q A ≤ c * P A)
-    (n : ℕ) : Q (⋃ i ∈ Finset.range n, B i) ≤ c * P (⋃ i ∈ Finset.range n, B i) := by
-  classical
-  let E (n : ℕ) : Set Ω := ⋃ i ∈ Finset.range n, B i
-  have hEm (n : ℕ) : MeasurableSet[F n] (E n) := by
-    apply MeasurableSet.iUnion
-    intro i
-    apply MeasurableSet.iUnion
-    intro hi
-    exact F.mono (Nat.le_of_lt (Finset.mem_range.mp hi)) _ (hBm i)
-  have hrec (n : ℕ) : E (n + 1) = E n ∪ (B n \ E n) := by
-    calc
-      E (n + 1) = B n ∪ E n := by
-        ext ω
-        simp only [E, Finset.range_add_one, Finset.mem_insert, mem_iUnion, mem_union]
-        aesop
-      _ = E n ∪ (B n \ E n) := by
-        ext ω
-        by_cases h : ω ∈ E n <;> simp [h]
-  change Q (E n) ≤ c * P (E n)
-  induction n with
-  | zero => simp [E]
-  | succ n ih =>
-      have hD := (hBm n).diff (hEm n)
-      have hd : Disjoint (E n) (B n \ E n) := disjoint_sdiff_right
-      rw [hrec n, measure_union hd (F.le n _ hD), measure_union hd (F.le n _ hD), mul_add]
-      exact add_le_add ih (hQ n _ hD sdiff_subset)
-
-lemma finite_density_crossing_bound_of_subset
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P Q : Measure Ω)
-    (F : Filtration ℕ mΩ) (B : ℕ → Set Ω)
-    (hBm : ∀ n, MeasurableSet[F n] (B n)) (c : ℝ≥0∞)
-    (hQ : ∀ n A, MeasurableSet[F n] A → A ⊆ B n → Q A ≤ c * P A)
-    (A : Set Ω) (hA : ∀ n, B n ⊆ A) (n : ℕ) :
-    Q (⋃ i ∈ Finset.range n, B i) ≤ c * P A := by
-  refine (finite_density_crossing_bound P Q F B hBm c hQ n).trans ?_
-  exact mul_le_mul (show c ≤ c from le_rfl)
-    (show P (⋃ i ∈ Finset.range n, B i) ≤ P A from
-      measure_mono (iUnion₂_subset fun i _ => hA i)) bot_le bot_le
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
-
-lemma limitRadiusKernel_pos_iff
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hu : 0 ≤ u)
-    (p α : ℝ) (ω : Ω) :
-    0 < limitRadiusKernel a p α u ω ↔ localizedRadiusStop a ω < ⊤ := by
-  have hx : (∃ n, max u 0 ≤ (a n).clock.S ((a n).τ ω) ω) ↔ ∃ n, ω ∈ (a n).hit := by
-    simp only [max_eq_left hu]
-    constructor
-    · rintro ⟨n, hn⟩
-      exact ⟨n, le_antisymm ((a n).clock_upper ω) hn⟩
-    · rintro ⟨n, hn⟩
-      exact ⟨n, le_of_eq hn.symm⟩
-  by_cases hh : ∃ n, ω ∈ (a n).hit
-  · have he := hx.mpr hh
-    obtain ⟨n, hn⟩ := hh
-    rw [localizedRadiusStop_eq_hit a ω n hn, limitRadiusKernel, dite_eq_left he,
-      AngularLocalization.radiusKernel, ite_eq_left (Nat.find_spec he)]
-    exact iff_of_true (Real.exp_pos _) (WithTop.coe_lt_top _)
-  · rw [localizedRadiusStop_eq_top a ω (not_exists.mp hh), limitRadiusKernel,
-      dite_eq_right (fun h => hh (hx.mp h))]
-    simp
-
-lemma finiteCenteredLaw_eq_radiusKernel
-    {P : Measure Ω} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) :
-    finiteCenteredLaw P κ B γ G z u = P.withDensity
-      (fun ω => ENNReal.ofReal (limitRadiusKernel a (codimension κ) (greenExponent κ) u ω /
-        initialGreen κ z)) := by
-  apply withDensity_congr_ae
-  filter_upwards [hw] with ω hω
-  rw [limitRadiusKernel_eq_stoppedGreen a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω]
-
-lemma radiusKernel_density_bound
-    {P : Measure Ω} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (δ : ℝ) (A : Set Ω) (hA : MeasurableSet A)
-    (hδ : ∀ ω ∈ A, limitRadiusKernel a (codimension κ) (greenExponent κ) u ω ≤ δ) :
-    finiteCenteredLaw P κ B γ G z u A ≤ ENNReal.ofReal (δ / initialGreen κ z) * P A := by
-  rw [finiteCenteredLaw_eq_radiusKernel a hε hεlim hBc hz hu γ G F hw, withDensity_apply _ hA]
-  calc
-    _ ≤ ∫⁻ ω in A, ENNReal.ofReal (δ / initialGreen κ z) ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem hA] with ω hω
-      exact ENNReal.ofReal_le_ofReal (div_le_div_of_nonneg_right (hδ ω hω) (initialGreen_pos κ z hz).le)
-    _ = _ := by simp
-
-lemma finiteCenteredLaw_consistent_at_radius
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) (hz : z ∈ upperHalfPlane)
-    {v : ℝ} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (b : ∀ n, AngularLocalization B hm κ z v (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hu : 0 ≤ u) (huv : u ≤ v)
-    (A : Set Ω) (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
-    finiteCenteredLaw P κ B γ G z v A = finiteCenteredLaw P κ B γ G z u A := by
-  have hAm := (localizedRadiusStop_stopping a).measurableSpace_le _ hA
-  rw [finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz v (hu.trans huv) A hAm,
-    finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz u hu A hAm,
-    finite_stopped_green_consistent_regular hB hBm hBc hB0 hκ hκ8 γ G F hw hz
-      a b hεlim hε hu huv A hA]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma finite_radius_density_crossing
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (v U : ℝ≥0) (s : ℕ → ℝ≥0) (hs : Monotone s)
-    (hvs : ∀ n, v ≤ s n) (hsU : ∀ n, s n ≤ U) (δ : ℝ) (N : ℕ) :
-    finiteCenteredLaw P κ B γ G z U
-      (⋃ i ∈ Finset.range N, {ω | 0 < limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ∧
-        limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ≤ δ}) ≤
-      ENNReal.ofReal (δ / initialGreen κ z) * P {ω | localizedRadiusStop (a v) ω < ⊤} := by
-  let K (n : ℕ) := limitRadiusKernel (a (s n)) (codimension κ) (greenExponent κ) (s n)
-  let E (n : ℕ) : Set Ω := {ω | 0 < K n ω ∧ K n ω ≤ δ}
-  let H : Filtration ℕ mΩ :=
-    { seq := fun n => radiusFiltration a (s n)
-      mono' := fun n m hnm => (radiusFiltration a).mono (hs hnm)
-      le' := fun n => (radiusFiltration a).le (s n) }
-  have hKm (n : ℕ) : Measurable[H n] (K n) :=
-    limitRadiusKernel_terminal_measurable (a (s n)) hε hBc (s n).coe_nonneg _ _
-  have hEm (n : ℕ) : MeasurableSet[H n] (E n) :=
-    measurableSet_Ioc.preimage (hKm n)
-  apply finite_density_crossing_bound_of_subset P (finiteCenteredLaw P κ B γ G z U)
-    H E hEm (ENNReal.ofReal (δ / initialGreen κ z))
-  · intro n A hA hAE
-    rw [finiteCenteredLaw_consistent_at_radius hB hBm hBc hB0 hκ hκ8 γ G F hw hz
-      (a (s n)) (a U) hε hεlim (s n).coe_nonneg (show (s n : ℝ) ≤ U from hsU n) A hA]
-    exact radiusKernel_density_bound (a (s n)) hε hεlim hBc hz (s n).coe_nonneg γ G F hw
-      δ A (H.le n _ hA) (fun ω hω => (hAE hω).2)
-  · intro n ω hω
-    have hp := (limitRadiusKernel_pos_iff (a (s n)) (s n).coe_nonneg
-      (codimension κ) (greenExponent κ) ω).mp hω.1
-    exact lt_of_le_of_lt (localizedRadiusStop_mono (a v) (a (s n))
-      (show (v : ℝ) ≤ s n from hvs n) ω) hp
-
-lemma radius_kernel_scale_cancel (κ v w L C : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (hL : L ≠ 0) :
-    (Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L / initialGreen κ z) *
-      (C * Real.exp (-codimension κ * v) * (Real.sin (Complex.arg z)) ^ greenExponent κ) =
-      C * Real.exp (codimension κ * (w - v)) / L := by
-  have hI := (initialGreen_pos κ z hz).ne'
-  have he : Real.exp (-codimension κ * v) * (Real.sin (Complex.arg z)) ^ greenExponent κ =
-      initialGreen κ z / Real.exp (codimension κ * (v - Real.log 2 - Real.log z.im)) :=
-    (initialGreen_div_zeroKernel κ v z hz).symm
-  rw [mul_assoc C, he]
-  rw [show (Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L / initialGreen κ z) *
-      (C * (initialGreen κ z / Real.exp (codimension κ * (v - Real.log 2 - Real.log z.im)))) =
-      C * (Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) /
-        Real.exp (codimension κ * (v - Real.log 2 - Real.log z.im))) / L by
-      field_simp]
-  rw [← Real.exp_sub]
-  congr 2
-  ring_nf
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma finite_radius_scaled_crossing
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (v U : ℝ≥0) (hv : 1 ≤ (v : ℝ)) (s : ℕ → ℝ≥0) (hs : Monotone s)
-    (hvs : ∀ n, v ≤ s n) (hsU : ∀ n, s n ≤ U) (w L : ℝ) (hL : 0 < L) (N : ℕ) :
-    finiteCenteredLaw P κ B γ G z U
-      (⋃ i ∈ Finset.range N, {ω | 0 < limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ∧
-        limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ≤
-          Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L}) ≤
-      ENNReal.ofReal (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ * (w - v)) / L) := by
-  let C := uniformInverseAngleConstant κ 1
-  have hC : 0 < C := uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
-  let Kv := Real.exp (codimension κ * ((v : ℝ) - Real.log 2 - Real.log z.im))
-  let δ := Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L
-  have hδ : 0 ≤ δ / initialGreen κ z := div_nonneg (div_pos (Real.exp_pos _) hL).le
-    (initialGreen_pos κ z hz).le
-  have hPe : P {ω | localizedRadiusStop (a v) ω < ⊤} =
-      P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞} := by
-    apply measure_congr
-    filter_upwards [hw] with ω hω
-    rw [localizedRadiusStop_eq_source (a v) hεlim hBc hz ω (γ ω) (G ω) (F ω) hω]
-    rfl
-  have hPr : (P {ω | localizedRadiusStop (a v) ω < ⊤}).toReal ≤ C * initialGreen κ z / Kv := by
-    rw [hPe]
-    refine (radius_approach_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz v hv).trans_eq ?_
-    dsimp only [C, Kv]
-    rw [mul_div_assoc, initialGreen_div_zeroKernel κ v z hz, mul_assoc]
-  have hPn : P {ω | localizedRadiusStop (a v) ω < ⊤} ≤ ENNReal.ofReal (C * initialGreen κ z / Kv) :=
-    (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (div_nonneg (mul_nonneg hC.le (initialGreen_pos κ z hz).le) (Real.exp_pos _).le)).mpr hPr
-  calc
-    _ ≤ ENNReal.ofReal (δ / initialGreen κ z) * P {ω | localizedRadiusStop (a v) ω < ⊤} :=
-      finite_radius_density_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim v U s hs hvs hsU δ N
-    _ ≤ ENNReal.ofReal (δ / initialGreen κ z) * ENNReal.ofReal (C * initialGreen κ z / Kv) :=
-      mul_le_mul le_rfl hPn bot_le bot_le
-    _ = ENNReal.ofReal ((δ / initialGreen κ z) * (C * initialGreen κ z / Kv)) :=
-      (ENNReal.ofReal_mul hδ).symm
-    _ = _ := by
-      congr 1
-      have he := radius_kernel_scale_cancel κ v w L C z hz hL.ne'
-      change (δ / initialGreen κ z) * (C * initialGreen κ z /
-        Real.exp (codimension κ * ((v : ℝ) - Real.log 2 - Real.log z.im))) = _
-      rw [mul_div_assoc C, initialGreen_div_zeroKernel κ v z hz, ← mul_assoc C]
-      exact he
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {u v : ℝ} {ε : ℕ → ℝ}
-
-lemma limitRadiusKernel_eq_of_surviving_cutoff
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (p α s : ℝ) (hs : 0 ≤ s) (ω : Ω) (n : ℕ)
-    (hn : s ≤ (a n).clock.S ((a n).τ ω) ω) :
-    limitRadiusKernel a p α s ω = (a n).radiusValue p α s ω := by
-  obtain ⟨m, hm, hlim⟩ := ((eventually_ge_atTop n).and (radiusKernel_eventually_limit a hε p α s ω)).exists
-  rw [max_eq_left hs] at hlim
-  rw [← hlim, ← (a n).radiusKernel_eq_of_live (a m) le_rfl (hε hm) p α s hs ω hn,
-    AngularLocalization.radiusKernel, ite_eq_left hn]
-
-lemma limitRadiusKernel_eq_below
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (b : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
-    (huv : u ≤ v) (p α s : ℝ) (hs : 0 ≤ s) (hsu : s ≤ u) (ω : Ω) :
-    limitRadiusKernel a p α s ω = limitRadiusKernel b p α s ω := by
-  obtain ⟨n, hn, hm⟩ := ((radiusKernel_eventually_limit a hε p α s ω).and
-    (radiusKernel_eventually_limit b hε p α s ω)).exists
-  rw [max_eq_left hs] at hn hm
-  rw [← hn, ← hm]
-  exact (a n).radiusKernel_eq_below (b n) huv p α s hs hsu ω
-
-lemma exists_cutoff_of_radiusStop_lt_top
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω)
-    (hω : localizedRadiusStop a ω < ⊤) : ∃ n, ω ∈ (a n).hit := by
-  by_contra hn
-  rw [localizedRadiusStop_eq_top a ω (not_exists.mp hn)] at hω
-  exact lt_irrefl _ hω
-
-lemma limitRadiusKernel_continuousOn_of_live
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω)
-    (hω : localizedRadiusStop a ω < ⊤) :
-    ContinuousOn (fun s => limitRadiusKernel a p α s ω) (Icc 0 u) := by
-  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top a ω hω
-  apply ((a n).radiusValue_continuous hBc p α ω).continuousOn.congr
-  intro s hs
-  exact limitRadiusKernel_eq_of_surviving_cutoff a hε p α s hs.1 ω n (hs.2.trans_eq hn.symm)
-
-lemma limitRadiusKernel_pos_of_live
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (p α s : ℝ) (hs : s ∈ Icc 0 u) (ω : Ω) (hω : localizedRadiusStop a ω < ⊤) :
-    0 < limitRadiusKernel a p α s ω := by
-  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top a ω hω
-  rw [limitRadiusKernel_eq_of_surviving_cutoff a hε p α s hs.1 ω n (hs.2.trans_eq hn.symm)]
-  exact powerLogDensity_pos _ _ _ _ _
-
-lemma radius_limit_clock_angle
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (p α s : ℝ) (hs : s ∈ Icc 0 u) (ω : Ω) (hω : localizedRadiusStop a ω < ⊤) :
-    limitRadiusKernel a p α s ω = Real.exp (p * (s - Real.log 2 - Real.log z.im)) *
-      (limitRadiusKernel a 0 2 s ω) ^ (α / 2) := by
-  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top a ω hω
-  have hs' : s ≤ (a n).clock.S ((a n).τ ω) ω := hs.2.trans_eq hn.symm
-  rw [limitRadiusKernel_eq_of_surviving_cutoff a hε p α s hs.1 ω n hs',
-    limitRadiusKernel_eq_of_surviving_cutoff a hε 0 2 s hs.1 ω n hs']
-  unfold AngularLocalization.radiusValue
-  rw [powerLogDensity_radius_angle κ p α (Real.log z.im),
-    powerLogDensity_radius_angle κ 0 2 (Real.log z.im)]
-  norm_num only [zero_mul, Real.exp_zero, one_mul, div_self (by norm_num : (2 : ℝ) ≠ 0), Real.rpow_one]
-  have he := (a n).clock.clock_inverse s ω
-  change (a n).clock.S ((a n).clock.inverse s ω) ω =
-    min (max s 0) ((a n).clock.S ((a n).τ ω) ω) at he
-  rw [max_eq_left hs.1, min_eq_left hs'] at he
-  change logRadiusClock (Real.log z.im)
-    ((a n).solution.Q ((a n).clock.inverse s ω) ω) = s at he
-  rw [he]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set MeasureTheory Classical
-open scoped ENNReal
-namespace SLEExactGauge
-
-lemma finite_grid_from_monotone
-    {Ω T : Type*} [MeasurableSpace Ω] [LinearOrder T]
-    (Q : Measure Ω) (E : T → Set Ω) (v U : T) (hvU : v ≤ U) (c : ℝ≥0∞)
-    (h : ∀ s : ℕ → T, Monotone s → (∀ n, v ≤ s n) → (∀ n, s n ≤ U) →
-      ∀ N, Q (⋃ i ∈ Finset.range N, E (s i)) ≤ c)
-    (t : Finset T) (ht : ∀ x ∈ t, x ∈ Icc v U) :
-    Q (⋃ x ∈ t, E x) ≤ c := by
-  let f := t.orderEmbOfFin rfl
-  let s (n : ℕ) : T := if hn : n < t.card then f ⟨n, hn⟩ else U
-  have hmem (i : Fin t.card) : f i ∈ t := t.orderEmbOfFin_mem rfl i
-  have hs : Monotone s := by
-    intro n m hnm
-    dsimp only [s]
-    split_ifs with hn hm hm
-    · exact f.monotone hnm
-    · exact (ht _ (hmem _)).2
-    · omega
-    · exact le_rfl
-  have hl (n : ℕ) : v ≤ s n := by
-    dsimp only [s]; split_ifs with hn
-    · exact (ht _ (hmem _)).1
-    · exact hvU
-  have hu (n : ℕ) : s n ≤ U := by
-    dsimp only [s]; split_ifs with hn
-    · exact (ht _ (hmem _)).2
-    · exact le_rfl
-  have he : (⋃ x ∈ t, E x) = ⋃ i ∈ Finset.range t.card, E (s i) := by
-    ext ω
-    simp only [mem_iUnion, Finset.mem_range]
-    constructor
-    · rintro ⟨x, hx, hω⟩
-      have hex : ∃ i, f i = x := by
-        change x ∈ Set.range f
-        rw [Finset.range_orderEmbOfFin]
-        exact hx
-      obtain ⟨i, rfl⟩ := hex
-      refine ⟨i.val, i.isLt, ?_⟩
-      simpa only [s, dite_eq_left i.isLt] using hω
-    · rintro ⟨i, hi, hω⟩
-      refine ⟨f ⟨i, hi⟩, hmem _, ?_⟩
-      simpa only [s, dite_eq_left hi] using hω
-  rw [he]
-  exact h s hs hl hu t.card
-
-lemma countable_grid_from_monotone
-    {Ω T : Type*} [MeasurableSpace Ω] [LinearOrder T]
-    (Q : Measure Ω) (E : T → Set Ω) (v U : T) (hvU : v ≤ U) (c : ℝ≥0∞)
-    (h : ∀ s : ℕ → T, Monotone s → (∀ n, v ≤ s n) → (∀ n, s n ≤ U) →
-      ∀ N, Q (⋃ i ∈ Finset.range N, E (s i)) ≤ c)
-    (s : ℕ → T) (hs : ∀ n, s n ∈ Icc v U) : Q (⋃ n, E (s n)) ≤ c := by
-  have hf (N : ℕ) : Q (⋃ i ∈ Finset.range N, E (s i)) ≤ c := by
-    have hh := finite_grid_from_monotone Q E v U hvU c h
-      ((Finset.range N).image s) (by simpa using (fun i (_ : i ∈ Finset.range N) => hs i))
-    simpa using hh
-  have he : (⋃ n, E (s n)) = ⋃ N, ⋃ i ∈ Finset.range N, E (s i) := by
-    ext ω
-    simp only [mem_iUnion, Finset.mem_range]
-    constructor
-    · rintro ⟨i, hi⟩; exact ⟨i+1, i, Nat.lt_succ_self i, hi⟩
-    · rintro ⟨N, i, _, hi⟩; exact ⟨i, hi⟩
-  have hmono : Monotone (fun N => ⋃ i ∈ Finset.range N, E (s i)) := by
-    intro n m hnm ω hω
-    obtain ⟨i, hi, hω⟩ := mem_iUnion₂.mp hω
-    exact mem_iUnion₂.mpr ⟨i, Finset.mem_range.mpr ((Finset.mem_range.mp hi).trans_le hnm), hω⟩
-  rw [he, hmono.measure_iUnion]
-  exact iSup_le hf
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set MeasureTheory Classical TopologicalSpace
-open scoped ENNReal Topology NNReal
-namespace SLEExactGauge
-
-lemma continuous_crossing_from_monotone
-    {Ω : Type*} [MeasurableSpace Ω]
-    (Q : Measure Ω) (K : ℝ≥0 → Ω → ℝ) (live : Set Ω)
-    (v w U : ℝ≥0) (hvw : v ≤ w) (hwU : w ≤ U) (δ : ℝ) (c : ℝ≥0∞)
-    (hc : ∀ ω ∈ live, ContinuousOn (fun s => K s ω) (Icc v w))
-    (hp : ∀ ω ∈ live, ∀ s ∈ Icc v w, 0 < K s ω)
-    (h : ∀ s : ℕ → ℝ≥0, Monotone s → (∀ n, v ≤ s n) → (∀ n, s n ≤ U) →
-      ∀ N, Q (⋃ i ∈ Finset.range N, {ω | 0 < K (s i) ω ∧ K (s i) ω ≤ δ}) ≤ c) :
-    Q {ω | ω ∈ live ∧ ∃ s ∈ Icc v w, K s ω < δ} ≤ c := by
-  let T := ↥(Icc v w)
-  let : Nonempty T := ⟨⟨v, le_rfl, hvw⟩⟩
-  let s (n : ℕ) : ℝ≥0 := (denseSeq T n).val
-  have hs (n : ℕ) : s n ∈ Icc v U :=
-    ⟨(denseSeq T n).property.1, (denseSeq T n).property.2.trans hwU⟩
-  refine (measure_mono (show {ω | ω ∈ live ∧ ∃ s ∈ Icc v w, K s ω < δ} ⊆
-    ⋃ n, {ω | 0 < K (s n) ω ∧ K (s n) ω ≤ δ} from ?_)).trans
-    (countable_grid_from_monotone Q (fun s => {ω | 0 < K s ω ∧ K s ω ≤ δ})
-      v U (hvw.trans hwU) c h s hs)
-  intro ω hω
-  have hcont : Continuous (fun t : T => K t.val ω) :=
-    (hc ω hω.1).domRestrict
-  obtain ⟨t, ht, hlt⟩ := hω.2
-  obtain ⟨n, hn⟩ := (denseRange_denseSeq T).exists_mem_open
-    (isOpen_lt hcont continuous_const) ⟨⟨t, ht⟩, hlt⟩
-  exact mem_iUnion.mpr ⟨n, hp ω hω.1 _ (denseSeq T n).property, hn.le⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
-
-lemma terminalRadiusKernel_continuousOn
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ)
-    (U : ℝ≥0) (ω : Ω) (hω : localizedRadiusStop (a U) ω < ⊤) :
-    ContinuousOn (fun s : ℝ≥0 => limitRadiusKernel (a s) p α s ω) (Icc 0 U) := by
-  have he (s : ℝ≥0) (hs : s ∈ Icc 0 U) :
-      limitRadiusKernel (a s) p α s ω = limitRadiusKernel (a U) p α s ω :=
-    limitRadiusKernel_eq_below (a s) (a U) hε (show (s : ℝ) ≤ U from hs.2) p α s
-      s.coe_nonneg le_rfl ω
-  have hc := (limitRadiusKernel_continuousOn_of_live (a U) hε hBc p α ω hω).comp
-    continuous_subtype_val.continuousOn (show MapsTo (fun s : ℝ≥0 => (s : ℝ)) (Icc 0 U) (Icc 0 (U : ℝ)) from
-      fun s hs => ⟨s.coe_nonneg, hs.2⟩)
-  exact hc.congr he
-
-lemma terminalRadiusKernel_pos
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (p α : ℝ) (U s : ℝ≥0) (hs : s ≤ U) (ω : Ω)
-    (hω : localizedRadiusStop (a U) ω < ⊤) :
-    0 < limitRadiusKernel (a s) p α s ω := by
-  rw [limitRadiusKernel_eq_below (a s) (a U) hε (show (s : ℝ) ≤ U from hs) p α s
-    s.coe_nonneg le_rfl ω]
-  exact limitRadiusKernel_pos_of_live (a U) hε p α s ⟨s.coe_nonneg, hs⟩ ω hω
-
-lemma continuous_radius_scaled_crossing
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hz : z ∈ upperHalfPlane)
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (v w U : ℝ≥0) (hv : 1 ≤ (v : ℝ)) (hvw : v ≤ w) (hwU : w ≤ U)
-    (L : ℝ) (hL : 0 < L) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w,
-        limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
-          Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L} ≤
-      ENNReal.ofReal (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ * ((w : ℝ) - v)) / L) := by
-  apply continuous_crossing_from_monotone _ _ _ v w U hvw hwU
-  · intro ω hω
-    exact (terminalRadiusKernel_continuousOn a hε hBc _ _ U ω hω).mono
-      (Icc_subset_Icc (show (0 : ℝ≥0) ≤ v from bot_le) hwU)
-  · intro ω hω s hs
-    exact terminalRadiusKernel_pos a hε _ _ U s (hs.2.trans hwU) ω hω
-  · intro s hs hvs hsU N
-    exact finite_radius_scaled_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz
-      a hε hεlim v U hv s hs hvs hsU w L hL N
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
-
-noncomputable def radiusInverseAngle
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (s : ℝ≥0) (ω : Ω) : ℝ :=
-  (limitRadiusKernel (a s) 0 2 s ω) ^ (-greenExponent κ / 2)
-
-lemma radiusInverseAngle_product
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (s : ℝ≥0) (ω : Ω) (hω : localizedRadiusStop (a s) ω < ⊤) :
-    limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω * radiusInverseAngle a s ω =
-      Real.exp (codimension κ * ((s : ℝ) - Real.log 2 - Real.log z.im)) := by
-  rw [radiusInverseAngle, ← limitRadiusKernel_inverse_factor (a s) hε, sub_self,
-    radius_limit_clock_angle (a s) hε (codimension κ) 0 s ⟨s.coe_nonneg, le_rfl⟩ ω hω]
-  simp
-
-lemma radiusInverseAngle_large_imp_crossing
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
-    (hκ8 : κ < 8) (s w U : ℝ≥0) (hsw : s ≤ w) (hwU : w ≤ U)
-    (L : ℝ) (hL : 0 < L) (ω : Ω) (hω : localizedRadiusStop (a U) ω < ⊤)
-    (hV : L < radiusInverseAngle a s ω) :
-    limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
-      Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L := by
-  have hls : localizedRadiusStop (a s) ω < ⊤ :=
-    lt_of_le_of_lt (localizedRadiusStop_mono (a s) (a U) (show (s : ℝ) ≤ U from hsw.trans hwU) ω) hω
-  have hK := terminalRadiusKernel_pos a hε (codimension κ) (greenExponent κ) U s (hsw.trans hwU) ω hω
-  apply (lt_div_iff₀ hL).mpr
-  calc
-    _ < limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω * radiusInverseAngle a s ω :=
-      mul_lt_mul_of_pos_left hV hK
-    _ = Real.exp (codimension κ * ((s : ℝ) - Real.log 2 - Real.log z.im)) :=
-      radiusInverseAngle_product a hε s ω hls
-    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (sub_le_sub_right (sub_le_sub_right (show (s : ℝ) ≤ w from hsw) _) _) (codimension_pos κ hκ8).le)
-
-lemma inverse_angle_window_bound
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hz : z ∈ upperHalfPlane)
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
-    (v w U : ℝ≥0) (hv : 1 ≤ (v : ℝ)) (hvw : v ≤ w) (hwU : w ≤ U)
-    (L : ℝ) (hL : 0 < L) :
-    finiteCenteredLaw P κ B γ G z U
-      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w, L < radiusInverseAngle a s ω} ≤
-      ENNReal.ofReal (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ * ((w : ℝ) - v)) / L) := by
-  refine (measure_mono (s := {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w, L < radiusInverseAngle a s ω})
-    (t := {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w,
-      limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
-        Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L}) ?_).trans
-    (continuous_radius_scaled_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim
-      v w U hv hvw hwU L hL)
-  rintro ω ⟨hω, s, hs, hV⟩
-  exact ⟨hω, s, hs, radiusInverseAngle_large_imp_crossing a hε hκ8 s w U hs.2 hwU L hL ω hω hV⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.InverseMoments
+import OAI.Probability.SLE.SausagePointBound
+import OAI.Probability.SLE.SpatialMeans
+
+namespace OAI
+
+/-! Measure Support for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable def spatialGreenMass (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
+    (G : ℝ≥0 → ℂ → ℂ) (r : ℝ) (A : Set ℂ) : ℝ≥0∞ :=
+  ∫⁻ z in A, ENNReal.ofReal (spatialGreenWeight κ U γ G z r)
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+lemma spatialGreenMass_measurable (r : ℝ) (A : Set ℂ) :
+    Measurable (fun ω : a.samples => spatialGreenMass κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A) :=
+  (a.spatialGreenWeight_measurable r).ennreal_ofReal.lintegral_prod_left'
+
+ 
+
+lemma spatialGreenMass_expectation [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, r ≤ z.im) :
+    (∫⁻ ω : a.samples, spatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
+      (γ ω) (a.G ω) r A ∂(P.comap Subtype.val)) =
+      ∫⁻ z in A, ENNReal.ofReal (initialGreen κ z) := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  simp only [spatialGreenMass]
+  rw [← lintegral_lintegral_swap (a.spatialGreenWeight_measurable r).ennreal_ofReal.aemeasurable]
+  apply setLIntegral_congr_fun hA
+  intro z hz
+  obtain ⟨hi, he⟩ := a.spatialGreen_expectation hκ hκ8 z r hr (hheight z hz)
+  change (∫⁻ ω : a.samples, ENNReal.ofReal (spatialGreenWeight κ
+    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) ∂μ) = ENNReal.ofReal (initialGreen κ z)
+  rw [← he]
+  exact (ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall fun ω => by
+    dsimp only [Pi.zero_apply, spatialGreenWeight]
+    split_ifs
+    · exact le_rfl
+    · exact greenDensity_nonneg (a.witness ω) κ _ z)).symm
+
+ 
+
+lemma spatialGreenMass_expectation_le [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, r ≤ z.im) :
+    (∫⁻ ω : a.samples, spatialGreenMass κ (fun t => Real.sqrt κ * a.B t ω)
+      (γ ω) (a.G ω) r A ∂(P.comap Subtype.val)) ≤
+      ENNReal.ofReal ((2 : ℝ) ^ (-codimension κ) * r ^ (-codimension κ)) * volume A := by
+  rw [a.spatialGreenMass_expectation hκ hκ8 r hr A hA hheight]
+  calc
+    _ ≤ ∫⁻ _z in A, ENNReal.ofReal ((2 : ℝ) ^ (-codimension κ) * r ^ (-codimension κ)) := by
+      apply setLIntegral_mono' hA
+      intro z hz
+      apply ENNReal.ofReal_le_ofReal
+      calc
+        _ ≤ (2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ) :=
+          initialGreen_le_height κ hκ hκ8 z (hr.trans_le (hheight z hz))
+        _ ≤ _ := mul_le_mul_of_nonneg_left
+          (Real.rpow_le_rpow_of_nonpos hr (hheight z hz) (neg_nonpos.mpr (codimension_pos κ hκ8).le))
+          (Real.rpow_nonneg (by norm_num) _)
+    _ = _ := by rw [lintegral_const, Measure.restrict_apply_univ]
+
+end RegularSLEPresentation
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma finite_density_crossing_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P Q : Measure Ω)
+    (F : Filtration ℕ mΩ) (B : ℕ → Set Ω)
+    (hBm : ∀ n, MeasurableSet[F n] (B n)) (c : ℝ≥0∞)
+    (hQ : ∀ n A, MeasurableSet[F n] A → A ⊆ B n → Q A ≤ c * P A)
+    (n : ℕ) : Q (⋃ i ∈ Finset.range n, B i) ≤ c * P (⋃ i ∈ Finset.range n, B i) := by
+  classical
+  let E (n : ℕ) : Set Ω := ⋃ i ∈ Finset.range n, B i
+  have hEm (n : ℕ) : MeasurableSet[F n] (E n) := by
+    apply MeasurableSet.iUnion
+    intro i
+    apply MeasurableSet.iUnion
+    intro hi
+    exact F.mono (Nat.le_of_lt (Finset.mem_range.mp hi)) _ (hBm i)
+  have hrec (n : ℕ) : E (n + 1) = E n ∪ (B n \ E n) := by
+    calc
+      E (n + 1) = B n ∪ E n := by
+        ext ω
+        simp only [E, Finset.range_add_one, Finset.mem_insert, mem_iUnion, mem_union]
+        aesop
+      _ = E n ∪ (B n \ E n) := by
+        ext ω
+        by_cases h : ω ∈ E n <;> simp [h]
+  change Q (E n) ≤ c * P (E n)
+  induction n with
+  | zero => simp [E]
+  | succ n ih =>
+      have hD := (hBm n).diff (hEm n)
+      have hd : Disjoint (E n) (B n \ E n) := disjoint_sdiff_right
+      rw [hrec n, measure_union hd (F.le n _ hD), measure_union hd (F.le n _ hD), mul_add]
+      exact add_le_add ih (hQ n _ hD sdiff_subset)
+
+lemma finite_density_crossing_bound_of_subset
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (P Q : Measure Ω)
+    (F : Filtration ℕ mΩ) (B : ℕ → Set Ω)
+    (hBm : ∀ n, MeasurableSet[F n] (B n)) (c : ℝ≥0∞)
+    (hQ : ∀ n A, MeasurableSet[F n] A → A ⊆ B n → Q A ≤ c * P A)
+    (A : Set Ω) (hA : ∀ n, B n ⊆ A) (n : ℕ) :
+    Q (⋃ i ∈ Finset.range n, B i) ≤ c * P A := by
+  refine (finite_density_crossing_bound P Q F B hBm c hQ n).trans ?_
+  exact mul_le_mul (show c ≤ c from le_rfl)
+    (show P (⋃ i ∈ Finset.range n, B i) ≤ P A from
+      measure_mono (iUnion₂_subset fun i _ => hA i)) bot_le bot_le
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u : ℝ} {ε : ℕ → ℝ}
+
+lemma limitRadiusKernel_pos_iff
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hu : 0 ≤ u)
+    (p α : ℝ) (ω : Ω) :
+    0 < limitRadiusKernel a p α u ω ↔ localizedRadiusStop a ω < ⊤ := by
+  have hx : (∃ n, max u 0 ≤ (a n).clock.S ((a n).τ ω) ω) ↔ ∃ n, ω ∈ (a n).hit := by
+    simp only [max_eq_left hu]
+    constructor
+    · rintro ⟨n, hn⟩
+      exact ⟨n, le_antisymm ((a n).clock_upper ω) hn⟩
+    · rintro ⟨n, hn⟩
+      exact ⟨n, le_of_eq hn.symm⟩
+  by_cases hh : ∃ n, ω ∈ (a n).hit
+  · have he := hx.mpr hh
+    obtain ⟨n, hn⟩ := hh
+    rw [localizedRadiusStop_eq_hit a ω n hn, limitRadiusKernel, dite_eq_left he,
+      AngularLocalization.radiusKernel, ite_eq_left (Nat.find_spec he)]
+    exact iff_of_true (Real.exp_pos _) (WithTop.coe_lt_top _)
+  · rw [localizedRadiusStop_eq_top a ω (not_exists.mp hh), limitRadiusKernel,
+      dite_eq_right (fun h => hh (hx.mp h))]
+    simp
+
+lemma finiteCenteredLaw_eq_radiusKernel
+    {P : Measure Ω} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) :
+    finiteCenteredLaw P κ B γ G z u = P.withDensity
+      (fun ω => ENNReal.ofReal (limitRadiusKernel a (codimension κ) (greenExponent κ) u ω /
+        initialGreen κ z)) := by
+  apply withDensity_congr_ae
+  filter_upwards [hw] with ω hω
+  rw [limitRadiusKernel_eq_stoppedGreen a hε hεlim hBc hz hu ω (γ ω) (G ω) (F ω) hω]
+
+lemma radiusKernel_density_bound
+    {P : Measure Ω} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hz : z ∈ upperHalfPlane) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (δ : ℝ) (A : Set Ω) (hA : MeasurableSet A)
+    (hδ : ∀ ω ∈ A, limitRadiusKernel a (codimension κ) (greenExponent κ) u ω ≤ δ) :
+    finiteCenteredLaw P κ B γ G z u A ≤ ENNReal.ofReal (δ / initialGreen κ z) * P A := by
+  rw [finiteCenteredLaw_eq_radiusKernel a hε hεlim hBc hz hu γ G F hw, withDensity_apply _ hA]
+  calc
+    _ ≤ ∫⁻ ω in A, ENNReal.ofReal (δ / initialGreen κ z) ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem hA] with ω hω
+      exact ENNReal.ofReal_le_ofReal (div_le_div_of_nonneg_right (hδ ω hω) (initialGreen_pos κ z hz).le)
+    _ = _ := by simp
+
+lemma finiteCenteredLaw_consistent_at_radius
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) (hz : z ∈ upperHalfPlane)
+    {v : ℝ} (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (b : ∀ n, AngularLocalization B hm κ z v (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hu : 0 ≤ u) (huv : u ≤ v)
+    (A : Set Ω) (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
+    finiteCenteredLaw P κ B γ G z v A = finiteCenteredLaw P κ B γ G z u A := by
+  have hAm := (localizedRadiusStop_stopping a).measurableSpace_le _ hA
+  rw [finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz v (hu.trans huv) A hAm,
+    finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz u hu A hAm,
+    finite_stopped_green_consistent_regular hB hBm hBc hB0 hκ hκ8 γ G F hw hz
+      a b hεlim hε hu huv A hA]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma finite_radius_density_crossing
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (v U : ℝ≥0) (s : ℕ → ℝ≥0) (hs : Monotone s)
+    (hvs : ∀ n, v ≤ s n) (hsU : ∀ n, s n ≤ U) (δ : ℝ) (N : ℕ) :
+    finiteCenteredLaw P κ B γ G z U
+      (⋃ i ∈ Finset.range N, {ω | 0 < limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ∧
+        limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ≤ δ}) ≤
+      ENNReal.ofReal (δ / initialGreen κ z) * P {ω | localizedRadiusStop (a v) ω < ⊤} := by
+  let K (n : ℕ) := limitRadiusKernel (a (s n)) (codimension κ) (greenExponent κ) (s n)
+  let E (n : ℕ) : Set Ω := {ω | 0 < K n ω ∧ K n ω ≤ δ}
+  let H : Filtration ℕ mΩ :=
+    { seq := fun n => radiusFiltration a (s n)
+      mono' := fun n m hnm => (radiusFiltration a).mono (hs hnm)
+      le' := fun n => (radiusFiltration a).le (s n) }
+  have hKm (n : ℕ) : Measurable[H n] (K n) :=
+    limitRadiusKernel_terminal_measurable (a (s n)) hε hBc (s n).coe_nonneg _ _
+  have hEm (n : ℕ) : MeasurableSet[H n] (E n) :=
+    measurableSet_Ioc.preimage (hKm n)
+  apply finite_density_crossing_bound_of_subset P (finiteCenteredLaw P κ B γ G z U)
+    H E hEm (ENNReal.ofReal (δ / initialGreen κ z))
+  · intro n A hA hAE
+    rw [finiteCenteredLaw_consistent_at_radius hB hBm hBc hB0 hκ hκ8 γ G F hw hz
+      (a (s n)) (a U) hε hεlim (s n).coe_nonneg (show (s n : ℝ) ≤ U from hsU n) A hA]
+    exact radiusKernel_density_bound (a (s n)) hε hεlim hBc hz (s n).coe_nonneg γ G F hw
+      δ A (H.le n _ hA) (fun ω hω => (hAE hω).2)
+  · intro n ω hω
+    have hp := (limitRadiusKernel_pos_iff (a (s n)) (s n).coe_nonneg
+      (codimension κ) (greenExponent κ) ω).mp hω.1
+    exact lt_of_le_of_lt (localizedRadiusStop_mono (a v) (a (s n))
+      (show (v : ℝ) ≤ s n from hvs n) ω) hp
+
+lemma radius_kernel_scale_cancel (κ v w L C : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (hL : L ≠ 0) :
+    (Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L / initialGreen κ z) *
+      (C * Real.exp (-codimension κ * v) * (Real.sin (Complex.arg z)) ^ greenExponent κ) =
+      C * Real.exp (codimension κ * (w - v)) / L := by
+  have hI := (initialGreen_pos κ z hz).ne'
+  have he : Real.exp (-codimension κ * v) * (Real.sin (Complex.arg z)) ^ greenExponent κ =
+      initialGreen κ z / Real.exp (codimension κ * (v - Real.log 2 - Real.log z.im)) :=
+    (initialGreen_div_zeroKernel κ v z hz).symm
+  rw [mul_assoc C, he]
+  rw [show (Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L / initialGreen κ z) *
+      (C * (initialGreen κ z / Real.exp (codimension κ * (v - Real.log 2 - Real.log z.im)))) =
+      C * (Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) /
+        Real.exp (codimension κ * (v - Real.log 2 - Real.log z.im))) / L by
+      field_simp]
+  rw [← Real.exp_sub]
+  congr 2
+  ring_nf
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma finite_radius_scaled_crossing
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    {z : ℂ} (hz : z ∈ upperHalfPlane) {ε : ℕ → ℝ}
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (v U : ℝ≥0) (hv : 1 ≤ (v : ℝ)) (s : ℕ → ℝ≥0) (hs : Monotone s)
+    (hvs : ∀ n, v ≤ s n) (hsU : ∀ n, s n ≤ U) (w L : ℝ) (hL : 0 < L) (N : ℕ) :
+    finiteCenteredLaw P κ B γ G z U
+      (⋃ i ∈ Finset.range N, {ω | 0 < limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ∧
+        limitRadiusKernel (a (s i)) (codimension κ) (greenExponent κ) (s i) ω ≤
+          Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L}) ≤
+      ENNReal.ofReal (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ * (w - v)) / L) := by
+  let C := uniformInverseAngleConstant κ 1
+  have hC : 0 < C := uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num)
+  let Kv := Real.exp (codimension κ * ((v : ℝ) - Real.log 2 - Real.log z.im))
+  let δ := Real.exp (codimension κ * (w - Real.log 2 - Real.log z.im)) / L
+  have hδ : 0 ≤ δ / initialGreen κ z := div_nonneg (div_pos (Real.exp_pos _) hL).le
+    (initialGreen_pos κ z hz).le
+  have hPe : P {ω | localizedRadiusStop (a v) ω < ⊤} =
+      P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞} := by
+    apply measure_congr
+    filter_upwards [hw] with ω hω
+    rw [localizedRadiusStop_eq_source (a v) hεlim hBc hz ω (γ ω) (G ω) (F ω) hω]
+    rfl
+  have hPr : (P {ω | localizedRadiusStop (a v) ω < ⊤}).toReal ≤ C * initialGreen κ z / Kv := by
+    rw [hPe]
+    refine (radius_approach_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz v hv).trans_eq ?_
+    dsimp only [C, Kv]
+    rw [mul_div_assoc, initialGreen_div_zeroKernel κ v z hz, mul_assoc]
+  have hPn : P {ω | localizedRadiusStop (a v) ω < ⊤} ≤ ENNReal.ofReal (C * initialGreen κ z / Kv) :=
+    (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (div_nonneg (mul_nonneg hC.le (initialGreen_pos κ z hz).le) (Real.exp_pos _).le)).mpr hPr
+  calc
+    _ ≤ ENNReal.ofReal (δ / initialGreen κ z) * P {ω | localizedRadiusStop (a v) ω < ⊤} :=
+      finite_radius_density_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim v U s hs hvs hsU δ N
+    _ ≤ ENNReal.ofReal (δ / initialGreen κ z) * ENNReal.ofReal (C * initialGreen κ z / Kv) :=
+      mul_le_mul le_rfl hPn bot_le bot_le
+    _ = ENNReal.ofReal ((δ / initialGreen κ z) * (C * initialGreen κ z / Kv)) :=
+      (ENNReal.ofReal_mul hδ).symm
+    _ = _ := by
+      congr 1
+      have he := radius_kernel_scale_cancel κ v w L C z hz hL.ne'
+      change (δ / initialGreen κ z) * (C * initialGreen κ z /
+        Real.exp (codimension κ * ((v : ℝ) - Real.log 2 - Real.log z.im))) = _
+      rw [mul_div_assoc C, initialGreen_div_zeroKernel κ v z hz, ← mul_assoc C]
+      exact he
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {u v : ℝ} {ε : ℕ → ℝ}
+
+lemma limitRadiusKernel_eq_of_surviving_cutoff
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (p α s : ℝ) (hs : 0 ≤ s) (ω : Ω) (n : ℕ)
+    (hn : s ≤ (a n).clock.S ((a n).τ ω) ω) :
+    limitRadiusKernel a p α s ω = (a n).radiusValue p α s ω := by
+  obtain ⟨m, hm, hlim⟩ := ((eventually_ge_atTop n).and (radiusKernel_eventually_limit a hε p α s ω)).exists
+  rw [max_eq_left hs] at hlim
+  rw [← hlim, ← (a n).radiusKernel_eq_of_live (a m) le_rfl (hε hm) p α s hs ω hn,
+    AngularLocalization.radiusKernel, ite_eq_left hn]
+
+lemma limitRadiusKernel_eq_below
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (b : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
+    (huv : u ≤ v) (p α s : ℝ) (hs : 0 ≤ s) (hsu : s ≤ u) (ω : Ω) :
+    limitRadiusKernel a p α s ω = limitRadiusKernel b p α s ω := by
+  obtain ⟨n, hn, hm⟩ := ((radiusKernel_eventually_limit a hε p α s ω).and
+    (radiusKernel_eventually_limit b hε p α s ω)).exists
+  rw [max_eq_left hs] at hn hm
+  rw [← hn, ← hm]
+  exact (a n).radiusKernel_eq_below (b n) huv p α s hs hsu ω
+
+lemma exists_cutoff_of_radiusStop_lt_top
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (ω : Ω)
+    (hω : localizedRadiusStop a ω < ⊤) : ∃ n, ω ∈ (a n).hit := by
+  by_contra hn
+  rw [localizedRadiusStop_eq_top a ω (not_exists.mp hn)] at hω
+  exact lt_irrefl _ hω
+
+lemma limitRadiusKernel_continuousOn_of_live
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ) (ω : Ω)
+    (hω : localizedRadiusStop a ω < ⊤) :
+    ContinuousOn (fun s => limitRadiusKernel a p α s ω) (Icc 0 u) := by
+  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top a ω hω
+  apply ((a n).radiusValue_continuous hBc p α ω).continuousOn.congr
+  intro s hs
+  exact limitRadiusKernel_eq_of_surviving_cutoff a hε p α s hs.1 ω n (hs.2.trans_eq hn.symm)
+
+lemma limitRadiusKernel_pos_of_live
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (p α s : ℝ) (hs : s ∈ Icc 0 u) (ω : Ω) (hω : localizedRadiusStop a ω < ⊤) :
+    0 < limitRadiusKernel a p α s ω := by
+  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top a ω hω
+  rw [limitRadiusKernel_eq_of_surviving_cutoff a hε p α s hs.1 ω n (hs.2.trans_eq hn.symm)]
+  exact powerLogDensity_pos _ _ _ _ _
+
+lemma radius_limit_clock_angle
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (p α s : ℝ) (hs : s ∈ Icc 0 u) (ω : Ω) (hω : localizedRadiusStop a ω < ⊤) :
+    limitRadiusKernel a p α s ω = Real.exp (p * (s - Real.log 2 - Real.log z.im)) *
+      (limitRadiusKernel a 0 2 s ω) ^ (α / 2) := by
+  obtain ⟨n, hn⟩ := exists_cutoff_of_radiusStop_lt_top a ω hω
+  have hs' : s ≤ (a n).clock.S ((a n).τ ω) ω := hs.2.trans_eq hn.symm
+  rw [limitRadiusKernel_eq_of_surviving_cutoff a hε p α s hs.1 ω n hs',
+    limitRadiusKernel_eq_of_surviving_cutoff a hε 0 2 s hs.1 ω n hs']
+  unfold AngularLocalization.radiusValue
+  rw [powerLogDensity_radius_angle κ p α (Real.log z.im),
+    powerLogDensity_radius_angle κ 0 2 (Real.log z.im)]
+  norm_num only [zero_mul, Real.exp_zero, one_mul, div_self (by norm_num : (2 : ℝ) ≠ 0), Real.rpow_one]
+  have he := (a n).clock.clock_inverse s ω
+  change (a n).clock.S ((a n).clock.inverse s ω) ω =
+    min (max s 0) ((a n).clock.S ((a n).τ ω) ω) at he
+  rw [max_eq_left hs.1, min_eq_left hs'] at he
+  change logRadiusClock (Real.log z.im)
+    ((a n).solution.Q ((a n).clock.inverse s ω) ω) = s at he
+  rw [he]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set MeasureTheory Classical
+open scoped ENNReal
+namespace SLEExactGauge
+
+lemma finite_grid_from_monotone
+    {Ω T : Type*} [MeasurableSpace Ω] [LinearOrder T]
+    (Q : Measure Ω) (E : T → Set Ω) (v U : T) (hvU : v ≤ U) (c : ℝ≥0∞)
+    (h : ∀ s : ℕ → T, Monotone s → (∀ n, v ≤ s n) → (∀ n, s n ≤ U) →
+      ∀ N, Q (⋃ i ∈ Finset.range N, E (s i)) ≤ c)
+    (t : Finset T) (ht : ∀ x ∈ t, x ∈ Icc v U) :
+    Q (⋃ x ∈ t, E x) ≤ c := by
+  let f := t.orderEmbOfFin rfl
+  let s (n : ℕ) : T := if hn : n < t.card then f ⟨n, hn⟩ else U
+  have hmem (i : Fin t.card) : f i ∈ t := t.orderEmbOfFin_mem rfl i
+  have hs : Monotone s := by
+    intro n m hnm
+    dsimp only [s]
+    split_ifs with hn hm hm
+    · exact f.monotone hnm
+    · exact (ht _ (hmem _)).2
+    · omega
+    · exact le_rfl
+  have hl (n : ℕ) : v ≤ s n := by
+    dsimp only [s]; split_ifs with hn
+    · exact (ht _ (hmem _)).1
+    · exact hvU
+  have hu (n : ℕ) : s n ≤ U := by
+    dsimp only [s]; split_ifs with hn
+    · exact (ht _ (hmem _)).2
+    · exact le_rfl
+  have he : (⋃ x ∈ t, E x) = ⋃ i ∈ Finset.range t.card, E (s i) := by
+    ext ω
+    simp only [mem_iUnion, Finset.mem_range]
+    constructor
+    · rintro ⟨x, hx, hω⟩
+      have hex : ∃ i, f i = x := by
+        change x ∈ Set.range f
+        rw [Finset.range_orderEmbOfFin]
+        exact hx
+      obtain ⟨i, rfl⟩ := hex
+      refine ⟨i.val, i.isLt, ?_⟩
+      simpa only [s, dite_eq_left i.isLt] using hω
+    · rintro ⟨i, hi, hω⟩
+      refine ⟨f ⟨i, hi⟩, hmem _, ?_⟩
+      simpa only [s, dite_eq_left hi] using hω
+  rw [he]
+  exact h s hs hl hu t.card
+
+lemma countable_grid_from_monotone
+    {Ω T : Type*} [MeasurableSpace Ω] [LinearOrder T]
+    (Q : Measure Ω) (E : T → Set Ω) (v U : T) (hvU : v ≤ U) (c : ℝ≥0∞)
+    (h : ∀ s : ℕ → T, Monotone s → (∀ n, v ≤ s n) → (∀ n, s n ≤ U) →
+      ∀ N, Q (⋃ i ∈ Finset.range N, E (s i)) ≤ c)
+    (s : ℕ → T) (hs : ∀ n, s n ∈ Icc v U) : Q (⋃ n, E (s n)) ≤ c := by
+  have hf (N : ℕ) : Q (⋃ i ∈ Finset.range N, E (s i)) ≤ c := by
+    have hh := finite_grid_from_monotone Q E v U hvU c h
+      ((Finset.range N).image s) (by simpa using (fun i (_ : i ∈ Finset.range N) => hs i))
+    simpa using hh
+  have he : (⋃ n, E (s n)) = ⋃ N, ⋃ i ∈ Finset.range N, E (s i) := by
+    ext ω
+    simp only [mem_iUnion, Finset.mem_range]
+    constructor
+    · rintro ⟨i, hi⟩; exact ⟨i+1, i, Nat.lt_succ_self i, hi⟩
+    · rintro ⟨N, i, _, hi⟩; exact ⟨i, hi⟩
+  have hmono : Monotone (fun N => ⋃ i ∈ Finset.range N, E (s i)) := by
+    intro n m hnm ω hω
+    obtain ⟨i, hi, hω⟩ := mem_iUnion₂.mp hω
+    exact mem_iUnion₂.mpr ⟨i, Finset.mem_range.mpr ((Finset.mem_range.mp hi).trans_le hnm), hω⟩
+  rw [he, hmono.measure_iUnion]
+  exact iSup_le hf
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set MeasureTheory Classical TopologicalSpace
+open scoped ENNReal Topology NNReal
+namespace SLEExactGauge
+
+lemma continuous_crossing_from_monotone
+    {Ω : Type*} [MeasurableSpace Ω]
+    (Q : Measure Ω) (K : ℝ≥0 → Ω → ℝ) (live : Set Ω)
+    (v w U : ℝ≥0) (hvw : v ≤ w) (hwU : w ≤ U) (δ : ℝ) (c : ℝ≥0∞)
+    (hc : ∀ ω ∈ live, ContinuousOn (fun s => K s ω) (Icc v w))
+    (hp : ∀ ω ∈ live, ∀ s ∈ Icc v w, 0 < K s ω)
+    (h : ∀ s : ℕ → ℝ≥0, Monotone s → (∀ n, v ≤ s n) → (∀ n, s n ≤ U) →
+      ∀ N, Q (⋃ i ∈ Finset.range N, {ω | 0 < K (s i) ω ∧ K (s i) ω ≤ δ}) ≤ c) :
+    Q {ω | ω ∈ live ∧ ∃ s ∈ Icc v w, K s ω < δ} ≤ c := by
+  let T := ↥(Icc v w)
+  let : Nonempty T := ⟨⟨v, le_rfl, hvw⟩⟩
+  let s (n : ℕ) : ℝ≥0 := (denseSeq T n).val
+  have hs (n : ℕ) : s n ∈ Icc v U :=
+    ⟨(denseSeq T n).property.1, (denseSeq T n).property.2.trans hwU⟩
+  refine (measure_mono (show {ω | ω ∈ live ∧ ∃ s ∈ Icc v w, K s ω < δ} ⊆
+    ⋃ n, {ω | 0 < K (s n) ω ∧ K (s n) ω ≤ δ} from ?_)).trans
+    (countable_grid_from_monotone Q (fun s => {ω | 0 < K s ω ∧ K s ω ≤ δ})
+      v U (hvw.trans hwU) c h s hs)
+  intro ω hω
+  have hcont : Continuous (fun t : T => K t.val ω) :=
+    (hc ω hω.1).domRestrict
+  obtain ⟨t, ht, hlt⟩ := hω.2
+  obtain ⟨n, hn⟩ := (denseRange_denseSeq T).exists_mem_open
+    (isOpen_lt hcont continuous_const) ⟨⟨t, ht⟩, hlt⟩
+  exact mem_iUnion.mpr ⟨n, hp ω hω.1 _ (denseSeq T n).property, hn.le⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
+
+lemma terminalRadiusKernel_continuousOn
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (p α : ℝ)
+    (U : ℝ≥0) (ω : Ω) (hω : localizedRadiusStop (a U) ω < ⊤) :
+    ContinuousOn (fun s : ℝ≥0 => limitRadiusKernel (a s) p α s ω) (Icc 0 U) := by
+  have he (s : ℝ≥0) (hs : s ∈ Icc 0 U) :
+      limitRadiusKernel (a s) p α s ω = limitRadiusKernel (a U) p α s ω :=
+    limitRadiusKernel_eq_below (a s) (a U) hε (show (s : ℝ) ≤ U from hs.2) p α s
+      s.coe_nonneg le_rfl ω
+  have hc := (limitRadiusKernel_continuousOn_of_live (a U) hε hBc p α ω hω).comp
+    continuous_subtype_val.continuousOn (show MapsTo (fun s : ℝ≥0 => (s : ℝ)) (Icc 0 U) (Icc 0 (U : ℝ)) from
+      fun s hs => ⟨s.coe_nonneg, hs.2⟩)
+  exact hc.congr he
+
+lemma terminalRadiusKernel_pos
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (p α : ℝ) (U s : ℝ≥0) (hs : s ≤ U) (ω : Ω)
+    (hω : localizedRadiusStop (a U) ω < ⊤) :
+    0 < limitRadiusKernel (a s) p α s ω := by
+  rw [limitRadiusKernel_eq_below (a s) (a U) hε (show (s : ℝ) ≤ U from hs) p α s
+    s.coe_nonneg le_rfl ω]
+  exact limitRadiusKernel_pos_of_live (a U) hε p α s ⟨s.coe_nonneg, hs⟩ ω hω
+
+lemma continuous_radius_scaled_crossing
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hz : z ∈ upperHalfPlane)
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (v w U : ℝ≥0) (hv : 1 ≤ (v : ℝ)) (hvw : v ≤ w) (hwU : w ≤ U)
+    (L : ℝ) (hL : 0 < L) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w,
+        limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
+          Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L} ≤
+      ENNReal.ofReal (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ * ((w : ℝ) - v)) / L) := by
+  apply continuous_crossing_from_monotone _ _ _ v w U hvw hwU
+  · intro ω hω
+    exact (terminalRadiusKernel_continuousOn a hε hBc _ _ U ω hω).mono
+      (Icc_subset_Icc (show (0 : ℝ≥0) ≤ v from bot_le) hwU)
+  · intro ω hω s hs
+    exact terminalRadiusKernel_pos a hε _ _ U s (hs.2.trans hwU) ω hω
+  · intro s hs hvs hsU N
+    exact finite_radius_scaled_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz
+      a hε hεlim v U hv s hs hvs hsU w L hL N
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
+
+noncomputable def radiusInverseAngle
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (s : ℝ≥0) (ω : Ω) : ℝ :=
+  (limitRadiusKernel (a s) 0 2 s ω) ^ (-greenExponent κ / 2)
+
+lemma radiusInverseAngle_product
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (s : ℝ≥0) (ω : Ω) (hω : localizedRadiusStop (a s) ω < ⊤) :
+    limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω * radiusInverseAngle a s ω =
+      Real.exp (codimension κ * ((s : ℝ) - Real.log 2 - Real.log z.im)) := by
+  rw [radiusInverseAngle, ← limitRadiusKernel_inverse_factor (a s) hε, sub_self,
+    radius_limit_clock_angle (a s) hε (codimension κ) 0 s ⟨s.coe_nonneg, le_rfl⟩ ω hω]
+  simp
+
+lemma radiusInverseAngle_large_imp_crossing
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n)) (hε : Antitone ε)
+    (hκ8 : κ < 8) (s w U : ℝ≥0) (hsw : s ≤ w) (hwU : w ≤ U)
+    (L : ℝ) (hL : 0 < L) (ω : Ω) (hω : localizedRadiusStop (a U) ω < ⊤)
+    (hV : L < radiusInverseAngle a s ω) :
+    limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
+      Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L := by
+  have hls : localizedRadiusStop (a s) ω < ⊤ :=
+    lt_of_le_of_lt (localizedRadiusStop_mono (a s) (a U) (show (s : ℝ) ≤ U from hsw.trans hwU) ω) hω
+  have hK := terminalRadiusKernel_pos a hε (codimension κ) (greenExponent κ) U s (hsw.trans hwU) ω hω
+  apply (lt_div_iff₀ hL).mpr
+  calc
+    _ < limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω * radiusInverseAngle a s ω :=
+      mul_lt_mul_of_pos_left hV hK
+    _ = Real.exp (codimension κ * ((s : ℝ) - Real.log 2 - Real.log z.im)) :=
+      radiusInverseAngle_product a hε s ω hls
+    _ ≤ _ := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (sub_le_sub_right (sub_le_sub_right (show (s : ℝ) ≤ w from hsw) _) _) (codimension_pos κ hκ8).le)
+
+lemma inverse_angle_window_bound
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hz : z ∈ upperHalfPlane)
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0))
+    (v w U : ℝ≥0) (hv : 1 ≤ (v : ℝ)) (hvw : v ≤ w) (hwU : w ≤ U)
+    (L : ℝ) (hL : 0 < L) :
+    finiteCenteredLaw P κ B γ G z U
+      {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w, L < radiusInverseAngle a s ω} ≤
+      ENNReal.ofReal (uniformInverseAngleConstant κ 1 * Real.exp (codimension κ * ((w : ℝ) - v)) / L) := by
+  refine (measure_mono (s := {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w, L < radiusInverseAngle a s ω})
+    (t := {ω | localizedRadiusStop (a U) ω < ⊤ ∧ ∃ s ∈ Icc v w,
+      limitRadiusKernel (a s) (codimension κ) (greenExponent κ) s ω <
+        Real.exp (codimension κ * ((w : ℝ) - Real.log 2 - Real.log z.im)) / L}) ?_).trans
+    (continuous_radius_scaled_crossing hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hεlim
+      v w U hv hvw hwU L hL)
+  rintro ω ⟨hω, s, hs, hV⟩
+  exact ⟨hω, s, hs, radiusInverseAngle_large_imp_crossing a hε hκ8 s w U hs.2 hwU L hL ω hω hV⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {ε : ℕ → ℝ}
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/MomentGauge.lean b/lean/OAI/Probability/SLE/MomentGauge.lean
index bc8285709d7280cc842022903a164f0508e46b43..d5c7be82449b3514eca5ea7cb26be9e7ff12a2fe 100644
--- a/lean/OAI/Probability/SLE/MomentGauge.lean
+++ b/lean/OAI/Probability/SLE/MomentGauge.lean
@@ -1,802 +1,802 @@
-import Mathlib
-
-namespace OAI
-
-/-! Quadratic moment envelopes, Hausdorff gauge admissibility and interval transfer. -/
-
-universe u
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-open scoped ContinuousMapZero
-abbrev Driver := C(ℝ≥0, ℝ)₀
-instance : MeasurableSpace Driver := borel Driver
-instance : BorelSpace Driver := ⟨rfl⟩
-abbrev WienerLaw := Measure Driver
-
-def IsWiener (W : WienerLaw) : Prop :=
-  ProbabilityTheory.IsBrownianReal (fun t b => b t) W
-
-def a (κ : ℝ) := 2 / κ
-def d (κ : ℝ) := 1 + 1 / (4 * a κ)
-def α (κ : ℝ) := 2 - d κ
-def p (κ : ℝ) := 4 * a κ - 1
-
-def driverValue (b : Driver) (t : ℝ) : ℝ := b t.toNNReal
-
-def cutoffField (κ : ℝ) (b : Driver) (j : ℕ) (t : ℝ) (z : ℂ) : ℂ :=
-  (a κ : ℂ) * star (z - driverValue b t) /
-    (max (‖z - driverValue b t‖ ^ 2) ((j : ℝ) ^ (-(4 : ℤ))) : ℝ)
-
-def mesh (N : ℕ) : ℝ := (2 : ℝ) ^ (-(N : ℤ))
-
-def eulerSteps (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) : ℕ → ℂ
-  | 0 => z
-  | k + 1 => eulerSteps κ b j N z k +
-      mesh N * cutoffField κ b j ((k : ℝ) * mesh N) (eulerSteps κ b j N z k)
-
-def eulerFlow (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
-  let k := ⌊t / mesh N⌋₊
-  let w :=  t / mesh N - (k : ℝ)
-  (1 - w : ℝ) * eulerSteps κ b j N z k + w * eulerSteps κ b j N z (k + 1)
-
-def cutoffFlow (κ : ℝ) (b : Driver) (j : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
-  limUnder atTop (fun N : ℕ => eulerFlow κ b j N z t)
-
-def Surviving (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Prop :=
-  ∃ j : ℕ, 1 ≤ j ∧ (j : ℝ)⁻¹ <
-    sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 t)
-
-instance (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Decidable (Surviving κ b z t) :=
-  Classical.propDecidable _
-
-def flow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
-  if h : Surviving κ b z t then cutoffFlow κ b (Nat.find h) z t else 0
-
-def IsHausdorffGauge (h : ℝ → ℝ) : Prop :=
-  ContinuousOn h (Ici 0) ∧ MonotoneOn h (Ici 0) ∧ h 0 = 0 ∧ ∀ r > 0, 0 < h r
-
-def hausdorff (h : ℝ → ℝ) : OuterMeasure ℂ :=
-  OuterMeasure.mkMetric (fun r => if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal))
-
-def inverseFlow (κ : ℝ) (b : Driver) (t : ℝ) (w : ℂ) : ℂ :=
-  @ite ℂ (∃ z : ℂ, Surviving κ b z t ∧ flow κ b z t = w) (Classical.propDecidable _)
-    (Classical.epsilon (fun z : ℂ => Surviving κ b z t ∧ flow κ b z t = w)) 0
-
-def trace (κ : ℝ) (b : Driver) (t : ℝ) : ℂ :=
-  limUnder (nhdsWithin 0 (Ioi 0)) (fun ε : ℝ =>
-    inverseFlow κ b (κ * t) (driverValue b (κ * t) + ε * Complex.I))
-
-def IsChordalSLE {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ)
-    (γ : Ω → ℝ → ℂ) (P : Measure Ω) : Prop :=
-  ∃ (W : WienerLaw) (B : Ω → Driver),
-    IsWiener W ∧ ProbabilityTheory.HasLaw B W P ∧
-      ∀ᵐ ω ∂P, ∀ t : ℝ, 0 ≤ t → γ ω t = trace κ (B ω) t
-
-end SLEExactGauge.Target
-
-namespace SLEExactGauge.QuadraticEnvelope
-
-variable {ι : Type*}
-
-def profile (b R : ι → ℝ) (d : ℝ) (i : ι) (r : ℝ) : ℝ :=
-  b i * R i ^ d * max 1 ((r / R i) ^ 2)
-
-def full (b R : ι → ℝ) (d : ℝ) (r : ℝ) : ℝ :=
-  if r ≤ 0 then 0 else sInf (range (fun i => profile b R d i r))
-
-variable {b R : ι → ℝ} {d : ℝ}
-
-lemma profile_nonneg (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (i : ι) (r : ℝ) : 0 ≤ profile b R d i r := by
-  unfold profile
-  exact mul_nonneg (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d))
-    (le_trans zero_le_one (le_max_left _ _))
-
-lemma base_le_profile (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (i : ι) (r : ℝ) : b i * R i ^ d ≤ profile b R d i r := by
-  unfold profile
-  exact le_mul_of_one_le_right
-    (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d)) (le_max_left _ _)
-
-lemma profile_mono (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (i : ι) {r s : ℝ} (hr : 0 ≤ r) (hrs : r ≤ s) :
-    profile b R d i r ≤ profile b R d i s := by
-  apply mul_le_mul_of_nonneg_left _
-    (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d))
-  exact max_le_max_left _ <| pow_le_pow_left₀ (div_nonneg hr (hR i).le)
-    (div_le_div_of_nonneg_right hrs (hR i).le) 2
-
-lemma profile_scale_le (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (i : ι) {C : ℝ} (hC : 1 ≤ C) (r : ℝ) :
-    profile b R d i (C * r) ≤ C ^ 2 * profile b R d i r := by
-  have hC2 : 1 ≤ C ^ 2 := by nlinarith
-  have hmax : max 1 (((C * r) / R i) ^ 2) ≤
-      C ^ 2 * max 1 ((r / R i) ^ 2) := by
-    apply max_le
-    · exact hC2.trans <| le_mul_of_one_le_right (sq_nonneg C) (le_max_left _ _)
-    · calc
-        (((C * r) / R i) ^ 2) = C ^ 2 * (r / R i) ^ 2 := by ring
-        _ ≤ C ^ 2 * max 1 ((r / R i) ^ 2) :=
-          mul_le_mul_of_nonneg_left (le_max_right _ _) (sq_nonneg C)
-  unfold profile
-  calc
-    _ ≤ b i * R i ^ d * (C ^ 2 * max 1 ((r / R i) ^ 2)) :=
-      mul_le_mul_of_nonneg_left hmax
-        (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d))
-    _ = _ := by ring
-
-@[simp] lemma full_zero : full b R d 0 = 0 := by simp [full]
-
-lemma full_of_nonpos {r : ℝ} (hr : r ≤ 0) : full b R d r = 0 := by
-  simp [full, hr]
-
-lemma full_of_pos {r : ℝ} (hr : 0 < r) :
-    full b R d r = sInf (range (fun i => profile b R d i r)) := by
-  simp [full, hr.not_ge]
-
-lemma profiles_bddBelow (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (r : ℝ) : BddBelow (range (fun i => profile b R d i r)) := by
-  refine ⟨0, ?_⟩
-  rintro _ ⟨i, rfl⟩
-  exact profile_nonneg hb hR i r
-
-variable [Nonempty ι]
-
-lemma le_full {r v : ℝ} (hr : 0 < r)
-    (hv : ∀ i, v ≤ profile b R d i r) : v ≤ full b R d r := by
-  rw [full_of_pos hr]
-  exact le_csInf (range_nonempty _) (by rintro _ ⟨i, rfl⟩; exact hv i)
-
-lemma full_nonneg (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (r : ℝ) : 0 ≤ full b R d r := by
-  by_cases hr : r ≤ 0
-  · rw [full_of_nonpos hr]
-  · exact le_full (lt_of_not_ge hr) (fun i => profile_nonneg hb hR i r)
-
-omit [Nonempty ι] in
-lemma full_le_profile (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (i : ι) (r : ℝ) : full b R d r ≤ profile b R d i r := by
-  by_cases hr : r ≤ 0
-  · rw [full_of_nonpos hr]
-    exact profile_nonneg hb hR i r
-  · rw [full_of_pos (lt_of_not_ge hr)]
-    exact csInf_le (profiles_bddBelow hb hR r) (mem_range_self i)
-
-lemma full_monoOn (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i) :
-    MonotoneOn (full b R d) (Ici 0) := by
-  intro r hr s hs hrs
-  by_cases hr0 : r ≤ 0
-  · rw [full_of_nonpos hr0]
-    exact full_nonneg hb hR s
-  · have hrpos := lt_of_not_ge hr0
-    apply le_full (hrpos.trans_le hrs)
-    intro i
-    exact (full_le_profile hb hR i r).trans (profile_mono hb hR i hr hrs)
-
-theorem full_scale_le (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    {C r : ℝ} (hC : 1 ≤ C) (hr : 0 ≤ r) :
-    full b R d (C * r) ≤ C ^ 2 * full b R d r := by
-  obtain rfl | hrpos := hr.eq_or_lt
-  · simp
-  have hCpos : 0 < C := zero_lt_one.trans_le hC
-  have hC2 : 0 < C ^ 2 := sq_pos_of_pos hCpos
-  have h : full b R d (C * r) / C ^ 2 ≤ full b R d r := by
-    apply le_full hrpos
-    intro i
-    apply (div_le_iff₀ hC2).2
-    calc
-      full b R d (C * r) ≤ profile b R d i (C * r) := full_le_profile hb hR i _
-      _ ≤ C ^ 2 * profile b R d i r := profile_scale_le hb hR i hC r
-      _ = profile b R d i r * C ^ 2 := mul_comm _ _
-  simpa [mul_comm] using (div_le_iff₀ hC2).1 h
-
-lemma power_lower_bound {c r : ℝ} (hc : 0 < c) (hb : ∀ i, c ≤ b i)
-    (hR : ∀ i, 0 < R i) (hd0 : 0 ≤ d) (hd2 : d ≤ 2) (hr : 0 < r) :
-    c * r ^ d ≤ full b R d r := by
-  apply le_full hr
-  intro i
-  have hshape : r ^ d ≤ R i ^ d * max 1 ((r / R i) ^ 2) := by
-    rcases le_total r (R i) with hsmall | hlarge
-    · exact (Real.rpow_le_rpow hr.le hsmall hd0).trans <|
-        le_mul_of_one_le_right (Real.rpow_nonneg (hR i).le d) (le_max_left _ _)
-    · have hratio : 1 ≤ r / R i := (le_div_iff₀ (hR i)).2 (by simpa using hlarge)
-      have hp : (r / R i) ^ d ≤ (r / R i) ^ 2 := by
-        exact_mod_cast Real.rpow_le_rpow_of_exponent_le hratio hd2
-      calc
-        r ^ d = R i ^ d * (r / R i) ^ d := by
-          rw [← Real.mul_rpow (hR i).le (div_nonneg hr.le (hR i).le)]
-          congr 1
-          field_simp [(hR i).ne']
-        _ ≤ R i ^ d * (r / R i) ^ 2 :=
-          mul_le_mul_of_nonneg_left hp (Real.rpow_nonneg (hR i).le d)
-        _ ≤ R i ^ d * max 1 ((r / R i) ^ 2) :=
-          mul_le_mul_of_nonneg_left (le_max_right _ _) (Real.rpow_nonneg (hR i).le d)
-  calc
-    c * r ^ d ≤ b i * r ^ d := mul_le_mul_of_nonneg_right (hb i) (Real.rpow_nonneg hr.le d)
-    _ ≤ b i * (R i ^ d * max 1 ((r / R i) ^ 2)) :=
-      mul_le_mul_of_nonneg_left hshape (hc.le.trans (hb i))
-    _ = profile b R d i r := by rw [profile]; ring
-
-lemma full_pos {c r : ℝ} (hc : 0 < c) (hb : ∀ i, c ≤ b i)
-    (hR : ∀ i, 0 < R i) (hd0 : 0 ≤ d) (hd2 : d ≤ 2) (hr : 0 < r) :
-    0 < full b R d r :=
-  (mul_pos hc (Real.rpow_pos_of_pos hr d)).trans_le
-    (power_lower_bound hc hb hR hd0 hd2 hr)
-
-lemma ratio_bounds (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    {r s : ℝ} (hr : 0 < r) (hs : 0 < s) :
-    min 1 ((s / r) ^ 2) * full b R d r ≤ full b R d s ∧
-      full b R d s ≤ max 1 ((s / r) ^ 2) * full b R d r := by
-  have hmono := full_monoOn (d := d) hb hR
-  have hnonneg := full_nonneg (d := d) hb hR r
-  constructor
-  · rcases le_total r s with hrs | hsr
-    · calc
-        _ ≤ 1 * full b R d r := mul_le_mul_of_nonneg_right (min_le_left _ _) hnonneg
-        _ = full b R d r := one_mul _
-        _ ≤ full b R d s := hmono hr.le hs.le hrs
-    · have hC : 1 ≤ r / s := (le_div_iff₀ hs).2 (by simpa using hsr)
-      have hscale := full_scale_le (d := d) hb hR hC hs.le
-      rw [div_mul_cancel₀ r hs.ne'] at hscale
-      calc
-        _ ≤ (s / r) ^ 2 * full b R d r :=
-          mul_le_mul_of_nonneg_right (min_le_right _ _) hnonneg
-        _ ≤ (s / r) ^ 2 * ((r / s) ^ 2 * full b R d s) :=
-          mul_le_mul_of_nonneg_left hscale (sq_nonneg _)
-        _ = full b R d s := by field_simp
-  · rcases le_total s r with hsr | hrs
-    · exact (hmono hs.le hr.le hsr).trans <| by
-        simpa using mul_le_mul_of_nonneg_right (le_max_left 1 ((s / r) ^ 2)) hnonneg
-    · have hC : 1 ≤ s / r := (le_div_iff₀ hr).2 (by simpa using hrs)
-      have hscale := full_scale_le (d := d) hb hR hC hr.le
-      rw [div_mul_cancel₀ s hr.ne'] at hscale
-      exact hscale.trans <| mul_le_mul_of_nonneg_right (le_max_right _ _) hnonneg
-
-lemma continuousAt_pos (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    {r : ℝ} (hr : 0 < r) : ContinuousAt (full b R d) r := by
-  have hlow : ContinuousAt (fun s : ℝ => min 1 ((s / r) ^ 2) * full b R d r) r := by
-    fun_prop
-  have hupp : ContinuousAt (fun s : ℝ => max 1 ((s / r) ^ 2) * full b R d r) r := by
-    fun_prop
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le'
-    (by simpa [hr.ne'] using hlow.tendsto)
-    (by simpa [hr.ne'] using hupp.tendsto)
-  · filter_upwards [Ioi_mem_nhds hr] with s hs
-    exact (ratio_bounds hb hR hr hs).1
-  · filter_upwards [Ioi_mem_nhds hr] with s hs
-    exact (ratio_bounds hb hR hr hs).2
-
-lemma continuousAt_zero (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
-    (hsmall : ∀ ε > 0, ∃ i, b i * R i ^ d < ε) :
-    ContinuousAt (full b R d) 0 := by
-  rw [Metric.continuousAt_iff]
-  intro ε hε
-  obtain ⟨i, hi⟩ := hsmall ε hε
-  refine ⟨R i, hR i, ?_⟩
-  intro x hx
-  rw [full_zero, Real.dist_eq, sub_zero, abs_of_nonneg (full_nonneg hb hR x)]
-  by_cases hx0 : x ≤ 0
-  · rw [full_of_nonpos hx0]
-    exact hε
-  · have hxpos : 0 < x := lt_of_not_ge hx0
-    have hxr : x < R i := by
-      simpa [Real.dist_eq, abs_of_pos hxpos] using hx
-    have hratio : 0 ≤ x / R i := div_nonneg hxpos.le (hR i).le
-    have hratio1 : x / R i ≤ 1 := (div_le_one (hR i)).2 hxr.le
-    have hp : (x / R i) ^ 2 ≤ 1 := by
-      simpa using (pow_le_pow_left₀ hratio hratio1 2)
-    calc
-      full b R d x ≤ profile b R d i x := full_le_profile hb hR i x
-      _ = b i * R i ^ d := by simp [profile, max_eq_left hp]
-      _ < ε := hi
-
-theorem isHausdorffGauge {c : ℝ} (hc : 0 < c) (hb : ∀ i, c ≤ b i)
-    (hR : ∀ i, 0 < R i) (hd0 : 0 ≤ d) (hd2 : d ≤ 2)
-    (hsmall : ∀ ε > 0, ∃ i, b i * R i ^ d < ε) :
-    SLEExactGauge.Target.IsHausdorffGauge (full b R d) := by
-  have hb0 : ∀ i, 0 ≤ b i := fun i => hc.le.trans (hb i)
-  refine ⟨?_, full_monoOn hb0 hR, full_zero, fun r hr => full_pos hc hb hR hd0 hd2 hr⟩
-  intro r hr
-  obtain rfl | hrpos := (show 0 ≤ r from hr).eq_or_lt
-  · exact (continuousAt_zero hb0 hR hsmall).continuousWithinAt
-  · exact (continuousAt_pos hb0 hR hrpos).continuousWithinAt
-
-theorem restore_finite (hb : ∀ i, 0 < b i) (hR : ∀ i, 0 < R i)
-    (discarded : Finset ι) (F : ℝ → ℝ) {M C : ℝ} (hM : 0 ≤ M)
-    (hbound : ∀ r > 0, F r ≤ M)
-    (htail : ∀ i, i ∉ discarded → ∀ r > 0, F r ≤ C * profile b R d i r) :
-    ∃ C' : ℝ, 0 < C' ∧ ∀ r > 0, F r ≤ C' * full b R d r := by
-  let C' := max C 0 + 1 + ∑ i ∈ discarded, M / (b i * R i ^ d)
-  have hbase : ∀ i, 0 < b i * R i ^ d :=
-    fun i => mul_pos (hb i) (Real.rpow_pos_of_pos (hR i) d)
-  have hterm : ∀ i, 0 ≤ M / (b i * R i ^ d) :=
-    fun i => div_nonneg hM (hbase i).le
-  have hsum : 0 ≤ ∑ i ∈ discarded, M / (b i * R i ^ d) :=
-    Finset.sum_nonneg (fun i _ => hterm i)
-  have hC' : 0 < C' := by
-    dsimp [C']
-    linarith [le_max_right C 0]
-  have hCC' : C ≤ C' := by
-    dsimp [C']
-    linarith [le_max_left C 0]
-  have hall : ∀ r > 0, ∀ i, F r ≤ C' * profile b R d i r := by
-    intro r hr i
-    by_cases hi : i ∈ discarded
-    · have hi_le : M / (b i * R i ^ d) ≤ C' := by
-        have hh := Finset.single_le_sum (fun i (_ : i ∈ discarded) => hterm i) hi
-        dsimp [C']
-        linarith [le_max_right C 0]
-      calc
-        F r ≤ M := hbound r hr
-        _ ≤ C' * (b i * R i ^ d) := (div_le_iff₀ (hbase i)).1 hi_le
-        _ ≤ C' * profile b R d i r := mul_le_mul_of_nonneg_left
-          (base_le_profile (fun i => (hb i).le) hR i r) hC'.le
-    · exact (htail i hi r hr).trans <| mul_le_mul_of_nonneg_right hCC'
-        (profile_nonneg (fun i => (hb i).le) hR i r)
-  refine ⟨C', hC', fun r hr => ?_⟩
-  have h : F r / C' ≤ full b R d r := by
-    apply le_full hr
-    intro i
-    exact (div_le_iff₀ hC').2 (by simpa [mul_comm] using hall r hr i)
-  simpa [mul_comm] using (div_le_iff₀ hC').1 h
-
-end SLEExactGauge.QuadraticEnvelope
-
-namespace SLEExactGauge.Target
-
-def diskMap (θ : ℝ) (z : ℂ) : ℂ :=
-  (z - Complex.exp (θ * Complex.I)) / (z - Complex.exp (-θ * Complex.I))
-
-def diskMapInv (θ : ℝ) (x : ℂ) : ℂ :=
-  (Complex.exp (θ * Complex.I) - x * Complex.exp (-θ * Complex.I)) / (1 - x)
-
-def centeredFlow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
-  flow κ b z t - driverValue b t
-
-def clock (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  ∫ u in Icc (0 : ℝ) t,
-    (flow κ b (diskMapInv θ x) u).im ^ 2 /
-      ‖centeredFlow κ b (diskMapInv θ x) u‖ ^ 4
-
-def initialRadius (θ : ℝ) (x : ℂ) : ℝ :=
-  2 * (diskMapInv θ x).im * ‖deriv (diskMap θ) (diskMapInv θ x)‖
-
-def radius (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  initialRadius θ x * Real.exp (-2 * a κ * clock κ θ b x t)
-
-def angle (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  Complex.arg (centeredFlow κ b (diskMapInv θ x) t)
-
-def green (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  if Surviving κ b (diskMapInv θ x) t then
-    radius κ θ b x t ^ (-α κ) * Real.sin (angle κ θ b x t) ^ p κ
-  else 0
-
-def levelTime (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ≥0∞ :=
-  sInf {t | ∃ q : ℚ, 0 < q ∧ t = ENNReal.ofReal (q : ℝ) ∧
-    Surviving κ b (diskMapInv θ 0) q ∧ ℓ < clock κ θ b 0 q}
-
-def diskK : Set ℂ := closedBall 0 (1 / 4)
-
-def potential (κ θ : ℝ) (b : Driver) (t : ℝ) : ℝ≥0∞ :=
-  ∫⁻ x in diskK, ENNReal.ofReal (green κ θ b x t)
-
-def concat (b c : Driver) (t : ℝ) : Driver where
-  toFun u := if u ≤ t.toNNReal then b u else b t.toNNReal + c (u - t.toNNReal)
-  continuous_toFun := by
-    apply Continuous.if_le (map_continuous b)
-      (continuous_const.add ((map_continuous c).comp (continuous_id.sub continuous_const)))
-      continuous_id continuous_const
-    intro u hu
-    change u = t.toNNReal at hu
-    subst u
-    simp
-  map_zero' := by simp
-
-def potentialDifference (κ θ : ℝ) (W : WienerLaw) (b : Driver) (h i : ℕ) : ℝ :=
-  let t := (i : ℝ) * mesh h
-  let v := potential κ θ b t
-  let w := ∫⁻ c, potential κ θ (concat b c t) (((i : ℝ) + 1) * mesh h) ∂W
-  if v < ∞ ∧ w < ∞ then v.toReal - w.toReal else 0
-
-def discreteMass (κ θ : ℝ) (W : WienerLaw) (b : Driver) (ℓ : ℝ) (h : ℕ) : ℝ :=
-  ∑ i ∈ Finset.range (h * 2 ^ h),
-    if ENNReal.ofReal (((i : ℝ) + 1) * mesh h) ≤ levelTime κ θ b ℓ
-    then potentialDifference κ θ W b h i else 0
-
-def levelDensity (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ :=
-  let τ := levelTime κ θ b ℓ
-  if τ < ∞ then Real.exp (2 * a κ * α κ * ℓ) *
-    Real.sin (angle κ θ b 0 τ.toReal) ^ p κ / Real.sin θ ^ p κ else 0
-
-def angleLaw (κ : ℝ) : Measure ℝ :=
-  let unnormalized := (volume.restrict (Ioo (0 : ℝ) Real.pi)).withDensity
-    (fun θ => ENNReal.ofReal (Real.sin θ ^ (4 * a κ)))
-  (unnormalized Set.univ)⁻¹ • unnormalized
-
-def clip (u x : ℝ) : ℝ := min (max x 0) u
-
-def finiteMoment (κ : ℝ) (W : WienerLaw) (n : ℕ) (ℓ : ℝ) : ℝ :=
-  limUnder atTop (fun u : ℕ => limUnder atTop (fun h : ℕ =>
-    ∫ θ, ∫ b, levelDensity κ θ b ℓ *
-      (clip u (discreteMass κ θ W b ℓ h)) ^ n ∂W ∂angleLaw κ))
-
-def moment (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
-  limUnder atTop (fun ℓ : ℕ => finiteMoment κ W n ℓ)
-
-def batchLevel (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℕ :=
-  1 + Nat.card {j : ℕ // 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4}
-
-def threshold (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
-  4 * moment κ W (n + 1) / moment κ W n
-
-def multiplicity (κ : ℝ) (W : WienerLaw) (n : ℕ) (k : ℤ) : ℕ :=
-  if (2 : ℝ) ^ k ≤ threshold κ W n then
-    ⌊(4 : ℝ) ^ n * ((2 : ℝ) ^ k) ^ n / moment κ W n⌋₊ else 0
-
-abbrev Entry := ℕ × Option (ℤ × ℕ)
-
-def ValidEntry (κ : ℝ) (W : WienerLaw) (e : Entry) : Prop :=
-  1 ≤ e.1 ∧ match e.2 with
-    | none => True
-    | some (k,j) => j < multiplicity κ W e.1 k
-
-def entryValue (e : Entry) : ℝ :=
-  match e.2 with
-    | none => e.1
-    | some (k,_) => (2 : ℝ) ^ k
-
-def entryPosition (κ : ℝ) (W : WienerLaw) (e : Entry) : ℕ :=
-  match e.2 with
-    | none => ∑' k : ℤ, multiplicity κ W e.1 k
-    | some (k,j) => (∑' l : ℤ, if l < k then multiplicity κ W e.1 l else 0) + j
-
-def entryRadius (κ : ℝ) (W : WienerLaw) (e : Entry) : ℝ :=
-  Real.exp (-(e.1 : ℝ) ^ 2 - 2 * a κ * (entryPosition κ W e : ℝ) *
-    ((batchLevel κ W e.1 : ℝ) + (e.1 : ℝ)))
-
-def entryCost (κ : ℝ) (W : WienerLaw) (e : Entry) (r : ℝ) : ℝ :=
-  entryValue e * entryRadius κ W e ^ d κ * max 1 ((r / entryRadius κ W e) ^ 2)
-
-def gauge (κ : ℝ) (W : WienerLaw) (r : ℝ) : ℝ :=
-  if r ≤ 0 then 0 else sInf {v | ∃ e : Entry, ValidEntry κ W e ∧ v = entryCost κ W e r}
-
-end SLEExactGauge.Target
-
-namespace SLEExactGauge.Intervals
-
-theorem all_real_of_all_rational {X : Type*} (μ : OuterMeasure X) (γ : ℝ → X)
-    (h : ∀ q r : ℚ, 0 < q → q < r →
-      0 < μ (γ '' Icc (q : ℝ) (r : ℝ)) ∧ μ (γ '' Icc (q : ℝ) (r : ℝ)) < ∞) :
-    ∀ s t : ℝ, 0 < s → s < t →
-      0 < μ (γ '' Icc s t) ∧ μ (γ '' Icc s t) < ∞ := by
-  intro s t hs hst
-  obtain ⟨q, hsq, hqt⟩ := exists_rat_btwn hst
-  obtain ⟨r, hqr, hrt⟩ := exists_rat_btwn hqt
-  have hq0 : 0 < q := by exact_mod_cast hs.trans hsq
-  have hqr' : q < r := by exact_mod_cast hqr
-  have hpos := (h q r hq0 hqr').1
-  have hlower : γ '' Icc (q : ℝ) (r : ℝ) ⊆ γ '' Icc s t :=
-    image_mono (Icc_subset_Icc hsq.le hrt.le)
-  obtain ⟨u, hu0, hus⟩ := exists_rat_btwn hs
-  obtain ⟨v, htv⟩ := exists_rat_gt t
-  have hu0' : 0 < u := by exact_mod_cast hu0
-  have huv : u < v := by exact_mod_cast hus.trans (hst.trans htv)
-  have hfinite := (h u v hu0' huv).2
-  have hupper : γ '' Icc s t ⊆ γ '' Icc (u : ℝ) (v : ℝ) :=
-    image_mono (Icc_subset_Icc hus.le htv.le)
-  exact ⟨hpos.trans_le (μ.mono hlower), (μ.mono hupper).trans_lt hfinite⟩
-
-theorem ae_all_real_of_ae_rational {Ω X : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (μ : OuterMeasure X) (γ : Ω → ℝ → X)
-    (h : ∀ q r : ℚ, 0 < q → q < r →
-      ∀ᵐ ω ∂P, 0 < μ (γ ω '' Icc (q : ℝ) (r : ℝ)) ∧
-        μ (γ ω '' Icc (q : ℝ) (r : ℝ)) < ∞) :
-    ∀ᵐ ω ∂P, ∀ s t : ℝ, 0 < s → s < t →
-      0 < μ (γ ω '' Icc s t) ∧ μ (γ ω '' Icc s t) < ∞ := by
-  have hall : ∀ᵐ ω ∂P, ∀ q r : ℚ, 0 < q → q < r →
-      0 < μ (γ ω '' Icc (q : ℝ) (r : ℝ)) ∧
-        μ (γ ω '' Icc (q : ℝ) (r : ℝ)) < ∞ := by
-    simpa only [ae_all_iff] using h
-  exact hall.mono (fun ω hω => all_real_of_all_rational μ (γ ω) hω)
-
-end SLEExactGauge.Intervals
-
-namespace SLEExactGauge.Target
-
-abbrev GaugeIndex (κ : ℝ) (W : WienerLaw) := {e : Entry // ValidEntry κ W e}
-
-def cleanupIndex (κ : ℝ) (W : WienerLaw) (n : ℕ) (hn : 1 ≤ n) : GaugeIndex κ W :=
-  ⟨(n, none), hn, trivial⟩
-
-instance (κ : ℝ) (W : WienerLaw) : Nonempty (GaugeIndex κ W) :=
-  ⟨cleanupIndex κ W 1 le_rfl⟩
-
-abbrev weights (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : ℝ := entryValue i.val
-abbrev radii (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : ℝ := entryRadius κ W i.val
-
-lemma weights_pos (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : 0 < weights κ W i := by
-  rcases i with ⟨⟨n, o⟩, hn, _⟩
-  cases o with
-  | none =>
-      change (0 : ℝ) < (n : ℝ)
-      exact Nat.cast_pos.mpr (Nat.succ_le_iff.mp hn)
-  | some kj =>
-      rcases kj with ⟨k, j⟩
-      exact zpow_pos (by norm_num : (0 : ℝ) < 2) k
-
-lemma radii_pos (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : 0 < radii κ W i :=
-  Real.exp_pos _
-
-lemma gauge_eq_full (κ : ℝ) (W : WienerLaw) :
-    gauge κ W = QuadraticEnvelope.full (weights κ W) (radii κ W) (d κ) := by
-  funext r
-  unfold gauge QuadraticEnvelope.full
-  split_ifs with hr
-  · rfl
-  · congr 1
-    ext v
-    constructor
-    · rintro ⟨e, he, hv⟩
-      exact ⟨⟨e, he⟩, hv.symm⟩
-    · rintro ⟨⟨e, he⟩, hv⟩
-      exact ⟨e, he, hv.symm⟩
-
-lemma internal_a_pos {κ : ℝ} (hκ : 0 < κ) : 0 < a κ := div_pos (by norm_num) hκ
-
-lemma dimension_pos {κ : ℝ} (hκ : 0 < κ) : 0 < d κ := by
-  have ha := internal_a_pos hκ
-  unfold d
-  positivity
-
-lemma cleanup_radius_le {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) (n : ℕ) (hn : 1 ≤ n) :
-    radii κ W (cleanupIndex κ W n hn) ≤ Real.exp (-(n : ℝ) ^ 2) := by
-  unfold radii cleanupIndex entryRadius
-  apply Real.exp_le_exp.mpr
-  dsimp only
-  have ha := internal_a_pos hκ
-  have hnonneg : 0 ≤ 2 * a κ * (entryPosition κ W (n, none) : ℝ) *
-      ((batchLevel κ W n : ℝ) + (n : ℝ)) := by positivity
-  linarith
-
-lemma cleanup_base_le {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) (n : ℕ) (hn : 1 ≤ n) :
-    weights κ W (cleanupIndex κ W n hn) * radii κ W (cleanupIndex κ W n hn) ^ d κ ≤
-      (n : ℝ) * Real.exp (-d κ * (n : ℝ) ^ 2) := by
-  have hd := dimension_pos hκ
-  change (n : ℝ) * radii κ W (cleanupIndex κ W n hn) ^ d κ ≤ _
-  calc
-    _ ≤ (n : ℝ) * Real.exp (-(n : ℝ) ^ 2) ^ d κ := by
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
-      exact Real.rpow_le_rpow (radii_pos κ W _).le (cleanup_radius_le hκ W n hn) hd.le
-    _ = _ := by rw [← Real.exp_mul]; congr 2; ring
-
-lemma cleanup_small {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) :
-    ∀ ε > 0, ∃ i : GaugeIndex κ W, weights κ W i * radii κ W i ^ d κ < ε := by
-  have hd := dimension_pos hκ
-  have hlim : Tendsto (fun n : ℕ => (n : ℝ) * Real.exp (-d κ * (n : ℝ))) atTop (𝓝 0) := by
-    convert (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 (d κ) hd).comp
-      (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop) using 1
-    ext n
-    simp only [Function.comp_apply, Real.rpow_one]
-  intro ε hε
-  obtain ⟨n, hnε, hn⟩ :=
-    ((hlim.eventually (gt_mem_nhds hε)).and (eventually_ge_atTop 1)).exists
-  refine ⟨cleanupIndex κ W n hn, lt_of_le_of_lt ?_ hnε⟩
-  have hnreal : 1 ≤ (n : ℝ) := by exact_mod_cast hn
-  have hsq : (n : ℝ) ≤ (n : ℝ) ^ 2 := by nlinarith
-  calc
-    _ ≤ (n : ℝ) * Real.exp (-d κ * (n : ℝ) ^ 2) := cleanup_base_le hκ W n hn
-    _ ≤ (n : ℝ) * Real.exp (-d κ * (n : ℝ)) := by
-      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
-      exact Real.exp_le_exp.mpr (mul_le_mul_of_nonpos_left hsq (neg_nonpos.mpr hd.le))
-
-theorem gauge_basic_properties {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) :
-    ContinuousOn (gauge κ W) (Ici 0) ∧
-    MonotoneOn (gauge κ W) (Ici 0) ∧
-    gauge κ W 0 = 0 ∧
-    (∀ r : ℝ, 0 ≤ gauge κ W r) ∧
-    ∀ C ≥ 1, ∀ r ≥ 0, gauge κ W (C * r) ≤ C ^ 2 * gauge κ W r := by
-  rw [gauge_eq_full]
-  have hb : ∀ i, 0 ≤ weights κ W i := fun i => (weights_pos κ W i).le
-  have hR := radii_pos κ W
-  refine ⟨?_, QuadraticEnvelope.full_monoOn hb hR, QuadraticEnvelope.full_zero,
-    QuadraticEnvelope.full_nonneg hb hR, fun _ hC _ hr => QuadraticEnvelope.full_scale_le hb hR hC hr⟩
-  intro r hr
-  obtain rfl | hrpos := (show 0 ≤ r from hr).eq_or_lt
-  · exact (QuadraticEnvelope.continuousAt_zero hb hR (cleanup_small hκ W)).continuousWithinAt
-  · exact (QuadraticEnvelope.continuousAt_pos hb hR hrpos).continuousWithinAt
-
-
-
-theorem hausdorff_arbitrary_covers (h : ℝ → ℝ) (h0 : h 0 = 0) (E : Set ℂ) :
-    hausdorff h E =
-      ⨆ (δ : ℝ≥0∞) (_ : 0 < δ),
-        ⨅ (U : ℕ → Set ℂ) (_ : E ⊆ ⋃ j, U j) (_ : ∀ j, Metric.ediam (U j) ≤ δ),
-          ∑' j, if Metric.ediam (U j) = ∞ then ∞
-            else ENNReal.ofReal (h (Metric.ediam (U j)).toReal) := by
-  let m : ℝ≥0∞ → ℝ≥0∞ := fun r => if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal)
-  have hm0 : m 0 = 0 := by simp [m, h0]
-  have hnonempty (U : Set ℂ) : (⨆ _ : U.Nonempty, m (Metric.ediam U)) = m (Metric.ediam U) := by
-    by_cases hU : U.Nonempty
-    · simp [hU]
-    · rw [not_nonempty_iff_eq_empty.mp hU]
-      simp [hm0]
-  change OuterMeasure.mkMetric m E = _
-  rw [OuterMeasure.coe_mkMetric, Measure.mkMetric_apply]
-  simp only [hnonempty]
-  rfl
-
-
-
-lemma trace_isChordalSLE (κ : ℝ) {W : WienerLaw} (hW : IsWiener W) :
-    IsChordalSLE κ (trace κ) W := by
-  refine ⟨W, id, hW, ProbabilityTheory.HasLaw.id, ?_⟩
-  exact ae_of_all _ (fun _ _ _ => rfl)
-
-theorem transfer_interval_estimates {κ : ℝ} {h : ℝ → ℝ}
-    (hcanonical : ∀ W : WienerLaw, IsWiener W →
-      ∀ᵐ b ∂W, ∀ s t : ℝ, 0 < s → s < t →
-        0 < hausdorff h (trace κ b '' Icc s t) ∧
-        hausdorff h (trace κ b '' Icc s t) < ∞)
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (γ : Ω → ℝ → ℂ)
-    (hγ : IsChordalSLE κ γ P) :
-    ∀ᵐ ω ∂P, ∀ s t : ℝ, 0 < s → s < t →
-      0 < hausdorff h (γ ω '' Icc s t) ∧
-      hausdorff h (γ ω '' Icc s t) < ∞ := by
-  obtain ⟨W, B, hW, hB, htrace⟩ := hγ
-  have hmap := hcanonical W hW
-  rw [← hB.map_eq] at hmap
-  have hpull := ae_of_ae_map hB.aemeasurable hmap
-  filter_upwards [hpull, htrace] with ω hω htr
-  intro s t hs hst
-  have heq : γ ω '' Icc s t = trace κ (B ω) '' Icc s t := by
-    apply Set.image_congr
-    intro v hv
-    exact htr v (hs.le.trans hv.1)
-  rw [heq]
-  exact hω s t hs hst
-
-
-
-lemma dimension_eq (κ : ℝ) : d κ = 1 + κ / 8 := by
-  unfold d a
-  field_simp
-  ring
-
-lemma dimension_lt_two {κ : ℝ} (hκ8 : κ < 8) : d κ < 2 := by
-  rw [dimension_eq κ]
-  linarith
-
-lemma multiplicity_positive_bounds (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1 ^ n ≤ moment κ W n) {k : ℤ}
-    (hNk : 0 < multiplicity κ W n k) :
-    moment κ W 1 / 4 ≤ (2 : ℝ) ^ k ∧ (2 : ℝ) ^ k ≤ threshold κ W n := by
-  have hcut : (2 : ℝ) ^ k ≤ threshold κ W n := by
-    by_contra hcut
-    simp [multiplicity, hcut] at hNk
-  have hmn : 0 < moment κ W n := (pow_pos hm1 n).trans_le hJ
-  have hfloor : 0 < ⌊(4 : ℝ) ^ n * ((2 : ℝ) ^ k) ^ n / moment κ W n⌋₊ := by
-    simpa [multiplicity, hcut] using hNk
-  have hw := Nat.floor_pos.mp hfloor
-  have hnum : moment κ W n ≤ ((4 : ℝ) * 2 ^ k) ^ n := by
-    simpa only [one_mul, mul_pow] using (le_div_iff₀ hmn).1 hw
-  have hb : moment κ W 1 ≤ (4 : ℝ) * 2 ^ k :=
-    le_of_pow_le_pow_left₀ (by omega) (by positivity) (hJ.trans hnum)
-  exact ⟨(div_le_iff₀ (by norm_num : (0 : ℝ) < 4)).2 (by linarith), hcut⟩
-
-lemma weights_lower_of_moment_inequalities (κ : ℝ) (W : WienerLaw)
-    (hm1 : 0 < moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n)
-    (i : GaugeIndex κ W) : min 1 (moment κ W 1 / 4) ≤ weights κ W i := by
-  rcases i with ⟨⟨n, o⟩, hn, ho⟩
-  cases o with
-  | none =>
-      change min 1 (moment κ W 1 / 4) ≤ (n : ℝ)
-      exact (min_le_left _ _).trans (by exact_mod_cast hn)
-  | some kj =>
-      rcases kj with ⟨k, j⟩
-      change j < multiplicity κ W n k at ho
-      change min 1 (moment κ W 1 / 4) ≤ (2 : ℝ) ^ k
-      exact (min_le_right _ _).trans
-        (multiplicity_positive_bounds κ W hn hm1 (hJ n hn)
-          ((Nat.zero_le j).trans_lt ho)).1
-
-theorem gauge_admissible_of_moment_inequalities {κ : ℝ}
-    (hκ0 : 0 < κ) (hκ8 : κ < 8) (W : WienerLaw)
-    (hm1 : 0 < moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n) :
-    IsHausdorffGauge (gauge κ W) := by
-  rw [gauge_eq_full]
-  exact QuadraticEnvelope.isHausdorffGauge
-    (c := min 1 (moment κ W 1 / 4))
-    (lt_min zero_lt_one (div_pos hm1 (by norm_num)))
-    (weights_lower_of_moment_inequalities κ W hm1 hJ)
-    (radii_pos κ W) (dimension_pos hκ0).le (dimension_lt_two hκ8).le
-    (cleanup_small hκ0 W)
-
-lemma multiplicity_support_finite (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1 ^ n ≤ moment κ W n) :
-    Set.Finite {k : ℤ | 0 < multiplicity κ W n k} := by
-  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
-  apply (Set.finite_Icc (⌈Real.log (moment κ W 1 / 4) / Real.log 2⌉ : ℤ)
-    (⌊Real.log (threshold κ W n) / Real.log 2⌋ : ℤ)).subset
-  intro k hk
-  obtain ⟨hlo, hhi⟩ := multiplicity_positive_bounds κ W hn hm1 hJ hk
-  constructor
-  · apply Int.ceil_le.mpr
-    apply (div_le_iff₀ hlog).2
-    exact (Real.le_zpow_iff_log_le (div_pos hm1 (by norm_num))
-      (by norm_num : (0 : ℝ) < 2)).1 hlo
-  · apply Int.le_floor.mpr
-    apply (le_div_iff₀ hlog).2
-    exact Real.le_log_of_zpow_le (by norm_num : (0 : ℝ) < 2) hhi
-
-lemma repeated_entries_finite (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1 ^ n ≤ moment κ W n) :
-    Set.Finite {kj : ℤ × ℕ | kj.2 < multiplicity κ W n kj.1} := by
-  have hfin := (multiplicity_support_finite κ W hn hm1 hJ).biUnion
-    (fun k _ => (Set.finite_Iio (multiplicity κ W n k)).image (fun j : ℕ => (k, j)))
-  apply hfin.subset
-  rintro ⟨k, j⟩ hj
-  exact mem_iUnion.2 ⟨k, mem_iUnion.2 ⟨(Nat.zero_le j).trans_lt hj, j, hj, rfl⟩⟩
-
-lemma batch_entries_finite (κ : ℝ) (W : WienerLaw) {n : ℕ}
-    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
-    (hJ : moment κ W 1 ^ n ≤ moment κ W n) :
-    Set.Finite {e : Entry | e.1 = n ∧ ValidEntry κ W e} := by
-  have hfin := ((repeated_entries_finite κ W hn hm1 hJ).image
-    (fun kj : ℤ × ℕ => (n, (some kj : Option (ℤ × ℕ))))).insert (n, none)
-  apply hfin.subset
-  rintro ⟨m, o⟩ ⟨heq, hm, ho⟩
-  dsimp at heq
-  subst m
-  cases o with
-  | none => exact Or.inl rfl
-  | some kj => exact Or.inr ⟨kj, ho, rfl⟩
-
-lemma early_entries_finite (κ : ℝ) (W : WienerLaw)
-    (hm1 : 0 < moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n)
-    (n₀ : ℕ) : Set.Finite {e : Entry | ValidEntry κ W e ∧ e.1 < n₀} := by
-  have hfin := (Set.finite_Ico 1 n₀).biUnion (fun n hn =>
-    batch_entries_finite κ W hn.1 hm1 (hJ n hn.1))
-  apply hfin.subset
-  intro e he
-  exact mem_iUnion.2 ⟨e.1, mem_iUnion.2 ⟨⟨he.1.1, he.2⟩, rfl, he.1⟩⟩
-
-theorem gauge_restore_early_batches (κ : ℝ) (W : WienerLaw)
-    (hm1 : 0 < moment κ W 1)
-    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n)
-    (n₀ : ℕ) (F : ℝ → ℝ) {M C : ℝ} (hM : 0 ≤ M)
-    (hbound : ∀ r > 0, F r ≤ M)
-    (htail : ∀ e : Entry, ValidEntry κ W e → n₀ ≤ e.1 →
-      ∀ r > 0, F r ≤ C * entryCost κ W e r) :
-    ∃ C' : ℝ, 0 < C' ∧ ∀ r > 0, F r ≤ C' * gauge κ W r := by
-  classical
-  have hfin : Set.Finite {i : GaugeIndex κ W | i.val.1 < n₀} := by
-    apply ((early_entries_finite κ W hm1 hJ n₀).preimage
-      (f := fun i : GaugeIndex κ W => i.val) Subtype.val_injective.injOn).subset
-    intro i hi
-    exact ⟨i.property, hi⟩
-  rw [gauge_eq_full]
-  apply QuadraticEnvelope.restore_finite (weights_pos κ W) (radii_pos κ W)
-    hfin.toFinset F hM hbound
-  intro i hi r hr
-  have hn : n₀ ≤ i.val.1 := by
-    have hnot : ¬ i.val.1 < n₀ := by simpa using hi
-    exact le_of_not_gt hnot
-  exact htail i.val i.property hn r hr
-
-end SLEExactGauge.Target
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Quadratic moment envelopes, Hausdorff gauge admissibility and interval transfer. -/
+
+universe u
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+open scoped ContinuousMapZero
+abbrev Driver := C(ℝ≥0, ℝ)₀
+instance : MeasurableSpace Driver := borel Driver
+instance : BorelSpace Driver := ⟨rfl⟩
+abbrev WienerLaw := Measure Driver
+
+def IsWiener (W : WienerLaw) : Prop :=
+  ProbabilityTheory.IsBrownianReal (fun t b => b t) W
+
+def a (κ : ℝ) := 2 / κ
+def d (κ : ℝ) := 1 + 1 / (4 * a κ)
+def α (κ : ℝ) := 2 - d κ
+def p (κ : ℝ) := 4 * a κ - 1
+
+def driverValue (b : Driver) (t : ℝ) : ℝ := b t.toNNReal
+
+def cutoffField (κ : ℝ) (b : Driver) (j : ℕ) (t : ℝ) (z : ℂ) : ℂ :=
+  (a κ : ℂ) * star (z - driverValue b t) /
+    (max (‖z - driverValue b t‖ ^ 2) ((j : ℝ) ^ (-(4 : ℤ))) : ℝ)
+
+def mesh (N : ℕ) : ℝ := (2 : ℝ) ^ (-(N : ℤ))
+
+def eulerSteps (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) : ℕ → ℂ
+  | 0 => z
+  | k + 1 => eulerSteps κ b j N z k +
+      mesh N * cutoffField κ b j ((k : ℝ) * mesh N) (eulerSteps κ b j N z k)
+
+def eulerFlow (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
+  let k := ⌊t / mesh N⌋₊
+  let w :=  t / mesh N - (k : ℝ)
+  (1 - w : ℝ) * eulerSteps κ b j N z k + w * eulerSteps κ b j N z (k + 1)
+
+def cutoffFlow (κ : ℝ) (b : Driver) (j : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
+  limUnder atTop (fun N : ℕ => eulerFlow κ b j N z t)
+
+def Surviving (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Prop :=
+  ∃ j : ℕ, 1 ≤ j ∧ (j : ℝ)⁻¹ <
+    sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 t)
+
+instance (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Decidable (Surviving κ b z t) :=
+  Classical.propDecidable _
+
+def flow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
+  if h : Surviving κ b z t then cutoffFlow κ b (Nat.find h) z t else 0
+
+def IsHausdorffGauge (h : ℝ → ℝ) : Prop :=
+  ContinuousOn h (Ici 0) ∧ MonotoneOn h (Ici 0) ∧ h 0 = 0 ∧ ∀ r > 0, 0 < h r
+
+def hausdorff (h : ℝ → ℝ) : OuterMeasure ℂ :=
+  OuterMeasure.mkMetric (fun r => if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal))
+
+def inverseFlow (κ : ℝ) (b : Driver) (t : ℝ) (w : ℂ) : ℂ :=
+  @ite ℂ (∃ z : ℂ, Surviving κ b z t ∧ flow κ b z t = w) (Classical.propDecidable _)
+    (Classical.epsilon (fun z : ℂ => Surviving κ b z t ∧ flow κ b z t = w)) 0
+
+def trace (κ : ℝ) (b : Driver) (t : ℝ) : ℂ :=
+  limUnder (nhdsWithin 0 (Ioi 0)) (fun ε : ℝ =>
+    inverseFlow κ b (κ * t) (driverValue b (κ * t) + ε * Complex.I))
+
+def IsChordalSLE {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ)
+    (γ : Ω → ℝ → ℂ) (P : Measure Ω) : Prop :=
+  ∃ (W : WienerLaw) (B : Ω → Driver),
+    IsWiener W ∧ ProbabilityTheory.HasLaw B W P ∧
+      ∀ᵐ ω ∂P, ∀ t : ℝ, 0 ≤ t → γ ω t = trace κ (B ω) t
+
+end SLEExactGauge.Target
+
+namespace SLEExactGauge.QuadraticEnvelope
+
+variable {ι : Type*}
+
+def profile (b R : ι → ℝ) (d : ℝ) (i : ι) (r : ℝ) : ℝ :=
+  b i * R i ^ d * max 1 ((r / R i) ^ 2)
+
+def full (b R : ι → ℝ) (d : ℝ) (r : ℝ) : ℝ :=
+  if r ≤ 0 then 0 else sInf (range (fun i => profile b R d i r))
+
+variable {b R : ι → ℝ} {d : ℝ}
+
+lemma profile_nonneg (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (i : ι) (r : ℝ) : 0 ≤ profile b R d i r := by
+  unfold profile
+  exact mul_nonneg (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d))
+    (le_trans zero_le_one (le_max_left _ _))
+
+lemma base_le_profile (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (i : ι) (r : ℝ) : b i * R i ^ d ≤ profile b R d i r := by
+  unfold profile
+  exact le_mul_of_one_le_right
+    (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d)) (le_max_left _ _)
+
+lemma profile_mono (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (i : ι) {r s : ℝ} (hr : 0 ≤ r) (hrs : r ≤ s) :
+    profile b R d i r ≤ profile b R d i s := by
+  apply mul_le_mul_of_nonneg_left _
+    (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d))
+  exact max_le_max_left _ <| pow_le_pow_left₀ (div_nonneg hr (hR i).le)
+    (div_le_div_of_nonneg_right hrs (hR i).le) 2
+
+lemma profile_scale_le (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (i : ι) {C : ℝ} (hC : 1 ≤ C) (r : ℝ) :
+    profile b R d i (C * r) ≤ C ^ 2 * profile b R d i r := by
+  have hC2 : 1 ≤ C ^ 2 := by nlinarith
+  have hmax : max 1 (((C * r) / R i) ^ 2) ≤
+      C ^ 2 * max 1 ((r / R i) ^ 2) := by
+    apply max_le
+    · exact hC2.trans <| le_mul_of_one_le_right (sq_nonneg C) (le_max_left _ _)
+    · calc
+        (((C * r) / R i) ^ 2) = C ^ 2 * (r / R i) ^ 2 := by ring
+        _ ≤ C ^ 2 * max 1 ((r / R i) ^ 2) :=
+          mul_le_mul_of_nonneg_left (le_max_right _ _) (sq_nonneg C)
+  unfold profile
+  calc
+    _ ≤ b i * R i ^ d * (C ^ 2 * max 1 ((r / R i) ^ 2)) :=
+      mul_le_mul_of_nonneg_left hmax
+        (mul_nonneg (hb i) (Real.rpow_nonneg (hR i).le d))
+    _ = _ := by ring
+
+@[simp] lemma full_zero : full b R d 0 = 0 := by simp [full]
+
+lemma full_of_nonpos {r : ℝ} (hr : r ≤ 0) : full b R d r = 0 := by
+  simp [full, hr]
+
+lemma full_of_pos {r : ℝ} (hr : 0 < r) :
+    full b R d r = sInf (range (fun i => profile b R d i r)) := by
+  simp [full, hr.not_ge]
+
+lemma profiles_bddBelow (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (r : ℝ) : BddBelow (range (fun i => profile b R d i r)) := by
+  refine ⟨0, ?_⟩
+  rintro _ ⟨i, rfl⟩
+  exact profile_nonneg hb hR i r
+
+variable [Nonempty ι]
+
+lemma le_full {r v : ℝ} (hr : 0 < r)
+    (hv : ∀ i, v ≤ profile b R d i r) : v ≤ full b R d r := by
+  rw [full_of_pos hr]
+  exact le_csInf (range_nonempty _) (by rintro _ ⟨i, rfl⟩; exact hv i)
+
+lemma full_nonneg (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (r : ℝ) : 0 ≤ full b R d r := by
+  by_cases hr : r ≤ 0
+  · rw [full_of_nonpos hr]
+  · exact le_full (lt_of_not_ge hr) (fun i => profile_nonneg hb hR i r)
+
+omit [Nonempty ι] in
+lemma full_le_profile (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (i : ι) (r : ℝ) : full b R d r ≤ profile b R d i r := by
+  by_cases hr : r ≤ 0
+  · rw [full_of_nonpos hr]
+    exact profile_nonneg hb hR i r
+  · rw [full_of_pos (lt_of_not_ge hr)]
+    exact csInf_le (profiles_bddBelow hb hR r) (mem_range_self i)
+
+lemma full_monoOn (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i) :
+    MonotoneOn (full b R d) (Ici 0) := by
+  intro r hr s hs hrs
+  by_cases hr0 : r ≤ 0
+  · rw [full_of_nonpos hr0]
+    exact full_nonneg hb hR s
+  · have hrpos := lt_of_not_ge hr0
+    apply le_full (hrpos.trans_le hrs)
+    intro i
+    exact (full_le_profile hb hR i r).trans (profile_mono hb hR i hr hrs)
+
+theorem full_scale_le (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    {C r : ℝ} (hC : 1 ≤ C) (hr : 0 ≤ r) :
+    full b R d (C * r) ≤ C ^ 2 * full b R d r := by
+  obtain rfl | hrpos := hr.eq_or_lt
+  · simp
+  have hCpos : 0 < C := zero_lt_one.trans_le hC
+  have hC2 : 0 < C ^ 2 := sq_pos_of_pos hCpos
+  have h : full b R d (C * r) / C ^ 2 ≤ full b R d r := by
+    apply le_full hrpos
+    intro i
+    apply (div_le_iff₀ hC2).2
+    calc
+      full b R d (C * r) ≤ profile b R d i (C * r) := full_le_profile hb hR i _
+      _ ≤ C ^ 2 * profile b R d i r := profile_scale_le hb hR i hC r
+      _ = profile b R d i r * C ^ 2 := mul_comm _ _
+  simpa [mul_comm] using (div_le_iff₀ hC2).1 h
+
+lemma power_lower_bound {c r : ℝ} (hc : 0 < c) (hb : ∀ i, c ≤ b i)
+    (hR : ∀ i, 0 < R i) (hd0 : 0 ≤ d) (hd2 : d ≤ 2) (hr : 0 < r) :
+    c * r ^ d ≤ full b R d r := by
+  apply le_full hr
+  intro i
+  have hshape : r ^ d ≤ R i ^ d * max 1 ((r / R i) ^ 2) := by
+    rcases le_total r (R i) with hsmall | hlarge
+    · exact (Real.rpow_le_rpow hr.le hsmall hd0).trans <|
+        le_mul_of_one_le_right (Real.rpow_nonneg (hR i).le d) (le_max_left _ _)
+    · have hratio : 1 ≤ r / R i := (le_div_iff₀ (hR i)).2 (by simpa using hlarge)
+      have hp : (r / R i) ^ d ≤ (r / R i) ^ 2 := by
+        exact_mod_cast Real.rpow_le_rpow_of_exponent_le hratio hd2
+      calc
+        r ^ d = R i ^ d * (r / R i) ^ d := by
+          rw [← Real.mul_rpow (hR i).le (div_nonneg hr.le (hR i).le)]
+          congr 1
+          field_simp [(hR i).ne']
+        _ ≤ R i ^ d * (r / R i) ^ 2 :=
+          mul_le_mul_of_nonneg_left hp (Real.rpow_nonneg (hR i).le d)
+        _ ≤ R i ^ d * max 1 ((r / R i) ^ 2) :=
+          mul_le_mul_of_nonneg_left (le_max_right _ _) (Real.rpow_nonneg (hR i).le d)
+  calc
+    c * r ^ d ≤ b i * r ^ d := mul_le_mul_of_nonneg_right (hb i) (Real.rpow_nonneg hr.le d)
+    _ ≤ b i * (R i ^ d * max 1 ((r / R i) ^ 2)) :=
+      mul_le_mul_of_nonneg_left hshape (hc.le.trans (hb i))
+    _ = profile b R d i r := by rw [profile]; ring
+
+lemma full_pos {c r : ℝ} (hc : 0 < c) (hb : ∀ i, c ≤ b i)
+    (hR : ∀ i, 0 < R i) (hd0 : 0 ≤ d) (hd2 : d ≤ 2) (hr : 0 < r) :
+    0 < full b R d r :=
+  (mul_pos hc (Real.rpow_pos_of_pos hr d)).trans_le
+    (power_lower_bound hc hb hR hd0 hd2 hr)
+
+lemma ratio_bounds (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    {r s : ℝ} (hr : 0 < r) (hs : 0 < s) :
+    min 1 ((s / r) ^ 2) * full b R d r ≤ full b R d s ∧
+      full b R d s ≤ max 1 ((s / r) ^ 2) * full b R d r := by
+  have hmono := full_monoOn (d := d) hb hR
+  have hnonneg := full_nonneg (d := d) hb hR r
+  constructor
+  · rcases le_total r s with hrs | hsr
+    · calc
+        _ ≤ 1 * full b R d r := mul_le_mul_of_nonneg_right (min_le_left _ _) hnonneg
+        _ = full b R d r := one_mul _
+        _ ≤ full b R d s := hmono hr.le hs.le hrs
+    · have hC : 1 ≤ r / s := (le_div_iff₀ hs).2 (by simpa using hsr)
+      have hscale := full_scale_le (d := d) hb hR hC hs.le
+      rw [div_mul_cancel₀ r hs.ne'] at hscale
+      calc
+        _ ≤ (s / r) ^ 2 * full b R d r :=
+          mul_le_mul_of_nonneg_right (min_le_right _ _) hnonneg
+        _ ≤ (s / r) ^ 2 * ((r / s) ^ 2 * full b R d s) :=
+          mul_le_mul_of_nonneg_left hscale (sq_nonneg _)
+        _ = full b R d s := by field_simp
+  · rcases le_total s r with hsr | hrs
+    · exact (hmono hs.le hr.le hsr).trans <| by
+        simpa using mul_le_mul_of_nonneg_right (le_max_left 1 ((s / r) ^ 2)) hnonneg
+    · have hC : 1 ≤ s / r := (le_div_iff₀ hr).2 (by simpa using hrs)
+      have hscale := full_scale_le (d := d) hb hR hC hr.le
+      rw [div_mul_cancel₀ s hr.ne'] at hscale
+      exact hscale.trans <| mul_le_mul_of_nonneg_right (le_max_right _ _) hnonneg
+
+lemma continuousAt_pos (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    {r : ℝ} (hr : 0 < r) : ContinuousAt (full b R d) r := by
+  have hlow : ContinuousAt (fun s : ℝ => min 1 ((s / r) ^ 2) * full b R d r) r := by
+    fun_prop
+  have hupp : ContinuousAt (fun s : ℝ => max 1 ((s / r) ^ 2) * full b R d r) r := by
+    fun_prop
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le'
+    (by simpa [hr.ne'] using hlow.tendsto)
+    (by simpa [hr.ne'] using hupp.tendsto)
+  · filter_upwards [Ioi_mem_nhds hr] with s hs
+    exact (ratio_bounds hb hR hr hs).1
+  · filter_upwards [Ioi_mem_nhds hr] with s hs
+    exact (ratio_bounds hb hR hr hs).2
+
+lemma continuousAt_zero (hb : ∀ i, 0 ≤ b i) (hR : ∀ i, 0 < R i)
+    (hsmall : ∀ ε > 0, ∃ i, b i * R i ^ d < ε) :
+    ContinuousAt (full b R d) 0 := by
+  rw [Metric.continuousAt_iff]
+  intro ε hε
+  obtain ⟨i, hi⟩ := hsmall ε hε
+  refine ⟨R i, hR i, ?_⟩
+  intro x hx
+  rw [full_zero, Real.dist_eq, sub_zero, abs_of_nonneg (full_nonneg hb hR x)]
+  by_cases hx0 : x ≤ 0
+  · rw [full_of_nonpos hx0]
+    exact hε
+  · have hxpos : 0 < x := lt_of_not_ge hx0
+    have hxr : x < R i := by
+      simpa [Real.dist_eq, abs_of_pos hxpos] using hx
+    have hratio : 0 ≤ x / R i := div_nonneg hxpos.le (hR i).le
+    have hratio1 : x / R i ≤ 1 := (div_le_one (hR i)).2 hxr.le
+    have hp : (x / R i) ^ 2 ≤ 1 := by
+      simpa using (pow_le_pow_left₀ hratio hratio1 2)
+    calc
+      full b R d x ≤ profile b R d i x := full_le_profile hb hR i x
+      _ = b i * R i ^ d := by simp [profile, max_eq_left hp]
+      _ < ε := hi
+
+theorem isHausdorffGauge {c : ℝ} (hc : 0 < c) (hb : ∀ i, c ≤ b i)
+    (hR : ∀ i, 0 < R i) (hd0 : 0 ≤ d) (hd2 : d ≤ 2)
+    (hsmall : ∀ ε > 0, ∃ i, b i * R i ^ d < ε) :
+    SLEExactGauge.Target.IsHausdorffGauge (full b R d) := by
+  have hb0 : ∀ i, 0 ≤ b i := fun i => hc.le.trans (hb i)
+  refine ⟨?_, full_monoOn hb0 hR, full_zero, fun r hr => full_pos hc hb hR hd0 hd2 hr⟩
+  intro r hr
+  obtain rfl | hrpos := (show 0 ≤ r from hr).eq_or_lt
+  · exact (continuousAt_zero hb0 hR hsmall).continuousWithinAt
+  · exact (continuousAt_pos hb0 hR hrpos).continuousWithinAt
+
+theorem restore_finite (hb : ∀ i, 0 < b i) (hR : ∀ i, 0 < R i)
+    (discarded : Finset ι) (F : ℝ → ℝ) {M C : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ r > 0, F r ≤ M)
+    (htail : ∀ i, i ∉ discarded → ∀ r > 0, F r ≤ C * profile b R d i r) :
+    ∃ C' : ℝ, 0 < C' ∧ ∀ r > 0, F r ≤ C' * full b R d r := by
+  let C' := max C 0 + 1 + ∑ i ∈ discarded, M / (b i * R i ^ d)
+  have hbase : ∀ i, 0 < b i * R i ^ d :=
+    fun i => mul_pos (hb i) (Real.rpow_pos_of_pos (hR i) d)
+  have hterm : ∀ i, 0 ≤ M / (b i * R i ^ d) :=
+    fun i => div_nonneg hM (hbase i).le
+  have hsum : 0 ≤ ∑ i ∈ discarded, M / (b i * R i ^ d) :=
+    Finset.sum_nonneg (fun i _ => hterm i)
+  have hC' : 0 < C' := by
+    dsimp [C']
+    linarith [le_max_right C 0]
+  have hCC' : C ≤ C' := by
+    dsimp [C']
+    linarith [le_max_left C 0]
+  have hall : ∀ r > 0, ∀ i, F r ≤ C' * profile b R d i r := by
+    intro r hr i
+    by_cases hi : i ∈ discarded
+    · have hi_le : M / (b i * R i ^ d) ≤ C' := by
+        have hh := Finset.single_le_sum (fun i (_ : i ∈ discarded) => hterm i) hi
+        dsimp [C']
+        linarith [le_max_right C 0]
+      calc
+        F r ≤ M := hbound r hr
+        _ ≤ C' * (b i * R i ^ d) := (div_le_iff₀ (hbase i)).1 hi_le
+        _ ≤ C' * profile b R d i r := mul_le_mul_of_nonneg_left
+          (base_le_profile (fun i => (hb i).le) hR i r) hC'.le
+    · exact (htail i hi r hr).trans <| mul_le_mul_of_nonneg_right hCC'
+        (profile_nonneg (fun i => (hb i).le) hR i r)
+  refine ⟨C', hC', fun r hr => ?_⟩
+  have h : F r / C' ≤ full b R d r := by
+    apply le_full hr
+    intro i
+    exact (div_le_iff₀ hC').2 (by simpa [mul_comm] using hall r hr i)
+  simpa [mul_comm] using (div_le_iff₀ hC').1 h
+
+end SLEExactGauge.QuadraticEnvelope
+
+namespace SLEExactGauge.Target
+
+def diskMap (θ : ℝ) (z : ℂ) : ℂ :=
+  (z - Complex.exp (θ * Complex.I)) / (z - Complex.exp (-θ * Complex.I))
+
+def diskMapInv (θ : ℝ) (x : ℂ) : ℂ :=
+  (Complex.exp (θ * Complex.I) - x * Complex.exp (-θ * Complex.I)) / (1 - x)
+
+def centeredFlow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
+  flow κ b z t - driverValue b t
+
+def clock (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  ∫ u in Icc (0 : ℝ) t,
+    (flow κ b (diskMapInv θ x) u).im ^ 2 /
+      ‖centeredFlow κ b (diskMapInv θ x) u‖ ^ 4
+
+def initialRadius (θ : ℝ) (x : ℂ) : ℝ :=
+  2 * (diskMapInv θ x).im * ‖deriv (diskMap θ) (diskMapInv θ x)‖
+
+def radius (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  initialRadius θ x * Real.exp (-2 * a κ * clock κ θ b x t)
+
+def angle (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  Complex.arg (centeredFlow κ b (diskMapInv θ x) t)
+
+def green (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  if Surviving κ b (diskMapInv θ x) t then
+    radius κ θ b x t ^ (-α κ) * Real.sin (angle κ θ b x t) ^ p κ
+  else 0
+
+def levelTime (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ≥0∞ :=
+  sInf {t | ∃ q : ℚ, 0 < q ∧ t = ENNReal.ofReal (q : ℝ) ∧
+    Surviving κ b (diskMapInv θ 0) q ∧ ℓ < clock κ θ b 0 q}
+
+def diskK : Set ℂ := closedBall 0 (1 / 4)
+
+def potential (κ θ : ℝ) (b : Driver) (t : ℝ) : ℝ≥0∞ :=
+  ∫⁻ x in diskK, ENNReal.ofReal (green κ θ b x t)
+
+def concat (b c : Driver) (t : ℝ) : Driver where
+  toFun u := if u ≤ t.toNNReal then b u else b t.toNNReal + c (u - t.toNNReal)
+  continuous_toFun := by
+    apply Continuous.if_le (map_continuous b)
+      (continuous_const.add ((map_continuous c).comp (continuous_id.sub continuous_const)))
+      continuous_id continuous_const
+    intro u hu
+    change u = t.toNNReal at hu
+    subst u
+    simp
+  map_zero' := by simp
+
+def potentialDifference (κ θ : ℝ) (W : WienerLaw) (b : Driver) (h i : ℕ) : ℝ :=
+  let t := (i : ℝ) * mesh h
+  let v := potential κ θ b t
+  let w := ∫⁻ c, potential κ θ (concat b c t) (((i : ℝ) + 1) * mesh h) ∂W
+  if v < ∞ ∧ w < ∞ then v.toReal - w.toReal else 0
+
+def discreteMass (κ θ : ℝ) (W : WienerLaw) (b : Driver) (ℓ : ℝ) (h : ℕ) : ℝ :=
+  ∑ i ∈ Finset.range (h * 2 ^ h),
+    if ENNReal.ofReal (((i : ℝ) + 1) * mesh h) ≤ levelTime κ θ b ℓ
+    then potentialDifference κ θ W b h i else 0
+
+def levelDensity (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ :=
+  let τ := levelTime κ θ b ℓ
+  if τ < ∞ then Real.exp (2 * a κ * α κ * ℓ) *
+    Real.sin (angle κ θ b 0 τ.toReal) ^ p κ / Real.sin θ ^ p κ else 0
+
+def angleLaw (κ : ℝ) : Measure ℝ :=
+  let unnormalized := (volume.restrict (Ioo (0 : ℝ) Real.pi)).withDensity
+    (fun θ => ENNReal.ofReal (Real.sin θ ^ (4 * a κ)))
+  (unnormalized Set.univ)⁻¹ • unnormalized
+
+def clip (u x : ℝ) : ℝ := min (max x 0) u
+
+def finiteMoment (κ : ℝ) (W : WienerLaw) (n : ℕ) (ℓ : ℝ) : ℝ :=
+  limUnder atTop (fun u : ℕ => limUnder atTop (fun h : ℕ =>
+    ∫ θ, ∫ b, levelDensity κ θ b ℓ *
+      (clip u (discreteMass κ θ W b ℓ h)) ^ n ∂W ∂angleLaw κ))
+
+def moment (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
+  limUnder atTop (fun ℓ : ℕ => finiteMoment κ W n ℓ)
+
+def batchLevel (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℕ :=
+  1 + Nat.card {j : ℕ // 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4}
+
+def threshold (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
+  4 * moment κ W (n + 1) / moment κ W n
+
+def multiplicity (κ : ℝ) (W : WienerLaw) (n : ℕ) (k : ℤ) : ℕ :=
+  if (2 : ℝ) ^ k ≤ threshold κ W n then
+    ⌊(4 : ℝ) ^ n * ((2 : ℝ) ^ k) ^ n / moment κ W n⌋₊ else 0
+
+abbrev Entry := ℕ × Option (ℤ × ℕ)
+
+def ValidEntry (κ : ℝ) (W : WienerLaw) (e : Entry) : Prop :=
+  1 ≤ e.1 ∧ match e.2 with
+    | none => True
+    | some (k,j) => j < multiplicity κ W e.1 k
+
+def entryValue (e : Entry) : ℝ :=
+  match e.2 with
+    | none => e.1
+    | some (k,_) => (2 : ℝ) ^ k
+
+def entryPosition (κ : ℝ) (W : WienerLaw) (e : Entry) : ℕ :=
+  match e.2 with
+    | none => ∑' k : ℤ, multiplicity κ W e.1 k
+    | some (k,j) => (∑' l : ℤ, if l < k then multiplicity κ W e.1 l else 0) + j
+
+def entryRadius (κ : ℝ) (W : WienerLaw) (e : Entry) : ℝ :=
+  Real.exp (-(e.1 : ℝ) ^ 2 - 2 * a κ * (entryPosition κ W e : ℝ) *
+    ((batchLevel κ W e.1 : ℝ) + (e.1 : ℝ)))
+
+def entryCost (κ : ℝ) (W : WienerLaw) (e : Entry) (r : ℝ) : ℝ :=
+  entryValue e * entryRadius κ W e ^ d κ * max 1 ((r / entryRadius κ W e) ^ 2)
+
+def gauge (κ : ℝ) (W : WienerLaw) (r : ℝ) : ℝ :=
+  if r ≤ 0 then 0 else sInf {v | ∃ e : Entry, ValidEntry κ W e ∧ v = entryCost κ W e r}
+
+end SLEExactGauge.Target
+
+namespace SLEExactGauge.Intervals
+
+theorem all_real_of_all_rational {X : Type*} (μ : OuterMeasure X) (γ : ℝ → X)
+    (h : ∀ q r : ℚ, 0 < q → q < r →
+      0 < μ (γ '' Icc (q : ℝ) (r : ℝ)) ∧ μ (γ '' Icc (q : ℝ) (r : ℝ)) < ∞) :
+    ∀ s t : ℝ, 0 < s → s < t →
+      0 < μ (γ '' Icc s t) ∧ μ (γ '' Icc s t) < ∞ := by
+  intro s t hs hst
+  obtain ⟨q, hsq, hqt⟩ := exists_rat_btwn hst
+  obtain ⟨r, hqr, hrt⟩ := exists_rat_btwn hqt
+  have hq0 : 0 < q := by exact_mod_cast hs.trans hsq
+  have hqr' : q < r := by exact_mod_cast hqr
+  have hpos := (h q r hq0 hqr').1
+  have hlower : γ '' Icc (q : ℝ) (r : ℝ) ⊆ γ '' Icc s t :=
+    image_mono (Icc_subset_Icc hsq.le hrt.le)
+  obtain ⟨u, hu0, hus⟩ := exists_rat_btwn hs
+  obtain ⟨v, htv⟩ := exists_rat_gt t
+  have hu0' : 0 < u := by exact_mod_cast hu0
+  have huv : u < v := by exact_mod_cast hus.trans (hst.trans htv)
+  have hfinite := (h u v hu0' huv).2
+  have hupper : γ '' Icc s t ⊆ γ '' Icc (u : ℝ) (v : ℝ) :=
+    image_mono (Icc_subset_Icc hus.le htv.le)
+  exact ⟨hpos.trans_le (μ.mono hlower), (μ.mono hupper).trans_lt hfinite⟩
+
+theorem ae_all_real_of_ae_rational {Ω X : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (μ : OuterMeasure X) (γ : Ω → ℝ → X)
+    (h : ∀ q r : ℚ, 0 < q → q < r →
+      ∀ᵐ ω ∂P, 0 < μ (γ ω '' Icc (q : ℝ) (r : ℝ)) ∧
+        μ (γ ω '' Icc (q : ℝ) (r : ℝ)) < ∞) :
+    ∀ᵐ ω ∂P, ∀ s t : ℝ, 0 < s → s < t →
+      0 < μ (γ ω '' Icc s t) ∧ μ (γ ω '' Icc s t) < ∞ := by
+  have hall : ∀ᵐ ω ∂P, ∀ q r : ℚ, 0 < q → q < r →
+      0 < μ (γ ω '' Icc (q : ℝ) (r : ℝ)) ∧
+        μ (γ ω '' Icc (q : ℝ) (r : ℝ)) < ∞ := by
+    simpa only [ae_all_iff] using h
+  exact hall.mono (fun ω hω => all_real_of_all_rational μ (γ ω) hω)
+
+end SLEExactGauge.Intervals
+
+namespace SLEExactGauge.Target
+
+abbrev GaugeIndex (κ : ℝ) (W : WienerLaw) := {e : Entry // ValidEntry κ W e}
+
+def cleanupIndex (κ : ℝ) (W : WienerLaw) (n : ℕ) (hn : 1 ≤ n) : GaugeIndex κ W :=
+  ⟨(n, none), hn, trivial⟩
+
+instance (κ : ℝ) (W : WienerLaw) : Nonempty (GaugeIndex κ W) :=
+  ⟨cleanupIndex κ W 1 le_rfl⟩
+
+abbrev weights (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : ℝ := entryValue i.val
+abbrev radii (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : ℝ := entryRadius κ W i.val
+
+lemma weights_pos (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : 0 < weights κ W i := by
+  rcases i with ⟨⟨n, o⟩, hn, _⟩
+  cases o with
+  | none =>
+      change (0 : ℝ) < (n : ℝ)
+      exact Nat.cast_pos.mpr (Nat.succ_le_iff.mp hn)
+  | some kj =>
+      rcases kj with ⟨k, j⟩
+      exact zpow_pos (by norm_num : (0 : ℝ) < 2) k
+
+lemma radii_pos (κ : ℝ) (W : WienerLaw) (i : GaugeIndex κ W) : 0 < radii κ W i :=
+  Real.exp_pos _
+
+lemma gauge_eq_full (κ : ℝ) (W : WienerLaw) :
+    gauge κ W = QuadraticEnvelope.full (weights κ W) (radii κ W) (d κ) := by
+  funext r
+  unfold gauge QuadraticEnvelope.full
+  split_ifs with hr
+  · rfl
+  · congr 1
+    ext v
+    constructor
+    · rintro ⟨e, he, hv⟩
+      exact ⟨⟨e, he⟩, hv.symm⟩
+    · rintro ⟨⟨e, he⟩, hv⟩
+      exact ⟨e, he, hv.symm⟩
+
+lemma internal_a_pos {κ : ℝ} (hκ : 0 < κ) : 0 < a κ := div_pos (by norm_num) hκ
+
+lemma dimension_pos {κ : ℝ} (hκ : 0 < κ) : 0 < d κ := by
+  have ha := internal_a_pos hκ
+  unfold d
+  positivity
+
+lemma cleanup_radius_le {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) (n : ℕ) (hn : 1 ≤ n) :
+    radii κ W (cleanupIndex κ W n hn) ≤ Real.exp (-(n : ℝ) ^ 2) := by
+  unfold radii cleanupIndex entryRadius
+  apply Real.exp_le_exp.mpr
+  dsimp only
+  have ha := internal_a_pos hκ
+  have hnonneg : 0 ≤ 2 * a κ * (entryPosition κ W (n, none) : ℝ) *
+      ((batchLevel κ W n : ℝ) + (n : ℝ)) := by positivity
+  linarith
+
+lemma cleanup_base_le {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) (n : ℕ) (hn : 1 ≤ n) :
+    weights κ W (cleanupIndex κ W n hn) * radii κ W (cleanupIndex κ W n hn) ^ d κ ≤
+      (n : ℝ) * Real.exp (-d κ * (n : ℝ) ^ 2) := by
+  have hd := dimension_pos hκ
+  change (n : ℝ) * radii κ W (cleanupIndex κ W n hn) ^ d κ ≤ _
+  calc
+    _ ≤ (n : ℝ) * Real.exp (-(n : ℝ) ^ 2) ^ d κ := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
+      exact Real.rpow_le_rpow (radii_pos κ W _).le (cleanup_radius_le hκ W n hn) hd.le
+    _ = _ := by rw [← Real.exp_mul]; congr 2; ring
+
+lemma cleanup_small {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) :
+    ∀ ε > 0, ∃ i : GaugeIndex κ W, weights κ W i * radii κ W i ^ d κ < ε := by
+  have hd := dimension_pos hκ
+  have hlim : Tendsto (fun n : ℕ => (n : ℝ) * Real.exp (-d κ * (n : ℝ))) atTop (𝓝 0) := by
+    convert (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 (d κ) hd).comp
+      (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop) using 1
+    ext n
+    simp only [Function.comp_apply, Real.rpow_one]
+  intro ε hε
+  obtain ⟨n, hnε, hn⟩ :=
+    ((hlim.eventually (gt_mem_nhds hε)).and (eventually_ge_atTop 1)).exists
+  refine ⟨cleanupIndex κ W n hn, lt_of_le_of_lt ?_ hnε⟩
+  have hnreal : 1 ≤ (n : ℝ) := by exact_mod_cast hn
+  have hsq : (n : ℝ) ≤ (n : ℝ) ^ 2 := by nlinarith
+  calc
+    _ ≤ (n : ℝ) * Real.exp (-d κ * (n : ℝ) ^ 2) := cleanup_base_le hκ W n hn
+    _ ≤ (n : ℝ) * Real.exp (-d κ * (n : ℝ)) := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
+      exact Real.exp_le_exp.mpr (mul_le_mul_of_nonpos_left hsq (neg_nonpos.mpr hd.le))
+
+theorem gauge_basic_properties {κ : ℝ} (hκ : 0 < κ) (W : WienerLaw) :
+    ContinuousOn (gauge κ W) (Ici 0) ∧
+    MonotoneOn (gauge κ W) (Ici 0) ∧
+    gauge κ W 0 = 0 ∧
+    (∀ r : ℝ, 0 ≤ gauge κ W r) ∧
+    ∀ C ≥ 1, ∀ r ≥ 0, gauge κ W (C * r) ≤ C ^ 2 * gauge κ W r := by
+  rw [gauge_eq_full]
+  have hb : ∀ i, 0 ≤ weights κ W i := fun i => (weights_pos κ W i).le
+  have hR := radii_pos κ W
+  refine ⟨?_, QuadraticEnvelope.full_monoOn hb hR, QuadraticEnvelope.full_zero,
+    QuadraticEnvelope.full_nonneg hb hR, fun _ hC _ hr => QuadraticEnvelope.full_scale_le hb hR hC hr⟩
+  intro r hr
+  obtain rfl | hrpos := (show 0 ≤ r from hr).eq_or_lt
+  · exact (QuadraticEnvelope.continuousAt_zero hb hR (cleanup_small hκ W)).continuousWithinAt
+  · exact (QuadraticEnvelope.continuousAt_pos hb hR hrpos).continuousWithinAt
+
+
+
+theorem hausdorff_arbitrary_covers (h : ℝ → ℝ) (h0 : h 0 = 0) (E : Set ℂ) :
+    hausdorff h E =
+      ⨆ (δ : ℝ≥0∞) (_ : 0 < δ),
+        ⨅ (U : ℕ → Set ℂ) (_ : E ⊆ ⋃ j, U j) (_ : ∀ j, Metric.ediam (U j) ≤ δ),
+          ∑' j, if Metric.ediam (U j) = ∞ then ∞
+            else ENNReal.ofReal (h (Metric.ediam (U j)).toReal) := by
+  let m : ℝ≥0∞ → ℝ≥0∞ := fun r => if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal)
+  have hm0 : m 0 = 0 := by simp [m, h0]
+  have hnonempty (U : Set ℂ) : (⨆ _ : U.Nonempty, m (Metric.ediam U)) = m (Metric.ediam U) := by
+    by_cases hU : U.Nonempty
+    · simp [hU]
+    · rw [not_nonempty_iff_eq_empty.mp hU]
+      simp [hm0]
+  change OuterMeasure.mkMetric m E = _
+  rw [OuterMeasure.coe_mkMetric, Measure.mkMetric_apply]
+  simp only [hnonempty]
+  rfl
+
+
+
+lemma trace_isChordalSLE (κ : ℝ) {W : WienerLaw} (hW : IsWiener W) :
+    IsChordalSLE κ (trace κ) W := by
+  refine ⟨W, id, hW, ProbabilityTheory.HasLaw.id, ?_⟩
+  exact ae_of_all _ (fun _ _ _ => rfl)
+
+theorem transfer_interval_estimates {κ : ℝ} {h : ℝ → ℝ}
+    (hcanonical : ∀ W : WienerLaw, IsWiener W →
+      ∀ᵐ b ∂W, ∀ s t : ℝ, 0 < s → s < t →
+        0 < hausdorff h (trace κ b '' Icc s t) ∧
+        hausdorff h (trace κ b '' Icc s t) < ∞)
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (γ : Ω → ℝ → ℂ)
+    (hγ : IsChordalSLE κ γ P) :
+    ∀ᵐ ω ∂P, ∀ s t : ℝ, 0 < s → s < t →
+      0 < hausdorff h (γ ω '' Icc s t) ∧
+      hausdorff h (γ ω '' Icc s t) < ∞ := by
+  obtain ⟨W, B, hW, hB, htrace⟩ := hγ
+  have hmap := hcanonical W hW
+  rw [← hB.map_eq] at hmap
+  have hpull := ae_of_ae_map hB.aemeasurable hmap
+  filter_upwards [hpull, htrace] with ω hω htr
+  intro s t hs hst
+  have heq : γ ω '' Icc s t = trace κ (B ω) '' Icc s t := by
+    apply Set.image_congr
+    intro v hv
+    exact htr v (hs.le.trans hv.1)
+  rw [heq]
+  exact hω s t hs hst
+
+
+
+lemma dimension_eq (κ : ℝ) : d κ = 1 + κ / 8 := by
+  unfold d a
+  field_simp
+  ring
+
+lemma dimension_lt_two {κ : ℝ} (hκ8 : κ < 8) : d κ < 2 := by
+  rw [dimension_eq κ]
+  linarith
+
+lemma multiplicity_positive_bounds (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1 ^ n ≤ moment κ W n) {k : ℤ}
+    (hNk : 0 < multiplicity κ W n k) :
+    moment κ W 1 / 4 ≤ (2 : ℝ) ^ k ∧ (2 : ℝ) ^ k ≤ threshold κ W n := by
+  have hcut : (2 : ℝ) ^ k ≤ threshold κ W n := by
+    by_contra hcut
+    simp [multiplicity, hcut] at hNk
+  have hmn : 0 < moment κ W n := (pow_pos hm1 n).trans_le hJ
+  have hfloor : 0 < ⌊(4 : ℝ) ^ n * ((2 : ℝ) ^ k) ^ n / moment κ W n⌋₊ := by
+    simpa [multiplicity, hcut] using hNk
+  have hw := Nat.floor_pos.mp hfloor
+  have hnum : moment κ W n ≤ ((4 : ℝ) * 2 ^ k) ^ n := by
+    simpa only [one_mul, mul_pow] using (le_div_iff₀ hmn).1 hw
+  have hb : moment κ W 1 ≤ (4 : ℝ) * 2 ^ k :=
+    le_of_pow_le_pow_left₀ (by omega) (by positivity) (hJ.trans hnum)
+  exact ⟨(div_le_iff₀ (by norm_num : (0 : ℝ) < 4)).2 (by linarith), hcut⟩
+
+lemma weights_lower_of_moment_inequalities (κ : ℝ) (W : WienerLaw)
+    (hm1 : 0 < moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n)
+    (i : GaugeIndex κ W) : min 1 (moment κ W 1 / 4) ≤ weights κ W i := by
+  rcases i with ⟨⟨n, o⟩, hn, ho⟩
+  cases o with
+  | none =>
+      change min 1 (moment κ W 1 / 4) ≤ (n : ℝ)
+      exact (min_le_left _ _).trans (by exact_mod_cast hn)
+  | some kj =>
+      rcases kj with ⟨k, j⟩
+      change j < multiplicity κ W n k at ho
+      change min 1 (moment κ W 1 / 4) ≤ (2 : ℝ) ^ k
+      exact (min_le_right _ _).trans
+        (multiplicity_positive_bounds κ W hn hm1 (hJ n hn)
+          ((Nat.zero_le j).trans_lt ho)).1
+
+theorem gauge_admissible_of_moment_inequalities {κ : ℝ}
+    (hκ0 : 0 < κ) (hκ8 : κ < 8) (W : WienerLaw)
+    (hm1 : 0 < moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n) :
+    IsHausdorffGauge (gauge κ W) := by
+  rw [gauge_eq_full]
+  exact QuadraticEnvelope.isHausdorffGauge
+    (c := min 1 (moment κ W 1 / 4))
+    (lt_min zero_lt_one (div_pos hm1 (by norm_num)))
+    (weights_lower_of_moment_inequalities κ W hm1 hJ)
+    (radii_pos κ W) (dimension_pos hκ0).le (dimension_lt_two hκ8).le
+    (cleanup_small hκ0 W)
+
+lemma multiplicity_support_finite (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1 ^ n ≤ moment κ W n) :
+    Set.Finite {k : ℤ | 0 < multiplicity κ W n k} := by
+  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  apply (Set.finite_Icc (⌈Real.log (moment κ W 1 / 4) / Real.log 2⌉ : ℤ)
+    (⌊Real.log (threshold κ W n) / Real.log 2⌋ : ℤ)).subset
+  intro k hk
+  obtain ⟨hlo, hhi⟩ := multiplicity_positive_bounds κ W hn hm1 hJ hk
+  constructor
+  · apply Int.ceil_le.mpr
+    apply (div_le_iff₀ hlog).2
+    exact (Real.le_zpow_iff_log_le (div_pos hm1 (by norm_num))
+      (by norm_num : (0 : ℝ) < 2)).1 hlo
+  · apply Int.le_floor.mpr
+    apply (le_div_iff₀ hlog).2
+    exact Real.le_log_of_zpow_le (by norm_num : (0 : ℝ) < 2) hhi
+
+lemma repeated_entries_finite (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1 ^ n ≤ moment κ W n) :
+    Set.Finite {kj : ℤ × ℕ | kj.2 < multiplicity κ W n kj.1} := by
+  have hfin := (multiplicity_support_finite κ W hn hm1 hJ).biUnion
+    (fun k _ => (Set.finite_Iio (multiplicity κ W n k)).image (fun j : ℕ => (k, j)))
+  apply hfin.subset
+  rintro ⟨k, j⟩ hj
+  exact mem_iUnion.2 ⟨k, mem_iUnion.2 ⟨(Nat.zero_le j).trans_lt hj, j, hj, rfl⟩⟩
+
+lemma batch_entries_finite (κ : ℝ) (W : WienerLaw) {n : ℕ}
+    (hn : 1 ≤ n) (hm1 : 0 < moment κ W 1)
+    (hJ : moment κ W 1 ^ n ≤ moment κ W n) :
+    Set.Finite {e : Entry | e.1 = n ∧ ValidEntry κ W e} := by
+  have hfin := ((repeated_entries_finite κ W hn hm1 hJ).image
+    (fun kj : ℤ × ℕ => (n, (some kj : Option (ℤ × ℕ))))).insert (n, none)
+  apply hfin.subset
+  rintro ⟨m, o⟩ ⟨heq, hm, ho⟩
+  dsimp at heq
+  subst m
+  cases o with
+  | none => exact Or.inl rfl
+  | some kj => exact Or.inr ⟨kj, ho, rfl⟩
+
+lemma early_entries_finite (κ : ℝ) (W : WienerLaw)
+    (hm1 : 0 < moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n)
+    (n₀ : ℕ) : Set.Finite {e : Entry | ValidEntry κ W e ∧ e.1 < n₀} := by
+  have hfin := (Set.finite_Ico 1 n₀).biUnion (fun n hn =>
+    batch_entries_finite κ W hn.1 hm1 (hJ n hn.1))
+  apply hfin.subset
+  intro e he
+  exact mem_iUnion.2 ⟨e.1, mem_iUnion.2 ⟨⟨he.1.1, he.2⟩, rfl, he.1⟩⟩
+
+theorem gauge_restore_early_batches (κ : ℝ) (W : WienerLaw)
+    (hm1 : 0 < moment κ W 1)
+    (hJ : ∀ n : ℕ, 1 ≤ n → moment κ W 1 ^ n ≤ moment κ W n)
+    (n₀ : ℕ) (F : ℝ → ℝ) {M C : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ r > 0, F r ≤ M)
+    (htail : ∀ e : Entry, ValidEntry κ W e → n₀ ≤ e.1 →
+      ∀ r > 0, F r ≤ C * entryCost κ W e r) :
+    ∃ C' : ℝ, 0 < C' ∧ ∀ r > 0, F r ≤ C' * gauge κ W r := by
+  classical
+  have hfin : Set.Finite {i : GaugeIndex κ W | i.val.1 < n₀} := by
+    apply ((early_entries_finite κ W hm1 hJ n₀).preimage
+      (f := fun i : GaugeIndex κ W => i.val) Subtype.val_injective.injOn).subset
+    intro i hi
+    exact ⟨i.property, hi⟩
+  rw [gauge_eq_full]
+  apply QuadraticEnvelope.restore_finite (weights_pos κ W) (radii_pos κ W)
+    hfin.toFinset F hM hbound
+  intro i hi r hr
+  have hn : n₀ ≤ i.val.1 := by
+    have hnot : ¬ i.val.1 < n₀ := by simpa using hi
+    exact le_of_not_gt hnot
+  exact htail i.val i.property hn r hr
+
+end SLEExactGauge.Target
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointAbsorption.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointAbsorption.lean
index 4acc0e27cc3533ba878cd0bea5e7b920fe3ab424..a3541c2ec02bc424ca5c177eda20ba027b6c2596 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointAbsorption.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointAbsorption.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Multipoint.MultipointYoung
-
-namespace OAI
-
-/-! Summation of the pairwise Young bounds. The active index set may
-change at the individual activation and completion times. -/
-
-noncomputable section
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-lemma offDiag_vertex_cost_le {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (A : ι → ℝ) (hA : ∀ i ∈ s, 0 ≤ A i) :
-    (∑ ij ∈ s.offDiag, (A ij.1 + A ij.2)) ≤
-      2 * (s.card : ℝ) * ∑ i ∈ s, A i := by
-  calc
-    _ ≤ ∑ ij ∈ s ×ˢ s, (A ij.1 + A ij.2) := by
-      apply Finset.sum_le_sum_of_subset_of_nonneg
-      · intro ij hij
-        obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
-        exact Finset.mem_product.mpr ⟨hi, hj⟩
-      · intro ij hij _
-        obtain ⟨hi, hj⟩ := Finset.mem_product.mp hij
-        exact add_nonneg (hA _ hi) (hA _ hj)
-    _ = _ := by
-      rw [Finset.sum_product]
-      simp only [Finset.sum_add_distrib, Finset.sum_const, nsmul_eq_mul,
-        ← Finset.mul_sum]
-      ring
-
-/-- Negative one-point drift absorbs every pair contribution. We bound
-the off-diagonal vertex count by the full square; this only changes the
-order-dependent choice of epsilon. -/
-theorem sum_drift_absorption {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (D A H : ι → ℝ) (cross weight : ι × ι → ℝ)
-    (C k ε L : ℝ) (hε : 0 ≤ ε) (hA : ∀ i ∈ s, 0 ≤ A i)
-    (hbudget : 2*ε*(s.card : ℝ) ≤ k)
-    (hD : ∀ i ∈ s, D i ≤ C*H i-k*A i)
-    (hcross : ∀ ij ∈ s.offDiag, cross ij ≤ ε*(A ij.1+A ij.2)+L*weight ij) :
-    (∑ i ∈ s, D i) + (∑ ij ∈ s.offDiag, cross ij) ≤
-      C*(∑ i ∈ s, H i) + L*(∑ ij ∈ s.offDiag, weight ij) := by
-  have hsumD := Finset.sum_le_sum hD
-  simp only [Finset.sum_sub_distrib, ← Finset.mul_sum] at hsumD
-  have hsumcross := Finset.sum_le_sum hcross
-  simp only [Finset.sum_add_distrib, ← Finset.mul_sum] at hsumcross
-  have hvertices := mul_le_mul_of_nonneg_left (offDiag_vertex_cost_le s A hA) hε
-  rw [Finset.sum_add_distrib] at hvertices
-  have hsumA : 0 ≤ ∑ i ∈ s, A i := Finset.sum_nonneg hA
-  have htotal := mul_le_mul_of_nonneg_right hbudget hsumA
-  nlinarith
-
-lemma absorption_epsilon_pos {k : ℝ} (hk : 0 < k) (m : ℕ) :
-    0 < k/(4*((m : ℝ)+1)) := by positivity
-
-lemma absorption_epsilon_budget {k : ℝ} (hk : 0 ≤ k) (m : ℕ) :
-    2*(k/(4*((m : ℝ)+1)))*(m : ℝ) ≤ k := by
-  have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg m
-  have hden : 0 < 4*((m : ℝ)+1) := by positivity
-  have he : 2*(k/(4*((m : ℝ)+1)))*(m : ℝ) =
-      (2*k*(m : ℝ))/(4*((m : ℝ)+1)) := by ring
-  rw [he]
-  apply (div_le_iff₀ hden).mpr
-  nlinarith
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointYoung
+
+namespace OAI
+
+/-! Summation of the pairwise Young bounds. The active index set may
+change at the individual activation and completion times. -/
+
+noncomputable section
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+lemma offDiag_vertex_cost_le {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (A : ι → ℝ) (hA : ∀ i ∈ s, 0 ≤ A i) :
+    (∑ ij ∈ s.offDiag, (A ij.1 + A ij.2)) ≤
+      2 * (s.card : ℝ) * ∑ i ∈ s, A i := by
+  calc
+    _ ≤ ∑ ij ∈ s ×ˢ s, (A ij.1 + A ij.2) := by
+      apply Finset.sum_le_sum_of_subset_of_nonneg
+      · intro ij hij
+        obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
+        exact Finset.mem_product.mpr ⟨hi, hj⟩
+      · intro ij hij _
+        obtain ⟨hi, hj⟩ := Finset.mem_product.mp hij
+        exact add_nonneg (hA _ hi) (hA _ hj)
+    _ = _ := by
+      rw [Finset.sum_product]
+      simp only [Finset.sum_add_distrib, Finset.sum_const, nsmul_eq_mul,
+        ← Finset.mul_sum]
+      ring
+
+/-- Negative one-point drift absorbs every pair contribution. We bound
+the off-diagonal vertex count by the full square; this only changes the
+order-dependent choice of epsilon. -/
+theorem sum_drift_absorption {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (D A H : ι → ℝ) (cross weight : ι × ι → ℝ)
+    (C k ε L : ℝ) (hε : 0 ≤ ε) (hA : ∀ i ∈ s, 0 ≤ A i)
+    (hbudget : 2*ε*(s.card : ℝ) ≤ k)
+    (hD : ∀ i ∈ s, D i ≤ C*H i-k*A i)
+    (hcross : ∀ ij ∈ s.offDiag, cross ij ≤ ε*(A ij.1+A ij.2)+L*weight ij) :
+    (∑ i ∈ s, D i) + (∑ ij ∈ s.offDiag, cross ij) ≤
+      C*(∑ i ∈ s, H i) + L*(∑ ij ∈ s.offDiag, weight ij) := by
+  have hsumD := Finset.sum_le_sum hD
+  simp only [Finset.sum_sub_distrib, ← Finset.mul_sum] at hsumD
+  have hsumcross := Finset.sum_le_sum hcross
+  simp only [Finset.sum_add_distrib, ← Finset.mul_sum] at hsumcross
+  have hvertices := mul_le_mul_of_nonneg_left (offDiag_vertex_cost_le s A hA) hε
+  rw [Finset.sum_add_distrib] at hvertices
+  have hsumA : 0 ≤ ∑ i ∈ s, A i := Finset.sum_nonneg hA
+  have htotal := mul_le_mul_of_nonneg_right hbudget hsumA
+  nlinarith
+
+lemma absorption_epsilon_pos {k : ℝ} (hk : 0 < k) (m : ℕ) :
+    0 < k/(4*((m : ℝ)+1)) := by positivity
+
+lemma absorption_epsilon_budget {k : ℝ} (hk : 0 ≤ k) (m : ℕ) :
+    2*(k/(4*((m : ℝ)+1)))*(m : ℝ) ≤ k := by
+  have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg m
+  have hden : 0 < 4*((m : ℝ)+1) := by positivity
+  have he : 2*(k/(4*((m : ℝ)+1)))*(m : ℝ) =
+      (2*k*(m : ℝ))/(4*((m : ℝ)+1)) := by ring
+  rw [he]
+  apply (div_le_iff₀ hden).mpr
+  nlinarith
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointActiveCosts.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointActiveCosts.lean
index d2a7a53b430c0885951cf9ff7f2302e467927e30..adbe65b1947f7cdaa48bf45f276491959d05daf1 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointActiveCosts.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointActiveCosts.lean
@@ -1,117 +1,117 @@
-import OAI.Probability.SLE.Multipoint.MultipointLogClockTransport
-import OAI.Probability.SLE.Multipoint.MultipointGlobalDiscount
-
-namespace OAI
-
-/-! The raw active costs and their bounds on individual completion
-intervals. Disjoint active intervals contribute zero pair cost. -/
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def actualSingleCost (κ β a U : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ) : ℝ :=
-  regulator β U (logFactorClock κ (logLoewnerCoordinates G z t)-a)*
-    logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2
-
-def actualPairCost (κ β az aw Uz Uw : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z w : ℂ) (t : ℝ) : ℝ :=
-  (regulator β Uz (logFactorClock κ (logLoewnerCoordinates G z t)-az)*
-    regulator β Uw (logFactorClock κ (logLoewnerCoordinates G w t)-aw))^(-exponent (greenExponent κ))*
-      (logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)*
-        logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal))
-
-lemma actualSingleCost_nonneg (κ β a U : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ) : 0 ≤ actualSingleCost κ β a U B G z t :=
-  mul_nonneg (regulator_pos _ _ _).le (sq_nonneg _)
-
-lemma actualPairCost_nonneg (κ β az aw Uz Uw : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z w : ℂ) (t : ℝ) :
-    0 ≤ actualPairCost κ β az aw Uz Uw B G z w t :=
-  mul_nonneg (Real.rpow_nonneg (mul_pos (regulator_pos _ _ _) (regulator_pos _ _ _)).le _)
-    (mul_pos (logVelocity_pos _ _ _) (logVelocity_pos _ _ _)).le
-
-theorem actualSingleCost_bound
-    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
-    (β a U : ℝ) (hβ : 0 < β) (hU : 0 ≤ U)
-    (hstart : σ < T → logFactorClock κ (logLoewnerCoordinates G z σ) = a)
-    (hend : σ < T → logFactorClock κ (logLoewnerCoordinates G z T)-a ≤ U) :
-    IntervalIntegrable (actualSingleCost κ β a U B G z) volume (σ : ℝ) (T : ℝ) ∧
-      (∫ t in (σ : ℝ)..(T : ℝ), actualSingleCost κ β a U B G z t) ≤ 2/(κ*β) := by
-  rcases lt_or_eq_of_le hσT with hlt | rfl
-  · have he : actualSingleCost κ β a U B G z = fun t =>
-        regulator β U (logFactorClock κ (logLoewnerCoordinates G z t)-
-          logFactorClock κ (logLoewnerCoordinates G z σ))*
-          logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2 := by
-      funext t
-      rw [actualSingleCost, hstart hlt]
-    rw [he]
-    refine ⟨hw.logFactor_single_intervalIntegrable hB σ T hσT z hz β U,
-      hw.logFactor_single_regulator_integral_le hκ hB σ T hσT z hz hβ hU ?_⟩
-    simpa only [hstart hlt] using hend hlt
-  · exact ⟨by simp, by simp only [intervalIntegral.integral_same]; positivity⟩
-
-def overlapStart (σz σw : ℝ≥0) : ℝ≥0 := max σz σw
-
-def overlapEnd (σz σw Tz Tw : ℝ≥0) : ℝ≥0 := max (overlapStart σz σw) (min Tz Tw)
-
-theorem actualPairCost_bound
-    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
-    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
-    (σz σw Tz Tw : ℝ≥0) (z w : ℂ)
-    (hz : z ∈ survivingDomain γ Tz) (hw' : w ∈ survivingDomain γ Tw) (hne : z ≠ w)
-    (β az aw Uz Uw : ℝ) (hβ : 0 ≤ β) (hM : 0 ≤ exponent (greenExponent κ))
-    (hdecay : 2*(exponent (greenExponent κ)*β) < 2/κ)
-    (hstartz : σz < Tz → logFactorClock κ (logLoewnerCoordinates G z σz) = az)
-    (hstartw : σw < Tw → logFactorClock κ (logLoewnerCoordinates G w σw) = aw)
-    (hgeoz : σz < Tz → σw < Tw → physicalConformalRadius J G σz z ≤ dist (J z) (J w))
-    (hgeow : σz < Tz → σw < Tw → physicalConformalRadius J G σw w ≤ dist (J z) (J w)) :
-    IntervalIntegrable (actualPairCost κ β az aw Uz Uw B G z w) volume
-      (overlapStart σz σw : ℝ) (overlapEnd σz σw Tz Tw : ℝ) ∧
-      (∫ t in (overlapStart σz σw : ℝ)..(overlapEnd σz σw Tz Tw : ℝ),
-        actualPairCost κ β az aw Uz Uw B G z w t) ≤
-          interactionBudget 2 (exponent (greenExponent κ)*(κ*β)) := by
-  have hadm : 2*(exponent (greenExponent κ)*(κ*β)) < 2 := by
-    have hh := (source_pair_exponent_admissible (c := exponent (greenExponent κ)*β) hκ).mpr hdecay
-    convert hh using 1
-    ring
-  by_cases ht : overlapStart σz σw < min Tz Tw
-  · have hzt : σz < Tz := (le_max_left σz σw).trans_lt (ht.trans_le (min_le_left _ _))
-    have hwt : σw < Tw := (le_max_right σz σw).trans_lt (ht.trans_le (min_le_right _ _))
-    have he : actualPairCost κ β az aw Uz Uw B G z w = fun t =>
-        (regulator β Uz (logFactorClock κ (logLoewnerCoordinates G z t)-
-          logFactorClock κ (logLoewnerCoordinates G z σz))*
-         regulator β Uw (logFactorClock κ (logLoewnerCoordinates G w t)-
-          logFactorClock κ (logLoewnerCoordinates G w σw)))^(-exponent (greenExponent κ))*
-         (logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)*
-          logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal)) := by
-      funext t
-      rw [actualPairCost, hstartz hzt, hstartw hwt]
-    rw [he, overlapEnd, max_eq_right ht.le]
-    have hzl := survivingDomain_antitone γ (min_le_left Tz Tw) hz
-    have hwl := survivingDomain_antitone γ (min_le_right Tz Tw) hw'
-    exact ⟨hw.logFactor_pair_intervalIntegrable hB σz σw (overlapStart σz σw) (min Tz Tw)
-        (le_max_left _ _) (le_max_right _ _) ht.le z w hzl hwl β _ Uz Uw,
-      hw.logFactor_pair_regulator_integral_le hκ hB hgrowth J hJ hJinj σz σw
-        (overlapStart σz σw) (min Tz Tw) (le_max_left _ _) (le_max_right _ _) ht.le
-        z w hzl hwl hne (hgeoz hzt hwt) (hgeow hzt hwt) hβ hM hdecay Uz Uw⟩
-  · have he : overlapEnd σz σw Tz Tw = overlapStart σz σw :=
-      max_eq_left (le_of_not_gt ht)
-    rw [he]
-    exact ⟨by simp, by simpa only [intervalIntegral.integral_same] using
-      (interactionBudget_pos (by norm_num : (0 : ℝ) < 2) hadm).le⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLogClockTransport
+import OAI.Probability.SLE.Multipoint.MultipointGlobalDiscount
+
+namespace OAI
+
+/-! The raw active costs and their bounds on individual completion
+intervals. Disjoint active intervals contribute zero pair cost. -/
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def actualSingleCost (κ β a U : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ) : ℝ :=
+  regulator β U (logFactorClock κ (logLoewnerCoordinates G z t)-a)*
+    logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2
+
+def actualPairCost (κ β az aw Uz Uw : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z w : ℂ) (t : ℝ) : ℝ :=
+  (regulator β Uz (logFactorClock κ (logLoewnerCoordinates G z t)-az)*
+    regulator β Uw (logFactorClock κ (logLoewnerCoordinates G w t)-aw))^(-exponent (greenExponent κ))*
+      (logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)*
+        logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal))
+
+lemma actualSingleCost_nonneg (κ β a U : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ) : 0 ≤ actualSingleCost κ β a U B G z t :=
+  mul_nonneg (regulator_pos _ _ _).le (sq_nonneg _)
+
+lemma actualPairCost_nonneg (κ β az aw Uz Uw : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z w : ℂ) (t : ℝ) :
+    0 ≤ actualPairCost κ β az aw Uz Uw B G z w t :=
+  mul_nonneg (Real.rpow_nonneg (mul_pos (regulator_pos _ _ _) (regulator_pos _ _ _)).le _)
+    (mul_pos (logVelocity_pos _ _ _) (logVelocity_pos _ _ _)).le
+
+theorem actualSingleCost_bound
+    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
+    (β a U : ℝ) (hβ : 0 < β) (hU : 0 ≤ U)
+    (hstart : σ < T → logFactorClock κ (logLoewnerCoordinates G z σ) = a)
+    (hend : σ < T → logFactorClock κ (logLoewnerCoordinates G z T)-a ≤ U) :
+    IntervalIntegrable (actualSingleCost κ β a U B G z) volume (σ : ℝ) (T : ℝ) ∧
+      (∫ t in (σ : ℝ)..(T : ℝ), actualSingleCost κ β a U B G z t) ≤ 2/(κ*β) := by
+  rcases lt_or_eq_of_le hσT with hlt | rfl
+  · have he : actualSingleCost κ β a U B G z = fun t =>
+        regulator β U (logFactorClock κ (logLoewnerCoordinates G z t)-
+          logFactorClock κ (logLoewnerCoordinates G z σ))*
+          logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2 := by
+      funext t
+      rw [actualSingleCost, hstart hlt]
+    rw [he]
+    refine ⟨hw.logFactor_single_intervalIntegrable hB σ T hσT z hz β U,
+      hw.logFactor_single_regulator_integral_le hκ hB σ T hσT z hz hβ hU ?_⟩
+    simpa only [hstart hlt] using hend hlt
+  · exact ⟨by simp, by simp only [intervalIntegral.integral_same]; positivity⟩
+
+def overlapStart (σz σw : ℝ≥0) : ℝ≥0 := max σz σw
+
+def overlapEnd (σz σw Tz Tw : ℝ≥0) : ℝ≥0 := max (overlapStart σz σw) (min Tz Tw)
+
+theorem actualPairCost_bound
+    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
+    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
+    (σz σw Tz Tw : ℝ≥0) (z w : ℂ)
+    (hz : z ∈ survivingDomain γ Tz) (hw' : w ∈ survivingDomain γ Tw) (hne : z ≠ w)
+    (β az aw Uz Uw : ℝ) (hβ : 0 ≤ β) (hM : 0 ≤ exponent (greenExponent κ))
+    (hdecay : 2*(exponent (greenExponent κ)*β) < 2/κ)
+    (hstartz : σz < Tz → logFactorClock κ (logLoewnerCoordinates G z σz) = az)
+    (hstartw : σw < Tw → logFactorClock κ (logLoewnerCoordinates G w σw) = aw)
+    (hgeoz : σz < Tz → σw < Tw → physicalConformalRadius J G σz z ≤ dist (J z) (J w))
+    (hgeow : σz < Tz → σw < Tw → physicalConformalRadius J G σw w ≤ dist (J z) (J w)) :
+    IntervalIntegrable (actualPairCost κ β az aw Uz Uw B G z w) volume
+      (overlapStart σz σw : ℝ) (overlapEnd σz σw Tz Tw : ℝ) ∧
+      (∫ t in (overlapStart σz σw : ℝ)..(overlapEnd σz σw Tz Tw : ℝ),
+        actualPairCost κ β az aw Uz Uw B G z w t) ≤
+          interactionBudget 2 (exponent (greenExponent κ)*(κ*β)) := by
+  have hadm : 2*(exponent (greenExponent κ)*(κ*β)) < 2 := by
+    have hh := (source_pair_exponent_admissible (c := exponent (greenExponent κ)*β) hκ).mpr hdecay
+    convert hh using 1
+    ring
+  by_cases ht : overlapStart σz σw < min Tz Tw
+  · have hzt : σz < Tz := (le_max_left σz σw).trans_lt (ht.trans_le (min_le_left _ _))
+    have hwt : σw < Tw := (le_max_right σz σw).trans_lt (ht.trans_le (min_le_right _ _))
+    have he : actualPairCost κ β az aw Uz Uw B G z w = fun t =>
+        (regulator β Uz (logFactorClock κ (logLoewnerCoordinates G z t)-
+          logFactorClock κ (logLoewnerCoordinates G z σz))*
+         regulator β Uw (logFactorClock κ (logLoewnerCoordinates G w t)-
+          logFactorClock κ (logLoewnerCoordinates G w σw)))^(-exponent (greenExponent κ))*
+         (logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)*
+          logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal)) := by
+      funext t
+      rw [actualPairCost, hstartz hzt, hstartw hwt]
+    rw [he, overlapEnd, max_eq_right ht.le]
+    have hzl := survivingDomain_antitone γ (min_le_left Tz Tw) hz
+    have hwl := survivingDomain_antitone γ (min_le_right Tz Tw) hw'
+    exact ⟨hw.logFactor_pair_intervalIntegrable hB σz σw (overlapStart σz σw) (min Tz Tw)
+        (le_max_left _ _) (le_max_right _ _) ht.le z w hzl hwl β _ Uz Uw,
+      hw.logFactor_pair_regulator_integral_le hκ hB hgrowth J hJ hJinj σz σw
+        (overlapStart σz σw) (min Tz Tw) (le_max_left _ _) (le_max_right _ _) ht.le
+        z w hzl hwl hne (hgeoz hzt hwt) (hgeow hzt hwt) hβ hM hdecay Uz Uw⟩
+  · have he : overlapEnd σz σw Tz Tw = overlapStart σz σw :=
+      max_eq_left (le_of_not_gt ht)
+    rw [he]
+    exact ⟨by simp, by simpa only [intervalIntegral.integral_same] using
+      (interactionBudget_pos (by norm_num : (0 : ℝ) < 2) hadm).le⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointActualBudget.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointActualBudget.lean
index bfc3574f3cb29a42ba5ebcb066c488323b54888a..a69e55962ccd71b30b811eaac2b02ea2738c8593 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointActualBudget.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointActualBudget.lean
@@ -1,146 +1,146 @@
-import OAI.Probability.SLE.Multipoint.MultipointActualExpectation
-import OAI.Probability.SLE.Multipoint.MultipointCostAssembly
-import OAI.Probability.SLE.Multipoint.MultipointInitialParameters
-
-namespace OAI
-
-/-! The uniform discount budget for the actual, individually stopped
-Loewner coordinates in an arbitrary conformal image. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem actual_localized_discount_le
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ β lo hi M : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hβ : 0 < β)
-    (hdecay : 2*(exponent (greenExponent κ)*β) < 2/κ)
-    {H : ℝ≥0} (z : ι → ℂ) (hne : Function.Injective z)
-    (u₀ u₂ : ι → ℝ) (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (S : LocalizedState B hm κ lo hi M H z
-      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G)
-    (horder : ∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
-    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
-    (R b : ι → ℝ) (hR : ∀ i, 0 < R i) (hb : ∀ i, 0 < b i)
-    (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
-    (hu₀ : ∀ i, u₀ i = Real.log (R i/b i))
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j)))
-    (r : ℝ≥0 → Ω → ℝ) (C : ℝ≥0)
-    (hrm : Measurable (Function.uncurry r)) (hrb : ∀ t ω, |r t ω| ≤ C)
-    (hcost : ∀ ω t, t < S.χ ω → r t ω =
-      productDriftCost (patternActiveSet (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω)) κ
-        (fun i => phaseRegulatorScale β (sourceClockLevel κ (z i) (u₀ i))
-          (κ/4*(u₂ i-u₀ i)) (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i))
-        (fun i => phaseRegulatorSlope β (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i))
-        (S.Q t ω) (B t ω)) (ω : Ω) :
-    discountClock r (S.χ ω) ω ≤ uniformDiscountBudget κ β (Finset.univ : Finset ι) := by
-  classical
-  let ρ := actualFactorStops γ G z u₀ u₂ H
-  let a := fun i => sourceClockLevel κ (z i) (u₀ i)
-  let U := fun i => κ/4*(u₂ i-u₀ i)
-  let σ := fun i => min (ρ (i,0) ω) (S.χ ω)
-  let T := fun i => min (ρ (i,2) ω) (S.χ ω)
-  have hσT (i : ι) : σ i ≤ T i := min_le_min (horder ω i (by decide : (0 : Fin 3) ≤ 2)) le_rfl
-  have hTH (i : ι) : T i ≤ S.χ ω := min_le_right _ _
-  have hlive (i : ι) : z i ∈ survivingDomain (γ ω) (T i) := (S.actual ω i).1
-  have hσeq (i : ι) (hlt : σ i < T i) : σ i = ρ (i,0) ω := by
-    apply min_eq_left
-    by_contra h
-    have he : σ i = S.χ ω := min_eq_right (le_of_not_ge h)
-    exact (not_lt_of_ge (hTH i)) (he ▸ hlt)
-  have hσχ (i : ι) : σ i ≤ S.χ ω := min_le_right _ _
-  have hστ (i : ι) : σ i ≤ ρ (i,2) ω :=
-    (min_le_left _ _).trans (horder ω i (by decide : (0 : Fin 3) ≤ 2))
-  have hstart (i : ι) (hlt : σ i < T i) :
-      logFactorClock κ (logLoewnerCoordinates (G ω) (z i) (σ i)) = a i := by
-    have he := (S.actual_at ω i (σ i) (hσχ i) (hστ i)).2
-    have hs : actualFactorStops γ G z u₀ u₂ H (i,0) ω = σ i := (hσeq i hlt).symm
-    have hc := actual_phase_switch_clock hBc hκ hw S horder ω i 0
-      (by rw [hs]; exact hσχ i)
-    rw [hs, he] at hc
-    exact sub_eq_zero.mp (by simpa only [ite_true] using hc)
-  have hend (i : ι) :
-      logFactorClock κ (logLoewnerCoordinates (G ω) (z i) (T i))-a i ≤ U i := by
-    have hc := cutoff_radius_strictly_reached_iff (continuous_const.mul (hBc ω))
-      (hw ω) hκ (z i) (u₂ i) ((h₀ i).trans (hu i)) H (T i)
-      ((hTH i).trans (S.bounded ω)) (hlive i) ω
-    have hle : logFactorClock κ (logLoewnerCoordinates (G ω) (z i) (T i)) ≤
-        sourceClockLevel κ (z i) (u₂ i) := by
-      apply le_of_not_gt
-      intro h
-      have ht : ρ (i,2) ω < T i := by
-        simpa only [ρ, actualFactorStops, factorRadiusLevel, Fin.reduceEq, ite_false, cutoffStop] using hc.mpr h
-      exact (not_lt_of_ge (min_le_left _ _)) ht
-    have he : sourceClockLevel κ (z i) (u₂ i) = a i+U i := by
-      dsimp [a,U,sourceClockLevel]
-      ring
-    rw [he] at hle
-    linarith
-  have hgeo (i j : ι) (hij : i ≠ j) (hlt : σ i < T i) :
-      physicalConformalRadius J (G ω) (σ i) (z i) ≤ dist (J (z i)) (J (z j)) := by
-    have hc := hstart i hlt
-    rw [logFactorClock_eq_logRadiusClock κ (Real.log (z i).im)] at hc
-    have he : logRadiusClock (Real.log (z i).im)
-        (logLoewnerCoordinates (G ω) (z i) (σ i)) = Real.log (R i/b i) := by
-      dsimp [a,sourceClockLevel] at hc
-      rw [← hu₀ i]
-      nlinarith [hκ]
-    rw [(hw ω).physical_radius_at_activation (continuous_const.mul (hBc ω)) J
-      (z i) (R i) (b i) (hR i) (hb i) (hRdef i) (σ i)
-      (S.actual_at ω i (σ i) (hσχ i) (hστ i)).1 he]
-    exact hsep i j hij
-  have hactive (t : ℝ) (ht : t ∈ Ico 0 (S.χ ω : ℝ)) (i : ι)
-      (hi : i ∈ patternActiveSet (phaseAt ρ t.toNNReal ω)) :
-      (σ i : ℝ) ≤ t ∧ t ≤ (T i : ℝ) := by
-    have ha := (phaseAt_active_iff ρ ω i (horder ω i) t.toNNReal).mp
-      (Finset.mem_filter.mp hi).2
-    have htχ : t.toNNReal ≤ S.χ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2.le
-    constructor
-    · have hh := (min_le_left (ρ (i,0) ω) (S.χ ω)).trans ha.1
-      simpa only [Real.coe_toNNReal t ht.1] using (NNReal.coe_le_coe.mpr hh)
-    · have hh : t.toNNReal ≤ T i := le_min ha.2.le htχ
-      simpa only [Real.coe_toNNReal t ht.1] using (NNReal.coe_le_coe.mpr hh)
-  have hQ (t : ℝ) (ht : t ∈ Ico 0 (S.χ ω : ℝ)) (i : ι)
-      (hi : i ∈ patternActiveSet (phaseAt ρ t.toNNReal ω)) :
-      S.Q t ω i = logLoewnerCoordinates (G ω) (z i) t := by
-    have ha := (phaseAt_active_iff ρ ω i (horder ω i) t.toNNReal).mp
-      (Finset.mem_filter.mp hi).2
-    have hh := (S.actual_at ω i t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2.le) ha.2.le).2
-    simpa only [Real.coe_toNNReal t ht.1] using hh
-  have hrt : ∀ ω, Measurable (fun t => r t ω) :=
-    fun ω => hrm.comp (measurable_id.prodMk measurable_const)
-  have hri : IntervalIntegrable (fun t : ℝ => r t.toNNReal ω) volume 0 (S.χ ω : ℝ) :=
-    ((bounded_rate_locallyIntegrable r hrt C hrb ω).integrableOn_isCompact isCompact_uIcc).intervalIntegrable
-  have hbound := actual_phase_cost_integral_le hκ hκ8 (hBc ω) (hw ω) hgrowth J hJ hJinj
-    (Finset.univ : Finset ι) z (fun _ _ _ _ he => hne he) β hβ hdecay a U
-    (fun i _ => mul_nonneg (div_nonneg hκ.le (by norm_num)) (sub_nonneg.mpr (hu i)))
-    σ T (S.χ ω) (fun i _ => hσT i) (fun i _ => hTH i) (fun i _ => hlive i)
-    (fun i _ => hstart i) (fun i _ _ => hend i)
-    (fun i _ j _ hij hi _ => hgeo i j hij hi)
-    (fun t => S.Q t ω) (fun t => phaseAt ρ t.toNNReal ω)
-    (fun _ _ => Finset.subset_univ _) hactive hQ
-    (fun t ht i hi => by
-      have hh := (actual_phase_regulator hBc hκ hβ.le h₀ hu hw S horder ω t.toNNReal
-        (Real.toNNReal_le_iff_le_coe.mpr ht.2.le) i (Finset.mem_filter.mp hi).2).1
-      simpa only [Real.coe_toNNReal t ht.1,hQ t ht i hi] using hh)
-    (fun t : ℝ => r t.toNNReal ω) hri
-    (fun t ht => by
-      have htχ : t.toNNReal < S.χ ω := by
-        apply NNReal.coe_lt_coe.mp
-        simpa only [Real.coe_toNNReal t ht.1] using ht.2
-      have hh := hcost ω t.toNNReal htχ
-      simpa only [Real.coe_toNNReal t ht.1] using hh)
-  simpa only [discountClock,Real.toNNReal_coe] using hbound
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointActualExpectation
+import OAI.Probability.SLE.Multipoint.MultipointCostAssembly
+import OAI.Probability.SLE.Multipoint.MultipointInitialParameters
+
+namespace OAI
+
+/-! The uniform discount budget for the actual, individually stopped
+Loewner coordinates in an arbitrary conformal image. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem actual_localized_discount_le
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ β lo hi M : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hβ : 0 < β)
+    (hdecay : 2*(exponent (greenExponent κ)*β) < 2/κ)
+    {H : ℝ≥0} (z : ι → ℂ) (hne : Function.Injective z)
+    (u₀ u₂ : ι → ℝ) (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (S : LocalizedState B hm κ lo hi M H z
+      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G)
+    (horder : ∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
+    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
+    (R b : ι → ℝ) (hR : ∀ i, 0 < R i) (hb : ∀ i, 0 < b i)
+    (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
+    (hu₀ : ∀ i, u₀ i = Real.log (R i/b i))
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j)))
+    (r : ℝ≥0 → Ω → ℝ) (C : ℝ≥0)
+    (hrm : Measurable (Function.uncurry r)) (hrb : ∀ t ω, |r t ω| ≤ C)
+    (hcost : ∀ ω t, t < S.χ ω → r t ω =
+      productDriftCost (patternActiveSet (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω)) κ
+        (fun i => phaseRegulatorScale β (sourceClockLevel κ (z i) (u₀ i))
+          (κ/4*(u₂ i-u₀ i)) (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i))
+        (fun i => phaseRegulatorSlope β (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i))
+        (S.Q t ω) (B t ω)) (ω : Ω) :
+    discountClock r (S.χ ω) ω ≤ uniformDiscountBudget κ β (Finset.univ : Finset ι) := by
+  classical
+  let ρ := actualFactorStops γ G z u₀ u₂ H
+  let a := fun i => sourceClockLevel κ (z i) (u₀ i)
+  let U := fun i => κ/4*(u₂ i-u₀ i)
+  let σ := fun i => min (ρ (i,0) ω) (S.χ ω)
+  let T := fun i => min (ρ (i,2) ω) (S.χ ω)
+  have hσT (i : ι) : σ i ≤ T i := min_le_min (horder ω i (by decide : (0 : Fin 3) ≤ 2)) le_rfl
+  have hTH (i : ι) : T i ≤ S.χ ω := min_le_right _ _
+  have hlive (i : ι) : z i ∈ survivingDomain (γ ω) (T i) := (S.actual ω i).1
+  have hσeq (i : ι) (hlt : σ i < T i) : σ i = ρ (i,0) ω := by
+    apply min_eq_left
+    by_contra h
+    have he : σ i = S.χ ω := min_eq_right (le_of_not_ge h)
+    exact (not_lt_of_ge (hTH i)) (he ▸ hlt)
+  have hσχ (i : ι) : σ i ≤ S.χ ω := min_le_right _ _
+  have hστ (i : ι) : σ i ≤ ρ (i,2) ω :=
+    (min_le_left _ _).trans (horder ω i (by decide : (0 : Fin 3) ≤ 2))
+  have hstart (i : ι) (hlt : σ i < T i) :
+      logFactorClock κ (logLoewnerCoordinates (G ω) (z i) (σ i)) = a i := by
+    have he := (S.actual_at ω i (σ i) (hσχ i) (hστ i)).2
+    have hs : actualFactorStops γ G z u₀ u₂ H (i,0) ω = σ i := (hσeq i hlt).symm
+    have hc := actual_phase_switch_clock hBc hκ hw S horder ω i 0
+      (by rw [hs]; exact hσχ i)
+    rw [hs, he] at hc
+    exact sub_eq_zero.mp (by simpa only [ite_true] using hc)
+  have hend (i : ι) :
+      logFactorClock κ (logLoewnerCoordinates (G ω) (z i) (T i))-a i ≤ U i := by
+    have hc := cutoff_radius_strictly_reached_iff (continuous_const.mul (hBc ω))
+      (hw ω) hκ (z i) (u₂ i) ((h₀ i).trans (hu i)) H (T i)
+      ((hTH i).trans (S.bounded ω)) (hlive i) ω
+    have hle : logFactorClock κ (logLoewnerCoordinates (G ω) (z i) (T i)) ≤
+        sourceClockLevel κ (z i) (u₂ i) := by
+      apply le_of_not_gt
+      intro h
+      have ht : ρ (i,2) ω < T i := by
+        simpa only [ρ, actualFactorStops, factorRadiusLevel, Fin.reduceEq, ite_false, cutoffStop] using hc.mpr h
+      exact (not_lt_of_ge (min_le_left _ _)) ht
+    have he : sourceClockLevel κ (z i) (u₂ i) = a i+U i := by
+      dsimp [a,U,sourceClockLevel]
+      ring
+    rw [he] at hle
+    linarith
+  have hgeo (i j : ι) (hij : i ≠ j) (hlt : σ i < T i) :
+      physicalConformalRadius J (G ω) (σ i) (z i) ≤ dist (J (z i)) (J (z j)) := by
+    have hc := hstart i hlt
+    rw [logFactorClock_eq_logRadiusClock κ (Real.log (z i).im)] at hc
+    have he : logRadiusClock (Real.log (z i).im)
+        (logLoewnerCoordinates (G ω) (z i) (σ i)) = Real.log (R i/b i) := by
+      dsimp [a,sourceClockLevel] at hc
+      rw [← hu₀ i]
+      nlinarith [hκ]
+    rw [(hw ω).physical_radius_at_activation (continuous_const.mul (hBc ω)) J
+      (z i) (R i) (b i) (hR i) (hb i) (hRdef i) (σ i)
+      (S.actual_at ω i (σ i) (hσχ i) (hστ i)).1 he]
+    exact hsep i j hij
+  have hactive (t : ℝ) (ht : t ∈ Ico 0 (S.χ ω : ℝ)) (i : ι)
+      (hi : i ∈ patternActiveSet (phaseAt ρ t.toNNReal ω)) :
+      (σ i : ℝ) ≤ t ∧ t ≤ (T i : ℝ) := by
+    have ha := (phaseAt_active_iff ρ ω i (horder ω i) t.toNNReal).mp
+      (Finset.mem_filter.mp hi).2
+    have htχ : t.toNNReal ≤ S.χ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2.le
+    constructor
+    · have hh := (min_le_left (ρ (i,0) ω) (S.χ ω)).trans ha.1
+      simpa only [Real.coe_toNNReal t ht.1] using (NNReal.coe_le_coe.mpr hh)
+    · have hh : t.toNNReal ≤ T i := le_min ha.2.le htχ
+      simpa only [Real.coe_toNNReal t ht.1] using (NNReal.coe_le_coe.mpr hh)
+  have hQ (t : ℝ) (ht : t ∈ Ico 0 (S.χ ω : ℝ)) (i : ι)
+      (hi : i ∈ patternActiveSet (phaseAt ρ t.toNNReal ω)) :
+      S.Q t ω i = logLoewnerCoordinates (G ω) (z i) t := by
+    have ha := (phaseAt_active_iff ρ ω i (horder ω i) t.toNNReal).mp
+      (Finset.mem_filter.mp hi).2
+    have hh := (S.actual_at ω i t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2.le) ha.2.le).2
+    simpa only [Real.coe_toNNReal t ht.1] using hh
+  have hrt : ∀ ω, Measurable (fun t => r t ω) :=
+    fun ω => hrm.comp (measurable_id.prodMk measurable_const)
+  have hri : IntervalIntegrable (fun t : ℝ => r t.toNNReal ω) volume 0 (S.χ ω : ℝ) :=
+    ((bounded_rate_locallyIntegrable r hrt C hrb ω).integrableOn_isCompact isCompact_uIcc).intervalIntegrable
+  have hbound := actual_phase_cost_integral_le hκ hκ8 (hBc ω) (hw ω) hgrowth J hJ hJinj
+    (Finset.univ : Finset ι) z (fun _ _ _ _ he => hne he) β hβ hdecay a U
+    (fun i _ => mul_nonneg (div_nonneg hκ.le (by norm_num)) (sub_nonneg.mpr (hu i)))
+    σ T (S.χ ω) (fun i _ => hσT i) (fun i _ => hTH i) (fun i _ => hlive i)
+    (fun i _ => hstart i) (fun i _ _ => hend i)
+    (fun i _ j _ hij hi _ => hgeo i j hij hi)
+    (fun t => S.Q t ω) (fun t => phaseAt ρ t.toNNReal ω)
+    (fun _ _ => Finset.subset_univ _) hactive hQ
+    (fun t ht i hi => by
+      have hh := (actual_phase_regulator hBc hκ hβ.le h₀ hu hw S horder ω t.toNNReal
+        (Real.toNNReal_le_iff_le_coe.mpr ht.2.le) i (Finset.mem_filter.mp hi).2).1
+      simpa only [Real.coe_toNNReal t ht.1,hQ t ht i hi] using hh)
+    (fun t : ℝ => r t.toNNReal ω) hri
+    (fun t ht => by
+      have htχ : t.toNNReal < S.χ ω := by
+        apply NNReal.coe_lt_coe.mp
+        simpa only [Real.coe_toNNReal t ht.1] using ht.2
+      have hh := hcost ω t.toNNReal htχ
+      simpa only [Real.coe_toNNReal t ht.1] using hh)
+  simpa only [discountClock,Real.toNNReal_coe] using hbound
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointActualCutoffs.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointActualCutoffs.lean
index 9a747fd36d3502062bce70728b8aa7c3c06130f8..e59e58cdb2e48148df5540a7103d9b76aca72514 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointActualCutoffs.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointActualCutoffs.lean
@@ -1,57 +1,57 @@
-import OAI.Probability.SLE.Multipoint.MultipointActualPhases
-
-namespace OAI
-
-/-! A countable family of concrete localizations, with the original
-radius stops capped beyond each integration horizon. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem exists_actual_localizations
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (u₀ u₂ : ι → ℝ) (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (A : ℝ) (hA : 0 ≤ A) (hzA : ∀ i, |Real.log (z i).im| ≤ A) :
-    Nonempty (∀ n : ℕ, LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
-      ((n : ℝ≥0)+1) z
-      (fun i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2)) γ G) := by
-  classical
-  have hn (n : ℕ) : 0 ≤ A+(n : ℝ)+1 := by positivity
-  have hex (n : ℕ) : Nonempty (LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
-      ((n : ℝ≥0)+1) z
-      (fun i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2)) γ G) := by
-    have ht := (actualFactorStops_regular hB hm hBm hBc hB0 κ hκ hκ8 γ G F hw
-      z hz u₀ u₂ h₀ hu ((n : ℝ≥0)+1)).1
-    apply exists_localizedState B hm hBm hBc κ (-(A+n+1)) (A+n+1) (A+n+1)
-      (by linarith [hn n]) ((n : ℝ≥0)+1) z hz
-    · intro i
-      have hi := (abs_le.mp (hzA i)).1
-      linarith [Nat.cast_nonneg (α := ℝ) n]
-    · intro i
-      have hi := (abs_le.mp (hzA i)).2
-      linarith [Nat.cast_nonneg (α := ℝ) n]
-    · intro ω
-      simpa only [hB0 ω,abs_zero] using hn n
-    · exact fun i => ht (i,2)
-    · exact hw
-  exact ⟨fun n => Classical.choice (hex n)⟩
-
-lemma initial_log_height_bound {ι : Type*} [Fintype ι] (z : ι → ℂ) (i : ι) :
-    |Real.log (z i).im| ≤ ‖fun j => Real.log (z j).im‖ := by
-  simpa only [Real.norm_eq_abs] using norm_le_pi_norm (fun j => Real.log (z j).im) i
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointActualPhases
+
+namespace OAI
+
+/-! A countable family of concrete localizations, with the original
+radius stops capped beyond each integration horizon. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem exists_actual_localizations
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (u₀ u₂ : ι → ℝ) (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (A : ℝ) (hA : 0 ≤ A) (hzA : ∀ i, |Real.log (z i).im| ≤ A) :
+    Nonempty (∀ n : ℕ, LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
+      ((n : ℝ≥0)+1) z
+      (fun i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2)) γ G) := by
+  classical
+  have hn (n : ℕ) : 0 ≤ A+(n : ℝ)+1 := by positivity
+  have hex (n : ℕ) : Nonempty (LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
+      ((n : ℝ≥0)+1) z
+      (fun i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2)) γ G) := by
+    have ht := (actualFactorStops_regular hB hm hBm hBc hB0 κ hκ hκ8 γ G F hw
+      z hz u₀ u₂ h₀ hu ((n : ℝ≥0)+1)).1
+    apply exists_localizedState B hm hBm hBc κ (-(A+n+1)) (A+n+1) (A+n+1)
+      (by linarith [hn n]) ((n : ℝ≥0)+1) z hz
+    · intro i
+      have hi := (abs_le.mp (hzA i)).1
+      linarith [Nat.cast_nonneg (α := ℝ) n]
+    · intro i
+      have hi := (abs_le.mp (hzA i)).2
+      linarith [Nat.cast_nonneg (α := ℝ) n]
+    · intro ω
+      simpa only [hB0 ω,abs_zero] using hn n
+    · exact fun i => ht (i,2)
+    · exact hw
+  exact ⟨fun n => Classical.choice (hex n)⟩
+
+lemma initial_log_height_bound {ι : Type*} [Fintype ι] (z : ι → ℂ) (i : ι) :
+    |Real.log (z i).im| ≤ ‖fun j => Real.log (z j).im‖ := by
+  simpa only [Real.norm_eq_abs] using norm_le_pi_norm (fun j => Real.log (z j).im) i
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointActualExpectation.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointActualExpectation.lean
index 23b93fca0c1b73bec38298b8f6358cbe6b4687f6..abdfec35e021597ad93fe734c62e4111f02b8138 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointActualExpectation.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointActualExpectation.lean
@@ -1,124 +1,124 @@
-import OAI.Probability.SLE.Multipoint.MultipointActualPhases
-import OAI.Probability.SLE.Multipoint.MultipointLocalizedExpectation
-import OAI.Probability.SLE.Multipoint.MultipointSelectedRate
-import OAI.Probability.SLE.Multipoint.MultipointStateBounds
-import OAI.Probability.SLE.Multipoint.MultipointTerminalProbability
-
-namespace OAI
-
-/-! The actual localized multipoint product has expectation at most one.
-The bounded discount process is constructed, not supplied as an input. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem actual_localized_product_expectation
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ β lo hi M : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hβ : 0 ≤ β) (hβ1 : β ≤ 1)
-    {H : ℝ≥0} (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (u₀ u₂ : ι → ℝ) (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (S : LocalizedState B hm κ lo hi M H z
-      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G) :
-    let ρ := actualFactorStops γ G z u₀ u₂ H
-    let a := fun i => sourceClockLevel κ (z i) (u₀ i)
-    let U := fun i => κ/4*(u₂ i-u₀ i)
-    ∃ (r : ℝ≥0 → Ω → ℝ) (C : ℝ≥0),
-      Measurable (Function.uncurry r) ∧
-      IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) r ∧
-      (∀ t ω, 0 ≤ r t ω ∧ |r t ω| ≤ C) ∧
-      (∀ ω t, t < S.χ ω → r t ω =
-        productDriftCost (patternActiveSet (phaseAt ρ t ω)) κ
-          (fun i => phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
-          (fun i => phaseRegulatorSlope β (phaseAt ρ t ω i)) (S.Q t ω) (B t ω)) ∧
-      Integrable (fun ω => discountedPatternProduct κ β a U (phaseAt ρ (S.χ ω) ω)
-        (S.Q (S.χ ω) ω,discountClock r (S.χ ω) ω) (B (S.χ ω) ω)) P ∧
-      (∫ ω, discountedPatternProduct κ β a U (phaseAt ρ (S.χ ω) ω)
-        (S.Q (S.χ ω) ω,discountClock r (S.χ ω) ω) (B (S.χ ω) ω) ∂P) ≤ 1 := by
-  classical
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  dsimp only
-  let ρ := actualFactorStops γ G z u₀ u₂ H
-  let a := fun i => sourceClockLevel κ (z i) (u₀ i)
-  let U := fun i => κ/4*(u₂ i-u₀ i)
-  let l := factorSwitchList ρ
-  obtain ⟨hρstop,horder,hρbound⟩ := actualFactorStops_regular hB hm hBm hBc hB0
-    κ hκ hκ8 γ G F hw z hz u₀ u₂ h₀ hu H
-  have hlstop := factorSwitchList_forall ρ
-    (fun τ => IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (by
-      simpa only [WithTop.coe_zero] using
-        (isStoppingTime_const (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-          (0 : ℝ≥0))) hρstop
-  have hlbound := factorSwitchList_forall ρ (fun τ => ∀ ω, τ ω ≤ H+1)
-    (fun _ => bot_le) hρbound
-  have hU (i : ι) : 0 ≤ U i := mul_nonneg (div_nonneg hκ.le (by norm_num)) (sub_nonneg.mpr (hu i))
-  let q₀ := fun i => initialLogCoordinates (z i)
-  have hQ0 (ω : Ω) : S.Q 0 ω = q₀ := funext (S.initial ω)
-  let Qbound := ‖q₀‖+(S.L : ℝ)*(H : ℝ)
-  have hQbound (ω : Ω) (t : ℝ≥0) (ht : t ≤ S.χ ω) : ‖S.Q t ω‖ ≤ Qbound := by
-    have hh := lipschitz_state_norm_le (fun u => S.Q u ω) S.L (S.lipschitz ω)
-      H t t.coe_nonneg (by exact_mod_cast ht.trans (S.bounded ω))
-    simpa only [hQ0 ω] using hh
-  let R₀ := max Qbound M
-  have hR₀ : 0 ≤ R₀ := (by positivity : (0 : ℝ) ≤ Qbound).trans (le_max_left _ _)
-  have hnorm₀ (ω : Ω) (t : ℝ≥0) (ht : t ≤ S.χ ω) : ‖(S.Q t ω,B t ω)‖ ≤ R₀ := by
-    rw [Prod.norm_def,Real.norm_eq_abs]
-    exact max_le_max (hQbound ω t ht) (S.driver_bound ω t ht)
-  obtain ⟨r,C,hrm,hrp,hrb,hcost⟩ := exists_localized_discount_rate B hm hBm hBc
-    S.Q S.measurable S.adapted (fun ω => (S.lipschitz ω).continuous) κ β hκ hκ8 a U
-    ρ l H hlstop hlbound (factorSwitchList_mem ρ) (factorSwitchList_zero ρ)
-    S.χ S.stopping S.bounded R₀ hR₀ hnorm₀
-  have hrt (ω : Ω) : Measurable (fun t => r t ω) :=
-    hrm.comp (measurable_id.prodMk measurable_const)
-  let R := max (max Qbound ((C : ℝ)*(H : ℝ))) M
-  have hR : 0 ≤ R := (by positivity : (0 : ℝ) ≤ Qbound).trans
-    ((le_max_left _ _).trans (le_max_left _ _))
-  have hnorm (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (htχ : t ≤ (S.χ ω : ℝ)) :
-      ‖((S.Q t ω,discountClock r t ω),B t.toNNReal ω)‖ ≤ R :=
-    discounted_state_norm_bound S.Q S.L S.lipschitz q₀ hQ0 r hrt C
-      (fun t ω => (hrb t ω).2) B S.χ H S.bounded M
-      (fun ω t ht => by simpa only [Real.norm_eq_abs] using S.driver_bound ω t ht) ω t ht htχ
-  have hclock (ω : Ω) (i : ι) (k : Fin 3) (hk : ρ (i,k) ω ≤ S.χ ω) :
-      logFactorClock κ (S.Q (ρ (i,k) ω) ω i)-a i =
-        if k = 0 then 0 else if k = 1 then U i/2 else U i :=
-    actual_phase_switch_clock hBc hκ hw S horder ω i k hk
-  obtain ⟨hI,hmean⟩ := localized_pattern_expectation_le hB hm hBm hBc S.Q S.V S.L
-    S.measurable S.adapted S.lipschitz S.velocity_measurable S.velocity_bound S.derivative
-    r hrm hrp C (fun t ω => (hrb t ω).2) κ β hκ hκ8 hβ hβ1 a U hU ρ l H
-    hlstop hlbound (factorSwitchList_mem ρ) horder (factorSwitchList_zero ρ)
-    S.χ S.stopping S.bounded R hR hnorm
-    (fun ω t ht i hi => (actual_phase_regulator hBc hκ hβ h₀ hu hw S horder ω t ht.le i
-      (Finset.mem_filter.mp hi).2).2)
-    (fun ω t ht i hi => S.physical_velocity ω t ht i
-      ((phaseAt_active_iff ρ ω i (horder ω i) t).mp (Finset.mem_filter.mp hi).2).2)
-    hcost hclock
-  refine ⟨r,C,hrm,hrp,hrb,hcost,hI,hmean.trans ?_⟩
-  have hinit (ω : Ω) : discountedPatternProduct κ β a U (phaseAt ρ 0 ω)
-      (S.Q 0 ω,discountClock r 0 ω) (B 0 ω) ≤ 1 := by
-    simp only [discountedPatternProduct, discountClock_initial, neg_zero, Real.exp_zero, one_mul]
-    apply initial_phase_product_le_one κ β hκ hκ8 a U hU ρ ω (horder ω) (S.Q 0 ω) (B 0 ω)
-    intro i k hk
-    have hh := hclock ω i k (by rw [hk]; exact bot_le)
-    simpa only [hk, NNReal.coe_zero] using hh
-  have hnonneg (ω : Ω) : 0 ≤ discountedPatternProduct κ β a U (phaseAt ρ 0 ω)
-      (S.Q 0 ω,discountClock r 0 ω) (B 0 ω) :=
-    mul_nonneg (Real.exp_pos _).le (Finset.prod_nonneg (fun i _ =>
-      (phaseFactor_pos κ β (a i) (U i) (phaseAt ρ 0 ω i) (S.Q 0 ω i) (B 0 ω)).le))
-  simpa only [integral_const,probReal_univ,one_smul] using
-    integral_mono_of_nonneg (μ := P) (ae_of_all P hnonneg)
-      (integrable_const (1 : ℝ)) (ae_of_all P hinit)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointActualPhases
+import OAI.Probability.SLE.Multipoint.MultipointLocalizedExpectation
+import OAI.Probability.SLE.Multipoint.MultipointSelectedRate
+import OAI.Probability.SLE.Multipoint.MultipointStateBounds
+import OAI.Probability.SLE.Multipoint.MultipointTerminalProbability
+
+namespace OAI
+
+/-! The actual localized multipoint product has expectation at most one.
+The bounded discount process is constructed, not supplied as an input. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem actual_localized_product_expectation
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ β lo hi M : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hβ : 0 ≤ β) (hβ1 : β ≤ 1)
+    {H : ℝ≥0} (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (u₀ u₂ : ι → ℝ) (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (S : LocalizedState B hm κ lo hi M H z
+      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G) :
+    let ρ := actualFactorStops γ G z u₀ u₂ H
+    let a := fun i => sourceClockLevel κ (z i) (u₀ i)
+    let U := fun i => κ/4*(u₂ i-u₀ i)
+    ∃ (r : ℝ≥0 → Ω → ℝ) (C : ℝ≥0),
+      Measurable (Function.uncurry r) ∧
+      IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) r ∧
+      (∀ t ω, 0 ≤ r t ω ∧ |r t ω| ≤ C) ∧
+      (∀ ω t, t < S.χ ω → r t ω =
+        productDriftCost (patternActiveSet (phaseAt ρ t ω)) κ
+          (fun i => phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
+          (fun i => phaseRegulatorSlope β (phaseAt ρ t ω i)) (S.Q t ω) (B t ω)) ∧
+      Integrable (fun ω => discountedPatternProduct κ β a U (phaseAt ρ (S.χ ω) ω)
+        (S.Q (S.χ ω) ω,discountClock r (S.χ ω) ω) (B (S.χ ω) ω)) P ∧
+      (∫ ω, discountedPatternProduct κ β a U (phaseAt ρ (S.χ ω) ω)
+        (S.Q (S.χ ω) ω,discountClock r (S.χ ω) ω) (B (S.χ ω) ω) ∂P) ≤ 1 := by
+  classical
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  dsimp only
+  let ρ := actualFactorStops γ G z u₀ u₂ H
+  let a := fun i => sourceClockLevel κ (z i) (u₀ i)
+  let U := fun i => κ/4*(u₂ i-u₀ i)
+  let l := factorSwitchList ρ
+  obtain ⟨hρstop,horder,hρbound⟩ := actualFactorStops_regular hB hm hBm hBc hB0
+    κ hκ hκ8 γ G F hw z hz u₀ u₂ h₀ hu H
+  have hlstop := factorSwitchList_forall ρ
+    (fun τ => IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (by
+      simpa only [WithTop.coe_zero] using
+        (isStoppingTime_const (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+          (0 : ℝ≥0))) hρstop
+  have hlbound := factorSwitchList_forall ρ (fun τ => ∀ ω, τ ω ≤ H+1)
+    (fun _ => bot_le) hρbound
+  have hU (i : ι) : 0 ≤ U i := mul_nonneg (div_nonneg hκ.le (by norm_num)) (sub_nonneg.mpr (hu i))
+  let q₀ := fun i => initialLogCoordinates (z i)
+  have hQ0 (ω : Ω) : S.Q 0 ω = q₀ := funext (S.initial ω)
+  let Qbound := ‖q₀‖+(S.L : ℝ)*(H : ℝ)
+  have hQbound (ω : Ω) (t : ℝ≥0) (ht : t ≤ S.χ ω) : ‖S.Q t ω‖ ≤ Qbound := by
+    have hh := lipschitz_state_norm_le (fun u => S.Q u ω) S.L (S.lipschitz ω)
+      H t t.coe_nonneg (by exact_mod_cast ht.trans (S.bounded ω))
+    simpa only [hQ0 ω] using hh
+  let R₀ := max Qbound M
+  have hR₀ : 0 ≤ R₀ := (by positivity : (0 : ℝ) ≤ Qbound).trans (le_max_left _ _)
+  have hnorm₀ (ω : Ω) (t : ℝ≥0) (ht : t ≤ S.χ ω) : ‖(S.Q t ω,B t ω)‖ ≤ R₀ := by
+    rw [Prod.norm_def,Real.norm_eq_abs]
+    exact max_le_max (hQbound ω t ht) (S.driver_bound ω t ht)
+  obtain ⟨r,C,hrm,hrp,hrb,hcost⟩ := exists_localized_discount_rate B hm hBm hBc
+    S.Q S.measurable S.adapted (fun ω => (S.lipschitz ω).continuous) κ β hκ hκ8 a U
+    ρ l H hlstop hlbound (factorSwitchList_mem ρ) (factorSwitchList_zero ρ)
+    S.χ S.stopping S.bounded R₀ hR₀ hnorm₀
+  have hrt (ω : Ω) : Measurable (fun t => r t ω) :=
+    hrm.comp (measurable_id.prodMk measurable_const)
+  let R := max (max Qbound ((C : ℝ)*(H : ℝ))) M
+  have hR : 0 ≤ R := (by positivity : (0 : ℝ) ≤ Qbound).trans
+    ((le_max_left _ _).trans (le_max_left _ _))
+  have hnorm (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (htχ : t ≤ (S.χ ω : ℝ)) :
+      ‖((S.Q t ω,discountClock r t ω),B t.toNNReal ω)‖ ≤ R :=
+    discounted_state_norm_bound S.Q S.L S.lipschitz q₀ hQ0 r hrt C
+      (fun t ω => (hrb t ω).2) B S.χ H S.bounded M
+      (fun ω t ht => by simpa only [Real.norm_eq_abs] using S.driver_bound ω t ht) ω t ht htχ
+  have hclock (ω : Ω) (i : ι) (k : Fin 3) (hk : ρ (i,k) ω ≤ S.χ ω) :
+      logFactorClock κ (S.Q (ρ (i,k) ω) ω i)-a i =
+        if k = 0 then 0 else if k = 1 then U i/2 else U i :=
+    actual_phase_switch_clock hBc hκ hw S horder ω i k hk
+  obtain ⟨hI,hmean⟩ := localized_pattern_expectation_le hB hm hBm hBc S.Q S.V S.L
+    S.measurable S.adapted S.lipschitz S.velocity_measurable S.velocity_bound S.derivative
+    r hrm hrp C (fun t ω => (hrb t ω).2) κ β hκ hκ8 hβ hβ1 a U hU ρ l H
+    hlstop hlbound (factorSwitchList_mem ρ) horder (factorSwitchList_zero ρ)
+    S.χ S.stopping S.bounded R hR hnorm
+    (fun ω t ht i hi => (actual_phase_regulator hBc hκ hβ h₀ hu hw S horder ω t ht.le i
+      (Finset.mem_filter.mp hi).2).2)
+    (fun ω t ht i hi => S.physical_velocity ω t ht i
+      ((phaseAt_active_iff ρ ω i (horder ω i) t).mp (Finset.mem_filter.mp hi).2).2)
+    hcost hclock
+  refine ⟨r,C,hrm,hrp,hrb,hcost,hI,hmean.trans ?_⟩
+  have hinit (ω : Ω) : discountedPatternProduct κ β a U (phaseAt ρ 0 ω)
+      (S.Q 0 ω,discountClock r 0 ω) (B 0 ω) ≤ 1 := by
+    simp only [discountedPatternProduct, discountClock_initial, neg_zero, Real.exp_zero, one_mul]
+    apply initial_phase_product_le_one κ β hκ hκ8 a U hU ρ ω (horder ω) (S.Q 0 ω) (B 0 ω)
+    intro i k hk
+    have hh := hclock ω i k (by rw [hk]; exact bot_le)
+    simpa only [hk, NNReal.coe_zero] using hh
+  have hnonneg (ω : Ω) : 0 ≤ discountedPatternProduct κ β a U (phaseAt ρ 0 ω)
+      (S.Q 0 ω,discountClock r 0 ω) (B 0 ω) :=
+    mul_nonneg (Real.exp_pos _).le (Finset.prod_nonneg (fun i _ =>
+      (phaseFactor_pos κ β (a i) (U i) (phaseAt ρ 0 ω i) (S.Q 0 ω i) (B 0 ω)).le))
+  simpa only [integral_const,probReal_univ,one_smul] using
+    integral_mono_of_nonneg (μ := P) (ae_of_all P hnonneg)
+      (integrable_const (1 : ℝ)) (ae_of_all P hinit)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointActualPhases.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointActualPhases.lean
index f12843a69982a3668e811c01090d95dd3e3a2957..0dc4f1220a1c25aa2beb6e9061f0185d5b10a455 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointActualPhases.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointActualPhases.lean
@@ -1,163 +1,163 @@
-import OAI.Probability.SLE.Multipoint.MultipointLocalizedState
-import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus
-import OAI.Probability.SLE.Multipoint.MultipointPhaseRegulator
-
-namespace OAI
-
-/-! The three actual radius stops and their phase identities along the
-constructed localized state. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def factorRadiusLevel (u₀ u₂ : ℝ) (k : Fin 3) : ℝ :=
-  if k = 0 then u₀ else if k = 1 then (u₀+u₂)/2 else u₂
-
-lemma factorRadiusLevel_nonneg {u₀ u₂ : ℝ} (h₀ : 0 ≤ u₀) (h₂ : 0 ≤ u₂) (k : Fin 3) :
-    0 ≤ factorRadiusLevel u₀ u₂ k := by
-  fin_cases k <;> norm_num [factorRadiusLevel] <;> positivity
-
-lemma factorRadiusLevel_monotone {u₀ u₂ : ℝ} (hu : u₀ ≤ u₂) :
-    Monotone (factorRadiusLevel u₀ u₂) := by
-  intro i j hij
-  fin_cases i <;> fin_cases j <;> norm_num [factorRadiusLevel] at * <;> linarith
-
-def actualFactorStops {Ω ι : Type*} (γ : Ω → ℝ≥0 → ℂ)
-    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ι → ℂ) (u₀ u₂ : ι → ℝ) (H : ℝ≥0)
-    (k : ι × Fin 3) : Ω → ℝ≥0 :=
-  cutoffStop (fun ω => radiusLevelTime (γ ω) (G ω) (z k.1)
-    (factorRadiusLevel (u₀ k.1) (u₂ k.1) k.2)) (H+1)
-
-def factorSwitchList {Ω ι : Type*} [Fintype ι]
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) : List (Ω → ℝ≥0) :=
-  (fun _ => 0) :: (Finset.univ.toList.map ρ)
-
-lemma factorSwitchList_zero {Ω ι : Type*} [Fintype ι]
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) : (fun _ : Ω => (0 : ℝ≥0)) ∈ factorSwitchList ρ := by
-  simp only [factorSwitchList, List.mem_cons, true_or]
-
-lemma factorSwitchList_mem {Ω ι : Type*} [Fintype ι]
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (k : ι × Fin 3) : ρ k ∈ factorSwitchList ρ := by
-  exact List.mem_cons_of_mem _ (List.mem_map.mpr ⟨k,Finset.mem_toList.mpr (Finset.mem_univ _),rfl⟩)
-
-lemma factorSwitchList_forall {Ω ι : Type*} [Fintype ι]
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (P : (Ω → ℝ≥0) → Prop)
-    (hzero : P (fun _ => 0)) (hρ : ∀ k, P (ρ k)) : ∀ τ ∈ factorSwitchList ρ, P τ := by
-  intro τ hτ
-  rcases List.mem_cons.mp hτ with rfl | hτ
-  · exact hzero
-  · obtain ⟨k,_,rfl⟩ := List.mem_map.mp hτ
-    exact hρ k
-
-theorem actualFactorStops_regular {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (u₀ u₂ : ι → ℝ)
-    (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i) (H : ℝ≥0) :
-    (∀ k, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (actualFactorStops γ G z u₀ u₂ H k ω : WithTop ℝ≥0))) ∧
-    (∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω)) ∧
-    (∀ k ω, actualFactorStops γ G z u₀ u₂ H k ω ≤ H+1) := by
-  refine ⟨?_,?_,?_⟩
-  · intro k
-    exact cutoffStop_stopping _ _ (actual_radiusLevelTime_stopping hB hm hBm hBc hB0 κ hκ hκ8
-      γ G F hw (z k.1) (hz k.1) _ (factorRadiusLevel_nonneg (h₀ k.1) ((h₀ k.1).trans (hu k.1)) k.2)) (H+1)
-  · intro ω i j k hjk
-    apply cutoffStop_mono _ (H+1) ω
-    intro ω'
-    exact radiusLevelTime_mono (continuous_const.mul (hBc ω')) (hw ω') (z i)
-      (factorRadiusLevel_nonneg (h₀ i) ((h₀ i).trans (hu i)) j)
-      (factorRadiusLevel_monotone (hu i) hjk)
-  · exact fun k ω => cutoffStop_le_horizon _ _ ω
-
-lemma LocalizedState.actual_at {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi M : ℝ} {H : ℝ≥0} {z : ι → ℂ} {τ : ι → Ω → ℝ≥0}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (S : LocalizedState B hm κ lo hi M H z τ γ G)
-    (ω : Ω) (i : ι) (t : ℝ≥0) (htχ : t ≤ S.χ ω) (htτ : t ≤ τ i ω) :
-    z i ∈ survivingDomain (γ ω) t ∧ S.Q t ω i = logLoewnerCoordinates (G ω) (z i) t := by
-  have ht : t ≤ min (τ i ω) (S.χ ω) := le_min htτ htχ
-  exact ⟨survivingDomain_antitone (γ ω) ht (S.actual ω i).1,
-    (S.actual ω i).2 ⟨t.coe_nonneg, by exact_mod_cast ht⟩⟩
-
-theorem actual_phase_regulator
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi M β : ℝ} (hκ : 0 < κ) (hβ : 0 ≤ β) {H : ℝ≥0}
-    {z : ι → ℂ} {u₀ u₂ : ι → ℝ} (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
-    {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ}
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (S : LocalizedState B hm κ lo hi M H z
-      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G)
-    (horder : ∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω))
-    (ω : Ω) (t : ℝ≥0) (htχ : t ≤ S.χ ω) (i : ι)
-    (hactive : phaseActive (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i)) :
-    let a := sourceClockLevel κ (z i) (u₀ i)
-    let U := κ/4*(u₂ i-u₀ i)
-    let j := phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i
-    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) (S.Q t ω i) =
-      regulator β U (logFactorClock κ (S.Q t ω i)-a) ∧
-    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) (S.Q t ω i) ≤ 1 := by
-  have htt := ((phaseAt_active_iff _ ω i (horder ω i) t).mp hactive).2.le
-  obtain ⟨hlive,he⟩ := S.actual_at ω i t htχ htt
-  have hh := cutoff_radius_factor_thresholds (continuous_const.mul (hBc ω)) (hw ω) hκ
-    (z i) (u₀ i) (u₂ i) (h₀ i) ((h₀ i).trans (hu i)) H t
-    (htχ.trans (S.bounded ω)) hlive ω
-  dsimp only at hh ⊢
-  apply phaseAt_regulator_eq_and_le_one _ ω i (horder ω i) t κ β
-    (sourceClockLevel κ (z i) (u₀ i)) (κ/4*(u₂ i-u₀ i)) hβ (S.Q t ω i) hactive
-  · intro ht
-    rw [he]
-    exact hh.1.mp ht
-  · intro ht
-    rw [he]
-    apply le_of_not_gt
-    intro h
-    exact (not_le_of_gt ht) (hh.2.1.mpr h.le)
-  · intro ht
-    rw [he]
-    exact hh.2.1.mp ht
-  · intro ht
-    rw [he]
-    apply le_of_not_gt
-    intro h
-    exact (not_le_of_gt ht) (hh.2.2.1.mpr h.le)
-
-theorem actual_phase_switch_clock
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi M : ℝ} (hκ : 0 < κ) {H : ℝ≥0}
-    {z : ι → ℂ} {u₀ u₂ : ι → ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ}
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (S : LocalizedState B hm κ lo hi M H z
-      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G)
-    (horder : ∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω))
-    (ω : Ω) (i : ι) (k : Fin 3)
-    (hk : actualFactorStops γ G z u₀ u₂ H (i,k) ω ≤ S.χ ω) :
-    logFactorClock κ (S.Q (actualFactorStops γ G z u₀ u₂ H (i,k) ω) ω i)-
-      sourceClockLevel κ (z i) (u₀ i) =
-      if k = 0 then 0 else if k = 1 then (κ/4*(u₂ i-u₀ i))/2 else κ/4*(u₂ i-u₀ i) := by
-  let t := actualFactorStops γ G z u₀ u₂ H (i,k) ω
-  have htτ : t ≤ actualFactorStops γ G z u₀ u₂ H (i,2) ω := horder ω i (Fin.le_last k)
-  obtain ⟨hlive,he⟩ := S.actual_at ω i t hk htτ
-  have hclock := (cutoff_radius_eq_iff (continuous_const.mul (hBc ω)) (hw ω) hκ
-    (z i) (factorRadiusLevel (u₀ i) (u₂ i) k) H t (hk.trans (S.bounded ω)) hlive ω).mp rfl
-  rw [he,hclock]
-  fin_cases k <;> norm_num [factorRadiusLevel, sourceClockLevel] <;> ring
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLocalizedState
+import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus
+import OAI.Probability.SLE.Multipoint.MultipointPhaseRegulator
+
+namespace OAI
+
+/-! The three actual radius stops and their phase identities along the
+constructed localized state. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def factorRadiusLevel (u₀ u₂ : ℝ) (k : Fin 3) : ℝ :=
+  if k = 0 then u₀ else if k = 1 then (u₀+u₂)/2 else u₂
+
+lemma factorRadiusLevel_nonneg {u₀ u₂ : ℝ} (h₀ : 0 ≤ u₀) (h₂ : 0 ≤ u₂) (k : Fin 3) :
+    0 ≤ factorRadiusLevel u₀ u₂ k := by
+  fin_cases k <;> norm_num [factorRadiusLevel] <;> positivity
+
+lemma factorRadiusLevel_monotone {u₀ u₂ : ℝ} (hu : u₀ ≤ u₂) :
+    Monotone (factorRadiusLevel u₀ u₂) := by
+  intro i j hij
+  fin_cases i <;> fin_cases j <;> norm_num [factorRadiusLevel] at * <;> linarith
+
+def actualFactorStops {Ω ι : Type*} (γ : Ω → ℝ≥0 → ℂ)
+    (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ι → ℂ) (u₀ u₂ : ι → ℝ) (H : ℝ≥0)
+    (k : ι × Fin 3) : Ω → ℝ≥0 :=
+  cutoffStop (fun ω => radiusLevelTime (γ ω) (G ω) (z k.1)
+    (factorRadiusLevel (u₀ k.1) (u₂ k.1) k.2)) (H+1)
+
+def factorSwitchList {Ω ι : Type*} [Fintype ι]
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) : List (Ω → ℝ≥0) :=
+  (fun _ => 0) :: (Finset.univ.toList.map ρ)
+
+lemma factorSwitchList_zero {Ω ι : Type*} [Fintype ι]
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) : (fun _ : Ω => (0 : ℝ≥0)) ∈ factorSwitchList ρ := by
+  simp only [factorSwitchList, List.mem_cons, true_or]
+
+lemma factorSwitchList_mem {Ω ι : Type*} [Fintype ι]
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (k : ι × Fin 3) : ρ k ∈ factorSwitchList ρ := by
+  exact List.mem_cons_of_mem _ (List.mem_map.mpr ⟨k,Finset.mem_toList.mpr (Finset.mem_univ _),rfl⟩)
+
+lemma factorSwitchList_forall {Ω ι : Type*} [Fintype ι]
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (P : (Ω → ℝ≥0) → Prop)
+    (hzero : P (fun _ => 0)) (hρ : ∀ k, P (ρ k)) : ∀ τ ∈ factorSwitchList ρ, P τ := by
+  intro τ hτ
+  rcases List.mem_cons.mp hτ with rfl | hτ
+  · exact hzero
+  · obtain ⟨k,_,rfl⟩ := List.mem_map.mp hτ
+    exact hρ k
+
+theorem actualFactorStops_regular {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (u₀ u₂ : ι → ℝ)
+    (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i) (H : ℝ≥0) :
+    (∀ k, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (actualFactorStops γ G z u₀ u₂ H k ω : WithTop ℝ≥0))) ∧
+    (∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω)) ∧
+    (∀ k ω, actualFactorStops γ G z u₀ u₂ H k ω ≤ H+1) := by
+  refine ⟨?_,?_,?_⟩
+  · intro k
+    exact cutoffStop_stopping _ _ (actual_radiusLevelTime_stopping hB hm hBm hBc hB0 κ hκ hκ8
+      γ G F hw (z k.1) (hz k.1) _ (factorRadiusLevel_nonneg (h₀ k.1) ((h₀ k.1).trans (hu k.1)) k.2)) (H+1)
+  · intro ω i j k hjk
+    apply cutoffStop_mono _ (H+1) ω
+    intro ω'
+    exact radiusLevelTime_mono (continuous_const.mul (hBc ω')) (hw ω') (z i)
+      (factorRadiusLevel_nonneg (h₀ i) ((h₀ i).trans (hu i)) j)
+      (factorRadiusLevel_monotone (hu i) hjk)
+  · exact fun k ω => cutoffStop_le_horizon _ _ ω
+
+lemma LocalizedState.actual_at {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi M : ℝ} {H : ℝ≥0} {z : ι → ℂ} {τ : ι → Ω → ℝ≥0}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (S : LocalizedState B hm κ lo hi M H z τ γ G)
+    (ω : Ω) (i : ι) (t : ℝ≥0) (htχ : t ≤ S.χ ω) (htτ : t ≤ τ i ω) :
+    z i ∈ survivingDomain (γ ω) t ∧ S.Q t ω i = logLoewnerCoordinates (G ω) (z i) t := by
+  have ht : t ≤ min (τ i ω) (S.χ ω) := le_min htτ htχ
+  exact ⟨survivingDomain_antitone (γ ω) ht (S.actual ω i).1,
+    (S.actual ω i).2 ⟨t.coe_nonneg, by exact_mod_cast ht⟩⟩
+
+theorem actual_phase_regulator
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi M β : ℝ} (hκ : 0 < κ) (hβ : 0 ≤ β) {H : ℝ≥0}
+    {z : ι → ℂ} {u₀ u₂ : ι → ℝ} (h₀ : ∀ i, 0 ≤ u₀ i) (hu : ∀ i, u₀ i ≤ u₂ i)
+    {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ}
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (S : LocalizedState B hm κ lo hi M H z
+      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G)
+    (horder : ∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω))
+    (ω : Ω) (t : ℝ≥0) (htχ : t ≤ S.χ ω) (i : ι)
+    (hactive : phaseActive (phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i)) :
+    let a := sourceClockLevel κ (z i) (u₀ i)
+    let U := κ/4*(u₂ i-u₀ i)
+    let j := phaseAt (actualFactorStops γ G z u₀ u₂ H) t ω i
+    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) (S.Q t ω i) =
+      regulator β U (logFactorClock κ (S.Q t ω i)-a) ∧
+    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) (S.Q t ω i) ≤ 1 := by
+  have htt := ((phaseAt_active_iff _ ω i (horder ω i) t).mp hactive).2.le
+  obtain ⟨hlive,he⟩ := S.actual_at ω i t htχ htt
+  have hh := cutoff_radius_factor_thresholds (continuous_const.mul (hBc ω)) (hw ω) hκ
+    (z i) (u₀ i) (u₂ i) (h₀ i) ((h₀ i).trans (hu i)) H t
+    (htχ.trans (S.bounded ω)) hlive ω
+  dsimp only at hh ⊢
+  apply phaseAt_regulator_eq_and_le_one _ ω i (horder ω i) t κ β
+    (sourceClockLevel κ (z i) (u₀ i)) (κ/4*(u₂ i-u₀ i)) hβ (S.Q t ω i) hactive
+  · intro ht
+    rw [he]
+    exact hh.1.mp ht
+  · intro ht
+    rw [he]
+    apply le_of_not_gt
+    intro h
+    exact (not_le_of_gt ht) (hh.2.1.mpr h.le)
+  · intro ht
+    rw [he]
+    exact hh.2.1.mp ht
+  · intro ht
+    rw [he]
+    apply le_of_not_gt
+    intro h
+    exact (not_le_of_gt ht) (hh.2.2.1.mpr h.le)
+
+theorem actual_phase_switch_clock
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi M : ℝ} (hκ : 0 < κ) {H : ℝ≥0}
+    {z : ι → ℂ} {u₀ u₂ : ι → ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ}
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (S : LocalizedState B hm κ lo hi M H z
+      (fun i => actualFactorStops γ G z u₀ u₂ H (i,2)) γ G)
+    (horder : ∀ ω i, Monotone (fun k => actualFactorStops γ G z u₀ u₂ H (i,k) ω))
+    (ω : Ω) (i : ι) (k : Fin 3)
+    (hk : actualFactorStops γ G z u₀ u₂ H (i,k) ω ≤ S.χ ω) :
+    logFactorClock κ (S.Q (actualFactorStops γ G z u₀ u₂ H (i,k) ω) ω i)-
+      sourceClockLevel κ (z i) (u₀ i) =
+      if k = 0 then 0 else if k = 1 then (κ/4*(u₂ i-u₀ i))/2 else κ/4*(u₂ i-u₀ i) := by
+  let t := actualFactorStops γ G z u₀ u₂ H (i,k) ω
+  have htτ : t ≤ actualFactorStops γ G z u₀ u₂ H (i,2) ω := horder ω i (Fin.le_last k)
+  obtain ⟨hlive,he⟩ := S.actual_at ω i t hk htτ
+  have hclock := (cutoff_radius_eq_iff (continuous_const.mul (hBc ω)) (hw ω) hκ
+    (z i) (factorRadiusLevel (u₀ i) (u₂ i) k) H t (hk.trans (S.bounded ω)) hlive ω).mp rfl
+  rw [he,hclock]
+  fin_cases k <;> norm_num [factorRadiusLevel, sourceClockLevel] <;> ring
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointActualProbability.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointActualProbability.lean
index a084c8d435887bc842d647b963c65fd26ccb8ae7..125b5cb8dc2c81eec2d324daa0b831869fc97a55 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointActualProbability.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointActualProbability.lean
@@ -1,86 +1,86 @@
-import OAI.Probability.SLE.Multipoint.MultipointActualBudget
-import OAI.Probability.SLE.Multipoint.MultipointActualCutoffs
-import OAI.Probability.SLE.Multipoint.MultipointCompletionExhaustion
-
-namespace OAI
-
-/-! The fixed-order radius estimate in an arbitrary conformal image.
-The only analytic input not proved here is the stated disk growth bound. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem actual_radius_multipoint_bound
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
-    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (R b e : ι → ℝ) (he : ∀ i, 0 < e i) (heb : ∀ i, e i ≤ b i)
-    (hbR : ∀ i, b i ≤ R i) (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
-    P {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (Real.log (R i/e i)) < ⊤} ≤
-      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
-        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
-        ∏ i, (e i/b i)^(1-κ/8)) := by
-  classical
-  let u₀ := fun i => Real.log (R i/b i)
-  let u₂ := fun i => Real.log (R i/e i)
-  let a := fun i => sourceClockLevel κ (z i) (u₀ i)
-  let U := fun i => κ/4*(u₂ i-u₀ i)
-  let A := ‖fun i => Real.log (z i).im‖
-  let K := uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)
-  have hb (i : ι) : 0 < b i := (he i).trans_le (heb i)
-  have hR (i : ι) : 0 < R i := (hb i).trans_le (hbR i)
-  have hparams (i : ι) := physical_radius_level_parameters hκ (z i)
-    (R i) (b i) (e i) (he i) (heb i) (hbR i)
-  have h₀ (i : ι) : 0 ≤ u₀ i := (hparams i).1
-  have hu (i : ι) : u₀ i ≤ u₂ i := (hparams i).2.1
-  have hU (i : ι) : U i = sourceTarget κ (b i) (e i) := by
-    have hh := (hparams i).2.2.2.1
-    dsimp [U,u₀,u₂,sourceClockLevel] at *
-    linarith
-  obtain ⟨_,hβ,hβ1,hdecay,_⟩ := source_parameters hκ hκ8
-  obtain ⟨S⟩ := exists_actual_localizations hB hm hBm hBc hB0 hκ hκ8 z hz u₀ u₂ h₀ hu
-    γ G F hw A (norm_nonneg _) (initial_log_height_bound z)
-  let ρ := fun n : ℕ => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1)
-  have horder (n : ℕ) := (actualFactorStops_regular hB hm hBm hBc hB0 κ hκ hκ8
-    γ G F hw z hz u₀ u₂ h₀ hu ((n : ℝ≥0)+1)).2.1
-  have hex (n : ℕ) := actual_localized_product_expectation hB hm hBm hBc hB0 hκ hκ8
-    hβ.le hβ1 z hz u₀ u₂ h₀ hu γ G F hw (S n)
-  choose r C hrm hrp hrb hcost hI hmean using hex
-  let E := {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) < ⊤}
-  let cutoff := fun n => {ω | phaseAt (ρ n) ((S n).χ ω) ω = fun _ => 3}
-  have hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n := by
-    intro ω hω
-    filter_upwards [eventually_all_completed_of_finite_radius_levels B hm hBc κ A
-      (norm_nonneg _) z u₀ u₂ γ G F hw S ω hω] with n hn
-    exact hn.2
-  have hbudget (n : ℕ) (ω : Ω) : discountClock (r n) ((S n).χ ω) ω ≤ K :=
-    actual_localized_discount_le hBc hκ hκ8 hβ hdecay z hne u₀ u₂ h₀ hu γ G F hw
-      (S n) (horder n) hgrowth J hJ hJinj R b hR hb hRdef (fun _ => rfl) hsep
-      (r n) (C n) (hrm n) (fun t ω => (hrb n t ω).2) (hcost n) ω
-  have hbound := measure_le_of_completed_phase_products P E cutoff κ (sourceBeta κ) a U K
-    (fun n ω => phaseAt (ρ n) ((S n).χ ω) ω)
-    (fun n ω => (S n).Q ((S n).χ ω) ω)
-    (fun n ω => discountClock (r n) ((S n).χ ω) ω)
-    (fun n ω => B ((S n).χ ω) ω)
-    hcover hI hmean (fun _ _ h => h) (fun n ω _ => hbudget n ω)
-  have hexp := source_radius_targets_exponential (Finset.univ : Finset ι) hκ b e
-    (fun i _ => hb i) (fun i _ => he i)
-  simpa only [E,u₂,K,show U = (fun i => sourceTarget κ (b i) (e i)) from funext hU,
-    hexp] using hbound
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointActualBudget
+import OAI.Probability.SLE.Multipoint.MultipointActualCutoffs
+import OAI.Probability.SLE.Multipoint.MultipointCompletionExhaustion
+
+namespace OAI
+
+/-! The fixed-order radius estimate in an arbitrary conformal image.
+The only analytic input not proved here is the stated disk growth bound. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem actual_radius_multipoint_bound
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
+    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (R b e : ι → ℝ) (he : ∀ i, 0 < e i) (heb : ∀ i, e i ≤ b i)
+    (hbR : ∀ i, b i ≤ R i) (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
+    P {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (Real.log (R i/e i)) < ⊤} ≤
+      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
+        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
+        ∏ i, (e i/b i)^(1-κ/8)) := by
+  classical
+  let u₀ := fun i => Real.log (R i/b i)
+  let u₂ := fun i => Real.log (R i/e i)
+  let a := fun i => sourceClockLevel κ (z i) (u₀ i)
+  let U := fun i => κ/4*(u₂ i-u₀ i)
+  let A := ‖fun i => Real.log (z i).im‖
+  let K := uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)
+  have hb (i : ι) : 0 < b i := (he i).trans_le (heb i)
+  have hR (i : ι) : 0 < R i := (hb i).trans_le (hbR i)
+  have hparams (i : ι) := physical_radius_level_parameters hκ (z i)
+    (R i) (b i) (e i) (he i) (heb i) (hbR i)
+  have h₀ (i : ι) : 0 ≤ u₀ i := (hparams i).1
+  have hu (i : ι) : u₀ i ≤ u₂ i := (hparams i).2.1
+  have hU (i : ι) : U i = sourceTarget κ (b i) (e i) := by
+    have hh := (hparams i).2.2.2.1
+    dsimp [U,u₀,u₂,sourceClockLevel] at *
+    linarith
+  obtain ⟨_,hβ,hβ1,hdecay,_⟩ := source_parameters hκ hκ8
+  obtain ⟨S⟩ := exists_actual_localizations hB hm hBm hBc hB0 hκ hκ8 z hz u₀ u₂ h₀ hu
+    γ G F hw A (norm_nonneg _) (initial_log_height_bound z)
+  let ρ := fun n : ℕ => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1)
+  have horder (n : ℕ) := (actualFactorStops_regular hB hm hBm hBc hB0 κ hκ hκ8
+    γ G F hw z hz u₀ u₂ h₀ hu ((n : ℝ≥0)+1)).2.1
+  have hex (n : ℕ) := actual_localized_product_expectation hB hm hBm hBc hB0 hκ hκ8
+    hβ.le hβ1 z hz u₀ u₂ h₀ hu γ G F hw (S n)
+  choose r C hrm hrp hrb hcost hI hmean using hex
+  let E := {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) < ⊤}
+  let cutoff := fun n => {ω | phaseAt (ρ n) ((S n).χ ω) ω = fun _ => 3}
+  have hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n := by
+    intro ω hω
+    filter_upwards [eventually_all_completed_of_finite_radius_levels B hm hBc κ A
+      (norm_nonneg _) z u₀ u₂ γ G F hw S ω hω] with n hn
+    exact hn.2
+  have hbudget (n : ℕ) (ω : Ω) : discountClock (r n) ((S n).χ ω) ω ≤ K :=
+    actual_localized_discount_le hBc hκ hκ8 hβ hdecay z hne u₀ u₂ h₀ hu γ G F hw
+      (S n) (horder n) hgrowth J hJ hJinj R b hR hb hRdef (fun _ => rfl) hsep
+      (r n) (C n) (hrm n) (fun t ω => (hrb n t ω).2) (hcost n) ω
+  have hbound := measure_le_of_completed_phase_products P E cutoff κ (sourceBeta κ) a U K
+    (fun n ω => phaseAt (ρ n) ((S n).χ ω) ω)
+    (fun n ω => (S n).Q ((S n).χ ω) ω)
+    (fun n ω => discountClock (r n) ((S n).χ ω) ω)
+    (fun n ω => B ((S n).χ ω) ω)
+    hcover hI hmean (fun _ _ h => h) (fun n ω _ => hbudget n ω)
+  have hexp := source_radius_targets_exponential (Finset.univ : Finset ι) hκ b e
+    (fun i _ => hb i) (fun i _ => he i)
+  simpa only [E,u₂,K,show U = (fun i => sourceTarget κ (b i) (e i)) from funext hU,
+    hexp] using hbound
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointAngular.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointAngular.lean
index 0b472ffc84e4520f8472b23f8d2e1d64c163c624..d6cfe0cb1565dc2e23201b2a9824406888dcf5f1 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointAngular.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointAngular.lean
@@ -1,289 +1,289 @@
-import OAI.Probability.SLE.Multipoint.MultipointRegularizer
-
-namespace OAI
-
-/-! The angular generator calculation for the regularized factors of the
-fixed-order multipoint radius estimate. The angular clock has generator
-`(1/2)∂² + ((1-p)/2) cot(θ) ∂`, with `p = 4a-1`. -/
-
-noncomputable section
-open Set
-
-namespace SLEExactGauge.Multipoint
-
-def sinePower (r θ : ℝ) : ℝ := Real.sin θ ^ r
-
-def sinePowerFirst (r θ : ℝ) : ℝ := r * Real.sin θ ^ (r-1) * Real.cos θ
-
-def sinePowerSecond (r θ : ℝ) : ℝ :=
-  r * (r-1) * Real.sin θ ^ (r-2) * Real.cos θ ^ 2 - r * Real.sin θ ^ r
-
-lemma sinePower_hasDerivAt {r θ : ℝ} (hθ : Real.sin θ ≠ 0) :
-    HasDerivAt (sinePower r) (sinePowerFirst r θ) θ := by
-  apply ((Real.hasDerivAt_sin θ).rpow_const (p := r) (Or.inl hθ)).congr_deriv
-  unfold sinePowerFirst
-  ring
-
-lemma sinePowerFirst_hasDerivAt {r θ : ℝ} (hθ : Real.sin θ ≠ 0) :
-    HasDerivAt (sinePowerFirst r) (sinePowerSecond r θ) θ := by
-  have hd := (((Real.hasDerivAt_sin θ).rpow_const (p := r-1) (Or.inl hθ)).const_mul r).mul
-    (Real.hasDerivAt_cos θ)
-  apply hd.congr_deriv
-  dsimp only [sinePowerSecond]
-  rw [show r-1-1 = r-2 by ring]
-  have hs : Real.sin θ ^ (r-1) * Real.sin θ = Real.sin θ ^ r := by
-    rw [← Real.rpow_add_one hθ]
-    congr 1
-    ring
-  calc
-    _ = r * (r-1) * Real.sin θ ^ (r-2) * Real.cos θ ^ 2 -
-        r * (Real.sin θ ^ (r-1) * Real.sin θ) := by ring
-    _ = _ := by rw [hs]
-
-def sineGenerator (p r θ : ℝ) : ℝ :=
-  sinePowerSecond r θ / 2 + (1-p)/2 * (Real.cos θ/Real.sin θ) * sinePowerFirst r θ
-
-lemma sineGenerator_eq {p r θ : ℝ} (hθ : 0 < Real.sin θ) :
-    sineGenerator p r θ = r*(r-p)/2 * Real.sin θ ^ (r-2) -
-      r*(r+1-p)/2 * Real.sin θ ^ r := by
-  unfold sineGenerator sinePowerSecond sinePowerFirst
-  rw [Real.rpow_sub hθ, Real.rpow_sub hθ, Real.rpow_two, Real.rpow_one]
-  have hcos : Real.cos θ ^ 2 = 1 - Real.sin θ ^ 2 := by
-    nlinarith [Real.sin_sq_add_cos_sq θ]
-  rw [hcos]
-  field_simp [hθ.ne']
-  ring_nf
-  rw [hcos]
-  ring
-
-lemma sineGenerator_eigen {p θ : ℝ} (hθ : 0 < Real.sin θ) :
-    sineGenerator p p θ + p/2 * sinePower p θ = 0 := by
-  rw [sineGenerator_eq hθ]
-  unfold sinePower
-  ring
-
-def angularFactor (p δ θ : ℝ) : ℝ := sinePower p θ + δ*(1+sinePower (q p) θ)
-
-def damping (p δ θ : ℝ) : ℝ := δ / angularFactor p δ θ * sinePower (q p-2) θ
-
-def negativeDrift (p : ℝ) : ℝ := q p * (p-q p)/2
-
-def driftAllowance (p : ℝ) : ℝ := p/2 + (p-q p)*(1+q p)/2 + 2
-
-lemma negativeDrift_pos {p : ℝ} (hp : 0 < p) : 0 < negativeDrift p := by
-  exact div_pos (mul_pos (q_pos hp) (sub_pos.mpr (q_lt_p hp))) (by norm_num)
-
-lemma driftAllowance_pos {p : ℝ} (hp : 0 < p) : 0 < driftAllowance p := by
-  have hq := q_pos hp
-  have hqp := q_lt_p hp
-  dsimp [driftAllowance]
-  positivity
-
-lemma angularFactor_pos {p δ θ : ℝ} (hδ : 0 < δ) (hθ : 0 < Real.sin θ) :
-    0 < angularFactor p δ θ := by
-  unfold angularFactor sinePower
-  positivity
-
-/-- Exact algebraic drift before estimating its negative singular term. -/
-lemma regularized_generator {p δ σ θ : ℝ} (hθ : 0 < Real.sin θ) :
-    sineGenerator p p θ + δ*sineGenerator p (q p) θ +
-      (p/2)*angularFactor p δ θ + δ*σ*(1+sinePower (q p) θ) =
-    δ*(p/2 + (p-q p)*(1+q p)/2*sinePower (q p) θ -
-      negativeDrift p*sinePower (q p-2) θ + σ*(1+sinePower (q p) θ)) := by
-  rw [sineGenerator_eq hθ, sineGenerator_eq hθ]
-  unfold angularFactor sinePower negativeDrift
-  ring
-
-lemma regularized_bracket_le {p σ θ : ℝ} (hp : 0 < p)
-    (hσ : σ ≤ 1) (hθ : 0 ≤ Real.sin θ) (hθ1 : Real.sin θ ≤ 1) :
-    p/2 + (p-q p)*(1+q p)/2*sinePower (q p) θ -
-      negativeDrift p*sinePower (q p-2) θ + σ*(1+sinePower (q p) θ) ≤
-    driftAllowance p - negativeDrift p*sinePower (q p-2) θ := by
-  have hs0 : 0 ≤ sinePower (q p) θ := Real.rpow_nonneg hθ _
-  have hs1 : sinePower (q p) θ ≤ 1 := Real.rpow_le_one hθ hθ1 (q_pos hp).le
-  have hk : 0 ≤ (p-q p)*(1+q p)/2 := by
-    have := q_lt_p hp
-    have := q_pos hp
-    positivity
-  have h1 := mul_le_mul_of_nonneg_left hs1 hk
-  have h2 := mul_le_mul_of_nonneg_right hσ (by linarith : 0 ≤ 1+sinePower (q p) θ)
-  unfold driftAllowance
-  nlinarith
-
-lemma angularFactor_ge_delta {p δ θ : ℝ} (hδ : 0 ≤ δ) (hθ : 0 ≤ Real.sin θ) :
-    δ ≤ angularFactor p δ θ := by
-  unfold angularFactor sinePower
-  have hp := Real.rpow_nonneg hθ p
-  have hq := Real.rpow_nonneg hθ (q p)
-  nlinarith
-
-lemma angularFactor_le_three {p δ θ : ℝ} (hp : 0 < p)
-    (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hθ : 0 ≤ Real.sin θ) :
-    angularFactor p δ θ ≤ 3 := by
-  have hp1 : sinePower p θ ≤ 1 := Real.rpow_le_one hθ (Real.sin_le_one θ) hp.le
-  have hq1 : sinePower (q p) θ ≤ 1 := Real.rpow_le_one hθ (Real.sin_le_one θ) (q_pos hp).le
-  have hq0 : 0 ≤ sinePower (q p) θ := Real.rpow_nonneg hθ _
-  unfold angularFactor
-  nlinarith
-
-lemma angularFactor_endpoint_bounds {p θ : ℝ} (hp : 0 < p) (hθ : 0 ≤ Real.sin θ) :
-    1 ≤ angularFactor p 1 θ ∧ angularFactor p 1 θ ≤ 3 :=
-  ⟨angularFactor_ge_delta zero_le_one hθ,
-    angularFactor_le_three hp zero_le_one le_rfl hθ⟩
-
-def angularFactorFirst (p δ θ : ℝ) : ℝ :=
-  sinePowerFirst p θ + δ*sinePowerFirst (q p) θ
-
-lemma angularFactor_hasDerivAt {p δ θ : ℝ} (hθ : Real.sin θ ≠ 0) :
-    HasDerivAt (angularFactor p δ) (angularFactorFirst p δ θ) θ := by
-  apply ((sinePower_hasDerivAt (r := p) hθ).add
-    (((hasDerivAt_const θ 1).add (sinePower_hasDerivAt (r := q p) hθ)).const_mul δ)).congr_deriv
-  simp only [angularFactorFirst, zero_add]
-
-lemma angularFactorFirst_bound {p δ θ : ℝ} (hp : 0 < p)
-    (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hθ : 0 < Real.sin θ) :
-    |angularFactorFirst p δ θ| ≤ (p+q p)*Real.sin θ ^ (q p-1) := by
-  have hpq : Real.sin θ ^ (p-1) ≤ Real.sin θ ^ (q p-1) :=
-    Real.rpow_le_rpow_of_exponent_ge hθ (Real.sin_le_one θ) (by linarith [q_lt_p hp])
-  have hq := (q_pos hp).le
-  have hpow : 0 ≤ Real.sin θ ^ (q p-1) := Real.rpow_nonneg hθ.le _
-  have hcos : |Real.cos θ| ≤ 1 := abs_le.mpr ⟨Real.neg_one_le_cos θ, Real.cos_le_one θ⟩
-  have hfirst (r : ℝ) (hr : 0 ≤ r) : |sinePowerFirst r θ| ≤ r * Real.sin θ ^ (r-1) := by
-    unfold sinePowerFirst
-    rw [abs_mul, abs_of_nonneg (mul_nonneg hr (Real.rpow_nonneg hθ.le _))]
-    exact mul_le_of_le_one_right (mul_nonneg hr (Real.rpow_nonneg hθ.le _)) hcos
-  calc
-    _ ≤ |sinePowerFirst p θ| + |δ*sinePowerFirst (q p) θ| := abs_add_le _ _
-    _ = |sinePowerFirst p θ| + δ*|sinePowerFirst (q p) θ| := by
-      rw [abs_mul, abs_of_nonneg hδ]
-    _ ≤ p*Real.sin θ ^ (p-1) + δ*(q p*Real.sin θ ^ (q p-1)) :=
-      add_le_add (hfirst p hp.le) (mul_le_mul_of_nonneg_left (hfirst (q p) hq) hδ)
-    _ ≤ p*Real.sin θ ^ (q p-1) + q p*Real.sin θ ^ (q p-1) :=
-      add_le_add (mul_le_mul_of_nonneg_left hpq hp.le)
-        (mul_le_of_le_one_left (mul_nonneg hq hpow) hδ1)
-    _ = _ := by ring
-
-/-- The singular angular derivative is controlled by the damping term.
-The exponent is chosen so that `eta * (q - 2) = q - 1`. -/
-lemma normalized_angularFactorFirst_bound {p δ θ : ℝ} (hp : 0 < p)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hθ : 0 < Real.sin θ) :
-    |angularFactorFirst p δ θ| / angularFactor p δ θ ≤
-      (p+q p)*3^(eta p)*δ^(-1-eta p)*(damping p δ θ)^(eta p) := by
-  let f := angularFactor p δ θ
-  let e := eta p
-  have hf : 0 < f := angularFactor_pos hδ hθ
-  have hdf : δ ≤ f := angularFactor_ge_delta hδ.le hθ.le
-  have hf3 : f ≤ 3 := angularFactor_le_three hp hδ.le hδ1 hθ.le
-  have he : 0 < e := eta_pos
-  have hD : 0 < damping p δ θ :=
-    mul_pos (div_pos hδ hf) (Real.rpow_pos_of_pos hθ _)
-  have hidentity : Real.sin θ ^ (q p-1) = (f/δ)^e * (damping p δ θ)^e := by
-    rw [← Real.mul_rpow (div_pos hf hδ).le hD.le]
-    have hcancel : f/δ*damping p δ θ = Real.sin θ ^ (q p-2) := by
-      dsimp [damping, sinePower, f]
-      field_simp [hδ.ne', (angularFactor_pos hδ hθ).ne']
-    rw [hcancel, ← Real.rpow_mul hθ.le]
-    congr 1
-    exact ((mul_comm _ _).trans (eta_identity p)).symm
-  have hpq : 0 ≤ p+q p := add_nonneg hp.le (q_pos hp).le
-  calc
-    _ ≤ ((p+q p)*Real.sin θ ^ (q p-1))/f :=
-      div_le_div_of_nonneg_right (angularFactorFirst_bound hp hδ.le hδ1 hθ) hf.le
-    _ ≤ ((p+q p)*Real.sin θ ^ (q p-1))/δ :=
-      div_le_div_of_nonneg_left (mul_nonneg hpq (Real.rpow_nonneg hθ.le _)) hδ hdf
-    _ = (p+q p)/δ * (f/δ)^e * (damping p δ θ)^e := by rw [hidentity]; ring
-    _ ≤ (p+q p)/δ * (3/δ)^e * (damping p δ θ)^e := by
-      apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hD.le _)
-      apply mul_le_mul_of_nonneg_left _ (div_nonneg hpq hδ.le)
-      exact Real.rpow_le_rpow (div_pos hf hδ).le
-        (div_le_div_of_nonneg_right hf3 hδ.le) he.le
-    _ = _ := by
-      rw [Real.div_rpow (by norm_num : (0:ℝ) ≤ 3) hδ.le,
-        Real.rpow_sub hδ, Real.rpow_neg_one]
-      ring
-
-def driftThreshold (p : ℝ) : ℝ :=
-  (negativeDrift p / (2*driftAllowance p)) ^ (1/(2-q p))
-
-lemma driftThreshold_pos {p : ℝ} (hp : 0 < p) : 0 < driftThreshold p := by
-  exact Real.rpow_pos_of_pos (div_pos (negativeDrift_pos hp)
-    (mul_pos (by norm_num) (driftAllowance_pos hp))) _
-
-lemma driftThreshold_lt_one {p : ℝ} (hp : 0 < p) : driftThreshold p < 1 := by
-  have hq := q_pos hp
-  have hq1 := q_lt_one (p := p)
-  have hqp := q_lt_p hp
-  have hk := negativeDrift_pos hp
-  have hA := driftAllowance_pos hp
-  have hden : 0 < 2-q p := by linarith
-  have hsmall : negativeDrift p < 2*driftAllowance p := by
-    unfold negativeDrift driftAllowance
-    nlinarith [mul_nonneg (sub_nonneg.mpr hq1.le) (sub_nonneg.mpr hqp.le)]
-  exact Real.rpow_lt_one (div_pos hk (by positivity)).le
-    ((div_lt_one (by positivity)).mpr hsmall) (by positivity : 0 < 1/(2-q p))
-
-lemma driftThreshold_identity {p : ℝ} (hp : 0 < p) :
-    negativeDrift p * driftThreshold p ^ (q p-2) = 2*driftAllowance p := by
-  have hk := negativeDrift_pos hp
-  have hA := driftAllowance_pos hp
-  have hq2 : 2-q p ≠ 0 := ne_of_gt (by linarith [q_lt_one (p := p)])
-  unfold driftThreshold
-  rw [← Real.rpow_mul (div_pos hk (by positivity)).le]
-  have he : (1/(2-q p))*(q p-2) = -1 := by field_simp; ring
-  rw [he, Real.rpow_neg_one]
-  field_simp
-
-/-- The negative singular part absorbs the bounded drift at small angles;
-away from the endpoints the denominator supplies a fixed lower bound. -/
-lemma regularized_drift_bound {p δ σ θ : ℝ} (hp : 0 < p)
-    (hδ : 0 < δ) (hσ : σ ≤ 1) (hθ : 0 < Real.sin θ) :
-    δ / angularFactor p δ θ *
-      (p/2 + (p-q p)*(1+q p)/2*sinePower (q p) θ -
-        negativeDrift p*sinePower (q p-2) θ + σ*(1+sinePower (q p) θ)) ≤
-    (driftAllowance p * driftThreshold p ^ (-p))*δ -
-      (negativeDrift p/2)*damping p δ θ := by
-  let f := angularFactor p δ θ
-  let S := sinePower (q p-2) θ
-  let A := driftAllowance p
-  let k := negativeDrift p
-  let h := driftThreshold p
-  have hf : 0 < f := angularFactor_pos hδ hθ
-  have hA : 0 < A := driftAllowance_pos hp
-  have hk : 0 < k := negativeDrift_pos hp
-  have hh : 0 < h := driftThreshold_pos hp
-  have hbr := regularized_bracket_le hp hσ hθ.le (Real.sin_le_one θ)
-  have hstart := mul_le_mul_of_nonneg_left hbr (div_pos hδ hf).le
-  change δ/f * _ ≤ A*h^(-p)*δ - (k/2)*(δ/f*S)
-  apply hstart.trans
-  change δ/f * (A-k*S) ≤ A*h^(-p)*δ - (k/2)*(δ/f*S)
-  by_cases hsmall : Real.sin θ ≤ h
-  · have hS : h^(q p-2) ≤ S :=
-      Real.rpow_le_rpow_of_nonpos hθ hsmall (by linarith [q_lt_one (p := p)])
-    have hcancel : A ≤ k/2*S := by
-      have he := driftThreshold_identity hp
-      change k*h^(q p-2) = 2*A at he
-      nlinarith [mul_le_mul_of_nonneg_left hS hk.le]
-    have hnonneg : 0 ≤ A*h^(-p)*δ := by positivity
-    have hneg := mul_nonpos_of_nonneg_of_nonpos (div_pos hδ hf).le
-      (sub_nonpos.mpr hcancel)
-    nlinarith
-  · have hsp : h^p ≤ Real.sin θ ^ p :=
-      Real.rpow_le_rpow hh.le (le_of_not_ge hsmall) hp.le
-    have hsf : Real.sin θ ^ p ≤ f := by
-      dsimp [f, angularFactor, sinePower]
-      exact le_add_of_nonneg_right (by positivity)
-    have hhpow : 0 < h^p := Real.rpow_pos_of_pos hh _
-    have hquot : δ/f ≤ δ/(h^p) := div_le_div_of_nonneg_left hδ.le hhpow (hsp.trans hsf)
-    have hmain := mul_le_mul_of_nonneg_left hquot hA.le
-    have hS0 : 0 ≤ S := Real.rpow_nonneg hθ.le _
-    have hnonneg : 0 ≤ (k/2)*(δ/f*S) := mul_nonneg (by positivity)
-      (mul_nonneg (div_pos hδ hf).le hS0)
-    rw [Real.rpow_neg hh.le]
-    change A*(δ/f) ≤ A*(δ*(h^p)⁻¹) at hmain
-    nlinarith
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointRegularizer
+
+namespace OAI
+
+/-! The angular generator calculation for the regularized factors of the
+fixed-order multipoint radius estimate. The angular clock has generator
+`(1/2)∂² + ((1-p)/2) cot(θ) ∂`, with `p = 4a-1`. -/
+
+noncomputable section
+open Set
+
+namespace SLEExactGauge.Multipoint
+
+def sinePower (r θ : ℝ) : ℝ := Real.sin θ ^ r
+
+def sinePowerFirst (r θ : ℝ) : ℝ := r * Real.sin θ ^ (r-1) * Real.cos θ
+
+def sinePowerSecond (r θ : ℝ) : ℝ :=
+  r * (r-1) * Real.sin θ ^ (r-2) * Real.cos θ ^ 2 - r * Real.sin θ ^ r
+
+lemma sinePower_hasDerivAt {r θ : ℝ} (hθ : Real.sin θ ≠ 0) :
+    HasDerivAt (sinePower r) (sinePowerFirst r θ) θ := by
+  apply ((Real.hasDerivAt_sin θ).rpow_const (p := r) (Or.inl hθ)).congr_deriv
+  unfold sinePowerFirst
+  ring
+
+lemma sinePowerFirst_hasDerivAt {r θ : ℝ} (hθ : Real.sin θ ≠ 0) :
+    HasDerivAt (sinePowerFirst r) (sinePowerSecond r θ) θ := by
+  have hd := (((Real.hasDerivAt_sin θ).rpow_const (p := r-1) (Or.inl hθ)).const_mul r).mul
+    (Real.hasDerivAt_cos θ)
+  apply hd.congr_deriv
+  dsimp only [sinePowerSecond]
+  rw [show r-1-1 = r-2 by ring]
+  have hs : Real.sin θ ^ (r-1) * Real.sin θ = Real.sin θ ^ r := by
+    rw [← Real.rpow_add_one hθ]
+    congr 1
+    ring
+  calc
+    _ = r * (r-1) * Real.sin θ ^ (r-2) * Real.cos θ ^ 2 -
+        r * (Real.sin θ ^ (r-1) * Real.sin θ) := by ring
+    _ = _ := by rw [hs]
+
+def sineGenerator (p r θ : ℝ) : ℝ :=
+  sinePowerSecond r θ / 2 + (1-p)/2 * (Real.cos θ/Real.sin θ) * sinePowerFirst r θ
+
+lemma sineGenerator_eq {p r θ : ℝ} (hθ : 0 < Real.sin θ) :
+    sineGenerator p r θ = r*(r-p)/2 * Real.sin θ ^ (r-2) -
+      r*(r+1-p)/2 * Real.sin θ ^ r := by
+  unfold sineGenerator sinePowerSecond sinePowerFirst
+  rw [Real.rpow_sub hθ, Real.rpow_sub hθ, Real.rpow_two, Real.rpow_one]
+  have hcos : Real.cos θ ^ 2 = 1 - Real.sin θ ^ 2 := by
+    nlinarith [Real.sin_sq_add_cos_sq θ]
+  rw [hcos]
+  field_simp [hθ.ne']
+  ring_nf
+  rw [hcos]
+  ring
+
+lemma sineGenerator_eigen {p θ : ℝ} (hθ : 0 < Real.sin θ) :
+    sineGenerator p p θ + p/2 * sinePower p θ = 0 := by
+  rw [sineGenerator_eq hθ]
+  unfold sinePower
+  ring
+
+def angularFactor (p δ θ : ℝ) : ℝ := sinePower p θ + δ*(1+sinePower (q p) θ)
+
+def damping (p δ θ : ℝ) : ℝ := δ / angularFactor p δ θ * sinePower (q p-2) θ
+
+def negativeDrift (p : ℝ) : ℝ := q p * (p-q p)/2
+
+def driftAllowance (p : ℝ) : ℝ := p/2 + (p-q p)*(1+q p)/2 + 2
+
+lemma negativeDrift_pos {p : ℝ} (hp : 0 < p) : 0 < negativeDrift p := by
+  exact div_pos (mul_pos (q_pos hp) (sub_pos.mpr (q_lt_p hp))) (by norm_num)
+
+lemma driftAllowance_pos {p : ℝ} (hp : 0 < p) : 0 < driftAllowance p := by
+  have hq := q_pos hp
+  have hqp := q_lt_p hp
+  dsimp [driftAllowance]
+  positivity
+
+lemma angularFactor_pos {p δ θ : ℝ} (hδ : 0 < δ) (hθ : 0 < Real.sin θ) :
+    0 < angularFactor p δ θ := by
+  unfold angularFactor sinePower
+  positivity
+
+/-- Exact algebraic drift before estimating its negative singular term. -/
+lemma regularized_generator {p δ σ θ : ℝ} (hθ : 0 < Real.sin θ) :
+    sineGenerator p p θ + δ*sineGenerator p (q p) θ +
+      (p/2)*angularFactor p δ θ + δ*σ*(1+sinePower (q p) θ) =
+    δ*(p/2 + (p-q p)*(1+q p)/2*sinePower (q p) θ -
+      negativeDrift p*sinePower (q p-2) θ + σ*(1+sinePower (q p) θ)) := by
+  rw [sineGenerator_eq hθ, sineGenerator_eq hθ]
+  unfold angularFactor sinePower negativeDrift
+  ring
+
+lemma regularized_bracket_le {p σ θ : ℝ} (hp : 0 < p)
+    (hσ : σ ≤ 1) (hθ : 0 ≤ Real.sin θ) (hθ1 : Real.sin θ ≤ 1) :
+    p/2 + (p-q p)*(1+q p)/2*sinePower (q p) θ -
+      negativeDrift p*sinePower (q p-2) θ + σ*(1+sinePower (q p) θ) ≤
+    driftAllowance p - negativeDrift p*sinePower (q p-2) θ := by
+  have hs0 : 0 ≤ sinePower (q p) θ := Real.rpow_nonneg hθ _
+  have hs1 : sinePower (q p) θ ≤ 1 := Real.rpow_le_one hθ hθ1 (q_pos hp).le
+  have hk : 0 ≤ (p-q p)*(1+q p)/2 := by
+    have := q_lt_p hp
+    have := q_pos hp
+    positivity
+  have h1 := mul_le_mul_of_nonneg_left hs1 hk
+  have h2 := mul_le_mul_of_nonneg_right hσ (by linarith : 0 ≤ 1+sinePower (q p) θ)
+  unfold driftAllowance
+  nlinarith
+
+lemma angularFactor_ge_delta {p δ θ : ℝ} (hδ : 0 ≤ δ) (hθ : 0 ≤ Real.sin θ) :
+    δ ≤ angularFactor p δ θ := by
+  unfold angularFactor sinePower
+  have hp := Real.rpow_nonneg hθ p
+  have hq := Real.rpow_nonneg hθ (q p)
+  nlinarith
+
+lemma angularFactor_le_three {p δ θ : ℝ} (hp : 0 < p)
+    (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hθ : 0 ≤ Real.sin θ) :
+    angularFactor p δ θ ≤ 3 := by
+  have hp1 : sinePower p θ ≤ 1 := Real.rpow_le_one hθ (Real.sin_le_one θ) hp.le
+  have hq1 : sinePower (q p) θ ≤ 1 := Real.rpow_le_one hθ (Real.sin_le_one θ) (q_pos hp).le
+  have hq0 : 0 ≤ sinePower (q p) θ := Real.rpow_nonneg hθ _
+  unfold angularFactor
+  nlinarith
+
+lemma angularFactor_endpoint_bounds {p θ : ℝ} (hp : 0 < p) (hθ : 0 ≤ Real.sin θ) :
+    1 ≤ angularFactor p 1 θ ∧ angularFactor p 1 θ ≤ 3 :=
+  ⟨angularFactor_ge_delta zero_le_one hθ,
+    angularFactor_le_three hp zero_le_one le_rfl hθ⟩
+
+def angularFactorFirst (p δ θ : ℝ) : ℝ :=
+  sinePowerFirst p θ + δ*sinePowerFirst (q p) θ
+
+lemma angularFactor_hasDerivAt {p δ θ : ℝ} (hθ : Real.sin θ ≠ 0) :
+    HasDerivAt (angularFactor p δ) (angularFactorFirst p δ θ) θ := by
+  apply ((sinePower_hasDerivAt (r := p) hθ).add
+    (((hasDerivAt_const θ 1).add (sinePower_hasDerivAt (r := q p) hθ)).const_mul δ)).congr_deriv
+  simp only [angularFactorFirst, zero_add]
+
+lemma angularFactorFirst_bound {p δ θ : ℝ} (hp : 0 < p)
+    (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hθ : 0 < Real.sin θ) :
+    |angularFactorFirst p δ θ| ≤ (p+q p)*Real.sin θ ^ (q p-1) := by
+  have hpq : Real.sin θ ^ (p-1) ≤ Real.sin θ ^ (q p-1) :=
+    Real.rpow_le_rpow_of_exponent_ge hθ (Real.sin_le_one θ) (by linarith [q_lt_p hp])
+  have hq := (q_pos hp).le
+  have hpow : 0 ≤ Real.sin θ ^ (q p-1) := Real.rpow_nonneg hθ.le _
+  have hcos : |Real.cos θ| ≤ 1 := abs_le.mpr ⟨Real.neg_one_le_cos θ, Real.cos_le_one θ⟩
+  have hfirst (r : ℝ) (hr : 0 ≤ r) : |sinePowerFirst r θ| ≤ r * Real.sin θ ^ (r-1) := by
+    unfold sinePowerFirst
+    rw [abs_mul, abs_of_nonneg (mul_nonneg hr (Real.rpow_nonneg hθ.le _))]
+    exact mul_le_of_le_one_right (mul_nonneg hr (Real.rpow_nonneg hθ.le _)) hcos
+  calc
+    _ ≤ |sinePowerFirst p θ| + |δ*sinePowerFirst (q p) θ| := abs_add_le _ _
+    _ = |sinePowerFirst p θ| + δ*|sinePowerFirst (q p) θ| := by
+      rw [abs_mul, abs_of_nonneg hδ]
+    _ ≤ p*Real.sin θ ^ (p-1) + δ*(q p*Real.sin θ ^ (q p-1)) :=
+      add_le_add (hfirst p hp.le) (mul_le_mul_of_nonneg_left (hfirst (q p) hq) hδ)
+    _ ≤ p*Real.sin θ ^ (q p-1) + q p*Real.sin θ ^ (q p-1) :=
+      add_le_add (mul_le_mul_of_nonneg_left hpq hp.le)
+        (mul_le_of_le_one_left (mul_nonneg hq hpow) hδ1)
+    _ = _ := by ring
+
+/-- The singular angular derivative is controlled by the damping term.
+The exponent is chosen so that `eta * (q - 2) = q - 1`. -/
+lemma normalized_angularFactorFirst_bound {p δ θ : ℝ} (hp : 0 < p)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hθ : 0 < Real.sin θ) :
+    |angularFactorFirst p δ θ| / angularFactor p δ θ ≤
+      (p+q p)*3^(eta p)*δ^(-1-eta p)*(damping p δ θ)^(eta p) := by
+  let f := angularFactor p δ θ
+  let e := eta p
+  have hf : 0 < f := angularFactor_pos hδ hθ
+  have hdf : δ ≤ f := angularFactor_ge_delta hδ.le hθ.le
+  have hf3 : f ≤ 3 := angularFactor_le_three hp hδ.le hδ1 hθ.le
+  have he : 0 < e := eta_pos
+  have hD : 0 < damping p δ θ :=
+    mul_pos (div_pos hδ hf) (Real.rpow_pos_of_pos hθ _)
+  have hidentity : Real.sin θ ^ (q p-1) = (f/δ)^e * (damping p δ θ)^e := by
+    rw [← Real.mul_rpow (div_pos hf hδ).le hD.le]
+    have hcancel : f/δ*damping p δ θ = Real.sin θ ^ (q p-2) := by
+      dsimp [damping, sinePower, f]
+      field_simp [hδ.ne', (angularFactor_pos hδ hθ).ne']
+    rw [hcancel, ← Real.rpow_mul hθ.le]
+    congr 1
+    exact ((mul_comm _ _).trans (eta_identity p)).symm
+  have hpq : 0 ≤ p+q p := add_nonneg hp.le (q_pos hp).le
+  calc
+    _ ≤ ((p+q p)*Real.sin θ ^ (q p-1))/f :=
+      div_le_div_of_nonneg_right (angularFactorFirst_bound hp hδ.le hδ1 hθ) hf.le
+    _ ≤ ((p+q p)*Real.sin θ ^ (q p-1))/δ :=
+      div_le_div_of_nonneg_left (mul_nonneg hpq (Real.rpow_nonneg hθ.le _)) hδ hdf
+    _ = (p+q p)/δ * (f/δ)^e * (damping p δ θ)^e := by rw [hidentity]; ring
+    _ ≤ (p+q p)/δ * (3/δ)^e * (damping p δ θ)^e := by
+      apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hD.le _)
+      apply mul_le_mul_of_nonneg_left _ (div_nonneg hpq hδ.le)
+      exact Real.rpow_le_rpow (div_pos hf hδ).le
+        (div_le_div_of_nonneg_right hf3 hδ.le) he.le
+    _ = _ := by
+      rw [Real.div_rpow (by norm_num : (0:ℝ) ≤ 3) hδ.le,
+        Real.rpow_sub hδ, Real.rpow_neg_one]
+      ring
+
+def driftThreshold (p : ℝ) : ℝ :=
+  (negativeDrift p / (2*driftAllowance p)) ^ (1/(2-q p))
+
+lemma driftThreshold_pos {p : ℝ} (hp : 0 < p) : 0 < driftThreshold p := by
+  exact Real.rpow_pos_of_pos (div_pos (negativeDrift_pos hp)
+    (mul_pos (by norm_num) (driftAllowance_pos hp))) _
+
+lemma driftThreshold_lt_one {p : ℝ} (hp : 0 < p) : driftThreshold p < 1 := by
+  have hq := q_pos hp
+  have hq1 := q_lt_one (p := p)
+  have hqp := q_lt_p hp
+  have hk := negativeDrift_pos hp
+  have hA := driftAllowance_pos hp
+  have hden : 0 < 2-q p := by linarith
+  have hsmall : negativeDrift p < 2*driftAllowance p := by
+    unfold negativeDrift driftAllowance
+    nlinarith [mul_nonneg (sub_nonneg.mpr hq1.le) (sub_nonneg.mpr hqp.le)]
+  exact Real.rpow_lt_one (div_pos hk (by positivity)).le
+    ((div_lt_one (by positivity)).mpr hsmall) (by positivity : 0 < 1/(2-q p))
+
+lemma driftThreshold_identity {p : ℝ} (hp : 0 < p) :
+    negativeDrift p * driftThreshold p ^ (q p-2) = 2*driftAllowance p := by
+  have hk := negativeDrift_pos hp
+  have hA := driftAllowance_pos hp
+  have hq2 : 2-q p ≠ 0 := ne_of_gt (by linarith [q_lt_one (p := p)])
+  unfold driftThreshold
+  rw [← Real.rpow_mul (div_pos hk (by positivity)).le]
+  have he : (1/(2-q p))*(q p-2) = -1 := by field_simp; ring
+  rw [he, Real.rpow_neg_one]
+  field_simp
+
+/-- The negative singular part absorbs the bounded drift at small angles;
+away from the endpoints the denominator supplies a fixed lower bound. -/
+lemma regularized_drift_bound {p δ σ θ : ℝ} (hp : 0 < p)
+    (hδ : 0 < δ) (hσ : σ ≤ 1) (hθ : 0 < Real.sin θ) :
+    δ / angularFactor p δ θ *
+      (p/2 + (p-q p)*(1+q p)/2*sinePower (q p) θ -
+        negativeDrift p*sinePower (q p-2) θ + σ*(1+sinePower (q p) θ)) ≤
+    (driftAllowance p * driftThreshold p ^ (-p))*δ -
+      (negativeDrift p/2)*damping p δ θ := by
+  let f := angularFactor p δ θ
+  let S := sinePower (q p-2) θ
+  let A := driftAllowance p
+  let k := negativeDrift p
+  let h := driftThreshold p
+  have hf : 0 < f := angularFactor_pos hδ hθ
+  have hA : 0 < A := driftAllowance_pos hp
+  have hk : 0 < k := negativeDrift_pos hp
+  have hh : 0 < h := driftThreshold_pos hp
+  have hbr := regularized_bracket_le hp hσ hθ.le (Real.sin_le_one θ)
+  have hstart := mul_le_mul_of_nonneg_left hbr (div_pos hδ hf).le
+  change δ/f * _ ≤ A*h^(-p)*δ - (k/2)*(δ/f*S)
+  apply hstart.trans
+  change δ/f * (A-k*S) ≤ A*h^(-p)*δ - (k/2)*(δ/f*S)
+  by_cases hsmall : Real.sin θ ≤ h
+  · have hS : h^(q p-2) ≤ S :=
+      Real.rpow_le_rpow_of_nonpos hθ hsmall (by linarith [q_lt_one (p := p)])
+    have hcancel : A ≤ k/2*S := by
+      have he := driftThreshold_identity hp
+      change k*h^(q p-2) = 2*A at he
+      nlinarith [mul_le_mul_of_nonneg_left hS hk.le]
+    have hnonneg : 0 ≤ A*h^(-p)*δ := by positivity
+    have hneg := mul_nonpos_of_nonneg_of_nonpos (div_pos hδ hf).le
+      (sub_nonpos.mpr hcancel)
+    nlinarith
+  · have hsp : h^p ≤ Real.sin θ ^ p :=
+      Real.rpow_le_rpow hh.le (le_of_not_ge hsmall) hp.le
+    have hsf : Real.sin θ ^ p ≤ f := by
+      dsimp [f, angularFactor, sinePower]
+      exact le_add_of_nonneg_right (by positivity)
+    have hhpow : 0 < h^p := Real.rpow_pos_of_pos hh _
+    have hquot : δ/f ≤ δ/(h^p) := div_le_div_of_nonneg_left hδ.le hhpow (hsp.trans hsf)
+    have hmain := mul_le_mul_of_nonneg_left hquot hA.le
+    have hS0 : 0 ≤ S := Real.rpow_nonneg hθ.le _
+    have hnonneg : 0 ≤ (k/2)*(δ/f*S) := mul_nonneg (by positivity)
+      (mul_nonneg (div_pos hδ hf).le hS0)
+    rw [Real.rpow_neg hh.le]
+    change A*(δ/f) ≤ A*(δ*(h^p)⁻¹) at hmain
+    nlinarith
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointBrownianCoefficient.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointBrownianCoefficient.lean
index f161c7f85376205379eb77f780760036cf89bde2..f2f82cef58adc3a58b69df10812330a753775a37 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointBrownianCoefficient.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointBrownianCoefficient.lean
@@ -1,81 +1,81 @@
-import OAI.Probability.SLE.Multipoint.MultipointAngular
-import OAI.Probability.SLE.Multipoint.MultipointYoung
-
-namespace OAI
-
-/-! The Brownian cross coefficient in the regularized multipoint product,
-including the singular powers when `p < 1`. -/
-
-noncomputable section
-
-namespace SLEExactGauge.Multipoint
-
-def angularCoefficientConstant (p : ℝ) : ℝ := (p+q p)*3^(eta p)
-
-lemma angularCoefficientConstant_pos {p : ℝ} (hp : 0 < p) :
-    0 < angularCoefficientConstant p := by
-  unfold angularCoefficientConstant
-  exact mul_pos (add_pos hp (q_pos hp)) (Real.rpow_pos_of_pos (by norm_num) _)
-
-lemma velocity_power_identity {D v e : ℝ} (hD : 0 ≤ D) (hv : 0 < v) :
-    (D*v^2)^e * v^(1-2*e) = D^e*v := by
-  rw [Real.mul_rpow hD (sq_nonneg v), ← Real.rpow_natCast v 2,
-    ← Real.rpow_mul hv.le, mul_assoc, ← Real.rpow_add hv]
-  norm_num
-
-lemma pair_velocity_power_identity {Di Dj vi vj e : ℝ}
-    (hDi : 0 ≤ Di) (hDj : 0 ≤ Dj) (hvi : 0 < vi) (hvj : 0 < vj) :
-    (Di*vi^2)^e*(Dj*vj^2)^e*(vi*vj)^(1-2*e) = Di^e*Dj^e*vi*vj := by
-  rw [Real.mul_rpow hvi.le hvj.le]
-  calc
-    _ = ((Di*vi^2)^e*vi^(1-2*e))*((Dj*vj^2)^e*vj^(1-2*e)) := by ring
-    _ = _ := by rw [velocity_power_identity hDi hvi, velocity_power_identity hDj hvj]; ring
-
-/-- The magnitude of one Brownian cross term is absorbed at its two
-endpoints, leaving exactly the inverse-regulator interaction cost. -/
-theorem angular_pair_absorption {p δi δj θi θj vi vj ε : ℝ}
-    (hp : 0 < p) (hδi : 0 < δi) (hδi1 : δi ≤ 1)
-    (hδj : 0 < δj) (hδj1 : δj ≤ 1)
-    (hθi : 0 < Real.sin θi) (hθj : 0 < Real.sin θj)
-    (hvi : 0 < vi) (hvj : 0 < vj) (hε : 0 < ε) :
-    (|angularFactorFirst p δi θi| / angularFactor p δi θi * vi) *
-      (|angularFactorFirst p δj θj| / angularFactor p δj θj * vj) ≤
-    ε*(damping p δi θi*vi^2+damping p δj θj*vj^2) +
-      youngConstant p (angularCoefficientConstant p ^ 2) ε *
-        (δi*δj)^(-exponent p)*(vi*vj) := by
-  let Di := damping p δi θi
-  let Dj := damping p δj θj
-  let C := angularCoefficientConstant p
-  have hDi : 0 < Di := mul_pos (div_pos hδi (angularFactor_pos hδi hθi))
-    (Real.rpow_pos_of_pos hθi _)
-  have hDj : 0 < Dj := mul_pos (div_pos hδj (angularFactor_pos hδj hθj))
-    (Real.rpow_pos_of_pos hθj _)
-  have hC : 0 < C := angularCoefficientConstant_pos hp
-  have hi := normalized_angularFactorFirst_bound hp hδi hδi1 hθi
-  have hj := normalized_angularFactorFirst_bound hp hδj hδj1 hθj
-  change _ ≤ C*δi^(-1-eta p)*Di^(eta p) at hi
-  change _ ≤ C*δj^(-1-eta p)*Dj^(eta p) at hj
-  have hmul := mul_le_mul hi hj (div_nonneg (abs_nonneg _) (angularFactor_pos hδj hθj).le)
-    (by positivity : 0 ≤ C*δi^(-1-eta p)*Di^(eta p))
-  have hmulv := mul_le_mul_of_nonneg_right hmul (mul_pos hvi hvj).le
-  calc
-    _ ≤ C^2*(δi*δj)^(-1-eta p)*Di^(eta p)*Dj^(eta p)*vi*vj := by
-      rw [Real.mul_rpow hδi.le hδj.le]
-      convert hmulv using 1 <;> ring
-    _ = C^2*(δi*δj)^(-1-eta p)*(Di*vi^2)^(eta p)*
-        (Dj*vj^2)^(eta p)*(vi*vj)^(1-2*eta p) := by
-      calc
-        _ = C^2*(δi*δj)^(-1-eta p)*(Di^(eta p)*Dj^(eta p)*vi*vj) := by ring
-        _ = C^2*(δi*δj)^(-1-eta p)*
-            ((Di*vi^2)^(eta p)*(Dj*vj^2)^(eta p)*(vi*vj)^(1-2*eta p)) := by
-          rw [pair_velocity_power_identity hDi.le hDj.le hvi hvj]
-        _ = _ := by ring
-    _ ≤ _ := weighted_pair_absorption hp (sq_pos_of_pos hC) hε (mul_pos hδi hδj)
-      (mul_nonneg hDi.le (sq_nonneg vi)) (mul_nonneg hDj.le (sq_nonneg vj))
-      (mul_pos hvi hvj).le
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointAngular
+import OAI.Probability.SLE.Multipoint.MultipointYoung
+
+namespace OAI
+
+/-! The Brownian cross coefficient in the regularized multipoint product,
+including the singular powers when `p < 1`. -/
+
+noncomputable section
+
+namespace SLEExactGauge.Multipoint
+
+def angularCoefficientConstant (p : ℝ) : ℝ := (p+q p)*3^(eta p)
+
+lemma angularCoefficientConstant_pos {p : ℝ} (hp : 0 < p) :
+    0 < angularCoefficientConstant p := by
+  unfold angularCoefficientConstant
+  exact mul_pos (add_pos hp (q_pos hp)) (Real.rpow_pos_of_pos (by norm_num) _)
+
+lemma velocity_power_identity {D v e : ℝ} (hD : 0 ≤ D) (hv : 0 < v) :
+    (D*v^2)^e * v^(1-2*e) = D^e*v := by
+  rw [Real.mul_rpow hD (sq_nonneg v), ← Real.rpow_natCast v 2,
+    ← Real.rpow_mul hv.le, mul_assoc, ← Real.rpow_add hv]
+  norm_num
+
+lemma pair_velocity_power_identity {Di Dj vi vj e : ℝ}
+    (hDi : 0 ≤ Di) (hDj : 0 ≤ Dj) (hvi : 0 < vi) (hvj : 0 < vj) :
+    (Di*vi^2)^e*(Dj*vj^2)^e*(vi*vj)^(1-2*e) = Di^e*Dj^e*vi*vj := by
+  rw [Real.mul_rpow hvi.le hvj.le]
+  calc
+    _ = ((Di*vi^2)^e*vi^(1-2*e))*((Dj*vj^2)^e*vj^(1-2*e)) := by ring
+    _ = _ := by rw [velocity_power_identity hDi hvi, velocity_power_identity hDj hvj]; ring
+
+/-- The magnitude of one Brownian cross term is absorbed at its two
+endpoints, leaving exactly the inverse-regulator interaction cost. -/
+theorem angular_pair_absorption {p δi δj θi θj vi vj ε : ℝ}
+    (hp : 0 < p) (hδi : 0 < δi) (hδi1 : δi ≤ 1)
+    (hδj : 0 < δj) (hδj1 : δj ≤ 1)
+    (hθi : 0 < Real.sin θi) (hθj : 0 < Real.sin θj)
+    (hvi : 0 < vi) (hvj : 0 < vj) (hε : 0 < ε) :
+    (|angularFactorFirst p δi θi| / angularFactor p δi θi * vi) *
+      (|angularFactorFirst p δj θj| / angularFactor p δj θj * vj) ≤
+    ε*(damping p δi θi*vi^2+damping p δj θj*vj^2) +
+      youngConstant p (angularCoefficientConstant p ^ 2) ε *
+        (δi*δj)^(-exponent p)*(vi*vj) := by
+  let Di := damping p δi θi
+  let Dj := damping p δj θj
+  let C := angularCoefficientConstant p
+  have hDi : 0 < Di := mul_pos (div_pos hδi (angularFactor_pos hδi hθi))
+    (Real.rpow_pos_of_pos hθi _)
+  have hDj : 0 < Dj := mul_pos (div_pos hδj (angularFactor_pos hδj hθj))
+    (Real.rpow_pos_of_pos hθj _)
+  have hC : 0 < C := angularCoefficientConstant_pos hp
+  have hi := normalized_angularFactorFirst_bound hp hδi hδi1 hθi
+  have hj := normalized_angularFactorFirst_bound hp hδj hδj1 hθj
+  change _ ≤ C*δi^(-1-eta p)*Di^(eta p) at hi
+  change _ ≤ C*δj^(-1-eta p)*Dj^(eta p) at hj
+  have hmul := mul_le_mul hi hj (div_nonneg (abs_nonneg _) (angularFactor_pos hδj hθj).le)
+    (by positivity : 0 ≤ C*δi^(-1-eta p)*Di^(eta p))
+  have hmulv := mul_le_mul_of_nonneg_right hmul (mul_pos hvi hvj).le
+  calc
+    _ ≤ C^2*(δi*δj)^(-1-eta p)*Di^(eta p)*Dj^(eta p)*vi*vj := by
+      rw [Real.mul_rpow hδi.le hδj.le]
+      convert hmulv using 1 <;> ring
+    _ = C^2*(δi*δj)^(-1-eta p)*(Di*vi^2)^(eta p)*
+        (Dj*vj^2)^(eta p)*(vi*vj)^(1-2*eta p) := by
+      calc
+        _ = C^2*(δi*δj)^(-1-eta p)*(Di^(eta p)*Dj^(eta p)*vi*vj) := by ring
+        _ = C^2*(δi*δj)^(-1-eta p)*
+            ((Di*vi^2)^(eta p)*(Dj*vj^2)^(eta p)*(vi*vj)^(1-2*eta p)) := by
+          rw [pair_velocity_power_identity hDi.le hDj.le hvi hvj]
+        _ = _ := by ring
+    _ ≤ _ := weighted_pair_absorption hp (sq_pos_of_pos hC) hε (mul_pos hδi hδj)
+      (mul_nonneg hDi.le (sq_nonneg vi)) (mul_nonneg hDj.le (sq_nonneg vj))
+      (mul_pos hvi hvj).le
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointBudgetMonotone.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointBudgetMonotone.lean
index 376d7f67c9d026a73273467d2b1b645e6f7d892a..432f48100890b9623407ec684b45d77099e2a522 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointBudgetMonotone.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointBudgetMonotone.lean
@@ -1,58 +1,58 @@
-import OAI.Probability.SLE.Multipoint.MultipointGlobalDiscount
-import OAI.Probability.SLE.Multipoint.MultipointInitialParameters
-
-namespace OAI
-
-/-! The fixed-order constant remains valid after discarding constraints
-whose targets already exceed their caps. -/
-
-noncomputable section
-open Set
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-lemma allPatternYoungConstant_mono {ι : Type*} [DecidableEq ι] {κ : ℝ}
-    (hκ : 0 < κ) (hκ8 : κ < 8) {I J : Finset ι} (hIJ : I ⊆ J) :
-    allPatternYoungConstant κ I ≤ allPatternYoungConstant κ J := by
-  apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.powerset_mono.mpr hIJ)
-  exact fun s _ _ => (patternYoungConstant_pos hκ hκ8 s).le
-
-lemma allPatternYoungConstant_map {ι ν : Type*} [DecidableEq ι] [DecidableEq ν]
-    (κ : ℝ) (f : ι ↪ ν) (I : Finset ι) :
-    allPatternYoungConstant κ (I.map f) = allPatternYoungConstant κ I := by
-  classical
-  unfold allPatternYoungConstant
-  rw [Finset.map_eq_image,Finset.powerset_image,Finset.sum_image]
-  · apply Finset.sum_congr rfl
-    intro s _
-    simp only [patternYoungConstant,Finset.card_image_of_injective s f.injective]
-  · exact Finset.image_injOn_powerset_of_injOn (fun _ _ _ _ h => f.injective h)
-
-lemma uniformDiscountBudget_map {ι ν : Type*} [DecidableEq ι] [DecidableEq ν]
-    (κ β : ℝ) (f : ι ↪ ν) (I : Finset ι) :
-    uniformDiscountBudget κ β (I.map f) = uniformDiscountBudget κ β I := by
-  simp only [uniformDiscountBudget,allPatternYoungConstant_map,Finset.offDiag_card,Finset.card_map]
-
-lemma uniformDiscountBudget_mono {ι : Type*} [DecidableEq ι] {κ : ℝ}
-    (hκ : 0 < κ) (hκ8 : κ < 8) {I J : Finset ι} (hIJ : I ⊆ J) :
-    uniformDiscountBudget κ (sourceBeta κ) I ≤ uniformDiscountBudget κ (sourceBeta κ) J := by
-  obtain ⟨_,hβ,_,_,hdecay⟩ := source_parameters hκ hκ8
-  have hpair := (interactionBudget_pos (by norm_num : (0 : ℝ) < 2) hdecay).le
-  have hcard : (I.card : ℝ) ≤ J.card := by exact_mod_cast Finset.card_le_card hIJ
-  have hoff : (I.offDiag.card : ℝ) ≤ J.offDiag.card := by
-    exact_mod_cast Finset.card_le_card (Finset.offDiag_mono hIJ)
-  unfold uniformDiscountBudget
-  apply mul_le_mul_of_nonneg_left _ hκ.le
-  apply add_le_add
-  · exact mul_le_mul_of_nonneg_left
-      (mul_le_mul_of_nonneg_right hcard (by positivity)) (singleDriftConstant_pos hκ hκ8).le
-  · exact mul_le_mul (allPatternYoungConstant_mono hκ hκ8 hIJ)
-      (mul_le_mul_of_nonneg_right hoff hpair) (mul_nonneg (Nat.cast_nonneg _) hpair)
-      (allPatternYoungConstant_pos hκ hκ8 J).le
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointGlobalDiscount
+import OAI.Probability.SLE.Multipoint.MultipointInitialParameters
+
+namespace OAI
+
+/-! The fixed-order constant remains valid after discarding constraints
+whose targets already exceed their caps. -/
+
+noncomputable section
+open Set
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+lemma allPatternYoungConstant_mono {ι : Type*} [DecidableEq ι] {κ : ℝ}
+    (hκ : 0 < κ) (hκ8 : κ < 8) {I J : Finset ι} (hIJ : I ⊆ J) :
+    allPatternYoungConstant κ I ≤ allPatternYoungConstant κ J := by
+  apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.powerset_mono.mpr hIJ)
+  exact fun s _ _ => (patternYoungConstant_pos hκ hκ8 s).le
+
+lemma allPatternYoungConstant_map {ι ν : Type*} [DecidableEq ι] [DecidableEq ν]
+    (κ : ℝ) (f : ι ↪ ν) (I : Finset ι) :
+    allPatternYoungConstant κ (I.map f) = allPatternYoungConstant κ I := by
+  classical
+  unfold allPatternYoungConstant
+  rw [Finset.map_eq_image,Finset.powerset_image,Finset.sum_image]
+  · apply Finset.sum_congr rfl
+    intro s _
+    simp only [patternYoungConstant,Finset.card_image_of_injective s f.injective]
+  · exact Finset.image_injOn_powerset_of_injOn (fun _ _ _ _ h => f.injective h)
+
+lemma uniformDiscountBudget_map {ι ν : Type*} [DecidableEq ι] [DecidableEq ν]
+    (κ β : ℝ) (f : ι ↪ ν) (I : Finset ι) :
+    uniformDiscountBudget κ β (I.map f) = uniformDiscountBudget κ β I := by
+  simp only [uniformDiscountBudget,allPatternYoungConstant_map,Finset.offDiag_card,Finset.card_map]
+
+lemma uniformDiscountBudget_mono {ι : Type*} [DecidableEq ι] {κ : ℝ}
+    (hκ : 0 < κ) (hκ8 : κ < 8) {I J : Finset ι} (hIJ : I ⊆ J) :
+    uniformDiscountBudget κ (sourceBeta κ) I ≤ uniformDiscountBudget κ (sourceBeta κ) J := by
+  obtain ⟨_,hβ,_,_,hdecay⟩ := source_parameters hκ hκ8
+  have hpair := (interactionBudget_pos (by norm_num : (0 : ℝ) < 2) hdecay).le
+  have hcard : (I.card : ℝ) ≤ J.card := by exact_mod_cast Finset.card_le_card hIJ
+  have hoff : (I.offDiag.card : ℝ) ≤ J.offDiag.card := by
+    exact_mod_cast Finset.card_le_card (Finset.offDiag_mono hIJ)
+  unfold uniformDiscountBudget
+  apply mul_le_mul_of_nonneg_left _ hκ.le
+  apply add_le_add
+  · exact mul_le_mul_of_nonneg_left
+      (mul_le_mul_of_nonneg_right hcard (by positivity)) (singleDriftConstant_pos hκ hκ8).le
+  · exact mul_le_mul (allPatternYoungConstant_mono hκ hκ8 hIJ)
+      (mul_le_mul_of_nonneg_right hoff hpair) (mul_nonneg (Nat.cast_nonneg _) hpair)
+      (allPatternYoungConstant_pos hκ hκ8 J).le
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCappedKernel.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCappedKernel.lean
index d933bbae846970151ae7309c973443da95d563bd..13fc23e71260b5502318a748ac302cf06e717b48 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCappedKernel.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCappedKernel.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Multipoint.MultipointKernelGraph
-import OAI.Probability.SLE.Multipoint.MultipointTupleNull
-
-namespace OAI
-
-/-! The exact fixed-cap, nearest-neighbor distance kernel from
-`mp:kernel-integral`, and its expansion into finite graph terms. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-def capEntry {ι : Type*} [DecidableEq ι] (b : ℝ) (x : ι → ℂ) (i : ι) : Option ι → ℝ
-  | none => b
-  | some j => if j=i then b else dist (x i) (x j)
-
-def cappedDistance {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (b : ℝ) (x : ι → ℂ) (i : ι) : ℝ :=
-  Finset.univ.inf' Finset.univ_nonempty (capEntry b x i)
-
-theorem cap_pos {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {b : ℝ} (hb : 0 < b) {x : ι → ℂ} (hx : Function.Injective x) (i : ι) :
-    0 < cappedDistance b x i := by
-  obtain ⟨o,_,ho⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (capEntry b x i)
-  change 0 < Finset.univ.inf' Finset.univ_nonempty (capEntry b x i)
-  rw [ho]
-  cases o with
-  | none => exact hb
-  | some j =>
-    dsimp only [capEntry]
-    split_ifs with hji
-    · exact hb
-    · exact dist_pos.mpr (fun h => hji (hx h.symm))
-
-theorem cap_le_fixed {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (b : ℝ) (x : ι → ℂ) (i : ι) : cappedDistance b x i ≤ b := by
-  exact Finset.inf'_le (capEntry b x i) (Finset.mem_univ none)
-
-theorem cap_le_dist {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (b : ℝ) (x : ι → ℂ) (i j : ι) (hij : i≠j) :
-    cappedDistance b x i ≤ dist (x i) (x j) := by
-  change Finset.univ.inf' Finset.univ_nonempty (capEntry b x i) ≤ _
-  have h := Finset.inf'_le (capEntry b x i) (Finset.mem_univ (some j))
-  simpa only [capEntry,ite_eq_right hij.symm] using h
-
-theorem continuous_cappedDistance {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (b : ℝ) (i : ι) : Continuous (fun x : ι → ℂ => cappedDistance b x i) := by
-  apply Continuous.finset_inf'_apply Finset.univ_nonempty
-  intro o _
-  cases o with
-  | none => exact continuous_const
-  | some j =>
-    dsimp only [capEntry]
-    split_ifs
-    · exact continuous_const
-    · fun_prop
-
-def cappedDistanceKernel {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) : ℝ≥0∞ :=
-  ∏ i, ENNReal.ofReal (cappedDistance b x i ^ (-α))
-
-lemma measurable_cappedDistanceKernel {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) : Measurable (cappedDistanceKernel (ι := ι) α b) := by
-  apply Finset.measurable_prod
-  intro i _
-  exact (((continuous_cappedDistance b i).measurable.pow_const (-α)).ennreal_ofReal)
-
-def rowChoice {ι : Type*} [DecidableEq ι] (α b : ℝ) (x : ι → ℂ) (i : ι) : Option ι → ℝ≥0∞
-  | none => ENNReal.ofReal (b^(-α))
-  | some j => if j=i then 0 else distanceKernel α (x i) (x j)
-
-lemma cap_reciprocal_le_row {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) (i : ι) :
-    ENNReal.ofReal (cappedDistance b x i ^ (-α)) ≤ ∑ o, rowChoice α b x i o := by
-  obtain ⟨o,_,ho⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (capEntry b x i)
-  change ENNReal.ofReal ((Finset.univ.inf' Finset.univ_nonempty (capEntry b x i))^(-α)) ≤ _
-  rw [ho]
-  cases o with
-  | none =>
-    exact Finset.single_le_sum (f := rowChoice α b x i)
-      (fun _ _ => bot_le) (Finset.mem_univ none)
-  | some j =>
-    by_cases hji : j=i
-    · simpa only [capEntry,ite_eq_left hji,rowChoice] using
-        (Finset.single_le_sum (f := rowChoice α b x i) (fun _ _ => bot_le) (Finset.mem_univ none))
-    · simpa only [capEntry,ite_eq_right hji,rowChoice,distanceKernel] using
-        (Finset.single_le_sum (f := rowChoice α b x i) (fun _ _ => bot_le) (Finset.mem_univ (some j)))
-
-lemma cappedDistanceKernel_le_expansion {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) :
-    cappedDistanceKernel α b x ≤ ∑ f : ι → Option ι, ∏ i, rowChoice α b x i (f i) := by
-  rw [← Fintype.prod_sum]
-  exact Finset.prod_le_prod (fun i _ => cap_reciprocal_le_row α b x i)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointKernelGraph
+import OAI.Probability.SLE.Multipoint.MultipointTupleNull
+
+namespace OAI
+
+/-! The exact fixed-cap, nearest-neighbor distance kernel from
+`mp:kernel-integral`, and its expansion into finite graph terms. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+def capEntry {ι : Type*} [DecidableEq ι] (b : ℝ) (x : ι → ℂ) (i : ι) : Option ι → ℝ
+  | none => b
+  | some j => if j=i then b else dist (x i) (x j)
+
+def cappedDistance {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (b : ℝ) (x : ι → ℂ) (i : ι) : ℝ :=
+  Finset.univ.inf' Finset.univ_nonempty (capEntry b x i)
+
+theorem cap_pos {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {b : ℝ} (hb : 0 < b) {x : ι → ℂ} (hx : Function.Injective x) (i : ι) :
+    0 < cappedDistance b x i := by
+  obtain ⟨o,_,ho⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (capEntry b x i)
+  change 0 < Finset.univ.inf' Finset.univ_nonempty (capEntry b x i)
+  rw [ho]
+  cases o with
+  | none => exact hb
+  | some j =>
+    dsimp only [capEntry]
+    split_ifs with hji
+    · exact hb
+    · exact dist_pos.mpr (fun h => hji (hx h.symm))
+
+theorem cap_le_fixed {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (b : ℝ) (x : ι → ℂ) (i : ι) : cappedDistance b x i ≤ b := by
+  exact Finset.inf'_le (capEntry b x i) (Finset.mem_univ none)
+
+theorem cap_le_dist {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (b : ℝ) (x : ι → ℂ) (i j : ι) (hij : i≠j) :
+    cappedDistance b x i ≤ dist (x i) (x j) := by
+  change Finset.univ.inf' Finset.univ_nonempty (capEntry b x i) ≤ _
+  have h := Finset.inf'_le (capEntry b x i) (Finset.mem_univ (some j))
+  simpa only [capEntry,ite_eq_right hij.symm] using h
+
+theorem continuous_cappedDistance {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (b : ℝ) (i : ι) : Continuous (fun x : ι → ℂ => cappedDistance b x i) := by
+  apply Continuous.finset_inf'_apply Finset.univ_nonempty
+  intro o _
+  cases o with
+  | none => exact continuous_const
+  | some j =>
+    dsimp only [capEntry]
+    split_ifs
+    · exact continuous_const
+    · fun_prop
+
+def cappedDistanceKernel {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) : ℝ≥0∞ :=
+  ∏ i, ENNReal.ofReal (cappedDistance b x i ^ (-α))
+
+lemma measurable_cappedDistanceKernel {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) : Measurable (cappedDistanceKernel (ι := ι) α b) := by
+  apply Finset.measurable_prod
+  intro i _
+  exact (((continuous_cappedDistance b i).measurable.pow_const (-α)).ennreal_ofReal)
+
+def rowChoice {ι : Type*} [DecidableEq ι] (α b : ℝ) (x : ι → ℂ) (i : ι) : Option ι → ℝ≥0∞
+  | none => ENNReal.ofReal (b^(-α))
+  | some j => if j=i then 0 else distanceKernel α (x i) (x j)
+
+lemma cap_reciprocal_le_row {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) (i : ι) :
+    ENNReal.ofReal (cappedDistance b x i ^ (-α)) ≤ ∑ o, rowChoice α b x i o := by
+  obtain ⟨o,_,ho⟩ := Finset.exists_mem_eq_inf' Finset.univ_nonempty (capEntry b x i)
+  change ENNReal.ofReal ((Finset.univ.inf' Finset.univ_nonempty (capEntry b x i))^(-α)) ≤ _
+  rw [ho]
+  cases o with
+  | none =>
+    exact Finset.single_le_sum (f := rowChoice α b x i)
+      (fun _ _ => bot_le) (Finset.mem_univ none)
+  | some j =>
+    by_cases hji : j=i
+    · simpa only [capEntry,ite_eq_left hji,rowChoice] using
+        (Finset.single_le_sum (f := rowChoice α b x i) (fun _ _ => bot_le) (Finset.mem_univ none))
+    · simpa only [capEntry,ite_eq_right hji,rowChoice,distanceKernel] using
+        (Finset.single_le_sum (f := rowChoice α b x i) (fun _ _ => bot_le) (Finset.mem_univ (some j)))
+
+lemma cappedDistanceKernel_le_expansion {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) :
+    cappedDistanceKernel α b x ≤ ∑ f : ι → Option ι, ∏ i, rowChoice α b x i (f i) := by
+  rw [← Fintype.prod_sum]
+  exact Finset.prod_le_prod (fun i _ => cap_reciprocal_le_row α b x i)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCenteredWeights.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCenteredWeights.lean
index ce54f8c6f71e96e6d789ca3b255aa9e97aec4914..8b5b3d37bcfa0c1c9333563976bb6d58984bc8dc 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCenteredWeights.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCenteredWeights.lean
@@ -1,192 +1,192 @@
-import OAI.Probability.SLE.Mass.LevelApproximants
-
-namespace OAI
-
-/-! A fixed additional approach event costs its target radius to the
-codimension power, uniformly in the levels of the integrated weights. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma lintegral_indicator_ofReal_prod_le_event
-    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
-    (P : Measure Ω) (E₀ E : Set Ω) (Y : ι → Ω → ℝ) (c : ι → ℝ)
-    (hY0 : ∀ i ω, 0 ≤ Y i ω)
-    (hY : ∀ ω ∈ E, ∀ i, Y i ω ≤ c i)
-    (hzero : ∀ ω, ω ∈ E₀ → ω ∉ E → ∃ i, Y i ω = 0) :
-    (∫⁻ ω, E₀.indicator (fun ω => ENNReal.ofReal (∏ i, Y i ω)) ω ∂P) ≤
-      ENNReal.ofReal (∏ i, c i) * P E := by
-  classical
-  apply (lintegral_mono (fun ω => ?_)).trans (lintegral_indicator_const_le E _)
-  by_cases hω₀ : ω ∈ E₀
-  · rw [indicator_of_mem hω₀]
-    by_cases hω : ω ∈ E
-    · rw [indicator_of_mem hω]
-      exact ENNReal.ofReal_le_ofReal (Finset.prod_le_prod₀ (fun i _ => hY0 i ω)
-        (fun i _ => hY ω hω i))
-    · obtain ⟨i,hi⟩ := hzero ω hω₀ hω
-      rw [Finset.prod_eq_zero (Finset.mem_univ i) hi,ENNReal.ofReal_zero]
-      exact bot_le
-  · rw [indicator_of_notMem hω₀]
-    exact bot_le
-
-theorem stoppedGreen_centered_product_expectation_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ)
-    (z : Fin (m+1) → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (hne : Function.Injective z) (b e : Fin (m+1) → ℝ)
-    (he : ∀ i, 0 < e i) (hb : ∀ i, 0 < b i)
-    (hbR : ∀ i, b i ≤ 2*(z i).im)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j)) :
-    (∫⁻ ω, {ω | ∃ t : ℝ≥0, z 0 ∈ survivingDomain (γ ω) t ∧
-        physicalConformalRadius (fun z : ℂ => z) (G ω) t (z 0) ≤ e 0}.indicator
-      (fun ω => ENNReal.ofReal (∏ i : Fin m, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-        (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ)))) ω ∂P) ≤
-      ENNReal.ofReal ((3 : ℝ)^(m+1) *
-        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset (Fin (m+1)))) *
-        (e 0)^(codimension κ) * ∏ i, (b i)^(-codimension κ)) := by
-  classical
-  let E₀ := {ω | ∃ t : ℝ≥0, z 0 ∈ survivingDomain (γ ω) t ∧
-    physicalConformalRadius (fun z : ℂ => z) (G ω) t (z 0) ≤ e 0}
-  let E := {ω | ω ∈ E₀ ∧ ∀ i : Fin m,
-    radiusLevelTime (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ)) < ∞}
-  let Y := fun (i : Fin m) ω => stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-    (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ))
-  let W := ∏ i : Fin m, (e i.succ)^(-codimension κ)
-  let C := (3 : ℝ)^(m+1) *
-    Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset (Fin (m+1))))
-  have hW : 0 ≤ W := Finset.prod_nonneg (fun i _ => Real.rpow_nonneg (he i.succ).le _)
-  have hC : 0 ≤ C := by dsimp [C]; positivity
-  have hint := lintegral_indicator_ofReal_prod_le_event P E₀ E Y
-    (fun i => (e i.succ)^(-codimension κ))
-    (fun i ω => stoppedGreen_nonneg (hw ω) κ _ _)
-    (fun ω _ i => stoppedGreen_level_bound (continuous_const.mul (hBc ω))
-      (hw ω) hκ hκ8 (z i.succ) (hz i.succ) (e i.succ) (he i.succ)) (fun ω hω hn => by
-      have hn' : ¬ ∀ i : Fin m,
-          radiusLevelTime (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ)) < ∞ :=
-        fun hi => hn ⟨hω,hi⟩
-      obtain ⟨i,hi⟩ := not_forall.mp hn'
-      exact ⟨i,by simp only [Y,stoppedGreen,ite_eq_right hi]⟩)
-  have hprob := capped_physical_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8
-    γ G F hw hgrowth (fun z : ℂ => z) differentiableOn_id (fun _ _ _ _ h => h)
-    z hz hne (fun i => 2*(z i).im) b e he hb
-    (fun i => by simpa only [deriv_id'',norm_one,mul_one] using hbR i)
-    (fun i => by simp only [deriv_id'',norm_one,mul_one]) hsep
-  have hsub : E ⊆ {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius (fun z : ℂ => z) (G ω) t (z i) ≤ e i} := by
-    intro ω hω
-    refine Fin.cases hω.1 (fun i => ?_)
-    obtain ⟨t,ht,hR⟩ := (physical_radius_eq_hit_iff_finite_level
-      (continuous_const.mul (hBc ω)) (hw ω) (fun z : ℂ => z) (z i.succ)
-      (2*(z i.succ).im) (e i.succ) (mul_pos (by norm_num) (hz i.succ)) (he i.succ)
-      (by simp only [deriv_id'',norm_one,mul_one])).mpr (hω.2 i)
-    exact ⟨t,ht,hR.le⟩
-  have hα : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  have hp : W * (∏ i, min 1 ((e i/b i)^(1-κ/8))) ≤
-      (e 0)^(codimension κ) * ∏ i, (b i)^(-codimension κ) := by
-    rw [← hα,Fin.prod_univ_succ]
-    have hroot : min 1 ((e 0/b 0)^codimension κ) ≤
-        (e 0)^codimension κ * (b 0)^(-codimension κ) := by
-      rw [Real.rpow_neg (hb 0).le,← div_eq_mul_inv,← Real.div_rpow (he 0).le (hb 0).le]
-      exact min_le_right _ _
-    have htail : W * (∏ i : Fin m, min 1 ((e i.succ/b i.succ)^codimension κ)) ≤
-        ∏ i : Fin m, (b i.succ)^(-codimension κ) := by
-      dsimp only [W]
-      rw [← Finset.prod_mul_distrib]
-      apply Finset.prod_le_prod₀
-      · intro i _
-        exact mul_nonneg (Real.rpow_nonneg (he i.succ).le _)
-          (le_min zero_le_one (Real.rpow_nonneg (div_nonneg (he i.succ).le (hb i.succ).le) _))
-      · exact fun i _ => target_weight_capped (hb i.succ) (he i.succ)
-    calc
-      _ = min 1 ((e 0/b 0)^codimension κ) *
-          (W * ∏ i : Fin m, min 1 ((e i.succ/b i.succ)^codimension κ)) := by ring
-      _ ≤ ((e 0)^codimension κ * (b 0)^(-codimension κ)) *
-          ∏ i : Fin m, (b i.succ)^(-codimension κ) :=
-        mul_le_mul hroot htail (mul_nonneg hW (Finset.prod_nonneg (fun i _ =>
-          le_min zero_le_one (Real.rpow_nonneg (div_nonneg (he i.succ).le (hb i.succ).le) _))))
-          (mul_nonneg (Real.rpow_nonneg (he 0).le _) (Real.rpow_nonneg (hb 0).le _))
-      _ = _ := by rw [Fin.prod_univ_succ]; ring
-  calc
-    _ ≤ ENNReal.ofReal W * P E := hint
-    _ ≤ ENNReal.ofReal W * ENNReal.ofReal (C * ∏ i, min 1 ((e i/b i)^(1-κ/8))) :=
-      mul_le_mul' le_rfl ((measure_mono hsub).trans (by simpa only [Fintype.card_fin] using hprob))
-    _ = ENNReal.ofReal (C * (W * ∏ i, min 1 ((e i/b i)^(1-κ/8)))) := by
-      rw [← ENNReal.ofReal_mul hW]
-      congr 1
-      ring
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (by
-      simpa only [C,mul_assoc] using (mul_le_mul_of_nonneg_left hp hC))
-
-lemma measurableSet_physical_radius_hit
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (e : ℝ) (he : 0 < e) (heR : e ≤ 2*z.im) :
-    MeasurableSet {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius (fun z : ℂ => z) (G ω) t z ≤ e} := by
-  have hu : 0 ≤ Real.log (2*z.im/e) :=
-    Real.log_nonneg ((one_le_div he).mpr heR)
-  have hj : Measurable (fun q : Ω × ℂ =>
-      radiusLevelTime (γ q.1) (G q.1) q.2 (Real.log (2*z.im/e))) :=
-    actual_radiusLevel_joint_measurable hm hBc κ (Real.log (2*z.im/e)) hu γ G F hw
-  have hp : Measurable (fun ω : Ω => (ω,z)) :=
-    (measurable_id : Measurable (fun ω : Ω => ω)).prodMk
-      (measurable_const : Measurable (fun _ : Ω => z))
-  have hc := hj.comp hp
-  have hmτ : Measurable (fun ω => radiusLevelTime (γ ω) (G ω) z (Real.log (2*z.im/e))) := by
-    simpa only [Function.comp_def] using hc
-  suffices hE : {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius (fun z : ℂ => z) (G ω) t z ≤ e} =
-      {ω | radiusLevelTime (γ ω) (G ω) z (Real.log (2*z.im/e)) < (⊤ : ℝ≥0∞)} by
-    rw [hE]
-    exact measurableSet_lt hmτ measurable_const
-  ext ω
-  exact physical_radius_hit_iff_finite_level (continuous_const.mul (hBc ω)) (hw ω)
-    (fun z : ℂ => z) z (2*z.im) e (mul_pos (by norm_num) hz) he heR
-    (by simp only [deriv_id'',norm_one,mul_one])
-
-theorem stoppedGreen_centered_common_level_bound
-    {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ)
-    (z : Fin (m+1) → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (hne : Function.Injective z) (b : Fin (m+1) → ℝ)
-    (hb : ∀ i, 0 < b i) (hbR : ∀ i, b i ≤ 2*(z i).im)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j))
-    (e₀ : ℝ) (he₀ : 0 < e₀) (u : ℝ) :
-    (∫⁻ ω, {ω | ∃ t : ℝ≥0, z 0 ∈ survivingDomain (γ ω) t ∧
-        physicalConformalRadius (fun z : ℂ => z) (G ω) t (z 0) ≤ e₀}.indicator
-      (fun ω => ENNReal.ofReal (∏ i : Fin m, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-        (γ ω) (G ω) (z i.succ) u)) ω ∂P) ≤
-      ENNReal.ofReal (multipointConstant κ (m+1) *
-        e₀^(codimension κ) * ∏ i, (b i)^(-codimension κ)) := by
-  let e : Fin (m+1) → ℝ := Fin.cons e₀ (fun i => 2*(z i.succ).im*Real.exp (-u))
-  have he : ∀ i, 0 < e i := by
-    refine Fin.cases he₀ (fun i => ?_)
-    exact mul_pos (mul_pos (by norm_num) (hz i.succ)) (Real.exp_pos _)
-  have h := stoppedGreen_centered_product_expectation_bound hB hm hBm hBc hB0 hκ hκ8
-    γ G F hw hgrowth m z hz hne b e he hb hbR hsep
-  simpa only [e,Fin.cons_zero,Fin.cons_succ,common_level_log_target _ (hz _) u,
-    multipointConstant] using h
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.LevelApproximants
+
+namespace OAI
+
+/-! A fixed additional approach event costs its target radius to the
+codimension power, uniformly in the levels of the integrated weights. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma lintegral_indicator_ofReal_prod_le_event
+    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
+    (P : Measure Ω) (E₀ E : Set Ω) (Y : ι → Ω → ℝ) (c : ι → ℝ)
+    (hY0 : ∀ i ω, 0 ≤ Y i ω)
+    (hY : ∀ ω ∈ E, ∀ i, Y i ω ≤ c i)
+    (hzero : ∀ ω, ω ∈ E₀ → ω ∉ E → ∃ i, Y i ω = 0) :
+    (∫⁻ ω, E₀.indicator (fun ω => ENNReal.ofReal (∏ i, Y i ω)) ω ∂P) ≤
+      ENNReal.ofReal (∏ i, c i) * P E := by
+  classical
+  apply (lintegral_mono (fun ω => ?_)).trans (lintegral_indicator_const_le E _)
+  by_cases hω₀ : ω ∈ E₀
+  · rw [indicator_of_mem hω₀]
+    by_cases hω : ω ∈ E
+    · rw [indicator_of_mem hω]
+      exact ENNReal.ofReal_le_ofReal (Finset.prod_le_prod₀ (fun i _ => hY0 i ω)
+        (fun i _ => hY ω hω i))
+    · obtain ⟨i,hi⟩ := hzero ω hω₀ hω
+      rw [Finset.prod_eq_zero (Finset.mem_univ i) hi,ENNReal.ofReal_zero]
+      exact bot_le
+  · rw [indicator_of_notMem hω₀]
+    exact bot_le
+
+theorem stoppedGreen_centered_product_expectation_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ)
+    (z : Fin (m+1) → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (hne : Function.Injective z) (b e : Fin (m+1) → ℝ)
+    (he : ∀ i, 0 < e i) (hb : ∀ i, 0 < b i)
+    (hbR : ∀ i, b i ≤ 2*(z i).im)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j)) :
+    (∫⁻ ω, {ω | ∃ t : ℝ≥0, z 0 ∈ survivingDomain (γ ω) t ∧
+        physicalConformalRadius (fun z : ℂ => z) (G ω) t (z 0) ≤ e 0}.indicator
+      (fun ω => ENNReal.ofReal (∏ i : Fin m, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+        (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ)))) ω ∂P) ≤
+      ENNReal.ofReal ((3 : ℝ)^(m+1) *
+        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset (Fin (m+1)))) *
+        (e 0)^(codimension κ) * ∏ i, (b i)^(-codimension κ)) := by
+  classical
+  let E₀ := {ω | ∃ t : ℝ≥0, z 0 ∈ survivingDomain (γ ω) t ∧
+    physicalConformalRadius (fun z : ℂ => z) (G ω) t (z 0) ≤ e 0}
+  let E := {ω | ω ∈ E₀ ∧ ∀ i : Fin m,
+    radiusLevelTime (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ)) < ∞}
+  let Y := fun (i : Fin m) ω => stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+    (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ))
+  let W := ∏ i : Fin m, (e i.succ)^(-codimension κ)
+  let C := (3 : ℝ)^(m+1) *
+    Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset (Fin (m+1))))
+  have hW : 0 ≤ W := Finset.prod_nonneg (fun i _ => Real.rpow_nonneg (he i.succ).le _)
+  have hC : 0 ≤ C := by dsimp [C]; positivity
+  have hint := lintegral_indicator_ofReal_prod_le_event P E₀ E Y
+    (fun i => (e i.succ)^(-codimension κ))
+    (fun i ω => stoppedGreen_nonneg (hw ω) κ _ _)
+    (fun ω _ i => stoppedGreen_level_bound (continuous_const.mul (hBc ω))
+      (hw ω) hκ hκ8 (z i.succ) (hz i.succ) (e i.succ) (he i.succ)) (fun ω hω hn => by
+      have hn' : ¬ ∀ i : Fin m,
+          radiusLevelTime (γ ω) (G ω) (z i.succ) (Real.log (2*(z i.succ).im/e i.succ)) < ∞ :=
+        fun hi => hn ⟨hω,hi⟩
+      obtain ⟨i,hi⟩ := not_forall.mp hn'
+      exact ⟨i,by simp only [Y,stoppedGreen,ite_eq_right hi]⟩)
+  have hprob := capped_physical_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8
+    γ G F hw hgrowth (fun z : ℂ => z) differentiableOn_id (fun _ _ _ _ h => h)
+    z hz hne (fun i => 2*(z i).im) b e he hb
+    (fun i => by simpa only [deriv_id'',norm_one,mul_one] using hbR i)
+    (fun i => by simp only [deriv_id'',norm_one,mul_one]) hsep
+  have hsub : E ⊆ {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius (fun z : ℂ => z) (G ω) t (z i) ≤ e i} := by
+    intro ω hω
+    refine Fin.cases hω.1 (fun i => ?_)
+    obtain ⟨t,ht,hR⟩ := (physical_radius_eq_hit_iff_finite_level
+      (continuous_const.mul (hBc ω)) (hw ω) (fun z : ℂ => z) (z i.succ)
+      (2*(z i.succ).im) (e i.succ) (mul_pos (by norm_num) (hz i.succ)) (he i.succ)
+      (by simp only [deriv_id'',norm_one,mul_one])).mpr (hω.2 i)
+    exact ⟨t,ht,hR.le⟩
+  have hα : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  have hp : W * (∏ i, min 1 ((e i/b i)^(1-κ/8))) ≤
+      (e 0)^(codimension κ) * ∏ i, (b i)^(-codimension κ) := by
+    rw [← hα,Fin.prod_univ_succ]
+    have hroot : min 1 ((e 0/b 0)^codimension κ) ≤
+        (e 0)^codimension κ * (b 0)^(-codimension κ) := by
+      rw [Real.rpow_neg (hb 0).le,← div_eq_mul_inv,← Real.div_rpow (he 0).le (hb 0).le]
+      exact min_le_right _ _
+    have htail : W * (∏ i : Fin m, min 1 ((e i.succ/b i.succ)^codimension κ)) ≤
+        ∏ i : Fin m, (b i.succ)^(-codimension κ) := by
+      dsimp only [W]
+      rw [← Finset.prod_mul_distrib]
+      apply Finset.prod_le_prod₀
+      · intro i _
+        exact mul_nonneg (Real.rpow_nonneg (he i.succ).le _)
+          (le_min zero_le_one (Real.rpow_nonneg (div_nonneg (he i.succ).le (hb i.succ).le) _))
+      · exact fun i _ => target_weight_capped (hb i.succ) (he i.succ)
+    calc
+      _ = min 1 ((e 0/b 0)^codimension κ) *
+          (W * ∏ i : Fin m, min 1 ((e i.succ/b i.succ)^codimension κ)) := by ring
+      _ ≤ ((e 0)^codimension κ * (b 0)^(-codimension κ)) *
+          ∏ i : Fin m, (b i.succ)^(-codimension κ) :=
+        mul_le_mul hroot htail (mul_nonneg hW (Finset.prod_nonneg (fun i _ =>
+          le_min zero_le_one (Real.rpow_nonneg (div_nonneg (he i.succ).le (hb i.succ).le) _))))
+          (mul_nonneg (Real.rpow_nonneg (he 0).le _) (Real.rpow_nonneg (hb 0).le _))
+      _ = _ := by rw [Fin.prod_univ_succ]; ring
+  calc
+    _ ≤ ENNReal.ofReal W * P E := hint
+    _ ≤ ENNReal.ofReal W * ENNReal.ofReal (C * ∏ i, min 1 ((e i/b i)^(1-κ/8))) :=
+      mul_le_mul' le_rfl ((measure_mono hsub).trans (by simpa only [Fintype.card_fin] using hprob))
+    _ = ENNReal.ofReal (C * (W * ∏ i, min 1 ((e i/b i)^(1-κ/8)))) := by
+      rw [← ENNReal.ofReal_mul hW]
+      congr 1
+      ring
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (by
+      simpa only [C,mul_assoc] using (mul_le_mul_of_nonneg_left hp hC))
+
+lemma measurableSet_physical_radius_hit
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (e : ℝ) (he : 0 < e) (heR : e ≤ 2*z.im) :
+    MeasurableSet {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius (fun z : ℂ => z) (G ω) t z ≤ e} := by
+  have hu : 0 ≤ Real.log (2*z.im/e) :=
+    Real.log_nonneg ((one_le_div he).mpr heR)
+  have hj : Measurable (fun q : Ω × ℂ =>
+      radiusLevelTime (γ q.1) (G q.1) q.2 (Real.log (2*z.im/e))) :=
+    actual_radiusLevel_joint_measurable hm hBc κ (Real.log (2*z.im/e)) hu γ G F hw
+  have hp : Measurable (fun ω : Ω => (ω,z)) :=
+    (measurable_id : Measurable (fun ω : Ω => ω)).prodMk
+      (measurable_const : Measurable (fun _ : Ω => z))
+  have hc := hj.comp hp
+  have hmτ : Measurable (fun ω => radiusLevelTime (γ ω) (G ω) z (Real.log (2*z.im/e))) := by
+    simpa only [Function.comp_def] using hc
+  suffices hE : {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius (fun z : ℂ => z) (G ω) t z ≤ e} =
+      {ω | radiusLevelTime (γ ω) (G ω) z (Real.log (2*z.im/e)) < (⊤ : ℝ≥0∞)} by
+    rw [hE]
+    exact measurableSet_lt hmτ measurable_const
+  ext ω
+  exact physical_radius_hit_iff_finite_level (continuous_const.mul (hBc ω)) (hw ω)
+    (fun z : ℂ => z) z (2*z.im) e (mul_pos (by norm_num) hz) he heR
+    (by simp only [deriv_id'',norm_one,mul_one])
+
+theorem stoppedGreen_centered_common_level_bound
+    {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ)
+    (z : Fin (m+1) → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (hne : Function.Injective z) (b : Fin (m+1) → ℝ)
+    (hb : ∀ i, 0 < b i) (hbR : ∀ i, b i ≤ 2*(z i).im)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j))
+    (e₀ : ℝ) (he₀ : 0 < e₀) (u : ℝ) :
+    (∫⁻ ω, {ω | ∃ t : ℝ≥0, z 0 ∈ survivingDomain (γ ω) t ∧
+        physicalConformalRadius (fun z : ℂ => z) (G ω) t (z 0) ≤ e₀}.indicator
+      (fun ω => ENNReal.ofReal (∏ i : Fin m, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+        (γ ω) (G ω) (z i.succ) u)) ω ∂P) ≤
+      ENNReal.ofReal (multipointConstant κ (m+1) *
+        e₀^(codimension κ) * ∏ i, (b i)^(-codimension κ)) := by
+  let e : Fin (m+1) → ℝ := Fin.cons e₀ (fun i => 2*(z i.succ).im*Real.exp (-u))
+  have he : ∀ i, 0 < e i := by
+    refine Fin.cases he₀ (fun i => ?_)
+    exact mul_pos (mul_pos (by norm_num) (hz i.succ)) (Real.exp_pos _)
+  have h := stoppedGreen_centered_product_expectation_bound hB hm hBm hBc hB0 hκ hκ8
+    γ G F hw hgrowth m z hz hne b e he hb hbR hsep
+  simpa only [e,Fin.cons_zero,Fin.cons_succ,common_level_log_target _ (hz _) u,
+    multipointConstant] using h
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointClippedState.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointClippedState.lean
index 53c37006c6bd5f34f3ca7077678e8049d2998714..254af3f89b68c00a6ba558264990a50d9b8fcd66 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointClippedState.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointClippedState.lean
@@ -1,109 +1,109 @@
-import OAI.Probability.SLE.BoundaryDensity
-
-namespace OAI
-
-/-! Simultaneous bounded adapted logarithmic Loewner coordinates. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma measurable_clipped_multipoint_velocity
-    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
-    (Q : ℝ → Ω → (ι → ℂ × ℝ)) (U : ℝ → Ω → ℝ)
-    (hQ : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hU : Measurable (fun p : ℝ × Ω => U p.1 p.2))
-    (lo hi : ℝ) (hle : lo ≤ hi) :
-    Measurable (fun p : ℝ × Ω => fun i =>
-      clippedLogLoewnerField lo hi hle (U p.1 p.2) (Q p.1 p.2 i)) := by
-  apply Measurable.of_eval
-  intro i
-  have hqi : Measurable (fun p : ℝ × Ω => Q p.1 p.2 i) :=
-    (measurable_pi_apply i).comp hQ
-  have hf : Measurable (fun p : ℝ × (ℂ × ℝ) =>
-      clippedLogLoewnerField lo hi hle p.1 p.2) :=
-    (continuous_clippedLogLoewnerField lo hi hle).measurable
-  have hp : Measurable (fun p : ℝ × Ω => (U p.1 p.2,Q p.1 p.2 i)) := hU.prodMk hqi
-  change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-    (fun p : ℝ × Ω => (U p.1 p.2,Q p.1 p.2 i)))
-  exact hf.comp hp
-
-theorem exists_clipped_multipoint_state
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ι → ℂ × ℝ) :
-    ∃ (L : ℝ≥0) (Q V : ℝ → Ω → (ι → ℂ × ℝ)),
-      Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧
-      Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun t : ℝ≥0 => Q t) ∧
-      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
-      Measurable (fun p : ℝ × Ω => V p.1 p.2) ∧
-      (∀ t ω, ‖V t ω‖ ≤ L) ∧
-      (∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t) ∧
-      (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun u => Q u ω) (V t ω) (Ici 0) t) ∧
-      (∀ ω, Q 0 ω = q₀) ∧
-      (∀ t ω i, V t ω i = clippedLogLoewnerField lo hi hle
-        (Real.sqrt κ*B t.toNNReal ω) (Q t ω i)) := by
-  classical
-  let c (i : ι) : ClippedLogSolution B hm κ lo hi hle (q₀ i) :=
-    Classical.choice (exists_clippedLogSolution B hm hBm hBc κ lo hi hle (q₀ i))
-  clear_value c
-  obtain ⟨L₀, K₀, hbound, _⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let L : ℝ≥0 := L₀ + ∑ i, (c i).L
-  let Q := fun t ω i => (c i).Q t ω
-  let V := fun t ω i => clippedLogLoewnerField lo hi hle
-    (Real.sqrt κ*B t.toNNReal ω) (Q t ω i)
-  have hLi (i : ι) : (c i).L ≤ L := by
-    change (c i).L ≤ L₀+∑ j, (c j).L
-    exact (Finset.single_le_sum (fun j _ => (zero_le : 0 ≤ (c j).L)) (Finset.mem_univ i)).trans
-      (le_add_left le_rfl)
-  have hL₀ : L₀ ≤ L := le_add_of_nonneg_right (by positivity)
-  have hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2) :=
-    Measurable.of_eval (fun i => (c i).measurable)
-  have hQadapt (t : ℝ≥0) (i : ι) :
-      @Measurable Ω (ℂ × ℝ)
-        ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t) _
-        (fun ω => Q t ω i) := (c i).adapted t
-  have hUm : Measurable (fun p : ℝ × Ω => Real.sqrt κ*B p.1.toNNReal p.2) :=
-    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)).const_mul _
-  refine ⟨L, Q, V, hQm, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
-  · intro t
-    let : MeasurableSpace Ω := (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) t
-    exact Measurable.of_eval (hQadapt t)
-  · intro ω
-    apply LipschitzWith.of_dist_le_mul
-    intro t u
-    apply (dist_pi_le_iff (mul_nonneg L.coe_nonneg dist_nonneg)).mpr
-    intro i
-    exact ((c i).lipschitz ω).dist_le_mul t u |>.trans
-      (mul_le_mul_of_nonneg_right (by exact_mod_cast hLi i) dist_nonneg)
-  · exact measurable_clipped_multipoint_velocity Q
-      (fun t ω => Real.sqrt κ * B t.toNNReal ω) hQm hUm lo hi hle
-  · intro t ω
-    apply (pi_norm_le_iff_of_nonneg L.coe_nonneg).mpr
-    intro i
-    change ‖clippedLogLoewnerField lo hi hle
-      (Real.sqrt κ * B t.toNNReal ω) (Q t ω i)‖ ≤ (L : ℝ)
-    have hb := hbound (Real.sqrt κ * B t.toNNReal ω) (Q t ω i)
-    exact hb.trans (by exact_mod_cast hL₀)
-  · intro ω
-    exact Eventually.of_forall fun t ht => hasDerivAt_pi.mpr fun i =>
-      ((c i).derivative ω t ht.le).hasDerivAt (Ici_mem_nhds ht)
-  · intro ω t ht
-    exact hasDerivWithinAt_pi.mpr (fun i => (c i).derivative ω t ht)
-  · intro ω
-    funext i
-    exact (c i).initial ω
-  · intro t ω i
-    rfl
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.BoundaryDensity
+
+namespace OAI
+
+/-! Simultaneous bounded adapted logarithmic Loewner coordinates. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma measurable_clipped_multipoint_velocity
+    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
+    (Q : ℝ → Ω → (ι → ℂ × ℝ)) (U : ℝ → Ω → ℝ)
+    (hQ : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hU : Measurable (fun p : ℝ × Ω => U p.1 p.2))
+    (lo hi : ℝ) (hle : lo ≤ hi) :
+    Measurable (fun p : ℝ × Ω => fun i =>
+      clippedLogLoewnerField lo hi hle (U p.1 p.2) (Q p.1 p.2 i)) := by
+  apply Measurable.of_eval
+  intro i
+  have hqi : Measurable (fun p : ℝ × Ω => Q p.1 p.2 i) :=
+    (measurable_pi_apply i).comp hQ
+  have hf : Measurable (fun p : ℝ × (ℂ × ℝ) =>
+      clippedLogLoewnerField lo hi hle p.1 p.2) :=
+    (continuous_clippedLogLoewnerField lo hi hle).measurable
+  have hp : Measurable (fun p : ℝ × Ω => (U p.1 p.2,Q p.1 p.2 i)) := hU.prodMk hqi
+  change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+    (fun p : ℝ × Ω => (U p.1 p.2,Q p.1 p.2 i)))
+  exact hf.comp hp
+
+theorem exists_clipped_multipoint_state
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ lo hi : ℝ) (hle : lo ≤ hi) (q₀ : ι → ℂ × ℝ) :
+    ∃ (L : ℝ≥0) (Q V : ℝ → Ω → (ι → ℂ × ℝ)),
+      Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧
+      Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun t : ℝ≥0 => Q t) ∧
+      (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧
+      Measurable (fun p : ℝ × Ω => V p.1 p.2) ∧
+      (∀ t ω, ‖V t ω‖ ≤ L) ∧
+      (∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t) ∧
+      (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun u => Q u ω) (V t ω) (Ici 0) t) ∧
+      (∀ ω, Q 0 ω = q₀) ∧
+      (∀ t ω i, V t ω i = clippedLogLoewnerField lo hi hle
+        (Real.sqrt κ*B t.toNNReal ω) (Q t ω i)) := by
+  classical
+  let c (i : ι) : ClippedLogSolution B hm κ lo hi hle (q₀ i) :=
+    Classical.choice (exists_clippedLogSolution B hm hBm hBc κ lo hi hle (q₀ i))
+  clear_value c
+  obtain ⟨L₀, K₀, hbound, _⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let L : ℝ≥0 := L₀ + ∑ i, (c i).L
+  let Q := fun t ω i => (c i).Q t ω
+  let V := fun t ω i => clippedLogLoewnerField lo hi hle
+    (Real.sqrt κ*B t.toNNReal ω) (Q t ω i)
+  have hLi (i : ι) : (c i).L ≤ L := by
+    change (c i).L ≤ L₀+∑ j, (c j).L
+    exact (Finset.single_le_sum (fun j _ => (zero_le : 0 ≤ (c j).L)) (Finset.mem_univ i)).trans
+      (le_add_left le_rfl)
+  have hL₀ : L₀ ≤ L := le_add_of_nonneg_right (by positivity)
+  have hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2) :=
+    Measurable.of_eval (fun i => (c i).measurable)
+  have hQadapt (t : ℝ≥0) (i : ι) :
+      @Measurable Ω (ℂ × ℝ)
+        ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t) _
+        (fun ω => Q t ω i) := (c i).adapted t
+  have hUm : Measurable (fun p : ℝ × Ω => Real.sqrt κ*B p.1.toNNReal p.2) :=
+    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)).const_mul _
+  refine ⟨L, Q, V, hQm, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · intro t
+    let : MeasurableSpace Ω := (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) t
+    exact Measurable.of_eval (hQadapt t)
+  · intro ω
+    apply LipschitzWith.of_dist_le_mul
+    intro t u
+    apply (dist_pi_le_iff (mul_nonneg L.coe_nonneg dist_nonneg)).mpr
+    intro i
+    exact ((c i).lipschitz ω).dist_le_mul t u |>.trans
+      (mul_le_mul_of_nonneg_right (by exact_mod_cast hLi i) dist_nonneg)
+  · exact measurable_clipped_multipoint_velocity Q
+      (fun t ω => Real.sqrt κ * B t.toNNReal ω) hQm hUm lo hi hle
+  · intro t ω
+    apply (pi_norm_le_iff_of_nonneg L.coe_nonneg).mpr
+    intro i
+    change ‖clippedLogLoewnerField lo hi hle
+      (Real.sqrt κ * B t.toNNReal ω) (Q t ω i)‖ ≤ (L : ℝ)
+    have hb := hbound (Real.sqrt κ * B t.toNNReal ω) (Q t ω i)
+    exact hb.trans (by exact_mod_cast hL₀)
+  · intro ω
+    exact Eventually.of_forall fun t ht => hasDerivAt_pi.mpr fun i =>
+      ((c i).derivative ω t ht.le).hasDerivAt (Ici_mem_nhds ht)
+  · intro ω t ht
+    exact hasDerivWithinAt_pi.mpr (fun i => (c i).derivative ω t ht)
+  · intro ω
+    funext i
+    exact (c i).initial ω
+  · intro t ω i
+    rfl
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCompactField.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCompactField.lean
index 1a832996ef7ce79f165d1912b7fe37eadf57da7b..7b9317bd153b127b05e74f30cc3a5f792d2a2665 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCompactField.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCompactField.lean
@@ -1,83 +1,83 @@
-import OAI.Probability.SLE.Multipoint.MultipointDiscountedField
-
-namespace OAI
-
-/-! Compact smooth tests agreeing with the discounted product on each
-bounded localization region. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology
-
-namespace SLEExactGauge.Multipoint
-
-theorem exists_compact_discounted_field {ι : Type*} [Fintype ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
-    (R : ℝ) (hR : 0 ≤ R) :
-    ∃ φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ,
-      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧ (∀ p, 0 ≤ φ p) ∧
-      (∀ z x, ‖(z,x)‖ ≤ R → φ (z,x) = discountedProduct s κ D σ z x) ∧
-      (∀ z x, ‖(z,x)‖ ≤ R →
-        fderiv ℝ φ (z,x) =
-          fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x)) ∧
-      (∀ z x, ‖(z,x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (z,y)) x =
-        iteratedDeriv 2 (discountedProduct s κ D σ z) x) := by
-  let b : ContDiffBump (0 : ((ι → ℂ × ℝ) × ℝ) × ℝ) :=
-    ⟨R+1, R+2, by linarith, by linarith⟩
-  let φ := fun p : ((ι → ℂ × ℝ) × ℝ) × ℝ => b p*discountedProduct s κ D σ p.1 p.2
-  have hnear (p : ((ι → ℂ × ℝ) × ℝ) × ℝ) (hp : ‖p‖ ≤ R) :
-      φ =ᶠ[𝓝 p] (fun q => discountedProduct s κ D σ q.1 q.2) := by
-    have hball : p ∈ Metric.ball (0 : ((ι → ℂ × ℝ) × ℝ) × ℝ) b.rIn := by
-      simp only [Metric.mem_ball, dist_zero_right]
-      change ‖p‖ < R+1
-      linarith
-    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with q hq
-    simp only [φ, hq, Pi.one_apply, one_mul]
-  refine ⟨φ, b.contDiff.mul (discountedProduct_contDiff s κ D σ), b.hasCompactSupport.mul_right,
-    ?_, (fun z x h => (hnear (z,x) h).eq_of_nhds), ?_, ?_⟩
-  · intro p
-    apply mul_nonneg b.nonneg
-    apply mul_nonneg (Real.exp_pos _).le
-    exact (Finset.prod_pos (fun i hi => regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) _ _)).le
-  · intro z x h
-    exact (hnear (z,x) h).fderiv_eq
-  · intro z x h
-    have he := (hnear (z,x) h).comp_tendsto
-      ((continuous_const.prodMk continuous_id).tendsto x)
-    exact he.iteratedDeriv_eq 2
-
-lemma discountedProduct_pair_directional {ι : Type*} [Fintype ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
-    (z v : (ι → ℂ × ℝ) × ℝ) (x : ℝ) :
-    fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x) (v,0) =
-      fderiv ℝ (fun q => discountedProduct s κ D σ q x) z v := by
-  have hd := ((discountedProduct_contDiff s κ D σ).differentiable (by simp)) (z,x)
-  have hp := (hasFDerivAt_id (𝕜 := ℝ) z).prodMk
-    (hasFDerivAt_const (𝕜 := ℝ) (E := (ι → ℂ × ℝ) × ℝ) x z)
-  have hc := (hd.hasFDerivAt.comp z hp).fderiv
-  have he := congrArg (fun A : ((ι → ℂ × ℝ) × ℝ) →L[ℝ] ℝ => A v) hc
-  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
-    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
-
-theorem compact_discounted_generator_nonpos {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (b x : ℝ)
-    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
-    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1)
-    (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ)
-    (hφD : fderiv ℝ φ ((z,b),x) =
-      fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) ((z,b),x))
-    (hφ2 : iteratedDeriv 2 (fun y => φ ((z,b),y)) x =
-      iteratedDeriv 2 (discountedProduct s κ D σ (z,b)) x) :
-    fderiv ℝ φ ((z,b),x)
-        (((fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)),
-          productDriftCost s κ D σ z x),0) +
-      (1/2 : ℝ)*iteratedDeriv 2 (fun y => φ ((z,b),y)) x ≤ 0 := by
-  rw [hφD, hφ2, discountedProduct_pair_directional]
-  exact discountedProduct_generator_nonpos s κ hκ hκ8 D σ z b x hD hσ hδ
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointDiscountedField
+
+namespace OAI
+
+/-! Compact smooth tests agreeing with the discounted product on each
+bounded localization region. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology
+
+namespace SLEExactGauge.Multipoint
+
+theorem exists_compact_discounted_field {ι : Type*} [Fintype ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
+    (R : ℝ) (hR : 0 ≤ R) :
+    ∃ φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ,
+      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧ (∀ p, 0 ≤ φ p) ∧
+      (∀ z x, ‖(z,x)‖ ≤ R → φ (z,x) = discountedProduct s κ D σ z x) ∧
+      (∀ z x, ‖(z,x)‖ ≤ R →
+        fderiv ℝ φ (z,x) =
+          fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x)) ∧
+      (∀ z x, ‖(z,x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (z,y)) x =
+        iteratedDeriv 2 (discountedProduct s κ D σ z) x) := by
+  let b : ContDiffBump (0 : ((ι → ℂ × ℝ) × ℝ) × ℝ) :=
+    ⟨R+1, R+2, by linarith, by linarith⟩
+  let φ := fun p : ((ι → ℂ × ℝ) × ℝ) × ℝ => b p*discountedProduct s κ D σ p.1 p.2
+  have hnear (p : ((ι → ℂ × ℝ) × ℝ) × ℝ) (hp : ‖p‖ ≤ R) :
+      φ =ᶠ[𝓝 p] (fun q => discountedProduct s κ D σ q.1 q.2) := by
+    have hball : p ∈ Metric.ball (0 : ((ι → ℂ × ℝ) × ℝ) × ℝ) b.rIn := by
+      simp only [Metric.mem_ball, dist_zero_right]
+      change ‖p‖ < R+1
+      linarith
+    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with q hq
+    simp only [φ, hq, Pi.one_apply, one_mul]
+  refine ⟨φ, b.contDiff.mul (discountedProduct_contDiff s κ D σ), b.hasCompactSupport.mul_right,
+    ?_, (fun z x h => (hnear (z,x) h).eq_of_nhds), ?_, ?_⟩
+  · intro p
+    apply mul_nonneg b.nonneg
+    apply mul_nonneg (Real.exp_pos _).le
+    exact (Finset.prod_pos (fun i hi => regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) _ _)).le
+  · intro z x h
+    exact (hnear (z,x) h).fderiv_eq
+  · intro z x h
+    have he := (hnear (z,x) h).comp_tendsto
+      ((continuous_const.prodMk continuous_id).tendsto x)
+    exact he.iteratedDeriv_eq 2
+
+lemma discountedProduct_pair_directional {ι : Type*} [Fintype ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
+    (z v : (ι → ℂ × ℝ) × ℝ) (x : ℝ) :
+    fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x) (v,0) =
+      fderiv ℝ (fun q => discountedProduct s κ D σ q x) z v := by
+  have hd := ((discountedProduct_contDiff s κ D σ).differentiable (by simp)) (z,x)
+  have hp := (hasFDerivAt_id (𝕜 := ℝ) z).prodMk
+    (hasFDerivAt_const (𝕜 := ℝ) (E := (ι → ℂ × ℝ) × ℝ) x z)
+  have hc := (hd.hasFDerivAt.comp z hp).fderiv
+  have he := congrArg (fun A : ((ι → ℂ × ℝ) × ℝ) →L[ℝ] ℝ => A v) hc
+  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
+    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
+
+theorem compact_discounted_generator_nonpos {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (b x : ℝ)
+    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
+    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1)
+    (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ)
+    (hφD : fderiv ℝ φ ((z,b),x) =
+      fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) ((z,b),x))
+    (hφ2 : iteratedDeriv 2 (fun y => φ ((z,b),y)) x =
+      iteratedDeriv 2 (discountedProduct s κ D σ (z,b)) x) :
+    fderiv ℝ φ ((z,b),x)
+        (((fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)),
+          productDriftCost s κ D σ z x),0) +
+      (1/2 : ℝ)*iteratedDeriv 2 (fun y => φ ((z,b),y)) x ≤ 0 := by
+  rw [hφD, hφ2, discountedProduct_pair_directional]
+  exact discountedProduct_generator_nonpos s κ hκ hκ8 D σ z b x hD hσ hδ
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCompletionExhaustion.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCompletionExhaustion.lean
index 6a36cc70cbff4e0be83b77ebcd46992021f0913d..36c7a84a6ff12752f2ece97db980971f43256d56 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCompletionExhaustion.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCompletionExhaustion.lean
@@ -1,81 +1,81 @@
-import OAI.Probability.SLE.Multipoint.MultipointCutoffExhaustion
-import OAI.Probability.SLE.Multipoint.MultipointActualPhases
-
-namespace OAI
-
-/-! On the literal event that every target radius is reached, every
-factor is eventually completed before the localization cutoff. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem eventually_all_completed_of_finite_radius_levels
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ A : ℝ) (hA : 0 ≤ A) (z : ι → ℂ) (u₀ u₂ : ι → ℝ)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) (F ω))
-    (S : ∀ n : ℕ, LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
-      ((n : ℝ≥0)+1) z
-      (fun i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2)) γ G)
-    (ω : Ω) (hfinish : ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) < ⊤) :
-    ∀ᶠ n : ℕ in atTop,
-      (∀ i, actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2) ω ≤ (S n).χ ω) ∧
-      phaseAt (actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1)) ((S n).χ ω) ω =
-        fun _ => 3 := by
-  classical
-  have hex (i : ι) : ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) = (t : ℝ≥0∞) := by
-    obtain ⟨t,ht,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top
-      (γ ω) (G ω) (z i) (u₂ i) (hfinish i)
-    exact ⟨t,ht,radiusLevelTime_eq_of_live_level
-      (continuous_const.mul (hBc ω)) (hw ω) (z i) (u₂ i) t ht hR⟩
-  choose σ hσlive hσeq using hex
-  let T : ℝ≥0 := (∑ i, σ i)+1
-  have hσT (i : ι) : σ i ≤ T := by
-    exact (Finset.single_le_sum (fun j _ => (show 0 ≤ σ j from bot_le)) (Finset.mem_univ i)).trans
-      (le_add_of_nonneg_right zero_le)
-  have hlarge : ∀ᶠ n : ℕ in atTop, (T : ℝ) < (n : ℝ) :=
-    (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).eventually
-      (eventually_gt_atTop (T : ℝ))
-  have hτ : ∀ᶠ n : ℕ in atTop,
-      ∀ i, actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2) ω = σ i := by
-    filter_upwards [hlarge] with n hn i
-    have hbound : σ i ≤ ((n : ℝ≥0)+1)+1 := by
-      have hsT : (σ i : ℝ) ≤ (T : ℝ) := by exact_mod_cast hσT i
-      exact_mod_cast (show (σ i : ℝ) ≤ ((n : ℝ)+1)+1 from by linarith)
-    change cutoffStop (fun ω => radiusLevelTime (γ ω) (G ω) (z i)
-      (factorRadiusLevel (u₀ i) (u₂ i) 2)) (((n : ℝ≥0)+1)+1) ω = σ i
-    have hfactor : factorRadiusLevel (u₀ i) (u₂ i) 2 = u₂ i := by
-      norm_num [factorRadiusLevel]
-    simp only [hfactor]
-    change (min (radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) : WithTop ℝ≥0)
-      ((((n : ℝ≥0)+1)+1 : ℝ≥0) : WithTop ℝ≥0)).untopD 0 = σ i
-    have he : (radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) : WithTop ℝ≥0) =
-        (σ i : WithTop ℝ≥0) := hσeq i
-    have hemin : min (radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) : WithTop ℝ≥0)
-        ((((n : ℝ≥0)+1)+1 : ℝ≥0) : WithTop ℝ≥0) = (σ i : WithTop ℝ≥0) :=
-      (congrArg (fun q : WithTop ℝ≥0 => min q
-        ((((n : ℝ≥0)+1)+1 : ℝ≥0) : WithTop ℝ≥0)) he).trans
-          (min_eq_left (WithTop.coe_le_coe.mpr hbound))
-    exact congrArg (WithTop.untopD 0) hemin
-  have hχ := eventually_localization_after_completed B hm hBc κ A hA z
-    (fun n i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2))
-    γ G F hw S ω σ T hσT hσlive hτ
-  filter_upwards [hτ,hχ] with n hτn hχn
-  have hdone (i : ι) :
-      actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2) ω ≤ (S n).χ ω := by
-    rw [hτn i]
-    exact (hσT i).trans hχn.le
-  exact ⟨hdone,funext (fun i => by simp only [phaseAt,ite_eq_left (hdone i)])⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointCutoffExhaustion
+import OAI.Probability.SLE.Multipoint.MultipointActualPhases
+
+namespace OAI
+
+/-! On the literal event that every target radius is reached, every
+factor is eventually completed before the localization cutoff. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem eventually_all_completed_of_finite_radius_levels
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ A : ℝ) (hA : 0 ≤ A) (z : ι → ℂ) (u₀ u₂ : ι → ℝ)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) (F ω))
+    (S : ∀ n : ℕ, LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
+      ((n : ℝ≥0)+1) z
+      (fun i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2)) γ G)
+    (ω : Ω) (hfinish : ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) < ⊤) :
+    ∀ᶠ n : ℕ in atTop,
+      (∀ i, actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2) ω ≤ (S n).χ ω) ∧
+      phaseAt (actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1)) ((S n).χ ω) ω =
+        fun _ => 3 := by
+  classical
+  have hex (i : ι) : ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) = (t : ℝ≥0∞) := by
+    obtain ⟨t,ht,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top
+      (γ ω) (G ω) (z i) (u₂ i) (hfinish i)
+    exact ⟨t,ht,radiusLevelTime_eq_of_live_level
+      (continuous_const.mul (hBc ω)) (hw ω) (z i) (u₂ i) t ht hR⟩
+  choose σ hσlive hσeq using hex
+  let T : ℝ≥0 := (∑ i, σ i)+1
+  have hσT (i : ι) : σ i ≤ T := by
+    exact (Finset.single_le_sum (fun j _ => (show 0 ≤ σ j from bot_le)) (Finset.mem_univ i)).trans
+      (le_add_of_nonneg_right zero_le)
+  have hlarge : ∀ᶠ n : ℕ in atTop, (T : ℝ) < (n : ℝ) :=
+    (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).eventually
+      (eventually_gt_atTop (T : ℝ))
+  have hτ : ∀ᶠ n : ℕ in atTop,
+      ∀ i, actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2) ω = σ i := by
+    filter_upwards [hlarge] with n hn i
+    have hbound : σ i ≤ ((n : ℝ≥0)+1)+1 := by
+      have hsT : (σ i : ℝ) ≤ (T : ℝ) := by exact_mod_cast hσT i
+      exact_mod_cast (show (σ i : ℝ) ≤ ((n : ℝ)+1)+1 from by linarith)
+    change cutoffStop (fun ω => radiusLevelTime (γ ω) (G ω) (z i)
+      (factorRadiusLevel (u₀ i) (u₂ i) 2)) (((n : ℝ≥0)+1)+1) ω = σ i
+    have hfactor : factorRadiusLevel (u₀ i) (u₂ i) 2 = u₂ i := by
+      norm_num [factorRadiusLevel]
+    simp only [hfactor]
+    change (min (radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) : WithTop ℝ≥0)
+      ((((n : ℝ≥0)+1)+1 : ℝ≥0) : WithTop ℝ≥0)).untopD 0 = σ i
+    have he : (radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) : WithTop ℝ≥0) =
+        (σ i : WithTop ℝ≥0) := hσeq i
+    have hemin : min (radiusLevelTime (γ ω) (G ω) (z i) (u₂ i) : WithTop ℝ≥0)
+        ((((n : ℝ≥0)+1)+1 : ℝ≥0) : WithTop ℝ≥0) = (σ i : WithTop ℝ≥0) :=
+      (congrArg (fun q : WithTop ℝ≥0 => min q
+        ((((n : ℝ≥0)+1)+1 : ℝ≥0) : WithTop ℝ≥0)) he).trans
+          (min_eq_left (WithTop.coe_le_coe.mpr hbound))
+    exact congrArg (WithTop.untopD 0) hemin
+  have hχ := eventually_localization_after_completed B hm hBc κ A hA z
+    (fun n i => actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2))
+    γ G F hw S ω σ T hσT hσlive hτ
+  filter_upwards [hτ,hχ] with n hτn hχn
+  have hdone (i : ι) :
+      actualFactorStops γ G z u₀ u₂ ((n : ℝ≥0)+1) (i,2) ω ≤ (S n).χ ω := by
+    rw [hτn i]
+    exact (hσT i).trans hχn.le
+  exact ⟨hdone,funext (fun i => by simp only [phaseAt,ite_eq_left (hdone i)])⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCostAssembly.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCostAssembly.lean
index 731e206d68fd4a215a6644a0714e4b593d910178..f4424323db0c4ab8adc6ed72e2f3483c0e3c30a7 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCostAssembly.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCostAssembly.lean
@@ -1,115 +1,115 @@
-import OAI.Probability.SLE.Multipoint.MultipointActiveCosts
-import OAI.Probability.SLE.Multipoint.MultipointPatternProduct
-
-namespace OAI
-
-/-! Integrating the actual phase-selected cost through the individual
-live intervals. No common survival endpoint is imposed on all points. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem actual_phase_cost_integral_le {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
-    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
-    (I : Finset ι) (z : ι → ℂ) (hne : Set.InjOn z (I : Set ι))
-    (β : ℝ) (hβ : 0 < β) (hdecay : 2*(exponent (greenExponent κ)*β) < 2/κ)
-    (a U : ι → ℝ) (hU : ∀ i ∈ I, 0 ≤ U i)
-    (σ T : ι → ℝ≥0) (H : ℝ≥0)
-    (hσT : ∀ i ∈ I, σ i ≤ T i) (hTH : ∀ i ∈ I, T i ≤ H)
-    (hz : ∀ i ∈ I, z i ∈ survivingDomain γ (T i))
-    (hstart : ∀ i ∈ I, σ i < T i → logFactorClock κ (logLoewnerCoordinates G (z i) (σ i)) = a i)
-    (hend : ∀ i ∈ I, σ i < T i → logFactorClock κ (logLoewnerCoordinates G (z i) (T i))-a i ≤ U i)
-    (hgeo : ∀ i ∈ I, ∀ j ∈ I, i ≠ j → σ i < T i → σ j < T j →
-      physicalConformalRadius J G (σ i) (z i) ≤ dist (J (z i)) (J (z j)))
-    (Q : ℝ → ι → ℂ × ℝ) (j : ℝ → ι → Fin 4)
-    (hj : ∀ t ∈ Ico 0 (H : ℝ), patternActiveSet (j t) ⊆ I)
-    (hactive : ∀ t ∈ Ico 0 (H : ℝ), ∀ i ∈ patternActiveSet (j t),
-      (σ i : ℝ) ≤ t ∧ t ≤ (T i : ℝ))
-    (hQ : ∀ t ∈ Ico 0 (H : ℝ), ∀ i ∈ patternActiveSet (j t),
-      Q t i = logLoewnerCoordinates G (z i) t)
-    (hreg : ∀ t ∈ Ico 0 (H : ℝ), ∀ i ∈ patternActiveSet (j t),
-      branchRegulator κ (phaseRegulatorScale β (a i) (U i) (j t i))
-        (phaseRegulatorSlope β (j t i)) (Q t i) =
-          regulator β (U i) (logFactorClock κ (logLoewnerCoordinates G (z i) t)-a i))
-    (R : ℝ → ℝ) (hR : IntervalIntegrable R volume 0 (H : ℝ))
-    (hcost : ∀ t ∈ Ico 0 (H : ℝ), R t =
-      productDriftCost (patternActiveSet (j t)) κ
-        (fun i => phaseRegulatorScale β (a i) (U i) (j t i))
-        (fun i => phaseRegulatorSlope β (j t i)) (Q t) (B t.toNNReal)) :
-    (∫ t in (0 : ℝ)..(H : ℝ), R t) ≤ uniformDiscountBudget κ β I := by
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  let f := fun i => actualSingleCost κ β (a i) (U i) B G (z i)
-  let g := fun ij : ι × ι => actualPairCost κ β (a ij.1) (a ij.2)
-    (U ij.1) (U ij.2) B G (z ij.1) (z ij.2)
-  let c := fun ij : ι × ι => overlapStart (σ ij.1) (σ ij.2)
-  let d := fun ij : ι × ι => overlapEnd (σ ij.1) (σ ij.2) (T ij.1) (T ij.2)
-  have hf (i) (hi : i ∈ I) := actualSingleCost_bound hκ hB hw (σ i) (T i) (hσT i hi)
-    (z i) (hz i hi) β (a i) (U i) hβ (hU i hi) (hstart i hi) (hend i hi)
-  have hg (ij : ι × ι) (hij : ij ∈ I.offDiag) :
-      IntervalIntegrable (g ij) volume (c ij : ℝ) (d ij : ℝ) ∧
-        (∫ t in (c ij : ℝ)..(d ij : ℝ), g ij t) ≤
-          interactionBudget 2 (exponent (greenExponent κ)*(κ*β)) := by
-    obtain ⟨hi,hj,hneij⟩ := Finset.mem_offDiag.mp hij
-    exact actualPairCost_bound hκ hB hw hgrowth J hJ hJinj (σ ij.1) (σ ij.2)
-      (T ij.1) (T ij.2) (z ij.1) (z ij.2) (hz _ hi) (hz _ hj)
-      (fun he => hneij (hne hi hj he)) β (a ij.1) (a ij.2) (U ij.1) (U ij.2)
-      hβ.le (exponent_pos hp).le hdecay (hstart _ hi) (hstart _ hj)
-      (hgeo _ hi _ hj hneij) (fun h₁ h₂ => by
-        simpa only [dist_comm] using hgeo _ hj _ hi hneij.symm h₂ h₁)
-  have hcH (ij : ι × ι) (hij : ij ∈ I.offDiag) : c ij ≤ H := by
-    obtain ⟨hi,hj,_⟩ := Finset.mem_offDiag.mp hij
-    exact max_le ((hσT _ hi).trans (hTH _ hi)) ((hσT _ hj).trans (hTH _ hj))
-  have hdH (ij : ι × ι) (hij : ij ∈ I.offDiag) : d ij ≤ H := by
-    obtain ⟨hi,_,_⟩ := Finset.mem_offDiag.mp hij
-    exact max_le (hcH ij hij) ((min_le_left _ _).trans (hTH _ hi))
-  apply integrated_uniform_discount_le hκ hκ8 β I (fun i => (σ i : ℝ))
-    (fun i => (T i : ℝ)) (fun ij => (c ij : ℝ)) (fun ij => (d ij : ℝ)) H H.coe_nonneg f g
-    (fun i _ => (σ i).coe_nonneg) (fun i hi => by exact_mod_cast hσT i hi)
-    (fun i hi => by exact_mod_cast hTH i hi)
-    (fun ij _ => (c ij).coe_nonneg) (fun ij _ => by exact_mod_cast le_max_left (c ij) (min (T ij.1) (T ij.2)))
-    (fun ij hij => by exact_mod_cast hdH ij hij)
-    (fun i hi => (hf i hi).1) (fun ij hij => (hg ij hij).1)
-    (fun i hi => (hf i hi).2) (fun ij hij => (hg ij hij).2) R hR
-  have hnσ : ∀ᵐ t : ℝ, ∀ i, t ≠ (σ i : ℝ) :=
-    ae_all_iff.mpr (fun i => volume.ae_ne (σ i : ℝ))
-  filter_upwards [ae_restrict_mem measurableSet_Icc,
-    ae_restrict_of_ae (volume.ae_ne (H : ℝ)), ae_restrict_of_ae hnσ] with t ht hneH hnσ
-  have ht' : t ∈ Ico 0 (H : ℝ) := ⟨ht.1, lt_of_le_of_ne ht.2 hneH⟩
-  rw [hcost t ht']
-  have his (i : ι) (hi : i ∈ patternActiveSet (j t)) : t ∈ Ioc (σ i : ℝ) (T i : ℝ) :=
-    ⟨lt_of_le_of_ne (hactive t ht' i hi).1 (hnσ i).symm, (hactive t ht' i hi).2⟩
-  apply productDriftCost_le_active_envelope hκ hκ8 (hj t ht')
-    (fun i => phaseRegulatorScale β (a i) (U i) (j t i))
-    (fun i => phaseRegulatorSlope β (j t i)) (fun i _ => phaseRegulatorScale_pos _ _ _ _)
-    (Q t) (B t.toNNReal) t (fun i => (σ i : ℝ)) (fun i => (T i : ℝ))
-    (fun ij => (c ij : ℝ)) (fun ij => (d ij : ℝ)) f g
-    (fun i _ => actualSingleCost_nonneg _ _ _ _ _ _ _ _)
-    (fun ij _ => actualPairCost_nonneg _ _ _ _ _ _ _ _ _ _ _) his
-  · intro ij hij
-    obtain ⟨hi,hj,_⟩ := Finset.mem_offDiag.mp hij
-    refine ⟨?_,?_⟩
-    · exact max_lt (his _ hi).1 (his _ hj).1
-    · exact (le_min (his _ hi).2 (his _ hj).2).trans (by exact_mod_cast le_max_right (c ij) (min (T ij.1) (T ij.2)))
-  · intro i hi
-    rw [hreg t ht' i hi, hQ t ht' i hi]
-    rfl
-  · intro ij hij
-    obtain ⟨hi,hj,_⟩ := Finset.mem_offDiag.mp hij
-    rw [hreg t ht' _ hi, hreg t ht' _ hj, hQ t ht' _ hi, hQ t ht' _ hj]
-    rfl
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointActiveCosts
+import OAI.Probability.SLE.Multipoint.MultipointPatternProduct
+
+namespace OAI
+
+/-! Integrating the actual phase-selected cost through the individual
+live intervals. No common survival endpoint is imposed on all points. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem actual_phase_cost_integral_le {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
+    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
+    (I : Finset ι) (z : ι → ℂ) (hne : Set.InjOn z (I : Set ι))
+    (β : ℝ) (hβ : 0 < β) (hdecay : 2*(exponent (greenExponent κ)*β) < 2/κ)
+    (a U : ι → ℝ) (hU : ∀ i ∈ I, 0 ≤ U i)
+    (σ T : ι → ℝ≥0) (H : ℝ≥0)
+    (hσT : ∀ i ∈ I, σ i ≤ T i) (hTH : ∀ i ∈ I, T i ≤ H)
+    (hz : ∀ i ∈ I, z i ∈ survivingDomain γ (T i))
+    (hstart : ∀ i ∈ I, σ i < T i → logFactorClock κ (logLoewnerCoordinates G (z i) (σ i)) = a i)
+    (hend : ∀ i ∈ I, σ i < T i → logFactorClock κ (logLoewnerCoordinates G (z i) (T i))-a i ≤ U i)
+    (hgeo : ∀ i ∈ I, ∀ j ∈ I, i ≠ j → σ i < T i → σ j < T j →
+      physicalConformalRadius J G (σ i) (z i) ≤ dist (J (z i)) (J (z j)))
+    (Q : ℝ → ι → ℂ × ℝ) (j : ℝ → ι → Fin 4)
+    (hj : ∀ t ∈ Ico 0 (H : ℝ), patternActiveSet (j t) ⊆ I)
+    (hactive : ∀ t ∈ Ico 0 (H : ℝ), ∀ i ∈ patternActiveSet (j t),
+      (σ i : ℝ) ≤ t ∧ t ≤ (T i : ℝ))
+    (hQ : ∀ t ∈ Ico 0 (H : ℝ), ∀ i ∈ patternActiveSet (j t),
+      Q t i = logLoewnerCoordinates G (z i) t)
+    (hreg : ∀ t ∈ Ico 0 (H : ℝ), ∀ i ∈ patternActiveSet (j t),
+      branchRegulator κ (phaseRegulatorScale β (a i) (U i) (j t i))
+        (phaseRegulatorSlope β (j t i)) (Q t i) =
+          regulator β (U i) (logFactorClock κ (logLoewnerCoordinates G (z i) t)-a i))
+    (R : ℝ → ℝ) (hR : IntervalIntegrable R volume 0 (H : ℝ))
+    (hcost : ∀ t ∈ Ico 0 (H : ℝ), R t =
+      productDriftCost (patternActiveSet (j t)) κ
+        (fun i => phaseRegulatorScale β (a i) (U i) (j t i))
+        (fun i => phaseRegulatorSlope β (j t i)) (Q t) (B t.toNNReal)) :
+    (∫ t in (0 : ℝ)..(H : ℝ), R t) ≤ uniformDiscountBudget κ β I := by
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  let f := fun i => actualSingleCost κ β (a i) (U i) B G (z i)
+  let g := fun ij : ι × ι => actualPairCost κ β (a ij.1) (a ij.2)
+    (U ij.1) (U ij.2) B G (z ij.1) (z ij.2)
+  let c := fun ij : ι × ι => overlapStart (σ ij.1) (σ ij.2)
+  let d := fun ij : ι × ι => overlapEnd (σ ij.1) (σ ij.2) (T ij.1) (T ij.2)
+  have hf (i) (hi : i ∈ I) := actualSingleCost_bound hκ hB hw (σ i) (T i) (hσT i hi)
+    (z i) (hz i hi) β (a i) (U i) hβ (hU i hi) (hstart i hi) (hend i hi)
+  have hg (ij : ι × ι) (hij : ij ∈ I.offDiag) :
+      IntervalIntegrable (g ij) volume (c ij : ℝ) (d ij : ℝ) ∧
+        (∫ t in (c ij : ℝ)..(d ij : ℝ), g ij t) ≤
+          interactionBudget 2 (exponent (greenExponent κ)*(κ*β)) := by
+    obtain ⟨hi,hj,hneij⟩ := Finset.mem_offDiag.mp hij
+    exact actualPairCost_bound hκ hB hw hgrowth J hJ hJinj (σ ij.1) (σ ij.2)
+      (T ij.1) (T ij.2) (z ij.1) (z ij.2) (hz _ hi) (hz _ hj)
+      (fun he => hneij (hne hi hj he)) β (a ij.1) (a ij.2) (U ij.1) (U ij.2)
+      hβ.le (exponent_pos hp).le hdecay (hstart _ hi) (hstart _ hj)
+      (hgeo _ hi _ hj hneij) (fun h₁ h₂ => by
+        simpa only [dist_comm] using hgeo _ hj _ hi hneij.symm h₂ h₁)
+  have hcH (ij : ι × ι) (hij : ij ∈ I.offDiag) : c ij ≤ H := by
+    obtain ⟨hi,hj,_⟩ := Finset.mem_offDiag.mp hij
+    exact max_le ((hσT _ hi).trans (hTH _ hi)) ((hσT _ hj).trans (hTH _ hj))
+  have hdH (ij : ι × ι) (hij : ij ∈ I.offDiag) : d ij ≤ H := by
+    obtain ⟨hi,_,_⟩ := Finset.mem_offDiag.mp hij
+    exact max_le (hcH ij hij) ((min_le_left _ _).trans (hTH _ hi))
+  apply integrated_uniform_discount_le hκ hκ8 β I (fun i => (σ i : ℝ))
+    (fun i => (T i : ℝ)) (fun ij => (c ij : ℝ)) (fun ij => (d ij : ℝ)) H H.coe_nonneg f g
+    (fun i _ => (σ i).coe_nonneg) (fun i hi => by exact_mod_cast hσT i hi)
+    (fun i hi => by exact_mod_cast hTH i hi)
+    (fun ij _ => (c ij).coe_nonneg) (fun ij _ => by exact_mod_cast le_max_left (c ij) (min (T ij.1) (T ij.2)))
+    (fun ij hij => by exact_mod_cast hdH ij hij)
+    (fun i hi => (hf i hi).1) (fun ij hij => (hg ij hij).1)
+    (fun i hi => (hf i hi).2) (fun ij hij => (hg ij hij).2) R hR
+  have hnσ : ∀ᵐ t : ℝ, ∀ i, t ≠ (σ i : ℝ) :=
+    ae_all_iff.mpr (fun i => volume.ae_ne (σ i : ℝ))
+  filter_upwards [ae_restrict_mem measurableSet_Icc,
+    ae_restrict_of_ae (volume.ae_ne (H : ℝ)), ae_restrict_of_ae hnσ] with t ht hneH hnσ
+  have ht' : t ∈ Ico 0 (H : ℝ) := ⟨ht.1, lt_of_le_of_ne ht.2 hneH⟩
+  rw [hcost t ht']
+  have his (i : ι) (hi : i ∈ patternActiveSet (j t)) : t ∈ Ioc (σ i : ℝ) (T i : ℝ) :=
+    ⟨lt_of_le_of_ne (hactive t ht' i hi).1 (hnσ i).symm, (hactive t ht' i hi).2⟩
+  apply productDriftCost_le_active_envelope hκ hκ8 (hj t ht')
+    (fun i => phaseRegulatorScale β (a i) (U i) (j t i))
+    (fun i => phaseRegulatorSlope β (j t i)) (fun i _ => phaseRegulatorScale_pos _ _ _ _)
+    (Q t) (B t.toNNReal) t (fun i => (σ i : ℝ)) (fun i => (T i : ℝ))
+    (fun ij => (c ij : ℝ)) (fun ij => (d ij : ℝ)) f g
+    (fun i _ => actualSingleCost_nonneg _ _ _ _ _ _ _ _)
+    (fun ij _ => actualPairCost_nonneg _ _ _ _ _ _ _ _ _ _ _) his
+  · intro ij hij
+    obtain ⟨hi,hj,_⟩ := Finset.mem_offDiag.mp hij
+    refine ⟨?_,?_⟩
+    · exact max_lt (his _ hi).1 (his _ hj).1
+    · exact (le_min (his _ hi).2 (his _ hj).2).trans (by exact_mod_cast le_max_right (c ij) (min (T ij.1) (T ij.2)))
+  · intro i hi
+    rw [hreg t ht' i hi, hQ t ht' i hi]
+    rfl
+  · intro ij hij
+    obtain ⟨hi,hj,_⟩ := Finset.mem_offDiag.mp hij
+    rw [hreg t ht' _ hi, hreg t ht' _ hj, hQ t ht' _ hi, hQ t ht' _ hj]
+    rfl
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCutoff.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCutoff.lean
index f7d7fc10cb002ff33c8530120aeca014124fdc46..33dd7f315da9eaa169eceb54d5e4fdeb6be9eecb 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCutoff.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCutoff.lean
@@ -1,83 +1,83 @@
-import Mathlib
-
-namespace OAI
-
-/-! Exhaustion of successful multipoint paths by nonsingular cutoffs.
-Each point is monitored only through its own completion time. Later
-swallowing of an already completed point does not enter this lemma. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma eventually_compact_positive_reciprocal {X : Type*} [TopologicalSpace X]
-    {K : Set X} (hK : IsCompact K) {f : X → ℝ} (hf : ContinuousOn f K)
-    (hpos : ∀ x ∈ K, 0 < f x) :
-    ∀ᶠ n : ℕ in atTop, ∀ x ∈ K, 1 / ((n : ℝ)+1) < f x := by
-  by_cases hne : K.Nonempty
-  · obtain ⟨x, hx, hmin⟩ := hK.exists_isMinOn hne hf
-    filter_upwards [(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
-      (gt_mem_nhds (hpos x hx))] with n hn
-    intro y hy
-    exact hn.trans_le (hmin hy)
-  · exact Eventually.of_forall (fun _ x hx => False.elim (hne ⟨x, hx⟩))
-
-/-- On a successful path, finitely many individually stopped points admit
-one common height, angle, and physical-time cutoff. -/
-theorem successful_path_eventually_inside_cutoff
-    {ι : Type*} [Finite ι] (T : ι → ℝ) (Y θ : ι → ℝ → ℝ)
-    (hY : ∀ i, ContinuousOn (Y i) (Icc 0 (T i)))
-    (hθ : ∀ i, ContinuousOn (θ i) (Icc 0 (T i)))
-    (hYpos : ∀ i t, t ∈ Icc 0 (T i) → 0 < Y i t)
-    (hθpos : ∀ i t, t ∈ Icc 0 (T i) → 0 < θ i t)
-    (hθpi : ∀ i t, t ∈ Icc 0 (T i) → θ i t < Real.pi) :
-    ∀ᶠ n : ℕ in atTop, ∀ i,
-      T i < (n : ℝ)+1 ∧ ∀ t ∈ Icc 0 (T i),
-        1 / ((n : ℝ)+1) < Y i t ∧
-        1 / ((n : ℝ)+1) < θ i t ∧
-        θ i t < Real.pi - 1 / ((n : ℝ)+1) := by
-  apply eventually_all.mpr
-  intro i
-  have htime : ∀ᶠ n : ℕ in atTop, T i < (n : ℝ)+1 := by
-    obtain ⟨N, hN⟩ := exists_nat_gt (T i)
-    filter_upwards [eventually_ge_atTop N] with n hn
-    have hcast : (N : ℝ) ≤ n := by exact_mod_cast hn
-    linarith
-  have hheight := eventually_compact_positive_reciprocal isCompact_Icc (hY i) (hYpos i)
-  have hangle := eventually_compact_positive_reciprocal isCompact_Icc (hθ i) (hθpos i)
-  have hupper := eventually_compact_positive_reciprocal isCompact_Icc
-    (continuousOn_const.sub (hθ i)) (fun t ht => sub_pos.mpr (hθpi i t ht))
-  filter_upwards [htime, hheight, hangle, hupper] with n hn hy ha hb
-  refine ⟨hn, fun t ht => ⟨hy t ht, ha t ht, ?_⟩⟩
-  have hupper' : 1 / ((n : ℝ)+1) < Real.pi - θ i t := hb t ht
-  linarith
-
-/-- Bounds for nonsingular stopped products pass to the successful event
-when each successful path eventually lies inside every cutoff. -/
-lemma measure_le_of_eventual_cutoffs {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (E : Set Ω) (cutoff : ℕ → Set Ω) (C : ℝ≥0∞)
-    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
-    (hbound : ∀ n, P (cutoff n) ≤ C) : P E ≤ C := by
-  let tail (N : ℕ) := ⋂ n ≥ N, cutoff n
-  have hmono : Monotone tail := by
-    intro N M hNM ω hω
-    simp only [tail, mem_iInter] at hω ⊢
-    exact fun n hn => hω n (hNM.trans hn)
-  have hsub : E ⊆ ⋃ N, tail N := by
-    intro ω hω
-    obtain ⟨N, hN⟩ := eventually_atTop.mp (hcover ω hω)
-    exact mem_iUnion.mpr ⟨N, mem_iInter₂.mpr hN⟩
-  apply (measure_mono hsub).trans
-  rw [hmono.measure_iUnion]
-  apply iSup_le
-  intro N
-  exact (measure_mono (show tail N ⊆ cutoff N from
-    fun ω hω => mem_iInter₂.mp hω N le_rfl)).trans (hbound N)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Exhaustion of successful multipoint paths by nonsingular cutoffs.
+Each point is monitored only through its own completion time. Later
+swallowing of an already completed point does not enter this lemma. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma eventually_compact_positive_reciprocal {X : Type*} [TopologicalSpace X]
+    {K : Set X} (hK : IsCompact K) {f : X → ℝ} (hf : ContinuousOn f K)
+    (hpos : ∀ x ∈ K, 0 < f x) :
+    ∀ᶠ n : ℕ in atTop, ∀ x ∈ K, 1 / ((n : ℝ)+1) < f x := by
+  by_cases hne : K.Nonempty
+  · obtain ⟨x, hx, hmin⟩ := hK.exists_isMinOn hne hf
+    filter_upwards [(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+      (gt_mem_nhds (hpos x hx))] with n hn
+    intro y hy
+    exact hn.trans_le (hmin hy)
+  · exact Eventually.of_forall (fun _ x hx => False.elim (hne ⟨x, hx⟩))
+
+/-- On a successful path, finitely many individually stopped points admit
+one common height, angle, and physical-time cutoff. -/
+theorem successful_path_eventually_inside_cutoff
+    {ι : Type*} [Finite ι] (T : ι → ℝ) (Y θ : ι → ℝ → ℝ)
+    (hY : ∀ i, ContinuousOn (Y i) (Icc 0 (T i)))
+    (hθ : ∀ i, ContinuousOn (θ i) (Icc 0 (T i)))
+    (hYpos : ∀ i t, t ∈ Icc 0 (T i) → 0 < Y i t)
+    (hθpos : ∀ i t, t ∈ Icc 0 (T i) → 0 < θ i t)
+    (hθpi : ∀ i t, t ∈ Icc 0 (T i) → θ i t < Real.pi) :
+    ∀ᶠ n : ℕ in atTop, ∀ i,
+      T i < (n : ℝ)+1 ∧ ∀ t ∈ Icc 0 (T i),
+        1 / ((n : ℝ)+1) < Y i t ∧
+        1 / ((n : ℝ)+1) < θ i t ∧
+        θ i t < Real.pi - 1 / ((n : ℝ)+1) := by
+  apply eventually_all.mpr
+  intro i
+  have htime : ∀ᶠ n : ℕ in atTop, T i < (n : ℝ)+1 := by
+    obtain ⟨N, hN⟩ := exists_nat_gt (T i)
+    filter_upwards [eventually_ge_atTop N] with n hn
+    have hcast : (N : ℝ) ≤ n := by exact_mod_cast hn
+    linarith
+  have hheight := eventually_compact_positive_reciprocal isCompact_Icc (hY i) (hYpos i)
+  have hangle := eventually_compact_positive_reciprocal isCompact_Icc (hθ i) (hθpos i)
+  have hupper := eventually_compact_positive_reciprocal isCompact_Icc
+    (continuousOn_const.sub (hθ i)) (fun t ht => sub_pos.mpr (hθpi i t ht))
+  filter_upwards [htime, hheight, hangle, hupper] with n hn hy ha hb
+  refine ⟨hn, fun t ht => ⟨hy t ht, ha t ht, ?_⟩⟩
+  have hupper' : 1 / ((n : ℝ)+1) < Real.pi - θ i t := hb t ht
+  linarith
+
+/-- Bounds for nonsingular stopped products pass to the successful event
+when each successful path eventually lies inside every cutoff. -/
+lemma measure_le_of_eventual_cutoffs {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (E : Set Ω) (cutoff : ℕ → Set Ω) (C : ℝ≥0∞)
+    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
+    (hbound : ∀ n, P (cutoff n) ≤ C) : P E ≤ C := by
+  let tail (N : ℕ) := ⋂ n ≥ N, cutoff n
+  have hmono : Monotone tail := by
+    intro N M hNM ω hω
+    simp only [tail, mem_iInter] at hω ⊢
+    exact fun n hn => hω n (hNM.trans hn)
+  have hsub : E ⊆ ⋃ N, tail N := by
+    intro ω hω
+    obtain ⟨N, hN⟩ := eventually_atTop.mp (hcover ω hω)
+    exact mem_iUnion.mpr ⟨N, mem_iInter₂.mpr hN⟩
+  apply (measure_mono hsub).trans
+  rw [hmono.measure_iUnion]
+  apply iSup_le
+  intro N
+  exact (measure_mono (show tail N ⊆ cutoff N from
+    fun ω hω => mem_iInter₂.mp hω N le_rfl)).trans (hbound N)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffExhaustion.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffExhaustion.lean
index f4a9f53073331d1e68f1f9886fa6f5124db1e001..d76c70e543146bbfe5c7edb95df6e5da57dd1b83 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffExhaustion.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffExhaustion.lean
@@ -1,95 +1,95 @@
-import OAI.Probability.SLE.Multipoint.MultipointCutoffSuccess
-
-namespace OAI
-
-/-! Integer localization bounds eventually contain every successful finite
-family of completion prefixes. Height monotonicity reduces the spatial
-bound to the two endpoints of each individual prefix. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem completion_prefixes_bounded_by_nat
-    {ι : Type*} [Fintype ι]
-    (B : ℝ≥0 → ℝ) (hB : Continuous B)
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F)
-    (z : ι → ℂ) (τ : ι → ℝ≥0)
-    (hlive : ∀ i, z i ∈ survivingDomain γ (τ i)) (T : ℝ≥0) :
-    ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
-      (∀ t : ℝ≥0, t ≤ T → |B t| < (n : ℝ)) ∧
-      (∀ i (t : ℝ≥0), t ≤ τ i →
-        (logLoewnerCoordinates G (z i) t).1.im ∈ Ioo (-(n : ℝ)) (n : ℝ)) := by
-  classical
-  obtain ⟨b,hb⟩ := continuous_path_bounded_by_nat B hB T
-  let C : ℝ := ∑ i, (|Real.log (G (τ i) (z i)).im| + |Real.log (z i).im|)
-  obtain ⟨k,hk⟩ := exists_nat_gt C
-  refine ⟨max (b+1) k, fun n hn => ⟨?_, ?_⟩⟩
-  · intro t ht
-    have hbn : (b : ℝ)+1 ≤ (n : ℝ) := by exact_mod_cast (le_max_left (b+1) k).trans hn
-    exact (hb t ht).trans_le hbn
-  · intro i t ht
-    have hi : |Real.log (G (τ i) (z i)).im| + |Real.log (z i).im| ≤ C := by
-      exact Finset.single_le_sum
-        (f := fun j => |Real.log (G (τ j) (z j)).im| + |Real.log (z j).im|)
-        (fun j _ => add_nonneg (abs_nonneg _) (abs_nonneg _)) (Finset.mem_univ i)
-    have hkn : (k : ℝ) ≤ (n : ℝ) := by exact_mod_cast (le_max_right (b+1) k).trans hn
-    have hh := hw.log_height_bounds (τ i) (z i) (hlive i) t
-      (show (t : ℝ) ∈ Icc 0 (τ i : ℝ) from ⟨t.coe_nonneg, by exact_mod_cast ht⟩)
-    simp only [Real.toNNReal_coe] at hh
-    have hl := neg_abs_le (Real.log (G (τ i) (z i)).im)
-    have hu := le_abs_self (Real.log (z i).im)
-    have hla := abs_nonneg (Real.log (G (τ i) (z i)).im)
-    have hua := abs_nonneg (Real.log (z i).im)
-    have hresult : Real.log (G t (z i)).im ∈ Ioo (-(n : ℝ)) (n : ℝ) := by
-      constructor <;> linarith
-    simpa only [logLoewnerCoordinates, Real.toNNReal_coe, Complex.add_im,
-      Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im,
-      Complex.I_re, mul_one, mul_zero, add_zero, zero_add] using hresult
-
-/-- The standard integer localizations eventually survive beyond any
-fixed successful completion horizon. Their frozen completion times may
-be horizon truncations, provided those truncations eventually agree. -/
-theorem eventually_localization_after_completed
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ A : ℝ) (hA : 0 ≤ A) (z : ι → ℂ)
-    (τ : ℕ → ι → Ω → ℝ≥0)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) (F ω))
-    (S : ∀ n : ℕ, LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
-      ((n : ℝ≥0)+1) z (τ n) γ G)
-    (ω : Ω) (σ : ι → ℝ≥0) (T : ℝ≥0)
-    (hσ : ∀ i, σ i ≤ T)
-    (hlive : ∀ i, z i ∈ survivingDomain (γ ω) (σ i))
-    (hτ : ∀ᶠ n : ℕ in atTop, ∀ i, τ n i ω = σ i) :
-    ∀ᶠ n : ℕ in atTop, T < (S n).χ ω := by
-  obtain ⟨N,hN⟩ := completion_prefixes_bounded_by_nat
-    (fun t => B t ω) (hBc ω) (hw ω) z σ hlive T
-  have hlarge : ∀ᶠ n : ℕ in atTop, (T : ℝ) < (n : ℝ) :=
-    (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).eventually
-      (eventually_gt_atTop (T : ℝ))
-  filter_upwards [eventually_ge_atTop N, hlarge, hτ] with n hn hT hτn
-  have hupper : (n : ℝ) ≤ A+n+1 := by linarith
-  apply (S n).cutoff_after_completed ω T
-  · exact_mod_cast (show (T : ℝ) < (n : ℝ)+1 from by linarith)
-  · intro i
-    rw [hτn i]
-    exact hσ i
-  · intro t ht
-    exact ((hN n hn).1 t ht).trans_le hupper
-  · intro i t ht
-    rw [hτn i] at ht
-    have hi := (hN n hn).2 i t ht
-    exact ⟨(neg_le_neg hupper).trans_lt hi.1, hi.2.trans_le hupper⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointCutoffSuccess
+
+namespace OAI
+
+/-! Integer localization bounds eventually contain every successful finite
+family of completion prefixes. Height monotonicity reduces the spatial
+bound to the two endpoints of each individual prefix. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem completion_prefixes_bounded_by_nat
+    {ι : Type*} [Fintype ι]
+    (B : ℝ≥0 → ℝ) (hB : Continuous B)
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F)
+    (z : ι → ℂ) (τ : ι → ℝ≥0)
+    (hlive : ∀ i, z i ∈ survivingDomain γ (τ i)) (T : ℝ≥0) :
+    ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
+      (∀ t : ℝ≥0, t ≤ T → |B t| < (n : ℝ)) ∧
+      (∀ i (t : ℝ≥0), t ≤ τ i →
+        (logLoewnerCoordinates G (z i) t).1.im ∈ Ioo (-(n : ℝ)) (n : ℝ)) := by
+  classical
+  obtain ⟨b,hb⟩ := continuous_path_bounded_by_nat B hB T
+  let C : ℝ := ∑ i, (|Real.log (G (τ i) (z i)).im| + |Real.log (z i).im|)
+  obtain ⟨k,hk⟩ := exists_nat_gt C
+  refine ⟨max (b+1) k, fun n hn => ⟨?_, ?_⟩⟩
+  · intro t ht
+    have hbn : (b : ℝ)+1 ≤ (n : ℝ) := by exact_mod_cast (le_max_left (b+1) k).trans hn
+    exact (hb t ht).trans_le hbn
+  · intro i t ht
+    have hi : |Real.log (G (τ i) (z i)).im| + |Real.log (z i).im| ≤ C := by
+      exact Finset.single_le_sum
+        (f := fun j => |Real.log (G (τ j) (z j)).im| + |Real.log (z j).im|)
+        (fun j _ => add_nonneg (abs_nonneg _) (abs_nonneg _)) (Finset.mem_univ i)
+    have hkn : (k : ℝ) ≤ (n : ℝ) := by exact_mod_cast (le_max_right (b+1) k).trans hn
+    have hh := hw.log_height_bounds (τ i) (z i) (hlive i) t
+      (show (t : ℝ) ∈ Icc 0 (τ i : ℝ) from ⟨t.coe_nonneg, by exact_mod_cast ht⟩)
+    simp only [Real.toNNReal_coe] at hh
+    have hl := neg_abs_le (Real.log (G (τ i) (z i)).im)
+    have hu := le_abs_self (Real.log (z i).im)
+    have hla := abs_nonneg (Real.log (G (τ i) (z i)).im)
+    have hua := abs_nonneg (Real.log (z i).im)
+    have hresult : Real.log (G t (z i)).im ∈ Ioo (-(n : ℝ)) (n : ℝ) := by
+      constructor <;> linarith
+    simpa only [logLoewnerCoordinates, Real.toNNReal_coe, Complex.add_im,
+      Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im,
+      Complex.I_re, mul_one, mul_zero, add_zero, zero_add] using hresult
+
+/-- The standard integer localizations eventually survive beyond any
+fixed successful completion horizon. Their frozen completion times may
+be horizon truncations, provided those truncations eventually agree. -/
+theorem eventually_localization_after_completed
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ A : ℝ) (hA : 0 ≤ A) (z : ι → ℂ)
+    (τ : ℕ → ι → Ω → ℝ≥0)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) (F ω))
+    (S : ∀ n : ℕ, LocalizedState B hm κ (-(A+n+1)) (A+n+1) (A+n+1)
+      ((n : ℝ≥0)+1) z (τ n) γ G)
+    (ω : Ω) (σ : ι → ℝ≥0) (T : ℝ≥0)
+    (hσ : ∀ i, σ i ≤ T)
+    (hlive : ∀ i, z i ∈ survivingDomain (γ ω) (σ i))
+    (hτ : ∀ᶠ n : ℕ in atTop, ∀ i, τ n i ω = σ i) :
+    ∀ᶠ n : ℕ in atTop, T < (S n).χ ω := by
+  obtain ⟨N,hN⟩ := completion_prefixes_bounded_by_nat
+    (fun t => B t ω) (hBc ω) (hw ω) z σ hlive T
+  have hlarge : ∀ᶠ n : ℕ in atTop, (T : ℝ) < (n : ℝ) :=
+    (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).eventually
+      (eventually_gt_atTop (T : ℝ))
+  filter_upwards [eventually_ge_atTop N, hlarge, hτ] with n hn hT hτn
+  have hupper : (n : ℝ) ≤ A+n+1 := by linarith
+  apply (S n).cutoff_after_completed ω T
+  · exact_mod_cast (show (T : ℝ) < (n : ℝ)+1 from by linarith)
+  · intro i
+    rw [hτn i]
+    exact hσ i
+  · intro t ht
+    exact ((hN n hn).1 t ht).trans_le hupper
+  · intro i t ht
+    rw [hτn i] at ht
+    have hi := (hN n hn).2 i t ht
+    exact ⟨(neg_le_neg hupper).trans_lt hi.1, hi.2.trans_le hupper⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffSuccess.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffSuccess.lean
index 386225aa50185b1122a3825b3a5c02e6da953c10..171c07f0d1937cd5a2f7e2ecf464afa820f17d7f 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffSuccess.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointCutoffSuccess.lean
@@ -1,62 +1,62 @@
-import OAI.Probability.SLE.Multipoint.MultipointLocalizedState
-
-namespace OAI
-
-/-! A successful path remains before the common localization boundary
-once its finitely many completion prefixes have strict bounds. -/
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma LocalizedState.at_cutoff_eq_actual
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi M : ℝ} {H : ℝ≥0} {z : ι → ℂ} {τ : ι → Ω → ℝ≥0}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (S : LocalizedState B hm κ lo hi M H z τ γ G) (ω : Ω) (i : ι) :
-    S.Q (S.χ ω) ω i =
-      logLoewnerCoordinates (G ω) (z i) (min (τ i ω) (S.χ ω) : ℝ≥0) := by
-  have ha := (S.actual ω i).2
-    (show ((min (τ i ω) (S.χ ω) : ℝ≥0) : ℝ) ∈
-      Icc (0 : ℝ) ((min (τ i ω) (S.χ ω) : ℝ≥0) : ℝ) from
-      ⟨(min (τ i ω) (S.χ ω)).coe_nonneg,le_rfl⟩)
-  by_cases hτχ : τ i ω ≤ S.χ ω
-  · rw [min_eq_left hτχ] at ha ⊢
-    exact (S.frozen ω (S.χ ω) i hτχ).trans ha
-  · have hχτ : S.χ ω ≤ τ i ω := le_of_not_ge hτχ
-    simpa only [min_eq_right hχτ] using ha
-
-/-- Every point is monitored only up to its own completion. Once all
-completion prefixes satisfy the strict bounds, the localization cannot
-occur before the common completion horizon. -/
-theorem LocalizedState.cutoff_after_completed
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi M : ℝ} {H : ℝ≥0} {z : ι → ℂ} {τ : ι → Ω → ℝ≥0}
-    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-    (S : LocalizedState B hm κ lo hi M H z τ γ G) (ω : Ω)
-    (T : ℝ≥0) (hT : T < H) (_hτ : ∀ i, τ i ω ≤ T)
-    (hB : ∀ t : ℝ≥0, t ≤ T → |B t ω| < M)
-    (hheight : ∀ i (t : ℝ≥0), t ≤ τ i ω →
-      (logLoewnerCoordinates (G ω) (z i) t).1.im ∈ Ioo lo hi) :
-    T < S.χ ω := by
-  by_contra hn
-  have hχT : S.χ ω ≤ T := le_of_not_gt hn
-  have hχH : S.χ ω < H := hχT.trans_lt hT
-  have hstrict (i : ι) : (S.Q (S.χ ω) ω i).1.im ∈ Ioo lo hi := by
-    rw [S.at_cutoff_eq_actual ω i]
-    exact hheight i _ (min_le_left _ _)
-  rcases S.boundary ω hχH with ⟨i,hi⟩ | hb
-  · rcases hi with hlow | hhigh
-    · exact (ne_of_gt (hstrict i).1) hlow
-    · exact (ne_of_lt (hstrict i).2) hhigh
-  · exact (ne_of_lt (hB (S.χ ω) hχT)) hb
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLocalizedState
+
+namespace OAI
+
+/-! A successful path remains before the common localization boundary
+once its finitely many completion prefixes have strict bounds. -/
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma LocalizedState.at_cutoff_eq_actual
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi M : ℝ} {H : ℝ≥0} {z : ι → ℂ} {τ : ι → Ω → ℝ≥0}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (S : LocalizedState B hm κ lo hi M H z τ γ G) (ω : Ω) (i : ι) :
+    S.Q (S.χ ω) ω i =
+      logLoewnerCoordinates (G ω) (z i) (min (τ i ω) (S.χ ω) : ℝ≥0) := by
+  have ha := (S.actual ω i).2
+    (show ((min (τ i ω) (S.χ ω) : ℝ≥0) : ℝ) ∈
+      Icc (0 : ℝ) ((min (τ i ω) (S.χ ω) : ℝ≥0) : ℝ) from
+      ⟨(min (τ i ω) (S.χ ω)).coe_nonneg,le_rfl⟩)
+  by_cases hτχ : τ i ω ≤ S.χ ω
+  · rw [min_eq_left hτχ] at ha ⊢
+    exact (S.frozen ω (S.χ ω) i hτχ).trans ha
+  · have hχτ : S.χ ω ≤ τ i ω := le_of_not_ge hτχ
+    simpa only [min_eq_right hχτ] using ha
+
+/-- Every point is monitored only up to its own completion. Once all
+completion prefixes satisfy the strict bounds, the localization cannot
+occur before the common completion horizon. -/
+theorem LocalizedState.cutoff_after_completed
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi M : ℝ} {H : ℝ≥0} {z : ι → ℂ} {τ : ι → Ω → ℝ≥0}
+    {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+    (S : LocalizedState B hm κ lo hi M H z τ γ G) (ω : Ω)
+    (T : ℝ≥0) (hT : T < H) (_hτ : ∀ i, τ i ω ≤ T)
+    (hB : ∀ t : ℝ≥0, t ≤ T → |B t ω| < M)
+    (hheight : ∀ i (t : ℝ≥0), t ≤ τ i ω →
+      (logLoewnerCoordinates (G ω) (z i) t).1.im ∈ Ioo lo hi) :
+    T < S.χ ω := by
+  by_contra hn
+  have hχT : S.χ ω ≤ T := le_of_not_gt hn
+  have hχH : S.χ ω < H := hχT.trans_lt hT
+  have hstrict (i : ι) : (S.Q (S.χ ω) ω i).1.im ∈ Ioo lo hi := by
+    rw [S.at_cutoff_eq_actual ω i]
+    exact hheight i _ (min_le_left _ _)
+  rcases S.boundary ω hχH with ⟨i,hi⟩ | hb
+  · rcases hi with hlow | hhigh
+    · exact (ne_of_gt (hstrict i).1) hlow
+    · exact (ne_of_lt (hstrict i).2) hhigh
+  · exact (ne_of_lt (hB (S.χ ω) hχT)) hb
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountClock.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountClock.lean
index 49c6ea815b7e154e5b28f325a911bb93a4ebebf2..b4ccd4c48ae26310d1dd61f7db73e21432c16ac9 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountClock.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountClock.lean
@@ -1,93 +1,93 @@
-import OAI.Probability.SLE.OptionalStopping
-
-namespace OAI
-
-/-! The bounded accumulated drift used as the product's discount state. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def discountClock {Ω : Type*} (g : ℝ≥0 → Ω → ℝ) (t : ℝ) (ω : Ω) : ℝ :=
-  ∫ u in (0 : ℝ)..(t.toNNReal : ℝ), g u.toNNReal ω
-
-lemma discountClock_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (g : ℝ≥0 → Ω → ℝ) (hg : Measurable (Function.uncurry g)) :
-    Measurable (fun p : ℝ × Ω => discountClock g p.1 p.2) := by
-  have hgr : Measurable (fun p : ℝ × Ω => g p.1.toNNReal p.2) :=
-    hg.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-  exact (cumulative_integral_measurable _ hgr).comp
-    ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-
-lemma discountClock_initial {Ω : Type*} (g : ℝ≥0 → Ω → ℝ) (ω : Ω) :
-    discountClock g 0 ω = 0 := by simp [discountClock]
-
-lemma discountClock_lipschitz {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Continuous (fun t => g t ω))
-    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
-    LipschitzWith C (fun t => discountClock g t ω) := by
-  have hc : Continuous (fun u : ℝ => g u.toNNReal ω) := (hg ω).comp continuous_real_toNNReal
-  apply LipschitzWith.of_dist_le_mul
-  intro s t
-  rw [Real.dist_eq, ← Real.norm_eq_abs]
-  unfold discountClock
-  rw [intervalIntegral.integral_interval_sub_left (hc.intervalIntegrable _ _) (hc.intervalIntegrable _ _)]
-  have hb := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (t.toNNReal : ℝ)) (b := (s.toNNReal : ℝ)) (C := (C : ℝ))
-    (f := fun u : ℝ => g u.toNNReal ω)
-    (fun u _ => by simpa only [Real.norm_eq_abs] using hC u.toNNReal ω)
-  apply hb.trans
-  apply mul_le_mul_of_nonneg_left _ C.coe_nonneg
-  have ht := Real.lipschitzWith_toNNReal.dist_le_mul s t
-  simpa only [NNReal.dist_eq, Real.dist_eq, NNReal.coe_one, one_mul] using ht
-
-lemma discountClock_derivative {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Continuous (fun t => g t ω))
-    (t : ℝ) (ht : 0 ≤ t) (ω : Ω) :
-    HasDerivWithinAt (fun u => discountClock g u ω) (g t.toNNReal ω) (Ici 0) t := by
-  have hc : Continuous (fun u : ℝ => g u.toNNReal ω) := (hg ω).comp continuous_real_toNNReal
-  apply (hc.integral_hasStrictDerivAt 0 t).hasDerivAt.hasDerivWithinAt.congr_of_mem
-  · intro u hu
-    simp only [discountClock, Real.coe_toNNReal u hu]
-  · exact ht
-
-lemma discountClock_nonneg {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ t ω, 0 ≤ g t ω) (t : ℝ) (ω : Ω) :
-    0 ≤ discountClock g t ω := by
-  apply intervalIntegral.integral_nonneg_of_forall t.toNNReal.coe_nonneg
-  exact fun u => hg u.toNNReal ω
-
-lemma discountClock_adapted {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (f : Filtration ℝ≥0 mΩ) (g : ℝ≥0 → Ω → ℝ)
-    (hga : Adapted f g) (hgc : ∀ ω, Continuous (fun t => g t ω)) :
-    Adapted f (fun t : ℝ≥0 => discountClock g t) := by
-  have hsa : StronglyAdapted f g := fun t => (hga t).stronglyMeasurable
-  have hp := hsa.isStronglyProgressive_of_continuous hgc
-  intro t
-  have hpt := (hp t).measurable
-  let : MeasurableSpace Ω := f t
-  have htime : Measurable (fun u : ℝ =>
-      (⟨min u.toNNReal t, by change min u.toNNReal t ≤ t; exact min_le_right _ _⟩ : Iic t)) :=
-    ((continuous_real_toNNReal.min continuous_const).measurable).subtype_mk
-  have hgm : Measurable (fun p : ℝ × Ω => g (min p.1.toNNReal t) p.2) :=
-    hpt.comp ((htime.comp measurable_fst).prodMk measurable_snd)
-  have hm := (cumulative_integral_measurable (fun u ω => g (min u.toNNReal t) ω) hgm).comp
-    (measurable_const.prodMk measurable_id : Measurable (fun ω : Ω => (t,ω)))
-  convert hm using 1
-  funext ω
-  simp only [discountClock, Real.toNNReal_coe, Function.comp_def]
-  apply intervalIntegral.integral_congr
-  intro u hu
-  rw [uIcc_of_le t.coe_nonneg] at hu
-  have hut : u.toNNReal ≤ t := by
-    simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hu.2
-  change g u.toNNReal ω = g (min u.toNNReal t) ω
-  rw [min_eq_left hut]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.OptionalStopping
+
+namespace OAI
+
+/-! The bounded accumulated drift used as the product's discount state. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def discountClock {Ω : Type*} (g : ℝ≥0 → Ω → ℝ) (t : ℝ) (ω : Ω) : ℝ :=
+  ∫ u in (0 : ℝ)..(t.toNNReal : ℝ), g u.toNNReal ω
+
+lemma discountClock_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (g : ℝ≥0 → Ω → ℝ) (hg : Measurable (Function.uncurry g)) :
+    Measurable (fun p : ℝ × Ω => discountClock g p.1 p.2) := by
+  have hgr : Measurable (fun p : ℝ × Ω => g p.1.toNNReal p.2) :=
+    hg.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+  exact (cumulative_integral_measurable _ hgr).comp
+    ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+
+lemma discountClock_initial {Ω : Type*} (g : ℝ≥0 → Ω → ℝ) (ω : Ω) :
+    discountClock g 0 ω = 0 := by simp [discountClock]
+
+lemma discountClock_lipschitz {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Continuous (fun t => g t ω))
+    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
+    LipschitzWith C (fun t => discountClock g t ω) := by
+  have hc : Continuous (fun u : ℝ => g u.toNNReal ω) := (hg ω).comp continuous_real_toNNReal
+  apply LipschitzWith.of_dist_le_mul
+  intro s t
+  rw [Real.dist_eq, ← Real.norm_eq_abs]
+  unfold discountClock
+  rw [intervalIntegral.integral_interval_sub_left (hc.intervalIntegrable _ _) (hc.intervalIntegrable _ _)]
+  have hb := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (t.toNNReal : ℝ)) (b := (s.toNNReal : ℝ)) (C := (C : ℝ))
+    (f := fun u : ℝ => g u.toNNReal ω)
+    (fun u _ => by simpa only [Real.norm_eq_abs] using hC u.toNNReal ω)
+  apply hb.trans
+  apply mul_le_mul_of_nonneg_left _ C.coe_nonneg
+  have ht := Real.lipschitzWith_toNNReal.dist_le_mul s t
+  simpa only [NNReal.dist_eq, Real.dist_eq, NNReal.coe_one, one_mul] using ht
+
+lemma discountClock_derivative {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Continuous (fun t => g t ω))
+    (t : ℝ) (ht : 0 ≤ t) (ω : Ω) :
+    HasDerivWithinAt (fun u => discountClock g u ω) (g t.toNNReal ω) (Ici 0) t := by
+  have hc : Continuous (fun u : ℝ => g u.toNNReal ω) := (hg ω).comp continuous_real_toNNReal
+  apply (hc.integral_hasStrictDerivAt 0 t).hasDerivAt.hasDerivWithinAt.congr_of_mem
+  · intro u hu
+    simp only [discountClock, Real.coe_toNNReal u hu]
+  · exact ht
+
+lemma discountClock_nonneg {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ t ω, 0 ≤ g t ω) (t : ℝ) (ω : Ω) :
+    0 ≤ discountClock g t ω := by
+  apply intervalIntegral.integral_nonneg_of_forall t.toNNReal.coe_nonneg
+  exact fun u => hg u.toNNReal ω
+
+lemma discountClock_adapted {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (f : Filtration ℝ≥0 mΩ) (g : ℝ≥0 → Ω → ℝ)
+    (hga : Adapted f g) (hgc : ∀ ω, Continuous (fun t => g t ω)) :
+    Adapted f (fun t : ℝ≥0 => discountClock g t) := by
+  have hsa : StronglyAdapted f g := fun t => (hga t).stronglyMeasurable
+  have hp := hsa.isStronglyProgressive_of_continuous hgc
+  intro t
+  have hpt := (hp t).measurable
+  let : MeasurableSpace Ω := f t
+  have htime : Measurable (fun u : ℝ =>
+      (⟨min u.toNNReal t, by change min u.toNNReal t ≤ t; exact min_le_right _ _⟩ : Iic t)) :=
+    ((continuous_real_toNNReal.min continuous_const).measurable).subtype_mk
+  have hgm : Measurable (fun p : ℝ × Ω => g (min p.1.toNNReal t) p.2) :=
+    hpt.comp ((htime.comp measurable_fst).prodMk measurable_snd)
+  have hm := (cumulative_integral_measurable (fun u ω => g (min u.toNNReal t) ω) hgm).comp
+    (measurable_const.prodMk measurable_id : Measurable (fun ω : Ω => (t,ω)))
+  convert hm using 1
+  funext ω
+  simp only [discountClock, Real.toNNReal_coe, Function.comp_def]
+  apply intervalIntegral.integral_congr
+  intro u hu
+  rw [uIcc_of_le t.coe_nonneg] at hu
+  have hut : u.toNNReal ≤ t := by
+    simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hu.2
+  change g u.toNNReal ω = g (min u.toNNReal t) ω
+  rw [min_eq_left hut]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountCost.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountCost.lean
index 3ee0fa96431e0e7666e68fed482ced3b604b9ba2..4238549751da07e1e6c57fad7efe30852f1c634c 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountCost.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountCost.lean
@@ -1,107 +1,107 @@
-import OAI.Probability.SLE.Multipoint.MultipointProductGenerator
-
-namespace OAI
-
-/-! Continuous nonnegative drift cost and its bounded localization. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology BigOperators NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma continuous_logVelocity (κ : ℝ) :
-    Continuous (fun p : (ℂ × ℝ) × ℝ => logVelocity κ p.1 p.2) := by
-  have hre : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
-    Complex.continuous_re.comp continuous_fst.fst
-  have him : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
-    Complex.continuous_im.comp continuous_fst.fst
-  have hN : Continuous (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) :=
-    ((hre.sub (continuous_const.mul continuous_snd)).pow 2).add
-      (Real.continuous_exp.comp (continuous_const.mul him))
-  exact (Real.continuous_exp.comp him).div hN (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
-
-lemma continuous_branchRegulator (κ D σ : ℝ) : Continuous (branchRegulator κ D σ) := by
-  unfold branchRegulator
-  have h : Continuous (fun z : ℂ × ℝ => (σ*κ/4)*(z.2-z.1.im)) :=
-    continuous_const.mul (continuous_snd.sub (Complex.continuous_im.comp continuous_fst))
-  exact continuous_const.mul (Real.continuous_exp.comp h)
-
-lemma productDriftCost_continuous {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) :
-    Continuous (fun p : (ι → ℂ × ℝ) × ℝ => productDriftCost s κ D σ p.1 p.2) := by
-  have hz (i : ι) : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => p.1 i) :=
-    (continuous_apply i).comp continuous_fst
-  have hv (i : ι) : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => logVelocity κ (p.1 i) p.2) :=
-    (continuous_logVelocity κ).comp ((hz i).prodMk continuous_snd)
-  have hd (i : ι) : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => branchRegulator κ (D i) (σ i) (p.1 i)) :=
-    (continuous_branchRegulator κ (D i) (σ i)).comp (hz i)
-  unfold productDriftCost
-  apply continuous_const.mul
-  apply Continuous.add <;> apply continuous_const.mul
-  · exact continuous_finsetSum s (fun i _ => (hd i).mul ((hv i).pow 2))
-  · apply continuous_finsetSum
-    intro ij hij
-    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
-    apply Continuous.mul
-    · apply ((hd ij.1).mul (hd ij.2)).rpow_const
-      intro p
-      exact Or.inl (mul_pos (mul_pos (hD _ hi) (Real.exp_pos _))
-        (mul_pos (hD _ hj) (Real.exp_pos _))).ne'
-    · exact (hv ij.1).mul (hv ij.2)
-
-lemma productDriftCost_nonneg {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (z : ι → ℂ × ℝ) (x : ℝ) :
-    0 ≤ productDriftCost s κ D σ z x := by
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  have hC : 0 < driftAllowance (greenExponent κ)*driftThreshold (greenExponent κ)^(-greenExponent κ) :=
-    mul_pos (driftAllowance_pos hp) (Real.rpow_pos_of_pos (driftThreshold_pos hp) _)
-  have he : 0 < (negativeDrift (greenExponent κ)/2)/(4*((s.card : ℝ)+1)) :=
-    absorption_epsilon_pos (div_pos (negativeDrift_pos hp) (by norm_num)) _
-  have hL := youngConstant_pos hp (sq_pos_of_pos (angularCoefficientConstant_pos hp)) he
-  unfold productDriftCost
-  apply mul_nonneg hκ.le
-  apply add_nonneg
-  · apply mul_nonneg hC.le
-    apply Finset.sum_nonneg
-    intro i hi
-    exact mul_nonneg (mul_pos (hD i hi) (Real.exp_pos _)).le (sq_nonneg _)
-  · apply mul_nonneg hL.le
-    apply Finset.sum_nonneg
-    intro ij hij
-    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
-    exact mul_nonneg (Real.rpow_nonneg (mul_nonneg
-      (mul_pos (hD _ hi) (Real.exp_pos _)).le (mul_pos (hD _ hj) (Real.exp_pos _)).le) _)
-      (mul_pos (logVelocity_pos κ (z ij.1) x) (logVelocity_pos κ (z ij.2) x)).le
-
-theorem exists_bounded_discount_cost {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (R : ℝ) (hR : 0 ≤ R) :
-    ∃ (G : (ι → ℂ × ℝ) × ℝ → ℝ) (C : ℝ≥0),
-      Continuous G ∧ (∀ p, 0 ≤ G p ∧ |G p| ≤ C) ∧
-      (∀ p, ‖p‖ ≤ R → G p = productDriftCost s κ D σ p.1 p.2) := by
-  let b : ContDiffBump (0 : (ι → ℂ × ℝ) × ℝ) := ⟨R+1, R+2, by linarith, by linarith⟩
-  let G := fun p : (ι → ℂ × ℝ) × ℝ => b p*productDriftCost s κ D σ p.1 p.2
-  have hGc : Continuous G := b.continuous.mul (productDriftCost_continuous s κ D σ hD)
-  have hGs : HasCompactSupport G := b.hasCompactSupport.mul_right
-  obtain ⟨A, hA⟩ := hGc.bounded_above_of_compact_support hGs
-  refine ⟨G, Real.toNNReal A, hGc, ?_, ?_⟩
-  · intro p
-    refine ⟨mul_nonneg b.nonneg (productDriftCost_nonneg s κ hκ hκ8 D σ hD p.1 p.2), ?_⟩
-    exact (show |G p| ≤ A by simpa only [Real.norm_eq_abs] using hA p).trans (Real.le_coe_toNNReal A)
-  · intro p hp
-    have hb : b p = 1 := b.one_of_mem_closedBall (by
-      simp only [Metric.mem_closedBall, dist_zero_right]
-      change ‖p‖ ≤ R+1
-      linarith)
-    simp only [G, hb, one_mul]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointProductGenerator
+
+namespace OAI
+
+/-! Continuous nonnegative drift cost and its bounded localization. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology BigOperators NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma continuous_logVelocity (κ : ℝ) :
+    Continuous (fun p : (ℂ × ℝ) × ℝ => logVelocity κ p.1 p.2) := by
+  have hre : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
+    Complex.continuous_re.comp continuous_fst.fst
+  have him : Continuous (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
+    Complex.continuous_im.comp continuous_fst.fst
+  have hN : Continuous (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) :=
+    ((hre.sub (continuous_const.mul continuous_snd)).pow 2).add
+      (Real.continuous_exp.comp (continuous_const.mul him))
+  exact (Real.continuous_exp.comp him).div hN (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
+
+lemma continuous_branchRegulator (κ D σ : ℝ) : Continuous (branchRegulator κ D σ) := by
+  unfold branchRegulator
+  have h : Continuous (fun z : ℂ × ℝ => (σ*κ/4)*(z.2-z.1.im)) :=
+    continuous_const.mul (continuous_snd.sub (Complex.continuous_im.comp continuous_fst))
+  exact continuous_const.mul (Real.continuous_exp.comp h)
+
+lemma productDriftCost_continuous {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) :
+    Continuous (fun p : (ι → ℂ × ℝ) × ℝ => productDriftCost s κ D σ p.1 p.2) := by
+  have hz (i : ι) : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => p.1 i) :=
+    (continuous_apply i).comp continuous_fst
+  have hv (i : ι) : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => logVelocity κ (p.1 i) p.2) :=
+    (continuous_logVelocity κ).comp ((hz i).prodMk continuous_snd)
+  have hd (i : ι) : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => branchRegulator κ (D i) (σ i) (p.1 i)) :=
+    (continuous_branchRegulator κ (D i) (σ i)).comp (hz i)
+  unfold productDriftCost
+  apply continuous_const.mul
+  apply Continuous.add <;> apply continuous_const.mul
+  · exact continuous_finsetSum s (fun i _ => (hd i).mul ((hv i).pow 2))
+  · apply continuous_finsetSum
+    intro ij hij
+    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
+    apply Continuous.mul
+    · apply ((hd ij.1).mul (hd ij.2)).rpow_const
+      intro p
+      exact Or.inl (mul_pos (mul_pos (hD _ hi) (Real.exp_pos _))
+        (mul_pos (hD _ hj) (Real.exp_pos _))).ne'
+    · exact (hv ij.1).mul (hv ij.2)
+
+lemma productDriftCost_nonneg {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (z : ι → ℂ × ℝ) (x : ℝ) :
+    0 ≤ productDriftCost s κ D σ z x := by
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  have hC : 0 < driftAllowance (greenExponent κ)*driftThreshold (greenExponent κ)^(-greenExponent κ) :=
+    mul_pos (driftAllowance_pos hp) (Real.rpow_pos_of_pos (driftThreshold_pos hp) _)
+  have he : 0 < (negativeDrift (greenExponent κ)/2)/(4*((s.card : ℝ)+1)) :=
+    absorption_epsilon_pos (div_pos (negativeDrift_pos hp) (by norm_num)) _
+  have hL := youngConstant_pos hp (sq_pos_of_pos (angularCoefficientConstant_pos hp)) he
+  unfold productDriftCost
+  apply mul_nonneg hκ.le
+  apply add_nonneg
+  · apply mul_nonneg hC.le
+    apply Finset.sum_nonneg
+    intro i hi
+    exact mul_nonneg (mul_pos (hD i hi) (Real.exp_pos _)).le (sq_nonneg _)
+  · apply mul_nonneg hL.le
+    apply Finset.sum_nonneg
+    intro ij hij
+    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
+    exact mul_nonneg (Real.rpow_nonneg (mul_nonneg
+      (mul_pos (hD _ hi) (Real.exp_pos _)).le (mul_pos (hD _ hj) (Real.exp_pos _)).le) _)
+      (mul_pos (logVelocity_pos κ (z ij.1) x) (logVelocity_pos κ (z ij.2) x)).le
+
+theorem exists_bounded_discount_cost {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (R : ℝ) (hR : 0 ≤ R) :
+    ∃ (G : (ι → ℂ × ℝ) × ℝ → ℝ) (C : ℝ≥0),
+      Continuous G ∧ (∀ p, 0 ≤ G p ∧ |G p| ≤ C) ∧
+      (∀ p, ‖p‖ ≤ R → G p = productDriftCost s κ D σ p.1 p.2) := by
+  let b : ContDiffBump (0 : (ι → ℂ × ℝ) × ℝ) := ⟨R+1, R+2, by linarith, by linarith⟩
+  let G := fun p : (ι → ℂ × ℝ) × ℝ => b p*productDriftCost s κ D σ p.1 p.2
+  have hGc : Continuous G := b.continuous.mul (productDriftCost_continuous s κ D σ hD)
+  have hGs : HasCompactSupport G := b.hasCompactSupport.mul_right
+  obtain ⟨A, hA⟩ := hGc.bounded_above_of_compact_support hGs
+  refine ⟨G, Real.toNNReal A, hGc, ?_, ?_⟩
+  · intro p
+    refine ⟨mul_nonneg b.nonneg (productDriftCost_nonneg s κ hκ hκ8 D σ hD p.1 p.2), ?_⟩
+    exact (show |G p| ≤ A by simpa only [Real.norm_eq_abs] using hA p).trans (Real.le_coe_toNNReal A)
+  · intro p hp
+    have hb : b p = 1 := b.one_of_mem_closedBall (by
+      simp only [Metric.mem_closedBall, dist_zero_right]
+      change ‖p‖ ≤ R+1
+      linarith)
+    simp only [G, hb, one_mul]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountProcess.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountProcess.lean
index 97ac841a691785ba3b38d65d33eea24ccbb09bfc..983200bdd7e4e321e1239fc87c7a885ad12c0952 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountProcess.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountProcess.lean
@@ -1,156 +1,156 @@
-import OAI.Probability.SLE.Stochastic.BrownianStateAE
-import OAI.Probability.SLE.Multipoint.MultipointCompactField
-import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
-import OAI.Probability.SLE.Multipoint.MultipointDiscountCost
-
-namespace OAI
-
-/-! The compact discounted product along adapted Loewner states. The
-discount rate may jump at stopping times. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem exists_bounded_discount_cost_process
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (Q : ℝ → Ω → (ι → ℂ × ℝ))
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q t))
-    (hQc : ∀ ω, Continuous (fun t => Q t ω))
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (R : ℝ) (hR : 0 ≤ R) :
-    ∃ (g : ℝ≥0 → Ω → ℝ) (C : ℝ≥0),
-      Measurable (Function.uncurry g) ∧
-      IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) g ∧
-      (∀ ω, Continuous (fun t => g t ω)) ∧
-      (∀ t ω, 0 ≤ g t ω ∧ |g t ω| ≤ C) ∧
-      (∀ (t : ℝ≥0) ω, ‖(Q t ω,B t ω)‖ ≤ R →
-        g t ω = productDriftCost s κ D σ (Q t ω) (B t ω)) := by
-  obtain ⟨G, C, hG, hGb, hGe⟩ := exists_bounded_discount_cost s κ hκ hκ8 D σ hD R hR
-  let g := fun (t : ℝ≥0) ω => G (Q t ω,B t ω)
-  have hQn : Measurable (fun p : ℝ≥0 × Ω => Q p.1 p.2) :=
-    hQm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hga : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) g :=
-    fun t => hG.measurable.comp ((hQa t).prodMk ((Filtration.stronglyAdapted_natural
-      (fun t => (hm t).stronglyMeasurable)).adapted t))
-  have hgc (ω : Ω) : Continuous (fun t => g t ω) :=
-    hG.comp (((hQc ω).comp NNReal.continuous_coe).prodMk (hBc ω))
-  refine ⟨g, C, hG.measurable.comp (hQn.prodMk hBm), ?_, hgc, ?_, ?_⟩
-  · have hs : StronglyAdapted (Filtration.natural B
-        (fun t => (hm t).stronglyMeasurable)) g := fun t => (hga t).stronglyMeasurable
-    exact (hs.isStronglyProgressive_of_continuous hgc).isProgressive
-  · exact fun t ω => hGb (Q t ω,B t ω)
-  · exact fun t ω ht => hGe (Q t ω,B t ω) ht
-
-theorem compact_discounted_smooth_field
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (Q V : ℝ → Ω → (ι → ℂ × ℝ)) (L : ℝ≥0)
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q t))
-    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
-    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
-    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
-    (r : ℝ≥0 → Ω → ℝ) (hrm : Measurable (Function.uncurry r))
-    (hrp : IsProgressive (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) r)
-    (C : ℝ≥0) (hrb : ∀ t ω, |r t ω| ≤ C)
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
-    (R : ℝ) (hR : 0 ≤ R) :
-    ∃ (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ) (K : ℝ≥0),
-      (∀ p, 0 ≤ φ p) ∧
-      (∀ z x, ‖(z,x)‖ ≤ R → φ (z,x) = discountedProduct s κ D σ z x) ∧
-      (∀ z x, ‖(z,x)‖ ≤ R → fderiv ℝ φ (z,x) =
-        fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x)) ∧
-      (∀ z x, ‖(z,x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (z,y)) x =
-        iteratedDeriv 2 (discountedProduct s κ D σ z) x) ∧
-      SmoothTimeBrownianField B hm K
-        (fun t ω x => φ ((Q t ω, discountClock r t ω),x))
-        (fun t ω x => fderiv ℝ φ ((Q t ω, discountClock r t ω),x)
-          ((V t ω, r t ω),0)) := by
-  obtain ⟨φ, hφ, hφc, hφ0, he, heD, he2⟩ :=
-    exists_compact_discounted_field s κ D σ hD R hR
-  have hrt (ω : Ω) : Measurable (fun t => r t ω) :=
-    hrm.comp (measurable_id.prodMk measurable_const)
-  have hcm := discountClock_measurable r hrm
-  have hca := discountClock_adapted_of_progressive _ r hrp
-  have hcl := discountClock_lipschitz_of_measurable r hrt C hrb
-  have hcd := discountClock_ae_derivative r hrt C hrb
-  let Q' := fun t ω => (Q t ω, discountClock r t ω)
-  let V' := fun t ω => (V t ω, r t.toNNReal ω)
-  have hQ'm : Measurable (fun p : ℝ × Ω => Q' p.1 p.2) := hQm.prodMk hcm
-  have hQ'a : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q' t) := fun t => (hQa t).prodMk (hca t)
-  have hQ'l (ω : Ω) : LipschitzWith (max L C) (fun t => Q' t ω) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro t u
-    rw [Prod.dist_eq]
-    exact max_le
-      (((hQl ω).dist_le_mul t u).trans
-        (mul_le_mul_of_nonneg_right (by exact_mod_cast le_max_left L C) dist_nonneg))
-      (((hcl ω).dist_le_mul t u).trans
-        (mul_le_mul_of_nonneg_right (by exact_mod_cast le_max_right L C) dist_nonneg))
-  have hrm' : Measurable (fun p : ℝ × Ω => r p.1.toNNReal p.2) :=
-    hrm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hV'm : Measurable (fun p : ℝ × Ω => V' p.1 p.2) := hVm.prodMk hrm'
-  have hV'b (t : ℝ) (ω : Ω) : ‖V' t ω‖ ≤ (max L C : ℝ≥0) := by
-    rw [Prod.norm_def]
-    have hrnorm : ‖r t.toNNReal ω‖ ≤ (C : ℝ) := by
-      simpa only [Real.norm_eq_abs] using hrb t.toNNReal ω
-    exact max_le
-      ((hVb t ω).trans (by exact_mod_cast le_max_left L C))
-      (hrnorm.trans (by exact_mod_cast le_max_right L C))
-  have hQ'd (ω : Ω) : ∀ᵐ t : ℝ, 0 < t →
-      HasDerivAt (fun u => Q' u ω) (V' t ω) t := by
-    filter_upwards [hQd ω, hcd ω] with t hq hc
-    intro ht
-    exact (hq ht).prodMk (hc ht)
-  obtain ⟨K, hK⟩ := compact_state_smooth_brownian_field_of_ae_deriv B hm Q' V'
-    (max L C) hQ'm hQ'a hQ'l hV'm hV'b hQ'd φ hφ hφc
-  refine ⟨φ, K, hφ0, he, heD, he2, ?_⟩
-  simpa only [Q', V', Real.toNNReal_coe] using hK
-
-lemma compact_discounted_process_drift_nonpos
-    {Ω ι : Type*} [Fintype ι] [DecidableEq ι]
-    (B : ℝ≥0 → Ω → ℝ) (Q V : ℝ → Ω → (ι → ℂ × ℝ))
-    (r : ℝ≥0 → Ω → ℝ) (s : Finset ι)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (D σ : ι → ℝ)
-    (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ) (t : ℝ) (ht : 0 ≤ t) (ω : Ω)
-    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
-    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (Q t ω i) ≤ 1)
-    (hV : ∀ i ∈ s, V t ω i = physicalLogLoewnerField
-      (Real.sqrt κ*B t.toNNReal ω) (Q t ω i))
-    (hr : r t.toNNReal ω = productDriftCost s κ D σ (Q t ω) (B t.toNNReal ω))
-    (hφD : fderiv ℝ φ ((Q t ω,discountClock r t ω),B t.toNNReal ω) =
-      fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2)
-        ((Q t ω,discountClock r t ω),B t.toNNReal ω))
-    (hφ2 : iteratedDeriv 2 (fun y => φ ((Q t ω,discountClock r t ω),y))
-      (B t.toNNReal ω) =
-      iteratedDeriv 2 (discountedProduct s κ D σ (Q t ω,discountClock r t ω))
-        (B t.toNNReal ω)) :
-    brownianFieldDrift B
-      (fun t ω x => φ ((Q t ω,discountClock r t ω),x))
-      (fun t ω x => fderiv ℝ φ ((Q t ω,discountClock r t ω),x)
-        ((V t ω,r t ω),0)) t ω ≤ 0 := by
-  unfold brownianFieldDrift
-  simp only [Real.coe_toNNReal t ht]
-  rw [hr, hφD, hφ2, discountedProduct_pair_directional,
-    discountedProduct_directional_congr s κ D σ hD (Q t ω) _ _
-      (discountClock r t ω) (productDriftCost s κ D σ (Q t ω) (B t.toNNReal ω))
-      (B t.toNNReal ω) hV]
-  exact discountedProduct_generator_nonpos s κ hκ hκ8 D σ (Q t ω)
-    (discountClock r t ω) (B t.toNNReal ω) hD hσ hδ
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.BrownianStateAE
+import OAI.Probability.SLE.Multipoint.MultipointCompactField
+import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
+import OAI.Probability.SLE.Multipoint.MultipointDiscountCost
+
+namespace OAI
+
+/-! The compact discounted product along adapted Loewner states. The
+discount rate may jump at stopping times. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem exists_bounded_discount_cost_process
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (Q : ℝ → Ω → (ι → ℂ × ℝ))
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q t))
+    (hQc : ∀ ω, Continuous (fun t => Q t ω))
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (R : ℝ) (hR : 0 ≤ R) :
+    ∃ (g : ℝ≥0 → Ω → ℝ) (C : ℝ≥0),
+      Measurable (Function.uncurry g) ∧
+      IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) g ∧
+      (∀ ω, Continuous (fun t => g t ω)) ∧
+      (∀ t ω, 0 ≤ g t ω ∧ |g t ω| ≤ C) ∧
+      (∀ (t : ℝ≥0) ω, ‖(Q t ω,B t ω)‖ ≤ R →
+        g t ω = productDriftCost s κ D σ (Q t ω) (B t ω)) := by
+  obtain ⟨G, C, hG, hGb, hGe⟩ := exists_bounded_discount_cost s κ hκ hκ8 D σ hD R hR
+  let g := fun (t : ℝ≥0) ω => G (Q t ω,B t ω)
+  have hQn : Measurable (fun p : ℝ≥0 × Ω => Q p.1 p.2) :=
+    hQm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hga : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) g :=
+    fun t => hG.measurable.comp ((hQa t).prodMk ((Filtration.stronglyAdapted_natural
+      (fun t => (hm t).stronglyMeasurable)).adapted t))
+  have hgc (ω : Ω) : Continuous (fun t => g t ω) :=
+    hG.comp (((hQc ω).comp NNReal.continuous_coe).prodMk (hBc ω))
+  refine ⟨g, C, hG.measurable.comp (hQn.prodMk hBm), ?_, hgc, ?_, ?_⟩
+  · have hs : StronglyAdapted (Filtration.natural B
+        (fun t => (hm t).stronglyMeasurable)) g := fun t => (hga t).stronglyMeasurable
+    exact (hs.isStronglyProgressive_of_continuous hgc).isProgressive
+  · exact fun t ω => hGb (Q t ω,B t ω)
+  · exact fun t ω ht => hGe (Q t ω,B t ω) ht
+
+theorem compact_discounted_smooth_field
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (Q V : ℝ → Ω → (ι → ℂ × ℝ)) (L : ℝ≥0)
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q t))
+    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
+    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
+    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
+    (r : ℝ≥0 → Ω → ℝ) (hrm : Measurable (Function.uncurry r))
+    (hrp : IsProgressive (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) r)
+    (C : ℝ≥0) (hrb : ∀ t ω, |r t ω| ≤ C)
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
+    (R : ℝ) (hR : 0 ≤ R) :
+    ∃ (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ) (K : ℝ≥0),
+      (∀ p, 0 ≤ φ p) ∧
+      (∀ z x, ‖(z,x)‖ ≤ R → φ (z,x) = discountedProduct s κ D σ z x) ∧
+      (∀ z x, ‖(z,x)‖ ≤ R → fderiv ℝ φ (z,x) =
+        fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x)) ∧
+      (∀ z x, ‖(z,x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (z,y)) x =
+        iteratedDeriv 2 (discountedProduct s κ D σ z) x) ∧
+      SmoothTimeBrownianField B hm K
+        (fun t ω x => φ ((Q t ω, discountClock r t ω),x))
+        (fun t ω x => fderiv ℝ φ ((Q t ω, discountClock r t ω),x)
+          ((V t ω, r t ω),0)) := by
+  obtain ⟨φ, hφ, hφc, hφ0, he, heD, he2⟩ :=
+    exists_compact_discounted_field s κ D σ hD R hR
+  have hrt (ω : Ω) : Measurable (fun t => r t ω) :=
+    hrm.comp (measurable_id.prodMk measurable_const)
+  have hcm := discountClock_measurable r hrm
+  have hca := discountClock_adapted_of_progressive _ r hrp
+  have hcl := discountClock_lipschitz_of_measurable r hrt C hrb
+  have hcd := discountClock_ae_derivative r hrt C hrb
+  let Q' := fun t ω => (Q t ω, discountClock r t ω)
+  let V' := fun t ω => (V t ω, r t.toNNReal ω)
+  have hQ'm : Measurable (fun p : ℝ × Ω => Q' p.1 p.2) := hQm.prodMk hcm
+  have hQ'a : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q' t) := fun t => (hQa t).prodMk (hca t)
+  have hQ'l (ω : Ω) : LipschitzWith (max L C) (fun t => Q' t ω) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro t u
+    rw [Prod.dist_eq]
+    exact max_le
+      (((hQl ω).dist_le_mul t u).trans
+        (mul_le_mul_of_nonneg_right (by exact_mod_cast le_max_left L C) dist_nonneg))
+      (((hcl ω).dist_le_mul t u).trans
+        (mul_le_mul_of_nonneg_right (by exact_mod_cast le_max_right L C) dist_nonneg))
+  have hrm' : Measurable (fun p : ℝ × Ω => r p.1.toNNReal p.2) :=
+    hrm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hV'm : Measurable (fun p : ℝ × Ω => V' p.1 p.2) := hVm.prodMk hrm'
+  have hV'b (t : ℝ) (ω : Ω) : ‖V' t ω‖ ≤ (max L C : ℝ≥0) := by
+    rw [Prod.norm_def]
+    have hrnorm : ‖r t.toNNReal ω‖ ≤ (C : ℝ) := by
+      simpa only [Real.norm_eq_abs] using hrb t.toNNReal ω
+    exact max_le
+      ((hVb t ω).trans (by exact_mod_cast le_max_left L C))
+      (hrnorm.trans (by exact_mod_cast le_max_right L C))
+  have hQ'd (ω : Ω) : ∀ᵐ t : ℝ, 0 < t →
+      HasDerivAt (fun u => Q' u ω) (V' t ω) t := by
+    filter_upwards [hQd ω, hcd ω] with t hq hc
+    intro ht
+    exact (hq ht).prodMk (hc ht)
+  obtain ⟨K, hK⟩ := compact_state_smooth_brownian_field_of_ae_deriv B hm Q' V'
+    (max L C) hQ'm hQ'a hQ'l hV'm hV'b hQ'd φ hφ hφc
+  refine ⟨φ, K, hφ0, he, heD, he2, ?_⟩
+  simpa only [Q', V', Real.toNNReal_coe] using hK
+
+lemma compact_discounted_process_drift_nonpos
+    {Ω ι : Type*} [Fintype ι] [DecidableEq ι]
+    (B : ℝ≥0 → Ω → ℝ) (Q V : ℝ → Ω → (ι → ℂ × ℝ))
+    (r : ℝ≥0 → Ω → ℝ) (s : Finset ι)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (D σ : ι → ℝ)
+    (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ) (t : ℝ) (ht : 0 ≤ t) (ω : Ω)
+    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
+    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (Q t ω i) ≤ 1)
+    (hV : ∀ i ∈ s, V t ω i = physicalLogLoewnerField
+      (Real.sqrt κ*B t.toNNReal ω) (Q t ω i))
+    (hr : r t.toNNReal ω = productDriftCost s κ D σ (Q t ω) (B t.toNNReal ω))
+    (hφD : fderiv ℝ φ ((Q t ω,discountClock r t ω),B t.toNNReal ω) =
+      fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2)
+        ((Q t ω,discountClock r t ω),B t.toNNReal ω))
+    (hφ2 : iteratedDeriv 2 (fun y => φ ((Q t ω,discountClock r t ω),y))
+      (B t.toNNReal ω) =
+      iteratedDeriv 2 (discountedProduct s κ D σ (Q t ω,discountClock r t ω))
+        (B t.toNNReal ω)) :
+    brownianFieldDrift B
+      (fun t ω x => φ ((Q t ω,discountClock r t ω),x))
+      (fun t ω x => fderiv ℝ φ ((Q t ω,discountClock r t ω),x)
+        ((V t ω,r t ω),0)) t ω ≤ 0 := by
+  unfold brownianFieldDrift
+  simp only [Real.coe_toNNReal t ht]
+  rw [hr, hφD, hφ2, discountedProduct_pair_directional,
+    discountedProduct_directional_congr s κ D σ hD (Q t ω) _ _
+      (discountClock r t ω) (productDriftCost s κ D σ (Q t ω) (B t.toNNReal ω))
+      (B t.toNNReal ω) hV]
+  exact discountedProduct_generator_nonpos s κ hκ hκ8 D σ (Q t ω)
+    (discountClock r t ω) (B t.toNNReal ω) hD hσ hδ
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountedField.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountedField.lean
index 8ca38e215edce9c7140f869e2df3fa033dcec508..7d63aafbd2ab588fc6345f8f77d2cf4e2215c713 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountedField.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointDiscountedField.lean
@@ -1,74 +1,74 @@
-import OAI.Probability.SLE.Multipoint.MultipointProductGenerator
-
-namespace OAI
-
-/-! The smooth discounted product used between consecutive switches. -/
-
-noncomputable section
-
-namespace SLEExactGauge.Multipoint
-
-def discountedProduct {ι : Type*} (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
-    (z : (ι → ℂ × ℝ) × ℝ) (x : ℝ) : ℝ :=
-  Real.exp (-z.2)*regularizedProduct s κ D σ z.1 x
-
-lemma discountedProduct_contDiff {ι : Type*} [Fintype ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞)
-      (fun p : ((ι → ℂ × ℝ) × ℝ) × ℝ => discountedProduct s κ D σ p.1 p.2) := by
-  unfold discountedProduct
-  exact (contDiff_fst.snd.neg.exp).mul ((regularizedProduct_contDiff s κ D σ).comp
-    (contDiff_fst.fst.prodMk contDiff_snd))
-
-lemma discountedProduct_directional {ι : Type*} [Fintype ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
-    (z v : ι → ℂ × ℝ) (b c x : ℝ) :
-    (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (v,c) =
-      Real.exp (-b)*((fderiv ℝ (fun q => regularizedProduct s κ D σ q x) z) v -
-        c*regularizedProduct s κ D σ z x) := by
-  have hp : ContDiff ℝ (⊤ : ℕ∞) (fun z => regularizedProduct s κ D σ z x) :=
-    (regularizedProduct_contDiff s κ D σ).comp (contDiff_id.prodMk contDiff_const)
-  have hd := ((hasFDerivAt_snd (𝕜 := ℝ) (p := (z,b))).neg.exp).mul
-    (((hp.differentiable (by simp)) z).hasFDerivAt.comp (z,b) (hasFDerivAt_fst (𝕜 := ℝ)))
-  have he := congrArg (fun f : ((ι → ℂ × ℝ) × ℝ) →L[ℝ] ℝ => f (v,c)) hd.fderiv
-  change (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (v,c) = _ at he
-  rw [he]
-  simp only [smul_apply, smul_eq_mul, add_apply, neg_apply, ContinuousLinearMap.comp_apply,
-    ContinuousLinearMap.coe_fst', ContinuousLinearMap.coe_snd', Function.comp_def,
-    Pi.neg_apply]
-  ring
-
-lemma discountedProduct_spatial_second {ι : Type*} [Fintype ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (b x : ℝ) :
-    iteratedDeriv 2 (discountedProduct s κ D σ (z,b)) x =
-      Real.exp (-b)*iteratedDeriv 2 (regularizedProduct s κ D σ z) x := by
-  apply iteratedDeriv_const_mul
-  exact (((regularizedProduct_contDiff s κ D σ).comp
-    (contDiff_const.prodMk contDiff_id)).of_le (by simp)).contDiffAt
-
-lemma discountedProduct_directional_congr {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
-    (z v w : ι → ℂ × ℝ) (b c x : ℝ) (hvw : ∀ i ∈ s, v i = w i) :
-    (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (v,c) =
-      (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (w,c) := by
-  rw [discountedProduct_directional, discountedProduct_directional,
-    regularizedProduct_directional_congr s κ D σ hD z v w x hvw]
-
-theorem discountedProduct_generator_nonpos {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (b x : ℝ)
-    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
-    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1) :
-    (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b))
-        ((fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)), productDriftCost s κ D σ z x) +
-      (1/2 : ℝ)*iteratedDeriv 2 (discountedProduct s κ D σ (z,b)) x ≤ 0 := by
-  rw [discountedProduct_directional, discountedProduct_spatial_second]
-  have h := mul_le_mul_of_nonneg_left
-    (regularizedProduct_generator_le s κ hκ hκ8 D σ z x hD hσ hδ) (Real.exp_pos (-b)).le
-  nlinarith only [h]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointProductGenerator
+
+namespace OAI
+
+/-! The smooth discounted product used between consecutive switches. -/
+
+noncomputable section
+
+namespace SLEExactGauge.Multipoint
+
+def discountedProduct {ι : Type*} (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
+    (z : (ι → ℂ × ℝ) × ℝ) (x : ℝ) : ℝ :=
+  Real.exp (-z.2)*regularizedProduct s κ D σ z.1 x
+
+lemma discountedProduct_contDiff {ι : Type*} [Fintype ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞)
+      (fun p : ((ι → ℂ × ℝ) × ℝ) × ℝ => discountedProduct s κ D σ p.1 p.2) := by
+  unfold discountedProduct
+  exact (contDiff_fst.snd.neg.exp).mul ((regularizedProduct_contDiff s κ D σ).comp
+    (contDiff_fst.fst.prodMk contDiff_snd))
+
+lemma discountedProduct_directional {ι : Type*} [Fintype ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
+    (z v : ι → ℂ × ℝ) (b c x : ℝ) :
+    (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (v,c) =
+      Real.exp (-b)*((fderiv ℝ (fun q => regularizedProduct s κ D σ q x) z) v -
+        c*regularizedProduct s κ D σ z x) := by
+  have hp : ContDiff ℝ (⊤ : ℕ∞) (fun z => regularizedProduct s κ D σ z x) :=
+    (regularizedProduct_contDiff s κ D σ).comp (contDiff_id.prodMk contDiff_const)
+  have hd := ((hasFDerivAt_snd (𝕜 := ℝ) (p := (z,b))).neg.exp).mul
+    (((hp.differentiable (by simp)) z).hasFDerivAt.comp (z,b) (hasFDerivAt_fst (𝕜 := ℝ)))
+  have he := congrArg (fun f : ((ι → ℂ × ℝ) × ℝ) →L[ℝ] ℝ => f (v,c)) hd.fderiv
+  change (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (v,c) = _ at he
+  rw [he]
+  simp only [smul_apply, smul_eq_mul, add_apply, neg_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.coe_fst', ContinuousLinearMap.coe_snd', Function.comp_def,
+    Pi.neg_apply]
+  ring
+
+lemma discountedProduct_spatial_second {ι : Type*} [Fintype ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (b x : ℝ) :
+    iteratedDeriv 2 (discountedProduct s κ D σ (z,b)) x =
+      Real.exp (-b)*iteratedDeriv 2 (regularizedProduct s κ D σ z) x := by
+  apply iteratedDeriv_const_mul
+  exact (((regularizedProduct_contDiff s κ D σ).comp
+    (contDiff_const.prodMk contDiff_id)).of_le (by simp)).contDiffAt
+
+lemma discountedProduct_directional_congr {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
+    (z v w : ι → ℂ × ℝ) (b c x : ℝ) (hvw : ∀ i ∈ s, v i = w i) :
+    (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (v,c) =
+      (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b)) (w,c) := by
+  rw [discountedProduct_directional, discountedProduct_directional,
+    regularizedProduct_directional_congr s κ D σ hD z v w x hvw]
+
+theorem discountedProduct_generator_nonpos {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (b x : ℝ)
+    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
+    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1) :
+    (fderiv ℝ (fun q => discountedProduct s κ D σ q x) (z,b))
+        ((fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)), productDriftCost s κ D σ z x) +
+      (1/2 : ℝ)*iteratedDeriv 2 (discountedProduct s κ D σ (z,b)) x ≤ 0 := by
+  rw [discountedProduct_directional, discountedProduct_spatial_second]
+  have h := mul_le_mul_of_nonneg_left
+    (regularizedProduct_generator_le s κ hκ hκ8 D σ z x hD hσ hδ) (Real.exp_pos (-b)).le
+  nlinarith only [h]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointDistanceKernel.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointDistanceKernel.lean
index 698f0cca4924481dd54f72565be25cebe2effc0e..44c99cf3895118a2a081628845d4bce8bb76a5ac 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointDistanceKernel.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointDistanceKernel.lean
@@ -1,140 +1,140 @@
-import Mathlib.Analysis.SpecialFunctions.Pow.Integral
-import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
-import Mathlib.MeasureTheory.Group.LIntegral
-import Mathlib.MeasureTheory.Integral.Marginal
-import Mathlib.MeasureTheory.Integral.Pi
-import Mathlib.LinearAlgebra.Complex.FiniteDimensional
-
-namespace OAI
-
-/-! The local planar singularities used to integrate the capped distance
-kernel. Diagonal values are set to zero; they have zero planar measure. -/
-
-noncomputable section
-open Set Metric Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-def distanceKernel (α : ℝ) (x y : ℂ) : ℝ≥0∞ :=
-  ENNReal.ofReal (dist x y ^ (-α))
-
-lemma distanceKernel_symm (α : ℝ) (x y : ℂ) :
-    distanceKernel α x y = distanceKernel α y x := by
-  simp only [distanceKernel,dist_comm]
-
-@[fun_prop]
-lemma measurable_distanceKernel (α : ℝ) :
-    Measurable (fun p : ℂ × ℂ => distanceKernel α p.1 p.2) := by
-  unfold distanceKernel
-  fun_prop
-
-lemma measurable_distanceKernel_left (α : ℝ) (y : ℂ) :
-    Measurable (fun x : ℂ => distanceKernel α x y) := by
-  unfold distanceKernel
-  fun_prop
-
-def radialKernelMass (α : ℝ) : ℝ≥0∞ :=
-  ∫⁻ x : ℂ in ball 0 1, ENNReal.ofReal (‖x‖ ^ (-α))
-
-lemma radialKernelMass_lt_top {α : ℝ} (hα : α < 2) : radialKernelMass α < ⊤ := by
-  have hi : IntegrableOn (fun x : ℂ => ‖x‖ ^ (-α)) (ball 0 1) := by
-    apply integrableOn_ball_of_norm_le_rpow (μ := volume) (C := 1)
-      (α := α) (by simp [Complex.finrank_real_complex])
-      (by simpa only [Complex.finrank_real_complex, Nat.cast_ofNat] using hα)
-    · filter_upwards [] with x
-      rw [Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg (norm_nonneg _) _),one_mul]
-    · exact ((continuous_norm.measurable).pow_const _).aestronglyMeasurable
-  exact (hasFiniteIntegral_iff_ofReal
-    (ae_of_all _ (fun x => Real.rpow_nonneg (norm_nonneg x) _))).mp hi.hasFiniteIntegral
-
-lemma distanceKernel_le_local_add_one {α : ℝ} (hα : 0 ≤ α) (x y : ℂ) :
-    distanceKernel α x y ≤
-      (ball (0 : ℂ) 1).indicator (fun z => ENNReal.ofReal (‖z‖ ^ (-α))) (x-y) + 1 := by
-  by_cases h : x-y ∈ ball (0 : ℂ) 1
-  · rw [indicator_of_mem h]
-    simpa only [distanceKernel,dist_eq_norm] using
-      (le_add_right (le_refl (ENNReal.ofReal (‖x-y‖ ^ (-α)))))
-  · rw [indicator_of_notMem h,zero_add]
-    have hd : 1 ≤ dist x y := by
-      simpa only [mem_ball_zero_iff,not_lt,← dist_eq_norm] using h
-    exact ENNReal.ofReal_le_one.mpr (Real.rpow_le_one_of_one_le_of_nonpos hd (neg_nonpos.mpr hα))
-
-theorem distanceKernel_lintegral_le {α : ℝ} (hα : 0 ≤ α)
-    (K : Set ℂ) (y : ℂ) :
-    (∫⁻ x in K, distanceKernel α x y) ≤ radialKernelMass α + volume K := by
-  let f : ℂ → ℝ≥0∞ := (ball (0 : ℂ) 1).indicator
-    (fun z => ENNReal.ofReal (‖z‖ ^ (-α)))
-  have hfm : Measurable f := by
-    apply Measurable.indicator _ measurableSet_ball
-    fun_prop
-  have hshiftm : Measurable (fun x : ℂ => f (x-y)) :=
-    hfm.comp (measurable_id.sub_const y)
-  calc
-    (∫⁻ x in K, distanceKernel α x y) ≤ ∫⁻ x in K, f (x-y)+1 :=
-      lintegral_mono (distanceKernel_le_local_add_one hα · y)
-    _ = (∫⁻ x in K, f (x-y)) + volume K := by
-      rw [lintegral_add_left hshiftm,lintegral_const]
-      simp only [Measure.restrict_apply_univ,one_mul]
-    _ ≤ (∫⁻ x, f (x-y)) + volume K :=
-      add_le_add (lintegral_mono' Measure.restrict_le_self (fun _ => le_rfl)) le_rfl
-    _ = radialKernelMass α + volume K := by
-      rw [lintegral_sub_right_eq_self]
-      simp only [f,lintegral_indicator measurableSet_ball,radialKernelMass]
-
-lemma distanceKernel_mul_le_squares (α : ℝ) (x y z : ℂ) :
-    distanceKernel α x y * distanceKernel α x z ≤
-      distanceKernel (2*α) x y + distanceKernel (2*α) x z := by
-  have hs (w : ℂ) : (dist x w ^ (-α))^2 = dist x w ^ (-(2*α)) := by
-    rw [← Real.rpow_natCast,← Real.rpow_mul dist_nonneg]
-    congr 1
-    ring
-  unfold distanceKernel
-  rw [← ENNReal.ofReal_mul (Real.rpow_nonneg dist_nonneg _),
-    ← ENNReal.ofReal_add (Real.rpow_nonneg dist_nonneg _) (Real.rpow_nonneg dist_nonneg _)]
-  apply ENNReal.ofReal_le_ofReal
-  rw [← hs y,← hs z]
-  nlinarith [sq_nonneg ((dist x y ^ (-α))-(dist x z ^ (-α)))]
-
-theorem distanceKernel_pair_lintegral_le {α : ℝ} (hα : 0 ≤ α)
-    (K : Set ℂ) (y z : ℂ) :
-    (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤
-      2*(radialKernelMass (2*α)+volume K) := by
-  calc
-    (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤
-        ∫⁻ x in K, distanceKernel (2*α) x y + distanceKernel (2*α) x z :=
-      lintegral_mono (distanceKernel_mul_le_squares α · y z)
-    _ = (∫⁻ x in K, distanceKernel (2*α) x y) +
-        (∫⁻ x in K, distanceKernel (2*α) x z) := by
-      apply lintegral_add_left
-      exact measurable_distanceKernel_left (2*α) y
-    _ ≤ (radialKernelMass (2*α)+volume K)+(radialKernelMass (2*α)+volume K) :=
-      add_le_add (distanceKernel_lintegral_le (by positivity) K y)
-        (distanceKernel_lintegral_le (by positivity) K z)
-    _ = _ := (two_mul _).symm
-
-theorem distanceKernel_uniform_local_bounds {α : ℝ} (hα : 0 < α) (hα1 : α < 1)
-    (K : Set ℂ) (hK : Bornology.IsBounded K) :
-    ∃ C : ℝ≥0∞, C < ⊤ ∧ 1 ≤ C ∧ volume K ≤ C ∧
-      (∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C) ∧
-      (∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C) := by
-  let C := 1+volume K+(radialKernelMass α+volume K)+2*(radialKernelMass (2*α)+volume K)
-  have hvol : volume K < ⊤ := hK.measure_lt_top
-  have hmass : radialKernelMass α < ⊤ := radialKernelMass_lt_top (by linarith)
-  have hmass2 : radialKernelMass (2*α) < ⊤ := radialKernelMass_lt_top (by linarith)
-  refine ⟨C,by dsimp [C]; finiteness,?_,?_,?_,?_⟩
-  · dsimp [C]
-    exact (le_add_right le_rfl).trans ((le_add_right le_rfl).trans (le_add_right le_rfl))
-  · dsimp [C]
-    exact (le_add_left le_rfl).trans ((le_add_right le_rfl).trans (le_add_right le_rfl))
-  · intro y
-    exact (distanceKernel_lintegral_le hα.le K y).trans
-      ((le_add_left le_rfl).trans (le_add_right le_rfl))
-  · intro y z
-    exact (distanceKernel_pair_lintegral_le hα.le K y z).trans (le_add_left le_rfl)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import Mathlib.Analysis.SpecialFunctions.Pow.Integral
+import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
+import Mathlib.MeasureTheory.Group.LIntegral
+import Mathlib.MeasureTheory.Integral.Marginal
+import Mathlib.MeasureTheory.Integral.Pi
+import Mathlib.LinearAlgebra.Complex.FiniteDimensional
+
+namespace OAI
+
+/-! The local planar singularities used to integrate the capped distance
+kernel. Diagonal values are set to zero; they have zero planar measure. -/
+
+noncomputable section
+open Set Metric Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+def distanceKernel (α : ℝ) (x y : ℂ) : ℝ≥0∞ :=
+  ENNReal.ofReal (dist x y ^ (-α))
+
+lemma distanceKernel_symm (α : ℝ) (x y : ℂ) :
+    distanceKernel α x y = distanceKernel α y x := by
+  simp only [distanceKernel,dist_comm]
+
+@[fun_prop]
+lemma measurable_distanceKernel (α : ℝ) :
+    Measurable (fun p : ℂ × ℂ => distanceKernel α p.1 p.2) := by
+  unfold distanceKernel
+  fun_prop
+
+lemma measurable_distanceKernel_left (α : ℝ) (y : ℂ) :
+    Measurable (fun x : ℂ => distanceKernel α x y) := by
+  unfold distanceKernel
+  fun_prop
+
+def radialKernelMass (α : ℝ) : ℝ≥0∞ :=
+  ∫⁻ x : ℂ in ball 0 1, ENNReal.ofReal (‖x‖ ^ (-α))
+
+lemma radialKernelMass_lt_top {α : ℝ} (hα : α < 2) : radialKernelMass α < ⊤ := by
+  have hi : IntegrableOn (fun x : ℂ => ‖x‖ ^ (-α)) (ball 0 1) := by
+    apply integrableOn_ball_of_norm_le_rpow (μ := volume) (C := 1)
+      (α := α) (by simp [Complex.finrank_real_complex])
+      (by simpa only [Complex.finrank_real_complex, Nat.cast_ofNat] using hα)
+    · filter_upwards [] with x
+      rw [Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg (norm_nonneg _) _),one_mul]
+    · exact ((continuous_norm.measurable).pow_const _).aestronglyMeasurable
+  exact (hasFiniteIntegral_iff_ofReal
+    (ae_of_all _ (fun x => Real.rpow_nonneg (norm_nonneg x) _))).mp hi.hasFiniteIntegral
+
+lemma distanceKernel_le_local_add_one {α : ℝ} (hα : 0 ≤ α) (x y : ℂ) :
+    distanceKernel α x y ≤
+      (ball (0 : ℂ) 1).indicator (fun z => ENNReal.ofReal (‖z‖ ^ (-α))) (x-y) + 1 := by
+  by_cases h : x-y ∈ ball (0 : ℂ) 1
+  · rw [indicator_of_mem h]
+    simpa only [distanceKernel,dist_eq_norm] using
+      (le_add_right (le_refl (ENNReal.ofReal (‖x-y‖ ^ (-α)))))
+  · rw [indicator_of_notMem h,zero_add]
+    have hd : 1 ≤ dist x y := by
+      simpa only [mem_ball_zero_iff,not_lt,← dist_eq_norm] using h
+    exact ENNReal.ofReal_le_one.mpr (Real.rpow_le_one_of_one_le_of_nonpos hd (neg_nonpos.mpr hα))
+
+theorem distanceKernel_lintegral_le {α : ℝ} (hα : 0 ≤ α)
+    (K : Set ℂ) (y : ℂ) :
+    (∫⁻ x in K, distanceKernel α x y) ≤ radialKernelMass α + volume K := by
+  let f : ℂ → ℝ≥0∞ := (ball (0 : ℂ) 1).indicator
+    (fun z => ENNReal.ofReal (‖z‖ ^ (-α)))
+  have hfm : Measurable f := by
+    apply Measurable.indicator _ measurableSet_ball
+    fun_prop
+  have hshiftm : Measurable (fun x : ℂ => f (x-y)) :=
+    hfm.comp (measurable_id.sub_const y)
+  calc
+    (∫⁻ x in K, distanceKernel α x y) ≤ ∫⁻ x in K, f (x-y)+1 :=
+      lintegral_mono (distanceKernel_le_local_add_one hα · y)
+    _ = (∫⁻ x in K, f (x-y)) + volume K := by
+      rw [lintegral_add_left hshiftm,lintegral_const]
+      simp only [Measure.restrict_apply_univ,one_mul]
+    _ ≤ (∫⁻ x, f (x-y)) + volume K :=
+      add_le_add (lintegral_mono' Measure.restrict_le_self (fun _ => le_rfl)) le_rfl
+    _ = radialKernelMass α + volume K := by
+      rw [lintegral_sub_right_eq_self]
+      simp only [f,lintegral_indicator measurableSet_ball,radialKernelMass]
+
+lemma distanceKernel_mul_le_squares (α : ℝ) (x y z : ℂ) :
+    distanceKernel α x y * distanceKernel α x z ≤
+      distanceKernel (2*α) x y + distanceKernel (2*α) x z := by
+  have hs (w : ℂ) : (dist x w ^ (-α))^2 = dist x w ^ (-(2*α)) := by
+    rw [← Real.rpow_natCast,← Real.rpow_mul dist_nonneg]
+    congr 1
+    ring
+  unfold distanceKernel
+  rw [← ENNReal.ofReal_mul (Real.rpow_nonneg dist_nonneg _),
+    ← ENNReal.ofReal_add (Real.rpow_nonneg dist_nonneg _) (Real.rpow_nonneg dist_nonneg _)]
+  apply ENNReal.ofReal_le_ofReal
+  rw [← hs y,← hs z]
+  nlinarith [sq_nonneg ((dist x y ^ (-α))-(dist x z ^ (-α)))]
+
+theorem distanceKernel_pair_lintegral_le {α : ℝ} (hα : 0 ≤ α)
+    (K : Set ℂ) (y z : ℂ) :
+    (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤
+      2*(radialKernelMass (2*α)+volume K) := by
+  calc
+    (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤
+        ∫⁻ x in K, distanceKernel (2*α) x y + distanceKernel (2*α) x z :=
+      lintegral_mono (distanceKernel_mul_le_squares α · y z)
+    _ = (∫⁻ x in K, distanceKernel (2*α) x y) +
+        (∫⁻ x in K, distanceKernel (2*α) x z) := by
+      apply lintegral_add_left
+      exact measurable_distanceKernel_left (2*α) y
+    _ ≤ (radialKernelMass (2*α)+volume K)+(radialKernelMass (2*α)+volume K) :=
+      add_le_add (distanceKernel_lintegral_le (by positivity) K y)
+        (distanceKernel_lintegral_le (by positivity) K z)
+    _ = _ := (two_mul _).symm
+
+theorem distanceKernel_uniform_local_bounds {α : ℝ} (hα : 0 < α) (hα1 : α < 1)
+    (K : Set ℂ) (hK : Bornology.IsBounded K) :
+    ∃ C : ℝ≥0∞, C < ⊤ ∧ 1 ≤ C ∧ volume K ≤ C ∧
+      (∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C) ∧
+      (∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C) := by
+  let C := 1+volume K+(radialKernelMass α+volume K)+2*(radialKernelMass (2*α)+volume K)
+  have hvol : volume K < ⊤ := hK.measure_lt_top
+  have hmass : radialKernelMass α < ⊤ := radialKernelMass_lt_top (by linarith)
+  have hmass2 : radialKernelMass (2*α) < ⊤ := radialKernelMass_lt_top (by linarith)
+  refine ⟨C,by dsimp [C]; finiteness,?_,?_,?_,?_⟩
+  · dsimp [C]
+    exact (le_add_right le_rfl).trans ((le_add_right le_rfl).trans (le_add_right le_rfl))
+  · dsimp [C]
+    exact (le_add_left le_rfl).trans ((le_add_right le_rfl).trans (le_add_right le_rfl))
+  · intro y
+    exact (distanceKernel_lintegral_le hα.le K y).trans
+      ((le_add_left le_rfl).trans (le_add_right le_rfl))
+  · intro y z
+    exact (distanceKernel_pair_lintegral_le hα.le K y z).trans (le_add_left le_rfl)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointFactorBranches.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointFactorBranches.lean
index 4eba71e455dedf8307044127b4f47a25e31867f8..933aae9a85adf6ea557c271567ee1086cfa082ff 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointFactorBranches.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointFactorBranches.lean
@@ -1,147 +1,147 @@
-import OAI.Probability.SLE.Multipoint.MultipointRegularizedField
-import OAI.Probability.SLE.Multipoint.MultipointSwitches
-
-namespace OAI
-
-/-! Exact first-half and second-half formulas for the active multipoint
-factor. The change of formula at the midpoint has no jump. -/
-
-noncomputable section
-open Set
-
-namespace SLEExactGauge.Multipoint
-
-/-- The source clock in logarithmic capacity-two coordinates. -/
-def logFactorClock (κ : ℝ) (z : ℂ × ℝ) : ℝ := κ / 4 * (z.2-z.1.im)
-
-def factorActivationScale (κ a : ℝ) : ℝ := Real.exp (-greenExponent κ*a/2)/3
-
-lemma factorActivationScale_pos (κ a : ℝ) : 0 < factorActivationScale κ a :=
-  div_pos (Real.exp_pos _) (by norm_num)
-
-lemma first_branch_regulator (κ β a : ℝ) (z : ℂ × ℝ) :
-    branchRegulator κ (Real.exp (β*a)) (-β) z =
-      Real.exp (-β*(logFactorClock κ z-a)) := by
-  unfold branchRegulator logFactorClock
-  rw [← Real.exp_add]
-  congr 1
-  ring
-
-lemma second_branch_regulator (κ β a U : ℝ) (z : ℂ × ℝ) :
-    branchRegulator κ (Real.exp (-β*(U+a))) β z =
-      Real.exp (-β*(U-(logFactorClock κ z-a))) := by
-  unfold branchRegulator logFactorClock
-  rw [← Real.exp_add]
-  congr 1
-  ring
-
-lemma first_branch_regulator_eq (κ β a U : ℝ) (z : ℂ × ℝ)
-    (hu : logFactorClock κ z-a ≤ U/2) :
-    branchRegulator κ (Real.exp (β*a)) (-β) z =
-      regulator β U (logFactorClock κ z-a) := by
-  rw [first_branch_regulator, regulator_eq_left hu]
-
-lemma second_branch_regulator_eq (κ β a U : ℝ) (z : ℂ × ℝ)
-    (hu : U/2 ≤ logFactorClock κ z-a) :
-    branchRegulator κ (Real.exp (-β*(U+a))) β z =
-      regulator β U (logFactorClock κ z-a) := by
-  rw [second_branch_regulator, regulator_eq_right hu]
-
-lemma regularizedLogFactor_normalized (κ D σ a : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    factorActivationScale κ a * regularizedLogFactor κ D σ z x =
-      Real.exp (greenExponent κ/2*(logFactorClock κ z-a))/3 *
-        angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by
-  rw [regularizedLogFactor_eq]
-  unfold factorActivationScale
-  have he : Real.exp (-greenExponent κ*a/2) *
-      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im)) =
-        Real.exp (greenExponent κ/2*(logFactorClock κ z-a)) := by
-    rw [← Real.exp_add]
-    congr 1
-    unfold logFactorClock
-    ring
-  calc
-    _ = (Real.exp (-greenExponent κ*a/2) *
-        Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im)))/3 *
-        angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by ring
-    _ = _ := by rw [he]
-
-lemma first_branch_factor_eq (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
-    (hu : logFactorClock κ z-a ≤ U/2) :
-    factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x =
-        activeFactor (greenExponent κ) β U (logFactorClock κ z-a) (logAngle κ z x) := by
-  rw [regularizedLogFactor_normalized, first_branch_regulator_eq κ β a U z hu]
-  rfl
-
-lemma second_branch_factor_eq (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
-    (hu : U/2 ≤ logFactorClock κ z-a) :
-    factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x =
-        activeFactor (greenExponent κ) β U (logFactorClock κ z-a) (logAngle κ z x) := by
-  rw [regularizedLogFactor_normalized, second_branch_regulator_eq κ β a U z hu]
-  rfl
-
-lemma first_branch_regulator_le_one (κ β a : ℝ) (z : ℂ × ℝ)
-    (hβ : 0 ≤ β) (hu : 0 ≤ logFactorClock κ z-a) :
-    branchRegulator κ (Real.exp (β*a)) (-β) z ≤ 1 := by
-  rw [first_branch_regulator]
-  exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hβ) hu)
-
-lemma second_branch_regulator_le_one (κ β a U : ℝ) (z : ℂ × ℝ)
-    (hβ : 0 ≤ β) (hu : logFactorClock κ z-a ≤ U) :
-    branchRegulator κ (Real.exp (-β*(U+a))) β z ≤ 1 := by
-  rw [second_branch_regulator]
-  exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg
-    (neg_nonpos.mpr hβ) (sub_nonneg.mpr hu))
-
-lemma first_branch_factor_pos (κ β a : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    0 < factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x :=
-  mul_pos (factorActivationScale_pos κ a)
-    (regularizedLogFactor_pos κ _ _ (Real.exp_pos _) z x)
-
-lemma second_branch_factor_pos (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    0 < factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x :=
-  mul_pos (factorActivationScale_pos κ a)
-    (regularizedLogFactor_pos κ _ _ (Real.exp_pos _) z x)
-
-lemma branch_midpoint_eq (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
-    (hu : logFactorClock κ z-a = U/2) :
-    regularizedLogFactor κ (Real.exp (β*a)) (-β) z x =
-      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x := by
-  rw [regularizedLogFactor_eq, regularizedLogFactor_eq,
-    first_branch_regulator_eq κ β a U z hu.le,
-    second_branch_regulator_eq κ β a U z hu.ge]
-
-private lemma factorExponent_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < greenExponent κ := by
-  unfold greenExponent
-  rw [sub_pos, lt_div_iff₀ hκ]
-  linarith
-
-/-- Activation changes the inactive factor one downward to the first
-smooth branch. -/
-lemma first_branch_activation_le (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = 0) :
-    factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x ≤ 1 := by
-  rw [first_branch_factor_eq κ β a U z x (by rw [hu]; linarith), hu]
-  exact activeFactor_activation_le (factorExponent_pos κ hκ hκ8) hU
-    (logAngle_sin_pos κ z x).le
-
-/-- Completion freezes at the angle-independent lower bound. -/
-lemma second_branch_completion_jump (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = U) :
-    Real.exp (greenExponent κ/2*U)/3 ≤ factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x := by
-  rw [second_branch_factor_eq κ β a U z x (by rw [hu]; linarith), hu]
-  exact activeFactor_completion_jump (factorExponent_pos κ hκ hκ8) hU
-    (logAngle_sin_pos κ z x).le
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointRegularizedField
+import OAI.Probability.SLE.Multipoint.MultipointSwitches
+
+namespace OAI
+
+/-! Exact first-half and second-half formulas for the active multipoint
+factor. The change of formula at the midpoint has no jump. -/
+
+noncomputable section
+open Set
+
+namespace SLEExactGauge.Multipoint
+
+/-- The source clock in logarithmic capacity-two coordinates. -/
+def logFactorClock (κ : ℝ) (z : ℂ × ℝ) : ℝ := κ / 4 * (z.2-z.1.im)
+
+def factorActivationScale (κ a : ℝ) : ℝ := Real.exp (-greenExponent κ*a/2)/3
+
+lemma factorActivationScale_pos (κ a : ℝ) : 0 < factorActivationScale κ a :=
+  div_pos (Real.exp_pos _) (by norm_num)
+
+lemma first_branch_regulator (κ β a : ℝ) (z : ℂ × ℝ) :
+    branchRegulator κ (Real.exp (β*a)) (-β) z =
+      Real.exp (-β*(logFactorClock κ z-a)) := by
+  unfold branchRegulator logFactorClock
+  rw [← Real.exp_add]
+  congr 1
+  ring
+
+lemma second_branch_regulator (κ β a U : ℝ) (z : ℂ × ℝ) :
+    branchRegulator κ (Real.exp (-β*(U+a))) β z =
+      Real.exp (-β*(U-(logFactorClock κ z-a))) := by
+  unfold branchRegulator logFactorClock
+  rw [← Real.exp_add]
+  congr 1
+  ring
+
+lemma first_branch_regulator_eq (κ β a U : ℝ) (z : ℂ × ℝ)
+    (hu : logFactorClock κ z-a ≤ U/2) :
+    branchRegulator κ (Real.exp (β*a)) (-β) z =
+      regulator β U (logFactorClock κ z-a) := by
+  rw [first_branch_regulator, regulator_eq_left hu]
+
+lemma second_branch_regulator_eq (κ β a U : ℝ) (z : ℂ × ℝ)
+    (hu : U/2 ≤ logFactorClock κ z-a) :
+    branchRegulator κ (Real.exp (-β*(U+a))) β z =
+      regulator β U (logFactorClock κ z-a) := by
+  rw [second_branch_regulator, regulator_eq_right hu]
+
+lemma regularizedLogFactor_normalized (κ D σ a : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    factorActivationScale κ a * regularizedLogFactor κ D σ z x =
+      Real.exp (greenExponent κ/2*(logFactorClock κ z-a))/3 *
+        angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by
+  rw [regularizedLogFactor_eq]
+  unfold factorActivationScale
+  have he : Real.exp (-greenExponent κ*a/2) *
+      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im)) =
+        Real.exp (greenExponent κ/2*(logFactorClock κ z-a)) := by
+    rw [← Real.exp_add]
+    congr 1
+    unfold logFactorClock
+    ring
+  calc
+    _ = (Real.exp (-greenExponent κ*a/2) *
+        Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im)))/3 *
+        angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by ring
+    _ = _ := by rw [he]
+
+lemma first_branch_factor_eq (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
+    (hu : logFactorClock κ z-a ≤ U/2) :
+    factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x =
+        activeFactor (greenExponent κ) β U (logFactorClock κ z-a) (logAngle κ z x) := by
+  rw [regularizedLogFactor_normalized, first_branch_regulator_eq κ β a U z hu]
+  rfl
+
+lemma second_branch_factor_eq (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
+    (hu : U/2 ≤ logFactorClock κ z-a) :
+    factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x =
+        activeFactor (greenExponent κ) β U (logFactorClock κ z-a) (logAngle κ z x) := by
+  rw [regularizedLogFactor_normalized, second_branch_regulator_eq κ β a U z hu]
+  rfl
+
+lemma first_branch_regulator_le_one (κ β a : ℝ) (z : ℂ × ℝ)
+    (hβ : 0 ≤ β) (hu : 0 ≤ logFactorClock κ z-a) :
+    branchRegulator κ (Real.exp (β*a)) (-β) z ≤ 1 := by
+  rw [first_branch_regulator]
+  exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hβ) hu)
+
+lemma second_branch_regulator_le_one (κ β a U : ℝ) (z : ℂ × ℝ)
+    (hβ : 0 ≤ β) (hu : logFactorClock κ z-a ≤ U) :
+    branchRegulator κ (Real.exp (-β*(U+a))) β z ≤ 1 := by
+  rw [second_branch_regulator]
+  exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg
+    (neg_nonpos.mpr hβ) (sub_nonneg.mpr hu))
+
+lemma first_branch_factor_pos (κ β a : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    0 < factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x :=
+  mul_pos (factorActivationScale_pos κ a)
+    (regularizedLogFactor_pos κ _ _ (Real.exp_pos _) z x)
+
+lemma second_branch_factor_pos (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    0 < factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x :=
+  mul_pos (factorActivationScale_pos κ a)
+    (regularizedLogFactor_pos κ _ _ (Real.exp_pos _) z x)
+
+lemma branch_midpoint_eq (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
+    (hu : logFactorClock κ z-a = U/2) :
+    regularizedLogFactor κ (Real.exp (β*a)) (-β) z x =
+      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x := by
+  rw [regularizedLogFactor_eq, regularizedLogFactor_eq,
+    first_branch_regulator_eq κ β a U z hu.le,
+    second_branch_regulator_eq κ β a U z hu.ge]
+
+private lemma factorExponent_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < greenExponent κ := by
+  unfold greenExponent
+  rw [sub_pos, lt_div_iff₀ hκ]
+  linarith
+
+/-- Activation changes the inactive factor one downward to the first
+smooth branch. -/
+lemma first_branch_activation_le (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = 0) :
+    factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x ≤ 1 := by
+  rw [first_branch_factor_eq κ β a U z x (by rw [hu]; linarith), hu]
+  exact activeFactor_activation_le (factorExponent_pos κ hκ hκ8) hU
+    (logAngle_sin_pos κ z x).le
+
+/-- Completion freezes at the angle-independent lower bound. -/
+lemma second_branch_completion_jump (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = U) :
+    Real.exp (greenExponent κ/2*U)/3 ≤ factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x := by
+  rw [second_branch_factor_eq κ β a U z x (by rw [hu]; linarith), hu]
+  exact activeFactor_completion_jump (factorExponent_pos κ hκ hκ8) hU
+    (logAngle_sin_pos κ z x).le
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointFieldBounds.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointFieldBounds.lean
index 087da90b408febbcabb2ccab972efb7dca3ae430..38ba9a2cc71ddfe72349aecafa9da84ccc62b5ca 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointFieldBounds.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointFieldBounds.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Multipoint.MultipointRegularizedField
-import OAI.Probability.SLE.Multipoint.MultipointBrownianCoefficient
-
-namespace OAI
-
-/-! Drift and Brownian cross bounds for the concrete regularized fields. -/
-
-noncomputable section
-
-namespace SLEExactGauge.Multipoint
-
-theorem regularizedLogFactor_relative_drift
-    (κ D σ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hD : 0 < D) (hσ : σ ≤ 1)
-    (z : ℂ × ℝ) (x : ℝ) :
-    logGenerator κ (regularizedLogFactor κ D σ) z x / regularizedLogFactor κ D σ z x ≤
-      κ*logVelocity κ z x^2 *
-        ((driftAllowance (greenExponent κ)*driftThreshold (greenExponent κ)^(-greenExponent κ))*
-          branchRegulator κ D σ z -
-          (negativeDrift (greenExponent κ)/2)*
-            damping (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x)) := by
-  have hδ : 0 < branchRegulator κ D σ z := mul_pos hD (Real.exp_pos _)
-  have hs := logAngle_sin_pos κ z x
-  have hf := angularFactor_pos (p := greenExponent κ) hδ hs
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  have h := regularized_drift_bound hp hδ hσ hs
-  have hm := mul_le_mul_of_nonneg_left h
-    (mul_nonneg hκ.le (sq_nonneg (logVelocity κ z x)))
-  rw [regularizedLogFactor_generator κ D σ hκ, regularizedLogFactor_eq]
-  convert hm using 1
-  field_simp [hf.ne', (Real.exp_pos (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))).ne']
-
-lemma regularizedLogFactor_relative_spatial
-    (κ D σ : ℝ) (hD : 0 < D) (z : ℂ × ℝ) (x : ℝ) :
-    deriv (regularizedLogFactor κ D σ z) x / regularizedLogFactor κ D σ z x =
-      Real.sqrt κ*logVelocity κ z x *
-        (angularFactorFirst (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) /
-          angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x)) := by
-  rw [regularizedLogFactor_spatial, regularizedLogFactor_eq]
-  have hf := angularFactor_pos (p := greenExponent κ) (mul_pos hD (Real.exp_pos ((σ*κ/4)*(z.2-z.1.im))))
-    (logAngle_sin_pos κ z x)
-  change 0 < angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) at hf
-  field_simp [hf.ne', (Real.exp_pos (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))).ne']
-
-theorem regularizedLogFactor_pair_absorption
-    (κ Di Dj σi σj ε : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hDi : 0 < Di) (hDj : 0 < Dj) (hε : 0 < ε)
-    (zi zj : ℂ × ℝ) (x : ℝ)
-    (hi1 : branchRegulator κ Di σi zi ≤ 1)
-    (hj1 : branchRegulator κ Dj σj zj ≤ 1) :
-    |(deriv (regularizedLogFactor κ Di σi zi) x / regularizedLogFactor κ Di σi zi x)*
-        (deriv (regularizedLogFactor κ Dj σj zj) x / regularizedLogFactor κ Dj σj zj x)| ≤
-      κ*(ε*(damping (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)*
-          logVelocity κ zi x^2 +
-        damping (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)*
-          logVelocity κ zj x^2) +
-        youngConstant (greenExponent κ) (angularCoefficientConstant (greenExponent κ)^2) ε *
-          (branchRegulator κ Di σi zi*branchRegulator κ Dj σj zj)^(-exponent (greenExponent κ))*
-            (logVelocity κ zi x*logVelocity κ zj x)) := by
-  have hi : 0 < branchRegulator κ Di σi zi := mul_pos hDi (Real.exp_pos _)
-  have hj : 0 < branchRegulator κ Dj σj zj := mul_pos hDj (Real.exp_pos _)
-  have hsi := logAngle_sin_pos κ zi x
-  have hsj := logAngle_sin_pos κ zj x
-  have hfi := angularFactor_pos (p := greenExponent κ) hi hsi
-  have hfj := angularFactor_pos (p := greenExponent κ) hj hsj
-  have hvi := logVelocity_pos κ zi x
-  have hvj := logVelocity_pos κ zj x
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  have h := angular_pair_absorption hp hi hi1 hj hj1 hsi hsj
-    hvi hvj hε
-  have hh := mul_le_mul_of_nonneg_left h hκ.le
-  rw [regularizedLogFactor_relative_spatial κ Di σi hDi,
-    regularizedLogFactor_relative_spatial κ Dj σj hDj]
-  calc
-    _ = κ*((|angularFactorFirst (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)| /
-        angularFactor (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)*logVelocity κ zi x)*
-      (|angularFactorFirst (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)| /
-        angularFactor (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)*logVelocity κ zj x)) := by
-      simp only [abs_mul, abs_div, abs_of_nonneg (Real.sqrt_nonneg κ),
-        abs_of_pos hvi, abs_of_pos hvj, abs_of_pos hfi, abs_of_pos hfj]
-      let ai := |angularFactorFirst (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)| /
-        angularFactor (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)
-      let aj := |angularFactorFirst (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)| /
-        angularFactor (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)
-      change Real.sqrt κ*logVelocity κ zi x*ai*(Real.sqrt κ*logVelocity κ zj x*aj) =
-        κ*((ai*logVelocity κ zi x)*(aj*logVelocity κ zj x))
-      calc
-        _ = (Real.sqrt κ)^2*((ai*logVelocity κ zi x)*(aj*logVelocity κ zj x)) := by ring
-        _ = _ := by rw [Real.sq_sqrt hκ.le]
-    _ ≤ _ := hh
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointRegularizedField
+import OAI.Probability.SLE.Multipoint.MultipointBrownianCoefficient
+
+namespace OAI
+
+/-! Drift and Brownian cross bounds for the concrete regularized fields. -/
+
+noncomputable section
+
+namespace SLEExactGauge.Multipoint
+
+theorem regularizedLogFactor_relative_drift
+    (κ D σ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hD : 0 < D) (hσ : σ ≤ 1)
+    (z : ℂ × ℝ) (x : ℝ) :
+    logGenerator κ (regularizedLogFactor κ D σ) z x / regularizedLogFactor κ D σ z x ≤
+      κ*logVelocity κ z x^2 *
+        ((driftAllowance (greenExponent κ)*driftThreshold (greenExponent κ)^(-greenExponent κ))*
+          branchRegulator κ D σ z -
+          (negativeDrift (greenExponent κ)/2)*
+            damping (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x)) := by
+  have hδ : 0 < branchRegulator κ D σ z := mul_pos hD (Real.exp_pos _)
+  have hs := logAngle_sin_pos κ z x
+  have hf := angularFactor_pos (p := greenExponent κ) hδ hs
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  have h := regularized_drift_bound hp hδ hσ hs
+  have hm := mul_le_mul_of_nonneg_left h
+    (mul_nonneg hκ.le (sq_nonneg (logVelocity κ z x)))
+  rw [regularizedLogFactor_generator κ D σ hκ, regularizedLogFactor_eq]
+  convert hm using 1
+  field_simp [hf.ne', (Real.exp_pos (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))).ne']
+
+lemma regularizedLogFactor_relative_spatial
+    (κ D σ : ℝ) (hD : 0 < D) (z : ℂ × ℝ) (x : ℝ) :
+    deriv (regularizedLogFactor κ D σ z) x / regularizedLogFactor κ D σ z x =
+      Real.sqrt κ*logVelocity κ z x *
+        (angularFactorFirst (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) /
+          angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x)) := by
+  rw [regularizedLogFactor_spatial, regularizedLogFactor_eq]
+  have hf := angularFactor_pos (p := greenExponent κ) (mul_pos hD (Real.exp_pos ((σ*κ/4)*(z.2-z.1.im))))
+    (logAngle_sin_pos κ z x)
+  change 0 < angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) at hf
+  field_simp [hf.ne', (Real.exp_pos (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))).ne']
+
+theorem regularizedLogFactor_pair_absorption
+    (κ Di Dj σi σj ε : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hDi : 0 < Di) (hDj : 0 < Dj) (hε : 0 < ε)
+    (zi zj : ℂ × ℝ) (x : ℝ)
+    (hi1 : branchRegulator κ Di σi zi ≤ 1)
+    (hj1 : branchRegulator κ Dj σj zj ≤ 1) :
+    |(deriv (regularizedLogFactor κ Di σi zi) x / regularizedLogFactor κ Di σi zi x)*
+        (deriv (regularizedLogFactor κ Dj σj zj) x / regularizedLogFactor κ Dj σj zj x)| ≤
+      κ*(ε*(damping (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)*
+          logVelocity κ zi x^2 +
+        damping (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)*
+          logVelocity κ zj x^2) +
+        youngConstant (greenExponent κ) (angularCoefficientConstant (greenExponent κ)^2) ε *
+          (branchRegulator κ Di σi zi*branchRegulator κ Dj σj zj)^(-exponent (greenExponent κ))*
+            (logVelocity κ zi x*logVelocity κ zj x)) := by
+  have hi : 0 < branchRegulator κ Di σi zi := mul_pos hDi (Real.exp_pos _)
+  have hj : 0 < branchRegulator κ Dj σj zj := mul_pos hDj (Real.exp_pos _)
+  have hsi := logAngle_sin_pos κ zi x
+  have hsj := logAngle_sin_pos κ zj x
+  have hfi := angularFactor_pos (p := greenExponent κ) hi hsi
+  have hfj := angularFactor_pos (p := greenExponent κ) hj hsj
+  have hvi := logVelocity_pos κ zi x
+  have hvj := logVelocity_pos κ zj x
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  have h := angular_pair_absorption hp hi hi1 hj hj1 hsi hsj
+    hvi hvj hε
+  have hh := mul_le_mul_of_nonneg_left h hκ.le
+  rw [regularizedLogFactor_relative_spatial κ Di σi hDi,
+    regularizedLogFactor_relative_spatial κ Dj σj hDj]
+  calc
+    _ = κ*((|angularFactorFirst (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)| /
+        angularFactor (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)*logVelocity κ zi x)*
+      (|angularFactorFirst (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)| /
+        angularFactor (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)*logVelocity κ zj x)) := by
+      simp only [abs_mul, abs_div, abs_of_nonneg (Real.sqrt_nonneg κ),
+        abs_of_pos hvi, abs_of_pos hvj, abs_of_pos hfi, abs_of_pos hfj]
+      let ai := |angularFactorFirst (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)| /
+        angularFactor (greenExponent κ) (branchRegulator κ Di σi zi) (logAngle κ zi x)
+      let aj := |angularFactorFirst (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)| /
+        angularFactor (greenExponent κ) (branchRegulator κ Dj σj zj) (logAngle κ zj x)
+      change Real.sqrt κ*logVelocity κ zi x*ai*(Real.sqrt κ*logVelocity κ zj x*aj) =
+        κ*((ai*logVelocity κ zi x)*(aj*logVelocity κ zj x))
+      calc
+        _ = (Real.sqrt κ)^2*((ai*logVelocity κ zi x)*(aj*logVelocity κ zj x)) := by ring
+        _ = _ := by rw [Real.sq_sqrt hκ.le]
+    _ ≤ _ := hh
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointFixedCenterKernel.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointFixedCenterKernel.lean
index 93b2e21a1e16031a511b81bb09f8a96cc61f2e82..7316f04fa787fcc22a2fae895135a052cfe8631f 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointFixedCenterKernel.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointFixedCenterKernel.lean
@@ -1,129 +1,129 @@
-import OAI.Probability.SLE.Multipoint.MultipointKernelExpansion
-import Mathlib.Data.Fin.Tuple.Basic
-
-namespace OAI
-
-/-! The fixed-center form of the capped kernel estimate. The constant
-is independent of the location of the fixed center. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-def fixedKernelBound (ι : Type*) [Fintype ι] [DecidableEq ι] (α b : ℝ) (C : ℝ≥0∞) : ℝ≥0∞ :=
-  ∑ _f : ι → Option ι, capCoefficient α b ^ Fintype.card ι * C^(Fintype.card ι-1)
-
-lemma fixedKernelBound_lt_top (ι : Type*) [Fintype ι] [DecidableEq ι] (α b : ℝ)
-    (C : ℝ≥0∞) (hC : C < ⊤) : fixedKernelBound ι α b C < ⊤ := by
-  apply ENNReal.sum_lt_top.mpr
-  intro f _
-  exact ENNReal.mul_lt_top (ENNReal.pow_lt_top (capCoefficient_finite α b))
-    (ENNReal.pow_lt_top hC)
-
-lemma cappedDistanceKernel_marginal_le {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
-    (hC0 : volume K ≤ C)
-    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
-    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
-    (root : ι) (x : ι → ℂ) :
-    lmarginal (fun _ : ι => volume.restrict K) (Finset.univ.erase root)
-      (cappedDistanceKernel α b) x ≤ fixedKernelBound ι α b C := by
-  let S : Finset ι := Finset.univ.erase root
-  have hg (f : ι → Option ι) :
-      lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α (selectedEdges f)) x ≤ C^S.card := by
-    apply graphKernel_marginal_le_with_root α K C hC hC0 hC1 hC2 root S (selectedEdges f)
-    · exact Finset.notMem_erase _ _
-    · intro e _
-      constructor <;> simp only [S,Finset.insert_erase (Finset.mem_univ root),Finset.mem_univ]
-    · exact selectedEdges_outgoing f
-    · exact selectedEdges_loopfree f
-  change lmarginal (fun _ : ι => volume.restrict K) S (cappedDistanceKernel α b) x ≤ _
-  calc
-    _ ≤ lmarginal (fun _ : ι => volume.restrict K) S
-        (fun y => ∑ f : ι → Option ι,
-          capCoefficient α b ^ Fintype.card ι * graphKernel α (selectedEdges f) y) x :=
-      lmarginal_mono (cappedDistanceKernel_le_graph_sum α b) x
-    _ = ∑ f : ι → Option ι, capCoefficient α b ^ Fintype.card ι *
-        lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α (selectedEdges f)) x := by
-      unfold lmarginal
-      rw [lintegral_finsetSum]
-      · apply Finset.sum_congr rfl
-        intro f _
-        exact lintegral_const_mul' _ _ (ENNReal.pow_ne_top (capCoefficient_finite α b).ne)
-      · intro f _
-        exact measurable_const.mul ((measurable_graphKernel α (selectedEdges f)).comp
-          measurable_updateFinset)
-    _ ≤ ∑ _f : ι → Option ι, capCoefficient α b ^ Fintype.card ι * C^S.card := by
-      apply Finset.sum_le_sum
-      intro f _
-      have hh := hg f
-      gcongr
-    _ = fixedKernelBound ι α b C := by
-      simp only [S,Finset.card_erase_of_mem (Finset.mem_univ root),Finset.card_univ,fixedKernelBound]
-
-lemma lintegral_fin_cons_eq_marginal (m : ℕ) (K : Set ℂ)
-    (f : (Fin (m+1) → ℂ) → ℝ≥0∞) (hf : Measurable f) (z : ℂ) :
-    (∫⁻ x, f (Fin.cons z x) ∂Measure.pi (fun _ : Fin m => volume.restrict K)) =
-      lmarginal (fun _ : Fin (m+1) => volume.restrict K) (Finset.univ.erase 0)
-        f (Fin.cons z (fun _ => 0)) := by
-  let g : (Fin m → ℂ) → ℝ≥0∞ := fun x => f (Fin.cons z x)
-  have hg : Measurable g := by
-    apply hf.comp
-    apply measurable_pi_iff.mpr
-    intro i
-    cases i using Fin.cases <;> simp only [Fin.cons_zero,Fin.cons_succ]
-    · exact measurable_const
-    · exact measurable_pi_apply _
-  have hcomp : (fun y : Fin (m+1) → ℂ => f (Function.update y 0 z)) =
-      g ∘ (fun y : Fin (m+1) → ℂ => y ∘ Fin.succ) := by
-    funext y
-    apply congrArg f
-    ext i
-    cases i using Fin.cases <;> simp [Function.comp_def]
-  have hset : (Finset.univ : Finset (Fin m)).image Fin.succ = Finset.univ.erase 0 := by
-    rw [Fin.image_succ_univ]
-    ext i
-    simp
-  calc
-    _ = lmarginal (fun _ : Fin m => volume.restrict K) Finset.univ g (fun _ => 0) := by
-      simp only [lmarginal_univ,g]
-    _ = lmarginal (fun _ : Fin (m+1) => volume.restrict K) (Finset.univ.image Fin.succ)
-        (g ∘ (fun y => y ∘ Fin.succ)) (Fin.cons z (fun _ => 0)) := by
-      exact (lmarginal_image (μ := fun _ : Fin (m+1) => volume.restrict K)
-        (Fin.succ_injective m) Finset.univ hg (Fin.cons z (fun _ => 0))).symm
-    _ = lmarginal (fun _ : Fin (m+1) => volume.restrict K) (Finset.univ.erase 0)
-        (fun y => f (Function.update y 0 z)) (Fin.cons z (fun _ => 0)) := by rw [hset,hcomp]
-    _ = _ := by
-      simpa only [Fin.update_cons_zero,Function.comp_def] using
-        (lmarginal_update_of_notMem (μ := fun _ : Fin (m+1) => volume.restrict K) hf
-          (Finset.notMem_erase 0 Finset.univ) (Fin.cons z (fun _ => 0)) z).symm
-
-/-- Uniform fixed-center kernel bound, with the center in coordinate zero
-and the free variables in the successor coordinates. -/
-theorem cappedDistanceKernel_fixed_center_bound (m : ℕ)
-    {α : ℝ} (hα : 0 < α) (hα1 : α < 1) (b : ℝ)
-    (K : Set ℂ) (hK : Bornology.IsBounded K) :
-    ∃ C : ℝ≥0∞, C < ⊤ ∧ ∀ z : ℂ,
-      (∫⁻ x, cappedDistanceKernel α b (Fin.cons z x)
-        ∂Measure.pi (fun _ : Fin m => volume.restrict K)) ≤ C := by
-  obtain ⟨C,hC,_,hC0,hC1,hC2⟩ := distanceKernel_uniform_local_bounds hα hα1 K hK
-  refine ⟨fixedKernelBound (Fin (m+1)) α b C,
-    fixedKernelBound_lt_top _ α b C hC,fun z => ?_⟩
-  rw [lintegral_fin_cons_eq_marginal m K _ (measurable_cappedDistanceKernel α b)]
-  exact cappedDistanceKernel_marginal_le α b K C hC hC0 hC1 hC2 0 _
-
-lemma ae_fin_cons_injective (m : ℕ) (K : Set ℂ) (z : ℂ) :
-    ∀ᵐ x : Fin m → ℂ ∂Measure.pi (fun _ : Fin m => volume.restrict K),
-      Function.Injective (Fin.cons z x) := by
-  filter_upwards [ae_tuple_injective K,ae_tuple_avoids K z] with x hx hz
-  apply Fin.cons_injective_of_injective _ hx
-  rintro ⟨i,hi⟩
-  exact hz i hi
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointKernelExpansion
+import Mathlib.Data.Fin.Tuple.Basic
+
+namespace OAI
+
+/-! The fixed-center form of the capped kernel estimate. The constant
+is independent of the location of the fixed center. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+def fixedKernelBound (ι : Type*) [Fintype ι] [DecidableEq ι] (α b : ℝ) (C : ℝ≥0∞) : ℝ≥0∞ :=
+  ∑ _f : ι → Option ι, capCoefficient α b ^ Fintype.card ι * C^(Fintype.card ι-1)
+
+lemma fixedKernelBound_lt_top (ι : Type*) [Fintype ι] [DecidableEq ι] (α b : ℝ)
+    (C : ℝ≥0∞) (hC : C < ⊤) : fixedKernelBound ι α b C < ⊤ := by
+  apply ENNReal.sum_lt_top.mpr
+  intro f _
+  exact ENNReal.mul_lt_top (ENNReal.pow_lt_top (capCoefficient_finite α b))
+    (ENNReal.pow_lt_top hC)
+
+lemma cappedDistanceKernel_marginal_le {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
+    (hC0 : volume K ≤ C)
+    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
+    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
+    (root : ι) (x : ι → ℂ) :
+    lmarginal (fun _ : ι => volume.restrict K) (Finset.univ.erase root)
+      (cappedDistanceKernel α b) x ≤ fixedKernelBound ι α b C := by
+  let S : Finset ι := Finset.univ.erase root
+  have hg (f : ι → Option ι) :
+      lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α (selectedEdges f)) x ≤ C^S.card := by
+    apply graphKernel_marginal_le_with_root α K C hC hC0 hC1 hC2 root S (selectedEdges f)
+    · exact Finset.notMem_erase _ _
+    · intro e _
+      constructor <;> simp only [S,Finset.insert_erase (Finset.mem_univ root),Finset.mem_univ]
+    · exact selectedEdges_outgoing f
+    · exact selectedEdges_loopfree f
+  change lmarginal (fun _ : ι => volume.restrict K) S (cappedDistanceKernel α b) x ≤ _
+  calc
+    _ ≤ lmarginal (fun _ : ι => volume.restrict K) S
+        (fun y => ∑ f : ι → Option ι,
+          capCoefficient α b ^ Fintype.card ι * graphKernel α (selectedEdges f) y) x :=
+      lmarginal_mono (cappedDistanceKernel_le_graph_sum α b) x
+    _ = ∑ f : ι → Option ι, capCoefficient α b ^ Fintype.card ι *
+        lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α (selectedEdges f)) x := by
+      unfold lmarginal
+      rw [lintegral_finsetSum]
+      · apply Finset.sum_congr rfl
+        intro f _
+        exact lintegral_const_mul' _ _ (ENNReal.pow_ne_top (capCoefficient_finite α b).ne)
+      · intro f _
+        exact measurable_const.mul ((measurable_graphKernel α (selectedEdges f)).comp
+          measurable_updateFinset)
+    _ ≤ ∑ _f : ι → Option ι, capCoefficient α b ^ Fintype.card ι * C^S.card := by
+      apply Finset.sum_le_sum
+      intro f _
+      have hh := hg f
+      gcongr
+    _ = fixedKernelBound ι α b C := by
+      simp only [S,Finset.card_erase_of_mem (Finset.mem_univ root),Finset.card_univ,fixedKernelBound]
+
+lemma lintegral_fin_cons_eq_marginal (m : ℕ) (K : Set ℂ)
+    (f : (Fin (m+1) → ℂ) → ℝ≥0∞) (hf : Measurable f) (z : ℂ) :
+    (∫⁻ x, f (Fin.cons z x) ∂Measure.pi (fun _ : Fin m => volume.restrict K)) =
+      lmarginal (fun _ : Fin (m+1) => volume.restrict K) (Finset.univ.erase 0)
+        f (Fin.cons z (fun _ => 0)) := by
+  let g : (Fin m → ℂ) → ℝ≥0∞ := fun x => f (Fin.cons z x)
+  have hg : Measurable g := by
+    apply hf.comp
+    apply measurable_pi_iff.mpr
+    intro i
+    cases i using Fin.cases <;> simp only [Fin.cons_zero,Fin.cons_succ]
+    · exact measurable_const
+    · exact measurable_pi_apply _
+  have hcomp : (fun y : Fin (m+1) → ℂ => f (Function.update y 0 z)) =
+      g ∘ (fun y : Fin (m+1) → ℂ => y ∘ Fin.succ) := by
+    funext y
+    apply congrArg f
+    ext i
+    cases i using Fin.cases <;> simp [Function.comp_def]
+  have hset : (Finset.univ : Finset (Fin m)).image Fin.succ = Finset.univ.erase 0 := by
+    rw [Fin.image_succ_univ]
+    ext i
+    simp
+  calc
+    _ = lmarginal (fun _ : Fin m => volume.restrict K) Finset.univ g (fun _ => 0) := by
+      simp only [lmarginal_univ,g]
+    _ = lmarginal (fun _ : Fin (m+1) => volume.restrict K) (Finset.univ.image Fin.succ)
+        (g ∘ (fun y => y ∘ Fin.succ)) (Fin.cons z (fun _ => 0)) := by
+      exact (lmarginal_image (μ := fun _ : Fin (m+1) => volume.restrict K)
+        (Fin.succ_injective m) Finset.univ hg (Fin.cons z (fun _ => 0))).symm
+    _ = lmarginal (fun _ : Fin (m+1) => volume.restrict K) (Finset.univ.erase 0)
+        (fun y => f (Function.update y 0 z)) (Fin.cons z (fun _ => 0)) := by rw [hset,hcomp]
+    _ = _ := by
+      simpa only [Fin.update_cons_zero,Function.comp_def] using
+        (lmarginal_update_of_notMem (μ := fun _ : Fin (m+1) => volume.restrict K) hf
+          (Finset.notMem_erase 0 Finset.univ) (Fin.cons z (fun _ => 0)) z).symm
+
+/-- Uniform fixed-center kernel bound, with the center in coordinate zero
+and the free variables in the successor coordinates. -/
+theorem cappedDistanceKernel_fixed_center_bound (m : ℕ)
+    {α : ℝ} (hα : 0 < α) (hα1 : α < 1) (b : ℝ)
+    (K : Set ℂ) (hK : Bornology.IsBounded K) :
+    ∃ C : ℝ≥0∞, C < ⊤ ∧ ∀ z : ℂ,
+      (∫⁻ x, cappedDistanceKernel α b (Fin.cons z x)
+        ∂Measure.pi (fun _ : Fin m => volume.restrict K)) ≤ C := by
+  obtain ⟨C,hC,_,hC0,hC1,hC2⟩ := distanceKernel_uniform_local_bounds hα hα1 K hK
+  refine ⟨fixedKernelBound (Fin (m+1)) α b C,
+    fixedKernelBound_lt_top _ α b C hC,fun z => ?_⟩
+  rw [lintegral_fin_cons_eq_marginal m K _ (measurable_cappedDistanceKernel α b)]
+  exact cappedDistanceKernel_marginal_le α b K C hC hC0 hC1 hC2 0 _
+
+lemma ae_fin_cons_injective (m : ℕ) (K : Set ℂ) (z : ℂ) :
+    ∀ᵐ x : Fin m → ℂ ∂Measure.pi (fun _ : Fin m => volume.restrict K),
+      Function.Injective (Fin.cons z x) := by
+  filter_upwards [ae_tuple_injective K,ae_tuple_avoids K z] with x hx hz
+  apply Fin.cons_injective_of_injective _ hx
+  rintro ⟨i,hi⟩
+  exact hz i hi
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointGeometry.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointGeometry.lean
index df50be933cbac19d5e61dfec6815b3b3a5ac3aa1..200cc7a4821944c36acdbf139ecd5499f9fa57e9 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointGeometry.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointGeometry.lean
@@ -1,105 +1,105 @@
-import OAI.Probability.SLE.Multipoint.MultipointInteraction
-
-namespace OAI
-
-/-! Geometric decay of the multipoint interaction follows from the
-univalent disk growth inequality in Duren, *Univalent Functions* (1983),
-Section 2.3. -/
-
-noncomputable section
-open Set Metric
-
-namespace SLEExactGauge
-
-def PublishedDiskUnivalentGrowth : Prop :=
-  ∀ f : ℂ → ℂ, DifferentiableOn ℂ f (ball 0 1) →
-    InjOn f (ball 0 1) → ∀ z ∈ ball (0 : ℂ) 1,
-      ‖f z-f 0‖ ≤ ‖deriv f 0‖ * ‖z‖ / (1-‖z‖)^2
-
-namespace Multipoint
-
-/-- The hyperbolic interaction is bounded by the square root of the
-ratio of conformal radius to point separation. -/
-lemma neg_log_sq_le_radius_ratio {ρ R D : ℝ}
-    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hD : 0 < D)
-    (hgrowth : D ≤ R*ρ/(1-ρ)^2) : (-Real.log ρ)^2 ≤ R/D := by
-  let L := -Real.log ρ
-  let E := Real.exp (L/2)
-  have hL : 0 ≤ L := neg_nonneg.mpr (Real.log_nonpos hρ.le hρ1.le)
-  have hE : 0 < E := Real.exp_pos _
-  have he2 : E^2*ρ = 1 := by
-    dsimp only [E]
-    rw [pow_two, ← Real.exp_add, show L/2+L/2 = L by ring]
-    dsimp only [L]
-    rw [Real.exp_neg, Real.exp_log hρ, inv_mul_cancel₀ hρ.ne']
-  have hminus : Real.exp (-(L/2)) = E*ρ := by
-    have hrho : ρ = Real.exp (-L) := by simp [L, Real.exp_log hρ]
-    rw [hrho]
-    dsimp only [E]
-    rw [← Real.exp_add]
-    congr 1
-    ring
-  have hs : L ≤ E-E*ρ := by
-    have hh := Real.self_le_sinh_iff.mpr (show 0 ≤ L/2 by positivity)
-    rw [Real.sinh_eq, hminus] at hh
-    change L/2 ≤ (E-E*ρ)/2 at hh
-    linarith
-  have hsquare : L^2 ≤ (E-E*ρ)^2 := by nlinarith
-  have hweighted := mul_le_mul_of_nonneg_right hsquare hρ.le
-  have hid : (E-E*ρ)^2*ρ = (1-ρ)^2 := by
-    calc
-      _ = (E^2*ρ)*(1-ρ)^2 := by ring
-      _ = _ := by rw [he2, one_mul]
-  rw [hid] at hweighted
-  have hcross := (le_div_iff₀ (sq_pos_of_pos (sub_pos.mpr hρ1))).mp hgrowth
-  have hratio : (1-ρ)^2/ρ ≤ R/D := by
-    apply (div_le_div_iff₀ hρ hD).mpr
-    nlinarith
-  exact ((le_div_iff₀ hρ).mpr hweighted).trans hratio
-
-lemma interaction_decay_of_growth {ρ b D a u : ℝ}
-    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hD : 0 < D) (hb : b ≤ D)
-    (hgrowth : D ≤ (b*Real.exp (-2*a*u))*ρ/(1-ρ)^2) :
-    -Real.log ρ ≤ Real.exp (-a*u) := by
-  have hsq := neg_log_sq_le_radius_ratio hρ hρ1 hD hgrowth
-  have hratio : (b*Real.exp (-2*a*u))/D ≤ Real.exp (-2*a*u) := by
-    apply (div_le_iff₀ hD).mpr
-    nlinarith [mul_le_mul_of_nonneg_right hb (Real.exp_pos (-2*a*u)).le]
-  have hexp : Real.exp (-2*a*u) = (Real.exp (-a*u))^2 := by
-    rw [pow_two, ← Real.exp_add]
-    congr 1
-    ring
-  have hfinal := hsq.trans hratio
-  rw [hexp] at hfinal
-  exact (sq_le_sq₀ (neg_nonneg.mpr (Real.log_nonpos hρ.le hρ1.le))
-    (Real.exp_pos _).le).mp hfinal
-
-/-- A physical disk uniformization supplies the growth hypothesis used
-above. No derivative of the original half-plane map is bounded. -/
-theorem disk_interaction_decay (hgrowth : PublishedDiskUnivalentGrowth)
-    (f : ℂ → ℂ) (hf : DifferentiableOn ℂ f (ball 0 1))
-    (hinj : InjOn f (ball 0 1)) (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1)
-    (hz0 : z ≠ 0) (b a u : ℝ) (hb : b ≤ ‖f z-f 0‖)
-    (hradius : ‖deriv f 0‖ = b*Real.exp (-2*a*u)) :
-    -Real.log ‖z‖ ≤ Real.exp (-a*u) := by
-  have hρ : 0 < ‖z‖ := norm_pos_iff.mpr hz0
-  have hρ1 : ‖z‖ < 1 := by simpa only [mem_ball, dist_zero_right] using hz
-  have hD : 0 < ‖f z-f 0‖ := norm_pos_iff.mpr (sub_ne_zero.mpr (by
-    intro he
-    exact hz0 (hinj hz (by simp) he)))
-  apply interaction_decay_of_growth hρ hρ1 hD hb
-  simpa only [hradius] using hgrowth f hf hinj z hz
-
-lemma interaction_decay_max {L a u v : ℝ}
-    (hu : L ≤ Real.exp (-a*u)) (hv : L ≤ Real.exp (-a*v)) :
-    L ≤ Real.exp (-a*max u v) := by
-  rcases le_total u v with h | h
-  · simpa only [max_eq_right h] using hv
-  · simpa only [max_eq_left h] using hu
-
-end Multipoint
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointInteraction
+
+namespace OAI
+
+/-! Geometric decay of the multipoint interaction follows from the
+univalent disk growth inequality in Duren, *Univalent Functions* (1983),
+Section 2.3. -/
+
+noncomputable section
+open Set Metric
+
+namespace SLEExactGauge
+
+def PublishedDiskUnivalentGrowth : Prop :=
+  ∀ f : ℂ → ℂ, DifferentiableOn ℂ f (ball 0 1) →
+    InjOn f (ball 0 1) → ∀ z ∈ ball (0 : ℂ) 1,
+      ‖f z-f 0‖ ≤ ‖deriv f 0‖ * ‖z‖ / (1-‖z‖)^2
+
+namespace Multipoint
+
+/-- The hyperbolic interaction is bounded by the square root of the
+ratio of conformal radius to point separation. -/
+lemma neg_log_sq_le_radius_ratio {ρ R D : ℝ}
+    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hD : 0 < D)
+    (hgrowth : D ≤ R*ρ/(1-ρ)^2) : (-Real.log ρ)^2 ≤ R/D := by
+  let L := -Real.log ρ
+  let E := Real.exp (L/2)
+  have hL : 0 ≤ L := neg_nonneg.mpr (Real.log_nonpos hρ.le hρ1.le)
+  have hE : 0 < E := Real.exp_pos _
+  have he2 : E^2*ρ = 1 := by
+    dsimp only [E]
+    rw [pow_two, ← Real.exp_add, show L/2+L/2 = L by ring]
+    dsimp only [L]
+    rw [Real.exp_neg, Real.exp_log hρ, inv_mul_cancel₀ hρ.ne']
+  have hminus : Real.exp (-(L/2)) = E*ρ := by
+    have hrho : ρ = Real.exp (-L) := by simp [L, Real.exp_log hρ]
+    rw [hrho]
+    dsimp only [E]
+    rw [← Real.exp_add]
+    congr 1
+    ring
+  have hs : L ≤ E-E*ρ := by
+    have hh := Real.self_le_sinh_iff.mpr (show 0 ≤ L/2 by positivity)
+    rw [Real.sinh_eq, hminus] at hh
+    change L/2 ≤ (E-E*ρ)/2 at hh
+    linarith
+  have hsquare : L^2 ≤ (E-E*ρ)^2 := by nlinarith
+  have hweighted := mul_le_mul_of_nonneg_right hsquare hρ.le
+  have hid : (E-E*ρ)^2*ρ = (1-ρ)^2 := by
+    calc
+      _ = (E^2*ρ)*(1-ρ)^2 := by ring
+      _ = _ := by rw [he2, one_mul]
+  rw [hid] at hweighted
+  have hcross := (le_div_iff₀ (sq_pos_of_pos (sub_pos.mpr hρ1))).mp hgrowth
+  have hratio : (1-ρ)^2/ρ ≤ R/D := by
+    apply (div_le_div_iff₀ hρ hD).mpr
+    nlinarith
+  exact ((le_div_iff₀ hρ).mpr hweighted).trans hratio
+
+lemma interaction_decay_of_growth {ρ b D a u : ℝ}
+    (hρ : 0 < ρ) (hρ1 : ρ < 1) (hD : 0 < D) (hb : b ≤ D)
+    (hgrowth : D ≤ (b*Real.exp (-2*a*u))*ρ/(1-ρ)^2) :
+    -Real.log ρ ≤ Real.exp (-a*u) := by
+  have hsq := neg_log_sq_le_radius_ratio hρ hρ1 hD hgrowth
+  have hratio : (b*Real.exp (-2*a*u))/D ≤ Real.exp (-2*a*u) := by
+    apply (div_le_iff₀ hD).mpr
+    nlinarith [mul_le_mul_of_nonneg_right hb (Real.exp_pos (-2*a*u)).le]
+  have hexp : Real.exp (-2*a*u) = (Real.exp (-a*u))^2 := by
+    rw [pow_two, ← Real.exp_add]
+    congr 1
+    ring
+  have hfinal := hsq.trans hratio
+  rw [hexp] at hfinal
+  exact (sq_le_sq₀ (neg_nonneg.mpr (Real.log_nonpos hρ.le hρ1.le))
+    (Real.exp_pos _).le).mp hfinal
+
+/-- A physical disk uniformization supplies the growth hypothesis used
+above. No derivative of the original half-plane map is bounded. -/
+theorem disk_interaction_decay (hgrowth : PublishedDiskUnivalentGrowth)
+    (f : ℂ → ℂ) (hf : DifferentiableOn ℂ f (ball 0 1))
+    (hinj : InjOn f (ball 0 1)) (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1)
+    (hz0 : z ≠ 0) (b a u : ℝ) (hb : b ≤ ‖f z-f 0‖)
+    (hradius : ‖deriv f 0‖ = b*Real.exp (-2*a*u)) :
+    -Real.log ‖z‖ ≤ Real.exp (-a*u) := by
+  have hρ : 0 < ‖z‖ := norm_pos_iff.mpr hz0
+  have hρ1 : ‖z‖ < 1 := by simpa only [mem_ball, dist_zero_right] using hz
+  have hD : 0 < ‖f z-f 0‖ := norm_pos_iff.mpr (sub_ne_zero.mpr (by
+    intro he
+    exact hz0 (hinj hz (by simp) he)))
+  apply interaction_decay_of_growth hρ hρ1 hD hb
+  simpa only [hradius] using hgrowth f hf hinj z hz
+
+lemma interaction_decay_max {L a u v : ℝ}
+    (hu : L ≤ Real.exp (-a*u)) (hv : L ≤ Real.exp (-a*v)) :
+    L ≤ Real.exp (-a*max u v) := by
+  rcases le_total u v with h | h
+  · simpa only [max_eq_right h] using hv
+  · simpa only [max_eq_left h] using hu
+
+end Multipoint
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointGlobalDiscount.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointGlobalDiscount.lean
index 793155090d50051672fffc41738dfabbcaa56a02..f1b9bc027f24181324162214225221ad6e3593c3 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointGlobalDiscount.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointGlobalDiscount.lean
@@ -1,121 +1,121 @@
-import OAI.Probability.SLE.Multipoint.MultipointUniformCost
-import OAI.Probability.SLE.Multipoint.MultipointIntegratedCost
-
-namespace OAI
-
-/-! A uniform discount budget from the individual live intervals. The
-pair sum is ordered, as in the product generator. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def singleDriftConstant (κ : ℝ) : ℝ :=
-  driftAllowance (greenExponent κ)*driftThreshold (greenExponent κ)^(-greenExponent κ)
-
-lemma singleDriftConstant_pos {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < singleDriftConstant κ := by
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  exact mul_pos (driftAllowance_pos hp) (Real.rpow_pos_of_pos (driftThreshold_pos hp) _)
-
-def uniformDiscountBudget {ι : Type*} [DecidableEq ι] (κ β : ℝ) (I : Finset ι) : ℝ :=
-  κ*(singleDriftConstant κ*((I.card : ℝ)*(2/(κ*β))) +
-    allPatternYoungConstant κ I*((I.offDiag.card : ℝ)*
-      interactionBudget 2 (exponent (greenExponent κ)*(κ*β))))
-
-lemma productDriftCost_le_active_envelope {ι : Type*} [DecidableEq ι]
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {I s : Finset ι} (hs : s ⊆ I)
-    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (z : ι → ℂ × ℝ) (x t : ℝ)
-    (a b : ι → ℝ) (c d : ι × ι → ℝ) (f : ι → ℝ → ℝ) (g : ι × ι → ℝ → ℝ)
-    (hf0 : ∀ i ∈ I, 0 ≤ f i t) (hg0 : ∀ ij ∈ I.offDiag, 0 ≤ g ij t)
-    (hfi : ∀ i ∈ s, t ∈ Ioc (a i) (b i))
-    (hgi : ∀ ij ∈ s.offDiag, t ∈ Ioc (c ij) (d ij))
-    (hf : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2 = f i t)
-    (hg : ∀ ij ∈ s.offDiag,
-      (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
-        branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent (greenExponent κ))*
-        (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x) = g ij t) :
-    productDriftCost s κ D σ z x ≤
-      κ*(singleDriftConstant κ*(∑ i ∈ I, (Ioc (a i) (b i)).indicator (f i) t) +
-        allPatternYoungConstant κ I*(∑ ij ∈ I.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t)) := by
-  apply (productDriftCost_le_uniform hκ hκ8 hs D σ hD z x).trans
-  apply mul_le_mul_of_nonneg_left _ hκ.le
-  apply add_le_add
-  · apply mul_le_mul_of_nonneg_left _ (singleDriftConstant_pos hκ hκ8).le
-    calc
-      _ = ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t := by
-        apply Finset.sum_congr rfl
-        intro i hi
-        rw [hf i hi, indicator_of_mem (hfi i hi)]
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg hs (fun i hi _ => by
-        by_cases ht : t ∈ Ioc (a i) (b i)
-        · simpa only [indicator_of_mem ht] using hf0 i hi
-        · simp only [indicator_of_notMem ht, le_refl])
-  · apply mul_le_mul_of_nonneg_left _ (allPatternYoungConstant_pos hκ hκ8 I).le
-    calc
-      _ = ∑ ij ∈ s.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t := by
-        apply Finset.sum_congr rfl
-        intro ij hij
-        rw [hg ij hij, indicator_of_mem (hgi ij hij)]
-      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.offDiag_mono hs) (fun ij hij _ => by
-        by_cases ht : t ∈ Ioc (c ij) (d ij)
-        · simpa only [indicator_of_mem ht] using hg0 ij hij
-        · simp only [indicator_of_notMem ht, le_refl])
-
-theorem integrated_uniform_discount_le {ι : Type*} [DecidableEq ι]
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (β : ℝ) (I : Finset ι)
-    (a b : ι → ℝ) (c d : ι × ι → ℝ) (H : ℝ) (hH : 0 ≤ H)
-    (f : ι → ℝ → ℝ) (g : ι × ι → ℝ → ℝ)
-    (ha : ∀ i ∈ I, 0 ≤ a i) (hab : ∀ i ∈ I, a i ≤ b i) (hb : ∀ i ∈ I, b i ≤ H)
-    (hc : ∀ ij ∈ I.offDiag, 0 ≤ c ij) (hcd : ∀ ij ∈ I.offDiag, c ij ≤ d ij)
-    (hd : ∀ ij ∈ I.offDiag, d ij ≤ H)
-    (hf : ∀ i ∈ I, IntervalIntegrable (f i) volume (a i) (b i))
-    (hg : ∀ ij ∈ I.offDiag, IntervalIntegrable (g ij) volume (c ij) (d ij))
-    (hfB : ∀ i ∈ I, (∫ t in a i..b i, f i t) ≤ 2/(κ*β))
-    (hgB : ∀ ij ∈ I.offDiag, (∫ t in c ij..d ij, g ij t) ≤
-      interactionBudget 2 (exponent (greenExponent κ)*(κ*β)))
-    (R : ℝ → ℝ) (hR : IntervalIntegrable R volume 0 H)
-    (hle : R ≤ᵐ[volume.restrict (Icc 0 H)] (fun t =>
-      κ*(singleDriftConstant κ*(∑ i ∈ I, (Ioc (a i) (b i)).indicator (f i) t) +
-        allPatternYoungConstant κ I*(∑ ij ∈ I.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t)))) :
-    (∫ t in 0..H, R t) ≤ uniformDiscountBudget κ β I := by
-  let F := fun t => ∑ i ∈ I, (Ioc (a i) (b i)).indicator (f i) t
-  let G := fun t => ∑ ij ∈ I.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t
-  have hFi : IntervalIntegrable F volume 0 H := by
-    convert (IntervalIntegrable.sum (a := 0) (b := H) I
-      (fun i hi => active_interval_integrable (f i) (hf i hi))) using 1
-    ext t
-    simp only [F, Finset.sum_apply]
-  have hGi : IntervalIntegrable G volume 0 H := by
-    convert (IntervalIntegrable.sum (a := 0) (b := H) I.offDiag
-      (fun i hi => active_interval_integrable (g i) (hg i hi))) using 1
-    ext t
-    simp only [G, Finset.sum_apply]
-  have hFB : (∫ t in 0..H, F t) ≤ (I.card : ℝ)*(2/(κ*β)) := by
-    simpa only [Finset.sum_const, nsmul_eq_mul] using
-      finite_active_cost_integral_le I a b (fun _ => 2/(κ*β)) f H ha hab hb hf hfB
-  have hGB : (∫ t in 0..H, G t) ≤ (I.offDiag.card : ℝ)*
-      interactionBudget 2 (exponent (greenExponent κ)*(κ*β)) := by
-    simpa only [Finset.sum_const, nsmul_eq_mul] using
-      finite_active_cost_integral_le I.offDiag c d
-        (fun _ => interactionBudget 2 (exponent (greenExponent κ)*(κ*β))) g H hc hcd hd hg hgB
-  apply (intervalIntegral.integral_mono_ae_restrict hH hR
-    (((hFi.const_mul _).add (hGi.const_mul _)).const_mul _) hle).trans
-  rw [intervalIntegral.integral_const_mul,
-    intervalIntegral.integral_add (hFi.const_mul _) (hGi.const_mul _),
-    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
-  apply mul_le_mul_of_nonneg_left _ hκ.le
-  exact add_le_add
-    (mul_le_mul_of_nonneg_left hFB (singleDriftConstant_pos hκ hκ8).le)
-    (mul_le_mul_of_nonneg_left hGB (allPatternYoungConstant_pos hκ hκ8 I).le)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointUniformCost
+import OAI.Probability.SLE.Multipoint.MultipointIntegratedCost
+
+namespace OAI
+
+/-! A uniform discount budget from the individual live intervals. The
+pair sum is ordered, as in the product generator. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def singleDriftConstant (κ : ℝ) : ℝ :=
+  driftAllowance (greenExponent κ)*driftThreshold (greenExponent κ)^(-greenExponent κ)
+
+lemma singleDriftConstant_pos {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < singleDriftConstant κ := by
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  exact mul_pos (driftAllowance_pos hp) (Real.rpow_pos_of_pos (driftThreshold_pos hp) _)
+
+def uniformDiscountBudget {ι : Type*} [DecidableEq ι] (κ β : ℝ) (I : Finset ι) : ℝ :=
+  κ*(singleDriftConstant κ*((I.card : ℝ)*(2/(κ*β))) +
+    allPatternYoungConstant κ I*((I.offDiag.card : ℝ)*
+      interactionBudget 2 (exponent (greenExponent κ)*(κ*β))))
+
+lemma productDriftCost_le_active_envelope {ι : Type*} [DecidableEq ι]
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {I s : Finset ι} (hs : s ⊆ I)
+    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (z : ι → ℂ × ℝ) (x t : ℝ)
+    (a b : ι → ℝ) (c d : ι × ι → ℝ) (f : ι → ℝ → ℝ) (g : ι × ι → ℝ → ℝ)
+    (hf0 : ∀ i ∈ I, 0 ≤ f i t) (hg0 : ∀ ij ∈ I.offDiag, 0 ≤ g ij t)
+    (hfi : ∀ i ∈ s, t ∈ Ioc (a i) (b i))
+    (hgi : ∀ ij ∈ s.offDiag, t ∈ Ioc (c ij) (d ij))
+    (hf : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2 = f i t)
+    (hg : ∀ ij ∈ s.offDiag,
+      (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
+        branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent (greenExponent κ))*
+        (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x) = g ij t) :
+    productDriftCost s κ D σ z x ≤
+      κ*(singleDriftConstant κ*(∑ i ∈ I, (Ioc (a i) (b i)).indicator (f i) t) +
+        allPatternYoungConstant κ I*(∑ ij ∈ I.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t)) := by
+  apply (productDriftCost_le_uniform hκ hκ8 hs D σ hD z x).trans
+  apply mul_le_mul_of_nonneg_left _ hκ.le
+  apply add_le_add
+  · apply mul_le_mul_of_nonneg_left _ (singleDriftConstant_pos hκ hκ8).le
+    calc
+      _ = ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t := by
+        apply Finset.sum_congr rfl
+        intro i hi
+        rw [hf i hi, indicator_of_mem (hfi i hi)]
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg hs (fun i hi _ => by
+        by_cases ht : t ∈ Ioc (a i) (b i)
+        · simpa only [indicator_of_mem ht] using hf0 i hi
+        · simp only [indicator_of_notMem ht, le_refl])
+  · apply mul_le_mul_of_nonneg_left _ (allPatternYoungConstant_pos hκ hκ8 I).le
+    calc
+      _ = ∑ ij ∈ s.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t := by
+        apply Finset.sum_congr rfl
+        intro ij hij
+        rw [hg ij hij, indicator_of_mem (hgi ij hij)]
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.offDiag_mono hs) (fun ij hij _ => by
+        by_cases ht : t ∈ Ioc (c ij) (d ij)
+        · simpa only [indicator_of_mem ht] using hg0 ij hij
+        · simp only [indicator_of_notMem ht, le_refl])
+
+theorem integrated_uniform_discount_le {ι : Type*} [DecidableEq ι]
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (β : ℝ) (I : Finset ι)
+    (a b : ι → ℝ) (c d : ι × ι → ℝ) (H : ℝ) (hH : 0 ≤ H)
+    (f : ι → ℝ → ℝ) (g : ι × ι → ℝ → ℝ)
+    (ha : ∀ i ∈ I, 0 ≤ a i) (hab : ∀ i ∈ I, a i ≤ b i) (hb : ∀ i ∈ I, b i ≤ H)
+    (hc : ∀ ij ∈ I.offDiag, 0 ≤ c ij) (hcd : ∀ ij ∈ I.offDiag, c ij ≤ d ij)
+    (hd : ∀ ij ∈ I.offDiag, d ij ≤ H)
+    (hf : ∀ i ∈ I, IntervalIntegrable (f i) volume (a i) (b i))
+    (hg : ∀ ij ∈ I.offDiag, IntervalIntegrable (g ij) volume (c ij) (d ij))
+    (hfB : ∀ i ∈ I, (∫ t in a i..b i, f i t) ≤ 2/(κ*β))
+    (hgB : ∀ ij ∈ I.offDiag, (∫ t in c ij..d ij, g ij t) ≤
+      interactionBudget 2 (exponent (greenExponent κ)*(κ*β)))
+    (R : ℝ → ℝ) (hR : IntervalIntegrable R volume 0 H)
+    (hle : R ≤ᵐ[volume.restrict (Icc 0 H)] (fun t =>
+      κ*(singleDriftConstant κ*(∑ i ∈ I, (Ioc (a i) (b i)).indicator (f i) t) +
+        allPatternYoungConstant κ I*(∑ ij ∈ I.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t)))) :
+    (∫ t in 0..H, R t) ≤ uniformDiscountBudget κ β I := by
+  let F := fun t => ∑ i ∈ I, (Ioc (a i) (b i)).indicator (f i) t
+  let G := fun t => ∑ ij ∈ I.offDiag, (Ioc (c ij) (d ij)).indicator (g ij) t
+  have hFi : IntervalIntegrable F volume 0 H := by
+    convert (IntervalIntegrable.sum (a := 0) (b := H) I
+      (fun i hi => active_interval_integrable (f i) (hf i hi))) using 1
+    ext t
+    simp only [F, Finset.sum_apply]
+  have hGi : IntervalIntegrable G volume 0 H := by
+    convert (IntervalIntegrable.sum (a := 0) (b := H) I.offDiag
+      (fun i hi => active_interval_integrable (g i) (hg i hi))) using 1
+    ext t
+    simp only [G, Finset.sum_apply]
+  have hFB : (∫ t in 0..H, F t) ≤ (I.card : ℝ)*(2/(κ*β)) := by
+    simpa only [Finset.sum_const, nsmul_eq_mul] using
+      finite_active_cost_integral_le I a b (fun _ => 2/(κ*β)) f H ha hab hb hf hfB
+  have hGB : (∫ t in 0..H, G t) ≤ (I.offDiag.card : ℝ)*
+      interactionBudget 2 (exponent (greenExponent κ)*(κ*β)) := by
+    simpa only [Finset.sum_const, nsmul_eq_mul] using
+      finite_active_cost_integral_le I.offDiag c d
+        (fun _ => interactionBudget 2 (exponent (greenExponent κ)*(κ*β))) g H hc hcd hd hg hgB
+  apply (intervalIntegral.integral_mono_ae_restrict hH hR
+    (((hFi.const_mul _).add (hGi.const_mul _)).const_mul _) hle).trans
+  rw [intervalIntegral.integral_const_mul,
+    intervalIntegral.integral_add (hFi.const_mul _) (hGi.const_mul _),
+    intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul]
+  apply mul_le_mul_of_nonneg_left _ hκ.le
+  exact add_le_add
+    (mul_le_mul_of_nonneg_left hFB (singleDriftConstant_pos hκ hκ8).le)
+    (mul_le_mul_of_nonneg_left hGB (allPatternYoungConstant_pos hκ hκ8 I).le)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointHorizonStages.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointHorizonStages.lean
index 397ce4ff70f39e2240a42e8c9de7560e16412960..d204863084410cac91392c0d4356a506fe120056 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointHorizonStages.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointHorizonStages.lean
@@ -1,117 +1,117 @@
-import OAI.Probability.SLE.Multipoint.MultipointPhaseProcess
-
-namespace OAI
-
-/-! Only the integration intervals are capped at the localization horizon.
-The phase continues to read the original switches, so an unreached switch
-is not mistaken for completion at the horizon. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def horizonStage {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0) (n : ℕ) (ω : Ω) : ℝ≥0 :=
-  min (sortedStopAt l (H+1) n ω) H
-
-lemma horizonStage_stopping {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (l : List (Ω → ℝ≥0))
-    (hl : ∀ τ ∈ l, IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    (H : ℝ≥0) (n : ℕ) :
-    IsStoppingTime ℱ (fun ω => (horizonStage l H n ω : WithTop ℝ≥0)) := by
-  simpa only [horizonStage, WithTop.coe_min] using
-    (sortedStopAt_stopping ℱ l hl (H+1) n).min_const H
-
-lemma horizonStage_monotone {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (ω : Ω) :
-    Monotone (fun n => horizonStage l H n ω) := by
-  intro i j hij
-  exact min_le_min_right H (sortedStopAt_monotone l (H+1) hl ω hij)
-
-lemma horizonStage_le {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0) (n : ℕ) (ω : Ω) :
-    horizonStage l H n ω ≤ H := min_le_right _ _
-
-lemma horizonStage_last {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0) (ω : Ω) :
-    horizonStage l H l.length ω = H := by
-  simp only [horizonStage, sortedStopAt_length]
-  exact min_eq_right (le_add_of_nonneg_right zero_le_one)
-
-lemma exists_finite_stage (τ : ℕ → ℝ≥0) (N : ℕ) (t : ℝ≥0)
-    (h0 : τ 0 ≤ t) (hN : t < τ N) :
-    ∃ n < N, τ n ≤ t ∧ t < τ (n+1) := by
-  induction N with
-  | zero => exact False.elim ((not_lt_of_ge h0) hN)
-  | succ N ih =>
-    by_cases h : t < τ N
-    · obtain ⟨n, hn, ht⟩ := ih h
-      exact ⟨n, Nat.lt_trans hn (Nat.lt_succ_self N), ht⟩
-    · exact ⟨N, Nat.lt_succ_self N, le_of_not_gt h, hN⟩
-
-lemma horizonStage_zero {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
-    (ω : Ω) : horizonStage l H 0 ω = 0 := by
-  rw [horizonStage, sortedStopAt_zero l (H+1) hl hzero ω]
-  exact min_eq_left bot_le
-
-lemma horizonStage_cover {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
-    (ω : Ω) (t : ℝ≥0) (ht : t < H) :
-    ∃ n < l.length, horizonStage l H n ω ≤ t ∧ t < horizonStage l H (n+1) ω := by
-  apply exists_finite_stage
-  · rw [horizonStage_zero l H hl hzero ω]
-    exact bot_le
-  · rwa [horizonStage_last]
-
-lemma horizonStage_nonempty {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (n : ℕ) (ω : Ω) (t : ℝ≥0)
-    (hnt : horizonStage l H n ω ≤ t) (htn : t < horizonStage l H (n+1) ω) :
-    sortedStopAt l (H+1) n ω ≤ t ∧ t < sortedStopAt l (H+1) (n+1) ω := by
-  have htH : t < H := htn.trans_le (horizonStage_le l H (n+1) ω)
-  have hnH : sortedStopAt l (H+1) n ω < H := by
-    by_contra h
-    have hn := le_of_not_gt h
-    have he : horizonStage l H n ω = H := min_eq_right hn
-    rw [he] at hnt
-    exact (not_le_of_gt htH) hnt
-  exact ⟨by simpa only [horizonStage, min_eq_left hnH.le] using hnt,
-    htn.trans_le (min_le_left _ _)⟩
-
-lemma phaseAt_const_horizon_interval {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
-    (n : ℕ) (ω : Ω) (t : ℝ≥0)
-    (hnt : horizonStage l H n ω ≤ t) (htn : t < horizonStage l H (n+1) ω) :
-    phaseAt ρ t ω = phaseAt ρ (horizonStage l H n ω) ω := by
-  obtain ⟨hleft, hright⟩ := horizonStage_nonempty l H n ω t hnt htn
-  have htH : t < H := htn.trans_le (horizonStage_le l H (n+1) ω)
-  have hnH' : sortedStopAt l (H+1) n ω ≤ H := hleft.trans htH.le
-  rw [horizonStage, min_eq_left hnH']
-  exact phaseAt_const_sorted_interval ρ l (H+1) hl hρ n ω t hleft hright
-
-lemma phaseAt_crossed_horizon_switch {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
-    (n : ℕ) (ω : Ω) (i : ι) (k : Fin 3)
-    (hord : Monotone (fun j => ρ (i,j) ω))
-    (hbefore : (phaseAt ρ (horizonStage l H n ω) ω i).val ≤ k.val)
-    (hafter : k.val+1 ≤ (phaseAt ρ (horizonStage l H (n+1) ω) ω i).val) :
-    ρ (i,k) ω = horizonStage l H (n+1) ω := by
-  have hupper := (phaseAt_ge_iff ρ ω i hord _ k).mp hafter
-  have hlt : horizonStage l H n ω < ρ (i,k) ω := by
-    by_contra h
-    have hc := (phaseAt_ge_iff ρ ω i hord _ k).mpr (le_of_not_gt h)
-    omega
-  have hsorted : sortedStopAt l (H+1) n ω < ρ (i,k) ω := by
-    by_contra h
-    have hp : ρ (i,k) ω ≤ sortedStopAt l (H+1) n ω := le_of_not_gt h
-    have hH : ρ (i,k) ω ≤ H := hupper.trans (horizonStage_le l H (n+1) ω)
-    exact (not_le_of_gt hlt) (le_min hp hH)
-  have hnext := sortedStopAt_no_intermediate l (H+1) hl _ (hρ (i,k)) n ω hsorted
-  exact le_antisymm hupper ((min_le_left _ _).trans hnext)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPhaseProcess
+
+namespace OAI
+
+/-! Only the integration intervals are capped at the localization horizon.
+The phase continues to read the original switches, so an unreached switch
+is not mistaken for completion at the horizon. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def horizonStage {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0) (n : ℕ) (ω : Ω) : ℝ≥0 :=
+  min (sortedStopAt l (H+1) n ω) H
+
+lemma horizonStage_stopping {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (l : List (Ω → ℝ≥0))
+    (hl : ∀ τ ∈ l, IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    (H : ℝ≥0) (n : ℕ) :
+    IsStoppingTime ℱ (fun ω => (horizonStage l H n ω : WithTop ℝ≥0)) := by
+  simpa only [horizonStage, WithTop.coe_min] using
+    (sortedStopAt_stopping ℱ l hl (H+1) n).min_const H
+
+lemma horizonStage_monotone {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (ω : Ω) :
+    Monotone (fun n => horizonStage l H n ω) := by
+  intro i j hij
+  exact min_le_min_right H (sortedStopAt_monotone l (H+1) hl ω hij)
+
+lemma horizonStage_le {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0) (n : ℕ) (ω : Ω) :
+    horizonStage l H n ω ≤ H := min_le_right _ _
+
+lemma horizonStage_last {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0) (ω : Ω) :
+    horizonStage l H l.length ω = H := by
+  simp only [horizonStage, sortedStopAt_length]
+  exact min_eq_right (le_add_of_nonneg_right zero_le_one)
+
+lemma exists_finite_stage (τ : ℕ → ℝ≥0) (N : ℕ) (t : ℝ≥0)
+    (h0 : τ 0 ≤ t) (hN : t < τ N) :
+    ∃ n < N, τ n ≤ t ∧ t < τ (n+1) := by
+  induction N with
+  | zero => exact False.elim ((not_lt_of_ge h0) hN)
+  | succ N ih =>
+    by_cases h : t < τ N
+    · obtain ⟨n, hn, ht⟩ := ih h
+      exact ⟨n, Nat.lt_trans hn (Nat.lt_succ_self N), ht⟩
+    · exact ⟨N, Nat.lt_succ_self N, le_of_not_gt h, hN⟩
+
+lemma horizonStage_zero {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
+    (ω : Ω) : horizonStage l H 0 ω = 0 := by
+  rw [horizonStage, sortedStopAt_zero l (H+1) hl hzero ω]
+  exact min_eq_left bot_le
+
+lemma horizonStage_cover {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
+    (ω : Ω) (t : ℝ≥0) (ht : t < H) :
+    ∃ n < l.length, horizonStage l H n ω ≤ t ∧ t < horizonStage l H (n+1) ω := by
+  apply exists_finite_stage
+  · rw [horizonStage_zero l H hl hzero ω]
+    exact bot_le
+  · rwa [horizonStage_last]
+
+lemma horizonStage_nonempty {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (n : ℕ) (ω : Ω) (t : ℝ≥0)
+    (hnt : horizonStage l H n ω ≤ t) (htn : t < horizonStage l H (n+1) ω) :
+    sortedStopAt l (H+1) n ω ≤ t ∧ t < sortedStopAt l (H+1) (n+1) ω := by
+  have htH : t < H := htn.trans_le (horizonStage_le l H (n+1) ω)
+  have hnH : sortedStopAt l (H+1) n ω < H := by
+    by_contra h
+    have hn := le_of_not_gt h
+    have he : horizonStage l H n ω = H := min_eq_right hn
+    rw [he] at hnt
+    exact (not_le_of_gt htH) hnt
+  exact ⟨by simpa only [horizonStage, min_eq_left hnH.le] using hnt,
+    htn.trans_le (min_le_left _ _)⟩
+
+lemma phaseAt_const_horizon_interval {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
+    (n : ℕ) (ω : Ω) (t : ℝ≥0)
+    (hnt : horizonStage l H n ω ≤ t) (htn : t < horizonStage l H (n+1) ω) :
+    phaseAt ρ t ω = phaseAt ρ (horizonStage l H n ω) ω := by
+  obtain ⟨hleft, hright⟩ := horizonStage_nonempty l H n ω t hnt htn
+  have htH : t < H := htn.trans_le (horizonStage_le l H (n+1) ω)
+  have hnH' : sortedStopAt l (H+1) n ω ≤ H := hleft.trans htH.le
+  rw [horizonStage, min_eq_left hnH']
+  exact phaseAt_const_sorted_interval ρ l (H+1) hl hρ n ω t hleft hright
+
+lemma phaseAt_crossed_horizon_switch {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
+    (n : ℕ) (ω : Ω) (i : ι) (k : Fin 3)
+    (hord : Monotone (fun j => ρ (i,j) ω))
+    (hbefore : (phaseAt ρ (horizonStage l H n ω) ω i).val ≤ k.val)
+    (hafter : k.val+1 ≤ (phaseAt ρ (horizonStage l H (n+1) ω) ω i).val) :
+    ρ (i,k) ω = horizonStage l H (n+1) ω := by
+  have hupper := (phaseAt_ge_iff ρ ω i hord _ k).mp hafter
+  have hlt : horizonStage l H n ω < ρ (i,k) ω := by
+    by_contra h
+    have hc := (phaseAt_ge_iff ρ ω i hord _ k).mpr (le_of_not_gt h)
+    omega
+  have hsorted : sortedStopAt l (H+1) n ω < ρ (i,k) ω := by
+    by_contra h
+    have hp : ρ (i,k) ω ≤ sortedStopAt l (H+1) n ω := le_of_not_gt h
+    have hH : ρ (i,k) ω ≤ H := hupper.trans (horizonStage_le l H (n+1) ω)
+    exact (not_le_of_gt hlt) (le_min hp hH)
+  have hnext := sortedStopAt_no_intermediate l (H+1) hl _ (hρ (i,k)) n ω hsorted
+  exact le_antisymm hupper ((min_le_left _ _).trans hnext)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointInitialParameters.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointInitialParameters.lean
index 9dd105dee9d0edfb205827de544b4792cc9c18b3..c0b7012759f70f3a93f255b5c10f0f6a60884be2 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointInitialParameters.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointInitialParameters.lean
@@ -1,121 +1,121 @@
-import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus
-import OAI.Probability.SLE.Multipoint.MultipointSuccess
-
-namespace OAI
-
-/-! The source-normalized parameters for the physical multipoint estimate.
-All constants depend only on the SLE parameter and the finite order. -/
-
-noncomputable section
-open Set
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-def sourceBeta (κ : ℝ) : ℝ := beta (2/κ) (greenExponent κ)
-
-def sourceTarget (κ b e : ℝ) : ℝ := κ/4*Real.log (b/e)
-
-lemma source_parameters {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < greenExponent κ ∧ 0 < sourceBeta κ ∧ sourceBeta κ ≤ 1 ∧
-      2*(exponent (greenExponent κ)*sourceBeta κ) < 2/κ ∧
-      2*(exponent (greenExponent κ)*(κ*sourceBeta κ)) < 2 := by
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos,lt_div_iff₀ hκ]
-    linarith
-  have ha : 0 < 2/κ := div_pos (by norm_num) hκ
-  have hdecay := twice_exponent_beta_lt ha hp
-  refine ⟨hp,beta_pos ha hp,beta_le_one _ _,hdecay,?_⟩
-  have hscaled := (source_pair_exponent_admissible (c := exponent (greenExponent κ)*sourceBeta κ) hκ).mpr hdecay
-  convert hscaled using 1
-  ring
-
-lemma sourceTarget_nonneg {κ b e : ℝ} (hκ : 0 ≤ κ) (he : 0 < e) (heb : e ≤ b) :
-    0 ≤ sourceTarget κ b e := by
-  exact mul_nonneg (div_nonneg hκ (by norm_num))
-    (Real.log_nonneg ((one_le_div he).mpr heb))
-
-lemma physical_radius_level_parameters {κ : ℝ} (hκ : 0 < κ)
-    (z : ℂ) (R b e : ℝ) (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) :
-    let u₀ := Real.log (R/b)
-    let u₂ := Real.log (R/e)
-    0 ≤ u₀ ∧ u₀ ≤ u₂ ∧ 0 ≤ sourceTarget κ b e ∧
-      sourceClockLevel κ z u₂ = sourceClockLevel κ z u₀ + sourceTarget κ b e ∧
-      sourceClockLevel κ z ((u₀+u₂)/2) =
-        sourceClockLevel κ z u₀ + sourceTarget κ b e/2 := by
-  dsimp only
-  have hb := he.trans_le heb
-  have hR := hb.trans_le hbR
-  have h₀ : 0 ≤ Real.log (R/b) := Real.log_nonneg ((one_le_div hb).mpr hbR)
-  have h₂ : Real.log (R/b) ≤ Real.log (R/e) :=
-    Real.log_le_log (div_pos hR hb) (div_le_div_of_nonneg_left hR.le he heb)
-  have hgap : Real.log (R/e)-Real.log (R/b) = Real.log (b/e) := by
-    rw [Real.log_div hR.ne' he.ne',Real.log_div hR.ne' hb.ne',Real.log_div hb.ne' he.ne']
-    ring
-  refine ⟨h₀,h₂,sourceTarget_nonneg hκ.le he heb,?_,?_⟩
-  · unfold sourceClockLevel sourceTarget
-    rw [← hgap]
-    ring
-  · unfold sourceClockLevel sourceTarget
-    rw [← hgap]
-    ring
-
-lemma physical_switch_level_parameters {κ : ℝ} (hκ : 0 < κ)
-    (z : ℂ) (R b e : ℝ) (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) :
-    (∀ j, 0 ≤ switchLevel R b e j) ∧ Monotone (switchLevel R b e) ∧
-      sourceClockLevel κ z (switchLevel R b e 1) =
-        sourceClockLevel κ z (switchLevel R b e 0) + sourceTarget κ b e/2 ∧
-      sourceClockLevel κ z (switchLevel R b e 2) =
-        sourceClockLevel κ z (switchLevel R b e 0) + sourceTarget κ b e := by
-  obtain ⟨_,_,_,hend,hmid⟩ := physical_radius_level_parameters hκ z R b e he heb hbR
-  refine ⟨fun j => switchLevel_nonneg he heb hbR j,
-    switchLevel_monotone he heb hbR,?_,?_⟩
-  · simpa [switchLevel] using hmid
-  · simpa [switchLevel] using hend
-
-lemma sourceClockLevel_physical_activation (κ : ℝ) (J : ℂ → ℂ) (z : ℂ)
-    (hz : 0 < z.im) (R b : ℝ) (hR : 0 < R) (hb : 0 < b)
-    (hRdef : R = 2*z.im*‖deriv J z‖) :
-    sourceClockLevel κ z (Real.log (R/b)) = κ/4*Real.log (2*‖deriv J z‖/b) := by
-  have hd : ‖deriv J z‖ ≠ 0 := by
-    intro he
-    rw [hRdef,he,mul_zero] at hR
-    exact (lt_irrefl 0 hR)
-  have h2 : (2 : ℝ) ≠ 0 := by norm_num
-  unfold sourceClockLevel
-  rw [hRdef,Real.log_div (mul_ne_zero (mul_ne_zero h2 hz.ne') hd) hb.ne',
-    Real.log_mul (mul_ne_zero h2 hz.ne') hd,Real.log_mul h2 hz.ne',
-    Real.log_div (mul_ne_zero h2 hd) hb.ne',Real.log_mul h2 hd]
-  ring
-
-lemma sourceTarget_coefficient {κ : ℝ} (hκ : 0 < κ) :
-    1/(2*(2/κ)) = κ/4 := by field_simp [hκ.ne']; ring
-
-lemma source_terminal_exponent {κ : ℝ} (hκ : 0 < κ) :
-    greenExponent κ/(4*(2/κ)) = 1-κ/8 := by
-  unfold greenExponent
-  field_simp [hκ.ne']
-  ring
-
-lemma source_radius_targets_exponential {ι : Type*} (s : Finset ι)
-    {κ : ℝ} (hκ : 0 < κ) (b e : ι → ℝ)
-    (hb : ∀ i ∈ s, 0 < b i) (he : ∀ i ∈ s, 0 < e i) :
-    Real.exp (-greenExponent κ/2 * ∑ i ∈ s, sourceTarget κ (b i) (e i)) =
-      ∏ i ∈ s, (e i/b i)^(1-κ/8) := by
-  have h := radius_targets_exponential s (a := 2/κ) (p := greenExponent κ)
-    (show 0 < (2 : ℝ)/κ from div_pos (by norm_num) hκ) b e hb he
-  simpa only [sourceTarget,sourceTarget_coefficient hκ,source_terminal_exponent hκ] using h
-
-lemma source_successBarrier_inv {ι : Type*} (s : Finset ι)
-    {κ : ℝ} (hκ : 0 < κ) (K : ℝ) (b e : ι → ℝ)
-    (hb : ∀ i ∈ s, 0 < b i) (he : ∀ i ∈ s, 0 < e i) :
-    (successBarrier (greenExponent κ) K s (fun i => sourceTarget κ (b i) (e i)))⁻¹ =
-      (3 : ℝ)^s.card * Real.exp K * ∏ i ∈ s, (e i/b i)^(1-κ/8) := by
-  rw [successBarrier_inv,source_radius_targets_exponential s hκ b e hb he]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus
+import OAI.Probability.SLE.Multipoint.MultipointSuccess
+
+namespace OAI
+
+/-! The source-normalized parameters for the physical multipoint estimate.
+All constants depend only on the SLE parameter and the finite order. -/
+
+noncomputable section
+open Set
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+def sourceBeta (κ : ℝ) : ℝ := beta (2/κ) (greenExponent κ)
+
+def sourceTarget (κ b e : ℝ) : ℝ := κ/4*Real.log (b/e)
+
+lemma source_parameters {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < greenExponent κ ∧ 0 < sourceBeta κ ∧ sourceBeta κ ≤ 1 ∧
+      2*(exponent (greenExponent κ)*sourceBeta κ) < 2/κ ∧
+      2*(exponent (greenExponent κ)*(κ*sourceBeta κ)) < 2 := by
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos,lt_div_iff₀ hκ]
+    linarith
+  have ha : 0 < 2/κ := div_pos (by norm_num) hκ
+  have hdecay := twice_exponent_beta_lt ha hp
+  refine ⟨hp,beta_pos ha hp,beta_le_one _ _,hdecay,?_⟩
+  have hscaled := (source_pair_exponent_admissible (c := exponent (greenExponent κ)*sourceBeta κ) hκ).mpr hdecay
+  convert hscaled using 1
+  ring
+
+lemma sourceTarget_nonneg {κ b e : ℝ} (hκ : 0 ≤ κ) (he : 0 < e) (heb : e ≤ b) :
+    0 ≤ sourceTarget κ b e := by
+  exact mul_nonneg (div_nonneg hκ (by norm_num))
+    (Real.log_nonneg ((one_le_div he).mpr heb))
+
+lemma physical_radius_level_parameters {κ : ℝ} (hκ : 0 < κ)
+    (z : ℂ) (R b e : ℝ) (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) :
+    let u₀ := Real.log (R/b)
+    let u₂ := Real.log (R/e)
+    0 ≤ u₀ ∧ u₀ ≤ u₂ ∧ 0 ≤ sourceTarget κ b e ∧
+      sourceClockLevel κ z u₂ = sourceClockLevel κ z u₀ + sourceTarget κ b e ∧
+      sourceClockLevel κ z ((u₀+u₂)/2) =
+        sourceClockLevel κ z u₀ + sourceTarget κ b e/2 := by
+  dsimp only
+  have hb := he.trans_le heb
+  have hR := hb.trans_le hbR
+  have h₀ : 0 ≤ Real.log (R/b) := Real.log_nonneg ((one_le_div hb).mpr hbR)
+  have h₂ : Real.log (R/b) ≤ Real.log (R/e) :=
+    Real.log_le_log (div_pos hR hb) (div_le_div_of_nonneg_left hR.le he heb)
+  have hgap : Real.log (R/e)-Real.log (R/b) = Real.log (b/e) := by
+    rw [Real.log_div hR.ne' he.ne',Real.log_div hR.ne' hb.ne',Real.log_div hb.ne' he.ne']
+    ring
+  refine ⟨h₀,h₂,sourceTarget_nonneg hκ.le he heb,?_,?_⟩
+  · unfold sourceClockLevel sourceTarget
+    rw [← hgap]
+    ring
+  · unfold sourceClockLevel sourceTarget
+    rw [← hgap]
+    ring
+
+lemma physical_switch_level_parameters {κ : ℝ} (hκ : 0 < κ)
+    (z : ℂ) (R b e : ℝ) (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) :
+    (∀ j, 0 ≤ switchLevel R b e j) ∧ Monotone (switchLevel R b e) ∧
+      sourceClockLevel κ z (switchLevel R b e 1) =
+        sourceClockLevel κ z (switchLevel R b e 0) + sourceTarget κ b e/2 ∧
+      sourceClockLevel κ z (switchLevel R b e 2) =
+        sourceClockLevel κ z (switchLevel R b e 0) + sourceTarget κ b e := by
+  obtain ⟨_,_,_,hend,hmid⟩ := physical_radius_level_parameters hκ z R b e he heb hbR
+  refine ⟨fun j => switchLevel_nonneg he heb hbR j,
+    switchLevel_monotone he heb hbR,?_,?_⟩
+  · simpa [switchLevel] using hmid
+  · simpa [switchLevel] using hend
+
+lemma sourceClockLevel_physical_activation (κ : ℝ) (J : ℂ → ℂ) (z : ℂ)
+    (hz : 0 < z.im) (R b : ℝ) (hR : 0 < R) (hb : 0 < b)
+    (hRdef : R = 2*z.im*‖deriv J z‖) :
+    sourceClockLevel κ z (Real.log (R/b)) = κ/4*Real.log (2*‖deriv J z‖/b) := by
+  have hd : ‖deriv J z‖ ≠ 0 := by
+    intro he
+    rw [hRdef,he,mul_zero] at hR
+    exact (lt_irrefl 0 hR)
+  have h2 : (2 : ℝ) ≠ 0 := by norm_num
+  unfold sourceClockLevel
+  rw [hRdef,Real.log_div (mul_ne_zero (mul_ne_zero h2 hz.ne') hd) hb.ne',
+    Real.log_mul (mul_ne_zero h2 hz.ne') hd,Real.log_mul h2 hz.ne',
+    Real.log_div (mul_ne_zero h2 hd) hb.ne',Real.log_mul h2 hd]
+  ring
+
+lemma sourceTarget_coefficient {κ : ℝ} (hκ : 0 < κ) :
+    1/(2*(2/κ)) = κ/4 := by field_simp [hκ.ne']; ring
+
+lemma source_terminal_exponent {κ : ℝ} (hκ : 0 < κ) :
+    greenExponent κ/(4*(2/κ)) = 1-κ/8 := by
+  unfold greenExponent
+  field_simp [hκ.ne']
+  ring
+
+lemma source_radius_targets_exponential {ι : Type*} (s : Finset ι)
+    {κ : ℝ} (hκ : 0 < κ) (b e : ι → ℝ)
+    (hb : ∀ i ∈ s, 0 < b i) (he : ∀ i ∈ s, 0 < e i) :
+    Real.exp (-greenExponent κ/2 * ∑ i ∈ s, sourceTarget κ (b i) (e i)) =
+      ∏ i ∈ s, (e i/b i)^(1-κ/8) := by
+  have h := radius_targets_exponential s (a := 2/κ) (p := greenExponent κ)
+    (show 0 < (2 : ℝ)/κ from div_pos (by norm_num) hκ) b e hb he
+  simpa only [sourceTarget,sourceTarget_coefficient hκ,source_terminal_exponent hκ] using h
+
+lemma source_successBarrier_inv {ι : Type*} (s : Finset ι)
+    {κ : ℝ} (hκ : 0 < κ) (K : ℝ) (b e : ι → ℝ)
+    (hb : ∀ i ∈ s, 0 < b i) (he : ∀ i ∈ s, 0 < e i) :
+    (successBarrier (greenExponent κ) K s (fun i => sourceTarget κ (b i) (e i)))⁻¹ =
+      (3 : ℝ)^s.card * Real.exp K * ∏ i ∈ s, (e i/b i)^(1-κ/8) := by
+  rw [successBarrier_inv,source_radius_targets_exponential s hκ b e hb he]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointIntegratedCost.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointIntegratedCost.lean
index e9cd1ec4e402d23bc0da97199f44bf6ef9136a04..f4aa57da331e90e76e26b4dd6aa42afa51c87960 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointIntegratedCost.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointIntegratedCost.lean
@@ -1,65 +1,65 @@
-import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
-
-namespace OAI
-
-/-! Integrating finitely many active-interval costs. The interval endpoints
-may be different for every point or pair. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-
-namespace SLEExactGauge.Multipoint
-
-lemma active_interval_integral (f : ℝ → ℝ) {a b H : ℝ}
-    (ha : 0 ≤ a) (hab : a ≤ b) (hb : b ≤ H) :
-    (∫ t in 0..H, (Ioc a b).indicator f t) = ∫ t in a..b, f t := by
-  have h0H := ha.trans (hab.trans hb)
-  rw [intervalIntegral.integral_of_le h0H, MeasureTheory.integral_indicator measurableSet_Ioc,
-    Measure.restrict_restrict measurableSet_Ioc]
-  rw [inter_eq_left.mpr (Ioc_subset_Ioc ha hb)]
-  exact (intervalIntegral.integral_of_le hab).symm
-
-lemma active_interval_integrable (f : ℝ → ℝ) {a b H : ℝ}
-    (hf : IntervalIntegrable f volume a b) :
-    IntervalIntegrable ((Ioc a b).indicator f) volume 0 H :=
-  ((integrable_indicator_iff measurableSet_Ioc).mpr hf.1).intervalIntegrable
-
-theorem finite_active_cost_integral_le {ι : Type*}
-    (s : Finset ι) (a b B : ι → ℝ) (f : ι → ℝ → ℝ) (H : ℝ)
-    (ha : ∀ i ∈ s, 0 ≤ a i) (hab : ∀ i ∈ s, a i ≤ b i)
-    (hb : ∀ i ∈ s, b i ≤ H)
-    (hf : ∀ i ∈ s, IntervalIntegrable (f i) volume (a i) (b i))
-    (hB : ∀ i ∈ s, (∫ t in a i..b i, f i t) ≤ B i) :
-    (∫ t in 0..H, ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t) ≤ ∑ i ∈ s, B i := by
-  rw [intervalIntegral.integral_finsetSum
-    (fun i hi => active_interval_integrable (f i) (hf i hi))]
-  apply Finset.sum_le_sum
-  intro i hi
-  rw [active_interval_integral (f i) (ha i hi) (hab i hi) (hb i hi)]
-  exact hB i hi
-
-theorem dominated_active_cost_integral_le {ι : Type*}
-    (s : Finset ι) (a b B : ι → ℝ) (f : ι → ℝ → ℝ) (H : ℝ) (hH : 0 ≤ H)
-    (ha : ∀ i ∈ s, 0 ≤ a i) (hab : ∀ i ∈ s, a i ≤ b i)
-    (hb : ∀ i ∈ s, b i ≤ H)
-    (hf : ∀ i ∈ s, IntervalIntegrable (f i) volume (a i) (b i))
-    (hB : ∀ i ∈ s, (∫ t in a i..b i, f i t) ≤ B i)
-    (R : ℝ → ℝ) (hR : IntervalIntegrable R volume 0 H)
-    (hle : R ≤ᵐ[volume.restrict (Icc 0 H)]
-      (fun t => ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t)) :
-    (∫ t in 0..H, R t) ≤ ∑ i ∈ s, B i := by
-  have hsum : IntervalIntegrable
-      (fun t => ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t) volume 0 H :=
-    by
-      convert (IntervalIntegrable.sum (a := 0) (b := H) s
-        (fun i hi => active_interval_integrable (f i) (hf i hi))) using 1
-      ext t
-      simp only [Finset.sum_apply]
-  exact (intervalIntegral.integral_mono_ae_restrict hH hR hsum hle).trans
-    (finite_active_cost_integral_le s a b B f H ha hab hb hf hB)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
+
+namespace OAI
+
+/-! Integrating finitely many active-interval costs. The interval endpoints
+may be different for every point or pair. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+
+namespace SLEExactGauge.Multipoint
+
+lemma active_interval_integral (f : ℝ → ℝ) {a b H : ℝ}
+    (ha : 0 ≤ a) (hab : a ≤ b) (hb : b ≤ H) :
+    (∫ t in 0..H, (Ioc a b).indicator f t) = ∫ t in a..b, f t := by
+  have h0H := ha.trans (hab.trans hb)
+  rw [intervalIntegral.integral_of_le h0H, MeasureTheory.integral_indicator measurableSet_Ioc,
+    Measure.restrict_restrict measurableSet_Ioc]
+  rw [inter_eq_left.mpr (Ioc_subset_Ioc ha hb)]
+  exact (intervalIntegral.integral_of_le hab).symm
+
+lemma active_interval_integrable (f : ℝ → ℝ) {a b H : ℝ}
+    (hf : IntervalIntegrable f volume a b) :
+    IntervalIntegrable ((Ioc a b).indicator f) volume 0 H :=
+  ((integrable_indicator_iff measurableSet_Ioc).mpr hf.1).intervalIntegrable
+
+theorem finite_active_cost_integral_le {ι : Type*}
+    (s : Finset ι) (a b B : ι → ℝ) (f : ι → ℝ → ℝ) (H : ℝ)
+    (ha : ∀ i ∈ s, 0 ≤ a i) (hab : ∀ i ∈ s, a i ≤ b i)
+    (hb : ∀ i ∈ s, b i ≤ H)
+    (hf : ∀ i ∈ s, IntervalIntegrable (f i) volume (a i) (b i))
+    (hB : ∀ i ∈ s, (∫ t in a i..b i, f i t) ≤ B i) :
+    (∫ t in 0..H, ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t) ≤ ∑ i ∈ s, B i := by
+  rw [intervalIntegral.integral_finsetSum
+    (fun i hi => active_interval_integrable (f i) (hf i hi))]
+  apply Finset.sum_le_sum
+  intro i hi
+  rw [active_interval_integral (f i) (ha i hi) (hab i hi) (hb i hi)]
+  exact hB i hi
+
+theorem dominated_active_cost_integral_le {ι : Type*}
+    (s : Finset ι) (a b B : ι → ℝ) (f : ι → ℝ → ℝ) (H : ℝ) (hH : 0 ≤ H)
+    (ha : ∀ i ∈ s, 0 ≤ a i) (hab : ∀ i ∈ s, a i ≤ b i)
+    (hb : ∀ i ∈ s, b i ≤ H)
+    (hf : ∀ i ∈ s, IntervalIntegrable (f i) volume (a i) (b i))
+    (hB : ∀ i ∈ s, (∫ t in a i..b i, f i t) ≤ B i)
+    (R : ℝ → ℝ) (hR : IntervalIntegrable R volume 0 H)
+    (hle : R ≤ᵐ[volume.restrict (Icc 0 H)]
+      (fun t => ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t)) :
+    (∫ t in 0..H, R t) ≤ ∑ i ∈ s, B i := by
+  have hsum : IntervalIntegrable
+      (fun t => ∑ i ∈ s, (Ioc (a i) (b i)).indicator (f i) t) volume 0 H :=
+    by
+      convert (IntervalIntegrable.sum (a := 0) (b := H) s
+        (fun i hi => active_interval_integrable (f i) (hf i hi))) using 1
+      ext t
+      simp only [Finset.sum_apply]
+  exact (intervalIntegral.integral_mono_ae_restrict hH hR hsum hle).trans
+    (finite_active_cost_integral_le s a b B f H ha hab hb hf hB)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointInteraction.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointInteraction.lean
index 3598ec5801aa79bcb94db7efd0fc2f9f7c43359e..3f40ed36cdae9668527733376479b5cf4f158485 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointInteraction.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointInteraction.lean
@@ -1,158 +1,158 @@
-import OAI.Probability.SLE.KernelConsistency
-
-namespace OAI
-
-/-! The interaction of two active points in the September 25 manuscript,
-equation `mp:interaction-derivative`.  Centering by a common real driver
-does not change the kernel, so differentiability of that driver is not
-needed for the ordinary Loewner-path calculation. -/
-
-noncomputable section
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def interaction (z w : ℂ) : ℝ := halfPlaneGreen z w
-
-lemma interaction_eq (z w : ℂ) :
-    interaction z w = Real.log ‖(z - conj w) / (z - w)‖ := rfl
-
-lemma interaction_nonneg {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    0 ≤ interaction z w := halfPlaneGreen_nonneg hz hw
-
-lemma interaction_center (z w : ℂ) (u : ℝ) :
-    interaction (z - (u : ℂ)) (w - (u : ℂ)) = interaction z w := by
-  have hn : z - (u : ℂ) - conj (w - (u : ℂ)) = z - conj w := by
-    simp only [map_sub, Complex.conj_ofReal]
-    ring
-  have hd : z - (u : ℂ) - (w - (u : ℂ)) = z - w := by ring
-  unfold interaction halfPlaneGreen
-  rw [hn, hd]
-
-/-- The real-part identity behind the interaction derivative. -/
-lemma interaction_velocity_algebra (a : ℝ) (z w : ℂ) :
-    (-(a : ℂ) / (z * conj w) + (a : ℂ) / (z * w)).re =
-      -2 * a * (z.im / ‖z‖ ^ 2) * (w.im / ‖w‖ ^ 2) := by
-  simp only [Complex.add_re, Complex.div_re, Complex.neg_re, Complex.neg_im,
-    Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re, Complex.conj_re,
-    Complex.conj_im, Complex.normSq_mul, Complex.normSq_conj, Complex.sq_norm]
-  ring
-
-lemma interaction_difference_algebra (a : ℝ) {z w : ℂ}
-    (hz : 0 < z.im) (hw : 0 < w.im) (hne : z ≠ w) :
-    (((a : ℂ) / z - conj ((a : ℂ) / w)) / (z - conj w) -
-      ((a : ℂ) / z - (a : ℂ) / w) / (z - w)).re =
-        -2 * a * (z.im / ‖z‖ ^ 2) * (w.im / ‖w‖ ^ 2) := by
-  have hz0 : z ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hz
-  have hw0 : w ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hw
-  have hcw0 : conj w ≠ 0 := by simpa using hw0
-  have hn : z - w ≠ 0 := sub_ne_zero.mpr hne
-  have hnc : z - conj w ≠ 0 := by
-    intro he
-    have hi := congrArg Complex.im (sub_eq_zero.mp he)
-    simp only [Complex.conj_im] at hi
-    linarith
-  have h₁ : ((a : ℂ) / z - conj ((a : ℂ) / w)) / (z - conj w) =
-      -(a : ℂ) / (z * conj w) := by
-    simp only [map_div₀, Complex.conj_ofReal]
-    field_simp
-    ring
-  have h₂ : ((a : ℂ) / z - (a : ℂ) / w) / (z - w) =
-      -(a : ℂ) / (z * w) := by
-    field_simp
-    ring
-  rw [h₁, h₂]
-  simpa only [neg_div, sub_neg_eq_add] using interaction_velocity_algebra a z w
-
-/-- Two differentiable paths with a common real drift have the source's
-interaction derivative; the common drift cancels in both differences. -/
-theorem interaction_hasDerivWithinAt
-    {f g : ℝ → ℂ} {S : Set ℝ} {t a b : ℝ}
-    (hf : HasDerivWithinAt f ((a : ℂ) / f t + (b : ℂ)) S t)
-    (hg : HasDerivWithinAt g ((a : ℂ) / g t + (b : ℂ)) S t)
-    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
-    HasDerivWithinAt (fun s => interaction (f s) (g s))
-      (-2 * a * ((f t).im / ‖f t‖ ^ 2) * ((g t).im / ‖g t‖ ^ 2)) S t := by
-  have hn : f t - g t ≠ 0 := sub_ne_zero.mpr hne
-  have hnc : f t - conj (g t) ≠ 0 := by
-    intro he
-    have hi := congrArg Complex.im (sub_eq_zero.mp he)
-    simp only [Complex.conj_im] at hi
-    linarith
-  have hd := hasDerivWithinAt_log_norm_div
-    (hf.sub (hasDerivWithinAt_conj hg)) (hf.sub hg) hnc hn
-  apply hd.congr_deriv
-  change (((a : ℂ) / f t + (b : ℂ) - conj ((a : ℂ) / g t + (b : ℂ))) /
-    (f t - conj (g t)) -
-    ((a : ℂ) / f t + (b : ℂ) - ((a : ℂ) / g t + (b : ℂ))) /
-      (f t - g t)).re = _
-  simp only [map_add, Complex.conj_ofReal]
-  have h₁ : (a : ℂ) / f t + (b : ℂ) - (conj ((a : ℂ) / g t) + (b : ℂ)) =
-      (a : ℂ) / f t - conj ((a : ℂ) / g t) := by ring
-  have h₂ : (a : ℂ) / f t + (b : ℂ) - ((a : ℂ) / g t + (b : ℂ)) =
-      (a : ℂ) / f t - (a : ℂ) / g t := by ring
-  rw [h₁, h₂]
-  exact interaction_difference_algebra a hfi hgi hne
-
-theorem interaction_hasDerivAt
-    {f g : ℝ → ℂ} {t a b : ℝ}
-    (hf : HasDerivAt f ((a : ℂ) / f t + (b : ℂ)) t)
-    (hg : HasDerivAt g ((a : ℂ) / g t + (b : ℂ)) t)
-    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
-    HasDerivAt (fun s => interaction (f s) (g s))
-      (-2 * a * ((f t).im / ‖f t‖ ^ 2) * ((g t).im / ‖g t‖ ^ 2)) t :=
-  (interaction_hasDerivWithinAt hf.hasDerivWithinAt hg.hasDerivWithinAt hfi hgi hne
-    (S := univ)).hasDerivAt (univ_mem)
-
-/-- The driver can be any real function here: only the two uncentered
-Loewner paths are differentiated. -/
-theorem centered_interaction_hasDerivWithinAt
-    {f g : ℝ → ℂ} (U : ℝ → ℝ) {S : Set ℝ} {t a : ℝ}
-    (hf : HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) S t)
-    (hg : HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) S t)
-    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
-    HasDerivWithinAt
-      (fun s => interaction (f s - (U s : ℂ)) (g s - (U s : ℂ)))
-      (-2 * a * ((f t).im / ‖f t - (U t : ℂ)‖ ^ 2) *
-        ((g t).im / ‖g t - (U t : ℂ)‖ ^ 2)) S t := by
-  have hn : f t - g t ≠ 0 := sub_ne_zero.mpr hne
-  have hnc : f t - conj (g t) ≠ 0 := by
-    intro he
-    have hi := congrArg Complex.im (sub_eq_zero.mp he)
-    simp only [Complex.conj_im] at hi
-    linarith
-  have hd := hasDerivWithinAt_log_norm_div
-    (hf.sub (hasDerivWithinAt_conj hg)) (hf.sub hg) hnc hn
-  have h₁ : f t - conj (g t) =
-      (f t - (U t : ℂ)) - conj (g t - (U t : ℂ)) := by
-    simp only [map_sub, Complex.conj_ofReal]
-    ring
-  have h₂ : f t - g t = (f t - (U t : ℂ)) - (g t - (U t : ℂ)) := by ring
-  have hd' : HasDerivWithinAt (fun s => interaction (f s) (g s))
-      (-2 * a * ((f t).im / ‖f t - (U t : ℂ)‖ ^ 2) *
-        ((g t).im / ‖g t - (U t : ℂ)‖ ^ 2)) S t := by
-    apply hd.congr_deriv
-    dsimp only [Pi.sub_apply]
-    rw [h₁, h₂, interaction_difference_algebra a (by simpa using hfi)
-      (by simpa using hgi) (sub_left_inj.not.mpr hne)]
-    simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
-  simpa only [interaction_center] using hd'
-
-theorem centered_interaction_hasDerivAt
-    {f g : ℝ → ℂ} (U : ℝ → ℝ) {t a : ℝ}
-    (hf : HasDerivAt f ((a : ℂ) / (f t - (U t : ℂ))) t)
-    (hg : HasDerivAt g ((a : ℂ) / (g t - (U t : ℂ))) t)
-    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
-    HasDerivAt
-      (fun s => interaction (f s - (U s : ℂ)) (g s - (U s : ℂ)))
-      (-2 * a * ((f t).im / ‖f t - (U t : ℂ)‖ ^ 2) *
-        ((g t).im / ‖g t - (U t : ℂ)‖ ^ 2)) t :=
-  (centered_interaction_hasDerivWithinAt U hf.hasDerivWithinAt hg.hasDerivWithinAt
-    hfi hgi hne (S := univ)).hasDerivAt univ_mem
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.KernelConsistency
+
+namespace OAI
+
+/-! The interaction of two active points in the September 25 manuscript,
+equation `mp:interaction-derivative`.  Centering by a common real driver
+does not change the kernel, so differentiability of that driver is not
+needed for the ordinary Loewner-path calculation. -/
+
+noncomputable section
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def interaction (z w : ℂ) : ℝ := halfPlaneGreen z w
+
+lemma interaction_eq (z w : ℂ) :
+    interaction z w = Real.log ‖(z - conj w) / (z - w)‖ := rfl
+
+lemma interaction_nonneg {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    0 ≤ interaction z w := halfPlaneGreen_nonneg hz hw
+
+lemma interaction_center (z w : ℂ) (u : ℝ) :
+    interaction (z - (u : ℂ)) (w - (u : ℂ)) = interaction z w := by
+  have hn : z - (u : ℂ) - conj (w - (u : ℂ)) = z - conj w := by
+    simp only [map_sub, Complex.conj_ofReal]
+    ring
+  have hd : z - (u : ℂ) - (w - (u : ℂ)) = z - w := by ring
+  unfold interaction halfPlaneGreen
+  rw [hn, hd]
+
+/-- The real-part identity behind the interaction derivative. -/
+lemma interaction_velocity_algebra (a : ℝ) (z w : ℂ) :
+    (-(a : ℂ) / (z * conj w) + (a : ℂ) / (z * w)).re =
+      -2 * a * (z.im / ‖z‖ ^ 2) * (w.im / ‖w‖ ^ 2) := by
+  simp only [Complex.add_re, Complex.div_re, Complex.neg_re, Complex.neg_im,
+    Complex.ofReal_re, Complex.ofReal_im, Complex.mul_re, Complex.conj_re,
+    Complex.conj_im, Complex.normSq_mul, Complex.normSq_conj, Complex.sq_norm]
+  ring
+
+lemma interaction_difference_algebra (a : ℝ) {z w : ℂ}
+    (hz : 0 < z.im) (hw : 0 < w.im) (hne : z ≠ w) :
+    (((a : ℂ) / z - conj ((a : ℂ) / w)) / (z - conj w) -
+      ((a : ℂ) / z - (a : ℂ) / w) / (z - w)).re =
+        -2 * a * (z.im / ‖z‖ ^ 2) * (w.im / ‖w‖ ^ 2) := by
+  have hz0 : z ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hz
+  have hw0 : w ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hw
+  have hcw0 : conj w ≠ 0 := by simpa using hw0
+  have hn : z - w ≠ 0 := sub_ne_zero.mpr hne
+  have hnc : z - conj w ≠ 0 := by
+    intro he
+    have hi := congrArg Complex.im (sub_eq_zero.mp he)
+    simp only [Complex.conj_im] at hi
+    linarith
+  have h₁ : ((a : ℂ) / z - conj ((a : ℂ) / w)) / (z - conj w) =
+      -(a : ℂ) / (z * conj w) := by
+    simp only [map_div₀, Complex.conj_ofReal]
+    field_simp
+    ring
+  have h₂ : ((a : ℂ) / z - (a : ℂ) / w) / (z - w) =
+      -(a : ℂ) / (z * w) := by
+    field_simp
+    ring
+  rw [h₁, h₂]
+  simpa only [neg_div, sub_neg_eq_add] using interaction_velocity_algebra a z w
+
+/-- Two differentiable paths with a common real drift have the source's
+interaction derivative; the common drift cancels in both differences. -/
+theorem interaction_hasDerivWithinAt
+    {f g : ℝ → ℂ} {S : Set ℝ} {t a b : ℝ}
+    (hf : HasDerivWithinAt f ((a : ℂ) / f t + (b : ℂ)) S t)
+    (hg : HasDerivWithinAt g ((a : ℂ) / g t + (b : ℂ)) S t)
+    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
+    HasDerivWithinAt (fun s => interaction (f s) (g s))
+      (-2 * a * ((f t).im / ‖f t‖ ^ 2) * ((g t).im / ‖g t‖ ^ 2)) S t := by
+  have hn : f t - g t ≠ 0 := sub_ne_zero.mpr hne
+  have hnc : f t - conj (g t) ≠ 0 := by
+    intro he
+    have hi := congrArg Complex.im (sub_eq_zero.mp he)
+    simp only [Complex.conj_im] at hi
+    linarith
+  have hd := hasDerivWithinAt_log_norm_div
+    (hf.sub (hasDerivWithinAt_conj hg)) (hf.sub hg) hnc hn
+  apply hd.congr_deriv
+  change (((a : ℂ) / f t + (b : ℂ) - conj ((a : ℂ) / g t + (b : ℂ))) /
+    (f t - conj (g t)) -
+    ((a : ℂ) / f t + (b : ℂ) - ((a : ℂ) / g t + (b : ℂ))) /
+      (f t - g t)).re = _
+  simp only [map_add, Complex.conj_ofReal]
+  have h₁ : (a : ℂ) / f t + (b : ℂ) - (conj ((a : ℂ) / g t) + (b : ℂ)) =
+      (a : ℂ) / f t - conj ((a : ℂ) / g t) := by ring
+  have h₂ : (a : ℂ) / f t + (b : ℂ) - ((a : ℂ) / g t + (b : ℂ)) =
+      (a : ℂ) / f t - (a : ℂ) / g t := by ring
+  rw [h₁, h₂]
+  exact interaction_difference_algebra a hfi hgi hne
+
+theorem interaction_hasDerivAt
+    {f g : ℝ → ℂ} {t a b : ℝ}
+    (hf : HasDerivAt f ((a : ℂ) / f t + (b : ℂ)) t)
+    (hg : HasDerivAt g ((a : ℂ) / g t + (b : ℂ)) t)
+    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
+    HasDerivAt (fun s => interaction (f s) (g s))
+      (-2 * a * ((f t).im / ‖f t‖ ^ 2) * ((g t).im / ‖g t‖ ^ 2)) t :=
+  (interaction_hasDerivWithinAt hf.hasDerivWithinAt hg.hasDerivWithinAt hfi hgi hne
+    (S := univ)).hasDerivAt (univ_mem)
+
+/-- The driver can be any real function here: only the two uncentered
+Loewner paths are differentiated. -/
+theorem centered_interaction_hasDerivWithinAt
+    {f g : ℝ → ℂ} (U : ℝ → ℝ) {S : Set ℝ} {t a : ℝ}
+    (hf : HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) S t)
+    (hg : HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) S t)
+    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
+    HasDerivWithinAt
+      (fun s => interaction (f s - (U s : ℂ)) (g s - (U s : ℂ)))
+      (-2 * a * ((f t).im / ‖f t - (U t : ℂ)‖ ^ 2) *
+        ((g t).im / ‖g t - (U t : ℂ)‖ ^ 2)) S t := by
+  have hn : f t - g t ≠ 0 := sub_ne_zero.mpr hne
+  have hnc : f t - conj (g t) ≠ 0 := by
+    intro he
+    have hi := congrArg Complex.im (sub_eq_zero.mp he)
+    simp only [Complex.conj_im] at hi
+    linarith
+  have hd := hasDerivWithinAt_log_norm_div
+    (hf.sub (hasDerivWithinAt_conj hg)) (hf.sub hg) hnc hn
+  have h₁ : f t - conj (g t) =
+      (f t - (U t : ℂ)) - conj (g t - (U t : ℂ)) := by
+    simp only [map_sub, Complex.conj_ofReal]
+    ring
+  have h₂ : f t - g t = (f t - (U t : ℂ)) - (g t - (U t : ℂ)) := by ring
+  have hd' : HasDerivWithinAt (fun s => interaction (f s) (g s))
+      (-2 * a * ((f t).im / ‖f t - (U t : ℂ)‖ ^ 2) *
+        ((g t).im / ‖g t - (U t : ℂ)‖ ^ 2)) S t := by
+    apply hd.congr_deriv
+    dsimp only [Pi.sub_apply]
+    rw [h₁, h₂, interaction_difference_algebra a (by simpa using hfi)
+      (by simpa using hgi) (sub_left_inj.not.mpr hne)]
+    simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
+  simpa only [interaction_center] using hd'
+
+theorem centered_interaction_hasDerivAt
+    {f g : ℝ → ℂ} (U : ℝ → ℝ) {t a : ℝ}
+    (hf : HasDerivAt f ((a : ℂ) / (f t - (U t : ℂ))) t)
+    (hg : HasDerivAt g ((a : ℂ) / (g t - (U t : ℂ))) t)
+    (hfi : 0 < (f t).im) (hgi : 0 < (g t).im) (hne : f t ≠ g t) :
+    HasDerivAt
+      (fun s => interaction (f s - (U s : ℂ)) (g s - (U s : ℂ)))
+      (-2 * a * ((f t).im / ‖f t - (U t : ℂ)‖ ^ 2) *
+        ((g t).im / ‖g t - (U t : ℂ)‖ ^ 2)) t :=
+  (centered_interaction_hasDerivWithinAt U hf.hasDerivWithinAt hg.hasDerivWithinAt
+    hfi hgi hne (S := univ)).hasDerivAt univ_mem
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointInteractionShells.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointInteractionShells.lean
index 52b45c69534b706653ccf74352e8998f0ac70ca7..aeeb7c6b01bba7d6eff6f10dfabe52fc1b6e3332 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointInteractionShells.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointInteractionShells.lean
@@ -1,242 +1,242 @@
-import OAI.Probability.SLE.Multipoint.MultipointInteraction
-import OAI.Probability.SLE.Multipoint.MultipointRegularizer
-
-namespace OAI
-
-/-! The Green drop bounds the mass of every clock shell.  The first point
-of the closed clock tail gives the bound even when a clock is constant on
-an interval.  Disjoint measurable shells then give the manuscript's
-weighted interaction budget. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal Function
-
-namespace SLEExactGauge.Multipoint
-
-def clockTail (s T : ℝ) (q : ℝ → ℝ) (k : ℕ) : Set ℝ :=
-  Icc s T ∩ q ⁻¹' Ici (k : ℝ)
-
-def clockShell (s T : ℝ) (q : ℝ → ℝ) (k : ℕ) : Set ℝ :=
-  clockTail s T q k \ clockTail s T q (k + 1)
-
-lemma clockTail_isClosed {s T : ℝ} {q : ℝ → ℝ}
-    (hq : ContinuousOn q (Icc s T)) (k : ℕ) : IsClosed (clockTail s T q k) :=
-  hq.preimage_isClosed_of_isClosed isClosed_Icc isClosed_Ici
-
-lemma clockTail_isCompact {s T : ℝ} {q : ℝ → ℝ}
-    (hq : ContinuousOn q (Icc s T)) (k : ℕ) : IsCompact (clockTail s T q k) :=
-  isCompact_Icc.of_isClosed_subset (clockTail_isClosed hq k) inter_subset_left
-
-lemma clockShell_measurableSet {s T : ℝ} {q : ℝ → ℝ}
-    (hq : ContinuousOn q (Icc s T)) (k : ℕ) : MeasurableSet (clockShell s T q k) :=
-  (clockTail_isClosed hq k).measurableSet.diff
-    (clockTail_isClosed hq (k + 1)).measurableSet
-
-lemma mem_clockShell {s T : ℝ} {q : ℝ → ℝ} {k : ℕ} {t : ℝ} :
-    t ∈ clockShell s T q k ↔ t ∈ Icc s T ∧ (k : ℝ) ≤ q t ∧ q t < k + 1 := by
-  simp only [clockShell, clockTail, Set.mem_sdiff, mem_inter_iff, mem_preimage, mem_Ici,
-    Nat.cast_add, Nat.cast_one]
-  constructor
-  · rintro ⟨⟨hI, hlo⟩, hn⟩
-    exact ⟨hI, hlo, lt_of_not_ge (fun hh => hn ⟨hI, hh⟩)⟩
-  · rintro ⟨hI, hlo, hhi⟩
-    exact ⟨⟨hI, hlo⟩, fun hh => (not_le_of_gt hhi) hh.2⟩
-
-lemma clockShell_pairwise (s T : ℝ) (q : ℝ → ℝ) :
-    Pairwise (Disjoint on clockShell s T q) := by
-  intro i j hij
-  apply disjoint_left.mpr
-  intro t hi hj
-  have hi' := mem_clockShell.mp hi
-  have hj' := mem_clockShell.mp hj
-  rcases lt_or_gt_of_ne hij with h | h
-  · have hcast : (i : ℝ) + 1 ≤ j := by exact_mod_cast Nat.succ_le_of_lt h
-    linarith
-  · have hcast : (j : ℝ) + 1 ≤ i := by exact_mod_cast Nat.succ_le_of_lt h
-    linarith
-
-lemma iUnion_clockShell {s T : ℝ} {q : ℝ → ℝ}
-    (h0 : ∀ t ∈ Icc s T, 0 ≤ q t) :
-    (⋃ k : ℕ, clockShell s T q k) = Icc s T := by
-  ext t
-  constructor
-  · intro ht
-    obtain ⟨k, hk⟩ := mem_iUnion.mp ht
-    exact (mem_clockShell.mp hk).1
-  · intro ht
-    refine mem_iUnion.mpr ⟨⌊q t⌋₊, mem_clockShell.mpr ⟨ht, Nat.floor_le (h0 t ht), ?_⟩⟩
-    exact Nat.lt_floor_add_one (q t)
-
-/-- A future Green-drop bound and the exponential interaction decay give
-the exact mass bound for a single shell of the increasing clock. -/
-theorem interaction_shell_mass_bound
-    {s T a M : ℝ} (ha : 0 < a) (hM : 0 ≤ M) (q R L : ℝ → ℝ)
-    (hq : ContinuousOn q (Icc s T)) (hR : ContinuousOn R (Icc s T))
-    (hR0 : ∀ t ∈ Icc s T, 0 ≤ R t)
-    (hdrop : ∀ t ∈ Icc s T, (∫ v in t..T, R v) ≤ L t / (2 * a))
-    (hdecay : ∀ t ∈ Icc s T, L t ≤ M * Real.exp (-a * q t)) (k : ℕ) :
-    (∫ t in clockShell s T q k, R t) ≤ M * Real.exp (-a * k) / (2 * a) := by
-  classical
-  by_cases hne : (clockTail s T q k).Nonempty
-  · obtain ⟨τ, hτ⟩ := (clockTail_isCompact hq k).exists_isLeast hne
-    have hτI : τ ∈ Icc s T := hτ.1.1
-    have hτq : (k : ℝ) ≤ q τ := hτ.1.2
-    have hsub : clockShell s T q k ⊆ Icc τ T := by
-      intro t ht
-      exact ⟨hτ.2 ht.1, ht.1.1.2⟩
-    have hI : Icc τ T ⊆ Icc s T := Icc_subset_Icc hτI.1 le_rfl
-    have hnonneg : 0 ≤ᵐ[volume.restrict (Icc τ T)] R := by
-      change ∀ᵐ t ∂volume.restrict (Icc τ T), 0 ≤ R t
-      rw [ae_restrict_iff' measurableSet_Icc]
-      exact Eventually.of_forall (fun t ht => hR0 t (hI ht))
-    calc
-      _ ≤ ∫ t in Icc τ T, R t :=
-        setIntegral_mono_set (hR.mono hI).integrableOn_Icc hnonneg
-          (Eventually.of_forall hsub)
-      _ = ∫ t in τ..T, R t := by
-        rw [integral_Icc_eq_integral_Ioc, intervalIntegral.integral_of_le hτI.2]
-      _ ≤ L τ / (2 * a) := hdrop τ hτI
-      _ ≤ M * Real.exp (-a * k) / (2 * a) := by
-        apply div_le_div_of_nonneg_right _ (by positivity)
-        exact (hdecay τ hτI).trans
-          (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (by nlinarith)) hM)
-  · have hs : clockShell s T q k = ∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      intro t ht
-      exact hne ⟨t, ht.1⟩
-    rw [hs]
-    simp only [Measure.restrict_empty, integral_zero_measure]
-    positivity
-
-lemma weighted_shell_sum_le_prefactor {a c M : ℝ}
-    (ha : 0 < a) (hc : 2 * c < a) (_hM : 0 ≤ M)
-    (mass : ℕ → ℝ) (h0 : ∀ k, 0 ≤ mass k)
-    (hmass : ∀ k : ℕ, mass k ≤ M * Real.exp (-a * k) / (2 * a)) :
-    Summable (fun k : ℕ => Real.exp (2 * c * ((k : ℝ) + 1)) * mass k) ∧
-      (∑' k : ℕ, Real.exp (2 * c * ((k : ℝ) + 1)) * mass k) ≤
-        M * interactionBudget a c := by
-  have hs := (interaction_shell_series ha hc).mul_left M
-  have hnonneg (k : ℕ) : 0 ≤ Real.exp (2 * c * ((k : ℝ) + 1)) * mass k :=
-    mul_nonneg (Real.exp_pos _).le (h0 k)
-  have hle (k : ℕ) : Real.exp (2 * c * ((k : ℝ) + 1)) * mass k ≤
-      M * (Real.exp (2 * c * ((k : ℝ) + 1)) * Real.exp (-a * k) / (2 * a)) := by
-    calc
-      _ ≤ Real.exp (2 * c * ((k : ℝ) + 1)) *
-          (M * Real.exp (-a * k) / (2 * a)) :=
-        mul_le_mul_of_nonneg_left (hmass k) (Real.exp_pos _).le
-      _ = _ := by ring
-  have hsum := Summable.of_nonneg_of_le hnonneg hle hs.summable
-  exact ⟨hsum, hs.tsum_eq ▸ hsum.tsum_le_tsum hle hs.summable⟩
-
-/-- The full weighted interaction bound, derived from the Green drop;
-no shell-mass estimate is assumed. -/
-theorem weighted_interaction_integral_le
-    {s T a c M : ℝ} (hsT : s ≤ T) (ha : 0 < a) (hM : 0 ≤ M)
-    (hc : 0 ≤ c) (hca : 2 * c < a)
-    (u v R L : ℝ → ℝ)
-    (hu : ContinuousOn u (Icc s T)) (hv : ContinuousOn v (Icc s T))
-    (hu0 : ∀ t ∈ Icc s T, 0 ≤ u t) (hv0 : ∀ t ∈ Icc s T, 0 ≤ v t)
-    (hR : ContinuousOn R (Icc s T)) (hR0 : ∀ t ∈ Icc s T, 0 ≤ R t)
-    (hdrop : ∀ t ∈ Icc s T, (∫ x in t..T, R x) ≤ L t / (2 * a))
-    (hdecay : ∀ t ∈ Icc s T, L t ≤ M * Real.exp (-a * max (u t) (v t))) :
-    (∫ t in s..T, Real.exp (c * (u t + v t)) * R t) ≤ M * interactionBudget a c := by
-  let q := fun t => max (u t) (v t)
-  let F := fun t => Real.exp (c * (u t + v t)) * R t
-  let mass := fun k => ∫ t in clockShell s T q k, R t
-  have hq : ContinuousOn q (Icc s T) := hu.sup hv
-  have hcov : (⋃ k : ℕ, clockShell s T q k) = Icc s T :=
-    iUnion_clockShell (fun t ht => (le_min (hu0 t ht) (hv0 t ht)).trans min_le_max)
-  have hFm : ContinuousOn F (Icc s T) :=
-    (Real.continuous_exp.comp_continuousOn (continuousOn_const.mul (hu.add hv))).mul hR
-  have hmass0 (k : ℕ) : 0 ≤ mass k := by
-    apply setIntegral_nonneg (clockShell_measurableSet hq k)
-    intro t ht
-    exact hR0 t (mem_clockShell.mp ht).1
-  have hmass (k : ℕ) : mass k ≤ M * Real.exp (-a * k) / (2 * a) :=
-    interaction_shell_mass_bound ha hM q R L hq hR hR0 hdrop hdecay k
-  obtain ⟨hsum, hsumle⟩ := weighted_shell_sum_le_prefactor ha hca hM mass hmass0 hmass
-  have hFint : IntegrableOn F (⋃ k : ℕ, clockShell s T q k) := by
-    rw [hcov]
-    exact hFm.integrableOn_Icc
-  have hpieces := hasSum_integral_iUnion (clockShell_measurableSet hq)
-    (clockShell_pairwise s T q) hFint
-  have hpiece (k : ℕ) : (∫ t in clockShell s T q k, F t) ≤
-      Real.exp (2 * c * ((k : ℝ) + 1)) * mass k := by
-    have hsub : clockShell s T q k ⊆ Icc s T := fun t ht => (mem_clockShell.mp ht).1
-    rw [← integral_const_mul]
-    apply setIntegral_mono_on (hFm.integrableOn_Icc.mono_set hsub)
-      ((hR.integrableOn_Icc.mono_set hsub).const_mul _)
-      (clockShell_measurableSet hq k)
-    intro t ht
-    have hkt := mem_clockShell.mp ht
-    apply mul_le_mul_of_nonneg_right _ (hR0 t hkt.1)
-    apply Real.exp_le_exp.mpr
-    have hsumq : u t + v t ≤ 2 * q t := by
-      have h₁ := le_max_left (u t) (v t)
-      have h₂ := le_max_right (u t) (v t)
-      dsimp only [q]
-      linarith
-    have hbound : u t + v t ≤ 2 * ((k : ℝ) + 1) := by linarith [hkt.2.2]
-    nlinarith
-  calc
-    _ = ∫ t in Icc s T, F t := by
-      rw [integral_Icc_eq_integral_Ioc, intervalIntegral.integral_of_le hsT]
-    _ = ∑' k : ℕ, ∫ t in clockShell s T q k, F t := by
-      rw [← hcov]
-      exact hpieces.tsum_eq.symm
-    _ ≤ ∑' k : ℕ, Real.exp (2 * c * ((k : ℝ) + 1)) * mass k :=
-      hpieces.summable.tsum_le_tsum hpiece hsum
-    _ ≤ M * interactionBudget a c := hsumle
-
-end SLEExactGauge.Multipoint
-
-namespace SLEExactGauge
-
-/-- The weighted-shell estimate for the actual capacity-two Loewner maps.
-The pair drop is supplied by the proved Green-kernel identity.  Its sole
-remaining geometric premise is the interaction decay during joint activity. -/
-theorem IsCapacityTwoWitness.weighted_interaction_integral_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (s T : ℝ≥0) (hsT : s ≤ T) (z w : ℂ)
-    (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T) (hne : z ≠ w)
-    {c M : ℝ} (hM : 0 ≤ M) (hc : 0 ≤ c) (hca : 2 * c < 2) (u v : ℝ → ℝ)
-    (hu : ContinuousOn u (Icc (s : ℝ) (T : ℝ)))
-    (hv : ContinuousOn v (Icc (s : ℝ) (T : ℝ)))
-    (hu0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ u t)
-    (hv0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ v t)
-    (hdecay : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      halfPlaneGreen (G t.toNNReal z) (G t.toNNReal w) ≤
-        M * Real.exp (-2 * max (u t) (v t))) :
-    (∫ t in (s : ℝ)..(T : ℝ), Real.exp (c * (u t + v t)) *
-      (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
-        M * Multipoint.interactionBudget 2 c := by
-  let R := fun t : ℝ => chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w
-  let L := fun t : ℝ => halfPlaneGreen (G t.toNNReal z) (G t.toNNReal w)
-  have hI : Icc (s : ℝ) (T : ℝ) ⊆ Icc 0 (T : ℝ) :=
-    Icc_subset_Icc s.coe_nonneg le_rfl
-  have hR : ContinuousOn R (Icc (s : ℝ) (T : ℝ)) :=
-    ((hw.poissonRate_continuousOn hU T z hz).mul
-      (hw.poissonRate_continuousOn hU T w hw')).mono hI
-  have hR0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ R t := by
-    intro t ht
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    exact mul_nonneg
-      (hw.poissonRate_nonneg _ z (survivingDomain_antitone γ htT hz))
-      (hw.poissonRate_nonneg _ w (survivingDomain_antitone γ htT hw'))
-  have hdrop : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      (∫ x in t..(T : ℝ), R x) ≤ L t / (2 * 2) := by
-    intro t ht
-    have ht0 : 0 ≤ t := s.coe_nonneg.trans ht.1
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    have hb := hw.green_pair_bound hU t.toNNReal T htT z w hz hw' hne
-    simpa only [R, L, Real.coe_toNNReal _ ht0, show (2 : ℝ) * 2 = 4 by norm_num] using hb
-  exact Multipoint.weighted_interaction_integral_le hsT (by norm_num) hM hc hca
-    u v R L hu hv hu0 hv0 hR hR0 hdrop hdecay
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointInteraction
+import OAI.Probability.SLE.Multipoint.MultipointRegularizer
+
+namespace OAI
+
+/-! The Green drop bounds the mass of every clock shell.  The first point
+of the closed clock tail gives the bound even when a clock is constant on
+an interval.  Disjoint measurable shells then give the manuscript's
+weighted interaction budget. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal Function
+
+namespace SLEExactGauge.Multipoint
+
+def clockTail (s T : ℝ) (q : ℝ → ℝ) (k : ℕ) : Set ℝ :=
+  Icc s T ∩ q ⁻¹' Ici (k : ℝ)
+
+def clockShell (s T : ℝ) (q : ℝ → ℝ) (k : ℕ) : Set ℝ :=
+  clockTail s T q k \ clockTail s T q (k + 1)
+
+lemma clockTail_isClosed {s T : ℝ} {q : ℝ → ℝ}
+    (hq : ContinuousOn q (Icc s T)) (k : ℕ) : IsClosed (clockTail s T q k) :=
+  hq.preimage_isClosed_of_isClosed isClosed_Icc isClosed_Ici
+
+lemma clockTail_isCompact {s T : ℝ} {q : ℝ → ℝ}
+    (hq : ContinuousOn q (Icc s T)) (k : ℕ) : IsCompact (clockTail s T q k) :=
+  isCompact_Icc.of_isClosed_subset (clockTail_isClosed hq k) inter_subset_left
+
+lemma clockShell_measurableSet {s T : ℝ} {q : ℝ → ℝ}
+    (hq : ContinuousOn q (Icc s T)) (k : ℕ) : MeasurableSet (clockShell s T q k) :=
+  (clockTail_isClosed hq k).measurableSet.diff
+    (clockTail_isClosed hq (k + 1)).measurableSet
+
+lemma mem_clockShell {s T : ℝ} {q : ℝ → ℝ} {k : ℕ} {t : ℝ} :
+    t ∈ clockShell s T q k ↔ t ∈ Icc s T ∧ (k : ℝ) ≤ q t ∧ q t < k + 1 := by
+  simp only [clockShell, clockTail, Set.mem_sdiff, mem_inter_iff, mem_preimage, mem_Ici,
+    Nat.cast_add, Nat.cast_one]
+  constructor
+  · rintro ⟨⟨hI, hlo⟩, hn⟩
+    exact ⟨hI, hlo, lt_of_not_ge (fun hh => hn ⟨hI, hh⟩)⟩
+  · rintro ⟨hI, hlo, hhi⟩
+    exact ⟨⟨hI, hlo⟩, fun hh => (not_le_of_gt hhi) hh.2⟩
+
+lemma clockShell_pairwise (s T : ℝ) (q : ℝ → ℝ) :
+    Pairwise (Disjoint on clockShell s T q) := by
+  intro i j hij
+  apply disjoint_left.mpr
+  intro t hi hj
+  have hi' := mem_clockShell.mp hi
+  have hj' := mem_clockShell.mp hj
+  rcases lt_or_gt_of_ne hij with h | h
+  · have hcast : (i : ℝ) + 1 ≤ j := by exact_mod_cast Nat.succ_le_of_lt h
+    linarith
+  · have hcast : (j : ℝ) + 1 ≤ i := by exact_mod_cast Nat.succ_le_of_lt h
+    linarith
+
+lemma iUnion_clockShell {s T : ℝ} {q : ℝ → ℝ}
+    (h0 : ∀ t ∈ Icc s T, 0 ≤ q t) :
+    (⋃ k : ℕ, clockShell s T q k) = Icc s T := by
+  ext t
+  constructor
+  · intro ht
+    obtain ⟨k, hk⟩ := mem_iUnion.mp ht
+    exact (mem_clockShell.mp hk).1
+  · intro ht
+    refine mem_iUnion.mpr ⟨⌊q t⌋₊, mem_clockShell.mpr ⟨ht, Nat.floor_le (h0 t ht), ?_⟩⟩
+    exact Nat.lt_floor_add_one (q t)
+
+/-- A future Green-drop bound and the exponential interaction decay give
+the exact mass bound for a single shell of the increasing clock. -/
+theorem interaction_shell_mass_bound
+    {s T a M : ℝ} (ha : 0 < a) (hM : 0 ≤ M) (q R L : ℝ → ℝ)
+    (hq : ContinuousOn q (Icc s T)) (hR : ContinuousOn R (Icc s T))
+    (hR0 : ∀ t ∈ Icc s T, 0 ≤ R t)
+    (hdrop : ∀ t ∈ Icc s T, (∫ v in t..T, R v) ≤ L t / (2 * a))
+    (hdecay : ∀ t ∈ Icc s T, L t ≤ M * Real.exp (-a * q t)) (k : ℕ) :
+    (∫ t in clockShell s T q k, R t) ≤ M * Real.exp (-a * k) / (2 * a) := by
+  classical
+  by_cases hne : (clockTail s T q k).Nonempty
+  · obtain ⟨τ, hτ⟩ := (clockTail_isCompact hq k).exists_isLeast hne
+    have hτI : τ ∈ Icc s T := hτ.1.1
+    have hτq : (k : ℝ) ≤ q τ := hτ.1.2
+    have hsub : clockShell s T q k ⊆ Icc τ T := by
+      intro t ht
+      exact ⟨hτ.2 ht.1, ht.1.1.2⟩
+    have hI : Icc τ T ⊆ Icc s T := Icc_subset_Icc hτI.1 le_rfl
+    have hnonneg : 0 ≤ᵐ[volume.restrict (Icc τ T)] R := by
+      change ∀ᵐ t ∂volume.restrict (Icc τ T), 0 ≤ R t
+      rw [ae_restrict_iff' measurableSet_Icc]
+      exact Eventually.of_forall (fun t ht => hR0 t (hI ht))
+    calc
+      _ ≤ ∫ t in Icc τ T, R t :=
+        setIntegral_mono_set (hR.mono hI).integrableOn_Icc hnonneg
+          (Eventually.of_forall hsub)
+      _ = ∫ t in τ..T, R t := by
+        rw [integral_Icc_eq_integral_Ioc, intervalIntegral.integral_of_le hτI.2]
+      _ ≤ L τ / (2 * a) := hdrop τ hτI
+      _ ≤ M * Real.exp (-a * k) / (2 * a) := by
+        apply div_le_div_of_nonneg_right _ (by positivity)
+        exact (hdecay τ hτI).trans
+          (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (by nlinarith)) hM)
+  · have hs : clockShell s T q k = ∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      intro t ht
+      exact hne ⟨t, ht.1⟩
+    rw [hs]
+    simp only [Measure.restrict_empty, integral_zero_measure]
+    positivity
+
+lemma weighted_shell_sum_le_prefactor {a c M : ℝ}
+    (ha : 0 < a) (hc : 2 * c < a) (_hM : 0 ≤ M)
+    (mass : ℕ → ℝ) (h0 : ∀ k, 0 ≤ mass k)
+    (hmass : ∀ k : ℕ, mass k ≤ M * Real.exp (-a * k) / (2 * a)) :
+    Summable (fun k : ℕ => Real.exp (2 * c * ((k : ℝ) + 1)) * mass k) ∧
+      (∑' k : ℕ, Real.exp (2 * c * ((k : ℝ) + 1)) * mass k) ≤
+        M * interactionBudget a c := by
+  have hs := (interaction_shell_series ha hc).mul_left M
+  have hnonneg (k : ℕ) : 0 ≤ Real.exp (2 * c * ((k : ℝ) + 1)) * mass k :=
+    mul_nonneg (Real.exp_pos _).le (h0 k)
+  have hle (k : ℕ) : Real.exp (2 * c * ((k : ℝ) + 1)) * mass k ≤
+      M * (Real.exp (2 * c * ((k : ℝ) + 1)) * Real.exp (-a * k) / (2 * a)) := by
+    calc
+      _ ≤ Real.exp (2 * c * ((k : ℝ) + 1)) *
+          (M * Real.exp (-a * k) / (2 * a)) :=
+        mul_le_mul_of_nonneg_left (hmass k) (Real.exp_pos _).le
+      _ = _ := by ring
+  have hsum := Summable.of_nonneg_of_le hnonneg hle hs.summable
+  exact ⟨hsum, hs.tsum_eq ▸ hsum.tsum_le_tsum hle hs.summable⟩
+
+/-- The full weighted interaction bound, derived from the Green drop;
+no shell-mass estimate is assumed. -/
+theorem weighted_interaction_integral_le
+    {s T a c M : ℝ} (hsT : s ≤ T) (ha : 0 < a) (hM : 0 ≤ M)
+    (hc : 0 ≤ c) (hca : 2 * c < a)
+    (u v R L : ℝ → ℝ)
+    (hu : ContinuousOn u (Icc s T)) (hv : ContinuousOn v (Icc s T))
+    (hu0 : ∀ t ∈ Icc s T, 0 ≤ u t) (hv0 : ∀ t ∈ Icc s T, 0 ≤ v t)
+    (hR : ContinuousOn R (Icc s T)) (hR0 : ∀ t ∈ Icc s T, 0 ≤ R t)
+    (hdrop : ∀ t ∈ Icc s T, (∫ x in t..T, R x) ≤ L t / (2 * a))
+    (hdecay : ∀ t ∈ Icc s T, L t ≤ M * Real.exp (-a * max (u t) (v t))) :
+    (∫ t in s..T, Real.exp (c * (u t + v t)) * R t) ≤ M * interactionBudget a c := by
+  let q := fun t => max (u t) (v t)
+  let F := fun t => Real.exp (c * (u t + v t)) * R t
+  let mass := fun k => ∫ t in clockShell s T q k, R t
+  have hq : ContinuousOn q (Icc s T) := hu.sup hv
+  have hcov : (⋃ k : ℕ, clockShell s T q k) = Icc s T :=
+    iUnion_clockShell (fun t ht => (le_min (hu0 t ht) (hv0 t ht)).trans min_le_max)
+  have hFm : ContinuousOn F (Icc s T) :=
+    (Real.continuous_exp.comp_continuousOn (continuousOn_const.mul (hu.add hv))).mul hR
+  have hmass0 (k : ℕ) : 0 ≤ mass k := by
+    apply setIntegral_nonneg (clockShell_measurableSet hq k)
+    intro t ht
+    exact hR0 t (mem_clockShell.mp ht).1
+  have hmass (k : ℕ) : mass k ≤ M * Real.exp (-a * k) / (2 * a) :=
+    interaction_shell_mass_bound ha hM q R L hq hR hR0 hdrop hdecay k
+  obtain ⟨hsum, hsumle⟩ := weighted_shell_sum_le_prefactor ha hca hM mass hmass0 hmass
+  have hFint : IntegrableOn F (⋃ k : ℕ, clockShell s T q k) := by
+    rw [hcov]
+    exact hFm.integrableOn_Icc
+  have hpieces := hasSum_integral_iUnion (clockShell_measurableSet hq)
+    (clockShell_pairwise s T q) hFint
+  have hpiece (k : ℕ) : (∫ t in clockShell s T q k, F t) ≤
+      Real.exp (2 * c * ((k : ℝ) + 1)) * mass k := by
+    have hsub : clockShell s T q k ⊆ Icc s T := fun t ht => (mem_clockShell.mp ht).1
+    rw [← integral_const_mul]
+    apply setIntegral_mono_on (hFm.integrableOn_Icc.mono_set hsub)
+      ((hR.integrableOn_Icc.mono_set hsub).const_mul _)
+      (clockShell_measurableSet hq k)
+    intro t ht
+    have hkt := mem_clockShell.mp ht
+    apply mul_le_mul_of_nonneg_right _ (hR0 t hkt.1)
+    apply Real.exp_le_exp.mpr
+    have hsumq : u t + v t ≤ 2 * q t := by
+      have h₁ := le_max_left (u t) (v t)
+      have h₂ := le_max_right (u t) (v t)
+      dsimp only [q]
+      linarith
+    have hbound : u t + v t ≤ 2 * ((k : ℝ) + 1) := by linarith [hkt.2.2]
+    nlinarith
+  calc
+    _ = ∫ t in Icc s T, F t := by
+      rw [integral_Icc_eq_integral_Ioc, intervalIntegral.integral_of_le hsT]
+    _ = ∑' k : ℕ, ∫ t in clockShell s T q k, F t := by
+      rw [← hcov]
+      exact hpieces.tsum_eq.symm
+    _ ≤ ∑' k : ℕ, Real.exp (2 * c * ((k : ℝ) + 1)) * mass k :=
+      hpieces.summable.tsum_le_tsum hpiece hsum
+    _ ≤ M * interactionBudget a c := hsumle
+
+end SLEExactGauge.Multipoint
+
+namespace SLEExactGauge
+
+/-- The weighted-shell estimate for the actual capacity-two Loewner maps.
+The pair drop is supplied by the proved Green-kernel identity.  Its sole
+remaining geometric premise is the interaction decay during joint activity. -/
+theorem IsCapacityTwoWitness.weighted_interaction_integral_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (s T : ℝ≥0) (hsT : s ≤ T) (z w : ℂ)
+    (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T) (hne : z ≠ w)
+    {c M : ℝ} (hM : 0 ≤ M) (hc : 0 ≤ c) (hca : 2 * c < 2) (u v : ℝ → ℝ)
+    (hu : ContinuousOn u (Icc (s : ℝ) (T : ℝ)))
+    (hv : ContinuousOn v (Icc (s : ℝ) (T : ℝ)))
+    (hu0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ u t)
+    (hv0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ v t)
+    (hdecay : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      halfPlaneGreen (G t.toNNReal z) (G t.toNNReal w) ≤
+        M * Real.exp (-2 * max (u t) (v t))) :
+    (∫ t in (s : ℝ)..(T : ℝ), Real.exp (c * (u t + v t)) *
+      (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
+        M * Multipoint.interactionBudget 2 c := by
+  let R := fun t : ℝ => chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w
+  let L := fun t : ℝ => halfPlaneGreen (G t.toNNReal z) (G t.toNNReal w)
+  have hI : Icc (s : ℝ) (T : ℝ) ⊆ Icc 0 (T : ℝ) :=
+    Icc_subset_Icc s.coe_nonneg le_rfl
+  have hR : ContinuousOn R (Icc (s : ℝ) (T : ℝ)) :=
+    ((hw.poissonRate_continuousOn hU T z hz).mul
+      (hw.poissonRate_continuousOn hU T w hw')).mono hI
+  have hR0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ R t := by
+    intro t ht
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    exact mul_nonneg
+      (hw.poissonRate_nonneg _ z (survivingDomain_antitone γ htT hz))
+      (hw.poissonRate_nonneg _ w (survivingDomain_antitone γ htT hw'))
+  have hdrop : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      (∫ x in t..(T : ℝ), R x) ≤ L t / (2 * 2) := by
+    intro t ht
+    have ht0 : 0 ≤ t := s.coe_nonneg.trans ht.1
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    have hb := hw.green_pair_bound hU t.toNNReal T htT z w hz hw' hne
+    simpa only [R, L, Real.coe_toNNReal _ ht0, show (2 : ℝ) * 2 = 4 by norm_num] using hb
+  exact Multipoint.weighted_interaction_integral_le hsT (by norm_num) hM hc hca
+    u v R L hu hv hu0 hv0 hR hR0 hdrop hdecay
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointKernelExpansion.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointKernelExpansion.lean
index 63c79773940795cd1deac63c973e4a40cff19b49..657e7e671578072ce116cc4616f0d892ce9986bd 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointKernelExpansion.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointKernelExpansion.lean
@@ -1,162 +1,162 @@
-import OAI.Probability.SLE.Multipoint.MultipointCappedKernel
-
-namespace OAI
-
-/-! Each term of the nearest-neighbor cap expansion is a loop-free graph
-with at most one outgoing edge at every vertex. Integrating a vertex of
-incidence at most two proves the fixed-order kernel estimate. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-def selectedTarget {ι : Type*} (f : ι → Option ι) (i : ι) : ι := (f i).getD i
-
-def selectedVertices {ι : Type*} [Fintype ι] [DecidableEq ι] (f : ι → Option ι) : Finset ι :=
-  Finset.univ.filter (fun i => selectedTarget f i ≠ i)
-
-def selectedEdges {ι : Type*} [Fintype ι] [DecidableEq ι] (f : ι → Option ι) : Finset (ι × ι) :=
-  (selectedVertices f).image (fun i => (i,selectedTarget f i))
-
-lemma selectedEdges_outgoing {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (f : ι → Option ι) :
-    ∀ e ∈ selectedEdges f, ∀ d ∈ selectedEdges f, e.1=d.1 → e=d := by
-  intro e he d hd h
-  obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp he
-  obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hd
-  exact congrArg (fun i => (i,selectedTarget f i)) h
-
-lemma selectedEdges_loopfree {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (f : ι → Option ι) : ∀ e ∈ selectedEdges f, e.1≠e.2 := by
-  intro e he
-  obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp he
-  exact (Finset.mem_filter.mp hi).2.symm
-
-lemma graphKernel_selectedEdges {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α : ℝ) (f : ι → Option ι) (x : ι → ℂ) :
-    graphKernel α (selectedEdges f) x =
-      ∏ i, if selectedTarget f i=i then 1 else distanceKernel α (x i) (x (selectedTarget f i)) := by
-  unfold graphKernel selectedEdges
-  rw [Finset.prod_image]
-  · simp only [selectedVertices,Finset.prod_filter]
-    apply Finset.prod_congr rfl
-    intro i _
-    by_cases h : selectedTarget f i=i <;> simp [h]
-  · intro i _ j _ h
-    exact congrArg Prod.fst h
-
-def capCoefficient (α b : ℝ) : ℝ≥0∞ := max 1 (ENNReal.ofReal (b^(-α)))
-
-lemma capCoefficient_finite (α b : ℝ) : capCoefficient α b < ⊤ := by
-  unfold capCoefficient
-  exact max_lt (by simp) ENNReal.ofReal_lt_top
-
-lemma rowChoice_le_graph_factor {ι : Type*} [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) (f : ι → Option ι) (i : ι) :
-    rowChoice α b x i (f i) ≤ capCoefficient α b *
-      (if selectedTarget f i=i then 1 else distanceKernel α (x i) (x (selectedTarget f i))) := by
-  cases h : f i with
-  | none =>
-    simp only [rowChoice,selectedTarget,h,Option.getD_none,ite_true,mul_one]
-    exact le_max_right _ _
-  | some j =>
-    simp only [rowChoice,selectedTarget,h,Option.getD_some]
-    by_cases hji : j=i
-    · simp only [hji,ite_true]
-      exact bot_le
-    · simp only [hji,ite_false]
-      have hc : 1 ≤ capCoefficient α b := le_max_left _ _
-      calc
-        distanceKernel α (x i) (x j) = 1 * distanceKernel α (x i) (x j) := (one_mul _).symm
-        _ ≤ capCoefficient α b * distanceKernel α (x i) (x j) := by gcongr
-
-lemma rowProduct_le_graph {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) (f : ι → Option ι) :
-    (∏ i, rowChoice α b x i (f i)) ≤
-      capCoefficient α b ^ Fintype.card ι * graphKernel α (selectedEdges f) x := by
-  rw [graphKernel_selectedEdges,← Finset.card_univ,← Finset.prod_const,← Finset.prod_mul_distrib]
-  exact Finset.prod_le_prod (fun i _ => rowChoice_le_graph_factor α b x f i)
-
-lemma cappedDistanceKernel_le_graph_sum {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) :
-    cappedDistanceKernel α b x ≤
-      ∑ f : ι → Option ι, capCoefficient α b ^ Fintype.card ι * graphKernel α (selectedEdges f) x :=
-  (cappedDistanceKernel_le_expansion α b x).trans
-    (Finset.sum_le_sum (fun f _ => rowProduct_le_graph α b x f))
-
-/-- Integrate all graph vertices. The empty index set is included. -/
-theorem graphKernel_marginal_le {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
-    (hC0 : volume K ≤ C)
-    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
-    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
-    (S : Finset ι) (E : Finset (ι × ι))
-    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
-    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
-    (hloop : ∀ e ∈ E, e.1≠e.2) (x : ι → ℂ) :
-    lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x ≤ C^S.card := by
-  induction S using Finset.strongInductionOn generalizing E with
-  | _ S ih =>
-    by_cases hempty : S=∅
-    · have hE : E=∅ := by
-        apply Finset.eq_empty_iff_forall_notMem.mpr
-        intro e he
-        simpa only [hempty,Finset.notMem_empty] using (hends e he).1
-      simp only [hempty,hE,lmarginal_empty,graphKernel,Finset.prod_empty,Finset.card_empty,pow_zero,le_rfl]
-    · obtain ⟨v,hv,hdegree⟩ := exists_vertex_incident_card_le_two S
-        (Finset.nonempty_iff_ne_empty.mpr hempty) E hends hout
-      obtain ⟨hends',hout',hloop'⟩ := eraseVertex_properties S E v hends hout hloop
-      have hsmall := ih (S.erase v) (Finset.erase_ssubset hv) (eraseVertex E v) hends' hout' hloop'
-      calc
-        lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x =
-            lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
-              (fun x => ∫⁻ y in K, graphKernel α E (Function.update x v y)) x := by
-          exact congrFun (lmarginal_erase' (μ := fun _ : ι => volume.restrict K)
-            (graphKernel α E) (measurable_graphKernel α E) hv) x
-        _ ≤ lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
-              (fun x => C*graphKernel α (eraseVertex E v) x) x :=
-          lmarginal_mono (fun x => graphKernel_section_le α K C hC0 hC1 hC2 E v x hdegree hloop) x
-        _ = C*lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
-              (graphKernel α (eraseVertex E v)) x := by
-          exact lintegral_const_mul' C _ hC.ne
-        _ ≤ C*C^(S.erase v).card := by gcongr
-        _ = C^S.card := by rw [← Finset.card_erase_add_one hv,pow_succ,mul_comm]
-
-lemma selectedGraph_integral_le {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
-    (hC0 : volume K ≤ C)
-    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
-    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
-    (f : ι → Option ι) :
-    (∫⁻ x, graphKernel α (selectedEdges f) x ∂Measure.pi (fun _ : ι => volume.restrict K)) ≤
-      C^Fintype.card ι := by
-  simpa only [lmarginal_univ,Finset.card_univ] using
-    graphKernel_marginal_le α K C hC hC0 hC1 hC2 Finset.univ (selectedEdges f)
-      (fun _ _ => ⟨Finset.mem_univ _,Finset.mem_univ _⟩)
-      (selectedEdges_outgoing f) (selectedEdges_loopfree f) (fun _ => 0)
-
-/-- The exact capped inverse-distance kernel has a finite integral on every
-bounded planar product set, for every fixed order. -/
-theorem cappedDistanceKernel_integral_lt_top {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {α : ℝ} (hα : 0 < α) (hα1 : α < 1) (b : ℝ)
-    (K : Set ℂ) (hK : Bornology.IsBounded K) :
-    (∫⁻ x, cappedDistanceKernel (ι := ι) α b x
-      ∂Measure.pi (fun _ : ι => volume.restrict K)) < ⊤ := by
-  obtain ⟨C,hC,_,hC0,hC1,hC2⟩ := distanceKernel_uniform_local_bounds hα hα1 K hK
-  apply lt_of_le_of_lt (lintegral_mono (cappedDistanceKernel_le_graph_sum α b))
-  rw [lintegral_finsetSum]
-  · apply ENNReal.sum_lt_top.mpr
-    intro f _
-    rw [lintegral_const_mul' _ _ (ENNReal.pow_ne_top (capCoefficient_finite α b).ne)]
-    exact lt_of_le_of_lt (by gcongr; exact selectedGraph_integral_le α K C hC hC0 hC1 hC2 f)
-      (ENNReal.mul_lt_top (ENNReal.pow_lt_top (capCoefficient_finite α b)) (ENNReal.pow_lt_top hC))
-  · intro f _
-    exact measurable_const.mul (measurable_graphKernel α (selectedEdges f))
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointCappedKernel
+
+namespace OAI
+
+/-! Each term of the nearest-neighbor cap expansion is a loop-free graph
+with at most one outgoing edge at every vertex. Integrating a vertex of
+incidence at most two proves the fixed-order kernel estimate. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+def selectedTarget {ι : Type*} (f : ι → Option ι) (i : ι) : ι := (f i).getD i
+
+def selectedVertices {ι : Type*} [Fintype ι] [DecidableEq ι] (f : ι → Option ι) : Finset ι :=
+  Finset.univ.filter (fun i => selectedTarget f i ≠ i)
+
+def selectedEdges {ι : Type*} [Fintype ι] [DecidableEq ι] (f : ι → Option ι) : Finset (ι × ι) :=
+  (selectedVertices f).image (fun i => (i,selectedTarget f i))
+
+lemma selectedEdges_outgoing {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (f : ι → Option ι) :
+    ∀ e ∈ selectedEdges f, ∀ d ∈ selectedEdges f, e.1=d.1 → e=d := by
+  intro e he d hd h
+  obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp he
+  obtain ⟨j,_,rfl⟩ := Finset.mem_image.mp hd
+  exact congrArg (fun i => (i,selectedTarget f i)) h
+
+lemma selectedEdges_loopfree {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (f : ι → Option ι) : ∀ e ∈ selectedEdges f, e.1≠e.2 := by
+  intro e he
+  obtain ⟨i,hi,rfl⟩ := Finset.mem_image.mp he
+  exact (Finset.mem_filter.mp hi).2.symm
+
+lemma graphKernel_selectedEdges {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α : ℝ) (f : ι → Option ι) (x : ι → ℂ) :
+    graphKernel α (selectedEdges f) x =
+      ∏ i, if selectedTarget f i=i then 1 else distanceKernel α (x i) (x (selectedTarget f i)) := by
+  unfold graphKernel selectedEdges
+  rw [Finset.prod_image]
+  · simp only [selectedVertices,Finset.prod_filter]
+    apply Finset.prod_congr rfl
+    intro i _
+    by_cases h : selectedTarget f i=i <;> simp [h]
+  · intro i _ j _ h
+    exact congrArg Prod.fst h
+
+def capCoefficient (α b : ℝ) : ℝ≥0∞ := max 1 (ENNReal.ofReal (b^(-α)))
+
+lemma capCoefficient_finite (α b : ℝ) : capCoefficient α b < ⊤ := by
+  unfold capCoefficient
+  exact max_lt (by simp) ENNReal.ofReal_lt_top
+
+lemma rowChoice_le_graph_factor {ι : Type*} [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) (f : ι → Option ι) (i : ι) :
+    rowChoice α b x i (f i) ≤ capCoefficient α b *
+      (if selectedTarget f i=i then 1 else distanceKernel α (x i) (x (selectedTarget f i))) := by
+  cases h : f i with
+  | none =>
+    simp only [rowChoice,selectedTarget,h,Option.getD_none,ite_true,mul_one]
+    exact le_max_right _ _
+  | some j =>
+    simp only [rowChoice,selectedTarget,h,Option.getD_some]
+    by_cases hji : j=i
+    · simp only [hji,ite_true]
+      exact bot_le
+    · simp only [hji,ite_false]
+      have hc : 1 ≤ capCoefficient α b := le_max_left _ _
+      calc
+        distanceKernel α (x i) (x j) = 1 * distanceKernel α (x i) (x j) := (one_mul _).symm
+        _ ≤ capCoefficient α b * distanceKernel α (x i) (x j) := by gcongr
+
+lemma rowProduct_le_graph {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) (f : ι → Option ι) :
+    (∏ i, rowChoice α b x i (f i)) ≤
+      capCoefficient α b ^ Fintype.card ι * graphKernel α (selectedEdges f) x := by
+  rw [graphKernel_selectedEdges,← Finset.card_univ,← Finset.prod_const,← Finset.prod_mul_distrib]
+  exact Finset.prod_le_prod (fun i _ => rowChoice_le_graph_factor α b x f i)
+
+lemma cappedDistanceKernel_le_graph_sum {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) :
+    cappedDistanceKernel α b x ≤
+      ∑ f : ι → Option ι, capCoefficient α b ^ Fintype.card ι * graphKernel α (selectedEdges f) x :=
+  (cappedDistanceKernel_le_expansion α b x).trans
+    (Finset.sum_le_sum (fun f _ => rowProduct_le_graph α b x f))
+
+/-- Integrate all graph vertices. The empty index set is included. -/
+theorem graphKernel_marginal_le {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
+    (hC0 : volume K ≤ C)
+    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
+    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
+    (S : Finset ι) (E : Finset (ι × ι))
+    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
+    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
+    (hloop : ∀ e ∈ E, e.1≠e.2) (x : ι → ℂ) :
+    lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x ≤ C^S.card := by
+  induction S using Finset.strongInductionOn generalizing E with
+  | _ S ih =>
+    by_cases hempty : S=∅
+    · have hE : E=∅ := by
+        apply Finset.eq_empty_iff_forall_notMem.mpr
+        intro e he
+        simpa only [hempty,Finset.notMem_empty] using (hends e he).1
+      simp only [hempty,hE,lmarginal_empty,graphKernel,Finset.prod_empty,Finset.card_empty,pow_zero,le_rfl]
+    · obtain ⟨v,hv,hdegree⟩ := exists_vertex_incident_card_le_two S
+        (Finset.nonempty_iff_ne_empty.mpr hempty) E hends hout
+      obtain ⟨hends',hout',hloop'⟩ := eraseVertex_properties S E v hends hout hloop
+      have hsmall := ih (S.erase v) (Finset.erase_ssubset hv) (eraseVertex E v) hends' hout' hloop'
+      calc
+        lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x =
+            lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
+              (fun x => ∫⁻ y in K, graphKernel α E (Function.update x v y)) x := by
+          exact congrFun (lmarginal_erase' (μ := fun _ : ι => volume.restrict K)
+            (graphKernel α E) (measurable_graphKernel α E) hv) x
+        _ ≤ lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
+              (fun x => C*graphKernel α (eraseVertex E v) x) x :=
+          lmarginal_mono (fun x => graphKernel_section_le α K C hC0 hC1 hC2 E v x hdegree hloop) x
+        _ = C*lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
+              (graphKernel α (eraseVertex E v)) x := by
+          exact lintegral_const_mul' C _ hC.ne
+        _ ≤ C*C^(S.erase v).card := by gcongr
+        _ = C^S.card := by rw [← Finset.card_erase_add_one hv,pow_succ,mul_comm]
+
+lemma selectedGraph_integral_le {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
+    (hC0 : volume K ≤ C)
+    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
+    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
+    (f : ι → Option ι) :
+    (∫⁻ x, graphKernel α (selectedEdges f) x ∂Measure.pi (fun _ : ι => volume.restrict K)) ≤
+      C^Fintype.card ι := by
+  simpa only [lmarginal_univ,Finset.card_univ] using
+    graphKernel_marginal_le α K C hC hC0 hC1 hC2 Finset.univ (selectedEdges f)
+      (fun _ _ => ⟨Finset.mem_univ _,Finset.mem_univ _⟩)
+      (selectedEdges_outgoing f) (selectedEdges_loopfree f) (fun _ => 0)
+
+/-- The exact capped inverse-distance kernel has a finite integral on every
+bounded planar product set, for every fixed order. -/
+theorem cappedDistanceKernel_integral_lt_top {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {α : ℝ} (hα : 0 < α) (hα1 : α < 1) (b : ℝ)
+    (K : Set ℂ) (hK : Bornology.IsBounded K) :
+    (∫⁻ x, cappedDistanceKernel (ι := ι) α b x
+      ∂Measure.pi (fun _ : ι => volume.restrict K)) < ⊤ := by
+  obtain ⟨C,hC,_,hC0,hC1,hC2⟩ := distanceKernel_uniform_local_bounds hα hα1 K hK
+  apply lt_of_le_of_lt (lintegral_mono (cappedDistanceKernel_le_graph_sum α b))
+  rw [lintegral_finsetSum]
+  · apply ENNReal.sum_lt_top.mpr
+    intro f _
+    rw [lintegral_const_mul' _ _ (ENNReal.pow_ne_top (capCoefficient_finite α b).ne)]
+    exact lt_of_le_of_lt (by gcongr; exact selectedGraph_integral_le α K C hC hC0 hC1 hC2 f)
+      (ENNReal.mul_lt_top (ENNReal.pow_lt_top (capCoefficient_finite α b)) (ENNReal.pow_lt_top hC))
+  · intro f _
+    exact measurable_const.mul (measurable_graphKernel α (selectedEdges f))
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointKernelGraph.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointKernelGraph.lean
index 07fe29a2840afdd98c17cc4da36035c87b98edeb..925adf250e9b1e95c3e0425a6e66367a8ebf08f5 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointKernelGraph.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointKernelGraph.lean
@@ -1,275 +1,275 @@
-import OAI.Probability.SLE.Multipoint.MultipointDistanceKernel
-
-namespace OAI
-
-/-! The finite graph elimination used in the capped distance kernel.
-An edge records one selected inverse-distance factor. Each vertex has
-at most one outgoing edge, including after deleting a vertex. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-def incidentEdges {ι : Type*} [DecidableEq ι] (E : Finset (ι × ι)) (v : ι) : Finset (ι × ι) :=
-  E.filter (fun e => e.1=v ∨ e.2=v)
-
-def eraseVertex {ι : Type*} [DecidableEq ι] (E : Finset (ι × ι)) (v : ι) : Finset (ι × ι) :=
-  E.filter (fun e => e.1≠v ∧ e.2≠v)
-
-def graphKernel {ι : Type*} (α : ℝ) (E : Finset (ι × ι)) (x : ι → ℂ) : ℝ≥0∞ :=
-  ∏ e ∈ E, distanceKernel α (x e.1) (x e.2)
-
-lemma measurable_graphKernel {ι : Type*} (α : ℝ) (E : Finset (ι × ι)) :
-    Measurable (graphKernel α E) := by
-  unfold graphKernel distanceKernel
-  fun_prop
-
-lemma exists_vertex_incident_card_le_two {ι : Type*} [DecidableEq ι]
-    (S : Finset ι) (hS : S.Nonempty) (E : Finset (ι × ι))
-    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
-    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f) :
-    ∃ v ∈ S, (incidentEdges E v).card ≤ 2 := by
-  have hcard : E.card ≤ S.card :=
-    Finset.card_le_card_of_injOn Prod.fst (fun e he => (hends e he).1)
-      (fun e he f hf hef => hout e he f hf hef)
-  have hsum : E.card = ∑ v ∈ S, (E.filter (fun e => e.2=v)).card :=
-    Finset.card_eq_sum_card_fiberwise (fun e he => (hends e he).2)
-  have hin : ∃ v ∈ S, (E.filter (fun e => e.2=v)).card ≤ 1 := by
-    by_contra! h
-    have hlt : S.card < ∑ v ∈ S, (E.filter (fun e => e.2=v)).card := by
-      calc
-        S.card = ∑ _v ∈ S, 1 := by simp
-        _ < _ := Finset.sum_lt_sum (fun v hv => (h v hv).le)
-          (by obtain ⟨v,hv⟩ := hS; exact ⟨v,hv,h v hv⟩)
-    omega
-  obtain ⟨v,hv,hin⟩ := hin
-  have houtcard : (E.filter (fun e => e.1=v)).card ≤ 1 := by
-    apply Finset.card_le_one.mpr
-    intro e he f hf
-    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
-      ((Finset.mem_filter.mp he).2.trans (Finset.mem_filter.mp hf).2.symm)
-  have heq : incidentEdges E v = (E.filter (fun e => e.1=v)) ∪ (E.filter (fun e => e.2=v)) := by
-    ext e
-    simp only [incidentEdges,Finset.mem_filter,Finset.mem_union]
-    tauto
-  refine ⟨v,hv,?_⟩
-  rw [heq]
-  exact (Finset.card_union_le _ _).trans (by omega)
-
-lemma incoming_card_le_erase {ι : Type*} [DecidableEq ι]
-    (S : Finset ι) (E : Finset (ι × ι)) (v : ι)
-    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
-    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
-    (hloop : ∀ e ∈ E, e.1≠e.2) :
-    (E.filter (fun e => e.2=v)).card ≤ (S.erase v).card := by
-  apply Finset.card_le_card_of_injOn Prod.fst
-  · intro e he
-    obtain ⟨he,hev⟩ := Finset.mem_filter.mp he
-    exact Finset.mem_erase.mpr ⟨by simpa only [← hev] using hloop e he,(hends e he).1⟩
-  · intro e he f hf hef
-    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1 hef
-
-/-- The low-incidence vertex can be chosen among the integration
-variables even when one additional vertex is fixed. -/
-lemma exists_free_vertex_incident_card_le_two {ι : Type*} [DecidableEq ι]
-    (S : Finset ι) (hS : S.Nonempty) (root : ι) (hr : root ∉ S)
-    (E : Finset (ι × ι))
-    (hends : ∀ e ∈ E, e.1 ∈ insert root S ∧ e.2 ∈ insert root S)
-    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
-    (hloop : ∀ e ∈ E, e.1≠e.2) :
-    ∃ v ∈ S, (incidentEdges E v).card ≤ 2 := by
-  have hcard : E.card ≤ S.card+1 := by
-    have hh := Finset.card_le_card_of_injOn Prod.fst (fun e he => (hends e he).1)
-      (fun e he f hf hef => hout e he f hf hef)
-    simpa only [Finset.card_insert_of_notMem hr] using hh
-  have hsum : (∑ v ∈ S, (E.filter (fun e => e.2=v)).card) ≤ E.card := by
-    rw [Finset.sum_card_fiberwise_eq_card_filter]
-    exact Finset.card_filter_le _ _
-  have hin : ∃ v ∈ S, (E.filter (fun e => e.2=v)).card ≤ 1 := by
-    by_cases hS1 : S.card=1
-    · obtain ⟨v,hv⟩ := hS
-      refine ⟨v,hv,?_⟩
-      have hh := incoming_card_le_erase (insert root S) E v hends hout hloop
-      have hv' : v ∈ insert root S := Finset.mem_insert_of_mem hv
-      simpa only [Finset.card_erase_of_mem hv',Finset.card_insert_of_notMem hr,hS1] using hh
-    · by_contra! hn
-      have htwo : S.card*2 ≤ ∑ v ∈ S, (E.filter (fun e => e.2=v)).card := by
-        simpa only [Finset.sum_const,smul_eq_mul] using
-          (Finset.sum_le_sum (s := S) (f := fun _ => (2 : ℕ))
-            (g := fun v => (E.filter (fun e : ι × ι => e.2=v)).card)
-            (fun v hv => Nat.succ_le_of_lt (hn v hv)))
-      have hpos := Finset.card_pos.mpr hS
-      omega
-  obtain ⟨v,hv,hin⟩ := hin
-  have houtcard : (E.filter (fun e => e.1=v)).card ≤ 1 := by
-    apply Finset.card_le_one.mpr
-    intro e he f hf
-    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
-      ((Finset.mem_filter.mp he).2.trans (Finset.mem_filter.mp hf).2.symm)
-  have heq : incidentEdges E v = (E.filter (fun e => e.1=v)) ∪ (E.filter (fun e => e.2=v)) := by
-    ext e
-    simp only [incidentEdges,Finset.mem_filter,Finset.mem_union]
-    tauto
-  refine ⟨v,hv,?_⟩
-  rw [heq]
-  exact (Finset.card_union_le _ _).trans (by omega)
-
-lemma eraseVertex_properties {ι : Type*} [DecidableEq ι]
-    (S : Finset ι) (E : Finset (ι × ι)) (v : ι)
-    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
-    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
-    (hloop : ∀ e ∈ E, e.1≠e.2) :
-    (∀ e ∈ eraseVertex E v, e.1 ∈ S.erase v ∧ e.2 ∈ S.erase v) ∧
-    (∀ e ∈ eraseVertex E v, ∀ f ∈ eraseVertex E v, e.1=f.1 → e=f) ∧
-    (∀ e ∈ eraseVertex E v, e.1≠e.2) := by
-  refine ⟨?_,?_,?_⟩
-  · intro e he
-    obtain ⟨he,h1,h2⟩ := Finset.mem_filter.mp he
-    exact ⟨Finset.mem_erase.mpr ⟨h1,(hends e he).1⟩,
-      Finset.mem_erase.mpr ⟨h2,(hends e he).2⟩⟩
-  · intro e he f hf
-    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
-  · intro e he
-    exact hloop e (Finset.mem_filter.mp he).1
-
-lemma graphKernel_split_vertex {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ) :
-    graphKernel α E x = graphKernel α (incidentEdges E v) x * graphKernel α (eraseVertex E v) x := by
-  unfold graphKernel incidentEdges eraseVertex
-  have he : E.filter (fun e => e.1≠v ∧ e.2≠v) = E.filter (fun e => ¬(e.1=v ∨ e.2=v)) := by
-    ext e
-    simp only [Finset.mem_filter,not_or]
-  rw [he]
-  exact (Finset.prod_filter_mul_prod_filter_not _ _ _).symm
-
-lemma graphKernel_erase_update {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ) (y : ℂ) :
-    graphKernel α (eraseVertex E v) (Function.update x v y) = graphKernel α (eraseVertex E v) x := by
-  apply Finset.prod_congr rfl
-  intro e he
-  obtain ⟨_,h1,h2⟩ := Finset.mem_filter.mp he
-  simp only [Function.update_of_ne h1,Function.update_of_ne h2]
-
-lemma incident_edge_section {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (e : ι × ι) (v : ι) (x : ι → ℂ)
-    (hinc : e.1=v ∨ e.2=v) (hloop : e.1≠e.2) :
-    ∃ c : ℂ, ∀ y, distanceKernel α ((Function.update x v y) e.1)
-      ((Function.update x v y) e.2) = distanceKernel α y c := by
-  rcases hinc with h1 | h2
-  · have h2 : e.2≠v := by rw [← h1]; exact hloop.symm
-    refine ⟨x e.2,fun y => ?_⟩
-    rw [h1,Function.update_self,Function.update_of_ne h2]
-  · have h1 : e.1≠v := by rw [← h2]; exact hloop
-    refine ⟨x e.1,fun y => ?_⟩
-    rw [h2,Function.update_self,Function.update_of_ne h1,distanceKernel_symm]
-
-lemma incident_kernel_section_le {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞)
-    (hC0 : volume K ≤ C)
-    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
-    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
-    (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ)
-    (hcard : E.card ≤ 2) (hinc : ∀ e ∈ E, e.1=v ∨ e.2=v)
-    (hloop : ∀ e ∈ E, e.1≠e.2) :
-    (∫⁻ y in K, graphKernel α E (Function.update x v y)) ≤ C := by
-  have hcases : E.card=0 ∨ E.card=1 ∨ E.card=2 := by omega
-  rcases hcases with h0 | h1 | h2
-  · have he : E=∅ := Finset.card_eq_zero.mp h0
-    simpa only [he,graphKernel,Finset.prod_empty,lintegral_const,
-      Measure.restrict_apply_univ,one_mul] using hC0
-  · obtain ⟨e,he⟩ := Finset.card_eq_one.mp h1
-    have hem : e ∈ E := he ▸ Finset.mem_singleton_self e
-    obtain ⟨c,hc⟩ := incident_edge_section α e v x (hinc e hem) (hloop e hem)
-    simpa only [he,graphKernel,Finset.prod_singleton,hc] using hC1 c
-  · obtain ⟨e,f,hef,he⟩ := Finset.card_eq_two.mp h2
-    have hem : e ∈ E := he ▸ Finset.mem_insert_self e {f}
-    have hfm : f ∈ E := he ▸ Finset.mem_insert_of_mem (Finset.mem_singleton_self f)
-    obtain ⟨c,hc⟩ := incident_edge_section α e v x (hinc e hem) (hloop e hem)
-    obtain ⟨d,hd⟩ := incident_edge_section α f v x (hinc f hfm) (hloop f hfm)
-    simpa only [he,graphKernel,Finset.prod_pair hef,hc,hd] using hC2 c d
-
-lemma graphKernel_section_le {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞)
-    (hC0 : volume K ≤ C)
-    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
-    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
-    (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ)
-    (hcard : (incidentEdges E v).card ≤ 2) (hloop : ∀ e ∈ E, e.1≠e.2) :
-    (∫⁻ y in K, graphKernel α E (Function.update x v y)) ≤
-      C*graphKernel α (eraseVertex E v) x := by
-  have he (y : ℂ) : graphKernel α E (Function.update x v y) =
-      graphKernel α (incidentEdges E v) (Function.update x v y) *
-        graphKernel α (eraseVertex E v) x := by
-    rw [graphKernel_split_vertex α E v,graphKernel_erase_update]
-  simp_rw [he]
-  have hu : Measurable (fun y : ℂ => Function.update x v y) := by fun_prop
-  have hm : Measurable (fun y : ℂ => graphKernel α (incidentEdges E v) (Function.update x v y)) :=
-    (measurable_graphKernel α (incidentEdges E v)).comp hu
-  rw [lintegral_mul_const _ hm]
-  have hh := incident_kernel_section_le α K C hC0 hC1 hC2 (incidentEdges E v) v x hcard
-    (fun e he => (Finset.mem_filter.mp he).2)
-    (fun e he => hloop e (Finset.mem_filter.mp he).1)
-  gcongr
-
-theorem graphKernel_marginal_le_with_root {ι : Type*} [DecidableEq ι]
-    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
-    (hC0 : volume K ≤ C)
-    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
-    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
-    (root : ι) (S : Finset ι) (E : Finset (ι × ι)) (hr : root ∉ S)
-    (hends : ∀ e ∈ E, e.1 ∈ insert root S ∧ e.2 ∈ insert root S)
-    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
-    (hloop : ∀ e ∈ E, e.1≠e.2) (x : ι → ℂ) :
-    lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x ≤ C^S.card := by
-  induction S using Finset.strongInductionOn generalizing E with
-  | _ S ih =>
-    by_cases hempty : S=∅
-    · have hE : E=∅ := by
-        apply Finset.eq_empty_iff_forall_notMem.mpr
-        intro e he
-        have hs := hends e he
-        simp only [hempty,Finset.mem_insert,Finset.notMem_empty,or_false] at hs
-        exact hloop e he (hs.1.trans hs.2.symm)
-      simp only [hempty,hE,lmarginal_empty,graphKernel,Finset.prod_empty,Finset.card_empty,pow_zero,le_rfl]
-    · obtain ⟨v,hv,hdegree⟩ := exists_free_vertex_incident_card_le_two S
-        (Finset.nonempty_iff_ne_empty.mpr hempty) root hr E hends hout hloop
-      have hr' : root ∉ S.erase v := fun h => hr (Finset.mem_of_mem_erase h)
-      have hends' : ∀ e ∈ eraseVertex E v,
-          e.1 ∈ insert root (S.erase v) ∧ e.2 ∈ insert root (S.erase v) := by
-        intro e he
-        obtain ⟨he,h1,h2⟩ := Finset.mem_filter.mp he
-        constructor
-        · rcases Finset.mem_insert.mp (hends e he).1 with h | h
-          · exact Finset.mem_insert.mpr (Or.inl h)
-          · exact Finset.mem_insert.mpr (Or.inr (Finset.mem_erase.mpr ⟨h1,h⟩))
-        · rcases Finset.mem_insert.mp (hends e he).2 with h | h
-          · exact Finset.mem_insert.mpr (Or.inl h)
-          · exact Finset.mem_insert.mpr (Or.inr (Finset.mem_erase.mpr ⟨h2,h⟩))
-      have hout' : ∀ e ∈ eraseVertex E v, ∀ f ∈ eraseVertex E v, e.1=f.1 → e=f :=
-        fun e he f hf => hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
-      have hloop' : ∀ e ∈ eraseVertex E v, e.1≠e.2 :=
-        fun e he => hloop e (Finset.mem_filter.mp he).1
-      have hsmall := ih (S.erase v) (Finset.erase_ssubset hv) (eraseVertex E v) hr' hends' hout' hloop'
-      calc
-        lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x =
-            lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
-              (fun x => ∫⁻ y in K, graphKernel α E (Function.update x v y)) x := by
-          exact congrFun (lmarginal_erase' (μ := fun _ : ι => volume.restrict K)
-            (graphKernel α E) (measurable_graphKernel α E) hv) x
-        _ ≤ lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
-              (fun x => C*graphKernel α (eraseVertex E v) x) x :=
-          lmarginal_mono (fun x => graphKernel_section_le α K C hC0 hC1 hC2 E v x hdegree hloop) x
-        _ = C*lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
-              (graphKernel α (eraseVertex E v)) x := by
-          exact lintegral_const_mul' C _ hC.ne
-        _ ≤ C*C^(S.erase v).card := by gcongr
-        _ = C^S.card := by
-          rw [← Finset.card_erase_add_one hv,pow_succ,mul_comm]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointDistanceKernel
+
+namespace OAI
+
+/-! The finite graph elimination used in the capped distance kernel.
+An edge records one selected inverse-distance factor. Each vertex has
+at most one outgoing edge, including after deleting a vertex. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+def incidentEdges {ι : Type*} [DecidableEq ι] (E : Finset (ι × ι)) (v : ι) : Finset (ι × ι) :=
+  E.filter (fun e => e.1=v ∨ e.2=v)
+
+def eraseVertex {ι : Type*} [DecidableEq ι] (E : Finset (ι × ι)) (v : ι) : Finset (ι × ι) :=
+  E.filter (fun e => e.1≠v ∧ e.2≠v)
+
+def graphKernel {ι : Type*} (α : ℝ) (E : Finset (ι × ι)) (x : ι → ℂ) : ℝ≥0∞ :=
+  ∏ e ∈ E, distanceKernel α (x e.1) (x e.2)
+
+lemma measurable_graphKernel {ι : Type*} (α : ℝ) (E : Finset (ι × ι)) :
+    Measurable (graphKernel α E) := by
+  unfold graphKernel distanceKernel
+  fun_prop
+
+lemma exists_vertex_incident_card_le_two {ι : Type*} [DecidableEq ι]
+    (S : Finset ι) (hS : S.Nonempty) (E : Finset (ι × ι))
+    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
+    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f) :
+    ∃ v ∈ S, (incidentEdges E v).card ≤ 2 := by
+  have hcard : E.card ≤ S.card :=
+    Finset.card_le_card_of_injOn Prod.fst (fun e he => (hends e he).1)
+      (fun e he f hf hef => hout e he f hf hef)
+  have hsum : E.card = ∑ v ∈ S, (E.filter (fun e => e.2=v)).card :=
+    Finset.card_eq_sum_card_fiberwise (fun e he => (hends e he).2)
+  have hin : ∃ v ∈ S, (E.filter (fun e => e.2=v)).card ≤ 1 := by
+    by_contra! h
+    have hlt : S.card < ∑ v ∈ S, (E.filter (fun e => e.2=v)).card := by
+      calc
+        S.card = ∑ _v ∈ S, 1 := by simp
+        _ < _ := Finset.sum_lt_sum (fun v hv => (h v hv).le)
+          (by obtain ⟨v,hv⟩ := hS; exact ⟨v,hv,h v hv⟩)
+    omega
+  obtain ⟨v,hv,hin⟩ := hin
+  have houtcard : (E.filter (fun e => e.1=v)).card ≤ 1 := by
+    apply Finset.card_le_one.mpr
+    intro e he f hf
+    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
+      ((Finset.mem_filter.mp he).2.trans (Finset.mem_filter.mp hf).2.symm)
+  have heq : incidentEdges E v = (E.filter (fun e => e.1=v)) ∪ (E.filter (fun e => e.2=v)) := by
+    ext e
+    simp only [incidentEdges,Finset.mem_filter,Finset.mem_union]
+    tauto
+  refine ⟨v,hv,?_⟩
+  rw [heq]
+  exact (Finset.card_union_le _ _).trans (by omega)
+
+lemma incoming_card_le_erase {ι : Type*} [DecidableEq ι]
+    (S : Finset ι) (E : Finset (ι × ι)) (v : ι)
+    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
+    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
+    (hloop : ∀ e ∈ E, e.1≠e.2) :
+    (E.filter (fun e => e.2=v)).card ≤ (S.erase v).card := by
+  apply Finset.card_le_card_of_injOn Prod.fst
+  · intro e he
+    obtain ⟨he,hev⟩ := Finset.mem_filter.mp he
+    exact Finset.mem_erase.mpr ⟨by simpa only [← hev] using hloop e he,(hends e he).1⟩
+  · intro e he f hf hef
+    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1 hef
+
+/-- The low-incidence vertex can be chosen among the integration
+variables even when one additional vertex is fixed. -/
+lemma exists_free_vertex_incident_card_le_two {ι : Type*} [DecidableEq ι]
+    (S : Finset ι) (hS : S.Nonempty) (root : ι) (hr : root ∉ S)
+    (E : Finset (ι × ι))
+    (hends : ∀ e ∈ E, e.1 ∈ insert root S ∧ e.2 ∈ insert root S)
+    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
+    (hloop : ∀ e ∈ E, e.1≠e.2) :
+    ∃ v ∈ S, (incidentEdges E v).card ≤ 2 := by
+  have hcard : E.card ≤ S.card+1 := by
+    have hh := Finset.card_le_card_of_injOn Prod.fst (fun e he => (hends e he).1)
+      (fun e he f hf hef => hout e he f hf hef)
+    simpa only [Finset.card_insert_of_notMem hr] using hh
+  have hsum : (∑ v ∈ S, (E.filter (fun e => e.2=v)).card) ≤ E.card := by
+    rw [Finset.sum_card_fiberwise_eq_card_filter]
+    exact Finset.card_filter_le _ _
+  have hin : ∃ v ∈ S, (E.filter (fun e => e.2=v)).card ≤ 1 := by
+    by_cases hS1 : S.card=1
+    · obtain ⟨v,hv⟩ := hS
+      refine ⟨v,hv,?_⟩
+      have hh := incoming_card_le_erase (insert root S) E v hends hout hloop
+      have hv' : v ∈ insert root S := Finset.mem_insert_of_mem hv
+      simpa only [Finset.card_erase_of_mem hv',Finset.card_insert_of_notMem hr,hS1] using hh
+    · by_contra! hn
+      have htwo : S.card*2 ≤ ∑ v ∈ S, (E.filter (fun e => e.2=v)).card := by
+        simpa only [Finset.sum_const,smul_eq_mul] using
+          (Finset.sum_le_sum (s := S) (f := fun _ => (2 : ℕ))
+            (g := fun v => (E.filter (fun e : ι × ι => e.2=v)).card)
+            (fun v hv => Nat.succ_le_of_lt (hn v hv)))
+      have hpos := Finset.card_pos.mpr hS
+      omega
+  obtain ⟨v,hv,hin⟩ := hin
+  have houtcard : (E.filter (fun e => e.1=v)).card ≤ 1 := by
+    apply Finset.card_le_one.mpr
+    intro e he f hf
+    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
+      ((Finset.mem_filter.mp he).2.trans (Finset.mem_filter.mp hf).2.symm)
+  have heq : incidentEdges E v = (E.filter (fun e => e.1=v)) ∪ (E.filter (fun e => e.2=v)) := by
+    ext e
+    simp only [incidentEdges,Finset.mem_filter,Finset.mem_union]
+    tauto
+  refine ⟨v,hv,?_⟩
+  rw [heq]
+  exact (Finset.card_union_le _ _).trans (by omega)
+
+lemma eraseVertex_properties {ι : Type*} [DecidableEq ι]
+    (S : Finset ι) (E : Finset (ι × ι)) (v : ι)
+    (hends : ∀ e ∈ E, e.1 ∈ S ∧ e.2 ∈ S)
+    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
+    (hloop : ∀ e ∈ E, e.1≠e.2) :
+    (∀ e ∈ eraseVertex E v, e.1 ∈ S.erase v ∧ e.2 ∈ S.erase v) ∧
+    (∀ e ∈ eraseVertex E v, ∀ f ∈ eraseVertex E v, e.1=f.1 → e=f) ∧
+    (∀ e ∈ eraseVertex E v, e.1≠e.2) := by
+  refine ⟨?_,?_,?_⟩
+  · intro e he
+    obtain ⟨he,h1,h2⟩ := Finset.mem_filter.mp he
+    exact ⟨Finset.mem_erase.mpr ⟨h1,(hends e he).1⟩,
+      Finset.mem_erase.mpr ⟨h2,(hends e he).2⟩⟩
+  · intro e he f hf
+    exact hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
+  · intro e he
+    exact hloop e (Finset.mem_filter.mp he).1
+
+lemma graphKernel_split_vertex {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ) :
+    graphKernel α E x = graphKernel α (incidentEdges E v) x * graphKernel α (eraseVertex E v) x := by
+  unfold graphKernel incidentEdges eraseVertex
+  have he : E.filter (fun e => e.1≠v ∧ e.2≠v) = E.filter (fun e => ¬(e.1=v ∨ e.2=v)) := by
+    ext e
+    simp only [Finset.mem_filter,not_or]
+  rw [he]
+  exact (Finset.prod_filter_mul_prod_filter_not _ _ _).symm
+
+lemma graphKernel_erase_update {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ) (y : ℂ) :
+    graphKernel α (eraseVertex E v) (Function.update x v y) = graphKernel α (eraseVertex E v) x := by
+  apply Finset.prod_congr rfl
+  intro e he
+  obtain ⟨_,h1,h2⟩ := Finset.mem_filter.mp he
+  simp only [Function.update_of_ne h1,Function.update_of_ne h2]
+
+lemma incident_edge_section {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (e : ι × ι) (v : ι) (x : ι → ℂ)
+    (hinc : e.1=v ∨ e.2=v) (hloop : e.1≠e.2) :
+    ∃ c : ℂ, ∀ y, distanceKernel α ((Function.update x v y) e.1)
+      ((Function.update x v y) e.2) = distanceKernel α y c := by
+  rcases hinc with h1 | h2
+  · have h2 : e.2≠v := by rw [← h1]; exact hloop.symm
+    refine ⟨x e.2,fun y => ?_⟩
+    rw [h1,Function.update_self,Function.update_of_ne h2]
+  · have h1 : e.1≠v := by rw [← h2]; exact hloop
+    refine ⟨x e.1,fun y => ?_⟩
+    rw [h2,Function.update_self,Function.update_of_ne h1,distanceKernel_symm]
+
+lemma incident_kernel_section_le {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞)
+    (hC0 : volume K ≤ C)
+    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
+    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
+    (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ)
+    (hcard : E.card ≤ 2) (hinc : ∀ e ∈ E, e.1=v ∨ e.2=v)
+    (hloop : ∀ e ∈ E, e.1≠e.2) :
+    (∫⁻ y in K, graphKernel α E (Function.update x v y)) ≤ C := by
+  have hcases : E.card=0 ∨ E.card=1 ∨ E.card=2 := by omega
+  rcases hcases with h0 | h1 | h2
+  · have he : E=∅ := Finset.card_eq_zero.mp h0
+    simpa only [he,graphKernel,Finset.prod_empty,lintegral_const,
+      Measure.restrict_apply_univ,one_mul] using hC0
+  · obtain ⟨e,he⟩ := Finset.card_eq_one.mp h1
+    have hem : e ∈ E := he ▸ Finset.mem_singleton_self e
+    obtain ⟨c,hc⟩ := incident_edge_section α e v x (hinc e hem) (hloop e hem)
+    simpa only [he,graphKernel,Finset.prod_singleton,hc] using hC1 c
+  · obtain ⟨e,f,hef,he⟩ := Finset.card_eq_two.mp h2
+    have hem : e ∈ E := he ▸ Finset.mem_insert_self e {f}
+    have hfm : f ∈ E := he ▸ Finset.mem_insert_of_mem (Finset.mem_singleton_self f)
+    obtain ⟨c,hc⟩ := incident_edge_section α e v x (hinc e hem) (hloop e hem)
+    obtain ⟨d,hd⟩ := incident_edge_section α f v x (hinc f hfm) (hloop f hfm)
+    simpa only [he,graphKernel,Finset.prod_pair hef,hc,hd] using hC2 c d
+
+lemma graphKernel_section_le {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞)
+    (hC0 : volume K ≤ C)
+    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
+    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
+    (E : Finset (ι × ι)) (v : ι) (x : ι → ℂ)
+    (hcard : (incidentEdges E v).card ≤ 2) (hloop : ∀ e ∈ E, e.1≠e.2) :
+    (∫⁻ y in K, graphKernel α E (Function.update x v y)) ≤
+      C*graphKernel α (eraseVertex E v) x := by
+  have he (y : ℂ) : graphKernel α E (Function.update x v y) =
+      graphKernel α (incidentEdges E v) (Function.update x v y) *
+        graphKernel α (eraseVertex E v) x := by
+    rw [graphKernel_split_vertex α E v,graphKernel_erase_update]
+  simp_rw [he]
+  have hu : Measurable (fun y : ℂ => Function.update x v y) := by fun_prop
+  have hm : Measurable (fun y : ℂ => graphKernel α (incidentEdges E v) (Function.update x v y)) :=
+    (measurable_graphKernel α (incidentEdges E v)).comp hu
+  rw [lintegral_mul_const _ hm]
+  have hh := incident_kernel_section_le α K C hC0 hC1 hC2 (incidentEdges E v) v x hcard
+    (fun e he => (Finset.mem_filter.mp he).2)
+    (fun e he => hloop e (Finset.mem_filter.mp he).1)
+  gcongr
+
+theorem graphKernel_marginal_le_with_root {ι : Type*} [DecidableEq ι]
+    (α : ℝ) (K : Set ℂ) (C : ℝ≥0∞) (hC : C < ⊤)
+    (hC0 : volume K ≤ C)
+    (hC1 : ∀ y, (∫⁻ x in K, distanceKernel α x y) ≤ C)
+    (hC2 : ∀ y z, (∫⁻ x in K, distanceKernel α x y * distanceKernel α x z) ≤ C)
+    (root : ι) (S : Finset ι) (E : Finset (ι × ι)) (hr : root ∉ S)
+    (hends : ∀ e ∈ E, e.1 ∈ insert root S ∧ e.2 ∈ insert root S)
+    (hout : ∀ e ∈ E, ∀ f ∈ E, e.1=f.1 → e=f)
+    (hloop : ∀ e ∈ E, e.1≠e.2) (x : ι → ℂ) :
+    lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x ≤ C^S.card := by
+  induction S using Finset.strongInductionOn generalizing E with
+  | _ S ih =>
+    by_cases hempty : S=∅
+    · have hE : E=∅ := by
+        apply Finset.eq_empty_iff_forall_notMem.mpr
+        intro e he
+        have hs := hends e he
+        simp only [hempty,Finset.mem_insert,Finset.notMem_empty,or_false] at hs
+        exact hloop e he (hs.1.trans hs.2.symm)
+      simp only [hempty,hE,lmarginal_empty,graphKernel,Finset.prod_empty,Finset.card_empty,pow_zero,le_rfl]
+    · obtain ⟨v,hv,hdegree⟩ := exists_free_vertex_incident_card_le_two S
+        (Finset.nonempty_iff_ne_empty.mpr hempty) root hr E hends hout hloop
+      have hr' : root ∉ S.erase v := fun h => hr (Finset.mem_of_mem_erase h)
+      have hends' : ∀ e ∈ eraseVertex E v,
+          e.1 ∈ insert root (S.erase v) ∧ e.2 ∈ insert root (S.erase v) := by
+        intro e he
+        obtain ⟨he,h1,h2⟩ := Finset.mem_filter.mp he
+        constructor
+        · rcases Finset.mem_insert.mp (hends e he).1 with h | h
+          · exact Finset.mem_insert.mpr (Or.inl h)
+          · exact Finset.mem_insert.mpr (Or.inr (Finset.mem_erase.mpr ⟨h1,h⟩))
+        · rcases Finset.mem_insert.mp (hends e he).2 with h | h
+          · exact Finset.mem_insert.mpr (Or.inl h)
+          · exact Finset.mem_insert.mpr (Or.inr (Finset.mem_erase.mpr ⟨h2,h⟩))
+      have hout' : ∀ e ∈ eraseVertex E v, ∀ f ∈ eraseVertex E v, e.1=f.1 → e=f :=
+        fun e he f hf => hout e (Finset.mem_filter.mp he).1 f (Finset.mem_filter.mp hf).1
+      have hloop' : ∀ e ∈ eraseVertex E v, e.1≠e.2 :=
+        fun e he => hloop e (Finset.mem_filter.mp he).1
+      have hsmall := ih (S.erase v) (Finset.erase_ssubset hv) (eraseVertex E v) hr' hends' hout' hloop'
+      calc
+        lmarginal (fun _ : ι => volume.restrict K) S (graphKernel α E) x =
+            lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
+              (fun x => ∫⁻ y in K, graphKernel α E (Function.update x v y)) x := by
+          exact congrFun (lmarginal_erase' (μ := fun _ : ι => volume.restrict K)
+            (graphKernel α E) (measurable_graphKernel α E) hv) x
+        _ ≤ lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
+              (fun x => C*graphKernel α (eraseVertex E v) x) x :=
+          lmarginal_mono (fun x => graphKernel_section_le α K C hC0 hC1 hC2 E v x hdegree hloop) x
+        _ = C*lmarginal (fun _ : ι => volume.restrict K) (S.erase v)
+              (graphKernel α (eraseVertex E v)) x := by
+          exact lintegral_const_mul' C _ hC.ne
+        _ ≤ C*C^(S.erase v).card := by gcongr
+        _ = C^S.card := by
+          rw [← Finset.card_erase_add_one hv,pow_succ,mul_comm]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLevelMass.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLevelMass.lean
index 73d8af9f52ea7069d1fbdd78b240d1bfaa9dfb81..9d5cae1328f9d4a86d6b2e4e6f5a8168c27e0899 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLevelMass.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLevelMass.lean
@@ -1,109 +1,109 @@
-import OAI.Probability.SLE.Multipoint.MultipointLevelWeights
-import OAI.Probability.SLE.Multipoint.MultipointMomentIntegration
-import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel
-
-namespace OAI
-
-/-! Jointly measurable actual level masses and the tensor identity used
-to integrate the multipoint estimate. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-
-lemma actual_stoppedGreen_joint_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ u : ℝ) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) :
-    Measurable (fun q : Ω × ℂ => stoppedGreen κ (fun t => Real.sqrt κ*B t q.1)
-      (γ q.1) (G q.1) q.2 u) := by
-  have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm
-  have h : Measurable (fun q : Ω × ℂ =>
-      canonicalStoppedGreen κ (u, q.2, sourceDriverPath B hBc q.1)) :=
-    (canonicalStoppedGreen_measurable κ).comp
-    (measurable_const.prodMk (measurable_snd.prodMk (hp.comp measurable_fst)))
-  convert h using 1
-  funext q
-  exact (canonicalStoppedGreen_eq κ (sourceDriverPath B hBc q.1) (hw q.1) u hu q.2).symm
-
-def levelMass {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
-    (u : ℝ) (K : Set ℂ) (ω : Ω) : ℝ≥0∞ :=
-  ∫⁻ z in K, ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-    (γ ω) (G ω) z u)
-
-lemma levelMass_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ u : ℝ) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (K : Set ℂ) : Measurable (levelMass κ B γ G u K) :=
-  (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal.lintegral_prod_right'
-
-theorem levelMass_moment_identity
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [SigmaFinite P]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (K : Set ℂ) (m : ℕ) :
-    (∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) =
-      ∫⁻ z : Fin m → ℂ, ∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ
-        (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (z i) u) ∂P
-        ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
-  have h := lintegral_mass_pow_eq P (volume.restrict K)
-    (fun ω z => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) z u))
-    (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal m
-  change (∫⁻ ω, (∫⁻ z in K, ENNReal.ofReal (stoppedGreen κ
-    (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u)) ^ m ∂P) = _
-  rw [h]
-  apply lintegral_congr
-  intro z
-  apply lintegral_congr
-  intro ω
-  exact (ENNReal.ofReal_prod_of_nonneg
-    (fun i _ => stoppedGreen_nonneg (hw ω) κ _ _)).symm
-
-lemma common_level_log_target (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) :
-    Real.log (2*z.im/(2*z.im*Real.exp (-u))) = u := by
-  have hratio : 2*z.im/(2*z.im*Real.exp (-u)) = Real.exp u := by
-    rw [Real.exp_neg]
-    field_simp [(show z.im ≠ 0 from ne_of_gt hz),(Real.exp_pos u).ne']
-  rw [hratio,Real.log_exp]
-
-theorem stoppedGreen_common_level_product_bound
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (b : ι → ℝ) (hb : ∀ i, 0 < b i) (hbR : ∀ i, b i ≤ 2*(z i).im)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j)) (u : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) (z i) u) ∂P) ≤
-      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
-        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
-        ∏ i, (b i)^(-codimension κ)) := by
-  have h := stoppedGreen_product_expectation_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw
-    hgrowth z hz hne b (fun i => 2*(z i).im*Real.exp (-u))
-    (fun i => mul_pos (mul_pos (by norm_num) (hz i)) (Real.exp_pos _)) hb hbR hsep
-  simpa only [common_level_log_target _ (hz _) u] using h
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLevelWeights
+import OAI.Probability.SLE.Multipoint.MultipointMomentIntegration
+import OAI.Probability.SLE.Loewner.CanonicalRadiusKernel
+
+namespace OAI
+
+/-! Jointly measurable actual level masses and the tensor identity used
+to integrate the multipoint estimate. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+
+lemma actual_stoppedGreen_joint_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ u : ℝ) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) :
+    Measurable (fun q : Ω × ℂ => stoppedGreen κ (fun t => Real.sqrt κ*B t q.1)
+      (γ q.1) (G q.1) q.2 u) := by
+  have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm
+  have h : Measurable (fun q : Ω × ℂ =>
+      canonicalStoppedGreen κ (u, q.2, sourceDriverPath B hBc q.1)) :=
+    (canonicalStoppedGreen_measurable κ).comp
+    (measurable_const.prodMk (measurable_snd.prodMk (hp.comp measurable_fst)))
+  convert h using 1
+  funext q
+  exact (canonicalStoppedGreen_eq κ (sourceDriverPath B hBc q.1) (hw q.1) u hu q.2).symm
+
+def levelMass {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ)
+    (u : ℝ) (K : Set ℂ) (ω : Ω) : ℝ≥0∞ :=
+  ∫⁻ z in K, ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+    (γ ω) (G ω) z u)
+
+lemma levelMass_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ u : ℝ) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (K : Set ℂ) : Measurable (levelMass κ B γ G u K) :=
+  (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal.lintegral_prod_right'
+
+theorem levelMass_moment_identity
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [SigmaFinite P]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (K : Set ℂ) (m : ℕ) :
+    (∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) =
+      ∫⁻ z : Fin m → ℂ, ∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ
+        (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (z i) u) ∂P
+        ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by
+  have h := lintegral_mass_pow_eq P (volume.restrict K)
+    (fun ω z => ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) z u))
+    (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal m
+  change (∫⁻ ω, (∫⁻ z in K, ENNReal.ofReal (stoppedGreen κ
+    (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u)) ^ m ∂P) = _
+  rw [h]
+  apply lintegral_congr
+  intro z
+  apply lintegral_congr
+  intro ω
+  exact (ENNReal.ofReal_prod_of_nonneg
+    (fun i _ => stoppedGreen_nonneg (hw ω) κ _ _)).symm
+
+lemma common_level_log_target (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) :
+    Real.log (2*z.im/(2*z.im*Real.exp (-u))) = u := by
+  have hratio : 2*z.im/(2*z.im*Real.exp (-u)) = Real.exp u := by
+    rw [Real.exp_neg]
+    field_simp [(show z.im ≠ 0 from ne_of_gt hz),(Real.exp_pos u).ne']
+  rw [hratio,Real.log_exp]
+
+theorem stoppedGreen_common_level_product_bound
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (b : ι → ℝ) (hb : ∀ i, 0 < b i) (hbR : ∀ i, b i ≤ 2*(z i).im)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j)) (u : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) (z i) u) ∂P) ≤
+      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
+        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
+        ∏ i, (b i)^(-codimension κ)) := by
+  have h := stoppedGreen_product_expectation_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw
+    hgrowth z hz hne b (fun i => 2*(z i).im*Real.exp (-u))
+    (fun i => mul_pos (mul_pos (by norm_num) (hz i)) (Real.exp_pos _)) hb hbR hsep
+  simpa only [common_level_log_target _ (hz _) u] using h
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLevelWeights.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLevelWeights.lean
index 584cca500094af2a37420b1195103db4037c31e0..b90f0a5308e26df1fcfcb6ed8623a9f4b19f3ef0 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLevelWeights.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLevelWeights.lean
@@ -1,130 +1,130 @@
-import OAI.Probability.SLE.Multipoint.MultipointOrdinaryProbability
-import OAI.Probability.SLE.FiniteStoppedLaw
-
-namespace OAI
-
-/-! From the multipoint probability bound to the product estimate for
-actual stopped Green weights. The estimate is uniform in their levels. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma target_weight_capped {α b e : ℝ} (hb : 0 < b) (he : 0 < e) :
-    e^(-α) * min 1 ((e/b)^α) ≤ b^(-α) := by
-  calc
-    _ ≤ e^(-α) * (e/b)^α :=
-      mul_le_mul_of_nonneg_left (min_le_right _ _) (Real.rpow_nonneg he.le _)
-    _ = b^(-α) := by
-      rw [Real.rpow_neg he.le,Real.rpow_neg hb.le,Real.div_rpow he.le hb.le]
-      field_simp [(Real.rpow_pos_of_pos he α).ne']
-
-lemma lintegral_ofReal_prod_le_event
-    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
-    (P : Measure Ω) (E : Set Ω) (Y : ι → Ω → ℝ) (c : ι → ℝ)
-    (hY0 : ∀ i ω, 0 ≤ Y i ω)
-    (hY : ∀ ω ∈ E, ∀ i, Y i ω ≤ c i)
-    (hzero : ∀ ω, ω ∉ E → ∃ i, Y i ω = 0) :
-    (∫⁻ ω, ENNReal.ofReal (∏ i, Y i ω) ∂P) ≤ ENNReal.ofReal (∏ i, c i) * P E := by
-  classical
-  apply (lintegral_mono (fun ω => ?_)).trans (lintegral_indicator_const_le E _)
-  by_cases hω : ω ∈ E
-  · rw [indicator_of_mem hω]
-    exact ENNReal.ofReal_le_ofReal (Finset.prod_le_prod₀ (fun i _ => hY0 i ω)
-      (fun i _ => hY ω hω i))
-  · obtain ⟨i,hi⟩ := hzero ω hω
-    rw [Finset.prod_eq_zero (Finset.mem_univ i) hi,ENNReal.ofReal_zero]
-    exact bot_le
-
-lemma stoppedGreen_level_bound
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (e : ℝ) (he : 0 < e) :
-    stoppedGreen κ U γ G z (Real.log (2*z.im/e)) ≤ e^(-codimension κ) := by
-  by_cases hf : radiusLevelTime γ G z (Real.log (2*z.im/e)) < ∞
-  · obtain ⟨t,ht,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z _ hf
-    have htarget : (2*z.im)*Real.exp (-Real.log (2*z.im/e)) = e := by
-      rw [Real.exp_neg,Real.exp_log (div_pos (mul_pos (by norm_num) hz) he)]
-      field_simp [he.ne',(show z.im ≠ 0 from ne_of_gt hz)]
-    rw [stoppedGreen_eq_of_live_level hU hw κ z _ t ht hR,greenDensity,ite_eq_left ht,
-      hR,htarget]
-    have hsin : 0 ≤ Real.sin (Complex.arg (G t z - (U t : ℂ))) := by
-      rw [Complex.sin_arg,Complex.sub_im,Complex.ofReal_im,sub_zero]
-      exact div_nonneg (hw.imag_positive t z ht).le (norm_nonneg _)
-    exact mul_le_of_le_one_right (Real.rpow_nonneg he.le _)
-      (Real.rpow_le_one hsin (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
-  · simp only [stoppedGreen,ite_eq_right hf]
-    exact Real.rpow_nonneg he.le _
-
-theorem stoppedGreen_product_expectation_bound
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (b e : ι → ℝ) (he : ∀ i, 0 < e i) (hb : ∀ i, 0 < b i)
-    (hbR : ∀ i, b i ≤ 2*(z i).im)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j)) :
-    (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-      (γ ω) (G ω) (z i) (Real.log (2*(z i).im/e i))) ∂P) ≤
-      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
-        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
-        ∏ i, (b i)^(-codimension κ)) := by
-  classical
-  let E := {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (Real.log (2*(z i).im/e i)) < ∞}
-  let Y := fun i ω => stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
-    (γ ω) (G ω) (z i) (Real.log (2*(z i).im/e i))
-  let W := ∏ i, (e i)^(-codimension κ)
-  let C := (3 : ℝ)^(Fintype.card ι) *
-    Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι))
-  have hW : 0 ≤ W := Finset.prod_nonneg (fun i _ => Real.rpow_nonneg (he i).le _)
-  have hC : 0 ≤ C := by dsimp [C]; positivity
-  have hint := lintegral_ofReal_prod_le_event P E Y (fun i => (e i)^(-codimension κ))
-    (fun i ω => stoppedGreen_nonneg (hw ω) κ _ _)
-    (fun ω _ i => stoppedGreen_level_bound (continuous_const.mul (hBc ω))
-      (hw ω) hκ hκ8 (z i) (hz i) (e i) (he i)) (fun ω hω => by
-      obtain ⟨i,hi⟩ := not_forall.mp hω
-      exact ⟨i,by simp only [Y,stoppedGreen,ite_eq_right hi]⟩)
-  have hprob := capped_physical_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8
-    γ G F hw hgrowth (fun z : ℂ => z) differentiableOn_id (fun _ _ _ _ h => h)
-    z hz hne (fun i => 2*(z i).im) b e he hb
-    (fun i => by simpa only [deriv_id'',norm_one,mul_one] using hbR i)
-    (fun i => by simp only [deriv_id'',norm_one,mul_one]) hsep
-  have hsub : E ⊆ {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius (fun z : ℂ => z) (G ω) t (z i) ≤ e i} := by
-    intro ω hω i
-    obtain ⟨t,ht,hR⟩ := (physical_radius_eq_hit_iff_finite_level
-      (continuous_const.mul (hBc ω)) (hw ω) (fun z : ℂ => z) (z i) (2*(z i).im) (e i)
-      (mul_pos (by norm_num) (hz i)) (he i)
-      (by simp only [deriv_id'',norm_one,mul_one])).mpr (hω i)
-    exact ⟨t,ht,hR.le⟩
-  have hα : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  have hp : W * (∏ i, min 1 ((e i/b i)^(1-κ/8))) ≤ ∏ i, (b i)^(-codimension κ) := by
-    dsimp only [W]
-    rw [← Finset.prod_mul_distrib,← hα]
-    apply Finset.prod_le_prod₀
-    · intro i _
-      exact mul_nonneg (Real.rpow_nonneg (he i).le _) (le_min zero_le_one (Real.rpow_nonneg (div_nonneg (he i).le (hb i).le) _))
-    · exact fun i _ => target_weight_capped (hb i) (he i)
-  calc
-    _ ≤ ENNReal.ofReal W * P E := hint
-    _ ≤ ENNReal.ofReal W * ENNReal.ofReal (C * ∏ i, min 1 ((e i/b i)^(1-κ/8))) :=
-      mul_le_mul' le_rfl ((measure_mono hsub).trans hprob)
-    _ = ENNReal.ofReal (C * (W * ∏ i, min 1 ((e i/b i)^(1-κ/8)))) := by
-      rw [← ENNReal.ofReal_mul hW]
-      congr 1
-      ring
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hp hC)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointOrdinaryProbability
+import OAI.Probability.SLE.FiniteStoppedLaw
+
+namespace OAI
+
+/-! From the multipoint probability bound to the product estimate for
+actual stopped Green weights. The estimate is uniform in their levels. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma target_weight_capped {α b e : ℝ} (hb : 0 < b) (he : 0 < e) :
+    e^(-α) * min 1 ((e/b)^α) ≤ b^(-α) := by
+  calc
+    _ ≤ e^(-α) * (e/b)^α :=
+      mul_le_mul_of_nonneg_left (min_le_right _ _) (Real.rpow_nonneg he.le _)
+    _ = b^(-α) := by
+      rw [Real.rpow_neg he.le,Real.rpow_neg hb.le,Real.div_rpow he.le hb.le]
+      field_simp [(Real.rpow_pos_of_pos he α).ne']
+
+lemma lintegral_ofReal_prod_le_event
+    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
+    (P : Measure Ω) (E : Set Ω) (Y : ι → Ω → ℝ) (c : ι → ℝ)
+    (hY0 : ∀ i ω, 0 ≤ Y i ω)
+    (hY : ∀ ω ∈ E, ∀ i, Y i ω ≤ c i)
+    (hzero : ∀ ω, ω ∉ E → ∃ i, Y i ω = 0) :
+    (∫⁻ ω, ENNReal.ofReal (∏ i, Y i ω) ∂P) ≤ ENNReal.ofReal (∏ i, c i) * P E := by
+  classical
+  apply (lintegral_mono (fun ω => ?_)).trans (lintegral_indicator_const_le E _)
+  by_cases hω : ω ∈ E
+  · rw [indicator_of_mem hω]
+    exact ENNReal.ofReal_le_ofReal (Finset.prod_le_prod₀ (fun i _ => hY0 i ω)
+      (fun i _ => hY ω hω i))
+  · obtain ⟨i,hi⟩ := hzero ω hω
+    rw [Finset.prod_eq_zero (Finset.mem_univ i) hi,ENNReal.ofReal_zero]
+    exact bot_le
+
+lemma stoppedGreen_level_bound
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (e : ℝ) (he : 0 < e) :
+    stoppedGreen κ U γ G z (Real.log (2*z.im/e)) ≤ e^(-codimension κ) := by
+  by_cases hf : radiusLevelTime γ G z (Real.log (2*z.im/e)) < ∞
+  · obtain ⟨t,ht,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z _ hf
+    have htarget : (2*z.im)*Real.exp (-Real.log (2*z.im/e)) = e := by
+      rw [Real.exp_neg,Real.exp_log (div_pos (mul_pos (by norm_num) hz) he)]
+      field_simp [he.ne',(show z.im ≠ 0 from ne_of_gt hz)]
+    rw [stoppedGreen_eq_of_live_level hU hw κ z _ t ht hR,greenDensity,ite_eq_left ht,
+      hR,htarget]
+    have hsin : 0 ≤ Real.sin (Complex.arg (G t z - (U t : ℂ))) := by
+      rw [Complex.sin_arg,Complex.sub_im,Complex.ofReal_im,sub_zero]
+      exact div_nonneg (hw.imag_positive t z ht).le (norm_nonneg _)
+    exact mul_le_of_le_one_right (Real.rpow_nonneg he.le _)
+      (Real.rpow_le_one hsin (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
+  · simp only [stoppedGreen,ite_eq_right hf]
+    exact Real.rpow_nonneg he.le _
+
+theorem stoppedGreen_product_expectation_bound
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (b e : ι → ℝ) (he : ∀ i, 0 < e i) (hb : ∀ i, 0 < b i)
+    (hbR : ∀ i, b i ≤ 2*(z i).im)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (z i) (z j)) :
+    (∫⁻ ω, ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+      (γ ω) (G ω) (z i) (Real.log (2*(z i).im/e i))) ∂P) ≤
+      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
+        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
+        ∏ i, (b i)^(-codimension κ)) := by
+  classical
+  let E := {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (Real.log (2*(z i).im/e i)) < ∞}
+  let Y := fun i ω => stoppedGreen κ (fun t => Real.sqrt κ*B t ω)
+    (γ ω) (G ω) (z i) (Real.log (2*(z i).im/e i))
+  let W := ∏ i, (e i)^(-codimension κ)
+  let C := (3 : ℝ)^(Fintype.card ι) *
+    Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι))
+  have hW : 0 ≤ W := Finset.prod_nonneg (fun i _ => Real.rpow_nonneg (he i).le _)
+  have hC : 0 ≤ C := by dsimp [C]; positivity
+  have hint := lintegral_ofReal_prod_le_event P E Y (fun i => (e i)^(-codimension κ))
+    (fun i ω => stoppedGreen_nonneg (hw ω) κ _ _)
+    (fun ω _ i => stoppedGreen_level_bound (continuous_const.mul (hBc ω))
+      (hw ω) hκ hκ8 (z i) (hz i) (e i) (he i)) (fun ω hω => by
+      obtain ⟨i,hi⟩ := not_forall.mp hω
+      exact ⟨i,by simp only [Y,stoppedGreen,ite_eq_right hi]⟩)
+  have hprob := capped_physical_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8
+    γ G F hw hgrowth (fun z : ℂ => z) differentiableOn_id (fun _ _ _ _ h => h)
+    z hz hne (fun i => 2*(z i).im) b e he hb
+    (fun i => by simpa only [deriv_id'',norm_one,mul_one] using hbR i)
+    (fun i => by simp only [deriv_id'',norm_one,mul_one]) hsep
+  have hsub : E ⊆ {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius (fun z : ℂ => z) (G ω) t (z i) ≤ e i} := by
+    intro ω hω i
+    obtain ⟨t,ht,hR⟩ := (physical_radius_eq_hit_iff_finite_level
+      (continuous_const.mul (hBc ω)) (hw ω) (fun z : ℂ => z) (z i) (2*(z i).im) (e i)
+      (mul_pos (by norm_num) (hz i)) (he i)
+      (by simp only [deriv_id'',norm_one,mul_one])).mpr (hω i)
+    exact ⟨t,ht,hR.le⟩
+  have hα : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  have hp : W * (∏ i, min 1 ((e i/b i)^(1-κ/8))) ≤ ∏ i, (b i)^(-codimension κ) := by
+    dsimp only [W]
+    rw [← Finset.prod_mul_distrib,← hα]
+    apply Finset.prod_le_prod₀
+    · intro i _
+      exact mul_nonneg (Real.rpow_nonneg (he i).le _) (le_min zero_le_one (Real.rpow_nonneg (div_nonneg (he i).le (hb i).le) _))
+    · exact fun i _ => target_weight_capped (hb i) (he i)
+  calc
+    _ ≤ ENNReal.ofReal W * P E := hint
+    _ ≤ ENNReal.ofReal W * ENNReal.ofReal (C * ∏ i, min 1 ((e i/b i)^(1-κ/8))) :=
+      mul_le_mul' le_rfl ((measure_mono hsub).trans hprob)
+    _ = ENNReal.ofReal (C * (W * ∏ i, min 1 ((e i/b i)^(1-κ/8)))) := by
+      rw [← ENNReal.ofReal_mul hW]
+      congr 1
+      ring
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hp hC)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalIdentification.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalIdentification.lean
index 9d56e03a723461c72252973cfeec80f130c7ccf8..bb4c3f14664ccc3c43b9be1f597434928dff0b26 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalIdentification.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalIdentification.lean
@@ -1,60 +1,60 @@
-import OAI.Probability.SLE.Multipoint.MultipointClippedState
-import OAI.Probability.SLE.Multipoint.MultipointStoppedCoordinates
-import OAI.Probability.SLE.FiniteRadiusStop
-
-namespace OAI
-
-/-! Safe clipped coordinates identify the actual Loewner flow separately
-up to each point's completion or the common localization time. -/
-
-noncomputable section
-open Set
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem clipped_multipoint_identification {ι : Type*} [Fintype ι]
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (lo hi : ℝ) (hle : lo ≤ hi) (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (T : ι → ℝ≥0) (Q : ℝ → ι → ℂ × ℝ)
-    (hQ0 : ∀ i, Q 0 i = initialLogCoordinates (z i))
-    (hQd : ∀ t, 0 ≤ t → HasDerivWithinAt Q
-      (fun i => clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t i)) (Ici 0) t)
-    (hheight : ∀ i t, t ∈ Icc 0 (T i : ℝ) → (Q t i).1.im ∈ Icc lo hi) :
-    ∀ i, z i ∈ survivingDomain γ (T i) ∧
-      EqOn (fun t => Q t i) (logLoewnerCoordinates G (z i)) (Icc 0 (T i : ℝ)) := by
-  intro i
-  apply clipped_log_identification hU hw (z i) (hz i) (T i) lo hi hle
-    (fun t => Q t i) (hQ0 i)
-  · intro t ht
-    exact ((hasDerivWithinAt_pi.mp (hQd t ht.1)) i).mono (fun _ hs => hs.1)
-  · exact hheight i
-
-theorem stopped_height_gives_local_identification
-    {Ω ι : Type*} [Fintype ι]
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (lo hi : ℝ) (hle : lo ≤ hi) (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (χ : Ω → ℝ≥0) (ω : Ω)
-    (hQ0 : ∀ i, Q 0 ω i = initialLogCoordinates (z i))
-    (hQd : ∀ t, 0 ≤ t → HasDerivWithinAt (fun t => Q t ω)
-      (fun i => clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t ω i)) (Ici 0) t)
-    (hheight : ∀ (t : ℝ≥0), t ≤ χ ω → ∀ i,
-      (stoppedCoordinateState Q τ t ω i).1.im ∈ Icc lo hi) :
-    ∀ i, z i ∈ survivingDomain γ (min (τ i ω) (χ ω)) ∧
-      EqOn (fun t => Q t ω i) (logLoewnerCoordinates G (z i))
-        (Icc 0 (min (τ i ω) (χ ω) : ℝ≥0)) := by
-  apply clipped_multipoint_identification hU hw lo hi hle z hz
-    (fun i => min (τ i ω) (χ ω)) (fun t => Q t ω) hQ0 hQd
-  intro i t ht
-  have htτ : t ≤ (τ i ω : ℝ) := ht.2.trans (by exact_mod_cast min_le_left (τ i ω) (χ ω))
-  have htχ : t ≤ (χ ω : ℝ) := ht.2.trans (by exact_mod_cast min_le_right (τ i ω) (χ ω))
-  have hh := hheight t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr htχ) i
-  simpa only [stoppedCoordinateState, Real.coe_toNNReal t ht.1, min_eq_left htτ] using hh
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointClippedState
+import OAI.Probability.SLE.Multipoint.MultipointStoppedCoordinates
+import OAI.Probability.SLE.FiniteRadiusStop
+
+namespace OAI
+
+/-! Safe clipped coordinates identify the actual Loewner flow separately
+up to each point's completion or the common localization time. -/
+
+noncomputable section
+open Set
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem clipped_multipoint_identification {ι : Type*} [Fintype ι]
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (lo hi : ℝ) (hle : lo ≤ hi) (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (T : ι → ℝ≥0) (Q : ℝ → ι → ℂ × ℝ)
+    (hQ0 : ∀ i, Q 0 i = initialLogCoordinates (z i))
+    (hQd : ∀ t, 0 ≤ t → HasDerivWithinAt Q
+      (fun i => clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t i)) (Ici 0) t)
+    (hheight : ∀ i t, t ∈ Icc 0 (T i : ℝ) → (Q t i).1.im ∈ Icc lo hi) :
+    ∀ i, z i ∈ survivingDomain γ (T i) ∧
+      EqOn (fun t => Q t i) (logLoewnerCoordinates G (z i)) (Icc 0 (T i : ℝ)) := by
+  intro i
+  apply clipped_log_identification hU hw (z i) (hz i) (T i) lo hi hle
+    (fun t => Q t i) (hQ0 i)
+  · intro t ht
+    exact ((hasDerivWithinAt_pi.mp (hQd t ht.1)) i).mono (fun _ hs => hs.1)
+  · exact hheight i
+
+theorem stopped_height_gives_local_identification
+    {Ω ι : Type*} [Fintype ι]
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (lo hi : ℝ) (hle : lo ≤ hi) (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (χ : Ω → ℝ≥0) (ω : Ω)
+    (hQ0 : ∀ i, Q 0 ω i = initialLogCoordinates (z i))
+    (hQd : ∀ t, 0 ≤ t → HasDerivWithinAt (fun t => Q t ω)
+      (fun i => clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t ω i)) (Ici 0) t)
+    (hheight : ∀ (t : ℝ≥0), t ≤ χ ω → ∀ i,
+      (stoppedCoordinateState Q τ t ω i).1.im ∈ Icc lo hi) :
+    ∀ i, z i ∈ survivingDomain γ (min (τ i ω) (χ ω)) ∧
+      EqOn (fun t => Q t ω i) (logLoewnerCoordinates G (z i))
+        (Icc 0 (min (τ i ω) (χ ω) : ℝ≥0)) := by
+  apply clipped_multipoint_identification hU hw lo hi hle z hz
+    (fun i => min (τ i ω) (χ ω)) (fun t => Q t ω) hQ0 hQd
+  intro i t ht
+  have htτ : t ≤ (τ i ω : ℝ) := ht.2.trans (by exact_mod_cast min_le_left (τ i ω) (χ ω))
+  have htχ : t ≤ (χ ω : ℝ) := ht.2.trans (by exact_mod_cast min_le_right (τ i ω) (χ ω))
+  have hh := hheight t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr htχ) i
+  simpa only [stoppedCoordinateState, Real.coe_toNNReal t ht.1, min_eq_left htτ] using hh
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalization.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalization.lean
index 59184839f082f7a6dc53d70b549fb63091bedc62..a5f5dae11da8d01d6f49955df4016d5832470f5f 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalization.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalization.lean
@@ -1,152 +1,152 @@
-import OAI.Probability.SLE.FiniteRadiusStop
-import OAI.Probability.SLE.Multipoint.MultipointStoppedCoordinates
-
-namespace OAI
-
-/-! A finite localization for the completion-frozen logarithmic coordinates.
-Only the log heights and the real driver are cut off; the smooth factors
-need no angular cutoff. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def localizationMarginTerm {ι : Type*} (lo hi M : ℝ)
-    (q : ι → ℂ × ℝ) (b : ℝ) : Option ι → ℝ
-  | none => M-|b|
-  | some i => min ((q i).1.im-lo) (hi-(q i).1.im)
-
-/-- The extra `none` index is the driver margin, so empty point families
-are allowed without an artificial height condition. -/
-def localizationMargin {ι : Type*} [Fintype ι] (lo hi M : ℝ)
-    (q : ι → ℂ × ℝ) (b : ℝ) : ℝ :=
-  Finset.univ.inf' Finset.univ_nonempty (localizationMarginTerm lo hi M q b)
-
-lemma localizationMargin_nonneg_iff {ι : Type*} [Fintype ι]
-    (lo hi M : ℝ) (q : ι → ℂ × ℝ) (b : ℝ) :
-    0 ≤ localizationMargin lo hi M q b ↔
-      (∀ i, lo ≤ (q i).1.im ∧ (q i).1.im ≤ hi) ∧ |b| ≤ M := by
-  classical
-  rw [localizationMargin, Finset.le_inf'_iff]
-  constructor
-  · intro h
-    refine ⟨fun i => ?_, ?_⟩
-    · have hi := h (some i) (Finset.mem_univ _)
-      simpa only [localizationMarginTerm, le_min_iff, sub_nonneg] using hi
-    · exact sub_nonneg.mp (h none (Finset.mem_univ _))
-  · rintro ⟨hq,hb⟩ i _
-    cases i with
-    | none => exact sub_nonneg.mpr hb
-    | some i =>
-      exact le_min (sub_nonneg.mpr (hq i).1) (sub_nonneg.mpr (hq i).2)
-
-lemma localizationMargin_pos_iff {ι : Type*} [Fintype ι]
-    (lo hi M : ℝ) (q : ι → ℂ × ℝ) (b : ℝ) :
-    0 < localizationMargin lo hi M q b ↔
-      (∀ i, lo < (q i).1.im ∧ (q i).1.im < hi) ∧ |b| < M := by
-  classical
-  rw [localizationMargin, Finset.lt_inf'_iff]
-  constructor
-  · intro h
-    refine ⟨fun i => ?_, ?_⟩
-    · have hi := h (some i) (Finset.mem_univ _)
-      simpa only [localizationMarginTerm, lt_min_iff, sub_pos] using hi
-    · exact sub_pos.mp (h none (Finset.mem_univ _))
-  · rintro ⟨hq,hb⟩ i _
-    cases i with
-    | none => exact sub_pos.mpr hb
-    | some i =>
-      exact lt_min (sub_pos.mpr (hq i).1) (sub_pos.mpr (hq i).2)
-
-lemma localizationMargin_continuous {ι : Type*} [Fintype ι]
-    (lo hi M : ℝ) :
-    Continuous (fun p : (ι → ℂ × ℝ) × ℝ => localizationMargin lo hi M p.1 p.2) := by
-  classical
-  unfold localizationMargin
-  apply Continuous.finset_inf'_apply
-  intro i _
-  cases i with
-  | none => exact continuous_const.sub continuous_snd.abs
-  | some i =>
-    have hc : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => (p.1 i).1.im) :=
-      Complex.continuous_im.comp (((continuous_apply i).comp continuous_fst).fst)
-    exact (hc.sub continuous_const).min (continuous_const.sub hc)
-
-lemma localizationMargin_boundary {ι : Type*} [Fintype ι]
-    (lo hi M : ℝ) (q : ι → ℂ × ℝ) (b : ℝ)
-    (hn : 0 ≤ localizationMargin lo hi M q b)
-    (hz : localizationMargin lo hi M q b = 0) :
-    (∃ i, (q i).1.im = lo ∨ (q i).1.im = hi) ∨ |b| = M := by
-  classical
-  obtain ⟨hq,hb⟩ := (localizationMargin_nonneg_iff lo hi M q b).mp hn
-  have hzle : localizationMargin lo hi M q b ≤ 0 := hz.le
-  rw [localizationMargin, Finset.inf'_le_iff] at hzle
-  obtain ⟨i, _, hterm⟩ := hzle
-  cases i with
-  | none =>
-    right
-    change M-|b| ≤ 0 at hterm
-    linarith
-  | some i =>
-    left
-    refine ⟨i, ?_⟩
-    change min ((q i).1.im-lo) (hi-(q i).1.im) ≤ 0 at hterm
-    rcases min_le_iff.mp hterm with hl | hh
-    · left
-      linarith [(hq i).1]
-    · right
-      linarith [(hq i).2]
-
-/-- Stop the continuous adapted state at the first height or driver
-boundary, capped at the deterministic horizon. -/
-theorem exists_multipoint_localization
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (ℱ : Filtration ℝ≥0 mΩ)
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (hQa : Adapted ℱ (fun t : ℝ≥0 => Q t))
-    (hQc : ∀ ω, Continuous (fun t : ℝ => Q t ω))
-    (hBa : Adapted ℱ B) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (lo hi M : ℝ) (H : ℝ≥0)
-    (hQ0 : ∀ ω i, lo ≤ (Q 0 ω i).1.im ∧ (Q 0 ω i).1.im ≤ hi)
-    (hB0 : ∀ ω, |B 0 ω| ≤ M) :
-    ∃ χ : Ω → ℝ≥0,
-      IsStoppingTime ℱ (fun ω => (χ ω : WithTop ℝ≥0)) ∧
-      (∀ ω, χ ω ≤ H) ∧
-      (∀ ω t, t ≤ χ ω →
-        (∀ i, lo ≤ (Q t ω i).1.im ∧ (Q t ω i).1.im ≤ hi) ∧ |B t ω| ≤ M) ∧
-      (∀ ω t, t < χ ω →
-        (∀ i, lo < (Q t ω i).1.im ∧ (Q t ω i).1.im < hi) ∧ |B t ω| < M) ∧
-      (∀ ω, χ ω < H →
-        (∃ i, (Q (χ ω) ω i).1.im = lo ∨ (Q (χ ω) ω i).1.im = hi) ∨
-          |B (χ ω) ω| = M) := by
-  let D := fun (t : ℝ≥0) ω => localizationMargin lo hi M (Q t ω) (B t ω)
-  have hmc : Continuous (fun p : (ι → ℂ × ℝ) × ℝ =>
-      localizationMargin lo hi M p.1 p.2) :=
-    localizationMargin_continuous (ι := ι) lo hi M
-  have hDa : Adapted ℱ D := by
-    intro t
-    change Measurable[ℱ t] ((fun p : (ι → ℂ × ℝ) × ℝ =>
-      localizationMargin lo hi M p.1 p.2) ∘ (fun ω => (Q t ω,B t ω)))
-    exact hmc.measurable.comp ((hQa t).prodMk (hBa t))
-  have hDc (ω : Ω) : Continuous (fun t => D t ω) :=
-    hmc.comp
-      (((hQc ω).comp NNReal.continuous_coe).prodMk (hBc ω))
-  have hD0 (ω : Ω) : 0 ≤ D 0 ω :=
-    (localizationMargin_nonneg_iff lo hi M (Q 0 ω) (B 0 ω)).mpr ⟨hQ0 ω,hB0 ω⟩
-  obtain ⟨χ,hχ,hH,hn,hp,hz⟩ := first_margin_stopping ℱ D hDa hDc hD0 H
-  refine ⟨χ,hχ,hH,?_,?_,?_⟩
-  · intro ω t ht
-    exact (localizationMargin_nonneg_iff lo hi M (Q t ω) (B t ω)).mp (hn ω t ht)
-  · intro ω t ht
-    exact (localizationMargin_pos_iff lo hi M (Q t ω) (B t ω)).mp (hp ω t ht)
-  · intro ω ht
-    exact localizationMargin_boundary lo hi M (Q (χ ω) ω) (B (χ ω) ω)
-      (hn ω (χ ω) le_rfl) (hz ω ht)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.FiniteRadiusStop
+import OAI.Probability.SLE.Multipoint.MultipointStoppedCoordinates
+
+namespace OAI
+
+/-! A finite localization for the completion-frozen logarithmic coordinates.
+Only the log heights and the real driver are cut off; the smooth factors
+need no angular cutoff. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def localizationMarginTerm {ι : Type*} (lo hi M : ℝ)
+    (q : ι → ℂ × ℝ) (b : ℝ) : Option ι → ℝ
+  | none => M-|b|
+  | some i => min ((q i).1.im-lo) (hi-(q i).1.im)
+
+/-- The extra `none` index is the driver margin, so empty point families
+are allowed without an artificial height condition. -/
+def localizationMargin {ι : Type*} [Fintype ι] (lo hi M : ℝ)
+    (q : ι → ℂ × ℝ) (b : ℝ) : ℝ :=
+  Finset.univ.inf' Finset.univ_nonempty (localizationMarginTerm lo hi M q b)
+
+lemma localizationMargin_nonneg_iff {ι : Type*} [Fintype ι]
+    (lo hi M : ℝ) (q : ι → ℂ × ℝ) (b : ℝ) :
+    0 ≤ localizationMargin lo hi M q b ↔
+      (∀ i, lo ≤ (q i).1.im ∧ (q i).1.im ≤ hi) ∧ |b| ≤ M := by
+  classical
+  rw [localizationMargin, Finset.le_inf'_iff]
+  constructor
+  · intro h
+    refine ⟨fun i => ?_, ?_⟩
+    · have hi := h (some i) (Finset.mem_univ _)
+      simpa only [localizationMarginTerm, le_min_iff, sub_nonneg] using hi
+    · exact sub_nonneg.mp (h none (Finset.mem_univ _))
+  · rintro ⟨hq,hb⟩ i _
+    cases i with
+    | none => exact sub_nonneg.mpr hb
+    | some i =>
+      exact le_min (sub_nonneg.mpr (hq i).1) (sub_nonneg.mpr (hq i).2)
+
+lemma localizationMargin_pos_iff {ι : Type*} [Fintype ι]
+    (lo hi M : ℝ) (q : ι → ℂ × ℝ) (b : ℝ) :
+    0 < localizationMargin lo hi M q b ↔
+      (∀ i, lo < (q i).1.im ∧ (q i).1.im < hi) ∧ |b| < M := by
+  classical
+  rw [localizationMargin, Finset.lt_inf'_iff]
+  constructor
+  · intro h
+    refine ⟨fun i => ?_, ?_⟩
+    · have hi := h (some i) (Finset.mem_univ _)
+      simpa only [localizationMarginTerm, lt_min_iff, sub_pos] using hi
+    · exact sub_pos.mp (h none (Finset.mem_univ _))
+  · rintro ⟨hq,hb⟩ i _
+    cases i with
+    | none => exact sub_pos.mpr hb
+    | some i =>
+      exact lt_min (sub_pos.mpr (hq i).1) (sub_pos.mpr (hq i).2)
+
+lemma localizationMargin_continuous {ι : Type*} [Fintype ι]
+    (lo hi M : ℝ) :
+    Continuous (fun p : (ι → ℂ × ℝ) × ℝ => localizationMargin lo hi M p.1 p.2) := by
+  classical
+  unfold localizationMargin
+  apply Continuous.finset_inf'_apply
+  intro i _
+  cases i with
+  | none => exact continuous_const.sub continuous_snd.abs
+  | some i =>
+    have hc : Continuous (fun p : (ι → ℂ × ℝ) × ℝ => (p.1 i).1.im) :=
+      Complex.continuous_im.comp (((continuous_apply i).comp continuous_fst).fst)
+    exact (hc.sub continuous_const).min (continuous_const.sub hc)
+
+lemma localizationMargin_boundary {ι : Type*} [Fintype ι]
+    (lo hi M : ℝ) (q : ι → ℂ × ℝ) (b : ℝ)
+    (hn : 0 ≤ localizationMargin lo hi M q b)
+    (hz : localizationMargin lo hi M q b = 0) :
+    (∃ i, (q i).1.im = lo ∨ (q i).1.im = hi) ∨ |b| = M := by
+  classical
+  obtain ⟨hq,hb⟩ := (localizationMargin_nonneg_iff lo hi M q b).mp hn
+  have hzle : localizationMargin lo hi M q b ≤ 0 := hz.le
+  rw [localizationMargin, Finset.inf'_le_iff] at hzle
+  obtain ⟨i, _, hterm⟩ := hzle
+  cases i with
+  | none =>
+    right
+    change M-|b| ≤ 0 at hterm
+    linarith
+  | some i =>
+    left
+    refine ⟨i, ?_⟩
+    change min ((q i).1.im-lo) (hi-(q i).1.im) ≤ 0 at hterm
+    rcases min_le_iff.mp hterm with hl | hh
+    · left
+      linarith [(hq i).1]
+    · right
+      linarith [(hq i).2]
+
+/-- Stop the continuous adapted state at the first height or driver
+boundary, capped at the deterministic horizon. -/
+theorem exists_multipoint_localization
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (ℱ : Filtration ℝ≥0 mΩ)
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (hQa : Adapted ℱ (fun t : ℝ≥0 => Q t))
+    (hQc : ∀ ω, Continuous (fun t : ℝ => Q t ω))
+    (hBa : Adapted ℱ B) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (lo hi M : ℝ) (H : ℝ≥0)
+    (hQ0 : ∀ ω i, lo ≤ (Q 0 ω i).1.im ∧ (Q 0 ω i).1.im ≤ hi)
+    (hB0 : ∀ ω, |B 0 ω| ≤ M) :
+    ∃ χ : Ω → ℝ≥0,
+      IsStoppingTime ℱ (fun ω => (χ ω : WithTop ℝ≥0)) ∧
+      (∀ ω, χ ω ≤ H) ∧
+      (∀ ω t, t ≤ χ ω →
+        (∀ i, lo ≤ (Q t ω i).1.im ∧ (Q t ω i).1.im ≤ hi) ∧ |B t ω| ≤ M) ∧
+      (∀ ω t, t < χ ω →
+        (∀ i, lo < (Q t ω i).1.im ∧ (Q t ω i).1.im < hi) ∧ |B t ω| < M) ∧
+      (∀ ω, χ ω < H →
+        (∃ i, (Q (χ ω) ω i).1.im = lo ∨ (Q (χ ω) ω i).1.im = hi) ∨
+          |B (χ ω) ω| = M) := by
+  let D := fun (t : ℝ≥0) ω => localizationMargin lo hi M (Q t ω) (B t ω)
+  have hmc : Continuous (fun p : (ι → ℂ × ℝ) × ℝ =>
+      localizationMargin lo hi M p.1 p.2) :=
+    localizationMargin_continuous (ι := ι) lo hi M
+  have hDa : Adapted ℱ D := by
+    intro t
+    change Measurable[ℱ t] ((fun p : (ι → ℂ × ℝ) × ℝ =>
+      localizationMargin lo hi M p.1 p.2) ∘ (fun ω => (Q t ω,B t ω)))
+    exact hmc.measurable.comp ((hQa t).prodMk (hBa t))
+  have hDc (ω : Ω) : Continuous (fun t => D t ω) :=
+    hmc.comp
+      (((hQc ω).comp NNReal.continuous_coe).prodMk (hBc ω))
+  have hD0 (ω : Ω) : 0 ≤ D 0 ω :=
+    (localizationMargin_nonneg_iff lo hi M (Q 0 ω) (B 0 ω)).mpr ⟨hQ0 ω,hB0 ω⟩
+  obtain ⟨χ,hχ,hH,hn,hp,hz⟩ := first_margin_stopping ℱ D hDa hDc hD0 H
+  refine ⟨χ,hχ,hH,?_,?_,?_⟩
+  · intro ω t ht
+    exact (localizationMargin_nonneg_iff lo hi M (Q t ω) (B t ω)).mp (hn ω t ht)
+  · intro ω t ht
+    exact (localizationMargin_pos_iff lo hi M (Q t ω) (B t ω)).mp (hp ω t ht)
+  · intro ω ht
+    exact localizationMargin_boundary lo hi M (Q (χ ω) ω) (B (χ ω) ω)
+      (hn ω (χ ω) le_rfl) (hz ω ht)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedExpectation.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedExpectation.lean
index ea06da96b47d9b5eb16db7f6f679b04973e0f7c7..94b189dc64844fae23855943ab0cbbed06e93377 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedExpectation.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedExpectation.lean
@@ -1,183 +1,183 @@
-import OAI.Probability.SLE.Multipoint.MultipointLocalizedStages
-import OAI.Probability.SLE.Multipoint.MultipointPatternExpectation
-
-namespace OAI
-
-/-! Optional stopping for the complete four-phase product, including
-simultaneous switches and a separate random localization time. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem localized_pattern_expectation_le
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (Q V : ℝ → Ω → (ι → ℂ × ℝ)) (L : ℝ≥0)
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q t))
-    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
-    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
-    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
-    (r : ℝ≥0 → Ω → ℝ) (hrm : Measurable (Function.uncurry r))
-    (hrp : IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) r)
-    (C : ℝ≥0) (hrb : ∀ t ω, |r t ω| ≤ C)
-    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hβ : 0 ≤ β) (hβ1 : β ≤ 1)
-    (a U : ι → ℝ) (hU : ∀ i, 0 ≤ U i)
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hlstop : ∀ σ ∈ l, IsStoppingTime
-      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hlbound : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρmem : ∀ k, ρ k ∈ l)
-    (hρorder : ∀ ω i, Monotone (fun k => ρ (i,k) ω))
-    (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
-    (χ : Ω → ℝ≥0)
-    (hχstop : IsStoppingTime
-      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (χ ω : WithTop ℝ≥0))) (hχbound : ∀ ω, χ ω ≤ H)
-    (R : ℝ) (hR : 0 ≤ R)
-    (hnorm : ∀ ω t, 0 ≤ t → t ≤ (χ ω : ℝ) →
-      ‖((Q t ω,discountClock r t ω),B t.toNNReal ω)‖ ≤ R)
-    (hδ : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i ∈ patternActiveSet (phaseAt ρ t ω),
-      branchRegulator κ (phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
-        (phaseRegulatorSlope β (phaseAt ρ t ω i)) (Q t ω i) ≤ 1)
-    (hV : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i ∈ patternActiveSet (phaseAt ρ t ω),
-      V t ω i = physicalLogLoewnerField (Real.sqrt κ*B t ω) (Q t ω i))
-    (hr : ∀ ω (t : ℝ≥0), t < χ ω → r t ω =
-      productDriftCost (patternActiveSet (phaseAt ρ t ω)) κ
-        (fun i => phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
-        (fun i => phaseRegulatorSlope β (phaseAt ρ t ω i)) (Q t ω) (B t ω))
-    (hclock : ∀ ω i (k : Fin 3), ρ (i,k) ω ≤ χ ω →
-      logFactorClock κ (Q (ρ (i,k) ω) ω i)-a i =
-        if k = 0 then 0 else if k = 1 then U i/2 else U i) :
-    Integrable (fun ω => discountedPatternProduct κ β a U (phaseAt ρ (χ ω) ω)
-      (Q (χ ω) ω,discountClock r (χ ω) ω) (B (χ ω) ω)) P ∧
-    (∫ ω, discountedPatternProduct κ β a U (phaseAt ρ (χ ω) ω)
-      (Q (χ ω) ω,discountClock r (χ ω) ω) (B (χ ω) ω) ∂P) ≤
-      ∫ ω, discountedPatternProduct κ β a U (phaseAt ρ 0 ω)
-        (Q 0 ω,discountClock r 0 ω) (B 0 ω) ∂P := by
-  classical
-  let τ := localizedStage l H χ
-  let j := fun n ω => phaseAt ρ (τ n ω) ω
-  let X := fun n ω => discountedPatternProduct κ β a U (j n ω)
-    (Q (τ n ω) ω,discountClock r (τ n ω) ω) (B (τ n ω) ω)
-  let Y := fun n ω => discountedPatternProduct κ β a U (j n ω)
-    (Q (τ (n+1) ω) ω,discountClock r (τ (n+1) ω) ω) (B (τ (n+1) ω) ω)
-  have hτ (n : ℕ) := localizedStage_stopping _ l hlstop H χ hχstop n
-  have hmono (ω : Ω) := localizedStage_monotone l H χ hlbound ω
-  have hρstop (k : ι × Fin 3) := hlstop (ρ k) (hρmem k)
-  have hstep (n : ℕ) : Integrable (Y n) P ∧ Integrable (X n) P ∧
-      (∫ ω, Y n ω ∂P) ≤ ∫ ω, X n ω ∂P := by
-    let F := fun (A : Finset (ι × Fin 3)) ω =>
-      discountedPatternProduct κ β a U (phaseOfPattern A)
-        (Q (τ (n+1) ω) ω,discountClock r (τ (n+1) ω) ω) (B (τ (n+1) ω) ω)
-    let G := fun (A : Finset (ι × Fin 3)) ω =>
-      discountedPatternProduct κ β a U (phaseOfPattern A)
-        (Q (τ n ω) ω,discountClock r (τ n ω) ω) (B (τ n ω) ω)
-    have hb (A : Finset (ι × Fin 3)) :
-        IntegrableOn (F A) (switchPatternEvent Finset.univ ρ (τ n) A) P ∧
-        IntegrableOn (G A) (switchPatternEvent Finset.univ ρ (τ n) A) P ∧
-        (∫ ω in switchPatternEvent Finset.univ ρ (τ n) A, F A ω ∂P) ≤
-          ∫ ω in switchPatternEvent Finset.univ ρ (τ n) A, G A ω ∂P := by
-      let s := patternActiveSet (phaseOfPattern A)
-      let D := fun i => phaseRegulatorScale β (a i) (U i) (phaseOfPattern A i)
-      let σ := fun i => phaseRegulatorSlope β (phaseOfPattern A i)
-      have hD (i) (_ : i ∈ s) : 0 < D i := phaseRegulatorScale_pos _ _ _ _
-      obtain ⟨φ,K,_,he,heD,he2,hfield⟩ := compact_discounted_smooth_field B hm
-        Q V L hQm hQa hQl hVm hVb hQd r hrm hrp C hrb s κ D σ hD R hR
-      have hphase (ω : Ω) (hω : ω ∈ switchPatternEvent Finset.univ ρ (τ n) A)
-          (u : ℝ) (hu : (τ n ω : ℝ) ≤ u) (hu' : u < (τ (n+1) ω : ℝ)) :
-          phaseAt ρ u.toNNReal ω = phaseOfPattern A := by
-        have hu0 := (τ n ω).coe_nonneg.trans hu
-        have ht : τ n ω ≤ u.toNNReal := by exact_mod_cast (show (τ n ω : ℝ) ≤
-          (u.toNNReal : ℝ) by simpa only [Real.coe_toNNReal u hu0] using hu)
-        have ht' : u.toNNReal < τ (n+1) ω := by exact_mod_cast (show
-          (u.toNNReal : ℝ) < (τ (n+1) ω : ℝ) by simpa only [Real.coe_toNNReal u hu0] using hu')
-        exact (phaseAt_const_localized_interval ρ l H χ hlbound hρmem n ω _ ht ht').trans
-          (phaseAt_of_pattern ρ (τ n) A hω)
-      have huχ (ω : Ω) (u : ℝ) (hu : (τ n ω : ℝ) ≤ u)
-          (hu' : u < (τ (n+1) ω : ℝ)) : u.toNNReal < χ ω := by
-        have hu0 := (τ n ω).coe_nonneg.trans hu
-        have hlt : (u.toNNReal : ℝ) < (χ ω : ℝ) := by
-          rw [Real.coe_toNNReal u hu0]
-          exact hu'.trans_le (by exact_mod_cast localizedStage_le l H χ (n+1) ω)
-        exact_mod_cast hlt
-      have hb := discounted_product_optional_interval_le hB hm hBm hBc Q V r s κ hκ hκ8
-        D σ hD (fun i _ => phaseRegulatorSlope_le_one hβ hβ1 _) φ K hfield R he heD he2
-        (τ n) (τ (n+1)) (hτ n) (hτ (n+1)) (fun ω => hmono ω (Nat.le_succ n)) H
-        (fun ω => (localizedStage_le l H χ (n+1) ω).trans (hχbound ω))
-        (switchPatternEvent_measurable _ Finset.univ ρ (τ n) (fun i _ => hρstop i) (hτ n) A)
-        (fun ω _ u hu hu' => hnorm ω u ((τ n ω).coe_nonneg.trans hu)
-          (hu'.trans (by exact_mod_cast localizedStage_le l H χ (n+1) ω)))
-        (fun ω hω u hu hu' => by
-          have hh := hδ ω u.toNNReal (huχ ω u hu hu')
-          rw [hphase ω hω u hu hu'] at hh
-          simpa only [Real.coe_toNNReal u ((τ n ω).coe_nonneg.trans hu)] using hh)
-        (fun ω hω u hu hu' => by
-          have hh := hV ω u.toNNReal (huχ ω u hu hu')
-          rw [hphase ω hω u hu hu'] at hh
-          simpa only [Real.coe_toNNReal u ((τ n ω).coe_nonneg.trans hu)] using hh)
-        (fun ω hω u hu hu' => by
-          have hh := hr ω u.toNNReal (huχ ω u hu hu')
-          rw [hphase ω hω u hu hu'] at hh
-          simpa only [Real.coe_toNNReal u ((τ n ω).coe_nonneg.trans hu)] using hh)
-      refine ⟨?_,?_,?_⟩
-      · simpa only [F,discountedPatternProduct_eq,s,D,σ,IntegrableOn] using hb.1.const_mul (patternWeight κ a U (phaseOfPattern A))
-      · simpa only [G,discountedPatternProduct_eq,s,D,σ,IntegrableOn] using hb.2.1.const_mul (patternWeight κ a U (phaseOfPattern A))
-      · simpa only [F,G,discountedPatternProduct_eq,integral_const_mul,s,D,σ] using
-          mul_le_mul_of_nonneg_left hb.2.2 (patternWeight_pos κ a U (phaseOfPattern A)).le
-    apply switchPattern_integral_le P Finset.univ ρ (τ n)
-      (fun A _ => (hτ n).measurableSpace_le _
-        (switchPatternEvent_measurable _ Finset.univ ρ (τ n) (fun i _ => hρstop i) (hτ n) A)) F G (Y n) (X n)
-      (fun A _ => (hb A).1) (fun A _ => (hb A).2.1)
-    · intro A _ ω hω
-      change discountedPatternProduct κ β a U (phaseOfPattern A) _ _ =
-        discountedPatternProduct κ β a U (phaseAt ρ (τ n ω) ω) _ _
-      rw [phaseAt_of_pattern ρ (τ n) A hω]
-    · intro A _ ω hω
-      change discountedPatternProduct κ β a U (phaseOfPattern A) _ _ =
-        discountedPatternProduct κ β a U (phaseAt ρ (τ n ω) ω) _ _
-      rw [phaseAt_of_pattern ρ (τ n) A hω]
-    · exact fun A _ => (hb A).2.2
-  have hjump (n : ℕ) (ω : Ω) : X (n+1) ω ≤ Y n ω := by
-    apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-    apply patternProduct_switch_le κ β hκ hκ8 a U hU _ _ _ _
-      (fun i => phaseAt_monotone ρ ω i (hmono ω (Nat.le_succ n)))
-    · intro i hbefore hafter
-      have hc := phaseAt_crossed_localized_switch ρ l H χ hlbound hρmem n ω i
-        (0: Fin 3) (hρorder ω i) hbefore hafter
-      have hh := hclock ω i (0: Fin 3) (hc ▸ localizedStage_le l H χ (n+1) ω)
-      rw [hc] at hh
-      simpa only [ite_true, ite_false, Fin.isValue, Fin.reduceEq] using hh
-    · intro i hbefore hafter
-      have hc := phaseAt_crossed_localized_switch ρ l H χ hlbound hρmem n ω i
-        (1: Fin 3) (hρorder ω i) hbefore hafter
-      have hh := hclock ω i (1: Fin 3) (hc ▸ localizedStage_le l H χ (n+1) ω)
-      rw [hc] at hh
-      simpa only [ite_true, ite_false, Fin.isValue, Fin.reduceEq] using hh
-    · intro i hbefore hafter
-      have hc := phaseAt_crossed_localized_switch ρ l H χ hlbound hρmem n ω i
-        (2: Fin 3) (hρorder ω i) hbefore hafter
-      have hh := hclock ω i (2: Fin 3) (hc ▸ localizedStage_le l H χ (n+1) ω)
-      rw [hc] at hh
-      simpa only [ite_true, ite_false, Fin.isValue, Fin.reduceEq] using hh
-  have hout := finite_switched_product_expectation_le P l.length X Y
-    (fun n _ => (hstep n).2.1) (fun n _ => (hstep n).1)
-    (fun n _ => (hstep n).2.2) (fun n _ => hjump n)
-  have hlast : ∀ ω, τ l.length ω = χ ω := localizedStage_last l H χ hχbound
-  have hfirst : ∀ ω, τ 0 ω = 0 := localizedStage_zero l H χ hlbound hzero
-  refine ⟨?_,?_⟩
-  · simpa only [X,j,hlast] using (hstep l.length).2.1
-  · simpa only [X,j,hlast,hfirst,NNReal.coe_zero] using hout
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLocalizedStages
+import OAI.Probability.SLE.Multipoint.MultipointPatternExpectation
+
+namespace OAI
+
+/-! Optional stopping for the complete four-phase product, including
+simultaneous switches and a separate random localization time. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem localized_pattern_expectation_le
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (Q V : ℝ → Ω → (ι → ℂ × ℝ)) (L : ℝ≥0)
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q t))
+    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
+    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
+    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
+    (r : ℝ≥0 → Ω → ℝ) (hrm : Measurable (Function.uncurry r))
+    (hrp : IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) r)
+    (C : ℝ≥0) (hrb : ∀ t ω, |r t ω| ≤ C)
+    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hβ : 0 ≤ β) (hβ1 : β ≤ 1)
+    (a U : ι → ℝ) (hU : ∀ i, 0 ≤ U i)
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hlstop : ∀ σ ∈ l, IsStoppingTime
+      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hlbound : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρmem : ∀ k, ρ k ∈ l)
+    (hρorder : ∀ ω i, Monotone (fun k => ρ (i,k) ω))
+    (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
+    (χ : Ω → ℝ≥0)
+    (hχstop : IsStoppingTime
+      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (χ ω : WithTop ℝ≥0))) (hχbound : ∀ ω, χ ω ≤ H)
+    (R : ℝ) (hR : 0 ≤ R)
+    (hnorm : ∀ ω t, 0 ≤ t → t ≤ (χ ω : ℝ) →
+      ‖((Q t ω,discountClock r t ω),B t.toNNReal ω)‖ ≤ R)
+    (hδ : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i ∈ patternActiveSet (phaseAt ρ t ω),
+      branchRegulator κ (phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
+        (phaseRegulatorSlope β (phaseAt ρ t ω i)) (Q t ω i) ≤ 1)
+    (hV : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i ∈ patternActiveSet (phaseAt ρ t ω),
+      V t ω i = physicalLogLoewnerField (Real.sqrt κ*B t ω) (Q t ω i))
+    (hr : ∀ ω (t : ℝ≥0), t < χ ω → r t ω =
+      productDriftCost (patternActiveSet (phaseAt ρ t ω)) κ
+        (fun i => phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
+        (fun i => phaseRegulatorSlope β (phaseAt ρ t ω i)) (Q t ω) (B t ω))
+    (hclock : ∀ ω i (k : Fin 3), ρ (i,k) ω ≤ χ ω →
+      logFactorClock κ (Q (ρ (i,k) ω) ω i)-a i =
+        if k = 0 then 0 else if k = 1 then U i/2 else U i) :
+    Integrable (fun ω => discountedPatternProduct κ β a U (phaseAt ρ (χ ω) ω)
+      (Q (χ ω) ω,discountClock r (χ ω) ω) (B (χ ω) ω)) P ∧
+    (∫ ω, discountedPatternProduct κ β a U (phaseAt ρ (χ ω) ω)
+      (Q (χ ω) ω,discountClock r (χ ω) ω) (B (χ ω) ω) ∂P) ≤
+      ∫ ω, discountedPatternProduct κ β a U (phaseAt ρ 0 ω)
+        (Q 0 ω,discountClock r 0 ω) (B 0 ω) ∂P := by
+  classical
+  let τ := localizedStage l H χ
+  let j := fun n ω => phaseAt ρ (τ n ω) ω
+  let X := fun n ω => discountedPatternProduct κ β a U (j n ω)
+    (Q (τ n ω) ω,discountClock r (τ n ω) ω) (B (τ n ω) ω)
+  let Y := fun n ω => discountedPatternProduct κ β a U (j n ω)
+    (Q (τ (n+1) ω) ω,discountClock r (τ (n+1) ω) ω) (B (τ (n+1) ω) ω)
+  have hτ (n : ℕ) := localizedStage_stopping _ l hlstop H χ hχstop n
+  have hmono (ω : Ω) := localizedStage_monotone l H χ hlbound ω
+  have hρstop (k : ι × Fin 3) := hlstop (ρ k) (hρmem k)
+  have hstep (n : ℕ) : Integrable (Y n) P ∧ Integrable (X n) P ∧
+      (∫ ω, Y n ω ∂P) ≤ ∫ ω, X n ω ∂P := by
+    let F := fun (A : Finset (ι × Fin 3)) ω =>
+      discountedPatternProduct κ β a U (phaseOfPattern A)
+        (Q (τ (n+1) ω) ω,discountClock r (τ (n+1) ω) ω) (B (τ (n+1) ω) ω)
+    let G := fun (A : Finset (ι × Fin 3)) ω =>
+      discountedPatternProduct κ β a U (phaseOfPattern A)
+        (Q (τ n ω) ω,discountClock r (τ n ω) ω) (B (τ n ω) ω)
+    have hb (A : Finset (ι × Fin 3)) :
+        IntegrableOn (F A) (switchPatternEvent Finset.univ ρ (τ n) A) P ∧
+        IntegrableOn (G A) (switchPatternEvent Finset.univ ρ (τ n) A) P ∧
+        (∫ ω in switchPatternEvent Finset.univ ρ (τ n) A, F A ω ∂P) ≤
+          ∫ ω in switchPatternEvent Finset.univ ρ (τ n) A, G A ω ∂P := by
+      let s := patternActiveSet (phaseOfPattern A)
+      let D := fun i => phaseRegulatorScale β (a i) (U i) (phaseOfPattern A i)
+      let σ := fun i => phaseRegulatorSlope β (phaseOfPattern A i)
+      have hD (i) (_ : i ∈ s) : 0 < D i := phaseRegulatorScale_pos _ _ _ _
+      obtain ⟨φ,K,_,he,heD,he2,hfield⟩ := compact_discounted_smooth_field B hm
+        Q V L hQm hQa hQl hVm hVb hQd r hrm hrp C hrb s κ D σ hD R hR
+      have hphase (ω : Ω) (hω : ω ∈ switchPatternEvent Finset.univ ρ (τ n) A)
+          (u : ℝ) (hu : (τ n ω : ℝ) ≤ u) (hu' : u < (τ (n+1) ω : ℝ)) :
+          phaseAt ρ u.toNNReal ω = phaseOfPattern A := by
+        have hu0 := (τ n ω).coe_nonneg.trans hu
+        have ht : τ n ω ≤ u.toNNReal := by exact_mod_cast (show (τ n ω : ℝ) ≤
+          (u.toNNReal : ℝ) by simpa only [Real.coe_toNNReal u hu0] using hu)
+        have ht' : u.toNNReal < τ (n+1) ω := by exact_mod_cast (show
+          (u.toNNReal : ℝ) < (τ (n+1) ω : ℝ) by simpa only [Real.coe_toNNReal u hu0] using hu')
+        exact (phaseAt_const_localized_interval ρ l H χ hlbound hρmem n ω _ ht ht').trans
+          (phaseAt_of_pattern ρ (τ n) A hω)
+      have huχ (ω : Ω) (u : ℝ) (hu : (τ n ω : ℝ) ≤ u)
+          (hu' : u < (τ (n+1) ω : ℝ)) : u.toNNReal < χ ω := by
+        have hu0 := (τ n ω).coe_nonneg.trans hu
+        have hlt : (u.toNNReal : ℝ) < (χ ω : ℝ) := by
+          rw [Real.coe_toNNReal u hu0]
+          exact hu'.trans_le (by exact_mod_cast localizedStage_le l H χ (n+1) ω)
+        exact_mod_cast hlt
+      have hb := discounted_product_optional_interval_le hB hm hBm hBc Q V r s κ hκ hκ8
+        D σ hD (fun i _ => phaseRegulatorSlope_le_one hβ hβ1 _) φ K hfield R he heD he2
+        (τ n) (τ (n+1)) (hτ n) (hτ (n+1)) (fun ω => hmono ω (Nat.le_succ n)) H
+        (fun ω => (localizedStage_le l H χ (n+1) ω).trans (hχbound ω))
+        (switchPatternEvent_measurable _ Finset.univ ρ (τ n) (fun i _ => hρstop i) (hτ n) A)
+        (fun ω _ u hu hu' => hnorm ω u ((τ n ω).coe_nonneg.trans hu)
+          (hu'.trans (by exact_mod_cast localizedStage_le l H χ (n+1) ω)))
+        (fun ω hω u hu hu' => by
+          have hh := hδ ω u.toNNReal (huχ ω u hu hu')
+          rw [hphase ω hω u hu hu'] at hh
+          simpa only [Real.coe_toNNReal u ((τ n ω).coe_nonneg.trans hu)] using hh)
+        (fun ω hω u hu hu' => by
+          have hh := hV ω u.toNNReal (huχ ω u hu hu')
+          rw [hphase ω hω u hu hu'] at hh
+          simpa only [Real.coe_toNNReal u ((τ n ω).coe_nonneg.trans hu)] using hh)
+        (fun ω hω u hu hu' => by
+          have hh := hr ω u.toNNReal (huχ ω u hu hu')
+          rw [hphase ω hω u hu hu'] at hh
+          simpa only [Real.coe_toNNReal u ((τ n ω).coe_nonneg.trans hu)] using hh)
+      refine ⟨?_,?_,?_⟩
+      · simpa only [F,discountedPatternProduct_eq,s,D,σ,IntegrableOn] using hb.1.const_mul (patternWeight κ a U (phaseOfPattern A))
+      · simpa only [G,discountedPatternProduct_eq,s,D,σ,IntegrableOn] using hb.2.1.const_mul (patternWeight κ a U (phaseOfPattern A))
+      · simpa only [F,G,discountedPatternProduct_eq,integral_const_mul,s,D,σ] using
+          mul_le_mul_of_nonneg_left hb.2.2 (patternWeight_pos κ a U (phaseOfPattern A)).le
+    apply switchPattern_integral_le P Finset.univ ρ (τ n)
+      (fun A _ => (hτ n).measurableSpace_le _
+        (switchPatternEvent_measurable _ Finset.univ ρ (τ n) (fun i _ => hρstop i) (hτ n) A)) F G (Y n) (X n)
+      (fun A _ => (hb A).1) (fun A _ => (hb A).2.1)
+    · intro A _ ω hω
+      change discountedPatternProduct κ β a U (phaseOfPattern A) _ _ =
+        discountedPatternProduct κ β a U (phaseAt ρ (τ n ω) ω) _ _
+      rw [phaseAt_of_pattern ρ (τ n) A hω]
+    · intro A _ ω hω
+      change discountedPatternProduct κ β a U (phaseOfPattern A) _ _ =
+        discountedPatternProduct κ β a U (phaseAt ρ (τ n ω) ω) _ _
+      rw [phaseAt_of_pattern ρ (τ n) A hω]
+    · exact fun A _ => (hb A).2.2
+  have hjump (n : ℕ) (ω : Ω) : X (n+1) ω ≤ Y n ω := by
+    apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+    apply patternProduct_switch_le κ β hκ hκ8 a U hU _ _ _ _
+      (fun i => phaseAt_monotone ρ ω i (hmono ω (Nat.le_succ n)))
+    · intro i hbefore hafter
+      have hc := phaseAt_crossed_localized_switch ρ l H χ hlbound hρmem n ω i
+        (0: Fin 3) (hρorder ω i) hbefore hafter
+      have hh := hclock ω i (0: Fin 3) (hc ▸ localizedStage_le l H χ (n+1) ω)
+      rw [hc] at hh
+      simpa only [ite_true, ite_false, Fin.isValue, Fin.reduceEq] using hh
+    · intro i hbefore hafter
+      have hc := phaseAt_crossed_localized_switch ρ l H χ hlbound hρmem n ω i
+        (1: Fin 3) (hρorder ω i) hbefore hafter
+      have hh := hclock ω i (1: Fin 3) (hc ▸ localizedStage_le l H χ (n+1) ω)
+      rw [hc] at hh
+      simpa only [ite_true, ite_false, Fin.isValue, Fin.reduceEq] using hh
+    · intro i hbefore hafter
+      have hc := phaseAt_crossed_localized_switch ρ l H χ hlbound hρmem n ω i
+        (2: Fin 3) (hρorder ω i) hbefore hafter
+      have hh := hclock ω i (2: Fin 3) (hc ▸ localizedStage_le l H χ (n+1) ω)
+      rw [hc] at hh
+      simpa only [ite_true, ite_false, Fin.isValue, Fin.reduceEq] using hh
+  have hout := finite_switched_product_expectation_le P l.length X Y
+    (fun n _ => (hstep n).2.1) (fun n _ => (hstep n).1)
+    (fun n _ => (hstep n).2.2) (fun n _ => hjump n)
+  have hlast : ∀ ω, τ l.length ω = χ ω := localizedStage_last l H χ hχbound
+  have hfirst : ∀ ω, τ 0 ω = 0 := localizedStage_zero l H χ hlbound hzero
+  refine ⟨?_,?_⟩
+  · simpa only [X,j,hlast] using (hstep l.length).2.1
+  · simpa only [X,j,hlast,hfirst,NNReal.coe_zero] using hout
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedStages.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedStages.lean
index 6e3b4163dc97002c300733e3821d0ea2878e76cb..cd96e12ce2592a7260521f4736bbfcc851820a6a 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedStages.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedStages.lean
@@ -1,100 +1,100 @@
-import OAI.Probability.SLE.Multipoint.MultipointHorizonStages
-
-namespace OAI
-
-/-! Stop the integration at a bounded random localization time, while
-continuing to detect phases from the original radius switches. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def localizedStage {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) : ℝ≥0 :=
-  min (sortedStopAt l (H+1) n ω) (χ ω)
-
-lemma localizedStage_stopping {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (l : List (Ω → ℝ≥0))
-    (hl : ∀ τ ∈ l, IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    (H : ℝ≥0) (χ : Ω → ℝ≥0)
-    (hχ : IsStoppingTime ℱ (fun ω => (χ ω : WithTop ℝ≥0))) (n : ℕ) :
-    IsStoppingTime ℱ (fun ω => (localizedStage l H χ n ω : WithTop ℝ≥0)) := by
-  simpa only [localizedStage, WithTop.coe_min] using
-    (sortedStopAt_stopping ℱ l hl (H+1) n).min hχ
-
-lemma localizedStage_monotone {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (ω : Ω) :
-    Monotone (fun n => localizedStage l H χ n ω) := by
-  intro i j hij
-  exact min_le_min_right (χ ω) (sortedStopAt_monotone l (H+1) hl ω hij)
-
-lemma localizedStage_le {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) : localizedStage l H χ n ω ≤ χ ω :=
-  min_le_right _ _
-
-lemma localizedStage_last {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (hχ : ∀ ω, χ ω ≤ H) (ω : Ω) :
-    localizedStage l H χ l.length ω = χ ω := by
-  simp only [localizedStage, sortedStopAt_length]
-  exact min_eq_right ((hχ ω).trans (le_add_of_nonneg_right zero_le_one))
-
-lemma localizedStage_zero {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1)
-    (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l) (ω : Ω) :
-    localizedStage l H χ 0 ω = 0 := by
-  rw [localizedStage, sortedStopAt_zero l (H+1) hl hzero ω]
-  exact min_eq_left bot_le
-
-lemma localizedStage_nonempty {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) (t : ℝ≥0)
-    (hnt : localizedStage l H χ n ω ≤ t) (htn : t < localizedStage l H χ (n+1) ω) :
-    sortedStopAt l (H+1) n ω ≤ t ∧ t < sortedStopAt l (H+1) (n+1) ω := by
-  have htχ : t < χ ω := htn.trans_le (localizedStage_le l H χ (n+1) ω)
-  have hnχ : sortedStopAt l (H+1) n ω < χ ω := by
-    by_contra h
-    have he : localizedStage l H χ n ω = χ ω := min_eq_right (le_of_not_gt h)
-    rw [he] at hnt
-    exact (not_le_of_gt htχ) hnt
-  exact ⟨by simpa only [localizedStage, min_eq_left hnχ.le] using hnt,
-    htn.trans_le (min_le_left _ _)⟩
-
-lemma phaseAt_const_localized_interval {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
-    (n : ℕ) (ω : Ω) (t : ℝ≥0)
-    (hnt : localizedStage l H χ n ω ≤ t) (htn : t < localizedStage l H χ (n+1) ω) :
-    phaseAt ρ t ω = phaseAt ρ (localizedStage l H χ n ω) ω := by
-  obtain ⟨hleft, hright⟩ := localizedStage_nonempty l H χ n ω t hnt htn
-  have hnχ : sortedStopAt l (H+1) n ω ≤ χ ω :=
-    hleft.trans ((htn.trans_le (localizedStage_le l H χ (n+1) ω)).le)
-  rw [localizedStage, min_eq_left hnχ]
-  exact phaseAt_const_sorted_interval ρ l (H+1) hl hρ n ω t hleft hright
-
-lemma phaseAt_crossed_localized_switch {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (χ : Ω → ℝ≥0) (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
-    (n : ℕ) (ω : Ω) (i : ι) (k : Fin 3)
-    (hord : Monotone (fun j => ρ (i,j) ω))
-    (hbefore : (phaseAt ρ (localizedStage l H χ n ω) ω i).val ≤ k.val)
-    (hafter : k.val+1 ≤ (phaseAt ρ (localizedStage l H χ (n+1) ω) ω i).val) :
-    ρ (i,k) ω = localizedStage l H χ (n+1) ω := by
-  have hupper := (phaseAt_ge_iff ρ ω i hord _ k).mp hafter
-  have hlt : localizedStage l H χ n ω < ρ (i,k) ω := by
-    by_contra h
-    have hc := (phaseAt_ge_iff ρ ω i hord _ k).mpr (le_of_not_gt h)
-    omega
-  have hsorted : sortedStopAt l (H+1) n ω < ρ (i,k) ω := by
-    by_contra h
-    have hp : ρ (i,k) ω ≤ sortedStopAt l (H+1) n ω := le_of_not_gt h
-    have hχ : ρ (i,k) ω ≤ χ ω := hupper.trans (localizedStage_le l H χ (n+1) ω)
-    exact (not_le_of_gt hlt) (le_min hp hχ)
-  have hnext := sortedStopAt_no_intermediate l (H+1) hl _ (hρ (i,k)) n ω hsorted
-  exact le_antisymm hupper ((min_le_left _ _).trans hnext)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointHorizonStages
+
+namespace OAI
+
+/-! Stop the integration at a bounded random localization time, while
+continuing to detect phases from the original radius switches. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def localizedStage {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) : ℝ≥0 :=
+  min (sortedStopAt l (H+1) n ω) (χ ω)
+
+lemma localizedStage_stopping {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (l : List (Ω → ℝ≥0))
+    (hl : ∀ τ ∈ l, IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    (H : ℝ≥0) (χ : Ω → ℝ≥0)
+    (hχ : IsStoppingTime ℱ (fun ω => (χ ω : WithTop ℝ≥0))) (n : ℕ) :
+    IsStoppingTime ℱ (fun ω => (localizedStage l H χ n ω : WithTop ℝ≥0)) := by
+  simpa only [localizedStage, WithTop.coe_min] using
+    (sortedStopAt_stopping ℱ l hl (H+1) n).min hχ
+
+lemma localizedStage_monotone {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1) (ω : Ω) :
+    Monotone (fun n => localizedStage l H χ n ω) := by
+  intro i j hij
+  exact min_le_min_right (χ ω) (sortedStopAt_monotone l (H+1) hl ω hij)
+
+lemma localizedStage_le {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) : localizedStage l H χ n ω ≤ χ ω :=
+  min_le_right _ _
+
+lemma localizedStage_last {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (hχ : ∀ ω, χ ω ≤ H) (ω : Ω) :
+    localizedStage l H χ l.length ω = χ ω := by
+  simp only [localizedStage, sortedStopAt_length]
+  exact min_eq_right ((hχ ω).trans (le_add_of_nonneg_right zero_le_one))
+
+lemma localizedStage_zero {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (hl : ∀ τ ∈ l, ∀ ω, τ ω ≤ H+1)
+    (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l) (ω : Ω) :
+    localizedStage l H χ 0 ω = 0 := by
+  rw [localizedStage, sortedStopAt_zero l (H+1) hl hzero ω]
+  exact min_eq_left bot_le
+
+lemma localizedStage_nonempty {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) (t : ℝ≥0)
+    (hnt : localizedStage l H χ n ω ≤ t) (htn : t < localizedStage l H χ (n+1) ω) :
+    sortedStopAt l (H+1) n ω ≤ t ∧ t < sortedStopAt l (H+1) (n+1) ω := by
+  have htχ : t < χ ω := htn.trans_le (localizedStage_le l H χ (n+1) ω)
+  have hnχ : sortedStopAt l (H+1) n ω < χ ω := by
+    by_contra h
+    have he : localizedStage l H χ n ω = χ ω := min_eq_right (le_of_not_gt h)
+    rw [he] at hnt
+    exact (not_le_of_gt htχ) hnt
+  exact ⟨by simpa only [localizedStage, min_eq_left hnχ.le] using hnt,
+    htn.trans_le (min_le_left _ _)⟩
+
+lemma phaseAt_const_localized_interval {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
+    (n : ℕ) (ω : Ω) (t : ℝ≥0)
+    (hnt : localizedStage l H χ n ω ≤ t) (htn : t < localizedStage l H χ (n+1) ω) :
+    phaseAt ρ t ω = phaseAt ρ (localizedStage l H χ n ω) ω := by
+  obtain ⟨hleft, hright⟩ := localizedStage_nonempty l H χ n ω t hnt htn
+  have hnχ : sortedStopAt l (H+1) n ω ≤ χ ω :=
+    hleft.trans ((htn.trans_le (localizedStage_le l H χ (n+1) ω)).le)
+  rw [localizedStage, min_eq_left hnχ]
+  exact phaseAt_const_sorted_interval ρ l (H+1) hl hρ n ω t hleft hright
+
+lemma phaseAt_crossed_localized_switch {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (χ : Ω → ℝ≥0) (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρ : ∀ k, ρ k ∈ l)
+    (n : ℕ) (ω : Ω) (i : ι) (k : Fin 3)
+    (hord : Monotone (fun j => ρ (i,j) ω))
+    (hbefore : (phaseAt ρ (localizedStage l H χ n ω) ω i).val ≤ k.val)
+    (hafter : k.val+1 ≤ (phaseAt ρ (localizedStage l H χ (n+1) ω) ω i).val) :
+    ρ (i,k) ω = localizedStage l H χ (n+1) ω := by
+  have hupper := (phaseAt_ge_iff ρ ω i hord _ k).mp hafter
+  have hlt : localizedStage l H χ n ω < ρ (i,k) ω := by
+    by_contra h
+    have hc := (phaseAt_ge_iff ρ ω i hord _ k).mpr (le_of_not_gt h)
+    omega
+  have hsorted : sortedStopAt l (H+1) n ω < ρ (i,k) ω := by
+    by_contra h
+    have hp : ρ (i,k) ω ≤ sortedStopAt l (H+1) n ω := le_of_not_gt h
+    have hχ : ρ (i,k) ω ≤ χ ω := hupper.trans (localizedStage_le l H χ (n+1) ω)
+    exact (not_le_of_gt hlt) (le_min hp hχ)
+  have hnext := sortedStopAt_no_intermediate l (H+1) hl _ (hρ (i,k)) n ω hsorted
+  exact le_antisymm hupper ((min_le_left _ _).trans hnext)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedState.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedState.lean
index bfb238a2d71eef51f9bcbb4a001b28dea0f5896e..192ee7d5edceaa25ad6e21ded1e3848495dd52ff 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedState.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLocalizedState.lean
@@ -1,116 +1,116 @@
-import OAI.Probability.SLE.Multipoint.MultipointLocalIdentification
-import OAI.Probability.SLE.Multipoint.MultipointLocalization
-
-namespace OAI
-
-/-! The concrete localized state used by the multipoint product. Each
-coordinate is frozen at its own completion; the common stop only bounds
-the driver and the remaining coordinate heights. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-structure LocalizedState {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (κ lo hi M : ℝ) (H : ℝ≥0) (z : ι → ℂ) (τ : ι → Ω → ℝ≥0)
-    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) where
-  L : ℝ≥0
-  Q : ℝ → Ω → ι → ℂ × ℝ
-  V : ℝ → Ω → ι → ℂ × ℝ
-  χ : Ω → ℝ≥0
-  measurable : Measurable (fun p : ℝ × Ω => Q p.1 p.2)
-  adapted : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun t : ℝ≥0 => Q t)
-  lipschitz : ∀ ω, LipschitzWith L (fun t => Q t ω)
-  velocity_measurable : Measurable (fun p : ℝ × Ω => V p.1 p.2)
-  velocity_bound : ∀ t ω, ‖V t ω‖ ≤ L
-  derivative : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t
-  initial : ∀ ω i, Q 0 ω i = initialLogCoordinates (z i)
-  frozen : ∀ ω (t : ℝ≥0) i, τ i ω ≤ t → Q t ω i = Q (τ i ω) ω i
-  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun ω => (χ ω : WithTop ℝ≥0))
-  bounded : ∀ ω, χ ω ≤ H
-  heights : ∀ ω (t : ℝ≥0), t ≤ χ ω → ∀ i, (Q t ω i).1.im ∈ Icc lo hi
-  driver_bound : ∀ ω t, t ≤ χ ω → |B t ω| ≤ M
-  strict_heights : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i, (Q t ω i).1.im ∈ Ioo lo hi
-  strict_driver : ∀ ω t, t < χ ω → |B t ω| < M
-  boundary : ∀ ω, χ ω < H →
-    (∃ i, (Q (χ ω) ω i).1.im = lo ∨ (Q (χ ω) ω i).1.im = hi) ∨ |B (χ ω) ω| = M
-  physical_velocity : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i, t < τ i ω →
-    V t ω i = physicalLogLoewnerField (Real.sqrt κ*B t ω) (Q t ω i)
-  actual : ∀ ω i, z i ∈ survivingDomain (γ ω) (min (τ i ω) (χ ω)) ∧
-    EqOn (fun t => Q t ω i) (logLoewnerCoordinates (G ω) (z i))
-      (Icc 0 (min (τ i ω) (χ ω) : ℝ≥0))
-
-theorem exists_localizedState
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ lo hi M : ℝ) (hle : lo ≤ hi) (H : ℝ≥0)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
-    (hzlo : ∀ i, lo ≤ Real.log (z i).im) (hzhi : ∀ i, Real.log (z i).im ≤ hi)
-    (hB0 : ∀ ω, |B 0 ω| ≤ M)
-    (τ : ι → Ω → ℝ≥0)
-    (hτ : ∀ i, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ i ω : WithTop ℝ≥0)))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) :
-    Nonempty (LocalizedState B hm κ lo hi M H z τ γ G) := by
-  classical
-  obtain ⟨L,Q,V,hQm,hQa,hQl,hVm,hVb,hQd,hQdw,hQ0,hfield⟩ :=
-    exists_clipped_multipoint_state B hm hBm hBc κ lo hi hle (fun i => initialLogCoordinates (z i))
-  obtain ⟨hSm,hSa,hSl,hSVm,hSVb,hSd⟩ := stoppedCoordinateState_regular _ Q V τ L
-    hQm hQa hQl hVm hVb hQd hτ
-  let S := stoppedCoordinateState Q τ
-  let W := stoppedCoordinateVelocity V τ
-  have hS0 (ω : Ω) (i : ι) : S 0 ω i = initialLogCoordinates (z i) := by
-    rw [show S 0 ω = Q 0 ω from stoppedCoordinateState_initial Q τ ω, hQ0 ω]
-  have hBa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  obtain ⟨χ,hχ,hχH,hbounds,hstrict,hboundary⟩ := exists_multipoint_localization _ S B hSa
-    (fun ω => (hSl ω).continuous) hBa hBc lo hi M H
-    (fun ω i => by simpa only [hS0 ω i, initialLogCoordinates, Complex.add_im,
-      Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, Complex.I_re,
-      mul_one, mul_zero, add_zero, zero_add] using And.intro (hzlo i) (hzhi i)) hB0
-  have hheight (ω : Ω) (t : ℝ≥0) (ht : t ≤ χ ω) (i : ι) :
-      (S t ω i).1.im ∈ Icc lo hi := (hbounds ω t ht).1 i
-  refine ⟨{
-    L := L, Q := S, V := W, χ := χ, measurable := hSm, adapted := hSa,
-    lipschitz := hSl, velocity_measurable := hSVm, velocity_bound := hSVb,
-    derivative := hSd, initial := hS0,
-    frozen := fun ω t i ht => by
-      have ht' : (τ i ω : ℝ) ≤ (t : ℝ) := by exact_mod_cast ht
-      simp only [S,stoppedCoordinateState,min_eq_right ht',min_self],
-    stopping := hχ, bounded := hχH,
-    heights := hheight, driver_bound := fun ω t ht => (hbounds ω t ht).2,
-    strict_heights := fun ω t ht i => (hstrict ω t ht).1 i,
-    strict_driver := fun ω t ht => (hstrict ω t ht).2, boundary := hboundary,
-    physical_velocity := ?_, actual := ?_ }⟩
-  · intro ω t ht i hti
-    have htir : (t : ℝ) < (τ i ω : ℝ) := by exact_mod_cast hti
-    have he : S t ω i = Q t ω i := by simp only [S,stoppedCoordinateState,min_eq_left htir.le]
-    change (if (t : ℝ) < (τ i ω : ℝ) then V t ω i else 0) = _
-    rw [ite_eq_left htir, hfield, ← he, Real.toNNReal_coe]
-    exact clippedLogLoewnerField_eq_physical lo hi hle _ _ (hheight ω t ht.le i)
-  · intro ω i
-    have hid := stopped_height_gives_local_identification (continuous_const.mul (hBc ω))
-      (hw ω) lo hi hle z hz Q τ χ ω (fun i => congrFun (hQ0 ω) i)
-      (fun t ht => by
-        have hv : V t ω = fun i => clippedLogLoewnerField lo hi hle
-            (Real.sqrt κ*B t.toNNReal ω) (Q t ω i) := funext (hfield t ω)
-        simpa only [hv, Pi.mul_apply] using hQdw ω t ht) (hheight ω)
-    refine ⟨(hid i).1, ?_⟩
-    intro t ht
-    have hti : t ≤ (τ i ω : ℝ) := ht.2.trans (by exact_mod_cast min_le_left (τ i ω) (χ ω))
-    change S t ω i = logLoewnerCoordinates (G ω) (z i) t
-    rw [show S t ω i = Q t ω i by simp only [S,stoppedCoordinateState,min_eq_left hti]]
-    exact (hid i).2 ht
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLocalIdentification
+import OAI.Probability.SLE.Multipoint.MultipointLocalization
+
+namespace OAI
+
+/-! The concrete localized state used by the multipoint product. Each
+coordinate is frozen at its own completion; the common stop only bounds
+the driver and the remaining coordinate heights. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+structure LocalizedState {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (κ lo hi M : ℝ) (H : ℝ≥0) (z : ι → ℂ) (τ : ι → Ω → ℝ≥0)
+    (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) where
+  L : ℝ≥0
+  Q : ℝ → Ω → ι → ℂ × ℝ
+  V : ℝ → Ω → ι → ℂ × ℝ
+  χ : Ω → ℝ≥0
+  measurable : Measurable (fun p : ℝ × Ω => Q p.1 p.2)
+  adapted : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun t : ℝ≥0 => Q t)
+  lipschitz : ∀ ω, LipschitzWith L (fun t => Q t ω)
+  velocity_measurable : Measurable (fun p : ℝ × Ω => V p.1 p.2)
+  velocity_bound : ∀ t ω, ‖V t ω‖ ≤ L
+  derivative : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t
+  initial : ∀ ω i, Q 0 ω i = initialLogCoordinates (z i)
+  frozen : ∀ ω (t : ℝ≥0) i, τ i ω ≤ t → Q t ω i = Q (τ i ω) ω i
+  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun ω => (χ ω : WithTop ℝ≥0))
+  bounded : ∀ ω, χ ω ≤ H
+  heights : ∀ ω (t : ℝ≥0), t ≤ χ ω → ∀ i, (Q t ω i).1.im ∈ Icc lo hi
+  driver_bound : ∀ ω t, t ≤ χ ω → |B t ω| ≤ M
+  strict_heights : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i, (Q t ω i).1.im ∈ Ioo lo hi
+  strict_driver : ∀ ω t, t < χ ω → |B t ω| < M
+  boundary : ∀ ω, χ ω < H →
+    (∃ i, (Q (χ ω) ω i).1.im = lo ∨ (Q (χ ω) ω i).1.im = hi) ∨ |B (χ ω) ω| = M
+  physical_velocity : ∀ ω (t : ℝ≥0), t < χ ω → ∀ i, t < τ i ω →
+    V t ω i = physicalLogLoewnerField (Real.sqrt κ*B t ω) (Q t ω i)
+  actual : ∀ ω i, z i ∈ survivingDomain (γ ω) (min (τ i ω) (χ ω)) ∧
+    EqOn (fun t => Q t ω i) (logLoewnerCoordinates (G ω) (z i))
+      (Icc 0 (min (τ i ω) (χ ω) : ℝ≥0))
+
+theorem exists_localizedState
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ lo hi M : ℝ) (hle : lo ≤ hi) (H : ℝ≥0)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane)
+    (hzlo : ∀ i, lo ≤ Real.log (z i).im) (hzhi : ∀ i, Real.log (z i).im ≤ hi)
+    (hB0 : ∀ ω, |B 0 ω| ≤ M)
+    (τ : ι → Ω → ℝ≥0)
+    (hτ : ∀ i, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ i ω : WithTop ℝ≥0)))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) :
+    Nonempty (LocalizedState B hm κ lo hi M H z τ γ G) := by
+  classical
+  obtain ⟨L,Q,V,hQm,hQa,hQl,hVm,hVb,hQd,hQdw,hQ0,hfield⟩ :=
+    exists_clipped_multipoint_state B hm hBm hBc κ lo hi hle (fun i => initialLogCoordinates (z i))
+  obtain ⟨hSm,hSa,hSl,hSVm,hSVb,hSd⟩ := stoppedCoordinateState_regular _ Q V τ L
+    hQm hQa hQl hVm hVb hQd hτ
+  let S := stoppedCoordinateState Q τ
+  let W := stoppedCoordinateVelocity V τ
+  have hS0 (ω : Ω) (i : ι) : S 0 ω i = initialLogCoordinates (z i) := by
+    rw [show S 0 ω = Q 0 ω from stoppedCoordinateState_initial Q τ ω, hQ0 ω]
+  have hBa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  obtain ⟨χ,hχ,hχH,hbounds,hstrict,hboundary⟩ := exists_multipoint_localization _ S B hSa
+    (fun ω => (hSl ω).continuous) hBa hBc lo hi M H
+    (fun ω i => by simpa only [hS0 ω i, initialLogCoordinates, Complex.add_im,
+      Complex.ofReal_im, Complex.mul_im, Complex.ofReal_re, Complex.I_im, Complex.I_re,
+      mul_one, mul_zero, add_zero, zero_add] using And.intro (hzlo i) (hzhi i)) hB0
+  have hheight (ω : Ω) (t : ℝ≥0) (ht : t ≤ χ ω) (i : ι) :
+      (S t ω i).1.im ∈ Icc lo hi := (hbounds ω t ht).1 i
+  refine ⟨{
+    L := L, Q := S, V := W, χ := χ, measurable := hSm, adapted := hSa,
+    lipschitz := hSl, velocity_measurable := hSVm, velocity_bound := hSVb,
+    derivative := hSd, initial := hS0,
+    frozen := fun ω t i ht => by
+      have ht' : (τ i ω : ℝ) ≤ (t : ℝ) := by exact_mod_cast ht
+      simp only [S,stoppedCoordinateState,min_eq_right ht',min_self],
+    stopping := hχ, bounded := hχH,
+    heights := hheight, driver_bound := fun ω t ht => (hbounds ω t ht).2,
+    strict_heights := fun ω t ht i => (hstrict ω t ht).1 i,
+    strict_driver := fun ω t ht => (hstrict ω t ht).2, boundary := hboundary,
+    physical_velocity := ?_, actual := ?_ }⟩
+  · intro ω t ht i hti
+    have htir : (t : ℝ) < (τ i ω : ℝ) := by exact_mod_cast hti
+    have he : S t ω i = Q t ω i := by simp only [S,stoppedCoordinateState,min_eq_left htir.le]
+    change (if (t : ℝ) < (τ i ω : ℝ) then V t ω i else 0) = _
+    rw [ite_eq_left htir, hfield, ← he, Real.toNNReal_coe]
+    exact clippedLogLoewnerField_eq_physical lo hi hle _ _ (hheight ω t ht.le i)
+  · intro ω i
+    have hid := stopped_height_gives_local_identification (continuous_const.mul (hBc ω))
+      (hw ω) lo hi hle z hz Q τ χ ω (fun i => congrFun (hQ0 ω) i)
+      (fun t ht => by
+        have hv : V t ω = fun i => clippedLogLoewnerField lo hi hle
+            (Real.sqrt κ*B t.toNNReal ω) (Q t ω i) := funext (hfield t ω)
+        simpa only [hv, Pi.mul_apply] using hQdw ω t ht) (hheight ω)
+    refine ⟨(hid i).1, ?_⟩
+    intro t ht
+    have hti : t ≤ (τ i ω : ℝ) := ht.2.trans (by exact_mod_cast min_le_left (τ i ω) (χ ω))
+    change S t ω i = logLoewnerCoordinates (G ω) (z i) t
+    rw [show S t ω i = Q t ω i by simp only [S,stoppedCoordinateState,min_eq_left hti]]
+    exact (hid i).2 ht
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLoewnerDrop.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLoewnerDrop.lean
index a7876f6eadc2d35529e09e6633a65a68e8e75d0e..679735ca29be0eb0c3dbe6e7eaf6aa898542f885 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLoewnerDrop.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLoewnerDrop.lean
@@ -1,132 +1,132 @@
-import OAI.Probability.SLE.Multipoint.MultipointInteractionShells
-
-namespace OAI
-
-/-! The source time normalization of the pair-interaction drop. The two
-uncentered Loewner paths have velocity `a/(f-U)`, while the common real
-driver is only assumed continuous. No derivative of Brownian motion is
-used. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge.Multipoint
-
-def loewnerPoissonRate (f : ℝ → ℂ) (U : ℝ → ℝ) (t : ℝ) : ℝ :=
-  (f t).im / ‖f t - (U t : ℂ)‖ ^ 2
-
-lemma loewnerPoissonRate_continuousOn
-    {s T : ℝ} {f : ℝ → ℂ} {U : ℝ → ℝ}
-    (hf : ContinuousOn f (Icc s T)) (hU : ContinuousOn U (Icc s T))
-    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im) :
-    ContinuousOn (loewnerPoissonRate f U) (Icc s T) := by
-  apply (Complex.continuous_im.comp_continuousOn hf).div
-    ((hf.sub (Complex.continuous_ofReal.comp_continuousOn hU)).norm.pow 2)
-  intro t ht
-  apply pow_ne_zero 2
-  apply norm_ne_zero_iff.mpr
-  intro he
-  have hi := congrArg Complex.im (sub_eq_zero.mp he)
-  have hp := hfi t ht
-  change (f t).im = 0 at hi
-  linarith
-
-lemma loewnerPoissonRate_nonneg
-    {f : ℝ → ℂ} {U : ℝ → ℝ} {t : ℝ} (hfi : 0 ≤ (f t).im) :
-    0 ≤ loewnerPoissonRate f U t := div_nonneg hfi (sq_nonneg _)
-
-/-- Exact fundamental-theorem-of-calculus form of the interaction identity
-in the manuscript's time normalization. -/
-theorem loewner_interaction_drop
-    {s T a : ℝ} (hs : 0 ≤ s) (hsT : s ≤ T)
-    (f g : ℝ → ℂ) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc s T))
-    (hf : ∀ t ∈ Icc s T, HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) (Ici 0) t)
-    (hg : ∀ t ∈ Icc s T, HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) (Ici 0) t)
-    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im)
-    (hgi : ∀ t ∈ Icc s T, 0 < (g t).im)
-    (hne : ∀ t ∈ Icc s T, f t ≠ g t) :
-    2 * a * (∫ t in s..T, loewnerPoissonRate f U t * loewnerPoissonRate g U t) =
-      interaction (f s) (g s) - interaction (f T) (g T) := by
-  let K := fun t => interaction (f t) (g t)
-  let R := fun t => loewnerPoissonRate f U t * loewnerPoissonRate g U t
-  have hfc : ContinuousOn f (Icc s T) := fun t ht =>
-    ((hf t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
-  have hgc : ContinuousOn g (Icc s T) := fun t ht =>
-    ((hg t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
-  have hRc : ContinuousOn R (Icc s T) :=
-    (loewnerPoissonRate_continuousOn hfc hU hfi).mul
-      (loewnerPoissonRate_continuousOn hgc hU hgi)
-  have hd : ∀ t ∈ Icc s T, HasDerivWithinAt K ((-2 * a) * R t) (Ici 0) t := by
-    intro t ht
-    simpa only [K, R, loewnerPoissonRate, interaction_center, mul_assoc] using
-      centered_interaction_hasDerivWithinAt U (hf t ht) (hg t ht)
-        (hfi t ht) (hgi t ht) (hne t ht)
-  have hKc : ContinuousOn K (Icc s T) := fun t ht =>
-    ((hd t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
-  have he := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le hsT hKc
-    (fun t ht => (hd t ⟨ht.1.le, ht.2.le⟩).mono
-      (fun x hx => hs.trans (ht.1.le.trans hx.le)))
-    ((continuousOn_const.mul hRc).intervalIntegrable_of_Icc hsT)
-  rw [intervalIntegral.integral_const_mul] at he
-  change (-2 * a) * (∫ t in s..T, R t) = K T - K s at he
-  change 2 * a * (∫ t in s..T, R t) = K s - K T
-  linarith
-
-theorem loewner_interaction_future_bound
-    {s T a : ℝ} (hs : 0 ≤ s) (hsT : s ≤ T) (ha : 0 < a)
-    (f g : ℝ → ℂ) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc s T))
-    (hf : ∀ t ∈ Icc s T, HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) (Ici 0) t)
-    (hg : ∀ t ∈ Icc s T, HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) (Ici 0) t)
-    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im)
-    (hgi : ∀ t ∈ Icc s T, 0 < (g t).im)
-    (hne : ∀ t ∈ Icc s T, f t ≠ g t) :
-    (∫ t in s..T, loewnerPoissonRate f U t * loewnerPoissonRate g U t) ≤
-      interaction (f s) (g s) / (2 * a) := by
-  have hd := loewner_interaction_drop hs hsT f g U hU hf hg hfi hgi hne
-  have hp := interaction_nonneg (hfi T ⟨hsT, le_rfl⟩) (hgi T ⟨hsT, le_rfl⟩)
-  apply (le_div_iff₀ (by positivity : 0 < 2 * a)).mpr
-  nlinarith
-
-/-- Source-normalized weighted interaction, with the interval drop
-discharged from the actual Loewner ODE. Only the geometric decay remains
-as a separate premise. -/
-theorem loewner_weighted_interaction_integral_le
-    {s T a c M : ℝ} (hs : 0 ≤ s) (hsT : s ≤ T) (ha : 0 < a)
-    (hM : 0 ≤ M) (hc : 0 ≤ c) (hca : 2 * c < a)
-    (f g : ℝ → ℂ) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc s T))
-    (hf : ∀ t ∈ Icc s T, HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) (Ici 0) t)
-    (hg : ∀ t ∈ Icc s T, HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) (Ici 0) t)
-    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im)
-    (hgi : ∀ t ∈ Icc s T, 0 < (g t).im)
-    (hne : ∀ t ∈ Icc s T, f t ≠ g t)
-    (u v : ℝ → ℝ) (hu : ContinuousOn u (Icc s T)) (hv : ContinuousOn v (Icc s T))
-    (hu0 : ∀ t ∈ Icc s T, 0 ≤ u t) (hv0 : ∀ t ∈ Icc s T, 0 ≤ v t)
-    (hdecay : ∀ t ∈ Icc s T, interaction (f t) (g t) ≤
-      M * Real.exp (-a * max (u t) (v t))) :
-    (∫ t in s..T, Real.exp (c * (u t + v t)) *
-      (loewnerPoissonRate f U t * loewnerPoissonRate g U t)) ≤
-        M * interactionBudget a c := by
-  have hfc : ContinuousOn f (Icc s T) := fun t ht =>
-    ((hf t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
-  have hgc : ContinuousOn g (Icc s T) := fun t ht =>
-    ((hg t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
-  apply weighted_interaction_integral_le hsT ha hM hc hca u v
-    (fun t => loewnerPoissonRate f U t * loewnerPoissonRate g U t)
-    (fun t => interaction (f t) (g t)) hu hv hu0 hv0
-    ((loewnerPoissonRate_continuousOn hfc hU hfi).mul
-      (loewnerPoissonRate_continuousOn hgc hU hgi))
-    (fun t ht => mul_nonneg (loewnerPoissonRate_nonneg (hfi t ht).le)
-      (loewnerPoissonRate_nonneg (hgi t ht).le)) ?_ hdecay
-  intro t ht
-  have hsub : Icc t T ⊆ Icc s T := Icc_subset_Icc ht.1 le_rfl
-  exact loewner_interaction_future_bound (hs.trans ht.1) ht.2 ha f g U (hU.mono hsub)
-    (fun x hx => hf x (hsub hx)) (fun x hx => hg x (hsub hx))
-    (fun x hx => hfi x (hsub hx)) (fun x hx => hgi x (hsub hx))
-    (fun x hx => hne x (hsub hx))
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointInteractionShells
+
+namespace OAI
+
+/-! The source time normalization of the pair-interaction drop. The two
+uncentered Loewner paths have velocity `a/(f-U)`, while the common real
+driver is only assumed continuous. No derivative of Brownian motion is
+used. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge.Multipoint
+
+def loewnerPoissonRate (f : ℝ → ℂ) (U : ℝ → ℝ) (t : ℝ) : ℝ :=
+  (f t).im / ‖f t - (U t : ℂ)‖ ^ 2
+
+lemma loewnerPoissonRate_continuousOn
+    {s T : ℝ} {f : ℝ → ℂ} {U : ℝ → ℝ}
+    (hf : ContinuousOn f (Icc s T)) (hU : ContinuousOn U (Icc s T))
+    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im) :
+    ContinuousOn (loewnerPoissonRate f U) (Icc s T) := by
+  apply (Complex.continuous_im.comp_continuousOn hf).div
+    ((hf.sub (Complex.continuous_ofReal.comp_continuousOn hU)).norm.pow 2)
+  intro t ht
+  apply pow_ne_zero 2
+  apply norm_ne_zero_iff.mpr
+  intro he
+  have hi := congrArg Complex.im (sub_eq_zero.mp he)
+  have hp := hfi t ht
+  change (f t).im = 0 at hi
+  linarith
+
+lemma loewnerPoissonRate_nonneg
+    {f : ℝ → ℂ} {U : ℝ → ℝ} {t : ℝ} (hfi : 0 ≤ (f t).im) :
+    0 ≤ loewnerPoissonRate f U t := div_nonneg hfi (sq_nonneg _)
+
+/-- Exact fundamental-theorem-of-calculus form of the interaction identity
+in the manuscript's time normalization. -/
+theorem loewner_interaction_drop
+    {s T a : ℝ} (hs : 0 ≤ s) (hsT : s ≤ T)
+    (f g : ℝ → ℂ) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc s T))
+    (hf : ∀ t ∈ Icc s T, HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) (Ici 0) t)
+    (hg : ∀ t ∈ Icc s T, HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) (Ici 0) t)
+    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im)
+    (hgi : ∀ t ∈ Icc s T, 0 < (g t).im)
+    (hne : ∀ t ∈ Icc s T, f t ≠ g t) :
+    2 * a * (∫ t in s..T, loewnerPoissonRate f U t * loewnerPoissonRate g U t) =
+      interaction (f s) (g s) - interaction (f T) (g T) := by
+  let K := fun t => interaction (f t) (g t)
+  let R := fun t => loewnerPoissonRate f U t * loewnerPoissonRate g U t
+  have hfc : ContinuousOn f (Icc s T) := fun t ht =>
+    ((hf t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
+  have hgc : ContinuousOn g (Icc s T) := fun t ht =>
+    ((hg t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
+  have hRc : ContinuousOn R (Icc s T) :=
+    (loewnerPoissonRate_continuousOn hfc hU hfi).mul
+      (loewnerPoissonRate_continuousOn hgc hU hgi)
+  have hd : ∀ t ∈ Icc s T, HasDerivWithinAt K ((-2 * a) * R t) (Ici 0) t := by
+    intro t ht
+    simpa only [K, R, loewnerPoissonRate, interaction_center, mul_assoc] using
+      centered_interaction_hasDerivWithinAt U (hf t ht) (hg t ht)
+        (hfi t ht) (hgi t ht) (hne t ht)
+  have hKc : ContinuousOn K (Icc s T) := fun t ht =>
+    ((hd t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
+  have he := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le hsT hKc
+    (fun t ht => (hd t ⟨ht.1.le, ht.2.le⟩).mono
+      (fun x hx => hs.trans (ht.1.le.trans hx.le)))
+    ((continuousOn_const.mul hRc).intervalIntegrable_of_Icc hsT)
+  rw [intervalIntegral.integral_const_mul] at he
+  change (-2 * a) * (∫ t in s..T, R t) = K T - K s at he
+  change 2 * a * (∫ t in s..T, R t) = K s - K T
+  linarith
+
+theorem loewner_interaction_future_bound
+    {s T a : ℝ} (hs : 0 ≤ s) (hsT : s ≤ T) (ha : 0 < a)
+    (f g : ℝ → ℂ) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc s T))
+    (hf : ∀ t ∈ Icc s T, HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) (Ici 0) t)
+    (hg : ∀ t ∈ Icc s T, HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) (Ici 0) t)
+    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im)
+    (hgi : ∀ t ∈ Icc s T, 0 < (g t).im)
+    (hne : ∀ t ∈ Icc s T, f t ≠ g t) :
+    (∫ t in s..T, loewnerPoissonRate f U t * loewnerPoissonRate g U t) ≤
+      interaction (f s) (g s) / (2 * a) := by
+  have hd := loewner_interaction_drop hs hsT f g U hU hf hg hfi hgi hne
+  have hp := interaction_nonneg (hfi T ⟨hsT, le_rfl⟩) (hgi T ⟨hsT, le_rfl⟩)
+  apply (le_div_iff₀ (by positivity : 0 < 2 * a)).mpr
+  nlinarith
+
+/-- Source-normalized weighted interaction, with the interval drop
+discharged from the actual Loewner ODE. Only the geometric decay remains
+as a separate premise. -/
+theorem loewner_weighted_interaction_integral_le
+    {s T a c M : ℝ} (hs : 0 ≤ s) (hsT : s ≤ T) (ha : 0 < a)
+    (hM : 0 ≤ M) (hc : 0 ≤ c) (hca : 2 * c < a)
+    (f g : ℝ → ℂ) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc s T))
+    (hf : ∀ t ∈ Icc s T, HasDerivWithinAt f ((a : ℂ) / (f t - (U t : ℂ))) (Ici 0) t)
+    (hg : ∀ t ∈ Icc s T, HasDerivWithinAt g ((a : ℂ) / (g t - (U t : ℂ))) (Ici 0) t)
+    (hfi : ∀ t ∈ Icc s T, 0 < (f t).im)
+    (hgi : ∀ t ∈ Icc s T, 0 < (g t).im)
+    (hne : ∀ t ∈ Icc s T, f t ≠ g t)
+    (u v : ℝ → ℝ) (hu : ContinuousOn u (Icc s T)) (hv : ContinuousOn v (Icc s T))
+    (hu0 : ∀ t ∈ Icc s T, 0 ≤ u t) (hv0 : ∀ t ∈ Icc s T, 0 ≤ v t)
+    (hdecay : ∀ t ∈ Icc s T, interaction (f t) (g t) ≤
+      M * Real.exp (-a * max (u t) (v t))) :
+    (∫ t in s..T, Real.exp (c * (u t + v t)) *
+      (loewnerPoissonRate f U t * loewnerPoissonRate g U t)) ≤
+        M * interactionBudget a c := by
+  have hfc : ContinuousOn f (Icc s T) := fun t ht =>
+    ((hf t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
+  have hgc : ContinuousOn g (Icc s T) := fun t ht =>
+    ((hg t ht).mono (fun x hx => hs.trans hx.1)).continuousWithinAt
+  apply weighted_interaction_integral_le hsT ha hM hc hca u v
+    (fun t => loewnerPoissonRate f U t * loewnerPoissonRate g U t)
+    (fun t => interaction (f t) (g t)) hu hv hu0 hv0
+    ((loewnerPoissonRate_continuousOn hfc hU hfi).mul
+      (loewnerPoissonRate_continuousOn hgc hU hgi))
+    (fun t ht => mul_nonneg (loewnerPoissonRate_nonneg (hfi t ht).le)
+      (loewnerPoissonRate_nonneg (hgi t ht).le)) ?_ hdecay
+  intro t ht
+  have hsub : Icc t T ⊆ Icc s T := Icc_subset_Icc ht.1 le_rfl
+  exact loewner_interaction_future_bound (hs.trans ht.1) ht.2 ha f g U (hU.mono hsub)
+    (fun x hx => hf x (hsub hx)) (fun x hx => hg x (hsub hx))
+    (fun x hx => hfi x (hsub hx)) (fun x hx => hgi x (hsub hx))
+    (fun x hx => hne x (hsub hx))
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLogAngular.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLogAngular.lean
index f004d5a652c9631eb464f48e004c787d5814fdec..9b3cff1c08a11d3b8aa48e0cecd40d0c7d6f8e80 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLogAngular.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLogAngular.lean
@@ -1,128 +1,128 @@
-import OAI.Probability.SLE.Multipoint.MultipointLogFactor
-import OAI.Probability.SLE.Multipoint.MultipointAngular
-
-namespace OAI
-
-/-! Identification of the smooth logarithmic monomials with the angular
-factors, in actual capacity-two coordinates. -/
-
-noncomputable section
-open Set
-
-namespace SLEExactGauge.Multipoint
-
-def logPoint (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℂ :=
-  ((q.1.re-Real.sqrt κ*x : ℝ) : ℂ) + (Real.exp q.1.im : ℂ)*Complex.I
-
-def logAngle (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ := (logPoint κ q x).arg
-
-def logVelocity (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  Real.exp q.1.im / greenLogNormSq κ q x
-
-lemma logPoint_re (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    (logPoint κ q x).re = q.1.re-Real.sqrt κ*x := by simp [logPoint]
-
-lemma logPoint_im (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    (logPoint κ q x).im = Real.exp q.1.im := by
-  simp only [logPoint, Complex.add_im, Complex.mul_im, Complex.ofReal_re,
-    Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add, add_zero]
-
-lemma logPoint_norm_sq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    ‖logPoint κ q x‖^2 = greenLogNormSq κ q x := by
-  rw [← Complex.normSq_eq_norm_sq, Complex.normSq_apply, logPoint_re, logPoint_im]
-  have he : Real.exp q.1.im * Real.exp q.1.im = Real.exp (2*q.1.im) := by
-    rw [← Real.exp_add]; congr 1; ring
-  simpa only [greenLogNormSq, pow_two] using congrArg
-    (fun y => (q.1.re-Real.sqrt κ*x)*(q.1.re-Real.sqrt κ*x)+y) he
-
-lemma logPoint_norm_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0 < ‖logPoint κ q x‖ := by
-  apply norm_pos_iff.mpr
-  intro h
-  have him := congrArg Complex.im h
-  rw [logPoint_im, Complex.zero_im] at him
-  exact (Real.exp_pos q.1.im).ne' him
-
-lemma logAngle_sin_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0 < Real.sin (logAngle κ q x) := by
-  rw [logAngle, Complex.sin_arg, logPoint_im]
-  exact div_pos (Real.exp_pos _) (logPoint_norm_pos κ q x)
-
-lemma logAngle_sin_sq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    Real.sin (logAngle κ q x)^2 = Real.exp (2*q.1.im)/greenLogNormSq κ q x := by
-  rw [logAngle, Complex.sin_arg, logPoint_im, div_pow, logPoint_norm_sq]
-  have he : Real.exp q.1.im^2 = Real.exp (2*q.1.im) := by
-    rw [pow_two, ← Real.exp_add]; congr 1; ring
-  rw [he]
-
-lemma logAngle_log_sin (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    Real.log (Real.sin (logAngle κ q x)) =
-      q.1.im - Real.log (greenLogNormSq κ q x)/2 := by
-  have hn := logPoint_norm_pos κ q x
-  have hN : Real.log (greenLogNormSq κ q x) = 2*Real.log ‖logPoint κ q x‖ := by
-    rw [← logPoint_norm_sq, Real.log_pow]
-    norm_num
-  rw [logAngle, Complex.sin_arg, logPoint_im, Real.log_div (Real.exp_ne_zero _) hn.ne',
-    Real.log_exp, hN]
-  ring
-
-lemma logFactor_eq_angular (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    logFactor κ c r q x = Real.exp ((c*κ/4)*(q.2-q.1.im))*
-      sinePower r (logAngle κ q x) := by
-  rw [sinePower, Real.rpow_def_of_pos (logAngle_sin_pos κ q x), ← Real.exp_add,
-    logAngle_log_sin]
-  unfold logFactor
-  congr 1
-  ring
-
-lemma logVelocity_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0 < logVelocity κ q x := div_pos (Real.exp_pos _) (greenLogNormSq_pos κ q x)
-
-lemma logFactor_spatial_eq_angular (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    deriv (logFactor κ c r q) x =
-      Real.sqrt κ*logVelocity κ q x*Real.exp ((c*κ/4)*(q.2-q.1.im))*
-        sinePowerFirst r (logAngle κ q x) := by
-  rw [(logFactor_spatial_derivative κ c r q x).deriv, logFactor_eq_angular]
-  unfold sinePowerFirst sinePower
-  rw [Real.rpow_sub (logAngle_sin_pos κ q x), Real.rpow_one]
-  have hn := logPoint_norm_pos κ q x
-  have hz : logPoint κ q x ≠ 0 := norm_pos_iff.mp hn
-  rw [logAngle, Complex.cos_arg hz, Complex.sin_arg, logPoint_re, logPoint_im]
-  unfold logVelocity
-  field_simp [hn.ne', Real.exp_ne_zero _, (greenLogNormSq_pos κ q x).ne']
-
-lemma sineGenerator_relative {p c r θ : ℝ} (hθ : 0 < Real.sin θ) :
-    sineGenerator p r θ+c*sinePower r θ =
-      sinePower r θ * ((r*(r-p)/2)*(1/Real.sin θ^2-1)+c-r/2) := by
-  rw [sineGenerator_eq hθ]
-  unfold sinePower
-  rw [Real.rpow_sub hθ, Real.rpow_two]
-  field_simp [hθ.ne']
-  ring
-
-theorem logFactor_generator_eq_angular (κ c r : ℝ) (hκ : 0 < κ)
-    (q : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun q => logFactor κ c r q x) q)
-        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-      (1/2 : ℝ)*iteratedDeriv 2 (logFactor κ c r q) x =
-    κ*logVelocity κ q x^2*Real.exp ((c*κ/4)*(q.2-q.1.im))*
-      (sineGenerator (greenExponent κ) r (logAngle κ q x)+
-        c*sinePower r (logAngle κ q x)) := by
-  rw [logFactor_physical_generator κ c r hκ,
-    sineGenerator_relative (logAngle_sin_pos κ q x), logAngle_sin_sq,
-    logFactor_eq_angular]
-  have hN := (greenLogNormSq_pos κ q x).ne'
-  have hY := Real.exp_ne_zero (2*q.1.im)
-  have he : Real.exp q.1.im^2 = Real.exp (2*q.1.im) := by
-    rw [pow_two, ← Real.exp_add]; congr 1; ring
-  unfold logVelocity
-  rw [div_pow, he]
-  field_simp [hN, hY]
-  dsimp only [greenLogNormSq]
-  ring_nf
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLogFactor
+import OAI.Probability.SLE.Multipoint.MultipointAngular
+
+namespace OAI
+
+/-! Identification of the smooth logarithmic monomials with the angular
+factors, in actual capacity-two coordinates. -/
+
+noncomputable section
+open Set
+
+namespace SLEExactGauge.Multipoint
+
+def logPoint (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℂ :=
+  ((q.1.re-Real.sqrt κ*x : ℝ) : ℂ) + (Real.exp q.1.im : ℂ)*Complex.I
+
+def logAngle (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ := (logPoint κ q x).arg
+
+def logVelocity (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  Real.exp q.1.im / greenLogNormSq κ q x
+
+lemma logPoint_re (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    (logPoint κ q x).re = q.1.re-Real.sqrt κ*x := by simp [logPoint]
+
+lemma logPoint_im (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    (logPoint κ q x).im = Real.exp q.1.im := by
+  simp only [logPoint, Complex.add_im, Complex.mul_im, Complex.ofReal_re,
+    Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_one, mul_zero, zero_add, add_zero]
+
+lemma logPoint_norm_sq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    ‖logPoint κ q x‖^2 = greenLogNormSq κ q x := by
+  rw [← Complex.normSq_eq_norm_sq, Complex.normSq_apply, logPoint_re, logPoint_im]
+  have he : Real.exp q.1.im * Real.exp q.1.im = Real.exp (2*q.1.im) := by
+    rw [← Real.exp_add]; congr 1; ring
+  simpa only [greenLogNormSq, pow_two] using congrArg
+    (fun y => (q.1.re-Real.sqrt κ*x)*(q.1.re-Real.sqrt κ*x)+y) he
+
+lemma logPoint_norm_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0 < ‖logPoint κ q x‖ := by
+  apply norm_pos_iff.mpr
+  intro h
+  have him := congrArg Complex.im h
+  rw [logPoint_im, Complex.zero_im] at him
+  exact (Real.exp_pos q.1.im).ne' him
+
+lemma logAngle_sin_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0 < Real.sin (logAngle κ q x) := by
+  rw [logAngle, Complex.sin_arg, logPoint_im]
+  exact div_pos (Real.exp_pos _) (logPoint_norm_pos κ q x)
+
+lemma logAngle_sin_sq (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    Real.sin (logAngle κ q x)^2 = Real.exp (2*q.1.im)/greenLogNormSq κ q x := by
+  rw [logAngle, Complex.sin_arg, logPoint_im, div_pow, logPoint_norm_sq]
+  have he : Real.exp q.1.im^2 = Real.exp (2*q.1.im) := by
+    rw [pow_two, ← Real.exp_add]; congr 1; ring
+  rw [he]
+
+lemma logAngle_log_sin (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    Real.log (Real.sin (logAngle κ q x)) =
+      q.1.im - Real.log (greenLogNormSq κ q x)/2 := by
+  have hn := logPoint_norm_pos κ q x
+  have hN : Real.log (greenLogNormSq κ q x) = 2*Real.log ‖logPoint κ q x‖ := by
+    rw [← logPoint_norm_sq, Real.log_pow]
+    norm_num
+  rw [logAngle, Complex.sin_arg, logPoint_im, Real.log_div (Real.exp_ne_zero _) hn.ne',
+    Real.log_exp, hN]
+  ring
+
+lemma logFactor_eq_angular (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    logFactor κ c r q x = Real.exp ((c*κ/4)*(q.2-q.1.im))*
+      sinePower r (logAngle κ q x) := by
+  rw [sinePower, Real.rpow_def_of_pos (logAngle_sin_pos κ q x), ← Real.exp_add,
+    logAngle_log_sin]
+  unfold logFactor
+  congr 1
+  ring
+
+lemma logVelocity_pos (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0 < logVelocity κ q x := div_pos (Real.exp_pos _) (greenLogNormSq_pos κ q x)
+
+lemma logFactor_spatial_eq_angular (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    deriv (logFactor κ c r q) x =
+      Real.sqrt κ*logVelocity κ q x*Real.exp ((c*κ/4)*(q.2-q.1.im))*
+        sinePowerFirst r (logAngle κ q x) := by
+  rw [(logFactor_spatial_derivative κ c r q x).deriv, logFactor_eq_angular]
+  unfold sinePowerFirst sinePower
+  rw [Real.rpow_sub (logAngle_sin_pos κ q x), Real.rpow_one]
+  have hn := logPoint_norm_pos κ q x
+  have hz : logPoint κ q x ≠ 0 := norm_pos_iff.mp hn
+  rw [logAngle, Complex.cos_arg hz, Complex.sin_arg, logPoint_re, logPoint_im]
+  unfold logVelocity
+  field_simp [hn.ne', Real.exp_ne_zero _, (greenLogNormSq_pos κ q x).ne']
+
+lemma sineGenerator_relative {p c r θ : ℝ} (hθ : 0 < Real.sin θ) :
+    sineGenerator p r θ+c*sinePower r θ =
+      sinePower r θ * ((r*(r-p)/2)*(1/Real.sin θ^2-1)+c-r/2) := by
+  rw [sineGenerator_eq hθ]
+  unfold sinePower
+  rw [Real.rpow_sub hθ, Real.rpow_two]
+  field_simp [hθ.ne']
+  ring
+
+theorem logFactor_generator_eq_angular (κ c r : ℝ) (hκ : 0 < κ)
+    (q : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun q => logFactor κ c r q x) q)
+        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+      (1/2 : ℝ)*iteratedDeriv 2 (logFactor κ c r q) x =
+    κ*logVelocity κ q x^2*Real.exp ((c*κ/4)*(q.2-q.1.im))*
+      (sineGenerator (greenExponent κ) r (logAngle κ q x)+
+        c*sinePower r (logAngle κ q x)) := by
+  rw [logFactor_physical_generator κ c r hκ,
+    sineGenerator_relative (logAngle_sin_pos κ q x), logAngle_sin_sq,
+    logFactor_eq_angular]
+  have hN := (greenLogNormSq_pos κ q x).ne'
+  have hY := Real.exp_ne_zero (2*q.1.im)
+  have he : Real.exp q.1.im^2 = Real.exp (2*q.1.im) := by
+    rw [pow_two, ← Real.exp_add]; congr 1; ring
+  unfold logVelocity
+  rw [div_pow, he]
+  field_simp [hN, hY]
+  dsimp only [greenLogNormSq]
+  ring_nf
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLogClockTransport.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLogClockTransport.lean
index 8eddb77d2cc0f6fe1e95affa1dc456f123318f02..ddf573e9b52fc05ee46d3ca309745959f3e8c7ba 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLogClockTransport.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLogClockTransport.lean
@@ -1,277 +1,277 @@
-import OAI.Probability.SLE.Multipoint.MultipointFactorBranches
-import OAI.Probability.SLE.Multipoint.MultipointRegulatorTransport
-import OAI.Probability.SLE.DiskHalfPlane
-
-namespace OAI
-
-/-! The smooth logarithmic-coordinate factors use precisely the source
-radius clock. These identities transport the proved one-point and pair
-costs to the normalized branches, in capacity-two physical time. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma logFactorClock_eq_logRadiusClock (κ η : ℝ) (z : ℂ × ℝ) :
-    logFactorClock κ z = κ/4*(logRadiusClock η z-η) := by
-  unfold logFactorClock logRadiusClock
-  ring
-
-lemma regulator_scaled_argument {κ : ℝ} (hκ : 0 < κ) (β U u : ℝ) :
-    regulator β U (κ*u) = regulator (κ*β) (U/κ) u := by
-  rw [regulator_scale_clock hκ.le]
-  have he : κ*(U/κ) = U := by field_simp [hκ.ne']
-  rw [he]
-
-lemma logVelocity_logCoordinates (κ : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ)
-    (hy : 0 < (G t.toNNReal z).im) :
-    logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) =
-      chartPoissonRate (fun u => Real.sqrt κ*B u) G t.toNNReal z := by
-  have he : logLoewnerCoordinates G z t = logLoewnerCoordinates G z (t.toNNReal : ℝ) := by
-    simp only [logLoewnerCoordinates, Real.toNNReal_coe]
-  rw [he, logVelocity, logCoordinates_im, Real.exp_log hy,
-    greenLogNormSq_logCoordinates κ (B t.toNNReal) G z t.toNNReal hy,
-    Complex.normSq_eq_norm_sq]
-  rfl
-
-lemma logFactor_single_integrand_nonneg (κ β target : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (σ : ℝ≥0) (t : ℝ) :
-    0 ≤ regulator β target
-      (logFactorClock κ (logLoewnerCoordinates G z t) -
-        logFactorClock κ (logLoewnerCoordinates G z σ)) *
-      logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2 :=
-  mul_nonneg (regulator_pos _ _ _).le (sq_nonneg _)
-
-lemma logFactor_pair_integrand_nonneg (κ β M Uz Uw : ℝ) (B : ℝ≥0 → ℝ)
-    (G : ℝ≥0 → ℂ → ℂ) (z w : ℂ) (σz σw : ℝ≥0) (t : ℝ) :
-    0 ≤ (regulator β Uz
-      (logFactorClock κ (logLoewnerCoordinates G z t) -
-        logFactorClock κ (logLoewnerCoordinates G z σz)) *
-      regulator β Uw
-      (logFactorClock κ (logLoewnerCoordinates G w t) -
-        logFactorClock κ (logLoewnerCoordinates G w σw)))^(-M) *
-      (logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) *
-        logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal)) :=
-  mul_nonneg (Real.rpow_nonneg (mul_nonneg (regulator_pos _ _ _).le
-    (regulator_pos _ _ _).le) _) (mul_pos (logVelocity_pos _ _ _) (logVelocity_pos _ _ _)).le
-
-end SLEExactGauge.Multipoint
-
-namespace SLEExactGauge
-
-lemma IsCapacityTwoWitness.logFactorClock_sub
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (κ : ℝ) (z : ℂ) (σ : ℝ≥0) (t : ℝ)
-    (hσ : z ∈ survivingDomain γ σ) (ht : z ∈ survivingDomain γ t.toNNReal) :
-    Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-      Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ) =
-        κ*Multipoint.activeRadiusClock G z σ t := by
-  have hLt : logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) =
-      Real.log ((2*z.im)/fullRadius G t.toNNReal z) := by
-    simpa only [logLoewnerCoordinates, Real.toNNReal_coe] using
-      logRadiusClock_logCoordinates hU hw z t.toNNReal ht
-  have hLs := logRadiusClock_logCoordinates hU hw z σ hσ
-  rw [Multipoint.logFactorClock_eq_logRadiusClock κ (Real.log z.im),
-    Multipoint.logFactorClock_eq_logRadiusClock κ (Real.log z.im), hLt, hLs]
-  unfold Multipoint.activeRadiusClock
-  ring
-
-/-- The one-point regulator cost in the smooth factor's clock units. -/
-theorem IsCapacityTwoWitness.logFactor_single_regulator_integral_le
-    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
-    {β target : ℝ} (hβ : 0 < β) (htarget : 0 ≤ target)
-    (hclock : Multipoint.logFactorClock κ (logLoewnerCoordinates G z T) -
-      Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ) ≤ target) :
-    (∫ t in (σ : ℝ)..(T : ℝ),
-      Multipoint.regulator β target
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) *
-        Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2) ≤
-          2/(κ*β) := by
-  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
-  have hzσ := survivingDomain_antitone γ hσT hz
-  have hclock' : Multipoint.activeRadiusClock G z σ T ≤ target/κ := by
-    apply (le_div_iff₀ hκ).mpr
-    have hc := hw.logFactorClock_sub hU κ z σ T hzσ (by simpa using hz)
-    rw [hc] at hclock
-    nlinarith only [hclock]
-  have hb := hw.active_regulator_integral_le hU σ T hσT z hz
-    (mul_pos hκ hβ) (div_nonneg htarget hκ.le) hclock'
-  convert hb using 1
-  apply intervalIntegral.integral_congr
-  intro t ht
-  rw [uIcc_of_le (show (σ : ℝ) ≤ T by exact_mod_cast hσT)] at ht
-  have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-  have hzt := survivingDomain_antitone γ htT hz
-  dsimp only
-  rw [hw.logFactorClock_sub hU κ z σ t hzσ hzt,
-    Multipoint.regulator_scaled_argument hκ,
-    Multipoint.logVelocity_logCoordinates κ B G z t (hw.imag_positive _ z hzt)]
-
-/-- Including the generator's factor `κ` gives exactly the source bound. -/
-theorem IsCapacityTwoWitness.logFactor_single_discount_le
-    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
-    {β target : ℝ} (hβ : 0 < β) (htarget : 0 ≤ target)
-    (hclock : Multipoint.logFactorClock κ (logLoewnerCoordinates G z T) -
-      Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ) ≤ target) :
-    κ*(∫ t in (σ : ℝ)..(T : ℝ),
-      Multipoint.regulator β target
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) *
-        Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2) ≤
-          2/β := by
-  have hb := mul_le_mul_of_nonneg_left
-    (hw.logFactor_single_regulator_integral_le hκ hB σ T hσT z hz hβ htarget hclock) hκ.le
-  have he : κ*(2/(κ*β)) = 2/β := by field_simp [hκ.ne', hβ.ne']
-  rwa [he] at hb
-
-/-- Pair interaction cost for the source-normalized regulator branches.
-The physical-time interaction budget remains in capacity-two units. -/
-theorem IsCapacityTwoWitness.logFactor_pair_regulator_integral_le
-    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (σz σw s T : ℝ≥0) (hσz : σz ≤ s) (hσw : σw ≤ s) (hsT : s ≤ T)
-    (z w : ℂ) (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T)
-    (hne : z ≠ w)
-    (hbz : physicalConformalRadius J G σz z ≤ dist (J z) (J w))
-    (hbw : physicalConformalRadius J G σw w ≤ dist (J z) (J w))
-    {β M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) (hdecay : 2*(M*β) < 2/κ)
-    (Uz Uw : ℝ) :
-    (∫ t in (s : ℝ)..(T : ℝ),
-      (Multipoint.regulator β Uz
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σz)) *
-       Multipoint.regulator β Uw
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G w t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G w σw)))^(-M) *
-      (Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) *
-        Multipoint.logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal))) ≤
-      Multipoint.interactionBudget 2 (M*(κ*β)) := by
-  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
-  have hzσ := survivingDomain_antitone γ (hσz.trans hsT) hz
-  have hwσ := survivingDomain_antitone γ (hσw.trans hsT) hw'
-  have hadm : 2*(M*(κ*β)) < 2 := by
-    have hh := (Multipoint.source_pair_exponent_admissible (c := M*β) hκ).mpr hdecay
-    convert hh using 1
-    ring
-  have hb := hw.actual_pair_regulator_integral_le hU hgrowth J hJ hJinj
-    σz σw s T hσz hσw hsT z w hz hw' hne hbz hbw (mul_nonneg hκ.le hβ) hM hadm
-    (Uz/κ) (Uw/κ)
-  convert hb using 1
-  apply intervalIntegral.integral_congr
-  intro t ht
-  rw [uIcc_of_le (show (s : ℝ) ≤ T by exact_mod_cast hsT)] at ht
-  have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-  have hzt := survivingDomain_antitone γ htT hz
-  have hwt := survivingDomain_antitone γ htT hw'
-  dsimp only
-  rw [hw.logFactorClock_sub hU κ z σz t hzσ hzt,
-    hw.logFactorClock_sub hU κ w σw t hwσ hwt,
-    Multipoint.regulator_scaled_argument hκ, Multipoint.regulator_scaled_argument hκ,
-    Multipoint.logVelocity_logCoordinates κ B G z t (hw.imag_positive _ z hzt),
-    Multipoint.logVelocity_logCoordinates κ B G w t (hw.imag_positive _ w hwt)]
-
-lemma IsCapacityTwoWitness.logFactorClock_sub_continuousOn
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (κ : ℝ) (σ T : ℝ≥0) (z : ℂ)
-    (hσ : z ∈ survivingDomain γ σ) (hz : z ∈ survivingDomain γ T) :
-    ContinuousOn (fun t : ℝ =>
-      Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-        Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) (Icc 0 (T : ℝ)) := by
-  apply (continuousOn_const.mul (hw.activeRadiusClock_continuousOn hU σ T z hz)).congr
-  intro t ht
-  exact hw.logFactorClock_sub hU κ z σ t hσ
-    (hw.live_at_real_time T z hz t ht.2)
-
-lemma IsCapacityTwoWitness.logVelocity_continuousOn
-    {κ : ℝ} {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
-    ContinuousOn (fun t : ℝ =>
-      Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal))
-      (Icc 0 (T : ℝ)) := by
-  apply (hw.poissonRate_continuousOn (continuous_const.mul hB) T z hz).congr
-  intro t ht
-  exact Multipoint.logVelocity_logCoordinates κ B G z t
-    (hw.imag_positive _ z (hw.live_at_real_time T z hz t ht.2))
-
-/-- Integrability for the exact source-clock single-point cost on its own
-active interval. The endpoint may be specific to this marked point. -/
-theorem IsCapacityTwoWitness.logFactor_single_intervalIntegrable
-    {κ : ℝ} {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
-    (β target : ℝ) :
-    IntervalIntegrable (fun t : ℝ =>
-      Multipoint.regulator β target
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) *
-        Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2)
-      volume (σ : ℝ) (T : ℝ) := by
-  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
-  have hσ := survivingDomain_antitone γ hσT hz
-  have hc := (Multipoint.regulator_continuous β target).comp_continuousOn
-    (hw.logFactorClock_sub_continuousOn hU κ σ T z hσ hz)
-  have hv := (hw.logVelocity_continuousOn hB T z hz).pow 2
-  have hsub : Icc (σ : ℝ) (T : ℝ) ⊆ Icc 0 (T : ℝ) :=
-    Icc_subset_Icc σ.coe_nonneg le_rfl
-  exact ((hc.mul hv).mono hsub).intervalIntegrable_of_Icc
-    (show (σ : ℝ) ≤ T by exact_mod_cast hσT)
-
-/-- Integrability for the exact source-clock pair cost on the overlap of
-the two active intervals; the common endpoint may be their minimum. -/
-theorem IsCapacityTwoWitness.logFactor_pair_intervalIntegrable
-    {κ : ℝ} {B : ℝ≥0 → ℝ} (hB : Continuous B)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
-    (σz σw s T : ℝ≥0) (hσz : σz ≤ s) (hσw : σw ≤ s) (hsT : s ≤ T)
-    (z w : ℂ) (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T)
-    (β M Uz Uw : ℝ) :
-    IntervalIntegrable (fun t : ℝ =>
-      (Multipoint.regulator β Uz
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σz)) *
-       Multipoint.regulator β Uw
-        (Multipoint.logFactorClock κ (logLoewnerCoordinates G w t) -
-          Multipoint.logFactorClock κ (logLoewnerCoordinates G w σw)))^(-M) *
-      (Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) *
-        Multipoint.logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal)))
-      volume (s : ℝ) (T : ℝ) := by
-  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
-  have hzσ := survivingDomain_antitone γ (hσz.trans hsT) hz
-  have hwσ := survivingDomain_antitone γ (hσw.trans hsT) hw'
-  have hzclock := hw.logFactorClock_sub_continuousOn hU κ σz T z hzσ hz
-  have hwclock := hw.logFactorClock_sub_continuousOn hU κ σw T w hwσ hw'
-  have hreg := ((Multipoint.regulator_continuous β Uz).comp_continuousOn hzclock).mul
-    ((Multipoint.regulator_continuous β Uw).comp_continuousOn hwclock)
-  have hpw := hreg.rpow_const (p := -M) fun t _ => Or.inl
-    (mul_pos (Multipoint.regulator_pos _ _ _) (Multipoint.regulator_pos _ _ _)).ne'
-  have hv := (hw.logVelocity_continuousOn hB T z hz).mul
-    (hw.logVelocity_continuousOn hB T w hw')
-  have hsub : Icc (s : ℝ) (T : ℝ) ⊆ Icc 0 (T : ℝ) :=
-    Icc_subset_Icc s.coe_nonneg le_rfl
-  exact ((hpw.mul hv).mono hsub).intervalIntegrable_of_Icc
-    (show (s : ℝ) ≤ T by exact_mod_cast hsT)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointFactorBranches
+import OAI.Probability.SLE.Multipoint.MultipointRegulatorTransport
+import OAI.Probability.SLE.DiskHalfPlane
+
+namespace OAI
+
+/-! The smooth logarithmic-coordinate factors use precisely the source
+radius clock. These identities transport the proved one-point and pair
+costs to the normalized branches, in capacity-two physical time. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma logFactorClock_eq_logRadiusClock (κ η : ℝ) (z : ℂ × ℝ) :
+    logFactorClock κ z = κ/4*(logRadiusClock η z-η) := by
+  unfold logFactorClock logRadiusClock
+  ring
+
+lemma regulator_scaled_argument {κ : ℝ} (hκ : 0 < κ) (β U u : ℝ) :
+    regulator β U (κ*u) = regulator (κ*β) (U/κ) u := by
+  rw [regulator_scale_clock hκ.le]
+  have he : κ*(U/κ) = U := by field_simp [hκ.ne']
+  rw [he]
+
+lemma logVelocity_logCoordinates (κ : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ)
+    (hy : 0 < (G t.toNNReal z).im) :
+    logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) =
+      chartPoissonRate (fun u => Real.sqrt κ*B u) G t.toNNReal z := by
+  have he : logLoewnerCoordinates G z t = logLoewnerCoordinates G z (t.toNNReal : ℝ) := by
+    simp only [logLoewnerCoordinates, Real.toNNReal_coe]
+  rw [he, logVelocity, logCoordinates_im, Real.exp_log hy,
+    greenLogNormSq_logCoordinates κ (B t.toNNReal) G z t.toNNReal hy,
+    Complex.normSq_eq_norm_sq]
+  rfl
+
+lemma logFactor_single_integrand_nonneg (κ β target : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (σ : ℝ≥0) (t : ℝ) :
+    0 ≤ regulator β target
+      (logFactorClock κ (logLoewnerCoordinates G z t) -
+        logFactorClock κ (logLoewnerCoordinates G z σ)) *
+      logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2 :=
+  mul_nonneg (regulator_pos _ _ _).le (sq_nonneg _)
+
+lemma logFactor_pair_integrand_nonneg (κ β M Uz Uw : ℝ) (B : ℝ≥0 → ℝ)
+    (G : ℝ≥0 → ℂ → ℂ) (z w : ℂ) (σz σw : ℝ≥0) (t : ℝ) :
+    0 ≤ (regulator β Uz
+      (logFactorClock κ (logLoewnerCoordinates G z t) -
+        logFactorClock κ (logLoewnerCoordinates G z σz)) *
+      regulator β Uw
+      (logFactorClock κ (logLoewnerCoordinates G w t) -
+        logFactorClock κ (logLoewnerCoordinates G w σw)))^(-M) *
+      (logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) *
+        logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal)) :=
+  mul_nonneg (Real.rpow_nonneg (mul_nonneg (regulator_pos _ _ _).le
+    (regulator_pos _ _ _).le) _) (mul_pos (logVelocity_pos _ _ _) (logVelocity_pos _ _ _)).le
+
+end SLEExactGauge.Multipoint
+
+namespace SLEExactGauge
+
+lemma IsCapacityTwoWitness.logFactorClock_sub
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (κ : ℝ) (z : ℂ) (σ : ℝ≥0) (t : ℝ)
+    (hσ : z ∈ survivingDomain γ σ) (ht : z ∈ survivingDomain γ t.toNNReal) :
+    Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+      Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ) =
+        κ*Multipoint.activeRadiusClock G z σ t := by
+  have hLt : logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) =
+      Real.log ((2*z.im)/fullRadius G t.toNNReal z) := by
+    simpa only [logLoewnerCoordinates, Real.toNNReal_coe] using
+      logRadiusClock_logCoordinates hU hw z t.toNNReal ht
+  have hLs := logRadiusClock_logCoordinates hU hw z σ hσ
+  rw [Multipoint.logFactorClock_eq_logRadiusClock κ (Real.log z.im),
+    Multipoint.logFactorClock_eq_logRadiusClock κ (Real.log z.im), hLt, hLs]
+  unfold Multipoint.activeRadiusClock
+  ring
+
+/-- The one-point regulator cost in the smooth factor's clock units. -/
+theorem IsCapacityTwoWitness.logFactor_single_regulator_integral_le
+    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
+    {β target : ℝ} (hβ : 0 < β) (htarget : 0 ≤ target)
+    (hclock : Multipoint.logFactorClock κ (logLoewnerCoordinates G z T) -
+      Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ) ≤ target) :
+    (∫ t in (σ : ℝ)..(T : ℝ),
+      Multipoint.regulator β target
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) *
+        Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2) ≤
+          2/(κ*β) := by
+  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
+  have hzσ := survivingDomain_antitone γ hσT hz
+  have hclock' : Multipoint.activeRadiusClock G z σ T ≤ target/κ := by
+    apply (le_div_iff₀ hκ).mpr
+    have hc := hw.logFactorClock_sub hU κ z σ T hzσ (by simpa using hz)
+    rw [hc] at hclock
+    nlinarith only [hclock]
+  have hb := hw.active_regulator_integral_le hU σ T hσT z hz
+    (mul_pos hκ hβ) (div_nonneg htarget hκ.le) hclock'
+  convert hb using 1
+  apply intervalIntegral.integral_congr
+  intro t ht
+  rw [uIcc_of_le (show (σ : ℝ) ≤ T by exact_mod_cast hσT)] at ht
+  have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+  have hzt := survivingDomain_antitone γ htT hz
+  dsimp only
+  rw [hw.logFactorClock_sub hU κ z σ t hzσ hzt,
+    Multipoint.regulator_scaled_argument hκ,
+    Multipoint.logVelocity_logCoordinates κ B G z t (hw.imag_positive _ z hzt)]
+
+/-- Including the generator's factor `κ` gives exactly the source bound. -/
+theorem IsCapacityTwoWitness.logFactor_single_discount_le
+    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
+    {β target : ℝ} (hβ : 0 < β) (htarget : 0 ≤ target)
+    (hclock : Multipoint.logFactorClock κ (logLoewnerCoordinates G z T) -
+      Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ) ≤ target) :
+    κ*(∫ t in (σ : ℝ)..(T : ℝ),
+      Multipoint.regulator β target
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) *
+        Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2) ≤
+          2/β := by
+  have hb := mul_le_mul_of_nonneg_left
+    (hw.logFactor_single_regulator_integral_le hκ hB σ T hσT z hz hβ htarget hclock) hκ.le
+  have he : κ*(2/(κ*β)) = 2/β := by field_simp [hκ.ne', hβ.ne']
+  rwa [he] at hb
+
+/-- Pair interaction cost for the source-normalized regulator branches.
+The physical-time interaction budget remains in capacity-two units. -/
+theorem IsCapacityTwoWitness.logFactor_pair_regulator_integral_le
+    {κ : ℝ} (hκ : 0 < κ) {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (σz σw s T : ℝ≥0) (hσz : σz ≤ s) (hσw : σw ≤ s) (hsT : s ≤ T)
+    (z w : ℂ) (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T)
+    (hne : z ≠ w)
+    (hbz : physicalConformalRadius J G σz z ≤ dist (J z) (J w))
+    (hbw : physicalConformalRadius J G σw w ≤ dist (J z) (J w))
+    {β M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) (hdecay : 2*(M*β) < 2/κ)
+    (Uz Uw : ℝ) :
+    (∫ t in (s : ℝ)..(T : ℝ),
+      (Multipoint.regulator β Uz
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σz)) *
+       Multipoint.regulator β Uw
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G w t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G w σw)))^(-M) *
+      (Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) *
+        Multipoint.logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal))) ≤
+      Multipoint.interactionBudget 2 (M*(κ*β)) := by
+  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
+  have hzσ := survivingDomain_antitone γ (hσz.trans hsT) hz
+  have hwσ := survivingDomain_antitone γ (hσw.trans hsT) hw'
+  have hadm : 2*(M*(κ*β)) < 2 := by
+    have hh := (Multipoint.source_pair_exponent_admissible (c := M*β) hκ).mpr hdecay
+    convert hh using 1
+    ring
+  have hb := hw.actual_pair_regulator_integral_le hU hgrowth J hJ hJinj
+    σz σw s T hσz hσw hsT z w hz hw' hne hbz hbw (mul_nonneg hκ.le hβ) hM hadm
+    (Uz/κ) (Uw/κ)
+  convert hb using 1
+  apply intervalIntegral.integral_congr
+  intro t ht
+  rw [uIcc_of_le (show (s : ℝ) ≤ T by exact_mod_cast hsT)] at ht
+  have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+  have hzt := survivingDomain_antitone γ htT hz
+  have hwt := survivingDomain_antitone γ htT hw'
+  dsimp only
+  rw [hw.logFactorClock_sub hU κ z σz t hzσ hzt,
+    hw.logFactorClock_sub hU κ w σw t hwσ hwt,
+    Multipoint.regulator_scaled_argument hκ, Multipoint.regulator_scaled_argument hκ,
+    Multipoint.logVelocity_logCoordinates κ B G z t (hw.imag_positive _ z hzt),
+    Multipoint.logVelocity_logCoordinates κ B G w t (hw.imag_positive _ w hwt)]
+
+lemma IsCapacityTwoWitness.logFactorClock_sub_continuousOn
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (κ : ℝ) (σ T : ℝ≥0) (z : ℂ)
+    (hσ : z ∈ survivingDomain γ σ) (hz : z ∈ survivingDomain γ T) :
+    ContinuousOn (fun t : ℝ =>
+      Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+        Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) (Icc 0 (T : ℝ)) := by
+  apply (continuousOn_const.mul (hw.activeRadiusClock_continuousOn hU σ T z hz)).congr
+  intro t ht
+  exact hw.logFactorClock_sub hU κ z σ t hσ
+    (hw.live_at_real_time T z hz t ht.2)
+
+lemma IsCapacityTwoWitness.logVelocity_continuousOn
+    {κ : ℝ} {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
+    ContinuousOn (fun t : ℝ =>
+      Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal))
+      (Icc 0 (T : ℝ)) := by
+  apply (hw.poissonRate_continuousOn (continuous_const.mul hB) T z hz).congr
+  intro t ht
+  exact Multipoint.logVelocity_logCoordinates κ B G z t
+    (hw.imag_positive _ z (hw.live_at_real_time T z hz t ht.2))
+
+/-- Integrability for the exact source-clock single-point cost on its own
+active interval. The endpoint may be specific to this marked point. -/
+theorem IsCapacityTwoWitness.logFactor_single_intervalIntegrable
+    {κ : ℝ} {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
+    (β target : ℝ) :
+    IntervalIntegrable (fun t : ℝ =>
+      Multipoint.regulator β target
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σ)) *
+        Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal)^2)
+      volume (σ : ℝ) (T : ℝ) := by
+  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
+  have hσ := survivingDomain_antitone γ hσT hz
+  have hc := (Multipoint.regulator_continuous β target).comp_continuousOn
+    (hw.logFactorClock_sub_continuousOn hU κ σ T z hσ hz)
+  have hv := (hw.logVelocity_continuousOn hB T z hz).pow 2
+  have hsub : Icc (σ : ℝ) (T : ℝ) ⊆ Icc 0 (T : ℝ) :=
+    Icc_subset_Icc σ.coe_nonneg le_rfl
+  exact ((hc.mul hv).mono hsub).intervalIntegrable_of_Icc
+    (show (σ : ℝ) ≤ T by exact_mod_cast hσT)
+
+/-- Integrability for the exact source-clock pair cost on the overlap of
+the two active intervals; the common endpoint may be their minimum. -/
+theorem IsCapacityTwoWitness.logFactor_pair_intervalIntegrable
+    {κ : ℝ} {B : ℝ≥0 → ℝ} (hB : Continuous B)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t) γ G F)
+    (σz σw s T : ℝ≥0) (hσz : σz ≤ s) (hσw : σw ≤ s) (hsT : s ≤ T)
+    (z w : ℂ) (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T)
+    (β M Uz Uw : ℝ) :
+    IntervalIntegrable (fun t : ℝ =>
+      (Multipoint.regulator β Uz
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G z t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G z σz)) *
+       Multipoint.regulator β Uw
+        (Multipoint.logFactorClock κ (logLoewnerCoordinates G w t) -
+          Multipoint.logFactorClock κ (logLoewnerCoordinates G w σw)))^(-M) *
+      (Multipoint.logVelocity κ (logLoewnerCoordinates G z t) (B t.toNNReal) *
+        Multipoint.logVelocity κ (logLoewnerCoordinates G w t) (B t.toNNReal)))
+      volume (s : ℝ) (T : ℝ) := by
+  have hU : Continuous (fun t => Real.sqrt κ*B t) := continuous_const.mul hB
+  have hzσ := survivingDomain_antitone γ (hσz.trans hsT) hz
+  have hwσ := survivingDomain_antitone γ (hσw.trans hsT) hw'
+  have hzclock := hw.logFactorClock_sub_continuousOn hU κ σz T z hzσ hz
+  have hwclock := hw.logFactorClock_sub_continuousOn hU κ σw T w hwσ hw'
+  have hreg := ((Multipoint.regulator_continuous β Uz).comp_continuousOn hzclock).mul
+    ((Multipoint.regulator_continuous β Uw).comp_continuousOn hwclock)
+  have hpw := hreg.rpow_const (p := -M) fun t _ => Or.inl
+    (mul_pos (Multipoint.regulator_pos _ _ _) (Multipoint.regulator_pos _ _ _)).ne'
+  have hv := (hw.logVelocity_continuousOn hB T z hz).mul
+    (hw.logVelocity_continuousOn hB T w hw')
+  have hsub : Icc (s : ℝ) (T : ℝ) ⊆ Icc 0 (T : ℝ) :=
+    Icc_subset_Icc s.coe_nonneg le_rfl
+  exact ((hpw.mul hv).mono hsub).intervalIntegrable_of_Icc
+    (show (s : ℝ) ≤ T by exact_mod_cast hsT)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointLogFactor.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointLogFactor.lean
index 419a3c19b6593e180fb7a4640e3e47309cc558e2..50fd81d9fbe7d3b2795861f585809e927bedea01 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointLogFactor.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointLogFactor.lean
@@ -1,140 +1,140 @@
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-/-! Smooth logarithmic-coordinate monomials for the multipoint factors.
-The three monomials in each regularized factor have this form. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge.Multipoint
-
-def logFactor (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  Real.exp ((c*κ/4)*q.2+(r-c*κ/4)*q.1.im-(r/2)*Real.log (greenLogNormSq κ q x))
-
-lemma logFactor_pos (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) : 0 < logFactor κ c r q x :=
-  Real.exp_pos _
-
-lemma logFactor_contDiff (κ c r : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => logFactor κ c r p.1 p.2) := by
-  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
-    Complex.reCLM.contDiff.comp contDiff_fst.fst
-  have him : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
-    Complex.imCLM.contDiff.comp contDiff_fst.fst
-  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) :=
-    ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add
-      ((contDiff_const.mul him).exp)
-  have hlog := hN.log (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
-  unfold logFactor
-  exact ((contDiff_const.mul contDiff_fst.snd).add
-    (contDiff_const.mul him) |>.sub (contDiff_const.mul hlog)).exp
-
-lemma logFactor_spatial_derivative (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    HasDerivAt (logFactor κ c r q)
-      (logFactor κ c r q x * (r*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x)) x := by
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
-    simp
-  have hlog := hN.log (greenLogNormSq_pos κ q x).ne'
-  have hd := ((hasDerivAt_const x ((c*κ/4)*q.2+(r-c*κ/4)*q.1.im)).sub
-    (hlog.const_mul (r/2))).exp
-  convert hd using 1 <;> try rfl
-  dsimp only [logFactor, Pi.sub_apply, Pi.mul_apply, Function.comp_def]
-  ring
-
-lemma logFactor_spatial_second (κ c r : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
-    iteratedDeriv 2 (logFactor κ c r q) x =
-      logFactor κ c r q x * κ*r*((r+1)*(q.1.re-Real.sqrt κ*x)^2-Real.exp (2*q.1.im)) /
-        (greenLogNormSq κ q x)^2 := by
-  have hd : deriv (logFactor κ c r q) = fun x =>
-      logFactor κ c r q x*(r*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x) :=
-    funext (fun x => (logFactor_spatial_derivative κ c r q x).deriv)
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
-    simp
-  have hcoef := (hX.const_mul (r*Real.sqrt κ)).div hN (greenLogNormSq_pos κ q x).ne'
-  have hsecond := (logFactor_spatial_derivative κ c r q x).mul hcoef
-  rw [show (2 : ℕ) = 1+1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
-  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, id_eq] at hsecond
-  change HasDerivAt (fun y => logFactor κ c r q y *
-    (r*Real.sqrt κ*(q.1.re-Real.sqrt κ*y)/greenLogNormSq κ q y)) _ x at hsecond
-  rw [hsecond.deriv]
-  dsimp only [greenLogNormSq]
-  generalize logFactor κ c r q x = f
-  generalize q.1.re-Real.sqrt κ*x = X
-  generalize Real.exp (2*q.1.im) = Y
-  have hs := Real.sq_sqrt hκ
-  generalize Real.sqrt κ = v at *
-  rw [← hs]
-  field_simp
-  ring
-
-lemma logFactor_directional (κ c r : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun q => logFactor κ c r q x) q) v =
-      logFactor κ c r q x*((c*κ/4)*v.2+(r-c*κ/4)*v.1.im-(r/2)*
-        ((2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im)/
-          greenLogNormSq κ q x)) := by
-  have hzre : HasDerivAt (fun t : ℝ => (q+t • v).1.re) v.1.re 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
-  have hzim : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
-  have hj : HasDerivAt (fun t : ℝ => (q+t • v).2) v.2 0 := by
-    simpa only [Prod.snd_add, Prod.smul_snd, smul_eq_mul, id_eq, one_mul]
-      using ((hasDerivAt_id (0 : ℝ)).mul_const v.2).const_add q.2
-  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q+t • v) x)
-      (2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im) 0 := by
-    convert ((hzre.sub_const (Real.sqrt κ*x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
-    simp only [zero_smul, add_zero, Nat.cast_ofNat]
-    ring
-  have hlog := hN.log (by simpa only [zero_smul, add_zero] using (greenLogNormSq_pos κ q x).ne')
-  have hcalc := ((hj.const_mul (c*κ/4)).add (hzim.const_mul (r-c*κ/4))).sub
-    (hlog.const_mul (r/2))
-  have hline : HasDerivAt (fun t : ℝ => logFactor κ c r (q+t • v) x)
-      (logFactor κ c r q x*((c*κ/4)*v.2+(r-c*κ/4)*v.1.im-(r/2)*
-        ((2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im)/
-          greenLogNormSq κ q x))) 0 := by
-    convert hcalc.exp using 1 <;> try rfl
-    simp only [Pi.sub_apply, Pi.add_apply, zero_smul, add_zero, logFactor]
-  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun q => logFactor κ c r q x) :=
-    (logFactor_contDiff κ c r).comp (contDiff_id.prodMk contDiff_const)
-  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
-    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
-  have hfd : HasFDerivAt (fun q => logFactor κ c r q x)
-      (fderiv ℝ (fun q => logFactor κ c r q x) q) (q+(0 : ℝ) • v) := by
-    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
-  have hchain := hfd.comp_hasDerivAt 0 haff
-  simpa only [Function.comp_def] using hchain.unique hline
-
-/-- The source angular generator in actual capacity-two coordinates. -/
-theorem logFactor_physical_generator (κ c r : ℝ) (hκ : 0 < κ)
-    (q : ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun q => logFactor κ c r q x) q)
-        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-      (1/2 : ℝ)*iteratedDeriv 2 (logFactor κ c r q) x =
-    logFactor κ c r q x * κ *
-      ((r*(r-greenExponent κ)/2)*(q.1.re-Real.sqrt κ*x)^2+
-        (c-r/2)*Real.exp (2*q.1.im)) / (greenLogNormSq κ q x)^2 := by
-  rw [logFactor_directional, logFactor_spatial_second κ c r hκ.le]
-  obtain ⟨hr, hi, hj⟩ := physicalLogLoewnerField_components κ q x
-  rw [hr, hi, hj]
-  have hp : κ*(greenExponent κ+1) = 8 := by
-    simp only [greenExponent, sub_add_cancel]
-    field_simp
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  field_simp [hn]
-  dsimp only [greenLogNormSq]
-  linear_combination 4*logFactor κ c r q x*r*(q.1.re-Real.sqrt κ*x)^2*hp
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+/-! Smooth logarithmic-coordinate monomials for the multipoint factors.
+The three monomials in each regularized factor have this form. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge.Multipoint
+
+def logFactor (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  Real.exp ((c*κ/4)*q.2+(r-c*κ/4)*q.1.im-(r/2)*Real.log (greenLogNormSq κ q x))
+
+lemma logFactor_pos (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) : 0 < logFactor κ c r q x :=
+  Real.exp_pos _
+
+lemma logFactor_contDiff (κ c r : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => logFactor κ c r p.1 p.2) := by
+  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.re) :=
+    Complex.reCLM.contDiff.comp contDiff_fst.fst
+  have him : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => p.1.1.im) :=
+    Complex.imCLM.contDiff.comp contDiff_fst.fst
+  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun p : (ℂ × ℝ) × ℝ => greenLogNormSq κ p.1 p.2) :=
+    ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add
+      ((contDiff_const.mul him).exp)
+  have hlog := hN.log (fun p => (greenLogNormSq_pos κ p.1 p.2).ne')
+  unfold logFactor
+  exact ((contDiff_const.mul contDiff_fst.snd).add
+    (contDiff_const.mul him) |>.sub (contDiff_const.mul hlog)).exp
+
+lemma logFactor_spatial_derivative (κ c r : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    HasDerivAt (logFactor κ c r q)
+      (logFactor κ c r q x * (r*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x)) x := by
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
+    simp
+  have hlog := hN.log (greenLogNormSq_pos κ q x).ne'
+  have hd := ((hasDerivAt_const x ((c*κ/4)*q.2+(r-c*κ/4)*q.1.im)).sub
+    (hlog.const_mul (r/2))).exp
+  convert hd using 1 <;> try rfl
+  dsimp only [logFactor, Pi.sub_apply, Pi.mul_apply, Function.comp_def]
+  ring
+
+lemma logFactor_spatial_second (κ c r : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
+    iteratedDeriv 2 (logFactor κ c r q) x =
+      logFactor κ c r q x * κ*r*((r+1)*(q.1.re-Real.sqrt κ*x)^2-Real.exp (2*q.1.im)) /
+        (greenLogNormSq κ q x)^2 := by
+  have hd : deriv (logFactor κ c r q) = fun x =>
+      logFactor κ c r q x*(r*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x) :=
+    funext (fun x => (logFactor_spatial_derivative κ c r q x).deriv)
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
+    simp
+  have hcoef := (hX.const_mul (r*Real.sqrt κ)).div hN (greenLogNormSq_pos κ q x).ne'
+  have hsecond := (logFactor_spatial_derivative κ c r q x).mul hcoef
+  rw [show (2 : ℕ) = 1+1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
+  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, id_eq] at hsecond
+  change HasDerivAt (fun y => logFactor κ c r q y *
+    (r*Real.sqrt κ*(q.1.re-Real.sqrt κ*y)/greenLogNormSq κ q y)) _ x at hsecond
+  rw [hsecond.deriv]
+  dsimp only [greenLogNormSq]
+  generalize logFactor κ c r q x = f
+  generalize q.1.re-Real.sqrt κ*x = X
+  generalize Real.exp (2*q.1.im) = Y
+  have hs := Real.sq_sqrt hκ
+  generalize Real.sqrt κ = v at *
+  rw [← hs]
+  field_simp
+  ring
+
+lemma logFactor_directional (κ c r : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun q => logFactor κ c r q x) q) v =
+      logFactor κ c r q x*((c*κ/4)*v.2+(r-c*κ/4)*v.1.im-(r/2)*
+        ((2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im)/
+          greenLogNormSq κ q x)) := by
+  have hzre : HasDerivAt (fun t : ℝ => (q+t • v).1.re) v.1.re 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
+  have hzim : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
+  have hj : HasDerivAt (fun t : ℝ => (q+t • v).2) v.2 0 := by
+    simpa only [Prod.snd_add, Prod.smul_snd, smul_eq_mul, id_eq, one_mul]
+      using ((hasDerivAt_id (0 : ℝ)).mul_const v.2).const_add q.2
+  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q+t • v) x)
+      (2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im) 0 := by
+    convert ((hzre.sub_const (Real.sqrt κ*x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
+    simp only [zero_smul, add_zero, Nat.cast_ofNat]
+    ring
+  have hlog := hN.log (by simpa only [zero_smul, add_zero] using (greenLogNormSq_pos κ q x).ne')
+  have hcalc := ((hj.const_mul (c*κ/4)).add (hzim.const_mul (r-c*κ/4))).sub
+    (hlog.const_mul (r/2))
+  have hline : HasDerivAt (fun t : ℝ => logFactor κ c r (q+t • v) x)
+      (logFactor κ c r q x*((c*κ/4)*v.2+(r-c*κ/4)*v.1.im-(r/2)*
+        ((2*(q.1.re-Real.sqrt κ*x)*v.1.re+2*Real.exp (2*q.1.im)*v.1.im)/
+          greenLogNormSq κ q x))) 0 := by
+    convert hcalc.exp using 1 <;> try rfl
+    simp only [Pi.sub_apply, Pi.add_apply, zero_smul, add_zero, logFactor]
+  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun q => logFactor κ c r q x) :=
+    (logFactor_contDiff κ c r).comp (contDiff_id.prodMk contDiff_const)
+  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
+    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
+  have hfd : HasFDerivAt (fun q => logFactor κ c r q x)
+      (fderiv ℝ (fun q => logFactor κ c r q x) q) (q+(0 : ℝ) • v) := by
+    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
+  have hchain := hfd.comp_hasDerivAt 0 haff
+  simpa only [Function.comp_def] using hchain.unique hline
+
+/-- The source angular generator in actual capacity-two coordinates. -/
+theorem logFactor_physical_generator (κ c r : ℝ) (hκ : 0 < κ)
+    (q : ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun q => logFactor κ c r q x) q)
+        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+      (1/2 : ℝ)*iteratedDeriv 2 (logFactor κ c r q) x =
+    logFactor κ c r q x * κ *
+      ((r*(r-greenExponent κ)/2)*(q.1.re-Real.sqrt κ*x)^2+
+        (c-r/2)*Real.exp (2*q.1.im)) / (greenLogNormSq κ q x)^2 := by
+  rw [logFactor_directional, logFactor_spatial_second κ c r hκ.le]
+  obtain ⟨hr, hi, hj⟩ := physicalLogLoewnerField_components κ q x
+  rw [hr, hi, hj]
+  have hp : κ*(greenExponent κ+1) = 8 := by
+    simp only [greenExponent, sub_add_cancel]
+    field_simp
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  field_simp [hn]
+  dsimp only [greenLogNormSq]
+  linear_combination 4*logFactor κ c r q x*r*(q.1.re-Real.sqrt κ*x)^2*hp
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointMomentIntegration.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointMomentIntegration.lean
index 5ae71c96940523978d91f3a9e7e908275fc8509e..e1c5aa2ad8d0e2d64109c216232d692a05616a90 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointMomentIntegration.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointMomentIntegration.lean
@@ -1,61 +1,61 @@
-import Mathlib.MeasureTheory.Integral.Pi
-
-namespace OAI
-
-/-! Finite products and Tonelli for the moments of a nonnegative spatial
-weight. These identities retain arbitrary finite orders. -/
-
-noncomputable section
-open Set MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma lintegral_fin_product
-    {X : Type*} [MeasurableSpace X] (μ : Measure X) [SigmaFinite μ]
-    (f : X → ℝ≥0∞) (hf : Measurable f) (n : ℕ) :
-    (∫⁻ x : Fin n → X, ∏ i, f (x i) ∂Measure.pi (fun _ => μ)) = (∫⁻ x, f x ∂μ)^n := by
-  induction n with
-  | zero => simp only [Finset.univ_eq_empty,Finset.prod_empty,lintegral_const,
-      one_mul,pow_zero,Measure.pi_empty_univ]
-  | succ n ih =>
-    rw [← (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => μ) 0).symm.lintegral_comp_emb
-      (MeasurableEquiv.measurableEmbedding _) (fun x : Fin (n+1) → X => ∏ i, f (x i))]
-    simp only [MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv,
-      Fin.prod_univ_succ,Fin.insertNth_zero,Fin.zero_succAbove,Equiv.coe_fn_mk,
-      Fin.cons_succ,cast_eq,Fin.cons_zero]
-    have hg : Measurable (fun x : Fin n → X => ∏ i, f (x i)) :=
-      Finset.measurable_prod Finset.univ (fun i _ => hf.comp (measurable_pi_apply i))
-    rw [lintegral_prod_mul (f := f) (g := fun x : Fin n → X => ∏ i, f (x i))
-      hf.aemeasurable hg.aemeasurable,ih,pow_succ,mul_comm]
-
-theorem lintegral_mass_pow_eq
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (P : Measure Ω) (μ : Measure X) [SigmaFinite P] [SigmaFinite μ]
-    (W : Ω → X → ℝ≥0∞) (hW : Measurable (Function.uncurry W)) (n : ℕ) :
-    (∫⁻ ω, (∫⁻ x, W ω x ∂μ)^n ∂P) =
-      ∫⁻ x : Fin n → X, ∫⁻ ω, ∏ i, W ω (x i) ∂P ∂Measure.pi (fun _ => μ) := by
-  have he (ω : Ω) := lintegral_fin_product μ (W ω)
-    (hW.comp (measurable_const.prodMk measurable_id)) n
-  simp_rw [← he]
-  apply lintegral_lintegral_swap
-  apply Measurable.aemeasurable
-  apply Finset.measurable_prod
-  intro i _
-  exact hW.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd))
-
-theorem lintegral_mass_pow_le
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (P : Measure Ω) (μ : Measure X) [SigmaFinite P] [SigmaFinite μ]
-    (W : Ω → X → ℝ≥0∞) (hW : Measurable (Function.uncurry W)) (n : ℕ)
-    (K : (Fin n → X) → ℝ≥0∞)
-    (hK : ∀ᵐ x ∂Measure.pi (fun _ : Fin n => μ), (∫⁻ ω, ∏ i, W ω (x i) ∂P) ≤ K x) :
-    (∫⁻ ω, (∫⁻ x, W ω x ∂μ)^n ∂P) ≤ ∫⁻ x, K x ∂Measure.pi (fun _ : Fin n => μ) := by
-  rw [lintegral_mass_pow_eq P μ W hW n]
-  exact lintegral_mono_ae hK
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import Mathlib.MeasureTheory.Integral.Pi
+
+namespace OAI
+
+/-! Finite products and Tonelli for the moments of a nonnegative spatial
+weight. These identities retain arbitrary finite orders. -/
+
+noncomputable section
+open Set MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma lintegral_fin_product
+    {X : Type*} [MeasurableSpace X] (μ : Measure X) [SigmaFinite μ]
+    (f : X → ℝ≥0∞) (hf : Measurable f) (n : ℕ) :
+    (∫⁻ x : Fin n → X, ∏ i, f (x i) ∂Measure.pi (fun _ => μ)) = (∫⁻ x, f x ∂μ)^n := by
+  induction n with
+  | zero => simp only [Finset.univ_eq_empty,Finset.prod_empty,lintegral_const,
+      one_mul,pow_zero,Measure.pi_empty_univ]
+  | succ n ih =>
+    rw [← (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => μ) 0).symm.lintegral_comp_emb
+      (MeasurableEquiv.measurableEmbedding _) (fun x : Fin (n+1) → X => ∏ i, f (x i))]
+    simp only [MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv,
+      Fin.prod_univ_succ,Fin.insertNth_zero,Fin.zero_succAbove,Equiv.coe_fn_mk,
+      Fin.cons_succ,cast_eq,Fin.cons_zero]
+    have hg : Measurable (fun x : Fin n → X => ∏ i, f (x i)) :=
+      Finset.measurable_prod Finset.univ (fun i _ => hf.comp (measurable_pi_apply i))
+    rw [lintegral_prod_mul (f := f) (g := fun x : Fin n → X => ∏ i, f (x i))
+      hf.aemeasurable hg.aemeasurable,ih,pow_succ,mul_comm]
+
+theorem lintegral_mass_pow_eq
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (P : Measure Ω) (μ : Measure X) [SigmaFinite P] [SigmaFinite μ]
+    (W : Ω → X → ℝ≥0∞) (hW : Measurable (Function.uncurry W)) (n : ℕ) :
+    (∫⁻ ω, (∫⁻ x, W ω x ∂μ)^n ∂P) =
+      ∫⁻ x : Fin n → X, ∫⁻ ω, ∏ i, W ω (x i) ∂P ∂Measure.pi (fun _ => μ) := by
+  have he (ω : Ω) := lintegral_fin_product μ (W ω)
+    (hW.comp (measurable_const.prodMk measurable_id)) n
+  simp_rw [← he]
+  apply lintegral_lintegral_swap
+  apply Measurable.aemeasurable
+  apply Finset.measurable_prod
+  intro i _
+  exact hW.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd))
+
+theorem lintegral_mass_pow_le
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (P : Measure Ω) (μ : Measure X) [SigmaFinite P] [SigmaFinite μ]
+    (W : Ω → X → ℝ≥0∞) (hW : Measurable (Function.uncurry W)) (n : ℕ)
+    (K : (Fin n → X) → ℝ≥0∞)
+    (hK : ∀ᵐ x ∂Measure.pi (fun _ : Fin n => μ), (∫⁻ ω, ∏ i, W ω (x i) ∂P) ≤ K x) :
+    (∫⁻ ω, (∫⁻ x, W ω x ∂μ)^n ∂P) ≤ ∫⁻ x, K x ∂Measure.pi (fun _ : Fin n => μ) := by
+  rw [lintegral_mass_pow_eq P μ W hW n]
+  exact lintegral_mono_ae hK
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointOrdinaryProbability.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointOrdinaryProbability.lean
index beb7102e58b506fe853d7ab5d2daf4eea8310ed5..7c5ad29c9b693cbca641f52e145a52b72e0e9b6c 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointOrdinaryProbability.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointOrdinaryProbability.lean
@@ -1,51 +1,51 @@
-import OAI.Probability.SLE.Multipoint.MultipointPhysicalProbability
-
-namespace OAI
-
-/-! The arbitrary-domain multipoint estimate for an ordinary chordal SLE
-in its regular Brownian presentation. The constant is fixed by the order
-and SLE parameter before choosing any domain, points, caps or targets. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge
-
-def multipointConstant (κ : ℝ) (m : ℕ) : ℝ :=
-  (3 : ℝ)^m * Real.exp (Multipoint.uniformDiscountBudget κ (Multipoint.sourceBeta κ)
-    (Finset.univ : Finset (Fin m)))
-
-lemma multipointConstant_pos (κ : ℝ) (m : ℕ) : 0 < multipointConstant κ m := by
-  unfold multipointConstant
-  positivity
-
-/-- The capped estimate of the first paper's multipoint proposition,
-for every fixed finite order and arbitrary physical conformal domain.
-The input `PublishedDiskUnivalentGrowth` is Duren (1983), §2.3. -/
-theorem RegularSLEPresentation.physical_multipoint_main
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hgrowth : PublishedDiskUnivalentGrowth)
-    (m : ℕ) (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane) (z : Fin m → ℂ)
-    (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (b e : Fin m → ℝ) (hb : ∀ i, 0 < b i) (he : ∀ i, 0 < e i)
-    (hbR : ∀ i, b i ≤ 2*(z i).im*‖deriv J (z i)‖)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
-    (P.comap (Subtype.val : a.samples → Ω))
-      {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-        physicalConformalRadius J (a.G ω) t (z i) ≤ e i} ≤
-      ENNReal.ofReal (multipointConstant κ m * ∏ i, min 1 ((e i/b i)^(1-κ/8))) := by
-  have hBm : Measurable (Function.uncurry a.B) :=
-    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
-  simpa only [multipointConstant,Fintype.card_fin] using
-    Multipoint.capped_physical_radius_multipoint_bound a.brownian a.measurable_B hBm
-      a.continuous_B a.zero_B hκ hκ8 (fun ω => γ ω) a.G a.F a.witness hgrowth J hJ hJinj
-      z hz hne (fun i => 2*(z i).im*‖deriv J (z i)‖) b e he hb hbR (fun _ => rfl) hsep
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPhysicalProbability
+
+namespace OAI
+
+/-! The arbitrary-domain multipoint estimate for an ordinary chordal SLE
+in its regular Brownian presentation. The constant is fixed by the order
+and SLE parameter before choosing any domain, points, caps or targets. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge
+
+def multipointConstant (κ : ℝ) (m : ℕ) : ℝ :=
+  (3 : ℝ)^m * Real.exp (Multipoint.uniformDiscountBudget κ (Multipoint.sourceBeta κ)
+    (Finset.univ : Finset (Fin m)))
+
+lemma multipointConstant_pos (κ : ℝ) (m : ℕ) : 0 < multipointConstant κ m := by
+  unfold multipointConstant
+  positivity
+
+/-- The capped estimate of the first paper's multipoint proposition,
+for every fixed finite order and arbitrary physical conformal domain.
+The input `PublishedDiskUnivalentGrowth` is Duren (1983), §2.3. -/
+theorem RegularSLEPresentation.physical_multipoint_main
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hgrowth : PublishedDiskUnivalentGrowth)
+    (m : ℕ) (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane) (z : Fin m → ℂ)
+    (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (b e : Fin m → ℝ) (hb : ∀ i, 0 < b i) (he : ∀ i, 0 < e i)
+    (hbR : ∀ i, b i ≤ 2*(z i).im*‖deriv J (z i)‖)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
+    (P.comap (Subtype.val : a.samples → Ω))
+      {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+        physicalConformalRadius J (a.G ω) t (z i) ≤ e i} ≤
+      ENNReal.ofReal (multipointConstant κ m * ∏ i, min 1 ((e i/b i)^(1-κ/8))) := by
+  have hBm : Measurable (Function.uncurry a.B) :=
+    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
+  simpa only [multipointConstant,Fintype.card_fin] using
+    Multipoint.capped_physical_radius_multipoint_bound a.brownian a.measurable_B hBm
+      a.continuous_B a.zero_B hκ hκ8 (fun ω => γ ω) a.G a.F a.witness hgrowth J hJ hJinj
+      z hz hne (fun i => 2*(z i).im*‖deriv J (z i)‖) b e he hb hbR (fun _ => rfl) hsep
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPatternExpectation.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPatternExpectation.lean
index 6805b1f209c8dd35b596a852cdf868948c932b82..84e208e4a0c77d97ab6da5eadd66e039ef73e5b7 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPatternExpectation.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPatternExpectation.lean
@@ -1,73 +1,73 @@
-import OAI.Probability.SLE.Multipoint.MultipointStoppedProduct
-import OAI.Probability.SLE.Multipoint.MultipointSwitchPatterns
-
-namespace OAI
-
-/-! Summing the conditional product estimate over the actual finite
-switch partition, then telescoping the downward product jumps. -/
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem switchPattern_integral_le
-    {Ω ι : Type*} [MeasurableSpace Ω] [DecidableEq ι] (P : Measure Ω)
-    (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0)
-    (hA : ∀ A ∈ s.powerset, MeasurableSet (switchPatternEvent s ρ τ A))
-    (F G : Finset ι → Ω → ℝ) (X Y : Ω → ℝ)
-    (hF : ∀ A ∈ s.powerset, IntegrableOn (F A) (switchPatternEvent s ρ τ A) P)
-    (hG : ∀ A ∈ s.powerset, IntegrableOn (G A) (switchPatternEvent s ρ τ A) P)
-    (hFX : ∀ A ∈ s.powerset, ∀ ω ∈ switchPatternEvent s ρ τ A, F A ω = X ω)
-    (hGY : ∀ A ∈ s.powerset, ∀ ω ∈ switchPatternEvent s ρ τ A, G A ω = Y ω)
-    (hle : ∀ A ∈ s.powerset,
-      (∫ ω in switchPatternEvent s ρ τ A, F A ω ∂P) ≤
-        ∫ ω in switchPatternEvent s ρ τ A, G A ω ∂P) :
-    Integrable X P ∧ Integrable Y P ∧ (∫ ω, X ω ∂P) ≤ ∫ ω, Y ω ∂P := by
-  have heX := sum_switchPattern_fields s ρ τ F X hFX
-  have heY := sum_switchPattern_fields s ρ τ G Y hGY
-  have hFi (A : Finset ι) (hAs : A ∈ s.powerset) :
-      Integrable ((switchPatternEvent s ρ τ A).indicator (F A)) P :=
-    (integrable_indicator_iff (hA A hAs)).mpr (hF A hAs)
-  have hGi (A : Finset ι) (hAs : A ∈ s.powerset) :
-      Integrable ((switchPatternEvent s ρ τ A).indicator (G A)) P :=
-    (integrable_indicator_iff (hA A hAs)).mpr (hG A hAs)
-  have hXi : Integrable X P := by
-    rw [← heX]
-    exact integrable_finsetSum _ hFi
-  have hYi : Integrable Y P := by
-    rw [← heY]
-    exact integrable_finsetSum _ hGi
-  refine ⟨hXi,hYi,?_⟩
-  rw [← heX, ← heY, integral_finsetSum _ hFi, integral_finsetSum _ hGi]
-  apply Finset.sum_le_sum
-  intro A hAs
-  rw [integral_indicator (hA A hAs), integral_indicator (hA A hAs)]
-  exact hle A hAs
-
-theorem finite_switched_product_expectation_le
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (N : ℕ)
-    (X Y : ℕ → Ω → ℝ)
-    (hX : ∀ n, n ≤ N → Integrable (X n) P)
-    (hY : ∀ n, n < N → Integrable (Y n) P)
-    (hflow : ∀ n, n < N → (∫ ω, Y n ω ∂P) ≤ ∫ ω, X n ω ∂P)
-    (hjump : ∀ n, n < N → ∀ ω, X (n+1) ω ≤ Y n ω) :
-    (∫ ω, X N ω ∂P) ≤ ∫ ω, X 0 ω ∂P := by
-  have hind : ∀ n, n ≤ N → (∫ ω, X n ω ∂P) ≤ ∫ ω, X 0 ω ∂P := by
-    intro n
-    induction n with
-    | zero => exact fun _ => le_rfl
-    | succ n ih =>
-      intro hn
-      have hnN : n < N := Nat.lt_of_succ_le hn
-      have hle : (∫ ω, X (n+1) ω ∂P) ≤ ∫ ω, Y n ω ∂P :=
-        integral_mono_ae (hX (n+1) hn) (hY n hnN) (ae_of_all _ (hjump n hnN))
-      exact hle.trans ((hflow n hnN).trans (ih (Nat.le_trans (Nat.le_succ n) hn)))
-  exact hind N le_rfl
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointStoppedProduct
+import OAI.Probability.SLE.Multipoint.MultipointSwitchPatterns
+
+namespace OAI
+
+/-! Summing the conditional product estimate over the actual finite
+switch partition, then telescoping the downward product jumps. -/
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem switchPattern_integral_le
+    {Ω ι : Type*} [MeasurableSpace Ω] [DecidableEq ι] (P : Measure Ω)
+    (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0)
+    (hA : ∀ A ∈ s.powerset, MeasurableSet (switchPatternEvent s ρ τ A))
+    (F G : Finset ι → Ω → ℝ) (X Y : Ω → ℝ)
+    (hF : ∀ A ∈ s.powerset, IntegrableOn (F A) (switchPatternEvent s ρ τ A) P)
+    (hG : ∀ A ∈ s.powerset, IntegrableOn (G A) (switchPatternEvent s ρ τ A) P)
+    (hFX : ∀ A ∈ s.powerset, ∀ ω ∈ switchPatternEvent s ρ τ A, F A ω = X ω)
+    (hGY : ∀ A ∈ s.powerset, ∀ ω ∈ switchPatternEvent s ρ τ A, G A ω = Y ω)
+    (hle : ∀ A ∈ s.powerset,
+      (∫ ω in switchPatternEvent s ρ τ A, F A ω ∂P) ≤
+        ∫ ω in switchPatternEvent s ρ τ A, G A ω ∂P) :
+    Integrable X P ∧ Integrable Y P ∧ (∫ ω, X ω ∂P) ≤ ∫ ω, Y ω ∂P := by
+  have heX := sum_switchPattern_fields s ρ τ F X hFX
+  have heY := sum_switchPattern_fields s ρ τ G Y hGY
+  have hFi (A : Finset ι) (hAs : A ∈ s.powerset) :
+      Integrable ((switchPatternEvent s ρ τ A).indicator (F A)) P :=
+    (integrable_indicator_iff (hA A hAs)).mpr (hF A hAs)
+  have hGi (A : Finset ι) (hAs : A ∈ s.powerset) :
+      Integrable ((switchPatternEvent s ρ τ A).indicator (G A)) P :=
+    (integrable_indicator_iff (hA A hAs)).mpr (hG A hAs)
+  have hXi : Integrable X P := by
+    rw [← heX]
+    exact integrable_finsetSum _ hFi
+  have hYi : Integrable Y P := by
+    rw [← heY]
+    exact integrable_finsetSum _ hGi
+  refine ⟨hXi,hYi,?_⟩
+  rw [← heX, ← heY, integral_finsetSum _ hFi, integral_finsetSum _ hGi]
+  apply Finset.sum_le_sum
+  intro A hAs
+  rw [integral_indicator (hA A hAs), integral_indicator (hA A hAs)]
+  exact hle A hAs
+
+theorem finite_switched_product_expectation_le
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (N : ℕ)
+    (X Y : ℕ → Ω → ℝ)
+    (hX : ∀ n, n ≤ N → Integrable (X n) P)
+    (hY : ∀ n, n < N → Integrable (Y n) P)
+    (hflow : ∀ n, n < N → (∫ ω, Y n ω ∂P) ≤ ∫ ω, X n ω ∂P)
+    (hjump : ∀ n, n < N → ∀ ω, X (n+1) ω ≤ Y n ω) :
+    (∫ ω, X N ω ∂P) ≤ ∫ ω, X 0 ω ∂P := by
+  have hind : ∀ n, n ≤ N → (∫ ω, X n ω ∂P) ≤ ∫ ω, X 0 ω ∂P := by
+    intro n
+    induction n with
+    | zero => exact fun _ => le_rfl
+    | succ n ih =>
+      intro hn
+      have hnN : n < N := Nat.lt_of_succ_le hn
+      have hle : (∫ ω, X (n+1) ω ∂P) ≤ ∫ ω, Y n ω ∂P :=
+        integral_mono_ae (hX (n+1) hn) (hY n hnN) (ae_of_all _ (hjump n hnN))
+      exact hle.trans ((hflow n hnN).trans (ih (Nat.le_trans (Nat.le_succ n) hn)))
+  exact hind N le_rfl
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPatternProduct.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPatternProduct.lean
index 8df5e0a353cd33d50392d7e3f28d0fee0ec00088..ec58baf9cb4b1a03fbf563606ec66c5dc93f32b0 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPatternProduct.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPatternProduct.lean
@@ -1,191 +1,191 @@
-import OAI.Probability.SLE.Multipoint.MultipointFactorBranches
-import OAI.Probability.SLE.Multipoint.MultipointDiscountedField
-import OAI.Probability.SLE.Multipoint.MultipointSuccess
-
-namespace OAI
-
-/-! The finite branch formula for inactive, first-half, second-half, and
-completed factors. Its constant part includes every completed factor. -/
-
-noncomputable section
-open Set
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-def phaseActive (j : Fin 4) : Prop := j = 1 ∨ j = 2
-
-instance (j : Fin 4) : Decidable (phaseActive j) := inferInstanceAs (Decidable (_ ∨ _))
-
-def phaseRegulatorScale (β a U : ℝ) (j : Fin 4) : ℝ :=
-  if j = 1 then Real.exp (β*a) else Real.exp (-β*(U+a))
-
-def phaseRegulatorSlope (β : ℝ) (j : Fin 4) : ℝ := if j = 1 then -β else β
-
-def phaseWeight (κ a U : ℝ) (j : Fin 4) : ℝ :=
-  if phaseActive j then factorActivationScale κ a
-  else if j = 3 then Real.exp (greenExponent κ/2*U)/3 else 1
-
-def phaseFactor (κ β a U : ℝ) (j : Fin 4) (z : ℂ × ℝ) (x : ℝ) : ℝ :=
-  phaseWeight κ a U j * if phaseActive j then
-    regularizedLogFactor κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) z x
-  else 1
-
-def patternActiveSet {ι : Type*} [Fintype ι] (j : ι → Fin 4) : Finset ι :=
-  Finset.univ.filter (fun i => phaseActive (j i))
-
-def patternWeight {ι : Type*} [Fintype ι] (κ : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) : ℝ :=
-  ∏ i, phaseWeight κ (a i) (U i) (j i)
-
-def patternProduct {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : ι → ℂ × ℝ) (x : ℝ) : ℝ :=
-  ∏ i, phaseFactor κ β (a i) (U i) (j i) (z i) x
-
-def discountedPatternProduct {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : (ι → ℂ × ℝ) × ℝ) (x : ℝ) : ℝ :=
-  Real.exp (-z.2)*patternProduct κ β a U j z.1 x
-
-lemma phaseRegulatorScale_pos (β a U : ℝ) (j : Fin 4) :
-    0 < phaseRegulatorScale β a U j := by
-  unfold phaseRegulatorScale
-  split_ifs <;> exact Real.exp_pos _
-
-lemma phaseRegulatorSlope_le_one {β : ℝ} (hβ : 0 ≤ β) (hβ1 : β ≤ 1) (j : Fin 4) :
-    phaseRegulatorSlope β j ≤ 1 := by
-  unfold phaseRegulatorSlope
-  split_ifs <;> linarith
-
-lemma phaseWeight_pos (κ a U : ℝ) (j : Fin 4) : 0 < phaseWeight κ a U j := by
-  unfold phaseWeight
-  split_ifs
-  · exact factorActivationScale_pos κ a
-  · positivity
-  · norm_num
-
-lemma phaseFactor_pos (κ β a U : ℝ) (j : Fin 4) (z : ℂ × ℝ) (x : ℝ) :
-    0 < phaseFactor κ β a U j z x := by
-  unfold phaseFactor
-  apply mul_pos (phaseWeight_pos κ a U j)
-  split_ifs
-  · exact regularizedLogFactor_pos κ _ _ (phaseRegulatorScale_pos β a U j) z x
-  · norm_num
-
-lemma patternWeight_pos {ι : Type*} [Fintype ι]
-    (κ : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) : 0 < patternWeight κ a U j :=
-  Finset.prod_pos (fun i _ => phaseWeight_pos κ (a i) (U i) (j i))
-
-lemma patternProduct_eq {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : ι → ℂ × ℝ) (x : ℝ) :
-    patternProduct κ β a U j z x = patternWeight κ a U j *
-      regularizedProduct (patternActiveSet j) κ
-        (fun i => phaseRegulatorScale β (a i) (U i) (j i))
-        (fun i => phaseRegulatorSlope β (j i)) z x := by
-  simp only [patternProduct, phaseFactor, Finset.prod_mul_distrib, patternWeight,
-    regularizedProduct, patternActiveSet, Finset.prod_filter]
-
-lemma discountedPatternProduct_eq {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : (ι → ℂ × ℝ) × ℝ) (x : ℝ) :
-    discountedPatternProduct κ β a U j z x = patternWeight κ a U j *
-      discountedProduct (patternActiveSet j) κ
-        (fun i => phaseRegulatorScale β (a i) (U i) (j i))
-        (fun i => phaseRegulatorSlope β (j i)) z x := by
-  rw [discountedPatternProduct, patternProduct_eq, discountedProduct]
-  ring
-
-lemma phaseFactor_inactive (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    phaseFactor κ β a U 0 z x = 1 := by
-  norm_num [phaseFactor, phaseWeight, phaseActive]
-
-lemma phaseFactor_first (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    phaseFactor κ β a U 1 z x = factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x := by
-  norm_num [phaseFactor, phaseWeight, phaseActive, phaseRegulatorScale, phaseRegulatorSlope]
-
-lemma phaseFactor_second (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    phaseFactor κ β a U 2 z x = factorActivationScale κ a *
-      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x := by
-  norm_num [phaseFactor, phaseWeight, phaseActive, phaseRegulatorScale, phaseRegulatorSlope]
-
-lemma phaseFactor_completed (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    phaseFactor κ β a U 3 z x = Real.exp (greenExponent κ/2*U)/3 := by
-  norm_num [phaseFactor, phaseWeight, phaseActive]
-
-lemma phaseFactor_activation (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = 0) :
-    phaseFactor κ β a U 1 z x ≤ phaseFactor κ β a U 0 z x := by
-  rw [phaseFactor_first, phaseFactor_inactive]
-  exact first_branch_activation_le κ β a U hκ hκ8 z x hU hu
-
-lemma phaseFactor_midpoint (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
-    (hu : logFactorClock κ z-a = U/2) :
-    phaseFactor κ β a U 2 z x = phaseFactor κ β a U 1 z x := by
-  rw [phaseFactor_first, phaseFactor_second, branch_midpoint_eq κ β a U z x hu]
-
-lemma phaseFactor_completion (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = U) :
-    phaseFactor κ β a U 3 z x ≤ phaseFactor κ β a U 2 z x := by
-  rw [phaseFactor_second, phaseFactor_completed]
-  exact second_branch_completion_jump κ β a U hκ hκ8 z x hU hu
-
-/-- Several coincident switches still move the factor downward. Each
-crossed threshold is imposed at its actual clock value. -/
-lemma phaseFactor_switch_le (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (j k : Fin 4) (hjk : j ≤ k)
-    (hstart : j ≤ 0 → 1 ≤ k → logFactorClock κ z-a = 0)
-    (hmid : j ≤ 1 → 2 ≤ k → logFactorClock κ z-a = U/2)
-    (hend : j ≤ 2 → 3 ≤ k → logFactorClock κ z-a = U) :
-    phaseFactor κ β a U k z x ≤ phaseFactor κ β a U j z x := by
-  fin_cases j <;> fin_cases k
-  all_goals norm_num at hjk
-  all_goals try exact le_rfl
-  · exact phaseFactor_activation κ β a U hκ hκ8 z x hU (hstart (by decide) (by decide))
-  · exact (phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le.trans
-      (phaseFactor_activation κ β a U hκ hκ8 z x hU (hstart (by decide) (by decide)))
-  · exact (phaseFactor_completion κ β a U hκ hκ8 z x hU (hend (by decide) (by decide))).trans
-      ((phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le.trans
-        (phaseFactor_activation κ β a U hκ hκ8 z x hU (hstart (by decide) (by decide))))
-  · exact (phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le
-  · exact (phaseFactor_completion κ β a U hκ hκ8 z x hU (hend (by decide) (by decide))).trans
-      (phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le
-  · exact phaseFactor_completion κ β a U hκ hκ8 z x hU (hend (by decide) (by decide))
-
-lemma patternProduct_switch_le {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (a U : ι → ℝ) (hU : ∀ i, 0 ≤ U i)
-    (j k : ι → Fin 4) (z : ι → ℂ × ℝ) (x : ℝ) (hjk : ∀ i, j i ≤ k i)
-    (hstart : ∀ i, j i ≤ 0 → 1 ≤ k i → logFactorClock κ (z i)-a i = 0)
-    (hmid : ∀ i, j i ≤ 1 → 2 ≤ k i → logFactorClock κ (z i)-a i = U i/2)
-    (hend : ∀ i, j i ≤ 2 → 3 ≤ k i → logFactorClock κ (z i)-a i = U i) :
-    patternProduct κ β a U k z x ≤ patternProduct κ β a U j z x := by
-  apply Finset.prod_le_prod₀ (fun i _ => (phaseFactor_pos κ β (a i) (U i) (k i) (z i) x).le)
-  exact fun i _ => phaseFactor_switch_le κ β (a i) (U i) hκ hκ8 (z i) x
-    (hU i) (j i) (k i) (hjk i) (hstart i) (hmid i) (hend i)
-
-lemma patternProduct_all_inactive {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ) :
-    patternProduct κ β a U (fun _ => 0) z x = 1 := by
-  simp only [patternProduct, phaseFactor_inactive, Finset.prod_const_one]
-
-lemma patternProduct_all_complete {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ) :
-    patternProduct κ β a U (fun _ => 3) z x =
-      Real.exp (greenExponent κ/2 * ∑ i, U i) / 3^(Fintype.card ι) := by
-  simp only [patternProduct, phaseFactor_completed, Finset.prod_div_distrib,
-    ← Real.exp_sum, Finset.prod_const, Finset.card_univ, Finset.mul_sum]
-
-lemma discountedPatternProduct_complete_lower {ι : Type*} [Fintype ι]
-    (κ β : ℝ) (a U : ι → ℝ) (z : ι → ℂ × ℝ) (b K x : ℝ) (hb : b ≤ K) :
-    successBarrier (greenExponent κ) K Finset.univ U ≤
-      discountedPatternProduct κ β a U (fun _ => 3) (z,b) x := by
-  rw [discountedPatternProduct, patternProduct_all_complete]
-  unfold successBarrier
-  simp only [Finset.card_univ, zpow_neg, zpow_natCast]
-  have he := Real.exp_le_exp.mpr (neg_le_neg hb)
-  have hp : 0 ≤ Real.exp (greenExponent κ/2 * ∑ i, U i) / 3^Fintype.card ι := by positivity
-  convert mul_le_mul_of_nonneg_right he hp using 1
-  ring
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointFactorBranches
+import OAI.Probability.SLE.Multipoint.MultipointDiscountedField
+import OAI.Probability.SLE.Multipoint.MultipointSuccess
+
+namespace OAI
+
+/-! The finite branch formula for inactive, first-half, second-half, and
+completed factors. Its constant part includes every completed factor. -/
+
+noncomputable section
+open Set
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+def phaseActive (j : Fin 4) : Prop := j = 1 ∨ j = 2
+
+instance (j : Fin 4) : Decidable (phaseActive j) := inferInstanceAs (Decidable (_ ∨ _))
+
+def phaseRegulatorScale (β a U : ℝ) (j : Fin 4) : ℝ :=
+  if j = 1 then Real.exp (β*a) else Real.exp (-β*(U+a))
+
+def phaseRegulatorSlope (β : ℝ) (j : Fin 4) : ℝ := if j = 1 then -β else β
+
+def phaseWeight (κ a U : ℝ) (j : Fin 4) : ℝ :=
+  if phaseActive j then factorActivationScale κ a
+  else if j = 3 then Real.exp (greenExponent κ/2*U)/3 else 1
+
+def phaseFactor (κ β a U : ℝ) (j : Fin 4) (z : ℂ × ℝ) (x : ℝ) : ℝ :=
+  phaseWeight κ a U j * if phaseActive j then
+    regularizedLogFactor κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) z x
+  else 1
+
+def patternActiveSet {ι : Type*} [Fintype ι] (j : ι → Fin 4) : Finset ι :=
+  Finset.univ.filter (fun i => phaseActive (j i))
+
+def patternWeight {ι : Type*} [Fintype ι] (κ : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) : ℝ :=
+  ∏ i, phaseWeight κ (a i) (U i) (j i)
+
+def patternProduct {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : ι → ℂ × ℝ) (x : ℝ) : ℝ :=
+  ∏ i, phaseFactor κ β (a i) (U i) (j i) (z i) x
+
+def discountedPatternProduct {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : (ι → ℂ × ℝ) × ℝ) (x : ℝ) : ℝ :=
+  Real.exp (-z.2)*patternProduct κ β a U j z.1 x
+
+lemma phaseRegulatorScale_pos (β a U : ℝ) (j : Fin 4) :
+    0 < phaseRegulatorScale β a U j := by
+  unfold phaseRegulatorScale
+  split_ifs <;> exact Real.exp_pos _
+
+lemma phaseRegulatorSlope_le_one {β : ℝ} (hβ : 0 ≤ β) (hβ1 : β ≤ 1) (j : Fin 4) :
+    phaseRegulatorSlope β j ≤ 1 := by
+  unfold phaseRegulatorSlope
+  split_ifs <;> linarith
+
+lemma phaseWeight_pos (κ a U : ℝ) (j : Fin 4) : 0 < phaseWeight κ a U j := by
+  unfold phaseWeight
+  split_ifs
+  · exact factorActivationScale_pos κ a
+  · positivity
+  · norm_num
+
+lemma phaseFactor_pos (κ β a U : ℝ) (j : Fin 4) (z : ℂ × ℝ) (x : ℝ) :
+    0 < phaseFactor κ β a U j z x := by
+  unfold phaseFactor
+  apply mul_pos (phaseWeight_pos κ a U j)
+  split_ifs
+  · exact regularizedLogFactor_pos κ _ _ (phaseRegulatorScale_pos β a U j) z x
+  · norm_num
+
+lemma patternWeight_pos {ι : Type*} [Fintype ι]
+    (κ : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) : 0 < patternWeight κ a U j :=
+  Finset.prod_pos (fun i _ => phaseWeight_pos κ (a i) (U i) (j i))
+
+lemma patternProduct_eq {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : ι → ℂ × ℝ) (x : ℝ) :
+    patternProduct κ β a U j z x = patternWeight κ a U j *
+      regularizedProduct (patternActiveSet j) κ
+        (fun i => phaseRegulatorScale β (a i) (U i) (j i))
+        (fun i => phaseRegulatorSlope β (j i)) z x := by
+  simp only [patternProduct, phaseFactor, Finset.prod_mul_distrib, patternWeight,
+    regularizedProduct, patternActiveSet, Finset.prod_filter]
+
+lemma discountedPatternProduct_eq {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (j : ι → Fin 4) (z : (ι → ℂ × ℝ) × ℝ) (x : ℝ) :
+    discountedPatternProduct κ β a U j z x = patternWeight κ a U j *
+      discountedProduct (patternActiveSet j) κ
+        (fun i => phaseRegulatorScale β (a i) (U i) (j i))
+        (fun i => phaseRegulatorSlope β (j i)) z x := by
+  rw [discountedPatternProduct, patternProduct_eq, discountedProduct]
+  ring
+
+lemma phaseFactor_inactive (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    phaseFactor κ β a U 0 z x = 1 := by
+  norm_num [phaseFactor, phaseWeight, phaseActive]
+
+lemma phaseFactor_first (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    phaseFactor κ β a U 1 z x = factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (β*a)) (-β) z x := by
+  norm_num [phaseFactor, phaseWeight, phaseActive, phaseRegulatorScale, phaseRegulatorSlope]
+
+lemma phaseFactor_second (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    phaseFactor κ β a U 2 z x = factorActivationScale κ a *
+      regularizedLogFactor κ (Real.exp (-β*(U+a))) β z x := by
+  norm_num [phaseFactor, phaseWeight, phaseActive, phaseRegulatorScale, phaseRegulatorSlope]
+
+lemma phaseFactor_completed (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    phaseFactor κ β a U 3 z x = Real.exp (greenExponent κ/2*U)/3 := by
+  norm_num [phaseFactor, phaseWeight, phaseActive]
+
+lemma phaseFactor_activation (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = 0) :
+    phaseFactor κ β a U 1 z x ≤ phaseFactor κ β a U 0 z x := by
+  rw [phaseFactor_first, phaseFactor_inactive]
+  exact first_branch_activation_le κ β a U hκ hκ8 z x hU hu
+
+lemma phaseFactor_midpoint (κ β a U : ℝ) (z : ℂ × ℝ) (x : ℝ)
+    (hu : logFactorClock κ z-a = U/2) :
+    phaseFactor κ β a U 2 z x = phaseFactor κ β a U 1 z x := by
+  rw [phaseFactor_first, phaseFactor_second, branch_midpoint_eq κ β a U z x hu]
+
+lemma phaseFactor_completion (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (hu : logFactorClock κ z-a = U) :
+    phaseFactor κ β a U 3 z x ≤ phaseFactor κ β a U 2 z x := by
+  rw [phaseFactor_second, phaseFactor_completed]
+  exact second_branch_completion_jump κ β a U hκ hκ8 z x hU hu
+
+/-- Several coincident switches still move the factor downward. Each
+crossed threshold is imposed at its actual clock value. -/
+lemma phaseFactor_switch_le (κ β a U : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ × ℝ) (x : ℝ) (hU : 0 ≤ U) (j k : Fin 4) (hjk : j ≤ k)
+    (hstart : j ≤ 0 → 1 ≤ k → logFactorClock κ z-a = 0)
+    (hmid : j ≤ 1 → 2 ≤ k → logFactorClock κ z-a = U/2)
+    (hend : j ≤ 2 → 3 ≤ k → logFactorClock κ z-a = U) :
+    phaseFactor κ β a U k z x ≤ phaseFactor κ β a U j z x := by
+  fin_cases j <;> fin_cases k
+  all_goals norm_num at hjk
+  all_goals try exact le_rfl
+  · exact phaseFactor_activation κ β a U hκ hκ8 z x hU (hstart (by decide) (by decide))
+  · exact (phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le.trans
+      (phaseFactor_activation κ β a U hκ hκ8 z x hU (hstart (by decide) (by decide)))
+  · exact (phaseFactor_completion κ β a U hκ hκ8 z x hU (hend (by decide) (by decide))).trans
+      ((phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le.trans
+        (phaseFactor_activation κ β a U hκ hκ8 z x hU (hstart (by decide) (by decide))))
+  · exact (phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le
+  · exact (phaseFactor_completion κ β a U hκ hκ8 z x hU (hend (by decide) (by decide))).trans
+      (phaseFactor_midpoint κ β a U z x (hmid (by decide) (by decide))).le
+  · exact phaseFactor_completion κ β a U hκ hκ8 z x hU (hend (by decide) (by decide))
+
+lemma patternProduct_switch_le {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (a U : ι → ℝ) (hU : ∀ i, 0 ≤ U i)
+    (j k : ι → Fin 4) (z : ι → ℂ × ℝ) (x : ℝ) (hjk : ∀ i, j i ≤ k i)
+    (hstart : ∀ i, j i ≤ 0 → 1 ≤ k i → logFactorClock κ (z i)-a i = 0)
+    (hmid : ∀ i, j i ≤ 1 → 2 ≤ k i → logFactorClock κ (z i)-a i = U i/2)
+    (hend : ∀ i, j i ≤ 2 → 3 ≤ k i → logFactorClock κ (z i)-a i = U i) :
+    patternProduct κ β a U k z x ≤ patternProduct κ β a U j z x := by
+  apply Finset.prod_le_prod₀ (fun i _ => (phaseFactor_pos κ β (a i) (U i) (k i) (z i) x).le)
+  exact fun i _ => phaseFactor_switch_le κ β (a i) (U i) hκ hκ8 (z i) x
+    (hU i) (j i) (k i) (hjk i) (hstart i) (hmid i) (hend i)
+
+lemma patternProduct_all_inactive {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ) :
+    patternProduct κ β a U (fun _ => 0) z x = 1 := by
+  simp only [patternProduct, phaseFactor_inactive, Finset.prod_const_one]
+
+lemma patternProduct_all_complete {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ) :
+    patternProduct κ β a U (fun _ => 3) z x =
+      Real.exp (greenExponent κ/2 * ∑ i, U i) / 3^(Fintype.card ι) := by
+  simp only [patternProduct, phaseFactor_completed, Finset.prod_div_distrib,
+    ← Real.exp_sum, Finset.prod_const, Finset.card_univ, Finset.mul_sum]
+
+lemma discountedPatternProduct_complete_lower {ι : Type*} [Fintype ι]
+    (κ β : ℝ) (a U : ι → ℝ) (z : ι → ℂ × ℝ) (b K x : ℝ) (hb : b ≤ K) :
+    successBarrier (greenExponent κ) K Finset.univ U ≤
+      discountedPatternProduct κ β a U (fun _ => 3) (z,b) x := by
+  rw [discountedPatternProduct, patternProduct_all_complete]
+  unfold successBarrier
+  simp only [Finset.card_univ, zpow_neg, zpow_natCast]
+  have he := Real.exp_le_exp.mpr (neg_le_neg hb)
+  have hp : 0 ≤ Real.exp (greenExponent κ/2 * ∑ i, U i) / 3^Fintype.card ι := by positivity
+  convert mul_le_mul_of_nonneg_right he hp using 1
+  ring
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseProcess.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseProcess.lean
index 7453b686c259e4233999ef0f3d3790909cbf1289..315f420c9b2b5bf81d639cb5d4843cb45c22c45e 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseProcess.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseProcess.lean
@@ -1,84 +1,84 @@
-import OAI.Probability.SLE.Multipoint.MultipointPatternProduct
-import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
-
-namespace OAI
-
-/-! The actual finite phase process read from the three stopping times of
-each marked point. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def phaseAt {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
-    (t : ℝ≥0) (ω : Ω) (i : ι) : Fin 4 :=
-  if ρ (i,2) ω ≤ t then 3 else if ρ (i,1) ω ≤ t then 2
-  else if ρ (i,0) ω ≤ t then 1 else 0
-
-def phaseOfPattern {ι : Type*} [DecidableEq ι] (A : Finset (ι × Fin 3)) (i : ι) : Fin 4 :=
-  if (i,2) ∈ A then 3 else if (i,1) ∈ A then 2 else if (i,0) ∈ A then 1 else 0
-
-lemma phaseAt_of_pattern {Ω ι : Type*} [Fintype ι] [DecidableEq ι]
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (A : Finset (ι × Fin 3))
-    {ω : Ω} (hω : ω ∈ switchPatternEvent Finset.univ ρ τ A) :
-    phaseAt ρ (τ ω) ω = phaseOfPattern A := by
-  funext i
-  simp only [phaseAt, phaseOfPattern, hω (i,2) (Finset.mem_univ _),
-    hω (i,1) (Finset.mem_univ _), hω (i,0) (Finset.mem_univ _)]
-
-lemma phaseAt_monotone {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
-    (ω : Ω) (i : ι) : Monotone (fun t => phaseAt ρ t ω i) := by
-  intro t u htu
-  have h0 : ρ (i,0) ω ≤ t → ρ (i,0) ω ≤ u := fun h => h.trans htu
-  have h1 : ρ (i,1) ω ≤ t → ρ (i,1) ω ≤ u := fun h => h.trans htu
-  have h2 : ρ (i,2) ω ≤ t → ρ (i,2) ω ≤ u := fun h => h.trans htu
-  dsimp only [phaseAt]
-  split_ifs <;> simp_all
-
-lemma phaseAt_ge_iff {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
-    (ω : Ω) (i : ι) (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) (k : Fin 3) :
-    k.val+1 ≤ (phaseAt ρ t ω i).val ↔ ρ (i,k) ω ≤ t := by
-  have h01 : ρ (i,0) ω ≤ ρ (i,1) ω := hρ (by decide)
-  have h12 : ρ (i,1) ω ≤ ρ (i,2) ω := hρ (by decide)
-  have h02 := h01.trans h12
-  fin_cases k <;> unfold phaseAt <;> split_ifs <;> simp_all <;> linarith
-
-lemma phaseAt_const_sorted_interval {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (hρ : ∀ k, ρ k ∈ l)
-    (n : ℕ) (ω : Ω) (t : ℝ≥0)
-    (hnt : sortedStopAt l H n ω ≤ t) (ht : t < sortedStopAt l H (n+1) ω) :
-    phaseAt ρ t ω = phaseAt ρ (sortedStopAt l H n ω) ω := by
-  funext i
-  simp only [phaseAt, sortedStopAt_reached_iff l H hl _ (hρ (i,2)) n ω t hnt ht,
-    sortedStopAt_reached_iff l H hl _ (hρ (i,1)) n ω t hnt ht,
-    sortedStopAt_reached_iff l H hl _ (hρ (i,0)) n ω t hnt ht]
-
-lemma phaseAt_crossed_sorted_switch {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (hρ : ∀ k, ρ k ∈ l)
-    (n : ℕ) (ω : Ω) (i : ι) (k : Fin 3)
-    (hord : Monotone (fun j => ρ (i,j) ω))
-    (hbefore : (phaseAt ρ (sortedStopAt l H n ω) ω i).val ≤ k.val)
-    (hafter : k.val+1 ≤ (phaseAt ρ (sortedStopAt l H (n+1) ω) ω i).val) :
-    ρ (i,k) ω = sortedStopAt l H (n+1) ω := by
-  have hlt : sortedStopAt l H n ω < ρ (i,k) ω := by
-    by_contra h
-    have hc := (phaseAt_ge_iff ρ ω i hord _ k).mpr (le_of_not_gt h)
-    omega
-  exact le_antisymm ((phaseAt_ge_iff ρ ω i hord _ k).mp hafter)
-    (sortedStopAt_no_intermediate l H hl _ (hρ (i,k)) n ω hlt)
-
-lemma phaseAt_horizon {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
-    (H : ℝ≥0) (hρ : ∀ k ω, ρ k ω ≤ H) (ω : Ω) :
-    phaseAt ρ H ω = fun _ => 3 := by
-  funext i
-  simp only [phaseAt, ite_eq_left (hρ (i,2) ω)]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPatternProduct
+import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
+
+namespace OAI
+
+/-! The actual finite phase process read from the three stopping times of
+each marked point. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def phaseAt {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
+    (t : ℝ≥0) (ω : Ω) (i : ι) : Fin 4 :=
+  if ρ (i,2) ω ≤ t then 3 else if ρ (i,1) ω ≤ t then 2
+  else if ρ (i,0) ω ≤ t then 1 else 0
+
+def phaseOfPattern {ι : Type*} [DecidableEq ι] (A : Finset (ι × Fin 3)) (i : ι) : Fin 4 :=
+  if (i,2) ∈ A then 3 else if (i,1) ∈ A then 2 else if (i,0) ∈ A then 1 else 0
+
+lemma phaseAt_of_pattern {Ω ι : Type*} [Fintype ι] [DecidableEq ι]
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (A : Finset (ι × Fin 3))
+    {ω : Ω} (hω : ω ∈ switchPatternEvent Finset.univ ρ τ A) :
+    phaseAt ρ (τ ω) ω = phaseOfPattern A := by
+  funext i
+  simp only [phaseAt, phaseOfPattern, hω (i,2) (Finset.mem_univ _),
+    hω (i,1) (Finset.mem_univ _), hω (i,0) (Finset.mem_univ _)]
+
+lemma phaseAt_monotone {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
+    (ω : Ω) (i : ι) : Monotone (fun t => phaseAt ρ t ω i) := by
+  intro t u htu
+  have h0 : ρ (i,0) ω ≤ t → ρ (i,0) ω ≤ u := fun h => h.trans htu
+  have h1 : ρ (i,1) ω ≤ t → ρ (i,1) ω ≤ u := fun h => h.trans htu
+  have h2 : ρ (i,2) ω ≤ t → ρ (i,2) ω ≤ u := fun h => h.trans htu
+  dsimp only [phaseAt]
+  split_ifs <;> simp_all
+
+lemma phaseAt_ge_iff {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
+    (ω : Ω) (i : ι) (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) (k : Fin 3) :
+    k.val+1 ≤ (phaseAt ρ t ω i).val ↔ ρ (i,k) ω ≤ t := by
+  have h01 : ρ (i,0) ω ≤ ρ (i,1) ω := hρ (by decide)
+  have h12 : ρ (i,1) ω ≤ ρ (i,2) ω := hρ (by decide)
+  have h02 := h01.trans h12
+  fin_cases k <;> unfold phaseAt <;> split_ifs <;> simp_all <;> linarith
+
+lemma phaseAt_const_sorted_interval {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (hρ : ∀ k, ρ k ∈ l)
+    (n : ℕ) (ω : Ω) (t : ℝ≥0)
+    (hnt : sortedStopAt l H n ω ≤ t) (ht : t < sortedStopAt l H (n+1) ω) :
+    phaseAt ρ t ω = phaseAt ρ (sortedStopAt l H n ω) ω := by
+  funext i
+  simp only [phaseAt, sortedStopAt_reached_iff l H hl _ (hρ (i,2)) n ω t hnt ht,
+    sortedStopAt_reached_iff l H hl _ (hρ (i,1)) n ω t hnt ht,
+    sortedStopAt_reached_iff l H hl _ (hρ (i,0)) n ω t hnt ht]
+
+lemma phaseAt_crossed_sorted_switch {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (hρ : ∀ k, ρ k ∈ l)
+    (n : ℕ) (ω : Ω) (i : ι) (k : Fin 3)
+    (hord : Monotone (fun j => ρ (i,j) ω))
+    (hbefore : (phaseAt ρ (sortedStopAt l H n ω) ω i).val ≤ k.val)
+    (hafter : k.val+1 ≤ (phaseAt ρ (sortedStopAt l H (n+1) ω) ω i).val) :
+    ρ (i,k) ω = sortedStopAt l H (n+1) ω := by
+  have hlt : sortedStopAt l H n ω < ρ (i,k) ω := by
+    by_contra h
+    have hc := (phaseAt_ge_iff ρ ω i hord _ k).mpr (le_of_not_gt h)
+    omega
+  exact le_antisymm ((phaseAt_ge_iff ρ ω i hord _ k).mp hafter)
+    (sortedStopAt_no_intermediate l H hl _ (hρ (i,k)) n ω hlt)
+
+lemma phaseAt_horizon {Ω ι : Type*} (ρ : (ι × Fin 3) → Ω → ℝ≥0)
+    (H : ℝ≥0) (hρ : ∀ k ω, ρ k ω ≤ H) (ω : Ω) :
+    phaseAt ρ H ω = fun _ => 3 := by
+  funext i
+  simp only [phaseAt, ite_eq_left (hρ (i,2) ω)]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseRegulator.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseRegulator.lean
index 7ba72fc5cf271cf419a6ce03ecb8c66539929e01..379f12b05f408ee7f708ef492524d1683d38fddd 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseRegulator.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPhaseRegulator.lean
@@ -1,168 +1,168 @@
-import OAI.Probability.SLE.Multipoint.MultipointPhaseProcess
-import OAI.Probability.SLE.Multipoint.MultipointStoppedCoordinates
-import OAI.Probability.SLE.FiniteRadiusStop
-
-namespace OAI
-
-/-! On each active interval the finite phase selects exactly the raw
-radius regulator. Completion freezes the coordinate and removes its rate. -/
-
-noncomputable section
-open Set
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma phaseActive_iff_val (j : Fin 4) :
-    phaseActive j ↔ 1 ≤ j.val ∧ j.val < 3 := by
-  fin_cases j <;> norm_num [phaseActive]
-
-lemma phaseAt_active_iff {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) :
-    phaseActive (phaseAt ρ t ω i) ↔ ρ (i,0) ω ≤ t ∧ t < ρ (i,2) ω := by
-  rw [phaseActive_iff_val]
-  have h0 := phaseAt_ge_iff ρ ω i hρ t 0
-  have h2 := phaseAt_ge_iff ρ ω i hρ t 2
-  simp only [Fin.val_zero] at h0 h2
-  constructor
-  · rintro ⟨ha,hb⟩
-    refine ⟨h0.mp ha, lt_of_not_ge ?_⟩
-    intro hc
-    have := h2.mpr hc
-    omega
-  · rintro ⟨ha,hb⟩
-    refine ⟨h0.mpr ha, ?_⟩
-    by_contra hc
-    have := h2.mp (by omega)
-    exact (not_le_of_gt hb) this
-
-lemma phaseAt_first_iff {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) :
-    phaseAt ρ t ω i = 1 ↔ ρ (i,0) ω ≤ t ∧ t < ρ (i,1) ω := by
-  have h0 := phaseAt_ge_iff ρ ω i hρ t 0
-  have h1 := phaseAt_ge_iff ρ ω i hρ t 1
-  simp only [Fin.val_zero, Fin.val_one] at h0 h1
-  constructor
-  · intro h
-    have hv : (phaseAt ρ t ω i).val = 1 := by rw [h]; rfl
-    refine ⟨h0.mp (by omega), lt_of_not_ge ?_⟩
-    intro hc
-    have := h1.mpr hc
-    omega
-  · rintro ⟨ha,hb⟩
-    have hlo := h0.mpr ha
-    have hhi : ¬ 2 ≤ (phaseAt ρ t ω i).val := by
-      intro hc
-      exact (not_le_of_gt hb) (h1.mp hc)
-    apply Fin.ext
-    change (phaseAt ρ t ω i).val = 1
-    omega
-
-lemma phaseAt_second_iff {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) :
-    phaseAt ρ t ω i = 2 ↔ ρ (i,1) ω ≤ t ∧ t < ρ (i,2) ω := by
-  have h1 := phaseAt_ge_iff ρ ω i hρ t 1
-  have h2 := phaseAt_ge_iff ρ ω i hρ t 2
-  simp only [Fin.val_one] at h1 h2
-  constructor
-  · intro h
-    have hv : (phaseAt ρ t ω i).val = 2 := by rw [h]; rfl
-    refine ⟨h1.mp (by omega), lt_of_not_ge ?_⟩
-    intro hc
-    have := h2.mpr hc
-    omega
-  · rintro ⟨ha,hb⟩
-    have hlo := h1.mpr ha
-    have hhi : ¬ 3 ≤ (phaseAt ρ t ω i).val := by
-      intro hc
-      exact (not_le_of_gt hb) (h2.mp hc)
-    apply Fin.ext
-    change (phaseAt ρ t ω i).val = 2
-    omega
-
-lemma phaseAt_active_stopped_state {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
-    (hactive : phaseActive (phaseAt ρ t ω i))
-    (Q : ℝ → Ω → ι → ℂ × ℝ) :
-    stoppedCoordinateState Q (fun i => ρ (i,2)) t ω i = Q t ω i := by
-  have ht := ((phaseAt_active_iff ρ ω i hρ t).mp hactive).2
-  have hle : (t : ℝ) ≤ (ρ (i,2) ω : ℝ) := by exact_mod_cast ht.le
-  simp only [stoppedCoordinateState, min_eq_left hle]
-
-lemma phaseAt_active_stopped_velocity {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
-    (hactive : phaseActive (phaseAt ρ t ω i))
-    (V : ℝ → Ω → ι → ℂ × ℝ) :
-    stoppedCoordinateVelocity V (fun i => ρ (i,2)) t ω i = V t ω i := by
-  have ht := ((phaseAt_active_iff ρ ω i hρ t).mp hactive).2
-  have hlt : (t : ℝ) < (ρ (i,2) ω : ℝ) := by exact_mod_cast ht
-  simp only [stoppedCoordinateVelocity, ite_eq_left hlt]
-
-/-- Agreement up to each point's own completion suffices on every active
-phase; no common live endpoint for all coordinates is required. -/
-lemma phaseAt_active_chart {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
-    (hactive : phaseActive (phaseAt ρ t ω i))
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ)
-    (hQ : EqOn (fun u => Q u ω i) (logLoewnerCoordinates G z)
-      (Icc 0 (ρ (i,2) ω : ℝ))) :
-    stoppedCoordinateState Q (fun i => ρ (i,2)) t ω i =
-      logLoewnerCoordinates G z t := by
-  rw [phaseAt_active_stopped_state ρ ω i hρ t hactive Q]
-  apply hQ
-  refine ⟨t.coe_nonneg, ?_⟩
-  exact_mod_cast ((phaseAt_active_iff ρ ω i hρ t).mp hactive).2.le
-
-/-- The selected active branch has the raw regulator and never exceeds one. -/
-theorem phaseRegulator_eq_and_le_one (κ β a U : ℝ) (hβ : 0 ≤ β)
-    (j : Fin 4) (hj : phaseActive j) (z : ℂ × ℝ)
-    (hfirst : j = 1 → 0 ≤ logFactorClock κ z-a ∧ logFactorClock κ z-a ≤ U/2)
-    (hsecond : j = 2 → U/2 ≤ logFactorClock κ z-a ∧ logFactorClock κ z-a ≤ U) :
-    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) z =
-      regulator β U (logFactorClock κ z-a) ∧
-    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) z ≤ 1 := by
-  rcases hj with rfl | rfl
-  · change branchRegulator κ (Real.exp (β*a)) (-β) z = _ ∧
-      branchRegulator κ (Real.exp (β*a)) (-β) z ≤ 1
-    exact ⟨first_branch_regulator_eq κ β a U z (hfirst rfl).2,
-      first_branch_regulator_le_one κ β a z hβ (hfirst rfl).1⟩
-  · change branchRegulator κ (Real.exp (-β*(U+a))) β z = _ ∧
-      branchRegulator κ (Real.exp (-β*(U+a))) β z ≤ 1
-    exact ⟨second_branch_regulator_eq κ β a U z (hsecond rfl).1,
-      second_branch_regulator_le_one κ β a U z hβ (hsecond rfl).2⟩
-
-/-- Threshold status for the actual stopping-time phase supplies the branch
-bounds; no separate regulator assumption is needed. -/
-theorem phaseAt_regulator_eq_and_le_one {Ω ι : Type*}
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
-    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
-    (κ β a U : ℝ) (hβ : 0 ≤ β) (z : ℂ × ℝ)
-    (hactive : phaseActive (phaseAt ρ t ω i))
-    (hstart : ρ (i,0) ω ≤ t → 0 ≤ logFactorClock κ z-a)
-    (hbefore : t < ρ (i,1) ω → logFactorClock κ z-a ≤ U/2)
-    (hafter : ρ (i,1) ω ≤ t → U/2 ≤ logFactorClock κ z-a)
-    (hend : t < ρ (i,2) ω → logFactorClock κ z-a ≤ U) :
-    branchRegulator κ (phaseRegulatorScale β a U (phaseAt ρ t ω i))
-      (phaseRegulatorSlope β (phaseAt ρ t ω i)) z =
-      regulator β U (logFactorClock κ z-a) ∧
-    branchRegulator κ (phaseRegulatorScale β a U (phaseAt ρ t ω i))
-      (phaseRegulatorSlope β (phaseAt ρ t ω i)) z ≤ 1 := by
-  apply phaseRegulator_eq_and_le_one κ β a U hβ _ hactive z
-  · intro h
-    obtain ⟨h0,h1⟩ := (phaseAt_first_iff ρ ω i hρ t).mp h
-    exact ⟨hstart h0,hbefore h1⟩
-  · intro h
-    obtain ⟨h1,h2⟩ := (phaseAt_second_iff ρ ω i hρ t).mp h
-    exact ⟨hafter h1,hend h2⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPhaseProcess
+import OAI.Probability.SLE.Multipoint.MultipointStoppedCoordinates
+import OAI.Probability.SLE.FiniteRadiusStop
+
+namespace OAI
+
+/-! On each active interval the finite phase selects exactly the raw
+radius regulator. Completion freezes the coordinate and removes its rate. -/
+
+noncomputable section
+open Set
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma phaseActive_iff_val (j : Fin 4) :
+    phaseActive j ↔ 1 ≤ j.val ∧ j.val < 3 := by
+  fin_cases j <;> norm_num [phaseActive]
+
+lemma phaseAt_active_iff {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) :
+    phaseActive (phaseAt ρ t ω i) ↔ ρ (i,0) ω ≤ t ∧ t < ρ (i,2) ω := by
+  rw [phaseActive_iff_val]
+  have h0 := phaseAt_ge_iff ρ ω i hρ t 0
+  have h2 := phaseAt_ge_iff ρ ω i hρ t 2
+  simp only [Fin.val_zero] at h0 h2
+  constructor
+  · rintro ⟨ha,hb⟩
+    refine ⟨h0.mp ha, lt_of_not_ge ?_⟩
+    intro hc
+    have := h2.mpr hc
+    omega
+  · rintro ⟨ha,hb⟩
+    refine ⟨h0.mpr ha, ?_⟩
+    by_contra hc
+    have := h2.mp (by omega)
+    exact (not_le_of_gt hb) this
+
+lemma phaseAt_first_iff {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) :
+    phaseAt ρ t ω i = 1 ↔ ρ (i,0) ω ≤ t ∧ t < ρ (i,1) ω := by
+  have h0 := phaseAt_ge_iff ρ ω i hρ t 0
+  have h1 := phaseAt_ge_iff ρ ω i hρ t 1
+  simp only [Fin.val_zero, Fin.val_one] at h0 h1
+  constructor
+  · intro h
+    have hv : (phaseAt ρ t ω i).val = 1 := by rw [h]; rfl
+    refine ⟨h0.mp (by omega), lt_of_not_ge ?_⟩
+    intro hc
+    have := h1.mpr hc
+    omega
+  · rintro ⟨ha,hb⟩
+    have hlo := h0.mpr ha
+    have hhi : ¬ 2 ≤ (phaseAt ρ t ω i).val := by
+      intro hc
+      exact (not_le_of_gt hb) (h1.mp hc)
+    apply Fin.ext
+    change (phaseAt ρ t ω i).val = 1
+    omega
+
+lemma phaseAt_second_iff {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0) :
+    phaseAt ρ t ω i = 2 ↔ ρ (i,1) ω ≤ t ∧ t < ρ (i,2) ω := by
+  have h1 := phaseAt_ge_iff ρ ω i hρ t 1
+  have h2 := phaseAt_ge_iff ρ ω i hρ t 2
+  simp only [Fin.val_one] at h1 h2
+  constructor
+  · intro h
+    have hv : (phaseAt ρ t ω i).val = 2 := by rw [h]; rfl
+    refine ⟨h1.mp (by omega), lt_of_not_ge ?_⟩
+    intro hc
+    have := h2.mpr hc
+    omega
+  · rintro ⟨ha,hb⟩
+    have hlo := h1.mpr ha
+    have hhi : ¬ 3 ≤ (phaseAt ρ t ω i).val := by
+      intro hc
+      exact (not_le_of_gt hb) (h2.mp hc)
+    apply Fin.ext
+    change (phaseAt ρ t ω i).val = 2
+    omega
+
+lemma phaseAt_active_stopped_state {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
+    (hactive : phaseActive (phaseAt ρ t ω i))
+    (Q : ℝ → Ω → ι → ℂ × ℝ) :
+    stoppedCoordinateState Q (fun i => ρ (i,2)) t ω i = Q t ω i := by
+  have ht := ((phaseAt_active_iff ρ ω i hρ t).mp hactive).2
+  have hle : (t : ℝ) ≤ (ρ (i,2) ω : ℝ) := by exact_mod_cast ht.le
+  simp only [stoppedCoordinateState, min_eq_left hle]
+
+lemma phaseAt_active_stopped_velocity {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
+    (hactive : phaseActive (phaseAt ρ t ω i))
+    (V : ℝ → Ω → ι → ℂ × ℝ) :
+    stoppedCoordinateVelocity V (fun i => ρ (i,2)) t ω i = V t ω i := by
+  have ht := ((phaseAt_active_iff ρ ω i hρ t).mp hactive).2
+  have hlt : (t : ℝ) < (ρ (i,2) ω : ℝ) := by exact_mod_cast ht
+  simp only [stoppedCoordinateVelocity, ite_eq_left hlt]
+
+/-- Agreement up to each point's own completion suffices on every active
+phase; no common live endpoint for all coordinates is required. -/
+lemma phaseAt_active_chart {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
+    (hactive : phaseActive (phaseAt ρ t ω i))
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ)
+    (hQ : EqOn (fun u => Q u ω i) (logLoewnerCoordinates G z)
+      (Icc 0 (ρ (i,2) ω : ℝ))) :
+    stoppedCoordinateState Q (fun i => ρ (i,2)) t ω i =
+      logLoewnerCoordinates G z t := by
+  rw [phaseAt_active_stopped_state ρ ω i hρ t hactive Q]
+  apply hQ
+  refine ⟨t.coe_nonneg, ?_⟩
+  exact_mod_cast ((phaseAt_active_iff ρ ω i hρ t).mp hactive).2.le
+
+/-- The selected active branch has the raw regulator and never exceeds one. -/
+theorem phaseRegulator_eq_and_le_one (κ β a U : ℝ) (hβ : 0 ≤ β)
+    (j : Fin 4) (hj : phaseActive j) (z : ℂ × ℝ)
+    (hfirst : j = 1 → 0 ≤ logFactorClock κ z-a ∧ logFactorClock κ z-a ≤ U/2)
+    (hsecond : j = 2 → U/2 ≤ logFactorClock κ z-a ∧ logFactorClock κ z-a ≤ U) :
+    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) z =
+      regulator β U (logFactorClock κ z-a) ∧
+    branchRegulator κ (phaseRegulatorScale β a U j) (phaseRegulatorSlope β j) z ≤ 1 := by
+  rcases hj with rfl | rfl
+  · change branchRegulator κ (Real.exp (β*a)) (-β) z = _ ∧
+      branchRegulator κ (Real.exp (β*a)) (-β) z ≤ 1
+    exact ⟨first_branch_regulator_eq κ β a U z (hfirst rfl).2,
+      first_branch_regulator_le_one κ β a z hβ (hfirst rfl).1⟩
+  · change branchRegulator κ (Real.exp (-β*(U+a))) β z = _ ∧
+      branchRegulator κ (Real.exp (-β*(U+a))) β z ≤ 1
+    exact ⟨second_branch_regulator_eq κ β a U z (hsecond rfl).1,
+      second_branch_regulator_le_one κ β a U z hβ (hsecond rfl).2⟩
+
+/-- Threshold status for the actual stopping-time phase supplies the branch
+bounds; no separate regulator assumption is needed. -/
+theorem phaseAt_regulator_eq_and_le_one {Ω ι : Type*}
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω) (i : ι)
+    (hρ : Monotone (fun k => ρ (i,k) ω)) (t : ℝ≥0)
+    (κ β a U : ℝ) (hβ : 0 ≤ β) (z : ℂ × ℝ)
+    (hactive : phaseActive (phaseAt ρ t ω i))
+    (hstart : ρ (i,0) ω ≤ t → 0 ≤ logFactorClock κ z-a)
+    (hbefore : t < ρ (i,1) ω → logFactorClock κ z-a ≤ U/2)
+    (hafter : ρ (i,1) ω ≤ t → U/2 ≤ logFactorClock κ z-a)
+    (hend : t < ρ (i,2) ω → logFactorClock κ z-a ≤ U) :
+    branchRegulator κ (phaseRegulatorScale β a U (phaseAt ρ t ω i))
+      (phaseRegulatorSlope β (phaseAt ρ t ω i)) z =
+      regulator β U (logFactorClock κ z-a) ∧
+    branchRegulator κ (phaseRegulatorScale β a U (phaseAt ρ t ω i))
+      (phaseRegulatorSlope β (phaseAt ρ t ω i)) z ≤ 1 := by
+  apply phaseRegulator_eq_and_le_one κ β a U hβ _ hactive z
+  · intro h
+    obtain ⟨h0,h1⟩ := (phaseAt_first_iff ρ ω i hρ t).mp h
+    exact ⟨hstart h0,hbefore h1⟩
+  · intro h
+    obtain ⟨h1,h2⟩ := (phaseAt_second_iff ρ ω i hρ t).mp h
+    exact ⟨hafter h1,hend h2⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalBudget.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalBudget.lean
index 084ad2b7eb730bca3ae5d6b3211fe77340ab8cd0..c0580c2d2219d2f1ca21160057a814f0f7aa1b82 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalBudget.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalBudget.lean
@@ -1,94 +1,94 @@
-import OAI.Probability.SLE.Multipoint.MultipointPhysicalGeometry
-import OAI.Probability.SLE.Multipoint.MultipointLoewnerDrop
-
-namespace OAI
-
-/-! The actual physical-domain interaction budget. This statement uses
-capacity-two time throughout: the radius clocks satisfy `R = b exp(-4u)`,
-and the Green derivative has coefficient four. No bound on the derivative
-of the original conformal map is assumed. -/
-
-noncomputable section
-open Set Metric Filter MeasureTheory Complex
-open scoped Topology NNReal ComplexConjugate
-
-namespace SLEExactGauge
-
-/-- A conformal-radius upper bound suffices for geometric interaction
-decay; the physical clock need not be specified by equality. -/
-theorem IsCapacityTwoWitness.physical_interaction_decay_of_radius_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (t : ℝ≥0) (z w : ℂ) (hz : z ∈ survivingDomain γ t)
-    (hw' : w ∈ survivingDomain γ t) (hne : z ≠ w)
-    (b a u : ℝ) (hb : b ≤ dist (J z) (J w))
-    (hR : physicalConformalRadius J G t z ≤ b * Real.exp (-2 * a * u)) :
-    halfPlaneGreen (G t z) (G t w) ≤ Real.exp (-a * u) := by
-  let b' := physicalConformalRadius J G t z / Real.exp (-2 * a * u)
-  have hb' : b' ≤ dist (J z) (J w) := by
-    apply (div_le_iff₀ (Real.exp_pos _)).mpr
-    exact hR.trans (mul_le_mul_of_nonneg_right hb (Real.exp_pos _).le)
-  apply hw.physical_interaction_decay hgrowth J hJ hJinj t z w hz hw' hne b' a u hb'
-  exact (div_mul_cancel₀ _ (Real.exp_ne_zero _)).symm
-
-lemma halfPlaneGreen_symm (z w : ℂ) : halfPlaneGreen z w = halfPlaneGreen w z := by
-  have hn : ‖z - conj w‖ = ‖w - conj z‖ := by
-    calc
-      _ = ‖conj (z - conj w)‖ := (Complex.norm_conj _).symm
-      _ = ‖-(w - conj z)‖ := by
-        congr 1
-        simp only [map_sub, Complex.conj_conj]
-        ring
-      _ = _ := norm_neg _
-  unfold halfPlaneGreen
-  rw [norm_div, norm_div, hn, norm_sub_rev z w]
-
-/-- Both radius caps yield the decay in the maximum of the two clocks;
-the proved Green drop then pays for every weighted interaction shell. -/
-theorem IsCapacityTwoWitness.physical_pair_interaction_budget
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (s T : ℝ≥0) (hsT : s ≤ T) (z w : ℂ)
-    (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T) (hne : z ≠ w)
-    (bz bw : ℝ) (hbz : bz ≤ dist (J z) (J w)) (hbw : bw ≤ dist (J z) (J w))
-    {c : ℝ} (hc : 0 ≤ c) (hca : 2 * c < 2) (u v : ℝ → ℝ)
-    (hu : ContinuousOn u (Icc (s : ℝ) (T : ℝ)))
-    (hv : ContinuousOn v (Icc (s : ℝ) (T : ℝ)))
-    (hu0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ u t)
-    (hv0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ v t)
-    (hRu : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      physicalConformalRadius J G t.toNNReal z ≤ bz * Real.exp (-4 * u t))
-    (hRv : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      physicalConformalRadius J G t.toNNReal w ≤ bw * Real.exp (-4 * v t)) :
-    (∫ t in (s : ℝ)..(T : ℝ), Real.exp (c * (u t + v t)) *
-      (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
-        Multipoint.interactionBudget 2 c := by
-  have hdecay : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      halfPlaneGreen (G t.toNNReal z) (G t.toNNReal w) ≤
-        Real.exp (-2 * max (u t) (v t)) := by
-    intro t ht
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    have hzt := survivingDomain_antitone γ htT hz
-    have hwt := survivingDomain_antitone γ htT hw'
-    have hdz := hw.physical_interaction_decay_of_radius_le hgrowth J hJ hJinj t.toNNReal z w
-      hzt hwt hne bz 2 (u t) hbz (by
-        simpa only [show (-2 : ℝ) * 2 = -4 by norm_num] using hRu t ht)
-    have hdw := hw.physical_interaction_decay_of_radius_le hgrowth J hJ hJinj t.toNNReal w z
-      hwt hzt hne.symm bw 2 (v t) (by simpa only [dist_comm] using hbw) (by
-        simpa only [show (-2 : ℝ) * 2 = -4 by norm_num] using hRv t ht)
-    rw [halfPlaneGreen_symm (G t.toNNReal w) (G t.toNNReal z)] at hdw
-    exact Multipoint.interaction_decay_max hdz hdw
-  simpa only [one_mul] using hw.weighted_interaction_integral_le hU s T hsT z w hz hw' hne
-    (M := 1) zero_le_one hc hca u v hu hv hu0 hv0
-    (by simpa only [one_mul] using hdecay)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPhysicalGeometry
+import OAI.Probability.SLE.Multipoint.MultipointLoewnerDrop
+
+namespace OAI
+
+/-! The actual physical-domain interaction budget. This statement uses
+capacity-two time throughout: the radius clocks satisfy `R = b exp(-4u)`,
+and the Green derivative has coefficient four. No bound on the derivative
+of the original conformal map is assumed. -/
+
+noncomputable section
+open Set Metric Filter MeasureTheory Complex
+open scoped Topology NNReal ComplexConjugate
+
+namespace SLEExactGauge
+
+/-- A conformal-radius upper bound suffices for geometric interaction
+decay; the physical clock need not be specified by equality. -/
+theorem IsCapacityTwoWitness.physical_interaction_decay_of_radius_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (t : ℝ≥0) (z w : ℂ) (hz : z ∈ survivingDomain γ t)
+    (hw' : w ∈ survivingDomain γ t) (hne : z ≠ w)
+    (b a u : ℝ) (hb : b ≤ dist (J z) (J w))
+    (hR : physicalConformalRadius J G t z ≤ b * Real.exp (-2 * a * u)) :
+    halfPlaneGreen (G t z) (G t w) ≤ Real.exp (-a * u) := by
+  let b' := physicalConformalRadius J G t z / Real.exp (-2 * a * u)
+  have hb' : b' ≤ dist (J z) (J w) := by
+    apply (div_le_iff₀ (Real.exp_pos _)).mpr
+    exact hR.trans (mul_le_mul_of_nonneg_right hb (Real.exp_pos _).le)
+  apply hw.physical_interaction_decay hgrowth J hJ hJinj t z w hz hw' hne b' a u hb'
+  exact (div_mul_cancel₀ _ (Real.exp_ne_zero _)).symm
+
+lemma halfPlaneGreen_symm (z w : ℂ) : halfPlaneGreen z w = halfPlaneGreen w z := by
+  have hn : ‖z - conj w‖ = ‖w - conj z‖ := by
+    calc
+      _ = ‖conj (z - conj w)‖ := (Complex.norm_conj _).symm
+      _ = ‖-(w - conj z)‖ := by
+        congr 1
+        simp only [map_sub, Complex.conj_conj]
+        ring
+      _ = _ := norm_neg _
+  unfold halfPlaneGreen
+  rw [norm_div, norm_div, hn, norm_sub_rev z w]
+
+/-- Both radius caps yield the decay in the maximum of the two clocks;
+the proved Green drop then pays for every weighted interaction shell. -/
+theorem IsCapacityTwoWitness.physical_pair_interaction_budget
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (s T : ℝ≥0) (hsT : s ≤ T) (z w : ℂ)
+    (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T) (hne : z ≠ w)
+    (bz bw : ℝ) (hbz : bz ≤ dist (J z) (J w)) (hbw : bw ≤ dist (J z) (J w))
+    {c : ℝ} (hc : 0 ≤ c) (hca : 2 * c < 2) (u v : ℝ → ℝ)
+    (hu : ContinuousOn u (Icc (s : ℝ) (T : ℝ)))
+    (hv : ContinuousOn v (Icc (s : ℝ) (T : ℝ)))
+    (hu0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ u t)
+    (hv0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ v t)
+    (hRu : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      physicalConformalRadius J G t.toNNReal z ≤ bz * Real.exp (-4 * u t))
+    (hRv : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      physicalConformalRadius J G t.toNNReal w ≤ bw * Real.exp (-4 * v t)) :
+    (∫ t in (s : ℝ)..(T : ℝ), Real.exp (c * (u t + v t)) *
+      (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
+        Multipoint.interactionBudget 2 c := by
+  have hdecay : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      halfPlaneGreen (G t.toNNReal z) (G t.toNNReal w) ≤
+        Real.exp (-2 * max (u t) (v t)) := by
+    intro t ht
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    have hzt := survivingDomain_antitone γ htT hz
+    have hwt := survivingDomain_antitone γ htT hw'
+    have hdz := hw.physical_interaction_decay_of_radius_le hgrowth J hJ hJinj t.toNNReal z w
+      hzt hwt hne bz 2 (u t) hbz (by
+        simpa only [show (-2 : ℝ) * 2 = -4 by norm_num] using hRu t ht)
+    have hdw := hw.physical_interaction_decay_of_radius_le hgrowth J hJ hJinj t.toNNReal w z
+      hwt hzt hne.symm bw 2 (v t) (by simpa only [dist_comm] using hbw) (by
+        simpa only [show (-2 : ℝ) * 2 = -4 by norm_num] using hRv t ht)
+    rw [halfPlaneGreen_symm (G t.toNNReal w) (G t.toNNReal z)] at hdw
+    exact Multipoint.interaction_decay_max hdz hdw
+  simpa only [one_mul] using hw.weighted_interaction_integral_le hU s T hsT z w hz hw' hne
+    (M := 1) zero_le_one hc hca u v hu hv hu0 hv0
+    (by simpa only [one_mul] using hdecay)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalGeometry.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalGeometry.lean
index 476677144817a37af1a8dadf9b88e14f4882277e..c45ab16ccfff93a2a6bce7979678370038beaf17 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalGeometry.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalGeometry.lean
@@ -1,117 +1,117 @@
-import OAI.Probability.SLE.Multipoint.MultipointGeometry
-
-namespace OAI
-
-/-! Disk uniformizations of the actual surviving physical domains.
-The radius includes the derivative of the original conformal map, with
-no uniform bound on that derivative. -/
-
-noncomputable section
-open Set Metric Filter
-open scoped Topology NNReal
-
-namespace SLEExactGauge
-
-def physicalConformalRadius (J : ℂ → ℂ) (G : ℝ≥0 → ℂ → ℂ)
-    (t : ℝ≥0) (z : ℂ) : ℝ := ‖deriv J z‖ * fullRadius G t z
-
-def physicalDiskChart (J : ℂ → ℂ) (F G : ℝ≥0 → ℂ → ℂ)
-    (t : ℝ≥0) (z : ℂ) : ℂ → ℂ := J ∘ F t ∘ diskToHalfPlane (G t z)
-
-theorem IsCapacityTwoWitness.physical_disk_chart
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) :
-    DifferentiableOn ℂ (physicalDiskChart J F G t z) (ball 0 1) ∧
-      InjOn (physicalDiskChart J F G t z) (ball 0 1) ∧
-      physicalDiskChart J F G t z 0 = J z ∧
-      ‖deriv (physicalDiskChart J F G t z) 0‖ = physicalConformalRadius J G t z ∧
-      ∀ w ∈ survivingDomain γ t,
-        physicalDiskChart J F G t z (halfPlaneToDisk (G t z) (G t w)) = J w := by
-  have hγ := h.1
-  rcases h with ⟨_, _, _, hG, hF, hGmap, hFmap, hinv, hinv', _, _, _, _⟩
-  have hy : 0 < (G t z).im := hGmap t hz
-  let f : ℂ → ℂ := F t ∘ diskToHalfPlane (G t z)
-  have hfd : DifferentiableOn ℂ f (ball 0 1) :=
-    (hF t).comp (diskToHalfPlane_differentiable _) (diskToHalfPlane_maps hy)
-  have hfmap : MapsTo f (ball 0 1) upperHalfPlane := by
-    intro v hv
-    exact (hFmap t (diskToHalfPlane_maps hy hv)).1.1
-  have hfi : InjOn f (ball 0 1) := by
-    intro v hv w hw he
-    apply diskToHalfPlane_injOn hy hv hw
-    have he' := congrArg (G t) he
-    change G t (F t (diskToHalfPlane (G t z) v)) =
-      G t (F t (diskToHalfPlane (G t z) w)) at he'
-    rwa [hinv' t _ (diskToHalfPlane_maps hy hv),
-      hinv' t _ (diskToHalfPlane_maps hy hw)] at he'
-  have hf0 : f 0 = z := by simp only [f, Function.comp_apply, diskToHalfPlane_zero, hinv t z hz]
-  have hdG := ((hG t).differentiableAt ((isOpen_survivingDomain hγ t).mem_nhds hz)).hasDerivAt
-  have hdF := ((hF t).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hGmap t hz))).hasDerivAt
-  have hident : (fun v => F t (G t v)) =ᶠ[𝓝 z] (fun v => v) := by
-    filter_upwards [(isOpen_survivingDomain hγ t).mem_nhds hz] with v hv
-    exact hinv t v hv
-  have hprod : deriv (F t) (G t z) * deriv (G t) z = 1 :=
-    ((hdF.comp z hdG).congr_of_eventuallyEq hident.symm).unique (hasDerivAt_id z)
-  have hnormprod : ‖deriv (F t) (G t z)‖ * ‖deriv (G t) z‖ = 1 := by
-    simpa only [norm_mul, norm_one] using congrArg norm hprod
-  have hgn : ‖deriv (G t) z‖ ≠ 0 := by
-    intro he
-    rw [he, mul_zero] at hnormprod
-    norm_num at hnormprod
-  have hdF' : HasDerivAt (F t) (deriv (F t) (G t z)) (diskToHalfPlane (G t z) 0) := by
-    simpa only [diskToHalfPlane_zero] using hdF
-  have hdf : HasDerivAt f (deriv (F t) (G t z) * (G t z-star (G t z))) 0 :=
-    hdF'.comp 0 (diskToHalfPlane_deriv_zero _)
-  have hfrad : ‖deriv f 0‖ = fullRadius G t z := by
-    rw [hdf.deriv, norm_mul, norm_sub_conj_of_im_pos hy, fullRadius]
-    apply (eq_div_iff hgn).mpr
-    nlinarith [hnormprod]
-  have hdJ := (hJ.differentiableAt (isOpen_upperHalfPlane.mem_nhds hz.1.1)).hasDerivAt
-  have hdcomp : HasDerivAt (J ∘ f) (deriv J z * deriv f 0) 0 := by
-    have hdJ' : HasDerivAt J (deriv J z) (f 0) := by rwa [hf0]
-    have hh := hdJ'.comp 0 hdf
-    simpa only [hdf.deriv] using hh
-  refine ⟨hJ.comp hfd hfmap, hJinj.comp hfi hfmap, ?_, ?_, ?_⟩
-  · exact congrArg J hf0
-  · change ‖deriv (J ∘ f) 0‖ = _
-    rw [hdcomp.deriv, norm_mul, hfrad]
-    rfl
-  · intro w hw
-    dsimp only [physicalDiskChart, Function.comp_apply]
-    rw [diskToHalfPlane_halfPlaneToDisk hy (hGmap t hw), hinv t w hw]
-
-theorem IsCapacityTwoWitness.physical_interaction_decay
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (t : ℝ≥0) (z w : ℂ) (hz : z ∈ survivingDomain γ t)
-    (hw : w ∈ survivingDomain γ t) (hne : z ≠ w)
-    (b a u : ℝ) (hb : b ≤ dist (J z) (J w))
-    (hR : physicalConformalRadius J G t z = b*Real.exp (-2*a*u)) :
-    halfPlaneGreen (G t z) (G t w) ≤ Real.exp (-a*u) := by
-  obtain ⟨hfd, hfi, hf0, hfrad, hfpoint⟩ := h.physical_disk_chart J hJ hJinj t z hz
-  let ξ := halfPlaneToDisk (G t z) (G t w)
-  have hξ : ξ ∈ ball (0 : ℂ) 1 :=
-    halfPlaneToDisk_maps (h.imag_positive t z hz) (h.imag_positive t w hw)
-  have hξ0 : ξ ≠ 0 := by
-    intro he
-    have hv := hfpoint w hw
-    change physicalDiskChart J F G t z ξ = J w at hv
-    rw [he, hf0] at hv
-    exact hne (hJinj hz.1.1 hw.1.1 hv)
-  rw [halfPlaneGreen_eq_disk]
-  apply Multipoint.disk_interaction_decay hgrowth _ hfd hfi ξ hξ hξ0 b a u
-  · rw [hfpoint w hw, hf0, norm_sub_rev, ← dist_eq_norm]
-    exact hb
-  · exact hfrad.trans hR
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointGeometry
+
+namespace OAI
+
+/-! Disk uniformizations of the actual surviving physical domains.
+The radius includes the derivative of the original conformal map, with
+no uniform bound on that derivative. -/
+
+noncomputable section
+open Set Metric Filter
+open scoped Topology NNReal
+
+namespace SLEExactGauge
+
+def physicalConformalRadius (J : ℂ → ℂ) (G : ℝ≥0 → ℂ → ℂ)
+    (t : ℝ≥0) (z : ℂ) : ℝ := ‖deriv J z‖ * fullRadius G t z
+
+def physicalDiskChart (J : ℂ → ℂ) (F G : ℝ≥0 → ℂ → ℂ)
+    (t : ℝ≥0) (z : ℂ) : ℂ → ℂ := J ∘ F t ∘ diskToHalfPlane (G t z)
+
+theorem IsCapacityTwoWitness.physical_disk_chart
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) :
+    DifferentiableOn ℂ (physicalDiskChart J F G t z) (ball 0 1) ∧
+      InjOn (physicalDiskChart J F G t z) (ball 0 1) ∧
+      physicalDiskChart J F G t z 0 = J z ∧
+      ‖deriv (physicalDiskChart J F G t z) 0‖ = physicalConformalRadius J G t z ∧
+      ∀ w ∈ survivingDomain γ t,
+        physicalDiskChart J F G t z (halfPlaneToDisk (G t z) (G t w)) = J w := by
+  have hγ := h.1
+  rcases h with ⟨_, _, _, hG, hF, hGmap, hFmap, hinv, hinv', _, _, _, _⟩
+  have hy : 0 < (G t z).im := hGmap t hz
+  let f : ℂ → ℂ := F t ∘ diskToHalfPlane (G t z)
+  have hfd : DifferentiableOn ℂ f (ball 0 1) :=
+    (hF t).comp (diskToHalfPlane_differentiable _) (diskToHalfPlane_maps hy)
+  have hfmap : MapsTo f (ball 0 1) upperHalfPlane := by
+    intro v hv
+    exact (hFmap t (diskToHalfPlane_maps hy hv)).1.1
+  have hfi : InjOn f (ball 0 1) := by
+    intro v hv w hw he
+    apply diskToHalfPlane_injOn hy hv hw
+    have he' := congrArg (G t) he
+    change G t (F t (diskToHalfPlane (G t z) v)) =
+      G t (F t (diskToHalfPlane (G t z) w)) at he'
+    rwa [hinv' t _ (diskToHalfPlane_maps hy hv),
+      hinv' t _ (diskToHalfPlane_maps hy hw)] at he'
+  have hf0 : f 0 = z := by simp only [f, Function.comp_apply, diskToHalfPlane_zero, hinv t z hz]
+  have hdG := ((hG t).differentiableAt ((isOpen_survivingDomain hγ t).mem_nhds hz)).hasDerivAt
+  have hdF := ((hF t).differentiableAt (isOpen_upperHalfPlane.mem_nhds (hGmap t hz))).hasDerivAt
+  have hident : (fun v => F t (G t v)) =ᶠ[𝓝 z] (fun v => v) := by
+    filter_upwards [(isOpen_survivingDomain hγ t).mem_nhds hz] with v hv
+    exact hinv t v hv
+  have hprod : deriv (F t) (G t z) * deriv (G t) z = 1 :=
+    ((hdF.comp z hdG).congr_of_eventuallyEq hident.symm).unique (hasDerivAt_id z)
+  have hnormprod : ‖deriv (F t) (G t z)‖ * ‖deriv (G t) z‖ = 1 := by
+    simpa only [norm_mul, norm_one] using congrArg norm hprod
+  have hgn : ‖deriv (G t) z‖ ≠ 0 := by
+    intro he
+    rw [he, mul_zero] at hnormprod
+    norm_num at hnormprod
+  have hdF' : HasDerivAt (F t) (deriv (F t) (G t z)) (diskToHalfPlane (G t z) 0) := by
+    simpa only [diskToHalfPlane_zero] using hdF
+  have hdf : HasDerivAt f (deriv (F t) (G t z) * (G t z-star (G t z))) 0 :=
+    hdF'.comp 0 (diskToHalfPlane_deriv_zero _)
+  have hfrad : ‖deriv f 0‖ = fullRadius G t z := by
+    rw [hdf.deriv, norm_mul, norm_sub_conj_of_im_pos hy, fullRadius]
+    apply (eq_div_iff hgn).mpr
+    nlinarith [hnormprod]
+  have hdJ := (hJ.differentiableAt (isOpen_upperHalfPlane.mem_nhds hz.1.1)).hasDerivAt
+  have hdcomp : HasDerivAt (J ∘ f) (deriv J z * deriv f 0) 0 := by
+    have hdJ' : HasDerivAt J (deriv J z) (f 0) := by rwa [hf0]
+    have hh := hdJ'.comp 0 hdf
+    simpa only [hdf.deriv] using hh
+  refine ⟨hJ.comp hfd hfmap, hJinj.comp hfi hfmap, ?_, ?_, ?_⟩
+  · exact congrArg J hf0
+  · change ‖deriv (J ∘ f) 0‖ = _
+    rw [hdcomp.deriv, norm_mul, hfrad]
+    rfl
+  · intro w hw
+    dsimp only [physicalDiskChart, Function.comp_apply]
+    rw [diskToHalfPlane_halfPlaneToDisk hy (hGmap t hw), hinv t w hw]
+
+theorem IsCapacityTwoWitness.physical_interaction_decay
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (t : ℝ≥0) (z w : ℂ) (hz : z ∈ survivingDomain γ t)
+    (hw : w ∈ survivingDomain γ t) (hne : z ≠ w)
+    (b a u : ℝ) (hb : b ≤ dist (J z) (J w))
+    (hR : physicalConformalRadius J G t z = b*Real.exp (-2*a*u)) :
+    halfPlaneGreen (G t z) (G t w) ≤ Real.exp (-a*u) := by
+  obtain ⟨hfd, hfi, hf0, hfrad, hfpoint⟩ := h.physical_disk_chart J hJ hJinj t z hz
+  let ξ := halfPlaneToDisk (G t z) (G t w)
+  have hξ : ξ ∈ ball (0 : ℂ) 1 :=
+    halfPlaneToDisk_maps (h.imag_positive t z hz) (h.imag_positive t w hw)
+  have hξ0 : ξ ≠ 0 := by
+    intro he
+    have hv := hfpoint w hw
+    change physicalDiskChart J F G t z ξ = J w at hv
+    rw [he, hf0] at hv
+    exact hne (hJinj hz.1.1 hw.1.1 hv)
+  rw [halfPlaneGreen_eq_disk]
+  apply Multipoint.disk_interaction_decay hgrowth _ hfd hfi ξ hξ hξ0 b a u
+  · rw [hfpoint w hw, hf0, norm_sub_rev, ← dist_eq_norm]
+    exact hb
+  · exact hfrad.trans hR
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalHits.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalHits.lean
index 1ed109d8bb522265cf61dee733ebe2fbb1162e97..23e12b1e4a113a8a6f7d34e7bbb5b870a4853dd1 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalHits.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalHits.lean
@@ -1,79 +1,79 @@
-import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus
-
-namespace OAI
-
-/-! The finite-radius-stop event is exactly the physical conformal-radius
-hit event in the original conformal domain. -/
-
-noncomputable section
-open Set Filter
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge
-
-private lemma physical_radius_level_value
-    (J : ℂ → ℂ) (z : ℂ) (R e : ℝ) (hR : 0 < R) (he : 0 < e)
-    (hRdef : R = 2*z.im*‖deriv J z‖) :
-    ‖deriv J z‖ * ((2*z.im)*Real.exp (-Real.log (R/e))) = e := by
-  have hexp : Real.exp (-Real.log (R/e)) = e/R := by
-    rw [Real.exp_neg, Real.exp_log (div_pos hR he)]
-    field_simp [hR.ne',he.ne']
-  have hcoef : ‖deriv J z‖*(2*z.im) = R := by rw [hRdef]; ring
-  rw [hexp,← mul_assoc,hcoef]
-  field_simp [hR.ne']
-
-theorem physical_radius_eq_hit_iff_finite_level
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (J : ℂ → ℂ) (z : ℂ) (R e : ℝ) (hR : 0 < R) (he : 0 < e)
-    (hRdef : R = 2*z.im*‖deriv J z‖) :
-    (∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧ physicalConformalRadius J G t z = e) ↔
-      radiusLevelTime γ G z (Real.log (R/e)) < ⊤ := by
-  have hc : 0 < ‖deriv J z‖ := by
-    by_contra hn
-    have hz : ‖deriv J z‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
-    rw [hz,mul_zero] at hRdef
-    linarith
-  have htarget := physical_radius_level_value J z R e hR he hRdef
-  constructor
-  · rintro ⟨t,ht,htR⟩
-    have hfull : fullRadius G t z = (2*z.im)*Real.exp (-Real.log (R/e)) :=
-      (mul_left_cancel₀ hc.ne') (htR.trans htarget.symm)
-    rw [radiusLevelTime_eq_of_live_level hU hw z (Real.log (R/e)) t ht hfull]
-    exact ENNReal.coe_lt_top
-  · intro hf
-    obtain ⟨t,ht,htR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z (Real.log (R/e)) hf
-    exact ⟨t,ht,by rw [physicalConformalRadius,htR]; exact htarget⟩
-
-theorem physical_radius_hit_iff_finite_level
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (J : ℂ → ℂ) (z : ℂ) (R e : ℝ) (hR : 0 < R) (he : 0 < e) (heR : e ≤ R)
-    (hRdef : R = 2*z.im*‖deriv J z‖) :
-    (∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧ physicalConformalRadius J G t z ≤ e) ↔
-      radiusLevelTime γ G z (Real.log (R/e)) < ⊤ := by
-  constructor
-  · rintro ⟨t,ht,htR⟩
-    rcases lt_or_eq_of_le htR with hlt | heq
-    · have hc : 0 < ‖deriv J z‖ := by
-        by_contra hn
-        have hz : ‖deriv J z‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
-        rw [hz,mul_zero] at hRdef
-        linarith
-      have hlog : 0 ≤ Real.log (R/e) := Real.log_nonneg
-        ((le_div_iff₀ he).mpr (by simpa only [one_mul] using heR))
-      have htarget := physical_radius_level_value J z R e hR he hRdef
-      apply radiusLevelTime_finite_of_live_radius_lt hU hw z (Real.log (R/e)) hlog t ht
-      apply (mul_lt_mul_iff_right₀ hc).mp
-      exact hlt.trans_eq htarget.symm
-    · exact (physical_radius_eq_hit_iff_finite_level hU hw J z R e hR he hRdef).mp
-        ⟨t,ht,heq⟩
-  · intro hf
-    obtain ⟨t,ht,hR⟩ := (physical_radius_eq_hit_iff_finite_level hU hw J z R e hR he hRdef).mpr hf
-    exact ⟨t,ht,hR.le⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus
+
+namespace OAI
+
+/-! The finite-radius-stop event is exactly the physical conformal-radius
+hit event in the original conformal domain. -/
+
+noncomputable section
+open Set Filter
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge
+
+private lemma physical_radius_level_value
+    (J : ℂ → ℂ) (z : ℂ) (R e : ℝ) (hR : 0 < R) (he : 0 < e)
+    (hRdef : R = 2*z.im*‖deriv J z‖) :
+    ‖deriv J z‖ * ((2*z.im)*Real.exp (-Real.log (R/e))) = e := by
+  have hexp : Real.exp (-Real.log (R/e)) = e/R := by
+    rw [Real.exp_neg, Real.exp_log (div_pos hR he)]
+    field_simp [hR.ne',he.ne']
+  have hcoef : ‖deriv J z‖*(2*z.im) = R := by rw [hRdef]; ring
+  rw [hexp,← mul_assoc,hcoef]
+  field_simp [hR.ne']
+
+theorem physical_radius_eq_hit_iff_finite_level
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (J : ℂ → ℂ) (z : ℂ) (R e : ℝ) (hR : 0 < R) (he : 0 < e)
+    (hRdef : R = 2*z.im*‖deriv J z‖) :
+    (∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧ physicalConformalRadius J G t z = e) ↔
+      radiusLevelTime γ G z (Real.log (R/e)) < ⊤ := by
+  have hc : 0 < ‖deriv J z‖ := by
+    by_contra hn
+    have hz : ‖deriv J z‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
+    rw [hz,mul_zero] at hRdef
+    linarith
+  have htarget := physical_radius_level_value J z R e hR he hRdef
+  constructor
+  · rintro ⟨t,ht,htR⟩
+    have hfull : fullRadius G t z = (2*z.im)*Real.exp (-Real.log (R/e)) :=
+      (mul_left_cancel₀ hc.ne') (htR.trans htarget.symm)
+    rw [radiusLevelTime_eq_of_live_level hU hw z (Real.log (R/e)) t ht hfull]
+    exact ENNReal.coe_lt_top
+  · intro hf
+    obtain ⟨t,ht,htR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z (Real.log (R/e)) hf
+    exact ⟨t,ht,by rw [physicalConformalRadius,htR]; exact htarget⟩
+
+theorem physical_radius_hit_iff_finite_level
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (J : ℂ → ℂ) (z : ℂ) (R e : ℝ) (hR : 0 < R) (he : 0 < e) (heR : e ≤ R)
+    (hRdef : R = 2*z.im*‖deriv J z‖) :
+    (∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧ physicalConformalRadius J G t z ≤ e) ↔
+      radiusLevelTime γ G z (Real.log (R/e)) < ⊤ := by
+  constructor
+  · rintro ⟨t,ht,htR⟩
+    rcases lt_or_eq_of_le htR with hlt | heq
+    · have hc : 0 < ‖deriv J z‖ := by
+        by_contra hn
+        have hz : ‖deriv J z‖ = 0 := le_antisymm (le_of_not_gt hn) (norm_nonneg _)
+        rw [hz,mul_zero] at hRdef
+        linarith
+      have hlog : 0 ≤ Real.log (R/e) := Real.log_nonneg
+        ((le_div_iff₀ he).mpr (by simpa only [one_mul] using heR))
+      have htarget := physical_radius_level_value J z R e hR he hRdef
+      apply radiusLevelTime_finite_of_live_radius_lt hU hw z (Real.log (R/e)) hlog t ht
+      apply (mul_lt_mul_iff_right₀ hc).mp
+      exact hlt.trans_eq htarget.symm
+    · exact (physical_radius_eq_hit_iff_finite_level hU hw J z R e hR he hRdef).mp
+        ⟨t,ht,heq⟩
+  · intro hf
+    obtain ⟨t,ht,hR⟩ := (physical_radius_eq_hit_iff_finite_level hU hw J z R e hR he hRdef).mpr hf
+    exact ⟨t,ht,hR.le⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalProbability.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalProbability.lean
index f9124d0ac23f3041735c639c32fda5a454327bd5..cc9d52f31e6e10a83a8606c972ed78e9fd9aff00 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalProbability.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPhysicalProbability.lean
@@ -1,113 +1,113 @@
-import OAI.Probability.SLE.Multipoint.MultipointActualProbability
-import OAI.Probability.SLE.Multipoint.MultipointPhysicalHits
-import OAI.Probability.SLE.Multipoint.MultipointBudgetMonotone
-
-namespace OAI
-
-/-! Physical-domain multipoint approach bounds, including arbitrary
-positive targets through the paper's capped product. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem physical_radius_multipoint_bound
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
-    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (R b e : ι → ℝ) (he : ∀ i, 0 < e i) (heb : ∀ i, e i ≤ b i)
-    (hbR : ∀ i, b i ≤ R i) (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
-    P {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius J (G ω) t (z i) ≤ e i} ≤
-      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
-        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
-        ∏ i, (e i/b i)^(1-κ/8)) := by
-  have heq : {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius J (G ω) t (z i) ≤ e i} =
-      {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (Real.log (R i/e i)) < ⊤} := by
-    ext ω
-    apply forall_congr'
-    intro i
-    exact physical_radius_hit_iff_finite_level (continuous_const.mul (hBc ω)) (hw ω)
-      J (z i) (R i) (e i) ((he i).trans_le ((heb i).trans (hbR i))) (he i)
-      ((heb i).trans (hbR i)) (hRdef i)
-  rw [heq]
-  exact actual_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth J
-    hJ hJinj z hz hne R b e he heb hbR hRdef hsep
-
-theorem capped_physical_radius_multipoint_bound
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
-    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
-    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
-    (R b e : ι → ℝ) (he : ∀ i, 0 < e i) (hb : ∀ i, 0 < b i)
-    (hbR : ∀ i, b i ≤ R i) (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
-    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
-    P {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius J (G ω) t (z i) ≤ e i} ≤
-      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
-        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
-        ∏ i, min 1 ((e i/b i)^(1-κ/8))) := by
-  classical
-  let I := Finset.univ.filter (fun i => e i ≤ b i)
-  let f : I ↪ ι := Function.Embedding.subtype _
-  have himage : (Finset.univ : Finset I).map f = I := by
-    ext i
-    simp only [Finset.mem_map,Finset.mem_univ,true_and]
-    exact ⟨fun ⟨j,hj⟩ => hj ▸ j.property,fun hi => ⟨⟨i,hi⟩,rfl⟩⟩
-  have hsmall := physical_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw
-    hgrowth J hJ hJinj (fun i : I => z i) (fun i => hz i)
-    (fun i j hij => Subtype.ext (hne hij)) (fun i : I => R i) (fun i : I => b i)
-    (fun i : I => e i) (fun i => he i) (fun i => (Finset.mem_filter.mp i.property).2)
-    (fun i => hbR i) (fun i => hRdef i)
-    (fun i j hij => hsep i j (fun h => hij (Subtype.ext h)))
-  have hsub : {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-      physicalConformalRadius J (G ω) t (z i) ≤ e i} ⊆
-      {ω | ∀ i : I, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
-        physicalConformalRadius J (G ω) t (z i) ≤ e i} := fun _ h i => h i
-  apply (measure_mono hsub).trans (hsmall.trans (ENNReal.ofReal_le_ofReal ?_))
-  have hK : uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset I) ≤
-      uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι) := by
-    rw [← uniformDiscountBudget_map κ (sourceBeta κ) f,himage]
-    exact uniformDiscountBudget_mono hκ hκ8 (Finset.subset_univ I)
-  have hcard : Fintype.card I ≤ Fintype.card ι := by
-    simpa only [Fintype.card_coe,Finset.card_univ] using Finset.card_le_card (Finset.subset_univ I)
-  have hα : 0 ≤ 1-κ/8 := by linarith
-  have hprod : (∏ i : I, (e i/b i)^(1-κ/8)) = ∏ i, min 1 ((e i/b i)^(1-κ/8)) := by
-    rw [Finset.prod_coe_sort I (fun i : ι => (e i/b i)^(1-κ/8))]
-    dsimp only [I]
-    rw [Finset.prod_filter]
-    apply Finset.prod_congr rfl
-    intro i _
-    by_cases hi : e i ≤ b i
-    · rw [ite_eq_left hi,min_eq_right (Real.rpow_le_one (div_nonneg (he i).le (hb i).le)
-        ((div_le_one (hb i)).mpr hi) hα)]
-    · rw [ite_eq_right hi,min_eq_left (Real.one_le_rpow
-        ((one_le_div (hb i)).mpr (le_of_not_ge hi)) hα)]
-  rw [← hprod]
-  apply mul_le_mul_of_nonneg_right _ (Finset.prod_nonneg (fun (i : I) _ => Real.rpow_nonneg
-    (div_nonneg (he i).le (hb i).le) _))
-  exact mul_le_mul (pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 3) hcard)
-    (Real.exp_le_exp.mpr hK) (Real.exp_pos _).le (by positivity)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointActualProbability
+import OAI.Probability.SLE.Multipoint.MultipointPhysicalHits
+import OAI.Probability.SLE.Multipoint.MultipointBudgetMonotone
+
+namespace OAI
+
+/-! Physical-domain multipoint approach bounds, including arbitrary
+positive targets through the paper's capped product. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem physical_radius_multipoint_bound
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
+    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (R b e : ι → ℝ) (he : ∀ i, 0 < e i) (heb : ∀ i, e i ≤ b i)
+    (hbR : ∀ i, b i ≤ R i) (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
+    P {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius J (G ω) t (z i) ≤ e i} ≤
+      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
+        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
+        ∏ i, (e i/b i)^(1-κ/8)) := by
+  have heq : {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius J (G ω) t (z i) ≤ e i} =
+      {ω | ∀ i, radiusLevelTime (γ ω) (G ω) (z i) (Real.log (R i/e i)) < ⊤} := by
+    ext ω
+    apply forall_congr'
+    intro i
+    exact physical_radius_hit_iff_finite_level (continuous_const.mul (hBc ω)) (hw ω)
+      J (z i) (R i) (e i) ((he i).trans_le ((heb i).trans (hbR i))) (he i)
+      ((heb i).trans (hbR i)) (hRdef i)
+  rw [heq]
+  exact actual_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth J
+    hJ hJinj z hz hne R b e he heb hbR hRdef hsep
+
+theorem capped_physical_radius_multipoint_bound
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (hgrowth : PublishedDiskUnivalentGrowth) (J : ℂ → ℂ)
+    (hJ : DifferentiableOn ℂ J upperHalfPlane) (hJinj : InjOn J upperHalfPlane)
+    (z : ι → ℂ) (hz : ∀ i, z i ∈ upperHalfPlane) (hne : Function.Injective z)
+    (R b e : ι → ℝ) (he : ∀ i, 0 < e i) (hb : ∀ i, 0 < b i)
+    (hbR : ∀ i, b i ≤ R i) (hRdef : ∀ i, R i = 2*(z i).im*‖deriv J (z i)‖)
+    (hsep : ∀ i j, i ≠ j → b i ≤ dist (J (z i)) (J (z j))) :
+    P {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius J (G ω) t (z i) ≤ e i} ≤
+      ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι) *
+        Real.exp (uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι)) *
+        ∏ i, min 1 ((e i/b i)^(1-κ/8))) := by
+  classical
+  let I := Finset.univ.filter (fun i => e i ≤ b i)
+  let f : I ↪ ι := Function.Embedding.subtype _
+  have himage : (Finset.univ : Finset I).map f = I := by
+    ext i
+    simp only [Finset.mem_map,Finset.mem_univ,true_and]
+    exact ⟨fun ⟨j,hj⟩ => hj ▸ j.property,fun hi => ⟨⟨i,hi⟩,rfl⟩⟩
+  have hsmall := physical_radius_multipoint_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw
+    hgrowth J hJ hJinj (fun i : I => z i) (fun i => hz i)
+    (fun i j hij => Subtype.ext (hne hij)) (fun i : I => R i) (fun i : I => b i)
+    (fun i : I => e i) (fun i => he i) (fun i => (Finset.mem_filter.mp i.property).2)
+    (fun i => hbR i) (fun i => hRdef i)
+    (fun i j hij => hsep i j (fun h => hij (Subtype.ext h)))
+  have hsub : {ω | ∀ i, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+      physicalConformalRadius J (G ω) t (z i) ≤ e i} ⊆
+      {ω | ∀ i : I, ∃ t : ℝ≥0, z i ∈ survivingDomain (γ ω) t ∧
+        physicalConformalRadius J (G ω) t (z i) ≤ e i} := fun _ h i => h i
+  apply (measure_mono hsub).trans (hsmall.trans (ENNReal.ofReal_le_ofReal ?_))
+  have hK : uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset I) ≤
+      uniformDiscountBudget κ (sourceBeta κ) (Finset.univ : Finset ι) := by
+    rw [← uniformDiscountBudget_map κ (sourceBeta κ) f,himage]
+    exact uniformDiscountBudget_mono hκ hκ8 (Finset.subset_univ I)
+  have hcard : Fintype.card I ≤ Fintype.card ι := by
+    simpa only [Fintype.card_coe,Finset.card_univ] using Finset.card_le_card (Finset.subset_univ I)
+  have hα : 0 ≤ 1-κ/8 := by linarith
+  have hprod : (∏ i : I, (e i/b i)^(1-κ/8)) = ∏ i, min 1 ((e i/b i)^(1-κ/8)) := by
+    rw [Finset.prod_coe_sort I (fun i : ι => (e i/b i)^(1-κ/8))]
+    dsimp only [I]
+    rw [Finset.prod_filter]
+    apply Finset.prod_congr rfl
+    intro i _
+    by_cases hi : e i ≤ b i
+    · rw [ite_eq_left hi,min_eq_right (Real.rpow_le_one (div_nonneg (he i).le (hb i).le)
+        ((div_le_one (hb i)).mpr hi) hα)]
+    · rw [ite_eq_right hi,min_eq_left (Real.one_le_rpow
+        ((one_le_div (hb i)).mpr (le_of_not_ge hi)) hα)]
+  rw [← hprod]
+  apply mul_le_mul_of_nonneg_right _ (Finset.prod_nonneg (fun (i : I) _ => Real.rpow_nonneg
+    (div_nonneg (he i).le (hb i).le) _))
+  exact mul_le_mul (pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 3) hcard)
+    (Real.exp_le_exp.mpr hK) (Real.exp_pos _).le (by positivity)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointPiecewiseClock.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointPiecewiseClock.lean
index f590ff289d2ccacf5d31728d1b87db518a2e0f02..4021b2a3ab87405d81188bd2c5a7ccee2d95f825 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointPiecewiseClock.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointPiecewiseClock.lean
@@ -1,294 +1,294 @@
-import OAI.Probability.SLE.Geometry.FiniteStopSorting
-import OAI.Probability.SLE.Multipoint.MultipointDiscountClock
-import OAI.Probability.SLE.Multipoint.MultipointSwitchPatterns
-import Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm
-
-namespace OAI
-
-/-! Bounded discount rates selected at finitely many stopping-time switches.
-The rate may jump at the switches; its accumulated clock is adapted and
-Lipschitz, and its derivative agrees with the rate almost everywhere. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-/-- Sorting all raw switches leaves no original switch strictly between
-successive entries, including when some entries coincide. -/
-lemma sortedStopAt_no_intermediate {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (σ : Ω → ℝ≥0) (hσ : σ ∈ l)
-    (n : ℕ) (ω : Ω) (hn : sortedStopAt l H n ω < σ ω) :
-    sortedStopAt l H (n+1) ω ≤ σ ω := by
-  obtain ⟨k, _, hk⟩ := sortedStopAt_covers l H σ hσ ω
-  have hnk : n+1 ≤ k := by
-    by_contra h
-    have hkn : k ≤ n := by omega
-    have hc := sortedStopAt_monotone l H hl ω hkn
-    change sortedStopAt l H k ω ≤ sortedStopAt l H n ω at hc
-    rw [hk] at hc
-    exact (not_le_of_gt hn) hc
-  exact (sortedStopAt_monotone l H hl ω hnk).trans_eq hk
-
-lemma sortedStopAt_reached_iff {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (σ : Ω → ℝ≥0) (hσ : σ ∈ l)
-    (n : ℕ) (ω : Ω) (t : ℝ≥0) (hnt : sortedStopAt l H n ω ≤ t)
-    (ht : t < sortedStopAt l H (n+1) ω) :
-    σ ω ≤ t ↔ σ ω ≤ sortedStopAt l H n ω := by
-  constructor
-  · intro hσt
-    by_contra h
-    have hnext := sortedStopAt_no_intermediate l H hl σ hσ n ω (lt_of_not_ge h)
-    exact (not_le_of_gt ht) (hnext.trans hσt)
-  · exact fun h => h.trans hnt
-
-def stoppedBranchRate {Ω : Type*} (σ τ : Ω → ℝ≥0) (A : Set Ω)
-    (g : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ := by
-  classical
-  exact if ω ∈ A ∧ σ ω ≤ t ∧ t < τ ω then g t ω else 0
-
-lemma stoppedBranchRate_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (σ τ : Ω → ℝ≥0) (hσ : Measurable σ) (hτ : Measurable τ)
-    (A : Set Ω) (hA : MeasurableSet A) (g : ℝ≥0 → Ω → ℝ)
-    (hg : Measurable (Function.uncurry g)) :
-    Measurable (Function.uncurry (stoppedBranchRate σ τ A g)) := by
-  have hstart : MeasurableSet {p : ℝ≥0 × Ω | σ p.2 ≤ p.1} :=
-    measurableSet_le (hσ.comp measurable_snd) measurable_fst
-  have hend : MeasurableSet {p : ℝ≥0 × Ω | p.1 < τ p.2} :=
-    measurableSet_lt measurable_fst (hτ.comp measurable_snd)
-  exact Measurable.ite ((hA.preimage measurable_snd).inter (hstart.inter hend))
-    hg measurable_const
-
-/-- A branch known at its starting stop gives a measurable stochastic
-interval on every finite deterministic horizon. -/
-lemma stoppedBranch_start_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
-    (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) (T : ℝ≥0) :
-    MeasurableSet[Subtype.instMeasurableSpace.prod (ℱ T)]
-      {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1} := by
-  let : MeasurableSpace Ω := ℱ T
-  have hAT : MeasurableSet (A ∩ {ω | σ ω ≤ T}) := by
-    simpa only [WithTop.coe_le_coe] using hA.2 T
-  have hσT : Measurable (fun ω => min (σ ω) T) := by
-    have hm := ((hσ.min_const T).measurable_of_le
-      (fun ω => min_le_right (σ ω : WithTop ℝ≥0) T)).untopD 0
-    simpa only [← WithTop.coe_min, WithTop.untopD_coe] using hm
-  have he : {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1} =
-      {p : Iic T × Ω | p.2 ∈ A ∩ {ω | σ ω ≤ T}} ∩
-        {p : Iic T × Ω | min (σ p.2) T ≤ p.1} := by
-    ext p
-    constructor
-    · rintro ⟨hA, hs⟩
-      exact ⟨⟨hA, hs.trans p.1.prop⟩, (min_le_left _ _).trans hs⟩
-    · rintro ⟨⟨hA, hsT⟩, hs⟩
-      change σ p.2 ≤ T at hsT
-      change min (σ p.2) T ≤ (p.1 : ℝ≥0) at hs
-      exact ⟨hA, by simpa only [min_eq_left hsT] using hs⟩
-  rw [he]
-  exact (hAT.preimage measurable_snd).inter
-    (measurableSet_le (hσT.comp measurable_snd)
-      (measurable_subtype_coe.comp measurable_fst))
-
-lemma stoppedBranchRate_progressive {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (σ τ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A)
-    (g : ℝ≥0 → Ω → ℝ) (hg : IsProgressive ℱ g) :
-    IsProgressive ℱ (stoppedBranchRate σ τ A g) := by
-  intro T
-  have hs := stoppedBranch_start_measurable ℱ σ hσ A hA T
-  have ht := stoppedBranch_start_measurable ℱ τ hτ univ MeasurableSet.univ T
-  have ht' : MeasurableSet[Subtype.instMeasurableSpace.prod (ℱ T)]
-      {p : Iic T × Ω | (p.1 : ℝ≥0) < τ p.2} := by
-    convert ht.compl using 1
-    ext p
-    simp only [mem_compl_iff, mem_ofPred_eq, mem_univ, true_and, not_le]
-  have he : {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1 ∧ (p.1 : ℝ≥0) < τ p.2} =
-      {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1} ∩
-        {p : Iic T × Ω | (p.1 : ℝ≥0) < τ p.2} := by
-    ext p
-    simp only [mem_inter_iff, mem_ofPred_eq, and_assoc]
-  apply Measurable.ite _ (hg T) measurable_const
-  rw [he]
-  exact hs.inter ht'
-
-lemma stoppedBranchRate_norm_le {Ω : Type*} (σ τ : Ω → ℝ≥0) (A : Set Ω)
-    (g : ℝ≥0 → Ω → ℝ) (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (t : ℝ≥0) (ω : Ω) :
-    |stoppedBranchRate σ τ A g t ω| ≤ C := by
-  unfold stoppedBranchRate
-  split_ifs
-  · exact hC t ω
-  · simpa only [abs_zero] using C.coe_nonneg
-
-lemma stoppedBranchRate_nonneg {Ω : Type*} (σ τ : Ω → ℝ≥0) (A : Set Ω)
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ t ω, 0 ≤ g t ω) (t : ℝ≥0) (ω : Ω) :
-    0 ≤ stoppedBranchRate σ τ A g t ω := by
-  unfold stoppedBranchRate
-  split_ifs
-  · exact hg t ω
-  · exact le_rfl
-
-/-- Local integrability of a bounded measurable rate, without continuity. -/
-lemma bounded_rate_locallyIntegrable {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
-    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
-    LocallyIntegrable (fun u : ℝ => g u.toNNReal ω) volume := by
-  have hc : LocallyIntegrable (fun _ : ℝ => (C : ℝ)) volume :=
-    continuous_const.locallyIntegrable
-  apply hc.mono ((hg ω).comp continuous_real_toNNReal.measurable).aestronglyMeasurable
-  filter_upwards [] with u
-  simpa only [Function.comp_apply, Real.norm_eq_abs, abs_of_nonneg C.coe_nonneg]
-    using hC u.toNNReal ω
-
-lemma discountClock_lipschitz_of_measurable {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
-    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
-    LipschitzWith C (fun t => discountClock g t ω) := by
-  have hi (a b : ℝ) : IntervalIntegrable (fun u : ℝ => g u.toNNReal ω) volume a b :=
-    ((bounded_rate_locallyIntegrable g hg C hC ω).integrableOn_isCompact
-      isCompact_uIcc).intervalIntegrable
-  apply LipschitzWith.of_dist_le_mul
-  intro s t
-  rw [Real.dist_eq, ← Real.norm_eq_abs]
-  unfold discountClock
-  rw [intervalIntegral.integral_interval_sub_left (hi _ _) (hi _ _)]
-  have hb := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (t.toNNReal : ℝ)) (b := (s.toNNReal : ℝ)) (C := (C : ℝ))
-    (f := fun u : ℝ => g u.toNNReal ω)
-    (fun u _ => by simpa only [Real.norm_eq_abs] using hC u.toNNReal ω)
-  apply hb.trans
-  apply mul_le_mul_of_nonneg_left _ C.coe_nonneg
-  have ht := Real.lipschitzWith_toNNReal.dist_le_mul s t
-  simpa only [NNReal.dist_eq, Real.dist_eq, NNReal.coe_one, one_mul] using ht
-
-lemma discountClock_increment_of_measurable {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
-    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (s t : ℝ≥0) (ω : Ω) :
-    discountClock g t ω - discountClock g s ω =
-      ∫ u in (s : ℝ)..(t : ℝ), g u.toNNReal ω := by
-  have hi (a b : ℝ) : IntervalIntegrable (fun u : ℝ => g u.toNNReal ω) volume a b :=
-    ((bounded_rate_locallyIntegrable g hg C hC ω).integrableOn_isCompact
-      isCompact_uIcc).intervalIntegrable
-  simp only [discountClock, Real.toNNReal_coe]
-  exact intervalIntegral.integral_interval_sub_left (hi _ _) (hi _ _)
-
-lemma discountClock_ae_derivative {Ω : Type*}
-    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
-    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
-    ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => discountClock g u ω) (g t.toNNReal ω) t := by
-  filter_upwards [LocallyIntegrable.ae_hasDerivAt_integral (bounded_rate_locallyIntegrable g hg C hC ω)]
-    with t ht
-  intro ht0
-  apply (ht 0).congr_of_eventuallyEq
-  filter_upwards [eventually_gt_nhds ht0] with u hu
-  simp only [discountClock, Real.coe_toNNReal u hu.le]
-
-lemma discountClock_adapted_of_progressive {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (g : ℝ≥0 → Ω → ℝ) (hg : IsProgressive ℱ g) :
-    Adapted ℱ (fun t : ℝ≥0 => discountClock g t) := by
-  intro t
-  let : MeasurableSpace Ω := ℱ t
-  have htime : Measurable (fun u : ℝ =>
-      (⟨min u.toNNReal t, by change min u.toNNReal t ≤ t; exact min_le_right _ _⟩ : Iic t)) :=
-    ((continuous_real_toNNReal.min continuous_const).measurable).subtype_mk
-  have hgm : Measurable (fun p : ℝ × Ω => g (min p.1.toNNReal t) p.2) :=
-    (hg t).comp ((htime.comp measurable_fst).prodMk measurable_snd)
-  have hm := (cumulative_integral_measurable (fun u ω => g (min u.toNNReal t) ω) hgm).comp
-    (measurable_const.prodMk measurable_id : Measurable (fun ω : Ω => (t, ω)))
-  convert hm using 1
-  funext ω
-  simp only [discountClock, Real.toNNReal_coe, Function.comp_def]
-  apply intervalIntegral.integral_congr
-  intro u hu
-  rw [uIcc_of_le t.coe_nonneg] at hu
-  have hut : u.toNNReal ≤ t := by
-    simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hu.2
-  change g u.toNNReal ω = g (min u.toNNReal t) ω
-  rw [min_eq_left hut]
-
-/-- The finite index may comprise a stage and its finite past-measurable
-pattern. Empty or simultaneous stages contribute zero. -/
-def finiteSwitchRate {Ω ι : Type*} (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
-    (A : ι → Set Ω) (g : ι → ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
-  ∑ i ∈ s, stoppedBranchRate (σ i) (τ i) (A i) (g i) t ω
-
-lemma finiteSwitchRate_measurable {Ω ι : Type*} [MeasurableSpace Ω]
-    (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
-    (hσ : ∀ i ∈ s, Measurable (σ i)) (hτ : ∀ i ∈ s, Measurable (τ i))
-    (A : ι → Set Ω) (hA : ∀ i ∈ s, MeasurableSet (A i))
-    (g : ι → ℝ≥0 → Ω → ℝ) (hg : ∀ i ∈ s, Measurable (Function.uncurry (g i))) :
-    Measurable (Function.uncurry (finiteSwitchRate s σ τ A g)) := by
-  exact Finset.measurable_sum s (fun i hi =>
-    stoppedBranchRate_measurable (σ i) (τ i) (hσ i hi) (hτ i hi) (A i) (hA i hi)
-      (g i) (hg i hi))
-
-lemma finiteSwitchRate_progressive {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
-    (hσ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (σ i ω : WithTop ℝ≥0)))
-    (hτ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0)))
-    (A : ι → Set Ω) (hA : ∀ (i) (hi : i ∈ s), MeasurableSet[(hσ i hi).measurableSpace] (A i))
-    (g : ι → ℝ≥0 → Ω → ℝ) (hg : ∀ i ∈ s, IsProgressive ℱ (g i)) :
-    IsProgressive ℱ (finiteSwitchRate s σ τ A g) := by
-  intro T
-  exact Finset.measurable_sum s (fun i hi =>
-    stoppedBranchRate_progressive ℱ (σ i) (τ i) (hσ i hi) (hτ i hi)
-      (A i) (hA i hi) (g i) (hg i hi) T)
-
-lemma finiteSwitchRate_norm_le {Ω ι : Type*} (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
-    (A : ι → Set Ω) (g : ι → ℝ≥0 → Ω → ℝ) (C : ι → ℝ≥0)
-    (hC : ∀ i ∈ s, ∀ t ω, |g i t ω| ≤ C i) (t : ℝ≥0) (ω : Ω) :
-    |finiteSwitchRate s σ τ A g t ω| ≤ (∑ i ∈ s, C i : ℝ≥0) := by
-  unfold finiteSwitchRate
-  apply (Finset.abs_sum_le_sum_abs _ _).trans
-  simpa only [NNReal.coe_sum] using Finset.sum_le_sum (fun i hi =>
-    stoppedBranchRate_norm_le (σ i) (τ i) (A i) (g i) (C i) (hC i hi) t ω)
-
-lemma finiteSwitchRate_nonneg {Ω ι : Type*} (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
-    (A : ι → Set Ω) (g : ι → ℝ≥0 → Ω → ℝ)
-    (hg : ∀ i ∈ s, ∀ t ω, 0 ≤ g i t ω) (t : ℝ≥0) (ω : Ω) :
-    0 ≤ finiteSwitchRate s σ τ A g t ω := by
-  exact Finset.sum_nonneg (fun i hi =>
-    stoppedBranchRate_nonneg (σ i) (τ i) (A i) (g i) (hg i hi) t ω)
-
-/-- The state used by the smooth-field adapter. No differentiability of
-rates at switching times is required. -/
-theorem finiteSwitchClock_regular {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
-    (hσ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (σ i ω : WithTop ℝ≥0)))
-    (hτ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0)))
-    (A : ι → Set Ω) (hA : ∀ (i) (hi : i ∈ s), MeasurableSet[(hσ i hi).measurableSpace] (A i))
-    (g : ι → ℝ≥0 → Ω → ℝ)
-    (hgm : ∀ i ∈ s, Measurable (Function.uncurry (g i)))
-    (hgp : ∀ i ∈ s, IsProgressive ℱ (g i))
-    (C : ι → ℝ≥0) (hC : ∀ i ∈ s, ∀ t ω, |g i t ω| ≤ C i) :
-    let R := finiteSwitchRate s σ τ A g
-    Measurable (fun p : ℝ × Ω => discountClock R p.1 p.2) ∧
-    Adapted ℱ (fun t : ℝ≥0 => discountClock R t) ∧
-    (∀ ω, LipschitzWith (∑ i ∈ s, C i) (fun t => discountClock R t ω)) ∧
-    (∀ ω, ∀ᵐ t : ℝ, 0 < t →
-      HasDerivAt (fun u => discountClock R u ω) (R t.toNNReal ω) t) ∧
-    ∀ ω, discountClock R 0 ω = 0 := by
-  let R := finiteSwitchRate s σ τ A g
-  have hAm (i : ι) (hi : i ∈ s) : MeasurableSet (A i) :=
-    (hσ i hi).measurableSpace_le _ (hA i hi)
-  have hRm : Measurable (Function.uncurry R) := finiteSwitchRate_measurable s σ τ
-    (fun i hi => stopping_time_measurable_nnreal (hσ i hi))
-    (fun i hi => stopping_time_measurable_nnreal (hτ i hi)) A hAm g hgm
-  have hRt : ∀ ω, Measurable (fun t => R t ω) := fun ω =>
-    hRm.comp (measurable_id.prodMk measurable_const)
-  have hbound := finiteSwitchRate_norm_le s σ τ A g C hC
-  exact ⟨discountClock_measurable R hRm,
-    discountClock_adapted_of_progressive ℱ R
-      (finiteSwitchRate_progressive ℱ s σ τ hσ hτ A hA g hgp),
-    discountClock_lipschitz_of_measurable R hRt _ hbound,
-    discountClock_ae_derivative R hRt _ hbound,
-    discountClock_initial R⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.FiniteStopSorting
+import OAI.Probability.SLE.Multipoint.MultipointDiscountClock
+import OAI.Probability.SLE.Multipoint.MultipointSwitchPatterns
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm
+
+namespace OAI
+
+/-! Bounded discount rates selected at finitely many stopping-time switches.
+The rate may jump at the switches; its accumulated clock is adapted and
+Lipschitz, and its derivative agrees with the rate almost everywhere. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+/-- Sorting all raw switches leaves no original switch strictly between
+successive entries, including when some entries coincide. -/
+lemma sortedStopAt_no_intermediate {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (σ : Ω → ℝ≥0) (hσ : σ ∈ l)
+    (n : ℕ) (ω : Ω) (hn : sortedStopAt l H n ω < σ ω) :
+    sortedStopAt l H (n+1) ω ≤ σ ω := by
+  obtain ⟨k, _, hk⟩ := sortedStopAt_covers l H σ hσ ω
+  have hnk : n+1 ≤ k := by
+    by_contra h
+    have hkn : k ≤ n := by omega
+    have hc := sortedStopAt_monotone l H hl ω hkn
+    change sortedStopAt l H k ω ≤ sortedStopAt l H n ω at hc
+    rw [hk] at hc
+    exact (not_le_of_gt hn) hc
+  exact (sortedStopAt_monotone l H hl ω hnk).trans_eq hk
+
+lemma sortedStopAt_reached_iff {Ω : Type*} (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hl : ∀ σ ∈ l, ∀ ω, σ ω ≤ H) (σ : Ω → ℝ≥0) (hσ : σ ∈ l)
+    (n : ℕ) (ω : Ω) (t : ℝ≥0) (hnt : sortedStopAt l H n ω ≤ t)
+    (ht : t < sortedStopAt l H (n+1) ω) :
+    σ ω ≤ t ↔ σ ω ≤ sortedStopAt l H n ω := by
+  constructor
+  · intro hσt
+    by_contra h
+    have hnext := sortedStopAt_no_intermediate l H hl σ hσ n ω (lt_of_not_ge h)
+    exact (not_le_of_gt ht) (hnext.trans hσt)
+  · exact fun h => h.trans hnt
+
+def stoppedBranchRate {Ω : Type*} (σ τ : Ω → ℝ≥0) (A : Set Ω)
+    (g : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ := by
+  classical
+  exact if ω ∈ A ∧ σ ω ≤ t ∧ t < τ ω then g t ω else 0
+
+lemma stoppedBranchRate_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (σ τ : Ω → ℝ≥0) (hσ : Measurable σ) (hτ : Measurable τ)
+    (A : Set Ω) (hA : MeasurableSet A) (g : ℝ≥0 → Ω → ℝ)
+    (hg : Measurable (Function.uncurry g)) :
+    Measurable (Function.uncurry (stoppedBranchRate σ τ A g)) := by
+  have hstart : MeasurableSet {p : ℝ≥0 × Ω | σ p.2 ≤ p.1} :=
+    measurableSet_le (hσ.comp measurable_snd) measurable_fst
+  have hend : MeasurableSet {p : ℝ≥0 × Ω | p.1 < τ p.2} :=
+    measurableSet_lt measurable_fst (hτ.comp measurable_snd)
+  exact Measurable.ite ((hA.preimage measurable_snd).inter (hstart.inter hend))
+    hg measurable_const
+
+/-- A branch known at its starting stop gives a measurable stochastic
+interval on every finite deterministic horizon. -/
+lemma stoppedBranch_start_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
+    (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) (T : ℝ≥0) :
+    MeasurableSet[Subtype.instMeasurableSpace.prod (ℱ T)]
+      {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1} := by
+  let : MeasurableSpace Ω := ℱ T
+  have hAT : MeasurableSet (A ∩ {ω | σ ω ≤ T}) := by
+    simpa only [WithTop.coe_le_coe] using hA.2 T
+  have hσT : Measurable (fun ω => min (σ ω) T) := by
+    have hm := ((hσ.min_const T).measurable_of_le
+      (fun ω => min_le_right (σ ω : WithTop ℝ≥0) T)).untopD 0
+    simpa only [← WithTop.coe_min, WithTop.untopD_coe] using hm
+  have he : {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1} =
+      {p : Iic T × Ω | p.2 ∈ A ∩ {ω | σ ω ≤ T}} ∩
+        {p : Iic T × Ω | min (σ p.2) T ≤ p.1} := by
+    ext p
+    constructor
+    · rintro ⟨hA, hs⟩
+      exact ⟨⟨hA, hs.trans p.1.prop⟩, (min_le_left _ _).trans hs⟩
+    · rintro ⟨⟨hA, hsT⟩, hs⟩
+      change σ p.2 ≤ T at hsT
+      change min (σ p.2) T ≤ (p.1 : ℝ≥0) at hs
+      exact ⟨hA, by simpa only [min_eq_left hsT] using hs⟩
+  rw [he]
+  exact (hAT.preimage measurable_snd).inter
+    (measurableSet_le (hσT.comp measurable_snd)
+      (measurable_subtype_coe.comp measurable_fst))
+
+lemma stoppedBranchRate_progressive {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (σ τ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A)
+    (g : ℝ≥0 → Ω → ℝ) (hg : IsProgressive ℱ g) :
+    IsProgressive ℱ (stoppedBranchRate σ τ A g) := by
+  intro T
+  have hs := stoppedBranch_start_measurable ℱ σ hσ A hA T
+  have ht := stoppedBranch_start_measurable ℱ τ hτ univ MeasurableSet.univ T
+  have ht' : MeasurableSet[Subtype.instMeasurableSpace.prod (ℱ T)]
+      {p : Iic T × Ω | (p.1 : ℝ≥0) < τ p.2} := by
+    convert ht.compl using 1
+    ext p
+    simp only [mem_compl_iff, mem_ofPred_eq, mem_univ, true_and, not_le]
+  have he : {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1 ∧ (p.1 : ℝ≥0) < τ p.2} =
+      {p : Iic T × Ω | p.2 ∈ A ∧ σ p.2 ≤ p.1} ∩
+        {p : Iic T × Ω | (p.1 : ℝ≥0) < τ p.2} := by
+    ext p
+    simp only [mem_inter_iff, mem_ofPred_eq, and_assoc]
+  apply Measurable.ite _ (hg T) measurable_const
+  rw [he]
+  exact hs.inter ht'
+
+lemma stoppedBranchRate_norm_le {Ω : Type*} (σ τ : Ω → ℝ≥0) (A : Set Ω)
+    (g : ℝ≥0 → Ω → ℝ) (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (t : ℝ≥0) (ω : Ω) :
+    |stoppedBranchRate σ τ A g t ω| ≤ C := by
+  unfold stoppedBranchRate
+  split_ifs
+  · exact hC t ω
+  · simpa only [abs_zero] using C.coe_nonneg
+
+lemma stoppedBranchRate_nonneg {Ω : Type*} (σ τ : Ω → ℝ≥0) (A : Set Ω)
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ t ω, 0 ≤ g t ω) (t : ℝ≥0) (ω : Ω) :
+    0 ≤ stoppedBranchRate σ τ A g t ω := by
+  unfold stoppedBranchRate
+  split_ifs
+  · exact hg t ω
+  · exact le_rfl
+
+/-- Local integrability of a bounded measurable rate, without continuity. -/
+lemma bounded_rate_locallyIntegrable {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
+    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
+    LocallyIntegrable (fun u : ℝ => g u.toNNReal ω) volume := by
+  have hc : LocallyIntegrable (fun _ : ℝ => (C : ℝ)) volume :=
+    continuous_const.locallyIntegrable
+  apply hc.mono ((hg ω).comp continuous_real_toNNReal.measurable).aestronglyMeasurable
+  filter_upwards [] with u
+  simpa only [Function.comp_apply, Real.norm_eq_abs, abs_of_nonneg C.coe_nonneg]
+    using hC u.toNNReal ω
+
+lemma discountClock_lipschitz_of_measurable {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
+    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
+    LipschitzWith C (fun t => discountClock g t ω) := by
+  have hi (a b : ℝ) : IntervalIntegrable (fun u : ℝ => g u.toNNReal ω) volume a b :=
+    ((bounded_rate_locallyIntegrable g hg C hC ω).integrableOn_isCompact
+      isCompact_uIcc).intervalIntegrable
+  apply LipschitzWith.of_dist_le_mul
+  intro s t
+  rw [Real.dist_eq, ← Real.norm_eq_abs]
+  unfold discountClock
+  rw [intervalIntegral.integral_interval_sub_left (hi _ _) (hi _ _)]
+  have hb := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (t.toNNReal : ℝ)) (b := (s.toNNReal : ℝ)) (C := (C : ℝ))
+    (f := fun u : ℝ => g u.toNNReal ω)
+    (fun u _ => by simpa only [Real.norm_eq_abs] using hC u.toNNReal ω)
+  apply hb.trans
+  apply mul_le_mul_of_nonneg_left _ C.coe_nonneg
+  have ht := Real.lipschitzWith_toNNReal.dist_le_mul s t
+  simpa only [NNReal.dist_eq, Real.dist_eq, NNReal.coe_one, one_mul] using ht
+
+lemma discountClock_increment_of_measurable {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
+    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (s t : ℝ≥0) (ω : Ω) :
+    discountClock g t ω - discountClock g s ω =
+      ∫ u in (s : ℝ)..(t : ℝ), g u.toNNReal ω := by
+  have hi (a b : ℝ) : IntervalIntegrable (fun u : ℝ => g u.toNNReal ω) volume a b :=
+    ((bounded_rate_locallyIntegrable g hg C hC ω).integrableOn_isCompact
+      isCompact_uIcc).intervalIntegrable
+  simp only [discountClock, Real.toNNReal_coe]
+  exact intervalIntegral.integral_interval_sub_left (hi _ _) (hi _ _)
+
+lemma discountClock_ae_derivative {Ω : Type*}
+    (g : ℝ≥0 → Ω → ℝ) (hg : ∀ ω, Measurable (fun t => g t ω))
+    (C : ℝ≥0) (hC : ∀ t ω, |g t ω| ≤ C) (ω : Ω) :
+    ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => discountClock g u ω) (g t.toNNReal ω) t := by
+  filter_upwards [LocallyIntegrable.ae_hasDerivAt_integral (bounded_rate_locallyIntegrable g hg C hC ω)]
+    with t ht
+  intro ht0
+  apply (ht 0).congr_of_eventuallyEq
+  filter_upwards [eventually_gt_nhds ht0] with u hu
+  simp only [discountClock, Real.coe_toNNReal u hu.le]
+
+lemma discountClock_adapted_of_progressive {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (g : ℝ≥0 → Ω → ℝ) (hg : IsProgressive ℱ g) :
+    Adapted ℱ (fun t : ℝ≥0 => discountClock g t) := by
+  intro t
+  let : MeasurableSpace Ω := ℱ t
+  have htime : Measurable (fun u : ℝ =>
+      (⟨min u.toNNReal t, by change min u.toNNReal t ≤ t; exact min_le_right _ _⟩ : Iic t)) :=
+    ((continuous_real_toNNReal.min continuous_const).measurable).subtype_mk
+  have hgm : Measurable (fun p : ℝ × Ω => g (min p.1.toNNReal t) p.2) :=
+    (hg t).comp ((htime.comp measurable_fst).prodMk measurable_snd)
+  have hm := (cumulative_integral_measurable (fun u ω => g (min u.toNNReal t) ω) hgm).comp
+    (measurable_const.prodMk measurable_id : Measurable (fun ω : Ω => (t, ω)))
+  convert hm using 1
+  funext ω
+  simp only [discountClock, Real.toNNReal_coe, Function.comp_def]
+  apply intervalIntegral.integral_congr
+  intro u hu
+  rw [uIcc_of_le t.coe_nonneg] at hu
+  have hut : u.toNNReal ≤ t := by
+    simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hu.2
+  change g u.toNNReal ω = g (min u.toNNReal t) ω
+  rw [min_eq_left hut]
+
+/-- The finite index may comprise a stage and its finite past-measurable
+pattern. Empty or simultaneous stages contribute zero. -/
+def finiteSwitchRate {Ω ι : Type*} (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
+    (A : ι → Set Ω) (g : ι → ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
+  ∑ i ∈ s, stoppedBranchRate (σ i) (τ i) (A i) (g i) t ω
+
+lemma finiteSwitchRate_measurable {Ω ι : Type*} [MeasurableSpace Ω]
+    (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
+    (hσ : ∀ i ∈ s, Measurable (σ i)) (hτ : ∀ i ∈ s, Measurable (τ i))
+    (A : ι → Set Ω) (hA : ∀ i ∈ s, MeasurableSet (A i))
+    (g : ι → ℝ≥0 → Ω → ℝ) (hg : ∀ i ∈ s, Measurable (Function.uncurry (g i))) :
+    Measurable (Function.uncurry (finiteSwitchRate s σ τ A g)) := by
+  exact Finset.measurable_sum s (fun i hi =>
+    stoppedBranchRate_measurable (σ i) (τ i) (hσ i hi) (hτ i hi) (A i) (hA i hi)
+      (g i) (hg i hi))
+
+lemma finiteSwitchRate_progressive {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
+    (hσ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (σ i ω : WithTop ℝ≥0)))
+    (hτ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0)))
+    (A : ι → Set Ω) (hA : ∀ (i) (hi : i ∈ s), MeasurableSet[(hσ i hi).measurableSpace] (A i))
+    (g : ι → ℝ≥0 → Ω → ℝ) (hg : ∀ i ∈ s, IsProgressive ℱ (g i)) :
+    IsProgressive ℱ (finiteSwitchRate s σ τ A g) := by
+  intro T
+  exact Finset.measurable_sum s (fun i hi =>
+    stoppedBranchRate_progressive ℱ (σ i) (τ i) (hσ i hi) (hτ i hi)
+      (A i) (hA i hi) (g i) (hg i hi) T)
+
+lemma finiteSwitchRate_norm_le {Ω ι : Type*} (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
+    (A : ι → Set Ω) (g : ι → ℝ≥0 → Ω → ℝ) (C : ι → ℝ≥0)
+    (hC : ∀ i ∈ s, ∀ t ω, |g i t ω| ≤ C i) (t : ℝ≥0) (ω : Ω) :
+    |finiteSwitchRate s σ τ A g t ω| ≤ (∑ i ∈ s, C i : ℝ≥0) := by
+  unfold finiteSwitchRate
+  apply (Finset.abs_sum_le_sum_abs _ _).trans
+  simpa only [NNReal.coe_sum] using Finset.sum_le_sum (fun i hi =>
+    stoppedBranchRate_norm_le (σ i) (τ i) (A i) (g i) (C i) (hC i hi) t ω)
+
+lemma finiteSwitchRate_nonneg {Ω ι : Type*} (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
+    (A : ι → Set Ω) (g : ι → ℝ≥0 → Ω → ℝ)
+    (hg : ∀ i ∈ s, ∀ t ω, 0 ≤ g i t ω) (t : ℝ≥0) (ω : Ω) :
+    0 ≤ finiteSwitchRate s σ τ A g t ω := by
+  exact Finset.sum_nonneg (fun i hi =>
+    stoppedBranchRate_nonneg (σ i) (τ i) (A i) (g i) (hg i hi) t ω)
+
+/-- The state used by the smooth-field adapter. No differentiability of
+rates at switching times is required. -/
+theorem finiteSwitchClock_regular {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (s : Finset ι) (σ τ : ι → Ω → ℝ≥0)
+    (hσ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (σ i ω : WithTop ℝ≥0)))
+    (hτ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0)))
+    (A : ι → Set Ω) (hA : ∀ (i) (hi : i ∈ s), MeasurableSet[(hσ i hi).measurableSpace] (A i))
+    (g : ι → ℝ≥0 → Ω → ℝ)
+    (hgm : ∀ i ∈ s, Measurable (Function.uncurry (g i)))
+    (hgp : ∀ i ∈ s, IsProgressive ℱ (g i))
+    (C : ι → ℝ≥0) (hC : ∀ i ∈ s, ∀ t ω, |g i t ω| ≤ C i) :
+    let R := finiteSwitchRate s σ τ A g
+    Measurable (fun p : ℝ × Ω => discountClock R p.1 p.2) ∧
+    Adapted ℱ (fun t : ℝ≥0 => discountClock R t) ∧
+    (∀ ω, LipschitzWith (∑ i ∈ s, C i) (fun t => discountClock R t ω)) ∧
+    (∀ ω, ∀ᵐ t : ℝ, 0 < t →
+      HasDerivAt (fun u => discountClock R u ω) (R t.toNNReal ω) t) ∧
+    ∀ ω, discountClock R 0 ω = 0 := by
+  let R := finiteSwitchRate s σ τ A g
+  have hAm (i : ι) (hi : i ∈ s) : MeasurableSet (A i) :=
+    (hσ i hi).measurableSpace_le _ (hA i hi)
+  have hRm : Measurable (Function.uncurry R) := finiteSwitchRate_measurable s σ τ
+    (fun i hi => stopping_time_measurable_nnreal (hσ i hi))
+    (fun i hi => stopping_time_measurable_nnreal (hτ i hi)) A hAm g hgm
+  have hRt : ∀ ω, Measurable (fun t => R t ω) := fun ω =>
+    hRm.comp (measurable_id.prodMk measurable_const)
+  have hbound := finiteSwitchRate_norm_le s σ τ A g C hC
+  exact ⟨discountClock_measurable R hRm,
+    discountClock_adapted_of_progressive ℱ R
+      (finiteSwitchRate_progressive ℱ s σ τ hσ hτ A hA g hgp),
+    discountClock_lipschitz_of_measurable R hRt _ hbound,
+    discountClock_ae_derivative R hRt _ hbound,
+    discountClock_initial R⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointProductCalculus.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointProductCalculus.lean
index bff46989797cc7c40511f114dcfbd9b72332514e..5740f9f41407c1cad321d3b55c1f0852c62596f2 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointProductCalculus.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointProductCalculus.lean
@@ -1,115 +1,115 @@
-import OAI.Probability.SLE.Multipoint.MultipointAbsorption
-
-namespace OAI
-
-/-! Relative derivatives of the finite product used in the multipoint
-supermartingale. The off-diagonal sum is ordered, so its generator carries
-the factor one half. -/
-
-noncomputable section
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-lemma finite_product_hasDerivAt_relative {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (f : ι → ℝ → ℝ) (df : ι → ℝ) (x : ℝ)
-    (hf : ∀ i ∈ s, HasDerivAt (f i) (df i) x)
-    (hne : ∀ i ∈ s, f i x ≠ 0) :
-    HasDerivAt (fun y => ∏ i ∈ s, f i y)
-      ((∏ i ∈ s, f i x)*(∑ i ∈ s, df i/f i x)) x := by
-  apply (HasDerivAt.fun_finsetProd hf).congr_deriv
-  rw [Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i hi
-  simp only [smul_eq_mul]
-  rw [← Finset.mul_prod_erase s (fun i => f i x) hi]
-  field_simp [hne i hi]
-
-lemma sum_offDiag_mul {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (b : ι → ℝ) :
-    (∑ ij ∈ s.offDiag, b ij.1*b ij.2) =
-      (∑ i ∈ s, b i)^2 - ∑ i ∈ s, (b i)^2 := by
-  have he := Finset.sum_union (f := fun ij : ι × ι => b ij.1*b ij.2)
-    (Finset.disjoint_diag_offDiag (s := s))
-  rw [Finset.diag_union_offDiag, Finset.sum_product, ← Finset.sum_mul_sum, ← pow_two] at he
-  have hdiag : (∑ ij ∈ s.diag, b ij.1*b ij.2) = ∑ i ∈ s, (b i)^2 := by
-    simp only [Finset.diag, Finset.sum_map, Function.Embedding.coeFn_mk, Function.diag_apply,
-      pow_two]
-  rw [hdiag] at he
-  linarith
-
-lemma finite_product_second_relative {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (f : ι → ℝ → ℝ)
-    (hf : ∀ i ∈ s, Differentiable ℝ (f i))
-    (hdf : ∀ i ∈ s, Differentiable ℝ (deriv (f i)))
-    (hne : ∀ y i, i ∈ s → f i y ≠ 0) (x : ℝ) :
-    iteratedDeriv 2 (fun y => ∏ i ∈ s, f i y) x =
-      (∏ i ∈ s, f i x)*((∑ i ∈ s, iteratedDeriv 2 (f i) x/f i x) +
-        ∑ ij ∈ s.offDiag, (deriv (f ij.1) x/f ij.1 x)*(deriv (f ij.2) x/f ij.2 x)) := by
-  let P := fun y => ∏ i ∈ s, f i y
-  let b := fun i y => deriv (f i) y/f i y
-  have hP (y : ℝ) : HasDerivAt P (P y*∑ i ∈ s, b i y) y :=
-    finite_product_hasDerivAt_relative s f (fun i => deriv (f i) y) y
-      (fun i hi => (hf i hi y).hasDerivAt) (hne y)
-  have hPeq : deriv P = fun y => P y*∑ i ∈ s, b i y :=
-    funext (fun y => (hP y).deriv)
-  have hb (i : ι) (hi : i ∈ s) : HasDerivAt (b i)
-      (iteratedDeriv 2 (f i) x/f i x-(b i x)^2) x := by
-    apply (((hdf i hi x).hasDerivAt).div (hf i hi x).hasDerivAt (hne x i hi)).congr_deriv
-    simp only [iteratedDeriv_succ, iteratedDeriv_zero, b]
-    field_simp [hne x i hi]
-  have hsum := HasDerivAt.fun_sum (fun i hi => hb i hi)
-  have hsecond := ((hP x).mul hsum).deriv
-  change deriv (fun y => P y * ∑ i ∈ s, b i y) x = _ at hsecond
-  change iteratedDeriv (1+1) P x = _
-  rw [iteratedDeriv_succ, iteratedDeriv_one]
-  rw [hPeq, hsecond]
-  rw [sum_offDiag_mul s (fun i => deriv (f i) x / f i x), Finset.sum_sub_distrib]
-  dsimp only [P, b]
-  ring
-
-lemma finite_product_directional_relative {ι E : Type*} [DecidableEq ι]
-    [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (s : Finset ι) (f : ι → E → ℝ) (z v : E)
-    (hf : ∀ i ∈ s, DifferentiableAt ℝ (f i) z)
-    (hne : ∀ i ∈ s, f i z ≠ 0) :
-    (fderiv ℝ (fun z => ∏ i ∈ s, f i z) z) v =
-      (∏ i ∈ s, f i z)*(∑ i ∈ s, (fderiv ℝ (f i) z) v/f i z) := by
-  rw [fderiv_finsetProd hf, sum_apply, Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro i hi
-  simp only [smul_apply, smul_eq_mul]
-  rw [← Finset.mul_prod_erase s (fun i => f i z) hi]
-  field_simp [hne i hi]
-
-/-- Generator identity before estimating the individual and cross terms. -/
-theorem finite_product_generator {ι E : Type*} [DecidableEq ι]
-    [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (s : Finset ι) (F : ι → E → ℝ → ℝ)
-    (hF : ∀ i ∈ s, ContDiff ℝ (⊤ : ℕ∞) (fun p : E × ℝ => F i p.1 p.2))
-    (hne : ∀ i ∈ s, ∀ z x, F i z x ≠ 0) (z v : E) (x : ℝ) :
-    (fderiv ℝ (fun z => ∏ i ∈ s, F i z x) z) v +
-        (1/2 : ℝ)*iteratedDeriv 2 (fun y => ∏ i ∈ s, F i z y) x =
-      (∏ i ∈ s, F i z x)*
-        ((∑ i ∈ s, ((fderiv ℝ (fun z => F i z x) z) v +
-          (1/2 : ℝ)*iteratedDeriv 2 (F i z) x)/F i z x) +
-        (1/2 : ℝ)*∑ ij ∈ s.offDiag,
-          (deriv (F ij.1 z) x/F ij.1 z x)*(deriv (F ij.2 z) x/F ij.2 z x)) := by
-  have hs (i : ι) (hi : i ∈ s) : ContDiff ℝ (⊤ : ℕ∞) (fun z => F i z x) :=
-    (hF i hi).comp (contDiff_id.prodMk contDiff_const)
-  have hx (i : ι) (hi : i ∈ s) : ContDiff ℝ (⊤ : ℕ∞) (F i z) :=
-    (hF i hi).comp (contDiff_const.prodMk contDiff_id)
-  rw [finite_product_directional_relative s (fun i z => F i z x) z v
-    (fun i hi => ((hs i hi).differentiable (by simp)) z) (fun i hi => hne i hi z x)]
-  rw [finite_product_second_relative s (fun i => F i z)
-    (fun i hi => (hx i hi).differentiable (by simp))
-    (fun i hi => ((hx i hi).of_le (by simp)).differentiable_deriv_two)
-    (fun y i hi => hne i hi z y) x]
-  simp only [add_div, mul_div_assoc, Finset.sum_add_distrib, ← Finset.mul_sum]
-  ring
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointAbsorption
+
+namespace OAI
+
+/-! Relative derivatives of the finite product used in the multipoint
+supermartingale. The off-diagonal sum is ordered, so its generator carries
+the factor one half. -/
+
+noncomputable section
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+lemma finite_product_hasDerivAt_relative {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (f : ι → ℝ → ℝ) (df : ι → ℝ) (x : ℝ)
+    (hf : ∀ i ∈ s, HasDerivAt (f i) (df i) x)
+    (hne : ∀ i ∈ s, f i x ≠ 0) :
+    HasDerivAt (fun y => ∏ i ∈ s, f i y)
+      ((∏ i ∈ s, f i x)*(∑ i ∈ s, df i/f i x)) x := by
+  apply (HasDerivAt.fun_finsetProd hf).congr_deriv
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i hi
+  simp only [smul_eq_mul]
+  rw [← Finset.mul_prod_erase s (fun i => f i x) hi]
+  field_simp [hne i hi]
+
+lemma sum_offDiag_mul {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (b : ι → ℝ) :
+    (∑ ij ∈ s.offDiag, b ij.1*b ij.2) =
+      (∑ i ∈ s, b i)^2 - ∑ i ∈ s, (b i)^2 := by
+  have he := Finset.sum_union (f := fun ij : ι × ι => b ij.1*b ij.2)
+    (Finset.disjoint_diag_offDiag (s := s))
+  rw [Finset.diag_union_offDiag, Finset.sum_product, ← Finset.sum_mul_sum, ← pow_two] at he
+  have hdiag : (∑ ij ∈ s.diag, b ij.1*b ij.2) = ∑ i ∈ s, (b i)^2 := by
+    simp only [Finset.diag, Finset.sum_map, Function.Embedding.coeFn_mk, Function.diag_apply,
+      pow_two]
+  rw [hdiag] at he
+  linarith
+
+lemma finite_product_second_relative {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (f : ι → ℝ → ℝ)
+    (hf : ∀ i ∈ s, Differentiable ℝ (f i))
+    (hdf : ∀ i ∈ s, Differentiable ℝ (deriv (f i)))
+    (hne : ∀ y i, i ∈ s → f i y ≠ 0) (x : ℝ) :
+    iteratedDeriv 2 (fun y => ∏ i ∈ s, f i y) x =
+      (∏ i ∈ s, f i x)*((∑ i ∈ s, iteratedDeriv 2 (f i) x/f i x) +
+        ∑ ij ∈ s.offDiag, (deriv (f ij.1) x/f ij.1 x)*(deriv (f ij.2) x/f ij.2 x)) := by
+  let P := fun y => ∏ i ∈ s, f i y
+  let b := fun i y => deriv (f i) y/f i y
+  have hP (y : ℝ) : HasDerivAt P (P y*∑ i ∈ s, b i y) y :=
+    finite_product_hasDerivAt_relative s f (fun i => deriv (f i) y) y
+      (fun i hi => (hf i hi y).hasDerivAt) (hne y)
+  have hPeq : deriv P = fun y => P y*∑ i ∈ s, b i y :=
+    funext (fun y => (hP y).deriv)
+  have hb (i : ι) (hi : i ∈ s) : HasDerivAt (b i)
+      (iteratedDeriv 2 (f i) x/f i x-(b i x)^2) x := by
+    apply (((hdf i hi x).hasDerivAt).div (hf i hi x).hasDerivAt (hne x i hi)).congr_deriv
+    simp only [iteratedDeriv_succ, iteratedDeriv_zero, b]
+    field_simp [hne x i hi]
+  have hsum := HasDerivAt.fun_sum (fun i hi => hb i hi)
+  have hsecond := ((hP x).mul hsum).deriv
+  change deriv (fun y => P y * ∑ i ∈ s, b i y) x = _ at hsecond
+  change iteratedDeriv (1+1) P x = _
+  rw [iteratedDeriv_succ, iteratedDeriv_one]
+  rw [hPeq, hsecond]
+  rw [sum_offDiag_mul s (fun i => deriv (f i) x / f i x), Finset.sum_sub_distrib]
+  dsimp only [P, b]
+  ring
+
+lemma finite_product_directional_relative {ι E : Type*} [DecidableEq ι]
+    [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (s : Finset ι) (f : ι → E → ℝ) (z v : E)
+    (hf : ∀ i ∈ s, DifferentiableAt ℝ (f i) z)
+    (hne : ∀ i ∈ s, f i z ≠ 0) :
+    (fderiv ℝ (fun z => ∏ i ∈ s, f i z) z) v =
+      (∏ i ∈ s, f i z)*(∑ i ∈ s, (fderiv ℝ (f i) z) v/f i z) := by
+  rw [fderiv_finsetProd hf, sum_apply, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i hi
+  simp only [smul_apply, smul_eq_mul]
+  rw [← Finset.mul_prod_erase s (fun i => f i z) hi]
+  field_simp [hne i hi]
+
+/-- Generator identity before estimating the individual and cross terms. -/
+theorem finite_product_generator {ι E : Type*} [DecidableEq ι]
+    [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (s : Finset ι) (F : ι → E → ℝ → ℝ)
+    (hF : ∀ i ∈ s, ContDiff ℝ (⊤ : ℕ∞) (fun p : E × ℝ => F i p.1 p.2))
+    (hne : ∀ i ∈ s, ∀ z x, F i z x ≠ 0) (z v : E) (x : ℝ) :
+    (fderiv ℝ (fun z => ∏ i ∈ s, F i z x) z) v +
+        (1/2 : ℝ)*iteratedDeriv 2 (fun y => ∏ i ∈ s, F i z y) x =
+      (∏ i ∈ s, F i z x)*
+        ((∑ i ∈ s, ((fderiv ℝ (fun z => F i z x) z) v +
+          (1/2 : ℝ)*iteratedDeriv 2 (F i z) x)/F i z x) +
+        (1/2 : ℝ)*∑ ij ∈ s.offDiag,
+          (deriv (F ij.1 z) x/F ij.1 z x)*(deriv (F ij.2 z) x/F ij.2 z x)) := by
+  have hs (i : ι) (hi : i ∈ s) : ContDiff ℝ (⊤ : ℕ∞) (fun z => F i z x) :=
+    (hF i hi).comp (contDiff_id.prodMk contDiff_const)
+  have hx (i : ι) (hi : i ∈ s) : ContDiff ℝ (⊤ : ℕ∞) (F i z) :=
+    (hF i hi).comp (contDiff_const.prodMk contDiff_id)
+  rw [finite_product_directional_relative s (fun i z => F i z x) z v
+    (fun i hi => ((hs i hi).differentiable (by simp)) z) (fun i hi => hne i hi z x)]
+  rw [finite_product_second_relative s (fun i => F i z)
+    (fun i hi => (hx i hi).differentiable (by simp))
+    (fun i hi => ((hx i hi).of_le (by simp)).differentiable_deriv_two)
+    (fun y i hi => hne i hi z y) x]
+  simp only [add_div, mul_div_assoc, Finset.sum_add_distrib, ← Finset.mul_sum]
+  ring
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointProductDrift.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointProductDrift.lean
index e6da3162e0b57cc143fb2340bba4545742a79794..d49a4c631a5e1ad2fee40d6f547a87d468ceff05 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointProductDrift.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointProductDrift.lean
@@ -1,95 +1,95 @@
-import OAI.Probability.SLE.Multipoint.MultipointFieldBounds
-import OAI.Probability.SLE.Multipoint.MultipointAbsorption
-
-namespace OAI
-
-/-! Absorption of the actual finite-product generator. -/
-
-noncomputable section
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-theorem regularized_product_relative_drift {ι : Type*} [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ)
-    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
-    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1) :
-    let p := greenExponent κ
-    let ε := (negativeDrift p/2)/(4*((s.card : ℝ)+1))
-    let C := driftAllowance p*driftThreshold p^(-p)
-    let L := youngConstant p (angularCoefficientConstant p^2) ε
-    (∑ i ∈ s, logGenerator κ (regularizedLogFactor κ (D i) (σ i)) (z i) x /
-        regularizedLogFactor κ (D i) (σ i) (z i) x) +
-      (1/2 : ℝ)*(∑ ij ∈ s.offDiag,
-        (deriv (regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1)) x /
-          regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1) x)*
-        (deriv (regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2)) x /
-          regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2) x)) ≤
-      κ*(C*(∑ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2) +
-        L*(∑ ij ∈ s.offDiag,
-          (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
-            branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent p)*
-          (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x))) := by
-  dsimp only
-  let p := greenExponent κ
-  let k := negativeDrift p/2
-  let ε := k/(4*((s.card : ℝ)+1))
-  let C := driftAllowance p*driftThreshold p^(-p)
-  let L := youngConstant p (angularCoefficientConstant p^2) ε
-  let δ := fun i => branchRegulator κ (D i) (σ i) (z i)
-  let A := fun i => damping p (δ i) (logAngle κ (z i) x)*logVelocity κ (z i) x^2
-  let H := fun i => δ i*logVelocity κ (z i) x^2
-  let b := fun i => deriv (regularizedLogFactor κ (D i) (σ i) (z i)) x /
-    regularizedLogFactor κ (D i) (σ i) (z i) x
-  let d := fun i => logGenerator κ (regularizedLogFactor κ (D i) (σ i)) (z i) x /
-    regularizedLogFactor κ (D i) (σ i) (z i) x
-  let w := fun ij : ι × ι => (δ ij.1*δ ij.2)^(-exponent p)*
-    (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x)
-  have hp : 0 < p := by
-    dsimp [p, greenExponent]
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  have hk : 0 < k := div_pos (negativeDrift_pos hp) (by norm_num)
-  have he : 0 < ε := absorption_epsilon_pos hk s.card
-  have hA (i : ι) (hi : i ∈ s) : 0 ≤ A i := by
-    have hδi : 0 < δ i := mul_pos (hD i hi) (Real.exp_pos _)
-    have hf := angularFactor_pos (p := p) hδi (logAngle_sin_pos κ (z i) x)
-    change 0 ≤ (δ i/angularFactor p (δ i) (logAngle κ (z i) x)*
-      sinePower (q p-2) (logAngle κ (z i) x))*logVelocity κ (z i) x^2
-    exact mul_nonneg (mul_nonneg (div_pos hδi hf).le
-      (Real.rpow_nonneg (logAngle_sin_pos κ (z i) x).le _)) (sq_nonneg _)
-  have hd (i : ι) (hi : i ∈ s) : d i ≤ (κ*C)*H i-(κ*k)*A i := by
-    have h := regularizedLogFactor_relative_drift κ (D i) (σ i) hκ hκ8
-      (hD i hi) (hσ i hi) (z i) x
-    convert h using 1
-    dsimp [d, C, H, k, A, δ, p]
-    ring
-  have hcross (ij : ι × ι) (hij : ij ∈ s.offDiag) :
-      (1/2 : ℝ)*(b ij.1*b ij.2) ≤ (κ*ε)*(A ij.1+A ij.2)+(κ*L)*w ij := by
-    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
-    have h := regularizedLogFactor_pair_absorption κ (D ij.1) (D ij.2) (σ ij.1) (σ ij.2)
-      ε hκ hκ8 (hD _ hi) (hD _ hj) he (z ij.1) (z ij.2) x (hδ _ hi) (hδ _ hj)
-    have hsmall : (1/2 : ℝ)*(b ij.1*b ij.2) ≤ |b ij.1*b ij.2| := by
-      nlinarith [le_abs_self (b ij.1*b ij.2), abs_nonneg (b ij.1*b ij.2)]
-    apply hsmall.trans
-    convert h using 1
-    dsimp [b, L, A, w, δ, p]
-    ring
-  have hbudget : 2*(κ*ε)*(s.card : ℝ) ≤ κ*k := by
-    have h := mul_le_mul_of_nonneg_left (absorption_epsilon_budget hk.le s.card) hκ.le
-    convert h using 1
-    dsimp [ε]
-    ring
-  have h := sum_drift_absorption s d A H (fun ij => (1/2 : ℝ)*(b ij.1*b ij.2)) w
-    (κ*C) (κ*k) (κ*ε) (κ*L) (mul_pos hκ he).le hA hbudget hd hcross
-  rw [← Finset.mul_sum] at h
-  convert h using 1
-  dsimp [d, b, C, L, H, w, ε, k, δ, p]
-  ring
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointFieldBounds
+import OAI.Probability.SLE.Multipoint.MultipointAbsorption
+
+namespace OAI
+
+/-! Absorption of the actual finite-product generator. -/
+
+noncomputable section
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+theorem regularized_product_relative_drift {ι : Type*} [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ)
+    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
+    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1) :
+    let p := greenExponent κ
+    let ε := (negativeDrift p/2)/(4*((s.card : ℝ)+1))
+    let C := driftAllowance p*driftThreshold p^(-p)
+    let L := youngConstant p (angularCoefficientConstant p^2) ε
+    (∑ i ∈ s, logGenerator κ (regularizedLogFactor κ (D i) (σ i)) (z i) x /
+        regularizedLogFactor κ (D i) (σ i) (z i) x) +
+      (1/2 : ℝ)*(∑ ij ∈ s.offDiag,
+        (deriv (regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1)) x /
+          regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1) x)*
+        (deriv (regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2)) x /
+          regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2) x)) ≤
+      κ*(C*(∑ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2) +
+        L*(∑ ij ∈ s.offDiag,
+          (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
+            branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent p)*
+          (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x))) := by
+  dsimp only
+  let p := greenExponent κ
+  let k := negativeDrift p/2
+  let ε := k/(4*((s.card : ℝ)+1))
+  let C := driftAllowance p*driftThreshold p^(-p)
+  let L := youngConstant p (angularCoefficientConstant p^2) ε
+  let δ := fun i => branchRegulator κ (D i) (σ i) (z i)
+  let A := fun i => damping p (δ i) (logAngle κ (z i) x)*logVelocity κ (z i) x^2
+  let H := fun i => δ i*logVelocity κ (z i) x^2
+  let b := fun i => deriv (regularizedLogFactor κ (D i) (σ i) (z i)) x /
+    regularizedLogFactor κ (D i) (σ i) (z i) x
+  let d := fun i => logGenerator κ (regularizedLogFactor κ (D i) (σ i)) (z i) x /
+    regularizedLogFactor κ (D i) (σ i) (z i) x
+  let w := fun ij : ι × ι => (δ ij.1*δ ij.2)^(-exponent p)*
+    (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x)
+  have hp : 0 < p := by
+    dsimp [p, greenExponent]
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  have hk : 0 < k := div_pos (negativeDrift_pos hp) (by norm_num)
+  have he : 0 < ε := absorption_epsilon_pos hk s.card
+  have hA (i : ι) (hi : i ∈ s) : 0 ≤ A i := by
+    have hδi : 0 < δ i := mul_pos (hD i hi) (Real.exp_pos _)
+    have hf := angularFactor_pos (p := p) hδi (logAngle_sin_pos κ (z i) x)
+    change 0 ≤ (δ i/angularFactor p (δ i) (logAngle κ (z i) x)*
+      sinePower (q p-2) (logAngle κ (z i) x))*logVelocity κ (z i) x^2
+    exact mul_nonneg (mul_nonneg (div_pos hδi hf).le
+      (Real.rpow_nonneg (logAngle_sin_pos κ (z i) x).le _)) (sq_nonneg _)
+  have hd (i : ι) (hi : i ∈ s) : d i ≤ (κ*C)*H i-(κ*k)*A i := by
+    have h := regularizedLogFactor_relative_drift κ (D i) (σ i) hκ hκ8
+      (hD i hi) (hσ i hi) (z i) x
+    convert h using 1
+    dsimp [d, C, H, k, A, δ, p]
+    ring
+  have hcross (ij : ι × ι) (hij : ij ∈ s.offDiag) :
+      (1/2 : ℝ)*(b ij.1*b ij.2) ≤ (κ*ε)*(A ij.1+A ij.2)+(κ*L)*w ij := by
+    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
+    have h := regularizedLogFactor_pair_absorption κ (D ij.1) (D ij.2) (σ ij.1) (σ ij.2)
+      ε hκ hκ8 (hD _ hi) (hD _ hj) he (z ij.1) (z ij.2) x (hδ _ hi) (hδ _ hj)
+    have hsmall : (1/2 : ℝ)*(b ij.1*b ij.2) ≤ |b ij.1*b ij.2| := by
+      nlinarith [le_abs_self (b ij.1*b ij.2), abs_nonneg (b ij.1*b ij.2)]
+    apply hsmall.trans
+    convert h using 1
+    dsimp [b, L, A, w, δ, p]
+    ring
+  have hbudget : 2*(κ*ε)*(s.card : ℝ) ≤ κ*k := by
+    have h := mul_le_mul_of_nonneg_left (absorption_epsilon_budget hk.le s.card) hκ.le
+    convert h using 1
+    dsimp [ε]
+    ring
+  have h := sum_drift_absorption s d A H (fun ij => (1/2 : ℝ)*(b ij.1*b ij.2)) w
+    (κ*C) (κ*k) (κ*ε) (κ*L) (mul_pos hκ he).le hA hbudget hd hcross
+  rw [← Finset.mul_sum] at h
+  convert h using 1
+  dsimp [d, b, C, L, H, w, ε, k, δ, p]
+  ring
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointProductGenerator.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointProductGenerator.lean
index 4212794719f9db53849b48755682f1324da23511..129575f4e88bea9fc86e40d61aca6f18042a3091 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointProductGenerator.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointProductGenerator.lean
@@ -1,115 +1,115 @@
-import OAI.Probability.SLE.Multipoint.MultipointProductDrift
-import OAI.Probability.SLE.Multipoint.MultipointProductCalculus
-
-namespace OAI
-
-/-! The generator of the actual regularized product in simultaneous
-Loewner logarithmic coordinates. -/
-
-noncomputable section
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-def regularizedProduct {ι : Type*} (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
-    (z : ι → ℂ × ℝ) (x : ℝ) : ℝ :=
-  ∏ i ∈ s, regularizedLogFactor κ (D i) (σ i) (z i) x
-
-def productDriftCost {ι : Type*} [DecidableEq ι] (s : Finset ι) (κ : ℝ)
-    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ) : ℝ :=
-  let p := greenExponent κ
-  let ε := (negativeDrift p/2)/(4*((s.card : ℝ)+1))
-  let C := driftAllowance p*driftThreshold p^(-p)
-  let L := youngConstant p (angularCoefficientConstant p^2) ε
-  κ*(C*(∑ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2) +
-    L*(∑ ij ∈ s.offDiag,
-      (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
-        branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent p)*
-      (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x)))
-
-lemma regularizedProduct_contDiff {ι : Type*} [Fintype ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ι → ℂ × ℝ) × ℝ => regularizedProduct s κ D σ p.1 p.2) := by
-  unfold regularizedProduct
-  apply contDiff_prod
-  intro i _
-  exact (regularizedLogFactor_contDiff κ (D i) (σ i)).comp
-    (((contDiff_apply ℝ (ℂ × ℝ) i).comp contDiff_fst).prodMk contDiff_snd)
-
-lemma regularizedFactor_state_directional {ι : Type*} [Fintype ι]
-    (i : ι) (κ D σ : ℝ) (z v : ι → ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun z => regularizedLogFactor κ D σ (z i) x) z) v =
-      (fderiv ℝ (fun q => regularizedLogFactor κ D σ q x) (z i)) (v i) := by
-  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun q => regularizedLogFactor κ D σ q x) :=
-    (regularizedLogFactor_contDiff κ D σ).comp (contDiff_id.prodMk contDiff_const)
-  have hd := ((hcd.differentiable (by simp)) (z i)).hasFDerivAt
-  have hproj : HasFDerivAt (fun q : ι → ℂ × ℝ => q i)
-      (ContinuousLinearMap.proj i) z := hasFDerivAt_apply (𝕜 := ℝ) i z
-  have he := hd.comp z hproj
-  have heq := congrArg (fun A : (ι → ℂ × ℝ) →L[ℝ] ℝ => A v) he.fderiv
-  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply,
-    ContinuousLinearMap.proj_apply] using heq
-
-/-- Completed and not-yet-active points impose no velocity condition. -/
-lemma regularizedProduct_directional_congr {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
-    (z v w : ι → ℂ × ℝ) (x : ℝ) (hvw : ∀ i ∈ s, v i = w i) :
-    (fderiv ℝ (fun q => regularizedProduct s κ D σ q x) z) v =
-      (fderiv ℝ (fun q => regularizedProduct s κ D σ q x) z) w := by
-  have hf (i : ι) : DifferentiableAt ℝ
-      (fun q : ι → ℂ × ℝ => regularizedLogFactor κ (D i) (σ i) (q i) x) z :=
-    (((regularizedLogFactor_contDiff κ (D i) (σ i)).comp
-      ((contDiff_apply ℝ (ℂ × ℝ) i).prodMk contDiff_const)).differentiable (by simp)) z
-  have hne (i : ι) (hi : i ∈ s) : regularizedLogFactor κ (D i) (σ i) (z i) x ≠ 0 :=
-    (regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) _ _).ne'
-  unfold regularizedProduct
-  rw [finite_product_directional_relative s _ z v (fun i _ => hf i) hne,
-    finite_product_directional_relative s _ z w (fun i _ => hf i) hne]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro i hi
-  rw [regularizedFactor_state_directional, regularizedFactor_state_directional, hvw i hi]
-
-theorem regularizedProduct_generator_identity {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
-    (z : ι → ℂ × ℝ) (x : ℝ) :
-    (fderiv ℝ (fun z => regularizedProduct s κ D σ z x) z)
-        (fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)) +
-      (1/2 : ℝ)*iteratedDeriv 2 (regularizedProduct s κ D σ z) x =
-    regularizedProduct s κ D σ z x *
-      ((∑ i ∈ s, logGenerator κ (regularizedLogFactor κ (D i) (σ i)) (z i) x /
-        regularizedLogFactor κ (D i) (σ i) (z i) x) +
-      (1/2 : ℝ)*(∑ ij ∈ s.offDiag,
-        (deriv (regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1)) x /
-          regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1) x)*
-        (deriv (regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2)) x /
-          regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2) x))) := by
-  have hcd (i : ι) : ContDiff ℝ (⊤ : ℕ∞)
-      (fun p : (ι → ℂ × ℝ) × ℝ => regularizedLogFactor κ (D i) (σ i) (p.1 i) p.2) :=
-    (regularizedLogFactor_contDiff κ (D i) (σ i)).comp
-      (((contDiff_apply ℝ (ℂ × ℝ) i).comp contDiff_fst).prodMk contDiff_snd)
-  have h := finite_product_generator s
-    (fun i z x => regularizedLogFactor κ (D i) (σ i) (z i) x) (fun i _ => hcd i)
-    (fun i hi z x => (regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) (z i) x).ne')
-    z (fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)) x
-  simp only [regularizedFactor_state_directional] at h
-  exact h
-
-theorem regularizedProduct_generator_le {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ)
-    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
-    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1) :
-    (fderiv ℝ (fun z => regularizedProduct s κ D σ z x) z)
-        (fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)) +
-      (1/2 : ℝ)*iteratedDeriv 2 (regularizedProduct s κ D σ z) x ≤
-    regularizedProduct s κ D σ z x * productDriftCost s κ D σ z x := by
-  rw [regularizedProduct_generator_identity s κ D σ hD]
-  apply mul_le_mul_of_nonneg_left (regularized_product_relative_drift s κ hκ hκ8 D σ z x hD hσ hδ)
-  exact (Finset.prod_pos (fun i hi => regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) (z i) x)).le
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointProductDrift
+import OAI.Probability.SLE.Multipoint.MultipointProductCalculus
+
+namespace OAI
+
+/-! The generator of the actual regularized product in simultaneous
+Loewner logarithmic coordinates. -/
+
+noncomputable section
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+def regularizedProduct {ι : Type*} (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ)
+    (z : ι → ℂ × ℝ) (x : ℝ) : ℝ :=
+  ∏ i ∈ s, regularizedLogFactor κ (D i) (σ i) (z i) x
+
+def productDriftCost {ι : Type*} [DecidableEq ι] (s : Finset ι) (κ : ℝ)
+    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ) : ℝ :=
+  let p := greenExponent κ
+  let ε := (negativeDrift p/2)/(4*((s.card : ℝ)+1))
+  let C := driftAllowance p*driftThreshold p^(-p)
+  let L := youngConstant p (angularCoefficientConstant p^2) ε
+  κ*(C*(∑ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2) +
+    L*(∑ ij ∈ s.offDiag,
+      (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
+        branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent p)*
+      (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x)))
+
+lemma regularizedProduct_contDiff {ι : Type*} [Fintype ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun p : (ι → ℂ × ℝ) × ℝ => regularizedProduct s κ D σ p.1 p.2) := by
+  unfold regularizedProduct
+  apply contDiff_prod
+  intro i _
+  exact (regularizedLogFactor_contDiff κ (D i) (σ i)).comp
+    (((contDiff_apply ℝ (ℂ × ℝ) i).comp contDiff_fst).prodMk contDiff_snd)
+
+lemma regularizedFactor_state_directional {ι : Type*} [Fintype ι]
+    (i : ι) (κ D σ : ℝ) (z v : ι → ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun z => regularizedLogFactor κ D σ (z i) x) z) v =
+      (fderiv ℝ (fun q => regularizedLogFactor κ D σ q x) (z i)) (v i) := by
+  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun q => regularizedLogFactor κ D σ q x) :=
+    (regularizedLogFactor_contDiff κ D σ).comp (contDiff_id.prodMk contDiff_const)
+  have hd := ((hcd.differentiable (by simp)) (z i)).hasFDerivAt
+  have hproj : HasFDerivAt (fun q : ι → ℂ × ℝ => q i)
+      (ContinuousLinearMap.proj i) z := hasFDerivAt_apply (𝕜 := ℝ) i z
+  have he := hd.comp z hproj
+  have heq := congrArg (fun A : (ι → ℂ × ℝ) →L[ℝ] ℝ => A v) he.fderiv
+  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.proj_apply] using heq
+
+/-- Completed and not-yet-active points impose no velocity condition. -/
+lemma regularizedProduct_directional_congr {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
+    (z v w : ι → ℂ × ℝ) (x : ℝ) (hvw : ∀ i ∈ s, v i = w i) :
+    (fderiv ℝ (fun q => regularizedProduct s κ D σ q x) z) v =
+      (fderiv ℝ (fun q => regularizedProduct s κ D σ q x) z) w := by
+  have hf (i : ι) : DifferentiableAt ℝ
+      (fun q : ι → ℂ × ℝ => regularizedLogFactor κ (D i) (σ i) (q i) x) z :=
+    (((regularizedLogFactor_contDiff κ (D i) (σ i)).comp
+      ((contDiff_apply ℝ (ℂ × ℝ) i).prodMk contDiff_const)).differentiable (by simp)) z
+  have hne (i : ι) (hi : i ∈ s) : regularizedLogFactor κ (D i) (σ i) (z i) x ≠ 0 :=
+    (regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) _ _).ne'
+  unfold regularizedProduct
+  rw [finite_product_directional_relative s _ z v (fun i _ => hf i) hne,
+    finite_product_directional_relative s _ z w (fun i _ => hf i) hne]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i hi
+  rw [regularizedFactor_state_directional, regularizedFactor_state_directional, hvw i hi]
+
+theorem regularizedProduct_generator_identity {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i)
+    (z : ι → ℂ × ℝ) (x : ℝ) :
+    (fderiv ℝ (fun z => regularizedProduct s κ D σ z x) z)
+        (fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)) +
+      (1/2 : ℝ)*iteratedDeriv 2 (regularizedProduct s κ D σ z) x =
+    regularizedProduct s κ D σ z x *
+      ((∑ i ∈ s, logGenerator κ (regularizedLogFactor κ (D i) (σ i)) (z i) x /
+        regularizedLogFactor κ (D i) (σ i) (z i) x) +
+      (1/2 : ℝ)*(∑ ij ∈ s.offDiag,
+        (deriv (regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1)) x /
+          regularizedLogFactor κ (D ij.1) (σ ij.1) (z ij.1) x)*
+        (deriv (regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2)) x /
+          regularizedLogFactor κ (D ij.2) (σ ij.2) (z ij.2) x))) := by
+  have hcd (i : ι) : ContDiff ℝ (⊤ : ℕ∞)
+      (fun p : (ι → ℂ × ℝ) × ℝ => regularizedLogFactor κ (D i) (σ i) (p.1 i) p.2) :=
+    (regularizedLogFactor_contDiff κ (D i) (σ i)).comp
+      (((contDiff_apply ℝ (ℂ × ℝ) i).comp contDiff_fst).prodMk contDiff_snd)
+  have h := finite_product_generator s
+    (fun i z x => regularizedLogFactor κ (D i) (σ i) (z i) x) (fun i _ => hcd i)
+    (fun i hi z x => (regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) (z i) x).ne')
+    z (fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)) x
+  simp only [regularizedFactor_state_directional] at h
+  exact h
+
+theorem regularizedProduct_generator_le {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (z : ι → ℂ × ℝ) (x : ℝ)
+    (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
+    (hδ : ∀ i ∈ s, branchRegulator κ (D i) (σ i) (z i) ≤ 1) :
+    (fderiv ℝ (fun z => regularizedProduct s κ D σ z x) z)
+        (fun i => physicalLogLoewnerField (Real.sqrt κ*x) (z i)) +
+      (1/2 : ℝ)*iteratedDeriv 2 (regularizedProduct s κ D σ z) x ≤
+    regularizedProduct s κ D σ z x * productDriftCost s κ D σ z x := by
+  rw [regularizedProduct_generator_identity s κ D σ hD]
+  apply mul_le_mul_of_nonneg_left (regularized_product_relative_drift s κ hκ hκ8 D σ z x hD hσ hδ)
+  exact (Finset.prod_pos (fun i hi => regularizedLogFactor_pos κ (D i) (σ i) (hD i hi) (z i) x)).le
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusStatus.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusStatus.lean
index e73ff05fbdb391d601849c4213ff346920641cf5..aaf7d48c7a82fb67529d2da0867f8167b2a9168c 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusStatus.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusStatus.lean
@@ -1,253 +1,253 @@
-import OAI.Probability.SLE.Loewner.ActualRadiusStops
-import OAI.Probability.SLE.Multipoint.MultipointLogClockTransport
-import OAI.Probability.SLE.Multipoint.MultipointRadiusSwitches
-
-namespace OAI
-
-/-! The literal radius stop is the inverse of the actual increasing radius
-clock. Horizon truncation preserves all threshold tests strictly before
-its cutoff, including equality at a crossed threshold. -/
-
-noncomputable section
-open Set
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma radiusLevelTime_le_iff_clock
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t) :
-    radiusLevelTime γ G z u ≤ (t : ℝ≥0∞) ↔
-      u ≤ logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
-  constructor
-  · intro hτ
-    obtain ⟨s,hs,hsR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u
-      (hτ.trans_lt ENNReal.coe_lt_top)
-    have hst : s ≤ t := by
-      rw [radiusLevelTime_eq_of_live_level hU hw z u s hs hsR] at hτ
-      exact_mod_cast hτ
-    have hmono := (hw.radius_clock_strictMonoOn hU z t ht).monotoneOn
-      (show (s : ℝ) ∈ Icc 0 (t : ℝ) from ⟨s.coe_nonneg,by exact_mod_cast hst⟩)
-      (show (t : ℝ) ∈ Icc 0 (t : ℝ) from ⟨t.coe_nonneg,le_rfl⟩)
-      (by exact_mod_cast hst)
-    have hsclock := (fullRadius_level_iff_logClock hU hw z s hs u).mp hsR
-    rw [Real.toNNReal_coe, Real.toNNReal_coe,
-      ← logRadiusClock_logCoordinates hU hw z s hs,
-      ← logRadiusClock_logCoordinates hU hw z t ht, hsclock] at hmono
-    exact hmono
-  · intro hc
-    have htR := (fullRadius_level_iff_logClock hU hw z t ht
-      (logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t))).mpr rfl
-    obtain ⟨s,hst,hs,hsR⟩ := live_radius_level_before hU hw z hu hc t ht htR
-    rw [radiusLevelTime_eq_of_live_level hU hw z u s hs hsR]
-    exact_mod_cast hst
-
-lemma radiusLevelTime_eq_iff_clock
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (u : ℝ) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t) :
-    radiusLevelTime γ G z u = (t : ℝ≥0∞) ↔
-      logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u := by
-  constructor
-  · intro hτ
-    obtain ⟨s,hs,hsR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u
-      (by rw [hτ]; exact ENNReal.coe_lt_top)
-    have hst : s = t := by
-      rw [radiusLevelTime_eq_of_live_level hU hw z u s hs hsR] at hτ
-      exact_mod_cast hτ
-    subst s
-    exact (fullRadius_level_iff_logClock hU hw z t ht u).mp hsR
-  · intro hc
-    exact radiusLevelTime_eq_of_live_level hU hw z u t ht
-      ((fullRadius_level_iff_logClock hU hw z t ht u).mpr hc)
-
-lemma radiusLevelTime_lt_iff_clock
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t) :
-    radiusLevelTime γ G z u < (t : ℝ≥0∞) ↔
-      u < logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
-  rw [lt_iff_le_and_ne, radiusLevelTime_le_iff_clock hU hw z u hu t ht]
-  constructor
-  · rintro ⟨hle,hne⟩
-    apply lt_of_le_of_ne hle
-    intro he
-    exact hne ((radiusLevelTime_eq_iff_clock hU hw z u t ht).mpr he.symm)
-  · intro h
-    refine ⟨h.le, ?_⟩
-    intro he
-    exact h.ne' ((radiusLevelTime_eq_iff_clock hU hw z u t ht).mp he)
-
-lemma cutoffStop_reached_before_horizon {Ω : Type*}
-    (τ : Ω → WithTop ℝ≥0) (H t : ℝ≥0) (ht : t ≤ H) (ω : Ω) :
-    cutoffStop τ (H+1) ω ≤ t ↔ τ ω ≤ (t : WithTop ℝ≥0) := by
-  have hHt : ¬ ((H+1 : ℝ≥0) : WithTop ℝ≥0) ≤ (t : WithTop ℝ≥0) := by
-    exact_mod_cast (not_le_of_gt (lt_of_le_of_lt ht (lt_add_one H)))
-  rw [← WithTop.coe_le_coe, cutoffStop_coe, min_le_iff]
-  exact or_iff_left hHt
-
-lemma cutoffStop_strictly_reached_before_horizon {Ω : Type*}
-    (τ : Ω → WithTop ℝ≥0) (H t : ℝ≥0) (ht : t ≤ H) (ω : Ω) :
-    cutoffStop τ (H+1) ω < t ↔ τ ω < (t : WithTop ℝ≥0) := by
-  have hHt : ¬ ((H+1 : ℝ≥0) : WithTop ℝ≥0) < (t : WithTop ℝ≥0) := by
-    exact_mod_cast (not_lt_of_ge (le_of_lt (lt_of_le_of_lt ht (lt_add_one H))))
-  rw [← WithTop.coe_lt_coe, cutoffStop_coe, min_lt_iff]
-  exact or_iff_left hHt
-
-lemma cutoffStop_eq_before_horizon {Ω : Type*}
-    (τ : Ω → WithTop ℝ≥0) (H t : ℝ≥0) (ht : t ≤ H) (ω : Ω) :
-    cutoffStop τ (H+1) ω = t ↔ τ ω = (t : WithTop ℝ≥0) := by
-  rw [eq_iff_le_not_lt, cutoffStop_reached_before_horizon τ H t ht ω,
-    cutoffStop_strictly_reached_before_horizon τ H t ht ω, eq_iff_le_not_lt]
-
-namespace Multipoint
-
-def sourceClockLevel (κ : ℝ) (z : ℂ) (u : ℝ) : ℝ := κ/4*(u-Real.log z.im)
-
-lemma cutoff_radius_reached_iff
-    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u : ℝ) (hu : 0 ≤ u)
-    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
-    cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω ≤ t ↔
-      sourceClockLevel κ z u ≤ logFactorClock κ (logLoewnerCoordinates G z t) := by
-  have hc : (radiusLevelTime γ G z u : WithTop ℝ≥0) ≤ (t : WithTop ℝ≥0) ↔
-      u ≤ logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
-    constructor
-    · intro h
-      apply (radiusLevelTime_le_iff_clock hU hw z u hu t ht).mp
-      exact_mod_cast h
-    · intro h
-      have h' := (radiusLevelTime_le_iff_clock hU hw z u hu t ht).mpr h
-      exact_mod_cast h'
-  have hs : (u ≤ logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t)) ↔
-      sourceClockLevel κ z u ≤ logFactorClock κ (logLoewnerCoordinates G z t) := by
-    rw [logFactorClock_eq_logRadiusClock κ (Real.log z.im)]
-    unfold sourceClockLevel
-    rw [mul_le_mul_iff_right₀ (div_pos hκ (by norm_num)), sub_le_sub_iff_right]
-  exact (cutoffStop_reached_before_horizon
-    (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) H t htH ω).trans (hc.trans hs)
-
-lemma cutoff_radius_strictly_reached_iff
-    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u : ℝ) (hu : 0 ≤ u)
-    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
-    cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω < t ↔
-      sourceClockLevel κ z u < logFactorClock κ (logLoewnerCoordinates G z t) := by
-  have hc : (radiusLevelTime γ G z u : WithTop ℝ≥0) < (t : WithTop ℝ≥0) ↔
-      u < logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
-    constructor
-    · intro h
-      apply (radiusLevelTime_lt_iff_clock hU hw z u hu t ht).mp
-      exact_mod_cast h
-    · intro h
-      have h' := (radiusLevelTime_lt_iff_clock hU hw z u hu t ht).mpr h
-      exact_mod_cast h'
-  have hs : (u < logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t)) ↔
-      sourceClockLevel κ z u < logFactorClock κ (logLoewnerCoordinates G z t) := by
-    rw [logFactorClock_eq_logRadiusClock κ (Real.log z.im)]
-    unfold sourceClockLevel
-    rw [mul_lt_mul_iff_right₀ (div_pos hκ (by norm_num)), sub_lt_sub_iff_right]
-  exact (cutoffStop_strictly_reached_before_horizon
-    (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) H t htH ω).trans (hc.trans hs)
-
-lemma cutoff_radius_eq_iff
-    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u : ℝ)
-    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
-    cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω = t ↔
-      logFactorClock κ (logLoewnerCoordinates G z t) = sourceClockLevel κ z u := by
-  have hc : (radiusLevelTime γ G z u : WithTop ℝ≥0) = (t : WithTop ℝ≥0) ↔
-      logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u := by
-    constructor
-    · intro h
-      apply (radiusLevelTime_eq_iff_clock hU hw z u t ht).mp
-      exact_mod_cast h
-    · intro h
-      have h' := (radiusLevelTime_eq_iff_clock hU hw z u t ht).mpr h
-      exact_mod_cast h'
-  have hs : (logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u) ↔
-      logFactorClock κ (logLoewnerCoordinates G z t) = sourceClockLevel κ z u := by
-    rw [logFactorClock_eq_logRadiusClock κ (Real.log z.im)]
-    unfold sourceClockLevel
-    rw [mul_right_inj' (div_pos hκ (by norm_num)).ne', sub_left_inj]
-  exact (cutoffStop_eq_before_horizon
-    (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) H t htH ω).trans (hc.trans hs)
-
-/-- The three concrete radius switches give exactly the source branch
-thresholds, also after truncation at `H + 1`. -/
-theorem cutoff_radius_factor_thresholds
-    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
-    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u₀ u₂ : ℝ) (h₀ : 0 ≤ u₀) (h₂ : 0 ≤ u₂)
-    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
-    let τ := fun u => cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω
-    let v := logFactorClock κ (logLoewnerCoordinates G z t)-sourceClockLevel κ z u₀
-    let V := κ/4*(u₂-u₀)
-    (τ u₀ ≤ t ↔ 0 ≤ v) ∧
-    (τ ((u₀+u₂)/2) ≤ t ↔ V/2 ≤ v) ∧
-    (τ u₂ ≤ t ↔ V ≤ v) ∧
-    (τ u₀ = t → v = 0) ∧
-    (τ ((u₀+u₂)/2) = t → v = V/2) ∧
-    (τ u₂ = t → v = V) := by
-  dsimp only
-  have hm : 0 ≤ (u₀+u₂)/2 := by positivity
-  have hstart := cutoff_radius_reached_iff hU hw hκ z u₀ h₀ H t htH ht ω
-  have hmid := cutoff_radius_reached_iff hU hw hκ z ((u₀+u₂)/2) hm H t htH ht ω
-  have hend := cutoff_radius_reached_iff hU hw hκ z u₂ h₂ H t htH ht ω
-  have hmidlevel : sourceClockLevel κ z ((u₀+u₂)/2) =
-      sourceClockLevel κ z u₀ + (κ/4*(u₂-u₀))/2 := by
-    unfold sourceClockLevel
-    ring
-  have hendlevel : sourceClockLevel κ z u₂ =
-      sourceClockLevel κ z u₀ + κ/4*(u₂-u₀) := by
-    unfold sourceClockLevel
-    ring
-  refine ⟨?_,?_,?_,?_,?_,?_⟩
-  · exact hstart.trans (sub_nonneg.symm)
-  · rw [hmid,hmidlevel]
-    constructor <;> intro h <;> linarith
-  · rw [hend,hendlevel]
-    constructor <;> intro h <;> linarith
-  · intro h
-    have he := (cutoff_radius_eq_iff hU hw hκ z u₀ H t htH ht ω).mp h
-    exact sub_eq_zero.mpr he
-  · intro h
-    have he := (cutoff_radius_eq_iff hU hw hκ z ((u₀+u₂)/2) H t htH ht ω).mp h
-    rw [hmidlevel] at he
-    linarith
-  · intro h
-    have he := (cutoff_radius_eq_iff hU hw hκ z u₂ H t htH ht ω).mp h
-    rw [hendlevel] at he
-    linarith
-
-end Multipoint
-
-/-- At the activation clock chosen from the physical initial radius, the
-physical conformal radius is exactly the prescribed activation radius. -/
-theorem IsCapacityTwoWitness.physical_radius_at_activation
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (J : ℂ → ℂ) (z : ℂ) (R b : ℝ) (hR : 0 < R) (hb : 0 < b)
-    (hRdef : R = 2*z.im*‖deriv J z‖)
-    (t : ℝ≥0) (ht : z ∈ survivingDomain γ t)
-    (hclock : logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) =
-      Real.log (R/b)) :
-    physicalConformalRadius J G t z = b := by
-  have hradius := (fullRadius_level_iff_logClock hU hw z t ht (Real.log (R/b))).mpr hclock
-  have hexp : Real.exp (-Real.log (R/b)) = b/R := by
-    rw [Real.exp_neg, Real.exp_log (div_pos hR hb)]
-    field_simp [hR.ne',hb.ne']
-  have hcoef : ‖deriv J z‖*(2*z.im) = R := by rw [hRdef]; ring
-  unfold physicalConformalRadius
-  rw [hradius,hexp,← mul_assoc,hcoef]
-  field_simp [hR.ne']
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.ActualRadiusStops
+import OAI.Probability.SLE.Multipoint.MultipointLogClockTransport
+import OAI.Probability.SLE.Multipoint.MultipointRadiusSwitches
+
+namespace OAI
+
+/-! The literal radius stop is the inverse of the actual increasing radius
+clock. Horizon truncation preserves all threshold tests strictly before
+its cutoff, including equality at a crossed threshold. -/
+
+noncomputable section
+open Set
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma radiusLevelTime_le_iff_clock
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t) :
+    radiusLevelTime γ G z u ≤ (t : ℝ≥0∞) ↔
+      u ≤ logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
+  constructor
+  · intro hτ
+    obtain ⟨s,hs,hsR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u
+      (hτ.trans_lt ENNReal.coe_lt_top)
+    have hst : s ≤ t := by
+      rw [radiusLevelTime_eq_of_live_level hU hw z u s hs hsR] at hτ
+      exact_mod_cast hτ
+    have hmono := (hw.radius_clock_strictMonoOn hU z t ht).monotoneOn
+      (show (s : ℝ) ∈ Icc 0 (t : ℝ) from ⟨s.coe_nonneg,by exact_mod_cast hst⟩)
+      (show (t : ℝ) ∈ Icc 0 (t : ℝ) from ⟨t.coe_nonneg,le_rfl⟩)
+      (by exact_mod_cast hst)
+    have hsclock := (fullRadius_level_iff_logClock hU hw z s hs u).mp hsR
+    rw [Real.toNNReal_coe, Real.toNNReal_coe,
+      ← logRadiusClock_logCoordinates hU hw z s hs,
+      ← logRadiusClock_logCoordinates hU hw z t ht, hsclock] at hmono
+    exact hmono
+  · intro hc
+    have htR := (fullRadius_level_iff_logClock hU hw z t ht
+      (logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t))).mpr rfl
+    obtain ⟨s,hst,hs,hsR⟩ := live_radius_level_before hU hw z hu hc t ht htR
+    rw [radiusLevelTime_eq_of_live_level hU hw z u s hs hsR]
+    exact_mod_cast hst
+
+lemma radiusLevelTime_eq_iff_clock
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (u : ℝ) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t) :
+    radiusLevelTime γ G z u = (t : ℝ≥0∞) ↔
+      logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u := by
+  constructor
+  · intro hτ
+    obtain ⟨s,hs,hsR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u
+      (by rw [hτ]; exact ENNReal.coe_lt_top)
+    have hst : s = t := by
+      rw [radiusLevelTime_eq_of_live_level hU hw z u s hs hsR] at hτ
+      exact_mod_cast hτ
+    subst s
+    exact (fullRadius_level_iff_logClock hU hw z t ht u).mp hsR
+  · intro hc
+    exact radiusLevelTime_eq_of_live_level hU hw z u t ht
+      ((fullRadius_level_iff_logClock hU hw z t ht u).mpr hc)
+
+lemma radiusLevelTime_lt_iff_clock
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (ht : z ∈ survivingDomain γ t) :
+    radiusLevelTime γ G z u < (t : ℝ≥0∞) ↔
+      u < logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
+  rw [lt_iff_le_and_ne, radiusLevelTime_le_iff_clock hU hw z u hu t ht]
+  constructor
+  · rintro ⟨hle,hne⟩
+    apply lt_of_le_of_ne hle
+    intro he
+    exact hne ((radiusLevelTime_eq_iff_clock hU hw z u t ht).mpr he.symm)
+  · intro h
+    refine ⟨h.le, ?_⟩
+    intro he
+    exact h.ne' ((radiusLevelTime_eq_iff_clock hU hw z u t ht).mp he)
+
+lemma cutoffStop_reached_before_horizon {Ω : Type*}
+    (τ : Ω → WithTop ℝ≥0) (H t : ℝ≥0) (ht : t ≤ H) (ω : Ω) :
+    cutoffStop τ (H+1) ω ≤ t ↔ τ ω ≤ (t : WithTop ℝ≥0) := by
+  have hHt : ¬ ((H+1 : ℝ≥0) : WithTop ℝ≥0) ≤ (t : WithTop ℝ≥0) := by
+    exact_mod_cast (not_le_of_gt (lt_of_le_of_lt ht (lt_add_one H)))
+  rw [← WithTop.coe_le_coe, cutoffStop_coe, min_le_iff]
+  exact or_iff_left hHt
+
+lemma cutoffStop_strictly_reached_before_horizon {Ω : Type*}
+    (τ : Ω → WithTop ℝ≥0) (H t : ℝ≥0) (ht : t ≤ H) (ω : Ω) :
+    cutoffStop τ (H+1) ω < t ↔ τ ω < (t : WithTop ℝ≥0) := by
+  have hHt : ¬ ((H+1 : ℝ≥0) : WithTop ℝ≥0) < (t : WithTop ℝ≥0) := by
+    exact_mod_cast (not_lt_of_ge (le_of_lt (lt_of_le_of_lt ht (lt_add_one H))))
+  rw [← WithTop.coe_lt_coe, cutoffStop_coe, min_lt_iff]
+  exact or_iff_left hHt
+
+lemma cutoffStop_eq_before_horizon {Ω : Type*}
+    (τ : Ω → WithTop ℝ≥0) (H t : ℝ≥0) (ht : t ≤ H) (ω : Ω) :
+    cutoffStop τ (H+1) ω = t ↔ τ ω = (t : WithTop ℝ≥0) := by
+  rw [eq_iff_le_not_lt, cutoffStop_reached_before_horizon τ H t ht ω,
+    cutoffStop_strictly_reached_before_horizon τ H t ht ω, eq_iff_le_not_lt]
+
+namespace Multipoint
+
+def sourceClockLevel (κ : ℝ) (z : ℂ) (u : ℝ) : ℝ := κ/4*(u-Real.log z.im)
+
+lemma cutoff_radius_reached_iff
+    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u : ℝ) (hu : 0 ≤ u)
+    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
+    cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω ≤ t ↔
+      sourceClockLevel κ z u ≤ logFactorClock κ (logLoewnerCoordinates G z t) := by
+  have hc : (radiusLevelTime γ G z u : WithTop ℝ≥0) ≤ (t : WithTop ℝ≥0) ↔
+      u ≤ logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
+    constructor
+    · intro h
+      apply (radiusLevelTime_le_iff_clock hU hw z u hu t ht).mp
+      exact_mod_cast h
+    · intro h
+      have h' := (radiusLevelTime_le_iff_clock hU hw z u hu t ht).mpr h
+      exact_mod_cast h'
+  have hs : (u ≤ logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t)) ↔
+      sourceClockLevel κ z u ≤ logFactorClock κ (logLoewnerCoordinates G z t) := by
+    rw [logFactorClock_eq_logRadiusClock κ (Real.log z.im)]
+    unfold sourceClockLevel
+    rw [mul_le_mul_iff_right₀ (div_pos hκ (by norm_num)), sub_le_sub_iff_right]
+  exact (cutoffStop_reached_before_horizon
+    (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) H t htH ω).trans (hc.trans hs)
+
+lemma cutoff_radius_strictly_reached_iff
+    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u : ℝ) (hu : 0 ≤ u)
+    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
+    cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω < t ↔
+      sourceClockLevel κ z u < logFactorClock κ (logLoewnerCoordinates G z t) := by
+  have hc : (radiusLevelTime γ G z u : WithTop ℝ≥0) < (t : WithTop ℝ≥0) ↔
+      u < logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) := by
+    constructor
+    · intro h
+      apply (radiusLevelTime_lt_iff_clock hU hw z u hu t ht).mp
+      exact_mod_cast h
+    · intro h
+      have h' := (radiusLevelTime_lt_iff_clock hU hw z u hu t ht).mpr h
+      exact_mod_cast h'
+  have hs : (u < logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t)) ↔
+      sourceClockLevel κ z u < logFactorClock κ (logLoewnerCoordinates G z t) := by
+    rw [logFactorClock_eq_logRadiusClock κ (Real.log z.im)]
+    unfold sourceClockLevel
+    rw [mul_lt_mul_iff_right₀ (div_pos hκ (by norm_num)), sub_lt_sub_iff_right]
+  exact (cutoffStop_strictly_reached_before_horizon
+    (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) H t htH ω).trans (hc.trans hs)
+
+lemma cutoff_radius_eq_iff
+    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u : ℝ)
+    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
+    cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω = t ↔
+      logFactorClock κ (logLoewnerCoordinates G z t) = sourceClockLevel κ z u := by
+  have hc : (radiusLevelTime γ G z u : WithTop ℝ≥0) = (t : WithTop ℝ≥0) ↔
+      logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u := by
+    constructor
+    · intro h
+      apply (radiusLevelTime_eq_iff_clock hU hw z u t ht).mp
+      exact_mod_cast h
+    · intro h
+      have h' := (radiusLevelTime_eq_iff_clock hU hw z u t ht).mpr h
+      exact_mod_cast h'
+  have hs : (logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) = u) ↔
+      logFactorClock κ (logLoewnerCoordinates G z t) = sourceClockLevel κ z u := by
+    rw [logFactorClock_eq_logRadiusClock κ (Real.log z.im)]
+    unfold sourceClockLevel
+    rw [mul_right_inj' (div_pos hκ (by norm_num)).ne', sub_left_inj]
+  exact (cutoffStop_eq_before_horizon
+    (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) H t htH ω).trans (hc.trans hs)
+
+/-- The three concrete radius switches give exactly the source branch
+thresholds, also after truncation at `H + 1`. -/
+theorem cutoff_radius_factor_thresholds
+    {Ω : Type*} {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F)
+    {κ : ℝ} (hκ : 0 < κ) (z : ℂ) (u₀ u₂ : ℝ) (h₀ : 0 ≤ u₀) (h₂ : 0 ≤ u₂)
+    (H t : ℝ≥0) (htH : t ≤ H) (ht : z ∈ survivingDomain γ t) (ω : Ω) :
+    let τ := fun u => cutoffStop (fun _ : Ω => (radiusLevelTime γ G z u : WithTop ℝ≥0)) (H+1) ω
+    let v := logFactorClock κ (logLoewnerCoordinates G z t)-sourceClockLevel κ z u₀
+    let V := κ/4*(u₂-u₀)
+    (τ u₀ ≤ t ↔ 0 ≤ v) ∧
+    (τ ((u₀+u₂)/2) ≤ t ↔ V/2 ≤ v) ∧
+    (τ u₂ ≤ t ↔ V ≤ v) ∧
+    (τ u₀ = t → v = 0) ∧
+    (τ ((u₀+u₂)/2) = t → v = V/2) ∧
+    (τ u₂ = t → v = V) := by
+  dsimp only
+  have hm : 0 ≤ (u₀+u₂)/2 := by positivity
+  have hstart := cutoff_radius_reached_iff hU hw hκ z u₀ h₀ H t htH ht ω
+  have hmid := cutoff_radius_reached_iff hU hw hκ z ((u₀+u₂)/2) hm H t htH ht ω
+  have hend := cutoff_radius_reached_iff hU hw hκ z u₂ h₂ H t htH ht ω
+  have hmidlevel : sourceClockLevel κ z ((u₀+u₂)/2) =
+      sourceClockLevel κ z u₀ + (κ/4*(u₂-u₀))/2 := by
+    unfold sourceClockLevel
+    ring
+  have hendlevel : sourceClockLevel κ z u₂ =
+      sourceClockLevel κ z u₀ + κ/4*(u₂-u₀) := by
+    unfold sourceClockLevel
+    ring
+  refine ⟨?_,?_,?_,?_,?_,?_⟩
+  · exact hstart.trans (sub_nonneg.symm)
+  · rw [hmid,hmidlevel]
+    constructor <;> intro h <;> linarith
+  · rw [hend,hendlevel]
+    constructor <;> intro h <;> linarith
+  · intro h
+    have he := (cutoff_radius_eq_iff hU hw hκ z u₀ H t htH ht ω).mp h
+    exact sub_eq_zero.mpr he
+  · intro h
+    have he := (cutoff_radius_eq_iff hU hw hκ z ((u₀+u₂)/2) H t htH ht ω).mp h
+    rw [hmidlevel] at he
+    linarith
+  · intro h
+    have he := (cutoff_radius_eq_iff hU hw hκ z u₂ H t htH ht ω).mp h
+    rw [hendlevel] at he
+    linarith
+
+end Multipoint
+
+/-- At the activation clock chosen from the physical initial radius, the
+physical conformal radius is exactly the prescribed activation radius. -/
+theorem IsCapacityTwoWitness.physical_radius_at_activation
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (J : ℂ → ℂ) (z : ℂ) (R b : ℝ) (hR : 0 < R) (hb : 0 < b)
+    (hRdef : R = 2*z.im*‖deriv J z‖)
+    (t : ℝ≥0) (ht : z ∈ survivingDomain γ t)
+    (hclock : logRadiusClock (Real.log z.im) (logLoewnerCoordinates G z t) =
+      Real.log (R/b)) :
+    physicalConformalRadius J G t z = b := by
+  have hradius := (fullRadius_level_iff_logClock hU hw z t ht (Real.log (R/b))).mpr hclock
+  have hexp : Real.exp (-Real.log (R/b)) = b/R := by
+    rw [Real.exp_neg, Real.exp_log (div_pos hR hb)]
+    field_simp [hR.ne',hb.ne']
+  have hcoef : ‖deriv J z‖*(2*z.im) = R := by rw [hRdef]; ring
+  unfold physicalConformalRadius
+  rw [hradius,hexp,← mul_assoc,hcoef]
+  field_simp [hR.ne']
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusSwitches.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusSwitches.lean
index 9b373832d47c02c56abab6b8d06730caa085d774..09138c293e6a2f41a1a9ae09152037fdc070a992 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusSwitches.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointRadiusSwitches.lean
@@ -1,103 +1,103 @@
-import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular
-import OAI.Probability.SLE.Gauge.ExtendedExperimentStop
-
-namespace OAI
-
-/-! The three actual radius switches of one multipoint factor: activation,
-the midpoint of its radius clock, and completion. They reuse the literal
-radius-level stopping times and are truncated only at the common finite
-localization horizon. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def switchLevel (R b e : ℝ) (j : Fin 3) : ℝ :=
-  if j = 0 then Real.log (R / b)
-  else if j = 1 then (Real.log (R / b) + Real.log (R / e)) / 2
-  else Real.log (R / e)
-
-lemma switchLevel_nonneg {R b e : ℝ} (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R)
-    (j : Fin 3) : 0 ≤ switchLevel R b e j := by
-  have hb := he.trans_le heb
-  have h₀ := Real.log_nonneg ((one_le_div hb).mpr hbR)
-  have h₂ := Real.log_nonneg ((one_le_div he).mpr (heb.trans hbR))
-  fin_cases j <;> norm_num [switchLevel] <;> linarith
-
-lemma switchLevel_monotone {R b e : ℝ} (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) :
-    Monotone (switchLevel R b e) := by
-  have hb := he.trans_le heb
-  have hR := hb.trans_le hbR
-  have hle : Real.log (R / b) ≤ Real.log (R / e) :=
-    Real.log_le_log (div_pos hR hb) (div_le_div_of_nonneg_left hR.le he heb)
-  intro i j hij
-  dsimp only [switchLevel]
-  split_ifs <;> first | omega | linarith
-
-lemma switchLevel_midpoint_offset {R b e : ℝ} (hR : 0 < R) (hb : 0 < b) (he : 0 < e) :
-    switchLevel R b e 1 - switchLevel R b e 0 = Real.log (b / e) / 2 := by
-  norm_num [switchLevel,
-    Real.log_div hR.ne' hb.ne', Real.log_div hR.ne' he.ne', Real.log_div hb.ne' he.ne']
-  ring
-
-lemma switchLevel_completion_offset {R b e : ℝ} (hR : 0 < R) (hb : 0 < b) (he : 0 < e) :
-    switchLevel R b e 2 - switchLevel R b e 0 = Real.log (b / e) := by
-  norm_num [switchLevel,
-    Real.log_div hR.ne' hb.ne', Real.log_div hR.ne' he.ne', Real.log_div hb.ne' he.ne']
-
-end SLEExactGauge.Multipoint
-
-namespace SLEExactGauge
-
-lemma cutoffStop_le_horizon {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) :
-    cutoffStop τ H ω ≤ H := by
-  apply WithTop.coe_le_coe.mp
-  rw [cutoffStop_coe]
-  exact min_le_right _ _
-
-lemma cutoffStop_mono {Ω : Type*} {σ τ : Ω → WithTop ℝ≥0}
-    (hστ : ∀ ω, σ ω ≤ τ ω) (H : ℝ≥0) (ω : Ω) :
-    cutoffStop σ H ω ≤ cutoffStop τ H ω := by
-  apply WithTop.coe_le_coe.mp
-  rw [cutoffStop_coe, cutoffStop_coe]
-  exact min_le_min (hστ ω) le_rfl
-
-/-- Concrete activation, midpoint, and completion stopping times for a
-regular actual SLE, all localized at the same horizon. -/
-theorem actual_radius_factor_switches
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane)
-    (R b e : ℝ) (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) (H : ℝ≥0) :
-    let τ := fun j : Fin 3 => cutoffStop
-      (fun ω => radiusLevelTime (γ ω) (G ω) z (Multipoint.switchLevel R b e j)) H
-    (∀ j, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ j ω : WithTop ℝ≥0))) ∧
-    (∀ ω, Monotone (fun j => τ j ω)) ∧ (∀ j ω, τ j ω ≤ H) := by
-  dsimp only
-  refine ⟨?_, ?_, ?_⟩
-  · intro j
-    exact cutoffStop_stopping _ _
-      (actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz
-        (Multipoint.switchLevel R b e j) (Multipoint.switchLevel_nonneg he heb hbR j)) H
-  · intro ω i j hij
-    apply cutoffStop_mono _ H ω
-    intro ω'
-    exact radiusLevelTime_mono ((hc ω').const_mul _) (hw ω') z
-      (Multipoint.switchLevel_nonneg he heb hbR i)
-      (Multipoint.switchLevel_monotone he heb hbR hij)
-  · exact fun j ω => cutoffStop_le_horizon _ H ω
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular
+import OAI.Probability.SLE.Gauge.ExtendedExperimentStop
+
+namespace OAI
+
+/-! The three actual radius switches of one multipoint factor: activation,
+the midpoint of its radius clock, and completion. They reuse the literal
+radius-level stopping times and are truncated only at the common finite
+localization horizon. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def switchLevel (R b e : ℝ) (j : Fin 3) : ℝ :=
+  if j = 0 then Real.log (R / b)
+  else if j = 1 then (Real.log (R / b) + Real.log (R / e)) / 2
+  else Real.log (R / e)
+
+lemma switchLevel_nonneg {R b e : ℝ} (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R)
+    (j : Fin 3) : 0 ≤ switchLevel R b e j := by
+  have hb := he.trans_le heb
+  have h₀ := Real.log_nonneg ((one_le_div hb).mpr hbR)
+  have h₂ := Real.log_nonneg ((one_le_div he).mpr (heb.trans hbR))
+  fin_cases j <;> norm_num [switchLevel] <;> linarith
+
+lemma switchLevel_monotone {R b e : ℝ} (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) :
+    Monotone (switchLevel R b e) := by
+  have hb := he.trans_le heb
+  have hR := hb.trans_le hbR
+  have hle : Real.log (R / b) ≤ Real.log (R / e) :=
+    Real.log_le_log (div_pos hR hb) (div_le_div_of_nonneg_left hR.le he heb)
+  intro i j hij
+  dsimp only [switchLevel]
+  split_ifs <;> first | omega | linarith
+
+lemma switchLevel_midpoint_offset {R b e : ℝ} (hR : 0 < R) (hb : 0 < b) (he : 0 < e) :
+    switchLevel R b e 1 - switchLevel R b e 0 = Real.log (b / e) / 2 := by
+  norm_num [switchLevel,
+    Real.log_div hR.ne' hb.ne', Real.log_div hR.ne' he.ne', Real.log_div hb.ne' he.ne']
+  ring
+
+lemma switchLevel_completion_offset {R b e : ℝ} (hR : 0 < R) (hb : 0 < b) (he : 0 < e) :
+    switchLevel R b e 2 - switchLevel R b e 0 = Real.log (b / e) := by
+  norm_num [switchLevel,
+    Real.log_div hR.ne' hb.ne', Real.log_div hR.ne' he.ne', Real.log_div hb.ne' he.ne']
+
+end SLEExactGauge.Multipoint
+
+namespace SLEExactGauge
+
+lemma cutoffStop_le_horizon {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (H : ℝ≥0) (ω : Ω) :
+    cutoffStop τ H ω ≤ H := by
+  apply WithTop.coe_le_coe.mp
+  rw [cutoffStop_coe]
+  exact min_le_right _ _
+
+lemma cutoffStop_mono {Ω : Type*} {σ τ : Ω → WithTop ℝ≥0}
+    (hστ : ∀ ω, σ ω ≤ τ ω) (H : ℝ≥0) (ω : Ω) :
+    cutoffStop σ H ω ≤ cutoffStop τ H ω := by
+  apply WithTop.coe_le_coe.mp
+  rw [cutoffStop_coe, cutoffStop_coe]
+  exact min_le_min (hστ ω) le_rfl
+
+/-- Concrete activation, midpoint, and completion stopping times for a
+regular actual SLE, all localized at the same horizon. -/
+theorem actual_radius_factor_switches
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane)
+    (R b e : ℝ) (he : 0 < e) (heb : e ≤ b) (hbR : b ≤ R) (H : ℝ≥0) :
+    let τ := fun j : Fin 3 => cutoffStop
+      (fun ω => radiusLevelTime (γ ω) (G ω) z (Multipoint.switchLevel R b e j)) H
+    (∀ j, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ j ω : WithTop ℝ≥0))) ∧
+    (∀ ω, Monotone (fun j => τ j ω)) ∧ (∀ j ω, τ j ω ≤ H) := by
+  dsimp only
+  refine ⟨?_, ?_, ?_⟩
+  · intro j
+    exact cutoffStop_stopping _ _
+      (actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz
+        (Multipoint.switchLevel R b e j) (Multipoint.switchLevel_nonneg he heb hbR j)) H
+  · intro ω i j hij
+    apply cutoffStop_mono _ H ω
+    intro ω'
+    exact radiusLevelTime_mono ((hc ω').const_mul _) (hw ω') z
+      (Multipoint.switchLevel_nonneg he heb hbR i)
+      (Multipoint.switchLevel_monotone he heb hbR hij)
+  · exact fun j ω => cutoffStop_le_horizon _ H ω
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizedField.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizedField.lean
index 4c1551f4c285f233954708d7e41675b33d6c02fd..dc557464803e3842b7f4bf743695f588a38b2447 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizedField.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizedField.lean
@@ -1,143 +1,143 @@
-import OAI.Probability.SLE.Multipoint.MultipointLogAngular
-
-namespace OAI
-
-/-! The actual smooth factor on either side of a regulator midpoint.
-Its only parameters are the constant and logarithmic slope of that branch. -/
-
-noncomputable section
-
-namespace SLEExactGauge.Multipoint
-
-def regularizedLogFactor (κ D σ : ℝ) (z : ℂ × ℝ) (x : ℝ) : ℝ :=
-  logFactor κ (greenExponent κ/2) (greenExponent κ) z x +
-    D*(logFactor κ (greenExponent κ/2+σ) 0 z x +
-      logFactor κ (greenExponent κ/2+σ) (q (greenExponent κ)) z x)
-
-def branchRegulator (κ D σ : ℝ) (z : ℂ × ℝ) : ℝ :=
-  D*Real.exp ((σ*κ/4)*(z.2-z.1.im))
-
-lemma regularizedLogFactor_contDiff (κ D σ : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => regularizedLogFactor κ D σ z.1 z.2) :=
-  (logFactor_contDiff κ _ _).add (contDiff_const.mul
-    ((logFactor_contDiff κ _ _).add (logFactor_contDiff κ _ _)))
-
-lemma regularizedLogFactor_eq (κ D σ : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    regularizedLogFactor κ D σ z x =
-      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
-        angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by
-  unfold regularizedLogFactor
-  rw [logFactor_eq_angular, logFactor_eq_angular, logFactor_eq_angular]
-  have he : Real.exp (((greenExponent κ/2+σ)*κ/4)*(z.2-z.1.im)) =
-      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
-        Real.exp ((σ*κ/4)*(z.2-z.1.im)) := by
-    rw [← Real.exp_add]; congr 1; ring
-  rw [he]
-  simp only [sinePower, Real.rpow_zero, angularFactor, branchRegulator]
-  ring
-
-lemma regularizedLogFactor_pos (κ D σ : ℝ) (hD : 0 < D) (z : ℂ × ℝ) (x : ℝ) :
-    0 < regularizedLogFactor κ D σ z x := by
-  rw [regularizedLogFactor_eq]
-  exact mul_pos (Real.exp_pos _) (angularFactor_pos (mul_pos hD (Real.exp_pos _))
-    (logAngle_sin_pos κ z x))
-
-lemma regularizedLogFactor_spatial (κ D σ : ℝ) (z : ℂ × ℝ) (x : ℝ) :
-    deriv (regularizedLogFactor κ D σ z) x =
-      Real.sqrt κ*logVelocity κ z x*
-        Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
-          angularFactorFirst (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by
-  have hd := (logFactor_spatial_derivative κ (greenExponent κ/2) (greenExponent κ) z x).add
-    (((logFactor_spatial_derivative κ (greenExponent κ/2+σ) 0 z x).add
-      (logFactor_spatial_derivative κ (greenExponent κ/2+σ) (q (greenExponent κ)) z x)).const_mul D)
-  change HasDerivAt (regularizedLogFactor κ D σ z) _ x at hd
-  rw [hd.deriv]
-  have hs (c r : ℝ) := logFactor_spatial_eq_angular κ c r z x
-  rw [← (logFactor_spatial_derivative κ (greenExponent κ/2) (greenExponent κ) z x).deriv,
-    ← (logFactor_spatial_derivative κ (greenExponent κ/2+σ) 0 z x).deriv,
-    ← (logFactor_spatial_derivative κ (greenExponent κ/2+σ) (q (greenExponent κ)) z x).deriv]
-  rw [hs, hs, hs]
-  have he : Real.exp (((greenExponent κ/2+σ)*κ/4)*(z.2-z.1.im)) =
-      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
-        Real.exp ((σ*κ/4)*(z.2-z.1.im)) := by
-    rw [← Real.exp_add]; congr 1; ring
-  rw [he]
-  simp only [sinePowerFirst, zero_mul, angularFactorFirst, branchRegulator]
-  ring
-
-def logGenerator (κ : ℝ) (F : (ℂ × ℝ) → ℝ → ℝ) (z : ℂ × ℝ) (x : ℝ) : ℝ :=
-  (fderiv ℝ (fun z => F z x) z) (physicalLogLoewnerField (Real.sqrt κ*x) z) +
-    (1/2 : ℝ)*iteratedDeriv 2 (F z) x
-
-lemma logGenerator_add (κ : ℝ) (F G : (ℂ × ℝ) → ℝ → ℝ)
-    (hF : ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => F z.1 z.2))
-    (hG : ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => G z.1 z.2))
-    (z : ℂ × ℝ) (x : ℝ) :
-    logGenerator κ (fun z x => F z x+G z x) z x =
-      logGenerator κ F z x+logGenerator κ G z x := by
-  have hFs : ContDiff ℝ (⊤ : ℕ∞) (fun z => F z x) :=
-    hF.comp (contDiff_id.prodMk contDiff_const)
-  have hGs : ContDiff ℝ (⊤ : ℕ∞) (fun z => G z x) :=
-    hG.comp (contDiff_id.prodMk contDiff_const)
-  have hFx : ContDiff ℝ (⊤ : ℕ∞) (F z) :=
-    hF.comp (contDiff_const.prodMk contDiff_id)
-  have hGx : ContDiff ℝ (⊤ : ℕ∞) (G z) :=
-    hG.comp (contDiff_const.prodMk contDiff_id)
-  unfold logGenerator
-  rw [fderiv_fun_add ((hFs.differentiable (by simp)) z) ((hGs.differentiable (by simp)) z),
-    iteratedDeriv_fun_add (hFx.of_le (by simp)).contDiffAt (hGx.of_le (by simp)).contDiffAt]
-  simp only [add_apply]
-  ring
-
-lemma logGenerator_const_mul (κ D : ℝ) (F : (ℂ × ℝ) → ℝ → ℝ)
-    (hF : ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => F z.1 z.2))
-    (z : ℂ × ℝ) (x : ℝ) :
-    logGenerator κ (fun z x => D*F z x) z x = D*logGenerator κ F z x := by
-  have hFs : ContDiff ℝ (⊤ : ℕ∞) (fun z => F z x) :=
-    hF.comp (contDiff_id.prodMk contDiff_const)
-  have hFx : ContDiff ℝ (⊤ : ℕ∞) (F z) :=
-    hF.comp (contDiff_const.prodMk contDiff_id)
-  unfold logGenerator
-  rw [fderiv_const_mul ((hFs.differentiable (by simp)) z),
-    iteratedDeriv_const_mul D (hFx.of_le (by simp)).contDiffAt]
-  simp only [smul_apply, smul_eq_mul]
-  ring
-
-theorem regularizedLogFactor_generator (κ D σ : ℝ) (hκ : 0 < κ)
-    (z : ℂ × ℝ) (x : ℝ) :
-    logGenerator κ (regularizedLogFactor κ D σ) z x =
-      κ*logVelocity κ z x^2*
-        Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
-        branchRegulator κ D σ z *
-          (greenExponent κ/2 +
-            (greenExponent κ-q (greenExponent κ))*(1+q (greenExponent κ))/2*
-              sinePower (q (greenExponent κ)) (logAngle κ z x) -
-            negativeDrift (greenExponent κ)*
-              sinePower (q (greenExponent κ)-2) (logAngle κ z x) +
-            σ*(1+sinePower (q (greenExponent κ)) (logAngle κ z x))) := by
-  unfold regularizedLogFactor
-  rw [logGenerator_add κ _ _ (logFactor_contDiff κ _ _)
-    (contDiff_const.mul ((logFactor_contDiff κ _ _).add (logFactor_contDiff κ _ _))),
-    logGenerator_const_mul κ D _ ((logFactor_contDiff κ _ _).add (logFactor_contDiff κ _ _)),
-    logGenerator_add κ _ _ (logFactor_contDiff κ _ _) (logFactor_contDiff κ _ _)]
-  simp only [logGenerator, logFactor_generator_eq_angular κ _ _ hκ]
-  have he : Real.exp (((greenExponent κ/2+σ)*κ/4)*(z.2-z.1.im)) =
-      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
-        Real.exp ((σ*κ/4)*(z.2-z.1.im)) := by
-    rw [← Real.exp_add]; congr 1; ring
-  rw [he]
-  have h0 : sineGenerator (greenExponent κ) 0 (logAngle κ z x) = 0 := by
-    simp [sineGenerator, sinePowerFirst, sinePowerSecond]
-  rw [h0]
-  have hreg := regularized_generator (p := greenExponent κ)
-    (δ := branchRegulator κ D σ z) (σ := σ) (logAngle_sin_pos κ z x)
-  simp only [sinePower, Real.rpow_zero, branchRegulator] at hreg ⊢
-  unfold angularFactor sinePower at hreg
-  linear_combination (κ*logVelocity κ z x^2*
-    Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im)))*hreg
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointLogAngular
+
+namespace OAI
+
+/-! The actual smooth factor on either side of a regulator midpoint.
+Its only parameters are the constant and logarithmic slope of that branch. -/
+
+noncomputable section
+
+namespace SLEExactGauge.Multipoint
+
+def regularizedLogFactor (κ D σ : ℝ) (z : ℂ × ℝ) (x : ℝ) : ℝ :=
+  logFactor κ (greenExponent κ/2) (greenExponent κ) z x +
+    D*(logFactor κ (greenExponent κ/2+σ) 0 z x +
+      logFactor κ (greenExponent κ/2+σ) (q (greenExponent κ)) z x)
+
+def branchRegulator (κ D σ : ℝ) (z : ℂ × ℝ) : ℝ :=
+  D*Real.exp ((σ*κ/4)*(z.2-z.1.im))
+
+lemma regularizedLogFactor_contDiff (κ D σ : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => regularizedLogFactor κ D σ z.1 z.2) :=
+  (logFactor_contDiff κ _ _).add (contDiff_const.mul
+    ((logFactor_contDiff κ _ _).add (logFactor_contDiff κ _ _)))
+
+lemma regularizedLogFactor_eq (κ D σ : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    regularizedLogFactor κ D σ z x =
+      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
+        angularFactor (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by
+  unfold regularizedLogFactor
+  rw [logFactor_eq_angular, logFactor_eq_angular, logFactor_eq_angular]
+  have he : Real.exp (((greenExponent κ/2+σ)*κ/4)*(z.2-z.1.im)) =
+      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
+        Real.exp ((σ*κ/4)*(z.2-z.1.im)) := by
+    rw [← Real.exp_add]; congr 1; ring
+  rw [he]
+  simp only [sinePower, Real.rpow_zero, angularFactor, branchRegulator]
+  ring
+
+lemma regularizedLogFactor_pos (κ D σ : ℝ) (hD : 0 < D) (z : ℂ × ℝ) (x : ℝ) :
+    0 < regularizedLogFactor κ D σ z x := by
+  rw [regularizedLogFactor_eq]
+  exact mul_pos (Real.exp_pos _) (angularFactor_pos (mul_pos hD (Real.exp_pos _))
+    (logAngle_sin_pos κ z x))
+
+lemma regularizedLogFactor_spatial (κ D σ : ℝ) (z : ℂ × ℝ) (x : ℝ) :
+    deriv (regularizedLogFactor κ D σ z) x =
+      Real.sqrt κ*logVelocity κ z x*
+        Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
+          angularFactorFirst (greenExponent κ) (branchRegulator κ D σ z) (logAngle κ z x) := by
+  have hd := (logFactor_spatial_derivative κ (greenExponent κ/2) (greenExponent κ) z x).add
+    (((logFactor_spatial_derivative κ (greenExponent κ/2+σ) 0 z x).add
+      (logFactor_spatial_derivative κ (greenExponent κ/2+σ) (q (greenExponent κ)) z x)).const_mul D)
+  change HasDerivAt (regularizedLogFactor κ D σ z) _ x at hd
+  rw [hd.deriv]
+  have hs (c r : ℝ) := logFactor_spatial_eq_angular κ c r z x
+  rw [← (logFactor_spatial_derivative κ (greenExponent κ/2) (greenExponent κ) z x).deriv,
+    ← (logFactor_spatial_derivative κ (greenExponent κ/2+σ) 0 z x).deriv,
+    ← (logFactor_spatial_derivative κ (greenExponent κ/2+σ) (q (greenExponent κ)) z x).deriv]
+  rw [hs, hs, hs]
+  have he : Real.exp (((greenExponent κ/2+σ)*κ/4)*(z.2-z.1.im)) =
+      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
+        Real.exp ((σ*κ/4)*(z.2-z.1.im)) := by
+    rw [← Real.exp_add]; congr 1; ring
+  rw [he]
+  simp only [sinePowerFirst, zero_mul, angularFactorFirst, branchRegulator]
+  ring
+
+def logGenerator (κ : ℝ) (F : (ℂ × ℝ) → ℝ → ℝ) (z : ℂ × ℝ) (x : ℝ) : ℝ :=
+  (fderiv ℝ (fun z => F z x) z) (physicalLogLoewnerField (Real.sqrt κ*x) z) +
+    (1/2 : ℝ)*iteratedDeriv 2 (F z) x
+
+lemma logGenerator_add (κ : ℝ) (F G : (ℂ × ℝ) → ℝ → ℝ)
+    (hF : ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => F z.1 z.2))
+    (hG : ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => G z.1 z.2))
+    (z : ℂ × ℝ) (x : ℝ) :
+    logGenerator κ (fun z x => F z x+G z x) z x =
+      logGenerator κ F z x+logGenerator κ G z x := by
+  have hFs : ContDiff ℝ (⊤ : ℕ∞) (fun z => F z x) :=
+    hF.comp (contDiff_id.prodMk contDiff_const)
+  have hGs : ContDiff ℝ (⊤ : ℕ∞) (fun z => G z x) :=
+    hG.comp (contDiff_id.prodMk contDiff_const)
+  have hFx : ContDiff ℝ (⊤ : ℕ∞) (F z) :=
+    hF.comp (contDiff_const.prodMk contDiff_id)
+  have hGx : ContDiff ℝ (⊤ : ℕ∞) (G z) :=
+    hG.comp (contDiff_const.prodMk contDiff_id)
+  unfold logGenerator
+  rw [fderiv_fun_add ((hFs.differentiable (by simp)) z) ((hGs.differentiable (by simp)) z),
+    iteratedDeriv_fun_add (hFx.of_le (by simp)).contDiffAt (hGx.of_le (by simp)).contDiffAt]
+  simp only [add_apply]
+  ring
+
+lemma logGenerator_const_mul (κ D : ℝ) (F : (ℂ × ℝ) → ℝ → ℝ)
+    (hF : ContDiff ℝ (⊤ : ℕ∞) (fun z : (ℂ × ℝ) × ℝ => F z.1 z.2))
+    (z : ℂ × ℝ) (x : ℝ) :
+    logGenerator κ (fun z x => D*F z x) z x = D*logGenerator κ F z x := by
+  have hFs : ContDiff ℝ (⊤ : ℕ∞) (fun z => F z x) :=
+    hF.comp (contDiff_id.prodMk contDiff_const)
+  have hFx : ContDiff ℝ (⊤ : ℕ∞) (F z) :=
+    hF.comp (contDiff_const.prodMk contDiff_id)
+  unfold logGenerator
+  rw [fderiv_const_mul ((hFs.differentiable (by simp)) z),
+    iteratedDeriv_const_mul D (hFx.of_le (by simp)).contDiffAt]
+  simp only [smul_apply, smul_eq_mul]
+  ring
+
+theorem regularizedLogFactor_generator (κ D σ : ℝ) (hκ : 0 < κ)
+    (z : ℂ × ℝ) (x : ℝ) :
+    logGenerator κ (regularizedLogFactor κ D σ) z x =
+      κ*logVelocity κ z x^2*
+        Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
+        branchRegulator κ D σ z *
+          (greenExponent κ/2 +
+            (greenExponent κ-q (greenExponent κ))*(1+q (greenExponent κ))/2*
+              sinePower (q (greenExponent κ)) (logAngle κ z x) -
+            negativeDrift (greenExponent κ)*
+              sinePower (q (greenExponent κ)-2) (logAngle κ z x) +
+            σ*(1+sinePower (q (greenExponent κ)) (logAngle κ z x))) := by
+  unfold regularizedLogFactor
+  rw [logGenerator_add κ _ _ (logFactor_contDiff κ _ _)
+    (contDiff_const.mul ((logFactor_contDiff κ _ _).add (logFactor_contDiff κ _ _))),
+    logGenerator_const_mul κ D _ ((logFactor_contDiff κ _ _).add (logFactor_contDiff κ _ _)),
+    logGenerator_add κ _ _ (logFactor_contDiff κ _ _) (logFactor_contDiff κ _ _)]
+  simp only [logGenerator, logFactor_generator_eq_angular κ _ _ hκ]
+  have he : Real.exp (((greenExponent κ/2+σ)*κ/4)*(z.2-z.1.im)) =
+      Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im))*
+        Real.exp ((σ*κ/4)*(z.2-z.1.im)) := by
+    rw [← Real.exp_add]; congr 1; ring
+  rw [he]
+  have h0 : sineGenerator (greenExponent κ) 0 (logAngle κ z x) = 0 := by
+    simp [sineGenerator, sinePowerFirst, sinePowerSecond]
+  rw [h0]
+  have hreg := regularized_generator (p := greenExponent κ)
+    (δ := branchRegulator κ D σ z) (σ := σ) (logAngle_sin_pos κ z x)
+  simp only [sinePower, Real.rpow_zero, branchRegulator] at hreg ⊢
+  unfold angularFactor sinePower at hreg
+  linear_combination (κ*logVelocity κ z x^2*
+    Real.exp (((greenExponent κ/2)*κ/4)*(z.2-z.1.im)))*hreg
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizer.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizer.lean
index 83a20caf2dc229c1a874cb084addb7329a559160..b6adf30e05c231ab7c80edaa4f3f9d7ace7c6023 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizer.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointRegularizer.lean
@@ -1,207 +1,207 @@
-import Mathlib
-
-namespace OAI
-
-/-! Parameters and the two-ended regularizer in the multipoint radius
-estimate, September 25 manuscript, Section 3. These bounds are uniform in
-all target levels; completed factors can therefore be frozen separately. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge.Multipoint
-
-def q (p : ℝ) : ℝ := min 1 p / 2
-
-def eta (p : ℝ) : ℝ := (1 - q p) / (2 - q p)
-
-def exponent (p : ℝ) : ℝ := (1 + eta p) / (1 - 2 * eta p)
-
-def beta (a p : ℝ) : ℝ := min 1 (a / (4 * exponent p))
-
-lemma q_pos {p : ℝ} (hp : 0 < p) : 0 < q p :=
-  div_pos (lt_min zero_lt_one hp) (by norm_num)
-
-lemma q_lt_one {p : ℝ} : q p < 1 := by
-  have h := min_le_left (1 : ℝ) p
-  dsimp [q]
-  linarith
-
-lemma q_lt_p {p : ℝ} (hp : 0 < p) : q p < p := by
-  have h := min_le_right (1 : ℝ) p
-  dsimp [q]
-  linarith
-
-lemma eta_pos {p : ℝ} : 0 < eta p := by
-  exact div_pos (sub_pos.mpr q_lt_one) (by linarith [q_lt_one (p := p)])
-
-lemma eta_lt_half {p : ℝ} (hp : 0 < p) : eta p < 1 / 2 := by
-  rw [eta, div_lt_iff₀ (by linarith [q_lt_one (p := p)])]
-  linarith [q_pos hp]
-
-lemma eta_identity (p : ℝ) : eta p * (q p - 2) = q p - 1 := by
-  have hn : 2 - q p ≠ 0 := ne_of_gt (by linarith [q_lt_one (p := p)])
-  dsimp [eta]
-  field_simp
-  ring
-
-lemma exponent_eq {p : ℝ} (hp : 0 < p) : exponent p = 3 / q p - 2 := by
-  have hq := q_pos hp
-  have hq2 : 2 - q p ≠ 0 := ne_of_gt (by linarith [q_lt_one (p := p)])
-  have he : 1 - 2 * eta p ≠ 0 := ne_of_gt (by linarith [eta_lt_half hp])
-  dsimp [exponent]
-  apply (div_eq_iff he).mpr
-  dsimp [eta]
-  field_simp
-  ring
-
-lemma exponent_pos {p : ℝ} (hp : 0 < p) : 0 < exponent p := by
-  exact div_pos (by linarith [eta_pos (p := p)]) (by linarith [eta_lt_half hp])
-
-lemma beta_pos {a p : ℝ} (ha : 0 < a) (hp : 0 < p) : 0 < beta a p := by
-  exact lt_min zero_lt_one (div_pos ha (mul_pos (by norm_num) (exponent_pos hp)))
-
-lemma beta_le_one (a p : ℝ) : beta a p ≤ 1 := min_le_left _ _
-
-lemma twice_exponent_beta_lt {a p : ℝ} (ha : 0 < a) (hp : 0 < p) :
-    2 * (exponent p * beta a p) < a := by
-  have hM := exponent_pos hp
-  have hb : beta a p ≤ a / (4 * exponent p) := min_le_right _ _
-  have h := (le_div_iff₀ (by positivity : 0 < 4 * exponent p)).mp hb
-  nlinarith
-
-/-- The regulator equals one at either endpoint of an active interval. -/
-def regulator (β U u : ℝ) : ℝ := Real.exp (-β * min u (U - u))
-
-@[simp] lemma regulator_zero (β U : ℝ) (hU : 0 ≤ U) : regulator β U 0 = 1 := by
-  simp [regulator, min_eq_left hU]
-
-@[simp] lemma regulator_end (β U : ℝ) (hU : 0 ≤ U) : regulator β U U = 1 := by
-  simp [regulator, min_eq_right hU]
-
-lemma regulator_pos (β U u : ℝ) : 0 < regulator β U u := Real.exp_pos _
-
-lemma regulator_le_one {β U u : ℝ} (hβ : 0 ≤ β) (hu : u ∈ Icc 0 U) :
-    regulator β U u ≤ 1 := by
-  apply Real.exp_le_one_iff.mpr
-  exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hβ)
-    (le_min hu.1 (sub_nonneg.mpr hu.2))
-
-lemma regulator_inv_power_le {β U u M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) :
-    regulator β U u ^ (-M) ≤ Real.exp (M * β * u) := by
-  rw [regulator, ← Real.exp_mul]
-  apply Real.exp_le_exp.mpr
-  have h := mul_le_mul_of_nonneg_left (min_le_left u (U-u)) (mul_nonneg hM hβ)
-  nlinarith
-
-lemma pair_regulator_inv_power_le {β U V u v M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) :
-    (regulator β U u * regulator β V v) ^ (-M) ≤ Real.exp (M * β * (u + v)) := by
-  rw [Real.mul_rpow (regulator_pos _ _ _).le (regulator_pos _ _ _).le]
-  calc
-    _ ≤ Real.exp (M * β * u) * Real.exp (M * β * v) :=
-      mul_le_mul (regulator_inv_power_le hβ hM) (regulator_inv_power_le hβ hM)
-        (Real.rpow_nonneg (regulator_pos _ _ _).le _) (Real.exp_pos _).le
-    _ = _ := by rw [← Real.exp_add]; congr 1; ring
-
-lemma regulator_eq_left {β U u : ℝ} (hu : u ≤ U / 2) :
-    regulator β U u = Real.exp (-β * u) := by
-  rw [regulator, min_eq_left (by linarith)]
-
-lemma regulator_eq_right {β U u : ℝ} (hu : U / 2 ≤ u) :
-    regulator β U u = Real.exp (-β * (U-u)) := by
-  rw [regulator, min_eq_right (by linarith)]
-
-lemma regulator_reflect (β U u : ℝ) : regulator β U (U-u) = regulator β U u := by
-  simp only [regulator, sub_sub_cancel, min_comm]
-
-lemma regulator_continuous (β U : ℝ) : Continuous (regulator β U) := by
-  exact Real.continuous_exp.comp (continuous_const.mul
-    (continuous_id.min (continuous_const.sub continuous_id)))
-
-/-- Exact total regulator cost, independent of the length of the active clock. -/
-lemma integral_regulator {β U : ℝ} (hβ : 0 < β) (hU : 0 ≤ U) :
-    (∫ u in (0 : ℝ)..U, regulator β U u) =
-      2 / β * (1 - Real.exp (-β * U / 2)) := by
-  have hi (a b : ℝ) : IntervalIntegrable (regulator β U) volume a b :=
-    (regulator_continuous β U).intervalIntegrable _ _
-  have he (a b : ℝ) :
-      (∫ u in a..b, Real.exp (-β * u)) =
-        (-β)⁻¹ * (Real.exp (-β * b) - Real.exp (-β * a)) := by
-    rw [intervalIntegral.integral_comp_mul_left _ (neg_ne_zero.mpr hβ.ne'),
-      integral_exp]
-    rfl
-  have hleft : (∫ u in (0 : ℝ)..U/2, regulator β U u) =
-      ∫ u in (0 : ℝ)..U/2, Real.exp (-β*u) := by
-    apply intervalIntegral.integral_congr
-    intro u hu
-    rw [uIcc_of_le (by linarith : (0 : ℝ) ≤ U/2)] at hu
-    exact regulator_eq_left hu.2
-  have hright : (∫ u in U/2..U, regulator β U u) =
-      ∫ u in (0 : ℝ)..U/2, Real.exp (-β*u) := by
-    calc
-      _ = ∫ u in U/2..U, Real.exp (-β*(U-u)) := by
-        apply intervalIntegral.integral_congr
-        intro u hu
-        rw [uIcc_of_le (by linarith : U/2 ≤ U)] at hu
-        exact regulator_eq_right hu.1
-      _ = _ := by
-        change (∫ u in U/2..U, (fun v : ℝ => Real.exp (-β*v)) (U-u)) = _
-        rw [intervalIntegral.integral_comp_sub_left (f := fun v : ℝ => Real.exp (-β*v)) U]
-        congr 1 <;> ring
-  rw [← intervalIntegral.integral_add_adjacent_intervals (hi 0 (U/2)) (hi (U/2) U),
-    hleft, hright, he]
-  simp only [mul_zero, Real.exp_zero]
-  field_simp [hβ.ne']
-  ring
-
-lemma integral_regulator_le {β U : ℝ} (hβ : 0 < β) (hU : 0 ≤ U) :
-    (∫ u in (0 : ℝ)..U, regulator β U u) ≤ 2 / β := by
-  rw [integral_regulator hβ hU]
-  exact mul_le_of_le_one_right (by positivity) (by linarith [Real.exp_pos (-β * U / 2)])
-
-/-- The deterministic pair-interaction allowance obtained by shells of
-`max uᵢ uⱼ`; see the interaction integral in the manuscript. -/
-def interactionBudget (a c : ℝ) : ℝ :=
-  Real.exp (2*c) / (2*a*(1-Real.exp (2*c-a)))
-
-lemma interactionBudget_pos {a c : ℝ} (ha : 0 < a) (hc : 2*c < a) :
-    0 < interactionBudget a c := by
-  unfold interactionBudget
-  exact div_pos (Real.exp_pos _) (mul_pos (by positivity)
-    (sub_pos.mpr (Real.exp_lt_one_iff.mpr (by linarith))))
-
-lemma interaction_shell_series {a c : ℝ} (_ha : 0 < a) (hc : 2*c < a) :
-    HasSum (fun k : ℕ => Real.exp (2*c*((k:ℝ)+1)) * Real.exp (-a*k) / (2*a))
-      (interactionBudget a c) := by
-  have hs := (hasSum_geometric_of_lt_one (Real.exp_pos (2*c-a)).le
-    (Real.exp_lt_one_iff.mpr (by linarith : 2*c-a < 0))).mul_left
-      (Real.exp (2*c)/(2*a))
-  convert hs using 1
-  · ext k
-    rw [← Real.exp_nat_mul, div_mul_eq_mul_div, ← Real.exp_add, ← Real.exp_add]
-    congr 2
-    ring
-  · simp only [interactionBudget, div_eq_mul_inv, mul_inv_rev]
-    ring
-
-lemma weighted_shell_sum_le {a c : ℝ} (ha : 0 < a) (hc : 2*c < a)
-    (mass : ℕ → ℝ) (h0 : ∀ k, 0 ≤ mass k)
-    (hmass : ∀ k : ℕ, mass k ≤ Real.exp (-a*k)/(2*a)) :
-    Summable (fun k : ℕ => Real.exp (2*c*((k:ℝ)+1))*mass k) ∧
-      (∑' k : ℕ, Real.exp (2*c*((k:ℝ)+1))*mass k) ≤ interactionBudget a c := by
-  have hs := interaction_shell_series ha hc
-  have hnonneg (k : ℕ) : 0 ≤ Real.exp (2*c*((k:ℝ)+1))*mass k :=
-    mul_nonneg (Real.exp_pos _).le (h0 k)
-  have hle (k : ℕ) : Real.exp (2*c*((k:ℝ)+1))*mass k ≤
-      Real.exp (2*c*((k:ℝ)+1))*Real.exp (-a*k)/(2*a) := by
-    simpa only [mul_div_assoc] using
-      mul_le_mul_of_nonneg_left (hmass k) (Real.exp_pos _).le
-  have hsum := Summable.of_nonneg_of_le hnonneg hle hs.summable
-  exact ⟨hsum, hs.tsum_eq ▸ hsum.tsum_le_tsum hle hs.summable⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Parameters and the two-ended regularizer in the multipoint radius
+estimate, September 25 manuscript, Section 3. These bounds are uniform in
+all target levels; completed factors can therefore be frozen separately. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge.Multipoint
+
+def q (p : ℝ) : ℝ := min 1 p / 2
+
+def eta (p : ℝ) : ℝ := (1 - q p) / (2 - q p)
+
+def exponent (p : ℝ) : ℝ := (1 + eta p) / (1 - 2 * eta p)
+
+def beta (a p : ℝ) : ℝ := min 1 (a / (4 * exponent p))
+
+lemma q_pos {p : ℝ} (hp : 0 < p) : 0 < q p :=
+  div_pos (lt_min zero_lt_one hp) (by norm_num)
+
+lemma q_lt_one {p : ℝ} : q p < 1 := by
+  have h := min_le_left (1 : ℝ) p
+  dsimp [q]
+  linarith
+
+lemma q_lt_p {p : ℝ} (hp : 0 < p) : q p < p := by
+  have h := min_le_right (1 : ℝ) p
+  dsimp [q]
+  linarith
+
+lemma eta_pos {p : ℝ} : 0 < eta p := by
+  exact div_pos (sub_pos.mpr q_lt_one) (by linarith [q_lt_one (p := p)])
+
+lemma eta_lt_half {p : ℝ} (hp : 0 < p) : eta p < 1 / 2 := by
+  rw [eta, div_lt_iff₀ (by linarith [q_lt_one (p := p)])]
+  linarith [q_pos hp]
+
+lemma eta_identity (p : ℝ) : eta p * (q p - 2) = q p - 1 := by
+  have hn : 2 - q p ≠ 0 := ne_of_gt (by linarith [q_lt_one (p := p)])
+  dsimp [eta]
+  field_simp
+  ring
+
+lemma exponent_eq {p : ℝ} (hp : 0 < p) : exponent p = 3 / q p - 2 := by
+  have hq := q_pos hp
+  have hq2 : 2 - q p ≠ 0 := ne_of_gt (by linarith [q_lt_one (p := p)])
+  have he : 1 - 2 * eta p ≠ 0 := ne_of_gt (by linarith [eta_lt_half hp])
+  dsimp [exponent]
+  apply (div_eq_iff he).mpr
+  dsimp [eta]
+  field_simp
+  ring
+
+lemma exponent_pos {p : ℝ} (hp : 0 < p) : 0 < exponent p := by
+  exact div_pos (by linarith [eta_pos (p := p)]) (by linarith [eta_lt_half hp])
+
+lemma beta_pos {a p : ℝ} (ha : 0 < a) (hp : 0 < p) : 0 < beta a p := by
+  exact lt_min zero_lt_one (div_pos ha (mul_pos (by norm_num) (exponent_pos hp)))
+
+lemma beta_le_one (a p : ℝ) : beta a p ≤ 1 := min_le_left _ _
+
+lemma twice_exponent_beta_lt {a p : ℝ} (ha : 0 < a) (hp : 0 < p) :
+    2 * (exponent p * beta a p) < a := by
+  have hM := exponent_pos hp
+  have hb : beta a p ≤ a / (4 * exponent p) := min_le_right _ _
+  have h := (le_div_iff₀ (by positivity : 0 < 4 * exponent p)).mp hb
+  nlinarith
+
+/-- The regulator equals one at either endpoint of an active interval. -/
+def regulator (β U u : ℝ) : ℝ := Real.exp (-β * min u (U - u))
+
+@[simp] lemma regulator_zero (β U : ℝ) (hU : 0 ≤ U) : regulator β U 0 = 1 := by
+  simp [regulator, min_eq_left hU]
+
+@[simp] lemma regulator_end (β U : ℝ) (hU : 0 ≤ U) : regulator β U U = 1 := by
+  simp [regulator, min_eq_right hU]
+
+lemma regulator_pos (β U u : ℝ) : 0 < regulator β U u := Real.exp_pos _
+
+lemma regulator_le_one {β U u : ℝ} (hβ : 0 ≤ β) (hu : u ∈ Icc 0 U) :
+    regulator β U u ≤ 1 := by
+  apply Real.exp_le_one_iff.mpr
+  exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hβ)
+    (le_min hu.1 (sub_nonneg.mpr hu.2))
+
+lemma regulator_inv_power_le {β U u M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) :
+    regulator β U u ^ (-M) ≤ Real.exp (M * β * u) := by
+  rw [regulator, ← Real.exp_mul]
+  apply Real.exp_le_exp.mpr
+  have h := mul_le_mul_of_nonneg_left (min_le_left u (U-u)) (mul_nonneg hM hβ)
+  nlinarith
+
+lemma pair_regulator_inv_power_le {β U V u v M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) :
+    (regulator β U u * regulator β V v) ^ (-M) ≤ Real.exp (M * β * (u + v)) := by
+  rw [Real.mul_rpow (regulator_pos _ _ _).le (regulator_pos _ _ _).le]
+  calc
+    _ ≤ Real.exp (M * β * u) * Real.exp (M * β * v) :=
+      mul_le_mul (regulator_inv_power_le hβ hM) (regulator_inv_power_le hβ hM)
+        (Real.rpow_nonneg (regulator_pos _ _ _).le _) (Real.exp_pos _).le
+    _ = _ := by rw [← Real.exp_add]; congr 1; ring
+
+lemma regulator_eq_left {β U u : ℝ} (hu : u ≤ U / 2) :
+    regulator β U u = Real.exp (-β * u) := by
+  rw [regulator, min_eq_left (by linarith)]
+
+lemma regulator_eq_right {β U u : ℝ} (hu : U / 2 ≤ u) :
+    regulator β U u = Real.exp (-β * (U-u)) := by
+  rw [regulator, min_eq_right (by linarith)]
+
+lemma regulator_reflect (β U u : ℝ) : regulator β U (U-u) = regulator β U u := by
+  simp only [regulator, sub_sub_cancel, min_comm]
+
+lemma regulator_continuous (β U : ℝ) : Continuous (regulator β U) := by
+  exact Real.continuous_exp.comp (continuous_const.mul
+    (continuous_id.min (continuous_const.sub continuous_id)))
+
+/-- Exact total regulator cost, independent of the length of the active clock. -/
+lemma integral_regulator {β U : ℝ} (hβ : 0 < β) (hU : 0 ≤ U) :
+    (∫ u in (0 : ℝ)..U, regulator β U u) =
+      2 / β * (1 - Real.exp (-β * U / 2)) := by
+  have hi (a b : ℝ) : IntervalIntegrable (regulator β U) volume a b :=
+    (regulator_continuous β U).intervalIntegrable _ _
+  have he (a b : ℝ) :
+      (∫ u in a..b, Real.exp (-β * u)) =
+        (-β)⁻¹ * (Real.exp (-β * b) - Real.exp (-β * a)) := by
+    rw [intervalIntegral.integral_comp_mul_left _ (neg_ne_zero.mpr hβ.ne'),
+      integral_exp]
+    rfl
+  have hleft : (∫ u in (0 : ℝ)..U/2, regulator β U u) =
+      ∫ u in (0 : ℝ)..U/2, Real.exp (-β*u) := by
+    apply intervalIntegral.integral_congr
+    intro u hu
+    rw [uIcc_of_le (by linarith : (0 : ℝ) ≤ U/2)] at hu
+    exact regulator_eq_left hu.2
+  have hright : (∫ u in U/2..U, regulator β U u) =
+      ∫ u in (0 : ℝ)..U/2, Real.exp (-β*u) := by
+    calc
+      _ = ∫ u in U/2..U, Real.exp (-β*(U-u)) := by
+        apply intervalIntegral.integral_congr
+        intro u hu
+        rw [uIcc_of_le (by linarith : U/2 ≤ U)] at hu
+        exact regulator_eq_right hu.1
+      _ = _ := by
+        change (∫ u in U/2..U, (fun v : ℝ => Real.exp (-β*v)) (U-u)) = _
+        rw [intervalIntegral.integral_comp_sub_left (f := fun v : ℝ => Real.exp (-β*v)) U]
+        congr 1 <;> ring
+  rw [← intervalIntegral.integral_add_adjacent_intervals (hi 0 (U/2)) (hi (U/2) U),
+    hleft, hright, he]
+  simp only [mul_zero, Real.exp_zero]
+  field_simp [hβ.ne']
+  ring
+
+lemma integral_regulator_le {β U : ℝ} (hβ : 0 < β) (hU : 0 ≤ U) :
+    (∫ u in (0 : ℝ)..U, regulator β U u) ≤ 2 / β := by
+  rw [integral_regulator hβ hU]
+  exact mul_le_of_le_one_right (by positivity) (by linarith [Real.exp_pos (-β * U / 2)])
+
+/-- The deterministic pair-interaction allowance obtained by shells of
+`max uᵢ uⱼ`; see the interaction integral in the manuscript. -/
+def interactionBudget (a c : ℝ) : ℝ :=
+  Real.exp (2*c) / (2*a*(1-Real.exp (2*c-a)))
+
+lemma interactionBudget_pos {a c : ℝ} (ha : 0 < a) (hc : 2*c < a) :
+    0 < interactionBudget a c := by
+  unfold interactionBudget
+  exact div_pos (Real.exp_pos _) (mul_pos (by positivity)
+    (sub_pos.mpr (Real.exp_lt_one_iff.mpr (by linarith))))
+
+lemma interaction_shell_series {a c : ℝ} (_ha : 0 < a) (hc : 2*c < a) :
+    HasSum (fun k : ℕ => Real.exp (2*c*((k:ℝ)+1)) * Real.exp (-a*k) / (2*a))
+      (interactionBudget a c) := by
+  have hs := (hasSum_geometric_of_lt_one (Real.exp_pos (2*c-a)).le
+    (Real.exp_lt_one_iff.mpr (by linarith : 2*c-a < 0))).mul_left
+      (Real.exp (2*c)/(2*a))
+  convert hs using 1
+  · ext k
+    rw [← Real.exp_nat_mul, div_mul_eq_mul_div, ← Real.exp_add, ← Real.exp_add]
+    congr 2
+    ring
+  · simp only [interactionBudget, div_eq_mul_inv, mul_inv_rev]
+    ring
+
+lemma weighted_shell_sum_le {a c : ℝ} (ha : 0 < a) (hc : 2*c < a)
+    (mass : ℕ → ℝ) (h0 : ∀ k, 0 ≤ mass k)
+    (hmass : ∀ k : ℕ, mass k ≤ Real.exp (-a*k)/(2*a)) :
+    Summable (fun k : ℕ => Real.exp (2*c*((k:ℝ)+1))*mass k) ∧
+      (∑' k : ℕ, Real.exp (2*c*((k:ℝ)+1))*mass k) ≤ interactionBudget a c := by
+  have hs := interaction_shell_series ha hc
+  have hnonneg (k : ℕ) : 0 ≤ Real.exp (2*c*((k:ℝ)+1))*mass k :=
+    mul_nonneg (Real.exp_pos _).le (h0 k)
+  have hle (k : ℕ) : Real.exp (2*c*((k:ℝ)+1))*mass k ≤
+      Real.exp (2*c*((k:ℝ)+1))*Real.exp (-a*k)/(2*a) := by
+    simpa only [mul_div_assoc] using
+      mul_le_mul_of_nonneg_left (hmass k) (Real.exp_pos _).le
+  have hsum := Summable.of_nonneg_of_le hnonneg hle hs.summable
+  exact ⟨hsum, hs.tsum_eq ▸ hsum.tsum_le_tsum hle hs.summable⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointRegulatorTransport.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointRegulatorTransport.lean
index 1ccfce13fa31f360dda623ebab2a15c84218f173..db16b70a239dff5677aa9cbf22cbfe5f336e3c65 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointRegulatorTransport.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointRegulatorTransport.lean
@@ -1,280 +1,280 @@
-import OAI.Probability.SLE.Multipoint.MultipointPhysicalBudget
-
-namespace OAI
-
-/-! Deterministic transport of the two-ended regulator to physical time.
-The single-point cost follows by change of variables along the actual
-radius clock, and the pair cost follows from geometric interaction decay. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-/-- Change of variables for a regulator on any subinterval of its clock.
-Only ordinary calculus is used here; no stochastic estimate is assumed. -/
-theorem regulator_clock_integral_le
-    {β U s T : ℝ} (hβ : 0 < β) (hU : 0 ≤ U) (hsT : s ≤ T)
-    (u speed : ℝ → ℝ) (hu : ContinuousOn u (Icc s T))
-    (hspeed : ContinuousOn speed (Icc s T))
-    (hdu : ∀ t ∈ Ioo s T, HasDerivWithinAt u (speed t) (Ioi t) t)
-    (hu0 : 0 ≤ u s) (huT : u T ≤ U) (hmono : u s ≤ u T) :
-    (∫ t in s..T, regulator β U (u t) * speed t) ≤ 2 / β := by
-  have he : (∫ t in s..T, regulator β U (u t) * speed t) =
-      ∫ v in u s..u T, regulator β U v := by
-    simpa only [Function.comp_apply] using
-      intervalIntegral.integral_comp_mul_deriv''
-        (by simpa only [uIcc_of_le hsT] using hu)
-        (by simpa only [min_eq_left hsT, max_eq_right hsT] using hdu)
-        (by simpa only [uIcc_of_le hsT] using hspeed)
-        (regulator_continuous β U).continuousOn
-  rw [he]
-  apply le_trans _ (integral_regulator_le hβ hU)
-  exact intervalIntegral.integral_mono_interval hu0 hmono huT
-    (ae_of_all _ fun v => (regulator_pos β U v).le)
-    ((regulator_continuous β U).intervalIntegrable 0 U)
-
-/-- The radius clock measured from activation, in capacity-two time.
-The clock derivative is the square of the Poisson rate. -/
-def activeRadiusClock (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (σ : ℝ≥0) (t : ℝ) : ℝ :=
-  (Real.log ((2 * z.im) / fullRadius G t.toNNReal z) -
-    Real.log ((2 * z.im) / fullRadius G σ z)) / 4
-
-@[simp] lemma activeRadiusClock_start (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (σ : ℝ≥0) :
-    activeRadiusClock G z σ σ = 0 := by
-  simp [activeRadiusClock]
-
-end SLEExactGauge.Multipoint
-
-namespace SLEExactGauge
-
-theorem IsCapacityTwoWitness.activeRadiusClock_hasDerivWithinAt
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (σ T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T)
-    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
-    HasDerivWithinAt (Multipoint.activeRadiusClock G z σ)
-      (chartPoissonRate U G t.toNNReal z ^ 2) (Icc 0 (T : ℝ)) t := by
-  refine (((loewner_radius_clock_derivative hU hw T z hz t ht).sub_const
-    (Real.log ((2 * z.im) / fullRadius G σ z))).div_const 4).congr_deriv ?_
-  simp only [chartPoissonRate, Complex.normSq_eq_norm_sq]
-  ring
-
-theorem IsCapacityTwoWitness.activeRadiusClock_continuousOn
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (σ T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
-    ContinuousOn (Multipoint.activeRadiusClock G z σ) (Icc 0 (T : ℝ)) := by
-  intro t ht
-  exact (hw.activeRadiusClock_hasDerivWithinAt hU σ T z hz t ht).continuousWithinAt
-
-theorem IsCapacityTwoWitness.activeRadiusClock_nonneg
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
-    0 ≤ Multipoint.activeRadiusClock G z σ T := by
-  have h := (hw.radius_clock_strictMonoOn hU z T hz).monotoneOn
-    (show (σ : ℝ) ∈ Icc 0 (T : ℝ) from ⟨σ.coe_nonneg, hσT⟩)
-    (show (T : ℝ) ∈ Icc 0 (T : ℝ) from ⟨T.coe_nonneg, le_rfl⟩) hσT
-  simp only [Real.toNNReal_coe] at h
-  change 0 ≤ (Real.log ((2 * z.im) / fullRadius G (T : ℝ).toNNReal z) -
-    Real.log ((2 * z.im) / fullRadius G σ z)) / 4
-  rw [Real.toNNReal_coe]
-  exact div_nonneg (sub_nonneg.mpr h) (by norm_num)
-
-/-- The physical conformal radius has the same ratio as the half-plane
-radius. In particular, no derivative bound on `J` is needed. -/
-theorem IsCapacityTwoWitness.physical_radius_active_clock
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (J : ℂ → ℂ) (σ t : ℝ≥0) (z : ℂ)
-    (hσ : z ∈ survivingDomain γ σ) (ht : z ∈ survivingDomain γ t) :
-    physicalConformalRadius J G t z = physicalConformalRadius J G σ z *
-      Real.exp (-4 * Multipoint.activeRadiusClock G z σ t) := by
-  have hRσ := hw.fullRadius_positive hU σ z hσ
-  have hRt := hw.fullRadius_positive hU t z ht
-  have hz0 : 2 * z.im ≠ 0 := ne_of_gt (mul_pos (by norm_num) ht.1.1)
-  have he : -4 * Multipoint.activeRadiusClock G z σ t =
-      Real.log (fullRadius G t z / fullRadius G σ z) := by
-    rw [Multipoint.activeRadiusClock, Real.toNNReal_coe,
-      Real.log_div hz0 hRt.ne', Real.log_div hz0 hRσ.ne',
-      Real.log_div hRt.ne' hRσ.ne']
-    ring
-  rw [he, Real.exp_log (div_pos hRt hRσ)]
-  unfold physicalConformalRadius
-  field_simp [hRσ.ne']
-
-/-- One complete or partial active interval has regulator cost at most
-`2 / β`, independently of its target-clock length. -/
-theorem IsCapacityTwoWitness.active_regulator_integral_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
-    {β target : ℝ} (hβ : 0 < β) (htarget : 0 ≤ target)
-    (hclock : Multipoint.activeRadiusClock G z σ T ≤ target) :
-    (∫ t in (σ : ℝ)..(T : ℝ),
-      Multipoint.regulator β target (Multipoint.activeRadiusClock G z σ t) *
-        chartPoissonRate U G t.toNNReal z ^ 2) ≤ 2 / β := by
-  have hsub : Icc (σ : ℝ) (T : ℝ) ⊆ Icc (0 : ℝ) (T : ℝ) :=
-    Icc_subset_Icc σ.coe_nonneg le_rfl
-  have hclockcont : ContinuousOn (Multipoint.activeRadiusClock G z σ)
-      (Icc (σ : ℝ) (T : ℝ)) := by
-    intro t ht
-    exact ((hw.activeRadiusClock_hasDerivWithinAt hU σ T z hz t (hsub ht)).mono
-      hsub).continuousWithinAt
-  apply Multipoint.regulator_clock_integral_le hβ htarget hσT
-    (Multipoint.activeRadiusClock G z σ) (fun t => chartPoissonRate U G t.toNNReal z ^ 2)
-    hclockcont ((hw.poissonRate_continuousOn hU T z hz).pow 2 |>.mono hsub)
-  · intro t ht
-    exact (hw.activeRadiusClock_hasDerivWithinAt hU σ T z hz t
-      ⟨σ.coe_nonneg.trans ht.1.le, ht.2.le⟩).hasDerivAt
-        (Icc_mem_nhds (σ.coe_nonneg.trans_lt ht.1) ht.2) |>.hasDerivWithinAt
-  · simp
-  · exact hclock
-  · simpa [Multipoint.activeRadiusClock] using
-      hw.activeRadiusClock_nonneg hU σ T hσT z hz
-
-/-- The inverse pair regulator is paid by the actual physical-domain
-interaction budget. The terminal clock levels can be arbitrary. -/
-theorem IsCapacityTwoWitness.physical_pair_regulator_integral_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (s T : ℝ≥0) (hsT : s ≤ T) (z w : ℂ)
-    (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T) (hne : z ≠ w)
-    (bz bw : ℝ) (hbz : bz ≤ dist (J z) (J w)) (hbw : bw ≤ dist (J z) (J w))
-    {β M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) (hdecay : 2 * (M * β) < 2)
-    (Uz Uw : ℝ) (u v : ℝ → ℝ)
-    (hu : ContinuousOn u (Icc (s : ℝ) (T : ℝ)))
-    (hv : ContinuousOn v (Icc (s : ℝ) (T : ℝ)))
-    (hu0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ u t)
-    (hv0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ v t)
-    (hRu : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      physicalConformalRadius J G t.toNNReal z ≤ bz * Real.exp (-4 * u t))
-    (hRv : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
-      physicalConformalRadius J G t.toNNReal w ≤ bw * Real.exp (-4 * v t)) :
-    (∫ t in (s : ℝ)..(T : ℝ),
-      (Multipoint.regulator β Uz (u t) * Multipoint.regulator β Uw (v t)) ^ (-M) *
-        (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
-      Multipoint.interactionBudget 2 (M * β) := by
-  have hsub : Icc (s : ℝ) (T : ℝ) ⊆ Icc (0 : ℝ) (T : ℝ) :=
-    Icc_subset_Icc s.coe_nonneg le_rfl
-  have hR := ((hw.poissonRate_continuousOn hU T z hz).mul
-    (hw.poissonRate_continuousOn hU T w hw')).mono hsub
-  have hreg := ((Multipoint.regulator_continuous β Uz).comp_continuousOn hu).mul
-    ((Multipoint.regulator_continuous β Uw).comp_continuousOn hv)
-  have hpw := hreg.rpow_const (p := -M) fun t _ => Or.inl
-    (mul_pos (Multipoint.regulator_pos _ _ _) (Multipoint.regulator_pos _ _ _)).ne'
-  apply le_trans (intervalIntegral.integral_mono_on hsT
-    ((hpw.mul hR).intervalIntegrable_of_Icc hsT)
-    (((Real.continuous_exp.comp_continuousOn
-      (continuousOn_const.mul (hu.add hv))).mul hR).intervalIntegrable_of_Icc hsT) ?_)
-    (hw.physical_pair_interaction_budget hU hgrowth J hJ hJinj s T hsT z w hz hw' hne
-      bz bw hbz hbw (mul_nonneg hM hβ) hdecay u v hu hv hu0 hv0 hRu hRv)
-  intro t ht
-  apply mul_le_mul_of_nonneg_right (Multipoint.pair_regulator_inv_power_le hβ hM)
-  have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-  exact mul_nonneg
-    (hw.poissonRate_nonneg _ z (survivingDomain_antitone γ htT hz))
-    (hw.poissonRate_nonneg _ w (survivingDomain_antitone γ htT hw'))
-
-/-- The pair discount on a joint active interval, expressed entirely in
-the actual radius clocks measured from the individual activation times. -/
-theorem IsCapacityTwoWitness.actual_pair_regulator_integral_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (hgrowth : PublishedDiskUnivalentGrowth)
-    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
-    (hJinj : InjOn J upperHalfPlane)
-    (σz σw s T : ℝ≥0) (hσz : σz ≤ s) (hσw : σw ≤ s) (hsT : s ≤ T)
-    (z w : ℂ) (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T)
-    (hne : z ≠ w)
-    (hbz : physicalConformalRadius J G σz z ≤ dist (J z) (J w))
-    (hbw : physicalConformalRadius J G σw w ≤ dist (J z) (J w))
-    {β M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) (hdecay : 2 * (M * β) < 2)
-    (Uz Uw : ℝ) :
-    (∫ t in (s : ℝ)..(T : ℝ),
-      (Multipoint.regulator β Uz (Multipoint.activeRadiusClock G z σz t) *
-        Multipoint.regulator β Uw (Multipoint.activeRadiusClock G w σw t)) ^ (-M) *
-        (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
-      Multipoint.interactionBudget 2 (M * β) := by
-  have hsub : Icc (s : ℝ) (T : ℝ) ⊆ Icc (0 : ℝ) (T : ℝ) :=
-    Icc_subset_Icc s.coe_nonneg le_rfl
-  have hzσ := survivingDomain_antitone γ (hσz.trans hsT) hz
-  have hwσ := survivingDomain_antitone γ (hσw.trans hsT) hw'
-  apply hw.physical_pair_regulator_integral_le hU hgrowth J hJ hJinj s T hsT z w hz hw'
-    hne (physicalConformalRadius J G σz z) (physicalConformalRadius J G σw w)
-    hbz hbw hβ hM hdecay Uz Uw
-    (Multipoint.activeRadiusClock G z σz) (Multipoint.activeRadiusClock G w σw)
-    ((hw.activeRadiusClock_continuousOn hU σz T z hz).mono hsub)
-    ((hw.activeRadiusClock_continuousOn hU σw T w hw').mono hsub)
-  · intro t ht
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    have hσzt : σz ≤ t.toNNReal := by
-      simpa only [Real.toNNReal_coe] using
-        Real.toNNReal_le_toNNReal ((show (σz : ℝ) ≤ s from hσz).trans ht.1)
-    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
-      hw.activeRadiusClock_nonneg hU σz t.toNNReal hσzt z
-        (survivingDomain_antitone γ htT hz)
-  · intro t ht
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    have hσwt : σw ≤ t.toNNReal := by
-      simpa only [Real.toNNReal_coe] using
-        Real.toNNReal_le_toNNReal ((show (σw : ℝ) ≤ s from hσw).trans ht.1)
-    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
-      hw.activeRadiusClock_nonneg hU σw t.toNNReal hσwt w
-        (survivingDomain_antitone γ htT hw')
-  · intro t ht
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
-      (hw.physical_radius_active_clock hU J σz t.toNNReal z hzσ
-        (survivingDomain_antitone γ htT hz)).le
-  · intro t ht
-    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
-      (hw.physical_radius_active_clock hU J σw t.toNNReal w hwσ
-        (survivingDomain_antitone γ htT hw')).le
-
-end SLEExactGauge
-
-namespace SLEExactGauge.Multipoint
-
-/-- The regulator is unchanged when moving from the capacity-two clock
-`u` to the manuscript clock `κ * u`, provided its decay rate is scaled
-by the inverse factor. -/
-lemma regulator_scale_clock {κ : ℝ} (hκ : 0 ≤ κ) (β U u : ℝ) :
-    regulator (κ * β) U u = regulator β (κ * U) (κ * u) := by
-  unfold regulator
-  rw [← mul_sub, ← mul_min_of_nonneg u (U-u) hκ]
-  congr 1
-  ring
-
-lemma source_radius_clock_scale {κ : ℝ} (hκ : 0 < κ) (b u : ℝ) :
-    b * Real.exp (-2 * (2 / κ) * (κ * u)) = b * Real.exp (-4 * u) := by
-  congr 2
-  field_simp [hκ.ne']
-  ring
-
-lemma source_pair_exponent_scale (κ c u v : ℝ) :
-    Real.exp (c * (κ * u + κ * v)) = Real.exp ((κ * c) * (u + v)) := by
-  congr 1
-  ring
-
-lemma source_pair_exponent_admissible {κ c : ℝ} (hκ : 0 < κ) :
-    2 * (κ * c) < 2 ↔ 2 * c < 2 / κ := by
-  rw [lt_div_iff₀ hκ]
-  constructor <;> intro h <;> nlinarith
-
-lemma source_terminal_factor_scale (κ p u : ℝ) :
-    Real.exp ((κ * p) / 2 * u) = Real.exp (p / 2 * (κ * u)) := by
-  congr 1
-  ring
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPhysicalBudget
+
+namespace OAI
+
+/-! Deterministic transport of the two-ended regulator to physical time.
+The single-point cost follows by change of variables along the actual
+radius clock, and the pair cost follows from geometric interaction decay. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+/-- Change of variables for a regulator on any subinterval of its clock.
+Only ordinary calculus is used here; no stochastic estimate is assumed. -/
+theorem regulator_clock_integral_le
+    {β U s T : ℝ} (hβ : 0 < β) (hU : 0 ≤ U) (hsT : s ≤ T)
+    (u speed : ℝ → ℝ) (hu : ContinuousOn u (Icc s T))
+    (hspeed : ContinuousOn speed (Icc s T))
+    (hdu : ∀ t ∈ Ioo s T, HasDerivWithinAt u (speed t) (Ioi t) t)
+    (hu0 : 0 ≤ u s) (huT : u T ≤ U) (hmono : u s ≤ u T) :
+    (∫ t in s..T, regulator β U (u t) * speed t) ≤ 2 / β := by
+  have he : (∫ t in s..T, regulator β U (u t) * speed t) =
+      ∫ v in u s..u T, regulator β U v := by
+    simpa only [Function.comp_apply] using
+      intervalIntegral.integral_comp_mul_deriv''
+        (by simpa only [uIcc_of_le hsT] using hu)
+        (by simpa only [min_eq_left hsT, max_eq_right hsT] using hdu)
+        (by simpa only [uIcc_of_le hsT] using hspeed)
+        (regulator_continuous β U).continuousOn
+  rw [he]
+  apply le_trans _ (integral_regulator_le hβ hU)
+  exact intervalIntegral.integral_mono_interval hu0 hmono huT
+    (ae_of_all _ fun v => (regulator_pos β U v).le)
+    ((regulator_continuous β U).intervalIntegrable 0 U)
+
+/-- The radius clock measured from activation, in capacity-two time.
+The clock derivative is the square of the Poisson rate. -/
+def activeRadiusClock (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (σ : ℝ≥0) (t : ℝ) : ℝ :=
+  (Real.log ((2 * z.im) / fullRadius G t.toNNReal z) -
+    Real.log ((2 * z.im) / fullRadius G σ z)) / 4
+
+@[simp] lemma activeRadiusClock_start (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (σ : ℝ≥0) :
+    activeRadiusClock G z σ σ = 0 := by
+  simp [activeRadiusClock]
+
+end SLEExactGauge.Multipoint
+
+namespace SLEExactGauge
+
+theorem IsCapacityTwoWitness.activeRadiusClock_hasDerivWithinAt
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (σ T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T)
+    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
+    HasDerivWithinAt (Multipoint.activeRadiusClock G z σ)
+      (chartPoissonRate U G t.toNNReal z ^ 2) (Icc 0 (T : ℝ)) t := by
+  refine (((loewner_radius_clock_derivative hU hw T z hz t ht).sub_const
+    (Real.log ((2 * z.im) / fullRadius G σ z))).div_const 4).congr_deriv ?_
+  simp only [chartPoissonRate, Complex.normSq_eq_norm_sq]
+  ring
+
+theorem IsCapacityTwoWitness.activeRadiusClock_continuousOn
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (σ T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
+    ContinuousOn (Multipoint.activeRadiusClock G z σ) (Icc 0 (T : ℝ)) := by
+  intro t ht
+  exact (hw.activeRadiusClock_hasDerivWithinAt hU σ T z hz t ht).continuousWithinAt
+
+theorem IsCapacityTwoWitness.activeRadiusClock_nonneg
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T) :
+    0 ≤ Multipoint.activeRadiusClock G z σ T := by
+  have h := (hw.radius_clock_strictMonoOn hU z T hz).monotoneOn
+    (show (σ : ℝ) ∈ Icc 0 (T : ℝ) from ⟨σ.coe_nonneg, hσT⟩)
+    (show (T : ℝ) ∈ Icc 0 (T : ℝ) from ⟨T.coe_nonneg, le_rfl⟩) hσT
+  simp only [Real.toNNReal_coe] at h
+  change 0 ≤ (Real.log ((2 * z.im) / fullRadius G (T : ℝ).toNNReal z) -
+    Real.log ((2 * z.im) / fullRadius G σ z)) / 4
+  rw [Real.toNNReal_coe]
+  exact div_nonneg (sub_nonneg.mpr h) (by norm_num)
+
+/-- The physical conformal radius has the same ratio as the half-plane
+radius. In particular, no derivative bound on `J` is needed. -/
+theorem IsCapacityTwoWitness.physical_radius_active_clock
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (J : ℂ → ℂ) (σ t : ℝ≥0) (z : ℂ)
+    (hσ : z ∈ survivingDomain γ σ) (ht : z ∈ survivingDomain γ t) :
+    physicalConformalRadius J G t z = physicalConformalRadius J G σ z *
+      Real.exp (-4 * Multipoint.activeRadiusClock G z σ t) := by
+  have hRσ := hw.fullRadius_positive hU σ z hσ
+  have hRt := hw.fullRadius_positive hU t z ht
+  have hz0 : 2 * z.im ≠ 0 := ne_of_gt (mul_pos (by norm_num) ht.1.1)
+  have he : -4 * Multipoint.activeRadiusClock G z σ t =
+      Real.log (fullRadius G t z / fullRadius G σ z) := by
+    rw [Multipoint.activeRadiusClock, Real.toNNReal_coe,
+      Real.log_div hz0 hRt.ne', Real.log_div hz0 hRσ.ne',
+      Real.log_div hRt.ne' hRσ.ne']
+    ring
+  rw [he, Real.exp_log (div_pos hRt hRσ)]
+  unfold physicalConformalRadius
+  field_simp [hRσ.ne']
+
+/-- One complete or partial active interval has regulator cost at most
+`2 / β`, independently of its target-clock length. -/
+theorem IsCapacityTwoWitness.active_regulator_integral_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (σ T : ℝ≥0) (hσT : σ ≤ T) (z : ℂ) (hz : z ∈ survivingDomain γ T)
+    {β target : ℝ} (hβ : 0 < β) (htarget : 0 ≤ target)
+    (hclock : Multipoint.activeRadiusClock G z σ T ≤ target) :
+    (∫ t in (σ : ℝ)..(T : ℝ),
+      Multipoint.regulator β target (Multipoint.activeRadiusClock G z σ t) *
+        chartPoissonRate U G t.toNNReal z ^ 2) ≤ 2 / β := by
+  have hsub : Icc (σ : ℝ) (T : ℝ) ⊆ Icc (0 : ℝ) (T : ℝ) :=
+    Icc_subset_Icc σ.coe_nonneg le_rfl
+  have hclockcont : ContinuousOn (Multipoint.activeRadiusClock G z σ)
+      (Icc (σ : ℝ) (T : ℝ)) := by
+    intro t ht
+    exact ((hw.activeRadiusClock_hasDerivWithinAt hU σ T z hz t (hsub ht)).mono
+      hsub).continuousWithinAt
+  apply Multipoint.regulator_clock_integral_le hβ htarget hσT
+    (Multipoint.activeRadiusClock G z σ) (fun t => chartPoissonRate U G t.toNNReal z ^ 2)
+    hclockcont ((hw.poissonRate_continuousOn hU T z hz).pow 2 |>.mono hsub)
+  · intro t ht
+    exact (hw.activeRadiusClock_hasDerivWithinAt hU σ T z hz t
+      ⟨σ.coe_nonneg.trans ht.1.le, ht.2.le⟩).hasDerivAt
+        (Icc_mem_nhds (σ.coe_nonneg.trans_lt ht.1) ht.2) |>.hasDerivWithinAt
+  · simp
+  · exact hclock
+  · simpa [Multipoint.activeRadiusClock] using
+      hw.activeRadiusClock_nonneg hU σ T hσT z hz
+
+/-- The inverse pair regulator is paid by the actual physical-domain
+interaction budget. The terminal clock levels can be arbitrary. -/
+theorem IsCapacityTwoWitness.physical_pair_regulator_integral_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (s T : ℝ≥0) (hsT : s ≤ T) (z w : ℂ)
+    (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T) (hne : z ≠ w)
+    (bz bw : ℝ) (hbz : bz ≤ dist (J z) (J w)) (hbw : bw ≤ dist (J z) (J w))
+    {β M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) (hdecay : 2 * (M * β) < 2)
+    (Uz Uw : ℝ) (u v : ℝ → ℝ)
+    (hu : ContinuousOn u (Icc (s : ℝ) (T : ℝ)))
+    (hv : ContinuousOn v (Icc (s : ℝ) (T : ℝ)))
+    (hu0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ u t)
+    (hv0 : ∀ t ∈ Icc (s : ℝ) (T : ℝ), 0 ≤ v t)
+    (hRu : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      physicalConformalRadius J G t.toNNReal z ≤ bz * Real.exp (-4 * u t))
+    (hRv : ∀ t ∈ Icc (s : ℝ) (T : ℝ),
+      physicalConformalRadius J G t.toNNReal w ≤ bw * Real.exp (-4 * v t)) :
+    (∫ t in (s : ℝ)..(T : ℝ),
+      (Multipoint.regulator β Uz (u t) * Multipoint.regulator β Uw (v t)) ^ (-M) *
+        (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
+      Multipoint.interactionBudget 2 (M * β) := by
+  have hsub : Icc (s : ℝ) (T : ℝ) ⊆ Icc (0 : ℝ) (T : ℝ) :=
+    Icc_subset_Icc s.coe_nonneg le_rfl
+  have hR := ((hw.poissonRate_continuousOn hU T z hz).mul
+    (hw.poissonRate_continuousOn hU T w hw')).mono hsub
+  have hreg := ((Multipoint.regulator_continuous β Uz).comp_continuousOn hu).mul
+    ((Multipoint.regulator_continuous β Uw).comp_continuousOn hv)
+  have hpw := hreg.rpow_const (p := -M) fun t _ => Or.inl
+    (mul_pos (Multipoint.regulator_pos _ _ _) (Multipoint.regulator_pos _ _ _)).ne'
+  apply le_trans (intervalIntegral.integral_mono_on hsT
+    ((hpw.mul hR).intervalIntegrable_of_Icc hsT)
+    (((Real.continuous_exp.comp_continuousOn
+      (continuousOn_const.mul (hu.add hv))).mul hR).intervalIntegrable_of_Icc hsT) ?_)
+    (hw.physical_pair_interaction_budget hU hgrowth J hJ hJinj s T hsT z w hz hw' hne
+      bz bw hbz hbw (mul_nonneg hM hβ) hdecay u v hu hv hu0 hv0 hRu hRv)
+  intro t ht
+  apply mul_le_mul_of_nonneg_right (Multipoint.pair_regulator_inv_power_le hβ hM)
+  have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+  exact mul_nonneg
+    (hw.poissonRate_nonneg _ z (survivingDomain_antitone γ htT hz))
+    (hw.poissonRate_nonneg _ w (survivingDomain_antitone γ htT hw'))
+
+/-- The pair discount on a joint active interval, expressed entirely in
+the actual radius clocks measured from the individual activation times. -/
+theorem IsCapacityTwoWitness.actual_pair_regulator_integral_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (hgrowth : PublishedDiskUnivalentGrowth)
+    (J : ℂ → ℂ) (hJ : DifferentiableOn ℂ J upperHalfPlane)
+    (hJinj : InjOn J upperHalfPlane)
+    (σz σw s T : ℝ≥0) (hσz : σz ≤ s) (hσw : σw ≤ s) (hsT : s ≤ T)
+    (z w : ℂ) (hz : z ∈ survivingDomain γ T) (hw' : w ∈ survivingDomain γ T)
+    (hne : z ≠ w)
+    (hbz : physicalConformalRadius J G σz z ≤ dist (J z) (J w))
+    (hbw : physicalConformalRadius J G σw w ≤ dist (J z) (J w))
+    {β M : ℝ} (hβ : 0 ≤ β) (hM : 0 ≤ M) (hdecay : 2 * (M * β) < 2)
+    (Uz Uw : ℝ) :
+    (∫ t in (s : ℝ)..(T : ℝ),
+      (Multipoint.regulator β Uz (Multipoint.activeRadiusClock G z σz t) *
+        Multipoint.regulator β Uw (Multipoint.activeRadiusClock G w σw t)) ^ (-M) *
+        (chartPoissonRate U G t.toNNReal z * chartPoissonRate U G t.toNNReal w)) ≤
+      Multipoint.interactionBudget 2 (M * β) := by
+  have hsub : Icc (s : ℝ) (T : ℝ) ⊆ Icc (0 : ℝ) (T : ℝ) :=
+    Icc_subset_Icc s.coe_nonneg le_rfl
+  have hzσ := survivingDomain_antitone γ (hσz.trans hsT) hz
+  have hwσ := survivingDomain_antitone γ (hσw.trans hsT) hw'
+  apply hw.physical_pair_regulator_integral_le hU hgrowth J hJ hJinj s T hsT z w hz hw'
+    hne (physicalConformalRadius J G σz z) (physicalConformalRadius J G σw w)
+    hbz hbw hβ hM hdecay Uz Uw
+    (Multipoint.activeRadiusClock G z σz) (Multipoint.activeRadiusClock G w σw)
+    ((hw.activeRadiusClock_continuousOn hU σz T z hz).mono hsub)
+    ((hw.activeRadiusClock_continuousOn hU σw T w hw').mono hsub)
+  · intro t ht
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    have hσzt : σz ≤ t.toNNReal := by
+      simpa only [Real.toNNReal_coe] using
+        Real.toNNReal_le_toNNReal ((show (σz : ℝ) ≤ s from hσz).trans ht.1)
+    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
+      hw.activeRadiusClock_nonneg hU σz t.toNNReal hσzt z
+        (survivingDomain_antitone γ htT hz)
+  · intro t ht
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    have hσwt : σw ≤ t.toNNReal := by
+      simpa only [Real.toNNReal_coe] using
+        Real.toNNReal_le_toNNReal ((show (σw : ℝ) ≤ s from hσw).trans ht.1)
+    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
+      hw.activeRadiusClock_nonneg hU σw t.toNNReal hσwt w
+        (survivingDomain_antitone γ htT hw')
+  · intro t ht
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
+      (hw.physical_radius_active_clock hU J σz t.toNNReal z hzσ
+        (survivingDomain_antitone γ htT hz)).le
+  · intro t ht
+    have htT : t.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    simpa only [Real.coe_toNNReal _ (s.coe_nonneg.trans ht.1)] using
+      (hw.physical_radius_active_clock hU J σw t.toNNReal w hwσ
+        (survivingDomain_antitone γ htT hw')).le
+
+end SLEExactGauge
+
+namespace SLEExactGauge.Multipoint
+
+/-- The regulator is unchanged when moving from the capacity-two clock
+`u` to the manuscript clock `κ * u`, provided its decay rate is scaled
+by the inverse factor. -/
+lemma regulator_scale_clock {κ : ℝ} (hκ : 0 ≤ κ) (β U u : ℝ) :
+    regulator (κ * β) U u = regulator β (κ * U) (κ * u) := by
+  unfold regulator
+  rw [← mul_sub, ← mul_min_of_nonneg u (U-u) hκ]
+  congr 1
+  ring
+
+lemma source_radius_clock_scale {κ : ℝ} (hκ : 0 < κ) (b u : ℝ) :
+    b * Real.exp (-2 * (2 / κ) * (κ * u)) = b * Real.exp (-4 * u) := by
+  congr 2
+  field_simp [hκ.ne']
+  ring
+
+lemma source_pair_exponent_scale (κ c u v : ℝ) :
+    Real.exp (c * (κ * u + κ * v)) = Real.exp ((κ * c) * (u + v)) := by
+  congr 1
+  ring
+
+lemma source_pair_exponent_admissible {κ c : ℝ} (hκ : 0 < κ) :
+    2 * (κ * c) < 2 ↔ 2 * c < 2 / κ := by
+  rw [lt_div_iff₀ hκ]
+  constructor <;> intro h <;> nlinarith
+
+lemma source_terminal_factor_scale (κ p u : ℝ) :
+    Real.exp ((κ * p) / 2 * u) = Real.exp (p / 2 * (κ * u)) := by
+  congr 1
+  ring
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointSelectedRate.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointSelectedRate.lean
index 4505c74667c5fe9c267ed8ff76a7ff6ddd11fa75..d4faa4da49b6294864d60c1d5018aa562ac29bd6 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointSelectedRate.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointSelectedRate.lean
@@ -1,84 +1,84 @@
-import OAI.Probability.SLE.Multipoint.MultipointDiscountProcess
-import OAI.Probability.SLE.Multipoint.MultipointStagedRate
-import OAI.Probability.SLE.Multipoint.MultipointLocalizedStages
-
-namespace OAI
-
-/-! A bounded progressive discount rate that agrees with the actual
-four-phase product cost before localization. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem exists_localized_discount_rate
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (Q : ℝ → Ω → ι → ℂ × ℝ)
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q t)) (hQc : ∀ ω, Continuous (fun t => Q t ω))
-    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (a U : ι → ℝ)
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
-    (hlstop : ∀ σ ∈ l, IsStoppingTime
-      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hlbound : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρmem : ∀ k, ρ k ∈ l)
-    (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
-    (χ : Ω → ℝ≥0)
-    (hχstop : IsStoppingTime
-      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (χ ω : WithTop ℝ≥0))) (hχbound : ∀ ω, χ ω ≤ H)
-    (R : ℝ) (hR : 0 ≤ R)
-    (hnorm : ∀ ω t, t ≤ χ ω → ‖(Q t ω,B t ω)‖ ≤ R) :
-    ∃ (r : ℝ≥0 → Ω → ℝ) (C : ℝ≥0),
-      Measurable (Function.uncurry r) ∧
-      IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) r ∧
-      (∀ t ω, 0 ≤ r t ω ∧ |r t ω| ≤ C) ∧
-      (∀ ω t, t < χ ω → r t ω =
-        productDriftCost (patternActiveSet (phaseAt ρ t ω)) κ
-          (fun i => phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
-          (fun i => phaseRegulatorSlope β (phaseAt ρ t ω i)) (Q t ω) (B t ω)) := by
-  classical
-  have hex (A : Finset (ι × Fin 3)) := exists_bounded_discount_cost_process B hm hBm hBc
-    Q hQm hQa hQc (patternActiveSet (phaseOfPattern A)) κ hκ hκ8
-    (fun i => phaseRegulatorScale β (a i) (U i) (phaseOfPattern A i))
-    (fun i => phaseRegulatorSlope β (phaseOfPattern A i))
-    (fun i _ => phaseRegulatorScale_pos _ _ _ _) R hR
-  choose g C hgm hgp hgc hgb hge using hex
-  let τ := localizedStage l H χ
-  have hτ (n : ℕ) := localizedStage_stopping _ l hlstop H χ hχstop n
-  have hρstop (k : ι × Fin 3) := hlstop (ρ k) (hρmem k)
-  let r := stagedPatternRate l.length Finset.univ ρ τ g
-  let C₀ : ℝ≥0 := ∑ k ∈ (Finset.range l.length).product (Finset.univ : Finset (ι × Fin 3)).powerset, C k.2
-  refine ⟨r,C₀,?_,?_,?_,?_⟩
-  · exact stagedPatternRate_measurable _ l.length Finset.univ ρ τ
-      (fun i _ => hρstop i) hτ g (fun A _ => hgm A)
-  · exact stagedPatternRate_progressive _ l.length Finset.univ ρ τ
-      (fun i _ => hρstop i) hτ g (fun A _ => hgp A)
-  · intro t ω
-    exact ⟨stagedPatternRate_nonneg l.length Finset.univ ρ τ g
-      (fun A _ u ω => (hgb A u ω).1) t ω,
-      stagedPatternRate_bound l.length Finset.univ ρ τ g C
-        (fun A _ u ω => (hgb A u ω).2) t ω⟩
-  · intro ω t htχ
-    obtain ⟨n,hn,hnt,htn⟩ := exists_finite_stage (fun n => τ n ω) l.length t
-      (by rw [show τ 0 ω = 0 from localizedStage_zero l H χ hlbound hzero ω]; exact bot_le)
-      (by rwa [show τ l.length ω = χ ω from localizedStage_last l H χ hχbound ω])
-    obtain ⟨A,hA,hω⟩ := switchPatternEvent_cover Finset.univ ρ (τ n) ω
-    have hphase : phaseAt ρ t ω = phaseOfPattern A :=
-      (phaseAt_const_localized_interval ρ l H χ hlbound hρmem n ω t hnt htn).trans
-        (phaseAt_of_pattern ρ (τ n) A hω)
-    change stagedPatternRate l.length Finset.univ ρ τ g t ω = _
-    rw [stagedPatternRate_eq l.length Finset.univ ρ τ g
-      (localizedStage_monotone l H χ hlbound) n hn A hA t ω hω hnt htn,
-      hge A t ω (hnorm ω t htχ.le), hphase]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointDiscountProcess
+import OAI.Probability.SLE.Multipoint.MultipointStagedRate
+import OAI.Probability.SLE.Multipoint.MultipointLocalizedStages
+
+namespace OAI
+
+/-! A bounded progressive discount rate that agrees with the actual
+four-phase product cost before localization. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem exists_localized_discount_rate
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (Q : ℝ → Ω → ι → ℂ × ℝ)
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q t)) (hQc : ∀ ω, Continuous (fun t => Q t ω))
+    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (a U : ι → ℝ)
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (H : ℝ≥0)
+    (hlstop : ∀ σ ∈ l, IsStoppingTime
+      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hlbound : ∀ σ ∈ l, ∀ ω, σ ω ≤ H+1) (hρmem : ∀ k, ρ k ∈ l)
+    (hzero : (fun _ : Ω => (0 : ℝ≥0)) ∈ l)
+    (χ : Ω → ℝ≥0)
+    (hχstop : IsStoppingTime
+      (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (χ ω : WithTop ℝ≥0))) (hχbound : ∀ ω, χ ω ≤ H)
+    (R : ℝ) (hR : 0 ≤ R)
+    (hnorm : ∀ ω t, t ≤ χ ω → ‖(Q t ω,B t ω)‖ ≤ R) :
+    ∃ (r : ℝ≥0 → Ω → ℝ) (C : ℝ≥0),
+      Measurable (Function.uncurry r) ∧
+      IsProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) r ∧
+      (∀ t ω, 0 ≤ r t ω ∧ |r t ω| ≤ C) ∧
+      (∀ ω t, t < χ ω → r t ω =
+        productDriftCost (patternActiveSet (phaseAt ρ t ω)) κ
+          (fun i => phaseRegulatorScale β (a i) (U i) (phaseAt ρ t ω i))
+          (fun i => phaseRegulatorSlope β (phaseAt ρ t ω i)) (Q t ω) (B t ω)) := by
+  classical
+  have hex (A : Finset (ι × Fin 3)) := exists_bounded_discount_cost_process B hm hBm hBc
+    Q hQm hQa hQc (patternActiveSet (phaseOfPattern A)) κ hκ hκ8
+    (fun i => phaseRegulatorScale β (a i) (U i) (phaseOfPattern A i))
+    (fun i => phaseRegulatorSlope β (phaseOfPattern A i))
+    (fun i _ => phaseRegulatorScale_pos _ _ _ _) R hR
+  choose g C hgm hgp hgc hgb hge using hex
+  let τ := localizedStage l H χ
+  have hτ (n : ℕ) := localizedStage_stopping _ l hlstop H χ hχstop n
+  have hρstop (k : ι × Fin 3) := hlstop (ρ k) (hρmem k)
+  let r := stagedPatternRate l.length Finset.univ ρ τ g
+  let C₀ : ℝ≥0 := ∑ k ∈ (Finset.range l.length).product (Finset.univ : Finset (ι × Fin 3)).powerset, C k.2
+  refine ⟨r,C₀,?_,?_,?_,?_⟩
+  · exact stagedPatternRate_measurable _ l.length Finset.univ ρ τ
+      (fun i _ => hρstop i) hτ g (fun A _ => hgm A)
+  · exact stagedPatternRate_progressive _ l.length Finset.univ ρ τ
+      (fun i _ => hρstop i) hτ g (fun A _ => hgp A)
+  · intro t ω
+    exact ⟨stagedPatternRate_nonneg l.length Finset.univ ρ τ g
+      (fun A _ u ω => (hgb A u ω).1) t ω,
+      stagedPatternRate_bound l.length Finset.univ ρ τ g C
+        (fun A _ u ω => (hgb A u ω).2) t ω⟩
+  · intro ω t htχ
+    obtain ⟨n,hn,hnt,htn⟩ := exists_finite_stage (fun n => τ n ω) l.length t
+      (by rw [show τ 0 ω = 0 from localizedStage_zero l H χ hlbound hzero ω]; exact bot_le)
+      (by rwa [show τ l.length ω = χ ω from localizedStage_last l H χ hχbound ω])
+    obtain ⟨A,hA,hω⟩ := switchPatternEvent_cover Finset.univ ρ (τ n) ω
+    have hphase : phaseAt ρ t ω = phaseOfPattern A :=
+      (phaseAt_const_localized_interval ρ l H χ hlbound hρmem n ω t hnt htn).trans
+        (phaseAt_of_pattern ρ (τ n) A hω)
+    change stagedPatternRate l.length Finset.univ ρ τ g t ω = _
+    rw [stagedPatternRate_eq l.length Finset.univ ρ τ g
+      (localizedStage_monotone l H χ hlbound) n hn A hA t ω hω hnt htn,
+      hge A t ω (hnorm ω t htχ.le), hphase]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointStagedRate.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointStagedRate.lean
index 42299590bc7e918465303ed534445d95f343c0da..9c0e145c0788dfde6957f46b7b9823a67c4b77ef 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointStagedRate.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointStagedRate.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
-
-namespace OAI
-
-/-! The single adapted discount rate obtained by reading the current
-finite stopping interval and its past-measurable switch pattern. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def stagedPatternRate {Ω ι : Type*} [DecidableEq ι]
-    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
-    (g : Finset ι → ℝ≥0 → Ω → ℝ) : ℝ≥0 → Ω → ℝ :=
-  finiteSwitchRate ((Finset.range N).product s.powerset)
-    (fun k => τ k.1) (fun k => τ (k.1+1))
-    (fun k => switchPatternEvent s ρ (τ k.1) k.2) (fun k => g k.2)
-
-theorem stagedPatternRate_eq {Ω ι : Type*} [DecidableEq ι]
-    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
-    (g : Finset ι → ℝ≥0 → Ω → ℝ) (hmono : ∀ ω, Monotone (fun n => τ n ω))
-    (n : ℕ) (hn : n < N) (A : Finset ι) (hA : A ∈ s.powerset)
-    (t : ℝ≥0) (ω : Ω) (hω : ω ∈ switchPatternEvent s ρ (τ n) A)
-    (hnt : τ n ω ≤ t) (htn : t < τ (n+1) ω) :
-    stagedPatternRate N s ρ τ g t ω = g A t ω := by
-  classical
-  unfold stagedPatternRate finiteSwitchRate
-  rw [Finset.sum_eq_single (n,A)]
-  · simp only [stoppedBranchRate, hω, hnt, htn, and_self, ite_true]
-  · rintro ⟨m,C⟩ hmem hneq
-    obtain ⟨_, hC⟩ := Finset.mem_product.mp hmem
-    change stoppedBranchRate (τ m) (τ (m+1)) (switchPatternEvent s ρ (τ m) C) (g C) t ω = 0
-    unfold stoppedBranchRate
-    apply ite_eq_right
-    rintro ⟨hωC, hmt, htm⟩
-    have hmn : m = n := by
-      rcases lt_trichotomy m n with h | h | h
-      · exact False.elim ((not_le_of_gt htm) ((hmono ω (Nat.succ_le_of_lt h)).trans hnt))
-      · exact h
-      · exact False.elim ((not_le_of_gt htn) ((hmono ω (Nat.succ_le_of_lt h)).trans hmt))
-    subst m
-    have hCA : C = A := switchPatternEvent_unique s ρ (τ n)
-      (Finset.mem_powerset.mp hC) (Finset.mem_powerset.mp hA) hωC hω
-    exact hneq (Prod.ext rfl hCA)
-  · intro h
-    exact False.elim (h (Finset.mem_product.mpr ⟨Finset.mem_range.mpr hn,hA⟩))
-
-theorem stagedPatternRate_progressive {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [DecidableEq ι]
-    (ℱ : Filtration ℝ≥0 mΩ) (N : ℕ) (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
-    (hρ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (ρ i ω : WithTop ℝ≥0)))
-    (hτ : ∀ n, IsStoppingTime ℱ (fun ω => (τ n ω : WithTop ℝ≥0)))
-    (g : Finset ι → ℝ≥0 → Ω → ℝ) (hg : ∀ A ∈ s.powerset, IsProgressive ℱ (g A)) :
-    IsProgressive ℱ (stagedPatternRate N s ρ τ g) := by
-  exact finiteSwitchRate_progressive ℱ ((Finset.range N).product s.powerset)
-    (fun k => τ k.1) (fun k => τ (k.1+1))
-    (fun k _ => hτ k.1) (fun k _ => hτ (k.1+1))
-    (fun k => switchPatternEvent s ρ (τ k.1) k.2)
-    (fun k _ => switchPatternEvent_measurable ℱ s ρ (τ k.1) hρ (hτ k.1) k.2)
-    (fun k => g k.2) (fun k hk => hg k.2 (Finset.mem_product.mp hk).2)
-
-theorem stagedPatternRate_measurable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [DecidableEq ι]
-    (ℱ : Filtration ℝ≥0 mΩ) (N : ℕ) (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
-    (hρ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (ρ i ω : WithTop ℝ≥0)))
-    (hτ : ∀ n, IsStoppingTime ℱ (fun ω => (τ n ω : WithTop ℝ≥0)))
-    (g : Finset ι → ℝ≥0 → Ω → ℝ)
-    (hg : ∀ A ∈ s.powerset, Measurable (Function.uncurry (g A))) :
-    Measurable (Function.uncurry (stagedPatternRate N s ρ τ g)) := by
-  apply finiteSwitchRate_measurable
-  · exact fun k _ => stopping_time_measurable_nnreal (hτ k.1)
-  · exact fun k _ => stopping_time_measurable_nnreal (hτ (k.1+1))
-  · exact fun k _ => (hτ k.1).measurableSpace_le _
-      (switchPatternEvent_measurable ℱ s ρ (τ k.1) hρ (hτ k.1) k.2)
-  · exact fun k hk => hg k.2 (Finset.mem_product.mp hk).2
-
-lemma stagedPatternRate_nonneg {Ω ι : Type*} [DecidableEq ι]
-    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
-    (g : Finset ι → ℝ≥0 → Ω → ℝ) (hg : ∀ A ∈ s.powerset, ∀ t ω, 0 ≤ g A t ω)
-    (t : ℝ≥0) (ω : Ω) : 0 ≤ stagedPatternRate N s ρ τ g t ω :=
-  finiteSwitchRate_nonneg _ _ _ _ _
-    (fun k hk => hg k.2 (Finset.mem_product.mp hk).2) t ω
-
-lemma stagedPatternRate_bound {Ω ι : Type*} [DecidableEq ι]
-    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
-    (g : Finset ι → ℝ≥0 → Ω → ℝ) (C : Finset ι → ℝ≥0)
-    (hg : ∀ A ∈ s.powerset, ∀ t ω, |g A t ω| ≤ C A) (t : ℝ≥0) (ω : Ω) :
-    |stagedPatternRate N s ρ τ g t ω| ≤
-      (∑ k ∈ (Finset.range N).product s.powerset, C k.2 : ℝ≥0) :=
-  finiteSwitchRate_norm_le ((Finset.range N).product s.powerset)
-    (fun k => τ k.1) (fun k => τ (k.1+1))
-    (fun k => switchPatternEvent s ρ (τ k.1) k.2) (fun k => g k.2) (fun k => C k.2)
-    (fun k hk => hg k.2 (Finset.mem_product.mp hk).2) t ω
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
+
+namespace OAI
+
+/-! The single adapted discount rate obtained by reading the current
+finite stopping interval and its past-measurable switch pattern. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def stagedPatternRate {Ω ι : Type*} [DecidableEq ι]
+    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
+    (g : Finset ι → ℝ≥0 → Ω → ℝ) : ℝ≥0 → Ω → ℝ :=
+  finiteSwitchRate ((Finset.range N).product s.powerset)
+    (fun k => τ k.1) (fun k => τ (k.1+1))
+    (fun k => switchPatternEvent s ρ (τ k.1) k.2) (fun k => g k.2)
+
+theorem stagedPatternRate_eq {Ω ι : Type*} [DecidableEq ι]
+    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
+    (g : Finset ι → ℝ≥0 → Ω → ℝ) (hmono : ∀ ω, Monotone (fun n => τ n ω))
+    (n : ℕ) (hn : n < N) (A : Finset ι) (hA : A ∈ s.powerset)
+    (t : ℝ≥0) (ω : Ω) (hω : ω ∈ switchPatternEvent s ρ (τ n) A)
+    (hnt : τ n ω ≤ t) (htn : t < τ (n+1) ω) :
+    stagedPatternRate N s ρ τ g t ω = g A t ω := by
+  classical
+  unfold stagedPatternRate finiteSwitchRate
+  rw [Finset.sum_eq_single (n,A)]
+  · simp only [stoppedBranchRate, hω, hnt, htn, and_self, ite_true]
+  · rintro ⟨m,C⟩ hmem hneq
+    obtain ⟨_, hC⟩ := Finset.mem_product.mp hmem
+    change stoppedBranchRate (τ m) (τ (m+1)) (switchPatternEvent s ρ (τ m) C) (g C) t ω = 0
+    unfold stoppedBranchRate
+    apply ite_eq_right
+    rintro ⟨hωC, hmt, htm⟩
+    have hmn : m = n := by
+      rcases lt_trichotomy m n with h | h | h
+      · exact False.elim ((not_le_of_gt htm) ((hmono ω (Nat.succ_le_of_lt h)).trans hnt))
+      · exact h
+      · exact False.elim ((not_le_of_gt htn) ((hmono ω (Nat.succ_le_of_lt h)).trans hmt))
+    subst m
+    have hCA : C = A := switchPatternEvent_unique s ρ (τ n)
+      (Finset.mem_powerset.mp hC) (Finset.mem_powerset.mp hA) hωC hω
+    exact hneq (Prod.ext rfl hCA)
+  · intro h
+    exact False.elim (h (Finset.mem_product.mpr ⟨Finset.mem_range.mpr hn,hA⟩))
+
+theorem stagedPatternRate_progressive {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [DecidableEq ι]
+    (ℱ : Filtration ℝ≥0 mΩ) (N : ℕ) (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
+    (hρ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (ρ i ω : WithTop ℝ≥0)))
+    (hτ : ∀ n, IsStoppingTime ℱ (fun ω => (τ n ω : WithTop ℝ≥0)))
+    (g : Finset ι → ℝ≥0 → Ω → ℝ) (hg : ∀ A ∈ s.powerset, IsProgressive ℱ (g A)) :
+    IsProgressive ℱ (stagedPatternRate N s ρ τ g) := by
+  exact finiteSwitchRate_progressive ℱ ((Finset.range N).product s.powerset)
+    (fun k => τ k.1) (fun k => τ (k.1+1))
+    (fun k _ => hτ k.1) (fun k _ => hτ (k.1+1))
+    (fun k => switchPatternEvent s ρ (τ k.1) k.2)
+    (fun k _ => switchPatternEvent_measurable ℱ s ρ (τ k.1) hρ (hτ k.1) k.2)
+    (fun k => g k.2) (fun k hk => hg k.2 (Finset.mem_product.mp hk).2)
+
+theorem stagedPatternRate_measurable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [DecidableEq ι]
+    (ℱ : Filtration ℝ≥0 mΩ) (N : ℕ) (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
+    (hρ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (ρ i ω : WithTop ℝ≥0)))
+    (hτ : ∀ n, IsStoppingTime ℱ (fun ω => (τ n ω : WithTop ℝ≥0)))
+    (g : Finset ι → ℝ≥0 → Ω → ℝ)
+    (hg : ∀ A ∈ s.powerset, Measurable (Function.uncurry (g A))) :
+    Measurable (Function.uncurry (stagedPatternRate N s ρ τ g)) := by
+  apply finiteSwitchRate_measurable
+  · exact fun k _ => stopping_time_measurable_nnreal (hτ k.1)
+  · exact fun k _ => stopping_time_measurable_nnreal (hτ (k.1+1))
+  · exact fun k _ => (hτ k.1).measurableSpace_le _
+      (switchPatternEvent_measurable ℱ s ρ (τ k.1) hρ (hτ k.1) k.2)
+  · exact fun k hk => hg k.2 (Finset.mem_product.mp hk).2
+
+lemma stagedPatternRate_nonneg {Ω ι : Type*} [DecidableEq ι]
+    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
+    (g : Finset ι → ℝ≥0 → Ω → ℝ) (hg : ∀ A ∈ s.powerset, ∀ t ω, 0 ≤ g A t ω)
+    (t : ℝ≥0) (ω : Ω) : 0 ≤ stagedPatternRate N s ρ τ g t ω :=
+  finiteSwitchRate_nonneg _ _ _ _ _
+    (fun k hk => hg k.2 (Finset.mem_product.mp hk).2) t ω
+
+lemma stagedPatternRate_bound {Ω ι : Type*} [DecidableEq ι]
+    (N : ℕ) (s : Finset ι) (ρ : ι → Ω → ℝ≥0) (τ : ℕ → Ω → ℝ≥0)
+    (g : Finset ι → ℝ≥0 → Ω → ℝ) (C : Finset ι → ℝ≥0)
+    (hg : ∀ A ∈ s.powerset, ∀ t ω, |g A t ω| ≤ C A) (t : ℝ≥0) (ω : Ω) :
+    |stagedPatternRate N s ρ τ g t ω| ≤
+      (∑ k ∈ (Finset.range N).product s.powerset, C k.2 : ℝ≥0) :=
+  finiteSwitchRate_norm_le ((Finset.range N).product s.powerset)
+    (fun k => τ k.1) (fun k => τ (k.1+1))
+    (fun k => switchPatternEvent s ρ (τ k.1) k.2) (fun k => g k.2) (fun k => C k.2)
+    (fun k hk => hg k.2 (Finset.mem_product.mp hk).2) t ω
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointStateBounds.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointStateBounds.lean
index a7329543b629d38ca039b5ff69144eae5ff048cb..cd4df526d7ff9730b9918dc03064be18e3dc1f4d 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointStateBounds.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointStateBounds.lean
@@ -1,50 +1,50 @@
-import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
-
-namespace OAI
-
-/-! Deterministic compact bounds for the adapted state and its discount.
-Only the Brownian coordinate needs a separate stopping bound. -/
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma lipschitz_state_norm_le {E : Type*} [NormedAddCommGroup E]
-    (Q : ℝ → E) (L : ℝ≥0) (hQ : LipschitzWith L Q) (H t : ℝ)
-    (ht : 0 ≤ t) (htH : t ≤ H) : ‖Q t‖ ≤ ‖Q 0‖+(L : ℝ)*H := by
-  have hd : ‖Q t-Q 0‖ ≤ (L : ℝ)*t := by
-    simpa only [dist_eq_norm, sub_zero, Real.norm_eq_abs, abs_of_nonneg ht] using hQ.dist_le_mul t 0
-  calc
-    ‖Q t‖ = ‖(Q t-Q 0)+Q 0‖ := by rw [sub_add_cancel]
-    _ ≤ ‖Q t-Q 0‖+‖Q 0‖ := norm_add_le _ _
-    _ ≤ (L : ℝ)*H+‖Q 0‖ := add_le_add
-      (hd.trans (mul_le_mul_of_nonneg_left htH L.coe_nonneg)) le_rfl
-    _ = _ := add_comm _ _
-
-lemma discounted_state_norm_bound {Ω E : Type*} [NormedAddCommGroup E]
-    (Q : ℝ → Ω → E) (L : ℝ≥0) (hQ : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (q₀ : E) (hQ0 : ∀ ω, Q 0 ω = q₀)
-    (r : ℝ≥0 → Ω → ℝ) (hr : ∀ ω, Measurable (fun t => r t ω))
-    (C : ℝ≥0) (hrb : ∀ t ω, |r t ω| ≤ C)
-    (B : ℝ≥0 → Ω → ℝ) (χ : Ω → ℝ≥0) (H : ℝ≥0) (hχ : ∀ ω, χ ω ≤ H)
-    (M : ℝ) (hB : ∀ ω t, t ≤ χ ω → ‖B t ω‖ ≤ M)
-    (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (htχ : t ≤ (χ ω : ℝ)) :
-    ‖((Q t ω,discountClock r t ω),B t.toNNReal ω)‖ ≤
-      max (max (‖q₀‖+(L : ℝ)*(H : ℝ)) ((C : ℝ)*(H : ℝ))) M := by
-  have htH : t ≤ (H : ℝ) := htχ.trans (by exact_mod_cast hχ ω)
-  have hq := lipschitz_state_norm_le (fun t => Q t ω) L (hQ ω) H t ht htH
-  rw [hQ0 ω] at hq
-  have hc := lipschitz_state_norm_le (fun t => discountClock r t ω) C
-    (discountClock_lipschitz_of_measurable r hr C hrb ω) H t ht htH
-  rw [discountClock_initial, norm_zero, zero_add] at hc
-  have hbt : t.toNNReal ≤ χ ω := Real.toNNReal_le_iff_le_coe.mpr htχ
-  simp only [Prod.norm_def]
-  exact max_le_max (max_le_max hq hc) (hB ω t.toNNReal hbt)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
+
+namespace OAI
+
+/-! Deterministic compact bounds for the adapted state and its discount.
+Only the Brownian coordinate needs a separate stopping bound. -/
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma lipschitz_state_norm_le {E : Type*} [NormedAddCommGroup E]
+    (Q : ℝ → E) (L : ℝ≥0) (hQ : LipschitzWith L Q) (H t : ℝ)
+    (ht : 0 ≤ t) (htH : t ≤ H) : ‖Q t‖ ≤ ‖Q 0‖+(L : ℝ)*H := by
+  have hd : ‖Q t-Q 0‖ ≤ (L : ℝ)*t := by
+    simpa only [dist_eq_norm, sub_zero, Real.norm_eq_abs, abs_of_nonneg ht] using hQ.dist_le_mul t 0
+  calc
+    ‖Q t‖ = ‖(Q t-Q 0)+Q 0‖ := by rw [sub_add_cancel]
+    _ ≤ ‖Q t-Q 0‖+‖Q 0‖ := norm_add_le _ _
+    _ ≤ (L : ℝ)*H+‖Q 0‖ := add_le_add
+      (hd.trans (mul_le_mul_of_nonneg_left htH L.coe_nonneg)) le_rfl
+    _ = _ := add_comm _ _
+
+lemma discounted_state_norm_bound {Ω E : Type*} [NormedAddCommGroup E]
+    (Q : ℝ → Ω → E) (L : ℝ≥0) (hQ : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (q₀ : E) (hQ0 : ∀ ω, Q 0 ω = q₀)
+    (r : ℝ≥0 → Ω → ℝ) (hr : ∀ ω, Measurable (fun t => r t ω))
+    (C : ℝ≥0) (hrb : ∀ t ω, |r t ω| ≤ C)
+    (B : ℝ≥0 → Ω → ℝ) (χ : Ω → ℝ≥0) (H : ℝ≥0) (hχ : ∀ ω, χ ω ≤ H)
+    (M : ℝ) (hB : ∀ ω t, t ≤ χ ω → ‖B t ω‖ ≤ M)
+    (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (htχ : t ≤ (χ ω : ℝ)) :
+    ‖((Q t ω,discountClock r t ω),B t.toNNReal ω)‖ ≤
+      max (max (‖q₀‖+(L : ℝ)*(H : ℝ)) ((C : ℝ)*(H : ℝ))) M := by
+  have htH : t ≤ (H : ℝ) := htχ.trans (by exact_mod_cast hχ ω)
+  have hq := lipschitz_state_norm_le (fun t => Q t ω) L (hQ ω) H t ht htH
+  rw [hQ0 ω] at hq
+  have hc := lipschitz_state_norm_le (fun t => discountClock r t ω) C
+    (discountClock_lipschitz_of_measurable r hr C hrb ω) H t ht htH
+  rw [discountClock_initial, norm_zero, zero_add] at hc
+  have hbt : t.toNNReal ≤ χ ω := Real.toNNReal_le_iff_le_coe.mpr htχ
+  simp only [Prod.norm_def]
+  exact max_le_max (max_le_max hq hc) (hB ω t.toNNReal hbt)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedCoordinates.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedCoordinates.lean
index bfb5770e9b92d3351dabf39a292a6d17a5780ec6..b11637c0ed66447ca26f77129b94db4f2e258059 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedCoordinates.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedCoordinates.lean
@@ -1,167 +1,167 @@
-import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
-
-namespace OAI
-
-/-! Each completed multipoint coordinate is frozen at its own stopping
-time. The finite vector remains an adapted Lipschitz state, with the
-expected stopped velocity almost everywhere in time. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-def stoppedCoordinateState {Ω ι : Type*}
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (t : ℝ) (ω : Ω) (i : ι) : ℂ × ℝ :=
-  Q (min t (τ i ω : ℝ)) ω i
-
-def stoppedCoordinateVelocity {Ω ι : Type*}
-    (V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (t : ℝ) (ω : Ω) (i : ι) : ℂ × ℝ :=
-  if t < (τ i ω : ℝ) then V t ω i else 0
-
-lemma stoppedCoordinateState_measurable {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
-    (hQ : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hτ : ∀ i, Measurable (τ i)) :
-    Measurable (fun p : ℝ × Ω => stoppedCoordinateState Q τ p.1 p.2) := by
-  apply Measurable.of_eval
-  intro i
-  have hQi : Measurable (fun p : ℝ × Ω => Q p.1 p.2 i) :=
-    (measurable_pi_apply i).comp hQ
-  have htime : Measurable (fun p : ℝ × Ω => min p.1 (τ i p.2 : ℝ)) :=
-    measurable_fst.min (NNReal.continuous_coe.measurable.comp ((hτ i).comp measurable_snd))
-  exact hQi.comp (htime.prodMk measurable_snd)
-
-lemma stoppedCoordinateVelocity_measurable {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
-    (V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
-    (hV : Measurable (fun p : ℝ × Ω => V p.1 p.2))
-    (hτ : ∀ i, Measurable (τ i)) :
-    Measurable (fun p : ℝ × Ω => stoppedCoordinateVelocity V τ p.1 p.2) := by
-  apply Measurable.of_eval
-  intro i
-  apply Measurable.ite _ ((measurable_pi_apply i).comp hV) measurable_const
-  exact measurableSet_lt measurable_fst
-    (NNReal.continuous_coe.measurable.comp ((hτ i).comp measurable_snd))
-
-lemma stoppedCoordinateState_lipschitz {Ω ι : Type*} [Fintype ι]
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
-    (hQ : ∀ ω, LipschitzWith L (fun t => Q t ω)) (ω : Ω) :
-    LipschitzWith L (fun t => stoppedCoordinateState Q τ t ω) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro t u
-  apply (dist_pi_le_iff (mul_nonneg L.coe_nonneg dist_nonneg)).mpr
-  intro i
-  have hmin := (LipschitzWith.id.min_const (τ i ω : ℝ)).dist_le_mul t u
-  calc
-    dist (Q (min t (τ i ω : ℝ)) ω i) (Q (min u (τ i ω : ℝ)) ω i) ≤
-        dist (Q (min t (τ i ω : ℝ)) ω) (Q (min u (τ i ω : ℝ)) ω) :=
-      dist_le_pi_dist _ _ i
-    _ ≤ (L : ℝ) * dist (min t (τ i ω : ℝ)) (min u (τ i ω : ℝ)) :=
-      (hQ ω).dist_le_mul _ _
-    _ ≤ (L : ℝ) * dist t u := by
-      apply mul_le_mul_of_nonneg_left _ L.coe_nonneg
-      simpa only [NNReal.coe_one, one_mul, id_eq] using hmin
-
-lemma stoppedCoordinateVelocity_bound {Ω ι : Type*} [Fintype ι]
-    (V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
-    (hV : ∀ t ω, ‖V t ω‖ ≤ L) (t : ℝ) (ω : Ω) :
-    ‖stoppedCoordinateVelocity V τ t ω‖ ≤ L := by
-  apply (pi_norm_le_iff_of_nonneg L.coe_nonneg).mpr
-  intro i
-  unfold stoppedCoordinateVelocity
-  split_ifs
-  · exact (norm_le_pi_norm (V t ω) i).trans (hV t ω)
-  · simpa only [norm_zero] using L.coe_nonneg
-
-lemma stoppedCoordinateState_ae_derivative {Ω ι : Type*} [Fintype ι]
-    (Q V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
-    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
-    (ω : Ω) : ∀ᵐ t : ℝ, 0 < t →
-      HasDerivAt (fun u => stoppedCoordinateState Q τ u ω)
-        (stoppedCoordinateVelocity V τ t ω) t := by
-  have hne : ∀ᵐ t : ℝ, ∀ i, t ≠ (τ i ω : ℝ) :=
-    ae_all_iff.mpr (fun i => volume.ae_ne (τ i ω : ℝ))
-  filter_upwards [hQd ω, hne] with t ht hne
-  intro ht0
-  apply hasDerivAt_pi.mpr
-  intro i
-  by_cases hti : t < (τ i ω : ℝ)
-  · have hd := hasDerivAt_pi.mp (ht ht0) i
-    have he : (fun u => stoppedCoordinateState Q τ u ω i) =ᶠ[𝓝 t] (fun u => Q u ω i) := by
-      filter_upwards [eventually_lt_nhds hti] with u hu
-      simp only [stoppedCoordinateState, min_eq_left hu.le]
-    simpa only [stoppedCoordinateVelocity, ite_eq_left hti] using hd.congr_of_eventuallyEq he
-  · have hit : (τ i ω : ℝ) < t := lt_of_le_of_ne (le_of_not_gt hti) (hne i).symm
-    have he : (fun u => stoppedCoordinateState Q τ u ω i) =ᶠ[𝓝 t]
-        (fun _ => Q (τ i ω : ℝ) ω i) := by
-      filter_upwards [eventually_gt_nhds hit] with u hu
-      simp only [stoppedCoordinateState, min_eq_right hu.le]
-    simpa only [stoppedCoordinateVelocity, ite_eq_right hti] using
-      (hasDerivAt_const t (Q (τ i ω : ℝ) ω i)).congr_of_eventuallyEq he
-
-lemma stoppedCoordinateState_adapted {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (ℱ : Filtration ℝ≥0 mΩ) (Q : ℝ → Ω → ι → ℂ × ℝ)
-    (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
-    (hQa : Adapted ℱ (fun t : ℝ≥0 => Q t))
-    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (hτ : ∀ i, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0))) :
-    Adapted ℱ (fun t : ℝ≥0 => stoppedCoordinateState Q τ t) := by
-  intro t
-  let : MeasurableSpace Ω := ℱ t
-  apply Measurable.of_eval
-  intro i
-  have hQi : StronglyAdapted ℱ (fun t : ℝ≥0 => fun ω => Q t ω i) := fun u =>
-    ((measurable_pi_apply i).comp (hQa u)).stronglyMeasurable
-  have hQc (ω : Ω) : Continuous (fun u : ℝ≥0 => Q u ω i) :=
-    ((continuous_apply i).comp (hQl ω).continuous).comp NNReal.continuous_coe
-  have hs := ((hQi.stoppedProcess hQc (hτ i)) t).measurable
-  convert hs using 1
-  funext ω
-  simp only [stoppedCoordinateState, MeasureTheory.stoppedProcess, ← WithTop.coe_min,
-    WithTop.untopA, WithTop.untopD_coe, NNReal.coe_min]
-
-@[simp] lemma stoppedCoordinateState_initial {Ω ι : Type*}
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (ω : Ω) :
-    stoppedCoordinateState Q τ 0 ω = Q 0 ω := by
-  funext i
-  simp only [stoppedCoordinateState, min_eq_left (τ i ω).coe_nonneg]
-
-lemma stoppedCoordinateState_frozen {Ω ι : Type*}
-    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
-    (t : ℝ) (ω : Ω) (i : ι) (ht : (τ i ω : ℝ) ≤ t) :
-    stoppedCoordinateState Q τ t ω i = Q (τ i ω : ℝ) ω i := by
-  rw [stoppedCoordinateState, min_eq_right ht]
-
-/-- Stop each coordinate separately while preserving all state hypotheses
-of the compact Brownian field adapter. -/
-theorem stoppedCoordinateState_regular {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
-    (ℱ : Filtration ℝ≥0 mΩ) (Q V : ℝ → Ω → ι → ℂ × ℝ)
-    (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted ℱ (fun t : ℝ≥0 => Q t))
-    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
-    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
-    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
-    (hτ : ∀ i, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0))) :
-    Measurable (fun p : ℝ × Ω => stoppedCoordinateState Q τ p.1 p.2) ∧
-    Adapted ℱ (fun t : ℝ≥0 => stoppedCoordinateState Q τ t) ∧
-    (∀ ω, LipschitzWith L (fun t => stoppedCoordinateState Q τ t ω)) ∧
-    Measurable (fun p : ℝ × Ω => stoppedCoordinateVelocity V τ p.1 p.2) ∧
-    (∀ t ω, ‖stoppedCoordinateVelocity V τ t ω‖ ≤ L) ∧
-    (∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => stoppedCoordinateState Q τ u ω)
-      (stoppedCoordinateVelocity V τ t ω) t) := by
-  have hτm (i : ι) : Measurable (τ i) := stopping_time_measurable_nnreal (hτ i)
-  exact ⟨stoppedCoordinateState_measurable Q τ hQm hτm,
-    stoppedCoordinateState_adapted ℱ Q τ L hQa hQl hτ,
-    stoppedCoordinateState_lipschitz Q τ L hQl,
-    stoppedCoordinateVelocity_measurable V τ hVm hτm,
-    stoppedCoordinateVelocity_bound V τ L hVb,
-    stoppedCoordinateState_ae_derivative Q V τ hQd⟩
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPiecewiseClock
+
+namespace OAI
+
+/-! Each completed multipoint coordinate is frozen at its own stopping
+time. The finite vector remains an adapted Lipschitz state, with the
+expected stopped velocity almost everywhere in time. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+def stoppedCoordinateState {Ω ι : Type*}
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (t : ℝ) (ω : Ω) (i : ι) : ℂ × ℝ :=
+  Q (min t (τ i ω : ℝ)) ω i
+
+def stoppedCoordinateVelocity {Ω ι : Type*}
+    (V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (t : ℝ) (ω : Ω) (i : ι) : ℂ × ℝ :=
+  if t < (τ i ω : ℝ) then V t ω i else 0
+
+lemma stoppedCoordinateState_measurable {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
+    (hQ : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hτ : ∀ i, Measurable (τ i)) :
+    Measurable (fun p : ℝ × Ω => stoppedCoordinateState Q τ p.1 p.2) := by
+  apply Measurable.of_eval
+  intro i
+  have hQi : Measurable (fun p : ℝ × Ω => Q p.1 p.2 i) :=
+    (measurable_pi_apply i).comp hQ
+  have htime : Measurable (fun p : ℝ × Ω => min p.1 (τ i p.2 : ℝ)) :=
+    measurable_fst.min (NNReal.continuous_coe.measurable.comp ((hτ i).comp measurable_snd))
+  exact hQi.comp (htime.prodMk measurable_snd)
+
+lemma stoppedCoordinateVelocity_measurable {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
+    (V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
+    (hV : Measurable (fun p : ℝ × Ω => V p.1 p.2))
+    (hτ : ∀ i, Measurable (τ i)) :
+    Measurable (fun p : ℝ × Ω => stoppedCoordinateVelocity V τ p.1 p.2) := by
+  apply Measurable.of_eval
+  intro i
+  apply Measurable.ite _ ((measurable_pi_apply i).comp hV) measurable_const
+  exact measurableSet_lt measurable_fst
+    (NNReal.continuous_coe.measurable.comp ((hτ i).comp measurable_snd))
+
+lemma stoppedCoordinateState_lipschitz {Ω ι : Type*} [Fintype ι]
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
+    (hQ : ∀ ω, LipschitzWith L (fun t => Q t ω)) (ω : Ω) :
+    LipschitzWith L (fun t => stoppedCoordinateState Q τ t ω) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro t u
+  apply (dist_pi_le_iff (mul_nonneg L.coe_nonneg dist_nonneg)).mpr
+  intro i
+  have hmin := (LipschitzWith.id.min_const (τ i ω : ℝ)).dist_le_mul t u
+  calc
+    dist (Q (min t (τ i ω : ℝ)) ω i) (Q (min u (τ i ω : ℝ)) ω i) ≤
+        dist (Q (min t (τ i ω : ℝ)) ω) (Q (min u (τ i ω : ℝ)) ω) :=
+      dist_le_pi_dist _ _ i
+    _ ≤ (L : ℝ) * dist (min t (τ i ω : ℝ)) (min u (τ i ω : ℝ)) :=
+      (hQ ω).dist_le_mul _ _
+    _ ≤ (L : ℝ) * dist t u := by
+      apply mul_le_mul_of_nonneg_left _ L.coe_nonneg
+      simpa only [NNReal.coe_one, one_mul, id_eq] using hmin
+
+lemma stoppedCoordinateVelocity_bound {Ω ι : Type*} [Fintype ι]
+    (V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
+    (hV : ∀ t ω, ‖V t ω‖ ≤ L) (t : ℝ) (ω : Ω) :
+    ‖stoppedCoordinateVelocity V τ t ω‖ ≤ L := by
+  apply (pi_norm_le_iff_of_nonneg L.coe_nonneg).mpr
+  intro i
+  unfold stoppedCoordinateVelocity
+  split_ifs
+  · exact (norm_le_pi_norm (V t ω) i).trans (hV t ω)
+  · simpa only [norm_zero] using L.coe_nonneg
+
+lemma stoppedCoordinateState_ae_derivative {Ω ι : Type*} [Fintype ι]
+    (Q V : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
+    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
+    (ω : Ω) : ∀ᵐ t : ℝ, 0 < t →
+      HasDerivAt (fun u => stoppedCoordinateState Q τ u ω)
+        (stoppedCoordinateVelocity V τ t ω) t := by
+  have hne : ∀ᵐ t : ℝ, ∀ i, t ≠ (τ i ω : ℝ) :=
+    ae_all_iff.mpr (fun i => volume.ae_ne (τ i ω : ℝ))
+  filter_upwards [hQd ω, hne] with t ht hne
+  intro ht0
+  apply hasDerivAt_pi.mpr
+  intro i
+  by_cases hti : t < (τ i ω : ℝ)
+  · have hd := hasDerivAt_pi.mp (ht ht0) i
+    have he : (fun u => stoppedCoordinateState Q τ u ω i) =ᶠ[𝓝 t] (fun u => Q u ω i) := by
+      filter_upwards [eventually_lt_nhds hti] with u hu
+      simp only [stoppedCoordinateState, min_eq_left hu.le]
+    simpa only [stoppedCoordinateVelocity, ite_eq_left hti] using hd.congr_of_eventuallyEq he
+  · have hit : (τ i ω : ℝ) < t := lt_of_le_of_ne (le_of_not_gt hti) (hne i).symm
+    have he : (fun u => stoppedCoordinateState Q τ u ω i) =ᶠ[𝓝 t]
+        (fun _ => Q (τ i ω : ℝ) ω i) := by
+      filter_upwards [eventually_gt_nhds hit] with u hu
+      simp only [stoppedCoordinateState, min_eq_right hu.le]
+    simpa only [stoppedCoordinateVelocity, ite_eq_right hti] using
+      (hasDerivAt_const t (Q (τ i ω : ℝ) ω i)).congr_of_eventuallyEq he
+
+lemma stoppedCoordinateState_adapted {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (ℱ : Filtration ℝ≥0 mΩ) (Q : ℝ → Ω → ι → ℂ × ℝ)
+    (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
+    (hQa : Adapted ℱ (fun t : ℝ≥0 => Q t))
+    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (hτ : ∀ i, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0))) :
+    Adapted ℱ (fun t : ℝ≥0 => stoppedCoordinateState Q τ t) := by
+  intro t
+  let : MeasurableSpace Ω := ℱ t
+  apply Measurable.of_eval
+  intro i
+  have hQi : StronglyAdapted ℱ (fun t : ℝ≥0 => fun ω => Q t ω i) := fun u =>
+    ((measurable_pi_apply i).comp (hQa u)).stronglyMeasurable
+  have hQc (ω : Ω) : Continuous (fun u : ℝ≥0 => Q u ω i) :=
+    ((continuous_apply i).comp (hQl ω).continuous).comp NNReal.continuous_coe
+  have hs := ((hQi.stoppedProcess hQc (hτ i)) t).measurable
+  convert hs using 1
+  funext ω
+  simp only [stoppedCoordinateState, MeasureTheory.stoppedProcess, ← WithTop.coe_min,
+    WithTop.untopA, WithTop.untopD_coe, NNReal.coe_min]
+
+@[simp] lemma stoppedCoordinateState_initial {Ω ι : Type*}
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0) (ω : Ω) :
+    stoppedCoordinateState Q τ 0 ω = Q 0 ω := by
+  funext i
+  simp only [stoppedCoordinateState, min_eq_left (τ i ω).coe_nonneg]
+
+lemma stoppedCoordinateState_frozen {Ω ι : Type*}
+    (Q : ℝ → Ω → ι → ℂ × ℝ) (τ : ι → Ω → ℝ≥0)
+    (t : ℝ) (ω : Ω) (i : ι) (ht : (τ i ω : ℝ) ≤ t) :
+    stoppedCoordinateState Q τ t ω i = Q (τ i ω : ℝ) ω i := by
+  rw [stoppedCoordinateState, min_eq_right ht]
+
+/-- Stop each coordinate separately while preserving all state hypotheses
+of the compact Brownian field adapter. -/
+theorem stoppedCoordinateState_regular {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
+    (ℱ : Filtration ℝ≥0 mΩ) (Q V : ℝ → Ω → ι → ℂ × ℝ)
+    (τ : ι → Ω → ℝ≥0) (L : ℝ≥0)
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted ℱ (fun t : ℝ≥0 => Q t))
+    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
+    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
+    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => Q u ω) (V t ω) t)
+    (hτ : ∀ i, IsStoppingTime ℱ (fun ω => (τ i ω : WithTop ℝ≥0))) :
+    Measurable (fun p : ℝ × Ω => stoppedCoordinateState Q τ p.1 p.2) ∧
+    Adapted ℱ (fun t : ℝ≥0 => stoppedCoordinateState Q τ t) ∧
+    (∀ ω, LipschitzWith L (fun t => stoppedCoordinateState Q τ t ω)) ∧
+    Measurable (fun p : ℝ × Ω => stoppedCoordinateVelocity V τ p.1 p.2) ∧
+    (∀ t ω, ‖stoppedCoordinateVelocity V τ t ω‖ ≤ L) ∧
+    (∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun u => stoppedCoordinateState Q τ u ω)
+      (stoppedCoordinateVelocity V τ t ω) t) := by
+  have hτm (i : ι) : Measurable (τ i) := stopping_time_measurable_nnreal (hτ i)
+  exact ⟨stoppedCoordinateState_measurable Q τ hQm hτm,
+    stoppedCoordinateState_adapted ℱ Q τ L hQa hQl hτ,
+    stoppedCoordinateState_lipschitz Q τ L hQl,
+    stoppedCoordinateVelocity_measurable V τ hVm hτm,
+    stoppedCoordinateVelocity_bound V τ L hVb,
+    stoppedCoordinateState_ae_derivative Q V τ hQd⟩
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedProduct.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedProduct.lean
index 7b9807b9ef7c1d45dbea270850c60f0287039a8c..5fefb1abf718d8f2e412c0abde2fb79b59b1fb30 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedProduct.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointStoppedProduct.lean
@@ -1,91 +1,91 @@
-import OAI.Probability.SLE.Multipoint.MultipointDiscountProcess
-import OAI.Probability.SLE.Stochastic.BrownianOptionalIntervals
-
-namespace OAI
-
-/-! The actual discounted regularized product has nonincreasing conditional
-expectation on each localized interval of constant active pattern. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal
-
-namespace SLEExactGauge.Multipoint
-
-theorem discounted_product_optional_interval_le
-    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (Q V : ℝ → Ω → (ι → ℂ × ℝ)) (r : ℝ≥0 → Ω → ℝ)
-    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
-    (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ) (K : ℝ≥0)
-    (hF : SmoothTimeBrownianField B hm K
-      (fun t ω x => φ ((Q t ω,discountClock r t ω),x))
-      (fun t ω x => fderiv ℝ φ ((Q t ω,discountClock r t ω),x)
-        ((V t ω,r t ω),0)))
-    (R : ℝ)
-    (he : ∀ z x, ‖(z,x)‖ ≤ R → φ (z,x) = discountedProduct s κ D σ z x)
-    (heD : ∀ z x, ‖(z,x)‖ ≤ R → fderiv ℝ φ (z,x) =
-      fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x))
-    (he2 : ∀ z x, ‖(z,x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (z,y)) x =
-      iteratedDeriv 2 (discountedProduct s κ D σ z) x)
-    (a b : Ω → ℝ≥0)
-    (ha : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (a ω : WithTop ℝ≥0)))
-    (hb : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (b ω : WithTop ℝ≥0)))
-    (hab : ∀ ω, a ω ≤ b ω) (T : ℝ≥0) (hbT : ∀ ω, b ω ≤ T)
-    {A : Set Ω} (hA : MeasurableSet[ha.measurableSpace] A)
-    (hnorm : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u ≤ (b ω : ℝ) →
-      ‖((Q u ω,discountClock r u ω),B u.toNNReal ω)‖ ≤ R)
-    (hδ : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u < (b ω : ℝ) →
-      ∀ i ∈ s, branchRegulator κ (D i) (σ i) (Q u ω i) ≤ 1)
-    (hV : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u < (b ω : ℝ) →
-      ∀ i ∈ s, V u ω i = physicalLogLoewnerField
-        (Real.sqrt κ*B u.toNNReal ω) (Q u ω i))
-    (hr : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u < (b ω : ℝ) →
-      r u.toNNReal ω = productDriftCost s κ D σ (Q u ω) (B u.toNNReal ω)) :
-    IntegrableOn (fun ω => discountedProduct s κ D σ
-      (Q (b ω) ω,discountClock r (b ω) ω) (B (b ω) ω)) A P ∧
-    IntegrableOn (fun ω => discountedProduct s κ D σ
-      (Q (a ω) ω,discountClock r (a ω) ω) (B (a ω) ω)) A P ∧
-    (∫ ω in A, discountedProduct s κ D σ
-      (Q (b ω) ω,discountClock r (b ω) ω) (B (b ω) ω) ∂P) ≤
-      ∫ ω in A, discountedProduct s κ D σ
-        (Q (a ω) ω,discountClock r (a ω) ω) (B (a ω) ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hAm : MeasurableSet A := ha.measurableSpace_le _ hA
-  have habe (ω : Ω) : (a ω : ℝ) ≤ (b ω : ℝ) := by exact_mod_cast hab ω
-  have hea : (fun ω => φ ((Q (a ω) ω,discountClock r (a ω) ω),B (a ω) ω))
-      =ᵐ[P.restrict A] (fun ω => discountedProduct s κ D σ
-        (Q (a ω) ω,discountClock r (a ω) ω) (B (a ω) ω)) := by
-    filter_upwards [ae_restrict_mem hAm] with ω hω
-    apply he
-    simpa only [Real.toNNReal_coe] using hnorm ω hω (a ω) le_rfl (habe ω)
-  have heb : (fun ω => φ ((Q (b ω) ω,discountClock r (b ω) ω),B (b ω) ω))
-      =ᵐ[P.restrict A] (fun ω => discountedProduct s κ D σ
-        (Q (b ω) ω,discountClock r (b ω) ω) (B (b ω) ω)) := by
-    filter_upwards [ae_restrict_mem hAm] with ω hω
-    apply he
-    simpa only [Real.toNNReal_coe] using hnorm ω hω (b ω) (habe ω) le_rfl
-  have hIb := hF.eval_stopped_integrable (P := P.restrict A) hBm
-    (stopping_time_measurable_nnreal hb)
-  have hIa := hF.eval_stopped_integrable (P := P.restrict A) hBm
-    (stopping_time_measurable_nnreal ha)
-  refine ⟨hIb.congr heb, hIa.congr hea, ?_⟩
-  rw [← integral_congr_ae heb, ← integral_congr_ae hea]
-  apply brownian_time_field_optional_interval_le hB hm hBm hBc hF a b ha hb hab T hbT hA
-  intro ω hω u hau hub
-  exact compact_discounted_process_drift_nonpos B Q V r s κ hκ hκ8 D σ φ u
-    (le_trans (a ω).coe_nonneg hau) ω hD hσ (hδ ω hω u hau hub)
-    (hV ω hω u hau hub) (hr ω hω u hau hub)
-    (heD _ _ (hnorm ω hω u hau hub.le)) (he2 _ _ (hnorm ω hω u hau hub.le))
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointDiscountProcess
+import OAI.Probability.SLE.Stochastic.BrownianOptionalIntervals
+
+namespace OAI
+
+/-! The actual discounted regularized product has nonincreasing conditional
+expectation on each localized interval of constant active pattern. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal
+
+namespace SLEExactGauge.Multipoint
+
+theorem discounted_product_optional_interval_le
+    {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] [DecidableEq ι]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (Q V : ℝ → Ω → (ι → ℂ × ℝ)) (r : ℝ≥0 → Ω → ℝ)
+    (s : Finset ι) (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (hσ : ∀ i ∈ s, σ i ≤ 1)
+    (φ : ((ι → ℂ × ℝ) × ℝ) × ℝ → ℝ) (K : ℝ≥0)
+    (hF : SmoothTimeBrownianField B hm K
+      (fun t ω x => φ ((Q t ω,discountClock r t ω),x))
+      (fun t ω x => fderiv ℝ φ ((Q t ω,discountClock r t ω),x)
+        ((V t ω,r t ω),0)))
+    (R : ℝ)
+    (he : ∀ z x, ‖(z,x)‖ ≤ R → φ (z,x) = discountedProduct s κ D σ z x)
+    (heD : ∀ z x, ‖(z,x)‖ ≤ R → fderiv ℝ φ (z,x) =
+      fderiv ℝ (fun p => discountedProduct s κ D σ p.1 p.2) (z,x))
+    (he2 : ∀ z x, ‖(z,x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (z,y)) x =
+      iteratedDeriv 2 (discountedProduct s κ D σ z) x)
+    (a b : Ω → ℝ≥0)
+    (ha : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (a ω : WithTop ℝ≥0)))
+    (hb : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (b ω : WithTop ℝ≥0)))
+    (hab : ∀ ω, a ω ≤ b ω) (T : ℝ≥0) (hbT : ∀ ω, b ω ≤ T)
+    {A : Set Ω} (hA : MeasurableSet[ha.measurableSpace] A)
+    (hnorm : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u ≤ (b ω : ℝ) →
+      ‖((Q u ω,discountClock r u ω),B u.toNNReal ω)‖ ≤ R)
+    (hδ : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u < (b ω : ℝ) →
+      ∀ i ∈ s, branchRegulator κ (D i) (σ i) (Q u ω i) ≤ 1)
+    (hV : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u < (b ω : ℝ) →
+      ∀ i ∈ s, V u ω i = physicalLogLoewnerField
+        (Real.sqrt κ*B u.toNNReal ω) (Q u ω i))
+    (hr : ∀ ω ∈ A, ∀ u, (a ω : ℝ) ≤ u → u < (b ω : ℝ) →
+      r u.toNNReal ω = productDriftCost s κ D σ (Q u ω) (B u.toNNReal ω)) :
+    IntegrableOn (fun ω => discountedProduct s κ D σ
+      (Q (b ω) ω,discountClock r (b ω) ω) (B (b ω) ω)) A P ∧
+    IntegrableOn (fun ω => discountedProduct s κ D σ
+      (Q (a ω) ω,discountClock r (a ω) ω) (B (a ω) ω)) A P ∧
+    (∫ ω in A, discountedProduct s κ D σ
+      (Q (b ω) ω,discountClock r (b ω) ω) (B (b ω) ω) ∂P) ≤
+      ∫ ω in A, discountedProduct s κ D σ
+        (Q (a ω) ω,discountClock r (a ω) ω) (B (a ω) ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hAm : MeasurableSet A := ha.measurableSpace_le _ hA
+  have habe (ω : Ω) : (a ω : ℝ) ≤ (b ω : ℝ) := by exact_mod_cast hab ω
+  have hea : (fun ω => φ ((Q (a ω) ω,discountClock r (a ω) ω),B (a ω) ω))
+      =ᵐ[P.restrict A] (fun ω => discountedProduct s κ D σ
+        (Q (a ω) ω,discountClock r (a ω) ω) (B (a ω) ω)) := by
+    filter_upwards [ae_restrict_mem hAm] with ω hω
+    apply he
+    simpa only [Real.toNNReal_coe] using hnorm ω hω (a ω) le_rfl (habe ω)
+  have heb : (fun ω => φ ((Q (b ω) ω,discountClock r (b ω) ω),B (b ω) ω))
+      =ᵐ[P.restrict A] (fun ω => discountedProduct s κ D σ
+        (Q (b ω) ω,discountClock r (b ω) ω) (B (b ω) ω)) := by
+    filter_upwards [ae_restrict_mem hAm] with ω hω
+    apply he
+    simpa only [Real.toNNReal_coe] using hnorm ω hω (b ω) (habe ω) le_rfl
+  have hIb := hF.eval_stopped_integrable (P := P.restrict A) hBm
+    (stopping_time_measurable_nnreal hb)
+  have hIa := hF.eval_stopped_integrable (P := P.restrict A) hBm
+    (stopping_time_measurable_nnreal ha)
+  refine ⟨hIb.congr heb, hIa.congr hea, ?_⟩
+  rw [← integral_congr_ae heb, ← integral_congr_ae hea]
+  apply brownian_time_field_optional_interval_le hB hm hBm hBc hF a b ha hb hab T hbT hA
+  intro ω hω u hau hub
+  exact compact_discounted_process_drift_nonpos B Q V r s κ hκ hκ8 D σ φ u
+    (le_trans (a ω).coe_nonneg hau) ω hD hσ (hδ ω hω u hau hub)
+    (hV ω hω u hau hub) (hr ω hω u hau hub)
+    (heD _ _ (hnorm ω hω u hau hub.le)) (he2 _ _ (hnorm ω hω u hau hub.le))
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointSuccess.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointSuccess.lean
index f4cd903721ea0b06c76af68056f24722c137d568..590c356519cb7f2d356795abb5065e5a47427be1 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointSuccess.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointSuccess.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Multipoint.MultipointCutoff
-
-namespace OAI
-
-/-! The terminal probability step for the multipoint product. Analytic
-estimates enter through the expected stopped product and its deterministic
-lower bound on success; completed points have already been frozen. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-lemma measure_le_of_eventual_stopped_products
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (E : Set Ω) (cutoff : ℕ → Set Ω) (X : ℕ → Ω → ℝ≥0∞)
-    (q : ℝ) (hq : 0 < q)
-    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
-    (hm : ∀ n, AEMeasurable (X n) P)
-    (hmean : ∀ n, (∫⁻ ω, X n ω ∂P) ≤ 1)
-    (hterminal : ∀ n ω, ω ∈ cutoff n → ENNReal.ofReal q ≤ X n ω) :
-    P E ≤ ENNReal.ofReal q⁻¹ := by
-  apply measure_le_of_eventual_cutoffs P E cutoff _ hcover
-  intro n
-  have hsub : cutoff n ⊆ {ω | ENNReal.ofReal q ≤ X n ω} := hterminal n
-  have hb := (measure_mono hsub).trans
-    (meas_ge_le_lintegral_div (hm n) (ne_of_gt (ENNReal.ofReal_pos.mpr hq)) ENNReal.ofReal_ne_top)
-  have hle : (∫⁻ ω, X n ω ∂P) / ENNReal.ofReal q ≤ 1 / ENNReal.ofReal q := by
-    exact mul_le_mul' (hmean n) le_rfl
-  apply hb.trans
-  simpa only [one_div, ENNReal.ofReal_inv_of_pos hq] using hle
-
-def successBarrier (p K : ℝ) {ι : Type*} (s : Finset ι) (U : ι → ℝ) : ℝ :=
-  Real.exp (-K) * (3 : ℝ)^(-(s.card : ℤ)) * Real.exp (p/2 * ∑ i ∈ s, U i)
-
-lemma successBarrier_pos (p K : ℝ) {ι : Type*} (s : Finset ι) (U : ι → ℝ) :
-    0 < successBarrier p K s U := by unfold successBarrier; positivity
-
-lemma successBarrier_inv (p K : ℝ) {ι : Type*} (s : Finset ι) (U : ι → ℝ) :
-    (successBarrier p K s U)⁻¹ =
-      (3 : ℝ)^s.card * Real.exp K * Real.exp (-p/2 * ∑ i ∈ s, U i) := by
-  unfold successBarrier
-  rw [mul_inv_rev, mul_inv_rev, ← Real.exp_neg, ← Real.exp_neg]
-  simp only [neg_neg, zpow_neg, inv_inv, zpow_natCast]
-  rw [show -(p/2 * ∑ i ∈ s, U i) = -p/2 * ∑ i ∈ s, U i by ring]
-  ring
-
-lemma radius_target_exponential {a p b e : ℝ} (ha : 0 < a)
-    (hb : 0 < b) (he : 0 < e) :
-    Real.exp (-p/2 * (1/(2*a)*Real.log (b/e))) = (e/b)^(p/(4*a)) := by
-  rw [Real.rpow_def_of_pos (div_pos he hb), Real.log_div he.ne' hb.ne',
-    Real.log_div hb.ne' he.ne']
-  congr 1
-  field_simp
-  ring
-
-lemma radius_targets_exponential {ι : Type*} (s : Finset ι)
-    {a p : ℝ} (ha : 0 < a) (b e : ι → ℝ)
-    (hb : ∀ i ∈ s, 0 < b i) (he : ∀ i ∈ s, 0 < e i) :
-    Real.exp (-p/2 * ∑ i ∈ s, (1/(2*a)*Real.log (b i/e i))) =
-      ∏ i ∈ s, (e i/b i)^(p/(4*a)) := by
-  rw [Finset.mul_sum, Real.exp_sum]
-  exact Finset.prod_congr rfl (fun i hi => radius_target_exponential ha (hb i hi) (he i hi))
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointCutoff
+
+namespace OAI
+
+/-! The terminal probability step for the multipoint product. Analytic
+estimates enter through the expected stopped product and its deterministic
+lower bound on success; completed points have already been frozen. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+lemma measure_le_of_eventual_stopped_products
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (E : Set Ω) (cutoff : ℕ → Set Ω) (X : ℕ → Ω → ℝ≥0∞)
+    (q : ℝ) (hq : 0 < q)
+    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
+    (hm : ∀ n, AEMeasurable (X n) P)
+    (hmean : ∀ n, (∫⁻ ω, X n ω ∂P) ≤ 1)
+    (hterminal : ∀ n ω, ω ∈ cutoff n → ENNReal.ofReal q ≤ X n ω) :
+    P E ≤ ENNReal.ofReal q⁻¹ := by
+  apply measure_le_of_eventual_cutoffs P E cutoff _ hcover
+  intro n
+  have hsub : cutoff n ⊆ {ω | ENNReal.ofReal q ≤ X n ω} := hterminal n
+  have hb := (measure_mono hsub).trans
+    (meas_ge_le_lintegral_div (hm n) (ne_of_gt (ENNReal.ofReal_pos.mpr hq)) ENNReal.ofReal_ne_top)
+  have hle : (∫⁻ ω, X n ω ∂P) / ENNReal.ofReal q ≤ 1 / ENNReal.ofReal q := by
+    exact mul_le_mul' (hmean n) le_rfl
+  apply hb.trans
+  simpa only [one_div, ENNReal.ofReal_inv_of_pos hq] using hle
+
+def successBarrier (p K : ℝ) {ι : Type*} (s : Finset ι) (U : ι → ℝ) : ℝ :=
+  Real.exp (-K) * (3 : ℝ)^(-(s.card : ℤ)) * Real.exp (p/2 * ∑ i ∈ s, U i)
+
+lemma successBarrier_pos (p K : ℝ) {ι : Type*} (s : Finset ι) (U : ι → ℝ) :
+    0 < successBarrier p K s U := by unfold successBarrier; positivity
+
+lemma successBarrier_inv (p K : ℝ) {ι : Type*} (s : Finset ι) (U : ι → ℝ) :
+    (successBarrier p K s U)⁻¹ =
+      (3 : ℝ)^s.card * Real.exp K * Real.exp (-p/2 * ∑ i ∈ s, U i) := by
+  unfold successBarrier
+  rw [mul_inv_rev, mul_inv_rev, ← Real.exp_neg, ← Real.exp_neg]
+  simp only [neg_neg, zpow_neg, inv_inv, zpow_natCast]
+  rw [show -(p/2 * ∑ i ∈ s, U i) = -p/2 * ∑ i ∈ s, U i by ring]
+  ring
+
+lemma radius_target_exponential {a p b e : ℝ} (ha : 0 < a)
+    (hb : 0 < b) (he : 0 < e) :
+    Real.exp (-p/2 * (1/(2*a)*Real.log (b/e))) = (e/b)^(p/(4*a)) := by
+  rw [Real.rpow_def_of_pos (div_pos he hb), Real.log_div he.ne' hb.ne',
+    Real.log_div hb.ne' he.ne']
+  congr 1
+  field_simp
+  ring
+
+lemma radius_targets_exponential {ι : Type*} (s : Finset ι)
+    {a p : ℝ} (ha : 0 < a) (b e : ι → ℝ)
+    (hb : ∀ i ∈ s, 0 < b i) (he : ∀ i ∈ s, 0 < e i) :
+    Real.exp (-p/2 * ∑ i ∈ s, (1/(2*a)*Real.log (b i/e i))) =
+      ∏ i ∈ s, (e i/b i)^(p/(4*a)) := by
+  rw [Finset.mul_sum, Real.exp_sum]
+  exact Finset.prod_congr rfl (fun i hi => radius_target_exponential ha (hb i hi) (he i hi))
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointSwitchPatterns.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointSwitchPatterns.lean
index b588451e9b5ea7adbfa0a336620d46fc1d1553ed..604aec83b5e41862c4eddf178e6f1dafb6955d67 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointSwitchPatterns.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointSwitchPatterns.lean
@@ -1,83 +1,83 @@
-import OAI.Probability.SLE.Multipoint.MultipointSwitches
-
-namespace OAI
-
-/-! The active/completed/midpoint pattern at a switch is a finite,
-stopped-past measurable partition. This supplies the actual branch events
-for the finite-field conditional optional inequality. -/
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory
-open scoped NNReal BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-def switchPatternEvent {Ω ι : Type*} (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (A : Finset ι) : Set Ω :=
-  {ω | ∀ i ∈ s, ρ i ω ≤ τ ω ↔ i ∈ A}
-
-theorem switchPatternEvent_measurable {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 mΩ) (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0)
-    (hρ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (ρ i ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) (A : Finset ι) :
-    MeasurableSet[hτ.measurableSpace] (switchPatternEvent s ρ τ A) := by
-  classical
-  have he : switchPatternEvent s ρ τ A = ⋂ i ∈ s,
-      if i ∈ A then {ω | ρ i ω ≤ τ ω} else {ω | ρ i ω ≤ τ ω}ᶜ := by
-    ext ω
-    simp only [switchPatternEvent, mem_ofPred_eq, mem_iInter]
-    apply forall_congr'
-    intro i
-    apply forall_congr'
-    intro _
-    by_cases hi : i ∈ A <;> simp [hi]
-  rw [he]
-  apply s.measurableSet_biInter
-  intro i hi
-  have hm : MeasurableSet[hτ.measurableSpace] {ω | ρ i ω ≤ τ ω} := by
-    simpa only [WithTop.coe_le_coe] using (hρ i hi).measurableSet_stopping_time_le hτ
-  split_ifs
-  · exact hm
-  · exact hm.compl
-
-theorem switchPatternEvent_unique {Ω ι : Type*} [DecidableEq ι] (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) {A B : Finset ι}
-    (hA : A ⊆ s) (hB : B ⊆ s) {ω : Ω}
-    (hωA : ω ∈ switchPatternEvent s ρ τ A)
-    (hωB : ω ∈ switchPatternEvent s ρ τ B) : A = B := by
-  apply Finset.ext
-  intro i
-  by_cases hi : i ∈ s
-  · exact (hωA i hi).symm.trans (hωB i hi)
-  · exact iff_of_false (fun h => hi (hA h)) (fun h => hi (hB h))
-
-theorem switchPatternEvent_cover {Ω ι : Type*} [DecidableEq ι] (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (ω : Ω) :
-    ∃ A ∈ s.powerset, ω ∈ switchPatternEvent s ρ τ A := by
-  refine ⟨s.filter (fun i => ρ i ω ≤ τ ω), Finset.mem_powerset.mpr (Finset.filter_subset _ _), ?_⟩
-  intro i hi
-  simp only [Finset.mem_filter, hi, true_and]
-
-theorem sum_switchPattern_fields {Ω ι : Type*} [DecidableEq ι] (s : Finset ι)
-    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (F : Finset ι → Ω → ℝ)
-    (X : Ω → ℝ)
-    (hF : ∀ A ∈ s.powerset, ∀ ω ∈ switchPatternEvent s ρ τ A, F A ω = X ω) :
-    (fun ω => ∑ A ∈ s.powerset, (switchPatternEvent s ρ τ A).indicator (F A) ω) = X := by
-  classical
-  funext ω
-  obtain ⟨A, hA, hωA⟩ := switchPatternEvent_cover s ρ τ ω
-  rw [Finset.sum_eq_single A]
-  · rw [indicator_of_mem hωA, hF A hA ω hωA]
-  · intro B hB hBA
-    apply indicator_of_notMem
-    intro hωB
-    exact hBA (switchPatternEvent_unique s ρ τ (Finset.mem_powerset.mp hB)
-      (Finset.mem_powerset.mp hA) hωB hωA)
-  · exact fun h => False.elim (h hA)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointSwitches
+
+namespace OAI
+
+/-! The active/completed/midpoint pattern at a switch is a finite,
+stopped-past measurable partition. This supplies the actual branch events
+for the finite-field conditional optional inequality. -/
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory
+open scoped NNReal BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+def switchPatternEvent {Ω ι : Type*} (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (A : Finset ι) : Set Ω :=
+  {ω | ∀ i ∈ s, ρ i ω ≤ τ ω ↔ i ∈ A}
+
+theorem switchPatternEvent_measurable {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 mΩ) (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0)
+    (hρ : ∀ i ∈ s, IsStoppingTime ℱ (fun ω => (ρ i ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) (A : Finset ι) :
+    MeasurableSet[hτ.measurableSpace] (switchPatternEvent s ρ τ A) := by
+  classical
+  have he : switchPatternEvent s ρ τ A = ⋂ i ∈ s,
+      if i ∈ A then {ω | ρ i ω ≤ τ ω} else {ω | ρ i ω ≤ τ ω}ᶜ := by
+    ext ω
+    simp only [switchPatternEvent, mem_ofPred_eq, mem_iInter]
+    apply forall_congr'
+    intro i
+    apply forall_congr'
+    intro _
+    by_cases hi : i ∈ A <;> simp [hi]
+  rw [he]
+  apply s.measurableSet_biInter
+  intro i hi
+  have hm : MeasurableSet[hτ.measurableSpace] {ω | ρ i ω ≤ τ ω} := by
+    simpa only [WithTop.coe_le_coe] using (hρ i hi).measurableSet_stopping_time_le hτ
+  split_ifs
+  · exact hm
+  · exact hm.compl
+
+theorem switchPatternEvent_unique {Ω ι : Type*} [DecidableEq ι] (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) {A B : Finset ι}
+    (hA : A ⊆ s) (hB : B ⊆ s) {ω : Ω}
+    (hωA : ω ∈ switchPatternEvent s ρ τ A)
+    (hωB : ω ∈ switchPatternEvent s ρ τ B) : A = B := by
+  apply Finset.ext
+  intro i
+  by_cases hi : i ∈ s
+  · exact (hωA i hi).symm.trans (hωB i hi)
+  · exact iff_of_false (fun h => hi (hA h)) (fun h => hi (hB h))
+
+theorem switchPatternEvent_cover {Ω ι : Type*} [DecidableEq ι] (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (ω : Ω) :
+    ∃ A ∈ s.powerset, ω ∈ switchPatternEvent s ρ τ A := by
+  refine ⟨s.filter (fun i => ρ i ω ≤ τ ω), Finset.mem_powerset.mpr (Finset.filter_subset _ _), ?_⟩
+  intro i hi
+  simp only [Finset.mem_filter, hi, true_and]
+
+theorem sum_switchPattern_fields {Ω ι : Type*} [DecidableEq ι] (s : Finset ι)
+    (ρ : ι → Ω → ℝ≥0) (τ : Ω → ℝ≥0) (F : Finset ι → Ω → ℝ)
+    (X : Ω → ℝ)
+    (hF : ∀ A ∈ s.powerset, ∀ ω ∈ switchPatternEvent s ρ τ A, F A ω = X ω) :
+    (fun ω => ∑ A ∈ s.powerset, (switchPatternEvent s ρ τ A).indicator (F A) ω) = X := by
+  classical
+  funext ω
+  obtain ⟨A, hA, hωA⟩ := switchPatternEvent_cover s ρ τ ω
+  rw [Finset.sum_eq_single A]
+  · rw [indicator_of_mem hωA, hF A hA ω hωA]
+  · intro B hB hBA
+    apply indicator_of_notMem
+    intro hωB
+    exact hBA (switchPatternEvent_unique s ρ τ (Finset.mem_powerset.mp hB)
+      (Finset.mem_powerset.mp hA) hωB hωA)
+  · exact fun h => False.elim (h hA)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointSwitches.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointSwitches.lean
index f1515aff858d4f7bff565eb83eb0a43299f607e2..06961d3b1e5895d6fdc6926e9f438e85e62966fe 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointSwitches.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointSwitches.lean
@@ -1,168 +1,168 @@
-import OAI.Probability.SLE.Stochastic.BrownianOptionalIntervals
-import OAI.Probability.SLE.Multipoint.MultipointAngular
-
-namespace OAI
-
-/-! Finite switching for the regularized multipoint product. Smooth
-pieces may change at bounded stopping times, provided the jumps are
-downward. A completed factor can therefore remain constant on all later
-pieces without any condition on its subsequent lifetime. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal BigOperators
-
-namespace SLEExactGauge
-
-/-- At a switching time the active/completed pattern has only finitely
-many possibilities. Apply the conditional interval estimate on each
-past-measurable branch and sum. The branch formula also covers several
-simultaneous switches. -/
-theorem brownian_finite_branch_optional_interval_le
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (σ τ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hστ : ∀ ω, σ ω ≤ τ ω) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    {ι : Type*} (s : Finset ι) (A : ι → Set Ω)
-    (hA : ∀ i ∈ s, MeasurableSet[hσ.measurableSpace] (A i))
-    (K : ι → ℝ≥0) (F a : ι → ℝ≥0 → Ω → ℝ → ℝ)
-    (hF : ∀ i ∈ s, SmoothTimeBrownianField B hm (K i) (F i) (a i))
-    (hdrift : ∀ i ∈ s, ∀ ω ∈ A i, ∀ u,
-      (σ ω : ℝ) ≤ u → u < (τ ω : ℝ) → brownianFieldDrift B (F i) (a i) u ω ≤ 0) :
-    (∫ ω, ∑ i ∈ s, (A i).indicator
-      (fun ω => F i (τ ω) ω (B (τ ω) ω)) ω ∂P) ≤
-    ∫ ω, ∑ i ∈ s, (A i).indicator
-      (fun ω => F i (σ ω) ω (B (σ ω) ω)) ω ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hAi (i : ι) (hi : i ∈ s) : MeasurableSet (A i) :=
-    hσ.measurableSpace_le _ (hA i hi)
-  have hleft (i : ι) (hi : i ∈ s) : Integrable ((A i).indicator
-      (fun ω => F i (τ ω) ω (B (τ ω) ω))) P :=
-    ((hF i hi).eval_stopped_integrable hBm (stopping_time_measurable_nnreal hτ)).indicator
-      (hAi i hi)
-  have hright (i : ι) (hi : i ∈ s) : Integrable ((A i).indicator
-      (fun ω => F i (σ ω) ω (B (σ ω) ω))) P :=
-    ((hF i hi).eval_stopped_integrable hBm (stopping_time_measurable_nnreal hσ)).indicator
-      (hAi i hi)
-  rw [integral_finsetSum _ hleft, integral_finsetSum _ hright]
-  apply Finset.sum_le_sum
-  intro i hi
-  rw [integral_indicator (hAi i hi), integral_indicator (hAi i hi)]
-  exact brownian_time_field_optional_interval_le hB hm hBm hBc (hF i hi)
-    σ τ hσ hτ hστ T hb (hA i hi) (hdrift i hi)
-
-/-- Patching finitely many bounded stopping intervals preserves the
-conditional expectation bound. Each field is required to have
-nonpositive drift only on its own interval. -/
-theorem brownian_finite_switch_optional_le
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (N : ℕ) (τ : ℕ → Ω → ℝ≥0)
-    (hτ : ∀ n, n ≤ N → IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ n ω : WithTop ℝ≥0)))
-    (hmono : ∀ n, n < N → ∀ ω, τ n ω ≤ τ (n+1) ω)
-    (T : ℝ≥0) (hbound : ∀ n, n ≤ N → ∀ ω, τ n ω ≤ T)
-    (K : ℕ → ℝ≥0) (F a : ℕ → ℝ≥0 → Ω → ℝ → ℝ)
-    (hF : ∀ n, n ≤ N → SmoothTimeBrownianField B hm (K n) (F n) (a n))
-    {A : Set Ω} (hA : MeasurableSet[(hτ 0 (Nat.zero_le N)).measurableSpace] A)
-    (hdrift : ∀ n, n < N → ∀ ω ∈ A, ∀ u,
-      (τ n ω : ℝ) ≤ u → u < (τ (n+1) ω : ℝ) →
-        brownianFieldDrift B (F n) (a n) u ω ≤ 0)
-    (hjump : ∀ n, n < N → ∀ ω ∈ A,
-      F (n+1) (τ (n+1) ω) ω (B (τ (n+1) ω) ω) ≤
-        F n (τ (n+1) ω) ω (B (τ (n+1) ω) ω)) :
-    (∫ ω in A, F N (τ N ω) ω (B (τ N ω) ω) ∂P) ≤
-      ∫ ω in A, F 0 (τ 0 ω) ω (B (τ 0 ω) ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hstart : ∀ n, n ≤ N → ∀ ω, τ 0 ω ≤ τ n ω := by
-    intro n
-    induction n with
-    | zero => intro _ _; exact le_rfl
-    | succ n ih =>
-      intro hn ω
-      exact (ih (Nat.le_trans (Nat.le_succ n) hn) ω).trans
-        (hmono n (Nat.lt_of_succ_le hn) ω)
-  have hAm (n : ℕ) (hn : n ≤ N) :
-      MeasurableSet[(hτ n hn).measurableSpace] A := by
-    apply IsStoppingTime.measurableSpace_mono (hτ 0 (Nat.zero_le N)) (hτ n hn)
-      (fun ω => WithTop.coe_le_coe.mpr (hstart n hn ω))
-    exact hA
-  have hind : ∀ n, n ≤ N →
-      (∫ ω in A, F n (τ n ω) ω (B (τ n ω) ω) ∂P) ≤
-        ∫ ω in A, F 0 (τ 0 ω) ω (B (τ 0 ω) ω) ∂P := by
-    intro n
-    induction n with
-    | zero => intro _; exact le_rfl
-    | succ n ih =>
-      intro hn
-      have hn' := Nat.le_trans (Nat.le_succ n) hn
-      have hnN := Nat.lt_of_succ_le hn
-      have hflow := brownian_time_field_optional_interval_le hB hm hBm hBc
-        (hF n hn') (τ n) (τ (n+1)) (hτ n hn') (hτ (n+1) hn)
-        (hmono n hnN) T (hbound (n+1) hn) (hAm n hn') (hdrift n hnN)
-      have hright := (hF n hn').eval_stopped_integrable hBm
-        (stopping_time_measurable_nnreal (hτ (n+1) hn)) (P := P.restrict A)
-      have hleft := (hF (n+1) hn).eval_stopped_integrable hBm
-        (stopping_time_measurable_nnreal (hτ (n+1) hn)) (P := P.restrict A)
-      have hswitch : (∫ ω in A, F (n+1) (τ (n+1) ω) ω (B (τ (n+1) ω) ω) ∂P) ≤
-          ∫ ω in A, F n (τ (n+1) ω) ω (B (τ (n+1) ω) ω) ∂P := by
-        apply integral_mono_ae hleft hright
-        filter_upwards [ae_restrict_mem ((hτ 0 (Nat.zero_le N)).measurableSpace_le _ hA)]
-          with ω hω
-        exact hjump n hnN ω hω
-      exact hswitch.trans (hflow.trans (ih hn'))
-  exact hind N le_rfl
-
-namespace Multipoint
-
-def activeFactor (p β U u θ : ℝ) : ℝ :=
-  Real.exp (p / 2 * u) / 3 * angularFactor p (regulator β U u) θ
-
-lemma activeFactor_activation_le {p β U θ : ℝ}
-    (hp : 0 < p) (hU : 0 ≤ U) (hθ : 0 ≤ Real.sin θ) :
-    activeFactor p β U 0 θ ≤ 1 := by
-  unfold activeFactor
-  rw [regulator_zero β U hU]
-  have h := (angularFactor_endpoint_bounds hp hθ).2
-  simp only [mul_zero, Real.exp_zero]
-  linarith
-
-/-- Completion freezes the factor at its angle-independent lower bound. -/
-lemma activeFactor_completion_jump {p β U θ : ℝ}
-    (hp : 0 < p) (hU : 0 ≤ U) (hθ : 0 ≤ Real.sin θ) :
-    Real.exp (p / 2 * U) / 3 ≤ activeFactor p β U U θ := by
-  unfold activeFactor
-  rw [regulator_end β U hU]
-  simpa only [mul_one] using mul_le_mul_of_nonneg_left
-    (angularFactor_endpoint_bounds hp hθ).1 (by positivity : 0 ≤ Real.exp (p / 2 * U) / 3)
-
-lemma activeFactor_nonneg {p β U u θ : ℝ} (hθ : 0 ≤ Real.sin θ) :
-    0 ≤ activeFactor p β U u θ := by
-  unfold activeFactor
-  exact mul_nonneg (by positivity)
-    ((regulator_pos β U u).le.trans (angularFactor_ge_delta (regulator_pos β U u).le hθ))
-
-/-- Simultaneous activation or completion of any subset still gives a
-downward jump for the finite product. Unchanged completed factors need
-only be nonnegative constants. -/
-lemma finite_factor_switch_le {ι : Type*} (s : Finset ι) (before after : ι → ℝ)
-    (hafter : ∀ i ∈ s, 0 ≤ after i) (hle : ∀ i ∈ s, after i ≤ before i) :
-    (∏ i ∈ s, after i) ≤ ∏ i ∈ s, before i :=
-  Finset.prod_le_prod₀ hafter hle
-
-end Multipoint
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.BrownianOptionalIntervals
+import OAI.Probability.SLE.Multipoint.MultipointAngular
+
+namespace OAI
+
+/-! Finite switching for the regularized multipoint product. Smooth
+pieces may change at bounded stopping times, provided the jumps are
+downward. A completed factor can therefore remain constant on all later
+pieces without any condition on its subsequent lifetime. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal BigOperators
+
+namespace SLEExactGauge
+
+/-- At a switching time the active/completed pattern has only finitely
+many possibilities. Apply the conditional interval estimate on each
+past-measurable branch and sum. The branch formula also covers several
+simultaneous switches. -/
+theorem brownian_finite_branch_optional_interval_le
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (σ τ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hστ : ∀ ω, σ ω ≤ τ ω) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    {ι : Type*} (s : Finset ι) (A : ι → Set Ω)
+    (hA : ∀ i ∈ s, MeasurableSet[hσ.measurableSpace] (A i))
+    (K : ι → ℝ≥0) (F a : ι → ℝ≥0 → Ω → ℝ → ℝ)
+    (hF : ∀ i ∈ s, SmoothTimeBrownianField B hm (K i) (F i) (a i))
+    (hdrift : ∀ i ∈ s, ∀ ω ∈ A i, ∀ u,
+      (σ ω : ℝ) ≤ u → u < (τ ω : ℝ) → brownianFieldDrift B (F i) (a i) u ω ≤ 0) :
+    (∫ ω, ∑ i ∈ s, (A i).indicator
+      (fun ω => F i (τ ω) ω (B (τ ω) ω)) ω ∂P) ≤
+    ∫ ω, ∑ i ∈ s, (A i).indicator
+      (fun ω => F i (σ ω) ω (B (σ ω) ω)) ω ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hAi (i : ι) (hi : i ∈ s) : MeasurableSet (A i) :=
+    hσ.measurableSpace_le _ (hA i hi)
+  have hleft (i : ι) (hi : i ∈ s) : Integrable ((A i).indicator
+      (fun ω => F i (τ ω) ω (B (τ ω) ω))) P :=
+    ((hF i hi).eval_stopped_integrable hBm (stopping_time_measurable_nnreal hτ)).indicator
+      (hAi i hi)
+  have hright (i : ι) (hi : i ∈ s) : Integrable ((A i).indicator
+      (fun ω => F i (σ ω) ω (B (σ ω) ω))) P :=
+    ((hF i hi).eval_stopped_integrable hBm (stopping_time_measurable_nnreal hσ)).indicator
+      (hAi i hi)
+  rw [integral_finsetSum _ hleft, integral_finsetSum _ hright]
+  apply Finset.sum_le_sum
+  intro i hi
+  rw [integral_indicator (hAi i hi), integral_indicator (hAi i hi)]
+  exact brownian_time_field_optional_interval_le hB hm hBm hBc (hF i hi)
+    σ τ hσ hτ hστ T hb (hA i hi) (hdrift i hi)
+
+/-- Patching finitely many bounded stopping intervals preserves the
+conditional expectation bound. Each field is required to have
+nonpositive drift only on its own interval. -/
+theorem brownian_finite_switch_optional_le
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (N : ℕ) (τ : ℕ → Ω → ℝ≥0)
+    (hτ : ∀ n, n ≤ N → IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ n ω : WithTop ℝ≥0)))
+    (hmono : ∀ n, n < N → ∀ ω, τ n ω ≤ τ (n+1) ω)
+    (T : ℝ≥0) (hbound : ∀ n, n ≤ N → ∀ ω, τ n ω ≤ T)
+    (K : ℕ → ℝ≥0) (F a : ℕ → ℝ≥0 → Ω → ℝ → ℝ)
+    (hF : ∀ n, n ≤ N → SmoothTimeBrownianField B hm (K n) (F n) (a n))
+    {A : Set Ω} (hA : MeasurableSet[(hτ 0 (Nat.zero_le N)).measurableSpace] A)
+    (hdrift : ∀ n, n < N → ∀ ω ∈ A, ∀ u,
+      (τ n ω : ℝ) ≤ u → u < (τ (n+1) ω : ℝ) →
+        brownianFieldDrift B (F n) (a n) u ω ≤ 0)
+    (hjump : ∀ n, n < N → ∀ ω ∈ A,
+      F (n+1) (τ (n+1) ω) ω (B (τ (n+1) ω) ω) ≤
+        F n (τ (n+1) ω) ω (B (τ (n+1) ω) ω)) :
+    (∫ ω in A, F N (τ N ω) ω (B (τ N ω) ω) ∂P) ≤
+      ∫ ω in A, F 0 (τ 0 ω) ω (B (τ 0 ω) ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hstart : ∀ n, n ≤ N → ∀ ω, τ 0 ω ≤ τ n ω := by
+    intro n
+    induction n with
+    | zero => intro _ _; exact le_rfl
+    | succ n ih =>
+      intro hn ω
+      exact (ih (Nat.le_trans (Nat.le_succ n) hn) ω).trans
+        (hmono n (Nat.lt_of_succ_le hn) ω)
+  have hAm (n : ℕ) (hn : n ≤ N) :
+      MeasurableSet[(hτ n hn).measurableSpace] A := by
+    apply IsStoppingTime.measurableSpace_mono (hτ 0 (Nat.zero_le N)) (hτ n hn)
+      (fun ω => WithTop.coe_le_coe.mpr (hstart n hn ω))
+    exact hA
+  have hind : ∀ n, n ≤ N →
+      (∫ ω in A, F n (τ n ω) ω (B (τ n ω) ω) ∂P) ≤
+        ∫ ω in A, F 0 (τ 0 ω) ω (B (τ 0 ω) ω) ∂P := by
+    intro n
+    induction n with
+    | zero => intro _; exact le_rfl
+    | succ n ih =>
+      intro hn
+      have hn' := Nat.le_trans (Nat.le_succ n) hn
+      have hnN := Nat.lt_of_succ_le hn
+      have hflow := brownian_time_field_optional_interval_le hB hm hBm hBc
+        (hF n hn') (τ n) (τ (n+1)) (hτ n hn') (hτ (n+1) hn)
+        (hmono n hnN) T (hbound (n+1) hn) (hAm n hn') (hdrift n hnN)
+      have hright := (hF n hn').eval_stopped_integrable hBm
+        (stopping_time_measurable_nnreal (hτ (n+1) hn)) (P := P.restrict A)
+      have hleft := (hF (n+1) hn).eval_stopped_integrable hBm
+        (stopping_time_measurable_nnreal (hτ (n+1) hn)) (P := P.restrict A)
+      have hswitch : (∫ ω in A, F (n+1) (τ (n+1) ω) ω (B (τ (n+1) ω) ω) ∂P) ≤
+          ∫ ω in A, F n (τ (n+1) ω) ω (B (τ (n+1) ω) ω) ∂P := by
+        apply integral_mono_ae hleft hright
+        filter_upwards [ae_restrict_mem ((hτ 0 (Nat.zero_le N)).measurableSpace_le _ hA)]
+          with ω hω
+        exact hjump n hnN ω hω
+      exact hswitch.trans (hflow.trans (ih hn'))
+  exact hind N le_rfl
+
+namespace Multipoint
+
+def activeFactor (p β U u θ : ℝ) : ℝ :=
+  Real.exp (p / 2 * u) / 3 * angularFactor p (regulator β U u) θ
+
+lemma activeFactor_activation_le {p β U θ : ℝ}
+    (hp : 0 < p) (hU : 0 ≤ U) (hθ : 0 ≤ Real.sin θ) :
+    activeFactor p β U 0 θ ≤ 1 := by
+  unfold activeFactor
+  rw [regulator_zero β U hU]
+  have h := (angularFactor_endpoint_bounds hp hθ).2
+  simp only [mul_zero, Real.exp_zero]
+  linarith
+
+/-- Completion freezes the factor at its angle-independent lower bound. -/
+lemma activeFactor_completion_jump {p β U θ : ℝ}
+    (hp : 0 < p) (hU : 0 ≤ U) (hθ : 0 ≤ Real.sin θ) :
+    Real.exp (p / 2 * U) / 3 ≤ activeFactor p β U U θ := by
+  unfold activeFactor
+  rw [regulator_end β U hU]
+  simpa only [mul_one] using mul_le_mul_of_nonneg_left
+    (angularFactor_endpoint_bounds hp hθ).1 (by positivity : 0 ≤ Real.exp (p / 2 * U) / 3)
+
+lemma activeFactor_nonneg {p β U u θ : ℝ} (hθ : 0 ≤ Real.sin θ) :
+    0 ≤ activeFactor p β U u θ := by
+  unfold activeFactor
+  exact mul_nonneg (by positivity)
+    ((regulator_pos β U u).le.trans (angularFactor_ge_delta (regulator_pos β U u).le hθ))
+
+/-- Simultaneous activation or completion of any subset still gives a
+downward jump for the finite product. Unchanged completed factors need
+only be nonnegative constants. -/
+lemma finite_factor_switch_le {ι : Type*} (s : Finset ι) (before after : ι → ℝ)
+    (hafter : ∀ i ∈ s, 0 ≤ after i) (hle : ∀ i ∈ s, after i ≤ before i) :
+    (∏ i ∈ s, after i) ≤ ∏ i ∈ s, before i :=
+  Finset.prod_le_prod₀ hafter hle
+
+end Multipoint
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointTerminalProbability.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointTerminalProbability.lean
index 5a71a1f1c8028acae99980865d30ed8b4428ef44..8b66ba3ced8441156030eb88178a44b45ce356b3 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointTerminalProbability.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointTerminalProbability.lean
@@ -1,81 +1,81 @@
-import OAI.Probability.SLE.Multipoint.MultipointPhaseProcess
-
-namespace OAI
-
-/-! Convert the real-valued stopped product estimate into the successful
-event probability, after exhaustion by the concrete localizations. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma initial_phase_product_le_one
-    {Ω ι : Type*} [Fintype ι]
-    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (a U : ι → ℝ) (hU : ∀ i, 0 ≤ U i)
-    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω)
-    (horder : ∀ i, Monotone (fun k => ρ (i,k) ω))
-    (z : ι → ℂ × ℝ) (x : ℝ)
-    (hclock : ∀ i (k : Fin 3), ρ (i,k) ω = 0 →
-      logFactorClock κ (z i)-a i = if k = 0 then 0 else if k = 1 then U i/2 else U i) :
-    patternProduct κ β a U (phaseAt ρ 0 ω) z x ≤ 1 := by
-  rw [← patternProduct_all_inactive κ β a U z x]
-  apply patternProduct_switch_le κ β hκ hκ8 a U hU _ _ _ _ (fun _ => bot_le)
-  · intro i _ hi
-    have hh := (phaseAt_ge_iff ρ ω i (horder i) 0 0).mp hi
-    simpa only [ite_true] using hclock i 0 (le_antisymm hh bot_le)
-  · intro i _ hi
-    have hh := (phaseAt_ge_iff ρ ω i (horder i) 0 1).mp hi
-    simpa only [Fin.reduceEq, ite_true, ite_false] using hclock i 1 (le_antisymm hh bot_le)
-  · intro i _ hi
-    have hh := (phaseAt_ge_iff ρ ω i (horder i) 0 2).mp hi
-    simpa only [Fin.reduceEq, ite_true, ite_false] using hclock i 2 (le_antisymm hh bot_le)
-
-lemma measure_le_of_eventual_real_products
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (E : Set Ω) (cutoff : ℕ → Set Ω) (X : ℕ → Ω → ℝ) (q : ℝ) (hq : 0 < q)
-    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
-    (hX : ∀ n, Integrable (X n) P) (hX0 : ∀ n ω, 0 ≤ X n ω)
-    (hmean : ∀ n, (∫ ω, X n ω ∂P) ≤ 1)
-    (hterminal : ∀ n ω, ω ∈ cutoff n → q ≤ X n ω) :
-    P E ≤ ENNReal.ofReal q⁻¹ := by
-  apply measure_le_of_eventual_stopped_products P E cutoff (fun n ω => ENNReal.ofReal (X n ω)) q hq hcover
-  · intro n
-    exact ENNReal.continuous_ofReal.measurable.comp_aemeasurable (hX n).aestronglyMeasurable.aemeasurable
-  · intro n
-    rw [← ofReal_integral_eq_lintegral_ofReal (hX n) (ae_of_all _ (hX0 n))]
-    exact (ENNReal.ofReal_le_ofReal (hmean n)).trans_eq ENNReal.ofReal_one
-  · exact fun n ω hω => ENNReal.ofReal_le_ofReal (hterminal n ω hω)
-
-theorem measure_le_of_completed_phase_products
-    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
-    (P : Measure Ω) (E : Set Ω) (cutoff : ℕ → Set Ω)
-    (κ β : ℝ) (a U : ι → ℝ) (K : ℝ)
-    (j : ℕ → Ω → ι → Fin 4) (Q : ℕ → Ω → ι → ℂ × ℝ)
-    (b x : ℕ → Ω → ℝ)
-    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
-    (hI : ∀ n, Integrable (fun ω => discountedPatternProduct κ β a U (j n ω)
-      (Q n ω,b n ω) (x n ω)) P)
-    (hmean : ∀ n, (∫ ω, discountedPatternProduct κ β a U (j n ω)
-      (Q n ω,b n ω) (x n ω) ∂P) ≤ 1)
-    (hcomplete : ∀ n ω, ω ∈ cutoff n → j n ω = fun _ => 3)
-    (hb : ∀ n ω, ω ∈ cutoff n → b n ω ≤ K) :
-    P E ≤ ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι)*Real.exp K*
-      Real.exp (-greenExponent κ/2 * ∑ i, U i)) := by
-  have hbound := measure_le_of_eventual_real_products P E cutoff
-    (fun n ω => discountedPatternProduct κ β a U (j n ω) (Q n ω,b n ω) (x n ω))
-    (successBarrier (greenExponent κ) K Finset.univ U)
-    (successBarrier_pos _ _ _ _) hcover hI
-    (fun n ω => mul_nonneg (Real.exp_pos _).le
-      (Finset.prod_nonneg (fun i _ => (phaseFactor_pos κ β (a i) (U i) (j n ω i) (Q n ω i) (x n ω)).le)))
-    hmean (fun n ω hω => by
-      rw [hcomplete n ω hω]
-      exact discountedPatternProduct_complete_lower κ β a U (Q n ω) (b n ω) K (x n ω) (hb n ω hω))
-  simpa only [successBarrier_inv, Finset.card_univ] using hbound
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointPhaseProcess
+
+namespace OAI
+
+/-! Convert the real-valued stopped product estimate into the successful
+event probability, after exhaustion by the concrete localizations. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma initial_phase_product_le_one
+    {Ω ι : Type*} [Fintype ι]
+    (κ β : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (a U : ι → ℝ) (hU : ∀ i, 0 ≤ U i)
+    (ρ : (ι × Fin 3) → Ω → ℝ≥0) (ω : Ω)
+    (horder : ∀ i, Monotone (fun k => ρ (i,k) ω))
+    (z : ι → ℂ × ℝ) (x : ℝ)
+    (hclock : ∀ i (k : Fin 3), ρ (i,k) ω = 0 →
+      logFactorClock κ (z i)-a i = if k = 0 then 0 else if k = 1 then U i/2 else U i) :
+    patternProduct κ β a U (phaseAt ρ 0 ω) z x ≤ 1 := by
+  rw [← patternProduct_all_inactive κ β a U z x]
+  apply patternProduct_switch_le κ β hκ hκ8 a U hU _ _ _ _ (fun _ => bot_le)
+  · intro i _ hi
+    have hh := (phaseAt_ge_iff ρ ω i (horder i) 0 0).mp hi
+    simpa only [ite_true] using hclock i 0 (le_antisymm hh bot_le)
+  · intro i _ hi
+    have hh := (phaseAt_ge_iff ρ ω i (horder i) 0 1).mp hi
+    simpa only [Fin.reduceEq, ite_true, ite_false] using hclock i 1 (le_antisymm hh bot_le)
+  · intro i _ hi
+    have hh := (phaseAt_ge_iff ρ ω i (horder i) 0 2).mp hi
+    simpa only [Fin.reduceEq, ite_true, ite_false] using hclock i 2 (le_antisymm hh bot_le)
+
+lemma measure_le_of_eventual_real_products
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (E : Set Ω) (cutoff : ℕ → Set Ω) (X : ℕ → Ω → ℝ) (q : ℝ) (hq : 0 < q)
+    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
+    (hX : ∀ n, Integrable (X n) P) (hX0 : ∀ n ω, 0 ≤ X n ω)
+    (hmean : ∀ n, (∫ ω, X n ω ∂P) ≤ 1)
+    (hterminal : ∀ n ω, ω ∈ cutoff n → q ≤ X n ω) :
+    P E ≤ ENNReal.ofReal q⁻¹ := by
+  apply measure_le_of_eventual_stopped_products P E cutoff (fun n ω => ENNReal.ofReal (X n ω)) q hq hcover
+  · intro n
+    exact ENNReal.continuous_ofReal.measurable.comp_aemeasurable (hX n).aestronglyMeasurable.aemeasurable
+  · intro n
+    rw [← ofReal_integral_eq_lintegral_ofReal (hX n) (ae_of_all _ (hX0 n))]
+    exact (ENNReal.ofReal_le_ofReal (hmean n)).trans_eq ENNReal.ofReal_one
+  · exact fun n ω hω => ENNReal.ofReal_le_ofReal (hterminal n ω hω)
+
+theorem measure_le_of_completed_phase_products
+    {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι]
+    (P : Measure Ω) (E : Set Ω) (cutoff : ℕ → Set Ω)
+    (κ β : ℝ) (a U : ι → ℝ) (K : ℝ)
+    (j : ℕ → Ω → ι → Fin 4) (Q : ℕ → Ω → ι → ℂ × ℝ)
+    (b x : ℕ → Ω → ℝ)
+    (hcover : ∀ ω ∈ E, ∀ᶠ n : ℕ in atTop, ω ∈ cutoff n)
+    (hI : ∀ n, Integrable (fun ω => discountedPatternProduct κ β a U (j n ω)
+      (Q n ω,b n ω) (x n ω)) P)
+    (hmean : ∀ n, (∫ ω, discountedPatternProduct κ β a U (j n ω)
+      (Q n ω,b n ω) (x n ω) ∂P) ≤ 1)
+    (hcomplete : ∀ n ω, ω ∈ cutoff n → j n ω = fun _ => 3)
+    (hb : ∀ n ω, ω ∈ cutoff n → b n ω ≤ K) :
+    P E ≤ ENNReal.ofReal ((3 : ℝ)^(Fintype.card ι)*Real.exp K*
+      Real.exp (-greenExponent κ/2 * ∑ i, U i)) := by
+  have hbound := measure_le_of_eventual_real_products P E cutoff
+    (fun n ω => discountedPatternProduct κ β a U (j n ω) (Q n ω,b n ω) (x n ω))
+    (successBarrier (greenExponent κ) K Finset.univ U)
+    (successBarrier_pos _ _ _ _) hcover hI
+    (fun n ω => mul_nonneg (Real.exp_pos _).le
+      (Finset.prod_nonneg (fun i _ => (phaseFactor_pos κ β (a i) (U i) (j n ω i) (Q n ω i) (x n ω)).le)))
+    hmean (fun n ω hω => by
+      rw [hcomplete n ω hω]
+      exact discountedPatternProduct_complete_lower κ β a U (Q n ω) (b n ω) K (x n ω) (hb n ω hω))
+  simpa only [successBarrier_inv, Finset.card_univ] using hbound
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointTupleNull.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointTupleNull.lean
index 772223b5dc3583cc715e7436f2c38fb539b1b50f..9bc87f026b034ce90fbdc32b287fccc305fd6b2a 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointTupleNull.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointTupleNull.lean
@@ -1,69 +1,69 @@
-import OAI.Probability.SLE.Multipoint.MultipointDistanceKernel
-
-namespace OAI
-
-/-! Distinctness of planar integration tuples. The values assigned to
-the distance kernel on collision diagonals do not affect its integral. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal
-
-namespace SLEExactGauge.Multipoint
-
-lemma ae_tuple_pair_ne {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (K : Set ℂ) (i j : ι) (hij : i≠j) :
-    ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), x i≠x j := by
-  let μ : ι → Measure ℂ := fun _ => volume.restrict K
-  let D : Set (ι → ℂ) := {x | x i=x j}
-  let f : (ι → ℂ) → ℝ≥0∞ := D.indicator (fun _ => 1)
-  have hD : MeasurableSet D := measurableSet_eq_fun (measurable_pi_apply i) (measurable_pi_apply j)
-  have hf : Measurable f := measurable_const.indicator hD
-  have hmarginal : lmarginal μ {i} f = lmarginal μ {i} (fun _ => 0) := by
-    rw [lmarginal_singleton,lmarginal_singleton]
-    funext x
-    have he : (fun y : ℂ => f (Function.update x i y)) =
-        ({x j} : Set ℂ).indicator (fun _ => (1 : ℝ≥0∞)) := by
-      funext y
-      simp only [f,D,indicator,mem_ofPred_eq,Function.update_self,
-        Function.update_of_ne hij.symm,mem_singleton_iff]
-    rw [he,lintegral_indicator (measurableSet_singleton _)]
-    simp only [lintegral_const,Measure.restrict_apply_univ,measure_singleton,one_mul,zero_mul]
-  have hint : (∫⁻ x, f x ∂Measure.pi μ) = 0 := by
-    have hh := lintegral_eq_of_lmarginal_eq {i} hf measurable_const hmarginal
-    simpa only [lintegral_zero] using hh
-  have hnull : Measure.pi μ D = 0 := by
-    simpa only [f,lintegral_indicator hD,lintegral_const,
-      Measure.restrict_apply_univ,one_mul] using hint
-  apply ae_iff.mpr
-  simpa only [not_not] using hnull
-
-theorem ae_tuple_injective {ι : Type*} [Fintype ι]
-    (K : Set ℂ) :
-    ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), Function.Injective x := by
-  classical
-  have hp : ∀ i j : ι,
-      ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), i≠j → x i≠x j := by
-    intro i j
-    by_cases hij : i=j
-    · exact ae_of_all _ (fun _ h => (h hij).elim)
-    · exact (ae_tuple_pair_ne K i j hij).mono (fun _ h _ => h)
-  filter_upwards [ae_all_iff.mpr (fun i => ae_all_iff.mpr (hp i))] with x hx
-  intro i j he
-  by_contra hij
-  exact hx i j hij he
-
-theorem ae_tuple_avoids {ι : Type*} [Fintype ι] (K : Set ℂ) (z : ℂ) :
-    ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), ∀ i, x i≠z := by
-  apply ae_all_iff.mpr
-  intro i
-  apply ae_iff.mpr
-  simpa only [not_not,Set.preimage,Function.eval,Set.mem_singleton_iff] using
-    Measure.pi_eval_preimage_null (i := i) (fun _ : ι => volume.restrict K)
-    (by simp only [measure_singleton] : (volume.restrict K) {z}=0)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointDistanceKernel
+
+namespace OAI
+
+/-! Distinctness of planar integration tuples. The values assigned to
+the distance kernel on collision diagonals do not affect its integral. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal
+
+namespace SLEExactGauge.Multipoint
+
+lemma ae_tuple_pair_ne {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (K : Set ℂ) (i j : ι) (hij : i≠j) :
+    ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), x i≠x j := by
+  let μ : ι → Measure ℂ := fun _ => volume.restrict K
+  let D : Set (ι → ℂ) := {x | x i=x j}
+  let f : (ι → ℂ) → ℝ≥0∞ := D.indicator (fun _ => 1)
+  have hD : MeasurableSet D := measurableSet_eq_fun (measurable_pi_apply i) (measurable_pi_apply j)
+  have hf : Measurable f := measurable_const.indicator hD
+  have hmarginal : lmarginal μ {i} f = lmarginal μ {i} (fun _ => 0) := by
+    rw [lmarginal_singleton,lmarginal_singleton]
+    funext x
+    have he : (fun y : ℂ => f (Function.update x i y)) =
+        ({x j} : Set ℂ).indicator (fun _ => (1 : ℝ≥0∞)) := by
+      funext y
+      simp only [f,D,indicator,mem_ofPred_eq,Function.update_self,
+        Function.update_of_ne hij.symm,mem_singleton_iff]
+    rw [he,lintegral_indicator (measurableSet_singleton _)]
+    simp only [lintegral_const,Measure.restrict_apply_univ,measure_singleton,one_mul,zero_mul]
+  have hint : (∫⁻ x, f x ∂Measure.pi μ) = 0 := by
+    have hh := lintegral_eq_of_lmarginal_eq {i} hf measurable_const hmarginal
+    simpa only [lintegral_zero] using hh
+  have hnull : Measure.pi μ D = 0 := by
+    simpa only [f,lintegral_indicator hD,lintegral_const,
+      Measure.restrict_apply_univ,one_mul] using hint
+  apply ae_iff.mpr
+  simpa only [not_not] using hnull
+
+theorem ae_tuple_injective {ι : Type*} [Fintype ι]
+    (K : Set ℂ) :
+    ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), Function.Injective x := by
+  classical
+  have hp : ∀ i j : ι,
+      ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), i≠j → x i≠x j := by
+    intro i j
+    by_cases hij : i=j
+    · exact ae_of_all _ (fun _ h => (h hij).elim)
+    · exact (ae_tuple_pair_ne K i j hij).mono (fun _ h _ => h)
+  filter_upwards [ae_all_iff.mpr (fun i => ae_all_iff.mpr (hp i))] with x hx
+  intro i j he
+  by_contra hij
+  exact hx i j hij he
+
+theorem ae_tuple_avoids {ι : Type*} [Fintype ι] (K : Set ℂ) (z : ℂ) :
+    ∀ᵐ x : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), ∀ i, x i≠z := by
+  apply ae_all_iff.mpr
+  intro i
+  apply ae_iff.mpr
+  simpa only [not_not,Set.preimage,Function.eval,Set.mem_singleton_iff] using
+    Measure.pi_eval_preimage_null (i := i) (fun _ : ι => volume.restrict K)
+    (by simp only [measure_singleton] : (volume.restrict K) {z}=0)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointUniformCost.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointUniformCost.lean
index ecf8a624c497d18577284a08048f4704a03b00e7..88e5e546dec5c77d725aa5dcc393baca452b9e6b 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointUniformCost.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointUniformCost.lean
@@ -1,79 +1,79 @@
-import OAI.Probability.SLE.Multipoint.MultipointDiscountCost
-
-namespace OAI
-
-/-! A single finite-order Young coefficient controls every active subset.
-The coefficient is chosen before the stopping-time pattern is known. -/
-
-noncomputable section
-open scoped BigOperators
-
-namespace SLEExactGauge.Multipoint
-
-/-- The Young coefficient associated with one active subset. -/
-def patternYoungConstant {ι : Type*} (κ : ℝ) (s : Finset ι) : ℝ :=
-  let p := greenExponent κ
-  youngConstant p (angularCoefficientConstant p^2)
-    ((negativeDrift p/2)/(4*((s.card : ℝ)+1)))
-
-/-- A common coefficient for all active subsets of a fixed finite family. -/
-def allPatternYoungConstant {ι : Type*} (κ : ℝ) (I : Finset ι) : ℝ :=
-  ∑ s ∈ I.powerset, patternYoungConstant κ s
-
-lemma patternYoungConstant_pos {ι : Type*} {κ : ℝ}
-    (hκ : 0 < κ) (hκ8 : κ < 8) (s : Finset ι) :
-    0 < patternYoungConstant κ s := by
-  have hp : 0 < greenExponent κ := by
-    unfold greenExponent
-    rw [sub_pos, lt_div_iff₀ hκ]
-    linarith
-  exact youngConstant_pos hp (sq_pos_of_pos (angularCoefficientConstant_pos hp))
-    (absorption_epsilon_pos (div_pos (negativeDrift_pos hp) (by norm_num)) s.card)
-
-lemma patternYoungConstant_le_all {ι : Type*} [DecidableEq ι] {κ : ℝ}
-    (hκ : 0 < κ) (hκ8 : κ < 8) {I s : Finset ι} (hs : s ⊆ I) :
-    patternYoungConstant κ s ≤ allPatternYoungConstant κ I := by
-  exact Finset.single_le_sum (fun t _ => (patternYoungConstant_pos hκ hκ8 t).le)
-    (Finset.mem_powerset.mpr hs)
-
-lemma allPatternYoungConstant_pos {ι : Type*} [DecidableEq ι] {κ : ℝ}
-    (hκ : 0 < κ) (hκ8 : κ < 8) (I : Finset ι) :
-    0 < allPatternYoungConstant κ I :=
-  (patternYoungConstant_pos hκ hκ8 (∅ : Finset ι)).trans_le
-    (patternYoungConstant_le_all hκ hκ8 (Finset.empty_subset I))
-
-/-- The actual drift cost is controlled by one deterministic coefficient,
-while keeping precisely the current active single and pair sums. -/
-theorem productDriftCost_le_uniform {ι : Type*} [DecidableEq ι]
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {I s : Finset ι} (hs : s ⊆ I)
-    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (z : ι → ℂ × ℝ) (x : ℝ) :
-    productDriftCost s κ D σ z x ≤
-      κ*((driftAllowance (greenExponent κ)*
-          driftThreshold (greenExponent κ)^(-greenExponent κ))*
-        (∑ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2) +
-        allPatternYoungConstant κ I *
-          (∑ ij ∈ s.offDiag,
-            (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
-              branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent (greenExponent κ))*
-            (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x))) := by
-  have hpair : 0 ≤ ∑ ij ∈ s.offDiag,
-      (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
-        branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent (greenExponent κ))*
-      (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x) := by
-    apply Finset.sum_nonneg
-    intro ij hij
-    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
-    exact mul_nonneg (Real.rpow_nonneg (mul_nonneg
-      (mul_pos (hD _ hi) (Real.exp_pos _)).le
-      (mul_pos (hD _ hj) (Real.exp_pos _)).le) _)
-      (mul_pos (logVelocity_pos κ (z ij.1) x) (logVelocity_pos κ (z ij.2) x)).le
-  unfold productDriftCost
-  apply mul_le_mul_of_nonneg_left _ hκ.le
-  exact add_le_add le_rfl
-    (mul_le_mul_of_nonneg_right (patternYoungConstant_le_all hκ hκ8 hs) hpair)
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointDiscountCost
+
+namespace OAI
+
+/-! A single finite-order Young coefficient controls every active subset.
+The coefficient is chosen before the stopping-time pattern is known. -/
+
+noncomputable section
+open scoped BigOperators
+
+namespace SLEExactGauge.Multipoint
+
+/-- The Young coefficient associated with one active subset. -/
+def patternYoungConstant {ι : Type*} (κ : ℝ) (s : Finset ι) : ℝ :=
+  let p := greenExponent κ
+  youngConstant p (angularCoefficientConstant p^2)
+    ((negativeDrift p/2)/(4*((s.card : ℝ)+1)))
+
+/-- A common coefficient for all active subsets of a fixed finite family. -/
+def allPatternYoungConstant {ι : Type*} (κ : ℝ) (I : Finset ι) : ℝ :=
+  ∑ s ∈ I.powerset, patternYoungConstant κ s
+
+lemma patternYoungConstant_pos {ι : Type*} {κ : ℝ}
+    (hκ : 0 < κ) (hκ8 : κ < 8) (s : Finset ι) :
+    0 < patternYoungConstant κ s := by
+  have hp : 0 < greenExponent κ := by
+    unfold greenExponent
+    rw [sub_pos, lt_div_iff₀ hκ]
+    linarith
+  exact youngConstant_pos hp (sq_pos_of_pos (angularCoefficientConstant_pos hp))
+    (absorption_epsilon_pos (div_pos (negativeDrift_pos hp) (by norm_num)) s.card)
+
+lemma patternYoungConstant_le_all {ι : Type*} [DecidableEq ι] {κ : ℝ}
+    (hκ : 0 < κ) (hκ8 : κ < 8) {I s : Finset ι} (hs : s ⊆ I) :
+    patternYoungConstant κ s ≤ allPatternYoungConstant κ I := by
+  exact Finset.single_le_sum (fun t _ => (patternYoungConstant_pos hκ hκ8 t).le)
+    (Finset.mem_powerset.mpr hs)
+
+lemma allPatternYoungConstant_pos {ι : Type*} [DecidableEq ι] {κ : ℝ}
+    (hκ : 0 < κ) (hκ8 : κ < 8) (I : Finset ι) :
+    0 < allPatternYoungConstant κ I :=
+  (patternYoungConstant_pos hκ hκ8 (∅ : Finset ι)).trans_le
+    (patternYoungConstant_le_all hκ hκ8 (Finset.empty_subset I))
+
+/-- The actual drift cost is controlled by one deterministic coefficient,
+while keeping precisely the current active single and pair sums. -/
+theorem productDriftCost_le_uniform {ι : Type*} [DecidableEq ι]
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {I s : Finset ι} (hs : s ⊆ I)
+    (D σ : ι → ℝ) (hD : ∀ i ∈ s, 0 < D i) (z : ι → ℂ × ℝ) (x : ℝ) :
+    productDriftCost s κ D σ z x ≤
+      κ*((driftAllowance (greenExponent κ)*
+          driftThreshold (greenExponent κ)^(-greenExponent κ))*
+        (∑ i ∈ s, branchRegulator κ (D i) (σ i) (z i)*logVelocity κ (z i) x^2) +
+        allPatternYoungConstant κ I *
+          (∑ ij ∈ s.offDiag,
+            (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
+              branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent (greenExponent κ))*
+            (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x))) := by
+  have hpair : 0 ≤ ∑ ij ∈ s.offDiag,
+      (branchRegulator κ (D ij.1) (σ ij.1) (z ij.1)*
+        branchRegulator κ (D ij.2) (σ ij.2) (z ij.2))^(-exponent (greenExponent κ))*
+      (logVelocity κ (z ij.1) x*logVelocity κ (z ij.2) x) := by
+    apply Finset.sum_nonneg
+    intro ij hij
+    obtain ⟨hi, hj, _⟩ := Finset.mem_offDiag.mp hij
+    exact mul_nonneg (Real.rpow_nonneg (mul_nonneg
+      (mul_pos (hD _ hi) (Real.exp_pos _)).le
+      (mul_pos (hD _ hj) (Real.exp_pos _)).le) _)
+      (mul_pos (logVelocity_pos κ (z ij.1) x) (logVelocity_pos κ (z ij.2) x)).le
+  unfold productDriftCost
+  apply mul_le_mul_of_nonneg_left _ hκ.le
+  exact add_le_add le_rfl
+    (mul_le_mul_of_nonneg_right (patternYoungConstant_le_all hκ hκ8 hs) hpair)
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/MultipointYoung.lean b/lean/OAI/Probability/SLE/Multipoint/MultipointYoung.lean
index b1f9549639ba649fdd3cc5f58f8d8ef89fce3fd5..819ed65497a24bb501166a57a740d95c778570f8 100644
--- a/lean/OAI/Probability/SLE/Multipoint/MultipointYoung.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/MultipointYoung.lean
@@ -1,90 +1,90 @@
-import OAI.Probability.SLE.Multipoint.MultipointRegularizer
-
-namespace OAI
-
-/-! Weighted Young absorption for the pair terms in the multipoint
-regularized product. The constant depends on the order and SLE parameter,
-not the two target levels or their current clocks. -/
-
-noncomputable section
-namespace SLEExactGauge.Multipoint
-
-def youngConstant (p C ε : ℝ) : ℝ :=
-  (1-2*eta p) * (C*(eta p/ε)^(2*eta p))^(1/(1-2*eta p))
-
-lemma youngConstant_pos {p C ε : ℝ} (hp : 0 < p) (hC : 0 < C) (hε : 0 < ε) :
-    0 < youngConstant p C ε := by
-  have he := eta_pos (p := p)
-  have hr : 0 < 1-2*eta p := by linarith [eta_lt_half hp]
-  unfold youngConstant
-  positivity
-
-lemma three_factor_young {p C ε x y z : ℝ} (hp : 0 < p)
-    (hC : 0 < C) (hε : 0 < ε) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
-    C*x^(eta p)*y^(eta p)*z^(1-2*eta p) ≤
-      ε*(x+y) + youngConstant p C ε*z := by
-  let e := eta p
-  let r := 1-2*e
-  let b := ε/e
-  let K := (C*(e/ε)^(2*e))^(1/r)
-  have he : 0 < e := eta_pos
-  have hr : 0 < r := by dsimp only [r, e]; linarith [eta_lt_half hp]
-  have hb : 0 < b := div_pos hε he
-  have hK : 0 < K := by dsimp only [K]; positivity
-  have hKr : K^r = C*(e/ε)^(2*e) := by
-    dsimp only [K]
-    rw [← Real.rpow_mul (by positivity), one_div_mul_cancel hr.ne', Real.rpow_one]
-  have hcoef : b^e*b^e*K^r = C := by
-    rw [← Real.rpow_add hb, show e+e = 2*e by ring, hKr]
-    calc
-      _ = C*(b^(2*e)*(e/ε)^(2*e)) := by ring
-      _ = C*(b*(e/ε))^(2*e) := by rw [Real.mul_rpow hb.le (div_pos he hε).le]
-      _ = C := by
-        have hinv : b*(e/ε) = 1 := by dsimp only [b]; field_simp
-        rw [hinv, Real.one_rpow, mul_one]
-  have hprod : (b*x)^e*(b*y)^e*(K*z)^r = C*x^e*y^e*z^r := by
-    rw [Real.mul_rpow hb.le hx, Real.mul_rpow hb.le hy, Real.mul_rpow hK.le hz]
-    calc
-      _ = (b^e*b^e*K^r)*(x^e*y^e*z^r) := by ring
-      _ = _ := by rw [hcoef]; ring
-  have hmean := Real.geom_mean_le_arith_mean3_weighted he.le he.le hr.le
-    (mul_nonneg hb.le hx) (mul_nonneg hb.le hy) (mul_nonneg hK.le hz)
-    (show e+e+r = 1 by dsimp only [r]; ring)
-  rw [hprod] at hmean
-  calc
-    _ ≤ e*(b*x)+e*(b*y)+r*(K*z) := hmean
-    _ = _ := by
-      change e*(b*x)+e*(b*y)+r*(K*z) = ε*(x+y)+(r*K)*z
-      dsimp only [b]
-      field_simp
-
-/-- The factor from the singular angular derivatives is absorbed with
-coefficient ε at both vertices and a deterministic inverse-regulator cost. -/
-theorem weighted_pair_absorption {p C ε D x y z : ℝ} (hp : 0 < p)
-    (hC : 0 < C) (hε : 0 < ε) (hD : 0 < D)
-    (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
-    C*D^(-1-eta p)*x^(eta p)*y^(eta p)*z^(1-2*eta p) ≤
-      ε*(x+y) + youngConstant p C ε*D^(-exponent p)*z := by
-  have he := eta_pos (p := p)
-  have hr : 0 < 1-2*eta p := by linarith [eta_lt_half hp]
-  have hCD : 0 < C*D^(-1-eta p) := mul_pos hC (Real.rpow_pos_of_pos hD _)
-  have hconst : youngConstant p (C*D^(-1-eta p)) ε =
-      youngConstant p C ε*D^(-exponent p) := by
-    unfold youngConstant
-    have hb : 0 < C*(eta p/ε)^(2*eta p) := by positivity
-    rw [show C*D^(-1-eta p)*(eta p/ε)^(2*eta p) =
-      (C*(eta p/ε)^(2*eta p))*D^(-1-eta p) by ring,
-      Real.mul_rpow hb.le (Real.rpow_pos_of_pos hD _).le,
-      ← Real.rpow_mul hD.le]
-    have hexp : (-1-eta p)*(1/(1-2*eta p)) = -exponent p := by
-      unfold exponent
-      ring
-    rw [hexp]
-    ring
-  simpa only [hconst, mul_assoc] using three_factor_young hp hCD hε hx hy hz
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointRegularizer
+
+namespace OAI
+
+/-! Weighted Young absorption for the pair terms in the multipoint
+regularized product. The constant depends on the order and SLE parameter,
+not the two target levels or their current clocks. -/
+
+noncomputable section
+namespace SLEExactGauge.Multipoint
+
+def youngConstant (p C ε : ℝ) : ℝ :=
+  (1-2*eta p) * (C*(eta p/ε)^(2*eta p))^(1/(1-2*eta p))
+
+lemma youngConstant_pos {p C ε : ℝ} (hp : 0 < p) (hC : 0 < C) (hε : 0 < ε) :
+    0 < youngConstant p C ε := by
+  have he := eta_pos (p := p)
+  have hr : 0 < 1-2*eta p := by linarith [eta_lt_half hp]
+  unfold youngConstant
+  positivity
+
+lemma three_factor_young {p C ε x y z : ℝ} (hp : 0 < p)
+    (hC : 0 < C) (hε : 0 < ε) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
+    C*x^(eta p)*y^(eta p)*z^(1-2*eta p) ≤
+      ε*(x+y) + youngConstant p C ε*z := by
+  let e := eta p
+  let r := 1-2*e
+  let b := ε/e
+  let K := (C*(e/ε)^(2*e))^(1/r)
+  have he : 0 < e := eta_pos
+  have hr : 0 < r := by dsimp only [r, e]; linarith [eta_lt_half hp]
+  have hb : 0 < b := div_pos hε he
+  have hK : 0 < K := by dsimp only [K]; positivity
+  have hKr : K^r = C*(e/ε)^(2*e) := by
+    dsimp only [K]
+    rw [← Real.rpow_mul (by positivity), one_div_mul_cancel hr.ne', Real.rpow_one]
+  have hcoef : b^e*b^e*K^r = C := by
+    rw [← Real.rpow_add hb, show e+e = 2*e by ring, hKr]
+    calc
+      _ = C*(b^(2*e)*(e/ε)^(2*e)) := by ring
+      _ = C*(b*(e/ε))^(2*e) := by rw [Real.mul_rpow hb.le (div_pos he hε).le]
+      _ = C := by
+        have hinv : b*(e/ε) = 1 := by dsimp only [b]; field_simp
+        rw [hinv, Real.one_rpow, mul_one]
+  have hprod : (b*x)^e*(b*y)^e*(K*z)^r = C*x^e*y^e*z^r := by
+    rw [Real.mul_rpow hb.le hx, Real.mul_rpow hb.le hy, Real.mul_rpow hK.le hz]
+    calc
+      _ = (b^e*b^e*K^r)*(x^e*y^e*z^r) := by ring
+      _ = _ := by rw [hcoef]; ring
+  have hmean := Real.geom_mean_le_arith_mean3_weighted he.le he.le hr.le
+    (mul_nonneg hb.le hx) (mul_nonneg hb.le hy) (mul_nonneg hK.le hz)
+    (show e+e+r = 1 by dsimp only [r]; ring)
+  rw [hprod] at hmean
+  calc
+    _ ≤ e*(b*x)+e*(b*y)+r*(K*z) := hmean
+    _ = _ := by
+      change e*(b*x)+e*(b*y)+r*(K*z) = ε*(x+y)+(r*K)*z
+      dsimp only [b]
+      field_simp
+
+/-- The factor from the singular angular derivatives is absorbed with
+coefficient ε at both vertices and a deterministic inverse-regulator cost. -/
+theorem weighted_pair_absorption {p C ε D x y z : ℝ} (hp : 0 < p)
+    (hC : 0 < C) (hε : 0 < ε) (hD : 0 < D)
+    (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
+    C*D^(-1-eta p)*x^(eta p)*y^(eta p)*z^(1-2*eta p) ≤
+      ε*(x+y) + youngConstant p C ε*D^(-exponent p)*z := by
+  have he := eta_pos (p := p)
+  have hr : 0 < 1-2*eta p := by linarith [eta_lt_half hp]
+  have hCD : 0 < C*D^(-1-eta p) := mul_pos hC (Real.rpow_pos_of_pos hD _)
+  have hconst : youngConstant p (C*D^(-1-eta p)) ε =
+      youngConstant p C ε*D^(-exponent p) := by
+    unfold youngConstant
+    have hb : 0 < C*(eta p/ε)^(2*eta p) := by positivity
+    rw [show C*D^(-1-eta p)*(eta p/ε)^(2*eta p) =
+      (C*(eta p/ε)^(2*eta p))*D^(-1-eta p) by ring,
+      Real.mul_rpow hb.le (Real.rpow_pos_of_pos hD _).le,
+      ← Real.rpow_mul hD.le]
+    have hexp : (-1-eta p)*(1/(1-2*eta p)) = -exponent p := by
+      unfold exponent
+      ring
+    rw [hexp]
+    ring
+  simpa only [hconst, mul_assoc] using three_factor_young hp hCD hε hx hy hz
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Multipoint/NearDiagonalKernel.lean b/lean/OAI/Probability/SLE/Multipoint/NearDiagonalKernel.lean
index 59416baabeefdf2d160cca8d98619d6689a3da9f..3a19465f22c333ef024dd4a3fa9c3ac28498d848 100644
--- a/lean/OAI/Probability/SLE/Multipoint/NearDiagonalKernel.lean
+++ b/lean/OAI/Probability/SLE/Multipoint/NearDiagonalKernel.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Multipoint.MultipointKernelExpansion
-import OAI.Probability.SLE.Multipoint.MultipointTupleNull
-import Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence
-
-namespace OAI
-
-/-! Integrable two-point kernels have vanishing mass near the diagonal.
-Finite cutoff sums with bounded overlap inherit this decay. -/
-
-noncomputable section
-open Set Filter MeasureTheory Classical
-open scoped ENNReal Topology
-
-namespace SLEExactGauge.Multipoint
-
-def nearDiagonalKernel (α b r : ℝ) (z : Fin 2 → ℂ) : ℝ≥0∞ :=
-  if dist (z 0) (z 1) ≤ r then cappedDistanceKernel α b z else 0
-
-lemma nearDiagonalKernel_measurable (α b r : ℝ) : Measurable (nearDiagonalKernel α b r) :=
-  (measurable_cappedDistanceKernel α b).ite
-    (measurableSet_le ((measurable_pi_apply 0).dist (measurable_pi_apply 1)) measurable_const)
-    measurable_const
-
-theorem nearDiagonalKernel_integral_tendsto_zero {α : ℝ} (hα : 0 < α) (hα1 : α < 1)
-    (b : ℝ) (K : Set ℂ) (hK : Bornology.IsBounded K)
-    (r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
-    Tendsto (fun n => ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel α b (r n) z
-      ∂Measure.pi (fun _ : Fin 2 => volume.restrict K)) atTop (𝓝 0) := by
-  have hh := tendsto_lintegral_of_dominated_convergence
-    (μ := Measure.pi (fun _ : Fin 2 => volume.restrict K))
-    (F := fun n => nearDiagonalKernel α b (r n)) (f := fun _ => 0)
-    (cappedDistanceKernel α b) (fun n => nearDiagonalKernel_measurable α b (r n))
-    (fun n => Eventually.of_forall (fun z => by
-      unfold nearDiagonalKernel
-      split_ifs <;> simp only [le_refl,zero_le]))
-    (cappedDistanceKernel_integral_lt_top hα hα1 b K hK).ne ?_
-  · simpa only [lintegral_zero] using hh
-  filter_upwards [ae_tuple_pair_ne K (0 : Fin 2) 1 (by decide)] with z hz
-  have hdist : 0 < dist (z 0) (z 1) := dist_pos.mpr hz
-  have he : ∀ᶠ n in atTop, nearDiagonalKernel α b (r n) z = 0 := by
-    filter_upwards [(tendsto_order.mp hr).2 _ hdist] with n hn
-    exact ite_eq_right (not_le.mpr hn)
-  exact tendsto_const_nhds.congr' (he.mono (fun _ hn => hn.symm))
-
-lemma sum_weighted_pair_kernel_le {ι : Type*} (I : Finset ι)
-    (f : ι → ℂ → ℝ≥0∞) (hf : ∀ i ∈ I, Measurable (f i))
-    (α b r : ℝ) (N : ℝ≥0∞)
-    (hcover : ∀ x y, (∑ i ∈ I, f i x * f i y) ≤ if dist x y ≤ r then N else 0)
-    (K : Set ℂ) :
-    (∑ i ∈ I, ∫⁻ z : Fin 2 → ℂ, (∏ j, f i (z j)) * cappedDistanceKernel α b z
-      ∂Measure.pi (fun _ : Fin 2 => volume.restrict K)) ≤
-      N * ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel α b r z
-        ∂Measure.pi (fun _ : Fin 2 => volume.restrict K) := by
-  rw [← lintegral_finsetSum I (f := fun i z => (∏ j : Fin 2, f i (z j)) * cappedDistanceKernel α b z)
-    (fun i hi => (Finset.measurable_prod _ (fun j _ => (hf i hi).comp (measurable_pi_apply j))).mul
-      (measurable_cappedDistanceKernel α b))]
-  rw [← lintegral_const_mul _ (nearDiagonalKernel_measurable α b r)]
-  apply lintegral_mono
-  intro z
-  simp only [Fin.prod_univ_two]
-  rw [← Finset.sum_mul]
-  apply (mul_le_mul' (hcover (z 0) (z 1)) le_rfl).trans_eq
-  unfold nearDiagonalKernel
-  split_ifs <;> simp only [zero_mul,mul_zero]
-
-end SLEExactGauge.Multipoint
-
-end
-
-end OAI
+import OAI.Probability.SLE.Multipoint.MultipointKernelExpansion
+import OAI.Probability.SLE.Multipoint.MultipointTupleNull
+import Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence
+
+namespace OAI
+
+/-! Integrable two-point kernels have vanishing mass near the diagonal.
+Finite cutoff sums with bounded overlap inherit this decay. -/
+
+noncomputable section
+open Set Filter MeasureTheory Classical
+open scoped ENNReal Topology
+
+namespace SLEExactGauge.Multipoint
+
+def nearDiagonalKernel (α b r : ℝ) (z : Fin 2 → ℂ) : ℝ≥0∞ :=
+  if dist (z 0) (z 1) ≤ r then cappedDistanceKernel α b z else 0
+
+lemma nearDiagonalKernel_measurable (α b r : ℝ) : Measurable (nearDiagonalKernel α b r) :=
+  (measurable_cappedDistanceKernel α b).ite
+    (measurableSet_le ((measurable_pi_apply 0).dist (measurable_pi_apply 1)) measurable_const)
+    measurable_const
+
+theorem nearDiagonalKernel_integral_tendsto_zero {α : ℝ} (hα : 0 < α) (hα1 : α < 1)
+    (b : ℝ) (K : Set ℂ) (hK : Bornology.IsBounded K)
+    (r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
+    Tendsto (fun n => ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel α b (r n) z
+      ∂Measure.pi (fun _ : Fin 2 => volume.restrict K)) atTop (𝓝 0) := by
+  have hh := tendsto_lintegral_of_dominated_convergence
+    (μ := Measure.pi (fun _ : Fin 2 => volume.restrict K))
+    (F := fun n => nearDiagonalKernel α b (r n)) (f := fun _ => 0)
+    (cappedDistanceKernel α b) (fun n => nearDiagonalKernel_measurable α b (r n))
+    (fun n => Eventually.of_forall (fun z => by
+      unfold nearDiagonalKernel
+      split_ifs <;> simp only [le_refl,zero_le]))
+    (cappedDistanceKernel_integral_lt_top hα hα1 b K hK).ne ?_
+  · simpa only [lintegral_zero] using hh
+  filter_upwards [ae_tuple_pair_ne K (0 : Fin 2) 1 (by decide)] with z hz
+  have hdist : 0 < dist (z 0) (z 1) := dist_pos.mpr hz
+  have he : ∀ᶠ n in atTop, nearDiagonalKernel α b (r n) z = 0 := by
+    filter_upwards [(tendsto_order.mp hr).2 _ hdist] with n hn
+    exact ite_eq_right (not_le.mpr hn)
+  exact tendsto_const_nhds.congr' (he.mono (fun _ hn => hn.symm))
+
+lemma sum_weighted_pair_kernel_le {ι : Type*} (I : Finset ι)
+    (f : ι → ℂ → ℝ≥0∞) (hf : ∀ i ∈ I, Measurable (f i))
+    (α b r : ℝ) (N : ℝ≥0∞)
+    (hcover : ∀ x y, (∑ i ∈ I, f i x * f i y) ≤ if dist x y ≤ r then N else 0)
+    (K : Set ℂ) :
+    (∑ i ∈ I, ∫⁻ z : Fin 2 → ℂ, (∏ j, f i (z j)) * cappedDistanceKernel α b z
+      ∂Measure.pi (fun _ : Fin 2 => volume.restrict K)) ≤
+      N * ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel α b r z
+        ∂Measure.pi (fun _ : Fin 2 => volume.restrict K) := by
+  rw [← lintegral_finsetSum I (f := fun i z => (∏ j : Fin 2, f i (z j)) * cappedDistanceKernel α b z)
+    (fun i hi => (Finset.measurable_prod _ (fun j _ => (hf i hi).comp (measurable_pi_apply j))).mul
+      (measurable_cappedDistanceKernel α b))]
+  rw [← lintegral_const_mul _ (nearDiagonalKernel_measurable α b r)]
+  apply lintegral_mono
+  intro z
+  simp only [Fin.prod_univ_two]
+  rw [← Finset.sum_mul]
+  apply (mul_le_mul' (hcover (z 0) (z 1)) le_rfl).trans_eq
+  unfold nearDiagonalKernel
+  split_ifs <;> simp only [zero_mul,mul_zero]
+
+end SLEExactGauge.Multipoint
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/NearRealContraction.lean b/lean/OAI/Probability/SLE/NearRealContraction.lean
index ca15b293220ae971fa7b0f236f1e02360eb55422..0b98aa4fab9402305bdaf4cf3946c4f0a7fd92e1 100644
--- a/lean/OAI/Probability/SLE/NearRealContraction.lean
+++ b/lean/OAI/Probability/SLE/NearRealContraction.lean
@@ -1,794 +1,794 @@
-import Mathlib
-import OAI.Probability.SLE.SwallowExit
-
-namespace OAI
-
-/-! Near Real Contraction for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma reciprocal_real_near_one {z : ℂ} {δ : ℝ}
-    (hδ : 0 ≤ δ) (hδ2 : δ ≤ 1/2) (hz : ‖z‖ ≤ δ) :
-    1 - 2*δ ≤ ((1+z)⁻¹).re := by
-  have hl : 1-δ ≤ ‖1+z‖ := by
-    have hh := norm_sub_le (1+z) z
-    simp only [add_sub_cancel_right, norm_one] at hh
-    linarith
-  have hn : 0 < ‖1+z‖ := by linarith
-  have hne : 1+z ≠ 0 := norm_pos_iff.mp hn
-  have he : (1+z)⁻¹ - 1 = -z/(1+z) := by field_simp; ring
-  have hb : ‖(1+z)⁻¹ - 1‖ ≤ 2*δ := by
-    rw [he,norm_div,norm_neg]
-    apply (div_le_iff₀ hn).mpr
-    nlinarith
-  have hr := Complex.abs_re_le_norm ((1+z)⁻¹-1)
-  simp only [sub_re,one_re] at hr
-  have := (abs_le.mp (hr.trans hb)).1
-  linarith
-
-lemma boundary_difference_coefficient {X δ : ℝ} {D : ℂ}
-    (hX : 0 < X) (hδ : 0 ≤ δ) (hδ2 : δ ≤ 1/2) (hD : ‖D‖ ≤ δ*X) :
-    (-2 / ((X:ℂ)*((X:ℂ)+D))).re ≤ (-2+4*δ)*X⁻¹^2 := by
-  have hx : (X:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hX.ne'
-  have hz : ‖D/(X:ℂ)‖ ≤ δ := by
-    rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hX]
-    exact (div_le_iff₀ hX).mpr hD
-  have hh := reciprocal_real_near_one hδ hδ2 hz
-  have he : (-2 / ((X:ℂ)*((X:ℂ)+D))) =
-      ((-2*X⁻¹^2:ℝ):ℂ)*(1+D/(X:ℂ))⁻¹ := by
-    push_cast
-    field_simp
-  rw [he,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
-  nlinarith [sq_nonneg X⁻¹]
-
- 
-
-theorem near_real_difference_bound
-    (T : ℝ) (hT : 0 ≤ T) (U Q : ℝ → ℝ) (Z : ℝ → ℂ)
-    (hU : ContinuousOn U (Icc 0 T)) (hQ : ContinuousOn Q (Icc 0 T))
-    (hZ : ContinuousOn Z (Icc 0 T))
-    (hpos : ∀ t ∈ Icc 0 T, 0 < Q t-U t)
-    (hQd : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q (2/(Q t-U t)) (Icc 0 T) t)
-    (hZd : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z (2/(Z t-(U t:ℂ))) (Icc 0 T) t)
-    (δ : ℝ) (hδ : 0 ≤ δ) (hδ2 : δ ≤ 1/2)
-    (hratio : ∀ t ∈ Icc 0 T, ‖Z t-(Q t:ℂ)‖ ≤ δ*(Q t-U t)) :
-    ‖Z T-(Q T:ℂ)‖ ≤ ‖Z 0-(Q 0:ℂ)‖ *
-      Real.exp ((-2+4*δ)*(∫ t in (0:ℝ)..T, (Q t-U t)⁻¹^2)) := by
-  let X (t : ℝ) := Q t-U t
-  let D (t : ℝ) := Z t-(Q t:ℂ)
-  have hXc : ContinuousOn X (Icc 0 T) := hQ.sub hU
-  have hDc : ContinuousOn D (Icc 0 T) := hZ.sub (Complex.continuous_ofReal.comp_continuousOn hQ)
-  have hne (t : ℝ) (ht : t ∈ Icc 0 T) : (X t:ℂ)+D t ≠ 0 := by
-    have hl : X t-‖D t‖ ≤ ‖(X t:ℂ)+D t‖ := by
-      have hh := norm_sub_le ((X t:ℂ)+D t) (D t)
-      simp only [add_sub_cancel_right,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (show 0 < X t from hpos t ht)] at hh
-      linarith
-    apply norm_pos_iff.mp
-    have hpp := hpos t ht
-    have hrr := hratio t ht
-    change ‖D t‖ ≤ δ*X t at hrr
-    change 0 < X t at hpp
-    nlinarith
-  let c (t : ℝ) := -2 / ((X t:ℂ)*((X t:ℂ)+D t))
-  have hcc : ContinuousOn c (Icc 0 T) := by
-    apply continuousOn_const.div
-      ((Complex.continuous_ofReal.comp_continuousOn hXc).mul
-        ((Complex.continuous_ofReal.comp_continuousOn hXc).add hDc))
-    intro t ht
-    exact mul_ne_zero (Complex.ofReal_ne_zero.mpr (hpos t ht).ne') (hne t ht)
-  have hd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt D (c t*D t) (Icc 0 T) t := by
-    have hh := (hZd t ht).sub (Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t (hQd t ht))
-    have halg : c t*D t = 2/(Z t-(U t:ℂ)) - ((2/(Q t-U t):ℝ):ℂ) := by
-      dsimp only [c,D,X]
-      have hxn := Complex.ofReal_ne_zero.mpr (hpos t ht).ne'
-      have hzn : Z t-(U t:ℂ) ≠ 0 := by
-        convert hne t ht using 1
-        simp only [X,D,Complex.ofReal_sub]
-        ring
-      simp only [Complex.ofReal_div, Complex.ofReal_ofNat, Complex.ofReal_sub]
-      rw [show (Q t:ℂ)-(U t:ℂ)+(Z t-(Q t:ℂ))=Z t-(U t:ℂ) by ring]
-      have hx' : (Q t:ℂ)-(U t:ℂ) ≠ 0 := by simpa only [Complex.ofReal_sub] using hxn
-      field_simp [hx',hzn]
-      ring
-    convert hh using 1 <;> try rfl
-    exact halg
-  have he := complex_linear_ode_exp T hT D c hcc hd
-  have hci : IntervalIntegrable (fun t => (c t).re) volume 0 T :=
-    (Complex.continuous_re.comp_continuousOn hcc).intervalIntegrable_of_Icc hT
-  have hxi : IntervalIntegrable (fun t => (-2+4*δ)*(X t)⁻¹^2) volume 0 T := by
-    exact (continuousOn_const.mul ((hXc.inv₀ (fun t ht => (hpos t ht).ne')).pow 2)).intervalIntegrable_of_Icc hT
-  have hi : (∫ t in (0:ℝ)..T, (c t).re) ≤
-      (-2+4*δ)*(∫ t in (0:ℝ)..T, (X t)⁻¹^2) := by
-    rw [← intervalIntegral.integral_const_mul]
-    apply intervalIntegral.integral_mono_on hT hci hxi
-    intro t ht
-    exact boundary_difference_coefficient (hpos t ht) hδ hδ2 (hratio t ht)
-  change ‖D T‖ ≤ _
-  have hir : (∫ t in (0:ℝ)..T, (c t).re) = (∫ t in (0:ℝ)..T, c t).re :=
-    Complex.reCLM.intervalIntegral_comp_comm (hcc.intervalIntegrable_of_Icc hT)
-  rw [he,norm_mul,Complex.norm_exp,← hir]
-  exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hi) (norm_nonneg _)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-def BoundaryComparisonFailure (U : ℝ≥0 → ℝ) (x r δ : ℝ) : Prop :=
-  ∃ (T : ℝ≥0) (Q Z : ℝ → ℂ), IsPositiveBoundaryClockPath U x T Q ∧
-    (∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Z
-      (2/(Z t-(U t.toNNReal:ℂ))) (Icc 0 (T:ℝ)) t) ∧
-    ‖Z 0-(x:ℂ)‖ ≤ r ∧
-    (∀ t ∈ Icc 0 (T:ℝ), ‖Z t-((Q t).re:ℂ)‖ ≤ δ*((Q t).re-U t.toNNReal)) ∧
-    δ*((Q T).re-U T) ≤ ‖Z T-((Q T).re:ℂ)‖
-
-lemma BoundaryComparisonFailure.density_crossing
-    {U : ℝ≥0 → ℝ} {x r δ β κ : ℝ}
-    (hU : Continuous U) (hr : 0 < r) (hδ : 0 < δ) (hδ2 : δ ≤ 1/2) (hβ : 0 < β)
-    (hfail : BoundaryComparisonFailure U x r δ)
-    (B : ℝ≥0 → ℝ) (heU : ∀ t, U t = Real.sqrt κ*B t) :
-    ∃ (T : ℝ≥0) (Q : ℝ → ℂ), IsPositiveBoundaryClockPath U x T Q ∧
-      Real.exp (β*Real.log (δ/r)) ≤ boundaryDensity κ (2-4*δ) β (Q T) (B T) := by
-  obtain ⟨T,Q,Z,hQ,hZd,hZ0,hratio,hend⟩ := hfail
-  have hQd (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt (fun u => (Q u).re)
-      (2/((Q t).re-U t.toNNReal)) (Icc 0 (T:ℝ)) t := by
-    simpa only [Function.comp_def,Complex.reCLM_apply,Complex.add_re,Complex.ofReal_re,
-      Complex.mul_re,Complex.I_re,mul_zero,Complex.ofReal_im,Complex.I_im,mul_one,
-      sub_zero,add_zero] using Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t (hQ.2.2 t ht)
-  have hn := near_real_difference_bound (T:ℝ) T.coe_nonneg
-    (fun t => U t.toNNReal) (fun t => (Q t).re) Z
-    (hU.comp continuous_real_toNNReal).continuousOn
-    (Complex.continuous_re.comp_continuousOn hQ.continuousOn)
-    (fun t ht => (hZd t ht).continuousWithinAt) hQ.2.1 hQd hZd δ hδ.le hδ2 hratio
-  have hc := hQ.clock_identity hU
-  have hq0 : ((Q 0).re:ℂ) = (x:ℂ) := by rw [hQ.1,Complex.ofReal_re]
-  rw [hq0,← hc] at hn
-  have hD : δ*((Q T).re-U T) ≤ r*Real.exp (-(2-4*δ)*(Q T).im) := by
-    have hh := hn.trans (mul_le_mul_of_nonneg_right hZ0 (Real.exp_pos _).le)
-    rw [show -2+4*δ = -(2-4*δ) by ring] at hh
-    exact hend.trans hh
-  have hX : 0 < (Q T).re-U T := by simpa only [Real.toNNReal_coe] using hQ.2.1 T ⟨T.coe_nonneg,le_rfl⟩
-  have hquot : δ/r ≤ Real.exp (-(2-4*δ)*(Q T).im)/((Q T).re-U T) :=
-    (div_le_div_iff₀ hr hX).mpr (by nlinarith [hD])
-  have hlog := Real.log_le_log (div_pos hδ hr) hquot
-  rw [Real.log_div (Real.exp_pos _).ne' hX.ne',Real.log_exp] at hlog
-  refine ⟨T,Q,hQ,?_⟩
-  unfold boundaryDensity
-  rw [← heU T]
-  apply Real.exp_le_exp.mpr
-  nlinarith [mul_le_mul_of_nonneg_left hlog hβ.le]
-
-lemma boundary_density_threshold_initial {κ α β x r δ : ℝ}
-    (hx : 0 < x) (hr : 0 < r) (_hδ : 0 < δ) (hβ : 0 ≤ β) (hrx : r ≤ δ*x) :
-    boundaryDensity κ α β (x:ℂ) 0 ≤ Real.exp (β*Real.log (δ/r)) := by
-  have hq : x⁻¹ ≤ δ/r := by
-    rw [inv_eq_one_div]
-    apply (div_le_div_iff₀ hx hr).mpr
-    simpa only [one_mul] using hrx
-  have hl := Real.log_le_log (inv_pos.mpr hx) hq
-  rw [Real.log_inv] at hl
-  simp only [boundaryDensity,Complex.ofReal_im,mul_zero,zero_sub,Complex.ofReal_re,sub_zero]
-  exact Real.exp_le_exp.mpr (by nlinarith [mul_le_mul_of_nonneg_left hl hβ])
-
-lemma boundary_density_threshold_quotient {κ α β x r δ : ℝ}
-    (hx : 0 < x) (hr : 0 < r) (hδ : 0 < δ) :
-    boundaryDensity κ α β (x:ℂ) 0 / Real.exp (β*Real.log (δ/r)) =
-      (r/(δ*x))^β := by
-  simp only [boundaryDensity,Complex.ofReal_im,mul_zero,zero_sub,Complex.ofReal_re,sub_zero]
-  rw [← Real.exp_sub,Real.rpow_def_of_pos (div_pos hr (mul_pos hδ hx)),
-    Real.log_div hr.ne' (mul_pos hδ hx).ne',Real.log_mul hδ.ne' hx.ne',
-    Real.log_div hδ.ne' hr.ne']
-  congr 1
-  ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-lemma near_real_first_contact {T : ℝ} (hT : 0 ≤ T) {f : ℝ → ℝ}
-    (hc : ContinuousOn f (Icc 0 T)) (h0 : f 0 < 0) (hTval : 0 ≤ f T) :
-    ∃ s ∈ Icc 0 T, f s = 0 ∧ ∀ t ∈ Icc 0 s, f t ≤ 0 := by
-  let F (t : ℝ) := f (projIcc 0 T hT t)
-  have hFc : Continuous F := hc.comp_continuous
-    (continuous_subtype_val.comp continuous_projIcc) (fun t => (projIcc 0 T hT t).property)
-  have hFe (t : ℝ) (ht : t ∈ Icc 0 T) : F t = f t := by
-    simp only [F,projIcc_of_mem hT ht]
-  obtain ⟨u,hu,hfu⟩ := intermediate_value_Icc hT hc ⟨h0.le,hTval⟩
-  let A := Icc 0 T ∩ F ⁻¹' {0}
-  have hAc : IsCompact A := isCompact_Icc.inter_right (isClosed_singleton.preimage hFc)
-  have hAn : A.Nonempty := ⟨u,hu,by simpa only [mem_preimage,mem_singleton_iff,hFe u hu] using hfu⟩
-  obtain ⟨s,hs,hmin⟩ := hAc.exists_isMinOn hAn continuousOn_id
-  have hfs : f s = 0 := (hFe s hs.1).symm.trans hs.2
-  refine ⟨s,hs.1,hfs,?_⟩
-  intro t ht
-  by_contra hn
-  have htp : 0 < f t := lt_of_not_ge hn
-  have hsub : Icc 0 t ⊆ Icc 0 T := fun v hv => ⟨hv.1,hv.2.trans (ht.2.trans hs.1.2)⟩
-  obtain ⟨v,hv,hfv⟩ := intermediate_value_Icc ht.1 (hc.mono hsub) ⟨h0.le,htp.le⟩
-  have hvA : v ∈ A := ⟨hsub hv,by simpa only [mem_preimage,mem_singleton_iff,hFe v (hsub hv)] using hfv⟩
-  have hst : s ≤ t := (hmin hvA).trans hv.2
-  have hts : t = s := le_antisymm ht.2 hst
-  exact htp.ne' (hts ▸ hfs)
-
-lemma rightField_im (a u : ℝ) (z : ℂ) :
-    (rightField a u z).im = (-2 / Complex.normSq (rightCenter a u z))*z.im := by
-  simp only [rightField,Complex.mul_im,Complex.re_ofNat,Complex.im_ofNat,
-    zero_mul,add_zero,Complex.inv_im]
-  have him : (rightCenter a u z).im = z.im := by simp [rightCenter]
-  rw [him]
-  ring
-
- 
-
-lemma right_flow_im_pos {U : ℝ → ℝ} (hU : Continuous U) {a : ℝ} (ha : 0 < a)
-    {Z : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z
-      (rightField a (U t) (Z t)) (Icc 0 T) t)
-    (h0 : 0 < (Z 0).im) : 0 < (Z T).im := by
-  have hZc : ContinuousOn Z (Icc 0 T) := fun t ht => (hd t ht).continuousWithinAt
-  let c (t : ℝ) := -2 / Complex.normSq (rightCenter a (U t) (Z t))
-  have hcenter : ContinuousOn (fun t => rightCenter a (U t) (Z t)) (Icc 0 T) := by
-    unfold rightCenter
-    fun_prop
-  have hcc : ContinuousOn c (Icc 0 T) :=
-    continuousOn_const.div (Complex.continuous_normSq.comp_continuousOn hcenter)
-      (fun t ht => (Complex.normSq_pos.mpr (rightCenter_ne_zero ha _ _)).ne')
-  have hfd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt
-      (fun s => ((Z s).im:ℂ)) ((c t:ℂ)*((Z t).im:ℂ)) (Icc 0 T) t := by
-    have hh := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t
-      (Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hd t ht))
-    simpa only [Function.comp_def,Complex.ofRealCLM_apply,Complex.imCLM_apply,
-      rightField_im,Complex.ofReal_mul,c] using hh
-  have he := complex_linear_ode_exp T hT (fun t => ((Z t).im:ℂ)) (fun t => (c t:ℂ))
-    (Complex.continuous_ofReal.comp_continuousOn hcc) hfd
-  have hint : (∫ t in (0:ℝ)..T, (c t:ℂ)) = ((∫ t in (0:ℝ)..T, c t : ℝ):ℂ) := by
-    exact Complex.ofRealCLM.intervalIntegral_comp_comm (hcc.intervalIntegrable_of_Icc hT)
-  rw [hint,← Complex.ofReal_exp,← Complex.ofReal_mul] at he
-  have heR : (Z T).im = (Z 0).im * Real.exp (∫ t in (0:ℝ)..T, c t) := Complex.ofReal_injective he
-  rw [heR]
-  exact mul_pos h0 (Real.exp_pos _)
-
-lemma right_flow_unclipped_of_ratio {a δ X u q : ℝ} {z : ℂ}
-    (haX : a ≤ X) (hX : 0 ≤ X) (hδ : δ ≤ 1/2)
-    (hXeq : X = q-u) (hratio : ‖z-(q:ℂ)‖ ≤ δ*X) :
-    a/2 ≤ z.re-u := by
-  have hn : q-z.re ≤ ‖z-(q:ℂ)‖ := by
-    have hh := Complex.re_le_norm (-(z-(q:ℂ)))
-    simpa only [Complex.neg_re,Complex.sub_re,Complex.ofReal_re,neg_sub,norm_neg,norm_sub_rev] using hh
-  have hmul := mul_le_mul_of_nonneg_right hδ hX
-  nlinarith
-
- 
-
-lemma no_boundary_failure_gives_complex_flow
-    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
-    {x r δ : ℝ} (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
-    (hno : ¬ BoundaryComparisonFailure U x r δ)
-    {T : ℝ≥0} {Q : ℝ → ℂ} (hQ : IsPositiveBoundaryClockPath U x T Q)
-    (w : ℂ) (hw : ‖w-(x:ℂ)‖ ≤ r) :
-    ∃ Z : ℝ → ℂ, Z 0 = w ∧
-      (∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Z
-        (2/(Z t-(U t.toNNReal:ℂ))) (Icc 0 (T:ℝ)) t) ∧
-      (∀ t ∈ Icc 0 (T:ℝ), ‖Z t-((Q t).re:ℂ)‖ < δ*((Q t).re-U t.toNNReal)) := by
-  obtain ⟨a,ha,hgap⟩ := hQ.positive_lower_bound hU
-  have hUc : Continuous (fun t : ℝ => U t.toNNReal) := hU.comp continuous_real_toNNReal
-  obtain ⟨Y,hYm,hY0,hYd⟩ := exists_measurable_right_flow
-    (fun t (_ : Unit) => U t.toNNReal)
-    (hUc.measurable.comp measurable_fst) (fun _ => hUc)
-    (fun _ => w) measurable_const (half_pos ha)
-  let Z (t : ℝ) := Y t ()
-  have hZ0 : Z 0 = w := hY0 ()
-  have hZd (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt Z
-      (rightField (a/2) (U t.toNNReal) (Z t)) (Ici 0) t := hYd () t ht
-  have hZc : ContinuousOn Z (Icc 0 (T:ℝ)) :=
-    fun t ht => ((hZd t ht.1).mono (fun s hs => hs.1)).continuousWithinAt
-  let f (t : ℝ) := ‖Z t-((Q t).re:ℂ)‖-δ*((Q t).re-U t.toNNReal)
-  have hfc : ContinuousOn f (Icc 0 (T:ℝ)) := by
-    exact (hZc.sub (Complex.continuous_ofReal.comp_continuousOn
-      (Complex.continuous_re.comp_continuousOn hQ.continuousOn))).norm.sub
-      (continuousOn_const.mul ((Complex.continuous_re.comp_continuousOn hQ.continuousOn).sub hUc.continuousOn))
-  have hf0 : f 0 < 0 := by
-    dsimp only [f]
-    rw [hZ0,hQ.1,Complex.ofReal_re,Real.toNNReal_zero,hU0,sub_zero]
-    linarith
-  have hratio (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : f t < 0 := by
-    by_contra hn
-    have hsub : Icc 0 t ⊆ Icc 0 (T:ℝ) := fun s hs => ⟨hs.1,hs.2.trans ht.2⟩
-    obtain ⟨s,hs,hfs,hbefore⟩ := near_real_first_contact ht.1 (hfc.mono hsub) hf0 (le_of_not_gt hn)
-    let S : ℝ≥0 := ⟨s,hs.1⟩
-    have hsubS : Icc 0 (S:ℝ) ⊆ Icc 0 (T:ℝ) := fun v hv => ⟨hv.1,hv.2.trans (hs.2.trans ht.2)⟩
-    have hcomp (v : ℝ) (hv : v ∈ Icc 0 (S:ℝ)) :
-        ‖Z v-((Q v).re:ℂ)‖ ≤ δ*((Q v).re-U v.toNNReal) := sub_nonpos.mp (hbefore v hv)
-    have hreal (v : ℝ) (hv : v ∈ Icc 0 (S:ℝ)) : a/2 ≤ (Z v).re-U v.toNNReal :=
-      right_flow_unclipped_of_ratio (hgap v (hsubS hv)) (hQ.2.1 v (hsubS hv)).le hδ2 rfl (hcomp v hv)
-    apply hno
-    refine ⟨S,Q,Z,⟨hQ.1,fun v hv => hQ.2.1 v (hsubS hv),
-      fun v hv => (hQ.2.2 v (hsubS hv)).mono hsubS⟩,?_,?_,hcomp,?_⟩
-    · intro v hv
-      rw [← rightField_eq (hreal v hv)]
-      exact (hZd v hv.1).mono (fun u hu => hu.1)
-    · simpa only [hZ0] using hw
-    · have hh : ‖Z s-((Q s).re:ℂ)‖ = δ*((Q s).re-U s.toNNReal) := sub_eq_zero.mp hfs
-      have hS : s.toNNReal = S := by
-        apply NNReal.eq
-        exact Real.coe_toNNReal s hs.1
-      change δ*((Q s).re-U S) ≤ ‖Z s-((Q s).re:ℂ)‖
-      rw [← hS]
-      exact hh.ge
-  have hreal (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a/2 ≤ (Z t).re-U t.toNNReal :=
-    right_flow_unclipped_of_ratio (hgap t ht) (hQ.2.1 t ht).le hδ2 rfl (sub_neg.mp (hratio t ht)).le
-  refine ⟨Z,hZ0,?_,fun t ht => sub_neg.mp (hratio t ht)⟩
-  intro t ht
-  rw [← rightField_eq (hreal t ht)]
-  exact (hZd t ht.1).mono (fun s hs => hs.1)
-
- 
-
-theorem IsCapacityTwoWitness.half_ball_survives_before_boundary_collision
-    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
-    {x r δ : ℝ} (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
-    (hno : ¬ BoundaryComparisonFailure U x r δ)
-    {T : ℝ≥0} {Q : ℝ → ℂ} (hQ : IsPositiveBoundaryClockPath U x T Q) :
-    ∀ w : ℂ, 0 < w.im → ‖w-(x:ℂ)‖ ≤ r →
-      w ∈ survivingDomain γ T ∧
-      ‖G T w-((Q T).re:ℂ)‖ < δ*((Q T).re-U T) := by
-  intro w hwp hw
-  obtain ⟨Z,hZ0,hZd,hratio⟩ := no_boundary_failure_gives_complex_flow hU hU0 hrx hδ2 hno hQ w hw
-  obtain ⟨a,ha,hgap⟩ := hQ.positive_lower_bound hU
-  have hclip (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a/2 ≤ (Z t).re-U t.toNNReal :=
-    right_flow_unclipped_of_ratio (hgap t ht) (hQ.2.1 t ht).le hδ2 rfl (hratio t ht).le
-  have hZp (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : 0 < (Z t).im := by
-    have hsub : Icc 0 t ⊆ Icc 0 (T:ℝ) := fun s hs => ⟨hs.1,hs.2.trans ht.2⟩
-    apply right_flow_im_pos (hU.comp continuous_real_toNNReal) (half_pos ha) ht.1
-    · intro s hs
-      dsimp only [Function.comp_def]
-      rw [rightField_eq (hclip s (hsub hs))]
-      exact (hZd s (hsub hs)).mono hsub
-    · simpa only [hZ0] using hwp
-  obtain ⟨hlive,he⟩ := h.forward_flow_survives T Z hZp hZd
-  rw [hZ0] at hlive he
-  refine ⟨hlive,?_⟩
-  rw [he]
-  simpa only [Real.toNNReal_coe] using hratio T ⟨T.coe_nonneg,le_rfl⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma upper_mobius_norm_lt_one {v w : ℂ} (hv : 0 < v.im) (hw : 0 < w.im) :
-    ‖(v - w) / (v - star w)‖ < 1 := by
-  have hd : v - star w ≠ 0 := by
-    intro he
-    have hh := congrArg Complex.im he
-    simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hh
-    linarith
-  rw [norm_div, div_lt_one (norm_pos_iff.mpr hd)]
-  have he : ‖v - w‖ ^ 2 < ‖v - star w‖ ^ 2 := by
-    simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im,
-      Complex.star_def, Complex.conj_re, Complex.conj_im]
-    nlinarith [mul_pos hv hw]
-  nlinarith [norm_nonneg (v - w), norm_nonneg (v - star w)]
-
- 
-
-lemma radius_of_halfplane_map {f : ℂ → ℂ} {z : ℂ} {r : ℝ} (hr : 0 < r)
-    (hd : DifferentiableOn ℂ f (ball z r))
-    (hm : MapsTo f (ball z r) upperHalfPlane) (hf : deriv f z ≠ 0) :
-    r ≤ 2 * (f z).im / ‖deriv f z‖ := by
-  have hz : z ∈ ball z r := mem_ball_self hr
-  have hy : 0 < (f z).im := hm hz
-  have hden (w : ℂ) (hw : w ∈ ball z r) : f w - star (f z) ≠ 0 := by
-    intro he
-    have hx := congrArg Complex.im he
-    simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hx
-    have hwyo : 0 < (f w).im := hm hw
-    linarith
-  let g := fun w => (f w - f z) / (f w - star (f z))
-  have hg0 : g z = 0 := by dsimp [g]; rw [sub_self, zero_div]
-  have hgd : DifferentiableOn ℂ g (ball z r) :=
-    (hd.sub_const _).div (hd.sub_const _) hden
-  have hgm : MapsTo g (ball z r) (closedBall (g z) 1) := by
-    intro w hw
-    rw [hg0, mem_closedBall_zero_iff]
-    exact (upper_mobius_norm_lt_one (hm hw) hy).le
-  have hdz : HasDerivAt f (deriv f z) z := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
-  have hgg : deriv g z = deriv f z / (f z - star (f z)) := by
-    calc
-      deriv g z = (deriv f z * (f z - star (f z)) - (f z - f z) * deriv f z) /
-          (f z - star (f z)) ^ 2 :=
-        ((hdz.sub_const (f z)).div (hdz.sub_const (star (f z))) (hden z hz)).deriv
-      _ = _ := by
-        rw [sub_self, zero_mul, sub_zero, pow_two, mul_div_mul_right _ _ (hden z hz)]
-  have hn : ‖f z - star (f z)‖ = 2 * (f z).im := by
-    rw [Complex.star_def, Complex.sub_conj, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-      abs_of_pos (mul_pos (by norm_num) hy), Complex.norm_I, mul_one]
-  have hh := Complex.norm_deriv_le_div_of_mapsTo_ball hgd hgm hr
-  rw [hgg, norm_div, hn] at hh
-  have hn' := norm_pos_iff.mpr hf
-  apply (le_div_iff₀ hn').mpr
-  have hq := (div_le_div_iff₀ (mul_pos (by norm_num) hy) hr).mp hh
-  nlinarith
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma ball_subset_survivingDomain {γ : ℝ≥0 → ℂ} {t : ℝ≥0} {z : ℂ} {r : ℝ}
-    (hr : 0 < r) (hz : z ∈ survivingDomain γ t)
-    (hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 t)) :
-    ball z r ⊆ survivingDomain γ t := by
-  have hcc := (convex_ball z r).isPreconnected.subset_connectedComponentIn (mem_ball_self hr) hfree
-  intro w hw
-  refine ⟨hfree hw, ?_⟩
-  rw [← connectedComponentIn_eq (hcc hw)]
-  exact hz.2
-
-lemma ball_subset_upperHalfPlane {z : ℂ} {r : ℝ} (hr : r ≤ z.im) :
-    ball z r ⊆ upperHalfPlane := by
-  intro w hw
-  have hh : ‖w - z‖ < r := mem_ball_iff_norm.mp hw
-  have hi := Complex.abs_im_le_norm (w - z)
-  simp only [Complex.sub_im] at hi
-  change 0 < w.im
-  have hx := neg_le_abs (w.im - z.im)
-  linarith
-
- 
-
-lemma IsCapacityTwoWitness.trace_approach_of_radius_lt
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) (r : ℝ)
-    (hr : 0 < r) (hy : r ≤ z.im) (hR : fullRadius G t z < r) :
-    ∃ s ∈ Icc 0 t, dist (γ s) z < r := by
-  by_contra hn
-  have hn' (s : ℝ≥0) (hs : s ∈ Icc 0 t) : r ≤ dist (γ s) z :=
-    le_of_not_gt (fun he => hn ⟨s, hs, he⟩)
-  have hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 t) := by
-    intro w hw
-    refine ⟨ball_subset_upperHalfPlane hy hw, ?_⟩
-    rintro ⟨s, hs, rfl⟩
-    exact not_lt_of_ge (hn' s hs) (mem_ball.mp hw)
-  have hb := ball_subset_survivingDomain hr hz hfree
-  have hd : DifferentiableOn ℂ (G t) (ball z r) := (h.2.2.2.1 t).mono hb
-  have hm : MapsTo (G t) (ball z r) upperHalfPlane := fun w hw => h.2.2.2.2.2.1 t (hb hw)
-  have hf : deriv (G t) z ≠ 0 := norm_pos_iff.mp (h.derivative_norm_positive hU t z hz)
-  exact not_lt_of_ge (radius_of_halfplane_map hr hd hm hf) hR
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma IsCapacityTwoWitness.trace_mem_closure_surviving
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (s t : ℝ≥0) (hst : s ≤ t) :
-    γ t ∈ closure (survivingDomain γ s) := by
-  have hfmap := h.2.2.2.2.2.2.1 t
-  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2 t
-  apply mem_closure_of_tendsto htip
-  filter_upwards [self_mem_nhdsWithin] with y hy
-  apply survivingDomain_antitone γ hst
-  apply hfmap
-  change 0 < (((U t : ℝ) : ℂ) + (y : ℂ) * Complex.I).im
-  simpa using hy
-
-lemma IsCapacityTwoWitness.surviving_of_free_ball_future_tip
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (s t : ℝ≥0) (hst : s ≤ t)
-    (z : ℂ) (r : ℝ) (hr : 0 < r)
-    (hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 s))
-    (htip : γ t ∈ ball z r) : z ∈ survivingDomain γ s := by
-  obtain ⟨w, hwb, hw⟩ := mem_closure_iff.mp (h.trace_mem_closure_surviving s t hst)
-    (ball z r) isOpen_ball htip
-  have hcc := (convex_ball z r).isPreconnected.subset_connectedComponentIn (mem_ball_self hr) hfree
-  refine ⟨hfree (mem_ball_self hr), ?_⟩
-  rw [connectedComponentIn_eq (hcc hwb)]
-  exact hw.2
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex Metric
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-lemma IsCapacityTwoWitness.boundary_path_of_half_ball_survivor
-    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
-    {x r δ : ℝ} (hx : 0 < x) (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
-    (hno : ¬ BoundaryComparisonFailure U x r δ)
-    (T : ℝ≥0) (w : ℂ) (hw : ‖w-(x:ℂ)‖ ≤ r)
-    (hlive : w ∈ survivingDomain γ T) :
-    ∃ Q : ℝ → ℂ, IsPositiveBoundaryClockPath U x T Q := by
-  let a := min (x/2) (G T w).im
-  have hy : 0 < (G T w).im := h.imag_positive T w hlive
-  have ha : 0 < a := lt_min (half_pos hx) hy
-  have hax : a < x := (min_le_left _ _).trans_lt (half_lt_self hx)
-  have hay : a ≤ (G T w).im := min_le_right _ _
-  obtain ⟨L,Y,hYm,hYl,hY0,hYd,hYa⟩ := boundary_clock_global_adapted_solution
-    (fun t (_ : Unit) => U t) (hU.measurable.comp measurable_fst) (fun _ => hU)
-    1 a ha (x:ℂ)
-  let Q (t : ℝ) := Y t ()
-  have hQ0 : Q 0 = (x:ℂ) := hY0 ()
-  have hQd (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt Q
-      (boundaryClockField a (U t.toNNReal) (Q t)) (Ici 0) t := by
-    simpa only [Real.sqrt_one,one_mul] using hYd () t ht
-  have hQc : Continuous Q := (hYl ()).continuous
-  let f (t : ℝ) := a-((Q t).re-U t.toNNReal)
-  have hfc : Continuous f := continuous_const.sub
-    ((Complex.continuous_re.comp hQc).sub (hU.comp continuous_real_toNNReal))
-  have hf0 : f 0 < 0 := by
-    dsimp only [f]
-    rw [hQ0,Complex.ofReal_re,Real.toNNReal_zero,hU0,sub_zero]
-    exact sub_neg.mpr hax
-  have hgap (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a < (Q t).re-U t.toNNReal := by
-    by_contra hn
-    obtain ⟨s,hs,hfs,hbefore⟩ := near_real_first_contact ht.1 hfc.continuousOn hf0
-      (sub_nonneg.mpr (le_of_not_gt hn))
-    let S : ℝ≥0 := ⟨s,hs.1⟩
-    have hS : s.toNNReal = S := by apply NNReal.eq; exact Real.coe_toNNReal s hs.1
-    have hST : S ≤ T := hs.2.trans ht.2
-    have hgapS (v : ℝ) (hv : v ∈ Icc 0 (S:ℝ)) : a ≤ (Q v).re-U v.toNNReal :=
-      sub_nonpos.mp (hbefore v hv)
-    have hpath : IsPositiveBoundaryClockPath U x S Q := by
-      refine ⟨hQ0,fun v hv => ha.trans_le (hgapS v hv),?_⟩
-      intro v hv
-      simpa only [boundaryClockField,positiveBoundaryField_eq (hgapS v hv),max_eq_right (hgapS v hv)] using
-        (hQd v hv.1).mono (show Icc (0:ℝ) (S:ℝ) ⊆ Ici 0 from fun u hu => hu.1)
-    have hratio := (h.half_ball_survives_before_boundary_collision hU hU0 hrx hδ2 hno hpath
-      w hlive.1.1 hw).2
-    have hgapEq : (Q S).re-U S = a := by
-      change (Q s).re-U S = a
-      rw [← hS]
-      exact (sub_eq_zero.mp hfs).symm
-    rw [hgapEq] at hratio
-    have him : (G S w).im ≤ ‖G S w-((Q S).re:ℂ)‖ := by
-      simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using
-        Complex.im_le_norm (G S w-((Q S).re:ℂ))
-    have hmono : (G T w).im ≤ (G S w).im := by
-      have hh := h.imag_antitoneOn T w hlive ⟨S.coe_nonneg,hST⟩ ⟨T.coe_nonneg,le_rfl⟩ hST
-      simpa only [Real.toNNReal_coe] using hh
-    have hδa : δ*a ≤ a/2 := by nlinarith [mul_le_mul_of_nonneg_right hδ2 ha.le]
-    linarith
-  refine ⟨Q,hQ0,fun t ht => ha.trans (hgap t ht),?_⟩
-  intro t ht
-  simpa only [boundaryClockField,positiveBoundaryField_eq (hgap t ht).le,max_eq_right (hgap t ht).le] using
-    (hQd t ht.1).mono (show Icc (0:ℝ) (T:ℝ) ⊆ Ici 0 from fun s hs => hs.1)
-
-lemma IsCapacityTwoWitness.tip_avoids_half_ball_of_boundary_path
-    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
-    {x r δ : ℝ} (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
-    (hno : ¬ BoundaryComparisonFailure U x r δ)
-    {T : ℝ≥0} {Q : ℝ → ℂ} (hQ : IsPositiveBoundaryClockPath U x T Q) :
-    r ≤ ‖γ T-(x:ℂ)‖ := by
-  by_contra hn
-  have hnear : ∀ᶠ y : ℝ in 𝓝[>] 0,
-      ‖F T ((U T:ℂ)+(y:ℂ)*I)-(x:ℂ)‖ < r := by
-    apply (tendsto_order.1 ((h.2.2.2.2.2.2.2.2.2.2.2.2 T).sub tendsto_const_nhds).norm).2
-    exact lt_of_not_ge hn
-  obtain ⟨y,hynear,hy⟩ := (hnear.and (show ∀ᶠ y : ℝ in 𝓝[>] 0, 0 < y from self_mem_nhdsWithin)).exists
-  have hupper : (U T:ℂ)+(y:ℂ)*I ∈ upperHalfPlane := by
-    change 0 < ((U T:ℂ)+(y:ℂ)*I).im
-    simpa only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
-      Complex.I_im,mul_one,Complex.I_re,mul_zero,add_zero,zero_add] using hy
-  have hlive := h.2.2.2.2.2.2.1 T hupper
-  have hratio := (h.half_ball_survives_before_boundary_collision hU hU0 hrx hδ2 hno hQ
-    _ hlive.1.1 hynear.le).2
-  have hg := h.2.2.2.2.2.2.2.2.1 T _ hupper
-  rw [hg] at hratio
-  have hX : 0 < (Q T).re-U T := by
-    simpa only [Real.toNNReal_coe] using hQ.2.1 T ⟨T.coe_nonneg,le_rfl⟩
-  have hmargin := right_flow_unclipped_of_ratio (a := (Q T).re-U T) le_rfl hX.le hδ2 rfl hratio.le
-  simp only [Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.I_re,mul_zero,
-    Complex.ofReal_im,Complex.I_im,zero_mul,add_zero,sub_self] at hmargin
-  linarith
-
- 
-
-theorem IsCapacityTwoWitness.all_time_real_center_avoidance
-    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
-    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
-    {x r δ : ℝ} (hx : 0 < x) (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
-    (hno : ¬ BoundaryComparisonFailure U x r δ) :
-    ∀ T : ℝ≥0, r ≤ ‖γ T-(x:ℂ)‖ := by
-  intro T
-  by_contra hn
-  have htip : γ T ∈ ball (x:ℂ) r := by
-    rw [mem_ball,dist_eq_norm]
-    exact lt_of_not_ge hn
-  obtain ⟨w,hwb,hw⟩ := mem_closure_iff.mp (h.trace_mem_closure_surviving T T le_rfl)
-    (ball (x:ℂ) r) isOpen_ball htip
-  have hwn : ‖w-(x:ℂ)‖ ≤ r := (by simpa only [mem_ball,dist_eq_norm] using hwb : ‖w-(x:ℂ)‖ < r).le
-  obtain ⟨Q,hQ⟩ := h.boundary_path_of_half_ball_survivor hU hU0 hx hrx hδ2 hno T w hwn hw
-  exact hn (h.tip_avoids_half_ball_of_boundary_path hU hU0 hrx hδ2 hno hQ)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma boundary_comparison_failure_probability_regular
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ β x r δ : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hr : 0 < r)
-    (hδ : 0 < δ) (hδ2 : δ ≤ 1/2) (hβ : 0 < β)
-    (heq : -(2-4*δ)-2+κ*(β+1)/2=0) :
-    P {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ≤
-      ENNReal.ofReal ((r/(δ*x))^β) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  by_cases hrx : r ≤ δ*x
-  · have hprob := boundary_maximal_density hB κ (2-4*δ) β x
-      (Real.exp (β*Real.log (δ/r))) hκ hx (Real.exp_pos _) heq
-        (boundary_density_threshold_initial hx hr hδ hβ.le hrx)
-    rw [boundary_density_threshold_quotient hx hr hδ] at hprob
-    apply (measure_mono (show {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ⊆
-      {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ), IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
-        Real.exp (β*Real.log (δ/r)) ≤ boundaryDensity κ (2-4*δ) β (Q T) (B T ω)} from ?_)).trans hprob
-    intro ω hω
-    exact hω.density_crossing ((hBc ω).const_mul _) hr hδ hδ2 hβ (fun t => B t ω) (fun t => rfl)
-  · have hq : 1 ≤ r/(δ*x) := (le_div_iff₀ (mul_pos hδ hx)).mpr (by linarith)
-    have hp := Real.one_le_rpow hq hβ.le
-    exact prob_le_one.trans (by simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal hp)
-
- 
-
-theorem boundary_comparison_failure_probability
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ β x r δ : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hr : 0 < r)
-    (hδ : 0 < δ) (hδ2 : δ ≤ 1/2) (hβ : 0 < β)
-    (heq : -(2-4*δ)-2+κ*(β+1)/2=0) :
-    P {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ≤
-      ENNReal.ofReal ((r/(δ*x))^β) := by
-  obtain ⟨C,hCm,hCc,hC0,hC,hCB⟩ := exists_regular_brownian_version hB
-  apply (measure_mono_ae (show {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ≤ᵐ[P]
-      {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*C t ω) x r δ} from ?_)).trans
-      (boundary_comparison_failure_probability_regular hC hCc κ β x r δ hκ hx hr hδ hδ2 hβ heq)
-  filter_upwards [hCB] with ω he
-  intro h
-  change BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ at h
-  simpa only [he] using h
-
- 
-theorem boundary_comparison_failure_sle_exponent
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ x r : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hx : 0 < x) (hr : 0 < r) :
-    P {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r ((8-κ)/32)} ≤
-      ENNReal.ofReal ((r/(((8-κ)/32)*x))^(3*(8-κ)/(4*κ))) := by
-  apply boundary_comparison_failure_probability hB κ _ x r _ hκ hx hr
-  · positivity
-  · linarith
-  · positivity
-  · field_simp
-    ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-variable {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω}
-  {ℱ : Filtration ℝ≥0 m} {τ : Ω → ℝ≥0}
-
- 
-
-lemma stopping_grid_piece_measurable
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) (n k : ℕ) :
-    MeasurableSet[ℱ (stopGridTime n k)] (A ∩ {ω | stopGridIndex n (τ ω) = k}) := by
-  have hle (j : ℕ) : MeasurableSet[ℱ (stopGridTime n j)]
-      (A ∩ {ω | stopGridIndex n (τ ω) ≤ j}) := by
-    simpa only [stopGridIndex_le, WithTop.coe_le_coe] using hA.2 (stopGridTime n j)
-  cases k with
-  | zero => simpa only [Nat.le_zero] using hle 0
-  | succ k =>
-    have he : A ∩ {ω | stopGridIndex n (τ ω) = k + 1} =
-        (A ∩ {ω | stopGridIndex n (τ ω) ≤ k + 1}) \ (A ∩ {ω | stopGridIndex n (τ ω) ≤ k}) := by
-      ext ω
-      simp only [mem_inter_iff, mem_ofPred_eq, Set.mem_sdiff]
-      constructor
-      · rintro ⟨hω, heq⟩
-        exact ⟨⟨hω, heq.le⟩, fun hb => by omega⟩
-      · rintro ⟨⟨hω, hle⟩, hn⟩
-        exact ⟨hω, by have hh : ¬ stopGridIndex n (τ ω) ≤ k := fun hk => hn ⟨hω, hk⟩; omega⟩
-    rw [he]
-    exact (hle (k + 1)).diff (ℱ.mono (monotone_stopGridTime n (Nat.le_succ k)) _ (hle k))
-
-omit m in
-lemma grid_partition_union (A : Set Ω) (n : ℕ) :
-    (⋃ k : ℕ, A ∩ {ω | stopGridIndex n (τ ω) = k}) = A := by
-  ext ω
-  simp only [mem_iUnion, mem_inter_iff, mem_ofPred_eq]
-  exact ⟨fun ⟨_, h, _⟩ => h, fun h => ⟨_, h, rfl⟩⟩
-
-omit m in
-lemma grid_partition_disjoint (A : Set Ω) (n : ℕ) :
-    Pairwise (fun i j : ℕ => Disjoint (A ∩ {ω | stopGridIndex n (τ ω) = i}) (A ∩ {ω | stopGridIndex n (τ ω) = j})) := by
-  intro i j hij
-  apply disjoint_left.mpr
-  intro ω hi hj
-  exact hij (hi.2.symm.trans hj.2)
-
- 
-
-lemma integral_stopping_grid_glue [IsFiniteMeasure P]
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) (n : ℕ)
-    (f : Ω → ℝ) (hf : Integrable f P) (c : ℝ)
-    (hpiece : ∀ k : ℕ, (∫ ω in A ∩ {ω | stopGridIndex n (τ ω) = k}, f ω ∂P) =
-      P.real (A ∩ {ω | stopGridIndex n (τ ω) = k}) * c) :
-    (∫ ω in A, f ω ∂P) = P.real A * c := by
-  have hm (k : ℕ) : MeasurableSet (A ∩ {ω | stopGridIndex n (τ ω) = k}) :=
-    ℱ.le _ _ (stopping_grid_piece_measurable hτ hA n k)
-  have hfSum := hasSum_integral_iUnion hm (grid_partition_disjoint (τ := τ) A n) hf.integrableOn
-  have hconstSum := hasSum_integral_iUnion hm (grid_partition_disjoint (τ := τ) A n)
-    (integrable_const (1 : ℝ) : Integrable (fun _ : Ω => (1 : ℝ)) P).integrableOn
-  simp only [grid_partition_union, hpiece] at hfSum
-  simp only [grid_partition_union, integral_const, smul_eq_mul, mul_one, measureReal_restrict_apply_univ] at hconstSum
-  exact HasSum.unique hfSum (hconstSum.mul_right c)
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.SwallowExit
+
+namespace OAI
+
+/-! Near Real Contraction for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma reciprocal_real_near_one {z : ℂ} {δ : ℝ}
+    (hδ : 0 ≤ δ) (hδ2 : δ ≤ 1/2) (hz : ‖z‖ ≤ δ) :
+    1 - 2*δ ≤ ((1+z)⁻¹).re := by
+  have hl : 1-δ ≤ ‖1+z‖ := by
+    have hh := norm_sub_le (1+z) z
+    simp only [add_sub_cancel_right, norm_one] at hh
+    linarith
+  have hn : 0 < ‖1+z‖ := by linarith
+  have hne : 1+z ≠ 0 := norm_pos_iff.mp hn
+  have he : (1+z)⁻¹ - 1 = -z/(1+z) := by field_simp; ring
+  have hb : ‖(1+z)⁻¹ - 1‖ ≤ 2*δ := by
+    rw [he,norm_div,norm_neg]
+    apply (div_le_iff₀ hn).mpr
+    nlinarith
+  have hr := Complex.abs_re_le_norm ((1+z)⁻¹-1)
+  simp only [sub_re,one_re] at hr
+  have := (abs_le.mp (hr.trans hb)).1
+  linarith
+
+lemma boundary_difference_coefficient {X δ : ℝ} {D : ℂ}
+    (hX : 0 < X) (hδ : 0 ≤ δ) (hδ2 : δ ≤ 1/2) (hD : ‖D‖ ≤ δ*X) :
+    (-2 / ((X:ℂ)*((X:ℂ)+D))).re ≤ (-2+4*δ)*X⁻¹^2 := by
+  have hx : (X:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hX.ne'
+  have hz : ‖D/(X:ℂ)‖ ≤ δ := by
+    rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hX]
+    exact (div_le_iff₀ hX).mpr hD
+  have hh := reciprocal_real_near_one hδ hδ2 hz
+  have he : (-2 / ((X:ℂ)*((X:ℂ)+D))) =
+      ((-2*X⁻¹^2:ℝ):ℂ)*(1+D/(X:ℂ))⁻¹ := by
+    push_cast
+    field_simp
+  rw [he,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  nlinarith [sq_nonneg X⁻¹]
+
+ 
+
+theorem near_real_difference_bound
+    (T : ℝ) (hT : 0 ≤ T) (U Q : ℝ → ℝ) (Z : ℝ → ℂ)
+    (hU : ContinuousOn U (Icc 0 T)) (hQ : ContinuousOn Q (Icc 0 T))
+    (hZ : ContinuousOn Z (Icc 0 T))
+    (hpos : ∀ t ∈ Icc 0 T, 0 < Q t-U t)
+    (hQd : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q (2/(Q t-U t)) (Icc 0 T) t)
+    (hZd : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z (2/(Z t-(U t:ℂ))) (Icc 0 T) t)
+    (δ : ℝ) (hδ : 0 ≤ δ) (hδ2 : δ ≤ 1/2)
+    (hratio : ∀ t ∈ Icc 0 T, ‖Z t-(Q t:ℂ)‖ ≤ δ*(Q t-U t)) :
+    ‖Z T-(Q T:ℂ)‖ ≤ ‖Z 0-(Q 0:ℂ)‖ *
+      Real.exp ((-2+4*δ)*(∫ t in (0:ℝ)..T, (Q t-U t)⁻¹^2)) := by
+  let X (t : ℝ) := Q t-U t
+  let D (t : ℝ) := Z t-(Q t:ℂ)
+  have hXc : ContinuousOn X (Icc 0 T) := hQ.sub hU
+  have hDc : ContinuousOn D (Icc 0 T) := hZ.sub (Complex.continuous_ofReal.comp_continuousOn hQ)
+  have hne (t : ℝ) (ht : t ∈ Icc 0 T) : (X t:ℂ)+D t ≠ 0 := by
+    have hl : X t-‖D t‖ ≤ ‖(X t:ℂ)+D t‖ := by
+      have hh := norm_sub_le ((X t:ℂ)+D t) (D t)
+      simp only [add_sub_cancel_right,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (show 0 < X t from hpos t ht)] at hh
+      linarith
+    apply norm_pos_iff.mp
+    have hpp := hpos t ht
+    have hrr := hratio t ht
+    change ‖D t‖ ≤ δ*X t at hrr
+    change 0 < X t at hpp
+    nlinarith
+  let c (t : ℝ) := -2 / ((X t:ℂ)*((X t:ℂ)+D t))
+  have hcc : ContinuousOn c (Icc 0 T) := by
+    apply continuousOn_const.div
+      ((Complex.continuous_ofReal.comp_continuousOn hXc).mul
+        ((Complex.continuous_ofReal.comp_continuousOn hXc).add hDc))
+    intro t ht
+    exact mul_ne_zero (Complex.ofReal_ne_zero.mpr (hpos t ht).ne') (hne t ht)
+  have hd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt D (c t*D t) (Icc 0 T) t := by
+    have hh := (hZd t ht).sub (Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t (hQd t ht))
+    have halg : c t*D t = 2/(Z t-(U t:ℂ)) - ((2/(Q t-U t):ℝ):ℂ) := by
+      dsimp only [c,D,X]
+      have hxn := Complex.ofReal_ne_zero.mpr (hpos t ht).ne'
+      have hzn : Z t-(U t:ℂ) ≠ 0 := by
+        convert hne t ht using 1
+        simp only [X,D,Complex.ofReal_sub]
+        ring
+      simp only [Complex.ofReal_div, Complex.ofReal_ofNat, Complex.ofReal_sub]
+      rw [show (Q t:ℂ)-(U t:ℂ)+(Z t-(Q t:ℂ))=Z t-(U t:ℂ) by ring]
+      have hx' : (Q t:ℂ)-(U t:ℂ) ≠ 0 := by simpa only [Complex.ofReal_sub] using hxn
+      field_simp [hx',hzn]
+      ring
+    convert hh using 1 <;> try rfl
+    exact halg
+  have he := complex_linear_ode_exp T hT D c hcc hd
+  have hci : IntervalIntegrable (fun t => (c t).re) volume 0 T :=
+    (Complex.continuous_re.comp_continuousOn hcc).intervalIntegrable_of_Icc hT
+  have hxi : IntervalIntegrable (fun t => (-2+4*δ)*(X t)⁻¹^2) volume 0 T := by
+    exact (continuousOn_const.mul ((hXc.inv₀ (fun t ht => (hpos t ht).ne')).pow 2)).intervalIntegrable_of_Icc hT
+  have hi : (∫ t in (0:ℝ)..T, (c t).re) ≤
+      (-2+4*δ)*(∫ t in (0:ℝ)..T, (X t)⁻¹^2) := by
+    rw [← intervalIntegral.integral_const_mul]
+    apply intervalIntegral.integral_mono_on hT hci hxi
+    intro t ht
+    exact boundary_difference_coefficient (hpos t ht) hδ hδ2 (hratio t ht)
+  change ‖D T‖ ≤ _
+  have hir : (∫ t in (0:ℝ)..T, (c t).re) = (∫ t in (0:ℝ)..T, c t).re :=
+    Complex.reCLM.intervalIntegral_comp_comm (hcc.intervalIntegrable_of_Icc hT)
+  rw [he,norm_mul,Complex.norm_exp,← hir]
+  exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hi) (norm_nonneg _)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+def BoundaryComparisonFailure (U : ℝ≥0 → ℝ) (x r δ : ℝ) : Prop :=
+  ∃ (T : ℝ≥0) (Q Z : ℝ → ℂ), IsPositiveBoundaryClockPath U x T Q ∧
+    (∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Z
+      (2/(Z t-(U t.toNNReal:ℂ))) (Icc 0 (T:ℝ)) t) ∧
+    ‖Z 0-(x:ℂ)‖ ≤ r ∧
+    (∀ t ∈ Icc 0 (T:ℝ), ‖Z t-((Q t).re:ℂ)‖ ≤ δ*((Q t).re-U t.toNNReal)) ∧
+    δ*((Q T).re-U T) ≤ ‖Z T-((Q T).re:ℂ)‖
+
+lemma BoundaryComparisonFailure.density_crossing
+    {U : ℝ≥0 → ℝ} {x r δ β κ : ℝ}
+    (hU : Continuous U) (hr : 0 < r) (hδ : 0 < δ) (hδ2 : δ ≤ 1/2) (hβ : 0 < β)
+    (hfail : BoundaryComparisonFailure U x r δ)
+    (B : ℝ≥0 → ℝ) (heU : ∀ t, U t = Real.sqrt κ*B t) :
+    ∃ (T : ℝ≥0) (Q : ℝ → ℂ), IsPositiveBoundaryClockPath U x T Q ∧
+      Real.exp (β*Real.log (δ/r)) ≤ boundaryDensity κ (2-4*δ) β (Q T) (B T) := by
+  obtain ⟨T,Q,Z,hQ,hZd,hZ0,hratio,hend⟩ := hfail
+  have hQd (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt (fun u => (Q u).re)
+      (2/((Q t).re-U t.toNNReal)) (Icc 0 (T:ℝ)) t := by
+    simpa only [Function.comp_def,Complex.reCLM_apply,Complex.add_re,Complex.ofReal_re,
+      Complex.mul_re,Complex.I_re,mul_zero,Complex.ofReal_im,Complex.I_im,mul_one,
+      sub_zero,add_zero] using Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t (hQ.2.2 t ht)
+  have hn := near_real_difference_bound (T:ℝ) T.coe_nonneg
+    (fun t => U t.toNNReal) (fun t => (Q t).re) Z
+    (hU.comp continuous_real_toNNReal).continuousOn
+    (Complex.continuous_re.comp_continuousOn hQ.continuousOn)
+    (fun t ht => (hZd t ht).continuousWithinAt) hQ.2.1 hQd hZd δ hδ.le hδ2 hratio
+  have hc := hQ.clock_identity hU
+  have hq0 : ((Q 0).re:ℂ) = (x:ℂ) := by rw [hQ.1,Complex.ofReal_re]
+  rw [hq0,← hc] at hn
+  have hD : δ*((Q T).re-U T) ≤ r*Real.exp (-(2-4*δ)*(Q T).im) := by
+    have hh := hn.trans (mul_le_mul_of_nonneg_right hZ0 (Real.exp_pos _).le)
+    rw [show -2+4*δ = -(2-4*δ) by ring] at hh
+    exact hend.trans hh
+  have hX : 0 < (Q T).re-U T := by simpa only [Real.toNNReal_coe] using hQ.2.1 T ⟨T.coe_nonneg,le_rfl⟩
+  have hquot : δ/r ≤ Real.exp (-(2-4*δ)*(Q T).im)/((Q T).re-U T) :=
+    (div_le_div_iff₀ hr hX).mpr (by nlinarith [hD])
+  have hlog := Real.log_le_log (div_pos hδ hr) hquot
+  rw [Real.log_div (Real.exp_pos _).ne' hX.ne',Real.log_exp] at hlog
+  refine ⟨T,Q,hQ,?_⟩
+  unfold boundaryDensity
+  rw [← heU T]
+  apply Real.exp_le_exp.mpr
+  nlinarith [mul_le_mul_of_nonneg_left hlog hβ.le]
+
+lemma boundary_density_threshold_initial {κ α β x r δ : ℝ}
+    (hx : 0 < x) (hr : 0 < r) (_hδ : 0 < δ) (hβ : 0 ≤ β) (hrx : r ≤ δ*x) :
+    boundaryDensity κ α β (x:ℂ) 0 ≤ Real.exp (β*Real.log (δ/r)) := by
+  have hq : x⁻¹ ≤ δ/r := by
+    rw [inv_eq_one_div]
+    apply (div_le_div_iff₀ hx hr).mpr
+    simpa only [one_mul] using hrx
+  have hl := Real.log_le_log (inv_pos.mpr hx) hq
+  rw [Real.log_inv] at hl
+  simp only [boundaryDensity,Complex.ofReal_im,mul_zero,zero_sub,Complex.ofReal_re,sub_zero]
+  exact Real.exp_le_exp.mpr (by nlinarith [mul_le_mul_of_nonneg_left hl hβ])
+
+lemma boundary_density_threshold_quotient {κ α β x r δ : ℝ}
+    (hx : 0 < x) (hr : 0 < r) (hδ : 0 < δ) :
+    boundaryDensity κ α β (x:ℂ) 0 / Real.exp (β*Real.log (δ/r)) =
+      (r/(δ*x))^β := by
+  simp only [boundaryDensity,Complex.ofReal_im,mul_zero,zero_sub,Complex.ofReal_re,sub_zero]
+  rw [← Real.exp_sub,Real.rpow_def_of_pos (div_pos hr (mul_pos hδ hx)),
+    Real.log_div hr.ne' (mul_pos hδ hx).ne',Real.log_mul hδ.ne' hx.ne',
+    Real.log_div hδ.ne' hr.ne']
+  congr 1
+  ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+lemma near_real_first_contact {T : ℝ} (hT : 0 ≤ T) {f : ℝ → ℝ}
+    (hc : ContinuousOn f (Icc 0 T)) (h0 : f 0 < 0) (hTval : 0 ≤ f T) :
+    ∃ s ∈ Icc 0 T, f s = 0 ∧ ∀ t ∈ Icc 0 s, f t ≤ 0 := by
+  let F (t : ℝ) := f (projIcc 0 T hT t)
+  have hFc : Continuous F := hc.comp_continuous
+    (continuous_subtype_val.comp continuous_projIcc) (fun t => (projIcc 0 T hT t).property)
+  have hFe (t : ℝ) (ht : t ∈ Icc 0 T) : F t = f t := by
+    simp only [F,projIcc_of_mem hT ht]
+  obtain ⟨u,hu,hfu⟩ := intermediate_value_Icc hT hc ⟨h0.le,hTval⟩
+  let A := Icc 0 T ∩ F ⁻¹' {0}
+  have hAc : IsCompact A := isCompact_Icc.inter_right (isClosed_singleton.preimage hFc)
+  have hAn : A.Nonempty := ⟨u,hu,by simpa only [mem_preimage,mem_singleton_iff,hFe u hu] using hfu⟩
+  obtain ⟨s,hs,hmin⟩ := hAc.exists_isMinOn hAn continuousOn_id
+  have hfs : f s = 0 := (hFe s hs.1).symm.trans hs.2
+  refine ⟨s,hs.1,hfs,?_⟩
+  intro t ht
+  by_contra hn
+  have htp : 0 < f t := lt_of_not_ge hn
+  have hsub : Icc 0 t ⊆ Icc 0 T := fun v hv => ⟨hv.1,hv.2.trans (ht.2.trans hs.1.2)⟩
+  obtain ⟨v,hv,hfv⟩ := intermediate_value_Icc ht.1 (hc.mono hsub) ⟨h0.le,htp.le⟩
+  have hvA : v ∈ A := ⟨hsub hv,by simpa only [mem_preimage,mem_singleton_iff,hFe v (hsub hv)] using hfv⟩
+  have hst : s ≤ t := (hmin hvA).trans hv.2
+  have hts : t = s := le_antisymm ht.2 hst
+  exact htp.ne' (hts ▸ hfs)
+
+lemma rightField_im (a u : ℝ) (z : ℂ) :
+    (rightField a u z).im = (-2 / Complex.normSq (rightCenter a u z))*z.im := by
+  simp only [rightField,Complex.mul_im,Complex.re_ofNat,Complex.im_ofNat,
+    zero_mul,add_zero,Complex.inv_im]
+  have him : (rightCenter a u z).im = z.im := by simp [rightCenter]
+  rw [him]
+  ring
+
+ 
+
+lemma right_flow_im_pos {U : ℝ → ℝ} (hU : Continuous U) {a : ℝ} (ha : 0 < a)
+    {Z : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z
+      (rightField a (U t) (Z t)) (Icc 0 T) t)
+    (h0 : 0 < (Z 0).im) : 0 < (Z T).im := by
+  have hZc : ContinuousOn Z (Icc 0 T) := fun t ht => (hd t ht).continuousWithinAt
+  let c (t : ℝ) := -2 / Complex.normSq (rightCenter a (U t) (Z t))
+  have hcenter : ContinuousOn (fun t => rightCenter a (U t) (Z t)) (Icc 0 T) := by
+    unfold rightCenter
+    fun_prop
+  have hcc : ContinuousOn c (Icc 0 T) :=
+    continuousOn_const.div (Complex.continuous_normSq.comp_continuousOn hcenter)
+      (fun t ht => (Complex.normSq_pos.mpr (rightCenter_ne_zero ha _ _)).ne')
+  have hfd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt
+      (fun s => ((Z s).im:ℂ)) ((c t:ℂ)*((Z t).im:ℂ)) (Icc 0 T) t := by
+    have hh := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t
+      (Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hd t ht))
+    simpa only [Function.comp_def,Complex.ofRealCLM_apply,Complex.imCLM_apply,
+      rightField_im,Complex.ofReal_mul,c] using hh
+  have he := complex_linear_ode_exp T hT (fun t => ((Z t).im:ℂ)) (fun t => (c t:ℂ))
+    (Complex.continuous_ofReal.comp_continuousOn hcc) hfd
+  have hint : (∫ t in (0:ℝ)..T, (c t:ℂ)) = ((∫ t in (0:ℝ)..T, c t : ℝ):ℂ) := by
+    exact Complex.ofRealCLM.intervalIntegral_comp_comm (hcc.intervalIntegrable_of_Icc hT)
+  rw [hint,← Complex.ofReal_exp,← Complex.ofReal_mul] at he
+  have heR : (Z T).im = (Z 0).im * Real.exp (∫ t in (0:ℝ)..T, c t) := Complex.ofReal_injective he
+  rw [heR]
+  exact mul_pos h0 (Real.exp_pos _)
+
+lemma right_flow_unclipped_of_ratio {a δ X u q : ℝ} {z : ℂ}
+    (haX : a ≤ X) (hX : 0 ≤ X) (hδ : δ ≤ 1/2)
+    (hXeq : X = q-u) (hratio : ‖z-(q:ℂ)‖ ≤ δ*X) :
+    a/2 ≤ z.re-u := by
+  have hn : q-z.re ≤ ‖z-(q:ℂ)‖ := by
+    have hh := Complex.re_le_norm (-(z-(q:ℂ)))
+    simpa only [Complex.neg_re,Complex.sub_re,Complex.ofReal_re,neg_sub,norm_neg,norm_sub_rev] using hh
+  have hmul := mul_le_mul_of_nonneg_right hδ hX
+  nlinarith
+
+ 
+
+lemma no_boundary_failure_gives_complex_flow
+    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
+    {x r δ : ℝ} (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
+    (hno : ¬ BoundaryComparisonFailure U x r δ)
+    {T : ℝ≥0} {Q : ℝ → ℂ} (hQ : IsPositiveBoundaryClockPath U x T Q)
+    (w : ℂ) (hw : ‖w-(x:ℂ)‖ ≤ r) :
+    ∃ Z : ℝ → ℂ, Z 0 = w ∧
+      (∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Z
+        (2/(Z t-(U t.toNNReal:ℂ))) (Icc 0 (T:ℝ)) t) ∧
+      (∀ t ∈ Icc 0 (T:ℝ), ‖Z t-((Q t).re:ℂ)‖ < δ*((Q t).re-U t.toNNReal)) := by
+  obtain ⟨a,ha,hgap⟩ := hQ.positive_lower_bound hU
+  have hUc : Continuous (fun t : ℝ => U t.toNNReal) := hU.comp continuous_real_toNNReal
+  obtain ⟨Y,hYm,hY0,hYd⟩ := exists_measurable_right_flow
+    (fun t (_ : Unit) => U t.toNNReal)
+    (hUc.measurable.comp measurable_fst) (fun _ => hUc)
+    (fun _ => w) measurable_const (half_pos ha)
+  let Z (t : ℝ) := Y t ()
+  have hZ0 : Z 0 = w := hY0 ()
+  have hZd (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt Z
+      (rightField (a/2) (U t.toNNReal) (Z t)) (Ici 0) t := hYd () t ht
+  have hZc : ContinuousOn Z (Icc 0 (T:ℝ)) :=
+    fun t ht => ((hZd t ht.1).mono (fun s hs => hs.1)).continuousWithinAt
+  let f (t : ℝ) := ‖Z t-((Q t).re:ℂ)‖-δ*((Q t).re-U t.toNNReal)
+  have hfc : ContinuousOn f (Icc 0 (T:ℝ)) := by
+    exact (hZc.sub (Complex.continuous_ofReal.comp_continuousOn
+      (Complex.continuous_re.comp_continuousOn hQ.continuousOn))).norm.sub
+      (continuousOn_const.mul ((Complex.continuous_re.comp_continuousOn hQ.continuousOn).sub hUc.continuousOn))
+  have hf0 : f 0 < 0 := by
+    dsimp only [f]
+    rw [hZ0,hQ.1,Complex.ofReal_re,Real.toNNReal_zero,hU0,sub_zero]
+    linarith
+  have hratio (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : f t < 0 := by
+    by_contra hn
+    have hsub : Icc 0 t ⊆ Icc 0 (T:ℝ) := fun s hs => ⟨hs.1,hs.2.trans ht.2⟩
+    obtain ⟨s,hs,hfs,hbefore⟩ := near_real_first_contact ht.1 (hfc.mono hsub) hf0 (le_of_not_gt hn)
+    let S : ℝ≥0 := ⟨s,hs.1⟩
+    have hsubS : Icc 0 (S:ℝ) ⊆ Icc 0 (T:ℝ) := fun v hv => ⟨hv.1,hv.2.trans (hs.2.trans ht.2)⟩
+    have hcomp (v : ℝ) (hv : v ∈ Icc 0 (S:ℝ)) :
+        ‖Z v-((Q v).re:ℂ)‖ ≤ δ*((Q v).re-U v.toNNReal) := sub_nonpos.mp (hbefore v hv)
+    have hreal (v : ℝ) (hv : v ∈ Icc 0 (S:ℝ)) : a/2 ≤ (Z v).re-U v.toNNReal :=
+      right_flow_unclipped_of_ratio (hgap v (hsubS hv)) (hQ.2.1 v (hsubS hv)).le hδ2 rfl (hcomp v hv)
+    apply hno
+    refine ⟨S,Q,Z,⟨hQ.1,fun v hv => hQ.2.1 v (hsubS hv),
+      fun v hv => (hQ.2.2 v (hsubS hv)).mono hsubS⟩,?_,?_,hcomp,?_⟩
+    · intro v hv
+      rw [← rightField_eq (hreal v hv)]
+      exact (hZd v hv.1).mono (fun u hu => hu.1)
+    · simpa only [hZ0] using hw
+    · have hh : ‖Z s-((Q s).re:ℂ)‖ = δ*((Q s).re-U s.toNNReal) := sub_eq_zero.mp hfs
+      have hS : s.toNNReal = S := by
+        apply NNReal.eq
+        exact Real.coe_toNNReal s hs.1
+      change δ*((Q s).re-U S) ≤ ‖Z s-((Q s).re:ℂ)‖
+      rw [← hS]
+      exact hh.ge
+  have hreal (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a/2 ≤ (Z t).re-U t.toNNReal :=
+    right_flow_unclipped_of_ratio (hgap t ht) (hQ.2.1 t ht).le hδ2 rfl (sub_neg.mp (hratio t ht)).le
+  refine ⟨Z,hZ0,?_,fun t ht => sub_neg.mp (hratio t ht)⟩
+  intro t ht
+  rw [← rightField_eq (hreal t ht)]
+  exact (hZd t ht.1).mono (fun s hs => hs.1)
+
+ 
+
+theorem IsCapacityTwoWitness.half_ball_survives_before_boundary_collision
+    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
+    {x r δ : ℝ} (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
+    (hno : ¬ BoundaryComparisonFailure U x r δ)
+    {T : ℝ≥0} {Q : ℝ → ℂ} (hQ : IsPositiveBoundaryClockPath U x T Q) :
+    ∀ w : ℂ, 0 < w.im → ‖w-(x:ℂ)‖ ≤ r →
+      w ∈ survivingDomain γ T ∧
+      ‖G T w-((Q T).re:ℂ)‖ < δ*((Q T).re-U T) := by
+  intro w hwp hw
+  obtain ⟨Z,hZ0,hZd,hratio⟩ := no_boundary_failure_gives_complex_flow hU hU0 hrx hδ2 hno hQ w hw
+  obtain ⟨a,ha,hgap⟩ := hQ.positive_lower_bound hU
+  have hclip (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a/2 ≤ (Z t).re-U t.toNNReal :=
+    right_flow_unclipped_of_ratio (hgap t ht) (hQ.2.1 t ht).le hδ2 rfl (hratio t ht).le
+  have hZp (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : 0 < (Z t).im := by
+    have hsub : Icc 0 t ⊆ Icc 0 (T:ℝ) := fun s hs => ⟨hs.1,hs.2.trans ht.2⟩
+    apply right_flow_im_pos (hU.comp continuous_real_toNNReal) (half_pos ha) ht.1
+    · intro s hs
+      dsimp only [Function.comp_def]
+      rw [rightField_eq (hclip s (hsub hs))]
+      exact (hZd s (hsub hs)).mono hsub
+    · simpa only [hZ0] using hwp
+  obtain ⟨hlive,he⟩ := h.forward_flow_survives T Z hZp hZd
+  rw [hZ0] at hlive he
+  refine ⟨hlive,?_⟩
+  rw [he]
+  simpa only [Real.toNNReal_coe] using hratio T ⟨T.coe_nonneg,le_rfl⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma upper_mobius_norm_lt_one {v w : ℂ} (hv : 0 < v.im) (hw : 0 < w.im) :
+    ‖(v - w) / (v - star w)‖ < 1 := by
+  have hd : v - star w ≠ 0 := by
+    intro he
+    have hh := congrArg Complex.im he
+    simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hh
+    linarith
+  rw [norm_div, div_lt_one (norm_pos_iff.mpr hd)]
+  have he : ‖v - w‖ ^ 2 < ‖v - star w‖ ^ 2 := by
+    simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im,
+      Complex.star_def, Complex.conj_re, Complex.conj_im]
+    nlinarith [mul_pos hv hw]
+  nlinarith [norm_nonneg (v - w), norm_nonneg (v - star w)]
+
+ 
+
+lemma radius_of_halfplane_map {f : ℂ → ℂ} {z : ℂ} {r : ℝ} (hr : 0 < r)
+    (hd : DifferentiableOn ℂ f (ball z r))
+    (hm : MapsTo f (ball z r) upperHalfPlane) (hf : deriv f z ≠ 0) :
+    r ≤ 2 * (f z).im / ‖deriv f z‖ := by
+  have hz : z ∈ ball z r := mem_ball_self hr
+  have hy : 0 < (f z).im := hm hz
+  have hden (w : ℂ) (hw : w ∈ ball z r) : f w - star (f z) ≠ 0 := by
+    intro he
+    have hx := congrArg Complex.im he
+    simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hx
+    have hwyo : 0 < (f w).im := hm hw
+    linarith
+  let g := fun w => (f w - f z) / (f w - star (f z))
+  have hg0 : g z = 0 := by dsimp [g]; rw [sub_self, zero_div]
+  have hgd : DifferentiableOn ℂ g (ball z r) :=
+    (hd.sub_const _).div (hd.sub_const _) hden
+  have hgm : MapsTo g (ball z r) (closedBall (g z) 1) := by
+    intro w hw
+    rw [hg0, mem_closedBall_zero_iff]
+    exact (upper_mobius_norm_lt_one (hm hw) hy).le
+  have hdz : HasDerivAt f (deriv f z) z := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
+  have hgg : deriv g z = deriv f z / (f z - star (f z)) := by
+    calc
+      deriv g z = (deriv f z * (f z - star (f z)) - (f z - f z) * deriv f z) /
+          (f z - star (f z)) ^ 2 :=
+        ((hdz.sub_const (f z)).div (hdz.sub_const (star (f z))) (hden z hz)).deriv
+      _ = _ := by
+        rw [sub_self, zero_mul, sub_zero, pow_two, mul_div_mul_right _ _ (hden z hz)]
+  have hn : ‖f z - star (f z)‖ = 2 * (f z).im := by
+    rw [Complex.star_def, Complex.sub_conj, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+      abs_of_pos (mul_pos (by norm_num) hy), Complex.norm_I, mul_one]
+  have hh := Complex.norm_deriv_le_div_of_mapsTo_ball hgd hgm hr
+  rw [hgg, norm_div, hn] at hh
+  have hn' := norm_pos_iff.mpr hf
+  apply (le_div_iff₀ hn').mpr
+  have hq := (div_le_div_iff₀ (mul_pos (by norm_num) hy) hr).mp hh
+  nlinarith
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma ball_subset_survivingDomain {γ : ℝ≥0 → ℂ} {t : ℝ≥0} {z : ℂ} {r : ℝ}
+    (hr : 0 < r) (hz : z ∈ survivingDomain γ t)
+    (hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 t)) :
+    ball z r ⊆ survivingDomain γ t := by
+  have hcc := (convex_ball z r).isPreconnected.subset_connectedComponentIn (mem_ball_self hr) hfree
+  intro w hw
+  refine ⟨hfree hw, ?_⟩
+  rw [← connectedComponentIn_eq (hcc hw)]
+  exact hz.2
+
+lemma ball_subset_upperHalfPlane {z : ℂ} {r : ℝ} (hr : r ≤ z.im) :
+    ball z r ⊆ upperHalfPlane := by
+  intro w hw
+  have hh : ‖w - z‖ < r := mem_ball_iff_norm.mp hw
+  have hi := Complex.abs_im_le_norm (w - z)
+  simp only [Complex.sub_im] at hi
+  change 0 < w.im
+  have hx := neg_le_abs (w.im - z.im)
+  linarith
+
+ 
+
+lemma IsCapacityTwoWitness.trace_approach_of_radius_lt
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) (r : ℝ)
+    (hr : 0 < r) (hy : r ≤ z.im) (hR : fullRadius G t z < r) :
+    ∃ s ∈ Icc 0 t, dist (γ s) z < r := by
+  by_contra hn
+  have hn' (s : ℝ≥0) (hs : s ∈ Icc 0 t) : r ≤ dist (γ s) z :=
+    le_of_not_gt (fun he => hn ⟨s, hs, he⟩)
+  have hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 t) := by
+    intro w hw
+    refine ⟨ball_subset_upperHalfPlane hy hw, ?_⟩
+    rintro ⟨s, hs, rfl⟩
+    exact not_lt_of_ge (hn' s hs) (mem_ball.mp hw)
+  have hb := ball_subset_survivingDomain hr hz hfree
+  have hd : DifferentiableOn ℂ (G t) (ball z r) := (h.2.2.2.1 t).mono hb
+  have hm : MapsTo (G t) (ball z r) upperHalfPlane := fun w hw => h.2.2.2.2.2.1 t (hb hw)
+  have hf : deriv (G t) z ≠ 0 := norm_pos_iff.mp (h.derivative_norm_positive hU t z hz)
+  exact not_lt_of_ge (radius_of_halfplane_map hr hd hm hf) hR
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma IsCapacityTwoWitness.trace_mem_closure_surviving
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (s t : ℝ≥0) (hst : s ≤ t) :
+    γ t ∈ closure (survivingDomain γ s) := by
+  have hfmap := h.2.2.2.2.2.2.1 t
+  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2 t
+  apply mem_closure_of_tendsto htip
+  filter_upwards [self_mem_nhdsWithin] with y hy
+  apply survivingDomain_antitone γ hst
+  apply hfmap
+  change 0 < (((U t : ℝ) : ℂ) + (y : ℂ) * Complex.I).im
+  simpa using hy
+
+lemma IsCapacityTwoWitness.surviving_of_free_ball_future_tip
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (s t : ℝ≥0) (hst : s ≤ t)
+    (z : ℂ) (r : ℝ) (hr : 0 < r)
+    (hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 s))
+    (htip : γ t ∈ ball z r) : z ∈ survivingDomain γ s := by
+  obtain ⟨w, hwb, hw⟩ := mem_closure_iff.mp (h.trace_mem_closure_surviving s t hst)
+    (ball z r) isOpen_ball htip
+  have hcc := (convex_ball z r).isPreconnected.subset_connectedComponentIn (mem_ball_self hr) hfree
+  refine ⟨hfree (mem_ball_self hr), ?_⟩
+  rw [connectedComponentIn_eq (hcc hwb)]
+  exact hw.2
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex Metric
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+lemma IsCapacityTwoWitness.boundary_path_of_half_ball_survivor
+    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
+    {x r δ : ℝ} (hx : 0 < x) (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
+    (hno : ¬ BoundaryComparisonFailure U x r δ)
+    (T : ℝ≥0) (w : ℂ) (hw : ‖w-(x:ℂ)‖ ≤ r)
+    (hlive : w ∈ survivingDomain γ T) :
+    ∃ Q : ℝ → ℂ, IsPositiveBoundaryClockPath U x T Q := by
+  let a := min (x/2) (G T w).im
+  have hy : 0 < (G T w).im := h.imag_positive T w hlive
+  have ha : 0 < a := lt_min (half_pos hx) hy
+  have hax : a < x := (min_le_left _ _).trans_lt (half_lt_self hx)
+  have hay : a ≤ (G T w).im := min_le_right _ _
+  obtain ⟨L,Y,hYm,hYl,hY0,hYd,hYa⟩ := boundary_clock_global_adapted_solution
+    (fun t (_ : Unit) => U t) (hU.measurable.comp measurable_fst) (fun _ => hU)
+    1 a ha (x:ℂ)
+  let Q (t : ℝ) := Y t ()
+  have hQ0 : Q 0 = (x:ℂ) := hY0 ()
+  have hQd (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt Q
+      (boundaryClockField a (U t.toNNReal) (Q t)) (Ici 0) t := by
+    simpa only [Real.sqrt_one,one_mul] using hYd () t ht
+  have hQc : Continuous Q := (hYl ()).continuous
+  let f (t : ℝ) := a-((Q t).re-U t.toNNReal)
+  have hfc : Continuous f := continuous_const.sub
+    ((Complex.continuous_re.comp hQc).sub (hU.comp continuous_real_toNNReal))
+  have hf0 : f 0 < 0 := by
+    dsimp only [f]
+    rw [hQ0,Complex.ofReal_re,Real.toNNReal_zero,hU0,sub_zero]
+    exact sub_neg.mpr hax
+  have hgap (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a < (Q t).re-U t.toNNReal := by
+    by_contra hn
+    obtain ⟨s,hs,hfs,hbefore⟩ := near_real_first_contact ht.1 hfc.continuousOn hf0
+      (sub_nonneg.mpr (le_of_not_gt hn))
+    let S : ℝ≥0 := ⟨s,hs.1⟩
+    have hS : s.toNNReal = S := by apply NNReal.eq; exact Real.coe_toNNReal s hs.1
+    have hST : S ≤ T := hs.2.trans ht.2
+    have hgapS (v : ℝ) (hv : v ∈ Icc 0 (S:ℝ)) : a ≤ (Q v).re-U v.toNNReal :=
+      sub_nonpos.mp (hbefore v hv)
+    have hpath : IsPositiveBoundaryClockPath U x S Q := by
+      refine ⟨hQ0,fun v hv => ha.trans_le (hgapS v hv),?_⟩
+      intro v hv
+      simpa only [boundaryClockField,positiveBoundaryField_eq (hgapS v hv),max_eq_right (hgapS v hv)] using
+        (hQd v hv.1).mono (show Icc (0:ℝ) (S:ℝ) ⊆ Ici 0 from fun u hu => hu.1)
+    have hratio := (h.half_ball_survives_before_boundary_collision hU hU0 hrx hδ2 hno hpath
+      w hlive.1.1 hw).2
+    have hgapEq : (Q S).re-U S = a := by
+      change (Q s).re-U S = a
+      rw [← hS]
+      exact (sub_eq_zero.mp hfs).symm
+    rw [hgapEq] at hratio
+    have him : (G S w).im ≤ ‖G S w-((Q S).re:ℂ)‖ := by
+      simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using
+        Complex.im_le_norm (G S w-((Q S).re:ℂ))
+    have hmono : (G T w).im ≤ (G S w).im := by
+      have hh := h.imag_antitoneOn T w hlive ⟨S.coe_nonneg,hST⟩ ⟨T.coe_nonneg,le_rfl⟩ hST
+      simpa only [Real.toNNReal_coe] using hh
+    have hδa : δ*a ≤ a/2 := by nlinarith [mul_le_mul_of_nonneg_right hδ2 ha.le]
+    linarith
+  refine ⟨Q,hQ0,fun t ht => ha.trans (hgap t ht),?_⟩
+  intro t ht
+  simpa only [boundaryClockField,positiveBoundaryField_eq (hgap t ht).le,max_eq_right (hgap t ht).le] using
+    (hQd t ht.1).mono (show Icc (0:ℝ) (T:ℝ) ⊆ Ici 0 from fun s hs => hs.1)
+
+lemma IsCapacityTwoWitness.tip_avoids_half_ball_of_boundary_path
+    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
+    {x r δ : ℝ} (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
+    (hno : ¬ BoundaryComparisonFailure U x r δ)
+    {T : ℝ≥0} {Q : ℝ → ℂ} (hQ : IsPositiveBoundaryClockPath U x T Q) :
+    r ≤ ‖γ T-(x:ℂ)‖ := by
+  by_contra hn
+  have hnear : ∀ᶠ y : ℝ in 𝓝[>] 0,
+      ‖F T ((U T:ℂ)+(y:ℂ)*I)-(x:ℂ)‖ < r := by
+    apply (tendsto_order.1 ((h.2.2.2.2.2.2.2.2.2.2.2.2 T).sub tendsto_const_nhds).norm).2
+    exact lt_of_not_ge hn
+  obtain ⟨y,hynear,hy⟩ := (hnear.and (show ∀ᶠ y : ℝ in 𝓝[>] 0, 0 < y from self_mem_nhdsWithin)).exists
+  have hupper : (U T:ℂ)+(y:ℂ)*I ∈ upperHalfPlane := by
+    change 0 < ((U T:ℂ)+(y:ℂ)*I).im
+    simpa only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
+      Complex.I_im,mul_one,Complex.I_re,mul_zero,add_zero,zero_add] using hy
+  have hlive := h.2.2.2.2.2.2.1 T hupper
+  have hratio := (h.half_ball_survives_before_boundary_collision hU hU0 hrx hδ2 hno hQ
+    _ hlive.1.1 hynear.le).2
+  have hg := h.2.2.2.2.2.2.2.2.1 T _ hupper
+  rw [hg] at hratio
+  have hX : 0 < (Q T).re-U T := by
+    simpa only [Real.toNNReal_coe] using hQ.2.1 T ⟨T.coe_nonneg,le_rfl⟩
+  have hmargin := right_flow_unclipped_of_ratio (a := (Q T).re-U T) le_rfl hX.le hδ2 rfl hratio.le
+  simp only [Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.I_re,mul_zero,
+    Complex.ofReal_im,Complex.I_im,zero_mul,add_zero,sub_self] at hmargin
+  linarith
+
+ 
+
+theorem IsCapacityTwoWitness.all_time_real_center_avoidance
+    {U : ℝ≥0 → ℝ} (hU : Continuous U) (hU0 : U 0 = 0)
+    {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
+    {x r δ : ℝ} (hx : 0 < x) (hrx : r < δ*x) (hδ2 : δ ≤ 1/2)
+    (hno : ¬ BoundaryComparisonFailure U x r δ) :
+    ∀ T : ℝ≥0, r ≤ ‖γ T-(x:ℂ)‖ := by
+  intro T
+  by_contra hn
+  have htip : γ T ∈ ball (x:ℂ) r := by
+    rw [mem_ball,dist_eq_norm]
+    exact lt_of_not_ge hn
+  obtain ⟨w,hwb,hw⟩ := mem_closure_iff.mp (h.trace_mem_closure_surviving T T le_rfl)
+    (ball (x:ℂ) r) isOpen_ball htip
+  have hwn : ‖w-(x:ℂ)‖ ≤ r := (by simpa only [mem_ball,dist_eq_norm] using hwb : ‖w-(x:ℂ)‖ < r).le
+  obtain ⟨Q,hQ⟩ := h.boundary_path_of_half_ball_survivor hU hU0 hx hrx hδ2 hno T w hwn hw
+  exact hn (h.tip_avoids_half_ball_of_boundary_path hU hU0 hrx hδ2 hno hQ)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma boundary_comparison_failure_probability_regular
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ β x r δ : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hr : 0 < r)
+    (hδ : 0 < δ) (hδ2 : δ ≤ 1/2) (hβ : 0 < β)
+    (heq : -(2-4*δ)-2+κ*(β+1)/2=0) :
+    P {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ≤
+      ENNReal.ofReal ((r/(δ*x))^β) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  by_cases hrx : r ≤ δ*x
+  · have hprob := boundary_maximal_density hB κ (2-4*δ) β x
+      (Real.exp (β*Real.log (δ/r))) hκ hx (Real.exp_pos _) heq
+        (boundary_density_threshold_initial hx hr hδ hβ.le hrx)
+    rw [boundary_density_threshold_quotient hx hr hδ] at hprob
+    apply (measure_mono (show {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ⊆
+      {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ), IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
+        Real.exp (β*Real.log (δ/r)) ≤ boundaryDensity κ (2-4*δ) β (Q T) (B T ω)} from ?_)).trans hprob
+    intro ω hω
+    exact hω.density_crossing ((hBc ω).const_mul _) hr hδ hδ2 hβ (fun t => B t ω) (fun t => rfl)
+  · have hq : 1 ≤ r/(δ*x) := (le_div_iff₀ (mul_pos hδ hx)).mpr (by linarith)
+    have hp := Real.one_le_rpow hq hβ.le
+    exact prob_le_one.trans (by simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal hp)
+
+ 
+
+theorem boundary_comparison_failure_probability
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ β x r δ : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hr : 0 < r)
+    (hδ : 0 < δ) (hδ2 : δ ≤ 1/2) (hβ : 0 < β)
+    (heq : -(2-4*δ)-2+κ*(β+1)/2=0) :
+    P {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ≤
+      ENNReal.ofReal ((r/(δ*x))^β) := by
+  obtain ⟨C,hCm,hCc,hC0,hC,hCB⟩ := exists_regular_brownian_version hB
+  apply (measure_mono_ae (show {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ} ≤ᵐ[P]
+      {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*C t ω) x r δ} from ?_)).trans
+      (boundary_comparison_failure_probability_regular hC hCc κ β x r δ hκ hx hr hδ hδ2 hβ heq)
+  filter_upwards [hCB] with ω he
+  intro h
+  change BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r δ at h
+  simpa only [he] using h
+
+ 
+theorem boundary_comparison_failure_sle_exponent
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ x r : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hx : 0 < x) (hr : 0 < r) :
+    P {ω | BoundaryComparisonFailure (fun t => Real.sqrt κ*B t ω) x r ((8-κ)/32)} ≤
+      ENNReal.ofReal ((r/(((8-κ)/32)*x))^(3*(8-κ)/(4*κ))) := by
+  apply boundary_comparison_failure_probability hB κ _ x r _ hκ hx hr
+  · positivity
+  · linarith
+  · positivity
+  · field_simp
+    ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+variable {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω}
+  {ℱ : Filtration ℝ≥0 m} {τ : Ω → ℝ≥0}
+
+ 
+
+lemma stopping_grid_piece_measurable
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) (n k : ℕ) :
+    MeasurableSet[ℱ (stopGridTime n k)] (A ∩ {ω | stopGridIndex n (τ ω) = k}) := by
+  have hle (j : ℕ) : MeasurableSet[ℱ (stopGridTime n j)]
+      (A ∩ {ω | stopGridIndex n (τ ω) ≤ j}) := by
+    simpa only [stopGridIndex_le, WithTop.coe_le_coe] using hA.2 (stopGridTime n j)
+  cases k with
+  | zero => simpa only [Nat.le_zero] using hle 0
+  | succ k =>
+    have he : A ∩ {ω | stopGridIndex n (τ ω) = k + 1} =
+        (A ∩ {ω | stopGridIndex n (τ ω) ≤ k + 1}) \ (A ∩ {ω | stopGridIndex n (τ ω) ≤ k}) := by
+      ext ω
+      simp only [mem_inter_iff, mem_ofPred_eq, Set.mem_sdiff]
+      constructor
+      · rintro ⟨hω, heq⟩
+        exact ⟨⟨hω, heq.le⟩, fun hb => by omega⟩
+      · rintro ⟨⟨hω, hle⟩, hn⟩
+        exact ⟨hω, by have hh : ¬ stopGridIndex n (τ ω) ≤ k := fun hk => hn ⟨hω, hk⟩; omega⟩
+    rw [he]
+    exact (hle (k + 1)).diff (ℱ.mono (monotone_stopGridTime n (Nat.le_succ k)) _ (hle k))
+
+omit m in
+lemma grid_partition_union (A : Set Ω) (n : ℕ) :
+    (⋃ k : ℕ, A ∩ {ω | stopGridIndex n (τ ω) = k}) = A := by
+  ext ω
+  simp only [mem_iUnion, mem_inter_iff, mem_ofPred_eq]
+  exact ⟨fun ⟨_, h, _⟩ => h, fun h => ⟨_, h, rfl⟩⟩
+
+omit m in
+lemma grid_partition_disjoint (A : Set Ω) (n : ℕ) :
+    Pairwise (fun i j : ℕ => Disjoint (A ∩ {ω | stopGridIndex n (τ ω) = i}) (A ∩ {ω | stopGridIndex n (τ ω) = j})) := by
+  intro i j hij
+  apply disjoint_left.mpr
+  intro ω hi hj
+  exact hij (hi.2.symm.trans hj.2)
+
+ 
+
+lemma integral_stopping_grid_glue [IsFiniteMeasure P]
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) (n : ℕ)
+    (f : Ω → ℝ) (hf : Integrable f P) (c : ℝ)
+    (hpiece : ∀ k : ℕ, (∫ ω in A ∩ {ω | stopGridIndex n (τ ω) = k}, f ω ∂P) =
+      P.real (A ∩ {ω | stopGridIndex n (τ ω) = k}) * c) :
+    (∫ ω in A, f ω ∂P) = P.real A * c := by
+  have hm (k : ℕ) : MeasurableSet (A ∩ {ω | stopGridIndex n (τ ω) = k}) :=
+    ℱ.le _ _ (stopping_grid_piece_measurable hτ hA n k)
+  have hfSum := hasSum_integral_iUnion hm (grid_partition_disjoint (τ := τ) A n) hf.integrableOn
+  have hconstSum := hasSum_integral_iUnion hm (grid_partition_disjoint (τ := τ) A n)
+    (integrable_const (1 : ℝ) : Integrable (fun _ : Ω => (1 : ℝ)) P).integrableOn
+  simp only [grid_partition_union, hpiece] at hfSum
+  simp only [grid_partition_union, integral_const, smul_eq_mul, mul_one, measureReal_restrict_apply_univ] at hconstSum
+  exact HasSum.unique hfSum (hconstSum.mul_right c)
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/NestedMeans.lean b/lean/OAI/Probability/SLE/NestedMeans.lean
index c601aef33013a0be49e1ba6596ba46ad1ee63289..9b0d9f9eae8603710a3a7fde06ac4cf29ada8410 100644
--- a/lean/OAI/Probability/SLE/NestedMeans.lean
+++ b/lean/OAI/Probability/SLE/NestedMeans.lean
@@ -1,842 +1,842 @@
-import Mathlib
-import OAI.Probability.SLE.RectangleMeans
-
-namespace OAI
-
-/-! Nested Means for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology ContDiff ENNReal BigOperators NNReal
-namespace SLEExactGauge
-universe u
-lemma slice_oscillation_le_integral {a b : ℝ} (hab : a < b) {f g : ℝ → ℝ}
-    (hf : ContinuousOn f (Icc a b)) (hl : LocallyLipschitzOn (Ioo a b) f)
-    (hg : IntegrableOn g (Icc a b)) (hgn : ∀ t ∈ Icc a b, 0 ≤ g t)
-    (hd : ∀ᵐ t ∂volume, t ∈ Ioo a b → ‖deriv f t‖ ≤ g t) :
-    ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, ‖f x-f y‖ ≤ ∫ t in Icc a b, g t := by
-  have hint (x y : ℝ) (hx : x ∈ Ioo a b) (hy : y ∈ Ioo a b) (hxy : x ≤ y) :
-      ‖f y-f x‖ ≤ ∫ t in Icc a b, g t := by
-    have hs : Icc x y ⊆ Ioo a b := by
-      intro t ht
-      exact ⟨hx.1.trans_le ht.1, ht.2.trans_lt hy.2⟩
-    obtain ⟨K,hK⟩ := LocallyLipschitzOn.exists_lipschitzOnWith_of_compact
-      isCompact_Icc (hl.mono hs)
-    have hAC : AbsolutelyContinuousOnInterval f x y :=
-      (by simpa [uIcc_of_le hxy] using hK : LipschitzOnWith K f (uIcc x y)).absolutelyContinuousOnInterval
-    rw [← hAC.integral_deriv_eq_sub, intervalIntegral.integral_of_le hxy]
-    have hdint : IntegrableOn (deriv f) (Ioc x y) :=
-      hAC.intervalIntegrable_deriv.1
-    have hgin : IntegrableOn g (Ioc x y) := hg.mono_set (by
-      intro t ht; exact ⟨hx.1.le.trans ht.1.le,ht.2.trans hy.2.le⟩)
-    calc
-      ‖∫ t in Ioc x y, deriv f t‖ ≤ ∫ t in Ioc x y, ‖deriv f t‖ := norm_integral_le_integral_norm _
-      _ ≤ ∫ t in Ioc x y, g t := integral_mono_ae hdint.norm hgin (by
-        filter_upwards [ae_restrict_of_ae hd,ae_restrict_mem measurableSet_Ioc] with t ht htS
-        exact ht (hs ⟨htS.1.le,htS.2⟩))
-      _ ≤ ∫ t in Icc a b, g t := setIntegral_mono_set hg
-        (by filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht; exact hgn t ht)
-        (Eventually.of_forall fun t ht => ⟨hx.1.le.trans ht.1.le,ht.2.trans hy.2.le⟩)
-  have hi (x : ℝ) (hx : x ∈ Ioo a b) (y : ℝ) (hy : y ∈ Ioo a b) :
-      ‖f x-f y‖ ≤ ∫ t in Icc a b, g t := by
-    rcases le_total x y with h | h
-    · simpa only [norm_sub_rev] using hint x y hx hy h
-    · exact hint y x hy hx h
-  have hc : closure (Ioo a b) = Icc a b := closure_Ioo hab.ne
-  have hyall (x : ℝ) (hx : x ∈ Ioo a b) :
-      ∀ y ∈ Icc a b, ‖f x-f y‖ ≤ ∫ t in Icc a b, g t := by
-    intro y hy
-    apply le_on_closure (fun y hy => hi x hx y hy) _ continuousOn_const (hc.symm ▸ hy)
-    rw [hc]
-    exact (continuousOn_const.sub hf).norm
-  intro x hx y hy
-  apply le_on_closure (fun x hx => hyall x hx y hy) _ continuousOn_const (hc.symm ▸ hx)
-  rw [hc]
-  exact (hf.sub continuousOn_const).norm
-
-lemma locallyLipschitz_ae_differentiable {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
-    (hl : LocallyLipschitzOn D u) :
-    ∀ᵐ z ∂volume, z ∈ D → DifferentiableAt ℝ u z := by
-  classical
-  have hloc (z : D) : ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ z.1 ∈ U ∧
-      LipschitzOnWith K u U := by
-    obtain ⟨K,t,ht,htl⟩ := hl z.2
-    rw [nhdsWithin_eq_nhds.mpr (hD.mem_nhds z.2)] at ht
-    obtain ⟨U,hUt,hUo,hzU⟩ := _root_.mem_nhds_iff.mp ht
-    exact ⟨K,U,hUo,hzU,htl.mono hUt⟩
-  choose K U hUo hxU hULip using hloc
-  have hcov : D ⊆ ⋃ x : D, U x := fun z hz => mem_iUnion.mpr ⟨⟨z,hz⟩,hxU ⟨z,hz⟩⟩
-  obtain ⟨t,ht,htcov⟩ := (HereditarilyLindelofSpace.isLindelof D).elim_countable_subcover U hUo hcov
-  have ha (x : D) : ∀ᵐ z ∂volume, z ∈ U x → DifferentiableAt ℝ u z := by
-    filter_upwards [(hULip x).ae_differentiableWithinAt_of_mem (μ := volume)] with z hz hzU
-    exact (hz hzU).differentiableAt ((hUo x).mem_nhds hzU)
-  filter_upwards [(ae_ball_iff ht).mpr (fun x _ => ha x)] with z hz hzD
-  obtain ⟨x,htx,hzx⟩ := mem_iUnion₂.mp (htcov hzD)
-  exact hz x htx hzx
-
-def complexCoordinates (p : ℝ × ℝ) : ℂ := p.1 + p.2*Complex.I
-
-lemma complexCoordinates_eq (p : ℝ × ℝ) :
-    complexCoordinates p = Complex.measurableEquivRealProd.symm p := by
-  apply Complex.ext <;> simp [complexCoordinates]
-
-lemma complexCoordinates_preserving : MeasurePreserving complexCoordinates volume volume := by
-  simpa only [funext complexCoordinates_eq] using
-    Complex.volume_preserving_equiv_real_prod.symm Complex.measurableEquivRealProd
-
-lemma complexCoordinates_continuous : Continuous complexCoordinates := by
-  unfold complexCoordinates
-  fun_prop
-
-lemma vertical_isometry (x : ℝ) : Isometry (fun y : ℝ => complexCoordinates (x,y)) := by
-  apply Isometry.of_dist_eq
-  intro y z
-  simp only [dist_eq_norm,complexCoordinates]
-  have he : ((x:ℂ)+(y:ℂ)*Complex.I)-((x:ℂ)+(z:ℂ)*Complex.I) =
-      ((y-z:ℝ):ℂ)*Complex.I := by push_cast; ring
-  rw [he,norm_mul,Complex.norm_I,mul_one,Complex.norm_real]
-
-lemma horizontal_isometry (y : ℝ) : Isometry (fun x : ℝ => complexCoordinates (x,y)) := by
-  apply Isometry.of_dist_eq
-  intro x z
-  simp [dist_eq_norm,complexCoordinates, ← Complex.ofReal_sub]
-
-lemma locallyLipschitz_comp_of_isometry {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
-    (hu : LocallyLipschitzOn D u) {f : ℝ → ℂ} (hf : Isometry f) {s : Set ℝ}
-    (hs : MapsTo f s D) : LocallyLipschitzOn s (u ∘ f) := by
-  intro x hx
-  obtain ⟨K,t,ht,hKt⟩ := hu (hs hx)
-  rw [nhdsWithin_eq_nhds.mpr (hD.mem_nhds (hs hx))] at ht
-  refine ⟨K*1,f ⁻¹' t,mem_nhdsWithin_of_mem_nhds (hf.continuous.continuousAt.preimage_mem_nhds ht),?_⟩
-  exact hKt.comp hf.lipschitzWith.lipschitzOnWith (fun _ h => h)
-
-lemma vertical_deriv_norm_le {u : ℂ → ℝ} {x y : ℝ}
-    (hu : DifferentiableAt ℝ u (complexCoordinates (x,y))) :
-    ‖deriv (fun t : ℝ => u (complexCoordinates (x,t))) y‖ ≤
-      ‖gradient u (complexCoordinates (x,y))‖ := by
-  have hi : HasDerivAt (fun t : ℝ => complexCoordinates (x,t)) Complex.I y := by
-    simpa [complexCoordinates] using
-      ((Complex.ofRealCLM.hasDerivAt (x := y)).mul_const Complex.I).const_add (x:ℂ)
-  have hh := hu.hasFDerivAt.comp_hasDerivAt y hi
-  change ‖deriv (u ∘ (fun t : ℝ => complexCoordinates (x,t))) y‖ ≤ _
-  rw [hh.deriv, ← inner_gradient_left]
-  simpa using norm_inner_le_norm (𝕜 := ℝ) (gradient u (complexCoordinates (x,y))) Complex.I
-
-lemma horizontal_deriv_norm_le {u : ℂ → ℝ} {x y : ℝ}
-    (hu : DifferentiableAt ℝ u (complexCoordinates (x,y))) :
-    ‖deriv (fun t : ℝ => u (complexCoordinates (t,y))) x‖ ≤
-      ‖gradient u (complexCoordinates (x,y))‖ := by
-  have hi : HasDerivAt (fun t : ℝ => complexCoordinates (t,y)) 1 x := by
-    simpa [complexCoordinates] using
-      (Complex.ofRealCLM.hasDerivAt (x := x)).add_const ((y:ℂ)*Complex.I)
-  have hh := hu.hasFDerivAt.comp_hasDerivAt x hi
-  change ‖deriv (u ∘ (fun t : ℝ => complexCoordinates (t,y))) x‖ ≤ _
-  rw [hh.deriv, ← inner_gradient_left]
-  simpa using norm_inner_le_norm (𝕜 := ℝ) (gradient u (complexCoordinates (x,y))) 1
-
-def cartesianRectangle (a b c d : ℝ) : Set ℂ :=
-  {z | a ≤ z.re ∧ z.re ≤ b ∧ c ≤ z.im ∧ z.im ≤ d}
-
-lemma cartesianRectangle_preimage (a b c d : ℝ) :
-    cartesianRectangle a b c d = Complex.measurableEquivRealProd ⁻¹' (Icc a b ×ˢ Icc c d) := by
-  ext z
-  simp only [cartesianRectangle, mem_ofPred_eq,mem_preimage,mem_prod,mem_Icc,Complex.measurableEquivRealProd_apply]
-  tauto
-
-lemma cartesianRectangle_measurable (a b c d : ℝ) : MeasurableSet (cartesianRectangle a b c d) := by
-  rw [cartesianRectangle_preimage]
-  exact (measurableSet_Icc.prod measurableSet_Icc).preimage
-    Complex.measurableEquivRealProd.measurable
-
-lemma complexCoordinates_rect_preserving (a b c d : ℝ) :
-    MeasurePreserving complexCoordinates
-      ((volume.restrict (Icc a b)).prod (volume.restrict (Icc c d)))
-      (volume.restrict (cartesianRectangle a b c d)) := by
-  have h := complexCoordinates_preserving.restrict_preimage (cartesianRectangle_measurable a b c d)
-  have he : complexCoordinates ⁻¹' cartesianRectangle a b c d = Icc a b ×ˢ Icc c d := by
-    ext p
-    simp only [mem_preimage,cartesianRectangle,mem_ofPred_eq,complexCoordinates,Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.I_re,Complex.ofReal_im,Complex.I_im,Complex.add_im,Complex.mul_im,mul_zero,sub_self,add_zero,mul_one,zero_add,mem_prod,mem_Icc]
-    tauto
-  rwa [he,Measure.volume_eq_prod, ← Measure.prod_restrict] at h
-
-lemma cartesianRectangle_compact (a b c d : ℝ) : IsCompact (cartesianRectangle a b c d) := by
-  have he : cartesianRectangle a b c d = complexCoordinates '' (Icc a b ×ˢ Icc c d) := by
-    ext z; constructor
-    · intro hz
-      refine ⟨(z.re,z.im),?_,?_⟩
-      · exact ⟨⟨hz.1,hz.2.1⟩,hz.2.2⟩
-      · apply Complex.ext <;> simp [complexCoordinates]
-    · rintro ⟨p,hp,rfl⟩
-      rcases hp with ⟨⟨ha,hb⟩,⟨hc,hd⟩⟩
-      simpa [cartesianRectangle,complexCoordinates] using And.intro ha (And.intro hb (And.intro hc hd))
-  rw [he]
-  exact (isCompact_Icc.prod isCompact_Icc).image complexCoordinates_continuous
-
-lemma cartesianRectangle_integral {a b c d : ℝ} {u : ℂ → ℝ}
-    (hu : IntegrableOn u (cartesianRectangle a b c d)) :
-    (∫ z in cartesianRectangle a b c d, u z) =
-      ∫ x in Icc a b, ∫ y in Icc c d, u (complexCoordinates (x,y)) := by
-  have hp := complexCoordinates_rect_preserving a b c d
-  have hemb : MeasurableEmbedding complexCoordinates := by
-    simpa only [funext complexCoordinates_eq] using Complex.measurableEquivRealProd.symm.measurableEmbedding
-  rw [← hp.integral_comp hemb]
-  exact integral_prod _ ((hp.integrable_comp hu.aestronglyMeasurable).mpr hu)
-
-def fullOpenStrip : Set ℂ := {z : ℂ | 0 < z.im ∧ z.im < Real.pi}
-def fullClosedStrip : Set ℂ := {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi}
-
-lemma fullOpenStrip_isOpen : IsOpen fullOpenStrip :=
-  (isOpen_lt continuous_const Complex.continuous_im).inter
-    (isOpen_lt Complex.continuous_im continuous_const)
-
-lemma cartesianRectangle_ae_strip (a b : ℝ) :
-    cartesianRectangle a b 0 Real.pi ≤ᵐ[volume] fullOpenStrip := by
-  have hf (c : ℝ) : volume {z : ℂ | z.im = c} = 0 := by
-    have hh := Complex.volume_preserving_equiv_real_prod.measure_preimage
-      (MeasurableSet.univ.prod (measurableSet_singleton c)).nullMeasurableSet
-    simpa [Set.preimage,Measure.volume_eq_prod,Measure.prod_prod] using hh
-  have h0 : ∀ᵐ z : ℂ ∂volume, z.im ≠ 0 := by
-    simpa only [ae_iff,not_not] using hf 0
-  have hp : ∀ᵐ z : ℂ ∂volume, z.im ≠ Real.pi := by
-    simpa only [ae_iff,not_not] using hf Real.pi
-  filter_upwards [h0,hp] with z hz0 hzp hzQ
-  exact ⟨lt_of_le_of_ne hzQ.2.2.1 hz0.symm,lt_of_le_of_ne hzQ.2.2.2 hzp⟩
-
-lemma cartesianRectangle_volume (a b c d : ℝ) :
-    volume (cartesianRectangle a b c d) = ENNReal.ofReal (b-a)*ENNReal.ofReal (d-c) := by
-  rw [cartesianRectangle_preimage,
-    Complex.volume_preserving_equiv_real_prod.measure_preimage
-      (measurableSet_Icc.prod measurableSet_Icc).nullMeasurableSet,
-    Measure.volume_eq_prod,Measure.prod_prod,Real.volume_Icc,Real.volume_Icc]
-
-lemma unitSlab_norm_integral_le {u : ℂ → ℝ} {a : ℝ}
-    (hg : MemLp (gradient u) 2 (volume.restrict (cartesianRectangle a (a+1) 0 Real.pi))) :
-    (∫ z in cartesianRectangle a (a+1) 0 Real.pi, ‖gradient u z‖) ≤
-      Real.sqrt (Real.pi * ∫ z in cartesianRectangle a (a+1) 0 Real.pi, ‖gradient u z‖^2) := by
-  let Q := cartesianRectangle a (a+1) 0 Real.pi
-  have hv : volume Q = ENNReal.ofReal Real.pi := by
-    simp [Q,cartesianRectangle_volume]
-  let : IsFiniteMeasure (volume.restrict Q) := ⟨by simp [hv]⟩
-  have hconst : MemLp (fun _ : ℂ => (1:ℝ)) 2 (volume.restrict Q) := memLp_const 1
-  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
-    (μ := volume.restrict Q) (f := fun z => ‖gradient u z‖) (g := fun _ => (1:ℝ))
-    (Eventually.of_forall fun z => norm_nonneg _) (Eventually.of_forall fun _ => zero_le_one)
-    (by simpa using hg.norm) (by simpa using hconst)
-  simp only [mul_one,Real.rpow_two,← Real.sqrt_eq_rpow,one_pow,integral_const,
-    Measure.real,Measure.restrict_apply_univ,smul_eq_mul,mul_one,hv,ENNReal.toReal_ofReal Real.pi_pos.le] at hh
-  simpa only [Real.sqrt_mul Real.pi_pos.le,mul_comm] using hh
-
-lemma vertical_derivative_bound_ae {u : ℂ → ℝ} (hu : LocallyLipschitzOn fullOpenStrip u) :
-    ∀ᵐ x : ℝ ∂volume, ∀ᵐ y : ℝ ∂volume, y ∈ Ioo 0 Real.pi →
-      ‖deriv (fun t : ℝ => u (complexCoordinates (x,t))) y‖ ≤
-        ‖gradient u (complexCoordinates (x,y))‖ := by
-  have hd := locallyLipschitz_ae_differentiable fullOpenStrip_isOpen hu
-  have hp := complexCoordinates_preserving.quasiMeasurePreserving.tendsto_ae.eventually hd
-  rw [Measure.volume_eq_prod] at hp
-  filter_upwards [Measure.ae_ae_of_ae_prod hp] with x hx
-  filter_upwards [hx] with y hy hyI
-  apply vertical_deriv_norm_le
-  apply hy
-  simpa [fullOpenStrip,complexCoordinates] using hyI
-
-lemma negativeSlab_mean_le {u : ℂ → ℝ}
-    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
-    (hz : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0)
-    {a : ℝ} (ha : a+1 ≤ 0) :
-    Real.pi⁻¹*(∫ z in cartesianRectangle a (a+1) 0 Real.pi, u z) ≤
-      Real.sqrt (Real.pi * ∫ z in cartesianRectangle a (a+1) 0 Real.pi, ‖gradient u z‖^2) := by
-  let Q := cartesianRectangle a (a+1) 0 Real.pi
-  have hQ : Q ⊆ fullClosedStrip := fun _ h => h.2.2
-  have hgQ : MemLp (gradient u) 2 (volume.restrict Q) :=
-    hg.mono_measure (Measure.restrict_mono_ae (cartesianRectangle_ae_strip a (a+1)))
-  have hv : volume Q < ⊤ := (cartesianRectangle_compact a (a+1) 0 Real.pi).measure_lt_top
-  let : IsFiniteMeasure (volume.restrict Q) := ⟨by simpa using hv⟩
-  have hgI : IntegrableOn (fun z => ‖gradient u z‖) Q := (hgQ.integrable (by norm_num)).norm
-  have huI : IntegrableOn u Q := (hu.mono hQ).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
-  have hp := complexCoordinates_rect_preserving a (a+1) 0 Real.pi
-  have hgp := (hp.integrable_comp hgI.aestronglyMeasurable).mpr hgI
-  have hup := (hp.integrable_comp huI.aestronglyMeasurable).mpr huI
-  have hder := vertical_derivative_bound_ae hl
-  have hslice : ∀ᵐ x ∂volume.restrict (Icc a (a+1)),
-      (∫ y in Icc 0 Real.pi, u (complexCoordinates (x,y))) ≤
-        Real.pi * ∫ y in Icc 0 Real.pi, ‖gradient u (complexCoordinates (x,y))‖ := by
-    filter_upwards [ae_restrict_of_ae hder,hgp.prod_right_ae,ae_restrict_mem measurableSet_Icc] with x hdx hgx hx
-    have hcont : ContinuousOn (fun y => u (complexCoordinates (x,y))) (Icc 0 Real.pi) :=
-      hu.comp (vertical_isometry x).continuous.continuousOn (by
-        intro y hy; simpa [fullClosedStrip,complexCoordinates] using hy)
-    have hlip : LocallyLipschitzOn (Ioo 0 Real.pi) (fun y => u (complexCoordinates (x,y))) :=
-      locallyLipschitz_comp_of_isometry fullOpenStrip_isOpen hl (vertical_isometry x) (by
-        intro y hy; simpa [fullOpenStrip,complexCoordinates] using hy)
-    obtain ⟨y0,hy0,huy0⟩ := hz x (hx.2.trans ha)
-    have hosc := slice_oscillation_le_integral Real.pi_pos hcont hlip hgx
-      (fun _ _ => norm_nonneg _) hdx
-    have hpoint (y : ℝ) (hy : y ∈ Icc 0 Real.pi) :
-        u (complexCoordinates (x,y)) ≤ ∫ t in Icc 0 Real.pi, ‖gradient u (complexCoordinates (x,t))‖ := by
-      have hh := hosc y hy y0 hy0
-      rw [huy0,sub_zero,Real.norm_eq_abs] at hh
-      exact (le_abs_self _).trans hh
-    have hci : IntegrableOn (fun _ : ℝ => ∫ t in Icc 0 Real.pi, ‖gradient u (complexCoordinates (x,t))‖) (Icc 0 Real.pi) := continuousOn_const.integrableOn_Icc
-    have hh := setIntegral_mono_on hcont.integrableOn_Icc hci measurableSet_Icc hpoint
-    simpa [integral_const,Real.volume_Icc,Real.pi_pos.le] using hh
-  have htotal := integral_mono_ae hup.integral_prod_left
-    (hgp.integral_prod_left.const_mul Real.pi) hslice
-  rw [integral_const_mul] at htotal
-  have huEq := cartesianRectangle_integral huI
-  have hgEq := cartesianRectangle_integral hgI
-  change (∫ z in Q,u z) = _ at huEq
-  change (∫ z in Q,‖gradient u z‖) = _ at hgEq
-  simp only [Function.comp_def] at htotal
-  rw [← huEq,← hgEq] at htotal
-  calc
-    Real.pi⁻¹*(∫ z in Q,u z) ≤ ∫ z in Q,‖gradient u z‖ := by
-      have hh := mul_le_mul_of_nonneg_left htotal (inv_nonneg.mpr Real.pi_pos.le)
-      simpa [← mul_assoc,Real.pi_pos.ne'] using hh
-    _ ≤ _ := unitSlab_norm_integral_le hgQ
-
-lemma interval_unit_mean_comparison {f g : ℝ → ℝ} {a : ℝ} (ha : 0 ≤ a)
-    (hf : ContinuousOn f (Icc (-1) (a+1)))
-    (hl : LocallyLipschitzOn (Ioo (-1) (a+1)) f)
-    (hg : IntegrableOn g (Icc (-1) (a+1))) (hgn : ∀ t ∈ Icc (-1) (a+1), 0 ≤ g t)
-    (hd : ∀ᵐ t ∂volume, t ∈ Ioo (-1) (a+1) → ‖deriv f t‖ ≤ g t) :
-    (∫ t in Icc a (a+1), f t) ≤ (∫ t in Icc (-1) 0, f t) + ∫ t in Icc (-1) (a+1), g t := by
-  have hA : Icc a (a+1) ⊆ Icc (-1) (a+1) := by
-    intro t ht; exact ⟨by linarith [ht.1],ht.2⟩
-  have hB : Icc (-1:ℝ) 0 ⊆ Icc (-1) (a+1) := by
-    intro t ht; exact ⟨ht.1,by linarith [ht.2]⟩
-  have hffA := (hf.mono hA).integrableOn_Icc (μ := volume)
-  have hffB := (hf.mono hB).integrableOn_Icc (μ := volume)
-  have hosc := slice_oscillation_le_integral (by linarith : (-1:ℝ)<a+1) hf hl hg hgn hd
-  let B := ∫ t in Icc (-1) (a+1),g t
-  have hpt (x : ℝ) (hx : x ∈ Icc a (a+1)) (y : ℝ) (hy : y ∈ Icc (-1:ℝ) 0) : f x ≤ f y+B := by
-    have hh := hosc x (hA hx) y (hB hy)
-    have hh' := (le_abs_self (f x-f y)).trans (by simpa [Real.norm_eq_abs] using hh)
-    dsimp [B]
-    linarith
-  have hxpt (x : ℝ) (hx : x ∈ Icc a (a+1)) : f x ≤ (∫ t in Icc (-1) 0,f t)+B := by
-    have hc : IntegrableOn (fun _ : ℝ => f x) (Icc (-1) 0) := continuousOn_const.integrableOn_Icc
-    have hh := setIntegral_mono_on hc (hffB.add (continuousOn_const.integrableOn_Icc))
-      measurableSet_Icc (hpt x hx)
-    simpa [integral_add hffB continuousOn_const.integrableOn_Icc,integral_const,Real.volume_Icc] using hh
-  have hc : IntegrableOn (fun _ : ℝ => (∫ t in Icc (-1) 0,f t)+B) (Icc a (a+1)) :=
-    continuousOn_const.integrableOn_Icc
-  have hh := setIntegral_mono_on hffA hc measurableSet_Icc hxpt
-  simpa [integral_const,Real.volume_Icc] using hh
-
-lemma cartesianRectangle_integral_symm {a b c d : ℝ} {u : ℂ → ℝ}
-    (hu : IntegrableOn u (cartesianRectangle a b c d)) :
-    (∫ z in cartesianRectangle a b c d, u z) =
-      ∫ y in Icc c d, ∫ x in Icc a b, u (complexCoordinates (x,y)) := by
-  have hp := complexCoordinates_rect_preserving a b c d
-  have hemb : MeasurableEmbedding complexCoordinates := by
-    simpa only [funext complexCoordinates_eq] using Complex.measurableEquivRealProd.symm.measurableEmbedding
-  rw [← hp.integral_comp hemb]
-  exact integral_prod_symm _ ((hp.integrable_comp hu.aestronglyMeasurable).mpr hu)
-
-lemma horizontal_derivative_bound_ae {u : ℂ → ℝ} (hu : LocallyLipschitzOn fullOpenStrip u) :
-    ∀ᵐ y : ℝ ∂volume, y ∈ Ioo 0 Real.pi → ∀ᵐ x : ℝ ∂volume,
-      ‖deriv (fun t : ℝ => u (complexCoordinates (t,y))) x‖ ≤
-        ‖gradient u (complexCoordinates (x,y))‖ := by
-  have hd := locallyLipschitz_ae_differentiable fullOpenStrip_isOpen hu
-  have hp := complexCoordinates_preserving.quasiMeasurePreserving.tendsto_ae.eventually hd
-  rw [Measure.volume_eq_prod] at hp
-  have hs := (Measure.measurePreserving_swap (μ := (volume : Measure ℝ)) (ν := volume)).quasiMeasurePreserving.tendsto_ae.eventually hp
-  filter_upwards [Measure.ae_ae_of_ae_prod hs] with y hy hyI
-  filter_upwards [hy] with x hx
-  apply horizontal_deriv_norm_le
-  apply hx
-  simpa [fullOpenStrip,complexCoordinates] using hyI
-
-lemma slab_integral_comparison {u : ℂ → ℝ}
-    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip)) {a : ℝ} (ha : 0 ≤ a) :
-    (∫ z in cartesianRectangle a (a+1) 0 Real.pi,u z) ≤
-      (∫ z in cartesianRectangle (-1) 0 0 Real.pi,u z) +
-      ∫ z in cartesianRectangle (-1) (a+1) 0 Real.pi, ‖gradient u z‖ := by
-  let R := cartesianRectangle (-1) (a+1) 0 Real.pi
-  have hR : R ⊆ fullClosedStrip := fun _ h => h.2.2
-  have huI : IntegrableOn u R := (hu.mono hR).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
-  have huA : IntegrableOn u (cartesianRectangle a (a+1) 0 Real.pi) :=
-    (hu.mono (fun _ h => h.2.2)).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
-  have huB : IntegrableOn u (cartesianRectangle (-1) 0 0 Real.pi) :=
-    (hu.mono (fun _ h => h.2.2)).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
-  have hgR : MemLp (gradient u) 2 (volume.restrict R) :=
-    hg.mono_measure (Measure.restrict_mono_ae (cartesianRectangle_ae_strip (-1) (a+1)))
-  let : IsFiniteMeasure (volume.restrict R) :=
-    ⟨by simpa using (cartesianRectangle_compact (-1) (a+1) 0 Real.pi).measure_lt_top⟩
-  have hgI : IntegrableOn (fun z => ‖gradient u z‖) R := (hgR.integrable (by norm_num)).norm
-  have hpR := complexCoordinates_rect_preserving (-1) (a+1) 0 Real.pi
-  have hpA := complexCoordinates_rect_preserving a (a+1) 0 Real.pi
-  have hpB := complexCoordinates_rect_preserving (-1) 0 0 Real.pi
-  have hgp := (hpR.integrable_comp hgI.aestronglyMeasurable).mpr hgI
-  have hupA := (hpA.integrable_comp huA.aestronglyMeasurable).mpr huA
-  have hupB := (hpB.integrable_comp huB.aestronglyMeasurable).mpr huB
-  have hder := horizontal_derivative_bound_ae hl
-  have hY : ∀ᵐ y : ℝ ∂volume.restrict (Icc 0 Real.pi), y ∈ Ioo 0 Real.pi := by
-    filter_upwards [ae_restrict_mem measurableSet_Icc, ae_restrict_of_ae (volume.ae_ne (0:ℝ)),
-      ae_restrict_of_ae (volume.ae_ne Real.pi)] with y hy hy0 hyp
-    exact ⟨lt_of_le_of_ne hy.1 hy0.symm,lt_of_le_of_ne hy.2 hyp⟩
-  have hslice : ∀ᵐ y ∂volume.restrict (Icc 0 Real.pi),
-      (∫ x in Icc a (a+1), u (complexCoordinates (x,y))) ≤
-        (∫ x in Icc (-1) 0, u (complexCoordinates (x,y)))+
-          ∫ x in Icc (-1) (a+1), ‖gradient u (complexCoordinates (x,y))‖ := by
-    filter_upwards [ae_restrict_of_ae hder,hgp.prod_left_ae,hY] with y hdy hgy hy
-    have hcont : ContinuousOn (fun x => u (complexCoordinates (x,y))) (Icc (-1) (a+1)) :=
-      hu.comp (horizontal_isometry y).continuous.continuousOn (by
-        intro x _; simpa [fullClosedStrip,complexCoordinates] using And.intro hy.1.le hy.2.le)
-    have hlip : LocallyLipschitzOn (Ioo (-1) (a+1)) (fun x => u (complexCoordinates (x,y))) :=
-      locallyLipschitz_comp_of_isometry fullOpenStrip_isOpen hl (horizontal_isometry y) (by
-        intro x _; simpa [fullOpenStrip,complexCoordinates] using hy)
-    exact interval_unit_mean_comparison ha hcont hlip hgy (fun _ _ => norm_nonneg _)
-      ((hdy hy).mono fun _ h _ => h)
-  have hh := integral_mono_ae hupA.integral_prod_right
-    (hupB.integral_prod_right.add hgp.integral_prod_right) hslice
-  simp only [Pi.add_apply] at hh
-  rw [integral_add hupB.integral_prod_right hgp.integral_prod_right] at hh
-  simp only [Function.comp_def] at hh
-  rwa [← cartesianRectangle_integral_symm huA,← cartesianRectangle_integral_symm huB,
-    ← cartesianRectangle_integral_symm hgI] at hh
-
-lemma rectangle_energy_le_total {u : ℂ → ℝ}
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip)) (a b : ℝ) :
-    (∫ z in cartesianRectangle a b 0 Real.pi, ‖gradient u z‖^2) ≤
-      ∫ z in fullOpenStrip, ‖gradient u z‖^2 := by
-  exact integral_mono_measure
-    (Measure.restrict_mono_ae (cartesianRectangle_ae_strip a b))
-    (Eventually.of_forall fun _ => sq_nonneg _) hg.norm.integrable_sq
-
-lemma rectangle_norm_integral_le {u : ℂ → ℝ} {a b : ℝ} (hab : a ≤ b)
-    (hg : MemLp (gradient u) 2 (volume.restrict (cartesianRectangle a b 0 Real.pi))) :
-    (∫ z in cartesianRectangle a b 0 Real.pi, ‖gradient u z‖) ≤
-      Real.sqrt ((b-a)*Real.pi * ∫ z in cartesianRectangle a b 0 Real.pi, ‖gradient u z‖^2) := by
-  let Q := cartesianRectangle a b 0 Real.pi
-  have harea : 0 ≤ (b-a)*Real.pi := mul_nonneg (sub_nonneg.mpr hab) Real.pi_pos.le
-  have hv : volume Q = ENNReal.ofReal ((b-a)*Real.pi) := by
-    simp [Q,cartesianRectangle_volume,ENNReal.ofReal_mul (sub_nonneg.mpr hab)]
-  let : IsFiniteMeasure (volume.restrict Q) := ⟨by simp [hv]⟩
-  have hconst : MemLp (fun _ : ℂ => (1:ℝ)) 2 (volume.restrict Q) := memLp_const 1
-  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
-    (μ := volume.restrict Q) (f := fun z => ‖gradient u z‖) (g := fun _ => (1:ℝ))
-    (Eventually.of_forall fun z => norm_nonneg _) (Eventually.of_forall fun _ => zero_le_one)
-    (by simpa using hg.norm) (by simpa using hconst)
-  simp only [mul_one,Real.rpow_two,← Real.sqrt_eq_rpow,one_pow,integral_const,
-    Measure.real,Measure.restrict_apply_univ,smul_eq_mul,mul_one,hv,ENNReal.toReal_ofReal harea] at hh
-  rw [Real.sqrt_mul harea]
-  simpa only [Q,mul_comm] using hh
-
-lemma negativeSlab_mean_small {u : ℂ → ℝ} {H a : ℝ} (hH : 1024 ≤ H)
-    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
-    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H)
-    (hz : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0)
-    (ha : a+1 ≤ 0) :
-    Real.pi⁻¹*(∫ z in cartesianRectangle a (a+1) 0 Real.pi, u z) ≤ H/4 := by
-  apply (negativeSlab_mean_le hu hl hg hz ha).trans
-  have hEq := (rectangle_energy_le_total hg a (a+1)).trans hE
-  apply (Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left hEq Real.pi_pos.le)).trans
-  apply Real.sqrt_le_iff.mpr
-  constructor
-  · linarith
-  · have hp := Real.pi_lt_four
-    have hps : Real.pi^2 ≤ 16 := by nlinarith [Real.pi_pos]
-    have hHn : 0 ≤ H := by linarith
-    nlinarith [mul_le_mul_of_nonneg_right hps hHn, mul_nonneg hHn (show 0 ≤ H-512 by linarith)]
-
-lemma positiveSlab_mean_small {u : ℂ → ℝ} {H a : ℝ} (hH : 1024 ≤ H)
-    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
-    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H)
-    (hz : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0)
-    (ha : 0 ≤ a) (haH : a ≤ H/128) :
-    Real.pi⁻¹*(∫ z in cartesianRectangle a (a+1) 0 Real.pi, u z) ≤ H/2 := by
-  have hh := mul_le_mul_of_nonneg_left (slab_integral_comparison hu hl hg ha)
-    (inv_nonneg.mpr Real.pi_pos.le)
-  have hm := negativeSlab_mean_small hH hu hl hg hE hz (a := -1) (by norm_num)
-  have hgR := hg.mono_measure (Measure.restrict_mono_ae (cartesianRectangle_ae_strip (-1) (a+1)))
-  have hn := rectangle_norm_integral_le (by linarith : (-1:ℝ) ≤ a+1) hgR
-  have hER := (rectangle_energy_le_total hg (-1) (a+1)).trans hE
-  have hn' : (∫ z in cartesianRectangle (-1) (a+1) 0 Real.pi, ‖gradient u z‖) ≤ Real.pi*H/4 := by
-    apply hn.trans
-    apply (Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left hER
-      (mul_nonneg (by linarith) Real.pi_pos.le))).trans
-    apply Real.sqrt_le_iff.mpr
-    constructor
-    · positivity
-    · have hHn : 0 ≤ H := by linarith
-      have hbase : 2*(a+2)*H ≤ H^2/16 := by
-        nlinarith [mul_nonneg hHn (show 0 ≤ H-128 by linarith),
-          mul_le_mul_of_nonneg_right haH hHn]
-      nlinarith [mul_le_mul_of_nonneg_right hbase (sq_nonneg Real.pi)]
-  have hn'' := mul_le_mul_of_nonneg_left hn' (inv_nonneg.mpr Real.pi_pos.le)
-  have he : Real.pi⁻¹*(Real.pi*H/4) = H/4 := by field_simp
-  rw [he] at hn''
-  simp only [mul_add] at hh
-  norm_num at hm
-  linarith
-
-def SourceSlabSmallMeanTarget : Prop :=
-  ∃ b H0 : ℝ, 0 < b ∧ 0 < H0 ∧ ∀ H : ℝ, H0 ≤ H → ∀ (u : ℂ → ℝ),
-    ContinuousOn u {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} →
-    LocallyLipschitzOn {z : ℂ | 0 < z.im ∧ z.im < Real.pi} u →
-    MemLp (gradient u) 2 (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) →
-    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi}, ‖gradient u z‖^2) ≤ 2*Real.pi*H →
-    (∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (x + y*Complex.I) = 0) →
-    ∀ j : ℤ, (j:ℝ) ≤ b*H →
-      (Real.pi)⁻¹ * ∫ z in {z : ℂ | (j:ℝ) ≤ z.re ∧ z.re ≤ (j:ℝ)+1 ∧
-          0 ≤ z.im ∧ z.im ≤ Real.pi}, u z ≤ H/2
-
-theorem sourceSlabSmallMean : SourceSlabSmallMeanTarget := by
-  refine ⟨1/128,1024,by norm_num,by norm_num,?_⟩
-  intro H hH u hu hl hg hE hz j hj
-  have hz' : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0 := hz
-  by_cases hj0 : j < 0
-  · exact (negativeSlab_mean_small hH hu hl hg hE hz' (a := (j:ℝ))
-      (by have hh : (j:ℝ) ≤ -1 := by exact_mod_cast (show j ≤ -1 by omega)
-          linarith)).trans (by linarith)
-  · apply positiveSlab_mean_small hH hu hl hg hE hz'
-    · exact_mod_cast (by omega : 0 ≤ j)
-    · linarith
-
-def closedIntegerStrip (j : ℤ) : Set ℂ :=
-  {z | (j:ℝ) ≤ z.re ∧ z.re ≤ (j:ℝ)+1 ∧ 0 ≤ z.im ∧ z.im ≤ Real.pi}
-
-def openIntegerStrip (j : ℤ) : Set ℂ :=
-  {z | (j:ℝ) < z.re ∧ z.re < (j:ℝ)+1 ∧ 0 < z.im ∧ z.im < Real.pi}
-
-lemma complex_re_fiber_null (a : ℝ) : volume {z : ℂ | z.re = a} = 0 := by
-  have he := Complex.volume_preserving_equiv_real_prod.measure_preimage
-    ((measurableSet_singleton a).prod MeasurableSet.univ).nullMeasurableSet
-  simpa [Set.preimage,Measure.volume_eq_prod,Measure.prod_prod] using he
-
-lemma complex_im_fiber_null (a : ℝ) : volume {z : ℂ | z.im = a} = 0 := by
-  have he := Complex.volume_preserving_equiv_real_prod.measure_preimage
-    (MeasurableSet.univ.prod (measurableSet_singleton a)).nullMeasurableSet
-  simpa [Set.preimage,Measure.volume_eq_prod,Measure.prod_prod] using he
-
-lemma closedIntegerStrip_ae_open (j : ℤ) : closedIntegerStrip j =ᵐ[volume] openIntegerStrip j := by
-  have hr0 : ∀ᵐ z : ℂ ∂volume, z.re ≠ (j:ℝ) := by
-    simpa only [ae_iff,not_not] using complex_re_fiber_null (j:ℝ)
-  have hr1 : ∀ᵐ z : ℂ ∂volume, z.re ≠ (j:ℝ)+1 := by
-    simpa only [ae_iff,not_not] using complex_re_fiber_null ((j:ℝ)+1)
-  have hi0 : ∀ᵐ z : ℂ ∂volume, z.im ≠ 0 := by
-    simpa only [ae_iff,not_not] using complex_im_fiber_null 0
-  have hi1 : ∀ᵐ z : ℂ ∂volume, z.im ≠ Real.pi := by
-    simpa only [ae_iff,not_not] using complex_im_fiber_null Real.pi
-  filter_upwards [hr0,hr1,hi0,hi1] with z hz0 hz1 hw0 hw1
-  apply propext
-  change ((j:ℝ) ≤ z.re ∧ z.re ≤ (j:ℝ)+1 ∧ 0 ≤ z.im ∧ z.im ≤ Real.pi) ↔
-    ((j:ℝ) < z.re ∧ z.re < (j:ℝ)+1 ∧ 0 < z.im ∧ z.im < Real.pi)
-  constructor
-  · rintro ⟨h0,h1,h2,h3⟩
-    exact ⟨lt_of_le_of_ne h0 hz0.symm,lt_of_le_of_ne h1 hz1,
-      lt_of_le_of_ne h2 hw0.symm,lt_of_le_of_ne h3 hw1⟩
-  · rintro ⟨h0,h1,h2,h3⟩; exact ⟨h0.le,h1.le,h2.le,h3.le⟩
-
-lemma openIntegerStrip_isOpen (j : ℤ) : IsOpen (openIntegerStrip j) :=
-  (isOpen_lt continuous_const Complex.continuous_re).inter
-    ((isOpen_lt Complex.continuous_re continuous_const).inter
-      ((isOpen_lt continuous_const Complex.continuous_im).inter
-        (isOpen_lt Complex.continuous_im continuous_const)))
-
-lemma openIntegerStrip_disjoint : Pairwise (fun i j => Disjoint (openIntegerStrip i) (openIntegerStrip j)) := by
-  intro i j hij
-  apply disjoint_left.mpr
-  intro z hi hj
-  rcases lt_or_gt_of_ne hij with h | h
-  · have hle : (i:ℝ)+1 ≤ (j:ℝ) := by exact_mod_cast (Int.add_one_le_iff.mpr h)
-    exact (not_lt_of_ge (hi.2.1.le.trans hle)) hj.1
-  · have hle : (j:ℝ)+1 ≤ (i:ℝ) := by exact_mod_cast (Int.add_one_le_iff.mpr h)
-    exact (not_lt_of_ge (hj.2.1.le.trans hle)) hi.1
-
-def slabShift (j : ℤ) (z : ℂ) : ℂ := (j:ℂ)+z
-
-lemma slabShift_continuous (j : ℤ) : Continuous (slabShift j) := by
-  unfold slabShift
-  fun_prop
-
-lemma slabShift_isometry (j : ℤ) : Isometry (slabShift j) :=
-  (IsometryEquiv.addLeft (j:ℂ)).isometry
-
-lemma slabShift_preimage_closed (j : ℤ) : slabShift j ⁻¹' closedIntegerStrip j =
-    cartesianRectangle 0 1 0 Real.pi := by
-  ext z
-  simp only [slabShift,closedIntegerStrip,cartesianRectangle,mem_preimage,mem_ofPred_eq,
-    Complex.add_re,Complex.add_im,Complex.intCast_re,Complex.intCast_im,zero_add]
-  constructor <;> rintro ⟨h1,h2,h3,h4⟩ <;> exact ⟨by linarith,by linarith,h3,h4⟩
-
-lemma slabShift_preimage_open (j : ℤ) : slabShift j ⁻¹' openIntegerStrip j = openUnitStrip := by
-  ext z
-  simp only [slabShift,openIntegerStrip,openUnitStrip,mem_preimage,mem_ofPred_eq,
-    Complex.add_re,Complex.add_im,Complex.intCast_re,Complex.intCast_im,zero_add]
-  constructor <;> rintro ⟨h1,h2,h3,h4⟩ <;> exact ⟨by linarith,by linarith,h3,h4⟩
-
-lemma slabShift_preserving_open (j : ℤ) : MeasurePreserving (slabShift j)
-    (volume.restrict openUnitStrip) (volume.restrict (openIntegerStrip j)) := by
-  have hh := (measurePreserving_add_left (volume : Measure ℂ) (j:ℂ)).restrict_preimage
-    (openIntegerStrip_isOpen j).measurableSet
-  change MeasurePreserving (slabShift j) (volume.restrict (slabShift j ⁻¹' openIntegerStrip j)) _ at hh
-  rwa [slabShift_preimage_open] at hh
-
-lemma slabShift_preserving_closed (j : ℤ) : MeasurePreserving (slabShift j)
-    (volume.restrict (cartesianRectangle 0 1 0 Real.pi)) (volume.restrict (closedIntegerStrip j)) := by
-  have hm : MeasurableSet (closedIntegerStrip j) := cartesianRectangle_measurable _ _ _ _
-  have hh := (measurePreserving_add_left (volume : Measure ℂ) (j:ℂ)).restrict_preimage hm
-  change MeasurePreserving (slabShift j) (volume.restrict (slabShift j ⁻¹' closedIntegerStrip j)) _ at hh
-  rwa [slabShift_preimage_closed] at hh
-
-lemma gradient_comp_slabShift (u : ℂ → ℝ) (j : ℤ) (z : ℂ) :
-    gradient (u ∘ slabShift j) z = gradient u (slabShift j z) := by
-  simp only [gradient,Function.comp_def,slabShift,fderiv_comp_add_left]
-
-lemma slabShift_integral (u : ℂ → ℝ) (j : ℤ) :
-    (∫ z in cartesianRectangle 0 1 0 Real.pi,u (slabShift j z)) =
-      ∫ z in closedIntegerStrip j,u z :=
-  (slabShift_preserving_closed j).integral_comp (IsometryEquiv.addLeft (j:ℂ)).toHomeomorph.measurableEmbedding u
-
-lemma slabShift_energy (u : ℂ → ℝ) (j : ℤ) :
-    stripDirichletEnergy (gradient (u ∘ slabShift j)) =
-      ∫ z in closedIntegerStrip j,‖gradient u z‖^2 := by
-  simp only [stripDirichletEnergy,gradient_comp_slabShift]
-  rw [(slabShift_preserving_open j).integral_comp (IsometryEquiv.addLeft (j:ℂ)).toHomeomorph.measurableEmbedding (fun z => ‖gradient u z‖^2)]
-  exact (setIntegral_congr_set (closedIntegerStrip_ae_open j)).symm
-
-lemma slabShift_mem_strip (j : ℤ) {z : ℂ} (hz : z ∈ openUnitStrip) :
-    slabShift j z ∈ fullOpenStrip := by
-  simpa [slabShift,fullOpenStrip] using hz.2.2
-
-lemma slabShift_locallyLipschitz {u : ℂ → ℝ}
-    (hl : LocallyLipschitzOn fullOpenStrip u) (j : ℤ) :
-    LocallyLipschitzOn openUnitStrip (u ∘ slabShift j) := by
-  intro z hz
-  obtain ⟨K,t,ht,hK⟩ := hl (slabShift_mem_strip j hz)
-  rw [nhdsWithin_eq_nhds.mpr (fullOpenStrip_isOpen.mem_nhds (slabShift_mem_strip j hz))] at ht
-  refine ⟨K,slabShift j ⁻¹' t,mem_nhdsWithin_of_mem_nhds
-    ((slabShift_continuous j).continuousAt.preimage_mem_nhds ht),?_⟩
-  intro x hx y hy
-  exact (hK hx hy).trans_eq (by rw [(slabShift_isometry j).edist_eq])
-
-lemma slabShift_gradient_memLp {u : ℂ → ℝ}
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip)) (j : ℤ) :
-    MemLp (gradient (u ∘ slabShift j)) 2 (volume.restrict openUnitStrip) := by
-  have hs : openIntegerStrip j ⊆ fullOpenStrip := fun _ hz => hz.2.2
-  have hh := (hg.mono_measure (Measure.restrict_mono hs le_rfl)).comp_measurePreserving
-    (slabShift_preserving_open j)
-  have he : gradient (u ∘ slabShift j) = gradient u ∘ slabShift j := funext (gradient_comp_slabShift u j)
-  rw [he]
-  exact hh
-
-lemma slabShift_source_weakGradient {u : ℂ → ℝ} {H : ℝ}
-    (hc : ContinuousOn u fullClosedStrip)
-    (hl : LocallyLipschitzOn fullOpenStrip u)
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
-    (hb : ∀ z, 0 ≤ u z ∧ u z ≤ H) (j : ℤ) :
-    SourceIsStripWeakGradient (u ∘ slabShift j) (gradient (u ∘ slabShift j)) := by
-  apply locallyLipschitz_source_weakGradient (slabShift_locallyLipschitz hl j) _ (slabShift_gradient_memLp hg j)
-  have hcont : ContinuousOn (u ∘ slabShift j) openUnitStrip :=
-    hc.comp (slabShift_continuous j).continuousOn (fun z hz =>
-      ⟨(slabShift_mem_strip j hz).1.le,(slabShift_mem_strip j hz).2.le⟩)
-  exact memLp_of_bounded (Eventually.of_forall fun z => hb (slabShift j z))
-    (hcont.aestronglyMeasurable openUnitStrip_isOpen.measurableSet) 2
-
-lemma integerSlab_energy_budget {u : ℂ → ℝ} {H : ℝ}
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
-    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H) :
-    (∑' j : ℤ, ENNReal.ofReal (∫ z in closedIntegerStrip j, ‖gradient u z‖^2)) ≤
-      ENNReal.ofReal (2*Real.pi*H) := by
-  have hint := hg.norm.integrable_sq
-  have hsub (j : ℤ) : openIntegerStrip j ⊆ fullOpenStrip := fun _ hz => hz.2.2
-  have hconv (j : ℤ) : ENNReal.ofReal (∫ z in closedIntegerStrip j, ‖gradient u z‖^2) =
-      ∫⁻ z in openIntegerStrip j, ENNReal.ofReal (‖gradient u z‖^2) := by
-    rw [setIntegral_congr_set (closedIntegerStrip_ae_open j)]
-    exact ofReal_integral_eq_lintegral_ofReal (hint.mono_measure (Measure.restrict_mono (hsub j) le_rfl))
-      (Eventually.of_forall fun z => sq_nonneg _)
-  simp_rw [hconv]
-  rw [← lintegral_iUnion (fun j => (openIntegerStrip_isOpen j).measurableSet) openIntegerStrip_disjoint]
-  calc
-    _ ≤ ∫⁻ z in fullOpenStrip, ENNReal.ofReal (‖gradient u z‖^2) :=
-      lintegral_mono_set (iUnion_subset hsub)
-    _ = ENNReal.ofReal (∫ z in fullOpenStrip, ‖gradient u z‖^2) :=
-      (ofReal_integral_eq_lintegral_ofReal hint (Eventually.of_forall fun z => sq_nonneg _)).symm
-    _ ≤ _ := ENNReal.ofReal_le_ofReal hE
-
-lemma abstract_slab_exponential_budget {E : ℤ → ℝ} {H C c : ℝ}
-    (hH : 0 < H) (hEn : ∀ j, 0 ≤ E j)
-    (hEs : (∑' j : ℤ, ENNReal.ofReal (E j)) ≤ ENNReal.ofReal (2*Real.pi*H))
-    (hC : 0 ≤ C) (hc : 0 < c) :
-    (∑' j : ℤ, ENNReal.ofReal (C*(E j/H^2)*Real.exp (-c*H^2/E j))) ≤
-      ENNReal.ofReal ((2*Real.pi*C/H)*Real.exp (-c*H/(2*Real.pi))) := by
-  have hv : 0 ≤ 2*Real.pi*H := by positivity
-  have hE (j : ℤ) : E j ≤ 2*Real.pi*H := by
-    exact (ENNReal.ofReal_le_ofReal_iff hv).mp ((ENNReal.le_tsum j).trans hEs)
-  let K := C/H^2*Real.exp (-c*H/(2*Real.pi))
-  have hK : 0 ≤ K := by dsimp [K]; positivity
-  have ht (j : ℤ) : C*(E j/H^2)*Real.exp (-c*H^2/E j) ≤ E j*K := by
-    by_cases hz : E j = 0
-    · simp [hz]
-    · have hj : 0 < E j := lt_of_le_of_ne (hEn j) (Ne.symm hz)
-      have hb : H/(2*Real.pi) ≤ H^2/E j := by
-        apply (le_div_iff₀ hj).mpr
-        rw [div_mul_eq_mul_div]
-        apply (div_le_iff₀ (by positivity : 0 < 2*Real.pi)).mpr
-        nlinarith [mul_le_mul_of_nonneg_left (hE j) hH.le]
-      have hex : Real.exp (-c*H^2/E j) ≤ Real.exp (-c*H/(2*Real.pi)) := by
-        apply Real.exp_le_exp.mpr
-        have hh := mul_le_mul_of_nonneg_left hb hc.le
-        calc
-          -c*H^2/E j = -(c*(H^2/E j)) := by ring
-          _ ≤ -(c*(H/(2*Real.pi))) := neg_le_neg hh
-          _ = -c*H/(2*Real.pi) := by ring
-      calc
-        _ ≤ C*(E j/H^2)*Real.exp (-c*H/(2*Real.pi)) :=
-          mul_le_mul_of_nonneg_left hex (mul_nonneg hC (div_nonneg hj.le (sq_nonneg _)))
-        _ = _ := by dsimp [K]; ring
-  calc
-    _ ≤ ∑' j : ℤ, ENNReal.ofReal (E j*K) := ENNReal.tsum_le_tsum fun j => ENNReal.ofReal_le_ofReal (ht j)
-    _ = (∑' j : ℤ, ENNReal.ofReal (E j))*ENNReal.ofReal K := by
-      calc
-        _ = ∑' j : ℤ, ENNReal.ofReal (E j)*ENNReal.ofReal K :=
-          tsum_congr (fun j => ENNReal.ofReal_mul (hEn j))
-        _ = _ := ENNReal.tsum_mul_right
-    _ ≤ ENNReal.ofReal (2*Real.pi*H)*ENNReal.ofReal K := by gcongr
-    _ = ENNReal.ofReal ((2*Real.pi*C/H)*Real.exp (-c*H/(2*Real.pi))) := by
-      rw [← ENNReal.ofReal_mul hv]
-      congr 1
-      dsimp [K]
-      field_simp [hH.ne']
-
- 
-
-def SourceStripLocalizationTarget : Prop :=
-  ∀ C0 α : ℝ, 0 < C0 → 0 < α → ∃ b C c H0 : ℝ,
-    0 < b ∧ 0 < C ∧ 0 < c ∧ 0 < H0 ∧
-    ∀ H : ℝ, H0 ≤ H → ∀ u : ℂ → ℝ,
-    ContinuousOn u {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} →
-    LocallyLipschitzOn {z : ℂ | 0 < z.im ∧ z.im < Real.pi} u →
-    (∀ z, 0 ≤ u z ∧ u z ≤ H) →
-    MemLp (gradient u) 2 (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) →
-    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi}, ‖gradient u z‖^2) ≤ 2*Real.pi*H →
-    (∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (x+y*Complex.I) = 0) →
-    ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ X : Ω → Set ℂ,
-      (∀ (j : ℤ) (k : ℕ) (i m : Fin (2^k)),
-        P {xi | (X xi ∩ dyadicStripRectangle j k i m).Nonempty} ≤
-          ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ)))) →
-      P {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*H ∧ u z = H} ≤
-        ENNReal.ofReal (C*Real.exp (-c*H))
-
-lemma stripRectangle_zero_eq_cartesian :
-    stripRectangle 0 0 1 = cartesianRectangle 0 1 0 Real.pi := by
-  ext z
-  simp [stripRectangle,cartesianRectangle]
-
-lemma slabShift_average (u : ℂ → ℝ) (j : ℤ) :
-    spatialAverage (u ∘ slabShift j) (stripRectangle 0 0 1) =
-      Real.pi⁻¹ * ∫ z in closedIntegerStrip j, u z := by
-  rw [spatialAverage,stripRectangle_volume]
-  simp only [mul_one,ENNReal.ofReal_one,one_mul,ENNReal.toReal_ofReal Real.pi_pos.le]
-  rw [stripRectangle_zero_eq_cartesian]
-  simp only [Function.comp_apply]
-  rw [slabShift_integral,div_eq_mul_inv,mul_comm]
-
-lemma slabShift_dyadic (j : ℤ) (k : ℕ) (i m : Fin (2^k)) :
-    slabShift j ⁻¹' dyadicStripRectangle j k i m = dyadicStripRectangle 0 k i m := by
-  ext z
-  simp only [dyadicStripRectangle,stripRectangle,slabShift,mem_preimage,mem_ofPred_eq,
-    Complex.add_re,Complex.add_im,Complex.intCast_re,Complex.intCast_im,zero_add,Int.cast_zero]
-  constructor <;> rintro ⟨h1,h2,h3,h4⟩ <;> exact ⟨by linarith,by linarith,h3,h4⟩
-
-lemma slabShift_continuousOn {u : ℂ → ℝ}
-    (hu : ContinuousOn u fullClosedStrip) (j : ℤ) :
-    ContinuousOn (u ∘ slabShift j) (stripRectangle 0 0 1) := by
-  apply hu.comp (slabShift_continuous j).continuousOn
-  intro z hz
-  change 0 ≤ (slabShift j z).im ∧ (slabShift j z).im ≤ Real.pi
-  simpa [slabShift,stripRectangle] using hz.2.2
-
-lemma point_integer_slab (z : ℂ) (hz : 0 ≤ z.im ∧ z.im ≤ Real.pi) :
-    ∃ w ∈ stripRectangle 0 0 1, slabShift ⌊z.re⌋ w = z := by
-  refine ⟨z-(⌊z.re⌋:ℂ),?_,by simp [slabShift]⟩
-  have h0 := Int.floor_le z.re
-  have h1 := Int.lt_floor_add_one z.re
-  simp only [stripRectangle,mem_ofPred_eq,Complex.sub_re,Complex.intCast_re,
-    Complex.sub_im,Complex.intCast_im,sub_zero,zero_add,mul_one]
-  exact ⟨by linarith,by linarith,hz.1,hz.2⟩
-
-lemma slab_energy_le_total {u : ℂ → ℝ} {H : ℝ}
-    (hH : 0 < H)
-    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
-    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H) (j : ℤ) :
-    (∫ z in closedIntegerStrip j, ‖gradient u z‖^2) ≤ 2*Real.pi*H := by
-  exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp
-    ((ENNReal.le_tsum j).trans (integerSlab_energy_budget hg hE))
-
- theorem sourceStripLocalization : SourceStripLocalizationTarget := by
-  intro C0 α hC0 hα
-  obtain ⟨C,c,c0,hC,hc,hc0,hrect⟩ := sourceStripRectangleEnergy C0 α hC0 hα
-  obtain ⟨b,Hm,hb,hHm,hmean⟩ := sourceSlabSmallMean
-  refine ⟨b,2*Real.pi*C,c/(2*Real.pi),max Hm (max 1 (2*Real.pi/c0)),hb,
-    by positivity,by positivity,lt_of_lt_of_le hHm (le_max_left _ _),?_⟩
-  intro H hH u hu hl huB hg hE hz Ω mΩ P hP X hhit
-  let := hP
-  have hHm' : Hm ≤ H := (le_max_left _ _).trans hH
-  have hH1 : 1 ≤ H := (le_max_left _ _).trans ((le_max_right _ _).trans hH)
-  have hH0 : 0 < H := by linarith
-  have hHc : 2*Real.pi/c0 ≤ H := (le_max_right _ _).trans ((le_max_right _ _).trans hH)
-  let E : ℤ → ℝ := fun j => ∫ z in closedIntegerStrip j, ‖gradient u z‖^2
-  let A : ℤ → Set Ω := fun j =>
-    {xi | (j:ℝ) ≤ b*H ∧ ((slabShift j ⁻¹' X xi) ∩
-      {z | z ∈ stripRectangle 0 0 1 ∧ (u ∘ slabShift j) z = H}).Nonempty}
-  have hEn (j : ℤ) : 0 ≤ E j := integral_nonneg fun z => sq_nonneg _
-  have hEj (j : ℤ) : E j ≤ 2*Real.pi*H := slab_energy_le_total hH0 hg hE j
-  have hsmall (j : ℤ) : E j ≤ c0*H^2 := by
-    have hhc := (div_le_iff₀ hc0).mp hHc
-    exact (hEj j).trans (by nlinarith [mul_le_mul_of_nonneg_right hhc hH0.le])
-  have hA (j : ℤ) : P (A j) ≤ ENNReal.ofReal (C*(E j/H^2)*Real.exp (-c*H^2/E j)) := by
-    by_cases hj : (j:ℝ) ≤ b*H
-    · have ht (k : ℕ) (i m : Fin (2^k)) :
-          P {xi | ((slabShift j ⁻¹' X xi) ∩ dyadicStripRectangle 0 k i m).Nonempty} ≤
-            ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ))) := by
-        apply (measure_mono ?_).trans (hhit j k i m)
-        rintro xi ⟨z,hzX,hzQ⟩
-        exact ⟨slabShift j z,hzX,by rwa [← slabShift_dyadic j k i m] at hzQ⟩
-      have hm : spatialAverage (u ∘ slabShift j) (stripRectangle 0 0 1) ≤ H/2 := by
-        rw [slabShift_average]
-        exact hmean H hHm' u hu hl hg hE hz j hj
-      have hh := hrect H hH0 (u ∘ slabShift j) (gradient (u ∘ slabShift j))
-        (slabShift_source_weakGradient hu hl hg huB j) (slabShift_continuousOn hu j)
-        (fun z _ => huB (slabShift j z)) hm Ω mΩ P hP (fun xi => slabShift j ⁻¹' X xi) ht
-        (by rw [slabShift_energy]; exact hsmall j)
-      rw [slabShift_energy] at hh
-      apply (measure_mono ?_).trans hh
-      exact fun xi hxi => hxi.2
-    · have he : A j = ∅ := by ext xi; simp [A,hj]
-      rw [he,measure_empty]
-      exact bot_le
-  have hcover : {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*H ∧ u z = H} ⊆
-      ⋃ j : ℤ, A j := by
-    rintro xi ⟨z,hzX,hz0,hzπ,hzr,hzu⟩
-    obtain ⟨w,hw,heq⟩ := point_integer_slab z ⟨hz0,hzπ⟩
-    apply mem_iUnion.mpr
-    refine ⟨⌊z.re⌋,⟨(Int.floor_le z.re).trans hzr,?_⟩⟩
-    exact ⟨w,by simpa only [mem_preimage,heq] using hzX,hw,by simpa [Function.comp_def,heq] using hzu⟩
-  calc
-    _ ≤ P (⋃ j : ℤ, A j) := measure_mono hcover
-    _ ≤ ∑' j : ℤ, P (A j) := measure_iUnion_le _
-    _ ≤ ∑' j : ℤ, ENNReal.ofReal (C*(E j/H^2)*Real.exp (-c*H^2/E j)) := ENNReal.tsum_le_tsum hA
-    _ ≤ ENNReal.ofReal ((2*Real.pi*C/H)*Real.exp (-c*H/(2*Real.pi))) :=
-      abstract_slab_exponential_budget hH0 hEn (integerSlab_energy_budget hg hE) hC.le hc
-    _ ≤ ENNReal.ofReal (2*Real.pi*C*Real.exp (-(c/(2*Real.pi))*H)) := by
-      apply ENNReal.ofReal_le_ofReal
-      have ht : 2*Real.pi*C/H ≤ 2*Real.pi*C := div_le_self (by positivity) hH1
-      have he : -c*H/(2*Real.pi) = -(c/(2*Real.pi))*H := by ring
-      rw [he]
-      exact mul_le_mul_of_nonneg_right ht (Real.exp_pos _).le
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.RectangleMeans
+
+namespace OAI
+
+/-! Nested Means for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology ContDiff ENNReal BigOperators NNReal
+namespace SLEExactGauge
+universe u
+lemma slice_oscillation_le_integral {a b : ℝ} (hab : a < b) {f g : ℝ → ℝ}
+    (hf : ContinuousOn f (Icc a b)) (hl : LocallyLipschitzOn (Ioo a b) f)
+    (hg : IntegrableOn g (Icc a b)) (hgn : ∀ t ∈ Icc a b, 0 ≤ g t)
+    (hd : ∀ᵐ t ∂volume, t ∈ Ioo a b → ‖deriv f t‖ ≤ g t) :
+    ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, ‖f x-f y‖ ≤ ∫ t in Icc a b, g t := by
+  have hint (x y : ℝ) (hx : x ∈ Ioo a b) (hy : y ∈ Ioo a b) (hxy : x ≤ y) :
+      ‖f y-f x‖ ≤ ∫ t in Icc a b, g t := by
+    have hs : Icc x y ⊆ Ioo a b := by
+      intro t ht
+      exact ⟨hx.1.trans_le ht.1, ht.2.trans_lt hy.2⟩
+    obtain ⟨K,hK⟩ := LocallyLipschitzOn.exists_lipschitzOnWith_of_compact
+      isCompact_Icc (hl.mono hs)
+    have hAC : AbsolutelyContinuousOnInterval f x y :=
+      (by simpa [uIcc_of_le hxy] using hK : LipschitzOnWith K f (uIcc x y)).absolutelyContinuousOnInterval
+    rw [← hAC.integral_deriv_eq_sub, intervalIntegral.integral_of_le hxy]
+    have hdint : IntegrableOn (deriv f) (Ioc x y) :=
+      hAC.intervalIntegrable_deriv.1
+    have hgin : IntegrableOn g (Ioc x y) := hg.mono_set (by
+      intro t ht; exact ⟨hx.1.le.trans ht.1.le,ht.2.trans hy.2.le⟩)
+    calc
+      ‖∫ t in Ioc x y, deriv f t‖ ≤ ∫ t in Ioc x y, ‖deriv f t‖ := norm_integral_le_integral_norm _
+      _ ≤ ∫ t in Ioc x y, g t := integral_mono_ae hdint.norm hgin (by
+        filter_upwards [ae_restrict_of_ae hd,ae_restrict_mem measurableSet_Ioc] with t ht htS
+        exact ht (hs ⟨htS.1.le,htS.2⟩))
+      _ ≤ ∫ t in Icc a b, g t := setIntegral_mono_set hg
+        (by filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht; exact hgn t ht)
+        (Eventually.of_forall fun t ht => ⟨hx.1.le.trans ht.1.le,ht.2.trans hy.2.le⟩)
+  have hi (x : ℝ) (hx : x ∈ Ioo a b) (y : ℝ) (hy : y ∈ Ioo a b) :
+      ‖f x-f y‖ ≤ ∫ t in Icc a b, g t := by
+    rcases le_total x y with h | h
+    · simpa only [norm_sub_rev] using hint x y hx hy h
+    · exact hint y x hy hx h
+  have hc : closure (Ioo a b) = Icc a b := closure_Ioo hab.ne
+  have hyall (x : ℝ) (hx : x ∈ Ioo a b) :
+      ∀ y ∈ Icc a b, ‖f x-f y‖ ≤ ∫ t in Icc a b, g t := by
+    intro y hy
+    apply le_on_closure (fun y hy => hi x hx y hy) _ continuousOn_const (hc.symm ▸ hy)
+    rw [hc]
+    exact (continuousOn_const.sub hf).norm
+  intro x hx y hy
+  apply le_on_closure (fun x hx => hyall x hx y hy) _ continuousOn_const (hc.symm ▸ hx)
+  rw [hc]
+  exact (hf.sub continuousOn_const).norm
+
+lemma locallyLipschitz_ae_differentiable {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
+    (hl : LocallyLipschitzOn D u) :
+    ∀ᵐ z ∂volume, z ∈ D → DifferentiableAt ℝ u z := by
+  classical
+  have hloc (z : D) : ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ z.1 ∈ U ∧
+      LipschitzOnWith K u U := by
+    obtain ⟨K,t,ht,htl⟩ := hl z.2
+    rw [nhdsWithin_eq_nhds.mpr (hD.mem_nhds z.2)] at ht
+    obtain ⟨U,hUt,hUo,hzU⟩ := _root_.mem_nhds_iff.mp ht
+    exact ⟨K,U,hUo,hzU,htl.mono hUt⟩
+  choose K U hUo hxU hULip using hloc
+  have hcov : D ⊆ ⋃ x : D, U x := fun z hz => mem_iUnion.mpr ⟨⟨z,hz⟩,hxU ⟨z,hz⟩⟩
+  obtain ⟨t,ht,htcov⟩ := (HereditarilyLindelofSpace.isLindelof D).elim_countable_subcover U hUo hcov
+  have ha (x : D) : ∀ᵐ z ∂volume, z ∈ U x → DifferentiableAt ℝ u z := by
+    filter_upwards [(hULip x).ae_differentiableWithinAt_of_mem (μ := volume)] with z hz hzU
+    exact (hz hzU).differentiableAt ((hUo x).mem_nhds hzU)
+  filter_upwards [(ae_ball_iff ht).mpr (fun x _ => ha x)] with z hz hzD
+  obtain ⟨x,htx,hzx⟩ := mem_iUnion₂.mp (htcov hzD)
+  exact hz x htx hzx
+
+def complexCoordinates (p : ℝ × ℝ) : ℂ := p.1 + p.2*Complex.I
+
+lemma complexCoordinates_eq (p : ℝ × ℝ) :
+    complexCoordinates p = Complex.measurableEquivRealProd.symm p := by
+  apply Complex.ext <;> simp [complexCoordinates]
+
+lemma complexCoordinates_preserving : MeasurePreserving complexCoordinates volume volume := by
+  simpa only [funext complexCoordinates_eq] using
+    Complex.volume_preserving_equiv_real_prod.symm Complex.measurableEquivRealProd
+
+lemma complexCoordinates_continuous : Continuous complexCoordinates := by
+  unfold complexCoordinates
+  fun_prop
+
+lemma vertical_isometry (x : ℝ) : Isometry (fun y : ℝ => complexCoordinates (x,y)) := by
+  apply Isometry.of_dist_eq
+  intro y z
+  simp only [dist_eq_norm,complexCoordinates]
+  have he : ((x:ℂ)+(y:ℂ)*Complex.I)-((x:ℂ)+(z:ℂ)*Complex.I) =
+      ((y-z:ℝ):ℂ)*Complex.I := by push_cast; ring
+  rw [he,norm_mul,Complex.norm_I,mul_one,Complex.norm_real]
+
+lemma horizontal_isometry (y : ℝ) : Isometry (fun x : ℝ => complexCoordinates (x,y)) := by
+  apply Isometry.of_dist_eq
+  intro x z
+  simp [dist_eq_norm,complexCoordinates, ← Complex.ofReal_sub]
+
+lemma locallyLipschitz_comp_of_isometry {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
+    (hu : LocallyLipschitzOn D u) {f : ℝ → ℂ} (hf : Isometry f) {s : Set ℝ}
+    (hs : MapsTo f s D) : LocallyLipschitzOn s (u ∘ f) := by
+  intro x hx
+  obtain ⟨K,t,ht,hKt⟩ := hu (hs hx)
+  rw [nhdsWithin_eq_nhds.mpr (hD.mem_nhds (hs hx))] at ht
+  refine ⟨K*1,f ⁻¹' t,mem_nhdsWithin_of_mem_nhds (hf.continuous.continuousAt.preimage_mem_nhds ht),?_⟩
+  exact hKt.comp hf.lipschitzWith.lipschitzOnWith (fun _ h => h)
+
+lemma vertical_deriv_norm_le {u : ℂ → ℝ} {x y : ℝ}
+    (hu : DifferentiableAt ℝ u (complexCoordinates (x,y))) :
+    ‖deriv (fun t : ℝ => u (complexCoordinates (x,t))) y‖ ≤
+      ‖gradient u (complexCoordinates (x,y))‖ := by
+  have hi : HasDerivAt (fun t : ℝ => complexCoordinates (x,t)) Complex.I y := by
+    simpa [complexCoordinates] using
+      ((Complex.ofRealCLM.hasDerivAt (x := y)).mul_const Complex.I).const_add (x:ℂ)
+  have hh := hu.hasFDerivAt.comp_hasDerivAt y hi
+  change ‖deriv (u ∘ (fun t : ℝ => complexCoordinates (x,t))) y‖ ≤ _
+  rw [hh.deriv, ← inner_gradient_left]
+  simpa using norm_inner_le_norm (𝕜 := ℝ) (gradient u (complexCoordinates (x,y))) Complex.I
+
+lemma horizontal_deriv_norm_le {u : ℂ → ℝ} {x y : ℝ}
+    (hu : DifferentiableAt ℝ u (complexCoordinates (x,y))) :
+    ‖deriv (fun t : ℝ => u (complexCoordinates (t,y))) x‖ ≤
+      ‖gradient u (complexCoordinates (x,y))‖ := by
+  have hi : HasDerivAt (fun t : ℝ => complexCoordinates (t,y)) 1 x := by
+    simpa [complexCoordinates] using
+      (Complex.ofRealCLM.hasDerivAt (x := x)).add_const ((y:ℂ)*Complex.I)
+  have hh := hu.hasFDerivAt.comp_hasDerivAt x hi
+  change ‖deriv (u ∘ (fun t : ℝ => complexCoordinates (t,y))) x‖ ≤ _
+  rw [hh.deriv, ← inner_gradient_left]
+  simpa using norm_inner_le_norm (𝕜 := ℝ) (gradient u (complexCoordinates (x,y))) 1
+
+def cartesianRectangle (a b c d : ℝ) : Set ℂ :=
+  {z | a ≤ z.re ∧ z.re ≤ b ∧ c ≤ z.im ∧ z.im ≤ d}
+
+lemma cartesianRectangle_preimage (a b c d : ℝ) :
+    cartesianRectangle a b c d = Complex.measurableEquivRealProd ⁻¹' (Icc a b ×ˢ Icc c d) := by
+  ext z
+  simp only [cartesianRectangle, mem_ofPred_eq,mem_preimage,mem_prod,mem_Icc,Complex.measurableEquivRealProd_apply]
+  tauto
+
+lemma cartesianRectangle_measurable (a b c d : ℝ) : MeasurableSet (cartesianRectangle a b c d) := by
+  rw [cartesianRectangle_preimage]
+  exact (measurableSet_Icc.prod measurableSet_Icc).preimage
+    Complex.measurableEquivRealProd.measurable
+
+lemma complexCoordinates_rect_preserving (a b c d : ℝ) :
+    MeasurePreserving complexCoordinates
+      ((volume.restrict (Icc a b)).prod (volume.restrict (Icc c d)))
+      (volume.restrict (cartesianRectangle a b c d)) := by
+  have h := complexCoordinates_preserving.restrict_preimage (cartesianRectangle_measurable a b c d)
+  have he : complexCoordinates ⁻¹' cartesianRectangle a b c d = Icc a b ×ˢ Icc c d := by
+    ext p
+    simp only [mem_preimage,cartesianRectangle,mem_ofPred_eq,complexCoordinates,Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.I_re,Complex.ofReal_im,Complex.I_im,Complex.add_im,Complex.mul_im,mul_zero,sub_self,add_zero,mul_one,zero_add,mem_prod,mem_Icc]
+    tauto
+  rwa [he,Measure.volume_eq_prod, ← Measure.prod_restrict] at h
+
+lemma cartesianRectangle_compact (a b c d : ℝ) : IsCompact (cartesianRectangle a b c d) := by
+  have he : cartesianRectangle a b c d = complexCoordinates '' (Icc a b ×ˢ Icc c d) := by
+    ext z; constructor
+    · intro hz
+      refine ⟨(z.re,z.im),?_,?_⟩
+      · exact ⟨⟨hz.1,hz.2.1⟩,hz.2.2⟩
+      · apply Complex.ext <;> simp [complexCoordinates]
+    · rintro ⟨p,hp,rfl⟩
+      rcases hp with ⟨⟨ha,hb⟩,⟨hc,hd⟩⟩
+      simpa [cartesianRectangle,complexCoordinates] using And.intro ha (And.intro hb (And.intro hc hd))
+  rw [he]
+  exact (isCompact_Icc.prod isCompact_Icc).image complexCoordinates_continuous
+
+lemma cartesianRectangle_integral {a b c d : ℝ} {u : ℂ → ℝ}
+    (hu : IntegrableOn u (cartesianRectangle a b c d)) :
+    (∫ z in cartesianRectangle a b c d, u z) =
+      ∫ x in Icc a b, ∫ y in Icc c d, u (complexCoordinates (x,y)) := by
+  have hp := complexCoordinates_rect_preserving a b c d
+  have hemb : MeasurableEmbedding complexCoordinates := by
+    simpa only [funext complexCoordinates_eq] using Complex.measurableEquivRealProd.symm.measurableEmbedding
+  rw [← hp.integral_comp hemb]
+  exact integral_prod _ ((hp.integrable_comp hu.aestronglyMeasurable).mpr hu)
+
+def fullOpenStrip : Set ℂ := {z : ℂ | 0 < z.im ∧ z.im < Real.pi}
+def fullClosedStrip : Set ℂ := {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi}
+
+lemma fullOpenStrip_isOpen : IsOpen fullOpenStrip :=
+  (isOpen_lt continuous_const Complex.continuous_im).inter
+    (isOpen_lt Complex.continuous_im continuous_const)
+
+lemma cartesianRectangle_ae_strip (a b : ℝ) :
+    cartesianRectangle a b 0 Real.pi ≤ᵐ[volume] fullOpenStrip := by
+  have hf (c : ℝ) : volume {z : ℂ | z.im = c} = 0 := by
+    have hh := Complex.volume_preserving_equiv_real_prod.measure_preimage
+      (MeasurableSet.univ.prod (measurableSet_singleton c)).nullMeasurableSet
+    simpa [Set.preimage,Measure.volume_eq_prod,Measure.prod_prod] using hh
+  have h0 : ∀ᵐ z : ℂ ∂volume, z.im ≠ 0 := by
+    simpa only [ae_iff,not_not] using hf 0
+  have hp : ∀ᵐ z : ℂ ∂volume, z.im ≠ Real.pi := by
+    simpa only [ae_iff,not_not] using hf Real.pi
+  filter_upwards [h0,hp] with z hz0 hzp hzQ
+  exact ⟨lt_of_le_of_ne hzQ.2.2.1 hz0.symm,lt_of_le_of_ne hzQ.2.2.2 hzp⟩
+
+lemma cartesianRectangle_volume (a b c d : ℝ) :
+    volume (cartesianRectangle a b c d) = ENNReal.ofReal (b-a)*ENNReal.ofReal (d-c) := by
+  rw [cartesianRectangle_preimage,
+    Complex.volume_preserving_equiv_real_prod.measure_preimage
+      (measurableSet_Icc.prod measurableSet_Icc).nullMeasurableSet,
+    Measure.volume_eq_prod,Measure.prod_prod,Real.volume_Icc,Real.volume_Icc]
+
+lemma unitSlab_norm_integral_le {u : ℂ → ℝ} {a : ℝ}
+    (hg : MemLp (gradient u) 2 (volume.restrict (cartesianRectangle a (a+1) 0 Real.pi))) :
+    (∫ z in cartesianRectangle a (a+1) 0 Real.pi, ‖gradient u z‖) ≤
+      Real.sqrt (Real.pi * ∫ z in cartesianRectangle a (a+1) 0 Real.pi, ‖gradient u z‖^2) := by
+  let Q := cartesianRectangle a (a+1) 0 Real.pi
+  have hv : volume Q = ENNReal.ofReal Real.pi := by
+    simp [Q,cartesianRectangle_volume]
+  let : IsFiniteMeasure (volume.restrict Q) := ⟨by simp [hv]⟩
+  have hconst : MemLp (fun _ : ℂ => (1:ℝ)) 2 (volume.restrict Q) := memLp_const 1
+  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
+    (μ := volume.restrict Q) (f := fun z => ‖gradient u z‖) (g := fun _ => (1:ℝ))
+    (Eventually.of_forall fun z => norm_nonneg _) (Eventually.of_forall fun _ => zero_le_one)
+    (by simpa using hg.norm) (by simpa using hconst)
+  simp only [mul_one,Real.rpow_two,← Real.sqrt_eq_rpow,one_pow,integral_const,
+    Measure.real,Measure.restrict_apply_univ,smul_eq_mul,mul_one,hv,ENNReal.toReal_ofReal Real.pi_pos.le] at hh
+  simpa only [Real.sqrt_mul Real.pi_pos.le,mul_comm] using hh
+
+lemma vertical_derivative_bound_ae {u : ℂ → ℝ} (hu : LocallyLipschitzOn fullOpenStrip u) :
+    ∀ᵐ x : ℝ ∂volume, ∀ᵐ y : ℝ ∂volume, y ∈ Ioo 0 Real.pi →
+      ‖deriv (fun t : ℝ => u (complexCoordinates (x,t))) y‖ ≤
+        ‖gradient u (complexCoordinates (x,y))‖ := by
+  have hd := locallyLipschitz_ae_differentiable fullOpenStrip_isOpen hu
+  have hp := complexCoordinates_preserving.quasiMeasurePreserving.tendsto_ae.eventually hd
+  rw [Measure.volume_eq_prod] at hp
+  filter_upwards [Measure.ae_ae_of_ae_prod hp] with x hx
+  filter_upwards [hx] with y hy hyI
+  apply vertical_deriv_norm_le
+  apply hy
+  simpa [fullOpenStrip,complexCoordinates] using hyI
+
+lemma negativeSlab_mean_le {u : ℂ → ℝ}
+    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
+    (hz : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0)
+    {a : ℝ} (ha : a+1 ≤ 0) :
+    Real.pi⁻¹*(∫ z in cartesianRectangle a (a+1) 0 Real.pi, u z) ≤
+      Real.sqrt (Real.pi * ∫ z in cartesianRectangle a (a+1) 0 Real.pi, ‖gradient u z‖^2) := by
+  let Q := cartesianRectangle a (a+1) 0 Real.pi
+  have hQ : Q ⊆ fullClosedStrip := fun _ h => h.2.2
+  have hgQ : MemLp (gradient u) 2 (volume.restrict Q) :=
+    hg.mono_measure (Measure.restrict_mono_ae (cartesianRectangle_ae_strip a (a+1)))
+  have hv : volume Q < ⊤ := (cartesianRectangle_compact a (a+1) 0 Real.pi).measure_lt_top
+  let : IsFiniteMeasure (volume.restrict Q) := ⟨by simpa using hv⟩
+  have hgI : IntegrableOn (fun z => ‖gradient u z‖) Q := (hgQ.integrable (by norm_num)).norm
+  have huI : IntegrableOn u Q := (hu.mono hQ).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
+  have hp := complexCoordinates_rect_preserving a (a+1) 0 Real.pi
+  have hgp := (hp.integrable_comp hgI.aestronglyMeasurable).mpr hgI
+  have hup := (hp.integrable_comp huI.aestronglyMeasurable).mpr huI
+  have hder := vertical_derivative_bound_ae hl
+  have hslice : ∀ᵐ x ∂volume.restrict (Icc a (a+1)),
+      (∫ y in Icc 0 Real.pi, u (complexCoordinates (x,y))) ≤
+        Real.pi * ∫ y in Icc 0 Real.pi, ‖gradient u (complexCoordinates (x,y))‖ := by
+    filter_upwards [ae_restrict_of_ae hder,hgp.prod_right_ae,ae_restrict_mem measurableSet_Icc] with x hdx hgx hx
+    have hcont : ContinuousOn (fun y => u (complexCoordinates (x,y))) (Icc 0 Real.pi) :=
+      hu.comp (vertical_isometry x).continuous.continuousOn (by
+        intro y hy; simpa [fullClosedStrip,complexCoordinates] using hy)
+    have hlip : LocallyLipschitzOn (Ioo 0 Real.pi) (fun y => u (complexCoordinates (x,y))) :=
+      locallyLipschitz_comp_of_isometry fullOpenStrip_isOpen hl (vertical_isometry x) (by
+        intro y hy; simpa [fullOpenStrip,complexCoordinates] using hy)
+    obtain ⟨y0,hy0,huy0⟩ := hz x (hx.2.trans ha)
+    have hosc := slice_oscillation_le_integral Real.pi_pos hcont hlip hgx
+      (fun _ _ => norm_nonneg _) hdx
+    have hpoint (y : ℝ) (hy : y ∈ Icc 0 Real.pi) :
+        u (complexCoordinates (x,y)) ≤ ∫ t in Icc 0 Real.pi, ‖gradient u (complexCoordinates (x,t))‖ := by
+      have hh := hosc y hy y0 hy0
+      rw [huy0,sub_zero,Real.norm_eq_abs] at hh
+      exact (le_abs_self _).trans hh
+    have hci : IntegrableOn (fun _ : ℝ => ∫ t in Icc 0 Real.pi, ‖gradient u (complexCoordinates (x,t))‖) (Icc 0 Real.pi) := continuousOn_const.integrableOn_Icc
+    have hh := setIntegral_mono_on hcont.integrableOn_Icc hci measurableSet_Icc hpoint
+    simpa [integral_const,Real.volume_Icc,Real.pi_pos.le] using hh
+  have htotal := integral_mono_ae hup.integral_prod_left
+    (hgp.integral_prod_left.const_mul Real.pi) hslice
+  rw [integral_const_mul] at htotal
+  have huEq := cartesianRectangle_integral huI
+  have hgEq := cartesianRectangle_integral hgI
+  change (∫ z in Q,u z) = _ at huEq
+  change (∫ z in Q,‖gradient u z‖) = _ at hgEq
+  simp only [Function.comp_def] at htotal
+  rw [← huEq,← hgEq] at htotal
+  calc
+    Real.pi⁻¹*(∫ z in Q,u z) ≤ ∫ z in Q,‖gradient u z‖ := by
+      have hh := mul_le_mul_of_nonneg_left htotal (inv_nonneg.mpr Real.pi_pos.le)
+      simpa [← mul_assoc,Real.pi_pos.ne'] using hh
+    _ ≤ _ := unitSlab_norm_integral_le hgQ
+
+lemma interval_unit_mean_comparison {f g : ℝ → ℝ} {a : ℝ} (ha : 0 ≤ a)
+    (hf : ContinuousOn f (Icc (-1) (a+1)))
+    (hl : LocallyLipschitzOn (Ioo (-1) (a+1)) f)
+    (hg : IntegrableOn g (Icc (-1) (a+1))) (hgn : ∀ t ∈ Icc (-1) (a+1), 0 ≤ g t)
+    (hd : ∀ᵐ t ∂volume, t ∈ Ioo (-1) (a+1) → ‖deriv f t‖ ≤ g t) :
+    (∫ t in Icc a (a+1), f t) ≤ (∫ t in Icc (-1) 0, f t) + ∫ t in Icc (-1) (a+1), g t := by
+  have hA : Icc a (a+1) ⊆ Icc (-1) (a+1) := by
+    intro t ht; exact ⟨by linarith [ht.1],ht.2⟩
+  have hB : Icc (-1:ℝ) 0 ⊆ Icc (-1) (a+1) := by
+    intro t ht; exact ⟨ht.1,by linarith [ht.2]⟩
+  have hffA := (hf.mono hA).integrableOn_Icc (μ := volume)
+  have hffB := (hf.mono hB).integrableOn_Icc (μ := volume)
+  have hosc := slice_oscillation_le_integral (by linarith : (-1:ℝ)<a+1) hf hl hg hgn hd
+  let B := ∫ t in Icc (-1) (a+1),g t
+  have hpt (x : ℝ) (hx : x ∈ Icc a (a+1)) (y : ℝ) (hy : y ∈ Icc (-1:ℝ) 0) : f x ≤ f y+B := by
+    have hh := hosc x (hA hx) y (hB hy)
+    have hh' := (le_abs_self (f x-f y)).trans (by simpa [Real.norm_eq_abs] using hh)
+    dsimp [B]
+    linarith
+  have hxpt (x : ℝ) (hx : x ∈ Icc a (a+1)) : f x ≤ (∫ t in Icc (-1) 0,f t)+B := by
+    have hc : IntegrableOn (fun _ : ℝ => f x) (Icc (-1) 0) := continuousOn_const.integrableOn_Icc
+    have hh := setIntegral_mono_on hc (hffB.add (continuousOn_const.integrableOn_Icc))
+      measurableSet_Icc (hpt x hx)
+    simpa [integral_add hffB continuousOn_const.integrableOn_Icc,integral_const,Real.volume_Icc] using hh
+  have hc : IntegrableOn (fun _ : ℝ => (∫ t in Icc (-1) 0,f t)+B) (Icc a (a+1)) :=
+    continuousOn_const.integrableOn_Icc
+  have hh := setIntegral_mono_on hffA hc measurableSet_Icc hxpt
+  simpa [integral_const,Real.volume_Icc] using hh
+
+lemma cartesianRectangle_integral_symm {a b c d : ℝ} {u : ℂ → ℝ}
+    (hu : IntegrableOn u (cartesianRectangle a b c d)) :
+    (∫ z in cartesianRectangle a b c d, u z) =
+      ∫ y in Icc c d, ∫ x in Icc a b, u (complexCoordinates (x,y)) := by
+  have hp := complexCoordinates_rect_preserving a b c d
+  have hemb : MeasurableEmbedding complexCoordinates := by
+    simpa only [funext complexCoordinates_eq] using Complex.measurableEquivRealProd.symm.measurableEmbedding
+  rw [← hp.integral_comp hemb]
+  exact integral_prod_symm _ ((hp.integrable_comp hu.aestronglyMeasurable).mpr hu)
+
+lemma horizontal_derivative_bound_ae {u : ℂ → ℝ} (hu : LocallyLipschitzOn fullOpenStrip u) :
+    ∀ᵐ y : ℝ ∂volume, y ∈ Ioo 0 Real.pi → ∀ᵐ x : ℝ ∂volume,
+      ‖deriv (fun t : ℝ => u (complexCoordinates (t,y))) x‖ ≤
+        ‖gradient u (complexCoordinates (x,y))‖ := by
+  have hd := locallyLipschitz_ae_differentiable fullOpenStrip_isOpen hu
+  have hp := complexCoordinates_preserving.quasiMeasurePreserving.tendsto_ae.eventually hd
+  rw [Measure.volume_eq_prod] at hp
+  have hs := (Measure.measurePreserving_swap (μ := (volume : Measure ℝ)) (ν := volume)).quasiMeasurePreserving.tendsto_ae.eventually hp
+  filter_upwards [Measure.ae_ae_of_ae_prod hs] with y hy hyI
+  filter_upwards [hy] with x hx
+  apply horizontal_deriv_norm_le
+  apply hx
+  simpa [fullOpenStrip,complexCoordinates] using hyI
+
+lemma slab_integral_comparison {u : ℂ → ℝ}
+    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip)) {a : ℝ} (ha : 0 ≤ a) :
+    (∫ z in cartesianRectangle a (a+1) 0 Real.pi,u z) ≤
+      (∫ z in cartesianRectangle (-1) 0 0 Real.pi,u z) +
+      ∫ z in cartesianRectangle (-1) (a+1) 0 Real.pi, ‖gradient u z‖ := by
+  let R := cartesianRectangle (-1) (a+1) 0 Real.pi
+  have hR : R ⊆ fullClosedStrip := fun _ h => h.2.2
+  have huI : IntegrableOn u R := (hu.mono hR).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
+  have huA : IntegrableOn u (cartesianRectangle a (a+1) 0 Real.pi) :=
+    (hu.mono (fun _ h => h.2.2)).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
+  have huB : IntegrableOn u (cartesianRectangle (-1) 0 0 Real.pi) :=
+    (hu.mono (fun _ h => h.2.2)).integrableOn_compact (cartesianRectangle_compact _ _ _ _)
+  have hgR : MemLp (gradient u) 2 (volume.restrict R) :=
+    hg.mono_measure (Measure.restrict_mono_ae (cartesianRectangle_ae_strip (-1) (a+1)))
+  let : IsFiniteMeasure (volume.restrict R) :=
+    ⟨by simpa using (cartesianRectangle_compact (-1) (a+1) 0 Real.pi).measure_lt_top⟩
+  have hgI : IntegrableOn (fun z => ‖gradient u z‖) R := (hgR.integrable (by norm_num)).norm
+  have hpR := complexCoordinates_rect_preserving (-1) (a+1) 0 Real.pi
+  have hpA := complexCoordinates_rect_preserving a (a+1) 0 Real.pi
+  have hpB := complexCoordinates_rect_preserving (-1) 0 0 Real.pi
+  have hgp := (hpR.integrable_comp hgI.aestronglyMeasurable).mpr hgI
+  have hupA := (hpA.integrable_comp huA.aestronglyMeasurable).mpr huA
+  have hupB := (hpB.integrable_comp huB.aestronglyMeasurable).mpr huB
+  have hder := horizontal_derivative_bound_ae hl
+  have hY : ∀ᵐ y : ℝ ∂volume.restrict (Icc 0 Real.pi), y ∈ Ioo 0 Real.pi := by
+    filter_upwards [ae_restrict_mem measurableSet_Icc, ae_restrict_of_ae (volume.ae_ne (0:ℝ)),
+      ae_restrict_of_ae (volume.ae_ne Real.pi)] with y hy hy0 hyp
+    exact ⟨lt_of_le_of_ne hy.1 hy0.symm,lt_of_le_of_ne hy.2 hyp⟩
+  have hslice : ∀ᵐ y ∂volume.restrict (Icc 0 Real.pi),
+      (∫ x in Icc a (a+1), u (complexCoordinates (x,y))) ≤
+        (∫ x in Icc (-1) 0, u (complexCoordinates (x,y)))+
+          ∫ x in Icc (-1) (a+1), ‖gradient u (complexCoordinates (x,y))‖ := by
+    filter_upwards [ae_restrict_of_ae hder,hgp.prod_left_ae,hY] with y hdy hgy hy
+    have hcont : ContinuousOn (fun x => u (complexCoordinates (x,y))) (Icc (-1) (a+1)) :=
+      hu.comp (horizontal_isometry y).continuous.continuousOn (by
+        intro x _; simpa [fullClosedStrip,complexCoordinates] using And.intro hy.1.le hy.2.le)
+    have hlip : LocallyLipschitzOn (Ioo (-1) (a+1)) (fun x => u (complexCoordinates (x,y))) :=
+      locallyLipschitz_comp_of_isometry fullOpenStrip_isOpen hl (horizontal_isometry y) (by
+        intro x _; simpa [fullOpenStrip,complexCoordinates] using hy)
+    exact interval_unit_mean_comparison ha hcont hlip hgy (fun _ _ => norm_nonneg _)
+      ((hdy hy).mono fun _ h _ => h)
+  have hh := integral_mono_ae hupA.integral_prod_right
+    (hupB.integral_prod_right.add hgp.integral_prod_right) hslice
+  simp only [Pi.add_apply] at hh
+  rw [integral_add hupB.integral_prod_right hgp.integral_prod_right] at hh
+  simp only [Function.comp_def] at hh
+  rwa [← cartesianRectangle_integral_symm huA,← cartesianRectangle_integral_symm huB,
+    ← cartesianRectangle_integral_symm hgI] at hh
+
+lemma rectangle_energy_le_total {u : ℂ → ℝ}
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip)) (a b : ℝ) :
+    (∫ z in cartesianRectangle a b 0 Real.pi, ‖gradient u z‖^2) ≤
+      ∫ z in fullOpenStrip, ‖gradient u z‖^2 := by
+  exact integral_mono_measure
+    (Measure.restrict_mono_ae (cartesianRectangle_ae_strip a b))
+    (Eventually.of_forall fun _ => sq_nonneg _) hg.norm.integrable_sq
+
+lemma rectangle_norm_integral_le {u : ℂ → ℝ} {a b : ℝ} (hab : a ≤ b)
+    (hg : MemLp (gradient u) 2 (volume.restrict (cartesianRectangle a b 0 Real.pi))) :
+    (∫ z in cartesianRectangle a b 0 Real.pi, ‖gradient u z‖) ≤
+      Real.sqrt ((b-a)*Real.pi * ∫ z in cartesianRectangle a b 0 Real.pi, ‖gradient u z‖^2) := by
+  let Q := cartesianRectangle a b 0 Real.pi
+  have harea : 0 ≤ (b-a)*Real.pi := mul_nonneg (sub_nonneg.mpr hab) Real.pi_pos.le
+  have hv : volume Q = ENNReal.ofReal ((b-a)*Real.pi) := by
+    simp [Q,cartesianRectangle_volume,ENNReal.ofReal_mul (sub_nonneg.mpr hab)]
+  let : IsFiniteMeasure (volume.restrict Q) := ⟨by simp [hv]⟩
+  have hconst : MemLp (fun _ : ℂ => (1:ℝ)) 2 (volume.restrict Q) := memLp_const 1
+  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
+    (μ := volume.restrict Q) (f := fun z => ‖gradient u z‖) (g := fun _ => (1:ℝ))
+    (Eventually.of_forall fun z => norm_nonneg _) (Eventually.of_forall fun _ => zero_le_one)
+    (by simpa using hg.norm) (by simpa using hconst)
+  simp only [mul_one,Real.rpow_two,← Real.sqrt_eq_rpow,one_pow,integral_const,
+    Measure.real,Measure.restrict_apply_univ,smul_eq_mul,mul_one,hv,ENNReal.toReal_ofReal harea] at hh
+  rw [Real.sqrt_mul harea]
+  simpa only [Q,mul_comm] using hh
+
+lemma negativeSlab_mean_small {u : ℂ → ℝ} {H a : ℝ} (hH : 1024 ≤ H)
+    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
+    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H)
+    (hz : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0)
+    (ha : a+1 ≤ 0) :
+    Real.pi⁻¹*(∫ z in cartesianRectangle a (a+1) 0 Real.pi, u z) ≤ H/4 := by
+  apply (negativeSlab_mean_le hu hl hg hz ha).trans
+  have hEq := (rectangle_energy_le_total hg a (a+1)).trans hE
+  apply (Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left hEq Real.pi_pos.le)).trans
+  apply Real.sqrt_le_iff.mpr
+  constructor
+  · linarith
+  · have hp := Real.pi_lt_four
+    have hps : Real.pi^2 ≤ 16 := by nlinarith [Real.pi_pos]
+    have hHn : 0 ≤ H := by linarith
+    nlinarith [mul_le_mul_of_nonneg_right hps hHn, mul_nonneg hHn (show 0 ≤ H-512 by linarith)]
+
+lemma positiveSlab_mean_small {u : ℂ → ℝ} {H a : ℝ} (hH : 1024 ≤ H)
+    (hu : ContinuousOn u fullClosedStrip) (hl : LocallyLipschitzOn fullOpenStrip u)
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
+    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H)
+    (hz : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0)
+    (ha : 0 ≤ a) (haH : a ≤ H/128) :
+    Real.pi⁻¹*(∫ z in cartesianRectangle a (a+1) 0 Real.pi, u z) ≤ H/2 := by
+  have hh := mul_le_mul_of_nonneg_left (slab_integral_comparison hu hl hg ha)
+    (inv_nonneg.mpr Real.pi_pos.le)
+  have hm := negativeSlab_mean_small hH hu hl hg hE hz (a := -1) (by norm_num)
+  have hgR := hg.mono_measure (Measure.restrict_mono_ae (cartesianRectangle_ae_strip (-1) (a+1)))
+  have hn := rectangle_norm_integral_le (by linarith : (-1:ℝ) ≤ a+1) hgR
+  have hER := (rectangle_energy_le_total hg (-1) (a+1)).trans hE
+  have hn' : (∫ z in cartesianRectangle (-1) (a+1) 0 Real.pi, ‖gradient u z‖) ≤ Real.pi*H/4 := by
+    apply hn.trans
+    apply (Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left hER
+      (mul_nonneg (by linarith) Real.pi_pos.le))).trans
+    apply Real.sqrt_le_iff.mpr
+    constructor
+    · positivity
+    · have hHn : 0 ≤ H := by linarith
+      have hbase : 2*(a+2)*H ≤ H^2/16 := by
+        nlinarith [mul_nonneg hHn (show 0 ≤ H-128 by linarith),
+          mul_le_mul_of_nonneg_right haH hHn]
+      nlinarith [mul_le_mul_of_nonneg_right hbase (sq_nonneg Real.pi)]
+  have hn'' := mul_le_mul_of_nonneg_left hn' (inv_nonneg.mpr Real.pi_pos.le)
+  have he : Real.pi⁻¹*(Real.pi*H/4) = H/4 := by field_simp
+  rw [he] at hn''
+  simp only [mul_add] at hh
+  norm_num at hm
+  linarith
+
+def SourceSlabSmallMeanTarget : Prop :=
+  ∃ b H0 : ℝ, 0 < b ∧ 0 < H0 ∧ ∀ H : ℝ, H0 ≤ H → ∀ (u : ℂ → ℝ),
+    ContinuousOn u {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} →
+    LocallyLipschitzOn {z : ℂ | 0 < z.im ∧ z.im < Real.pi} u →
+    MemLp (gradient u) 2 (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) →
+    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi}, ‖gradient u z‖^2) ≤ 2*Real.pi*H →
+    (∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (x + y*Complex.I) = 0) →
+    ∀ j : ℤ, (j:ℝ) ≤ b*H →
+      (Real.pi)⁻¹ * ∫ z in {z : ℂ | (j:ℝ) ≤ z.re ∧ z.re ≤ (j:ℝ)+1 ∧
+          0 ≤ z.im ∧ z.im ≤ Real.pi}, u z ≤ H/2
+
+theorem sourceSlabSmallMean : SourceSlabSmallMeanTarget := by
+  refine ⟨1/128,1024,by norm_num,by norm_num,?_⟩
+  intro H hH u hu hl hg hE hz j hj
+  have hz' : ∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (complexCoordinates (x,y)) = 0 := hz
+  by_cases hj0 : j < 0
+  · exact (negativeSlab_mean_small hH hu hl hg hE hz' (a := (j:ℝ))
+      (by have hh : (j:ℝ) ≤ -1 := by exact_mod_cast (show j ≤ -1 by omega)
+          linarith)).trans (by linarith)
+  · apply positiveSlab_mean_small hH hu hl hg hE hz'
+    · exact_mod_cast (by omega : 0 ≤ j)
+    · linarith
+
+def closedIntegerStrip (j : ℤ) : Set ℂ :=
+  {z | (j:ℝ) ≤ z.re ∧ z.re ≤ (j:ℝ)+1 ∧ 0 ≤ z.im ∧ z.im ≤ Real.pi}
+
+def openIntegerStrip (j : ℤ) : Set ℂ :=
+  {z | (j:ℝ) < z.re ∧ z.re < (j:ℝ)+1 ∧ 0 < z.im ∧ z.im < Real.pi}
+
+lemma complex_re_fiber_null (a : ℝ) : volume {z : ℂ | z.re = a} = 0 := by
+  have he := Complex.volume_preserving_equiv_real_prod.measure_preimage
+    ((measurableSet_singleton a).prod MeasurableSet.univ).nullMeasurableSet
+  simpa [Set.preimage,Measure.volume_eq_prod,Measure.prod_prod] using he
+
+lemma complex_im_fiber_null (a : ℝ) : volume {z : ℂ | z.im = a} = 0 := by
+  have he := Complex.volume_preserving_equiv_real_prod.measure_preimage
+    (MeasurableSet.univ.prod (measurableSet_singleton a)).nullMeasurableSet
+  simpa [Set.preimage,Measure.volume_eq_prod,Measure.prod_prod] using he
+
+lemma closedIntegerStrip_ae_open (j : ℤ) : closedIntegerStrip j =ᵐ[volume] openIntegerStrip j := by
+  have hr0 : ∀ᵐ z : ℂ ∂volume, z.re ≠ (j:ℝ) := by
+    simpa only [ae_iff,not_not] using complex_re_fiber_null (j:ℝ)
+  have hr1 : ∀ᵐ z : ℂ ∂volume, z.re ≠ (j:ℝ)+1 := by
+    simpa only [ae_iff,not_not] using complex_re_fiber_null ((j:ℝ)+1)
+  have hi0 : ∀ᵐ z : ℂ ∂volume, z.im ≠ 0 := by
+    simpa only [ae_iff,not_not] using complex_im_fiber_null 0
+  have hi1 : ∀ᵐ z : ℂ ∂volume, z.im ≠ Real.pi := by
+    simpa only [ae_iff,not_not] using complex_im_fiber_null Real.pi
+  filter_upwards [hr0,hr1,hi0,hi1] with z hz0 hz1 hw0 hw1
+  apply propext
+  change ((j:ℝ) ≤ z.re ∧ z.re ≤ (j:ℝ)+1 ∧ 0 ≤ z.im ∧ z.im ≤ Real.pi) ↔
+    ((j:ℝ) < z.re ∧ z.re < (j:ℝ)+1 ∧ 0 < z.im ∧ z.im < Real.pi)
+  constructor
+  · rintro ⟨h0,h1,h2,h3⟩
+    exact ⟨lt_of_le_of_ne h0 hz0.symm,lt_of_le_of_ne h1 hz1,
+      lt_of_le_of_ne h2 hw0.symm,lt_of_le_of_ne h3 hw1⟩
+  · rintro ⟨h0,h1,h2,h3⟩; exact ⟨h0.le,h1.le,h2.le,h3.le⟩
+
+lemma openIntegerStrip_isOpen (j : ℤ) : IsOpen (openIntegerStrip j) :=
+  (isOpen_lt continuous_const Complex.continuous_re).inter
+    ((isOpen_lt Complex.continuous_re continuous_const).inter
+      ((isOpen_lt continuous_const Complex.continuous_im).inter
+        (isOpen_lt Complex.continuous_im continuous_const)))
+
+lemma openIntegerStrip_disjoint : Pairwise (fun i j => Disjoint (openIntegerStrip i) (openIntegerStrip j)) := by
+  intro i j hij
+  apply disjoint_left.mpr
+  intro z hi hj
+  rcases lt_or_gt_of_ne hij with h | h
+  · have hle : (i:ℝ)+1 ≤ (j:ℝ) := by exact_mod_cast (Int.add_one_le_iff.mpr h)
+    exact (not_lt_of_ge (hi.2.1.le.trans hle)) hj.1
+  · have hle : (j:ℝ)+1 ≤ (i:ℝ) := by exact_mod_cast (Int.add_one_le_iff.mpr h)
+    exact (not_lt_of_ge (hj.2.1.le.trans hle)) hi.1
+
+def slabShift (j : ℤ) (z : ℂ) : ℂ := (j:ℂ)+z
+
+lemma slabShift_continuous (j : ℤ) : Continuous (slabShift j) := by
+  unfold slabShift
+  fun_prop
+
+lemma slabShift_isometry (j : ℤ) : Isometry (slabShift j) :=
+  (IsometryEquiv.addLeft (j:ℂ)).isometry
+
+lemma slabShift_preimage_closed (j : ℤ) : slabShift j ⁻¹' closedIntegerStrip j =
+    cartesianRectangle 0 1 0 Real.pi := by
+  ext z
+  simp only [slabShift,closedIntegerStrip,cartesianRectangle,mem_preimage,mem_ofPred_eq,
+    Complex.add_re,Complex.add_im,Complex.intCast_re,Complex.intCast_im,zero_add]
+  constructor <;> rintro ⟨h1,h2,h3,h4⟩ <;> exact ⟨by linarith,by linarith,h3,h4⟩
+
+lemma slabShift_preimage_open (j : ℤ) : slabShift j ⁻¹' openIntegerStrip j = openUnitStrip := by
+  ext z
+  simp only [slabShift,openIntegerStrip,openUnitStrip,mem_preimage,mem_ofPred_eq,
+    Complex.add_re,Complex.add_im,Complex.intCast_re,Complex.intCast_im,zero_add]
+  constructor <;> rintro ⟨h1,h2,h3,h4⟩ <;> exact ⟨by linarith,by linarith,h3,h4⟩
+
+lemma slabShift_preserving_open (j : ℤ) : MeasurePreserving (slabShift j)
+    (volume.restrict openUnitStrip) (volume.restrict (openIntegerStrip j)) := by
+  have hh := (measurePreserving_add_left (volume : Measure ℂ) (j:ℂ)).restrict_preimage
+    (openIntegerStrip_isOpen j).measurableSet
+  change MeasurePreserving (slabShift j) (volume.restrict (slabShift j ⁻¹' openIntegerStrip j)) _ at hh
+  rwa [slabShift_preimage_open] at hh
+
+lemma slabShift_preserving_closed (j : ℤ) : MeasurePreserving (slabShift j)
+    (volume.restrict (cartesianRectangle 0 1 0 Real.pi)) (volume.restrict (closedIntegerStrip j)) := by
+  have hm : MeasurableSet (closedIntegerStrip j) := cartesianRectangle_measurable _ _ _ _
+  have hh := (measurePreserving_add_left (volume : Measure ℂ) (j:ℂ)).restrict_preimage hm
+  change MeasurePreserving (slabShift j) (volume.restrict (slabShift j ⁻¹' closedIntegerStrip j)) _ at hh
+  rwa [slabShift_preimage_closed] at hh
+
+lemma gradient_comp_slabShift (u : ℂ → ℝ) (j : ℤ) (z : ℂ) :
+    gradient (u ∘ slabShift j) z = gradient u (slabShift j z) := by
+  simp only [gradient,Function.comp_def,slabShift,fderiv_comp_add_left]
+
+lemma slabShift_integral (u : ℂ → ℝ) (j : ℤ) :
+    (∫ z in cartesianRectangle 0 1 0 Real.pi,u (slabShift j z)) =
+      ∫ z in closedIntegerStrip j,u z :=
+  (slabShift_preserving_closed j).integral_comp (IsometryEquiv.addLeft (j:ℂ)).toHomeomorph.measurableEmbedding u
+
+lemma slabShift_energy (u : ℂ → ℝ) (j : ℤ) :
+    stripDirichletEnergy (gradient (u ∘ slabShift j)) =
+      ∫ z in closedIntegerStrip j,‖gradient u z‖^2 := by
+  simp only [stripDirichletEnergy,gradient_comp_slabShift]
+  rw [(slabShift_preserving_open j).integral_comp (IsometryEquiv.addLeft (j:ℂ)).toHomeomorph.measurableEmbedding (fun z => ‖gradient u z‖^2)]
+  exact (setIntegral_congr_set (closedIntegerStrip_ae_open j)).symm
+
+lemma slabShift_mem_strip (j : ℤ) {z : ℂ} (hz : z ∈ openUnitStrip) :
+    slabShift j z ∈ fullOpenStrip := by
+  simpa [slabShift,fullOpenStrip] using hz.2.2
+
+lemma slabShift_locallyLipschitz {u : ℂ → ℝ}
+    (hl : LocallyLipschitzOn fullOpenStrip u) (j : ℤ) :
+    LocallyLipschitzOn openUnitStrip (u ∘ slabShift j) := by
+  intro z hz
+  obtain ⟨K,t,ht,hK⟩ := hl (slabShift_mem_strip j hz)
+  rw [nhdsWithin_eq_nhds.mpr (fullOpenStrip_isOpen.mem_nhds (slabShift_mem_strip j hz))] at ht
+  refine ⟨K,slabShift j ⁻¹' t,mem_nhdsWithin_of_mem_nhds
+    ((slabShift_continuous j).continuousAt.preimage_mem_nhds ht),?_⟩
+  intro x hx y hy
+  exact (hK hx hy).trans_eq (by rw [(slabShift_isometry j).edist_eq])
+
+lemma slabShift_gradient_memLp {u : ℂ → ℝ}
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip)) (j : ℤ) :
+    MemLp (gradient (u ∘ slabShift j)) 2 (volume.restrict openUnitStrip) := by
+  have hs : openIntegerStrip j ⊆ fullOpenStrip := fun _ hz => hz.2.2
+  have hh := (hg.mono_measure (Measure.restrict_mono hs le_rfl)).comp_measurePreserving
+    (slabShift_preserving_open j)
+  have he : gradient (u ∘ slabShift j) = gradient u ∘ slabShift j := funext (gradient_comp_slabShift u j)
+  rw [he]
+  exact hh
+
+lemma slabShift_source_weakGradient {u : ℂ → ℝ} {H : ℝ}
+    (hc : ContinuousOn u fullClosedStrip)
+    (hl : LocallyLipschitzOn fullOpenStrip u)
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
+    (hb : ∀ z, 0 ≤ u z ∧ u z ≤ H) (j : ℤ) :
+    SourceIsStripWeakGradient (u ∘ slabShift j) (gradient (u ∘ slabShift j)) := by
+  apply locallyLipschitz_source_weakGradient (slabShift_locallyLipschitz hl j) _ (slabShift_gradient_memLp hg j)
+  have hcont : ContinuousOn (u ∘ slabShift j) openUnitStrip :=
+    hc.comp (slabShift_continuous j).continuousOn (fun z hz =>
+      ⟨(slabShift_mem_strip j hz).1.le,(slabShift_mem_strip j hz).2.le⟩)
+  exact memLp_of_bounded (Eventually.of_forall fun z => hb (slabShift j z))
+    (hcont.aestronglyMeasurable openUnitStrip_isOpen.measurableSet) 2
+
+lemma integerSlab_energy_budget {u : ℂ → ℝ} {H : ℝ}
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
+    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H) :
+    (∑' j : ℤ, ENNReal.ofReal (∫ z in closedIntegerStrip j, ‖gradient u z‖^2)) ≤
+      ENNReal.ofReal (2*Real.pi*H) := by
+  have hint := hg.norm.integrable_sq
+  have hsub (j : ℤ) : openIntegerStrip j ⊆ fullOpenStrip := fun _ hz => hz.2.2
+  have hconv (j : ℤ) : ENNReal.ofReal (∫ z in closedIntegerStrip j, ‖gradient u z‖^2) =
+      ∫⁻ z in openIntegerStrip j, ENNReal.ofReal (‖gradient u z‖^2) := by
+    rw [setIntegral_congr_set (closedIntegerStrip_ae_open j)]
+    exact ofReal_integral_eq_lintegral_ofReal (hint.mono_measure (Measure.restrict_mono (hsub j) le_rfl))
+      (Eventually.of_forall fun z => sq_nonneg _)
+  simp_rw [hconv]
+  rw [← lintegral_iUnion (fun j => (openIntegerStrip_isOpen j).measurableSet) openIntegerStrip_disjoint]
+  calc
+    _ ≤ ∫⁻ z in fullOpenStrip, ENNReal.ofReal (‖gradient u z‖^2) :=
+      lintegral_mono_set (iUnion_subset hsub)
+    _ = ENNReal.ofReal (∫ z in fullOpenStrip, ‖gradient u z‖^2) :=
+      (ofReal_integral_eq_lintegral_ofReal hint (Eventually.of_forall fun z => sq_nonneg _)).symm
+    _ ≤ _ := ENNReal.ofReal_le_ofReal hE
+
+lemma abstract_slab_exponential_budget {E : ℤ → ℝ} {H C c : ℝ}
+    (hH : 0 < H) (hEn : ∀ j, 0 ≤ E j)
+    (hEs : (∑' j : ℤ, ENNReal.ofReal (E j)) ≤ ENNReal.ofReal (2*Real.pi*H))
+    (hC : 0 ≤ C) (hc : 0 < c) :
+    (∑' j : ℤ, ENNReal.ofReal (C*(E j/H^2)*Real.exp (-c*H^2/E j))) ≤
+      ENNReal.ofReal ((2*Real.pi*C/H)*Real.exp (-c*H/(2*Real.pi))) := by
+  have hv : 0 ≤ 2*Real.pi*H := by positivity
+  have hE (j : ℤ) : E j ≤ 2*Real.pi*H := by
+    exact (ENNReal.ofReal_le_ofReal_iff hv).mp ((ENNReal.le_tsum j).trans hEs)
+  let K := C/H^2*Real.exp (-c*H/(2*Real.pi))
+  have hK : 0 ≤ K := by dsimp [K]; positivity
+  have ht (j : ℤ) : C*(E j/H^2)*Real.exp (-c*H^2/E j) ≤ E j*K := by
+    by_cases hz : E j = 0
+    · simp [hz]
+    · have hj : 0 < E j := lt_of_le_of_ne (hEn j) (Ne.symm hz)
+      have hb : H/(2*Real.pi) ≤ H^2/E j := by
+        apply (le_div_iff₀ hj).mpr
+        rw [div_mul_eq_mul_div]
+        apply (div_le_iff₀ (by positivity : 0 < 2*Real.pi)).mpr
+        nlinarith [mul_le_mul_of_nonneg_left (hE j) hH.le]
+      have hex : Real.exp (-c*H^2/E j) ≤ Real.exp (-c*H/(2*Real.pi)) := by
+        apply Real.exp_le_exp.mpr
+        have hh := mul_le_mul_of_nonneg_left hb hc.le
+        calc
+          -c*H^2/E j = -(c*(H^2/E j)) := by ring
+          _ ≤ -(c*(H/(2*Real.pi))) := neg_le_neg hh
+          _ = -c*H/(2*Real.pi) := by ring
+      calc
+        _ ≤ C*(E j/H^2)*Real.exp (-c*H/(2*Real.pi)) :=
+          mul_le_mul_of_nonneg_left hex (mul_nonneg hC (div_nonneg hj.le (sq_nonneg _)))
+        _ = _ := by dsimp [K]; ring
+  calc
+    _ ≤ ∑' j : ℤ, ENNReal.ofReal (E j*K) := ENNReal.tsum_le_tsum fun j => ENNReal.ofReal_le_ofReal (ht j)
+    _ = (∑' j : ℤ, ENNReal.ofReal (E j))*ENNReal.ofReal K := by
+      calc
+        _ = ∑' j : ℤ, ENNReal.ofReal (E j)*ENNReal.ofReal K :=
+          tsum_congr (fun j => ENNReal.ofReal_mul (hEn j))
+        _ = _ := ENNReal.tsum_mul_right
+    _ ≤ ENNReal.ofReal (2*Real.pi*H)*ENNReal.ofReal K := by gcongr
+    _ = ENNReal.ofReal ((2*Real.pi*C/H)*Real.exp (-c*H/(2*Real.pi))) := by
+      rw [← ENNReal.ofReal_mul hv]
+      congr 1
+      dsimp [K]
+      field_simp [hH.ne']
+
+ 
+
+def SourceStripLocalizationTarget : Prop :=
+  ∀ C0 α : ℝ, 0 < C0 → 0 < α → ∃ b C c H0 : ℝ,
+    0 < b ∧ 0 < C ∧ 0 < c ∧ 0 < H0 ∧
+    ∀ H : ℝ, H0 ≤ H → ∀ u : ℂ → ℝ,
+    ContinuousOn u {z : ℂ | 0 ≤ z.im ∧ z.im ≤ Real.pi} →
+    LocallyLipschitzOn {z : ℂ | 0 < z.im ∧ z.im < Real.pi} u →
+    (∀ z, 0 ≤ u z ∧ u z ≤ H) →
+    MemLp (gradient u) 2 (volume.restrict {z : ℂ | 0 < z.im ∧ z.im < Real.pi}) →
+    (∫ z in {z : ℂ | 0 < z.im ∧ z.im < Real.pi}, ‖gradient u z‖^2) ≤ 2*Real.pi*H →
+    (∀ x : ℝ, x ≤ 0 → ∃ y ∈ Icc (0:ℝ) Real.pi, u (x+y*Complex.I) = 0) →
+    ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ X : Ω → Set ℂ,
+      (∀ (j : ℤ) (k : ℕ) (i m : Fin (2^k)),
+        P {xi | (X xi ∩ dyadicStripRectangle j k i m).Nonempty} ≤
+          ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ)))) →
+      P {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*H ∧ u z = H} ≤
+        ENNReal.ofReal (C*Real.exp (-c*H))
+
+lemma stripRectangle_zero_eq_cartesian :
+    stripRectangle 0 0 1 = cartesianRectangle 0 1 0 Real.pi := by
+  ext z
+  simp [stripRectangle,cartesianRectangle]
+
+lemma slabShift_average (u : ℂ → ℝ) (j : ℤ) :
+    spatialAverage (u ∘ slabShift j) (stripRectangle 0 0 1) =
+      Real.pi⁻¹ * ∫ z in closedIntegerStrip j, u z := by
+  rw [spatialAverage,stripRectangle_volume]
+  simp only [mul_one,ENNReal.ofReal_one,one_mul,ENNReal.toReal_ofReal Real.pi_pos.le]
+  rw [stripRectangle_zero_eq_cartesian]
+  simp only [Function.comp_apply]
+  rw [slabShift_integral,div_eq_mul_inv,mul_comm]
+
+lemma slabShift_dyadic (j : ℤ) (k : ℕ) (i m : Fin (2^k)) :
+    slabShift j ⁻¹' dyadicStripRectangle j k i m = dyadicStripRectangle 0 k i m := by
+  ext z
+  simp only [dyadicStripRectangle,stripRectangle,slabShift,mem_preimage,mem_ofPred_eq,
+    Complex.add_re,Complex.add_im,Complex.intCast_re,Complex.intCast_im,zero_add,Int.cast_zero]
+  constructor <;> rintro ⟨h1,h2,h3,h4⟩ <;> exact ⟨by linarith,by linarith,h3,h4⟩
+
+lemma slabShift_continuousOn {u : ℂ → ℝ}
+    (hu : ContinuousOn u fullClosedStrip) (j : ℤ) :
+    ContinuousOn (u ∘ slabShift j) (stripRectangle 0 0 1) := by
+  apply hu.comp (slabShift_continuous j).continuousOn
+  intro z hz
+  change 0 ≤ (slabShift j z).im ∧ (slabShift j z).im ≤ Real.pi
+  simpa [slabShift,stripRectangle] using hz.2.2
+
+lemma point_integer_slab (z : ℂ) (hz : 0 ≤ z.im ∧ z.im ≤ Real.pi) :
+    ∃ w ∈ stripRectangle 0 0 1, slabShift ⌊z.re⌋ w = z := by
+  refine ⟨z-(⌊z.re⌋:ℂ),?_,by simp [slabShift]⟩
+  have h0 := Int.floor_le z.re
+  have h1 := Int.lt_floor_add_one z.re
+  simp only [stripRectangle,mem_ofPred_eq,Complex.sub_re,Complex.intCast_re,
+    Complex.sub_im,Complex.intCast_im,sub_zero,zero_add,mul_one]
+  exact ⟨by linarith,by linarith,hz.1,hz.2⟩
+
+lemma slab_energy_le_total {u : ℂ → ℝ} {H : ℝ}
+    (hH : 0 < H)
+    (hg : MemLp (gradient u) 2 (volume.restrict fullOpenStrip))
+    (hE : (∫ z in fullOpenStrip, ‖gradient u z‖^2) ≤ 2*Real.pi*H) (j : ℤ) :
+    (∫ z in closedIntegerStrip j, ‖gradient u z‖^2) ≤ 2*Real.pi*H := by
+  exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp
+    ((ENNReal.le_tsum j).trans (integerSlab_energy_budget hg hE))
+
+ theorem sourceStripLocalization : SourceStripLocalizationTarget := by
+  intro C0 α hC0 hα
+  obtain ⟨C,c,c0,hC,hc,hc0,hrect⟩ := sourceStripRectangleEnergy C0 α hC0 hα
+  obtain ⟨b,Hm,hb,hHm,hmean⟩ := sourceSlabSmallMean
+  refine ⟨b,2*Real.pi*C,c/(2*Real.pi),max Hm (max 1 (2*Real.pi/c0)),hb,
+    by positivity,by positivity,lt_of_lt_of_le hHm (le_max_left _ _),?_⟩
+  intro H hH u hu hl huB hg hE hz Ω mΩ P hP X hhit
+  let := hP
+  have hHm' : Hm ≤ H := (le_max_left _ _).trans hH
+  have hH1 : 1 ≤ H := (le_max_left _ _).trans ((le_max_right _ _).trans hH)
+  have hH0 : 0 < H := by linarith
+  have hHc : 2*Real.pi/c0 ≤ H := (le_max_right _ _).trans ((le_max_right _ _).trans hH)
+  let E : ℤ → ℝ := fun j => ∫ z in closedIntegerStrip j, ‖gradient u z‖^2
+  let A : ℤ → Set Ω := fun j =>
+    {xi | (j:ℝ) ≤ b*H ∧ ((slabShift j ⁻¹' X xi) ∩
+      {z | z ∈ stripRectangle 0 0 1 ∧ (u ∘ slabShift j) z = H}).Nonempty}
+  have hEn (j : ℤ) : 0 ≤ E j := integral_nonneg fun z => sq_nonneg _
+  have hEj (j : ℤ) : E j ≤ 2*Real.pi*H := slab_energy_le_total hH0 hg hE j
+  have hsmall (j : ℤ) : E j ≤ c0*H^2 := by
+    have hhc := (div_le_iff₀ hc0).mp hHc
+    exact (hEj j).trans (by nlinarith [mul_le_mul_of_nonneg_right hhc hH0.le])
+  have hA (j : ℤ) : P (A j) ≤ ENNReal.ofReal (C*(E j/H^2)*Real.exp (-c*H^2/E j)) := by
+    by_cases hj : (j:ℝ) ≤ b*H
+    · have ht (k : ℕ) (i m : Fin (2^k)) :
+          P {xi | ((slabShift j ⁻¹' X xi) ∩ dyadicStripRectangle 0 k i m).Nonempty} ≤
+            ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ))) := by
+        apply (measure_mono ?_).trans (hhit j k i m)
+        rintro xi ⟨z,hzX,hzQ⟩
+        exact ⟨slabShift j z,hzX,by rwa [← slabShift_dyadic j k i m] at hzQ⟩
+      have hm : spatialAverage (u ∘ slabShift j) (stripRectangle 0 0 1) ≤ H/2 := by
+        rw [slabShift_average]
+        exact hmean H hHm' u hu hl hg hE hz j hj
+      have hh := hrect H hH0 (u ∘ slabShift j) (gradient (u ∘ slabShift j))
+        (slabShift_source_weakGradient hu hl hg huB j) (slabShift_continuousOn hu j)
+        (fun z _ => huB (slabShift j z)) hm Ω mΩ P hP (fun xi => slabShift j ⁻¹' X xi) ht
+        (by rw [slabShift_energy]; exact hsmall j)
+      rw [slabShift_energy] at hh
+      apply (measure_mono ?_).trans hh
+      exact fun xi hxi => hxi.2
+    · have he : A j = ∅ := by ext xi; simp [A,hj]
+      rw [he,measure_empty]
+      exact bot_le
+  have hcover : {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*H ∧ u z = H} ⊆
+      ⋃ j : ℤ, A j := by
+    rintro xi ⟨z,hzX,hz0,hzπ,hzr,hzu⟩
+    obtain ⟨w,hw,heq⟩ := point_integer_slab z ⟨hz0,hzπ⟩
+    apply mem_iUnion.mpr
+    refine ⟨⌊z.re⌋,⟨(Int.floor_le z.re).trans hzr,?_⟩⟩
+    exact ⟨w,by simpa only [mem_preimage,heq] using hzX,hw,by simpa [Function.comp_def,heq] using hzu⟩
+  calc
+    _ ≤ P (⋃ j : ℤ, A j) := measure_mono hcover
+    _ ≤ ∑' j : ℤ, P (A j) := measure_iUnion_le _
+    _ ≤ ∑' j : ℤ, ENNReal.ofReal (C*(E j/H^2)*Real.exp (-c*H^2/E j)) := ENNReal.tsum_le_tsum hA
+    _ ≤ ENNReal.ofReal ((2*Real.pi*C/H)*Real.exp (-c*H/(2*Real.pi))) :=
+      abstract_slab_exponential_budget hH0 hEn (integerSlab_energy_budget hg hE) hC.le hc
+    _ ≤ ENNReal.ofReal (2*Real.pi*C*Real.exp (-(c/(2*Real.pi))*H)) := by
+      apply ENNReal.ofReal_le_ofReal
+      have ht : 2*Real.pi*C/H ≤ 2*Real.pi*C := div_le_self (by positivity) hH1
+      have he : -c*H/(2*Real.pi) = -(c/(2*Real.pi))*H := by ring
+      rw [he]
+      exact mul_le_mul_of_nonneg_right ht (Real.exp_pos _).le
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/OccupationBound.lean b/lean/OAI/Probability/SLE/OccupationBound.lean
index 5af7a13e9e6e3212647cfedee1f88c451c0701a9..864788dc4dddcfce29c3e1723eb81739076dcafe 100644
--- a/lean/OAI/Probability/SLE/OccupationBound.lean
+++ b/lean/OAI/Probability/SLE/OccupationBound.lean
@@ -1,731 +1,731 @@
-import Mathlib
-import OAI.Probability.SLE.FiniteStoppedLaw
-
-namespace OAI
-
-/-! Occupation Bound for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge
-
-lemma occupation_difference {F H : ℝ → ℝ} {v A C L : ℝ}
-    (hv : 0 ≤ v) (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
-    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ s in (0 : ℝ)..u, H s) =
-      L + A * (∫ s in (0 : ℝ)..u, F s))
-    {s t : ℝ} (hs : s ∈ Icc 0 v) (ht : t ∈ Icc 0 v) :
-    F t + C * (∫ u in s..t, H u) = F s + A * (∫ u in s..t, F u) := by
-  have sub (x : ℝ) (hx : x ∈ Icc 0 v) : uIcc 0 x ⊆ uIcc 0 v := by
-    simpa only [uIcc_of_le hx.1, uIcc_of_le hv] using Icc_subset_Icc le_rfl hx.2
-  rw [← intervalIntegral.integral_interval_sub_left (hHi.mono_set (sub t ht)) (hHi.mono_set (sub s hs)),
-    ← intervalIntegral.integral_interval_sub_left (hFi.mono_set (sub t ht)) (hFi.mono_set (sub s hs))]
-  linarith [hbal s hs, hbal t ht]
-
-lemma high_occupation_le {F H : ℝ → ℝ} {v A C L D s t : ℝ}
-    (hv : 0 ≤ v) (hA : 0 ≤ A) (hC : 0 < C)
-    (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
-    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ x in (0 : ℝ)..u, H x) =
-      L + A * (∫ x in (0 : ℝ)..u, F x))
-    (hs : s ∈ Icc 0 v) (ht : t ∈ Icc 0 v) (hst : s ≤ t)
-    (hn : 0 ≤ F t) (hb : ∀ x ∈ Icc s t, F x ≤ D) :
-    (∫ x in s..t, H x) ≤ D * (1 + A * (t - s)) / C := by
-  have sub : uIcc s t ⊆ uIcc 0 v := by
-    simpa only [uIcc_of_le hst, uIcc_of_le hv] using Icc_subset_Icc hs.1 ht.2
-  have hl : (∫ x in s..t, F x) ≤ (t - s) * D := by
-    have h := intervalIntegral.integral_mono_on hst (hFi.mono_set sub) intervalIntegrable_const hb
-    simpa only [intervalIntegral.integral_const, smul_eq_mul] using h
-  have he := occupation_difference hv hFi hHi hbal hs ht
-  have hh := mul_le_mul_of_nonneg_left hl hA
-  apply (le_div_iff₀ hC).mpr
-  have hFs := hb s (left_mem_Icc.mpr hst)
-  nlinarith
-
-lemma point_le_occupation {H : ℝ → ℝ} {s t G Q : ℝ}
-    (hst : s < t) (hi : IntervalIntegrable H volume s t)
-    (hd : ∀ᵐ x ∂volume.restrict (Icc s t), G ≤ Q * H x) :
-    G ≤ Q * (∫ x in s..t, H x) / (t - s) := by
-  have hh := intervalIntegral.integral_mono_ae_restrict hst.le
-    (intervalIntegrable_const (c := G)) (hi.const_mul Q) hd
-  rw [intervalIntegral.integral_const, smul_eq_mul, intervalIntegral.integral_const_mul] at hh
-  exact (le_div_iff₀ (sub_pos.mpr hst)).mpr (by nlinarith)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
-
-lemma radiusMoment_zero_growth
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (hκ8 : κ < 8)
-    (hz : z ∈ upperHalfPlane) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) :
-    radiusMoment P c (greenExponent κ) t ≤ Real.exp (codimension κ * (t - s)) *
-      radiusMoment P c (greenExponent κ) s := by
-  have hh := div_le_div_of_nonneg_right
-    (integral_limitRadiusKernel_zero_growth (P := P) c hε hBm (codimension κ) s t (codimension_pos κ hκ8).le hs hst)
-    (initialGreen_pos κ z hz).le
-  simpa only [radiusMoment, sub_self, mul_div_assoc] using hh
-
-lemma radiusMoment_zero_le_high
-    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (hκ8 : κ < 8)
-    (hz : z ∈ upperHalfPlane) (b s : ℝ) (hb : greenExponent κ ≤ b + 2)
-    (hh : Integrable (limitRadiusKernel c (codimension κ) (greenExponent κ - b - 2) s) P) :
-    radiusMoment P c (greenExponent κ) s ≤ radiusMoment P c (b + 2) s := by
-  apply div_le_div_of_nonneg_right _ (initialGreen_pos κ z hz).le
-  rw [sub_self, sub_add_eq_sub_sub]
-  apply integral_limitRadiusKernel_antitone_power c hε (codimension κ) (greenExponent κ - b - 2) 0 s
-    (by linarith) hh
-  exact limitRadiusKernel_integrable c hε hBm (codimension κ) 0 (codimension_pos κ hκ8).le le_rfl s
-
-lemma radiusMoment_high_integrable
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
-    IntervalIntegrable (radiusMoment P (a v) (b + 2)) volume 0 v := by
-  let := hB.isGaussianProcess.isProbabilityMeasure
-  apply radiusMoment_intervalIntegrable _ v.coe_nonneg
-  simpa only [sub_add_eq_sub_sub] using
-    (limit_power_balance (a v) (a v) hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb v.coe_nonneg le_rfl).1
-
-lemma radiusMoment_high_occupation
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ)
-    (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) (hηv : 2 * η ≤ (v : ℝ)) :
-    (∫ s in (v : ℝ) - η..v, radiusMoment P (a v) (b + 2) s) ≤
-      angularMomentBound κ b η * (1 + lyapunovA κ b * η) / lyapunovC κ b := by
-  let := hB.isGaussianProcess.isProbabilityMeasure
-  have hFi := radiusMoment_intervalIntegrable (P := P) (a v) v.coe_nonneg
-    (limitRadiusKernel_region_integrable (a v) hε hBm (codimension κ) (greenExponent κ - b)
-      (codimension_pos κ hκ8).le (sub_pos.mpr hb).le v)
-  have hHi := radiusMoment_high_integrable a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb (v := v)
-  have hbal := radiusMoment_balance a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb (v := v)
-  have hbound := radiusMoment_bound a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz b hb0 hb η hη hη1 (v := v)
-  have hh := high_occupation_le v.coe_nonneg (lyapunovA_nonneg hκ hb0 hb) (lyapunovC_pos hκ hb0 hb)
-    hFi hHi hbal (s := (v : ℝ) - η) (t := v) (D := angularMomentBound κ b η)
-    (by constructor <;> linarith) ⟨v.coe_nonneg, le_rfl⟩ (by linarith)
-    (radiusMoment_nonneg _ (initialGreen_pos κ z hz).le b v)
-    (fun x hx => hbound x ⟨by linarith [hx.1], hx.2⟩)
-  simpa only [sub_sub_cancel] using hh
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def inverseAngleBound (κ b η : ℝ) : ℝ :=
-  Real.exp (codimension κ * η) *
-    (angularMomentBound κ b η * (1 + lyapunovA κ b * η) / lyapunovC κ b) / η
-
-lemma inverseAngleBound_pos {κ b η : ℝ} (hκ : 0 < κ) (hb0 : 0 < b)
-    (hb : b < greenExponent κ) (hη : 0 < η) : 0 < inverseAngleBound κ b η := by
-  have hD := angularMomentBound_pos hκ hb0 hb hη
-  have hA := lyapunovA_nonneg hκ hb0 hb
-  have hC := lyapunovC_pos hκ hb0 hb
-  dsimp only [inverseAngleBound]
-  positivity
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
-
-lemma radiusMoment_zero_growth_high_ae
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) (hb2 : greenExponent κ ≤ b + 2)
-    (η : ℝ) (_hη : 0 < η) (hηv : η ≤ (v : ℝ)) :
-    ∀ᵐ s ∂volume.restrict (Icc ((v : ℝ) - η) v),
-      radiusMoment P (a v) (greenExponent κ) v ≤
-        Real.exp (codimension κ * η) * radiusMoment P (a v) (b + 2) s := by
-  let := hB.isGaussianProcess.isProbabilityMeasure
-  have hh := (limit_power_balance (a v) (a v) hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb v.coe_nonneg le_rfl).1.prod_right_ae
-  rw [restrict_Ioc_eq_restrict_Icc] at hh
-  have hs : Icc ((v : ℝ) - η) v ⊆ Icc 0 v := Icc_subset_Icc (by linarith) le_rfl
-  filter_upwards [ae_restrict_of_ae_restrict_of_subset hs hh, ae_restrict_mem measurableSet_Icc] with s hsi hsm
-  have hs0 : 0 ≤ s := (hs hsm).1
-  have hg := radiusMoment_zero_growth (a v) hε hBm hκ8 hz hs0 hsm.2 (P := P)
-  have hdom := radiusMoment_zero_le_high (a v) hε hBm hκ8 hz b s hb2 hsi (P := P)
-  have he : Real.exp (codimension κ * ((v : ℝ) - s)) ≤ Real.exp (codimension κ * η) := by
-    apply Real.exp_le_exp.mpr
-    exact mul_le_mul_of_nonneg_left (by linarith [hsm.1]) (codimension_pos κ hκ8).le
-  exact hg.trans ((mul_le_mul_of_nonneg_right he
-    (radiusMoment_nonneg _ (initialGreen_pos κ z hz).le _ _)).trans
-      (mul_le_mul_of_nonneg_left hdom (Real.exp_pos _).le))
-
- 
-
-theorem radiusMoment_inverseAngle_bound
-    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
-    {P : Measure Ω} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
-    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) (hb2 : greenExponent κ ≤ b + 2)
-    (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) (hηv : 2 * η ≤ (v : ℝ)) :
-    radiusMoment P (a v) (greenExponent κ) v ≤ inverseAngleBound κ b η := by
-  have hd := radiusMoment_zero_growth_high_ae a hε hεlim hε0 hB hBm hBc hB0
-    hκ hκ8 hz b hb0 hb hb2 η hη (by linarith : η ≤ (v : ℝ))
-  have hi := radiusMoment_high_integrable a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb (v := v)
-  have his : IntervalIntegrable (radiusMoment P (a v) (b + 2)) volume ((v : ℝ) - η) v := by
-    apply hi.mono_set
-    simp only [uIcc_of_le (by linarith : (v : ℝ) - η ≤ v), uIcc_of_le v.coe_nonneg]
-    exact Icc_subset_Icc (by linarith) le_rfl
-  have hp := point_le_occupation (by linarith : (v : ℝ) - η < v) his hd
-  have hocc := radiusMoment_high_occupation a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz b hb0 hb η hη hη1 hηv
-  simp only [sub_sub_cancel] at hp
-  exact hp.trans (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hocc (Real.exp_pos _).le) hη.le)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  (hB : IsBrownianReal B P)
-  (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-  (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-  {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-  (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-    (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-  {z : ℂ} (hz : z ∈ upperHalfPlane) {u v : ℝ} {ε : ℕ → ℝ}
-
-include hB hBm hBc hB0 hκ hκ8 hw hz
-
-lemma localization_integral_tendsto
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Tendsto ε atTop (𝓝 0)) (A : Set Ω) :
-    Tendsto (fun n => ∫ ω in A, (a n).weight ω ∂P) atTop
-      (𝓝 (∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P)) := by
-  have hi (n : ℕ) := ((a n).mean_green hB hBm hBc hB0 hκ hκ8 hz).1
-  apply tendsto_integral_of_dominated_convergence
-    (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-    (fun n => (hi n).integrableOn.aestronglyMeasurable) (integrable_const _)
-  · intro n
-    apply Eventually.of_forall
-    intro ω
-    change ‖(a n).weight ω‖ ≤ _
-    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).weight_bound hκ hκ8 ω).1]
-    exact ((a n).weight_bound hκ hκ8 ω).2
-  · filter_upwards [ae_restrict_of_ae hw] with ω hω
-    exact localization_weight_tendsto a hε hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω
-
-omit hw hz [IsProbabilityMeasure P] in
-lemma localization_history_finer
-    {η δ : ℝ} (a : AngularLocalization B hm κ z u η)
-    (b : AngularLocalization B hm κ z v δ) (huv : u ≤ v) (hδη : δ ≤ η)
-    (A : Set Ω) (hA : MeasurableSet[a.stopping.measurableSpace] A) :
-    (∫ ω in A, b.weight ω ∂P) = ∫ ω in A, a.weight ω ∂P := by
-  have hle := a.stopping_le b huv hδη
-  have he := b.solution.stopped_green_history hB hBm hBc hB0 hκ hκ8
-    (Real.log z.im) v a.τ b.τ a.stopping b.stopping hle b.T b.bounded
-    (fun ω t ht => ⟨(b.inside ω t ht).1, (b.inside ω t ht).2.2⟩) A hA
-  refine he.trans (integral_congr_ae (Eventually.of_forall fun ω => ?_))
-  dsimp only
-  rw [← a.synchronize b ω (a.τ ω) le_rfl (hle ω)]
-  rfl
-
-lemma finite_history_on_hit
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (b : ∀ n, AngularLocalization B hm κ z v (ε n))
-    (hε : Tendsto ε atTop (𝓝 0)) (hεanti : Antitone ε) (huv : u ≤ v)
-    (n : ℕ) (A : Set Ω)
-    (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
-    (∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z v ∂P) =
-      ∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P := by
-  have hAn := localizedRadiusStop_history_hit a n A hA
-  have he (m : ℕ) := localization_history_finer hB hBm hBc hB0 hκ hκ8
-    (a n) (b (m + n)) huv (hεanti (Nat.le_add_left n m)) (A ∩ (a n).hit) hAn
-  have hlim := (localization_integral_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw hz b hε
-    (A ∩ (a n).hit)).comp (tendsto_add_atTop_nat n)
-  have hx : (∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z v ∂P) = ∫ ω in A ∩ (a n).hit, (a n).weight ω ∂P := by
-    apply tendsto_nhds_unique hlim
-    simpa only [Function.comp_def, he] using
-      (tendsto_const_nhds : Tendsto (fun _ : ℕ => ∫ ω in A ∩ (a n).hit, (a n).weight ω ∂P) atTop (𝓝 _))
-  refine hx.trans (integral_congr_ae ?_)
-  have hAnm : MeasurableSet (A ∩ (a n).hit) := (a n).stopping.measurableSpace_le _ hAn
-  filter_upwards [ae_restrict_of_ae hw, ae_restrict_mem hAnm] with ω hω hωA
-  exact (a n).weight_eq_stoppedGreen_of_hit hBc hz ω (γ ω) (G ω) (F ω) hω hωA.2
-
-lemma hit_integral_tendsto
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (hε : Tendsto ε atTop (𝓝 0)) (hu : 0 ≤ u) (A : Set Ω) :
-    Tendsto (fun n => ∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) z u ∂P) atTop
-      (𝓝 (∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P)) := by
-  let g (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
-  have hi := (finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu).1
-  have hhit (n : ℕ) : MeasurableSet (a n).hit := (a n).stopping.measurableSpace_le _ (a n).measurable_hit
-  have hlim : Tendsto (fun n => ∫ ω in A, (a n).hit.indicator g ω ∂P) atTop (𝓝 (∫ ω in A, g ω ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence (fun ω => ‖g ω‖)
-      (fun n => (hi.indicator (hhit n)).integrableOn.aestronglyMeasurable) hi.norm.integrableOn
-    · intro n
-      exact Eventually.of_forall fun ω => norm_indicator_le_norm_self _ _
-    · filter_upwards [ae_restrict_of_ae hw] with ω hω
-      change Tendsto (fun n => (a n).hit.indicator g ω) atTop (𝓝 (g ω))
-      by_cases hh : ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
-          fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u)
-      · obtain ⟨T, hT, hR⟩ := hh
-        apply tendsto_const_nhds.congr'
-        filter_upwards [eventually_hit_of_live_level a hε hBc hz ω (γ ω) (G ω) (F ω) hω T hT hR] with n hn
-        simp only [Set.indicator_of_mem hn.1]
-      · have hg : g ω = 0 := stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hh
-        have he (n : ℕ) : (a n).hit.indicator g ω = 0 := by
-          by_cases hn : ω ∈ (a n).hit <;> simp [hn, hg]
-        simpa only [he, hg] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))
-  simpa only [integral_indicator (hhit _), Measure.restrict_restrict (hhit _), Set.inter_comm] using hlim
-
- 
-
-theorem finite_stopped_green_consistent_regular
-    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
-    (b : ∀ n, AngularLocalization B hm κ z v (ε n))
-    (hε : Tendsto ε atTop (𝓝 0)) (hεanti : Antitone ε)
-    (hu : 0 ≤ u) (huv : u ≤ v) (A : Set Ω)
-    (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
-    (∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z v ∂P) =
-      ∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P := by
-  let gu (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
-  let gv (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z v
-  obtain ⟨hiu, heu⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu
-  obtain ⟨hiv, hev⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz v (hu.trans huv)
-  have hgv : 0 ≤ᵐ[P] gv := hw.mono fun ω hω => stoppedGreen_nonneg hω κ z v
-  have hmono (C : Set Ω) (hC : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] C) :
-      (∫ ω in C, gu ω ∂P) ≤ ∫ ω in C, gv ω ∂P := by
-    have hlim := hit_integral_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hu C
-    apply le_of_tendsto hlim
-    apply Eventually.of_forall
-    intro n
-    rw [← finite_history_on_hit hB hBm hBc hB0 hκ hκ8 γ G F hw hz a b hε hεanti huv n C hC]
-    exact setIntegral_mono_set hiv.integrableOn (ae_restrict_of_ae hgv)
-      (Eventually.of_forall fun ω hω => hω.1)
-  have h1 := hmono A hA
-  have h2 := hmono Aᶜ hA.compl
-  have hAm : MeasurableSet A := (localizedRadiusStop_stopping a).measurableSpace_le _ hA
-  have hsumu := integral_add_compl hAm hiu
-  have hsumv := integral_add_compl hAm hiv
-  change (∫ ω in A, gu ω ∂P) + (∫ ω in Aᶜ, gu ω ∂P) = _ at hsumu
-  change (∫ ω in A, gv ω ∂P) + (∫ ω in Aᶜ, gv ω ∂P) = _ at hsumv
-  rw [heu] at hsumu
-  rw [hev] at hsumv
-  change (∫ ω in A, gv ω ∂P) = ∫ ω in A, gu ω ∂P
-  linarith
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma finiteCenteredLaw_apply
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
-    (A : Set Ω) (hA : MeasurableSet A) :
-    finiteCenteredLaw P κ B γ G z u A = ENNReal.ofReal
-      ((∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P) /
-        initialGreen κ z) := by
-  have hi := (finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu).1
-  rw [finiteCenteredLaw, withDensity_apply _ hA,
-    ← ofReal_integral_eq_lintegral_ofReal ((hi.div_const _).integrableOn) ?_, integral_div]
-  filter_upwards [ae_restrict_of_ae hw] with ω hω
-  exact div_nonneg (stoppedGreen_nonneg hω κ z u) (initialGreen_pos κ z hz).le
-
- 
-
-theorem exists_consistent_finiteCenteredLaw_regular
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    ∃ (ρ : ℝ≥0 → Ω → WithTop ℝ≥0)
-      (hρ : ∀ u, IsStoppingTime (Filtration.natural B
-        (fun _ => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) (ρ u)),
-      (∀ u, ∀ᵐ ω ∂P, ρ u ω = radiusLevelTime (γ ω) (G ω) z u) ∧
-      (∀ (u : ℝ≥0), IsProbabilityMeasure (finiteCenteredLaw P κ B γ G z u)) ∧
-      (∀ (u v : ℝ≥0), u ≤ v → ∀ A : Set Ω, MeasurableSet[(hρ u).measurableSpace] A →
-        finiteCenteredLaw P κ B γ G z v A = finiteCenteredLaw P κ B γ G z u A) := by
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
-  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
-  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
-  have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
-  have hε₀ (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
-    simpa only [mul_one_div, mul_zero] using hh
-  have hεanti : Antitone ε := by
-    intro n m hnm
-    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
-  let a (u : ℝ≥0) (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
-    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
-      u.coe_nonneg (hε n) (hε₀ n))
-  refine ⟨fun u => localizedRadiusStop (a u), fun u => localizedRadiusStop_stopping (a u), ?_, ?_, ?_⟩
-  · intro u
-    filter_upwards [hw] with ω hω
-    exact localizedRadiusStop_eq_source (a u) hεlim hBc hz ω (γ ω) (G ω) (F ω) hω
-  · intro u
-    exact finiteCenteredLaw_probability hB κ hκ hκ8 γ G F hw z hz u u.coe_nonneg
-  · intro u v huv A hA
-    have hAm : MeasurableSet A := (localizedRadiusStop_stopping (a u)).measurableSpace_le _ hA
-    rw [finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz v v.coe_nonneg A hAm,
-      finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz u u.coe_nonneg A hAm,
-      finite_stopped_green_consistent_regular hB hBm hBc hB0 hκ hκ8 γ G F hw hz
-        (a u) (a v) hεlim hεanti u.coe_nonneg huv A hA]
-
-lemma finiteCenteredLaw_congr_driver
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {B C : ℝ≥0 → Ω → ℝ}
-    (hCB : ∀ᵐ ω ∂P, ∀ t, C t ω = B t ω)
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (u : ℝ) :
-    finiteCenteredLaw P κ C γ G z u = finiteCenteredLaw P κ B γ G z u := by
-  apply withDensity_congr_ae
-  filter_upwards [hCB] with ω hω
-  simp only [hω]
-
- 
-
-theorem exists_consistent_finiteCenteredLaw
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    ∃ (C : ℝ≥0 → Ω → ℝ) (hCm : ∀ t, Measurable (C t))
-      (ρ : ℝ≥0 → Ω → WithTop ℝ≥0)
-      (hρ : ∀ u, IsStoppingTime (Filtration.natural C (fun t => (hCm t).stronglyMeasurable)) (ρ u)),
-      (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) ∧
-      (∀ u, ∀ᵐ ω ∂P, ρ u ω = radiusLevelTime (γ ω) (G ω) z u) ∧
-      (∀ (u : ℝ≥0), IsProbabilityMeasure (finiteCenteredLaw P κ B γ G z u)) ∧
-      (∀ (u v : ℝ≥0), u ≤ v → ∀ A : Set Ω, MeasurableSet[(hρ u).measurableSpace] A →
-        finiteCenteredLaw P κ B γ G z v A = finiteCenteredLaw P κ B γ G z u A) := by
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
-    filter_upwards [hw, hCB] with ω hω he
-    simpa only [he] using hω
-  obtain ⟨ρ, hρ, hρeq, hprob, hcon⟩ := exists_consistent_finiteCenteredLaw_regular
-    hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz
-  have he (u : ℝ≥0) := finiteCenteredLaw_congr_driver hCB κ γ G z u
-  refine ⟨C, fun t => hCm.comp (measurable_const.prodMk measurable_id), ρ, hρ, hCB, hρeq, ?_, ?_⟩
-  · intro u
-    rw [← he u]
-    exact hprob u
-  · intro u v huv A hA
-    rw [← he u, ← he v]
-    exact hcon u v huv A hA
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def inverseAngleSubcritical (κ : ℝ) : ℝ :=
-  greenExponent κ - min (greenExponent κ / 2) 1
-noncomputable def inverseAngleBuffer (u₀ : ℝ) : ℝ := min (u₀ / 2) 1
-noncomputable def uniformInverseAngleConstant (κ u₀ : ℝ) : ℝ :=
-  inverseAngleBound κ (inverseAngleSubcritical κ) (inverseAngleBuffer u₀)
-
-lemma inverseAngleSubcritical_bounds (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < inverseAngleSubcritical κ ∧ inverseAngleSubcritical κ < greenExponent κ ∧
-      greenExponent κ ≤ inverseAngleSubcritical κ + 2 := by
-  have ha := greenExponent_pos κ hκ hκ8
-  unfold inverseAngleSubcritical
-  have hm0 : 0 < min (greenExponent κ / 2) 1 := lt_min (by positivity) (by norm_num)
-  have hm := min_le_left (greenExponent κ / 2) 1
-  have hm1 := min_le_right (greenExponent κ / 2) 1
-  constructor
-  · linarith
-  constructor <;> linarith
-
-lemma uniformInverseAngleConstant_pos (κ u₀ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hu₀ : 0 < u₀) :
-    0 < uniformInverseAngleConstant κ u₀ := by
-  have hb := inverseAngleSubcritical_bounds κ hκ hκ8
-  exact inverseAngleBound_pos hκ hb.1 hb.2.1 (lt_min (by positivity) (by norm_num))
-
- 
-
-theorem uniform_inverse_angle_regular
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u₀ : ℝ) (hu₀ : 0 < u₀)
-    (v : ℝ≥0) (hv : u₀ ≤ v) :
-    Integrable (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ))
-      (finiteCenteredLaw P κ B γ G z v) ∧
-    (∫ ω, stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ)
-      ∂finiteCenteredLaw P κ B γ G z v) ≤ uniformInverseAngleConstant κ u₀ := by
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
-  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
-  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
-  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
-  have hεA (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
-    simpa only [mul_one_div, mul_zero] using hh
-  have hεanti : Antitone ε := by
-    intro n m hnm
-    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
-  let a (u : ℝ≥0) (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
-    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
-      u.coe_nonneg (hε0 n) (hεA n))
-  obtain ⟨hi, he⟩ := finiteCenteredLaw_inverseAngle_moment hB hBm hBc κ hκ hκ8 γ G F hw z hz v v.coe_nonneg
-    (a v) hεanti hεlim (greenExponent κ) le_rfl
-  refine ⟨hi, he.trans_le ?_⟩
-  have hb := inverseAngleSubcritical_bounds κ hκ hκ8
-  apply radiusMoment_inverseAngle_bound a hεanti hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz
-    (inverseAngleSubcritical κ) hb.1 hb.2.1 hb.2.2
-    (inverseAngleBuffer u₀) (lt_min (by positivity) (by norm_num)) (min_le_right _ _)
-  have hη := min_le_left (u₀ / 2) 1
-  dsimp only [inverseAngleBuffer]
-  linarith
-
- 
-
-theorem uniform_inverse_angle
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (u₀ : ℝ) (hu₀ : 0 < u₀)
-    (v : ℝ≥0) (hv : u₀ ≤ v) :
-    Integrable (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ))
-      (finiteCenteredLaw P κ B γ G z v) ∧
-    (∫ ω, stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ)
-      ∂finiteCenteredLaw P κ B γ G z v) ≤ uniformInverseAngleConstant κ u₀ := by
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
-    filter_upwards [hw, hCB] with ω hω he
-    simpa only [he] using hω
-  obtain ⟨hi, hb⟩ := uniform_inverse_angle_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz u₀ hu₀ v hv
-  rw [finiteCenteredLaw_congr_driver hCB κ γ G z v] at hi hb
-  have he : (fun ω => stoppedInverseAngle κ (fun t => C t ω) (γ ω) (G ω) z v (greenExponent κ)) =ᵐ[finiteCenteredLaw P κ B γ G z v]
-      (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ)) := by
-    have hCB' : ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z v, ∀ t, C t ω = B t ω :=
-      (withDensity_absolutelyContinuous P _).ae_le hCB
-    filter_upwards [hCB'] with ω he
-    simp only [stoppedInverseAngle, stoppedAngleSq, he]
-  exact ⟨hi.congr he, (integral_congr_ae he).symm.trans_le hb⟩
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma radius_approach_regular
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (v : ℝ≥0) (hv : 1 ≤ (v : ℝ)) :
-    (P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞}).toReal ≤
-      uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
-        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
-  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
-  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
-  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
-  have hεA (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have hεlim : Tendsto ε atTop (𝓝 0) := by
-    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
-    simpa only [mul_one_div, mul_zero] using hh
-  have hεanti : Antitone ε := by
-    intro n m hnm
-    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
-  let a (n : ℕ) : AngularLocalization B hm κ z v (ε n) :=
-    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z v (ε n)
-      v.coe_nonneg (hε0 n) (hεA n))
-  have hbound := (uniform_inverse_angle_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz
-    1 (by norm_num) v hv).2
-  have he := (finiteCenteredLaw_inverseAngle_moment hB hBm hBc κ hκ hκ8 γ G F hw z hz v v.coe_nonneg
-    a hεanti hεlim (greenExponent κ) le_rfl).2
-  rw [he, radiusMoment, sub_self,
-    integral_limitRadiusKernel_zero_eq_probability a hεanti hεlim hBm hBc hz v.coe_nonneg γ G F hw] at hbound
-  have hmult := (div_le_iff₀ (initialGreen_pos κ z hz)).mp hbound
-  calc
-    (P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞}).toReal ≤
-        uniformInverseAngleConstant κ 1 * initialGreen κ z /
-          Real.exp (codimension κ * ((v : ℝ) - Real.log 2 - Real.log z.im)) := by
-      apply (le_div_iff₀ (Real.exp_pos _)).mpr
-      simpa only [mul_comm] using hmult
-    _ = uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
-        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
-      rw [mul_div_assoc, initialGreen_div_zeroKernel κ v z hz, mul_assoc]
-
- 
-
-theorem radius_approach
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (v : ℝ≥0) (hv : 1 ≤ (v : ℝ)) :
-    (P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞}).toReal ≤
-      uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
-        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
-    filter_upwards [hw, hCB] with ω hω he
-    simpa only [he] using hω
-  exact radius_approach_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz v hv
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem euclidean_approach_at_level
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (v : ℝ≥0) (hv : 1 ≤ (v : ℝ))
-    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
-    (hscale : (128 * Real.pi) * r ≤ (2 * z.im) * Real.exp (-v)) :
-    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-      uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
-        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
-  apply le_trans _ (radius_approach hB κ hκ hκ8 γ G F hw z hz v hv)
-  apply ENNReal.toReal_mono (measure_ne_top _ _)
-  apply measure_mono_ae
-  filter_upwards [hw, hB.cont] with ω hω hc
-  rintro ⟨t, ht⟩
-  have hU : Continuous (fun s => Real.sqrt κ * B s ω) := hc.const_mul _
-  obtain ⟨s, _, hsl, hsR⟩ := hω.radius_approach_of_trace_approach hU z r hr hy t ht
-  exact radiusLevelTime_finite_of_live_radius_lt hU hω z v v.coe_nonneg s hsl
-    (hsR.trans_le hscale)
-
- 
-
-theorem euclidean_approach
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (r : ℝ) (hr : 0 < r)
-    (hy : r ≤ z.im)
-    (hsmall : 1 ≤ Real.log ((2 * z.im) / ((128 * Real.pi) * r))) :
-    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-      (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
-        initialGreen κ z * r ^ codimension κ := by
-  let u := Real.log ((2 * z.im) / ((128 * Real.pi) * r))
-  have hu : 0 ≤ u := (by norm_num : (0 : ℝ) ≤ 1).trans hsmall
-  have hC : 0 < 128 * Real.pi := mul_pos (by norm_num) Real.pi_pos
-  have hy0 : 0 < 2 * z.im := mul_pos (by norm_num) hz
-  have hratio : 0 < (2 * z.im) / ((128 * Real.pi) * r) := div_pos hy0 (mul_pos hC hr)
-  have he : (2 * z.im) * Real.exp (-u) = (128 * Real.pi) * r := by
-    dsimp only [u]
-    rw [Real.exp_neg, Real.exp_log hratio]
-    field_simp [(show 0 < z.im from hz).ne']
-  have hb := euclidean_approach_at_level hB κ hκ hκ8 γ G F hw z hz
-    ⟨u, hu⟩ hsmall r hr hy he.ge
-  have hexp : Real.exp (-codimension κ * u) =
-      (128 * Real.pi) ^ codimension κ * r ^ codimension κ *
-        (2 * z.im) ^ (-codimension κ) := by
-    rw [Real.rpow_def_of_pos hC, Real.rpow_def_of_pos hr,
-      Real.rpow_def_of_pos hy0, ← Real.exp_add, ← Real.exp_add]
-    congr 1
-    dsimp only [u]
-    rw [Real.log_div hy0.ne' (mul_pos hC hr).ne', Real.log_mul hC.ne' hr.ne']
-    ring
-  change (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-    uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * u) *
-      (Real.sin (Complex.arg z)) ^ greenExponent κ at hb
-  rw [hexp] at hb
-  simpa only [initialGreen, mul_assoc, mul_comm, mul_left_comm] using hb
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.FiniteStoppedLaw
+
+namespace OAI
+
+/-! Occupation Bound for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge
+
+lemma occupation_difference {F H : ℝ → ℝ} {v A C L : ℝ}
+    (hv : 0 ≤ v) (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
+    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ s in (0 : ℝ)..u, H s) =
+      L + A * (∫ s in (0 : ℝ)..u, F s))
+    {s t : ℝ} (hs : s ∈ Icc 0 v) (ht : t ∈ Icc 0 v) :
+    F t + C * (∫ u in s..t, H u) = F s + A * (∫ u in s..t, F u) := by
+  have sub (x : ℝ) (hx : x ∈ Icc 0 v) : uIcc 0 x ⊆ uIcc 0 v := by
+    simpa only [uIcc_of_le hx.1, uIcc_of_le hv] using Icc_subset_Icc le_rfl hx.2
+  rw [← intervalIntegral.integral_interval_sub_left (hHi.mono_set (sub t ht)) (hHi.mono_set (sub s hs)),
+    ← intervalIntegral.integral_interval_sub_left (hFi.mono_set (sub t ht)) (hFi.mono_set (sub s hs))]
+  linarith [hbal s hs, hbal t ht]
+
+lemma high_occupation_le {F H : ℝ → ℝ} {v A C L D s t : ℝ}
+    (hv : 0 ≤ v) (hA : 0 ≤ A) (hC : 0 < C)
+    (hFi : IntervalIntegrable F volume 0 v) (hHi : IntervalIntegrable H volume 0 v)
+    (hbal : ∀ u ∈ Icc 0 v, F u + C * (∫ x in (0 : ℝ)..u, H x) =
+      L + A * (∫ x in (0 : ℝ)..u, F x))
+    (hs : s ∈ Icc 0 v) (ht : t ∈ Icc 0 v) (hst : s ≤ t)
+    (hn : 0 ≤ F t) (hb : ∀ x ∈ Icc s t, F x ≤ D) :
+    (∫ x in s..t, H x) ≤ D * (1 + A * (t - s)) / C := by
+  have sub : uIcc s t ⊆ uIcc 0 v := by
+    simpa only [uIcc_of_le hst, uIcc_of_le hv] using Icc_subset_Icc hs.1 ht.2
+  have hl : (∫ x in s..t, F x) ≤ (t - s) * D := by
+    have h := intervalIntegral.integral_mono_on hst (hFi.mono_set sub) intervalIntegrable_const hb
+    simpa only [intervalIntegral.integral_const, smul_eq_mul] using h
+  have he := occupation_difference hv hFi hHi hbal hs ht
+  have hh := mul_le_mul_of_nonneg_left hl hA
+  apply (le_div_iff₀ hC).mpr
+  have hFs := hb s (left_mem_Icc.mpr hst)
+  nlinarith
+
+lemma point_le_occupation {H : ℝ → ℝ} {s t G Q : ℝ}
+    (hst : s < t) (hi : IntervalIntegrable H volume s t)
+    (hd : ∀ᵐ x ∂volume.restrict (Icc s t), G ≤ Q * H x) :
+    G ≤ Q * (∫ x in s..t, H x) / (t - s) := by
+  have hh := intervalIntegral.integral_mono_ae_restrict hst.le
+    (intervalIntegrable_const (c := G)) (hi.const_mul Q) hd
+  rw [intervalIntegral.integral_const, smul_eq_mul, intervalIntegral.integral_const_mul] at hh
+  exact (le_div_iff₀ (sub_pos.mpr hst)).mpr (by nlinarith)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
+
+lemma radiusMoment_zero_growth
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (hκ8 : κ < 8)
+    (hz : z ∈ upperHalfPlane) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) :
+    radiusMoment P c (greenExponent κ) t ≤ Real.exp (codimension κ * (t - s)) *
+      radiusMoment P c (greenExponent κ) s := by
+  have hh := div_le_div_of_nonneg_right
+    (integral_limitRadiusKernel_zero_growth (P := P) c hε hBm (codimension κ) s t (codimension_pos κ hκ8).le hs hst)
+    (initialGreen_pos κ z hz).le
+  simpa only [radiusMoment, sub_self, mul_div_assoc] using hh
+
+lemma radiusMoment_zero_le_high
+    (c : ∀ n, AngularLocalization B hm κ z v (ε n)) (hε : Antitone ε)
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (hκ8 : κ < 8)
+    (hz : z ∈ upperHalfPlane) (b s : ℝ) (hb : greenExponent κ ≤ b + 2)
+    (hh : Integrable (limitRadiusKernel c (codimension κ) (greenExponent κ - b - 2) s) P) :
+    radiusMoment P c (greenExponent κ) s ≤ radiusMoment P c (b + 2) s := by
+  apply div_le_div_of_nonneg_right _ (initialGreen_pos κ z hz).le
+  rw [sub_self, sub_add_eq_sub_sub]
+  apply integral_limitRadiusKernel_antitone_power c hε (codimension κ) (greenExponent κ - b - 2) 0 s
+    (by linarith) hh
+  exact limitRadiusKernel_integrable c hε hBm (codimension κ) 0 (codimension_pos κ hκ8).le le_rfl s
+
+lemma radiusMoment_high_integrable
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) :
+    IntervalIntegrable (radiusMoment P (a v) (b + 2)) volume 0 v := by
+  let := hB.isGaussianProcess.isProbabilityMeasure
+  apply radiusMoment_intervalIntegrable _ v.coe_nonneg
+  simpa only [sub_add_eq_sub_sub] using
+    (limit_power_balance (a v) (a v) hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb v.coe_nonneg le_rfl).1
+
+lemma radiusMoment_high_occupation
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ)
+    (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) (hηv : 2 * η ≤ (v : ℝ)) :
+    (∫ s in (v : ℝ) - η..v, radiusMoment P (a v) (b + 2) s) ≤
+      angularMomentBound κ b η * (1 + lyapunovA κ b * η) / lyapunovC κ b := by
+  let := hB.isGaussianProcess.isProbabilityMeasure
+  have hFi := radiusMoment_intervalIntegrable (P := P) (a v) v.coe_nonneg
+    (limitRadiusKernel_region_integrable (a v) hε hBm (codimension κ) (greenExponent κ - b)
+      (codimension_pos κ hκ8).le (sub_pos.mpr hb).le v)
+  have hHi := radiusMoment_high_integrable a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb (v := v)
+  have hbal := radiusMoment_balance a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb (v := v)
+  have hbound := radiusMoment_bound a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz b hb0 hb η hη hη1 (v := v)
+  have hh := high_occupation_le v.coe_nonneg (lyapunovA_nonneg hκ hb0 hb) (lyapunovC_pos hκ hb0 hb)
+    hFi hHi hbal (s := (v : ℝ) - η) (t := v) (D := angularMomentBound κ b η)
+    (by constructor <;> linarith) ⟨v.coe_nonneg, le_rfl⟩ (by linarith)
+    (radiusMoment_nonneg _ (initialGreen_pos κ z hz).le b v)
+    (fun x hx => hbound x ⟨by linarith [hx.1], hx.2⟩)
+  simpa only [sub_sub_cancel] using hh
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def inverseAngleBound (κ b η : ℝ) : ℝ :=
+  Real.exp (codimension κ * η) *
+    (angularMomentBound κ b η * (1 + lyapunovA κ b * η) / lyapunovC κ b) / η
+
+lemma inverseAngleBound_pos {κ b η : ℝ} (hκ : 0 < κ) (hb0 : 0 < b)
+    (hb : b < greenExponent κ) (hη : 0 < η) : 0 < inverseAngleBound κ b η := by
+  have hD := angularMomentBound_pos hκ hb0 hb hη
+  have hA := lyapunovA_nonneg hκ hb0 hb
+  have hC := lyapunovC_pos hκ hb0 hb
+  dsimp only [inverseAngleBound]
+  positivity
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ : ℝ} {z : ℂ} {v : ℝ≥0} {ε : ℕ → ℝ}
+
+lemma radiusMoment_zero_growth_high_ae
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) (hb2 : greenExponent κ ≤ b + 2)
+    (η : ℝ) (_hη : 0 < η) (hηv : η ≤ (v : ℝ)) :
+    ∀ᵐ s ∂volume.restrict (Icc ((v : ℝ) - η) v),
+      radiusMoment P (a v) (greenExponent κ) v ≤
+        Real.exp (codimension κ * η) * radiusMoment P (a v) (b + 2) s := by
+  let := hB.isGaussianProcess.isProbabilityMeasure
+  have hh := (limit_power_balance (a v) (a v) hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb v.coe_nonneg le_rfl).1.prod_right_ae
+  rw [restrict_Ioc_eq_restrict_Icc] at hh
+  have hs : Icc ((v : ℝ) - η) v ⊆ Icc 0 v := Icc_subset_Icc (by linarith) le_rfl
+  filter_upwards [ae_restrict_of_ae_restrict_of_subset hs hh, ae_restrict_mem measurableSet_Icc] with s hsi hsm
+  have hs0 : 0 ≤ s := (hs hsm).1
+  have hg := radiusMoment_zero_growth (a v) hε hBm hκ8 hz hs0 hsm.2 (P := P)
+  have hdom := radiusMoment_zero_le_high (a v) hε hBm hκ8 hz b s hb2 hsi (P := P)
+  have he : Real.exp (codimension κ * ((v : ℝ) - s)) ≤ Real.exp (codimension κ * η) := by
+    apply Real.exp_le_exp.mpr
+    exact mul_le_mul_of_nonneg_left (by linarith [hsm.1]) (codimension_pos κ hκ8).le
+  exact hg.trans ((mul_le_mul_of_nonneg_right he
+    (radiusMoment_nonneg _ (initialGreen_pos κ z hz).le _ _)).trans
+      (mul_le_mul_of_nonneg_left hdom (Real.exp_pos _).le))
+
+ 
+
+theorem radiusMoment_inverseAngle_bound
+    (a : ∀ u : ℝ≥0, ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Antitone ε) (hεlim : Tendsto ε atTop (𝓝 0)) (hε0 : ∀ n, 0 < ε n)
+    {P : Measure Ω} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hz : z ∈ upperHalfPlane)
+    (b : ℝ) (hb0 : 0 < b) (hb : b < greenExponent κ) (hb2 : greenExponent κ ≤ b + 2)
+    (η : ℝ) (hη : 0 < η) (hη1 : η ≤ 1) (hηv : 2 * η ≤ (v : ℝ)) :
+    radiusMoment P (a v) (greenExponent κ) v ≤ inverseAngleBound κ b η := by
+  have hd := radiusMoment_zero_growth_high_ae a hε hεlim hε0 hB hBm hBc hB0
+    hκ hκ8 hz b hb0 hb hb2 η hη (by linarith : η ≤ (v : ℝ))
+  have hi := radiusMoment_high_integrable a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 b hb0 hb (v := v)
+  have his : IntervalIntegrable (radiusMoment P (a v) (b + 2)) volume ((v : ℝ) - η) v := by
+    apply hi.mono_set
+    simp only [uIcc_of_le (by linarith : (v : ℝ) - η ≤ v), uIcc_of_le v.coe_nonneg]
+    exact Icc_subset_Icc (by linarith) le_rfl
+  have hp := point_le_occupation (by linarith : (v : ℝ) - η < v) his hd
+  have hocc := radiusMoment_high_occupation a hε hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz b hb0 hb η hη hη1 hηv
+  simp only [sub_sub_cancel] at hp
+  exact hp.trans (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hocc (Real.exp_pos _).le) hη.le)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  (hB : IsBrownianReal B P)
+  (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+  (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+  {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+  (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+  (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+    (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+  {z : ℂ} (hz : z ∈ upperHalfPlane) {u v : ℝ} {ε : ℕ → ℝ}
+
+include hB hBm hBc hB0 hκ hκ8 hw hz
+
+lemma localization_integral_tendsto
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Tendsto ε atTop (𝓝 0)) (A : Set Ω) :
+    Tendsto (fun n => ∫ ω in A, (a n).weight ω ∂P) atTop
+      (𝓝 (∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P)) := by
+  have hi (n : ℕ) := ((a n).mean_green hB hBm hBc hB0 hκ hκ8 hz).1
+  apply tendsto_integral_of_dominated_convergence
+    (fun _ => Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+    (fun n => (hi n).integrableOn.aestronglyMeasurable) (integrable_const _)
+  · intro n
+    apply Eventually.of_forall
+    intro ω
+    change ‖(a n).weight ω‖ ≤ _
+    rw [Real.norm_eq_abs, abs_of_nonneg ((a n).weight_bound hκ hκ8 ω).1]
+    exact ((a n).weight_bound hκ hκ8 ω).2
+  · filter_upwards [ae_restrict_of_ae hw] with ω hω
+    exact localization_weight_tendsto a hε hBc hκ hκ8 hz ω (γ ω) (G ω) (F ω) hω
+
+omit hw hz [IsProbabilityMeasure P] in
+lemma localization_history_finer
+    {η δ : ℝ} (a : AngularLocalization B hm κ z u η)
+    (b : AngularLocalization B hm κ z v δ) (huv : u ≤ v) (hδη : δ ≤ η)
+    (A : Set Ω) (hA : MeasurableSet[a.stopping.measurableSpace] A) :
+    (∫ ω in A, b.weight ω ∂P) = ∫ ω in A, a.weight ω ∂P := by
+  have hle := a.stopping_le b huv hδη
+  have he := b.solution.stopped_green_history hB hBm hBc hB0 hκ hκ8
+    (Real.log z.im) v a.τ b.τ a.stopping b.stopping hle b.T b.bounded
+    (fun ω t ht => ⟨(b.inside ω t ht).1, (b.inside ω t ht).2.2⟩) A hA
+  refine he.trans (integral_congr_ae (Eventually.of_forall fun ω => ?_))
+  dsimp only
+  rw [← a.synchronize b ω (a.τ ω) le_rfl (hle ω)]
+  rfl
+
+lemma finite_history_on_hit
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (b : ∀ n, AngularLocalization B hm κ z v (ε n))
+    (hε : Tendsto ε atTop (𝓝 0)) (hεanti : Antitone ε) (huv : u ≤ v)
+    (n : ℕ) (A : Set Ω)
+    (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
+    (∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z v ∂P) =
+      ∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P := by
+  have hAn := localizedRadiusStop_history_hit a n A hA
+  have he (m : ℕ) := localization_history_finer hB hBm hBc hB0 hκ hκ8
+    (a n) (b (m + n)) huv (hεanti (Nat.le_add_left n m)) (A ∩ (a n).hit) hAn
+  have hlim := (localization_integral_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw hz b hε
+    (A ∩ (a n).hit)).comp (tendsto_add_atTop_nat n)
+  have hx : (∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z v ∂P) = ∫ ω in A ∩ (a n).hit, (a n).weight ω ∂P := by
+    apply tendsto_nhds_unique hlim
+    simpa only [Function.comp_def, he] using
+      (tendsto_const_nhds : Tendsto (fun _ : ℕ => ∫ ω in A ∩ (a n).hit, (a n).weight ω ∂P) atTop (𝓝 _))
+  refine hx.trans (integral_congr_ae ?_)
+  have hAnm : MeasurableSet (A ∩ (a n).hit) := (a n).stopping.measurableSpace_le _ hAn
+  filter_upwards [ae_restrict_of_ae hw, ae_restrict_mem hAnm] with ω hω hωA
+  exact (a n).weight_eq_stoppedGreen_of_hit hBc hz ω (γ ω) (G ω) (F ω) hω hωA.2
+
+lemma hit_integral_tendsto
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (hε : Tendsto ε atTop (𝓝 0)) (hu : 0 ≤ u) (A : Set Ω) :
+    Tendsto (fun n => ∫ ω in A ∩ (a n).hit, stoppedGreen κ (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) z u ∂P) atTop
+      (𝓝 (∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P)) := by
+  let g (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
+  have hi := (finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu).1
+  have hhit (n : ℕ) : MeasurableSet (a n).hit := (a n).stopping.measurableSpace_le _ (a n).measurable_hit
+  have hlim : Tendsto (fun n => ∫ ω in A, (a n).hit.indicator g ω ∂P) atTop (𝓝 (∫ ω in A, g ω ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence (fun ω => ‖g ω‖)
+      (fun n => (hi.indicator (hhit n)).integrableOn.aestronglyMeasurable) hi.norm.integrableOn
+    · intro n
+      exact Eventually.of_forall fun ω => norm_indicator_le_norm_self _ _
+    · filter_upwards [ae_restrict_of_ae hw] with ω hω
+      change Tendsto (fun n => (a n).hit.indicator g ω) atTop (𝓝 (g ω))
+      by_cases hh : ∃ T : ℝ≥0, z ∈ survivingDomain (γ ω) T ∧
+          fullRadius (G ω) T z = (2 * z.im) * Real.exp (-u)
+      · obtain ⟨T, hT, hR⟩ := hh
+        apply tendsto_const_nhds.congr'
+        filter_upwards [eventually_hit_of_live_level a hε hBc hz ω (γ ω) (G ω) (F ω) hω T hT hR] with n hn
+        simp only [Set.indicator_of_mem hn.1]
+      · have hg : g ω = 0 := stoppedGreen_eq_zero_of_no_level κ _ _ _ z u hh
+        have he (n : ℕ) : (a n).hit.indicator g ω = 0 := by
+          by_cases hn : ω ∈ (a n).hit <;> simp [hn, hg]
+        simpa only [he, hg] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))
+  simpa only [integral_indicator (hhit _), Measure.restrict_restrict (hhit _), Set.inter_comm] using hlim
+
+ 
+
+theorem finite_stopped_green_consistent_regular
+    (a : ∀ n, AngularLocalization B hm κ z u (ε n))
+    (b : ∀ n, AngularLocalization B hm κ z v (ε n))
+    (hε : Tendsto ε atTop (𝓝 0)) (hεanti : Antitone ε)
+    (hu : 0 ≤ u) (huv : u ≤ v) (A : Set Ω)
+    (hA : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] A) :
+    (∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z v ∂P) =
+      ∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P := by
+  let gu (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u
+  let gv (ω : Ω) := stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z v
+  obtain ⟨hiu, heu⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu
+  obtain ⟨hiv, hev⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz v (hu.trans huv)
+  have hgv : 0 ≤ᵐ[P] gv := hw.mono fun ω hω => stoppedGreen_nonneg hω κ z v
+  have hmono (C : Set Ω) (hC : MeasurableSet[(localizedRadiusStop_stopping a).measurableSpace] C) :
+      (∫ ω in C, gu ω ∂P) ≤ ∫ ω in C, gv ω ∂P := by
+    have hlim := hit_integral_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw hz a hε hu C
+    apply le_of_tendsto hlim
+    apply Eventually.of_forall
+    intro n
+    rw [← finite_history_on_hit hB hBm hBc hB0 hκ hκ8 γ G F hw hz a b hε hεanti huv n C hC]
+    exact setIntegral_mono_set hiv.integrableOn (ae_restrict_of_ae hgv)
+      (Eventually.of_forall fun ω hω => hω.1)
+  have h1 := hmono A hA
+  have h2 := hmono Aᶜ hA.compl
+  have hAm : MeasurableSet A := (localizedRadiusStop_stopping a).measurableSpace_le _ hA
+  have hsumu := integral_add_compl hAm hiu
+  have hsumv := integral_add_compl hAm hiv
+  change (∫ ω in A, gu ω ∂P) + (∫ ω in Aᶜ, gu ω ∂P) = _ at hsumu
+  change (∫ ω in A, gv ω ∂P) + (∫ ω in Aᶜ, gv ω ∂P) = _ at hsumv
+  rw [heu] at hsumu
+  rw [hev] at hsumv
+  change (∫ ω in A, gv ω ∂P) = ∫ ω in A, gu ω ∂P
+  linarith
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma finiteCenteredLaw_apply
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u)
+    (A : Set Ω) (hA : MeasurableSet A) :
+    finiteCenteredLaw P κ B γ G z u A = ENNReal.ofReal
+      ((∫ ω in A, stoppedGreen κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z u ∂P) /
+        initialGreen κ z) := by
+  have hi := (finite_stopped_green_of_witness hB κ hκ hκ8 γ G F hw z hz u hu).1
+  rw [finiteCenteredLaw, withDensity_apply _ hA,
+    ← ofReal_integral_eq_lintegral_ofReal ((hi.div_const _).integrableOn) ?_, integral_div]
+  filter_upwards [ae_restrict_of_ae hw] with ω hω
+  exact div_nonneg (stoppedGreen_nonneg hω κ z u) (initialGreen_pos κ z hz).le
+
+ 
+
+theorem exists_consistent_finiteCenteredLaw_regular
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    ∃ (ρ : ℝ≥0 → Ω → WithTop ℝ≥0)
+      (hρ : ∀ u, IsStoppingTime (Filtration.natural B
+        (fun _ => (hBm.comp (measurable_const.prodMk measurable_id)).stronglyMeasurable)) (ρ u)),
+      (∀ u, ∀ᵐ ω ∂P, ρ u ω = radiusLevelTime (γ ω) (G ω) z u) ∧
+      (∀ (u : ℝ≥0), IsProbabilityMeasure (finiteCenteredLaw P κ B γ G z u)) ∧
+      (∀ (u v : ℝ≥0), u ≤ v → ∀ A : Set Ω, MeasurableSet[(hρ u).measurableSpace] A →
+        finiteCenteredLaw P κ B γ G z v A = finiteCenteredLaw P κ B γ G z u A) := by
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
+  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
+  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
+  have hε (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
+  have hε₀ (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
+    simpa only [mul_one_div, mul_zero] using hh
+  have hεanti : Antitone ε := by
+    intro n m hnm
+    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
+  let a (u : ℝ≥0) (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
+    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
+      u.coe_nonneg (hε n) (hε₀ n))
+  refine ⟨fun u => localizedRadiusStop (a u), fun u => localizedRadiusStop_stopping (a u), ?_, ?_, ?_⟩
+  · intro u
+    filter_upwards [hw] with ω hω
+    exact localizedRadiusStop_eq_source (a u) hεlim hBc hz ω (γ ω) (G ω) (F ω) hω
+  · intro u
+    exact finiteCenteredLaw_probability hB κ hκ hκ8 γ G F hw z hz u u.coe_nonneg
+  · intro u v huv A hA
+    have hAm : MeasurableSet A := (localizedRadiusStop_stopping (a u)).measurableSpace_le _ hA
+    rw [finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz v v.coe_nonneg A hAm,
+      finiteCenteredLaw_apply hB κ hκ hκ8 γ G F hw z hz u u.coe_nonneg A hAm,
+      finite_stopped_green_consistent_regular hB hBm hBc hB0 hκ hκ8 γ G F hw hz
+        (a u) (a v) hεlim hεanti u.coe_nonneg huv A hA]
+
+lemma finiteCenteredLaw_congr_driver
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {B C : ℝ≥0 → Ω → ℝ}
+    (hCB : ∀ᵐ ω ∂P, ∀ t, C t ω = B t ω)
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (z : ℂ) (u : ℝ) :
+    finiteCenteredLaw P κ C γ G z u = finiteCenteredLaw P κ B γ G z u := by
+  apply withDensity_congr_ae
+  filter_upwards [hCB] with ω hω
+  simp only [hω]
+
+ 
+
+theorem exists_consistent_finiteCenteredLaw
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    ∃ (C : ℝ≥0 → Ω → ℝ) (hCm : ∀ t, Measurable (C t))
+      (ρ : ℝ≥0 → Ω → WithTop ℝ≥0)
+      (hρ : ∀ u, IsStoppingTime (Filtration.natural C (fun t => (hCm t).stronglyMeasurable)) (ρ u)),
+      (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) ∧
+      (∀ u, ∀ᵐ ω ∂P, ρ u ω = radiusLevelTime (γ ω) (G ω) z u) ∧
+      (∀ (u : ℝ≥0), IsProbabilityMeasure (finiteCenteredLaw P κ B γ G z u)) ∧
+      (∀ (u v : ℝ≥0), u ≤ v → ∀ A : Set Ω, MeasurableSet[(hρ u).measurableSpace] A →
+        finiteCenteredLaw P κ B γ G z v A = finiteCenteredLaw P κ B γ G z u A) := by
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
+    filter_upwards [hw, hCB] with ω hω he
+    simpa only [he] using hω
+  obtain ⟨ρ, hρ, hρeq, hprob, hcon⟩ := exists_consistent_finiteCenteredLaw_regular
+    hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz
+  have he (u : ℝ≥0) := finiteCenteredLaw_congr_driver hCB κ γ G z u
+  refine ⟨C, fun t => hCm.comp (measurable_const.prodMk measurable_id), ρ, hρ, hCB, hρeq, ?_, ?_⟩
+  · intro u
+    rw [← he u]
+    exact hprob u
+  · intro u v huv A hA
+    rw [← he u, ← he v]
+    exact hcon u v huv A hA
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def inverseAngleSubcritical (κ : ℝ) : ℝ :=
+  greenExponent κ - min (greenExponent κ / 2) 1
+noncomputable def inverseAngleBuffer (u₀ : ℝ) : ℝ := min (u₀ / 2) 1
+noncomputable def uniformInverseAngleConstant (κ u₀ : ℝ) : ℝ :=
+  inverseAngleBound κ (inverseAngleSubcritical κ) (inverseAngleBuffer u₀)
+
+lemma inverseAngleSubcritical_bounds (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < inverseAngleSubcritical κ ∧ inverseAngleSubcritical κ < greenExponent κ ∧
+      greenExponent κ ≤ inverseAngleSubcritical κ + 2 := by
+  have ha := greenExponent_pos κ hκ hκ8
+  unfold inverseAngleSubcritical
+  have hm0 : 0 < min (greenExponent κ / 2) 1 := lt_min (by positivity) (by norm_num)
+  have hm := min_le_left (greenExponent κ / 2) 1
+  have hm1 := min_le_right (greenExponent κ / 2) 1
+  constructor
+  · linarith
+  constructor <;> linarith
+
+lemma uniformInverseAngleConstant_pos (κ u₀ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) (hu₀ : 0 < u₀) :
+    0 < uniformInverseAngleConstant κ u₀ := by
+  have hb := inverseAngleSubcritical_bounds κ hκ hκ8
+  exact inverseAngleBound_pos hκ hb.1 hb.2.1 (lt_min (by positivity) (by norm_num))
+
+ 
+
+theorem uniform_inverse_angle_regular
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u₀ : ℝ) (hu₀ : 0 < u₀)
+    (v : ℝ≥0) (hv : u₀ ≤ v) :
+    Integrable (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ))
+      (finiteCenteredLaw P κ B γ G z v) ∧
+    (∫ ω, stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ)
+      ∂finiteCenteredLaw P κ B γ G z v) ≤ uniformInverseAngleConstant κ u₀ := by
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
+  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
+  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
+  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
+  have hεA (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
+    simpa only [mul_one_div, mul_zero] using hh
+  have hεanti : Antitone ε := by
+    intro n m hnm
+    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
+  let a (u : ℝ≥0) (n : ℕ) : AngularLocalization B hm κ z u (ε n) :=
+    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z u (ε n)
+      u.coe_nonneg (hε0 n) (hεA n))
+  obtain ⟨hi, he⟩ := finiteCenteredLaw_inverseAngle_moment hB hBm hBc κ hκ hκ8 γ G F hw z hz v v.coe_nonneg
+    (a v) hεanti hεlim (greenExponent κ) le_rfl
+  refine ⟨hi, he.trans_le ?_⟩
+  have hb := inverseAngleSubcritical_bounds κ hκ hκ8
+  apply radiusMoment_inverseAngle_bound a hεanti hεlim hε0 hB hBm hBc hB0 hκ hκ8 hz
+    (inverseAngleSubcritical κ) hb.1 hb.2.1 hb.2.2
+    (inverseAngleBuffer u₀) (lt_min (by positivity) (by norm_num)) (min_le_right _ _)
+  have hη := min_le_left (u₀ / 2) 1
+  dsimp only [inverseAngleBuffer]
+  linarith
+
+ 
+
+theorem uniform_inverse_angle
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (u₀ : ℝ) (hu₀ : 0 < u₀)
+    (v : ℝ≥0) (hv : u₀ ≤ v) :
+    Integrable (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ))
+      (finiteCenteredLaw P κ B γ G z v) ∧
+    (∫ ω, stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ)
+      ∂finiteCenteredLaw P κ B γ G z v) ≤ uniformInverseAngleConstant κ u₀ := by
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
+    filter_upwards [hw, hCB] with ω hω he
+    simpa only [he] using hω
+  obtain ⟨hi, hb⟩ := uniform_inverse_angle_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz u₀ hu₀ v hv
+  rw [finiteCenteredLaw_congr_driver hCB κ γ G z v] at hi hb
+  have he : (fun ω => stoppedInverseAngle κ (fun t => C t ω) (γ ω) (G ω) z v (greenExponent κ)) =ᵐ[finiteCenteredLaw P κ B γ G z v]
+      (fun ω => stoppedInverseAngle κ (fun t => B t ω) (γ ω) (G ω) z v (greenExponent κ)) := by
+    have hCB' : ∀ᵐ ω ∂finiteCenteredLaw P κ B γ G z v, ∀ t, C t ω = B t ω :=
+      (withDensity_absolutelyContinuous P _).ae_le hCB
+    filter_upwards [hCB'] with ω he
+    simp only [stoppedInverseAngle, stoppedAngleSq, he]
+  exact ⟨hi.congr he, (integral_congr_ae he).symm.trans_le hb⟩
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma radius_approach_regular
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (v : ℝ≥0) (hv : 1 ≤ (v : ℝ)) :
+    (P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞}).toReal ≤
+      uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
+        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let A₀ := logAngleSq κ (initialLogCoordinates z) 0
+  have hA₀ : 0 < A₀ := logAngleSq_pos _ _ _
+  let ε (n : ℕ) := A₀ / ((n : ℝ) + 1)
+  have hε0 (n : ℕ) : 0 < ε n := by dsimp [ε]; positivity
+  have hεA (n : ℕ) : ε n ≤ A₀ := div_le_self hA₀.le (by have h := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have hεlim : Tendsto ε atTop (𝓝 0) := by
+    have hh := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul A₀
+    simpa only [mul_one_div, mul_zero] using hh
+  have hεanti : Antitone ε := by
+    intro n m hnm
+    exact div_le_div_of_nonneg_left hA₀.le (by positivity) (by exact_mod_cast Nat.add_le_add_right hnm 1)
+  let a (n : ℕ) : AngularLocalization B hm κ z v (ε n) :=
+    Classical.choice (exists_angularLocalization hB hm hBm hBc hB0 κ hκ hκ8 z v (ε n)
+      v.coe_nonneg (hε0 n) (hεA n))
+  have hbound := (uniform_inverse_angle_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz
+    1 (by norm_num) v hv).2
+  have he := (finiteCenteredLaw_inverseAngle_moment hB hBm hBc κ hκ hκ8 γ G F hw z hz v v.coe_nonneg
+    a hεanti hεlim (greenExponent κ) le_rfl).2
+  rw [he, radiusMoment, sub_self,
+    integral_limitRadiusKernel_zero_eq_probability a hεanti hεlim hBm hBc hz v.coe_nonneg γ G F hw] at hbound
+  have hmult := (div_le_iff₀ (initialGreen_pos κ z hz)).mp hbound
+  calc
+    (P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞}).toReal ≤
+        uniformInverseAngleConstant κ 1 * initialGreen κ z /
+          Real.exp (codimension κ * ((v : ℝ) - Real.log 2 - Real.log z.im)) := by
+      apply (le_div_iff₀ (Real.exp_pos _)).mpr
+      simpa only [mul_comm] using hmult
+    _ = uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
+        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
+      rw [mul_div_assoc, initialGreen_div_zeroKernel κ v z hz, mul_assoc]
+
+ 
+
+theorem radius_approach
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (v : ℝ≥0) (hv : 1 ≤ (v : ℝ)) :
+    (P {ω | radiusLevelTime (γ ω) (G ω) z v < ∞}).toReal ≤
+      uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
+        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
+    filter_upwards [hw, hCB] with ω hω he
+    simpa only [he] using hω
+  exact radius_approach_regular hC hCm hCc hC0 κ hκ hκ8 γ G F hwC z hz v hv
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem euclidean_approach_at_level
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (v : ℝ≥0) (hv : 1 ≤ (v : ℝ))
+    (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
+    (hscale : (128 * Real.pi) * r ≤ (2 * z.im) * Real.exp (-v)) :
+    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+      uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * v) *
+        (Real.sin (Complex.arg z)) ^ greenExponent κ := by
+  apply le_trans _ (radius_approach hB κ hκ hκ8 γ G F hw z hz v hv)
+  apply ENNReal.toReal_mono (measure_ne_top _ _)
+  apply measure_mono_ae
+  filter_upwards [hw, hB.cont] with ω hω hc
+  rintro ⟨t, ht⟩
+  have hU : Continuous (fun s => Real.sqrt κ * B s ω) := hc.const_mul _
+  obtain ⟨s, _, hsl, hsR⟩ := hω.radius_approach_of_trace_approach hU z r hr hy t ht
+  exact radiusLevelTime_finite_of_live_radius_lt hU hω z v v.coe_nonneg s hsl
+    (hsR.trans_le hscale)
+
+ 
+
+theorem euclidean_approach
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (r : ℝ) (hr : 0 < r)
+    (hy : r ≤ z.im)
+    (hsmall : 1 ≤ Real.log ((2 * z.im) / ((128 * Real.pi) * r))) :
+    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+      (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
+        initialGreen κ z * r ^ codimension κ := by
+  let u := Real.log ((2 * z.im) / ((128 * Real.pi) * r))
+  have hu : 0 ≤ u := (by norm_num : (0 : ℝ) ≤ 1).trans hsmall
+  have hC : 0 < 128 * Real.pi := mul_pos (by norm_num) Real.pi_pos
+  have hy0 : 0 < 2 * z.im := mul_pos (by norm_num) hz
+  have hratio : 0 < (2 * z.im) / ((128 * Real.pi) * r) := div_pos hy0 (mul_pos hC hr)
+  have he : (2 * z.im) * Real.exp (-u) = (128 * Real.pi) * r := by
+    dsimp only [u]
+    rw [Real.exp_neg, Real.exp_log hratio]
+    field_simp [(show 0 < z.im from hz).ne']
+  have hb := euclidean_approach_at_level hB κ hκ hκ8 γ G F hw z hz
+    ⟨u, hu⟩ hsmall r hr hy he.ge
+  have hexp : Real.exp (-codimension κ * u) =
+      (128 * Real.pi) ^ codimension κ * r ^ codimension κ *
+        (2 * z.im) ^ (-codimension κ) := by
+    rw [Real.rpow_def_of_pos hC, Real.rpow_def_of_pos hr,
+      Real.rpow_def_of_pos hy0, ← Real.exp_add, ← Real.exp_add]
+    congr 1
+    dsimp only [u]
+    rw [Real.log_div hy0.ne' (mul_pos hC hr).ne', Real.log_mul hC.ne' hr.ne']
+    ring
+  change (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+    uniformInverseAngleConstant κ 1 * Real.exp (-codimension κ * u) *
+      (Real.sin (Complex.arg z)) ^ greenExponent κ at hb
+  rw [hexp] at hb
+  simpa only [initialGreen, mul_assoc, mul_comm, mul_left_comm] using hb
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/OptionalStopping.lean b/lean/OAI/Probability/SLE/OptionalStopping.lean
index c1388a1ec952689d837f9fc082a3e2f4120eda26..4f8f45df145b078e191aa3b2ab4a3518e6f88dda 100644
--- a/lean/OAI/Probability/SLE/OptionalStopping.lean
+++ b/lean/OAI/Probability/SLE/OptionalStopping.lean
@@ -1,675 +1,675 @@
-import Mathlib
-import OAI.Probability.SLE.BrownianGenerators
-
-namespace OAI
-
-/-! Optional Stopping for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-lemma eq_of_uniform_small_increment
-    (f : ℝ≥0 → ℝ) (rate : ℝ≥0 → ℝ) (C : ℝ)
-    (hrate : Tendsto rate (𝓝 0) (𝓝 0))
-    (hb : ∀ s δ, |f (s + δ) - f s| ≤ C * (δ : ℝ) * rate δ)
-    (s t : ℝ≥0) (hst : s ≤ t) : f t = f s := by
-  have mesh (n : ℕ) : |f t - f s| ≤ C * ((t : ℝ) - s) * rate ((t - s) / (n + 1)) := by
-    let δ : ℝ≥0 := (t - s) / (n + 1)
-    let u (i : ℕ) : ℝ≥0 := s + i * δ
-    have hn : (n + 1 : ℝ≥0) ≠ 0 := by positivity
-    have hlast : u (n + 1) = t := by
-      dsimp only [u, δ]
-      push_cast
-      rw [mul_div_cancel₀ _ hn, add_tsub_cancel_of_le hst]
-    have hstep (i : ℕ) : u (i + 1) = u i + δ := by
-      dsimp [u]; push_cast; ring
-    have htel : (∑ i ∈ Finset.range (n + 1), (f (u (i + 1)) - f (u i))) = f t - f s := by
-      rw [Finset.sum_range_sub (fun i => f (u i))]
-      simp only [hlast, u, Nat.cast_zero, zero_mul, add_zero]
-    have hδ : ((n : ℝ) + 1) * (δ : ℝ) = (t : ℝ) - s := by
-      dsimp only [δ]
-      rw [NNReal.coe_div, NNReal.coe_sub hst]
-      push_cast
-      exact mul_div_cancel₀ _ (by positivity)
-    calc
-      |f t - f s| = |∑ i ∈ Finset.range (n + 1), (f (u (i + 1)) - f (u i))| := by rw [htel]
-      _ ≤ ∑ i ∈ Finset.range (n + 1), |f (u (i + 1)) - f (u i)| :=
-        Finset.abs_sum_le_sum_abs _ _
-      _ ≤ ∑ _i ∈ Finset.range (n + 1), C * (δ : ℝ) * rate δ := by
-        apply Finset.sum_le_sum
-        intro i hi
-        rw [hstep]
-        exact hb (u i) δ
-      _ = C * ((t : ℝ) - s) * rate δ := by
-        simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_add, Nat.cast_one]
-        calc
-          ((n : ℝ) + 1) * (C * (δ : ℝ) * rate δ) = C * (((n : ℝ) + 1) * (δ : ℝ)) * rate δ := by ring
-          _ = _ := by rw [hδ]
-  have hmesh : Tendsto (fun n : ℕ => (t - s) / (n + 1 : ℝ≥0)) atTop (𝓝 0) := by
-    apply NNReal.tendsto_coe.mp
-    simp only [NNReal.coe_div, NNReal.coe_add, NNReal.coe_natCast, NNReal.coe_one, NNReal.coe_zero]
-    apply tendsto_const_nhds.div_atTop
-    simpa only [Function.comp_def, Nat.cast_add, Nat.cast_one] using
-      (tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1)
-  have hl : Tendsto (fun n : ℕ => C * ((t : ℝ) - s) * rate ((t - s) / (n + 1))) atTop (𝓝 0) := by
-    simpa only [mul_zero, Function.comp_apply] using tendsto_const_nhds.mul (hrate.comp hmesh)
-  have he : |f t - f s| ≤ 0 := le_of_tendsto_of_tendsto tendsto_const_nhds hl
-    (Eventually.of_forall mesh)
-  exact sub_eq_zero.mp (abs_nonpos_iff.mp he)
-
-lemma bounded_interval_integral_expectation_sub
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (a b c : ℝ) (hab : a ≤ b) (hbc : b ≤ c) (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hGb : ∀ u ω, |G u ω| ≤ C) :
-    (∫ ω, (∫ u in a..c, G u ω) ∂P) - (∫ ω, (∫ u in a..b, G u ω) ∂P) =
-      ∫ ω, (∫ u in b..c, G u ω) ∂P := by
-  have hi₁ := bounded_interval_integral_integrable a b hab G hG C hGb (P := P)
-  have hi₂ := bounded_interval_integral_integrable b c hbc G hG C hGb (P := P)
-  have he : (∫ ω, (∫ u in a..c, G u ω) ∂P) =
-      (∫ ω, (∫ u in a..b, G u ω) ∂P) + (∫ ω, (∫ u in b..c, G u ω) ∂P) := by
-    rw [← integral_add hi₁ hi₂]
-    apply integral_congr_ae
-    exact Eventually.of_forall fun ω => (intervalIntegral.integral_add_adjacent_intervals
-      (bounded_time_section_intervalIntegrable a b hab G hG C hGb ω)
-      (bounded_time_section_intervalIntegrable b c hbc G hG C hGb ω)).symm
-  rw [he, add_sub_cancel_left]
-
-lemma SmoothTimeBrownianField.generator_measurable
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
-    Measurable (Function.uncurry (brownianFieldDrift B F a)) := by
-  change Measurable (fun q : ℝ × Ω => a q.1.toNNReal q.2 (B q.1.toNNReal q.2) +
-    (1 / 2 : ℝ) * iteratedDeriv 2 (F q.1.toNNReal q.2) (B q.1.toNNReal q.2))
-  exact (hF.drift_real_joint hBm).add (measurable_const.mul (hF.deriv_real_joint hBm 2 (by decide)))
-
-lemma SmoothTimeBrownianField.generator_bound
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (u : ℝ) (ω : Ω) :
-    |brownianFieldDrift B F a u ω| ≤ 2 * (K : ℝ) := by
-  unfold brownianFieldDrift
-  have hb := hF.bound u.toNNReal ω (B u.toNNReal ω) 2 (by decide)
-  have ha := hF.drift_bound u.toNNReal ω (B u.toNNReal ω)
-  have ht := abs_add_le (a u.toNNReal ω (B u.toNNReal ω))
-    ((1 / 2 : ℝ) * iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω))
-  rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2)] at ht
-  linarith [K.coe_nonneg]
-
- 
-
-theorem brownian_time_field_dynkin
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (s t : ℝ≥0) (hst : s ≤ t) {E : Set Ω}
-    (hE : MeasurableSet[Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable) s] E) :
-    (∫ ω in E, F t ω (B t ω) - F s ω (B s ω) ∂P) =
-      ∫ ω in E, (∫ u in (s : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let c (r : ℝ≥0) := ∫ ω in E, (∫ u in (s : ℝ)..(s + r : ℝ≥0), brownianFieldDrift B F a u ω) ∂P
-  let v (r : ℝ≥0) := ∫ ω in E, F r ω (B r ω) ∂P
-  let f (r : ℝ≥0) := v (s + r) - c r
-  have hI (r : ℝ≥0) : Integrable (fun ω => F r ω (B r ω)) (P.restrict E) := by
-    simpa only [iteratedDeriv_zero] using hF.eval_integrable r 0 (by decide) (hm r) (P := P.restrict E)
-  have hinc (r δ : ℝ≥0) : |f (r + δ) - f r| ≤ (K : ℝ) * (δ : ℝ) * dynkinRate δ := by
-    have hsr : s ≤ s + r := le_add_of_nonneg_right (zero_le)
-    have hE' : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (s + r)] E :=
-      (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono hsr E hE
-    have h := smooth_field_one_step_error hB hm hBm K hF (s + r) δ hE'
-    have hcomp : c (r + δ) - c r =
-        ∫ ω in E, (∫ u in (s + r : ℝ≥0)..((s + r) + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P := by
-      dsimp only [c]
-      rw [← add_assoc]
-      apply bounded_interval_integral_expectation_sub _ _ _
-        (by exact_mod_cast hsr) (by simp only [NNReal.coe_add]; linarith [δ.coe_nonneg]) _
-        (hF.generator_measurable hBm) (2 * (K : ℝ)) (hF.generator_bound)
-    have hv : v ((s + r) + δ) - v (s + r) =
-        ∫ ω in E, F ((s + r) + δ) ω (B ((s + r) + δ) ω) - F (s + r) ω (B (s + r) ω) ∂P :=
-      (integral_sub (hI _) (hI _)).symm
-    have he : f (r + δ) - f r =
-        (∫ ω in E, F ((s + r) + δ) ω (B ((s + r) + δ) ω) - F (s + r) ω (B (s + r) ω) ∂P) -
-        (∫ ω in E, (∫ u in (s + r : ℝ≥0)..((s + r) + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P) := by
-      dsimp only [f]
-      rw [← add_assoc]
-      linarith
-    rw [he]
-    exact h
-  have heq := eq_of_uniform_small_increment f dynkinRate K dynkinRate_tendsto_zero hinc 0 (t - s) (zero_le)
-  have hc0 : c 0 = 0 := by simp [c]
-  have hst' : s + (t - s) = t := add_tsub_cancel_of_le hst
-  change v (s + (t - s)) - c (t - s) = v (s + 0) - c 0 at heq
-  rw [hst', add_zero, hc0, sub_zero] at heq
-  have hc : c (t - s) = ∫ ω in E, (∫ u in (s : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω) ∂P := by
-    dsimp only [c]
-    rw [hst']
-  rw [integral_sub (hI _) (hI _)]
-  change v t - v s = _
-  linarith
-
- 
-noncomputable def stopGridIndex (n : ℕ) (t : ℝ≥0) : ℕ :=
-  Nat.ceil (((n : ℝ≥0) + 1) * t)
-
-noncomputable def stopGridTime (n k : ℕ) : ℝ≥0 :=
-  (k : ℝ≥0) / ((n : ℝ≥0) + 1)
-
-noncomputable def stopGrid (n : ℕ) (t : ℝ≥0) : ℝ≥0 :=
-  stopGridTime n (stopGridIndex n t)
-
-lemma stopGridIndex_le (n k : ℕ) (t : ℝ≥0) :
-    stopGridIndex n t ≤ k ↔ t ≤ stopGridTime n k := by
-  have hd : 0 < (n : ℝ≥0) + 1 := by positivity
-  simp only [stopGridIndex, Nat.ceil_le, stopGridTime, le_div_iff₀ hd, mul_comm]
-
-lemma stopGrid_bounds (n : ℕ) (t : ℝ≥0) :
-    t ≤ stopGrid n t ∧ stopGrid n t ≤ t + 1 / ((n : ℝ≥0) + 1) := by
-  have hd : 0 < (n : ℝ≥0) + 1 := by positivity
-  constructor
-  · exact (stopGridIndex_le n _ t).mp le_rfl
-  · dsimp [stopGrid, stopGridTime, stopGridIndex]
-    rw [div_le_iff₀ hd]
-    calc
-      _ ≤ ((n : ℝ≥0) + 1) * t + 1 :=
-        (Nat.ceil_lt_add_one (by positivity)).le
-      _ = (t + 1 / ((n : ℝ≥0) + 1)) * ((n : ℝ≥0) + 1) := by
-        conv_rhs => rw [add_mul, div_mul_cancel₀ _ hd.ne']
-        ac_rfl
-
-lemma stopGrid_tendsto (t : ℝ≥0) : Tendsto (fun n => stopGrid n t) atTop (𝓝 t) := by
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
-    (by simpa using (tendsto_const_nhds.add
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ≥0))))
-  · exact fun n => (stopGrid_bounds n t).1
-  · intro n
-    simpa only [one_div] using (stopGrid_bounds n t).2
-
-lemma measurable_discrete_eval {Ω : Type*} [MeasurableSpace Ω]
-    (f : ℕ → Ω → ℝ) (hf : ∀ k, Measurable (f k))
-    {j : Ω → ℕ} (hj : Measurable j) : Measurable (fun ω => f (j ω) ω) := by
-  exact (measurable_from_prod_countable_right (f := fun p : ℕ × Ω => f p.1 p.2) hf).comp (hj.prodMk measurable_id)
-
-lemma stopping_time_measurable_nnreal {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {ℱ : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) : Measurable τ := by
-  simpa only [WithTop.untopD_coe] using hτ.measurable'.untopD 0
-
-lemma stopGridIndex_measurable {Ω : Type*} [MeasurableSpace Ω]
-    {τ : Ω → ℝ≥0} (hτ : Measurable τ) (n : ℕ) :
-    Measurable (fun ω => stopGridIndex n (τ ω)) := by
-  exact (measurable_const.mul hτ).nat_ceil
-
-lemma SmoothTimeBrownianField.time_increment_all
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s t : ℝ≥0) (ω : Ω) (x : ℝ) :
-    F t ω x - F s ω x = ∫ u in (s : ℝ)..(t : ℝ), a u.toNNReal ω x := by
-  rcases le_total s t with h | h
-  · exact hF.time_increment s t h ω x
-  · rw [intervalIntegral.integral_symm, ← hF.time_increment t s h ω x]
-    ring
-
-lemma SmoothTimeBrownianField.time_dist_le
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s t : ℝ≥0) (ω : Ω) (x : ℝ) :
-    dist (F t ω x) (F s ω x) ≤ (K : ℝ) * dist t s := by
-  rw [Real.dist_eq, hF.time_increment_all]
-  have hb := intervalIntegral.norm_integral_le_of_norm_le_const (a := (s : ℝ)) (b := (t : ℝ))
-    (f := fun u => a u.toNNReal ω x) (C := (K : ℝ))
-    (fun u hu => by simpa only [Real.norm_eq_abs] using hF.drift_bound u.toNNReal ω x)
-  change |(∫ u in (s : ℝ)..(t : ℝ), a u.toNNReal ω x)| ≤ (K : ℝ) * |(t : ℝ) - s|
-  simpa only [Real.norm_eq_abs] using hb
-
-lemma SmoothTimeBrownianField.space_lipschitz
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (ω : Ω) :
-    LipschitzWith K (F t ω) := by
-  apply lipschitzWith_of_nnnorm_deriv_le ((hF.smooth t ω).differentiable (by norm_num))
-  intro x
-  change ‖deriv (F t ω) x‖ ≤ (K : ℝ)
-  simpa only [iteratedDeriv_one, Real.norm_eq_abs] using hF.bound t ω x 1 (by decide)
-
-lemma SmoothTimeBrownianField.path_continuous
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
-    Continuous (fun t => F t ω (B t ω)) := by
-  have hpair : LipschitzWith (2 * K) (fun p : ℝ≥0 × ℝ => F p.1 ω p.2) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro p q
-    calc
-      dist (F p.1 ω p.2) (F q.1 ω q.2) ≤
-        dist (F p.1 ω p.2) (F q.1 ω p.2) + dist (F q.1 ω p.2) (F q.1 ω q.2) := dist_triangle _ _ _
-      _ ≤ (K : ℝ) * dist p.1 q.1 + (K : ℝ) * dist p.2 q.2 :=
-        add_le_add (hF.time_dist_le q.1 p.1 ω p.2) ((hF.space_lipschitz q.1 ω).dist_le_mul p.2 q.2)
-      _ ≤ (K : ℝ) * dist p q + (K : ℝ) * dist p q := by
-        gcongr
-        · exact le_max_left _ _
-        · exact le_max_right _ _
-      _ = _ := by change _ = (2 * (K : ℝ)) * dist p q; ring
-  exact hpair.continuous.comp (continuous_id.prodMk (hc ω))
-
-lemma cumulative_integral_measurable
-    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) :
-    Measurable (fun q : ℝ≥0 × Ω => ∫ u in (0 : ℝ)..(q.1 : ℝ), G u q.2) := by
-  have hf : Measurable (fun q : (ℝ≥0 × Ω) × ℝ => G q.2 q.1.2) :=
-    hG.comp (measurable_snd.prodMk measurable_fst.snd)
-  let A : Set ((ℝ≥0 × Ω) × ℝ) := {q | 0 < q.2 ∧ q.2 ≤ (q.1.1 : ℝ)}
-  have hA : MeasurableSet A :=
-    (measurableSet_lt measurable_const measurable_snd).inter
-      (measurableSet_le measurable_snd (measurable_coe_nnreal_real.comp measurable_fst.fst))
-  have hm := ((hf.indicator hA).stronglyMeasurable.integral_prod_right' (ν := volume)).measurable
-  convert hm using 1
-  funext q
-  rw [intervalIntegral.integral_of_le q.1.coe_nonneg, ← integral_indicator measurableSet_Ioc]
-  rfl
-
-noncomputable def brownianFieldCompensated {Ω : Type*}
-    (B : ℝ≥0 → Ω → ℝ) (F a : ℝ≥0 → Ω → ℝ → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
-  F t ω (B t ω) - ∫ u in (0 : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω
-
-lemma SmoothTimeBrownianField.compensated_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (t : ℝ≥0) :
-    Integrable (brownianFieldCompensated B F a t) P := by
-  have hi : Integrable (fun ω => F t ω (B t ω)) P :=
-    hF.eval_integrable t 0 (by decide) (hm t)
-  exact hi.sub (bounded_interval_integral_integrable 0 t t.coe_nonneg _
-    (hF.generator_measurable hBm) (2 * (K : ℝ)) hF.generator_bound)
-
-lemma SmoothTimeBrownianField.compensated_past_integral_zero
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a) (s t : ℝ≥0) (hst : s ≤ t)
-    {E : Set Ω} (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
-    (∫ ω in E, brownianFieldCompensated B F a t ω - brownianFieldCompensated B F a s ω ∂P) = 0 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hGm := hF.generator_measurable hBm
-  have hGi (a b : ℝ) (h : a ≤ b) (ω : Ω) :=
-    bounded_time_section_intervalIntegrable a b h _ hGm (2 * (K : ℝ)) hF.generator_bound ω
-  have hp (ω : Ω) : brownianFieldCompensated B F a t ω - brownianFieldCompensated B F a s ω =
-      (F t ω (B t ω) - F s ω (B s ω)) - ∫ u in (s : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω := by
-    have hh := intervalIntegral.integral_add_adjacent_intervals (hGi 0 s s.coe_nonneg ω) (hGi s t hst ω)
-    dsimp only [brownianFieldCompensated]
-    linarith
-  simp_rw [hp]
-  have hf : Integrable (fun ω => F t ω (B t ω) - F s ω (B s ω)) (P.restrict E) :=
-    (hF.eval_integrable t 0 (by decide) (hm t)).sub (hF.eval_integrable s 0 (by decide) (hm s))
-  rw [integral_sub hf (bounded_interval_integral_integrable s t hst _ hGm (2 * (K : ℝ)) hF.generator_bound),
-    brownian_time_field_dynkin hB hm hBm K F a hF s t hst hE, sub_self]
-
-lemma sum_stopped_increments (f : ℕ → ℝ) (k N : ℕ) (hk : k ≤ N) :
-    (∑ j ∈ Finset.range N, if j < k then f (j + 1) - f j else 0) = f k - f 0 := by
-  have hs : (∑ j ∈ Finset.range k, if j < k then f (j + 1) - f j else 0) =
-      ∑ j ∈ Finset.range N, if j < k then f (j + 1) - f j else 0 := by
-    apply Finset.sum_subset (Finset.range_mono hk)
-    intro j hj hjk
-    exact ite_eq_right (fun hh => hjk (Finset.mem_range.mpr hh))
-  rw [← hs]
-  calc
-    _ = ∑ j ∈ Finset.range k, (f (j + 1) - f j) := by
-      apply Finset.sum_congr rfl
-      intro j hj
-      exact ite_eq_left (Finset.mem_range.mp hj)
-    _ = _ := Finset.sum_range_sub f k
-
- 
-
-lemma finite_grid_stopped_integral_eq
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    (ℱ : Filtration ℕ mΩ) (C : ℕ → Ω → ℝ)
-    (hi : ∀ j, Integrable (C j) P) (σ : Ω → ℕ) (N : ℕ)
-    (hb : ∀ ω, σ ω ≤ N)
-    (hE : ∀ j, MeasurableSet[ℱ j] {ω | j < σ ω})
-    (hzero : ∀ j (E : Set Ω), MeasurableSet[ℱ j] E →
-      (∫ ω in E, C (j + 1) ω - C j ω ∂P) = 0) :
-    (∫ ω, C (σ ω) ω ∂P) = ∫ ω, C 0 ω ∂P := by
-  classical
-  let D (j : ℕ) : Ω → ℝ := {ω | j < σ ω}.indicator (fun ω => C (j + 1) ω - C j ω)
-  have hD (j : ℕ) : Integrable (D j) P :=
-    ((hi (j + 1)).sub (hi j)).indicator (ℱ.le j _ (hE j))
-  have hDi (j : ℕ) : (∫ ω, D j ω ∂P) = 0 := by
-    change (∫ ω, {ω | j < σ ω}.indicator (fun ω => C (j + 1) ω - C j ω) ω ∂P) = 0
-    rw [integral_indicator (ℱ.le j _ (hE j))]
-    exact hzero j _ (hE j)
-  have hpoint (ω : Ω) : C (σ ω) ω = C 0 ω + ∑ j ∈ Finset.range N, D j ω := by
-    have hh := sum_stopped_increments (fun j => C j ω) (σ ω) N (hb ω)
-    have hsum : (∑ j ∈ Finset.range N, D j ω) =
-        ∑ j ∈ Finset.range N, if j < σ ω then C (j + 1) ω - C j ω else 0 := by
-      apply Finset.sum_congr rfl
-      intro index _
-      by_cases hindex : index < σ ω <;> simp [D, hindex]
-    rw [hsum, hh]
-    ring
-  simp_rw [hpoint]
-  rw [integral_add (hi 0) (integrable_finsetSum _ (fun j hj => hD j)), integral_finsetSum]
-  · simp only [hDi, Finset.sum_const_zero, add_zero]
-  · intro j hj; exact hD j
-
-lemma monotone_stopGridTime (n : ℕ) : Monotone (stopGridTime n) := by
-  intro i j hij
-  dsimp only [stopGridTime]
-  exact div_le_div_of_nonneg_right (Nat.cast_le.mpr hij) (by positivity)
-
-noncomputable def gridFiltration {Ω : Type*} {mΩ : MeasurableSpace Ω}
-    (ℱ : Filtration ℝ≥0 mΩ) (n : ℕ) : Filtration ℕ mΩ where
-  seq j := ℱ (stopGridTime n j)
-  mono' := fun _ _ hij => ℱ.mono (monotone_stopGridTime n hij)
-  le' j := ℱ.le (stopGridTime n j)
-
-lemma brownian_compensated_grid_integral
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) (n : ℕ) :
-    (∫ ω, brownianFieldCompensated B F a (stopGrid n (τ ω)) ω ∂P) =
-      ∫ ω, F 0 ω (B 0 ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let ℱn := gridFiltration ℱ n
-  have hbound (ω : Ω) : stopGridIndex n (τ ω) ≤ stopGridIndex n T := by
-    apply (stopGridIndex_le n _ _).mpr
-    exact (hb ω).trans (stopGrid_bounds n T).1
-  have hE (j : ℕ) : MeasurableSet[ℱn j] {ω | j < stopGridIndex n (τ ω)} := by
-    have hle : MeasurableSet[ℱ (stopGridTime n j)] {ω | τ ω ≤ stopGridTime n j} := by
-      simpa only [WithTop.coe_le_coe] using hτ (stopGridTime n j)
-    have he : {ω | j < stopGridIndex n (τ ω)} = {ω | τ ω ≤ stopGridTime n j}ᶜ := by
-      ext ω
-      simp only [mem_compl_iff, mem_ofPred_eq, ← stopGridIndex_le, not_le]
-    rw [he]
-    exact hle.compl
-  have he := finite_grid_stopped_integral_eq ℱn
-    (fun j => brownianFieldCompensated B F a (stopGridTime n j))
-    (fun j => hF.compensated_integrable hBm _)
-    (fun ω => stopGridIndex n (τ ω)) (stopGridIndex n T) hbound hE
-    (fun j E hE => hF.compensated_past_integral_zero hB hm hBm _ _
-      (monotone_stopGridTime n (Nat.le_succ j)) hE)
-  simpa only [stopGrid, stopGridTime, Nat.cast_zero, zero_div,
-    brownianFieldCompensated, NNReal.coe_zero, intervalIntegral.integral_same, sub_zero] using he
-
-lemma bounded_cumulative_at_grid
-    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hC : 0 ≤ C)
-    (hb : ∀ u ω, |G u ω| ≤ C) (τ : Ω → ℝ≥0)
-    (hz : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → G u ω = 0) (n : ℕ) (ω : Ω) :
-    |∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), G u ω| ≤ C * (1 / ((n : ℝ) + 1)) := by
-  have h0 : (∫ u in (0 : ℝ)..(τ ω : ℝ), G u ω) = 0 := by
-    calc
-      _ = ∫ u in (0 : ℝ)..(τ ω : ℝ), (0 : ℝ) :=
-        intervalIntegral.integral_congr_Ioo_of_le (τ ω).coe_nonneg (fun u hu => hz ω u hu.1.le hu.2)
-      _ = _ := by simp
-  have hle : (τ ω : ℝ) ≤ stopGrid n (τ ω) := (stopGrid_bounds n (τ ω)).1
-  have hsplit := intervalIntegral.integral_add_adjacent_intervals
-    (bounded_time_section_intervalIntegrable 0 (τ ω) (τ ω).coe_nonneg G hG C hb ω)
-    (bounded_time_section_intervalIntegrable (τ ω) (stopGrid n (τ ω)) hle G hG C hb ω)
-  rw [h0, zero_add] at hsplit
-  rw [← hsplit]
-  have hbnd := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (τ ω : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := C) (f := fun u => G u ω)
-    (fun u hu => by simpa only [Real.norm_eq_abs] using hb u ω)
-  rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hle)] at hbnd
-  refine hbnd.trans (mul_le_mul_of_nonneg_left ?_ hC)
-  have hupper := (stopGrid_bounds n (τ ω)).2
-  have hu : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ) + 1 / ((n : ℝ) + 1) := by exact_mod_cast hupper
-  linarith
-
- 
-
-theorem brownian_time_field_optional
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hzero : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → brownianFieldDrift B F a u ω = 0) :
-    (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P) = (∫ ω, F 0 ω (B 0 ω) ∂P) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hτm := stopping_time_measurable_nnreal hτ
-  let V (n : ℕ) (ω : Ω) := F (stopGrid n (τ ω)) ω (B (stopGrid n (τ ω)) ω)
-  let H (n : ℕ) (ω : Ω) := ∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), brownianFieldDrift B F a u ω
-  have hVm (n : ℕ) : Measurable (V n) :=
-    measurable_discrete_eval (fun k ω => F (stopGridTime n k) ω (B (stopGridTime n k) ω))
-      (fun k => hF.eval_measurable _ 0 (by decide) (hm _)) (stopGridIndex_measurable hτm n)
-  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) :=
-    hF.bound _ ω _ 0 (by decide)
-  have hVi (n : ℕ) : Integrable (V n) P :=
-    Integrable.of_bound (hVm n).aestronglyMeasurable (K : ℝ)
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-  have hGm := hF.generator_measurable hBm
-  have hHjoint := cumulative_integral_measurable _ hGm
-  have hHm (n : ℕ) : Measurable (H n) :=
-    measurable_discrete_eval (fun k ω => ∫ u in (0 : ℝ)..(stopGridTime n k : ℝ), brownianFieldDrift B F a u ω)
-      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
-  have hHb (n : ℕ) (ω : Ω) : |H n ω| ≤ (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) :=
-    bounded_cumulative_at_grid _ hGm (2 * (K : ℝ)) (by positivity) hF.generator_bound τ hzero n ω
-  have hHi (n : ℕ) : Integrable (H n) P :=
-    Integrable.of_bound (hHm n).aestronglyMeasurable _
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
-  have hmean (n : ℕ) : (∫ ω, V n ω ∂P) - (∫ ω, F 0 ω (B 0 ω) ∂P) = ∫ ω, H n ω ∂P := by
-    have hh := brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
-    change (∫ ω, V n ω - H n ω ∂P) = _ at hh
-    rw [integral_sub (hVi n) (hHi n)] at hh
-    linarith
-  have herror (n : ℕ) : dist (∫ ω, V n ω ∂P) (∫ ω, F 0 ω (B 0 ω) ∂P) ≤
-      (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) := by
-    rw [Real.dist_eq, hmean]
-    have hh := norm_integral_le_of_norm_le_const (μ := P) (Eventually.of_forall fun ω =>
-      show ‖H n ω‖ ≤ (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) from by
-        simpa only [Real.norm_eq_abs] using hHb n ω)
-    simpa only [Real.norm_eq_abs, Measure.real, measure_univ, ENNReal.toReal_one, mul_one] using hh
-  have hlim : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop (𝓝 (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => (K : ℝ))
-      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-    exact Eventually.of_forall fun ω => (hF.path_continuous hBc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω))
-  have hlim0 : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop (𝓝 (∫ ω, F 0 ω (B 0 ω) ∂P)) := by
-    apply tendsto_iff_dist_tendsto_zero.mpr
-    apply squeeze_zero (fun n => dist_nonneg) herror
-    simpa only [mul_zero] using
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2 * (K : ℝ))
-  exact tendsto_nhds_unique hlim hlim0
-
-noncomputable def picardApprox
-    {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (V : ℝ → Ω → E → E) (x₀ : E) : ℕ → ℝ → Ω → E
-  | 0, _, _ => x₀
-  | n + 1, t, ω => x₀ + ∫ u in Ioc (0 : ℝ) t, V u ω (picardApprox V x₀ n u ω)
-
-lemma picardApprox_measurable
-    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
-    (V : ℝ → Ω → E → E) (x₀ : E)
-    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2)) (n : ℕ) :
-    Measurable (fun q : ℝ × Ω => picardApprox V x₀ n q.1 q.2) := by
-  induction n with
-  | zero => exact measurable_const
-  | succ n ih =>
-    have hf : Measurable (fun q : (ℝ × Ω) × ℝ => V q.2 q.1.2 (picardApprox V x₀ n q.2 q.1.2)) :=
-      hV.comp ((measurable_snd.prodMk measurable_fst.snd).prodMk
-        (ih.comp (measurable_snd.prodMk measurable_fst.snd)))
-    let A : Set ((ℝ × Ω) × ℝ) := {q | 0 < q.2 ∧ q.2 ≤ q.1.1}
-    have hA : MeasurableSet A :=
-      (measurableSet_lt measurable_const measurable_snd).inter
-        (measurableSet_le measurable_snd measurable_fst.fst)
-    have hm := ((hf.indicator hA).stronglyMeasurable.integral_prod_right' (ν := volume)).measurable
-    apply measurable_const.add
-    convert hm using 1
-    funext q
-    rw [← integral_indicator measurableSet_Ioc]
-    rfl
-
-lemma picardApprox_eq_iterate
-    {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {V : ℝ → Ω → E → E} {x₀ : E} {T L K : ℝ≥0} (ω : Ω)
-    (hf : IsPicardLindelof (fun t x => V t ω x)
-      (⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩ : Icc (0 : ℝ) T) x₀ (L * T) 0 L K)
-    (α₀ : ODE.FunSpace (⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩ : Icc (0 : ℝ) T) x₀ 0 L)
-    (hα₀ : ∀ t, α₀ t = x₀) (n : ℕ) (t : Icc (0 : ℝ) T) :
-    picardApprox V x₀ n t ω =
-      ((ODE.FunSpace.next hf (Metric.mem_closedBall_self (le_refl (0 : ℝ))))^[n] α₀) t := by
-  induction n generalizing t with
-  | zero => exact (hα₀ t).symm
-  | succ n ih =>
-    rw [Function.iterate_succ_apply', ODE.FunSpace.next_apply, ODE.picard_apply]
-    dsimp only [picardApprox]
-    congr 1
-    rw [intervalIntegral.integral_of_le t.property.1]
-    apply setIntegral_congr_fun measurableSet_Ioc
-    intro u hu
-    have hu' : u ∈ Icc (0 : ℝ) T := ⟨hu.1.le, hu.2.trans t.property.2⟩
-    exact congrArg (V u ω) ((ih ⟨u, hu'⟩).trans (ODE.FunSpace.compProj_of_mem hu').symm)
-
- 
-
-theorem exists_measurable_bounded_ode
-    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
-    (V : ℝ → Ω → E → E) (x₀ : E) (T L K : ℝ≥0)
-    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2))
-    (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x))
-    (hl : ∀ ω t, LipschitzWith K (V t ω))
-    (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) :
-    ∃ X : ℝ → Ω → E,
-      Measurable (fun q : ℝ × Ω => X q.1 q.2) ∧
-      (∀ ω, LipschitzWith L (fun t => X t ω)) ∧
-      (∀ ω, X 0 ω = x₀) ∧
-      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
-        HasDerivWithinAt (fun u => X u ω) (V t ω (X t ω)) (Icc 0 (T : ℝ)) t) := by
-  let t₀ : Icc (0 : ℝ) T := ⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩
-  have hPL (ω : Ω) : IsPicardLindelof (fun t x => V t ω x) t₀ x₀ (L * T) 0 L K := by
-    refine ⟨fun t ht => (hl ω t).lipschitzOnWith, fun x hx => (hc ω x).continuousOn,
-      fun t ht x hx => hb ω t x, ?_⟩
-    simp only [t₀, sub_zero, sub_self, max_eq_left T.coe_nonneg, NNReal.coe_mul,
-      NNReal.coe_zero, le_refl]
-  have hx : x₀ ∈ Metric.closedBall x₀ (0 : ℝ) := Metric.mem_closedBall_self le_rfl
-  choose α hα using fun ω => ODE.FunSpace.exists_isFixedPt_next (hPL ω) hx
-  let X (t : ℝ) (ω : Ω) := (α ω).compProj t
-  let α₀ : ODE.FunSpace t₀ x₀ 0 L :=
-    ⟨fun _ => x₀, (LipschitzWith.const x₀).weaken (show (0 : ℝ≥0) ≤ L from bot_le), hx⟩
-  have hα₀ (t : Icc (0 : ℝ) T) : α₀ t = x₀ := rfl
-  have hconv (t : Icc (0 : ℝ) T) (ω : Ω) :
-      Tendsto (fun n => picardApprox V x₀ n t ω) atTop (𝓝 ((α ω) t)) := by
-    apply tendsto_iff_dist_tendsto_zero.mpr
-    apply squeeze_zero (fun _ => dist_nonneg)
-      (fun n => ?_) (by simpa only [zero_mul] using (FloorSemiring.tendsto_pow_div_factorial_atTop
-        ((K : ℝ) * |(t : ℝ)|)).mul_const (dist α₀ (α ω)))
-    have hh := ODE.FunSpace.dist_iterate_next_apply_le (hPL ω) hx α₀ (α ω) n t
-    rw [(hα ω).iterate n] at hh
-    simpa only [t₀, sub_zero, picardApprox_eq_iterate ω (hPL ω) α₀ hα₀ n t] using hh
-  refine ⟨X, ?_, ?_, ?_, ?_⟩
-  · apply measurable_of_tendsto_metrizable
-      (f := fun n q => picardApprox V x₀ n (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2)
-    · intro n
-      exact (picardApprox_measurable V x₀ hV n).comp
-        (((continuous_subtype_val.comp (continuous_projIcc (h := T.coe_nonneg))).measurable.comp measurable_fst).prodMk measurable_snd)
-    · apply tendsto_pi_nhds.mpr
-      intro q
-      exact hconv (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2
-  · intro ω
-    simpa only [mul_one, X, ODE.FunSpace.compProj, Function.comp_def] using (α ω).lipschitzWith.comp (LipschitzWith.projIcc T.coe_nonneg)
-  · intro ω
-    change (α ω).compProj 0 = x₀
-    rw [ODE.FunSpace.compProj_of_mem t₀.property]
-    exact ODE.FunSpace.apply_of_zero (α ω)
-  · intro ω t ht
-    apply ODE.hasDerivWithinAt_picard_Icc t₀.property (hPL ω).continuousOn_uncurry
-      (α ω).continuous_compProj.continuousOn
-      (fun s hs => (α ω).compProj_mem_closedBall (t := s) (hPL ω).mul_max_le) x₀ ht
-      |>.congr_of_mem _ ht
-    intro s hs
-    change (α ω).compProj s = ODE.picard (fun t x => V t ω x) t₀ x₀ (α ω).compProj s
-    rw [ODE.FunSpace.compProj_of_mem hs]
-    nth_rw 1 [← hα ω]
-    exact ODE.FunSpace.next_apply (hPL ω) hx (α ω)
-
- 
-
-noncomputable def clippedHeight (lo hi : ℝ) (hle : lo ≤ hi) (q : ℂ × ℝ) : ℝ :=
-  Real.exp (projIcc lo hi hle q.1.im)
-
-noncomputable def clippedCenter (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) : ℂ :=
-  ((q.1.re - u : ℝ) : ℂ) + (clippedHeight lo hi hle q : ℂ) * Complex.I
-
-noncomputable def clippedLogLoewnerField (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ)
-    (q : ℂ × ℝ) : ℂ × ℝ :=
-  let v := (clippedCenter lo hi hle u q)⁻¹
-  (((2 * v.re : ℝ) : ℂ) + ((-2 * Complex.normSq v : ℝ) : ℂ) * Complex.I,
-    (-2 * v ^ 2).re)
-
-lemma clippedHeight_pos (lo hi : ℝ) (hle : lo ≤ hi) (q : ℂ × ℝ) :
-    0 < clippedHeight lo hi hle q := Real.exp_pos _
-
-lemma clippedHeight_bounds (lo hi : ℝ) (hle : lo ≤ hi) (q : ℂ × ℝ) :
-    Real.exp lo ≤ clippedHeight lo hi hle q ∧ clippedHeight lo hi hle q ≤ Real.exp hi := by
-  exact ⟨Real.exp_le_exp.mpr (projIcc lo hi hle q.1.im).property.1,
-    Real.exp_le_exp.mpr (projIcc lo hi hle q.1.im).property.2⟩
-
-lemma clippedCenter_im (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
-    (clippedCenter lo hi hle u q).im = clippedHeight lo hi hle q := by
-  simp [clippedCenter]
-
-lemma clippedCenter_ne_zero (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
-    clippedCenter lo hi hle u q ≠ 0 := by
-  intro hh
-  have he := congrArg Complex.im hh
-  rw [clippedCenter_im] at he
-  exact (clippedHeight_pos lo hi hle q).ne' he
-
-lemma exp_clamp_lipschitz (lo hi : ℝ) (hle : lo ≤ hi) :
-    LipschitzWith ⟨Real.exp hi, (Real.exp_pos hi).le⟩
-      (fun x : ℝ => Real.exp (projIcc lo hi hle x)) := by
-  have hf : LipschitzOnWith ⟨Real.exp hi, (Real.exp_pos hi).le⟩ Real.exp (Icc lo hi) := by
-    apply (convex_Icc lo hi).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
-      (fun x hx => (Real.hasDerivAt_exp x).hasDerivWithinAt)
-    intro x hx
-    change ‖Real.exp x‖ ≤ Real.exp hi
-    rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos x)]
-    exact Real.exp_le_exp.mpr hx.2
-  apply LipschitzWith.of_dist_le_mul
-  intro x y
-  calc
-    _ ≤ Real.exp hi * dist (projIcc lo hi hle x : ℝ) (projIcc lo hi hle y : ℝ) :=
-      hf.dist_le_mul (projIcc lo hi hle x) (projIcc lo hi hle x).property
-        (projIcc lo hi hle y) (projIcc lo hi hle y).property
-    _ ≤ Real.exp hi * dist x y := by
-      apply mul_le_mul_of_nonneg_left _ (Real.exp_pos hi).le
-      have hp := (LipschitzWith.projIcc hle).dist_le_mul x y
-      change dist (projIcc lo hi hle x : ℝ) (projIcc lo hi hle y : ℝ) ≤ (1 : ℝ) * dist x y at hp
-      simpa only [one_mul] using hp
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.BrownianGenerators
+
+namespace OAI
+
+/-! Optional Stopping for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+lemma eq_of_uniform_small_increment
+    (f : ℝ≥0 → ℝ) (rate : ℝ≥0 → ℝ) (C : ℝ)
+    (hrate : Tendsto rate (𝓝 0) (𝓝 0))
+    (hb : ∀ s δ, |f (s + δ) - f s| ≤ C * (δ : ℝ) * rate δ)
+    (s t : ℝ≥0) (hst : s ≤ t) : f t = f s := by
+  have mesh (n : ℕ) : |f t - f s| ≤ C * ((t : ℝ) - s) * rate ((t - s) / (n + 1)) := by
+    let δ : ℝ≥0 := (t - s) / (n + 1)
+    let u (i : ℕ) : ℝ≥0 := s + i * δ
+    have hn : (n + 1 : ℝ≥0) ≠ 0 := by positivity
+    have hlast : u (n + 1) = t := by
+      dsimp only [u, δ]
+      push_cast
+      rw [mul_div_cancel₀ _ hn, add_tsub_cancel_of_le hst]
+    have hstep (i : ℕ) : u (i + 1) = u i + δ := by
+      dsimp [u]; push_cast; ring
+    have htel : (∑ i ∈ Finset.range (n + 1), (f (u (i + 1)) - f (u i))) = f t - f s := by
+      rw [Finset.sum_range_sub (fun i => f (u i))]
+      simp only [hlast, u, Nat.cast_zero, zero_mul, add_zero]
+    have hδ : ((n : ℝ) + 1) * (δ : ℝ) = (t : ℝ) - s := by
+      dsimp only [δ]
+      rw [NNReal.coe_div, NNReal.coe_sub hst]
+      push_cast
+      exact mul_div_cancel₀ _ (by positivity)
+    calc
+      |f t - f s| = |∑ i ∈ Finset.range (n + 1), (f (u (i + 1)) - f (u i))| := by rw [htel]
+      _ ≤ ∑ i ∈ Finset.range (n + 1), |f (u (i + 1)) - f (u i)| :=
+        Finset.abs_sum_le_sum_abs _ _
+      _ ≤ ∑ _i ∈ Finset.range (n + 1), C * (δ : ℝ) * rate δ := by
+        apply Finset.sum_le_sum
+        intro i hi
+        rw [hstep]
+        exact hb (u i) δ
+      _ = C * ((t : ℝ) - s) * rate δ := by
+        simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_add, Nat.cast_one]
+        calc
+          ((n : ℝ) + 1) * (C * (δ : ℝ) * rate δ) = C * (((n : ℝ) + 1) * (δ : ℝ)) * rate δ := by ring
+          _ = _ := by rw [hδ]
+  have hmesh : Tendsto (fun n : ℕ => (t - s) / (n + 1 : ℝ≥0)) atTop (𝓝 0) := by
+    apply NNReal.tendsto_coe.mp
+    simp only [NNReal.coe_div, NNReal.coe_add, NNReal.coe_natCast, NNReal.coe_one, NNReal.coe_zero]
+    apply tendsto_const_nhds.div_atTop
+    simpa only [Function.comp_def, Nat.cast_add, Nat.cast_one] using
+      (tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1)
+  have hl : Tendsto (fun n : ℕ => C * ((t : ℝ) - s) * rate ((t - s) / (n + 1))) atTop (𝓝 0) := by
+    simpa only [mul_zero, Function.comp_apply] using tendsto_const_nhds.mul (hrate.comp hmesh)
+  have he : |f t - f s| ≤ 0 := le_of_tendsto_of_tendsto tendsto_const_nhds hl
+    (Eventually.of_forall mesh)
+  exact sub_eq_zero.mp (abs_nonpos_iff.mp he)
+
+lemma bounded_interval_integral_expectation_sub
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (a b c : ℝ) (hab : a ≤ b) (hbc : b ≤ c) (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hGb : ∀ u ω, |G u ω| ≤ C) :
+    (∫ ω, (∫ u in a..c, G u ω) ∂P) - (∫ ω, (∫ u in a..b, G u ω) ∂P) =
+      ∫ ω, (∫ u in b..c, G u ω) ∂P := by
+  have hi₁ := bounded_interval_integral_integrable a b hab G hG C hGb (P := P)
+  have hi₂ := bounded_interval_integral_integrable b c hbc G hG C hGb (P := P)
+  have he : (∫ ω, (∫ u in a..c, G u ω) ∂P) =
+      (∫ ω, (∫ u in a..b, G u ω) ∂P) + (∫ ω, (∫ u in b..c, G u ω) ∂P) := by
+    rw [← integral_add hi₁ hi₂]
+    apply integral_congr_ae
+    exact Eventually.of_forall fun ω => (intervalIntegral.integral_add_adjacent_intervals
+      (bounded_time_section_intervalIntegrable a b hab G hG C hGb ω)
+      (bounded_time_section_intervalIntegrable b c hbc G hG C hGb ω)).symm
+  rw [he, add_sub_cancel_left]
+
+lemma SmoothTimeBrownianField.generator_measurable
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
+    Measurable (Function.uncurry (brownianFieldDrift B F a)) := by
+  change Measurable (fun q : ℝ × Ω => a q.1.toNNReal q.2 (B q.1.toNNReal q.2) +
+    (1 / 2 : ℝ) * iteratedDeriv 2 (F q.1.toNNReal q.2) (B q.1.toNNReal q.2))
+  exact (hF.drift_real_joint hBm).add (measurable_const.mul (hF.deriv_real_joint hBm 2 (by decide)))
+
+lemma SmoothTimeBrownianField.generator_bound
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (u : ℝ) (ω : Ω) :
+    |brownianFieldDrift B F a u ω| ≤ 2 * (K : ℝ) := by
+  unfold brownianFieldDrift
+  have hb := hF.bound u.toNNReal ω (B u.toNNReal ω) 2 (by decide)
+  have ha := hF.drift_bound u.toNNReal ω (B u.toNNReal ω)
+  have ht := abs_add_le (a u.toNNReal ω (B u.toNNReal ω))
+    ((1 / 2 : ℝ) * iteratedDeriv 2 (F u.toNNReal ω) (B u.toNNReal ω))
+  rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2)] at ht
+  linarith [K.coe_nonneg]
+
+ 
+
+theorem brownian_time_field_dynkin
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (s t : ℝ≥0) (hst : s ≤ t) {E : Set Ω}
+    (hE : MeasurableSet[Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable) s] E) :
+    (∫ ω in E, F t ω (B t ω) - F s ω (B s ω) ∂P) =
+      ∫ ω in E, (∫ u in (s : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let c (r : ℝ≥0) := ∫ ω in E, (∫ u in (s : ℝ)..(s + r : ℝ≥0), brownianFieldDrift B F a u ω) ∂P
+  let v (r : ℝ≥0) := ∫ ω in E, F r ω (B r ω) ∂P
+  let f (r : ℝ≥0) := v (s + r) - c r
+  have hI (r : ℝ≥0) : Integrable (fun ω => F r ω (B r ω)) (P.restrict E) := by
+    simpa only [iteratedDeriv_zero] using hF.eval_integrable r 0 (by decide) (hm r) (P := P.restrict E)
+  have hinc (r δ : ℝ≥0) : |f (r + δ) - f r| ≤ (K : ℝ) * (δ : ℝ) * dynkinRate δ := by
+    have hsr : s ≤ s + r := le_add_of_nonneg_right (zero_le)
+    have hE' : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (s + r)] E :=
+      (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono hsr E hE
+    have h := smooth_field_one_step_error hB hm hBm K hF (s + r) δ hE'
+    have hcomp : c (r + δ) - c r =
+        ∫ ω in E, (∫ u in (s + r : ℝ≥0)..((s + r) + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P := by
+      dsimp only [c]
+      rw [← add_assoc]
+      apply bounded_interval_integral_expectation_sub _ _ _
+        (by exact_mod_cast hsr) (by simp only [NNReal.coe_add]; linarith [δ.coe_nonneg]) _
+        (hF.generator_measurable hBm) (2 * (K : ℝ)) (hF.generator_bound)
+    have hv : v ((s + r) + δ) - v (s + r) =
+        ∫ ω in E, F ((s + r) + δ) ω (B ((s + r) + δ) ω) - F (s + r) ω (B (s + r) ω) ∂P :=
+      (integral_sub (hI _) (hI _)).symm
+    have he : f (r + δ) - f r =
+        (∫ ω in E, F ((s + r) + δ) ω (B ((s + r) + δ) ω) - F (s + r) ω (B (s + r) ω) ∂P) -
+        (∫ ω in E, (∫ u in (s + r : ℝ≥0)..((s + r) + δ : ℝ≥0), brownianFieldDrift B F a u ω) ∂P) := by
+      dsimp only [f]
+      rw [← add_assoc]
+      linarith
+    rw [he]
+    exact h
+  have heq := eq_of_uniform_small_increment f dynkinRate K dynkinRate_tendsto_zero hinc 0 (t - s) (zero_le)
+  have hc0 : c 0 = 0 := by simp [c]
+  have hst' : s + (t - s) = t := add_tsub_cancel_of_le hst
+  change v (s + (t - s)) - c (t - s) = v (s + 0) - c 0 at heq
+  rw [hst', add_zero, hc0, sub_zero] at heq
+  have hc : c (t - s) = ∫ ω in E, (∫ u in (s : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω) ∂P := by
+    dsimp only [c]
+    rw [hst']
+  rw [integral_sub (hI _) (hI _)]
+  change v t - v s = _
+  linarith
+
+ 
+noncomputable def stopGridIndex (n : ℕ) (t : ℝ≥0) : ℕ :=
+  Nat.ceil (((n : ℝ≥0) + 1) * t)
+
+noncomputable def stopGridTime (n k : ℕ) : ℝ≥0 :=
+  (k : ℝ≥0) / ((n : ℝ≥0) + 1)
+
+noncomputable def stopGrid (n : ℕ) (t : ℝ≥0) : ℝ≥0 :=
+  stopGridTime n (stopGridIndex n t)
+
+lemma stopGridIndex_le (n k : ℕ) (t : ℝ≥0) :
+    stopGridIndex n t ≤ k ↔ t ≤ stopGridTime n k := by
+  have hd : 0 < (n : ℝ≥0) + 1 := by positivity
+  simp only [stopGridIndex, Nat.ceil_le, stopGridTime, le_div_iff₀ hd, mul_comm]
+
+lemma stopGrid_bounds (n : ℕ) (t : ℝ≥0) :
+    t ≤ stopGrid n t ∧ stopGrid n t ≤ t + 1 / ((n : ℝ≥0) + 1) := by
+  have hd : 0 < (n : ℝ≥0) + 1 := by positivity
+  constructor
+  · exact (stopGridIndex_le n _ t).mp le_rfl
+  · dsimp [stopGrid, stopGridTime, stopGridIndex]
+    rw [div_le_iff₀ hd]
+    calc
+      _ ≤ ((n : ℝ≥0) + 1) * t + 1 :=
+        (Nat.ceil_lt_add_one (by positivity)).le
+      _ = (t + 1 / ((n : ℝ≥0) + 1)) * ((n : ℝ≥0) + 1) := by
+        conv_rhs => rw [add_mul, div_mul_cancel₀ _ hd.ne']
+        ac_rfl
+
+lemma stopGrid_tendsto (t : ℝ≥0) : Tendsto (fun n => stopGrid n t) atTop (𝓝 t) := by
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
+    (by simpa using (tendsto_const_nhds.add
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ≥0))))
+  · exact fun n => (stopGrid_bounds n t).1
+  · intro n
+    simpa only [one_div] using (stopGrid_bounds n t).2
+
+lemma measurable_discrete_eval {Ω : Type*} [MeasurableSpace Ω]
+    (f : ℕ → Ω → ℝ) (hf : ∀ k, Measurable (f k))
+    {j : Ω → ℕ} (hj : Measurable j) : Measurable (fun ω => f (j ω) ω) := by
+  exact (measurable_from_prod_countable_right (f := fun p : ℕ × Ω => f p.1 p.2) hf).comp (hj.prodMk measurable_id)
+
+lemma stopping_time_measurable_nnreal {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {ℱ : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) : Measurable τ := by
+  simpa only [WithTop.untopD_coe] using hτ.measurable'.untopD 0
+
+lemma stopGridIndex_measurable {Ω : Type*} [MeasurableSpace Ω]
+    {τ : Ω → ℝ≥0} (hτ : Measurable τ) (n : ℕ) :
+    Measurable (fun ω => stopGridIndex n (τ ω)) := by
+  exact (measurable_const.mul hτ).nat_ceil
+
+lemma SmoothTimeBrownianField.time_increment_all
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s t : ℝ≥0) (ω : Ω) (x : ℝ) :
+    F t ω x - F s ω x = ∫ u in (s : ℝ)..(t : ℝ), a u.toNNReal ω x := by
+  rcases le_total s t with h | h
+  · exact hF.time_increment s t h ω x
+  · rw [intervalIntegral.integral_symm, ← hF.time_increment t s h ω x]
+    ring
+
+lemma SmoothTimeBrownianField.time_dist_le
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s t : ℝ≥0) (ω : Ω) (x : ℝ) :
+    dist (F t ω x) (F s ω x) ≤ (K : ℝ) * dist t s := by
+  rw [Real.dist_eq, hF.time_increment_all]
+  have hb := intervalIntegral.norm_integral_le_of_norm_le_const (a := (s : ℝ)) (b := (t : ℝ))
+    (f := fun u => a u.toNNReal ω x) (C := (K : ℝ))
+    (fun u hu => by simpa only [Real.norm_eq_abs] using hF.drift_bound u.toNNReal ω x)
+  change |(∫ u in (s : ℝ)..(t : ℝ), a u.toNNReal ω x)| ≤ (K : ℝ) * |(t : ℝ) - s|
+  simpa only [Real.norm_eq_abs] using hb
+
+lemma SmoothTimeBrownianField.space_lipschitz
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (t : ℝ≥0) (ω : Ω) :
+    LipschitzWith K (F t ω) := by
+  apply lipschitzWith_of_nnnorm_deriv_le ((hF.smooth t ω).differentiable (by norm_num))
+  intro x
+  change ‖deriv (F t ω) x‖ ≤ (K : ℝ)
+  simpa only [iteratedDeriv_one, Real.norm_eq_abs] using hF.bound t ω x 1 (by decide)
+
+lemma SmoothTimeBrownianField.path_continuous
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
+    Continuous (fun t => F t ω (B t ω)) := by
+  have hpair : LipschitzWith (2 * K) (fun p : ℝ≥0 × ℝ => F p.1 ω p.2) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro p q
+    calc
+      dist (F p.1 ω p.2) (F q.1 ω q.2) ≤
+        dist (F p.1 ω p.2) (F q.1 ω p.2) + dist (F q.1 ω p.2) (F q.1 ω q.2) := dist_triangle _ _ _
+      _ ≤ (K : ℝ) * dist p.1 q.1 + (K : ℝ) * dist p.2 q.2 :=
+        add_le_add (hF.time_dist_le q.1 p.1 ω p.2) ((hF.space_lipschitz q.1 ω).dist_le_mul p.2 q.2)
+      _ ≤ (K : ℝ) * dist p q + (K : ℝ) * dist p q := by
+        gcongr
+        · exact le_max_left _ _
+        · exact le_max_right _ _
+      _ = _ := by change _ = (2 * (K : ℝ)) * dist p q; ring
+  exact hpair.continuous.comp (continuous_id.prodMk (hc ω))
+
+lemma cumulative_integral_measurable
+    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) :
+    Measurable (fun q : ℝ≥0 × Ω => ∫ u in (0 : ℝ)..(q.1 : ℝ), G u q.2) := by
+  have hf : Measurable (fun q : (ℝ≥0 × Ω) × ℝ => G q.2 q.1.2) :=
+    hG.comp (measurable_snd.prodMk measurable_fst.snd)
+  let A : Set ((ℝ≥0 × Ω) × ℝ) := {q | 0 < q.2 ∧ q.2 ≤ (q.1.1 : ℝ)}
+  have hA : MeasurableSet A :=
+    (measurableSet_lt measurable_const measurable_snd).inter
+      (measurableSet_le measurable_snd (measurable_coe_nnreal_real.comp measurable_fst.fst))
+  have hm := ((hf.indicator hA).stronglyMeasurable.integral_prod_right' (ν := volume)).measurable
+  convert hm using 1
+  funext q
+  rw [intervalIntegral.integral_of_le q.1.coe_nonneg, ← integral_indicator measurableSet_Ioc]
+  rfl
+
+noncomputable def brownianFieldCompensated {Ω : Type*}
+    (B : ℝ≥0 → Ω → ℝ) (F a : ℝ≥0 → Ω → ℝ → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
+  F t ω (B t ω) - ∫ u in (0 : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω
+
+lemma SmoothTimeBrownianField.compensated_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) (t : ℝ≥0) :
+    Integrable (brownianFieldCompensated B F a t) P := by
+  have hi : Integrable (fun ω => F t ω (B t ω)) P :=
+    hF.eval_integrable t 0 (by decide) (hm t)
+  exact hi.sub (bounded_interval_integral_integrable 0 t t.coe_nonneg _
+    (hF.generator_measurable hBm) (2 * (K : ℝ)) hF.generator_bound)
+
+lemma SmoothTimeBrownianField.compensated_past_integral_zero
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a) (s t : ℝ≥0) (hst : s ≤ t)
+    {E : Set Ω} (hE : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) s] E) :
+    (∫ ω in E, brownianFieldCompensated B F a t ω - brownianFieldCompensated B F a s ω ∂P) = 0 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hGm := hF.generator_measurable hBm
+  have hGi (a b : ℝ) (h : a ≤ b) (ω : Ω) :=
+    bounded_time_section_intervalIntegrable a b h _ hGm (2 * (K : ℝ)) hF.generator_bound ω
+  have hp (ω : Ω) : brownianFieldCompensated B F a t ω - brownianFieldCompensated B F a s ω =
+      (F t ω (B t ω) - F s ω (B s ω)) - ∫ u in (s : ℝ)..(t : ℝ), brownianFieldDrift B F a u ω := by
+    have hh := intervalIntegral.integral_add_adjacent_intervals (hGi 0 s s.coe_nonneg ω) (hGi s t hst ω)
+    dsimp only [brownianFieldCompensated]
+    linarith
+  simp_rw [hp]
+  have hf : Integrable (fun ω => F t ω (B t ω) - F s ω (B s ω)) (P.restrict E) :=
+    (hF.eval_integrable t 0 (by decide) (hm t)).sub (hF.eval_integrable s 0 (by decide) (hm s))
+  rw [integral_sub hf (bounded_interval_integral_integrable s t hst _ hGm (2 * (K : ℝ)) hF.generator_bound),
+    brownian_time_field_dynkin hB hm hBm K F a hF s t hst hE, sub_self]
+
+lemma sum_stopped_increments (f : ℕ → ℝ) (k N : ℕ) (hk : k ≤ N) :
+    (∑ j ∈ Finset.range N, if j < k then f (j + 1) - f j else 0) = f k - f 0 := by
+  have hs : (∑ j ∈ Finset.range k, if j < k then f (j + 1) - f j else 0) =
+      ∑ j ∈ Finset.range N, if j < k then f (j + 1) - f j else 0 := by
+    apply Finset.sum_subset (Finset.range_mono hk)
+    intro j hj hjk
+    exact ite_eq_right (fun hh => hjk (Finset.mem_range.mpr hh))
+  rw [← hs]
+  calc
+    _ = ∑ j ∈ Finset.range k, (f (j + 1) - f j) := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      exact ite_eq_left (Finset.mem_range.mp hj)
+    _ = _ := Finset.sum_range_sub f k
+
+ 
+
+lemma finite_grid_stopped_integral_eq
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    (ℱ : Filtration ℕ mΩ) (C : ℕ → Ω → ℝ)
+    (hi : ∀ j, Integrable (C j) P) (σ : Ω → ℕ) (N : ℕ)
+    (hb : ∀ ω, σ ω ≤ N)
+    (hE : ∀ j, MeasurableSet[ℱ j] {ω | j < σ ω})
+    (hzero : ∀ j (E : Set Ω), MeasurableSet[ℱ j] E →
+      (∫ ω in E, C (j + 1) ω - C j ω ∂P) = 0) :
+    (∫ ω, C (σ ω) ω ∂P) = ∫ ω, C 0 ω ∂P := by
+  classical
+  let D (j : ℕ) : Ω → ℝ := {ω | j < σ ω}.indicator (fun ω => C (j + 1) ω - C j ω)
+  have hD (j : ℕ) : Integrable (D j) P :=
+    ((hi (j + 1)).sub (hi j)).indicator (ℱ.le j _ (hE j))
+  have hDi (j : ℕ) : (∫ ω, D j ω ∂P) = 0 := by
+    change (∫ ω, {ω | j < σ ω}.indicator (fun ω => C (j + 1) ω - C j ω) ω ∂P) = 0
+    rw [integral_indicator (ℱ.le j _ (hE j))]
+    exact hzero j _ (hE j)
+  have hpoint (ω : Ω) : C (σ ω) ω = C 0 ω + ∑ j ∈ Finset.range N, D j ω := by
+    have hh := sum_stopped_increments (fun j => C j ω) (σ ω) N (hb ω)
+    have hsum : (∑ j ∈ Finset.range N, D j ω) =
+        ∑ j ∈ Finset.range N, if j < σ ω then C (j + 1) ω - C j ω else 0 := by
+      apply Finset.sum_congr rfl
+      intro index _
+      by_cases hindex : index < σ ω <;> simp [D, hindex]
+    rw [hsum, hh]
+    ring
+  simp_rw [hpoint]
+  rw [integral_add (hi 0) (integrable_finsetSum _ (fun j hj => hD j)), integral_finsetSum]
+  · simp only [hDi, Finset.sum_const_zero, add_zero]
+  · intro j hj; exact hD j
+
+lemma monotone_stopGridTime (n : ℕ) : Monotone (stopGridTime n) := by
+  intro i j hij
+  dsimp only [stopGridTime]
+  exact div_le_div_of_nonneg_right (Nat.cast_le.mpr hij) (by positivity)
+
+noncomputable def gridFiltration {Ω : Type*} {mΩ : MeasurableSpace Ω}
+    (ℱ : Filtration ℝ≥0 mΩ) (n : ℕ) : Filtration ℕ mΩ where
+  seq j := ℱ (stopGridTime n j)
+  mono' := fun _ _ hij => ℱ.mono (monotone_stopGridTime n hij)
+  le' j := ℱ.le (stopGridTime n j)
+
+lemma brownian_compensated_grid_integral
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) (n : ℕ) :
+    (∫ ω, brownianFieldCompensated B F a (stopGrid n (τ ω)) ω ∂P) =
+      ∫ ω, F 0 ω (B 0 ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let ℱn := gridFiltration ℱ n
+  have hbound (ω : Ω) : stopGridIndex n (τ ω) ≤ stopGridIndex n T := by
+    apply (stopGridIndex_le n _ _).mpr
+    exact (hb ω).trans (stopGrid_bounds n T).1
+  have hE (j : ℕ) : MeasurableSet[ℱn j] {ω | j < stopGridIndex n (τ ω)} := by
+    have hle : MeasurableSet[ℱ (stopGridTime n j)] {ω | τ ω ≤ stopGridTime n j} := by
+      simpa only [WithTop.coe_le_coe] using hτ (stopGridTime n j)
+    have he : {ω | j < stopGridIndex n (τ ω)} = {ω | τ ω ≤ stopGridTime n j}ᶜ := by
+      ext ω
+      simp only [mem_compl_iff, mem_ofPred_eq, ← stopGridIndex_le, not_le]
+    rw [he]
+    exact hle.compl
+  have he := finite_grid_stopped_integral_eq ℱn
+    (fun j => brownianFieldCompensated B F a (stopGridTime n j))
+    (fun j => hF.compensated_integrable hBm _)
+    (fun ω => stopGridIndex n (τ ω)) (stopGridIndex n T) hbound hE
+    (fun j E hE => hF.compensated_past_integral_zero hB hm hBm _ _
+      (monotone_stopGridTime n (Nat.le_succ j)) hE)
+  simpa only [stopGrid, stopGridTime, Nat.cast_zero, zero_div,
+    brownianFieldCompensated, NNReal.coe_zero, intervalIntegral.integral_same, sub_zero] using he
+
+lemma bounded_cumulative_at_grid
+    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hC : 0 ≤ C)
+    (hb : ∀ u ω, |G u ω| ≤ C) (τ : Ω → ℝ≥0)
+    (hz : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → G u ω = 0) (n : ℕ) (ω : Ω) :
+    |∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), G u ω| ≤ C * (1 / ((n : ℝ) + 1)) := by
+  have h0 : (∫ u in (0 : ℝ)..(τ ω : ℝ), G u ω) = 0 := by
+    calc
+      _ = ∫ u in (0 : ℝ)..(τ ω : ℝ), (0 : ℝ) :=
+        intervalIntegral.integral_congr_Ioo_of_le (τ ω).coe_nonneg (fun u hu => hz ω u hu.1.le hu.2)
+      _ = _ := by simp
+  have hle : (τ ω : ℝ) ≤ stopGrid n (τ ω) := (stopGrid_bounds n (τ ω)).1
+  have hsplit := intervalIntegral.integral_add_adjacent_intervals
+    (bounded_time_section_intervalIntegrable 0 (τ ω) (τ ω).coe_nonneg G hG C hb ω)
+    (bounded_time_section_intervalIntegrable (τ ω) (stopGrid n (τ ω)) hle G hG C hb ω)
+  rw [h0, zero_add] at hsplit
+  rw [← hsplit]
+  have hbnd := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (τ ω : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := C) (f := fun u => G u ω)
+    (fun u hu => by simpa only [Real.norm_eq_abs] using hb u ω)
+  rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hle)] at hbnd
+  refine hbnd.trans (mul_le_mul_of_nonneg_left ?_ hC)
+  have hupper := (stopGrid_bounds n (τ ω)).2
+  have hu : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ) + 1 / ((n : ℝ) + 1) := by exact_mod_cast hupper
+  linarith
+
+ 
+
+theorem brownian_time_field_optional
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hzero : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → brownianFieldDrift B F a u ω = 0) :
+    (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P) = (∫ ω, F 0 ω (B 0 ω) ∂P) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hτm := stopping_time_measurable_nnreal hτ
+  let V (n : ℕ) (ω : Ω) := F (stopGrid n (τ ω)) ω (B (stopGrid n (τ ω)) ω)
+  let H (n : ℕ) (ω : Ω) := ∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), brownianFieldDrift B F a u ω
+  have hVm (n : ℕ) : Measurable (V n) :=
+    measurable_discrete_eval (fun k ω => F (stopGridTime n k) ω (B (stopGridTime n k) ω))
+      (fun k => hF.eval_measurable _ 0 (by decide) (hm _)) (stopGridIndex_measurable hτm n)
+  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) :=
+    hF.bound _ ω _ 0 (by decide)
+  have hVi (n : ℕ) : Integrable (V n) P :=
+    Integrable.of_bound (hVm n).aestronglyMeasurable (K : ℝ)
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+  have hGm := hF.generator_measurable hBm
+  have hHjoint := cumulative_integral_measurable _ hGm
+  have hHm (n : ℕ) : Measurable (H n) :=
+    measurable_discrete_eval (fun k ω => ∫ u in (0 : ℝ)..(stopGridTime n k : ℝ), brownianFieldDrift B F a u ω)
+      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
+  have hHb (n : ℕ) (ω : Ω) : |H n ω| ≤ (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) :=
+    bounded_cumulative_at_grid _ hGm (2 * (K : ℝ)) (by positivity) hF.generator_bound τ hzero n ω
+  have hHi (n : ℕ) : Integrable (H n) P :=
+    Integrable.of_bound (hHm n).aestronglyMeasurable _
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
+  have hmean (n : ℕ) : (∫ ω, V n ω ∂P) - (∫ ω, F 0 ω (B 0 ω) ∂P) = ∫ ω, H n ω ∂P := by
+    have hh := brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
+    change (∫ ω, V n ω - H n ω ∂P) = _ at hh
+    rw [integral_sub (hVi n) (hHi n)] at hh
+    linarith
+  have herror (n : ℕ) : dist (∫ ω, V n ω ∂P) (∫ ω, F 0 ω (B 0 ω) ∂P) ≤
+      (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) := by
+    rw [Real.dist_eq, hmean]
+    have hh := norm_integral_le_of_norm_le_const (μ := P) (Eventually.of_forall fun ω =>
+      show ‖H n ω‖ ≤ (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) from by
+        simpa only [Real.norm_eq_abs] using hHb n ω)
+    simpa only [Real.norm_eq_abs, Measure.real, measure_univ, ENNReal.toReal_one, mul_one] using hh
+  have hlim : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop (𝓝 (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => (K : ℝ))
+      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+    exact Eventually.of_forall fun ω => (hF.path_continuous hBc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω))
+  have hlim0 : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop (𝓝 (∫ ω, F 0 ω (B 0 ω) ∂P)) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun n => dist_nonneg) herror
+    simpa only [mul_zero] using
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2 * (K : ℝ))
+  exact tendsto_nhds_unique hlim hlim0
+
+noncomputable def picardApprox
+    {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (V : ℝ → Ω → E → E) (x₀ : E) : ℕ → ℝ → Ω → E
+  | 0, _, _ => x₀
+  | n + 1, t, ω => x₀ + ∫ u in Ioc (0 : ℝ) t, V u ω (picardApprox V x₀ n u ω)
+
+lemma picardApprox_measurable
+    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
+    (V : ℝ → Ω → E → E) (x₀ : E)
+    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2)) (n : ℕ) :
+    Measurable (fun q : ℝ × Ω => picardApprox V x₀ n q.1 q.2) := by
+  induction n with
+  | zero => exact measurable_const
+  | succ n ih =>
+    have hf : Measurable (fun q : (ℝ × Ω) × ℝ => V q.2 q.1.2 (picardApprox V x₀ n q.2 q.1.2)) :=
+      hV.comp ((measurable_snd.prodMk measurable_fst.snd).prodMk
+        (ih.comp (measurable_snd.prodMk measurable_fst.snd)))
+    let A : Set ((ℝ × Ω) × ℝ) := {q | 0 < q.2 ∧ q.2 ≤ q.1.1}
+    have hA : MeasurableSet A :=
+      (measurableSet_lt measurable_const measurable_snd).inter
+        (measurableSet_le measurable_snd measurable_fst.fst)
+    have hm := ((hf.indicator hA).stronglyMeasurable.integral_prod_right' (ν := volume)).measurable
+    apply measurable_const.add
+    convert hm using 1
+    funext q
+    rw [← integral_indicator measurableSet_Ioc]
+    rfl
+
+lemma picardApprox_eq_iterate
+    {Ω E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {V : ℝ → Ω → E → E} {x₀ : E} {T L K : ℝ≥0} (ω : Ω)
+    (hf : IsPicardLindelof (fun t x => V t ω x)
+      (⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩ : Icc (0 : ℝ) T) x₀ (L * T) 0 L K)
+    (α₀ : ODE.FunSpace (⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩ : Icc (0 : ℝ) T) x₀ 0 L)
+    (hα₀ : ∀ t, α₀ t = x₀) (n : ℕ) (t : Icc (0 : ℝ) T) :
+    picardApprox V x₀ n t ω =
+      ((ODE.FunSpace.next hf (Metric.mem_closedBall_self (le_refl (0 : ℝ))))^[n] α₀) t := by
+  induction n generalizing t with
+  | zero => exact (hα₀ t).symm
+  | succ n ih =>
+    rw [Function.iterate_succ_apply', ODE.FunSpace.next_apply, ODE.picard_apply]
+    dsimp only [picardApprox]
+    congr 1
+    rw [intervalIntegral.integral_of_le t.property.1]
+    apply setIntegral_congr_fun measurableSet_Ioc
+    intro u hu
+    have hu' : u ∈ Icc (0 : ℝ) T := ⟨hu.1.le, hu.2.trans t.property.2⟩
+    exact congrArg (V u ω) ((ih ⟨u, hu'⟩).trans (ODE.FunSpace.compProj_of_mem hu').symm)
+
+ 
+
+theorem exists_measurable_bounded_ode
+    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
+    (V : ℝ → Ω → E → E) (x₀ : E) (T L K : ℝ≥0)
+    (hV : Measurable (fun q : (ℝ × Ω) × E => V q.1.1 q.1.2 q.2))
+    (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x))
+    (hl : ∀ ω t, LipschitzWith K (V t ω))
+    (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) :
+    ∃ X : ℝ → Ω → E,
+      Measurable (fun q : ℝ × Ω => X q.1 q.2) ∧
+      (∀ ω, LipschitzWith L (fun t => X t ω)) ∧
+      (∀ ω, X 0 ω = x₀) ∧
+      (∀ ω t, t ∈ Icc 0 (T : ℝ) →
+        HasDerivWithinAt (fun u => X u ω) (V t ω (X t ω)) (Icc 0 (T : ℝ)) t) := by
+  let t₀ : Icc (0 : ℝ) T := ⟨0, ⟨le_rfl, T.coe_nonneg⟩⟩
+  have hPL (ω : Ω) : IsPicardLindelof (fun t x => V t ω x) t₀ x₀ (L * T) 0 L K := by
+    refine ⟨fun t ht => (hl ω t).lipschitzOnWith, fun x hx => (hc ω x).continuousOn,
+      fun t ht x hx => hb ω t x, ?_⟩
+    simp only [t₀, sub_zero, sub_self, max_eq_left T.coe_nonneg, NNReal.coe_mul,
+      NNReal.coe_zero, le_refl]
+  have hx : x₀ ∈ Metric.closedBall x₀ (0 : ℝ) := Metric.mem_closedBall_self le_rfl
+  choose α hα using fun ω => ODE.FunSpace.exists_isFixedPt_next (hPL ω) hx
+  let X (t : ℝ) (ω : Ω) := (α ω).compProj t
+  let α₀ : ODE.FunSpace t₀ x₀ 0 L :=
+    ⟨fun _ => x₀, (LipschitzWith.const x₀).weaken (show (0 : ℝ≥0) ≤ L from bot_le), hx⟩
+  have hα₀ (t : Icc (0 : ℝ) T) : α₀ t = x₀ := rfl
+  have hconv (t : Icc (0 : ℝ) T) (ω : Ω) :
+      Tendsto (fun n => picardApprox V x₀ n t ω) atTop (𝓝 ((α ω) t)) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun _ => dist_nonneg)
+      (fun n => ?_) (by simpa only [zero_mul] using (FloorSemiring.tendsto_pow_div_factorial_atTop
+        ((K : ℝ) * |(t : ℝ)|)).mul_const (dist α₀ (α ω)))
+    have hh := ODE.FunSpace.dist_iterate_next_apply_le (hPL ω) hx α₀ (α ω) n t
+    rw [(hα ω).iterate n] at hh
+    simpa only [t₀, sub_zero, picardApprox_eq_iterate ω (hPL ω) α₀ hα₀ n t] using hh
+  refine ⟨X, ?_, ?_, ?_, ?_⟩
+  · apply measurable_of_tendsto_metrizable
+      (f := fun n q => picardApprox V x₀ n (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2)
+    · intro n
+      exact (picardApprox_measurable V x₀ hV n).comp
+        (((continuous_subtype_val.comp (continuous_projIcc (h := T.coe_nonneg))).measurable.comp measurable_fst).prodMk measurable_snd)
+    · apply tendsto_pi_nhds.mpr
+      intro q
+      exact hconv (projIcc 0 (T : ℝ) T.coe_nonneg q.1) q.2
+  · intro ω
+    simpa only [mul_one, X, ODE.FunSpace.compProj, Function.comp_def] using (α ω).lipschitzWith.comp (LipschitzWith.projIcc T.coe_nonneg)
+  · intro ω
+    change (α ω).compProj 0 = x₀
+    rw [ODE.FunSpace.compProj_of_mem t₀.property]
+    exact ODE.FunSpace.apply_of_zero (α ω)
+  · intro ω t ht
+    apply ODE.hasDerivWithinAt_picard_Icc t₀.property (hPL ω).continuousOn_uncurry
+      (α ω).continuous_compProj.continuousOn
+      (fun s hs => (α ω).compProj_mem_closedBall (t := s) (hPL ω).mul_max_le) x₀ ht
+      |>.congr_of_mem _ ht
+    intro s hs
+    change (α ω).compProj s = ODE.picard (fun t x => V t ω x) t₀ x₀ (α ω).compProj s
+    rw [ODE.FunSpace.compProj_of_mem hs]
+    nth_rw 1 [← hα ω]
+    exact ODE.FunSpace.next_apply (hPL ω) hx (α ω)
+
+ 
+
+noncomputable def clippedHeight (lo hi : ℝ) (hle : lo ≤ hi) (q : ℂ × ℝ) : ℝ :=
+  Real.exp (projIcc lo hi hle q.1.im)
+
+noncomputable def clippedCenter (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) : ℂ :=
+  ((q.1.re - u : ℝ) : ℂ) + (clippedHeight lo hi hle q : ℂ) * Complex.I
+
+noncomputable def clippedLogLoewnerField (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ)
+    (q : ℂ × ℝ) : ℂ × ℝ :=
+  let v := (clippedCenter lo hi hle u q)⁻¹
+  (((2 * v.re : ℝ) : ℂ) + ((-2 * Complex.normSq v : ℝ) : ℂ) * Complex.I,
+    (-2 * v ^ 2).re)
+
+lemma clippedHeight_pos (lo hi : ℝ) (hle : lo ≤ hi) (q : ℂ × ℝ) :
+    0 < clippedHeight lo hi hle q := Real.exp_pos _
+
+lemma clippedHeight_bounds (lo hi : ℝ) (hle : lo ≤ hi) (q : ℂ × ℝ) :
+    Real.exp lo ≤ clippedHeight lo hi hle q ∧ clippedHeight lo hi hle q ≤ Real.exp hi := by
+  exact ⟨Real.exp_le_exp.mpr (projIcc lo hi hle q.1.im).property.1,
+    Real.exp_le_exp.mpr (projIcc lo hi hle q.1.im).property.2⟩
+
+lemma clippedCenter_im (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
+    (clippedCenter lo hi hle u q).im = clippedHeight lo hi hle q := by
+  simp [clippedCenter]
+
+lemma clippedCenter_ne_zero (lo hi : ℝ) (hle : lo ≤ hi) (u : ℝ) (q : ℂ × ℝ) :
+    clippedCenter lo hi hle u q ≠ 0 := by
+  intro hh
+  have he := congrArg Complex.im hh
+  rw [clippedCenter_im] at he
+  exact (clippedHeight_pos lo hi hle q).ne' he
+
+lemma exp_clamp_lipschitz (lo hi : ℝ) (hle : lo ≤ hi) :
+    LipschitzWith ⟨Real.exp hi, (Real.exp_pos hi).le⟩
+      (fun x : ℝ => Real.exp (projIcc lo hi hle x)) := by
+  have hf : LipschitzOnWith ⟨Real.exp hi, (Real.exp_pos hi).le⟩ Real.exp (Icc lo hi) := by
+    apply (convex_Icc lo hi).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
+      (fun x hx => (Real.hasDerivAt_exp x).hasDerivWithinAt)
+    intro x hx
+    change ‖Real.exp x‖ ≤ Real.exp hi
+    rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos x)]
+    exact Real.exp_le_exp.mpr hx.2
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  calc
+    _ ≤ Real.exp hi * dist (projIcc lo hi hle x : ℝ) (projIcc lo hi hle y : ℝ) :=
+      hf.dist_le_mul (projIcc lo hi hle x) (projIcc lo hi hle x).property
+        (projIcc lo hi hle y) (projIcc lo hi hle y).property
+    _ ≤ Real.exp hi * dist x y := by
+      apply mul_le_mul_of_nonneg_left _ (Real.exp_pos hi).le
+      have hp := (LipschitzWith.projIcc hle).dist_le_mul x y
+      change dist (projIcc lo hi hle x : ℝ) (projIcc lo hi hle y : ℝ) ≤ (1 : ℝ) * dist x y at hp
+      simpa only [one_mul] using hp
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/PhysicalKernelUtilities.lean b/lean/OAI/Probability/SLE/PhysicalKernelUtilities.lean
index 203f44e746e996e02133a14972a70df16acdf4a9..fd231ac92634ced16c51cd76b4252db5c6eeee4b 100644
--- a/lean/OAI/Probability/SLE/PhysicalKernelUtilities.lean
+++ b/lean/OAI/Probability/SLE/PhysicalKernelUtilities.lean
@@ -1,764 +1,764 @@
-import Mathlib
-import OAI.Probability.SLE.ChartMeasurability
-import OAI.Probability.SLE.InteriorDiskProof
-import OAI.Probability.SLE.PhysicalTransport
-
-namespace OAI
-
-/-! Physical Kernel Utilities for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance physicalDiskKernelTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance physicalDiskKernelTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-local instance physicalDiskKernelChartMeasurableSpace : MeasurableSpace C(upperHalfPlane,ℂ) := borel _
-local instance physicalDiskKernelChartBorelSpace : BorelSpace C(upperHalfPlane,ℂ) := ⟨rfl⟩
-
-def sourcePhysicalDiskKernel (p : SourceDiskParameter × C(ℝ≥0,ℂ)) : ℝ :=
-  if ∃ t : ℝ≥0, p.1.1.2 ∈ survivingDomain p.2 t ∧
-    ∃ h : p.2 t ∈ upperHalfPlane, p.1.2.2 < ‖p.1.1.1 ⟨p.2 t,h⟩-p.1.2.1‖ then 1 else 0
-
-lemma sourcePhysicalDiskKernel_measurable : Measurable sourcePhysicalDiskKernel :=
-  Measurable.ite source_physical_disk_kernel_open.measurableSet measurable_const measurable_const
-
-lemma sourcePhysicalDiskKernel_norm_le (p : SourceDiskParameter × C(ℝ≥0,ℂ)) :
-    ‖sourcePhysicalDiskKernel p‖ ≤ 1 := by
-  unfold sourcePhysicalDiskKernel
-  split_ifs <;> norm_num
-
-lemma source_physical_exit_open (z : ℂ) (R : ℝ) :
-    IsOpen {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
-      z ∈ survivingDomain p.1 (p.2+t) ∧ R < dist (p.1 (p.2+t)) z} := by
-  have ht (t : ℝ≥0) : IsOpen {p : C(ℝ≥0,ℂ) × ℝ≥0 |
-      z ∈ survivingDomain p.1 (p.2+t) ∧ R < dist (p.1 (p.2+t)) z} := by
-    have hc : Continuous (fun p : C(ℝ≥0,ℂ) × ℝ≥0 => (p.1,p.2+t)) :=
-      continuous_fst.prodMk (continuous_snd.add continuous_const)
-    exact (source_survival_open.preimage (hc.prodMk continuous_const)).inter
-      (isOpen_lt continuous_const ((continuous_eval.comp hc).dist continuous_const))
-  simpa only [ofPred_exists] using isOpen_iUnion ht
-
-lemma source_restarted_interior_dense
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgm : Measurable (sourceTracePath γ (fun ω => (hw ω).1)))
-    (β : Ω → C(ℝ≥0,ℂ)) (hβm : Measurable β)
-    (hlaw : P.map β = P.map (sourceTracePath γ (fun ω => (hw ω).1))) :
-    ∀ᵐ ω ∂P, SourcePathInteriorDense (β ω) := by
-  apply (ae_map_iff hβm.aemeasurable source_path_interior_dense_measurable).mp
-  rw [hlaw]
-  apply (ae_map_iff hgm.aemeasurable source_path_interior_dense_measurable).mpr
-  exact Eventually.of_forall fun ω => sourceInteriorTraceDensity_proved
-    (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)
-    (continuous_const.mul (hc ω)) (hw ω)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma source_firsthit_inverse_access
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hUc : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (s τ : ℝ≥0) (z : ℂ) (r : ℝ) (hr : 0 < r)
-    (hfree : Metric.closedBall z r ⊆ survivingDomain γ s) (hsτ : s ≤ τ)
-    (hhit : dist (γ τ) z = r)
-    (hfirst : ∀ t ∈ Ico s τ, dist (γ t) z ≠ r)
-    (hz : z ∈ survivingDomain γ τ) :
-    ∃ η : ℝ → ℂ, ContinuousOn η (Icc 0 1) ∧ η 0 = 0 ∧
-      η 1 = G τ z - (U τ : ℂ) ∧
-      (∀ v ∈ Ioc 0 1, 0 < (η v).im) ∧
-      (∀ v ∈ Ioc 0 1, ‖F τ (η v + (U τ : ℂ))-z‖ ≤ r) := by
-  let α : ℝ → ℂ := fun v => (1-v) • γ τ + v • z
-  let f : ℝ → ℂ := fun v => G τ (α v) - (U τ : ℂ)
-  let η := Function.update f 0 0
-  have hα : Continuous α := by dsimp [α]; fun_prop
-  have hαD := (sourceFirstHitPhysicalAccess γ h.1 s τ z r hr hfree hsτ hhit hfirst hz).2.2
-  have hf : ContinuousOn f (Icc 0 1 \ {0}) := by
-    refine ((h.2.2.2.1 τ).continuousOn.comp hα.continuousOn ?_).sub continuousOn_const
-    intro v hv
-    exact hαD v ⟨lt_of_le_of_ne hv.1.1 (Ne.symm hv.2),hv.1.2⟩
-  have hη : ContinuousOn η (Icc 0 1) := by
-    apply continuousOn_update_iff.mpr
-    refine ⟨hf,fun _ => ?_⟩
-    apply (sourceFirstHitChartAccess_proved U γ G F hUc h s τ z r hr hfree hsτ hhit hfirst hz).mono_left
-    apply nhdsWithin_mono
-    intro v hv
-    exact lt_of_le_of_ne hv.1.1 (Ne.symm hv.2)
-  have hηv (v : ℝ) (hv : v ∈ Ioc 0 1) : η v = f v :=
-    Function.update_of_ne (ne_of_gt hv.1) _ _
-  refine ⟨η,hη,by simp [η],?_,?_,?_⟩
-  · rw [hηv 1 (by norm_num)]
-    simp [f,α]
-  · intro v hv
-    rw [hηv v hv]
-    simpa only [f,α,upperHalfPlane,Set.mem_ofPred_eq,sub_im,ofReal_im,sub_zero] using h.2.2.2.2.2.1 τ (hαD v hv)
-  · intro v hv
-    rw [hηv v hv]
-    simp only [f,sub_add_cancel,h.2.2.2.2.2.2.2.1 τ (α v) (hαD v hv)]
-    simpa only [dist_eq_norm] using
-      (show dist (α v) z < r from radialAccess_mem_ball (γ τ) z r hr hhit v hv).le
-
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-lemma source_physical_exit_interior_inclusion
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (β : C(ℝ≥0,ℂ))
-    (hdense : SourcePathInteriorDense β)
-    (hid : ∀ t : ℝ≥0, β t ∈ upperHalfPlane → γ (T+t) = F T (β t+(U T:ℂ)))
-    (z : ℂ) (R : ℝ) (hR : dist (γ T) z ≤ R) :
-    (∃ t : ℝ≥0, z ∈ survivingDomain γ (T+t) ∧ R < dist (γ (T+t)) z) →
-    ∃ t : ℝ≥0, G T z-(U T:ℂ) ∈ survivingDomain β t ∧
-      ∃ _ht : β t ∈ upperHalfPlane, R < ‖F T (β t+(U T:ℂ))-z‖ := by
-  rintro ⟨t,hl,he⟩
-  have ht : 0 < t := by
-    by_contra hn
-    have ht0 : t = 0 := le_antisymm (not_lt.mp hn) bot_le
-    simp only [ht0,add_zero] at he
-    exact (not_lt_of_ge hR) he
-  have hc : Continuous (fun u : ℝ≥0 => dist (γ (T+u)) z) :=
-    (h.1.comp (continuous_const.add continuous_id)).dist continuous_const
-  have hevent : {u : ℝ≥0 | R < dist (γ (T+u)) z} ∈ 𝓝 t :=
-    (isOpen_lt continuous_const hc).mem_nhds he
-  obtain ⟨b,hbt,hb⟩ := exists_Ioc_subset_of_mem_nhds hevent ⟨0,ht⟩
-  obtain ⟨u,hu,hupos⟩ := hdense b t hbt
-  have hlu : z ∈ survivingDomain γ (T+u) :=
-    survivingDomain_antitone γ (by simpa only [add_comm] using add_le_add_left hu.2.le T) hl
-  refine ⟨u,source_stopped_live_transfer h T u β (fun v _ => hid v) z hlu,hupos,?_⟩
-  rw [← hid u hupos,← dist_eq_norm]
-  exact hb ⟨hu.1,hu.2.le⟩
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance finitePhysicalDiskTargetTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance finitePhysicalDiskTargetTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-local instance finitePhysicalDiskTargetDriverMeasurableSpace : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance finitePhysicalDiskTargetDriverBorelSpace : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance finitePhysicalDiskTargetChartMeasurableSpace : MeasurableSpace C(upperHalfPlane,ℂ) := borel _
-local instance finitePhysicalDiskTargetChartBorelSpace : BorelSpace C(upperHalfPlane,ℂ) := ⟨rfl⟩
- 
-
-def SourceFinitePhysicalDiskTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_m : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
-  ∀ (hm : ∀ t, Measurable (B t)) (_hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (_hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) (z : ℂ) (r : ℝ), 0 < r →
-  ∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
-    (∀ ω ∈ A, ∃ s : ℝ≥0, s ≤ τ ω ∧
-      Metric.closedBall z r ⊆ survivingDomain (γ ω) s ∧
-      dist (γ ω (τ ω)) z = r ∧
-      (∀ t ∈ Ico s (τ ω), dist (γ ω t) z ≠ r) ∧
-      z ∈ survivingDomain (γ ω) (τ ω)) →
-  ∀ L : ℝ, 1 ≤ L →
-    ∀ᵐ ω ∂P, ω ∈ A →
-      P[(fun ω => if ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) (τ ω+t) ∧
-        L*r < dist (γ ω (τ ω+t)) z then (1 : ℝ) else 0) | hτ.measurableSpace] ω ≤ C*L^(-c)
-
-lemma source_physical_kernel_mean_le
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (gp : Ω → C(ℝ≥0,ℂ)) (hgm : Measurable gp)
-    (Y : SourceDiskParameter) (Ψ : ℂ → ℂ)
-    (hΨ : ∀ w : upperHalfPlane, Y.1.1 w = Ψ w)
-    (b : ℝ) (hb : 0 ≤ b)
-    (hbound : P {v | ∃ t : ℝ≥0, Y.1.2 ∈ survivingDomain (gp v) t ∧
-      0 < (gp v t).im ∧ Y.2.2 < ‖Ψ (gp v t)-Y.2.1‖} ≤ ENNReal.ofReal b) :
-    (∫ p, sourcePhysicalDiskKernel (Y,p) ∂P.map gp) ≤ b := by
-  let S : Set Ω := {v | ∃ t : ℝ≥0, Y.1.2 ∈ survivingDomain (gp v) t ∧
-      0 < (gp v t).im ∧ Y.2.2 < ‖Ψ (gp v t)-Y.2.1‖}
-  have he (v : Ω) : sourcePhysicalDiskKernel (Y,gp v) = S.indicator (fun _ => (1:ℝ)) v := by
-    unfold sourcePhysicalDiskKernel
-    have hi : (∃ t : ℝ≥0, Y.1.2 ∈ survivingDomain (gp v) t ∧
-        ∃ ht : gp v t ∈ upperHalfPlane,
-          Y.2.2 < ‖Y.1.1 ⟨gp v t,ht⟩-Y.2.1‖) ↔ v ∈ S := by
-      constructor
-      · rintro ⟨t,ht,hi,hf⟩
-        exact ⟨t,ht,hi,by simpa only [hΨ] using hf⟩
-      · rintro ⟨t,ht,hi,hf⟩
-        exact ⟨t,ht,hi,by simpa only [hΨ] using hf⟩
-    simp only [hi,Set.indicator_apply]
-  have hfun : Measurable (fun p : C(ℝ≥0,ℂ) => sourcePhysicalDiskKernel (Y,p)) :=
-    sourcePhysicalDiskKernel_measurable.comp (measurable_const.prodMk measurable_id)
-  have hS : MeasurableSet S := by
-    have hh := measurableSet_lt (measurable_const (a := (0:ℝ))) (hfun.comp hgm)
-    convert hh using 1
-    ext v
-    change v ∈ S ↔ 0 < sourcePhysicalDiskKernel (Y,gp v)
-    rw [he]
-    by_cases hv : v ∈ S <;> simp [hv]
-  have hKi : (∫ p, sourcePhysicalDiskKernel (Y,p) ∂P.map gp) = (P S).toReal := by
-    rw [integral_map hgm.aemeasurable hfun.aestronglyMeasurable]
-    simp_rw [he]
-    simp only [integral_indicator_const _ hS,smul_eq_mul,mul_one,measureReal_def]
-  rw [hKi]
-  exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr hbound |>.trans
-    (le_of_eq (ENNReal.toReal_ofReal hb))
-
-lemma source_firsthit_kernel_mean_le
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (gp : Ω → C(ℝ≥0,ℂ)) (hgm : Measurable gp)
-    (U : ℝ≥0 → ℝ) (hU : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) (hw : IsCapacityTwoWitness U γ G F)
-    (s T : ℝ≥0) (z : ℂ) (r : ℝ) (hr : 0 < r)
-    (hs : s ≤ T) (hfree : Metric.closedBall z r ⊆ survivingDomain γ s)
-    (hhit : dist (γ T) z = r)
-    (hfirst : ∀ t ∈ Ico s T, dist (γ t) z ≠ r) (hz : z ∈ survivingDomain γ T)
-    (ψ : C(upperHalfPlane,ℂ)) (hψ : ∀ w, ψ w = F T (w+(U T:ℂ)))
-    (L b : ℝ) (hb : 0 ≤ b)
-    (hbound : ∀ Ψ : ℂ → ℂ, DifferentiableOn ℂ Ψ upperHalfPlane → InjOn Ψ upperHalfPlane →
-      ∀ η : ℝ → ℂ, ContinuousOn η (Icc 0 1) → η 0 = 0 → Ψ (η 1) = z →
-      (∀ v ∈ Ioc 0 1, 0 < (η v).im) → (∀ v ∈ Ioc 0 1, ‖Ψ (η v)-z‖ ≤ r) →
-      P {v | ∃ t : ℝ≥0, η 1 ∈ survivingDomain (gp v) t ∧
-        0 < (gp v t).im ∧ L*r < ‖Ψ (gp v t)-z‖} ≤ ENNReal.ofReal b) :
-    (∫ p, sourcePhysicalDiskKernel (((ψ,G T z-(U T:ℂ)),(z,L*r)),p) ∂P.map gp) ≤ b := by
-  let Ψ : ℂ → ℂ := fun w => F T (w+(U T:ℂ))
-  have hmap : MapsTo (fun w : ℂ => w+(U T:ℂ)) upperHalfPlane upperHalfPlane := by
-    intro w hw
-    simpa [upperHalfPlane] using hw
-  have hΨd : DifferentiableOn ℂ Ψ upperHalfPlane :=
-    (hw.2.2.2.2.1 T).comp (differentiable_id.add_const _).differentiableOn hmap
-  have hΨi : InjOn Ψ upperHalfPlane := by
-    intro a ha b hb he
-    have ha' := hw.2.2.2.2.2.2.2.2.1 T (a+(U T:ℂ)) (hmap ha)
-    have hb' := hw.2.2.2.2.2.2.2.2.1 T (b+(U T:ℂ)) (hmap hb)
-    have hh := congrArg (G T) he
-    dsimp only [Ψ] at hh
-    rw [ha',hb'] at hh
-    exact add_right_cancel hh
-  obtain ⟨η,hη,hη0,hη1,hηup,hηpath⟩ := source_firsthit_inverse_access
-    hU hw s T z r hr hfree hs hhit hfirst hz
-  have hq : Ψ (η 1) = z := by
-    rw [hη1]
-    simp only [Ψ,sub_add_cancel,hw.2.2.2.2.2.2.2.1 T z hz]
-  have hprob := hbound Ψ hΨd hΨi η hη hη0 hq hηup hηpath
-  apply source_physical_kernel_mean_le gp hgm _ Ψ hψ b hb
-  simpa only [hη1] using hprob
-
-lemma source_physical_exit_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (gp : Ω → C(ℝ≥0,ℂ)) (hgp : Measurable gp)
-    (τ : Ω → ℝ≥0) (hτ : Measurable τ) (z : ℂ) (R : ℝ) :
-    MeasurableSet {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (gp ω) (τ ω+t) ∧
-      R < dist (gp ω (τ ω+t)) z} := by
-  have hp : Measurable (fun ω => (gp ω,τ ω)) := hgp.prodMk hτ
-  have he : MeasurableSet {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
-      z ∈ survivingDomain p.1 (p.2+t) ∧ R < dist (p.1 (p.2+t)) z} :=
-    (source_physical_exit_open z R).measurableSet
-  have hh := he.preimage hp
-  simpa only [Set.preimage_ofPred_eq] using hh
-
-theorem sourceFinitePhysicalDisk_proved : SourceFinitePhysicalDiskTarget := by
-  intro κ hκ hκ8
-  obtain ⟨C,c,hC,hc,hbound⟩ := source_interior_disk_unnormalized κ hκ hκ8
-  refine ⟨C,c,hC,hc,?_⟩
-  intro Ω m P hP B hB hm hBc γ G F hw τ hτ z r hr A hA hgeom L hL
-  let : IsProbabilityMeasure P := hP
-  let gp := sourceTracePath γ (fun ω => (hw ω).1)
-  obtain ⟨Φ,hΦm,hΦeq,hΦ⟩ := sourceStoppedTraceLaw κ (by linarith) hκ8 B hB hm hBc γ G F hw
-  have hgm : Measurable gp := by
-    have he : gp = fun ω => Φ (sourceDriverPath B hBc ω) := funext fun ω => (hΦeq ω).symm
-    rw [he]
-    exact hΦm.comp (source_path_measurable _ hm)
-  let ν := P.map gp
-  let : IsProbabilityMeasure ν := inferInstance
-  have hγsle : IsOrdinaryChordalSLE κ γ P :=
-    ⟨fun t => (continuous_eval_const t).measurable.comp hgm,
-      B,hB,Eventually.of_forall fun ω => ⟨G ω,F ω,hw ω⟩⟩
-  obtain ⟨hlaw,hid⟩ := hΦ τ hτ
-  let rp := sourceRestartDriverPath B hBc τ
-  let β (ω : Ω) := Φ (rp ω)
-  have hrm : Measurable rp := source_path_measurable rp (fun t =>
-    (measurable_pi_apply t).comp (brownianRestart_measurable hm hBc (stopping_time_measurable_nnreal hτ)))
-  have hβm : Measurable β := hΦm.comp hrm
-  have hβlaw : P.map β = ν := by simpa only [Measure.restrict_univ,measure_univ,one_smul] using hlaw univ MeasurableSet.univ
-  have hdense : ∀ᵐ ω ∂P, SourcePathInteriorDense (β ω) :=
-    source_restarted_interior_dense B hBc κ γ G F hw hgm β hβm hβlaw
-  let ξ (ω : Ω) :=
-    (if z ∈ survivingDomain (γ ω) (τ ω) then G ω (τ ω) z else 0) -
-      (Real.sqrt κ * B (τ ω) ω : ℝ)
-  let Y (ω : Ω) : SourceDiskParameter :=
-    ((sourceStoppedInversePath B κ γ G F hw τ ω,ξ ω),(z,L*r))
-  have hξm : Measurable[hτ.measurableSpace] ξ := by
-    have hGm := source_killedChart_stopped_measurable hm hBc κ γ G F hw z τ hτ
-    have hBm := measurable_stoppedValue
-      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hBc) hτ
-    exact hGm.sub (Complex.measurable_ofReal.comp (measurable_const.mul hBm))
-  have hYm : Measurable[hτ.measurableSpace] Y :=
-    ((source_stopped_inversePath_measurable B hm hBc κ γ G F hw τ hτ).prodMk hξm).prodMk measurable_const
-  let H (ω : Ω) := sourcePhysicalDiskKernel (Y ω,β ω)
-  let K (ω : Ω) := ∫ p, sourcePhysicalDiskKernel (Y ω,p) ∂ν
-  have hHm : Measurable H := sourcePhysicalDiskKernel_measurable.comp
-    ((hYm.mono hτ.measurableSpace_le le_rfl).prodMk hβm)
-  have hHi : Integrable H P :=
-    (integrable_const (1:ℝ)).mono' hHm.aestronglyMeasurable
-      (Eventually.of_forall fun ω => sourcePhysicalDiskKernel_norm_le (Y ω,β ω))
-  have hcond : P[H | hτ.measurableSpace] =ᵐ[P] K :=
-    source_parameter_condExp hτ.measurableSpace_le hβm hYm hlaw sourcePhysicalDiskKernel
-      sourcePhysicalDiskKernel_measurable sourcePhysicalDiskKernel_norm_le
-  have hmean (ω : Ω) (hω : ω ∈ A) : K ω ≤ C*L^(-c) := by
-    obtain ⟨s,hs,hfree,hhit,hfirst,hz⟩ := hgeom ω hω
-    dsimp only [K,Y,ξ]
-    rw [ite_eq_left hz]
-    apply source_firsthit_kernel_mean_le gp hgm (fun t => Real.sqrt κ*B t ω)
-      (continuous_const.mul (hBc ω)) (γ ω) (G ω) (F ω) (hw ω)
-      s (τ ω) z r hr hs hfree hhit hfirst hz _ (fun _ => rfl)
-      L (C*L^(-c)) (mul_nonneg hC.le (Real.rpow_nonneg (by linarith) _))
-    intro Ψ hΨd hΨi η hη hη0 hq hηup hηpath
-    exact hbound Ω m P hP γ hγsle Ψ hΨd hΨi z r hr η 1 (by norm_num)
-      hη hη0 hq hηup hηpath L hL
-  let E : Set Ω := {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) (τ ω+t) ∧
-      L*r < dist (γ ω (τ ω+t)) z}
-  let f : Ω → ℝ := E.indicator (fun _ => 1)
-  have hE : MeasurableSet E :=
-    source_physical_exit_measurable gp hgm τ (stopping_time_measurable_nnreal hτ) z (L*r)
-  have hfi : Integrable f P := (integrable_const (1:ℝ)).indicator hE
-  have hAi : Integrable (A.indicator f) P := hfi.indicator (hτ.measurableSpace_le _ hA)
-  have hAH : Integrable (A.indicator H) P := hHi.indicator (hτ.measurableSpace_le _ hA)
-  have hdom : A.indicator f ≤ᵐ[P] A.indicator H := by
-    filter_upwards [hid,hdense] with ω hωid hωdense
-    by_cases hω : ω ∈ A
-    · simp only [Set.indicator_of_mem hω]
-      by_cases he : ω ∈ E
-      · obtain ⟨s,hs,hfree,hhit,hfirst,hz⟩ := hgeom ω hω
-        have hR : dist (γ ω (τ ω)) z ≤ L*r := by rw [hhit]; nlinarith
-        obtain ⟨t,ht,htH,hfar⟩ := source_physical_exit_interior_inclusion (hw ω) (τ ω) (β ω)
-          hωdense hωid z (L*r) hR he
-        have hkernel : H ω = 1 := by
-          apply ite_eq_left
-          refine ⟨t,?_,htH,?_⟩
-          · simpa only [Y,ξ,ite_eq_left hz] using ht
-          · exact hfar
-        simp only [f,Set.indicator_of_mem he,hkernel,le_refl]
-      · simp only [f,Set.indicator_of_notMem he]
-        dsimp [H,sourcePhysicalDiskKernel]
-        split_ifs <;> norm_num
-    · simp only [Set.indicator_of_notMem hω,le_refl]
-  have hmono := condExp_mono (m := hτ.measurableSpace) hAi hAH hdom
-  filter_upwards [hmono,condExp_indicator hfi hA,condExp_indicator hHi hA,hcond] with ω hm hf hH hK
-  intro hω
-  rw [hf,hH,Set.indicator_of_mem hω,Set.indicator_of_mem hω,hK] at hm
-  exact hm.trans (hmean ω hω)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-noncomputable def sourceStopTruncate {Ω : Type*} (τ : Ω → WithTop ℝ≥0)
-    (t : ℝ≥0) (ω : Ω) : ℝ≥0 := (min (τ ω) t).untopD 0
-
-lemma sourceStopTruncate_coe {Ω : Type*} (τ : Ω → WithTop ℝ≥0)
-    (t : ℝ≥0) (ω : Ω) :
-    (sourceStopTruncate τ t ω : WithTop ℝ≥0) = min (τ ω) t := by
-  have hn : min (τ ω) (t : WithTop ℝ≥0) ≠ ⊤ :=
-    ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_right _ _)
-  obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp hn
-  change (((min (τ ω) (t : WithTop ℝ≥0)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = _
-  rw [← hv, WithTop.untopD_coe]
-
-lemma sourceStopTruncate_stopping {Ω : Type*} [MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›) {τ : Ω → WithTop ℝ≥0}
-    (hτ : IsStoppingTime ℱ τ) (t : ℝ≥0) :
-    IsStoppingTime ℱ (fun ω => (sourceStopTruncate τ t ω : WithTop ℝ≥0)) := by
-  simpa only [sourceStopTruncate_coe] using hτ.min_const t
-
-lemma sourceStop_indicator_tendsto (τ : WithTop ℝ≥0) (v : ℝ)
-    (hzero : τ = ⊤ → v = 0) :
-    Tendsto (fun n : ℕ => if τ ≤ (n : ℝ≥0) then v else 0) atTop (𝓝 v) := by
-  by_cases ht : τ = ⊤
-  · simp only [ht, hzero ht, ite_self]
-    exact tendsto_const_nhds
-  obtain ⟨t, rfl⟩ := WithTop.ne_top_iff_exists.mp ht
-  apply tendsto_const_nhds.congr'
-  filter_upwards [eventually_ge_atTop (⌈(t : ℝ)⌉₊)] with n hn
-  have htn : t ≤ (n : ℝ≥0) := by exact_mod_cast (Nat.le_of_ceil_le hn)
-  simp only [WithTop.coe_le_coe, ite_eq_left htn]
-
-lemma source_unbounded_setIntegral_le
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    (ℱ : Filtration ℝ≥0 mΩ) {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime ℱ τ)
-    (f g : Ω → ℝ) (hf : Integrable f μ) (hg : Integrable g μ)
-    (hz : ∀ ω, τ ω = ⊤ → f ω = 0 ∧ g ω = 0)
-    (hloc : ∀ (n : ℕ) (A : Set Ω),
-      MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace] A →
-      (∀ ω ∈ A, τ ω = (sourceStopTruncate τ n ω : WithTop ℝ≥0)) →
-      (∫ ω in A, f ω ∂μ) ≤ ∫ ω in A, g ω ∂μ)
-    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
-    (∫ ω in A, f ω ∂μ) ≤ ∫ ω in A, g ω ∂μ := by
-  have hAm (n : ℕ) : MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace]
-      (A ∩ {ω | τ ω ≤ (n : ℝ≥0)}) := by
-    have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ (n : ℝ≥0)) A hA
-    convert hh using 1
-    congr 1
-    funext ω
-    exact sourceStopTruncate_coe τ n ω
-  have hset (n : ℕ) :
-      (∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, f ω ∂μ) ≤
-        ∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, g ω ∂μ := by
-    apply hloc n _ (hAm n)
-    intro ω hω
-    rw [sourceStopTruncate_coe, min_eq_left hω.2]
-  have hlim (v : Ω → ℝ) (hv : Integrable v μ) (hzero : ∀ ω, τ ω = ⊤ → v ω = 0) :
-      Tendsto (fun n : ℕ => ∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, v ω ∂μ) atTop
-        (𝓝 (∫ ω in A, v ω ∂μ)) := by
-    have he (n : ℕ) : (∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, v ω ∂μ) =
-        ∫ ω in A, {ω | τ ω ≤ (n : ℝ≥0)}.indicator v ω ∂μ :=
-      (setIntegral_indicator (μ := μ) (s := A) (f := v)
-        ((ℱ.le (n : ℝ≥0)) _ (hτ n))).symm
-    simp_rw [he]
-    apply tendsto_integral_of_dominated_convergence (fun ω => ‖v ω‖)
-      (fun n => (hv.indicator ((ℱ.le (n : ℝ≥0)) _ (hτ n))).integrableOn.aestronglyMeasurable)
-      hv.norm.integrableOn
-    · intro n
-      exact Eventually.of_forall fun ω => norm_indicator_le_norm_self _ _
-    · apply Eventually.of_forall
-      intro ω
-      apply (sourceStop_indicator_tendsto (τ ω) (v ω) (hzero ω)).congr'
-      apply Eventually.of_forall
-      intro n
-      by_cases hn : τ ω ≤ (n : WithTop ℝ≥0) <;> simp [Set.indicator, hn]
-  exact le_of_tendsto_of_tendsto' (hlim f hf (fun ω ht => (hz ω ht).1))
-    (hlim g hg (fun ω ht => (hz ω ht).2)) hset
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal
-namespace SLEExactGauge
-
-lemma source_condExp_le_of_setIntegral_le
-    {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
-    {μ : Measure Ω} [IsFiniteMeasure μ] (hm : m ≤ mΩ)
-    {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
-    (hgm : StronglyMeasurable[m] g)
-    (h : ∀ A : Set Ω, MeasurableSet[m] A → (∫ ω in A, f ω ∂μ) ≤ ∫ ω in A, g ω ∂μ) :
-    μ[f | m] ≤ᵐ[μ] g := by
-  have : IsFiniteMeasure (μ.trim hm) := isFiniteMeasure_trim hm
-  apply ae_of_ae_trim hm
-  apply ae_le_of_forall_setIntegral_le
-    (Integrable.trim hm integrable_condExp stronglyMeasurable_condExp)
-    (Integrable.trim hm hg hgm)
-  intro A hA _
-  rw [← setIntegral_trim hm stronglyMeasurable_condExp hA,
-    ← setIntegral_trim hm hgm hA, setIntegral_condExp hm hf hA]
-  exact h A hA
-
-end SLEExactGauge
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal NNReal
-namespace SLEExactGauge
-lemma source_unbounded_conditional_bound_on
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    (ℱ : Filtration ℝ≥0 mΩ) {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime ℱ τ)
-    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A)
-    (hAfin : ∀ ω ∈ A, τ ω ≠ ⊤)
-    (f : Ω → ℝ) (hf : Integrable f μ) (b : ℝ)
-    (hloc : ∀ (n : ℕ) (E : Set Ω),
-      MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace] E →
-      E ⊆ A ∩ {ω | τ ω ≤ (n : ℝ≥0)} →
-      (∫ ω in E, f ω ∂μ) ≤ ∫ _ in E, b ∂μ) :
-    ∀ᵐ ω ∂μ, ω ∈ A → μ[f | hτ.measurableSpace] ω ≤ b := by
-  have hAm : MeasurableSet A := hτ.measurableSpace_le _ hA
-  have hfi : Integrable (A.indicator f) μ := hf.indicator hAm
-  have hbi : Integrable (A.indicator (fun _ => b)) μ := (integrable_const _).indicator hAm
-  have hbound : μ[A.indicator f | hτ.measurableSpace] ≤ᵐ[μ] A.indicator (fun _ => b) := by
-    apply source_condExp_le_of_setIntegral_le hτ.measurableSpace_le hfi hbi
-      (stronglyMeasurable_const.indicator hA)
-    intro S hS
-    apply source_unbounded_setIntegral_le ℱ hτ _ _ hfi hbi ?_ ?_ S hS
-    · intro ω ht
-      have hnot : ω ∉ A := fun hω => hAfin ω hω ht
-      simp only [Set.indicator_of_notMem hnot, and_self]
-    · intro n E hE hτE
-      have hen (ω : Ω) (hω : ω ∈ E) : τ ω ≤ (n : ℝ≥0) := by
-        rw [hτE ω hω,sourceStopTruncate_coe]
-        exact min_le_right _ _
-      have hAn : MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace]
-          (A ∩ {ω | τ ω ≤ (n : ℝ≥0)}) := by
-        have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ (n : ℝ≥0)) A hA
-        convert hh using 1
-        congr 1
-        funext ω
-        exact sourceStopTruncate_coe τ n ω
-      have heq : E ∩ A = E ∩ (A ∩ {ω | τ ω ≤ (n : ℝ≥0)}) := by
-        ext ω
-        exact ⟨fun h => ⟨h.1,h.2,hen ω h.1⟩,fun h => ⟨h.1,h.2.1⟩⟩
-      have hEA : MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace] (E ∩ A) := by
-        rw [heq]
-        exact hE.inter hAn
-      rw [setIntegral_indicator hAm,setIntegral_indicator hAm]
-      exact hloc n (E ∩ A) hEA (fun ω hω => ⟨hω.2,hen ω hω.1⟩)
-  filter_upwards [hbound,condExp_indicator hf hA] with ω hb he
-  intro hω
-  rw [he,Set.indicator_of_mem hω,Set.indicator_of_mem hω] at hb
-  exact hb
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance physicalClosedPrefixTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance physicalClosedPrefixTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-lemma source_physical_closed_prefix_isClosed (z : ℂ) (R : ℝ) (v : ℝ≥0) :
-    IsClosed {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t ∈ Icc (0 : ℝ≥0) v,
-      R ≤ dist (p.1 (p.2+t)) z} := by
-  let E := {p : (C(ℝ≥0,ℂ) × ℝ≥0) × Icc (0 : ℝ≥0) v |
-    R ≤ dist (p.1.1 (p.1.2+p.2)) z}
-  have he : IsClosed E := by
-    apply isClosed_le continuous_const
-    exact (continuous_eval.comp ((continuous_fst.fst).prodMk
-      (continuous_fst.snd.add (continuous_subtype_val.comp continuous_snd)))).dist continuous_const
-  have hc := isClosedMap_fst_of_compactSpace E he
-  convert hc using 1
-  ext p
-  constructor
-  · rintro ⟨t,ht,he⟩
-    exact ⟨(p,⟨t,ht⟩),he,rfl⟩
-  · rintro ⟨⟨p',t⟩,ht,rfl⟩
-    exact ⟨t,t.2,ht⟩
-
-lemma source_physical_closed_exit_measurableSet (z : ℂ) (R : ℝ) :
-    MeasurableSet {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
-      z ∈ survivingDomain p.1 (p.2+t) ∧ R ≤ dist (p.1 (p.2+t)) z} := by
-  have heq : {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
-      z ∈ survivingDomain p.1 (p.2+t) ∧ R ≤ dist (p.1 (p.2+t)) z} =
-      {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ q : ℚ,
-        z ∈ survivingDomain p.1 (p.2+(q:ℝ).toNNReal) ∧
-        ∃ t ∈ Icc (0 : ℝ≥0) (q:ℝ).toNNReal, R ≤ dist (p.1 (p.2+t)) z} := by
-    ext p
-    constructor
-    · rintro ⟨t,hl,he⟩
-      have ho : IsOpen {u : ℝ≥0 | z ∈ survivingDomain p.1 (p.2+u)} :=
-        source_survival_open.preimage
-          ((continuous_const.prodMk (continuous_const.add continuous_id)).prodMk continuous_const)
-      obtain ⟨b,htb,hb⟩ := exists_Ico_subset_of_mem_nhds (ho.mem_nhds hl) ⟨t+1,by simp⟩
-      obtain ⟨q,htq,hqb⟩ := exists_rat_btwn (show (t:ℝ) < b from htb)
-      have hqn : 0 ≤ (q:ℝ) := le_trans t.coe_nonneg htq.le
-      have htu : t ≤ (q:ℝ).toNNReal := by
-        change (t:ℝ) ≤ ((q:ℝ).toNNReal:ℝ)
-        rw [Real.coe_toNNReal _ hqn]
-        exact htq.le
-      have hub : (q:ℝ).toNNReal < b := by
-        change ((q:ℝ).toNNReal:ℝ) < (b:ℝ)
-        rw [Real.coe_toNNReal _ hqn]
-        exact hqb
-      exact ⟨q,hb ⟨htu,hub⟩,t,⟨bot_le,htu⟩,he⟩
-    · rintro ⟨q,hl,t,ht,he⟩
-      refine ⟨t,?_,he⟩
-      exact survivingDomain_antitone p.1 (by simpa only [add_comm] using add_le_add_left ht.2 p.2) hl
-  rw [heq]
-  simp only [ofPred_exists]
-  apply MeasurableSet.iUnion
-  intro q
-  have ho : IsOpen {p : C(ℝ≥0,ℂ) × ℝ≥0 |
-      z ∈ survivingDomain p.1 (p.2+(q:ℝ).toNNReal)} :=
-    source_survival_open.preimage
-      ((continuous_fst.prodMk (continuous_snd.add continuous_const)).prodMk continuous_const)
-  exact ho.measurableSet.inter (source_physical_closed_prefix_isClosed z R (q:ℝ).toNNReal).measurableSet
-
-lemma source_physical_closed_exit_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (gp : Ω → C(ℝ≥0,ℂ)) (hgp : Measurable gp)
-    (τ : Ω → ℝ≥0) (hτ : Measurable τ) (z : ℂ) (R : ℝ) :
-    MeasurableSet {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (gp ω) (τ ω+t) ∧
-      R ≤ dist (gp ω (τ ω+t)) z} := by
-  have hp : Measurable (fun ω => (gp ω,τ ω)) := hgp.prodMk hτ
-  have hh := (source_physical_closed_exit_measurableSet z R).preimage hp
-  simpa only [Set.preimage_ofPred_eq] using hh
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped ENNReal NNReal
-namespace SLEExactGauge
-lemma source_physical_slack_large (C c L : ℝ) (_hC : 0 ≤ C) (hL : 0 < L) :
-    C * (L/2)^(-c) ≤ ((C+1)*2^c)*L^(-c) := by
-  have he : (L/2)^(-c) = 2^c*L^(-c) := by
-    rw [Real.div_rpow hL.le (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2)]
-    simp only [div_inv_eq_mul,mul_comm]
-  rw [he]
-  have hh := Real.rpow_nonneg (by norm_num : (0:ℝ) ≤ 2) c
-  have hh' := Real.rpow_nonneg hL.le (-c)
-  nlinarith [mul_nonneg hh hh']
-
-lemma source_physical_slack_small (C c L : ℝ) (hC : 0 ≤ C) (hc : 0 < c)
-    (hL : 0 < L) (hL2 : L ≤ 2) :
-    1 ≤ ((C+1)*2^c)*L^(-c) := by
-  have h2 : (0:ℝ) < 2^c := Real.rpow_pos_of_pos (by norm_num) _
-  have hrec : 2^(-c) ≤ L^(-c) := Real.rpow_le_rpow_of_nonpos hL hL2 (by linarith)
-  have he : (2:ℝ)^c * 2^(-c) = 1 := by
-    rw [← Real.rpow_add (by norm_num : (0:ℝ) < 2)]
-    simp
-  have hh := mul_le_mul_of_nonneg_left hrec h2.le
-  rw [he] at hh
-  have hp : 0 ≤ 2^c * L^(-c) := le_trans (by norm_num) hh
-  nlinarith [mul_nonneg hC hp]
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance physicalDiskTargetTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance physicalDiskTargetTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-local instance physicalDiskTargetDriverMeasurableSpace : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance physicalDiskTargetDriverBorelSpace : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
- 
-
-def SourcePhysicalDiskTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_m : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
-  ∀ (hm : ∀ t, Measurable (B t)) (_hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (_hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)) (τ : Ω → WithTop ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) τ)
-    (z : ℂ) (r : ℝ), 0 < r →
-  ∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
-    (∀ ω ∈ A, ∃ T s : ℝ≥0, τ ω = T ∧ s ≤ T ∧
-      Metric.closedBall z r ⊆ survivingDomain (γ ω) s ∧
-      dist (γ ω T) z = r ∧
-      (∀ t ∈ Ico s T, dist (γ ω t) z ≠ r) ∧
-      z ∈ survivingDomain (γ ω) T) →
-  ∀ L : ℝ, 1 ≤ L →
-    ∀ᵐ ω ∂P, ω ∈ A →
-      P[(fun ω => if τ ω ≠ ⊤ ∧ ∃ t : ℝ≥0,
-        z ∈ survivingDomain (γ ω) ((τ ω).untopD 0+t) ∧
-        L*r ≤ dist (γ ω ((τ ω).untopD 0+t)) z
-        then (1 : ℝ) else 0) | hτ.measurableSpace] ω ≤ C*L^(-c)
-
-theorem sourcePhysicalDisk_proved : SourcePhysicalDiskTarget := by
-  intro κ hκ hκ8
-  obtain ⟨C,c,hC,hc,hbound⟩ := sourceFinitePhysicalDisk_proved κ hκ hκ8
-  refine ⟨(C+1)*2^c,c,mul_pos (by linarith) (Real.rpow_pos_of_pos (by norm_num) _),hc,?_⟩
-  intro Ω m P hP B hB hm hBc γ G F hw τ hτ z r hr A hA hgeom L hL
-  let : IsProbabilityMeasure P := hP
-  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let gp := sourceTracePath γ (fun ω => (hw ω).1)
-  obtain ⟨Φ,hΦm,hΦeq,hΦ⟩ := sourceStoppedTraceLaw κ (by linarith) hκ8 B hB hm hBc γ G F hw
-  have hgm : Measurable gp := by
-    have he : gp = fun ω => Φ (sourceDriverPath B hBc ω) := funext fun ω => (hΦeq ω).symm
-    rw [he]
-    exact hΦm.comp (source_path_measurable _ hm)
-  let H := {ω | τ ω ≠ ⊤ ∧ ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) ((τ ω).untopD 0+t) ∧
-    L*r ≤ dist (γ ω ((τ ω).untopD 0+t)) z}
-  let f : Ω → ℝ := H.indicator (fun _ => 1)
-  have hH : MeasurableSet H := by
-    have hfin : MeasurableSet {ω | τ ω ≠ ⊤} := (measurableSet_eq_fun hτ.measurable' measurable_const).compl
-    have he := source_physical_closed_exit_measurable gp hgm (fun ω => (τ ω).untopD 0)
-      (hτ.measurable'.untopD 0) z (L*r)
-    exact hfin.inter he
-  have hf : Integrable f P := (integrable_const (1:ℝ)).indicator hH
-  have hfeq : f = (fun ω => if ω ∈ H then (1:ℝ) else 0) := by
-    funext ω
-    exact Set.indicator_apply _ _ _
-  suffices hh : ∀ᵐ ω ∂P, ω ∈ A → P[f | hτ.measurableSpace] ω ≤ ((C+1)*2^c)*L^(-c) by
-    simpa only [hfeq,H,Set.mem_ofPred_eq] using hh
-  by_cases hL2 : L ≤ 2
-  · have hfle : f ≤ᵐ[P] fun _ => (1:ℝ) := Eventually.of_forall fun ω => by
-      dsimp [f,Set.indicator]; split_ifs <;> norm_num
-    have hmono := condExp_mono (m := hτ.measurableSpace) hf (integrable_const (1:ℝ)) hfle
-    have he := condExp_const (μ := P) hτ.measurableSpace_le (1:ℝ)
-    filter_upwards [hmono] with ω hω
-    intro _
-    rw [he] at hω
-    exact hω.trans (source_physical_slack_small C c L hC.le hc (by linarith) hL2)
-  have hAfin : ∀ ω ∈ A, τ ω ≠ ⊤ := by
-    intro ω hω
-    obtain ⟨T,s,hT,_⟩ := hgeom ω hω
-    rw [hT]
-    exact WithTop.coe_ne_top
-  apply source_unbounded_conditional_bound_on ℱ hτ A hA hAfin f hf _
-  intro n E hE hEA
-  let ρ := sourceStopTruncate τ n
-  have hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0)) := sourceStopTruncate_stopping ℱ hτ n
-  have heq (ω : Ω) (hω : ω ∈ E) : τ ω = (ρ ω : WithTop ℝ≥0) := by
-    rw [sourceStopTruncate_coe,min_eq_left (hEA hω).2]
-  have hgeomE : ∀ ω ∈ E, ∃ s : ℝ≥0, s ≤ ρ ω ∧
-      Metric.closedBall z r ⊆ survivingDomain (γ ω) s ∧ dist (γ ω (ρ ω)) z = r ∧
-      (∀ t ∈ Ico s (ρ ω), dist (γ ω t) z ≠ r) ∧ z ∈ survivingDomain (γ ω) (ρ ω) := by
-    intro ω hω
-    obtain ⟨T,s,hT,hs,hfree,hhit,hfirst,hz⟩ := hgeom ω (hEA hω).1
-    have hTr : T = ρ ω := WithTop.coe_injective (hT.symm.trans (heq ω hω))
-    exact ⟨s,hTr ▸ hs,hfree,hTr ▸ hhit,hTr ▸ hfirst,hTr ▸ hz⟩
-  have hLhalf : 1 ≤ L/2 := by linarith
-  have hlocal := hbound Ω m P hP B hB hm hBc γ G F hw ρ hρ z r hr E hE hgeomE (L/2) hLhalf
-  let K := {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) (ρ ω+t) ∧
-    (L/2)*r < dist (γ ω (ρ ω+t)) z}
-  let g : Ω → ℝ := K.indicator (fun _ => 1)
-  have hK : MeasurableSet K := source_physical_exit_measurable gp hgm ρ
-    (stopping_time_measurable_nnreal hρ) z ((L/2)*r)
-  have hg : Integrable g P := (integrable_const (1:ℝ)).indicator hK
-  have hgeq : g = (fun ω => if ω ∈ K then (1:ℝ) else 0) := by
-    funext ω
-    exact Set.indicator_apply _ _ _
-  have hfg : ∀ ω ∈ E, f ω ≤ g ω := by
-    intro ω hω
-    by_cases hh : ω ∈ H
-    · obtain ⟨hfin,t,ht,hd⟩ := (show τ ω ≠ ⊤ ∧ ∃ t : ℝ≥0,
-        z ∈ survivingDomain (γ ω) ((τ ω).untopD 0+t) ∧
-        L*r ≤ dist (γ ω ((τ ω).untopD 0+t)) z from hh)
-      have hτr : (τ ω).untopD 0 = ρ ω := by rw [heq ω hω,WithTop.untopD_coe]
-      have hk : ω ∈ K := ⟨t,by rwa [hτr] at ht,by rw [hτr] at hd; nlinarith⟩
-      simp only [f,g,Set.indicator_of_mem hh,Set.indicator_of_mem hk,le_refl]
-    · simp only [f,Set.indicator_of_notMem hh]
-      dsimp [g,Set.indicator]; split_ifs <;> norm_num
-  have hEm : MeasurableSet E := hρ.measurableSpace_le _ hE
-  have hi1 : (∫ ω in E, f ω ∂P) ≤ ∫ ω in E, g ω ∂P :=
-    setIntegral_mono_on hf.integrableOn hg.integrableOn hEm hfg
-  have hi2 : (∫ ω in E, g ω ∂P) ≤ ∫ ω in E, C*(L/2)^(-c) ∂P := by
-    rw [← setIntegral_condExp hρ.measurableSpace_le hg hE]
-    apply setIntegral_mono_on_ae integrable_condExp.integrableOn (integrable_const _).integrableOn hEm
-    filter_upwards [hlocal] with ω hω
-    simpa only [hgeq,K,Set.mem_ofPred_eq] using hω
-  have hi3 : (∫ ω in E, C*(L/2)^(-c) ∂P) ≤ ∫ ω in E, ((C+1)*2^c)*L^(-c) ∂P := by
-    exact setIntegral_mono_on (integrable_const _).integrableOn (integrable_const _).integrableOn hEm
-      (fun _ _ => source_physical_slack_large C c L hC.le (by linarith))
-  exact hi1.trans (hi2.trans hi3)
-
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.ChartMeasurability
+import OAI.Probability.SLE.InteriorDiskProof
+import OAI.Probability.SLE.PhysicalTransport
+
+namespace OAI
+
+/-! Physical Kernel Utilities for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance physicalDiskKernelTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance physicalDiskKernelTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+local instance physicalDiskKernelChartMeasurableSpace : MeasurableSpace C(upperHalfPlane,ℂ) := borel _
+local instance physicalDiskKernelChartBorelSpace : BorelSpace C(upperHalfPlane,ℂ) := ⟨rfl⟩
+
+def sourcePhysicalDiskKernel (p : SourceDiskParameter × C(ℝ≥0,ℂ)) : ℝ :=
+  if ∃ t : ℝ≥0, p.1.1.2 ∈ survivingDomain p.2 t ∧
+    ∃ h : p.2 t ∈ upperHalfPlane, p.1.2.2 < ‖p.1.1.1 ⟨p.2 t,h⟩-p.1.2.1‖ then 1 else 0
+
+lemma sourcePhysicalDiskKernel_measurable : Measurable sourcePhysicalDiskKernel :=
+  Measurable.ite source_physical_disk_kernel_open.measurableSet measurable_const measurable_const
+
+lemma sourcePhysicalDiskKernel_norm_le (p : SourceDiskParameter × C(ℝ≥0,ℂ)) :
+    ‖sourcePhysicalDiskKernel p‖ ≤ 1 := by
+  unfold sourcePhysicalDiskKernel
+  split_ifs <;> norm_num
+
+lemma source_physical_exit_open (z : ℂ) (R : ℝ) :
+    IsOpen {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
+      z ∈ survivingDomain p.1 (p.2+t) ∧ R < dist (p.1 (p.2+t)) z} := by
+  have ht (t : ℝ≥0) : IsOpen {p : C(ℝ≥0,ℂ) × ℝ≥0 |
+      z ∈ survivingDomain p.1 (p.2+t) ∧ R < dist (p.1 (p.2+t)) z} := by
+    have hc : Continuous (fun p : C(ℝ≥0,ℂ) × ℝ≥0 => (p.1,p.2+t)) :=
+      continuous_fst.prodMk (continuous_snd.add continuous_const)
+    exact (source_survival_open.preimage (hc.prodMk continuous_const)).inter
+      (isOpen_lt continuous_const ((continuous_eval.comp hc).dist continuous_const))
+  simpa only [ofPred_exists] using isOpen_iUnion ht
+
+lemma source_restarted_interior_dense
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgm : Measurable (sourceTracePath γ (fun ω => (hw ω).1)))
+    (β : Ω → C(ℝ≥0,ℂ)) (hβm : Measurable β)
+    (hlaw : P.map β = P.map (sourceTracePath γ (fun ω => (hw ω).1))) :
+    ∀ᵐ ω ∂P, SourcePathInteriorDense (β ω) := by
+  apply (ae_map_iff hβm.aemeasurable source_path_interior_dense_measurable).mp
+  rw [hlaw]
+  apply (ae_map_iff hgm.aemeasurable source_path_interior_dense_measurable).mpr
+  exact Eventually.of_forall fun ω => sourceInteriorTraceDensity_proved
+    (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)
+    (continuous_const.mul (hc ω)) (hw ω)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma source_firsthit_inverse_access
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hUc : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (s τ : ℝ≥0) (z : ℂ) (r : ℝ) (hr : 0 < r)
+    (hfree : Metric.closedBall z r ⊆ survivingDomain γ s) (hsτ : s ≤ τ)
+    (hhit : dist (γ τ) z = r)
+    (hfirst : ∀ t ∈ Ico s τ, dist (γ t) z ≠ r)
+    (hz : z ∈ survivingDomain γ τ) :
+    ∃ η : ℝ → ℂ, ContinuousOn η (Icc 0 1) ∧ η 0 = 0 ∧
+      η 1 = G τ z - (U τ : ℂ) ∧
+      (∀ v ∈ Ioc 0 1, 0 < (η v).im) ∧
+      (∀ v ∈ Ioc 0 1, ‖F τ (η v + (U τ : ℂ))-z‖ ≤ r) := by
+  let α : ℝ → ℂ := fun v => (1-v) • γ τ + v • z
+  let f : ℝ → ℂ := fun v => G τ (α v) - (U τ : ℂ)
+  let η := Function.update f 0 0
+  have hα : Continuous α := by dsimp [α]; fun_prop
+  have hαD := (sourceFirstHitPhysicalAccess γ h.1 s τ z r hr hfree hsτ hhit hfirst hz).2.2
+  have hf : ContinuousOn f (Icc 0 1 \ {0}) := by
+    refine ((h.2.2.2.1 τ).continuousOn.comp hα.continuousOn ?_).sub continuousOn_const
+    intro v hv
+    exact hαD v ⟨lt_of_le_of_ne hv.1.1 (Ne.symm hv.2),hv.1.2⟩
+  have hη : ContinuousOn η (Icc 0 1) := by
+    apply continuousOn_update_iff.mpr
+    refine ⟨hf,fun _ => ?_⟩
+    apply (sourceFirstHitChartAccess_proved U γ G F hUc h s τ z r hr hfree hsτ hhit hfirst hz).mono_left
+    apply nhdsWithin_mono
+    intro v hv
+    exact lt_of_le_of_ne hv.1.1 (Ne.symm hv.2)
+  have hηv (v : ℝ) (hv : v ∈ Ioc 0 1) : η v = f v :=
+    Function.update_of_ne (ne_of_gt hv.1) _ _
+  refine ⟨η,hη,by simp [η],?_,?_,?_⟩
+  · rw [hηv 1 (by norm_num)]
+    simp [f,α]
+  · intro v hv
+    rw [hηv v hv]
+    simpa only [f,α,upperHalfPlane,Set.mem_ofPred_eq,sub_im,ofReal_im,sub_zero] using h.2.2.2.2.2.1 τ (hαD v hv)
+  · intro v hv
+    rw [hηv v hv]
+    simp only [f,sub_add_cancel,h.2.2.2.2.2.2.2.1 τ (α v) (hαD v hv)]
+    simpa only [dist_eq_norm] using
+      (show dist (α v) z < r from radialAccess_mem_ball (γ τ) z r hr hhit v hv).le
+
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+lemma source_physical_exit_interior_inclusion
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (β : C(ℝ≥0,ℂ))
+    (hdense : SourcePathInteriorDense β)
+    (hid : ∀ t : ℝ≥0, β t ∈ upperHalfPlane → γ (T+t) = F T (β t+(U T:ℂ)))
+    (z : ℂ) (R : ℝ) (hR : dist (γ T) z ≤ R) :
+    (∃ t : ℝ≥0, z ∈ survivingDomain γ (T+t) ∧ R < dist (γ (T+t)) z) →
+    ∃ t : ℝ≥0, G T z-(U T:ℂ) ∈ survivingDomain β t ∧
+      ∃ _ht : β t ∈ upperHalfPlane, R < ‖F T (β t+(U T:ℂ))-z‖ := by
+  rintro ⟨t,hl,he⟩
+  have ht : 0 < t := by
+    by_contra hn
+    have ht0 : t = 0 := le_antisymm (not_lt.mp hn) bot_le
+    simp only [ht0,add_zero] at he
+    exact (not_lt_of_ge hR) he
+  have hc : Continuous (fun u : ℝ≥0 => dist (γ (T+u)) z) :=
+    (h.1.comp (continuous_const.add continuous_id)).dist continuous_const
+  have hevent : {u : ℝ≥0 | R < dist (γ (T+u)) z} ∈ 𝓝 t :=
+    (isOpen_lt continuous_const hc).mem_nhds he
+  obtain ⟨b,hbt,hb⟩ := exists_Ioc_subset_of_mem_nhds hevent ⟨0,ht⟩
+  obtain ⟨u,hu,hupos⟩ := hdense b t hbt
+  have hlu : z ∈ survivingDomain γ (T+u) :=
+    survivingDomain_antitone γ (by simpa only [add_comm] using add_le_add_left hu.2.le T) hl
+  refine ⟨u,source_stopped_live_transfer h T u β (fun v _ => hid v) z hlu,hupos,?_⟩
+  rw [← hid u hupos,← dist_eq_norm]
+  exact hb ⟨hu.1,hu.2.le⟩
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance finitePhysicalDiskTargetTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance finitePhysicalDiskTargetTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+local instance finitePhysicalDiskTargetDriverMeasurableSpace : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance finitePhysicalDiskTargetDriverBorelSpace : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance finitePhysicalDiskTargetChartMeasurableSpace : MeasurableSpace C(upperHalfPlane,ℂ) := borel _
+local instance finitePhysicalDiskTargetChartBorelSpace : BorelSpace C(upperHalfPlane,ℂ) := ⟨rfl⟩
+ 
+
+def SourceFinitePhysicalDiskTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_m : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
+  ∀ (hm : ∀ t, Measurable (B t)) (_hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (_hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) (z : ℂ) (r : ℝ), 0 < r →
+  ∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
+    (∀ ω ∈ A, ∃ s : ℝ≥0, s ≤ τ ω ∧
+      Metric.closedBall z r ⊆ survivingDomain (γ ω) s ∧
+      dist (γ ω (τ ω)) z = r ∧
+      (∀ t ∈ Ico s (τ ω), dist (γ ω t) z ≠ r) ∧
+      z ∈ survivingDomain (γ ω) (τ ω)) →
+  ∀ L : ℝ, 1 ≤ L →
+    ∀ᵐ ω ∂P, ω ∈ A →
+      P[(fun ω => if ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) (τ ω+t) ∧
+        L*r < dist (γ ω (τ ω+t)) z then (1 : ℝ) else 0) | hτ.measurableSpace] ω ≤ C*L^(-c)
+
+lemma source_physical_kernel_mean_le
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (gp : Ω → C(ℝ≥0,ℂ)) (hgm : Measurable gp)
+    (Y : SourceDiskParameter) (Ψ : ℂ → ℂ)
+    (hΨ : ∀ w : upperHalfPlane, Y.1.1 w = Ψ w)
+    (b : ℝ) (hb : 0 ≤ b)
+    (hbound : P {v | ∃ t : ℝ≥0, Y.1.2 ∈ survivingDomain (gp v) t ∧
+      0 < (gp v t).im ∧ Y.2.2 < ‖Ψ (gp v t)-Y.2.1‖} ≤ ENNReal.ofReal b) :
+    (∫ p, sourcePhysicalDiskKernel (Y,p) ∂P.map gp) ≤ b := by
+  let S : Set Ω := {v | ∃ t : ℝ≥0, Y.1.2 ∈ survivingDomain (gp v) t ∧
+      0 < (gp v t).im ∧ Y.2.2 < ‖Ψ (gp v t)-Y.2.1‖}
+  have he (v : Ω) : sourcePhysicalDiskKernel (Y,gp v) = S.indicator (fun _ => (1:ℝ)) v := by
+    unfold sourcePhysicalDiskKernel
+    have hi : (∃ t : ℝ≥0, Y.1.2 ∈ survivingDomain (gp v) t ∧
+        ∃ ht : gp v t ∈ upperHalfPlane,
+          Y.2.2 < ‖Y.1.1 ⟨gp v t,ht⟩-Y.2.1‖) ↔ v ∈ S := by
+      constructor
+      · rintro ⟨t,ht,hi,hf⟩
+        exact ⟨t,ht,hi,by simpa only [hΨ] using hf⟩
+      · rintro ⟨t,ht,hi,hf⟩
+        exact ⟨t,ht,hi,by simpa only [hΨ] using hf⟩
+    simp only [hi,Set.indicator_apply]
+  have hfun : Measurable (fun p : C(ℝ≥0,ℂ) => sourcePhysicalDiskKernel (Y,p)) :=
+    sourcePhysicalDiskKernel_measurable.comp (measurable_const.prodMk measurable_id)
+  have hS : MeasurableSet S := by
+    have hh := measurableSet_lt (measurable_const (a := (0:ℝ))) (hfun.comp hgm)
+    convert hh using 1
+    ext v
+    change v ∈ S ↔ 0 < sourcePhysicalDiskKernel (Y,gp v)
+    rw [he]
+    by_cases hv : v ∈ S <;> simp [hv]
+  have hKi : (∫ p, sourcePhysicalDiskKernel (Y,p) ∂P.map gp) = (P S).toReal := by
+    rw [integral_map hgm.aemeasurable hfun.aestronglyMeasurable]
+    simp_rw [he]
+    simp only [integral_indicator_const _ hS,smul_eq_mul,mul_one,measureReal_def]
+  rw [hKi]
+  exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr hbound |>.trans
+    (le_of_eq (ENNReal.toReal_ofReal hb))
+
+lemma source_firsthit_kernel_mean_le
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (gp : Ω → C(ℝ≥0,ℂ)) (hgm : Measurable gp)
+    (U : ℝ≥0 → ℝ) (hU : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) (hw : IsCapacityTwoWitness U γ G F)
+    (s T : ℝ≥0) (z : ℂ) (r : ℝ) (hr : 0 < r)
+    (hs : s ≤ T) (hfree : Metric.closedBall z r ⊆ survivingDomain γ s)
+    (hhit : dist (γ T) z = r)
+    (hfirst : ∀ t ∈ Ico s T, dist (γ t) z ≠ r) (hz : z ∈ survivingDomain γ T)
+    (ψ : C(upperHalfPlane,ℂ)) (hψ : ∀ w, ψ w = F T (w+(U T:ℂ)))
+    (L b : ℝ) (hb : 0 ≤ b)
+    (hbound : ∀ Ψ : ℂ → ℂ, DifferentiableOn ℂ Ψ upperHalfPlane → InjOn Ψ upperHalfPlane →
+      ∀ η : ℝ → ℂ, ContinuousOn η (Icc 0 1) → η 0 = 0 → Ψ (η 1) = z →
+      (∀ v ∈ Ioc 0 1, 0 < (η v).im) → (∀ v ∈ Ioc 0 1, ‖Ψ (η v)-z‖ ≤ r) →
+      P {v | ∃ t : ℝ≥0, η 1 ∈ survivingDomain (gp v) t ∧
+        0 < (gp v t).im ∧ L*r < ‖Ψ (gp v t)-z‖} ≤ ENNReal.ofReal b) :
+    (∫ p, sourcePhysicalDiskKernel (((ψ,G T z-(U T:ℂ)),(z,L*r)),p) ∂P.map gp) ≤ b := by
+  let Ψ : ℂ → ℂ := fun w => F T (w+(U T:ℂ))
+  have hmap : MapsTo (fun w : ℂ => w+(U T:ℂ)) upperHalfPlane upperHalfPlane := by
+    intro w hw
+    simpa [upperHalfPlane] using hw
+  have hΨd : DifferentiableOn ℂ Ψ upperHalfPlane :=
+    (hw.2.2.2.2.1 T).comp (differentiable_id.add_const _).differentiableOn hmap
+  have hΨi : InjOn Ψ upperHalfPlane := by
+    intro a ha b hb he
+    have ha' := hw.2.2.2.2.2.2.2.2.1 T (a+(U T:ℂ)) (hmap ha)
+    have hb' := hw.2.2.2.2.2.2.2.2.1 T (b+(U T:ℂ)) (hmap hb)
+    have hh := congrArg (G T) he
+    dsimp only [Ψ] at hh
+    rw [ha',hb'] at hh
+    exact add_right_cancel hh
+  obtain ⟨η,hη,hη0,hη1,hηup,hηpath⟩ := source_firsthit_inverse_access
+    hU hw s T z r hr hfree hs hhit hfirst hz
+  have hq : Ψ (η 1) = z := by
+    rw [hη1]
+    simp only [Ψ,sub_add_cancel,hw.2.2.2.2.2.2.2.1 T z hz]
+  have hprob := hbound Ψ hΨd hΨi η hη hη0 hq hηup hηpath
+  apply source_physical_kernel_mean_le gp hgm _ Ψ hψ b hb
+  simpa only [hη1] using hprob
+
+lemma source_physical_exit_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (gp : Ω → C(ℝ≥0,ℂ)) (hgp : Measurable gp)
+    (τ : Ω → ℝ≥0) (hτ : Measurable τ) (z : ℂ) (R : ℝ) :
+    MeasurableSet {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (gp ω) (τ ω+t) ∧
+      R < dist (gp ω (τ ω+t)) z} := by
+  have hp : Measurable (fun ω => (gp ω,τ ω)) := hgp.prodMk hτ
+  have he : MeasurableSet {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
+      z ∈ survivingDomain p.1 (p.2+t) ∧ R < dist (p.1 (p.2+t)) z} :=
+    (source_physical_exit_open z R).measurableSet
+  have hh := he.preimage hp
+  simpa only [Set.preimage_ofPred_eq] using hh
+
+theorem sourceFinitePhysicalDisk_proved : SourceFinitePhysicalDiskTarget := by
+  intro κ hκ hκ8
+  obtain ⟨C,c,hC,hc,hbound⟩ := source_interior_disk_unnormalized κ hκ hκ8
+  refine ⟨C,c,hC,hc,?_⟩
+  intro Ω m P hP B hB hm hBc γ G F hw τ hτ z r hr A hA hgeom L hL
+  let : IsProbabilityMeasure P := hP
+  let gp := sourceTracePath γ (fun ω => (hw ω).1)
+  obtain ⟨Φ,hΦm,hΦeq,hΦ⟩ := sourceStoppedTraceLaw κ (by linarith) hκ8 B hB hm hBc γ G F hw
+  have hgm : Measurable gp := by
+    have he : gp = fun ω => Φ (sourceDriverPath B hBc ω) := funext fun ω => (hΦeq ω).symm
+    rw [he]
+    exact hΦm.comp (source_path_measurable _ hm)
+  let ν := P.map gp
+  let : IsProbabilityMeasure ν := inferInstance
+  have hγsle : IsOrdinaryChordalSLE κ γ P :=
+    ⟨fun t => (continuous_eval_const t).measurable.comp hgm,
+      B,hB,Eventually.of_forall fun ω => ⟨G ω,F ω,hw ω⟩⟩
+  obtain ⟨hlaw,hid⟩ := hΦ τ hτ
+  let rp := sourceRestartDriverPath B hBc τ
+  let β (ω : Ω) := Φ (rp ω)
+  have hrm : Measurable rp := source_path_measurable rp (fun t =>
+    (measurable_pi_apply t).comp (brownianRestart_measurable hm hBc (stopping_time_measurable_nnreal hτ)))
+  have hβm : Measurable β := hΦm.comp hrm
+  have hβlaw : P.map β = ν := by simpa only [Measure.restrict_univ,measure_univ,one_smul] using hlaw univ MeasurableSet.univ
+  have hdense : ∀ᵐ ω ∂P, SourcePathInteriorDense (β ω) :=
+    source_restarted_interior_dense B hBc κ γ G F hw hgm β hβm hβlaw
+  let ξ (ω : Ω) :=
+    (if z ∈ survivingDomain (γ ω) (τ ω) then G ω (τ ω) z else 0) -
+      (Real.sqrt κ * B (τ ω) ω : ℝ)
+  let Y (ω : Ω) : SourceDiskParameter :=
+    ((sourceStoppedInversePath B κ γ G F hw τ ω,ξ ω),(z,L*r))
+  have hξm : Measurable[hτ.measurableSpace] ξ := by
+    have hGm := source_killedChart_stopped_measurable hm hBc κ γ G F hw z τ hτ
+    have hBm := measurable_stoppedValue
+      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hBc) hτ
+    exact hGm.sub (Complex.measurable_ofReal.comp (measurable_const.mul hBm))
+  have hYm : Measurable[hτ.measurableSpace] Y :=
+    ((source_stopped_inversePath_measurable B hm hBc κ γ G F hw τ hτ).prodMk hξm).prodMk measurable_const
+  let H (ω : Ω) := sourcePhysicalDiskKernel (Y ω,β ω)
+  let K (ω : Ω) := ∫ p, sourcePhysicalDiskKernel (Y ω,p) ∂ν
+  have hHm : Measurable H := sourcePhysicalDiskKernel_measurable.comp
+    ((hYm.mono hτ.measurableSpace_le le_rfl).prodMk hβm)
+  have hHi : Integrable H P :=
+    (integrable_const (1:ℝ)).mono' hHm.aestronglyMeasurable
+      (Eventually.of_forall fun ω => sourcePhysicalDiskKernel_norm_le (Y ω,β ω))
+  have hcond : P[H | hτ.measurableSpace] =ᵐ[P] K :=
+    source_parameter_condExp hτ.measurableSpace_le hβm hYm hlaw sourcePhysicalDiskKernel
+      sourcePhysicalDiskKernel_measurable sourcePhysicalDiskKernel_norm_le
+  have hmean (ω : Ω) (hω : ω ∈ A) : K ω ≤ C*L^(-c) := by
+    obtain ⟨s,hs,hfree,hhit,hfirst,hz⟩ := hgeom ω hω
+    dsimp only [K,Y,ξ]
+    rw [ite_eq_left hz]
+    apply source_firsthit_kernel_mean_le gp hgm (fun t => Real.sqrt κ*B t ω)
+      (continuous_const.mul (hBc ω)) (γ ω) (G ω) (F ω) (hw ω)
+      s (τ ω) z r hr hs hfree hhit hfirst hz _ (fun _ => rfl)
+      L (C*L^(-c)) (mul_nonneg hC.le (Real.rpow_nonneg (by linarith) _))
+    intro Ψ hΨd hΨi η hη hη0 hq hηup hηpath
+    exact hbound Ω m P hP γ hγsle Ψ hΨd hΨi z r hr η 1 (by norm_num)
+      hη hη0 hq hηup hηpath L hL
+  let E : Set Ω := {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) (τ ω+t) ∧
+      L*r < dist (γ ω (τ ω+t)) z}
+  let f : Ω → ℝ := E.indicator (fun _ => 1)
+  have hE : MeasurableSet E :=
+    source_physical_exit_measurable gp hgm τ (stopping_time_measurable_nnreal hτ) z (L*r)
+  have hfi : Integrable f P := (integrable_const (1:ℝ)).indicator hE
+  have hAi : Integrable (A.indicator f) P := hfi.indicator (hτ.measurableSpace_le _ hA)
+  have hAH : Integrable (A.indicator H) P := hHi.indicator (hτ.measurableSpace_le _ hA)
+  have hdom : A.indicator f ≤ᵐ[P] A.indicator H := by
+    filter_upwards [hid,hdense] with ω hωid hωdense
+    by_cases hω : ω ∈ A
+    · simp only [Set.indicator_of_mem hω]
+      by_cases he : ω ∈ E
+      · obtain ⟨s,hs,hfree,hhit,hfirst,hz⟩ := hgeom ω hω
+        have hR : dist (γ ω (τ ω)) z ≤ L*r := by rw [hhit]; nlinarith
+        obtain ⟨t,ht,htH,hfar⟩ := source_physical_exit_interior_inclusion (hw ω) (τ ω) (β ω)
+          hωdense hωid z (L*r) hR he
+        have hkernel : H ω = 1 := by
+          apply ite_eq_left
+          refine ⟨t,?_,htH,?_⟩
+          · simpa only [Y,ξ,ite_eq_left hz] using ht
+          · exact hfar
+        simp only [f,Set.indicator_of_mem he,hkernel,le_refl]
+      · simp only [f,Set.indicator_of_notMem he]
+        dsimp [H,sourcePhysicalDiskKernel]
+        split_ifs <;> norm_num
+    · simp only [Set.indicator_of_notMem hω,le_refl]
+  have hmono := condExp_mono (m := hτ.measurableSpace) hAi hAH hdom
+  filter_upwards [hmono,condExp_indicator hfi hA,condExp_indicator hHi hA,hcond] with ω hm hf hH hK
+  intro hω
+  rw [hf,hH,Set.indicator_of_mem hω,Set.indicator_of_mem hω,hK] at hm
+  exact hm.trans (hmean ω hω)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+noncomputable def sourceStopTruncate {Ω : Type*} (τ : Ω → WithTop ℝ≥0)
+    (t : ℝ≥0) (ω : Ω) : ℝ≥0 := (min (τ ω) t).untopD 0
+
+lemma sourceStopTruncate_coe {Ω : Type*} (τ : Ω → WithTop ℝ≥0)
+    (t : ℝ≥0) (ω : Ω) :
+    (sourceStopTruncate τ t ω : WithTop ℝ≥0) = min (τ ω) t := by
+  have hn : min (τ ω) (t : WithTop ℝ≥0) ≠ ⊤ :=
+    ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_right _ _)
+  obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp hn
+  change (((min (τ ω) (t : WithTop ℝ≥0)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = _
+  rw [← hv, WithTop.untopD_coe]
+
+lemma sourceStopTruncate_stopping {Ω : Type*} [MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›) {τ : Ω → WithTop ℝ≥0}
+    (hτ : IsStoppingTime ℱ τ) (t : ℝ≥0) :
+    IsStoppingTime ℱ (fun ω => (sourceStopTruncate τ t ω : WithTop ℝ≥0)) := by
+  simpa only [sourceStopTruncate_coe] using hτ.min_const t
+
+lemma sourceStop_indicator_tendsto (τ : WithTop ℝ≥0) (v : ℝ)
+    (hzero : τ = ⊤ → v = 0) :
+    Tendsto (fun n : ℕ => if τ ≤ (n : ℝ≥0) then v else 0) atTop (𝓝 v) := by
+  by_cases ht : τ = ⊤
+  · simp only [ht, hzero ht, ite_self]
+    exact tendsto_const_nhds
+  obtain ⟨t, rfl⟩ := WithTop.ne_top_iff_exists.mp ht
+  apply tendsto_const_nhds.congr'
+  filter_upwards [eventually_ge_atTop (⌈(t : ℝ)⌉₊)] with n hn
+  have htn : t ≤ (n : ℝ≥0) := by exact_mod_cast (Nat.le_of_ceil_le hn)
+  simp only [WithTop.coe_le_coe, ite_eq_left htn]
+
+lemma source_unbounded_setIntegral_le
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    (ℱ : Filtration ℝ≥0 mΩ) {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime ℱ τ)
+    (f g : Ω → ℝ) (hf : Integrable f μ) (hg : Integrable g μ)
+    (hz : ∀ ω, τ ω = ⊤ → f ω = 0 ∧ g ω = 0)
+    (hloc : ∀ (n : ℕ) (A : Set Ω),
+      MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace] A →
+      (∀ ω ∈ A, τ ω = (sourceStopTruncate τ n ω : WithTop ℝ≥0)) →
+      (∫ ω in A, f ω ∂μ) ≤ ∫ ω in A, g ω ∂μ)
+    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
+    (∫ ω in A, f ω ∂μ) ≤ ∫ ω in A, g ω ∂μ := by
+  have hAm (n : ℕ) : MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace]
+      (A ∩ {ω | τ ω ≤ (n : ℝ≥0)}) := by
+    have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ (n : ℝ≥0)) A hA
+    convert hh using 1
+    congr 1
+    funext ω
+    exact sourceStopTruncate_coe τ n ω
+  have hset (n : ℕ) :
+      (∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, f ω ∂μ) ≤
+        ∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, g ω ∂μ := by
+    apply hloc n _ (hAm n)
+    intro ω hω
+    rw [sourceStopTruncate_coe, min_eq_left hω.2]
+  have hlim (v : Ω → ℝ) (hv : Integrable v μ) (hzero : ∀ ω, τ ω = ⊤ → v ω = 0) :
+      Tendsto (fun n : ℕ => ∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, v ω ∂μ) atTop
+        (𝓝 (∫ ω in A, v ω ∂μ)) := by
+    have he (n : ℕ) : (∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, v ω ∂μ) =
+        ∫ ω in A, {ω | τ ω ≤ (n : ℝ≥0)}.indicator v ω ∂μ :=
+      (setIntegral_indicator (μ := μ) (s := A) (f := v)
+        ((ℱ.le (n : ℝ≥0)) _ (hτ n))).symm
+    simp_rw [he]
+    apply tendsto_integral_of_dominated_convergence (fun ω => ‖v ω‖)
+      (fun n => (hv.indicator ((ℱ.le (n : ℝ≥0)) _ (hτ n))).integrableOn.aestronglyMeasurable)
+      hv.norm.integrableOn
+    · intro n
+      exact Eventually.of_forall fun ω => norm_indicator_le_norm_self _ _
+    · apply Eventually.of_forall
+      intro ω
+      apply (sourceStop_indicator_tendsto (τ ω) (v ω) (hzero ω)).congr'
+      apply Eventually.of_forall
+      intro n
+      by_cases hn : τ ω ≤ (n : WithTop ℝ≥0) <;> simp [Set.indicator, hn]
+  exact le_of_tendsto_of_tendsto' (hlim f hf (fun ω ht => (hz ω ht).1))
+    (hlim g hg (fun ω ht => (hz ω ht).2)) hset
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal
+namespace SLEExactGauge
+
+lemma source_condExp_le_of_setIntegral_le
+    {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
+    {μ : Measure Ω} [IsFiniteMeasure μ] (hm : m ≤ mΩ)
+    {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
+    (hgm : StronglyMeasurable[m] g)
+    (h : ∀ A : Set Ω, MeasurableSet[m] A → (∫ ω in A, f ω ∂μ) ≤ ∫ ω in A, g ω ∂μ) :
+    μ[f | m] ≤ᵐ[μ] g := by
+  have : IsFiniteMeasure (μ.trim hm) := isFiniteMeasure_trim hm
+  apply ae_of_ae_trim hm
+  apply ae_le_of_forall_setIntegral_le
+    (Integrable.trim hm integrable_condExp stronglyMeasurable_condExp)
+    (Integrable.trim hm hg hgm)
+  intro A hA _
+  rw [← setIntegral_trim hm stronglyMeasurable_condExp hA,
+    ← setIntegral_trim hm hgm hA, setIntegral_condExp hm hf hA]
+  exact h A hA
+
+end SLEExactGauge
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal NNReal
+namespace SLEExactGauge
+lemma source_unbounded_conditional_bound_on
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    (ℱ : Filtration ℝ≥0 mΩ) {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime ℱ τ)
+    (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A)
+    (hAfin : ∀ ω ∈ A, τ ω ≠ ⊤)
+    (f : Ω → ℝ) (hf : Integrable f μ) (b : ℝ)
+    (hloc : ∀ (n : ℕ) (E : Set Ω),
+      MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace] E →
+      E ⊆ A ∩ {ω | τ ω ≤ (n : ℝ≥0)} →
+      (∫ ω in E, f ω ∂μ) ≤ ∫ _ in E, b ∂μ) :
+    ∀ᵐ ω ∂μ, ω ∈ A → μ[f | hτ.measurableSpace] ω ≤ b := by
+  have hAm : MeasurableSet A := hτ.measurableSpace_le _ hA
+  have hfi : Integrable (A.indicator f) μ := hf.indicator hAm
+  have hbi : Integrable (A.indicator (fun _ => b)) μ := (integrable_const _).indicator hAm
+  have hbound : μ[A.indicator f | hτ.measurableSpace] ≤ᵐ[μ] A.indicator (fun _ => b) := by
+    apply source_condExp_le_of_setIntegral_le hτ.measurableSpace_le hfi hbi
+      (stronglyMeasurable_const.indicator hA)
+    intro S hS
+    apply source_unbounded_setIntegral_le ℱ hτ _ _ hfi hbi ?_ ?_ S hS
+    · intro ω ht
+      have hnot : ω ∉ A := fun hω => hAfin ω hω ht
+      simp only [Set.indicator_of_notMem hnot, and_self]
+    · intro n E hE hτE
+      have hen (ω : Ω) (hω : ω ∈ E) : τ ω ≤ (n : ℝ≥0) := by
+        rw [hτE ω hω,sourceStopTruncate_coe]
+        exact min_le_right _ _
+      have hAn : MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace]
+          (A ∩ {ω | τ ω ≤ (n : ℝ≥0)}) := by
+        have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ (n : ℝ≥0)) A hA
+        convert hh using 1
+        congr 1
+        funext ω
+        exact sourceStopTruncate_coe τ n ω
+      have heq : E ∩ A = E ∩ (A ∩ {ω | τ ω ≤ (n : ℝ≥0)}) := by
+        ext ω
+        exact ⟨fun h => ⟨h.1,h.2,hen ω h.1⟩,fun h => ⟨h.1,h.2.1⟩⟩
+      have hEA : MeasurableSet[(sourceStopTruncate_stopping ℱ hτ n).measurableSpace] (E ∩ A) := by
+        rw [heq]
+        exact hE.inter hAn
+      rw [setIntegral_indicator hAm,setIntegral_indicator hAm]
+      exact hloc n (E ∩ A) hEA (fun ω hω => ⟨hω.2,hen ω hω.1⟩)
+  filter_upwards [hbound,condExp_indicator hf hA] with ω hb he
+  intro hω
+  rw [he,Set.indicator_of_mem hω,Set.indicator_of_mem hω] at hb
+  exact hb
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance physicalClosedPrefixTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance physicalClosedPrefixTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+lemma source_physical_closed_prefix_isClosed (z : ℂ) (R : ℝ) (v : ℝ≥0) :
+    IsClosed {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t ∈ Icc (0 : ℝ≥0) v,
+      R ≤ dist (p.1 (p.2+t)) z} := by
+  let E := {p : (C(ℝ≥0,ℂ) × ℝ≥0) × Icc (0 : ℝ≥0) v |
+    R ≤ dist (p.1.1 (p.1.2+p.2)) z}
+  have he : IsClosed E := by
+    apply isClosed_le continuous_const
+    exact (continuous_eval.comp ((continuous_fst.fst).prodMk
+      (continuous_fst.snd.add (continuous_subtype_val.comp continuous_snd)))).dist continuous_const
+  have hc := isClosedMap_fst_of_compactSpace E he
+  convert hc using 1
+  ext p
+  constructor
+  · rintro ⟨t,ht,he⟩
+    exact ⟨(p,⟨t,ht⟩),he,rfl⟩
+  · rintro ⟨⟨p',t⟩,ht,rfl⟩
+    exact ⟨t,t.2,ht⟩
+
+lemma source_physical_closed_exit_measurableSet (z : ℂ) (R : ℝ) :
+    MeasurableSet {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
+      z ∈ survivingDomain p.1 (p.2+t) ∧ R ≤ dist (p.1 (p.2+t)) z} := by
+  have heq : {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ t : ℝ≥0,
+      z ∈ survivingDomain p.1 (p.2+t) ∧ R ≤ dist (p.1 (p.2+t)) z} =
+      {p : C(ℝ≥0,ℂ) × ℝ≥0 | ∃ q : ℚ,
+        z ∈ survivingDomain p.1 (p.2+(q:ℝ).toNNReal) ∧
+        ∃ t ∈ Icc (0 : ℝ≥0) (q:ℝ).toNNReal, R ≤ dist (p.1 (p.2+t)) z} := by
+    ext p
+    constructor
+    · rintro ⟨t,hl,he⟩
+      have ho : IsOpen {u : ℝ≥0 | z ∈ survivingDomain p.1 (p.2+u)} :=
+        source_survival_open.preimage
+          ((continuous_const.prodMk (continuous_const.add continuous_id)).prodMk continuous_const)
+      obtain ⟨b,htb,hb⟩ := exists_Ico_subset_of_mem_nhds (ho.mem_nhds hl) ⟨t+1,by simp⟩
+      obtain ⟨q,htq,hqb⟩ := exists_rat_btwn (show (t:ℝ) < b from htb)
+      have hqn : 0 ≤ (q:ℝ) := le_trans t.coe_nonneg htq.le
+      have htu : t ≤ (q:ℝ).toNNReal := by
+        change (t:ℝ) ≤ ((q:ℝ).toNNReal:ℝ)
+        rw [Real.coe_toNNReal _ hqn]
+        exact htq.le
+      have hub : (q:ℝ).toNNReal < b := by
+        change ((q:ℝ).toNNReal:ℝ) < (b:ℝ)
+        rw [Real.coe_toNNReal _ hqn]
+        exact hqb
+      exact ⟨q,hb ⟨htu,hub⟩,t,⟨bot_le,htu⟩,he⟩
+    · rintro ⟨q,hl,t,ht,he⟩
+      refine ⟨t,?_,he⟩
+      exact survivingDomain_antitone p.1 (by simpa only [add_comm] using add_le_add_left ht.2 p.2) hl
+  rw [heq]
+  simp only [ofPred_exists]
+  apply MeasurableSet.iUnion
+  intro q
+  have ho : IsOpen {p : C(ℝ≥0,ℂ) × ℝ≥0 |
+      z ∈ survivingDomain p.1 (p.2+(q:ℝ).toNNReal)} :=
+    source_survival_open.preimage
+      ((continuous_fst.prodMk (continuous_snd.add continuous_const)).prodMk continuous_const)
+  exact ho.measurableSet.inter (source_physical_closed_prefix_isClosed z R (q:ℝ).toNNReal).measurableSet
+
+lemma source_physical_closed_exit_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (gp : Ω → C(ℝ≥0,ℂ)) (hgp : Measurable gp)
+    (τ : Ω → ℝ≥0) (hτ : Measurable τ) (z : ℂ) (R : ℝ) :
+    MeasurableSet {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (gp ω) (τ ω+t) ∧
+      R ≤ dist (gp ω (τ ω+t)) z} := by
+  have hp : Measurable (fun ω => (gp ω,τ ω)) := hgp.prodMk hτ
+  have hh := (source_physical_closed_exit_measurableSet z R).preimage hp
+  simpa only [Set.preimage_ofPred_eq] using hh
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped ENNReal NNReal
+namespace SLEExactGauge
+lemma source_physical_slack_large (C c L : ℝ) (_hC : 0 ≤ C) (hL : 0 < L) :
+    C * (L/2)^(-c) ≤ ((C+1)*2^c)*L^(-c) := by
+  have he : (L/2)^(-c) = 2^c*L^(-c) := by
+    rw [Real.div_rpow hL.le (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2)]
+    simp only [div_inv_eq_mul,mul_comm]
+  rw [he]
+  have hh := Real.rpow_nonneg (by norm_num : (0:ℝ) ≤ 2) c
+  have hh' := Real.rpow_nonneg hL.le (-c)
+  nlinarith [mul_nonneg hh hh']
+
+lemma source_physical_slack_small (C c L : ℝ) (hC : 0 ≤ C) (hc : 0 < c)
+    (hL : 0 < L) (hL2 : L ≤ 2) :
+    1 ≤ ((C+1)*2^c)*L^(-c) := by
+  have h2 : (0:ℝ) < 2^c := Real.rpow_pos_of_pos (by norm_num) _
+  have hrec : 2^(-c) ≤ L^(-c) := Real.rpow_le_rpow_of_nonpos hL hL2 (by linarith)
+  have he : (2:ℝ)^c * 2^(-c) = 1 := by
+    rw [← Real.rpow_add (by norm_num : (0:ℝ) < 2)]
+    simp
+  have hh := mul_le_mul_of_nonneg_left hrec h2.le
+  rw [he] at hh
+  have hp : 0 ≤ 2^c * L^(-c) := le_trans (by norm_num) hh
+  nlinarith [mul_nonneg hC hp]
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance physicalDiskTargetTraceMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance physicalDiskTargetTraceBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+local instance physicalDiskTargetDriverMeasurableSpace : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance physicalDiskTargetDriverBorelSpace : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+ 
+
+def SourcePhysicalDiskTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_m : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
+  ∀ (hm : ∀ t, Measurable (B t)) (_hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (_hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)) (τ : Ω → WithTop ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) τ)
+    (z : ℂ) (r : ℝ), 0 < r →
+  ∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
+    (∀ ω ∈ A, ∃ T s : ℝ≥0, τ ω = T ∧ s ≤ T ∧
+      Metric.closedBall z r ⊆ survivingDomain (γ ω) s ∧
+      dist (γ ω T) z = r ∧
+      (∀ t ∈ Ico s T, dist (γ ω t) z ≠ r) ∧
+      z ∈ survivingDomain (γ ω) T) →
+  ∀ L : ℝ, 1 ≤ L →
+    ∀ᵐ ω ∂P, ω ∈ A →
+      P[(fun ω => if τ ω ≠ ⊤ ∧ ∃ t : ℝ≥0,
+        z ∈ survivingDomain (γ ω) ((τ ω).untopD 0+t) ∧
+        L*r ≤ dist (γ ω ((τ ω).untopD 0+t)) z
+        then (1 : ℝ) else 0) | hτ.measurableSpace] ω ≤ C*L^(-c)
+
+theorem sourcePhysicalDisk_proved : SourcePhysicalDiskTarget := by
+  intro κ hκ hκ8
+  obtain ⟨C,c,hC,hc,hbound⟩ := sourceFinitePhysicalDisk_proved κ hκ hκ8
+  refine ⟨(C+1)*2^c,c,mul_pos (by linarith) (Real.rpow_pos_of_pos (by norm_num) _),hc,?_⟩
+  intro Ω m P hP B hB hm hBc γ G F hw τ hτ z r hr A hA hgeom L hL
+  let : IsProbabilityMeasure P := hP
+  let ℱ := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let gp := sourceTracePath γ (fun ω => (hw ω).1)
+  obtain ⟨Φ,hΦm,hΦeq,hΦ⟩ := sourceStoppedTraceLaw κ (by linarith) hκ8 B hB hm hBc γ G F hw
+  have hgm : Measurable gp := by
+    have he : gp = fun ω => Φ (sourceDriverPath B hBc ω) := funext fun ω => (hΦeq ω).symm
+    rw [he]
+    exact hΦm.comp (source_path_measurable _ hm)
+  let H := {ω | τ ω ≠ ⊤ ∧ ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) ((τ ω).untopD 0+t) ∧
+    L*r ≤ dist (γ ω ((τ ω).untopD 0+t)) z}
+  let f : Ω → ℝ := H.indicator (fun _ => 1)
+  have hH : MeasurableSet H := by
+    have hfin : MeasurableSet {ω | τ ω ≠ ⊤} := (measurableSet_eq_fun hτ.measurable' measurable_const).compl
+    have he := source_physical_closed_exit_measurable gp hgm (fun ω => (τ ω).untopD 0)
+      (hτ.measurable'.untopD 0) z (L*r)
+    exact hfin.inter he
+  have hf : Integrable f P := (integrable_const (1:ℝ)).indicator hH
+  have hfeq : f = (fun ω => if ω ∈ H then (1:ℝ) else 0) := by
+    funext ω
+    exact Set.indicator_apply _ _ _
+  suffices hh : ∀ᵐ ω ∂P, ω ∈ A → P[f | hτ.measurableSpace] ω ≤ ((C+1)*2^c)*L^(-c) by
+    simpa only [hfeq,H,Set.mem_ofPred_eq] using hh
+  by_cases hL2 : L ≤ 2
+  · have hfle : f ≤ᵐ[P] fun _ => (1:ℝ) := Eventually.of_forall fun ω => by
+      dsimp [f,Set.indicator]; split_ifs <;> norm_num
+    have hmono := condExp_mono (m := hτ.measurableSpace) hf (integrable_const (1:ℝ)) hfle
+    have he := condExp_const (μ := P) hτ.measurableSpace_le (1:ℝ)
+    filter_upwards [hmono] with ω hω
+    intro _
+    rw [he] at hω
+    exact hω.trans (source_physical_slack_small C c L hC.le hc (by linarith) hL2)
+  have hAfin : ∀ ω ∈ A, τ ω ≠ ⊤ := by
+    intro ω hω
+    obtain ⟨T,s,hT,_⟩ := hgeom ω hω
+    rw [hT]
+    exact WithTop.coe_ne_top
+  apply source_unbounded_conditional_bound_on ℱ hτ A hA hAfin f hf _
+  intro n E hE hEA
+  let ρ := sourceStopTruncate τ n
+  have hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0)) := sourceStopTruncate_stopping ℱ hτ n
+  have heq (ω : Ω) (hω : ω ∈ E) : τ ω = (ρ ω : WithTop ℝ≥0) := by
+    rw [sourceStopTruncate_coe,min_eq_left (hEA hω).2]
+  have hgeomE : ∀ ω ∈ E, ∃ s : ℝ≥0, s ≤ ρ ω ∧
+      Metric.closedBall z r ⊆ survivingDomain (γ ω) s ∧ dist (γ ω (ρ ω)) z = r ∧
+      (∀ t ∈ Ico s (ρ ω), dist (γ ω t) z ≠ r) ∧ z ∈ survivingDomain (γ ω) (ρ ω) := by
+    intro ω hω
+    obtain ⟨T,s,hT,hs,hfree,hhit,hfirst,hz⟩ := hgeom ω (hEA hω).1
+    have hTr : T = ρ ω := WithTop.coe_injective (hT.symm.trans (heq ω hω))
+    exact ⟨s,hTr ▸ hs,hfree,hTr ▸ hhit,hTr ▸ hfirst,hTr ▸ hz⟩
+  have hLhalf : 1 ≤ L/2 := by linarith
+  have hlocal := hbound Ω m P hP B hB hm hBc γ G F hw ρ hρ z r hr E hE hgeomE (L/2) hLhalf
+  let K := {ω | ∃ t : ℝ≥0, z ∈ survivingDomain (γ ω) (ρ ω+t) ∧
+    (L/2)*r < dist (γ ω (ρ ω+t)) z}
+  let g : Ω → ℝ := K.indicator (fun _ => 1)
+  have hK : MeasurableSet K := source_physical_exit_measurable gp hgm ρ
+    (stopping_time_measurable_nnreal hρ) z ((L/2)*r)
+  have hg : Integrable g P := (integrable_const (1:ℝ)).indicator hK
+  have hgeq : g = (fun ω => if ω ∈ K then (1:ℝ) else 0) := by
+    funext ω
+    exact Set.indicator_apply _ _ _
+  have hfg : ∀ ω ∈ E, f ω ≤ g ω := by
+    intro ω hω
+    by_cases hh : ω ∈ H
+    · obtain ⟨hfin,t,ht,hd⟩ := (show τ ω ≠ ⊤ ∧ ∃ t : ℝ≥0,
+        z ∈ survivingDomain (γ ω) ((τ ω).untopD 0+t) ∧
+        L*r ≤ dist (γ ω ((τ ω).untopD 0+t)) z from hh)
+      have hτr : (τ ω).untopD 0 = ρ ω := by rw [heq ω hω,WithTop.untopD_coe]
+      have hk : ω ∈ K := ⟨t,by rwa [hτr] at ht,by rw [hτr] at hd; nlinarith⟩
+      simp only [f,g,Set.indicator_of_mem hh,Set.indicator_of_mem hk,le_refl]
+    · simp only [f,Set.indicator_of_notMem hh]
+      dsimp [g,Set.indicator]; split_ifs <;> norm_num
+  have hEm : MeasurableSet E := hρ.measurableSpace_le _ hE
+  have hi1 : (∫ ω in E, f ω ∂P) ≤ ∫ ω in E, g ω ∂P :=
+    setIntegral_mono_on hf.integrableOn hg.integrableOn hEm hfg
+  have hi2 : (∫ ω in E, g ω ∂P) ≤ ∫ ω in E, C*(L/2)^(-c) ∂P := by
+    rw [← setIntegral_condExp hρ.measurableSpace_le hg hE]
+    apply setIntegral_mono_on_ae integrable_condExp.integrableOn (integrable_const _).integrableOn hEm
+    filter_upwards [hlocal] with ω hω
+    simpa only [hgeq,K,Set.mem_ofPred_eq] using hω
+  have hi3 : (∫ ω in E, C*(L/2)^(-c) ∂P) ≤ ∫ ω in E, ((C+1)*2^c)*L^(-c) ∂P := by
+    exact setIntegral_mono_on (integrable_const _).integrableOn (integrable_const _).integrableOn hEm
+      (fun _ _ => source_physical_slack_large C c L hC.le (by linarith))
+  exact hi1.trans (hi2.trans hi3)
+
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/PhysicalMeshPast.lean b/lean/OAI/Probability/SLE/PhysicalMeshPast.lean
index 5c269672962ee3d22d7a4f3030bfa387307c3562..32150f5c5137d588cf48f22eedfc0057c2c16749 100644
--- a/lean/OAI/Probability/SLE/PhysicalMeshPast.lean
+++ b/lean/OAI/Probability/SLE/PhysicalMeshPast.lean
@@ -1,782 +1,782 @@
-import Mathlib
-import OAI.Probability.SLE.PhysicalMeshWindowCore
-import OAI.Probability.SLE.WindowBGeometry
-
-namespace OAI
-
-/-! Physical Mesh Past for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_shift_integral (c : ℂ) (D : Set ℂ) (f : ℂ → ℝ) :
-    (∫ z in ((fun z : ℂ => z+c) ⁻¹' D), f (z+c)) = ∫ z in D, f z := by
-  exact (measurePreserving_add_right volume c).setIntegral_preimage_emb
-    (MeasurableEquiv.addRight c).measurableEmbedding f D
-
-lemma source_mesh_shifted_weighted_tail
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ} (c : ℂ)
-    (h : SourceDisjointWindows P ℱ ((fun z : ℂ => z+c) ⁻¹' D) r R K γ
-      (fun z => σ (z+c)) (fun z => τ (z+c)) (fun z => Y (z+c)) (fun z => μ (z+c)))
-    (hs : ∀ z, IsStoppingTime ℱ (σ z))
-    (hK : 0 < K) {b : ℝ} (hb : 0 < b)
-    {W : Ω → ℝ} (hWa : Measurable[mΩ] W)
-    (hWs : ∀ z, Measurable[(hs z).measurableSpace] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω*(if b < ∫ z in D, (Y z ω-μ z ω) then (1:ℝ) else 0) ∂P) ≤
-      (∫ ω, W ω ∂P)*Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2)) := by
-  have hh := window_source_weighted_tail h hK hb hWa (fun z => hWs (z+c)) hW0 hWb
-  have he (ω : Ω) : (∫ z in ((fun z : ℂ => z+c) ⁻¹' D), (Y (z+c) ω-μ (z+c) ω)) =
-      ∫ z in D, (Y z ω-μ z ω) := source_mesh_shift_integral c D (fun z => Y z ω-μ z ω)
-  simpa only [he] using hh
-
-lemma source_mesh_center_before
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (c : ℂ) (hD : D ⊆ ball c 1) (η r : ℝ)
-    (hη : 0 < η) (hr : r < η/2) (hc : 1+η ≤ c.im)
-    (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples) :
-    spatialHitTime (γ ω) c (1+η/2) ≤ sourceCollarStart D (γ ω) z r := by
-  by_cases hz : z ∈ D
-  · simp only [sourceCollarStart,ite_eq_left hz]
-    by_cases hn : liveSpatialHitTime (γ ω) z r = ⊤
-    · rw [hn]; exact le_top
-    obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hn
-    have hcontact := (source_liveHit_mem (a.trace_continuous ω) z r
-      (source_window_origin_bound a z r (hy z hz) ω) T hT.symm).2
-    have hd : dist (γ ω T) c < 1+η/2 := by
-      have hz' := hD hz
-      change dist z c < 1 at hz'
-      have ht := dist_triangle (γ ω T) z c
-      linarith
-    have hstart : 1+η/2 ≤ dist (γ ω 0) c := by
-      rw [(a.witness ω).2.1,dist_zero_left]
-      have hi : c.im ≤ ‖c‖ := (le_abs_self _).trans (Complex.abs_im_le_norm c)
-      linarith
-    rw [← hT]
-    exact (spatialHitTime_lt_of_dist_lt (a.trace_continuous ω) c _ hstart T hd).le
-  · rw [sourceCollarStart,ite_eq_right hz]
-    exact le_top
-
-lemma source_mesh_center_stopping
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (c : ℂ) (η : ℝ) :
-    IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) c (1+η/2)) := by
-  let X (t : ℝ≥0) (ω : a.samples) := dist (γ ω t) c - (1+η/2)
-  have ha : Adapted a.filtration X := by
-    intro t
-    let : MeasurableSpace a.samples := a.filtration t
-    exact ((a.trace_adapted t).dist measurable_const).sub_const _
-  exact continuous_zero_hit_stopping a.filtration X ha
-    (fun ω => ((a.trace_continuous ω).dist continuous_const).sub continuous_const) 0
-
-lemma source_mesh_free_im {γ : ℝ≥0 → ℂ} {s : ℝ≥0} {c : ℂ} {η : ℝ}
-    (hη : 0 < η) (hfree : closedBall c (1+η) ⊆ survivingDomain γ s) :
-    1+η < c.im := by
-  let z := c-((1+η:ℝ):ℂ)*Complex.I
-  have hz : z ∈ closedBall c (1+η) := by
-    rw [mem_closedBall,dist_eq_norm]
-    simp only [z,sub_sub_cancel_left,norm_neg,norm_mul,Complex.norm_real,Complex.norm_I,mul_one]
-    rw [Real.norm_eq_abs,abs_of_pos (by linarith)]
-  have hi := (hfree hz).1.1
-  change 0 < z.im at hi
-  simpa [z] using hi
-
-lemma source_mesh_free_before_center {γ : ℝ≥0 → ℂ} (hγ : Continuous γ)
-    (s : ℝ≥0) (c : ℂ) (η : ℝ) (hη : 0 < η)
-    (hfree : closedBall c (1+η) ⊆ survivingDomain γ s) :
-    (s : WithTop ℝ≥0) ≤ spatialHitTime γ c (1+η/2) := by
-  by_contra! hn
-  obtain ⟨t,ht,hcontact⟩ := (spatialHitTime_le_iff hγ c (1+η/2) s).mp hn.le
-  have hin : γ t ∈ closedBall c (1+η) := by
-    rw [mem_closedBall,hcontact]; linarith
-  exact (hfree hin).1.2 (mem_image_of_mem γ ht)
-
-lemma source_past_indicator_at_start
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ}
-    {ρ τ σ : Ω → WithTop ℝ≥0} (hρ : IsStoppingTime ℱ ρ)
-    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) (hle : ∀ ω, τ ω ≤ σ ω)
-    (A : Set Ω) (hA : MeasurableSet[hρ.measurableSpace] A) :
-    Measurable[hσ.measurableSpace] (fun ω => if ω ∈ A ∧ ρ ω ≤ τ ω then (1:ℝ) else 0) := by
-  have hB : MeasurableSet[hσ.measurableSpace] {ω | ρ ω ≤ τ ω} :=
-    (hτ.measurableSpace_mono hσ hle) _ (hρ.measurableSet_stopping_time_le hτ)
-  have hI : Measurable[hρ.measurableSpace] (fun ω => if ω ∈ A then (1:ℝ) else 0) :=
-    Measurable.ite hA measurable_const measurable_const
-  have hh := window_measurable_on_event hσ hρ hI hB (fun ω hω => hω.trans (hle ω)) 0
-  convert hh using 1
-  funext ω
-  by_cases ha : ω ∈ A <;> by_cases hb : ρ ω ≤ τ ω <;> simp [ha,hb]
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_integrated_step
-    {X : Type*} [MeasurableSpace X] (ν : Measure X)
-    (W W' I I' H V B : X → ℝ) (p q r C A b : ℝ)
-    (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hp : 0 < p)
-    (_hW : Integrable W ν) (hW' : Integrable W' ν)
-    (_hI : Integrable I ν) (hI' : Integrable I' ν)
-    (hH : Integrable H ν) (hV : Integrable V ν) (hB : Integrable B ν)
-    (hb : 0 ≤ b)
-    (hmissI : ∀ᵐ z ∂ν, I' z ≤ H z+B z)
-    (hmissW : ∀ᵐ z ∂ν, W' z ≤ r^(-p)*V z+(q*r)^(-p)*B z)
-    (hmeanH : (∫ z, H z ∂ν) ≤ C*(q*r)^p*(∫ z, W z ∂ν)+b)
-    (hmeanV : (∫ z, V z ∂ν) ≤ r^p*(∫ z, W z ∂ν)+b)
-    (hmeanB : (∫ z, B z ∂ν) ≤ A*(∫ z, I z ∂ν)+b) :
-    (∫ z, W' z ∂ν) ≤ (∫ z, W z ∂ν)+(A*q^(-p))*(r^(-p)*(∫ z, I z ∂ν))+
-       (2*q^(-p))*r^(-p)*b ∧
-    (q*r)^(-p)*(∫ z, I' z ∂ν) ≤ C*(∫ z, W z ∂ν)+
-       (A*q^(-p))*(r^(-p)*(∫ z, I z ∂ν))+(2*q^(-p))*r^(-p)*b := by
-  have hrp : 0 ≤ r^(-p) := (Real.rpow_pos_of_pos hr _).le
-  have hqp : 0 ≤ (q*r)^(-p) := (Real.rpow_pos_of_pos (mul_pos hq hr) _).le
-  have hq1p : 1 ≤ q^(-p) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos hq hq1.le (neg_nonpos.mpr hp.le)
-  have hcancel : r^(-p)*r^p = 1 := by rw [←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero]
-  have hcancelq : (q*r)^(-p)*(q*r)^p = 1 := by rw [←Real.rpow_add (mul_pos hq hr),neg_add_cancel,Real.rpow_zero]
-  have hmul : (q*r)^(-p) = q^(-p)*r^(-p) := Real.mul_rpow hq.le hr.le
-  constructor
-  · calc
-      _ ≤ ∫ z, (r^(-p)*V z+(q*r)^(-p)*B z) ∂ν :=
-        integral_mono_ae hW' ((hV.const_mul _).add (hB.const_mul _)) hmissW
-      _ = r^(-p)*(∫ z, V z ∂ν)+(q*r)^(-p)*(∫ z, B z ∂ν) := by
-        rw [integral_add (hV.const_mul _) (hB.const_mul _),integral_const_mul,integral_const_mul]
-      _ ≤ r^(-p)*(r^p*(∫ z, W z ∂ν)+b)+(q*r)^(-p)*(A*(∫ z,I z ∂ν)+b) :=
-        add_le_add (mul_le_mul_of_nonneg_left hmeanV hrp) (mul_le_mul_of_nonneg_left hmeanB hqp)
-      _ ≤ _ := by rw [mul_add,←mul_assoc,hcancel,one_mul,hmul]; nlinarith [mul_nonneg (sub_nonneg.mpr hq1p) (mul_nonneg hrp hb)]
-  · calc
-      _ ≤ (q*r)^(-p)*(∫ z, (H z+B z) ∂ν) := mul_le_mul_of_nonneg_left
-        (integral_mono_ae hI' (hH.add hB) hmissI) hqp
-      _ = (q*r)^(-p)*((∫ z,H z ∂ν)+(∫ z,B z ∂ν)) := by rw [integral_add hH hB]
-      _ ≤ (q*r)^(-p)*(C*(q*r)^p*(∫ z,W z ∂ν)+b+(A*(∫ z,I z ∂ν)+b)) :=
-        mul_le_mul_of_nonneg_left (add_le_add hmeanH hmeanB) hqp
-      _ = _ := by
-        calc
-          _ = C*((q*r)^(-p)*(q*r)^p)*(∫ z,W z ∂ν)+(q*r)^(-p)*(A*(∫ z,I z ∂ν))+2*(q*r)^(-p)*b := by ring
-          _ = _ := by rw [hcancelq,hmul]; ring
-
-lemma source_collar_missed
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (D : Set ℂ) (q r L : ℝ)
-    (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hL : 1 < L)
-    (hy : ∀ z ∈ D, r ≤ z.im) (ω : a.samples) (z : ℂ) (hz : z ∈ D) :
-    (if sourceCollarStart D (γ ω) z (q*r) = ⊤ then (0:ℝ) else 1) ≤
-      sourceCollarH D (γ ω) z r (q*r) (L*r)+sourceCollarB D (γ ω) z r (q*r) (L*r) ∧
-    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r) ≤
-      r^(-codimension κ)*(r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceCollarH D (γ ω) z r (q*r) (L*r))+
-      (q*r)^(-codimension κ)*sourceCollarB D (γ ω) z r (q*r) (L*r) := by
-  have hh := sourceSpatialWindowGeometry_proved (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) (a.F ω) ((a.continuous_B ω).const_mul _) (a.witness ω) κ hκ hκ8
-    z (q*r) r (L*r) (mul_pos hq hr) (by nlinarith) (by nlinarith) (hy z hz)
-  simp only [sourceCollarStart,sourceCollarH,sourceCollarB,ite_eq_left hz]
-  refine ⟨hh.2.2.2.2.2.2.2.2.2.2.1,?_⟩
-  have he : r^(-codimension κ)*r^codimension κ = 1 := by
-    rw [←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero]
-  simpa only [←mul_assoc,he,one_mul] using hh.2.2.2.2.2.2.2.2.2.2.2
-
-lemma source_mesh_bounded_integrable {X : Type*} [MeasurableSpace X]
-    {ν : Measure X} [IsFiniteMeasure ν] {f : X → ℝ} {K : ℝ}
-    (hm : Measurable f) (hb : ∀ x, 0 ≤ f x ∧ f x ≤ K) : Integrable f ν := by
-  apply Integrable.of_bound hm.aestronglyMeasurable K
-  exact ae_of_all _ (fun x => by simpa only [Real.norm_eq_abs,abs_of_nonneg (hb x).1] using (hb x).2)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_shifted_integral_measurable
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ} (c : ℂ)
-    (h : SourceDisjointWindows P ℱ ((fun z : ℂ => z+c) ⁻¹' D) r R K γ
-      (fun z => σ (z+c)) (fun z => τ (z+c)) (fun z => Y (z+c)) (fun z => μ (z+c))) :
-    Measurable (fun ω => ∫ z in D, (Y z ω-μ z ω)) := by
-  have hm := window_source_integral_measurable h
-  have he (ω : Ω) : (∫ z in ((fun z : ℂ => z+c) ⁻¹' D), (Y (z+c) ω-μ (z+c) ω)) =
-      ∫ z in D, (Y z ω-μ z ω) := source_mesh_shift_integral c D (fun z => Y z ω-μ z ω)
-  simpa only [he] using hm
-
-lemma source_mesh_shifted_restricted_tail
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ} (c : ℂ)
-    (h : SourceDisjointWindows P ℱ ((fun z : ℂ => z+c) ⁻¹' D) r R K γ
-      (fun z => σ (z+c)) (fun z => τ (z+c)) (fun z => Y (z+c)) (fun z => μ (z+c)))
-    (hs : ∀ z, IsStoppingTime ℱ (σ z))
-    (hK : 0 < K) {b : ℝ} (hb : 0 < b)
-    (E : Set Ω) (hE : MeasurableSet[mΩ] E) (hEs : ∀ z, MeasurableSet[(hs z).measurableSpace] E) :
-    (P.restrict E) {ω | b < ∫ z in D, (Y z ω-μ z ω)} ≤
-      ENNReal.ofReal (P.real E*Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2))) := by
-  let F := {ω | b < ∫ z in D, (Y z ω-μ z ω)}
-  have hF : MeasurableSet F := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable c h)
-  have hWm : Measurable (fun ω => if ω ∈ E then (1:ℝ) else 0) :=
-    Measurable.ite hE measurable_const measurable_const
-  have hWs : ∀ z, Measurable[(hs z).measurableSpace] (fun ω => if ω ∈ E then (1:ℝ) else 0) :=
-    fun z => Measurable.ite (hEs z) measurable_const measurable_const
-  have hh := source_mesh_shifted_weighted_tail c h hs hK hb hWm hWs
-    (fun ω => by split_ifs <;> norm_num) ⟨1,fun ω => by split_ifs <;> norm_num⟩
-  have hi : (fun ω => (if ω ∈ E then (1:ℝ) else 0)*(if ω ∈ F then (1:ℝ) else 0)) =
-      (F∩E).indicator 1 := by
-    funext ω; by_cases he : ω ∈ E <;> by_cases hf : ω ∈ F <;> simp [he,hf]
-  change (∫ ω, (if ω ∈ E then (1:ℝ) else 0)*(if ω ∈ F then (1:ℝ) else 0) ∂P) ≤ _ at hh
-  rw [hi,integral_indicator_one (hF.inter hE)] at hh
-  have hiE : (fun ω => if ω ∈ E then (1:ℝ) else 0) = E.indicator 1 := rfl
-  rw [hiE,integral_indicator_one hE] at hh
-  rw [Measure.restrict_apply hF]
-  exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity)).mpr hh
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-noncomputable def sourceMeshCount (p : ℝ) (D : Set ℂ) (γ : ℝ≥0 → ℂ) (r : ℝ) : ℝ :=
-  r^(-p)*∫ z in D, if sourceCollarStart D γ z r = ⊤ then (0:ℝ) else 1
-
-lemma source_mesh_I_integrable
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (hD : MeasurableSet D) [IsFiniteMeasure (volume.restrict D)]
-    (v : ℝ) (hy : ∀ z ∈ D, v ≤ z.im) (ω : a.samples) :
-    IntegrableOn (fun z => if sourceCollarStart D (γ ω) z v = ⊤ then (0:ℝ) else 1) D := by
-  have hm : Measurable (fun p : ℂ × a.samples =>
-      if sourceCollarStart D (γ p.2) p.1 v = ⊤ then (0:ℝ) else 1) :=
-    Measurable.ite (measurableSet_eq_fun (source_collarStart_joint a D hD v hy) measurable_const)
-      measurable_const measurable_const
-  exact source_mesh_bounded_integrable (K := 1)
-    (hm.comp (measurable_id.prodMk (measurable_const : Measurable (fun _ : ℂ => ω)))) (fun z => by split_ifs <;> norm_num)
-
-lemma source_mesh_integral_recurrences
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
-    (q r L C Cbad c b : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hL : 1 < L) (hb : 0 ≤ b) (hy : ∀ z ∈ D, r ≤ z.im) (ω : a.samples) :
-    let p := codimension κ
-    let W := fun v z => spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
-    let Ic := fun v z => if sourceCollarStart D (γ ω) z v = ⊤ then (0:ℝ) else 1
-    let H := fun z => sourceCollarH D (γ ω) z r (q*r) (L*r)
-    let B := fun z => sourceCollarB D (γ ω) z r (q*r) (L*r)
-    let V := fun z => r^p*W (q*r) z*H z
-    let μH := fun z => C*(q*r)^p*(if z ∈ D then W r z else 0)
-    let μV := fun z => r^p*(if z ∈ D then W r z else 0)
-    let μB := fun z => Cbad*L^(-c)*Ic r z
-    (∫ z in D, (H z-μH z)) ≤ b → (∫ z in D, (V z-μV z)) ≤ b → (∫ z in D, (B z-μB z)) ≤ b →
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (q*r) D ≤
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
-        (Cbad*L^(-c)*q^(-p))*sourceMeshCount p D (γ ω) r+(2*q^(-p))*r^(-p)*b ∧
-      sourceMeshCount p D (γ ω) (q*r) ≤
-        C*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
-        (Cbad*L^(-c)*q^(-p))*sourceMeshCount p D (γ ω) r+(2*q^(-p))*r^(-p)*b := by
-  intro p W Ic H B V μH μV μB hFH hFV hFB
-  change (∫ z in D, W (q*r) z) ≤ (∫ z in D, W r z)+
-    (Cbad*L^(-c)*q^(-p))*(r^(-p)*(∫ z in D, Ic r z))+(2*q^(-p))*r^(-p)*b ∧
-    (q*r)^(-p)*(∫ z in D, Ic (q*r) z) ≤ C*(∫ z in D, W r z)+
-    (Cbad*L^(-c)*q^(-p))*(r^(-p)*(∫ z in D, Ic r z))+(2*q^(-p))*r^(-p)*b
-  have hp : 0 < p := by dsimp [p,codimension,dimension]; linarith
-  have hfin : volume D ≠ ∞ := ne_top_of_le_ne_top (by finiteness : volume (ball c₀ 1) ≠ ∞) (measure_mono hDc)
-  let : IsFiniteMeasure (volume.restrict D) := ⟨by simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
-  have hqr : q*r ≤ r := (mul_le_mul_of_nonneg_right hq1.le hr.le).trans_eq (one_mul _)
-  have hWi (v : ℝ) (hv : 0 < v) (hvr : v ≤ r) : IntegrableOn (fun z => W v z) D :=
-    a.spatialGreenWeight_spatial_integrable hκ hκ8 v hv D hD hfin (fun z hz => hvr.trans (hy z hz)) ω
-  have hIi (v : ℝ) (hvr : v ≤ r) : IntegrableOn (fun z => Ic v z) D :=
-    source_mesh_I_integrable a D hD v (fun z hz => hvr.trans (hy z hz)) ω
-  have hHi : IntegrableOn (fun z => H z) D := source_mesh_bounded_integrable
-    ((source_collarHB_joint a D hD r (q*r) (L*r) hy).1.comp (measurable_id.prodMk (measurable_const : Measurable (fun _ : ℂ => ω))))
-    (fun z => (source_collar_HB_bounds D (γ ω) z r (q*r) (L*r)).1)
-  have hBi : IntegrableOn (fun z => B z) D := source_mesh_bounded_integrable
-    ((source_collarHB_joint a D hD r (q*r) (L*r) hy).2.comp (measurable_id.prodMk (measurable_const : Measurable (fun _ : ℂ => ω))))
-    (fun z => (source_collar_HB_bounds D (γ ω) z r (q*r) (L*r)).2)
-  have hVm : Measurable (fun p : ℂ × a.samples => r^codimension κ*spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t p.2) (γ p.2) (a.G p.2) p.1 (q*r)*sourceCollarH D (γ p.2) p.1 r (q*r) (L*r)) := by
-    simpa only [add_zero] using source_collar_V_joint a D hD 0 q r L hy
-  have hmsec : Measurable (fun z : ℂ => (z,ω)) := measurable_id.prodMk measurable_const
-  have hVsec := hVm.comp hmsec
-  simp only [Function.comp_def] at hVsec
-  have hVi : IntegrableOn V D := source_mesh_bounded_integrable hVsec
-    (fun z => source_collar_V_bounds a hκ hκ8 D q r (L*r) hq hq1 hr hy z ω)
-  have hWceq : (fun z => if z ∈ D then W r z else 0) =ᵐ[volume.restrict D] (fun z => W r z) := by
-    filter_upwards [ae_restrict_mem hD] with z hz; simp only [ite_eq_left hz]
-  have hWci : IntegrableOn (fun z => if z ∈ D then W r z else 0) D := (hWi r hr le_rfl).congr hWceq.symm
-  have hiH : (∫ z in D, μH z) = C*(q*r)^p*(∫ z in D, W r z) := by
-    rw [show (∫ z in D, μH z) = C*(q*r)^p*(∫ z in D, if z ∈ D then W r z else 0) from integral_const_mul _ _,integral_congr_ae hWceq]
-  have hiV : (∫ z in D, μV z) = r^p*(∫ z in D, W r z) := by
-    rw [show (∫ z in D, μV z) = r^p*(∫ z in D, if z ∈ D then W r z else 0) from integral_const_mul _ _,integral_congr_ae hWceq]
-  have hiB : (∫ z in D, μB z) = Cbad*L^(-c)*(∫ z in D, Ic r z) := integral_const_mul _ _
-  rw [integral_sub hHi (hWci.const_mul _),hiH] at hFH
-  rw [integral_sub hVi (hWci.const_mul _),hiV] at hFV
-  rw [integral_sub hBi ((hIi r le_rfl).const_mul _),hiB] at hFB
-  have hmiss : ∀ᵐ z ∂volume.restrict D,
-      Ic (q*r) z ≤ H z+B z ∧ W (q*r) z ≤ r^(-p)*V z+(q*r)^(-p)*B z := by
-    filter_upwards [ae_restrict_mem hD] with z hz
-    exact source_collar_missed a hκ hκ8 D q r L hq hq1 hr hL hy ω z hz
-  exact source_mesh_integrated_step (volume.restrict D) (fun z => W r z) (fun z => W (q*r) z)
-    (fun z => Ic r z) (fun z => Ic (q*r) z) (fun z => H z) (fun z => V z) (fun z => B z)
-    p q r C (Cbad*L^(-c)) b hq hq1 hr hp (hWi r hr le_rfl) (hWi (q*r) (mul_pos hq hr) hqr)
-    (hIi r le_rfl) (hIi (q*r) hqr) hHi hVi hBi hb (hmiss.mono fun _ h => h.1) (hmiss.mono fun _ h => h.2)
-    (sub_le_iff_le_add'.mp hFH) (sub_le_iff_le_add'.mp hFV) (sub_le_iff_le_add'.mp hFB)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_step
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
-    (q r L C Cbad c b : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hL : 1 < L) (hR : 3*(L*r) ≤ 1) (hC : 0 ≤ C) (hCb : 0 ≤ Cbad) (hb : 0 < b)
-    (hy : ∀ z ∈ D, r ≤ z.im)
-    (hmean : ∀ z (hz : z ∈ D),
-      let μ := P.comap (Subtype.val : a.samples → Ω)
-      let W := fun v (ω : a.samples) => spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
-      let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r (q*r) (L*r)
-      let B := fun ω : a.samples => sourceSpatialWindowB (γ ω) z r (q*r) (L*r)
-      let hs := source_liveHit_stopping a z r (hy z hz)
-      μ[H | hs.measurableSpace] ≤ᵐ[μ] (fun ω => C*(q*r)^codimension κ*W r ω) ∧
-      μ[(fun ω => r^codimension κ*W (q*r) ω*H ω) | hs.measurableSpace] ≤ᵐ[μ]
-        (fun ω => r^codimension κ*W r ω) ∧
-      μ[B | hs.measurableSpace] ≤ᵐ[μ]
-        (fun ω => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1)))
-    (E : Set a.samples) (hE : MeasurableSet E)
-    (hEs : ∀ z, MeasurableSet[(source_collarStart_stopping a D r hy z).measurableSpace] E) :
-    ∃ F : Set a.samples, MeasurableSet F ∧
-      ((P.comap Subtype.val).restrict E) F ≤ ENNReal.ofReal
-        (3*(P.comap Subtype.val).real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-codimension κ))^2*(L*r)^2))) ∧
-      ∀ ω ∉ F,
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (q*r) D ≤
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
-        (Cbad*L^(-c)*q^(-codimension κ))*sourceMeshCount (codimension κ) D (γ ω) r+
-        (2*q^(-codimension κ))*r^(-codimension κ)*b ∧
-      sourceMeshCount (codimension κ) D (γ ω) (q*r) ≤
-        C*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
-        (Cbad*L^(-c)*q^(-codimension κ))*sourceMeshCount (codimension κ) D (γ ω) r+
-        (2*q^(-codimension κ))*r^(-codimension κ)*b := by
-  let p := codimension κ
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let W (v : ℝ) (z : ℂ) (ω : a.samples) := spatialGreenWeight κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
-  let Ic (v : ℝ) (z : ℂ) (ω : a.samples) : ℝ := if sourceCollarStart D (γ ω) z v = ⊤ then 0 else 1
-  let H (z : ℂ) (ω : a.samples) := sourceCollarH D (γ ω) z r (q*r) (L*r)
-  let B (z : ℂ) (ω : a.samples) := sourceCollarB D (γ ω) z r (q*r) (L*r)
-  let V (z : ℂ) (ω : a.samples) := r^p*W (q*r) z ω*H z ω
-  let μH (z : ℂ) (ω : a.samples) := C*(q*r)^p*(if z ∈ D then W r z ω else 0)
-  let μV (z : ℂ) (ω : a.samples) := r^p*(if z ∈ D then W r z ω else 0)
-  let μB (z : ℂ) (ω : a.samples) := Cbad*L^(-c)*Ic r z ω
-  let FH := {ω | b < ∫ z in D, (H z ω-μH z ω)}
-  let FV := {ω | b < ∫ z in D, (V z ω-μV z ω)}
-  let FB := {ω | b < ∫ z in D, (B z ω-μB z ω)}
-  let σ := fun z (ω : a.samples) => sourceCollarStart D (γ ω) z r
-  let τ := fun z (ω : a.samples) => sourceCollarFinish D (γ ω) z r (q*r) (L*r)
-  have hWinH := source_mesh_H_window a hκ hκ8 D hD c₀ hDc q r L C hq hq1 hr hL hR hC hy (fun z hz => (hmean z hz).1)
-  have hWinV := source_mesh_V_window a hκ hκ8 D hD c₀ hDc q r L hq hq1 hr hL hR hy (fun z hz => (hmean z hz).2.1)
-  have hWinB := source_mesh_B_window a D hD c₀ hDc q r L Cbad c hq hq1 hr hL hR hCb hy (fun z hz => (hmean z hz).2.2)
-  have hmH : MeasurableSet FH := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable (σ := σ) (τ := τ) (Y := H) (μ := μH) c₀ hWinH)
-  have hmV : MeasurableSet FV := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable (σ := σ) (τ := τ) (Y := V) (μ := μV) c₀ hWinV)
-  have hmB : MeasurableSet FB := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable (σ := σ) (τ := τ) (Y := B) (μ := μB) c₀ hWinB)
-  have hp : 0 < p := by dsimp [p,codimension,dimension]; linarith
-  have hK1 : 1 ≤ q^(-p) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos hq hq1.le (neg_nonpos.mpr hp.le)
-  have hK : 0 < q^(-p) := Real.rpow_pos_of_pos hq _
-  have hTailH := source_mesh_shifted_restricted_tail (σ := σ) (τ := τ) (Y := H) (μ := μH) c₀ hWinH
-    (source_collarStart_stopping a D r hy) zero_lt_one hb E hE hEs
-  have hTailV := source_mesh_shifted_restricted_tail (σ := σ) (τ := τ) (Y := V) (μ := μV) c₀ hWinV
-    (source_collarStart_stopping a D r hy) hK hb E hE hEs
-  have hTailB := source_mesh_shifted_restricted_tail (σ := σ) (τ := τ) (Y := B) (μ := μB) c₀ hWinB
-    (source_collarStart_stopping a D r hy) zero_lt_one hb E hE hEs
-  have hexp : Real.exp (-(1/512:ℝ)*b^2/(1^2*(L*r)^2)) ≤
-      Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2)) := by
-    apply Real.exp_le_exp.mpr
-    have hr0 : 0 < L*r := mul_pos (by linarith) hr
-    have hden : (L*r)^2 ≤ (q^(-p))^2*(L*r)^2 := by nlinarith [sq_nonneg (q^(-p)-1),sq_nonneg (L*r)]
-    have hh := div_le_div_of_nonneg_left (show 0 ≤ (1/512:ℝ)*b^2 by positivity) (sq_pos_of_pos hr0) hden
-    simpa only [one_pow,one_mul,neg_mul,neg_div] using neg_le_neg hh
-  have hTail (F : Set a.samples) (hh : (μ.restrict E) F ≤ ENNReal.ofReal
-      (μ.real E*Real.exp (-(1/512:ℝ)*b^2/(1^2*(L*r)^2)))) :
-      (μ.restrict E) F ≤ ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2))) :=
-    hh.trans (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hexp (measureReal_nonneg)))
-  refine ⟨FH∪FV∪FB,(hmH.union hmV).union hmB,?_,?_⟩
-  · calc
-      _ ≤ (μ.restrict E) FH+(μ.restrict E) FV+(μ.restrict E) FB :=
-        (measure_union_le _ _).trans (add_le_add (measure_union_le _ _) le_rfl)
-      _ ≤ ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2)))+
-           ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2)))+
-           ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2))) :=
-        add_le_add (add_le_add (hTail FH hTailH) hTailV) (hTail FB hTailB)
-      _ = _ := by rw [←ENNReal.ofReal_add (by positivity) (by positivity),←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
-  · intro ω hω
-    have hFH : (∫ z in D, (H z ω-μH z ω)) ≤ b := le_of_not_gt (fun hh => hω (Or.inl (Or.inl hh)))
-    have hFV : (∫ z in D, (V z ω-μV z ω)) ≤ b := le_of_not_gt (fun hh => hω (Or.inl (Or.inr hh)))
-    have hFB : (∫ z in D, (B z ω-μB z ω)) ≤ b := le_of_not_gt (fun hh => hω (Or.inr hh))
-    exact source_mesh_integral_recurrences a hκ hκ8 D hD c₀ hDc q r L C Cbad c b
-      hq hq1 hr hL hb.le hy ω hFH hFV hFB
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-lemma source_mesh_power_growth {p q : ℝ} (hp : 0 < p) (hp1 : p < 1)
-    (hq : 0 < q) (hq1 : q < 1) :
-    ∃ u β : ℝ, 1 < u ∧ u*q < 1 ∧ 1 < q^p/(u*q) ∧ β = (q^p/(u*q))^2 ∧ 1 < β ∧
-      ∀ c : ℝ, 0 < c → 0 < u^(-c) ∧ u^(-c) < 1 := by
-  have hqp : q < q^p := by
-    have hh := Real.rpow_lt_rpow_of_exponent_gt hq hq1 hp1
-    simpa only [Real.rpow_one] using hh
-  have hqp1 : q^p < 1 := Real.rpow_lt_one hq.le hq1 hp
-  let u := (1+q^p/q)/2
-  have hu : 1 < u := by dsimp [u]; have : 1 < q^p/q := (lt_div_iff₀ hq).mpr (by simpa using hqp); linarith
-  have huq : u*q < q^p := by dsimp [u]; field_simp; nlinarith
-  have hratio : 1 < q^p/(u*q) := (lt_div_iff₀ (mul_pos (by linarith) hq)).mpr (by simpa using huq)
-  refine ⟨u,(q^p/(u*q))^2,hu,huq.trans hqp1,hratio,rfl,by nlinarith,?_⟩
-  intro c hc
-  refine ⟨Real.rpow_pos_of_pos (by linarith) _,?_⟩
-  exact Real.rpow_lt_one_of_one_lt_of_neg hu (neg_neg_of_pos hc)
-
-lemma source_mesh_scale_cancel {p r δ : ℝ} (hr : 0 < r) (_hδ : 0 < δ) (n : ℕ) :
-    r^(-p)*(δ^(-p)*r^p/(n+1:ℝ)^2) = δ^(-p)/(n+1:ℝ)^2 := by
-  have he : r^(-p)*r^p = 1 := by rw [←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero]
-  calc
-    _ = δ^(-p)*(r^(-p)*r^p)/(n+1:ℝ)^2 := by ring
-    _ = _ := by rw [he,mul_one]
-
-lemma source_mesh_radius_bound {q u L r δ : ℝ} (hq : 0 < q) (hu : 0 < u)
-    (huq : u*q ≤ 1) (hL : 0 < L) (hr : 0 < r) (hrδ : r ≤ δ) (n : ℕ) :
-    (L*u^n)*(r*q^n) ≤ L*δ := by
-  have hh : (u*q)^n ≤ 1 := pow_le_one₀ (mul_nonneg hu.le hq.le) huq
-  calc
-    _ = (L*r)*(u*q)^n := by rw [mul_pow]; ring
-    _ ≤ (L*r)*1 := mul_le_mul_of_nonneg_left hh (mul_nonneg hL.le hr.le)
-    _ ≤ L*δ := by rw [mul_one]; exact mul_le_mul_of_nonneg_left hrδ hL.le
-
-lemma source_mesh_alpha {c u L : ℝ} (hL : 0 < L) (hu : 0 < u) (n : ℕ) :
-    (L*u^n)^(-c) = L^(-c)*(u^(-c))^n := by
-  rw [Real.mul_rpow hL.le (pow_nonneg hu.le _),Real.rpow_pow_comm hu.le]
-
-lemma source_mesh_failure_exponent {p q u L r δ K : ℝ}
-    (hp1 : p < 1) (hq : 0 < q) (hu : 0 < u) (hL : 0 < L)
-    (hr : 0 < r) (hrδ : r ≤ δ) (_hK : 0 < K) (n : ℕ) :
-    δ^(-2:ℝ)/(K^2*L^2)*((q^p/(u*q))^2)^n/(n+1:ℝ)^4 ≤
-    (δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)^2/(K^2*((L*u^n)*(r*q^n))^2) := by
-  have hδ : 0 < δ := hr.trans_le hrδ
-  have hn : 0 < (n+1:ℝ) := by positivity
-  have hR : 0 < (L*u^n)*(r*q^n) := by positivity
-  have hrq : 0 < r*q^n := by positivity
-  have hid : (δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)) =
-      (δ^(-p)*r^(p-1))/L*(q^p/(u*q))^n/(n+1:ℝ)^2 := by
-    rw [Real.mul_rpow hr.le (pow_nonneg hq.le _),←Real.rpow_pow_comm hq.le,
-      Real.rpow_sub hr,Real.rpow_one,div_pow,mul_pow]
-    field_simp
-  have hant := Real.rpow_le_rpow_of_nonpos hr hrδ (by linarith : p-1 ≤ 0)
-  have hc : δ⁻¹ ≤ δ^(-p)*r^(p-1) := by
-    calc
-      δ⁻¹ = δ^(-p)*δ^(p-1) := by rw [←Real.rpow_add hδ]; ring_nf; rw [Real.rpow_neg_one]
-      _ ≤ _ := mul_le_mul_of_nonneg_left hant (Real.rpow_nonneg hδ.le _)
-  have ratio : δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2 ≤
-      (δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)) := by
-    rw [hid]
-    exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right
-      (div_le_div_of_nonneg_right hc hL.le) (by positivity)) (by positivity)
-  have hleft : 0 ≤ δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2 := by positivity
-  have hsq : (δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2)^2 ≤
-      ((δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)))^2 := by nlinarith
-  calc
-    _ = (δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2)^2/K^2 := by
-      rw [Real.rpow_neg hδ.le,Real.rpow_two]
-      simp only [div_pow,mul_pow,inv_pow,←pow_mul]
-      ring
-    _ ≤ ((δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)))^2/K^2 :=
-      div_le_div_of_nonneg_right hsq (sq_nonneg K)
-    _ = _ := by rw [div_pow]; ring
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-lemma source_mesh_choose_large {c A : ℝ} (hc : 0 < c) :
-    ∃ L : ℝ, 1 < L ∧ A*L^(-c) ≤ 1/2 := by
-  have ht : Tendsto (fun L : ℝ => A*L^(-c)) atTop (𝓝 0) := by
-    simpa only [mul_zero] using (tendsto_rpow_neg_atTop hc).const_mul A
-  have hh := ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1/2))
-  obtain ⟨N,hN⟩ := eventually_atTop.mp hh
-  refine ⟨max N 2,lt_of_lt_of_le (by norm_num) (le_max_right _ _),(hN _ (le_max_left _ _)).le⟩
-
-lemma source_mesh_select_start {q rf δ : ℝ} (hq : 0 < q) (hq1 : q < 1)
-    (hf : 0 < rf) (hfδ : rf ≤ δ) :
-    ∃ r : ℝ, ∃ n : ℕ, 0 < r ∧ q*δ ≤ r ∧ r ≤ δ ∧ rf = r*q^n := by
-  have hδ : 0 < δ := hf.trans_le hfδ
-  obtain ⟨n,hn1,hn2⟩ := exists_nat_pow_near_of_lt_one (div_pos hf hδ)
-    ((div_le_one₀ hδ).mpr hfδ) hq hq1
-  have hqn : 0 < q^n := pow_pos hq _
-  refine ⟨rf/q^n,n,div_pos hf hqn,?_,?_,?_⟩
-  · apply (le_div_iff₀ hqn).mpr
-    have hh := (lt_div_iff₀ hδ).mp hn1
-    rw [pow_succ] at hh
-    nlinarith
-  · apply (div_le_iff₀ hqn).mpr
-    have hh := (div_le_iff₀ hδ).mp hn2
-    nlinarith
-  · symm; exact div_mul_cancel₀ _ hqn.ne'
-
-lemma source_mesh_height {z c : ℂ} {η r : ℝ} (hc : 1+η ≤ c.im)
-    (hz : z ∈ ball c 1) (hr : r < η) : r < z.im := by
-  have hdist : abs (z.im-c.im) ≤ dist z c := by
-    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-c)
-  have hz' : dist z c < 1 := hz
-  have hh := neg_abs_le (z.im-c.im)
-  linarith
-
-lemma source_mesh_count_bound
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (hD : MeasurableSet D) (hfin : volume D ≠ ∞)
-    (p r : ℝ) (hr : 0 < r) (hy : ∀ z ∈ D, r ≤ z.im) (ω : a.samples) :
-    0 ≤ r^(-p)*(∫ z in D, if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) ∧
-    r^(-p)*(∫ z in D, if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) ≤
-      (volume D).toReal*r^(-p) := by
-  let : IsFiniteMeasure (volume.restrict D) := ⟨by simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
-  have hi : IntegrableOn (fun z => if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) D := by
-    apply source_mesh_bounded_integrable (K := 1)
-    · have hm : Measurable (fun p : ℂ × a.samples => if sourceCollarStart D (γ p.2) p.1 r = ⊤ then (0:ℝ) else 1) :=
-        Measurable.ite (measurableSet_eq_fun (source_collarStart_joint a D hD r hy) measurable_const) measurable_const measurable_const
-      exact hm.comp (measurable_id.prodMk measurable_const)
-    · intro z; split_ifs <;> norm_num
-  have hb : (∫ z in D, if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) ≤ (volume D).toReal := by
-    calc
-      _ ≤ ∫ _z in D, (1:ℝ) := integral_mono hi (integrable_const _) (fun z => by split_ifs <;> norm_num)
-      _ = _ := by simp only [integral_const,Measure.restrict_apply_univ,smul_eq_mul,mul_one,measureReal_def]
-  refine ⟨mul_nonneg (Real.rpow_nonneg hr.le _) (integral_nonneg (fun z => by split_ifs <;> norm_num)),?_⟩
-  simpa only [mul_comm] using mul_le_mul_of_nonneg_left hb (Real.rpow_nonneg hr.le (-p))
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_mesh_grid_exponent {p q u L r δ K : ℝ}
-    (hp1 : p < 1) (hq : 0 < q) (hu : 0 < u) (hL : 0 < L)
-    (hr : 0 < r) (hrδ : r ≤ δ) (hK : 0 < K) (n : ℕ) :
-    -(1/512:ℝ)*(δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)^2/(K^2*((L*u^n)*(r*q^n))^2) ≤
-      -(1/(512*K^2*L^2))*δ^(-2:ℝ)*(((q^p/(u*q))^2)^n/(n+1:ℝ)^4) := by
-  have hh := source_mesh_failure_exponent hp1 hq hu hL hr hrδ hK n
-  calc
-    _ = -(1/512:ℝ)*((δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)^2/(K^2*((L*u^n)*(r*q^n))^2)) := by ring
-    _ ≤ -(1/512:ℝ)*(δ^(-2:ℝ)/(K^2*L^2)*((q^p/(u*q))^2)^n/(n+1:ℝ)^4) :=
-      mul_le_mul_of_nonpos_left hh (by norm_num)
-    _ = _ := by ring
-
-lemma source_mesh_grid_error {p q r δ E : ℝ} (hq : 0 < q) (hr : 0 < r) (hδ : 0 < δ) (n : ℕ) :
-    E*(r*q^n)^(-p)*(δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2) = E*δ^(-p)/(n+1:ℝ)^2 := by
-  rw [mul_assoc,source_mesh_scale_cancel (mul_pos hr (pow_pos hq _)) hδ]
-  ring
-
-lemma source_mesh_grid_coefficient {c u L Cbad q p : ℝ} (hL : 0 < L) (hu : 0 < u) (n : ℕ) :
-    Cbad*(L*u^n)^(-c)*q^(-p) = (Cbad*L^(-c)*q^(-p))*(u^(-c))^n := by
-  rw [source_mesh_alpha hL hu]; ring
-
-lemma source_mesh_grid_radius_le {q r δ : ℝ} (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hrδ : r ≤ δ) (n : ℕ) : r*q^n ≤ δ :=
-  (mul_le_mul_of_nonneg_left (pow_le_one₀ hq.le hq1.le) hr.le).trans (by simpa using hrδ)
-
-lemma source_mesh_initial_scale {p q r δ : ℝ} (hp : 0 < p) (hq : 0 < q) (hδ : 0 < δ)
-    (hr : q*δ ≤ r) : r^(-p) ≤ q^(-p)*δ^(-p) := by
-  rw [←Real.mul_rpow hq.le hδ.le]
-  exact Real.rpow_le_rpow_of_nonpos (mul_pos hq hδ) hr (neg_nonpos.mpr hp.le)
-
-lemma source_mesh_ball_volume (c : ℂ) : volume (ball c 1) = volume (ball (0:ℂ) 1) := by
-  rw [Complex.volume_ball,Complex.volume_ball]
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-structure SourceMeshParameters (κ : ℝ) where
-  C : ℝ
-  Cbad : ℝ
-  c : ℝ
-  q : ℝ
-  u : ℝ
-  L : ℝ
-  hC : 0 < C
-  hCb : 0 < Cbad
-  hc : 0 < c
-  hq : 0 < q
-  hq1 : q < 1
-  hu : 1 < u
-  huq : u*q < 1
-  hratio : 1 < q^codimension κ/(u*q)
-  hL : 1 < L
-  halpha : Cbad*L^(-c)*q^(-codimension κ) ≤ 1/2
-  means :
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-      (z : ℂ) (ε r L : ℝ), 0 < ε → ε ≤ q*r → r < z.im → 1 < L →
-    ∃ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
-      let μ := P.comap (Subtype.val : a.samples → Ω)
-      let W := fun v (ω : a.samples) => spatialGreenWeight κ
-        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z v
-      let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε (L*r)
-      let B := fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)
-      μ[H | hσ.measurableSpace] ≤ᵐ[μ] (fun ω => C*ε^codimension κ*W r ω) ∧
-      μ[(fun ω => r^codimension κ*W ε ω*H ω) | hσ.measurableSpace] ≤ᵐ[μ]
-        (fun ω => r^codimension κ*W r ω) ∧
-      μ[B | hσ.measurableSpace] ≤ᵐ[μ]
-        (fun ω => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1))
-
-lemma source_mesh_parameters (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
-    Nonempty (SourceMeshParameters κ) := by
-  obtain ⟨C,Cbad,c,q,hC,hCb,hc,hq,hq1,hmean⟩ := sourceSpatialWindowMeans_proved κ hκ hκ8
-  have hp : 0 < codimension κ := by dsimp [codimension,dimension]; linarith
-  have hp1 : codimension κ < 1 := by dsimp [codimension,dimension]; linarith
-  obtain ⟨u,β,hu,huq,hratio,hβdef,hβ,htheta⟩ := source_mesh_power_growth hp hp1 hq hq1
-  obtain ⟨L,hL,hlarge⟩ := source_mesh_choose_large (A := Cbad*q^(-codimension κ)) hc
-  refine ⟨⟨C,Cbad,c,q,u,L,hC,hCb,hc,hq,hq1,hu,huq,hratio,hL,?_,hmean⟩⟩
-  calc
-    Cbad*L^(-c)*q^(-codimension κ) = (Cbad*q^(-codimension κ))*L^(-c) := by ring
-    _ ≤ 1/2 := hlarge
-
-lemma source_mesh_choose_delta (η L : ℝ) (hη : 0 < η) (hL : 0 < L) :
-    ∃ δ₀ : ℝ, 0 < δ₀ ∧ δ₀ < η/2 ∧ δ₀ ≤ 1 ∧ 3*L*δ₀ ≤ 1 := by
-  let δ₀ := min (min (η/4) (1/2)) (1/(6*L))
-  have hpos : 0 < δ₀ := by dsimp [δ₀]; positivity
-  refine ⟨δ₀,hpos,?_,?_,?_⟩
-  · have := (min_le_left (min (η/4) (1/2)) (1/(6*L))).trans (min_le_left _ _)
-    change δ₀ ≤ η/4 at this
-    linarith
-  · exact ((min_le_left _ _).trans (min_le_right _ _)).trans (by norm_num)
-  · have hh : δ₀ ≤ 1/(6*L) := min_le_right _ _
-    have hh' := (le_div_iff₀ (by positivity : 0 < 6*L)).mp hh
-    nlinarith
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_grid_step
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) (m : SourceMeshParameters κ)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
-    (r δ : ℝ) (hr : 0 < r) (hrδ : r ≤ δ) (hR : 3*m.L*δ ≤ 1)
-    (hy : ∀ z ∈ D, δ < z.im) (n : ℕ)
-    (E : Set a.samples) (hE : MeasurableSet E)
-    (hEs : ∀ z, MeasurableSet[(source_collarStart_stopping a D (r*m.q^n)
-      (fun z hz => (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n).trans (hy z hz).le) z).measurableSpace] E) :
-    ∃ F : Set a.samples, MeasurableSet F ∧
-      ((P.comap Subtype.val).restrict E) F ≤ ENNReal.ofReal
-        (3*(P.comap Subtype.val).real E*Real.exp (-(1/(512*(m.q^(-codimension κ))^2*m.L^2))*
-          δ^(-2:ℝ)*(((m.q^codimension κ/(m.u*m.q))^2)^n/(n+1:ℝ)^4))) ∧
-      ∀ ω ∉ F,
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^(n+1)) D ≤
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^n) D+
-        ((m.Cbad*m.L^(-m.c)*m.q^(-codimension κ))*(m.u^(-m.c))^n)*sourceMeshCount (codimension κ) D (γ ω) (r*m.q^n)+
-        (2*m.q^(-codimension κ))*δ^(-codimension κ)/(n+1:ℝ)^2 ∧
-      sourceMeshCount (codimension κ) D (γ ω) (r*m.q^(n+1)) ≤
-        m.C*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^n) D+
-        ((m.Cbad*m.L^(-m.c)*m.q^(-codimension κ))*(m.u^(-m.c))^n)*sourceMeshCount (codimension κ) D (γ ω) (r*m.q^n)+
-        (2*m.q^(-codimension κ))*δ^(-codimension κ)/(n+1:ℝ)^2 := by
-  have hp1 : codimension κ < 1 := by dsimp [codimension,dimension]; linarith
-  have hk : 0 < κ := by linarith
-  have hδ : 0 < δ := hr.trans_le hrδ
-  have hL : 0 < m.L := lt_trans zero_lt_one m.hL
-  have hu : 0 < m.u := lt_trans zero_lt_one m.hu
-  have hrn : 0 < r*m.q^n := mul_pos hr (pow_pos m.hq _)
-  have hy' (z : ℂ) (hz : z ∈ D) : r*m.q^n < z.im :=
-    (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n).trans_lt (hy z hz)
-  have hLn : 1 < m.L*m.u^n := m.hL.trans_le
-    (by simpa using mul_le_mul_of_nonneg_left (one_le_pow₀ m.hu.le) hL.le)
-  have hRn : 3*((m.L*m.u^n)*(r*m.q^n)) ≤ 1 :=
-    (mul_le_mul_of_nonneg_left (source_mesh_radius_bound m.hq hu m.huq.le hL hr hrδ n) (by norm_num)).trans (by simpa only [mul_assoc] using hR)
-  obtain ⟨F,hFm,hFt,hFg⟩ := source_mesh_step a hk hκ8 D hD c₀ hDc m.q (r*m.q^n) (m.L*m.u^n)
-    m.C m.Cbad m.c (δ^(-codimension κ)*(r*m.q^n)^codimension κ/(n+1:ℝ)^2)
-    m.hq m.hq1 hrn hLn hRn m.hC.le m.hCb.le (by positivity) (fun z hz => (hy' z hz).le)
-    (fun z hz => by
-      obtain ⟨hs,hH,hV,hB⟩ := m.means Ω ‹MeasurableSpace Ω› P inferInstance γ a z
-        (m.q*(r*m.q^n)) (r*m.q^n) (m.L*m.u^n) (mul_pos m.hq hrn) le_rfl (hy' z hz) hLn
-      exact ⟨hH,hV,hB⟩) E hE hEs
-  refine ⟨F,hFm,hFt.trans ?_,?_⟩
-  · apply ENNReal.ofReal_le_ofReal
-    apply mul_le_mul_of_nonneg_left _ (by positivity)
-    apply Real.exp_le_exp.mpr
-    exact source_mesh_grid_exponent hp1 m.hq hu hL hr hrδ (Real.rpow_pos_of_pos m.hq _) n
-  · intro ω hω
-    have hh := hFg ω hω
-    have hrnext : m.q*(r*m.q^n) = r*m.q^(n+1) := by rw [pow_succ]; ring
-    rwa [hrnext,source_mesh_grid_coefficient hL hu,
-      source_mesh_grid_error m.hq hr hδ] at hh
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.PhysicalMeshWindowCore
+import OAI.Probability.SLE.WindowBGeometry
+
+namespace OAI
+
+/-! Physical Mesh Past for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_shift_integral (c : ℂ) (D : Set ℂ) (f : ℂ → ℝ) :
+    (∫ z in ((fun z : ℂ => z+c) ⁻¹' D), f (z+c)) = ∫ z in D, f z := by
+  exact (measurePreserving_add_right volume c).setIntegral_preimage_emb
+    (MeasurableEquiv.addRight c).measurableEmbedding f D
+
+lemma source_mesh_shifted_weighted_tail
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ} (c : ℂ)
+    (h : SourceDisjointWindows P ℱ ((fun z : ℂ => z+c) ⁻¹' D) r R K γ
+      (fun z => σ (z+c)) (fun z => τ (z+c)) (fun z => Y (z+c)) (fun z => μ (z+c)))
+    (hs : ∀ z, IsStoppingTime ℱ (σ z))
+    (hK : 0 < K) {b : ℝ} (hb : 0 < b)
+    {W : Ω → ℝ} (hWa : Measurable[mΩ] W)
+    (hWs : ∀ z, Measurable[(hs z).measurableSpace] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω*(if b < ∫ z in D, (Y z ω-μ z ω) then (1:ℝ) else 0) ∂P) ≤
+      (∫ ω, W ω ∂P)*Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2)) := by
+  have hh := window_source_weighted_tail h hK hb hWa (fun z => hWs (z+c)) hW0 hWb
+  have he (ω : Ω) : (∫ z in ((fun z : ℂ => z+c) ⁻¹' D), (Y (z+c) ω-μ (z+c) ω)) =
+      ∫ z in D, (Y z ω-μ z ω) := source_mesh_shift_integral c D (fun z => Y z ω-μ z ω)
+  simpa only [he] using hh
+
+lemma source_mesh_center_before
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (c : ℂ) (hD : D ⊆ ball c 1) (η r : ℝ)
+    (hη : 0 < η) (hr : r < η/2) (hc : 1+η ≤ c.im)
+    (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples) :
+    spatialHitTime (γ ω) c (1+η/2) ≤ sourceCollarStart D (γ ω) z r := by
+  by_cases hz : z ∈ D
+  · simp only [sourceCollarStart,ite_eq_left hz]
+    by_cases hn : liveSpatialHitTime (γ ω) z r = ⊤
+    · rw [hn]; exact le_top
+    obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hn
+    have hcontact := (source_liveHit_mem (a.trace_continuous ω) z r
+      (source_window_origin_bound a z r (hy z hz) ω) T hT.symm).2
+    have hd : dist (γ ω T) c < 1+η/2 := by
+      have hz' := hD hz
+      change dist z c < 1 at hz'
+      have ht := dist_triangle (γ ω T) z c
+      linarith
+    have hstart : 1+η/2 ≤ dist (γ ω 0) c := by
+      rw [(a.witness ω).2.1,dist_zero_left]
+      have hi : c.im ≤ ‖c‖ := (le_abs_self _).trans (Complex.abs_im_le_norm c)
+      linarith
+    rw [← hT]
+    exact (spatialHitTime_lt_of_dist_lt (a.trace_continuous ω) c _ hstart T hd).le
+  · rw [sourceCollarStart,ite_eq_right hz]
+    exact le_top
+
+lemma source_mesh_center_stopping
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (c : ℂ) (η : ℝ) :
+    IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) c (1+η/2)) := by
+  let X (t : ℝ≥0) (ω : a.samples) := dist (γ ω t) c - (1+η/2)
+  have ha : Adapted a.filtration X := by
+    intro t
+    let : MeasurableSpace a.samples := a.filtration t
+    exact ((a.trace_adapted t).dist measurable_const).sub_const _
+  exact continuous_zero_hit_stopping a.filtration X ha
+    (fun ω => ((a.trace_continuous ω).dist continuous_const).sub continuous_const) 0
+
+lemma source_mesh_free_im {γ : ℝ≥0 → ℂ} {s : ℝ≥0} {c : ℂ} {η : ℝ}
+    (hη : 0 < η) (hfree : closedBall c (1+η) ⊆ survivingDomain γ s) :
+    1+η < c.im := by
+  let z := c-((1+η:ℝ):ℂ)*Complex.I
+  have hz : z ∈ closedBall c (1+η) := by
+    rw [mem_closedBall,dist_eq_norm]
+    simp only [z,sub_sub_cancel_left,norm_neg,norm_mul,Complex.norm_real,Complex.norm_I,mul_one]
+    rw [Real.norm_eq_abs,abs_of_pos (by linarith)]
+  have hi := (hfree hz).1.1
+  change 0 < z.im at hi
+  simpa [z] using hi
+
+lemma source_mesh_free_before_center {γ : ℝ≥0 → ℂ} (hγ : Continuous γ)
+    (s : ℝ≥0) (c : ℂ) (η : ℝ) (hη : 0 < η)
+    (hfree : closedBall c (1+η) ⊆ survivingDomain γ s) :
+    (s : WithTop ℝ≥0) ≤ spatialHitTime γ c (1+η/2) := by
+  by_contra! hn
+  obtain ⟨t,ht,hcontact⟩ := (spatialHitTime_le_iff hγ c (1+η/2) s).mp hn.le
+  have hin : γ t ∈ closedBall c (1+η) := by
+    rw [mem_closedBall,hcontact]; linarith
+  exact (hfree hin).1.2 (mem_image_of_mem γ ht)
+
+lemma source_past_indicator_at_start
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ}
+    {ρ τ σ : Ω → WithTop ℝ≥0} (hρ : IsStoppingTime ℱ ρ)
+    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) (hle : ∀ ω, τ ω ≤ σ ω)
+    (A : Set Ω) (hA : MeasurableSet[hρ.measurableSpace] A) :
+    Measurable[hσ.measurableSpace] (fun ω => if ω ∈ A ∧ ρ ω ≤ τ ω then (1:ℝ) else 0) := by
+  have hB : MeasurableSet[hσ.measurableSpace] {ω | ρ ω ≤ τ ω} :=
+    (hτ.measurableSpace_mono hσ hle) _ (hρ.measurableSet_stopping_time_le hτ)
+  have hI : Measurable[hρ.measurableSpace] (fun ω => if ω ∈ A then (1:ℝ) else 0) :=
+    Measurable.ite hA measurable_const measurable_const
+  have hh := window_measurable_on_event hσ hρ hI hB (fun ω hω => hω.trans (hle ω)) 0
+  convert hh using 1
+  funext ω
+  by_cases ha : ω ∈ A <;> by_cases hb : ρ ω ≤ τ ω <;> simp [ha,hb]
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_integrated_step
+    {X : Type*} [MeasurableSpace X] (ν : Measure X)
+    (W W' I I' H V B : X → ℝ) (p q r C A b : ℝ)
+    (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hp : 0 < p)
+    (_hW : Integrable W ν) (hW' : Integrable W' ν)
+    (_hI : Integrable I ν) (hI' : Integrable I' ν)
+    (hH : Integrable H ν) (hV : Integrable V ν) (hB : Integrable B ν)
+    (hb : 0 ≤ b)
+    (hmissI : ∀ᵐ z ∂ν, I' z ≤ H z+B z)
+    (hmissW : ∀ᵐ z ∂ν, W' z ≤ r^(-p)*V z+(q*r)^(-p)*B z)
+    (hmeanH : (∫ z, H z ∂ν) ≤ C*(q*r)^p*(∫ z, W z ∂ν)+b)
+    (hmeanV : (∫ z, V z ∂ν) ≤ r^p*(∫ z, W z ∂ν)+b)
+    (hmeanB : (∫ z, B z ∂ν) ≤ A*(∫ z, I z ∂ν)+b) :
+    (∫ z, W' z ∂ν) ≤ (∫ z, W z ∂ν)+(A*q^(-p))*(r^(-p)*(∫ z, I z ∂ν))+
+       (2*q^(-p))*r^(-p)*b ∧
+    (q*r)^(-p)*(∫ z, I' z ∂ν) ≤ C*(∫ z, W z ∂ν)+
+       (A*q^(-p))*(r^(-p)*(∫ z, I z ∂ν))+(2*q^(-p))*r^(-p)*b := by
+  have hrp : 0 ≤ r^(-p) := (Real.rpow_pos_of_pos hr _).le
+  have hqp : 0 ≤ (q*r)^(-p) := (Real.rpow_pos_of_pos (mul_pos hq hr) _).le
+  have hq1p : 1 ≤ q^(-p) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos hq hq1.le (neg_nonpos.mpr hp.le)
+  have hcancel : r^(-p)*r^p = 1 := by rw [←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero]
+  have hcancelq : (q*r)^(-p)*(q*r)^p = 1 := by rw [←Real.rpow_add (mul_pos hq hr),neg_add_cancel,Real.rpow_zero]
+  have hmul : (q*r)^(-p) = q^(-p)*r^(-p) := Real.mul_rpow hq.le hr.le
+  constructor
+  · calc
+      _ ≤ ∫ z, (r^(-p)*V z+(q*r)^(-p)*B z) ∂ν :=
+        integral_mono_ae hW' ((hV.const_mul _).add (hB.const_mul _)) hmissW
+      _ = r^(-p)*(∫ z, V z ∂ν)+(q*r)^(-p)*(∫ z, B z ∂ν) := by
+        rw [integral_add (hV.const_mul _) (hB.const_mul _),integral_const_mul,integral_const_mul]
+      _ ≤ r^(-p)*(r^p*(∫ z, W z ∂ν)+b)+(q*r)^(-p)*(A*(∫ z,I z ∂ν)+b) :=
+        add_le_add (mul_le_mul_of_nonneg_left hmeanV hrp) (mul_le_mul_of_nonneg_left hmeanB hqp)
+      _ ≤ _ := by rw [mul_add,←mul_assoc,hcancel,one_mul,hmul]; nlinarith [mul_nonneg (sub_nonneg.mpr hq1p) (mul_nonneg hrp hb)]
+  · calc
+      _ ≤ (q*r)^(-p)*(∫ z, (H z+B z) ∂ν) := mul_le_mul_of_nonneg_left
+        (integral_mono_ae hI' (hH.add hB) hmissI) hqp
+      _ = (q*r)^(-p)*((∫ z,H z ∂ν)+(∫ z,B z ∂ν)) := by rw [integral_add hH hB]
+      _ ≤ (q*r)^(-p)*(C*(q*r)^p*(∫ z,W z ∂ν)+b+(A*(∫ z,I z ∂ν)+b)) :=
+        mul_le_mul_of_nonneg_left (add_le_add hmeanH hmeanB) hqp
+      _ = _ := by
+        calc
+          _ = C*((q*r)^(-p)*(q*r)^p)*(∫ z,W z ∂ν)+(q*r)^(-p)*(A*(∫ z,I z ∂ν))+2*(q*r)^(-p)*b := by ring
+          _ = _ := by rw [hcancelq,hmul]; ring
+
+lemma source_collar_missed
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (D : Set ℂ) (q r L : ℝ)
+    (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hL : 1 < L)
+    (hy : ∀ z ∈ D, r ≤ z.im) (ω : a.samples) (z : ℂ) (hz : z ∈ D) :
+    (if sourceCollarStart D (γ ω) z (q*r) = ⊤ then (0:ℝ) else 1) ≤
+      sourceCollarH D (γ ω) z r (q*r) (L*r)+sourceCollarB D (γ ω) z r (q*r) (L*r) ∧
+    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r) ≤
+      r^(-codimension κ)*(r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceCollarH D (γ ω) z r (q*r) (L*r))+
+      (q*r)^(-codimension κ)*sourceCollarB D (γ ω) z r (q*r) (L*r) := by
+  have hh := sourceSpatialWindowGeometry_proved (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) (a.F ω) ((a.continuous_B ω).const_mul _) (a.witness ω) κ hκ hκ8
+    z (q*r) r (L*r) (mul_pos hq hr) (by nlinarith) (by nlinarith) (hy z hz)
+  simp only [sourceCollarStart,sourceCollarH,sourceCollarB,ite_eq_left hz]
+  refine ⟨hh.2.2.2.2.2.2.2.2.2.2.1,?_⟩
+  have he : r^(-codimension κ)*r^codimension κ = 1 := by
+    rw [←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero]
+  simpa only [←mul_assoc,he,one_mul] using hh.2.2.2.2.2.2.2.2.2.2.2
+
+lemma source_mesh_bounded_integrable {X : Type*} [MeasurableSpace X]
+    {ν : Measure X} [IsFiniteMeasure ν] {f : X → ℝ} {K : ℝ}
+    (hm : Measurable f) (hb : ∀ x, 0 ≤ f x ∧ f x ≤ K) : Integrable f ν := by
+  apply Integrable.of_bound hm.aestronglyMeasurable K
+  exact ae_of_all _ (fun x => by simpa only [Real.norm_eq_abs,abs_of_nonneg (hb x).1] using (hb x).2)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_shifted_integral_measurable
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ} (c : ℂ)
+    (h : SourceDisjointWindows P ℱ ((fun z : ℂ => z+c) ⁻¹' D) r R K γ
+      (fun z => σ (z+c)) (fun z => τ (z+c)) (fun z => Y (z+c)) (fun z => μ (z+c))) :
+    Measurable (fun ω => ∫ z in D, (Y z ω-μ z ω)) := by
+  have hm := window_source_integral_measurable h
+  have he (ω : Ω) : (∫ z in ((fun z : ℂ => z+c) ⁻¹' D), (Y (z+c) ω-μ (z+c) ω)) =
+      ∫ z in D, (Y z ω-μ z ω) := source_mesh_shift_integral c D (fun z => Y z ω-μ z ω)
+  simpa only [he] using hm
+
+lemma source_mesh_shifted_restricted_tail
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ} (c : ℂ)
+    (h : SourceDisjointWindows P ℱ ((fun z : ℂ => z+c) ⁻¹' D) r R K γ
+      (fun z => σ (z+c)) (fun z => τ (z+c)) (fun z => Y (z+c)) (fun z => μ (z+c)))
+    (hs : ∀ z, IsStoppingTime ℱ (σ z))
+    (hK : 0 < K) {b : ℝ} (hb : 0 < b)
+    (E : Set Ω) (hE : MeasurableSet[mΩ] E) (hEs : ∀ z, MeasurableSet[(hs z).measurableSpace] E) :
+    (P.restrict E) {ω | b < ∫ z in D, (Y z ω-μ z ω)} ≤
+      ENNReal.ofReal (P.real E*Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2))) := by
+  let F := {ω | b < ∫ z in D, (Y z ω-μ z ω)}
+  have hF : MeasurableSet F := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable c h)
+  have hWm : Measurable (fun ω => if ω ∈ E then (1:ℝ) else 0) :=
+    Measurable.ite hE measurable_const measurable_const
+  have hWs : ∀ z, Measurable[(hs z).measurableSpace] (fun ω => if ω ∈ E then (1:ℝ) else 0) :=
+    fun z => Measurable.ite (hEs z) measurable_const measurable_const
+  have hh := source_mesh_shifted_weighted_tail c h hs hK hb hWm hWs
+    (fun ω => by split_ifs <;> norm_num) ⟨1,fun ω => by split_ifs <;> norm_num⟩
+  have hi : (fun ω => (if ω ∈ E then (1:ℝ) else 0)*(if ω ∈ F then (1:ℝ) else 0)) =
+      (F∩E).indicator 1 := by
+    funext ω; by_cases he : ω ∈ E <;> by_cases hf : ω ∈ F <;> simp [he,hf]
+  change (∫ ω, (if ω ∈ E then (1:ℝ) else 0)*(if ω ∈ F then (1:ℝ) else 0) ∂P) ≤ _ at hh
+  rw [hi,integral_indicator_one (hF.inter hE)] at hh
+  have hiE : (fun ω => if ω ∈ E then (1:ℝ) else 0) = E.indicator 1 := rfl
+  rw [hiE,integral_indicator_one hE] at hh
+  rw [Measure.restrict_apply hF]
+  exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity)).mpr hh
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+noncomputable def sourceMeshCount (p : ℝ) (D : Set ℂ) (γ : ℝ≥0 → ℂ) (r : ℝ) : ℝ :=
+  r^(-p)*∫ z in D, if sourceCollarStart D γ z r = ⊤ then (0:ℝ) else 1
+
+lemma source_mesh_I_integrable
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (hD : MeasurableSet D) [IsFiniteMeasure (volume.restrict D)]
+    (v : ℝ) (hy : ∀ z ∈ D, v ≤ z.im) (ω : a.samples) :
+    IntegrableOn (fun z => if sourceCollarStart D (γ ω) z v = ⊤ then (0:ℝ) else 1) D := by
+  have hm : Measurable (fun p : ℂ × a.samples =>
+      if sourceCollarStart D (γ p.2) p.1 v = ⊤ then (0:ℝ) else 1) :=
+    Measurable.ite (measurableSet_eq_fun (source_collarStart_joint a D hD v hy) measurable_const)
+      measurable_const measurable_const
+  exact source_mesh_bounded_integrable (K := 1)
+    (hm.comp (measurable_id.prodMk (measurable_const : Measurable (fun _ : ℂ => ω)))) (fun z => by split_ifs <;> norm_num)
+
+lemma source_mesh_integral_recurrences
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
+    (q r L C Cbad c b : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hL : 1 < L) (hb : 0 ≤ b) (hy : ∀ z ∈ D, r ≤ z.im) (ω : a.samples) :
+    let p := codimension κ
+    let W := fun v z => spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
+    let Ic := fun v z => if sourceCollarStart D (γ ω) z v = ⊤ then (0:ℝ) else 1
+    let H := fun z => sourceCollarH D (γ ω) z r (q*r) (L*r)
+    let B := fun z => sourceCollarB D (γ ω) z r (q*r) (L*r)
+    let V := fun z => r^p*W (q*r) z*H z
+    let μH := fun z => C*(q*r)^p*(if z ∈ D then W r z else 0)
+    let μV := fun z => r^p*(if z ∈ D then W r z else 0)
+    let μB := fun z => Cbad*L^(-c)*Ic r z
+    (∫ z in D, (H z-μH z)) ≤ b → (∫ z in D, (V z-μV z)) ≤ b → (∫ z in D, (B z-μB z)) ≤ b →
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (q*r) D ≤
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
+        (Cbad*L^(-c)*q^(-p))*sourceMeshCount p D (γ ω) r+(2*q^(-p))*r^(-p)*b ∧
+      sourceMeshCount p D (γ ω) (q*r) ≤
+        C*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
+        (Cbad*L^(-c)*q^(-p))*sourceMeshCount p D (γ ω) r+(2*q^(-p))*r^(-p)*b := by
+  intro p W Ic H B V μH μV μB hFH hFV hFB
+  change (∫ z in D, W (q*r) z) ≤ (∫ z in D, W r z)+
+    (Cbad*L^(-c)*q^(-p))*(r^(-p)*(∫ z in D, Ic r z))+(2*q^(-p))*r^(-p)*b ∧
+    (q*r)^(-p)*(∫ z in D, Ic (q*r) z) ≤ C*(∫ z in D, W r z)+
+    (Cbad*L^(-c)*q^(-p))*(r^(-p)*(∫ z in D, Ic r z))+(2*q^(-p))*r^(-p)*b
+  have hp : 0 < p := by dsimp [p,codimension,dimension]; linarith
+  have hfin : volume D ≠ ∞ := ne_top_of_le_ne_top (by finiteness : volume (ball c₀ 1) ≠ ∞) (measure_mono hDc)
+  let : IsFiniteMeasure (volume.restrict D) := ⟨by simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
+  have hqr : q*r ≤ r := (mul_le_mul_of_nonneg_right hq1.le hr.le).trans_eq (one_mul _)
+  have hWi (v : ℝ) (hv : 0 < v) (hvr : v ≤ r) : IntegrableOn (fun z => W v z) D :=
+    a.spatialGreenWeight_spatial_integrable hκ hκ8 v hv D hD hfin (fun z hz => hvr.trans (hy z hz)) ω
+  have hIi (v : ℝ) (hvr : v ≤ r) : IntegrableOn (fun z => Ic v z) D :=
+    source_mesh_I_integrable a D hD v (fun z hz => hvr.trans (hy z hz)) ω
+  have hHi : IntegrableOn (fun z => H z) D := source_mesh_bounded_integrable
+    ((source_collarHB_joint a D hD r (q*r) (L*r) hy).1.comp (measurable_id.prodMk (measurable_const : Measurable (fun _ : ℂ => ω))))
+    (fun z => (source_collar_HB_bounds D (γ ω) z r (q*r) (L*r)).1)
+  have hBi : IntegrableOn (fun z => B z) D := source_mesh_bounded_integrable
+    ((source_collarHB_joint a D hD r (q*r) (L*r) hy).2.comp (measurable_id.prodMk (measurable_const : Measurable (fun _ : ℂ => ω))))
+    (fun z => (source_collar_HB_bounds D (γ ω) z r (q*r) (L*r)).2)
+  have hVm : Measurable (fun p : ℂ × a.samples => r^codimension κ*spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t p.2) (γ p.2) (a.G p.2) p.1 (q*r)*sourceCollarH D (γ p.2) p.1 r (q*r) (L*r)) := by
+    simpa only [add_zero] using source_collar_V_joint a D hD 0 q r L hy
+  have hmsec : Measurable (fun z : ℂ => (z,ω)) := measurable_id.prodMk measurable_const
+  have hVsec := hVm.comp hmsec
+  simp only [Function.comp_def] at hVsec
+  have hVi : IntegrableOn V D := source_mesh_bounded_integrable hVsec
+    (fun z => source_collar_V_bounds a hκ hκ8 D q r (L*r) hq hq1 hr hy z ω)
+  have hWceq : (fun z => if z ∈ D then W r z else 0) =ᵐ[volume.restrict D] (fun z => W r z) := by
+    filter_upwards [ae_restrict_mem hD] with z hz; simp only [ite_eq_left hz]
+  have hWci : IntegrableOn (fun z => if z ∈ D then W r z else 0) D := (hWi r hr le_rfl).congr hWceq.symm
+  have hiH : (∫ z in D, μH z) = C*(q*r)^p*(∫ z in D, W r z) := by
+    rw [show (∫ z in D, μH z) = C*(q*r)^p*(∫ z in D, if z ∈ D then W r z else 0) from integral_const_mul _ _,integral_congr_ae hWceq]
+  have hiV : (∫ z in D, μV z) = r^p*(∫ z in D, W r z) := by
+    rw [show (∫ z in D, μV z) = r^p*(∫ z in D, if z ∈ D then W r z else 0) from integral_const_mul _ _,integral_congr_ae hWceq]
+  have hiB : (∫ z in D, μB z) = Cbad*L^(-c)*(∫ z in D, Ic r z) := integral_const_mul _ _
+  rw [integral_sub hHi (hWci.const_mul _),hiH] at hFH
+  rw [integral_sub hVi (hWci.const_mul _),hiV] at hFV
+  rw [integral_sub hBi ((hIi r le_rfl).const_mul _),hiB] at hFB
+  have hmiss : ∀ᵐ z ∂volume.restrict D,
+      Ic (q*r) z ≤ H z+B z ∧ W (q*r) z ≤ r^(-p)*V z+(q*r)^(-p)*B z := by
+    filter_upwards [ae_restrict_mem hD] with z hz
+    exact source_collar_missed a hκ hκ8 D q r L hq hq1 hr hL hy ω z hz
+  exact source_mesh_integrated_step (volume.restrict D) (fun z => W r z) (fun z => W (q*r) z)
+    (fun z => Ic r z) (fun z => Ic (q*r) z) (fun z => H z) (fun z => V z) (fun z => B z)
+    p q r C (Cbad*L^(-c)) b hq hq1 hr hp (hWi r hr le_rfl) (hWi (q*r) (mul_pos hq hr) hqr)
+    (hIi r le_rfl) (hIi (q*r) hqr) hHi hVi hBi hb (hmiss.mono fun _ h => h.1) (hmiss.mono fun _ h => h.2)
+    (sub_le_iff_le_add'.mp hFH) (sub_le_iff_le_add'.mp hFV) (sub_le_iff_le_add'.mp hFB)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_step
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
+    (q r L C Cbad c b : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hL : 1 < L) (hR : 3*(L*r) ≤ 1) (hC : 0 ≤ C) (hCb : 0 ≤ Cbad) (hb : 0 < b)
+    (hy : ∀ z ∈ D, r ≤ z.im)
+    (hmean : ∀ z (hz : z ∈ D),
+      let μ := P.comap (Subtype.val : a.samples → Ω)
+      let W := fun v (ω : a.samples) => spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
+      let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r (q*r) (L*r)
+      let B := fun ω : a.samples => sourceSpatialWindowB (γ ω) z r (q*r) (L*r)
+      let hs := source_liveHit_stopping a z r (hy z hz)
+      μ[H | hs.measurableSpace] ≤ᵐ[μ] (fun ω => C*(q*r)^codimension κ*W r ω) ∧
+      μ[(fun ω => r^codimension κ*W (q*r) ω*H ω) | hs.measurableSpace] ≤ᵐ[μ]
+        (fun ω => r^codimension κ*W r ω) ∧
+      μ[B | hs.measurableSpace] ≤ᵐ[μ]
+        (fun ω => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1)))
+    (E : Set a.samples) (hE : MeasurableSet E)
+    (hEs : ∀ z, MeasurableSet[(source_collarStart_stopping a D r hy z).measurableSpace] E) :
+    ∃ F : Set a.samples, MeasurableSet F ∧
+      ((P.comap Subtype.val).restrict E) F ≤ ENNReal.ofReal
+        (3*(P.comap Subtype.val).real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-codimension κ))^2*(L*r)^2))) ∧
+      ∀ ω ∉ F,
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (q*r) D ≤
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
+        (Cbad*L^(-c)*q^(-codimension κ))*sourceMeshCount (codimension κ) D (γ ω) r+
+        (2*q^(-codimension κ))*r^(-codimension κ)*b ∧
+      sourceMeshCount (codimension κ) D (γ ω) (q*r) ≤
+        C*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r D+
+        (Cbad*L^(-c)*q^(-codimension κ))*sourceMeshCount (codimension κ) D (γ ω) r+
+        (2*q^(-codimension κ))*r^(-codimension κ)*b := by
+  let p := codimension κ
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let W (v : ℝ) (z : ℂ) (ω : a.samples) := spatialGreenWeight κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
+  let Ic (v : ℝ) (z : ℂ) (ω : a.samples) : ℝ := if sourceCollarStart D (γ ω) z v = ⊤ then 0 else 1
+  let H (z : ℂ) (ω : a.samples) := sourceCollarH D (γ ω) z r (q*r) (L*r)
+  let B (z : ℂ) (ω : a.samples) := sourceCollarB D (γ ω) z r (q*r) (L*r)
+  let V (z : ℂ) (ω : a.samples) := r^p*W (q*r) z ω*H z ω
+  let μH (z : ℂ) (ω : a.samples) := C*(q*r)^p*(if z ∈ D then W r z ω else 0)
+  let μV (z : ℂ) (ω : a.samples) := r^p*(if z ∈ D then W r z ω else 0)
+  let μB (z : ℂ) (ω : a.samples) := Cbad*L^(-c)*Ic r z ω
+  let FH := {ω | b < ∫ z in D, (H z ω-μH z ω)}
+  let FV := {ω | b < ∫ z in D, (V z ω-μV z ω)}
+  let FB := {ω | b < ∫ z in D, (B z ω-μB z ω)}
+  let σ := fun z (ω : a.samples) => sourceCollarStart D (γ ω) z r
+  let τ := fun z (ω : a.samples) => sourceCollarFinish D (γ ω) z r (q*r) (L*r)
+  have hWinH := source_mesh_H_window a hκ hκ8 D hD c₀ hDc q r L C hq hq1 hr hL hR hC hy (fun z hz => (hmean z hz).1)
+  have hWinV := source_mesh_V_window a hκ hκ8 D hD c₀ hDc q r L hq hq1 hr hL hR hy (fun z hz => (hmean z hz).2.1)
+  have hWinB := source_mesh_B_window a D hD c₀ hDc q r L Cbad c hq hq1 hr hL hR hCb hy (fun z hz => (hmean z hz).2.2)
+  have hmH : MeasurableSet FH := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable (σ := σ) (τ := τ) (Y := H) (μ := μH) c₀ hWinH)
+  have hmV : MeasurableSet FV := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable (σ := σ) (τ := τ) (Y := V) (μ := μV) c₀ hWinV)
+  have hmB : MeasurableSet FB := measurableSet_lt measurable_const (source_mesh_shifted_integral_measurable (σ := σ) (τ := τ) (Y := B) (μ := μB) c₀ hWinB)
+  have hp : 0 < p := by dsimp [p,codimension,dimension]; linarith
+  have hK1 : 1 ≤ q^(-p) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos hq hq1.le (neg_nonpos.mpr hp.le)
+  have hK : 0 < q^(-p) := Real.rpow_pos_of_pos hq _
+  have hTailH := source_mesh_shifted_restricted_tail (σ := σ) (τ := τ) (Y := H) (μ := μH) c₀ hWinH
+    (source_collarStart_stopping a D r hy) zero_lt_one hb E hE hEs
+  have hTailV := source_mesh_shifted_restricted_tail (σ := σ) (τ := τ) (Y := V) (μ := μV) c₀ hWinV
+    (source_collarStart_stopping a D r hy) hK hb E hE hEs
+  have hTailB := source_mesh_shifted_restricted_tail (σ := σ) (τ := τ) (Y := B) (μ := μB) c₀ hWinB
+    (source_collarStart_stopping a D r hy) zero_lt_one hb E hE hEs
+  have hexp : Real.exp (-(1/512:ℝ)*b^2/(1^2*(L*r)^2)) ≤
+      Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2)) := by
+    apply Real.exp_le_exp.mpr
+    have hr0 : 0 < L*r := mul_pos (by linarith) hr
+    have hden : (L*r)^2 ≤ (q^(-p))^2*(L*r)^2 := by nlinarith [sq_nonneg (q^(-p)-1),sq_nonneg (L*r)]
+    have hh := div_le_div_of_nonneg_left (show 0 ≤ (1/512:ℝ)*b^2 by positivity) (sq_pos_of_pos hr0) hden
+    simpa only [one_pow,one_mul,neg_mul,neg_div] using neg_le_neg hh
+  have hTail (F : Set a.samples) (hh : (μ.restrict E) F ≤ ENNReal.ofReal
+      (μ.real E*Real.exp (-(1/512:ℝ)*b^2/(1^2*(L*r)^2)))) :
+      (μ.restrict E) F ≤ ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2))) :=
+    hh.trans (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hexp (measureReal_nonneg)))
+  refine ⟨FH∪FV∪FB,(hmH.union hmV).union hmB,?_,?_⟩
+  · calc
+      _ ≤ (μ.restrict E) FH+(μ.restrict E) FV+(μ.restrict E) FB :=
+        (measure_union_le _ _).trans (add_le_add (measure_union_le _ _) le_rfl)
+      _ ≤ ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2)))+
+           ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2)))+
+           ENNReal.ofReal (μ.real E*Real.exp (-(1/512:ℝ)*b^2/((q^(-p))^2*(L*r)^2))) :=
+        add_le_add (add_le_add (hTail FH hTailH) hTailV) (hTail FB hTailB)
+      _ = _ := by rw [←ENNReal.ofReal_add (by positivity) (by positivity),←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
+  · intro ω hω
+    have hFH : (∫ z in D, (H z ω-μH z ω)) ≤ b := le_of_not_gt (fun hh => hω (Or.inl (Or.inl hh)))
+    have hFV : (∫ z in D, (V z ω-μV z ω)) ≤ b := le_of_not_gt (fun hh => hω (Or.inl (Or.inr hh)))
+    have hFB : (∫ z in D, (B z ω-μB z ω)) ≤ b := le_of_not_gt (fun hh => hω (Or.inr hh))
+    exact source_mesh_integral_recurrences a hκ hκ8 D hD c₀ hDc q r L C Cbad c b
+      hq hq1 hr hL hb.le hy ω hFH hFV hFB
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+lemma source_mesh_power_growth {p q : ℝ} (hp : 0 < p) (hp1 : p < 1)
+    (hq : 0 < q) (hq1 : q < 1) :
+    ∃ u β : ℝ, 1 < u ∧ u*q < 1 ∧ 1 < q^p/(u*q) ∧ β = (q^p/(u*q))^2 ∧ 1 < β ∧
+      ∀ c : ℝ, 0 < c → 0 < u^(-c) ∧ u^(-c) < 1 := by
+  have hqp : q < q^p := by
+    have hh := Real.rpow_lt_rpow_of_exponent_gt hq hq1 hp1
+    simpa only [Real.rpow_one] using hh
+  have hqp1 : q^p < 1 := Real.rpow_lt_one hq.le hq1 hp
+  let u := (1+q^p/q)/2
+  have hu : 1 < u := by dsimp [u]; have : 1 < q^p/q := (lt_div_iff₀ hq).mpr (by simpa using hqp); linarith
+  have huq : u*q < q^p := by dsimp [u]; field_simp; nlinarith
+  have hratio : 1 < q^p/(u*q) := (lt_div_iff₀ (mul_pos (by linarith) hq)).mpr (by simpa using huq)
+  refine ⟨u,(q^p/(u*q))^2,hu,huq.trans hqp1,hratio,rfl,by nlinarith,?_⟩
+  intro c hc
+  refine ⟨Real.rpow_pos_of_pos (by linarith) _,?_⟩
+  exact Real.rpow_lt_one_of_one_lt_of_neg hu (neg_neg_of_pos hc)
+
+lemma source_mesh_scale_cancel {p r δ : ℝ} (hr : 0 < r) (_hδ : 0 < δ) (n : ℕ) :
+    r^(-p)*(δ^(-p)*r^p/(n+1:ℝ)^2) = δ^(-p)/(n+1:ℝ)^2 := by
+  have he : r^(-p)*r^p = 1 := by rw [←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero]
+  calc
+    _ = δ^(-p)*(r^(-p)*r^p)/(n+1:ℝ)^2 := by ring
+    _ = _ := by rw [he,mul_one]
+
+lemma source_mesh_radius_bound {q u L r δ : ℝ} (hq : 0 < q) (hu : 0 < u)
+    (huq : u*q ≤ 1) (hL : 0 < L) (hr : 0 < r) (hrδ : r ≤ δ) (n : ℕ) :
+    (L*u^n)*(r*q^n) ≤ L*δ := by
+  have hh : (u*q)^n ≤ 1 := pow_le_one₀ (mul_nonneg hu.le hq.le) huq
+  calc
+    _ = (L*r)*(u*q)^n := by rw [mul_pow]; ring
+    _ ≤ (L*r)*1 := mul_le_mul_of_nonneg_left hh (mul_nonneg hL.le hr.le)
+    _ ≤ L*δ := by rw [mul_one]; exact mul_le_mul_of_nonneg_left hrδ hL.le
+
+lemma source_mesh_alpha {c u L : ℝ} (hL : 0 < L) (hu : 0 < u) (n : ℕ) :
+    (L*u^n)^(-c) = L^(-c)*(u^(-c))^n := by
+  rw [Real.mul_rpow hL.le (pow_nonneg hu.le _),Real.rpow_pow_comm hu.le]
+
+lemma source_mesh_failure_exponent {p q u L r δ K : ℝ}
+    (hp1 : p < 1) (hq : 0 < q) (hu : 0 < u) (hL : 0 < L)
+    (hr : 0 < r) (hrδ : r ≤ δ) (_hK : 0 < K) (n : ℕ) :
+    δ^(-2:ℝ)/(K^2*L^2)*((q^p/(u*q))^2)^n/(n+1:ℝ)^4 ≤
+    (δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)^2/(K^2*((L*u^n)*(r*q^n))^2) := by
+  have hδ : 0 < δ := hr.trans_le hrδ
+  have hn : 0 < (n+1:ℝ) := by positivity
+  have hR : 0 < (L*u^n)*(r*q^n) := by positivity
+  have hrq : 0 < r*q^n := by positivity
+  have hid : (δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)) =
+      (δ^(-p)*r^(p-1))/L*(q^p/(u*q))^n/(n+1:ℝ)^2 := by
+    rw [Real.mul_rpow hr.le (pow_nonneg hq.le _),←Real.rpow_pow_comm hq.le,
+      Real.rpow_sub hr,Real.rpow_one,div_pow,mul_pow]
+    field_simp
+  have hant := Real.rpow_le_rpow_of_nonpos hr hrδ (by linarith : p-1 ≤ 0)
+  have hc : δ⁻¹ ≤ δ^(-p)*r^(p-1) := by
+    calc
+      δ⁻¹ = δ^(-p)*δ^(p-1) := by rw [←Real.rpow_add hδ]; ring_nf; rw [Real.rpow_neg_one]
+      _ ≤ _ := mul_le_mul_of_nonneg_left hant (Real.rpow_nonneg hδ.le _)
+  have ratio : δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2 ≤
+      (δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)) := by
+    rw [hid]
+    exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right
+      (div_le_div_of_nonneg_right hc hL.le) (by positivity)) (by positivity)
+  have hleft : 0 ≤ δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2 := by positivity
+  have hsq : (δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2)^2 ≤
+      ((δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)))^2 := by nlinarith
+  calc
+    _ = (δ⁻¹/L*(q^p/(u*q))^n/(n+1:ℝ)^2)^2/K^2 := by
+      rw [Real.rpow_neg hδ.le,Real.rpow_two]
+      simp only [div_pow,mul_pow,inv_pow,←pow_mul]
+      ring
+    _ ≤ ((δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)/((L*u^n)*(r*q^n)))^2/K^2 :=
+      div_le_div_of_nonneg_right hsq (sq_nonneg K)
+    _ = _ := by rw [div_pow]; ring
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+lemma source_mesh_choose_large {c A : ℝ} (hc : 0 < c) :
+    ∃ L : ℝ, 1 < L ∧ A*L^(-c) ≤ 1/2 := by
+  have ht : Tendsto (fun L : ℝ => A*L^(-c)) atTop (𝓝 0) := by
+    simpa only [mul_zero] using (tendsto_rpow_neg_atTop hc).const_mul A
+  have hh := ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1/2))
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hh
+  refine ⟨max N 2,lt_of_lt_of_le (by norm_num) (le_max_right _ _),(hN _ (le_max_left _ _)).le⟩
+
+lemma source_mesh_select_start {q rf δ : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hf : 0 < rf) (hfδ : rf ≤ δ) :
+    ∃ r : ℝ, ∃ n : ℕ, 0 < r ∧ q*δ ≤ r ∧ r ≤ δ ∧ rf = r*q^n := by
+  have hδ : 0 < δ := hf.trans_le hfδ
+  obtain ⟨n,hn1,hn2⟩ := exists_nat_pow_near_of_lt_one (div_pos hf hδ)
+    ((div_le_one₀ hδ).mpr hfδ) hq hq1
+  have hqn : 0 < q^n := pow_pos hq _
+  refine ⟨rf/q^n,n,div_pos hf hqn,?_,?_,?_⟩
+  · apply (le_div_iff₀ hqn).mpr
+    have hh := (lt_div_iff₀ hδ).mp hn1
+    rw [pow_succ] at hh
+    nlinarith
+  · apply (div_le_iff₀ hqn).mpr
+    have hh := (div_le_iff₀ hδ).mp hn2
+    nlinarith
+  · symm; exact div_mul_cancel₀ _ hqn.ne'
+
+lemma source_mesh_height {z c : ℂ} {η r : ℝ} (hc : 1+η ≤ c.im)
+    (hz : z ∈ ball c 1) (hr : r < η) : r < z.im := by
+  have hdist : abs (z.im-c.im) ≤ dist z c := by
+    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-c)
+  have hz' : dist z c < 1 := hz
+  have hh := neg_abs_le (z.im-c.im)
+  linarith
+
+lemma source_mesh_count_bound
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (hD : MeasurableSet D) (hfin : volume D ≠ ∞)
+    (p r : ℝ) (hr : 0 < r) (hy : ∀ z ∈ D, r ≤ z.im) (ω : a.samples) :
+    0 ≤ r^(-p)*(∫ z in D, if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) ∧
+    r^(-p)*(∫ z in D, if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) ≤
+      (volume D).toReal*r^(-p) := by
+  let : IsFiniteMeasure (volume.restrict D) := ⟨by simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
+  have hi : IntegrableOn (fun z => if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) D := by
+    apply source_mesh_bounded_integrable (K := 1)
+    · have hm : Measurable (fun p : ℂ × a.samples => if sourceCollarStart D (γ p.2) p.1 r = ⊤ then (0:ℝ) else 1) :=
+        Measurable.ite (measurableSet_eq_fun (source_collarStart_joint a D hD r hy) measurable_const) measurable_const measurable_const
+      exact hm.comp (measurable_id.prodMk measurable_const)
+    · intro z; split_ifs <;> norm_num
+  have hb : (∫ z in D, if sourceCollarStart D (γ ω) z r = ⊤ then (0:ℝ) else 1) ≤ (volume D).toReal := by
+    calc
+      _ ≤ ∫ _z in D, (1:ℝ) := integral_mono hi (integrable_const _) (fun z => by split_ifs <;> norm_num)
+      _ = _ := by simp only [integral_const,Measure.restrict_apply_univ,smul_eq_mul,mul_one,measureReal_def]
+  refine ⟨mul_nonneg (Real.rpow_nonneg hr.le _) (integral_nonneg (fun z => by split_ifs <;> norm_num)),?_⟩
+  simpa only [mul_comm] using mul_le_mul_of_nonneg_left hb (Real.rpow_nonneg hr.le (-p))
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_mesh_grid_exponent {p q u L r δ K : ℝ}
+    (hp1 : p < 1) (hq : 0 < q) (hu : 0 < u) (hL : 0 < L)
+    (hr : 0 < r) (hrδ : r ≤ δ) (hK : 0 < K) (n : ℕ) :
+    -(1/512:ℝ)*(δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)^2/(K^2*((L*u^n)*(r*q^n))^2) ≤
+      -(1/(512*K^2*L^2))*δ^(-2:ℝ)*(((q^p/(u*q))^2)^n/(n+1:ℝ)^4) := by
+  have hh := source_mesh_failure_exponent hp1 hq hu hL hr hrδ hK n
+  calc
+    _ = -(1/512:ℝ)*((δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2)^2/(K^2*((L*u^n)*(r*q^n))^2)) := by ring
+    _ ≤ -(1/512:ℝ)*(δ^(-2:ℝ)/(K^2*L^2)*((q^p/(u*q))^2)^n/(n+1:ℝ)^4) :=
+      mul_le_mul_of_nonpos_left hh (by norm_num)
+    _ = _ := by ring
+
+lemma source_mesh_grid_error {p q r δ E : ℝ} (hq : 0 < q) (hr : 0 < r) (hδ : 0 < δ) (n : ℕ) :
+    E*(r*q^n)^(-p)*(δ^(-p)*(r*q^n)^p/(n+1:ℝ)^2) = E*δ^(-p)/(n+1:ℝ)^2 := by
+  rw [mul_assoc,source_mesh_scale_cancel (mul_pos hr (pow_pos hq _)) hδ]
+  ring
+
+lemma source_mesh_grid_coefficient {c u L Cbad q p : ℝ} (hL : 0 < L) (hu : 0 < u) (n : ℕ) :
+    Cbad*(L*u^n)^(-c)*q^(-p) = (Cbad*L^(-c)*q^(-p))*(u^(-c))^n := by
+  rw [source_mesh_alpha hL hu]; ring
+
+lemma source_mesh_grid_radius_le {q r δ : ℝ} (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hrδ : r ≤ δ) (n : ℕ) : r*q^n ≤ δ :=
+  (mul_le_mul_of_nonneg_left (pow_le_one₀ hq.le hq1.le) hr.le).trans (by simpa using hrδ)
+
+lemma source_mesh_initial_scale {p q r δ : ℝ} (hp : 0 < p) (hq : 0 < q) (hδ : 0 < δ)
+    (hr : q*δ ≤ r) : r^(-p) ≤ q^(-p)*δ^(-p) := by
+  rw [←Real.mul_rpow hq.le hδ.le]
+  exact Real.rpow_le_rpow_of_nonpos (mul_pos hq hδ) hr (neg_nonpos.mpr hp.le)
+
+lemma source_mesh_ball_volume (c : ℂ) : volume (ball c 1) = volume (ball (0:ℂ) 1) := by
+  rw [Complex.volume_ball,Complex.volume_ball]
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+structure SourceMeshParameters (κ : ℝ) where
+  C : ℝ
+  Cbad : ℝ
+  c : ℝ
+  q : ℝ
+  u : ℝ
+  L : ℝ
+  hC : 0 < C
+  hCb : 0 < Cbad
+  hc : 0 < c
+  hq : 0 < q
+  hq1 : q < 1
+  hu : 1 < u
+  huq : u*q < 1
+  hratio : 1 < q^codimension κ/(u*q)
+  hL : 1 < L
+  halpha : Cbad*L^(-c)*q^(-codimension κ) ≤ 1/2
+  means :
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+      (z : ℂ) (ε r L : ℝ), 0 < ε → ε ≤ q*r → r < z.im → 1 < L →
+    ∃ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
+      let μ := P.comap (Subtype.val : a.samples → Ω)
+      let W := fun v (ω : a.samples) => spatialGreenWeight κ
+        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z v
+      let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε (L*r)
+      let B := fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)
+      μ[H | hσ.measurableSpace] ≤ᵐ[μ] (fun ω => C*ε^codimension κ*W r ω) ∧
+      μ[(fun ω => r^codimension κ*W ε ω*H ω) | hσ.measurableSpace] ≤ᵐ[μ]
+        (fun ω => r^codimension κ*W r ω) ∧
+      μ[B | hσ.measurableSpace] ≤ᵐ[μ]
+        (fun ω => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1))
+
+lemma source_mesh_parameters (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
+    Nonempty (SourceMeshParameters κ) := by
+  obtain ⟨C,Cbad,c,q,hC,hCb,hc,hq,hq1,hmean⟩ := sourceSpatialWindowMeans_proved κ hκ hκ8
+  have hp : 0 < codimension κ := by dsimp [codimension,dimension]; linarith
+  have hp1 : codimension κ < 1 := by dsimp [codimension,dimension]; linarith
+  obtain ⟨u,β,hu,huq,hratio,hβdef,hβ,htheta⟩ := source_mesh_power_growth hp hp1 hq hq1
+  obtain ⟨L,hL,hlarge⟩ := source_mesh_choose_large (A := Cbad*q^(-codimension κ)) hc
+  refine ⟨⟨C,Cbad,c,q,u,L,hC,hCb,hc,hq,hq1,hu,huq,hratio,hL,?_,hmean⟩⟩
+  calc
+    Cbad*L^(-c)*q^(-codimension κ) = (Cbad*q^(-codimension κ))*L^(-c) := by ring
+    _ ≤ 1/2 := hlarge
+
+lemma source_mesh_choose_delta (η L : ℝ) (hη : 0 < η) (hL : 0 < L) :
+    ∃ δ₀ : ℝ, 0 < δ₀ ∧ δ₀ < η/2 ∧ δ₀ ≤ 1 ∧ 3*L*δ₀ ≤ 1 := by
+  let δ₀ := min (min (η/4) (1/2)) (1/(6*L))
+  have hpos : 0 < δ₀ := by dsimp [δ₀]; positivity
+  refine ⟨δ₀,hpos,?_,?_,?_⟩
+  · have := (min_le_left (min (η/4) (1/2)) (1/(6*L))).trans (min_le_left _ _)
+    change δ₀ ≤ η/4 at this
+    linarith
+  · exact ((min_le_left _ _).trans (min_le_right _ _)).trans (by norm_num)
+  · have hh : δ₀ ≤ 1/(6*L) := min_le_right _ _
+    have hh' := (le_div_iff₀ (by positivity : 0 < 6*L)).mp hh
+    nlinarith
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_grid_step
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) (m : SourceMeshParameters κ)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
+    (r δ : ℝ) (hr : 0 < r) (hrδ : r ≤ δ) (hR : 3*m.L*δ ≤ 1)
+    (hy : ∀ z ∈ D, δ < z.im) (n : ℕ)
+    (E : Set a.samples) (hE : MeasurableSet E)
+    (hEs : ∀ z, MeasurableSet[(source_collarStart_stopping a D (r*m.q^n)
+      (fun z hz => (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n).trans (hy z hz).le) z).measurableSpace] E) :
+    ∃ F : Set a.samples, MeasurableSet F ∧
+      ((P.comap Subtype.val).restrict E) F ≤ ENNReal.ofReal
+        (3*(P.comap Subtype.val).real E*Real.exp (-(1/(512*(m.q^(-codimension κ))^2*m.L^2))*
+          δ^(-2:ℝ)*(((m.q^codimension κ/(m.u*m.q))^2)^n/(n+1:ℝ)^4))) ∧
+      ∀ ω ∉ F,
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^(n+1)) D ≤
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^n) D+
+        ((m.Cbad*m.L^(-m.c)*m.q^(-codimension κ))*(m.u^(-m.c))^n)*sourceMeshCount (codimension κ) D (γ ω) (r*m.q^n)+
+        (2*m.q^(-codimension κ))*δ^(-codimension κ)/(n+1:ℝ)^2 ∧
+      sourceMeshCount (codimension κ) D (γ ω) (r*m.q^(n+1)) ≤
+        m.C*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^n) D+
+        ((m.Cbad*m.L^(-m.c)*m.q^(-codimension κ))*(m.u^(-m.c))^n)*sourceMeshCount (codimension κ) D (γ ω) (r*m.q^n)+
+        (2*m.q^(-codimension κ))*δ^(-codimension κ)/(n+1:ℝ)^2 := by
+  have hp1 : codimension κ < 1 := by dsimp [codimension,dimension]; linarith
+  have hk : 0 < κ := by linarith
+  have hδ : 0 < δ := hr.trans_le hrδ
+  have hL : 0 < m.L := lt_trans zero_lt_one m.hL
+  have hu : 0 < m.u := lt_trans zero_lt_one m.hu
+  have hrn : 0 < r*m.q^n := mul_pos hr (pow_pos m.hq _)
+  have hy' (z : ℂ) (hz : z ∈ D) : r*m.q^n < z.im :=
+    (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n).trans_lt (hy z hz)
+  have hLn : 1 < m.L*m.u^n := m.hL.trans_le
+    (by simpa using mul_le_mul_of_nonneg_left (one_le_pow₀ m.hu.le) hL.le)
+  have hRn : 3*((m.L*m.u^n)*(r*m.q^n)) ≤ 1 :=
+    (mul_le_mul_of_nonneg_left (source_mesh_radius_bound m.hq hu m.huq.le hL hr hrδ n) (by norm_num)).trans (by simpa only [mul_assoc] using hR)
+  obtain ⟨F,hFm,hFt,hFg⟩ := source_mesh_step a hk hκ8 D hD c₀ hDc m.q (r*m.q^n) (m.L*m.u^n)
+    m.C m.Cbad m.c (δ^(-codimension κ)*(r*m.q^n)^codimension κ/(n+1:ℝ)^2)
+    m.hq m.hq1 hrn hLn hRn m.hC.le m.hCb.le (by positivity) (fun z hz => (hy' z hz).le)
+    (fun z hz => by
+      obtain ⟨hs,hH,hV,hB⟩ := m.means Ω ‹MeasurableSpace Ω› P inferInstance γ a z
+        (m.q*(r*m.q^n)) (r*m.q^n) (m.L*m.u^n) (mul_pos m.hq hrn) le_rfl (hy' z hz) hLn
+      exact ⟨hH,hV,hB⟩) E hE hEs
+  refine ⟨F,hFm,hFt.trans ?_,?_⟩
+  · apply ENNReal.ofReal_le_ofReal
+    apply mul_le_mul_of_nonneg_left _ (by positivity)
+    apply Real.exp_le_exp.mpr
+    exact source_mesh_grid_exponent hp1 m.hq hu hL hr hrδ (Real.rpow_pos_of_pos m.hq _) n
+  · intro ω hω
+    have hh := hFg ω hω
+    have hrnext : m.q*(r*m.q^n) = r*m.q^(n+1) := by rw [pow_succ]; ring
+    rwa [hrnext,source_mesh_grid_coefficient hL hu,
+      source_mesh_grid_error m.hq hr hδ] at hh
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/PhysicalMeshWindowCore.lean b/lean/OAI/Probability/SLE/PhysicalMeshWindowCore.lean
index 2bb60cecb5678eadcd26b0ab2499b24cdf60967a..623d47051d563d2ff203486c1559fd90ae779896 100644
--- a/lean/OAI/Probability/SLE/PhysicalMeshWindowCore.lean
+++ b/lean/OAI/Probability/SLE/PhysicalMeshWindowCore.lean
@@ -1,770 +1,770 @@
-import Mathlib
-import OAI.Probability.SLE.WindowAssembly
-import OAI.Probability.SLE.WindowTails
-
-namespace OAI
-
-/-! Physical Mesh Window Core for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
- 
-
-structure SourceWindowCore {Ω : Type*} [MeasurableSpace Ω]
-    (ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›) (D : Set ℂ) (r R : ℝ)
-    (γ : ℝ≥0 → Ω → ℂ) (σ τ : ℂ → Ω → WithTop ℝ≥0) : Prop where
-  measurable_D : MeasurableSet D
-  bounded_D : D ⊆ ball 0 1
-  radius_pos : 0 < r
-  radius_le : r ≤ R
-  radius_small : 3*R ≤ 1
-  continuous_trace : ∀ ω, Continuous (fun t => γ t ω)
-  adapted_trace : ∀ t, Measurable[ℱ t] (γ t)
-  start : ∀ z, IsStoppingTime ℱ (σ z)
-  finish : ∀ z, IsStoppingTime ℱ (τ z)
-  time_le : ∀ z ω, σ z ω ≤ τ z ω
-  contact : ∀ z ω, σ z ω ≠ ⊤ → dist (γ ((σ z ω).untopD 0) ω) z = r
-  confined : ∀ z ω (t : ℝ≥0), σ z ω ≤ (t : WithTop ℝ≥0) → (t : WithTop ℝ≥0) < τ z ω →
-    dist (γ t ω) z < R
-  outside : ∀ z, z ∉ D → ∀ ω, σ z ω = ⊤
-
-lemma SourceWindowCore.attach {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} {D : Set ℂ} {r R K : ℝ}
-    {γ : ℝ≥0 → Ω → ℂ} {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceWindowCore ℱ D r R γ σ τ) (hK : 0 ≤ K)
-    (hYm : ∀ z, Measurable[(h.finish z).measurableSpace] (Y z))
-    (hμm : ∀ z, Measurable[(h.start z).measurableSpace] (μ z))
-    (hY : ∀ z ω, 0 ≤ Y z ω ∧ Y z ω ≤ K)
-    (hμ : ∀ z ω, 0 ≤ μ z ω) (hμb : ∃ M : ℝ, ∀ z ω, μ z ω ≤ M)
-    (hmean : ∀ z, ∀ᵐ ω ∂P, σ z ω ≠ ⊤ → P[Y z | (h.start z).measurableSpace] ω ≤ μ z ω)
-    (hns : ∀ z ω, σ z ω = ⊤ → Y z ω = 0 ∧ μ z ω = 0)
-    (hne : ∀ z ω, τ z ω = ⊤ → Y z ω = 0)
-    (hYj : Measurable (fun p : ℂ × Ω => Y p.1 p.2))
-    (hμj : Measurable (fun p : ℂ × Ω => μ p.1 p.2)) :
-    SourceDisjointWindows P ℱ D r R K γ σ τ Y μ :=
-  ⟨h.measurable_D,h.bounded_D,h.radius_pos,h.radius_le,h.radius_small,hK,
-    h.continuous_trace,h.adapted_trace,h.start,h.finish,h.time_le,h.contact,h.confined,
-    hYm,hμm,hY,hμ,hμb,hmean,hns,hne,
-    fun z hz ω => ⟨h.outside z hz ω,hns z ω (h.outside z hz ω)⟩,hYj,hμj⟩
-
-lemma source_mesh_window_core
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (hD : MeasurableSet D) (c : ℂ) (hDc : D ⊆ ball c 1)
-    (ε r R : ℝ) (hr : 0 < r) (hrR : r < R) (hR : 3*R ≤ 1)
-    (hy : ∀ z ∈ D, r ≤ z.im) :
-    SourceWindowCore a.filtration ((fun z : ℂ => z+c) ⁻¹' D) r R
-      (fun t (ω : a.samples) => γ ω t-c)
-      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c) r)
-      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c) r ε R) := by
-  have hd (x z : ℂ) : dist (x-c) z = dist x (z+c) := by
-    simp only [dist_eq_norm]; congr 1; ring
-  refine ⟨hD.preimage (measurable_id.add_const c),?_,hr,hrR.le,hR,
-    fun ω => (a.trace_continuous ω).sub continuous_const,
-    fun t => (a.trace_adapted t).sub_const c,
-    fun z => source_collarStart_stopping a D r hy (z+c),
-    fun z => source_collarFinish_stopping a D r ε R hy (z+c),?_,?_,?_,?_⟩
-  · intro z hz
-    have hh := hDc hz
-    simpa only [mem_ball,dist_eq_norm,add_sub_cancel_right,sub_zero] using hh
-  · intro z ω
-    by_cases hz : z+c ∈ D
-    · simpa only [sourceCollarStart,sourceCollarFinish,ite_eq_left hz] using
-        source_window_start_le (a.trace_continuous ω) (z+c) r ε R
-    · simp only [sourceCollarStart,sourceCollarFinish,ite_eq_right hz,le_refl]
-  · intro z ω hn
-    have hz : z+c ∈ D := by
-      by_contra hh
-      exact hn (by simp only [sourceCollarStart,ite_eq_right hh])
-    simp only [sourceCollarStart,ite_eq_left hz] at hn ⊢
-    obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hn
-    rw [← hT,WithTop.untopD_coe,hd]
-    exact (source_liveHit_mem (a.trace_continuous ω) (z+c) r
-      (source_window_origin_bound a (z+c) r (hy (z+c) hz) ω) T hT.symm).2
-  · intro z ω t hst htt
-    have hz : z+c ∈ D := by
-      by_contra hh
-      simp only [sourceCollarStart,ite_eq_right hh] at hst
-      exact (WithTop.not_top_le_coe t) hst
-    simp only [sourceCollarStart,sourceCollarFinish,ite_eq_left hz] at hst htt
-    rw [hd]
-    exact (source_before_window (γ ω) (z+c) r ε R t hst htt).2.2
-  · intro z hz ω
-    exact ite_eq_right hz
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_if_bound {p : Prop} [Decidable p] {x K : ℝ}
-    (hK : 0 ≤ K) (hx : p → 0 ≤ x ∧ x ≤ K) :
-    0 ≤ (if p then x else 0) ∧ (if p then x else 0) ≤ K := by
-  by_cases hp : p
-  · simpa only [ite_eq_left hp] using hx hp
-  · simp only [ite_eq_right hp,le_refl,true_and]; exact hK
-
-lemma source_collar_HB_bounds (D : Set ℂ) (γ : ℝ≥0 → ℂ)
-    (z : ℂ) (r ε R : ℝ) :
-    (0 ≤ sourceCollarH D γ z r ε R ∧ sourceCollarH D γ z r ε R ≤ 1) ∧
-    (0 ≤ sourceCollarB D γ z r ε R ∧ sourceCollarB D γ z r ε R ≤ 1) := by
-  constructor
-  · apply source_if_bound zero_le_one
-    intro _; rcases (source_windowHB_range γ z r ε R).1 with h|h <;> simp [h]
-  · apply source_if_bound zero_le_one
-    intro _; rcases (source_windowHB_range γ z r ε R).2 with h|h <;> simp [h]
-
-lemma source_collar_end_zero (D : Set ℂ) (γ : ℝ≥0 → ℂ)
-    (z : ℂ) (r ε R : ℝ)
-    (ht : sourceCollarFinish D γ z r ε R = ⊤) :
-    sourceCollarH D γ z r ε R = 0 ∧ sourceCollarB D γ z r ε R = 0 := by
-  by_cases hz : z ∈ D
-  · simp only [sourceCollarFinish,ite_eq_left hz] at ht
-    simp only [sourceCollarH,sourceCollarB,ite_eq_left hz,sourceSpatialWindowH,sourceSpatialWindowB,
-      ht,ne_eq,not_true_eq_false,false_and,ite_false,and_self]
-  · simp only [sourceCollarH,sourceCollarB,ite_eq_right hz,and_self]
-
-lemma source_collar_start_zero (D : Set ℂ) {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ)
-    (ht : sourceCollarStart D γ z r = ⊤) :
-    sourceCollarH D γ z r ε R = 0 ∧ sourceCollarB D γ z r ε R = 0 := by
-  apply source_collar_end_zero
-  by_cases hz : z ∈ D
-  · simp only [sourceCollarStart,ite_eq_left hz] at ht
-    simp only [sourceCollarFinish,ite_eq_left hz]
-    exact top_le_iff.mp (by simpa only [ht] using source_window_start_le hc z r ε R)
-  · exact ite_eq_right hz
-
-lemma source_collar_weight_bounds
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (D : Set ℂ) (r : ℝ) (hr : 0 < r)
-    (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples) :
-    0 ≤ (if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-      (γ ω) (a.G ω) z r else 0) ∧
-    (if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-      (γ ω) (a.G ω) z r else 0) ≤ r^(-codimension κ) := by
-  apply source_if_bound (Real.rpow_nonneg hr.le _)
-  exact fun hz => (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _)
-    hκ hκ8 hr z (hy z hz)
-
-lemma source_collar_weight_zero
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples)
-    (hn : sourceCollarStart D (γ ω) z r = ⊤) :
-    (if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-      (γ ω) (a.G ω) z r else 0) = 0 := by
-  by_cases hz : z ∈ D
-  · simp only [sourceCollarStart,ite_eq_left hz] at hn
-    rw [ite_eq_left hz]
-    exact source_window_weight_zero a z r (hy z hz) ω hn
-  · exact ite_eq_right hz
-
-lemma source_collar_weight_joint
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (hD : MeasurableSet D) (r : ℝ) :
-    Measurable (fun p : ℂ × a.samples =>
-      if p.1 ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t p.2)
-        (γ p.2) (a.G p.2) p.1 r else 0) :=
-  Measurable.ite (hD.preimage measurable_fst) (a.spatialGreenWeight_measurable r) measurable_const
-
-lemma source_collar_mean_transfer
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im)
-    (f g : ℂ → a.samples → ℝ)
-    (hmean : ∀ z (hz : z ∈ D),
-      (P.comap Subtype.val)[f z | (source_liveHit_stopping a z r (hy z hz)).measurableSpace]
-        ≤ᵐ[P.comap Subtype.val] g z) (z : ℂ) :
-    (P.comap Subtype.val)[(fun ω => if z ∈ D then f z ω else 0) |
-      (source_collarStart_stopping a D r hy z).measurableSpace] ≤ᵐ[P.comap Subtype.val]
-      (fun ω => if z ∈ D then g z ω else 0) := by
-  by_cases hz : z ∈ D
-  · simp only [ite_eq_left hz]
-    have hs := source_stop_space_congr (source_collarStart_stopping a D r hy z)
-      (source_liveHit_stopping a z r (hy z hz))
-      (funext fun ω => by simp only [sourceCollarStart,ite_eq_left hz])
-    rw [hs]
-    exact hmean z hz
-  · simp only [ite_eq_right hz]
-    change (P.comap Subtype.val)[(0 : a.samples → ℝ) |
-      (source_collarStart_stopping a D r hy z).measurableSpace] ≤ᵐ[P.comap Subtype.val] 0
-    rw [condExp_zero]
-
-lemma source_window_scaled_bound {p q r : ℝ} (hq : 0 < q) (hr : 0 < r) :
-    r^p * (q*r)^(-p) = q^(-p) := by
-  rw [Real.mul_rpow hq.le hr.le]
-  calc
-    r^p * (q^(-p)*r^(-p)) = q^(-p)*(r^p*r^(-p)) := by ring
-    _ = q^(-p) := by rw [← Real.rpow_add hr,add_neg_cancel,Real.rpow_zero,mul_one]
-
-lemma source_collar_V_bounds
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (D : Set ℂ) (q r R : ℝ)
-    (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples) :
-    0 ≤ r^codimension κ * spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-      (γ ω) (a.G ω) z (q*r) * sourceCollarH D (γ ω) z r (q*r) R ∧
-    r^codimension κ * spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-      (γ ω) (a.G ω) z (q*r) * sourceCollarH D (γ ω) z r (q*r) R ≤ q^(-codimension κ) := by
-  by_cases hz : z ∈ D
-  · have hw := (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _)
-      hκ hκ8 (mul_pos hq hr) z ((show q*r ≤ r by nlinarith).trans (hy z hz))
-    have hH := (source_collar_HB_bounds D (γ ω) z r (q*r) R).1
-    have hnon := mul_nonneg (Real.rpow_nonneg hr.le (codimension κ)) hw.1
-    refine ⟨mul_nonneg hnon hH.1,?_⟩
-    calc
-      _ ≤ r^codimension κ * (q*r)^(-codimension κ) * 1 :=
-        mul_le_mul (mul_le_mul_of_nonneg_left hw.2 (Real.rpow_nonneg hr.le _)) hH.2 hH.1
-          (mul_nonneg (Real.rpow_nonneg hr.le _) (Real.rpow_nonneg (mul_pos hq hr).le _))
-      _ = _ := by rw [mul_one,source_window_scaled_bound hq hr]
-  · simp only [sourceCollarH,ite_eq_right hz,mul_zero,le_refl,true_and]
-    exact Real.rpow_nonneg hq.le _
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_H_window
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
-    (q r L C : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hL : 1 < L) (hR : 3*(L*r) ≤ 1) (hC : 0 ≤ C)
-    (hy : ∀ z ∈ D, r ≤ z.im)
-    (hmean : ∀ z (hz : z ∈ D),
-      (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowH (γ ω) z r (q*r) (L*r)) |
-        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
-      (fun ω : a.samples => C*(q*r)^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) :
-    SourceDisjointWindows (P.comap Subtype.val) a.filtration
-      ((fun z : ℂ => z+c₀) ⁻¹' D) r (L*r) 1
-      (fun t (ω : a.samples) => γ ω t-c₀)
-      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c₀) r)
-      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c₀) r (q*r) (L*r))
-      (fun z (ω : a.samples) => sourceCollarH D (γ ω) (z+c₀) r (q*r) (L*r))
-      (fun z (ω : a.samples) => C*(q*r)^codimension κ*
-        (if z+c₀ ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-          (γ ω) (a.G ω) (z+c₀) r else 0)) := by
-  let core := source_mesh_window_core a D hD c₀ hDc (q*r) r (L*r) hr (by nlinarith) hR hy
-  have hs : q*r ≤ r := by nlinarith
-  have hfac : 0 ≤ C*(q*r)^codimension κ := mul_nonneg hC (Real.rpow_nonneg (mul_pos hq hr).le _)
-  apply core.attach zero_le_one
-  · intro z
-    exact (source_collarHBV_stopped a D r (q*r) (L*r) hs hy (z+c₀)).1
-  · intro z
-    exact (source_collarWeight_stopped a D r hy (z+c₀)).const_mul _
-  · intro z ω
-    exact (source_collar_HB_bounds D (γ ω) (z+c₀) r (q*r) (L*r)).1
-  · intro z ω
-    exact mul_nonneg hfac (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).1
-  · refine ⟨C*(q*r)^codimension κ*r^(-codimension κ),?_⟩
-    intro z ω
-    exact mul_le_mul_of_nonneg_left (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).2 hfac
-  · intro z
-    have hh := source_collar_mean_transfer a D r hy
-      (fun z (ω : a.samples) => sourceSpatialWindowH (γ ω) z r (q*r) (L*r))
-      (fun z (ω : a.samples) => C*(q*r)^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r) hmean (z+c₀)
-    have hg : (fun ω : a.samples => if z+c₀ ∈ D then C*(q*r)^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (z+c₀) r else 0) =
-        (fun ω : a.samples => C*(q*r)^codimension κ*(if z+c₀ ∈ D then spatialGreenWeight κ
-          (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (z+c₀) r else 0)) := by
-      funext ω; rw [mul_ite,mul_zero]
-    rw [hg] at hh
-    exact hh.mono (fun _ h _ => h)
-  · intro z ω hn
-    exact ⟨(source_collar_start_zero D (a.trace_continuous ω) (z+c₀) r (q*r) (L*r) hn).1,
-      by rw [source_collar_weight_zero a D r hy (z+c₀) ω hn,mul_zero]⟩
-  · intro z ω hn
-    exact (source_collar_end_zero D (γ ω) (z+c₀) r (q*r) (L*r) hn).1
-  · exact (source_collarHB_joint a D hD r (q*r) (L*r) hy).1.comp
-      ((measurable_fst.add_const c₀).prodMk measurable_snd)
-  · exact ((source_collar_weight_joint a D hD r).const_mul (C*(q*r)^codimension κ)).comp
-      ((measurable_fst.add_const c₀).prodMk measurable_snd)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_collar_V_mean
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (q r L : ℝ) (hy : ∀ z ∈ D, r ≤ z.im)
-    (hmean : ∀ z (hz : z ∈ D),
-      (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*
-        sourceSpatialWindowH (γ ω) z r (q*r) (L*r)) |
-        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
-      (fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) (z : ℂ) :
-    (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceCollarH D (γ ω) z r (q*r) (L*r)) |
-      (source_collarStart_stopping a D r hy z).measurableSpace] ≤ᵐ[P.comap Subtype.val]
-    (fun ω : a.samples => r^codimension κ*(if z ∈ D then spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0)) := by
-  have hh := source_collar_mean_transfer a D r hy
-    (fun z (ω : a.samples) => r^codimension κ*spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceSpatialWindowH (γ ω) z r (q*r) (L*r))
-    (fun z (ω : a.samples) => r^codimension κ*spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r) hmean z
-  have hf : (fun ω : a.samples => if z ∈ D then r^codimension κ*spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceSpatialWindowH (γ ω) z r (q*r) (L*r) else 0) =
-      (fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceCollarH D (γ ω) z r (q*r) (L*r)) := by
-    funext ω; simp only [sourceCollarH,mul_ite,mul_zero]
-  have hg : (fun ω : a.samples => if z ∈ D then r^codimension κ*spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0) =
-      (fun ω : a.samples => r^codimension κ*(if z ∈ D then spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0)) := by
-    funext ω; rw [mul_ite,mul_zero]
-  rw [hf,hg] at hh
-  exact hh.mono (fun _ h => h)
-
-lemma source_collar_V_joint
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (q r L : ℝ)
-    (hy : ∀ z ∈ D, r ≤ z.im) :
-    Measurable (fun p : ℂ × a.samples => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t p.2) (γ p.2) (a.G p.2) (p.1+c₀) (q*r)*
-        sourceCollarH D (γ p.2) (p.1+c₀) r (q*r) (L*r)) := by
-  have hj := (((a.spatialGreenWeight_measurable (q*r)).const_mul (r^codimension κ)).mul
-      (source_collarHB_joint a D hD r (q*r) (L*r) hy).1)
-  have ht : Measurable (fun p : ℂ × a.samples => (p.1+c₀,p.2)) :=
-    (measurable_fst.add_const c₀).prodMk measurable_snd
-  have hh := hj.comp ht
-  dsimp only [Function.comp_def] at hh
-  exact hh
-
-lemma source_mesh_V_window
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
-    (q r L : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hL : 1 < L) (hR : 3*(L*r) ≤ 1)
-    (hy : ∀ z ∈ D, r ≤ z.im)
-    (hmean : ∀ z (hz : z ∈ D),
-      (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*
-        sourceSpatialWindowH (γ ω) z r (q*r) (L*r)) |
-        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
-      (fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) :
-    SourceDisjointWindows (P.comap Subtype.val) a.filtration
-      ((fun z : ℂ => z+c₀) ⁻¹' D) r (L*r) (q^(-codimension κ))
-      (fun t (ω : a.samples) => γ ω t-c₀)
-      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c₀) r)
-      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c₀) r (q*r) (L*r))
-      (fun z (ω : a.samples) => r^codimension κ*spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (z+c₀) (q*r)*
-        sourceCollarH D (γ ω) (z+c₀) r (q*r) (L*r))
-      (fun z (ω : a.samples) => r^codimension κ*
-        (if z+c₀ ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
-          (γ ω) (a.G ω) (z+c₀) r else 0)) := by
-  let core := source_mesh_window_core a D hD c₀ hDc (q*r) r (L*r) hr (by nlinarith) hR hy
-  have hs : q*r ≤ r := by nlinarith
-  have hfac : 0 ≤ r^codimension κ := Real.rpow_nonneg hr.le _
-  apply core.attach (Real.rpow_nonneg hq.le _)
-  · intro z
-    simpa only [mul_assoc] using (source_collarHBV_stopped a D r (q*r) (L*r) hs hy (z+c₀)).2.2.const_mul (r^codimension κ)
-  · intro z
-    exact (source_collarWeight_stopped a D r hy (z+c₀)).const_mul _
-  · intro z ω
-    exact source_collar_V_bounds a hκ hκ8 D q r (L*r) hq hq1 hr hy (z+c₀) ω
-  · intro z ω
-    exact mul_nonneg hfac (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).1
-  · refine ⟨r^codimension κ*r^(-codimension κ),?_⟩
-    intro z ω
-    exact mul_le_mul_of_nonneg_left (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).2 hfac
-  · intro z
-    exact (source_collar_V_mean a D q r L hy hmean (z+c₀)).mono (fun _ h _ => h)
-  · intro z ω hn
-    refine ⟨?_,?_⟩
-    · rw [(source_collar_start_zero D (a.trace_continuous ω) (z+c₀) r (q*r) (L*r) hn).1,mul_zero]
-    · rw [source_collar_weight_zero a D r hy (z+c₀) ω hn,mul_zero]
-  · intro z ω hn
-    rw [(source_collar_end_zero D (γ ω) (z+c₀) r (q*r) (L*r) hn).1,mul_zero]
-  · exact source_collar_V_joint a D hD c₀ q r L hy
-  · exact ((source_collar_weight_joint a D hD r).const_mul (r^codimension κ)).comp
-      ((measurable_fst.add_const c₀).prodMk measurable_snd)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_B_window
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
-    (q r L Cbad c : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
-    (hL : 1 < L) (hR : 3*(L*r) ≤ 1) (hCb : 0 ≤ Cbad)
-    (hy : ∀ z ∈ D, r ≤ z.im)
-    (hmean : ∀ z (hz : z ∈ D),
-      (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowB (γ ω) z r (q*r) (L*r)) |
-        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
-      (fun ω : a.samples => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1))) :
-    SourceDisjointWindows (P.comap Subtype.val) a.filtration
-      ((fun z : ℂ => z+c₀) ⁻¹' D) r (L*r) 1
-      (fun t (ω : a.samples) => γ ω t-c₀)
-      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c₀) r)
-      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c₀) r (q*r) (L*r))
-      (fun z (ω : a.samples) => sourceCollarB D (γ ω) (z+c₀) r (q*r) (L*r))
-      (fun z (ω : a.samples) => Cbad*L^(-c)*
-        (if sourceCollarStart D (γ ω) (z+c₀) r = ⊤ then 0 else 1)) := by
-  let core := source_mesh_window_core a D hD c₀ hDc (q*r) r (L*r) hr (by nlinarith) hR hy
-  have hs : q*r ≤ r := by nlinarith
-  have hfac : 0 ≤ Cbad*L^(-c) := mul_nonneg hCb (Real.rpow_nonneg (by linarith) _)
-  apply core.attach zero_le_one
-  · intro z
-    exact (source_collarHBV_stopped a D r (q*r) (L*r) hs hy (z+c₀)).2.1
-  · intro z
-    exact (Measurable.ite (measurableSet_eq_fun
-      (source_collarStart_stopping a D r hy (z+c₀)).measurable measurable_const)
-      measurable_const measurable_const).const_mul _
-  · intro z ω
-    exact (source_collar_HB_bounds D (γ ω) (z+c₀) r (q*r) (L*r)).2
-  · intro z ω
-    exact mul_nonneg hfac (by split_ifs <;> norm_num)
-  · refine ⟨Cbad*L^(-c),?_⟩
-    intro z ω
-    exact (mul_le_mul_of_nonneg_left (show (if sourceCollarStart D (γ ω) (z+c₀) r = ⊤ then (0:ℝ) else 1) ≤ 1 by split_ifs <;> norm_num) hfac).trans_eq (mul_one _)
-  · intro z
-    have hh := source_collar_mean_transfer a D r hy
-      (fun z (ω : a.samples) => sourceSpatialWindowB (γ ω) z r (q*r) (L*r))
-      (fun z (ω : a.samples) => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1)) hmean (z+c₀)
-    have hg : (fun ω : a.samples => if z+c₀ ∈ D then Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) (z+c₀) r = ⊤ then 0 else 1) else 0) =
-        (fun ω : a.samples => Cbad*L^(-c)*(if sourceCollarStart D (γ ω) (z+c₀) r = ⊤ then 0 else 1)) := by
-      funext ω; by_cases hz : z+c₀ ∈ D <;> simp [sourceCollarStart,hz]
-    rw [hg] at hh
-    exact hh.mono (fun _ h _ => h)
-  · intro z ω hn
-    exact ⟨(source_collar_start_zero D (a.trace_continuous ω) (z+c₀) r (q*r) (L*r) hn).2,
-      by rw [ite_eq_left hn,mul_zero]⟩
-  · intro z ω hn
-    exact (source_collar_end_zero D (γ ω) (z+c₀) r (q*r) (L*r) hn).2
-  · exact (source_collarHB_joint a D hD r (q*r) (L*r) hy).2.comp
-      ((measurable_fst.add_const c₀).prodMk measurable_snd)
-  · exact ((Measurable.ite (measurableSet_eq_fun
-      (source_collarStart_joint a D hD r hy) measurable_const)
-      measurable_const measurable_const).const_mul (Cbad*L^(-c))).comp
-      ((measurable_fst.add_const c₀).prodMk measurable_snd)
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma window_finite_mgf_at_starts {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
-    {ℱ : Filtration ℝ≥0 mΩ} {σ τ : ι → Ω → WithTop ℝ≥0}
-    (hs : ∀ i, IsStoppingTime ℱ (σ i)) (ht : ∀ i, IsStoppingTime ℱ (τ i))
-    (horder : ∀ i j ω, windowBefore σ j i ω → τ j ω ≤ σ i ω)
-    {X : ι → Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
-    (hX : ∀ i, Measurable[(ht i).measurableSpace] (X i))
-    (hXb : ∀ i ω, |X i ω| ≤ K)
-    (hmean : ∀ i, P[X i|(hs i).measurableSpace] ≤ᵐ[P] 0)
-    {W : Ω → ℝ} (hWa : Measurable[mΩ] W) (hWstart : ∀ i, Measurable[(hs i).measurableSpace] W) (hW0 : ∀ ω, 0 ≤ W ω)
-    (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω * Real.exp (θ*∑ i, X i ω) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
-  classical
-  let q := (θ*K)^2/2
-  let g (i : ι) (ω : Ω) := Real.exp (θ*X i ω-q)
-  let pre (i : ι) (ω : Ω) := ∏ j, if windowBefore σ j i ω then g j ω else 1
-  have hg (i : ι) : Measurable[(ht i).measurableSpace] (g i) :=
-    (((hX i).const_mul θ).sub_const q).exp
-  have hga (i : ι) : Measurable[mΩ] (g i) := (hg i).mono (ht i).measurableSpace_le le_rfl
-  have hg0 (i : ι) (ω : Ω) : 0 ≤ g i ω := (Real.exp_pos _).le
-  have hgB (i : ι) (ω : Ω) : g i ω ≤ Real.exp (θ*K) := by
-    apply Real.exp_le_exp.mpr
-    have := mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hXb i ω)) hθ
-    dsimp [q]
-    nlinarith [sq_nonneg (θ*K)]
-  have hp (i : ι) : Measurable[(hs i).measurableSpace] (pre i) := by
-    apply Finset.measurable_prod
-    intro j hj
-    exact window_measurable_on_event (hs i) (ht j) (hg j)
-      (window_before_measurable hs i j) (fun ω hω => horder i j ω hω) 1
-  have hp0 (i : ι) (ω : Ω) : 0 ≤ pre i ω := by
-    apply Finset.prod_nonneg
-    intro j hj
-    split_ifs <;> positivity
-  have hpB (i : ι) (ω : Ω) : pre i ω ≤ Real.exp (θ*K)^Fintype.card ι := by
-    calc
-      _ ≤ ∏ _j : ι, Real.exp (θ*K) := by
-        apply Finset.prod_le_prod₀
-        · intro j hj; split_ifs <;> positivity
-        · intro j hj; split_ifs
-          · exact hgB j ω
-          · exact Real.one_le_exp (mul_nonneg hθ hK.le)
-      _ = _ := by simp
-  obtain ⟨M,hM⟩ := hWb
-  have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
-  have hHi (i : ι) : Integrable (fun ω => W ω*pre i ω*(g i ω-1)) P := by
-    apply window_integrable_bounded (C := (M*Real.exp (θ*K)^Fintype.card ι)*(Real.exp (θ*K)+1)) (((hWa.mul ((hp i).mono (hs i).measurableSpace_le le_rfl))).mul
-      ((hga i).sub_const 1))
-    intro ω
-    change ‖W ω*pre i ω*(g i ω-1)‖ ≤ _
-    rw [norm_mul, norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω),
-      Real.norm_eq_abs, abs_of_nonneg (hp0 i ω)]
-    apply mul_le_mul
-    · exact mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))
-    · calc
-        _ ≤ ‖g i ω‖ + ‖(1 : ℝ)‖ := norm_sub_le _ _
-        _ = g i ω + 1 := by rw [Real.norm_eq_abs, abs_of_nonneg (hg0 i ω), norm_one]
-        _ ≤ _ := add_le_add (hgB i ω) le_rfl
-    · exact norm_nonneg _
-    · exact mul_nonneg ((hW0 ω).trans (hM ω)) (by positivity)
-  have hHi0 (i : ι) : (∫ ω, W ω*pre i ω*(g i ω-1) ∂P) ≤ 0 := by
-    exact window_weighted_normalized (hs i).measurableSpace_le hK hθ
-      ((hX i).mono (ht i).measurableSpace_le le_rfl) (hXb i) (hmean i)
-      ((hWstart i).mul (hp i)) (fun ω => mul_nonneg (hW0 ω) (hp0 i ω))
-      ⟨M*Real.exp (θ*K)^Fintype.card ι, fun ω =>
-        mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))⟩
-  have hnorm : (∫ ω, W ω * ∏ i, g i ω ∂P) ≤ ∫ ω, W ω ∂P := by
-    have hid (ω : Ω) := window_prod_before σ ω (fun i => g i ω)
-    simp_rw [hid, mul_add, mul_one, Finset.mul_sum]
-    change (∫ ω, W ω + ∑ i, W ω*((g i ω-1)*pre i ω) ∂P) ≤ _
-    simp_rw [show ∀ i ω, W ω*((g i ω-1)*pre i ω) = W ω*pre i ω*(g i ω-1)
-      from fun _ _ => by ring]
-    rw [integral_add hWi (integrable_finsetSum _ (fun i hi => hHi i)), integral_finsetSum]
-    · exact add_le_of_nonpos_right (Finset.sum_nonpos fun i hi => hHi0 i)
-    · exact fun i hi => hHi i
-  have hprod (ω : Ω) : (∏ i, g i ω) =
-      Real.exp (θ*∑ i, X i ω) / Real.exp (Fintype.card ι * q) := by
-    change (∏ i, Real.exp (θ*X i ω-q)) = _
-    rw [← Real.exp_sum]
-    simp only [Finset.sum_sub_distrib, ← Finset.mul_sum, Finset.sum_const,
-      Finset.card_univ, nsmul_eq_mul]
-    exact Real.exp_sub _ _
-  simp only [hprod, ← mul_div_assoc] at hnorm
-  rw [integral_div] at hnorm
-  simpa only [q, mul_div_assoc] using (div_le_iff₀ (Real.exp_pos _)).mp hnorm
-
- 
-
-lemma window_source_finite_mgf_at_starts {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
-    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    {z : ι → ℂ} (hz : ∀ i j, i ≠ j → 2*R < dist (z i) (z j))
-    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hWa : Measurable[mΩ] W) (hWs : ∀ i, Measurable[(h.start (z i)).measurableSpace] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
-    (∫ ω, W ω * Real.exp (θ*∑ i, (Y (z i) ω-μ (z i) ω)) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
-  let X (i : ι) (ω : Ω) := Y (z i) ω-min (μ (z i) ω) K
-  have hYa (i : ι) : Measurable[mΩ] (Y (z i)) :=
-    (h.output_measurable (z i)).mono (h.finish (z i)).measurableSpace_le le_rfl
-  have hμa (i : ι) : Measurable[mΩ] (μ (z i)) :=
-    (h.mean_measurable (z i)).mono (h.start (z i)).measurableSpace_le le_rfl
-  have hX (i : ι) : Measurable[(h.finish (z i)).measurableSpace] (X i) :=
-    (h.output_measurable (z i)).sub (((h.mean_measurable (z i)).mono
-      ((h.start (z i)).measurableSpace_mono (h.finish (z i)) (h.time_le (z i))) le_rfl).min measurable_const)
-  have hXb (i : ι) (ω : Ω) : |X i ω| ≤ K := by
-    rw [abs_le]
-    have := h.output_bound (z i) ω
-    have := le_min (h.mean_nonneg (z i) ω) hK.le
-    have := min_le_right (μ (z i) ω) K
-    dsimp [X]
-    constructor <;> linarith
-  have hm (i : ι) : P[X i|(h.start (z i)).measurableSpace] ≤ᵐ[P] 0 :=
-    window_capped_cond_mean (h.start (z i)).measurableSpace_le hK.le (hYa i)
-      (h.output_bound (z i)) (h.mean_measurable (z i)) (h.mean_nonneg (z i))
-      (window_cond_mean_at_infinity (h.start (z i))
-        (window_integrable_nonneg (hYa i) (fun ω => (h.output_bound (z i) ω).1)
-          (fun ω => (h.output_bound (z i) ω).2)) (h.upper_mean (z i)) (h.no_start (z i)))
-  have ht := window_finite_mgf_at_starts (fun i => h.start (z i)) (fun i => h.finish (z i))
-    (window_geometric_order h hz) hK hθ hX hXb hm hWa hWs hW0 hWb
-  apply le_trans _ ht
-  obtain ⟨M,hM⟩ := hWb
-  obtain ⟨L,hL⟩ := h.mean_bounded
-  have hib (f : ι → Ω → ℝ) (hf : ∀ i, Measurable[mΩ] (f i))
-      (hb : ∀ i ω, f i ω ≤ K) : Integrable (fun ω => W ω*Real.exp (θ*∑ i, f i ω)) P :=
-    window_integrable_nonneg (C := M*Real.exp (θ*(Fintype.card ι*K))) (hWa.mul ((Finset.measurable_sum _ (fun i _ => hf i)).const_mul θ).exp)
-      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
-      (fun ω => mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-        ((Finset.sum_le_sum fun i hi => hb i ω).trans_eq (by simp)) hθ))
-        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
-  apply integral_mono
-  · exact hib (fun i ω => Y (z i) ω-μ (z i) ω) (fun i => (hYa i).sub (hμa i))
-      (fun i ω => sub_le_self _ (h.mean_nonneg (z i) ω) |>.trans (h.output_bound (z i) ω).2)
-  · exact hib X (fun i => (hX i).mono (h.finish (z i)).measurableSpace_le le_rfl)
-      (fun i ω => le_abs_self _ |>.trans (hXb i ω))
-  · intro ω
-    apply mul_le_mul_of_nonneg_left _ (hW0 ω)
-    apply Real.exp_le_exp.mpr
-    apply mul_le_mul_of_nonneg_left _ hθ
-    exact Finset.sum_le_sum fun i hi => sub_le_sub_left (min_le_left _ _) _
-
-lemma window_source_mgf_pos_at_starts {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hWa : Measurable[mΩ] W) (hWs : ∀ z, Measurable[(h.start z).measurableSpace] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
-    (∫ ω, W ω * Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
-  obtain ⟨n,a,ha,han,hsep,haR⟩ := window_choose_grid (h.radius_pos.trans_le h.radius_le) h.radius_small
-  let : LinearOrder (Fin n × Fin n) := LinearOrder.lift' toLex toLex.injective
-  let F (ξ : ℝ × ℝ) (ω : Ω) := a^2 * ∑ i : Fin n × Fin n,
-    (Y (windowCenter a ξ i) ω - μ (windowCenter a ξ i) ω)
-  have hcenter (i : Fin n × Fin n) : Measurable (fun ξ => windowCenter a ξ i) :=
-    Complex.measurableEquivRealProd.symm.measurable.comp
-      (((measurable_fst.add_const _).sub_const _).prodMk ((measurable_snd.add_const _).sub_const _))
-  have hmeas : Measurable (Function.uncurry F) := by
-    apply Measurable.const_mul
-    apply Finset.measurable_sum
-    intro i hi
-    exact (h.joint_output.sub h.joint_mean).comp
-      (((hcenter i).comp measurable_fst).prodMk measurable_snd)
-  obtain ⟨L,hL⟩ := h.mean_bounded
-  have hnorm (z : ℂ) (ω : Ω) : ‖Y z ω-μ z ω‖ ≤ K+L := by
-    calc
-      _ ≤ ‖Y z ω‖+‖μ z ω‖ := norm_sub_le _ _
-      _ = Y z ω+μ z ω := by rw [Real.norm_eq_abs, abs_of_nonneg (h.output_bound z ω).1,
-        Real.norm_eq_abs, abs_of_nonneg (h.mean_nonneg z ω)]
-      _ ≤ _ := add_le_add (h.output_bound z ω).2 (hL z ω)
-  have hFb (ξ : ℝ × ℝ) (ω : Ω) : ‖F ξ ω‖ ≤ a^2*((n:ℝ)^2*(K+L)) := by
-    dsimp only [F]
-    rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (sq_nonneg a)]
-    apply mul_le_mul_of_nonneg_left _ (sq_nonneg a)
-    exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun i hi => hnorm _ ω).trans_eq (by simp [sq]; ring))
-  have havg (ω : Ω) : (∫ ξ, F ξ ω ∂windowShiftLaw a) = ∫ z in D, (Y z ω-μ z ω) := by
-    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by
-      obtain ⟨_,hy,hμ⟩ := h.outside z hz ω
-      simp only [hy,hμ,sub_self])]
-    exact window_grid_average ((h.joint_output.sub h.joint_mean).comp
-      (measurable_id.prodMk measurable_const)) (fun z => hnorm z ω) ha han
-      (fun z hz => by
-        obtain ⟨_,hy,hμ⟩ := h.outside z (fun hzD => hz (h.bounded_D hzD)) ω
-        change Y z ω-μ z ω=0
-        simp only [hy,hμ,sub_self])
-  let : IsProbabilityMeasure (windowShiftLaw a) := windowShiftLaw_probability ha
-  have hfinite (ξ : ℝ × ℝ) :
-      (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
-    have ht := window_source_finite_mgf_at_starts h (window_centers_separated (n := n) hsep ha.le ξ)
-      (mul_nonneg hθ (sq_nonneg a)) hWa (fun i => hWs _) hW0 hWb hK
-    have he : (Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2 ≤ 128*K^2*θ^2*R^2 := by
-      simp only [Fintype.card_prod, Fintype.card_fin, Nat.cast_mul]
-      have hs : a^2 ≤ 16*R^2 := by nlinarith [h.radius_pos.trans_le h.radius_le]
-      calc
-        _ = (a*n)^2*a^2*θ^2*K^2/2 := by ring
-        _ = 8*a^2*θ^2*K^2 := by rw [han]; ring
-        _ ≤ 8*(16*R^2)*θ^2*K^2 := by gcongr
-        _ = _ := by ring
-    have ht' : (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
-        (∫ ω, W ω ∂P) * Real.exp ((Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2) := by
-      simpa only [F, mul_assoc] using ht
-    exact ht'.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr he) (integral_nonneg hW0))
-  have ht := window_jensen_mgf (Q := windowShiftLaw a) hmeas hFb hθ hWa hW0 hWb hfinite
-  simpa only [havg] using ht
-
-lemma window_source_weighted_tail {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    (hK : 0 < K) {b : ℝ} (hb : 0 < b)
-    {W : Ω → ℝ} (hWa : Measurable[mΩ] W)
-    (hWs : ∀ z, Measurable[(h.start z).measurableSpace] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω*(if b < ∫ z in D, (Y z ω-μ z ω) then (1:ℝ) else 0) ∂P) ≤
-      (∫ ω, W ω ∂P)*Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2)) := by
-  classical
-  let X (ω : Ω) := ∫ z in D, (Y z ω-μ z ω)
-  let θ := b/(256*K^2*R^2)
-  have hR : 0 < R := h.radius_pos.trans_le h.radius_le
-  have hθ : 0 ≤ θ := by dsimp [θ]; positivity
-  have hXm : Measurable X := window_source_integral_measurable h
-  obtain ⟨M,hM⟩ := hWb
-  obtain ⟨B,hB⟩ := window_source_integral_bounded h
-  have hEi : Integrable (fun ω => W ω*Real.exp (θ*X ω)) P :=
-    window_integrable_nonneg (hWa.mul (hXm.const_mul θ).exp)
-      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
-      (fun ω => mul_le_mul (hM ω) (Real.exp_le_exp.mpr
-        (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hB ω)) hθ))
-        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
-  have hIi : Integrable (fun ω => W ω*(if b<X ω then (1:ℝ) else 0)) P :=
-    window_integrable_nonneg (hWa.mul (Measurable.ite (measurableSet_lt measurable_const hXm)
-      measurable_const measurable_const))
-      (fun ω => mul_nonneg (hW0 ω) (by split_ifs <;> norm_num))
-      (fun ω => by split_ifs <;> simp only [mul_one,mul_zero]; exact hM ω; exact (hW0 ω).trans (hM ω))
-  have hpoint (ω : Ω) : W ω*(if b<X ω then (1:ℝ) else 0) ≤
-      Real.exp (-θ*b)*(W ω*Real.exp (θ*X ω)) := by
-    rw [mul_left_comm,← Real.exp_add]
-    apply mul_le_mul_of_nonneg_left _ (hW0 ω)
-    by_cases hh : b<X ω
-    · rw [ite_eq_left hh]
-      apply Real.one_le_exp
-      nlinarith [mul_nonneg hθ (sub_nonneg.mpr hh.le)]
-    · rw [ite_eq_right hh]
-      exact (Real.exp_pos _).le
-  have hmgf := window_source_mgf_pos_at_starts h hθ hWa hWs hW0 ⟨M,hM⟩ hK
-  calc
-    _ ≤ ∫ ω, Real.exp (-θ*b)*(W ω*Real.exp (θ*X ω)) ∂P :=
-      integral_mono hIi (hEi.const_mul _) hpoint
-    _ = Real.exp (-θ*b)*(∫ ω, W ω*Real.exp (θ*X ω) ∂P) := integral_const_mul _ _
-    _ ≤ Real.exp (-θ*b)*((∫ ω, W ω ∂P)*Real.exp (128*K^2*θ^2*R^2)) :=
-      mul_le_mul_of_nonneg_left hmgf (Real.exp_pos _).le
-    _ = _ := by
-      rw [mul_left_comm,← Real.exp_add]
-      congr 2
-      dsimp [θ]
-      field_simp
-      ring
-
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.WindowAssembly
+import OAI.Probability.SLE.WindowTails
+
+namespace OAI
+
+/-! Physical Mesh Window Core for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+ 
+
+structure SourceWindowCore {Ω : Type*} [MeasurableSpace Ω]
+    (ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›) (D : Set ℂ) (r R : ℝ)
+    (γ : ℝ≥0 → Ω → ℂ) (σ τ : ℂ → Ω → WithTop ℝ≥0) : Prop where
+  measurable_D : MeasurableSet D
+  bounded_D : D ⊆ ball 0 1
+  radius_pos : 0 < r
+  radius_le : r ≤ R
+  radius_small : 3*R ≤ 1
+  continuous_trace : ∀ ω, Continuous (fun t => γ t ω)
+  adapted_trace : ∀ t, Measurable[ℱ t] (γ t)
+  start : ∀ z, IsStoppingTime ℱ (σ z)
+  finish : ∀ z, IsStoppingTime ℱ (τ z)
+  time_le : ∀ z ω, σ z ω ≤ τ z ω
+  contact : ∀ z ω, σ z ω ≠ ⊤ → dist (γ ((σ z ω).untopD 0) ω) z = r
+  confined : ∀ z ω (t : ℝ≥0), σ z ω ≤ (t : WithTop ℝ≥0) → (t : WithTop ℝ≥0) < τ z ω →
+    dist (γ t ω) z < R
+  outside : ∀ z, z ∉ D → ∀ ω, σ z ω = ⊤
+
+lemma SourceWindowCore.attach {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} {D : Set ℂ} {r R K : ℝ}
+    {γ : ℝ≥0 → Ω → ℂ} {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceWindowCore ℱ D r R γ σ τ) (hK : 0 ≤ K)
+    (hYm : ∀ z, Measurable[(h.finish z).measurableSpace] (Y z))
+    (hμm : ∀ z, Measurable[(h.start z).measurableSpace] (μ z))
+    (hY : ∀ z ω, 0 ≤ Y z ω ∧ Y z ω ≤ K)
+    (hμ : ∀ z ω, 0 ≤ μ z ω) (hμb : ∃ M : ℝ, ∀ z ω, μ z ω ≤ M)
+    (hmean : ∀ z, ∀ᵐ ω ∂P, σ z ω ≠ ⊤ → P[Y z | (h.start z).measurableSpace] ω ≤ μ z ω)
+    (hns : ∀ z ω, σ z ω = ⊤ → Y z ω = 0 ∧ μ z ω = 0)
+    (hne : ∀ z ω, τ z ω = ⊤ → Y z ω = 0)
+    (hYj : Measurable (fun p : ℂ × Ω => Y p.1 p.2))
+    (hμj : Measurable (fun p : ℂ × Ω => μ p.1 p.2)) :
+    SourceDisjointWindows P ℱ D r R K γ σ τ Y μ :=
+  ⟨h.measurable_D,h.bounded_D,h.radius_pos,h.radius_le,h.radius_small,hK,
+    h.continuous_trace,h.adapted_trace,h.start,h.finish,h.time_le,h.contact,h.confined,
+    hYm,hμm,hY,hμ,hμb,hmean,hns,hne,
+    fun z hz ω => ⟨h.outside z hz ω,hns z ω (h.outside z hz ω)⟩,hYj,hμj⟩
+
+lemma source_mesh_window_core
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (hD : MeasurableSet D) (c : ℂ) (hDc : D ⊆ ball c 1)
+    (ε r R : ℝ) (hr : 0 < r) (hrR : r < R) (hR : 3*R ≤ 1)
+    (hy : ∀ z ∈ D, r ≤ z.im) :
+    SourceWindowCore a.filtration ((fun z : ℂ => z+c) ⁻¹' D) r R
+      (fun t (ω : a.samples) => γ ω t-c)
+      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c) r)
+      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c) r ε R) := by
+  have hd (x z : ℂ) : dist (x-c) z = dist x (z+c) := by
+    simp only [dist_eq_norm]; congr 1; ring
+  refine ⟨hD.preimage (measurable_id.add_const c),?_,hr,hrR.le,hR,
+    fun ω => (a.trace_continuous ω).sub continuous_const,
+    fun t => (a.trace_adapted t).sub_const c,
+    fun z => source_collarStart_stopping a D r hy (z+c),
+    fun z => source_collarFinish_stopping a D r ε R hy (z+c),?_,?_,?_,?_⟩
+  · intro z hz
+    have hh := hDc hz
+    simpa only [mem_ball,dist_eq_norm,add_sub_cancel_right,sub_zero] using hh
+  · intro z ω
+    by_cases hz : z+c ∈ D
+    · simpa only [sourceCollarStart,sourceCollarFinish,ite_eq_left hz] using
+        source_window_start_le (a.trace_continuous ω) (z+c) r ε R
+    · simp only [sourceCollarStart,sourceCollarFinish,ite_eq_right hz,le_refl]
+  · intro z ω hn
+    have hz : z+c ∈ D := by
+      by_contra hh
+      exact hn (by simp only [sourceCollarStart,ite_eq_right hh])
+    simp only [sourceCollarStart,ite_eq_left hz] at hn ⊢
+    obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hn
+    rw [← hT,WithTop.untopD_coe,hd]
+    exact (source_liveHit_mem (a.trace_continuous ω) (z+c) r
+      (source_window_origin_bound a (z+c) r (hy (z+c) hz) ω) T hT.symm).2
+  · intro z ω t hst htt
+    have hz : z+c ∈ D := by
+      by_contra hh
+      simp only [sourceCollarStart,ite_eq_right hh] at hst
+      exact (WithTop.not_top_le_coe t) hst
+    simp only [sourceCollarStart,sourceCollarFinish,ite_eq_left hz] at hst htt
+    rw [hd]
+    exact (source_before_window (γ ω) (z+c) r ε R t hst htt).2.2
+  · intro z hz ω
+    exact ite_eq_right hz
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_if_bound {p : Prop} [Decidable p] {x K : ℝ}
+    (hK : 0 ≤ K) (hx : p → 0 ≤ x ∧ x ≤ K) :
+    0 ≤ (if p then x else 0) ∧ (if p then x else 0) ≤ K := by
+  by_cases hp : p
+  · simpa only [ite_eq_left hp] using hx hp
+  · simp only [ite_eq_right hp,le_refl,true_and]; exact hK
+
+lemma source_collar_HB_bounds (D : Set ℂ) (γ : ℝ≥0 → ℂ)
+    (z : ℂ) (r ε R : ℝ) :
+    (0 ≤ sourceCollarH D γ z r ε R ∧ sourceCollarH D γ z r ε R ≤ 1) ∧
+    (0 ≤ sourceCollarB D γ z r ε R ∧ sourceCollarB D γ z r ε R ≤ 1) := by
+  constructor
+  · apply source_if_bound zero_le_one
+    intro _; rcases (source_windowHB_range γ z r ε R).1 with h|h <;> simp [h]
+  · apply source_if_bound zero_le_one
+    intro _; rcases (source_windowHB_range γ z r ε R).2 with h|h <;> simp [h]
+
+lemma source_collar_end_zero (D : Set ℂ) (γ : ℝ≥0 → ℂ)
+    (z : ℂ) (r ε R : ℝ)
+    (ht : sourceCollarFinish D γ z r ε R = ⊤) :
+    sourceCollarH D γ z r ε R = 0 ∧ sourceCollarB D γ z r ε R = 0 := by
+  by_cases hz : z ∈ D
+  · simp only [sourceCollarFinish,ite_eq_left hz] at ht
+    simp only [sourceCollarH,sourceCollarB,ite_eq_left hz,sourceSpatialWindowH,sourceSpatialWindowB,
+      ht,ne_eq,not_true_eq_false,false_and,ite_false,and_self]
+  · simp only [sourceCollarH,sourceCollarB,ite_eq_right hz,and_self]
+
+lemma source_collar_start_zero (D : Set ℂ) {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ)
+    (ht : sourceCollarStart D γ z r = ⊤) :
+    sourceCollarH D γ z r ε R = 0 ∧ sourceCollarB D γ z r ε R = 0 := by
+  apply source_collar_end_zero
+  by_cases hz : z ∈ D
+  · simp only [sourceCollarStart,ite_eq_left hz] at ht
+    simp only [sourceCollarFinish,ite_eq_left hz]
+    exact top_le_iff.mp (by simpa only [ht] using source_window_start_le hc z r ε R)
+  · exact ite_eq_right hz
+
+lemma source_collar_weight_bounds
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (D : Set ℂ) (r : ℝ) (hr : 0 < r)
+    (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples) :
+    0 ≤ (if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+      (γ ω) (a.G ω) z r else 0) ∧
+    (if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+      (γ ω) (a.G ω) z r else 0) ≤ r^(-codimension κ) := by
+  apply source_if_bound (Real.rpow_nonneg hr.le _)
+  exact fun hz => (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _)
+    hκ hκ8 hr z (hy z hz)
+
+lemma source_collar_weight_zero
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples)
+    (hn : sourceCollarStart D (γ ω) z r = ⊤) :
+    (if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+      (γ ω) (a.G ω) z r else 0) = 0 := by
+  by_cases hz : z ∈ D
+  · simp only [sourceCollarStart,ite_eq_left hz] at hn
+    rw [ite_eq_left hz]
+    exact source_window_weight_zero a z r (hy z hz) ω hn
+  · exact ite_eq_right hz
+
+lemma source_collar_weight_joint
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (hD : MeasurableSet D) (r : ℝ) :
+    Measurable (fun p : ℂ × a.samples =>
+      if p.1 ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t p.2)
+        (γ p.2) (a.G p.2) p.1 r else 0) :=
+  Measurable.ite (hD.preimage measurable_fst) (a.spatialGreenWeight_measurable r) measurable_const
+
+lemma source_collar_mean_transfer
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im)
+    (f g : ℂ → a.samples → ℝ)
+    (hmean : ∀ z (hz : z ∈ D),
+      (P.comap Subtype.val)[f z | (source_liveHit_stopping a z r (hy z hz)).measurableSpace]
+        ≤ᵐ[P.comap Subtype.val] g z) (z : ℂ) :
+    (P.comap Subtype.val)[(fun ω => if z ∈ D then f z ω else 0) |
+      (source_collarStart_stopping a D r hy z).measurableSpace] ≤ᵐ[P.comap Subtype.val]
+      (fun ω => if z ∈ D then g z ω else 0) := by
+  by_cases hz : z ∈ D
+  · simp only [ite_eq_left hz]
+    have hs := source_stop_space_congr (source_collarStart_stopping a D r hy z)
+      (source_liveHit_stopping a z r (hy z hz))
+      (funext fun ω => by simp only [sourceCollarStart,ite_eq_left hz])
+    rw [hs]
+    exact hmean z hz
+  · simp only [ite_eq_right hz]
+    change (P.comap Subtype.val)[(0 : a.samples → ℝ) |
+      (source_collarStart_stopping a D r hy z).measurableSpace] ≤ᵐ[P.comap Subtype.val] 0
+    rw [condExp_zero]
+
+lemma source_window_scaled_bound {p q r : ℝ} (hq : 0 < q) (hr : 0 < r) :
+    r^p * (q*r)^(-p) = q^(-p) := by
+  rw [Real.mul_rpow hq.le hr.le]
+  calc
+    r^p * (q^(-p)*r^(-p)) = q^(-p)*(r^p*r^(-p)) := by ring
+    _ = q^(-p) := by rw [← Real.rpow_add hr,add_neg_cancel,Real.rpow_zero,mul_one]
+
+lemma source_collar_V_bounds
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (D : Set ℂ) (q r R : ℝ)
+    (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) (ω : a.samples) :
+    0 ≤ r^codimension κ * spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+      (γ ω) (a.G ω) z (q*r) * sourceCollarH D (γ ω) z r (q*r) R ∧
+    r^codimension κ * spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+      (γ ω) (a.G ω) z (q*r) * sourceCollarH D (γ ω) z r (q*r) R ≤ q^(-codimension κ) := by
+  by_cases hz : z ∈ D
+  · have hw := (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _)
+      hκ hκ8 (mul_pos hq hr) z ((show q*r ≤ r by nlinarith).trans (hy z hz))
+    have hH := (source_collar_HB_bounds D (γ ω) z r (q*r) R).1
+    have hnon := mul_nonneg (Real.rpow_nonneg hr.le (codimension κ)) hw.1
+    refine ⟨mul_nonneg hnon hH.1,?_⟩
+    calc
+      _ ≤ r^codimension κ * (q*r)^(-codimension κ) * 1 :=
+        mul_le_mul (mul_le_mul_of_nonneg_left hw.2 (Real.rpow_nonneg hr.le _)) hH.2 hH.1
+          (mul_nonneg (Real.rpow_nonneg hr.le _) (Real.rpow_nonneg (mul_pos hq hr).le _))
+      _ = _ := by rw [mul_one,source_window_scaled_bound hq hr]
+  · simp only [sourceCollarH,ite_eq_right hz,mul_zero,le_refl,true_and]
+    exact Real.rpow_nonneg hq.le _
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_H_window
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
+    (q r L C : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hL : 1 < L) (hR : 3*(L*r) ≤ 1) (hC : 0 ≤ C)
+    (hy : ∀ z ∈ D, r ≤ z.im)
+    (hmean : ∀ z (hz : z ∈ D),
+      (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowH (γ ω) z r (q*r) (L*r)) |
+        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
+      (fun ω : a.samples => C*(q*r)^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) :
+    SourceDisjointWindows (P.comap Subtype.val) a.filtration
+      ((fun z : ℂ => z+c₀) ⁻¹' D) r (L*r) 1
+      (fun t (ω : a.samples) => γ ω t-c₀)
+      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c₀) r)
+      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c₀) r (q*r) (L*r))
+      (fun z (ω : a.samples) => sourceCollarH D (γ ω) (z+c₀) r (q*r) (L*r))
+      (fun z (ω : a.samples) => C*(q*r)^codimension κ*
+        (if z+c₀ ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+          (γ ω) (a.G ω) (z+c₀) r else 0)) := by
+  let core := source_mesh_window_core a D hD c₀ hDc (q*r) r (L*r) hr (by nlinarith) hR hy
+  have hs : q*r ≤ r := by nlinarith
+  have hfac : 0 ≤ C*(q*r)^codimension κ := mul_nonneg hC (Real.rpow_nonneg (mul_pos hq hr).le _)
+  apply core.attach zero_le_one
+  · intro z
+    exact (source_collarHBV_stopped a D r (q*r) (L*r) hs hy (z+c₀)).1
+  · intro z
+    exact (source_collarWeight_stopped a D r hy (z+c₀)).const_mul _
+  · intro z ω
+    exact (source_collar_HB_bounds D (γ ω) (z+c₀) r (q*r) (L*r)).1
+  · intro z ω
+    exact mul_nonneg hfac (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).1
+  · refine ⟨C*(q*r)^codimension κ*r^(-codimension κ),?_⟩
+    intro z ω
+    exact mul_le_mul_of_nonneg_left (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).2 hfac
+  · intro z
+    have hh := source_collar_mean_transfer a D r hy
+      (fun z (ω : a.samples) => sourceSpatialWindowH (γ ω) z r (q*r) (L*r))
+      (fun z (ω : a.samples) => C*(q*r)^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r) hmean (z+c₀)
+    have hg : (fun ω : a.samples => if z+c₀ ∈ D then C*(q*r)^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (z+c₀) r else 0) =
+        (fun ω : a.samples => C*(q*r)^codimension κ*(if z+c₀ ∈ D then spatialGreenWeight κ
+          (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (z+c₀) r else 0)) := by
+      funext ω; rw [mul_ite,mul_zero]
+    rw [hg] at hh
+    exact hh.mono (fun _ h _ => h)
+  · intro z ω hn
+    exact ⟨(source_collar_start_zero D (a.trace_continuous ω) (z+c₀) r (q*r) (L*r) hn).1,
+      by rw [source_collar_weight_zero a D r hy (z+c₀) ω hn,mul_zero]⟩
+  · intro z ω hn
+    exact (source_collar_end_zero D (γ ω) (z+c₀) r (q*r) (L*r) hn).1
+  · exact (source_collarHB_joint a D hD r (q*r) (L*r) hy).1.comp
+      ((measurable_fst.add_const c₀).prodMk measurable_snd)
+  · exact ((source_collar_weight_joint a D hD r).const_mul (C*(q*r)^codimension κ)).comp
+      ((measurable_fst.add_const c₀).prodMk measurable_snd)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_collar_V_mean
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (q r L : ℝ) (hy : ∀ z ∈ D, r ≤ z.im)
+    (hmean : ∀ z (hz : z ∈ D),
+      (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*
+        sourceSpatialWindowH (γ ω) z r (q*r) (L*r)) |
+        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
+      (fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) (z : ℂ) :
+    (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceCollarH D (γ ω) z r (q*r) (L*r)) |
+      (source_collarStart_stopping a D r hy z).measurableSpace] ≤ᵐ[P.comap Subtype.val]
+    (fun ω : a.samples => r^codimension κ*(if z ∈ D then spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0)) := by
+  have hh := source_collar_mean_transfer a D r hy
+    (fun z (ω : a.samples) => r^codimension κ*spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceSpatialWindowH (γ ω) z r (q*r) (L*r))
+    (fun z (ω : a.samples) => r^codimension κ*spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r) hmean z
+  have hf : (fun ω : a.samples => if z ∈ D then r^codimension κ*spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceSpatialWindowH (γ ω) z r (q*r) (L*r) else 0) =
+      (fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*sourceCollarH D (γ ω) z r (q*r) (L*r)) := by
+    funext ω; simp only [sourceCollarH,mul_ite,mul_zero]
+  have hg : (fun ω : a.samples => if z ∈ D then r^codimension κ*spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0) =
+      (fun ω : a.samples => r^codimension κ*(if z ∈ D then spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r else 0)) := by
+    funext ω; rw [mul_ite,mul_zero]
+  rw [hf,hg] at hh
+  exact hh.mono (fun _ h => h)
+
+lemma source_collar_V_joint
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (q r L : ℝ)
+    (hy : ∀ z ∈ D, r ≤ z.im) :
+    Measurable (fun p : ℂ × a.samples => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t p.2) (γ p.2) (a.G p.2) (p.1+c₀) (q*r)*
+        sourceCollarH D (γ p.2) (p.1+c₀) r (q*r) (L*r)) := by
+  have hj := (((a.spatialGreenWeight_measurable (q*r)).const_mul (r^codimension κ)).mul
+      (source_collarHB_joint a D hD r (q*r) (L*r) hy).1)
+  have ht : Measurable (fun p : ℂ × a.samples => (p.1+c₀,p.2)) :=
+    (measurable_fst.add_const c₀).prodMk measurable_snd
+  have hh := hj.comp ht
+  dsimp only [Function.comp_def] at hh
+  exact hh
+
+lemma source_mesh_V_window
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
+    (q r L : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hL : 1 < L) (hR : 3*(L*r) ≤ 1)
+    (hy : ∀ z ∈ D, r ≤ z.im)
+    (hmean : ∀ z (hz : z ∈ D),
+      (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z (q*r)*
+        sourceSpatialWindowH (γ ω) z r (q*r) (L*r)) |
+        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
+      (fun ω : a.samples => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) :
+    SourceDisjointWindows (P.comap Subtype.val) a.filtration
+      ((fun z : ℂ => z+c₀) ⁻¹' D) r (L*r) (q^(-codimension κ))
+      (fun t (ω : a.samples) => γ ω t-c₀)
+      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c₀) r)
+      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c₀) r (q*r) (L*r))
+      (fun z (ω : a.samples) => r^codimension κ*spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (z+c₀) (q*r)*
+        sourceCollarH D (γ ω) (z+c₀) r (q*r) (L*r))
+      (fun z (ω : a.samples) => r^codimension κ*
+        (if z+c₀ ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω)
+          (γ ω) (a.G ω) (z+c₀) r else 0)) := by
+  let core := source_mesh_window_core a D hD c₀ hDc (q*r) r (L*r) hr (by nlinarith) hR hy
+  have hs : q*r ≤ r := by nlinarith
+  have hfac : 0 ≤ r^codimension κ := Real.rpow_nonneg hr.le _
+  apply core.attach (Real.rpow_nonneg hq.le _)
+  · intro z
+    simpa only [mul_assoc] using (source_collarHBV_stopped a D r (q*r) (L*r) hs hy (z+c₀)).2.2.const_mul (r^codimension κ)
+  · intro z
+    exact (source_collarWeight_stopped a D r hy (z+c₀)).const_mul _
+  · intro z ω
+    exact source_collar_V_bounds a hκ hκ8 D q r (L*r) hq hq1 hr hy (z+c₀) ω
+  · intro z ω
+    exact mul_nonneg hfac (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).1
+  · refine ⟨r^codimension κ*r^(-codimension κ),?_⟩
+    intro z ω
+    exact mul_le_mul_of_nonneg_left (source_collar_weight_bounds a hκ hκ8 D r hr hy (z+c₀) ω).2 hfac
+  · intro z
+    exact (source_collar_V_mean a D q r L hy hmean (z+c₀)).mono (fun _ h _ => h)
+  · intro z ω hn
+    refine ⟨?_,?_⟩
+    · rw [(source_collar_start_zero D (a.trace_continuous ω) (z+c₀) r (q*r) (L*r) hn).1,mul_zero]
+    · rw [source_collar_weight_zero a D r hy (z+c₀) ω hn,mul_zero]
+  · intro z ω hn
+    rw [(source_collar_end_zero D (γ ω) (z+c₀) r (q*r) (L*r) hn).1,mul_zero]
+  · exact source_collar_V_joint a D hD c₀ q r L hy
+  · exact ((source_collar_weight_joint a D hD r).const_mul (r^codimension κ)).comp
+      ((measurable_fst.add_const c₀).prodMk measurable_snd)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_B_window
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ) (hD : MeasurableSet D) (c₀ : ℂ) (hDc : D ⊆ ball c₀ 1)
+    (q r L Cbad c : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r)
+    (hL : 1 < L) (hR : 3*(L*r) ≤ 1) (hCb : 0 ≤ Cbad)
+    (hy : ∀ z ∈ D, r ≤ z.im)
+    (hmean : ∀ z (hz : z ∈ D),
+      (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowB (γ ω) z r (q*r) (L*r)) |
+        (source_liveHit_stopping a z r (hy z hz)).measurableSpace] ≤ᵐ[P.comap Subtype.val]
+      (fun ω : a.samples => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1))) :
+    SourceDisjointWindows (P.comap Subtype.val) a.filtration
+      ((fun z : ℂ => z+c₀) ⁻¹' D) r (L*r) 1
+      (fun t (ω : a.samples) => γ ω t-c₀)
+      (fun z (ω : a.samples) => sourceCollarStart D (γ ω) (z+c₀) r)
+      (fun z (ω : a.samples) => sourceCollarFinish D (γ ω) (z+c₀) r (q*r) (L*r))
+      (fun z (ω : a.samples) => sourceCollarB D (γ ω) (z+c₀) r (q*r) (L*r))
+      (fun z (ω : a.samples) => Cbad*L^(-c)*
+        (if sourceCollarStart D (γ ω) (z+c₀) r = ⊤ then 0 else 1)) := by
+  let core := source_mesh_window_core a D hD c₀ hDc (q*r) r (L*r) hr (by nlinarith) hR hy
+  have hs : q*r ≤ r := by nlinarith
+  have hfac : 0 ≤ Cbad*L^(-c) := mul_nonneg hCb (Real.rpow_nonneg (by linarith) _)
+  apply core.attach zero_le_one
+  · intro z
+    exact (source_collarHBV_stopped a D r (q*r) (L*r) hs hy (z+c₀)).2.1
+  · intro z
+    exact (Measurable.ite (measurableSet_eq_fun
+      (source_collarStart_stopping a D r hy (z+c₀)).measurable measurable_const)
+      measurable_const measurable_const).const_mul _
+  · intro z ω
+    exact (source_collar_HB_bounds D (γ ω) (z+c₀) r (q*r) (L*r)).2
+  · intro z ω
+    exact mul_nonneg hfac (by split_ifs <;> norm_num)
+  · refine ⟨Cbad*L^(-c),?_⟩
+    intro z ω
+    exact (mul_le_mul_of_nonneg_left (show (if sourceCollarStart D (γ ω) (z+c₀) r = ⊤ then (0:ℝ) else 1) ≤ 1 by split_ifs <;> norm_num) hfac).trans_eq (mul_one _)
+  · intro z
+    have hh := source_collar_mean_transfer a D r hy
+      (fun z (ω : a.samples) => sourceSpatialWindowB (γ ω) z r (q*r) (L*r))
+      (fun z (ω : a.samples) => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1)) hmean (z+c₀)
+    have hg : (fun ω : a.samples => if z+c₀ ∈ D then Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) (z+c₀) r = ⊤ then 0 else 1) else 0) =
+        (fun ω : a.samples => Cbad*L^(-c)*(if sourceCollarStart D (γ ω) (z+c₀) r = ⊤ then 0 else 1)) := by
+      funext ω; by_cases hz : z+c₀ ∈ D <;> simp [sourceCollarStart,hz]
+    rw [hg] at hh
+    exact hh.mono (fun _ h _ => h)
+  · intro z ω hn
+    exact ⟨(source_collar_start_zero D (a.trace_continuous ω) (z+c₀) r (q*r) (L*r) hn).2,
+      by rw [ite_eq_left hn,mul_zero]⟩
+  · intro z ω hn
+    exact (source_collar_end_zero D (γ ω) (z+c₀) r (q*r) (L*r) hn).2
+  · exact (source_collarHB_joint a D hD r (q*r) (L*r) hy).2.comp
+      ((measurable_fst.add_const c₀).prodMk measurable_snd)
+  · exact ((Measurable.ite (measurableSet_eq_fun
+      (source_collarStart_joint a D hD r hy) measurable_const)
+      measurable_const measurable_const).const_mul (Cbad*L^(-c))).comp
+      ((measurable_fst.add_const c₀).prodMk measurable_snd)
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma window_finite_mgf_at_starts {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
+    {ℱ : Filtration ℝ≥0 mΩ} {σ τ : ι → Ω → WithTop ℝ≥0}
+    (hs : ∀ i, IsStoppingTime ℱ (σ i)) (ht : ∀ i, IsStoppingTime ℱ (τ i))
+    (horder : ∀ i j ω, windowBefore σ j i ω → τ j ω ≤ σ i ω)
+    {X : ι → Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
+    (hX : ∀ i, Measurable[(ht i).measurableSpace] (X i))
+    (hXb : ∀ i ω, |X i ω| ≤ K)
+    (hmean : ∀ i, P[X i|(hs i).measurableSpace] ≤ᵐ[P] 0)
+    {W : Ω → ℝ} (hWa : Measurable[mΩ] W) (hWstart : ∀ i, Measurable[(hs i).measurableSpace] W) (hW0 : ∀ ω, 0 ≤ W ω)
+    (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω * Real.exp (θ*∑ i, X i ω) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
+  classical
+  let q := (θ*K)^2/2
+  let g (i : ι) (ω : Ω) := Real.exp (θ*X i ω-q)
+  let pre (i : ι) (ω : Ω) := ∏ j, if windowBefore σ j i ω then g j ω else 1
+  have hg (i : ι) : Measurable[(ht i).measurableSpace] (g i) :=
+    (((hX i).const_mul θ).sub_const q).exp
+  have hga (i : ι) : Measurable[mΩ] (g i) := (hg i).mono (ht i).measurableSpace_le le_rfl
+  have hg0 (i : ι) (ω : Ω) : 0 ≤ g i ω := (Real.exp_pos _).le
+  have hgB (i : ι) (ω : Ω) : g i ω ≤ Real.exp (θ*K) := by
+    apply Real.exp_le_exp.mpr
+    have := mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hXb i ω)) hθ
+    dsimp [q]
+    nlinarith [sq_nonneg (θ*K)]
+  have hp (i : ι) : Measurable[(hs i).measurableSpace] (pre i) := by
+    apply Finset.measurable_prod
+    intro j hj
+    exact window_measurable_on_event (hs i) (ht j) (hg j)
+      (window_before_measurable hs i j) (fun ω hω => horder i j ω hω) 1
+  have hp0 (i : ι) (ω : Ω) : 0 ≤ pre i ω := by
+    apply Finset.prod_nonneg
+    intro j hj
+    split_ifs <;> positivity
+  have hpB (i : ι) (ω : Ω) : pre i ω ≤ Real.exp (θ*K)^Fintype.card ι := by
+    calc
+      _ ≤ ∏ _j : ι, Real.exp (θ*K) := by
+        apply Finset.prod_le_prod₀
+        · intro j hj; split_ifs <;> positivity
+        · intro j hj; split_ifs
+          · exact hgB j ω
+          · exact Real.one_le_exp (mul_nonneg hθ hK.le)
+      _ = _ := by simp
+  obtain ⟨M,hM⟩ := hWb
+  have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
+  have hHi (i : ι) : Integrable (fun ω => W ω*pre i ω*(g i ω-1)) P := by
+    apply window_integrable_bounded (C := (M*Real.exp (θ*K)^Fintype.card ι)*(Real.exp (θ*K)+1)) (((hWa.mul ((hp i).mono (hs i).measurableSpace_le le_rfl))).mul
+      ((hga i).sub_const 1))
+    intro ω
+    change ‖W ω*pre i ω*(g i ω-1)‖ ≤ _
+    rw [norm_mul, norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω),
+      Real.norm_eq_abs, abs_of_nonneg (hp0 i ω)]
+    apply mul_le_mul
+    · exact mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))
+    · calc
+        _ ≤ ‖g i ω‖ + ‖(1 : ℝ)‖ := norm_sub_le _ _
+        _ = g i ω + 1 := by rw [Real.norm_eq_abs, abs_of_nonneg (hg0 i ω), norm_one]
+        _ ≤ _ := add_le_add (hgB i ω) le_rfl
+    · exact norm_nonneg _
+    · exact mul_nonneg ((hW0 ω).trans (hM ω)) (by positivity)
+  have hHi0 (i : ι) : (∫ ω, W ω*pre i ω*(g i ω-1) ∂P) ≤ 0 := by
+    exact window_weighted_normalized (hs i).measurableSpace_le hK hθ
+      ((hX i).mono (ht i).measurableSpace_le le_rfl) (hXb i) (hmean i)
+      ((hWstart i).mul (hp i)) (fun ω => mul_nonneg (hW0 ω) (hp0 i ω))
+      ⟨M*Real.exp (θ*K)^Fintype.card ι, fun ω =>
+        mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))⟩
+  have hnorm : (∫ ω, W ω * ∏ i, g i ω ∂P) ≤ ∫ ω, W ω ∂P := by
+    have hid (ω : Ω) := window_prod_before σ ω (fun i => g i ω)
+    simp_rw [hid, mul_add, mul_one, Finset.mul_sum]
+    change (∫ ω, W ω + ∑ i, W ω*((g i ω-1)*pre i ω) ∂P) ≤ _
+    simp_rw [show ∀ i ω, W ω*((g i ω-1)*pre i ω) = W ω*pre i ω*(g i ω-1)
+      from fun _ _ => by ring]
+    rw [integral_add hWi (integrable_finsetSum _ (fun i hi => hHi i)), integral_finsetSum]
+    · exact add_le_of_nonpos_right (Finset.sum_nonpos fun i hi => hHi0 i)
+    · exact fun i hi => hHi i
+  have hprod (ω : Ω) : (∏ i, g i ω) =
+      Real.exp (θ*∑ i, X i ω) / Real.exp (Fintype.card ι * q) := by
+    change (∏ i, Real.exp (θ*X i ω-q)) = _
+    rw [← Real.exp_sum]
+    simp only [Finset.sum_sub_distrib, ← Finset.mul_sum, Finset.sum_const,
+      Finset.card_univ, nsmul_eq_mul]
+    exact Real.exp_sub _ _
+  simp only [hprod, ← mul_div_assoc] at hnorm
+  rw [integral_div] at hnorm
+  simpa only [q, mul_div_assoc] using (div_le_iff₀ (Real.exp_pos _)).mp hnorm
+
+ 
+
+lemma window_source_finite_mgf_at_starts {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
+    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    {z : ι → ℂ} (hz : ∀ i j, i ≠ j → 2*R < dist (z i) (z j))
+    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hWa : Measurable[mΩ] W) (hWs : ∀ i, Measurable[(h.start (z i)).measurableSpace] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
+    (∫ ω, W ω * Real.exp (θ*∑ i, (Y (z i) ω-μ (z i) ω)) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
+  let X (i : ι) (ω : Ω) := Y (z i) ω-min (μ (z i) ω) K
+  have hYa (i : ι) : Measurable[mΩ] (Y (z i)) :=
+    (h.output_measurable (z i)).mono (h.finish (z i)).measurableSpace_le le_rfl
+  have hμa (i : ι) : Measurable[mΩ] (μ (z i)) :=
+    (h.mean_measurable (z i)).mono (h.start (z i)).measurableSpace_le le_rfl
+  have hX (i : ι) : Measurable[(h.finish (z i)).measurableSpace] (X i) :=
+    (h.output_measurable (z i)).sub (((h.mean_measurable (z i)).mono
+      ((h.start (z i)).measurableSpace_mono (h.finish (z i)) (h.time_le (z i))) le_rfl).min measurable_const)
+  have hXb (i : ι) (ω : Ω) : |X i ω| ≤ K := by
+    rw [abs_le]
+    have := h.output_bound (z i) ω
+    have := le_min (h.mean_nonneg (z i) ω) hK.le
+    have := min_le_right (μ (z i) ω) K
+    dsimp [X]
+    constructor <;> linarith
+  have hm (i : ι) : P[X i|(h.start (z i)).measurableSpace] ≤ᵐ[P] 0 :=
+    window_capped_cond_mean (h.start (z i)).measurableSpace_le hK.le (hYa i)
+      (h.output_bound (z i)) (h.mean_measurable (z i)) (h.mean_nonneg (z i))
+      (window_cond_mean_at_infinity (h.start (z i))
+        (window_integrable_nonneg (hYa i) (fun ω => (h.output_bound (z i) ω).1)
+          (fun ω => (h.output_bound (z i) ω).2)) (h.upper_mean (z i)) (h.no_start (z i)))
+  have ht := window_finite_mgf_at_starts (fun i => h.start (z i)) (fun i => h.finish (z i))
+    (window_geometric_order h hz) hK hθ hX hXb hm hWa hWs hW0 hWb
+  apply le_trans _ ht
+  obtain ⟨M,hM⟩ := hWb
+  obtain ⟨L,hL⟩ := h.mean_bounded
+  have hib (f : ι → Ω → ℝ) (hf : ∀ i, Measurable[mΩ] (f i))
+      (hb : ∀ i ω, f i ω ≤ K) : Integrable (fun ω => W ω*Real.exp (θ*∑ i, f i ω)) P :=
+    window_integrable_nonneg (C := M*Real.exp (θ*(Fintype.card ι*K))) (hWa.mul ((Finset.measurable_sum _ (fun i _ => hf i)).const_mul θ).exp)
+      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
+      (fun ω => mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+        ((Finset.sum_le_sum fun i hi => hb i ω).trans_eq (by simp)) hθ))
+        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
+  apply integral_mono
+  · exact hib (fun i ω => Y (z i) ω-μ (z i) ω) (fun i => (hYa i).sub (hμa i))
+      (fun i ω => sub_le_self _ (h.mean_nonneg (z i) ω) |>.trans (h.output_bound (z i) ω).2)
+  · exact hib X (fun i => (hX i).mono (h.finish (z i)).measurableSpace_le le_rfl)
+      (fun i ω => le_abs_self _ |>.trans (hXb i ω))
+  · intro ω
+    apply mul_le_mul_of_nonneg_left _ (hW0 ω)
+    apply Real.exp_le_exp.mpr
+    apply mul_le_mul_of_nonneg_left _ hθ
+    exact Finset.sum_le_sum fun i hi => sub_le_sub_left (min_le_left _ _) _
+
+lemma window_source_mgf_pos_at_starts {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hWa : Measurable[mΩ] W) (hWs : ∀ z, Measurable[(h.start z).measurableSpace] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
+    (∫ ω, W ω * Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
+  obtain ⟨n,a,ha,han,hsep,haR⟩ := window_choose_grid (h.radius_pos.trans_le h.radius_le) h.radius_small
+  let : LinearOrder (Fin n × Fin n) := LinearOrder.lift' toLex toLex.injective
+  let F (ξ : ℝ × ℝ) (ω : Ω) := a^2 * ∑ i : Fin n × Fin n,
+    (Y (windowCenter a ξ i) ω - μ (windowCenter a ξ i) ω)
+  have hcenter (i : Fin n × Fin n) : Measurable (fun ξ => windowCenter a ξ i) :=
+    Complex.measurableEquivRealProd.symm.measurable.comp
+      (((measurable_fst.add_const _).sub_const _).prodMk ((measurable_snd.add_const _).sub_const _))
+  have hmeas : Measurable (Function.uncurry F) := by
+    apply Measurable.const_mul
+    apply Finset.measurable_sum
+    intro i hi
+    exact (h.joint_output.sub h.joint_mean).comp
+      (((hcenter i).comp measurable_fst).prodMk measurable_snd)
+  obtain ⟨L,hL⟩ := h.mean_bounded
+  have hnorm (z : ℂ) (ω : Ω) : ‖Y z ω-μ z ω‖ ≤ K+L := by
+    calc
+      _ ≤ ‖Y z ω‖+‖μ z ω‖ := norm_sub_le _ _
+      _ = Y z ω+μ z ω := by rw [Real.norm_eq_abs, abs_of_nonneg (h.output_bound z ω).1,
+        Real.norm_eq_abs, abs_of_nonneg (h.mean_nonneg z ω)]
+      _ ≤ _ := add_le_add (h.output_bound z ω).2 (hL z ω)
+  have hFb (ξ : ℝ × ℝ) (ω : Ω) : ‖F ξ ω‖ ≤ a^2*((n:ℝ)^2*(K+L)) := by
+    dsimp only [F]
+    rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (sq_nonneg a)]
+    apply mul_le_mul_of_nonneg_left _ (sq_nonneg a)
+    exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun i hi => hnorm _ ω).trans_eq (by simp [sq]; ring))
+  have havg (ω : Ω) : (∫ ξ, F ξ ω ∂windowShiftLaw a) = ∫ z in D, (Y z ω-μ z ω) := by
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by
+      obtain ⟨_,hy,hμ⟩ := h.outside z hz ω
+      simp only [hy,hμ,sub_self])]
+    exact window_grid_average ((h.joint_output.sub h.joint_mean).comp
+      (measurable_id.prodMk measurable_const)) (fun z => hnorm z ω) ha han
+      (fun z hz => by
+        obtain ⟨_,hy,hμ⟩ := h.outside z (fun hzD => hz (h.bounded_D hzD)) ω
+        change Y z ω-μ z ω=0
+        simp only [hy,hμ,sub_self])
+  let : IsProbabilityMeasure (windowShiftLaw a) := windowShiftLaw_probability ha
+  have hfinite (ξ : ℝ × ℝ) :
+      (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
+    have ht := window_source_finite_mgf_at_starts h (window_centers_separated (n := n) hsep ha.le ξ)
+      (mul_nonneg hθ (sq_nonneg a)) hWa (fun i => hWs _) hW0 hWb hK
+    have he : (Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2 ≤ 128*K^2*θ^2*R^2 := by
+      simp only [Fintype.card_prod, Fintype.card_fin, Nat.cast_mul]
+      have hs : a^2 ≤ 16*R^2 := by nlinarith [h.radius_pos.trans_le h.radius_le]
+      calc
+        _ = (a*n)^2*a^2*θ^2*K^2/2 := by ring
+        _ = 8*a^2*θ^2*K^2 := by rw [han]; ring
+        _ ≤ 8*(16*R^2)*θ^2*K^2 := by gcongr
+        _ = _ := by ring
+    have ht' : (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
+        (∫ ω, W ω ∂P) * Real.exp ((Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2) := by
+      simpa only [F, mul_assoc] using ht
+    exact ht'.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr he) (integral_nonneg hW0))
+  have ht := window_jensen_mgf (Q := windowShiftLaw a) hmeas hFb hθ hWa hW0 hWb hfinite
+  simpa only [havg] using ht
+
+lemma window_source_weighted_tail {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    (hK : 0 < K) {b : ℝ} (hb : 0 < b)
+    {W : Ω → ℝ} (hWa : Measurable[mΩ] W)
+    (hWs : ∀ z, Measurable[(h.start z).measurableSpace] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω*(if b < ∫ z in D, (Y z ω-μ z ω) then (1:ℝ) else 0) ∂P) ≤
+      (∫ ω, W ω ∂P)*Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2)) := by
+  classical
+  let X (ω : Ω) := ∫ z in D, (Y z ω-μ z ω)
+  let θ := b/(256*K^2*R^2)
+  have hR : 0 < R := h.radius_pos.trans_le h.radius_le
+  have hθ : 0 ≤ θ := by dsimp [θ]; positivity
+  have hXm : Measurable X := window_source_integral_measurable h
+  obtain ⟨M,hM⟩ := hWb
+  obtain ⟨B,hB⟩ := window_source_integral_bounded h
+  have hEi : Integrable (fun ω => W ω*Real.exp (θ*X ω)) P :=
+    window_integrable_nonneg (hWa.mul (hXm.const_mul θ).exp)
+      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
+      (fun ω => mul_le_mul (hM ω) (Real.exp_le_exp.mpr
+        (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hB ω)) hθ))
+        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
+  have hIi : Integrable (fun ω => W ω*(if b<X ω then (1:ℝ) else 0)) P :=
+    window_integrable_nonneg (hWa.mul (Measurable.ite (measurableSet_lt measurable_const hXm)
+      measurable_const measurable_const))
+      (fun ω => mul_nonneg (hW0 ω) (by split_ifs <;> norm_num))
+      (fun ω => by split_ifs <;> simp only [mul_one,mul_zero]; exact hM ω; exact (hW0 ω).trans (hM ω))
+  have hpoint (ω : Ω) : W ω*(if b<X ω then (1:ℝ) else 0) ≤
+      Real.exp (-θ*b)*(W ω*Real.exp (θ*X ω)) := by
+    rw [mul_left_comm,← Real.exp_add]
+    apply mul_le_mul_of_nonneg_left _ (hW0 ω)
+    by_cases hh : b<X ω
+    · rw [ite_eq_left hh]
+      apply Real.one_le_exp
+      nlinarith [mul_nonneg hθ (sub_nonneg.mpr hh.le)]
+    · rw [ite_eq_right hh]
+      exact (Real.exp_pos _).le
+  have hmgf := window_source_mgf_pos_at_starts h hθ hWa hWs hW0 ⟨M,hM⟩ hK
+  calc
+    _ ≤ ∫ ω, Real.exp (-θ*b)*(W ω*Real.exp (θ*X ω)) ∂P :=
+      integral_mono hIi (hEi.const_mul _) hpoint
+    _ = Real.exp (-θ*b)*(∫ ω, W ω*Real.exp (θ*X ω) ∂P) := integral_const_mul _ _
+    _ ≤ Real.exp (-θ*b)*((∫ ω, W ω ∂P)*Real.exp (128*K^2*θ^2*R^2)) :=
+      mul_le_mul_of_nonneg_left hmgf (Real.exp_pos _).le
+    _ = _ := by
+      rw [mul_left_comm,← Real.exp_add]
+      congr 2
+      dsimp [θ]
+      field_simp
+      ring
+
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/PhysicalTransport.lean b/lean/OAI/Probability/SLE/PhysicalTransport.lean
index 7ea3aaf4f52b2568b8896aaa76cef15338be5b0c..c0d46f2f5a07fc4ae928d394609b30d6d75f1110 100644
--- a/lean/OAI/Probability/SLE/PhysicalTransport.lean
+++ b/lean/OAI/Probability/SLE/PhysicalTransport.lean
@@ -1,394 +1,394 @@
-import Mathlib
-import OAI.Probability.SLE.DiskHalfPlane
-import OAI.Probability.SLE.ChartMeasurability
-
-namespace OAI
-
-/-! Physical Transport for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-def SourceInteriorTraceDensityTarget : Prop :=
-  ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
-  Continuous U → IsCapacityTwoWitness U γ G F →
-  ∀ s t : ℝ≥0, s < t → ∃ u ∈ Ioo s t, 0 < (γ u).im
-
-lemma IsCapacityTwoWitness.survivingDomain_injective
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) :
-    Function.Injective (survivingDomain γ) := by
-  intro s t he
-  have hz : F s Complex.I ∈ survivingDomain γ s := by
-    rcases h with ⟨hγ,h0,hupper,hG,hF,hGm,hFm,hrest⟩
-    exact hFm s (by simp [upperHalfPlane])
-  have ht : F s Complex.I ∈ survivingDomain γ t := he ▸ hz
-  apply live_radius_unique hU h (F s Complex.I) s t hz ht
-  exact le_antisymm (h.radius_le_of_domain_subset hU s t he.subset _ hz)
-    (h.radius_le_of_domain_subset hU t s he.symm.subset _ ht)
-
-theorem sourceInteriorTraceDensity_proved : SourceInteriorTraceDensityTarget := by
-  intro U γ G F hU h s t hst
-  by_contra hnone
-  obtain ⟨b,hsb,hbt⟩ := exists_between hst
-  obtain ⟨a,hsa,hab⟩ := exists_between hsb
-  have hbase : upperHalfPlane \ (γ '' Icc 0 a) = upperHalfPlane \ (γ '' Icc 0 b) := by
-    apply Subset.antisymm
-    · intro z hz
-      refine ⟨hz.1,?_⟩
-      rintro ⟨u,hu,huz⟩
-      by_cases hua : u ≤ a
-      · exact hz.2 ⟨u,⟨hu.1,hua⟩,huz⟩
-      · apply hnone
-        refine ⟨u,⟨hsa.trans (not_le.mp hua),hu.2.trans_lt hbt⟩,?_⟩
-        rw [huz]
-        exact hz.1
-    · intro z hz
-      refine ⟨hz.1,?_⟩
-      rintro ⟨u,hu,huz⟩
-      exact hz.2 ⟨u,⟨hu.1,hu.2.trans hab.le⟩,huz⟩
-  have he : survivingDomain γ a = survivingDomain γ b := by
-    unfold survivingDomain
-    rw [hbase]
-  exact hab.ne (h.survivingDomain_injective hU he)
-
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
-lemma source_flow_fence {g dg : ℝ → ℂ} {u : ℝ → ℝ} {a b c δ : ℝ}
-    (hab : a ≤ b) (hδ : 0 ≤ δ)
-    (hg : ContinuousOn g (Icc a b))
-    (hd : ∀ t ∈ Ico a b, HasDerivWithinAt g (dg t) (Ici t) t)
-    (hu : ∀ t ∈ Ico a b, |u t-c| ≤ δ)
-    (heq : ∀ t ∈ Ico a b, ‖dg t‖ ≤ 2 / ‖g t-(u t:ℂ)‖) :
-    ‖g b-(c:ℂ)‖^2 ≤ ‖g a-(c:ℂ)‖^2+4*δ^2+9*(b-a) := by
-  let f (t : ℝ) := ‖g t-(c:ℂ)‖^2
-  let df (t : ℝ) := 2 * inner ℝ (g t-(c:ℂ)) (dg t)
-  have hf : ContinuousOn f (Icc a b) := (hg.sub continuousOn_const).norm.pow 2
-  have hdf (t : ℝ) (ht : t ∈ Ico a b) :
-      HasDerivWithinAt f (df t) (Ici t) t := (hd t ht |>.sub_const (c:ℂ)).norm_sq
-  have hn (t : ℝ) (ht : t ∈ Ico a b) (hh : 4*δ^2 < f t) : df t < 9 := by
-    have hr : 2*δ < ‖g t-(c:ℂ)‖ := by
-      dsimp [f] at hh
-      nlinarith [norm_nonneg (g t-(c:ℂ))]
-    have htri : ‖g t-(c:ℂ)‖ ≤ ‖g t-(u t:ℂ)‖+δ := by
-      calc
-        _ ≤ ‖g t-(u t:ℂ)‖+‖(u t:ℂ)-(c:ℂ)‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
-        _ ≤ _ := by
-          have hc : ‖(u t:ℂ)-(c:ℂ)‖ ≤ δ := by simpa only [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs] using hu t ht
-          linarith
-    have hp : 0 < ‖g t-(u t:ℂ)‖ := by nlinarith
-    have hb : ‖g t-(c:ℂ)‖ ≤ 2*‖g t-(u t:ℂ)‖ := by linarith
-    have hi := real_inner_le_norm (g t-(c:ℂ)) (dg t)
-    have hm := mul_le_mul_of_nonneg_left (heq t ht) (norm_nonneg (g t-(c:ℂ)))
-    have hx : ‖g t-(c:ℂ)‖ * (2 / ‖g t-(u t:ℂ)‖) ≤ 4 := by
-      rw [← mul_div_assoc]
-      exact (div_le_iff₀ hp).mpr (by linarith)
-    change 2 * inner ℝ (g t-(c:ℂ)) (dg t) < 9
-    linarith
-  have hbound (ε : ℝ) (hε : 0 < ε) :
-      f b ≤ f a+4*δ^2+ε+9*(b-a) := by
-    apply image_le_of_deriv_right_lt_deriv_boundary hf hdf
-      (B := fun t => f a+4*δ^2+ε+9*(t-a)) (B' := fun _ => 9)
-      (by nlinarith [sq_nonneg δ])
-      (fun t => by convert (((hasDerivAt_id t).sub_const a).const_mul 9).const_add (f a+4*δ^2+ε) using 1 <;> first | rfl | norm_num)
-      (fun t ht he => ?_) ⟨hab,le_rfl⟩
-    apply hn t ht
-    have : 0 ≤ f a := sq_nonneg _
-    rw [he]
-    nlinarith [ht.1]
-  apply le_of_forall_pos_le_add
-  intro ε hε
-  simpa only [f] using (hbound ε hε).trans_eq (by ring)
-end SLEExactGauge
-
-end
-
-section
-namespace SLEExactGauge
-lemma IsCapacityTwoWitness.short_flow_bound
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) {s T : ℝ≥0} (hs : s ≤ T)
-    {z : ℂ} (hz : z ∈ survivingDomain γ T) {c δ : ℝ} (hδ : 0 ≤ δ)
-    (hU : ∀ t ∈ Icc s T, |U t-c| ≤ δ) :
-    ‖G T z-(c:ℂ)‖^2 ≤ ‖G s z-(c:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s) := by
-  have hsub : Icc (s:ℝ) T ⊆ Icc (0:ℝ) T := Icc_subset_Icc_left s.coe_nonneg
-  have hb := source_flow_fence (g := fun u => G u.toNNReal z)
-    (dg := fun u => 2/(G u.toNNReal z-(U u.toNNReal:ℂ)))
-    (u := fun u => U u.toNNReal) (a := (s:ℝ)) (b := (T:ℝ)) (c := c)
-    (NNReal.coe_le_coe.mpr hs) hδ ((h.time_continuousOn T z hz).mono hsub)
-    (fun u hu => hasDerivWithinAt_Ici_of_Icc hu ((h.time_derivative T z hz u
-      (hsub ⟨hu.1,hu.2.le⟩)).mono hsub))
-    (fun u hu => hU u.toNNReal ⟨by
-      apply NNReal.coe_le_coe.mp
-      rw [Real.coe_toNNReal _ (s.coe_nonneg.trans hu.1)]
-      exact hu.1, Real.toNNReal_le_iff_le_coe.mpr hu.2.le⟩)
-    (fun u hu => by simp)
-  simpa only [Real.toNNReal_coe] using hb
-
-lemma IsCapacityTwoWitness.short_reverse_bound
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) {s T : ℝ≥0} (hs : s ≤ T)
-    {z : ℂ} (hz : z ∈ survivingDomain γ T) {c δ : ℝ} (hδ : 0 ≤ δ)
-    (hU : ∀ t ∈ Icc s T, |U t-c| ≤ δ) :
-    ‖G s z-(c:ℂ)‖^2 ≤ ‖G T z-(c:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s) := by
-  have hm (u : ℝ) (hu : u ∈ Icc (0:ℝ) ((T:ℝ)-s)) :
-      (T:ℝ)-u ∈ Icc (s:ℝ) T := by constructor <;> linarith [hu.1,hu.2]
-  have hm' (u : ℝ) (hu : u ∈ Icc (0:ℝ) ((T:ℝ)-s)) :
-      (T:ℝ)-u ∈ Icc (0:ℝ) T := ⟨s.coe_nonneg.trans (hm u hu).1,(hm u hu).2⟩
-  have hd (u : ℝ) (hu : u ∈ Icc (0:ℝ) ((T:ℝ)-s)) :
-      HasDerivWithinAt (fun v : ℝ => G ((T:ℝ)-v).toNNReal z)
-        (-(2/(G ((T:ℝ)-u).toNNReal z-(U ((T:ℝ)-u).toNNReal:ℂ))))
-        (Icc 0 ((T:ℝ)-s)) u := by
-    have hi : HasDerivWithinAt (fun v : ℝ => (T:ℝ)-v) (-1)
-        (Icc 0 ((T:ℝ)-s)) u := by
-      convert ((hasDerivAt_id u).const_sub (T:ℝ)).hasDerivWithinAt using 1
-      rfl
-    have hh := (h.time_derivative T z hz _ (hm' u hu)).scomp u hi
-      (fun v hv => hm' v hv)
-    simpa only [neg_smul, one_smul, Function.comp_def] using hh
-  have hc : ContinuousOn (fun u : ℝ => G ((T:ℝ)-u).toNNReal z)
-      (Icc (0:ℝ) ((T:ℝ)-s)) :=
-    (h.time_continuousOn T z hz).comp (continuousOn_const.sub continuousOn_id) hm'
-  have hb := source_flow_fence (g := fun u => G ((T:ℝ)-u).toNNReal z)
-    (dg := fun u => -(2/(G ((T:ℝ)-u).toNNReal z-(U ((T:ℝ)-u).toNNReal:ℂ))))
-    (u := fun u => U ((T:ℝ)-u).toNNReal) (a := 0) (b := (T:ℝ)-s) (c := c)
-    (sub_nonneg.mpr (NNReal.coe_le_coe.mpr hs)) hδ hc
-    (fun u hu => hasDerivWithinAt_Ici_of_Icc hu (hd u ⟨hu.1,hu.2.le⟩))
-    (fun u hu => hU _ (by
-      constructor
-      · apply NNReal.coe_le_coe.mp
-        rw [Real.coe_toNNReal _ (hm' u ⟨hu.1,hu.2.le⟩).1]
-        exact (hm u ⟨hu.1,hu.2.le⟩).1
-      · exact Real.toNNReal_le_iff_le_coe.mpr (hm u ⟨hu.1,hu.2.le⟩).2))
-    (fun u hu => by simp)
-  simpa only [sub_sub_cancel,sub_zero, Real.toNNReal_coe] using hb
-end SLEExactGauge
-
-end
-
-section
-namespace SLEExactGauge
-lemma IsCapacityTwoWitness.earlier_tip_bound
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) {s T : ℝ≥0} (hs : s ≤ T)
-    (ha : γ T ∈ survivingDomain γ s) {δ : ℝ} (hδ : 0 ≤ δ)
-    (hU : ∀ t ∈ Icc s T, |U t-U T| ≤ δ) :
-    ‖G s (γ T)-(U T:ℂ)‖^2 ≤ 4*δ^2+9*((T:ℝ)-s) := by
-  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2
-  have hlim := (h.spatial_differentiableAt s (γ T) ha).continuousAt.tendsto.comp (htip T)
-  have hleft := ((hlim.sub (tendsto_const_nhds (x := (U T:ℂ)))).norm).pow 2
-  have hright : Tendsto (fun y : ℝ =>
-      ‖((U T:ℂ)+(y:ℂ)*Complex.I)-(U T:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s))
-      (𝓝[>] 0) (𝓝 (4*δ^2+9*((T:ℝ)-s))) := by
-    have hc : Continuous (fun y : ℝ =>
-      ‖((U T:ℂ)+(y:ℂ)*Complex.I)-(U T:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s)) := by fun_prop
-    simpa using (hc.continuousAt (x := 0)).tendsto.mono_left nhdsWithin_le_nhds
-  apply le_of_tendsto_of_tendsto hleft hright
-  filter_upwards [self_mem_nhdsWithin] with y hy
-  have hiy : (U T:ℂ)+(y:ℂ)*Complex.I ∈ upperHalfPlane := by
-    simpa [upperHalfPlane] using hy
-  have hf := h.2.2.2.2.2.2.1 T hiy
-  have hi := h.2.2.2.2.2.2.2.2.1 T _ hiy
-  convert h.short_reverse_bound hs hf hδ hU using 1 <;> first | rfl | simp only [hi]
-end SLEExactGauge
-
-end
-
-section
-namespace SLEExactGauge
-lemma IsCapacityTwoWitness.fresh_tip_chart_tendsto
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (hUc : Continuous U)
-    {T : ℝ≥0} (hT : 0 < T)
-    (ha : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) :
-    Tendsto (G T) (𝓝[survivingDomain γ T] (γ T)) (𝓝 (U T:ℂ)) := by
-  rw [Metric.tendsto_nhds]
-  intro ε hε
-  obtain ⟨ρ,hρ,hosc⟩ := Metric.continuousAt_iff.mp (hUc.continuousAt (x := T)) (ε/10) (by positivity)
-  let η : ℝ := min ((T:ℝ)/2) (min (ρ/2) (ε^2/100))
-  have hη : 0 < η := lt_min (by exact div_pos (NNReal.coe_pos.mpr hT) (by norm_num))
-    (lt_min (by positivity) (by positivity))
-  have hηT : η ≤ (T:ℝ)/2 := min_le_left _ _
-  have hηρ : η ≤ ρ/2 := (min_le_right _ _).trans (min_le_left _ _)
-  have hηε : η ≤ ε^2/100 := (min_le_right _ _).trans (min_le_right _ _)
-  let s : ℝ≥0 := ⟨(T:ℝ)-η,by linarith [T.coe_nonneg]⟩
-  have hs : s < T := by change (T:ℝ)-η < T; linarith
-  have hdiff : (T:ℝ)-s = η := by change (T:ℝ)-((T:ℝ)-η)=η; ring
-  have hosc' (t : ℝ≥0) (ht : t ∈ Icc s T) : |U t-U T| ≤ ε/10 := by
-    have hdist : dist t T < ρ := by
-      rw [NNReal.dist_eq,abs_of_nonpos (sub_nonpos.mpr (NNReal.coe_le_coe.mpr ht.2))]
-      have ht' : (T:ℝ)-η ≤ t := NNReal.coe_le_coe.mpr ht.1
-      linarith
-    simpa only [Real.dist_eq] using (hosc hdist).le
-  have hbound := h.earlier_tip_bound hs.le (ha s hs) (by positivity : 0 ≤ ε/10) hosc'
-  rw [hdiff] at hbound
-  have hnorm : ‖G s (γ T)-(U T:ℂ)‖ < ε/2 := by
-    nlinarith [norm_nonneg (G s (γ T)-(U T:ℂ))]
-  have hev : ∀ᶠ w in 𝓝 (γ T), ‖G s w-(U T:ℂ)‖ < ε/2 :=
-    ((h.spatial_differentiableAt s (γ T) (ha s hs)).continuousAt.sub continuousAt_const).norm
-      |>.eventually (isOpen_Iio.mem_nhds hnorm)
-  filter_upwards [hev.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with w hw hwt
-  have hh := h.short_flow_bound hs.le hwt (by positivity : 0 ≤ ε/10) hosc'
-  rw [hdiff] at hh
-  rw [dist_eq_norm]
-  nlinarith [norm_nonneg (G s w-(U T:ℂ)), norm_nonneg (G T w-(U T:ℂ))]
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
- 
-
-def SourceFirstHitPhysicalAccessTarget : Prop :=
-  ∀ (γ : ℝ≥0 → ℂ), Continuous γ →
-  ∀ (s τ : ℝ≥0) (z : ℂ) (r : ℝ), 0 < r →
-    Metric.closedBall z r ⊆ survivingDomain γ s → s ≤ τ →
-    dist (γ τ) z = r →
-    (∀ t ∈ Ico s τ, dist (γ t) z ≠ r) → z ∈ survivingDomain γ τ →
-    (∀ t : ℝ≥0, t < τ → γ t ≠ γ τ) ∧
-    Metric.ball z r ⊆ survivingDomain γ τ ∧
-    (∀ v ∈ Ioc (0:ℝ) 1, (1-v) • γ τ + v • z ∈ survivingDomain γ τ)
-
-lemma disk_distance_gt_before_first_hit (γ : ℝ≥0 → ℂ) (hc : Continuous γ)
-    (s τ : ℝ≥0) (z : ℂ) (r : ℝ)
-    (hfree : Metric.closedBall z r ⊆ survivingDomain γ s)
-    (hfirst : ∀ t ∈ Ico s τ, dist (γ t) z ≠ r)
-    (t : ℝ≥0) (ht : t < τ) : r < dist (γ t) z := by
-  have hold (u : ℝ≥0) (hu : u ≤ s) : r < dist (γ u) z := by
-    by_contra hn
-    have hu' : γ u ∈ Metric.closedBall z r := by
-      simpa only [Metric.mem_closedBall] using not_lt.mp hn
-    have hh := hfree hu'
-    exact hh.1.2 ⟨u,⟨zero_le,hu⟩,rfl⟩
-  by_cases hts : t ≤ s
-  · exact hold t hts
-  · by_contra hn
-    obtain ⟨u,hu,he⟩ := intermediate_value_Icc' (not_le.mp hts).le
-      ((hc.dist continuous_const).continuousOn) (show r ∈ Icc (dist (γ t) z) (dist (γ s) z)
-        from ⟨not_lt.mp hn,(hold s le_rfl).le⟩)
-    exact hfirst u ⟨hu.1,hu.2.trans_lt ht⟩ he
-
-lemma radialAccess_mem_ball (a z : ℂ) (r : ℝ) (hr : 0 < r)
-    (ha : dist a z = r) (v : ℝ) (hv : v ∈ Ioc (0:ℝ) 1) :
-    (1-v) • a + v • z ∈ Metric.ball z r := by
-  rw [Metric.mem_ball,dist_eq_norm]
-  have he : (1-v) • a + v • z-z = (1-v) • (a-z) := by module
-  rw [he,norm_smul,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hv.2),← dist_eq_norm,ha]
-  nlinarith [hv.1]
-
-theorem sourceFirstHitPhysicalAccess : SourceFirstHitPhysicalAccessTarget := by
-  intro γ hc s τ z r hr hfree hsτ hhit hfirst hz
-  have hprior := disk_distance_gt_before_first_hit γ hc s τ z r hfree hfirst
-  have hball : Metric.ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 τ) := by
-    intro w hw
-    refine ⟨(hfree (Metric.ball_subset_closedBall hw)).1.1,?_⟩
-    rintro ⟨t,ht,rfl⟩
-    have hlt : dist (γ t) z < r := hw
-    rcases lt_or_eq_of_le ht.2 with htτ | rfl
-    · exact (not_lt_of_ge (hprior t htτ).le) hlt
-    · rw [hhit] at hlt
-      exact lt_irrefl _ hlt
-  have hcc := (convex_ball z r).isPreconnected.subset_connectedComponentIn
-    (Metric.mem_ball_self hr) hball
-  have hsurv : Metric.ball z r ⊆ survivingDomain γ τ := by
-    intro w hw
-    refine ⟨hball hw,?_⟩
-    rw [← connectedComponentIn_eq (hcc hw)]
-    exact hz.2
-  refine ⟨?_,hsurv,?_⟩
-  · intro t ht he
-    have hh := hprior t ht
-    rw [he,hhit] at hh
-    exact lt_irrefl _ hh
-  · intro v hv
-    exact hsurv (radialAccess_mem_ball (γ τ) z r hr hhit v hv)
-
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
- 
-
-def SourceFirstHitChartAccessTarget : Prop :=
-  ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
-    Continuous U → IsCapacityTwoWitness U γ G F →
-    ∀ (s τ : ℝ≥0) (z : ℂ) (r : ℝ), 0 < r →
-      Metric.closedBall z r ⊆ survivingDomain γ s → s ≤ τ →
-      dist (γ τ) z = r → (∀ t ∈ Ico s τ, dist (γ t) z ≠ r) →
-      z ∈ survivingDomain γ τ →
-      Tendsto (fun v : ℝ => G τ ((1-v) • γ τ + v • z) - (U τ : ℂ))
-        (𝓝[>] 0) (𝓝 0)
-
-end SLEExactGauge
-
-end
-
-section
-namespace SLEExactGauge
-lemma first_hit_tip_survives_earlier (γ : ℝ≥0 → ℂ) (hc : Continuous γ)
-    (s τ : ℝ≥0) (z : ℂ) (r : ℝ) (hr : 0 < r)
-    (hfree : Metric.closedBall z r ⊆ survivingDomain γ s)
-    (hhit : dist (γ τ) z = r)
-    (hfirst : ∀ t ∈ Ico s τ, dist (γ t) z ≠ r)
-    (hz : z ∈ survivingDomain γ τ) :
-    ∀ u : ℝ≥0, u < τ → γ τ ∈ survivingDomain γ u := by
-  intro u hu
-  have hprior := disk_distance_gt_before_first_hit γ hc s τ z r hfree hfirst
-  have hclosed : Metric.closedBall z r ⊆ upperHalfPlane \ (γ '' Icc 0 u) := by
-    intro w hw
-    refine ⟨(hfree hw).1.1,?_⟩
-    rintro ⟨t,ht,rfl⟩
-    exact (not_le_of_gt (hprior t (ht.2.trans_lt hu))) (Metric.mem_closedBall.mp hw)
-  have hcc := (convex_closedBall z r).isPreconnected.subset_connectedComponentIn
-    (Metric.mem_closedBall_self hr.le) hclosed
-  have hmem : γ τ ∈ Metric.closedBall z r := Metric.mem_closedBall.mpr hhit.le
-  refine ⟨hclosed hmem,?_⟩
-  rw [← connectedComponentIn_eq (hcc hmem)]
-  exact (survivingDomain_antitone γ hu.le hz).2
-
-theorem sourceFirstHitChartAccess_proved : SourceFirstHitChartAccessTarget := by
-  intro U γ G F hUc h s τ z r hr hfree hsτ hhit hfirst hz
-  have hslt : s < τ := by
-    apply lt_of_le_of_ne hsτ
-    intro he
-    have hm := hfree (Metric.mem_closedBall.mpr hhit.le)
-    exact hm.1.2 ⟨τ,⟨zero_le,by rw [he]⟩,rfl⟩
-  have hT : 0 < τ := lt_of_le_of_lt (show (0:ℝ≥0) ≤ s from zero_le) hslt
-  have ha := first_hit_tip_survives_earlier γ h.1 s τ z r hr hfree hhit hfirst hz
-  have hlim := h.fresh_tip_chart_tendsto hUc hT ha
-  have hradial := (sourceFirstHitPhysicalAccess γ h.1 s τ z r hr hfree hsτ hhit hfirst hz).2.2
-  have hcont : Continuous (fun v : ℝ => (1-v) • γ τ + v • z) := by fun_prop
-  have hpath : Tendsto (fun v : ℝ => (1-v) • γ τ + v • z) (𝓝[>] 0)
-      (𝓝[survivingDomain γ τ] (γ τ)) := by
-    apply tendsto_nhdsWithin_iff.mpr
-    constructor
-    · simpa using (hcont.continuousAt (x := 0)).tendsto.mono_left nhdsWithin_le_nhds
-    · have hev : ∀ᶠ v : ℝ in 𝓝[>] 0, v < 1 :=
-        (eventually_lt_nhds (by norm_num : (0:ℝ) < 1)).filter_mono nhdsWithin_le_nhds
-      filter_upwards [self_mem_nhdsWithin,hev] with v hv hv'
-      exact hradial v ⟨hv,hv'.le⟩
-  simpa only [Function.comp_def,sub_self] using
-    (hlim.comp hpath).sub (tendsto_const_nhds (x := (U τ:ℂ)))
-end SLEExactGauge
-
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.DiskHalfPlane
+import OAI.Probability.SLE.ChartMeasurability
+
+namespace OAI
+
+/-! Physical Transport for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+def SourceInteriorTraceDensityTarget : Prop :=
+  ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
+  Continuous U → IsCapacityTwoWitness U γ G F →
+  ∀ s t : ℝ≥0, s < t → ∃ u ∈ Ioo s t, 0 < (γ u).im
+
+lemma IsCapacityTwoWitness.survivingDomain_injective
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) :
+    Function.Injective (survivingDomain γ) := by
+  intro s t he
+  have hz : F s Complex.I ∈ survivingDomain γ s := by
+    rcases h with ⟨hγ,h0,hupper,hG,hF,hGm,hFm,hrest⟩
+    exact hFm s (by simp [upperHalfPlane])
+  have ht : F s Complex.I ∈ survivingDomain γ t := he ▸ hz
+  apply live_radius_unique hU h (F s Complex.I) s t hz ht
+  exact le_antisymm (h.radius_le_of_domain_subset hU s t he.subset _ hz)
+    (h.radius_le_of_domain_subset hU t s he.symm.subset _ ht)
+
+theorem sourceInteriorTraceDensity_proved : SourceInteriorTraceDensityTarget := by
+  intro U γ G F hU h s t hst
+  by_contra hnone
+  obtain ⟨b,hsb,hbt⟩ := exists_between hst
+  obtain ⟨a,hsa,hab⟩ := exists_between hsb
+  have hbase : upperHalfPlane \ (γ '' Icc 0 a) = upperHalfPlane \ (γ '' Icc 0 b) := by
+    apply Subset.antisymm
+    · intro z hz
+      refine ⟨hz.1,?_⟩
+      rintro ⟨u,hu,huz⟩
+      by_cases hua : u ≤ a
+      · exact hz.2 ⟨u,⟨hu.1,hua⟩,huz⟩
+      · apply hnone
+        refine ⟨u,⟨hsa.trans (not_le.mp hua),hu.2.trans_lt hbt⟩,?_⟩
+        rw [huz]
+        exact hz.1
+    · intro z hz
+      refine ⟨hz.1,?_⟩
+      rintro ⟨u,hu,huz⟩
+      exact hz.2 ⟨u,⟨hu.1,hu.2.trans hab.le⟩,huz⟩
+  have he : survivingDomain γ a = survivingDomain γ b := by
+    unfold survivingDomain
+    rw [hbase]
+  exact hab.ne (h.survivingDomain_injective hU he)
+
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+lemma source_flow_fence {g dg : ℝ → ℂ} {u : ℝ → ℝ} {a b c δ : ℝ}
+    (hab : a ≤ b) (hδ : 0 ≤ δ)
+    (hg : ContinuousOn g (Icc a b))
+    (hd : ∀ t ∈ Ico a b, HasDerivWithinAt g (dg t) (Ici t) t)
+    (hu : ∀ t ∈ Ico a b, |u t-c| ≤ δ)
+    (heq : ∀ t ∈ Ico a b, ‖dg t‖ ≤ 2 / ‖g t-(u t:ℂ)‖) :
+    ‖g b-(c:ℂ)‖^2 ≤ ‖g a-(c:ℂ)‖^2+4*δ^2+9*(b-a) := by
+  let f (t : ℝ) := ‖g t-(c:ℂ)‖^2
+  let df (t : ℝ) := 2 * inner ℝ (g t-(c:ℂ)) (dg t)
+  have hf : ContinuousOn f (Icc a b) := (hg.sub continuousOn_const).norm.pow 2
+  have hdf (t : ℝ) (ht : t ∈ Ico a b) :
+      HasDerivWithinAt f (df t) (Ici t) t := (hd t ht |>.sub_const (c:ℂ)).norm_sq
+  have hn (t : ℝ) (ht : t ∈ Ico a b) (hh : 4*δ^2 < f t) : df t < 9 := by
+    have hr : 2*δ < ‖g t-(c:ℂ)‖ := by
+      dsimp [f] at hh
+      nlinarith [norm_nonneg (g t-(c:ℂ))]
+    have htri : ‖g t-(c:ℂ)‖ ≤ ‖g t-(u t:ℂ)‖+δ := by
+      calc
+        _ ≤ ‖g t-(u t:ℂ)‖+‖(u t:ℂ)-(c:ℂ)‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
+        _ ≤ _ := by
+          have hc : ‖(u t:ℂ)-(c:ℂ)‖ ≤ δ := by simpa only [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs] using hu t ht
+          linarith
+    have hp : 0 < ‖g t-(u t:ℂ)‖ := by nlinarith
+    have hb : ‖g t-(c:ℂ)‖ ≤ 2*‖g t-(u t:ℂ)‖ := by linarith
+    have hi := real_inner_le_norm (g t-(c:ℂ)) (dg t)
+    have hm := mul_le_mul_of_nonneg_left (heq t ht) (norm_nonneg (g t-(c:ℂ)))
+    have hx : ‖g t-(c:ℂ)‖ * (2 / ‖g t-(u t:ℂ)‖) ≤ 4 := by
+      rw [← mul_div_assoc]
+      exact (div_le_iff₀ hp).mpr (by linarith)
+    change 2 * inner ℝ (g t-(c:ℂ)) (dg t) < 9
+    linarith
+  have hbound (ε : ℝ) (hε : 0 < ε) :
+      f b ≤ f a+4*δ^2+ε+9*(b-a) := by
+    apply image_le_of_deriv_right_lt_deriv_boundary hf hdf
+      (B := fun t => f a+4*δ^2+ε+9*(t-a)) (B' := fun _ => 9)
+      (by nlinarith [sq_nonneg δ])
+      (fun t => by convert (((hasDerivAt_id t).sub_const a).const_mul 9).const_add (f a+4*δ^2+ε) using 1 <;> first | rfl | norm_num)
+      (fun t ht he => ?_) ⟨hab,le_rfl⟩
+    apply hn t ht
+    have : 0 ≤ f a := sq_nonneg _
+    rw [he]
+    nlinarith [ht.1]
+  apply le_of_forall_pos_le_add
+  intro ε hε
+  simpa only [f] using (hbound ε hε).trans_eq (by ring)
+end SLEExactGauge
+
+end
+
+section
+namespace SLEExactGauge
+lemma IsCapacityTwoWitness.short_flow_bound
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) {s T : ℝ≥0} (hs : s ≤ T)
+    {z : ℂ} (hz : z ∈ survivingDomain γ T) {c δ : ℝ} (hδ : 0 ≤ δ)
+    (hU : ∀ t ∈ Icc s T, |U t-c| ≤ δ) :
+    ‖G T z-(c:ℂ)‖^2 ≤ ‖G s z-(c:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s) := by
+  have hsub : Icc (s:ℝ) T ⊆ Icc (0:ℝ) T := Icc_subset_Icc_left s.coe_nonneg
+  have hb := source_flow_fence (g := fun u => G u.toNNReal z)
+    (dg := fun u => 2/(G u.toNNReal z-(U u.toNNReal:ℂ)))
+    (u := fun u => U u.toNNReal) (a := (s:ℝ)) (b := (T:ℝ)) (c := c)
+    (NNReal.coe_le_coe.mpr hs) hδ ((h.time_continuousOn T z hz).mono hsub)
+    (fun u hu => hasDerivWithinAt_Ici_of_Icc hu ((h.time_derivative T z hz u
+      (hsub ⟨hu.1,hu.2.le⟩)).mono hsub))
+    (fun u hu => hU u.toNNReal ⟨by
+      apply NNReal.coe_le_coe.mp
+      rw [Real.coe_toNNReal _ (s.coe_nonneg.trans hu.1)]
+      exact hu.1, Real.toNNReal_le_iff_le_coe.mpr hu.2.le⟩)
+    (fun u hu => by simp)
+  simpa only [Real.toNNReal_coe] using hb
+
+lemma IsCapacityTwoWitness.short_reverse_bound
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) {s T : ℝ≥0} (hs : s ≤ T)
+    {z : ℂ} (hz : z ∈ survivingDomain γ T) {c δ : ℝ} (hδ : 0 ≤ δ)
+    (hU : ∀ t ∈ Icc s T, |U t-c| ≤ δ) :
+    ‖G s z-(c:ℂ)‖^2 ≤ ‖G T z-(c:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s) := by
+  have hm (u : ℝ) (hu : u ∈ Icc (0:ℝ) ((T:ℝ)-s)) :
+      (T:ℝ)-u ∈ Icc (s:ℝ) T := by constructor <;> linarith [hu.1,hu.2]
+  have hm' (u : ℝ) (hu : u ∈ Icc (0:ℝ) ((T:ℝ)-s)) :
+      (T:ℝ)-u ∈ Icc (0:ℝ) T := ⟨s.coe_nonneg.trans (hm u hu).1,(hm u hu).2⟩
+  have hd (u : ℝ) (hu : u ∈ Icc (0:ℝ) ((T:ℝ)-s)) :
+      HasDerivWithinAt (fun v : ℝ => G ((T:ℝ)-v).toNNReal z)
+        (-(2/(G ((T:ℝ)-u).toNNReal z-(U ((T:ℝ)-u).toNNReal:ℂ))))
+        (Icc 0 ((T:ℝ)-s)) u := by
+    have hi : HasDerivWithinAt (fun v : ℝ => (T:ℝ)-v) (-1)
+        (Icc 0 ((T:ℝ)-s)) u := by
+      convert ((hasDerivAt_id u).const_sub (T:ℝ)).hasDerivWithinAt using 1
+      rfl
+    have hh := (h.time_derivative T z hz _ (hm' u hu)).scomp u hi
+      (fun v hv => hm' v hv)
+    simpa only [neg_smul, one_smul, Function.comp_def] using hh
+  have hc : ContinuousOn (fun u : ℝ => G ((T:ℝ)-u).toNNReal z)
+      (Icc (0:ℝ) ((T:ℝ)-s)) :=
+    (h.time_continuousOn T z hz).comp (continuousOn_const.sub continuousOn_id) hm'
+  have hb := source_flow_fence (g := fun u => G ((T:ℝ)-u).toNNReal z)
+    (dg := fun u => -(2/(G ((T:ℝ)-u).toNNReal z-(U ((T:ℝ)-u).toNNReal:ℂ))))
+    (u := fun u => U ((T:ℝ)-u).toNNReal) (a := 0) (b := (T:ℝ)-s) (c := c)
+    (sub_nonneg.mpr (NNReal.coe_le_coe.mpr hs)) hδ hc
+    (fun u hu => hasDerivWithinAt_Ici_of_Icc hu (hd u ⟨hu.1,hu.2.le⟩))
+    (fun u hu => hU _ (by
+      constructor
+      · apply NNReal.coe_le_coe.mp
+        rw [Real.coe_toNNReal _ (hm' u ⟨hu.1,hu.2.le⟩).1]
+        exact (hm u ⟨hu.1,hu.2.le⟩).1
+      · exact Real.toNNReal_le_iff_le_coe.mpr (hm u ⟨hu.1,hu.2.le⟩).2))
+    (fun u hu => by simp)
+  simpa only [sub_sub_cancel,sub_zero, Real.toNNReal_coe] using hb
+end SLEExactGauge
+
+end
+
+section
+namespace SLEExactGauge
+lemma IsCapacityTwoWitness.earlier_tip_bound
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) {s T : ℝ≥0} (hs : s ≤ T)
+    (ha : γ T ∈ survivingDomain γ s) {δ : ℝ} (hδ : 0 ≤ δ)
+    (hU : ∀ t ∈ Icc s T, |U t-U T| ≤ δ) :
+    ‖G s (γ T)-(U T:ℂ)‖^2 ≤ 4*δ^2+9*((T:ℝ)-s) := by
+  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2
+  have hlim := (h.spatial_differentiableAt s (γ T) ha).continuousAt.tendsto.comp (htip T)
+  have hleft := ((hlim.sub (tendsto_const_nhds (x := (U T:ℂ)))).norm).pow 2
+  have hright : Tendsto (fun y : ℝ =>
+      ‖((U T:ℂ)+(y:ℂ)*Complex.I)-(U T:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s))
+      (𝓝[>] 0) (𝓝 (4*δ^2+9*((T:ℝ)-s))) := by
+    have hc : Continuous (fun y : ℝ =>
+      ‖((U T:ℂ)+(y:ℂ)*Complex.I)-(U T:ℂ)‖^2+4*δ^2+9*((T:ℝ)-s)) := by fun_prop
+    simpa using (hc.continuousAt (x := 0)).tendsto.mono_left nhdsWithin_le_nhds
+  apply le_of_tendsto_of_tendsto hleft hright
+  filter_upwards [self_mem_nhdsWithin] with y hy
+  have hiy : (U T:ℂ)+(y:ℂ)*Complex.I ∈ upperHalfPlane := by
+    simpa [upperHalfPlane] using hy
+  have hf := h.2.2.2.2.2.2.1 T hiy
+  have hi := h.2.2.2.2.2.2.2.2.1 T _ hiy
+  convert h.short_reverse_bound hs hf hδ hU using 1 <;> first | rfl | simp only [hi]
+end SLEExactGauge
+
+end
+
+section
+namespace SLEExactGauge
+lemma IsCapacityTwoWitness.fresh_tip_chart_tendsto
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (hUc : Continuous U)
+    {T : ℝ≥0} (hT : 0 < T)
+    (ha : ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s) :
+    Tendsto (G T) (𝓝[survivingDomain γ T] (γ T)) (𝓝 (U T:ℂ)) := by
+  rw [Metric.tendsto_nhds]
+  intro ε hε
+  obtain ⟨ρ,hρ,hosc⟩ := Metric.continuousAt_iff.mp (hUc.continuousAt (x := T)) (ε/10) (by positivity)
+  let η : ℝ := min ((T:ℝ)/2) (min (ρ/2) (ε^2/100))
+  have hη : 0 < η := lt_min (by exact div_pos (NNReal.coe_pos.mpr hT) (by norm_num))
+    (lt_min (by positivity) (by positivity))
+  have hηT : η ≤ (T:ℝ)/2 := min_le_left _ _
+  have hηρ : η ≤ ρ/2 := (min_le_right _ _).trans (min_le_left _ _)
+  have hηε : η ≤ ε^2/100 := (min_le_right _ _).trans (min_le_right _ _)
+  let s : ℝ≥0 := ⟨(T:ℝ)-η,by linarith [T.coe_nonneg]⟩
+  have hs : s < T := by change (T:ℝ)-η < T; linarith
+  have hdiff : (T:ℝ)-s = η := by change (T:ℝ)-((T:ℝ)-η)=η; ring
+  have hosc' (t : ℝ≥0) (ht : t ∈ Icc s T) : |U t-U T| ≤ ε/10 := by
+    have hdist : dist t T < ρ := by
+      rw [NNReal.dist_eq,abs_of_nonpos (sub_nonpos.mpr (NNReal.coe_le_coe.mpr ht.2))]
+      have ht' : (T:ℝ)-η ≤ t := NNReal.coe_le_coe.mpr ht.1
+      linarith
+    simpa only [Real.dist_eq] using (hosc hdist).le
+  have hbound := h.earlier_tip_bound hs.le (ha s hs) (by positivity : 0 ≤ ε/10) hosc'
+  rw [hdiff] at hbound
+  have hnorm : ‖G s (γ T)-(U T:ℂ)‖ < ε/2 := by
+    nlinarith [norm_nonneg (G s (γ T)-(U T:ℂ))]
+  have hev : ∀ᶠ w in 𝓝 (γ T), ‖G s w-(U T:ℂ)‖ < ε/2 :=
+    ((h.spatial_differentiableAt s (γ T) (ha s hs)).continuousAt.sub continuousAt_const).norm
+      |>.eventually (isOpen_Iio.mem_nhds hnorm)
+  filter_upwards [hev.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with w hw hwt
+  have hh := h.short_flow_bound hs.le hwt (by positivity : 0 ≤ ε/10) hosc'
+  rw [hdiff] at hh
+  rw [dist_eq_norm]
+  nlinarith [norm_nonneg (G s w-(U T:ℂ)), norm_nonneg (G T w-(U T:ℂ))]
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+ 
+
+def SourceFirstHitPhysicalAccessTarget : Prop :=
+  ∀ (γ : ℝ≥0 → ℂ), Continuous γ →
+  ∀ (s τ : ℝ≥0) (z : ℂ) (r : ℝ), 0 < r →
+    Metric.closedBall z r ⊆ survivingDomain γ s → s ≤ τ →
+    dist (γ τ) z = r →
+    (∀ t ∈ Ico s τ, dist (γ t) z ≠ r) → z ∈ survivingDomain γ τ →
+    (∀ t : ℝ≥0, t < τ → γ t ≠ γ τ) ∧
+    Metric.ball z r ⊆ survivingDomain γ τ ∧
+    (∀ v ∈ Ioc (0:ℝ) 1, (1-v) • γ τ + v • z ∈ survivingDomain γ τ)
+
+lemma disk_distance_gt_before_first_hit (γ : ℝ≥0 → ℂ) (hc : Continuous γ)
+    (s τ : ℝ≥0) (z : ℂ) (r : ℝ)
+    (hfree : Metric.closedBall z r ⊆ survivingDomain γ s)
+    (hfirst : ∀ t ∈ Ico s τ, dist (γ t) z ≠ r)
+    (t : ℝ≥0) (ht : t < τ) : r < dist (γ t) z := by
+  have hold (u : ℝ≥0) (hu : u ≤ s) : r < dist (γ u) z := by
+    by_contra hn
+    have hu' : γ u ∈ Metric.closedBall z r := by
+      simpa only [Metric.mem_closedBall] using not_lt.mp hn
+    have hh := hfree hu'
+    exact hh.1.2 ⟨u,⟨zero_le,hu⟩,rfl⟩
+  by_cases hts : t ≤ s
+  · exact hold t hts
+  · by_contra hn
+    obtain ⟨u,hu,he⟩ := intermediate_value_Icc' (not_le.mp hts).le
+      ((hc.dist continuous_const).continuousOn) (show r ∈ Icc (dist (γ t) z) (dist (γ s) z)
+        from ⟨not_lt.mp hn,(hold s le_rfl).le⟩)
+    exact hfirst u ⟨hu.1,hu.2.trans_lt ht⟩ he
+
+lemma radialAccess_mem_ball (a z : ℂ) (r : ℝ) (hr : 0 < r)
+    (ha : dist a z = r) (v : ℝ) (hv : v ∈ Ioc (0:ℝ) 1) :
+    (1-v) • a + v • z ∈ Metric.ball z r := by
+  rw [Metric.mem_ball,dist_eq_norm]
+  have he : (1-v) • a + v • z-z = (1-v) • (a-z) := by module
+  rw [he,norm_smul,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hv.2),← dist_eq_norm,ha]
+  nlinarith [hv.1]
+
+theorem sourceFirstHitPhysicalAccess : SourceFirstHitPhysicalAccessTarget := by
+  intro γ hc s τ z r hr hfree hsτ hhit hfirst hz
+  have hprior := disk_distance_gt_before_first_hit γ hc s τ z r hfree hfirst
+  have hball : Metric.ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 τ) := by
+    intro w hw
+    refine ⟨(hfree (Metric.ball_subset_closedBall hw)).1.1,?_⟩
+    rintro ⟨t,ht,rfl⟩
+    have hlt : dist (γ t) z < r := hw
+    rcases lt_or_eq_of_le ht.2 with htτ | rfl
+    · exact (not_lt_of_ge (hprior t htτ).le) hlt
+    · rw [hhit] at hlt
+      exact lt_irrefl _ hlt
+  have hcc := (convex_ball z r).isPreconnected.subset_connectedComponentIn
+    (Metric.mem_ball_self hr) hball
+  have hsurv : Metric.ball z r ⊆ survivingDomain γ τ := by
+    intro w hw
+    refine ⟨hball hw,?_⟩
+    rw [← connectedComponentIn_eq (hcc hw)]
+    exact hz.2
+  refine ⟨?_,hsurv,?_⟩
+  · intro t ht he
+    have hh := hprior t ht
+    rw [he,hhit] at hh
+    exact lt_irrefl _ hh
+  · intro v hv
+    exact hsurv (radialAccess_mem_ball (γ τ) z r hr hhit v hv)
+
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+ 
+
+def SourceFirstHitChartAccessTarget : Prop :=
+  ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
+    Continuous U → IsCapacityTwoWitness U γ G F →
+    ∀ (s τ : ℝ≥0) (z : ℂ) (r : ℝ), 0 < r →
+      Metric.closedBall z r ⊆ survivingDomain γ s → s ≤ τ →
+      dist (γ τ) z = r → (∀ t ∈ Ico s τ, dist (γ t) z ≠ r) →
+      z ∈ survivingDomain γ τ →
+      Tendsto (fun v : ℝ => G τ ((1-v) • γ τ + v • z) - (U τ : ℂ))
+        (𝓝[>] 0) (𝓝 0)
+
+end SLEExactGauge
+
+end
+
+section
+namespace SLEExactGauge
+lemma first_hit_tip_survives_earlier (γ : ℝ≥0 → ℂ) (hc : Continuous γ)
+    (s τ : ℝ≥0) (z : ℂ) (r : ℝ) (hr : 0 < r)
+    (hfree : Metric.closedBall z r ⊆ survivingDomain γ s)
+    (hhit : dist (γ τ) z = r)
+    (hfirst : ∀ t ∈ Ico s τ, dist (γ t) z ≠ r)
+    (hz : z ∈ survivingDomain γ τ) :
+    ∀ u : ℝ≥0, u < τ → γ τ ∈ survivingDomain γ u := by
+  intro u hu
+  have hprior := disk_distance_gt_before_first_hit γ hc s τ z r hfree hfirst
+  have hclosed : Metric.closedBall z r ⊆ upperHalfPlane \ (γ '' Icc 0 u) := by
+    intro w hw
+    refine ⟨(hfree hw).1.1,?_⟩
+    rintro ⟨t,ht,rfl⟩
+    exact (not_le_of_gt (hprior t (ht.2.trans_lt hu))) (Metric.mem_closedBall.mp hw)
+  have hcc := (convex_closedBall z r).isPreconnected.subset_connectedComponentIn
+    (Metric.mem_closedBall_self hr.le) hclosed
+  have hmem : γ τ ∈ Metric.closedBall z r := Metric.mem_closedBall.mpr hhit.le
+  refine ⟨hclosed hmem,?_⟩
+  rw [← connectedComponentIn_eq (hcc hmem)]
+  exact (survivingDomain_antitone γ hu.le hz).2
+
+theorem sourceFirstHitChartAccess_proved : SourceFirstHitChartAccessTarget := by
+  intro U γ G F hUc h s τ z r hr hfree hsτ hhit hfirst hz
+  have hslt : s < τ := by
+    apply lt_of_le_of_ne hsτ
+    intro he
+    have hm := hfree (Metric.mem_closedBall.mpr hhit.le)
+    exact hm.1.2 ⟨τ,⟨zero_le,by rw [he]⟩,rfl⟩
+  have hT : 0 < τ := lt_of_le_of_lt (show (0:ℝ≥0) ≤ s from zero_le) hslt
+  have ha := first_hit_tip_survives_earlier γ h.1 s τ z r hr hfree hhit hfirst hz
+  have hlim := h.fresh_tip_chart_tendsto hUc hT ha
+  have hradial := (sourceFirstHitPhysicalAccess γ h.1 s τ z r hr hfree hsτ hhit hfirst hz).2.2
+  have hcont : Continuous (fun v : ℝ => (1-v) • γ τ + v • z) := by fun_prop
+  have hpath : Tendsto (fun v : ℝ => (1-v) • γ τ + v • z) (𝓝[>] 0)
+      (𝓝[survivingDomain γ τ] (γ τ)) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    constructor
+    · simpa using (hcont.continuousAt (x := 0)).tendsto.mono_left nhdsWithin_le_nhds
+    · have hev : ∀ᶠ v : ℝ in 𝓝[>] 0, v < 1 :=
+        (eventually_lt_nhds (by norm_num : (0:ℝ) < 1)).filter_mono nhdsWithin_le_nhds
+      filter_upwards [self_mem_nhdsWithin,hev] with v hv hv'
+      exact hradial v ⟨hv,hv'.le⟩
+  simpa only [Function.comp_def,sub_self] using
+    (hlim.comp hpath).sub (tendsto_const_nhds (x := (U τ:ℂ)))
+end SLEExactGauge
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/RadiusClocks.lean b/lean/OAI/Probability/SLE/RadiusClocks.lean
index 2e1b2b2a09730ba467b97e38454ce118bd44ae24..49a87ff7ba54aa2a08fa40e0179c7a795adfcba7 100644
--- a/lean/OAI/Probability/SLE/RadiusClocks.lean
+++ b/lean/OAI/Probability/SLE/RadiusClocks.lean
@@ -1,676 +1,676 @@
-import Mathlib
-import OAI.Probability.SLE.BrownianStrongMarkov
-
-namespace OAI
-
-/-! Radius Clocks for chordal SLE. -/
-
-section
-section
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
-lemma measurableSet_exists_zero_compact
-    {Ω K : Type*} [MeasurableSpace Ω] [TopologicalSpace K]
-    [CompactSpace K] [TopologicalSpace.SeparableSpace K] [Nonempty K]
-    (f : K → Ω → ℝ) (hm : ∀ t, Measurable (f t))
-    (hc : ∀ ω, Continuous (fun t => f t ω)) :
-    MeasurableSet {ω | ∃ t, f t ω = 0} := by
-  obtain ⟨D, hDc, hDd⟩ := TopologicalSpace.exists_countable_dense K
-  let : Countable D := hDc.to_subtype
-  have hi : Measurable (fun ω => ⨅ t : D, |f t ω|) :=
-    Measurable.iInf (fun t => (hm t).abs)
-  have he (ω : Ω) : (∃ t, f t ω = 0) ↔ (⨅ t : D, |f t ω|) = 0 := by
-    have hb : BddBelow (range (fun t => |f t ω|)) := ⟨0, by rintro _ ⟨t, rfl⟩; exact abs_nonneg _⟩
-    rw [hDd.ciInf (hc ω).abs hb]
-    obtain ⟨t, _, ht⟩ := isCompact_univ.exists_isMinOn (Set.univ_nonempty : (univ : Set K).Nonempty)
-      (hc ω).abs.continuousOn
-    have hv : (⨅ s, |f s ω|) = |f t ω| :=
-      le_antisymm (ciInf_le hb t) (le_ciInf (fun s => ht (mem_univ s)))
-    rw [hv]
-    constructor
-    · rintro ⟨s, hs⟩
-      have ht' : |f t ω| ≤ |f s ω| := ht (mem_univ s)
-      rw [hs, abs_zero] at ht'
-      exact le_antisymm ht' (abs_nonneg _)
-    · intro h
-      exact ⟨t, abs_eq_zero.mp h⟩
-  have hs : MeasurableSet {ω | (⨅ t : D, |f t ω|) = 0} :=
-    measurableSet_eq_fun hi measurable_const
-  have heSet : {ω | ∃ t, f t ω = 0} = {ω | (⨅ t : D, |f t ω|) = 0} :=
-    Set.ext he
-  rw [heSet]
-  exact hs
-
-lemma hittingAfter_zero_le_iff
-    {Ω : Type*} (X : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => X t ω)) (n T : ℝ≥0) (ω : Ω) :
-    hittingAfter X {0} n ω ≤ (T : WithTop ℝ≥0) ↔
-      ∃ t ∈ Icc n T, X t ω = 0 := by
-  classical
-  let V : Set ℝ≥0 := {t | n ≤ t ∧ X t ω = 0}
-  have hV : IsClosed V := isClosed_Ici.inter (isClosed_eq (hc ω) continuous_const)
-  have hb : BddBelow V := ⟨0, fun t _ => bot_le⟩
-  change (if V.Nonempty then ((sInf V : ℝ≥0) : WithTop ℝ≥0) else ⊤) ≤ (T : WithTop ℝ≥0) ↔ _
-  by_cases hv : V.Nonempty
-  · obtain ⟨v, hv⟩ := hv
-    have he : ∃ t, n ≤ t ∧ X t ω ∈ ({0} : Set ℝ) := ⟨v, hv.1, hv.2⟩
-    have hin : sInf V ∈ V := hV.csInf_mem ⟨v, hv⟩ hb
-    rw [ite_eq_left (show V.Nonempty from ⟨v, hv⟩), WithTop.coe_le_coe]
-    constructor
-    · intro h
-      exact ⟨sInf V, ⟨hin.1, h⟩, hin.2⟩
-    · rintro ⟨t, ht, hx⟩
-      exact (csInf_le hb ⟨ht.1, hx⟩).trans ht.2
-  · have he : ¬ ∃ t, n ≤ t ∧ X t ω ∈ ({0} : Set ℝ) := by
-      exact hv
-    rw [ite_eq_right hv]
-    simp only [WithTop.not_top_le_coe, false_iff]
-    rintro ⟨t, ht, hx⟩
-    exact hv ⟨t, ht.1, hx⟩
-
- 
-
-theorem continuous_zero_hit_stopping
-    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
-    (X : ℝ≥0 → Ω → ℝ) (hXa : Adapted ℱ X)
-    (hXc : ∀ ω, Continuous (fun t => X t ω)) (n : ℝ≥0) :
-    IsStoppingTime ℱ (hittingAfter X {0} n) := by
-  intro T
-  have he : {ω | hittingAfter X {0} n ω ≤ (T : WithTop ℝ≥0)} =
-      {ω | ∃ t : Icc n T, X t ω = 0} := by
-    ext ω
-    simp only [mem_ofPred_eq, hittingAfter_zero_le_iff X hXc, Subtype.exists, exists_prop]
-  rw [he]
-  by_cases hnT : n ≤ T
-  · let : Nonempty (Icc n T) := ⟨⟨n, le_rfl, hnT⟩⟩
-    have hm (t : Icc n T) : @Measurable Ω ℝ (ℱ T) (borel ℝ) (X t) :=
-      (hXa t).mono (ℱ.mono t.property.2) le_rfl
-    let : MeasurableSpace Ω := ℱ T
-    exact measurableSet_exists_zero_compact (fun t : Icc n T => X t) hm
-      (fun ω => (hXc ω).comp continuous_subtype_val)
-  · have hnil : IsEmpty (Icc n T) := ⟨fun t => hnT (t.property.1.trans t.property.2)⟩
-    have : {ω | ∃ t : Icc n T, X t ω = 0} = ∅ := by ext ω; simp
-    rw [this]
-    exact @MeasurableSet.empty Ω (ℱ T)
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma stopped_min_measurable {Ω : Type*} [m : MeasurableSpace Ω]
-    {ℱ : Filtration ℝ≥0 m} {ρ : Ω → ℝ≥0}
-    (hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0))) (T : ℝ≥0) :
-    Measurable[ℱ T] (fun ω => min (ρ ω) T) := by
-  have h := ((hρ.min_const T).measurable_of_le (fun _ => min_le_right _ _)).untopD 0
-  simpa only [← WithTop.coe_min, WithTop.untopD_coe] using h
-
- 
-
-theorem continuous_zero_hit_after_stopping
-    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
-    (X : ℝ≥0 → Ω → ℝ) (hXa : Adapted ℱ X)
-    (hXc : ∀ ω, Continuous (fun t => X t ω)) (ρ : Ω → ℝ≥0)
-    (hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0))) :
-    IsStoppingTime ℱ (fun ω => hittingAfter X {0} (ρ ω) ω) := by
-  intro T
-  let f : Icc (0 : ℝ≥0) T → Ω → ℝ :=
-    fun t ω => |X t ω| + max (((min (ρ ω) T : ℝ≥0) : ℝ) - (t : ℝ≥0)) 0
-  have he : {ω | hittingAfter X {0} (ρ ω) ω ≤ (T : WithTop ℝ≥0)} =
-      {ω | ρ ω ≤ T} ∩ {ω | ∃ t : Icc (0 : ℝ≥0) T, f t ω = 0} := by
-    ext ω
-    rw [mem_ofPred_eq, hittingAfter_zero_le_iff X hXc]
-    constructor
-    · rintro ⟨t, ht, hx⟩
-      have hr : ρ ω ≤ T := ht.1.trans ht.2
-      refine ⟨hr, ⟨⟨t, bot_le, ht.2⟩, ?_⟩⟩
-      dsimp only [f]
-      rw [hx, abs_zero, min_eq_left hr, max_eq_right (sub_nonpos.mpr (show (ρ ω : ℝ) ≤ (t : ℝ) from ht.1))]
-      exact zero_add 0
-    · rintro ⟨hr, t, ht⟩
-      have hp : 0 ≤ max (((min (ρ ω) T : ℝ≥0) : ℝ) - (t : ℝ≥0)) 0 := le_max_right _ _
-      have hx : X t ω = 0 := abs_eq_zero.mp (by dsimp only [f] at ht; linarith [abs_nonneg (X t ω)])
-      have hle : (ρ ω : ℝ) ≤ (t : ℝ≥0) := by
-        dsimp only [f] at ht
-        rw [min_eq_left hr] at ht
-        have hh := le_max_left ((ρ ω : ℝ) - (t : ℝ≥0)) 0
-        linarith [abs_nonneg (X t ω)]
-      exact ⟨t, ⟨hle, t.property.2⟩, hx⟩
-  rw [he]
-  have hmρ := stopped_min_measurable hρ T
-  have hmX (t : Icc (0 : ℝ≥0) T) : Measurable[ℱ T] (X t) :=
-    (hXa t).mono (ℱ.mono t.property.2) le_rfl
-  let : MeasurableSpace Ω := ℱ T
-  have hm (t : Icc (0 : ℝ≥0) T) : Measurable (f t) :=
-    (hmX t).abs.add (((NNReal.continuous_coe.measurable.comp hmρ).sub measurable_const).max measurable_const)
-  have hme : MeasurableSet {ω | ∃ t : Icc (0 : ℝ≥0) T, f t ω = 0} := by
-    let : Nonempty (Icc (0 : ℝ≥0) T) := ⟨⟨0, le_rfl, bot_le⟩⟩
-    apply measurableSet_exists_zero_compact f hm
-    intro ω
-    exact ((hXc ω).comp continuous_subtype_val).abs.add
-      ((continuous_const.sub (NNReal.continuous_coe.comp continuous_subtype_val)).max continuous_const)
-  have hmle : MeasurableSet {ω | ρ ω ≤ T} := by
-    simpa only [WithTop.coe_le_coe] using hρ T
-  exact hmle.inter hme
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def diskHitAfter {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
-    (ρ : Ω → ℝ≥0) (z : ℂ) (r : ℝ) (ω : Ω) : WithTop ℝ≥0 :=
-  hittingAfter (fun t ω => dist (γ ω t) z - r) {0} (ρ ω) ω
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
-lemma diskHitAfter_stopping (ρ : a.samples → ℝ≥0)
-    (hρ : IsStoppingTime a.filtration (fun ω => (ρ ω : WithTop ℝ≥0)))
-    (z : ℂ) (r : ℝ) :
-    IsStoppingTime a.filtration (diskHitAfter (γ ∘ Subtype.val) ρ z r) := by
-  apply continuous_zero_hit_after_stopping a.filtration
-    (fun t ω => dist (γ ω t) z - r) ?_ ?_ ρ hρ
-  · intro t
-    have hm := a.trace_adapted t
-    let : MeasurableSpace a.samples := a.filtration t
-    exact (hm.dist measurable_const).sub measurable_const
-  · intro ω
-    exact ((a.trace_continuous ω).dist continuous_const).sub continuous_const
-
-noncomputable def boundedDiskHit (ρ : a.samples → ℝ≥0) (z : ℂ) (r : ℝ)
-    (T : ℝ≥0) (ω : a.samples) : ℝ≥0 :=
-  (min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) T).untopD 0
-
-lemma boundedDiskHit_coe (ρ : a.samples → ℝ≥0) (z : ℂ) (r : ℝ) (T : ℝ≥0)
-    (ω : a.samples) :
-    (a.boundedDiskHit ρ z r T ω : WithTop ℝ≥0) =
-      min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) T := by
-  have hn : min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) (T : WithTop ℝ≥0) ≠ ⊤ :=
-    ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_right _ _)
-  obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp hn
-  change (((min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) (T : WithTop ℝ≥0)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = _
-  rw [← hv, WithTop.untopD_coe]
-
-lemma boundedDiskHit_stopping (ρ : a.samples → ℝ≥0)
-    (hρ : IsStoppingTime a.filtration (fun ω => (ρ ω : WithTop ℝ≥0)))
-    (z : ℂ) (r : ℝ) (T : ℝ≥0) :
-    IsStoppingTime a.filtration (fun ω => (a.boundedDiskHit ρ z r T ω : WithTop ℝ≥0)) := by
-  simpa only [a.boundedDiskHit_coe] using (a.diskHitAfter_stopping ρ hρ z r).min_const T
-
- 
-
-theorem boundedDiskHit_restart_law [IsProbabilityMeasure P]
-    (ρ : a.samples → ℝ≥0)
-    (hρ : IsStoppingTime a.filtration (fun ω => (ρ ω : WithTop ℝ≥0)))
-    (z : ℂ) (r : ℝ) (T : ℝ≥0) {A : Set a.samples}
-    (hA : MeasurableSet[(a.boundedDiskHit_stopping ρ hρ z r T).measurableSpace] A) :
-    ((P.comap (Subtype.val : a.samples → Ω)).restrict A).map
-      (fun ω t => brownianRestart a.B (a.boundedDiskHit ρ z r T) t ω) =
-    (P.comap (Subtype.val : a.samples → Ω)) A •
-      (P.comap (Subtype.val : a.samples → Ω)).map (fun ω t => a.B t ω) :=
-  brownian_restart_path_law a.brownian a.measurable_B a.continuous_B
-    (a.boundedDiskHit_stopping ρ hρ z r T) hA
-
-end RegularSLEPresentation
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-namespace SLEExactGauge
-
-lemma complex_bilinear_rotation_trace (B : ℂ →L[ℝ] ℂ →L[ℝ] ℝ) (e : ℂ) :
-    B e e + B (e * I) (e * I) = ‖e‖ ^ 2 * (B 1 1 + B I I) := by
-  have he : e = e.re • (1 : ℂ) + e.im • I := by apply Complex.ext <;> simp
-  have heI : e * I = (-e.im) • (1 : ℂ) + e.re • I := by apply Complex.ext <;> simp
-  calc
-    _ = B (e.re • 1 + e.im • I) (e.re • 1 + e.im • I) +
-      B ((-e.im) • 1 + e.re • I) ((-e.im) • 1 + e.re • I) := by rw [← he, ← heI]
-    _ = _ := by
-      simp only [map_add, map_smul, add_apply, smul_apply, smul_eq_mul]
-      rw [Complex.sq_norm]
-      simp only [Complex.normSq_apply]
-      ring
-
-lemma complex_hessian_rotation {u : ℂ → ℝ} (z e : ℂ) :
-    fderiv ℝ (fderiv ℝ u) z e e + fderiv ℝ (fderiv ℝ u) z (e * I) (e * I) =
-      ‖e‖ ^ 2 * (Δ u) z := by
-  rw [complex_bilinear_rotation_trace]
-  simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
-    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
-
-noncomputable def polarPoint (r θ : ℝ) : ℂ := r • circleMap 0 1 θ
-
-lemma polarPoint_eq_circleMap (r θ : ℝ) : polarPoint r θ = circleMap 0 r θ := by
-  simp [polarPoint, circleMap]
-
-lemma polarPoint_norm (r θ : ℝ) : ‖polarPoint r θ‖ = |r| := by
-  simp [polarPoint, norm_circleMap_zero]
-
-lemma polarPoint_radial_hasDeriv (r θ : ℝ) :
-    HasDerivAt (fun s => polarPoint s θ) (circleMap 0 1 θ) r := by
-  simpa [polarPoint] using (hasDerivAt_id r).smul_const (circleMap 0 1 θ)
-
-lemma polarPoint_angular_hasDeriv (r θ : ℝ) :
-    HasDerivAt (polarPoint r) (polarPoint r θ * I) θ := by
-  simpa only [funext (polarPoint_eq_circleMap r), polarPoint_eq_circleMap] using
-    hasDerivAt_circleMap 0 r θ
-
-lemma radial_partial_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
-    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    HasDerivAt (fun s => u (polarPoint s θ))
-      (fderiv ℝ u (polarPoint r θ) (circleMap 0 1 θ)) r :=
-  (hu.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt r
-    (polarPoint_radial_hasDeriv r θ)
-
-lemma radial_second_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
-    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    HasDerivAt (fun s => fderiv ℝ u (polarPoint s θ) (circleMap 0 1 θ))
-      (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ) (circleMap 0 1 θ)) r := by
-  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).hasFDerivAt
-  have hc : HasDerivAt (fun s => fderiv ℝ u (polarPoint s θ))
-      (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ)) r :=
-    hd.comp_hasDerivAt r (polarPoint_radial_hasDeriv r θ)
-  have ha := hc.clm_apply (hasDerivAt_const r (circleMap 0 1 θ))
-  simpa only [map_zero, add_zero] using ha
-
-lemma angular_partial_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
-    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    HasDerivAt (fun φ => u (polarPoint r φ))
-      (fderiv ℝ u (polarPoint r θ) (polarPoint r θ * I)) θ :=
-  (hu.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt θ
-    (polarPoint_angular_hasDeriv r θ)
-
-lemma angular_second_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
-    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    HasDerivAt (fun φ => fderiv ℝ u (polarPoint r φ) (polarPoint r φ * I))
-      (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (polarPoint r θ * I) (polarPoint r θ * I)
-        - fderiv ℝ u (polarPoint r θ) (polarPoint r θ)) θ := by
-  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).hasFDerivAt
-  have ht := (polarPoint_angular_hasDeriv r θ).mul_const I
-  have hmul : (polarPoint r θ * I) * I = -polarPoint r θ := by rw [mul_assoc, I_mul_I, mul_neg_one]
-  rw [hmul] at ht
-  simpa [sub_eq_add_neg] using
-    (hd.comp_hasDerivAt θ (polarPoint_angular_hasDeriv r θ)).clm_apply ht
-
- 
-
-lemma polar_laplacian_identity {u : ℂ → ℝ} (r θ : ℝ) :
-    r ^ 2 * (Δ u) (polarPoint r θ) =
-      r ^ 2 * fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ) (circleMap 0 1 θ) +
-      fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (polarPoint r θ * I) (polarPoint r θ * I) := by
-  have hh := complex_hessian_rotation (u := u) (polarPoint r θ) (circleMap 0 1 θ)
-  have he : polarPoint r θ * I = r • (circleMap 0 1 θ * I) := by
-    simp [polarPoint, mul_assoc]
-  rw [he]
-  simp only [map_smul, smul_apply, smul_eq_mul]
-  simp only [norm_circleMap_zero, abs_one, one_pow, one_mul] at hh
-  nlinarith
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian
-namespace SLEExactGauge
-
-noncomputable def polarRadialFlux (u : ℂ → ℝ) (R : ℝ) (p : ℝ × ℝ) : ℝ :=
-  p.1 * (Real.log R - Real.log p.1) *
-    fderiv ℝ u (polarPoint p.1 p.2) (circleMap 0 1 p.2) + u (polarPoint p.1 p.2)
-
-noncomputable def polarAngularFlux (u : ℂ → ℝ) (R : ℝ) (p : ℝ × ℝ) : ℝ :=
-  (Real.log R - Real.log p.1) / p.1 *
-    fderiv ℝ u (polarPoint p.1 p.2) (polarPoint p.1 p.2 * I)
-
-lemma polarPoint_contDiff : ContDiff ℝ ⊤ (fun p : ℝ × ℝ => polarPoint p.1 p.2) := by
-  have hc : ContDiff ℝ ⊤ (fun p : ℝ × ℝ => (p.2 : ℂ)) :=
-    Complex.ofRealCLM.contDiff.comp contDiff_snd
-  have he : ContDiff ℝ ⊤ (fun p : ℝ × ℝ => circleMap 0 1 p.2) := by
-    have hExp : ContDiff ℝ ⊤ Complex.exp := Complex.contDiff_exp
-    simpa only [circleMap, Complex.ofReal_one, one_mul, zero_add, Function.comp_def] using
-      hExp.comp (hc.mul (contDiff_const (c := I)))
-  exact contDiff_fst.smul he
-
-lemma polarRadialFlux_differentiableAt {u : ℂ → ℝ} {R r θ : ℝ}
-    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    DifferentiableAt ℝ (polarRadialFlux u R) (r, θ) := by
-  have hp := polarPoint_contDiff.differentiable (by simp) (r, θ)
-  have hv := (hu.differentiableAt (by norm_num)).comp (r, θ) hp
-  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).comp (r, θ) hp
-  unfold polarRadialFlux
-  exact ((differentiableAt_fst.mul ((differentiableAt_const (Real.log R)).sub
-    (differentiableAt_fst.log hr))).mul (hd.clm_apply (by unfold circleMap; fun_prop))).add hv
-
-lemma polarAngularFlux_differentiableAt {u : ℂ → ℝ} {R r θ : ℝ}
-    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    DifferentiableAt ℝ (polarAngularFlux u R) (r, θ) := by
-  have hp := polarPoint_contDiff.differentiable (by simp) (r, θ)
-  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).comp (r, θ) hp
-  have hfst : DifferentiableAt ℝ (Prod.fst : ℝ × ℝ → ℝ) (r, θ) := differentiableAt_fst
-  have hl : DifferentiableAt ℝ (fun p : ℝ × ℝ => (Real.log R - Real.log p.1) / p.1) (r, θ) :=
-    by
-      simp only [div_eq_mul_inv]
-      exact ((differentiableAt_const (Real.log R)).sub (hfst.log hr)).mul (hfst.inv hr)
-  exact hl.mul (hd.clm_apply (hp.mul_const I))
-
-lemma polarRadialFlux_radial_hasDeriv {u : ℂ → ℝ} {R r θ : ℝ}
-    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    HasDerivAt (fun s => polarRadialFlux u R (s, θ))
-      ((Real.log R - Real.log r) * fderiv ℝ u (polarPoint r θ) (circleMap 0 1 θ) +
-        r * (Real.log R - Real.log r) *
-          fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ) (circleMap 0 1 θ)) r := by
-  have hl := (Real.hasDerivAt_log hr).const_sub (Real.log R)
-  have hh := (((hasDerivAt_id r).mul hl).mul (radial_second_hasDeriv hu)).add
-    (radial_partial_hasDeriv hu)
-  convert! hh using 1
-  simp only [id_eq, Pi.mul_apply]
-  field_simp [hr]
-  ring
-
-lemma polarAngularFlux_angular_hasDeriv {u : ℂ → ℝ} {R r θ : ℝ}
-    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    HasDerivAt (fun φ => polarAngularFlux u R (r, φ))
-      ((Real.log R - Real.log r) / r *
-        (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (polarPoint r θ * I) (polarPoint r θ * I) -
-          fderiv ℝ u (polarPoint r θ) (polarPoint r θ))) θ :=
-  (angular_second_hasDeriv hu).const_mul ((Real.log R - Real.log r) / r)
-
-lemma polarGreen_divergence {u : ℂ → ℝ} {R r θ : ℝ}
-    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
-    fderiv ℝ (polarRadialFlux u R) (r, θ) (1, 0) +
-      fderiv ℝ (polarAngularFlux u R) (r, θ) (0, 1) =
-        r * (Real.log R - Real.log r) * (Δ u) (polarPoint r θ) := by
-  have hf := (polarRadialFlux_differentiableAt (R := R) hr hu).hasFDerivAt.comp_hasDerivAt r
-    ((hasDerivAt_id r).prodMk (hasDerivAt_const r θ))
-  have hg := (polarAngularFlux_differentiableAt (R := R) hr hu).hasFDerivAt.comp_hasDerivAt θ
-    ((hasDerivAt_const θ r).prodMk (hasDerivAt_id θ))
-  rw [hf.unique (polarRadialFlux_radial_hasDeriv hr hu),
-    hg.unique (polarAngularFlux_angular_hasDeriv hu)]
-  have he : fderiv ℝ u (polarPoint r θ) (polarPoint r θ) =
-      r * fderiv ℝ u (polarPoint r θ) (circleMap 0 1 θ) := by
-    conv_lhs => arg 2; rw [polarPoint]
-    simp only [map_smul, smul_eq_mul]
-  rw [he]
-  have hh := polar_laplacian_identity (u := u) r θ
-  apply (mul_right_cancel₀ hr)
-  field_simp [hr]
-  have hmul := congrArg (fun x : ℝ => (Real.log R - Real.log r) * x) hh
-  nlinarith only [hmul]
-
-lemma polarPoint_endpoints (r : ℝ) : polarPoint r (-Real.pi) = polarPoint r Real.pi := by
-  have hp := periodic_circleMap 0 1 (-Real.pi)
-  have he : -Real.pi + 2 * Real.pi = Real.pi := by ring
-  rw [he] at hp
-  exact congrArg (fun z : ℂ => r • z) hp.symm
-
-lemma polarAngularFlux_endpoints (u : ℂ → ℝ) (R r : ℝ) :
-    polarAngularFlux u R (r, -Real.pi) = polarAngularFlux u R (r, Real.pi) := by
-  simp only [polarAngularFlux, polarPoint_endpoints]
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian
-namespace SLEExactGauge
-
-lemma plane_laplacian_continuousAt {u : ℂ → ℝ} {z : ℂ} (hu : ContDiffAt ℝ 2 u z) :
-    ContinuousAt (Δ u) z := by
-  have hh := ((hu.fderiv_right (m := 1) (by norm_num)).fderiv_right (m := 0)
-    (by norm_num)).continuousAt
-  have h1 := (hh.clm_apply (continuousAt_const (y := (1 : ℂ)))).clm_apply
-    (continuousAt_const (y := (1 : ℂ)))
-  have hI := (hh.clm_apply (continuousAt_const (y := I))).clm_apply
-    (continuousAt_const (y := I))
-  simpa only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
-    Fin.isValue, Pi.add_def, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
-    using h1.add hI
-
-lemma polarPoint_mem_ball {a R r θ : ℝ} (ha : 0 < a) (hR : R < 1)
-    (hr : r ∈ Icc a R) : polarPoint r θ ∈ ball (0 : ℂ) 1 := by
-  rw [mem_ball_zero_iff, polarPoint_norm, abs_of_pos (ha.trans_le hr.1)]
-  exact hr.2.trans_lt hR
-
- 
-
-lemma annulus_green_identity {u : ℂ → ℝ} {a R : ℝ}
-    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
-    (∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
-      r * (Real.log R - Real.log r) * (Δ u) (polarPoint r θ)) =
-      (∫ θ in -Real.pi..Real.pi, u (polarPoint R θ)) -
-        ∫ θ in -Real.pi..Real.pi, polarRadialFlux u R (a, θ) := by
-  let S := uIcc a R ×ˢ uIcc (-Real.pi) Real.pi
-  have hmem (p : ℝ × ℝ) (hp : p ∈ S) : p.1 ≠ 0 ∧ polarPoint p.1 p.2 ∈ ball (0 : ℂ) 1 := by
-    have hr : p.1 ∈ Icc a R := by simpa only [uIcc_of_le haR] using hp.1
-    exact ⟨(ha.trans_le hr.1).ne', polarPoint_mem_ball ha hR hr⟩
-  have hf : ContinuousOn (polarRadialFlux u R) S := fun p hp =>
-    (polarRadialFlux_differentiableAt (hmem p hp).1 (hu _ (hmem p hp).2)).continuousAt.continuousWithinAt
-  have hg : ContinuousOn (polarAngularFlux u R) S := fun p hp =>
-    (polarAngularFlux_differentiableAt (hmem p hp).1 (hu _ (hmem p hp).2)).continuousAt.continuousWithinAt
-  have hl : ContinuousOn (fun p : ℝ × ℝ =>
-      p.1 * (Real.log R - Real.log p.1) * (Δ u) (polarPoint p.1 p.2)) S := by
-    intro p hp
-    exact ((continuousAt_fst.mul (continuousAt_const.sub (continuousAt_fst.log (hmem p hp).1))).mul
-      ((plane_laplacian_continuousAt (hu _ (hmem p hp).2)).comp (f := fun p : ℝ × ℝ => polarPoint p.1 p.2)
-        polarPoint_contDiff.continuous.continuousAt)).continuousWithinAt
-  have he : EqOn (fun p : ℝ × ℝ => fderiv ℝ (polarRadialFlux u R) p (1, 0) +
-      fderiv ℝ (polarAngularFlux u R) p (0, 1))
-      (fun p => p.1 * (Real.log R - Real.log p.1) * (Δ u) (polarPoint p.1 p.2)) S :=
-    fun p hp => polarGreen_divergence (hmem p hp).1 (hu _ (hmem p hp).2)
-  have hi : IntegrableOn (fun p : ℝ × ℝ => fderiv ℝ (polarRadialFlux u R) p (1, 0) +
-      fderiv ℝ (polarAngularFlux u R) p (0, 1)) S :=
-    (hl.integrableOn_compact (isCompact_uIcc.prod isCompact_uIcc)).congr_fun
-      (fun p hp => (he hp).symm) (measurableSet_uIcc.prod measurableSet_uIcc)
-  have hm : Ioo (min a R) (max a R) ×ˢ Ioo (min (-Real.pi) Real.pi) (max (-Real.pi) Real.pi) ⊆ S :=
-    prod_mono Ioo_subset_Icc_self Ioo_subset_Icc_self
-  have hh := integral2_divergence_prod_of_hasFDerivAt (polarRadialFlux u R) (polarAngularFlux u R)
-    (fderiv ℝ (polarRadialFlux u R)) (fderiv ℝ (polarAngularFlux u R)) a (-Real.pi) R Real.pi hf hg
-    (fun p hp => (polarRadialFlux_differentiableAt (hmem p (hm hp)).1
-      (hu _ (hmem p (hm hp)).2)).hasFDerivAt)
-    (fun p hp => (polarAngularFlux_differentiableAt (hmem p (hm hp)).1
-      (hu _ (hmem p (hm hp)).2)).hasFDerivAt) hi
-  have hi_eq : (∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
-      fderiv ℝ (polarRadialFlux u R) (r, θ) (1, 0) +
-        fderiv ℝ (polarAngularFlux u R) (r, θ) (0, 1)) =
-      ∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
-        r * (Real.log R - Real.log r) * (Δ u) (polarPoint r θ) := by
-    apply intervalIntegral.integral_congr
-    intro r hr
-    apply intervalIntegral.integral_congr
-    intro θ hθ
-    exact he ⟨hr, hθ⟩
-  rw [hi_eq] at hh
-  simpa only [polarAngularFlux_endpoints, sub_self, zero_add,
-    polarRadialFlux, sub_self, mul_zero, zero_mul, zero_add] using hh
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian
-namespace SLEExactGauge
-
-lemma annulus_green_le {u : ℂ → ℝ} {a R C M : ℝ}
-    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
-    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
-    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
-    (hM : 0 ≤ M) (hgrad : ∀ θ : ℝ, ‖fderiv ℝ u (polarPoint a θ)‖ ≤ M) :
-    (∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
-      r * (Real.log R - Real.log r) * (-(Δ u) (polarPoint r θ))) ≤
-        2 * Real.pi * (C + M * (-a * Real.log a)) := by
-  have hRpos := ha.trans_le haR
-  have ha_mem (θ : ℝ) : polarPoint a θ ∈ ball (0 : ℂ) 1 :=
-    polarPoint_mem_ball ha hR ⟨le_rfl, haR⟩
-  have hR_mem (θ : ℝ) : polarPoint R θ ∈ ball (0 : ℂ) 1 :=
-    polarPoint_mem_ball ha hR ⟨haR, le_rfl⟩
-  have hL : 0 ≤ Real.log R - Real.log a := sub_nonneg.mpr (Real.log_le_log ha haR)
-  have hlogR : Real.log R ≤ 0 := Real.log_nonpos hRpos.le hR.le
-  have hbound (θ : ℝ) : polarRadialFlux u R (a, θ) ≤ C + M * (-a * Real.log a) := by
-    have hb : fderiv ℝ u (polarPoint a θ) (circleMap 0 1 θ) ≤ M := by
-      apply (le_abs_self _).trans
-      calc
-        _ = ‖fderiv ℝ u (polarPoint a θ) (circleMap 0 1 θ)‖ := (Real.norm_eq_abs _).symm
-        _ ≤ ‖fderiv ℝ u (polarPoint a θ)‖ * ‖circleMap 0 1 θ‖ :=
-          (fderiv ℝ u (polarPoint a θ)).le_opNorm _
-        _ ≤ M := by simpa using hgrad θ
-    have hm := mul_le_mul_of_nonneg_left hb (mul_nonneg ha.le hL)
-    have hn := mul_nonpos_of_nonneg_of_nonpos (mul_nonneg ha.le hM) hlogR
-    unfold polarRadialFlux
-    nlinarith [hub _ (ha_mem θ)]
-  have hf : Continuous (fun θ => polarRadialFlux u R (a, θ)) :=
-    continuous_iff_continuousAt.mpr fun θ =>
-      (polarRadialFlux_differentiableAt ha.ne' (hu _ (ha_mem θ))).continuousAt.comp
-        (continuousAt_const.prodMk continuousAt_id)
-  have hf_int := hf.intervalIntegrable (μ := volume) (-Real.pi) Real.pi
-  have hlo_pi : -Real.pi ≤ Real.pi := by nlinarith [Real.pi_pos]
-  have hf_le := intervalIntegral.integral_mono_on hlo_pi hf_int
-    (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => C + M * (-a * Real.log a)) volume
-      (-Real.pi) Real.pi) (fun θ _ => hbound θ)
-  simp only [intervalIntegral.integral_const, smul_eq_mul] at hf_le
-  have huR := intervalIntegral.integral_nonneg_of_forall (μ := volume) hlo_pi (fun θ => hun _ (hR_mem θ))
-  have hh := annulus_green_identity ha haR hR hu
-  simp only [mul_neg, intervalIntegral.integral_neg]
-  linarith
-
-lemma derivative_locally_bounded {u : ℂ → ℝ}
-    (hu : ContDiffAt ℝ 2 u 0) :
-    ∃ δ M : ℝ, 0 < δ ∧ 0 ≤ M ∧ ∀ z : ℂ, ‖z‖ < δ → ‖fderiv ℝ u z‖ ≤ M := by
-  have hc := (hu.fderiv_right (m := 1) (by norm_num)).continuousAt.norm
-  have he : ∀ᶠ z in 𝓝 (0 : ℂ), ‖fderiv ℝ u z‖ < ‖fderiv ℝ u 0‖ + 1 :=
-    hc (Iio_mem_nhds (show ‖fderiv ℝ u 0‖ < ‖fderiv ℝ u 0‖ + 1 by linarith))
-  rcases Metric.eventually_nhds_iff.mp he with ⟨δ, hδ, hd⟩
-  refine ⟨δ, ‖fderiv ℝ u 0‖ + 1, hδ, by positivity, ?_⟩
-  intro z hz
-  exact (hd (by simpa only [dist_zero_right] using hz)).le
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian ENNReal
-namespace SLEExactGauge
-
-noncomputable def annulusDensity (u : ℂ → ℝ) (R : ℝ) (p : ℝ × ℝ) : ℝ :=
-  p.1 * (Real.log R - Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2))
-
-def greenRectangle (a R : ℝ) : Set (ℝ × ℝ) := Icc a R ×ˢ Icc (-Real.pi) Real.pi
-
-noncomputable def greenTruncation (u : ℂ → ℝ) (a R : ℝ) : ℝ × ℝ → ℝ :=
-  (greenRectangle a R).indicator (annulusDensity u R)
-
-lemma annulusDensity_continuousOn {u : ℂ → ℝ} {a R : ℝ}
-    (ha : 0 < a) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
-    ContinuousOn (annulusDensity u R) (greenRectangle a R) := by
-  intro p hp
-  have hr : p.1 ≠ 0 := (ha.trans_le hp.1.1).ne'
-  have hz := polarPoint_mem_ball (θ := p.2) ha hR hp.1
-  exact ((continuousAt_fst.mul (continuousAt_const.sub (continuousAt_fst.log hr))).mul
-    ((plane_laplacian_continuousAt (hu _ hz)).neg.comp (f := fun p : ℝ × ℝ => polarPoint p.1 p.2)
-      polarPoint_contDiff.continuous.continuousAt)).continuousWithinAt
-
-lemma annulusDensity_integrableOn {u : ℂ → ℝ} {a R : ℝ}
-    (ha : 0 < a) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
-    IntegrableOn (annulusDensity u R) (greenRectangle a R) :=
-  (annulusDensity_continuousOn ha hR hu).integrableOn_compact (isCompact_Icc.prod isCompact_Icc)
-
-lemma greenTruncation_integrable {u : ℂ → ℝ} {a R : ℝ}
-    (ha : 0 < a) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
-    Integrable (greenTruncation u a R) :=
-  (integrable_indicator_iff (measurableSet_Icc.prod measurableSet_Icc)).mpr
-    (annulusDensity_integrableOn ha hR hu)
-
-lemma greenTruncation_nonneg {u : ℂ → ℝ} {a R : ℝ}
-    (ha : 0 < a) (hR : R < 1)
-    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0) (p : ℝ × ℝ) :
-    0 ≤ greenTruncation u a R p := by
-  by_cases hp : p ∈ greenRectangle a R
-  · rw [greenTruncation, indicator_of_mem hp]
-    have hr := ha.trans_le hp.1.1
-    exact mul_nonneg (mul_nonneg hr.le (sub_nonneg.mpr (Real.log_le_log hr hp.1.2)))
-      (neg_nonneg.mpr (hΔ _ (polarPoint_mem_ball ha hR hp.1)))
-  · simp [greenTruncation, indicator_of_notMem hp]
-
-lemma greenTruncation_integral {u : ℂ → ℝ} {a R : ℝ}
-    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
-    (∫ p, greenTruncation u a R p) =
-      ∫ r in a..R, ∫ θ in -Real.pi..Real.pi, annulusDensity u R (r, θ) := by
-  rw [greenTruncation, greenRectangle, integral_indicator (measurableSet_Icc.prod measurableSet_Icc)]
-  have hf := annulusDensity_integrableOn ha hR hu
-  rw [Measure.volume_eq_prod] at hf ⊢
-  rw [setIntegral_prod _ hf]
-  have hpi : -Real.pi ≤ Real.pi := by linarith [Real.pi_pos]
-  simp only [intervalIntegral.integral_of_le haR, intervalIntegral.integral_of_le hpi,
-    integral_Icc_eq_integral_Ioc]
-
-lemma greenTruncation_lintegral_le {u : ℂ → ℝ} {a R C M : ℝ}
-    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
-    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
-    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
-    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
-    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0)
-    (hM : 0 ≤ M) (hgrad : ∀ θ : ℝ, ‖fderiv ℝ u (polarPoint a θ)‖ ≤ M) :
-    (∫⁻ p, ENNReal.ofReal (greenTruncation u a R p)) ≤
-      ENNReal.ofReal (2 * Real.pi * (C + M * (-a * Real.log a))) := by
-  rw [← ofReal_integral_eq_lintegral_ofReal (greenTruncation_integrable ha hR hu)
-    (Eventually.of_forall (greenTruncation_nonneg ha hR hΔ)), greenTruncation_integral ha haR hR hu]
-  exact ENNReal.ofReal_le_ofReal (annulus_green_le ha haR hR hu hun hub hM hgrad)
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.BrownianStrongMarkov
+
+namespace OAI
+
+/-! Radius Clocks for chordal SLE. -/
+
+section
+section
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+lemma measurableSet_exists_zero_compact
+    {Ω K : Type*} [MeasurableSpace Ω] [TopologicalSpace K]
+    [CompactSpace K] [TopologicalSpace.SeparableSpace K] [Nonempty K]
+    (f : K → Ω → ℝ) (hm : ∀ t, Measurable (f t))
+    (hc : ∀ ω, Continuous (fun t => f t ω)) :
+    MeasurableSet {ω | ∃ t, f t ω = 0} := by
+  obtain ⟨D, hDc, hDd⟩ := TopologicalSpace.exists_countable_dense K
+  let : Countable D := hDc.to_subtype
+  have hi : Measurable (fun ω => ⨅ t : D, |f t ω|) :=
+    Measurable.iInf (fun t => (hm t).abs)
+  have he (ω : Ω) : (∃ t, f t ω = 0) ↔ (⨅ t : D, |f t ω|) = 0 := by
+    have hb : BddBelow (range (fun t => |f t ω|)) := ⟨0, by rintro _ ⟨t, rfl⟩; exact abs_nonneg _⟩
+    rw [hDd.ciInf (hc ω).abs hb]
+    obtain ⟨t, _, ht⟩ := isCompact_univ.exists_isMinOn (Set.univ_nonempty : (univ : Set K).Nonempty)
+      (hc ω).abs.continuousOn
+    have hv : (⨅ s, |f s ω|) = |f t ω| :=
+      le_antisymm (ciInf_le hb t) (le_ciInf (fun s => ht (mem_univ s)))
+    rw [hv]
+    constructor
+    · rintro ⟨s, hs⟩
+      have ht' : |f t ω| ≤ |f s ω| := ht (mem_univ s)
+      rw [hs, abs_zero] at ht'
+      exact le_antisymm ht' (abs_nonneg _)
+    · intro h
+      exact ⟨t, abs_eq_zero.mp h⟩
+  have hs : MeasurableSet {ω | (⨅ t : D, |f t ω|) = 0} :=
+    measurableSet_eq_fun hi measurable_const
+  have heSet : {ω | ∃ t, f t ω = 0} = {ω | (⨅ t : D, |f t ω|) = 0} :=
+    Set.ext he
+  rw [heSet]
+  exact hs
+
+lemma hittingAfter_zero_le_iff
+    {Ω : Type*} (X : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => X t ω)) (n T : ℝ≥0) (ω : Ω) :
+    hittingAfter X {0} n ω ≤ (T : WithTop ℝ≥0) ↔
+      ∃ t ∈ Icc n T, X t ω = 0 := by
+  classical
+  let V : Set ℝ≥0 := {t | n ≤ t ∧ X t ω = 0}
+  have hV : IsClosed V := isClosed_Ici.inter (isClosed_eq (hc ω) continuous_const)
+  have hb : BddBelow V := ⟨0, fun t _ => bot_le⟩
+  change (if V.Nonempty then ((sInf V : ℝ≥0) : WithTop ℝ≥0) else ⊤) ≤ (T : WithTop ℝ≥0) ↔ _
+  by_cases hv : V.Nonempty
+  · obtain ⟨v, hv⟩ := hv
+    have he : ∃ t, n ≤ t ∧ X t ω ∈ ({0} : Set ℝ) := ⟨v, hv.1, hv.2⟩
+    have hin : sInf V ∈ V := hV.csInf_mem ⟨v, hv⟩ hb
+    rw [ite_eq_left (show V.Nonempty from ⟨v, hv⟩), WithTop.coe_le_coe]
+    constructor
+    · intro h
+      exact ⟨sInf V, ⟨hin.1, h⟩, hin.2⟩
+    · rintro ⟨t, ht, hx⟩
+      exact (csInf_le hb ⟨ht.1, hx⟩).trans ht.2
+  · have he : ¬ ∃ t, n ≤ t ∧ X t ω ∈ ({0} : Set ℝ) := by
+      exact hv
+    rw [ite_eq_right hv]
+    simp only [WithTop.not_top_le_coe, false_iff]
+    rintro ⟨t, ht, hx⟩
+    exact hv ⟨t, ht.1, hx⟩
+
+ 
+
+theorem continuous_zero_hit_stopping
+    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
+    (X : ℝ≥0 → Ω → ℝ) (hXa : Adapted ℱ X)
+    (hXc : ∀ ω, Continuous (fun t => X t ω)) (n : ℝ≥0) :
+    IsStoppingTime ℱ (hittingAfter X {0} n) := by
+  intro T
+  have he : {ω | hittingAfter X {0} n ω ≤ (T : WithTop ℝ≥0)} =
+      {ω | ∃ t : Icc n T, X t ω = 0} := by
+    ext ω
+    simp only [mem_ofPred_eq, hittingAfter_zero_le_iff X hXc, Subtype.exists, exists_prop]
+  rw [he]
+  by_cases hnT : n ≤ T
+  · let : Nonempty (Icc n T) := ⟨⟨n, le_rfl, hnT⟩⟩
+    have hm (t : Icc n T) : @Measurable Ω ℝ (ℱ T) (borel ℝ) (X t) :=
+      (hXa t).mono (ℱ.mono t.property.2) le_rfl
+    let : MeasurableSpace Ω := ℱ T
+    exact measurableSet_exists_zero_compact (fun t : Icc n T => X t) hm
+      (fun ω => (hXc ω).comp continuous_subtype_val)
+  · have hnil : IsEmpty (Icc n T) := ⟨fun t => hnT (t.property.1.trans t.property.2)⟩
+    have : {ω | ∃ t : Icc n T, X t ω = 0} = ∅ := by ext ω; simp
+    rw [this]
+    exact @MeasurableSet.empty Ω (ℱ T)
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma stopped_min_measurable {Ω : Type*} [m : MeasurableSpace Ω]
+    {ℱ : Filtration ℝ≥0 m} {ρ : Ω → ℝ≥0}
+    (hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0))) (T : ℝ≥0) :
+    Measurable[ℱ T] (fun ω => min (ρ ω) T) := by
+  have h := ((hρ.min_const T).measurable_of_le (fun _ => min_le_right _ _)).untopD 0
+  simpa only [← WithTop.coe_min, WithTop.untopD_coe] using h
+
+ 
+
+theorem continuous_zero_hit_after_stopping
+    {Ω : Type*} [m : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 m)
+    (X : ℝ≥0 → Ω → ℝ) (hXa : Adapted ℱ X)
+    (hXc : ∀ ω, Continuous (fun t => X t ω)) (ρ : Ω → ℝ≥0)
+    (hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0))) :
+    IsStoppingTime ℱ (fun ω => hittingAfter X {0} (ρ ω) ω) := by
+  intro T
+  let f : Icc (0 : ℝ≥0) T → Ω → ℝ :=
+    fun t ω => |X t ω| + max (((min (ρ ω) T : ℝ≥0) : ℝ) - (t : ℝ≥0)) 0
+  have he : {ω | hittingAfter X {0} (ρ ω) ω ≤ (T : WithTop ℝ≥0)} =
+      {ω | ρ ω ≤ T} ∩ {ω | ∃ t : Icc (0 : ℝ≥0) T, f t ω = 0} := by
+    ext ω
+    rw [mem_ofPred_eq, hittingAfter_zero_le_iff X hXc]
+    constructor
+    · rintro ⟨t, ht, hx⟩
+      have hr : ρ ω ≤ T := ht.1.trans ht.2
+      refine ⟨hr, ⟨⟨t, bot_le, ht.2⟩, ?_⟩⟩
+      dsimp only [f]
+      rw [hx, abs_zero, min_eq_left hr, max_eq_right (sub_nonpos.mpr (show (ρ ω : ℝ) ≤ (t : ℝ) from ht.1))]
+      exact zero_add 0
+    · rintro ⟨hr, t, ht⟩
+      have hp : 0 ≤ max (((min (ρ ω) T : ℝ≥0) : ℝ) - (t : ℝ≥0)) 0 := le_max_right _ _
+      have hx : X t ω = 0 := abs_eq_zero.mp (by dsimp only [f] at ht; linarith [abs_nonneg (X t ω)])
+      have hle : (ρ ω : ℝ) ≤ (t : ℝ≥0) := by
+        dsimp only [f] at ht
+        rw [min_eq_left hr] at ht
+        have hh := le_max_left ((ρ ω : ℝ) - (t : ℝ≥0)) 0
+        linarith [abs_nonneg (X t ω)]
+      exact ⟨t, ⟨hle, t.property.2⟩, hx⟩
+  rw [he]
+  have hmρ := stopped_min_measurable hρ T
+  have hmX (t : Icc (0 : ℝ≥0) T) : Measurable[ℱ T] (X t) :=
+    (hXa t).mono (ℱ.mono t.property.2) le_rfl
+  let : MeasurableSpace Ω := ℱ T
+  have hm (t : Icc (0 : ℝ≥0) T) : Measurable (f t) :=
+    (hmX t).abs.add (((NNReal.continuous_coe.measurable.comp hmρ).sub measurable_const).max measurable_const)
+  have hme : MeasurableSet {ω | ∃ t : Icc (0 : ℝ≥0) T, f t ω = 0} := by
+    let : Nonempty (Icc (0 : ℝ≥0) T) := ⟨⟨0, le_rfl, bot_le⟩⟩
+    apply measurableSet_exists_zero_compact f hm
+    intro ω
+    exact ((hXc ω).comp continuous_subtype_val).abs.add
+      ((continuous_const.sub (NNReal.continuous_coe.comp continuous_subtype_val)).max continuous_const)
+  have hmle : MeasurableSet {ω | ρ ω ≤ T} := by
+    simpa only [WithTop.coe_le_coe] using hρ T
+  exact hmle.inter hme
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def diskHitAfter {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
+    (ρ : Ω → ℝ≥0) (z : ℂ) (r : ℝ) (ω : Ω) : WithTop ℝ≥0 :=
+  hittingAfter (fun t ω => dist (γ ω t) z - r) {0} (ρ ω) ω
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+lemma diskHitAfter_stopping (ρ : a.samples → ℝ≥0)
+    (hρ : IsStoppingTime a.filtration (fun ω => (ρ ω : WithTop ℝ≥0)))
+    (z : ℂ) (r : ℝ) :
+    IsStoppingTime a.filtration (diskHitAfter (γ ∘ Subtype.val) ρ z r) := by
+  apply continuous_zero_hit_after_stopping a.filtration
+    (fun t ω => dist (γ ω t) z - r) ?_ ?_ ρ hρ
+  · intro t
+    have hm := a.trace_adapted t
+    let : MeasurableSpace a.samples := a.filtration t
+    exact (hm.dist measurable_const).sub measurable_const
+  · intro ω
+    exact ((a.trace_continuous ω).dist continuous_const).sub continuous_const
+
+noncomputable def boundedDiskHit (ρ : a.samples → ℝ≥0) (z : ℂ) (r : ℝ)
+    (T : ℝ≥0) (ω : a.samples) : ℝ≥0 :=
+  (min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) T).untopD 0
+
+lemma boundedDiskHit_coe (ρ : a.samples → ℝ≥0) (z : ℂ) (r : ℝ) (T : ℝ≥0)
+    (ω : a.samples) :
+    (a.boundedDiskHit ρ z r T ω : WithTop ℝ≥0) =
+      min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) T := by
+  have hn : min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) (T : WithTop ℝ≥0) ≠ ⊤ :=
+    ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_right _ _)
+  obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp hn
+  change (((min (diskHitAfter (γ ∘ Subtype.val) ρ z r ω) (T : WithTop ℝ≥0)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = _
+  rw [← hv, WithTop.untopD_coe]
+
+lemma boundedDiskHit_stopping (ρ : a.samples → ℝ≥0)
+    (hρ : IsStoppingTime a.filtration (fun ω => (ρ ω : WithTop ℝ≥0)))
+    (z : ℂ) (r : ℝ) (T : ℝ≥0) :
+    IsStoppingTime a.filtration (fun ω => (a.boundedDiskHit ρ z r T ω : WithTop ℝ≥0)) := by
+  simpa only [a.boundedDiskHit_coe] using (a.diskHitAfter_stopping ρ hρ z r).min_const T
+
+ 
+
+theorem boundedDiskHit_restart_law [IsProbabilityMeasure P]
+    (ρ : a.samples → ℝ≥0)
+    (hρ : IsStoppingTime a.filtration (fun ω => (ρ ω : WithTop ℝ≥0)))
+    (z : ℂ) (r : ℝ) (T : ℝ≥0) {A : Set a.samples}
+    (hA : MeasurableSet[(a.boundedDiskHit_stopping ρ hρ z r T).measurableSpace] A) :
+    ((P.comap (Subtype.val : a.samples → Ω)).restrict A).map
+      (fun ω t => brownianRestart a.B (a.boundedDiskHit ρ z r T) t ω) =
+    (P.comap (Subtype.val : a.samples → Ω)) A •
+      (P.comap (Subtype.val : a.samples → Ω)).map (fun ω t => a.B t ω) :=
+  brownian_restart_path_law a.brownian a.measurable_B a.continuous_B
+    (a.boundedDiskHit_stopping ρ hρ z r T) hA
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+namespace SLEExactGauge
+
+lemma complex_bilinear_rotation_trace (B : ℂ →L[ℝ] ℂ →L[ℝ] ℝ) (e : ℂ) :
+    B e e + B (e * I) (e * I) = ‖e‖ ^ 2 * (B 1 1 + B I I) := by
+  have he : e = e.re • (1 : ℂ) + e.im • I := by apply Complex.ext <;> simp
+  have heI : e * I = (-e.im) • (1 : ℂ) + e.re • I := by apply Complex.ext <;> simp
+  calc
+    _ = B (e.re • 1 + e.im • I) (e.re • 1 + e.im • I) +
+      B ((-e.im) • 1 + e.re • I) ((-e.im) • 1 + e.re • I) := by rw [← he, ← heI]
+    _ = _ := by
+      simp only [map_add, map_smul, add_apply, smul_apply, smul_eq_mul]
+      rw [Complex.sq_norm]
+      simp only [Complex.normSq_apply]
+      ring
+
+lemma complex_hessian_rotation {u : ℂ → ℝ} (z e : ℂ) :
+    fderiv ℝ (fderiv ℝ u) z e e + fderiv ℝ (fderiv ℝ u) z (e * I) (e * I) =
+      ‖e‖ ^ 2 * (Δ u) z := by
+  rw [complex_bilinear_rotation_trace]
+  simp only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
+    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+
+noncomputable def polarPoint (r θ : ℝ) : ℂ := r • circleMap 0 1 θ
+
+lemma polarPoint_eq_circleMap (r θ : ℝ) : polarPoint r θ = circleMap 0 r θ := by
+  simp [polarPoint, circleMap]
+
+lemma polarPoint_norm (r θ : ℝ) : ‖polarPoint r θ‖ = |r| := by
+  simp [polarPoint, norm_circleMap_zero]
+
+lemma polarPoint_radial_hasDeriv (r θ : ℝ) :
+    HasDerivAt (fun s => polarPoint s θ) (circleMap 0 1 θ) r := by
+  simpa [polarPoint] using (hasDerivAt_id r).smul_const (circleMap 0 1 θ)
+
+lemma polarPoint_angular_hasDeriv (r θ : ℝ) :
+    HasDerivAt (polarPoint r) (polarPoint r θ * I) θ := by
+  simpa only [funext (polarPoint_eq_circleMap r), polarPoint_eq_circleMap] using
+    hasDerivAt_circleMap 0 r θ
+
+lemma radial_partial_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
+    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    HasDerivAt (fun s => u (polarPoint s θ))
+      (fderiv ℝ u (polarPoint r θ) (circleMap 0 1 θ)) r :=
+  (hu.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt r
+    (polarPoint_radial_hasDeriv r θ)
+
+lemma radial_second_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
+    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    HasDerivAt (fun s => fderiv ℝ u (polarPoint s θ) (circleMap 0 1 θ))
+      (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ) (circleMap 0 1 θ)) r := by
+  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).hasFDerivAt
+  have hc : HasDerivAt (fun s => fderiv ℝ u (polarPoint s θ))
+      (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ)) r :=
+    hd.comp_hasDerivAt r (polarPoint_radial_hasDeriv r θ)
+  have ha := hc.clm_apply (hasDerivAt_const r (circleMap 0 1 θ))
+  simpa only [map_zero, add_zero] using ha
+
+lemma angular_partial_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
+    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    HasDerivAt (fun φ => u (polarPoint r φ))
+      (fderiv ℝ u (polarPoint r θ) (polarPoint r θ * I)) θ :=
+  (hu.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt θ
+    (polarPoint_angular_hasDeriv r θ)
+
+lemma angular_second_hasDeriv {u : ℂ → ℝ} {r θ : ℝ}
+    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    HasDerivAt (fun φ => fderiv ℝ u (polarPoint r φ) (polarPoint r φ * I))
+      (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (polarPoint r θ * I) (polarPoint r θ * I)
+        - fderiv ℝ u (polarPoint r θ) (polarPoint r θ)) θ := by
+  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).hasFDerivAt
+  have ht := (polarPoint_angular_hasDeriv r θ).mul_const I
+  have hmul : (polarPoint r θ * I) * I = -polarPoint r θ := by rw [mul_assoc, I_mul_I, mul_neg_one]
+  rw [hmul] at ht
+  simpa [sub_eq_add_neg] using
+    (hd.comp_hasDerivAt θ (polarPoint_angular_hasDeriv r θ)).clm_apply ht
+
+ 
+
+lemma polar_laplacian_identity {u : ℂ → ℝ} (r θ : ℝ) :
+    r ^ 2 * (Δ u) (polarPoint r θ) =
+      r ^ 2 * fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ) (circleMap 0 1 θ) +
+      fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (polarPoint r θ * I) (polarPoint r θ * I) := by
+  have hh := complex_hessian_rotation (u := u) (polarPoint r θ) (circleMap 0 1 θ)
+  have he : polarPoint r θ * I = r • (circleMap 0 1 θ * I) := by
+    simp [polarPoint, mul_assoc]
+  rw [he]
+  simp only [map_smul, smul_apply, smul_eq_mul]
+  simp only [norm_circleMap_zero, abs_one, one_pow, one_mul] at hh
+  nlinarith
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian
+namespace SLEExactGauge
+
+noncomputable def polarRadialFlux (u : ℂ → ℝ) (R : ℝ) (p : ℝ × ℝ) : ℝ :=
+  p.1 * (Real.log R - Real.log p.1) *
+    fderiv ℝ u (polarPoint p.1 p.2) (circleMap 0 1 p.2) + u (polarPoint p.1 p.2)
+
+noncomputable def polarAngularFlux (u : ℂ → ℝ) (R : ℝ) (p : ℝ × ℝ) : ℝ :=
+  (Real.log R - Real.log p.1) / p.1 *
+    fderiv ℝ u (polarPoint p.1 p.2) (polarPoint p.1 p.2 * I)
+
+lemma polarPoint_contDiff : ContDiff ℝ ⊤ (fun p : ℝ × ℝ => polarPoint p.1 p.2) := by
+  have hc : ContDiff ℝ ⊤ (fun p : ℝ × ℝ => (p.2 : ℂ)) :=
+    Complex.ofRealCLM.contDiff.comp contDiff_snd
+  have he : ContDiff ℝ ⊤ (fun p : ℝ × ℝ => circleMap 0 1 p.2) := by
+    have hExp : ContDiff ℝ ⊤ Complex.exp := Complex.contDiff_exp
+    simpa only [circleMap, Complex.ofReal_one, one_mul, zero_add, Function.comp_def] using
+      hExp.comp (hc.mul (contDiff_const (c := I)))
+  exact contDiff_fst.smul he
+
+lemma polarRadialFlux_differentiableAt {u : ℂ → ℝ} {R r θ : ℝ}
+    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    DifferentiableAt ℝ (polarRadialFlux u R) (r, θ) := by
+  have hp := polarPoint_contDiff.differentiable (by simp) (r, θ)
+  have hv := (hu.differentiableAt (by norm_num)).comp (r, θ) hp
+  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).comp (r, θ) hp
+  unfold polarRadialFlux
+  exact ((differentiableAt_fst.mul ((differentiableAt_const (Real.log R)).sub
+    (differentiableAt_fst.log hr))).mul (hd.clm_apply (by unfold circleMap; fun_prop))).add hv
+
+lemma polarAngularFlux_differentiableAt {u : ℂ → ℝ} {R r θ : ℝ}
+    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    DifferentiableAt ℝ (polarAngularFlux u R) (r, θ) := by
+  have hp := polarPoint_contDiff.differentiable (by simp) (r, θ)
+  have hd := ((hu.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num)).comp (r, θ) hp
+  have hfst : DifferentiableAt ℝ (Prod.fst : ℝ × ℝ → ℝ) (r, θ) := differentiableAt_fst
+  have hl : DifferentiableAt ℝ (fun p : ℝ × ℝ => (Real.log R - Real.log p.1) / p.1) (r, θ) :=
+    by
+      simp only [div_eq_mul_inv]
+      exact ((differentiableAt_const (Real.log R)).sub (hfst.log hr)).mul (hfst.inv hr)
+  exact hl.mul (hd.clm_apply (hp.mul_const I))
+
+lemma polarRadialFlux_radial_hasDeriv {u : ℂ → ℝ} {R r θ : ℝ}
+    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    HasDerivAt (fun s => polarRadialFlux u R (s, θ))
+      ((Real.log R - Real.log r) * fderiv ℝ u (polarPoint r θ) (circleMap 0 1 θ) +
+        r * (Real.log R - Real.log r) *
+          fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (circleMap 0 1 θ) (circleMap 0 1 θ)) r := by
+  have hl := (Real.hasDerivAt_log hr).const_sub (Real.log R)
+  have hh := (((hasDerivAt_id r).mul hl).mul (radial_second_hasDeriv hu)).add
+    (radial_partial_hasDeriv hu)
+  convert! hh using 1
+  simp only [id_eq, Pi.mul_apply]
+  field_simp [hr]
+  ring
+
+lemma polarAngularFlux_angular_hasDeriv {u : ℂ → ℝ} {R r θ : ℝ}
+    (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    HasDerivAt (fun φ => polarAngularFlux u R (r, φ))
+      ((Real.log R - Real.log r) / r *
+        (fderiv ℝ (fderiv ℝ u) (polarPoint r θ) (polarPoint r θ * I) (polarPoint r θ * I) -
+          fderiv ℝ u (polarPoint r θ) (polarPoint r θ))) θ :=
+  (angular_second_hasDeriv hu).const_mul ((Real.log R - Real.log r) / r)
+
+lemma polarGreen_divergence {u : ℂ → ℝ} {R r θ : ℝ}
+    (hr : r ≠ 0) (hu : ContDiffAt ℝ 2 u (polarPoint r θ)) :
+    fderiv ℝ (polarRadialFlux u R) (r, θ) (1, 0) +
+      fderiv ℝ (polarAngularFlux u R) (r, θ) (0, 1) =
+        r * (Real.log R - Real.log r) * (Δ u) (polarPoint r θ) := by
+  have hf := (polarRadialFlux_differentiableAt (R := R) hr hu).hasFDerivAt.comp_hasDerivAt r
+    ((hasDerivAt_id r).prodMk (hasDerivAt_const r θ))
+  have hg := (polarAngularFlux_differentiableAt (R := R) hr hu).hasFDerivAt.comp_hasDerivAt θ
+    ((hasDerivAt_const θ r).prodMk (hasDerivAt_id θ))
+  rw [hf.unique (polarRadialFlux_radial_hasDeriv hr hu),
+    hg.unique (polarAngularFlux_angular_hasDeriv hu)]
+  have he : fderiv ℝ u (polarPoint r θ) (polarPoint r θ) =
+      r * fderiv ℝ u (polarPoint r θ) (circleMap 0 1 θ) := by
+    conv_lhs => arg 2; rw [polarPoint]
+    simp only [map_smul, smul_eq_mul]
+  rw [he]
+  have hh := polar_laplacian_identity (u := u) r θ
+  apply (mul_right_cancel₀ hr)
+  field_simp [hr]
+  have hmul := congrArg (fun x : ℝ => (Real.log R - Real.log r) * x) hh
+  nlinarith only [hmul]
+
+lemma polarPoint_endpoints (r : ℝ) : polarPoint r (-Real.pi) = polarPoint r Real.pi := by
+  have hp := periodic_circleMap 0 1 (-Real.pi)
+  have he : -Real.pi + 2 * Real.pi = Real.pi := by ring
+  rw [he] at hp
+  exact congrArg (fun z : ℂ => r • z) hp.symm
+
+lemma polarAngularFlux_endpoints (u : ℂ → ℝ) (R r : ℝ) :
+    polarAngularFlux u R (r, -Real.pi) = polarAngularFlux u R (r, Real.pi) := by
+  simp only [polarAngularFlux, polarPoint_endpoints]
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian
+namespace SLEExactGauge
+
+lemma plane_laplacian_continuousAt {u : ℂ → ℝ} {z : ℂ} (hu : ContDiffAt ℝ 2 u z) :
+    ContinuousAt (Δ u) z := by
+  have hh := ((hu.fderiv_right (m := 1) (by norm_num)).fderiv_right (m := 0)
+    (by norm_num)).continuousAt
+  have h1 := (hh.clm_apply (continuousAt_const (y := (1 : ℂ)))).clm_apply
+    (continuousAt_const (y := (1 : ℂ)))
+  have hI := (hh.clm_apply (continuousAt_const (y := I))).clm_apply
+    (continuousAt_const (y := I))
+  simpa only [laplacian_eq_iteratedFDeriv_complexPlane, iteratedFDeriv_two_apply,
+    Fin.isValue, Pi.add_def, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+    using h1.add hI
+
+lemma polarPoint_mem_ball {a R r θ : ℝ} (ha : 0 < a) (hR : R < 1)
+    (hr : r ∈ Icc a R) : polarPoint r θ ∈ ball (0 : ℂ) 1 := by
+  rw [mem_ball_zero_iff, polarPoint_norm, abs_of_pos (ha.trans_le hr.1)]
+  exact hr.2.trans_lt hR
+
+ 
+
+lemma annulus_green_identity {u : ℂ → ℝ} {a R : ℝ}
+    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
+    (∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
+      r * (Real.log R - Real.log r) * (Δ u) (polarPoint r θ)) =
+      (∫ θ in -Real.pi..Real.pi, u (polarPoint R θ)) -
+        ∫ θ in -Real.pi..Real.pi, polarRadialFlux u R (a, θ) := by
+  let S := uIcc a R ×ˢ uIcc (-Real.pi) Real.pi
+  have hmem (p : ℝ × ℝ) (hp : p ∈ S) : p.1 ≠ 0 ∧ polarPoint p.1 p.2 ∈ ball (0 : ℂ) 1 := by
+    have hr : p.1 ∈ Icc a R := by simpa only [uIcc_of_le haR] using hp.1
+    exact ⟨(ha.trans_le hr.1).ne', polarPoint_mem_ball ha hR hr⟩
+  have hf : ContinuousOn (polarRadialFlux u R) S := fun p hp =>
+    (polarRadialFlux_differentiableAt (hmem p hp).1 (hu _ (hmem p hp).2)).continuousAt.continuousWithinAt
+  have hg : ContinuousOn (polarAngularFlux u R) S := fun p hp =>
+    (polarAngularFlux_differentiableAt (hmem p hp).1 (hu _ (hmem p hp).2)).continuousAt.continuousWithinAt
+  have hl : ContinuousOn (fun p : ℝ × ℝ =>
+      p.1 * (Real.log R - Real.log p.1) * (Δ u) (polarPoint p.1 p.2)) S := by
+    intro p hp
+    exact ((continuousAt_fst.mul (continuousAt_const.sub (continuousAt_fst.log (hmem p hp).1))).mul
+      ((plane_laplacian_continuousAt (hu _ (hmem p hp).2)).comp (f := fun p : ℝ × ℝ => polarPoint p.1 p.2)
+        polarPoint_contDiff.continuous.continuousAt)).continuousWithinAt
+  have he : EqOn (fun p : ℝ × ℝ => fderiv ℝ (polarRadialFlux u R) p (1, 0) +
+      fderiv ℝ (polarAngularFlux u R) p (0, 1))
+      (fun p => p.1 * (Real.log R - Real.log p.1) * (Δ u) (polarPoint p.1 p.2)) S :=
+    fun p hp => polarGreen_divergence (hmem p hp).1 (hu _ (hmem p hp).2)
+  have hi : IntegrableOn (fun p : ℝ × ℝ => fderiv ℝ (polarRadialFlux u R) p (1, 0) +
+      fderiv ℝ (polarAngularFlux u R) p (0, 1)) S :=
+    (hl.integrableOn_compact (isCompact_uIcc.prod isCompact_uIcc)).congr_fun
+      (fun p hp => (he hp).symm) (measurableSet_uIcc.prod measurableSet_uIcc)
+  have hm : Ioo (min a R) (max a R) ×ˢ Ioo (min (-Real.pi) Real.pi) (max (-Real.pi) Real.pi) ⊆ S :=
+    prod_mono Ioo_subset_Icc_self Ioo_subset_Icc_self
+  have hh := integral2_divergence_prod_of_hasFDerivAt (polarRadialFlux u R) (polarAngularFlux u R)
+    (fderiv ℝ (polarRadialFlux u R)) (fderiv ℝ (polarAngularFlux u R)) a (-Real.pi) R Real.pi hf hg
+    (fun p hp => (polarRadialFlux_differentiableAt (hmem p (hm hp)).1
+      (hu _ (hmem p (hm hp)).2)).hasFDerivAt)
+    (fun p hp => (polarAngularFlux_differentiableAt (hmem p (hm hp)).1
+      (hu _ (hmem p (hm hp)).2)).hasFDerivAt) hi
+  have hi_eq : (∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
+      fderiv ℝ (polarRadialFlux u R) (r, θ) (1, 0) +
+        fderiv ℝ (polarAngularFlux u R) (r, θ) (0, 1)) =
+      ∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
+        r * (Real.log R - Real.log r) * (Δ u) (polarPoint r θ) := by
+    apply intervalIntegral.integral_congr
+    intro r hr
+    apply intervalIntegral.integral_congr
+    intro θ hθ
+    exact he ⟨hr, hθ⟩
+  rw [hi_eq] at hh
+  simpa only [polarAngularFlux_endpoints, sub_self, zero_add,
+    polarRadialFlux, sub_self, mul_zero, zero_mul, zero_add] using hh
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian
+namespace SLEExactGauge
+
+lemma annulus_green_le {u : ℂ → ℝ} {a R C M : ℝ}
+    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
+    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
+    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
+    (hM : 0 ≤ M) (hgrad : ∀ θ : ℝ, ‖fderiv ℝ u (polarPoint a θ)‖ ≤ M) :
+    (∫ r in a..R, ∫ θ in -Real.pi..Real.pi,
+      r * (Real.log R - Real.log r) * (-(Δ u) (polarPoint r θ))) ≤
+        2 * Real.pi * (C + M * (-a * Real.log a)) := by
+  have hRpos := ha.trans_le haR
+  have ha_mem (θ : ℝ) : polarPoint a θ ∈ ball (0 : ℂ) 1 :=
+    polarPoint_mem_ball ha hR ⟨le_rfl, haR⟩
+  have hR_mem (θ : ℝ) : polarPoint R θ ∈ ball (0 : ℂ) 1 :=
+    polarPoint_mem_ball ha hR ⟨haR, le_rfl⟩
+  have hL : 0 ≤ Real.log R - Real.log a := sub_nonneg.mpr (Real.log_le_log ha haR)
+  have hlogR : Real.log R ≤ 0 := Real.log_nonpos hRpos.le hR.le
+  have hbound (θ : ℝ) : polarRadialFlux u R (a, θ) ≤ C + M * (-a * Real.log a) := by
+    have hb : fderiv ℝ u (polarPoint a θ) (circleMap 0 1 θ) ≤ M := by
+      apply (le_abs_self _).trans
+      calc
+        _ = ‖fderiv ℝ u (polarPoint a θ) (circleMap 0 1 θ)‖ := (Real.norm_eq_abs _).symm
+        _ ≤ ‖fderiv ℝ u (polarPoint a θ)‖ * ‖circleMap 0 1 θ‖ :=
+          (fderiv ℝ u (polarPoint a θ)).le_opNorm _
+        _ ≤ M := by simpa using hgrad θ
+    have hm := mul_le_mul_of_nonneg_left hb (mul_nonneg ha.le hL)
+    have hn := mul_nonpos_of_nonneg_of_nonpos (mul_nonneg ha.le hM) hlogR
+    unfold polarRadialFlux
+    nlinarith [hub _ (ha_mem θ)]
+  have hf : Continuous (fun θ => polarRadialFlux u R (a, θ)) :=
+    continuous_iff_continuousAt.mpr fun θ =>
+      (polarRadialFlux_differentiableAt ha.ne' (hu _ (ha_mem θ))).continuousAt.comp
+        (continuousAt_const.prodMk continuousAt_id)
+  have hf_int := hf.intervalIntegrable (μ := volume) (-Real.pi) Real.pi
+  have hlo_pi : -Real.pi ≤ Real.pi := by nlinarith [Real.pi_pos]
+  have hf_le := intervalIntegral.integral_mono_on hlo_pi hf_int
+    (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => C + M * (-a * Real.log a)) volume
+      (-Real.pi) Real.pi) (fun θ _ => hbound θ)
+  simp only [intervalIntegral.integral_const, smul_eq_mul] at hf_le
+  have huR := intervalIntegral.integral_nonneg_of_forall (μ := volume) hlo_pi (fun θ => hun _ (hR_mem θ))
+  have hh := annulus_green_identity ha haR hR hu
+  simp only [mul_neg, intervalIntegral.integral_neg]
+  linarith
+
+lemma derivative_locally_bounded {u : ℂ → ℝ}
+    (hu : ContDiffAt ℝ 2 u 0) :
+    ∃ δ M : ℝ, 0 < δ ∧ 0 ≤ M ∧ ∀ z : ℂ, ‖z‖ < δ → ‖fderiv ℝ u z‖ ≤ M := by
+  have hc := (hu.fderiv_right (m := 1) (by norm_num)).continuousAt.norm
+  have he : ∀ᶠ z in 𝓝 (0 : ℂ), ‖fderiv ℝ u z‖ < ‖fderiv ℝ u 0‖ + 1 :=
+    hc (Iio_mem_nhds (show ‖fderiv ℝ u 0‖ < ‖fderiv ℝ u 0‖ + 1 by linarith))
+  rcases Metric.eventually_nhds_iff.mp he with ⟨δ, hδ, hd⟩
+  refine ⟨δ, ‖fderiv ℝ u 0‖ + 1, hδ, by positivity, ?_⟩
+  intro z hz
+  exact (hd (by simpa only [dist_zero_right] using hz)).le
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian ENNReal
+namespace SLEExactGauge
+
+noncomputable def annulusDensity (u : ℂ → ℝ) (R : ℝ) (p : ℝ × ℝ) : ℝ :=
+  p.1 * (Real.log R - Real.log p.1) * (-(Δ u) (polarPoint p.1 p.2))
+
+def greenRectangle (a R : ℝ) : Set (ℝ × ℝ) := Icc a R ×ˢ Icc (-Real.pi) Real.pi
+
+noncomputable def greenTruncation (u : ℂ → ℝ) (a R : ℝ) : ℝ × ℝ → ℝ :=
+  (greenRectangle a R).indicator (annulusDensity u R)
+
+lemma annulusDensity_continuousOn {u : ℂ → ℝ} {a R : ℝ}
+    (ha : 0 < a) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
+    ContinuousOn (annulusDensity u R) (greenRectangle a R) := by
+  intro p hp
+  have hr : p.1 ≠ 0 := (ha.trans_le hp.1.1).ne'
+  have hz := polarPoint_mem_ball (θ := p.2) ha hR hp.1
+  exact ((continuousAt_fst.mul (continuousAt_const.sub (continuousAt_fst.log hr))).mul
+    ((plane_laplacian_continuousAt (hu _ hz)).neg.comp (f := fun p : ℝ × ℝ => polarPoint p.1 p.2)
+      polarPoint_contDiff.continuous.continuousAt)).continuousWithinAt
+
+lemma annulusDensity_integrableOn {u : ℂ → ℝ} {a R : ℝ}
+    (ha : 0 < a) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
+    IntegrableOn (annulusDensity u R) (greenRectangle a R) :=
+  (annulusDensity_continuousOn ha hR hu).integrableOn_compact (isCompact_Icc.prod isCompact_Icc)
+
+lemma greenTruncation_integrable {u : ℂ → ℝ} {a R : ℝ}
+    (ha : 0 < a) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
+    Integrable (greenTruncation u a R) :=
+  (integrable_indicator_iff (measurableSet_Icc.prod measurableSet_Icc)).mpr
+    (annulusDensity_integrableOn ha hR hu)
+
+lemma greenTruncation_nonneg {u : ℂ → ℝ} {a R : ℝ}
+    (ha : 0 < a) (hR : R < 1)
+    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0) (p : ℝ × ℝ) :
+    0 ≤ greenTruncation u a R p := by
+  by_cases hp : p ∈ greenRectangle a R
+  · rw [greenTruncation, indicator_of_mem hp]
+    have hr := ha.trans_le hp.1.1
+    exact mul_nonneg (mul_nonneg hr.le (sub_nonneg.mpr (Real.log_le_log hr hp.1.2)))
+      (neg_nonneg.mpr (hΔ _ (polarPoint_mem_ball ha hR hp.1)))
+  · simp [greenTruncation, indicator_of_notMem hp]
+
+lemma greenTruncation_integral {u : ℂ → ℝ} {a R : ℝ}
+    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z) :
+    (∫ p, greenTruncation u a R p) =
+      ∫ r in a..R, ∫ θ in -Real.pi..Real.pi, annulusDensity u R (r, θ) := by
+  rw [greenTruncation, greenRectangle, integral_indicator (measurableSet_Icc.prod measurableSet_Icc)]
+  have hf := annulusDensity_integrableOn ha hR hu
+  rw [Measure.volume_eq_prod] at hf ⊢
+  rw [setIntegral_prod _ hf]
+  have hpi : -Real.pi ≤ Real.pi := by linarith [Real.pi_pos]
+  simp only [intervalIntegral.integral_of_le haR, intervalIntegral.integral_of_le hpi,
+    integral_Icc_eq_integral_Ioc]
+
+lemma greenTruncation_lintegral_le {u : ℂ → ℝ} {a R C M : ℝ}
+    (ha : 0 < a) (haR : a ≤ R) (hR : R < 1)
+    (hu : ∀ z ∈ ball (0 : ℂ) 1, ContDiffAt ℝ 2 u z)
+    (hun : ∀ z ∈ ball (0 : ℂ) 1, 0 ≤ u z)
+    (hub : ∀ z ∈ ball (0 : ℂ) 1, u z ≤ C)
+    (hΔ : ∀ z ∈ ball (0 : ℂ) 1, (Δ u) z ≤ 0)
+    (hM : 0 ≤ M) (hgrad : ∀ θ : ℝ, ‖fderiv ℝ u (polarPoint a θ)‖ ≤ M) :
+    (∫⁻ p, ENNReal.ofReal (greenTruncation u a R p)) ≤
+      ENNReal.ofReal (2 * Real.pi * (C + M * (-a * Real.log a))) := by
+  rw [← ofReal_integral_eq_lintegral_ofReal (greenTruncation_integrable ha hR hu)
+    (Eventually.of_forall (greenTruncation_nonneg ha hR hΔ)), greenTruncation_integral ha haR hR hu]
+  exact ENNReal.ofReal_le_ofReal (annulus_green_le ha haR hR hu hun hub hM hgrad)
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/RectangleMeans.lean b/lean/OAI/Probability/SLE/RectangleMeans.lean
index 8e0d6f5dd68b0f677837b72b04622dd4f06cdd72..a7cc5c93f544de608d35710c38a6409536c56ab6 100644
--- a/lean/OAI/Probability/SLE/RectangleMeans.lean
+++ b/lean/OAI/Probability/SLE/RectangleMeans.lean
@@ -1,659 +1,659 @@
-import Mathlib
-import OAI.Probability.SLE.StripKernels
-
-namespace OAI
-
-/-! Rectangle Means for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology ContDiff ENNReal BigOperators NNReal
-namespace SLEExactGauge
-noncomputable def cellOrigin (k : ℕ) (i : Fin (2^k)) : ℝ := (i:ℝ)*scaleLength k
-noncomputable def dyadicCell (k : ℕ) (p : Fin (2^k) × Fin (2^k)) : Set ℂ :=
-  stripRectangle (cellOrigin k p.1) (Real.pi*cellOrigin k p.2) (scaleLength k)
-noncomputable def dyadicInterior (k : ℕ) (p : Fin (2^k) × Fin (2^k)) : Set ℂ :=
-  openStripRectangle (cellOrigin k p.1) (Real.pi*cellOrigin k p.2) (scaleLength k)
-
-lemma dyadicCell_eq (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
-    dyadicCell k p = dyadicStripRectangle 0 k p.1 p.2 := by
-  simp [dyadicCell, dyadicStripRectangle, cellOrigin, scaleLength, div_eq_mul_inv]
-lemma cellOrigin_nonneg (k : ℕ) (i : Fin (2^k)) : 0 ≤ cellOrigin k i := by
-  exact mul_nonneg (Nat.cast_nonneg _) (scaleLength_pos k).le
-lemma cellOrigin_right (k : ℕ) (i : Fin (2^k)) : cellOrigin k i + scaleLength k ≤ 1 := by
-  have hh : (i:ℝ)+1 ≤ (2:ℝ)^k := by exact_mod_cast i.isLt
-  dsimp [cellOrigin, scaleLength]
-  calc
-    _ = ((i:ℝ)+1)/(2:ℝ)^k := by ring
-    _ ≤ 1 := (div_le_one (by positivity)).2 hh
-lemma dyadicInterior_subset_unit (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
-    dyadicInterior k p ⊆ openUnitStrip := by
-  intro z hz
-  have hr := cellOrigin_right k p.1
-  have hi := mul_le_mul_of_nonneg_left (cellOrigin_right k p.2) Real.pi_pos.le
-  have h0 := cellOrigin_nonneg k p.1
-  have h0' := mul_nonneg Real.pi_pos.le (cellOrigin_nonneg k p.2)
-  change _ < z.re ∧ z.re < _ ∧ _ < z.im ∧ z.im < _ at hz
-  exact ⟨h0.trans_lt hz.1, hz.2.1.trans_le hr,
-    h0'.trans_lt hz.2.2.1, by nlinarith [hz.2.2.2]⟩
-lemma dyadicCell_subset_root (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
-    dyadicCell k p ⊆ stripRectangle 0 0 1 := by
-  intro z hz
-  have hr := cellOrigin_right k p.1
-  have hi := mul_le_mul_of_nonneg_left (cellOrigin_right k p.2) Real.pi_pos.le
-  have h0 := cellOrigin_nonneg k p.1
-  have h0' := mul_nonneg Real.pi_pos.le (cellOrigin_nonneg k p.2)
-  change _ ≤ z.re ∧ z.re ≤ _ ∧ _ ≤ z.im ∧ z.im ≤ _ at hz
-  change 0 ≤ z.re ∧ z.re ≤ 0+1 ∧ 0 ≤ z.im ∧ z.im ≤ 0+Real.pi*1
-  exact ⟨h0.trans hz.1, by linarith [hz.2.1],
-    h0'.trans hz.2.2.1, by nlinarith [hz.2.2.2]⟩
-
-lemma cellOrigin_separated {k : ℕ} {i j : Fin (2^k)} (hij : i < j) :
-    cellOrigin k i + scaleLength k ≤ cellOrigin k j := by
-  have hh : (i:ℝ)+1 ≤ (j:ℝ) := by exact_mod_cast hij
-  have hm := mul_le_mul_of_nonneg_right hh (scaleLength_pos k).le
-  dsimp [cellOrigin]
-  nlinarith
-
-lemma dyadicInterior_disjoint (k : ℕ) : Pairwise (fun p q => Disjoint (dyadicInterior k p) (dyadicInterior k q)) := by
-  intro p q hpq
-  rw [Set.disjoint_left]
-  intro z hz hz'
-  change _ < z.re ∧ z.re < _ ∧ _ < z.im ∧ z.im < _ at hz hz'
-  by_cases h : p.1 = q.1
-  · have hne : p.2 ≠ q.2 := by intro h'; exact hpq (Prod.ext h h')
-    rcases lt_or_gt_of_ne hne with hlt | hlt
-    · have hh := mul_le_mul_of_nonneg_left (cellOrigin_separated hlt) Real.pi_pos.le
-      nlinarith [hz.2.2.2, hz'.2.2.1]
-    · have hh := mul_le_mul_of_nonneg_left (cellOrigin_separated hlt) Real.pi_pos.le
-      nlinarith [hz'.2.2.2, hz.2.2.1]
-  · rcases lt_or_gt_of_ne h with hlt | hlt
-    · have hh := cellOrigin_separated hlt
-      linarith [hz.2.1, hz'.1]
-    · have hh := cellOrigin_separated hlt
-      linarith [hz'.2.1, hz.1]
-
-def leftChild (k : ℕ) (i : Fin (2^k)) : Fin (2^(k+1)) :=
-  ⟨2*i.val, by have := i.isLt; rw [pow_succ]; omega⟩
-def rightChild (k : ℕ) (i : Fin (2^k)) : Fin (2^(k+1)) :=
-  ⟨2*i.val+1, by have := i.isLt; rw [pow_succ]; omega⟩
-lemma cellOrigin_leftChild (k : ℕ) (i : Fin (2^k)) :
-    cellOrigin (k+1) (leftChild k i) = cellOrigin k i := by
-  simp only [cellOrigin, leftChild, Fin.val_mk, Nat.cast_mul, Nat.cast_ofNat, scaleLength_succ]
-  ring
-lemma cellOrigin_rightChild (k : ℕ) (i : Fin (2^k)) :
-    cellOrigin (k+1) (rightChild k i) = cellOrigin k i + scaleLength k/2 := by
-  simp only [cellOrigin, rightChild, Fin.val_mk, Nat.cast_add, Nat.cast_mul,
-    Nat.cast_ofNat, Nat.cast_one, scaleLength_succ]
-  ring
-
-noncomputable def branchIndex (x : ℝ) : (k : ℕ) → Fin (2^k)
-  | 0 => 0
-  | k+1 => if x ≤ cellOrigin k (branchIndex x k)+scaleLength k/2 then
-      leftChild k (branchIndex x k) else rightChild k (branchIndex x k)
-
-lemma branchIndex_step (x : ℝ) (k : ℕ) :
-    cellOrigin k (branchIndex x k) ≤ cellOrigin (k+1) (branchIndex x (k+1)) ∧
-    cellOrigin (k+1) (branchIndex x (k+1))+scaleLength k/2 ≤
-      cellOrigin k (branchIndex x k)+scaleLength k := by
-  rw [branchIndex]
-  split
-  · rw [cellOrigin_leftChild]
-    constructor
-    · rfl
-    · linarith [scaleLength_pos k]
-  · rw [cellOrigin_rightChild]
-    constructor <;> linarith [scaleLength_pos k]
-
-lemma branchIndex_contains {x : ℝ} (hx : 0 ≤ x) (hx1 : x ≤ 1) (k : ℕ) :
-    cellOrigin k (branchIndex x k) ≤ x ∧
-    x ≤ cellOrigin k (branchIndex x k)+scaleLength k := by
-  induction k with
-  | zero => simpa [branchIndex, cellOrigin, scaleLength] using And.intro hx hx1
-  | succ k ih =>
-    rw [branchIndex]
-    split_ifs with h
-    · rw [cellOrigin_leftChild, scaleLength_succ]
-      exact ⟨ih.1,h⟩
-    · rw [cellOrigin_rightChild, scaleLength_succ]
-      exact ⟨(not_le.mp h).le, by linarith [ih.2]⟩
-
-noncomputable def pointBranch (z : ℂ) (k : ℕ) : Fin (2^k) × Fin (2^k) :=
-  (branchIndex z.re k, branchIndex (z.im/Real.pi) k)
-lemma pointBranch_contains {z : ℂ} (hz : z ∈ stripRectangle 0 0 1) (k : ℕ) :
-    z ∈ dyadicCell k (pointBranch z k) := by
-  have hzr : 0 ≤ z.re ∧ z.re ≤ 1 := by simpa [stripRectangle] using And.intro hz.1 hz.2.1
-  have hzi : 0 ≤ z.im/Real.pi ∧ z.im/Real.pi ≤ 1 := by
-    refine ⟨div_nonneg hz.2.2.1 Real.pi_pos.le, (div_le_one Real.pi_pos).2 ?_⟩
-    simpa using hz.2.2.2
-  have hr := branchIndex_contains hzr.1 hzr.2 k
-  have hi := branchIndex_contains hzi.1 hzi.2 k
-  have hi₁ := mul_le_mul_of_nonneg_left hi.1 Real.pi_pos.le
-  have hi₂ := mul_le_mul_of_nonneg_left hi.2 Real.pi_pos.le
-  rw [mul_div_cancel₀ _ Real.pi_ne_zero] at hi₁ hi₂
-  change _ ≤ z.re ∧ z.re ≤ _ ∧ Real.pi*cellOrigin k (branchIndex (z.im/Real.pi) k) ≤ z.im ∧
-    z.im ≤ Real.pi*cellOrigin k (branchIndex (z.im/Real.pi) k)+Real.pi*scaleLength k
-  exact ⟨hr.1, hr.2, hi₁, by nlinarith⟩
-
-lemma stripRectangle_convex (a b l : ℝ) : Convex ℝ (stripRectangle a b l) := by
-  have he : stripRectangle a b l =
-      Complex.reCLM ⁻¹' Icc a (a+l) ∩ Complex.imCLM ⁻¹' Icc b (b+Real.pi*l) := by
-    ext z
-    simp only [stripRectangle, Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_preimage,
-      Set.mem_Icc, Complex.reCLM_apply, Complex.imCLM_apply]
-    tauto
-  rw [he]
-  exact ((convex_Icc a (a+l)).linear_preimage Complex.reCLM.toLinearMap).inter
-    ((convex_Icc b (b+Real.pi*l)).linear_preimage Complex.imCLM.toLinearMap)
-
-lemma stripRectangle_compact (a b l : ℝ) : IsCompact (stripRectangle a b l) := by
-  rw [stripRectangle_eq_preimage]
-  change IsCompact (Complex.equivRealProdCLM.toHomeomorph ⁻¹' _)
-  exact Complex.equivRealProdCLM.toHomeomorph.isCompact_preimage.mpr
-    (isCompact_Icc.prod isCompact_Icc)
-
-lemma stripRectangle_nonempty (a b l : ℝ) (hl : 0 ≤ l) : (stripRectangle a b l).Nonempty := by
-  refine ⟨⟨a,b⟩, ?_⟩
-  change a ≤ a ∧ a ≤ a+l ∧ b ≤ b ∧ b ≤ b+Real.pi*l
-  exact ⟨le_rfl,le_add_of_nonneg_right hl,le_rfl,
-    le_add_of_nonneg_right (mul_nonneg Real.pi_pos.le hl)⟩
-
-lemma spatialAverage_eq_setAverage (u : ℂ → ℝ) (Q : Set ℂ) :
-    spatialAverage u Q = ⨍ z in Q, u z := by
-  rw [spatialAverage, setAverage_eq, Measure.real]
-  simp only [smul_eq_mul, div_eq_mul_inv, mul_comm]
-
-lemma stripRectangle_average_attained {u : ℂ → ℝ} {a b l : ℝ}
-    (hu : ContinuousOn u (stripRectangle a b l)) (hl : 0 < l) :
-    ∃ z ∈ stripRectangle a b l, u z = spatialAverage u (stripRectangle a b l) := by
-  rw [spatialAverage_eq_setAverage]
-  apply exists_eq_setAverage ((stripRectangle_convex a b l).isConnected
-    (stripRectangle_nonempty a b l hl.le)) hu
-    (hu.integrableOn_compact (stripRectangle_compact a b l))
-    (stripRectangle_compact a b l).measure_lt_top.ne
-  rw [stripRectangle_volume]
-  exact mul_ne_zero (ENNReal.ofReal_ne_zero_iff.mpr hl)
-    (ENNReal.ofReal_ne_zero_iff.mpr (mul_pos Real.pi_pos hl))
-
-lemma dyadicCell_dist_le {k : ℕ} {p : Fin (2^k) × Fin (2^k)} {z w : ℂ}
-    (hz : z ∈ dyadicCell k p) (hw : w ∈ dyadicCell k p) :
-    dist z w ≤ 5*scaleLength k := by
-  change _ ≤ z.re ∧ z.re ≤ _ ∧ _ ≤ z.im ∧ z.im ≤ _ at hz
-  change _ ≤ w.re ∧ w.re ≤ _ ∧ _ ≤ w.im ∧ w.im ≤ _ at hw
-  have hre : |z.re-w.re| ≤ scaleLength k := by rw [abs_le]; constructor <;> linarith
-  have him : |z.im-w.im| ≤ Real.pi*scaleLength k := by rw [abs_le]; constructor <;> linarith
-  rw [dist_eq_norm]
-  have hn := Complex.norm_le_abs_re_add_abs_im (z-w)
-  simp only [Complex.sub_re, Complex.sub_im] at hn
-  nlinarith [scaleLength_pos k, Real.pi_lt_four]
-
-lemma scaleLength_tendsto : Tendsto scaleLength atTop (𝓝 0) := by
-  have hh := tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2)
-    (by norm_num : (1:ℝ)/2 < 1)
-  change Tendsto (fun n => 1/(2:ℝ)^n) atTop (𝓝 0)
-  simpa only [div_pow, one_pow] using hh
-
-lemma branch_means_tendsto {u : ℂ → ℝ} (hu : ContinuousOn u (stripRectangle 0 0 1))
-    {z : ℂ} (hz : z ∈ stripRectangle 0 0 1) :
-    Tendsto (fun k => spatialAverage u (dyadicCell k (pointBranch z k))) atTop (𝓝 (u z)) := by
-  rw [Metric.tendsto_atTop]
-  intro ε hε
-  obtain ⟨δ, hδ, huδ⟩ := (Metric.continuousWithinAt_iff.1 (hu z hz)) ε hε
-  have hs : ∀ᶠ k in atTop, 5*scaleLength k < δ :=
-    ((scaleLength_tendsto.const_mul 5).eventually (eventually_lt_nhds (by simpa using hδ)))
-  obtain ⟨N,hN⟩ := eventually_atTop.1 hs
-  refine ⟨N,fun k hk => ?_⟩
-  obtain ⟨w,hw,hew⟩ := stripRectangle_average_attained
-    (hu.mono (dyadicCell_subset_root k (pointBranch z k))) (scaleLength_pos k)
-  change u w = spatialAverage u (dyadicCell k (pointBranch z k)) at hew
-  rw [← hew]
-  apply huδ (dyadicCell_subset_root k (pointBranch z k) hw)
-  exact (dyadicCell_dist_le hw (pointBranch_contains hz k)).trans_lt (hN k hk)
-
-noncomputable def cellEnergy (D : ℂ → ℂ) (k : ℕ) (p : Fin (2^k) × Fin (2^k)) : ℝ :=
-  ∫ z in dyadicInterior k p, ‖D z‖^2
-lemma cellEnergy_nonneg (D : ℂ → ℂ) (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
-    0 ≤ cellEnergy D k p := integral_nonneg (fun _ => sq_nonneg _)
-
-lemma branch_mean_step_le {u : ℂ → ℝ} {D : ℂ → ℂ}
-    (hu : SourceIsStripWeakGradient u D) (z : ℂ) (k : ℕ) :
-    |spatialAverage u (dyadicCell (k+1) (pointBranch z (k+1))) -
-      spatialAverage u (dyadicCell k (pointBranch z k))| ≤
-      12*Real.sqrt (cellEnergy D k (pointBranch z k)) := by
-  have hs₁ := branchIndex_step z.re k
-  have hs₂ := branchIndex_step (z.im/Real.pi) k
-  have hh := local_means_le hu
-    (cellOrigin k (branchIndex z.re k)) (Real.pi*cellOrigin k (branchIndex (z.im/Real.pi) k))
-    (scaleLength k) (scaleLength_pos k)
-    (cellOrigin_nonneg ..) (cellOrigin_right ..)
-    (mul_nonneg Real.pi_pos.le (cellOrigin_nonneg ..))
-    (by nlinarith [cellOrigin_right k (branchIndex (z.im/Real.pi) k), Real.pi_pos])
-    (cellOrigin (k+1) (branchIndex z.re (k+1)))
-    (Real.pi*cellOrigin (k+1) (branchIndex (z.im/Real.pi) (k+1)))
-    hs₁.1 hs₁.2 (mul_le_mul_of_nonneg_left hs₂.1 Real.pi_pos.le)
-    (by have := mul_le_mul_of_nonneg_left hs₂.2 Real.pi_pos.le; nlinarith)
-  simpa only [dyadicCell, pointBranch, cellEnergy, dyadicInterior, scaleLength_succ] using hh
-
-lemma sum_cellEnergy_le {D : ℂ → ℂ} (hD : MemLp D 2 (volume.restrict openUnitStrip)) (k : ℕ) :
-    ∑ p : Fin (2^k) × Fin (2^k), cellEnergy D k p ≤ stripDirichletEnergy D := by
-  have hs : (⋃ p : Fin (2^k) × Fin (2^k), dyadicInterior k p) ⊆ openUnitStrip :=
-    iUnion_subset (fun p => dyadicInterior_subset_unit k p)
-  have hi : IntegrableOn (fun z => ‖D z‖^2) openUnitStrip volume := hD.norm.integrable_sq
-  have he := integral_iUnion
-    (fun p : Fin (2^k) × Fin (2^k) => openStripRectangle_measurable _ _ _)
-    (dyadicInterior_disjoint k) (hi.mono_set hs)
-  rw [tsum_fintype] at he
-  change (∫ z in ⋃ p : Fin (2^k) × Fin (2^k), dyadicInterior k p, ‖D z‖^2) =
-    ∑ p : Fin (2^k) × Fin (2^k), ∫ z in dyadicInterior k p, ‖D z‖^2 at he
-  change (∑ p : Fin (2^k) × Fin (2^k), ∫ z in dyadicInterior k p, ‖D z‖^2) ≤ _
-  rw [← he]
-  exact setIntegral_mono_set hi (Eventually.of_forall (fun _ => sq_nonneg _))
-    (Eventually.of_forall (fun z hz => hs hz))
-
-lemma exists_large_increment {a : ℕ → ℝ} {H : ℝ} (hH : 0 < H)
-    (ha : Tendsto a atTop (𝓝 H)) (ha0 : a 0 ≤ 3*H/4) :
-    ∃ j : ℕ, H/(8*((j:ℝ)+1)*((j:ℝ)+2)) < |a (j+1)-a j| := by
-  by_contra! hn
-  have hpartial (n : ℕ) : a n ≤ a 0 + H/8*(1-1/((n:ℝ)+1)) := by
-    induction n with
-    | zero => norm_num
-    | succ n ih =>
-      have hstep := (le_abs_self (a (n+1)-a n)).trans (hn n)
-      have htel : H/8*(1-1/((n:ℝ)+1)) + H/(8*((n:ℝ)+1)*((n:ℝ)+2)) =
-          H/8*(1-1/(((n+1:ℕ):ℝ)+1)) := by
-        push_cast
-        field_simp
-        ring
-      linarith
-  have hb (n : ℕ) : a n ≤ a 0+H/8 := by
-    have hp : 0 ≤ H/8*(1/((n:ℝ)+1)) := by positivity
-    have hh := hpartial n
-    nlinarith
-  have hh := le_of_tendsto ha (Eventually.of_forall hb)
-  linarith
-
-lemma large_increment_energy {H e : ℝ} (hH : 0 < H) (he : 0 ≤ e) (j : ℕ)
-    (h : H/(8*((j:ℝ)+1)*((j:ℝ)+2)) ≤ 12*Real.sqrt e) :
-    H^2/(9216*((j:ℝ)+2)^4) ≤ e := by
-  have hd : 0 < 8*((j:ℝ)+1)*((j:ℝ)+2) := by positivity
-  have hh := (div_le_iff₀ hd).1 h
-  have hj : ((j:ℝ)+1)*((j:ℝ)+2) ≤ ((j:ℝ)+2)^2 := by nlinarith [Nat.cast_nonneg (α := ℝ) j]
-  have hm := mul_le_mul_of_nonneg_right hj (show 0 ≤ 96*Real.sqrt e by positivity)
-  have hlinear : H ≤ 96*((j:ℝ)+2)^2*Real.sqrt e := by nlinarith
-  have hsq := (sq_le_sq₀ hH.le (by positivity)).2 hlinear
-  rw [mul_pow, mul_pow, Real.sq_sqrt he] at hsq
-  apply (div_le_iff₀ (by positivity)).2
-  nlinarith [show (((j:ℝ)+2)^2)^2 = ((j:ℝ)+2)^4 by ring]
-
-noncomputable def exceptionalParents (D : ℂ → ℂ) (H : ℝ) (N j : ℕ) :
-    Finset (Fin (2^(N+j)) × Fin (2^(N+j))) := by
-  classical
-  exact Finset.univ.filter (fun p => H^2/(9216*((j:ℝ)+2)^4) ≤ cellEnergy D (N+j) p)
-
-lemma mem_exceptionalParents (D : ℂ → ℂ) (H : ℝ) (N j : ℕ)
-    (p : Fin (2^(N+j)) × Fin (2^(N+j))) :
-    p ∈ exceptionalParents D H N j ↔ H^2/(9216*((j:ℝ)+2)^4) ≤ cellEnergy D (N+j) p := by
-  classical
-  simp [exceptionalParents]
-
-lemma exceptional_cover {u : ℂ → ℝ} {D : ℂ → ℂ} {H : ℝ} (hH : 0 < H)
-    (hu : SourceIsStripWeakGradient u D) (hc : ContinuousOn u (stripRectangle 0 0 1))
-    (N : ℕ) (hN : ∀ p, spatialAverage u (dyadicCell N p) ≤ 3*H/4) :
-    {z | z ∈ stripRectangle 0 0 1 ∧ u z = H} ⊆
-      ⋃ j : ℕ, ⋃ p ∈ exceptionalParents D H N j, dyadicCell (N+j) p := by
-  intro z hz
-  have ht : Tendsto (fun j => spatialAverage u (dyadicCell (N+j) (pointBranch z (N+j))))
-      atTop (𝓝 H) := by
-    simpa only [Function.comp_def, Nat.add_comm, hz.2] using
-      (branch_means_tendsto hc hz.1).comp (tendsto_add_atTop_nat N)
-  obtain ⟨j,hj⟩ := exists_large_increment hH ht (by simpa using hN (pointBranch z N))
-  have hstep := branch_mean_step_le hu z (N+j)
-  have he : H^2/(9216*((j:ℝ)+2)^4) ≤ cellEnergy D (N+j) (pointBranch z (N+j)) := by
-    apply large_increment_energy hH (cellEnergy_nonneg ..) j
-    have hnj : N+(j+1) = N+j+1 := by omega
-    rw [hnj] at hj
-    exact hj.le.trans hstep
-  exact mem_iUnion.2 ⟨j, mem_iUnion.2 ⟨pointBranch z (N+j), mem_iUnion.2
-    ⟨(mem_exceptionalParents ..).2 he, pointBranch_contains hz.1 (N+j)⟩⟩⟩
-
-lemma exceptional_count {D : ℂ → ℂ} (hD : MemLp D 2 (volume.restrict openUnitStrip))
-    {H : ℝ} (hH : 0 < H) (N j : ℕ) :
-    ((exceptionalParents D H N j).card : ℝ) ≤
-      9216*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4 := by
-  classical
-  have hl := Finset.card_nsmul_le_sum (exceptionalParents D H N j)
-    (cellEnergy D (N+j)) (H^2/(9216*((j:ℝ)+2)^4))
-    (fun p hp => (mem_exceptionalParents ..).1 hp)
-  simp only [nsmul_eq_mul] at hl
-  have hs : ∑ p ∈ exceptionalParents D H N j, cellEnergy D (N+j) p ≤
-      stripDirichletEnergy D := by
-    apply le_trans _ (sum_cellEnergy_le hD (N+j))
-    exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
-      (fun p _ _ => cellEnergy_nonneg ..)
-  have hd : 0 < 9216*((j:ℝ)+2)^4 := by positivity
-  have hh : ((exceptionalParents D H N j).card : ℝ)*H^2 ≤
-      stripDirichletEnergy D*(9216*((j:ℝ)+2)^4) := by
-    apply (div_le_iff₀ hd).1
-    simpa only [mul_div_assoc] using hl.trans hs
-  calc
-    ((exceptionalParents D H N j).card : ℝ) ≤
-        (stripDirichletEnergy D*(9216*((j:ℝ)+2)^4))/H^2 :=
-      (le_div_iff₀ (sq_pos_of_pos hH)).2 hh
-    _ = 9216*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4 := by ring
-
-noncomputable def polyGeometricSum (q : ℝ) : ℝ := ∑' j : ℕ, ((j:ℝ)+2)^4*q^j
-
-lemma summable_polyGeometric {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
-    Summable (fun j : ℕ => ((j:ℝ)+2)^4*q^j) := by
-  have hs₀ : Summable (fun n : ℕ => (n:ℝ)^4*q^n) :=
-    summable_pow_mul_geometric_of_norm_lt_one 4 (by simpa [Real.norm_eq_abs, abs_of_pos hq] using hq1)
-  have hs : Summable (fun n : ℕ => ((n+2:ℕ):ℝ)^4*q^(n+2)) :=
-    (summable_nat_add_iff 2).2 hs₀
-  apply (hs.mul_right (q^2)⁻¹).congr
-  intro j
-  push_cast
-  rw [pow_add]
-  field_simp
-
-lemma polyGeometricSum_pos {q : ℝ} (hq : 0 < q) (hq1 : q < 1) : 0 < polyGeometricSum q := by
-  have hh := (summable_polyGeometric hq hq1).le_tsum 0
-    (fun j _ => by positivity : ∀ j : ℕ, j ≠ 0 → 0 ≤ ((j:ℝ)+2)^4*q^j)
-  norm_num at hh
-  exact lt_of_lt_of_le (by norm_num : (0:ℝ) < 16) hh
-
-lemma decay_factor (α : ℝ) (N j : ℕ) :
-    (2:ℝ)^(-α*((N+j:ℕ):ℝ)) = ((2:ℝ)^(-α))^N * ((2:ℝ)^(-α))^j := by
-  push_cast
-  rw [mul_add, Real.rpow_add (by norm_num), Real.rpow_mul_natCast (by norm_num),
-    Real.rpow_mul_natCast (by norm_num)]
-
-lemma summable_weighted_decay (α : ℝ) (hα : 0 < α) (N : ℕ) (A : ℝ) :
-    Summable (fun j : ℕ => A*((j:ℝ)+2)^4*(2:ℝ)^(-α*((N+j:ℕ):ℝ))) := by
-  have hq : 0 < (2:ℝ)^(-α) := Real.rpow_pos_of_pos (by norm_num) _
-  have hq1 : (2:ℝ)^(-α) < 1 :=
-    Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (by linarith)
-  apply ((summable_polyGeometric hq hq1).mul_left (A*((2:ℝ)^(-α))^N)).congr
-  intro j
-  rw [decay_factor]
-  ring
-
-lemma tsum_weighted_decay (α : ℝ) (_hα : 0 < α) (N : ℕ) (A : ℝ) :
-    (∑' j : ℕ, A*((j:ℝ)+2)^4*(2:ℝ)^(-α*((N+j:ℕ):ℝ))) =
-      A*polyGeometricSum ((2:ℝ)^(-α))*((2:ℝ)^(-α))^N := by
-  simp_rw [decay_factor]
-  have he : (fun j : ℕ => A*((j:ℝ)+2)^4*(((2:ℝ)^(-α))^N*((2:ℝ)^(-α))^j)) =
-      (fun j : ℕ => (A*((2:ℝ)^(-α))^N)*(((j:ℝ)+2)^4*((2:ℝ)^(-α))^j)) := by
-    ext j
-    ring
-  rw [he, tsum_mul_left]
-  dsimp [polyGeometricSum]
-  ring
-
-lemma finite_union_hit_bound {Ω I : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (Q : I → Set ℂ) (X : Ω → Set ℂ) (S : Finset I) {B : ℝ} (hB : 0 ≤ B)
-    (h : ∀ p ∈ S, P {ξ | (X ξ ∩ Q p).Nonempty} ≤ ENNReal.ofReal B) :
-    P (⋃ p ∈ S, {ξ | (X ξ ∩ Q p).Nonempty}) ≤ ENNReal.ofReal ((S.card:ℝ)*B) := by
-  calc
-    _ ≤ ∑ p ∈ S, P {ξ | (X ξ ∩ Q p).Nonempty} := measure_biUnion_finset_le S _
-    _ ≤ ∑ p ∈ S, ENNReal.ofReal B := Finset.sum_le_sum h
-    _ = ENNReal.ofReal ((S.card:ℝ)*B) := by
-      rw [← ENNReal.ofReal_sum_of_nonneg (fun _ _ => hB)]
-      simp
-
-lemma rectangle_bound_at_depth {u : ℂ → ℝ} {D : ℂ → ℂ} {H C0 α : ℝ}
-    (hH : 0 < H) (hC0 : 0 < C0) (hα : 0 < α)
-    (hu : SourceIsStripWeakGradient u D) (hc : ContinuousOn u (stripRectangle 0 0 1))
-    (N : ℕ) (hN : ∀ p, spatialAverage u (dyadicCell N p) ≤ 3*H/4)
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (X : Ω → Set ℂ)
-    (hhit : ∀ (k : ℕ) (p : Fin (2^k) × Fin (2^k)),
-      P {ξ | (X ξ ∩ dyadicCell k p).Nonempty} ≤ ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ)))) :
-    P {ξ | (X ξ ∩ {z | z ∈ stripRectangle 0 0 1 ∧ u z = H}).Nonempty} ≤
-      ENNReal.ofReal (9216*C0*(stripDirichletEnergy D/H^2)*
-        polyGeometricSum ((2:ℝ)^(-α))*((2:ℝ)^(-α))^N) := by
-  have hE : 0 ≤ stripDirichletEnergy D := integral_nonneg (fun _ => sq_nonneg _)
-  have hsub : {ξ | (X ξ ∩ {z | z ∈ stripRectangle 0 0 1 ∧ u z = H}).Nonempty} ⊆
-      ⋃ j : ℕ, ⋃ p ∈ exceptionalParents D H N j, {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty} := by
-    intro ξ hξ
-    obtain ⟨z, hzX, hz⟩ := hξ
-    have hz' := exceptional_cover hH hu hc N hN hz
-    simp only [mem_iUnion] at hz' ⊢
-    obtain ⟨j,p,hp,hzp⟩ := hz'
-    exact ⟨j,p,hp, z,hzX,hzp⟩
-  have hlevel (j : ℕ) :
-      P (⋃ p ∈ exceptionalParents D H N j, {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty}) ≤
-      ENNReal.ofReal (9216*C0*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4*
-        (2:ℝ)^(-α*((N+j:ℕ):ℝ))) := by
-    apply (finite_union_hit_bound P (dyadicCell (N+j)) X (exceptionalParents D H N j)
-      (by positivity) (fun p _ => hhit (N+j) p)).trans
-    apply ENNReal.ofReal_le_ofReal
-    have hh := mul_le_mul_of_nonneg_right (exceptional_count hu.2.1 hH N j)
-      (show 0 ≤ C0*(2:ℝ)^(-α*((N+j:ℕ):ℝ)) by positivity)
-    calc
-      _ ≤ (9216*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4)*
-          (C0*(2:ℝ)^(-α*((N+j:ℕ):ℝ))) := hh
-      _ = _ := by ring
-  calc
-    _ ≤ P (⋃ j : ℕ, ⋃ p ∈ exceptionalParents D H N j,
-        {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty}) := measure_mono hsub
-    _ ≤ ∑' j : ℕ, P (⋃ p ∈ exceptionalParents D H N j,
-        {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty}) := measure_iUnion_le _
-    _ ≤ ∑' j : ℕ, ENNReal.ofReal (9216*C0*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4*
-        (2:ℝ)^(-α*((N+j:ℕ):ℝ))) := ENNReal.tsum_le_tsum hlevel
-    _ = _ := by
-      rw [← ENNReal.ofReal_tsum_of_nonneg (fun _ => by positivity)
-        (summable_weighted_decay α hα N (9216*C0*(stripDirichletEnergy D/H^2))),
-        tsum_weighted_decay α hα N]
-
-noncomputable def energyDepth (H E : ℝ) : ℕ := ⌊H^2/(25600*E)⌋₊
-
-lemma energyDepth_mean_bound {H E : ℝ} (hH : 0 < H) (hE : 0 < E)
-    (hsmall : E ≤ H^2/51200) :
-    20*Real.sqrt ((((energyDepth H E:ℕ):ℝ)+1)*E) ≤ H/4 := by
-  have hf : ((energyDepth H E:ℕ):ℝ) ≤ H^2/(25600*E) := Nat.floor_le (by positivity)
-  have hm := (le_div_iff₀ (show 0 < 25600*E by positivity)).1 hf
-  have hn : 0 ≤ (((energyDepth H E:ℕ):ℝ)+1)*E := by positivity
-  change 20*Real.sqrt ((((energyDepth H E:ℕ):ℝ)+1)*E) ≤ H/4
-  apply (sq_le_sq₀ (by positivity) (by positivity)).1
-  rw [mul_pow, Real.sq_sqrt hn, div_pow]
-  norm_num
-  nlinarith [sq_nonneg H]
-
-lemma energyDepth_lower {H E : ℝ} (_hH : 0 < H) (hE : 0 < E)
-    (hsmall : E ≤ H^2/51200) :
-    H^2/(51200*E) ≤ ((energyDepth H E:ℕ):ℝ) := by
-  have hf : H^2/(25600*E) < ((energyDepth H E:ℕ):ℝ)+1 := Nat.lt_floor_add_one _
-  have hx : 2 ≤ H^2/(25600*E) := by
-    apply (le_div_iff₀ (by positivity)).2
-    linarith
-  have he : H^2/(51200*E) = (H^2/(25600*E))/2 := by ring
-  rw [he]
-  linarith
-
-lemma energyDepth_decay {H E α : ℝ} (hH : 0 < H) (hE : 0 < E) (hα : 0 < α)
-    (hsmall : E ≤ H^2/51200) :
-    ((2:ℝ)^(-α))^(energyDepth H E) ≤ Real.exp (-(α*Real.log 2/51200)*H^2/E) := by
-  rw [← Real.rpow_mul_natCast (by norm_num : (0:ℝ) ≤ 2), Real.rpow_def_of_pos (by norm_num)]
-  apply Real.exp_le_exp.mpr
-  have hn := energyDepth_lower hH hE hsmall
-  have hh := mul_le_mul_of_nonneg_left hn (show 0 ≤ α*Real.log 2 by positivity)
-  calc
-    Real.log 2 * (-α*((energyDepth H E:ℕ):ℝ)) = -(α*Real.log 2*((energyDepth H E:ℕ):ℝ)) := by ring
-    _ ≤ -(α*Real.log 2*(H^2/(51200*E))) := neg_le_neg hh
-    _ = _ := by ring
-
-universe u
-
- 
-
-def SourceStripRectangleEnergyTarget : Prop :=
-  ∀ C0 α : ℝ, 0 < C0 → 0 < α →
-    ∃ C c c0 : ℝ, 0 < C ∧ 0 < c ∧ 0 < c0 ∧
-      ∀ (H : ℝ), 0 < H → ∀ (u : ℂ → ℝ) (D : ℂ → ℂ),
-      SourceIsStripWeakGradient u D → ContinuousOn u (stripRectangle 0 0 1) →
-      (∀ z ∈ stripRectangle 0 0 1, 0 ≤ u z ∧ u z ≤ H) →
-      spatialAverage u (stripRectangle 0 0 1) ≤ H/2 →
-      ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω),
-      IsProbabilityMeasure P → ∀ X : Ω → Set ℂ,
-      (∀ (k : ℕ) (i m : Fin (2^k)),
-        P {ξ | (X ξ ∩ dyadicStripRectangle 0 k i m).Nonempty} ≤
-          ENNReal.ofReal (C0 * (2:ℝ)^(-α*(k:ℝ)))) →
-      stripDirichletEnergy D ≤ c0*H^2 →
-      P {ξ | (X ξ ∩ {z | z ∈ stripRectangle 0 0 1 ∧ u z = H}).Nonempty} ≤
-        ENNReal.ofReal (C * (stripDirichletEnergy D/H^2) *
-          Real.exp (-c*H^2/stripDirichletEnergy D))
-
-lemma critical_cell_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
-    (hu : SourceIsStripWeakGradient u D) (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
-    |spatialAverage u (dyadicCell k p) - spatialAverage u (stripRectangle 0 0 1)| ≤
-      20*Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D) := by
-  rw [dyadicCell_eq]
-  by_cases hk : k = 0
-  · subst k
-    have hi : p.1 = 0 := by apply Fin.ext; have := p.1.isLt; norm_num at this ⊢
-    have hm : p.2 = 0 := by apply Fin.ext; have := p.2.isLt; norm_num at this ⊢
-    simp [hi, hm, dyadicStripRectangle]
-  · obtain ⟨t, ht, ht1, hit⟩ := dyadic_position_param k hk p.1
-    obtain ⟨s, hs, hs1, hms⟩ := dyadic_position_param k hk p.2
-    have hh := chain_means_le hu ht ht1 hs hs1 k
-    simpa only [dyadicStripRectangle, Int.cast_zero, zero_add, hit, hms, scaleLength] using hh
-
-lemma high_set_empty_of_energy_zero {u : ℂ → ℝ} {D : ℂ → ℂ} {H : ℝ}
-    (hH : 0 < H) (hu : SourceIsStripWeakGradient u D)
-    (hc : ContinuousOn u (stripRectangle 0 0 1))
-    (hmean : spatialAverage u (stripRectangle 0 0 1) ≤ H/2)
-    (hzero : stripDirichletEnergy D = 0) :
-    {z | z ∈ stripRectangle 0 0 1 ∧ u z = H} = ∅ := by
-  apply Set.eq_empty_iff_forall_notMem.mpr
-  intro z hz
-  have hb (k : ℕ) : spatialAverage u (dyadicCell k (pointBranch z k)) ≤ H/2 := by
-    have hh := critical_cell_means_le hu k (pointBranch z k)
-    rw [hzero, mul_zero, Real.sqrt_zero, mul_zero] at hh
-    have hh' := (le_abs_self _).trans hh
-    linarith
-  have hh := le_of_tendsto (branch_means_tendsto hc hz.1) (Eventually.of_forall hb)
-  rw [hz.2] at hh
-  linarith
-
-theorem sourceStripRectangleEnergy : SourceStripRectangleEnergyTarget := by
-  intro C0 α hC0 hα
-  have hq : 0 < (2:ℝ)^(-α) := Real.rpow_pos_of_pos (by norm_num) _
-  have hq1 : (2:ℝ)^(-α) < 1 :=
-    Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (by linarith)
-  refine ⟨9216*C0*polyGeometricSum ((2:ℝ)^(-α)), α*Real.log 2/51200, 1/51200,
-    by positivity [polyGeometricSum_pos hq hq1], by positivity, by norm_num, ?_⟩
-  intro H hH u D hu hc _hrange hmean Ω mΩ P _hP X hhit hsmall
-  have hE : 0 ≤ stripDirichletEnergy D := integral_nonneg (fun _ => sq_nonneg _)
-  have hsmall' : stripDirichletEnergy D ≤ H^2/51200 := by linarith
-  by_cases hz : stripDirichletEnergy D = 0
-  · rw [high_set_empty_of_energy_zero hH hu hc hmean hz]
-    simp [hz]
-  · have hEpos : 0 < stripDirichletEnergy D := lt_of_le_of_ne hE (Ne.symm hz)
-    let N := energyDepth H (stripDirichletEnergy D)
-    have hN : ∀ p, spatialAverage u (dyadicCell N p) ≤ 3*H/4 := by
-      intro p
-      have hh := critical_cell_means_le hu N p
-      have hs := energyDepth_mean_bound hH hEpos hsmall'
-      change 20*Real.sqrt (((N:ℝ)+1)*stripDirichletEnergy D) ≤ H/4 at hs
-      have hh' := (le_abs_self _).trans (hh.trans hs)
-      linarith
-    have hh := rectangle_bound_at_depth hH hC0 hα hu hc N hN P X
-      (fun k p => by simpa only [dyadicCell_eq] using hhit k p.1 p.2)
-    apply hh.trans
-    apply ENNReal.ofReal_le_ofReal
-    have hl := mul_le_mul_of_nonneg_left (energyDepth_decay hH hEpos hα hsmall')
-      (show 0 ≤ 9216*C0*(stripDirichletEnergy D/H^2)*polyGeometricSum ((2:ℝ)^(-α)) by
-        exact mul_nonneg (by positivity) (polyGeometricSum_pos hq hq1).le)
-    calc
-      _ ≤ (9216*C0*(stripDirichletEnergy D/H^2)*polyGeometricSum ((2:ℝ)^(-α)))*
-          Real.exp (-(α*Real.log 2/51200)*H^2/stripDirichletEnergy D) := hl
-      _ = _ := by ring
-
-lemma source_local_lipschitz_weak_derivative {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
-    (hlocal : ∀ x ∈ D, ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ x ∈ U ∧ LipschitzOnWith K u U)
-    (φ : ℂ → ℝ) (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
-    (hsφ : tsupport φ ⊆ D) (v : ℂ) :
-    (∫ z in D, u z * (fderiv ℝ φ z) v) =
-      -(∫ z in D, (fderiv ℝ u z) v * φ z) := by
-  obtain ⟨S,hSc,hφS,hSD⟩ := exists_compact_between hcφ hD hsφ
-  have huS : LocallyLipschitzOn S u := by
-    intro x hx
-    obtain ⟨K,U,hUo,hxU,hU⟩ := hlocal x (hSD hx)
-    exact ⟨K,U,mem_nhdsWithin_of_mem_nhds (hUo.mem_nhds hxU),hU⟩
-  obtain ⟨K,hK⟩ := huS.exists_lipschitzOnWith_of_compact hSc
-  obtain ⟨g,hglip,hug⟩ := hK.extend_real
-  obtain ⟨C,hφlip⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hcφ hφ (by simp)
-  have hφdiff : ∀ z, DifferentiableAt ℝ φ z := fun z => (hφ.differentiable (by simp)).differentiableAt
-  have ht (z : ℂ) (hz : z ∈ tsupport φ) : u =ᶠ[𝓝 z] g := by
-    filter_upwards [mem_interior_iff_mem_nhds.mp (hφS hz)] with w hw
-    exact hug hw
-  have hid := hglip.integral_lineDeriv_mul_eq (μ := volume) hφlip hcφ v
-  have hfull : (∫ z, (fderiv ℝ g z) v * φ z) = -(∫ z, g z * (fderiv ℝ φ z) v) := by
-    calc
-      _ = ∫ z, lineDeriv ℝ g z v * φ z := by
-        apply integral_congr_ae
-        filter_upwards [hglip.ae_differentiableAt (μ := volume)] with z hz
-        rw [hz.lineDeriv_eq_fderiv]
-      _ = ∫ z, lineDeriv ℝ φ z (-v) * g z := hid
-      _ = _ := by
-        simp_rw [(hφdiff _).lineDeriv_eq_fderiv,map_neg,neg_mul,mul_comm _ (g _)]
-        rw [integral_neg]
-  have he1 : (fun z => u z * (fderiv ℝ φ z) v) = (fun z => g z * (fderiv ℝ φ z) v) := by
-    funext z
-    by_cases hz : z ∈ tsupport φ
-    · rw [hug (interior_subset (hφS hz))]
-    · simp [fderiv_of_notMem_tsupport ℝ hz]
-  have he2 : (fun z => (fderiv ℝ u z) v * φ z) = (fun z => (fderiv ℝ g z) v * φ z) := by
-    funext z
-    by_cases hz : z ∈ tsupport φ
-    · rw [(ht z hz).fderiv_eq]
-    · simp [image_eq_zero_of_notMem_tsupport hz]
-  have hs1 : (∫ z in D, u z * (fderiv ℝ φ z) v) = ∫ z, u z * (fderiv ℝ φ z) v := by
-    apply setIntegral_eq_integral_of_forall_compl_eq_zero
-    intro z hz
-    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
-    simp [fderiv_of_notMem_tsupport ℝ hn]
-  have hs2 : (∫ z in D, (fderiv ℝ u z) v * φ z) = ∫ z, (fderiv ℝ u z) v * φ z := by
-    apply setIntegral_eq_integral_of_forall_compl_eq_zero
-    intro z hz
-    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
-    simp [image_eq_zero_of_notMem_tsupport hn]
-  rw [hs1,hs2,he1,he2,hfull]
-  simp
-
-lemma real_gradient_re (u : ℂ → ℝ) (z : ℂ) : (gradient u z).re = (fderiv ℝ u z) (1:ℂ) := by
-  have hh := inner_gradient_left (f := u) (x := z) (y := (1:ℂ))
-  rw [Complex.inner] at hh
-  simpa using hh
-
-lemma real_gradient_im (u : ℂ → ℝ) (z : ℂ) : (gradient u z).im = (fderiv ℝ u z) Complex.I := by
-  have hh := inner_gradient_left (f := u) (x := z) (y := Complex.I)
-  rw [Complex.inner] at hh
-  simpa using hh
-
-lemma locallyLipschitz_source_weakGradient {u : ℂ → ℝ}
-    (hl : LocallyLipschitzOn openUnitStrip u)
-    (hu : MemLp u 2 (volume.restrict openUnitStrip))
-    (hg : MemLp (gradient u) 2 (volume.restrict openUnitStrip)) :
-    SourceIsStripWeakGradient u (gradient u) := by
-  refine ⟨hu,hg,?_⟩
-  have hlocal (z : ℂ) (hz : z ∈ openUnitStrip) :
-      ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ z ∈ U ∧ LipschitzOnWith K u U := by
-    obtain ⟨K,t,ht,htl⟩ := hl hz
-    rw [nhdsWithin_eq_nhds.mpr (openUnitStrip_isOpen.mem_nhds hz)] at ht
-    obtain ⟨U,hUt,hUo,hzU⟩ := _root_.mem_nhds_iff.mp ht
-    exact ⟨K,U,hUo,hzU,htl.mono hUt⟩
-  intro φ hφ hc hs
-  constructor
-  · simpa only [real_gradient_re] using
-      source_local_lipschitz_weak_derivative openUnitStrip_isOpen hlocal φ hφ hc hs 1
-  · simpa only [real_gradient_im] using
-      source_local_lipschitz_weak_derivative openUnitStrip_isOpen hlocal φ hφ hc hs Complex.I
-
- 
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.StripKernels
+
+namespace OAI
+
+/-! Rectangle Means for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology ContDiff ENNReal BigOperators NNReal
+namespace SLEExactGauge
+noncomputable def cellOrigin (k : ℕ) (i : Fin (2^k)) : ℝ := (i:ℝ)*scaleLength k
+noncomputable def dyadicCell (k : ℕ) (p : Fin (2^k) × Fin (2^k)) : Set ℂ :=
+  stripRectangle (cellOrigin k p.1) (Real.pi*cellOrigin k p.2) (scaleLength k)
+noncomputable def dyadicInterior (k : ℕ) (p : Fin (2^k) × Fin (2^k)) : Set ℂ :=
+  openStripRectangle (cellOrigin k p.1) (Real.pi*cellOrigin k p.2) (scaleLength k)
+
+lemma dyadicCell_eq (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
+    dyadicCell k p = dyadicStripRectangle 0 k p.1 p.2 := by
+  simp [dyadicCell, dyadicStripRectangle, cellOrigin, scaleLength, div_eq_mul_inv]
+lemma cellOrigin_nonneg (k : ℕ) (i : Fin (2^k)) : 0 ≤ cellOrigin k i := by
+  exact mul_nonneg (Nat.cast_nonneg _) (scaleLength_pos k).le
+lemma cellOrigin_right (k : ℕ) (i : Fin (2^k)) : cellOrigin k i + scaleLength k ≤ 1 := by
+  have hh : (i:ℝ)+1 ≤ (2:ℝ)^k := by exact_mod_cast i.isLt
+  dsimp [cellOrigin, scaleLength]
+  calc
+    _ = ((i:ℝ)+1)/(2:ℝ)^k := by ring
+    _ ≤ 1 := (div_le_one (by positivity)).2 hh
+lemma dyadicInterior_subset_unit (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
+    dyadicInterior k p ⊆ openUnitStrip := by
+  intro z hz
+  have hr := cellOrigin_right k p.1
+  have hi := mul_le_mul_of_nonneg_left (cellOrigin_right k p.2) Real.pi_pos.le
+  have h0 := cellOrigin_nonneg k p.1
+  have h0' := mul_nonneg Real.pi_pos.le (cellOrigin_nonneg k p.2)
+  change _ < z.re ∧ z.re < _ ∧ _ < z.im ∧ z.im < _ at hz
+  exact ⟨h0.trans_lt hz.1, hz.2.1.trans_le hr,
+    h0'.trans_lt hz.2.2.1, by nlinarith [hz.2.2.2]⟩
+lemma dyadicCell_subset_root (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
+    dyadicCell k p ⊆ stripRectangle 0 0 1 := by
+  intro z hz
+  have hr := cellOrigin_right k p.1
+  have hi := mul_le_mul_of_nonneg_left (cellOrigin_right k p.2) Real.pi_pos.le
+  have h0 := cellOrigin_nonneg k p.1
+  have h0' := mul_nonneg Real.pi_pos.le (cellOrigin_nonneg k p.2)
+  change _ ≤ z.re ∧ z.re ≤ _ ∧ _ ≤ z.im ∧ z.im ≤ _ at hz
+  change 0 ≤ z.re ∧ z.re ≤ 0+1 ∧ 0 ≤ z.im ∧ z.im ≤ 0+Real.pi*1
+  exact ⟨h0.trans hz.1, by linarith [hz.2.1],
+    h0'.trans hz.2.2.1, by nlinarith [hz.2.2.2]⟩
+
+lemma cellOrigin_separated {k : ℕ} {i j : Fin (2^k)} (hij : i < j) :
+    cellOrigin k i + scaleLength k ≤ cellOrigin k j := by
+  have hh : (i:ℝ)+1 ≤ (j:ℝ) := by exact_mod_cast hij
+  have hm := mul_le_mul_of_nonneg_right hh (scaleLength_pos k).le
+  dsimp [cellOrigin]
+  nlinarith
+
+lemma dyadicInterior_disjoint (k : ℕ) : Pairwise (fun p q => Disjoint (dyadicInterior k p) (dyadicInterior k q)) := by
+  intro p q hpq
+  rw [Set.disjoint_left]
+  intro z hz hz'
+  change _ < z.re ∧ z.re < _ ∧ _ < z.im ∧ z.im < _ at hz hz'
+  by_cases h : p.1 = q.1
+  · have hne : p.2 ≠ q.2 := by intro h'; exact hpq (Prod.ext h h')
+    rcases lt_or_gt_of_ne hne with hlt | hlt
+    · have hh := mul_le_mul_of_nonneg_left (cellOrigin_separated hlt) Real.pi_pos.le
+      nlinarith [hz.2.2.2, hz'.2.2.1]
+    · have hh := mul_le_mul_of_nonneg_left (cellOrigin_separated hlt) Real.pi_pos.le
+      nlinarith [hz'.2.2.2, hz.2.2.1]
+  · rcases lt_or_gt_of_ne h with hlt | hlt
+    · have hh := cellOrigin_separated hlt
+      linarith [hz.2.1, hz'.1]
+    · have hh := cellOrigin_separated hlt
+      linarith [hz'.2.1, hz.1]
+
+def leftChild (k : ℕ) (i : Fin (2^k)) : Fin (2^(k+1)) :=
+  ⟨2*i.val, by have := i.isLt; rw [pow_succ]; omega⟩
+def rightChild (k : ℕ) (i : Fin (2^k)) : Fin (2^(k+1)) :=
+  ⟨2*i.val+1, by have := i.isLt; rw [pow_succ]; omega⟩
+lemma cellOrigin_leftChild (k : ℕ) (i : Fin (2^k)) :
+    cellOrigin (k+1) (leftChild k i) = cellOrigin k i := by
+  simp only [cellOrigin, leftChild, Fin.val_mk, Nat.cast_mul, Nat.cast_ofNat, scaleLength_succ]
+  ring
+lemma cellOrigin_rightChild (k : ℕ) (i : Fin (2^k)) :
+    cellOrigin (k+1) (rightChild k i) = cellOrigin k i + scaleLength k/2 := by
+  simp only [cellOrigin, rightChild, Fin.val_mk, Nat.cast_add, Nat.cast_mul,
+    Nat.cast_ofNat, Nat.cast_one, scaleLength_succ]
+  ring
+
+noncomputable def branchIndex (x : ℝ) : (k : ℕ) → Fin (2^k)
+  | 0 => 0
+  | k+1 => if x ≤ cellOrigin k (branchIndex x k)+scaleLength k/2 then
+      leftChild k (branchIndex x k) else rightChild k (branchIndex x k)
+
+lemma branchIndex_step (x : ℝ) (k : ℕ) :
+    cellOrigin k (branchIndex x k) ≤ cellOrigin (k+1) (branchIndex x (k+1)) ∧
+    cellOrigin (k+1) (branchIndex x (k+1))+scaleLength k/2 ≤
+      cellOrigin k (branchIndex x k)+scaleLength k := by
+  rw [branchIndex]
+  split
+  · rw [cellOrigin_leftChild]
+    constructor
+    · rfl
+    · linarith [scaleLength_pos k]
+  · rw [cellOrigin_rightChild]
+    constructor <;> linarith [scaleLength_pos k]
+
+lemma branchIndex_contains {x : ℝ} (hx : 0 ≤ x) (hx1 : x ≤ 1) (k : ℕ) :
+    cellOrigin k (branchIndex x k) ≤ x ∧
+    x ≤ cellOrigin k (branchIndex x k)+scaleLength k := by
+  induction k with
+  | zero => simpa [branchIndex, cellOrigin, scaleLength] using And.intro hx hx1
+  | succ k ih =>
+    rw [branchIndex]
+    split_ifs with h
+    · rw [cellOrigin_leftChild, scaleLength_succ]
+      exact ⟨ih.1,h⟩
+    · rw [cellOrigin_rightChild, scaleLength_succ]
+      exact ⟨(not_le.mp h).le, by linarith [ih.2]⟩
+
+noncomputable def pointBranch (z : ℂ) (k : ℕ) : Fin (2^k) × Fin (2^k) :=
+  (branchIndex z.re k, branchIndex (z.im/Real.pi) k)
+lemma pointBranch_contains {z : ℂ} (hz : z ∈ stripRectangle 0 0 1) (k : ℕ) :
+    z ∈ dyadicCell k (pointBranch z k) := by
+  have hzr : 0 ≤ z.re ∧ z.re ≤ 1 := by simpa [stripRectangle] using And.intro hz.1 hz.2.1
+  have hzi : 0 ≤ z.im/Real.pi ∧ z.im/Real.pi ≤ 1 := by
+    refine ⟨div_nonneg hz.2.2.1 Real.pi_pos.le, (div_le_one Real.pi_pos).2 ?_⟩
+    simpa using hz.2.2.2
+  have hr := branchIndex_contains hzr.1 hzr.2 k
+  have hi := branchIndex_contains hzi.1 hzi.2 k
+  have hi₁ := mul_le_mul_of_nonneg_left hi.1 Real.pi_pos.le
+  have hi₂ := mul_le_mul_of_nonneg_left hi.2 Real.pi_pos.le
+  rw [mul_div_cancel₀ _ Real.pi_ne_zero] at hi₁ hi₂
+  change _ ≤ z.re ∧ z.re ≤ _ ∧ Real.pi*cellOrigin k (branchIndex (z.im/Real.pi) k) ≤ z.im ∧
+    z.im ≤ Real.pi*cellOrigin k (branchIndex (z.im/Real.pi) k)+Real.pi*scaleLength k
+  exact ⟨hr.1, hr.2, hi₁, by nlinarith⟩
+
+lemma stripRectangle_convex (a b l : ℝ) : Convex ℝ (stripRectangle a b l) := by
+  have he : stripRectangle a b l =
+      Complex.reCLM ⁻¹' Icc a (a+l) ∩ Complex.imCLM ⁻¹' Icc b (b+Real.pi*l) := by
+    ext z
+    simp only [stripRectangle, Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_preimage,
+      Set.mem_Icc, Complex.reCLM_apply, Complex.imCLM_apply]
+    tauto
+  rw [he]
+  exact ((convex_Icc a (a+l)).linear_preimage Complex.reCLM.toLinearMap).inter
+    ((convex_Icc b (b+Real.pi*l)).linear_preimage Complex.imCLM.toLinearMap)
+
+lemma stripRectangle_compact (a b l : ℝ) : IsCompact (stripRectangle a b l) := by
+  rw [stripRectangle_eq_preimage]
+  change IsCompact (Complex.equivRealProdCLM.toHomeomorph ⁻¹' _)
+  exact Complex.equivRealProdCLM.toHomeomorph.isCompact_preimage.mpr
+    (isCompact_Icc.prod isCompact_Icc)
+
+lemma stripRectangle_nonempty (a b l : ℝ) (hl : 0 ≤ l) : (stripRectangle a b l).Nonempty := by
+  refine ⟨⟨a,b⟩, ?_⟩
+  change a ≤ a ∧ a ≤ a+l ∧ b ≤ b ∧ b ≤ b+Real.pi*l
+  exact ⟨le_rfl,le_add_of_nonneg_right hl,le_rfl,
+    le_add_of_nonneg_right (mul_nonneg Real.pi_pos.le hl)⟩
+
+lemma spatialAverage_eq_setAverage (u : ℂ → ℝ) (Q : Set ℂ) :
+    spatialAverage u Q = ⨍ z in Q, u z := by
+  rw [spatialAverage, setAverage_eq, Measure.real]
+  simp only [smul_eq_mul, div_eq_mul_inv, mul_comm]
+
+lemma stripRectangle_average_attained {u : ℂ → ℝ} {a b l : ℝ}
+    (hu : ContinuousOn u (stripRectangle a b l)) (hl : 0 < l) :
+    ∃ z ∈ stripRectangle a b l, u z = spatialAverage u (stripRectangle a b l) := by
+  rw [spatialAverage_eq_setAverage]
+  apply exists_eq_setAverage ((stripRectangle_convex a b l).isConnected
+    (stripRectangle_nonempty a b l hl.le)) hu
+    (hu.integrableOn_compact (stripRectangle_compact a b l))
+    (stripRectangle_compact a b l).measure_lt_top.ne
+  rw [stripRectangle_volume]
+  exact mul_ne_zero (ENNReal.ofReal_ne_zero_iff.mpr hl)
+    (ENNReal.ofReal_ne_zero_iff.mpr (mul_pos Real.pi_pos hl))
+
+lemma dyadicCell_dist_le {k : ℕ} {p : Fin (2^k) × Fin (2^k)} {z w : ℂ}
+    (hz : z ∈ dyadicCell k p) (hw : w ∈ dyadicCell k p) :
+    dist z w ≤ 5*scaleLength k := by
+  change _ ≤ z.re ∧ z.re ≤ _ ∧ _ ≤ z.im ∧ z.im ≤ _ at hz
+  change _ ≤ w.re ∧ w.re ≤ _ ∧ _ ≤ w.im ∧ w.im ≤ _ at hw
+  have hre : |z.re-w.re| ≤ scaleLength k := by rw [abs_le]; constructor <;> linarith
+  have him : |z.im-w.im| ≤ Real.pi*scaleLength k := by rw [abs_le]; constructor <;> linarith
+  rw [dist_eq_norm]
+  have hn := Complex.norm_le_abs_re_add_abs_im (z-w)
+  simp only [Complex.sub_re, Complex.sub_im] at hn
+  nlinarith [scaleLength_pos k, Real.pi_lt_four]
+
+lemma scaleLength_tendsto : Tendsto scaleLength atTop (𝓝 0) := by
+  have hh := tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2)
+    (by norm_num : (1:ℝ)/2 < 1)
+  change Tendsto (fun n => 1/(2:ℝ)^n) atTop (𝓝 0)
+  simpa only [div_pow, one_pow] using hh
+
+lemma branch_means_tendsto {u : ℂ → ℝ} (hu : ContinuousOn u (stripRectangle 0 0 1))
+    {z : ℂ} (hz : z ∈ stripRectangle 0 0 1) :
+    Tendsto (fun k => spatialAverage u (dyadicCell k (pointBranch z k))) atTop (𝓝 (u z)) := by
+  rw [Metric.tendsto_atTop]
+  intro ε hε
+  obtain ⟨δ, hδ, huδ⟩ := (Metric.continuousWithinAt_iff.1 (hu z hz)) ε hε
+  have hs : ∀ᶠ k in atTop, 5*scaleLength k < δ :=
+    ((scaleLength_tendsto.const_mul 5).eventually (eventually_lt_nhds (by simpa using hδ)))
+  obtain ⟨N,hN⟩ := eventually_atTop.1 hs
+  refine ⟨N,fun k hk => ?_⟩
+  obtain ⟨w,hw,hew⟩ := stripRectangle_average_attained
+    (hu.mono (dyadicCell_subset_root k (pointBranch z k))) (scaleLength_pos k)
+  change u w = spatialAverage u (dyadicCell k (pointBranch z k)) at hew
+  rw [← hew]
+  apply huδ (dyadicCell_subset_root k (pointBranch z k) hw)
+  exact (dyadicCell_dist_le hw (pointBranch_contains hz k)).trans_lt (hN k hk)
+
+noncomputable def cellEnergy (D : ℂ → ℂ) (k : ℕ) (p : Fin (2^k) × Fin (2^k)) : ℝ :=
+  ∫ z in dyadicInterior k p, ‖D z‖^2
+lemma cellEnergy_nonneg (D : ℂ → ℂ) (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
+    0 ≤ cellEnergy D k p := integral_nonneg (fun _ => sq_nonneg _)
+
+lemma branch_mean_step_le {u : ℂ → ℝ} {D : ℂ → ℂ}
+    (hu : SourceIsStripWeakGradient u D) (z : ℂ) (k : ℕ) :
+    |spatialAverage u (dyadicCell (k+1) (pointBranch z (k+1))) -
+      spatialAverage u (dyadicCell k (pointBranch z k))| ≤
+      12*Real.sqrt (cellEnergy D k (pointBranch z k)) := by
+  have hs₁ := branchIndex_step z.re k
+  have hs₂ := branchIndex_step (z.im/Real.pi) k
+  have hh := local_means_le hu
+    (cellOrigin k (branchIndex z.re k)) (Real.pi*cellOrigin k (branchIndex (z.im/Real.pi) k))
+    (scaleLength k) (scaleLength_pos k)
+    (cellOrigin_nonneg ..) (cellOrigin_right ..)
+    (mul_nonneg Real.pi_pos.le (cellOrigin_nonneg ..))
+    (by nlinarith [cellOrigin_right k (branchIndex (z.im/Real.pi) k), Real.pi_pos])
+    (cellOrigin (k+1) (branchIndex z.re (k+1)))
+    (Real.pi*cellOrigin (k+1) (branchIndex (z.im/Real.pi) (k+1)))
+    hs₁.1 hs₁.2 (mul_le_mul_of_nonneg_left hs₂.1 Real.pi_pos.le)
+    (by have := mul_le_mul_of_nonneg_left hs₂.2 Real.pi_pos.le; nlinarith)
+  simpa only [dyadicCell, pointBranch, cellEnergy, dyadicInterior, scaleLength_succ] using hh
+
+lemma sum_cellEnergy_le {D : ℂ → ℂ} (hD : MemLp D 2 (volume.restrict openUnitStrip)) (k : ℕ) :
+    ∑ p : Fin (2^k) × Fin (2^k), cellEnergy D k p ≤ stripDirichletEnergy D := by
+  have hs : (⋃ p : Fin (2^k) × Fin (2^k), dyadicInterior k p) ⊆ openUnitStrip :=
+    iUnion_subset (fun p => dyadicInterior_subset_unit k p)
+  have hi : IntegrableOn (fun z => ‖D z‖^2) openUnitStrip volume := hD.norm.integrable_sq
+  have he := integral_iUnion
+    (fun p : Fin (2^k) × Fin (2^k) => openStripRectangle_measurable _ _ _)
+    (dyadicInterior_disjoint k) (hi.mono_set hs)
+  rw [tsum_fintype] at he
+  change (∫ z in ⋃ p : Fin (2^k) × Fin (2^k), dyadicInterior k p, ‖D z‖^2) =
+    ∑ p : Fin (2^k) × Fin (2^k), ∫ z in dyadicInterior k p, ‖D z‖^2 at he
+  change (∑ p : Fin (2^k) × Fin (2^k), ∫ z in dyadicInterior k p, ‖D z‖^2) ≤ _
+  rw [← he]
+  exact setIntegral_mono_set hi (Eventually.of_forall (fun _ => sq_nonneg _))
+    (Eventually.of_forall (fun z hz => hs hz))
+
+lemma exists_large_increment {a : ℕ → ℝ} {H : ℝ} (hH : 0 < H)
+    (ha : Tendsto a atTop (𝓝 H)) (ha0 : a 0 ≤ 3*H/4) :
+    ∃ j : ℕ, H/(8*((j:ℝ)+1)*((j:ℝ)+2)) < |a (j+1)-a j| := by
+  by_contra! hn
+  have hpartial (n : ℕ) : a n ≤ a 0 + H/8*(1-1/((n:ℝ)+1)) := by
+    induction n with
+    | zero => norm_num
+    | succ n ih =>
+      have hstep := (le_abs_self (a (n+1)-a n)).trans (hn n)
+      have htel : H/8*(1-1/((n:ℝ)+1)) + H/(8*((n:ℝ)+1)*((n:ℝ)+2)) =
+          H/8*(1-1/(((n+1:ℕ):ℝ)+1)) := by
+        push_cast
+        field_simp
+        ring
+      linarith
+  have hb (n : ℕ) : a n ≤ a 0+H/8 := by
+    have hp : 0 ≤ H/8*(1/((n:ℝ)+1)) := by positivity
+    have hh := hpartial n
+    nlinarith
+  have hh := le_of_tendsto ha (Eventually.of_forall hb)
+  linarith
+
+lemma large_increment_energy {H e : ℝ} (hH : 0 < H) (he : 0 ≤ e) (j : ℕ)
+    (h : H/(8*((j:ℝ)+1)*((j:ℝ)+2)) ≤ 12*Real.sqrt e) :
+    H^2/(9216*((j:ℝ)+2)^4) ≤ e := by
+  have hd : 0 < 8*((j:ℝ)+1)*((j:ℝ)+2) := by positivity
+  have hh := (div_le_iff₀ hd).1 h
+  have hj : ((j:ℝ)+1)*((j:ℝ)+2) ≤ ((j:ℝ)+2)^2 := by nlinarith [Nat.cast_nonneg (α := ℝ) j]
+  have hm := mul_le_mul_of_nonneg_right hj (show 0 ≤ 96*Real.sqrt e by positivity)
+  have hlinear : H ≤ 96*((j:ℝ)+2)^2*Real.sqrt e := by nlinarith
+  have hsq := (sq_le_sq₀ hH.le (by positivity)).2 hlinear
+  rw [mul_pow, mul_pow, Real.sq_sqrt he] at hsq
+  apply (div_le_iff₀ (by positivity)).2
+  nlinarith [show (((j:ℝ)+2)^2)^2 = ((j:ℝ)+2)^4 by ring]
+
+noncomputable def exceptionalParents (D : ℂ → ℂ) (H : ℝ) (N j : ℕ) :
+    Finset (Fin (2^(N+j)) × Fin (2^(N+j))) := by
+  classical
+  exact Finset.univ.filter (fun p => H^2/(9216*((j:ℝ)+2)^4) ≤ cellEnergy D (N+j) p)
+
+lemma mem_exceptionalParents (D : ℂ → ℂ) (H : ℝ) (N j : ℕ)
+    (p : Fin (2^(N+j)) × Fin (2^(N+j))) :
+    p ∈ exceptionalParents D H N j ↔ H^2/(9216*((j:ℝ)+2)^4) ≤ cellEnergy D (N+j) p := by
+  classical
+  simp [exceptionalParents]
+
+lemma exceptional_cover {u : ℂ → ℝ} {D : ℂ → ℂ} {H : ℝ} (hH : 0 < H)
+    (hu : SourceIsStripWeakGradient u D) (hc : ContinuousOn u (stripRectangle 0 0 1))
+    (N : ℕ) (hN : ∀ p, spatialAverage u (dyadicCell N p) ≤ 3*H/4) :
+    {z | z ∈ stripRectangle 0 0 1 ∧ u z = H} ⊆
+      ⋃ j : ℕ, ⋃ p ∈ exceptionalParents D H N j, dyadicCell (N+j) p := by
+  intro z hz
+  have ht : Tendsto (fun j => spatialAverage u (dyadicCell (N+j) (pointBranch z (N+j))))
+      atTop (𝓝 H) := by
+    simpa only [Function.comp_def, Nat.add_comm, hz.2] using
+      (branch_means_tendsto hc hz.1).comp (tendsto_add_atTop_nat N)
+  obtain ⟨j,hj⟩ := exists_large_increment hH ht (by simpa using hN (pointBranch z N))
+  have hstep := branch_mean_step_le hu z (N+j)
+  have he : H^2/(9216*((j:ℝ)+2)^4) ≤ cellEnergy D (N+j) (pointBranch z (N+j)) := by
+    apply large_increment_energy hH (cellEnergy_nonneg ..) j
+    have hnj : N+(j+1) = N+j+1 := by omega
+    rw [hnj] at hj
+    exact hj.le.trans hstep
+  exact mem_iUnion.2 ⟨j, mem_iUnion.2 ⟨pointBranch z (N+j), mem_iUnion.2
+    ⟨(mem_exceptionalParents ..).2 he, pointBranch_contains hz.1 (N+j)⟩⟩⟩
+
+lemma exceptional_count {D : ℂ → ℂ} (hD : MemLp D 2 (volume.restrict openUnitStrip))
+    {H : ℝ} (hH : 0 < H) (N j : ℕ) :
+    ((exceptionalParents D H N j).card : ℝ) ≤
+      9216*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4 := by
+  classical
+  have hl := Finset.card_nsmul_le_sum (exceptionalParents D H N j)
+    (cellEnergy D (N+j)) (H^2/(9216*((j:ℝ)+2)^4))
+    (fun p hp => (mem_exceptionalParents ..).1 hp)
+  simp only [nsmul_eq_mul] at hl
+  have hs : ∑ p ∈ exceptionalParents D H N j, cellEnergy D (N+j) p ≤
+      stripDirichletEnergy D := by
+    apply le_trans _ (sum_cellEnergy_le hD (N+j))
+    exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _)
+      (fun p _ _ => cellEnergy_nonneg ..)
+  have hd : 0 < 9216*((j:ℝ)+2)^4 := by positivity
+  have hh : ((exceptionalParents D H N j).card : ℝ)*H^2 ≤
+      stripDirichletEnergy D*(9216*((j:ℝ)+2)^4) := by
+    apply (div_le_iff₀ hd).1
+    simpa only [mul_div_assoc] using hl.trans hs
+  calc
+    ((exceptionalParents D H N j).card : ℝ) ≤
+        (stripDirichletEnergy D*(9216*((j:ℝ)+2)^4))/H^2 :=
+      (le_div_iff₀ (sq_pos_of_pos hH)).2 hh
+    _ = 9216*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4 := by ring
+
+noncomputable def polyGeometricSum (q : ℝ) : ℝ := ∑' j : ℕ, ((j:ℝ)+2)^4*q^j
+
+lemma summable_polyGeometric {q : ℝ} (hq : 0 < q) (hq1 : q < 1) :
+    Summable (fun j : ℕ => ((j:ℝ)+2)^4*q^j) := by
+  have hs₀ : Summable (fun n : ℕ => (n:ℝ)^4*q^n) :=
+    summable_pow_mul_geometric_of_norm_lt_one 4 (by simpa [Real.norm_eq_abs, abs_of_pos hq] using hq1)
+  have hs : Summable (fun n : ℕ => ((n+2:ℕ):ℝ)^4*q^(n+2)) :=
+    (summable_nat_add_iff 2).2 hs₀
+  apply (hs.mul_right (q^2)⁻¹).congr
+  intro j
+  push_cast
+  rw [pow_add]
+  field_simp
+
+lemma polyGeometricSum_pos {q : ℝ} (hq : 0 < q) (hq1 : q < 1) : 0 < polyGeometricSum q := by
+  have hh := (summable_polyGeometric hq hq1).le_tsum 0
+    (fun j _ => by positivity : ∀ j : ℕ, j ≠ 0 → 0 ≤ ((j:ℝ)+2)^4*q^j)
+  norm_num at hh
+  exact lt_of_lt_of_le (by norm_num : (0:ℝ) < 16) hh
+
+lemma decay_factor (α : ℝ) (N j : ℕ) :
+    (2:ℝ)^(-α*((N+j:ℕ):ℝ)) = ((2:ℝ)^(-α))^N * ((2:ℝ)^(-α))^j := by
+  push_cast
+  rw [mul_add, Real.rpow_add (by norm_num), Real.rpow_mul_natCast (by norm_num),
+    Real.rpow_mul_natCast (by norm_num)]
+
+lemma summable_weighted_decay (α : ℝ) (hα : 0 < α) (N : ℕ) (A : ℝ) :
+    Summable (fun j : ℕ => A*((j:ℝ)+2)^4*(2:ℝ)^(-α*((N+j:ℕ):ℝ))) := by
+  have hq : 0 < (2:ℝ)^(-α) := Real.rpow_pos_of_pos (by norm_num) _
+  have hq1 : (2:ℝ)^(-α) < 1 :=
+    Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (by linarith)
+  apply ((summable_polyGeometric hq hq1).mul_left (A*((2:ℝ)^(-α))^N)).congr
+  intro j
+  rw [decay_factor]
+  ring
+
+lemma tsum_weighted_decay (α : ℝ) (_hα : 0 < α) (N : ℕ) (A : ℝ) :
+    (∑' j : ℕ, A*((j:ℝ)+2)^4*(2:ℝ)^(-α*((N+j:ℕ):ℝ))) =
+      A*polyGeometricSum ((2:ℝ)^(-α))*((2:ℝ)^(-α))^N := by
+  simp_rw [decay_factor]
+  have he : (fun j : ℕ => A*((j:ℝ)+2)^4*(((2:ℝ)^(-α))^N*((2:ℝ)^(-α))^j)) =
+      (fun j : ℕ => (A*((2:ℝ)^(-α))^N)*(((j:ℝ)+2)^4*((2:ℝ)^(-α))^j)) := by
+    ext j
+    ring
+  rw [he, tsum_mul_left]
+  dsimp [polyGeometricSum]
+  ring
+
+lemma finite_union_hit_bound {Ω I : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (Q : I → Set ℂ) (X : Ω → Set ℂ) (S : Finset I) {B : ℝ} (hB : 0 ≤ B)
+    (h : ∀ p ∈ S, P {ξ | (X ξ ∩ Q p).Nonempty} ≤ ENNReal.ofReal B) :
+    P (⋃ p ∈ S, {ξ | (X ξ ∩ Q p).Nonempty}) ≤ ENNReal.ofReal ((S.card:ℝ)*B) := by
+  calc
+    _ ≤ ∑ p ∈ S, P {ξ | (X ξ ∩ Q p).Nonempty} := measure_biUnion_finset_le S _
+    _ ≤ ∑ p ∈ S, ENNReal.ofReal B := Finset.sum_le_sum h
+    _ = ENNReal.ofReal ((S.card:ℝ)*B) := by
+      rw [← ENNReal.ofReal_sum_of_nonneg (fun _ _ => hB)]
+      simp
+
+lemma rectangle_bound_at_depth {u : ℂ → ℝ} {D : ℂ → ℂ} {H C0 α : ℝ}
+    (hH : 0 < H) (hC0 : 0 < C0) (hα : 0 < α)
+    (hu : SourceIsStripWeakGradient u D) (hc : ContinuousOn u (stripRectangle 0 0 1))
+    (N : ℕ) (hN : ∀ p, spatialAverage u (dyadicCell N p) ≤ 3*H/4)
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (X : Ω → Set ℂ)
+    (hhit : ∀ (k : ℕ) (p : Fin (2^k) × Fin (2^k)),
+      P {ξ | (X ξ ∩ dyadicCell k p).Nonempty} ≤ ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ)))) :
+    P {ξ | (X ξ ∩ {z | z ∈ stripRectangle 0 0 1 ∧ u z = H}).Nonempty} ≤
+      ENNReal.ofReal (9216*C0*(stripDirichletEnergy D/H^2)*
+        polyGeometricSum ((2:ℝ)^(-α))*((2:ℝ)^(-α))^N) := by
+  have hE : 0 ≤ stripDirichletEnergy D := integral_nonneg (fun _ => sq_nonneg _)
+  have hsub : {ξ | (X ξ ∩ {z | z ∈ stripRectangle 0 0 1 ∧ u z = H}).Nonempty} ⊆
+      ⋃ j : ℕ, ⋃ p ∈ exceptionalParents D H N j, {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty} := by
+    intro ξ hξ
+    obtain ⟨z, hzX, hz⟩ := hξ
+    have hz' := exceptional_cover hH hu hc N hN hz
+    simp only [mem_iUnion] at hz' ⊢
+    obtain ⟨j,p,hp,hzp⟩ := hz'
+    exact ⟨j,p,hp, z,hzX,hzp⟩
+  have hlevel (j : ℕ) :
+      P (⋃ p ∈ exceptionalParents D H N j, {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty}) ≤
+      ENNReal.ofReal (9216*C0*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4*
+        (2:ℝ)^(-α*((N+j:ℕ):ℝ))) := by
+    apply (finite_union_hit_bound P (dyadicCell (N+j)) X (exceptionalParents D H N j)
+      (by positivity) (fun p _ => hhit (N+j) p)).trans
+    apply ENNReal.ofReal_le_ofReal
+    have hh := mul_le_mul_of_nonneg_right (exceptional_count hu.2.1 hH N j)
+      (show 0 ≤ C0*(2:ℝ)^(-α*((N+j:ℕ):ℝ)) by positivity)
+    calc
+      _ ≤ (9216*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4)*
+          (C0*(2:ℝ)^(-α*((N+j:ℕ):ℝ))) := hh
+      _ = _ := by ring
+  calc
+    _ ≤ P (⋃ j : ℕ, ⋃ p ∈ exceptionalParents D H N j,
+        {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty}) := measure_mono hsub
+    _ ≤ ∑' j : ℕ, P (⋃ p ∈ exceptionalParents D H N j,
+        {ξ | (X ξ ∩ dyadicCell (N+j) p).Nonempty}) := measure_iUnion_le _
+    _ ≤ ∑' j : ℕ, ENNReal.ofReal (9216*C0*(stripDirichletEnergy D/H^2)*((j:ℝ)+2)^4*
+        (2:ℝ)^(-α*((N+j:ℕ):ℝ))) := ENNReal.tsum_le_tsum hlevel
+    _ = _ := by
+      rw [← ENNReal.ofReal_tsum_of_nonneg (fun _ => by positivity)
+        (summable_weighted_decay α hα N (9216*C0*(stripDirichletEnergy D/H^2))),
+        tsum_weighted_decay α hα N]
+
+noncomputable def energyDepth (H E : ℝ) : ℕ := ⌊H^2/(25600*E)⌋₊
+
+lemma energyDepth_mean_bound {H E : ℝ} (hH : 0 < H) (hE : 0 < E)
+    (hsmall : E ≤ H^2/51200) :
+    20*Real.sqrt ((((energyDepth H E:ℕ):ℝ)+1)*E) ≤ H/4 := by
+  have hf : ((energyDepth H E:ℕ):ℝ) ≤ H^2/(25600*E) := Nat.floor_le (by positivity)
+  have hm := (le_div_iff₀ (show 0 < 25600*E by positivity)).1 hf
+  have hn : 0 ≤ (((energyDepth H E:ℕ):ℝ)+1)*E := by positivity
+  change 20*Real.sqrt ((((energyDepth H E:ℕ):ℝ)+1)*E) ≤ H/4
+  apply (sq_le_sq₀ (by positivity) (by positivity)).1
+  rw [mul_pow, Real.sq_sqrt hn, div_pow]
+  norm_num
+  nlinarith [sq_nonneg H]
+
+lemma energyDepth_lower {H E : ℝ} (_hH : 0 < H) (hE : 0 < E)
+    (hsmall : E ≤ H^2/51200) :
+    H^2/(51200*E) ≤ ((energyDepth H E:ℕ):ℝ) := by
+  have hf : H^2/(25600*E) < ((energyDepth H E:ℕ):ℝ)+1 := Nat.lt_floor_add_one _
+  have hx : 2 ≤ H^2/(25600*E) := by
+    apply (le_div_iff₀ (by positivity)).2
+    linarith
+  have he : H^2/(51200*E) = (H^2/(25600*E))/2 := by ring
+  rw [he]
+  linarith
+
+lemma energyDepth_decay {H E α : ℝ} (hH : 0 < H) (hE : 0 < E) (hα : 0 < α)
+    (hsmall : E ≤ H^2/51200) :
+    ((2:ℝ)^(-α))^(energyDepth H E) ≤ Real.exp (-(α*Real.log 2/51200)*H^2/E) := by
+  rw [← Real.rpow_mul_natCast (by norm_num : (0:ℝ) ≤ 2), Real.rpow_def_of_pos (by norm_num)]
+  apply Real.exp_le_exp.mpr
+  have hn := energyDepth_lower hH hE hsmall
+  have hh := mul_le_mul_of_nonneg_left hn (show 0 ≤ α*Real.log 2 by positivity)
+  calc
+    Real.log 2 * (-α*((energyDepth H E:ℕ):ℝ)) = -(α*Real.log 2*((energyDepth H E:ℕ):ℝ)) := by ring
+    _ ≤ -(α*Real.log 2*(H^2/(51200*E))) := neg_le_neg hh
+    _ = _ := by ring
+
+universe u
+
+ 
+
+def SourceStripRectangleEnergyTarget : Prop :=
+  ∀ C0 α : ℝ, 0 < C0 → 0 < α →
+    ∃ C c c0 : ℝ, 0 < C ∧ 0 < c ∧ 0 < c0 ∧
+      ∀ (H : ℝ), 0 < H → ∀ (u : ℂ → ℝ) (D : ℂ → ℂ),
+      SourceIsStripWeakGradient u D → ContinuousOn u (stripRectangle 0 0 1) →
+      (∀ z ∈ stripRectangle 0 0 1, 0 ≤ u z ∧ u z ≤ H) →
+      spatialAverage u (stripRectangle 0 0 1) ≤ H/2 →
+      ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω),
+      IsProbabilityMeasure P → ∀ X : Ω → Set ℂ,
+      (∀ (k : ℕ) (i m : Fin (2^k)),
+        P {ξ | (X ξ ∩ dyadicStripRectangle 0 k i m).Nonempty} ≤
+          ENNReal.ofReal (C0 * (2:ℝ)^(-α*(k:ℝ)))) →
+      stripDirichletEnergy D ≤ c0*H^2 →
+      P {ξ | (X ξ ∩ {z | z ∈ stripRectangle 0 0 1 ∧ u z = H}).Nonempty} ≤
+        ENNReal.ofReal (C * (stripDirichletEnergy D/H^2) *
+          Real.exp (-c*H^2/stripDirichletEnergy D))
+
+lemma critical_cell_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
+    (hu : SourceIsStripWeakGradient u D) (k : ℕ) (p : Fin (2^k) × Fin (2^k)) :
+    |spatialAverage u (dyadicCell k p) - spatialAverage u (stripRectangle 0 0 1)| ≤
+      20*Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D) := by
+  rw [dyadicCell_eq]
+  by_cases hk : k = 0
+  · subst k
+    have hi : p.1 = 0 := by apply Fin.ext; have := p.1.isLt; norm_num at this ⊢
+    have hm : p.2 = 0 := by apply Fin.ext; have := p.2.isLt; norm_num at this ⊢
+    simp [hi, hm, dyadicStripRectangle]
+  · obtain ⟨t, ht, ht1, hit⟩ := dyadic_position_param k hk p.1
+    obtain ⟨s, hs, hs1, hms⟩ := dyadic_position_param k hk p.2
+    have hh := chain_means_le hu ht ht1 hs hs1 k
+    simpa only [dyadicStripRectangle, Int.cast_zero, zero_add, hit, hms, scaleLength] using hh
+
+lemma high_set_empty_of_energy_zero {u : ℂ → ℝ} {D : ℂ → ℂ} {H : ℝ}
+    (hH : 0 < H) (hu : SourceIsStripWeakGradient u D)
+    (hc : ContinuousOn u (stripRectangle 0 0 1))
+    (hmean : spatialAverage u (stripRectangle 0 0 1) ≤ H/2)
+    (hzero : stripDirichletEnergy D = 0) :
+    {z | z ∈ stripRectangle 0 0 1 ∧ u z = H} = ∅ := by
+  apply Set.eq_empty_iff_forall_notMem.mpr
+  intro z hz
+  have hb (k : ℕ) : spatialAverage u (dyadicCell k (pointBranch z k)) ≤ H/2 := by
+    have hh := critical_cell_means_le hu k (pointBranch z k)
+    rw [hzero, mul_zero, Real.sqrt_zero, mul_zero] at hh
+    have hh' := (le_abs_self _).trans hh
+    linarith
+  have hh := le_of_tendsto (branch_means_tendsto hc hz.1) (Eventually.of_forall hb)
+  rw [hz.2] at hh
+  linarith
+
+theorem sourceStripRectangleEnergy : SourceStripRectangleEnergyTarget := by
+  intro C0 α hC0 hα
+  have hq : 0 < (2:ℝ)^(-α) := Real.rpow_pos_of_pos (by norm_num) _
+  have hq1 : (2:ℝ)^(-α) < 1 :=
+    Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (by linarith)
+  refine ⟨9216*C0*polyGeometricSum ((2:ℝ)^(-α)), α*Real.log 2/51200, 1/51200,
+    by positivity [polyGeometricSum_pos hq hq1], by positivity, by norm_num, ?_⟩
+  intro H hH u D hu hc _hrange hmean Ω mΩ P _hP X hhit hsmall
+  have hE : 0 ≤ stripDirichletEnergy D := integral_nonneg (fun _ => sq_nonneg _)
+  have hsmall' : stripDirichletEnergy D ≤ H^2/51200 := by linarith
+  by_cases hz : stripDirichletEnergy D = 0
+  · rw [high_set_empty_of_energy_zero hH hu hc hmean hz]
+    simp [hz]
+  · have hEpos : 0 < stripDirichletEnergy D := lt_of_le_of_ne hE (Ne.symm hz)
+    let N := energyDepth H (stripDirichletEnergy D)
+    have hN : ∀ p, spatialAverage u (dyadicCell N p) ≤ 3*H/4 := by
+      intro p
+      have hh := critical_cell_means_le hu N p
+      have hs := energyDepth_mean_bound hH hEpos hsmall'
+      change 20*Real.sqrt (((N:ℝ)+1)*stripDirichletEnergy D) ≤ H/4 at hs
+      have hh' := (le_abs_self _).trans (hh.trans hs)
+      linarith
+    have hh := rectangle_bound_at_depth hH hC0 hα hu hc N hN P X
+      (fun k p => by simpa only [dyadicCell_eq] using hhit k p.1 p.2)
+    apply hh.trans
+    apply ENNReal.ofReal_le_ofReal
+    have hl := mul_le_mul_of_nonneg_left (energyDepth_decay hH hEpos hα hsmall')
+      (show 0 ≤ 9216*C0*(stripDirichletEnergy D/H^2)*polyGeometricSum ((2:ℝ)^(-α)) by
+        exact mul_nonneg (by positivity) (polyGeometricSum_pos hq hq1).le)
+    calc
+      _ ≤ (9216*C0*(stripDirichletEnergy D/H^2)*polyGeometricSum ((2:ℝ)^(-α)))*
+          Real.exp (-(α*Real.log 2/51200)*H^2/stripDirichletEnergy D) := hl
+      _ = _ := by ring
+
+lemma source_local_lipschitz_weak_derivative {D : Set ℂ} (hD : IsOpen D) {u : ℂ → ℝ}
+    (hlocal : ∀ x ∈ D, ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ x ∈ U ∧ LipschitzOnWith K u U)
+    (φ : ℂ → ℝ) (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ D) (v : ℂ) :
+    (∫ z in D, u z * (fderiv ℝ φ z) v) =
+      -(∫ z in D, (fderiv ℝ u z) v * φ z) := by
+  obtain ⟨S,hSc,hφS,hSD⟩ := exists_compact_between hcφ hD hsφ
+  have huS : LocallyLipschitzOn S u := by
+    intro x hx
+    obtain ⟨K,U,hUo,hxU,hU⟩ := hlocal x (hSD hx)
+    exact ⟨K,U,mem_nhdsWithin_of_mem_nhds (hUo.mem_nhds hxU),hU⟩
+  obtain ⟨K,hK⟩ := huS.exists_lipschitzOnWith_of_compact hSc
+  obtain ⟨g,hglip,hug⟩ := hK.extend_real
+  obtain ⟨C,hφlip⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hcφ hφ (by simp)
+  have hφdiff : ∀ z, DifferentiableAt ℝ φ z := fun z => (hφ.differentiable (by simp)).differentiableAt
+  have ht (z : ℂ) (hz : z ∈ tsupport φ) : u =ᶠ[𝓝 z] g := by
+    filter_upwards [mem_interior_iff_mem_nhds.mp (hφS hz)] with w hw
+    exact hug hw
+  have hid := hglip.integral_lineDeriv_mul_eq (μ := volume) hφlip hcφ v
+  have hfull : (∫ z, (fderiv ℝ g z) v * φ z) = -(∫ z, g z * (fderiv ℝ φ z) v) := by
+    calc
+      _ = ∫ z, lineDeriv ℝ g z v * φ z := by
+        apply integral_congr_ae
+        filter_upwards [hglip.ae_differentiableAt (μ := volume)] with z hz
+        rw [hz.lineDeriv_eq_fderiv]
+      _ = ∫ z, lineDeriv ℝ φ z (-v) * g z := hid
+      _ = _ := by
+        simp_rw [(hφdiff _).lineDeriv_eq_fderiv,map_neg,neg_mul,mul_comm _ (g _)]
+        rw [integral_neg]
+  have he1 : (fun z => u z * (fderiv ℝ φ z) v) = (fun z => g z * (fderiv ℝ φ z) v) := by
+    funext z
+    by_cases hz : z ∈ tsupport φ
+    · rw [hug (interior_subset (hφS hz))]
+    · simp [fderiv_of_notMem_tsupport ℝ hz]
+  have he2 : (fun z => (fderiv ℝ u z) v * φ z) = (fun z => (fderiv ℝ g z) v * φ z) := by
+    funext z
+    by_cases hz : z ∈ tsupport φ
+    · rw [(ht z hz).fderiv_eq]
+    · simp [image_eq_zero_of_notMem_tsupport hz]
+  have hs1 : (∫ z in D, u z * (fderiv ℝ φ z) v) = ∫ z, u z * (fderiv ℝ φ z) v := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro z hz
+    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
+    simp [fderiv_of_notMem_tsupport ℝ hn]
+  have hs2 : (∫ z in D, (fderiv ℝ u z) v * φ z) = ∫ z, (fderiv ℝ u z) v * φ z := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro z hz
+    have hn : z ∉ tsupport φ := fun h => hz (hsφ h)
+    simp [image_eq_zero_of_notMem_tsupport hn]
+  rw [hs1,hs2,he1,he2,hfull]
+  simp
+
+lemma real_gradient_re (u : ℂ → ℝ) (z : ℂ) : (gradient u z).re = (fderiv ℝ u z) (1:ℂ) := by
+  have hh := inner_gradient_left (f := u) (x := z) (y := (1:ℂ))
+  rw [Complex.inner] at hh
+  simpa using hh
+
+lemma real_gradient_im (u : ℂ → ℝ) (z : ℂ) : (gradient u z).im = (fderiv ℝ u z) Complex.I := by
+  have hh := inner_gradient_left (f := u) (x := z) (y := Complex.I)
+  rw [Complex.inner] at hh
+  simpa using hh
+
+lemma locallyLipschitz_source_weakGradient {u : ℂ → ℝ}
+    (hl : LocallyLipschitzOn openUnitStrip u)
+    (hu : MemLp u 2 (volume.restrict openUnitStrip))
+    (hg : MemLp (gradient u) 2 (volume.restrict openUnitStrip)) :
+    SourceIsStripWeakGradient u (gradient u) := by
+  refine ⟨hu,hg,?_⟩
+  have hlocal (z : ℂ) (hz : z ∈ openUnitStrip) :
+      ∃ (K : ℝ≥0) (U : Set ℂ), IsOpen U ∧ z ∈ U ∧ LipschitzOnWith K u U := by
+    obtain ⟨K,t,ht,htl⟩ := hl hz
+    rw [nhdsWithin_eq_nhds.mpr (openUnitStrip_isOpen.mem_nhds hz)] at ht
+    obtain ⟨U,hUt,hUo,hzU⟩ := _root_.mem_nhds_iff.mp ht
+    exact ⟨K,U,hUo,hzU,htl.mono hUt⟩
+  intro φ hφ hc hs
+  constructor
+  · simpa only [real_gradient_re] using
+      source_local_lipschitz_weak_derivative openUnitStrip_isOpen hlocal φ hφ hc hs 1
+  · simpa only [real_gradient_im] using
+      source_local_lipschitz_weak_derivative openUnitStrip_isOpen hlocal φ hφ hc hs Complex.I
+
+ 
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/RetainedDyadicGeometry.lean b/lean/OAI/Probability/SLE/RetainedDyadicGeometry.lean
index 8c8be2bc99162218b8c15c99156aab6148e812d9..1afed174dba407c6869ada0996774c1356c0d50d 100644
--- a/lean/OAI/Probability/SLE/RetainedDyadicGeometry.lean
+++ b/lean/OAI/Probability/SLE/RetainedDyadicGeometry.lean
@@ -1,298 +1,298 @@
-import Mathlib
-import OAI.Probability.SLE.RetainedSeedMoments
-
-namespace OAI
-
-/-! Retained Dyadic Geometry for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
- 
-def retainedDyadicSquare (k : ℕ) (ij : ℤ × ℤ) : Set ℂ :=
-  {z | (ij.1 : ℝ) ≤ (2:ℝ)^k*z.re ∧ (2:ℝ)^k*z.re < (ij.1 : ℝ)+1 ∧
-    (ij.2 : ℝ) ≤ (2:ℝ)^k*z.im ∧ (2:ℝ)^k*z.im < (ij.2 : ℝ)+1}
-noncomputable def retainedDyadicAddress (k : ℕ) (z : ℂ) : ℤ × ℤ :=
-  (⌊(2:ℝ)^k*z.re⌋, ⌊(2:ℝ)^k*z.im⌋)
-noncomputable def retainedCandidates (M k : ℕ) : Finset (ℤ × ℤ) :=
-  (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)).product
-    (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k))
-
-lemma retainedSquare_measurable (k : ℕ) (ij : ℤ × ℤ) :
-    MeasurableSet (retainedDyadicSquare k ij) := by
-  have hre : Measurable (fun z : ℂ => (2:ℝ)^k*z.re) := by fun_prop
-  have him : Measurable (fun z : ℂ => (2:ℝ)^k*z.im) := by fun_prop
-  exact (measurableSet_le measurable_const hre).inter
-    ((measurableSet_lt hre measurable_const).inter
-    ((measurableSet_le measurable_const him).inter (measurableSet_lt him measurable_const)))
-
-lemma retainedSquare_iff_address (k : ℕ) (ij : ℤ × ℤ) (z : ℂ) :
-    z ∈ retainedDyadicSquare k ij ↔ retainedDyadicAddress k z = ij := by
-  simp only [retainedDyadicSquare, mem_ofPred_eq, retainedDyadicAddress,
-    Prod.ext_iff, Int.floor_eq_iff]
-  tauto
-
-lemma retainedSquare_address (k : ℕ) (z : ℂ) :
-    z ∈ retainedDyadicSquare k (retainedDyadicAddress k z) :=
-  (retainedSquare_iff_address _ _ _).mpr rfl
-
-lemma retainedAddress_mem_candidates (M k : ℕ) {z : ℂ} (hz : ‖z‖ ≤ M) :
-    retainedDyadicAddress k z ∈ retainedCandidates M k := by
-  have hn : 0 ≤ (2:ℝ)^k := by positivity
-  have hr : |z.re| ≤ (M:ℝ) := Complex.abs_re_le_norm z |>.trans hz
-  have hi : |z.im| ≤ (M:ℝ) := Complex.abs_im_le_norm z |>.trans hz
-  have bound (x : ℝ) (hx : |x| ≤ M) :
-      -(M:ℤ)*2^k ≤ ⌊(2:ℝ)^k*x⌋ ∧ ⌊(2:ℝ)^k*x⌋ ≤ (M:ℤ)*2^k := by
-    constructor
-    · apply Int.le_floor.mpr
-      push_cast
-      nlinarith [mul_le_mul_of_nonneg_left (abs_le.mp hx).1 hn]
-    · apply Int.floor_le_iff.mpr
-      push_cast
-      nlinarith [mul_le_mul_of_nonneg_left (abs_le.mp hx).2 hn]
-  change (⌊(2:ℝ)^k*z.re⌋, ⌊(2:ℝ)^k*z.im⌋) ∈
-    (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)) ×ˢ
-      (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k))
-  simpa only [Finset.mem_product, Finset.mem_Icc] using
-    And.intro (bound z.re hr) (bound z.im hi)
-
-lemma compact_retained_candidate_radius {K : Set ℂ} (hK : IsCompact K) :
-    ∃ M : ℕ, ∀ z ∈ K, ‖z‖ ≤ M := by
-  obtain ⟨R, hR⟩ := hK.isBounded.subset_closedBall (0:ℂ)
-  obtain ⟨M, hM⟩ := exists_nat_ge R
-  refine ⟨M, fun z hz => ?_⟩
-  have := hR hz
-  simpa using (Metric.mem_closedBall.mp this).trans hM
-
-lemma retainedCandidates_cover (M k : ℕ) {K : Set ℂ}
-    (hK : ∀ z ∈ K, ‖z‖ ≤ M) :
-    K ⊆ ⋃ ij ∈ (retainedCandidates M k : Set (ℤ × ℤ)), retainedDyadicSquare k ij := by
-  intro z hz
-  exact mem_iUnion.mpr ⟨retainedDyadicAddress k z, mem_iUnion.mpr
-    ⟨retainedAddress_mem_candidates M k (hK z hz), retainedSquare_address k z⟩⟩
-
-lemma retainedCandidates_card (M k : ℕ) :
-    ((retainedCandidates M k).card : ℝ) = (2*(M:ℝ)*(2:ℝ)^k+1)^2 := by
-  have hi : -(M:ℤ)*2^k ≤ (M:ℤ)*2^k + 1 := by
-    have hh : (0:ℤ) ≤ (M:ℤ)*2^k := by positivity
-    linarith
-  have he : ((Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)).card : ℝ) =
-      (M:ℝ)*(2:ℝ)^k + 1 - (-(M:ℝ)*(2:ℝ)^k) := by
-    exact_mod_cast Int.card_Icc_of_le _ _ hi
-  change (((Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)) ×ˢ
-    (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k))).card : ℝ) = _
-  rw [Finset.card_product, Nat.cast_mul, he]
-  ring
-
-lemma retainedCandidates_scale_card (M k : ℕ) :
-    ((retainedCandidates M k).card : ℝ) * ((2:ℝ)^(-(k:ℝ)))^2 ≤ (2*(M:ℝ)+1)^2 := by
-  rw [retainedCandidates_card, Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2), Real.rpow_natCast]
-  have hn : 1 ≤ (2:ℝ)^k := one_le_pow₀ (by norm_num)
-  have hn0 : 0 < (2:ℝ)^k := by positivity
-  have hinv : ((2:ℝ)^k)⁻¹ ≤ 1 := inv_le_one_of_one_le₀ hn
-  have hid : (2*(M:ℝ)*(2:ℝ)^k+1)^2 * ((2:ℝ)^k)⁻¹^2 =
-      (2*(M:ℝ) + ((2:ℝ)^k)⁻¹)^2 := by
-        field_simp
-  rw [hid]
-  gcongr
-
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter Metric Complex
-open scoped Topology
-namespace SLEExactGauge
-noncomputable section
-def sourceDyadicCenter (k : ℕ) (ij : ℤ × ℤ) : ℂ :=
-  ⟨((ij.1:ℝ)+1/2)*(2:ℝ)^(-(k:ℝ)),((ij.2:ℝ)+1/2)*(2:ℝ)^(-(k:ℝ))⟩
-lemma source_dyadic_side_eq (k : ℕ) : (2:ℝ)^(-(k:ℝ)) = (1/2:ℝ)^k := by
-  rw [Real.rpow_neg (by norm_num),Real.rpow_natCast,←inv_pow]
-  norm_num
-lemma source_dyadic_side_pos (k : ℕ) : 0 < (2:ℝ)^(-(k:ℝ)) := Real.rpow_pos_of_pos (by norm_num) _
-lemma source_dyadic_side_le (k N : ℕ) (h : k ≤ N) : (2:ℝ)^(-(N:ℝ)) ≤ (2:ℝ)^(-(k:ℝ)) := by
-  apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
-  exact neg_le_neg (Nat.cast_le.mpr h)
-lemma source_square_subset_ball (k : ℕ) (ij : ℤ × ℤ) :
-    retainedDyadicSquare k ij ⊆ ball (sourceDyadicCenter k ij) ((2:ℝ)^(-(k:ℝ))) := by
-  let l := (2:ℝ)^(-(k:ℝ))
-  have hl : 0 < l := source_dyadic_side_pos k
-  have hinv : l*(2:ℝ)^k = 1 := by
-    dsimp [l]
-    rw [Real.rpow_neg (by norm_num),Real.rpow_natCast,inv_mul_cancel₀ (by positivity)]
-  intro z ⟨hr1,hr2,hi1,hi2⟩
-  have bound (j x : ℝ) (h1 : j ≤ (2:ℝ)^k*x) (h2 : (2:ℝ)^k*x < j+1) :
-      (x-(j+1/2)*l)^2 ≤ (l/2)^2 := by
-    have ha := mul_le_mul_of_nonneg_left h1 hl.le
-    have hb := mul_lt_mul_of_pos_left h2 hl
-    have he : l*((2:ℝ)^k*x) = x := by rw [←mul_assoc,hinv,one_mul]
-    rw [he] at ha hb
-    nlinarith [sq_nonneg (x-(j+1/2)*l+l/2),sq_nonneg (x-(j+1/2)*l-l/2),
-      mul_nonneg (show 0 ≤ x-j*l by nlinarith) (show 0 ≤ (j+1)*l-x by nlinarith)]
-  have hr := bound ij.1 z.re hr1 hr2
-  have hi := bound ij.2 z.im hi1 hi2
-  have hn := norm_nonneg (z-sourceDyadicCenter k ij)
-  have hs : ‖z-sourceDyadicCenter k ij‖^2 =
-      (z.re-((ij.1:ℝ)+1/2)*l)^2+(z.im-((ij.2:ℝ)+1/2)*l)^2 := by
-    rw [Complex.sq_norm]
-    simp [Complex.normSq_apply,sourceDyadicCenter,l,pow_two]
-  rw [Metric.mem_ball,dist_eq_norm]
-  change ‖z-sourceDyadicCenter k ij‖ < l
-  nlinarith [sq_pos_of_pos hl]
-lemma source_square_center_height (z₀ : ℂ) (k : ℕ) (ij : ℤ × ℤ)
-    (hm : (closedBall z₀ 1 ∩ retainedDyadicSquare k ij).Nonempty) :
-    z₀.im-2 ≤ (sourceDyadicCenter k ij).im := by
-  obtain ⟨z,hzK,hzQ⟩ := hm
-  have hd : dist z (sourceDyadicCenter k ij) < (2:ℝ)^(-(k:ℝ)) := source_square_subset_ball k ij hzQ
-  have hs : (2:ℝ)^(-(k:ℝ)) ≤ 1 := by
-    rw [source_dyadic_side_eq]
-    exact pow_le_one₀ (by norm_num) (by norm_num)
-  have hK : dist z z₀ ≤ 1 := hzK
-  have ha : abs (z.im-z₀.im) ≤ dist z z₀ := by
-    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
-  have hb : abs (z.im-(sourceDyadicCenter k ij).im) ≤ dist z (sourceDyadicCenter k ij) := by
-    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-sourceDyadicCenter k ij)
-  have := neg_abs_le (z.im-z₀.im)
-  have := le_abs_self (z.im-(sourceDyadicCenter k ij).im)
-  linarith
-lemma source_small_dyadic_level (H : ℝ) (hH : 0 < H) :
-    ∃ kmin : ℕ, ∀ k : ℕ, kmin ≤ k →
-      4*(2:ℝ)^(-(k:ℝ)) ≤ H ∧
-      Real.exp 1*((128*Real.pi)*(4*(2:ℝ)^(-(k:ℝ)))) ≤ 2*H := by
-  have ht : Tendsto (fun k : ℕ => (2:ℝ)^(-(k:ℝ))) atTop (𝓝 0) := by
-    simp_rw [source_dyadic_side_eq]
-    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-  have he : ∀ᶠ k : ℕ in atTop, (2:ℝ)^(-(k:ℝ)) <
-      min (H/4) (H/(256*Real.pi*Real.exp 1)) :=
-    ht.eventually (gt_mem_nhds (lt_min (by positivity) (by positivity)))
-  obtain ⟨kmin,he⟩ := eventually_atTop.mp he
-  refine ⟨kmin,fun k hk => ?_⟩
-  have ha := (lt_min_iff.mp (he k hk)).1
-  have hb := (lt_min_iff.mp (he k hk)).2
-  constructor
-  · linarith
-  · have hh := (lt_div_iff₀ (by positivity : 0 < 256*Real.pi*Real.exp 1)).mp hb
-    nlinarith
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_finite_mass_le_ball {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (z₀ : ℂ) (r rf : ℝ)
-    (hr : 0 < r) (hy : 4*r ≤ z₀.im) (hf : 0 < rf) (hfr : rf ≤ r)
-    (T : ℝ≥0) (K : Set ℂ) (hK : K ⊆ ball z₀ r) (ω : a.samples) :
-    (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T K).toReal ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ r) := by
-  have hheight (z : ℂ) (hz : z ∈ ball z₀ r) : rf ≤ z.im := by
-    have ha : abs (z.im-z₀.im) ≤ dist z z₀ := by
-      simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
-    have hb := neg_abs_le (z.im-z₀.im)
-    have hd : dist z z₀ < r := hz
-    linarith
-  have hi := a.spatialGreenWeight_spatial_integrable hκ hκ8 rf hf (ball z₀ r)
-    measurableSet_ball (measure_ball_lt_top (μ := volume) (x := z₀) (r := r)).ne hheight ω
-  have hn (z : ℂ) : 0 ≤ spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z rf := by
-    dsimp [spatialGreenWeight]; split_ifs
-    · exact le_rfl
-    · exact greenDensity_nonneg (a.witness ω) κ _ z
-  have hbound : finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T K ≤
-      ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ r)) := by
-    rw [spatialRealMass,ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall hn)]
-    apply (lintegral_mono_set hK).trans
-    apply lintegral_mono
-    intro z
-    apply ENNReal.ofReal_le_ofReal
-    split_ifs
-    · exact le_rfl
-    · exact hn z
-  have hrn : 0 ≤ spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ r) :=
-    integral_nonneg hn
-  simpa only [ENNReal.toReal_ofReal hrn] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourceActualSquareTailTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-  ∃ (A c : ℝ) (kmin : ℕ), 0 < A ∧ 0 < c ∧
-  ∀ (T : ℝ≥0) (N k : ℕ), kmin ≤ k → k ≤ N → ∀ ij : ℤ × ℤ, ∀ u : ℝ, 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
-      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω)
-        ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ≤
-    ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ)))
-theorem sourceActualSquareTail_proved : SourceActualSquareTailTarget := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀
-  obtain ⟨C,c,hC,hc,ht⟩ := sourcePhysicalBallAllTail_proved κ hκ hκ8
-  let H := z₀.im-2
-  have hH : 0 < H := sub_pos.mpr hz₀
-  let G := (2:ℝ)^(-codimension κ)*H^(-codimension κ)
-  have hG : 0 < G := mul_pos (Real.rpow_pos_of_pos (by norm_num) _) (Real.rpow_pos_of_pos hH _)
-  obtain ⟨kmin,hkmin⟩ := source_small_dyadic_level H hH
-  refine ⟨C*G,c,kmin,mul_pos hC hG,hc,?_⟩
-  intro T N k hk hkN ij u hu
-  let l := (2:ℝ)^(-(k:ℝ))
-  let rf := (2:ℝ)^(-(N:ℝ))
-  have hl : 0 < l := source_dyadic_side_pos k
-  have hf : 0 < rf := source_dyadic_side_pos N
-  have hfl : rf ≤ l := source_dyadic_side_le k N hkN
-  have hpos : 0 < l^dimension κ*u := mul_pos (Real.rpow_pos_of_pos hl _) (zero_lt_one.trans_le hu)
-  by_cases hmeet : (closedBall z₀ 1 ∩ retainedDyadicSquare k ij).Nonempty
-  · let z := sourceDyadicCenter k ij
-    have hheight : H ≤ z.im := source_square_center_height z₀ k ij hmeet
-    have hy : 4*l ≤ z.im := (hkmin k hk).1.trans hheight
-    have hsmall : 1 ≤ Real.log ((2*z.im)/((128*Real.pi)*(4*l))) := by
-      apply (Real.le_log_iff_exp_le (div_pos (by linarith) (by positivity))).mpr
-      apply (le_div_iff₀ (by positivity : 0 < (128*Real.pi)*(4*l))).mpr
-      exact (hkmin k hk).2.trans (mul_le_mul_of_nonneg_left hheight (by norm_num))
-    have hgreen : initialGreen κ z ≤ G := by
-      apply (initialGreen_le_height κ (by linarith) hκ8 z (hH.trans_le hheight)).trans
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
-      exact Real.rpow_le_rpow_of_nonpos hH hheight (neg_nonpos.mpr (codimension_pos κ hκ8).le)
-    have hh := ht Ω m P hP γ a z l hl hy hsmall rf u hf hfl hu
-    apply le_trans (measure_mono (show {ω : a.samples | l^dimension κ*u <
-      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
-        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ⊆
-      {ω | l^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) rf (ball z l)} from ?_)) (hh.trans ?_)
-    · intro ω hω
-      exact hω.trans_le (source_finite_mass_le_ball a (by linarith) hκ8 z l rf hl hy hf hfl T
-        _ (fun w hw => source_square_subset_ball k ij hw.2) ω)
-    · apply ENNReal.ofReal_le_ofReal
-      calc
-        C*l^codimension κ*initialGreen κ z*Real.exp (-c*u^(2/codimension κ)) ≤
-            C*l^codimension κ*G*Real.exp (-c*u^(2/codimension κ)) := by gcongr
-        _ = _ := by ring
-  · have he : closedBall z₀ 1 ∩ retainedDyadicSquare k ij = ∅ := not_nonempty_iff_eq_empty.mp hmeet
-    have hempty : {ω : a.samples | l^dimension κ*u <
-      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
-        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} = ∅ := by
-      simp only [he,finiteSpatialGreenMass,Measure.restrict_empty,lintegral_zero_measure,ENNReal.toReal_zero,
-        not_lt_of_ge hpos.le,Set.ofPred_false]
-    change (P.comap Subtype.val) _ ≤ _
-    rw [hempty,measure_empty]
-    exact bot_le
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.RetainedSeedMoments
+
+namespace OAI
+
+/-! Retained Dyadic Geometry for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+ 
+def retainedDyadicSquare (k : ℕ) (ij : ℤ × ℤ) : Set ℂ :=
+  {z | (ij.1 : ℝ) ≤ (2:ℝ)^k*z.re ∧ (2:ℝ)^k*z.re < (ij.1 : ℝ)+1 ∧
+    (ij.2 : ℝ) ≤ (2:ℝ)^k*z.im ∧ (2:ℝ)^k*z.im < (ij.2 : ℝ)+1}
+noncomputable def retainedDyadicAddress (k : ℕ) (z : ℂ) : ℤ × ℤ :=
+  (⌊(2:ℝ)^k*z.re⌋, ⌊(2:ℝ)^k*z.im⌋)
+noncomputable def retainedCandidates (M k : ℕ) : Finset (ℤ × ℤ) :=
+  (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)).product
+    (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k))
+
+lemma retainedSquare_measurable (k : ℕ) (ij : ℤ × ℤ) :
+    MeasurableSet (retainedDyadicSquare k ij) := by
+  have hre : Measurable (fun z : ℂ => (2:ℝ)^k*z.re) := by fun_prop
+  have him : Measurable (fun z : ℂ => (2:ℝ)^k*z.im) := by fun_prop
+  exact (measurableSet_le measurable_const hre).inter
+    ((measurableSet_lt hre measurable_const).inter
+    ((measurableSet_le measurable_const him).inter (measurableSet_lt him measurable_const)))
+
+lemma retainedSquare_iff_address (k : ℕ) (ij : ℤ × ℤ) (z : ℂ) :
+    z ∈ retainedDyadicSquare k ij ↔ retainedDyadicAddress k z = ij := by
+  simp only [retainedDyadicSquare, mem_ofPred_eq, retainedDyadicAddress,
+    Prod.ext_iff, Int.floor_eq_iff]
+  tauto
+
+lemma retainedSquare_address (k : ℕ) (z : ℂ) :
+    z ∈ retainedDyadicSquare k (retainedDyadicAddress k z) :=
+  (retainedSquare_iff_address _ _ _).mpr rfl
+
+lemma retainedAddress_mem_candidates (M k : ℕ) {z : ℂ} (hz : ‖z‖ ≤ M) :
+    retainedDyadicAddress k z ∈ retainedCandidates M k := by
+  have hn : 0 ≤ (2:ℝ)^k := by positivity
+  have hr : |z.re| ≤ (M:ℝ) := Complex.abs_re_le_norm z |>.trans hz
+  have hi : |z.im| ≤ (M:ℝ) := Complex.abs_im_le_norm z |>.trans hz
+  have bound (x : ℝ) (hx : |x| ≤ M) :
+      -(M:ℤ)*2^k ≤ ⌊(2:ℝ)^k*x⌋ ∧ ⌊(2:ℝ)^k*x⌋ ≤ (M:ℤ)*2^k := by
+    constructor
+    · apply Int.le_floor.mpr
+      push_cast
+      nlinarith [mul_le_mul_of_nonneg_left (abs_le.mp hx).1 hn]
+    · apply Int.floor_le_iff.mpr
+      push_cast
+      nlinarith [mul_le_mul_of_nonneg_left (abs_le.mp hx).2 hn]
+  change (⌊(2:ℝ)^k*z.re⌋, ⌊(2:ℝ)^k*z.im⌋) ∈
+    (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)) ×ˢ
+      (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k))
+  simpa only [Finset.mem_product, Finset.mem_Icc] using
+    And.intro (bound z.re hr) (bound z.im hi)
+
+lemma compact_retained_candidate_radius {K : Set ℂ} (hK : IsCompact K) :
+    ∃ M : ℕ, ∀ z ∈ K, ‖z‖ ≤ M := by
+  obtain ⟨R, hR⟩ := hK.isBounded.subset_closedBall (0:ℂ)
+  obtain ⟨M, hM⟩ := exists_nat_ge R
+  refine ⟨M, fun z hz => ?_⟩
+  have := hR hz
+  simpa using (Metric.mem_closedBall.mp this).trans hM
+
+lemma retainedCandidates_cover (M k : ℕ) {K : Set ℂ}
+    (hK : ∀ z ∈ K, ‖z‖ ≤ M) :
+    K ⊆ ⋃ ij ∈ (retainedCandidates M k : Set (ℤ × ℤ)), retainedDyadicSquare k ij := by
+  intro z hz
+  exact mem_iUnion.mpr ⟨retainedDyadicAddress k z, mem_iUnion.mpr
+    ⟨retainedAddress_mem_candidates M k (hK z hz), retainedSquare_address k z⟩⟩
+
+lemma retainedCandidates_card (M k : ℕ) :
+    ((retainedCandidates M k).card : ℝ) = (2*(M:ℝ)*(2:ℝ)^k+1)^2 := by
+  have hi : -(M:ℤ)*2^k ≤ (M:ℤ)*2^k + 1 := by
+    have hh : (0:ℤ) ≤ (M:ℤ)*2^k := by positivity
+    linarith
+  have he : ((Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)).card : ℝ) =
+      (M:ℝ)*(2:ℝ)^k + 1 - (-(M:ℝ)*(2:ℝ)^k) := by
+    exact_mod_cast Int.card_Icc_of_le _ _ hi
+  change (((Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k)) ×ˢ
+    (Finset.Icc (-(M:ℤ)*2^k) ((M:ℤ)*2^k))).card : ℝ) = _
+  rw [Finset.card_product, Nat.cast_mul, he]
+  ring
+
+lemma retainedCandidates_scale_card (M k : ℕ) :
+    ((retainedCandidates M k).card : ℝ) * ((2:ℝ)^(-(k:ℝ)))^2 ≤ (2*(M:ℝ)+1)^2 := by
+  rw [retainedCandidates_card, Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2), Real.rpow_natCast]
+  have hn : 1 ≤ (2:ℝ)^k := one_le_pow₀ (by norm_num)
+  have hn0 : 0 < (2:ℝ)^k := by positivity
+  have hinv : ((2:ℝ)^k)⁻¹ ≤ 1 := inv_le_one_of_one_le₀ hn
+  have hid : (2*(M:ℝ)*(2:ℝ)^k+1)^2 * ((2:ℝ)^k)⁻¹^2 =
+      (2*(M:ℝ) + ((2:ℝ)^k)⁻¹)^2 := by
+        field_simp
+  rw [hid]
+  gcongr
+
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter Metric Complex
+open scoped Topology
+namespace SLEExactGauge
+noncomputable section
+def sourceDyadicCenter (k : ℕ) (ij : ℤ × ℤ) : ℂ :=
+  ⟨((ij.1:ℝ)+1/2)*(2:ℝ)^(-(k:ℝ)),((ij.2:ℝ)+1/2)*(2:ℝ)^(-(k:ℝ))⟩
+lemma source_dyadic_side_eq (k : ℕ) : (2:ℝ)^(-(k:ℝ)) = (1/2:ℝ)^k := by
+  rw [Real.rpow_neg (by norm_num),Real.rpow_natCast,←inv_pow]
+  norm_num
+lemma source_dyadic_side_pos (k : ℕ) : 0 < (2:ℝ)^(-(k:ℝ)) := Real.rpow_pos_of_pos (by norm_num) _
+lemma source_dyadic_side_le (k N : ℕ) (h : k ≤ N) : (2:ℝ)^(-(N:ℝ)) ≤ (2:ℝ)^(-(k:ℝ)) := by
+  apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
+  exact neg_le_neg (Nat.cast_le.mpr h)
+lemma source_square_subset_ball (k : ℕ) (ij : ℤ × ℤ) :
+    retainedDyadicSquare k ij ⊆ ball (sourceDyadicCenter k ij) ((2:ℝ)^(-(k:ℝ))) := by
+  let l := (2:ℝ)^(-(k:ℝ))
+  have hl : 0 < l := source_dyadic_side_pos k
+  have hinv : l*(2:ℝ)^k = 1 := by
+    dsimp [l]
+    rw [Real.rpow_neg (by norm_num),Real.rpow_natCast,inv_mul_cancel₀ (by positivity)]
+  intro z ⟨hr1,hr2,hi1,hi2⟩
+  have bound (j x : ℝ) (h1 : j ≤ (2:ℝ)^k*x) (h2 : (2:ℝ)^k*x < j+1) :
+      (x-(j+1/2)*l)^2 ≤ (l/2)^2 := by
+    have ha := mul_le_mul_of_nonneg_left h1 hl.le
+    have hb := mul_lt_mul_of_pos_left h2 hl
+    have he : l*((2:ℝ)^k*x) = x := by rw [←mul_assoc,hinv,one_mul]
+    rw [he] at ha hb
+    nlinarith [sq_nonneg (x-(j+1/2)*l+l/2),sq_nonneg (x-(j+1/2)*l-l/2),
+      mul_nonneg (show 0 ≤ x-j*l by nlinarith) (show 0 ≤ (j+1)*l-x by nlinarith)]
+  have hr := bound ij.1 z.re hr1 hr2
+  have hi := bound ij.2 z.im hi1 hi2
+  have hn := norm_nonneg (z-sourceDyadicCenter k ij)
+  have hs : ‖z-sourceDyadicCenter k ij‖^2 =
+      (z.re-((ij.1:ℝ)+1/2)*l)^2+(z.im-((ij.2:ℝ)+1/2)*l)^2 := by
+    rw [Complex.sq_norm]
+    simp [Complex.normSq_apply,sourceDyadicCenter,l,pow_two]
+  rw [Metric.mem_ball,dist_eq_norm]
+  change ‖z-sourceDyadicCenter k ij‖ < l
+  nlinarith [sq_pos_of_pos hl]
+lemma source_square_center_height (z₀ : ℂ) (k : ℕ) (ij : ℤ × ℤ)
+    (hm : (closedBall z₀ 1 ∩ retainedDyadicSquare k ij).Nonempty) :
+    z₀.im-2 ≤ (sourceDyadicCenter k ij).im := by
+  obtain ⟨z,hzK,hzQ⟩ := hm
+  have hd : dist z (sourceDyadicCenter k ij) < (2:ℝ)^(-(k:ℝ)) := source_square_subset_ball k ij hzQ
+  have hs : (2:ℝ)^(-(k:ℝ)) ≤ 1 := by
+    rw [source_dyadic_side_eq]
+    exact pow_le_one₀ (by norm_num) (by norm_num)
+  have hK : dist z z₀ ≤ 1 := hzK
+  have ha : abs (z.im-z₀.im) ≤ dist z z₀ := by
+    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
+  have hb : abs (z.im-(sourceDyadicCenter k ij).im) ≤ dist z (sourceDyadicCenter k ij) := by
+    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-sourceDyadicCenter k ij)
+  have := neg_abs_le (z.im-z₀.im)
+  have := le_abs_self (z.im-(sourceDyadicCenter k ij).im)
+  linarith
+lemma source_small_dyadic_level (H : ℝ) (hH : 0 < H) :
+    ∃ kmin : ℕ, ∀ k : ℕ, kmin ≤ k →
+      4*(2:ℝ)^(-(k:ℝ)) ≤ H ∧
+      Real.exp 1*((128*Real.pi)*(4*(2:ℝ)^(-(k:ℝ)))) ≤ 2*H := by
+  have ht : Tendsto (fun k : ℕ => (2:ℝ)^(-(k:ℝ))) atTop (𝓝 0) := by
+    simp_rw [source_dyadic_side_eq]
+    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+  have he : ∀ᶠ k : ℕ in atTop, (2:ℝ)^(-(k:ℝ)) <
+      min (H/4) (H/(256*Real.pi*Real.exp 1)) :=
+    ht.eventually (gt_mem_nhds (lt_min (by positivity) (by positivity)))
+  obtain ⟨kmin,he⟩ := eventually_atTop.mp he
+  refine ⟨kmin,fun k hk => ?_⟩
+  have ha := (lt_min_iff.mp (he k hk)).1
+  have hb := (lt_min_iff.mp (he k hk)).2
+  constructor
+  · linarith
+  · have hh := (lt_div_iff₀ (by positivity : 0 < 256*Real.pi*Real.exp 1)).mp hb
+    nlinarith
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_finite_mass_le_ball {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (z₀ : ℂ) (r rf : ℝ)
+    (hr : 0 < r) (hy : 4*r ≤ z₀.im) (hf : 0 < rf) (hfr : rf ≤ r)
+    (T : ℝ≥0) (K : Set ℂ) (hK : K ⊆ ball z₀ r) (ω : a.samples) :
+    (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T K).toReal ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ r) := by
+  have hheight (z : ℂ) (hz : z ∈ ball z₀ r) : rf ≤ z.im := by
+    have ha : abs (z.im-z₀.im) ≤ dist z z₀ := by
+      simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
+    have hb := neg_abs_le (z.im-z₀.im)
+    have hd : dist z z₀ < r := hz
+    linarith
+  have hi := a.spatialGreenWeight_spatial_integrable hκ hκ8 rf hf (ball z₀ r)
+    measurableSet_ball (measure_ball_lt_top (μ := volume) (x := z₀) (r := r)).ne hheight ω
+  have hn (z : ℂ) : 0 ≤ spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z rf := by
+    dsimp [spatialGreenWeight]; split_ifs
+    · exact le_rfl
+    · exact greenDensity_nonneg (a.witness ω) κ _ z
+  have hbound : finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T K ≤
+      ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ r)) := by
+    rw [spatialRealMass,ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall hn)]
+    apply (lintegral_mono_set hK).trans
+    apply lintegral_mono
+    intro z
+    apply ENNReal.ofReal_le_ofReal
+    split_ifs
+    · exact le_rfl
+    · exact hn z
+  have hrn : 0 ≤ spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ r) :=
+    integral_nonneg hn
+  simpa only [ENNReal.toReal_ofReal hrn] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourceActualSquareTailTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+  ∃ (A c : ℝ) (kmin : ℕ), 0 < A ∧ 0 < c ∧
+  ∀ (T : ℝ≥0) (N k : ℕ), kmin ≤ k → k ≤ N → ∀ ij : ℤ × ℤ, ∀ u : ℝ, 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
+      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω)
+        ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ≤
+    ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ)))
+theorem sourceActualSquareTail_proved : SourceActualSquareTailTarget := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀
+  obtain ⟨C,c,hC,hc,ht⟩ := sourcePhysicalBallAllTail_proved κ hκ hκ8
+  let H := z₀.im-2
+  have hH : 0 < H := sub_pos.mpr hz₀
+  let G := (2:ℝ)^(-codimension κ)*H^(-codimension κ)
+  have hG : 0 < G := mul_pos (Real.rpow_pos_of_pos (by norm_num) _) (Real.rpow_pos_of_pos hH _)
+  obtain ⟨kmin,hkmin⟩ := source_small_dyadic_level H hH
+  refine ⟨C*G,c,kmin,mul_pos hC hG,hc,?_⟩
+  intro T N k hk hkN ij u hu
+  let l := (2:ℝ)^(-(k:ℝ))
+  let rf := (2:ℝ)^(-(N:ℝ))
+  have hl : 0 < l := source_dyadic_side_pos k
+  have hf : 0 < rf := source_dyadic_side_pos N
+  have hfl : rf ≤ l := source_dyadic_side_le k N hkN
+  have hpos : 0 < l^dimension κ*u := mul_pos (Real.rpow_pos_of_pos hl _) (zero_lt_one.trans_le hu)
+  by_cases hmeet : (closedBall z₀ 1 ∩ retainedDyadicSquare k ij).Nonempty
+  · let z := sourceDyadicCenter k ij
+    have hheight : H ≤ z.im := source_square_center_height z₀ k ij hmeet
+    have hy : 4*l ≤ z.im := (hkmin k hk).1.trans hheight
+    have hsmall : 1 ≤ Real.log ((2*z.im)/((128*Real.pi)*(4*l))) := by
+      apply (Real.le_log_iff_exp_le (div_pos (by linarith) (by positivity))).mpr
+      apply (le_div_iff₀ (by positivity : 0 < (128*Real.pi)*(4*l))).mpr
+      exact (hkmin k hk).2.trans (mul_le_mul_of_nonneg_left hheight (by norm_num))
+    have hgreen : initialGreen κ z ≤ G := by
+      apply (initialGreen_le_height κ (by linarith) hκ8 z (hH.trans_le hheight)).trans
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
+      exact Real.rpow_le_rpow_of_nonpos hH hheight (neg_nonpos.mpr (codimension_pos κ hκ8).le)
+    have hh := ht Ω m P hP γ a z l hl hy hsmall rf u hf hfl hu
+    apply le_trans (measure_mono (show {ω : a.samples | l^dimension κ*u <
+      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
+        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ⊆
+      {ω | l^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) rf (ball z l)} from ?_)) (hh.trans ?_)
+    · intro ω hω
+      exact hω.trans_le (source_finite_mass_le_ball a (by linarith) hκ8 z l rf hl hy hf hfl T
+        _ (fun w hw => source_square_subset_ball k ij hw.2) ω)
+    · apply ENNReal.ofReal_le_ofReal
+      calc
+        C*l^codimension κ*initialGreen κ z*Real.exp (-c*u^(2/codimension κ)) ≤
+            C*l^codimension κ*G*Real.exp (-c*u^(2/codimension κ)) := by gcongr
+        _ = _ := by ring
+  · have he : closedBall z₀ 1 ∩ retainedDyadicSquare k ij = ∅ := not_nonempty_iff_eq_empty.mp hmeet
+    have hempty : {ω : a.samples | l^dimension κ*u <
+      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
+        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} = ∅ := by
+      simp only [he,finiteSpatialGreenMass,Measure.restrict_empty,lintegral_zero_measure,ENNReal.toReal_zero,
+        not_lt_of_ge hpos.le,Set.ofPred_false]
+    change (P.comap Subtype.val) _ ≤ _
+    rw [hempty,measure_empty]
+    exact bot_le
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/RetainedSeedMoments.lean b/lean/OAI/Probability/SLE/RetainedSeedMoments.lean
index 163982be5a108f62dd9f0c107fd51ebfd8f2d4ca..a066b49eff14051d0eb1743965233bb3af46e284 100644
--- a/lean/OAI/Probability/SLE/RetainedSeedMoments.lean
+++ b/lean/OAI/Probability/SLE/RetainedSeedMoments.lean
@@ -1,840 +1,840 @@
-import Mathlib
-import OAI.Probability.SLE.FiniteMass
-import OAI.Probability.SLE.WindowPropagation
-
-namespace OAI
-
-/-! Retained Seed Moments for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_unit_ball_height {z₀ z : ℂ} (hz₀ : 2 < z₀.im) (hz : z ∈ closedBall z₀ 1) :
-    1 < z.im := by
-  have hdist : abs (z.im-z₀.im) ≤ dist z z₀ := by
-    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
-  have hh : dist z z₀ ≤ 1 := hz
-  have ha := neg_abs_le (z.im-z₀.im)
-  linarith
-
-lemma source_open_closed_ball_ae (z₀ : ℂ) :
-    closedBall z₀ 1 =ᵐ[volume] ball z₀ 1 := by
-  have he : ∀ᵐ z : ℂ ∂volume, z ∉ sphere z₀ 1 := by
-    rw [ae_iff]
-    simpa [Metric.sphere, dist_eq_norm] using Measure.addHaar_sphere volume z₀ (1:ℝ)
-  filter_upwards [he] with z hz
-  apply propext
-  change dist z z₀ ≤ 1 ↔ dist z z₀ < 1
-  change dist z z₀ ≠ 1 at hz
-  exact le_iff_lt_or_eq.trans (or_iff_left hz)
-
-lemma source_finite_mass_le_unit_mass {Ω : Type} [MeasurableSpace Ω]
-    {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 2 < z₀.im) (rf : ℝ) (hf : 0 < rf) (hf1 : rf ≤ 1)
-    (T : ℝ≥0) (ω : a.samples) :
-    finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T (closedBall z₀ 1) ≤
-      ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)) := by
-  have hi := a.spatialGreenWeight_spatial_integrable hκ hκ8 rf hf
-    (ball z₀ 1) measurableSet_ball (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne
-    (fun z hz' => hf1.trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le) ω
-  rw [spatialRealMass,ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall (fun z => by
-    dsimp [spatialGreenWeight]; split_ifs
-    · exact le_rfl
-    · exact greenDensity_nonneg (a.witness ω) κ _ z))]
-  rw [finiteSpatialGreenMass,Measure.restrict_congr_set (source_open_closed_ball_ae z₀)]
-  apply lintegral_mono
-  intro z
-  apply ENNReal.ofReal_le_ofReal
-  split_ifs
-  · exact le_rfl
-  · dsimp [spatialGreenWeight]; split_ifs
-    · exact le_rfl
-    · exact greenDensity_nonneg (a.witness ω) κ _ z
-
-lemma source_unit_mesh_second_moment (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
-    ∃ d : ℝ, 0 < d ∧ d < 1 ∧
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-    ∃ m₂ : ℝ, 0 < m₂ ∧ ∀ rf : ℝ, 0 < rf → rf ≤ d →
-      (∫⁻ ω : a.samples, ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) rf (ball z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
-  obtain ⟨C,c,d,hC,hc,hd,hd1,ht⟩ := source_initial_unit_mesh_tail κ hκ hκ8
-  refine ⟨d,hd,hd1,?_⟩
-  intro Ω m P hP γ a z₀ hz
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let v := (volume (ball z₀ 1)).toReal
-  let M := max C v*d^(-codimension κ)
-  let b := c*d^(-2:ℝ)
-  let A := max 1 C*Real.exp b
-  let α := b/M
-  have hk : 0 < κ := by linarith
-  have hp : 0 < codimension κ := codimension_pos κ hκ8
-  have hp1 : codimension κ < 1 := by dsimp [codimension, dimension]; linarith
-  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hd _)
-  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hd _)
-  have hA : 0 < A := mul_pos (zero_lt_one.trans_le (le_max_left _ _)) (Real.exp_pos _)
-  have hα : 0 < α := div_pos hb hM
-  refine ⟨8*A/α^2,by positivity,?_⟩
-  intro rf hf hfd
-  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) rf (ball z₀ 1)
-  have hx (ω : a.samples) : 0 ≤ X ω ∧ X ω ≤ v*rf^(-codimension κ) :=
-    a.spatialRealMass_bound hk hκ8 rf hf (ball z₀ 1) measurableSet_ball
-      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne
-      (fun z hz' => (hfd.trans hd1.le).trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le) ω
-  have hlinear : ∀ t : ℝ, 1 ≤ t → μ {ω | M*t < X ω} ≤ ENNReal.ofReal (C*Real.exp (-b*t)) :=
-    source_mesh_tail_to_linear μ X (codimension κ) C c d v rf hp hp1.le hC hc hd
-      ENNReal.toReal_nonneg hf (fun ω => (hx ω).2)
-      (fun δ hfd hd' => ht Ω m P hP γ a z₀ hz rf δ hf hfd hd')
-  exact source_exponential_second_moment μ X (a.spatialRealMass_measurable _ _) (fun ω => (hx ω).1)
-    A α hA.le hα (source_linear_tail_all X M C b hM hC hb hlinear)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourceRetainedSeedTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (z₀ : ℂ), 2 < z₀.im →
-  ∃ (T : ℝ≥0) (e₀ m₀ m₂ : ℝ),
-    0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧ 0 < m₂ ∧
-    ∀ r : ℝ, 0 < r → r ≤ e₀ →
-      let X := fun ω : a.samples => finiteSpatialGreenMass κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r T (closedBall z₀ 1)
-      ENNReal.ofReal m₀ ≤ ∫⁻ ω, X ω ∂(P.comap Subtype.val) ∧
-      (∫⁻ ω, (X ω)^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂
-theorem sourceRetainedSeed_proved : SourceRetainedSeedTarget := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  have hK : closedBall z₀ 1 ⊆ upperHalfPlane :=
-    fun z hz' => zero_lt_one.trans (source_unit_ball_height hz hz')
-  obtain ⟨T,e₀,m₀,hT,he,hm,hfirst⟩ := a.source_finite_time_first_moment hk hκ8 z₀ 1 (by norm_num) hK
-  obtain ⟨d,hd,hd1,hsecond⟩ := source_unit_mesh_second_moment κ hκ hκ8
-  obtain ⟨m₂,hm₂,hsecond⟩ := hsecond Ω m P hP γ a z₀ hz
-  refine ⟨T,min e₀ d,m₀,m₂,hT,lt_min he hd,hm,hm₂,?_⟩
-  intro r hr hre
-  refine ⟨hfirst r hr (hre.trans (min_le_left _ _)),?_⟩
-  have hrd : r ≤ d := hre.trans (min_le_right _ _)
-  apply le_trans (lintegral_mono (fun ω => pow_le_pow_left' (source_finite_mass_le_unit_mass
-    a hk hκ8 z₀ hz r hr (hrd.trans hd1.le) T ω) 2))
-  exact hsecond r hr hrd
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_first_sphere_free {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hy : r ≤ z.im) (hs : r ≤ dist (γ 0) z)
-    (s : ℝ≥0) (hfirst : spatialHitTime γ z r = s) :
-    ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by
-  intro w hw
-  refine ⟨ball_subset_upperHalfPlane hy hw,?_⟩
-  rintro ⟨u,hu,rfl⟩
-  have hh := spatialHitTime_lt_of_dist_lt hc z r hs u hw
-  rw [hfirst] at hh
-  exact (not_lt_of_ge hu.2) (WithTop.coe_lt_coe.mp hh)
-
-lemma source_live_hit_spatial_eq {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z)
-    (s : ℝ≥0) (hh : liveSpatialHitTime γ z r = s) :
-    spatialHitTime γ z r = s := by
-  rw [liveSpatialHitTime_eq hc z r hs] at hh
-  split_ifs at hh with ht hl
-  · exact (WithTop.coe_ne_top hh.symm).elim
-  · exact hh
-  · exact (WithTop.coe_ne_top hh.symm).elim
-
-lemma source_live_collar_free {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (hy : 4 ≤ z.im) (hs : 3 ≤ dist (γ 0) z)
-    (s : ℝ≥0) (hh : liveSpatialHitTime γ z 3 = s) :
-    closedBall z 2 ⊆ survivingDomain γ s := by
-  have hm := source_liveHit_mem hc z 3 hs s hh
-  have hball := ball_subset_survivingDomain (by norm_num : (0:ℝ)<3) hm.1
-    (source_first_sphere_free hc z 3 (by linarith) hs s
-      (source_live_hit_spatial_eq hc z 3 hs s hh))
-  exact fun w hw => hball (by change dist w z < 3; have : dist w z ≤ 2 := hw; linarith)
-
-lemma source_enter_implies_live_contact {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ}
-    {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
-    (z : ℂ) (hy : 4 ≤ z.im) (t : ℝ≥0) (ht : dist (γ t) z < 3) :
-    liveSpatialHitTime γ z 3 ≠ ⊤ := by
-  have hs : 3 ≤ dist (γ 0) z := by
-    rw [h.2.1,dist_zero_left]
-    have := (le_abs_self z.im).trans (Complex.abs_im_le_norm z)
-    linarith
-  have hh := spatialHitTime_lt_of_dist_lt h.1 z 3 hs t ht
-  have hn : spatialHitTime γ z 3 ≠ ⊤ := ne_of_lt (hh.trans_le le_top)
-  obtain ⟨s,he⟩ := WithTop.ne_top_iff_exists.mp hn
-  have hst : s ≤ t := (WithTop.coe_lt_coe.mp (he ▸ hh)).le
-  have hfree := source_first_sphere_free h.1 z 3 (by linarith) hs s he.symm
-  have hl := h.surviving_of_free_ball_future_tip s t hst z 3 (by norm_num) hfree ht
-  have hm := spatialHitTime_mem h.1 z 3 s he.symm
-  exact ne_of_lt ((source_liveHit_le γ z 3 s hl hm.le).trans_lt (WithTop.coe_lt_top s))
-
-lemma source_weight_zero_no_unit_entrance {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ}
-    {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
-    (κ : ℝ) (z₀ : ℂ) (hy : 4 ≤ z₀.im) (rf : ℝ) (hf1 : rf ≤ 1)
-    (hn : liveSpatialHitTime γ z₀ 3 = ⊤) (z : ℂ) (hz : z ∈ ball z₀ 1) :
-    spatialGreenWeight κ U γ G z rf = 0 := by
-  unfold spatialGreenWeight
-  split_ifs with hb
-  · rfl
-  · have hn' : spatialHitTime γ z rf ≠ ⊤ := fun hh => hb hh
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn'
-    have hm := spatialHitTime_mem h.1 z rf t ht.symm
-    have hd : dist (γ t) z₀ < 3 := by
-      have hz' : dist z z₀ < 1 := hz
-      have hh := dist_triangle (γ t) z z₀
-      linarith
-    exact (source_enter_implies_live_contact h z₀ hy t hd hn).elim
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_event_bound_from_past {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
-    {μ : Measure Ω} [IsFiniteMeasure μ] (hm : m ≤ mΩ) (A E : Set Ω)
-    (hA : MeasurableSet[m] A) (hE : MeasurableSet E) (hEA : E ⊆ A)
-    (K : ℝ) (hK : 0 ≤ K)
-    (ht : ∀ᵐ ω ∂μ, ω ∈ A → μ[(fun ω => if ω ∈ E then (1:ℝ) else 0) | m] ω ≤ K) :
-    μ E ≤ ENNReal.ofReal (μ.real A*K) := by
-  have hi : Integrable (fun ω => if ω ∈ E then (1:ℝ) else 0) μ :=
-    (integrable_const (1:ℝ)).indicator hE
-  have hb := setIntegral_mono_ae_restrict (s := A) integrable_condExp.integrableOn
-    (integrable_const K).integrableOn ((ae_restrict_iff' (hm _ hA)).mpr ht)
-  rw [setIntegral_condExp hm hi hA] at hb
-  change (∫ ω in A, E.indicator (fun _ => (1:ℝ)) ω ∂μ) ≤ _ at hb
-  rw [setIntegral_indicator hE,inter_eq_right.mpr hEA,setIntegral_const,setIntegral_const] at hb
-  simp only [smul_eq_mul,mul_one] at hb
-  exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (mul_nonneg ENNReal.toReal_nonneg hK)).mpr hb
-
-lemma source_live_unit_mesh_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
-    ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < 1 ∧
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 4 ≤ z₀.im →
-    ∀ (rf δ : ℝ), 0 < rf → rf ≤ δ → δ ≤ δ₀ →
-      (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) <
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
-      ENNReal.ofReal ((P.comap Subtype.val).real {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} *
-        (C*Real.exp (-c*δ^(-2:ℝ)))) := by
-  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := sourcePhysicalMeshTail_proved κ hκ hκ8 1 (by norm_num)
-  refine ⟨C,c,δ₀,hC,hc,hδ,hδ1,?_⟩
-  intro Ω m P hP γ a z₀ hz rf δ hf hfd hd
-  let : IsProbabilityMeasure P := hP
-  let ρ := fun ω : a.samples => liveSpatialHitTime (γ ω) z₀ 3
-  have hρ : IsStoppingTime a.filtration ρ := source_liveHit_stopping a z₀ 3 (by linarith)
-  let A := {ω : a.samples | ρ ω ≠ ⊤}
-  have hA : MeasurableSet[hρ.measurableSpace] A := (measurableSet_eq_fun hρ.measurable measurable_const).compl
-  apply source_event_bound_from_past hρ.measurableSpace_le A _ hA
-    (measurableSet_lt measurable_const (a.spatialRealMass_measurable _ _))
-  · intro ω hω
-    by_contra hn
-    have htop : liveSpatialHitTime (γ ω) z₀ 3 = ⊤ := by simpa only [A,ρ,mem_ofPred_eq,not_not] using hn
-    have hzX : spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1) = 0 := by
-      apply setIntegral_eq_zero_of_forall_eq_zero
-      intro z hz'
-      exact source_weight_zero_no_unit_entrance (a.witness ω) κ z₀ hz rf (hfd.trans (hd.trans hδ1.le)) htop z hz'
-    change C*δ^(-codimension κ) < _ at hω
-    rw [hzX] at hω
-    exact (not_lt_of_ge (mul_nonneg hC.le (Real.rpow_nonneg (hf.trans_le hfd).le _))) hω
-  · positivity
-  · have hh := ht Ω m P hP γ a ρ hρ z₀ rf δ hf hfd hd
-    filter_upwards [hh] with ω hω
-    intro hωA
-    apply hω hωA
-    have hs : 3 ≤ dist (γ ω 0) z₀ := by
-      rw [(a.witness ω).2.1,dist_zero_left]
-      have := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
-      linarith
-    have he : liveSpatialHitTime (γ ω) z₀ 3 = ((ρ ω).untopD 0 : ℝ≥0) := by
-      change ρ ω = _
-      obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hωA
-      rw [← ht]
-      rfl
-    change closedBall z₀ (1+1) ⊆ _
-    convert source_live_collar_free (a.trace_continuous ω) z₀ hz hs ((ρ ω).untopD 0) he using 1
-    norm_num
-
-lemma source_live_entrance_bound {Ω : Type} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 4 ≤ z₀.im) (hsmall : 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4))) :
-    (P.comap Subtype.val).real {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} ≤
-      (uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ)*initialGreen κ z₀ := by
-  apply le_trans (measureReal_mono (show {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} ⊆
-      {ω | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4} from ?_))
-  · have hh := euclidean_approach a.brownian κ hκ hκ8 (fun ω : a.samples => γ ω) a.G a.F
-      (Eventually.of_forall a.witness) z₀ (by change 0 < z₀.im; linarith) 4 (by norm_num) hz hsmall
-    change ((P.comap Subtype.val) {ω : a.samples | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4}).toReal ≤ _
-    calc
-      _ ≤ _ := hh
-      _ = _ := by ring
-  · intro ω hω
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hω
-    have hs : 3 ≤ dist (γ ω 0) z₀ := by
-      rw [(a.witness ω).2.1,dist_zero_left]
-      have := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
-      linarith
-    exact ⟨t,by rw [(source_liveHit_mem (a.trace_continuous ω) z₀ 3 hs t ht.symm).2]; norm_num⟩
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_mesh_tail_to_stretched {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (X : Ω → ℝ) (p C c δ₀ v rf a : ℝ) (hp : 0 < p)
-    (_hC : 0 < C) (_hc : 0 < c) (hδ : 0 < δ₀) (hv : 0 ≤ v) (_hf : 0 < rf) (_ha : 0 ≤ a)
-    (hbound : ∀ ω, X ω ≤ v*rf^(-p))
-    (ht : ∀ δ : ℝ, rf ≤ δ → δ ≤ δ₀ → μ {ω | C*δ^(-p) < X ω} ≤
-      ENNReal.ofReal (a*(C*Real.exp (-c*δ^(-2:ℝ))))) :
-    ∀ u : ℝ, 1 ≤ u →
-      μ {ω | max C v*δ₀^(-p)*u < X ω} ≤
-        ENNReal.ofReal (a*C*Real.exp (-(c*δ₀^(-2:ℝ))*u^(2/p))) := by
-  intro u hu1
-  have hu : 0 < u := zero_lt_one.trans_le hu1
-  let δ := δ₀*u^(-1/p)
-  have he0 : -1/p ≤ 0 := div_nonpos_of_nonpos_of_nonneg (by norm_num) hp.le
-  have hδp : 0 < δ := mul_pos hδ (Real.rpow_pos_of_pos hu _)
-  have hδd : δ ≤ δ₀ := by
-    dsimp [δ]
-    exact mul_le_of_le_one_right hδ.le (Real.rpow_le_one_of_one_le_of_nonpos hu1 he0)
-  have he : δ^(-p) = δ₀^(-p)*u := by
-    dsimp [δ]
-    rw [Real.mul_rpow hδ.le (Real.rpow_nonneg hu.le _),← Real.rpow_mul hu.le]
-    have he : (-1/p)*(-p) = 1 := by field_simp
-    rw [he,Real.rpow_one]
-  have hex : δ^(-2:ℝ) = δ₀^(-2:ℝ)*u^(2/p) := by
-    dsimp [δ]
-    rw [Real.mul_rpow hδ.le (Real.rpow_nonneg hu.le _),← Real.rpow_mul hu.le]
-    congr 2
-    ring
-  have hn (b : ℝ) (hb : b ≤ max C v) : b*δ^(-p) ≤ max C v*δ₀^(-p)*u := by
-    rw [he,←mul_assoc]
-    exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hb (Real.rpow_nonneg hδ.le _)) hu.le
-  by_cases hfd : rf ≤ δ
-  · apply (measure_mono (show {ω | max C v*δ₀^(-p)*u < X ω} ⊆
-        {ω | C*δ^(-p) < X ω} from fun ω hω => (hn C (le_max_left _ _)).trans_lt hω)).trans
-    calc
-      _ ≤ _ := ht δ hfd hδd
-      _ = _ := by
-        rw [hex]
-        have he : -c*(δ₀^(-2:ℝ)*u^(2/p)) = -(c*δ₀^(-2:ℝ))*u^(2/p) := by ring
-        rw [he,mul_assoc]
-  · have hx (ω : Ω) : X ω ≤ max C v*δ₀^(-p)*u := by
-      calc
-        X ω ≤ v*rf^(-p) := hbound ω
-        _ ≤ v*δ^(-p) := mul_le_mul_of_nonneg_left
-          (Real.rpow_le_rpow_of_nonpos hδp (le_of_not_ge hfd) (by linarith)) hv
-        _ ≤ _ := hn v (le_max_right _ _)
-    have he : {ω | max C v*δ₀^(-p)*u < X ω} = ∅ := by
-      ext ω; simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]; exact not_lt_of_ge (hx ω)
-    rw [he,measure_empty]
-    exact bot_le
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourceLiveEntranceTailTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
-    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | M*u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ 1)} ≤
-      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
-theorem sourceLiveEntranceTail_proved : SourceLiveEntranceTailTarget := by
-  intro κ hκ hκ8
-  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := source_live_unit_mesh_tail κ hκ hκ8
-  let v := (volume (ball (0:ℂ) 1)).toReal
-  let M := max C v*δ₀^(-codimension κ)
-  let b := c*δ₀^(-2:ℝ)
-  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
-  have hk : 0 < κ := by linarith
-  have hp : 0 < codimension κ := codimension_pos κ hκ8
-  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hδ _)
-  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hδ _)
-  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
-    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
-  refine ⟨M,A*C,b,hM,mul_pos hA hC,hb,?_⟩
-  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) rf (ball z₀ 1)
-  have hx (ω : a.samples) : X ω ≤ v*rf^(-codimension κ) := by
-    have hh := (a.spatialRealMass_bound hk hκ8 rf hf (ball z₀ 1) measurableSet_ball
-      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne (fun z hz' => ?_) ω).2
-    · simpa only [v,Complex.volume_ball] using hh
-    · have hd : abs (z.im-z₀.im) ≤ dist z z₀ := by
-        simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
-      have hh : dist z z₀ < 1 := hz'
-      have ha := neg_abs_le (z.im-z₀.im)
-      linarith
-  have ht' := source_mesh_tail_to_stretched μ X (codimension κ) C c δ₀ v rf
-    (μ.real {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤}) hp hC hc hδ
-    ENNReal.toReal_nonneg hf ENNReal.toReal_nonneg hx
-    (fun δ hfd hd => ht Ω m P hP γ a z₀ hz rf δ hf hfd hd) u hu
-  apply ht'.trans
-  apply ENNReal.ofReal_le_ofReal
-  calc
-    _ ≤ (A*initialGreen κ z₀)*C*Real.exp (-b*u^(2/codimension κ)) :=
-      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
-        (source_live_entrance_bound a hk hκ8 z₀ hz hsmall) hC.le) (Real.exp_pos _).le
-    _ = _ := by ring
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_scaled_radius_hit_iff (c : ℝ≥0) (hc : 0 < c)
-    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (t : ℝ≥0) :
-    dist (sourceScaleTrace c γ t) ((c : ℂ) * z) - (c : ℝ) * r = 0 ↔
-      dist (γ (sourceScaleTime c t)) z - r = 0 := by
-  have hcR : 0 < (c : ℝ) := hc
-  rw [sourceScaleTrace, dist_eq_norm, ← mul_sub, norm_mul, Complex.norm_real,
-    Real.norm_eq_abs, abs_of_pos hcR, ← dist_eq_norm, ← mul_sub]
-  exact mul_eq_zero.trans (or_iff_right hcR.ne')
-
-lemma source_scale_spatial_hit (c : ℝ≥0) (hc : 0 < c)
-    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
-    spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r) =
-      WithTop.map (fun t : ℝ≥0 => c ^ 2 * t) (spatialHitTime γ z r) := by
-  classical
-  let S : Set ℝ≥0 := {t | dist (γ t) z - r = 0}
-  let Sc : Set ℝ≥0 := {t | dist (sourceScaleTrace c γ t) ((c : ℂ) * z) - (c : ℝ) * r = 0}
-  let e : ℝ≥0 ≃o ℝ≥0 := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2)
-  have he (x : ℝ≥0) : e x = c ^ 2 * x := rfl
-  have htime (x : ℝ≥0) : sourceScaleTime c (c ^ 2 * x) = x := by
-    simp [sourceScaleTime, hc.ne']
-  have hset : e '' S = Sc := by
-    ext t
-    constructor
-    · rintro ⟨u, hu, rfl⟩
-      exact (source_scaled_radius_hit_iff c hc γ z r _).2 (by simpa only [he, htime] using (show dist (γ u) z - r = 0 from hu))
-    · intro ht
-      refine ⟨sourceScaleTime c t, (source_scaled_radius_hit_iff c hc γ z r t).1 ht, ?_⟩
-      simp [he, sourceScaleTime, mul_div_cancel₀ _ (pow_ne_zero 2 hc.ne')]
-  unfold spatialHitTime hittingAfter
-  simp only [zero_le, true_and, Set.mem_singleton_iff]
-  change (if Sc.Nonempty then ((sInf Sc : ℝ≥0) : WithTop ℝ≥0) else ⊤) =
-    WithTop.map (fun t : ℝ≥0 => c^2*t) (if S.Nonempty then ((sInf S : ℝ≥0) : WithTop ℝ≥0) else ⊤)
-  have hh : Sc.Nonempty ↔ S.Nonempty := by rw [← hset]; exact Set.image_nonempty
-  by_cases hS : S.Nonempty
-  · rw [ite_eq_left hS, ite_eq_left (hh.mpr hS), WithTop.map_coe]
-    congr 1
-    rw [← he, e.map_csInf' hS (OrderBot.bddBelow S), hset]
-  · rw [ite_eq_right hS, ite_eq_right (mt hh.mp hS), WithTop.map_top]
-
-lemma source_scale_spatial_hit_top (c : ℝ≥0) (hc : 0 < c)
-    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
-    spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r) = ⊤ ↔
-      spatialHitTime γ z r = ⊤ := by
-  rw [source_scale_spatial_hit c hc]
-  simp
-
-lemma source_scale_spatial_hit_value (c : ℝ≥0) (hc : 0 < c)
-    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
-    (spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r)).untopD 0 =
-      c ^ 2 * (spatialHitTime γ z r).untopD 0 := by
-  rw [source_scale_spatial_hit c hc]
-  cases hh : spatialHitTime γ z r
-  · simp
-  · rfl
-
-lemma source_scale_spatial_hit_le (c : ℝ≥0) (hc : 0 < c)
-    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (T : ℝ≥0) :
-    spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r) ≤ (c ^ 2 * T : ℝ≥0) ↔
-      spatialHitTime γ z r ≤ T := by
-  rw [source_scale_spatial_hit c hc]
-  cases hh : spatialHitTime γ z r
-  · exact iff_of_false (WithTop.not_top_le_coe _) (WithTop.not_top_le_coe _)
-  · simp only [WithTop.map_coe, WithTop.coe_le_coe]
-    exact mul_le_mul_iff_right₀ (pow_pos hc 2)
-
-lemma source_scale_chart_deriv (c : ℝ≥0) (hc : 0 < c)
-    (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) :
-    deriv (sourceScaleChart c G t) ((c : ℂ) * z) = deriv (G (sourceScaleTime c t)) z := by
-  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
-  change deriv (fun w : ℂ => (c : ℂ) * G (sourceScaleTime c t) (w / (c : ℂ))) _ = _
-  rw [deriv_const_mul_field]
-  simp_rw [div_eq_mul_inv, mul_comm _ ((c : ℂ)⁻¹)]
-  rw [deriv_comp_mul_left]
-  simp only [smul_eq_mul]
-  simp [hcC]
-
-lemma source_scale_fullRadius (c : ℝ≥0) (hc : 0 < c)
-    (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) :
-    fullRadius (sourceScaleChart c G) t ((c : ℂ) * z) =
-      (c : ℝ) * fullRadius G (sourceScaleTime c t) z := by
-  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
-  unfold fullRadius
-  rw [source_scale_chart_deriv c hc]
-  simp only [sourceScaleChart, mul_div_cancel_left₀ _ hcC, Complex.mul_im, Complex.ofReal_re,
-    Complex.ofReal_im, zero_mul, add_zero]
-  ring
-
-lemma source_scale_greenDensity (κ : ℝ) (c : ℝ≥0) (hc : 0 < c)
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) :
-    greenDensity κ (fun u => (c : ℝ) * U (sourceScaleTime c u))
-      (sourceScaleTrace c γ) (sourceScaleChart c G) t ((c : ℂ) * z) =
-    ((c : ℝ) ^ (-codimension κ)) * greenDensity κ U γ G (sourceScaleTime c t) z := by
-  have hcR : 0 < (c : ℝ) := hc
-  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
-  have hlive : (c : ℂ) * z ∈ survivingDomain (sourceScaleTrace c γ) t ↔
-      z ∈ survivingDomain γ (sourceScaleTime c t) := by
-    simpa [hcC] using source_scale_domain_iff c t hc γ ((c : ℂ) * z)
-  by_cases hz : z ∈ survivingDomain γ (sourceScaleTime c t)
-  · simp only [greenDensity,ite_eq_left hz,ite_eq_left (hlive.mpr hz)]
-    rw [source_scale_fullRadius c hc]
-    have hR : 0 ≤ fullRadius G (sourceScaleTime c t) z := by
-      exact div_nonneg (mul_nonneg (by norm_num) (hw.2.2.2.2.2.1 _ hz).le) (norm_nonneg _)
-    rw [Real.mul_rpow hcR.le hR]
-    simp only [sourceScaleChart, mul_div_cancel_left₀ _ hcC, Complex.ofReal_mul, ← mul_sub]
-    rw [Complex.arg_real_mul _ hcR]
-    ring
-  · simp only [greenDensity,ite_eq_right hz,ite_eq_right (mt hlive.mp hz),mul_zero]
-lemma source_scale_spatial_greenWeight (κ : ℝ) (c : ℝ≥0) (hc : 0 < c)
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (z : ℂ) (r : ℝ) :
-    spatialGreenWeight κ (fun u => (c : ℝ) * U (sourceScaleTime c u))
-      (sourceScaleTrace c γ) (sourceScaleChart c G) ((c : ℂ) * z) ((c : ℝ) * r) =
-    ((c : ℝ) ^ (-codimension κ)) * spatialGreenWeight κ U γ G z r := by
-  unfold spatialGreenWeight
-  simp only [source_scale_spatial_hit_top c hc]
-  split_ifs with ht
-  · simp
-  · rw [source_scale_greenDensity κ c hc U γ G F hw, source_scale_spatial_hit_value c hc]
-    simp [sourceScaleTime, hc.ne']
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-def sourceScaled (a : RegularSLEPresentation P κ γ) (c : ℝ≥0) (hc : 0 < c) :
-    RegularSLEPresentation P κ (fun ω => sourceScaleTrace c (γ ω)) where
-  samples := a.samples
-  measurable_samples := a.measurable_samples
-  full_samples := a.full_samples
-  B := fun t ω => (c:ℝ)*a.B (sourceScaleTime c t) ω
-  measurable_B := fun t => measurable_const.mul (a.measurable_B _)
-  continuous_B := fun ω => continuous_const.mul ((a.continuous_B ω).comp (by unfold sourceScaleTime; fun_prop))
-  zero_B := fun ω => by simp [sourceScaleTime,a.zero_B ω]
-  brownian := by
-    have hh := a.brownian.smul (c := (c^2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hc.ne'))
-    simpa only [sourceScaleTime,NNReal.coe_inv,NNReal.coe_pow,Real.sqrt_inv,
-      Real.sqrt_sq c.coe_nonneg,inv_inv,div_eq_mul_inv,mul_comm] using hh
-  G := fun ω => sourceScaleChart c (a.G ω)
-  F := fun ω => sourceScaleChart c (a.F ω)
-  witness := fun ω => by
-    have hh := source_scale_witness c hc (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω)
-    simpa only [mul_left_comm (c:ℝ) (Real.sqrt κ)] using hh
-end RegularSLEPresentation
-
-lemma source_scale_initialGreen (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) (z : ℂ) (hz : 0 ≤ z.im) :
-    initialGreen κ ((c:ℂ)*z) = (c:ℝ)^(-codimension κ)*initialGreen κ z := by
-  have hcR : 0 < (c:ℝ) := hc
-  unfold initialGreen
-  rw [Complex.arg_real_mul _ hcR]
-  have him : 2*((c:ℂ)*z).im = (c:ℝ)*(2*z.im) := by simp [Complex.mul_im]; ring
-  rw [him,Real.mul_rpow hcR.le (mul_nonneg (by norm_num) hz),mul_assoc]
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_scale_area_real_integral (c : ℝ≥0) (hc : 0 < c)
-    (f : ℂ → ℝ) (K : Set ℂ) :
-    (∫ z in (fun z : ℂ => (c : ℂ) * z) '' K, f z) =
-      ((c : ℝ)^2) * ∫ z in K, f ((c : ℂ) * z) := by
-  have hcR : 0 < (c : ℝ) := hc
-  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
-  let e := Homeomorph.mulLeft₀ (c : ℂ) hcC
-  have he (z : ℂ) : e z = (c : ℂ) * z := rfl
-  have hvol (s : Set ℂ) : volume (e ⁻¹' s) =
-      ENNReal.ofReal (((c : ℝ)^2)⁻¹) * volume s := by
-    change volume ((fun z : ℂ => (c : ℂ) * z) ⁻¹' s) = _
-    simpa only [Complex.real_smul, smul_eq_mul, Complex.finrank_real_complex,
-      abs_of_nonneg (inv_nonneg.mpr (sq_nonneg (c : ℝ)))] using
-      (volume : Measure ℂ).addHaar_preimage_smul hcR.ne' s
-  have hp : MeasurePreserving e (ENNReal.ofReal ((c : ℝ)^2) • (volume : Measure ℂ)) volume := by
-    refine ⟨e.continuous.measurable, ?_⟩
-    apply Measure.ext
-    intro s hs
-    rw [e.measurableEmbedding.map_apply, Measure.smul_apply, smul_eq_mul, hvol, ← mul_assoc,
-      ← ENNReal.ofReal_mul (sq_nonneg (c : ℝ)), mul_inv_cancel₀ (pow_ne_zero 2 hcR.ne'),
-      ENNReal.ofReal_one, one_mul]
-  have hh := hp.setIntegral_image_emb e.measurableEmbedding f K
-  rw [Measure.restrict_smul,integral_smul_measure] at hh
-  simpa only [he,ENNReal.toReal_ofReal (sq_nonneg (c:ℝ)),smul_eq_mul] using hh
-
-lemma source_scale_spatialRealMass (κ : ℝ) (c : ℝ≥0) (hc : 0 < c)
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (r : ℝ) (K : Set ℂ) :
-    spatialRealMass κ (fun t => (c:ℝ)*U (sourceScaleTime c t))
-      (sourceScaleTrace c γ) (sourceScaleChart c G) ((c:ℝ)*r)
-      ((fun z : ℂ => (c:ℂ)*z) '' K) =
-      (c:ℝ)^dimension κ*spatialRealMass κ U γ G r K := by
-  unfold spatialRealMass
-  rw [source_scale_area_real_integral c hc]
-  simp_rw [source_scale_spatial_greenWeight κ c hc U γ G F hw]
-  rw [integral_const_mul,←mul_assoc]
-  congr 1
-  rw [←Real.rpow_natCast (c:ℝ) 2,←Real.rpow_add (show 0 < (c:ℝ) from hc)]
-  congr 1
-  simp [codimension]
-
-lemma source_scale_ball (c : ℝ≥0) (hc : 0 < c) (z : ℂ) (r : ℝ) :
-    (fun w : ℂ => (c:ℂ)*w) '' Metric.ball z r = Metric.ball ((c:ℂ)*z) ((c:ℝ)*r) := by
-  have hcR : 0 < (c:ℝ) := hc
-  simpa only [Complex.real_smul,smul_eq_mul,Real.norm_eq_abs,abs_of_pos hcR] using
-    Metric.smul_image_ball (α := ℝ) hcR.ne' z r
-
-lemma RegularSLEPresentation.sourceScaled_mass {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (c : ℝ≥0) (hc : 0 < c) (z : ℂ) (r rf : ℝ) (ω : a.samples) :
-    spatialRealMass κ (fun t => Real.sqrt κ*(a.sourceScaled c hc).B t ω)
-      (sourceScaleTrace c (γ ω)) ((a.sourceScaled c hc).G ω) ((c:ℝ)*rf)
-      (Metric.ball ((c:ℂ)*z) ((c:ℝ)*r)) =
-      (c:ℝ)^dimension κ*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z r) := by
-  rw [←source_scale_ball c hc]
-  simpa only [RegularSLEPresentation.sourceScaled,mul_left_comm (Real.sqrt κ) (c:ℝ)] using
-    source_scale_spatialRealMass κ c hc (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) rf (Metric.ball z r)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set MeasureTheory
-lemma source_tail_threshold_normalize {Ω : Type*} [MeasurableSpace Ω]
-    (μ : Measure Ω) (X : Ω → ℝ) (M C A b q K : ℝ)
-    (hM : 0 < M) (hC : 0 < C) (_hA : 0 < A) (hb : 0 < b) (hq : 0 < q) (hK : 0 ≤ K)
-    (hzero : μ {ω | 0 < X ω} ≤ ENNReal.ofReal (A*K))
-    (ht : ∀ u : ℝ, 1 ≤ u → μ {ω | M*u < X ω} ≤ ENNReal.ofReal (C*K*Real.exp (-b*u^q)))
-    (u : ℝ) (hu : 1 ≤ u) :
-    μ {ω | u < X ω} ≤ ENNReal.ofReal
-      ((max C A*Real.exp b)*K*Real.exp (-(b/(max M 1)^q)*u^q)) := by
-  let R := max M 1
-  have hR : 0 < R := hM.trans_le (le_max_left _ _)
-  have hu0 : 0 < u := zero_lt_one.trans_le hu
-  have he : -(b/R^q)*u^q = -b*(u/R)^q := by
-    rw [Real.div_rpow hu0.le hR.le]
-    ring
-  rw [he]
-  by_cases hbig : R ≤ u
-  · have hv : 1 ≤ u/R := (le_div_iff₀ hR).mpr (by simpa using hbig)
-    have hth : M*(u/R) ≤ u := by
-      calc
-        _ ≤ R*(u/R) := mul_le_mul_of_nonneg_right (le_max_left _ _) (div_nonneg hu0.le hR.le)
-        _ = _ := by field_simp
-    apply (measure_mono (show {ω | u < X ω} ⊆ {ω | M*(u/R) < X ω} from
-      fun ω hω => hth.trans_lt hω)).trans (ht (u/R) hv |>.trans _)
-    apply ENNReal.ofReal_le_ofReal
-    gcongr
-    calc
-      C ≤ max C A := le_max_left _ _
-      _ ≤ max C A*Real.exp b := le_mul_of_one_le_right (hC.le.trans (le_max_left _ _))
-        (Real.one_le_exp_iff.mpr hb.le)
-  · have hv : 0 ≤ u/R := div_nonneg hu0.le hR.le
-    have hv1 : u/R ≤ 1 := (div_le_one hR).mpr (le_of_not_ge hbig)
-    have hp : (u/R)^q ≤ 1 := Real.rpow_le_one hv hv1 hq.le
-    have hexp : 1 ≤ Real.exp b*Real.exp (-b*(u/R)^q) := by
-      rw [←Real.exp_add]
-      apply Real.one_le_exp_iff.mpr
-      nlinarith
-    apply (measure_mono (show {ω | u < X ω} ⊆ {ω | 0 < X ω} from
-      fun ω hω => hu0.trans hω)).trans (hzero.trans _)
-    apply ENNReal.ofReal_le_ofReal
-    calc
-      A*K ≤ max C A*K := mul_le_mul_of_nonneg_right (le_max_right _ _) hK
-      _ ≤ (max C A*K)*(Real.exp b*Real.exp (-b*(u/R)^q)) :=
-        le_mul_of_one_le_right (mul_nonneg (hC.le.trans (le_max_left _ _)) hK) hexp
-      _ = _ := by ring
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_live_entrance_all_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
-  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
-    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
-      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ))) := by
-  obtain ⟨M,C,b,hM,hC,hb,ht⟩ := sourceLiveEntranceTail_proved κ hκ hκ8
-  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
-  have hk : 0 < κ := by linarith
-  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
-    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
-  refine ⟨max C A*Real.exp b,b/(max M 1)^(2/codimension κ),
-    mul_pos (hC.trans_le (le_max_left _ _)) (Real.exp_pos _),
-    div_pos hb (Real.rpow_pos_of_pos (hM.trans_le (le_max_left _ _)) _),?_⟩
-  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) rf (ball z₀ 1)
-  have hK : 0 < initialGreen κ z₀ := initialGreen_pos κ z₀ (by change 0 < z₀.im; linarith)
-  have hzero : μ {ω | 0 < X ω} ≤ ENNReal.ofReal (A*initialGreen κ z₀) := by
-    apply le_trans (measure_mono (show {ω | 0 < X ω} ⊆
-      {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} from ?_))
-    · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (mul_pos hA hK).le).mpr
-      exact source_live_entrance_bound a hk hκ8 z₀ hz hsmall
-    · intro ω hω hn
-      have hx : X ω = 0 := by
-        apply setIntegral_eq_zero_of_forall_eq_zero
-        intro z hz'
-        exact source_weight_zero_no_unit_entrance (a.witness ω) κ z₀ hz rf hf1 hn z hz'
-      change 0 < X ω at hω
-      rw [hx] at hω
-      exact (lt_irrefl 0) hω
-  exact source_tail_threshold_normalize μ X M C A b (2/codimension κ) (initialGreen κ z₀)
-    hM hC hA hb (div_pos (by norm_num) (codimension_pos κ hκ8)) hK.le hzero
-    (fun v hv => ht Ω m P hP γ a z₀ hz hsmall rf v hf hf1 hv) u hu
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourcePhysicalBallAllTailTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (r : ℝ),
-    0 < r → 4*r ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*(4*r))) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ r → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | r^dimension κ*u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ r)} ≤
-      ENNReal.ofReal (C*r^codimension κ*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
-theorem sourcePhysicalBallAllTail_proved : SourcePhysicalBallAllTailTarget := by
-  intro κ hκ hκ8
-  obtain ⟨C,b,hC,hb,ht⟩ := source_live_entrance_all_tail κ hκ hκ8
-  refine ⟨C,b,hC,hb,?_⟩
-  intro Ω m P hP γ a z r hr hz hsmall rf u hf hfr hu
-  let : IsProbabilityMeasure P := hP
-  let c : ℝ≥0 := ⟨r⁻¹,(inv_pos.mpr hr).le⟩
-  have hc : 0 < c := inv_pos.mpr hr
-  have hcR : (c:ℝ) = r⁻¹ := rfl
-  let a' := a.sourceScaled c hc
-  have him : ((c:ℂ)*z).im = r⁻¹*z.im := by simp [hcR,Complex.mul_im]
-  have hcr : (c:ℝ)*r = 1 := by rw [hcR,inv_mul_cancel₀ hr.ne']
-  have hheight : 4 ≤ ((c:ℂ)*z).im := by
-    rw [him,mul_comm,←div_eq_mul_inv]
-    exact (le_div_iff₀ hr).mpr hz
-  have hsmall' : 1 ≤ Real.log ((2*((c:ℂ)*z).im)/((128*Real.pi)*4)) := by
-    have he : (2*((c:ℂ)*z).im)/((128*Real.pi)*4) = (2*z.im)/((128*Real.pi)*(4*r)) := by
-      rw [him]
-      field_simp
-    rwa [he]
-  have hf' : 0 < (c:ℝ)*rf := mul_pos hc hf
-  have hfr' : (c:ℝ)*rf ≤ 1 := by rw [←hcr]; exact mul_le_mul_of_nonneg_left hfr hc.le
-  have htail := ht Ω m P hP (fun ω => sourceScaleTrace c (γ ω)) a' ((c:ℂ)*z)
-    hheight hsmall' ((c:ℝ)*rf) u hf' hfr' hu
-  change (P.comap (Subtype.val : a.samples → Ω)) {ω : a.samples | u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a'.B t ω) (sourceScaleTrace c (γ ω)) (a'.G ω)
-      ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} ≤
-      ENNReal.ofReal (C*initialGreen κ ((c:ℂ)*z)*Real.exp (-b*u^(2/codimension κ))) at htail
-  have hp : 0 < (c:ℝ)^dimension κ := Real.rpow_pos_of_pos hc _
-  have hrp : 0 < r^dimension κ := Real.rpow_pos_of_pos hr _
-  have hmr : (c:ℝ)^dimension κ*(r^dimension κ*u) = u := by
-    rw [hcR,Real.inv_rpow hr.le]
-    field_simp
-  have hx (ω : a.samples) : spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
-      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1) =
-      (c:ℝ)^dimension κ*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z r) := by
-    simpa only [hcr] using a.sourceScaled_mass c hc z r rf ω
-  have he : {ω : a.samples | u < spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
-      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} =
-      {ω : a.samples | r^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) rf (ball z r)} := by
-    ext ω
-    change (u < _) ↔ _
-    rw [hx ω]
-    conv_lhs => rw [←hmr]
-    exact mul_lt_mul_iff_right₀ hp
-  rw [he] at htail
-  have hg : initialGreen κ ((c:ℂ)*z) = r^codimension κ*initialGreen κ z := by
-    rw [source_scale_initialGreen κ c hc z (by nlinarith),hcR,Real.inv_rpow hr.le,Real.rpow_neg hr.le,inv_inv]
-  rw [hg] at htail
-  convert htail using 1
-  congr 1
-  ring
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.FiniteMass
+import OAI.Probability.SLE.WindowPropagation
+
+namespace OAI
+
+/-! Retained Seed Moments for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_unit_ball_height {z₀ z : ℂ} (hz₀ : 2 < z₀.im) (hz : z ∈ closedBall z₀ 1) :
+    1 < z.im := by
+  have hdist : abs (z.im-z₀.im) ≤ dist z z₀ := by
+    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
+  have hh : dist z z₀ ≤ 1 := hz
+  have ha := neg_abs_le (z.im-z₀.im)
+  linarith
+
+lemma source_open_closed_ball_ae (z₀ : ℂ) :
+    closedBall z₀ 1 =ᵐ[volume] ball z₀ 1 := by
+  have he : ∀ᵐ z : ℂ ∂volume, z ∉ sphere z₀ 1 := by
+    rw [ae_iff]
+    simpa [Metric.sphere, dist_eq_norm] using Measure.addHaar_sphere volume z₀ (1:ℝ)
+  filter_upwards [he] with z hz
+  apply propext
+  change dist z z₀ ≤ 1 ↔ dist z z₀ < 1
+  change dist z z₀ ≠ 1 at hz
+  exact le_iff_lt_or_eq.trans (or_iff_left hz)
+
+lemma source_finite_mass_le_unit_mass {Ω : Type} [MeasurableSpace Ω]
+    {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 2 < z₀.im) (rf : ℝ) (hf : 0 < rf) (hf1 : rf ≤ 1)
+    (T : ℝ≥0) (ω : a.samples) :
+    finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T (closedBall z₀ 1) ≤
+      ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)) := by
+  have hi := a.spatialGreenWeight_spatial_integrable hκ hκ8 rf hf
+    (ball z₀ 1) measurableSet_ball (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne
+    (fun z hz' => hf1.trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le) ω
+  rw [spatialRealMass,ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall (fun z => by
+    dsimp [spatialGreenWeight]; split_ifs
+    · exact le_rfl
+    · exact greenDensity_nonneg (a.witness ω) κ _ z))]
+  rw [finiteSpatialGreenMass,Measure.restrict_congr_set (source_open_closed_ball_ae z₀)]
+  apply lintegral_mono
+  intro z
+  apply ENNReal.ofReal_le_ofReal
+  split_ifs
+  · exact le_rfl
+  · dsimp [spatialGreenWeight]; split_ifs
+    · exact le_rfl
+    · exact greenDensity_nonneg (a.witness ω) κ _ z
+
+lemma source_unit_mesh_second_moment (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
+    ∃ d : ℝ, 0 < d ∧ d < 1 ∧
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+    ∃ m₂ : ℝ, 0 < m₂ ∧ ∀ rf : ℝ, 0 < rf → rf ≤ d →
+      (∫⁻ ω : a.samples, ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) rf (ball z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
+  obtain ⟨C,c,d,hC,hc,hd,hd1,ht⟩ := source_initial_unit_mesh_tail κ hκ hκ8
+  refine ⟨d,hd,hd1,?_⟩
+  intro Ω m P hP γ a z₀ hz
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let v := (volume (ball z₀ 1)).toReal
+  let M := max C v*d^(-codimension κ)
+  let b := c*d^(-2:ℝ)
+  let A := max 1 C*Real.exp b
+  let α := b/M
+  have hk : 0 < κ := by linarith
+  have hp : 0 < codimension κ := codimension_pos κ hκ8
+  have hp1 : codimension κ < 1 := by dsimp [codimension, dimension]; linarith
+  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hd _)
+  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hd _)
+  have hA : 0 < A := mul_pos (zero_lt_one.trans_le (le_max_left _ _)) (Real.exp_pos _)
+  have hα : 0 < α := div_pos hb hM
+  refine ⟨8*A/α^2,by positivity,?_⟩
+  intro rf hf hfd
+  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) rf (ball z₀ 1)
+  have hx (ω : a.samples) : 0 ≤ X ω ∧ X ω ≤ v*rf^(-codimension κ) :=
+    a.spatialRealMass_bound hk hκ8 rf hf (ball z₀ 1) measurableSet_ball
+      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne
+      (fun z hz' => (hfd.trans hd1.le).trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le) ω
+  have hlinear : ∀ t : ℝ, 1 ≤ t → μ {ω | M*t < X ω} ≤ ENNReal.ofReal (C*Real.exp (-b*t)) :=
+    source_mesh_tail_to_linear μ X (codimension κ) C c d v rf hp hp1.le hC hc hd
+      ENNReal.toReal_nonneg hf (fun ω => (hx ω).2)
+      (fun δ hfd hd' => ht Ω m P hP γ a z₀ hz rf δ hf hfd hd')
+  exact source_exponential_second_moment μ X (a.spatialRealMass_measurable _ _) (fun ω => (hx ω).1)
+    A α hA.le hα (source_linear_tail_all X M C b hM hC hb hlinear)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourceRetainedSeedTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (z₀ : ℂ), 2 < z₀.im →
+  ∃ (T : ℝ≥0) (e₀ m₀ m₂ : ℝ),
+    0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧ 0 < m₂ ∧
+    ∀ r : ℝ, 0 < r → r ≤ e₀ →
+      let X := fun ω : a.samples => finiteSpatialGreenMass κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r T (closedBall z₀ 1)
+      ENNReal.ofReal m₀ ≤ ∫⁻ ω, X ω ∂(P.comap Subtype.val) ∧
+      (∫⁻ ω, (X ω)^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂
+theorem sourceRetainedSeed_proved : SourceRetainedSeedTarget := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  have hK : closedBall z₀ 1 ⊆ upperHalfPlane :=
+    fun z hz' => zero_lt_one.trans (source_unit_ball_height hz hz')
+  obtain ⟨T,e₀,m₀,hT,he,hm,hfirst⟩ := a.source_finite_time_first_moment hk hκ8 z₀ 1 (by norm_num) hK
+  obtain ⟨d,hd,hd1,hsecond⟩ := source_unit_mesh_second_moment κ hκ hκ8
+  obtain ⟨m₂,hm₂,hsecond⟩ := hsecond Ω m P hP γ a z₀ hz
+  refine ⟨T,min e₀ d,m₀,m₂,hT,lt_min he hd,hm,hm₂,?_⟩
+  intro r hr hre
+  refine ⟨hfirst r hr (hre.trans (min_le_left _ _)),?_⟩
+  have hrd : r ≤ d := hre.trans (min_le_right _ _)
+  apply le_trans (lintegral_mono (fun ω => pow_le_pow_left' (source_finite_mass_le_unit_mass
+    a hk hκ8 z₀ hz r hr (hrd.trans hd1.le) T ω) 2))
+  exact hsecond r hr hrd
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_first_sphere_free {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hy : r ≤ z.im) (hs : r ≤ dist (γ 0) z)
+    (s : ℝ≥0) (hfirst : spatialHitTime γ z r = s) :
+    ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by
+  intro w hw
+  refine ⟨ball_subset_upperHalfPlane hy hw,?_⟩
+  rintro ⟨u,hu,rfl⟩
+  have hh := spatialHitTime_lt_of_dist_lt hc z r hs u hw
+  rw [hfirst] at hh
+  exact (not_lt_of_ge hu.2) (WithTop.coe_lt_coe.mp hh)
+
+lemma source_live_hit_spatial_eq {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z)
+    (s : ℝ≥0) (hh : liveSpatialHitTime γ z r = s) :
+    spatialHitTime γ z r = s := by
+  rw [liveSpatialHitTime_eq hc z r hs] at hh
+  split_ifs at hh with ht hl
+  · exact (WithTop.coe_ne_top hh.symm).elim
+  · exact hh
+  · exact (WithTop.coe_ne_top hh.symm).elim
+
+lemma source_live_collar_free {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (hy : 4 ≤ z.im) (hs : 3 ≤ dist (γ 0) z)
+    (s : ℝ≥0) (hh : liveSpatialHitTime γ z 3 = s) :
+    closedBall z 2 ⊆ survivingDomain γ s := by
+  have hm := source_liveHit_mem hc z 3 hs s hh
+  have hball := ball_subset_survivingDomain (by norm_num : (0:ℝ)<3) hm.1
+    (source_first_sphere_free hc z 3 (by linarith) hs s
+      (source_live_hit_spatial_eq hc z 3 hs s hh))
+  exact fun w hw => hball (by change dist w z < 3; have : dist w z ≤ 2 := hw; linarith)
+
+lemma source_enter_implies_live_contact {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ}
+    {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
+    (z : ℂ) (hy : 4 ≤ z.im) (t : ℝ≥0) (ht : dist (γ t) z < 3) :
+    liveSpatialHitTime γ z 3 ≠ ⊤ := by
+  have hs : 3 ≤ dist (γ 0) z := by
+    rw [h.2.1,dist_zero_left]
+    have := (le_abs_self z.im).trans (Complex.abs_im_le_norm z)
+    linarith
+  have hh := spatialHitTime_lt_of_dist_lt h.1 z 3 hs t ht
+  have hn : spatialHitTime γ z 3 ≠ ⊤ := ne_of_lt (hh.trans_le le_top)
+  obtain ⟨s,he⟩ := WithTop.ne_top_iff_exists.mp hn
+  have hst : s ≤ t := (WithTop.coe_lt_coe.mp (he ▸ hh)).le
+  have hfree := source_first_sphere_free h.1 z 3 (by linarith) hs s he.symm
+  have hl := h.surviving_of_free_ball_future_tip s t hst z 3 (by norm_num) hfree ht
+  have hm := spatialHitTime_mem h.1 z 3 s he.symm
+  exact ne_of_lt ((source_liveHit_le γ z 3 s hl hm.le).trans_lt (WithTop.coe_lt_top s))
+
+lemma source_weight_zero_no_unit_entrance {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ}
+    {G F : ℝ≥0 → ℂ → ℂ} (h : IsCapacityTwoWitness U γ G F)
+    (κ : ℝ) (z₀ : ℂ) (hy : 4 ≤ z₀.im) (rf : ℝ) (hf1 : rf ≤ 1)
+    (hn : liveSpatialHitTime γ z₀ 3 = ⊤) (z : ℂ) (hz : z ∈ ball z₀ 1) :
+    spatialGreenWeight κ U γ G z rf = 0 := by
+  unfold spatialGreenWeight
+  split_ifs with hb
+  · rfl
+  · have hn' : spatialHitTime γ z rf ≠ ⊤ := fun hh => hb hh
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn'
+    have hm := spatialHitTime_mem h.1 z rf t ht.symm
+    have hd : dist (γ t) z₀ < 3 := by
+      have hz' : dist z z₀ < 1 := hz
+      have hh := dist_triangle (γ t) z z₀
+      linarith
+    exact (source_enter_implies_live_contact h z₀ hy t hd hn).elim
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_event_bound_from_past {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
+    {μ : Measure Ω} [IsFiniteMeasure μ] (hm : m ≤ mΩ) (A E : Set Ω)
+    (hA : MeasurableSet[m] A) (hE : MeasurableSet E) (hEA : E ⊆ A)
+    (K : ℝ) (hK : 0 ≤ K)
+    (ht : ∀ᵐ ω ∂μ, ω ∈ A → μ[(fun ω => if ω ∈ E then (1:ℝ) else 0) | m] ω ≤ K) :
+    μ E ≤ ENNReal.ofReal (μ.real A*K) := by
+  have hi : Integrable (fun ω => if ω ∈ E then (1:ℝ) else 0) μ :=
+    (integrable_const (1:ℝ)).indicator hE
+  have hb := setIntegral_mono_ae_restrict (s := A) integrable_condExp.integrableOn
+    (integrable_const K).integrableOn ((ae_restrict_iff' (hm _ hA)).mpr ht)
+  rw [setIntegral_condExp hm hi hA] at hb
+  change (∫ ω in A, E.indicator (fun _ => (1:ℝ)) ω ∂μ) ≤ _ at hb
+  rw [setIntegral_indicator hE,inter_eq_right.mpr hEA,setIntegral_const,setIntegral_const] at hb
+  simp only [smul_eq_mul,mul_one] at hb
+  exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (mul_nonneg ENNReal.toReal_nonneg hK)).mpr hb
+
+lemma source_live_unit_mesh_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
+    ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < 1 ∧
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 4 ≤ z₀.im →
+    ∀ (rf δ : ℝ), 0 < rf → rf ≤ δ → δ ≤ δ₀ →
+      (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) <
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
+      ENNReal.ofReal ((P.comap Subtype.val).real {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} *
+        (C*Real.exp (-c*δ^(-2:ℝ)))) := by
+  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := sourcePhysicalMeshTail_proved κ hκ hκ8 1 (by norm_num)
+  refine ⟨C,c,δ₀,hC,hc,hδ,hδ1,?_⟩
+  intro Ω m P hP γ a z₀ hz rf δ hf hfd hd
+  let : IsProbabilityMeasure P := hP
+  let ρ := fun ω : a.samples => liveSpatialHitTime (γ ω) z₀ 3
+  have hρ : IsStoppingTime a.filtration ρ := source_liveHit_stopping a z₀ 3 (by linarith)
+  let A := {ω : a.samples | ρ ω ≠ ⊤}
+  have hA : MeasurableSet[hρ.measurableSpace] A := (measurableSet_eq_fun hρ.measurable measurable_const).compl
+  apply source_event_bound_from_past hρ.measurableSpace_le A _ hA
+    (measurableSet_lt measurable_const (a.spatialRealMass_measurable _ _))
+  · intro ω hω
+    by_contra hn
+    have htop : liveSpatialHitTime (γ ω) z₀ 3 = ⊤ := by simpa only [A,ρ,mem_ofPred_eq,not_not] using hn
+    have hzX : spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1) = 0 := by
+      apply setIntegral_eq_zero_of_forall_eq_zero
+      intro z hz'
+      exact source_weight_zero_no_unit_entrance (a.witness ω) κ z₀ hz rf (hfd.trans (hd.trans hδ1.le)) htop z hz'
+    change C*δ^(-codimension κ) < _ at hω
+    rw [hzX] at hω
+    exact (not_lt_of_ge (mul_nonneg hC.le (Real.rpow_nonneg (hf.trans_le hfd).le _))) hω
+  · positivity
+  · have hh := ht Ω m P hP γ a ρ hρ z₀ rf δ hf hfd hd
+    filter_upwards [hh] with ω hω
+    intro hωA
+    apply hω hωA
+    have hs : 3 ≤ dist (γ ω 0) z₀ := by
+      rw [(a.witness ω).2.1,dist_zero_left]
+      have := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
+      linarith
+    have he : liveSpatialHitTime (γ ω) z₀ 3 = ((ρ ω).untopD 0 : ℝ≥0) := by
+      change ρ ω = _
+      obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hωA
+      rw [← ht]
+      rfl
+    change closedBall z₀ (1+1) ⊆ _
+    convert source_live_collar_free (a.trace_continuous ω) z₀ hz hs ((ρ ω).untopD 0) he using 1
+    norm_num
+
+lemma source_live_entrance_bound {Ω : Type} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 4 ≤ z₀.im) (hsmall : 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4))) :
+    (P.comap Subtype.val).real {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} ≤
+      (uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ)*initialGreen κ z₀ := by
+  apply le_trans (measureReal_mono (show {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} ⊆
+      {ω | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4} from ?_))
+  · have hh := euclidean_approach a.brownian κ hκ hκ8 (fun ω : a.samples => γ ω) a.G a.F
+      (Eventually.of_forall a.witness) z₀ (by change 0 < z₀.im; linarith) 4 (by norm_num) hz hsmall
+    change ((P.comap Subtype.val) {ω : a.samples | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4}).toReal ≤ _
+    calc
+      _ ≤ _ := hh
+      _ = _ := by ring
+  · intro ω hω
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hω
+    have hs : 3 ≤ dist (γ ω 0) z₀ := by
+      rw [(a.witness ω).2.1,dist_zero_left]
+      have := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
+      linarith
+    exact ⟨t,by rw [(source_liveHit_mem (a.trace_continuous ω) z₀ 3 hs t ht.symm).2]; norm_num⟩
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_mesh_tail_to_stretched {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (X : Ω → ℝ) (p C c δ₀ v rf a : ℝ) (hp : 0 < p)
+    (_hC : 0 < C) (_hc : 0 < c) (hδ : 0 < δ₀) (hv : 0 ≤ v) (_hf : 0 < rf) (_ha : 0 ≤ a)
+    (hbound : ∀ ω, X ω ≤ v*rf^(-p))
+    (ht : ∀ δ : ℝ, rf ≤ δ → δ ≤ δ₀ → μ {ω | C*δ^(-p) < X ω} ≤
+      ENNReal.ofReal (a*(C*Real.exp (-c*δ^(-2:ℝ))))) :
+    ∀ u : ℝ, 1 ≤ u →
+      μ {ω | max C v*δ₀^(-p)*u < X ω} ≤
+        ENNReal.ofReal (a*C*Real.exp (-(c*δ₀^(-2:ℝ))*u^(2/p))) := by
+  intro u hu1
+  have hu : 0 < u := zero_lt_one.trans_le hu1
+  let δ := δ₀*u^(-1/p)
+  have he0 : -1/p ≤ 0 := div_nonpos_of_nonpos_of_nonneg (by norm_num) hp.le
+  have hδp : 0 < δ := mul_pos hδ (Real.rpow_pos_of_pos hu _)
+  have hδd : δ ≤ δ₀ := by
+    dsimp [δ]
+    exact mul_le_of_le_one_right hδ.le (Real.rpow_le_one_of_one_le_of_nonpos hu1 he0)
+  have he : δ^(-p) = δ₀^(-p)*u := by
+    dsimp [δ]
+    rw [Real.mul_rpow hδ.le (Real.rpow_nonneg hu.le _),← Real.rpow_mul hu.le]
+    have he : (-1/p)*(-p) = 1 := by field_simp
+    rw [he,Real.rpow_one]
+  have hex : δ^(-2:ℝ) = δ₀^(-2:ℝ)*u^(2/p) := by
+    dsimp [δ]
+    rw [Real.mul_rpow hδ.le (Real.rpow_nonneg hu.le _),← Real.rpow_mul hu.le]
+    congr 2
+    ring
+  have hn (b : ℝ) (hb : b ≤ max C v) : b*δ^(-p) ≤ max C v*δ₀^(-p)*u := by
+    rw [he,←mul_assoc]
+    exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hb (Real.rpow_nonneg hδ.le _)) hu.le
+  by_cases hfd : rf ≤ δ
+  · apply (measure_mono (show {ω | max C v*δ₀^(-p)*u < X ω} ⊆
+        {ω | C*δ^(-p) < X ω} from fun ω hω => (hn C (le_max_left _ _)).trans_lt hω)).trans
+    calc
+      _ ≤ _ := ht δ hfd hδd
+      _ = _ := by
+        rw [hex]
+        have he : -c*(δ₀^(-2:ℝ)*u^(2/p)) = -(c*δ₀^(-2:ℝ))*u^(2/p) := by ring
+        rw [he,mul_assoc]
+  · have hx (ω : Ω) : X ω ≤ max C v*δ₀^(-p)*u := by
+      calc
+        X ω ≤ v*rf^(-p) := hbound ω
+        _ ≤ v*δ^(-p) := mul_le_mul_of_nonneg_left
+          (Real.rpow_le_rpow_of_nonpos hδp (le_of_not_ge hfd) (by linarith)) hv
+        _ ≤ _ := hn v (le_max_right _ _)
+    have he : {ω | max C v*δ₀^(-p)*u < X ω} = ∅ := by
+      ext ω; simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]; exact not_lt_of_ge (hx ω)
+    rw [he,measure_empty]
+    exact bot_le
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourceLiveEntranceTailTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
+    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | M*u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ 1)} ≤
+      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
+theorem sourceLiveEntranceTail_proved : SourceLiveEntranceTailTarget := by
+  intro κ hκ hκ8
+  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := source_live_unit_mesh_tail κ hκ hκ8
+  let v := (volume (ball (0:ℂ) 1)).toReal
+  let M := max C v*δ₀^(-codimension κ)
+  let b := c*δ₀^(-2:ℝ)
+  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
+  have hk : 0 < κ := by linarith
+  have hp : 0 < codimension κ := codimension_pos κ hκ8
+  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hδ _)
+  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hδ _)
+  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
+    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
+  refine ⟨M,A*C,b,hM,mul_pos hA hC,hb,?_⟩
+  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) rf (ball z₀ 1)
+  have hx (ω : a.samples) : X ω ≤ v*rf^(-codimension κ) := by
+    have hh := (a.spatialRealMass_bound hk hκ8 rf hf (ball z₀ 1) measurableSet_ball
+      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne (fun z hz' => ?_) ω).2
+    · simpa only [v,Complex.volume_ball] using hh
+    · have hd : abs (z.im-z₀.im) ≤ dist z z₀ := by
+        simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
+      have hh : dist z z₀ < 1 := hz'
+      have ha := neg_abs_le (z.im-z₀.im)
+      linarith
+  have ht' := source_mesh_tail_to_stretched μ X (codimension κ) C c δ₀ v rf
+    (μ.real {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤}) hp hC hc hδ
+    ENNReal.toReal_nonneg hf ENNReal.toReal_nonneg hx
+    (fun δ hfd hd => ht Ω m P hP γ a z₀ hz rf δ hf hfd hd) u hu
+  apply ht'.trans
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ ≤ (A*initialGreen κ z₀)*C*Real.exp (-b*u^(2/codimension κ)) :=
+      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
+        (source_live_entrance_bound a hk hκ8 z₀ hz hsmall) hC.le) (Real.exp_pos _).le
+    _ = _ := by ring
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_scaled_radius_hit_iff (c : ℝ≥0) (hc : 0 < c)
+    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (t : ℝ≥0) :
+    dist (sourceScaleTrace c γ t) ((c : ℂ) * z) - (c : ℝ) * r = 0 ↔
+      dist (γ (sourceScaleTime c t)) z - r = 0 := by
+  have hcR : 0 < (c : ℝ) := hc
+  rw [sourceScaleTrace, dist_eq_norm, ← mul_sub, norm_mul, Complex.norm_real,
+    Real.norm_eq_abs, abs_of_pos hcR, ← dist_eq_norm, ← mul_sub]
+  exact mul_eq_zero.trans (or_iff_right hcR.ne')
+
+lemma source_scale_spatial_hit (c : ℝ≥0) (hc : 0 < c)
+    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
+    spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r) =
+      WithTop.map (fun t : ℝ≥0 => c ^ 2 * t) (spatialHitTime γ z r) := by
+  classical
+  let S : Set ℝ≥0 := {t | dist (γ t) z - r = 0}
+  let Sc : Set ℝ≥0 := {t | dist (sourceScaleTrace c γ t) ((c : ℂ) * z) - (c : ℝ) * r = 0}
+  let e : ℝ≥0 ≃o ℝ≥0 := OrderIso.mulLeft₀ (c ^ 2) (pow_pos hc 2)
+  have he (x : ℝ≥0) : e x = c ^ 2 * x := rfl
+  have htime (x : ℝ≥0) : sourceScaleTime c (c ^ 2 * x) = x := by
+    simp [sourceScaleTime, hc.ne']
+  have hset : e '' S = Sc := by
+    ext t
+    constructor
+    · rintro ⟨u, hu, rfl⟩
+      exact (source_scaled_radius_hit_iff c hc γ z r _).2 (by simpa only [he, htime] using (show dist (γ u) z - r = 0 from hu))
+    · intro ht
+      refine ⟨sourceScaleTime c t, (source_scaled_radius_hit_iff c hc γ z r t).1 ht, ?_⟩
+      simp [he, sourceScaleTime, mul_div_cancel₀ _ (pow_ne_zero 2 hc.ne')]
+  unfold spatialHitTime hittingAfter
+  simp only [zero_le, true_and, Set.mem_singleton_iff]
+  change (if Sc.Nonempty then ((sInf Sc : ℝ≥0) : WithTop ℝ≥0) else ⊤) =
+    WithTop.map (fun t : ℝ≥0 => c^2*t) (if S.Nonempty then ((sInf S : ℝ≥0) : WithTop ℝ≥0) else ⊤)
+  have hh : Sc.Nonempty ↔ S.Nonempty := by rw [← hset]; exact Set.image_nonempty
+  by_cases hS : S.Nonempty
+  · rw [ite_eq_left hS, ite_eq_left (hh.mpr hS), WithTop.map_coe]
+    congr 1
+    rw [← he, e.map_csInf' hS (OrderBot.bddBelow S), hset]
+  · rw [ite_eq_right hS, ite_eq_right (mt hh.mp hS), WithTop.map_top]
+
+lemma source_scale_spatial_hit_top (c : ℝ≥0) (hc : 0 < c)
+    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
+    spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r) = ⊤ ↔
+      spatialHitTime γ z r = ⊤ := by
+  rw [source_scale_spatial_hit c hc]
+  simp
+
+lemma source_scale_spatial_hit_value (c : ℝ≥0) (hc : 0 < c)
+    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
+    (spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r)).untopD 0 =
+      c ^ 2 * (spatialHitTime γ z r).untopD 0 := by
+  rw [source_scale_spatial_hit c hc]
+  cases hh : spatialHitTime γ z r
+  · simp
+  · rfl
+
+lemma source_scale_spatial_hit_le (c : ℝ≥0) (hc : 0 < c)
+    (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (T : ℝ≥0) :
+    spatialHitTime (sourceScaleTrace c γ) ((c : ℂ) * z) ((c : ℝ) * r) ≤ (c ^ 2 * T : ℝ≥0) ↔
+      spatialHitTime γ z r ≤ T := by
+  rw [source_scale_spatial_hit c hc]
+  cases hh : spatialHitTime γ z r
+  · exact iff_of_false (WithTop.not_top_le_coe _) (WithTop.not_top_le_coe _)
+  · simp only [WithTop.map_coe, WithTop.coe_le_coe]
+    exact mul_le_mul_iff_right₀ (pow_pos hc 2)
+
+lemma source_scale_chart_deriv (c : ℝ≥0) (hc : 0 < c)
+    (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) :
+    deriv (sourceScaleChart c G t) ((c : ℂ) * z) = deriv (G (sourceScaleTime c t)) z := by
+  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
+  change deriv (fun w : ℂ => (c : ℂ) * G (sourceScaleTime c t) (w / (c : ℂ))) _ = _
+  rw [deriv_const_mul_field]
+  simp_rw [div_eq_mul_inv, mul_comm _ ((c : ℂ)⁻¹)]
+  rw [deriv_comp_mul_left]
+  simp only [smul_eq_mul]
+  simp [hcC]
+
+lemma source_scale_fullRadius (c : ℝ≥0) (hc : 0 < c)
+    (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) :
+    fullRadius (sourceScaleChart c G) t ((c : ℂ) * z) =
+      (c : ℝ) * fullRadius G (sourceScaleTime c t) z := by
+  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
+  unfold fullRadius
+  rw [source_scale_chart_deriv c hc]
+  simp only [sourceScaleChart, mul_div_cancel_left₀ _ hcC, Complex.mul_im, Complex.ofReal_re,
+    Complex.ofReal_im, zero_mul, add_zero]
+  ring
+
+lemma source_scale_greenDensity (κ : ℝ) (c : ℝ≥0) (hc : 0 < c)
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) :
+    greenDensity κ (fun u => (c : ℝ) * U (sourceScaleTime c u))
+      (sourceScaleTrace c γ) (sourceScaleChart c G) t ((c : ℂ) * z) =
+    ((c : ℝ) ^ (-codimension κ)) * greenDensity κ U γ G (sourceScaleTime c t) z := by
+  have hcR : 0 < (c : ℝ) := hc
+  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
+  have hlive : (c : ℂ) * z ∈ survivingDomain (sourceScaleTrace c γ) t ↔
+      z ∈ survivingDomain γ (sourceScaleTime c t) := by
+    simpa [hcC] using source_scale_domain_iff c t hc γ ((c : ℂ) * z)
+  by_cases hz : z ∈ survivingDomain γ (sourceScaleTime c t)
+  · simp only [greenDensity,ite_eq_left hz,ite_eq_left (hlive.mpr hz)]
+    rw [source_scale_fullRadius c hc]
+    have hR : 0 ≤ fullRadius G (sourceScaleTime c t) z := by
+      exact div_nonneg (mul_nonneg (by norm_num) (hw.2.2.2.2.2.1 _ hz).le) (norm_nonneg _)
+    rw [Real.mul_rpow hcR.le hR]
+    simp only [sourceScaleChart, mul_div_cancel_left₀ _ hcC, Complex.ofReal_mul, ← mul_sub]
+    rw [Complex.arg_real_mul _ hcR]
+    ring
+  · simp only [greenDensity,ite_eq_right hz,ite_eq_right (mt hlive.mp hz),mul_zero]
+lemma source_scale_spatial_greenWeight (κ : ℝ) (c : ℝ≥0) (hc : 0 < c)
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (z : ℂ) (r : ℝ) :
+    spatialGreenWeight κ (fun u => (c : ℝ) * U (sourceScaleTime c u))
+      (sourceScaleTrace c γ) (sourceScaleChart c G) ((c : ℂ) * z) ((c : ℝ) * r) =
+    ((c : ℝ) ^ (-codimension κ)) * spatialGreenWeight κ U γ G z r := by
+  unfold spatialGreenWeight
+  simp only [source_scale_spatial_hit_top c hc]
+  split_ifs with ht
+  · simp
+  · rw [source_scale_greenDensity κ c hc U γ G F hw, source_scale_spatial_hit_value c hc]
+    simp [sourceScaleTime, hc.ne']
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+def sourceScaled (a : RegularSLEPresentation P κ γ) (c : ℝ≥0) (hc : 0 < c) :
+    RegularSLEPresentation P κ (fun ω => sourceScaleTrace c (γ ω)) where
+  samples := a.samples
+  measurable_samples := a.measurable_samples
+  full_samples := a.full_samples
+  B := fun t ω => (c:ℝ)*a.B (sourceScaleTime c t) ω
+  measurable_B := fun t => measurable_const.mul (a.measurable_B _)
+  continuous_B := fun ω => continuous_const.mul ((a.continuous_B ω).comp (by unfold sourceScaleTime; fun_prop))
+  zero_B := fun ω => by simp [sourceScaleTime,a.zero_B ω]
+  brownian := by
+    have hh := a.brownian.smul (c := (c^2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hc.ne'))
+    simpa only [sourceScaleTime,NNReal.coe_inv,NNReal.coe_pow,Real.sqrt_inv,
+      Real.sqrt_sq c.coe_nonneg,inv_inv,div_eq_mul_inv,mul_comm] using hh
+  G := fun ω => sourceScaleChart c (a.G ω)
+  F := fun ω => sourceScaleChart c (a.F ω)
+  witness := fun ω => by
+    have hh := source_scale_witness c hc (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω)
+    simpa only [mul_left_comm (c:ℝ) (Real.sqrt κ)] using hh
+end RegularSLEPresentation
+
+lemma source_scale_initialGreen (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) (z : ℂ) (hz : 0 ≤ z.im) :
+    initialGreen κ ((c:ℂ)*z) = (c:ℝ)^(-codimension κ)*initialGreen κ z := by
+  have hcR : 0 < (c:ℝ) := hc
+  unfold initialGreen
+  rw [Complex.arg_real_mul _ hcR]
+  have him : 2*((c:ℂ)*z).im = (c:ℝ)*(2*z.im) := by simp [Complex.mul_im]; ring
+  rw [him,Real.mul_rpow hcR.le (mul_nonneg (by norm_num) hz),mul_assoc]
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_scale_area_real_integral (c : ℝ≥0) (hc : 0 < c)
+    (f : ℂ → ℝ) (K : Set ℂ) :
+    (∫ z in (fun z : ℂ => (c : ℂ) * z) '' K, f z) =
+      ((c : ℝ)^2) * ∫ z in K, f ((c : ℂ) * z) := by
+  have hcR : 0 < (c : ℝ) := hc
+  have hcC : (c : ℂ) ≠ 0 := by exact_mod_cast hc.ne'
+  let e := Homeomorph.mulLeft₀ (c : ℂ) hcC
+  have he (z : ℂ) : e z = (c : ℂ) * z := rfl
+  have hvol (s : Set ℂ) : volume (e ⁻¹' s) =
+      ENNReal.ofReal (((c : ℝ)^2)⁻¹) * volume s := by
+    change volume ((fun z : ℂ => (c : ℂ) * z) ⁻¹' s) = _
+    simpa only [Complex.real_smul, smul_eq_mul, Complex.finrank_real_complex,
+      abs_of_nonneg (inv_nonneg.mpr (sq_nonneg (c : ℝ)))] using
+      (volume : Measure ℂ).addHaar_preimage_smul hcR.ne' s
+  have hp : MeasurePreserving e (ENNReal.ofReal ((c : ℝ)^2) • (volume : Measure ℂ)) volume := by
+    refine ⟨e.continuous.measurable, ?_⟩
+    apply Measure.ext
+    intro s hs
+    rw [e.measurableEmbedding.map_apply, Measure.smul_apply, smul_eq_mul, hvol, ← mul_assoc,
+      ← ENNReal.ofReal_mul (sq_nonneg (c : ℝ)), mul_inv_cancel₀ (pow_ne_zero 2 hcR.ne'),
+      ENNReal.ofReal_one, one_mul]
+  have hh := hp.setIntegral_image_emb e.measurableEmbedding f K
+  rw [Measure.restrict_smul,integral_smul_measure] at hh
+  simpa only [he,ENNReal.toReal_ofReal (sq_nonneg (c:ℝ)),smul_eq_mul] using hh
+
+lemma source_scale_spatialRealMass (κ : ℝ) (c : ℝ≥0) (hc : 0 < c)
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (r : ℝ) (K : Set ℂ) :
+    spatialRealMass κ (fun t => (c:ℝ)*U (sourceScaleTime c t))
+      (sourceScaleTrace c γ) (sourceScaleChart c G) ((c:ℝ)*r)
+      ((fun z : ℂ => (c:ℂ)*z) '' K) =
+      (c:ℝ)^dimension κ*spatialRealMass κ U γ G r K := by
+  unfold spatialRealMass
+  rw [source_scale_area_real_integral c hc]
+  simp_rw [source_scale_spatial_greenWeight κ c hc U γ G F hw]
+  rw [integral_const_mul,←mul_assoc]
+  congr 1
+  rw [←Real.rpow_natCast (c:ℝ) 2,←Real.rpow_add (show 0 < (c:ℝ) from hc)]
+  congr 1
+  simp [codimension]
+
+lemma source_scale_ball (c : ℝ≥0) (hc : 0 < c) (z : ℂ) (r : ℝ) :
+    (fun w : ℂ => (c:ℂ)*w) '' Metric.ball z r = Metric.ball ((c:ℂ)*z) ((c:ℝ)*r) := by
+  have hcR : 0 < (c:ℝ) := hc
+  simpa only [Complex.real_smul,smul_eq_mul,Real.norm_eq_abs,abs_of_pos hcR] using
+    Metric.smul_image_ball (α := ℝ) hcR.ne' z r
+
+lemma RegularSLEPresentation.sourceScaled_mass {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (c : ℝ≥0) (hc : 0 < c) (z : ℂ) (r rf : ℝ) (ω : a.samples) :
+    spatialRealMass κ (fun t => Real.sqrt κ*(a.sourceScaled c hc).B t ω)
+      (sourceScaleTrace c (γ ω)) ((a.sourceScaled c hc).G ω) ((c:ℝ)*rf)
+      (Metric.ball ((c:ℂ)*z) ((c:ℝ)*r)) =
+      (c:ℝ)^dimension κ*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z r) := by
+  rw [←source_scale_ball c hc]
+  simpa only [RegularSLEPresentation.sourceScaled,mul_left_comm (Real.sqrt κ) (c:ℝ)] using
+    source_scale_spatialRealMass κ c hc (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) rf (Metric.ball z r)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set MeasureTheory
+lemma source_tail_threshold_normalize {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Measure Ω) (X : Ω → ℝ) (M C A b q K : ℝ)
+    (hM : 0 < M) (hC : 0 < C) (_hA : 0 < A) (hb : 0 < b) (hq : 0 < q) (hK : 0 ≤ K)
+    (hzero : μ {ω | 0 < X ω} ≤ ENNReal.ofReal (A*K))
+    (ht : ∀ u : ℝ, 1 ≤ u → μ {ω | M*u < X ω} ≤ ENNReal.ofReal (C*K*Real.exp (-b*u^q)))
+    (u : ℝ) (hu : 1 ≤ u) :
+    μ {ω | u < X ω} ≤ ENNReal.ofReal
+      ((max C A*Real.exp b)*K*Real.exp (-(b/(max M 1)^q)*u^q)) := by
+  let R := max M 1
+  have hR : 0 < R := hM.trans_le (le_max_left _ _)
+  have hu0 : 0 < u := zero_lt_one.trans_le hu
+  have he : -(b/R^q)*u^q = -b*(u/R)^q := by
+    rw [Real.div_rpow hu0.le hR.le]
+    ring
+  rw [he]
+  by_cases hbig : R ≤ u
+  · have hv : 1 ≤ u/R := (le_div_iff₀ hR).mpr (by simpa using hbig)
+    have hth : M*(u/R) ≤ u := by
+      calc
+        _ ≤ R*(u/R) := mul_le_mul_of_nonneg_right (le_max_left _ _) (div_nonneg hu0.le hR.le)
+        _ = _ := by field_simp
+    apply (measure_mono (show {ω | u < X ω} ⊆ {ω | M*(u/R) < X ω} from
+      fun ω hω => hth.trans_lt hω)).trans (ht (u/R) hv |>.trans _)
+    apply ENNReal.ofReal_le_ofReal
+    gcongr
+    calc
+      C ≤ max C A := le_max_left _ _
+      _ ≤ max C A*Real.exp b := le_mul_of_one_le_right (hC.le.trans (le_max_left _ _))
+        (Real.one_le_exp_iff.mpr hb.le)
+  · have hv : 0 ≤ u/R := div_nonneg hu0.le hR.le
+    have hv1 : u/R ≤ 1 := (div_le_one hR).mpr (le_of_not_ge hbig)
+    have hp : (u/R)^q ≤ 1 := Real.rpow_le_one hv hv1 hq.le
+    have hexp : 1 ≤ Real.exp b*Real.exp (-b*(u/R)^q) := by
+      rw [←Real.exp_add]
+      apply Real.one_le_exp_iff.mpr
+      nlinarith
+    apply (measure_mono (show {ω | u < X ω} ⊆ {ω | 0 < X ω} from
+      fun ω hω => hu0.trans hω)).trans (hzero.trans _)
+    apply ENNReal.ofReal_le_ofReal
+    calc
+      A*K ≤ max C A*K := mul_le_mul_of_nonneg_right (le_max_right _ _) hK
+      _ ≤ (max C A*K)*(Real.exp b*Real.exp (-b*(u/R)^q)) :=
+        le_mul_of_one_le_right (mul_nonneg (hC.le.trans (le_max_left _ _)) hK) hexp
+      _ = _ := by ring
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_live_entrance_all_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
+  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
+    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
+      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ))) := by
+  obtain ⟨M,C,b,hM,hC,hb,ht⟩ := sourceLiveEntranceTail_proved κ hκ hκ8
+  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
+  have hk : 0 < κ := by linarith
+  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
+    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
+  refine ⟨max C A*Real.exp b,b/(max M 1)^(2/codimension κ),
+    mul_pos (hC.trans_le (le_max_left _ _)) (Real.exp_pos _),
+    div_pos hb (Real.rpow_pos_of_pos (hM.trans_le (le_max_left _ _)) _),?_⟩
+  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) rf (ball z₀ 1)
+  have hK : 0 < initialGreen κ z₀ := initialGreen_pos κ z₀ (by change 0 < z₀.im; linarith)
+  have hzero : μ {ω | 0 < X ω} ≤ ENNReal.ofReal (A*initialGreen κ z₀) := by
+    apply le_trans (measure_mono (show {ω | 0 < X ω} ⊆
+      {ω : a.samples | liveSpatialHitTime (γ ω) z₀ 3 ≠ ⊤} from ?_))
+    · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (mul_pos hA hK).le).mpr
+      exact source_live_entrance_bound a hk hκ8 z₀ hz hsmall
+    · intro ω hω hn
+      have hx : X ω = 0 := by
+        apply setIntegral_eq_zero_of_forall_eq_zero
+        intro z hz'
+        exact source_weight_zero_no_unit_entrance (a.witness ω) κ z₀ hz rf hf1 hn z hz'
+      change 0 < X ω at hω
+      rw [hx] at hω
+      exact (lt_irrefl 0) hω
+  exact source_tail_threshold_normalize μ X M C A b (2/codimension κ) (initialGreen κ z₀)
+    hM hC hA hb (div_pos (by norm_num) (codimension_pos κ hκ8)) hK.le hzero
+    (fun v hv => ht Ω m P hP γ a z₀ hz hsmall rf v hf hf1 hv) u hu
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourcePhysicalBallAllTailTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (r : ℝ),
+    0 < r → 4*r ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*(4*r))) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ r → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | r^dimension κ*u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ r)} ≤
+      ENNReal.ofReal (C*r^codimension κ*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
+theorem sourcePhysicalBallAllTail_proved : SourcePhysicalBallAllTailTarget := by
+  intro κ hκ hκ8
+  obtain ⟨C,b,hC,hb,ht⟩ := source_live_entrance_all_tail κ hκ hκ8
+  refine ⟨C,b,hC,hb,?_⟩
+  intro Ω m P hP γ a z r hr hz hsmall rf u hf hfr hu
+  let : IsProbabilityMeasure P := hP
+  let c : ℝ≥0 := ⟨r⁻¹,(inv_pos.mpr hr).le⟩
+  have hc : 0 < c := inv_pos.mpr hr
+  have hcR : (c:ℝ) = r⁻¹ := rfl
+  let a' := a.sourceScaled c hc
+  have him : ((c:ℂ)*z).im = r⁻¹*z.im := by simp [hcR,Complex.mul_im]
+  have hcr : (c:ℝ)*r = 1 := by rw [hcR,inv_mul_cancel₀ hr.ne']
+  have hheight : 4 ≤ ((c:ℂ)*z).im := by
+    rw [him,mul_comm,←div_eq_mul_inv]
+    exact (le_div_iff₀ hr).mpr hz
+  have hsmall' : 1 ≤ Real.log ((2*((c:ℂ)*z).im)/((128*Real.pi)*4)) := by
+    have he : (2*((c:ℂ)*z).im)/((128*Real.pi)*4) = (2*z.im)/((128*Real.pi)*(4*r)) := by
+      rw [him]
+      field_simp
+    rwa [he]
+  have hf' : 0 < (c:ℝ)*rf := mul_pos hc hf
+  have hfr' : (c:ℝ)*rf ≤ 1 := by rw [←hcr]; exact mul_le_mul_of_nonneg_left hfr hc.le
+  have htail := ht Ω m P hP (fun ω => sourceScaleTrace c (γ ω)) a' ((c:ℂ)*z)
+    hheight hsmall' ((c:ℝ)*rf) u hf' hfr' hu
+  change (P.comap (Subtype.val : a.samples → Ω)) {ω : a.samples | u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a'.B t ω) (sourceScaleTrace c (γ ω)) (a'.G ω)
+      ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} ≤
+      ENNReal.ofReal (C*initialGreen κ ((c:ℂ)*z)*Real.exp (-b*u^(2/codimension κ))) at htail
+  have hp : 0 < (c:ℝ)^dimension κ := Real.rpow_pos_of_pos hc _
+  have hrp : 0 < r^dimension κ := Real.rpow_pos_of_pos hr _
+  have hmr : (c:ℝ)^dimension κ*(r^dimension κ*u) = u := by
+    rw [hcR,Real.inv_rpow hr.le]
+    field_simp
+  have hx (ω : a.samples) : spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
+      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1) =
+      (c:ℝ)^dimension κ*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z r) := by
+    simpa only [hcr] using a.sourceScaled_mass c hc z r rf ω
+  have he : {ω : a.samples | u < spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
+      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} =
+      {ω : a.samples | r^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) rf (ball z r)} := by
+    ext ω
+    change (u < _) ↔ _
+    rw [hx ω]
+    conv_lhs => rw [←hmr]
+    exact mul_lt_mul_iff_right₀ hp
+  rw [he] at htail
+  have hg : initialGreen κ ((c:ℂ)*z) = r^codimension κ*initialGreen κ z := by
+    rw [source_scale_initialGreen κ c hc z (by nlinarith),hcR,Real.inv_rpow hr.le,Real.rpow_neg hr.le,inv_inv]
+  rw [hg] at htail
+  convert htail using 1
+  congr 1
+  ring
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SausageCharge.lean b/lean/OAI/Probability/SLE/SausageCharge.lean
index bd936bbd4b80acd2568fea9b659abc14eb45f170..f38feedfefca8bf5890090db2134a2b0ef2e789d 100644
--- a/lean/OAI/Probability/SLE/SausageCharge.lean
+++ b/lean/OAI/Probability/SLE/SausageCharge.lean
@@ -1,223 +1,223 @@
-import Mathlib
-import OAI.Probability.SLE.TraceMeasurability
-
-namespace OAI
-
-/-! Sausage Charge for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance wholeSausageChargePathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance wholeSausageChargePathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
- 
-def wholeSausageSet (γ : ℝ≥0 → ℂ) (v : ℝ) : Set ℂ :=
-  {z | ∃ t : ℝ≥0, dist (γ t) z < v}
-
- 
-def wholeSausageCharge (κ : ℝ) (γ : ℝ≥0 → ℂ) (v : ℝ) (A : Set ℂ) : ℝ≥0∞ :=
-  ENNReal.ofReal (v ^ (-codimension κ)) * volume (A ∩ wholeSausageSet γ v)
-
-lemma wholeSausageSet_isOpen (γ : ℝ≥0 → ℂ) (v : ℝ) :
-    IsOpen (wholeSausageSet γ v) := by
-  have he : wholeSausageSet γ v = ⋃ t : ℝ≥0, ball (γ t) v := by
-    ext z
-    simp [wholeSausageSet, dist_comm]
-  rw [he]
-  exact isOpen_iUnion fun _ => isOpen_ball
-
-lemma wholeSausage_joint_open (v : ℝ) :
-    IsOpen {p : C(ℝ≥0, ℂ) × ℂ | p.2 ∈ wholeSausageSet p.1 v} := by
-  have he : {p : C(ℝ≥0, ℂ) × ℂ | p.2 ∈ wholeSausageSet p.1 v} =
-      ⋃ t : ℝ≥0, {p : C(ℝ≥0, ℂ) × ℂ | dist (p.1 t) p.2 < v} := by
-    ext p
-    simp [wholeSausageSet]
-  rw [he]
-  exact isOpen_iUnion fun t => isOpen_lt
-    (((continuous_eval_const t).comp continuous_fst).dist continuous_snd) continuous_const
-
-lemma wholeSausageCharge_measurable (κ v : ℝ) (A : Set ℂ) :
-    Measurable (fun g : C(ℝ≥0, ℂ) => wholeSausageCharge κ g v A) := by
-  classical
-  have hm : Measurable (fun p : C(ℝ≥0, ℂ) × ℂ =>
-      if p.2 ∈ wholeSausageSet p.1 v then (1 : ℝ≥0∞) else 0) :=
-    Measurable.ite (wholeSausage_joint_open v).measurableSet measurable_const measurable_const
-  apply measurable_const.mul
-  have hh := hm.lintegral_prod_right' (ν := volume.restrict A)
-  convert hh using 1
-  ext g
-  change volume (A ∩ wholeSausageSet g v) =
-    ∫⁻ z, (wholeSausageSet g v).indicator (fun _ => 1) z ∂(volume.restrict A)
-  rw [lintegral_indicator (wholeSausageSet_isOpen g v).measurableSet]
-  simp [Measure.restrict_apply (wholeSausageSet_isOpen g v).measurableSet, inter_comm]
-
- 
-lemma wholeSausageCharge_expectation
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [SFinite P]
-    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (κ v : ℝ) (A : Set ℂ) :
-    (∫⁻ ω, wholeSausageCharge κ (f ω) v A ∂P) =
-      ∫⁻ z in A, ENNReal.ofReal (v ^ (-codimension κ)) *
-        P {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} := by
-  classical
-  let I : Ω × ℂ → ℝ≥0∞ := fun p =>
-    if p.2 ∈ wholeSausageSet (f p.1) v then 1 else 0
-  have hset : MeasurableSet {p : Ω × ℂ | p.2 ∈ wholeSausageSet (f p.1) v} :=
-    (wholeSausage_joint_open v).measurableSet.preimage
-      ((hf.comp measurable_fst).prodMk measurable_snd)
-  have hI : Measurable I := Measurable.ite hset measurable_const measurable_const
-  have hre (ω : Ω) : wholeSausageCharge κ (f ω) v A =
-      ENNReal.ofReal (v ^ (-codimension κ)) *
-        ∫⁻ z, I (ω, z) ∂(volume.restrict A) := by
-    unfold wholeSausageCharge
-    congr 1
-    change volume (A ∩ wholeSausageSet (f ω) v) =
-      ∫⁻ z, (wholeSausageSet (f ω) v).indicator (fun _ => 1) z ∂(volume.restrict A)
-    rw [lintegral_indicator (wholeSausageSet_isOpen (f ω) v).measurableSet]
-    simp [Measure.restrict_apply (wholeSausageSet_isOpen (f ω) v).measurableSet, inter_comm]
-  have hem (z : ℂ) : MeasurableSet {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} :=
-    hset.preimage (measurable_id.prodMk measurable_const)
-  have hei (z : ℂ) : (∫⁻ ω, I (ω, z) ∂P) =
-      P {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} := by
-    exact lintegral_indicator_one (hem z)
-  simp_rw [hre]
-  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,
-    lintegral_lintegral_swap hI.aemeasurable]
-  simp_rw [hei]
-  exact (lintegral_const_mul' _ _ ENNReal.ofReal_ne_top).symm
-
- 
-def wholeSausageChargeBound (κ R : ℝ) : ℝ≥0∞ :=
-  ∫⁻ z in closedBall (0 : ℂ) R,
-    ENNReal.ofReal (sausagePointConstant κ * |z.im| ^ (-codimension κ))
-
-lemma wholeSausageChargeBound_lt_top (κ R : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    wholeSausageChargeBound κ R < ∞ := by
-  have hi : Integrable (fun z : ℂ =>
-      sausagePointConstant κ * |z.im| ^ (-codimension κ))
-      (volume.restrict (closedBall (0 : ℂ) R)) :=
-    (integrable_height_power_ball (codimension κ) R (codimension_lt_one κ hκ)).const_mul _
-  exact (hasFiniteIntegral_iff_ofReal (Eventually.of_forall fun (z : ℂ) =>
-    mul_nonneg (sausagePointConstant_pos κ hκ hκ8).le
-      (Real.rpow_nonneg (abs_nonneg z.im) _))).mp hi.2
-
-lemma regularSLE_wholeSausageCharge_bound
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (hc : ∀ ω, Continuous (γ ω)) (R v : ℝ) (hv : 0 < v) :
-    (∫⁻ ω, wholeSausageCharge κ (γ ω) v (closedBall 0 R) ∂P) ≤
-      wholeSausageChargeBound κ R := by
-  let f : Ω → C(ℝ≥0, ℂ) := fun ω => ⟨γ ω, hc ω⟩
-  have hf : Measurable f := source_whole_path_measurable f hSLE.1
-  change (∫⁻ ω, wholeSausageCharge κ (f ω) v (closedBall 0 R) ∂P) ≤ _
-  rw [wholeSausageCharge_expectation P f hf]
-  obtain ⟨B, G, F, hB, hw⟩ := hSLE.exists_charts
-  apply lintegral_mono_ae
-  filter_upwards [ae_restrict_of_ae ae_im_ne_zero] with z hz
-  have hb := normalized_visit_bound hB κ hκ hκ8 γ G F hw z hz v hv
-  have he : P {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} =
-      ENNReal.ofReal (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < v}).toReal :=
-    (ENNReal.ofReal_toReal (measure_ne_top P _)).symm
-  rw [he, ← ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _)]
-  exact ENNReal.ofReal_le_ofReal hb
-
- 
-
-theorem source_cover_charge
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) (R : ℝ) :
-    wholeSausageChargeBound κ R < ∞ ∧
-    ∀ v : ℝ, 0 < v →
-      AEMeasurable (fun ω => wholeSausageCharge κ (γ ω) v (closedBall 0 R)) P ∧
-      (∫⁻ ω, wholeSausageCharge κ (γ ω) v (closedBall 0 R) ∂P) ≤
-        wholeSausageChargeBound κ R := by
-  classical
-  refine ⟨wholeSausageChargeBound_lt_top κ R hκ hκ8, ?_⟩
-  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  have : IsProbabilityMeasure Q := a.probability
-  let g : a.samples → ℝ≥0 → ℂ := fun ω => γ ω
-  have hsle : IsOrdinaryChordalSLE κ g Q := by
-    refine ⟨fun t => (hSLE.1 t).comp measurable_subtype_coe, a.B, a.brownian, ?_⟩
-    exact Eventually.of_forall fun ω => ⟨a.G ω, a.F ω, a.witness ω⟩
-  let f : a.samples → C(ℝ≥0, ℂ) := fun ω => ⟨g ω, a.trace_continuous ω⟩
-  have hfm : Measurable f := source_whole_path_measurable f hsle.1
-  intro v hv
-  have hm := (wholeSausageCharge_measurable κ v (closedBall (0 : ℂ) R)).comp hfm
-  change Measurable (fun ω : a.samples => wholeSausageCharge κ (g ω) v (closedBall 0 R)) at hm
-  have hb := regularSLE_wholeSausageCharge_bound Q κ hκ hκ8 g hsle a.trace_continuous R v hv
-  have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) :=
-    MeasurableEmbedding.subtype_coe a.measurable_samples
-  constructor
-  · exact (a.measurePreserving_val.aemeasurable_comp_iff hemb).mp hm.aemeasurable
-  · have he := a.measurePreserving_val.lintegral_comp_emb hemb
-      (fun ω => wholeSausageCharge κ (γ ω) v (closedBall 0 R))
-    change (∫⁻ ω, wholeSausageCharge κ (g ω) v (closedBall 0 R) ∂Q) = _ at he
-    rw [← he]
-    exact hb
-
- 
-
-lemma source_cover_charge_sup
-    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (R : ℝ) (_hR : 0 < R) :
-    ∃ C : ℝ≥0, (⨆ (v : ℝ) (_ : 0 < v) (_ : v ≤ 1),
-      ∫⁻ ω, wholeSausageCharge κ (γ ω) v (ball 0 R) ∂P) ≤ C := by
-  obtain ⟨hfinite, hbound⟩ := source_cover_charge P κ hκ hκ8 γ hSLE R
-  refine ⟨(wholeSausageChargeBound κ R).toNNReal, ?_⟩
-  rw [ENNReal.coe_toNNReal hfinite.ne]
-  refine iSup_le fun v => iSup_le fun hv => iSup_le fun _hv1 => ?_
-  apply le_trans _ (hbound v hv).2
-  apply lintegral_mono
-  intro ω
-  apply mul_le_mul' le_rfl
-  exact measure_mono (inter_subset_inter_left _ ball_subset_closedBall)
-end
-end SLEExactGauge
-end
-
- 
-
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.TraceMeasurability
+
+namespace OAI
+
+/-! Sausage Charge for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance wholeSausageChargePathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance wholeSausageChargePathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+ 
+def wholeSausageSet (γ : ℝ≥0 → ℂ) (v : ℝ) : Set ℂ :=
+  {z | ∃ t : ℝ≥0, dist (γ t) z < v}
+
+ 
+def wholeSausageCharge (κ : ℝ) (γ : ℝ≥0 → ℂ) (v : ℝ) (A : Set ℂ) : ℝ≥0∞ :=
+  ENNReal.ofReal (v ^ (-codimension κ)) * volume (A ∩ wholeSausageSet γ v)
+
+lemma wholeSausageSet_isOpen (γ : ℝ≥0 → ℂ) (v : ℝ) :
+    IsOpen (wholeSausageSet γ v) := by
+  have he : wholeSausageSet γ v = ⋃ t : ℝ≥0, ball (γ t) v := by
+    ext z
+    simp [wholeSausageSet, dist_comm]
+  rw [he]
+  exact isOpen_iUnion fun _ => isOpen_ball
+
+lemma wholeSausage_joint_open (v : ℝ) :
+    IsOpen {p : C(ℝ≥0, ℂ) × ℂ | p.2 ∈ wholeSausageSet p.1 v} := by
+  have he : {p : C(ℝ≥0, ℂ) × ℂ | p.2 ∈ wholeSausageSet p.1 v} =
+      ⋃ t : ℝ≥0, {p : C(ℝ≥0, ℂ) × ℂ | dist (p.1 t) p.2 < v} := by
+    ext p
+    simp [wholeSausageSet]
+  rw [he]
+  exact isOpen_iUnion fun t => isOpen_lt
+    (((continuous_eval_const t).comp continuous_fst).dist continuous_snd) continuous_const
+
+lemma wholeSausageCharge_measurable (κ v : ℝ) (A : Set ℂ) :
+    Measurable (fun g : C(ℝ≥0, ℂ) => wholeSausageCharge κ g v A) := by
+  classical
+  have hm : Measurable (fun p : C(ℝ≥0, ℂ) × ℂ =>
+      if p.2 ∈ wholeSausageSet p.1 v then (1 : ℝ≥0∞) else 0) :=
+    Measurable.ite (wholeSausage_joint_open v).measurableSet measurable_const measurable_const
+  apply measurable_const.mul
+  have hh := hm.lintegral_prod_right' (ν := volume.restrict A)
+  convert hh using 1
+  ext g
+  change volume (A ∩ wholeSausageSet g v) =
+    ∫⁻ z, (wholeSausageSet g v).indicator (fun _ => 1) z ∂(volume.restrict A)
+  rw [lintegral_indicator (wholeSausageSet_isOpen g v).measurableSet]
+  simp [Measure.restrict_apply (wholeSausageSet_isOpen g v).measurableSet, inter_comm]
+
+ 
+lemma wholeSausageCharge_expectation
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [SFinite P]
+    (f : Ω → C(ℝ≥0, ℂ)) (hf : Measurable f) (κ v : ℝ) (A : Set ℂ) :
+    (∫⁻ ω, wholeSausageCharge κ (f ω) v A ∂P) =
+      ∫⁻ z in A, ENNReal.ofReal (v ^ (-codimension κ)) *
+        P {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} := by
+  classical
+  let I : Ω × ℂ → ℝ≥0∞ := fun p =>
+    if p.2 ∈ wholeSausageSet (f p.1) v then 1 else 0
+  have hset : MeasurableSet {p : Ω × ℂ | p.2 ∈ wholeSausageSet (f p.1) v} :=
+    (wholeSausage_joint_open v).measurableSet.preimage
+      ((hf.comp measurable_fst).prodMk measurable_snd)
+  have hI : Measurable I := Measurable.ite hset measurable_const measurable_const
+  have hre (ω : Ω) : wholeSausageCharge κ (f ω) v A =
+      ENNReal.ofReal (v ^ (-codimension κ)) *
+        ∫⁻ z, I (ω, z) ∂(volume.restrict A) := by
+    unfold wholeSausageCharge
+    congr 1
+    change volume (A ∩ wholeSausageSet (f ω) v) =
+      ∫⁻ z, (wholeSausageSet (f ω) v).indicator (fun _ => 1) z ∂(volume.restrict A)
+    rw [lintegral_indicator (wholeSausageSet_isOpen (f ω) v).measurableSet]
+    simp [Measure.restrict_apply (wholeSausageSet_isOpen (f ω) v).measurableSet, inter_comm]
+  have hem (z : ℂ) : MeasurableSet {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} :=
+    hset.preimage (measurable_id.prodMk measurable_const)
+  have hei (z : ℂ) : (∫⁻ ω, I (ω, z) ∂P) =
+      P {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} := by
+    exact lintegral_indicator_one (hem z)
+  simp_rw [hre]
+  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,
+    lintegral_lintegral_swap hI.aemeasurable]
+  simp_rw [hei]
+  exact (lintegral_const_mul' _ _ ENNReal.ofReal_ne_top).symm
+
+ 
+def wholeSausageChargeBound (κ R : ℝ) : ℝ≥0∞ :=
+  ∫⁻ z in closedBall (0 : ℂ) R,
+    ENNReal.ofReal (sausagePointConstant κ * |z.im| ^ (-codimension κ))
+
+lemma wholeSausageChargeBound_lt_top (κ R : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    wholeSausageChargeBound κ R < ∞ := by
+  have hi : Integrable (fun z : ℂ =>
+      sausagePointConstant κ * |z.im| ^ (-codimension κ))
+      (volume.restrict (closedBall (0 : ℂ) R)) :=
+    (integrable_height_power_ball (codimension κ) R (codimension_lt_one κ hκ)).const_mul _
+  exact (hasFiniteIntegral_iff_ofReal (Eventually.of_forall fun (z : ℂ) =>
+    mul_nonneg (sausagePointConstant_pos κ hκ hκ8).le
+      (Real.rpow_nonneg (abs_nonneg z.im) _))).mp hi.2
+
+lemma regularSLE_wholeSausageCharge_bound
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (hc : ∀ ω, Continuous (γ ω)) (R v : ℝ) (hv : 0 < v) :
+    (∫⁻ ω, wholeSausageCharge κ (γ ω) v (closedBall 0 R) ∂P) ≤
+      wholeSausageChargeBound κ R := by
+  let f : Ω → C(ℝ≥0, ℂ) := fun ω => ⟨γ ω, hc ω⟩
+  have hf : Measurable f := source_whole_path_measurable f hSLE.1
+  change (∫⁻ ω, wholeSausageCharge κ (f ω) v (closedBall 0 R) ∂P) ≤ _
+  rw [wholeSausageCharge_expectation P f hf]
+  obtain ⟨B, G, F, hB, hw⟩ := hSLE.exists_charts
+  apply lintegral_mono_ae
+  filter_upwards [ae_restrict_of_ae ae_im_ne_zero] with z hz
+  have hb := normalized_visit_bound hB κ hκ hκ8 γ G F hw z hz v hv
+  have he : P {ω | ∃ t : ℝ≥0, dist (f ω t) z < v} =
+      ENNReal.ofReal (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < v}).toReal :=
+    (ENNReal.ofReal_toReal (measure_ne_top P _)).symm
+  rw [he, ← ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _)]
+  exact ENNReal.ofReal_le_ofReal hb
+
+ 
+
+theorem source_cover_charge
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) (R : ℝ) :
+    wholeSausageChargeBound κ R < ∞ ∧
+    ∀ v : ℝ, 0 < v →
+      AEMeasurable (fun ω => wholeSausageCharge κ (γ ω) v (closedBall 0 R)) P ∧
+      (∫⁻ ω, wholeSausageCharge κ (γ ω) v (closedBall 0 R) ∂P) ≤
+        wholeSausageChargeBound κ R := by
+  classical
+  refine ⟨wholeSausageChargeBound_lt_top κ R hκ hκ8, ?_⟩
+  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  have : IsProbabilityMeasure Q := a.probability
+  let g : a.samples → ℝ≥0 → ℂ := fun ω => γ ω
+  have hsle : IsOrdinaryChordalSLE κ g Q := by
+    refine ⟨fun t => (hSLE.1 t).comp measurable_subtype_coe, a.B, a.brownian, ?_⟩
+    exact Eventually.of_forall fun ω => ⟨a.G ω, a.F ω, a.witness ω⟩
+  let f : a.samples → C(ℝ≥0, ℂ) := fun ω => ⟨g ω, a.trace_continuous ω⟩
+  have hfm : Measurable f := source_whole_path_measurable f hsle.1
+  intro v hv
+  have hm := (wholeSausageCharge_measurable κ v (closedBall (0 : ℂ) R)).comp hfm
+  change Measurable (fun ω : a.samples => wholeSausageCharge κ (g ω) v (closedBall 0 R)) at hm
+  have hb := regularSLE_wholeSausageCharge_bound Q κ hκ hκ8 g hsle a.trace_continuous R v hv
+  have hemb : MeasurableEmbedding (Subtype.val : a.samples → Ω) :=
+    MeasurableEmbedding.subtype_coe a.measurable_samples
+  constructor
+  · exact (a.measurePreserving_val.aemeasurable_comp_iff hemb).mp hm.aemeasurable
+  · have he := a.measurePreserving_val.lintegral_comp_emb hemb
+      (fun ω => wholeSausageCharge κ (γ ω) v (closedBall 0 R))
+    change (∫⁻ ω, wholeSausageCharge κ (g ω) v (closedBall 0 R) ∂Q) = _ at he
+    rw [← he]
+    exact hb
+
+ 
+
+lemma source_cover_charge_sup
+    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (R : ℝ) (_hR : 0 < R) :
+    ∃ C : ℝ≥0, (⨆ (v : ℝ) (_ : 0 < v) (_ : v ≤ 1),
+      ∫⁻ ω, wholeSausageCharge κ (γ ω) v (ball 0 R) ∂P) ≤ C := by
+  obtain ⟨hfinite, hbound⟩ := source_cover_charge P κ hκ hκ8 γ hSLE R
+  refine ⟨(wholeSausageChargeBound κ R).toNNReal, ?_⟩
+  rw [ENNReal.coe_toNNReal hfinite.ne]
+  refine iSup_le fun v => iSup_le fun hv => iSup_le fun _hv1 => ?_
+  apply le_trans _ (hbound v hv).2
+  apply lintegral_mono
+  intro ω
+  apply mul_le_mul' le_rfl
+  exact measure_mono (inter_subset_inter_left _ ball_subset_closedBall)
+end
+end SLEExactGauge
+end
+
+ 
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SausagePointBound.lean b/lean/OAI/Probability/SLE/SausagePointBound.lean
index 4d8ab0856e2bae0fca71a941386ed44fbfbb9565..47d1b429cb53d938acd7662a8f9c506b83575d04 100644
--- a/lean/OAI/Probability/SLE/SausagePointBound.lean
+++ b/lean/OAI/Probability/SLE/SausagePointBound.lean
@@ -1,582 +1,582 @@
-import Mathlib
-import OAI.Probability.SLE.OccupationBound
-import OAI.Probability.SLE.StripKernels
-
-namespace OAI
-
-/-! Sausage Point Bound for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def spatialSmallnessConstant : ℝ := 128 * Real.pi * Real.exp 1 + 1
-noncomputable def spatialOnePointConstant (κ : ℝ) : ℝ :=
-  uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ
-
-lemma spatialSmallnessConstant_gt_one : 1 < spatialSmallnessConstant := by
-  dsimp [spatialSmallnessConstant]
-  have := mul_pos (mul_pos (by norm_num : (0 : ℝ) < 128) Real.pi_pos) (Real.exp_pos 1)
-  linarith
-
-lemma spatial_smallness (z : ℂ) (r : ℝ) (hr : 0 < r)
-    (hy : spatialSmallnessConstant * r < z.im) :
-    0 < z.im ∧ r ≤ z.im ∧ 1 ≤ Real.log ((2 * z.im) / ((128 * Real.pi) * r)) := by
-  have hA : 1 ≤ spatialSmallnessConstant := spatialSmallnessConstant_gt_one.le
-  have hry : r < z.im := (le_mul_of_one_le_left hr.le hA).trans_lt hy
-  refine ⟨hr.trans hry, hry.le, ?_⟩
-  have hC : 0 < (128 * Real.pi) * r := mul_pos (mul_pos (by norm_num) Real.pi_pos) hr
-  have hquot : 0 < (2 * z.im) / ((128 * Real.pi) * r) :=
-    div_pos (mul_pos (by norm_num) (hr.trans hry)) hC
-  apply (Real.le_log_iff_exp_le hquot).mpr
-  apply (le_div_iff₀ hC).mpr
-  dsimp [spatialSmallnessConstant] at hy
-  nlinarith [Real.exp_pos 1]
-
-lemma initialGreen_le_height (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    initialGreen κ z ≤ (2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ) := by
-  have hs : 0 ≤ Real.sin (Complex.arg z) := by
-    rw [Complex.sin_arg]; exact div_nonneg hz.le (norm_nonneg _)
-  calc
-    initialGreen κ z ≤ (2 * z.im) ^ (-codimension κ) * 1 :=
-      mul_le_mul_of_nonneg_left (Real.rpow_le_one hs (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
-        (Real.rpow_nonneg (mul_nonneg (by norm_num) hz.le) _)
-    _ = (2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ) := by
-      rw [mul_one, Real.mul_rpow (by norm_num) hz.le]
-
-noncomputable def sausagePointConstant (κ : ℝ) : ℝ :=
-  max (spatialOnePointConstant κ * (2 : ℝ) ^ (-codimension κ))
-    (spatialSmallnessConstant ^ codimension κ)
-
-lemma spatialOnePointConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < spatialOnePointConstant κ :=
-  mul_pos (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num))
-    (Real.rpow_pos_of_pos (mul_pos (by norm_num) Real.pi_pos) _)
-
-lemma sausagePointConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < sausagePointConstant κ :=
-  (mul_pos (spatialOnePointConstant_pos κ hκ hκ8) (Real.rpow_pos_of_pos (by norm_num) _)).trans_le
-    (le_max_left _ _)
-
-lemma negative_power_scale (x r A p : ℝ) (hx : 0 < x) (hr : 0 < r)
-    (hA : 0 < A) (hp : 0 ≤ p) (hxr : x ≤ A * r) :
-    r ^ (-p) ≤ A ^ p * x ^ (-p) := by
-  have hh := Real.rpow_le_rpow hx.le hxr hp
-  rw [Real.mul_rpow hA.le hr.le] at hh
-  rw [Real.rpow_neg hr.le, Real.rpow_neg hx.le, ← div_eq_mul_inv]
-  apply (le_div_iff₀ (Real.rpow_pos_of_pos hx p)).mpr
-  exact (inv_mul_le_iff₀ (Real.rpow_pos_of_pos hr p)).mpr (by simpa [mul_comm] using hh)
-
- 
-
-theorem normalized_visit_bound
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z.im ≠ 0) (r : ℝ) (hr : 0 < r) :
-    r ^ (-codimension κ) * (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-      sausagePointConstant κ * |z.im| ^ (-codimension κ) := by
-  have hp := codimension_pos κ hκ8
-  have hA : 0 < spatialSmallnessConstant := zero_lt_one.trans spatialSmallnessConstant_gt_one
-  have hscale (hy : |z.im| ≤ spatialSmallnessConstant * r) :
-      r ^ (-codimension κ) * (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-        sausagePointConstant κ * |z.im| ^ (-codimension κ) := by
-    calc
-      _ ≤ r ^ (-codimension κ) * 1 := mul_le_mul_of_nonneg_left measureReal_le_one
-        (Real.rpow_nonneg hr.le _)
-      _ ≤ spatialSmallnessConstant ^ codimension κ * |z.im| ^ (-codimension κ) := by
-        rw [mul_one]
-        exact negative_power_scale _ _ _ _ (abs_pos.mpr hz) hr hA hp.le hy
-      _ ≤ _ := mul_le_mul_of_nonneg_right (le_max_right _ _)
-        (Real.rpow_nonneg (abs_nonneg _) _)
-  by_cases hsmall : spatialSmallnessConstant * r < z.im
-  · obtain ⟨hy, hry, hlog⟩ := spatial_smallness z r hr hsmall
-    have hh := euclidean_approach hB κ hκ hκ8 γ G F hw z hy r hr hry hlog
-    have hS := (spatialOnePointConstant_pos κ hκ hκ8).le
-    change (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-      spatialOnePointConstant κ * initialGreen κ z * r ^ codimension κ at hh
-    have hcancel : r ^ (-codimension κ) * r ^ codimension κ = 1 := by
-      rw [← Real.rpow_add hr]; simp
-    calc
-      _ ≤ r ^ (-codimension κ) * (spatialOnePointConstant κ * initialGreen κ z * r ^ codimension κ) :=
-        mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg hr.le _)
-      _ = spatialOnePointConstant κ * initialGreen κ z := by
-        calc
-          _ = (spatialOnePointConstant κ * initialGreen κ z) *
-              (r ^ (-codimension κ) * r ^ codimension κ) := by ring
-          _ = _ := by rw [hcancel, mul_one]
-      _ ≤ spatialOnePointConstant κ * ((2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ)) :=
-        mul_le_mul_of_nonneg_left (initialGreen_le_height κ hκ hκ8 z hy) hS
-      _ ≤ _ := by
-        rw [abs_of_pos hy, ← mul_assoc]
-        exact mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg hy.le _)
-  · by_cases hy : 0 ≤ z.im
-    · apply hscale
-      rw [abs_of_nonneg hy]
-      exact le_of_not_gt hsmall
-    · by_cases hzr : -r < z.im
-      · apply hscale
-        rw [abs_of_neg (lt_of_not_ge hy)]
-        exact (by linarith : -z.im ≤ r).trans (le_mul_of_one_le_left hr.le spatialSmallnessConstant_gt_one.le)
-      · have hzero : P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r} = 0 := by
-          have hn : ∀ᵐ ω ∂P, ¬ (∃ t : ℝ≥0, dist (γ ω t) z < r) := by
-            filter_upwards [hw] with ω hω
-            rintro ⟨t, ht⟩
-            have hi := Complex.im_le_norm (γ ω t - z)
-            rw [Complex.sub_im, ← dist_eq_norm] at hi
-            have hnon := hω.2.2.1 t
-            linarith [le_of_not_gt hzr]
-          simpa only [not_not, ofPred_mem_eq] using ae_iff.mp hn
-        rw [hzero, ENNReal.toReal_zero, mul_zero]
-        exact mul_nonneg (sausagePointConstant_pos κ hκ hκ8).le (Real.rpow_nonneg (abs_nonneg _) _)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma IsOrdinaryChordalSLE.exists_charts
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) :
-    ∃ (B : ℝ≥0 → Ω → ℝ) (G F : Ω → ℝ≥0 → ℂ → ℂ),
-      IsBrownianReal B P ∧ ∀ᵐ ω ∂P, IsCapacityTwoWitness
-        (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω) := by
-  classical
-  obtain ⟨B,hB,htrace⟩ := hSLE.2
-  have hchoice (ω : Ω) : ∃ G F : ℝ≥0 → ℂ → ℂ,
-      IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω) →
-        IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) G F := by
-    by_cases h : IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω)
-    · obtain ⟨G,F,hw⟩ := h
-      exact ⟨G,F,fun _ => hw⟩
-    · exact ⟨fun _ _ => 0,fun _ _ => 0,fun hh => False.elim (h hh)⟩
-  choose G F hw using hchoice
-  exact ⟨B,G,F,hB,htrace.mono fun ω hω => hw ω hω⟩
-
- 
-
-lemma sle_high_center_approach
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : Real.sqrt r ≤ z.im) :
-    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-      sausagePointConstant κ * r^(codimension κ/2) := by
-  obtain ⟨B,G,F,hB,hw⟩ := hSLE.exists_charts
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hp := codimension_pos κ hκ8
-  have hs := Real.sqrt_pos.mpr hr
-  have hz : 0 < z.im := hs.trans_le hy
-  have hb := normalized_visit_bound hB κ hκ hκ8 γ G F hw z hz.ne' r hr
-  have hb' := mul_le_mul_of_nonneg_left hb (Real.rpow_nonneg hr.le (codimension κ))
-  rw [← mul_assoc,← Real.rpow_add hr,add_neg_cancel,Real.rpow_zero,one_mul,abs_of_pos hz] at hb'
-  have hpow : z.im^(-codimension κ) ≤ (Real.sqrt r)^(-codimension κ) :=
-    Real.rpow_le_rpow_of_nonpos hs hy (neg_nonpos.mpr hp.le)
-  have hid : r^codimension κ * (Real.sqrt r)^(-codimension κ) = r^(codimension κ/2) := by
-    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hr.le,← Real.rpow_add hr]
-    congr 1
-    ring
-  calc
-    _ ≤ r^codimension κ * (sausagePointConstant κ * (Real.sqrt r)^(-codimension κ)) :=
-      hb'.trans (mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left hpow (sausagePointConstant_pos κ hκ hκ8).le)
-        (Real.rpow_nonneg hr.le _))
-    _ = _ := by rw [← mul_assoc,mul_comm (r^codimension κ),mul_assoc,hid]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def boundaryApproachPower (κ : ℝ) : ℝ := 3*(8-κ)/(4*κ)
-noncomputable def annularApproachPower (κ : ℝ) : ℝ :=
-  min (boundaryApproachPower κ/2) (codimension κ/2)
-noncomputable def annularApproachConstant (κ : ℝ) : ℝ :=
-  (8/((8-κ)/32))^boundaryApproachPower κ + sausagePointConstant κ
-
-lemma annularApproachPower_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < annularApproachPower κ := by
-  have hp := codimension_pos κ hκ8
-  unfold annularApproachPower boundaryApproachPower
-  exact lt_min (by positivity) (by positivity)
-
-lemma annularApproachConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < annularApproachConstant κ := by
-  have hh := sausagePointConstant_pos κ hκ hκ8
-  unfold annularApproachConstant
-  exact add_pos (Real.rpow_pos_of_pos (by positivity) _) hh
-
- 
-
-lemma sle_low_center_approach
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (z : ℂ) (hz : 0 ≤ z.im) (hn : 1/2 ≤ ‖z‖)
-    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) (hy : z.im ≤ Real.sqrt r) :
-    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
-      (8/((8-κ)/32))^boundaryApproachPower κ * r^(boundaryApproachPower κ/2) := by
-  obtain ⟨B,hB,_⟩ := hSLE.2
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hδ : 0 < (8-κ)/32 := by positivity
-  have hβ : 0 < boundaryApproachPower κ := by unfold boundaryApproachPower; positivity
-  have hs := Real.sqrt_pos.mpr hr
-  have hsq := Real.sq_sqrt hr.le
-  have hs4 : Real.sqrt r < 1/4 := by nlinarith [Real.sqrt_nonneg r]
-  have hrs : r ≤ Real.sqrt r := by nlinarith [Real.sqrt_nonneg r]
-  have hx : 1/4 ≤ |z.re| := by
-    have hn' := Complex.norm_le_abs_re_add_abs_im z
-    rw [abs_of_nonneg hz] at hn'
-    linarith
-  have hx0 : z.re ≠ 0 := by
-    intro he
-    rw [he,abs_zero] at hx
-    norm_num at hx
-  have hd : dist z (z.re:ℂ) = z.im := by
-    rw [dist_eq_norm]
-    have he : z-(z.re:ℂ) = (z.im:ℂ)*Complex.I := by apply Complex.ext <;> simp
-    rw [he,norm_mul,Complex.norm_I,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hz]
-  have hb := sle_real_center_approach κ hκ hκ8 γ hSLE z.re (2*Real.sqrt r) hx0 (by positivity)
-  have he : P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r} ≤
-      P {ω | ∃ t : ℝ≥0, dist (γ ω t) (z.re:ℂ) < 2*Real.sqrt r} := by
-    apply measure_mono
-    rintro ω ⟨t,ht⟩
-    refine ⟨t,?_⟩
-    have hh := dist_triangle (γ ω t) z (z.re:ℂ)
-    rw [hd] at hh
-    linarith
-  have hb' := ENNReal.toReal_mono ENNReal.ofReal_ne_top (he.trans hb)
-  rw [ENNReal.toReal_ofReal (Real.rpow_nonneg (by positivity) _)] at hb'
-  have hq : (2*Real.sqrt r)/(((8-κ)/32)*|z.re|) ≤ (8/((8-κ)/32))*Real.sqrt r := by
-    calc
-      _ ≤ (2*Real.sqrt r)/(((8-κ)/32)*(1/4)) :=
-        div_le_div_of_nonneg_left (by positivity) (by positivity) (mul_le_mul_of_nonneg_left hx hδ.le)
-      _ = _ := by field_simp; ring
-  have hsβ : (Real.sqrt r)^boundaryApproachPower κ = r^(boundaryApproachPower κ/2) := by
-    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hr.le]
-    congr 1
-    ring
-  calc
-    _ ≤ ((8/((8-κ)/32))*Real.sqrt r)^boundaryApproachPower κ :=
-      hb'.trans (Real.rpow_le_rpow (by positivity) hq hβ.le)
-    _ = _ := by rw [Real.mul_rpow (by positivity) hs.le,hsβ]
-
- 
-
-theorem sle_annular_small_target
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (z : ℂ) (hz : 0 ≤ z.im) (hn : 1/2 ≤ ‖z‖) (_hn2 : ‖z‖ ≤ 2)
-    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) :
-    P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r} ≤
-      ENNReal.ofReal (annularApproachConstant κ * r^annularApproachPower κ) := by
-  obtain ⟨B,hB,_⟩ := hSLE.2
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hC := annularApproachConstant_pos κ hκ hκ8
-  apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity)).mpr
-  have hr1 : r ≤ 1 := by linarith
-  by_cases hy : z.im ≤ Real.sqrt r
-  · have hb := sle_low_center_approach κ hκ hκ8 γ hSLE z hz hn r hr hr16 hy
-    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_left
-      (boundaryApproachPower κ/2) (codimension κ/2))
-    have hcoeff : (8/((8-κ)/32))^boundaryApproachPower κ ≤ annularApproachConstant κ := by
-      exact le_add_of_nonneg_right (sausagePointConstant_pos κ hκ hκ8).le
-    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
-  · have hb := sle_high_center_approach κ hκ hκ8 γ hSLE z r hr (le_of_not_ge hy)
-    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_right
-      (boundaryApproachPower κ/2) (codimension κ/2))
-    have hcoeff : sausagePointConstant κ ≤ annularApproachConstant κ := by
-      exact le_add_of_nonneg_left (Real.rpow_nonneg (by positivity) _)
-    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma sle_high_center_approach_scaled
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (z : ℂ) (R r : ℝ) (hR : 0 < R) (hr : 0 < r) (hy : R*Real.sqrt r ≤ z.im) :
-    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r}).toReal ≤
-      sausagePointConstant κ * r^(codimension κ/2) := by
-  obtain ⟨B,G,F,hB,hw⟩ := hSLE.exists_charts
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hp := codimension_pos κ hκ8
-  have hs := Real.sqrt_pos.mpr hr
-  have hz : 0 < z.im := (mul_pos hR hs).trans_le hy
-  have hb := normalized_visit_bound hB κ hκ hκ8 γ G F hw z hz.ne' (R*r) (mul_pos hR hr)
-  have hb' := mul_le_mul_of_nonneg_left hb (Real.rpow_nonneg (mul_nonneg hR.le hr.le) (codimension κ))
-  rw [← mul_assoc,← Real.rpow_add (mul_pos hR hr),add_neg_cancel,Real.rpow_zero,one_mul,abs_of_pos hz] at hb'
-  have hpow : z.im^(-codimension κ) ≤ (R*Real.sqrt r)^(-codimension κ) :=
-    Real.rpow_le_rpow_of_nonpos (mul_pos hR hs) hy (neg_nonpos.mpr hp.le)
-  have hid : (R*r)^codimension κ * (R*Real.sqrt r)^(-codimension κ) = r^(codimension κ/2) := by
-    rw [Real.mul_rpow hR.le hr.le,Real.mul_rpow hR.le hs.le]
-    calc
-      _ = (R^codimension κ*R^(-codimension κ)) *
-          (r^codimension κ*(Real.sqrt r)^(-codimension κ)) := by ring
-      _ = r^(codimension κ/2) := by
-        rw [← Real.rpow_add hR,add_neg_cancel,Real.rpow_zero,one_mul,
-          Real.sqrt_eq_rpow,← Real.rpow_mul hr.le,← Real.rpow_add hr]
-        congr 1
-        ring
-  calc
-    _ ≤ (R*r)^codimension κ * (sausagePointConstant κ * (R*Real.sqrt r)^(-codimension κ)) :=
-      hb'.trans (mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left hpow (sausagePointConstant_pos κ hκ hκ8).le)
-        (Real.rpow_nonneg (mul_nonneg hR.le hr.le) _))
-    _ = _ := by rw [← mul_assoc,mul_comm ((R*r)^codimension κ),mul_assoc,hid]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma sle_low_center_approach_scaled
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (z : ℂ) (R : ℝ) (hR : 0 < R) (hz : 0 ≤ z.im) (hn : R/2 ≤ ‖z‖)
-    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) (hy : z.im ≤ R*Real.sqrt r) :
-    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r}).toReal ≤
-      (8/((8-κ)/32))^boundaryApproachPower κ * r^(boundaryApproachPower κ/2) := by
-  obtain ⟨B,hB,_⟩ := hSLE.2
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hδ : 0 < (8-κ)/32 := by positivity
-  have hβ : 0 < boundaryApproachPower κ := by unfold boundaryApproachPower; positivity
-  have hs := Real.sqrt_pos.mpr hr
-  have hsq := Real.sq_sqrt hr.le
-  have hs4 : Real.sqrt r < 1/4 := by nlinarith [Real.sqrt_nonneg r]
-  have hrs : r ≤ Real.sqrt r := by nlinarith [Real.sqrt_nonneg r]
-  have hx : R/4 ≤ |z.re| := by
-    have hn' := Complex.norm_le_abs_re_add_abs_im z
-    rw [abs_of_nonneg hz] at hn'
-    nlinarith [mul_pos hR (sub_pos.mpr hs4)]
-  have hx0 : z.re ≠ 0 := by
-    intro he
-    rw [he,abs_zero] at hx
-    linarith
-  have hd : dist z (z.re:ℂ) = z.im := by
-    rw [dist_eq_norm]
-    have he : z-(z.re:ℂ) = (z.im:ℂ)*Complex.I := by apply Complex.ext <;> simp
-    rw [he,norm_mul,Complex.norm_I,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hz]
-  have hb := sle_real_center_approach κ hκ hκ8 γ hSLE z.re (2*R*Real.sqrt r) hx0 (by positivity)
-  have he : P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r} ≤
-      P {ω | ∃ t : ℝ≥0, dist (γ ω t) (z.re:ℂ) < 2*R*Real.sqrt r} := by
-    apply measure_mono
-    rintro ω ⟨t,ht⟩
-    refine ⟨t,?_⟩
-    have hh := dist_triangle (γ ω t) z (z.re:ℂ)
-    rw [hd] at hh
-    nlinarith [mul_nonneg hR.le (sub_nonneg.mpr hrs)]
-  have hb' := ENNReal.toReal_mono ENNReal.ofReal_ne_top (he.trans hb)
-  rw [ENNReal.toReal_ofReal (Real.rpow_nonneg (by positivity) _)] at hb'
-  have hq : (2*R*Real.sqrt r)/(((8-κ)/32)*|z.re|) ≤ (8/((8-κ)/32))*Real.sqrt r := by
-    calc
-      _ ≤ (2*R*Real.sqrt r)/(((8-κ)/32)*(R/4)) :=
-        div_le_div_of_nonneg_left (by positivity) (by positivity) (mul_le_mul_of_nonneg_left (show R/4 ≤ |z.re| from hx) hδ.le)
-      _ = _ := by field_simp; ring
-  have hsβ : (Real.sqrt r)^boundaryApproachPower κ = r^(boundaryApproachPower κ/2) := by
-    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hr.le]
-    congr 1
-    ring
-  calc
-    _ ≤ ((8/((8-κ)/32))*Real.sqrt r)^boundaryApproachPower κ :=
-      hb'.trans (Real.rpow_le_rpow (by positivity) hq hβ.le)
-    _ = _ := by rw [Real.mul_rpow (by positivity) hs.le,hsβ]
-
- 
-
-theorem sle_annular_small_target_scaled
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (z : ℂ) (R : ℝ) (hR : 0 < R) (hz : 0 ≤ z.im)
-    (hn : R/2 ≤ ‖z‖) (_hn2 : ‖z‖ ≤ 2*R)
-    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) :
-    P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r} ≤
-      ENNReal.ofReal (annularApproachConstant κ * r^annularApproachPower κ) := by
-  obtain ⟨B,hB,_⟩ := hSLE.2
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hC := annularApproachConstant_pos κ hκ hκ8
-  apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity)).mpr
-  have hr1 : r ≤ 1 := by linarith
-  by_cases hy : z.im ≤ R*Real.sqrt r
-  · have hb := sle_low_center_approach_scaled κ hκ hκ8 γ hSLE z R hR hz hn r hr hr16 hy
-    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_left
-      (boundaryApproachPower κ/2) (codimension κ/2))
-    have hcoeff : (8/((8-κ)/32))^boundaryApproachPower κ ≤ annularApproachConstant κ := by
-      exact le_add_of_nonneg_right (sausagePointConstant_pos κ hκ hκ8).le
-    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
-  · have hb := sle_high_center_approach_scaled κ hκ hκ8 γ hSLE z R r hR hr (le_of_not_ge hy)
-    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_right
-      (boundaryApproachPower κ/2) (codimension κ/2))
-    have hcoeff : sausagePointConstant κ ≤ annularApproachConstant κ := by
-      exact le_add_of_nonneg_left (Real.rpow_nonneg (by positivity) _)
-    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-lemma stripRectangle_norm_sub_le (a b l : ℝ) (w : ℂ)
-    (hw : w ∈ stripRectangle a b l) :
-    ‖w-((a:ℂ)+(b:ℂ)*Complex.I)‖ ≤ (1+Real.pi)*l := by
-  rcases hw with ⟨hwr,hwr',hwi,hwi'⟩
-  have hr : 0 ≤ (w-((a:ℂ)+(b:ℂ)*Complex.I)).re := by simpa using sub_nonneg.mpr hwr
-  have hi : 0 ≤ (w-((a:ℂ)+(b:ℂ)*Complex.I)).im := by simpa using sub_nonneg.mpr hwi
-  have hh := Complex.norm_le_abs_re_add_abs_im (w-((a:ℂ)+(b:ℂ)*Complex.I))
-  rw [abs_of_nonneg hr,abs_of_nonneg hi] at hh
-  simp only [Complex.sub_re,Complex.add_re,Complex.ofReal_re,Complex.mul_re,
-    Complex.ofReal_im,Complex.I_re,Complex.I_im, mul_zero, mul_one,
-    sub_self,add_zero,Complex.sub_im,Complex.add_im,Complex.mul_im,
-    zero_add] at hh
-  nlinarith
-
-lemma stripRectangle_exp_distance (a b l : ℝ) (hl : 0 < l)
-    (hs : (1+Real.pi)*l ≤ 1) (w : ℂ) (hw : w ∈ stripRectangle a b l) :
-    dist (Complex.exp w) (Complex.exp ((a:ℂ)+(b:ℂ)*Complex.I)) <
-      Real.exp a * (3*(1+Real.pi)*l) := by
-  let c : ℂ := (a:ℂ)+(b:ℂ)*Complex.I
-  have hn := stripRectangle_norm_sub_le a b l w hw
-  have he : Complex.exp w - Complex.exp c =
-      Complex.exp c * (Complex.exp (w-c)-1) := by
-    rw [mul_sub,mul_one,← Complex.exp_add]
-    congr 2
-    ring
-  have hc : ‖Complex.exp c‖ = Real.exp a := by simp [c,Complex.norm_exp]
-  rw [dist_eq_norm,show ((a:ℂ)+(b:ℂ)*Complex.I) = c from rfl,he,norm_mul,hc]
-  calc
-    _ ≤ Real.exp a * (2*‖w-c‖) := mul_le_mul_of_nonneg_left
-      (Complex.norm_exp_sub_one_le (hn.trans hs)) (Real.exp_pos a).le
-    _ ≤ Real.exp a * (2*((1+Real.pi)*l)) := by gcongr
-    _ < _ := by
-      apply mul_lt_mul_of_pos_left _ (Real.exp_pos a)
-      have hh : 0 < (1+Real.pi)*l := mul_pos (by linarith [Real.pi_pos]) hl
-      nlinarith
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def stripHittingConstant (κ : ℝ) : ℝ :=
-  (annularApproachConstant κ+1) * (64*(1+Real.pi))^annularApproachPower κ
-
-lemma stripHittingConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < stripHittingConstant κ := by
-  have hC := annularApproachConstant_pos κ hκ hκ8
-  have hp := Real.pi_pos
-  unfold stripHittingConstant
-  positivity
-
- 
-
-theorem sle_exp_strip_rectangle_hit
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (a b l : ℝ) (hb0 : 0 ≤ b) (hbpi : b ≤ Real.pi) (hl : 0 < l) :
-    P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' stripRectangle a b l} ≤
-      ENNReal.ofReal (stripHittingConstant κ * l^annularApproachPower κ) := by
-  obtain ⟨B,hB,_⟩ := hSLE.2
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hC := annularApproachConstant_pos κ hκ hκ8
-  have hα := annularApproachPower_pos κ hκ hκ8
-  have hD : 0 < 1+Real.pi := by have hp := Real.pi_pos; linarith
-  have h64 : 0 < 64*(1+Real.pi) := by positivity
-  by_cases hs : l < 1/(64*(1+Real.pi))
-  · have hs' : 64*(1+Real.pi)*l < 1 := by
-      have hh := (lt_div_iff₀ h64).mp hs
-      nlinarith
-    let c : ℂ := Complex.exp ((a:ℂ)+(b:ℂ)*Complex.I)
-    have hc : ‖c‖ = Real.exp a := by simp [c,Complex.norm_exp]
-    have hci : 0 ≤ c.im := by
-      simp only [c,Complex.exp_im,Complex.add_re,Complex.ofReal_re,Complex.mul_re,
-        Complex.ofReal_im,Complex.I_re,Complex.I_im,mul_zero,mul_one,
-        sub_self,add_zero,Complex.add_im,Complex.mul_im,zero_add]
-      exact mul_nonneg (Real.exp_pos a).le (Real.sin_nonneg_of_mem_Icc ⟨hb0,hbpi⟩)
-    have hb := sle_annular_small_target_scaled κ hκ hκ8 γ hSLE c (Real.exp a)
-      (Real.exp_pos a) hci (by rw [hc]; linarith [Real.exp_pos a])
-      (by rw [hc]; linarith [Real.exp_pos a]) (3*(1+Real.pi)*l)
-      (by positivity) (by nlinarith)
-    have hsub : P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' stripRectangle a b l} ≤
-        P {ω | ∃ t : ℝ≥0, dist (γ ω t) c < Real.exp a * (3*(1+Real.pi)*l)} := by
-      apply measure_mono
-      rintro ω ⟨t,w,hw,he⟩
-      refine ⟨t,?_⟩
-      rw [← he]
-      exact stripRectangle_exp_distance a b l hl (by nlinarith) w hw
-    refine (hsub.trans hb).trans (ENNReal.ofReal_le_ofReal ?_)
-    rw [Real.mul_rpow (by positivity : 0 ≤ 3*(1+Real.pi)) hl.le]
-    unfold stripHittingConstant
-    have hp : (3*(1+Real.pi))^annularApproachPower κ ≤
-        (64*(1+Real.pi))^annularApproachPower κ := by
-      apply Real.rpow_le_rpow (by positivity) (by nlinarith) hα.le
-    have hcoeff : annularApproachConstant κ * (3*(1+Real.pi))^annularApproachPower κ ≤
-        (annularApproachConstant κ+1) * (64*(1+Real.pi))^annularApproachPower κ :=
-      mul_le_mul (by linarith) hp (Real.rpow_nonneg (by positivity) _) (by linarith)
-    nlinarith [mul_le_mul_of_nonneg_right hcoeff (Real.rpow_nonneg hl.le (annularApproachPower κ))]
-  · have hs' : 1 ≤ 64*(1+Real.pi)*l := by
-      have hh := (div_le_iff₀ h64).mp (le_of_not_gt hs)
-      nlinarith
-    have hp : 1 ≤ (64*(1+Real.pi)*l)^annularApproachPower κ :=
-      Real.one_le_rpow hs' hα.le
-    rw [Real.mul_rpow h64.le hl.le] at hp
-    have hc : 1 ≤ stripHittingConstant κ * l^annularApproachPower κ := by
-      unfold stripHittingConstant
-      nlinarith [mul_nonneg hC.le (sub_nonneg.mpr hp)]
-    calc
-      _ ≤ 1 := prob_le_one
-      _ ≤ _ := by simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal hc
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.OccupationBound
+import OAI.Probability.SLE.StripKernels
+
+namespace OAI
+
+/-! Sausage Point Bound for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def spatialSmallnessConstant : ℝ := 128 * Real.pi * Real.exp 1 + 1
+noncomputable def spatialOnePointConstant (κ : ℝ) : ℝ :=
+  uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ
+
+lemma spatialSmallnessConstant_gt_one : 1 < spatialSmallnessConstant := by
+  dsimp [spatialSmallnessConstant]
+  have := mul_pos (mul_pos (by norm_num : (0 : ℝ) < 128) Real.pi_pos) (Real.exp_pos 1)
+  linarith
+
+lemma spatial_smallness (z : ℂ) (r : ℝ) (hr : 0 < r)
+    (hy : spatialSmallnessConstant * r < z.im) :
+    0 < z.im ∧ r ≤ z.im ∧ 1 ≤ Real.log ((2 * z.im) / ((128 * Real.pi) * r)) := by
+  have hA : 1 ≤ spatialSmallnessConstant := spatialSmallnessConstant_gt_one.le
+  have hry : r < z.im := (le_mul_of_one_le_left hr.le hA).trans_lt hy
+  refine ⟨hr.trans hry, hry.le, ?_⟩
+  have hC : 0 < (128 * Real.pi) * r := mul_pos (mul_pos (by norm_num) Real.pi_pos) hr
+  have hquot : 0 < (2 * z.im) / ((128 * Real.pi) * r) :=
+    div_pos (mul_pos (by norm_num) (hr.trans hry)) hC
+  apply (Real.le_log_iff_exp_le hquot).mpr
+  apply (le_div_iff₀ hC).mpr
+  dsimp [spatialSmallnessConstant] at hy
+  nlinarith [Real.exp_pos 1]
+
+lemma initialGreen_le_height (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    initialGreen κ z ≤ (2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ) := by
+  have hs : 0 ≤ Real.sin (Complex.arg z) := by
+    rw [Complex.sin_arg]; exact div_nonneg hz.le (norm_nonneg _)
+  calc
+    initialGreen κ z ≤ (2 * z.im) ^ (-codimension κ) * 1 :=
+      mul_le_mul_of_nonneg_left (Real.rpow_le_one hs (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
+        (Real.rpow_nonneg (mul_nonneg (by norm_num) hz.le) _)
+    _ = (2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ) := by
+      rw [mul_one, Real.mul_rpow (by norm_num) hz.le]
+
+noncomputable def sausagePointConstant (κ : ℝ) : ℝ :=
+  max (spatialOnePointConstant κ * (2 : ℝ) ^ (-codimension κ))
+    (spatialSmallnessConstant ^ codimension κ)
+
+lemma spatialOnePointConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < spatialOnePointConstant κ :=
+  mul_pos (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num))
+    (Real.rpow_pos_of_pos (mul_pos (by norm_num) Real.pi_pos) _)
+
+lemma sausagePointConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < sausagePointConstant κ :=
+  (mul_pos (spatialOnePointConstant_pos κ hκ hκ8) (Real.rpow_pos_of_pos (by norm_num) _)).trans_le
+    (le_max_left _ _)
+
+lemma negative_power_scale (x r A p : ℝ) (hx : 0 < x) (hr : 0 < r)
+    (hA : 0 < A) (hp : 0 ≤ p) (hxr : x ≤ A * r) :
+    r ^ (-p) ≤ A ^ p * x ^ (-p) := by
+  have hh := Real.rpow_le_rpow hx.le hxr hp
+  rw [Real.mul_rpow hA.le hr.le] at hh
+  rw [Real.rpow_neg hr.le, Real.rpow_neg hx.le, ← div_eq_mul_inv]
+  apply (le_div_iff₀ (Real.rpow_pos_of_pos hx p)).mpr
+  exact (inv_mul_le_iff₀ (Real.rpow_pos_of_pos hr p)).mpr (by simpa [mul_comm] using hh)
+
+ 
+
+theorem normalized_visit_bound
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z.im ≠ 0) (r : ℝ) (hr : 0 < r) :
+    r ^ (-codimension κ) * (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+      sausagePointConstant κ * |z.im| ^ (-codimension κ) := by
+  have hp := codimension_pos κ hκ8
+  have hA : 0 < spatialSmallnessConstant := zero_lt_one.trans spatialSmallnessConstant_gt_one
+  have hscale (hy : |z.im| ≤ spatialSmallnessConstant * r) :
+      r ^ (-codimension κ) * (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+        sausagePointConstant κ * |z.im| ^ (-codimension κ) := by
+    calc
+      _ ≤ r ^ (-codimension κ) * 1 := mul_le_mul_of_nonneg_left measureReal_le_one
+        (Real.rpow_nonneg hr.le _)
+      _ ≤ spatialSmallnessConstant ^ codimension κ * |z.im| ^ (-codimension κ) := by
+        rw [mul_one]
+        exact negative_power_scale _ _ _ _ (abs_pos.mpr hz) hr hA hp.le hy
+      _ ≤ _ := mul_le_mul_of_nonneg_right (le_max_right _ _)
+        (Real.rpow_nonneg (abs_nonneg _) _)
+  by_cases hsmall : spatialSmallnessConstant * r < z.im
+  · obtain ⟨hy, hry, hlog⟩ := spatial_smallness z r hr hsmall
+    have hh := euclidean_approach hB κ hκ hκ8 γ G F hw z hy r hr hry hlog
+    have hS := (spatialOnePointConstant_pos κ hκ hκ8).le
+    change (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+      spatialOnePointConstant κ * initialGreen κ z * r ^ codimension κ at hh
+    have hcancel : r ^ (-codimension κ) * r ^ codimension κ = 1 := by
+      rw [← Real.rpow_add hr]; simp
+    calc
+      _ ≤ r ^ (-codimension κ) * (spatialOnePointConstant κ * initialGreen κ z * r ^ codimension κ) :=
+        mul_le_mul_of_nonneg_left hh (Real.rpow_nonneg hr.le _)
+      _ = spatialOnePointConstant κ * initialGreen κ z := by
+        calc
+          _ = (spatialOnePointConstant κ * initialGreen κ z) *
+              (r ^ (-codimension κ) * r ^ codimension κ) := by ring
+          _ = _ := by rw [hcancel, mul_one]
+      _ ≤ spatialOnePointConstant κ * ((2 : ℝ) ^ (-codimension κ) * z.im ^ (-codimension κ)) :=
+        mul_le_mul_of_nonneg_left (initialGreen_le_height κ hκ hκ8 z hy) hS
+      _ ≤ _ := by
+        rw [abs_of_pos hy, ← mul_assoc]
+        exact mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg hy.le _)
+  · by_cases hy : 0 ≤ z.im
+    · apply hscale
+      rw [abs_of_nonneg hy]
+      exact le_of_not_gt hsmall
+    · by_cases hzr : -r < z.im
+      · apply hscale
+        rw [abs_of_neg (lt_of_not_ge hy)]
+        exact (by linarith : -z.im ≤ r).trans (le_mul_of_one_le_left hr.le spatialSmallnessConstant_gt_one.le)
+      · have hzero : P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r} = 0 := by
+          have hn : ∀ᵐ ω ∂P, ¬ (∃ t : ℝ≥0, dist (γ ω t) z < r) := by
+            filter_upwards [hw] with ω hω
+            rintro ⟨t, ht⟩
+            have hi := Complex.im_le_norm (γ ω t - z)
+            rw [Complex.sub_im, ← dist_eq_norm] at hi
+            have hnon := hω.2.2.1 t
+            linarith [le_of_not_gt hzr]
+          simpa only [not_not, ofPred_mem_eq] using ae_iff.mp hn
+        rw [hzero, ENNReal.toReal_zero, mul_zero]
+        exact mul_nonneg (sausagePointConstant_pos κ hκ hκ8).le (Real.rpow_nonneg (abs_nonneg _) _)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma IsOrdinaryChordalSLE.exists_charts
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) :
+    ∃ (B : ℝ≥0 → Ω → ℝ) (G F : Ω → ℝ≥0 → ℂ → ℂ),
+      IsBrownianReal B P ∧ ∀ᵐ ω ∂P, IsCapacityTwoWitness
+        (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω) := by
+  classical
+  obtain ⟨B,hB,htrace⟩ := hSLE.2
+  have hchoice (ω : Ω) : ∃ G F : ℝ≥0 → ℂ → ℂ,
+      IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω) →
+        IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) G F := by
+    by_cases h : IsCapacityTwoTrace (fun t => Real.sqrt κ*B t ω) (γ ω)
+    · obtain ⟨G,F,hw⟩ := h
+      exact ⟨G,F,fun _ => hw⟩
+    · exact ⟨fun _ _ => 0,fun _ _ => 0,fun hh => False.elim (h hh)⟩
+  choose G F hw using hchoice
+  exact ⟨B,G,F,hB,htrace.mono fun ω hω => hw ω hω⟩
+
+ 
+
+lemma sle_high_center_approach
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : Real.sqrt r ≤ z.im) :
+    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+      sausagePointConstant κ * r^(codimension κ/2) := by
+  obtain ⟨B,G,F,hB,hw⟩ := hSLE.exists_charts
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hp := codimension_pos κ hκ8
+  have hs := Real.sqrt_pos.mpr hr
+  have hz : 0 < z.im := hs.trans_le hy
+  have hb := normalized_visit_bound hB κ hκ hκ8 γ G F hw z hz.ne' r hr
+  have hb' := mul_le_mul_of_nonneg_left hb (Real.rpow_nonneg hr.le (codimension κ))
+  rw [← mul_assoc,← Real.rpow_add hr,add_neg_cancel,Real.rpow_zero,one_mul,abs_of_pos hz] at hb'
+  have hpow : z.im^(-codimension κ) ≤ (Real.sqrt r)^(-codimension κ) :=
+    Real.rpow_le_rpow_of_nonpos hs hy (neg_nonpos.mpr hp.le)
+  have hid : r^codimension κ * (Real.sqrt r)^(-codimension κ) = r^(codimension κ/2) := by
+    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hr.le,← Real.rpow_add hr]
+    congr 1
+    ring
+  calc
+    _ ≤ r^codimension κ * (sausagePointConstant κ * (Real.sqrt r)^(-codimension κ)) :=
+      hb'.trans (mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left hpow (sausagePointConstant_pos κ hκ hκ8).le)
+        (Real.rpow_nonneg hr.le _))
+    _ = _ := by rw [← mul_assoc,mul_comm (r^codimension κ),mul_assoc,hid]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def boundaryApproachPower (κ : ℝ) : ℝ := 3*(8-κ)/(4*κ)
+noncomputable def annularApproachPower (κ : ℝ) : ℝ :=
+  min (boundaryApproachPower κ/2) (codimension κ/2)
+noncomputable def annularApproachConstant (κ : ℝ) : ℝ :=
+  (8/((8-κ)/32))^boundaryApproachPower κ + sausagePointConstant κ
+
+lemma annularApproachPower_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < annularApproachPower κ := by
+  have hp := codimension_pos κ hκ8
+  unfold annularApproachPower boundaryApproachPower
+  exact lt_min (by positivity) (by positivity)
+
+lemma annularApproachConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < annularApproachConstant κ := by
+  have hh := sausagePointConstant_pos κ hκ hκ8
+  unfold annularApproachConstant
+  exact add_pos (Real.rpow_pos_of_pos (by positivity) _) hh
+
+ 
+
+lemma sle_low_center_approach
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (z : ℂ) (hz : 0 ≤ z.im) (hn : 1/2 ≤ ‖z‖)
+    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) (hy : z.im ≤ Real.sqrt r) :
+    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r}).toReal ≤
+      (8/((8-κ)/32))^boundaryApproachPower κ * r^(boundaryApproachPower κ/2) := by
+  obtain ⟨B,hB,_⟩ := hSLE.2
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hδ : 0 < (8-κ)/32 := by positivity
+  have hβ : 0 < boundaryApproachPower κ := by unfold boundaryApproachPower; positivity
+  have hs := Real.sqrt_pos.mpr hr
+  have hsq := Real.sq_sqrt hr.le
+  have hs4 : Real.sqrt r < 1/4 := by nlinarith [Real.sqrt_nonneg r]
+  have hrs : r ≤ Real.sqrt r := by nlinarith [Real.sqrt_nonneg r]
+  have hx : 1/4 ≤ |z.re| := by
+    have hn' := Complex.norm_le_abs_re_add_abs_im z
+    rw [abs_of_nonneg hz] at hn'
+    linarith
+  have hx0 : z.re ≠ 0 := by
+    intro he
+    rw [he,abs_zero] at hx
+    norm_num at hx
+  have hd : dist z (z.re:ℂ) = z.im := by
+    rw [dist_eq_norm]
+    have he : z-(z.re:ℂ) = (z.im:ℂ)*Complex.I := by apply Complex.ext <;> simp
+    rw [he,norm_mul,Complex.norm_I,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hz]
+  have hb := sle_real_center_approach κ hκ hκ8 γ hSLE z.re (2*Real.sqrt r) hx0 (by positivity)
+  have he : P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r} ≤
+      P {ω | ∃ t : ℝ≥0, dist (γ ω t) (z.re:ℂ) < 2*Real.sqrt r} := by
+    apply measure_mono
+    rintro ω ⟨t,ht⟩
+    refine ⟨t,?_⟩
+    have hh := dist_triangle (γ ω t) z (z.re:ℂ)
+    rw [hd] at hh
+    linarith
+  have hb' := ENNReal.toReal_mono ENNReal.ofReal_ne_top (he.trans hb)
+  rw [ENNReal.toReal_ofReal (Real.rpow_nonneg (by positivity) _)] at hb'
+  have hq : (2*Real.sqrt r)/(((8-κ)/32)*|z.re|) ≤ (8/((8-κ)/32))*Real.sqrt r := by
+    calc
+      _ ≤ (2*Real.sqrt r)/(((8-κ)/32)*(1/4)) :=
+        div_le_div_of_nonneg_left (by positivity) (by positivity) (mul_le_mul_of_nonneg_left hx hδ.le)
+      _ = _ := by field_simp; ring
+  have hsβ : (Real.sqrt r)^boundaryApproachPower κ = r^(boundaryApproachPower κ/2) := by
+    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hr.le]
+    congr 1
+    ring
+  calc
+    _ ≤ ((8/((8-κ)/32))*Real.sqrt r)^boundaryApproachPower κ :=
+      hb'.trans (Real.rpow_le_rpow (by positivity) hq hβ.le)
+    _ = _ := by rw [Real.mul_rpow (by positivity) hs.le,hsβ]
+
+ 
+
+theorem sle_annular_small_target
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (z : ℂ) (hz : 0 ≤ z.im) (hn : 1/2 ≤ ‖z‖) (_hn2 : ‖z‖ ≤ 2)
+    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) :
+    P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < r} ≤
+      ENNReal.ofReal (annularApproachConstant κ * r^annularApproachPower κ) := by
+  obtain ⟨B,hB,_⟩ := hSLE.2
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hC := annularApproachConstant_pos κ hκ hκ8
+  apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity)).mpr
+  have hr1 : r ≤ 1 := by linarith
+  by_cases hy : z.im ≤ Real.sqrt r
+  · have hb := sle_low_center_approach κ hκ hκ8 γ hSLE z hz hn r hr hr16 hy
+    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_left
+      (boundaryApproachPower κ/2) (codimension κ/2))
+    have hcoeff : (8/((8-κ)/32))^boundaryApproachPower κ ≤ annularApproachConstant κ := by
+      exact le_add_of_nonneg_right (sausagePointConstant_pos κ hκ hκ8).le
+    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
+  · have hb := sle_high_center_approach κ hκ hκ8 γ hSLE z r hr (le_of_not_ge hy)
+    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_right
+      (boundaryApproachPower κ/2) (codimension κ/2))
+    have hcoeff : sausagePointConstant κ ≤ annularApproachConstant κ := by
+      exact le_add_of_nonneg_left (Real.rpow_nonneg (by positivity) _)
+    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma sle_high_center_approach_scaled
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (z : ℂ) (R r : ℝ) (hR : 0 < R) (hr : 0 < r) (hy : R*Real.sqrt r ≤ z.im) :
+    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r}).toReal ≤
+      sausagePointConstant κ * r^(codimension κ/2) := by
+  obtain ⟨B,G,F,hB,hw⟩ := hSLE.exists_charts
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hp := codimension_pos κ hκ8
+  have hs := Real.sqrt_pos.mpr hr
+  have hz : 0 < z.im := (mul_pos hR hs).trans_le hy
+  have hb := normalized_visit_bound hB κ hκ hκ8 γ G F hw z hz.ne' (R*r) (mul_pos hR hr)
+  have hb' := mul_le_mul_of_nonneg_left hb (Real.rpow_nonneg (mul_nonneg hR.le hr.le) (codimension κ))
+  rw [← mul_assoc,← Real.rpow_add (mul_pos hR hr),add_neg_cancel,Real.rpow_zero,one_mul,abs_of_pos hz] at hb'
+  have hpow : z.im^(-codimension κ) ≤ (R*Real.sqrt r)^(-codimension κ) :=
+    Real.rpow_le_rpow_of_nonpos (mul_pos hR hs) hy (neg_nonpos.mpr hp.le)
+  have hid : (R*r)^codimension κ * (R*Real.sqrt r)^(-codimension κ) = r^(codimension κ/2) := by
+    rw [Real.mul_rpow hR.le hr.le,Real.mul_rpow hR.le hs.le]
+    calc
+      _ = (R^codimension κ*R^(-codimension κ)) *
+          (r^codimension κ*(Real.sqrt r)^(-codimension κ)) := by ring
+      _ = r^(codimension κ/2) := by
+        rw [← Real.rpow_add hR,add_neg_cancel,Real.rpow_zero,one_mul,
+          Real.sqrt_eq_rpow,← Real.rpow_mul hr.le,← Real.rpow_add hr]
+        congr 1
+        ring
+  calc
+    _ ≤ (R*r)^codimension κ * (sausagePointConstant κ * (R*Real.sqrt r)^(-codimension κ)) :=
+      hb'.trans (mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left hpow (sausagePointConstant_pos κ hκ hκ8).le)
+        (Real.rpow_nonneg (mul_nonneg hR.le hr.le) _))
+    _ = _ := by rw [← mul_assoc,mul_comm ((R*r)^codimension κ),mul_assoc,hid]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma sle_low_center_approach_scaled
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (z : ℂ) (R : ℝ) (hR : 0 < R) (hz : 0 ≤ z.im) (hn : R/2 ≤ ‖z‖)
+    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) (hy : z.im ≤ R*Real.sqrt r) :
+    (P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r}).toReal ≤
+      (8/((8-κ)/32))^boundaryApproachPower κ * r^(boundaryApproachPower κ/2) := by
+  obtain ⟨B,hB,_⟩ := hSLE.2
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hδ : 0 < (8-κ)/32 := by positivity
+  have hβ : 0 < boundaryApproachPower κ := by unfold boundaryApproachPower; positivity
+  have hs := Real.sqrt_pos.mpr hr
+  have hsq := Real.sq_sqrt hr.le
+  have hs4 : Real.sqrt r < 1/4 := by nlinarith [Real.sqrt_nonneg r]
+  have hrs : r ≤ Real.sqrt r := by nlinarith [Real.sqrt_nonneg r]
+  have hx : R/4 ≤ |z.re| := by
+    have hn' := Complex.norm_le_abs_re_add_abs_im z
+    rw [abs_of_nonneg hz] at hn'
+    nlinarith [mul_pos hR (sub_pos.mpr hs4)]
+  have hx0 : z.re ≠ 0 := by
+    intro he
+    rw [he,abs_zero] at hx
+    linarith
+  have hd : dist z (z.re:ℂ) = z.im := by
+    rw [dist_eq_norm]
+    have he : z-(z.re:ℂ) = (z.im:ℂ)*Complex.I := by apply Complex.ext <;> simp
+    rw [he,norm_mul,Complex.norm_I,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hz]
+  have hb := sle_real_center_approach κ hκ hκ8 γ hSLE z.re (2*R*Real.sqrt r) hx0 (by positivity)
+  have he : P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r} ≤
+      P {ω | ∃ t : ℝ≥0, dist (γ ω t) (z.re:ℂ) < 2*R*Real.sqrt r} := by
+    apply measure_mono
+    rintro ω ⟨t,ht⟩
+    refine ⟨t,?_⟩
+    have hh := dist_triangle (γ ω t) z (z.re:ℂ)
+    rw [hd] at hh
+    nlinarith [mul_nonneg hR.le (sub_nonneg.mpr hrs)]
+  have hb' := ENNReal.toReal_mono ENNReal.ofReal_ne_top (he.trans hb)
+  rw [ENNReal.toReal_ofReal (Real.rpow_nonneg (by positivity) _)] at hb'
+  have hq : (2*R*Real.sqrt r)/(((8-κ)/32)*|z.re|) ≤ (8/((8-κ)/32))*Real.sqrt r := by
+    calc
+      _ ≤ (2*R*Real.sqrt r)/(((8-κ)/32)*(R/4)) :=
+        div_le_div_of_nonneg_left (by positivity) (by positivity) (mul_le_mul_of_nonneg_left (show R/4 ≤ |z.re| from hx) hδ.le)
+      _ = _ := by field_simp; ring
+  have hsβ : (Real.sqrt r)^boundaryApproachPower κ = r^(boundaryApproachPower κ/2) := by
+    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hr.le]
+    congr 1
+    ring
+  calc
+    _ ≤ ((8/((8-κ)/32))*Real.sqrt r)^boundaryApproachPower κ :=
+      hb'.trans (Real.rpow_le_rpow (by positivity) hq hβ.le)
+    _ = _ := by rw [Real.mul_rpow (by positivity) hs.le,hsβ]
+
+ 
+
+theorem sle_annular_small_target_scaled
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (z : ℂ) (R : ℝ) (hR : 0 < R) (hz : 0 ≤ z.im)
+    (hn : R/2 ≤ ‖z‖) (_hn2 : ‖z‖ ≤ 2*R)
+    (r : ℝ) (hr : 0 < r) (hr16 : r < 1/16) :
+    P {ω | ∃ t : ℝ≥0, dist (γ ω t) z < R*r} ≤
+      ENNReal.ofReal (annularApproachConstant κ * r^annularApproachPower κ) := by
+  obtain ⟨B,hB,_⟩ := hSLE.2
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hC := annularApproachConstant_pos κ hκ hκ8
+  apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity)).mpr
+  have hr1 : r ≤ 1 := by linarith
+  by_cases hy : z.im ≤ R*Real.sqrt r
+  · have hb := sle_low_center_approach_scaled κ hκ hκ8 γ hSLE z R hR hz hn r hr hr16 hy
+    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_left
+      (boundaryApproachPower κ/2) (codimension κ/2))
+    have hcoeff : (8/((8-κ)/32))^boundaryApproachPower κ ≤ annularApproachConstant κ := by
+      exact le_add_of_nonneg_right (sausagePointConstant_pos κ hκ hκ8).le
+    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
+  · have hb := sle_high_center_approach_scaled κ hκ hκ8 γ hSLE z R r hR hr (le_of_not_ge hy)
+    have hp := Real.rpow_le_rpow_of_exponent_ge hr hr1 (min_le_right
+      (boundaryApproachPower κ/2) (codimension κ/2))
+    have hcoeff : sausagePointConstant κ ≤ annularApproachConstant κ := by
+      exact le_add_of_nonneg_left (Real.rpow_nonneg (by positivity) _)
+    exact hb.trans (mul_le_mul hcoeff hp (Real.rpow_nonneg hr.le _) hC.le)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+lemma stripRectangle_norm_sub_le (a b l : ℝ) (w : ℂ)
+    (hw : w ∈ stripRectangle a b l) :
+    ‖w-((a:ℂ)+(b:ℂ)*Complex.I)‖ ≤ (1+Real.pi)*l := by
+  rcases hw with ⟨hwr,hwr',hwi,hwi'⟩
+  have hr : 0 ≤ (w-((a:ℂ)+(b:ℂ)*Complex.I)).re := by simpa using sub_nonneg.mpr hwr
+  have hi : 0 ≤ (w-((a:ℂ)+(b:ℂ)*Complex.I)).im := by simpa using sub_nonneg.mpr hwi
+  have hh := Complex.norm_le_abs_re_add_abs_im (w-((a:ℂ)+(b:ℂ)*Complex.I))
+  rw [abs_of_nonneg hr,abs_of_nonneg hi] at hh
+  simp only [Complex.sub_re,Complex.add_re,Complex.ofReal_re,Complex.mul_re,
+    Complex.ofReal_im,Complex.I_re,Complex.I_im, mul_zero, mul_one,
+    sub_self,add_zero,Complex.sub_im,Complex.add_im,Complex.mul_im,
+    zero_add] at hh
+  nlinarith
+
+lemma stripRectangle_exp_distance (a b l : ℝ) (hl : 0 < l)
+    (hs : (1+Real.pi)*l ≤ 1) (w : ℂ) (hw : w ∈ stripRectangle a b l) :
+    dist (Complex.exp w) (Complex.exp ((a:ℂ)+(b:ℂ)*Complex.I)) <
+      Real.exp a * (3*(1+Real.pi)*l) := by
+  let c : ℂ := (a:ℂ)+(b:ℂ)*Complex.I
+  have hn := stripRectangle_norm_sub_le a b l w hw
+  have he : Complex.exp w - Complex.exp c =
+      Complex.exp c * (Complex.exp (w-c)-1) := by
+    rw [mul_sub,mul_one,← Complex.exp_add]
+    congr 2
+    ring
+  have hc : ‖Complex.exp c‖ = Real.exp a := by simp [c,Complex.norm_exp]
+  rw [dist_eq_norm,show ((a:ℂ)+(b:ℂ)*Complex.I) = c from rfl,he,norm_mul,hc]
+  calc
+    _ ≤ Real.exp a * (2*‖w-c‖) := mul_le_mul_of_nonneg_left
+      (Complex.norm_exp_sub_one_le (hn.trans hs)) (Real.exp_pos a).le
+    _ ≤ Real.exp a * (2*((1+Real.pi)*l)) := by gcongr
+    _ < _ := by
+      apply mul_lt_mul_of_pos_left _ (Real.exp_pos a)
+      have hh : 0 < (1+Real.pi)*l := mul_pos (by linarith [Real.pi_pos]) hl
+      nlinarith
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def stripHittingConstant (κ : ℝ) : ℝ :=
+  (annularApproachConstant κ+1) * (64*(1+Real.pi))^annularApproachPower κ
+
+lemma stripHittingConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < stripHittingConstant κ := by
+  have hC := annularApproachConstant_pos κ hκ hκ8
+  have hp := Real.pi_pos
+  unfold stripHittingConstant
+  positivity
+
+ 
+
+theorem sle_exp_strip_rectangle_hit
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (a b l : ℝ) (hb0 : 0 ≤ b) (hbpi : b ≤ Real.pi) (hl : 0 < l) :
+    P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' stripRectangle a b l} ≤
+      ENNReal.ofReal (stripHittingConstant κ * l^annularApproachPower κ) := by
+  obtain ⟨B,hB,_⟩ := hSLE.2
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hC := annularApproachConstant_pos κ hκ hκ8
+  have hα := annularApproachPower_pos κ hκ hκ8
+  have hD : 0 < 1+Real.pi := by have hp := Real.pi_pos; linarith
+  have h64 : 0 < 64*(1+Real.pi) := by positivity
+  by_cases hs : l < 1/(64*(1+Real.pi))
+  · have hs' : 64*(1+Real.pi)*l < 1 := by
+      have hh := (lt_div_iff₀ h64).mp hs
+      nlinarith
+    let c : ℂ := Complex.exp ((a:ℂ)+(b:ℂ)*Complex.I)
+    have hc : ‖c‖ = Real.exp a := by simp [c,Complex.norm_exp]
+    have hci : 0 ≤ c.im := by
+      simp only [c,Complex.exp_im,Complex.add_re,Complex.ofReal_re,Complex.mul_re,
+        Complex.ofReal_im,Complex.I_re,Complex.I_im,mul_zero,mul_one,
+        sub_self,add_zero,Complex.add_im,Complex.mul_im,zero_add]
+      exact mul_nonneg (Real.exp_pos a).le (Real.sin_nonneg_of_mem_Icc ⟨hb0,hbpi⟩)
+    have hb := sle_annular_small_target_scaled κ hκ hκ8 γ hSLE c (Real.exp a)
+      (Real.exp_pos a) hci (by rw [hc]; linarith [Real.exp_pos a])
+      (by rw [hc]; linarith [Real.exp_pos a]) (3*(1+Real.pi)*l)
+      (by positivity) (by nlinarith)
+    have hsub : P {ω | ∃ t : ℝ≥0, γ ω t ∈ Complex.exp '' stripRectangle a b l} ≤
+        P {ω | ∃ t : ℝ≥0, dist (γ ω t) c < Real.exp a * (3*(1+Real.pi)*l)} := by
+      apply measure_mono
+      rintro ω ⟨t,w,hw,he⟩
+      refine ⟨t,?_⟩
+      rw [← he]
+      exact stripRectangle_exp_distance a b l hl (by nlinarith) w hw
+    refine (hsub.trans hb).trans (ENNReal.ofReal_le_ofReal ?_)
+    rw [Real.mul_rpow (by positivity : 0 ≤ 3*(1+Real.pi)) hl.le]
+    unfold stripHittingConstant
+    have hp : (3*(1+Real.pi))^annularApproachPower κ ≤
+        (64*(1+Real.pi))^annularApproachPower κ := by
+      apply Real.rpow_le_rpow (by positivity) (by nlinarith) hα.le
+    have hcoeff : annularApproachConstant κ * (3*(1+Real.pi))^annularApproachPower κ ≤
+        (annularApproachConstant κ+1) * (64*(1+Real.pi))^annularApproachPower κ :=
+      mul_le_mul (by linarith) hp (Real.rpow_nonneg (by positivity) _) (by linarith)
+    nlinarith [mul_le_mul_of_nonneg_right hcoeff (Real.rpow_nonneg hl.le (annularApproachPower κ))]
+  · have hs' : 1 ≤ 64*(1+Real.pi)*l := by
+      have hh := (div_le_iff₀ h64).mp (le_of_not_gt hs)
+      nlinarith
+    have hp : 1 ≤ (64*(1+Real.pi)*l)^annularApproachPower κ :=
+      Real.one_le_rpow hs' hα.le
+    rw [Real.mul_rpow h64.le hl.le] at hp
+    have hc : 1 ≤ stripHittingConstant κ * l^annularApproachPower κ := by
+      unfold stripHittingConstant
+      nlinarith [mul_nonneg hC.le (sub_nonneg.mpr hp)]
+    calc
+      _ ≤ 1 := prob_le_one
+      _ ≤ _ := by simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal hc
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SausageScaling.lean b/lean/OAI/Probability/SLE/SausageScaling.lean
index c88d10caa1aefe83f1a99b6c7db9450f10ceec71..bcb2666a952e6a19386cabd56f30291ce3b903e8 100644
--- a/lean/OAI/Probability/SLE/SausageScaling.lean
+++ b/lean/OAI/Probability/SLE/SausageScaling.lean
@@ -1,621 +1,621 @@
-import Mathlib
-import OAI.Probability.SLE.TracePosCombined
-
-namespace OAI
-
-/-! Sausage Scaling for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance SausageScaling_local1 : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance SausageScaling_local2 : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma source_sausage_dilation {c : ℝ≥0} (hc : 0 < c) (η : ℝ≥0 → ℂ)
-    (T δ : ℝ≥0) (ht : sourceScaleTime c δ = T) (K : Set ℂ) (e : ℝ) :
-    ((fun z : ℂ => (c:ℂ)*z) '' (K ∩ sourceSausageSet η 0 T (e/(c:ℝ)))) ⊆
-      ((fun z : ℂ => (c:ℂ)*z) '' K) ∩ sourceSausageSet (sourceScaleTrace c η) 0 δ e := by
-  rintro _ ⟨z,hz,rfl⟩
-  refine ⟨⟨z,hz.1,rfl⟩,?_⟩
-  obtain ⟨u,hu,hzu⟩ := hz.2
-  have hc2 : c^2 ≠ 0 := pow_ne_zero _ hc.ne'
-  have htu : c^2*u ≤ δ := by
-    have hh := hu.2
-    rw [←ht,sourceScaleTime,le_div_iff₀ (pow_pos hc _)] at hh
-    simpa only [mul_comm] using hh
-  refine ⟨c^2*u,⟨bot_le,htu⟩,?_⟩
-  have he : sourceScaleTime c (c^2*u) = u := by
-    unfold sourceScaleTime; field_simp
-  simp only [sourceScaleTrace,he,dist_eq_norm,←mul_sub,norm_mul,norm_real,Real.norm_eq_abs,abs_of_nonneg c.coe_nonneg] 
-  have hh : ‖z-η u‖ < e/(c:ℝ) := by simpa only [dist_eq_norm] using hzu
-  have hcR : 0 < (c:ℝ) := hc
-  nlinarith [(lt_div_iff₀ hcR).mp hh]
-lemma source_volume_dilation_lower {c : ℝ≥0} (hc : 0 < c) (S : Set ℂ) :
-    volume S ≤ ((c⁻¹:ℝ≥0):ℝ≥0∞)^2 * volume ((fun z : ℂ => (c:ℂ)*z) '' S) := by
-  have hLip : LipschitzWith c⁻¹ (fun z : ℂ => (c⁻¹:ℂ)*z) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro x y
-    simp [dist_eq_norm,←mul_sub]
-  have hh := source_volume_lipschitz_image (hLip.lipschitzOnWith (s := (fun z : ℂ => (c:ℂ)*z) '' S))
-  have hi : (fun z : ℂ => (c⁻¹:ℂ)*z) '' ((fun z : ℂ => (c:ℂ)*z) '' S) = S := by
-    rw [←image_comp]
-    have he : (fun z : ℂ => (c⁻¹:ℂ)*((c:ℂ)*z)) = id := by
-      funext z
-      simp only [←mul_assoc,inv_mul_cancel₀ (by exact_mod_cast hc.ne' : (c:ℂ) ≠ 0),one_mul,id_eq]
-    simp only [Function.comp_def,he,image_id]
-  rwa [hi] at hh
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance SausageUniformSeed_local1 : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance SausageUniformSeed_local2 : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma source_sausage_scale_mass_bound (κ : ℝ) {c : ℝ≥0} (hc : 0 < c)
-    (η : ℝ≥0 → ℂ) (T δ : ℝ≥0) (ht : sourceScaleTime c δ = T)
-    (K : Set ℂ) (e : ℝ) (he : 0 < e) :
-    ENNReal.ofReal ((e/(c:ℝ))^(-codimension κ))*volume (K ∩ sourceSausageSet η 0 T (e/(c:ℝ))) ≤
-      ENNReal.ofReal (((c:ℝ)⁻¹)^(-codimension κ)*((c:ℝ)⁻¹)^2)*
-        (ENNReal.ofReal (e^(-codimension κ))*volume (((fun z : ℂ => (c:ℂ)*z) '' K) ∩
-          sourceSausageSet (sourceScaleTrace c η) 0 δ e)) := by
-  have hvol := (source_volume_dilation_lower hc (K ∩ sourceSausageSet η 0 T (e/(c:ℝ)))).trans (mul_le_mul' le_rfl (measure_mono (source_sausage_dilation hc η T δ ht K e)))
-  have hmul : (e/(c:ℝ))^(-codimension κ) = ((c:ℝ)⁻¹)^(-codimension κ)*e^(-codimension κ) := by
-    rw [show e/(c:ℝ) = (c:ℝ)⁻¹*e by ring,Real.mul_rpow (by positivity) he.le]
-  have hh := mul_le_mul' (le_refl (ENNReal.ofReal ((e/(c:ℝ))^(-codimension κ)))) hvol
-  apply hh.trans_eq
-  rw [hmul]
-  have hcI : 0 ≤ (c:ℝ)⁻¹ := inv_nonneg.mpr c.coe_nonneg
-  have hcP : 0 ≤ ((c:ℝ)⁻¹)^(-codimension κ) := Real.rpow_nonneg hcI _
-  rw [ENNReal.ofReal_mul hcP,ENNReal.ofReal_mul hcP,ENNReal.ofReal_pow hcI]
-  simp only [ENNReal.ofReal_inv_of_pos (show 0 < (c:ℝ) from hc),ENNReal.ofReal_coe_nnreal,ENNReal.coe_inv hc.ne']
-  ac_rfl
-lemma RegularSLEPresentation.source_uniform_sausage_seed
-    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) :
-    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ δ : ℝ≥0, 0 < δ →
-      ∃ (K : Set ℂ) (b e₀ : ℝ), IsCompact K ∧ K ⊆ upperHalfPlane ∧ 0 < b ∧ 0 < e₀ ∧
-        ∀ e : ℝ, 0 < e → e < e₀ →
-          p ≤ (P.comap (Subtype.val : a.samples → Ω))
-            {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
-              volume (K ∩ sourceSausageSet (γ ω) 0 δ e)} := by
-  have hk : 0 < κ := by linarith
-  let z₀ : ℂ := 3*I
-  have hz₀ : 2 < z₀.im := by norm_num [z₀]
-  obtain ⟨T,p,b,e₀,hT,hp,hb,he₀,hseed⟩ := a.source_positive_probability_initial_sausage hκ hκ8 z₀ hz₀
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  have hp1 : ENNReal.ofReal p ≤ 1 := (hseed (e₀/2) (by positivity) (by linarith)).trans prob_le_one
-  refine ⟨ENNReal.ofReal p,ENNReal.ofReal_pos.mpr hp,hp1,?_⟩
-  intro δ hδ
-  let c := NNReal.sqrt (δ/T)
-  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
-  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
-  have hcR : 0 < (c:ℝ) := hc
-  have ht : sourceScaleTime c δ = T := by
-    unfold sourceScaleTime
-    rw [hc2]
-    field_simp
-  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
-  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
-  have hKH : K ⊆ upperHalfPlane := by
-    rintro _ ⟨z,hz,rfl⟩
-    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
-    change 0 < ((c:ℂ)*z).im
-    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos hcR hz'
-  let C := ((c:ℝ)⁻¹)^(-codimension κ)*((c:ℝ)⁻¹)^2
-  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (inv_pos.mpr hcR) _) (sq_pos_of_pos (inv_pos.mpr hcR))
-  refine ⟨K,b/C,(c:ℝ)*e₀,hK,hKH,div_pos hb hC,mul_pos hcR he₀,?_⟩
-  intro e he hecut
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω)) (a.sourceScaled c hc).trace_continuous
-  let A : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal (b/C) ≤ ENNReal.ofReal (e^(-codimension κ))*volume (K ∩ sourceSausageSet g 0 δ e)}
-  have hA : MeasurableSet A := measurableSet_le measurable_const
-    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ e K))
-  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law hk hκ8 c hc)
-  have hgm : Measurable gp := a.source_trace_path_measurable
-  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable
-  rw [Measure.map_apply hgm hA,Measure.map_apply hsm hA] at hmap
-  change ENNReal.ofReal p ≤ Q (gp ⁻¹' A)
-  rw [hmap]
-  have hmesh : e/(c:ℝ) < e₀ := (div_lt_iff₀ hcR).mpr (by simpa only [mul_comm] using hecut)
-  apply (hseed (e/(c:ℝ)) (div_pos he hcR) hmesh).trans
-  apply measure_mono
-  intro ω hω
-  have hh := hω.trans (source_sausage_scale_mass_bound κ hc (γ ω) T δ ht (closedBall z₀ 1) e he)
-  change ENNReal.ofReal (b/C) ≤ _
-  rw [ENNReal.ofReal_div_of_pos hC]
-  apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
-  simpa only [C,K,sp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hh
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_volume_le_bilip_image {f : ℂ → ℂ} {S : Set ℂ} {L : ℝ≥0}
-    (hf : ∀ x ∈ S, ∀ y ∈ S, dist x y ≤ (L:ℝ)*dist (f x) (f y)) :
-    volume S ≤ (L:ℝ≥0∞)^2*volume (f '' S) := by
-  have hinj : InjOn f S := by
-    intro x hx y hy he
-    have hh := hf x hx y hy
-    rw [he,dist_self,mul_zero] at hh
-    exact dist_le_zero.mp hh
-  let g := Function.invFunOn f S
-  have hleft : LeftInvOn g f S := hinj.leftInvOn_invFunOn
-  have hLip : LipschitzOnWith L g (f '' S) := by
-    apply LipschitzOnWith.of_dist_le_mul
-    rintro _ ⟨x,hx,rfl⟩ _ ⟨y,hy,rfl⟩
-    rw [hleft hx,hleft hy]
-    exact hf x hx y hy
-  have hh := source_volume_lipschitz_image hLip
-  rw [hleft.image_image] at hh
-  exact hh
-lemma source_chart_sausage_area {f : ℂ → ℂ} {K : Set ℂ} {r q : ℝ} {L : ℝ≥0}
-    (hL : 0 < L) (_hr : 0 < r) (hq : 0 < q) (hqr : q/(L:ℝ) ≤ r)
-    (hf : LipschitzOnWith L f (cthickening r K))
-    (hlo : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ (L:ℝ)*dist (f x) (f y))
-    (η γ : ℝ≥0 → ℂ) (s δ : ℝ≥0)
-    (hid : ∀ u ∈ Icc 0 δ, η u ∈ cthickening r K → γ (s+u) = f (η u)) :
-    volume (K ∩ sourceSausageSet η 0 δ (q/(L:ℝ))) ≤
-      (L:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) q) := by
-  let S := K ∩ sourceSausageSet η 0 δ (q/(L:ℝ))
-  have hin : f '' S ⊆ sourceSausageSet γ s (s+δ) q := by
-    rintro _ ⟨z,hz,rfl⟩
-    obtain ⟨u,hu,hzu⟩ := hz.2
-    have huz : η u ∈ cthickening r K := by
-      apply mem_cthickening_of_dist_le (η u) z r K hz.1
-      exact (by simpa only [dist_comm] using hzu.le.trans hqr)
-    have hh := hf.dist_le_mul z (self_subset_cthickening (δ := r) K hz.1) (η u) huz
-    have hdist : dist (f z) (γ (s+u)) < q := by
-      rw [hid u hu huz]
-      apply hh.trans_lt
-      exact (mul_lt_mul_of_pos_left hzu (show 0 < (L:ℝ) from hL)).trans_eq (by field_simp)
-    exact ⟨s+u,⟨by simp,add_le_add le_rfl hu.2⟩,hdist⟩
-  exact (source_volume_le_bilip_image (fun x hx y hy => hlo x hx.1 y hy.1)).trans
-    (mul_le_mul' le_rfl (measure_mono hin))
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_inverse_translated_chart_compact_bounds
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (s : ℝ≥0)
-    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) :
-    ∃ (r : ℝ) (L : ℝ≥0), 0 < r ∧ 0 < L ∧
-      cthickening r K ⊆ upperHalfPlane ∧
-      LipschitzOnWith L (fun z => F s (z+(U s:ℂ))) (cthickening r K) ∧
-      ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ (L:ℝ)*dist (F s (x+(U s:ℂ))) (F s (y+(U s:ℂ))) := by
-  obtain ⟨r,hr,hrK⟩ := hK.exists_cthickening_subset_open isOpen_upperHalfPlane hKH
-  have hKr : IsCompact (cthickening r K) := hK.cthickening
-  let Z := (fun z : ℂ => z+(U s:ℂ)) '' cthickening r K
-  have hZ : IsCompact Z := hKr.image (continuous_id.add_const _)
-  have hZH : Z ⊆ upperHalfPlane := by
-    rintro _ ⟨z,hz,rfl⟩
-    simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,add_zero] using hrK hz
-  obtain ⟨L,hL,hLip,hLo⟩ := source_inverse_chart_compact_bilipschitz U γ G F hw s hZ hZH
-  refine ⟨r,L,hr,hL,hrK,?_,?_⟩
-  · apply LipschitzOnWith.of_dist_le_mul
-    intro x hx y hy
-    simpa only [dist_add_right] using hLip.dist_le_mul (x+(U s:ℂ)) ⟨x,hx,rfl⟩ (y+(U s:ℂ)) ⟨y,hy,rfl⟩
-  · intro x hx y hy
-    simpa only [dist_add_right] using hLo (x+(U s:ℂ)) ⟨x,self_subset_cthickening K hx,rfl⟩
-      (y+(U s:ℂ)) ⟨y,self_subset_cthickening K hy,rfl⟩
-lemma source_restart_sausage_countable_bound
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (s δ : ℝ≥0) (η : ℝ≥0 → ℂ)
-    (hid : ∀ u, η u ∈ upperHalfPlane → γ (s+u) = F s (η u+(U s:ℂ)))
-    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) :
-    ∃ N : ℕ, 1 ≤ N ∧ ∀ q : ℝ, 0 < q → q < 1/(N:ℝ) →
-      volume (K ∩ sourceSausageSet η 0 δ (q/(N:ℝ))) ≤
-        (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) q) := by
-  obtain ⟨r,L,hr,hL,hrK,hLip,hLo⟩ := source_inverse_translated_chart_compact_bounds U γ G F hw s hK hKH
-  obtain ⟨N,hN⟩ := exists_nat_gt (max 1 (max (L:ℝ) r⁻¹))
-  have hN1 : 1 ≤ N := by exact_mod_cast (le_max_left 1 (max (L:ℝ) r⁻¹)).trans hN.le
-  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN1)
-  have hLN : (L:ℝ) ≤ N := (le_max_left (L:ℝ) r⁻¹).trans ((le_max_right _ _).trans hN.le)
-  have hrN : 1/(N:ℝ) ≤ r := by
-    have h := (le_max_right (L:ℝ) r⁻¹).trans ((le_max_right 1 _).trans hN.le)
-    apply (div_le_iff₀ hNp).mpr
-    have hh := (inv_le_iff_one_le_mul₀ hr).mp h
-    simpa only [mul_comm] using hh
-  refine ⟨N,hN1,?_⟩
-  intro q hq hqN
-  have hqdiv : q/(N:ℝ) ≤ r := by
-    have hh : q/(N:ℝ) ≤ q := div_le_self hq.le (by exact_mod_cast hN1)
-    exact hh.trans (hqN.le.trans hrN)
-  exact source_chart_sausage_area (L := (N:ℝ≥0)) (by exact_mod_cast hNp) hr hq hqdiv
-    (hLip.weaken (by exact_mod_cast hLN))
-    (fun x hx y hy => (hLo x hx y hy).trans (mul_le_mul_of_nonneg_right hLN dist_nonneg))
-    η γ s δ (fun u _ hu => hid u (hrK hu))
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance sausageScalingMeshPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance sausageScalingMeshPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma source_sausage_mono_mesh (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) {r e : ℝ} (hle : r ≤ e) :
-    sourceSausageSet γ s t r ⊆ sourceSausageSet γ s t e := by
-  rintro z ⟨u,hu,hz⟩
-  exact ⟨u,hu,hz.trans_le hle⟩
-def sourceSausageChartGood (η γ : ℝ≥0 → ℂ) (K : Set ℂ) (s δ : ℝ≥0) (N : ℕ) : Prop :=
-  1 ≤ N ∧ ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) < 1/(N:ℝ) →
-    volume (K ∩ sourceSausageSet η 0 δ ((q:ℝ)/(N:ℝ))) ≤
-      (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) (q:ℝ))
-lemma source_chart_good_mono (η γ : ℝ≥0 → ℂ) (K : Set ℂ) (s δ : ℝ≥0) :
-    Monotone (fun N : ℕ => sourceSausageChartGood η γ K s δ N) := by
-  intro M N hMN hM
-  refine ⟨hM.1.trans hMN,?_⟩
-  intro q hq hqN
-  have hMp : 0 < (M:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hM.1)
-  have hMrN : (M:ℝ) ≤ N := by exact_mod_cast hMN
-  have hqM : (q:ℝ) < 1/(M:ℝ) := hqN.trans_le (one_div_le_one_div_of_le hMp hMrN)
-  calc
-    _ ≤ volume (K ∩ sourceSausageSet η 0 δ ((q:ℝ)/(M:ℝ))) :=
-      measure_mono (inter_subset_inter_right K (source_sausage_mono_mesh η 0 δ (div_le_div_of_nonneg_left hq.le hMp hMrN)))
-    _ ≤ (M:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) (q:ℝ)) := hM.2 q hq hqM
-    _ ≤ _ := by gcongr
-lemma source_chart_good_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (η γ : Ω → C(ℝ≥0,ℂ)) (hmη : Measurable η) (hmγ : Measurable γ)
-    (K : Set ℂ) (s δ : ℝ≥0) (N : ℕ) :
-    MeasurableSet {ω | sourceSausageChartGood (η ω) (γ ω) K s δ N} := by
-  simp only [sourceSausageChartGood,ofPred_and,ofPred_forall]
-  apply MeasurableSet.inter (by measurability)
-  apply MeasurableSet.iInter; intro q
-  apply MeasurableSet.iInter; intro hq
-  apply MeasurableSet.iInter; intro hqN
-  apply measurableSet_le
-  · exact (source_sausage_local_volume_measurable 0 δ ((q:ℝ)/(N:ℝ)) K).comp hmη
-  · have hmvol : Measurable (fun g : C(ℝ≥0,ℂ) => volume (sourceSausageSet g s (s+δ) (q:ℝ))) := by
-      simpa only [univ_inter] using source_sausage_local_volume_measurable s (s+δ) (q:ℝ) univ
-    exact measurable_const.mul (hmvol.comp hmγ)
-lemma source_chart_good_real_bound (η γ : ℝ≥0 → ℂ) (K : Set ℂ) (s δ : ℝ≥0) (N : ℕ)
-    (hN : sourceSausageChartGood η γ K s δ N) {e : ℝ} (he : 0 < e) (heN : e < 1/(N:ℝ)) :
-    volume (K ∩ sourceSausageSet η 0 δ (e/(2*(N:ℝ)))) ≤
-      (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) e) := by
-  obtain ⟨q,hq1,hq2⟩ := exists_rat_btwn (show e/2 < e by linarith)
-  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN.1)
-  have hmesh : e/(2*(N:ℝ)) ≤ (q:ℝ)/(N:ℝ) := by rw [←div_div]; exact div_le_div_of_nonneg_right hq1.le hNp.le
-  calc
-    _ ≤ volume (K ∩ sourceSausageSet η 0 δ ((q:ℝ)/(N:ℝ))) :=
-      measure_mono (inter_subset_inter_right K (source_sausage_mono_mesh η 0 δ hmesh))
-    _ ≤ (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) (q:ℝ)) := hN.2 q (by linarith) (hq2.trans heN)
-    _ ≤ _ := mul_le_mul' le_rfl (measure_mono (source_sausage_mono_mesh γ s (s+δ) hq2.le))
-lemma source_chart_good_high_probability {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsProbabilityMeasure P]
-    (η γ : Ω → C(ℝ≥0,ℂ)) (hmη : Measurable η) (hmγ : Measurable γ)
-    (K : Set ℂ) (s δ : ℝ≥0)
-    (hfull : ∀ᵐ ω ∂P, ∃ N : ℕ, sourceSausageChartGood (η ω) (γ ω) K s δ N)
-    (ε : ℝ) (hε : 0 < ε) :
-    ∃ N : ℕ, 1 ≤ N ∧ P {ω | ¬sourceSausageChartGood (η ω) (γ ω) K s δ N} < ENNReal.ofReal ε := by
-  let B := fun N : ℕ => {ω | ¬sourceSausageChartGood (η ω) (γ ω) K s δ N}
-  have hB : ∀ N, MeasurableSet (B N) := fun N => (source_chart_good_measurable η γ hmη hmγ K s δ N).compl
-  have hanti : Antitone B := by
-    intro M N hMN ω hω hM
-    exact hω (source_chart_good_mono (η ω) (γ ω) K s δ hMN hM)
-  have hz : P (⋂ N, B N) = 0 := by
-    have hh := ae_iff.mp hfull
-    simpa only [B,not_exists,ofPred_forall] using hh
-  have ht := tendsto_measure_iInter_atTop (fun N => (hB N).nullMeasurableSet) hanti
-    ⟨0,measure_ne_top P _⟩
-  rw [hz] at ht
-  have hevv : ∀ᶠ N in atTop, P (B N) < ENNReal.ofReal ε :=
-    ht.eventually (gt_mem_nhds (ENNReal.ofReal_pos.mpr hε))
-  obtain ⟨N,hN,hp⟩ := ((eventually_ge_atTop 1).and hevv).exists
-  exact ⟨N,hN,hp⟩
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance sausageScalingIntervalPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance sausageScalingIntervalPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma source_sausage_mono_interval (γ : ℝ≥0 → ℂ) (e : ℝ) {s t u v : ℝ≥0}
-    (hI : Icc s t ⊆ Icc u v) : sourceSausageSet γ s t e ⊆ sourceSausageSet γ u v e := by
-  rintro z ⟨w,hw,hz⟩
-  exact ⟨w,hI hw,hz⟩
-lemma source_sausage_mass_mono_interval (κ : ℝ) (γ : ℝ≥0 → ℂ) (e : ℝ) {s t u v : ℝ≥0}
-    (hI : Icc s t ⊆ Icc u v) : sourceSausageMass κ γ s t e ≤ sourceSausageMass κ γ u v e :=
-  mul_le_mul' le_rfl (measure_mono (source_sausage_mono_interval γ e hI))
-lemma RegularSLEPresentation.source_sausage_mass_measurable_at
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (s t : ℝ≥0) (e : ℝ) :
-    @Measurable a.samples ℝ≥0∞ (a.filtration t) inferInstance (fun ω => sourceSausageMass κ (γ ω) s t e) := by
-  let f : a.samples → C(ℝ≥0,ℂ) := fun ω =>
-    ⟨fun u => γ ω (min u t), (a.trace_continuous ω).comp (continuous_id.min continuous_const)⟩
-  have hm : @Measurable a.samples C(ℝ≥0,ℂ) (a.filtration t) (borel _) f := by
-    let : MeasurableSpace a.samples := a.filtration t
-    apply source_path_measurable
-    intro u
-    exact (a.trace_adapted (min u t)).mono (a.filtration.mono (min_le_right u t)) le_rfl
-  have he (ω : a.samples) : sourceSausageSet (γ ω) s t e = sourceSausageSet (f ω) s t e := by
-    ext z
-    constructor
-    · rintro ⟨u,hu,huz⟩
-      exact ⟨u,hu,by simpa only [f,ContinuousMap.coe_mk,min_eq_left hu.2] using huz⟩
-    · rintro ⟨u,hu,huz⟩
-      exact ⟨u,hu,by simpa only [f,ContinuousMap.coe_mk,min_eq_left hu.2] using huz⟩
-  have he' : (fun ω : a.samples => sourceSausageMass κ (γ ω) s t e) =
-      (fun ω => sourceSausageMass κ (f ω) s t e) := by
-    funext ω; unfold sourceSausageMass; rw [he]
-  rw [he']
-  exact (source_sausage_mass_measurable κ s t e).comp hm
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_chart_good_mass_bound (κ : ℝ) (η γ : ℝ≥0 → ℂ) (K : Set ℂ)
-    (s δ : ℝ≥0) (N : ℕ) (hN : sourceSausageChartGood η γ K s δ N)
-    {e : ℝ} (he : 0 < e) (heN : e < 1/(N:ℝ)) :
-    ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-      volume (K ∩ sourceSausageSet η 0 δ (e/(2*(N:ℝ)))) ≤
-    ENNReal.ofReal (((2*(N:ℝ))⁻¹)^(-codimension κ)*(N:ℝ)^2)*sourceSausageMass κ γ s (s+δ) e := by
-  have hvol := source_chart_good_real_bound η γ K s δ N hN he heN
-  have hnp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN.1)
-  have hbase : 0 ≤ (2*(N:ℝ))⁻¹ := by positivity
-  have hmul : (e/(2*(N:ℝ)))^(-codimension κ) = ((2*(N:ℝ))⁻¹)^(-codimension κ)*e^(-codimension κ) := by
-    rw [show e/(2*(N:ℝ)) = (2*(N:ℝ))⁻¹*e by ring,Real.mul_rpow hbase he.le]
-  have hh := mul_le_mul' (le_refl (ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ)))) hvol
-  apply hh.trans_eq
-  rw [hmul,ENNReal.ofReal_mul (Real.rpow_nonneg hbase _),
-    ENNReal.ofReal_mul (Real.rpow_nonneg hbase _),ENNReal.ofReal_pow (Nat.cast_nonneg N)]
-  simp only [ENNReal.ofReal_natCast,sourceSausageMass]
-  ac_rfl
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance SausageConditional_local1 : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance SausageConditional_local2 : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance SausageConditional_local3 : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance SausageConditional_local4 : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma RegularSLEPresentation.source_uniform_sausage_slot_bound
-    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) :
-    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ →
-      ∀ ε : ℝ, 0 < ε → ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧
-        ∀ e : ℝ, 0 < e → e < e₀ →
-        ∀ A : Set a.samples, MeasurableSet[a.filtration s] A →
-          (P.comap (Subtype.val : a.samples → Ω))
-            (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal b}) ≤
-          (1-p)*(P.comap (Subtype.val : a.samples → Ω)) A + ENNReal.ofReal ε := by
-  have hk : 0 < κ := by linarith
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨p,hp,hp1,hseed⟩ := a.source_uniform_sausage_seed hκ hκ8
-  refine ⟨p,hp,hp1,?_⟩
-  intro s δ hδ ε hε
-  obtain ⟨K,b,e₀,hK,hKH,hb,he₀,hKP⟩ := hseed δ hδ
-  obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian
-    a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness
-  let τ : a.samples → ℝ≥0 := fun _ => s
-  have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s
-  obtain ⟨hlaw,hident⟩ := hrestart τ hτ
-  let rp := sourceRestartDriverPath a.B a.continuous_B τ
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let η := fun ω : a.samples => Φ (rp ω)
-  have hrm : Measurable rp := by
-    apply source_path_measurable
-    intro t
-    exact (a.measurable_B (s+t)).sub (a.measurable_B s)
-  have hem : Measurable η := hΦm.comp hrm
-  have hgm : Measurable gp := a.source_trace_path_measurable
-  have hfull : ∀ᵐ ω ∂Q, ∃ N : ℕ, sourceSausageChartGood (η ω) (gp ω) K s δ N := by
-    filter_upwards [hident] with ω hω
-    obtain ⟨N,hN,hbound⟩ := source_restart_sausage_countable_bound
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) s δ (η ω) hω hK hKH
-    exact ⟨N,hN,fun q hq hqN => hbound q hq hqN⟩
-  obtain ⟨N,hN,hbad⟩ := source_chart_good_high_probability Q η gp hem hgm K s δ hfull ε hε
-  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN)
-  let C := ((2*(N:ℝ))⁻¹)^(-codimension κ)*(N:ℝ)^2
-  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (sq_pos_of_pos hNp)
-  refine ⟨b/C,min (1/(N:ℝ)) (2*(N:ℝ)*e₀),div_pos hb hC,lt_min (by positivity) (by positivity),?_⟩
-  intro e he hecut A hA
-  have hmesh : e/(2*(N:ℝ)) < e₀ := (div_lt_iff₀ (by positivity)).mpr
-    (by simpa only [mul_comm] using hecut.trans_le (min_le_right _ _))
-  let D : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-      volume (K ∩ sourceSausageSet g 0 δ (e/(2*(N:ℝ))))}
-  let E : Set a.samples := η ⁻¹' D
-  let B : Set a.samples := {ω | ¬sourceSausageChartGood (η ω) (gp ω) K s δ N}
-  have hD : MeasurableSet D := measurableSet_le measurable_const
-    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ (e/(2*(N:ℝ))) K))
-  have hEm : MeasurableSet E := hD.preimage hem
-  have hAm : MeasurableSet A := (a.filtration.le s) A hA
-  have hAstop : MeasurableSet[hτ.measurableSpace] A := by
-    change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A
-    rw [IsStoppingTime.measurableSpace_const]
-    exact hA
-  have hprob : p*Q A ≤ Q (A ∩ E) := by
-    have hEq := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ D) (hlaw A hAstop)
-    rw [Measure.map_apply hem hD,Measure.smul_apply,Measure.map_apply hgm hD,
-      Measure.restrict_apply (hD.preimage hem)] at hEq
-    change Q (E ∩ A) = Q A*Q {ω | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-      volume (K ∩ sourceSausageSet (γ ω) 0 δ (e/(2*(N:ℝ))))} at hEq
-    rw [inter_comm] at hEq
-    rw [hEq,mul_comm p (Q A)]
-    exact mul_le_mul' le_rfl (hKP (e/(2*(N:ℝ))) (by positivity) hmesh)
-  have hins : A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)} ⊆ (A \ E) ∪ B := by
-    intro ω hω
-    by_cases hG : sourceSausageChartGood (η ω) (gp ω) K s δ N
-    · left
-      refine ⟨hω.1,?_⟩
-      intro hE
-      have hEE : ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-          volume (K ∩ sourceSausageSet (η ω) 0 δ (e/(2*(N:ℝ)))) := hE
-      have hmass := hEE.trans (source_chart_good_mass_bound κ (η ω) (gp ω) K s δ N hG he
-        (hecut.trans_le (min_le_left _ _)))
-      have hlow : ENNReal.ofReal (b/C) ≤ sourceSausageMass κ (γ ω) s (s+δ) e := by
-        rw [ENNReal.ofReal_div_of_pos hC]
-        apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
-        simpa only [C,gp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hmass
-      exact (not_lt_of_ge hlow) hω.2
-    · exact Or.inr hG
-  have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by
-    have heq : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto
-    rw [heq]
-    exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _)
-  calc
-    Q (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)})
-        ≤ Q ((A \ E) ∪ B) := measure_mono hins
-    _ ≤ Q (A \ E) + Q B := measure_union_le _ _
-    _ ≤ (Q A-p*Q A)+ENNReal.ofReal ε := by rw [hdiff]; exact add_le_add (tsub_le_tsub_left hprob _) hbad.le
-    _ = (1-p)*Q A+ENNReal.ofReal ε := by
-      rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul]
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma RegularSLEPresentation.source_sausage_nslot_bound
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) :
-    ∃ c : ℝ≥0∞, c < 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ → ∀ ν : ℝ, 0 < ν →
-      ∀ n : ℕ, ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-        (P.comap (Subtype.val : a.samples → Ω))
-          {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b} ≤
-        c^n + (n:ℝ≥0∞)*ENNReal.ofReal ν := by
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_sausage_slot_bound hκ hκ8
-  let c : ℝ≥0∞ := 1-p
-  have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne'
-  refine ⟨c,hc,?_⟩
-  intro s δ hδ ν hν n
-  induction n with
-  | zero =>
-    refine ⟨1,1,by norm_num,by norm_num,?_⟩
-    intro e he hel
-    simpa only [pow_zero,Nat.cast_zero,zero_mul,add_zero] using
-      (prob_le_one (μ := Q) (s := {ω | sourceSausageMass κ (γ ω) s (s+(0:ℝ≥0)*δ) e < ENNReal.ofReal 1}))
-  | succ n ih =>
-    obtain ⟨b₀,e₀,hb₀,he₀,hprev⟩ := ih
-    obtain ⟨b₁,e₁,hb₁,he₁,hstep⟩ := hslot (s+(n:ℝ≥0)*δ) δ hδ ν hν
-    refine ⟨min b₀ b₁,min e₀ e₁,lt_min hb₀ hb₁,lt_min he₀ he₁,?_⟩
-    intro e he hel
-    have hend : s+((n+1:ℕ):ℝ≥0)*δ = (s+(n:ℝ≥0)*δ)+δ := by push_cast; ring
-    let A : Set a.samples := {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b₀}
-    have hA : MeasurableSet[a.filtration (s+(n:ℝ≥0)*δ)] A :=
-      measurableSet_lt (a.source_sausage_mass_measurable_at s _ e) measurable_const
-    rw [hend]
-    calc
-      Q {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal (min b₀ b₁)}
-          ≤ Q (A ∩ {ω | sourceSausageMass κ (γ ω) (s+(n:ℝ≥0)*δ) (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal b₁}) := by
-          apply measure_mono
-          intro ω hω
-          constructor
-          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
-              (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))).trans_lt hω).trans_le
-                (ENNReal.ofReal_le_ofReal (min_le_left _ _))
-          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
-              (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)).trans_lt hω).trans_le
-                (ENNReal.ofReal_le_ofReal (min_le_right _ _))
-      _ ≤ c*Q A+ENNReal.ofReal ν := hstep e he (hel.trans_le (min_le_right _ _)) A hA
-      _ ≤ c*(c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν)+ENNReal.ofReal ν := by
-          exact add_le_add (mul_le_mul' le_rfl (hprev e he (hel.trans_le (min_le_left _ _)))) le_rfl
-      _ ≤ c^(n+1)+((n+1:ℕ):ℝ≥0∞)*ENNReal.ofReal ν := by
-          rw [mul_add,pow_succ,mul_comm (c^n) c,Nat.cast_add,Nat.cast_one,add_mul,one_mul]
-          rw [←add_assoc]
-          exact add_le_add (add_le_add le_rfl (mul_le_of_le_one_left' hc.le)) le_rfl
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma RegularSLEPresentation.source_small_sausage
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) (s t : ℝ≥0) (hst : s < t) (ε : ℝ) (hε : 0 < ε) :
-    ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-      (P.comap (Subtype.val : a.samples → Ω))
-        {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal b} < ENNReal.ofReal ε := by
-  obtain ⟨c,hc,hsteps⟩ := a.source_sausage_nslot_bound hκ hκ8
-  have hev : ∀ᶠ n : ℕ in atTop, c^n < ENNReal.ofReal (ε/4) :=
-    (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc).eventually
-      (gt_mem_nhds (ENNReal.ofReal_pos.mpr (by positivity)))
-  obtain ⟨n,hn,hcn⟩ := ((eventually_ge_atTop 1).and hev).exists
-  have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega)
-  let δ : ℝ≥0 := (t-s)/(n:ℝ≥0)
-  have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0
-  have hend : s+(n:ℝ≥0)*δ = t := by
-    dsimp [δ]
-    rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le]
-  let ν : ℝ := ε/(4*((n:ℝ)+1))
-  have hν : 0 < ν := by dsimp [ν]; positivity
-  obtain ⟨b,e₀,hb,he₀,hbound⟩ := hsteps s δ hδ ν hν n
-  have hnν : (n:ℝ≥0∞)*ENNReal.ofReal ν ≤ ENNReal.ofReal (ε/4) := by
-    rw [←ENNReal.ofReal_natCast,←ENNReal.ofReal_mul (Nat.cast_nonneg n)]
-    apply ENNReal.ofReal_le_ofReal
-    calc (n:ℝ)*ν ≤ ((n:ℝ)+1)*ν := mul_le_mul_of_nonneg_right (by linarith) hν.le
-      _ = ε/4 := by dsimp [ν]; field_simp
-  refine ⟨b,e₀,hb,he₀,?_⟩
-  intro e he hel
-  have hh := hbound e he hel
-  rw [hend] at hh
-  apply hh.trans_lt
-  calc c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν
-      ≤ ENNReal.ofReal (ε/4)+ENNReal.ofReal (ε/4) := add_le_add hcn.le hnν
-    _ = ENNReal.ofReal (ε/2) := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
-    _ < ENNReal.ofReal ε := (ENNReal.ofReal_lt_ofReal_iff hε).mpr (by linarith)
-theorem sourceSmallSausage_proved : SourceSmallSausageTarget := by
-  intro κ hκ hκ8 Ω m P hP γ hSLE s t hst ε hε
-  let : IsProbabilityMeasure P := hP
-  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
-  obtain ⟨b,e₀,hb,he₀,hbound⟩ := a.source_small_sausage hκ hκ8 s t hst ε hε
-  refine ⟨b,e₀,hb,he₀,?_⟩
-  intro e he hel
-  have hh := hbound e he hel
-  rw [←a.measurePreserving_val.map_eq]
-  rwa [(MeasurableEmbedding.subtype_coe a.measurable_samples).map_apply] 
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.TracePosCombined
+
+namespace OAI
+
+/-! Sausage Scaling for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance SausageScaling_local1 : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance SausageScaling_local2 : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma source_sausage_dilation {c : ℝ≥0} (hc : 0 < c) (η : ℝ≥0 → ℂ)
+    (T δ : ℝ≥0) (ht : sourceScaleTime c δ = T) (K : Set ℂ) (e : ℝ) :
+    ((fun z : ℂ => (c:ℂ)*z) '' (K ∩ sourceSausageSet η 0 T (e/(c:ℝ)))) ⊆
+      ((fun z : ℂ => (c:ℂ)*z) '' K) ∩ sourceSausageSet (sourceScaleTrace c η) 0 δ e := by
+  rintro _ ⟨z,hz,rfl⟩
+  refine ⟨⟨z,hz.1,rfl⟩,?_⟩
+  obtain ⟨u,hu,hzu⟩ := hz.2
+  have hc2 : c^2 ≠ 0 := pow_ne_zero _ hc.ne'
+  have htu : c^2*u ≤ δ := by
+    have hh := hu.2
+    rw [←ht,sourceScaleTime,le_div_iff₀ (pow_pos hc _)] at hh
+    simpa only [mul_comm] using hh
+  refine ⟨c^2*u,⟨bot_le,htu⟩,?_⟩
+  have he : sourceScaleTime c (c^2*u) = u := by
+    unfold sourceScaleTime; field_simp
+  simp only [sourceScaleTrace,he,dist_eq_norm,←mul_sub,norm_mul,norm_real,Real.norm_eq_abs,abs_of_nonneg c.coe_nonneg] 
+  have hh : ‖z-η u‖ < e/(c:ℝ) := by simpa only [dist_eq_norm] using hzu
+  have hcR : 0 < (c:ℝ) := hc
+  nlinarith [(lt_div_iff₀ hcR).mp hh]
+lemma source_volume_dilation_lower {c : ℝ≥0} (hc : 0 < c) (S : Set ℂ) :
+    volume S ≤ ((c⁻¹:ℝ≥0):ℝ≥0∞)^2 * volume ((fun z : ℂ => (c:ℂ)*z) '' S) := by
+  have hLip : LipschitzWith c⁻¹ (fun z : ℂ => (c⁻¹:ℂ)*z) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    simp [dist_eq_norm,←mul_sub]
+  have hh := source_volume_lipschitz_image (hLip.lipschitzOnWith (s := (fun z : ℂ => (c:ℂ)*z) '' S))
+  have hi : (fun z : ℂ => (c⁻¹:ℂ)*z) '' ((fun z : ℂ => (c:ℂ)*z) '' S) = S := by
+    rw [←image_comp]
+    have he : (fun z : ℂ => (c⁻¹:ℂ)*((c:ℂ)*z)) = id := by
+      funext z
+      simp only [←mul_assoc,inv_mul_cancel₀ (by exact_mod_cast hc.ne' : (c:ℂ) ≠ 0),one_mul,id_eq]
+    simp only [Function.comp_def,he,image_id]
+  rwa [hi] at hh
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance SausageUniformSeed_local1 : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance SausageUniformSeed_local2 : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma source_sausage_scale_mass_bound (κ : ℝ) {c : ℝ≥0} (hc : 0 < c)
+    (η : ℝ≥0 → ℂ) (T δ : ℝ≥0) (ht : sourceScaleTime c δ = T)
+    (K : Set ℂ) (e : ℝ) (he : 0 < e) :
+    ENNReal.ofReal ((e/(c:ℝ))^(-codimension κ))*volume (K ∩ sourceSausageSet η 0 T (e/(c:ℝ))) ≤
+      ENNReal.ofReal (((c:ℝ)⁻¹)^(-codimension κ)*((c:ℝ)⁻¹)^2)*
+        (ENNReal.ofReal (e^(-codimension κ))*volume (((fun z : ℂ => (c:ℂ)*z) '' K) ∩
+          sourceSausageSet (sourceScaleTrace c η) 0 δ e)) := by
+  have hvol := (source_volume_dilation_lower hc (K ∩ sourceSausageSet η 0 T (e/(c:ℝ)))).trans (mul_le_mul' le_rfl (measure_mono (source_sausage_dilation hc η T δ ht K e)))
+  have hmul : (e/(c:ℝ))^(-codimension κ) = ((c:ℝ)⁻¹)^(-codimension κ)*e^(-codimension κ) := by
+    rw [show e/(c:ℝ) = (c:ℝ)⁻¹*e by ring,Real.mul_rpow (by positivity) he.le]
+  have hh := mul_le_mul' (le_refl (ENNReal.ofReal ((e/(c:ℝ))^(-codimension κ)))) hvol
+  apply hh.trans_eq
+  rw [hmul]
+  have hcI : 0 ≤ (c:ℝ)⁻¹ := inv_nonneg.mpr c.coe_nonneg
+  have hcP : 0 ≤ ((c:ℝ)⁻¹)^(-codimension κ) := Real.rpow_nonneg hcI _
+  rw [ENNReal.ofReal_mul hcP,ENNReal.ofReal_mul hcP,ENNReal.ofReal_pow hcI]
+  simp only [ENNReal.ofReal_inv_of_pos (show 0 < (c:ℝ) from hc),ENNReal.ofReal_coe_nnreal,ENNReal.coe_inv hc.ne']
+  ac_rfl
+lemma RegularSLEPresentation.source_uniform_sausage_seed
+    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ δ : ℝ≥0, 0 < δ →
+      ∃ (K : Set ℂ) (b e₀ : ℝ), IsCompact K ∧ K ⊆ upperHalfPlane ∧ 0 < b ∧ 0 < e₀ ∧
+        ∀ e : ℝ, 0 < e → e < e₀ →
+          p ≤ (P.comap (Subtype.val : a.samples → Ω))
+            {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
+              volume (K ∩ sourceSausageSet (γ ω) 0 δ e)} := by
+  have hk : 0 < κ := by linarith
+  let z₀ : ℂ := 3*I
+  have hz₀ : 2 < z₀.im := by norm_num [z₀]
+  obtain ⟨T,p,b,e₀,hT,hp,hb,he₀,hseed⟩ := a.source_positive_probability_initial_sausage hκ hκ8 z₀ hz₀
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  have hp1 : ENNReal.ofReal p ≤ 1 := (hseed (e₀/2) (by positivity) (by linarith)).trans prob_le_one
+  refine ⟨ENNReal.ofReal p,ENNReal.ofReal_pos.mpr hp,hp1,?_⟩
+  intro δ hδ
+  let c := NNReal.sqrt (δ/T)
+  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
+  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
+  have hcR : 0 < (c:ℝ) := hc
+  have ht : sourceScaleTime c δ = T := by
+    unfold sourceScaleTime
+    rw [hc2]
+    field_simp
+  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
+  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
+  have hKH : K ⊆ upperHalfPlane := by
+    rintro _ ⟨z,hz,rfl⟩
+    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
+    change 0 < ((c:ℂ)*z).im
+    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos hcR hz'
+  let C := ((c:ℝ)⁻¹)^(-codimension κ)*((c:ℝ)⁻¹)^2
+  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (inv_pos.mpr hcR) _) (sq_pos_of_pos (inv_pos.mpr hcR))
+  refine ⟨K,b/C,(c:ℝ)*e₀,hK,hKH,div_pos hb hC,mul_pos hcR he₀,?_⟩
+  intro e he hecut
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω)) (a.sourceScaled c hc).trace_continuous
+  let A : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal (b/C) ≤ ENNReal.ofReal (e^(-codimension κ))*volume (K ∩ sourceSausageSet g 0 δ e)}
+  have hA : MeasurableSet A := measurableSet_le measurable_const
+    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ e K))
+  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law hk hκ8 c hc)
+  have hgm : Measurable gp := a.source_trace_path_measurable
+  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable
+  rw [Measure.map_apply hgm hA,Measure.map_apply hsm hA] at hmap
+  change ENNReal.ofReal p ≤ Q (gp ⁻¹' A)
+  rw [hmap]
+  have hmesh : e/(c:ℝ) < e₀ := (div_lt_iff₀ hcR).mpr (by simpa only [mul_comm] using hecut)
+  apply (hseed (e/(c:ℝ)) (div_pos he hcR) hmesh).trans
+  apply measure_mono
+  intro ω hω
+  have hh := hω.trans (source_sausage_scale_mass_bound κ hc (γ ω) T δ ht (closedBall z₀ 1) e he)
+  change ENNReal.ofReal (b/C) ≤ _
+  rw [ENNReal.ofReal_div_of_pos hC]
+  apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
+  simpa only [C,K,sp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hh
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_volume_le_bilip_image {f : ℂ → ℂ} {S : Set ℂ} {L : ℝ≥0}
+    (hf : ∀ x ∈ S, ∀ y ∈ S, dist x y ≤ (L:ℝ)*dist (f x) (f y)) :
+    volume S ≤ (L:ℝ≥0∞)^2*volume (f '' S) := by
+  have hinj : InjOn f S := by
+    intro x hx y hy he
+    have hh := hf x hx y hy
+    rw [he,dist_self,mul_zero] at hh
+    exact dist_le_zero.mp hh
+  let g := Function.invFunOn f S
+  have hleft : LeftInvOn g f S := hinj.leftInvOn_invFunOn
+  have hLip : LipschitzOnWith L g (f '' S) := by
+    apply LipschitzOnWith.of_dist_le_mul
+    rintro _ ⟨x,hx,rfl⟩ _ ⟨y,hy,rfl⟩
+    rw [hleft hx,hleft hy]
+    exact hf x hx y hy
+  have hh := source_volume_lipschitz_image hLip
+  rw [hleft.image_image] at hh
+  exact hh
+lemma source_chart_sausage_area {f : ℂ → ℂ} {K : Set ℂ} {r q : ℝ} {L : ℝ≥0}
+    (hL : 0 < L) (_hr : 0 < r) (hq : 0 < q) (hqr : q/(L:ℝ) ≤ r)
+    (hf : LipschitzOnWith L f (cthickening r K))
+    (hlo : ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ (L:ℝ)*dist (f x) (f y))
+    (η γ : ℝ≥0 → ℂ) (s δ : ℝ≥0)
+    (hid : ∀ u ∈ Icc 0 δ, η u ∈ cthickening r K → γ (s+u) = f (η u)) :
+    volume (K ∩ sourceSausageSet η 0 δ (q/(L:ℝ))) ≤
+      (L:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) q) := by
+  let S := K ∩ sourceSausageSet η 0 δ (q/(L:ℝ))
+  have hin : f '' S ⊆ sourceSausageSet γ s (s+δ) q := by
+    rintro _ ⟨z,hz,rfl⟩
+    obtain ⟨u,hu,hzu⟩ := hz.2
+    have huz : η u ∈ cthickening r K := by
+      apply mem_cthickening_of_dist_le (η u) z r K hz.1
+      exact (by simpa only [dist_comm] using hzu.le.trans hqr)
+    have hh := hf.dist_le_mul z (self_subset_cthickening (δ := r) K hz.1) (η u) huz
+    have hdist : dist (f z) (γ (s+u)) < q := by
+      rw [hid u hu huz]
+      apply hh.trans_lt
+      exact (mul_lt_mul_of_pos_left hzu (show 0 < (L:ℝ) from hL)).trans_eq (by field_simp)
+    exact ⟨s+u,⟨by simp,add_le_add le_rfl hu.2⟩,hdist⟩
+  exact (source_volume_le_bilip_image (fun x hx y hy => hlo x hx.1 y hy.1)).trans
+    (mul_le_mul' le_rfl (measure_mono hin))
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_inverse_translated_chart_compact_bounds
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (s : ℝ≥0)
+    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) :
+    ∃ (r : ℝ) (L : ℝ≥0), 0 < r ∧ 0 < L ∧
+      cthickening r K ⊆ upperHalfPlane ∧
+      LipschitzOnWith L (fun z => F s (z+(U s:ℂ))) (cthickening r K) ∧
+      ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ (L:ℝ)*dist (F s (x+(U s:ℂ))) (F s (y+(U s:ℂ))) := by
+  obtain ⟨r,hr,hrK⟩ := hK.exists_cthickening_subset_open isOpen_upperHalfPlane hKH
+  have hKr : IsCompact (cthickening r K) := hK.cthickening
+  let Z := (fun z : ℂ => z+(U s:ℂ)) '' cthickening r K
+  have hZ : IsCompact Z := hKr.image (continuous_id.add_const _)
+  have hZH : Z ⊆ upperHalfPlane := by
+    rintro _ ⟨z,hz,rfl⟩
+    simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,add_zero] using hrK hz
+  obtain ⟨L,hL,hLip,hLo⟩ := source_inverse_chart_compact_bilipschitz U γ G F hw s hZ hZH
+  refine ⟨r,L,hr,hL,hrK,?_,?_⟩
+  · apply LipschitzOnWith.of_dist_le_mul
+    intro x hx y hy
+    simpa only [dist_add_right] using hLip.dist_le_mul (x+(U s:ℂ)) ⟨x,hx,rfl⟩ (y+(U s:ℂ)) ⟨y,hy,rfl⟩
+  · intro x hx y hy
+    simpa only [dist_add_right] using hLo (x+(U s:ℂ)) ⟨x,self_subset_cthickening K hx,rfl⟩
+      (y+(U s:ℂ)) ⟨y,self_subset_cthickening K hy,rfl⟩
+lemma source_restart_sausage_countable_bound
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (s δ : ℝ≥0) (η : ℝ≥0 → ℂ)
+    (hid : ∀ u, η u ∈ upperHalfPlane → γ (s+u) = F s (η u+(U s:ℂ)))
+    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) :
+    ∃ N : ℕ, 1 ≤ N ∧ ∀ q : ℝ, 0 < q → q < 1/(N:ℝ) →
+      volume (K ∩ sourceSausageSet η 0 δ (q/(N:ℝ))) ≤
+        (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) q) := by
+  obtain ⟨r,L,hr,hL,hrK,hLip,hLo⟩ := source_inverse_translated_chart_compact_bounds U γ G F hw s hK hKH
+  obtain ⟨N,hN⟩ := exists_nat_gt (max 1 (max (L:ℝ) r⁻¹))
+  have hN1 : 1 ≤ N := by exact_mod_cast (le_max_left 1 (max (L:ℝ) r⁻¹)).trans hN.le
+  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN1)
+  have hLN : (L:ℝ) ≤ N := (le_max_left (L:ℝ) r⁻¹).trans ((le_max_right _ _).trans hN.le)
+  have hrN : 1/(N:ℝ) ≤ r := by
+    have h := (le_max_right (L:ℝ) r⁻¹).trans ((le_max_right 1 _).trans hN.le)
+    apply (div_le_iff₀ hNp).mpr
+    have hh := (inv_le_iff_one_le_mul₀ hr).mp h
+    simpa only [mul_comm] using hh
+  refine ⟨N,hN1,?_⟩
+  intro q hq hqN
+  have hqdiv : q/(N:ℝ) ≤ r := by
+    have hh : q/(N:ℝ) ≤ q := div_le_self hq.le (by exact_mod_cast hN1)
+    exact hh.trans (hqN.le.trans hrN)
+  exact source_chart_sausage_area (L := (N:ℝ≥0)) (by exact_mod_cast hNp) hr hq hqdiv
+    (hLip.weaken (by exact_mod_cast hLN))
+    (fun x hx y hy => (hLo x hx y hy).trans (mul_le_mul_of_nonneg_right hLN dist_nonneg))
+    η γ s δ (fun u _ hu => hid u (hrK hu))
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance sausageScalingMeshPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance sausageScalingMeshPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma source_sausage_mono_mesh (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) {r e : ℝ} (hle : r ≤ e) :
+    sourceSausageSet γ s t r ⊆ sourceSausageSet γ s t e := by
+  rintro z ⟨u,hu,hz⟩
+  exact ⟨u,hu,hz.trans_le hle⟩
+def sourceSausageChartGood (η γ : ℝ≥0 → ℂ) (K : Set ℂ) (s δ : ℝ≥0) (N : ℕ) : Prop :=
+  1 ≤ N ∧ ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) < 1/(N:ℝ) →
+    volume (K ∩ sourceSausageSet η 0 δ ((q:ℝ)/(N:ℝ))) ≤
+      (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) (q:ℝ))
+lemma source_chart_good_mono (η γ : ℝ≥0 → ℂ) (K : Set ℂ) (s δ : ℝ≥0) :
+    Monotone (fun N : ℕ => sourceSausageChartGood η γ K s δ N) := by
+  intro M N hMN hM
+  refine ⟨hM.1.trans hMN,?_⟩
+  intro q hq hqN
+  have hMp : 0 < (M:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hM.1)
+  have hMrN : (M:ℝ) ≤ N := by exact_mod_cast hMN
+  have hqM : (q:ℝ) < 1/(M:ℝ) := hqN.trans_le (one_div_le_one_div_of_le hMp hMrN)
+  calc
+    _ ≤ volume (K ∩ sourceSausageSet η 0 δ ((q:ℝ)/(M:ℝ))) :=
+      measure_mono (inter_subset_inter_right K (source_sausage_mono_mesh η 0 δ (div_le_div_of_nonneg_left hq.le hMp hMrN)))
+    _ ≤ (M:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) (q:ℝ)) := hM.2 q hq hqM
+    _ ≤ _ := by gcongr
+lemma source_chart_good_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (η γ : Ω → C(ℝ≥0,ℂ)) (hmη : Measurable η) (hmγ : Measurable γ)
+    (K : Set ℂ) (s δ : ℝ≥0) (N : ℕ) :
+    MeasurableSet {ω | sourceSausageChartGood (η ω) (γ ω) K s δ N} := by
+  simp only [sourceSausageChartGood,ofPred_and,ofPred_forall]
+  apply MeasurableSet.inter (by measurability)
+  apply MeasurableSet.iInter; intro q
+  apply MeasurableSet.iInter; intro hq
+  apply MeasurableSet.iInter; intro hqN
+  apply measurableSet_le
+  · exact (source_sausage_local_volume_measurable 0 δ ((q:ℝ)/(N:ℝ)) K).comp hmη
+  · have hmvol : Measurable (fun g : C(ℝ≥0,ℂ) => volume (sourceSausageSet g s (s+δ) (q:ℝ))) := by
+      simpa only [univ_inter] using source_sausage_local_volume_measurable s (s+δ) (q:ℝ) univ
+    exact measurable_const.mul (hmvol.comp hmγ)
+lemma source_chart_good_real_bound (η γ : ℝ≥0 → ℂ) (K : Set ℂ) (s δ : ℝ≥0) (N : ℕ)
+    (hN : sourceSausageChartGood η γ K s δ N) {e : ℝ} (he : 0 < e) (heN : e < 1/(N:ℝ)) :
+    volume (K ∩ sourceSausageSet η 0 δ (e/(2*(N:ℝ)))) ≤
+      (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) e) := by
+  obtain ⟨q,hq1,hq2⟩ := exists_rat_btwn (show e/2 < e by linarith)
+  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN.1)
+  have hmesh : e/(2*(N:ℝ)) ≤ (q:ℝ)/(N:ℝ) := by rw [←div_div]; exact div_le_div_of_nonneg_right hq1.le hNp.le
+  calc
+    _ ≤ volume (K ∩ sourceSausageSet η 0 δ ((q:ℝ)/(N:ℝ))) :=
+      measure_mono (inter_subset_inter_right K (source_sausage_mono_mesh η 0 δ hmesh))
+    _ ≤ (N:ℝ≥0∞)^2*volume (sourceSausageSet γ s (s+δ) (q:ℝ)) := hN.2 q (by linarith) (hq2.trans heN)
+    _ ≤ _ := mul_le_mul' le_rfl (measure_mono (source_sausage_mono_mesh γ s (s+δ) hq2.le))
+lemma source_chart_good_high_probability {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsProbabilityMeasure P]
+    (η γ : Ω → C(ℝ≥0,ℂ)) (hmη : Measurable η) (hmγ : Measurable γ)
+    (K : Set ℂ) (s δ : ℝ≥0)
+    (hfull : ∀ᵐ ω ∂P, ∃ N : ℕ, sourceSausageChartGood (η ω) (γ ω) K s δ N)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∃ N : ℕ, 1 ≤ N ∧ P {ω | ¬sourceSausageChartGood (η ω) (γ ω) K s δ N} < ENNReal.ofReal ε := by
+  let B := fun N : ℕ => {ω | ¬sourceSausageChartGood (η ω) (γ ω) K s δ N}
+  have hB : ∀ N, MeasurableSet (B N) := fun N => (source_chart_good_measurable η γ hmη hmγ K s δ N).compl
+  have hanti : Antitone B := by
+    intro M N hMN ω hω hM
+    exact hω (source_chart_good_mono (η ω) (γ ω) K s δ hMN hM)
+  have hz : P (⋂ N, B N) = 0 := by
+    have hh := ae_iff.mp hfull
+    simpa only [B,not_exists,ofPred_forall] using hh
+  have ht := tendsto_measure_iInter_atTop (fun N => (hB N).nullMeasurableSet) hanti
+    ⟨0,measure_ne_top P _⟩
+  rw [hz] at ht
+  have hevv : ∀ᶠ N in atTop, P (B N) < ENNReal.ofReal ε :=
+    ht.eventually (gt_mem_nhds (ENNReal.ofReal_pos.mpr hε))
+  obtain ⟨N,hN,hp⟩ := ((eventually_ge_atTop 1).and hevv).exists
+  exact ⟨N,hN,hp⟩
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance sausageScalingIntervalPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance sausageScalingIntervalPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma source_sausage_mono_interval (γ : ℝ≥0 → ℂ) (e : ℝ) {s t u v : ℝ≥0}
+    (hI : Icc s t ⊆ Icc u v) : sourceSausageSet γ s t e ⊆ sourceSausageSet γ u v e := by
+  rintro z ⟨w,hw,hz⟩
+  exact ⟨w,hI hw,hz⟩
+lemma source_sausage_mass_mono_interval (κ : ℝ) (γ : ℝ≥0 → ℂ) (e : ℝ) {s t u v : ℝ≥0}
+    (hI : Icc s t ⊆ Icc u v) : sourceSausageMass κ γ s t e ≤ sourceSausageMass κ γ u v e :=
+  mul_le_mul' le_rfl (measure_mono (source_sausage_mono_interval γ e hI))
+lemma RegularSLEPresentation.source_sausage_mass_measurable_at
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (s t : ℝ≥0) (e : ℝ) :
+    @Measurable a.samples ℝ≥0∞ (a.filtration t) inferInstance (fun ω => sourceSausageMass κ (γ ω) s t e) := by
+  let f : a.samples → C(ℝ≥0,ℂ) := fun ω =>
+    ⟨fun u => γ ω (min u t), (a.trace_continuous ω).comp (continuous_id.min continuous_const)⟩
+  have hm : @Measurable a.samples C(ℝ≥0,ℂ) (a.filtration t) (borel _) f := by
+    let : MeasurableSpace a.samples := a.filtration t
+    apply source_path_measurable
+    intro u
+    exact (a.trace_adapted (min u t)).mono (a.filtration.mono (min_le_right u t)) le_rfl
+  have he (ω : a.samples) : sourceSausageSet (γ ω) s t e = sourceSausageSet (f ω) s t e := by
+    ext z
+    constructor
+    · rintro ⟨u,hu,huz⟩
+      exact ⟨u,hu,by simpa only [f,ContinuousMap.coe_mk,min_eq_left hu.2] using huz⟩
+    · rintro ⟨u,hu,huz⟩
+      exact ⟨u,hu,by simpa only [f,ContinuousMap.coe_mk,min_eq_left hu.2] using huz⟩
+  have he' : (fun ω : a.samples => sourceSausageMass κ (γ ω) s t e) =
+      (fun ω => sourceSausageMass κ (f ω) s t e) := by
+    funext ω; unfold sourceSausageMass; rw [he]
+  rw [he']
+  exact (source_sausage_mass_measurable κ s t e).comp hm
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_chart_good_mass_bound (κ : ℝ) (η γ : ℝ≥0 → ℂ) (K : Set ℂ)
+    (s δ : ℝ≥0) (N : ℕ) (hN : sourceSausageChartGood η γ K s δ N)
+    {e : ℝ} (he : 0 < e) (heN : e < 1/(N:ℝ)) :
+    ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+      volume (K ∩ sourceSausageSet η 0 δ (e/(2*(N:ℝ)))) ≤
+    ENNReal.ofReal (((2*(N:ℝ))⁻¹)^(-codimension κ)*(N:ℝ)^2)*sourceSausageMass κ γ s (s+δ) e := by
+  have hvol := source_chart_good_real_bound η γ K s δ N hN he heN
+  have hnp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN.1)
+  have hbase : 0 ≤ (2*(N:ℝ))⁻¹ := by positivity
+  have hmul : (e/(2*(N:ℝ)))^(-codimension κ) = ((2*(N:ℝ))⁻¹)^(-codimension κ)*e^(-codimension κ) := by
+    rw [show e/(2*(N:ℝ)) = (2*(N:ℝ))⁻¹*e by ring,Real.mul_rpow hbase he.le]
+  have hh := mul_le_mul' (le_refl (ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ)))) hvol
+  apply hh.trans_eq
+  rw [hmul,ENNReal.ofReal_mul (Real.rpow_nonneg hbase _),
+    ENNReal.ofReal_mul (Real.rpow_nonneg hbase _),ENNReal.ofReal_pow (Nat.cast_nonneg N)]
+  simp only [ENNReal.ofReal_natCast,sourceSausageMass]
+  ac_rfl
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance SausageConditional_local1 : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance SausageConditional_local2 : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance SausageConditional_local3 : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance SausageConditional_local4 : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma RegularSLEPresentation.source_uniform_sausage_slot_bound
+    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ →
+      ∀ ε : ℝ, 0 < ε → ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧
+        ∀ e : ℝ, 0 < e → e < e₀ →
+        ∀ A : Set a.samples, MeasurableSet[a.filtration s] A →
+          (P.comap (Subtype.val : a.samples → Ω))
+            (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal b}) ≤
+          (1-p)*(P.comap (Subtype.val : a.samples → Ω)) A + ENNReal.ofReal ε := by
+  have hk : 0 < κ := by linarith
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨p,hp,hp1,hseed⟩ := a.source_uniform_sausage_seed hκ hκ8
+  refine ⟨p,hp,hp1,?_⟩
+  intro s δ hδ ε hε
+  obtain ⟨K,b,e₀,hK,hKH,hb,he₀,hKP⟩ := hseed δ hδ
+  obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian
+    a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness
+  let τ : a.samples → ℝ≥0 := fun _ => s
+  have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s
+  obtain ⟨hlaw,hident⟩ := hrestart τ hτ
+  let rp := sourceRestartDriverPath a.B a.continuous_B τ
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let η := fun ω : a.samples => Φ (rp ω)
+  have hrm : Measurable rp := by
+    apply source_path_measurable
+    intro t
+    exact (a.measurable_B (s+t)).sub (a.measurable_B s)
+  have hem : Measurable η := hΦm.comp hrm
+  have hgm : Measurable gp := a.source_trace_path_measurable
+  have hfull : ∀ᵐ ω ∂Q, ∃ N : ℕ, sourceSausageChartGood (η ω) (gp ω) K s δ N := by
+    filter_upwards [hident] with ω hω
+    obtain ⟨N,hN,hbound⟩ := source_restart_sausage_countable_bound
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) s δ (η ω) hω hK hKH
+    exact ⟨N,hN,fun q hq hqN => hbound q hq hqN⟩
+  obtain ⟨N,hN,hbad⟩ := source_chart_good_high_probability Q η gp hem hgm K s δ hfull ε hε
+  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN)
+  let C := ((2*(N:ℝ))⁻¹)^(-codimension κ)*(N:ℝ)^2
+  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (sq_pos_of_pos hNp)
+  refine ⟨b/C,min (1/(N:ℝ)) (2*(N:ℝ)*e₀),div_pos hb hC,lt_min (by positivity) (by positivity),?_⟩
+  intro e he hecut A hA
+  have hmesh : e/(2*(N:ℝ)) < e₀ := (div_lt_iff₀ (by positivity)).mpr
+    (by simpa only [mul_comm] using hecut.trans_le (min_le_right _ _))
+  let D : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+      volume (K ∩ sourceSausageSet g 0 δ (e/(2*(N:ℝ))))}
+  let E : Set a.samples := η ⁻¹' D
+  let B : Set a.samples := {ω | ¬sourceSausageChartGood (η ω) (gp ω) K s δ N}
+  have hD : MeasurableSet D := measurableSet_le measurable_const
+    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ (e/(2*(N:ℝ))) K))
+  have hEm : MeasurableSet E := hD.preimage hem
+  have hAm : MeasurableSet A := (a.filtration.le s) A hA
+  have hAstop : MeasurableSet[hτ.measurableSpace] A := by
+    change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A
+    rw [IsStoppingTime.measurableSpace_const]
+    exact hA
+  have hprob : p*Q A ≤ Q (A ∩ E) := by
+    have hEq := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ D) (hlaw A hAstop)
+    rw [Measure.map_apply hem hD,Measure.smul_apply,Measure.map_apply hgm hD,
+      Measure.restrict_apply (hD.preimage hem)] at hEq
+    change Q (E ∩ A) = Q A*Q {ω | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+      volume (K ∩ sourceSausageSet (γ ω) 0 δ (e/(2*(N:ℝ))))} at hEq
+    rw [inter_comm] at hEq
+    rw [hEq,mul_comm p (Q A)]
+    exact mul_le_mul' le_rfl (hKP (e/(2*(N:ℝ))) (by positivity) hmesh)
+  have hins : A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)} ⊆ (A \ E) ∪ B := by
+    intro ω hω
+    by_cases hG : sourceSausageChartGood (η ω) (gp ω) K s δ N
+    · left
+      refine ⟨hω.1,?_⟩
+      intro hE
+      have hEE : ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+          volume (K ∩ sourceSausageSet (η ω) 0 δ (e/(2*(N:ℝ)))) := hE
+      have hmass := hEE.trans (source_chart_good_mass_bound κ (η ω) (gp ω) K s δ N hG he
+        (hecut.trans_le (min_le_left _ _)))
+      have hlow : ENNReal.ofReal (b/C) ≤ sourceSausageMass κ (γ ω) s (s+δ) e := by
+        rw [ENNReal.ofReal_div_of_pos hC]
+        apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
+        simpa only [C,gp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hmass
+      exact (not_lt_of_ge hlow) hω.2
+    · exact Or.inr hG
+  have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by
+    have heq : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto
+    rw [heq]
+    exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _)
+  calc
+    Q (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)})
+        ≤ Q ((A \ E) ∪ B) := measure_mono hins
+    _ ≤ Q (A \ E) + Q B := measure_union_le _ _
+    _ ≤ (Q A-p*Q A)+ENNReal.ofReal ε := by rw [hdiff]; exact add_le_add (tsub_le_tsub_left hprob _) hbad.le
+    _ = (1-p)*Q A+ENNReal.ofReal ε := by
+      rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul]
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma RegularSLEPresentation.source_sausage_nslot_bound
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) :
+    ∃ c : ℝ≥0∞, c < 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ → ∀ ν : ℝ, 0 < ν →
+      ∀ n : ℕ, ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+        (P.comap (Subtype.val : a.samples → Ω))
+          {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b} ≤
+        c^n + (n:ℝ≥0∞)*ENNReal.ofReal ν := by
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_sausage_slot_bound hκ hκ8
+  let c : ℝ≥0∞ := 1-p
+  have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne'
+  refine ⟨c,hc,?_⟩
+  intro s δ hδ ν hν n
+  induction n with
+  | zero =>
+    refine ⟨1,1,by norm_num,by norm_num,?_⟩
+    intro e he hel
+    simpa only [pow_zero,Nat.cast_zero,zero_mul,add_zero] using
+      (prob_le_one (μ := Q) (s := {ω | sourceSausageMass κ (γ ω) s (s+(0:ℝ≥0)*δ) e < ENNReal.ofReal 1}))
+  | succ n ih =>
+    obtain ⟨b₀,e₀,hb₀,he₀,hprev⟩ := ih
+    obtain ⟨b₁,e₁,hb₁,he₁,hstep⟩ := hslot (s+(n:ℝ≥0)*δ) δ hδ ν hν
+    refine ⟨min b₀ b₁,min e₀ e₁,lt_min hb₀ hb₁,lt_min he₀ he₁,?_⟩
+    intro e he hel
+    have hend : s+((n+1:ℕ):ℝ≥0)*δ = (s+(n:ℝ≥0)*δ)+δ := by push_cast; ring
+    let A : Set a.samples := {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b₀}
+    have hA : MeasurableSet[a.filtration (s+(n:ℝ≥0)*δ)] A :=
+      measurableSet_lt (a.source_sausage_mass_measurable_at s _ e) measurable_const
+    rw [hend]
+    calc
+      Q {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal (min b₀ b₁)}
+          ≤ Q (A ∩ {ω | sourceSausageMass κ (γ ω) (s+(n:ℝ≥0)*δ) (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal b₁}) := by
+          apply measure_mono
+          intro ω hω
+          constructor
+          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
+              (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))).trans_lt hω).trans_le
+                (ENNReal.ofReal_le_ofReal (min_le_left _ _))
+          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
+              (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)).trans_lt hω).trans_le
+                (ENNReal.ofReal_le_ofReal (min_le_right _ _))
+      _ ≤ c*Q A+ENNReal.ofReal ν := hstep e he (hel.trans_le (min_le_right _ _)) A hA
+      _ ≤ c*(c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν)+ENNReal.ofReal ν := by
+          exact add_le_add (mul_le_mul' le_rfl (hprev e he (hel.trans_le (min_le_left _ _)))) le_rfl
+      _ ≤ c^(n+1)+((n+1:ℕ):ℝ≥0∞)*ENNReal.ofReal ν := by
+          rw [mul_add,pow_succ,mul_comm (c^n) c,Nat.cast_add,Nat.cast_one,add_mul,one_mul]
+          rw [←add_assoc]
+          exact add_le_add (add_le_add le_rfl (mul_le_of_le_one_left' hc.le)) le_rfl
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma RegularSLEPresentation.source_small_sausage
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) (s t : ℝ≥0) (hst : s < t) (ε : ℝ) (hε : 0 < ε) :
+    ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+      (P.comap (Subtype.val : a.samples → Ω))
+        {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal b} < ENNReal.ofReal ε := by
+  obtain ⟨c,hc,hsteps⟩ := a.source_sausage_nslot_bound hκ hκ8
+  have hev : ∀ᶠ n : ℕ in atTop, c^n < ENNReal.ofReal (ε/4) :=
+    (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc).eventually
+      (gt_mem_nhds (ENNReal.ofReal_pos.mpr (by positivity)))
+  obtain ⟨n,hn,hcn⟩ := ((eventually_ge_atTop 1).and hev).exists
+  have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega)
+  let δ : ℝ≥0 := (t-s)/(n:ℝ≥0)
+  have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0
+  have hend : s+(n:ℝ≥0)*δ = t := by
+    dsimp [δ]
+    rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le]
+  let ν : ℝ := ε/(4*((n:ℝ)+1))
+  have hν : 0 < ν := by dsimp [ν]; positivity
+  obtain ⟨b,e₀,hb,he₀,hbound⟩ := hsteps s δ hδ ν hν n
+  have hnν : (n:ℝ≥0∞)*ENNReal.ofReal ν ≤ ENNReal.ofReal (ε/4) := by
+    rw [←ENNReal.ofReal_natCast,←ENNReal.ofReal_mul (Nat.cast_nonneg n)]
+    apply ENNReal.ofReal_le_ofReal
+    calc (n:ℝ)*ν ≤ ((n:ℝ)+1)*ν := mul_le_mul_of_nonneg_right (by linarith) hν.le
+      _ = ε/4 := by dsimp [ν]; field_simp
+  refine ⟨b,e₀,hb,he₀,?_⟩
+  intro e he hel
+  have hh := hbound e he hel
+  rw [hend] at hh
+  apply hh.trans_lt
+  calc c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν
+      ≤ ENNReal.ofReal (ε/4)+ENNReal.ofReal (ε/4) := add_le_add hcn.le hnν
+    _ = ENNReal.ofReal (ε/2) := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
+    _ < ENNReal.ofReal ε := (ENNReal.ofReal_lt_ofReal_iff hε).mpr (by linarith)
+theorem sourceSmallSausage_proved : SourceSmallSausageTarget := by
+  intro κ hκ hκ8 Ω m P hP γ hSLE s t hst ε hε
+  let : IsProbabilityMeasure P := hP
+  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
+  obtain ⟨b,e₀,hb,he₀,hbound⟩ := a.source_small_sausage hκ hκ8 s t hst ε hε
+  refine ⟨b,e₀,hb,he₀,?_⟩
+  intro e he hel
+  have hh := hbound e he hel
+  rw [←a.measurePreserving_val.map_eq]
+  rwa [(MeasurableEmbedding.subtype_coe a.measurable_samples).map_apply] 
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SausageSeedAll.lean b/lean/OAI/Probability/SLE/SausageSeedAll.lean
index d0e1f3bd399300de231477fe8e4bce4f60e79656..6399a294224f5e9273ddbf8ef4f1e3b9a4253d33 100644
--- a/lean/OAI/Probability/SLE/SausageSeedAll.lean
+++ b/lean/OAI/Probability/SLE/SausageSeedAll.lean
@@ -1,649 +1,649 @@
-import Mathlib
-import OAI.Probability.SLE.LowerPositivity
-import OAI.Probability.SLE.SausageScaling
-
-namespace OAI
-
-/-! Sausage Seed All for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-namespace RegularSLEPresentation
-variable {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-variable {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-lemma source_positive_probability_initial_sausage_all (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 2 < z₀.im) :
-    ∃ (T : ℝ≥0) (p b e₀ : ℝ), 0 < T ∧ 0 < p ∧ 0 < b ∧ 0 < e₀ ∧
-      ∀ e : ℝ, 0 < e → e < e₀ →
-        ENNReal.ofReal p ≤ (P.comap (Subtype.val : a.samples → Ω))
-          {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
-            volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e)} := by
-  have hk : 0 < κ := by linarith
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨T,e₀,m₀,M,hT,he₀,hm₀,hM,hMom⟩ := source_retained_seed_all κ hκ hκ8 Ω m P inferInstance γ a z₀ hz
-  let C := ((1:ℝ)/2)^(-codimension κ)
-  have hC : 0 < C := Real.rpow_pos_of_pos (by norm_num) _
-  refine ⟨T,(m₀/2)^2/M,(m₀/2)/C,min 1 (2*e₀),hT,by positivity,by positivity,lt_min (by norm_num) (by positivity),?_⟩
-  intro e he hecut
-  let X := fun ω : a.samples => finiteSpatialGreenMass κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (e/2) T (closedBall z₀ 1)
-  have hXm : Measurable X := (a.sourceFiniteGreenDensity_measurable (e/2) T).lintegral_prod_right'
-  have hhalf : e/2 ≤ e₀ := by have := hecut.trans_le (min_le_right _ _); linarith
-  have hXX := hMom (e/2) (by positivity) hhalf
-  apply (source_second_moment_seed Q X hXm m₀ M hm₀ hM hXX.1 hXX.2).trans
-  apply measure_mono
-  intro ω hω
-  have hbound := a.source_finite_mass_le_sausage hk hκ8 z₀ hz T he
-    (hecut.le.trans (min_le_left _ _)) ω
-  have heq : (e/2)^(-codimension κ) = C*e^(-codimension κ) := by
-    rw [show e/2 = (1/2)*e by ring,Real.mul_rpow (by norm_num) he.le]
-  rw [heq,ENNReal.ofReal_mul hC.le,mul_assoc] at hbound
-  have hdiv : ENNReal.ofReal (m₀/2) / ENNReal.ofReal C ≤
-      ENNReal.ofReal (e^(-codimension κ))*volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e) := by
-    apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
-    simpa only [mul_comm] using (hω.trans hbound)
-  simpa only [mem_ofPred_eq,ENNReal.ofReal_div_of_pos hC] using hdiv
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance SausageUniformSeedAllI5  : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance SausageUniformSeedAllI6  : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma RegularSLEPresentation.source_uniform_sausage_seed_all
-    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ δ : ℝ≥0, 0 < δ →
-      ∃ (K : Set ℂ) (b e₀ : ℝ), IsCompact K ∧ K ⊆ upperHalfPlane ∧ 0 < b ∧ 0 < e₀ ∧
-        ∀ e : ℝ, 0 < e → e < e₀ →
-          p ≤ (P.comap (Subtype.val : a.samples → Ω))
-            {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
-              volume (K ∩ sourceSausageSet (γ ω) 0 δ e)} := by
-  have hk : 0 < κ := by linarith
-  let z₀ : ℂ := 3*I
-  have hz₀ : 2 < z₀.im := by norm_num [z₀]
-  obtain ⟨T,p,b,e₀,hT,hp,hb,he₀,hseed⟩ := a.source_positive_probability_initial_sausage_all hκ hκ8 z₀ hz₀
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  have hp1 : ENNReal.ofReal p ≤ 1 := (hseed (e₀/2) (by positivity) (by linarith)).trans prob_le_one
-  refine ⟨ENNReal.ofReal p,ENNReal.ofReal_pos.mpr hp,hp1,?_⟩
-  intro δ hδ
-  let c := NNReal.sqrt (δ/T)
-  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
-  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
-  have hcR : 0 < (c:ℝ) := hc
-  have ht : sourceScaleTime c δ = T := by
-    unfold sourceScaleTime
-    rw [hc2]
-    field_simp
-  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
-  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
-  have hKH : K ⊆ upperHalfPlane := by
-    rintro _ ⟨z,hz,rfl⟩
-    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
-    change 0 < ((c:ℂ)*z).im
-    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos hcR hz'
-  let C := ((c:ℝ)⁻¹)^(-codimension κ)*((c:ℝ)⁻¹)^2
-  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (inv_pos.mpr hcR) _) (sq_pos_of_pos (inv_pos.mpr hcR))
-  refine ⟨K,b/C,(c:ℝ)*e₀,hK,hKH,div_pos hb hC,mul_pos hcR he₀,?_⟩
-  intro e he hecut
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω)) (a.sourceScaled c hc).trace_continuous
-  let A : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal (b/C) ≤ ENNReal.ofReal (e^(-codimension κ))*volume (K ∩ sourceSausageSet g 0 δ e)}
-  have hA : MeasurableSet A := measurableSet_le measurable_const
-    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ e K))
-  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law hk hκ8 c hc)
-  have hgm : Measurable gp := a.source_trace_path_measurable
-  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable
-  rw [Measure.map_apply hgm hA,Measure.map_apply hsm hA] at hmap
-  change ENNReal.ofReal p ≤ Q (gp ⁻¹' A)
-  rw [hmap]
-  have hmesh : e/(c:ℝ) < e₀ := (div_lt_iff₀ hcR).mpr (by simpa only [mul_comm] using hecut)
-  apply (hseed (e/(c:ℝ)) (div_pos he hcR) hmesh).trans
-  apply measure_mono
-  intro ω hω
-  have hh := hω.trans (source_sausage_scale_mass_bound κ hc (γ ω) T δ ht (closedBall z₀ 1) e he)
-  change ENNReal.ofReal (b/C) ≤ _
-  rw [ENNReal.ofReal_div_of_pos hC]
-  apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
-  simpa only [C,K,sp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hh
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance SausageConditionalAllI6  : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance SausageConditionalAllI7  : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance SausageConditionalAllI8  : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance SausageConditionalAllI9  : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma RegularSLEPresentation.source_uniform_sausage_slot_bound_all
-    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ →
-      ∀ ε : ℝ, 0 < ε → ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧
-        ∀ e : ℝ, 0 < e → e < e₀ →
-        ∀ A : Set a.samples, MeasurableSet[a.filtration s] A →
-          (P.comap (Subtype.val : a.samples → Ω))
-            (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal b}) ≤
-          (1-p)*(P.comap (Subtype.val : a.samples → Ω)) A + ENNReal.ofReal ε := by
-  have hk : 0 < κ := by linarith
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨p,hp,hp1,hseed⟩ := a.source_uniform_sausage_seed_all hκ hκ8
-  refine ⟨p,hp,hp1,?_⟩
-  intro s δ hδ ε hε
-  obtain ⟨K,b,e₀,hK,hKH,hb,he₀,hKP⟩ := hseed δ hδ
-  obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian
-    a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness
-  let τ : a.samples → ℝ≥0 := fun _ => s
-  have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s
-  obtain ⟨hlaw,hident⟩ := hrestart τ hτ
-  let rp := sourceRestartDriverPath a.B a.continuous_B τ
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let η := fun ω : a.samples => Φ (rp ω)
-  have hrm : Measurable rp := by
-    apply source_path_measurable
-    intro t
-    exact (a.measurable_B (s+t)).sub (a.measurable_B s)
-  have hem : Measurable η := hΦm.comp hrm
-  have hgm : Measurable gp := a.source_trace_path_measurable
-  have hfull : ∀ᵐ ω ∂Q, ∃ N : ℕ, sourceSausageChartGood (η ω) (gp ω) K s δ N := by
-    filter_upwards [hident] with ω hω
-    obtain ⟨N,hN,hbound⟩ := source_restart_sausage_countable_bound
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) s δ (η ω) hω hK hKH
-    exact ⟨N,hN,fun q hq hqN => hbound q hq hqN⟩
-  obtain ⟨N,hN,hbad⟩ := source_chart_good_high_probability Q η gp hem hgm K s δ hfull ε hε
-  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN)
-  let C := ((2*(N:ℝ))⁻¹)^(-codimension κ)*(N:ℝ)^2
-  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (sq_pos_of_pos hNp)
-  refine ⟨b/C,min (1/(N:ℝ)) (2*(N:ℝ)*e₀),div_pos hb hC,lt_min (by positivity) (by positivity),?_⟩
-  intro e he hecut A hA
-  have hmesh : e/(2*(N:ℝ)) < e₀ := (div_lt_iff₀ (by positivity)).mpr
-    (by simpa only [mul_comm] using hecut.trans_le (min_le_right _ _))
-  let D : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-      volume (K ∩ sourceSausageSet g 0 δ (e/(2*(N:ℝ))))}
-  let E : Set a.samples := η ⁻¹' D
-  let B : Set a.samples := {ω | ¬sourceSausageChartGood (η ω) (gp ω) K s δ N}
-  have hD : MeasurableSet D := measurableSet_le measurable_const
-    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ (e/(2*(N:ℝ))) K))
-  have hEm : MeasurableSet E := hD.preimage hem
-  have hAm : MeasurableSet A := (a.filtration.le s) A hA
-  have hAstop : MeasurableSet[hτ.measurableSpace] A := by
-    change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A
-    rw [IsStoppingTime.measurableSpace_const]
-    exact hA
-  have hprob : p*Q A ≤ Q (A ∩ E) := by
-    have hEq := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ D) (hlaw A hAstop)
-    rw [Measure.map_apply hem hD,Measure.smul_apply,Measure.map_apply hgm hD,
-      Measure.restrict_apply (hD.preimage hem)] at hEq
-    change Q (E ∩ A) = Q A*Q {ω | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-      volume (K ∩ sourceSausageSet (γ ω) 0 δ (e/(2*(N:ℝ))))} at hEq
-    rw [inter_comm] at hEq
-    rw [hEq,mul_comm p (Q A)]
-    exact mul_le_mul' le_rfl (hKP (e/(2*(N:ℝ))) (by positivity) hmesh)
-  have hins : A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)} ⊆ (A \ E) ∪ B := by
-    intro ω hω
-    by_cases hG : sourceSausageChartGood (η ω) (gp ω) K s δ N
-    · left
-      refine ⟨hω.1,?_⟩
-      intro hE
-      have hEE : ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
-          volume (K ∩ sourceSausageSet (η ω) 0 δ (e/(2*(N:ℝ)))) := hE
-      have hmass := hEE.trans (source_chart_good_mass_bound κ (η ω) (gp ω) K s δ N hG he
-        (hecut.trans_le (min_le_left _ _)))
-      have hlow : ENNReal.ofReal (b/C) ≤ sourceSausageMass κ (γ ω) s (s+δ) e := by
-        rw [ENNReal.ofReal_div_of_pos hC]
-        apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
-        simpa only [C,gp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hmass
-      exact (not_lt_of_ge hlow) hω.2
-    · exact Or.inr hG
-  have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by
-    have heq : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto
-    rw [heq]
-    exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _)
-  calc
-    Q (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)})
-        ≤ Q ((A \ E) ∪ B) := measure_mono hins
-    _ ≤ Q (A \ E) + Q B := measure_union_le _ _
-    _ ≤ (Q A-p*Q A)+ENNReal.ofReal ε := by rw [hdiff]; exact add_le_add (tsub_le_tsub_left hprob _) hbad.le
-    _ = (1-p)*Q A+ENNReal.ofReal ε := by
-      rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul]
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-lemma RegularSLEPresentation.source_sausage_nslot_bound_all
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) :
-    ∃ c : ℝ≥0∞, c < 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ → ∀ ν : ℝ, 0 < ν →
-      ∀ n : ℕ, ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-        (P.comap (Subtype.val : a.samples → Ω))
-          {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b} ≤
-        c^n + (n:ℝ≥0∞)*ENNReal.ofReal ν := by
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_sausage_slot_bound_all hκ hκ8
-  let c : ℝ≥0∞ := 1-p
-  have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne'
-  refine ⟨c,hc,?_⟩
-  intro s δ hδ ν hν n
-  induction n with
-  | zero =>
-    refine ⟨1,1,by norm_num,by norm_num,?_⟩
-    intro e he hel
-    simpa only [pow_zero,Nat.cast_zero,zero_mul,add_zero] using
-      (prob_le_one (μ := Q) (s := {ω | sourceSausageMass κ (γ ω) s (s+(0:ℝ≥0)*δ) e < ENNReal.ofReal 1}))
-  | succ n ih =>
-    obtain ⟨b₀,e₀,hb₀,he₀,hprev⟩ := ih
-    obtain ⟨b₁,e₁,hb₁,he₁,hstep⟩ := hslot (s+(n:ℝ≥0)*δ) δ hδ ν hν
-    refine ⟨min b₀ b₁,min e₀ e₁,lt_min hb₀ hb₁,lt_min he₀ he₁,?_⟩
-    intro e he hel
-    have hend : s+((n+1:ℕ):ℝ≥0)*δ = (s+(n:ℝ≥0)*δ)+δ := by push_cast; ring
-    let A : Set a.samples := {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b₀}
-    have hA : MeasurableSet[a.filtration (s+(n:ℝ≥0)*δ)] A :=
-      measurableSet_lt (a.source_sausage_mass_measurable_at s _ e) measurable_const
-    rw [hend]
-    calc
-      Q {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal (min b₀ b₁)}
-          ≤ Q (A ∩ {ω | sourceSausageMass κ (γ ω) (s+(n:ℝ≥0)*δ) (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal b₁}) := by
-          apply measure_mono
-          intro ω hω
-          constructor
-          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
-              (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))).trans_lt hω).trans_le
-                (ENNReal.ofReal_le_ofReal (min_le_left _ _))
-          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
-              (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)).trans_lt hω).trans_le
-                (ENNReal.ofReal_le_ofReal (min_le_right _ _))
-      _ ≤ c*Q A+ENNReal.ofReal ν := hstep e he (hel.trans_le (min_le_right _ _)) A hA
-      _ ≤ c*(c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν)+ENNReal.ofReal ν := by
-          exact add_le_add (mul_le_mul' le_rfl (hprev e he (hel.trans_le (min_le_left _ _)))) le_rfl
-      _ ≤ c^(n+1)+((n+1:ℕ):ℝ≥0∞)*ENNReal.ofReal ν := by
-          rw [mul_add,pow_succ,mul_comm (c^n) c,Nat.cast_add,Nat.cast_one,add_mul,one_mul]
-          rw [←add_assoc]
-          exact add_le_add (add_le_add le_rfl (mul_le_of_le_one_left' hc.le)) le_rfl
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-def SourceSmallSausageAllTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : MeasureTheory.Measure Ω),
-    MeasureTheory.IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-  ∀ s t : ℝ≥0, s < t → ∀ ε : ℝ, 0 < ε →
-    ∃ a e₀ : ℝ, 0 < a ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-      P {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal a} < ENNReal.ofReal ε
-
-lemma RegularSLEPresentation.source_small_sausage_all
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ8 : κ < 8) (s t : ℝ≥0) (hst : s < t) (ε : ℝ) (hε : 0 < ε) :
-    ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-      (P.comap (Subtype.val : a.samples → Ω))
-        {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal b} < ENNReal.ofReal ε := by
-  obtain ⟨c,hc,hsteps⟩ := a.source_sausage_nslot_bound_all hκ hκ8
-  have hev : ∀ᶠ n : ℕ in atTop, c^n < ENNReal.ofReal (ε/4) :=
-    (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc).eventually
-      (gt_mem_nhds (ENNReal.ofReal_pos.mpr (by positivity)))
-  obtain ⟨n,hn,hcn⟩ := ((eventually_ge_atTop 1).and hev).exists
-  have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega)
-  let δ : ℝ≥0 := (t-s)/(n:ℝ≥0)
-  have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0
-  have hend : s+(n:ℝ≥0)*δ = t := by
-    dsimp [δ]
-    rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le]
-  let ν : ℝ := ε/(4*((n:ℝ)+1))
-  have hν : 0 < ν := by dsimp [ν]; positivity
-  obtain ⟨b,e₀,hb,he₀,hbound⟩ := hsteps s δ hδ ν hν n
-  have hnν : (n:ℝ≥0∞)*ENNReal.ofReal ν ≤ ENNReal.ofReal (ε/4) := by
-    rw [←ENNReal.ofReal_natCast,←ENNReal.ofReal_mul (Nat.cast_nonneg n)]
-    apply ENNReal.ofReal_le_ofReal
-    calc (n:ℝ)*ν ≤ ((n:ℝ)+1)*ν := mul_le_mul_of_nonneg_right (by linarith) hν.le
-      _ = ε/4 := by dsimp [ν]; field_simp
-  refine ⟨b,e₀,hb,he₀,?_⟩
-  intro e he hel
-  have hh := hbound e he hel
-  rw [hend] at hh
-  apply hh.trans_lt
-  calc c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν
-      ≤ ENNReal.ofReal (ε/4)+ENNReal.ofReal (ε/4) := add_le_add hcn.le hnν
-    _ = ENNReal.ofReal (ε/2) := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
-    _ < ENNReal.ofReal ε := (ENNReal.ofReal_lt_ofReal_iff hε).mpr (by linarith)
-theorem sourceSmallSausageAll_proved : SourceSmallSausageAllTarget := by
-  intro κ hκ hκ8 Ω m P hP γ hSLE s t hst ε hε
-  let : IsProbabilityMeasure P := hP
-  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
-  obtain ⟨b,e₀,hb,he₀,hbound⟩ := a.source_small_sausage_all hκ hκ8 s t hst ε hε
-  refine ⟨b,e₀,hb,he₀,?_⟩
-  intro e he hel
-  have hh := hbound e he hel
-  rw [←a.measurePreserving_val.map_eq]
-  rwa [(MeasurableEmbedding.subtype_coe a.measurable_samples).map_apply] 
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance sausageSeedAllUniverseDriverMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance sausageSeedAllUniverseTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance sausageSeedAllUniverseDriverBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance sausageSeedAllUniverseTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_sausage_universe_single {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P)
-    (s t : ℝ≥0) (hst : s < t) (ε : ℝ) (hε : 0 < ε) :
-    ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-      P {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal b} < ENNReal.ofReal ε := by
-  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let : IsProbabilityMeasure μ := a.probability
-  let bp := sourceDriverPath a.B a.continuous_B
-  let gp := sourceTracePath (γ ∘ Subtype.val) a.trace_continuous
-  have hbm : Measurable bp := source_path_measurable bp a.measurable_B
-  have hgm : Measurable gp := source_path_measurable gp (fun t => (hSLE.1 t).comp measurable_subtype_coe)
-  let pair : a.samples → C(ℝ≥0,ℝ) × C(ℝ≥0,ℂ) := fun ω => (bp ω, gp ω)
-  have hpair : Measurable pair := hbm.prodMk hgm
-  let X := range pair
-  let f : a.samples → X := fun ω => ⟨pair ω, mem_range_self ω⟩
-  have hfm : Measurable f := hpair.subtype_mk
-  let Q := μ.map f
-  let : IsProbabilityMeasure Q := inferInstance
-  let B : ℝ≥0 → X → ℝ := fun t x => x.val.1 t
-  let η : X → ℝ≥0 → ℂ := fun x t => x.val.2 t
-  have hBm (t : ℝ≥0) : Measurable (B t) :=
-    (continuous_eval_const t).measurable.comp (measurable_fst.comp measurable_subtype_coe)
-  have hηm (t : ℝ≥0) : Measurable (fun x => η x t) :=
-    (continuous_eval_const t).measurable.comp (measurable_snd.comp measurable_subtype_coe)
-  have hBrownian : IsBrownianReal B Q := by
-    constructor
-    · constructor
-      intro I
-      have hmI : Measurable (fun x : X => I.restrict (B · x)) :=
-        Measurable.of_eval (fun index => hBm index)
-      refine ⟨hmI.aemeasurable, ?_⟩
-      rw [show Q = μ.map f from rfl, Measure.map_map hmI hfm]
-      exact (a.brownian.hasLaw I).map_eq
-    · exact Eventually.of_forall (fun x => x.val.1.continuous)
-  have hη : IsOrdinaryChordalSLE κ η Q := by
-    refine ⟨hηm, B, hBrownian, Eventually.of_forall ?_⟩
-    intro x
-    obtain ⟨ω, hω⟩ := x.property
-    refine ⟨a.G ω, a.F ω, ?_⟩
-    have hp : pair ω = x.val := hω
-    have heB : ∀ t, B t x = a.B t ω := fun t => congrArg (fun z => z.1 t) hp.symm
-    have heη : η x = γ ω := by
-      exact congrArg (fun z => (z.2 : ℝ≥0 → ℂ)) hp.symm
-    simpa only [heB,heη] using a.witness ω
-  obtain ⟨b,e₀,hb,he₀,hbound⟩ :=
-    sourceSmallSausageAll_proved κ hκ hκ8 X inferInstance Q inferInstance η hη s t hst ε hε
-  refine ⟨b,e₀,hb,he₀,?_⟩
-  intro e he hel
-  have hset : MeasurableSet {x : X | sourceSausageMass κ (η x) s t e < ENNReal.ofReal b} := by
-    exact measurableSet_lt ((source_sausage_mass_measurable κ s t e).comp
-      (measurable_snd.comp measurable_subtype_coe)) measurable_const
-  have hh := hbound e he hel
-  change (μ.map f) _ < _ at hh
-  rw [Measure.map_apply hfm hset] at hh
-  rw [←a.measurePreserving_val.map_eq,
-    (MeasurableEmbedding.subtype_coe a.measurable_samples).map_apply]
-  exact hh
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-universe v_sausage
- 
-
-def SourceSmallSausageMainTarget : Prop :=
-  ∀ (Ω : Type v_sausage) (_ : MeasurableSpace Ω) (P : MeasureTheory.Measure Ω),
-    MeasureTheory.IsProbabilityMeasure P →
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-  ∀ I : Finset (ℝ≥0 × ℝ≥0), (∀ J ∈ I, J.1 < J.2) →
-  ∀ ε : ℝ, 0 < ε → ∃ a e₀ : ℝ, 0 < a ∧ 0 < e₀ ∧
-    ∀ e : ℝ, 0 < e → e < e₀ →
-      P {ω | ∃ J ∈ I, sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a} < ENNReal.ofReal ε
-end SLEExactGauge
-namespace SLEExactGauge
-theorem sourceSmallSausageMain_proved : SourceSmallSausageMainTarget := by
-  classical
-  intro Ω m P hP κ hκ hκ8 γ hSLE I hI
-  let : IsProbabilityMeasure P := hP
-  revert hI
-  induction I using Finset.induction_on with
-  | empty =>
-      intro _ ε hε
-      refine ⟨1,1,by norm_num,by norm_num,?_⟩
-      intro e he hel
-      simpa using ENNReal.ofReal_pos.mpr hε
-  | @insert J I hJI ih =>
-      intro hI ε hε
-      have hε2 : 0 < ε/2 := by positivity
-      obtain ⟨a₁,e₁,ha₁,he₁,hb₁⟩ := source_sausage_universe_single hκ hκ8 hSLE
-        J.1 J.2 (hI J (Finset.mem_insert_self _ _)) (ε/2) hε2
-      obtain ⟨a₂,e₂,ha₂,he₂,hb₂⟩ := ih
-        (fun K hK => hI K (Finset.mem_insert_of_mem hK)) (ε/2) hε2
-      refine ⟨min a₁ a₂,min e₁ e₂,lt_min ha₁ ha₂,lt_min he₁ he₂,?_⟩
-      intro e he hel
-      have hsub : {ω | ∃ K ∈ insert J I,
-          sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal (min a₁ a₂)} ⊆
-          {ω | sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a₁} ∪
-          {ω | ∃ K ∈ I, sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal a₂} := by
-        rintro ω ⟨K,hK,hm⟩
-        rcases Finset.mem_insert.mp hK with rfl | hK
-        · exact Or.inl (hm.trans_le (ENNReal.ofReal_le_ofReal (min_le_left _ _)))
-        · exact Or.inr ⟨K,hK,hm.trans_le (ENNReal.ofReal_le_ofReal (min_le_right _ _))⟩
-      calc
-        P {ω | ∃ K ∈ insert J I,
-            sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal (min a₁ a₂)} ≤
-            P ({ω | sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a₁} ∪
-              {ω | ∃ K ∈ I, sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal a₂}) := measure_mono hsub
-        _ ≤ P {ω | sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a₁} +
-            P {ω | ∃ K ∈ I, sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal a₂} := measure_union_le _ _
-        _ < ENNReal.ofReal (ε/2) + ENNReal.ofReal (ε/2) := ENNReal.add_lt_add
-          (hb₁ e he (hel.trans_le (min_le_left _ _)))
-          (hb₂ e he (hel.trans_le (min_le_right _ _)))
-        _ = ENNReal.ofReal ε := by
-          rw [←ENNReal.ofReal_add hε2.le hε2.le]
-          congr 1
-          ring
-end SLEExactGauge
-end
-
-section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
- 
-def wholeTrace (γ : ℝ≥0 → ℂ) : Set ℂ := range γ
-
- 
-def integerRectangle (m : ℕ) : Set ℂ :=
-  {z | -(m : ℝ) ≤ z.re ∧ z.re ≤ m ∧ 0 ≤ z.im ∧ z.im ≤ m}
-
- 
-
-def MainEvent {Ω : Type*} (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : Set Ω :=
-  {ω | (∀ s t : ℝ≥0, 0 < s → s < t →
-      0 < hausdorffGauge h (segment (γ ω) s t) ∧
-      hausdorffGauge h (segment (γ ω) s t) < ∞) ∧
-    (∀ m : ℕ, 0 < m →
-      hausdorffGauge h (wholeTrace (γ ω) ∩ integerRectangle m) < ∞)}
-
- 
-def MainConclusion {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : Prop :=
-  (∀ᵐ ω ∂P, ω ∈ MainEvent h γ) ∧
-  (∀ R : ℝ, 0 < R →
-    AEMeasurable (fun ω => hausdorffGauge h
-      (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P ∧
-    ∃ C : ℝ≥0, (∫⁻ ω, hausdorffGauge h
-      (wholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P) ≤ C)
-
-end SLEExactGauge
-end
- 
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance sausageSeedAllOpenCoverTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance sausageSeedAllOpenCoverTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_compact_finite_open_cover_lt (h : ℝ → ℝ) (hh : IsGauge h)
-    (K : Set ℂ) (hK : IsCompact K) (r : ℝ) (hr : 0 < r)
-    (ε : ℝ≥0∞) (hε : hausdorffGauge h K < ε) :
-    sourceFiniteOpenCover h (ENNReal.ofReal r) ε K := by
-  obtain ⟨b,hb,hbε⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hε
-  obtain ⟨δ,hδ,hbδ⟩ := ENNReal.lt_iff_exists_add_pos_lt.mp hbε
-  have hsmall :
-      (⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n)
-        (_ : ∀ n, ediam (U n) ≤ ENNReal.ofReal (r/2)),
-          ∑' n, extendedGauge h (ediam (U n))) < (b : ℝ≥0∞) := by
-    apply lt_of_le_of_lt (b := hausdorffGauge h K) _ hb
-    rw [source_gauge_cover_formula h hh]
-    exact le_iSup_of_le (ENNReal.ofReal (r/2))
-      (le_iSup_of_le (ENNReal.ofReal_pos.mpr (by positivity)) le_rfl)
-  obtain ⟨U,hU,hdiam,hcost⟩ := (by simpa only [iInf_lt_iff] using hsmall)
-  obtain ⟨e,he,hes⟩ := ENNReal.exists_pos_sum_of_countable
-    (ENNReal.coe_pos.mpr hδ).ne' ℕ
-  have hfin (n : ℕ) : ediam (U n) ≠ ∞ :=
-    ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hdiam n)
-  have hrd (n : ℕ) : (ediam (U n)).toReal < r := by
-    have hb := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hdiam n)
-    rw [ENNReal.toReal_ofReal (by positivity)] at hb
-    linarith
-  choose V hopen hUV hVdiam hVcost using fun n =>
-    source_gauge_open_enlargement h hh (U n) r (hrd n) (hfin n) (e n) (he n)
-  obtain ⟨I,hI⟩ := hK.elim_finite_subcover V hopen (hU.trans (iUnion_mono hUV))
-  refine ⟨V,I,fun n _ => hopen n,fun n _ => (hVdiam n).le,hI,?_⟩
-  calc
-    (∑ n ∈ I, extendedGauge h (ediam (V n))) ≤ ∑' n, extendedGauge h (ediam (V n)) :=
-      ENNReal.sum_le_tsum I
-    _ ≤ ∑' n, (extendedGauge h (ediam (U n)) + e n) := ENNReal.tsum_le_tsum hVcost
-    _ = (∑' n, extendedGauge h (ediam (U n))) + ∑' n, (e n : ℝ≥0∞) := ENNReal.tsum_add
-    _ < (b : ℝ≥0∞) + δ := ENNReal.add_lt_add hcost hes
-    _ < ε := hbδ
-
-def sourceOpenGaugeContent (h : ℝ → ℝ) (r : ℝ≥0∞) (K : Set ℂ) : ℝ≥0∞ :=
-  ⨅ ε : ℝ≥0∞, ⨅ (_ : sourceFiniteOpenCover h r ε K), ε
-lemma source_open_content_measurable (h : ℝ → ℝ) (C : Set ℂ) (hC : IsClosed C)
-    (s t : ℝ≥0) (r : ℝ≥0∞) :
-    Measurable (fun γ : C(ℝ≥0,ℂ) => sourceOpenGaugeContent h r (C ∩ (γ '' Icc s t))) := by
-  apply measurable_of_Iio
-  intro a
-  have heq : (fun γ : C(ℝ≥0,ℂ) => sourceOpenGaugeContent h r (C ∩ (γ '' Icc s t))) ⁻¹' Iio a =
-      ⋃ ε : ℝ≥0∞, ⋃ (_ : ε < a), {γ : C(ℝ≥0,ℂ) | sourceFiniteOpenCover h r ε (C ∩ (γ '' Icc s t))} := by
-    ext γ
-    simp only [mem_preimage,mem_Iio,sourceOpenGaugeContent,iInf_lt_iff,mem_iUnion,mem_ofPred_eq]
-    tauto
-  rw [heq]
-  exact (isOpen_iUnion (fun ε => isOpen_iUnion (fun _ =>
-    source_finite_open_cover_trace_isOpen h C hC s t r ε))).measurableSet
-lemma source_compact_gauge_sup_open (h : ℝ → ℝ) (hh : IsGauge h)
-    (K : Set ℂ) (hK : IsCompact K) :
-    hausdorffGauge h K = ⨆ n : ℕ, sourceOpenGaugeContent h (sourceCoverScale n) K := by
-  apply le_antisymm
-  · rw [source_gauge_cover_formula h hh]
-    refine iSup_le fun R => iSup_le fun hR => ?_
-    obtain ⟨n,hn⟩ := (source_coverScale_tendsto.eventually (gt_mem_nhds hR)).exists
-    apply le_iSup_of_le n
-    change sourceFineGaugeContent h R K ≤ sourceOpenGaugeContent h (sourceCoverScale n) K
-    unfold sourceOpenGaugeContent
-    exact le_iInf fun ε => le_iInf fun hcov => (source_fine_content_finite_cover h hh hn.le hcov).le
-  · refine iSup_le fun n => ?_
-    apply le_of_forall_gt_imp_ge_of_dense
-    intro ε hε
-    unfold sourceOpenGaugeContent
-    exact iInf_le_of_le ε (iInf_le_of_le
-      (source_compact_finite_open_cover_lt h hh K hK (1/((n:ℝ)+1)) (by positivity) ε hε) le_rfl)
-lemma source_compact_trace_measure_measurable (h : ℝ → ℝ) (hh : IsGauge h)
-    (C : Set ℂ) (hC : IsClosed C) (s t : ℝ≥0) :
-    Measurable (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (C ∩ (γ '' Icc s t))) := by
-  have heq : (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (C ∩ (γ '' Icc s t))) =
-      fun γ : C(ℝ≥0,ℂ) => ⨆ n : ℕ, sourceOpenGaugeContent h (sourceCoverScale n) (C ∩ (γ '' Icc s t)) := by
-    funext γ
-    exact source_compact_gauge_sup_open h hh _ ((isCompact_Icc.image γ.continuous).inter_left hC)
-  rw [heq]
-  exact Measurable.iSup (fun n => source_open_content_measurable h C hC s t _)
-end
-end SLEExactGauge
-end
-
- 
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.LowerPositivity
+import OAI.Probability.SLE.SausageScaling
+
+namespace OAI
+
+/-! Sausage Seed All for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+namespace RegularSLEPresentation
+variable {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+variable {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+lemma source_positive_probability_initial_sausage_all (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 2 < z₀.im) :
+    ∃ (T : ℝ≥0) (p b e₀ : ℝ), 0 < T ∧ 0 < p ∧ 0 < b ∧ 0 < e₀ ∧
+      ∀ e : ℝ, 0 < e → e < e₀ →
+        ENNReal.ofReal p ≤ (P.comap (Subtype.val : a.samples → Ω))
+          {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
+            volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e)} := by
+  have hk : 0 < κ := by linarith
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨T,e₀,m₀,M,hT,he₀,hm₀,hM,hMom⟩ := source_retained_seed_all κ hκ hκ8 Ω m P inferInstance γ a z₀ hz
+  let C := ((1:ℝ)/2)^(-codimension κ)
+  have hC : 0 < C := Real.rpow_pos_of_pos (by norm_num) _
+  refine ⟨T,(m₀/2)^2/M,(m₀/2)/C,min 1 (2*e₀),hT,by positivity,by positivity,lt_min (by norm_num) (by positivity),?_⟩
+  intro e he hecut
+  let X := fun ω : a.samples => finiteSpatialGreenMass κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (e/2) T (closedBall z₀ 1)
+  have hXm : Measurable X := (a.sourceFiniteGreenDensity_measurable (e/2) T).lintegral_prod_right'
+  have hhalf : e/2 ≤ e₀ := by have := hecut.trans_le (min_le_right _ _); linarith
+  have hXX := hMom (e/2) (by positivity) hhalf
+  apply (source_second_moment_seed Q X hXm m₀ M hm₀ hM hXX.1 hXX.2).trans
+  apply measure_mono
+  intro ω hω
+  have hbound := a.source_finite_mass_le_sausage hk hκ8 z₀ hz T he
+    (hecut.le.trans (min_le_left _ _)) ω
+  have heq : (e/2)^(-codimension κ) = C*e^(-codimension κ) := by
+    rw [show e/2 = (1/2)*e by ring,Real.mul_rpow (by norm_num) he.le]
+  rw [heq,ENNReal.ofReal_mul hC.le,mul_assoc] at hbound
+  have hdiv : ENNReal.ofReal (m₀/2) / ENNReal.ofReal C ≤
+      ENNReal.ofReal (e^(-codimension κ))*volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e) := by
+    apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
+    simpa only [mul_comm] using (hω.trans hbound)
+  simpa only [mem_ofPred_eq,ENNReal.ofReal_div_of_pos hC] using hdiv
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance SausageUniformSeedAllI5  : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance SausageUniformSeedAllI6  : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma RegularSLEPresentation.source_uniform_sausage_seed_all
+    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ δ : ℝ≥0, 0 < δ →
+      ∃ (K : Set ℂ) (b e₀ : ℝ), IsCompact K ∧ K ⊆ upperHalfPlane ∧ 0 < b ∧ 0 < e₀ ∧
+        ∀ e : ℝ, 0 < e → e < e₀ →
+          p ≤ (P.comap (Subtype.val : a.samples → Ω))
+            {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
+              volume (K ∩ sourceSausageSet (γ ω) 0 δ e)} := by
+  have hk : 0 < κ := by linarith
+  let z₀ : ℂ := 3*I
+  have hz₀ : 2 < z₀.im := by norm_num [z₀]
+  obtain ⟨T,p,b,e₀,hT,hp,hb,he₀,hseed⟩ := a.source_positive_probability_initial_sausage_all hκ hκ8 z₀ hz₀
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  have hp1 : ENNReal.ofReal p ≤ 1 := (hseed (e₀/2) (by positivity) (by linarith)).trans prob_le_one
+  refine ⟨ENNReal.ofReal p,ENNReal.ofReal_pos.mpr hp,hp1,?_⟩
+  intro δ hδ
+  let c := NNReal.sqrt (δ/T)
+  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
+  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
+  have hcR : 0 < (c:ℝ) := hc
+  have ht : sourceScaleTime c δ = T := by
+    unfold sourceScaleTime
+    rw [hc2]
+    field_simp
+  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
+  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
+  have hKH : K ⊆ upperHalfPlane := by
+    rintro _ ⟨z,hz,rfl⟩
+    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
+    change 0 < ((c:ℂ)*z).im
+    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos hcR hz'
+  let C := ((c:ℝ)⁻¹)^(-codimension κ)*((c:ℝ)⁻¹)^2
+  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (inv_pos.mpr hcR) _) (sq_pos_of_pos (inv_pos.mpr hcR))
+  refine ⟨K,b/C,(c:ℝ)*e₀,hK,hKH,div_pos hb hC,mul_pos hcR he₀,?_⟩
+  intro e he hecut
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω)) (a.sourceScaled c hc).trace_continuous
+  let A : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal (b/C) ≤ ENNReal.ofReal (e^(-codimension κ))*volume (K ∩ sourceSausageSet g 0 δ e)}
+  have hA : MeasurableSet A := measurableSet_le measurable_const
+    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ e K))
+  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law hk hκ8 c hc)
+  have hgm : Measurable gp := a.source_trace_path_measurable
+  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable
+  rw [Measure.map_apply hgm hA,Measure.map_apply hsm hA] at hmap
+  change ENNReal.ofReal p ≤ Q (gp ⁻¹' A)
+  rw [hmap]
+  have hmesh : e/(c:ℝ) < e₀ := (div_lt_iff₀ hcR).mpr (by simpa only [mul_comm] using hecut)
+  apply (hseed (e/(c:ℝ)) (div_pos he hcR) hmesh).trans
+  apply measure_mono
+  intro ω hω
+  have hh := hω.trans (source_sausage_scale_mass_bound κ hc (γ ω) T δ ht (closedBall z₀ 1) e he)
+  change ENNReal.ofReal (b/C) ≤ _
+  rw [ENNReal.ofReal_div_of_pos hC]
+  apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
+  simpa only [C,K,sp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hh
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance SausageConditionalAllI6  : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance SausageConditionalAllI7  : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance SausageConditionalAllI8  : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance SausageConditionalAllI9  : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma RegularSLEPresentation.source_uniform_sausage_slot_bound_all
+    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ →
+      ∀ ε : ℝ, 0 < ε → ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧
+        ∀ e : ℝ, 0 < e → e < e₀ →
+        ∀ A : Set a.samples, MeasurableSet[a.filtration s] A →
+          (P.comap (Subtype.val : a.samples → Ω))
+            (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal b}) ≤
+          (1-p)*(P.comap (Subtype.val : a.samples → Ω)) A + ENNReal.ofReal ε := by
+  have hk : 0 < κ := by linarith
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨p,hp,hp1,hseed⟩ := a.source_uniform_sausage_seed_all hκ hκ8
+  refine ⟨p,hp,hp1,?_⟩
+  intro s δ hδ ε hε
+  obtain ⟨K,b,e₀,hK,hKH,hb,he₀,hKP⟩ := hseed δ hδ
+  obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian
+    a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness
+  let τ : a.samples → ℝ≥0 := fun _ => s
+  have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s
+  obtain ⟨hlaw,hident⟩ := hrestart τ hτ
+  let rp := sourceRestartDriverPath a.B a.continuous_B τ
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let η := fun ω : a.samples => Φ (rp ω)
+  have hrm : Measurable rp := by
+    apply source_path_measurable
+    intro t
+    exact (a.measurable_B (s+t)).sub (a.measurable_B s)
+  have hem : Measurable η := hΦm.comp hrm
+  have hgm : Measurable gp := a.source_trace_path_measurable
+  have hfull : ∀ᵐ ω ∂Q, ∃ N : ℕ, sourceSausageChartGood (η ω) (gp ω) K s δ N := by
+    filter_upwards [hident] with ω hω
+    obtain ⟨N,hN,hbound⟩ := source_restart_sausage_countable_bound
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) s δ (η ω) hω hK hKH
+    exact ⟨N,hN,fun q hq hqN => hbound q hq hqN⟩
+  obtain ⟨N,hN,hbad⟩ := source_chart_good_high_probability Q η gp hem hgm K s δ hfull ε hε
+  have hNp : 0 < (N:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hN)
+  let C := ((2*(N:ℝ))⁻¹)^(-codimension κ)*(N:ℝ)^2
+  have hC : 0 < C := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (sq_pos_of_pos hNp)
+  refine ⟨b/C,min (1/(N:ℝ)) (2*(N:ℝ)*e₀),div_pos hb hC,lt_min (by positivity) (by positivity),?_⟩
+  intro e he hecut A hA
+  have hmesh : e/(2*(N:ℝ)) < e₀ := (div_lt_iff₀ (by positivity)).mpr
+    (by simpa only [mul_comm] using hecut.trans_le (min_le_right _ _))
+  let D : Set C(ℝ≥0,ℂ) := {g | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+      volume (K ∩ sourceSausageSet g 0 δ (e/(2*(N:ℝ))))}
+  let E : Set a.samples := η ⁻¹' D
+  let B : Set a.samples := {ω | ¬sourceSausageChartGood (η ω) (gp ω) K s δ N}
+  have hD : MeasurableSet D := measurableSet_le measurable_const
+    (measurable_const.mul (source_sausage_local_volume_measurable 0 δ (e/(2*(N:ℝ))) K))
+  have hEm : MeasurableSet E := hD.preimage hem
+  have hAm : MeasurableSet A := (a.filtration.le s) A hA
+  have hAstop : MeasurableSet[hτ.measurableSpace] A := by
+    change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A
+    rw [IsStoppingTime.measurableSpace_const]
+    exact hA
+  have hprob : p*Q A ≤ Q (A ∩ E) := by
+    have hEq := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ D) (hlaw A hAstop)
+    rw [Measure.map_apply hem hD,Measure.smul_apply,Measure.map_apply hgm hD,
+      Measure.restrict_apply (hD.preimage hem)] at hEq
+    change Q (E ∩ A) = Q A*Q {ω | ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+      volume (K ∩ sourceSausageSet (γ ω) 0 δ (e/(2*(N:ℝ))))} at hEq
+    rw [inter_comm] at hEq
+    rw [hEq,mul_comm p (Q A)]
+    exact mul_le_mul' le_rfl (hKP (e/(2*(N:ℝ))) (by positivity) hmesh)
+  have hins : A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)} ⊆ (A \ E) ∪ B := by
+    intro ω hω
+    by_cases hG : sourceSausageChartGood (η ω) (gp ω) K s δ N
+    · left
+      refine ⟨hω.1,?_⟩
+      intro hE
+      have hEE : ENNReal.ofReal b ≤ ENNReal.ofReal ((e/(2*(N:ℝ)))^(-codimension κ))*
+          volume (K ∩ sourceSausageSet (η ω) 0 δ (e/(2*(N:ℝ)))) := hE
+      have hmass := hEE.trans (source_chart_good_mass_bound κ (η ω) (gp ω) K s δ N hG he
+        (hecut.trans_le (min_le_left _ _)))
+      have hlow : ENNReal.ofReal (b/C) ≤ sourceSausageMass κ (γ ω) s (s+δ) e := by
+        rw [ENNReal.ofReal_div_of_pos hC]
+        apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
+        simpa only [C,gp,sourceTracePath,ContinuousMap.coe_mk,mul_comm] using hmass
+      exact (not_lt_of_ge hlow) hω.2
+    · exact Or.inr hG
+  have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by
+    have heq : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto
+    rw [heq]
+    exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _)
+  calc
+    Q (A ∩ {ω | sourceSausageMass κ (γ ω) s (s+δ) e < ENNReal.ofReal (b/C)})
+        ≤ Q ((A \ E) ∪ B) := measure_mono hins
+    _ ≤ Q (A \ E) + Q B := measure_union_le _ _
+    _ ≤ (Q A-p*Q A)+ENNReal.ofReal ε := by rw [hdiff]; exact add_le_add (tsub_le_tsub_left hprob _) hbad.le
+    _ = (1-p)*Q A+ENNReal.ofReal ε := by
+      rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul]
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+lemma RegularSLEPresentation.source_sausage_nslot_bound_all
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) :
+    ∃ c : ℝ≥0∞, c < 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ → ∀ ν : ℝ, 0 < ν →
+      ∀ n : ℕ, ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+        (P.comap (Subtype.val : a.samples → Ω))
+          {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b} ≤
+        c^n + (n:ℝ≥0∞)*ENNReal.ofReal ν := by
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_sausage_slot_bound_all hκ hκ8
+  let c : ℝ≥0∞ := 1-p
+  have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne'
+  refine ⟨c,hc,?_⟩
+  intro s δ hδ ν hν n
+  induction n with
+  | zero =>
+    refine ⟨1,1,by norm_num,by norm_num,?_⟩
+    intro e he hel
+    simpa only [pow_zero,Nat.cast_zero,zero_mul,add_zero] using
+      (prob_le_one (μ := Q) (s := {ω | sourceSausageMass κ (γ ω) s (s+(0:ℝ≥0)*δ) e < ENNReal.ofReal 1}))
+  | succ n ih =>
+    obtain ⟨b₀,e₀,hb₀,he₀,hprev⟩ := ih
+    obtain ⟨b₁,e₁,hb₁,he₁,hstep⟩ := hslot (s+(n:ℝ≥0)*δ) δ hδ ν hν
+    refine ⟨min b₀ b₁,min e₀ e₁,lt_min hb₀ hb₁,lt_min he₀ he₁,?_⟩
+    intro e he hel
+    have hend : s+((n+1:ℕ):ℝ≥0)*δ = (s+(n:ℝ≥0)*δ)+δ := by push_cast; ring
+    let A : Set a.samples := {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ) e < ENNReal.ofReal b₀}
+    have hA : MeasurableSet[a.filtration (s+(n:ℝ≥0)*δ)] A :=
+      measurableSet_lt (a.source_sausage_mass_measurable_at s _ e) measurable_const
+    rw [hend]
+    calc
+      Q {ω | sourceSausageMass κ (γ ω) s (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal (min b₀ b₁)}
+          ≤ Q (A ∩ {ω | sourceSausageMass κ (γ ω) (s+(n:ℝ≥0)*δ) (s+(n:ℝ≥0)*δ+δ) e < ENNReal.ofReal b₁}) := by
+          apply measure_mono
+          intro ω hω
+          constructor
+          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
+              (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))).trans_lt hω).trans_le
+                (ENNReal.ofReal_le_ofReal (min_le_left _ _))
+          · exact ((source_sausage_mass_mono_interval κ (γ ω) e
+              (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)).trans_lt hω).trans_le
+                (ENNReal.ofReal_le_ofReal (min_le_right _ _))
+      _ ≤ c*Q A+ENNReal.ofReal ν := hstep e he (hel.trans_le (min_le_right _ _)) A hA
+      _ ≤ c*(c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν)+ENNReal.ofReal ν := by
+          exact add_le_add (mul_le_mul' le_rfl (hprev e he (hel.trans_le (min_le_left _ _)))) le_rfl
+      _ ≤ c^(n+1)+((n+1:ℕ):ℝ≥0∞)*ENNReal.ofReal ν := by
+          rw [mul_add,pow_succ,mul_comm (c^n) c,Nat.cast_add,Nat.cast_one,add_mul,one_mul]
+          rw [←add_assoc]
+          exact add_le_add (add_le_add le_rfl (mul_le_of_le_one_left' hc.le)) le_rfl
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+def SourceSmallSausageAllTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : MeasureTheory.Measure Ω),
+    MeasureTheory.IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+  ∀ s t : ℝ≥0, s < t → ∀ ε : ℝ, 0 < ε →
+    ∃ a e₀ : ℝ, 0 < a ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+      P {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal a} < ENNReal.ofReal ε
+
+lemma RegularSLEPresentation.source_small_sausage_all
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ8 : κ < 8) (s t : ℝ≥0) (hst : s < t) (ε : ℝ) (hε : 0 < ε) :
+    ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+      (P.comap (Subtype.val : a.samples → Ω))
+        {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal b} < ENNReal.ofReal ε := by
+  obtain ⟨c,hc,hsteps⟩ := a.source_sausage_nslot_bound_all hκ hκ8
+  have hev : ∀ᶠ n : ℕ in atTop, c^n < ENNReal.ofReal (ε/4) :=
+    (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc).eventually
+      (gt_mem_nhds (ENNReal.ofReal_pos.mpr (by positivity)))
+  obtain ⟨n,hn,hcn⟩ := ((eventually_ge_atTop 1).and hev).exists
+  have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega)
+  let δ : ℝ≥0 := (t-s)/(n:ℝ≥0)
+  have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0
+  have hend : s+(n:ℝ≥0)*δ = t := by
+    dsimp [δ]
+    rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le]
+  let ν : ℝ := ε/(4*((n:ℝ)+1))
+  have hν : 0 < ν := by dsimp [ν]; positivity
+  obtain ⟨b,e₀,hb,he₀,hbound⟩ := hsteps s δ hδ ν hν n
+  have hnν : (n:ℝ≥0∞)*ENNReal.ofReal ν ≤ ENNReal.ofReal (ε/4) := by
+    rw [←ENNReal.ofReal_natCast,←ENNReal.ofReal_mul (Nat.cast_nonneg n)]
+    apply ENNReal.ofReal_le_ofReal
+    calc (n:ℝ)*ν ≤ ((n:ℝ)+1)*ν := mul_le_mul_of_nonneg_right (by linarith) hν.le
+      _ = ε/4 := by dsimp [ν]; field_simp
+  refine ⟨b,e₀,hb,he₀,?_⟩
+  intro e he hel
+  have hh := hbound e he hel
+  rw [hend] at hh
+  apply hh.trans_lt
+  calc c^n+(n:ℝ≥0∞)*ENNReal.ofReal ν
+      ≤ ENNReal.ofReal (ε/4)+ENNReal.ofReal (ε/4) := add_le_add hcn.le hnν
+    _ = ENNReal.ofReal (ε/2) := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
+    _ < ENNReal.ofReal ε := (ENNReal.ofReal_lt_ofReal_iff hε).mpr (by linarith)
+theorem sourceSmallSausageAll_proved : SourceSmallSausageAllTarget := by
+  intro κ hκ hκ8 Ω m P hP γ hSLE s t hst ε hε
+  let : IsProbabilityMeasure P := hP
+  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
+  obtain ⟨b,e₀,hb,he₀,hbound⟩ := a.source_small_sausage_all hκ hκ8 s t hst ε hε
+  refine ⟨b,e₀,hb,he₀,?_⟩
+  intro e he hel
+  have hh := hbound e he hel
+  rw [←a.measurePreserving_val.map_eq]
+  rwa [(MeasurableEmbedding.subtype_coe a.measurable_samples).map_apply] 
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance sausageSeedAllUniverseDriverMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance sausageSeedAllUniverseTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance sausageSeedAllUniverseDriverBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance sausageSeedAllUniverseTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_sausage_universe_single {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P)
+    (s t : ℝ≥0) (hst : s < t) (ε : ℝ) (hε : 0 < ε) :
+    ∃ b e₀ : ℝ, 0 < b ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+      P {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal b} < ENNReal.ofReal ε := by
+  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let : IsProbabilityMeasure μ := a.probability
+  let bp := sourceDriverPath a.B a.continuous_B
+  let gp := sourceTracePath (γ ∘ Subtype.val) a.trace_continuous
+  have hbm : Measurable bp := source_path_measurable bp a.measurable_B
+  have hgm : Measurable gp := source_path_measurable gp (fun t => (hSLE.1 t).comp measurable_subtype_coe)
+  let pair : a.samples → C(ℝ≥0,ℝ) × C(ℝ≥0,ℂ) := fun ω => (bp ω, gp ω)
+  have hpair : Measurable pair := hbm.prodMk hgm
+  let X := range pair
+  let f : a.samples → X := fun ω => ⟨pair ω, mem_range_self ω⟩
+  have hfm : Measurable f := hpair.subtype_mk
+  let Q := μ.map f
+  let : IsProbabilityMeasure Q := inferInstance
+  let B : ℝ≥0 → X → ℝ := fun t x => x.val.1 t
+  let η : X → ℝ≥0 → ℂ := fun x t => x.val.2 t
+  have hBm (t : ℝ≥0) : Measurable (B t) :=
+    (continuous_eval_const t).measurable.comp (measurable_fst.comp measurable_subtype_coe)
+  have hηm (t : ℝ≥0) : Measurable (fun x => η x t) :=
+    (continuous_eval_const t).measurable.comp (measurable_snd.comp measurable_subtype_coe)
+  have hBrownian : IsBrownianReal B Q := by
+    constructor
+    · constructor
+      intro I
+      have hmI : Measurable (fun x : X => I.restrict (B · x)) :=
+        Measurable.of_eval (fun index => hBm index)
+      refine ⟨hmI.aemeasurable, ?_⟩
+      rw [show Q = μ.map f from rfl, Measure.map_map hmI hfm]
+      exact (a.brownian.hasLaw I).map_eq
+    · exact Eventually.of_forall (fun x => x.val.1.continuous)
+  have hη : IsOrdinaryChordalSLE κ η Q := by
+    refine ⟨hηm, B, hBrownian, Eventually.of_forall ?_⟩
+    intro x
+    obtain ⟨ω, hω⟩ := x.property
+    refine ⟨a.G ω, a.F ω, ?_⟩
+    have hp : pair ω = x.val := hω
+    have heB : ∀ t, B t x = a.B t ω := fun t => congrArg (fun z => z.1 t) hp.symm
+    have heη : η x = γ ω := by
+      exact congrArg (fun z => (z.2 : ℝ≥0 → ℂ)) hp.symm
+    simpa only [heB,heη] using a.witness ω
+  obtain ⟨b,e₀,hb,he₀,hbound⟩ :=
+    sourceSmallSausageAll_proved κ hκ hκ8 X inferInstance Q inferInstance η hη s t hst ε hε
+  refine ⟨b,e₀,hb,he₀,?_⟩
+  intro e he hel
+  have hset : MeasurableSet {x : X | sourceSausageMass κ (η x) s t e < ENNReal.ofReal b} := by
+    exact measurableSet_lt ((source_sausage_mass_measurable κ s t e).comp
+      (measurable_snd.comp measurable_subtype_coe)) measurable_const
+  have hh := hbound e he hel
+  change (μ.map f) _ < _ at hh
+  rw [Measure.map_apply hfm hset] at hh
+  rw [←a.measurePreserving_val.map_eq,
+    (MeasurableEmbedding.subtype_coe a.measurable_samples).map_apply]
+  exact hh
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+universe v_sausage
+ 
+
+def SourceSmallSausageMainTarget : Prop :=
+  ∀ (Ω : Type v_sausage) (_ : MeasurableSpace Ω) (P : MeasureTheory.Measure Ω),
+    MeasureTheory.IsProbabilityMeasure P →
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+  ∀ I : Finset (ℝ≥0 × ℝ≥0), (∀ J ∈ I, J.1 < J.2) →
+  ∀ ε : ℝ, 0 < ε → ∃ a e₀ : ℝ, 0 < a ∧ 0 < e₀ ∧
+    ∀ e : ℝ, 0 < e → e < e₀ →
+      P {ω | ∃ J ∈ I, sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a} < ENNReal.ofReal ε
+end SLEExactGauge
+namespace SLEExactGauge
+theorem sourceSmallSausageMain_proved : SourceSmallSausageMainTarget := by
+  classical
+  intro Ω m P hP κ hκ hκ8 γ hSLE I hI
+  let : IsProbabilityMeasure P := hP
+  revert hI
+  induction I using Finset.induction_on with
+  | empty =>
+      intro _ ε hε
+      refine ⟨1,1,by norm_num,by norm_num,?_⟩
+      intro e he hel
+      simpa using ENNReal.ofReal_pos.mpr hε
+  | @insert J I hJI ih =>
+      intro hI ε hε
+      have hε2 : 0 < ε/2 := by positivity
+      obtain ⟨a₁,e₁,ha₁,he₁,hb₁⟩ := source_sausage_universe_single hκ hκ8 hSLE
+        J.1 J.2 (hI J (Finset.mem_insert_self _ _)) (ε/2) hε2
+      obtain ⟨a₂,e₂,ha₂,he₂,hb₂⟩ := ih
+        (fun K hK => hI K (Finset.mem_insert_of_mem hK)) (ε/2) hε2
+      refine ⟨min a₁ a₂,min e₁ e₂,lt_min ha₁ ha₂,lt_min he₁ he₂,?_⟩
+      intro e he hel
+      have hsub : {ω | ∃ K ∈ insert J I,
+          sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal (min a₁ a₂)} ⊆
+          {ω | sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a₁} ∪
+          {ω | ∃ K ∈ I, sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal a₂} := by
+        rintro ω ⟨K,hK,hm⟩
+        rcases Finset.mem_insert.mp hK with rfl | hK
+        · exact Or.inl (hm.trans_le (ENNReal.ofReal_le_ofReal (min_le_left _ _)))
+        · exact Or.inr ⟨K,hK,hm.trans_le (ENNReal.ofReal_le_ofReal (min_le_right _ _))⟩
+      calc
+        P {ω | ∃ K ∈ insert J I,
+            sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal (min a₁ a₂)} ≤
+            P ({ω | sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a₁} ∪
+              {ω | ∃ K ∈ I, sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal a₂}) := measure_mono hsub
+        _ ≤ P {ω | sourceSausageMass κ (γ ω) J.1 J.2 e < ENNReal.ofReal a₁} +
+            P {ω | ∃ K ∈ I, sourceSausageMass κ (γ ω) K.1 K.2 e < ENNReal.ofReal a₂} := measure_union_le _ _
+        _ < ENNReal.ofReal (ε/2) + ENNReal.ofReal (ε/2) := ENNReal.add_lt_add
+          (hb₁ e he (hel.trans_le (min_le_left _ _)))
+          (hb₂ e he (hel.trans_le (min_le_right _ _)))
+        _ = ENNReal.ofReal ε := by
+          rw [←ENNReal.ofReal_add hε2.le hε2.le]
+          congr 1
+          ring
+end SLEExactGauge
+end
+
+section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+ 
+def wholeTrace (γ : ℝ≥0 → ℂ) : Set ℂ := range γ
+
+ 
+def integerRectangle (m : ℕ) : Set ℂ :=
+  {z | -(m : ℝ) ≤ z.re ∧ z.re ≤ m ∧ 0 ≤ z.im ∧ z.im ≤ m}
+
+ 
+
+def MainEvent {Ω : Type*} (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : Set Ω :=
+  {ω | (∀ s t : ℝ≥0, 0 < s → s < t →
+      0 < hausdorffGauge h (segment (γ ω) s t) ∧
+      hausdorffGauge h (segment (γ ω) s t) < ∞) ∧
+    (∀ m : ℕ, 0 < m →
+      hausdorffGauge h (wholeTrace (γ ω) ∩ integerRectangle m) < ∞)}
+
+ 
+def MainConclusion {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : Prop :=
+  (∀ᵐ ω ∂P, ω ∈ MainEvent h γ) ∧
+  (∀ R : ℝ, 0 < R →
+    AEMeasurable (fun ω => hausdorffGauge h
+      (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P ∧
+    ∃ C : ℝ≥0, (∫⁻ ω, hausdorffGauge h
+      (wholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P) ≤ C)
+
+end SLEExactGauge
+end
+ 
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance sausageSeedAllOpenCoverTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance sausageSeedAllOpenCoverTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_compact_finite_open_cover_lt (h : ℝ → ℝ) (hh : IsGauge h)
+    (K : Set ℂ) (hK : IsCompact K) (r : ℝ) (hr : 0 < r)
+    (ε : ℝ≥0∞) (hε : hausdorffGauge h K < ε) :
+    sourceFiniteOpenCover h (ENNReal.ofReal r) ε K := by
+  obtain ⟨b,hb,hbε⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hε
+  obtain ⟨δ,hδ,hbδ⟩ := ENNReal.lt_iff_exists_add_pos_lt.mp hbε
+  have hsmall :
+      (⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n)
+        (_ : ∀ n, ediam (U n) ≤ ENNReal.ofReal (r/2)),
+          ∑' n, extendedGauge h (ediam (U n))) < (b : ℝ≥0∞) := by
+    apply lt_of_le_of_lt (b := hausdorffGauge h K) _ hb
+    rw [source_gauge_cover_formula h hh]
+    exact le_iSup_of_le (ENNReal.ofReal (r/2))
+      (le_iSup_of_le (ENNReal.ofReal_pos.mpr (by positivity)) le_rfl)
+  obtain ⟨U,hU,hdiam,hcost⟩ := (by simpa only [iInf_lt_iff] using hsmall)
+  obtain ⟨e,he,hes⟩ := ENNReal.exists_pos_sum_of_countable
+    (ENNReal.coe_pos.mpr hδ).ne' ℕ
+  have hfin (n : ℕ) : ediam (U n) ≠ ∞ :=
+    ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hdiam n)
+  have hrd (n : ℕ) : (ediam (U n)).toReal < r := by
+    have hb := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hdiam n)
+    rw [ENNReal.toReal_ofReal (by positivity)] at hb
+    linarith
+  choose V hopen hUV hVdiam hVcost using fun n =>
+    source_gauge_open_enlargement h hh (U n) r (hrd n) (hfin n) (e n) (he n)
+  obtain ⟨I,hI⟩ := hK.elim_finite_subcover V hopen (hU.trans (iUnion_mono hUV))
+  refine ⟨V,I,fun n _ => hopen n,fun n _ => (hVdiam n).le,hI,?_⟩
+  calc
+    (∑ n ∈ I, extendedGauge h (ediam (V n))) ≤ ∑' n, extendedGauge h (ediam (V n)) :=
+      ENNReal.sum_le_tsum I
+    _ ≤ ∑' n, (extendedGauge h (ediam (U n)) + e n) := ENNReal.tsum_le_tsum hVcost
+    _ = (∑' n, extendedGauge h (ediam (U n))) + ∑' n, (e n : ℝ≥0∞) := ENNReal.tsum_add
+    _ < (b : ℝ≥0∞) + δ := ENNReal.add_lt_add hcost hes
+    _ < ε := hbδ
+
+def sourceOpenGaugeContent (h : ℝ → ℝ) (r : ℝ≥0∞) (K : Set ℂ) : ℝ≥0∞ :=
+  ⨅ ε : ℝ≥0∞, ⨅ (_ : sourceFiniteOpenCover h r ε K), ε
+lemma source_open_content_measurable (h : ℝ → ℝ) (C : Set ℂ) (hC : IsClosed C)
+    (s t : ℝ≥0) (r : ℝ≥0∞) :
+    Measurable (fun γ : C(ℝ≥0,ℂ) => sourceOpenGaugeContent h r (C ∩ (γ '' Icc s t))) := by
+  apply measurable_of_Iio
+  intro a
+  have heq : (fun γ : C(ℝ≥0,ℂ) => sourceOpenGaugeContent h r (C ∩ (γ '' Icc s t))) ⁻¹' Iio a =
+      ⋃ ε : ℝ≥0∞, ⋃ (_ : ε < a), {γ : C(ℝ≥0,ℂ) | sourceFiniteOpenCover h r ε (C ∩ (γ '' Icc s t))} := by
+    ext γ
+    simp only [mem_preimage,mem_Iio,sourceOpenGaugeContent,iInf_lt_iff,mem_iUnion,mem_ofPred_eq]
+    tauto
+  rw [heq]
+  exact (isOpen_iUnion (fun ε => isOpen_iUnion (fun _ =>
+    source_finite_open_cover_trace_isOpen h C hC s t r ε))).measurableSet
+lemma source_compact_gauge_sup_open (h : ℝ → ℝ) (hh : IsGauge h)
+    (K : Set ℂ) (hK : IsCompact K) :
+    hausdorffGauge h K = ⨆ n : ℕ, sourceOpenGaugeContent h (sourceCoverScale n) K := by
+  apply le_antisymm
+  · rw [source_gauge_cover_formula h hh]
+    refine iSup_le fun R => iSup_le fun hR => ?_
+    obtain ⟨n,hn⟩ := (source_coverScale_tendsto.eventually (gt_mem_nhds hR)).exists
+    apply le_iSup_of_le n
+    change sourceFineGaugeContent h R K ≤ sourceOpenGaugeContent h (sourceCoverScale n) K
+    unfold sourceOpenGaugeContent
+    exact le_iInf fun ε => le_iInf fun hcov => (source_fine_content_finite_cover h hh hn.le hcov).le
+  · refine iSup_le fun n => ?_
+    apply le_of_forall_gt_imp_ge_of_dense
+    intro ε hε
+    unfold sourceOpenGaugeContent
+    exact iInf_le_of_le ε (iInf_le_of_le
+      (source_compact_finite_open_cover_lt h hh K hK (1/((n:ℝ)+1)) (by positivity) ε hε) le_rfl)
+lemma source_compact_trace_measure_measurable (h : ℝ → ℝ) (hh : IsGauge h)
+    (C : Set ℂ) (hC : IsClosed C) (s t : ℝ≥0) :
+    Measurable (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (C ∩ (γ '' Icc s t))) := by
+  have heq : (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (C ∩ (γ '' Icc s t))) =
+      fun γ : C(ℝ≥0,ℂ) => ⨆ n : ℕ, sourceOpenGaugeContent h (sourceCoverScale n) (C ∩ (γ '' Icc s t)) := by
+    funext γ
+    exact source_compact_gauge_sup_open h hh _ ((isCompact_Icc.image γ.continuous).inter_left hC)
+  rw [heq]
+  exact Measurable.iSup (fun n => source_open_content_measurable h C hC s t _)
+end
+end SLEExactGauge
+end
+
+ 
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SimpleFields.lean b/lean/OAI/Probability/SLE/SimpleFields.lean
index b8d917ac787910f9588063e0772594a436d389ae..64502bc41acea3cd8b508096910886390c953d1f 100644
--- a/lean/OAI/Probability/SLE/SimpleFields.lean
+++ b/lean/OAI/Probability/SLE/SimpleFields.lean
@@ -1,720 +1,720 @@
-import Mathlib
-import OAI.Probability.SLE.ItoIsometry
-
-namespace OAI
-
-/-! Simple Fields for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianIto TaylorMeshes
-lemma stoppedDyadicIndex_bounds {T : ℝ≥0} (hT : 0 < T) (n : ℕ) (t : ℝ≥0) :
-    t ≤ dyadicTime T n (stoppedDyadicIndex T n t) ∧
-    dyadicTime T n (stoppedDyadicIndex T n t) ≤ t + T / (2 : ℝ≥0)^n := by
-  have hd : 0 < T / (2 : ℝ≥0)^n := by positivity
-  constructor
-  · apply (div_le_iff₀ hd).mp
-    exact Nat.le_ceil _
-  · change (Nat.ceil (t / (T / (2 : ℝ≥0)^n)) : ℝ≥0) * (T / (2 : ℝ≥0)^n) ≤ _
-    calc
-      _ ≤ (t / (T / (2 : ℝ≥0)^n) + 1) * (T / (2 : ℝ≥0)^n) :=
-        mul_le_mul_of_nonneg_right (Nat.ceil_lt_add_one (show (0 : ℝ≥0) ≤ _ from zero_le)).le hd.le
-      _ = _ := by rw [add_mul, div_mul_cancel₀ _ hd.ne', one_mul]
-
-lemma stoppedDyadicIndex_tendsto {T : ℝ≥0} (hT : 0 < T) (t : ℝ≥0) :
-    Tendsto (fun n => mesh T n (stoppedDyadicIndex T n t)) atTop (𝓝 (t : ℝ)) := by
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
-    (by simpa only [add_zero] using (tendsto_const_nhds (x := (t : ℝ))).add (mesh_step_tendsto T))
-  · intro n
-    simpa only [← coe_dyadicTime] using
-      (show (t : ℝ) ≤ (dyadicTime T n (stoppedDyadicIndex T n t) : ℝ) by
-        exact_mod_cast (stoppedDyadicIndex_bounds hT n t).1)
-  · intro n
-    have hh := (stoppedDyadicIndex_bounds hT n t).2
-    simpa only [← coe_dyadicTime, NNReal.coe_add, NNReal.coe_div, NNReal.coe_pow, NNReal.coe_ofNat]
-      using (show (dyadicTime T n (stoppedDyadicIndex T n t) : ℝ) ≤
-        ((t + T / (2 : ℝ≥0)^n : ℝ≥0) : ℝ) by exact_mod_cast hh)
-
-lemma stopping_before_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {f : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0))) (t : ℝ≥0) :
-    MeasurableSet[f t] {ω | t < τ ω} := by
-  simpa only [Set.compl_ofPred, WithTop.coe_le_coe, not_le] using (hτ.measurableSet_le t).compl
-
-lemma stopped_coefficient_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {f : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
-    (hτ : IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0))) (t : ℝ≥0)
-    {C : Ω → ℝ} (hC : Measurable[f t] C) :
-    Measurable[f t] (fun ω => if t < τ ω then C ω else 0) := by
-  classical
-  exact hC.ite (stopping_before_measurable hτ t) measurable_const
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes TimeField
-
-lemma prefix_mask_sum {α : Type*} [AddCommMonoid α] (f : ℕ → α) {k N : ℕ} (hk : k ≤ N) :
-    (∑ i ∈ Finset.range N, if i < k then f i else 0) = ∑ i ∈ Finset.range k, f i := by
-  classical
-  calc
-    _ = ∑ i ∈ Finset.range k, if i < k then f i else 0 := by
-      symm
-      apply Finset.sum_subset (Finset.range_mono hk)
-      intro i hi hni
-      simp only [Finset.mem_range] at hni
-      simp [hni]
-    _ = _ := Finset.sum_congr rfl fun i hi => ite_eq_left (Finset.mem_range.mp hi)
-
-noncomputable def stoppedFieldSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
-    (F : Ω → ℝ → ℝ → ℝ) (τ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) : ℝ :=
-  ∑ i ∈ Finset.range (2^n), (if dyadicTime T n i < τ ω then
-    deriv (F ω (mesh T n i)) (B (dyadicTime T n i) ω) else 0) *
-      (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)
-
-lemma stoppedFieldSum_eq_prefix {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) {T : ℝ≥0}
-    (hT : 0 < T) (F : Ω → ℝ → ℝ → ℝ) (τ : Ω → ℝ≥0) (hτ : ∀ ω, τ ω ≤ T)
-    (n : ℕ) (ω : Ω) :
-    stoppedFieldSum B T F τ n ω =
-      ∑ i ∈ Finset.range (stoppedDyadicIndex T n (τ ω)),
-        deriv (F ω (mesh T n i)) (B (dyadicTime T n i) ω) *
-          (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω) := by
-  unfold stoppedFieldSum
-  simp_rw [← lt_stoppedDyadicIndex hT, ite_mul, zero_mul]
-  exact prefix_mask_sum _ (stoppedDyadicIndex_le hT n (hτ ω))
-
- 
-
-theorem stopped_fieldSum_tendsto {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ s, Measurable (B s))
-    (T : ℝ≥0) (F V : Ω → ℝ → ℝ → ℝ) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T)
-    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) T)
-    (hcont : ∀ᵐ ω ∂P, ContinuousOn (fun t => F ω t (B t.toNNReal ω)) (Icc 0 T))
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)]
-      (fun ω => iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω)))
-    (hC2 : ∀ n i, MemLp
-      (fun ω => iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω)) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ ω ∂P,
-      |iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω)| ≤ K₂) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => stoppedFieldSum B T F τ n ω) atTop
-      (𝓝 (F ω (τ ω) (B (τ ω) ω) - F ω 0 (B 0 ω) -
-        (∫ t in (0 : ℝ)..(τ ω : ℝ), V ω t (B t.toNNReal ω)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ),
-          iteratedDeriv 2 (F ω t) (B t.toNNReal ω))) := by
-  classical
-  rcases eq_or_lt_of_le (zero_le : 0 ≤ T) with hT | hT
-  · subst T
-    have hτ0 (ω : Ω) : τ ω = 0 := le_antisymm (hbτ ω) zero_le
-    simp [stoppedFieldSum, dyadicTime, hτ0]
-  let C (n i : ℕ) (ω : Ω) := if dyadicTime T n i < τ ω then
-    iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω) else 0
-  have hs (n i : ℕ) : MeasurableSet[pastSigma B (dyadicTime T n i)]
-      {ω | dyadicTime T n i < τ ω} := by
-    have he := stopping_before_measurable hτ (dyadicTime T n i)
-    exact (le_of_eq (Filtration.natural_eq_comap B (fun t => (hm t).stronglyMeasurable) _)) _ he
-  have hCp (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) :=
-    (hC n i).ite (hs n i) measurable_const
-  have hC2p (n i : ℕ) : MemLp (C n i) 2 P := by
-    have hma := (hCp n i).mono
-      (le_trans (le_of_eq (Filtration.natural_eq_comap B
-        (fun t => (hm t).stronglyMeasurable) _).symm) (Filtration.le _ _)) le_rfl
-    apply MemLp.of_bound hma.aestronglyMeasurable K₂
-    filter_upwards [hbound n i] with ω hω
-    dsimp only [C]
-    split_ifs <;> simpa only [Real.norm_eq_abs, abs_zero] using (by assumption : _ ≤ K₂)
-  have hbp (n i : ℕ) : ∀ᵐ ω ∂P, |C n i ω| ≤ K₂ := by
-    filter_upwards [hbound n i] with ω hω
-    dsimp only [C]
-    split_ifs <;> simp_all
-  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T C hCp hC2p K₂ hK₂ hbp
-  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hcont] with ω hcω hQω heω hrω hfω
-  let b : ℝ → ℝ := fun t => B t.toNNReal ω
-  let k : ℕ → ℕ := fun n => stoppedDyadicIndex T n (τ ω)
-  have hk (n : ℕ) : k n ≤ 2^n := stoppedDyadicIndex_le hT n (hbτ ω)
-  have hS : (τ ω : ℝ) ∈ Icc (0 : ℝ) T := ⟨(τ ω).coe_nonneg, by exact_mod_cast hbτ ω⟩
-  have hlim : Tendsto (fun n => mesh T n (k n)) atTop (𝓝 (τ ω : ℝ)) := stoppedDyadicIndex_tendsto hT _
-  have hb : Continuous b := hcω.comp continuous_real_toNNReal
-  have hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
-    simpa only [b, mesh_toNNReal, quadraticSum] using hQω
-  obtain ⟨K₃, hK₃, h₃⟩ := hrω.third_bound
-  have hr := mesh_prefix_remainder_tendsto
-    (f := fun n i => F ω (mesh T n i)) (fun _ _ => hrω.smooth _) T.coe_nonneg
-    hb.continuousOn hK₃ (fun _ _ x => h₃ _ x) k hk hQ
-  have hriem := mesh_prefix_riemann_tendsto T.coe_nonneg hrω.second_cont k hk hS hlim
-  have htime := mesh_prefix_time_increment_tendsto (b := b) T.coe_nonneg hrω.time_cont
-    hrω.time_deriv hrω.drift_cont k hk hS hlim
-  let Q (n : ℕ) := ∑ i ∈ Finset.range (k n),
-    iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-      (b (mesh T n (i+1))-b (mesh T n i))^2
-  have hQweighted : Tendsto Q atTop
-      (𝓝 (∫ t in (0 : ℝ)..(τ ω : ℝ), iteratedDeriv 2 (F ω t) (b t))) := by
-    have heq (n : ℕ) : Q n = dyadicQuadraticError B T C n ω +
-        ∑ i ∈ Finset.range (k n), iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-          ((T : ℝ) / (2 : ℝ)^n) := by
-      have hmask : dyadicQuadraticError B T C n ω =
-          ∑ i ∈ Finset.range (k n),
-          iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ) / (2 : ℝ)^n) := by
-        simp only [dyadicQuadraticError, C, b, mesh_toNNReal]
-        simp_rw [← lt_stoppedDyadicIndex hT, ite_mul, zero_mul]
-        exact prefix_mask_sum _ (hk n)
-      rw [hmask, ← Finset.sum_add_distrib]
-      apply Finset.sum_congr rfl
-      intro i hi
-      ring
-    simpa only [b, ← heq, zero_add] using heω.add hriem
-  have hTel (n : ℕ) : ∑ i ∈ Finset.range (k n),
-      (F ω (mesh T n (i+1)) (b (mesh T n (i+1))) - F ω (mesh T n i) (b (mesh T n i))) =
-        F ω (mesh T n (k n)) (b (mesh T n (k n))) - F ω 0 (B 0 ω) := by
-    rw [Finset.sum_range_sub (fun i => F ω (mesh T n i) (b (mesh T n i))) (k n), mesh_zero]
-    simp only [b, Real.toNNReal_zero]
-  have hIdentity (n : ℕ) : stoppedFieldSum B T F τ n ω =
-      F ω (mesh T n (k n)) (b (mesh T n (k n))) - F ω 0 (B 0 ω) -
-        (∑ i ∈ Finset.range (k n),
-          (F ω (mesh T n (i+1)) (b (mesh T n (i+1))) -
-            F ω (mesh T n i) (b (mesh T n (i+1))))) - (1/2 : ℝ) * Q n -
-        ∑ i ∈ Finset.range (k n), remainder₂ (F ω (mesh T n i))
-          (b (mesh T n i)) (b (mesh T n (i+1))) := by
-    rw [← hTel n]
-    simp only [Q, Finset.mul_sum, ← Finset.sum_sub_distrib]
-    rw [stoppedFieldSum_eq_prefix B hT F τ hbτ]
-    apply Finset.sum_congr rfl
-    intro i hi
-    simp only [remainder₂, iteratedDeriv_succ, iteratedDeriv_zero, b, mesh_toNNReal]
-    ring
-  have hend := (hfω _ hS).tendsto.comp (tendsto_nhdsWithin_iff.mpr
-    ⟨hlim, Eventually.of_forall fun n => mesh_mem T T.coe_nonneg n (k n) (hk n)⟩)
-  have ht := (((hend.sub_const (F ω 0 (B 0 ω))).sub htime).sub
-    (hQweighted.const_mul (1/2 : ℝ))).sub hr
-  simpa only [Function.comp_apply, b, Real.toNNReal_coe, ← hIdentity, sub_zero] using ht
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-
-theorem ClippedLogSolution.compact_stopped_field_ito
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (φ : (ℂ × ℝ) × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ)
-    (hφc : HasCompactSupport φ) (T : ℝ≥0) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => stoppedFieldSum B T (fun ω t x => φ (c.Q t ω, x)) τ n ω) atTop
-      (𝓝 (φ (c.Q (τ ω) ω, B (τ ω) ω) - φ (c.Q 0 ω, B 0 ω) -
-        (∫ t in (0 : ℝ)..(τ ω : ℝ), fderiv ℝ φ (c.Q t ω, B t.toNNReal ω) (c.velocity t ω, 0)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ),
-          iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t.toNNReal ω))) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨K, hK, _⟩ := compact_smooth_bounded_lipschitz (spatialPartial φ 2)
-    (spatialPartial_smooth φ hφ 2) (spatialPartial_compact φ hφc 2)
-  have hb (t : ℝ≥0) (ω : Ω) :
-      |iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t ω)| ≤ K := by
-    simpa only [← spatialPartial_eq_iteratedDeriv φ hφ, Real.norm_eq_abs] using hK (c.Q t ω, B t ω)
-  have hma (t : ℝ≥0) := c.partial_adapted φ hφ 2 t
-  have hp (t : ℝ≥0) : MemLp
-      (fun ω => iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t ω)) 2 P := by
-    exact MemLp.of_bound ((hma t).mono (Filtration.le _ _) le_rfl).aestronglyMeasurable K
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hb t ω)
-  apply stopped_fieldSum_tendsto hB hm T (fun ω t x => φ (c.Q t ω, x))
-    (fun ω t x => fderiv ℝ φ (c.Q t ω, x) (c.velocity t ω, 0)) τ hτ hbτ
-    (Eventually.of_forall (c.compact_field_regular hBc φ hφ hφc T))
-    (Eventually.of_forall fun ω =>
-      (hφ.continuous.comp ((c.lipschitz ω).continuous.prodMk
-        ((hBc ω).comp continuous_real_toNNReal))).continuousOn)
-    (fun n i => ?_) (fun n i => ?_) K K.coe_nonneg (fun n i => ?_)
-  · have hx := (hma (dyadicTime T n i)).mono
-      (le_of_eq (natural_eq_pastSigma (B := B) (hm := hm) (dyadicTime T n i))) le_rfl
-    simpa only [mesh_eq_dyadicTime] using hx
-  · simpa only [mesh_eq_dyadicTime] using hp (dyadicTime T n i)
-  · exact Eventually.of_forall fun ω => by
-      simpa only [mesh_eq_dyadicTime] using hb (dyadicTime T n i) ω
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes
-
-lemma deriv_fiber_fderiv (φ : (ℂ × ℝ) × ℝ → ℝ)
-    (hφ : Differentiable ℝ φ) (q : ℂ × ℝ) (x : ℝ) :
-    deriv (fun y => φ (q, y)) x = fderiv ℝ φ (q, x) (0, 1) := by
-  exact ((hφ (q, x)).hasFDerivAt.comp_hasDerivAt x
-    ((hasDerivAt_const x q).prodMk (hasDerivAt_id x))).deriv
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-
- 
-noncomputable def greenLogCoefficient (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  greenLogDensity κ q x * (greenExponent κ * Real.sqrt κ *
-    (q.1.re - Real.sqrt κ * x) / greenLogNormSq κ q x)
-
- 
-
-theorem ClippedLogSolution.stopped_green_ito
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (R : ℝ) (hR : 0 ≤ R)
-    (T : ℝ≥0) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (if dyadicTime T n i < τ ω then
-        greenLogCoefficient κ (c.Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω)
-      else 0) * (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) atTop
-      (𝓝 (greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) - greenLogDensity κ q₀ 0)) := by
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
-  have hφ1 (q : ℂ × ℝ) (x : ℝ) (hq : ‖(q, x)‖ ≤ R) :
-      deriv (fun y => φ (q, y)) x = greenLogCoefficient κ q x := by
-    rw [deriv_fiber_fderiv φ (hφ.differentiable (by simp)), hφD q x hq,
-      ← deriv_fiber_fderiv _ ((greenLogDensity_contDiff κ).differentiable (by simp))]
-    exact (greenLogDensity_spatial_derivative κ q x).deriv
-  have hrep := c.compact_stopped_field_ito hB hBc φ hφ hφc T τ hτ hbτ
-  filter_upwards [hrep, hB.eval_zero_ae_eq_zero] with ω hrepω hB0
-  have heq (n : ℕ) : stoppedFieldSum B T (fun ω t x => φ (c.Q t ω, x)) τ n ω =
-      ∑ i ∈ Finset.range (2^n),
-        (if dyadicTime T n i < τ ω then
-          greenLogCoefficient κ (c.Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω)
-        else 0) * (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω) := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [mesh_eq_dyadicTime]
-    split_ifs with ht
-    · rw [hφ1 _ _ (hinside ω _ ht.le).1]
-    · rfl
-  let V (t : ℝ) := fderiv ℝ φ (c.Q t ω, B t.toNNReal ω) (c.velocity t ω, 0)
-  let D (t : ℝ) := iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t.toNNReal ω)
-  have hpair : Continuous (fun t : ℝ => (c.Q t ω, B t.toNNReal ω)) :=
-    (c.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal)
-  have hVc : Continuous V := ((hφ.continuous_fderiv (by simp)).comp hpair).clm_apply
-    ((c.velocity_continuous hBc ω).prodMk continuous_const)
-  have hDc : Continuous D := by
-    simpa only [Function.comp_def, spatialPartial_eq_iteratedDeriv φ hφ] using
-      (spatialPartial_smooth φ hφ 2).continuous.comp hpair
-  have hz : (∫ t in (0 : ℝ)..(τ ω : ℝ), V t) +
-      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ), D t = 0 := by
-    rw [← intervalIntegral.integral_const_mul,
-      ← intervalIntegral.integral_add (hVc.intervalIntegrable _ _) ((hDc.const_mul _).intervalIntegrable _ _)]
-    calc
-      _ = ∫ t in (0 : ℝ)..(τ ω : ℝ), (0 : ℝ) := by
-        apply intervalIntegral.integral_congr
-        intro t ht
-        rw [uIcc_of_le (τ ω).coe_nonneg] at ht
-        have htn : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
-        obtain ⟨hball, hlo, hhi⟩ := hinside ω t.toNNReal htn
-        rw [Real.coe_toNNReal _ ht.1] at hball hlo hhi
-        exact compact_green_drift_zero κ hκ hκ8 lo hi hle φ _ _
-          (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
-      _ = 0 := by simp
-  have hend := hφeq _ _ (hinside ω (τ ω) le_rfl).1
-  have hstart := hφeq _ _ (hinside ω 0 zero_le).1
-  simp only [NNReal.coe_zero, c.initial ω, hB0] at hstart
-  have hval : φ (c.Q (τ ω) ω, B (τ ω) ω) - φ (c.Q 0 ω, B 0 ω) -
-      (∫ t in (0 : ℝ)..(τ ω : ℝ), V t) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ), D t =
-        greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) - greenLogDensity κ q₀ 0 := by
-    rw [hend, c.initial ω, hB0, hstart]
-    linarith
-  simpa only [heq, ← hval, V, D] using hrepω
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes
-
-lemma stopped_energy_tendsto {T : ℝ≥0} {S : ℝ≥0} (hS : S ≤ T)
-    {f : ℝ → ℝ} (hf : ContinuousOn f (Icc 0 T)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (if dyadicTime T n i < S then f (dyadicTime T n i) else 0) *
-        ((dyadicTime T n (i+1) : ℝ) - dyadicTime T n i)) atTop
-      (𝓝 (∫ t in (0 : ℝ)..(S : ℝ), f t)) := by
-  classical
-  rcases eq_or_lt_of_le (zero_le : 0 ≤ T) with hT | hT
-  · subst T
-    have hS0 : S = 0 := le_antisymm hS zero_le
-    simp [hS0, dyadicTime]
-  · have hh := mesh_prefix_riemann_tendsto T.coe_nonneg hf
-      (fun n => stoppedDyadicIndex T n S)
-      (fun n => stoppedDyadicIndex_le hT n hS)
-      (show (S : ℝ) ∈ Icc (0 : ℝ) T from ⟨S.coe_nonneg, by exact_mod_cast hS⟩)
-      (stoppedDyadicIndex_tendsto hT S)
-    have heq (n : ℕ) : (∑ i ∈ Finset.range (2^n),
-        (if dyadicTime T n i < S then f (dyadicTime T n i) else 0) *
-          ((dyadicTime T n (i+1) : ℝ) - dyadicTime T n i)) =
-        ∑ i ∈ Finset.range (stoppedDyadicIndex T n S), f (mesh T n i) *
-          ((T : ℝ) / (2 : ℝ)^n) := by
-      simp_rw [dyadicTime_step, ← lt_stoppedDyadicIndex hT, ite_mul, zero_mul]
-      rw [SLEExactGauge.prefix_mask_sum _ (stoppedDyadicIndex_le hT n hS)]
-      simp only [mesh_eq_dyadicTime]
-    simpa only [heq] using hh
-
-end SLEExactGauge
-
- 
-
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge
-
-lemma gaussian_exponential_lintegral (v : ℝ≥0) (c : ℝ) :
-    (∫⁻ x : ℝ, ENNReal.ofReal (Real.exp (c * x)) ∂gaussianReal 0 v) =
-      ENNReal.ofReal (Real.exp ((v : ℝ) * c ^ 2 / 2)) := by
-  rw [← ofReal_integral_eq_lintegral_ofReal (integrable_exp_mul_gaussianReal c)
-    (Eventually.of_forall fun _ => (Real.exp_pos _).le)]
-  congr 1
-  simpa only [mgf, zero_mul, zero_add] using congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := v)) c
-
-lemma gaussian_compensated_exponential (v : ℝ≥0) (a c : ℝ) :
-    (∫⁻ x : ℝ, ENNReal.ofReal (Real.exp (a + c * x - (v : ℝ) * c ^ 2 / 2)) ∂gaussianReal 0 v) =
-      ENNReal.ofReal (Real.exp a) := by
-  have he (x : ℝ) : Real.exp (a + c * x - (v : ℝ) * c ^ 2 / 2) =
-      Real.exp (a - (v : ℝ) * c ^ 2 / 2) * Real.exp (c * x) := by
-    rw [← Real.exp_add]
-    congr 1
-    ring
-  simp_rw [he, ENNReal.ofReal_mul (Real.exp_pos _).le]
-  rw [lintegral_const_mul _ (by fun_prop), gaussian_exponential_lintegral,
-    ← ENNReal.ofReal_mul (Real.exp_pos _).le, ← Real.exp_add]
-  congr 2
-  ring
-
- 
-
-lemma independent_gaussian_exponential
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    {X : Ω → ℝ × ℝ} {Y : Ω → ℝ} {v : ℝ≥0}
-    (hX : Measurable X) (hY : HasLaw Y (gaussianReal 0 v) μ)
-    (hi : IndepFun X Y μ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp ((X ω).1 + (X ω).2 * Y ω -
-      (v : ℝ) * (X ω).2 ^ 2 / 2)) ∂μ) =
-      ∫⁻ ω, ENNReal.ofReal (Real.exp (X ω).1) ∂μ := by
-  let f (p : (ℝ × ℝ) × ℝ) := ENNReal.ofReal
-    (Real.exp (p.1.1 + p.1.2 * p.2 - (v : ℝ) * p.1.2 ^ 2 / 2))
-  have hf : Measurable f := by unfold f; fun_prop
-  have hmap : μ.map (fun ω => (X ω, Y ω)) = (μ.map X).prod (gaussianReal 0 v) := by
-    rw [IndepFun.map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable hi, hY.map_eq]
-  calc
-    _ = ∫⁻ p, f p ∂μ.map (fun ω => (X ω, Y ω)) :=
-      (lintegral_map' hf.aemeasurable (hX.aemeasurable.prodMk hY.aemeasurable)).symm
-    _ = ∫⁻ p, ∫⁻ x, f (p, x) ∂gaussianReal 0 v ∂μ.map X := by
-      rw [hmap, lintegral_prod f hf.aemeasurable]
-    _ = ∫⁻ p, ENNReal.ofReal (Real.exp p.1) ∂μ.map X :=
-      lintegral_congr fun p => gaussian_compensated_exponential v p.1 p.2
-    _ = _ := lintegral_map' (by fun_prop) hX.aemeasurable
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def driverPredictableSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (t : ℕ → ℝ≥0) (C : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
-  ∑ i ∈ Finset.range n, C i ω * (B (t (i+1)) ω - B (t i) ω)
-
-noncomputable def driverPredictableEnergy {Ω : Type*} (t : ℕ → ℝ≥0)
-    (C : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
-  ∑ i ∈ Finset.range n, C i ω ^ 2 * ((t (i+1) : ℝ) - t i)
-
-lemma driverPredictableSum_measurable_past
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ s, Measurable (B s))
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t i)] (C i))
-    (n : ℕ) : Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t n)]
-      (driverPredictableSum B t C n) := by
-  have hB := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted
-  apply Finset.measurable_sum
-  intro i hi
-  have hin : i < n := Finset.mem_range.mp hi
-  exact ((hC i).mono (Filtration.mono _ (ht hin.le)) le_rfl).mul
-    ((hB.measurable_le (ht (Nat.succ_le_iff.mpr hin))).sub (hB.measurable_le (ht hin.le)))
-
-lemma driverPredictableEnergy_measurable_past
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ s, Measurable (B s))
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t i)] (C i))
-    (n : ℕ) : Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t n)]
-      (driverPredictableEnergy t C n) := by
-  apply Finset.measurable_sum
-  intro i hi
-  exact (((hC i).mono (Filtration.mono _ (ht (Finset.mem_range.mp hi).le)) le_rfl).pow_const 2).mul_const _
-
- 
-
-theorem driverPredictable_exponential
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ s, Measurable (B s))
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t i)] (C i))
-    (θ : ℝ) (n : ℕ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * driverPredictableSum B t C n ω -
-      θ ^ 2 * driverPredictableEnergy t C n ω / 2)) ∂P) = 1 := by
-  let := hB.isGaussianProcess.isProbabilityMeasure
-  induction n with
-  | zero => simp [driverPredictableSum, driverPredictableEnergy]
-  | succ n hprev =>
-    let a (ω : Ω) := θ * driverPredictableSum B t C n ω -
-      θ ^ 2 * driverPredictableEnergy t C n ω / 2
-    let X (ω : Ω) : ℝ × ℝ := (a ω, θ * C n ω)
-    let Y (ω : Ω) := B (t (n+1)) ω - B (t n) ω
-    let v := t (n+1) - t n
-    have hXp : Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t n)] X :=
-      (((driverPredictableSum_measurable_past B hm t ht C hC n).const_mul θ).sub
-        (((driverPredictableEnergy_measurable_past B hm t ht C hC n).const_mul (θ^2)).div_const 2)).prodMk
-          ((hC n).const_mul θ)
-    have hX : Measurable X := hXp.mono (Filtration.le _ _) le_rfl
-    have hY : HasLaw Y (gaussianReal 0 v) P := by
-      simpa only [Y, v, add_tsub_cancel_of_le (ht (Nat.le_succ n))] using
-        (hB.toIsPreBrownianReal.shift (t n)).hasLaw_eval v
-    have hi : IndepFun X Y P := by
-      rw [IndepFun_iff_Indep]
-      have hb := brownian_increment_indep_natural hB hm (t n) v
-      dsimp only [v] at hb
-      rw [add_tsub_cancel_of_le (ht (Nat.le_succ n))] at hb
-      exact indep_of_indep_of_le hb hXp.comap_le le_rfl
-    have hvc : (v : ℝ) = (t (n+1) : ℝ) - t n :=
-      NNReal.coe_sub (ht (Nat.le_succ n))
-    have he (ω : Ω) : θ * driverPredictableSum B t C (n+1) ω -
-        θ ^ 2 * driverPredictableEnergy t C (n+1) ω / 2 =
-        (X ω).1 + (X ω).2 * Y ω - (v : ℝ) * (X ω).2 ^ 2 / 2 := by
-      simp only [driverPredictableSum, driverPredictableEnergy, Finset.sum_range_succ,
-        X, a, Y, hvc]
-      ring
-    simp_rw [he]
-    rw [independent_gaussian_exponential hX hY hi]
-    exact hprev
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge
-
- 
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes
-
-lemma greenLogCoefficient_eq_deriv (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    greenLogCoefficient κ q x = deriv (greenLogDensity κ q) x :=
-  (greenLogDensity_spatial_derivative κ q x).deriv.symm
-
-lemma greenLogCoefficient_continuous (κ : ℝ) :
-    Continuous (fun p : (ℂ × ℝ) × ℝ => greenLogCoefficient κ p.1 p.2) := by
-  have hc := (spatialPartial_smooth
-    (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (greenLogDensity_contDiff κ) 1).continuous
-  convert hc using 1
-  funext p
-  rw [spatialPartial_eq_iteratedDeriv _ (greenLogDensity_contDiff κ), iteratedDeriv_one]
-  exact greenLogCoefficient_eq_deriv κ p.1 p.2
-
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-
-lemma ClippedLogSolution.green_coefficient_adapted
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (t : ℝ≥0) :
-    Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) t]
-      (fun ω => greenLogCoefficient κ (c.Q t ω) (B t ω)) := by
-  simpa only [iteratedDeriv_one, greenLogCoefficient_eq_deriv] using
-    c.partial_adapted (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2)
-      (greenLogDensity_contDiff κ) 1 t
-
- 
-
-theorem ClippedLogSolution.stopped_green_exponential
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (R : ℝ) (hR : 0 ≤ R)
-    (T : ℝ≥0) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi)
-    (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp
-      (θ * (greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) - greenLogDensity κ q₀ 0) -
-        θ ^ 2 * (∫ t in (0 : ℝ)..(τ ω : ℝ),
-          (greenLogCoefficient κ (c.Q t ω) (B t.toNNReal ω)) ^ 2) / 2)) ∂P) ≤ 1 := by
-  classical
-  let C (n i : ℕ) (ω : Ω) := if dyadicTime T n i < τ ω then
-    greenLogCoefficient κ (c.Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0
-  have hCa (n i : ℕ) : Measurable[Filtration.natural B
-      (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)] (C n i) :=
-    (c.green_coefficient_adapted _).ite (stopping_before_measurable hτ _) measurable_const
-  let X (n : ℕ) := driverPredictableSum B (dyadicTime T n) (C n) (2^n)
-  let E (n : ℕ) := driverPredictableEnergy (dyadicTime T n) (C n) (2^n)
-  have hXm (n : ℕ) : Measurable (X n) :=
-    (driverPredictableSum_measurable_past B hm _ (dyadicTime_mono T n) _ (hCa n) _).mono
-      (Filtration.le _ _) le_rfl
-  have hEm (n : ℕ) : Measurable (E n) :=
-    (driverPredictableEnergy_measurable_past B hm _ (dyadicTime_mono T n) _ (hCa n) _).mono
-      (Filtration.le _ _) le_rfl
-  have hx := c.stopped_green_ito hB hBc hκ hκ8 R hR T τ hτ hbτ hinside
-  have he : ∀ᵐ ω ∂P, Tendsto (fun n => E n ω) atTop
-      (𝓝 (∫ t in (0 : ℝ)..(τ ω : ℝ),
-        (greenLogCoefficient κ (c.Q t ω) (B t.toNNReal ω)) ^ 2)) := by
-    apply Eventually.of_forall
-    intro ω
-    have hc : Continuous (fun t : ℝ => greenLogCoefficient κ (c.Q t ω) (B t.toNNReal ω)) :=
-      (greenLogCoefficient_continuous κ).comp
-        ((c.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal))
-    have hl := stopped_energy_tendsto (hbτ ω) (hc.pow 2).continuousOn
-    simpa only [E, driverPredictableEnergy, C, ite_pow, zero_pow (by decide : 2 ≠ 0),
-      Pi.pow_apply, Real.toNNReal_coe] using hl
-  exact compensated_exponential_limit_energy X E _ _ θ
-    (fun n => (((hXm n).const_mul θ).sub (((hEm n).const_mul (θ^2)).div_const 2)).exp.ennreal_ofReal.aemeasurable)
-    hx he (fun n => (driverPredictable_exponential hB hm _ (dyadicTime_mono T n) _ (hCa n) θ _).le)
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ lo hi : ℝ} {hle : lo ≤ hi}
-
- 
-
-lemma clipped_family_initial_continuous
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (t : ℝ≥0) (ω : Ω) : Continuous (fun q => (c q).Q t ω) := by
-  obtain ⟨L, K, hb, hl⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let D : ℝ≥0 := ⟨Real.exp ((K : ℝ) * (t : ℝ)), (Real.exp_pos _).le⟩
-  apply (show LipschitzWith D (fun q => (c q).Q t ω) from ?_).continuous
-  apply LipschitzWith.of_dist_le_mul
-  intro q q'
-  have hd (x : ℂ × ℝ) (s : ℝ) (hs : s ∈ Ico (0 : ℝ) (t : ℝ)) :
-      HasDerivWithinAt (fun v => (c x).Q v ω)
-        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * B s.toNNReal ω) ((c x).Q s ω)) (Ici s) s :=
-    ((c x).derivative ω s hs.1).mono (Ici_subset_Ici.mpr hs.1)
-  have hh := dist_le_of_trajectories_ODE
-    (fun s => hl (Real.sqrt κ * B s.toNNReal ω))
-    ((c q).lipschitz ω).continuous.continuousOn (hd q)
-    ((c q').lipschitz ω).continuous.continuousOn (hd q')
-    (show dist ((c q).Q 0 ω) ((c q').Q 0 ω) ≤ dist q q' by rw [(c q).initial, (c q').initial])
-    (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩
-  change dist ((c q).Q t ω) ((c q').Q t ω) ≤ Real.exp ((K : ℝ) * (t : ℝ)) * dist q q'
-  simpa only [sub_zero, mul_comm] using hh
-
-lemma clipped_family_past_measurable
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q) (t : ℝ≥0) :
-    @Measurable ((ℂ × ℝ) × Ω) (ℂ × ℝ)
-      ((inferInstance : MeasurableSpace (ℂ × ℝ)).prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
-      (inferInstance : MeasurableSpace (ℂ × ℝ)) (fun p => (c p.1).Q t p.2) := by
-  have hc := clipped_family_initial_continuous c t
-  have ha (q : ℂ × ℝ) := (c q).adapted t
-  let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) t
-  exact measurable_uncurry_of_continuous_of_measurable hc ha
-
-lemma clipped_family_joint_measurable
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q) :
-    Measurable (fun p : ℝ≥0 × ((ℂ × ℝ) × Ω) => (c p.2.1).Q p.1 p.2.2) := by
-  change Measurable (Function.uncurry (fun (t : ℝ≥0) (p : (ℂ × ℝ) × Ω) => (c p.1).Q t p.2))
-  apply measurable_uncurry_of_continuous_of_measurable
-  · intro p
-    exact ((c p.1).lipschitz p.2).continuous.comp NNReal.continuous_coe
-  · intro t
-    have h := clipped_family_past_measurable c t
-    exact h.mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (Filtration.le _ _))) le_rfl
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-open L2Limit
-
-end SLEExactGauge
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.ItoIsometry
+
+namespace OAI
+
+/-! Simple Fields for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianIto TaylorMeshes
+lemma stoppedDyadicIndex_bounds {T : ℝ≥0} (hT : 0 < T) (n : ℕ) (t : ℝ≥0) :
+    t ≤ dyadicTime T n (stoppedDyadicIndex T n t) ∧
+    dyadicTime T n (stoppedDyadicIndex T n t) ≤ t + T / (2 : ℝ≥0)^n := by
+  have hd : 0 < T / (2 : ℝ≥0)^n := by positivity
+  constructor
+  · apply (div_le_iff₀ hd).mp
+    exact Nat.le_ceil _
+  · change (Nat.ceil (t / (T / (2 : ℝ≥0)^n)) : ℝ≥0) * (T / (2 : ℝ≥0)^n) ≤ _
+    calc
+      _ ≤ (t / (T / (2 : ℝ≥0)^n) + 1) * (T / (2 : ℝ≥0)^n) :=
+        mul_le_mul_of_nonneg_right (Nat.ceil_lt_add_one (show (0 : ℝ≥0) ≤ _ from zero_le)).le hd.le
+      _ = _ := by rw [add_mul, div_mul_cancel₀ _ hd.ne', one_mul]
+
+lemma stoppedDyadicIndex_tendsto {T : ℝ≥0} (hT : 0 < T) (t : ℝ≥0) :
+    Tendsto (fun n => mesh T n (stoppedDyadicIndex T n t)) atTop (𝓝 (t : ℝ)) := by
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
+    (by simpa only [add_zero] using (tendsto_const_nhds (x := (t : ℝ))).add (mesh_step_tendsto T))
+  · intro n
+    simpa only [← coe_dyadicTime] using
+      (show (t : ℝ) ≤ (dyadicTime T n (stoppedDyadicIndex T n t) : ℝ) by
+        exact_mod_cast (stoppedDyadicIndex_bounds hT n t).1)
+  · intro n
+    have hh := (stoppedDyadicIndex_bounds hT n t).2
+    simpa only [← coe_dyadicTime, NNReal.coe_add, NNReal.coe_div, NNReal.coe_pow, NNReal.coe_ofNat]
+      using (show (dyadicTime T n (stoppedDyadicIndex T n t) : ℝ) ≤
+        ((t + T / (2 : ℝ≥0)^n : ℝ≥0) : ℝ) by exact_mod_cast hh)
+
+lemma stopping_before_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {f : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0))) (t : ℝ≥0) :
+    MeasurableSet[f t] {ω | t < τ ω} := by
+  simpa only [Set.compl_ofPred, WithTop.coe_le_coe, not_le] using (hτ.measurableSet_le t).compl
+
+lemma stopped_coefficient_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {f : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
+    (hτ : IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0))) (t : ℝ≥0)
+    {C : Ω → ℝ} (hC : Measurable[f t] C) :
+    Measurable[f t] (fun ω => if t < τ ω then C ω else 0) := by
+  classical
+  exact hC.ite (stopping_before_measurable hτ t) measurable_const
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes TimeField
+
+lemma prefix_mask_sum {α : Type*} [AddCommMonoid α] (f : ℕ → α) {k N : ℕ} (hk : k ≤ N) :
+    (∑ i ∈ Finset.range N, if i < k then f i else 0) = ∑ i ∈ Finset.range k, f i := by
+  classical
+  calc
+    _ = ∑ i ∈ Finset.range k, if i < k then f i else 0 := by
+      symm
+      apply Finset.sum_subset (Finset.range_mono hk)
+      intro i hi hni
+      simp only [Finset.mem_range] at hni
+      simp [hni]
+    _ = _ := Finset.sum_congr rfl fun i hi => ite_eq_left (Finset.mem_range.mp hi)
+
+noncomputable def stoppedFieldSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
+    (F : Ω → ℝ → ℝ → ℝ) (τ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) : ℝ :=
+  ∑ i ∈ Finset.range (2^n), (if dyadicTime T n i < τ ω then
+    deriv (F ω (mesh T n i)) (B (dyadicTime T n i) ω) else 0) *
+      (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)
+
+lemma stoppedFieldSum_eq_prefix {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) {T : ℝ≥0}
+    (hT : 0 < T) (F : Ω → ℝ → ℝ → ℝ) (τ : Ω → ℝ≥0) (hτ : ∀ ω, τ ω ≤ T)
+    (n : ℕ) (ω : Ω) :
+    stoppedFieldSum B T F τ n ω =
+      ∑ i ∈ Finset.range (stoppedDyadicIndex T n (τ ω)),
+        deriv (F ω (mesh T n i)) (B (dyadicTime T n i) ω) *
+          (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω) := by
+  unfold stoppedFieldSum
+  simp_rw [← lt_stoppedDyadicIndex hT, ite_mul, zero_mul]
+  exact prefix_mask_sum _ (stoppedDyadicIndex_le hT n (hτ ω))
+
+ 
+
+theorem stopped_fieldSum_tendsto {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ s, Measurable (B s))
+    (T : ℝ≥0) (F V : Ω → ℝ → ℝ → ℝ) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T)
+    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) T)
+    (hcont : ∀ᵐ ω ∂P, ContinuousOn (fun t => F ω t (B t.toNNReal ω)) (Icc 0 T))
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)]
+      (fun ω => iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω)))
+    (hC2 : ∀ n i, MemLp
+      (fun ω => iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω)) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ ω ∂P,
+      |iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω)| ≤ K₂) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => stoppedFieldSum B T F τ n ω) atTop
+      (𝓝 (F ω (τ ω) (B (τ ω) ω) - F ω 0 (B 0 ω) -
+        (∫ t in (0 : ℝ)..(τ ω : ℝ), V ω t (B t.toNNReal ω)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ),
+          iteratedDeriv 2 (F ω t) (B t.toNNReal ω))) := by
+  classical
+  rcases eq_or_lt_of_le (zero_le : 0 ≤ T) with hT | hT
+  · subst T
+    have hτ0 (ω : Ω) : τ ω = 0 := le_antisymm (hbτ ω) zero_le
+    simp [stoppedFieldSum, dyadicTime, hτ0]
+  let C (n i : ℕ) (ω : Ω) := if dyadicTime T n i < τ ω then
+    iteratedDeriv 2 (F ω (mesh T n i)) (B (dyadicTime T n i) ω) else 0
+  have hs (n i : ℕ) : MeasurableSet[pastSigma B (dyadicTime T n i)]
+      {ω | dyadicTime T n i < τ ω} := by
+    have he := stopping_before_measurable hτ (dyadicTime T n i)
+    exact (le_of_eq (Filtration.natural_eq_comap B (fun t => (hm t).stronglyMeasurable) _)) _ he
+  have hCp (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) :=
+    (hC n i).ite (hs n i) measurable_const
+  have hC2p (n i : ℕ) : MemLp (C n i) 2 P := by
+    have hma := (hCp n i).mono
+      (le_trans (le_of_eq (Filtration.natural_eq_comap B
+        (fun t => (hm t).stronglyMeasurable) _).symm) (Filtration.le _ _)) le_rfl
+    apply MemLp.of_bound hma.aestronglyMeasurable K₂
+    filter_upwards [hbound n i] with ω hω
+    dsimp only [C]
+    split_ifs <;> simpa only [Real.norm_eq_abs, abs_zero] using (by assumption : _ ≤ K₂)
+  have hbp (n i : ℕ) : ∀ᵐ ω ∂P, |C n i ω| ≤ K₂ := by
+    filter_upwards [hbound n i] with ω hω
+    dsimp only [C]
+    split_ifs <;> simp_all
+  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T C hCp hC2p K₂ hK₂ hbp
+  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hcont] with ω hcω hQω heω hrω hfω
+  let b : ℝ → ℝ := fun t => B t.toNNReal ω
+  let k : ℕ → ℕ := fun n => stoppedDyadicIndex T n (τ ω)
+  have hk (n : ℕ) : k n ≤ 2^n := stoppedDyadicIndex_le hT n (hbτ ω)
+  have hS : (τ ω : ℝ) ∈ Icc (0 : ℝ) T := ⟨(τ ω).coe_nonneg, by exact_mod_cast hbτ ω⟩
+  have hlim : Tendsto (fun n => mesh T n (k n)) atTop (𝓝 (τ ω : ℝ)) := stoppedDyadicIndex_tendsto hT _
+  have hb : Continuous b := hcω.comp continuous_real_toNNReal
+  have hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
+    simpa only [b, mesh_toNNReal, quadraticSum] using hQω
+  obtain ⟨K₃, hK₃, h₃⟩ := hrω.third_bound
+  have hr := mesh_prefix_remainder_tendsto
+    (f := fun n i => F ω (mesh T n i)) (fun _ _ => hrω.smooth _) T.coe_nonneg
+    hb.continuousOn hK₃ (fun _ _ x => h₃ _ x) k hk hQ
+  have hriem := mesh_prefix_riemann_tendsto T.coe_nonneg hrω.second_cont k hk hS hlim
+  have htime := mesh_prefix_time_increment_tendsto (b := b) T.coe_nonneg hrω.time_cont
+    hrω.time_deriv hrω.drift_cont k hk hS hlim
+  let Q (n : ℕ) := ∑ i ∈ Finset.range (k n),
+    iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+      (b (mesh T n (i+1))-b (mesh T n i))^2
+  have hQweighted : Tendsto Q atTop
+      (𝓝 (∫ t in (0 : ℝ)..(τ ω : ℝ), iteratedDeriv 2 (F ω t) (b t))) := by
+    have heq (n : ℕ) : Q n = dyadicQuadraticError B T C n ω +
+        ∑ i ∈ Finset.range (k n), iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+          ((T : ℝ) / (2 : ℝ)^n) := by
+      have hmask : dyadicQuadraticError B T C n ω =
+          ∑ i ∈ Finset.range (k n),
+          iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ) / (2 : ℝ)^n) := by
+        simp only [dyadicQuadraticError, C, b, mesh_toNNReal]
+        simp_rw [← lt_stoppedDyadicIndex hT, ite_mul, zero_mul]
+        exact prefix_mask_sum _ (hk n)
+      rw [hmask, ← Finset.sum_add_distrib]
+      apply Finset.sum_congr rfl
+      intro i hi
+      ring
+    simpa only [b, ← heq, zero_add] using heω.add hriem
+  have hTel (n : ℕ) : ∑ i ∈ Finset.range (k n),
+      (F ω (mesh T n (i+1)) (b (mesh T n (i+1))) - F ω (mesh T n i) (b (mesh T n i))) =
+        F ω (mesh T n (k n)) (b (mesh T n (k n))) - F ω 0 (B 0 ω) := by
+    rw [Finset.sum_range_sub (fun i => F ω (mesh T n i) (b (mesh T n i))) (k n), mesh_zero]
+    simp only [b, Real.toNNReal_zero]
+  have hIdentity (n : ℕ) : stoppedFieldSum B T F τ n ω =
+      F ω (mesh T n (k n)) (b (mesh T n (k n))) - F ω 0 (B 0 ω) -
+        (∑ i ∈ Finset.range (k n),
+          (F ω (mesh T n (i+1)) (b (mesh T n (i+1))) -
+            F ω (mesh T n i) (b (mesh T n (i+1))))) - (1/2 : ℝ) * Q n -
+        ∑ i ∈ Finset.range (k n), remainder₂ (F ω (mesh T n i))
+          (b (mesh T n i)) (b (mesh T n (i+1))) := by
+    rw [← hTel n]
+    simp only [Q, Finset.mul_sum, ← Finset.sum_sub_distrib]
+    rw [stoppedFieldSum_eq_prefix B hT F τ hbτ]
+    apply Finset.sum_congr rfl
+    intro i hi
+    simp only [remainder₂, iteratedDeriv_succ, iteratedDeriv_zero, b, mesh_toNNReal]
+    ring
+  have hend := (hfω _ hS).tendsto.comp (tendsto_nhdsWithin_iff.mpr
+    ⟨hlim, Eventually.of_forall fun n => mesh_mem T T.coe_nonneg n (k n) (hk n)⟩)
+  have ht := (((hend.sub_const (F ω 0 (B 0 ω))).sub htime).sub
+    (hQweighted.const_mul (1/2 : ℝ))).sub hr
+  simpa only [Function.comp_apply, b, Real.toNNReal_coe, ← hIdentity, sub_zero] using ht
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+
+theorem ClippedLogSolution.compact_stopped_field_ito
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (φ : (ℂ × ℝ) × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ)
+    (hφc : HasCompactSupport φ) (T : ℝ≥0) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => stoppedFieldSum B T (fun ω t x => φ (c.Q t ω, x)) τ n ω) atTop
+      (𝓝 (φ (c.Q (τ ω) ω, B (τ ω) ω) - φ (c.Q 0 ω, B 0 ω) -
+        (∫ t in (0 : ℝ)..(τ ω : ℝ), fderiv ℝ φ (c.Q t ω, B t.toNNReal ω) (c.velocity t ω, 0)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ),
+          iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t.toNNReal ω))) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨K, hK, _⟩ := compact_smooth_bounded_lipschitz (spatialPartial φ 2)
+    (spatialPartial_smooth φ hφ 2) (spatialPartial_compact φ hφc 2)
+  have hb (t : ℝ≥0) (ω : Ω) :
+      |iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t ω)| ≤ K := by
+    simpa only [← spatialPartial_eq_iteratedDeriv φ hφ, Real.norm_eq_abs] using hK (c.Q t ω, B t ω)
+  have hma (t : ℝ≥0) := c.partial_adapted φ hφ 2 t
+  have hp (t : ℝ≥0) : MemLp
+      (fun ω => iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t ω)) 2 P := by
+    exact MemLp.of_bound ((hma t).mono (Filtration.le _ _) le_rfl).aestronglyMeasurable K
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hb t ω)
+  apply stopped_fieldSum_tendsto hB hm T (fun ω t x => φ (c.Q t ω, x))
+    (fun ω t x => fderiv ℝ φ (c.Q t ω, x) (c.velocity t ω, 0)) τ hτ hbτ
+    (Eventually.of_forall (c.compact_field_regular hBc φ hφ hφc T))
+    (Eventually.of_forall fun ω =>
+      (hφ.continuous.comp ((c.lipschitz ω).continuous.prodMk
+        ((hBc ω).comp continuous_real_toNNReal))).continuousOn)
+    (fun n i => ?_) (fun n i => ?_) K K.coe_nonneg (fun n i => ?_)
+  · have hx := (hma (dyadicTime T n i)).mono
+      (le_of_eq (natural_eq_pastSigma (B := B) (hm := hm) (dyadicTime T n i))) le_rfl
+    simpa only [mesh_eq_dyadicTime] using hx
+  · simpa only [mesh_eq_dyadicTime] using hp (dyadicTime T n i)
+  · exact Eventually.of_forall fun ω => by
+      simpa only [mesh_eq_dyadicTime] using hb (dyadicTime T n i) ω
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes
+
+lemma deriv_fiber_fderiv (φ : (ℂ × ℝ) × ℝ → ℝ)
+    (hφ : Differentiable ℝ φ) (q : ℂ × ℝ) (x : ℝ) :
+    deriv (fun y => φ (q, y)) x = fderiv ℝ φ (q, x) (0, 1) := by
+  exact ((hφ (q, x)).hasFDerivAt.comp_hasDerivAt x
+    ((hasDerivAt_const x q).prodMk (hasDerivAt_id x))).deriv
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+
+ 
+noncomputable def greenLogCoefficient (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  greenLogDensity κ q x * (greenExponent κ * Real.sqrt κ *
+    (q.1.re - Real.sqrt κ * x) / greenLogNormSq κ q x)
+
+ 
+
+theorem ClippedLogSolution.stopped_green_ito
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (R : ℝ) (hR : 0 ≤ R)
+    (T : ℝ≥0) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (if dyadicTime T n i < τ ω then
+        greenLogCoefficient κ (c.Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω)
+      else 0) * (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) atTop
+      (𝓝 (greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) - greenLogDensity κ q₀ 0)) := by
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
+  have hφ1 (q : ℂ × ℝ) (x : ℝ) (hq : ‖(q, x)‖ ≤ R) :
+      deriv (fun y => φ (q, y)) x = greenLogCoefficient κ q x := by
+    rw [deriv_fiber_fderiv φ (hφ.differentiable (by simp)), hφD q x hq,
+      ← deriv_fiber_fderiv _ ((greenLogDensity_contDiff κ).differentiable (by simp))]
+    exact (greenLogDensity_spatial_derivative κ q x).deriv
+  have hrep := c.compact_stopped_field_ito hB hBc φ hφ hφc T τ hτ hbτ
+  filter_upwards [hrep, hB.eval_zero_ae_eq_zero] with ω hrepω hB0
+  have heq (n : ℕ) : stoppedFieldSum B T (fun ω t x => φ (c.Q t ω, x)) τ n ω =
+      ∑ i ∈ Finset.range (2^n),
+        (if dyadicTime T n i < τ ω then
+          greenLogCoefficient κ (c.Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω)
+        else 0) * (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω) := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [mesh_eq_dyadicTime]
+    split_ifs with ht
+    · rw [hφ1 _ _ (hinside ω _ ht.le).1]
+    · rfl
+  let V (t : ℝ) := fderiv ℝ φ (c.Q t ω, B t.toNNReal ω) (c.velocity t ω, 0)
+  let D (t : ℝ) := iteratedDeriv 2 (fun x => φ (c.Q t ω, x)) (B t.toNNReal ω)
+  have hpair : Continuous (fun t : ℝ => (c.Q t ω, B t.toNNReal ω)) :=
+    (c.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal)
+  have hVc : Continuous V := ((hφ.continuous_fderiv (by simp)).comp hpair).clm_apply
+    ((c.velocity_continuous hBc ω).prodMk continuous_const)
+  have hDc : Continuous D := by
+    simpa only [Function.comp_def, spatialPartial_eq_iteratedDeriv φ hφ] using
+      (spatialPartial_smooth φ hφ 2).continuous.comp hpair
+  have hz : (∫ t in (0 : ℝ)..(τ ω : ℝ), V t) +
+      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ), D t = 0 := by
+    rw [← intervalIntegral.integral_const_mul,
+      ← intervalIntegral.integral_add (hVc.intervalIntegrable _ _) ((hDc.const_mul _).intervalIntegrable _ _)]
+    calc
+      _ = ∫ t in (0 : ℝ)..(τ ω : ℝ), (0 : ℝ) := by
+        apply intervalIntegral.integral_congr
+        intro t ht
+        rw [uIcc_of_le (τ ω).coe_nonneg] at ht
+        have htn : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
+        obtain ⟨hball, hlo, hhi⟩ := hinside ω t.toNNReal htn
+        rw [Real.coe_toNNReal _ ht.1] at hball hlo hhi
+        exact compact_green_drift_zero κ hκ hκ8 lo hi hle φ _ _
+          (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
+      _ = 0 := by simp
+  have hend := hφeq _ _ (hinside ω (τ ω) le_rfl).1
+  have hstart := hφeq _ _ (hinside ω 0 zero_le).1
+  simp only [NNReal.coe_zero, c.initial ω, hB0] at hstart
+  have hval : φ (c.Q (τ ω) ω, B (τ ω) ω) - φ (c.Q 0 ω, B 0 ω) -
+      (∫ t in (0 : ℝ)..(τ ω : ℝ), V t) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω : ℝ), D t =
+        greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) - greenLogDensity κ q₀ 0 := by
+    rw [hend, c.initial ω, hB0, hstart]
+    linarith
+  simpa only [heq, ← hval, V, D] using hrepω
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes
+
+lemma stopped_energy_tendsto {T : ℝ≥0} {S : ℝ≥0} (hS : S ≤ T)
+    {f : ℝ → ℝ} (hf : ContinuousOn f (Icc 0 T)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (if dyadicTime T n i < S then f (dyadicTime T n i) else 0) *
+        ((dyadicTime T n (i+1) : ℝ) - dyadicTime T n i)) atTop
+      (𝓝 (∫ t in (0 : ℝ)..(S : ℝ), f t)) := by
+  classical
+  rcases eq_or_lt_of_le (zero_le : 0 ≤ T) with hT | hT
+  · subst T
+    have hS0 : S = 0 := le_antisymm hS zero_le
+    simp [hS0, dyadicTime]
+  · have hh := mesh_prefix_riemann_tendsto T.coe_nonneg hf
+      (fun n => stoppedDyadicIndex T n S)
+      (fun n => stoppedDyadicIndex_le hT n hS)
+      (show (S : ℝ) ∈ Icc (0 : ℝ) T from ⟨S.coe_nonneg, by exact_mod_cast hS⟩)
+      (stoppedDyadicIndex_tendsto hT S)
+    have heq (n : ℕ) : (∑ i ∈ Finset.range (2^n),
+        (if dyadicTime T n i < S then f (dyadicTime T n i) else 0) *
+          ((dyadicTime T n (i+1) : ℝ) - dyadicTime T n i)) =
+        ∑ i ∈ Finset.range (stoppedDyadicIndex T n S), f (mesh T n i) *
+          ((T : ℝ) / (2 : ℝ)^n) := by
+      simp_rw [dyadicTime_step, ← lt_stoppedDyadicIndex hT, ite_mul, zero_mul]
+      rw [SLEExactGauge.prefix_mask_sum _ (stoppedDyadicIndex_le hT n hS)]
+      simp only [mesh_eq_dyadicTime]
+    simpa only [heq] using hh
+
+end SLEExactGauge
+
+ 
+
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge
+
+lemma gaussian_exponential_lintegral (v : ℝ≥0) (c : ℝ) :
+    (∫⁻ x : ℝ, ENNReal.ofReal (Real.exp (c * x)) ∂gaussianReal 0 v) =
+      ENNReal.ofReal (Real.exp ((v : ℝ) * c ^ 2 / 2)) := by
+  rw [← ofReal_integral_eq_lintegral_ofReal (integrable_exp_mul_gaussianReal c)
+    (Eventually.of_forall fun _ => (Real.exp_pos _).le)]
+  congr 1
+  simpa only [mgf, zero_mul, zero_add] using congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := v)) c
+
+lemma gaussian_compensated_exponential (v : ℝ≥0) (a c : ℝ) :
+    (∫⁻ x : ℝ, ENNReal.ofReal (Real.exp (a + c * x - (v : ℝ) * c ^ 2 / 2)) ∂gaussianReal 0 v) =
+      ENNReal.ofReal (Real.exp a) := by
+  have he (x : ℝ) : Real.exp (a + c * x - (v : ℝ) * c ^ 2 / 2) =
+      Real.exp (a - (v : ℝ) * c ^ 2 / 2) * Real.exp (c * x) := by
+    rw [← Real.exp_add]
+    congr 1
+    ring
+  simp_rw [he, ENNReal.ofReal_mul (Real.exp_pos _).le]
+  rw [lintegral_const_mul _ (by fun_prop), gaussian_exponential_lintegral,
+    ← ENNReal.ofReal_mul (Real.exp_pos _).le, ← Real.exp_add]
+  congr 2
+  ring
+
+ 
+
+lemma independent_gaussian_exponential
+    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    {X : Ω → ℝ × ℝ} {Y : Ω → ℝ} {v : ℝ≥0}
+    (hX : Measurable X) (hY : HasLaw Y (gaussianReal 0 v) μ)
+    (hi : IndepFun X Y μ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp ((X ω).1 + (X ω).2 * Y ω -
+      (v : ℝ) * (X ω).2 ^ 2 / 2)) ∂μ) =
+      ∫⁻ ω, ENNReal.ofReal (Real.exp (X ω).1) ∂μ := by
+  let f (p : (ℝ × ℝ) × ℝ) := ENNReal.ofReal
+    (Real.exp (p.1.1 + p.1.2 * p.2 - (v : ℝ) * p.1.2 ^ 2 / 2))
+  have hf : Measurable f := by unfold f; fun_prop
+  have hmap : μ.map (fun ω => (X ω, Y ω)) = (μ.map X).prod (gaussianReal 0 v) := by
+    rw [IndepFun.map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable hi, hY.map_eq]
+  calc
+    _ = ∫⁻ p, f p ∂μ.map (fun ω => (X ω, Y ω)) :=
+      (lintegral_map' hf.aemeasurable (hX.aemeasurable.prodMk hY.aemeasurable)).symm
+    _ = ∫⁻ p, ∫⁻ x, f (p, x) ∂gaussianReal 0 v ∂μ.map X := by
+      rw [hmap, lintegral_prod f hf.aemeasurable]
+    _ = ∫⁻ p, ENNReal.ofReal (Real.exp p.1) ∂μ.map X :=
+      lintegral_congr fun p => gaussian_compensated_exponential v p.1 p.2
+    _ = _ := lintegral_map' (by fun_prop) hX.aemeasurable
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def driverPredictableSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (t : ℕ → ℝ≥0) (C : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
+  ∑ i ∈ Finset.range n, C i ω * (B (t (i+1)) ω - B (t i) ω)
+
+noncomputable def driverPredictableEnergy {Ω : Type*} (t : ℕ → ℝ≥0)
+    (C : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
+  ∑ i ∈ Finset.range n, C i ω ^ 2 * ((t (i+1) : ℝ) - t i)
+
+lemma driverPredictableSum_measurable_past
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ s, Measurable (B s))
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t i)] (C i))
+    (n : ℕ) : Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t n)]
+      (driverPredictableSum B t C n) := by
+  have hB := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted
+  apply Finset.measurable_sum
+  intro i hi
+  have hin : i < n := Finset.mem_range.mp hi
+  exact ((hC i).mono (Filtration.mono _ (ht hin.le)) le_rfl).mul
+    ((hB.measurable_le (ht (Nat.succ_le_iff.mpr hin))).sub (hB.measurable_le (ht hin.le)))
+
+lemma driverPredictableEnergy_measurable_past
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ s, Measurable (B s))
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t i)] (C i))
+    (n : ℕ) : Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t n)]
+      (driverPredictableEnergy t C n) := by
+  apply Finset.measurable_sum
+  intro i hi
+  exact (((hC i).mono (Filtration.mono _ (ht (Finset.mem_range.mp hi).le)) le_rfl).pow_const 2).mul_const _
+
+ 
+
+theorem driverPredictable_exponential
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ s, Measurable (B s))
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t i)] (C i))
+    (θ : ℝ) (n : ℕ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * driverPredictableSum B t C n ω -
+      θ ^ 2 * driverPredictableEnergy t C n ω / 2)) ∂P) = 1 := by
+  let := hB.isGaussianProcess.isProbabilityMeasure
+  induction n with
+  | zero => simp [driverPredictableSum, driverPredictableEnergy]
+  | succ n hprev =>
+    let a (ω : Ω) := θ * driverPredictableSum B t C n ω -
+      θ ^ 2 * driverPredictableEnergy t C n ω / 2
+    let X (ω : Ω) : ℝ × ℝ := (a ω, θ * C n ω)
+    let Y (ω : Ω) := B (t (n+1)) ω - B (t n) ω
+    let v := t (n+1) - t n
+    have hXp : Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) (t n)] X :=
+      (((driverPredictableSum_measurable_past B hm t ht C hC n).const_mul θ).sub
+        (((driverPredictableEnergy_measurable_past B hm t ht C hC n).const_mul (θ^2)).div_const 2)).prodMk
+          ((hC n).const_mul θ)
+    have hX : Measurable X := hXp.mono (Filtration.le _ _) le_rfl
+    have hY : HasLaw Y (gaussianReal 0 v) P := by
+      simpa only [Y, v, add_tsub_cancel_of_le (ht (Nat.le_succ n))] using
+        (hB.toIsPreBrownianReal.shift (t n)).hasLaw_eval v
+    have hi : IndepFun X Y P := by
+      rw [IndepFun_iff_Indep]
+      have hb := brownian_increment_indep_natural hB hm (t n) v
+      dsimp only [v] at hb
+      rw [add_tsub_cancel_of_le (ht (Nat.le_succ n))] at hb
+      exact indep_of_indep_of_le hb hXp.comap_le le_rfl
+    have hvc : (v : ℝ) = (t (n+1) : ℝ) - t n :=
+      NNReal.coe_sub (ht (Nat.le_succ n))
+    have he (ω : Ω) : θ * driverPredictableSum B t C (n+1) ω -
+        θ ^ 2 * driverPredictableEnergy t C (n+1) ω / 2 =
+        (X ω).1 + (X ω).2 * Y ω - (v : ℝ) * (X ω).2 ^ 2 / 2 := by
+      simp only [driverPredictableSum, driverPredictableEnergy, Finset.sum_range_succ,
+        X, a, Y, hvc]
+      ring
+    simp_rw [he]
+    rw [independent_gaussian_exponential hX hY hi]
+    exact hprev
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge
+
+ 
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes
+
+lemma greenLogCoefficient_eq_deriv (κ : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    greenLogCoefficient κ q x = deriv (greenLogDensity κ q) x :=
+  (greenLogDensity_spatial_derivative κ q x).deriv.symm
+
+lemma greenLogCoefficient_continuous (κ : ℝ) :
+    Continuous (fun p : (ℂ × ℝ) × ℝ => greenLogCoefficient κ p.1 p.2) := by
+  have hc := (spatialPartial_smooth
+    (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2) (greenLogDensity_contDiff κ) 1).continuous
+  convert hc using 1
+  funext p
+  rw [spatialPartial_eq_iteratedDeriv _ (greenLogDensity_contDiff κ), iteratedDeriv_one]
+  exact greenLogCoefficient_eq_deriv κ p.1 p.2
+
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+
+lemma ClippedLogSolution.green_coefficient_adapted
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (t : ℝ≥0) :
+    Measurable[Filtration.natural B (fun s => (hm s).stronglyMeasurable) t]
+      (fun ω => greenLogCoefficient κ (c.Q t ω) (B t ω)) := by
+  simpa only [iteratedDeriv_one, greenLogCoefficient_eq_deriv] using
+    c.partial_adapted (fun p : (ℂ × ℝ) × ℝ => greenLogDensity κ p.1 p.2)
+      (greenLogDensity_contDiff κ) 1 t
+
+ 
+
+theorem ClippedLogSolution.stopped_green_exponential
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (R : ℝ) (hR : 0 ≤ R)
+    (T : ℝ≥0) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0))) (hbτ : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi)
+    (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp
+      (θ * (greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) - greenLogDensity κ q₀ 0) -
+        θ ^ 2 * (∫ t in (0 : ℝ)..(τ ω : ℝ),
+          (greenLogCoefficient κ (c.Q t ω) (B t.toNNReal ω)) ^ 2) / 2)) ∂P) ≤ 1 := by
+  classical
+  let C (n i : ℕ) (ω : Ω) := if dyadicTime T n i < τ ω then
+    greenLogCoefficient κ (c.Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0
+  have hCa (n i : ℕ) : Measurable[Filtration.natural B
+      (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)] (C n i) :=
+    (c.green_coefficient_adapted _).ite (stopping_before_measurable hτ _) measurable_const
+  let X (n : ℕ) := driverPredictableSum B (dyadicTime T n) (C n) (2^n)
+  let E (n : ℕ) := driverPredictableEnergy (dyadicTime T n) (C n) (2^n)
+  have hXm (n : ℕ) : Measurable (X n) :=
+    (driverPredictableSum_measurable_past B hm _ (dyadicTime_mono T n) _ (hCa n) _).mono
+      (Filtration.le _ _) le_rfl
+  have hEm (n : ℕ) : Measurable (E n) :=
+    (driverPredictableEnergy_measurable_past B hm _ (dyadicTime_mono T n) _ (hCa n) _).mono
+      (Filtration.le _ _) le_rfl
+  have hx := c.stopped_green_ito hB hBc hκ hκ8 R hR T τ hτ hbτ hinside
+  have he : ∀ᵐ ω ∂P, Tendsto (fun n => E n ω) atTop
+      (𝓝 (∫ t in (0 : ℝ)..(τ ω : ℝ),
+        (greenLogCoefficient κ (c.Q t ω) (B t.toNNReal ω)) ^ 2)) := by
+    apply Eventually.of_forall
+    intro ω
+    have hc : Continuous (fun t : ℝ => greenLogCoefficient κ (c.Q t ω) (B t.toNNReal ω)) :=
+      (greenLogCoefficient_continuous κ).comp
+        ((c.lipschitz ω).continuous.prodMk ((hBc ω).comp continuous_real_toNNReal))
+    have hl := stopped_energy_tendsto (hbτ ω) (hc.pow 2).continuousOn
+    simpa only [E, driverPredictableEnergy, C, ite_pow, zero_pow (by decide : 2 ≠ 0),
+      Pi.pow_apply, Real.toNNReal_coe] using hl
+  exact compensated_exponential_limit_energy X E _ _ θ
+    (fun n => (((hXm n).const_mul θ).sub (((hEm n).const_mul (θ^2)).div_const 2)).exp.ennreal_ofReal.aemeasurable)
+    hx he (fun n => (driverPredictable_exponential hB hm _ (dyadicTime_mono T n) _ (hCa n) θ _).le)
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ lo hi : ℝ} {hle : lo ≤ hi}
+
+ 
+
+lemma clipped_family_initial_continuous
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (t : ℝ≥0) (ω : Ω) : Continuous (fun q => (c q).Q t ω) := by
+  obtain ⟨L, K, hb, hl⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let D : ℝ≥0 := ⟨Real.exp ((K : ℝ) * (t : ℝ)), (Real.exp_pos _).le⟩
+  apply (show LipschitzWith D (fun q => (c q).Q t ω) from ?_).continuous
+  apply LipschitzWith.of_dist_le_mul
+  intro q q'
+  have hd (x : ℂ × ℝ) (s : ℝ) (hs : s ∈ Ico (0 : ℝ) (t : ℝ)) :
+      HasDerivWithinAt (fun v => (c x).Q v ω)
+        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * B s.toNNReal ω) ((c x).Q s ω)) (Ici s) s :=
+    ((c x).derivative ω s hs.1).mono (Ici_subset_Ici.mpr hs.1)
+  have hh := dist_le_of_trajectories_ODE
+    (fun s => hl (Real.sqrt κ * B s.toNNReal ω))
+    ((c q).lipschitz ω).continuous.continuousOn (hd q)
+    ((c q').lipschitz ω).continuous.continuousOn (hd q')
+    (show dist ((c q).Q 0 ω) ((c q').Q 0 ω) ≤ dist q q' by rw [(c q).initial, (c q').initial])
+    (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩
+  change dist ((c q).Q t ω) ((c q').Q t ω) ≤ Real.exp ((K : ℝ) * (t : ℝ)) * dist q q'
+  simpa only [sub_zero, mul_comm] using hh
+
+lemma clipped_family_past_measurable
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q) (t : ℝ≥0) :
+    @Measurable ((ℂ × ℝ) × Ω) (ℂ × ℝ)
+      ((inferInstance : MeasurableSpace (ℂ × ℝ)).prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
+      (inferInstance : MeasurableSpace (ℂ × ℝ)) (fun p => (c p.1).Q t p.2) := by
+  have hc := clipped_family_initial_continuous c t
+  have ha (q : ℂ × ℝ) := (c q).adapted t
+  let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) t
+  exact measurable_uncurry_of_continuous_of_measurable hc ha
+
+lemma clipped_family_joint_measurable
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q) :
+    Measurable (fun p : ℝ≥0 × ((ℂ × ℝ) × Ω) => (c p.2.1).Q p.1 p.2.2) := by
+  change Measurable (Function.uncurry (fun (t : ℝ≥0) (p : (ℂ × ℝ) × Ω) => (c p.1).Q t p.2))
+  apply measurable_uncurry_of_continuous_of_measurable
+  · intro p
+    exact ((c p.1).lipschitz p.2).continuous.comp NNReal.continuous_coe
+  · intro t
+    have h := clipped_family_past_measurable c t
+    exact h.mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (Filtration.le _ _))) le_rfl
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+open L2Limit
+
+end SLEExactGauge
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SimplePastMesh.lean b/lean/OAI/Probability/SLE/SimplePastMesh.lean
index a544c7c25d84a02545d7fa4b6a6ab29ed026b953..cd46e3de4db5512e8081bde603791ac4e50df344 100644
--- a/lean/OAI/Probability/SLE/SimplePastMesh.lean
+++ b/lean/OAI/Probability/SLE/SimplePastMesh.lean
@@ -1,789 +1,789 @@
-import Mathlib
-import OAI.Probability.SLE.SmallKappaSausageMoments
-import OAI.Probability.SLE.TracePosCombined
-
-namespace OAI
-
-/-! Simple Past Mesh for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem source_past_half_plane_mesh_tail [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
-    (q ε δ R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hε : 0 < ε)
-    (hεδ : ε ≤ δ) (hδR : δ ≤ R)
-    (hheight : ∀ z ∈ A, δ ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
-    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z δ)
-    :
-    ((P.comap Subtype.val)[|D]) {ω : a.samples |
-      ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A} ≤
-      ENNReal.ofReal (Real.exp (-(1/(2*(meshTailAmplitude κ q A)^2))*δ^(-2 : ℝ))) := by
-  obtain ⟨N,r, hqr, hrδ, he⟩ := exists_exact_geometric_mesh hq hq1 hε hεδ
-  have hδ : 0 < δ := hε.trans_le hεδ
-  have hr : 0 < r := (mul_pos hq hδ).trans hqr
-  have hκ8 : κ < 8 := by linarith
-  have hp : 0 < codimension κ := codimension_pos κ hκ8
-  have hα : 0 < 1-codimension κ := sub_pos.mpr (codimension_lt_one κ hκ)
-  have hd : 0 < 1-q^(1-codimension κ) := sub_pos.mpr (Real.rpow_lt_one hq.le hq1 hα)
-  have hB := meshTailAmplitude_pos A hκ hq hq1
-  have ham : slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)) ≤
-      meshTailAmplitude κ q A*δ^(1-codimension κ) := by
-    calc
-      _ ≤ slabAmplitude κ q A*δ^(1-codimension κ)/(1-q^(1-codimension κ)) :=
-        div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left
-          (Real.rpow_le_rpow hr.le hrδ hα.le) (slabAmplitude_nonneg κ q A)) hd.le
-      _ ≤ _ := by
-        dsimp [meshTailAmplitude]
-        have hx := (Real.rpow_pos_of_pos hδ (1-codimension κ)).le
-        rw [add_mul, div_mul_eq_mul_div]
-        nlinarith
-  let B : ℝ≥0 := NNReal.mk (meshTailAmplitude κ q A*δ^(1-codimension κ))
-    (mul_nonneg hB.le (Real.rpow_nonneg hδ.le _))
-  have hsg := a.source_past_spatial_geometric_subgaussian hκ hκ4 A hA q r R hq hq1 hr
-    (hrδ.trans hδR) (fun z hz => hrδ.trans (hheight z hz)) hbound N s D hD (fun ω hω z hz => (hbefore ω hω z hz).trans
-      (spatialHitTime_antitone_radius (a.trace_continuous ω) z hrδ (by
-        rw [(a.witness ω).2.1,dist_zero_left]
-        exact (hheight z hz).trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z)))))
-  have hsg' : HasSubgaussianMGF (fun ω : a.samples =>
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
-      (B^2) ((P.comap Subtype.val)[|D]) := by
-    rw [he]
-    apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsg
-    apply pow_le_pow_left₀ (by positivity)
-    exact NNReal.coe_le_coe.mp ham
-  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ closedBall 0 R from
-    fun z hz => by simpa only [mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
-      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
-  have hstart (ω : a.samples) :
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A ≤
-      (volume A).toReal*q^(-codimension κ)*δ^(-codimension κ) := by
-    apply (a.spatialRealMass_bound hκ hκ8 r hr A hA hfin
-      (fun z hz => hrδ.trans (hheight z hz)) ω).2.trans
-    have hh := Real.rpow_le_rpow_of_nonpos (mul_pos hq hδ) hqr.le (neg_nonpos.mpr hp.le)
-    rw [Real.mul_rpow hq.le hδ.le] at hh
-    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh ENNReal.toReal_nonneg
-  apply (measure_mono (show _ ⊆ {ω : a.samples | δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A} from ?_)).trans
-  · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (Real.exp_pos _).le).mpr
-    have ht := hsg'.measure_ge_le (Real.rpow_nonneg hδ.le (-codimension κ))
-    change ((P.comap Subtype.val)[|D]).real _ ≤ _
-    simpa only [B, NNReal.coe_pow, NNReal.coe_mk, mesh_tail_exponent hδ hB] using ht
-  · intro ω hω
-    have := hstart ω
-    change ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A at hω
-    change δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A
-    nlinarith
-
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourceSimpleEntranceMeshTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ ≤ 4 →
-  ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < 1 ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 4 ≤ z₀.im →
-  ∀ rf δ : ℝ, 0 < rf → rf ≤ δ → δ ≤ δ₀ →
-    (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ 1)} ≤
-      (P.comap Subtype.val) {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} *
-        ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ)))
-theorem sourceSimpleEntranceMesh_proved : SourceSimpleEntranceMeshTarget := by
-  intro κ hκ hκ4
-  let C := max 1 ((volume (ball (0:ℂ) 1)).toReal*(1/2:ℝ)^(-codimension κ)+1)
-  let c := 1/(2*(meshTailAmplitude κ (1/2) (ball (0:ℂ) 1))^2)
-  have hC1 : 1 ≤ C := le_max_left _ _
-  have hC : 0 < C := zero_lt_one.trans_le hC1
-  have hc : 0 < c := by
-    have hh := meshTailAmplitude_pos (κ := κ) (q := (1/2:ℝ)) (ball (0:ℂ) 1) hκ (by norm_num) (by norm_num)
-    exact div_pos zero_lt_one (mul_pos (by norm_num) (sq_pos_of_pos hh))
-  refine ⟨C,c,1/2,hC,hc,by norm_num,by norm_num,?_⟩
-  intro Ω m P hP γ a z₀ hz₀ rf δ hf hfd hd
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let H := {ω : a.samples | C*δ^(-codimension κ) < spatialRealMass κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)}
-  have hH : MeasurableSet H := measurableSet_lt measurable_const (a.spatialRealMass_measurable _ _)
-  have hδ : 0 < δ := hf.trans_le hfd
-  have hδ1 : δ ≤ 1 := hd.trans (by norm_num)
-  have hrf1 : rf ≤ 1 := hfd.trans hδ1
-  have hHA : H ⊆ {ω | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} := by
-    intro ω hω hn
-    have hm : spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1) = 0 := by
-      unfold spatialRealMass
-      apply setIntegral_eq_zero_of_forall_eq_zero
-      exact fun z hz => a.source_simple_weight_zero_no_entrance z₀ hz₀ rf hrf1 ω hn z hz
-    change C*δ^(-codimension κ) < _ at hω
-    rw [hm] at hω
-    exact (not_lt_of_ge (mul_nonneg hC.le (Real.rpow_nonneg hδ.le _))) hω
-  apply source_between_stopping_tail (a.source_simple_sphere_stopping z₀ 3)
-    (a.source_simple_sphere_stopping z₀ 2) H hH hHA
-    (a.source_simple_spheres_separated z₀ hz₀) (ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))))
-  intro s D hD hbefore
-  have hDm : MeasurableSet D := (a.filtration.le s) _ hD
-  have hheight : ∀ z ∈ ball z₀ 1, δ ≤ z.im := by
-    intro z hz
-    have he := (le_abs_self (z₀.im-z.im)).trans (Complex.abs_im_le_norm (z₀-z))
-    have hn : ‖z₀-z‖ < 1 := by simpa only [mem_ball,dist_eq_norm, norm_sub_rev] using hz
-    linarith
-  have hbound : ∀ z ∈ ball z₀ 1, ‖z‖ ≤ ‖z₀‖+1 := by
-    intro z hz
-    have hh := norm_le_insert z₀ z
-    rw [norm_sub_rev z₀ z] at hh
-    have hn : ‖z-z₀‖ < 1 := by simpa only [mem_ball, dist_eq_norm] using hz
-    linarith
-  have hR : δ ≤ ‖z₀‖+1 := hδ1.trans (by linarith [norm_nonneg z₀])
-  have ht := a.source_past_half_plane_mesh_tail hκ hκ4 (ball z₀ 1) measurableSet_ball
-    (1/2) rf δ (‖z₀‖+1) (by norm_num) (by norm_num) hf hfd hR hheight hbound s D hD
-    (fun ω hω z hz => (hbefore ω hω).trans (a.source_simple_center_before z₀ hz₀ δ hδ1 ω z hz))
-  have hv : volume (ball z₀ 1) = volume (ball (0:ℂ) 1) := by rw [Complex.volume_ball,Complex.volume_ball]
-  have ha : meshTailAmplitude κ (1/2) (ball z₀ 1) = meshTailAmplitude κ (1/2) (ball (0:ℂ) 1) := by
-    simp only [meshTailAmplitude,slabAmplitude,hv]
-  have hcoeff : (volume (ball z₀ 1)).toReal*(1/2:ℝ)^(-codimension κ)+1 ≤ C := by
-    rw [hv]; exact le_max_right _ _
-  have ht' : μ[H|D] ≤ ENNReal.ofReal (Real.exp (-c*δ^(-2:ℝ))) := by
-    apply (measure_mono (show H ⊆ _ from ?_)).trans
-    · simpa only [ha] using ht
-    · intro ω hω
-      exact (mul_le_mul_of_nonneg_right hcoeff (Real.rpow_nonneg hδ.le _)).trans hω.le
-  calc
-    μ (D ∩ H) = μ[H|D]*μ D := (cond_mul_eq_inter hDm H μ).symm
-    _ ≤ ENNReal.ofReal (Real.exp (-c*δ^(-2:ℝ)))*μ D := mul_le_mul' ht' le_rfl
-    _ ≤ μ D*ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
-      rw [mul_comm]
-      apply mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal ?_)
-      exact le_mul_of_one_le_left (Real.exp_pos _).le hC1
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_simple_entrance_bound {Ω : Type} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 4 ≤ z₀.im) (hsmall : 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4))) :
-    (P.comap Subtype.val).real {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} ≤
-      (uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ)*initialGreen κ z₀ := by
-  apply le_trans (measureReal_mono (show {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} ⊆
-      {ω | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4} from ?_))
-  · have hh := euclidean_approach a.brownian κ hκ hκ8 (fun ω : a.samples => γ ω) a.G a.F
-      (Eventually.of_forall a.witness) z₀ (by change 0 < z₀.im; linarith) 4 (by norm_num) hz hsmall
-    change ((P.comap Subtype.val) {ω : a.samples | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4}).toReal ≤ _
-    calc
-      _ ≤ _ := hh
-      _ = _ := by ring
-  · intro ω hω
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hω
-    have hs : 3 ≤ dist (γ ω 0) z₀ := by
-      rw [(a.witness ω).2.1,dist_zero_left]
-      have := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
-      linarith
-    exact ⟨t,by rw [spatialHitTime_mem (a.trace_continuous ω) z₀ 3 t ht.symm]; norm_num⟩
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-def SourceSimpleStretchedTailTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ ≤ 4 →
-  ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
-    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | M*u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ 1)} ≤
-      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
-theorem sourceSimpleStretchedTail_proved : SourceSimpleStretchedTailTarget := by
-  intro κ hκ hκ4
-  have hκ8 : κ < 8 := by linarith
-  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := sourceSimpleEntranceMesh_proved κ hκ hκ4
-  let v := (volume (ball (0:ℂ) 1)).toReal
-  let M := max C v*δ₀^(-codimension κ)
-  let b := c*δ₀^(-2:ℝ)
-  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
-  have hk : 0 < κ := by linarith
-  have hp : 0 < codimension κ := codimension_pos κ hκ8
-  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hδ _)
-  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hδ _)
-  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
-    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
-  refine ⟨M,A*C,b,hM,mul_pos hA hC,hb,?_⟩
-  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) rf (ball z₀ 1)
-  have hx (ω : a.samples) : X ω ≤ v*rf^(-codimension κ) := by
-    have hh := (a.spatialRealMass_bound hk hκ8 rf hf (ball z₀ 1) measurableSet_ball
-      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne (fun z hz' => ?_) ω).2
-    · simpa only [v,Complex.volume_ball] using hh
-    · have hd : abs (z.im-z₀.im) ≤ dist z z₀ := by
-        simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
-      have hh : dist z z₀ < 1 := hz'
-      have ha := neg_abs_le (z.im-z₀.im)
-      linarith
-  have ht' := source_mesh_tail_to_stretched μ X (codimension κ) C c δ₀ v rf
-    (μ.real {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤}) hp hC hc hδ
-    ENNReal.toReal_nonneg hf ENNReal.toReal_nonneg hx
-    (fun δ hfd hd => by
-      have hh := ht Ω m P hP γ a z₀ hz rf δ hf hfd hd
-      simpa only [Measure.real, ENNReal.ofReal_mul ENNReal.toReal_nonneg,
-        ENNReal.ofReal_toReal (measure_ne_top _ _)] using hh) u hu
-  apply ht'.trans
-  apply ENNReal.ofReal_le_ofReal
-  calc
-    _ ≤ (A*initialGreen κ z₀)*C*Real.exp (-b*u^(2/codimension κ)) :=
-      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
-        (source_simple_entrance_bound a hk hκ8 z₀ hz hsmall) hC.le) (Real.exp_pos _).le
-    _ = _ := by ring
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_simple_entrance_all_tail (κ : ℝ) (hκ : 0 < κ) (hκ4 : κ ≤ 4) :
-  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
-    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
-      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ))) := by
-  have hκ8 : κ < 8 := by linarith
-  obtain ⟨M,C,b,hM,hC,hb,ht⟩ := sourceSimpleStretchedTail_proved κ hκ hκ4
-  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
-  have hk : 0 < κ := by linarith
-  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
-    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
-  refine ⟨max C A*Real.exp b,b/(max M 1)^(2/codimension κ),
-    mul_pos (hC.trans_le (le_max_left _ _)) (Real.exp_pos _),
-    div_pos hb (Real.rpow_pos_of_pos (hM.trans_le (le_max_left _ _)) _),?_⟩
-  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) rf (ball z₀ 1)
-  have hK : 0 < initialGreen κ z₀ := initialGreen_pos κ z₀ (by change 0 < z₀.im; linarith)
-  have hzero : μ {ω | 0 < X ω} ≤ ENNReal.ofReal (A*initialGreen κ z₀) := by
-    apply le_trans (measure_mono (show {ω | 0 < X ω} ⊆
-      {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} from ?_))
-    · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (mul_pos hA hK).le).mpr
-      exact source_simple_entrance_bound a hk hκ8 z₀ hz hsmall
-    · intro ω hω hn
-      have hx : X ω = 0 := by
-        apply setIntegral_eq_zero_of_forall_eq_zero
-        intro z hz'
-        exact a.source_simple_weight_zero_no_entrance z₀ hz rf hf1 ω hn z hz'
-      change 0 < X ω at hω
-      rw [hx] at hω
-      exact (lt_irrefl 0) hω
-  exact source_tail_threshold_normalize μ X M C A b (2/codimension κ) (initialGreen κ z₀)
-    hM hC hA hb (div_pos (by norm_num) (codimension_pos κ hκ8)) hK.le hzero
-    (fun v hv => ht Ω m P hP γ a z₀ hz hsmall rf v hf hf1 hv) u hu
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourceSimplePhysicalBallAllTailTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ ≤ 4 →
-  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (r : ℝ),
-    0 < r → 4*r ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*(4*r))) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ r → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | r^dimension κ*u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ r)} ≤
-      ENNReal.ofReal (C*r^codimension κ*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
-theorem sourceSimplePhysicalBallAllTail_proved : SourceSimplePhysicalBallAllTailTarget := by
-  intro κ hκ hκ4
-  have hκ8 : κ < 8 := by linarith
-  obtain ⟨C,b,hC,hb,ht⟩ := source_simple_entrance_all_tail κ hκ hκ4
-  refine ⟨C,b,hC,hb,?_⟩
-  intro Ω m P hP γ a z r hr hz hsmall rf u hf hfr hu
-  let : IsProbabilityMeasure P := hP
-  let c : ℝ≥0 := ⟨r⁻¹,(inv_pos.mpr hr).le⟩
-  have hc : 0 < c := inv_pos.mpr hr
-  have hcR : (c:ℝ) = r⁻¹ := rfl
-  let a' := a.sourceScaled c hc
-  have him : ((c:ℂ)*z).im = r⁻¹*z.im := by simp [hcR,Complex.mul_im]
-  have hcr : (c:ℝ)*r = 1 := by rw [hcR,inv_mul_cancel₀ hr.ne']
-  have hheight : 4 ≤ ((c:ℂ)*z).im := by
-    rw [him,mul_comm,←div_eq_mul_inv]
-    exact (le_div_iff₀ hr).mpr hz
-  have hsmall' : 1 ≤ Real.log ((2*((c:ℂ)*z).im)/((128*Real.pi)*4)) := by
-    have he : (2*((c:ℂ)*z).im)/((128*Real.pi)*4) = (2*z.im)/((128*Real.pi)*(4*r)) := by
-      rw [him]
-      field_simp
-    rwa [he]
-  have hf' : 0 < (c:ℝ)*rf := mul_pos hc hf
-  have hfr' : (c:ℝ)*rf ≤ 1 := by rw [←hcr]; exact mul_le_mul_of_nonneg_left hfr hc.le
-  have htail := ht Ω m P hP (fun ω => sourceScaleTrace c (γ ω)) a' ((c:ℂ)*z)
-    hheight hsmall' ((c:ℝ)*rf) u hf' hfr' hu
-  change (P.comap (Subtype.val : a.samples → Ω)) {ω : a.samples | u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a'.B t ω) (sourceScaleTrace c (γ ω)) (a'.G ω)
-      ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} ≤
-      ENNReal.ofReal (C*initialGreen κ ((c:ℂ)*z)*Real.exp (-b*u^(2/codimension κ))) at htail
-  have hp : 0 < (c:ℝ)^dimension κ := Real.rpow_pos_of_pos hc _
-  have hrp : 0 < r^dimension κ := Real.rpow_pos_of_pos hr _
-  have hmr : (c:ℝ)^dimension κ*(r^dimension κ*u) = u := by
-    rw [hcR,Real.inv_rpow hr.le]
-    field_simp
-  have hx (ω : a.samples) : spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
-      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1) =
-      (c:ℝ)^dimension κ*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z r) := by
-    simpa only [hcr] using a.sourceScaled_mass c hc z r rf ω
-  have he : {ω : a.samples | u < spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
-      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} =
-      {ω : a.samples | r^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) rf (ball z r)} := by
-    ext ω
-    change (u < _) ↔ _
-    rw [hx ω]
-    conv_lhs => rw [←hmr]
-    exact mul_lt_mul_iff_right₀ hp
-  rw [he] at htail
-  have hg : initialGreen κ ((c:ℂ)*z) = r^codimension κ*initialGreen κ z := by
-    rw [source_scale_initialGreen κ c hc z (by nlinarith),hcR,Real.inv_rpow hr.le,Real.rpow_neg hr.le,inv_inv]
-  rw [hg] at htail
-  convert htail using 1
-  congr 1
-  ring
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-def SourcePhysicalBallEveryKappaTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (r : ℝ),
-    0 < r → 4*r ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*(4*r))) →
-  ∀ (rf u : ℝ), 0 < rf → rf ≤ r → 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | r^dimension κ*u < spatialRealMass κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ r)} ≤
-      ENNReal.ofReal (C*r^codimension κ*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
-theorem sourcePhysicalBallEveryKappa_proved : SourcePhysicalBallEveryKappaTarget := by
-  intro κ hκ hκ8
-  by_cases hκ4 : κ ≤ 4
-  · exact sourceSimplePhysicalBallAllTail_proved κ hκ hκ4
-  · exact sourcePhysicalBallAllTail_proved κ (lt_of_not_ge hκ4) hκ8
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-def SourceActualSquareTailTarget_allKappa : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-  ∃ (A c : ℝ) (kmin : ℕ), 0 < A ∧ 0 < c ∧
-  ∀ (T : ℝ≥0) (N k : ℕ), kmin ≤ k → k ≤ N → ∀ ij : ℤ × ℤ, ∀ u : ℝ, 1 ≤ u →
-    (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
-      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω)
-        ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ≤
-    ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ)))
-theorem sourceActualSquareTail_proved_allKappa : SourceActualSquareTailTarget_allKappa := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀
-  obtain ⟨C,c,hC,hc,ht⟩ := sourcePhysicalBallEveryKappa_proved κ hκ hκ8
-  let H := z₀.im-2
-  have hH : 0 < H := sub_pos.mpr hz₀
-  let G := (2:ℝ)^(-codimension κ)*H^(-codimension κ)
-  have hG : 0 < G := mul_pos (Real.rpow_pos_of_pos (by norm_num) _) (Real.rpow_pos_of_pos hH _)
-  obtain ⟨kmin,hkmin⟩ := source_small_dyadic_level H hH
-  refine ⟨C*G,c,kmin,mul_pos hC hG,hc,?_⟩
-  intro T N k hk hkN ij u hu
-  let l := (2:ℝ)^(-(k:ℝ))
-  let rf := (2:ℝ)^(-(N:ℝ))
-  have hl : 0 < l := source_dyadic_side_pos k
-  have hf : 0 < rf := source_dyadic_side_pos N
-  have hfl : rf ≤ l := source_dyadic_side_le k N hkN
-  have hpos : 0 < l^dimension κ*u := mul_pos (Real.rpow_pos_of_pos hl _) (zero_lt_one.trans_le hu)
-  by_cases hmeet : (closedBall z₀ 1 ∩ retainedDyadicSquare k ij).Nonempty
-  · let z := sourceDyadicCenter k ij
-    have hheight : H ≤ z.im := source_square_center_height z₀ k ij hmeet
-    have hy : 4*l ≤ z.im := (hkmin k hk).1.trans hheight
-    have hsmall : 1 ≤ Real.log ((2*z.im)/((128*Real.pi)*(4*l))) := by
-      apply (Real.le_log_iff_exp_le (div_pos (by linarith) (by positivity))).mpr
-      apply (le_div_iff₀ (by positivity : 0 < (128*Real.pi)*(4*l))).mpr
-      exact (hkmin k hk).2.trans (mul_le_mul_of_nonneg_left hheight (by norm_num))
-    have hgreen : initialGreen κ z ≤ G := by
-      apply (initialGreen_le_height κ (by linarith) hκ8 z (hH.trans_le hheight)).trans
-      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
-      exact Real.rpow_le_rpow_of_nonpos hH hheight (neg_nonpos.mpr (codimension_pos κ hκ8).le)
-    have hh := ht Ω m P hP γ a z l hl hy hsmall rf u hf hfl hu
-    apply le_trans (measure_mono (show {ω : a.samples | l^dimension κ*u <
-      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
-        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ⊆
-      {ω | l^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) rf (ball z l)} from ?_)) (hh.trans ?_)
-    · intro ω hω
-      exact hω.trans_le (source_finite_mass_le_ball a (by linarith) hκ8 z l rf hl hy hf hfl T
-        _ (fun w hw => source_square_subset_ball k ij hw.2) ω)
-    · apply ENNReal.ofReal_le_ofReal
-      calc
-        C*l^codimension κ*initialGreen κ z*Real.exp (-c*u^(2/codimension κ)) ≤
-            C*l^codimension κ*G*Real.exp (-c*u^(2/codimension κ)) := by gcongr
-        _ = _ := by ring
-  · have he : closedBall z₀ 1 ∩ retainedDyadicSquare k ij = ∅ := not_nonempty_iff_eq_empty.mp hmeet
-    have hempty : {ω : a.samples | l^dimension κ*u <
-      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
-        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} = ∅ := by
-      simp only [he,finiteSpatialGreenMass,Measure.restrict_empty,lintegral_zero_measure,ENNReal.toReal_zero,
-        not_lt_of_ge hpos.le,Set.ofPred_false]
-    change (P.comap Subtype.val) _ ≤ _
-    rw [hempty,measure_empty]
-    exact bot_le
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-variable (a : RegularSLEPresentation P κ γ)
-lemma source_actual_retained_experiment_allKappa [IsProbabilityMeasure P] (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz₀ : 2 < z₀.im) :
-    ∃ (T : ℝ≥0) (D b p : ℝ) (k₀ : ℕ), 0 < T ∧ 1 ≤ D ∧ 0 < b ∧ 0 < p ∧
-      ENNReal.ofReal p ≤ (P.comap Subtype.val)
-        {ω : a.samples | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-          retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)} := by
-  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
-  have hk : 0 < κ := by linarith
-  obtain ⟨T,e₀,m₀,m₂,hT,he,hm₀,hm₂,hm⟩ := source_retained_seed_all κ hκ hκ8
-    Ω inferInstance P inferInstance γ a z₀ hz₀
-  obtain ⟨A,c,k₁,hA,hc,ht⟩ := sourceActualSquareTail_proved_allKappa κ hκ hκ8
-    Ω inferInstance P inferInstance γ a z₀ hz₀
-  obtain ⟨k₂,hk₂⟩ := source_dyadic_eventually_le he
-  let kmin := max k₁ k₂
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  have hν (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
-  have hfirst (N : ℕ) (hN : kmin ≤ N) :
-      ENNReal.ofReal m₀ ≤ ∫⁻ ω, ν N ω (closedBall z₀ 1) ∂(P.comap Subtype.val) := by
-    simp only [ν,a.sourceFiniteGreenKernel_K]
-    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).1
-  have hsecond (N : ℕ) (hN : kmin ≤ N) :
-      (∫⁻ ω, (ν N ω (closedBall z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
-    simp only [ν,a.sourceFiniteGreenKernel_K]
-    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).2
-  have htail (N k : ℕ) (hkmin : kmin ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) (u : ℝ) (hu : 1 ≤ u) :
-      (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
-        (ν N ω (retainedDyadicSquare k ij)).toReal} ≤
-        ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ))) := by
-    simp only [ν,a.sourceFiniteGreenKernel_apply z₀ T N _ _ (retainedSquare_measurable k ij)]
-    exact ht T N k ((le_max_left _ _).trans hkmin) hkN ij u hu
-  obtain ⟨D,b,p,B,k₀,hD,hb,hp,hB,hk₀,hall,hlim⟩ := sourceRetainedConstruction a.samples
-    inferInstance (P.comap Subtype.val) inferInstance κ hk hκ8 ν hν
-    (closedBall z₀ 1) (isCompact_closedBall _ _) (fun N ω => a.sourceFiniteGreenKernel_supported z₀ T N ω)
-    A c m₀ m₂ hA hc hm₀ hm₂ kmin hfirst hsecond htail
-  exact ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
- 
-
-def SourceActualRetainedLowerTarget_allKappa : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
-    0 < (P.comap Subtype.val) E ∧
-    ∀ ω ∈ E, 0 < hausdorffGauge h (γ ω '' Set.Icc 0 T)
-theorem sourceActualRetainedLower_proved_allKappa : SourceActualRetainedLowerTarget_allKappa := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment_allKappa hκ hκ8 z₀ hz₀
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
-  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
-  have hE : MeasurableSet E := by
-    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
-        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
-      ext ω
-      simp [E,frequently_atTop]
-    rw [heq]
-    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
-      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
-  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
-  intro ω hω
-  exact a.source_actual_retained_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
-    k₀ h hh hf ω hω
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
- 
-
-def SourceInteriorRetainedTransferTarget_allKappa : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
-    0 < (P.comap Subtype.val) E ∧ ∀ ω ∈ E,
-      0 < hausdorffGauge h (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)) ∧
-      ∀ (U : ℝ≥0 → ℝ) (η : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
-        IsCapacityTwoWitness U η G F → ∀ s : ℝ≥0,
-          0 < hausdorffGauge h (F s '' (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)))
-
-theorem sourceInteriorRetainedTransfer_proved_allKappa : SourceInteriorRetainedTransferTarget_allKappa := by
-  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment_allKappa hκ hκ8 z₀ hz₀
-  let ν := a.sourceFiniteGreenKernel z₀ T
-  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
-  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
-    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
-  have hE : MeasurableSet E := by
-    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
-        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
-      ext ω
-      simp [E,frequently_atTop]
-    rw [heq]
-    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
-      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
-  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
-  intro ω hω
-  have hpos := a.source_actual_retained_interior_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
-    k₀ h hh hf ω hω
-  refine ⟨hpos,?_⟩
-  intro U η G F hw s
-  exact source_inverse_chart_gauge_positive κ hk hκ8 h hh hf U η G F hw s
-    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed)
-    (fun z hz => zero_lt_one.trans (source_unit_ball_height hz₀ hz.1)) hpos
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance simplePastMeshDriverMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance simplePastMeshTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance simplePastMeshDriverBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance simplePastMeshTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_dilation_gauge_positive_allKappa (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    {c : ℂ} (hc : c ≠ 0) {S : Set ℂ} (hS : IsCompact S)
-    (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => c*z) '' S) := by
-  by_contra hn
-  have hz : hausdorffGauge h ((fun z => c*z) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le
-  have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
-    (D := univ) (f := fun z => c⁻¹*z) isOpen_univ (hS.image (continuous_const.mul continuous_id))
-    (subset_univ _) (by fun_prop) hz
-  have hi : (fun z => c⁻¹*z) '' ((fun z => c*z) '' S) = S := by
-    rw [←image_comp]
-    simp only [Function.comp_def,←mul_assoc,inv_mul_cancel₀ hc,one_mul,image_id']
-  change hausdorffGauge h ((fun z => c⁻¹*z) '' ((fun z => c*z) '' S)) = 0 at he
-  rw [hi] at he
-  exact hp.ne' he
-
-lemma RegularSLEPresentation.source_trace_path_measurable_allKappa
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) :
-    Measurable (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) := by
-  apply source_path_measurable
-  intro t
-  exact (a.trace_adapted t).mono (a.filtration.le t) le_rfl
-
-lemma RegularSLEPresentation.source_scale_trace_law_allKappa
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (c : ℝ≥0) (hc : 0 < c) :
-    (P.comap (Subtype.val : a.samples → Ω)).map
-      (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) =
-    (P.comap (Subtype.val : a.samples → Ω)).map
-      (sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
-        (a.sourceScaled c hc).trace_continuous) := by
-  exact sourceTraceLawUnique_proved κ hκ hκ8 a.B (a.sourceScaled c hc).B
-    a.brownian (a.sourceScaled c hc).brownian a.measurable_B (a.sourceScaled c hc).measurable_B
-    a.continuous_B (a.sourceScaled c hc).continuous_B _ _ a.G a.F
-    (a.sourceScaled c hc).G (a.sourceScaled c hc).F a.witness (a.sourceScaled c hc).witness
- 
-
-def SourceUniformInteriorSeedTarget_allKappa : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (h : ℝ → ℝ), IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∃ p : ℝ≥0∞, 0 < p ∧ ∀ δ : ℝ≥0, 0 < δ →
-    ∃ K : Set ℂ, IsCompact K ∧ K ⊆ upperHalfPlane ∧
-      p ≤ (P.comap (Subtype.val : a.samples → Ω))
-        {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Set.Icc 0 δ))}
-theorem sourceUniformInteriorSeed_proved_allKappa : SourceUniformInteriorSeedTarget_allKappa := by
-  intro κ hκ hκ8 Ω m P hP γ a h hh hf
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  let z₀ : ℂ := 3*I
-  have hz₀ : 2 < z₀.im := by norm_num [z₀]
-  obtain ⟨T,E,hT,hE,hp,hseed⟩ :=
-    sourceInteriorRetainedTransfer_proved_allKappa κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  refine ⟨Q E,hp,?_⟩
-  intro δ hδ
-  let c := NNReal.sqrt (δ/T)
-  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
-  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
-  have ht : sourceScaleTime c δ = T := by
-    unfold sourceScaleTime
-    rw [hc2]
-    field_simp
-  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
-  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
-  have hKH : K ⊆ upperHalfPlane := by
-    rintro _ ⟨z,hz,rfl⟩
-    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
-    change 0 < ((c:ℂ)*z).im
-    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos (show 0 < (c:ℝ) from hc) hz'
-  refine ⟨K,hK,hKH,?_⟩
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
-    (a.sourceScaled c hc).trace_continuous
-  let A : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))}
-  have hA : MeasurableSet A := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ
-  have hm : Measurable gp := a.source_trace_path_measurable_allKappa
-  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable_allKappa
-  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law_allKappa hk hκ8 c hc)
-  rw [Measure.map_apply hm hA,Measure.map_apply hsm hA] at hmap
-  change Q E ≤ Q (gp ⁻¹' A)
-  rw [hmap]
-  apply measure_mono
-  intro ω hω
-  have hpos := (hseed ω hω).1
-  have heq : K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ) =
-      (fun z : ℂ => (c:ℂ)*z) '' (closedBall z₀ 1 ∩ (γ ω '' Icc 0 T)) := by
-    rw [source_scale_past c δ hc,ht, image_inter (mul_right_injective₀ (by exact_mod_cast hc.ne' : (c:ℂ) ≠ 0))]
-  change 0 < hausdorffGauge h (K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ))
-  rw [heq]
-  exact source_dilation_gauge_positive_allKappa κ hk hκ8 h hh hf (by exact_mod_cast hc.ne')
-    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed) hpos
-end
-end SLEExactGauge
-end
-
- 
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.SmallKappaSausageMoments
+import OAI.Probability.SLE.TracePosCombined
+
+namespace OAI
+
+/-! Simple Past Mesh for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem source_past_half_plane_mesh_tail [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
+    (q ε δ R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hε : 0 < ε)
+    (hεδ : ε ≤ δ) (hδR : δ ≤ R)
+    (hheight : ∀ z ∈ A, δ ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
+    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z δ)
+    :
+    ((P.comap Subtype.val)[|D]) {ω : a.samples |
+      ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A} ≤
+      ENNReal.ofReal (Real.exp (-(1/(2*(meshTailAmplitude κ q A)^2))*δ^(-2 : ℝ))) := by
+  obtain ⟨N,r, hqr, hrδ, he⟩ := exists_exact_geometric_mesh hq hq1 hε hεδ
+  have hδ : 0 < δ := hε.trans_le hεδ
+  have hr : 0 < r := (mul_pos hq hδ).trans hqr
+  have hκ8 : κ < 8 := by linarith
+  have hp : 0 < codimension κ := codimension_pos κ hκ8
+  have hα : 0 < 1-codimension κ := sub_pos.mpr (codimension_lt_one κ hκ)
+  have hd : 0 < 1-q^(1-codimension κ) := sub_pos.mpr (Real.rpow_lt_one hq.le hq1 hα)
+  have hB := meshTailAmplitude_pos A hκ hq hq1
+  have ham : slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)) ≤
+      meshTailAmplitude κ q A*δ^(1-codimension κ) := by
+    calc
+      _ ≤ slabAmplitude κ q A*δ^(1-codimension κ)/(1-q^(1-codimension κ)) :=
+        div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left
+          (Real.rpow_le_rpow hr.le hrδ hα.le) (slabAmplitude_nonneg κ q A)) hd.le
+      _ ≤ _ := by
+        dsimp [meshTailAmplitude]
+        have hx := (Real.rpow_pos_of_pos hδ (1-codimension κ)).le
+        rw [add_mul, div_mul_eq_mul_div]
+        nlinarith
+  let B : ℝ≥0 := NNReal.mk (meshTailAmplitude κ q A*δ^(1-codimension κ))
+    (mul_nonneg hB.le (Real.rpow_nonneg hδ.le _))
+  have hsg := a.source_past_spatial_geometric_subgaussian hκ hκ4 A hA q r R hq hq1 hr
+    (hrδ.trans hδR) (fun z hz => hrδ.trans (hheight z hz)) hbound N s D hD (fun ω hω z hz => (hbefore ω hω z hz).trans
+      (spatialHitTime_antitone_radius (a.trace_continuous ω) z hrδ (by
+        rw [(a.witness ω).2.1,dist_zero_left]
+        exact (hheight z hz).trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z)))))
+  have hsg' : HasSubgaussianMGF (fun ω : a.samples =>
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
+      (B^2) ((P.comap Subtype.val)[|D]) := by
+    rw [he]
+    apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsg
+    apply pow_le_pow_left₀ (by positivity)
+    exact NNReal.coe_le_coe.mp ham
+  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ closedBall 0 R from
+    fun z hz => by simpa only [mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
+      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
+  have hstart (ω : a.samples) :
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A ≤
+      (volume A).toReal*q^(-codimension κ)*δ^(-codimension κ) := by
+    apply (a.spatialRealMass_bound hκ hκ8 r hr A hA hfin
+      (fun z hz => hrδ.trans (hheight z hz)) ω).2.trans
+    have hh := Real.rpow_le_rpow_of_nonpos (mul_pos hq hδ) hqr.le (neg_nonpos.mpr hp.le)
+    rw [Real.mul_rpow hq.le hδ.le] at hh
+    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh ENNReal.toReal_nonneg
+  apply (measure_mono (show _ ⊆ {ω : a.samples | δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A} from ?_)).trans
+  · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (Real.exp_pos _).le).mpr
+    have ht := hsg'.measure_ge_le (Real.rpow_nonneg hδ.le (-codimension κ))
+    change ((P.comap Subtype.val)[|D]).real _ ≤ _
+    simpa only [B, NNReal.coe_pow, NNReal.coe_mk, mesh_tail_exponent hδ hB] using ht
+  · intro ω hω
+    have := hstart ω
+    change ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A at hω
+    change δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A
+    nlinarith
+
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourceSimpleEntranceMeshTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ ≤ 4 →
+  ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < 1 ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 4 ≤ z₀.im →
+  ∀ rf δ : ℝ, 0 < rf → rf ≤ δ → δ ≤ δ₀ →
+    (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ 1)} ≤
+      (P.comap Subtype.val) {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} *
+        ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ)))
+theorem sourceSimpleEntranceMesh_proved : SourceSimpleEntranceMeshTarget := by
+  intro κ hκ hκ4
+  let C := max 1 ((volume (ball (0:ℂ) 1)).toReal*(1/2:ℝ)^(-codimension κ)+1)
+  let c := 1/(2*(meshTailAmplitude κ (1/2) (ball (0:ℂ) 1))^2)
+  have hC1 : 1 ≤ C := le_max_left _ _
+  have hC : 0 < C := zero_lt_one.trans_le hC1
+  have hc : 0 < c := by
+    have hh := meshTailAmplitude_pos (κ := κ) (q := (1/2:ℝ)) (ball (0:ℂ) 1) hκ (by norm_num) (by norm_num)
+    exact div_pos zero_lt_one (mul_pos (by norm_num) (sq_pos_of_pos hh))
+  refine ⟨C,c,1/2,hC,hc,by norm_num,by norm_num,?_⟩
+  intro Ω m P hP γ a z₀ hz₀ rf δ hf hfd hd
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let H := {ω : a.samples | C*δ^(-codimension κ) < spatialRealMass κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)}
+  have hH : MeasurableSet H := measurableSet_lt measurable_const (a.spatialRealMass_measurable _ _)
+  have hδ : 0 < δ := hf.trans_le hfd
+  have hδ1 : δ ≤ 1 := hd.trans (by norm_num)
+  have hrf1 : rf ≤ 1 := hfd.trans hδ1
+  have hHA : H ⊆ {ω | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} := by
+    intro ω hω hn
+    have hm : spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1) = 0 := by
+      unfold spatialRealMass
+      apply setIntegral_eq_zero_of_forall_eq_zero
+      exact fun z hz => a.source_simple_weight_zero_no_entrance z₀ hz₀ rf hrf1 ω hn z hz
+    change C*δ^(-codimension κ) < _ at hω
+    rw [hm] at hω
+    exact (not_lt_of_ge (mul_nonneg hC.le (Real.rpow_nonneg hδ.le _))) hω
+  apply source_between_stopping_tail (a.source_simple_sphere_stopping z₀ 3)
+    (a.source_simple_sphere_stopping z₀ 2) H hH hHA
+    (a.source_simple_spheres_separated z₀ hz₀) (ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))))
+  intro s D hD hbefore
+  have hDm : MeasurableSet D := (a.filtration.le s) _ hD
+  have hheight : ∀ z ∈ ball z₀ 1, δ ≤ z.im := by
+    intro z hz
+    have he := (le_abs_self (z₀.im-z.im)).trans (Complex.abs_im_le_norm (z₀-z))
+    have hn : ‖z₀-z‖ < 1 := by simpa only [mem_ball,dist_eq_norm, norm_sub_rev] using hz
+    linarith
+  have hbound : ∀ z ∈ ball z₀ 1, ‖z‖ ≤ ‖z₀‖+1 := by
+    intro z hz
+    have hh := norm_le_insert z₀ z
+    rw [norm_sub_rev z₀ z] at hh
+    have hn : ‖z-z₀‖ < 1 := by simpa only [mem_ball, dist_eq_norm] using hz
+    linarith
+  have hR : δ ≤ ‖z₀‖+1 := hδ1.trans (by linarith [norm_nonneg z₀])
+  have ht := a.source_past_half_plane_mesh_tail hκ hκ4 (ball z₀ 1) measurableSet_ball
+    (1/2) rf δ (‖z₀‖+1) (by norm_num) (by norm_num) hf hfd hR hheight hbound s D hD
+    (fun ω hω z hz => (hbefore ω hω).trans (a.source_simple_center_before z₀ hz₀ δ hδ1 ω z hz))
+  have hv : volume (ball z₀ 1) = volume (ball (0:ℂ) 1) := by rw [Complex.volume_ball,Complex.volume_ball]
+  have ha : meshTailAmplitude κ (1/2) (ball z₀ 1) = meshTailAmplitude κ (1/2) (ball (0:ℂ) 1) := by
+    simp only [meshTailAmplitude,slabAmplitude,hv]
+  have hcoeff : (volume (ball z₀ 1)).toReal*(1/2:ℝ)^(-codimension κ)+1 ≤ C := by
+    rw [hv]; exact le_max_right _ _
+  have ht' : μ[H|D] ≤ ENNReal.ofReal (Real.exp (-c*δ^(-2:ℝ))) := by
+    apply (measure_mono (show H ⊆ _ from ?_)).trans
+    · simpa only [ha] using ht
+    · intro ω hω
+      exact (mul_le_mul_of_nonneg_right hcoeff (Real.rpow_nonneg hδ.le _)).trans hω.le
+  calc
+    μ (D ∩ H) = μ[H|D]*μ D := (cond_mul_eq_inter hDm H μ).symm
+    _ ≤ ENNReal.ofReal (Real.exp (-c*δ^(-2:ℝ)))*μ D := mul_le_mul' ht' le_rfl
+    _ ≤ μ D*ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
+      rw [mul_comm]
+      apply mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal ?_)
+      exact le_mul_of_one_le_left (Real.exp_pos _).le hC1
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_simple_entrance_bound {Ω : Type} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 4 ≤ z₀.im) (hsmall : 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4))) :
+    (P.comap Subtype.val).real {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} ≤
+      (uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ)*initialGreen κ z₀ := by
+  apply le_trans (measureReal_mono (show {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} ⊆
+      {ω | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4} from ?_))
+  · have hh := euclidean_approach a.brownian κ hκ hκ8 (fun ω : a.samples => γ ω) a.G a.F
+      (Eventually.of_forall a.witness) z₀ (by change 0 < z₀.im; linarith) 4 (by norm_num) hz hsmall
+    change ((P.comap Subtype.val) {ω : a.samples | ∃ t : ℝ≥0, dist (γ ω t) z₀ < 4}).toReal ≤ _
+    calc
+      _ ≤ _ := hh
+      _ = _ := by ring
+  · intro ω hω
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hω
+    have hs : 3 ≤ dist (γ ω 0) z₀ := by
+      rw [(a.witness ω).2.1,dist_zero_left]
+      have := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
+      linarith
+    exact ⟨t,by rw [spatialHitTime_mem (a.trace_continuous ω) z₀ 3 t ht.symm]; norm_num⟩
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+def SourceSimpleStretchedTailTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ ≤ 4 →
+  ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
+    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | M*u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ 1)} ≤
+      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
+theorem sourceSimpleStretchedTail_proved : SourceSimpleStretchedTailTarget := by
+  intro κ hκ hκ4
+  have hκ8 : κ < 8 := by linarith
+  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := sourceSimpleEntranceMesh_proved κ hκ hκ4
+  let v := (volume (ball (0:ℂ) 1)).toReal
+  let M := max C v*δ₀^(-codimension κ)
+  let b := c*δ₀^(-2:ℝ)
+  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
+  have hk : 0 < κ := by linarith
+  have hp : 0 < codimension κ := codimension_pos κ hκ8
+  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hδ _)
+  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hδ _)
+  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
+    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
+  refine ⟨M,A*C,b,hM,mul_pos hA hC,hb,?_⟩
+  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) rf (ball z₀ 1)
+  have hx (ω : a.samples) : X ω ≤ v*rf^(-codimension κ) := by
+    have hh := (a.spatialRealMass_bound hk hκ8 rf hf (ball z₀ 1) measurableSet_ball
+      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne (fun z hz' => ?_) ω).2
+    · simpa only [v,Complex.volume_ball] using hh
+    · have hd : abs (z.im-z₀.im) ≤ dist z z₀ := by
+        simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
+      have hh : dist z z₀ < 1 := hz'
+      have ha := neg_abs_le (z.im-z₀.im)
+      linarith
+  have ht' := source_mesh_tail_to_stretched μ X (codimension κ) C c δ₀ v rf
+    (μ.real {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤}) hp hC hc hδ
+    ENNReal.toReal_nonneg hf ENNReal.toReal_nonneg hx
+    (fun δ hfd hd => by
+      have hh := ht Ω m P hP γ a z₀ hz rf δ hf hfd hd
+      simpa only [Measure.real, ENNReal.ofReal_mul ENNReal.toReal_nonneg,
+        ENNReal.ofReal_toReal (measure_ne_top _ _)] using hh) u hu
+  apply ht'.trans
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ ≤ (A*initialGreen κ z₀)*C*Real.exp (-b*u^(2/codimension κ)) :=
+      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
+        (source_simple_entrance_bound a hk hκ8 z₀ hz hsmall) hC.le) (Real.exp_pos _).le
+    _ = _ := by ring
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_simple_entrance_all_tail (κ : ℝ) (hκ : 0 < κ) (hκ4 : κ ≤ 4) :
+  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ),
+    4 ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*4)) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ 1 → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
+      ENNReal.ofReal (C*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ))) := by
+  have hκ8 : κ < 8 := by linarith
+  obtain ⟨M,C,b,hM,hC,hb,ht⟩ := sourceSimpleStretchedTail_proved κ hκ hκ4
+  let A := uniformInverseAngleConstant κ 1*(128*Real.pi)^codimension κ*4^codimension κ
+  have hk : 0 < κ := by linarith
+  have hA : 0 < A := mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hk hκ8 zero_lt_one)
+    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by norm_num) _)
+  refine ⟨max C A*Real.exp b,b/(max M 1)^(2/codimension κ),
+    mul_pos (hC.trans_le (le_max_left _ _)) (Real.exp_pos _),
+    div_pos hb (Real.rpow_pos_of_pos (hM.trans_le (le_max_left _ _)) _),?_⟩
+  intro Ω m P hP γ a z₀ hz hsmall rf u hf hf1 hu
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) rf (ball z₀ 1)
+  have hK : 0 < initialGreen κ z₀ := initialGreen_pos κ z₀ (by change 0 < z₀.im; linarith)
+  have hzero : μ {ω | 0 < X ω} ≤ ENNReal.ofReal (A*initialGreen κ z₀) := by
+    apply le_trans (measure_mono (show {ω | 0 < X ω} ⊆
+      {ω : a.samples | spatialHitTime (γ ω) z₀ 3 ≠ ⊤} from ?_))
+    · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (mul_pos hA hK).le).mpr
+      exact source_simple_entrance_bound a hk hκ8 z₀ hz hsmall
+    · intro ω hω hn
+      have hx : X ω = 0 := by
+        apply setIntegral_eq_zero_of_forall_eq_zero
+        intro z hz'
+        exact a.source_simple_weight_zero_no_entrance z₀ hz rf hf1 ω hn z hz'
+      change 0 < X ω at hω
+      rw [hx] at hω
+      exact (lt_irrefl 0) hω
+  exact source_tail_threshold_normalize μ X M C A b (2/codimension κ) (initialGreen κ z₀)
+    hM hC hA hb (div_pos (by norm_num) (codimension_pos κ hκ8)) hK.le hzero
+    (fun v hv => ht Ω m P hP γ a z₀ hz hsmall rf v hf hf1 hv) u hu
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourceSimplePhysicalBallAllTailTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ ≤ 4 →
+  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (r : ℝ),
+    0 < r → 4*r ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*(4*r))) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ r → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | r^dimension κ*u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ r)} ≤
+      ENNReal.ofReal (C*r^codimension κ*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
+theorem sourceSimplePhysicalBallAllTail_proved : SourceSimplePhysicalBallAllTailTarget := by
+  intro κ hκ hκ4
+  have hκ8 : κ < 8 := by linarith
+  obtain ⟨C,b,hC,hb,ht⟩ := source_simple_entrance_all_tail κ hκ hκ4
+  refine ⟨C,b,hC,hb,?_⟩
+  intro Ω m P hP γ a z r hr hz hsmall rf u hf hfr hu
+  let : IsProbabilityMeasure P := hP
+  let c : ℝ≥0 := ⟨r⁻¹,(inv_pos.mpr hr).le⟩
+  have hc : 0 < c := inv_pos.mpr hr
+  have hcR : (c:ℝ) = r⁻¹ := rfl
+  let a' := a.sourceScaled c hc
+  have him : ((c:ℂ)*z).im = r⁻¹*z.im := by simp [hcR,Complex.mul_im]
+  have hcr : (c:ℝ)*r = 1 := by rw [hcR,inv_mul_cancel₀ hr.ne']
+  have hheight : 4 ≤ ((c:ℂ)*z).im := by
+    rw [him,mul_comm,←div_eq_mul_inv]
+    exact (le_div_iff₀ hr).mpr hz
+  have hsmall' : 1 ≤ Real.log ((2*((c:ℂ)*z).im)/((128*Real.pi)*4)) := by
+    have he : (2*((c:ℂ)*z).im)/((128*Real.pi)*4) = (2*z.im)/((128*Real.pi)*(4*r)) := by
+      rw [him]
+      field_simp
+    rwa [he]
+  have hf' : 0 < (c:ℝ)*rf := mul_pos hc hf
+  have hfr' : (c:ℝ)*rf ≤ 1 := by rw [←hcr]; exact mul_le_mul_of_nonneg_left hfr hc.le
+  have htail := ht Ω m P hP (fun ω => sourceScaleTrace c (γ ω)) a' ((c:ℂ)*z)
+    hheight hsmall' ((c:ℝ)*rf) u hf' hfr' hu
+  change (P.comap (Subtype.val : a.samples → Ω)) {ω : a.samples | u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a'.B t ω) (sourceScaleTrace c (γ ω)) (a'.G ω)
+      ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} ≤
+      ENNReal.ofReal (C*initialGreen κ ((c:ℂ)*z)*Real.exp (-b*u^(2/codimension κ))) at htail
+  have hp : 0 < (c:ℝ)^dimension κ := Real.rpow_pos_of_pos hc _
+  have hrp : 0 < r^dimension κ := Real.rpow_pos_of_pos hr _
+  have hmr : (c:ℝ)^dimension κ*(r^dimension κ*u) = u := by
+    rw [hcR,Real.inv_rpow hr.le]
+    field_simp
+  have hx (ω : a.samples) : spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
+      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1) =
+      (c:ℝ)^dimension κ*spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z r) := by
+    simpa only [hcr] using a.sourceScaled_mass c hc z r rf ω
+  have he : {ω : a.samples | u < spatialRealMass κ (fun t => Real.sqrt κ*a'.B t ω)
+      (sourceScaleTrace c (γ ω)) (a'.G ω) ((c:ℝ)*rf) (ball ((c:ℂ)*z) 1)} =
+      {ω : a.samples | r^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) rf (ball z r)} := by
+    ext ω
+    change (u < _) ↔ _
+    rw [hx ω]
+    conv_lhs => rw [←hmr]
+    exact mul_lt_mul_iff_right₀ hp
+  rw [he] at htail
+  have hg : initialGreen κ ((c:ℂ)*z) = r^codimension κ*initialGreen κ z := by
+    rw [source_scale_initialGreen κ c hc z (by nlinarith),hcR,Real.inv_rpow hr.le,Real.rpow_neg hr.le,inv_inv]
+  rw [hg] at htail
+  convert htail using 1
+  congr 1
+  ring
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+def SourcePhysicalBallEveryKappaTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (r : ℝ),
+    0 < r → 4*r ≤ z₀.im → 1 ≤ Real.log ((2*z₀.im)/((128*Real.pi)*(4*r))) →
+  ∀ (rf u : ℝ), 0 < rf → rf ≤ r → 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | r^dimension κ*u < spatialRealMass κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (Metric.ball z₀ r)} ≤
+      ENNReal.ofReal (C*r^codimension κ*initialGreen κ z₀*Real.exp (-c*u^(2/codimension κ)))
+theorem sourcePhysicalBallEveryKappa_proved : SourcePhysicalBallEveryKappaTarget := by
+  intro κ hκ hκ8
+  by_cases hκ4 : κ ≤ 4
+  · exact sourceSimplePhysicalBallAllTail_proved κ hκ hκ4
+  · exact sourcePhysicalBallAllTail_proved κ (lt_of_not_ge hκ4) hκ8
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+def SourceActualSquareTailTarget_allKappa : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+  ∃ (A c : ℝ) (kmin : ℕ), 0 < A ∧ 0 < c ∧
+  ∀ (T : ℝ≥0) (N k : ℕ), kmin ≤ k → k ≤ N → ∀ ij : ℤ × ℤ, ∀ u : ℝ, 1 ≤ u →
+    (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
+      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω)
+        ((2:ℝ)^(-(N:ℝ))) T (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ≤
+    ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ)))
+theorem sourceActualSquareTail_proved_allKappa : SourceActualSquareTailTarget_allKappa := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀
+  obtain ⟨C,c,hC,hc,ht⟩ := sourcePhysicalBallEveryKappa_proved κ hκ hκ8
+  let H := z₀.im-2
+  have hH : 0 < H := sub_pos.mpr hz₀
+  let G := (2:ℝ)^(-codimension κ)*H^(-codimension κ)
+  have hG : 0 < G := mul_pos (Real.rpow_pos_of_pos (by norm_num) _) (Real.rpow_pos_of_pos hH _)
+  obtain ⟨kmin,hkmin⟩ := source_small_dyadic_level H hH
+  refine ⟨C*G,c,kmin,mul_pos hC hG,hc,?_⟩
+  intro T N k hk hkN ij u hu
+  let l := (2:ℝ)^(-(k:ℝ))
+  let rf := (2:ℝ)^(-(N:ℝ))
+  have hl : 0 < l := source_dyadic_side_pos k
+  have hf : 0 < rf := source_dyadic_side_pos N
+  have hfl : rf ≤ l := source_dyadic_side_le k N hkN
+  have hpos : 0 < l^dimension κ*u := mul_pos (Real.rpow_pos_of_pos hl _) (zero_lt_one.trans_le hu)
+  by_cases hmeet : (closedBall z₀ 1 ∩ retainedDyadicSquare k ij).Nonempty
+  · let z := sourceDyadicCenter k ij
+    have hheight : H ≤ z.im := source_square_center_height z₀ k ij hmeet
+    have hy : 4*l ≤ z.im := (hkmin k hk).1.trans hheight
+    have hsmall : 1 ≤ Real.log ((2*z.im)/((128*Real.pi)*(4*l))) := by
+      apply (Real.le_log_iff_exp_le (div_pos (by linarith) (by positivity))).mpr
+      apply (le_div_iff₀ (by positivity : 0 < (128*Real.pi)*(4*l))).mpr
+      exact (hkmin k hk).2.trans (mul_le_mul_of_nonneg_left hheight (by norm_num))
+    have hgreen : initialGreen κ z ≤ G := by
+      apply (initialGreen_le_height κ (by linarith) hκ8 z (hH.trans_le hheight)).trans
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
+      exact Real.rpow_le_rpow_of_nonpos hH hheight (neg_nonpos.mpr (codimension_pos κ hκ8).le)
+    have hh := ht Ω m P hP γ a z l hl hy hsmall rf u hf hfl hu
+    apply le_trans (measure_mono (show {ω : a.samples | l^dimension κ*u <
+      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
+        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} ⊆
+      {ω | l^dimension κ*u < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) rf (ball z l)} from ?_)) (hh.trans ?_)
+    · intro ω hω
+      exact hω.trans_le (source_finite_mass_le_ball a (by linarith) hκ8 z l rf hl hy hf hfl T
+        _ (fun w hw => source_square_subset_ball k ij hw.2) ω)
+    · apply ENNReal.ofReal_le_ofReal
+      calc
+        C*l^codimension κ*initialGreen κ z*Real.exp (-c*u^(2/codimension κ)) ≤
+            C*l^codimension κ*G*Real.exp (-c*u^(2/codimension κ)) := by gcongr
+        _ = _ := by ring
+  · have he : closedBall z₀ 1 ∩ retainedDyadicSquare k ij = ∅ := not_nonempty_iff_eq_empty.mp hmeet
+    have hempty : {ω : a.samples | l^dimension κ*u <
+      (finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf T
+        (closedBall z₀ 1 ∩ retainedDyadicSquare k ij)).toReal} = ∅ := by
+      simp only [he,finiteSpatialGreenMass,Measure.restrict_empty,lintegral_zero_measure,ENNReal.toReal_zero,
+        not_lt_of_ge hpos.le,Set.ofPred_false]
+    change (P.comap Subtype.val) _ ≤ _
+    rw [hempty,measure_empty]
+    exact bot_le
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+variable (a : RegularSLEPresentation P κ γ)
+lemma source_actual_retained_experiment_allKappa [IsProbabilityMeasure P] (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz₀ : 2 < z₀.im) :
+    ∃ (T : ℝ≥0) (D b p : ℝ) (k₀ : ℕ), 0 < T ∧ 1 ≤ D ∧ 0 < b ∧ 0 < p ∧
+      ENNReal.ofReal p ≤ (P.comap Subtype.val)
+        {ω : a.samples | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+          retainedMeasure κ D (a.sourceFiniteGreenKernel z₀ T) k₀ N ω (closedBall z₀ 1)} := by
+  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
+  have hk : 0 < κ := by linarith
+  obtain ⟨T,e₀,m₀,m₂,hT,he,hm₀,hm₂,hm⟩ := source_retained_seed_all κ hκ hκ8
+    Ω inferInstance P inferInstance γ a z₀ hz₀
+  obtain ⟨A,c,k₁,hA,hc,ht⟩ := sourceActualSquareTail_proved_allKappa κ hκ hκ8
+    Ω inferInstance P inferInstance γ a z₀ hz₀
+  obtain ⟨k₂,hk₂⟩ := source_dyadic_eventually_le he
+  let kmin := max k₁ k₂
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  have hν (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
+  have hfirst (N : ℕ) (hN : kmin ≤ N) :
+      ENNReal.ofReal m₀ ≤ ∫⁻ ω, ν N ω (closedBall z₀ 1) ∂(P.comap Subtype.val) := by
+    simp only [ν,a.sourceFiniteGreenKernel_K]
+    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).1
+  have hsecond (N : ℕ) (hN : kmin ≤ N) :
+      (∫⁻ ω, (ν N ω (closedBall z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
+    simp only [ν,a.sourceFiniteGreenKernel_K]
+    exact (hm _ (source_dyadic_side_pos N) (hk₂ N ((le_max_right _ _).trans hN))).2
+  have htail (N k : ℕ) (hkmin : kmin ≤ k) (hkN : k ≤ N) (ij : ℤ × ℤ) (u : ℝ) (hu : 1 ≤ u) :
+      (P.comap Subtype.val) {ω : a.samples | ((2:ℝ)^(-(k:ℝ)))^dimension κ*u <
+        (ν N ω (retainedDyadicSquare k ij)).toReal} ≤
+        ENNReal.ofReal (A*((2:ℝ)^(-(k:ℝ)))^codimension κ*Real.exp (-c*u^(2/codimension κ))) := by
+    simp only [ν,a.sourceFiniteGreenKernel_apply z₀ T N _ _ (retainedSquare_measurable k ij)]
+    exact ht T N k ((le_max_left _ _).trans hkmin) hkN ij u hu
+  obtain ⟨D,b,p,B,k₀,hD,hb,hp,hB,hk₀,hall,hlim⟩ := sourceRetainedConstruction a.samples
+    inferInstance (P.comap Subtype.val) inferInstance κ hk hκ8 ν hν
+    (closedBall z₀ 1) (isCompact_closedBall _ _) (fun N ω => a.sourceFiniteGreenKernel_supported z₀ T N ω)
+    A c m₀ m₂ hA hc hm₀ hm₂ kmin hfirst hsecond htail
+  exact ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+ 
+
+def SourceActualRetainedLowerTarget_allKappa : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
+    0 < (P.comap Subtype.val) E ∧
+    ∀ ω ∈ E, 0 < hausdorffGauge h (γ ω '' Set.Icc 0 T)
+theorem sourceActualRetainedLower_proved_allKappa : SourceActualRetainedLowerTarget_allKappa := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment_allKappa hκ hκ8 z₀ hz₀
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
+  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
+  have hE : MeasurableSet E := by
+    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
+        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
+      ext ω
+      simp [E,frequently_atTop]
+    rw [heq]
+    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
+      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
+  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
+  intro ω hω
+  exact a.source_actual_retained_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
+    k₀ h hh hf ω hω
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+ 
+
+def SourceInteriorRetainedTransferTarget_allKappa : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∃ (T : ℝ≥0) (E : Set a.samples), 0 < T ∧ MeasurableSet E ∧
+    0 < (P.comap Subtype.val) E ∧ ∀ ω ∈ E,
+      0 < hausdorffGauge h (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)) ∧
+      ∀ (U : ℝ≥0 → ℝ) (η : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
+        IsCapacityTwoWitness U η G F → ∀ s : ℝ≥0,
+          0 < hausdorffGauge h (F s '' (Metric.closedBall z₀ 1 ∩ (γ ω '' Set.Icc 0 T)))
+
+theorem sourceInteriorRetainedTransfer_proved_allKappa : SourceInteriorRetainedTransferTarget_allKappa := by
+  intro κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  obtain ⟨T,D,b,p,k₀,hT,hD,hb,hp,hlim⟩ := a.source_actual_retained_experiment_allKappa hκ hκ8 z₀ hz₀
+  let ν := a.sourceFiniteGreenKernel z₀ T
+  let (N : ℕ) : IsFiniteKernel (ν N) := a.sourceFiniteGreenKernel_finite hk hκ8 z₀ hz₀ T N
+  let E : Set a.samples := {ω | ∃ᶠ N in atTop, ENNReal.ofReal b ≤
+    retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)}
+  have hE : MeasurableSet E := by
+    have heq : E = ⋂ n : ℕ, ⋃ N ≥ n,
+        {ω | ENNReal.ofReal b ≤ retainedMeasure κ D ν k₀ N ω (closedBall z₀ 1)} := by
+      ext ω
+      simp [E,frequently_atTop]
+    rw [heq]
+    exact MeasurableSet.iInter (fun n => MeasurableSet.iUnion (fun N => MeasurableSet.iUnion
+      (fun _ => measurableSet_le measurable_const (retainedMeasure_measurable κ D ν k₀ N measurableSet_closedBall))))
+  refine ⟨T,E,hT,hE,(ENNReal.ofReal_pos.mpr hp).trans_le hlim,?_⟩
+  intro ω hω
+  have hpos := a.source_actual_retained_interior_pathwise hk hκ8 z₀ hz₀ T D b (zero_lt_one.trans_le hD) hb
+    k₀ h hh hf ω hω
+  refine ⟨hpos,?_⟩
+  intro U η G F hw s
+  exact source_inverse_chart_gauge_positive κ hk hκ8 h hh hf U η G F hw s
+    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed)
+    (fun z hz => zero_lt_one.trans (source_unit_ball_height hz₀ hz.1)) hpos
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance simplePastMeshDriverMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance simplePastMeshTraceMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance simplePastMeshDriverBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance simplePastMeshTraceBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_dilation_gauge_positive_allKappa (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    {c : ℂ} (hc : c ≠ 0) {S : Set ℂ} (hS : IsCompact S)
+    (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => c*z) '' S) := by
+  by_contra hn
+  have hz : hausdorffGauge h ((fun z => c*z) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le
+  have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
+    (D := univ) (f := fun z => c⁻¹*z) isOpen_univ (hS.image (continuous_const.mul continuous_id))
+    (subset_univ _) (by fun_prop) hz
+  have hi : (fun z => c⁻¹*z) '' ((fun z => c*z) '' S) = S := by
+    rw [←image_comp]
+    simp only [Function.comp_def,←mul_assoc,inv_mul_cancel₀ hc,one_mul,image_id']
+  change hausdorffGauge h ((fun z => c⁻¹*z) '' ((fun z => c*z) '' S)) = 0 at he
+  rw [hi] at he
+  exact hp.ne' he
+
+lemma RegularSLEPresentation.source_trace_path_measurable_allKappa
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) :
+    Measurable (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) := by
+  apply source_path_measurable
+  intro t
+  exact (a.trace_adapted t).mono (a.filtration.le t) le_rfl
+
+lemma RegularSLEPresentation.source_scale_trace_law_allKappa
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (c : ℝ≥0) (hc : 0 < c) :
+    (P.comap (Subtype.val : a.samples → Ω)).map
+      (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) =
+    (P.comap (Subtype.val : a.samples → Ω)).map
+      (sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
+        (a.sourceScaled c hc).trace_continuous) := by
+  exact sourceTraceLawUnique_proved κ hκ hκ8 a.B (a.sourceScaled c hc).B
+    a.brownian (a.sourceScaled c hc).brownian a.measurable_B (a.sourceScaled c hc).measurable_B
+    a.continuous_B (a.sourceScaled c hc).continuous_B _ _ a.G a.F
+    (a.sourceScaled c hc).G (a.sourceScaled c hc).F a.witness (a.sourceScaled c hc).witness
+ 
+
+def SourceUniformInteriorSeedTarget_allKappa : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (h : ℝ → ℝ), IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∃ p : ℝ≥0∞, 0 < p ∧ ∀ δ : ℝ≥0, 0 < δ →
+    ∃ K : Set ℂ, IsCompact K ∧ K ⊆ upperHalfPlane ∧
+      p ≤ (P.comap (Subtype.val : a.samples → Ω))
+        {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Set.Icc 0 δ))}
+theorem sourceUniformInteriorSeed_proved_allKappa : SourceUniformInteriorSeedTarget_allKappa := by
+  intro κ hκ hκ8 Ω m P hP γ a h hh hf
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  let z₀ : ℂ := 3*I
+  have hz₀ : 2 < z₀.im := by norm_num [z₀]
+  obtain ⟨T,E,hT,hE,hp,hseed⟩ :=
+    sourceInteriorRetainedTransfer_proved_allKappa κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  refine ⟨Q E,hp,?_⟩
+  intro δ hδ
+  let c := NNReal.sqrt (δ/T)
+  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
+  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
+  have ht : sourceScaleTime c δ = T := by
+    unfold sourceScaleTime
+    rw [hc2]
+    field_simp
+  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
+  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
+  have hKH : K ⊆ upperHalfPlane := by
+    rintro _ ⟨z,hz,rfl⟩
+    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
+    change 0 < ((c:ℂ)*z).im
+    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos (show 0 < (c:ℝ) from hc) hz'
+  refine ⟨K,hK,hKH,?_⟩
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
+    (a.sourceScaled c hc).trace_continuous
+  let A : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))}
+  have hA : MeasurableSet A := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ
+  have hm : Measurable gp := a.source_trace_path_measurable_allKappa
+  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable_allKappa
+  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law_allKappa hk hκ8 c hc)
+  rw [Measure.map_apply hm hA,Measure.map_apply hsm hA] at hmap
+  change Q E ≤ Q (gp ⁻¹' A)
+  rw [hmap]
+  apply measure_mono
+  intro ω hω
+  have hpos := (hseed ω hω).1
+  have heq : K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ) =
+      (fun z : ℂ => (c:ℂ)*z) '' (closedBall z₀ 1 ∩ (γ ω '' Icc 0 T)) := by
+    rw [source_scale_past c δ hc,ht, image_inter (mul_right_injective₀ (by exact_mod_cast hc.ne' : (c:ℂ) ≠ 0))]
+  change 0 < hausdorffGauge h (K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ))
+  rw [heq]
+  exact source_dilation_gauge_positive_allKappa κ hk hκ8 h hh hf (by exact_mod_cast hc.ne')
+    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed) hpos
+end
+end SLEExactGauge
+end
+
+ 
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SimpleTails.lean b/lean/OAI/Probability/SLE/SimpleTails.lean
index 3f67ff0189a4eb4f9429e45d4871978b0bfc593b..8a2f5c2b1ec901f5334d0c47ba4595c7bb8a9155 100644
--- a/lean/OAI/Probability/SLE/SimpleTails.lean
+++ b/lean/OAI/Probability/SLE/SimpleTails.lean
@@ -1,407 +1,407 @@
-import Mathlib
-import OAI.Probability.SLE.GreenCovariance
-
-namespace OAI
-
-/-! Simple Tails for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric
-namespace SLEExactGauge
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem spatial_slab_subgaussian [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    (q ε r R : ℝ) (hq : 0 < q) (hε : 0 < ε) (hqr : q*r ≤ ε)
-    (hεr : ε ≤ r) (hrR : r ≤ R)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) :
-    HasSubgaussianMGF (fun ω : a.samples => spatialSlabIncrement κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) A ε r)
-      (⟨slabAmplitude κ q A * r ^ (1 - codimension κ),
-        mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hε.trans_le hεr).le _)⟩ ^ 2)
-      (P.comap Subtype.val) := by
-  apply hasSubgaussianMGF_of_compensated (a.spatialSlabIncrement_measurable A ε r).aemeasurable
-  intro θ
-  have hv := slabVariance_scale_le A hκ (by linarith) hq (hε.trans_le hεr) hqr
-  apply le_trans _ (a.spatial_slab_terminal_exponential hκ hκ4 A hA ε r R hε hεr hrR hheight hbound θ)
-  apply lintegral_mono
-  intro ω
-  apply ENNReal.ofReal_le_ofReal
-  apply Real.exp_le_exp.mpr
-  change θ * _ - θ^2 * (slabAmplitude κ q A * r ^ (1 - codimension κ))^2 / 2 ≤ _
-  nlinarith [mul_le_mul_of_nonneg_left hv (sq_nonneg θ)]
-
-end RegularSLEPresentation
-end SLEExactGauge
-
- 
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-end SLEExactGauge
-
- 
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-end SLEExactGauge
-
- 
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-end SLEExactGauge
-
- 
-
-open Set Filter
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-lemma codimension_lt_one (κ : ℝ) (hκ : 0 < κ) : codimension κ < 1 := by
-  dsimp [codimension, dimension]
-  linarith
-
-lemma integrableOn_abs_heightPower (p R : ℝ) (hp : p < 1) :
-    IntegrableOn (fun y : ℝ => |y| ^ (-p)) (Icc (-R) R) volume := by
-  have hm : Measurable (fun y : ℝ => |y| ^ (-p)) := measurable_id.abs.pow measurable_const
-  have hl : LocallyIntegrable (fun y : ℝ => |y| ^ (-p)) volume := by
-    apply locallyIntegrable_of_norm_le_rpow (C := 1) (α := p) (by simp) (by simpa using hp) _ hm.aestronglyMeasurable
-    filter_upwards [] with y
-    simp only [Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg (abs_nonneg y) (-p)), one_mul, le_refl]
-  exact hl.integrableOn_isCompact isCompact_Icc
-
- 
-lemma integrable_height_power_box (p R : ℝ) (hp : p < 1) :
-    IntegrableOn (fun z : ℂ => |z.im| ^ (-p))
-      (Complex.measurableEquivRealProd ⁻¹' (Icc (-R) R ×ˢ Icc (-R) R)) volume := by
-  have h1 := (integrableOn_abs_heightPower p R hp).comp_snd (volume.restrict (Icc (-R) R))
-  rw [Measure.prod_restrict] at h1
-  have hpres := Complex.volume_preserving_equiv_real_prod.restrict_preimage
-    (measurableSet_Icc.prod measurableSet_Icc : MeasurableSet (Icc (-R) R ×ˢ Icc (-R) R))
-  exact (hpres.integrable_comp_emb Complex.measurableEquivRealProd.measurableEmbedding).mpr h1
-
-lemma integrable_height_power_ball (p R : ℝ) (hp : p < 1) :
-    IntegrableOn (fun z : ℂ => |z.im| ^ (-p)) (Metric.closedBall 0 R) volume := by
-  apply (integrable_height_power_box p R hp).mono_set
-  intro z hz
-  have hn : ‖z‖ ≤ R := by simpa only [Metric.mem_closedBall, dist_zero_right] using hz
-  have hre := (abs_le.mp ((Complex.abs_re_le_norm z).trans hn))
-  have him := (abs_le.mp ((Complex.abs_im_le_norm z).trans hn))
-  exact ⟨hre, him⟩
-
-lemma ae_im_ne_zero : ∀ᵐ z : ℂ ∂volume, z.im ≠ 0 := by
-  apply ae_iff.mpr
-  simp only [not_not]
-  have hproper : (LinearMap.ker Complex.imLm) ≠ ⊤ := by
-    intro h
-    have hi : Complex.I ∈ LinearMap.ker Complex.imLm := by rw [h]; trivial
-    simp at hi
-  exact Measure.addHaar_submodule volume (LinearMap.ker Complex.imLm) hproper
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal BigOperators
-namespace SLEExactGauge
-
-lemma geometric_amplitude_sum (C r q α : ℝ) (hC : 0 ≤ C) (hr : 0 < r)
-    (hq : 0 < q) (hq1 : q < 1) (hα : 0 < α) (N : ℕ) :
-    ∑ n ∈ Finset.range N, C * (r*q^n)^α ≤ C*r^α / (1-q^α) := by
-  have hs : q^α < 1 := Real.rpow_lt_one hq.le hq1 hα
-  have he (n : ℕ) : C*(r*q^n)^α = C*r^α*(q^α)^n := by
-    rw [Real.mul_rpow hr.le (pow_nonneg hq.le _), ← Real.rpow_pow_comm hq.le]
-    ring
-  simp_rw [he]
-  rw [← Finset.mul_sum]
-  have hgeom : ∑ n ∈ Finset.range N, (q^α)^n ≤ (1-q^α)⁻¹ := by
-    rw [geom_sum_eq (ne_of_lt hs)]
-    have hpos : 0 < 1-q^α := sub_pos.mpr hs
-    rw [show ((q^α)^N-1)/(q^α-1) = (1-(q^α)^N)/(1-q^α) by rw [div_eq_div_iff (sub_ne_zero.mpr (ne_of_lt hs)) hpos.ne']; ring]
-    rw [inv_eq_one_div]
-    exact div_le_div_of_nonneg_right (by have := pow_nonneg (Real.rpow_nonneg hq.le α) N; linarith) hpos.le
-  simpa only [div_eq_mul_inv] using mul_le_mul_of_nonneg_left hgeom
-    (mul_nonneg hC (Real.rpow_nonneg hr.le _))
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
- 
-
-theorem spatial_geometric_subgaussian [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
-    (q r R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hrR : r ≤ R)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) (N : ℕ) :
-    HasSubgaussianMGF (fun ω : a.samples =>
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
-      (NNReal.mk (slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)))
-        (div_nonneg (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg hr.le _))
-          (sub_nonneg.mpr (Real.rpow_lt_one hq.le hq1 (sub_pos.mpr (codimension_lt_one κ hκ))).le)) ^ 2)
-      (P.comap Subtype.val) := by
-  let rj : ℕ → ℝ := fun n => r*q^n
-  have hp (n : ℕ) : 0 < rj n := mul_pos hr (pow_pos hq n)
-  have hl (n : ℕ) : rj n ≤ r := by
-    simpa only [mul_one] using mul_le_mul_of_nonneg_left (pow_le_one₀ (n := n) hq.le hq1.le) hr.le
-  have he (n : ℕ) : rj (n+1) = q*rj n := by dsimp [rj]; rw [pow_succ]; ring
-  let X : ℕ → a.samples → ℝ := fun n ω => spatialSlabIncrement κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) A (rj (n+1)) (rj n)
-  let b : ℕ → ℝ≥0 := fun n => NNReal.mk (slabAmplitude κ q A*(rj n)^(1-codimension κ))
-    (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hp n).le _))
-  have hsub (n : ℕ) : HasSubgaussianMGF (X n) (b n^2) (P.comap Subtype.val) := by
-    apply a.spatial_slab_subgaussian hκ hκ4 A hA q (rj (n+1)) (rj n) R hq (hp _) (he n).symm.le
-    · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
-    · exact (hl n).trans hrR
-    · exact fun z hz => (hl n).trans (hheight z hz)
-    · exact hbound
-  have hsum := subgaussian_sum_amplitudes X b N (fun n _ => hsub n)
-  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ Metric.closedBall 0 R from
-    fun z hz => by simpa only [Metric.mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
-      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
-  have htel (ω : a.samples) : (∑ n ∈ Finset.range N, X n ω) =
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A := by
-    have hm (n : ℕ) : X n ω =
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj (n+1)) A -
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A := by
-      symm
-      apply a.spatialRealMass_sub_eq_slab hκ (by linarith) A hA hfin _ _ (hp _)
-      · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
-      · exact fun z hz => (hl n).trans (hheight z hz)
-    simp_rw [hm]
-    simpa only [rj, pow_zero, mul_one] using Finset.sum_range_sub
-      (fun n => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A) N
-  simp_rw [htel] at hsum
-  apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsum
-  apply pow_le_pow_left₀ (by positivity)
-  apply NNReal.coe_le_coe.mp
-  simpa only [NNReal.coe_sum, b, rj, NNReal.coe_mk] using geometric_amplitude_sum (slabAmplitude κ q A) r q (1-codimension κ)
-    (slabAmplitude_nonneg κ q A) hr hq hq1 (sub_pos.mpr (codimension_lt_one κ hκ)) N
-
-end RegularSLEPresentation
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma exists_exact_geometric_mesh {q δ ε : ℝ} (hq : 0 < q) (hq1 : q < 1)
-    (hε : 0 < ε) (hεδ : ε ≤ δ) :
-    ∃ (N : ℕ) (r : ℝ), q*δ < r ∧ r ≤ δ ∧ ε = r*q^N := by
-  have hδ : 0 < δ := hε.trans_le hεδ
-  obtain ⟨N, hN1, hN2⟩ := exists_nat_pow_near_of_lt_one (div_pos hε hδ)
-    ((div_le_one hδ).mpr hεδ) hq hq1
-  refine ⟨N, ε/q^N, ?_, ?_, ?_⟩
-  · apply (lt_div_iff₀ (pow_pos hq N)).mpr
-    have hh := (lt_div_iff₀ hδ).mp hN1
-    rw [pow_succ] at hh
-    nlinarith
-  · apply (div_le_iff₀ (pow_pos hq N)).mpr
-    simpa only [mul_comm] using (div_le_iff₀ hδ).mp hN2
-  · field_simp
-
- 
-
-lemma mesh_tail_exponent {δ B p : ℝ} (hδ : 0 < δ) (hB : 0 < B) :
-    -(δ^(-p))^2 / (2*(B*δ^(1-p))^2) = -(1/(2*B^2))*δ^(-2 : ℝ) := by
-  have he : δ ^ (1-p) = δ*δ^(-p) := by
-    rw [sub_eq_add_neg, Real.rpow_add hδ, Real.rpow_one]
-  rw [he, show δ^(-2 : ℝ) = (δ^2)⁻¹ by rw [Real.rpow_neg hδ.le, Real.rpow_two]]
-  have hp : δ^(-p) ≠ 0 := (Real.rpow_pos_of_pos hδ _).ne'
-  field_simp [hδ.ne', hB.ne', hp]
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def meshTailAmplitude (κ q : ℝ) (A : Set ℂ) : ℝ :=
-  1 + slabAmplitude κ q A / (1-q^(1-codimension κ))
-
-lemma meshTailAmplitude_pos {κ q : ℝ} (A : Set ℂ)
-    (hκ : 0 < κ) (hq : 0 < q) (hq1 : q < 1) : 0 < meshTailAmplitude κ q A := by
-  have hd : 0 < 1-q^(1-codimension κ) :=
-    sub_pos.mpr (Real.rpow_lt_one hq.le hq1 (sub_pos.mpr (codimension_lt_one κ hκ)))
-  have := div_nonneg (slabAmplitude_nonneg κ q A) hd.le
-  dsimp [meshTailAmplitude]
-  linarith
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
- 
-
-theorem half_plane_mesh_tail [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
-    (q ε δ R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hε : 0 < ε)
-    (hεδ : ε ≤ δ) (hδR : δ ≤ R)
-    (hheight : ∀ z ∈ A, δ ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) :
-    (P.comap Subtype.val) {ω : a.samples |
-      ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A} ≤
-      ENNReal.ofReal (Real.exp (-(1/(2*(meshTailAmplitude κ q A)^2))*δ^(-2 : ℝ))) := by
-  obtain ⟨N,r, hqr, hrδ, he⟩ := exists_exact_geometric_mesh hq hq1 hε hεδ
-  have hδ : 0 < δ := hε.trans_le hεδ
-  have hr : 0 < r := (mul_pos hq hδ).trans hqr
-  have hκ8 : κ < 8 := by linarith
-  have hp : 0 < codimension κ := codimension_pos κ hκ8
-  have hα : 0 < 1-codimension κ := sub_pos.mpr (codimension_lt_one κ hκ)
-  have hd : 0 < 1-q^(1-codimension κ) := sub_pos.mpr (Real.rpow_lt_one hq.le hq1 hα)
-  have hB := meshTailAmplitude_pos A hκ hq hq1
-  have ham : slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)) ≤
-      meshTailAmplitude κ q A*δ^(1-codimension κ) := by
-    calc
-      _ ≤ slabAmplitude κ q A*δ^(1-codimension κ)/(1-q^(1-codimension κ)) :=
-        div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left
-          (Real.rpow_le_rpow hr.le hrδ hα.le) (slabAmplitude_nonneg κ q A)) hd.le
-      _ ≤ _ := by
-        dsimp [meshTailAmplitude]
-        have hx := (Real.rpow_pos_of_pos hδ (1-codimension κ)).le
-        rw [add_mul, div_mul_eq_mul_div]
-        nlinarith
-  let B : ℝ≥0 := NNReal.mk (meshTailAmplitude κ q A*δ^(1-codimension κ))
-    (mul_nonneg hB.le (Real.rpow_nonneg hδ.le _))
-  have hsg := a.spatial_geometric_subgaussian hκ hκ4 A hA q r R hq hq1 hr
-    (hrδ.trans hδR) (fun z hz => hrδ.trans (hheight z hz)) hbound N
-  have hsg' : HasSubgaussianMGF (fun ω : a.samples =>
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
-      (B^2) (P.comap Subtype.val) := by
-    rw [he]
-    apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsg
-    apply pow_le_pow_left₀ (by positivity)
-    exact NNReal.coe_le_coe.mp ham
-  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ closedBall 0 R from
-    fun z hz => by simpa only [mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
-      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
-  have hstart (ω : a.samples) :
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A ≤
-      (volume A).toReal*q^(-codimension κ)*δ^(-codimension κ) := by
-    apply (a.spatialRealMass_bound hκ hκ8 r hr A hA hfin
-      (fun z hz => hrδ.trans (hheight z hz)) ω).2.trans
-    have hh := Real.rpow_le_rpow_of_nonpos (mul_pos hq hδ) hqr.le (neg_nonpos.mpr hp.le)
-    rw [Real.mul_rpow hq.le hδ.le] at hh
-    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh ENNReal.toReal_nonneg
-  apply (measure_mono (show _ ⊆ {ω : a.samples | δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A} from ?_)).trans
-  · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (Real.exp_pos _).le).mpr
-    have ht := hsg'.measure_ge_le (Real.rpow_nonneg hδ.le (-codimension κ))
-    change (P.comap Subtype.val).real _ ≤ _
-    simpa only [B, NNReal.coe_pow, NNReal.coe_mk, mesh_tail_exponent hδ hB] using ht
-  · intro ω hω
-    have := hstart ω
-    change ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A at hω
-    change δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A
-    nlinarith
-
-end RegularSLEPresentation
-end SLEExactGauge
-
-end
-end
-
- 
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.GreenCovariance
+
+namespace OAI
+
+/-! Simple Tails for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric
+namespace SLEExactGauge
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem spatial_slab_subgaussian [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    (q ε r R : ℝ) (hq : 0 < q) (hε : 0 < ε) (hqr : q*r ≤ ε)
+    (hεr : ε ≤ r) (hrR : r ≤ R)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) :
+    HasSubgaussianMGF (fun ω : a.samples => spatialSlabIncrement κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) A ε r)
+      (⟨slabAmplitude κ q A * r ^ (1 - codimension κ),
+        mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hε.trans_le hεr).le _)⟩ ^ 2)
+      (P.comap Subtype.val) := by
+  apply hasSubgaussianMGF_of_compensated (a.spatialSlabIncrement_measurable A ε r).aemeasurable
+  intro θ
+  have hv := slabVariance_scale_le A hκ (by linarith) hq (hε.trans_le hεr) hqr
+  apply le_trans _ (a.spatial_slab_terminal_exponential hκ hκ4 A hA ε r R hε hεr hrR hheight hbound θ)
+  apply lintegral_mono
+  intro ω
+  apply ENNReal.ofReal_le_ofReal
+  apply Real.exp_le_exp.mpr
+  change θ * _ - θ^2 * (slabAmplitude κ q A * r ^ (1 - codimension κ))^2 / 2 ≤ _
+  nlinarith [mul_le_mul_of_nonneg_left hv (sq_nonneg θ)]
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+ 
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+end SLEExactGauge
+
+ 
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+end SLEExactGauge
+
+ 
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+end SLEExactGauge
+
+ 
+
+open Set Filter
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+lemma codimension_lt_one (κ : ℝ) (hκ : 0 < κ) : codimension κ < 1 := by
+  dsimp [codimension, dimension]
+  linarith
+
+lemma integrableOn_abs_heightPower (p R : ℝ) (hp : p < 1) :
+    IntegrableOn (fun y : ℝ => |y| ^ (-p)) (Icc (-R) R) volume := by
+  have hm : Measurable (fun y : ℝ => |y| ^ (-p)) := measurable_id.abs.pow measurable_const
+  have hl : LocallyIntegrable (fun y : ℝ => |y| ^ (-p)) volume := by
+    apply locallyIntegrable_of_norm_le_rpow (C := 1) (α := p) (by simp) (by simpa using hp) _ hm.aestronglyMeasurable
+    filter_upwards [] with y
+    simp only [Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg (abs_nonneg y) (-p)), one_mul, le_refl]
+  exact hl.integrableOn_isCompact isCompact_Icc
+
+ 
+lemma integrable_height_power_box (p R : ℝ) (hp : p < 1) :
+    IntegrableOn (fun z : ℂ => |z.im| ^ (-p))
+      (Complex.measurableEquivRealProd ⁻¹' (Icc (-R) R ×ˢ Icc (-R) R)) volume := by
+  have h1 := (integrableOn_abs_heightPower p R hp).comp_snd (volume.restrict (Icc (-R) R))
+  rw [Measure.prod_restrict] at h1
+  have hpres := Complex.volume_preserving_equiv_real_prod.restrict_preimage
+    (measurableSet_Icc.prod measurableSet_Icc : MeasurableSet (Icc (-R) R ×ˢ Icc (-R) R))
+  exact (hpres.integrable_comp_emb Complex.measurableEquivRealProd.measurableEmbedding).mpr h1
+
+lemma integrable_height_power_ball (p R : ℝ) (hp : p < 1) :
+    IntegrableOn (fun z : ℂ => |z.im| ^ (-p)) (Metric.closedBall 0 R) volume := by
+  apply (integrable_height_power_box p R hp).mono_set
+  intro z hz
+  have hn : ‖z‖ ≤ R := by simpa only [Metric.mem_closedBall, dist_zero_right] using hz
+  have hre := (abs_le.mp ((Complex.abs_re_le_norm z).trans hn))
+  have him := (abs_le.mp ((Complex.abs_im_le_norm z).trans hn))
+  exact ⟨hre, him⟩
+
+lemma ae_im_ne_zero : ∀ᵐ z : ℂ ∂volume, z.im ≠ 0 := by
+  apply ae_iff.mpr
+  simp only [not_not]
+  have hproper : (LinearMap.ker Complex.imLm) ≠ ⊤ := by
+    intro h
+    have hi : Complex.I ∈ LinearMap.ker Complex.imLm := by rw [h]; trivial
+    simp at hi
+  exact Measure.addHaar_submodule volume (LinearMap.ker Complex.imLm) hproper
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal BigOperators
+namespace SLEExactGauge
+
+lemma geometric_amplitude_sum (C r q α : ℝ) (hC : 0 ≤ C) (hr : 0 < r)
+    (hq : 0 < q) (hq1 : q < 1) (hα : 0 < α) (N : ℕ) :
+    ∑ n ∈ Finset.range N, C * (r*q^n)^α ≤ C*r^α / (1-q^α) := by
+  have hs : q^α < 1 := Real.rpow_lt_one hq.le hq1 hα
+  have he (n : ℕ) : C*(r*q^n)^α = C*r^α*(q^α)^n := by
+    rw [Real.mul_rpow hr.le (pow_nonneg hq.le _), ← Real.rpow_pow_comm hq.le]
+    ring
+  simp_rw [he]
+  rw [← Finset.mul_sum]
+  have hgeom : ∑ n ∈ Finset.range N, (q^α)^n ≤ (1-q^α)⁻¹ := by
+    rw [geom_sum_eq (ne_of_lt hs)]
+    have hpos : 0 < 1-q^α := sub_pos.mpr hs
+    rw [show ((q^α)^N-1)/(q^α-1) = (1-(q^α)^N)/(1-q^α) by rw [div_eq_div_iff (sub_ne_zero.mpr (ne_of_lt hs)) hpos.ne']; ring]
+    rw [inv_eq_one_div]
+    exact div_le_div_of_nonneg_right (by have := pow_nonneg (Real.rpow_nonneg hq.le α) N; linarith) hpos.le
+  simpa only [div_eq_mul_inv] using mul_le_mul_of_nonneg_left hgeom
+    (mul_nonneg hC (Real.rpow_nonneg hr.le _))
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+ 
+
+theorem spatial_geometric_subgaussian [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
+    (q r R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hrR : r ≤ R)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) (N : ℕ) :
+    HasSubgaussianMGF (fun ω : a.samples =>
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
+      (NNReal.mk (slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)))
+        (div_nonneg (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg hr.le _))
+          (sub_nonneg.mpr (Real.rpow_lt_one hq.le hq1 (sub_pos.mpr (codimension_lt_one κ hκ))).le)) ^ 2)
+      (P.comap Subtype.val) := by
+  let rj : ℕ → ℝ := fun n => r*q^n
+  have hp (n : ℕ) : 0 < rj n := mul_pos hr (pow_pos hq n)
+  have hl (n : ℕ) : rj n ≤ r := by
+    simpa only [mul_one] using mul_le_mul_of_nonneg_left (pow_le_one₀ (n := n) hq.le hq1.le) hr.le
+  have he (n : ℕ) : rj (n+1) = q*rj n := by dsimp [rj]; rw [pow_succ]; ring
+  let X : ℕ → a.samples → ℝ := fun n ω => spatialSlabIncrement κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) A (rj (n+1)) (rj n)
+  let b : ℕ → ℝ≥0 := fun n => NNReal.mk (slabAmplitude κ q A*(rj n)^(1-codimension κ))
+    (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hp n).le _))
+  have hsub (n : ℕ) : HasSubgaussianMGF (X n) (b n^2) (P.comap Subtype.val) := by
+    apply a.spatial_slab_subgaussian hκ hκ4 A hA q (rj (n+1)) (rj n) R hq (hp _) (he n).symm.le
+    · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
+    · exact (hl n).trans hrR
+    · exact fun z hz => (hl n).trans (hheight z hz)
+    · exact hbound
+  have hsum := subgaussian_sum_amplitudes X b N (fun n _ => hsub n)
+  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ Metric.closedBall 0 R from
+    fun z hz => by simpa only [Metric.mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
+      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
+  have htel (ω : a.samples) : (∑ n ∈ Finset.range N, X n ω) =
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A := by
+    have hm (n : ℕ) : X n ω =
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj (n+1)) A -
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A := by
+      symm
+      apply a.spatialRealMass_sub_eq_slab hκ (by linarith) A hA hfin _ _ (hp _)
+      · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
+      · exact fun z hz => (hl n).trans (hheight z hz)
+    simp_rw [hm]
+    simpa only [rj, pow_zero, mul_one] using Finset.sum_range_sub
+      (fun n => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A) N
+  simp_rw [htel] at hsum
+  apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsum
+  apply pow_le_pow_left₀ (by positivity)
+  apply NNReal.coe_le_coe.mp
+  simpa only [NNReal.coe_sum, b, rj, NNReal.coe_mk] using geometric_amplitude_sum (slabAmplitude κ q A) r q (1-codimension κ)
+    (slabAmplitude_nonneg κ q A) hr hq hq1 (sub_pos.mpr (codimension_lt_one κ hκ)) N
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma exists_exact_geometric_mesh {q δ ε : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hε : 0 < ε) (hεδ : ε ≤ δ) :
+    ∃ (N : ℕ) (r : ℝ), q*δ < r ∧ r ≤ δ ∧ ε = r*q^N := by
+  have hδ : 0 < δ := hε.trans_le hεδ
+  obtain ⟨N, hN1, hN2⟩ := exists_nat_pow_near_of_lt_one (div_pos hε hδ)
+    ((div_le_one hδ).mpr hεδ) hq hq1
+  refine ⟨N, ε/q^N, ?_, ?_, ?_⟩
+  · apply (lt_div_iff₀ (pow_pos hq N)).mpr
+    have hh := (lt_div_iff₀ hδ).mp hN1
+    rw [pow_succ] at hh
+    nlinarith
+  · apply (div_le_iff₀ (pow_pos hq N)).mpr
+    simpa only [mul_comm] using (div_le_iff₀ hδ).mp hN2
+  · field_simp
+
+ 
+
+lemma mesh_tail_exponent {δ B p : ℝ} (hδ : 0 < δ) (hB : 0 < B) :
+    -(δ^(-p))^2 / (2*(B*δ^(1-p))^2) = -(1/(2*B^2))*δ^(-2 : ℝ) := by
+  have he : δ ^ (1-p) = δ*δ^(-p) := by
+    rw [sub_eq_add_neg, Real.rpow_add hδ, Real.rpow_one]
+  rw [he, show δ^(-2 : ℝ) = (δ^2)⁻¹ by rw [Real.rpow_neg hδ.le, Real.rpow_two]]
+  have hp : δ^(-p) ≠ 0 := (Real.rpow_pos_of_pos hδ _).ne'
+  field_simp [hδ.ne', hB.ne', hp]
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def meshTailAmplitude (κ q : ℝ) (A : Set ℂ) : ℝ :=
+  1 + slabAmplitude κ q A / (1-q^(1-codimension κ))
+
+lemma meshTailAmplitude_pos {κ q : ℝ} (A : Set ℂ)
+    (hκ : 0 < κ) (hq : 0 < q) (hq1 : q < 1) : 0 < meshTailAmplitude κ q A := by
+  have hd : 0 < 1-q^(1-codimension κ) :=
+    sub_pos.mpr (Real.rpow_lt_one hq.le hq1 (sub_pos.mpr (codimension_lt_one κ hκ)))
+  have := div_nonneg (slabAmplitude_nonneg κ q A) hd.le
+  dsimp [meshTailAmplitude]
+  linarith
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+ 
+
+theorem half_plane_mesh_tail [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
+    (q ε δ R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hε : 0 < ε)
+    (hεδ : ε ≤ δ) (hδR : δ ≤ R)
+    (hheight : ∀ z ∈ A, δ ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) :
+    (P.comap Subtype.val) {ω : a.samples |
+      ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A} ≤
+      ENNReal.ofReal (Real.exp (-(1/(2*(meshTailAmplitude κ q A)^2))*δ^(-2 : ℝ))) := by
+  obtain ⟨N,r, hqr, hrδ, he⟩ := exists_exact_geometric_mesh hq hq1 hε hεδ
+  have hδ : 0 < δ := hε.trans_le hεδ
+  have hr : 0 < r := (mul_pos hq hδ).trans hqr
+  have hκ8 : κ < 8 := by linarith
+  have hp : 0 < codimension κ := codimension_pos κ hκ8
+  have hα : 0 < 1-codimension κ := sub_pos.mpr (codimension_lt_one κ hκ)
+  have hd : 0 < 1-q^(1-codimension κ) := sub_pos.mpr (Real.rpow_lt_one hq.le hq1 hα)
+  have hB := meshTailAmplitude_pos A hκ hq hq1
+  have ham : slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)) ≤
+      meshTailAmplitude κ q A*δ^(1-codimension κ) := by
+    calc
+      _ ≤ slabAmplitude κ q A*δ^(1-codimension κ)/(1-q^(1-codimension κ)) :=
+        div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left
+          (Real.rpow_le_rpow hr.le hrδ hα.le) (slabAmplitude_nonneg κ q A)) hd.le
+      _ ≤ _ := by
+        dsimp [meshTailAmplitude]
+        have hx := (Real.rpow_pos_of_pos hδ (1-codimension κ)).le
+        rw [add_mul, div_mul_eq_mul_div]
+        nlinarith
+  let B : ℝ≥0 := NNReal.mk (meshTailAmplitude κ q A*δ^(1-codimension κ))
+    (mul_nonneg hB.le (Real.rpow_nonneg hδ.le _))
+  have hsg := a.spatial_geometric_subgaussian hκ hκ4 A hA q r R hq hq1 hr
+    (hrδ.trans hδR) (fun z hz => hrδ.trans (hheight z hz)) hbound N
+  have hsg' : HasSubgaussianMGF (fun ω : a.samples =>
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
+      (B^2) (P.comap Subtype.val) := by
+    rw [he]
+    apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsg
+    apply pow_le_pow_left₀ (by positivity)
+    exact NNReal.coe_le_coe.mp ham
+  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ closedBall 0 R from
+    fun z hz => by simpa only [mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
+      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
+  have hstart (ω : a.samples) :
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A ≤
+      (volume A).toReal*q^(-codimension κ)*δ^(-codimension κ) := by
+    apply (a.spatialRealMass_bound hκ hκ8 r hr A hA hfin
+      (fun z hz => hrδ.trans (hheight z hz)) ω).2.trans
+    have hh := Real.rpow_le_rpow_of_nonpos (mul_pos hq hδ) hqr.le (neg_nonpos.mpr hp.le)
+    rw [Real.mul_rpow hq.le hδ.le] at hh
+    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh ENNReal.toReal_nonneg
+  apply (measure_mono (show _ ⊆ {ω : a.samples | δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A} from ?_)).trans
+  · apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (Real.exp_pos _).le).mpr
+    have ht := hsg'.measure_ge_le (Real.rpow_nonneg hδ.le (-codimension κ))
+    change (P.comap Subtype.val).real _ ≤ _
+    simpa only [B, NNReal.coe_pow, NNReal.coe_mk, mesh_tail_exponent hδ hB] using ht
+  · intro ω hω
+    have := hstart ω
+    change ((volume A).toReal*q^(-codimension κ)+1)*δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A at hω
+    change δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) ε A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A
+    nlinarith
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+end
+end
+
+ 
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SmallKappaSausageMoments.lean b/lean/OAI/Probability/SLE/SmallKappaSausageMoments.lean
index aad37a8b5fb08649836e076ddb0a367b0a7ca9dc..2bf8ca9ba0c1051e356447496a69684d5575468d 100644
--- a/lean/OAI/Probability/SLE/SmallKappaSausageMoments.lean
+++ b/lean/OAI/Probability/SLE/SmallKappaSausageMoments.lean
@@ -1,845 +1,845 @@
-import Mathlib
-import OAI.Probability.SLE.RetainedSeedMoments
-import OAI.Probability.SLE.SimpleTails
-
-namespace OAI
-
-/-! Small Kappa Sausage Moments for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma RegularSLEPresentation.source_initial_unit_mesh_tail_small
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (z₀ : ℂ) (hz : 2 < z₀.im) :
-    ∃ C c d : ℝ, 0 < C ∧ 0 < c ∧ 0 < d ∧ d < 1 ∧
-      ∀ rf δ : ℝ, 0 < rf → rf ≤ δ → δ ≤ d →
-      (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) <
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
-        ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
-  let A := ball z₀ 1
-  let B := meshTailAmplitude κ (1/2) A
-  let C := max 1 ((volume A).toReal*(1/2:ℝ)^(-codimension κ)+1)
-  have hB : 0 < B := meshTailAmplitude_pos A hκ (by norm_num) (by norm_num)
-  have hC : 0 < C := zero_lt_one.trans_le (le_max_left _ _)
-  refine ⟨C,1/(2*B^2),1/2,hC,by positivity,by norm_num,by norm_num,?_⟩
-  intro rf δ hrf hrδ hδd
-  have hδ : 0 < δ := hrf.trans_le hrδ
-  have hR : δ ≤ ‖z₀‖+2 := by have := norm_nonneg z₀; linarith
-  have hh := a.half_plane_mesh_tail hκ hκ4 A measurableSet_ball (1/2) rf δ (‖z₀‖+2)
-    (by norm_num) (by norm_num) hrf hrδ hR
-    (fun z hz' => (show δ ≤ 1 by linarith).trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le)
-    (fun z hz' => by
-      have hz'' : dist z z₀ < 1 := hz'
-      have htri := dist_triangle z z₀ 0
-      simp only [dist_zero_right] at htri
-      linarith)
-  apply (measure_mono (show {ω : a.samples | C*δ^(-codimension κ) <
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf A} ⊆
-      {ω : a.samples | ((volume A).toReal*(1/2:ℝ)^(-codimension κ)+1)*δ^(-codimension κ) ≤
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf A} from
-      fun ω hω => ((mul_le_mul_of_nonneg_right (le_max_right _ _) (Real.rpow_nonneg hδ.le _)).trans_lt hω).le)).trans
-  apply hh.trans
-  apply ENNReal.ofReal_le_ofReal
-  have he := mul_le_mul_of_nonneg_right (show 1 ≤ C from le_max_left _ _)
-    (Real.exp_pos (-(1/(2*B^2))*δ^(-2:ℝ))).le
-  simpa only [one_mul] using he
-
-lemma RegularSLEPresentation.source_unit_mesh_second_moment_small
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (z₀ : ℂ) (hz : 2 < z₀.im) :
-    ∃ d m₂ : ℝ, 0 < d ∧ d < 1 ∧ 0 < m₂ ∧ ∀ rf : ℝ, 0 < rf → rf ≤ d →
-      (∫⁻ ω : a.samples, ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) rf (ball z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
-  obtain ⟨C,c,d,hC,hc,hd,hd1,ht⟩ := a.source_initial_unit_mesh_tail_small hκ hκ4 z₀ hz
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let v := (volume (ball z₀ 1)).toReal
-  let M := max C v*d^(-codimension κ)
-  let b := c*d^(-2:ℝ)
-  let A := max 1 C*Real.exp b
-  let α := b/M
-  have hκ8 : κ < 8 := by linarith
-  have hp : 0 < codimension κ := codimension_pos κ hκ8
-  have hp1 : codimension κ < 1 := by dsimp [codimension, dimension]; linarith
-  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hd _)
-  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hd _)
-  have hA : 0 < A := mul_pos (zero_lt_one.trans_le (le_max_left _ _)) (Real.exp_pos _)
-  have hα : 0 < α := div_pos hb hM
-  refine ⟨d,8*A/α^2,hd,hd1,by positivity,?_⟩
-  intro rf hf hfd
-  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) rf (ball z₀ 1)
-  have hx (ω : a.samples) : 0 ≤ X ω ∧ X ω ≤ v*rf^(-codimension κ) :=
-    a.spatialRealMass_bound hκ hκ8 rf hf (ball z₀ 1) measurableSet_ball
-      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne
-      (fun z hz' => (hfd.trans hd1.le).trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le) ω
-  have hlinear : ∀ t : ℝ, 1 ≤ t → μ {ω | M*t < X ω} ≤ ENNReal.ofReal (C*Real.exp (-b*t)) :=
-    source_mesh_tail_to_linear μ X (codimension κ) C c d v rf hp hp1.le hC hc hd
-      ENNReal.toReal_nonneg hf (fun ω => (hx ω).2)
-      (fun δ hfd hd' => ht rf δ hf hfd hd')
-  exact source_exponential_second_moment μ X (a.spatialRealMass_measurable _ _) (fun ω => (hx ω).1)
-    A α hA.le hα (source_linear_tail_all X M C b hM hC hb hlinear)
-
-lemma source_retained_seed_all (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (Ω : Type) (m : MeasurableSpace Ω) (P : Measure Ω) (hP : IsProbabilityMeasure P)
-    (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (hz : 2 < z₀.im) :
-    ∃ (T : ℝ≥0) (e₀ m₀ m₂ : ℝ), 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧ 0 < m₂ ∧
-      ∀ r : ℝ, 0 < r → r ≤ e₀ →
-      let X := fun ω : a.samples => finiteSpatialGreenMass κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r T (closedBall z₀ 1)
-      ENNReal.ofReal m₀ ≤ ∫⁻ ω, X ω ∂(P.comap Subtype.val) ∧
-      (∫⁻ ω, (X ω)^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
-  let : IsProbabilityMeasure P := hP
-  by_cases hκ4 : 4 < κ
-  · exact sourceRetainedSeed_proved κ hκ4 hκ8 Ω m P hP γ a z₀ hz
-  have hk4 : κ ≤ 4 := le_of_not_gt hκ4
-  have hK : closedBall z₀ 1 ⊆ upperHalfPlane :=
-    fun z hz' => zero_lt_one.trans (source_unit_ball_height hz hz')
-  obtain ⟨T,e₀,m₀,hT,he,hm,hfirst⟩ := a.source_finite_time_first_moment hκ hκ8 z₀ 1 (by norm_num) hK
-  obtain ⟨d,m₂,hd,hd1,hm₂,hsecond⟩ := a.source_unit_mesh_second_moment_small hκ hk4 z₀ hz
-  refine ⟨T,min e₀ d,m₀,m₂,hT,lt_min he hd,hm,hm₂,?_⟩
-  intro r hr hre
-  refine ⟨hfirst r hr (hre.trans (min_le_left _ _)),?_⟩
-  have hrd : r ≤ d := hre.trans (min_le_right _ _)
-  apply le_trans (lintegral_mono (fun ω => pow_le_pow_left' (source_finite_mass_le_unit_mass
-    a hκ hκ8 z₀ hz r hr (hrd.trans hd1.le) T ω) 2))
-  exact hsecond r hr hrd
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem source_localized_spatial_slab_past_exponential
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    [IsFiniteMeasure (volume.comap (Subtype.val : A → ℂ))]
-    (hfin : volume A ≠ ∞)
-    (ε r : ℝ) (hε : 0 < ε) (hεr : ε ≤ r)
-    (hheight : ∀ z ∈ A, r ≤ z.im)
-    (u δ : ℝ) (b : SpatialAngularFamily a.B a.measurable_B κ
-      (Subtype.val : A → ℂ) u δ)
-    (M : ℝ) (hM : ∀ z : A, ‖initialLogCoordinates z‖ ≤ M)
-    (S s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z : A, s ≤ boundedSpatialHit (γ ω) z r S)
-    (θ : ℝ) :
-    (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal (Real.exp
-      (θ * (∫ z : A,
-        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
-        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω
-        ∂(volume.comap (Subtype.val : A → ℂ))) -
-        θ^2 * slabVariance κ ε r A / 2)) else 1 ∂(P.comap Subtype.val)) ≤ 1 := by
-  classical
-  let ν := volume.comap (Subtype.val : A → ℂ)
-  classical
-  let σ₀ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 r S
-  let τ₀ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 ε S
-  let σ₁ (p : A × a.samples) := max (σ₀ p) s
-  let τ₁ (p : A × a.samples) := if p.2 ∈ D then max (τ₀ p) s else σ₁ p
-  let σ (p : A × a.samples) := min (σ₁ p) (b.τ p)
-  let τ (p : A × a.samples) := min (τ₁ p) (b.τ p)
-  have hσ₀ := a.joint_boundedSpatialHit_stopping (Subtype.val : A → ℂ) measurable_subtype_coe r S
-  have hτ₀ := a.joint_boundedSpatialHit_stopping (Subtype.val : A → ℂ) measurable_subtype_coe ε S
-  have hs := isStoppingTime_const (spatialFiltration (Z := A) a.filtration) s
-  have hσ₁ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-      (fun p => (σ₁ p : WithTop ℝ≥0)) := by
-    simpa only [σ₁, σ₀, WithTop.coe_max] using hσ₀.max hs
-  have hDm : MeasurableSet[spatialFiltration (Z := A) a.filtration s] {p : A × a.samples | p.2 ∈ D} := by
-    let : MeasurableSpace a.samples := a.filtration s
-    exact hD.preimage measurable_snd
-  have hτ₁ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-      (fun p => (τ₁ p : WithTop ℝ≥0)) := by
-    have htmax : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-        (fun p => ((max (τ₀ p) s : ℝ≥0) : WithTop ℝ≥0)) := by
-      simpa only [τ₀, WithTop.coe_max] using hτ₀.max hs
-    have hh := htmax.piecewise_of_le hσ₁
-      (fun p => WithTop.coe_le_coe.mpr (le_max_right (τ₀ p) s))
-      (fun p => WithTop.coe_le_coe.mpr (le_max_right (σ₀ p) s)) hDm
-    convert hh using 1
-    funext p
-    by_cases hp : p.2 ∈ D <;> simp [τ₁, Set.piecewise, hp]
-    rfl
-  have hσ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-      (fun p => (σ p : WithTop ℝ≥0)) := by
-    simpa only [σ, WithTop.coe_min] using hσ₁.min b.stopping
-  have hτ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-      (fun p => (τ p : WithTop ℝ≥0)) := by
-    simpa only [τ, WithTop.coe_min] using hτ₁.min b.stopping
-  have hord (p : A × a.samples) : σ₀ p ≤ τ₀ p := by
-    apply boundedSpatialHit_antitone_radius (a.trace_continuous p.2) p.1 hεr
-    rw [(a.witness p.2).2.1, dist_zero_left]
-    exact (hheight p.1 p.1.2).trans ((le_abs_self p.1.val.im).trans (Complex.abs_im_le_norm p.1))
-  have hστ (p : A × a.samples) : σ p ≤ τ p := by
-    apply min_le_min _ le_rfl
-    dsimp only [τ₁]
-    split_ifs
-    · exact max_le_max (hord p) le_rfl
-    · exact le_rfl
-  have hsD (ω : a.samples) (hω : ω ∈ D) (z : A) :
-      σ (z,ω) = min (boundedSpatialHit (γ ω) z r S) (b.τ (z,ω)) := by
-    simp only [σ, σ₁, σ₀, max_eq_left (hbefore ω hω z)]
-  have htD (ω : a.samples) (hω : ω ∈ D) (z : A) :
-      τ (z,ω) = min (boundedSpatialHit (γ ω) z ε S) (b.τ (z,ω)) := by
-    have hh := (hbefore ω hω z).trans (hord (z,ω))
-    simp only [τ, τ₁, τ₀, hω, ↓reduceIte, max_eq_left hh]
-  have hnotD (ω : a.samples) (hω : ω ∉ D) (z : A) : τ (z,ω) = σ (z,ω) := by
-    simp only [τ, τ₁, hω, ↓reduceIte, σ]
-  have htbound (p : A × a.samples) : τ p ≤ b.T := (min_le_right _ _).trans (b.bounded p)
-  have hin (z : A) (ω : a.samples) (t : ℝ≥0) (ht : t ≤ τ (z,ω)) :
-      b.lo ≤ ((b.c (initialLogCoordinates z)).Q t ω).1.im ∧
-        ((b.c (initialLogCoordinates z)).Q t ω).1.im ≤ b.hi :=
-    (b.inside z ω t (ht.trans (min_le_right _ _))).2.2
-  have hexp := clipped_spatial_green_window_unbounded_driver (ν := ν) a.measurable_B a.brownian
-    a.continuous_B a.zero_B κ b.lo b.hi b.le hκ (by linarith) b.c
-    (fun z : A => initialLogCoordinates z) (measurable_initialLogCoordinates.comp measurable_subtype_coe)
-    M hM b.T σ τ hσ hτ hστ htbound hin θ
-  let E (ω : a.samples) := ∫ t in (0 : ℝ)..(b.T : ℝ),
-    (∫ z : A, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
-      greenLogCoefficient κ ((b.c (initialLogCoordinates z)).Q t ω) (a.B t.toNNReal ω) else 0 ∂ν)^2
-  have hE (ω : a.samples) : E ω ≤ slabVariance κ ε r A := by
-    apply interval_square_energy_bound _ b.T _ (slabVariance_nonneg _ _ _ _)
-    rw [ofReal_slabVariance _ _ _ _ hfin]
-    apply (a.witness ω).spatial_masked_bracket_bound ((a.continuous_B ω).const_mul _)
-      hκ hκ4 hε (hε.trans_le hεr) A hA (fun z hz => hεr.trans (hheight z hz))
-    intro t ht z
-    have hb := clipped_spatial_window_coefficient_le b.c a.continuous_B z
-      (show z.val ∈ upperHalfPlane from hε.trans_le (hεr.trans (hheight z z.2))) ω (b.τ (z,ω)) S
-      (γ ω) (a.G ω) (a.F ω) (a.witness ω)
-      (fun s hs => (b.inside z ω s hs).2.2) r ε t.toNNReal
-    by_cases hω : ω ∈ D
-    · simpa only [hsD ω hω z, htD ω hω z, ← NNReal.coe_le_coe,
-        ← NNReal.coe_lt_coe, Real.coe_toNNReal _ ht] using hb
-    · simp only [hnotD ω hω z]
-      have hn (v : ℝ) : ¬ ((σ (z,ω) : ℝ) ≤ v ∧ v < (σ (z,ω) : ℝ)) := fun hh => (not_lt_of_ge hh.1) hh.2
-      simp only [hn t, ↓reduceIte, abs_zero]
-      exact (abs_nonneg _)
-
-  have hEzero (ω : a.samples) (hω : ω ∉ D) : E ω = 0 := by
-    have hf (t : ℝ) (z : A) : ¬ ((σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ)) := by
-      rw [hnotD ω hω z]
-      exact fun hh => (not_lt_of_ge hh.1) hh.2
-    simp only [E, hf, ↓reduceIte, integral_zero, zero_pow (by norm_num : 2 ≠ 0), intervalIntegral.integral_zero]
-  apply le_trans _ hexp
-  apply lintegral_mono
-  intro ω
-  by_cases hω : ω ∈ D
-  · simp only [hω, ↓reduceIte]
-    apply ENNReal.ofReal_le_ofReal
-    apply Real.exp_le_exp.mpr
-    change θ * _ - θ^2 * slabVariance κ ε r A / 2 ≤ θ * _ - θ^2 * E ω / 2
-    have hid : (∫ z : A,
-        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
-        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω ∂ν) =
-        ∫ z : A, greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (τ (z,ω)) ω) (a.B (τ (z,ω)) ω) -
-          greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (σ (z,ω)) ω) (a.B (σ (z,ω)) ω) ∂ν := by
-      apply integral_congr_ae
-      exact Eventually.of_forall fun z => by
-        simp only [AngularLocalization.physicalWeight, SpatialAngularFamily.section,
-          hsD ω hω z, htD ω hω z, NNReal.coe_min]
-    rw [hid]
-    nlinarith [mul_le_mul_of_nonneg_left (hE ω) (sq_nonneg θ)]
-  · simp only [hω, ↓reduceIte]
-    change (1:ℝ≥0∞) ≤ ENNReal.ofReal (Real.exp (θ*_ - θ^2*E ω/2))
-    simp only [hnotD ω hω, sub_self, integral_zero, hEzero ω hω, mul_zero,
-      zero_div, Real.exp_zero, ENNReal.ofReal_one, le_refl]
-
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_past_exponential_fatou
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (D : Set Ω) (hD : MeasurableSet D) (X : ℕ → Ω → ℝ) (Y : Ω → ℝ)
-    (θ V : ℝ) (hm : ∀ n, Measurable (X n))
-    (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
-    (hexp : ∀ n, (∫⁻ ω, if ω ∈ D then ENNReal.ofReal
-      (Real.exp (θ*X n ω-θ^2*V/2)) else 1 ∂P) ≤ 1) :
-    (∫⁻ ω, if ω ∈ D then ENNReal.ofReal
-      (Real.exp (θ*Y ω-θ^2*V/2)) else 1 ∂P) ≤ 1 := by
-  let f (n : ℕ) (ω : Ω) := if ω ∈ D then
-    ENNReal.ofReal (Real.exp (θ*X n ω-θ^2*V/2)) else 1
-  have hfm (n : ℕ) : Measurable (f n) :=
-    (((((hm n).const_mul θ).sub_const (θ^2*V/2)).exp).ennreal_ofReal).ite hD measurable_const
-  have hl : ∀ᵐ ω ∂P, (if ω ∈ D then ENNReal.ofReal
-      (Real.exp (θ*Y ω-θ^2*V/2)) else 1) = liminf (fun n => f n ω) atTop := by
-    filter_upwards [hlim] with ω hω
-    by_cases hmem : ω ∈ D
-    · simp only [hmem, ↓reduceIte, f]
-      exact ((ENNReal.continuous_ofReal.tendsto _).comp
-        ((Real.continuous_exp.tendsto _).comp
-          ((hω.const_mul θ).sub_const (θ^2*V/2)))).liminf_eq.symm
-    · simp only [hmem, ↓reduceIte, f, liminf_const]
-  calc
-    _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := lintegral_congr_ae hl
-    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' (fun n => (hfm n).aemeasurable)
-    _ ≤ 1 := liminf_le_of_frequently_le (Frequently.of_forall hexp)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem source_past_spatial_slab_finite_exponential_of_localizations
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    [IsFiniteMeasure (volume.comap (Subtype.val : A → ℂ))]
-    (hfin : volume A ≠ ∞)
-    (ε r : ℝ) (hε : 0 < ε) (hεr : ε ≤ r)
-    (hheight : ∀ z ∈ A, r ≤ z.im)
-    (u : ℝ) (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0))
-    (b : ∀ n, SpatialAngularFamily a.B a.measurable_B κ (Subtype.val : A → ℂ) u (δ n))
-    (l : ℝ) (hl : ∀ z : A, l ≤ Real.log z.val.im)
-    (hlevel : ∀ z : A, (2 * z.val.im) * Real.exp (-u) < ε)
-    (M : ℝ) (hM : ∀ z : A, ‖initialLogCoordinates z‖ ≤ M)
-    (S s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z : A, s ≤ boundedSpatialHit (γ ω) z r S)
-    (θ : ℝ) :
-    (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal (Real.exp
-      (θ * (∫ z : A,
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z
-        ∂(volume.comap (Subtype.val : A → ℂ))) -
-        θ^2 * slabVariance κ ε r A / 2)) else 1 ∂(P.comap Subtype.val)) ≤ 1 := by
-  let ν := volume.comap (Subtype.val : A → ℂ)
-  let σ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 r S
-  let τ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 ε S
-  have hσ : Measurable σ := stopping_time_measurable_nnreal
-    (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe r S)
-  have hτ : Measurable τ := stopping_time_measurable_nnreal
-    (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe ε S)
-  let X (n : ℕ) (ω : a.samples) := ∫ z : A,
-        ((b n).section z).physicalWeight (τ (z,ω)) ω -
-        ((b n).section z).physicalWeight (σ (z,ω)) ω ∂ν
-  let Y (ω : a.samples) := ∫ z : A,
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (τ (z,ω)) z -
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (σ (z,ω)) z ∂ν
-  have hXm (n : ℕ) : Measurable (X n) :=
-    (((b n).physicalWeight_joint_measurable a.continuous_B measurable_subtype_coe τ hτ).sub
-      ((b n).physicalWeight_joint_measurable a.continuous_B measurable_subtype_coe σ hσ)).stronglyMeasurable.integral_prod_left'.measurable
-  have hlim (ω : a.samples) : Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)) := by
-    have hh := spatial_angular_weight_integral_tendsto (ν := ν) b hδ a.continuous_B
-      measurable_subtype_coe hκ (by linarith) (fun z => hε.trans_le (hεr.trans (hheight z z.2)))
-      l hl ω (γ ω) (a.G ω) (a.F ω) (a.witness ω) σ τ hσ hτ
-    convert hh using 1
-    congr 1
-    apply integral_congr_ae
-    apply Eventually.of_forall
-    intro z
-    dsimp only
-    rw [(a.witness ω).physicalGreen_eq_before_spatial ((a.continuous_B ω).const_mul _)
-      κ z ε hε (hεr.trans (hheight z z.2)) u (hlevel z) (τ (z,ω)) (boundedSpatialHit_le _ _ _ _),
-      (a.witness ω).physicalGreen_eq_before_spatial ((a.continuous_B ω).const_mul _)
-      κ z r (hε.trans_le hεr) (hheight z z.2) u ((hlevel z).trans_le hεr)
-        (σ (z,ω)) (boundedSpatialHit_le _ _ _ _)]
-  apply source_past_exponential_fatou D ((a.filtration.le s) _ hD) X Y θ
-    (slabVariance κ ε r A) hXm (Eventually.of_forall hlim)
-  intro n
-  exact a.source_localized_spatial_slab_past_exponential hκ hκ4 A hA hfin ε r hε hεr hheight u (δ n) (b n) M hM S s D hD hbefore θ
-
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem source_past_spatial_slab_finite_exponential
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    (ε r R : ℝ) (hε : 0 < ε) (hεr : ε ≤ r) (hrR : r ≤ R)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
-    (S s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z : A, s ≤ boundedSpatialHit (γ ω) z r S)
-    (θ : ℝ) :
-    (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal (Real.exp
-      (θ * (∫ z in A,
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z) -
-        θ^2 * slabVariance κ ε r A / 2)) else 1 ∂(P.comap Subtype.val)) ≤ 1 := by
-  have hr : 0 < r := hε.trans_le hεr
-  have hR : 0 < R := hr.trans_le hrR
-  have hs : A ⊆ closedBall 0 R := by
-    intro z hz
-    simpa only [mem_closedBall, dist_zero_right] using hbound z hz
-  have hfin : volume A ≠ ∞ := ne_of_lt
-    ((measure_mono hs).trans_lt (isCompact_closedBall (0 : ℂ) R).measure_lt_top)
-  let ν := volume.comap (Subtype.val : A → ℂ)
-  let : IsFiniteMeasure ν := ⟨by
-    have hh := Measure.comap_apply_le (Subtype.val : A → ℂ) volume
-      (show NullMeasurableSet (univ : Set A) ν from MeasurableSet.univ.nullMeasurableSet)
-    apply lt_of_le_of_lt hh
-    simpa only [Set.image_univ, Subtype.range_coe] using lt_top_iff_ne_top.mpr hfin⟩
-  obtain ⟨u, hu, hul⟩ := exists_uniform_radius_level ε R hε hR.le
-  let α := r^2 / R^2
-  have hα : 0 < α := by dsimp [α]; positivity
-  let δ (n : ℕ) := α / ((n : ℝ) + 1)
-  have hδpos (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity
-  have hδle (n : ℕ) : δ n ≤ α := div_le_self hα.le (by linarith [Nat.cast_nonneg (α := ℝ) n])
-  have hδlim : Tendsto δ atTop (𝓝 0) := by
-    simpa only [mul_one_div, mul_zero] using
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul α
-  have hbounds (z : A) := initial_spatial_bounds κ r R hr hR z (hheight z z.2) (hbound z z.2)
-  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
-  let b (n : ℕ) : SpatialAngularFamily a.B a.measurable_B κ (Subtype.val : A → ℂ) u (δ n) :=
-    (exists_spatial_angular_family a.measurable_B hBm a.continuous_B a.zero_B κ Subtype.val
-      measurable_subtype_coe (Real.log r) (Real.log R) (Real.log_le_log hr hrR)
-      (fun z => (hbounds z).1) u (δ n) hu (hδpos n)
-      (fun z => (hδle n).trans (hbounds z).2.1)).some
-  have hlevel (z : A) : (2 * z.val.im) * Real.exp (-u) < ε := by
-    have hi : z.val.im ≤ R := (le_abs_self _).trans ((Complex.abs_im_le_norm z).trans (hbound z z.2))
-    exact (mul_le_mul_of_nonneg_right (by linarith : 2 * z.val.im ≤ 2 * R) (Real.exp_pos _).le).trans_lt hul
-  have hh := a.source_past_spatial_slab_finite_exponential_of_localizations hκ hκ4 A hA hfin ε r hε hεr
-    hheight u δ hδlim b (Real.log r) (fun z => (hbounds z).1.1) hlevel
-    (R + |Real.log r| + |Real.log R|) (fun z => (hbounds z).2.2) S s D hD hbefore θ
-  have he (ω : a.samples) :
-      (∫ z : A,
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z
-        ∂(volume.comap (Subtype.val : A → ℂ))) =
-      ∫ z in A,
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
-        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z :=
-    integral_subtype_comap (μ := volume) hA (fun z =>
-      greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
-      greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z)
-  simpa only [he] using hh
-
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-lemma source_past_spatial_slab_terminal_exponential [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    (ε r R : ℝ) (hε : 0 < ε) (hεr : ε ≤ r) (hrR : r ≤ R)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
-    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z r)
-    (θ : ℝ) :
-    (∫⁻ ω in D, ENNReal.ofReal (Real.exp
-      (θ*spatialSlabIncrement κ (fun t => Real.sqrt κ*a.B t ω)
-        (γ ω) (a.G ω) A ε r - θ^2*slabVariance κ ε r A/2))
-      ∂(P.comap Subtype.val)) ≤ (P.comap Subtype.val) D := by
-  have hs : A ⊆ closedBall 0 R := by
-    intro z hz
-    simpa only [mem_closedBall, dist_zero_right] using hbound z hz
-  have hfin : volume A ≠ ∞ := ne_of_lt
-    ((measure_mono hs).trans_lt (isCompact_closedBall (0 : ℂ) R).measure_lt_top)
-  obtain ⟨N,hN⟩ := exists_nat_ge s
-  let X (n : ℕ) (ω : a.samples) := ∫ z in A,
-    greenDensity κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε (n+N)) z -
-    greenDensity κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r (n+N)) z
-  let Y (ω : a.samples) := spatialSlabIncrement κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) A ε r
-  have hXm (n : ℕ) : Measurable (X n) :=
-    ((a.boundedSpatialGreen_measurable ε (n+N)).sub
-      (a.boundedSpatialGreen_measurable r (n+N))).stronglyMeasurable.integral_prod_left'.measurable
-  have hDm : MeasurableSet D := (a.filtration.le s) _ hD
-  have hmask : (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal
-      (Real.exp (θ*Y ω-θ^2*slabVariance κ ε r A/2)) else 1 ∂P.comap Subtype.val) ≤ 1 := by
-    apply source_past_exponential_fatou D hDm X Y θ (slabVariance κ ε r A) hXm
-    · filter_upwards [a.spatial_slab_area_tendsto hκ (by linarith) A hA hfin ε r R hε hεr hrR hheight hbound] with ω hω
-      simpa only [X, Y, spatialSlabIncrement, Function.comp_def, Nat.cast_add] using hω.comp (tendsto_add_atTop_nat N)
-    · intro n
-      apply a.source_past_spatial_slab_finite_exponential hκ hκ4 A hA ε r R hε hεr hrR hheight hbound (n+N) s D hD
-      intro ω hω z
-      apply WithTop.coe_le_coe.mp
-      rw [boundedSpatialHit_coe]
-      apply le_min (hbefore ω hω z z.2)
-      exact WithTop.coe_le_coe.mpr (hN.trans (le_add_of_nonneg_left (by positivity)))
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let f (ω : a.samples) := ENNReal.ofReal (Real.exp (θ*Y ω-θ^2*slabVariance κ ε r A/2))
-  change (∫⁻ ω, if ω ∈ D then f ω else 1 ∂μ) ≤ 1 at hmask
-  rw [←lintegral_add_compl _ hDm] at hmask
-  have hleft : (∫⁻ ω in D, if ω ∈ D then f ω else 1 ∂μ) = ∫⁻ ω in D, f ω ∂μ :=
-    setLIntegral_congr_fun hDm (fun ω hω => ite_eq_left hω)
-  have hright : (∫⁻ ω in Dᶜ, if ω ∈ D then f ω else 1 ∂μ) = μ Dᶜ := by
-    rw [setLIntegral_congr_fun hDm.compl (fun ω hω => ite_eq_right hω)]
-    simp
-  rw [hleft,hright] at hmask
-  have hsum : μ D+μ Dᶜ = 1 := by rw [measure_add_measure_compl hDm,measure_univ]
-  exact (ENNReal.add_le_add_iff_right (measure_ne_top μ Dᶜ)).mp (hmask.trans_eq hsum.symm)
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_infinite_stopping_grid_piece
-    {Ω : Type*} [m : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 m}
-    {ρ : Ω → WithTop ℝ≥0} (hρ : IsStoppingTime ℱ ρ) (n k : ℕ) :
-    MeasurableSet[ℱ (stopGridTime n k)]
-      ({ω | ρ ω ≠ ⊤} ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) = k}) := by
-  let A := {ω | ρ ω ≠ ⊤}
-  have hle (j : ℕ) : MeasurableSet[ℱ (stopGridTime n j)]
-      (A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) ≤ j}) := by
-    convert hρ (stopGridTime n j) using 1
-    ext ω
-    cases he : ρ ω with
-    | top => simp only [A,mem_inter_iff,mem_ofPred_eq,he,ne_eq,not_true_eq_false,false_and,WithTop.not_top_le_coe]
-    | coe t => simp only [A,mem_inter_iff,mem_ofPred_eq,he,WithTop.coe_ne_top,ne_eq,not_false_eq_true,true_and,WithTop.untopD_coe,stopGridIndex_le,WithTop.coe_le_coe]
-  cases k with
-  | zero => simpa only [Nat.le_zero] using hle 0
-  | succ k =>
-    have he : A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) = k+1} =
-        (A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) ≤ k+1}) \
-          (A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) ≤ k}) := by
-      ext ω
-      simp only [mem_inter_iff,mem_ofPred_eq,Set.mem_sdiff]
-      constructor
-      · rintro ⟨hω,he⟩; exact ⟨⟨hω,he.le⟩,fun hn => by omega⟩
-      · rintro ⟨⟨hω,hle⟩,hn⟩
-        exact ⟨hω,by have hh : ¬stopGridIndex n ((ρ ω).untopD 0) ≤ k := fun hk => hn ⟨hω,hk⟩; omega⟩
-    change MeasurableSet[ℱ (stopGridTime n (k+1))] (A ∩ _)
-    rw [he]
-    exact (hle (k+1)).diff (ℱ.mono (monotone_stopGridTime n (Nat.le_succ k)) _ (hle k))
-
- 
-
-lemma source_between_stopping_tail
-    {Ω : Type*} [m : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    {ℱ : Filtration ℝ≥0 m} {ρ τ : Ω → WithTop ℝ≥0}
-    (hρ : IsStoppingTime ℱ ρ) (hτ : IsStoppingTime ℱ τ)
-    (H : Set Ω) (hH : MeasurableSet H)
-    (hHA : H ⊆ {ω | ρ ω ≠ ⊤})
-    (hsep : ∀ ω, ρ ω ≠ ⊤ → ρ ω < τ ω)
-    (K : ℝ≥0∞)
-    (ht : ∀ (s : ℝ≥0) (D : Set Ω), MeasurableSet[ℱ s] D →
-      (∀ ω ∈ D, (s : WithTop ℝ≥0) ≤ τ ω) → μ (D ∩ H) ≤ μ D*K) :
-    μ H ≤ μ {ω | ρ ω ≠ ⊤}*K := by
-  let A := {ω | ρ ω ≠ ⊤}
-  let Q (n k : ℕ) := A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) = k}
-  let D (n k : ℕ) := Q n k ∩ {ω | (stopGridTime n k : WithTop ℝ≥0) < τ ω}
-  let E (n : ℕ) := ⋃ k, D n k
-  have hQ (n k : ℕ) : MeasurableSet[ℱ (stopGridTime n k)] (Q n k) :=
-    source_infinite_stopping_grid_piece hρ n k
-  have hD (n k : ℕ) : MeasurableSet[ℱ (stopGridTime n k)] (D n k) :=
-    (hQ n k).inter (by simpa only [compl_ofPred,not_le] using (hτ (stopGridTime n k)).compl)
-  have hDm (n k : ℕ) : MeasurableSet (D n k) := (ℱ.le _) _ (hD n k)
-  have hEm (n : ℕ) : MeasurableSet (E n) := MeasurableSet.iUnion (hDm n)
-  have hEq (n : ℕ) : ⋃ k, Q n k = A := grid_partition_union A n
-  have hEd (n : ℕ) : Pairwise (fun i j => Disjoint (Q n i) (Q n j)) := grid_partition_disjoint A n
-  have hb (n : ℕ) : μ (E n ∩ H) ≤ μ A*K := by
-    calc
-      μ (E n ∩ H) = μ (⋃ k, D n k ∩ H) := by simp only [E,iUnion_inter]
-      _ ≤ ∑' k, μ (D n k ∩ H) := measure_iUnion_le _
-      _ ≤ ∑' k, μ (Q n k)*K := ENNReal.tsum_le_tsum fun k =>
-        (ht (stopGridTime n k) (D n k) (hD n k) (fun ω hω => hω.2.le)).trans
-          (mul_le_mul' (measure_mono inter_subset_left) le_rfl)
-      _ = (∑' k, μ (Q n k))*K := ENNReal.tsum_mul_right
-      _ = μ A*K := by rw [←measure_iUnion (hEd n) (fun k => (ℱ.le _) _ (hQ n k)),hEq]
-  have hevent (ω : Ω) (hω : ω ∈ A) : ∀ᶠ n in atTop, ω ∈ E n := by
-    obtain ⟨t,ht'⟩ := WithTop.ne_top_iff_exists.mp hω
-    have ht : ρ ω = (t : WithTop ℝ≥0) := ht'.symm
-    have hlt := hsep ω hω
-    rw [ht] at hlt
-    have hgrid : ∀ᶠ n in atTop, (stopGrid n t : WithTop ℝ≥0) < τ ω := by
-      cases hv : τ ω with
-      | top => exact Eventually.of_forall fun n => WithTop.coe_lt_top _
-      | coe v =>
-        rw [hv] at hlt
-        have hh : ∀ᶠ n in atTop, stopGrid n t < v := (stopGrid_tendsto t) (Iio_mem_nhds (WithTop.coe_lt_coe.mp hlt))
-        exact hh.mono fun n hn => by simpa only [hv,WithTop.coe_lt_coe] using hn
-    filter_upwards [hgrid] with n hn
-    refine mem_iUnion.mpr ⟨stopGridIndex n t, ?_⟩
-    exact ⟨⟨hω,by change stopGridIndex n ((ρ ω).untopD 0) = stopGridIndex n t; rw [ht]; rfl⟩,hn⟩
-  let f (n : ℕ) := (E n ∩ H).indicator (fun _ : Ω => (1 : ℝ≥0∞))
-  have hfm (n : ℕ) : Measurable (f n) := measurable_const.indicator ((hEm n).inter hH)
-  have hp : H.indicator (fun _ : Ω => (1 : ℝ≥0∞)) ≤ fun ω => liminf (fun n => f n ω) atTop := by
-    intro ω
-    by_cases hω : ω ∈ H
-    · rw [indicator_of_mem hω]
-      apply le_liminf_of_le (by isBoundedDefault)
-      filter_upwards [hevent ω (hHA hω)] with n hn
-      simp only [f,indicator_of_mem (show ω ∈ E n ∩ H from ⟨hn,hω⟩),le_refl]
-    · simp only [indicator_of_notMem hω]
-      exact bot_le
-  calc
-    μ H = ∫⁻ ω, H.indicator (fun _ => (1 : ℝ≥0∞)) ω ∂μ := by simp [hH]
-    _ ≤ ∫⁻ ω, liminf (fun n => f n ω) atTop ∂μ := lintegral_mono hp
-    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂μ) atTop := lintegral_liminf_le' (fun n => (hfm n).aemeasurable)
-    _ ≤ μ A*K := liminf_le_of_frequently_le (Frequently.of_forall fun n => by
-      simpa only [f,lintegral_indicator ((hEm n).inter hH),lintegral_const,Measure.restrict_apply_univ,one_mul] using hb n)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-lemma source_simple_sphere_stopping (z : ℂ) (r : ℝ) :
-    IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) := by
-  let X (t : ℝ≥0) (ω : a.samples) := dist (γ ω t) z-r
-  have ha : Adapted a.filtration X := by
-    intro t
-    let : MeasurableSpace a.samples := a.filtration t
-    exact ((a.trace_adapted t).dist measurable_const).sub_const _
-  exact continuous_zero_hit_stopping a.filtration X ha
-    (fun ω => ((a.trace_continuous ω).dist continuous_const).sub continuous_const) 0
-lemma source_simple_center_before (z₀ : ℂ) (hz : 4 ≤ z₀.im) (r : ℝ) (hr : r ≤ 1)
-    (ω : a.samples) (z : ℂ) (hzz : z ∈ ball z₀ 1) :
-    spatialHitTime (γ ω) z₀ 2 ≤ spatialHitTime (γ ω) z r := by
-  by_cases hn : spatialHitTime (γ ω) z r = ⊤
-  · rw [hn]; exact le_top
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-  rw [←ht]
-  have hm := spatialHitTime_mem (a.trace_continuous ω) z r t ht.symm
-  have hs : 2 ≤ dist (γ ω 0) z₀ := by
-    rw [(a.witness ω).2.1,dist_zero_left]
-    have hh := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
-    linarith
-  apply (spatialHitTime_lt_of_dist_lt (a.trace_continuous ω) z₀ 2 hs t ?_).le
-  have hh : dist z z₀ < 1 := hzz
-  have ht' := dist_triangle (γ ω t) z z₀
-  linarith
-lemma source_simple_spheres_separated (z : ℂ) (hz : 4 ≤ z.im) (ω : a.samples)
-    (hn : spatialHitTime (γ ω) z 3 ≠ ⊤) :
-    spatialHitTime (γ ω) z 3 < spatialHitTime (γ ω) z 2 := by
-  have hs : 3 ≤ dist (γ ω 0) z := by
-    rw [(a.witness ω).2.1,dist_zero_left]
-    have hh := (le_abs_self z.im).trans (Complex.abs_im_le_norm z)
-    linarith
-  apply lt_of_le_of_ne (spatialHitTime_antitone_radius (a.trace_continuous ω) z (by norm_num : (2:ℝ) ≤ 3) hs)
-  intro he
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-  have h3 := spatialHitTime_mem (a.trace_continuous ω) z 3 t ht.symm
-  have h2 := spatialHitTime_mem (a.trace_continuous ω) z 2 t (he.symm.trans ht.symm)
-  linarith
-lemma source_simple_weight_zero_no_entrance (z₀ : ℂ) (hz : 4 ≤ z₀.im) (r : ℝ) (hr : r ≤ 1)
-    (ω : a.samples) (hn : spatialHitTime (γ ω) z₀ 3 = ⊤) (z : ℂ) (hzz : z ∈ ball z₀ 1) :
-    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r = 0 := by
-  unfold spatialGreenWeight
-  split_ifs with hb
-  · rfl
-  · have hn' : spatialHitTime (γ ω) z r ≠ ⊤ := fun hh => hb hh
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn'
-    have hm := spatialHitTime_mem (a.trace_continuous ω) z r t ht.symm
-    have hs : 3 ≤ dist (γ ω 0) z₀ := by
-      rw [(a.witness ω).2.1,dist_zero_left]
-      have hh := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
-      linarith
-    have hd : dist (γ ω t) z₀ < 3 := by
-      have hh : dist z z₀ < 1 := hzz
-      have ht' := dist_triangle (γ ω t) z z₀
-      linarith
-    have hh := spatialHitTime_lt_of_dist_lt (a.trace_continuous ω) z₀ 3 hs t hd
-    rw [hn] at hh
-    exact (not_lt_of_ge le_top hh).elim
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem source_past_spatial_slab_subgaussian [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    (q ε r R : ℝ) (hq : 0 < q) (hε : 0 < ε) (hqr : q*r ≤ ε)
-    (hεr : ε ≤ r) (hrR : r ≤ R)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
-    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z r)
-    :
-    HasSubgaussianMGF (fun ω : a.samples => spatialSlabIncrement κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) A ε r)
-      (⟨slabAmplitude κ q A * r ^ (1 - codimension κ),
-        mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hε.trans_le hεr).le _)⟩ ^ 2)
-      ((P.comap Subtype.val)[|D]) := by
-  apply hasSubgaussianMGF_of_compensated (a.spatialSlabIncrement_measurable A ε r).aemeasurable
-  intro θ
-  rw [ProbabilityTheory.cond, lintegral_smul_measure]
-  apply (mul_le_mul' (le_refl (((P.comap Subtype.val) D)⁻¹)) ?_).trans (ENNReal.inv_mul_le_one _)
-  have hv := slabVariance_scale_le A hκ (by linarith) hq (hε.trans_le hεr) hqr
-  apply le_trans _ (a.source_past_spatial_slab_terminal_exponential hκ hκ4 A hA ε r R hε hεr hrR hheight hbound s D hD hbefore θ)
-  apply lintegral_mono
-  intro ω
-  apply ENNReal.ofReal_le_ofReal
-  apply Real.exp_le_exp.mpr
-  change θ * _ - θ^2 * (slabAmplitude κ q A * r ^ (1 - codimension κ))^2 / 2 ≤ _
-  nlinarith [mul_le_mul_of_nonneg_left hv (sq_nonneg θ)]
-
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-theorem source_past_spatial_geometric_subgaussian [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
-    (q r R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hrR : r ≤ R)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) (N : ℕ)
-    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
-    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z r)
-    :
-    HasSubgaussianMGF (fun ω : a.samples =>
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
-      (NNReal.mk (slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)))
-        (div_nonneg (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg hr.le _))
-          (sub_nonneg.mpr (Real.rpow_lt_one hq.le hq1 (sub_pos.mpr (codimension_lt_one κ hκ))).le)) ^ 2)
-      ((P.comap Subtype.val)[|D]) := by
-  let rj : ℕ → ℝ := fun n => r*q^n
-  have hp (n : ℕ) : 0 < rj n := mul_pos hr (pow_pos hq n)
-  have hl (n : ℕ) : rj n ≤ r := by
-    simpa only [mul_one] using mul_le_mul_of_nonneg_left (pow_le_one₀ (n := n) hq.le hq1.le) hr.le
-  have he (n : ℕ) : rj (n+1) = q*rj n := by dsimp [rj]; rw [pow_succ]; ring
-  let X : ℕ → a.samples → ℝ := fun n ω => spatialSlabIncrement κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) A (rj (n+1)) (rj n)
-  let b : ℕ → ℝ≥0 := fun n => NNReal.mk (slabAmplitude κ q A*(rj n)^(1-codimension κ))
-    (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hp n).le _))
-  have hsub (n : ℕ) : HasSubgaussianMGF (X n) (b n^2) ((P.comap Subtype.val)[|D]) := by
-    apply a.source_past_spatial_slab_subgaussian hκ hκ4 A hA q (rj (n+1)) (rj n) R hq (hp _) (he n).symm.le (s := s) (D := D)
-    · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
-    · exact (hl n).trans hrR
-    · exact fun z hz => (hl n).trans (hheight z hz)
-    · exact hbound
-    · exact hD
-    · intro ω hω z hz
-      apply (hbefore ω hω z hz).trans
-      apply spatialHitTime_antitone_radius (a.trace_continuous ω) z (hl n)
-      rw [(a.witness ω).2.1,dist_zero_left]
-      exact (hheight z hz).trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  have hsum := subgaussian_sum_amplitudes X b N (fun n _ => hsub n)
-  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ Metric.closedBall 0 R from
-    fun z hz => by simpa only [Metric.mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
-      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
-  have htel (ω : a.samples) : (∑ n ∈ Finset.range N, X n ω) =
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
-      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A := by
-    have hm (n : ℕ) : X n ω =
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj (n+1)) A -
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A := by
-      symm
-      apply a.spatialRealMass_sub_eq_slab hκ (by linarith) A hA hfin _ _ (hp _)
-      · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
-      · exact fun z hz => (hl n).trans (hheight z hz)
-    simp_rw [hm]
-    simpa only [rj, pow_zero, mul_one] using Finset.sum_range_sub
-      (fun n => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A) N
-  simp_rw [htel] at hsum
-  apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsum
-  apply pow_le_pow_left₀ (by positivity)
-  apply NNReal.coe_le_coe.mp
-  simpa only [NNReal.coe_sum, b, rj, NNReal.coe_mk] using geometric_amplitude_sum (slabAmplitude κ q A) r q (1-codimension κ)
-    (slabAmplitude_nonneg κ q A) hr hq hq1 (sub_pos.mpr (codimension_lt_one κ hκ)) N
-
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.RetainedSeedMoments
+import OAI.Probability.SLE.SimpleTails
+
+namespace OAI
+
+/-! Small Kappa Sausage Moments for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma RegularSLEPresentation.source_initial_unit_mesh_tail_small
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (z₀ : ℂ) (hz : 2 < z₀.im) :
+    ∃ C c d : ℝ, 0 < C ∧ 0 < c ∧ 0 < d ∧ d < 1 ∧
+      ∀ rf δ : ℝ, 0 < rf → rf ≤ δ → δ ≤ d →
+      (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) <
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
+        ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
+  let A := ball z₀ 1
+  let B := meshTailAmplitude κ (1/2) A
+  let C := max 1 ((volume A).toReal*(1/2:ℝ)^(-codimension κ)+1)
+  have hB : 0 < B := meshTailAmplitude_pos A hκ (by norm_num) (by norm_num)
+  have hC : 0 < C := zero_lt_one.trans_le (le_max_left _ _)
+  refine ⟨C,1/(2*B^2),1/2,hC,by positivity,by norm_num,by norm_num,?_⟩
+  intro rf δ hrf hrδ hδd
+  have hδ : 0 < δ := hrf.trans_le hrδ
+  have hR : δ ≤ ‖z₀‖+2 := by have := norm_nonneg z₀; linarith
+  have hh := a.half_plane_mesh_tail hκ hκ4 A measurableSet_ball (1/2) rf δ (‖z₀‖+2)
+    (by norm_num) (by norm_num) hrf hrδ hR
+    (fun z hz' => (show δ ≤ 1 by linarith).trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le)
+    (fun z hz' => by
+      have hz'' : dist z z₀ < 1 := hz'
+      have htri := dist_triangle z z₀ 0
+      simp only [dist_zero_right] at htri
+      linarith)
+  apply (measure_mono (show {ω : a.samples | C*δ^(-codimension κ) <
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf A} ⊆
+      {ω : a.samples | ((volume A).toReal*(1/2:ℝ)^(-codimension κ)+1)*δ^(-codimension κ) ≤
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf A} from
+      fun ω hω => ((mul_le_mul_of_nonneg_right (le_max_right _ _) (Real.rpow_nonneg hδ.le _)).trans_lt hω).le)).trans
+  apply hh.trans
+  apply ENNReal.ofReal_le_ofReal
+  have he := mul_le_mul_of_nonneg_right (show 1 ≤ C from le_max_left _ _)
+    (Real.exp_pos (-(1/(2*B^2))*δ^(-2:ℝ))).le
+  simpa only [one_mul] using he
+
+lemma RegularSLEPresentation.source_unit_mesh_second_moment_small
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (z₀ : ℂ) (hz : 2 < z₀.im) :
+    ∃ d m₂ : ℝ, 0 < d ∧ d < 1 ∧ 0 < m₂ ∧ ∀ rf : ℝ, 0 < rf → rf ≤ d →
+      (∫⁻ ω : a.samples, ENNReal.ofReal (spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) rf (ball z₀ 1))^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
+  obtain ⟨C,c,d,hC,hc,hd,hd1,ht⟩ := a.source_initial_unit_mesh_tail_small hκ hκ4 z₀ hz
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let v := (volume (ball z₀ 1)).toReal
+  let M := max C v*d^(-codimension κ)
+  let b := c*d^(-2:ℝ)
+  let A := max 1 C*Real.exp b
+  let α := b/M
+  have hκ8 : κ < 8 := by linarith
+  have hp : 0 < codimension κ := codimension_pos κ hκ8
+  have hp1 : codimension κ < 1 := by dsimp [codimension, dimension]; linarith
+  have hM : 0 < M := mul_pos (hC.trans_le (le_max_left _ _)) (Real.rpow_pos_of_pos hd _)
+  have hb : 0 < b := mul_pos hc (Real.rpow_pos_of_pos hd _)
+  have hA : 0 < A := mul_pos (zero_lt_one.trans_le (le_max_left _ _)) (Real.exp_pos _)
+  have hα : 0 < α := div_pos hb hM
+  refine ⟨d,8*A/α^2,hd,hd1,by positivity,?_⟩
+  intro rf hf hfd
+  let X := fun ω : a.samples => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) rf (ball z₀ 1)
+  have hx (ω : a.samples) : 0 ≤ X ω ∧ X ω ≤ v*rf^(-codimension κ) :=
+    a.spatialRealMass_bound hκ hκ8 rf hf (ball z₀ 1) measurableSet_ball
+      (measure_ball_lt_top (μ := volume) (x := z₀) (r := 1)).ne
+      (fun z hz' => (hfd.trans hd1.le).trans (source_unit_ball_height hz (ball_subset_closedBall hz')).le) ω
+  have hlinear : ∀ t : ℝ, 1 ≤ t → μ {ω | M*t < X ω} ≤ ENNReal.ofReal (C*Real.exp (-b*t)) :=
+    source_mesh_tail_to_linear μ X (codimension κ) C c d v rf hp hp1.le hC hc hd
+      ENNReal.toReal_nonneg hf (fun ω => (hx ω).2)
+      (fun δ hfd hd' => ht rf δ hf hfd hd')
+  exact source_exponential_second_moment μ X (a.spatialRealMass_measurable _ _) (fun ω => (hx ω).1)
+    A α hA.le hα (source_linear_tail_all X M C b hM hC hb hlinear)
+
+lemma source_retained_seed_all (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (Ω : Type) (m : MeasurableSpace Ω) (P : Measure Ω) (hP : IsProbabilityMeasure P)
+    (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ) (hz : 2 < z₀.im) :
+    ∃ (T : ℝ≥0) (e₀ m₀ m₂ : ℝ), 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧ 0 < m₂ ∧
+      ∀ r : ℝ, 0 < r → r ≤ e₀ →
+      let X := fun ω : a.samples => finiteSpatialGreenMass κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r T (closedBall z₀ 1)
+      ENNReal.ofReal m₀ ≤ ∫⁻ ω, X ω ∂(P.comap Subtype.val) ∧
+      (∫⁻ ω, (X ω)^2 ∂(P.comap Subtype.val)) ≤ ENNReal.ofReal m₂ := by
+  let : IsProbabilityMeasure P := hP
+  by_cases hκ4 : 4 < κ
+  · exact sourceRetainedSeed_proved κ hκ4 hκ8 Ω m P hP γ a z₀ hz
+  have hk4 : κ ≤ 4 := le_of_not_gt hκ4
+  have hK : closedBall z₀ 1 ⊆ upperHalfPlane :=
+    fun z hz' => zero_lt_one.trans (source_unit_ball_height hz hz')
+  obtain ⟨T,e₀,m₀,hT,he,hm,hfirst⟩ := a.source_finite_time_first_moment hκ hκ8 z₀ 1 (by norm_num) hK
+  obtain ⟨d,m₂,hd,hd1,hm₂,hsecond⟩ := a.source_unit_mesh_second_moment_small hκ hk4 z₀ hz
+  refine ⟨T,min e₀ d,m₀,m₂,hT,lt_min he hd,hm,hm₂,?_⟩
+  intro r hr hre
+  refine ⟨hfirst r hr (hre.trans (min_le_left _ _)),?_⟩
+  have hrd : r ≤ d := hre.trans (min_le_right _ _)
+  apply le_trans (lintegral_mono (fun ω => pow_le_pow_left' (source_finite_mass_le_unit_mass
+    a hκ hκ8 z₀ hz r hr (hrd.trans hd1.le) T ω) 2))
+  exact hsecond r hr hrd
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem source_localized_spatial_slab_past_exponential
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    [IsFiniteMeasure (volume.comap (Subtype.val : A → ℂ))]
+    (hfin : volume A ≠ ∞)
+    (ε r : ℝ) (hε : 0 < ε) (hεr : ε ≤ r)
+    (hheight : ∀ z ∈ A, r ≤ z.im)
+    (u δ : ℝ) (b : SpatialAngularFamily a.B a.measurable_B κ
+      (Subtype.val : A → ℂ) u δ)
+    (M : ℝ) (hM : ∀ z : A, ‖initialLogCoordinates z‖ ≤ M)
+    (S s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z : A, s ≤ boundedSpatialHit (γ ω) z r S)
+    (θ : ℝ) :
+    (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal (Real.exp
+      (θ * (∫ z : A,
+        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
+        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω
+        ∂(volume.comap (Subtype.val : A → ℂ))) -
+        θ^2 * slabVariance κ ε r A / 2)) else 1 ∂(P.comap Subtype.val)) ≤ 1 := by
+  classical
+  let ν := volume.comap (Subtype.val : A → ℂ)
+  classical
+  let σ₀ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 r S
+  let τ₀ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 ε S
+  let σ₁ (p : A × a.samples) := max (σ₀ p) s
+  let τ₁ (p : A × a.samples) := if p.2 ∈ D then max (τ₀ p) s else σ₁ p
+  let σ (p : A × a.samples) := min (σ₁ p) (b.τ p)
+  let τ (p : A × a.samples) := min (τ₁ p) (b.τ p)
+  have hσ₀ := a.joint_boundedSpatialHit_stopping (Subtype.val : A → ℂ) measurable_subtype_coe r S
+  have hτ₀ := a.joint_boundedSpatialHit_stopping (Subtype.val : A → ℂ) measurable_subtype_coe ε S
+  have hs := isStoppingTime_const (spatialFiltration (Z := A) a.filtration) s
+  have hσ₁ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+      (fun p => (σ₁ p : WithTop ℝ≥0)) := by
+    simpa only [σ₁, σ₀, WithTop.coe_max] using hσ₀.max hs
+  have hDm : MeasurableSet[spatialFiltration (Z := A) a.filtration s] {p : A × a.samples | p.2 ∈ D} := by
+    let : MeasurableSpace a.samples := a.filtration s
+    exact hD.preimage measurable_snd
+  have hτ₁ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+      (fun p => (τ₁ p : WithTop ℝ≥0)) := by
+    have htmax : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+        (fun p => ((max (τ₀ p) s : ℝ≥0) : WithTop ℝ≥0)) := by
+      simpa only [τ₀, WithTop.coe_max] using hτ₀.max hs
+    have hh := htmax.piecewise_of_le hσ₁
+      (fun p => WithTop.coe_le_coe.mpr (le_max_right (τ₀ p) s))
+      (fun p => WithTop.coe_le_coe.mpr (le_max_right (σ₀ p) s)) hDm
+    convert hh using 1
+    funext p
+    by_cases hp : p.2 ∈ D <;> simp [τ₁, Set.piecewise, hp]
+    rfl
+  have hσ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+      (fun p => (σ p : WithTop ℝ≥0)) := by
+    simpa only [σ, WithTop.coe_min] using hσ₁.min b.stopping
+  have hτ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+      (fun p => (τ p : WithTop ℝ≥0)) := by
+    simpa only [τ, WithTop.coe_min] using hτ₁.min b.stopping
+  have hord (p : A × a.samples) : σ₀ p ≤ τ₀ p := by
+    apply boundedSpatialHit_antitone_radius (a.trace_continuous p.2) p.1 hεr
+    rw [(a.witness p.2).2.1, dist_zero_left]
+    exact (hheight p.1 p.1.2).trans ((le_abs_self p.1.val.im).trans (Complex.abs_im_le_norm p.1))
+  have hστ (p : A × a.samples) : σ p ≤ τ p := by
+    apply min_le_min _ le_rfl
+    dsimp only [τ₁]
+    split_ifs
+    · exact max_le_max (hord p) le_rfl
+    · exact le_rfl
+  have hsD (ω : a.samples) (hω : ω ∈ D) (z : A) :
+      σ (z,ω) = min (boundedSpatialHit (γ ω) z r S) (b.τ (z,ω)) := by
+    simp only [σ, σ₁, σ₀, max_eq_left (hbefore ω hω z)]
+  have htD (ω : a.samples) (hω : ω ∈ D) (z : A) :
+      τ (z,ω) = min (boundedSpatialHit (γ ω) z ε S) (b.τ (z,ω)) := by
+    have hh := (hbefore ω hω z).trans (hord (z,ω))
+    simp only [τ, τ₁, τ₀, hω, ↓reduceIte, max_eq_left hh]
+  have hnotD (ω : a.samples) (hω : ω ∉ D) (z : A) : τ (z,ω) = σ (z,ω) := by
+    simp only [τ, τ₁, hω, ↓reduceIte, σ]
+  have htbound (p : A × a.samples) : τ p ≤ b.T := (min_le_right _ _).trans (b.bounded p)
+  have hin (z : A) (ω : a.samples) (t : ℝ≥0) (ht : t ≤ τ (z,ω)) :
+      b.lo ≤ ((b.c (initialLogCoordinates z)).Q t ω).1.im ∧
+        ((b.c (initialLogCoordinates z)).Q t ω).1.im ≤ b.hi :=
+    (b.inside z ω t (ht.trans (min_le_right _ _))).2.2
+  have hexp := clipped_spatial_green_window_unbounded_driver (ν := ν) a.measurable_B a.brownian
+    a.continuous_B a.zero_B κ b.lo b.hi b.le hκ (by linarith) b.c
+    (fun z : A => initialLogCoordinates z) (measurable_initialLogCoordinates.comp measurable_subtype_coe)
+    M hM b.T σ τ hσ hτ hστ htbound hin θ
+  let E (ω : a.samples) := ∫ t in (0 : ℝ)..(b.T : ℝ),
+    (∫ z : A, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
+      greenLogCoefficient κ ((b.c (initialLogCoordinates z)).Q t ω) (a.B t.toNNReal ω) else 0 ∂ν)^2
+  have hE (ω : a.samples) : E ω ≤ slabVariance κ ε r A := by
+    apply interval_square_energy_bound _ b.T _ (slabVariance_nonneg _ _ _ _)
+    rw [ofReal_slabVariance _ _ _ _ hfin]
+    apply (a.witness ω).spatial_masked_bracket_bound ((a.continuous_B ω).const_mul _)
+      hκ hκ4 hε (hε.trans_le hεr) A hA (fun z hz => hεr.trans (hheight z hz))
+    intro t ht z
+    have hb := clipped_spatial_window_coefficient_le b.c a.continuous_B z
+      (show z.val ∈ upperHalfPlane from hε.trans_le (hεr.trans (hheight z z.2))) ω (b.τ (z,ω)) S
+      (γ ω) (a.G ω) (a.F ω) (a.witness ω)
+      (fun s hs => (b.inside z ω s hs).2.2) r ε t.toNNReal
+    by_cases hω : ω ∈ D
+    · simpa only [hsD ω hω z, htD ω hω z, ← NNReal.coe_le_coe,
+        ← NNReal.coe_lt_coe, Real.coe_toNNReal _ ht] using hb
+    · simp only [hnotD ω hω z]
+      have hn (v : ℝ) : ¬ ((σ (z,ω) : ℝ) ≤ v ∧ v < (σ (z,ω) : ℝ)) := fun hh => (not_lt_of_ge hh.1) hh.2
+      simp only [hn t, ↓reduceIte, abs_zero]
+      exact (abs_nonneg _)
+
+  have hEzero (ω : a.samples) (hω : ω ∉ D) : E ω = 0 := by
+    have hf (t : ℝ) (z : A) : ¬ ((σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ)) := by
+      rw [hnotD ω hω z]
+      exact fun hh => (not_lt_of_ge hh.1) hh.2
+    simp only [E, hf, ↓reduceIte, integral_zero, zero_pow (by norm_num : 2 ≠ 0), intervalIntegral.integral_zero]
+  apply le_trans _ hexp
+  apply lintegral_mono
+  intro ω
+  by_cases hω : ω ∈ D
+  · simp only [hω, ↓reduceIte]
+    apply ENNReal.ofReal_le_ofReal
+    apply Real.exp_le_exp.mpr
+    change θ * _ - θ^2 * slabVariance κ ε r A / 2 ≤ θ * _ - θ^2 * E ω / 2
+    have hid : (∫ z : A,
+        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
+        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω ∂ν) =
+        ∫ z : A, greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (τ (z,ω)) ω) (a.B (τ (z,ω)) ω) -
+          greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (σ (z,ω)) ω) (a.B (σ (z,ω)) ω) ∂ν := by
+      apply integral_congr_ae
+      exact Eventually.of_forall fun z => by
+        simp only [AngularLocalization.physicalWeight, SpatialAngularFamily.section,
+          hsD ω hω z, htD ω hω z, NNReal.coe_min]
+    rw [hid]
+    nlinarith [mul_le_mul_of_nonneg_left (hE ω) (sq_nonneg θ)]
+  · simp only [hω, ↓reduceIte]
+    change (1:ℝ≥0∞) ≤ ENNReal.ofReal (Real.exp (θ*_ - θ^2*E ω/2))
+    simp only [hnotD ω hω, sub_self, integral_zero, hEzero ω hω, mul_zero,
+      zero_div, Real.exp_zero, ENNReal.ofReal_one, le_refl]
+
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_past_exponential_fatou
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (D : Set Ω) (hD : MeasurableSet D) (X : ℕ → Ω → ℝ) (Y : Ω → ℝ)
+    (θ V : ℝ) (hm : ∀ n, Measurable (X n))
+    (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
+    (hexp : ∀ n, (∫⁻ ω, if ω ∈ D then ENNReal.ofReal
+      (Real.exp (θ*X n ω-θ^2*V/2)) else 1 ∂P) ≤ 1) :
+    (∫⁻ ω, if ω ∈ D then ENNReal.ofReal
+      (Real.exp (θ*Y ω-θ^2*V/2)) else 1 ∂P) ≤ 1 := by
+  let f (n : ℕ) (ω : Ω) := if ω ∈ D then
+    ENNReal.ofReal (Real.exp (θ*X n ω-θ^2*V/2)) else 1
+  have hfm (n : ℕ) : Measurable (f n) :=
+    (((((hm n).const_mul θ).sub_const (θ^2*V/2)).exp).ennreal_ofReal).ite hD measurable_const
+  have hl : ∀ᵐ ω ∂P, (if ω ∈ D then ENNReal.ofReal
+      (Real.exp (θ*Y ω-θ^2*V/2)) else 1) = liminf (fun n => f n ω) atTop := by
+    filter_upwards [hlim] with ω hω
+    by_cases hmem : ω ∈ D
+    · simp only [hmem, ↓reduceIte, f]
+      exact ((ENNReal.continuous_ofReal.tendsto _).comp
+        ((Real.continuous_exp.tendsto _).comp
+          ((hω.const_mul θ).sub_const (θ^2*V/2)))).liminf_eq.symm
+    · simp only [hmem, ↓reduceIte, f, liminf_const]
+  calc
+    _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := lintegral_congr_ae hl
+    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' (fun n => (hfm n).aemeasurable)
+    _ ≤ 1 := liminf_le_of_frequently_le (Frequently.of_forall hexp)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem source_past_spatial_slab_finite_exponential_of_localizations
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    [IsFiniteMeasure (volume.comap (Subtype.val : A → ℂ))]
+    (hfin : volume A ≠ ∞)
+    (ε r : ℝ) (hε : 0 < ε) (hεr : ε ≤ r)
+    (hheight : ∀ z ∈ A, r ≤ z.im)
+    (u : ℝ) (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0))
+    (b : ∀ n, SpatialAngularFamily a.B a.measurable_B κ (Subtype.val : A → ℂ) u (δ n))
+    (l : ℝ) (hl : ∀ z : A, l ≤ Real.log z.val.im)
+    (hlevel : ∀ z : A, (2 * z.val.im) * Real.exp (-u) < ε)
+    (M : ℝ) (hM : ∀ z : A, ‖initialLogCoordinates z‖ ≤ M)
+    (S s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z : A, s ≤ boundedSpatialHit (γ ω) z r S)
+    (θ : ℝ) :
+    (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal (Real.exp
+      (θ * (∫ z : A,
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z
+        ∂(volume.comap (Subtype.val : A → ℂ))) -
+        θ^2 * slabVariance κ ε r A / 2)) else 1 ∂(P.comap Subtype.val)) ≤ 1 := by
+  let ν := volume.comap (Subtype.val : A → ℂ)
+  let σ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 r S
+  let τ (p : A × a.samples) := boundedSpatialHit (γ p.2) p.1 ε S
+  have hσ : Measurable σ := stopping_time_measurable_nnreal
+    (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe r S)
+  have hτ : Measurable τ := stopping_time_measurable_nnreal
+    (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe ε S)
+  let X (n : ℕ) (ω : a.samples) := ∫ z : A,
+        ((b n).section z).physicalWeight (τ (z,ω)) ω -
+        ((b n).section z).physicalWeight (σ (z,ω)) ω ∂ν
+  let Y (ω : a.samples) := ∫ z : A,
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (τ (z,ω)) z -
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (σ (z,ω)) z ∂ν
+  have hXm (n : ℕ) : Measurable (X n) :=
+    (((b n).physicalWeight_joint_measurable a.continuous_B measurable_subtype_coe τ hτ).sub
+      ((b n).physicalWeight_joint_measurable a.continuous_B measurable_subtype_coe σ hσ)).stronglyMeasurable.integral_prod_left'.measurable
+  have hlim (ω : a.samples) : Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)) := by
+    have hh := spatial_angular_weight_integral_tendsto (ν := ν) b hδ a.continuous_B
+      measurable_subtype_coe hκ (by linarith) (fun z => hε.trans_le (hεr.trans (hheight z z.2)))
+      l hl ω (γ ω) (a.G ω) (a.F ω) (a.witness ω) σ τ hσ hτ
+    convert hh using 1
+    congr 1
+    apply integral_congr_ae
+    apply Eventually.of_forall
+    intro z
+    dsimp only
+    rw [(a.witness ω).physicalGreen_eq_before_spatial ((a.continuous_B ω).const_mul _)
+      κ z ε hε (hεr.trans (hheight z z.2)) u (hlevel z) (τ (z,ω)) (boundedSpatialHit_le _ _ _ _),
+      (a.witness ω).physicalGreen_eq_before_spatial ((a.continuous_B ω).const_mul _)
+      κ z r (hε.trans_le hεr) (hheight z z.2) u ((hlevel z).trans_le hεr)
+        (σ (z,ω)) (boundedSpatialHit_le _ _ _ _)]
+  apply source_past_exponential_fatou D ((a.filtration.le s) _ hD) X Y θ
+    (slabVariance κ ε r A) hXm (Eventually.of_forall hlim)
+  intro n
+  exact a.source_localized_spatial_slab_past_exponential hκ hκ4 A hA hfin ε r hε hεr hheight u (δ n) (b n) M hM S s D hD hbefore θ
+
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem source_past_spatial_slab_finite_exponential
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    (ε r R : ℝ) (hε : 0 < ε) (hεr : ε ≤ r) (hrR : r ≤ R)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
+    (S s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z : A, s ≤ boundedSpatialHit (γ ω) z r S)
+    (θ : ℝ) :
+    (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal (Real.exp
+      (θ * (∫ z in A,
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z) -
+        θ^2 * slabVariance κ ε r A / 2)) else 1 ∂(P.comap Subtype.val)) ≤ 1 := by
+  have hr : 0 < r := hε.trans_le hεr
+  have hR : 0 < R := hr.trans_le hrR
+  have hs : A ⊆ closedBall 0 R := by
+    intro z hz
+    simpa only [mem_closedBall, dist_zero_right] using hbound z hz
+  have hfin : volume A ≠ ∞ := ne_of_lt
+    ((measure_mono hs).trans_lt (isCompact_closedBall (0 : ℂ) R).measure_lt_top)
+  let ν := volume.comap (Subtype.val : A → ℂ)
+  let : IsFiniteMeasure ν := ⟨by
+    have hh := Measure.comap_apply_le (Subtype.val : A → ℂ) volume
+      (show NullMeasurableSet (univ : Set A) ν from MeasurableSet.univ.nullMeasurableSet)
+    apply lt_of_le_of_lt hh
+    simpa only [Set.image_univ, Subtype.range_coe] using lt_top_iff_ne_top.mpr hfin⟩
+  obtain ⟨u, hu, hul⟩ := exists_uniform_radius_level ε R hε hR.le
+  let α := r^2 / R^2
+  have hα : 0 < α := by dsimp [α]; positivity
+  let δ (n : ℕ) := α / ((n : ℝ) + 1)
+  have hδpos (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity
+  have hδle (n : ℕ) : δ n ≤ α := div_le_self hα.le (by linarith [Nat.cast_nonneg (α := ℝ) n])
+  have hδlim : Tendsto δ atTop (𝓝 0) := by
+    simpa only [mul_one_div, mul_zero] using
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul α
+  have hbounds (z : A) := initial_spatial_bounds κ r R hr hR z (hheight z z.2) (hbound z z.2)
+  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
+  let b (n : ℕ) : SpatialAngularFamily a.B a.measurable_B κ (Subtype.val : A → ℂ) u (δ n) :=
+    (exists_spatial_angular_family a.measurable_B hBm a.continuous_B a.zero_B κ Subtype.val
+      measurable_subtype_coe (Real.log r) (Real.log R) (Real.log_le_log hr hrR)
+      (fun z => (hbounds z).1) u (δ n) hu (hδpos n)
+      (fun z => (hδle n).trans (hbounds z).2.1)).some
+  have hlevel (z : A) : (2 * z.val.im) * Real.exp (-u) < ε := by
+    have hi : z.val.im ≤ R := (le_abs_self _).trans ((Complex.abs_im_le_norm z).trans (hbound z z.2))
+    exact (mul_le_mul_of_nonneg_right (by linarith : 2 * z.val.im ≤ 2 * R) (Real.exp_pos _).le).trans_lt hul
+  have hh := a.source_past_spatial_slab_finite_exponential_of_localizations hκ hκ4 A hA hfin ε r hε hεr
+    hheight u δ hδlim b (Real.log r) (fun z => (hbounds z).1.1) hlevel
+    (R + |Real.log r| + |Real.log R|) (fun z => (hbounds z).2.2) S s D hD hbefore θ
+  have he (ω : a.samples) :
+      (∫ z : A,
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z
+        ∂(volume.comap (Subtype.val : A → ℂ))) =
+      ∫ z in A,
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
+        greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z :=
+    integral_subtype_comap (μ := volume) hA (fun z =>
+      greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε S) z -
+      greenDensity κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r S) z)
+  simpa only [he] using hh
+
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+lemma source_past_spatial_slab_terminal_exponential [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    (ε r R : ℝ) (hε : 0 < ε) (hεr : ε ≤ r) (hrR : r ≤ R)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
+    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z r)
+    (θ : ℝ) :
+    (∫⁻ ω in D, ENNReal.ofReal (Real.exp
+      (θ*spatialSlabIncrement κ (fun t => Real.sqrt κ*a.B t ω)
+        (γ ω) (a.G ω) A ε r - θ^2*slabVariance κ ε r A/2))
+      ∂(P.comap Subtype.val)) ≤ (P.comap Subtype.val) D := by
+  have hs : A ⊆ closedBall 0 R := by
+    intro z hz
+    simpa only [mem_closedBall, dist_zero_right] using hbound z hz
+  have hfin : volume A ≠ ∞ := ne_of_lt
+    ((measure_mono hs).trans_lt (isCompact_closedBall (0 : ℂ) R).measure_lt_top)
+  obtain ⟨N,hN⟩ := exists_nat_ge s
+  let X (n : ℕ) (ω : a.samples) := ∫ z in A,
+    greenDensity κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z ε (n+N)) z -
+    greenDensity κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r (n+N)) z
+  let Y (ω : a.samples) := spatialSlabIncrement κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) A ε r
+  have hXm (n : ℕ) : Measurable (X n) :=
+    ((a.boundedSpatialGreen_measurable ε (n+N)).sub
+      (a.boundedSpatialGreen_measurable r (n+N))).stronglyMeasurable.integral_prod_left'.measurable
+  have hDm : MeasurableSet D := (a.filtration.le s) _ hD
+  have hmask : (∫⁻ ω : a.samples, if ω ∈ D then ENNReal.ofReal
+      (Real.exp (θ*Y ω-θ^2*slabVariance κ ε r A/2)) else 1 ∂P.comap Subtype.val) ≤ 1 := by
+    apply source_past_exponential_fatou D hDm X Y θ (slabVariance κ ε r A) hXm
+    · filter_upwards [a.spatial_slab_area_tendsto hκ (by linarith) A hA hfin ε r R hε hεr hrR hheight hbound] with ω hω
+      simpa only [X, Y, spatialSlabIncrement, Function.comp_def, Nat.cast_add] using hω.comp (tendsto_add_atTop_nat N)
+    · intro n
+      apply a.source_past_spatial_slab_finite_exponential hκ hκ4 A hA ε r R hε hεr hrR hheight hbound (n+N) s D hD
+      intro ω hω z
+      apply WithTop.coe_le_coe.mp
+      rw [boundedSpatialHit_coe]
+      apply le_min (hbefore ω hω z z.2)
+      exact WithTop.coe_le_coe.mpr (hN.trans (le_add_of_nonneg_left (by positivity)))
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let f (ω : a.samples) := ENNReal.ofReal (Real.exp (θ*Y ω-θ^2*slabVariance κ ε r A/2))
+  change (∫⁻ ω, if ω ∈ D then f ω else 1 ∂μ) ≤ 1 at hmask
+  rw [←lintegral_add_compl _ hDm] at hmask
+  have hleft : (∫⁻ ω in D, if ω ∈ D then f ω else 1 ∂μ) = ∫⁻ ω in D, f ω ∂μ :=
+    setLIntegral_congr_fun hDm (fun ω hω => ite_eq_left hω)
+  have hright : (∫⁻ ω in Dᶜ, if ω ∈ D then f ω else 1 ∂μ) = μ Dᶜ := by
+    rw [setLIntegral_congr_fun hDm.compl (fun ω hω => ite_eq_right hω)]
+    simp
+  rw [hleft,hright] at hmask
+  have hsum : μ D+μ Dᶜ = 1 := by rw [measure_add_measure_compl hDm,measure_univ]
+  exact (ENNReal.add_le_add_iff_right (measure_ne_top μ Dᶜ)).mp (hmask.trans_eq hsum.symm)
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_infinite_stopping_grid_piece
+    {Ω : Type*} [m : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 m}
+    {ρ : Ω → WithTop ℝ≥0} (hρ : IsStoppingTime ℱ ρ) (n k : ℕ) :
+    MeasurableSet[ℱ (stopGridTime n k)]
+      ({ω | ρ ω ≠ ⊤} ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) = k}) := by
+  let A := {ω | ρ ω ≠ ⊤}
+  have hle (j : ℕ) : MeasurableSet[ℱ (stopGridTime n j)]
+      (A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) ≤ j}) := by
+    convert hρ (stopGridTime n j) using 1
+    ext ω
+    cases he : ρ ω with
+    | top => simp only [A,mem_inter_iff,mem_ofPred_eq,he,ne_eq,not_true_eq_false,false_and,WithTop.not_top_le_coe]
+    | coe t => simp only [A,mem_inter_iff,mem_ofPred_eq,he,WithTop.coe_ne_top,ne_eq,not_false_eq_true,true_and,WithTop.untopD_coe,stopGridIndex_le,WithTop.coe_le_coe]
+  cases k with
+  | zero => simpa only [Nat.le_zero] using hle 0
+  | succ k =>
+    have he : A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) = k+1} =
+        (A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) ≤ k+1}) \
+          (A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) ≤ k}) := by
+      ext ω
+      simp only [mem_inter_iff,mem_ofPred_eq,Set.mem_sdiff]
+      constructor
+      · rintro ⟨hω,he⟩; exact ⟨⟨hω,he.le⟩,fun hn => by omega⟩
+      · rintro ⟨⟨hω,hle⟩,hn⟩
+        exact ⟨hω,by have hh : ¬stopGridIndex n ((ρ ω).untopD 0) ≤ k := fun hk => hn ⟨hω,hk⟩; omega⟩
+    change MeasurableSet[ℱ (stopGridTime n (k+1))] (A ∩ _)
+    rw [he]
+    exact (hle (k+1)).diff (ℱ.mono (monotone_stopGridTime n (Nat.le_succ k)) _ (hle k))
+
+ 
+
+lemma source_between_stopping_tail
+    {Ω : Type*} [m : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    {ℱ : Filtration ℝ≥0 m} {ρ τ : Ω → WithTop ℝ≥0}
+    (hρ : IsStoppingTime ℱ ρ) (hτ : IsStoppingTime ℱ τ)
+    (H : Set Ω) (hH : MeasurableSet H)
+    (hHA : H ⊆ {ω | ρ ω ≠ ⊤})
+    (hsep : ∀ ω, ρ ω ≠ ⊤ → ρ ω < τ ω)
+    (K : ℝ≥0∞)
+    (ht : ∀ (s : ℝ≥0) (D : Set Ω), MeasurableSet[ℱ s] D →
+      (∀ ω ∈ D, (s : WithTop ℝ≥0) ≤ τ ω) → μ (D ∩ H) ≤ μ D*K) :
+    μ H ≤ μ {ω | ρ ω ≠ ⊤}*K := by
+  let A := {ω | ρ ω ≠ ⊤}
+  let Q (n k : ℕ) := A ∩ {ω | stopGridIndex n ((ρ ω).untopD 0) = k}
+  let D (n k : ℕ) := Q n k ∩ {ω | (stopGridTime n k : WithTop ℝ≥0) < τ ω}
+  let E (n : ℕ) := ⋃ k, D n k
+  have hQ (n k : ℕ) : MeasurableSet[ℱ (stopGridTime n k)] (Q n k) :=
+    source_infinite_stopping_grid_piece hρ n k
+  have hD (n k : ℕ) : MeasurableSet[ℱ (stopGridTime n k)] (D n k) :=
+    (hQ n k).inter (by simpa only [compl_ofPred,not_le] using (hτ (stopGridTime n k)).compl)
+  have hDm (n k : ℕ) : MeasurableSet (D n k) := (ℱ.le _) _ (hD n k)
+  have hEm (n : ℕ) : MeasurableSet (E n) := MeasurableSet.iUnion (hDm n)
+  have hEq (n : ℕ) : ⋃ k, Q n k = A := grid_partition_union A n
+  have hEd (n : ℕ) : Pairwise (fun i j => Disjoint (Q n i) (Q n j)) := grid_partition_disjoint A n
+  have hb (n : ℕ) : μ (E n ∩ H) ≤ μ A*K := by
+    calc
+      μ (E n ∩ H) = μ (⋃ k, D n k ∩ H) := by simp only [E,iUnion_inter]
+      _ ≤ ∑' k, μ (D n k ∩ H) := measure_iUnion_le _
+      _ ≤ ∑' k, μ (Q n k)*K := ENNReal.tsum_le_tsum fun k =>
+        (ht (stopGridTime n k) (D n k) (hD n k) (fun ω hω => hω.2.le)).trans
+          (mul_le_mul' (measure_mono inter_subset_left) le_rfl)
+      _ = (∑' k, μ (Q n k))*K := ENNReal.tsum_mul_right
+      _ = μ A*K := by rw [←measure_iUnion (hEd n) (fun k => (ℱ.le _) _ (hQ n k)),hEq]
+  have hevent (ω : Ω) (hω : ω ∈ A) : ∀ᶠ n in atTop, ω ∈ E n := by
+    obtain ⟨t,ht'⟩ := WithTop.ne_top_iff_exists.mp hω
+    have ht : ρ ω = (t : WithTop ℝ≥0) := ht'.symm
+    have hlt := hsep ω hω
+    rw [ht] at hlt
+    have hgrid : ∀ᶠ n in atTop, (stopGrid n t : WithTop ℝ≥0) < τ ω := by
+      cases hv : τ ω with
+      | top => exact Eventually.of_forall fun n => WithTop.coe_lt_top _
+      | coe v =>
+        rw [hv] at hlt
+        have hh : ∀ᶠ n in atTop, stopGrid n t < v := (stopGrid_tendsto t) (Iio_mem_nhds (WithTop.coe_lt_coe.mp hlt))
+        exact hh.mono fun n hn => by simpa only [hv,WithTop.coe_lt_coe] using hn
+    filter_upwards [hgrid] with n hn
+    refine mem_iUnion.mpr ⟨stopGridIndex n t, ?_⟩
+    exact ⟨⟨hω,by change stopGridIndex n ((ρ ω).untopD 0) = stopGridIndex n t; rw [ht]; rfl⟩,hn⟩
+  let f (n : ℕ) := (E n ∩ H).indicator (fun _ : Ω => (1 : ℝ≥0∞))
+  have hfm (n : ℕ) : Measurable (f n) := measurable_const.indicator ((hEm n).inter hH)
+  have hp : H.indicator (fun _ : Ω => (1 : ℝ≥0∞)) ≤ fun ω => liminf (fun n => f n ω) atTop := by
+    intro ω
+    by_cases hω : ω ∈ H
+    · rw [indicator_of_mem hω]
+      apply le_liminf_of_le (by isBoundedDefault)
+      filter_upwards [hevent ω (hHA hω)] with n hn
+      simp only [f,indicator_of_mem (show ω ∈ E n ∩ H from ⟨hn,hω⟩),le_refl]
+    · simp only [indicator_of_notMem hω]
+      exact bot_le
+  calc
+    μ H = ∫⁻ ω, H.indicator (fun _ => (1 : ℝ≥0∞)) ω ∂μ := by simp [hH]
+    _ ≤ ∫⁻ ω, liminf (fun n => f n ω) atTop ∂μ := lintegral_mono hp
+    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂μ) atTop := lintegral_liminf_le' (fun n => (hfm n).aemeasurable)
+    _ ≤ μ A*K := liminf_le_of_frequently_le (Frequently.of_forall fun n => by
+      simpa only [f,lintegral_indicator ((hEm n).inter hH),lintegral_const,Measure.restrict_apply_univ,one_mul] using hb n)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+lemma source_simple_sphere_stopping (z : ℂ) (r : ℝ) :
+    IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) := by
+  let X (t : ℝ≥0) (ω : a.samples) := dist (γ ω t) z-r
+  have ha : Adapted a.filtration X := by
+    intro t
+    let : MeasurableSpace a.samples := a.filtration t
+    exact ((a.trace_adapted t).dist measurable_const).sub_const _
+  exact continuous_zero_hit_stopping a.filtration X ha
+    (fun ω => ((a.trace_continuous ω).dist continuous_const).sub continuous_const) 0
+lemma source_simple_center_before (z₀ : ℂ) (hz : 4 ≤ z₀.im) (r : ℝ) (hr : r ≤ 1)
+    (ω : a.samples) (z : ℂ) (hzz : z ∈ ball z₀ 1) :
+    spatialHitTime (γ ω) z₀ 2 ≤ spatialHitTime (γ ω) z r := by
+  by_cases hn : spatialHitTime (γ ω) z r = ⊤
+  · rw [hn]; exact le_top
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+  rw [←ht]
+  have hm := spatialHitTime_mem (a.trace_continuous ω) z r t ht.symm
+  have hs : 2 ≤ dist (γ ω 0) z₀ := by
+    rw [(a.witness ω).2.1,dist_zero_left]
+    have hh := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
+    linarith
+  apply (spatialHitTime_lt_of_dist_lt (a.trace_continuous ω) z₀ 2 hs t ?_).le
+  have hh : dist z z₀ < 1 := hzz
+  have ht' := dist_triangle (γ ω t) z z₀
+  linarith
+lemma source_simple_spheres_separated (z : ℂ) (hz : 4 ≤ z.im) (ω : a.samples)
+    (hn : spatialHitTime (γ ω) z 3 ≠ ⊤) :
+    spatialHitTime (γ ω) z 3 < spatialHitTime (γ ω) z 2 := by
+  have hs : 3 ≤ dist (γ ω 0) z := by
+    rw [(a.witness ω).2.1,dist_zero_left]
+    have hh := (le_abs_self z.im).trans (Complex.abs_im_le_norm z)
+    linarith
+  apply lt_of_le_of_ne (spatialHitTime_antitone_radius (a.trace_continuous ω) z (by norm_num : (2:ℝ) ≤ 3) hs)
+  intro he
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+  have h3 := spatialHitTime_mem (a.trace_continuous ω) z 3 t ht.symm
+  have h2 := spatialHitTime_mem (a.trace_continuous ω) z 2 t (he.symm.trans ht.symm)
+  linarith
+lemma source_simple_weight_zero_no_entrance (z₀ : ℂ) (hz : 4 ≤ z₀.im) (r : ℝ) (hr : r ≤ 1)
+    (ω : a.samples) (hn : spatialHitTime (γ ω) z₀ 3 = ⊤) (z : ℂ) (hzz : z ∈ ball z₀ 1) :
+    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r = 0 := by
+  unfold spatialGreenWeight
+  split_ifs with hb
+  · rfl
+  · have hn' : spatialHitTime (γ ω) z r ≠ ⊤ := fun hh => hb hh
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn'
+    have hm := spatialHitTime_mem (a.trace_continuous ω) z r t ht.symm
+    have hs : 3 ≤ dist (γ ω 0) z₀ := by
+      rw [(a.witness ω).2.1,dist_zero_left]
+      have hh := (le_abs_self z₀.im).trans (Complex.abs_im_le_norm z₀)
+      linarith
+    have hd : dist (γ ω t) z₀ < 3 := by
+      have hh : dist z z₀ < 1 := hzz
+      have ht' := dist_triangle (γ ω t) z z₀
+      linarith
+    have hh := spatialHitTime_lt_of_dist_lt (a.trace_continuous ω) z₀ 3 hs t hd
+    rw [hn] at hh
+    exact (not_lt_of_ge le_top hh).elim
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem source_past_spatial_slab_subgaussian [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    (q ε r R : ℝ) (hq : 0 < q) (hε : 0 < ε) (hqr : q*r ≤ ε)
+    (hεr : ε ≤ r) (hrR : r ≤ R)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R)
+    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z r)
+    :
+    HasSubgaussianMGF (fun ω : a.samples => spatialSlabIncrement κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) A ε r)
+      (⟨slabAmplitude κ q A * r ^ (1 - codimension κ),
+        mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hε.trans_le hεr).le _)⟩ ^ 2)
+      ((P.comap Subtype.val)[|D]) := by
+  apply hasSubgaussianMGF_of_compensated (a.spatialSlabIncrement_measurable A ε r).aemeasurable
+  intro θ
+  rw [ProbabilityTheory.cond, lintegral_smul_measure]
+  apply (mul_le_mul' (le_refl (((P.comap Subtype.val) D)⁻¹)) ?_).trans (ENNReal.inv_mul_le_one _)
+  have hv := slabVariance_scale_le A hκ (by linarith) hq (hε.trans_le hεr) hqr
+  apply le_trans _ (a.source_past_spatial_slab_terminal_exponential hκ hκ4 A hA ε r R hε hεr hrR hheight hbound s D hD hbefore θ)
+  apply lintegral_mono
+  intro ω
+  apply ENNReal.ofReal_le_ofReal
+  apply Real.exp_le_exp.mpr
+  change θ * _ - θ^2 * (slabAmplitude κ q A * r ^ (1 - codimension κ))^2 / 2 ≤ _
+  nlinarith [mul_le_mul_of_nonneg_left hv (sq_nonneg θ)]
+
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+theorem source_past_spatial_geometric_subgaussian [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4) (A : Set ℂ) (hA : MeasurableSet A)
+    (q r R : ℝ) (hq : 0 < q) (hq1 : q < 1) (hr : 0 < r) (hrR : r ≤ R)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (hbound : ∀ z ∈ A, ‖z‖ ≤ R) (N : ℕ)
+    (s : ℝ≥0) (D : Set a.samples) (hD : MeasurableSet[a.filtration s] D)
+    (hbefore : ∀ ω ∈ D, ∀ z ∈ A, (s : WithTop ℝ≥0) ≤ spatialHitTime (γ ω) z r)
+    :
+    HasSubgaussianMGF (fun ω : a.samples =>
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A)
+      (NNReal.mk (slabAmplitude κ q A*r^(1-codimension κ)/(1-q^(1-codimension κ)))
+        (div_nonneg (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg hr.le _))
+          (sub_nonneg.mpr (Real.rpow_lt_one hq.le hq1 (sub_pos.mpr (codimension_lt_one κ hκ))).le)) ^ 2)
+      ((P.comap Subtype.val)[|D]) := by
+  let rj : ℕ → ℝ := fun n => r*q^n
+  have hp (n : ℕ) : 0 < rj n := mul_pos hr (pow_pos hq n)
+  have hl (n : ℕ) : rj n ≤ r := by
+    simpa only [mul_one] using mul_le_mul_of_nonneg_left (pow_le_one₀ (n := n) hq.le hq1.le) hr.le
+  have he (n : ℕ) : rj (n+1) = q*rj n := by dsimp [rj]; rw [pow_succ]; ring
+  let X : ℕ → a.samples → ℝ := fun n ω => spatialSlabIncrement κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) A (rj (n+1)) (rj n)
+  let b : ℕ → ℝ≥0 := fun n => NNReal.mk (slabAmplitude κ q A*(rj n)^(1-codimension κ))
+    (mul_nonneg (slabAmplitude_nonneg κ q A) (Real.rpow_nonneg (hp n).le _))
+  have hsub (n : ℕ) : HasSubgaussianMGF (X n) (b n^2) ((P.comap Subtype.val)[|D]) := by
+    apply a.source_past_spatial_slab_subgaussian hκ hκ4 A hA q (rj (n+1)) (rj n) R hq (hp _) (he n).symm.le (s := s) (D := D)
+    · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
+    · exact (hl n).trans hrR
+    · exact fun z hz => (hl n).trans (hheight z hz)
+    · exact hbound
+    · exact hD
+    · intro ω hω z hz
+      apply (hbefore ω hω z hz).trans
+      apply spatialHitTime_antitone_radius (a.trace_continuous ω) z (hl n)
+      rw [(a.witness ω).2.1,dist_zero_left]
+      exact (hheight z hz).trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  have hsum := subgaussian_sum_amplitudes X b N (fun n _ => hsub n)
+  have hfin : volume A ≠ ∞ := (measure_mono (show A ⊆ Metric.closedBall 0 R from
+    fun z hz => by simpa only [Metric.mem_closedBall, dist_zero_right] using hbound z hz)).trans_lt
+      (measure_closedBall_lt_top (μ := volume) (x := (0 : ℂ)) (r := R)) |>.ne
+  have htel (ω : a.samples) : (∑ n ∈ Finset.range N, X n ω) =
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*q^N) A -
+      spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r A := by
+    have hm (n : ℕ) : X n ω =
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj (n+1)) A -
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A := by
+      symm
+      apply a.spatialRealMass_sub_eq_slab hκ (by linarith) A hA hfin _ _ (hp _)
+      · rw [he]; exact mul_le_of_le_one_left (hp n).le hq1.le
+      · exact fun z hz => (hl n).trans (hheight z hz)
+    simp_rw [hm]
+    simpa only [rj, pow_zero, mul_one] using Finset.sum_range_sub
+      (fun n => spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (rj n) A) N
+  simp_rw [htel] at hsum
+  apply SLEExactGauge.HasSubgaussianMGF.mono_parameter hsum
+  apply pow_le_pow_left₀ (by positivity)
+  apply NNReal.coe_le_coe.mp
+  simpa only [NNReal.coe_sum, b, rj, NNReal.coe_mk] using geometric_amplitude_sum (slabAmplitude κ q A) r q (1-codimension κ)
+    (slabAmplitude_nonneg κ q A) hr hq hq1 (sub_pos.mpr (codimension_lt_one κ hκ)) N
+
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SpatialMeans.lean b/lean/OAI/Probability/SLE/SpatialMeans.lean
index 4b08697ce1222accd71c36e6cb8f5f72925c8dfa..dd473eea9ff8d53c120b96b9fbd949d6bc5ce27d 100644
--- a/lean/OAI/Probability/SLE/SpatialMeans.lean
+++ b/lean/OAI/Probability/SLE/SpatialMeans.lean
@@ -1,720 +1,720 @@
-import Mathlib
-import OAI.Probability.SLE.InverseMoments
-
-namespace OAI
-
-/-! Spatial Means for chordal SLE. -/
-
-section
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace SpatialClippedFamily
-variable {Ω : Type*} [MeasurableSpace Ω] {U : ℝ≥0 → Ω → ℝ}
-  (c : SpatialClippedFamily U)
-noncomputable def jointChart (p : ℝ≥0 × (ℂ × Ω)) : ℂ :=
-  limUnder atTop (fun n => c.chartApprox n p)
-
-lemma jointChart_measurable : Measurable c.jointChart :=
-  (StronglyMeasurable.limUnder (fun n => (c.chartApprox_measurable n).stronglyMeasurable)).measurable
-
-lemma jointChart_eq {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (ω : Ω)
-    (hUc : Continuous (fun t => U t ω))
-    (hw : IsCapacityTwoWitness (fun t => U t ω) γ G F)
-    (t : ℝ≥0) (z : ℂ) :
-    c.jointChart (t, z, ω) = if z ∈ survivingDomain γ t then G t z else 0 := by
-  classical
-  unfold jointChart
-  apply Filter.Tendsto.limUnder_eq
-  apply tendsto_nhds_of_eventually_eq
-  by_cases hz : z ∈ survivingDomain γ t
-  · filter_upwards [c.eventually_valid_agrees ω hUc hw t z hz] with n hn
-    rw [ite_eq_left hz, chartApprox, ite_eq_left hn.1, hn.2, logCoordinates_re, logCoordinates_im,
-      Real.exp_log (hw.imag_positive t z hz)]
-    exact Complex.re_add_im _
-  · filter_upwards [] with n
-    have hn : ¬ c.valid n t z ω := fun hv => hz (c.live_of_valid ω hUc hw n t z hv)
-    simp only [chartApprox, ite_eq_right hn, ite_eq_right hz]
-
-include c
-
- 
-
-lemma live_measurable
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hUc : ∀ ω, Continuous (fun t => U t ω))
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => U t ω) (γ ω) (G ω) (F ω)) :
-    MeasurableSet {p : ℝ≥0 × (ℂ × Ω) | p.2.1 ∈ survivingDomain (γ p.2.2) p.1} := by
-  have he : {p : ℝ≥0 × (ℂ × Ω) | p.2.1 ∈ survivingDomain (γ p.2.2) p.1} =
-      ⋃ n : ℕ, {p | c.valid n p.1 p.2.1 p.2.2} := by
-    ext p
-    constructor
-    · intro hp
-      obtain ⟨n, hn⟩ := (c.eventually_valid_agrees p.2.2 (hUc _) (hw _) p.1 p.2.1 hp).exists
-      exact mem_iUnion.mpr ⟨n, hn.1⟩
-    · intro hp
-      obtain ⟨n, hn⟩ := mem_iUnion.mp hp
-      exact c.live_of_valid p.2.2 (hUc _) (hw _) n p.1 p.2.1 hn
-  rw [he]
-  exact MeasurableSet.iUnion (c.valid_measurable)
-
-end SpatialClippedFamily
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma spatialHitTime_mem {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (t : ℝ≥0) (ht : spatialHitTime γ z r = t) :
-    dist (γ t) z = r := by
-  obtain ⟨s, hs, he⟩ := (spatialHitTime_le_iff hc z r t).mp ht.le
-  have hle := (spatialHitTime_le_iff hc z r s).mpr ⟨s, ⟨bot_le, le_rfl⟩, he⟩
-  rw [ht] at hle
-  have hst : s = t := le_antisymm hs.2 (WithTop.coe_le_coe.mp hle)
-  exact hst ▸ he
-
- 
-
-noncomputable def liveSpatialHitTime (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) : WithTop ℝ≥0 :=
-  if ({t : ℝ≥0 | z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r}).Nonempty then
-    ↑(sInf {t : ℝ≥0 | z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r}) else ⊤
-
-lemma spatialHitTime_le_of_dist_le {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z)
-    (t : ℝ≥0) (ht : dist (γ t) z ≤ r) : spatialHitTime γ z r ≤ t := by
-  obtain ⟨s, hst, he⟩ := intermediate_value_Icc' (show (0 : ℝ≥0) ≤ t from bot_le)
-    (hc.dist continuous_const).continuousOn ⟨ht, hs⟩
-  exact (spatialHitTime_le_iff hc z r t).mpr ⟨s, hst, he⟩
-
-lemma liveSpatialHitTime_eq {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) :
-    liveSpatialHitTime γ z r =
-      if spatialHitTime γ z r = ⊤ then ⊤ else
-        if z ∈ survivingDomain γ ((spatialHitTime γ z r).untopD 0) then
-          spatialHitTime γ z r else ⊤ := by
-  let V := {t : ℝ≥0 | z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r}
-  by_cases hv : V.Nonempty
-  · obtain ⟨t, ht⟩ := hv
-    have hhit := spatialHitTime_le_of_dist_le hc z r hs t ht.2
-    have hn : spatialHitTime γ z r ≠ ⊤ := ne_top_of_le_ne_top WithTop.coe_ne_top hhit
-    obtain ⟨v, he⟩ := WithTop.ne_top_iff_exists.mp hn
-    have hvle : v ≤ t := WithTop.coe_le_coe.mp (he ▸ hhit)
-    have hvl : z ∈ survivingDomain γ v := survivingDomain_antitone γ hvle ht.1
-    have hvm : v ∈ V := ⟨hvl, (spatialHitTime_mem hc z r v he.symm).le⟩
-    have hmin : sInf V = v := by
-      apply le_antisymm (csInf_le ⟨0, fun _ _ => bot_le⟩ hvm)
-      apply le_csInf ⟨v, hvm⟩
-      intro s hs'
-      exact WithTop.coe_le_coe.mp (he ▸ spatialHitTime_le_of_dist_le hc z r hs s hs'.2)
-    change (if V.Nonempty then ((sInf V : ℝ≥0) : WithTop ℝ≥0) else ⊤) = _
-    rw [ite_eq_left ⟨v, hvm⟩, hmin, ite_eq_right hn, ← he, WithTop.untopD_coe, ite_eq_left hvl]
-  · change (if V.Nonempty then ((sInf V : ℝ≥0) : WithTop ℝ≥0) else ⊤) = _
-    rw [ite_eq_right hv]
-    by_cases hn : spatialHitTime γ z r = ⊤
-    · rw [ite_eq_left hn]
-    · rw [ite_eq_right hn]
-      obtain ⟨v, he⟩ := WithTop.ne_top_iff_exists.mp hn
-      rw [← he, WithTop.untopD_coe]
-      have hnl : z ∉ survivingDomain γ v := fun h =>
-        hv ⟨v, h, (spatialHitTime_mem hc z r v he.symm).le⟩
-      rw [ite_eq_right hnl]
-
- 
-
-lemma spatialGreenWeight_eq_live {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (κ : ℝ) (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) :
-    spatialGreenWeight κ U γ G z r =
-      if liveSpatialHitTime γ z r = ⊤ then 0 else
-        greenDensity κ U γ G ((liveSpatialHitTime γ z r).untopD 0) z := by
-  rw [liveSpatialHitTime_eq hc z r hs]
-  by_cases hn : spatialHitTime γ z r = ⊤
-  · simp only [spatialGreenWeight, ite_eq_left hn, ite_true]
-  · rw [ite_eq_right hn]
-    by_cases hl : z ∈ survivingDomain γ ((spatialHitTime γ z r).untopD 0)
-    · simp only [ite_eq_left hl, ite_eq_right hn, spatialGreenWeight]
-    · simp only [spatialGreenWeight, ite_eq_right hn,
-        greenDensity, ite_eq_right hl, ite_true]
-
-lemma spatialHitTime_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (γ : Ω → ℝ≥0 → ℂ) (hc : ∀ ω, Continuous (γ ω))
-    (hm : ∀ t, Measurable (fun ω => γ ω t)) (r : ℝ) :
-    Measurable (fun p : ℂ × Ω => spatialHitTime (γ p.2) p.1 r) := by
-  let X (t : ℝ≥0) (p : ℂ × Ω) : ℝ := dist (γ p.2 t) p.1 - r
-  have hXa : Adapted (⊤ : Filtration ℝ≥0 (inferInstance : MeasurableSpace (ℂ × Ω))) X := by
-    intro t
-    exact (((hm t).comp measurable_snd).dist measurable_fst).sub_const r
-  have hXc (p : ℂ × Ω) : Continuous (fun t => X t p) :=
-    ((hc p.2).dist continuous_const).sub continuous_const
-  have h := continuous_zero_hit_stopping ⊤ X hXa hXc 0
-  exact h.measurable.mono h.measurableSpace_le le_rfl
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
-lemma jointGreen_measurable :
-    Measurable (fun p : ℝ≥0 × (ℂ × a.samples) => greenDensity κ
-      (fun t => Real.sqrt κ * a.B t p.2.2) (γ p.2.2) (a.G p.2.2) p.1 p.2.1) := by
-  have hm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
-  let c := spatialClippedFamily (fun t ω => Real.sqrt κ * a.B t ω)
-    (hm.const_mul _) (fun ω => (a.continuous_B ω).const_mul _)
-  convert c.jointGreen_measurable κ a.B hm using 1
-  funext p
-  exact (c.jointGreen_eq κ a.B p.2.2 ((a.continuous_B _).const_mul _)
-    (fun _ => rfl) (a.witness _) p.1 p.2.1).symm
-
-lemma spatialGreenWeight_measurable (r : ℝ) :
-    Measurable (fun p : ℂ × a.samples => spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t p.2) (γ p.2) (a.G p.2) p.1 r) := by
-  classical
-  have hm (t : ℝ≥0) : Measurable (fun ω : a.samples => γ ω t) :=
-    (a.trace_adapted t).mono (a.filtration.le t) le_rfl
-  have hc (ω : a.samples) : Continuous (γ ω) := a.trace_continuous ω
-  have ht : Measurable (fun p : ℂ × a.samples => spatialHitTime (γ p.2) p.1 r) :=
-    spatialHitTime_measurable (fun ω : a.samples => γ ω) hc hm r
-  have he : MeasurableSet {p : ℂ × a.samples | spatialHitTime (γ p.2) p.1 r = ⊤} :=
-    measurableSet_eq_fun ht measurable_const
-  have hg := a.jointGreen_measurable.comp ((ht.untopD 0).prodMk measurable_id)
-  have hh := @Measurable.ite _ _ _ _ _ _ _ (fun _ => Classical.propDecidable _) he
-    (measurable_const (a := (0 : ℝ))) hg
-  simpa only [spatialGreenWeight, Function.comp_def, id_eq] using hh
-
-end RegularSLEPresentation
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def spatialSlabCoefficient (κ : ℝ) (U : ℝ≥0 → ℝ)
-    (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ) (r ε : ℝ) (t : ℝ≥0) (z : ℂ) : ℝ :=
-  if spatialHitTime γ z r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧
-      z ∈ survivingDomain γ t then greenBrownianCoefficient κ U γ G t z else 0
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
-lemma IsCapacityTwoWitness.radius_before_spatial_hit
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im) (t : ℝ≥0)
-    (hz : z ∈ survivingDomain γ t) (ht : (t : WithTop ℝ≥0) ≤ spatialHitTime γ z r) :
-    r ≤ fullRadius G t z := by
-  apply h.radius_ge_of_no_trace_approach hU t z hz hr hy
-  intro s hs
-  by_contra hn
-  have hstart : r ≤ dist (γ 0) z := by
-    rw [h.2.1, dist_zero_left]
-    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  have hhit := spatialHitTime_lt_of_dist_lt h.1 z r hstart s (lt_of_not_ge hn)
-  exact (not_lt_of_ge ht) (hhit.trans_le (WithTop.coe_le_coe.mpr hs.2))
-
-lemma IsCapacityTwoWitness.slab_coefficient_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
-    (r : ℝ) (z : ℂ) (hy : ε ≤ z.im) (t : ℝ≥0) :
-    |spatialSlabCoefficient κ U γ G r ε t z| ≤
-      if z ∈ survivingDomain γ t then
-        (Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) * chartPoissonRate U G t z
-      else 0 := by
-  by_cases ha : spatialHitTime γ z r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧
-      z ∈ survivingDomain γ t
-  · rw [spatialSlabCoefficient, ite_eq_left ha, ite_eq_left ha.2.2]
-    exact h.green_coefficient_le_poisson hU hκ hκ4 hε t z ha.2.2
-      (h.radius_before_spatial_hit hU z ε hε hy t ha.2.2 ha.2.1)
-  · rw [spatialSlabCoefficient, ite_eq_right ha, abs_zero]
-    split_ifs with hz
-    · exact mul_nonneg (by
-        have hg := greenExponent_ge_one hκ hκ4
-        exact mul_nonneg (mul_nonneg (Real.sqrt_nonneg _) (by linarith))
-          (Real.rpow_nonneg hε.le _)) (h.poissonRate_nonneg t z hz)
-    · exact le_rfl
-
- 
-
-theorem IsCapacityTwoWitness.slab_pair_later_start
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
-    (r : ℝ) (z w : ℂ) (hzy : ε ≤ z.im) (hwy : ε ≤ w.im) (hne : z ≠ w)
-    (s : ℝ≥0) (hs : spatialHitTime γ z r = s) :
-    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-      spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2) *
-        ENNReal.ofReal (halfPlaneGreen (G s z) (G s w) / 4) := by
-  let C := Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)
-  let S := {t : ℝ | (s : ℝ) ≤ t ∧ z ∈ survivingDomain γ t.toNNReal ∧
-    w ∈ survivingDomain γ t.toNNReal}
-  let f := fun t : ℝ => ENNReal.ofReal (chartPoissonRate U G t.toNNReal z *
-    chartPoissonRate U G t.toNNReal w)
-  have hC : 0 ≤ C := by
-    have hg := greenExponent_ge_one hκ hκ4
-    exact mul_nonneg (mul_nonneg (Real.sqrt_nonneg _) (by linarith)) (Real.rpow_nonneg hε.le _)
-  have hpoint (t : ℝ) (ht0 : 0 ≤ t) : ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-      spatialSlabCoefficient κ U γ G r ε t.toNNReal w| ≤
-      S.indicator (fun t => ENNReal.ofReal (C ^ 2) * f t) t := by
-    by_cases ha : spatialHitTime γ z r ≤ t.toNNReal ∧
-        (t.toNNReal : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧ z ∈ survivingDomain γ t.toNNReal
-    · by_cases hb : spatialHitTime γ w r ≤ t.toNNReal ∧
-          (t.toNNReal : WithTop ℝ≥0) ≤ spatialHitTime γ w ε ∧ w ∈ survivingDomain γ t.toNNReal
-      · have hst : (s : ℝ) ≤ t := by
-          have hst' : s ≤ t.toNNReal := WithTop.coe_le_coe.mp (hs ▸ ha.1)
-          rw [← Real.coe_toNNReal t ht0]
-          exact hst'
-        rw [Set.indicator_of_mem (show t ∈ S from ⟨hst, ha.2.2, hb.2.2⟩),
-          ← ENNReal.ofReal_mul (sq_nonneg C)]
-        apply ENNReal.ofReal_le_ofReal
-        rw [abs_mul]
-        have hz := h.slab_coefficient_bound hU hκ hκ4 hε r z hzy t.toNNReal
-        have hw := h.slab_coefficient_bound hU hκ hκ4 hε r w hwy t.toNNReal
-        rw [ite_eq_left ha.2.2] at hz
-        rw [ite_eq_left hb.2.2] at hw
-        calc
-          _ ≤ (C * chartPoissonRate U G t.toNNReal z) *
-              (C * chartPoissonRate U G t.toNNReal w) :=
-            mul_le_mul hz hw (abs_nonneg _) (mul_nonneg hC (h.poissonRate_nonneg _ z ha.2.2))
-          _ = _ := by ring
-      · simp only [spatialSlabCoefficient, ite_eq_right hb, mul_zero, abs_zero, ENNReal.ofReal_zero, zero_le]
-    · simp only [spatialSlabCoefficient, ite_eq_right ha, zero_mul, abs_zero, ENNReal.ofReal_zero, zero_le]
-  calc
-    _ ≤ ∫⁻ t in Ici (0 : ℝ), S.indicator (fun t => ENNReal.ofReal (C ^ 2) * f t) t := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem measurableSet_Ici] with t ht
-      exact hpoint t ht
-    _ ≤ ∫⁻ t, S.indicator (fun t => ENNReal.ofReal (C ^ 2) * f t) t :=
-      lintegral_mono' Measure.restrict_le_self le_rfl
-    _ ≤ ∫⁻ t in S, ENNReal.ofReal (C ^ 2) * f t := lintegral_indicator_le _ _
-    _ = ENNReal.ofReal (C ^ 2) * ∫⁻ t in S, f t := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-    _ ≤ _ := mul_le_mul le_rfl (h.live_green_pair_bound hU s z w hne) (by positivity) (by positivity)
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
-lemma IsCapacityTwoWitness.green_before_spatial_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ r : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hr : 0 < r)
-    (z : ℂ) (hy : r ≤ z.im) (t : ℝ≥0)
-    (ht : (t : WithTop ℝ≥0) ≤ spatialHitTime γ z r) :
-    greenDensity κ U γ G t z ≤ r ^ (-codimension κ) := by
-  by_cases hz : z ∈ survivingDomain γ t
-  · have hR := h.radius_before_spatial_hit hU z r hr hy t hz ht
-    have hs : 0 ≤ Real.sin (Complex.arg (G t z - (U t : ℂ))) := by
-      rw [Complex.sin_arg]
-      have hi := (h.imag_positive t z hz).le
-      simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
-      exact div_nonneg hi (norm_nonneg _)
-    rw [greenDensity, ite_eq_left hz]
-    calc
-      _ ≤ r ^ (-codimension κ) * 1 := mul_le_mul
-        (Real.rpow_le_rpow_of_nonpos hr hR (neg_nonpos.mpr (codimension_pos κ hκ8).le))
-        (Real.rpow_le_one hs (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
-        (Real.rpow_nonneg hs _) (Real.rpow_nonneg hr.le _)
-      _ = _ := mul_one _
-  · rw [greenDensity, ite_eq_right hz]
-    exact Real.rpow_nonneg hr.le _
-
-lemma IsCapacityTwoWitness.spatialGreenWeight_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ r : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hr : 0 < r)
-    (z : ℂ) (hy : r ≤ z.im) :
-    0 ≤ spatialGreenWeight κ U γ G z r ∧
-      spatialGreenWeight κ U γ G z r ≤ r ^ (-codimension κ) := by
-  by_cases ht : spatialHitTime γ z r = ⊤
-  · simp only [spatialGreenWeight, ite_eq_left ht]
-    exact ⟨le_rfl, Real.rpow_nonneg hr.le _⟩
-  · rw [spatialGreenWeight, ite_eq_right ht]
-    refine ⟨greenDensity_nonneg h κ _ z, h.green_before_spatial_bound hU hκ hκ8 hr z hy _ ?_⟩
-    exact WithTop.coe_untopD_le _ _
-
-noncomputable def spatialRealMass (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
-    (G : ℝ≥0 → ℂ → ℂ) (r : ℝ) (A : Set ℂ) : ℝ :=
-  ∫ z in A, spatialGreenWeight κ U γ G z r
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
-lemma spatialRealMass_measurable (A : Set ℂ) (r : ℝ) :
-    Measurable (fun ω : a.samples => spatialRealMass κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A) :=
-  (a.spatialGreenWeight_measurable r).stronglyMeasurable.integral_prod_left'.measurable
-
-lemma spatialGreenWeight_spatial_integrable
-    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hfin : volume A ≠ ∞)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (ω : a.samples) :
-    IntegrableOn (fun z => spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) A := by
-  let : IsFiniteMeasure (volume.restrict A) := ⟨by
-    simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
-  apply Integrable.of_bound
-    ((a.spatialGreenWeight_measurable r).comp (measurable_id.prodMk measurable_const)).aestronglyMeasurable
-    (r ^ (-codimension κ))
-  filter_upwards [ae_restrict_mem hA] with z hz
-  have h := (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hκ hκ8 hr z (hheight z hz)
-  simpa only [Function.comp_apply, id_eq, Real.norm_eq_abs, abs_of_nonneg h.1] using h.2
-
-lemma spatialRealMass_bound
-    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hfin : volume A ≠ ∞)
-    (hheight : ∀ z ∈ A, r ≤ z.im) (ω : a.samples) :
-    0 ≤ spatialRealMass κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A ∧
-      spatialRealMass κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A ≤
-        (volume A).toReal * r ^ (-codimension κ) := by
-  let : IsFiniteMeasure (volume.restrict A) := ⟨by
-    simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
-  have hi := a.spatialGreenWeight_spatial_integrable hκ hκ8 r hr A hA hfin hheight ω
-  have hb : ∀ᵐ z ∂(volume.restrict A), 0 ≤ spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r ∧
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r ≤ r ^ (-codimension κ) := by
-    filter_upwards [ae_restrict_mem hA] with z hz
-    exact (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hκ hκ8 hr z (hheight z hz)
-  refine ⟨integral_nonneg_of_ae (hb.mono fun _ h => h.1), ?_⟩
-  calc
-    _ ≤ ∫ _z in A, r ^ (-codimension κ) := integral_mono_ae hi (integrable_const _) (hb.mono fun _ h => h.2)
-    _ = _ := by simp only [integral_const, Measure.restrict_apply_univ, smul_eq_mul, measureReal_def]
-
-end RegularSLEPresentation
-end SLEExactGauge
-
-end
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-universe u_source_stopped
-
- 
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-local instance spatialMeansStoppedRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance spatialMeansStoppedComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance spatialMeansStoppedRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance spatialMeansStoppedComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance spatialMeansDrivingRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance spatialMeansDrivingRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance spatialMeansSourceRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance spatialMeansSourceComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance spatialMeansSourceRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance spatialMeansSourceComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance spatialMeansMapRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance spatialMeansMapComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance spatialMeansMapRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance spatialMeansMapComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance spatialMeansTraceRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance spatialMeansTraceComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance spatialMeansTraceRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance spatialMeansTraceComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-end
-end SLEExactGauge
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
- 
-
-lemma source_halfplane_map_local_bound {f : ℂ → ℂ} {z w : ℂ} {r ε : ℝ}
-    (hr : 0 < r) (hε : 0 ≤ ε) (hεr : ε ≤ r / 2)
-    (hd : DifferentiableOn ℂ f (ball z r))
-    (hm : MapsTo f (ball z r) upperHalfPlane) (hw : dist w z ≤ ε) :
-    dist (f w) (f z) ≤ 4 * (f z).im * ε / r := by
-  have hz : z ∈ ball z r := mem_ball_self hr
-  have hwr : w ∈ ball z r := mem_ball.mpr (hw.trans_lt (by linarith))
-  have hy : 0 < (f z).im := hm hz
-  have hden (v : ℂ) (hv : v ∈ ball z r) : f v - star (f z) ≠ 0 := by
-    intro he
-    have hx := congrArg Complex.im he
-    simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hx
-    have hvy : 0 < (f v).im := hm hv
-    linarith
-  let g := fun v => (f v - f z) / (f v - star (f z))
-  have hg0 : g z = 0 := by simp [g]
-  have hgd : DifferentiableOn ℂ g (ball z r) :=
-    (hd.sub_const _).div (hd.sub_const _) hden
-  have hgm : MapsTo g (ball z r) (closedBall (g z) 1) := by
-    intro v hv
-    rw [hg0, mem_closedBall_zero_iff]
-    exact (upper_mobius_norm_lt_one (hm hv) hy).le
-  have hsch := Complex.dist_le_div_mul_dist_of_mapsTo_ball hgd hgm hwr
-  rw [hg0, dist_zero_right] at hsch
-  have hsch' : ‖g w‖ ≤ ε / r :=
-    hsch.trans (by simpa only [one_div, div_eq_mul_inv, mul_comm, mul_one] using
-      div_le_div_of_nonneg_right hw hr.le)
-  have hmult : ‖f w - f z‖ ≤ (ε / r) * ‖f w - star (f z)‖ := by
-    simpa only [g, norm_div, div_le_iff₀ (norm_pos_iff.mpr (hden w hwr))] using hsch'
-  have hn : ‖f z - star (f z)‖ = 2 * (f z).im := by
-    rw [Complex.star_def, Complex.sub_conj, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-      abs_of_pos (mul_pos (by norm_num) hy), Complex.norm_I, mul_one]
-  have htri : ‖f w - star (f z)‖ ≤ ‖f w - f z‖ + 2 * (f z).im := by
-    rw [← hn]
-    exact norm_sub_le_norm_sub_add_norm_sub _ _ _
-  have ha : 0 ≤ ε / r := div_nonneg hε hr.le
-  have hb : ε / r ≤ 1 / 2 := (div_le_iff₀ hr).mpr (by linarith)
-  have hmain := hmult.trans (mul_le_mul_of_nonneg_left htri ha)
-  rw [dist_eq_norm]
-  have hp := mul_nonneg (sub_nonneg.mpr hb) (norm_nonneg (f w - f z))
-  calc
-    ‖f w - f z‖ ≤ 4 * (f z).im * (ε / r) := by nlinarith
-    _ = _ := by ring
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal
-namespace SLEExactGauge
-
- 
-
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
- 
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-theorem source_killedGreen_progressive
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)) (z : ℂ) :
-    IsStronglyProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) := by
-  classical
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable hc hm
-  have hnn (n : ℕ) : -(n : ℝ) ≤ n := by linarith [Nat.cast_nonneg (α := ℝ) n]
-  let c (n : ℕ) := (exists_clippedLogSolution B hm hBm hc κ (-(n : ℝ)) n (hnn n)
-    (initialLogCoordinates z)).some
-  let v (n : ℕ) (t : ℝ≥0) (ω : Ω) : Prop :=
-    0 < z.im ∧ Real.log z.im ≤ (n : ℝ) ∧ -(n : ℝ) ≤ ((c n).Q t ω).1.im
-  let A (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ :=
-    if v n t ω then greenLogDensity κ ((c n).Q t ω) (B t ω) else 0
-  have hQp (n : ℕ) : IsStronglyProgressive f (fun t : ℝ≥0 => (c n).Q t) :=
-    (show StronglyAdapted f (fun t : ℝ≥0 => (c n).Q t) from
-      fun t => ((c n).adapted t).stronglyMeasurable).isStronglyProgressive_of_continuous
-      (fun ω => ((c n).lipschitz ω).continuous.comp NNReal.continuous_coe)
-  have hAp (n : ℕ) : IsStronglyProgressive f (A n) := by
-    intro T
-    have hQ := (hQp n T).measurable
-    let : MeasurableSpace Ω := f T
-    have hv : MeasurableSet {p : Iic T × Ω | v n p.1 p.2} :=
-      (MeasurableSet.const _).inter ((MeasurableSet.const _).inter
-        (measurableSet_le measurable_const (Complex.measurable_im.comp hQ.fst)))
-    have hBp := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc
-    have hBm := (hBp T).measurable
-    exact (Measurable.ite hv ((greenLogDensity_contDiff κ).continuous.measurable.comp
-      (hQ.prodMk hBm)) measurable_const).stronglyMeasurable
-  have hlim (t : ℝ≥0) (ω : Ω) : Tendsto (fun n => A n t ω) atTop
-      (𝓝 (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z)) := by
-    apply tendsto_nhds_of_eventually_eq
-    have hU : Continuous (fun s => Real.sqrt κ * B s ω) := (hc ω).const_mul _
-    have hd (n : ℕ) (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :=
-      ((c n).derivative ω s hs.1).mono (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self)
-    by_cases hz : z ∈ survivingDomain (γ ω) t
-    · obtain ⟨N, hN⟩ := exists_nat_ge (max (-Real.log (G ω t z).im) (Real.log z.im))
-      filter_upwards [eventually_ge_atTop N] with n hn
-      have hNn : (N : ℝ) ≤ n := by exact_mod_cast hn
-      have hlo : -(n : ℝ) ≤ Real.log (G ω t z).im := by
-        have hh := (le_max_left (-Real.log (G ω t z).im) (Real.log z.im)).trans (hN.trans hNn)
-        linarith
-      have hhi : Real.log z.im ≤ (n : ℝ) := (le_max_right _ _).trans (hN.trans hNn)
-      have he : (c n).Q t ω = logLoewnerCoordinates (G ω) z t :=
-        clipped_log_loewner_agrees hU (hw ω) t z hz _ _ _ hlo hhi
-          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) ⟨t.coe_nonneg, le_rfl⟩
-      have hv : v n t ω := ⟨hz.1.1, hhi, by simpa only [he, logCoordinates_im] using hlo⟩
-      simp only [A, ite_eq_left hv, he]
-      exact greenLogDensity_logCoordinates hU (hw ω) κ (B t ω) z t hz rfl
-    · filter_upwards [] with n
-      have hv : ¬ v n t ω := by
-        intro hv
-        exact hz (clipped_log_identification_of_endpoint hU (hw ω) t z hv.1 _ _ _
-          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) hv.2.2 hv.2.1).1
-      simp only [A, ite_eq_right hv, greenDensity, ite_eq_right hz]
-  exact isStronglyProgressive_of_tendsto hAp
-    (tendsto_pi_nhds.mpr (fun t => tendsto_pi_nhds.mpr (hlim t)))
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance sourceApproachComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance sourceApproachComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-noncomputable def sourceApproachKernel (ε r : ℝ) (p : ℂ × C(ℝ≥0, ℂ)) : ℝ :=
-  if ∃ t : ℝ≥0, dist (p.2 t) p.1 < 8 * p.1.im * ε / r then 1 else 0
-
-lemma sourceApproachKernel_measurable (ε r : ℝ) : Measurable (sourceApproachKernel ε r) := by
-  have he (p : ℂ × C(ℝ≥0, ℂ)) :
-      (∃ t : ℝ≥0, dist (p.2 t) p.1 < 8 * p.1.im * ε / r) ↔
-      ∃ n : ℕ, dist (p.2 (denseSeq ℝ≥0 n)) p.1 < 8 * p.1.im * ε / r := by
-    constructor
-    · intro h
-      exact (denseRange_denseSeq ℝ≥0).exists_mem_open
-        (isOpen_lt (p.2.continuous.dist continuous_const) continuous_const) h
-    · rintro ⟨n, hn⟩
-      exact ⟨_, hn⟩
-  apply Measurable.ite _ measurable_const measurable_const
-  simp_rw [he]
-  rw [ofPred_exists]
-  apply MeasurableSet.iUnion
-  intro n
-  exact measurableSet_lt
-    (((continuous_eval_const _).comp continuous_snd).dist continuous_fst).measurable
-    ((((continuous_im.comp continuous_fst).const_mul 8).mul_const ε).div_const r).measurable
-
-lemma sourceApproachKernel_norm_le (ε r : ℝ) (p : ℂ × C(ℝ≥0, ℂ)) :
-    ‖sourceApproachKernel ε r p‖ ≤ 1 := by
-  unfold sourceApproachKernel
-  split_ifs <;> norm_num
-
-lemma sourceApproachKernel_zero {ε r : ℝ} (hε : 0 ≤ ε) (hr : 0 ≤ r)
-    {p : ℂ × C(ℝ≥0, ℂ)} (hp : p.1.im ≤ 0) : sourceApproachKernel ε r p = 0 := by
-  apply ite_eq_right
-  rintro ⟨t, ht⟩
-  have hh : 8 * p.1.im * ε / r ≤ 0 :=
-    div_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonpos_of_nonneg
-      (mul_nonpos_of_nonneg_of_nonpos (by norm_num) hp) hε) hr
-  exact not_lt_of_ge (dist_nonneg.trans' hh) ht
-
-lemma source_rpow_scale (p y ε r R : ℝ) (hy : 0 < y) (he : 0 < ε)
-    (hr : 0 < r) (hR : 0 < R) :
-    (2 * y) ^ (-p) * (8 * y * ε / r) ^ p =
-      (4 * R / r) ^ p * ε ^ p * R ^ (-p) := by
-  rw [Real.rpow_def_of_pos (by positivity : 0 < 2*y),
-    Real.rpow_def_of_pos (by positivity : 0 < 8*y*ε/r),
-    Real.rpow_def_of_pos (by positivity : 0 < 4*R/r),
-    Real.rpow_def_of_pos he, Real.rpow_def_of_pos hR,
-    ← Real.exp_add, ← Real.exp_add, ← Real.exp_add]
-  congr 1
-  rw [Real.log_div (by positivity) hr.ne', Real.log_div (by positivity) hr.ne',
-    Real.log_mul (by norm_num : (2:ℝ) ≠ 0) hy.ne',
-    Real.log_mul (by positivity : 8*y ≠ 0) he.ne',
-    Real.log_mul (by norm_num : (8:ℝ) ≠ 0) hy.ne',
-    Real.log_mul (by norm_num : (4:ℝ) ≠ 0) hR.ne']
-  have h8 : Real.log 8 = 3 * Real.log 2 := by
-    rw [show (8:ℝ)=2^3 by norm_num, Real.log_pow]; ring
-  have h4 : Real.log 4 = 2 * Real.log 2 := by
-    rw [show (4:ℝ)=2^2 by norm_num, Real.log_pow]; ring
-  rw [h8, h4]
-  ring
-end
-end SLEExactGauge
-
-end
-
- 
-section
-open Set Filter MeasureTheory Metric Bornology
-open scoped Topology NNReal
-namespace SLEExactGauge
- 
-
-end SLEExactGauge
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-
-end
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.InverseMoments
+
+namespace OAI
+
+/-! Spatial Means for chordal SLE. -/
+
+section
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace SpatialClippedFamily
+variable {Ω : Type*} [MeasurableSpace Ω] {U : ℝ≥0 → Ω → ℝ}
+  (c : SpatialClippedFamily U)
+noncomputable def jointChart (p : ℝ≥0 × (ℂ × Ω)) : ℂ :=
+  limUnder atTop (fun n => c.chartApprox n p)
+
+lemma jointChart_measurable : Measurable c.jointChart :=
+  (StronglyMeasurable.limUnder (fun n => (c.chartApprox_measurable n).stronglyMeasurable)).measurable
+
+lemma jointChart_eq {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} (ω : Ω)
+    (hUc : Continuous (fun t => U t ω))
+    (hw : IsCapacityTwoWitness (fun t => U t ω) γ G F)
+    (t : ℝ≥0) (z : ℂ) :
+    c.jointChart (t, z, ω) = if z ∈ survivingDomain γ t then G t z else 0 := by
+  classical
+  unfold jointChart
+  apply Filter.Tendsto.limUnder_eq
+  apply tendsto_nhds_of_eventually_eq
+  by_cases hz : z ∈ survivingDomain γ t
+  · filter_upwards [c.eventually_valid_agrees ω hUc hw t z hz] with n hn
+    rw [ite_eq_left hz, chartApprox, ite_eq_left hn.1, hn.2, logCoordinates_re, logCoordinates_im,
+      Real.exp_log (hw.imag_positive t z hz)]
+    exact Complex.re_add_im _
+  · filter_upwards [] with n
+    have hn : ¬ c.valid n t z ω := fun hv => hz (c.live_of_valid ω hUc hw n t z hv)
+    simp only [chartApprox, ite_eq_right hn, ite_eq_right hz]
+
+include c
+
+ 
+
+lemma live_measurable
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hUc : ∀ ω, Continuous (fun t => U t ω))
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => U t ω) (γ ω) (G ω) (F ω)) :
+    MeasurableSet {p : ℝ≥0 × (ℂ × Ω) | p.2.1 ∈ survivingDomain (γ p.2.2) p.1} := by
+  have he : {p : ℝ≥0 × (ℂ × Ω) | p.2.1 ∈ survivingDomain (γ p.2.2) p.1} =
+      ⋃ n : ℕ, {p | c.valid n p.1 p.2.1 p.2.2} := by
+    ext p
+    constructor
+    · intro hp
+      obtain ⟨n, hn⟩ := (c.eventually_valid_agrees p.2.2 (hUc _) (hw _) p.1 p.2.1 hp).exists
+      exact mem_iUnion.mpr ⟨n, hn.1⟩
+    · intro hp
+      obtain ⟨n, hn⟩ := mem_iUnion.mp hp
+      exact c.live_of_valid p.2.2 (hUc _) (hw _) n p.1 p.2.1 hn
+  rw [he]
+  exact MeasurableSet.iUnion (c.valid_measurable)
+
+end SpatialClippedFamily
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma spatialHitTime_mem {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (t : ℝ≥0) (ht : spatialHitTime γ z r = t) :
+    dist (γ t) z = r := by
+  obtain ⟨s, hs, he⟩ := (spatialHitTime_le_iff hc z r t).mp ht.le
+  have hle := (spatialHitTime_le_iff hc z r s).mpr ⟨s, ⟨bot_le, le_rfl⟩, he⟩
+  rw [ht] at hle
+  have hst : s = t := le_antisymm hs.2 (WithTop.coe_le_coe.mp hle)
+  exact hst ▸ he
+
+ 
+
+noncomputable def liveSpatialHitTime (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) : WithTop ℝ≥0 :=
+  if ({t : ℝ≥0 | z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r}).Nonempty then
+    ↑(sInf {t : ℝ≥0 | z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r}) else ⊤
+
+lemma spatialHitTime_le_of_dist_le {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z)
+    (t : ℝ≥0) (ht : dist (γ t) z ≤ r) : spatialHitTime γ z r ≤ t := by
+  obtain ⟨s, hst, he⟩ := intermediate_value_Icc' (show (0 : ℝ≥0) ≤ t from bot_le)
+    (hc.dist continuous_const).continuousOn ⟨ht, hs⟩
+  exact (spatialHitTime_le_iff hc z r t).mpr ⟨s, hst, he⟩
+
+lemma liveSpatialHitTime_eq {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) :
+    liveSpatialHitTime γ z r =
+      if spatialHitTime γ z r = ⊤ then ⊤ else
+        if z ∈ survivingDomain γ ((spatialHitTime γ z r).untopD 0) then
+          spatialHitTime γ z r else ⊤ := by
+  let V := {t : ℝ≥0 | z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r}
+  by_cases hv : V.Nonempty
+  · obtain ⟨t, ht⟩ := hv
+    have hhit := spatialHitTime_le_of_dist_le hc z r hs t ht.2
+    have hn : spatialHitTime γ z r ≠ ⊤ := ne_top_of_le_ne_top WithTop.coe_ne_top hhit
+    obtain ⟨v, he⟩ := WithTop.ne_top_iff_exists.mp hn
+    have hvle : v ≤ t := WithTop.coe_le_coe.mp (he ▸ hhit)
+    have hvl : z ∈ survivingDomain γ v := survivingDomain_antitone γ hvle ht.1
+    have hvm : v ∈ V := ⟨hvl, (spatialHitTime_mem hc z r v he.symm).le⟩
+    have hmin : sInf V = v := by
+      apply le_antisymm (csInf_le ⟨0, fun _ _ => bot_le⟩ hvm)
+      apply le_csInf ⟨v, hvm⟩
+      intro s hs'
+      exact WithTop.coe_le_coe.mp (he ▸ spatialHitTime_le_of_dist_le hc z r hs s hs'.2)
+    change (if V.Nonempty then ((sInf V : ℝ≥0) : WithTop ℝ≥0) else ⊤) = _
+    rw [ite_eq_left ⟨v, hvm⟩, hmin, ite_eq_right hn, ← he, WithTop.untopD_coe, ite_eq_left hvl]
+  · change (if V.Nonempty then ((sInf V : ℝ≥0) : WithTop ℝ≥0) else ⊤) = _
+    rw [ite_eq_right hv]
+    by_cases hn : spatialHitTime γ z r = ⊤
+    · rw [ite_eq_left hn]
+    · rw [ite_eq_right hn]
+      obtain ⟨v, he⟩ := WithTop.ne_top_iff_exists.mp hn
+      rw [← he, WithTop.untopD_coe]
+      have hnl : z ∉ survivingDomain γ v := fun h =>
+        hv ⟨v, h, (spatialHitTime_mem hc z r v he.symm).le⟩
+      rw [ite_eq_right hnl]
+
+ 
+
+lemma spatialGreenWeight_eq_live {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (κ : ℝ) (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) :
+    spatialGreenWeight κ U γ G z r =
+      if liveSpatialHitTime γ z r = ⊤ then 0 else
+        greenDensity κ U γ G ((liveSpatialHitTime γ z r).untopD 0) z := by
+  rw [liveSpatialHitTime_eq hc z r hs]
+  by_cases hn : spatialHitTime γ z r = ⊤
+  · simp only [spatialGreenWeight, ite_eq_left hn, ite_true]
+  · rw [ite_eq_right hn]
+    by_cases hl : z ∈ survivingDomain γ ((spatialHitTime γ z r).untopD 0)
+    · simp only [ite_eq_left hl, ite_eq_right hn, spatialGreenWeight]
+    · simp only [spatialGreenWeight, ite_eq_right hn,
+        greenDensity, ite_eq_right hl, ite_true]
+
+lemma spatialHitTime_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (γ : Ω → ℝ≥0 → ℂ) (hc : ∀ ω, Continuous (γ ω))
+    (hm : ∀ t, Measurable (fun ω => γ ω t)) (r : ℝ) :
+    Measurable (fun p : ℂ × Ω => spatialHitTime (γ p.2) p.1 r) := by
+  let X (t : ℝ≥0) (p : ℂ × Ω) : ℝ := dist (γ p.2 t) p.1 - r
+  have hXa : Adapted (⊤ : Filtration ℝ≥0 (inferInstance : MeasurableSpace (ℂ × Ω))) X := by
+    intro t
+    exact (((hm t).comp measurable_snd).dist measurable_fst).sub_const r
+  have hXc (p : ℂ × Ω) : Continuous (fun t => X t p) :=
+    ((hc p.2).dist continuous_const).sub continuous_const
+  have h := continuous_zero_hit_stopping ⊤ X hXa hXc 0
+  exact h.measurable.mono h.measurableSpace_le le_rfl
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+lemma jointGreen_measurable :
+    Measurable (fun p : ℝ≥0 × (ℂ × a.samples) => greenDensity κ
+      (fun t => Real.sqrt κ * a.B t p.2.2) (γ p.2.2) (a.G p.2.2) p.1 p.2.1) := by
+  have hm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
+  let c := spatialClippedFamily (fun t ω => Real.sqrt κ * a.B t ω)
+    (hm.const_mul _) (fun ω => (a.continuous_B ω).const_mul _)
+  convert c.jointGreen_measurable κ a.B hm using 1
+  funext p
+  exact (c.jointGreen_eq κ a.B p.2.2 ((a.continuous_B _).const_mul _)
+    (fun _ => rfl) (a.witness _) p.1 p.2.1).symm
+
+lemma spatialGreenWeight_measurable (r : ℝ) :
+    Measurable (fun p : ℂ × a.samples => spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t p.2) (γ p.2) (a.G p.2) p.1 r) := by
+  classical
+  have hm (t : ℝ≥0) : Measurable (fun ω : a.samples => γ ω t) :=
+    (a.trace_adapted t).mono (a.filtration.le t) le_rfl
+  have hc (ω : a.samples) : Continuous (γ ω) := a.trace_continuous ω
+  have ht : Measurable (fun p : ℂ × a.samples => spatialHitTime (γ p.2) p.1 r) :=
+    spatialHitTime_measurable (fun ω : a.samples => γ ω) hc hm r
+  have he : MeasurableSet {p : ℂ × a.samples | spatialHitTime (γ p.2) p.1 r = ⊤} :=
+    measurableSet_eq_fun ht measurable_const
+  have hg := a.jointGreen_measurable.comp ((ht.untopD 0).prodMk measurable_id)
+  have hh := @Measurable.ite _ _ _ _ _ _ _ (fun _ => Classical.propDecidable _) he
+    (measurable_const (a := (0 : ℝ))) hg
+  simpa only [spatialGreenWeight, Function.comp_def, id_eq] using hh
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def spatialSlabCoefficient (κ : ℝ) (U : ℝ≥0 → ℝ)
+    (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ) (r ε : ℝ) (t : ℝ≥0) (z : ℂ) : ℝ :=
+  if spatialHitTime γ z r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧
+      z ∈ survivingDomain γ t then greenBrownianCoefficient κ U γ G t z else 0
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+lemma IsCapacityTwoWitness.radius_before_spatial_hit
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im) (t : ℝ≥0)
+    (hz : z ∈ survivingDomain γ t) (ht : (t : WithTop ℝ≥0) ≤ spatialHitTime γ z r) :
+    r ≤ fullRadius G t z := by
+  apply h.radius_ge_of_no_trace_approach hU t z hz hr hy
+  intro s hs
+  by_contra hn
+  have hstart : r ≤ dist (γ 0) z := by
+    rw [h.2.1, dist_zero_left]
+    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  have hhit := spatialHitTime_lt_of_dist_lt h.1 z r hstart s (lt_of_not_ge hn)
+  exact (not_lt_of_ge ht) (hhit.trans_le (WithTop.coe_le_coe.mpr hs.2))
+
+lemma IsCapacityTwoWitness.slab_coefficient_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
+    (r : ℝ) (z : ℂ) (hy : ε ≤ z.im) (t : ℝ≥0) :
+    |spatialSlabCoefficient κ U γ G r ε t z| ≤
+      if z ∈ survivingDomain γ t then
+        (Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) * chartPoissonRate U G t z
+      else 0 := by
+  by_cases ha : spatialHitTime γ z r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧
+      z ∈ survivingDomain γ t
+  · rw [spatialSlabCoefficient, ite_eq_left ha, ite_eq_left ha.2.2]
+    exact h.green_coefficient_le_poisson hU hκ hκ4 hε t z ha.2.2
+      (h.radius_before_spatial_hit hU z ε hε hy t ha.2.2 ha.2.1)
+  · rw [spatialSlabCoefficient, ite_eq_right ha, abs_zero]
+    split_ifs with hz
+    · exact mul_nonneg (by
+        have hg := greenExponent_ge_one hκ hκ4
+        exact mul_nonneg (mul_nonneg (Real.sqrt_nonneg _) (by linarith))
+          (Real.rpow_nonneg hε.le _)) (h.poissonRate_nonneg t z hz)
+    · exact le_rfl
+
+ 
+
+theorem IsCapacityTwoWitness.slab_pair_later_start
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
+    (r : ℝ) (z w : ℂ) (hzy : ε ≤ z.im) (hwy : ε ≤ w.im) (hne : z ≠ w)
+    (s : ℝ≥0) (hs : spatialHitTime γ z r = s) :
+    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+      spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2) *
+        ENNReal.ofReal (halfPlaneGreen (G s z) (G s w) / 4) := by
+  let C := Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)
+  let S := {t : ℝ | (s : ℝ) ≤ t ∧ z ∈ survivingDomain γ t.toNNReal ∧
+    w ∈ survivingDomain γ t.toNNReal}
+  let f := fun t : ℝ => ENNReal.ofReal (chartPoissonRate U G t.toNNReal z *
+    chartPoissonRate U G t.toNNReal w)
+  have hC : 0 ≤ C := by
+    have hg := greenExponent_ge_one hκ hκ4
+    exact mul_nonneg (mul_nonneg (Real.sqrt_nonneg _) (by linarith)) (Real.rpow_nonneg hε.le _)
+  have hpoint (t : ℝ) (ht0 : 0 ≤ t) : ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+      spatialSlabCoefficient κ U γ G r ε t.toNNReal w| ≤
+      S.indicator (fun t => ENNReal.ofReal (C ^ 2) * f t) t := by
+    by_cases ha : spatialHitTime γ z r ≤ t.toNNReal ∧
+        (t.toNNReal : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧ z ∈ survivingDomain γ t.toNNReal
+    · by_cases hb : spatialHitTime γ w r ≤ t.toNNReal ∧
+          (t.toNNReal : WithTop ℝ≥0) ≤ spatialHitTime γ w ε ∧ w ∈ survivingDomain γ t.toNNReal
+      · have hst : (s : ℝ) ≤ t := by
+          have hst' : s ≤ t.toNNReal := WithTop.coe_le_coe.mp (hs ▸ ha.1)
+          rw [← Real.coe_toNNReal t ht0]
+          exact hst'
+        rw [Set.indicator_of_mem (show t ∈ S from ⟨hst, ha.2.2, hb.2.2⟩),
+          ← ENNReal.ofReal_mul (sq_nonneg C)]
+        apply ENNReal.ofReal_le_ofReal
+        rw [abs_mul]
+        have hz := h.slab_coefficient_bound hU hκ hκ4 hε r z hzy t.toNNReal
+        have hw := h.slab_coefficient_bound hU hκ hκ4 hε r w hwy t.toNNReal
+        rw [ite_eq_left ha.2.2] at hz
+        rw [ite_eq_left hb.2.2] at hw
+        calc
+          _ ≤ (C * chartPoissonRate U G t.toNNReal z) *
+              (C * chartPoissonRate U G t.toNNReal w) :=
+            mul_le_mul hz hw (abs_nonneg _) (mul_nonneg hC (h.poissonRate_nonneg _ z ha.2.2))
+          _ = _ := by ring
+      · simp only [spatialSlabCoefficient, ite_eq_right hb, mul_zero, abs_zero, ENNReal.ofReal_zero, zero_le]
+    · simp only [spatialSlabCoefficient, ite_eq_right ha, zero_mul, abs_zero, ENNReal.ofReal_zero, zero_le]
+  calc
+    _ ≤ ∫⁻ t in Ici (0 : ℝ), S.indicator (fun t => ENNReal.ofReal (C ^ 2) * f t) t := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem measurableSet_Ici] with t ht
+      exact hpoint t ht
+    _ ≤ ∫⁻ t, S.indicator (fun t => ENNReal.ofReal (C ^ 2) * f t) t :=
+      lintegral_mono' Measure.restrict_le_self le_rfl
+    _ ≤ ∫⁻ t in S, ENNReal.ofReal (C ^ 2) * f t := lintegral_indicator_le _ _
+    _ = ENNReal.ofReal (C ^ 2) * ∫⁻ t in S, f t := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+    _ ≤ _ := mul_le_mul le_rfl (h.live_green_pair_bound hU s z w hne) (by positivity) (by positivity)
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+lemma IsCapacityTwoWitness.green_before_spatial_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ r : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hr : 0 < r)
+    (z : ℂ) (hy : r ≤ z.im) (t : ℝ≥0)
+    (ht : (t : WithTop ℝ≥0) ≤ spatialHitTime γ z r) :
+    greenDensity κ U γ G t z ≤ r ^ (-codimension κ) := by
+  by_cases hz : z ∈ survivingDomain γ t
+  · have hR := h.radius_before_spatial_hit hU z r hr hy t hz ht
+    have hs : 0 ≤ Real.sin (Complex.arg (G t z - (U t : ℂ))) := by
+      rw [Complex.sin_arg]
+      have hi := (h.imag_positive t z hz).le
+      simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
+      exact div_nonneg hi (norm_nonneg _)
+    rw [greenDensity, ite_eq_left hz]
+    calc
+      _ ≤ r ^ (-codimension κ) * 1 := mul_le_mul
+        (Real.rpow_le_rpow_of_nonpos hr hR (neg_nonpos.mpr (codimension_pos κ hκ8).le))
+        (Real.rpow_le_one hs (Real.sin_le_one _) (greenExponent_pos κ hκ hκ8).le)
+        (Real.rpow_nonneg hs _) (Real.rpow_nonneg hr.le _)
+      _ = _ := mul_one _
+  · rw [greenDensity, ite_eq_right hz]
+    exact Real.rpow_nonneg hr.le _
+
+lemma IsCapacityTwoWitness.spatialGreenWeight_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ r : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) (hr : 0 < r)
+    (z : ℂ) (hy : r ≤ z.im) :
+    0 ≤ spatialGreenWeight κ U γ G z r ∧
+      spatialGreenWeight κ U γ G z r ≤ r ^ (-codimension κ) := by
+  by_cases ht : spatialHitTime γ z r = ⊤
+  · simp only [spatialGreenWeight, ite_eq_left ht]
+    exact ⟨le_rfl, Real.rpow_nonneg hr.le _⟩
+  · rw [spatialGreenWeight, ite_eq_right ht]
+    refine ⟨greenDensity_nonneg h κ _ z, h.green_before_spatial_bound hU hκ hκ8 hr z hy _ ?_⟩
+    exact WithTop.coe_untopD_le _ _
+
+noncomputable def spatialRealMass (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
+    (G : ℝ≥0 → ℂ → ℂ) (r : ℝ) (A : Set ℂ) : ℝ :=
+  ∫ z in A, spatialGreenWeight κ U γ G z r
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+lemma spatialRealMass_measurable (A : Set ℂ) (r : ℝ) :
+    Measurable (fun ω : a.samples => spatialRealMass κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A) :=
+  (a.spatialGreenWeight_measurable r).stronglyMeasurable.integral_prod_left'.measurable
+
+lemma spatialGreenWeight_spatial_integrable
+    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hfin : volume A ≠ ∞)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (ω : a.samples) :
+    IntegrableOn (fun z => spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) A := by
+  let : IsFiniteMeasure (volume.restrict A) := ⟨by
+    simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
+  apply Integrable.of_bound
+    ((a.spatialGreenWeight_measurable r).comp (measurable_id.prodMk measurable_const)).aestronglyMeasurable
+    (r ^ (-codimension κ))
+  filter_upwards [ae_restrict_mem hA] with z hz
+  have h := (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hκ hκ8 hr z (hheight z hz)
+  simpa only [Function.comp_apply, id_eq, Real.norm_eq_abs, abs_of_nonneg h.1] using h.2
+
+lemma spatialRealMass_bound
+    (hκ : 0 < κ) (hκ8 : κ < 8) (r : ℝ) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hfin : volume A ≠ ∞)
+    (hheight : ∀ z ∈ A, r ≤ z.im) (ω : a.samples) :
+    0 ≤ spatialRealMass κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A ∧
+      spatialRealMass κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) r A ≤
+        (volume A).toReal * r ^ (-codimension κ) := by
+  let : IsFiniteMeasure (volume.restrict A) := ⟨by
+    simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hfin⟩
+  have hi := a.spatialGreenWeight_spatial_integrable hκ hκ8 r hr A hA hfin hheight ω
+  have hb : ∀ᵐ z ∂(volume.restrict A), 0 ≤ spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r ∧
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r ≤ r ^ (-codimension κ) := by
+    filter_upwards [ae_restrict_mem hA] with z hz
+    exact (a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hκ hκ8 hr z (hheight z hz)
+  refine ⟨integral_nonneg_of_ae (hb.mono fun _ h => h.1), ?_⟩
+  calc
+    _ ≤ ∫ _z in A, r ^ (-codimension κ) := integral_mono_ae hi (integrable_const _) (hb.mono fun _ h => h.2)
+    _ = _ := by simp only [integral_const, Measure.restrict_apply_univ, smul_eq_mul, measureReal_def]
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+end
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+universe u_source_stopped
+
+ 
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+local instance spatialMeansStoppedRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance spatialMeansStoppedComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance spatialMeansStoppedRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance spatialMeansStoppedComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance spatialMeansDrivingRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance spatialMeansDrivingRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance spatialMeansSourceRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance spatialMeansSourceComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance spatialMeansSourceRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance spatialMeansSourceComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance spatialMeansMapRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance spatialMeansMapComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance spatialMeansMapRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance spatialMeansMapComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance spatialMeansTraceRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance spatialMeansTraceComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance spatialMeansTraceRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance spatialMeansTraceComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+end
+end SLEExactGauge
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+ 
+
+lemma source_halfplane_map_local_bound {f : ℂ → ℂ} {z w : ℂ} {r ε : ℝ}
+    (hr : 0 < r) (hε : 0 ≤ ε) (hεr : ε ≤ r / 2)
+    (hd : DifferentiableOn ℂ f (ball z r))
+    (hm : MapsTo f (ball z r) upperHalfPlane) (hw : dist w z ≤ ε) :
+    dist (f w) (f z) ≤ 4 * (f z).im * ε / r := by
+  have hz : z ∈ ball z r := mem_ball_self hr
+  have hwr : w ∈ ball z r := mem_ball.mpr (hw.trans_lt (by linarith))
+  have hy : 0 < (f z).im := hm hz
+  have hden (v : ℂ) (hv : v ∈ ball z r) : f v - star (f z) ≠ 0 := by
+    intro he
+    have hx := congrArg Complex.im he
+    simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hx
+    have hvy : 0 < (f v).im := hm hv
+    linarith
+  let g := fun v => (f v - f z) / (f v - star (f z))
+  have hg0 : g z = 0 := by simp [g]
+  have hgd : DifferentiableOn ℂ g (ball z r) :=
+    (hd.sub_const _).div (hd.sub_const _) hden
+  have hgm : MapsTo g (ball z r) (closedBall (g z) 1) := by
+    intro v hv
+    rw [hg0, mem_closedBall_zero_iff]
+    exact (upper_mobius_norm_lt_one (hm hv) hy).le
+  have hsch := Complex.dist_le_div_mul_dist_of_mapsTo_ball hgd hgm hwr
+  rw [hg0, dist_zero_right] at hsch
+  have hsch' : ‖g w‖ ≤ ε / r :=
+    hsch.trans (by simpa only [one_div, div_eq_mul_inv, mul_comm, mul_one] using
+      div_le_div_of_nonneg_right hw hr.le)
+  have hmult : ‖f w - f z‖ ≤ (ε / r) * ‖f w - star (f z)‖ := by
+    simpa only [g, norm_div, div_le_iff₀ (norm_pos_iff.mpr (hden w hwr))] using hsch'
+  have hn : ‖f z - star (f z)‖ = 2 * (f z).im := by
+    rw [Complex.star_def, Complex.sub_conj, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+      abs_of_pos (mul_pos (by norm_num) hy), Complex.norm_I, mul_one]
+  have htri : ‖f w - star (f z)‖ ≤ ‖f w - f z‖ + 2 * (f z).im := by
+    rw [← hn]
+    exact norm_sub_le_norm_sub_add_norm_sub _ _ _
+  have ha : 0 ≤ ε / r := div_nonneg hε hr.le
+  have hb : ε / r ≤ 1 / 2 := (div_le_iff₀ hr).mpr (by linarith)
+  have hmain := hmult.trans (mul_le_mul_of_nonneg_left htri ha)
+  rw [dist_eq_norm]
+  have hp := mul_nonneg (sub_nonneg.mpr hb) (norm_nonneg (f w - f z))
+  calc
+    ‖f w - f z‖ ≤ 4 * (f z).im * (ε / r) := by nlinarith
+    _ = _ := by ring
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal
+namespace SLEExactGauge
+
+ 
+
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+ 
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+theorem source_killedGreen_progressive
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)) (z : ℂ) :
+    IsStronglyProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) := by
+  classical
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable hc hm
+  have hnn (n : ℕ) : -(n : ℝ) ≤ n := by linarith [Nat.cast_nonneg (α := ℝ) n]
+  let c (n : ℕ) := (exists_clippedLogSolution B hm hBm hc κ (-(n : ℝ)) n (hnn n)
+    (initialLogCoordinates z)).some
+  let v (n : ℕ) (t : ℝ≥0) (ω : Ω) : Prop :=
+    0 < z.im ∧ Real.log z.im ≤ (n : ℝ) ∧ -(n : ℝ) ≤ ((c n).Q t ω).1.im
+  let A (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ :=
+    if v n t ω then greenLogDensity κ ((c n).Q t ω) (B t ω) else 0
+  have hQp (n : ℕ) : IsStronglyProgressive f (fun t : ℝ≥0 => (c n).Q t) :=
+    (show StronglyAdapted f (fun t : ℝ≥0 => (c n).Q t) from
+      fun t => ((c n).adapted t).stronglyMeasurable).isStronglyProgressive_of_continuous
+      (fun ω => ((c n).lipschitz ω).continuous.comp NNReal.continuous_coe)
+  have hAp (n : ℕ) : IsStronglyProgressive f (A n) := by
+    intro T
+    have hQ := (hQp n T).measurable
+    let : MeasurableSpace Ω := f T
+    have hv : MeasurableSet {p : Iic T × Ω | v n p.1 p.2} :=
+      (MeasurableSet.const _).inter ((MeasurableSet.const _).inter
+        (measurableSet_le measurable_const (Complex.measurable_im.comp hQ.fst)))
+    have hBp := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc
+    have hBm := (hBp T).measurable
+    exact (Measurable.ite hv ((greenLogDensity_contDiff κ).continuous.measurable.comp
+      (hQ.prodMk hBm)) measurable_const).stronglyMeasurable
+  have hlim (t : ℝ≥0) (ω : Ω) : Tendsto (fun n => A n t ω) atTop
+      (𝓝 (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z)) := by
+    apply tendsto_nhds_of_eventually_eq
+    have hU : Continuous (fun s => Real.sqrt κ * B s ω) := (hc ω).const_mul _
+    have hd (n : ℕ) (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :=
+      ((c n).derivative ω s hs.1).mono (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self)
+    by_cases hz : z ∈ survivingDomain (γ ω) t
+    · obtain ⟨N, hN⟩ := exists_nat_ge (max (-Real.log (G ω t z).im) (Real.log z.im))
+      filter_upwards [eventually_ge_atTop N] with n hn
+      have hNn : (N : ℝ) ≤ n := by exact_mod_cast hn
+      have hlo : -(n : ℝ) ≤ Real.log (G ω t z).im := by
+        have hh := (le_max_left (-Real.log (G ω t z).im) (Real.log z.im)).trans (hN.trans hNn)
+        linarith
+      have hhi : Real.log z.im ≤ (n : ℝ) := (le_max_right _ _).trans (hN.trans hNn)
+      have he : (c n).Q t ω = logLoewnerCoordinates (G ω) z t :=
+        clipped_log_loewner_agrees hU (hw ω) t z hz _ _ _ hlo hhi
+          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) ⟨t.coe_nonneg, le_rfl⟩
+      have hv : v n t ω := ⟨hz.1.1, hhi, by simpa only [he, logCoordinates_im] using hlo⟩
+      simp only [A, ite_eq_left hv, he]
+      exact greenLogDensity_logCoordinates hU (hw ω) κ (B t ω) z t hz rfl
+    · filter_upwards [] with n
+      have hv : ¬ v n t ω := by
+        intro hv
+        exact hz (clipped_log_identification_of_endpoint hU (hw ω) t z hv.1 _ _ _
+          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) hv.2.2 hv.2.1).1
+      simp only [A, ite_eq_right hv, greenDensity, ite_eq_right hz]
+  exact isStronglyProgressive_of_tendsto hAp
+    (tendsto_pi_nhds.mpr (fun t => tendsto_pi_nhds.mpr (hlim t)))
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance sourceApproachComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance sourceApproachComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+noncomputable def sourceApproachKernel (ε r : ℝ) (p : ℂ × C(ℝ≥0, ℂ)) : ℝ :=
+  if ∃ t : ℝ≥0, dist (p.2 t) p.1 < 8 * p.1.im * ε / r then 1 else 0
+
+lemma sourceApproachKernel_measurable (ε r : ℝ) : Measurable (sourceApproachKernel ε r) := by
+  have he (p : ℂ × C(ℝ≥0, ℂ)) :
+      (∃ t : ℝ≥0, dist (p.2 t) p.1 < 8 * p.1.im * ε / r) ↔
+      ∃ n : ℕ, dist (p.2 (denseSeq ℝ≥0 n)) p.1 < 8 * p.1.im * ε / r := by
+    constructor
+    · intro h
+      exact (denseRange_denseSeq ℝ≥0).exists_mem_open
+        (isOpen_lt (p.2.continuous.dist continuous_const) continuous_const) h
+    · rintro ⟨n, hn⟩
+      exact ⟨_, hn⟩
+  apply Measurable.ite _ measurable_const measurable_const
+  simp_rw [he]
+  rw [ofPred_exists]
+  apply MeasurableSet.iUnion
+  intro n
+  exact measurableSet_lt
+    (((continuous_eval_const _).comp continuous_snd).dist continuous_fst).measurable
+    ((((continuous_im.comp continuous_fst).const_mul 8).mul_const ε).div_const r).measurable
+
+lemma sourceApproachKernel_norm_le (ε r : ℝ) (p : ℂ × C(ℝ≥0, ℂ)) :
+    ‖sourceApproachKernel ε r p‖ ≤ 1 := by
+  unfold sourceApproachKernel
+  split_ifs <;> norm_num
+
+lemma sourceApproachKernel_zero {ε r : ℝ} (hε : 0 ≤ ε) (hr : 0 ≤ r)
+    {p : ℂ × C(ℝ≥0, ℂ)} (hp : p.1.im ≤ 0) : sourceApproachKernel ε r p = 0 := by
+  apply ite_eq_right
+  rintro ⟨t, ht⟩
+  have hh : 8 * p.1.im * ε / r ≤ 0 :=
+    div_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonpos_of_nonneg
+      (mul_nonpos_of_nonneg_of_nonpos (by norm_num) hp) hε) hr
+  exact not_lt_of_ge (dist_nonneg.trans' hh) ht
+
+lemma source_rpow_scale (p y ε r R : ℝ) (hy : 0 < y) (he : 0 < ε)
+    (hr : 0 < r) (hR : 0 < R) :
+    (2 * y) ^ (-p) * (8 * y * ε / r) ^ p =
+      (4 * R / r) ^ p * ε ^ p * R ^ (-p) := by
+  rw [Real.rpow_def_of_pos (by positivity : 0 < 2*y),
+    Real.rpow_def_of_pos (by positivity : 0 < 8*y*ε/r),
+    Real.rpow_def_of_pos (by positivity : 0 < 4*R/r),
+    Real.rpow_def_of_pos he, Real.rpow_def_of_pos hR,
+    ← Real.exp_add, ← Real.exp_add, ← Real.exp_add]
+  congr 1
+  rw [Real.log_div (by positivity) hr.ne', Real.log_div (by positivity) hr.ne',
+    Real.log_mul (by norm_num : (2:ℝ) ≠ 0) hy.ne',
+    Real.log_mul (by positivity : 8*y ≠ 0) he.ne',
+    Real.log_mul (by norm_num : (8:ℝ) ≠ 0) hy.ne',
+    Real.log_mul (by norm_num : (4:ℝ) ≠ 0) hR.ne']
+  have h8 : Real.log 8 = 3 * Real.log 2 := by
+    rw [show (8:ℝ)=2^3 by norm_num, Real.log_pow]; ring
+  have h4 : Real.log 4 = 2 * Real.log 2 := by
+    rw [show (4:ℝ)=2^2 by norm_num, Real.log_pow]; ring
+  rw [h8, h4]
+  ring
+end
+end SLEExactGauge
+
+end
+
+ 
+section
+open Set Filter MeasureTheory Metric Bornology
+open scoped Topology NNReal
+namespace SLEExactGauge
+ 
+
+end SLEExactGauge
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+
+end
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteSmallBall.lean b/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteSmallBall.lean
index 21a025ef177c030d635678c89625071d0e6f74ca..3e98794c1639beff3065ea0f6f28ecea97daff0b 100644
--- a/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteSmallBall.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteSmallBall.lean
@@ -1,81 +1,81 @@
-import OAI.Probability.SLE.Angle.TargetBrownianNormalization
-import OAI.Probability.SLE.Stochastic.WienerRealSupport
-
-namespace OAI
-
-/-! Positive probability of a uniformly small real Brownian driver on any
-fixed finite time interval, using its actual based path law. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance smallBallPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance smallBallPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-def driverUnitSmallBall (ε : ℝ) : Set Target.Driver :=
-  {b | ∀ t : ℝ≥0, t ≤ 1 → |b t| < ε}
-
-lemma isOpen_driverUnitSmallBall (ε : ℝ) : IsOpen (driverUnitSmallBall ε) := by
-  have ho := ContinuousMap.isOpen_setOfPred_mapsTo
-    (isCompact_Icc : IsCompact (Icc (0:ℝ≥0) 1))
-    (isOpen_Ioo : IsOpen (Ioo (-ε) ε))
-  have he : driverUnitSmallBall ε =
-      (fun b : Target.Driver => b.toContinuousMap) ⁻¹'
-        {p : C(ℝ≥0,ℝ) | MapsTo p (Icc 0 1) (Ioo (-ε) ε)} := by
-    ext b
-    simp only [driverUnitSmallBall,mem_ofPred_eq,mem_preimage,MapsTo,mem_Icc,mem_Ioo]
-    constructor
-    · intro hb t ht
-      exact abs_lt.mp (hb t ht.2)
-    · intro hb t ht
-      exact abs_lt.mpr (hb ⟨zero_le,ht⟩)
-  rw [he]
-  exact ho.preimage ContinuousMapZero.isClosedEmbedding_toContinuousMap.continuous
-
-lemma driverUnitSmallBall_positive (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    {ε : ℝ} (hε : 0 < ε) : 0 < W (driverUnitSmallBall ε) := by
-  have hp := wienerReal_uniform_tube_pos W hW (fun _ => 0) continuous_const rfl ε hε
-  apply hp.trans_le
-  apply measure_mono
-  intro b hb t ht
-  convert! hb (⟨(t:ℝ),t.coe_nonneg,by exact_mod_cast ht⟩ : unitInterval) using 1
-  simp only [sub_zero]
-  exact congrArg (fun u : ℝ≥0 => |b u|) (Subtype.ext rfl)
-
-theorem brownian_finite_smallBall_positive
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (T : ℝ≥0) (hT : 0 < T) {ε : ℝ} (hε : 0 < ε) :
-    0 < P {ω | ∀ t ∈ Icc 0 T, |B t ω| < ε} := by
-  let C (t : ℝ≥0) (ω : Ω) := (Real.sqrt T)⁻¹*B (T*t) ω
-  have hC : IsBrownianReal C P := hB.smul hT.ne'
-  have hCm : ∀ t, Measurable (C t) := fun t => measurable_const.mul (hm _)
-  have hCc : ∀ ω, Continuous (fun t => C t ω) :=
-    fun ω => continuous_const.mul ((hc ω).comp (continuous_const.mul continuous_id))
-  have hC0 : ∀ ω, C 0 ω=0 := by intro ω; simp [C,h0]
-  let b := Target.normalizedDriver 1 C hCc hC0
-  have hb : Measurable b := Target.normalizedDriver_measurable 1 C hCm hCc hC0
-  have hW := Target.normalizedDriver_isWiener 1 (by norm_num) C hC hCm hCc hC0
-  have hsqrt : 0 < Real.sqrt T := Real.sqrt_pos.mpr (by exact_mod_cast hT)
-  have hp := driverUnitSmallBall_positive (P.map b) hW (div_pos hε hsqrt)
-  rw [Measure.map_apply hb (isOpen_driverUnitSmallBall _).measurableSet] at hp
-  apply hp.trans_le (measure_mono ?_)
-  intro ω hω t ht
-  let q : ℝ≥0 := t/T
-  have hq : q ≤ 1 := (div_le_one₀ hT).mpr ht.2
-  have hbq := hω q hq
-  change |Real.sqrt 1*C (q/(1:ℝ).toNNReal) ω| < ε/Real.sqrt T at hbq
-  simp only [Real.sqrt_one,Real.toNNReal_one,div_one,one_mul] at hbq
-  have htq : T*q=t := by dsimp [q]; exact mul_div_cancel₀ _ hT.ne'
-  dsimp only [C] at hbq
-  rw [htq,abs_mul,abs_inv,abs_of_pos hsqrt,← div_eq_inv_mul] at hbq
-  exact (div_lt_div_iff_of_pos_right hsqrt).mp hbq
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Angle.TargetBrownianNormalization
+import OAI.Probability.SLE.Stochastic.WienerRealSupport
+
+namespace OAI
+
+/-! Positive probability of a uniformly small real Brownian driver on any
+fixed finite time interval, using its actual based path law. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance smallBallPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance smallBallPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+def driverUnitSmallBall (ε : ℝ) : Set Target.Driver :=
+  {b | ∀ t : ℝ≥0, t ≤ 1 → |b t| < ε}
+
+lemma isOpen_driverUnitSmallBall (ε : ℝ) : IsOpen (driverUnitSmallBall ε) := by
+  have ho := ContinuousMap.isOpen_setOfPred_mapsTo
+    (isCompact_Icc : IsCompact (Icc (0:ℝ≥0) 1))
+    (isOpen_Ioo : IsOpen (Ioo (-ε) ε))
+  have he : driverUnitSmallBall ε =
+      (fun b : Target.Driver => b.toContinuousMap) ⁻¹'
+        {p : C(ℝ≥0,ℝ) | MapsTo p (Icc 0 1) (Ioo (-ε) ε)} := by
+    ext b
+    simp only [driverUnitSmallBall,mem_ofPred_eq,mem_preimage,MapsTo,mem_Icc,mem_Ioo]
+    constructor
+    · intro hb t ht
+      exact abs_lt.mp (hb t ht.2)
+    · intro hb t ht
+      exact abs_lt.mpr (hb ⟨zero_le,ht⟩)
+  rw [he]
+  exact ho.preimage ContinuousMapZero.isClosedEmbedding_toContinuousMap.continuous
+
+lemma driverUnitSmallBall_positive (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    {ε : ℝ} (hε : 0 < ε) : 0 < W (driverUnitSmallBall ε) := by
+  have hp := wienerReal_uniform_tube_pos W hW (fun _ => 0) continuous_const rfl ε hε
+  apply hp.trans_le
+  apply measure_mono
+  intro b hb t ht
+  convert! hb (⟨(t:ℝ),t.coe_nonneg,by exact_mod_cast ht⟩ : unitInterval) using 1
+  simp only [sub_zero]
+  exact congrArg (fun u : ℝ≥0 => |b u|) (Subtype.ext rfl)
+
+theorem brownian_finite_smallBall_positive
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (T : ℝ≥0) (hT : 0 < T) {ε : ℝ} (hε : 0 < ε) :
+    0 < P {ω | ∀ t ∈ Icc 0 T, |B t ω| < ε} := by
+  let C (t : ℝ≥0) (ω : Ω) := (Real.sqrt T)⁻¹*B (T*t) ω
+  have hC : IsBrownianReal C P := hB.smul hT.ne'
+  have hCm : ∀ t, Measurable (C t) := fun t => measurable_const.mul (hm _)
+  have hCc : ∀ ω, Continuous (fun t => C t ω) :=
+    fun ω => continuous_const.mul ((hc ω).comp (continuous_const.mul continuous_id))
+  have hC0 : ∀ ω, C 0 ω=0 := by intro ω; simp [C,h0]
+  let b := Target.normalizedDriver 1 C hCc hC0
+  have hb : Measurable b := Target.normalizedDriver_measurable 1 C hCm hCc hC0
+  have hW := Target.normalizedDriver_isWiener 1 (by norm_num) C hC hCm hCc hC0
+  have hsqrt : 0 < Real.sqrt T := Real.sqrt_pos.mpr (by exact_mod_cast hT)
+  have hp := driverUnitSmallBall_positive (P.map b) hW (div_pos hε hsqrt)
+  rw [Measure.map_apply hb (isOpen_driverUnitSmallBall _).measurableSet] at hp
+  apply hp.trans_le (measure_mono ?_)
+  intro ω hω t ht
+  let q : ℝ≥0 := t/T
+  have hq : q ≤ 1 := (div_le_one₀ hT).mpr ht.2
+  have hbq := hω q hq
+  change |Real.sqrt 1*C (q/(1:ℝ).toNNReal) ω| < ε/Real.sqrt T at hbq
+  simp only [Real.sqrt_one,Real.toNNReal_one,div_one,one_mul] at hbq
+  have htq : T*q=t := by dsimp [q]; exact mul_div_cancel₀ _ hT.ne'
+  dsimp only [C] at hbq
+  rw [htq,abs_mul,abs_inv,abs_of_pos hsqrt,← div_eq_inv_mul] at hbq
+  exact (div_lt_div_iff_of_pos_right hsqrt).mp hbq
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteTube.lean b/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteTube.lean
index 44013a746f5e65ea7c9da0631a7a54a72b028bd9..74eb81d59316c017e7115a8312241575867892f2 100644
--- a/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteTube.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/BrownianFiniteTube.lean
@@ -1,84 +1,84 @@
-import OAI.Probability.SLE.Stochastic.BrownianFiniteSmallBall
-
-namespace OAI
-
-/-! Finite-window support around any prescribed continuous based real
-driver, transported from the actual unit-window Wiener law. -/
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance finiteTubePathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance finiteTubePathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-def driverUnitTube (f : C(ℝ≥0,ℝ)) (ε : ℝ) : Set Target.Driver :=
-  {b | ∀ t : ℝ≥0, t ≤ 1 → |b t-f t| < ε}
-
-lemma isOpen_driverUnitTube (f : C(ℝ≥0,ℝ)) (ε : ℝ) : IsOpen (driverUnitTube f ε) := by
-  have ho := ContinuousMap.isOpen_setOfPred_mapsTo
-    (isCompact_Icc : IsCompact (Icc (0:ℝ≥0) 1))
-    (isOpen_Ioo : IsOpen (Ioo (-ε) ε))
-  have he : driverUnitTube f ε =
-      (fun b : Target.Driver => b.toContinuousMap-f) ⁻¹'
-        {p : C(ℝ≥0,ℝ) | MapsTo p (Icc 0 1) (Ioo (-ε) ε)} := by
-    ext b
-    simp only [driverUnitTube,mem_ofPred_eq,mem_preimage,MapsTo,mem_Icc,mem_Ioo,
-      ContinuousMap.sub_apply]
-    constructor
-    · intro hb t ht
-      exact abs_lt.mp (hb t ht.2)
-    · intro hb t ht
-      exact abs_lt.mpr (hb ⟨zero_le,ht⟩)
-  rw [he]
-  exact ho.preimage (ContinuousMapZero.isClosedEmbedding_toContinuousMap.continuous.sub
-    continuous_const)
-
-lemma driverUnitTube_positive (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (f : C(ℝ≥0,ℝ)) (hf0 : f 0=0) {ε : ℝ} (hε : 0 < ε) :
-    0 < W (driverUnitTube f ε) := by
-  have hp := wienerReal_uniform_tube_pos W hW f f.continuous hf0 ε hε
-  apply hp.trans_le (measure_mono ?_)
-  intro b hb t ht
-  convert! hb (⟨(t:ℝ),t.coe_nonneg,by exact_mod_cast ht⟩ : unitInterval) using 1
-
-theorem brownian_finite_uniform_tube_positive
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (h0 : ∀ ω, B 0 ω=0) (T : ℝ≥0) (hT : 0 < T)
-    (f : C(ℝ≥0,ℝ)) (hf0 : f 0=0) {ε : ℝ} (hε : 0 < ε) :
-    0 < P {ω | ∀ t ∈ Icc 0 T, |B t ω-f t| < ε} := by
-  let C (t : ℝ≥0) (ω : Ω) := (Real.sqrt T)⁻¹*B (T*t) ω
-  have hC : IsBrownianReal C P := hB.smul hT.ne'
-  have hCm : ∀ t, Measurable (C t) := fun t => measurable_const.mul (hm _)
-  have hCc : ∀ ω, Continuous (fun t => C t ω) :=
-    fun ω => continuous_const.mul ((hc ω).comp (continuous_const.mul continuous_id))
-  have hC0 : ∀ ω, C 0 ω=0 := by intro ω; simp [C,h0]
-  let g : C(ℝ≥0,ℝ) := ⟨fun t => (Real.sqrt T)⁻¹*f (T*t),
-    continuous_const.mul (f.continuous.comp (continuous_const.mul continuous_id))⟩
-  have hg0 : g 0=0 := by simp [g,hf0]
-  let b := Target.normalizedDriver 1 C hCc hC0
-  have hb : Measurable b := Target.normalizedDriver_measurable 1 C hCm hCc hC0
-  have hW := Target.normalizedDriver_isWiener 1 (by norm_num) C hC hCm hCc hC0
-  have hsqrt : 0 < Real.sqrt T := Real.sqrt_pos.mpr (by exact_mod_cast hT)
-  have hp := driverUnitTube_positive (P.map b) hW g hg0 (div_pos hε hsqrt)
-  rw [Measure.map_apply hb (isOpen_driverUnitTube _ _).measurableSet] at hp
-  apply hp.trans_le (measure_mono ?_)
-  intro ω hω t ht
-  let q : ℝ≥0 := t/T
-  have hq : q ≤ 1 := (div_le_one₀ hT).mpr ht.2
-  have hbq := hω q hq
-  change |Real.sqrt 1*C (q/(1:ℝ).toNNReal) ω-g q| < ε/Real.sqrt T at hbq
-  simp only [Real.sqrt_one,Real.toNNReal_one,div_one,one_mul] at hbq
-  have htq : T*q=t := by dsimp [q]; exact mul_div_cancel₀ _ hT.ne'
-  change |(Real.sqrt T)⁻¹*B (T*q) ω-(Real.sqrt T)⁻¹*f (T*q)| < _ at hbq
-  rw [htq,← mul_sub,abs_mul,abs_inv,abs_of_pos hsqrt,← div_eq_inv_mul] at hbq
-  exact (div_lt_div_iff_of_pos_right hsqrt).mp hbq
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.BrownianFiniteSmallBall
+
+namespace OAI
+
+/-! Finite-window support around any prescribed continuous based real
+driver, transported from the actual unit-window Wiener law. -/
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance finiteTubePathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance finiteTubePathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+def driverUnitTube (f : C(ℝ≥0,ℝ)) (ε : ℝ) : Set Target.Driver :=
+  {b | ∀ t : ℝ≥0, t ≤ 1 → |b t-f t| < ε}
+
+lemma isOpen_driverUnitTube (f : C(ℝ≥0,ℝ)) (ε : ℝ) : IsOpen (driverUnitTube f ε) := by
+  have ho := ContinuousMap.isOpen_setOfPred_mapsTo
+    (isCompact_Icc : IsCompact (Icc (0:ℝ≥0) 1))
+    (isOpen_Ioo : IsOpen (Ioo (-ε) ε))
+  have he : driverUnitTube f ε =
+      (fun b : Target.Driver => b.toContinuousMap-f) ⁻¹'
+        {p : C(ℝ≥0,ℝ) | MapsTo p (Icc 0 1) (Ioo (-ε) ε)} := by
+    ext b
+    simp only [driverUnitTube,mem_ofPred_eq,mem_preimage,MapsTo,mem_Icc,mem_Ioo,
+      ContinuousMap.sub_apply]
+    constructor
+    · intro hb t ht
+      exact abs_lt.mp (hb t ht.2)
+    · intro hb t ht
+      exact abs_lt.mpr (hb ⟨zero_le,ht⟩)
+  rw [he]
+  exact ho.preimage (ContinuousMapZero.isClosedEmbedding_toContinuousMap.continuous.sub
+    continuous_const)
+
+lemma driverUnitTube_positive (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (f : C(ℝ≥0,ℝ)) (hf0 : f 0=0) {ε : ℝ} (hε : 0 < ε) :
+    0 < W (driverUnitTube f ε) := by
+  have hp := wienerReal_uniform_tube_pos W hW f f.continuous hf0 ε hε
+  apply hp.trans_le (measure_mono ?_)
+  intro b hb t ht
+  convert! hb (⟨(t:ℝ),t.coe_nonneg,by exact_mod_cast ht⟩ : unitInterval) using 1
+
+theorem brownian_finite_uniform_tube_positive
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (h0 : ∀ ω, B 0 ω=0) (T : ℝ≥0) (hT : 0 < T)
+    (f : C(ℝ≥0,ℝ)) (hf0 : f 0=0) {ε : ℝ} (hε : 0 < ε) :
+    0 < P {ω | ∀ t ∈ Icc 0 T, |B t ω-f t| < ε} := by
+  let C (t : ℝ≥0) (ω : Ω) := (Real.sqrt T)⁻¹*B (T*t) ω
+  have hC : IsBrownianReal C P := hB.smul hT.ne'
+  have hCm : ∀ t, Measurable (C t) := fun t => measurable_const.mul (hm _)
+  have hCc : ∀ ω, Continuous (fun t => C t ω) :=
+    fun ω => continuous_const.mul ((hc ω).comp (continuous_const.mul continuous_id))
+  have hC0 : ∀ ω, C 0 ω=0 := by intro ω; simp [C,h0]
+  let g : C(ℝ≥0,ℝ) := ⟨fun t => (Real.sqrt T)⁻¹*f (T*t),
+    continuous_const.mul (f.continuous.comp (continuous_const.mul continuous_id))⟩
+  have hg0 : g 0=0 := by simp [g,hf0]
+  let b := Target.normalizedDriver 1 C hCc hC0
+  have hb : Measurable b := Target.normalizedDriver_measurable 1 C hCm hCc hC0
+  have hW := Target.normalizedDriver_isWiener 1 (by norm_num) C hC hCm hCc hC0
+  have hsqrt : 0 < Real.sqrt T := Real.sqrt_pos.mpr (by exact_mod_cast hT)
+  have hp := driverUnitTube_positive (P.map b) hW g hg0 (div_pos hε hsqrt)
+  rw [Measure.map_apply hb (isOpen_driverUnitTube _ _).measurableSet] at hp
+  apply hp.trans_le (measure_mono ?_)
+  intro ω hω t ht
+  let q : ℝ≥0 := t/T
+  have hq : q ≤ 1 := (div_le_one₀ hT).mpr ht.2
+  have hbq := hω q hq
+  change |Real.sqrt 1*C (q/(1:ℝ).toNNReal) ω-g q| < ε/Real.sqrt T at hbq
+  simp only [Real.sqrt_one,Real.toNNReal_one,div_one,one_mul] at hbq
+  have htq : T*q=t := by dsimp [q]; exact mul_div_cancel₀ _ hT.ne'
+  change |(Real.sqrt T)⁻¹*B (T*q) ω-(Real.sqrt T)⁻¹*f (T*q)| < _ at hbq
+  rw [htq,← mul_sub,abs_mul,abs_inv,abs_of_pos hsqrt,← div_eq_inv_mul] at hbq
+  exact (div_lt_div_iff_of_pos_right hsqrt).mp hbq
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/BrownianGerm.lean b/lean/OAI/Probability/SLE/Stochastic/BrownianGerm.lean
index 863a32b56d341db24514a252a5d128c66f4d9c6b..cc7a02e011e9afc71c4ada1625bcc0c1bdcdca5d 100644
--- a/lean/OAI/Probability/SLE/Stochastic/BrownianGerm.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/BrownianGerm.lean
@@ -1,104 +1,104 @@
-import OAI.Probability.SLE.BrownianStrongMarkov
-
-namespace OAI
-
-/-! The Brownian germ zero-one argument needed for positive canonical
-mass before every positive time. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {B : ℝ≥0 → Ω → ℝ}
-
-lemma brownian_germ_fdd_law
-    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (ε : ℕ → ℝ≥0) (hε : Tendsto ε atTop (𝓝 0))
-    (E : Set Ω)
-    (hE : ∀ n, MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε n)] E)
-    (I : Finset ℝ≥0) :
-    (P.restrict E).map (fun ω => I.restrict (fun t => B t ω)) =
-      P E • BrownianReal.projectiveFamily I := by
-  let : IsFiniteMeasure (P E • BrownianReal.projectiveFamily I) :=
-    Measure.smul_finite _ (measure_ne_top P E)
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro φ
-  have hXm : Measurable (fun ω => I.restrict (fun t => B t ω)) :=
-    (Finset.measurable_restrict I).comp (Measurable.of_eval hm)
-  rw [integral_map hXm.aemeasurable φ.continuous.aestronglyMeasurable, integral_smul_measure]
-  have ht (ω : Ω) : Tendsto (fun n => shiftedFDD B I (ε n) ω) atTop
-      (𝓝 (I.restrict (fun t => B t ω))) := by
-    apply tendsto_pi_nhds.mpr
-    intro i
-    change Tendsto (fun n => B (ε n+(i:ℝ≥0)) ω-B (ε n) ω) atTop (𝓝 (B (i:ℝ≥0) ω))
-    have hh := ((hc ω).continuousAt.tendsto.comp (hε.add_const (i:ℝ≥0))).sub
-      ((hc ω).continuousAt.tendsto.comp hε)
-    simpa only [Function.comp_def,zero_add,h0 ω,sub_zero] using hh
-  have hi := tendsto_integral_of_dominated_convergence (μ:=P.restrict E)
-    (F:=fun n ω => φ (shiftedFDD B I (ε n) ω))
-    (f:=fun ω => φ (I.restrict (fun t => B t ω))) (fun _ => ‖φ‖)
-    (fun n => (φ.continuous.measurable.comp (shiftedFDD_measurable hm I (ε n))).aestronglyMeasurable)
-    (integrable_const _) (fun _ => Eventually.of_forall (fun _ => φ.norm_coe_le_norm _))
-    (Eventually.of_forall (fun ω => φ.continuous.continuousAt.tendsto.comp (ht ω)))
-  simp only [shiftedFDD_setIntegral hB hm I _ φ (hE _)] at hi
-  exact tendsto_nhds_unique hi tendsto_const_nhds
-
-lemma brownian_germ_path_law
-    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (ε : ℕ → ℝ≥0) (hε : Tendsto ε atTop (𝓝 0))
-    (E : Set Ω)
-    (hE : ∀ n, MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε n)] E) :
-    (P.restrict E).map (fun ω t => B t ω) = P E • P.map (fun ω t => B t ω) := by
-  let (I : Finset ℝ≥0) : IsFiniteMeasure (P E • BrownianReal.projectiveFamily I) :=
-    Measure.smul_finite _ (measure_ne_top P E)
-  have hL : IsProjectiveLimit ((P.restrict E).map (fun ω t => B t ω))
-      (fun I => P E • BrownianReal.projectiveFamily I) := by
-    intro I
-    rw [Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm)]
-    exact brownian_germ_fdd_law hB hm hc h0 ε hε E hE I
-  have hR : IsProjectiveLimit (P E • P.map (fun ω t => B t ω))
-      (fun I => P E • BrownianReal.projectiveFamily I) := by
-    intro I
-    rw [Measure.map_smul _ (Finset.measurable_restrict I).aemeasurable,
-      Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm)]
-    exact congrArg (fun μ => P E • μ) (hB.hasLaw I).map_eq
-  exact hL.unique hR
-
-/-- An event observable along times decreasing to zero has probability
-zero or one. Only the continuous based Brownian path is needed. -/
-theorem brownian_germ_zero_one
-    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (ε : ℕ → ℝ≥0) (hε : Tendsto ε atTop (𝓝 0))
-    (E : Set Ω)
-    (hE : ∀ n, MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε n)] E) :
-    P E=0 ∨ P E=1 := by
-  have hevent := hE 0
-  rw [Filtration.natural_eq_comap B _ (ε 0)] at hevent
-  obtain ⟨D,hD,heD⟩ := hevent
-  let A : Set (ℝ≥0 → ℝ) := (fun b => fun s : Iic (ε 0) => b s) ⁻¹' D
-  have hr : Measurable (fun b : ℝ≥0 → ℝ => fun s : Iic (ε 0) => b (s:ℝ≥0)) :=
-    Measurable.of_eval (fun s => measurable_pi_apply (s:ℝ≥0))
-  have hA : MeasurableSet A := hr hD
-  have hpre : (fun ω t => B t ω) ⁻¹' A=E := heD
-  have hEm : MeasurableSet E := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le (ε 0) E (hE 0)
-  have he := congrArg (fun μ : Measure (ℝ≥0 → ℝ) => μ A)
-    (brownian_germ_path_law hB hm hc h0 ε hε E hE)
-  rw [Measure.map_apply (Measurable.of_eval hm) hA,Measure.smul_apply,
-    Measure.map_apply (Measurable.of_eval hm) hA,hpre,Measure.restrict_apply hEm,inter_self] at he
-  by_cases hz : P E=0
-  · exact Or.inl hz
-  · right
-    apply (ENNReal.mul_left_inj hz (measure_ne_top P E)).mp
-    simpa only [smul_eq_mul,one_mul] using he.symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.BrownianStrongMarkov
+
+namespace OAI
+
+/-! The Brownian germ zero-one argument needed for positive canonical
+mass before every positive time. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {B : ℝ≥0 → Ω → ℝ}
+
+lemma brownian_germ_fdd_law
+    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (ε : ℕ → ℝ≥0) (hε : Tendsto ε atTop (𝓝 0))
+    (E : Set Ω)
+    (hE : ∀ n, MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε n)] E)
+    (I : Finset ℝ≥0) :
+    (P.restrict E).map (fun ω => I.restrict (fun t => B t ω)) =
+      P E • BrownianReal.projectiveFamily I := by
+  let : IsFiniteMeasure (P E • BrownianReal.projectiveFamily I) :=
+    Measure.smul_finite _ (measure_ne_top P E)
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro φ
+  have hXm : Measurable (fun ω => I.restrict (fun t => B t ω)) :=
+    (Finset.measurable_restrict I).comp (Measurable.of_eval hm)
+  rw [integral_map hXm.aemeasurable φ.continuous.aestronglyMeasurable, integral_smul_measure]
+  have ht (ω : Ω) : Tendsto (fun n => shiftedFDD B I (ε n) ω) atTop
+      (𝓝 (I.restrict (fun t => B t ω))) := by
+    apply tendsto_pi_nhds.mpr
+    intro i
+    change Tendsto (fun n => B (ε n+(i:ℝ≥0)) ω-B (ε n) ω) atTop (𝓝 (B (i:ℝ≥0) ω))
+    have hh := ((hc ω).continuousAt.tendsto.comp (hε.add_const (i:ℝ≥0))).sub
+      ((hc ω).continuousAt.tendsto.comp hε)
+    simpa only [Function.comp_def,zero_add,h0 ω,sub_zero] using hh
+  have hi := tendsto_integral_of_dominated_convergence (μ:=P.restrict E)
+    (F:=fun n ω => φ (shiftedFDD B I (ε n) ω))
+    (f:=fun ω => φ (I.restrict (fun t => B t ω))) (fun _ => ‖φ‖)
+    (fun n => (φ.continuous.measurable.comp (shiftedFDD_measurable hm I (ε n))).aestronglyMeasurable)
+    (integrable_const _) (fun _ => Eventually.of_forall (fun _ => φ.norm_coe_le_norm _))
+    (Eventually.of_forall (fun ω => φ.continuous.continuousAt.tendsto.comp (ht ω)))
+  simp only [shiftedFDD_setIntegral hB hm I _ φ (hE _)] at hi
+  exact tendsto_nhds_unique hi tendsto_const_nhds
+
+lemma brownian_germ_path_law
+    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (ε : ℕ → ℝ≥0) (hε : Tendsto ε atTop (𝓝 0))
+    (E : Set Ω)
+    (hE : ∀ n, MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε n)] E) :
+    (P.restrict E).map (fun ω t => B t ω) = P E • P.map (fun ω t => B t ω) := by
+  let (I : Finset ℝ≥0) : IsFiniteMeasure (P E • BrownianReal.projectiveFamily I) :=
+    Measure.smul_finite _ (measure_ne_top P E)
+  have hL : IsProjectiveLimit ((P.restrict E).map (fun ω t => B t ω))
+      (fun I => P E • BrownianReal.projectiveFamily I) := by
+    intro I
+    rw [Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm)]
+    exact brownian_germ_fdd_law hB hm hc h0 ε hε E hE I
+  have hR : IsProjectiveLimit (P E • P.map (fun ω t => B t ω))
+      (fun I => P E • BrownianReal.projectiveFamily I) := by
+    intro I
+    rw [Measure.map_smul _ (Finset.measurable_restrict I).aemeasurable,
+      Measure.map_map (Finset.measurable_restrict I) (Measurable.of_eval hm)]
+    exact congrArg (fun μ => P E • μ) (hB.hasLaw I).map_eq
+  exact hL.unique hR
+
+/-- An event observable along times decreasing to zero has probability
+zero or one. Only the continuous based Brownian path is needed. -/
+theorem brownian_germ_zero_one
+    (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (ε : ℕ → ℝ≥0) (hε : Tendsto ε atTop (𝓝 0))
+    (E : Set Ω)
+    (hE : ∀ n, MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε n)] E) :
+    P E=0 ∨ P E=1 := by
+  have hevent := hE 0
+  rw [Filtration.natural_eq_comap B _ (ε 0)] at hevent
+  obtain ⟨D,hD,heD⟩ := hevent
+  let A : Set (ℝ≥0 → ℝ) := (fun b => fun s : Iic (ε 0) => b s) ⁻¹' D
+  have hr : Measurable (fun b : ℝ≥0 → ℝ => fun s : Iic (ε 0) => b (s:ℝ≥0)) :=
+    Measurable.of_eval (fun s => measurable_pi_apply (s:ℝ≥0))
+  have hA : MeasurableSet A := hr hD
+  have hpre : (fun ω t => B t ω) ⁻¹' A=E := heD
+  have hEm : MeasurableSet E := (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le (ε 0) E (hE 0)
+  have he := congrArg (fun μ : Measure (ℝ≥0 → ℝ) => μ A)
+    (brownian_germ_path_law hB hm hc h0 ε hε E hE)
+  rw [Measure.map_apply (Measurable.of_eval hm) hA,Measure.smul_apply,
+    Measure.map_apply (Measurable.of_eval hm) hA,hpre,Measure.restrict_apply hEm,inter_self] at he
+  by_cases hz : P E=0
+  · exact Or.inl hz
+  · right
+    apply (ENNReal.mul_left_inj hz (measure_ne_top P E)).mp
+    simpa only [smul_eq_mul,one_mul] using he.symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalInequality.lean b/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalInequality.lean
index 75d2805e8e38c9b005a1698db3e6a6e0928146c6..79ea52eba27044c11cba2873a66dc35d7a57c276 100644
--- a/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalInequality.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalInequality.lean
@@ -1,118 +1,118 @@
-import OAI.Probability.SLE.OptionalStopping
-
-namespace OAI
-
-/-! A one-sided optional-stopping estimate for the discounted multipoint
-product, obtained from the centered grid identity and the sign of the drift. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma bounded_cumulative_at_grid_le
-    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hC : 0 ≤ C)
-    (hb : ∀ u ω, |G u ω| ≤ C) (τ : Ω → ℝ≥0)
-    (hnonpos : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → G u ω ≤ 0)
-    (n : ℕ) (ω : Ω) :
-    (∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), G u ω) ≤
-      C*(1/((n : ℝ)+1)) := by
-  have hi₀ := bounded_time_section_intervalIntegrable 0 (τ ω) (τ ω).coe_nonneg G hG C hb ω
-  have h0 : (∫ u in (0 : ℝ)..(τ ω : ℝ), G u ω) ≤ 0 := by
-    have hh := intervalIntegral.integral_mono_on_of_le_Ioo (τ ω).coe_nonneg hi₀
-      (intervalIntegrable_const (c := (0 : ℝ)))
-      (fun u hu => hnonpos ω u hu.1.le hu.2)
-    simpa only [intervalIntegral.integral_zero] using hh
-  have hle : (τ ω : ℝ) ≤ stopGrid n (τ ω) := (stopGrid_bounds n (τ ω)).1
-  have hsplit := intervalIntegral.integral_add_adjacent_intervals hi₀
-    (bounded_time_section_intervalIntegrable (τ ω) (stopGrid n (τ ω)) hle G hG C hb ω)
-  have hbnd := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (τ ω : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := C)
-    (f := fun u => G u ω) (fun u _ => by simpa only [Real.norm_eq_abs] using hb u ω)
-  rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hle)] at hbnd
-  have hupper := (stopGrid_bounds n (τ ω)).2
-  have hu : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ)+1/((n : ℝ)+1) := by exact_mod_cast hupper
-  have htail := (le_abs_self _).trans hbnd
-  have hrange := mul_le_mul_of_nonneg_left (show (stopGrid n (τ ω) : ℝ)-(τ ω : ℝ) ≤
-    1/((n : ℝ)+1) by linarith) hC
-  linarith
-
-theorem brownian_time_field_optional_le
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hnonpos : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → brownianFieldDrift B F a u ω ≤ 0) :
-    (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P) ≤ (∫ ω, F 0 ω (B 0 ω) ∂P) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hτm := stopping_time_measurable_nnreal hτ
-  let V (n : ℕ) (ω : Ω) := F (stopGrid n (τ ω)) ω (B (stopGrid n (τ ω)) ω)
-  let H (n : ℕ) (ω : Ω) := ∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), brownianFieldDrift B F a u ω
-  have hVm (n : ℕ) : Measurable (V n) :=
-    measurable_discrete_eval (fun k ω => F (stopGridTime n k) ω (B (stopGridTime n k) ω))
-      (fun k => hF.eval_measurable _ 0 (by decide) (hm _)) (stopGridIndex_measurable hτm n)
-  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) := hF.bound _ ω _ 0 (by decide)
-  have hVi (n : ℕ) : Integrable (V n) P :=
-    Integrable.of_bound (hVm n).aestronglyMeasurable (K : ℝ)
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-  have hGm := hF.generator_measurable hBm
-  have hHjoint := cumulative_integral_measurable _ hGm
-  have hHm (n : ℕ) : Measurable (H n) :=
-    measurable_discrete_eval
-      (fun k ω => ∫ u in (0 : ℝ)..(stopGridTime n k : ℝ), brownianFieldDrift B F a u ω)
-      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
-  have hHb (n : ℕ) (ω : Ω) : |H n ω| ≤ 2*(K : ℝ)*((T : ℝ)+1) := by
-    have hbound := intervalIntegral.norm_integral_le_of_norm_le_const
-      (a := (0 : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := 2*(K : ℝ))
-      (f := fun u => brownianFieldDrift B F a u ω)
-      (fun u _ => by simpa only [Real.norm_eq_abs] using hF.generator_bound u ω)
-    rw [Real.norm_eq_abs, sub_zero, abs_of_nonneg (stopGrid n (τ ω)).coe_nonneg] at hbound
-    have hupper := (stopGrid_bounds n (τ ω)).2
-    have hgrid : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ)+1/((n : ℝ)+1) := by exact_mod_cast hupper
-    have hT : (τ ω : ℝ) ≤ T := hb ω
-    have hone : 1/((n : ℝ)+1) ≤ 1 := by
-      apply (div_le_iff₀ (by positivity : 0 < (n : ℝ)+1)).mpr
-      nlinarith [Nat.cast_nonneg (α := ℝ) n]
-    exact hbound.trans (mul_le_mul_of_nonneg_left (by linarith) (by positivity))
-  have hHi (n : ℕ) : Integrable (H n) P :=
-    Integrable.of_bound (hHm n).aestronglyMeasurable _
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
-  have hHle (n : ℕ) (ω : Ω) : H n ω ≤ 2*(K : ℝ)*(1/((n : ℝ)+1)) :=
-    bounded_cumulative_at_grid_le _ hGm (2*(K : ℝ)) (by positivity)
-      hF.generator_bound τ hnonpos n ω
-  have hmean (n : ℕ) : (∫ ω, V n ω ∂P) ≤
-      (∫ ω, F 0 ω (B 0 ω) ∂P)+2*(K : ℝ)*(1/((n : ℝ)+1)) := by
-    have he := brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
-    change (∫ ω, V n ω-H n ω ∂P) = _ at he
-    rw [integral_sub (hVi n) (hHi n)] at he
-    have hi := integral_mono_ae (hHi n) (integrable_const (2*(K : ℝ)*(1/((n : ℝ)+1))))
-      (Eventually.of_forall (hHle n))
-    simp only [integral_const, measureReal_def, measure_univ, ENNReal.toReal_one, one_smul] at hi
-    linarith
-  have hlim : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
-      (𝓝 (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => (K : ℝ))
-      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-    exact Eventually.of_forall fun ω =>
-      (hF.path_continuous hBc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω))
-  have hlimUpper : Tendsto (fun n : ℕ => (∫ ω, F 0 ω (B 0 ω) ∂P)+
-      2*(K : ℝ)*(1/((n : ℝ)+1))) atTop (𝓝 (∫ ω, F 0 ω (B 0 ω) ∂P)) := by
-    simpa only [mul_zero, add_zero] using tendsto_const_nhds.add
-      ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2*(K : ℝ)))
-  exact le_of_tendsto_of_tendsto hlim hlimUpper (Eventually.of_forall hmean)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.OptionalStopping
+
+namespace OAI
+
+/-! A one-sided optional-stopping estimate for the discounted multipoint
+product, obtained from the centered grid identity and the sign of the drift. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma bounded_cumulative_at_grid_le
+    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hC : 0 ≤ C)
+    (hb : ∀ u ω, |G u ω| ≤ C) (τ : Ω → ℝ≥0)
+    (hnonpos : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → G u ω ≤ 0)
+    (n : ℕ) (ω : Ω) :
+    (∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), G u ω) ≤
+      C*(1/((n : ℝ)+1)) := by
+  have hi₀ := bounded_time_section_intervalIntegrable 0 (τ ω) (τ ω).coe_nonneg G hG C hb ω
+  have h0 : (∫ u in (0 : ℝ)..(τ ω : ℝ), G u ω) ≤ 0 := by
+    have hh := intervalIntegral.integral_mono_on_of_le_Ioo (τ ω).coe_nonneg hi₀
+      (intervalIntegrable_const (c := (0 : ℝ)))
+      (fun u hu => hnonpos ω u hu.1.le hu.2)
+    simpa only [intervalIntegral.integral_zero] using hh
+  have hle : (τ ω : ℝ) ≤ stopGrid n (τ ω) := (stopGrid_bounds n (τ ω)).1
+  have hsplit := intervalIntegral.integral_add_adjacent_intervals hi₀
+    (bounded_time_section_intervalIntegrable (τ ω) (stopGrid n (τ ω)) hle G hG C hb ω)
+  have hbnd := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (τ ω : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := C)
+    (f := fun u => G u ω) (fun u _ => by simpa only [Real.norm_eq_abs] using hb u ω)
+  rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hle)] at hbnd
+  have hupper := (stopGrid_bounds n (τ ω)).2
+  have hu : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ)+1/((n : ℝ)+1) := by exact_mod_cast hupper
+  have htail := (le_abs_self _).trans hbnd
+  have hrange := mul_le_mul_of_nonneg_left (show (stopGrid n (τ ω) : ℝ)-(τ ω : ℝ) ≤
+    1/((n : ℝ)+1) by linarith) hC
+  linarith
+
+theorem brownian_time_field_optional_le
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hnonpos : ∀ ω u, 0 ≤ u → u < (τ ω : ℝ) → brownianFieldDrift B F a u ω ≤ 0) :
+    (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P) ≤ (∫ ω, F 0 ω (B 0 ω) ∂P) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hτm := stopping_time_measurable_nnreal hτ
+  let V (n : ℕ) (ω : Ω) := F (stopGrid n (τ ω)) ω (B (stopGrid n (τ ω)) ω)
+  let H (n : ℕ) (ω : Ω) := ∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), brownianFieldDrift B F a u ω
+  have hVm (n : ℕ) : Measurable (V n) :=
+    measurable_discrete_eval (fun k ω => F (stopGridTime n k) ω (B (stopGridTime n k) ω))
+      (fun k => hF.eval_measurable _ 0 (by decide) (hm _)) (stopGridIndex_measurable hτm n)
+  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) := hF.bound _ ω _ 0 (by decide)
+  have hVi (n : ℕ) : Integrable (V n) P :=
+    Integrable.of_bound (hVm n).aestronglyMeasurable (K : ℝ)
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+  have hGm := hF.generator_measurable hBm
+  have hHjoint := cumulative_integral_measurable _ hGm
+  have hHm (n : ℕ) : Measurable (H n) :=
+    measurable_discrete_eval
+      (fun k ω => ∫ u in (0 : ℝ)..(stopGridTime n k : ℝ), brownianFieldDrift B F a u ω)
+      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
+  have hHb (n : ℕ) (ω : Ω) : |H n ω| ≤ 2*(K : ℝ)*((T : ℝ)+1) := by
+    have hbound := intervalIntegral.norm_integral_le_of_norm_le_const
+      (a := (0 : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := 2*(K : ℝ))
+      (f := fun u => brownianFieldDrift B F a u ω)
+      (fun u _ => by simpa only [Real.norm_eq_abs] using hF.generator_bound u ω)
+    rw [Real.norm_eq_abs, sub_zero, abs_of_nonneg (stopGrid n (τ ω)).coe_nonneg] at hbound
+    have hupper := (stopGrid_bounds n (τ ω)).2
+    have hgrid : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ)+1/((n : ℝ)+1) := by exact_mod_cast hupper
+    have hT : (τ ω : ℝ) ≤ T := hb ω
+    have hone : 1/((n : ℝ)+1) ≤ 1 := by
+      apply (div_le_iff₀ (by positivity : 0 < (n : ℝ)+1)).mpr
+      nlinarith [Nat.cast_nonneg (α := ℝ) n]
+    exact hbound.trans (mul_le_mul_of_nonneg_left (by linarith) (by positivity))
+  have hHi (n : ℕ) : Integrable (H n) P :=
+    Integrable.of_bound (hHm n).aestronglyMeasurable _
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
+  have hHle (n : ℕ) (ω : Ω) : H n ω ≤ 2*(K : ℝ)*(1/((n : ℝ)+1)) :=
+    bounded_cumulative_at_grid_le _ hGm (2*(K : ℝ)) (by positivity)
+      hF.generator_bound τ hnonpos n ω
+  have hmean (n : ℕ) : (∫ ω, V n ω ∂P) ≤
+      (∫ ω, F 0 ω (B 0 ω) ∂P)+2*(K : ℝ)*(1/((n : ℝ)+1)) := by
+    have he := brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
+    change (∫ ω, V n ω-H n ω ∂P) = _ at he
+    rw [integral_sub (hVi n) (hHi n)] at he
+    have hi := integral_mono_ae (hHi n) (integrable_const (2*(K : ℝ)*(1/((n : ℝ)+1))))
+      (Eventually.of_forall (hHle n))
+    simp only [integral_const, measureReal_def, measure_univ, ENNReal.toReal_one, one_smul] at hi
+    linarith
+  have hlim : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
+      (𝓝 (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => (K : ℝ))
+      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+    exact Eventually.of_forall fun ω =>
+      (hF.path_continuous hBc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω))
+  have hlimUpper : Tendsto (fun n : ℕ => (∫ ω, F 0 ω (B 0 ω) ∂P)+
+      2*(K : ℝ)*(1/((n : ℝ)+1))) atTop (𝓝 (∫ ω, F 0 ω (B 0 ω) ∂P)) := by
+    simpa only [mul_zero, add_zero] using tendsto_const_nhds.add
+      ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2*(K : ℝ)))
+  exact le_of_tendsto_of_tendsto hlim hlimUpper (Eventually.of_forall hmean)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalIntervals.lean b/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalIntervals.lean
index c83a2046187ca84d6cc3f16b981aeddb9c1da21b..8e81bfc81a4e49623efe8341f6c16a4d4fac23c4 100644
--- a/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalIntervals.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/BrownianOptionalIntervals.lean
@@ -1,273 +1,273 @@
-import OAI.Probability.SLE.Stochastic.BrownianOptionalInequality
-
-namespace OAI
-
-/-! Conditional optional stopping between two bounded stopping times.
-The compensated identity follows from the discrete grid identity. The drift is required to have its sign only between the two
-stops, as needed between multipoint activation and completion switches. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma SmoothTimeBrownianField.eval_joint_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
-    {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
-    Measurable (fun q : ℝ≥0 × Ω => F q.1 q.2 (B q.1 q.2)) := by
-  convert (hF.joint 0 (by decide)).comp (measurable_id.prodMk hBm) using 1
-  funext q
-  simp only [iteratedDeriv_zero, Function.comp_apply, id_eq]
-
-lemma SmoothTimeBrownianField.compensated_joint_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
-    {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
-    Measurable (fun q : ℝ≥0 × Ω => brownianFieldCompensated B F a q.1 q.2) :=
-  (hF.eval_joint_measurable hBm).sub
-    (cumulative_integral_measurable _ (hF.generator_measurable hBm))
-
-lemma SmoothTimeBrownianField.compensated_bound
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
-    {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (T t : ℝ≥0) (ht : t ≤ T) (ω : Ω) :
-    |brownianFieldCompensated B F a t ω| ≤ (K : ℝ) + 2 * K * T := by
-  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (0 : ℝ)) (b := (t : ℝ)) (C := 2 * (K : ℝ))
-    (f := fun u => brownianFieldDrift B F a u ω)
-    (fun u _ => by simpa only [Real.norm_eq_abs] using hF.generator_bound u ω)
-  rw [Real.norm_eq_abs, sub_zero, abs_of_nonneg t.coe_nonneg] at hi
-  exact (abs_sub _ _).trans (add_le_add (hF.bound t ω (B t ω) 0 (by decide))
-    (hi.trans (mul_le_mul_of_nonneg_left ht (by positivity))))
-
-lemma SmoothTimeBrownianField.compensated_path_continuous
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
-    {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
-    Continuous (fun t => brownianFieldCompensated B F a t ω) := by
-  have hi (s t : ℝ) : IntervalIntegrable (fun u => brownianFieldDrift B F a u ω)
-      volume s t := by
-    rcases le_total s t with h | h
-    · exact bounded_time_section_intervalIntegrable s t h _ (hF.generator_measurable hBm)
-        (2 * (K : ℝ)) hF.generator_bound ω
-    · exact (bounded_time_section_intervalIntegrable t s h _ (hF.generator_measurable hBm)
-        (2 * (K : ℝ)) hF.generator_bound ω).symm
-  exact (hF.path_continuous hBc ω).sub
-    ((intervalIntegral.continuous_primitive hi 0).comp continuous_subtype_val)
-
-lemma SmoothTimeBrownianField.eval_stopped_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
-    {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {τ : Ω → ℝ≥0} (hτm : Measurable τ) :
-    Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
-  apply Integrable.of_bound
-    ((hF.eval_joint_measurable hBm).comp (hτm.prodMk measurable_id)).aestronglyMeasurable
-    (K : ℝ)
-  exact ae_of_all _ fun ω => by
-    simpa only [Real.norm_eq_abs, Function.comp_apply, id_eq, iteratedDeriv_zero] using
-      hF.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
-
-lemma SmoothTimeBrownianField.compensated_stopped_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
-    {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {τ : Ω → ℝ≥0} (hτm : Measurable τ) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
-    Integrable (fun ω => brownianFieldCompensated B F a (τ ω) ω) P := by
-  apply Integrable.of_bound
-    ((hF.compensated_joint_measurable hBm).comp (hτm.prodMk measurable_id)).aestronglyMeasurable
-    ((K : ℝ) + 2 * K * T)
-  exact ae_of_all _ fun ω => by
-    simpa only [Real.norm_eq_abs, Function.comp_apply, id_eq] using
-      hF.compensated_bound T (τ ω) (hb ω) ω
-
-/-- The full compensated optional identity. No sign restriction on the
-drift is needed for this identity. -/
-theorem brownian_compensated_optional_integral
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
-    (∫ ω, brownianFieldCompensated B F a (τ ω) ω ∂P) =
-      ∫ ω, F 0 ω (B 0 ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hCm := hF.compensated_joint_measurable hBm
-  let V (n : ℕ) (ω : Ω) := brownianFieldCompensated B F a (stopGrid n (τ ω)) ω
-  have hVm (n : ℕ) : Measurable (V n) :=
-    measurable_discrete_eval (fun k ω => brownianFieldCompensated B F a (stopGridTime n k) ω)
-      (fun k => hCm.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
-  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) + 2 * K * (T + 1) := by
-    apply hF.compensated_bound (T + 1) _ _ ω
-    have hgrid := (stopGrid_bounds n (τ ω)).2
-    have hone : (1 : ℝ≥0) / ((n : ℝ≥0) + 1) ≤ 1 := by
-      apply (div_le_iff₀ (by positivity)).mpr
-      simp
-    exact hgrid.trans (add_le_add (hb ω) hone)
-  have hlim : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
-      (𝓝 (∫ ω, brownianFieldCompensated B F a (τ ω) ω ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence
-      (fun _ => (K : ℝ) + 2 * K * (T + 1))
-      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-    exact ae_of_all _ fun ω =>
-      (hF.compensated_path_continuous hBm hBc ω).continuousAt.tendsto.comp
-        (stopGrid_tendsto (τ ω))
-  have he (n : ℕ) : (∫ ω, V n ω ∂P) = ∫ ω, F 0 ω (B 0 ω) ∂P :=
-    brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
-  have hlim' : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
-      (𝓝 (∫ ω, F 0 ω (B 0 ω) ∂P)) := by
-    have heq : (fun n => ∫ ω, V n ω ∂P) = fun _ : ℕ => ∫ ω, F 0 ω (B 0 ω) ∂P :=
-      funext he
-    rw [heq]
-    exact tendsto_const_nhds
-  exact tendsto_nhds_unique hlim hlim'
-
-lemma stopping_time_piecewise_after
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
-    {σ τ : Ω → ℝ≥0}
-    (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hστ : ∀ ω, σ ω ≤ τ ω) {A : Set Ω} [DecidablePred (· ∈ A)]
-    (hA : MeasurableSet[hσ.measurableSpace] A) :
-    IsStoppingTime ℱ (fun ω => ((if ω ∈ A then τ ω else σ ω) : WithTop ℝ≥0)) := by
-  classical
-  intro t
-  have hAt := (hσ.measurableSet A).mp hA |>.2 t
-  have hAct := (hσ.measurableSet Aᶜ).mp hA.compl |>.2 t
-  convert (hAt.inter (hτ t)).union hAct using 1
-  ext ω
-  by_cases hω : ω ∈ A
-  · simp only [hω, ite_true, mem_union, mem_inter_iff, mem_ofPred_eq, mem_compl_iff,
-      not_true_eq_false, false_and, or_false, true_and, WithTop.coe_le_coe]
-    exact ⟨fun h => ⟨(hστ ω).trans h, h⟩, fun h => h.2⟩
-  · simp [hω]
-
-/-- Conditional compensated equality between two bounded stopping times. -/
-theorem brownian_compensated_optional_interval
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (σ τ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hστ : ∀ ω, σ ω ≤ τ ω) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    {A : Set Ω} (hA : MeasurableSet[hσ.measurableSpace] A) :
-    (∫ ω in A, brownianFieldCompensated B F a (τ ω) ω -
-      brownianFieldCompensated B F a (σ ω) ω ∂P) = 0 := by
-  classical
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let π := fun ω => if ω ∈ A then τ ω else σ ω
-  have hπ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (π ω : WithTop ℝ≥0)) := by
-    convert stopping_time_piecewise_after _ hσ hτ hστ hA using 1
-    funext ω
-    by_cases hω : ω ∈ A <;> simp [π, hω] <;> rfl
-  have hbσ : ∀ ω, σ ω ≤ T := fun ω => (hστ ω).trans (hb ω)
-  have hbπ : ∀ ω, π ω ≤ T := by
-    intro ω
-    dsimp only [π]
-    split_ifs
-    · exact hb ω
-    · exact hbσ ω
-  have hπi := hF.compensated_stopped_integrable hBm (stopping_time_measurable_nnreal hπ) T hbπ
-    (P := P)
-  have hσi := hF.compensated_stopped_integrable hBm (stopping_time_measurable_nnreal hσ) T hbσ
-    (P := P)
-  have heπ := brownian_compensated_optional_integral hB hm hBm hBc hF π hπ T hbπ
-  have heσ := brownian_compensated_optional_integral hB hm hBm hBc hF σ hσ T hbσ
-  have hpoint (ω : Ω) : brownianFieldCompensated B F a (π ω) ω -
-      brownianFieldCompensated B F a (σ ω) ω =
-      A.indicator (fun ω => brownianFieldCompensated B F a (τ ω) ω -
-        brownianFieldCompensated B F a (σ ω) ω) ω := by
-    by_cases hω : ω ∈ A <;> simp [π, hω]
-  have he : (∫ ω, brownianFieldCompensated B F a (π ω) ω -
-      brownianFieldCompensated B F a (σ ω) ω ∂P) = 0 := by
-    rw [integral_sub hπi hσi, heπ, heσ, sub_self]
-  simpa only [hpoint, integral_indicator (hσ.measurableSpace_le _ hA)] using he
-
-/-- Only the drift between the stops needs to be nonpositive, and the
-inequality remains valid after restricting to any event known at the
-first stop. -/
-theorem brownian_time_field_optional_interval_le
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (σ τ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hστ : ∀ ω, σ ω ≤ τ ω) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    {A : Set Ω} (hA : MeasurableSet[hσ.measurableSpace] A)
-    (hnonpos : ∀ ω ∈ A, ∀ u, (σ ω : ℝ) ≤ u → u < (τ ω : ℝ) →
-      brownianFieldDrift B F a u ω ≤ 0) :
-    (∫ ω in A, F (τ ω) ω (B (τ ω) ω) ∂P) ≤
-      ∫ ω in A, F (σ ω) ω (B (σ ω) ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hbσ : ∀ ω, σ ω ≤ T := fun ω => (hστ ω).trans (hb ω)
-  have hσm := stopping_time_measurable_nnreal hσ
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hFσ := hF.eval_stopped_integrable hBm hσm (P := P.restrict A)
-  have hFτ := hF.eval_stopped_integrable hBm hτm (P := P.restrict A)
-  have hCσ := hF.compensated_stopped_integrable hBm hσm T hbσ (P := P.restrict A)
-  have hCτ := hF.compensated_stopped_integrable hBm hτm T hb (P := P.restrict A)
-  have hCm := brownian_compensated_optional_interval hB hm hBm hBc hF σ τ hσ hτ hστ T hb hA
-  have hGm := hF.generator_measurable hBm
-  have hi (s t : ℝ) (hst : s ≤ t) (ω : Ω) :=
-    bounded_time_section_intervalIntegrable s t hst _ hGm (2 * (K : ℝ)) hF.generator_bound ω
-  have hle : (∫ ω in A, F (τ ω) ω (B (τ ω) ω) - F (σ ω) ω (B (σ ω) ω) ∂P) ≤
-      ∫ ω in A, brownianFieldCompensated B F a (τ ω) ω -
-        brownianFieldCompensated B F a (σ ω) ω ∂P := by
-    apply integral_mono_ae (hFτ.sub hFσ) (hCτ.sub hCσ)
-    filter_upwards [ae_restrict_mem (hσ.measurableSpace_le _ hA)] with ω hω
-    have hστreal : (σ ω : ℝ) ≤ (τ ω : ℝ) := hστ ω
-    have hneg : (∫ u in (σ ω : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a u ω) ≤ 0 := by
-      have hh := intervalIntegral.integral_mono_on_of_le_Ioo hστreal
-        (hi (σ ω) (τ ω) hστreal ω) (intervalIntegrable_const (c := (0 : ℝ)))
-        (fun u hu => hnonpos ω hω u hu.1.le hu.2)
-      simpa only [intervalIntegral.integral_zero] using hh
-    have hsplit := intervalIntegral.integral_add_adjacent_intervals
-      (hi 0 (σ ω) (σ ω).coe_nonneg ω) (hi (σ ω) (τ ω) (hστ ω) ω)
-    dsimp only [brownianFieldCompensated, Pi.sub_apply]
-    linarith
-  rw [hCm, integral_sub hFτ hFσ] at hle
-  exact sub_nonpos.mp hle
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.BrownianOptionalInequality
+
+namespace OAI
+
+/-! Conditional optional stopping between two bounded stopping times.
+The compensated identity follows from the discrete grid identity. The drift is required to have its sign only between the two
+stops, as needed between multipoint activation and completion switches. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma SmoothTimeBrownianField.eval_joint_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
+    {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
+    Measurable (fun q : ℝ≥0 × Ω => F q.1 q.2 (B q.1 q.2)) := by
+  convert (hF.joint 0 (by decide)).comp (measurable_id.prodMk hBm) using 1
+  funext q
+  simp only [iteratedDeriv_zero, Function.comp_apply, id_eq]
+
+lemma SmoothTimeBrownianField.compensated_joint_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
+    {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2)) :
+    Measurable (fun q : ℝ≥0 × Ω => brownianFieldCompensated B F a q.1 q.2) :=
+  (hF.eval_joint_measurable hBm).sub
+    (cumulative_integral_measurable _ (hF.generator_measurable hBm))
+
+lemma SmoothTimeBrownianField.compensated_bound
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
+    {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (T t : ℝ≥0) (ht : t ≤ T) (ω : Ω) :
+    |brownianFieldCompensated B F a t ω| ≤ (K : ℝ) + 2 * K * T := by
+  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (0 : ℝ)) (b := (t : ℝ)) (C := 2 * (K : ℝ))
+    (f := fun u => brownianFieldDrift B F a u ω)
+    (fun u _ => by simpa only [Real.norm_eq_abs] using hF.generator_bound u ω)
+  rw [Real.norm_eq_abs, sub_zero, abs_of_nonneg t.coe_nonneg] at hi
+  exact (abs_sub _ _).trans (add_le_add (hF.bound t ω (B t ω) 0 (by decide))
+    (hi.trans (mul_le_mul_of_nonneg_left ht (by positivity))))
+
+lemma SmoothTimeBrownianField.compensated_path_continuous
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
+    {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
+    Continuous (fun t => brownianFieldCompensated B F a t ω) := by
+  have hi (s t : ℝ) : IntervalIntegrable (fun u => brownianFieldDrift B F a u ω)
+      volume s t := by
+    rcases le_total s t with h | h
+    · exact bounded_time_section_intervalIntegrable s t h _ (hF.generator_measurable hBm)
+        (2 * (K : ℝ)) hF.generator_bound ω
+    · exact (bounded_time_section_intervalIntegrable t s h _ (hF.generator_measurable hBm)
+        (2 * (K : ℝ)) hF.generator_bound ω).symm
+  exact (hF.path_continuous hBc ω).sub
+    ((intervalIntegral.continuous_primitive hi 0).comp continuous_subtype_val)
+
+lemma SmoothTimeBrownianField.eval_stopped_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
+    {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {τ : Ω → ℝ≥0} (hτm : Measurable τ) :
+    Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
+  apply Integrable.of_bound
+    ((hF.eval_joint_measurable hBm).comp (hτm.prodMk measurable_id)).aestronglyMeasurable
+    (K : ℝ)
+  exact ae_of_all _ fun ω => by
+    simpa only [Real.norm_eq_abs, Function.comp_apply, id_eq, iteratedDeriv_zero] using
+      hF.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
+
+lemma SmoothTimeBrownianField.compensated_stopped_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)} {K : ℝ≥0}
+    {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {τ : Ω → ℝ≥0} (hτm : Measurable τ) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
+    Integrable (fun ω => brownianFieldCompensated B F a (τ ω) ω) P := by
+  apply Integrable.of_bound
+    ((hF.compensated_joint_measurable hBm).comp (hτm.prodMk measurable_id)).aestronglyMeasurable
+    ((K : ℝ) + 2 * K * T)
+  exact ae_of_all _ fun ω => by
+    simpa only [Real.norm_eq_abs, Function.comp_apply, id_eq] using
+      hF.compensated_bound T (τ ω) (hb ω) ω
+
+/-- The full compensated optional identity. No sign restriction on the
+drift is needed for this identity. -/
+theorem brownian_compensated_optional_integral
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
+    (∫ ω, brownianFieldCompensated B F a (τ ω) ω ∂P) =
+      ∫ ω, F 0 ω (B 0 ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hCm := hF.compensated_joint_measurable hBm
+  let V (n : ℕ) (ω : Ω) := brownianFieldCompensated B F a (stopGrid n (τ ω)) ω
+  have hVm (n : ℕ) : Measurable (V n) :=
+    measurable_discrete_eval (fun k ω => brownianFieldCompensated B F a (stopGridTime n k) ω)
+      (fun k => hCm.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
+  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) + 2 * K * (T + 1) := by
+    apply hF.compensated_bound (T + 1) _ _ ω
+    have hgrid := (stopGrid_bounds n (τ ω)).2
+    have hone : (1 : ℝ≥0) / ((n : ℝ≥0) + 1) ≤ 1 := by
+      apply (div_le_iff₀ (by positivity)).mpr
+      simp
+    exact hgrid.trans (add_le_add (hb ω) hone)
+  have hlim : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
+      (𝓝 (∫ ω, brownianFieldCompensated B F a (τ ω) ω ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence
+      (fun _ => (K : ℝ) + 2 * K * (T + 1))
+      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+    exact ae_of_all _ fun ω =>
+      (hF.compensated_path_continuous hBm hBc ω).continuousAt.tendsto.comp
+        (stopGrid_tendsto (τ ω))
+  have he (n : ℕ) : (∫ ω, V n ω ∂P) = ∫ ω, F 0 ω (B 0 ω) ∂P :=
+    brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
+  have hlim' : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
+      (𝓝 (∫ ω, F 0 ω (B 0 ω) ∂P)) := by
+    have heq : (fun n => ∫ ω, V n ω ∂P) = fun _ : ℕ => ∫ ω, F 0 ω (B 0 ω) ∂P :=
+      funext he
+    rw [heq]
+    exact tendsto_const_nhds
+  exact tendsto_nhds_unique hlim hlim'
+
+lemma stopping_time_piecewise_after
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ)
+    {σ τ : Ω → ℝ≥0}
+    (hσ : IsStoppingTime ℱ (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hστ : ∀ ω, σ ω ≤ τ ω) {A : Set Ω} [DecidablePred (· ∈ A)]
+    (hA : MeasurableSet[hσ.measurableSpace] A) :
+    IsStoppingTime ℱ (fun ω => ((if ω ∈ A then τ ω else σ ω) : WithTop ℝ≥0)) := by
+  classical
+  intro t
+  have hAt := (hσ.measurableSet A).mp hA |>.2 t
+  have hAct := (hσ.measurableSet Aᶜ).mp hA.compl |>.2 t
+  convert (hAt.inter (hτ t)).union hAct using 1
+  ext ω
+  by_cases hω : ω ∈ A
+  · simp only [hω, ite_true, mem_union, mem_inter_iff, mem_ofPred_eq, mem_compl_iff,
+      not_true_eq_false, false_and, or_false, true_and, WithTop.coe_le_coe]
+    exact ⟨fun h => ⟨(hστ ω).trans h, h⟩, fun h => h.2⟩
+  · simp [hω]
+
+/-- Conditional compensated equality between two bounded stopping times. -/
+theorem brownian_compensated_optional_interval
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (σ τ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hστ : ∀ ω, σ ω ≤ τ ω) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    {A : Set Ω} (hA : MeasurableSet[hσ.measurableSpace] A) :
+    (∫ ω in A, brownianFieldCompensated B F a (τ ω) ω -
+      brownianFieldCompensated B F a (σ ω) ω ∂P) = 0 := by
+  classical
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let π := fun ω => if ω ∈ A then τ ω else σ ω
+  have hπ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (π ω : WithTop ℝ≥0)) := by
+    convert stopping_time_piecewise_after _ hσ hτ hστ hA using 1
+    funext ω
+    by_cases hω : ω ∈ A <;> simp [π, hω] <;> rfl
+  have hbσ : ∀ ω, σ ω ≤ T := fun ω => (hστ ω).trans (hb ω)
+  have hbπ : ∀ ω, π ω ≤ T := by
+    intro ω
+    dsimp only [π]
+    split_ifs
+    · exact hb ω
+    · exact hbσ ω
+  have hπi := hF.compensated_stopped_integrable hBm (stopping_time_measurable_nnreal hπ) T hbπ
+    (P := P)
+  have hσi := hF.compensated_stopped_integrable hBm (stopping_time_measurable_nnreal hσ) T hbσ
+    (P := P)
+  have heπ := brownian_compensated_optional_integral hB hm hBm hBc hF π hπ T hbπ
+  have heσ := brownian_compensated_optional_integral hB hm hBm hBc hF σ hσ T hbσ
+  have hpoint (ω : Ω) : brownianFieldCompensated B F a (π ω) ω -
+      brownianFieldCompensated B F a (σ ω) ω =
+      A.indicator (fun ω => brownianFieldCompensated B F a (τ ω) ω -
+        brownianFieldCompensated B F a (σ ω) ω) ω := by
+    by_cases hω : ω ∈ A <;> simp [π, hω]
+  have he : (∫ ω, brownianFieldCompensated B F a (π ω) ω -
+      brownianFieldCompensated B F a (σ ω) ω ∂P) = 0 := by
+    rw [integral_sub hπi hσi, heπ, heσ, sub_self]
+  simpa only [hpoint, integral_indicator (hσ.measurableSpace_le _ hA)] using he
+
+/-- Only the drift between the stops needs to be nonpositive, and the
+inequality remains valid after restricting to any event known at the
+first stop. -/
+theorem brownian_time_field_optional_interval_le
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {K : ℝ≥0} {F a : ℝ≥0 → Ω → ℝ → ℝ}
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (σ τ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hστ : ∀ ω, σ ω ≤ τ ω) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    {A : Set Ω} (hA : MeasurableSet[hσ.measurableSpace] A)
+    (hnonpos : ∀ ω ∈ A, ∀ u, (σ ω : ℝ) ≤ u → u < (τ ω : ℝ) →
+      brownianFieldDrift B F a u ω ≤ 0) :
+    (∫ ω in A, F (τ ω) ω (B (τ ω) ω) ∂P) ≤
+      ∫ ω in A, F (σ ω) ω (B (σ ω) ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hbσ : ∀ ω, σ ω ≤ T := fun ω => (hστ ω).trans (hb ω)
+  have hσm := stopping_time_measurable_nnreal hσ
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hFσ := hF.eval_stopped_integrable hBm hσm (P := P.restrict A)
+  have hFτ := hF.eval_stopped_integrable hBm hτm (P := P.restrict A)
+  have hCσ := hF.compensated_stopped_integrable hBm hσm T hbσ (P := P.restrict A)
+  have hCτ := hF.compensated_stopped_integrable hBm hτm T hb (P := P.restrict A)
+  have hCm := brownian_compensated_optional_interval hB hm hBm hBc hF σ τ hσ hτ hστ T hb hA
+  have hGm := hF.generator_measurable hBm
+  have hi (s t : ℝ) (hst : s ≤ t) (ω : Ω) :=
+    bounded_time_section_intervalIntegrable s t hst _ hGm (2 * (K : ℝ)) hF.generator_bound ω
+  have hle : (∫ ω in A, F (τ ω) ω (B (τ ω) ω) - F (σ ω) ω (B (σ ω) ω) ∂P) ≤
+      ∫ ω in A, brownianFieldCompensated B F a (τ ω) ω -
+        brownianFieldCompensated B F a (σ ω) ω ∂P := by
+    apply integral_mono_ae (hFτ.sub hFσ) (hCτ.sub hCσ)
+    filter_upwards [ae_restrict_mem (hσ.measurableSpace_le _ hA)] with ω hω
+    have hστreal : (σ ω : ℝ) ≤ (τ ω : ℝ) := hστ ω
+    have hneg : (∫ u in (σ ω : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a u ω) ≤ 0 := by
+      have hh := intervalIntegral.integral_mono_on_of_le_Ioo hστreal
+        (hi (σ ω) (τ ω) hστreal ω) (intervalIntegrable_const (c := (0 : ℝ)))
+        (fun u hu => hnonpos ω hω u hu.1.le hu.2)
+      simpa only [intervalIntegral.integral_zero] using hh
+    have hsplit := intervalIntegral.integral_add_adjacent_intervals
+      (hi 0 (σ ω) (σ ω).coe_nonneg ω) (hi (σ ω) (τ ω) (hστ ω) ω)
+    dsimp only [brownianFieldCompensated, Pi.sub_apply]
+    linarith
+  rw [hCm, integral_sub hFτ hFσ] at hle
+  exact sub_nonpos.mp hle
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/BrownianStateAE.lean b/lean/OAI/Probability/SLE/Stochastic/BrownianStateAE.lean
index 425d3bef256b035b58af47a882d2f89a4b523575..c6b0e917c3143799c01867e9b537e71e4392cf90 100644
--- a/lean/OAI/Probability/SLE/Stochastic/BrownianStateAE.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/BrownianStateAE.lean
@@ -1,142 +1,142 @@
-import OAI.Probability.SLE.ClippedCoordinates
-import Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun
-
-namespace OAI
-
-/-! Compact Brownian test fields with absolutely continuous adapted states.
-The state derivative may fail at the finitely many multipoint switches. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem compact_state_smooth_brownian_field_of_ae_deriv
-    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (Q V : ℝ → Ω → E) (L : ℝ≥0)
-    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
-    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t : ℝ≥0 => Q t))
-    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
-    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
-    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
-    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun s => Q s ω) (V t ω) t)
-    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hφc : HasCompactSupport φ) :
-    ∃ K : ℝ≥0, SmoothTimeBrownianField B hm K
-      (fun t ω x => φ (Q t ω, x))
-      (fun t ω x => fderiv ℝ φ (Q t ω, x) (V t ω, 0)) := by
-
-  classical
-  have hr (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial φ j)
-    (spatialPartial_smooth φ hφ j) (spatialPartial_compact φ hφc j)
-  choose C hCb hCl using hr
-  let C₀ : ℝ≥0 := ∑ j ∈ Finset.range 4, C j
-  have hC (j : ℕ) (hj : j ≤ 3) : C j ≤ C₀ :=
-    Finset.single_le_sum (fun i _ => (zero_le : (0 : ℝ≥0) ≤ C i)) (Finset.mem_range.mpr (by omega))
-  have hCb' (j : ℕ) (hj : j ≤ 3) (p : E × ℝ) : |spatialPartial φ j p| ≤ C₀ := by
-    have hh := (hCb j p).trans (show (C j : ℝ) ≤ C₀ by exact_mod_cast hC j hj)
-    simpa only [Real.norm_eq_abs] using hh
-  have hC2 : LipschitzWith C₀ (spatialPartial φ 2) := (hCl 2).weaken (hC 2 (by decide))
-  obtain ⟨D, hDb, hDl⟩ := compact_smooth_bounded_lipschitz (fderiv ℝ φ)
-    (hφ.fderiv_right (by simp)) (hφc.fderiv ℝ)
-  let K : ℝ≥0 := (C₀ + D) * (L + 1)
-  have hKC : (C₀ : ℝ) ≤ (K : ℝ) := by
-    change (C₀ : ℝ) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
-    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
-  have hKDL : (D : ℝ) * L ≤ (K : ℝ) := by
-    change (D : ℝ) * L ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
-    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
-  have hKCL : (C₀ : ℝ) * ((L : ℝ) + 1) ≤ (K : ℝ) := by
-    change (C₀ : ℝ) * ((L : ℝ) + 1) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
-    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
-  have hQn : Measurable (fun p : ℝ≥0 × Ω => Q p.1 p.2) :=
-    hQm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hVn : Measurable (fun p : ℝ≥0 × Ω => V p.1 p.2) :=
-    hVm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
-  have happ : Continuous (fun p : (E × ℝ) × (E × ℝ) => fderiv ℝ φ p.1 p.2) :=
-    ((hφ.continuous_fderiv (by simp)).comp continuous_fst).clm_apply continuous_snd
-  have hameas : Measurable (fun p : (ℝ≥0 × Ω) × ℝ => fderiv ℝ φ (Q p.1.1 p.1.2, p.2) (V p.1.1 p.1.2, 0)) :=
-    happ.measurable.comp (((hQn.comp measurable_fst).prodMk measurable_snd).prodMk
-      ((hVn.comp measurable_fst).prodMk measurable_const))
-  have habound (t : ℝ≥0) (ω : Ω) (x : ℝ) :
-      |fderiv ℝ φ (Q t ω, x) (V t ω, 0)| ≤ K := by
-    calc
-      _ = ‖fderiv ℝ φ (Q t ω, x) (V t ω, 0)‖ := (Real.norm_eq_abs _).symm
-      _ ≤ ‖fderiv ℝ φ (Q t ω, x)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
-      _ = ‖fderiv ℝ φ (Q t ω, x)‖ * ‖V t ω‖ := by simp
-      _ ≤ (D : ℝ) * L := mul_le_mul (hDb _) (hVb t ω) (norm_nonneg _) D.coe_nonneg
-      _ ≤ K := hKDL
-  have haLip (t : ℝ≥0) (ω : Ω) (x y : ℝ) :
-      |fderiv ℝ φ (Q t ω, x) (V t ω, 0) - fderiv ℝ φ (Q t ω, y) (V t ω, 0)| ≤
-        (K : ℝ) * |x - y| := by
-    have hd : ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ ≤ (D : ℝ) * |x - y| := by
-      simpa [dist_eq_norm, Prod.norm_def, Real.norm_eq_abs] using hDl.dist_le_mul (Q t ω, x) (Q t ω, y)
-    calc
-      _ = ‖(fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)) (V t ω, 0)‖ := by
-        rw [sub_apply, Real.norm_eq_abs]
-      _ ≤ ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
-      _ = ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖V t ω‖ := by simp
-      _ ≤ ((D : ℝ) * |x - y|) * L := mul_le_mul hd (hVb t ω) (norm_nonneg _) (by positivity)
-      _ = ((D : ℝ) * L) * |x - y| := by ring
-      _ ≤ (K : ℝ) * |x - y| := mul_le_mul_of_nonneg_right hKDL (abs_nonneg _)
-  refine ⟨K, ⟨?_, ?_, ?_, hameas, ?_, habound, haLip, ?_, ?_⟩⟩
-  · intro t ω
-    have hc : ContDiff ℝ (⊤ : ℕ∞) (fun x : ℝ => φ (Q t ω, x)) :=
-      hφ.comp (contDiff_const.prodMk contDiff_id)
-    exact hc.of_le (WithTop.coe_le_coe.mpr (show (3 : ℕ∞) ≤ ⊤ from le_top))
-  · intro t j hj
-    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
-    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
-      (((hQa t).comp measurable_fst).prodMk measurable_snd)
-  · intro j hj
-    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
-    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
-      ((hQn.comp measurable_fst).prodMk measurable_snd)
-  · intro t ω x j hj
-    rw [← spatialPartial_eq_iteratedDeriv φ hφ]
-    exact (hCb' j hj _).trans hKC
-  · intro s t ω x y
-    rw [← spatialPartial_eq_iteratedDeriv φ hφ, ← spatialPartial_eq_iteratedDeriv φ hφ]
-    have hQdist := (hQl ω).dist_le_mul (t : ℝ) (s : ℝ)
-    rw [Real.dist_eq] at hQdist
-    calc
-      _ = dist (spatialPartial φ 2 (Q t ω, x)) (spatialPartial φ 2 (Q s ω, y)) := (Real.dist_eq _ _).symm
-      _ ≤ (C₀ : ℝ) * dist (Q t ω, x) (Q s ω, y) := hC2.dist_le_mul _ _
-      _ ≤ (C₀ : ℝ) * (dist (Q t ω) (Q s ω) + |x - y|) := by
-        apply mul_le_mul_of_nonneg_left _ C₀.coe_nonneg
-        rw [Prod.dist_eq, Real.dist_eq]
-        exact max_le (le_add_of_nonneg_right (abs_nonneg _)) (le_add_of_nonneg_left dist_nonneg)
-      _ ≤ (C₀ : ℝ) * ((L : ℝ) * |(t : ℝ) - s| + |x - y|) :=
-        mul_le_mul_of_nonneg_left (show dist (Q (t : ℝ) ω) (Q (s : ℝ) ω) + |x - y| ≤
-          (L : ℝ) * |(t : ℝ) - s| + |x - y| by linarith) C₀.coe_nonneg
-      _ ≤ ((C₀ : ℝ) * ((L : ℝ) + 1)) * (|(t : ℝ) - s| + |x - y|) := by
-        nlinarith [mul_nonneg C₀.coe_nonneg (abs_nonneg ((t : ℝ) - s)),
-          mul_nonneg (mul_nonneg C₀.coe_nonneg L.coe_nonneg) (abs_nonneg (x - y))]
-      _ ≤ (K : ℝ) * (|(t : ℝ) - s| + |x - y|) :=
-        mul_le_mul_of_nonneg_right hKCL (by positivity)
-  · intro s t hst ω x
-    have hst' : (s : ℝ) ≤ (t : ℝ) := by exact_mod_cast hst
-    have hpair : LipschitzWith L (fun u : ℝ => (Q u ω, x)) := by
-      apply LipschitzWith.of_dist_le_mul
-      intro u v
-      simpa only [Prod.dist_eq, dist_self, max_eq_left dist_nonneg] using (hQl ω).dist_le_mul u v
-    have hlip : LipschitzWith (C 0*L) (fun u : ℝ => φ (Q u ω,x)) := (hCl 0).comp hpair
-    have hAC := hlip.lipschitzOnWith.absolutelyContinuousOnInterval (a := (s : ℝ)) (b := (t : ℝ))
-    symm
-    rw [← hAC.integral_deriv_eq_sub]
-    apply intervalIntegral.integral_congr_ae
-    filter_upwards [hQd ω] with u hu huu
-    rw [uIoc_of_le hst'] at huu
-    have hu0 : 0 < u := lt_of_le_of_lt s.coe_nonneg huu.1
-    have hd := (hu hu0).prodMk (hasDerivAt_const u x)
-    have hc := ((hφ.differentiable (by simp)) (Q u ω,x)).hasFDerivAt.comp_hasDerivAt u hd
-    simpa only [Function.comp_def, Real.coe_toNNReal _ hu0.le] using hc.deriv.symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.ClippedCoordinates
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun
+
+namespace OAI
+
+/-! Compact Brownian test fields with absolutely continuous adapted states.
+The state derivative may fail at the finitely many multipoint switches. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem compact_state_smooth_brownian_field_of_ae_deriv
+    {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (Q V : ℝ → Ω → E) (L : ℝ≥0)
+    (hQm : Measurable (fun p : ℝ × Ω => Q p.1 p.2))
+    (hQa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t : ℝ≥0 => Q t))
+    (hQl : ∀ ω, LipschitzWith L (fun t => Q t ω))
+    (hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2))
+    (hVb : ∀ t ω, ‖V t ω‖ ≤ L)
+    (hQd : ∀ ω, ∀ᵐ t : ℝ, 0 < t → HasDerivAt (fun s => Q s ω) (V t ω) t)
+    (φ : E × ℝ → ℝ) (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hφc : HasCompactSupport φ) :
+    ∃ K : ℝ≥0, SmoothTimeBrownianField B hm K
+      (fun t ω x => φ (Q t ω, x))
+      (fun t ω x => fderiv ℝ φ (Q t ω, x) (V t ω, 0)) := by
+
+  classical
+  have hr (j : ℕ) := compact_smooth_bounded_lipschitz (spatialPartial φ j)
+    (spatialPartial_smooth φ hφ j) (spatialPartial_compact φ hφc j)
+  choose C hCb hCl using hr
+  let C₀ : ℝ≥0 := ∑ j ∈ Finset.range 4, C j
+  have hC (j : ℕ) (hj : j ≤ 3) : C j ≤ C₀ :=
+    Finset.single_le_sum (fun i _ => (zero_le : (0 : ℝ≥0) ≤ C i)) (Finset.mem_range.mpr (by omega))
+  have hCb' (j : ℕ) (hj : j ≤ 3) (p : E × ℝ) : |spatialPartial φ j p| ≤ C₀ := by
+    have hh := (hCb j p).trans (show (C j : ℝ) ≤ C₀ by exact_mod_cast hC j hj)
+    simpa only [Real.norm_eq_abs] using hh
+  have hC2 : LipschitzWith C₀ (spatialPartial φ 2) := (hCl 2).weaken (hC 2 (by decide))
+  obtain ⟨D, hDb, hDl⟩ := compact_smooth_bounded_lipschitz (fderiv ℝ φ)
+    (hφ.fderiv_right (by simp)) (hφc.fderiv ℝ)
+  let K : ℝ≥0 := (C₀ + D) * (L + 1)
+  have hKC : (C₀ : ℝ) ≤ (K : ℝ) := by
+    change (C₀ : ℝ) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
+    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
+  have hKDL : (D : ℝ) * L ≤ (K : ℝ) := by
+    change (D : ℝ) * L ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
+    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
+  have hKCL : (C₀ : ℝ) * ((L : ℝ) + 1) ≤ (K : ℝ) := by
+    change (C₀ : ℝ) * ((L : ℝ) + 1) ≤ ((C₀ : ℝ) + D) * ((L : ℝ) + 1)
+    nlinarith [C₀.coe_nonneg, D.coe_nonneg, L.coe_nonneg]
+  have hQn : Measurable (fun p : ℝ≥0 × Ω => Q p.1 p.2) :=
+    hQm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hVn : Measurable (fun p : ℝ≥0 × Ω => V p.1 p.2) :=
+    hVm.comp ((NNReal.continuous_coe.measurable.comp measurable_fst).prodMk measurable_snd)
+  have happ : Continuous (fun p : (E × ℝ) × (E × ℝ) => fderiv ℝ φ p.1 p.2) :=
+    ((hφ.continuous_fderiv (by simp)).comp continuous_fst).clm_apply continuous_snd
+  have hameas : Measurable (fun p : (ℝ≥0 × Ω) × ℝ => fderiv ℝ φ (Q p.1.1 p.1.2, p.2) (V p.1.1 p.1.2, 0)) :=
+    happ.measurable.comp (((hQn.comp measurable_fst).prodMk measurable_snd).prodMk
+      ((hVn.comp measurable_fst).prodMk measurable_const))
+  have habound (t : ℝ≥0) (ω : Ω) (x : ℝ) :
+      |fderiv ℝ φ (Q t ω, x) (V t ω, 0)| ≤ K := by
+    calc
+      _ = ‖fderiv ℝ φ (Q t ω, x) (V t ω, 0)‖ := (Real.norm_eq_abs _).symm
+      _ ≤ ‖fderiv ℝ φ (Q t ω, x)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
+      _ = ‖fderiv ℝ φ (Q t ω, x)‖ * ‖V t ω‖ := by simp
+      _ ≤ (D : ℝ) * L := mul_le_mul (hDb _) (hVb t ω) (norm_nonneg _) D.coe_nonneg
+      _ ≤ K := hKDL
+  have haLip (t : ℝ≥0) (ω : Ω) (x y : ℝ) :
+      |fderiv ℝ φ (Q t ω, x) (V t ω, 0) - fderiv ℝ φ (Q t ω, y) (V t ω, 0)| ≤
+        (K : ℝ) * |x - y| := by
+    have hd : ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ ≤ (D : ℝ) * |x - y| := by
+      simpa [dist_eq_norm, Prod.norm_def, Real.norm_eq_abs] using hDl.dist_le_mul (Q t ω, x) (Q t ω, y)
+    calc
+      _ = ‖(fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)) (V t ω, 0)‖ := by
+        rw [sub_apply, Real.norm_eq_abs]
+      _ ≤ ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖(V t ω, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm _ _
+      _ = ‖fderiv ℝ φ (Q t ω, x) - fderiv ℝ φ (Q t ω, y)‖ * ‖V t ω‖ := by simp
+      _ ≤ ((D : ℝ) * |x - y|) * L := mul_le_mul hd (hVb t ω) (norm_nonneg _) (by positivity)
+      _ = ((D : ℝ) * L) * |x - y| := by ring
+      _ ≤ (K : ℝ) * |x - y| := mul_le_mul_of_nonneg_right hKDL (abs_nonneg _)
+  refine ⟨K, ⟨?_, ?_, ?_, hameas, ?_, habound, haLip, ?_, ?_⟩⟩
+  · intro t ω
+    have hc : ContDiff ℝ (⊤ : ℕ∞) (fun x : ℝ => φ (Q t ω, x)) :=
+      hφ.comp (contDiff_const.prodMk contDiff_id)
+    exact hc.of_le (WithTop.coe_le_coe.mpr (show (3 : ℕ∞) ≤ ⊤ from le_top))
+  · intro t j hj
+    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
+    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
+      (((hQa t).comp measurable_fst).prodMk measurable_snd)
+  · intro j hj
+    simp_rw [← spatialPartial_eq_iteratedDeriv φ hφ]
+    exact (spatialPartial_smooth φ hφ j).continuous.measurable.comp
+      ((hQn.comp measurable_fst).prodMk measurable_snd)
+  · intro t ω x j hj
+    rw [← spatialPartial_eq_iteratedDeriv φ hφ]
+    exact (hCb' j hj _).trans hKC
+  · intro s t ω x y
+    rw [← spatialPartial_eq_iteratedDeriv φ hφ, ← spatialPartial_eq_iteratedDeriv φ hφ]
+    have hQdist := (hQl ω).dist_le_mul (t : ℝ) (s : ℝ)
+    rw [Real.dist_eq] at hQdist
+    calc
+      _ = dist (spatialPartial φ 2 (Q t ω, x)) (spatialPartial φ 2 (Q s ω, y)) := (Real.dist_eq _ _).symm
+      _ ≤ (C₀ : ℝ) * dist (Q t ω, x) (Q s ω, y) := hC2.dist_le_mul _ _
+      _ ≤ (C₀ : ℝ) * (dist (Q t ω) (Q s ω) + |x - y|) := by
+        apply mul_le_mul_of_nonneg_left _ C₀.coe_nonneg
+        rw [Prod.dist_eq, Real.dist_eq]
+        exact max_le (le_add_of_nonneg_right (abs_nonneg _)) (le_add_of_nonneg_left dist_nonneg)
+      _ ≤ (C₀ : ℝ) * ((L : ℝ) * |(t : ℝ) - s| + |x - y|) :=
+        mul_le_mul_of_nonneg_left (show dist (Q (t : ℝ) ω) (Q (s : ℝ) ω) + |x - y| ≤
+          (L : ℝ) * |(t : ℝ) - s| + |x - y| by linarith) C₀.coe_nonneg
+      _ ≤ ((C₀ : ℝ) * ((L : ℝ) + 1)) * (|(t : ℝ) - s| + |x - y|) := by
+        nlinarith [mul_nonneg C₀.coe_nonneg (abs_nonneg ((t : ℝ) - s)),
+          mul_nonneg (mul_nonneg C₀.coe_nonneg L.coe_nonneg) (abs_nonneg (x - y))]
+      _ ≤ (K : ℝ) * (|(t : ℝ) - s| + |x - y|) :=
+        mul_le_mul_of_nonneg_right hKCL (by positivity)
+  · intro s t hst ω x
+    have hst' : (s : ℝ) ≤ (t : ℝ) := by exact_mod_cast hst
+    have hpair : LipschitzWith L (fun u : ℝ => (Q u ω, x)) := by
+      apply LipschitzWith.of_dist_le_mul
+      intro u v
+      simpa only [Prod.dist_eq, dist_self, max_eq_left dist_nonneg] using (hQl ω).dist_le_mul u v
+    have hlip : LipschitzWith (C 0*L) (fun u : ℝ => φ (Q u ω,x)) := (hCl 0).comp hpair
+    have hAC := hlip.lipschitzOnWith.absolutelyContinuousOnInterval (a := (s : ℝ)) (b := (t : ℝ))
+    symm
+    rw [← hAC.integral_deriv_eq_sub]
+    apply intervalIntegral.integral_congr_ae
+    filter_upwards [hQd ω] with u hu huu
+    rw [uIoc_of_le hst'] at huu
+    have hu0 : 0 < u := lt_of_le_of_lt s.coe_nonneg huu.1
+    have hd := (hu hu0).prodMk (hasDerivAt_const u x)
+    have hc := ((hφ.differentiable (by simp)) (Q u ω,x)).hasFDerivAt.comp_hasDerivAt u hd
+    simpa only [Function.comp_def, Real.coe_toNNReal _ hu0.le] using hc.deriv.symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/DriverLawKernelTransfer.lean b/lean/OAI/Probability/SLE/Stochastic/DriverLawKernelTransfer.lean
index 11021d9d83d00e20a63985821264016173c164f3..f3de8c00b640a79019aa4c8924e111423a226a34 100644
--- a/lean/OAI/Probability/SLE/Stochastic/DriverLawKernelTransfer.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/DriverLawKernelTransfer.lean
@@ -1,90 +1,90 @@
-import OAI.Probability.SLE.Geometry.RestartContinuationKernel
-import OAI.Probability.SLE.Loewner.ScaleTraceLaw
-
-namespace OAI
-
-/-! A deterministic-time kernel specified by the original Brownian past
-transfers to another Brownian realization with the same continuous path law. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-local instance driverLawKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance driverLawKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-  {B B' : ℝ≥0 → Ω → ℝ}
-
-lemma source_driver_integral_of_same_law_kernel
-    (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P)
-    (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t))
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (hc' : ∀ ω, Continuous (fun t => B' t ω))
-    (hlaw : P.map (sourceDriverPath B hc)=P.map (sourceDriverPath B' hc'))
-    (t : ℝ≥0) (f K : C(ℝ≥0,ℝ) → ℝ≥0∞) (hf : Measurable f)
-    (hK : Measurable[driverPathFiltration t] K)
-    (hmean : ∀ A : Set Ω,
-      MeasurableSet[(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t] A →
-      (∫⁻ ω in A, f (sourceDriverPath B hc ω) ∂P)=
-        ∫⁻ ω in A, K (sourceDriverPath B hc ω) ∂P)
-    (E : Set Ω)
-    (hE : MeasurableSet[(Filtration.natural B' (fun s => (hm' s).stronglyMeasurable)) t] E) :
-    (∫⁻ ω in E, f (sourceDriverPath B' hc' ω) ∂P)=
-      ∫⁻ ω in E, K (sourceDriverPath B' hc' ω) ∂P := by
-  have hs := driverContinuationMean_eq_of_past_integrals hB hm hc t f K hf hK hmean
-  have hKm := hK.mono (driverPathFiltration.le t) le_rfl
-  have hsec : MeasurableSet {p | driverContinuationMean (P.map (sourceDriverPath B' hc'))
-      t f p=K p} := measurableSet_eq_fun
-    ((driverContinuationMean_measurable _ t f hf).mono (driverPathFiltration.le t) le_rfl) hKm
-  rw [hlaw] at hs
-  have hp := (ae_map_iff (source_path_measurable (sourceDriverPath B' hc') hm').aemeasurable hsec).mp hs
-  have he := (source_driver_continuation_condLExp hB' hm' hc' t f hf).symm.trans hp
-  calc
-    _ = ∫⁻ ω in E, P⁻[(fun ω => f (sourceDriverPath B' hc' ω)) |
-        (Filtration.natural B' (fun s => (hm' s).stronglyMeasurable)) t] ω ∂P :=
-      (setLIntegral_condLExp (Filtration.le _ t) P _ hE).symm
-    _ = _ := lintegral_congr_ae (ae_restrict_of_ae he)
-
-lemma source_driver_bind_of_same_law_kernel
-    (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P)
-    (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t))
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (hc' : ∀ ω, Continuous (fun t => B' t ω))
-    (hlaw : P.map (sourceDriverPath B hc)=P.map (sourceDriverPath B' hc'))
-    (t : ℝ≥0) (N K : C(ℝ≥0,ℝ) → Measure upperHalfPlane) (hN : Measurable N)
-    (hK : Measurable[driverPathFiltration t] K)
-    (hmean : ∀ A : Set Ω,
-      MeasurableSet[(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t] A →
-      (P.restrict A).bind (fun ω => N (sourceDriverPath B hc ω))=
-        (P.restrict A).bind (fun ω => K (sourceDriverPath B hc ω)))
-    (E : Set Ω)
-    (hE : MeasurableSet[(Filtration.natural B' (fun s => (hm' s).stronglyMeasurable)) t] E) :
-    (P.restrict E).bind (fun ω => N (sourceDriverPath B' hc' ω))=
-      (P.restrict E).bind (fun ω => K (sourceDriverPath B' hc' ω)) := by
-  have hp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
-  have hp' : Measurable (sourceDriverPath B' hc') := source_path_measurable _ hm'
-  have hKm := hK.mono (driverPathFiltration.le t) le_rfl
-  have hNB : Measurable (fun ω => N (sourceDriverPath B hc ω)) := hN.comp hp
-  have hKB : Measurable (fun ω => K (sourceDriverPath B hc ω)) := hKm.comp hp
-  have hNB' : Measurable (fun ω => N (sourceDriverPath B' hc' ω)) := hN.comp hp'
-  have hKB' : Measurable (fun ω => K (sourceDriverPath B' hc' ω)) := hKm.comp hp'
-  apply Measure.ext
-  intro S hS
-  rw [Measure.bind_apply hS hNB'.aemeasurable,
-    Measure.bind_apply hS hKB'.aemeasurable]
-  apply source_driver_integral_of_same_law_kernel hB hB' hm hm' hc hc' hlaw t
-    (fun p => N p S) (fun p => K p S)
-    ((Measure.measurable_coe hS).comp hN) ((Measure.measurable_coe hS).comp hK) ?_ E hE
-  intro A hA
-  have he := congrArg (fun μ : Measure upperHalfPlane => μ S) (hmean A hA)
-  simpa only [Measure.bind_apply hS hNB.aemeasurable,
-    Measure.bind_apply hS hKB.aemeasurable] using he
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.RestartContinuationKernel
+import OAI.Probability.SLE.Loewner.ScaleTraceLaw
+
+namespace OAI
+
+/-! A deterministic-time kernel specified by the original Brownian past
+transfers to another Brownian realization with the same continuous path law. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+local instance driverLawKernelMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance driverLawKernelBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+  {B B' : ℝ≥0 → Ω → ℝ}
+
+lemma source_driver_integral_of_same_law_kernel
+    (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P)
+    (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t))
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (hc' : ∀ ω, Continuous (fun t => B' t ω))
+    (hlaw : P.map (sourceDriverPath B hc)=P.map (sourceDriverPath B' hc'))
+    (t : ℝ≥0) (f K : C(ℝ≥0,ℝ) → ℝ≥0∞) (hf : Measurable f)
+    (hK : Measurable[driverPathFiltration t] K)
+    (hmean : ∀ A : Set Ω,
+      MeasurableSet[(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t] A →
+      (∫⁻ ω in A, f (sourceDriverPath B hc ω) ∂P)=
+        ∫⁻ ω in A, K (sourceDriverPath B hc ω) ∂P)
+    (E : Set Ω)
+    (hE : MeasurableSet[(Filtration.natural B' (fun s => (hm' s).stronglyMeasurable)) t] E) :
+    (∫⁻ ω in E, f (sourceDriverPath B' hc' ω) ∂P)=
+      ∫⁻ ω in E, K (sourceDriverPath B' hc' ω) ∂P := by
+  have hs := driverContinuationMean_eq_of_past_integrals hB hm hc t f K hf hK hmean
+  have hKm := hK.mono (driverPathFiltration.le t) le_rfl
+  have hsec : MeasurableSet {p | driverContinuationMean (P.map (sourceDriverPath B' hc'))
+      t f p=K p} := measurableSet_eq_fun
+    ((driverContinuationMean_measurable _ t f hf).mono (driverPathFiltration.le t) le_rfl) hKm
+  rw [hlaw] at hs
+  have hp := (ae_map_iff (source_path_measurable (sourceDriverPath B' hc') hm').aemeasurable hsec).mp hs
+  have he := (source_driver_continuation_condLExp hB' hm' hc' t f hf).symm.trans hp
+  calc
+    _ = ∫⁻ ω in E, P⁻[(fun ω => f (sourceDriverPath B' hc' ω)) |
+        (Filtration.natural B' (fun s => (hm' s).stronglyMeasurable)) t] ω ∂P :=
+      (setLIntegral_condLExp (Filtration.le _ t) P _ hE).symm
+    _ = _ := lintegral_congr_ae (ae_restrict_of_ae he)
+
+lemma source_driver_bind_of_same_law_kernel
+    (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P)
+    (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t))
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (hc' : ∀ ω, Continuous (fun t => B' t ω))
+    (hlaw : P.map (sourceDriverPath B hc)=P.map (sourceDriverPath B' hc'))
+    (t : ℝ≥0) (N K : C(ℝ≥0,ℝ) → Measure upperHalfPlane) (hN : Measurable N)
+    (hK : Measurable[driverPathFiltration t] K)
+    (hmean : ∀ A : Set Ω,
+      MeasurableSet[(Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t] A →
+      (P.restrict A).bind (fun ω => N (sourceDriverPath B hc ω))=
+        (P.restrict A).bind (fun ω => K (sourceDriverPath B hc ω)))
+    (E : Set Ω)
+    (hE : MeasurableSet[(Filtration.natural B' (fun s => (hm' s).stronglyMeasurable)) t] E) :
+    (P.restrict E).bind (fun ω => N (sourceDriverPath B' hc' ω))=
+      (P.restrict E).bind (fun ω => K (sourceDriverPath B' hc' ω)) := by
+  have hp : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
+  have hp' : Measurable (sourceDriverPath B' hc') := source_path_measurable _ hm'
+  have hKm := hK.mono (driverPathFiltration.le t) le_rfl
+  have hNB : Measurable (fun ω => N (sourceDriverPath B hc ω)) := hN.comp hp
+  have hKB : Measurable (fun ω => K (sourceDriverPath B hc ω)) := hKm.comp hp
+  have hNB' : Measurable (fun ω => N (sourceDriverPath B' hc' ω)) := hN.comp hp'
+  have hKB' : Measurable (fun ω => K (sourceDriverPath B' hc' ω)) := hKm.comp hp'
+  apply Measure.ext
+  intro S hS
+  rw [Measure.bind_apply hS hNB'.aemeasurable,
+    Measure.bind_apply hS hKB'.aemeasurable]
+  apply source_driver_integral_of_same_law_kernel hB hB' hm hm' hc hc' hlaw t
+    (fun p => N p S) (fun p => K p S)
+    ((Measure.measurable_coe hS).comp hN) ((Measure.measurable_coe hS).comp hK) ?_ E hE
+  intro A hA
+  have he := congrArg (fun μ : Measure upperHalfPlane => μ S) (hmean A hA)
+  simpa only [Measure.bind_apply hS hNB.aemeasurable,
+    Measure.bind_apply hS hKB.aemeasurable] using he
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/DriverPastFactorization.lean b/lean/OAI/Probability/SLE/Stochastic/DriverPastFactorization.lean
index b10ba0b89d147d0212dfeb7f9de000892f1f5fca..30108290b0413cd3edb8884ac6630fa01bae08e4 100644
--- a/lean/OAI/Probability/SLE/Stochastic/DriverPastFactorization.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/DriverPastFactorization.lean
@@ -1,80 +1,80 @@
-import OAI.Probability.SLE.Mass.RadonFactorization
-import OAI.Probability.SLE.StoppedMeasures
-import OAI.Probability.SLE.Mass.RationalTraceMassAdapted
-
-namespace OAI
-
-/-! Real continuous driver prefixes realize the natural filtration.
-Radon kernels adapted to that filtration factor through each stopped prefix. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal CompactlySupported
-
-namespace SLEExactGauge
-
-local instance driverPastPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
-local instance driverPastPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
-local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
-
-def driverPathStopped (t : ℝ≥0) (p : C(ℝ≥0,ℝ)) : C(ℝ≥0,ℝ) :=
-  ⟨fun s => p (min s t),p.continuous.comp (continuous_id.min continuous_const)⟩
-
-def driverPathFiltration : Filtration ℝ≥0 (borel C(ℝ≥0,ℝ)) :=
-  Filtration.natural (fun t (p : C(ℝ≥0,ℝ)) => p t)
-    (fun t => (continuous_eval_const t).measurable.stronglyMeasurable)
-
-lemma driverPathStopped_eq_of_agree (T t : ℝ≥0) (ht : t ≤ T)
-    (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ T, p s=q s) :
-    driverPathStopped t p = driverPathStopped t q := by
-  ext s
-  exact hpq (min s t) ((min_le_right s t).trans ht)
-
-lemma driverPathStopped_measurable_past (t : ℝ≥0) :
-    Measurable[driverPathFiltration t] (driverPathStopped t) := by
-  have he (s : ℝ≥0) : Measurable[driverPathFiltration t]
-      (fun p : C(ℝ≥0,ℝ) => p (min s t)) := by
-    have hm (r : ℝ≥0) : StronglyMeasurable (fun p : C(ℝ≥0,ℝ) => p r) :=
-      (continuous_eval_const r).measurable.stronglyMeasurable
-    have h := (Filtration.stronglyAdapted_natural hm) (min s t)
-    exact (h.mono (driverPathFiltration.mono (min_le_right s t))).measurable
-  exact @source_path_measurable C(ℝ≥0,ℝ) ℝ (driverPathFiltration t)
-    _ _ _ _ (borel _) driverPastPathBorel _ (driverPathStopped t) he
-
-lemma natural_le_stoppedDriverPath_comap
-    {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) (t : ℝ≥0) :
-    (Filtration.natural B (fun r => (hm r).stronglyMeasurable)) t ≤
-      MeasurableSpace.comap (fun ω => driverPathStopped t (sourceDriverPath B hc ω)) (borel _) := by
-  have hb (s : ℝ≥0) (hs : s ≤ t) :
-      Measurable[MeasurableSpace.comap (fun ω => driverPathStopped t (sourceDriverPath B hc ω))
-        (borel C(ℝ≥0,ℝ))] (B s) := by
-    have h := (continuous_eval_const s).measurable.comp
-      (Measurable.of_comap_le (le_refl (MeasurableSpace.comap
-        (fun ω => driverPathStopped t (sourceDriverPath B hc ω)) (borel C(ℝ≥0,ℝ)))))
-    convert h using 1
-    funext ω
-    simp only [Function.comp_def,driverPathStopped,sourceDriverPath,ContinuousMap.coe_mk,min_eq_left hs]
-  change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ _
-  exact iSup_le fun s => iSup_le fun hs => measurable_iff_comap_le.mp (hb s hs)
-
-lemma exists_stopped_driver_radon_factor
-    {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (t : ℝ≥0) (N : Ω → Measure upperHalfPlane) (hNr : ∀ ω, (N ω).Regular)
-    (hNm : Measurable[(Filtration.natural B (fun r => (hm r).stronglyMeasurable)) t] N) :
-    ∃ R : C(ℝ≥0,ℝ) → Measure upperHalfPlane, Measurable R ∧
-      (∀ p, (R p).Regular) ∧ ∀ ω, R (driverPathStopped t (sourceDriverPath B hc ω)) = N ω := by
-  apply (localTestFamily upperHalfPlane).measurable_radon_factor
-    (fun ω => driverPathStopped t (sourceDriverPath B hc ω)) N hNr
-  intro j
-  have ht := @measurable_halfPlane_compact_test_integral Ω
-    ((Filtration.natural B (fun r => (hm r).stronglyMeasurable)) t)
-    N hNm hNr ((localTestFamily upperHalfPlane).test j)
-  exact ht.mono (natural_le_stoppedDriverPath_comap B hm hc t) le_rfl
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Mass.RadonFactorization
+import OAI.Probability.SLE.StoppedMeasures
+import OAI.Probability.SLE.Mass.RationalTraceMassAdapted
+
+namespace OAI
+
+/-! Real continuous driver prefixes realize the natural filtration.
+Radon kernels adapted to that filtration factor through each stopped prefix. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal CompactlySupported
+
+namespace SLEExactGauge
+
+local instance driverPastPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _
+local instance driverPastPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩
+local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace
+
+def driverPathStopped (t : ℝ≥0) (p : C(ℝ≥0,ℝ)) : C(ℝ≥0,ℝ) :=
+  ⟨fun s => p (min s t),p.continuous.comp (continuous_id.min continuous_const)⟩
+
+def driverPathFiltration : Filtration ℝ≥0 (borel C(ℝ≥0,ℝ)) :=
+  Filtration.natural (fun t (p : C(ℝ≥0,ℝ)) => p t)
+    (fun t => (continuous_eval_const t).measurable.stronglyMeasurable)
+
+lemma driverPathStopped_eq_of_agree (T t : ℝ≥0) (ht : t ≤ T)
+    (p q : C(ℝ≥0,ℝ)) (hpq : ∀ s ≤ T, p s=q s) :
+    driverPathStopped t p = driverPathStopped t q := by
+  ext s
+  exact hpq (min s t) ((min_le_right s t).trans ht)
+
+lemma driverPathStopped_measurable_past (t : ℝ≥0) :
+    Measurable[driverPathFiltration t] (driverPathStopped t) := by
+  have he (s : ℝ≥0) : Measurable[driverPathFiltration t]
+      (fun p : C(ℝ≥0,ℝ) => p (min s t)) := by
+    have hm (r : ℝ≥0) : StronglyMeasurable (fun p : C(ℝ≥0,ℝ) => p r) :=
+      (continuous_eval_const r).measurable.stronglyMeasurable
+    have h := (Filtration.stronglyAdapted_natural hm) (min s t)
+    exact (h.mono (driverPathFiltration.mono (min_le_right s t))).measurable
+  exact @source_path_measurable C(ℝ≥0,ℝ) ℝ (driverPathFiltration t)
+    _ _ _ _ (borel _) driverPastPathBorel _ (driverPathStopped t) he
+
+lemma natural_le_stoppedDriverPath_comap
+    {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) (t : ℝ≥0) :
+    (Filtration.natural B (fun r => (hm r).stronglyMeasurable)) t ≤
+      MeasurableSpace.comap (fun ω => driverPathStopped t (sourceDriverPath B hc ω)) (borel _) := by
+  have hb (s : ℝ≥0) (hs : s ≤ t) :
+      Measurable[MeasurableSpace.comap (fun ω => driverPathStopped t (sourceDriverPath B hc ω))
+        (borel C(ℝ≥0,ℝ))] (B s) := by
+    have h := (continuous_eval_const s).measurable.comp
+      (Measurable.of_comap_le (le_refl (MeasurableSpace.comap
+        (fun ω => driverPathStopped t (sourceDriverPath B hc ω)) (borel C(ℝ≥0,ℝ)))))
+    convert h using 1
+    funext ω
+    simp only [Function.comp_def,driverPathStopped,sourceDriverPath,ContinuousMap.coe_mk,min_eq_left hs]
+  change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ _
+  exact iSup_le fun s => iSup_le fun hs => measurable_iff_comap_le.mp (hb s hs)
+
+lemma exists_stopped_driver_radon_factor
+    {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (t : ℝ≥0) (N : Ω → Measure upperHalfPlane) (hNr : ∀ ω, (N ω).Regular)
+    (hNm : Measurable[(Filtration.natural B (fun r => (hm r).stronglyMeasurable)) t] N) :
+    ∃ R : C(ℝ≥0,ℝ) → Measure upperHalfPlane, Measurable R ∧
+      (∀ p, (R p).Regular) ∧ ∀ ω, R (driverPathStopped t (sourceDriverPath B hc ω)) = N ω := by
+  apply (localTestFamily upperHalfPlane).measurable_radon_factor
+    (fun ω => driverPathStopped t (sourceDriverPath B hc ω)) N hNr
+  intro j
+  have ht := @measurable_halfPlane_compact_test_integral Ω
+    ((Filtration.natural B (fun r => (hm r).stronglyMeasurable)) t)
+    N hNm hNr ((localTestFamily upperHalfPlane).test j)
+  exact ht.mono (natural_le_stoppedDriverPath_comap B hm hc t) le_rfl
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/DrivingEndpointUniqueness.lean b/lean/OAI/Probability/SLE/Stochastic/DrivingEndpointUniqueness.lean
index 0d373ceac9d9a7805a7cdd85ca35efaa5a7d4a58..ba6f415376cdd2c41dcaea0976e1fff2084a7d99 100644
--- a/lean/OAI/Probability/SLE/Stochastic/DrivingEndpointUniqueness.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/DrivingEndpointUniqueness.lean
@@ -1,72 +1,72 @@
-import OAI.Probability.SLE.Geometry.ShrinkingPhysicalSegments
-import OAI.Probability.SLE.Loewner.SlitBoundaryFiber
-
-namespace OAI
-
-/-! The two ends of a shrinking mapped future cannot have different
-real limits: a separating real interval would collapse in the inverse. -/
-noncomputable section
-open Set Filter Function Complex Metric MeasureTheory
-open scoped Topology
-
-namespace SLEExactGauge
-
-theorem initialDriving_shrinking_endpoint_unique (p : C(unitInterval,ℂ)) (hi : Injective p)
-    (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane)
-    {R : ℝ} (hR : 0<R) (hp : ∀ t : unitInterval, ‖p t‖<R)
-    {s : unitInterval} (hs : s≠0) {t r : ℕ → unitInterval}
-    (ht : Tendsto t atTop (𝓝 s)) (hr : Tendsto r atTop (𝓝 s))
-    (htr : ∀ n, t n<r n) (hr1 : ∀ n, r n<1) {a b : ℝ}
-    (ha : Tendsto (fun n => initialDrivingValue p hi h0 hH (t n) ((htr n).trans (hr1 n)))
-      atTop (𝓝 a))
-    (hb : Tendsto (fun n => initialForwardChart p hi h0 hH (t n) (p (r n)))
-      atTop (𝓝 (b:ℂ))) : a=b := by
-  by_contra hab
-  have hb' := Complex.continuous_re.continuousAt.tendsto.comp hb
-  simp only [ofReal_re] at hb'
-  have hex : ∃ A B : ℝ, A<B ∧ ∀ᶠ n in atTop, ∀ x∈Ioo A B,
-      x∈Ioo (initialDrivingValue p hi h0 hH (t n) ((htr n).trans (hr1 n)))
-        (initialForwardChart p hi h0 hH (t n) (p (r n))).re ∨
-      x∈Ioo (initialForwardChart p hi h0 hH (t n) (p (r n))).re
-        (initialDrivingValue p hi h0 hH (t n) ((htr n).trans (hr1 n))) := by
-    rcases lt_or_gt_of_ne hab with hab | hba
-    · refine ⟨(2*a+b)/3,(a+2*b)/3,by linarith,?_⟩
-      filter_upwards [ha.eventually (gt_mem_nhds (show a<(2*a+b)/3 by linarith)),
-        hb'.eventually (lt_mem_nhds (show (a+2*b)/3<b by linarith))] with n hn hm
-      intro x hx
-      exact Or.inl ⟨hn.trans hx.1,hx.2.trans hm⟩
-    · refine ⟨(2*b+a)/3,(b+2*a)/3,by linarith,?_⟩
-      filter_upwards [hb'.eventually (gt_mem_nhds (show b<(2*b+a)/3 by linarith)),
-        ha.eventually (lt_mem_nhds (show (b+2*a)/3<a by linarith))] with n hn hm
-      intro x hx
-      exact Or.inr ⟨hn.trans hx.1,hx.2.trans hm⟩
-  obtain ⟨A,B,hAB,hcross⟩ := hex
-  obtain ⟨ε,hε,hbound⟩ := shrinking_physical_segment_bound p ht hr (fun n => (htr n).le)
-  let η := fun n => Real.sqrt (2*(initialCapacity p hi h0 hH (r n)-
-    initialCapacity p hi h0 hH (t n)))
-  have hη : Tendsto η atTop (𝓝 0) := by
-    have hcap := ((initialCapacity_continuous p hi h0 hH).tendsto s).comp hr
-    have hcap' := ((initialCapacity_continuous p hi h0 hH).tendsto s).comp ht
-    simpa only [sub_self,mul_zero,Real.sqrt_zero,Function.comp_def,η] using
-      (hcap.sub hcap').const_mul 2 |>.sqrt
-  obtain ⟨Fbar,hFbar,hFe⟩ := (initialSlitChart p hi h0 hH s hs).exists_boundary_extension
-  have hFi : ∀ z∈upperHalfPlane, Fbar z=initialInverseChart p hi h0 hH s z := by
-    intro z hz
-    simpa only [initialInverseChart,dite_eq_right hs] using hFe z hz
-  have hlow : ∀ᶠ n in atTop, ∀ x∈Ioo A B, ∃ y∈Ioc 0 (η n),
-      ‖initialInverseChart p hi h0 hH (t n) ((x:ℂ)+(y:ℂ)*I)-p s‖≤ε n := by
-    filter_upwards [hcross] with n hn
-    intro x hx
-    exact mappedFutureArc_thin_crossing p hi h0 hH (htr n) (hr1 n) (p s) (hn x hx)
-      (fun v hv => hbound n v ⟨hv.1.le,hv.2⟩)
-  have hconst := initialInverse_thin_crossing_collapse p hi h0 hH hR hp ht hFbar hFi hε hη hlow
-  apply (initialSlitChart p hi h0 hH s hs).boundary_not_constant_on_interval
-    hFbar hFe ((A+B)/2) (show 0<(B-A)/2 by linarith) (p s)
-  intro x hx
-  exact hconst x ⟨by linarith [hx.1],by linarith [hx.2]⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.ShrinkingPhysicalSegments
+import OAI.Probability.SLE.Loewner.SlitBoundaryFiber
+
+namespace OAI
+
+/-! The two ends of a shrinking mapped future cannot have different
+real limits: a separating real interval would collapse in the inverse. -/
+noncomputable section
+open Set Filter Function Complex Metric MeasureTheory
+open scoped Topology
+
+namespace SLEExactGauge
+
+theorem initialDriving_shrinking_endpoint_unique (p : C(unitInterval,ℂ)) (hi : Injective p)
+    (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane)
+    {R : ℝ} (hR : 0<R) (hp : ∀ t : unitInterval, ‖p t‖<R)
+    {s : unitInterval} (hs : s≠0) {t r : ℕ → unitInterval}
+    (ht : Tendsto t atTop (𝓝 s)) (hr : Tendsto r atTop (𝓝 s))
+    (htr : ∀ n, t n<r n) (hr1 : ∀ n, r n<1) {a b : ℝ}
+    (ha : Tendsto (fun n => initialDrivingValue p hi h0 hH (t n) ((htr n).trans (hr1 n)))
+      atTop (𝓝 a))
+    (hb : Tendsto (fun n => initialForwardChart p hi h0 hH (t n) (p (r n)))
+      atTop (𝓝 (b:ℂ))) : a=b := by
+  by_contra hab
+  have hb' := Complex.continuous_re.continuousAt.tendsto.comp hb
+  simp only [ofReal_re] at hb'
+  have hex : ∃ A B : ℝ, A<B ∧ ∀ᶠ n in atTop, ∀ x∈Ioo A B,
+      x∈Ioo (initialDrivingValue p hi h0 hH (t n) ((htr n).trans (hr1 n)))
+        (initialForwardChart p hi h0 hH (t n) (p (r n))).re ∨
+      x∈Ioo (initialForwardChart p hi h0 hH (t n) (p (r n))).re
+        (initialDrivingValue p hi h0 hH (t n) ((htr n).trans (hr1 n))) := by
+    rcases lt_or_gt_of_ne hab with hab | hba
+    · refine ⟨(2*a+b)/3,(a+2*b)/3,by linarith,?_⟩
+      filter_upwards [ha.eventually (gt_mem_nhds (show a<(2*a+b)/3 by linarith)),
+        hb'.eventually (lt_mem_nhds (show (a+2*b)/3<b by linarith))] with n hn hm
+      intro x hx
+      exact Or.inl ⟨hn.trans hx.1,hx.2.trans hm⟩
+    · refine ⟨(2*b+a)/3,(b+2*a)/3,by linarith,?_⟩
+      filter_upwards [hb'.eventually (gt_mem_nhds (show b<(2*b+a)/3 by linarith)),
+        ha.eventually (lt_mem_nhds (show (b+2*a)/3<a by linarith))] with n hn hm
+      intro x hx
+      exact Or.inr ⟨hn.trans hx.1,hx.2.trans hm⟩
+  obtain ⟨A,B,hAB,hcross⟩ := hex
+  obtain ⟨ε,hε,hbound⟩ := shrinking_physical_segment_bound p ht hr (fun n => (htr n).le)
+  let η := fun n => Real.sqrt (2*(initialCapacity p hi h0 hH (r n)-
+    initialCapacity p hi h0 hH (t n)))
+  have hη : Tendsto η atTop (𝓝 0) := by
+    have hcap := ((initialCapacity_continuous p hi h0 hH).tendsto s).comp hr
+    have hcap' := ((initialCapacity_continuous p hi h0 hH).tendsto s).comp ht
+    simpa only [sub_self,mul_zero,Real.sqrt_zero,Function.comp_def,η] using
+      (hcap.sub hcap').const_mul 2 |>.sqrt
+  obtain ⟨Fbar,hFbar,hFe⟩ := (initialSlitChart p hi h0 hH s hs).exists_boundary_extension
+  have hFi : ∀ z∈upperHalfPlane, Fbar z=initialInverseChart p hi h0 hH s z := by
+    intro z hz
+    simpa only [initialInverseChart,dite_eq_right hs] using hFe z hz
+  have hlow : ∀ᶠ n in atTop, ∀ x∈Ioo A B, ∃ y∈Ioc 0 (η n),
+      ‖initialInverseChart p hi h0 hH (t n) ((x:ℂ)+(y:ℂ)*I)-p s‖≤ε n := by
+    filter_upwards [hcross] with n hn
+    intro x hx
+    exact mappedFutureArc_thin_crossing p hi h0 hH (htr n) (hr1 n) (p s) (hn x hx)
+      (fun v hv => hbound n v ⟨hv.1.le,hv.2⟩)
+  have hconst := initialInverse_thin_crossing_collapse p hi h0 hH hR hp ht hFbar hFi hε hη hlow
+  apply (initialSlitChart p hi h0 hH s hs).boundary_not_constant_on_interval
+    hFbar hFe ((A+B)/2) (show 0<(B-A)/2 by linarith) (p s)
+  intro x hx
+  exact hconst x ⟨by linarith [hx.1],by linarith [hx.2]⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/DrivingLeftCluster.lean b/lean/OAI/Probability/SLE/Stochastic/DrivingLeftCluster.lean
index 7e4dd10f5ed661248c68ec34e0ed2a9643dac183..3c67ec53f6d6a1946eec38da1e17bc80a3fd9656 100644
--- a/lean/OAI/Probability/SLE/Stochastic/DrivingLeftCluster.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/DrivingLeftCluster.lean
@@ -1,70 +1,70 @@
-import OAI.Probability.SLE.Geometry.DiagonalComplexLimit
-import OAI.Probability.SLE.Loewner.InitialDrivingRightContinuity
-import OAI.Probability.SLE.Loewner.InitialForwardTimeContinuity
-
-namespace OAI
-
-/-! Every left-hand cluster of the driver is the exact tip value. -/
-noncomputable section
-open Set Filter Function Complex Metric
-open scoped Topology
-
-namespace SLEExactGauge
-
-theorem initialDriving_left_cluster (p : C(unitInterval,ℂ)) (hi : Injective p)
-    (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane)
-    {R : ℝ} (hR : 0<R) (hp : ∀ t : unitInterval, ‖p t‖<R)
-    {s : unitInterval} (hs0 : s≠0) (hs : s<1)
-    {t : ℕ → unitInterval} (hts : ∀ n, t n<s) (ht : Tendsto t atTop (𝓝 s))
-    {a : ℝ} (ha : Tendsto (fun n => initialDriving p hi h0 hH (t n)) atTop (𝓝 a)) :
-    a=initialDriving p hi h0 hH s := by
-  let D : ℝ := (1-(s:ℝ))/2
-  have hD : 0<D := half_pos (sub_pos.mpr hs)
-  let d := fun n : ℕ => D*(1/((n:ℝ)+1))
-  have hd (n : ℕ) : d n∈Ioo 0 (1-(s:ℝ)) := by
-    have hn : (0:ℝ)<(n:ℝ)+1 := by positivity
-    have hb : 1/((n:ℝ)+1)≤1 := (div_le_one hn).mpr (by linarith [Nat.cast_nonneg (α := ℝ) n])
-    refine ⟨mul_pos hD (one_div_pos.mpr hn),?_⟩
-    have hle := mul_le_mul_of_nonneg_left hb hD.le
-    dsimp only [D] at hD hle
-    dsimp only [d,D]
-    linarith
-  let q := fun n => slitFutureParameter s (d n)
-  have hsq (n : ℕ) : s<q n := by
-    change (s:ℝ)<(slitFutureParameter s (d n):ℝ)
-    rw [slitFutureParameter_coe s (hd n)]
-    linarith [(hd n).1]
-  have hq1 (n : ℕ) : q n<1 := by
-    change (slitFutureParameter s (d n):ℝ)<1
-    rw [slitFutureParameter_coe s (hd n)]
-    linarith [(hd n).2]
-  have hdlim : Tendsto d atTop (𝓝 0) := by
-    simpa only [mul_zero,d] using tendsto_one_div_add_atTop_nhds_zero_nat.const_mul D
-  have hqlim : Tendsto q atTop (𝓝 s) := by
-    simpa only [slitFutureParameter_zero,Function.comp_def,q] using
-      ((slitFutureParameter_continuous s).tendsto 0).comp hdlim
-  have hg : Tendsto (fun n => initialForwardChart p hi h0 hH s (p (q n)))
-      atTop (𝓝 (initialDrivingValue p hi h0 hH s hs:ℂ)) :=
-    (initialDrivingValue_spec p hi h0 hH s hs).comp
-      (tendsto_nhdsWithin_iff.mpr ⟨hdlim,Eventually.of_forall fun n => (hd n).1⟩)
-  have hf (k : ℕ) : Tendsto (fun n => initialForwardChart p hi h0 hH (t n) (p (q k)))
-      atTop (𝓝 (initialForwardChart p hi h0 hH s (p (q k)))) :=
-    initialForwardChart_tendsto p hi h0 hH ht tendsto_const_nhds
-      (point_after_prefix_mem_domain p hi hH (hsq k))
-      (fun n => point_after_prefix_mem_domain p hi hH ((hts n).trans (hsq k)))
-  obtain ⟨φ,hφ,hφlim⟩ := diagonal_complex_limit hf hg
-  have hφtop : Tendsto φ atTop atTop := tendsto_atTop_mono hφ tendsto_id
-  have ha' : Tendsto (fun n => initialDrivingValue p hi h0 hH (t (φ n))
-      (((hts (φ n)).trans (hsq n)).trans (hq1 n))) atTop (𝓝 a) := by
-    convert ha.comp hφtop using 1
-    funext n
-    exact (initialDriving_of_lt p hi h0 hH ((hts (φ n)).trans hs)).symm
-  have he := initialDriving_shrinking_endpoint_unique p hi h0 hH hR hp hs0
-    (ht.comp hφtop) hqlim (fun n => (hts (φ n)).trans (hsq n)) hq1 ha' hφlim
-  exact he.trans (initialDriving_of_lt p hi h0 hH hs).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.DiagonalComplexLimit
+import OAI.Probability.SLE.Loewner.InitialDrivingRightContinuity
+import OAI.Probability.SLE.Loewner.InitialForwardTimeContinuity
+
+namespace OAI
+
+/-! Every left-hand cluster of the driver is the exact tip value. -/
+noncomputable section
+open Set Filter Function Complex Metric
+open scoped Topology
+
+namespace SLEExactGauge
+
+theorem initialDriving_left_cluster (p : C(unitInterval,ℂ)) (hi : Injective p)
+    (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane)
+    {R : ℝ} (hR : 0<R) (hp : ∀ t : unitInterval, ‖p t‖<R)
+    {s : unitInterval} (hs0 : s≠0) (hs : s<1)
+    {t : ℕ → unitInterval} (hts : ∀ n, t n<s) (ht : Tendsto t atTop (𝓝 s))
+    {a : ℝ} (ha : Tendsto (fun n => initialDriving p hi h0 hH (t n)) atTop (𝓝 a)) :
+    a=initialDriving p hi h0 hH s := by
+  let D : ℝ := (1-(s:ℝ))/2
+  have hD : 0<D := half_pos (sub_pos.mpr hs)
+  let d := fun n : ℕ => D*(1/((n:ℝ)+1))
+  have hd (n : ℕ) : d n∈Ioo 0 (1-(s:ℝ)) := by
+    have hn : (0:ℝ)<(n:ℝ)+1 := by positivity
+    have hb : 1/((n:ℝ)+1)≤1 := (div_le_one hn).mpr (by linarith [Nat.cast_nonneg (α := ℝ) n])
+    refine ⟨mul_pos hD (one_div_pos.mpr hn),?_⟩
+    have hle := mul_le_mul_of_nonneg_left hb hD.le
+    dsimp only [D] at hD hle
+    dsimp only [d,D]
+    linarith
+  let q := fun n => slitFutureParameter s (d n)
+  have hsq (n : ℕ) : s<q n := by
+    change (s:ℝ)<(slitFutureParameter s (d n):ℝ)
+    rw [slitFutureParameter_coe s (hd n)]
+    linarith [(hd n).1]
+  have hq1 (n : ℕ) : q n<1 := by
+    change (slitFutureParameter s (d n):ℝ)<1
+    rw [slitFutureParameter_coe s (hd n)]
+    linarith [(hd n).2]
+  have hdlim : Tendsto d atTop (𝓝 0) := by
+    simpa only [mul_zero,d] using tendsto_one_div_add_atTop_nhds_zero_nat.const_mul D
+  have hqlim : Tendsto q atTop (𝓝 s) := by
+    simpa only [slitFutureParameter_zero,Function.comp_def,q] using
+      ((slitFutureParameter_continuous s).tendsto 0).comp hdlim
+  have hg : Tendsto (fun n => initialForwardChart p hi h0 hH s (p (q n)))
+      atTop (𝓝 (initialDrivingValue p hi h0 hH s hs:ℂ)) :=
+    (initialDrivingValue_spec p hi h0 hH s hs).comp
+      (tendsto_nhdsWithin_iff.mpr ⟨hdlim,Eventually.of_forall fun n => (hd n).1⟩)
+  have hf (k : ℕ) : Tendsto (fun n => initialForwardChart p hi h0 hH (t n) (p (q k)))
+      atTop (𝓝 (initialForwardChart p hi h0 hH s (p (q k)))) :=
+    initialForwardChart_tendsto p hi h0 hH ht tendsto_const_nhds
+      (point_after_prefix_mem_domain p hi hH (hsq k))
+      (fun n => point_after_prefix_mem_domain p hi hH ((hts n).trans (hsq k)))
+  obtain ⟨φ,hφ,hφlim⟩ := diagonal_complex_limit hf hg
+  have hφtop : Tendsto φ atTop atTop := tendsto_atTop_mono hφ tendsto_id
+  have ha' : Tendsto (fun n => initialDrivingValue p hi h0 hH (t (φ n))
+      (((hts (φ n)).trans (hsq n)).trans (hq1 n))) atTop (𝓝 a) := by
+    convert ha.comp hφtop using 1
+    funext n
+    exact (initialDriving_of_lt p hi h0 hH ((hts (φ n)).trans hs)).symm
+  have he := initialDriving_shrinking_endpoint_unique p hi h0 hH hR hp hs0
+    (ht.comp hφtop) hqlim (fun n => (hts (φ n)).trans (hsq n)) hq1 ha' hφlim
+  exact he.trans (initialDriving_of_lt p hi h0 hH hs).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/GaussianTaylorMean.lean b/lean/OAI/Probability/SLE/Stochastic/GaussianTaylorMean.lean
index 81ef4f694e9bfa2a7a18522dd6d5ff4d72aeeab3..14963d754d1ada87466cd0df2e3f641b1b4937c9 100644
--- a/lean/OAI/Probability/SLE/Stochastic/GaussianTaylorMean.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/GaussianTaylorMean.lean
@@ -1,66 +1,66 @@
-import OAI.Probability.SLE.BrownianGenerators
-
-namespace OAI
-
-/-! A scalar Gaussian Taylor estimate with finite derivative bounds for
-the planar Brownian harmonic-mean argument. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma gaussian_second_taylor_integral
-    (f : ℝ → ℝ) (hf : ContDiff ℝ 3 f) (K : ℝ≥0)
-    (hbound : ∀ x j, j ≤ 3 → |iteratedDeriv j f x| ≤ K)
-    (δ : ℝ≥0) (x : ℝ) :
-    |(∫ y, f (x+y) ∂gaussianReal 0 δ)-f x-
-      (1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 f x| ≤
-      (K:ℝ)*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 3 := by
-  let μ := gaussianReal 0 δ
-  let : IsProbabilityMeasure μ := by dsimp [μ]; infer_instance
-  have hμ : HasLaw (fun y : ℝ => y) μ μ := ⟨measurable_id.aemeasurable, by simp⟩
-  have hi : Integrable (fun y => f (x+y)) μ := by
-    apply Integrable.of_bound (hf.continuous.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
-    filter_upwards [] with y
-    simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs,iteratedDeriv_zero]
-      using hbound (x+y) 0 (by decide)
-  have hyi : Integrable (fun y : ℝ => y) μ := hμ.hasGaussianLaw.integrable
-  have hysi : Integrable (fun y : ℝ => y^2) μ := hμ.hasGaussianLaw.memLp_two.integrable_sq
-  have hlin : (∫ y : ℝ, deriv f x*y ∂μ)=0 := by
-    rw [integral_const_mul]
-    simp [μ]
-  have hquad : (∫ y : ℝ, (1/2:ℝ)*iteratedDeriv 2 f x*y^2 ∂μ) =
-      (1/2:ℝ)*iteratedDeriv 2 f x*(δ:ℝ) := by
-    rw [integral_const_mul,gaussian_square_integral]
-  let R (y : ℝ) := f (x+y)-f x-deriv f x*y-(1/2:ℝ)*iteratedDeriv 2 f x*y^2
-  have hRi : Integrable R μ :=
-    ((hi.sub (integrable_const _)).sub (hyi.const_mul _)).sub (hysi.const_mul _)
-  have hR : ∀ y, |R y| ≤ (K:ℝ)*|y|^3 :=
-    fun y => real_second_taylor_bound hf K (fun z => hbound z 3 (by decide)) x y
-  have he : (∫ y, R y ∂μ) = (∫ y, f (x+y) ∂μ)-f x-
-      (1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 f x := by
-    have hci : Integrable (fun _ : ℝ => f x) μ := integrable_const _
-    have hli : Integrable (fun y => deriv f x*y) μ := hyi.const_mul _
-    have hqi : Integrable (fun y => (1/2:ℝ)*iteratedDeriv 2 f x*y^2) μ := hysi.const_mul _
-    have h0i : Integrable (fun y => f (x+y)-f x) μ := hi.sub hci
-    have h1i : Integrable (fun y => f (x+y)-f x-deriv f x*y) μ := h0i.sub hli
-    dsimp only [R]
-    rw [integral_sub h1i hqi,integral_sub h0i hli,integral_sub hi hci,hlin,hquad]
-    simp only [integral_const,probReal_univ,one_smul,sub_zero]
-    ring
-  rw [← he]
-  calc
-    |∫ y, R y ∂μ| ≤ ∫ y, |R y| ∂μ := by
-      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := μ) R
-    _ ≤ ∫ y, (K:ℝ)*|y|^3 ∂μ := integral_mono hRi.abs
-      ((gaussian_abs_moment_integrable hμ 3 (by decide)).const_mul _) hR
-    _ = (K:ℝ)*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 3 := by
-      rw [integral_const_mul,gaussian_abs_cube hμ]
-      ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.BrownianGenerators
+
+namespace OAI
+
+/-! A scalar Gaussian Taylor estimate with finite derivative bounds for
+the planar Brownian harmonic-mean argument. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma gaussian_second_taylor_integral
+    (f : ℝ → ℝ) (hf : ContDiff ℝ 3 f) (K : ℝ≥0)
+    (hbound : ∀ x j, j ≤ 3 → |iteratedDeriv j f x| ≤ K)
+    (δ : ℝ≥0) (x : ℝ) :
+    |(∫ y, f (x+y) ∂gaussianReal 0 δ)-f x-
+      (1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 f x| ≤
+      (K:ℝ)*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 3 := by
+  let μ := gaussianReal 0 δ
+  let : IsProbabilityMeasure μ := by dsimp [μ]; infer_instance
+  have hμ : HasLaw (fun y : ℝ => y) μ μ := ⟨measurable_id.aemeasurable, by simp⟩
+  have hi : Integrable (fun y => f (x+y)) μ := by
+    apply Integrable.of_bound (hf.continuous.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
+    filter_upwards [] with y
+    simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs,iteratedDeriv_zero]
+      using hbound (x+y) 0 (by decide)
+  have hyi : Integrable (fun y : ℝ => y) μ := hμ.hasGaussianLaw.integrable
+  have hysi : Integrable (fun y : ℝ => y^2) μ := hμ.hasGaussianLaw.memLp_two.integrable_sq
+  have hlin : (∫ y : ℝ, deriv f x*y ∂μ)=0 := by
+    rw [integral_const_mul]
+    simp [μ]
+  have hquad : (∫ y : ℝ, (1/2:ℝ)*iteratedDeriv 2 f x*y^2 ∂μ) =
+      (1/2:ℝ)*iteratedDeriv 2 f x*(δ:ℝ) := by
+    rw [integral_const_mul,gaussian_square_integral]
+  let R (y : ℝ) := f (x+y)-f x-deriv f x*y-(1/2:ℝ)*iteratedDeriv 2 f x*y^2
+  have hRi : Integrable R μ :=
+    ((hi.sub (integrable_const _)).sub (hyi.const_mul _)).sub (hysi.const_mul _)
+  have hR : ∀ y, |R y| ≤ (K:ℝ)*|y|^3 :=
+    fun y => real_second_taylor_bound hf K (fun z => hbound z 3 (by decide)) x y
+  have he : (∫ y, R y ∂μ) = (∫ y, f (x+y) ∂μ)-f x-
+      (1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 f x := by
+    have hci : Integrable (fun _ : ℝ => f x) μ := integrable_const _
+    have hli : Integrable (fun y => deriv f x*y) μ := hyi.const_mul _
+    have hqi : Integrable (fun y => (1/2:ℝ)*iteratedDeriv 2 f x*y^2) μ := hysi.const_mul _
+    have h0i : Integrable (fun y => f (x+y)-f x) μ := hi.sub hci
+    have h1i : Integrable (fun y => f (x+y)-f x-deriv f x*y) μ := h0i.sub hli
+    dsimp only [R]
+    rw [integral_sub h1i hqi,integral_sub h0i hli,integral_sub hi hci,hlin,hquad]
+    simp only [integral_const,probReal_univ,one_smul,sub_zero]
+    ring
+  rw [← he]
+  calc
+    |∫ y, R y ∂μ| ≤ ∫ y, |R y| ∂μ := by
+      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (μ := μ) R
+    _ ≤ ∫ y, (K:ℝ)*|y|^3 ∂μ := integral_mono hRi.abs
+      ((gaussian_abs_moment_integrable hμ 3 (by decide)).const_mul _) hR
+    _ = (K:ℝ)*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 3 := by
+      rw [integral_const_mul,gaussian_abs_cube hμ]
+      ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/IndependentPlanarIncrements.lean b/lean/OAI/Probability/SLE/Stochastic/IndependentPlanarIncrements.lean
index e59fd37305558e64d7a2433e0ad59e44ffcf2d2c..db8372b9a94326c8d12de9f35bec264aef1c2760 100644
--- a/lean/OAI/Probability/SLE/Stochastic/IndependentPlanarIncrements.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/IndependentPlanarIncrements.lean
@@ -1,66 +1,66 @@
-import OAI.Probability.SLE.BrownianGenerators
-import OAI.Probability.SLE.Geometry.PublishedWienerSupport
-
-namespace OAI
-
-/-! Independence of the two-coordinate Wiener increment from its full
-original past. Product coordinates retain both original pasts. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-private lemma measurePreserving_shuffle
-    {A B C D : Type*} [MeasurableSpace A] [MeasurableSpace B]
-    [MeasurableSpace C] [MeasurableSpace D]
-    (μ : Measure A) (ν : Measure B) (ρ : Measure C) (τ : Measure D)
-    [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
-    [IsProbabilityMeasure ρ] [IsProbabilityMeasure τ] :
-    MeasurePreserving (fun p : (A×B)×(C×D) => ((p.1.1,p.2.1),(p.1.2,p.2.2)))
-      ((μ.prod ν).prod (ρ.prod τ)) ((μ.prod ρ).prod (ν.prod τ)) := by
-  have h1 := measurePreserving_prodAssoc μ ν (ρ.prod τ)
-  have h2 := (MeasurePreserving.id μ).prod (measurePreserving_prodAssoc ν ρ τ).symm
-  have h3 := (MeasurePreserving.id μ).prod
-    ((Measure.measurePreserving_swap (μ := ν) (ν := ρ)).prod (MeasurePreserving.id τ))
-  have h4 := (MeasurePreserving.id μ).prod (measurePreserving_prodAssoc ρ ν τ)
-  have h5 := (measurePreserving_prodAssoc μ ρ (ν.prod τ)).symm
-  exact h5.comp (h4.comp (h3.comp (h2.comp h1)))
-
-lemma independent_product_pairs
-    {Ω Ω' A B C D : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']
-    [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] [MeasurableSpace D]
-    {μ : Measure Ω} {ν : Measure Ω'} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
-    {X : Ω → A} {Y : Ω → B} {X' : Ω' → C} {Y' : Ω' → D}
-    (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')
-    (hXY : IndepFun X Y μ) (hXY' : IndepFun X' Y' ν) :
-    IndepFun (fun p : Ω×Ω' => (X p.1,X' p.2))
-      (fun p : Ω×Ω' => (Y p.1,Y' p.2)) (μ.prod ν) := by
-  have hx : HasLaw X (μ.map X) μ := ⟨hX.aemeasurable,rfl⟩
-  have hy : HasLaw Y (μ.map Y) μ := ⟨hY.aemeasurable,rfl⟩
-  have hx' : HasLaw X' (ν.map X') ν := ⟨hX'.aemeasurable,rfl⟩
-  have hy' : HasLaw Y' (ν.map Y') ν := ⟨hY'.aemeasurable,rfl⟩
-  have h1 : MeasurePreserving (fun ω => (X ω,Y ω)) μ ((μ.map X).prod (μ.map Y)) :=
-    ⟨hX.prodMk hY,(hXY.hasLaw_prod hx hy).map_eq⟩
-  have h2 : MeasurePreserving (fun ω => (X' ω,Y' ω)) ν ((ν.map X').prod (ν.map Y')) :=
-    ⟨hX'.prodMk hY',(hXY'.hasLaw_prod hx' hy').map_eq⟩
-  have h3 := (measurePreserving_shuffle (μ.map X) (μ.map Y) (ν.map X') (ν.map Y')).comp
-    (h1.prod h2)
-  have hpair : HasLaw (fun p : Ω×Ω' => ((X p.1,X' p.2),(Y p.1,Y' p.2)))
-      (((μ.map X).prod (ν.map X')).prod ((μ.map Y).prod (ν.map Y'))) (μ.prod ν) :=
-    ⟨h3.measurable.aemeasurable,h3.map_eq⟩
-  have ha : HasLaw (fun p : Ω×Ω' => (X p.1,X' p.2))
-      ((μ.map X).prod (ν.map X')) (μ.prod ν) := by
-    exact (measurePreserving_fst (μ := (μ.map X).prod (ν.map X'))
-      (ν := (μ.map Y).prod (ν.map Y'))).hasLaw.comp hpair
-  have hb : HasLaw (fun p : Ω×Ω' => (Y p.1,Y' p.2))
-      ((μ.map Y).prod (ν.map Y')) (μ.prod ν) := by
-    exact (measurePreserving_snd (μ := (μ.map X).prod (ν.map X'))
-      (ν := (μ.map Y).prod (ν.map Y'))).hasLaw.comp hpair
-  exact (indepFun_iff_hasLaw_prodMk_prod ha hb).mpr hpair
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.BrownianGenerators
+import OAI.Probability.SLE.Geometry.PublishedWienerSupport
+
+namespace OAI
+
+/-! Independence of the two-coordinate Wiener increment from its full
+original past. Product coordinates retain both original pasts. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+private lemma measurePreserving_shuffle
+    {A B C D : Type*} [MeasurableSpace A] [MeasurableSpace B]
+    [MeasurableSpace C] [MeasurableSpace D]
+    (μ : Measure A) (ν : Measure B) (ρ : Measure C) (τ : Measure D)
+    [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    [IsProbabilityMeasure ρ] [IsProbabilityMeasure τ] :
+    MeasurePreserving (fun p : (A×B)×(C×D) => ((p.1.1,p.2.1),(p.1.2,p.2.2)))
+      ((μ.prod ν).prod (ρ.prod τ)) ((μ.prod ρ).prod (ν.prod τ)) := by
+  have h1 := measurePreserving_prodAssoc μ ν (ρ.prod τ)
+  have h2 := (MeasurePreserving.id μ).prod (measurePreserving_prodAssoc ν ρ τ).symm
+  have h3 := (MeasurePreserving.id μ).prod
+    ((Measure.measurePreserving_swap (μ := ν) (ν := ρ)).prod (MeasurePreserving.id τ))
+  have h4 := (MeasurePreserving.id μ).prod (measurePreserving_prodAssoc ρ ν τ)
+  have h5 := (measurePreserving_prodAssoc μ ρ (ν.prod τ)).symm
+  exact h5.comp (h4.comp (h3.comp (h2.comp h1)))
+
+lemma independent_product_pairs
+    {Ω Ω' A B C D : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']
+    [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] [MeasurableSpace D]
+    {μ : Measure Ω} {ν : Measure Ω'} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    {X : Ω → A} {Y : Ω → B} {X' : Ω' → C} {Y' : Ω' → D}
+    (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')
+    (hXY : IndepFun X Y μ) (hXY' : IndepFun X' Y' ν) :
+    IndepFun (fun p : Ω×Ω' => (X p.1,X' p.2))
+      (fun p : Ω×Ω' => (Y p.1,Y' p.2)) (μ.prod ν) := by
+  have hx : HasLaw X (μ.map X) μ := ⟨hX.aemeasurable,rfl⟩
+  have hy : HasLaw Y (μ.map Y) μ := ⟨hY.aemeasurable,rfl⟩
+  have hx' : HasLaw X' (ν.map X') ν := ⟨hX'.aemeasurable,rfl⟩
+  have hy' : HasLaw Y' (ν.map Y') ν := ⟨hY'.aemeasurable,rfl⟩
+  have h1 : MeasurePreserving (fun ω => (X ω,Y ω)) μ ((μ.map X).prod (μ.map Y)) :=
+    ⟨hX.prodMk hY,(hXY.hasLaw_prod hx hy).map_eq⟩
+  have h2 : MeasurePreserving (fun ω => (X' ω,Y' ω)) ν ((ν.map X').prod (ν.map Y')) :=
+    ⟨hX'.prodMk hY',(hXY'.hasLaw_prod hx' hy').map_eq⟩
+  have h3 := (measurePreserving_shuffle (μ.map X) (μ.map Y) (ν.map X') (ν.map Y')).comp
+    (h1.prod h2)
+  have hpair : HasLaw (fun p : Ω×Ω' => ((X p.1,X' p.2),(Y p.1,Y' p.2)))
+      (((μ.map X).prod (ν.map X')).prod ((μ.map Y).prod (ν.map Y'))) (μ.prod ν) :=
+    ⟨h3.measurable.aemeasurable,h3.map_eq⟩
+  have ha : HasLaw (fun p : Ω×Ω' => (X p.1,X' p.2))
+      ((μ.map X).prod (ν.map X')) (μ.prod ν) := by
+    exact (measurePreserving_fst (μ := (μ.map X).prod (ν.map X'))
+      (ν := (μ.map Y).prod (ν.map Y'))).hasLaw.comp hpair
+  have hb : HasLaw (fun p : Ω×Ω' => (Y p.1,Y' p.2))
+      ((μ.map Y).prod (ν.map Y')) (μ.prod ν) := by
+    exact (measurePreserving_snd (μ := (μ.map X).prod (ν.map X'))
+      (ν := (μ.map Y).prod (ν.map Y'))).hasLaw.comp hpair
+  exact (indepFun_iff_hasLaw_prodMk_prod ha hb).mpr hpair
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/LogRatioPaths.lean b/lean/OAI/Probability/SLE/Stochastic/LogRatioPaths.lean
index 0ba3f68daa259496f1cb8f3cba4fc38b1ecd8e8f..ca0b51656d998a5c33eb7dc918ef0abe532b4737 100644
--- a/lean/OAI/Probability/SLE/Stochastic/LogRatioPaths.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/LogRatioPaths.lean
@@ -1,58 +1,58 @@
-import OAI.Probability.SLE.Stochastic.PlanarLogLifting
-
-namespace OAI
-
-/-! Continuous logarithms used in the local crosscut separation argument.
-Moving a puncture along a path gives a logarithm of the endpoint ratio on
-the complement of that path. -/
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma continuous_log_ratio_off_path {a b : ℂ} (p : Path a b) :
-    ∃ L : C({z : ℂ // z∉range p},ℂ),
-      ∀ z, Complex.exp (L z)=((z:ℂ)-b)/((z:ℂ)-a) := by
-  let D := {z : ℂ // z∉range p}
-  have hn (z : D) (t : unitInterval) : (z:ℂ)-p t≠0 := by
-    intro h
-    exact z.property ⟨t,(sub_eq_zero.mp h).symm⟩
-  have hden (z : D) : (z:ℂ)-a≠0 := by simpa using hn z 0
-  let H : C(unitInterval × D,ℂ) :=
-    ⟨fun q => ((q.2:ℂ)-p q.1)/((q.2:ℂ)-a),
-      ((continuous_subtype_val.comp continuous_snd).sub (p.continuous.comp continuous_fst)).div
-        ((continuous_subtype_val.comp continuous_snd).sub continuous_const) (fun q => hden q.2)⟩
-  obtain ⟨L,hL⟩ := continuous_log_at_end_of_homotopy H
-    (fun q => div_ne_zero (hn q.2 q.1) (hden q.2)) (ContinuousMap.const D 0)
-    (fun z => by simp [H,hden z])
-  exact ⟨L,fun z => by simpa [H] using hL z⟩
-
-lemma continuous_log_on_ball_away (c a : ℂ) (r : ℝ) (hr : 0<r)
-    (ha : a∉ball c r) :
-    ∃ L : C(ball c r,ℂ), ∀ z, Complex.exp (L z)=(z:ℂ)-a := by
-  have hca : c-a≠0 := by
-    intro h
-    exact ha (sub_eq_zero.mp h ▸ mem_ball_self hr)
-  let H : C(unitInterval × ball c r,ℂ) :=
-    ⟨fun q => c+(q.1:ℂ)*((q.2:ℂ)-c)-a,by fun_prop⟩
-  have hn (q : unitInterval × ball c r) : H q≠0 := by
-    have hm : c+(q.1:ℂ)*((q.2:ℂ)-c)∈ball c r := by
-      rw [mem_ball,dist_eq_norm]
-      rw [show c+(q.1:ℂ)*((q.2:ℂ)-c)-c=(q.1:ℂ)*((q.2:ℂ)-c) by ring]
-      rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg q.1.property.1]
-      exact (mul_le_of_le_one_left (norm_nonneg _) q.1.property.2).trans_lt
-        (by simpa only [mem_ball,dist_eq_norm] using q.2.property)
-    intro h
-    exact ha (sub_eq_zero.mp h ▸ hm)
-  obtain ⟨L,hL⟩ := continuous_log_at_end_of_homotopy H hn
-    (ContinuousMap.const (ball c r) (Complex.log (c-a)))
-    (fun z => by simp [H,Complex.exp_log hca])
-  refine ⟨L,fun z => ?_⟩
-  simpa [H] using hL z
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarLogLifting
+
+namespace OAI
+
+/-! Continuous logarithms used in the local crosscut separation argument.
+Moving a puncture along a path gives a logarithm of the endpoint ratio on
+the complement of that path. -/
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma continuous_log_ratio_off_path {a b : ℂ} (p : Path a b) :
+    ∃ L : C({z : ℂ // z∉range p},ℂ),
+      ∀ z, Complex.exp (L z)=((z:ℂ)-b)/((z:ℂ)-a) := by
+  let D := {z : ℂ // z∉range p}
+  have hn (z : D) (t : unitInterval) : (z:ℂ)-p t≠0 := by
+    intro h
+    exact z.property ⟨t,(sub_eq_zero.mp h).symm⟩
+  have hden (z : D) : (z:ℂ)-a≠0 := by simpa using hn z 0
+  let H : C(unitInterval × D,ℂ) :=
+    ⟨fun q => ((q.2:ℂ)-p q.1)/((q.2:ℂ)-a),
+      ((continuous_subtype_val.comp continuous_snd).sub (p.continuous.comp continuous_fst)).div
+        ((continuous_subtype_val.comp continuous_snd).sub continuous_const) (fun q => hden q.2)⟩
+  obtain ⟨L,hL⟩ := continuous_log_at_end_of_homotopy H
+    (fun q => div_ne_zero (hn q.2 q.1) (hden q.2)) (ContinuousMap.const D 0)
+    (fun z => by simp [H,hden z])
+  exact ⟨L,fun z => by simpa [H] using hL z⟩
+
+lemma continuous_log_on_ball_away (c a : ℂ) (r : ℝ) (hr : 0<r)
+    (ha : a∉ball c r) :
+    ∃ L : C(ball c r,ℂ), ∀ z, Complex.exp (L z)=(z:ℂ)-a := by
+  have hca : c-a≠0 := by
+    intro h
+    exact ha (sub_eq_zero.mp h ▸ mem_ball_self hr)
+  let H : C(unitInterval × ball c r,ℂ) :=
+    ⟨fun q => c+(q.1:ℂ)*((q.2:ℂ)-c)-a,by fun_prop⟩
+  have hn (q : unitInterval × ball c r) : H q≠0 := by
+    have hm : c+(q.1:ℂ)*((q.2:ℂ)-c)∈ball c r := by
+      rw [mem_ball,dist_eq_norm]
+      rw [show c+(q.1:ℂ)*((q.2:ℂ)-c)-c=(q.1:ℂ)*((q.2:ℂ)-c) by ring]
+      rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg q.1.property.1]
+      exact (mul_le_of_le_one_left (norm_nonneg _) q.1.property.2).trans_lt
+        (by simpa only [mem_ball,dist_eq_norm] using q.2.property)
+    intro h
+    exact ha (sub_eq_zero.mp h ▸ hm)
+  obtain ⟨L,hL⟩ := continuous_log_at_end_of_homotopy H hn
+    (ContinuousMap.const (ball c r) (Complex.log (c-a)))
+    (fun z => by simp [H,Complex.exp_log hca])
+  refine ⟨L,fun z => ?_⟩
+  simpa [H] using hL z
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarCompensated.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarCompensated.lean
index b255a135460c6385b57d4aef0975f56d49f880f6..c1bcd901bac83d785d878f90f07b1cf6a72e4f07 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarCompensated.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarCompensated.lean
@@ -1,87 +1,87 @@
-import OAI.Probability.SLE.Stochastic.PlanarDynkin
-
-namespace OAI
-
-/-! The compact-test planar compensated process has zero mean increments
-on every event in the original past, including its stopped finite grids. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-def planarCompensated (f : ℝ×ℝ → ℝ) (t : ℝ≥0) (b : Driver×Driver) : ℝ :=
-  f (planarWiener t b)-∫ u in (0:ℝ)..(t:ℝ), planarGenerator f (planarWiener u.toNNReal b)
-
-lemma planar_compensated_integrable
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) (t : ℝ≥0) :
-    Integrable (planarCompensated f t) (W.prod W) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
-  have hi : Integrable (fun b => f (planarWiener t b)) (W.prod W) := by
-    simpa only [Measure.restrict_univ] using planar_compact_eval_integrable W hW f hf hc univ t
-  exact hi.sub (bounded_interval_integral_integrable 0 t t.coe_nonneg _
-    (planar_generator_joint_measurable f hf) D (fun u b => hD _))
-
-lemma planar_compensated_past_integral_zero
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (s t : ℝ≥0) (hst : s ≤ t) (A : Set (Driver×Driver))
-    (hA : MeasurableSet[planarWienerFiltration s] A) :
-    (∫ b in A, planarCompensated f t b-planarCompensated f s b ∂W.prod W)=0 := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
-  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
-  have hG : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
-  have hGb (u b) : |G u b| ≤ D := hD _
-  have hGi (a c : ℝ) (h : a ≤ c) (b : Driver×Driver) :=
-    bounded_time_section_intervalIntegrable a c h G hG D hGb b
-  have hp (b : Driver×Driver) : planarCompensated f t b-planarCompensated f s b =
-      (f (planarWiener t b)-f (planarWiener s b))-∫ u in (s:ℝ)..(t:ℝ), G u b := by
-    have h := intervalIntegral.integral_add_adjacent_intervals (hGi 0 s s.coe_nonneg b) (hGi s t hst b)
-    dsimp only [planarCompensated]
-    change _ = _ at h
-    linarith
-  simp_rw [hp]
-  have hfi := (planar_compact_eval_integrable W hW f hf hc A t).sub
-    (planar_compact_eval_integrable W hW f hf hc A s)
-  have hdi := bounded_interval_integral_integrable (P := (W.prod W).restrict A)
-    s t hst G hG D hGb
-  have hfi' : Integrable (fun b => f (planarWiener t b)-f (planarWiener s b))
-      ((W.prod W).restrict A) := hfi
-  rw [integral_sub hfi' hdi,planarWiener_dynkin W hW f hf hc s t hst A hA,sub_self]
-
-lemma planar_compensated_grid_integral
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (τ : Driver×Driver → ℝ≥0)
-    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ b, τ b ≤ T) (n : ℕ) :
-    (∫ b, planarCompensated f (stopGrid n (τ b)) b ∂W.prod W)=
-      ∫ b, f (planarWiener 0 b) ∂W.prod W := by
-  let ℱn := gridFiltration planarWienerFiltration n
-  have hbound (b) : stopGridIndex n (τ b) ≤ stopGridIndex n T := by
-    exact (stopGridIndex_le n _ _).mpr ((hb b).trans (stopGrid_bounds n T).1)
-  have hE (j : ℕ) : MeasurableSet[ℱn j] {b | j<stopGridIndex n (τ b)} := by
-    have hle : MeasurableSet[planarWienerFiltration (stopGridTime n j)] {b | τ b ≤ stopGridTime n j} := by
-      simpa only [WithTop.coe_le_coe] using hτ (stopGridTime n j)
-    have he : {b | j<stopGridIndex n (τ b)}={b | τ b ≤ stopGridTime n j}ᶜ := by
-      ext b
-      simp only [mem_compl_iff,mem_ofPred_eq,← stopGridIndex_le,not_le]
-    rw [he]
-    exact hle.compl
-  have he := finite_grid_stopped_integral_eq ℱn (fun j => planarCompensated f (stopGridTime n j))
-    (fun j => planar_compensated_integrable W hW f hf hc _)
-    (fun b => stopGridIndex n (τ b)) (stopGridIndex n T) hbound hE
-    (fun j A hA => planar_compensated_past_integral_zero W hW f hf hc _ _
-      (monotone_stopGridTime n (Nat.le_succ j)) A hA)
-  simpa only [stopGrid,stopGridTime,Nat.cast_zero,zero_div,planarCompensated,
-    NNReal.coe_zero,intervalIntegral.integral_same,sub_zero] using he
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarDynkin
+
+namespace OAI
+
+/-! The compact-test planar compensated process has zero mean increments
+on every event in the original past, including its stopped finite grids. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+def planarCompensated (f : ℝ×ℝ → ℝ) (t : ℝ≥0) (b : Driver×Driver) : ℝ :=
+  f (planarWiener t b)-∫ u in (0:ℝ)..(t:ℝ), planarGenerator f (planarWiener u.toNNReal b)
+
+lemma planar_compensated_integrable
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) (t : ℝ≥0) :
+    Integrable (planarCompensated f t) (W.prod W) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
+  have hi : Integrable (fun b => f (planarWiener t b)) (W.prod W) := by
+    simpa only [Measure.restrict_univ] using planar_compact_eval_integrable W hW f hf hc univ t
+  exact hi.sub (bounded_interval_integral_integrable 0 t t.coe_nonneg _
+    (planar_generator_joint_measurable f hf) D (fun u b => hD _))
+
+lemma planar_compensated_past_integral_zero
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (s t : ℝ≥0) (hst : s ≤ t) (A : Set (Driver×Driver))
+    (hA : MeasurableSet[planarWienerFiltration s] A) :
+    (∫ b in A, planarCompensated f t b-planarCompensated f s b ∂W.prod W)=0 := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
+  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
+  have hG : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
+  have hGb (u b) : |G u b| ≤ D := hD _
+  have hGi (a c : ℝ) (h : a ≤ c) (b : Driver×Driver) :=
+    bounded_time_section_intervalIntegrable a c h G hG D hGb b
+  have hp (b : Driver×Driver) : planarCompensated f t b-planarCompensated f s b =
+      (f (planarWiener t b)-f (planarWiener s b))-∫ u in (s:ℝ)..(t:ℝ), G u b := by
+    have h := intervalIntegral.integral_add_adjacent_intervals (hGi 0 s s.coe_nonneg b) (hGi s t hst b)
+    dsimp only [planarCompensated]
+    change _ = _ at h
+    linarith
+  simp_rw [hp]
+  have hfi := (planar_compact_eval_integrable W hW f hf hc A t).sub
+    (planar_compact_eval_integrable W hW f hf hc A s)
+  have hdi := bounded_interval_integral_integrable (P := (W.prod W).restrict A)
+    s t hst G hG D hGb
+  have hfi' : Integrable (fun b => f (planarWiener t b)-f (planarWiener s b))
+      ((W.prod W).restrict A) := hfi
+  rw [integral_sub hfi' hdi,planarWiener_dynkin W hW f hf hc s t hst A hA,sub_self]
+
+lemma planar_compensated_grid_integral
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (τ : Driver×Driver → ℝ≥0)
+    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ b, τ b ≤ T) (n : ℕ) :
+    (∫ b, planarCompensated f (stopGrid n (τ b)) b ∂W.prod W)=
+      ∫ b, f (planarWiener 0 b) ∂W.prod W := by
+  let ℱn := gridFiltration planarWienerFiltration n
+  have hbound (b) : stopGridIndex n (τ b) ≤ stopGridIndex n T := by
+    exact (stopGridIndex_le n _ _).mpr ((hb b).trans (stopGrid_bounds n T).1)
+  have hE (j : ℕ) : MeasurableSet[ℱn j] {b | j<stopGridIndex n (τ b)} := by
+    have hle : MeasurableSet[planarWienerFiltration (stopGridTime n j)] {b | τ b ≤ stopGridTime n j} := by
+      simpa only [WithTop.coe_le_coe] using hτ (stopGridTime n j)
+    have he : {b | j<stopGridIndex n (τ b)}={b | τ b ≤ stopGridTime n j}ᶜ := by
+      ext b
+      simp only [mem_compl_iff,mem_ofPred_eq,← stopGridIndex_le,not_le]
+    rw [he]
+    exact hle.compl
+  have he := finite_grid_stopped_integral_eq ℱn (fun j => planarCompensated f (stopGridTime n j))
+    (fun j => planar_compensated_integrable W hW f hf hc _)
+    (fun b => stopGridIndex n (τ b)) (stopGridIndex n T) hbound hE
+    (fun j A hA => planar_compensated_past_integral_zero W hW f hf hc _ _
+      (monotone_stopGridTime n (Nat.le_succ j)) A hA)
+  simpa only [stopGrid,stopGridTime,Nat.cast_zero,zero_div,planarCompensated,
+    NNReal.coe_zero,intervalIntegral.integral_same,sub_zero] using he
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarCoordinatePassage.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarCoordinatePassage.lean
index 71f640ba08be79674ee6dc0d0b3e4f4f34114516..230c34c0b4c224b2a2c6e768d986a4581772e6a0 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarCoordinatePassage.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarCoordinatePassage.lean
@@ -1,167 +1,167 @@
-import OAI.Probability.SLE.Gauge.ActualSequencePassageRegions
-import OAI.Probability.SLE.Gauge.RibbonLimitSplice
-import OAI.Probability.SLE.Stochastic.PlanarPassageNeighborhood
-
-namespace OAI
-
-/-! Passage along an actual convergent family of normalized charts and
-ribbon coordinate maps. Every continuation tube, early-containment event,
-and moving mass region is derived from the limiting coordinate guide. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.SequencePassage
-open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess RibbonCoordinates
-
-/-- A convergent recentered admissible family has a positive one-step passage-and-mass
-probability at every sufficiently small mesh in a prescribed vanishing
-sequence. The mass lies in the actual moving coordinate middle strip. -/
-theorem planar_coordinate_limit_passage
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P)
-    (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω)
-      (γ ω) (G ω) (F₀ ω))
-    (hboundary : PublishedLoewnerBoundaryExtension)
-    (hsupport : PublishedGeometricTubeSupport P γ)
-    (hplanar : PublishedPlanarDiskOpenness)
-    (Qₙ : ℕ → ℂ → ℂ) (Q : ℂ → ℂ) (K : Set ℂ) (C Lq : ℝ≥0)
-    (hC : 0 < C) (hLq : 0 < Lq)
-    (hQu : ∀ n, LipschitzOnWith C (Qₙ n) K)
-    (hQl : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y))
-    (hQconv : TendstoUniformlyOn Qₙ Q atTop K)
-    (hQ : ContinuousOn Q K)
-    (hbase : (1 : ℂ) ∈ K)
-    (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) (F : ℂ → ℂ)
-    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
-    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
-    (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane)
-    (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ))
-    (hF : DifferentiableOn ℂ F upperHalfPlane) (hiF : InjOn F upperHalfPlane)
-    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
-    (hFconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
-    (αₙ βₙ : ℕ → ℝ → ℂ) (α β : ℝ → ℂ) (Cg : ℝ≥0)
-    (hαL : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2))
-    (hαconv : TendstoUniformlyOn αₙ α atTop (Icc 0 2))
-    (hα : ContinuousOn α (Icc 0 2))
-    (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 2)) (hβ0 : ∀ n, βₙ n 0 = 0)
-    (hβHn : ∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im)
-    (hmap : ∀ n t, t ∈ Icc (0 : ℝ) 2 → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ))
-    (hβconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2))
-    (hβ : ContinuousOn β (Icc 0 2)) (hβlim0 : β 0 = 0) (hβi : InjOn β (Icc 0 2))
-    (hβH : MapsTo β (Ioc 0 2) upperHalfPlane)
-    (hrel : EqOn (F ∘ β) α (Ioc 0 2)) (homit : α 0 ∉ F '' upperHalfPlane)
-    (x ε ηtol : ℝ) (hx : |x| < ε) (hηtol : 0 < ηtol)
-    (hshape : EqOn α (fun t => Q (guideCoordinate 0 x t)) (Icc 0 2))
-    (hmem : ∀ t ∈ Icc (0 : ℝ) (3 / 2), guideCoordinate 0 x t ∈ interior K)
-    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
-    (η : ℕ → Ω → ℝ≥0 → ℂ) (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
-    (hηbar : ∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) :
-    ∃ B : ℝ, 0 < B ∧ ∃ Tc : ℝ≥0, 0 < Tc ∧ ∃ c q : ℝ, 0 < c ∧ 0 < q ∧
-      ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω |
-        ω ∈ FrontSuccess η (fun n => longitudinalCoordinate (Qₙ n) K 0)
-          (fun ω => boundedFirstHeightTime (γ ω) B Tc)
-          (fun n => physicalPassageRegion (Qₙ n) K C ε ηtol) (fun n => Qₙ n 1) ηtol n ∧
-        ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω)
-          (boundedFirstHeightTime (γ ω) B Tc)
-          (frontTime (fun t => longitudinalCoordinate (Qₙ n) K 0 (η n ω t)) 1)
-          (v n) (physicalMiddleRegion (Qₙ n) K 0 ε)} := by
-  let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure
-  let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable)
-  have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw
-  have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl
-  have hSLE : IsOrdinaryChordalSLE κ γ P :=
-    ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩
-  let pₙ := fun n => longitudinalCoordinate (Qₙ n) K 0
-  let p := longitudinalCoordinate Q K 0
-  let Vₙ := fun n => physicalPassageRegion (Qₙ n) K C ε ηtol
-  let Rₙ := fun n => physicalMiddleRegion (Qₙ n) K 0 ε
-  let aₙ := fun n => αₙ n 0
-  obtain ⟨U, hU, hαU, _, _, hp, hpn, hpconv, hVlate, hterminal, hregion⟩ :=
-    limiting_guide_passage_neighborhood_plane hplanar Qₙ Q K C Lq hC hLq hQu hQl hQconv
-      hQ α hα x ε ηtol hx hηtol hshape hmem hbase
-  have hprogress : ∀ t ∈ Icc (0 : ℝ) (3 / 2), p (α t) = t := by
-    intro t ht
-    have hiQ := lower_lipschitz_injOn Q K Lq
-      (lower_lipschitz_uniform_limit Qₙ Q K Lq hQconv hQl)
-    dsimp only [p]
-    rw [hshape ⟨ht.1, ht.2.trans (by norm_num)⟩]
-    simp only [longitudinalCoordinate, coordinateInverse_left hiQ (interior_subset (hmem t ht)),
-      guideCoordinate_re, zero_add, sub_zero]
-  have haU : α 0 ∈ U := hαU (mem_image_of_mem α ⟨le_rfl, by norm_num⟩)
-  obtain ⟨r, hr, hball, hstartBall, hearly⟩ := eventual_early_coordinate_ball pₙ p U hU hp hpconv
-    aₙ (α 0) haU (hprogress _ ⟨le_rfl, by norm_num⟩)
-    (hαconv.tendsto_at ⟨le_rfl, by norm_num⟩)
-  obtain ⟨T₀, hT₀, q₀, hq₀, hheight⟩ := uniform_bounded_firstHeight_real_probability
-    κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport
-  obtain ⟨b, A, d, L₀, hb, hA, hd, hL₀, hloc⟩ :=
-    common_access_gives_firstHeight_localization κ hκ hκ8
-  have ht : Tendsto (fun L : ℝ => A * L ^ (-d)) atTop (𝓝 0) := by
-    simpa only [mul_zero] using (tendsto_rpow_neg_atTop hd).const_mul A
-  obtain ⟨L, hL, hsmall⟩ := ((eventually_ge_atTop L₀).and
-    (ht.eventually (gt_mem_nhds (half_pos hq₀)))).exists
-  have hLp : 0 < L := (zero_lt_one.trans hL₀).trans_le hL
-  let r₀ := r / L
-  have hr₀ : 0 < r₀ := div_pos hr hLp
-  obtain ⟨δ, hδ, hδ2, hδsmall, hz, hpoint⟩ :=
-    small_initial_access_segment Cg r₀ hr₀ βₙ β hβlim0 hβi hβconv
-  have hδsub : Icc (0 : ℝ) δ ⊆ Icc 0 2 := Icc_subset_Icc_right hδ2
-  obtain ⟨b₀, hb₀, hlocseq⟩ := hloc Ω P γ hSLE (fun ω => (hw ω).1)
-    (fun ω => (hw ω).2.1) (fun ω t => (hw ω).2.2.1 t)
-    Fₙ Fbar hFn hinj hbar hext δ hδ βₙ (fun n => (hβc n).mono hδsub) hβ0
-    (fun n t ht => hβHn n t (hδsub ht)) (β δ) hz hpoint αₙ aₙ Cg
-    (fun n => (hαL n).mono hδsub) (fun _ => rfl)
-    (fun n t ht => hmap n t (hδsub ht)) r₀ hr₀ hδsmall
-  have hβsub : Icc (0 : ℝ) (3 / 2) ⊆ Icc 0 2 := Icc_subset_Icc_right (by norm_num)
-  obtain ⟨B, tB, hB, hBm, htB, htBT, hcross, hafter, hcircle, hsector, htail, hW,
-      hend, hWfront⟩ := limiting_guide_splice_geometry F hF hiF α β (hα.mono hβsub)
-    (hβ.mono hβsub) hβlim0 (fun t ht => hβH ⟨ht.1, ht.2.trans (by norm_num)⟩)
-    (hrel.mono (Ioc_subset_Ioc le_rfl (by norm_num))) homit U hU p hp hαU hprogress
-    r (b₀ * L ^ b) hr (mul_pos hb₀ (Real.rpow_pos_of_pos hLp _))
-  let Tc := T₀ * (Real.toNNReal B) ^ 2
-  have hTc : 0 < Tc := mul_pos hT₀ (pow_pos (Real.toNNReal_pos.mpr hB) 2)
-  have hheightB := hheight B hB
-  have hpositive : 0 < P (stoppedNarrowCertificate γ
-      (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) :=
-    (ENNReal.ofReal_pos.mpr hq₀).trans_le hheightB
-  have hLr : L * r₀ = r := by
-    dsimp [r₀]
-    exact mul_div_cancel₀ r hLp.ne'
-  have hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ
-      (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \
-        stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc)
-          (Fbar n) (aₙ n) r) ≤
-      P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2 := by
-    filter_upwards [hlocseq] with n hn
-    have hbound := hn L hL B hB hBm Tc
-    rw [hLr] at hbound
-    calc
-      _ ≤ ENNReal.ofReal (A * L ^ (-d)) := hbound
-      _ ≤ ENNReal.ofReal (q₀ / 2) := ENNReal.ofReal_le_ofReal hsmall.le
-      _ = ENNReal.ofReal q₀ / 2 := by
-        rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)]
-        norm_num
-      _ ≤ _ := ENNReal.div_le_div_right hheightB 2
-  have hearlyAllowed : ∀ᶠ n in atTop, ball (aₙ n) r ⊆ Vₙ n := by
-    filter_upwards [hearly, hVlate] with n hn hV
-    exact (hn.trans inter_subset_left).trans hV
-  obtain ⟨c, q, hcpos, hqpos, hprob⟩ := bounded_splice_sequence_success_middle_regions
-    κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport B r hB hr Tc F hF (α 0)
-    U (limitingFrontTarget F p U) hU hW β tB (3 / 2) htB htBT
-    (hβ.mono hβsub) (hβi.mono hβsub)
-    (fun t ht => hβH ⟨ht.1, ht.2.trans (by norm_num)⟩)
-    hcross hafter hcircle hsector htail hend Fbar hbar aₙ r Fₙ hFn hinj hne hFconv hext
-    pₙ p U hU hp hpn hpconv (union_subset hball Subset.rfl) hWfront hstartBall hearly
-    U hU (union_subset hball Subset.rfl) Vₙ hVlate (fun n => Qₙ n 1) ηtol hterminal
-    hearlyAllowed v hv hlim η hηc hηbar Rₙ hregion hpositive hmissing
-  exact ⟨B, hB, Tc, hTc, c, q, hcpos, hqpos, hprob⟩
-
-end SLEExactGauge.SequencePassage
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.ActualSequencePassageRegions
+import OAI.Probability.SLE.Gauge.RibbonLimitSplice
+import OAI.Probability.SLE.Stochastic.PlanarPassageNeighborhood
+
+namespace OAI
+
+/-! Passage along an actual convergent family of normalized charts and
+ribbon coordinate maps. Every continuation tube, early-containment event,
+and moving mass region is derived from the limiting coordinate guide. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.SequencePassage
+open FirstHeightSupport EarlyPassageLocalization FirstFrontPassage PassageSuccess RibbonCoordinates
+
+/-- A convergent recentered admissible family has a positive one-step passage-and-mass
+probability at every sufficiently small mesh in a prescribed vanishing
+sequence. The mass lies in the actual moving coordinate middle strip. -/
+theorem planar_coordinate_limit_passage
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (B₀ : ℝ≥0 → Ω → ℝ) (hB₀ : IsBrownianReal B₀ P)
+    (hm : ∀ t, Measurable (B₀ t)) (hc : ∀ ω, Continuous (fun t => B₀ t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F₀ : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω)
+      (γ ω) (G ω) (F₀ ω))
+    (hboundary : PublishedLoewnerBoundaryExtension)
+    (hsupport : PublishedGeometricTubeSupport P γ)
+    (hplanar : PublishedPlanarDiskOpenness)
+    (Qₙ : ℕ → ℂ → ℂ) (Q : ℂ → ℂ) (K : Set ℂ) (C Lq : ℝ≥0)
+    (hC : 0 < C) (hLq : 0 < Lq)
+    (hQu : ∀ n, LipschitzOnWith C (Qₙ n) K)
+    (hQl : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ Lq * dist (Qₙ n x) (Qₙ n y))
+    (hQconv : TendstoUniformlyOn Qₙ Q atTop K)
+    (hQ : ContinuousOn Q K)
+    (hbase : (1 : ℂ) ∈ K)
+    (Fₙ : ℕ → ℂ → ℂ) (Fbar : ℕ → ℂ → OnePoint ℂ) (F : ℂ → ℂ)
+    (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane)
+    (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane)
+    (hbar : ∀ n, ContinuousOn (Fbar n) closedHalfPlane)
+    (hext : ∀ n z, z ∈ upperHalfPlane → Fbar n z = (Fₙ n z : OnePoint ℂ))
+    (hF : DifferentiableOn ℂ F upperHalfPlane) (hiF : InjOn F upperHalfPlane)
+    (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0)
+    (hFconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane)
+    (αₙ βₙ : ℕ → ℝ → ℂ) (α β : ℝ → ℂ) (Cg : ℝ≥0)
+    (hαL : ∀ n, LipschitzOnWith Cg (αₙ n) (Icc 0 2))
+    (hαconv : TendstoUniformlyOn αₙ α atTop (Icc 0 2))
+    (hα : ContinuousOn α (Icc 0 2))
+    (hβc : ∀ n, ContinuousOn (βₙ n) (Icc 0 2)) (hβ0 : ∀ n, βₙ n 0 = 0)
+    (hβHn : ∀ n t, t ∈ Icc (0 : ℝ) 2 → 0 ≤ (βₙ n t).im)
+    (hmap : ∀ n t, t ∈ Icc (0 : ℝ) 2 → Fbar n (βₙ n t) = (αₙ n t : OnePoint ℂ))
+    (hβconv : TendstoUniformlyOn βₙ β atTop (Icc 0 2))
+    (hβ : ContinuousOn β (Icc 0 2)) (hβlim0 : β 0 = 0) (hβi : InjOn β (Icc 0 2))
+    (hβH : MapsTo β (Ioc 0 2) upperHalfPlane)
+    (hrel : EqOn (F ∘ β) α (Ioc 0 2)) (homit : α 0 ∉ F '' upperHalfPlane)
+    (x ε ηtol : ℝ) (hx : |x| < ε) (hηtol : 0 < ηtol)
+    (hshape : EqOn α (fun t => Q (guideCoordinate 0 x t)) (Icc 0 2))
+    (hmem : ∀ t ∈ Icc (0 : ℝ) (3 / 2), guideCoordinate 0 x t ∈ interior K)
+    (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0))
+    (η : ℕ → Ω → ℝ≥0 → ℂ) (hηc : ∀ n, ∀ᵐ ω ∂P, Continuous (η n ω))
+    (hηbar : ∀ n, ∀ᵐ ω ∂P, ∀ t, Fbar n (γ ω t) = (η n ω t : OnePoint ℂ)) :
+    ∃ B : ℝ, 0 < B ∧ ∃ Tc : ℝ≥0, 0 < Tc ∧ ∃ c q : ℝ, 0 < c ∧ 0 < q ∧
+      ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P {ω |
+        ω ∈ FrontSuccess η (fun n => longitudinalCoordinate (Qₙ n) K 0)
+          (fun ω => boundedFirstHeightTime (γ ω) B Tc)
+          (fun n => physicalPassageRegion (Qₙ n) K C ε ηtol) (fun n => Qₙ n 1) ηtol n ∧
+        ENNReal.ofReal c ≤ segmentSausageCharge κ (η n ω)
+          (boundedFirstHeightTime (γ ω) B Tc)
+          (frontTime (fun t => longitudinalCoordinate (Qₙ n) K 0 (η n ω t)) 1)
+          (v n) (physicalMiddleRegion (Qₙ n) K 0 ε)} := by
+  let : IsProbabilityMeasure P := hB₀.isGaussianProcess.isProbabilityMeasure
+  let ℱ := Filtration.natural B₀ (fun t => (hm t).stronglyMeasurable)
+  have ha := brownian_capacity_trace_adapted B₀ hm hc κ γ G F₀ hw
+  have hγm (t : ℝ≥0) : Measurable (fun ω => γ ω t) := (ha t).mono (ℱ.le t) le_rfl
+  have hSLE : IsOrdinaryChordalSLE κ γ P :=
+    ⟨hγm, B₀, hB₀, Eventually.of_forall (fun ω => ⟨G ω, F₀ ω, hw ω⟩)⟩
+  let pₙ := fun n => longitudinalCoordinate (Qₙ n) K 0
+  let p := longitudinalCoordinate Q K 0
+  let Vₙ := fun n => physicalPassageRegion (Qₙ n) K C ε ηtol
+  let Rₙ := fun n => physicalMiddleRegion (Qₙ n) K 0 ε
+  let aₙ := fun n => αₙ n 0
+  obtain ⟨U, hU, hαU, _, _, hp, hpn, hpconv, hVlate, hterminal, hregion⟩ :=
+    limiting_guide_passage_neighborhood_plane hplanar Qₙ Q K C Lq hC hLq hQu hQl hQconv
+      hQ α hα x ε ηtol hx hηtol hshape hmem hbase
+  have hprogress : ∀ t ∈ Icc (0 : ℝ) (3 / 2), p (α t) = t := by
+    intro t ht
+    have hiQ := lower_lipschitz_injOn Q K Lq
+      (lower_lipschitz_uniform_limit Qₙ Q K Lq hQconv hQl)
+    dsimp only [p]
+    rw [hshape ⟨ht.1, ht.2.trans (by norm_num)⟩]
+    simp only [longitudinalCoordinate, coordinateInverse_left hiQ (interior_subset (hmem t ht)),
+      guideCoordinate_re, zero_add, sub_zero]
+  have haU : α 0 ∈ U := hαU (mem_image_of_mem α ⟨le_rfl, by norm_num⟩)
+  obtain ⟨r, hr, hball, hstartBall, hearly⟩ := eventual_early_coordinate_ball pₙ p U hU hp hpconv
+    aₙ (α 0) haU (hprogress _ ⟨le_rfl, by norm_num⟩)
+    (hαconv.tendsto_at ⟨le_rfl, by norm_num⟩)
+  obtain ⟨T₀, hT₀, q₀, hq₀, hheight⟩ := uniform_bounded_firstHeight_real_probability
+    κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hsupport
+  obtain ⟨b, A, d, L₀, hb, hA, hd, hL₀, hloc⟩ :=
+    common_access_gives_firstHeight_localization κ hκ hκ8
+  have ht : Tendsto (fun L : ℝ => A * L ^ (-d)) atTop (𝓝 0) := by
+    simpa only [mul_zero] using (tendsto_rpow_neg_atTop hd).const_mul A
+  obtain ⟨L, hL, hsmall⟩ := ((eventually_ge_atTop L₀).and
+    (ht.eventually (gt_mem_nhds (half_pos hq₀)))).exists
+  have hLp : 0 < L := (zero_lt_one.trans hL₀).trans_le hL
+  let r₀ := r / L
+  have hr₀ : 0 < r₀ := div_pos hr hLp
+  obtain ⟨δ, hδ, hδ2, hδsmall, hz, hpoint⟩ :=
+    small_initial_access_segment Cg r₀ hr₀ βₙ β hβlim0 hβi hβconv
+  have hδsub : Icc (0 : ℝ) δ ⊆ Icc 0 2 := Icc_subset_Icc_right hδ2
+  obtain ⟨b₀, hb₀, hlocseq⟩ := hloc Ω P γ hSLE (fun ω => (hw ω).1)
+    (fun ω => (hw ω).2.1) (fun ω t => (hw ω).2.2.1 t)
+    Fₙ Fbar hFn hinj hbar hext δ hδ βₙ (fun n => (hβc n).mono hδsub) hβ0
+    (fun n t ht => hβHn n t (hδsub ht)) (β δ) hz hpoint αₙ aₙ Cg
+    (fun n => (hαL n).mono hδsub) (fun _ => rfl)
+    (fun n t ht => hmap n t (hδsub ht)) r₀ hr₀ hδsmall
+  have hβsub : Icc (0 : ℝ) (3 / 2) ⊆ Icc 0 2 := Icc_subset_Icc_right (by norm_num)
+  obtain ⟨B, tB, hB, hBm, htB, htBT, hcross, hafter, hcircle, hsector, htail, hW,
+      hend, hWfront⟩ := limiting_guide_splice_geometry F hF hiF α β (hα.mono hβsub)
+    (hβ.mono hβsub) hβlim0 (fun t ht => hβH ⟨ht.1, ht.2.trans (by norm_num)⟩)
+    (hrel.mono (Ioc_subset_Ioc le_rfl (by norm_num))) homit U hU p hp hαU hprogress
+    r (b₀ * L ^ b) hr (mul_pos hb₀ (Real.rpow_pos_of_pos hLp _))
+  let Tc := T₀ * (Real.toNNReal B) ^ 2
+  have hTc : 0 < Tc := mul_pos hT₀ (pow_pos (Real.toNNReal_pos.mpr hB) 2)
+  have hheightB := hheight B hB
+  have hpositive : 0 < P (stoppedNarrowCertificate γ
+      (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) :=
+    (ENNReal.ofReal_pos.mpr hq₀).trans_le hheightB
+  have hLr : L * r₀ = r := by
+    dsimp [r₀]
+    exact mul_div_cancel₀ r hLp.ne'
+  have hmissing : ∀ᶠ n in atTop, P (stoppedNarrowCertificate γ
+      (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc \
+        stoppedImageEvent γ (fun ω => boundedFirstHeightTime (γ ω) B Tc)
+          (Fbar n) (aₙ n) r) ≤
+      P (stoppedNarrowCertificate γ (fun ω => boundedFirstHeightTime (γ ω) B Tc) B Tc) / 2 := by
+    filter_upwards [hlocseq] with n hn
+    have hbound := hn L hL B hB hBm Tc
+    rw [hLr] at hbound
+    calc
+      _ ≤ ENNReal.ofReal (A * L ^ (-d)) := hbound
+      _ ≤ ENNReal.ofReal (q₀ / 2) := ENNReal.ofReal_le_ofReal hsmall.le
+      _ = ENNReal.ofReal q₀ / 2 := by
+        rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)]
+        norm_num
+      _ ≤ _ := ENNReal.div_le_div_right hheightB 2
+  have hearlyAllowed : ∀ᶠ n in atTop, ball (aₙ n) r ⊆ Vₙ n := by
+    filter_upwards [hearly, hVlate] with n hn hV
+    exact (hn.trans inter_subset_left).trans hV
+  obtain ⟨c, q, hcpos, hqpos, hprob⟩ := bounded_splice_sequence_success_middle_regions
+    κ hκ hκ8 B₀ hB₀ hm hc γ G F₀ hw hboundary hsupport B r hB hr Tc F hF (α 0)
+    U (limitingFrontTarget F p U) hU hW β tB (3 / 2) htB htBT
+    (hβ.mono hβsub) (hβi.mono hβsub)
+    (fun t ht => hβH ⟨ht.1, ht.2.trans (by norm_num)⟩)
+    hcross hafter hcircle hsector htail hend Fbar hbar aₙ r Fₙ hFn hinj hne hFconv hext
+    pₙ p U hU hp hpn hpconv (union_subset hball Subset.rfl) hWfront hstartBall hearly
+    U hU (union_subset hball Subset.rfl) Vₙ hVlate (fun n => Qₙ n 1) ηtol hterminal
+    hearlyAllowed v hv hlim η hηc hηbar Rₙ hregion hpositive hmissing
+  exact ⟨B, hB, Tc, hTc, c, q, hcpos, hqpos, hprob⟩
+
+end SLEExactGauge.SequencePassage
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarCutoffStop.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarCutoffStop.lean
index e903e037e10b25d543539f1327ef1709c90f87f5..e7d7cd378868e69a2efd7afad6c5ad9a91d04823 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarCutoffStop.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarCutoffStop.lean
@@ -1,83 +1,83 @@
-import OAI.Probability.SLE.Stochastic.PlanarHarmonicLocalization
-import OAI.Probability.SLE.Geometry.ExtendedStopLocalization
-
-namespace OAI
-
-/-! Concrete compact localization stops for planar Brownian motion. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planarWiener_zero (b : Driver×Driver) : planarWiener 0 b=0 := by
-  simp [planarWiener]
-
-lemma planarWiener_adapted : Adapted planarWienerFiltration planarWiener :=
-  (Filtration.stronglyAdapted_natural (fun t => (measurable_planarWiener t).stronglyMeasurable)).adapted
-
-def planarCutoffHit (χ : ℝ×ℝ → ℝ) : Driver×Driver → WithTop ℝ≥0 :=
-  hittingAfter (fun t b => χ (planarWiener t b)) {0} 0
-
-def planarCutoffStop (χ : ℝ×ℝ → ℝ) (T : ℝ≥0) : Driver×Driver → ℝ≥0 :=
-  cutoffStop (planarCutoffHit χ) T
-
-lemma planarCutoffHit_stopping (χ : ℝ×ℝ → ℝ) (hχ : Continuous χ) :
-    IsStoppingTime planarWienerFiltration (planarCutoffHit χ) :=
-  continuous_zero_hit_stopping planarWienerFiltration (fun t b => χ (planarWiener t b))
-    (fun t => hχ.measurable.comp (planarWiener_adapted t))
-    (fun b => hχ.comp (planarWiener_continuous b)) 0
-
-lemma planarCutoffStop_stopping (χ : ℝ×ℝ → ℝ) (hχ : Continuous χ) (T : ℝ≥0) :
-    IsStoppingTime planarWienerFiltration (fun b => (planarCutoffStop χ T b : WithTop ℝ≥0)) :=
-  cutoffStop_stopping planarWienerFiltration _ (planarCutoffHit_stopping χ hχ) T
-
-lemma planarCutoffStop_le (χ : ℝ×ℝ → ℝ) (T : ℝ≥0) (b : Driver×Driver) :
-    planarCutoffStop χ T b ≤ T := cutoffStop_le_horizon _ _ _
-
-lemma planarCutoffStop_before (χ : ℝ×ℝ → ℝ) (T : ℝ≥0) (b : Driver×Driver)
-    (t : ℝ≥0) (ht : t < planarCutoffStop χ T b) : χ (planarWiener t b) ≠ 0 := by
-  have hhit : (t : WithTop ℝ≥0)<planarCutoffHit χ b :=
-    (WithTop.coe_lt_coe.mpr ht).trans_le (cutoffStop_le_stop _ _ _)
-  exact fun hz => notMem_of_lt_hittingAfter hhit (zero_le : (0:ℝ≥0) ≤ t) hz
-
-lemma planarCutoffStop_in_support (χ : ℝ×ℝ → ℝ) (h0 : χ 0≠0)
-    (T : ℝ≥0) (b : Driver×Driver) (t : ℝ≥0) (ht : t ≤ planarCutoffStop χ T b) :
-    planarWiener t b ∈ tsupport χ := by
-  by_cases hlt : t < planarCutoffStop χ T b
-  · exact subset_tsupport χ (planarCutoffStop_before χ T b t hlt)
-  have he : t=planarCutoffStop χ T b := le_antisymm ht (le_of_not_gt hlt)
-  by_cases ht0 : t=0
-  · rw [ht0,planarWiener_zero]
-    exact subset_tsupport χ h0
-  have htpos : 0<t := pos_iff_ne_zero.mpr ht0
-  have hcl : t∈closure (Iio t) := by
-    rw [closure_Iio' (show (Iio t).Nonempty from ⟨0,htpos⟩)]
-    change t ≤ t
-    exact le_rfl
-  have hm : MapsTo (fun s => planarWiener s b) (Iio t) (tsupport χ) := by
-    intro s hs
-    exact subset_tsupport χ (planarCutoffStop_before χ T b s (he ▸ hs))
-  have hx := (planarWiener_continuous b).continuousWithinAt.mem_closure hcl hm
-  simpa only [(isClosed_tsupport χ).closure_eq] using hx
-
-lemma planarCutoffStop_ge_of_nonzero (χ : ℝ×ℝ → ℝ) (hχ : Continuous χ)
-    (T t : ℝ≥0) (ht : t ≤ T) (b : Driver×Driver)
-    (hn : ∀ s ≤ t, χ (planarWiener s b)≠0) : t ≤ planarCutoffStop χ T b := by
-  apply WithTop.coe_le_coe.mp
-  rw [planarCutoffStop,cutoffStop_coe,le_min_iff]
-  refine ⟨?_,WithTop.coe_le_coe.mpr ht⟩
-  apply le_of_lt
-  apply lt_of_not_ge
-  intro hh
-  have hc : ∀ b : Driver×Driver, Continuous (fun s => χ (planarWiener s b)) :=
-    fun b => hχ.comp (planarWiener_continuous b)
-  obtain ⟨s,hs,hz⟩ := (hittingAfter_zero_le_iff (fun s b => χ (planarWiener s b)) hc 0 t b).mp hh
-  exact hn s hs.2 hz
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarHarmonicLocalization
+import OAI.Probability.SLE.Geometry.ExtendedStopLocalization
+
+namespace OAI
+
+/-! Concrete compact localization stops for planar Brownian motion. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planarWiener_zero (b : Driver×Driver) : planarWiener 0 b=0 := by
+  simp [planarWiener]
+
+lemma planarWiener_adapted : Adapted planarWienerFiltration planarWiener :=
+  (Filtration.stronglyAdapted_natural (fun t => (measurable_planarWiener t).stronglyMeasurable)).adapted
+
+def planarCutoffHit (χ : ℝ×ℝ → ℝ) : Driver×Driver → WithTop ℝ≥0 :=
+  hittingAfter (fun t b => χ (planarWiener t b)) {0} 0
+
+def planarCutoffStop (χ : ℝ×ℝ → ℝ) (T : ℝ≥0) : Driver×Driver → ℝ≥0 :=
+  cutoffStop (planarCutoffHit χ) T
+
+lemma planarCutoffHit_stopping (χ : ℝ×ℝ → ℝ) (hχ : Continuous χ) :
+    IsStoppingTime planarWienerFiltration (planarCutoffHit χ) :=
+  continuous_zero_hit_stopping planarWienerFiltration (fun t b => χ (planarWiener t b))
+    (fun t => hχ.measurable.comp (planarWiener_adapted t))
+    (fun b => hχ.comp (planarWiener_continuous b)) 0
+
+lemma planarCutoffStop_stopping (χ : ℝ×ℝ → ℝ) (hχ : Continuous χ) (T : ℝ≥0) :
+    IsStoppingTime planarWienerFiltration (fun b => (planarCutoffStop χ T b : WithTop ℝ≥0)) :=
+  cutoffStop_stopping planarWienerFiltration _ (planarCutoffHit_stopping χ hχ) T
+
+lemma planarCutoffStop_le (χ : ℝ×ℝ → ℝ) (T : ℝ≥0) (b : Driver×Driver) :
+    planarCutoffStop χ T b ≤ T := cutoffStop_le_horizon _ _ _
+
+lemma planarCutoffStop_before (χ : ℝ×ℝ → ℝ) (T : ℝ≥0) (b : Driver×Driver)
+    (t : ℝ≥0) (ht : t < planarCutoffStop χ T b) : χ (planarWiener t b) ≠ 0 := by
+  have hhit : (t : WithTop ℝ≥0)<planarCutoffHit χ b :=
+    (WithTop.coe_lt_coe.mpr ht).trans_le (cutoffStop_le_stop _ _ _)
+  exact fun hz => notMem_of_lt_hittingAfter hhit (zero_le : (0:ℝ≥0) ≤ t) hz
+
+lemma planarCutoffStop_in_support (χ : ℝ×ℝ → ℝ) (h0 : χ 0≠0)
+    (T : ℝ≥0) (b : Driver×Driver) (t : ℝ≥0) (ht : t ≤ planarCutoffStop χ T b) :
+    planarWiener t b ∈ tsupport χ := by
+  by_cases hlt : t < planarCutoffStop χ T b
+  · exact subset_tsupport χ (planarCutoffStop_before χ T b t hlt)
+  have he : t=planarCutoffStop χ T b := le_antisymm ht (le_of_not_gt hlt)
+  by_cases ht0 : t=0
+  · rw [ht0,planarWiener_zero]
+    exact subset_tsupport χ h0
+  have htpos : 0<t := pos_iff_ne_zero.mpr ht0
+  have hcl : t∈closure (Iio t) := by
+    rw [closure_Iio' (show (Iio t).Nonempty from ⟨0,htpos⟩)]
+    change t ≤ t
+    exact le_rfl
+  have hm : MapsTo (fun s => planarWiener s b) (Iio t) (tsupport χ) := by
+    intro s hs
+    exact subset_tsupport χ (planarCutoffStop_before χ T b s (he ▸ hs))
+  have hx := (planarWiener_continuous b).continuousWithinAt.mem_closure hcl hm
+  simpa only [(isClosed_tsupport χ).closure_eq] using hx
+
+lemma planarCutoffStop_ge_of_nonzero (χ : ℝ×ℝ → ℝ) (hχ : Continuous χ)
+    (T t : ℝ≥0) (ht : t ≤ T) (b : Driver×Driver)
+    (hn : ∀ s ≤ t, χ (planarWiener s b)≠0) : t ≤ planarCutoffStop χ T b := by
+  apply WithTop.coe_le_coe.mp
+  rw [planarCutoffStop,cutoffStop_coe,le_min_iff]
+  refine ⟨?_,WithTop.coe_le_coe.mpr ht⟩
+  apply le_of_lt
+  apply lt_of_not_ge
+  intro hh
+  have hc : ∀ b : Driver×Driver, Continuous (fun s => χ (planarWiener s b)) :=
+    fun b => hχ.comp (planarWiener_continuous b)
+  obtain ⟨s,hs,hz⟩ := (hittingAfter_zero_le_iff (fun s b => χ (planarWiener s b)) hc 0 t b).mp hh
+  exact hn s hs.2 hz
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarDiskImage.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarDiskImage.lean
index 8358c038e72177573a09884bf6d8dd4d7dc04da2..8fdd0b7a7f4991167cd975c4ca3b85c3b818a3b5 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarDiskImage.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarDiskImage.lean
@@ -1,102 +1,102 @@
-import OAI.Probability.SLE.Stochastic.PlanarLogLifting
-
-namespace OAI
-
-/-! A continuous injection of a closed planar disk contains a neighborhood
-of its center image. The proof uses exponential-covering homotopy lifting. -/
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma planar_closedBall_image_contains_ball
-    (f : ℂ → ℂ) (r δ : ℝ) (hr : 0<r)
-    (hf : ContinuousOn f (closedBall 0 r)) (hi : InjOn f (closedBall 0 r))
-    (hsep : ∀ z∈sphere (0:ℂ) r, δ≤‖f z-f 0‖) :
-    ball (f 0) δ ⊆ f '' closedBall 0 r := by
-  intro y hy
-  by_contra hmiss
-  have hyδ : ‖y-f 0‖<δ := by simpa only [mem_ball,dist_eq_norm] using hy
-  have hn (z : ℂ) (hz : z∈closedBall 0 r) : f z-y≠0 := by
-    intro he
-    exact hmiss ⟨z,hz,sub_eq_zero.mp he⟩
-  let S := sphere (0:ℂ) r
-  have hS : IsPreconnected S := isPreconnected_sphere (by simp) 0 r
-  have : PreconnectedSpace S := isPreconnected_iff_preconnectedSpace.mp hS
-  have hnorm (z : S) : ‖(z:ℂ)‖=r := by
-    simpa only [S,mem_sphere,dist_zero_right] using z.property
-  have hc : Continuous (fun z : S => f z) :=
-    hf.comp_continuous continuous_subtype_val (fun z => sphere_subset_closedBall z.property)
-  have hscale (q : unitInterval × S) : ((q.1:ℝ):ℂ)*(q.2:ℂ)∈closedBall (0:ℂ) r := by
-    rw [mem_closedBall,dist_zero_right,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-      abs_of_nonneg q.1.property.1,hnorm]
-    simpa only [one_mul] using mul_le_mul_of_nonneg_right q.1.property.2 hr.le
-  have hnscale (q : unitInterval × S) : -(((q.1:ℝ):ℂ)*(q.2:ℂ))∈closedBall (0:ℂ) r := by
-    simpa only [mem_closedBall,dist_zero_right,norm_neg] using hscale q
-  have hsc : Continuous (fun q : unitInterval × S => f (((q.1:ℝ):ℂ)*(q.2:ℂ))) :=
-    hf.comp_continuous (by fun_prop) hscale
-  have hsn : Continuous (fun q : unitInterval × S => f (-(((q.1:ℝ):ℂ)*(q.2:ℂ)))) :=
-    hf.comp_continuous (by fun_prop) hnscale
-  let H₁ : C(unitInterval × S,ℂ) := ⟨fun q => f (((q.1:ℝ):ℂ)*(q.2:ℂ))-y,
-    hsc.sub continuous_const⟩
-  obtain ⟨L₁,hL₁⟩ := continuous_log_at_end_of_homotopy H₁
-    (fun q => hn _ (hscale q)) (.const S (Complex.log (f 0-y))) (by
-      intro z
-      simpa [H₁] using Complex.exp_log (hn 0 (mem_closedBall_self hr.le)))
-  have hL₁' (z : S) : Complex.exp (L₁ z)=f z-y := by simpa [H₁] using hL₁ z
-  let H₂ : C(unitInterval × S,ℂ) :=
-    ⟨fun q => f q.2-f 0-(1-((q.1:ℝ):ℂ))*(y-f 0),by
-      exact (hc.comp continuous_snd).sub continuous_const |>.sub (by fun_prop)⟩
-  have hH₂ (q : unitInterval × S) : H₂ q≠0 := by
-    intro he
-    have heq : f q.2-f 0=(1-((q.1:ℝ):ℂ))*(y-f 0) := sub_eq_zero.mp he
-    have ht : ‖1-((q.1:ℝ):ℂ)‖≤1 := by
-      rw [← Complex.ofReal_one,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs,
-        abs_of_nonneg (sub_nonneg.mpr q.1.property.2)]
-      linarith [q.1.property.1]
-    have hsmall : ‖f q.2-f 0‖<δ := by
-      rw [heq,norm_mul]
-      exact (mul_le_of_le_one_left (norm_nonneg _) ht).trans_lt hyδ
-    exact not_lt_of_ge (hsep q.2 q.2.property) hsmall
-  obtain ⟨L₂,hL₂⟩ := continuous_log_at_end_of_homotopy H₂ hH₂ L₁ (by
-    intro z
-    rw [hL₁']
-    dsimp [H₂]
-    ring)
-  have hL₂' (z : S) : Complex.exp (L₂ z)=f z-f 0 := by simpa [H₂] using hL₂ z
-  let H₃ : C(unitInterval × S,ℂ) := ⟨fun q => f q.2-f (-(((q.1:ℝ):ℂ)*(q.2:ℂ))),
-    (hc.comp continuous_snd).sub hsn⟩
-  have hH₃ (q : unitInterval × S) : H₃ q≠0 := by
-    intro he
-    have heq := hi (sphere_subset_closedBall q.2.property) (hnscale q) (sub_eq_zero.mp he)
-    have hz0 : (q.2:ℂ)≠0 := by
-      intro hz
-      have := hnorm q.2
-      rw [hz,norm_zero] at this
-      exact hr.ne' this.symm
-    have hp : (1+((q.1:ℝ):ℂ))*(q.2:ℂ)=0 := by linear_combination heq
-    have hp0 := (mul_eq_zero.mp hp).resolve_right hz0
-    have hpR : (1:ℝ)+(q.1:ℝ)=0 := by exact_mod_cast hp0
-    linarith [q.1.property.1]
-  obtain ⟨L₃,hL₃⟩ := continuous_log_at_end_of_homotopy H₃ hH₃ L₂ (by
-    intro z
-    simpa [H₃] using hL₂' z)
-  let J : S → S := fun z => ⟨-z,by simpa only [S,mem_sphere,dist_zero_right,norm_neg] using z.property⟩
-  have hJ : Continuous J := (continuous_subtype_val.neg).subtype_mk _
-  have hJi : Involutive J := fun z => Subtype.ext (neg_neg z.1)
-  let F : C(S,ℂ) := ⟨fun z => f z-f (-z),hc.sub (hf.comp_continuous
-    (continuous_subtype_val.neg) (fun z => by
-      simpa only [mem_closedBall,dist_zero_right,norm_neg] using sphere_subset_closedBall z.property))⟩
-  have hF (z : S) : F z≠0 := by simpa [F,H₃] using hH₃ (1,z)
-  have hodd (z : S) : F (J z)= -F z := by dsimp [F,J]; rw [neg_neg]; ring
-  have hlog (z : S) : Complex.exp (L₃ z)=F z := by simpa [F,H₃] using hL₃ z
-  have ha : (r:ℂ)∈S := by
-    simp only [S,mem_sphere,dist_zero_right,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hr]
-  exact no_continuous_log_of_odd_involution J hJ hJi F L₃ hF hodd hlog ⟨r,ha⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarLogLifting
+
+namespace OAI
+
+/-! A continuous injection of a closed planar disk contains a neighborhood
+of its center image. The proof uses exponential-covering homotopy lifting. -/
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma planar_closedBall_image_contains_ball
+    (f : ℂ → ℂ) (r δ : ℝ) (hr : 0<r)
+    (hf : ContinuousOn f (closedBall 0 r)) (hi : InjOn f (closedBall 0 r))
+    (hsep : ∀ z∈sphere (0:ℂ) r, δ≤‖f z-f 0‖) :
+    ball (f 0) δ ⊆ f '' closedBall 0 r := by
+  intro y hy
+  by_contra hmiss
+  have hyδ : ‖y-f 0‖<δ := by simpa only [mem_ball,dist_eq_norm] using hy
+  have hn (z : ℂ) (hz : z∈closedBall 0 r) : f z-y≠0 := by
+    intro he
+    exact hmiss ⟨z,hz,sub_eq_zero.mp he⟩
+  let S := sphere (0:ℂ) r
+  have hS : IsPreconnected S := isPreconnected_sphere (by simp) 0 r
+  have : PreconnectedSpace S := isPreconnected_iff_preconnectedSpace.mp hS
+  have hnorm (z : S) : ‖(z:ℂ)‖=r := by
+    simpa only [S,mem_sphere,dist_zero_right] using z.property
+  have hc : Continuous (fun z : S => f z) :=
+    hf.comp_continuous continuous_subtype_val (fun z => sphere_subset_closedBall z.property)
+  have hscale (q : unitInterval × S) : ((q.1:ℝ):ℂ)*(q.2:ℂ)∈closedBall (0:ℂ) r := by
+    rw [mem_closedBall,dist_zero_right,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+      abs_of_nonneg q.1.property.1,hnorm]
+    simpa only [one_mul] using mul_le_mul_of_nonneg_right q.1.property.2 hr.le
+  have hnscale (q : unitInterval × S) : -(((q.1:ℝ):ℂ)*(q.2:ℂ))∈closedBall (0:ℂ) r := by
+    simpa only [mem_closedBall,dist_zero_right,norm_neg] using hscale q
+  have hsc : Continuous (fun q : unitInterval × S => f (((q.1:ℝ):ℂ)*(q.2:ℂ))) :=
+    hf.comp_continuous (by fun_prop) hscale
+  have hsn : Continuous (fun q : unitInterval × S => f (-(((q.1:ℝ):ℂ)*(q.2:ℂ)))) :=
+    hf.comp_continuous (by fun_prop) hnscale
+  let H₁ : C(unitInterval × S,ℂ) := ⟨fun q => f (((q.1:ℝ):ℂ)*(q.2:ℂ))-y,
+    hsc.sub continuous_const⟩
+  obtain ⟨L₁,hL₁⟩ := continuous_log_at_end_of_homotopy H₁
+    (fun q => hn _ (hscale q)) (.const S (Complex.log (f 0-y))) (by
+      intro z
+      simpa [H₁] using Complex.exp_log (hn 0 (mem_closedBall_self hr.le)))
+  have hL₁' (z : S) : Complex.exp (L₁ z)=f z-y := by simpa [H₁] using hL₁ z
+  let H₂ : C(unitInterval × S,ℂ) :=
+    ⟨fun q => f q.2-f 0-(1-((q.1:ℝ):ℂ))*(y-f 0),by
+      exact (hc.comp continuous_snd).sub continuous_const |>.sub (by fun_prop)⟩
+  have hH₂ (q : unitInterval × S) : H₂ q≠0 := by
+    intro he
+    have heq : f q.2-f 0=(1-((q.1:ℝ):ℂ))*(y-f 0) := sub_eq_zero.mp he
+    have ht : ‖1-((q.1:ℝ):ℂ)‖≤1 := by
+      rw [← Complex.ofReal_one,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs,
+        abs_of_nonneg (sub_nonneg.mpr q.1.property.2)]
+      linarith [q.1.property.1]
+    have hsmall : ‖f q.2-f 0‖<δ := by
+      rw [heq,norm_mul]
+      exact (mul_le_of_le_one_left (norm_nonneg _) ht).trans_lt hyδ
+    exact not_lt_of_ge (hsep q.2 q.2.property) hsmall
+  obtain ⟨L₂,hL₂⟩ := continuous_log_at_end_of_homotopy H₂ hH₂ L₁ (by
+    intro z
+    rw [hL₁']
+    dsimp [H₂]
+    ring)
+  have hL₂' (z : S) : Complex.exp (L₂ z)=f z-f 0 := by simpa [H₂] using hL₂ z
+  let H₃ : C(unitInterval × S,ℂ) := ⟨fun q => f q.2-f (-(((q.1:ℝ):ℂ)*(q.2:ℂ))),
+    (hc.comp continuous_snd).sub hsn⟩
+  have hH₃ (q : unitInterval × S) : H₃ q≠0 := by
+    intro he
+    have heq := hi (sphere_subset_closedBall q.2.property) (hnscale q) (sub_eq_zero.mp he)
+    have hz0 : (q.2:ℂ)≠0 := by
+      intro hz
+      have := hnorm q.2
+      rw [hz,norm_zero] at this
+      exact hr.ne' this.symm
+    have hp : (1+((q.1:ℝ):ℂ))*(q.2:ℂ)=0 := by linear_combination heq
+    have hp0 := (mul_eq_zero.mp hp).resolve_right hz0
+    have hpR : (1:ℝ)+(q.1:ℝ)=0 := by exact_mod_cast hp0
+    linarith [q.1.property.1]
+  obtain ⟨L₃,hL₃⟩ := continuous_log_at_end_of_homotopy H₃ hH₃ L₂ (by
+    intro z
+    simpa [H₃] using hL₂' z)
+  let J : S → S := fun z => ⟨-z,by simpa only [S,mem_sphere,dist_zero_right,norm_neg] using z.property⟩
+  have hJ : Continuous J := (continuous_subtype_val.neg).subtype_mk _
+  have hJi : Involutive J := fun z => Subtype.ext (neg_neg z.1)
+  let F : C(S,ℂ) := ⟨fun z => f z-f (-z),hc.sub (hf.comp_continuous
+    (continuous_subtype_val.neg) (fun z => by
+      simpa only [mem_closedBall,dist_zero_right,norm_neg] using sphere_subset_closedBall z.property))⟩
+  have hF (z : S) : F z≠0 := by simpa [F,H₃] using hH₃ (1,z)
+  have hodd (z : S) : F (J z)= -F z := by dsimp [F,J]; rw [neg_neg]; ring
+  have hlog (z : S) : Complex.exp (L₃ z)=F z := by simpa [F,H₃] using hL₃ z
+  have ha : (r:ℂ)∈S := by
+    simp only [S,mem_sphere,dist_zero_right,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hr]
+  exact no_continuous_log_of_odd_involution J hJ hJi F L₃ hF hodd hlog ⟨r,ha⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarDiskOpenness.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarDiskOpenness.lean
index 9a3910f8c93b753b8c69167604344d3cfa6cea4b..ca432c2972e54208665d0f696489048c8416db74 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarDiskOpenness.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarDiskOpenness.lean
@@ -1,46 +1,46 @@
-import OAI.Probability.SLE.Stochastic.PlanarDiskImage
-
-namespace OAI
-
-/-! The planar open-disk specialization of invariance of domain, proved
-from compact boundary separation and the exponential-covering argument. -/
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-
-namespace SLEExactGauge
-
-theorem actual_planarDiskOpenness : PublishedPlanarDiskOpenness := by
-  intro f x r _hr hf hi
-  apply Metric.isOpen_iff.mpr
-  rintro _ ⟨a,ha,rfl⟩
-  obtain ⟨R,hR,hRa⟩ := Metric.nhds_basis_closedBall.mem_iff.mp (isOpen_ball.mem_nhds ha)
-  let F : ℂ → ℂ := fun z => f (a+z)
-  have htrans (z : ℂ) (hz : z∈closedBall 0 R) : a+z∈ball x r := by
-    apply hRa
-    simpa only [mem_closedBall,dist_eq_norm,add_sub_cancel_left,sub_zero] using hz
-  have hFc : ContinuousOn F (closedBall 0 R) :=
-    hf.comp (continuous_const.add continuous_id).continuousOn htrans
-  have hFi : InjOn F (closedBall 0 R) := by
-    intro z hz w hw he
-    exact add_left_cancel (hi (htrans z hz) (htrans w hw) he)
-  have hpos (z : ℂ) (hz : z∈sphere (0:ℂ) R) : 0<‖F z-F 0‖ := by
-    apply norm_pos_iff.mpr
-    intro he
-    have hez : z=0 := hFi (sphere_subset_closedBall hz) (mem_closedBall_self hR.le)
-      (sub_eq_zero.mp he)
-    have hzr : ‖z‖=R := by simpa only [mem_sphere,dist_zero_right] using hz
-    rw [hez,norm_zero] at hzr
-    exact hR.ne' hzr.symm
-  obtain ⟨δ,hδ,hsep⟩ := (isCompact_sphere (0:ℂ) R).exists_forall_le'
-    ((hFc.mono sphere_subset_closedBall).sub continuousOn_const).norm hpos
-  refine ⟨δ,hδ,fun y hy => ?_⟩
-  have hfy : y∈ball (F 0) δ := by simpa only [F,add_zero] using hy
-  obtain ⟨z,hz,hFz⟩ := planar_closedBall_image_contains_ball F R δ hR hFc hFi hsep hfy
-  exact ⟨a+z,htrans z hz,hFz⟩
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarDiskImage
+
+namespace OAI
+
+/-! The planar open-disk specialization of invariance of domain, proved
+from compact boundary separation and the exponential-covering argument. -/
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+
+namespace SLEExactGauge
+
+theorem actual_planarDiskOpenness : PublishedPlanarDiskOpenness := by
+  intro f x r _hr hf hi
+  apply Metric.isOpen_iff.mpr
+  rintro _ ⟨a,ha,rfl⟩
+  obtain ⟨R,hR,hRa⟩ := Metric.nhds_basis_closedBall.mem_iff.mp (isOpen_ball.mem_nhds ha)
+  let F : ℂ → ℂ := fun z => f (a+z)
+  have htrans (z : ℂ) (hz : z∈closedBall 0 R) : a+z∈ball x r := by
+    apply hRa
+    simpa only [mem_closedBall,dist_eq_norm,add_sub_cancel_left,sub_zero] using hz
+  have hFc : ContinuousOn F (closedBall 0 R) :=
+    hf.comp (continuous_const.add continuous_id).continuousOn htrans
+  have hFi : InjOn F (closedBall 0 R) := by
+    intro z hz w hw he
+    exact add_left_cancel (hi (htrans z hz) (htrans w hw) he)
+  have hpos (z : ℂ) (hz : z∈sphere (0:ℂ) R) : 0<‖F z-F 0‖ := by
+    apply norm_pos_iff.mpr
+    intro he
+    have hez : z=0 := hFi (sphere_subset_closedBall hz) (mem_closedBall_self hR.le)
+      (sub_eq_zero.mp he)
+    have hzr : ‖z‖=R := by simpa only [mem_sphere,dist_zero_right] using hz
+    rw [hez,norm_zero] at hzr
+    exact hR.ne' hzr.symm
+  obtain ⟨δ,hδ,hsep⟩ := (isCompact_sphere (0:ℂ) R).exists_forall_le'
+    ((hFc.mono sphere_subset_closedBall).sub continuousOn_const).norm hpos
+  refine ⟨δ,hδ,fun y hy => ?_⟩
+  have hfy : y∈ball (F 0) δ := by simpa only [F,add_zero] using hy
+  obtain ⟨z,hz,hFz⟩ := planar_closedBall_image_contains_ball F R δ hR hFc hFi hsep hfy
+  exact ⟨a+z,htrans z hz,hFz⟩
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarDomainStops.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarDomainStops.lean
index fdddce1dd850bffcea9f341a879a1725c29064bc..1be9cebbab7e052debc3fbf4463fe75f93ed29ef 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarDomainStops.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarDomainStops.lean
@@ -1,88 +1,88 @@
-import OAI.Probability.SLE.Stochastic.PlanarCutoffStop
-import OAI.Probability.SLE.Loewner.SmoothDomainExhaustion
-
-namespace OAI
-
-/-! A nested sequence of actual stopping times exhausts each finite exit
-of a continuous planar Wiener path from an open domain. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Function
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-namespace SmoothDomainExhaustion
-variable {U : Set (ℝ×ℝ)} (e : SmoothDomainExhaustion U 0)
-
-def stop (T : ℝ≥0) (n : ℕ) : Driver×Driver → ℝ≥0 := planarCutoffStop (e.cutoff n) T
-
-lemma stop_stopping (T : ℝ≥0) (n : ℕ) :
-    IsStoppingTime planarWienerFiltration (fun b => (e.stop T n b : WithTop ℝ≥0)) :=
-  planarCutoffStop_stopping _ (e.smooth n).continuous T
-
-lemma stop_le (T : ℝ≥0) (n : ℕ) (b : Driver×Driver) : e.stop T n b ≤ T :=
-  planarCutoffStop_le _ _ _
-
-lemma stop_mem_support (T : ℝ≥0) (n : ℕ) (b : Driver×Driver) (t : ℝ≥0)
-    (ht : t ≤ e.stop T n b) : planarWiener t b∈tsupport (e.cutoff n) :=
-  planarCutoffStop_in_support _ (by rw [e.based n]; exact one_ne_zero) _ _ _ ht
-
-lemma stop_mem_domain (T : ℝ≥0) (n : ℕ) (b : Driver×Driver) (t : ℝ≥0)
-    (ht : t ≤ e.stop T n b) : planarWiener t b∈U :=
-  e.supported n (e.stop_mem_support T n b t ht)
-
-lemma stop_monotone (T : ℝ≥0) (b : Driver×Driver) : Monotone (fun n => e.stop T n b) := by
-  intro n m hnm
-  have hh : planarCutoffHit (e.cutoff n) b ≤ planarCutoffHit (e.cutoff m) b := by
-    change hittingAfter planarWiener {z | e.cutoff n z=0} 0 b ≤
-      hittingAfter planarWiener {z | e.cutoff m z=0} 0 b
-    apply hittingAfter_anti planarWiener 0
-    intro z hz
-    by_contra hn
-    exact (e.increasing hnm hn) hz
-  apply WithTop.coe_le_coe.mp
-  change (cutoffStop _ _ _ : WithTop ℝ≥0) ≤ (cutoffStop _ _ _ : WithTop ℝ≥0)
-  rw [cutoffStop_coe,cutoffStop_coe]
-  exact min_le_min_right _ hh
-
-lemma eventually_le_stop (T t : ℝ≥0) (ht : t ≤ T) (b : Driver×Driver)
-    (hpath : ∀ s ≤ t, planarWiener s b∈U) : ∀ᶠ n in atTop, t ≤ e.stop T n b := by
-  let K := (fun s => planarWiener s b) '' Icc 0 t
-  have hK : IsCompact K := isCompact_Icc.image (planarWiener_continuous b)
-  have hKU : K ⊆ U := by
-    rintro z ⟨s,hs,rfl⟩
-    exact hpath s hs.2
-  filter_upwards [e.exhausts K hK hKU] with n hn
-  apply planarCutoffStop_ge_of_nonzero _ (e.smooth n).continuous T t ht b
-  intro s hs
-  exact hn ⟨s,⟨zero_le,hs⟩,rfl⟩
-
-lemma stop_tendsto_finite_exit (T τ : ℝ≥0) (hτT : τ ≤ T) (b : Driver×Driver)
-    (hpath : ∀ s < τ, planarWiener s b∈U) (hexit : planarWiener τ b∉U) :
-    Tendsto (fun n => e.stop T n b) atTop (𝓝 τ) := by
-  have hb (n) : e.stop T n b ≤ τ := by
-    have hz : e.cutoff n (planarWiener τ b)=0 := by
-      by_contra hn
-      exact hexit (e.supported n (subset_tsupport _ hn))
-    have hh : planarCutoffHit (e.cutoff n) b ≤ (τ : WithTop ℝ≥0) :=
-      hittingAfter_le_of_mem (zero_le : (0:ℝ≥0) ≤ τ) hz
-    apply WithTop.coe_le_coe.mp
-    change (cutoffStop _ _ _ : WithTop ℝ≥0) ≤ _
-    rw [cutoffStop_coe]
-    exact (min_le_left _ _).trans hh
-  apply tendsto_order.2
-  constructor
-  · intro a ha
-    obtain ⟨t,hat,htτ⟩ := exists_between ha
-    filter_upwards [e.eventually_le_stop T t (htτ.le.trans hτT) b
-      (fun s hs => hpath s (hs.trans_lt htτ))] with n hn
-    exact hat.trans_le hn
-  · intro a ha
-    exact Eventually.of_forall fun n => (hb n).trans_lt ha
-
-end SmoothDomainExhaustion
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarCutoffStop
+import OAI.Probability.SLE.Loewner.SmoothDomainExhaustion
+
+namespace OAI
+
+/-! A nested sequence of actual stopping times exhausts each finite exit
+of a continuous planar Wiener path from an open domain. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Function
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+namespace SmoothDomainExhaustion
+variable {U : Set (ℝ×ℝ)} (e : SmoothDomainExhaustion U 0)
+
+def stop (T : ℝ≥0) (n : ℕ) : Driver×Driver → ℝ≥0 := planarCutoffStop (e.cutoff n) T
+
+lemma stop_stopping (T : ℝ≥0) (n : ℕ) :
+    IsStoppingTime planarWienerFiltration (fun b => (e.stop T n b : WithTop ℝ≥0)) :=
+  planarCutoffStop_stopping _ (e.smooth n).continuous T
+
+lemma stop_le (T : ℝ≥0) (n : ℕ) (b : Driver×Driver) : e.stop T n b ≤ T :=
+  planarCutoffStop_le _ _ _
+
+lemma stop_mem_support (T : ℝ≥0) (n : ℕ) (b : Driver×Driver) (t : ℝ≥0)
+    (ht : t ≤ e.stop T n b) : planarWiener t b∈tsupport (e.cutoff n) :=
+  planarCutoffStop_in_support _ (by rw [e.based n]; exact one_ne_zero) _ _ _ ht
+
+lemma stop_mem_domain (T : ℝ≥0) (n : ℕ) (b : Driver×Driver) (t : ℝ≥0)
+    (ht : t ≤ e.stop T n b) : planarWiener t b∈U :=
+  e.supported n (e.stop_mem_support T n b t ht)
+
+lemma stop_monotone (T : ℝ≥0) (b : Driver×Driver) : Monotone (fun n => e.stop T n b) := by
+  intro n m hnm
+  have hh : planarCutoffHit (e.cutoff n) b ≤ planarCutoffHit (e.cutoff m) b := by
+    change hittingAfter planarWiener {z | e.cutoff n z=0} 0 b ≤
+      hittingAfter planarWiener {z | e.cutoff m z=0} 0 b
+    apply hittingAfter_anti planarWiener 0
+    intro z hz
+    by_contra hn
+    exact (e.increasing hnm hn) hz
+  apply WithTop.coe_le_coe.mp
+  change (cutoffStop _ _ _ : WithTop ℝ≥0) ≤ (cutoffStop _ _ _ : WithTop ℝ≥0)
+  rw [cutoffStop_coe,cutoffStop_coe]
+  exact min_le_min_right _ hh
+
+lemma eventually_le_stop (T t : ℝ≥0) (ht : t ≤ T) (b : Driver×Driver)
+    (hpath : ∀ s ≤ t, planarWiener s b∈U) : ∀ᶠ n in atTop, t ≤ e.stop T n b := by
+  let K := (fun s => planarWiener s b) '' Icc 0 t
+  have hK : IsCompact K := isCompact_Icc.image (planarWiener_continuous b)
+  have hKU : K ⊆ U := by
+    rintro z ⟨s,hs,rfl⟩
+    exact hpath s hs.2
+  filter_upwards [e.exhausts K hK hKU] with n hn
+  apply planarCutoffStop_ge_of_nonzero _ (e.smooth n).continuous T t ht b
+  intro s hs
+  exact hn ⟨s,⟨zero_le,hs⟩,rfl⟩
+
+lemma stop_tendsto_finite_exit (T τ : ℝ≥0) (hτT : τ ≤ T) (b : Driver×Driver)
+    (hpath : ∀ s < τ, planarWiener s b∈U) (hexit : planarWiener τ b∉U) :
+    Tendsto (fun n => e.stop T n b) atTop (𝓝 τ) := by
+  have hb (n) : e.stop T n b ≤ τ := by
+    have hz : e.cutoff n (planarWiener τ b)=0 := by
+      by_contra hn
+      exact hexit (e.supported n (subset_tsupport _ hn))
+    have hh : planarCutoffHit (e.cutoff n) b ≤ (τ : WithTop ℝ≥0) :=
+      hittingAfter_le_of_mem (zero_le : (0:ℝ≥0) ≤ τ) hz
+    apply WithTop.coe_le_coe.mp
+    change (cutoffStop _ _ _ : WithTop ℝ≥0) ≤ _
+    rw [cutoffStop_coe]
+    exact (min_le_left _ _).trans hh
+  apply tendsto_order.2
+  constructor
+  · intro a ha
+    obtain ⟨t,hat,htτ⟩ := exists_between ha
+    filter_upwards [e.eventually_le_stop T t (htτ.le.trans hτT) b
+      (fun s hs => hpath s (hs.trans_lt htτ))] with n hn
+    exact hat.trans_le hn
+  · intro a ha
+    exact Eventually.of_forall fun n => (hb n).trans_lt ha
+
+end SmoothDomainExhaustion
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarDriftError.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarDriftError.lean
index 614a9848538711655ad7e671302263994fd2db72..ad6faefd3383e617f4ec5009f85b70be234745cf 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarDriftError.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarDriftError.lean
@@ -1,102 +1,102 @@
-import OAI.Probability.SLE.Stochastic.PlanarWienerMoments
-
-namespace OAI
-
-/-! Riemann errors for bounded Lipschitz tests along planar Wiener paths. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planarWiener_lipschitz_mean_error
-    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) (A : Set (Driver×Driver))
-    (f : ℝ×ℝ → ℝ) (K L : ℝ≥0) (hb : ∀ z, |f z|≤K) (hf : LipschitzWith L f) :
-    |∫ b in A, f (planarWiener s b)-f (planarWiener (s+δ) b) ∂W.prod W| ≤
-      (L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1 := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hi (t : ℝ≥0) : Integrable (fun b => f (planarWiener t b)) ((W.prod W).restrict A) :=
-    Integrable.of_bound (hf.continuous.measurable.comp (measurable_planarWiener t)).aestronglyMeasurable K
-      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb _)
-  have hni := planarWiener_increment_integrable W hW s δ
-  calc
-    _ ≤ ∫ b in A, |f (planarWiener s b)-f (planarWiener (s+δ) b)| ∂W.prod W := by
-      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm
-        (μ := (W.prod W).restrict A) (fun b => f (planarWiener s b)-f (planarWiener (s+δ) b))
-    _ ≤ ∫ b in A, (L:ℝ)*‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W := by
-      apply integral_mono ((hi s).sub (hi (s+δ))).abs (hni.norm.restrict.const_mul _)
-      intro b
-      change |f (planarWiener s b)-f (planarWiener (s+δ) b)|≤_
-      simpa only [Real.dist_eq,dist_eq_norm,Real.norm_eq_abs,abs_sub_comm] using
-        hf.dist_le_mul (planarWiener (s+δ) b) (planarWiener s b)
-    _ = (L:ℝ)*(∫ b in A, ‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W) := integral_const_mul _ _
-    _ ≤ (L:ℝ)*(∫ b, ‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W) := by
-      exact mul_le_mul_of_nonneg_left
-        (setIntegral_le_integral hni.norm (Eventually.of_forall fun b => norm_nonneg _)) L.coe_nonneg
-    _ ≤ (L:ℝ)*(2*Real.sqrt δ*standardGaussianAbsMoment 1) :=
-      mul_le_mul_of_nonneg_left (planarWiener_increment_norm_mean W hW s δ) L.coe_nonneg
-    _ = _ := by ring
-
-lemma planarWiener_integrated_drift_error
-    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) (A : Set (Driver×Driver))
-    (f : ℝ×ℝ → ℝ) (K L : ℝ≥0) (hb : ∀ z, |f z|≤K) (hf : LipschitzWith L f) :
-    |(δ:ℝ)*(∫ b in A, f (planarWiener s b) ∂W.prod W)-
-      ∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), f (planarWiener u.toNNReal b)) ∂W.prod W| ≤
-      (L:ℝ)*2*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  let C (u : ℝ) (b : Driver×Driver) := f (planarWiener s b)
-  let D (u : ℝ) (b : Driver×Driver) := f (planarWiener u.toNNReal b)
-  let G (u : ℝ) (b : Driver×Driver) := C u b-D u b
-  have hC : Measurable (Function.uncurry C) :=
-    (hf.continuous.measurable.comp (measurable_planarWiener s)).comp measurable_snd
-  have ht : Measurable (fun q : ℝ×(Driver×Driver) => (q.1.toNNReal,q.2)) := by fun_prop
-  have hD : Measurable (Function.uncurry D) :=
-    hf.continuous.measurable.comp (planarWiener_joint_measurable.comp ht)
-  have hG : Measurable (Function.uncurry G) := hC.sub hD
-  have hCb (u b) : |C u b|≤K := hb _
-  have hDb (u b) : |D u b|≤K := hb _
-  have hGb (u b) : |G u b|≤2*(K:ℝ) := (abs_sub _ _).trans (by linarith [hCb u b,hDb u b])
-  have hle : (s:ℝ)≤(s+δ:ℝ≥0) := by simp only [NNReal.coe_add]; linarith [δ.coe_nonneg]
-  have hCI := bounded_interval_integral_integrable (P := (W.prod W).restrict A) _ _ hle C hC K hCb
-  have hDI := bounded_interval_integral_integrable (P := (W.prod W).restrict A) _ _ hle D hD K hDb
-  have he : (δ:ℝ)*(∫ b in A, C 0 b ∂W.prod W)-
-      (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), D u b) ∂W.prod W) =
-      ∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), G u b) ∂W.prod W := by
-    have hc : (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), C u b) ∂W.prod W) =
-        (δ:ℝ)*∫ b in A, C 0 b ∂W.prod W := by
-      simp only [C,intervalIntegral.integral_const,NNReal.coe_add,add_sub_cancel_left,
-        smul_eq_mul,integral_const_mul]
-    rw [← hc,← integral_sub hCI hDI]
-    apply integral_congr_ae
-    exact Eventually.of_forall fun b => (intervalIntegral.integral_sub
-      (bounded_time_section_intervalIntegrable _ _ hle C hC K hCb b)
-      (bounded_time_section_intervalIntegrable _ _ hle D hD K hDb b)).symm
-  change |(δ:ℝ)*(∫ b in A, C 0 b ∂W.prod W)-
-    (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), D u b) ∂W.prod W)|≤_
-  rw [he]
-  have hmean (u : ℝ) (hu : u ∈ Ioc (s:ℝ) (s+δ:ℝ≥0)) :
-      |∫ b in A, G u b ∂W.prod W|≤(L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1 := by
-    have hus : s≤u.toNNReal := by
-      exact_mod_cast (hu.1.le.trans (Real.le_coe_toNNReal u))
-    have hud : u.toNNReal-s≤δ := by
-      apply tsub_le_iff_right.mpr
-      exact (Real.toNNReal_le_iff_le_coe).mpr (by simpa only [add_comm] using hu.2)
-    have ht : s+(u.toNNReal-s)=u.toNNReal := add_tsub_cancel_of_le hus
-    have hh := planarWiener_lipschitz_mean_error W hW s (u.toNNReal-s) A f K L hb hf
-    rw [ht] at hh
-    apply hh.trans
-    apply mul_le_mul_of_nonneg_right _ (standardGaussianAbsMoment_nonneg 1)
-    apply mul_le_mul_of_nonneg_left _ (by positivity)
-    exact Real.sqrt_le_sqrt (by exact_mod_cast hud)
-  have hh := bounded_interval_integral_expectation_bound (P := (W.prod W).restrict A)
-    _ _ hle G hG (2*(K:ℝ)) ((L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1) hGb hmean
-  calc
-    _ ≤ (L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1*((s+δ:ℝ≥0)-s) := hh
-    _ = _ := by simp only [NNReal.coe_add]; ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarWienerMoments
+
+namespace OAI
+
+/-! Riemann errors for bounded Lipschitz tests along planar Wiener paths. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planarWiener_lipschitz_mean_error
+    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) (A : Set (Driver×Driver))
+    (f : ℝ×ℝ → ℝ) (K L : ℝ≥0) (hb : ∀ z, |f z|≤K) (hf : LipschitzWith L f) :
+    |∫ b in A, f (planarWiener s b)-f (planarWiener (s+δ) b) ∂W.prod W| ≤
+      (L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1 := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hi (t : ℝ≥0) : Integrable (fun b => f (planarWiener t b)) ((W.prod W).restrict A) :=
+    Integrable.of_bound (hf.continuous.measurable.comp (measurable_planarWiener t)).aestronglyMeasurable K
+      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb _)
+  have hni := planarWiener_increment_integrable W hW s δ
+  calc
+    _ ≤ ∫ b in A, |f (planarWiener s b)-f (planarWiener (s+δ) b)| ∂W.prod W := by
+      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm
+        (μ := (W.prod W).restrict A) (fun b => f (planarWiener s b)-f (planarWiener (s+δ) b))
+    _ ≤ ∫ b in A, (L:ℝ)*‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W := by
+      apply integral_mono ((hi s).sub (hi (s+δ))).abs (hni.norm.restrict.const_mul _)
+      intro b
+      change |f (planarWiener s b)-f (planarWiener (s+δ) b)|≤_
+      simpa only [Real.dist_eq,dist_eq_norm,Real.norm_eq_abs,abs_sub_comm] using
+        hf.dist_le_mul (planarWiener (s+δ) b) (planarWiener s b)
+    _ = (L:ℝ)*(∫ b in A, ‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W) := integral_const_mul _ _
+    _ ≤ (L:ℝ)*(∫ b, ‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W) := by
+      exact mul_le_mul_of_nonneg_left
+        (setIntegral_le_integral hni.norm (Eventually.of_forall fun b => norm_nonneg _)) L.coe_nonneg
+    _ ≤ (L:ℝ)*(2*Real.sqrt δ*standardGaussianAbsMoment 1) :=
+      mul_le_mul_of_nonneg_left (planarWiener_increment_norm_mean W hW s δ) L.coe_nonneg
+    _ = _ := by ring
+
+lemma planarWiener_integrated_drift_error
+    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) (A : Set (Driver×Driver))
+    (f : ℝ×ℝ → ℝ) (K L : ℝ≥0) (hb : ∀ z, |f z|≤K) (hf : LipschitzWith L f) :
+    |(δ:ℝ)*(∫ b in A, f (planarWiener s b) ∂W.prod W)-
+      ∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), f (planarWiener u.toNNReal b)) ∂W.prod W| ≤
+      (L:ℝ)*2*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  let C (u : ℝ) (b : Driver×Driver) := f (planarWiener s b)
+  let D (u : ℝ) (b : Driver×Driver) := f (planarWiener u.toNNReal b)
+  let G (u : ℝ) (b : Driver×Driver) := C u b-D u b
+  have hC : Measurable (Function.uncurry C) :=
+    (hf.continuous.measurable.comp (measurable_planarWiener s)).comp measurable_snd
+  have ht : Measurable (fun q : ℝ×(Driver×Driver) => (q.1.toNNReal,q.2)) := by fun_prop
+  have hD : Measurable (Function.uncurry D) :=
+    hf.continuous.measurable.comp (planarWiener_joint_measurable.comp ht)
+  have hG : Measurable (Function.uncurry G) := hC.sub hD
+  have hCb (u b) : |C u b|≤K := hb _
+  have hDb (u b) : |D u b|≤K := hb _
+  have hGb (u b) : |G u b|≤2*(K:ℝ) := (abs_sub _ _).trans (by linarith [hCb u b,hDb u b])
+  have hle : (s:ℝ)≤(s+δ:ℝ≥0) := by simp only [NNReal.coe_add]; linarith [δ.coe_nonneg]
+  have hCI := bounded_interval_integral_integrable (P := (W.prod W).restrict A) _ _ hle C hC K hCb
+  have hDI := bounded_interval_integral_integrable (P := (W.prod W).restrict A) _ _ hle D hD K hDb
+  have he : (δ:ℝ)*(∫ b in A, C 0 b ∂W.prod W)-
+      (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), D u b) ∂W.prod W) =
+      ∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), G u b) ∂W.prod W := by
+    have hc : (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), C u b) ∂W.prod W) =
+        (δ:ℝ)*∫ b in A, C 0 b ∂W.prod W := by
+      simp only [C,intervalIntegral.integral_const,NNReal.coe_add,add_sub_cancel_left,
+        smul_eq_mul,integral_const_mul]
+    rw [← hc,← integral_sub hCI hDI]
+    apply integral_congr_ae
+    exact Eventually.of_forall fun b => (intervalIntegral.integral_sub
+      (bounded_time_section_intervalIntegrable _ _ hle C hC K hCb b)
+      (bounded_time_section_intervalIntegrable _ _ hle D hD K hDb b)).symm
+  change |(δ:ℝ)*(∫ b in A, C 0 b ∂W.prod W)-
+    (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), D u b) ∂W.prod W)|≤_
+  rw [he]
+  have hmean (u : ℝ) (hu : u ∈ Ioc (s:ℝ) (s+δ:ℝ≥0)) :
+      |∫ b in A, G u b ∂W.prod W|≤(L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1 := by
+    have hus : s≤u.toNNReal := by
+      exact_mod_cast (hu.1.le.trans (Real.le_coe_toNNReal u))
+    have hud : u.toNNReal-s≤δ := by
+      apply tsub_le_iff_right.mpr
+      exact (Real.toNNReal_le_iff_le_coe).mpr (by simpa only [add_comm] using hu.2)
+    have ht : s+(u.toNNReal-s)=u.toNNReal := add_tsub_cancel_of_le hus
+    have hh := planarWiener_lipschitz_mean_error W hW s (u.toNNReal-s) A f K L hb hf
+    rw [ht] at hh
+    apply hh.trans
+    apply mul_le_mul_of_nonneg_right _ (standardGaussianAbsMoment_nonneg 1)
+    apply mul_le_mul_of_nonneg_left _ (by positivity)
+    exact Real.sqrt_le_sqrt (by exact_mod_cast hud)
+  have hh := bounded_interval_integral_expectation_bound (P := (W.prod W).restrict A)
+    _ _ hle G hG (2*(K:ℝ)) ((L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1) hGb hmean
+  calc
+    _ ≤ (L:ℝ)*2*Real.sqrt δ*standardGaussianAbsMoment 1*((s+δ:ℝ≥0)-s) := hh
+    _ = _ := by simp only [NNReal.coe_add]; ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarDynkin.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarDynkin.lean
index 28ced28a15212e7484a9a792f6ec1570c9526c54..58c28edeee22229784bb3847fadeb4f0b8ab4a17 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarDynkin.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarDynkin.lean
@@ -1,89 +1,89 @@
-import OAI.Probability.SLE.Stochastic.PlanarDynkinStep
-import OAI.Probability.SLE.OptionalStopping
-
-namespace OAI
-
-/-! Dynkin's formula for compact smooth tests of the actual planar Wiener
-path, relative to its full two-coordinate past. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planar_compact_eval_integrable
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (A : Set (Driver×Driver)) (t : ℝ≥0) :
-    Integrable (fun b => f (planarWiener t b)) ((W.prod W).restrict A) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
-  exact Integrable.of_bound (hf.continuous.measurable.comp (measurable_planarWiener t)).aestronglyMeasurable K
-    (Eventually.of_forall fun b => hK _)
-
-lemma planar_generator_joint_measurable
-    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) :
-    Measurable (fun p : ℝ×(Driver×Driver) => planarGenerator f (planarWiener p.1.toNNReal p.2)) := by
-  have ht : Measurable (fun p : ℝ×(Driver×Driver) => (p.1.toNNReal,p.2)) := by fun_prop
-  exact (planarGenerator_smooth f hf).continuous.measurable.comp (planarWiener_joint_measurable.comp ht)
-
-theorem planarWiener_dynkin
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (s t : ℝ≥0) (hst : s≤t) (A : Set (Driver×Driver))
-    (hA : MeasurableSet[planarWienerFiltration s] A) :
-    (∫ b in A, f (planarWiener t b)-f (planarWiener s b) ∂W.prod W) =
-      ∫ b in A, (∫ u in (s:ℝ)..(t:ℝ), planarGenerator f (planarWiener u.toNNReal b)) ∂W.prod W := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨C,hC⟩ := planarWiener_one_step_error W hW f hf hc
-  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
-  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
-  have hG : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
-  have hGb (u b) : |G u b|≤D := hD _
-  let c (r : ℝ≥0) := ∫ b in A, (∫ u in (s:ℝ)..(s+r:ℝ≥0), G u b) ∂W.prod W
-  let v (r : ℝ≥0) := ∫ b in A, f (planarWiener r b) ∂W.prod W
-  let F (r : ℝ≥0) := v (s+r)-c r
-  have hI (r) := planar_compact_eval_integrable W hW f hf hc A r
-  have hinc (r δ : ℝ≥0) : |F (r+δ)-F r|≤(C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
-    have hsr : s ≤ s+r := le_add_of_nonneg_right zero_le
-    have hA' : MeasurableSet[planarWienerFiltration (s+r)] A :=
-      planarWienerFiltration.mono hsr A hA
-    have h := hC (s+r) δ A hA'
-    have hc' : c (r+δ)-c r =
-        ∫ b in A, (∫ u in (s+r:ℝ≥0)..((s+r)+δ:ℝ≥0), G u b) ∂W.prod W := by
-      dsimp only [c]
-      rw [← add_assoc]
-      exact bounded_interval_integral_expectation_sub _ _ _
-        (by exact_mod_cast hsr) (by simp only [NNReal.coe_add]; linarith [δ.coe_nonneg]) G hG D hGb
-    have hv : v ((s+r)+δ)-v (s+r) =
-        ∫ b in A, f (planarWiener ((s+r)+δ) b)-f (planarWiener (s+r) b) ∂W.prod W :=
-      (integral_sub (hI _) (hI _)).symm
-    have he : F (r+δ)-F r =
-        (∫ b in A, f (planarWiener ((s+r)+δ) b)-f (planarWiener (s+r) b) ∂W.prod W)-
-        (∫ b in A, (∫ u in (s+r:ℝ≥0)..((s+r)+δ:ℝ≥0), G u b) ∂W.prod W) := by
-      dsimp only [F]
-      rw [← add_assoc]
-      linarith
-    rw [he]
-    exact h
-  have hsqrt : Tendsto (fun δ : ℝ≥0 => Real.sqrt δ) (𝓝 0) (𝓝 0) := by
-    have hh : Continuous (fun δ : ℝ≥0 => Real.sqrt δ) := by fun_prop
-    simpa only [ContinuousAt,NNReal.coe_zero,Real.sqrt_zero] using hh.continuousAt (x := 0)
-  have heq := eq_of_uniform_small_increment F (fun δ => Real.sqrt δ) C hsqrt hinc 0 (t-s) zero_le
-  have hc0 : c 0=0 := by simp [c]
-  have ht : s+(t-s)=t := add_tsub_cancel_of_le hst
-  change v (s+(t-s))-c (t-s)=v (s+0)-c 0 at heq
-  rw [ht,add_zero,hc0,sub_zero] at heq
-  have hc' : c (t-s)=∫ b in A, (∫ u in (s:ℝ)..(t:ℝ), G u b) ∂W.prod W := by
-    dsimp only [c]
-    rw [ht]
-  rw [integral_sub (hI _) (hI _)]
-  change v t-v s=_
-  linarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarDynkinStep
+import OAI.Probability.SLE.OptionalStopping
+
+namespace OAI
+
+/-! Dynkin's formula for compact smooth tests of the actual planar Wiener
+path, relative to its full two-coordinate past. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planar_compact_eval_integrable
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (A : Set (Driver×Driver)) (t : ℝ≥0) :
+    Integrable (fun b => f (planarWiener t b)) ((W.prod W).restrict A) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
+  exact Integrable.of_bound (hf.continuous.measurable.comp (measurable_planarWiener t)).aestronglyMeasurable K
+    (Eventually.of_forall fun b => hK _)
+
+lemma planar_generator_joint_measurable
+    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) :
+    Measurable (fun p : ℝ×(Driver×Driver) => planarGenerator f (planarWiener p.1.toNNReal p.2)) := by
+  have ht : Measurable (fun p : ℝ×(Driver×Driver) => (p.1.toNNReal,p.2)) := by fun_prop
+  exact (planarGenerator_smooth f hf).continuous.measurable.comp (planarWiener_joint_measurable.comp ht)
+
+theorem planarWiener_dynkin
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (s t : ℝ≥0) (hst : s≤t) (A : Set (Driver×Driver))
+    (hA : MeasurableSet[planarWienerFiltration s] A) :
+    (∫ b in A, f (planarWiener t b)-f (planarWiener s b) ∂W.prod W) =
+      ∫ b in A, (∫ u in (s:ℝ)..(t:ℝ), planarGenerator f (planarWiener u.toNNReal b)) ∂W.prod W := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨C,hC⟩ := planarWiener_one_step_error W hW f hf hc
+  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
+  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
+  have hG : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
+  have hGb (u b) : |G u b|≤D := hD _
+  let c (r : ℝ≥0) := ∫ b in A, (∫ u in (s:ℝ)..(s+r:ℝ≥0), G u b) ∂W.prod W
+  let v (r : ℝ≥0) := ∫ b in A, f (planarWiener r b) ∂W.prod W
+  let F (r : ℝ≥0) := v (s+r)-c r
+  have hI (r) := planar_compact_eval_integrable W hW f hf hc A r
+  have hinc (r δ : ℝ≥0) : |F (r+δ)-F r|≤(C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
+    have hsr : s ≤ s+r := le_add_of_nonneg_right zero_le
+    have hA' : MeasurableSet[planarWienerFiltration (s+r)] A :=
+      planarWienerFiltration.mono hsr A hA
+    have h := hC (s+r) δ A hA'
+    have hc' : c (r+δ)-c r =
+        ∫ b in A, (∫ u in (s+r:ℝ≥0)..((s+r)+δ:ℝ≥0), G u b) ∂W.prod W := by
+      dsimp only [c]
+      rw [← add_assoc]
+      exact bounded_interval_integral_expectation_sub _ _ _
+        (by exact_mod_cast hsr) (by simp only [NNReal.coe_add]; linarith [δ.coe_nonneg]) G hG D hGb
+    have hv : v ((s+r)+δ)-v (s+r) =
+        ∫ b in A, f (planarWiener ((s+r)+δ) b)-f (planarWiener (s+r) b) ∂W.prod W :=
+      (integral_sub (hI _) (hI _)).symm
+    have he : F (r+δ)-F r =
+        (∫ b in A, f (planarWiener ((s+r)+δ) b)-f (planarWiener (s+r) b) ∂W.prod W)-
+        (∫ b in A, (∫ u in (s+r:ℝ≥0)..((s+r)+δ:ℝ≥0), G u b) ∂W.prod W) := by
+      dsimp only [F]
+      rw [← add_assoc]
+      linarith
+    rw [he]
+    exact h
+  have hsqrt : Tendsto (fun δ : ℝ≥0 => Real.sqrt δ) (𝓝 0) (𝓝 0) := by
+    have hh : Continuous (fun δ : ℝ≥0 => Real.sqrt δ) := by fun_prop
+    simpa only [ContinuousAt,NNReal.coe_zero,Real.sqrt_zero] using hh.continuousAt (x := 0)
+  have heq := eq_of_uniform_small_increment F (fun δ => Real.sqrt δ) C hsqrt hinc 0 (t-s) zero_le
+  have hc0 : c 0=0 := by simp [c]
+  have ht : s+(t-s)=t := add_tsub_cancel_of_le hst
+  change v (s+(t-s))-c (t-s)=v (s+0)-c 0 at heq
+  rw [ht,add_zero,hc0,sub_zero] at heq
+  have hc' : c (t-s)=∫ b in A, (∫ u in (s:ℝ)..(t:ℝ), G u b) ∂W.prod W := by
+    dsimp only [c]
+    rw [ht]
+  rw [integral_sub (hI _) (hI _)]
+  change v t-v s=_
+  linarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarDynkinStep.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarDynkinStep.lean
index 33d67c7fcf1fa111c12b34b59e1cb858c8f55a17..2d22bdb46537dd2d7443339e5df168e47d380587 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarDynkinStep.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarDynkinStep.lean
@@ -1,100 +1,100 @@
-import OAI.Probability.SLE.Stochastic.PlanarWienerAverage
-import OAI.Probability.SLE.Stochastic.PlanarGaussianRegularity
-import OAI.Probability.SLE.Stochastic.PlanarDriftError
-import OAI.Probability.SLE.Stochastic.PlanarGeneratorRegularity
-
-namespace OAI
-
-/-! Uniform one-step error for compact smooth planar Brownian tests. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planarWiener_taylor_set_error
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
-    ∃ C : ℝ≥0, ∀ (s δ : ℝ≥0) (A : Set (Driver×Driver)),
-      MeasurableSet[planarWienerFiltration s] A →
-      |(∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
-        (δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W)| ≤
-        (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨C,hC⟩ := compact_planar_gaussian_generator f hf hc
-  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
-  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
-  have hKb (z) : |f z|≤K := by simpa only [Real.norm_eq_abs] using hK z
-  refine ⟨C,?_⟩
-  intro s δ A hA
-  have hI (t) : Integrable (fun b => f (planarWiener t b)) ((W.prod W).restrict A) :=
-    Integrable.of_bound (hf.continuous.measurable.comp (measurable_planarWiener t)).aestronglyMeasurable K
-      (Eventually.of_forall fun b => hK _)
-  have hGI : Integrable (fun b => planarGenerator f (planarWiener s b)) ((W.prod W).restrict A) :=
-    Integrable.of_bound ((planarGenerator_smooth f hf).continuous.measurable.comp
-      (measurable_planarWiener s)).aestronglyMeasurable D
-        (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hD _)
-  have hMI : Integrable (fun b => planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2)
-      ((W.prod W).restrict A) :=
-    Integrable.of_bound ((planarGaussianMean_measurable δ f hf.continuous K hKb).comp
-      (measurable_planarWiener s)).aestronglyMeasurable K
-        (Eventually.of_forall fun b => by
-          simpa only [Real.norm_eq_abs] using
-            (planarGaussianMean_bound δ f hf.continuous K hKb (planarWiener s b)))
-  let R (b : Driver×Driver) := planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2-
-    f (planarWiener s b)-(δ:ℝ)*planarGenerator f (planarWiener s b)
-  have he : (∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
-      (δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W) =
-      ∫ b in A, R b ∂W.prod W := by
-    rw [integral_sub (hI _) (hI _),planarWiener_set_integral_average W hW s δ A hA f hf.continuous K hKb]
-    dsimp only [R]
-    have hSI : Integrable (fun b => planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2-
-        f (planarWiener s b)) ((W.prod W).restrict A) := hMI.sub (hI s)
-    have hDI : Integrable (fun b => (δ:ℝ)*planarGenerator f (planarWiener s b))
-        ((W.prod W).restrict A) := hGI.const_mul _
-    rw [integral_sub hSI hDI,integral_sub hMI (hI s),integral_const_mul]
-  rw [he]
-  have hb := norm_integral_le_of_norm_le_const (μ := (W.prod W).restrict A)
-    (f := R) (C := (C:ℝ)*(δ:ℝ)*Real.sqrt δ)
-    (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hC δ (planarWiener s b))
-  calc
-    _ ≤ (C:ℝ)*(δ:ℝ)*Real.sqrt δ*((W.prod W).restrict A).real univ := by
-      simpa only [Real.norm_eq_abs] using hb
-    _ ≤ (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
-      have hm : ((W.prod W).restrict A).real univ ≤ 1 := by
-        simpa only [measureReal_def,Measure.restrict_apply_univ] using
-          (measureReal_le_one (μ := W.prod W) (s := A))
-      exact mul_le_of_le_one_right (by positivity) hm
-
-lemma planarWiener_one_step_error
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
-    ∃ C : ℝ≥0, ∀ (s δ : ℝ≥0) (A : Set (Driver×Driver)),
-      MeasurableSet[planarWienerFiltration s] A →
-      |(∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
-        (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), planarGenerator f (planarWiener u.toNNReal b)) ∂W.prod W)| ≤
-        (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
-  obtain ⟨C,hC⟩ := planarWiener_taylor_set_error W hW f hf hc
-  obtain ⟨L,hL,hLip⟩ := planarGenerator_bounded_lipschitz f hf hc
-  let M : ℝ≥0 := ⟨standardGaussianAbsMoment 1,standardGaussianAbsMoment_nonneg 1⟩
-  refine ⟨C+2*L*M,?_⟩
-  intro s δ A hA
-  have he := planarWiener_integrated_drift_error W hW s δ A (planarGenerator f) L L hL hLip
-  calc
-    _ ≤ |(∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
-          (δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W)|+
-        |(δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W)-
-          (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), planarGenerator f (planarWiener u.toNNReal b)) ∂W.prod W)| :=
-      abs_sub_le _ _ _
-    _ ≤ (C:ℝ)*(δ:ℝ)*Real.sqrt δ+(L:ℝ)*2*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 :=
-      add_le_add (hC s δ A hA) he
-    _ = _ := by
-      change _ = ((C:ℝ)+2*(L:ℝ)*standardGaussianAbsMoment 1)*(δ:ℝ)*Real.sqrt δ
-      ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarWienerAverage
+import OAI.Probability.SLE.Stochastic.PlanarGaussianRegularity
+import OAI.Probability.SLE.Stochastic.PlanarDriftError
+import OAI.Probability.SLE.Stochastic.PlanarGeneratorRegularity
+
+namespace OAI
+
+/-! Uniform one-step error for compact smooth planar Brownian tests. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planarWiener_taylor_set_error
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
+    ∃ C : ℝ≥0, ∀ (s δ : ℝ≥0) (A : Set (Driver×Driver)),
+      MeasurableSet[planarWienerFiltration s] A →
+      |(∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
+        (δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W)| ≤
+        (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨C,hC⟩ := compact_planar_gaussian_generator f hf hc
+  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
+  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
+  have hKb (z) : |f z|≤K := by simpa only [Real.norm_eq_abs] using hK z
+  refine ⟨C,?_⟩
+  intro s δ A hA
+  have hI (t) : Integrable (fun b => f (planarWiener t b)) ((W.prod W).restrict A) :=
+    Integrable.of_bound (hf.continuous.measurable.comp (measurable_planarWiener t)).aestronglyMeasurable K
+      (Eventually.of_forall fun b => hK _)
+  have hGI : Integrable (fun b => planarGenerator f (planarWiener s b)) ((W.prod W).restrict A) :=
+    Integrable.of_bound ((planarGenerator_smooth f hf).continuous.measurable.comp
+      (measurable_planarWiener s)).aestronglyMeasurable D
+        (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hD _)
+  have hMI : Integrable (fun b => planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2)
+      ((W.prod W).restrict A) :=
+    Integrable.of_bound ((planarGaussianMean_measurable δ f hf.continuous K hKb).comp
+      (measurable_planarWiener s)).aestronglyMeasurable K
+        (Eventually.of_forall fun b => by
+          simpa only [Real.norm_eq_abs] using
+            (planarGaussianMean_bound δ f hf.continuous K hKb (planarWiener s b)))
+  let R (b : Driver×Driver) := planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2-
+    f (planarWiener s b)-(δ:ℝ)*planarGenerator f (planarWiener s b)
+  have he : (∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
+      (δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W) =
+      ∫ b in A, R b ∂W.prod W := by
+    rw [integral_sub (hI _) (hI _),planarWiener_set_integral_average W hW s δ A hA f hf.continuous K hKb]
+    dsimp only [R]
+    have hSI : Integrable (fun b => planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2-
+        f (planarWiener s b)) ((W.prod W).restrict A) := hMI.sub (hI s)
+    have hDI : Integrable (fun b => (δ:ℝ)*planarGenerator f (planarWiener s b))
+        ((W.prod W).restrict A) := hGI.const_mul _
+    rw [integral_sub hSI hDI,integral_sub hMI (hI s),integral_const_mul]
+  rw [he]
+  have hb := norm_integral_le_of_norm_le_const (μ := (W.prod W).restrict A)
+    (f := R) (C := (C:ℝ)*(δ:ℝ)*Real.sqrt δ)
+    (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hC δ (planarWiener s b))
+  calc
+    _ ≤ (C:ℝ)*(δ:ℝ)*Real.sqrt δ*((W.prod W).restrict A).real univ := by
+      simpa only [Real.norm_eq_abs] using hb
+    _ ≤ (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
+      have hm : ((W.prod W).restrict A).real univ ≤ 1 := by
+        simpa only [measureReal_def,Measure.restrict_apply_univ] using
+          (measureReal_le_one (μ := W.prod W) (s := A))
+      exact mul_le_of_le_one_right (by positivity) hm
+
+lemma planarWiener_one_step_error
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
+    ∃ C : ℝ≥0, ∀ (s δ : ℝ≥0) (A : Set (Driver×Driver)),
+      MeasurableSet[planarWienerFiltration s] A →
+      |(∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
+        (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), planarGenerator f (planarWiener u.toNNReal b)) ∂W.prod W)| ≤
+        (C:ℝ)*(δ:ℝ)*Real.sqrt δ := by
+  obtain ⟨C,hC⟩ := planarWiener_taylor_set_error W hW f hf hc
+  obtain ⟨L,hL,hLip⟩ := planarGenerator_bounded_lipschitz f hf hc
+  let M : ℝ≥0 := ⟨standardGaussianAbsMoment 1,standardGaussianAbsMoment_nonneg 1⟩
+  refine ⟨C+2*L*M,?_⟩
+  intro s δ A hA
+  have he := planarWiener_integrated_drift_error W hW s δ A (planarGenerator f) L L hL hLip
+  calc
+    _ ≤ |(∫ b in A, f (planarWiener (s+δ) b)-f (planarWiener s b) ∂W.prod W)-
+          (δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W)|+
+        |(δ:ℝ)*(∫ b in A, planarGenerator f (planarWiener s b) ∂W.prod W)-
+          (∫ b in A, (∫ u in (s:ℝ)..(s+δ:ℝ≥0), planarGenerator f (planarWiener u.toNNReal b)) ∂W.prod W)| :=
+      abs_sub_le _ _ _
+    _ ≤ (C:ℝ)*(δ:ℝ)*Real.sqrt δ+(L:ℝ)*2*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 :=
+      add_le_add (hC s δ A hA) he
+    _ = _ := by
+      change _ = ((C:ℝ)+2*(L:ℝ)*standardGaussianAbsMoment 1)*(δ:ℝ)*Real.sqrt δ
+      ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianMean.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianMean.lean
index 6e18f6c36eac462949697210f3b6df5848004fe3..77bc37ea7794f6bc1caf75da61eefd1f692fb68b 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianMean.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianMean.lean
@@ -1,115 +1,115 @@
-import OAI.Probability.SLE.Stochastic.GaussianTaylorMean
-
-namespace OAI
-
-/-! The two-coordinate Gaussian generator estimate. Its error is uniform
-under the compact smooth test bounds used for stopped planar Brownian motion. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def planarGaussianMean (δ : ℝ≥0) (f : ℝ × ℝ → ℝ) (a b : ℝ) : ℝ :=
-  ∫ y, ∫ x, f (a+x,b+y) ∂gaussianReal 0 δ ∂gaussianReal 0 δ
-
-lemma planar_gaussian_generator_error
-    (f : ℝ × ℝ → ℝ) (hf : Continuous f) (K : ℝ≥0)
-    (hx : ∀ b, ContDiff ℝ 3 (fun a => f (a,b)))
-    (hy : ∀ a, ContDiff ℝ 3 (fun b => f (a,b)))
-    (hxb : ∀ a b j, j ≤ 3 → |iteratedDeriv j (fun x => f (x,b)) a| ≤ K)
-    (hyb : ∀ a b j, j ≤ 3 → |iteratedDeriv j (fun y => f (a,y)) b| ≤ K)
-    (hxy : ∀ a, LipschitzWith K (fun b => iteratedDeriv 2 (fun x => f (x,b)) a))
-    (δ : ℝ≥0) (a b : ℝ) :
-    |planarGaussianMean δ f a b-f (a,b)-(1/2:ℝ)*(δ:ℝ)*
-      (iteratedDeriv 2 (fun x => f (x,b)) a+iteratedDeriv 2 (fun y => f (a,y)) b)| ≤
-      (K:ℝ)*(δ:ℝ)*Real.sqrt δ*
-        (2*standardGaussianAbsMoment 3+(1/2:ℝ)*standardGaussianAbsMoment 1) := by
-  let μ := gaussianReal 0 δ
-  let : IsProbabilityMeasure μ := by dsimp [μ]; infer_instance
-  let A (y : ℝ) := ∫ x, f (a+x,b+y) ∂μ
-  let D (y : ℝ) := iteratedDeriv 2 (fun x => f (x,y)) a
-  let R (y : ℝ) := A y-f (a,b+y)-(1/2:ℝ)*(δ:ℝ)*D (b+y)
-  let e := (K:ℝ)*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 3
-  have hμ : HasLaw (fun y : ℝ => y) μ μ := ⟨measurable_id.aemeasurable, by simp⟩
-  have hfbound (x y : ℝ) : |f (x,y)| ≤ K := by
-    simpa only [iteratedDeriv_zero] using hxb x y 0 (by decide)
-  have hAi : Integrable A μ := by
-    have hprod : Integrable (fun p : ℝ × ℝ => f (a+p.2,b+p.1)) (μ.prod μ) := by
-      apply Integrable.of_bound (hf.comp (by fun_prop)).aestronglyMeasurable K
-      filter_upwards [] with p
-      simpa only [Function.comp_apply,Real.norm_eq_abs] using hfbound (a+p.2) (b+p.1)
-    exact hprod.integral_prod_left
-  have hfi : Integrable (fun y => f (a,b+y)) μ := by
-    apply Integrable.of_bound ((hy a).continuous.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
-    filter_upwards [] with y
-    simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs] using hfbound a (b+y)
-  have hDi : Integrable (fun y => D (b+y)) μ := by
-    apply Integrable.of_bound ((hxy a).continuous.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
-    filter_upwards [] with y
-    simpa only [Function.comp_apply,Pi.add_apply,id_eq,D,Real.norm_eq_abs] using hxb a (b+y) 2 (by decide)
-  have hRi : Integrable R μ := (hAi.sub hfi).sub (hDi.const_mul _)
-  have hR (y : ℝ) : |R y| ≤ e :=
-    gaussian_second_taylor_integral (fun x => f (x,b+y)) (hx (b+y)) K
-      (fun x j hj => hxb x (b+y) j hj) δ a
-  have hRint : |∫ y, R y ∂μ| ≤ e := by
-    simpa only [probReal_univ,mul_one,Real.norm_eq_abs] using
-      norm_integral_le_of_norm_le_const (μ := μ) (f := R) (C := e)
-        (Eventually.of_forall fun y => by simpa only [Real.norm_eq_abs] using hR y)
-  have hY : |(∫ y, f (a,b+y) ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*
-      iteratedDeriv 2 (fun y => f (a,y)) b| ≤ e :=
-    gaussian_second_taylor_integral (fun y => f (a,y)) (hy a) K
-      (fun y j hj => hyb a y j hj) δ b
-  have hDD : Integrable (fun y => D (b+y)-D b) μ := hDi.sub (integrable_const _)
-  have hDDb (y : ℝ) : |D (b+y)-D b| ≤ (K:ℝ)*|y| := by
-    simpa only [D,Real.dist_eq,add_sub_cancel_left] using (hxy a).dist_le_mul (b+y) b
-  have hDDint : |(∫ y, D (b+y) ∂μ)-D b| ≤
-      (K:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 := by
-    have he : (∫ y, D (b+y)-D b ∂μ)=(∫ y, D (b+y) ∂μ)-D b := by
-      rw [integral_sub hDi (integrable_const _)]
-      simp
-    rw [← he]
-    calc
-      |∫ y, D (b+y)-D b ∂μ| ≤ ∫ y, |D (b+y)-D b| ∂μ := by
-        simpa only [Real.norm_eq_abs] using
-          norm_integral_le_integral_norm (μ := μ) (fun y => D (b+y)-D b)
-      _ ≤ ∫ y, (K:ℝ)*|y| ∂μ := integral_mono hDD.abs
-        (by simpa only [pow_one] using (gaussian_abs_moment_integrable hμ 1 (by decide)).const_mul (K:ℝ)) hDDb
-      _ = (K:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 := by
-        rw [integral_const_mul]
-        have he := gaussian_abs_moment hμ 1
-        simp only [pow_one] at he
-        rw [he]
-        ring
-  have he : planarGaussianMean δ f a b-f (a,b)-(1/2:ℝ)*(δ:ℝ)*
-      (D b+iteratedDeriv 2 (fun y => f (a,y)) b) =
-      (∫ y, R y ∂μ)+
-      ((∫ y, f (a,b+y) ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 (fun y => f (a,y)) b)+
-      (1/2:ℝ)*(δ:ℝ)*((∫ y, D (b+y) ∂μ)-D b) := by
-    have h0i : Integrable (fun y => A y-f (a,b+y)) μ := hAi.sub hfi
-    have h1i : Integrable (fun y => (1/2:ℝ)*(δ:ℝ)*D (b+y)) μ := hDi.const_mul _
-    have hRcalc : (∫ y, R y ∂μ)=(∫ y, A y ∂μ)-(∫ y, f (a,b+y) ∂μ)-
-        (1/2:ℝ)*(δ:ℝ)*(∫ y, D (b+y) ∂μ) := by
-      dsimp only [R]
-      rw [integral_sub h0i h1i,integral_sub hAi hfi,integral_const_mul]
-    rw [hRcalc]
-    change (∫ y, A y ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*(D b+_)=_
-    ring
-  change |planarGaussianMean δ f a b-f (a,b)-(1/2:ℝ)*(δ:ℝ)*(D b+_)| ≤ _
-  rw [he]
-  calc
-    _ ≤ |∫ y, R y ∂μ|+
-        |(∫ y, f (a,b+y) ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 (fun y => f (a,y)) b|+
-        |(1/2:ℝ)*(δ:ℝ)*((∫ y, D (b+y) ∂μ)-D b)| :=
-      (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
-    _ ≤ e+e+(1/2:ℝ)*(δ:ℝ)*((K:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1) := by
-      apply add_le_add (add_le_add hRint hY)
-      rw [abs_mul,abs_of_nonneg (by positivity : 0≤(1/2:ℝ)*(δ:ℝ))]
-      exact mul_le_mul_of_nonneg_left hDDint (by positivity)
-    _ = _ := by dsimp [e]; ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.GaussianTaylorMean
+
+namespace OAI
+
+/-! The two-coordinate Gaussian generator estimate. Its error is uniform
+under the compact smooth test bounds used for stopped planar Brownian motion. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def planarGaussianMean (δ : ℝ≥0) (f : ℝ × ℝ → ℝ) (a b : ℝ) : ℝ :=
+  ∫ y, ∫ x, f (a+x,b+y) ∂gaussianReal 0 δ ∂gaussianReal 0 δ
+
+lemma planar_gaussian_generator_error
+    (f : ℝ × ℝ → ℝ) (hf : Continuous f) (K : ℝ≥0)
+    (hx : ∀ b, ContDiff ℝ 3 (fun a => f (a,b)))
+    (hy : ∀ a, ContDiff ℝ 3 (fun b => f (a,b)))
+    (hxb : ∀ a b j, j ≤ 3 → |iteratedDeriv j (fun x => f (x,b)) a| ≤ K)
+    (hyb : ∀ a b j, j ≤ 3 → |iteratedDeriv j (fun y => f (a,y)) b| ≤ K)
+    (hxy : ∀ a, LipschitzWith K (fun b => iteratedDeriv 2 (fun x => f (x,b)) a))
+    (δ : ℝ≥0) (a b : ℝ) :
+    |planarGaussianMean δ f a b-f (a,b)-(1/2:ℝ)*(δ:ℝ)*
+      (iteratedDeriv 2 (fun x => f (x,b)) a+iteratedDeriv 2 (fun y => f (a,y)) b)| ≤
+      (K:ℝ)*(δ:ℝ)*Real.sqrt δ*
+        (2*standardGaussianAbsMoment 3+(1/2:ℝ)*standardGaussianAbsMoment 1) := by
+  let μ := gaussianReal 0 δ
+  let : IsProbabilityMeasure μ := by dsimp [μ]; infer_instance
+  let A (y : ℝ) := ∫ x, f (a+x,b+y) ∂μ
+  let D (y : ℝ) := iteratedDeriv 2 (fun x => f (x,y)) a
+  let R (y : ℝ) := A y-f (a,b+y)-(1/2:ℝ)*(δ:ℝ)*D (b+y)
+  let e := (K:ℝ)*(δ:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 3
+  have hμ : HasLaw (fun y : ℝ => y) μ μ := ⟨measurable_id.aemeasurable, by simp⟩
+  have hfbound (x y : ℝ) : |f (x,y)| ≤ K := by
+    simpa only [iteratedDeriv_zero] using hxb x y 0 (by decide)
+  have hAi : Integrable A μ := by
+    have hprod : Integrable (fun p : ℝ × ℝ => f (a+p.2,b+p.1)) (μ.prod μ) := by
+      apply Integrable.of_bound (hf.comp (by fun_prop)).aestronglyMeasurable K
+      filter_upwards [] with p
+      simpa only [Function.comp_apply,Real.norm_eq_abs] using hfbound (a+p.2) (b+p.1)
+    exact hprod.integral_prod_left
+  have hfi : Integrable (fun y => f (a,b+y)) μ := by
+    apply Integrable.of_bound ((hy a).continuous.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
+    filter_upwards [] with y
+    simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs] using hfbound a (b+y)
+  have hDi : Integrable (fun y => D (b+y)) μ := by
+    apply Integrable.of_bound ((hxy a).continuous.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
+    filter_upwards [] with y
+    simpa only [Function.comp_apply,Pi.add_apply,id_eq,D,Real.norm_eq_abs] using hxb a (b+y) 2 (by decide)
+  have hRi : Integrable R μ := (hAi.sub hfi).sub (hDi.const_mul _)
+  have hR (y : ℝ) : |R y| ≤ e :=
+    gaussian_second_taylor_integral (fun x => f (x,b+y)) (hx (b+y)) K
+      (fun x j hj => hxb x (b+y) j hj) δ a
+  have hRint : |∫ y, R y ∂μ| ≤ e := by
+    simpa only [probReal_univ,mul_one,Real.norm_eq_abs] using
+      norm_integral_le_of_norm_le_const (μ := μ) (f := R) (C := e)
+        (Eventually.of_forall fun y => by simpa only [Real.norm_eq_abs] using hR y)
+  have hY : |(∫ y, f (a,b+y) ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*
+      iteratedDeriv 2 (fun y => f (a,y)) b| ≤ e :=
+    gaussian_second_taylor_integral (fun y => f (a,y)) (hy a) K
+      (fun y j hj => hyb a y j hj) δ b
+  have hDD : Integrable (fun y => D (b+y)-D b) μ := hDi.sub (integrable_const _)
+  have hDDb (y : ℝ) : |D (b+y)-D b| ≤ (K:ℝ)*|y| := by
+    simpa only [D,Real.dist_eq,add_sub_cancel_left] using (hxy a).dist_le_mul (b+y) b
+  have hDDint : |(∫ y, D (b+y) ∂μ)-D b| ≤
+      (K:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 := by
+    have he : (∫ y, D (b+y)-D b ∂μ)=(∫ y, D (b+y) ∂μ)-D b := by
+      rw [integral_sub hDi (integrable_const _)]
+      simp
+    rw [← he]
+    calc
+      |∫ y, D (b+y)-D b ∂μ| ≤ ∫ y, |D (b+y)-D b| ∂μ := by
+        simpa only [Real.norm_eq_abs] using
+          norm_integral_le_integral_norm (μ := μ) (fun y => D (b+y)-D b)
+      _ ≤ ∫ y, (K:ℝ)*|y| ∂μ := integral_mono hDD.abs
+        (by simpa only [pow_one] using (gaussian_abs_moment_integrable hμ 1 (by decide)).const_mul (K:ℝ)) hDDb
+      _ = (K:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1 := by
+        rw [integral_const_mul]
+        have he := gaussian_abs_moment hμ 1
+        simp only [pow_one] at he
+        rw [he]
+        ring
+  have he : planarGaussianMean δ f a b-f (a,b)-(1/2:ℝ)*(δ:ℝ)*
+      (D b+iteratedDeriv 2 (fun y => f (a,y)) b) =
+      (∫ y, R y ∂μ)+
+      ((∫ y, f (a,b+y) ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 (fun y => f (a,y)) b)+
+      (1/2:ℝ)*(δ:ℝ)*((∫ y, D (b+y) ∂μ)-D b) := by
+    have h0i : Integrable (fun y => A y-f (a,b+y)) μ := hAi.sub hfi
+    have h1i : Integrable (fun y => (1/2:ℝ)*(δ:ℝ)*D (b+y)) μ := hDi.const_mul _
+    have hRcalc : (∫ y, R y ∂μ)=(∫ y, A y ∂μ)-(∫ y, f (a,b+y) ∂μ)-
+        (1/2:ℝ)*(δ:ℝ)*(∫ y, D (b+y) ∂μ) := by
+      dsimp only [R]
+      rw [integral_sub h0i h1i,integral_sub hAi hfi,integral_const_mul]
+    rw [hRcalc]
+    change (∫ y, A y ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*(D b+_)=_
+    ring
+  change |planarGaussianMean δ f a b-f (a,b)-(1/2:ℝ)*(δ:ℝ)*(D b+_)| ≤ _
+  rw [he]
+  calc
+    _ ≤ |∫ y, R y ∂μ|+
+        |(∫ y, f (a,b+y) ∂μ)-f (a,b)-(1/2:ℝ)*(δ:ℝ)*iteratedDeriv 2 (fun y => f (a,y)) b|+
+        |(1/2:ℝ)*(δ:ℝ)*((∫ y, D (b+y) ∂μ)-D b)| :=
+      (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
+    _ ≤ e+e+(1/2:ℝ)*(δ:ℝ)*((K:ℝ)*Real.sqrt δ*standardGaussianAbsMoment 1) := by
+      apply add_le_add (add_le_add hRint hY)
+      rw [abs_mul,abs_of_nonneg (by positivity : 0≤(1/2:ℝ)*(δ:ℝ))]
+      exact mul_le_mul_of_nonneg_left hDDint (by positivity)
+    _ = _ := by dsimp [e]; ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianRegularity.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianRegularity.lean
index 233b0b591f0a0e6eb685a330946c38739c8687d1..f3132650491593b1230b3ed4ba1ca053496b56c3 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianRegularity.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarGaussianRegularity.lean
@@ -1,45 +1,45 @@
-import OAI.Probability.SLE.Stochastic.PlanarKernelMean
-
-namespace OAI
-
-/-! Measurability and boundedness of the planar Gaussian averaging operator. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma planarGaussianMean_eq_integral (δ : ℝ≥0) (f : ℝ×ℝ → ℝ)
-    (hf : Continuous f) (K : ℝ≥0) (hb : ∀ z, |f z|≤K) (z : ℝ×ℝ) :
-    planarGaussianMean δ f z.1 z.2 =
-      ∫ y, f (z+y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ) := by
-  have hi : Integrable (fun y : ℝ×ℝ => f (z+y))
-      ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) :=
-    Integrable.of_bound (hf.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
-      (Eventually.of_forall fun y => by
-        simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs] using hb (z+y))
-  exact (integral_prod_symm _ hi).symm
-
-lemma planarGaussianMean_measurable (δ : ℝ≥0) (f : ℝ×ℝ → ℝ)
-    (hf : Continuous f) (K : ℝ≥0) (hb : ∀ z, |f z|≤K) :
-    Measurable (fun z : ℝ×ℝ => planarGaussianMean δ f z.1 z.2) := by
-  simp_rw [planarGaussianMean_eq_integral δ f hf K hb]
-  have hm : Measurable (fun p : (ℝ×ℝ)×(ℝ×ℝ) => f (p.1+p.2)) :=
-    hf.measurable.comp (measurable_fst.add measurable_snd)
-  exact hm.stronglyMeasurable.integral_prod_right'.measurable
-
-lemma planarGaussianMean_bound (δ : ℝ≥0) (f : ℝ×ℝ → ℝ)
-    (hf : Continuous f) (K : ℝ≥0) (hb : ∀ z, |f z|≤K) (z : ℝ×ℝ) :
-    |planarGaussianMean δ f z.1 z.2|≤K := by
-  rw [planarGaussianMean_eq_integral δ f hf K hb]
-  simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using
-    norm_integral_le_of_norm_le_const
-      (μ := (gaussianReal 0 δ).prod (gaussianReal 0 δ))
-      (f := fun y => f (z+y)) (C := (K:ℝ))
-      (Eventually.of_forall fun y => by simpa only [Real.norm_eq_abs] using hb (z+y))
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarKernelMean
+
+namespace OAI
+
+/-! Measurability and boundedness of the planar Gaussian averaging operator. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma planarGaussianMean_eq_integral (δ : ℝ≥0) (f : ℝ×ℝ → ℝ)
+    (hf : Continuous f) (K : ℝ≥0) (hb : ∀ z, |f z|≤K) (z : ℝ×ℝ) :
+    planarGaussianMean δ f z.1 z.2 =
+      ∫ y, f (z+y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ) := by
+  have hi : Integrable (fun y : ℝ×ℝ => f (z+y))
+      ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) :=
+    Integrable.of_bound (hf.comp (continuous_const.add continuous_id)).aestronglyMeasurable K
+      (Eventually.of_forall fun y => by
+        simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs] using hb (z+y))
+  exact (integral_prod_symm _ hi).symm
+
+lemma planarGaussianMean_measurable (δ : ℝ≥0) (f : ℝ×ℝ → ℝ)
+    (hf : Continuous f) (K : ℝ≥0) (hb : ∀ z, |f z|≤K) :
+    Measurable (fun z : ℝ×ℝ => planarGaussianMean δ f z.1 z.2) := by
+  simp_rw [planarGaussianMean_eq_integral δ f hf K hb]
+  have hm : Measurable (fun p : (ℝ×ℝ)×(ℝ×ℝ) => f (p.1+p.2)) :=
+    hf.measurable.comp (measurable_fst.add measurable_snd)
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+
+lemma planarGaussianMean_bound (δ : ℝ≥0) (f : ℝ×ℝ → ℝ)
+    (hf : Continuous f) (K : ℝ≥0) (hb : ∀ z, |f z|≤K) (z : ℝ×ℝ) :
+    |planarGaussianMean δ f z.1 z.2|≤K := by
+  rw [planarGaussianMean_eq_integral δ f hf K hb]
+  simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using
+    norm_integral_le_of_norm_le_const
+      (μ := (gaussianReal 0 δ).prod (gaussianReal 0 δ))
+      (f := fun y => f (z+y)) (C := (K:ℝ))
+      (Eventually.of_forall fun y => by simpa only [Real.norm_eq_abs] using hb (z+y))
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarGeneratorRegularity.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarGeneratorRegularity.lean
index e9e234e3b5e96be1c1e7a411585658495f352483..7e3050981b7aed05e9f8a6661158ece9f18baeec 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarGeneratorRegularity.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarGeneratorRegularity.lean
@@ -1,52 +1,52 @@
-import OAI.Probability.SLE.Geometry.CompactPlanarGaussian
-
-namespace OAI
-
-/-! Compact smooth tests retain bounded smoothness and Lipschitz control
-under the planar Brownian generator. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma planarGenerator_partial_formula
-    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) :
-    planarGenerator f = fun z => (1/2:ℝ)*
-      (spatialPartial (f ∘ Prod.swap) 2 z.swap+spatialPartial f 2 z) := by
-  funext z
-  have hg : ContDiff ℝ (⊤ : ℕ∞) (f ∘ Prod.swap) :=
-    hf.comp (contDiff_snd.prodMk contDiff_fst)
-  rcases z with ⟨a,b⟩
-  rw [spatialPartial_eq_iteratedDeriv _ hg,spatialPartial_eq_iteratedDeriv _ hf]
-  rfl
-
-lemma planarGenerator_smooth
-    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) :
-    ContDiff ℝ (⊤ : ℕ∞) (planarGenerator f) := by
-  rw [planarGenerator_partial_formula f hf]
-  have hg : ContDiff ℝ (⊤ : ℕ∞) (f ∘ Prod.swap) :=
-    hf.comp (contDiff_snd.prodMk contDiff_fst)
-  exact contDiff_const.mul
-    (((spatialPartial_smooth _ hg 2).comp (contDiff_snd.prodMk contDiff_fst)).add
-      (spatialPartial_smooth f hf 2))
-
-lemma planarGenerator_compact
-    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
-    HasCompactSupport (planarGenerator f) := by
-  rw [planarGenerator_partial_formula f hf]
-  have hg : HasCompactSupport (f ∘ Prod.swap) := hc.comp_homeomorph (Homeomorph.prodComm ℝ ℝ)
-  have hh := (spatialPartial_compact _ hg 2).comp_homeomorph (Homeomorph.prodComm ℝ ℝ)
-  exact (hh.add (spatialPartial_compact f hc 2)).mul_left
-
-lemma planarGenerator_bounded_lipschitz
-    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
-    ∃ C : ℝ≥0, (∀ z, |planarGenerator f z|≤C) ∧ LipschitzWith C (planarGenerator f) := by
-  simpa only [Real.norm_eq_abs] using compact_smooth_bounded_lipschitz (planarGenerator f)
-    (planarGenerator_smooth f hf) (planarGenerator_compact f hf hc)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CompactPlanarGaussian
+
+namespace OAI
+
+/-! Compact smooth tests retain bounded smoothness and Lipschitz control
+under the planar Brownian generator. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma planarGenerator_partial_formula
+    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) :
+    planarGenerator f = fun z => (1/2:ℝ)*
+      (spatialPartial (f ∘ Prod.swap) 2 z.swap+spatialPartial f 2 z) := by
+  funext z
+  have hg : ContDiff ℝ (⊤ : ℕ∞) (f ∘ Prod.swap) :=
+    hf.comp (contDiff_snd.prodMk contDiff_fst)
+  rcases z with ⟨a,b⟩
+  rw [spatialPartial_eq_iteratedDeriv _ hg,spatialPartial_eq_iteratedDeriv _ hf]
+  rfl
+
+lemma planarGenerator_smooth
+    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) :
+    ContDiff ℝ (⊤ : ℕ∞) (planarGenerator f) := by
+  rw [planarGenerator_partial_formula f hf]
+  have hg : ContDiff ℝ (⊤ : ℕ∞) (f ∘ Prod.swap) :=
+    hf.comp (contDiff_snd.prodMk contDiff_fst)
+  exact contDiff_const.mul
+    (((spatialPartial_smooth _ hg 2).comp (contDiff_snd.prodMk contDiff_fst)).add
+      (spatialPartial_smooth f hf 2))
+
+lemma planarGenerator_compact
+    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
+    HasCompactSupport (planarGenerator f) := by
+  rw [planarGenerator_partial_formula f hf]
+  have hg : HasCompactSupport (f ∘ Prod.swap) := hc.comp_homeomorph (Homeomorph.prodComm ℝ ℝ)
+  have hh := (spatialPartial_compact _ hg 2).comp_homeomorph (Homeomorph.prodComm ℝ ℝ)
+  exact (hh.add (spatialPartial_compact f hc 2)).mul_left
+
+lemma planarGenerator_bounded_lipschitz
+    (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
+    ∃ C : ℝ≥0, (∀ z, |planarGenerator f z|≤C) ∧ LipschitzWith C (planarGenerator f) := by
+  simpa only [Real.norm_eq_abs] using compact_smooth_bounded_lipschitz (planarGenerator f)
+    (planarGenerator_smooth f hf) (planarGenerator_compact f hf hc)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicLocalization.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicLocalization.lean
index 992a2705cae41b6c53d63bca0b2d2e5d6201422a..7130359db1c41b392862f018e24bbd70d566d26e 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicLocalization.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicLocalization.lean
@@ -1,52 +1,52 @@
-import OAI.Probability.SLE.Stochastic.PlanarOptional
-import OAI.Probability.SLE.Geometry.CompactSmoothExtension
-
-namespace OAI
-
-/-! Compact localization of harmonic tests uses actual stopped planar
-Wiener paths, retaining the original two-coordinate filtration. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planarGenerator_congr_germ {f g : ℝ×ℝ → ℝ} {z : ℝ×ℝ}
-    (h : f =ᶠ[𝓝 z] g) : planarGenerator f z=planarGenerator g z := by
-  have hx : (fun x => f (x,z.2)) =ᶠ[𝓝 z.1] (fun x => g (x,z.2)) :=
-    h.comp_tendsto (continuous_id.prodMk continuous_const).continuousAt.tendsto
-  have hy : (fun y => f (z.1,y)) =ᶠ[𝓝 z.2] (fun y => g (z.1,y)) :=
-    h.comp_tendsto (continuous_const.prodMk continuous_id).continuousAt.tendsto
-  unfold planarGenerator
-  rw [hx.iteratedDeriv_eq 2,hy.iteratedDeriv_eq 2]
-
-theorem planar_local_harmonic_optional
-    (W : WienerLaw) (hW : IsWiener W) (U K : Set (ℝ×ℝ))
-    (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U)
-    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
-    (hhar : ∀ z∈U, planarGenerator u z=0)
-    (τ : Driver×Driver → ℝ≥0)
-    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
-    (hpath : ∀ b t, t ≤ τ b → planarWiener t b∈K) :
-    (∫ b, u (planarWiener (τ b) b) ∂W.prod W)=∫ b, u (planarWiener 0 b) ∂W.prod W := by
-  obtain ⟨f,hf,hfc,_,he,_,_⟩ := exists_compact_smooth_extension U K hU hK hKU u hu
-  have hzero (b : Driver×Driver) (v : ℝ) (hv : 0 ≤ v) (hvτ : v<(τ b:ℝ)) :
-      planarGenerator f (planarWiener v.toNNReal b)=0 := by
-    have ht : v.toNNReal ≤ τ b := (Real.toNNReal_le_iff_le_coe).mpr hvτ.le
-    have hz := hpath b v.toNNReal ht
-    rw [planarGenerator_congr_germ (he _ hz)]
-    exact hhar _ (hKU hz)
-  have hh := planar_compact_optional W hW f hf hfc τ hτ T hb hzero
-  have ht (b) : f (planarWiener (τ b) b)=u (planarWiener (τ b) b) :=
-    (he _ (hpath b (τ b) le_rfl)).self_of_nhds
-  have h0 (b) : f (planarWiener 0 b)=u (planarWiener 0 b) :=
-    (he _ (hpath b 0 zero_le)).self_of_nhds
-  simpa only [ht,h0] using hh
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarOptional
+import OAI.Probability.SLE.Geometry.CompactSmoothExtension
+
+namespace OAI
+
+/-! Compact localization of harmonic tests uses actual stopped planar
+Wiener paths, retaining the original two-coordinate filtration. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planarGenerator_congr_germ {f g : ℝ×ℝ → ℝ} {z : ℝ×ℝ}
+    (h : f =ᶠ[𝓝 z] g) : planarGenerator f z=planarGenerator g z := by
+  have hx : (fun x => f (x,z.2)) =ᶠ[𝓝 z.1] (fun x => g (x,z.2)) :=
+    h.comp_tendsto (continuous_id.prodMk continuous_const).continuousAt.tendsto
+  have hy : (fun y => f (z.1,y)) =ᶠ[𝓝 z.2] (fun y => g (z.1,y)) :=
+    h.comp_tendsto (continuous_const.prodMk continuous_id).continuousAt.tendsto
+  unfold planarGenerator
+  rw [hx.iteratedDeriv_eq 2,hy.iteratedDeriv_eq 2]
+
+theorem planar_local_harmonic_optional
+    (W : WienerLaw) (hW : IsWiener W) (U K : Set (ℝ×ℝ))
+    (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U)
+    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
+    (hhar : ∀ z∈U, planarGenerator u z=0)
+    (τ : Driver×Driver → ℝ≥0)
+    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
+    (hpath : ∀ b t, t ≤ τ b → planarWiener t b∈K) :
+    (∫ b, u (planarWiener (τ b) b) ∂W.prod W)=∫ b, u (planarWiener 0 b) ∂W.prod W := by
+  obtain ⟨f,hf,hfc,_,he,_,_⟩ := exists_compact_smooth_extension U K hU hK hKU u hu
+  have hzero (b : Driver×Driver) (v : ℝ) (hv : 0 ≤ v) (hvτ : v<(τ b:ℝ)) :
+      planarGenerator f (planarWiener v.toNNReal b)=0 := by
+    have ht : v.toNNReal ≤ τ b := (Real.toNNReal_le_iff_le_coe).mpr hvτ.le
+    have hz := hpath b v.toNNReal ht
+    rw [planarGenerator_congr_germ (he _ hz)]
+    exact hhar _ (hKU hz)
+  have hh := planar_compact_optional W hW f hf hfc τ hτ T hb hzero
+  have ht (b) : f (planarWiener (τ b) b)=u (planarWiener (τ b) b) :=
+    (he _ (hpath b (τ b) le_rfl)).self_of_nhds
+  have h0 (b) : f (planarWiener 0 b)=u (planarWiener 0 b) :=
+    (he _ (hpath b 0 zero_le)).self_of_nhds
+  simpa only [ht,h0] using hh
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicSampling.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicSampling.lean
index cf3765c513ef86cae4e6f1a7e409ed7dcc577f45..59dee952be9cc76c2cbd212d9046541d8a144e6e 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicSampling.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarHarmonicSampling.lean
@@ -1,105 +1,105 @@
-import OAI.Probability.SLE.Stochastic.PlanarSamplingLimit
-import OAI.Probability.SLE.Stochastic.PlanarDomainStops
-
-namespace OAI
-
-/-! Conditional harmonic sampling on the actual compact domain stops. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-theorem planar_local_harmonic_optional_between
-    (W : WienerLaw) (hW : IsWiener W) (U K : Set (ℝ×ℝ))
-    (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U)
-    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
-    (hhar : ∀ z∈U, planarGenerator u z=0)
-    (σ τ : Driver×Driver → ℝ≥0)
-    (hσ : IsStoppingTime planarWienerFiltration (fun b => (σ b : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
-    (hστ : ∀ b, σ b ≤ τ b) (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
-    (A : Set (Driver×Driver)) (hA : MeasurableSet[hσ.measurableSpace] A)
-    (hpath : ∀ b t, t ≤ τ b → planarWiener t b∈K) :
-    (∫ b in A, u (planarWiener (σ b) b) ∂W.prod W)=
-      ∫ b in A, u (planarWiener (τ b) b) ∂W.prod W := by
-  obtain ⟨f,hf,hfc,_,he,_,_⟩ := exists_compact_smooth_extension U K hU hK hKU u hu
-  have hzero (b : Driver×Driver) (v : ℝ) (_hv : 0 ≤ v) (hvτ : v<(τ b:ℝ)) :
-      planarGenerator f (planarWiener v.toNNReal b)=0 := by
-    have ht : v.toNNReal ≤ τ b := Real.toNNReal_le_iff_le_coe.mpr hvτ.le
-    have hz := hpath b v.toNNReal ht
-    rw [planarGenerator_congr_germ (he _ hz)]
-    exact hhar _ (hKU hz)
-  have hh := planar_compact_optional_between W hW f hf hfc σ τ hσ hτ hστ T hb A hA hzero
-  have ht (b) : f (planarWiener (τ b) b)=u (planarWiener (τ b) b) :=
-    (he _ (hpath b (τ b) le_rfl)).self_of_nhds
-  have hs (b) : f (planarWiener (σ b) b)=u (planarWiener (σ b) b) :=
-    (he _ (hpath b (σ b) (hστ b))).self_of_nhds
-  simpa only [ht,hs] using hh
-
-namespace SmoothDomainExhaustion
-variable {U : Set (ℝ×ℝ)} (e : SmoothDomainExhaustion U 0)
-
-def stopFiltration (T : ℝ≥0) : Filtration ℕ (inferInstance : MeasurableSpace (Driver×Driver)) where
-  seq n := (e.stop_stopping T n).measurableSpace
-  mono' := fun n m hnm => (e.stop_stopping T n).measurableSpace_mono (e.stop_stopping T m)
-    (fun b => WithTop.coe_le_coe.mpr (e.stop_monotone T b hnm))
-  le' n := (e.stop_stopping T n).measurableSpace_le
-
-lemma harmonic_stopped_adapted (T : ℝ≥0) (hU : IsOpen U)
-    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U) :
-    StronglyAdapted (e.stopFiltration T) (fun n b => u (planarWiener (e.stop T n b) b)) := by
-  intro n
-  obtain ⟨f,hf,_,_,he,_,_⟩ := exists_compact_smooth_extension U (tsupport (e.cutoff n))
-    hU (e.compact n) (e.supported n) u hu
-  have hb := measurable_stoppedValue ((Filtration.stronglyAdapted_natural (fun t =>
-    (measurable_planarWiener t).stronglyMeasurable)).isStronglyProgressive_of_continuous
-      planarWiener_continuous) (e.stop_stopping T n)
-  have hm := hf.continuous.measurable.comp hb
-  have heq (b) : f (planarWiener (e.stop T n b) b)=u (planarWiener (e.stop T n b) b) :=
-    (he _ (e.stop_mem_support T n b _ le_rfl)).self_of_nhds
-  change Measurable[(e.stopFiltration T) n] (fun b => f (planarWiener (e.stop T n b) b)) at hm
-  simpa only [heq] using hm.stronglyMeasurable
-
-theorem harmonic_stopped_martingale
-    (W : WienerLaw) (hW : IsWiener W) (T : ℝ≥0) (hU : IsOpen U)
-    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
-    (hhar : ∀ z∈U, planarGenerator u z=0)
-    (C : ℝ≥0) (huC : ∀ z∈U, |u z|≤C) :
-    Martingale (fun n b => u (planarWiener (e.stop T n b) b)) (e.stopFiltration T) (W.prod W) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have ha := e.harmonic_stopped_adapted T hU u hu
-  apply martingale_of_setIntegral_eq_succ ha
-  · intro n
-    exact Integrable.of_bound ((ha n).mono ((e.stopFiltration T).le n)).aestronglyMeasurable C
-      (Eventually.of_forall fun b => by
-        simpa only [Real.norm_eq_abs] using huC _ (e.stop_mem_domain T n b _ le_rfl))
-  · intro n A hA
-    exact planar_local_harmonic_optional_between W hW U (tsupport (e.cutoff (n+1)))
-      hU (e.compact (n+1)) (e.supported (n+1)) u hu hhar (e.stop T n) (e.stop T (n+1))
-      (e.stop_stopping T n) (e.stop_stopping T (n+1)) (fun b => e.stop_monotone T b (Nat.le_succ n))
-      T (e.stop_le T (n+1)) A hA (e.stop_mem_support T (n+1))
-
-theorem harmonic_stopped_converges
-    (W : WienerLaw) (hW : IsWiener W) (T : ℝ≥0) (hU : IsOpen U)
-    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
-    (hhar : ∀ z∈U, planarGenerator u z=0)
-    (C : ℝ≥0) (huC : ∀ z∈U, |u z|≤C) :
-    ∀ᵐ b ∂W.prod W, ∃ c : ℝ, Tendsto (fun n => u (planarWiener (e.stop T n b) b)) atTop (𝓝 c) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hm := e.harmonic_stopped_martingale W hW T hU u hu hhar C huC
-  apply hm.submartingale.exists_ae_tendsto_of_bdd (R := C)
-  intro n
-  have hb : ∀ᵐ b ∂W.prod W, ‖u (planarWiener (e.stop T n b) b)‖ ≤ C :=
-    Eventually.of_forall fun b => by
-      simpa only [Real.norm_eq_abs] using huC _ (e.stop_mem_domain T n b _ le_rfl)
-  simpa using eLpNorm_le_of_ae_bound (p := 1)
-    (((hm.1 n).mono ((e.stopFiltration T).le n)).aestronglyMeasurable) hb
-
-end SmoothDomainExhaustion
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarSamplingLimit
+import OAI.Probability.SLE.Stochastic.PlanarDomainStops
+
+namespace OAI
+
+/-! Conditional harmonic sampling on the actual compact domain stops. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+theorem planar_local_harmonic_optional_between
+    (W : WienerLaw) (hW : IsWiener W) (U K : Set (ℝ×ℝ))
+    (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U)
+    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
+    (hhar : ∀ z∈U, planarGenerator u z=0)
+    (σ τ : Driver×Driver → ℝ≥0)
+    (hσ : IsStoppingTime planarWienerFiltration (fun b => (σ b : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
+    (hστ : ∀ b, σ b ≤ τ b) (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
+    (A : Set (Driver×Driver)) (hA : MeasurableSet[hσ.measurableSpace] A)
+    (hpath : ∀ b t, t ≤ τ b → planarWiener t b∈K) :
+    (∫ b in A, u (planarWiener (σ b) b) ∂W.prod W)=
+      ∫ b in A, u (planarWiener (τ b) b) ∂W.prod W := by
+  obtain ⟨f,hf,hfc,_,he,_,_⟩ := exists_compact_smooth_extension U K hU hK hKU u hu
+  have hzero (b : Driver×Driver) (v : ℝ) (_hv : 0 ≤ v) (hvτ : v<(τ b:ℝ)) :
+      planarGenerator f (planarWiener v.toNNReal b)=0 := by
+    have ht : v.toNNReal ≤ τ b := Real.toNNReal_le_iff_le_coe.mpr hvτ.le
+    have hz := hpath b v.toNNReal ht
+    rw [planarGenerator_congr_germ (he _ hz)]
+    exact hhar _ (hKU hz)
+  have hh := planar_compact_optional_between W hW f hf hfc σ τ hσ hτ hστ T hb A hA hzero
+  have ht (b) : f (planarWiener (τ b) b)=u (planarWiener (τ b) b) :=
+    (he _ (hpath b (τ b) le_rfl)).self_of_nhds
+  have hs (b) : f (planarWiener (σ b) b)=u (planarWiener (σ b) b) :=
+    (he _ (hpath b (σ b) (hστ b))).self_of_nhds
+  simpa only [ht,hs] using hh
+
+namespace SmoothDomainExhaustion
+variable {U : Set (ℝ×ℝ)} (e : SmoothDomainExhaustion U 0)
+
+def stopFiltration (T : ℝ≥0) : Filtration ℕ (inferInstance : MeasurableSpace (Driver×Driver)) where
+  seq n := (e.stop_stopping T n).measurableSpace
+  mono' := fun n m hnm => (e.stop_stopping T n).measurableSpace_mono (e.stop_stopping T m)
+    (fun b => WithTop.coe_le_coe.mpr (e.stop_monotone T b hnm))
+  le' n := (e.stop_stopping T n).measurableSpace_le
+
+lemma harmonic_stopped_adapted (T : ℝ≥0) (hU : IsOpen U)
+    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U) :
+    StronglyAdapted (e.stopFiltration T) (fun n b => u (planarWiener (e.stop T n b) b)) := by
+  intro n
+  obtain ⟨f,hf,_,_,he,_,_⟩ := exists_compact_smooth_extension U (tsupport (e.cutoff n))
+    hU (e.compact n) (e.supported n) u hu
+  have hb := measurable_stoppedValue ((Filtration.stronglyAdapted_natural (fun t =>
+    (measurable_planarWiener t).stronglyMeasurable)).isStronglyProgressive_of_continuous
+      planarWiener_continuous) (e.stop_stopping T n)
+  have hm := hf.continuous.measurable.comp hb
+  have heq (b) : f (planarWiener (e.stop T n b) b)=u (planarWiener (e.stop T n b) b) :=
+    (he _ (e.stop_mem_support T n b _ le_rfl)).self_of_nhds
+  change Measurable[(e.stopFiltration T) n] (fun b => f (planarWiener (e.stop T n b) b)) at hm
+  simpa only [heq] using hm.stronglyMeasurable
+
+theorem harmonic_stopped_martingale
+    (W : WienerLaw) (hW : IsWiener W) (T : ℝ≥0) (hU : IsOpen U)
+    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
+    (hhar : ∀ z∈U, planarGenerator u z=0)
+    (C : ℝ≥0) (huC : ∀ z∈U, |u z|≤C) :
+    Martingale (fun n b => u (planarWiener (e.stop T n b) b)) (e.stopFiltration T) (W.prod W) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have ha := e.harmonic_stopped_adapted T hU u hu
+  apply martingale_of_setIntegral_eq_succ ha
+  · intro n
+    exact Integrable.of_bound ((ha n).mono ((e.stopFiltration T).le n)).aestronglyMeasurable C
+      (Eventually.of_forall fun b => by
+        simpa only [Real.norm_eq_abs] using huC _ (e.stop_mem_domain T n b _ le_rfl))
+  · intro n A hA
+    exact planar_local_harmonic_optional_between W hW U (tsupport (e.cutoff (n+1)))
+      hU (e.compact (n+1)) (e.supported (n+1)) u hu hhar (e.stop T n) (e.stop T (n+1))
+      (e.stop_stopping T n) (e.stop_stopping T (n+1)) (fun b => e.stop_monotone T b (Nat.le_succ n))
+      T (e.stop_le T (n+1)) A hA (e.stop_mem_support T (n+1))
+
+theorem harmonic_stopped_converges
+    (W : WienerLaw) (hW : IsWiener W) (T : ℝ≥0) (hU : IsOpen U)
+    (u : ℝ×ℝ → ℝ) (hu : ContDiffOn ℝ (⊤ : ℕ∞) u U)
+    (hhar : ∀ z∈U, planarGenerator u z=0)
+    (C : ℝ≥0) (huC : ∀ z∈U, |u z|≤C) :
+    ∀ᵐ b ∂W.prod W, ∃ c : ℝ, Tendsto (fun n => u (planarWiener (e.stop T n b) b)) atTop (𝓝 c) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hm := e.harmonic_stopped_martingale W hW T hU u hu hhar C huC
+  apply hm.submartingale.exists_ae_tendsto_of_bdd (R := C)
+  intro n
+  have hb : ∀ᵐ b ∂W.prod W, ‖u (planarWiener (e.stop T n b) b)‖ ≤ C :=
+    Eventually.of_forall fun b => by
+      simpa only [Real.norm_eq_abs] using huC _ (e.stop_mem_domain T n b _ le_rfl)
+  simpa using eLpNorm_le_of_ae_bound (p := 1)
+    (((hm.1 n).mono ((e.stopFiltration T).le n)).aestronglyMeasurable) hb
+
+end SmoothDomainExhaustion
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarHolomorphicTests.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarHolomorphicTests.lean
index 447525c709177b423016a551236602a5c940f124..e0fc1386ce599729892a9d926a3ddb9490f70431 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarHolomorphicTests.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarHolomorphicTests.lean
@@ -1,82 +1,82 @@
-import OAI.Probability.SLE.Stochastic.PlanarHarmonicSampling
-import OAI.Probability.SLE.Geometry.ComplexHarmonicGenerator
-
-namespace OAI
-
-/-! Bounded holomorphic observables on the physical planar domain give
-the two real harmonic tests used for conformal exit limits. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-def planarTranslatedDomain (D : Set ℂ) (w : ℂ) : Set (ℝ×ℝ) :=
-  {z | w+Complex.equivRealProdCLM.symm z∈D}
-
-lemma planarTranslatedDomain_open {D : Set ℂ} (hD : IsOpen D) (w : ℂ) :
-    IsOpen (planarTranslatedDomain D w) :=
-  hD.preimage (continuous_const.add Complex.equivRealProdCLM.symm.continuous)
-
-lemma holomorphic_planar_components {D : Set ℂ} (hD : IsOpen D)
-    {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F D) (w : ℂ) :
-    (ContDiffOn ℝ (⊤ : ℕ∞) (fun z => (F (w+Complex.equivRealProdCLM.symm z)).re)
-      (planarTranslatedDomain D w) ∧
-      ∀ z∈planarTranslatedDomain D w,
-        planarGenerator (fun z => (F (w+Complex.equivRealProdCLM.symm z)).re) z=0) ∧
-    (ContDiffOn ℝ (⊤ : ℕ∞) (fun z => (F (w+Complex.equivRealProdCLM.symm z)).im)
-      (planarTranslatedDomain D w) ∧
-      ∀ z∈planarTranslatedDomain D w,
-        planarGenerator (fun z => (F (w+Complex.equivRealProdCLM.symm z)).im) z=0) := by
-  let G (z : ℂ) := F (w+z)
-  have hG (z : ℂ) (hz : w+z∈D) : AnalyticAt ℂ G z :=
-    (hF.analyticOnNhd hD _ hz).comp (analyticAt_const.add analyticAt_id)
-  have hr (z) (hz : z∈planarTranslatedDomain D w) :
-      ContDiffAt ℝ (⊤ : ℕ∞) (fun z => (G (Complex.equivRealProdCLM.symm z)).re) z :=
-    Complex.reCLM.contDiff.contDiffAt.comp z
-      (((hG _ hz).contDiffAt.restrict_scalars ℝ).comp z
-        Complex.equivRealProdCLM.symm.contDiff.contDiffAt)
-  have hi (z) (hz : z∈planarTranslatedDomain D w) :
-      ContDiffAt ℝ (⊤ : ℕ∞) (fun z => (G (Complex.equivRealProdCLM.symm z)).im) z :=
-    Complex.imCLM.contDiff.contDiffAt.comp z
-      (((hG _ hz).contDiffAt.restrict_scalars ℝ).comp z
-        Complex.equivRealProdCLM.symm.contDiff.contDiffAt)
-  refine ⟨⟨?_,?_⟩,⟨?_,?_⟩⟩
-  · exact fun z hz => (hr z hz).contDiffWithinAt
-  · intro z hz
-    exact planarGenerator_harmonicAt (fun v => (G v).re) z (hG _ hz).harmonicAt_re
-  · exact fun z hz => (hi z hz).contDiffWithinAt
-  · intro z hz
-    exact planarGenerator_harmonicAt (fun v => (G v).im) z (hG _ hz).harmonicAt_im
-
-theorem holomorphic_stopped_converges
-    (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D)
-    (w : ℂ) {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F D)
-    (e : SmoothDomainExhaustion (planarTranslatedDomain D w) 0)
-    (T : ℝ≥0) (C : ℝ≥0) (hC : ∀ z∈D, ‖F z‖ ≤ C) :
-    ∀ᵐ b ∂W.prod W, ∃ c : ℂ,
-      Tendsto (fun n => F (w+Complex.equivRealProdCLM.symm (planarWiener (e.stop T n b) b)))
-        atTop (𝓝 c) := by
-  obtain ⟨⟨hr,hr0⟩,⟨hi,hi0⟩⟩ := holomorphic_planar_components hD hF w
-  have hreal := e.harmonic_stopped_converges W hW T (planarTranslatedDomain_open hD w)
-    _ hr hr0 C (fun z hz => (Complex.abs_re_le_norm _).trans (hC _ hz))
-  have himag := e.harmonic_stopped_converges W hW T (planarTranslatedDomain_open hD w)
-    _ hi hi0 C (fun z hz => (Complex.abs_im_le_norm _).trans (hC _ hz))
-  filter_upwards [hreal,himag] with b hb hi
-  obtain ⟨a,ha⟩ := hb
-  obtain ⟨c,hc⟩ := hi
-  refine ⟨Complex.equivRealProdCLM.symm (a,c),?_⟩
-  have hh := Complex.equivRealProdCLM.symm.continuous.continuousAt.tendsto.comp
-    (ha.prodMk_nhds hc)
-  change Tendsto (fun n => Complex.equivRealProdCLM.symm
-    ((F (w+Complex.equivRealProdCLM.symm (planarWiener (e.stop T n b) b))).re,
-     (F (w+Complex.equivRealProdCLM.symm (planarWiener (e.stop T n b) b))).im))
-    atTop (𝓝 (Complex.equivRealProdCLM.symm (a,c))) at hh
-  simpa only [Complex.equivRealProdCLM_symm_apply,Complex.re_add_im] using hh
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarHarmonicSampling
+import OAI.Probability.SLE.Geometry.ComplexHarmonicGenerator
+
+namespace OAI
+
+/-! Bounded holomorphic observables on the physical planar domain give
+the two real harmonic tests used for conformal exit limits. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+def planarTranslatedDomain (D : Set ℂ) (w : ℂ) : Set (ℝ×ℝ) :=
+  {z | w+Complex.equivRealProdCLM.symm z∈D}
+
+lemma planarTranslatedDomain_open {D : Set ℂ} (hD : IsOpen D) (w : ℂ) :
+    IsOpen (planarTranslatedDomain D w) :=
+  hD.preimage (continuous_const.add Complex.equivRealProdCLM.symm.continuous)
+
+lemma holomorphic_planar_components {D : Set ℂ} (hD : IsOpen D)
+    {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F D) (w : ℂ) :
+    (ContDiffOn ℝ (⊤ : ℕ∞) (fun z => (F (w+Complex.equivRealProdCLM.symm z)).re)
+      (planarTranslatedDomain D w) ∧
+      ∀ z∈planarTranslatedDomain D w,
+        planarGenerator (fun z => (F (w+Complex.equivRealProdCLM.symm z)).re) z=0) ∧
+    (ContDiffOn ℝ (⊤ : ℕ∞) (fun z => (F (w+Complex.equivRealProdCLM.symm z)).im)
+      (planarTranslatedDomain D w) ∧
+      ∀ z∈planarTranslatedDomain D w,
+        planarGenerator (fun z => (F (w+Complex.equivRealProdCLM.symm z)).im) z=0) := by
+  let G (z : ℂ) := F (w+z)
+  have hG (z : ℂ) (hz : w+z∈D) : AnalyticAt ℂ G z :=
+    (hF.analyticOnNhd hD _ hz).comp (analyticAt_const.add analyticAt_id)
+  have hr (z) (hz : z∈planarTranslatedDomain D w) :
+      ContDiffAt ℝ (⊤ : ℕ∞) (fun z => (G (Complex.equivRealProdCLM.symm z)).re) z :=
+    Complex.reCLM.contDiff.contDiffAt.comp z
+      (((hG _ hz).contDiffAt.restrict_scalars ℝ).comp z
+        Complex.equivRealProdCLM.symm.contDiff.contDiffAt)
+  have hi (z) (hz : z∈planarTranslatedDomain D w) :
+      ContDiffAt ℝ (⊤ : ℕ∞) (fun z => (G (Complex.equivRealProdCLM.symm z)).im) z :=
+    Complex.imCLM.contDiff.contDiffAt.comp z
+      (((hG _ hz).contDiffAt.restrict_scalars ℝ).comp z
+        Complex.equivRealProdCLM.symm.contDiff.contDiffAt)
+  refine ⟨⟨?_,?_⟩,⟨?_,?_⟩⟩
+  · exact fun z hz => (hr z hz).contDiffWithinAt
+  · intro z hz
+    exact planarGenerator_harmonicAt (fun v => (G v).re) z (hG _ hz).harmonicAt_re
+  · exact fun z hz => (hi z hz).contDiffWithinAt
+  · intro z hz
+    exact planarGenerator_harmonicAt (fun v => (G v).im) z (hG _ hz).harmonicAt_im
+
+theorem holomorphic_stopped_converges
+    (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D)
+    (w : ℂ) {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F D)
+    (e : SmoothDomainExhaustion (planarTranslatedDomain D w) 0)
+    (T : ℝ≥0) (C : ℝ≥0) (hC : ∀ z∈D, ‖F z‖ ≤ C) :
+    ∀ᵐ b ∂W.prod W, ∃ c : ℂ,
+      Tendsto (fun n => F (w+Complex.equivRealProdCLM.symm (planarWiener (e.stop T n b) b)))
+        atTop (𝓝 c) := by
+  obtain ⟨⟨hr,hr0⟩,⟨hi,hi0⟩⟩ := holomorphic_planar_components hD hF w
+  have hreal := e.harmonic_stopped_converges W hW T (planarTranslatedDomain_open hD w)
+    _ hr hr0 C (fun z hz => (Complex.abs_re_le_norm _).trans (hC _ hz))
+  have himag := e.harmonic_stopped_converges W hW T (planarTranslatedDomain_open hD w)
+    _ hi hi0 C (fun z hz => (Complex.abs_im_le_norm _).trans (hC _ hz))
+  filter_upwards [hreal,himag] with b hb hi
+  obtain ⟨a,ha⟩ := hb
+  obtain ⟨c,hc⟩ := hi
+  refine ⟨Complex.equivRealProdCLM.symm (a,c),?_⟩
+  have hh := Complex.equivRealProdCLM.symm.continuous.continuousAt.tendsto.comp
+    (ha.prodMk_nhds hc)
+  change Tendsto (fun n => Complex.equivRealProdCLM.symm
+    ((F (w+Complex.equivRealProdCLM.symm (planarWiener (e.stop T n b) b))).re,
+     (F (w+Complex.equivRealProdCLM.symm (planarWiener (e.stop T n b) b))).im))
+    atTop (𝓝 (Complex.equivRealProdCLM.symm (a,c))) at hh
+  simpa only [Complex.equivRealProdCLM_symm_apply,Complex.re_add_im] using hh
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarKernelMean.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarKernelMean.lean
index b0e84710c098d1b37aea0763b2e29cd7a905075c..36beb617e629cfafc7fba3a72b3f4b63c11bf454 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarKernelMean.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarKernelMean.lean
@@ -1,40 +1,40 @@
-import OAI.Probability.SLE.Geometry.CompactPlanarGaussian
-import OAI.Probability.SLE.Stochastic.PlanarWienerIncrements
-
-namespace OAI
-
-/-! Conditional averaging against the actual product Gaussian increment.
-This is the integral form needed for the planar Brownian generator. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma independent_planar_gaussian_integral
-    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (Z : Ω → X) (Y : Ω → ℝ×ℝ) (hZ : Measurable Z) (_hY : Measurable Y)
-    (δ : ℝ≥0)
-    (hLaw : HasLaw Y ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) P)
-    (hind : IndepFun Z Y P)
-    (H : X×(ℝ×ℝ) → ℝ) (hH : Measurable H) (C : ℝ)
-    (hbound : ∀ p, |H p| ≤ C) :
-    (∫ ω, H (Z ω,Y ω) ∂P) =
-      ∫ ω, ∫ y, H (Z ω,y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ) ∂P := by
-  let ν := (gaussianReal 0 δ).prod (gaussianReal 0 δ)
-  have hZLaw : HasLaw Z (P.map Z) P := ⟨hZ.aemeasurable,rfl⟩
-  have hj := hind.hasLaw_prod hZLaw hLaw
-  have hHi : Integrable H ((P.map Z).prod ν) :=
-    Integrable.of_bound hH.aestronglyMeasurable C
-      (Eventually.of_forall fun p => by simpa only [Real.norm_eq_abs] using hbound p)
-  calc
-    _ = ∫ p, H p ∂(P.map Z).prod ν := hj.integral_comp hH.aestronglyMeasurable
-    _ = ∫ x, ∫ y, H (x,y) ∂ν ∂P.map Z := integral_prod H hHi
-    _ = _ := (hZLaw.integral_comp hHi.integral_prod_left.aestronglyMeasurable).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.CompactPlanarGaussian
+import OAI.Probability.SLE.Stochastic.PlanarWienerIncrements
+
+namespace OAI
+
+/-! Conditional averaging against the actual product Gaussian increment.
+This is the integral form needed for the planar Brownian generator. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma independent_planar_gaussian_integral
+    {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (Z : Ω → X) (Y : Ω → ℝ×ℝ) (hZ : Measurable Z) (_hY : Measurable Y)
+    (δ : ℝ≥0)
+    (hLaw : HasLaw Y ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) P)
+    (hind : IndepFun Z Y P)
+    (H : X×(ℝ×ℝ) → ℝ) (hH : Measurable H) (C : ℝ)
+    (hbound : ∀ p, |H p| ≤ C) :
+    (∫ ω, H (Z ω,Y ω) ∂P) =
+      ∫ ω, ∫ y, H (Z ω,y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ) ∂P := by
+  let ν := (gaussianReal 0 δ).prod (gaussianReal 0 δ)
+  have hZLaw : HasLaw Z (P.map Z) P := ⟨hZ.aemeasurable,rfl⟩
+  have hj := hind.hasLaw_prod hZLaw hLaw
+  have hHi : Integrable H ((P.map Z).prod ν) :=
+    Integrable.of_bound hH.aestronglyMeasurable C
+      (Eventually.of_forall fun p => by simpa only [Real.norm_eq_abs] using hbound p)
+  calc
+    _ = ∫ p, H p ∂(P.map Z).prod ν := hj.integral_comp hH.aestronglyMeasurable
+    _ = ∫ x, ∫ y, H (x,y) ∂ν ∂P.map Z := integral_prod H hHi
+    _ = _ := (hZLaw.integral_comp hHi.integral_prod_left.aestronglyMeasurable).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarLogLifting.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarLogLifting.lean
index ea5c19ba9ff36681a4ef88d7c388d1900c9cd278..e7324b3227a6401a71482f67ce352cd3a4151eaf 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarLogLifting.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarLogLifting.lean
@@ -1,55 +1,55 @@
-import OAI.Probability.SLE.Geometry.PublishedPlanarDiskOpenness
-import Mathlib.Analysis.Complex.CoveringMap
-import Mathlib.Topology.Homotopy.Lifting
-
-namespace OAI
-
-/-! Elementary lifting facts used to prove the planar disk openness input.
-An odd nonzero function on a connected antipodal space cannot have a
-continuous logarithm. -/
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma continuous_log_at_end_of_homotopy {A : Type*} [TopologicalSpace A]
-    (H : C(unitInterval × A,ℂ)) (hH : ∀ q, H q≠0)
-    (L : C(A,ℂ)) (hL : ∀ a, Complex.exp (L a)=H (0,a)) :
-    ∃ M : C(A,ℂ), ∀ a, Complex.exp (M a)=H (1,a) := by
-  let H' : C(unitInterval × A,{z : ℂ // z≠0}) :=
-    ⟨fun q => ⟨H q,hH q⟩, H.continuous.subtype_mk _⟩
-  have h0 (a : A) : H' (0,a)=⟨Complex.exp (L a),Complex.exp_ne_zero _⟩ :=
-    Subtype.ext (hL a).symm
-  let G := Complex.isCoveringMap_exp.liftHomotopy H' L h0
-  refine ⟨G.comp ⟨fun a => (1,a),continuous_const.prodMk continuous_id⟩,fun a => ?_⟩
-  have he := congr_fun (Complex.isCoveringMap_exp.liftHomotopy_lifts H' L h0) (1,a)
-  exact congrArg Subtype.val he
-
-lemma no_continuous_log_of_odd_involution {A : Type*} [TopologicalSpace A]
-    [PreconnectedSpace A] (J : A → A) (hJ : Continuous J) (hinv : Involutive J)
-    (F L : C(A,ℂ)) (hne : ∀ a, F a≠0) (hodd : ∀ a, F (J a)= -F a)
-    (hL : ∀ a, Complex.exp (L a)=F a) (a : A) : False := by
-  let d : A → ℂ := fun x => L (J x)-L x
-  have hd : Continuous d := (L.continuous.comp hJ).sub L.continuous
-  have he (x : A) : Complex.exp (d x)= -1 := by
-    simp only [d,Complex.exp_sub,hL,hodd]
-    exact neg_div_self (hne x)
-  have hconst (x y : A) : d x=d y :=
-    Complex.isCoveringMap_exp.const_of_comp hd (by
-      intro x y
-      apply Subtype.ext
-      exact (he x).trans (he y).symm) x y
-  have hi : d (J a)= -d a := by simp only [d,hinv a]; ring
-  have hz : d a=0 := by
-    have heq := (hconst a (J a)).trans hi
-    linear_combination (1/2:ℂ)*heq
-  have := he a
-  rw [hz,Complex.exp_zero] at this
-  norm_num at this
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.PublishedPlanarDiskOpenness
+import Mathlib.Analysis.Complex.CoveringMap
+import Mathlib.Topology.Homotopy.Lifting
+
+namespace OAI
+
+/-! Elementary lifting facts used to prove the planar disk openness input.
+An odd nonzero function on a connected antipodal space cannot have a
+continuous logarithm. -/
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma continuous_log_at_end_of_homotopy {A : Type*} [TopologicalSpace A]
+    (H : C(unitInterval × A,ℂ)) (hH : ∀ q, H q≠0)
+    (L : C(A,ℂ)) (hL : ∀ a, Complex.exp (L a)=H (0,a)) :
+    ∃ M : C(A,ℂ), ∀ a, Complex.exp (M a)=H (1,a) := by
+  let H' : C(unitInterval × A,{z : ℂ // z≠0}) :=
+    ⟨fun q => ⟨H q,hH q⟩, H.continuous.subtype_mk _⟩
+  have h0 (a : A) : H' (0,a)=⟨Complex.exp (L a),Complex.exp_ne_zero _⟩ :=
+    Subtype.ext (hL a).symm
+  let G := Complex.isCoveringMap_exp.liftHomotopy H' L h0
+  refine ⟨G.comp ⟨fun a => (1,a),continuous_const.prodMk continuous_id⟩,fun a => ?_⟩
+  have he := congr_fun (Complex.isCoveringMap_exp.liftHomotopy_lifts H' L h0) (1,a)
+  exact congrArg Subtype.val he
+
+lemma no_continuous_log_of_odd_involution {A : Type*} [TopologicalSpace A]
+    [PreconnectedSpace A] (J : A → A) (hJ : Continuous J) (hinv : Involutive J)
+    (F L : C(A,ℂ)) (hne : ∀ a, F a≠0) (hodd : ∀ a, F (J a)= -F a)
+    (hL : ∀ a, Complex.exp (L a)=F a) (a : A) : False := by
+  let d : A → ℂ := fun x => L (J x)-L x
+  have hd : Continuous d := (L.continuous.comp hJ).sub L.continuous
+  have he (x : A) : Complex.exp (d x)= -1 := by
+    simp only [d,Complex.exp_sub,hL,hodd]
+    exact neg_div_self (hne x)
+  have hconst (x y : A) : d x=d y :=
+    Complex.isCoveringMap_exp.const_of_comp hd (by
+      intro x y
+      apply Subtype.ext
+      exact (he x).trans (he y).symm) x y
+  have hi : d (J a)= -d a := by simp only [d,hinv a]; ring
+  have hz : d a=0 := by
+    have heq := (hconst a (J a)).trans hi
+    linear_combination (1/2:ℂ)*heq
+  have := he a
+  rw [hz,Complex.exp_zero] at this
+  norm_num at this
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarMartingale.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarMartingale.lean
index 055371ca4a0b28f39a5baf4ef9288164549c8f5c..4d9f5406839bc99c928ea4d32fee3fa2ce6f20df 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarMartingale.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarMartingale.lean
@@ -1,77 +1,77 @@
-import OAI.Probability.SLE.Stochastic.PlanarCompensated
-import OAI.Probability.SLE.Stochastic.PlanarCutoffStop
-
-namespace OAI
-
-/-! The full-past martingale associated with each compact smooth planar
-test. This supplies conditional optional sampling at localization times. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planarWiener_frozen_measurable (T : ℝ≥0) :
-    @Measurable (ℝ×(Driver×Driver)) (ℝ×ℝ)
-      ((borel ℝ).prod (planarWienerFiltration T)) inferInstance
-      (fun p => planarWiener (min p.1.toNNReal T) p.2) := by
-  have hp := (Filtration.stronglyAdapted_natural (fun t =>
-    (measurable_planarWiener t).stronglyMeasurable)).isStronglyProgressive_of_continuous
-      planarWiener_continuous
-  let : MeasurableSpace (Driver×Driver) := planarWienerFiltration T
-  have ht : Measurable (fun p : ℝ×(Driver×Driver) =>
-      ((⟨min p.1.toNNReal T,mem_Iic.mpr (min_le_right _ _)⟩ : Iic T),p.2)) :=
-    ((measurable_fst.real_toNNReal.min measurable_const).subtype_mk).prodMk measurable_snd
-  exact (hp T).measurable.comp ht
-
-lemma planar_compensator_adapted (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f)
-    (T : ℝ≥0) : Measurable[planarWienerFiltration T]
-      (fun b => ∫ u in (0:ℝ)..(T:ℝ), planarGenerator f (planarWiener u.toNNReal b)) := by
-  let : MeasurableSpace (Driver×Driver) := planarWienerFiltration T
-  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener (min u.toNNReal T) b)
-  have hm : Measurable (Function.uncurry G) :=
-    (planarGenerator_smooth f hf).continuous.measurable.comp (planarWiener_frozen_measurable T)
-  have hi := (cumulative_integral_measurable G hm).comp
-    ((measurable_const (a := T)).prodMk measurable_id)
-  have he (b : Driver×Driver) : (∫ u in (0:ℝ)..(T:ℝ), G u b)=
-      ∫ u in (0:ℝ)..(T:ℝ), planarGenerator f (planarWiener u.toNNReal b) := by
-    apply intervalIntegral.integral_congr_Ioo_of_le T.coe_nonneg
-    intro u hu
-    have huT : u.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr hu.2.le
-    simp only [G,min_eq_left huT]
-  change Measurable (fun b => ∫ u in (0:ℝ)..(T:ℝ), G u b) at hi
-  simpa only [he] using hi
-
-lemma planar_compensated_stronglyAdapted (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) : StronglyAdapted planarWienerFiltration (planarCompensated f) := by
-  intro t
-  exact ((hf.continuous.measurable.comp (planarWiener_adapted t)).sub
-    (planar_compensator_adapted f hf t)).stronglyMeasurable
-
-theorem planar_compensated_martingale
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
-    Martingale (planarCompensated f) planarWienerFiltration (W.prod W) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have ha := planar_compensated_stronglyAdapted f hf
-  refine ⟨ha,fun s t hst => ?_⟩
-  apply Filter.EventuallyEq.symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq (planarWienerFiltration.le s)
-    (planar_compensated_integrable W hW f hf hc t)
-    (fun A _ _ => (planar_compensated_integrable W hW f hf hc s).integrableOn)
-    ?_ (ha s).aestronglyMeasurable
-  intro A hA _
-  have hh := planar_compensated_past_integral_zero W hW f hf hc s t hst A hA
-  have hi : Integrable (fun b => planarCompensated f t b) ((W.prod W).restrict A) :=
-    (planar_compensated_integrable W hW f hf hc t).restrict
-  have hj : Integrable (fun b => planarCompensated f s b) ((W.prod W).restrict A) :=
-    (planar_compensated_integrable W hW f hf hc s).restrict
-  rw [integral_sub hi hj] at hh
-  exact (sub_eq_zero.mp hh).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarCompensated
+import OAI.Probability.SLE.Stochastic.PlanarCutoffStop
+
+namespace OAI
+
+/-! The full-past martingale associated with each compact smooth planar
+test. This supplies conditional optional sampling at localization times. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planarWiener_frozen_measurable (T : ℝ≥0) :
+    @Measurable (ℝ×(Driver×Driver)) (ℝ×ℝ)
+      ((borel ℝ).prod (planarWienerFiltration T)) inferInstance
+      (fun p => planarWiener (min p.1.toNNReal T) p.2) := by
+  have hp := (Filtration.stronglyAdapted_natural (fun t =>
+    (measurable_planarWiener t).stronglyMeasurable)).isStronglyProgressive_of_continuous
+      planarWiener_continuous
+  let : MeasurableSpace (Driver×Driver) := planarWienerFiltration T
+  have ht : Measurable (fun p : ℝ×(Driver×Driver) =>
+      ((⟨min p.1.toNNReal T,mem_Iic.mpr (min_le_right _ _)⟩ : Iic T),p.2)) :=
+    ((measurable_fst.real_toNNReal.min measurable_const).subtype_mk).prodMk measurable_snd
+  exact (hp T).measurable.comp ht
+
+lemma planar_compensator_adapted (f : ℝ×ℝ → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f)
+    (T : ℝ≥0) : Measurable[planarWienerFiltration T]
+      (fun b => ∫ u in (0:ℝ)..(T:ℝ), planarGenerator f (planarWiener u.toNNReal b)) := by
+  let : MeasurableSpace (Driver×Driver) := planarWienerFiltration T
+  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener (min u.toNNReal T) b)
+  have hm : Measurable (Function.uncurry G) :=
+    (planarGenerator_smooth f hf).continuous.measurable.comp (planarWiener_frozen_measurable T)
+  have hi := (cumulative_integral_measurable G hm).comp
+    ((measurable_const (a := T)).prodMk measurable_id)
+  have he (b : Driver×Driver) : (∫ u in (0:ℝ)..(T:ℝ), G u b)=
+      ∫ u in (0:ℝ)..(T:ℝ), planarGenerator f (planarWiener u.toNNReal b) := by
+    apply intervalIntegral.integral_congr_Ioo_of_le T.coe_nonneg
+    intro u hu
+    have huT : u.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr hu.2.le
+    simp only [G,min_eq_left huT]
+  change Measurable (fun b => ∫ u in (0:ℝ)..(T:ℝ), G u b) at hi
+  simpa only [he] using hi
+
+lemma planar_compensated_stronglyAdapted (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) : StronglyAdapted planarWienerFiltration (planarCompensated f) := by
+  intro t
+  exact ((hf.continuous.measurable.comp (planarWiener_adapted t)).sub
+    (planar_compensator_adapted f hf t)).stronglyMeasurable
+
+theorem planar_compensated_martingale
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f) :
+    Martingale (planarCompensated f) planarWienerFiltration (W.prod W) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have ha := planar_compensated_stronglyAdapted f hf
+  refine ⟨ha,fun s t hst => ?_⟩
+  apply Filter.EventuallyEq.symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq (planarWienerFiltration.le s)
+    (planar_compensated_integrable W hW f hf hc t)
+    (fun A _ _ => (planar_compensated_integrable W hW f hf hc s).integrableOn)
+    ?_ (ha s).aestronglyMeasurable
+  intro A hA _
+  have hh := planar_compensated_past_integral_zero W hW f hf hc s t hst A hA
+  have hi : Integrable (fun b => planarCompensated f t b) ((W.prod W).restrict A) :=
+    (planar_compensated_integrable W hW f hf hc t).restrict
+  have hj : Integrable (fun b => planarCompensated f s b) ((W.prod W).restrict A) :=
+    (planar_compensated_integrable W hW f hf hc s).restrict
+  rw [integral_sub hi hj] at hh
+  exact (sub_eq_zero.mp hh).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarOptional.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarOptional.lean
index 3e55d0c5322c692e0f5e45f70ae514cc34206719..4c96cd8822eeab93c5a399c59a7c7a0b92ea03a0 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarOptional.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarOptional.lean
@@ -1,80 +1,80 @@
-import OAI.Probability.SLE.Stochastic.PlanarCompensated
-
-namespace OAI
-
-/-! Bounded optional stopping for compact smooth planar harmonic tests. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-theorem planar_compact_optional
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (τ : Driver×Driver → ℝ≥0)
-    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ b, τ b≤T)
-    (hz : ∀ b u, 0≤u → u<(τ b:ℝ) → planarGenerator f (planarWiener u.toNNReal b)=0) :
-    (∫ b, f (planarWiener (τ b) b) ∂W.prod W)=∫ b, f (planarWiener 0 b) ∂W.prod W := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
-  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
-  have hτm := stopping_time_measurable_nnreal hτ
-  let V (n : ℕ) (b : Driver×Driver) := f (planarWiener (stopGrid n (τ b)) b)
-  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
-  let H (n : ℕ) (b : Driver×Driver) := ∫ u in (0:ℝ)..(stopGrid n (τ b):ℝ), G u b
-  have hVm (n : ℕ) : Measurable (V n) :=
-    measurable_discrete_eval (fun k b => f (planarWiener (stopGridTime n k) b))
-      (fun k => hf.continuous.measurable.comp (measurable_planarWiener _))
-      (stopGridIndex_measurable hτm n)
-  have hVb (n : ℕ) (b : Driver×Driver) : |V n b|≤K := by
-    simpa only [Real.norm_eq_abs] using hK (planarWiener (stopGrid n (τ b)) b)
-  have hVi (n : ℕ) : Integrable (V n) (W.prod W) :=
-    Integrable.of_bound (hVm n).aestronglyMeasurable K
-      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hVb n b)
-  have hGm : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
-  have hGb (u b) : |G u b|≤D := hD _
-  have hHjoint := cumulative_integral_measurable G hGm
-  have hHm (n : ℕ) : Measurable (H n) :=
-    measurable_discrete_eval (fun k b => ∫ u in (0:ℝ)..(stopGridTime n k:ℝ), G u b)
-      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
-  have hHb (n : ℕ) (b : Driver×Driver) : |H n b|≤(D:ℝ)*(1/((n:ℝ)+1)) :=
-    bounded_cumulative_at_grid G hGm D D.coe_nonneg hGb τ hz n b
-  have hHi (n : ℕ) : Integrable (H n) (W.prod W) :=
-    Integrable.of_bound (hHm n).aestronglyMeasurable _
-      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
-  have hmean (n : ℕ) : (∫ b, V n b ∂W.prod W)-(∫ b, f (planarWiener 0 b) ∂W.prod W)=
-      ∫ b, H n b ∂W.prod W := by
-    have hh := planar_compensated_grid_integral W hW f hf hc τ hτ T hb n
-    change (∫ b, V n b-H n b ∂W.prod W)=_ at hh
-    rw [integral_sub (hVi n) (hHi n)] at hh
-    linarith
-  have herror (n : ℕ) : dist (∫ b, V n b ∂W.prod W) (∫ b, f (planarWiener 0 b) ∂W.prod W)≤
-      (D:ℝ)*(1/((n:ℝ)+1)) := by
-    rw [Real.dist_eq,hmean]
-    have hh := norm_integral_le_of_norm_le_const (μ := W.prod W)
-      (f := H n) (C := (D:ℝ)*(1/((n:ℝ)+1)))
-      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
-    simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using hh
-  have hlim : Tendsto (fun n => ∫ b, V n b ∂W.prod W) atTop
-      (𝓝 (∫ b, f (planarWiener (τ b) b) ∂W.prod W)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => (K:ℝ))
-      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hVb n b)
-    exact Eventually.of_forall fun b =>
-      (hf.continuous.comp (planarWiener_continuous b)).continuousAt.tendsto.comp (stopGrid_tendsto (τ b))
-  have hlim0 : Tendsto (fun n => ∫ b, V n b ∂W.prod W) atTop
-      (𝓝 (∫ b, f (planarWiener 0 b) ∂W.prod W)) := by
-    apply tendsto_iff_dist_tendsto_zero.mpr
-    apply squeeze_zero (fun n => dist_nonneg) herror
-    simpa only [mul_zero] using
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (D:ℝ)
-  exact tendsto_nhds_unique hlim hlim0
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarCompensated
+
+namespace OAI
+
+/-! Bounded optional stopping for compact smooth planar harmonic tests. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+theorem planar_compact_optional
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (τ : Driver×Driver → ℝ≥0)
+    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ b, τ b≤T)
+    (hz : ∀ b u, 0≤u → u<(τ b:ℝ) → planarGenerator f (planarWiener u.toNNReal b)=0) :
+    (∫ b, f (planarWiener (τ b) b) ∂W.prod W)=∫ b, f (planarWiener 0 b) ∂W.prod W := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
+  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
+  have hτm := stopping_time_measurable_nnreal hτ
+  let V (n : ℕ) (b : Driver×Driver) := f (planarWiener (stopGrid n (τ b)) b)
+  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
+  let H (n : ℕ) (b : Driver×Driver) := ∫ u in (0:ℝ)..(stopGrid n (τ b):ℝ), G u b
+  have hVm (n : ℕ) : Measurable (V n) :=
+    measurable_discrete_eval (fun k b => f (planarWiener (stopGridTime n k) b))
+      (fun k => hf.continuous.measurable.comp (measurable_planarWiener _))
+      (stopGridIndex_measurable hτm n)
+  have hVb (n : ℕ) (b : Driver×Driver) : |V n b|≤K := by
+    simpa only [Real.norm_eq_abs] using hK (planarWiener (stopGrid n (τ b)) b)
+  have hVi (n : ℕ) : Integrable (V n) (W.prod W) :=
+    Integrable.of_bound (hVm n).aestronglyMeasurable K
+      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hVb n b)
+  have hGm : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
+  have hGb (u b) : |G u b|≤D := hD _
+  have hHjoint := cumulative_integral_measurable G hGm
+  have hHm (n : ℕ) : Measurable (H n) :=
+    measurable_discrete_eval (fun k b => ∫ u in (0:ℝ)..(stopGridTime n k:ℝ), G u b)
+      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
+  have hHb (n : ℕ) (b : Driver×Driver) : |H n b|≤(D:ℝ)*(1/((n:ℝ)+1)) :=
+    bounded_cumulative_at_grid G hGm D D.coe_nonneg hGb τ hz n b
+  have hHi (n : ℕ) : Integrable (H n) (W.prod W) :=
+    Integrable.of_bound (hHm n).aestronglyMeasurable _
+      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
+  have hmean (n : ℕ) : (∫ b, V n b ∂W.prod W)-(∫ b, f (planarWiener 0 b) ∂W.prod W)=
+      ∫ b, H n b ∂W.prod W := by
+    have hh := planar_compensated_grid_integral W hW f hf hc τ hτ T hb n
+    change (∫ b, V n b-H n b ∂W.prod W)=_ at hh
+    rw [integral_sub (hVi n) (hHi n)] at hh
+    linarith
+  have herror (n : ℕ) : dist (∫ b, V n b ∂W.prod W) (∫ b, f (planarWiener 0 b) ∂W.prod W)≤
+      (D:ℝ)*(1/((n:ℝ)+1)) := by
+    rw [Real.dist_eq,hmean]
+    have hh := norm_integral_le_of_norm_le_const (μ := W.prod W)
+      (f := H n) (C := (D:ℝ)*(1/((n:ℝ)+1)))
+      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
+    simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using hh
+  have hlim : Tendsto (fun n => ∫ b, V n b ∂W.prod W) atTop
+      (𝓝 (∫ b, f (planarWiener (τ b) b) ∂W.prod W)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => (K:ℝ))
+      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hVb n b)
+    exact Eventually.of_forall fun b =>
+      (hf.continuous.comp (planarWiener_continuous b)).continuousAt.tendsto.comp (stopGrid_tendsto (τ b))
+  have hlim0 : Tendsto (fun n => ∫ b, V n b ∂W.prod W) atTop
+      (𝓝 (∫ b, f (planarWiener 0 b) ∂W.prod W)) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun n => dist_nonneg) herror
+    simpa only [mul_zero] using
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (D:ℝ)
+  exact tendsto_nhds_unique hlim hlim0
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarPassageNeighborhood.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarPassageNeighborhood.lean
index 5e2094d026e1e80683514e5d0816621561f70a2c..127c7ae424c90a7ff7a644b6767896f28dfddfd9 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarPassageNeighborhood.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarPassageNeighborhood.lean
@@ -1,82 +1,82 @@
-import OAI.Probability.SLE.Gauge.RibbonLimitPassageGeometry
-
-namespace OAI
-
-/-! Passage neighborhoods in the entire physical plane. Recentring a
-physical ribbon does not preserve the upper half-plane, and this
-construction uses only its embedded coordinate patch. -/
-
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-
-namespace SLEExactGauge.RibbonCoordinates
-
-/-- The actual limiting guide supplies a common physical neighborhood and
-all varying-region conditions used by the retained-mass passage theorem. -/
-theorem limiting_guide_passage_neighborhood_plane
-    (hpublished : PublishedPlanarDiskOpenness)
-    (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (C L : ℝ≥0)
-    (hC : 0 < C) (hL : 0 < L)
-    (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K)
-    (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y))
-    (hconv : TendstoUniformlyOn Pₙ P atTop K)
-    (hP : ContinuousOn P K)
-    (α : ℝ → ℂ) (hα : ContinuousOn α (Icc 0 2)) (x ε η : ℝ)
-    (hx : |x| < ε) (hη : 0 < η)
-    (hshape : EqOn α (fun t => P (guideCoordinate 0 x t)) (Icc 0 2))
-    (hmem : ∀ t ∈ Icc (0 : ℝ) (3 / 2), guideCoordinate 0 x t ∈ interior K)
-    (hbase : (1 : ℂ) ∈ K) :
-    ∃ U : Set ℂ, IsOpen U ∧ α '' Icc (0 : ℝ) (3 / 2) ⊆ U ∧
-      IsCompact (closure U) ∧ closure U ⊆ P '' interior K ∧
-      ContinuousOn (longitudinalCoordinate P K 0) U ∧
-      (∀ᶠ n in atTop, ContinuousOn (longitudinalCoordinate (Pₙ n) K 0) U) ∧
-      TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K 0)
-        (longitudinalCoordinate P K 0) atTop U ∧
-      (∀ᶠ n in atTop, U ⊆ physicalPassageRegion (Pₙ n) K C ε η) ∧
-      (∀ n z, z ∈ physicalPassageRegion (Pₙ n) K C ε η →
-        longitudinalCoordinate (Pₙ n) K 0 z = 1 → ‖z - Pₙ n 1‖ < η / 4) ∧
-      (∀ᶠ n in atTop,
-        U ∩ {z | (1 / 3 : ℝ) < longitudinalCoordinate (Pₙ n) K 0 z ∧
-          longitudinalCoordinate (Pₙ n) K 0 z < 2 / 3} ⊆ physicalMiddleRegion (Pₙ n) K 0 ε) := by
-  let V := passageCoordinateRegion C ε η
-  let A := α '' Icc (0 : ℝ) (3 / 2)
-  have hA : IsCompact A := isCompact_Icc.image_of_continuousOn
-    (hα.mono (Icc_subset_Icc_right (by norm_num)))
-  have hAV : A ⊆ P '' (interior K ∩ V) := by
-    rintro _ ⟨t, ht, rfl⟩
-    exact ⟨guideCoordinate 0 x t, ⟨hmem t ht,
-      guideCoordinate_mem_passageRegion C hC hx hη ht⟩,
-      (hshape ⟨ht.1, ht.2.trans (by norm_num)⟩).symm⟩
-  obtain ⟨U, hU, hAU, hUc, hUPV, hVlate, hKlate, _⟩ :=
-    common_coordinate_region_neighborhood hpublished Pₙ P K A V L hL hP
-      (fun n => (hupper n).continuousOn) hconv hlower hA
-      (passageCoordinateRegion_isOpen C ε η) hAV
-  have hUP : closure U ⊆ P '' interior K := hUPV.trans (image_mono inter_subset_left)
-  have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower
-  have hi := lower_lipschitz_injOn P K L hlim
-  have hwidth : ∀ z ∈ closure U, |(coordinateInverse P K z).im| < ε := by
-    intro z hz
-    obtain ⟨w, hw, he⟩ := hUPV hz
-    rw [← he, coordinateInverse_left hi (interior_subset hw.1)]
-    exact hw.2.2.2.1
-  refine ⟨U, hU, hAU, hUc, hUP, ?_, ?_, ?_, ?_, ?_, ?_⟩
-  · exact (longitudinalCoordinate_continuousOn P K L 0 hlim).mono
-      (subset_closure.trans (hUP.trans (image_mono interior_subset)))
-  · filter_upwards [hKlate] with n hn
-    exact (longitudinalCoordinate_continuousOn (Pₙ n) K L 0 (hlower n)).mono
-      (subset_closure.trans (hn.trans (image_mono interior_subset)))
-  · exact (longitudinalCoordinate_locally_uniform hpublished Pₙ P K L 0 hL hP
-      (fun n => (hupper n).continuousOn) hconv hlower).mono (subset_closure.trans hUP)
-  · exact hVlate.mono (fun _ hn => subset_closure.trans hn)
-  · exact fun n z hz hp => physicalPassageRegion_terminal (Pₙ n) K C L ε η hC
-      (hupper n) (hlower n) hbase hz hp
-  · exact eventual_middle_region_inclusion Pₙ P K U L 0 ε hconv hlower hUc
-      (hUP.trans (image_mono interior_subset))
-      (hKlate.mono (fun _ hn => hn.trans (image_mono interior_subset))) hwidth
-
-end SLEExactGauge.RibbonCoordinates
-
-end
-
-end OAI
+import OAI.Probability.SLE.Gauge.RibbonLimitPassageGeometry
+
+namespace OAI
+
+/-! Passage neighborhoods in the entire physical plane. Recentring a
+physical ribbon does not preserve the upper half-plane, and this
+construction uses only its embedded coordinate patch. -/
+
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+
+namespace SLEExactGauge.RibbonCoordinates
+
+/-- The actual limiting guide supplies a common physical neighborhood and
+all varying-region conditions used by the retained-mass passage theorem. -/
+theorem limiting_guide_passage_neighborhood_plane
+    (hpublished : PublishedPlanarDiskOpenness)
+    (Pₙ : ℕ → ℂ → ℂ) (P : ℂ → ℂ) (K : Set ℂ) (C L : ℝ≥0)
+    (hC : 0 < C) (hL : 0 < L)
+    (hupper : ∀ n, LipschitzOnWith C (Pₙ n) K)
+    (hlower : ∀ n x, x ∈ K → ∀ y, y ∈ K → dist x y ≤ L * dist (Pₙ n x) (Pₙ n y))
+    (hconv : TendstoUniformlyOn Pₙ P atTop K)
+    (hP : ContinuousOn P K)
+    (α : ℝ → ℂ) (hα : ContinuousOn α (Icc 0 2)) (x ε η : ℝ)
+    (hx : |x| < ε) (hη : 0 < η)
+    (hshape : EqOn α (fun t => P (guideCoordinate 0 x t)) (Icc 0 2))
+    (hmem : ∀ t ∈ Icc (0 : ℝ) (3 / 2), guideCoordinate 0 x t ∈ interior K)
+    (hbase : (1 : ℂ) ∈ K) :
+    ∃ U : Set ℂ, IsOpen U ∧ α '' Icc (0 : ℝ) (3 / 2) ⊆ U ∧
+      IsCompact (closure U) ∧ closure U ⊆ P '' interior K ∧
+      ContinuousOn (longitudinalCoordinate P K 0) U ∧
+      (∀ᶠ n in atTop, ContinuousOn (longitudinalCoordinate (Pₙ n) K 0) U) ∧
+      TendstoLocallyUniformlyOn (fun n => longitudinalCoordinate (Pₙ n) K 0)
+        (longitudinalCoordinate P K 0) atTop U ∧
+      (∀ᶠ n in atTop, U ⊆ physicalPassageRegion (Pₙ n) K C ε η) ∧
+      (∀ n z, z ∈ physicalPassageRegion (Pₙ n) K C ε η →
+        longitudinalCoordinate (Pₙ n) K 0 z = 1 → ‖z - Pₙ n 1‖ < η / 4) ∧
+      (∀ᶠ n in atTop,
+        U ∩ {z | (1 / 3 : ℝ) < longitudinalCoordinate (Pₙ n) K 0 z ∧
+          longitudinalCoordinate (Pₙ n) K 0 z < 2 / 3} ⊆ physicalMiddleRegion (Pₙ n) K 0 ε) := by
+  let V := passageCoordinateRegion C ε η
+  let A := α '' Icc (0 : ℝ) (3 / 2)
+  have hA : IsCompact A := isCompact_Icc.image_of_continuousOn
+    (hα.mono (Icc_subset_Icc_right (by norm_num)))
+  have hAV : A ⊆ P '' (interior K ∩ V) := by
+    rintro _ ⟨t, ht, rfl⟩
+    exact ⟨guideCoordinate 0 x t, ⟨hmem t ht,
+      guideCoordinate_mem_passageRegion C hC hx hη ht⟩,
+      (hshape ⟨ht.1, ht.2.trans (by norm_num)⟩).symm⟩
+  obtain ⟨U, hU, hAU, hUc, hUPV, hVlate, hKlate, _⟩ :=
+    common_coordinate_region_neighborhood hpublished Pₙ P K A V L hL hP
+      (fun n => (hupper n).continuousOn) hconv hlower hA
+      (passageCoordinateRegion_isOpen C ε η) hAV
+  have hUP : closure U ⊆ P '' interior K := hUPV.trans (image_mono inter_subset_left)
+  have hlim := lower_lipschitz_uniform_limit Pₙ P K L hconv hlower
+  have hi := lower_lipschitz_injOn P K L hlim
+  have hwidth : ∀ z ∈ closure U, |(coordinateInverse P K z).im| < ε := by
+    intro z hz
+    obtain ⟨w, hw, he⟩ := hUPV hz
+    rw [← he, coordinateInverse_left hi (interior_subset hw.1)]
+    exact hw.2.2.2.1
+  refine ⟨U, hU, hAU, hUc, hUP, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · exact (longitudinalCoordinate_continuousOn P K L 0 hlim).mono
+      (subset_closure.trans (hUP.trans (image_mono interior_subset)))
+  · filter_upwards [hKlate] with n hn
+    exact (longitudinalCoordinate_continuousOn (Pₙ n) K L 0 (hlower n)).mono
+      (subset_closure.trans (hn.trans (image_mono interior_subset)))
+  · exact (longitudinalCoordinate_locally_uniform hpublished Pₙ P K L 0 hL hP
+      (fun n => (hupper n).continuousOn) hconv hlower).mono (subset_closure.trans hUP)
+  · exact hVlate.mono (fun _ hn => subset_closure.trans hn)
+  · exact fun n z hz hp => physicalPassageRegion_terminal (Pₙ n) K C L ε η hC
+      (hupper n) (hlower n) hbase hz hp
+  · exact eventual_middle_region_inclusion Pₙ P K U L 0 ε hconv hlower hUc
+      (hUP.trans (image_mono interior_subset))
+      (hKlate.mono (fun _ hn => hn.trans (image_mono interior_subset))) hwidth
+
+end SLEExactGauge.RibbonCoordinates
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarRetractComplement.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarRetractComplement.lean
index 9e1ee27270e2a588a39fd3084ebc762c0d138a80..24dd2c337001d2806ab6dd92033dbc275f6574b4 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarRetractComplement.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarRetractComplement.lean
@@ -1,87 +1,87 @@
-import OAI.Probability.SLE.Geometry.LocalLogObstruction
-
-namespace OAI
-
-/-! A closed planar retract has no bounded complementary component, since
-the identity on a circle has no continuous logarithm. -/
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-
-namespace SLEExactGauge
-
-lemma no_point_avoiding_identity_outside_ball (z : ℂ) {R : ℝ} (hR : 0 < R)
-    (f : ℂ → ℂ) (hf : Continuous f) (hne : ∀ w, f w ≠ z)
-    (hout : ∀ w, R ≤ dist w z → f w=w) : False := by
-  let S := sphere (0 : ℂ) R
-  let H : C(unitInterval × S,ℂ) :=
-    ⟨fun p => f (z+(p.1 : ℂ)*(p.2 : ℂ))-z,by fun_prop⟩
-  let L : C(S,ℂ) := ContinuousMap.const _ (Complex.log (f z-z))
-  have hn : f z-z ≠ 0 := sub_ne_zero.mpr (hne z)
-  obtain ⟨M,hM⟩ := continuous_log_at_end_of_homotopy H
-    (fun p => sub_ne_zero.mpr (hne _)) L (fun a => by
-      simpa [H,L] using Complex.exp_log hn)
-  apply univalent_boundary_no_continuous_log id R hR continuous_id.continuousOn
-    injective_id.injOn M
-  intro a
-  have ha : R ≤ dist (z+(a : ℂ)) z := by
-    simpa only [dist_eq_norm,add_sub_cancel_left,sub_zero] using a.property.ge
-  simpa [H,hout _ ha] using hM a
-
-theorem bounded_open_frontier_not_retract
-    {D K : Set ℂ} (hDb : Bornology.IsBounded D)
-    {z : ℂ} (hz : z ∈ D) (hdisj : Disjoint D K)
-    (hfrontier : frontier D ⊆ K)
-    (r : ℂ → ℂ) (hr : Continuous r) (hrK : ∀ w, r w ∈ K)
-    (hrfix : ∀ w ∈ K, r w=w) : False := by
-  classical
-  let f : ℂ → ℂ := D.piecewise r id
-  have hf : Continuous f := hr.piecewise
-    (fun w hw => hrfix w (hfrontier hw)) continuous_id
-  have hne (w : ℂ) : f w ≠ z := by
-    change (D.piecewise r id) w ≠ z
-    by_cases hw : w ∈ D
-    · rw [piecewise_eq_of_mem D r id hw]
-      intro he
-      exact Set.disjoint_left.mp hdisj hz (he ▸ hrK w)
-    · rw [piecewise_eq_of_notMem D r id hw]
-      intro he
-      have he' : w=z := he
-      exact hw (he'.symm ▸ hz)
-  obtain ⟨R,hR,hball⟩ := hDb.subset_ball_lt 0 z
-  apply no_point_avoiding_identity_outside_ball z hR f hf hne
-  intro w hw
-  have hn : w ∉ D := fun hd => (hball hd).not_ge hw
-  exact piecewise_eq_of_notMem D r id hn
-
-lemma frontier_connectedComponentIn_subset_compl {U : Set ℂ} (hU : IsOpen U)
-    {z : ℂ} : frontier (connectedComponentIn U z) ⊆ Uᶜ := by
-  intro w hw hwin
-  obtain ⟨ε,hε,hball⟩ := Metric.isOpen_iff.mp hU w hwin
-  obtain ⟨v,hvball,hvC⟩ := mem_closure_iff_nhds.mp hw.1 _ (ball_mem_nhds w hε)
-  have hsub : ball w ε ⊆ connectedComponentIn U v :=
-    isPreconnected_ball.subset_connectedComponentIn hvball hball
-  have hwC : w ∈ connectedComponentIn U z := by
-    rw [connectedComponentIn_eq hvC]
-    exact hsub (mem_ball_self hε)
-  exact hw.2 (by simpa only [(hU.connectedComponentIn).interior_eq] using hwC)
-
-theorem retract_complement_component_unbounded
-    {K : Set ℂ} (hK : IsClosed K)
-    (r : ℂ → ℂ) (hr : Continuous r) (hrK : ∀ w, r w ∈ K)
-    (hrfix : ∀ w ∈ K, r w=w) {z : ℂ} (hz : z ∉ K) :
-    ¬ Bornology.IsBounded (connectedComponentIn Kᶜ z) := by
-  intro hb
-  have hfront : frontier (connectedComponentIn Kᶜ z) ⊆ K := by
-    simpa only [compl_compl] using
-      (frontier_connectedComponentIn_subset_compl hK.isOpen_compl (z := z))
-  apply bounded_open_frontier_not_retract hb
-    (mem_connectedComponentIn hz) (disjoint_left.mpr (fun w hw hwin =>
-      connectedComponentIn_subset (Kᶜ) z hw hwin))
-    hfront r hr hrK hrfix
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.LocalLogObstruction
+
+namespace OAI
+
+/-! A closed planar retract has no bounded complementary component, since
+the identity on a circle has no continuous logarithm. -/
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+
+namespace SLEExactGauge
+
+lemma no_point_avoiding_identity_outside_ball (z : ℂ) {R : ℝ} (hR : 0 < R)
+    (f : ℂ → ℂ) (hf : Continuous f) (hne : ∀ w, f w ≠ z)
+    (hout : ∀ w, R ≤ dist w z → f w=w) : False := by
+  let S := sphere (0 : ℂ) R
+  let H : C(unitInterval × S,ℂ) :=
+    ⟨fun p => f (z+(p.1 : ℂ)*(p.2 : ℂ))-z,by fun_prop⟩
+  let L : C(S,ℂ) := ContinuousMap.const _ (Complex.log (f z-z))
+  have hn : f z-z ≠ 0 := sub_ne_zero.mpr (hne z)
+  obtain ⟨M,hM⟩ := continuous_log_at_end_of_homotopy H
+    (fun p => sub_ne_zero.mpr (hne _)) L (fun a => by
+      simpa [H,L] using Complex.exp_log hn)
+  apply univalent_boundary_no_continuous_log id R hR continuous_id.continuousOn
+    injective_id.injOn M
+  intro a
+  have ha : R ≤ dist (z+(a : ℂ)) z := by
+    simpa only [dist_eq_norm,add_sub_cancel_left,sub_zero] using a.property.ge
+  simpa [H,hout _ ha] using hM a
+
+theorem bounded_open_frontier_not_retract
+    {D K : Set ℂ} (hDb : Bornology.IsBounded D)
+    {z : ℂ} (hz : z ∈ D) (hdisj : Disjoint D K)
+    (hfrontier : frontier D ⊆ K)
+    (r : ℂ → ℂ) (hr : Continuous r) (hrK : ∀ w, r w ∈ K)
+    (hrfix : ∀ w ∈ K, r w=w) : False := by
+  classical
+  let f : ℂ → ℂ := D.piecewise r id
+  have hf : Continuous f := hr.piecewise
+    (fun w hw => hrfix w (hfrontier hw)) continuous_id
+  have hne (w : ℂ) : f w ≠ z := by
+    change (D.piecewise r id) w ≠ z
+    by_cases hw : w ∈ D
+    · rw [piecewise_eq_of_mem D r id hw]
+      intro he
+      exact Set.disjoint_left.mp hdisj hz (he ▸ hrK w)
+    · rw [piecewise_eq_of_notMem D r id hw]
+      intro he
+      have he' : w=z := he
+      exact hw (he'.symm ▸ hz)
+  obtain ⟨R,hR,hball⟩ := hDb.subset_ball_lt 0 z
+  apply no_point_avoiding_identity_outside_ball z hR f hf hne
+  intro w hw
+  have hn : w ∉ D := fun hd => (hball hd).not_ge hw
+  exact piecewise_eq_of_notMem D r id hn
+
+lemma frontier_connectedComponentIn_subset_compl {U : Set ℂ} (hU : IsOpen U)
+    {z : ℂ} : frontier (connectedComponentIn U z) ⊆ Uᶜ := by
+  intro w hw hwin
+  obtain ⟨ε,hε,hball⟩ := Metric.isOpen_iff.mp hU w hwin
+  obtain ⟨v,hvball,hvC⟩ := mem_closure_iff_nhds.mp hw.1 _ (ball_mem_nhds w hε)
+  have hsub : ball w ε ⊆ connectedComponentIn U v :=
+    isPreconnected_ball.subset_connectedComponentIn hvball hball
+  have hwC : w ∈ connectedComponentIn U z := by
+    rw [connectedComponentIn_eq hvC]
+    exact hsub (mem_ball_self hε)
+  exact hw.2 (by simpa only [(hU.connectedComponentIn).interior_eq] using hwC)
+
+theorem retract_complement_component_unbounded
+    {K : Set ℂ} (hK : IsClosed K)
+    (r : ℂ → ℂ) (hr : Continuous r) (hrK : ∀ w, r w ∈ K)
+    (hrfix : ∀ w ∈ K, r w=w) {z : ℂ} (hz : z ∉ K) :
+    ¬ Bornology.IsBounded (connectedComponentIn Kᶜ z) := by
+  intro hb
+  have hfront : frontier (connectedComponentIn Kᶜ z) ⊆ K := by
+    simpa only [compl_compl] using
+      (frontier_connectedComponentIn_subset_compl hK.isOpen_compl (z := z))
+  apply bounded_open_frontier_not_retract hb
+    (mem_connectedComponentIn hz) (disjoint_left.mpr (fun w hw hwin =>
+      connectedComponentIn_subset (Kᶜ) z hw hwin))
+    hfront r hr hrK hrfix
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingGrid.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingGrid.lean
index f21d09e703c1897401b5c0a988528b434814191c..a1848b1b9f1bcc1a2671d91460850ad1baf12e18 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingGrid.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingGrid.lean
@@ -1,94 +1,94 @@
-import OAI.Probability.SLE.Stochastic.PlanarMartingale
-
-namespace OAI
-
-/-! Conditional optional sampling on the finite grids used to localize
-planar harmonic tests. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma stopGridIndex_stopping {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {ℱ : Filtration ℝ≥0 mΩ} {σ : Ω → ℝ≥0}
-    (hσ : IsStoppingTime ℱ (fun b => (σ b : WithTop ℝ≥0))) (n : ℕ) :
-    IsStoppingTime (gridFiltration ℱ n) (fun b => (stopGridIndex n (σ b) : WithTop ℕ)) := by
-  intro j
-  change MeasurableSet[ℱ (stopGridTime n j)]
-    {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (j : WithTop ℕ)}
-  have he : {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (j : WithTop ℕ)}=
-      {b | σ b ≤ stopGridTime n j} := by
-    ext b
-    exact WithTop.coe_le_coe.trans (stopGridIndex_le n j (σ b))
-  rw [he]
-  simpa only [WithTop.coe_le_coe] using hσ (stopGridTime n j)
-
-lemma measurableSet_stopGridIndex {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {ℱ : Filtration ℝ≥0 mΩ} {σ : Ω → ℝ≥0}
-    (hσ : IsStoppingTime ℱ (fun b => (σ b : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ b, σ b ≤ T) (n : ℕ) (A : Set Ω)
-    (hA : MeasurableSet[hσ.measurableSpace] A) :
-    MeasurableSet[(stopGridIndex_stopping hσ n).measurableSpace] A := by
-  have hAj (j : ℕ) : MeasurableSet[(gridFiltration ℱ n) j]
-      (A ∩ {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ j}) := by
-    have he : {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (j : WithTop ℕ)}=
-        {b | σ b ≤ stopGridTime n j} := by
-      ext b
-      exact WithTop.coe_le_coe.trans (stopGridIndex_le n j (σ b))
-    rw [he]
-    change MeasurableSet[ℱ (stopGridTime n j)] (A ∩ {b | σ b ≤ stopGridTime n j})
-    simpa only [WithTop.coe_le_coe] using hA.2 (stopGridTime n j)
-  have hn (b) : stopGridIndex n (σ b) ≤ stopGridIndex n T :=
-    (stopGridIndex_le n _ _).mpr ((hb b).trans (stopGrid_bounds n T).1)
-  have hAN : MeasurableSet[(gridFiltration ℱ n) (stopGridIndex n T)] A := by
-    have he : {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (stopGridIndex n T : ℕ)}=univ := by
-      ext b
-      simp only [mem_ofPred_eq,mem_univ,iff_true]
-      exact WithTop.coe_le_coe.mpr (hn b)
-    simpa only [he,inter_univ] using hAj (stopGridIndex n T)
-  exact ⟨le_iSup (gridFiltration ℱ n) (stopGridIndex n T) A hAN,hAj⟩
-
-lemma planar_compensated_grid_sampling
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (σ τ : Driver×Driver → ℝ≥0)
-    (hσ : IsStoppingTime planarWienerFiltration (fun b => (σ b : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
-    (hστ : ∀ b, σ b ≤ τ b) (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
-    (A : Set (Driver×Driver)) (hA : MeasurableSet[hσ.measurableSpace] A) (n : ℕ) :
-    (∫ b in A, planarCompensated f (stopGrid n (σ b)) b ∂W.prod W)=
-      ∫ b in A, planarCompensated f (stopGrid n (τ b)) b ∂W.prod W := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  let ℱn := gridFiltration planarWienerFiltration n
-  let C (j : ℕ) := planarCompensated f (stopGridTime n j)
-  have hM : Martingale C ℱn (W.prod W) := by
-    have hm := planar_compensated_martingale W hW f hf hc
-    exact ⟨fun j => hm.1 _,fun i j hij => hm.2 _ _ (monotone_stopGridTime n hij)⟩
-  have hσn := stopGridIndex_stopping hσ n
-  have hτn := stopGridIndex_stopping hτ n
-  have hle : (fun b => (stopGridIndex n (σ b) : WithTop ℕ)) ≤
-      (fun b => (stopGridIndex n (τ b) : WithTop ℕ)) := by
-    intro b
-    apply WithTop.coe_le_coe.mpr
-    exact (stopGridIndex_le n _ _).mpr ((hστ b).trans (stopGrid_bounds n (τ b)).1)
-  have hN (b) : (stopGridIndex n (τ b) : WithTop ℕ) ≤ (stopGridIndex n T : ℕ) := by
-    apply WithTop.coe_le_coe.mpr
-    exact (stopGridIndex_le n _ _).mpr ((hb b).trans (stopGrid_bounds n T).1)
-  have hCE := hM.stoppedValue_ae_eq_condExp_of_le hτn hσn hle hN
-  have hAn := measurableSet_stopGridIndex hσ T (fun b => (hστ b).trans (hb b)) n A hA
-  have hI := integrable_stoppedValue ℕ hτn hM.integrable hN
-  have he : (∫ b in A, stoppedValue C (fun b => (stopGridIndex n (σ b) : WithTop ℕ)) b ∂W.prod W)=
-      ∫ b in A, stoppedValue C (fun b => (stopGridIndex n (τ b) : WithTop ℕ)) b ∂W.prod W := by
-    calc
-      _ = ∫ b in A, (W.prod W)[stoppedValue C (fun b => (stopGridIndex n (τ b) : WithTop ℕ)) | hσn.measurableSpace] b ∂W.prod W :=
-        integral_congr_ae (ae_restrict_of_ae hCE)
-      _ = _ := setIntegral_condExp hσn.measurableSpace_le hI hAn
-  exact he
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarMartingale
+
+namespace OAI
+
+/-! Conditional optional sampling on the finite grids used to localize
+planar harmonic tests. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma stopGridIndex_stopping {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {ℱ : Filtration ℝ≥0 mΩ} {σ : Ω → ℝ≥0}
+    (hσ : IsStoppingTime ℱ (fun b => (σ b : WithTop ℝ≥0))) (n : ℕ) :
+    IsStoppingTime (gridFiltration ℱ n) (fun b => (stopGridIndex n (σ b) : WithTop ℕ)) := by
+  intro j
+  change MeasurableSet[ℱ (stopGridTime n j)]
+    {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (j : WithTop ℕ)}
+  have he : {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (j : WithTop ℕ)}=
+      {b | σ b ≤ stopGridTime n j} := by
+    ext b
+    exact WithTop.coe_le_coe.trans (stopGridIndex_le n j (σ b))
+  rw [he]
+  simpa only [WithTop.coe_le_coe] using hσ (stopGridTime n j)
+
+lemma measurableSet_stopGridIndex {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {ℱ : Filtration ℝ≥0 mΩ} {σ : Ω → ℝ≥0}
+    (hσ : IsStoppingTime ℱ (fun b => (σ b : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ b, σ b ≤ T) (n : ℕ) (A : Set Ω)
+    (hA : MeasurableSet[hσ.measurableSpace] A) :
+    MeasurableSet[(stopGridIndex_stopping hσ n).measurableSpace] A := by
+  have hAj (j : ℕ) : MeasurableSet[(gridFiltration ℱ n) j]
+      (A ∩ {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ j}) := by
+    have he : {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (j : WithTop ℕ)}=
+        {b | σ b ≤ stopGridTime n j} := by
+      ext b
+      exact WithTop.coe_le_coe.trans (stopGridIndex_le n j (σ b))
+    rw [he]
+    change MeasurableSet[ℱ (stopGridTime n j)] (A ∩ {b | σ b ≤ stopGridTime n j})
+    simpa only [WithTop.coe_le_coe] using hA.2 (stopGridTime n j)
+  have hn (b) : stopGridIndex n (σ b) ≤ stopGridIndex n T :=
+    (stopGridIndex_le n _ _).mpr ((hb b).trans (stopGrid_bounds n T).1)
+  have hAN : MeasurableSet[(gridFiltration ℱ n) (stopGridIndex n T)] A := by
+    have he : {b | (stopGridIndex n (σ b) : WithTop ℕ) ≤ (stopGridIndex n T : ℕ)}=univ := by
+      ext b
+      simp only [mem_ofPred_eq,mem_univ,iff_true]
+      exact WithTop.coe_le_coe.mpr (hn b)
+    simpa only [he,inter_univ] using hAj (stopGridIndex n T)
+  exact ⟨le_iSup (gridFiltration ℱ n) (stopGridIndex n T) A hAN,hAj⟩
+
+lemma planar_compensated_grid_sampling
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (σ τ : Driver×Driver → ℝ≥0)
+    (hσ : IsStoppingTime planarWienerFiltration (fun b => (σ b : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
+    (hστ : ∀ b, σ b ≤ τ b) (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
+    (A : Set (Driver×Driver)) (hA : MeasurableSet[hσ.measurableSpace] A) (n : ℕ) :
+    (∫ b in A, planarCompensated f (stopGrid n (σ b)) b ∂W.prod W)=
+      ∫ b in A, planarCompensated f (stopGrid n (τ b)) b ∂W.prod W := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  let ℱn := gridFiltration planarWienerFiltration n
+  let C (j : ℕ) := planarCompensated f (stopGridTime n j)
+  have hM : Martingale C ℱn (W.prod W) := by
+    have hm := planar_compensated_martingale W hW f hf hc
+    exact ⟨fun j => hm.1 _,fun i j hij => hm.2 _ _ (monotone_stopGridTime n hij)⟩
+  have hσn := stopGridIndex_stopping hσ n
+  have hτn := stopGridIndex_stopping hτ n
+  have hle : (fun b => (stopGridIndex n (σ b) : WithTop ℕ)) ≤
+      (fun b => (stopGridIndex n (τ b) : WithTop ℕ)) := by
+    intro b
+    apply WithTop.coe_le_coe.mpr
+    exact (stopGridIndex_le n _ _).mpr ((hστ b).trans (stopGrid_bounds n (τ b)).1)
+  have hN (b) : (stopGridIndex n (τ b) : WithTop ℕ) ≤ (stopGridIndex n T : ℕ) := by
+    apply WithTop.coe_le_coe.mpr
+    exact (stopGridIndex_le n _ _).mpr ((hb b).trans (stopGrid_bounds n T).1)
+  have hCE := hM.stoppedValue_ae_eq_condExp_of_le hτn hσn hle hN
+  have hAn := measurableSet_stopGridIndex hσ T (fun b => (hστ b).trans (hb b)) n A hA
+  have hI := integrable_stoppedValue ℕ hτn hM.integrable hN
+  have he : (∫ b in A, stoppedValue C (fun b => (stopGridIndex n (σ b) : WithTop ℕ)) b ∂W.prod W)=
+      ∫ b in A, stoppedValue C (fun b => (stopGridIndex n (τ b) : WithTop ℕ)) b ∂W.prod W := by
+    calc
+      _ = ∫ b in A, (W.prod W)[stoppedValue C (fun b => (stopGridIndex n (τ b) : WithTop ℕ)) | hσn.measurableSpace] b ∂W.prod W :=
+        integral_congr_ae (ae_restrict_of_ae hCE)
+      _ = _ := setIntegral_condExp hσn.measurableSpace_le hI hAn
+  exact he
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingLimit.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingLimit.lean
index ab472b4486dfd2f32a4404d546fb942ba9351fea..d2db4d32dfb126b3c6a7622c7a80115bd178ddc5 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingLimit.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarSamplingLimit.lean
@@ -1,99 +1,99 @@
-import OAI.Probability.SLE.Stochastic.PlanarSamplingGrid
-
-namespace OAI
-
-/-! Passing finite-grid conditional sampling to bounded stopping times
-for compact smooth planar harmonic tests. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planar_compensated_grid_tendsto
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (ρ : Driver×Driver → ℝ≥0) (hρ : Measurable ρ) (A : Set (Driver×Driver))
-    (hz : ∀ b u, 0 ≤ u → u<(ρ b:ℝ) → planarGenerator f (planarWiener u.toNNReal b)=0) :
-    Tendsto (fun n => ∫ b in A, planarCompensated f (stopGrid n (ρ b)) b ∂W.prod W)
-      atTop (𝓝 (∫ b in A, f (planarWiener (ρ b) b) ∂W.prod W)) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
-  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
-  let V (n : ℕ) (b : Driver×Driver) := f (planarWiener (stopGrid n (ρ b)) b)
-  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
-  let H (n : ℕ) (b : Driver×Driver) := ∫ u in (0:ℝ)..(stopGrid n (ρ b):ℝ), G u b
-  have hVm (n : ℕ) : Measurable (V n) :=
-    measurable_discrete_eval (fun k b => f (planarWiener (stopGridTime n k) b))
-      (fun k => hf.continuous.measurable.comp (measurable_planarWiener _))
-      (stopGridIndex_measurable hρ n)
-  have hVi (n : ℕ) : Integrable (V n) ((W.prod W).restrict A) :=
-    Integrable.of_bound (hVm n).aestronglyMeasurable K (Eventually.of_forall fun b => hK _)
-  have hVlim : Tendsto (fun n => ∫ b in A, V n b ∂W.prod W) atTop
-      (𝓝 (∫ b in A, f (planarWiener (ρ b) b) ∂W.prod W)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => (K:ℝ))
-      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => Eventually.of_forall fun b => hK _)
-    exact Eventually.of_forall fun b =>
-      (hf.continuous.comp (planarWiener_continuous b)).continuousAt.tendsto.comp (stopGrid_tendsto (ρ b))
-  have hGm : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
-  have hGb (u b) : |G u b|≤D := hD _
-  have hHjoint := cumulative_integral_measurable G hGm
-  have hHm (n : ℕ) : Measurable (H n) :=
-    measurable_discrete_eval (fun k b => ∫ u in (0:ℝ)..(stopGridTime n k:ℝ), G u b)
-      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hρ n)
-  have hHb (n : ℕ) (b : Driver×Driver) : |H n b|≤(D:ℝ)*(1/((n:ℝ)+1)) :=
-    bounded_cumulative_at_grid G hGm D D.coe_nonneg hGb ρ hz n b
-  have hHi (n : ℕ) : Integrable (H n) ((W.prod W).restrict A) :=
-    Integrable.of_bound (hHm n).aestronglyMeasurable _
-      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
-  have hHbound (n : ℕ) : |∫ b in A, H n b ∂W.prod W|≤(D:ℝ)*(1/((n:ℝ)+1)) := by
-    have hh := norm_integral_le_of_norm_le_const (μ := (W.prod W).restrict A)
-      (f := H n) (C := (D:ℝ)*(1/((n:ℝ)+1)))
-      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
-    have hm : ((W.prod W).restrict A).real univ≤1 := by
-      simpa only [measureReal_def,Measure.restrict_apply_univ] using
-        (measureReal_le_one (μ := W.prod W) (s := A))
-    have hh' : |∫ b in A, H n b ∂W.prod W| ≤
-        (D:ℝ)*(1/((n:ℝ)+1))*((W.prod W).restrict A).real univ := by
-      simpa only [Real.norm_eq_abs] using hh
-    exact hh'.trans
-      (mul_le_of_le_one_right (by positivity) hm)
-  have hHlim : Tendsto (fun n => ∫ b in A, H n b ∂W.prod W) atTop (𝓝 0) := by
-    apply tendsto_iff_norm_sub_tendsto_zero.mpr
-    simp only [sub_zero,Real.norm_eq_abs]
-    apply squeeze_zero (fun n => abs_nonneg _) hHbound
-    simpa only [mul_zero] using
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (D:ℝ)
-  have he (n) : (∫ b in A, planarCompensated f (stopGrid n (ρ b)) b ∂W.prod W)=
-      (∫ b in A, V n b ∂W.prod W)-(∫ b in A, H n b ∂W.prod W) :=
-    integral_sub (hVi n) (hHi n)
-  simpa only [← he,sub_zero] using hVlim.sub hHlim
-
-theorem planar_compact_optional_between
-    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
-    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
-    (σ τ : Driver×Driver → ℝ≥0)
-    (hσ : IsStoppingTime planarWienerFiltration (fun b => (σ b : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
-    (hστ : ∀ b, σ b ≤ τ b) (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
-    (A : Set (Driver×Driver)) (hA : MeasurableSet[hσ.measurableSpace] A)
-    (hz : ∀ b u, 0 ≤ u → u<(τ b:ℝ) → planarGenerator f (planarWiener u.toNNReal b)=0) :
-    (∫ b in A, f (planarWiener (σ b) b) ∂W.prod W)=
-      ∫ b in A, f (planarWiener (τ b) b) ∂W.prod W := by
-  have hs := planar_compensated_grid_tendsto W hW f hf hc σ
-    (stopping_time_measurable_nnreal hσ) A (fun b u hu h => hz b u hu (h.trans_le (hστ b)))
-  have ht := planar_compensated_grid_tendsto W hW f hf hc τ
-    (stopping_time_measurable_nnreal hτ) A hz
-  have he (n) := planar_compensated_grid_sampling W hW f hf hc σ τ hσ hτ hστ T hb A hA n
-  have ht' : Tendsto (fun n => ∫ b in A, planarCompensated f (stopGrid n (σ b)) b ∂W.prod W)
-      atTop (𝓝 (∫ b in A, f (planarWiener (τ b) b) ∂W.prod W)) := by
-    simpa only [he] using ht
-  exact tendsto_nhds_unique hs ht'
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarSamplingGrid
+
+namespace OAI
+
+/-! Passing finite-grid conditional sampling to bounded stopping times
+for compact smooth planar harmonic tests. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planar_compensated_grid_tendsto
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (ρ : Driver×Driver → ℝ≥0) (hρ : Measurable ρ) (A : Set (Driver×Driver))
+    (hz : ∀ b u, 0 ≤ u → u<(ρ b:ℝ) → planarGenerator f (planarWiener u.toNNReal b)=0) :
+    Tendsto (fun n => ∫ b in A, planarCompensated f (stopGrid n (ρ b)) b ∂W.prod W)
+      atTop (𝓝 (∫ b in A, f (planarWiener (ρ b) b) ∂W.prod W)) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨K,hK,_⟩ := compact_smooth_bounded_lipschitz f hf hc
+  obtain ⟨D,hD,_⟩ := planarGenerator_bounded_lipschitz f hf hc
+  let V (n : ℕ) (b : Driver×Driver) := f (planarWiener (stopGrid n (ρ b)) b)
+  let G (u : ℝ) (b : Driver×Driver) := planarGenerator f (planarWiener u.toNNReal b)
+  let H (n : ℕ) (b : Driver×Driver) := ∫ u in (0:ℝ)..(stopGrid n (ρ b):ℝ), G u b
+  have hVm (n : ℕ) : Measurable (V n) :=
+    measurable_discrete_eval (fun k b => f (planarWiener (stopGridTime n k) b))
+      (fun k => hf.continuous.measurable.comp (measurable_planarWiener _))
+      (stopGridIndex_measurable hρ n)
+  have hVi (n : ℕ) : Integrable (V n) ((W.prod W).restrict A) :=
+    Integrable.of_bound (hVm n).aestronglyMeasurable K (Eventually.of_forall fun b => hK _)
+  have hVlim : Tendsto (fun n => ∫ b in A, V n b ∂W.prod W) atTop
+      (𝓝 (∫ b in A, f (planarWiener (ρ b) b) ∂W.prod W)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => (K:ℝ))
+      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun b => hK _)
+    exact Eventually.of_forall fun b =>
+      (hf.continuous.comp (planarWiener_continuous b)).continuousAt.tendsto.comp (stopGrid_tendsto (ρ b))
+  have hGm : Measurable (Function.uncurry G) := planar_generator_joint_measurable f hf
+  have hGb (u b) : |G u b|≤D := hD _
+  have hHjoint := cumulative_integral_measurable G hGm
+  have hHm (n : ℕ) : Measurable (H n) :=
+    measurable_discrete_eval (fun k b => ∫ u in (0:ℝ)..(stopGridTime n k:ℝ), G u b)
+      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hρ n)
+  have hHb (n : ℕ) (b : Driver×Driver) : |H n b|≤(D:ℝ)*(1/((n:ℝ)+1)) :=
+    bounded_cumulative_at_grid G hGm D D.coe_nonneg hGb ρ hz n b
+  have hHi (n : ℕ) : Integrable (H n) ((W.prod W).restrict A) :=
+    Integrable.of_bound (hHm n).aestronglyMeasurable _
+      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
+  have hHbound (n : ℕ) : |∫ b in A, H n b ∂W.prod W|≤(D:ℝ)*(1/((n:ℝ)+1)) := by
+    have hh := norm_integral_le_of_norm_le_const (μ := (W.prod W).restrict A)
+      (f := H n) (C := (D:ℝ)*(1/((n:ℝ)+1)))
+      (Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hHb n b)
+    have hm : ((W.prod W).restrict A).real univ≤1 := by
+      simpa only [measureReal_def,Measure.restrict_apply_univ] using
+        (measureReal_le_one (μ := W.prod W) (s := A))
+    have hh' : |∫ b in A, H n b ∂W.prod W| ≤
+        (D:ℝ)*(1/((n:ℝ)+1))*((W.prod W).restrict A).real univ := by
+      simpa only [Real.norm_eq_abs] using hh
+    exact hh'.trans
+      (mul_le_of_le_one_right (by positivity) hm)
+  have hHlim : Tendsto (fun n => ∫ b in A, H n b ∂W.prod W) atTop (𝓝 0) := by
+    apply tendsto_iff_norm_sub_tendsto_zero.mpr
+    simp only [sub_zero,Real.norm_eq_abs]
+    apply squeeze_zero (fun n => abs_nonneg _) hHbound
+    simpa only [mul_zero] using
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (D:ℝ)
+  have he (n) : (∫ b in A, planarCompensated f (stopGrid n (ρ b)) b ∂W.prod W)=
+      (∫ b in A, V n b ∂W.prod W)-(∫ b in A, H n b ∂W.prod W) :=
+    integral_sub (hVi n) (hHi n)
+  simpa only [← he,sub_zero] using hVlim.sub hHlim
+
+theorem planar_compact_optional_between
+    (W : WienerLaw) (hW : IsWiener W) (f : ℝ×ℝ → ℝ)
+    (hf : ContDiff ℝ (⊤ : ℕ∞) f) (hc : HasCompactSupport f)
+    (σ τ : Driver×Driver → ℝ≥0)
+    (hσ : IsStoppingTime planarWienerFiltration (fun b => (σ b : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime planarWienerFiltration (fun b => (τ b : WithTop ℝ≥0)))
+    (hστ : ∀ b, σ b ≤ τ b) (T : ℝ≥0) (hb : ∀ b, τ b ≤ T)
+    (A : Set (Driver×Driver)) (hA : MeasurableSet[hσ.measurableSpace] A)
+    (hz : ∀ b u, 0 ≤ u → u<(τ b:ℝ) → planarGenerator f (planarWiener u.toNNReal b)=0) :
+    (∫ b in A, f (planarWiener (σ b) b) ∂W.prod W)=
+      ∫ b in A, f (planarWiener (τ b) b) ∂W.prod W := by
+  have hs := planar_compensated_grid_tendsto W hW f hf hc σ
+    (stopping_time_measurable_nnreal hσ) A (fun b u hu h => hz b u hu (h.trans_le (hστ b)))
+  have ht := planar_compensated_grid_tendsto W hW f hf hc τ
+    (stopping_time_measurable_nnreal hτ) A hz
+  have he (n) := planar_compensated_grid_sampling W hW f hf hc σ τ hσ hτ hστ T hb A hA n
+  have ht' : Tendsto (fun n => ∫ b in A, planarCompensated f (stopGrid n (σ b)) b ∂W.prod W)
+      atTop (𝓝 (∫ b in A, f (planarWiener (τ b) b) ∂W.prod W)) := by
+    simpa only [he] using ht
+  exact tendsto_nhds_unique hs ht'
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerAverage.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerAverage.lean
index 2f98aaacc528e5d5168d31dad311bf6ea9be50b6..fb24959762d6a0df6846eeb841abf42d0843798c 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerAverage.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerAverage.lean
@@ -1,71 +1,71 @@
-import OAI.Probability.SLE.Stochastic.PlanarKernelMean
-import OAI.Probability.SLE.Stochastic.PlanarWienerFiltration
-
-namespace OAI
-
-/-! Exact Gaussian averaging for every event in the planar Wiener past. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-lemma planarWiener_set_integral_average
-    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0)
-    (A : Set (Driver×Driver)) (hA : MeasurableSet[planarWienerFiltration s] A)
-    (f : ℝ×ℝ → ℝ) (hf : Continuous f) (C : ℝ≥0) (hb : ∀ z, |f z|≤C) :
-    (∫ b in A, f (planarWiener (s+δ) b) ∂W.prod W) =
-      ∫ b in A, planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2 ∂W.prod W := by
-  classical
-  let P := W.prod W
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  let Z (b : Driver×Driver) : (ℝ×ℝ)×Bool := (planarWiener s b,if b∈A then true else false)
-  let Y (b : Driver×Driver) := planarWiener (s+δ) b-planarWiener s b
-  have hZm : Measurable[planarWienerFiltration s] Z :=
-    ((Filtration.stronglyAdapted_natural (fun t =>
-      (measurable_planarWiener t).stronglyMeasurable)).adapted s).prodMk
-      (Measurable.ite hA measurable_const measurable_const)
-  have hZ : Measurable Z := hZm.mono (planarWienerFiltration.le s) le_rfl
-  have hY : Measurable Y := (measurable_planarWiener _).sub (measurable_planarWiener _)
-  have hind : IndepFun Z Y P := by
-    apply (IndepFun_iff_Indep _ _ _).mpr
-    exact indep_of_indep_of_le (planarWiener_increment_indep_filtration W hW s δ)
-      hZm.comap_le le_rfl
-  let H (p : ((ℝ×ℝ)×Bool)×(ℝ×ℝ)) : ℝ := if p.1.2=true then f (p.1.1+p.2) else 0
-  have hH : Measurable H := Measurable.ite
-    (measurableSet_eq_fun (measurable_snd.comp measurable_fst) measurable_const)
-    (hf.measurable.comp ((measurable_fst.comp measurable_fst).add measurable_snd)) measurable_const
-  have hHb (p : ((ℝ×ℝ)×Bool)×(ℝ×ℝ)) : |H p|≤C := by
-    dsimp only [H]
-    split_ifs
-    · exact hb _
-    · simp
-  have hi := independent_planar_gaussian_integral Z Y hZ hY δ
-    (planarWiener_increment_law W hW s δ) hind H hH C hHb
-  have hG (z : ℝ×ℝ) :
-      (∫ y, f (z+y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ)) =
-      planarGaussianMean δ f z.1 z.2 := by
-    have hfi : Integrable (fun y : ℝ×ℝ => f (z+y))
-        ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) :=
-      Integrable.of_bound (hf.comp (continuous_const.add continuous_id)).aestronglyMeasurable C
-        (Eventually.of_forall fun y => by simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs] using hb (z+y))
-    exact integral_prod_symm _ hfi
-  have hleft (b : Driver×Driver) : H (Z b,Y b)=A.indicator (fun b => f (planarWiener (s+δ) b)) b := by
-    by_cases ha : b∈A
-    · simp only [H,Z,Y,ha,ite_true,Set.indicator_of_mem,add_sub_cancel]
-    · simp [H,Z,ha]
-  have hright (b : Driver×Driver) :
-      (∫ y, H (Z b,y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ)) =
-        A.indicator (fun b => planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2) b := by
-    by_cases ha : b∈A
-    · simpa only [H,Z,ha,ite_true,Set.indicator_of_mem] using hG (planarWiener s b)
-    · simp [H,Z,ha]
-  have hAm : MeasurableSet A := planarWienerFiltration.le s A hA
-  simpa only [hleft,hright,integral_indicator hAm] using hi
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarKernelMean
+import OAI.Probability.SLE.Stochastic.PlanarWienerFiltration
+
+namespace OAI
+
+/-! Exact Gaussian averaging for every event in the planar Wiener past. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+lemma planarWiener_set_integral_average
+    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0)
+    (A : Set (Driver×Driver)) (hA : MeasurableSet[planarWienerFiltration s] A)
+    (f : ℝ×ℝ → ℝ) (hf : Continuous f) (C : ℝ≥0) (hb : ∀ z, |f z|≤C) :
+    (∫ b in A, f (planarWiener (s+δ) b) ∂W.prod W) =
+      ∫ b in A, planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2 ∂W.prod W := by
+  classical
+  let P := W.prod W
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  let Z (b : Driver×Driver) : (ℝ×ℝ)×Bool := (planarWiener s b,if b∈A then true else false)
+  let Y (b : Driver×Driver) := planarWiener (s+δ) b-planarWiener s b
+  have hZm : Measurable[planarWienerFiltration s] Z :=
+    ((Filtration.stronglyAdapted_natural (fun t =>
+      (measurable_planarWiener t).stronglyMeasurable)).adapted s).prodMk
+      (Measurable.ite hA measurable_const measurable_const)
+  have hZ : Measurable Z := hZm.mono (planarWienerFiltration.le s) le_rfl
+  have hY : Measurable Y := (measurable_planarWiener _).sub (measurable_planarWiener _)
+  have hind : IndepFun Z Y P := by
+    apply (IndepFun_iff_Indep _ _ _).mpr
+    exact indep_of_indep_of_le (planarWiener_increment_indep_filtration W hW s δ)
+      hZm.comap_le le_rfl
+  let H (p : ((ℝ×ℝ)×Bool)×(ℝ×ℝ)) : ℝ := if p.1.2=true then f (p.1.1+p.2) else 0
+  have hH : Measurable H := Measurable.ite
+    (measurableSet_eq_fun (measurable_snd.comp measurable_fst) measurable_const)
+    (hf.measurable.comp ((measurable_fst.comp measurable_fst).add measurable_snd)) measurable_const
+  have hHb (p : ((ℝ×ℝ)×Bool)×(ℝ×ℝ)) : |H p|≤C := by
+    dsimp only [H]
+    split_ifs
+    · exact hb _
+    · simp
+  have hi := independent_planar_gaussian_integral Z Y hZ hY δ
+    (planarWiener_increment_law W hW s δ) hind H hH C hHb
+  have hG (z : ℝ×ℝ) :
+      (∫ y, f (z+y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ)) =
+      planarGaussianMean δ f z.1 z.2 := by
+    have hfi : Integrable (fun y : ℝ×ℝ => f (z+y))
+        ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) :=
+      Integrable.of_bound (hf.comp (continuous_const.add continuous_id)).aestronglyMeasurable C
+        (Eventually.of_forall fun y => by simpa only [Function.comp_apply,Pi.add_apply,id_eq,Real.norm_eq_abs] using hb (z+y))
+    exact integral_prod_symm _ hfi
+  have hleft (b : Driver×Driver) : H (Z b,Y b)=A.indicator (fun b => f (planarWiener (s+δ) b)) b := by
+    by_cases ha : b∈A
+    · simp only [H,Z,Y,ha,ite_true,Set.indicator_of_mem,add_sub_cancel]
+    · simp [H,Z,ha]
+  have hright (b : Driver×Driver) :
+      (∫ y, H (Z b,y) ∂(gaussianReal 0 δ).prod (gaussianReal 0 δ)) =
+        A.indicator (fun b => planarGaussianMean δ f (planarWiener s b).1 (planarWiener s b).2) b := by
+    by_cases ha : b∈A
+    · simpa only [H,Z,ha,ite_true,Set.indicator_of_mem] using hG (planarWiener s b)
+    · simp [H,Z,ha]
+  have hAm : MeasurableSet A := planarWienerFiltration.le s A hA
+  simpa only [hleft,hright,integral_indicator hAm] using hi
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerFiltration.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerFiltration.lean
index 9ad993cf171a15ed0208cba5478aa15f253451bd..dca34a078188cacce9c9f1d8e4602630fe67e5e0 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerFiltration.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerFiltration.lean
@@ -1,59 +1,59 @@
-import OAI.Probability.SLE.Stochastic.PlanarWienerIncrements
-
-namespace OAI
-
-/-! Full-past filtration and restricted Gaussian increment laws for the
-actual two-coordinate Wiener path. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-def planarWienerFiltration : Filtration ℝ≥0 (inferInstance : MeasurableSpace (Driver×Driver)) :=
-  Filtration.natural planarWiener (fun t => (measurable_planarWiener t).stronglyMeasurable)
-
-lemma planarWienerFiltration_le_past (s : ℝ≥0) :
-    planarWienerFiltration s ≤ MeasurableSpace.comap (planarWienerPast s) inferInstance := by
-  rw [planarWienerFiltration,Filtration.natural_eq_comap]
-  have hp : Measurable[MeasurableSpace.comap (planarWienerPast s) inferInstance]
-      (planarWienerPast s) := measurable_iff_comap_le.mpr le_rfl
-  have hm : Measurable[MeasurableSpace.comap (planarWienerPast s) inferInstance]
-      (fun b : Driver×Driver => fun t : Iic s => planarWiener t b) := by
-    let : MeasurableSpace (Driver×Driver) :=
-      MeasurableSpace.comap (planarWienerPast s) inferInstance
-    apply measurable_pi_iff.mpr
-    intro t
-    exact ((measurable_pi_apply t).comp (measurable_fst.comp hp)).prodMk
-      ((measurable_pi_apply t).comp (measurable_snd.comp hp))
-  exact hm.comap_le
-
-lemma planarWiener_increment_indep_filtration
-    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
-    Indep (planarWienerFiltration s)
-      (MeasurableSpace.comap (fun b => planarWiener (s+δ) b-planarWiener s b) inferInstance)
-      (W.prod W) := by
-  have hi := (IndepFun_iff_Indep _ _ _).mp (planarWiener_increment_indep_past W hW s δ)
-  exact indep_of_indep_of_le hi (planarWienerFiltration_le_past s) le_rfl
-
-lemma planarWiener_increment_restrict_law
-    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0)
-    (A : Set (Driver×Driver)) (hA : MeasurableSet[planarWienerFiltration s] A) :
-    ((W.prod W).restrict A).map (fun b => planarWiener (s+δ) b-planarWiener s b) =
-      (W.prod W) A • (gaussianReal 0 δ).prod (gaussianReal 0 δ) := by
-  have hm : Measurable (fun b => planarWiener (s+δ) b-planarWiener s b) :=
-    (measurable_planarWiener (s+δ)).sub (measurable_planarWiener s)
-  have hi := planarWiener_increment_indep_filtration W hW s δ
-  have hL := planarWiener_increment_law W hW s δ
-  ext S hS
-  rw [Measure.map_apply hm hS,Measure.restrict_apply (hm hS),Measure.smul_apply,smul_eq_mul]
-  rw [inter_comm]
-  rw [(hi.indepSet_of_measurableSet hA ⟨S,hS,rfl⟩).measure_inter_eq_mul]
-  exact congrArg (fun x => (W.prod W) A*x) (hL.measure_eq hS)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarWienerIncrements
+
+namespace OAI
+
+/-! Full-past filtration and restricted Gaussian increment laws for the
+actual two-coordinate Wiener path. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+def planarWienerFiltration : Filtration ℝ≥0 (inferInstance : MeasurableSpace (Driver×Driver)) :=
+  Filtration.natural planarWiener (fun t => (measurable_planarWiener t).stronglyMeasurable)
+
+lemma planarWienerFiltration_le_past (s : ℝ≥0) :
+    planarWienerFiltration s ≤ MeasurableSpace.comap (planarWienerPast s) inferInstance := by
+  rw [planarWienerFiltration,Filtration.natural_eq_comap]
+  have hp : Measurable[MeasurableSpace.comap (planarWienerPast s) inferInstance]
+      (planarWienerPast s) := measurable_iff_comap_le.mpr le_rfl
+  have hm : Measurable[MeasurableSpace.comap (planarWienerPast s) inferInstance]
+      (fun b : Driver×Driver => fun t : Iic s => planarWiener t b) := by
+    let : MeasurableSpace (Driver×Driver) :=
+      MeasurableSpace.comap (planarWienerPast s) inferInstance
+    apply measurable_pi_iff.mpr
+    intro t
+    exact ((measurable_pi_apply t).comp (measurable_fst.comp hp)).prodMk
+      ((measurable_pi_apply t).comp (measurable_snd.comp hp))
+  exact hm.comap_le
+
+lemma planarWiener_increment_indep_filtration
+    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
+    Indep (planarWienerFiltration s)
+      (MeasurableSpace.comap (fun b => planarWiener (s+δ) b-planarWiener s b) inferInstance)
+      (W.prod W) := by
+  have hi := (IndepFun_iff_Indep _ _ _).mp (planarWiener_increment_indep_past W hW s δ)
+  exact indep_of_indep_of_le hi (planarWienerFiltration_le_past s) le_rfl
+
+lemma planarWiener_increment_restrict_law
+    (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0)
+    (A : Set (Driver×Driver)) (hA : MeasurableSet[planarWienerFiltration s] A) :
+    ((W.prod W).restrict A).map (fun b => planarWiener (s+δ) b-planarWiener s b) =
+      (W.prod W) A • (gaussianReal 0 δ).prod (gaussianReal 0 δ) := by
+  have hm : Measurable (fun b => planarWiener (s+δ) b-planarWiener s b) :=
+    (measurable_planarWiener (s+δ)).sub (measurable_planarWiener s)
+  have hi := planarWiener_increment_indep_filtration W hW s δ
+  have hL := planarWiener_increment_law W hW s δ
+  ext S hS
+  rw [Measure.map_apply hm hS,Measure.restrict_apply (hm hS),Measure.smul_apply,smul_eq_mul]
+  rw [inter_comm]
+  rw [(hi.indepSet_of_measurableSet hA ⟨S,hS,rfl⟩).measure_inter_eq_mul]
+  exact congrArg (fun x => (W.prod W) A*x) (hL.measure_eq hS)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerIncrements.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerIncrements.lean
index cc77b8dfd115112c3229e8f4126ca667a449d5b1..5afb45022800c06d198b6ff5d70b267316856443 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerIncrements.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerIncrements.lean
@@ -1,59 +1,59 @@
-import OAI.Probability.SLE.Stochastic.IndependentPlanarIncrements
-
-namespace OAI
-
-/-! The two-coordinate Wiener increment has its exact product Gaussian
-law and is independent of both coordinate pasts. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-def planarWiener (t : ℝ≥0) (b : Driver×Driver) : ℝ×ℝ := (b.1 t,b.2 t)
-
-def planarWienerPast (s : ℝ≥0) (b : Driver×Driver) :
-    (Iic s → ℝ)×(Iic s → ℝ) := (fun t => b.1 t,fun t => b.2 t)
-
-lemma measurable_planarWiener (t : ℝ≥0) : Measurable (planarWiener t) := by
-  exact ((continuous_eval_const t).measurable.comp measurable_fst).prodMk
-    ((continuous_eval_const t).measurable.comp measurable_snd)
-
-lemma measurable_planarWienerPast (s : ℝ≥0) : Measurable (planarWienerPast s) := by
-  apply Measurable.prodMk
-  · exact measurable_pi_iff.mpr (fun t => (continuous_eval_const (t:ℝ≥0)).measurable.comp measurable_fst)
-  · exact measurable_pi_iff.mpr (fun t => (continuous_eval_const (t:ℝ≥0)).measurable.comp measurable_snd)
-
-lemma planarWiener_increment_law (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
-    HasLaw (fun b => planarWiener (s+δ) b-planarWiener s b)
-      ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) (W.prod W) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hm : Measurable (fun b : Driver => b (s+δ)-b s) :=
-    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
-  have hL : HasLaw (fun b : Driver => b (s+δ)-b s) (gaussianReal 0 δ) W :=
-    (hW.shift s).hasLaw_eval δ
-  have h1 : HasLaw (fun b : Driver×Driver => b.1 (s+δ)-b.1 s)
-      (gaussianReal 0 δ) (W.prod W) := hL.comp measurePreserving_fst.hasLaw
-  have h2 : HasLaw (fun b : Driver×Driver => b.2 (s+δ)-b.2 s)
-      (gaussianReal 0 δ) (W.prod W) := hL.comp measurePreserving_snd.hasLaw
-  exact (indepFun_prod hm hm).hasLaw_prod h1 h2
-
-lemma planarWiener_increment_indep_past (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
-    IndepFun (planarWienerPast s)
-      (fun b => planarWiener (s+δ) b-planarWiener s b) (W.prod W) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hp : Measurable (fun b : Driver => fun t : Iic s => b t) :=
-    measurable_pi_iff.mpr (fun t => (continuous_eval_const (t:ℝ≥0)).measurable)
-  have hi : Measurable (fun b : Driver => b (s+δ)-b s) :=
-    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
-  have hind : IndepFun (fun b : Driver => fun t : Iic s => b t)
-      (fun b : Driver => b (s+δ)-b s) W := by
-    exact ((hW.indepFun_shift s).comp (measurable_pi_apply δ) measurable_id).symm
-  exact independent_product_pairs hp hi hp hi hind hind
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.IndependentPlanarIncrements
+
+namespace OAI
+
+/-! The two-coordinate Wiener increment has its exact product Gaussian
+law and is independent of both coordinate pasts. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+def planarWiener (t : ℝ≥0) (b : Driver×Driver) : ℝ×ℝ := (b.1 t,b.2 t)
+
+def planarWienerPast (s : ℝ≥0) (b : Driver×Driver) :
+    (Iic s → ℝ)×(Iic s → ℝ) := (fun t => b.1 t,fun t => b.2 t)
+
+lemma measurable_planarWiener (t : ℝ≥0) : Measurable (planarWiener t) := by
+  exact ((continuous_eval_const t).measurable.comp measurable_fst).prodMk
+    ((continuous_eval_const t).measurable.comp measurable_snd)
+
+lemma measurable_planarWienerPast (s : ℝ≥0) : Measurable (planarWienerPast s) := by
+  apply Measurable.prodMk
+  · exact measurable_pi_iff.mpr (fun t => (continuous_eval_const (t:ℝ≥0)).measurable.comp measurable_fst)
+  · exact measurable_pi_iff.mpr (fun t => (continuous_eval_const (t:ℝ≥0)).measurable.comp measurable_snd)
+
+lemma planarWiener_increment_law (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
+    HasLaw (fun b => planarWiener (s+δ) b-planarWiener s b)
+      ((gaussianReal 0 δ).prod (gaussianReal 0 δ)) (W.prod W) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hm : Measurable (fun b : Driver => b (s+δ)-b s) :=
+    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
+  have hL : HasLaw (fun b : Driver => b (s+δ)-b s) (gaussianReal 0 δ) W :=
+    (hW.shift s).hasLaw_eval δ
+  have h1 : HasLaw (fun b : Driver×Driver => b.1 (s+δ)-b.1 s)
+      (gaussianReal 0 δ) (W.prod W) := hL.comp measurePreserving_fst.hasLaw
+  have h2 : HasLaw (fun b : Driver×Driver => b.2 (s+δ)-b.2 s)
+      (gaussianReal 0 δ) (W.prod W) := hL.comp measurePreserving_snd.hasLaw
+  exact (indepFun_prod hm hm).hasLaw_prod h1 h2
+
+lemma planarWiener_increment_indep_past (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
+    IndepFun (planarWienerPast s)
+      (fun b => planarWiener (s+δ) b-planarWiener s b) (W.prod W) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hp : Measurable (fun b : Driver => fun t : Iic s => b t) :=
+    measurable_pi_iff.mpr (fun t => (continuous_eval_const (t:ℝ≥0)).measurable)
+  have hi : Measurable (fun b : Driver => b (s+δ)-b s) :=
+    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
+  have hind : IndepFun (fun b : Driver => fun t : Iic s => b t)
+      (fun b : Driver => b (s+δ)-b s) W := by
+    exact ((hW.indepFun_shift s).comp (measurable_pi_apply δ) measurable_id).symm
+  exact independent_product_pairs hp hi hp hi hind hind
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerMoments.lean b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerMoments.lean
index d3d8e39d2be97e56ecb8309fbc6cbdc1e10e7812..a068e633da62d9cf8022ef0d8077d13dee6189f3 100644
--- a/lean/OAI/Probability/SLE/Stochastic/PlanarWienerMoments.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/PlanarWienerMoments.lean
@@ -1,65 +1,65 @@
-import OAI.Probability.SLE.Stochastic.PlanarWienerIncrements
-
-namespace OAI
-
-/-! First-moment control of the actual two-coordinate Wiener increments. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-open Target
-
-local instance : SecondCountableTopology Driver :=
-  ContinuousMapZero.isEmbedding_toContinuousMap.secondCountableTopology
-
-lemma planarWiener_increment_integrable (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
-    Integrable (fun b => planarWiener (s+δ) b-planarWiener s b) (W.prod W) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hL := planarWiener_increment_law W hW s δ
-  have hx := (measurePreserving_fst (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
-  have hy := (measurePreserving_snd (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
-  exact (hx.hasGaussianLaw.integrable).prodMk hy.hasGaussianLaw.integrable
-
-lemma planarWiener_increment_norm_mean (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
-    (∫ b, ‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W) ≤
-      2*Real.sqrt δ*standardGaussianAbsMoment 1 := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hL := planarWiener_increment_law W hW s δ
-  have hx := (measurePreserving_fst (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
-  have hy := (measurePreserving_snd (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
-  have hxi : Integrable (fun b => |(planarWiener (s+δ) b-planarWiener s b).1|) (W.prod W) := by
-    simpa only [Function.comp_apply,Real.norm_eq_abs] using hx.hasGaussianLaw.integrable.norm
-  have hyi : Integrable (fun b => |(planarWiener (s+δ) b-planarWiener s b).2|) (W.prod W) := by
-    simpa only [Function.comp_apply,Real.norm_eq_abs] using hy.hasGaussianLaw.integrable.norm
-  calc
-    _ ≤ ∫ b, |(planarWiener (s+δ) b-planarWiener s b).1|+
-        |(planarWiener (s+δ) b-planarWiener s b).2| ∂W.prod W := by
-      apply integral_mono (planarWiener_increment_integrable W hW s δ).norm (hxi.add hyi)
-      intro b
-      change ‖planarWiener (s+δ) b-planarWiener s b‖ ≤
-        |(planarWiener (s+δ) b-planarWiener s b).1|+|(planarWiener (s+δ) b-planarWiener s b).2|
-      rw [Prod.norm_def,Real.norm_eq_abs,Real.norm_eq_abs]
-      exact max_le (le_add_of_nonneg_right (abs_nonneg _))
-        (le_add_of_nonneg_left (abs_nonneg _))
-    _ = _ := by
-      rw [integral_add hxi hyi]
-      have ex := gaussian_abs_moment hx 1
-      have ey := gaussian_abs_moment hy 1
-      simp only [pow_one,Function.comp_apply] at ex ey
-      rw [ex,ey]
-      ring
-
-lemma planarWiener_continuous (b : Driver×Driver) : Continuous (fun t => planarWiener t b) :=
-  b.1.continuous.prodMk b.2.continuous
-
-lemma planarWiener_joint_measurable :
-    Measurable (fun p : ℝ≥0×(Driver×Driver) => planarWiener p.1 p.2) := by
-  exact ((continuous_eval.comp (continuous_snd.fst.prodMk continuous_fst)).prodMk
-    (continuous_eval.comp (continuous_snd.snd.prodMk continuous_fst))).measurable
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.PlanarWienerIncrements
+
+namespace OAI
+
+/-! First-moment control of the actual two-coordinate Wiener increments. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+open Target
+
+local instance : SecondCountableTopology Driver :=
+  ContinuousMapZero.isEmbedding_toContinuousMap.secondCountableTopology
+
+lemma planarWiener_increment_integrable (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
+    Integrable (fun b => planarWiener (s+δ) b-planarWiener s b) (W.prod W) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hL := planarWiener_increment_law W hW s δ
+  have hx := (measurePreserving_fst (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
+  have hy := (measurePreserving_snd (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
+  exact (hx.hasGaussianLaw.integrable).prodMk hy.hasGaussianLaw.integrable
+
+lemma planarWiener_increment_norm_mean (W : WienerLaw) (hW : IsWiener W) (s δ : ℝ≥0) :
+    (∫ b, ‖planarWiener (s+δ) b-planarWiener s b‖ ∂W.prod W) ≤
+      2*Real.sqrt δ*standardGaussianAbsMoment 1 := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hL := planarWiener_increment_law W hW s δ
+  have hx := (measurePreserving_fst (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
+  have hy := (measurePreserving_snd (μ := gaussianReal 0 δ) (ν := gaussianReal 0 δ)).hasLaw.comp hL
+  have hxi : Integrable (fun b => |(planarWiener (s+δ) b-planarWiener s b).1|) (W.prod W) := by
+    simpa only [Function.comp_apply,Real.norm_eq_abs] using hx.hasGaussianLaw.integrable.norm
+  have hyi : Integrable (fun b => |(planarWiener (s+δ) b-planarWiener s b).2|) (W.prod W) := by
+    simpa only [Function.comp_apply,Real.norm_eq_abs] using hy.hasGaussianLaw.integrable.norm
+  calc
+    _ ≤ ∫ b, |(planarWiener (s+δ) b-planarWiener s b).1|+
+        |(planarWiener (s+δ) b-planarWiener s b).2| ∂W.prod W := by
+      apply integral_mono (planarWiener_increment_integrable W hW s δ).norm (hxi.add hyi)
+      intro b
+      change ‖planarWiener (s+δ) b-planarWiener s b‖ ≤
+        |(planarWiener (s+δ) b-planarWiener s b).1|+|(planarWiener (s+δ) b-planarWiener s b).2|
+      rw [Prod.norm_def,Real.norm_eq_abs,Real.norm_eq_abs]
+      exact max_le (le_add_of_nonneg_right (abs_nonneg _))
+        (le_add_of_nonneg_left (abs_nonneg _))
+    _ = _ := by
+      rw [integral_add hxi hyi]
+      have ex := gaussian_abs_moment hx 1
+      have ey := gaussian_abs_moment hy 1
+      simp only [pow_one,Function.comp_apply] at ex ey
+      rw [ex,ey]
+      ring
+
+lemma planarWiener_continuous (b : Driver×Driver) : Continuous (fun t => planarWiener t b) :=
+  b.1.continuous.prodMk b.2.continuous
+
+lemma planarWiener_joint_measurable :
+    Measurable (fun p : ℝ≥0×(Driver×Driver) => planarWiener p.1 p.2) := by
+  exact ((continuous_eval.comp (continuous_snd.fst.prodMk continuous_fst)).prodMk
+    (continuous_eval.comp (continuous_snd.snd.prodMk continuous_fst))).measurable
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/ProgressiveParameterTime.lean b/lean/OAI/Probability/SLE/Stochastic/ProgressiveParameterTime.lean
index 3b0b6401f119f7f16d08630c156e94e49aa951f5..b57d695fa0a14e2538a4b3e3a44489e2deeb491a 100644
--- a/lean/OAI/Probability/SLE/Stochastic/ProgressiveParameterTime.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/ProgressiveParameterTime.lean
@@ -1,46 +1,46 @@
-import OAI.Probability.SLE.Stochastic.ProgressiveRandomTime
-
-namespace OAI
-
-/-! A parameter may choose both the sample and the evaluation time of a
-progressive process. Finite truncation keeps the original raw sigma algebras. -/
-
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped NNReal ENNReal
-
-namespace MeasureTheory.IsProgressive
-open scoped _root_.MeasureTheory _root_.MeasureTheory.IsProgressive
-
-theorem parameter_time_measurable
-    {Ω Ξ X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace Ξ] [MeasurableSpace X]
-    {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X} (hN : _root_.MeasureTheory.IsProgressive ℱ N)
-    (g : Ξ → Ω) (hg : Measurable g) (T : Ξ → ℝ≥0) (hT : Measurable T) :
-    Measurable (fun x => N (T x) (g x)) := by
-  let f (n : ℕ) (x : Ξ) := N (min (T x) (n:ℝ≥0)) (g x)
-  have hf (n : ℕ) : Measurable (f n) := by
-    have ht : Measurable (fun x => (⟨min (T x) (n:ℝ≥0),
-        Set.mem_Iic.mpr (min_le_right _ _)⟩ : Iic (n:ℝ≥0))) :=
-      (hT.min measurable_const).subtype_mk
-    have hgn : @Measurable Ξ Ω _ (ℱ (n:ℝ≥0)) g := hg.mono le_rfl (ℱ.le _)
-    exact (hN (n:ℝ≥0)).comp (ht.prodMk hgn)
-  intro A hA
-  have he : (fun x => N (T x) (g x)) ⁻¹' A =
-      ⋃ n : ℕ, {x | T x ≤ (n:ℝ≥0)} ∩ (f n) ⁻¹' A := by
-    ext x
-    simp only [mem_preimage,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
-    constructor
-    · intro hx
-      obtain ⟨n,hn⟩ := exists_nat_ge (T x : ℝ)
-      have htn : T x ≤ (n:ℝ≥0) := by exact_mod_cast hn
-      exact ⟨n,htn,by simpa only [f,min_eq_left htn] using hx⟩
-    · rintro ⟨n,hn,hx⟩
-      simpa only [f,min_eq_left hn] using hx
-  rw [he]
-  exact MeasurableSet.iUnion (fun n => (measurableSet_le hT measurable_const).inter (hf n hA))
-
-end MeasureTheory.IsProgressive
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.ProgressiveRandomTime
+
+namespace OAI
+
+/-! A parameter may choose both the sample and the evaluation time of a
+progressive process. Finite truncation keeps the original raw sigma algebras. -/
+
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped NNReal ENNReal
+
+namespace MeasureTheory.IsProgressive
+open scoped _root_.MeasureTheory _root_.MeasureTheory.IsProgressive
+
+theorem parameter_time_measurable
+    {Ω Ξ X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace Ξ] [MeasurableSpace X]
+    {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X} (hN : _root_.MeasureTheory.IsProgressive ℱ N)
+    (g : Ξ → Ω) (hg : Measurable g) (T : Ξ → ℝ≥0) (hT : Measurable T) :
+    Measurable (fun x => N (T x) (g x)) := by
+  let f (n : ℕ) (x : Ξ) := N (min (T x) (n:ℝ≥0)) (g x)
+  have hf (n : ℕ) : Measurable (f n) := by
+    have ht : Measurable (fun x => (⟨min (T x) (n:ℝ≥0),
+        Set.mem_Iic.mpr (min_le_right _ _)⟩ : Iic (n:ℝ≥0))) :=
+      (hT.min measurable_const).subtype_mk
+    have hgn : @Measurable Ξ Ω _ (ℱ (n:ℝ≥0)) g := hg.mono le_rfl (ℱ.le _)
+    exact (hN (n:ℝ≥0)).comp (ht.prodMk hgn)
+  intro A hA
+  have he : (fun x => N (T x) (g x)) ⁻¹' A =
+      ⋃ n : ℕ, {x | T x ≤ (n:ℝ≥0)} ∩ (f n) ⁻¹' A := by
+    ext x
+    simp only [mem_preimage,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
+    constructor
+    · intro hx
+      obtain ⟨n,hn⟩ := exists_nat_ge (T x : ℝ)
+      have htn : T x ≤ (n:ℝ≥0) := by exact_mod_cast hn
+      exact ⟨n,htn,by simpa only [f,min_eq_left htn] using hx⟩
+    · rintro ⟨n,hn,hx⟩
+      simpa only [f,min_eq_left hn] using hx
+  rw [he]
+  exact MeasurableSet.iUnion (fun n => (measurableSet_le hT measurable_const).inter (hf n hA))
+
+end MeasureTheory.IsProgressive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/ProgressiveRandomTime.lean b/lean/OAI/Probability/SLE/Stochastic/ProgressiveRandomTime.lean
index e70202d2317b512e0a0caaedc4325bae7993bde8..68157f60a87904ecaaa3cf4cfc3da0e58ee1d95b 100644
--- a/lean/OAI/Probability/SLE/Stochastic/ProgressiveRandomTime.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/ProgressiveRandomTime.lean
@@ -1,59 +1,59 @@
-import Mathlib.Probability.Process.Adapted
-import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
-
-namespace OAI
-
-/-! Measurable evaluation of a progressive process at an arbitrary finite
-random time. The target needs only a measurable-space structure. -/
-
-noncomputable section
-open Set MeasureTheory
-open scoped NNReal ENNReal
-
-namespace MeasureTheory.IsProgressive
-open scoped _root_.MeasureTheory _root_.MeasureTheory.IsProgressive
-
-theorem random_time_measurable
-    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
-    {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X}
-    (hN : _root_.MeasureTheory.IsProgressive ℱ N) (T : Ω → ℝ≥0) (hT : Measurable T) :
-    Measurable (fun ω => N (T ω) ω) := by
-  let f (n : ℕ) (ω : Ω) := N (min (T ω) (n:ℝ≥0)) ω
-  have hf (n : ℕ) : Measurable (f n) := by
-    have ht : Measurable (fun ω => (⟨min (T ω) (n:ℝ≥0),Set.mem_Iic.mpr (min_le_right _ _)⟩ : Iic (n:ℝ≥0))) :=
-      (hT.min measurable_const).subtype_mk
-    have hω : @Measurable Ω Ω mΩ (ℱ (n:ℝ≥0)) id := by
-      intro A hA
-      exact ℱ.le (n:ℝ≥0) A hA
-    have hp : @Measurable Ω (Iic (n:ℝ≥0) × Ω) mΩ
-        (Subtype.instMeasurableSpace.prod (ℱ (n:ℝ≥0)))
-        (fun ω => (⟨min (T ω) (n:ℝ≥0),Set.mem_Iic.mpr (min_le_right _ _)⟩,ω)) := ht.prodMk hω
-    exact (hN (n:ℝ≥0)).comp hp
-  intro A hA
-  have he : (fun ω => N (T ω) ω) ⁻¹' A =
-      ⋃ n : ℕ, {ω | T ω ≤ (n:ℝ≥0)} ∩ (f n) ⁻¹' A := by
-    ext ω
-    simp only [mem_preimage,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
-    constructor
-    · intro hω
-      obtain ⟨n,hn⟩ := exists_nat_ge (T ω : ℝ)
-      have htn : T ω ≤ (n:ℝ≥0) := by exact_mod_cast hn
-      exact ⟨n,htn,by simpa only [f,min_eq_left htn] using hω⟩
-    · rintro ⟨n,hn,hω⟩
-      simpa only [f,min_eq_left hn] using hω
-  rw [he]
-  exact MeasurableSet.iUnion (fun n =>
-    (measurableSet_le hT measurable_const).inter (hf n hA))
-
-theorem extended_random_time_measurable
-    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
-    {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X}
-    (hN : _root_.MeasureTheory.IsProgressive ℱ N) (S : Ω → ℝ≥0∞) (hS : Measurable S) :
-    Measurable (fun ω => N (S ω).toNNReal ω) :=
-  _root_.OAI.MeasureTheory.IsProgressive.random_time_measurable hN (fun ω => (S ω).toNNReal) (ENNReal.measurable_toNNReal.comp hS)
-
-end MeasureTheory.IsProgressive
-
-end
-
-end OAI
+import Mathlib.Probability.Process.Adapted
+import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
+
+namespace OAI
+
+/-! Measurable evaluation of a progressive process at an arbitrary finite
+random time. The target needs only a measurable-space structure. -/
+
+noncomputable section
+open Set MeasureTheory
+open scoped NNReal ENNReal
+
+namespace MeasureTheory.IsProgressive
+open scoped _root_.MeasureTheory _root_.MeasureTheory.IsProgressive
+
+theorem random_time_measurable
+    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
+    {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X}
+    (hN : _root_.MeasureTheory.IsProgressive ℱ N) (T : Ω → ℝ≥0) (hT : Measurable T) :
+    Measurable (fun ω => N (T ω) ω) := by
+  let f (n : ℕ) (ω : Ω) := N (min (T ω) (n:ℝ≥0)) ω
+  have hf (n : ℕ) : Measurable (f n) := by
+    have ht : Measurable (fun ω => (⟨min (T ω) (n:ℝ≥0),Set.mem_Iic.mpr (min_le_right _ _)⟩ : Iic (n:ℝ≥0))) :=
+      (hT.min measurable_const).subtype_mk
+    have hω : @Measurable Ω Ω mΩ (ℱ (n:ℝ≥0)) id := by
+      intro A hA
+      exact ℱ.le (n:ℝ≥0) A hA
+    have hp : @Measurable Ω (Iic (n:ℝ≥0) × Ω) mΩ
+        (Subtype.instMeasurableSpace.prod (ℱ (n:ℝ≥0)))
+        (fun ω => (⟨min (T ω) (n:ℝ≥0),Set.mem_Iic.mpr (min_le_right _ _)⟩,ω)) := ht.prodMk hω
+    exact (hN (n:ℝ≥0)).comp hp
+  intro A hA
+  have he : (fun ω => N (T ω) ω) ⁻¹' A =
+      ⋃ n : ℕ, {ω | T ω ≤ (n:ℝ≥0)} ∩ (f n) ⁻¹' A := by
+    ext ω
+    simp only [mem_preimage,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
+    constructor
+    · intro hω
+      obtain ⟨n,hn⟩ := exists_nat_ge (T ω : ℝ)
+      have htn : T ω ≤ (n:ℝ≥0) := by exact_mod_cast hn
+      exact ⟨n,htn,by simpa only [f,min_eq_left htn] using hω⟩
+    · rintro ⟨n,hn,hω⟩
+      simpa only [f,min_eq_left hn] using hω
+  rw [he]
+  exact MeasurableSet.iUnion (fun n =>
+    (measurableSet_le hT measurable_const).inter (hf n hA))
+
+theorem extended_random_time_measurable
+    {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X]
+    {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X}
+    (hN : _root_.MeasureTheory.IsProgressive ℱ N) (S : Ω → ℝ≥0∞) (hS : Measurable S) :
+    Measurable (fun ω => N (S ω).toNNReal ω) :=
+  _root_.OAI.MeasureTheory.IsProgressive.random_time_measurable hN (fun ω => (S ω).toNNReal) (ENNReal.measurable_toNNReal.comp hS)
+
+end MeasureTheory.IsProgressive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/ProgressiveStoppedTime.lean b/lean/OAI/Probability/SLE/Stochastic/ProgressiveStoppedTime.lean
index 52e64527394d78d519098b2c380dcce9e85ac9c4..be7cbd619875647d46eaccb2c1d9662eb8be1589 100644
--- a/lean/OAI/Probability/SLE/Stochastic/ProgressiveStoppedTime.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/ProgressiveStoppedTime.lean
@@ -1,88 +1,88 @@
-import OAI.Probability.SLE.Stochastic.ProgressiveRandomTime
-import OAI.Probability.SLE.Stochastic.ShiftedStoppingFiltration
-
-namespace OAI
-
-/-! Stopped-value measurability for a progressive process with a merely
-measurable target, in particular a measure-valued canonical mass. -/
-
-noncomputable section
-open Set _root_.MeasureTheory _root_.OAI.MeasureTheory
-open scoped NNReal ENNReal
-
-namespace MeasureTheory.IsProgressive
-open scoped _root_.MeasureTheory _root_.MeasureTheory.IsProgressive
-
-variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
-  {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X}
-
-lemma bounded_stopped_measurable (hN : _root_.MeasureTheory.IsProgressive ℱ N)
-    (τ : Ω → ℝ≥0)
-    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
-    (H : ℝ≥0) (hH : ∀ ω, τ ω ≤ H) :
-    Measurable[ℱ H] (fun ω => N (τ ω) ω) := by
-  have htime : Measurable[ℱ H] τ := by
-    have h := (hτ.measurable_of_le (fun ω => WithTop.coe_le_coe.mpr (hH ω))).untopA
-    convert h using 1
-    funext ω
-    rw [WithTop.untopA_eq_untop WithTop.coe_ne_top]
-    rfl
-  have hp : Measurable[ℱ H,Subtype.instMeasurableSpace.prod (ℱ H)]
-      (fun ω => ((⟨τ ω,hH ω⟩ : Iic H),ω)) := by
-    exact htime.subtype_mk.prodMk measurable_id
-  exact (hN H).comp hp
-
-lemma truncated_stopped_measurable (hN : _root_.MeasureTheory.IsProgressive ℱ N)
-    (τ : Ω → ℝ≥0)
-    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) (H : ℝ≥0) :
-    Measurable[ℱ H] (fun ω => N (min (τ ω) H) ω) := by
-  have hmin : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => ((min (τ ω) H : ℝ≥0) : WithTop ℝ≥0)) := by
-    simpa only [WithTop.coe_min] using hτ.min_const H
-  exact _root_.OAI.MeasureTheory.IsProgressive.bounded_stopped_measurable hN _ hmin H (fun _ => min_le_right _ _)
-
-/-- This requires no topology on the target. -/
-theorem stopped_time_measurable (hN : _root_.MeasureTheory.IsProgressive ℱ N)
-    (τ : Ω → ℝ≥0)
-    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) :
-    Measurable[hτ.measurableSpace] (fun ω => N (τ ω) ω) := by
-  have hslice (S : Set X) (hS : MeasurableSet S) (H : ℝ≥0) :
-      MeasurableSet[ℱ H] ((fun ω => N (τ ω) ω) ⁻¹' S ∩ {ω | τ ω ≤ H}) := by
-    have he : (fun ω => N (τ ω) ω) ⁻¹' S ∩ {ω | τ ω ≤ H} =
-        (fun ω => N (min (τ ω) H) ω) ⁻¹' S ∩ {ω | τ ω ≤ H} := by
-      ext ω
-      simp only [mem_inter_iff,mem_preimage,mem_ofPred_eq]
-      constructor <;> rintro ⟨hω,ht⟩ <;> exact ⟨by simpa only [min_eq_left ht] using hω,ht⟩
-    rw [he]
-    exact ((_root_.OAI.MeasureTheory.IsProgressive.truncated_stopped_measurable hN τ hτ H) hS).inter (by simpa only [WithTop.coe_le_coe] using hτ H)
-  intro S hS
-  constructor
-  · have he : (fun ω => N (τ ω) ω) ⁻¹' S =
-        ⋃ n : ℕ, (fun ω => N (τ ω) ω) ⁻¹' S ∩ {ω | τ ω ≤ (n:ℝ≥0)} := by
-      ext ω
-      simp only [mem_preimage,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
-      constructor
-      · intro hω
-        obtain ⟨n,hn⟩ := exists_nat_ge (τ ω : ℝ)
-        exact ⟨n,hω,by exact_mod_cast hn⟩
-      · rintro ⟨n,hω,_⟩
-        exact hω
-    rw [he]
-    exact MeasurableSet.iUnion (fun n => le_iSup ℱ (n:ℝ≥0) _ (hslice S hS n))
-  · intro H
-    simpa only [WithTop.coe_le_coe] using hslice S hS H
-
-/-- The original measure-valued process evaluated after a finite stopping
-time is adapted to the full shifted stopped-past filtration. -/
-theorem shifted_adapted (hN : _root_.MeasureTheory.IsProgressive ℱ N)
-    (τ : Ω → ℝ≥0)
-    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) :
-    _root_.MeasureTheory.Adapted (SLEExactGauge.shiftedStoppingFiltration hτ)
-      (fun t ω => N (τ ω+t) ω) := by
-  intro t
-  exact _root_.OAI.MeasureTheory.IsProgressive.stopped_time_measurable hN _ (SLEExactGauge.stopping_time_add_nnreal hτ t)
-
-end MeasureTheory.IsProgressive
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.ProgressiveRandomTime
+import OAI.Probability.SLE.Stochastic.ShiftedStoppingFiltration
+
+namespace OAI
+
+/-! Stopped-value measurability for a progressive process with a merely
+measurable target, in particular a measure-valued canonical mass. -/
+
+noncomputable section
+open Set _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped NNReal ENNReal
+
+namespace MeasureTheory.IsProgressive
+open scoped _root_.MeasureTheory _root_.MeasureTheory.IsProgressive
+
+variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
+  {ℱ : _root_.MeasureTheory.Filtration ℝ≥0 mΩ} {N : ℝ≥0 → Ω → X}
+
+lemma bounded_stopped_measurable (hN : _root_.MeasureTheory.IsProgressive ℱ N)
+    (τ : Ω → ℝ≥0)
+    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0)))
+    (H : ℝ≥0) (hH : ∀ ω, τ ω ≤ H) :
+    Measurable[ℱ H] (fun ω => N (τ ω) ω) := by
+  have htime : Measurable[ℱ H] τ := by
+    have h := (hτ.measurable_of_le (fun ω => WithTop.coe_le_coe.mpr (hH ω))).untopA
+    convert h using 1
+    funext ω
+    rw [WithTop.untopA_eq_untop WithTop.coe_ne_top]
+    rfl
+  have hp : Measurable[ℱ H,Subtype.instMeasurableSpace.prod (ℱ H)]
+      (fun ω => ((⟨τ ω,hH ω⟩ : Iic H),ω)) := by
+    exact htime.subtype_mk.prodMk measurable_id
+  exact (hN H).comp hp
+
+lemma truncated_stopped_measurable (hN : _root_.MeasureTheory.IsProgressive ℱ N)
+    (τ : Ω → ℝ≥0)
+    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) (H : ℝ≥0) :
+    Measurable[ℱ H] (fun ω => N (min (τ ω) H) ω) := by
+  have hmin : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => ((min (τ ω) H : ℝ≥0) : WithTop ℝ≥0)) := by
+    simpa only [WithTop.coe_min] using hτ.min_const H
+  exact _root_.OAI.MeasureTheory.IsProgressive.bounded_stopped_measurable hN _ hmin H (fun _ => min_le_right _ _)
+
+/-- This requires no topology on the target. -/
+theorem stopped_time_measurable (hN : _root_.MeasureTheory.IsProgressive ℱ N)
+    (τ : Ω → ℝ≥0)
+    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) :
+    Measurable[hτ.measurableSpace] (fun ω => N (τ ω) ω) := by
+  have hslice (S : Set X) (hS : MeasurableSet S) (H : ℝ≥0) :
+      MeasurableSet[ℱ H] ((fun ω => N (τ ω) ω) ⁻¹' S ∩ {ω | τ ω ≤ H}) := by
+    have he : (fun ω => N (τ ω) ω) ⁻¹' S ∩ {ω | τ ω ≤ H} =
+        (fun ω => N (min (τ ω) H) ω) ⁻¹' S ∩ {ω | τ ω ≤ H} := by
+      ext ω
+      simp only [mem_inter_iff,mem_preimage,mem_ofPred_eq]
+      constructor <;> rintro ⟨hω,ht⟩ <;> exact ⟨by simpa only [min_eq_left ht] using hω,ht⟩
+    rw [he]
+    exact ((_root_.OAI.MeasureTheory.IsProgressive.truncated_stopped_measurable hN τ hτ H) hS).inter (by simpa only [WithTop.coe_le_coe] using hτ H)
+  intro S hS
+  constructor
+  · have he : (fun ω => N (τ ω) ω) ⁻¹' S =
+        ⋃ n : ℕ, (fun ω => N (τ ω) ω) ⁻¹' S ∩ {ω | τ ω ≤ (n:ℝ≥0)} := by
+      ext ω
+      simp only [mem_preimage,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
+      constructor
+      · intro hω
+        obtain ⟨n,hn⟩ := exists_nat_ge (τ ω : ℝ)
+        exact ⟨n,hω,by exact_mod_cast hn⟩
+      · rintro ⟨n,hω,_⟩
+        exact hω
+    rw [he]
+    exact MeasurableSet.iUnion (fun n => le_iSup ℱ (n:ℝ≥0) _ (hslice S hS n))
+  · intro H
+    simpa only [WithTop.coe_le_coe] using hslice S hS H
+
+/-- The original measure-valued process evaluated after a finite stopping
+time is adapted to the full shifted stopped-past filtration. -/
+theorem shifted_adapted (hN : _root_.MeasureTheory.IsProgressive ℱ N)
+    (τ : Ω → ℝ≥0)
+    (hτ : _root_.MeasureTheory.IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) :
+    _root_.MeasureTheory.Adapted (SLEExactGauge.shiftedStoppingFiltration hτ)
+      (fun t ω => N (τ ω+t) ω) := by
+  intro t
+  exact _root_.OAI.MeasureTheory.IsProgressive.stopped_time_measurable hN _ (SLEExactGauge.stopping_time_add_nnreal hτ t)
+
+end MeasureTheory.IsProgressive
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/ScalarExponentialLower.lean b/lean/OAI/Probability/SLE/Stochastic/ScalarExponentialLower.lean
index f26d6f0434386006fdc8baaf72cadf533b3917b9..8bc025d12a81be8803192fe1805a82823eaf7fc0 100644
--- a/lean/OAI/Probability/SLE/Stochastic/ScalarExponentialLower.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/ScalarExponentialLower.lean
@@ -1,45 +1,45 @@
-import Mathlib
-
-namespace OAI
-
-/-! A lower exponential comparison for the positive forward height. -/
-noncomputable section
-open Set
-
-namespace SLEExactGauge
-
-lemma scalar_exponential_lower {a b C : ℝ} (y y' : ℝ → ℝ)
-    (hc : ContinuousOn y (Icc a b))
-    (hd : ∀ t ∈ Ico a b, HasDerivWithinAt y (y' t) (Ici t) t)
-    (hb : ∀ t ∈ Ico a b, -C*y t ≤ y' t) :
-    ∀ t ∈ Icc a b, y a * Real.exp (-C*(t-a)) ≤ y t := by
-  let f (t : ℝ) := -(y t*Real.exp (C*(t-a)))
-  have hfc : ContinuousOn f (Icc a b) :=
-    (hc.mul (Real.continuous_exp.comp
-      (continuous_const.mul (continuous_id.sub continuous_const))).continuousOn).neg
-  have hfd (t : ℝ) (ht : t ∈ Ico a b) : HasDerivWithinAt f
-      (-(y' t+C*y t)*Real.exp (C*(t-a))) (Ici t) t := by
-    have he : HasDerivAt (fun s : ℝ => Real.exp (C*(s-a)))
-        (C*Real.exp (C*(t-a))) t := by
-      convert! (((hasDerivAt_id t).sub_const a).const_mul C).exp using 1
-      simp only [id_eq]
-      ring
-    convert! ((hd t ht).mul he.hasDerivWithinAt).neg using 1
-    ring
-  have he := image_le_of_deriv_right_le_deriv_boundary hfc hfd
-    (B := fun _ => -y a) (B' := fun _ => 0)
-    (by simp [f]) continuousOn_const
-    (fun t _ => hasDerivWithinAt_const t (Ici t) (-y a))
-    (fun t ht => mul_nonpos_of_nonpos_of_nonneg (by linarith [hb t ht]) (Real.exp_pos _).le)
-  intro t ht
-  have h := he ht
-  change -(y t*Real.exp (C*(t-a))) ≤ -y a at h
-  have hh : y a ≤ y t*Real.exp (C*(t-a)) := by linarith
-  have hdv := (div_le_iff₀ (Real.exp_pos (C*(t-a)))).mpr hh
-  simpa only [div_eq_mul_inv,← Real.exp_neg,neg_mul] using hdv
-
-end SLEExactGauge
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! A lower exponential comparison for the positive forward height. -/
+noncomputable section
+open Set
+
+namespace SLEExactGauge
+
+lemma scalar_exponential_lower {a b C : ℝ} (y y' : ℝ → ℝ)
+    (hc : ContinuousOn y (Icc a b))
+    (hd : ∀ t ∈ Ico a b, HasDerivWithinAt y (y' t) (Ici t) t)
+    (hb : ∀ t ∈ Ico a b, -C*y t ≤ y' t) :
+    ∀ t ∈ Icc a b, y a * Real.exp (-C*(t-a)) ≤ y t := by
+  let f (t : ℝ) := -(y t*Real.exp (C*(t-a)))
+  have hfc : ContinuousOn f (Icc a b) :=
+    (hc.mul (Real.continuous_exp.comp
+      (continuous_const.mul (continuous_id.sub continuous_const))).continuousOn).neg
+  have hfd (t : ℝ) (ht : t ∈ Ico a b) : HasDerivWithinAt f
+      (-(y' t+C*y t)*Real.exp (C*(t-a))) (Ici t) t := by
+    have he : HasDerivAt (fun s : ℝ => Real.exp (C*(s-a)))
+        (C*Real.exp (C*(t-a))) t := by
+      convert! (((hasDerivAt_id t).sub_const a).const_mul C).exp using 1
+      simp only [id_eq]
+      ring
+    convert! ((hd t ht).mul he.hasDerivWithinAt).neg using 1
+    ring
+  have he := image_le_of_deriv_right_le_deriv_boundary hfc hfd
+    (B := fun _ => -y a) (B' := fun _ => 0)
+    (by simp [f]) continuousOn_const
+    (fun t _ => hasDerivWithinAt_const t (Ici t) (-y a))
+    (fun t ht => mul_nonpos_of_nonpos_of_nonneg (by linarith [hb t ht]) (Real.exp_pos _).le)
+  intro t ht
+  have h := he ht
+  change -(y t*Real.exp (C*(t-a))) ≤ -y a at h
+  have hh : y a ≤ y t*Real.exp (C*(t-a)) := by linarith
+  have hdv := (div_le_iff₀ (Real.exp_pos (C*(t-a)))).mpr hh
+  simpa only [div_eq_mul_inv,← Real.exp_neg,neg_mul] using hdv
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/ScaledDriverFiltration.lean b/lean/OAI/Probability/SLE/Stochastic/ScaledDriverFiltration.lean
index 2eb4487bf14b16cfdba68c5ce829610fc2febc29..84701cb35fb192873165dbeafcc1834930028286 100644
--- a/lean/OAI/Probability/SLE/Stochastic/ScaledDriverFiltration.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/ScaledDriverFiltration.lean
@@ -1,83 +1,83 @@
-import OAI.Probability.SLE.Loewner.RestartDriverScaling
-
-namespace OAI
-
-/-! The natural past of the capacity-scaled Brownian driver is exactly
-the original past after the deterministic quadratic time change. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-variable {Ω : Type*} [MeasurableSpace Ω]
-
-def scaledBrownianDriver (c : ℝ≥0) (B : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
-  (c:ℝ)*B (sourceScaleTime c t) ω
-
-lemma scaledBrownianDriver_measurable (c : ℝ≥0) (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, Measurable (B t)) (t : ℝ≥0) :
-    Measurable (scaledBrownianDriver c B t) := measurable_const.mul (hm _)
-
-omit [MeasurableSpace Ω] in
-lemma scaledBrownianDriver_continuous (c : ℝ≥0) (B : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
-    Continuous (fun t => scaledBrownianDriver c B t ω) :=
-  continuous_const.mul ((hc ω).comp (continuous_id.div_const (c^2)))
-
-lemma scaledBrownianDriver_brownian {P : Measure Ω} (c : ℝ≥0) (hc : 0 < c)
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
-    IsBrownianReal (scaledBrownianDriver c B) P := by
-  change IsBrownianReal (fun t ω => (c:ℝ)*B (t/(c^2)) ω) P
-  have h := hB.smul (c := (c^2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hc.ne'))
-  simpa only [scaledBrownianDriver,sourceScaleTime,NNReal.coe_inv,NNReal.coe_pow,
-    Real.sqrt_inv,Real.sqrt_sq c.coe_nonneg,inv_inv,div_eq_mul_inv,mul_comm] using h
-
-lemma scaledBrownianDriver_filtration (c : ℝ≥0) (hc : 0 < c)
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (t : ℝ≥0) :
-    (Filtration.natural (scaledBrownianDriver c B)
-      (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable)) (c^2*t)=
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t := by
-  let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable)
-  let 𝒢 := Filtration.natural (scaledBrownianDriver c B)
-    (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable)
-  have hc2 : 0 < c^2 := pow_pos hc 2
-  apply le_antisymm
-  · change (⨆ s ≤ c^2*t,
-      MeasurableSpace.comap (scaledBrownianDriver c B s) (borel ℝ)) ≤ ℱ t
-    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
-    have hst : sourceScaleTime c s ≤ t := by
-      exact (div_le_iff₀ hc2).mpr (by simpa only [mul_comm] using hs)
-    have hBm : Measurable[ℱ t] (B (sourceScaleTime c s)) :=
-      ((Filtration.stronglyAdapted_natural (fun r => (hm r).stronglyMeasurable)
-        (sourceScaleTime c s)).mono (ℱ.mono hst)).measurable
-    exact measurable_const.mul hBm
-  · change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ 𝒢 (c^2*t)
-    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
-    have hst : c^2*s ≤ c^2*t := mul_le_mul_of_nonneg_left hs zero_le
-    have hYm : Measurable[𝒢 (c^2*t)] (scaledBrownianDriver c B (c^2*s)) :=
-      ((Filtration.stronglyAdapted_natural (fun r =>
-        (scaledBrownianDriver_measurable c B hm r).stronglyMeasurable)
-        (c^2*s)).mono (𝒢.mono hst)).measurable
-    have he : B s=(fun ω => (c:ℝ)⁻¹*scaledBrownianDriver c B (c^2*s) ω) := by
-      funext ω
-      change B s ω=(c:ℝ)⁻¹*((c:ℝ)*B ((c^2*s)/c^2) ω)
-      rw [mul_div_cancel_left₀ _ hc2.ne',inv_mul_cancel_left₀ (show (c:ℝ) ≠ 0 by exact_mod_cast hc.ne')]
-    rw [he]
-    exact measurable_const.mul hYm
-
-lemma scaledBrownianDriver_path_law {P : Measure Ω}
-    (c : ℝ≥0) (hc : 0 < c) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hcont : ∀ ω, Continuous (fun t => B t ω)) :
-    P.map (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hcont))=
-      P.map (sourceDriverPath B hcont) :=
-  (ScaleTraceLaw.brownian_continuous_path_law B (scaledBrownianDriver c B)
-    hB (scaledBrownianDriver_brownian c hc hB) hm (scaledBrownianDriver_measurable c B hm)
-    hcont (scaledBrownianDriver_continuous c B hcont)).symm
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Loewner.RestartDriverScaling
+
+namespace OAI
+
+/-! The natural past of the capacity-scaled Brownian driver is exactly
+the original past after the deterministic quadratic time change. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+variable {Ω : Type*} [MeasurableSpace Ω]
+
+def scaledBrownianDriver (c : ℝ≥0) (B : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
+  (c:ℝ)*B (sourceScaleTime c t) ω
+
+lemma scaledBrownianDriver_measurable (c : ℝ≥0) (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, Measurable (B t)) (t : ℝ≥0) :
+    Measurable (scaledBrownianDriver c B t) := measurable_const.mul (hm _)
+
+omit [MeasurableSpace Ω] in
+lemma scaledBrownianDriver_continuous (c : ℝ≥0) (B : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) :
+    Continuous (fun t => scaledBrownianDriver c B t ω) :=
+  continuous_const.mul ((hc ω).comp (continuous_id.div_const (c^2)))
+
+lemma scaledBrownianDriver_brownian {P : Measure Ω} (c : ℝ≥0) (hc : 0 < c)
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
+    IsBrownianReal (scaledBrownianDriver c B) P := by
+  change IsBrownianReal (fun t ω => (c:ℝ)*B (t/(c^2)) ω) P
+  have h := hB.smul (c := (c^2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hc.ne'))
+  simpa only [scaledBrownianDriver,sourceScaleTime,NNReal.coe_inv,NNReal.coe_pow,
+    Real.sqrt_inv,Real.sqrt_sq c.coe_nonneg,inv_inv,div_eq_mul_inv,mul_comm] using h
+
+lemma scaledBrownianDriver_filtration (c : ℝ≥0) (hc : 0 < c)
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (t : ℝ≥0) :
+    (Filtration.natural (scaledBrownianDriver c B)
+      (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable)) (c^2*t)=
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t := by
+  let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable)
+  let 𝒢 := Filtration.natural (scaledBrownianDriver c B)
+    (fun s => (scaledBrownianDriver_measurable c B hm s).stronglyMeasurable)
+  have hc2 : 0 < c^2 := pow_pos hc 2
+  apply le_antisymm
+  · change (⨆ s ≤ c^2*t,
+      MeasurableSpace.comap (scaledBrownianDriver c B s) (borel ℝ)) ≤ ℱ t
+    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
+    have hst : sourceScaleTime c s ≤ t := by
+      exact (div_le_iff₀ hc2).mpr (by simpa only [mul_comm] using hs)
+    have hBm : Measurable[ℱ t] (B (sourceScaleTime c s)) :=
+      ((Filtration.stronglyAdapted_natural (fun r => (hm r).stronglyMeasurable)
+        (sourceScaleTime c s)).mono (ℱ.mono hst)).measurable
+    exact measurable_const.mul hBm
+  · change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ 𝒢 (c^2*t)
+    refine iSup₂_le fun s hs => measurable_iff_comap_le.mp ?_
+    have hst : c^2*s ≤ c^2*t := mul_le_mul_of_nonneg_left hs zero_le
+    have hYm : Measurable[𝒢 (c^2*t)] (scaledBrownianDriver c B (c^2*s)) :=
+      ((Filtration.stronglyAdapted_natural (fun r =>
+        (scaledBrownianDriver_measurable c B hm r).stronglyMeasurable)
+        (c^2*s)).mono (𝒢.mono hst)).measurable
+    have he : B s=(fun ω => (c:ℝ)⁻¹*scaledBrownianDriver c B (c^2*s) ω) := by
+      funext ω
+      change B s ω=(c:ℝ)⁻¹*((c:ℝ)*B ((c^2*s)/c^2) ω)
+      rw [mul_div_cancel_left₀ _ hc2.ne',inv_mul_cancel_left₀ (show (c:ℝ) ≠ 0 by exact_mod_cast hc.ne')]
+    rw [he]
+    exact measurable_const.mul hYm
+
+lemma scaledBrownianDriver_path_law {P : Measure Ω}
+    (c : ℝ≥0) (hc : 0 < c) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hcont : ∀ ω, Continuous (fun t => B t ω)) :
+    P.map (sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hcont))=
+      P.map (sourceDriverPath B hcont) :=
+  (ScaleTraceLaw.brownian_continuous_path_law B (scaledBrownianDriver c B)
+    hB (scaledBrownianDriver_brownian c hc hB) hm (scaledBrownianDriver_measurable c B hm)
+    hcont (scaledBrownianDriver_continuous c B hcont)).symm
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/ShiftedStoppingFiltration.lean b/lean/OAI/Probability/SLE/Stochastic/ShiftedStoppingFiltration.lean
index 31d2df6fda0c35ea89dae4c4b892441a6942d1e7..fbf671e49813813baaef97b4c147deab7f686a27 100644
--- a/lean/OAI/Probability/SLE/Stochastic/ShiftedStoppingFiltration.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/ShiftedStoppingFiltration.lean
@@ -1,67 +1,67 @@
-import OAI.Probability.SLE.BrownianStrongMarkov
-
-namespace OAI
-
-/-! The original stopped-past filtration after a finite stopping time.
-This keeps the restart comparison on raw sigma algebras. -/
-
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-  {ℱ : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
-
-theorem stopping_time_add_nnreal
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω:WithTop ℝ≥0))) (t : ℝ≥0) :
-    IsStoppingTime ℱ (fun ω => ((τ ω+t:ℝ≥0):WithTop ℝ≥0)) := by
-  intro r
-  by_cases htr : t ≤ r
-  · have he : {ω | ((τ ω+t:ℝ≥0):WithTop ℝ≥0) ≤ r} =
-        {ω | (τ ω:WithTop ℝ≥0) ≤ ((r-t:ℝ≥0):WithTop ℝ≥0)} := by
-      ext ω
-      simp only [mem_ofPred_eq,WithTop.coe_le_coe]
-      exact (le_tsub_iff_right htr).symm
-    rw [he]
-    exact ℱ.mono (tsub_le_self : r-t ≤ r) _ (hτ (r-t))
-  · have he : {ω | ((τ ω+t:ℝ≥0):WithTop ℝ≥0) ≤ r} = ∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      intro ω hω
-      have hle : τ ω+t ≤ r := by exact_mod_cast hω
-      exact htr ((le_add_left le_rfl).trans hle)
-    rw [he]
-    exact @MeasurableSet.empty Ω (ℱ r)
-
-def shiftedStoppingFiltration
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω:WithTop ℝ≥0))) : Filtration ℝ≥0 mΩ where
-  seq t := (stopping_time_add_nnreal hτ t).measurableSpace
-  mono' := by
-    intro s t hst
-    apply IsStoppingTime.measurableSpace_mono
-      (stopping_time_add_nnreal hτ s) (stopping_time_add_nnreal hτ t)
-    intro ω
-    change ((τ ω+s:ℝ≥0):WithTop ℝ≥0) ≤ ((τ ω+t:ℝ≥0):WithTop ℝ≥0)
-    exact WithTop.coe_le_coe.mpr (add_le_add le_rfl hst)
-  le' t := (stopping_time_add_nnreal hτ t).measurableSpace_le
-
-theorem shiftedStopping_adapted
-    (hτ : IsStoppingTime ℱ (fun ω => (τ ω:WithTop ℝ≥0)))
-    (X : ℝ≥0 → Ω → ℝ) (hX : IsStronglyProgressive ℱ X) :
-    StronglyAdapted (shiftedStoppingFiltration hτ) (fun t ω => X (τ ω+t) ω) := by
-  intro t
-  have h := measurable_stoppedValue hX (stopping_time_add_nnreal hτ t)
-  change StronglyMeasurable[(stopping_time_add_nnreal hτ t).measurableSpace]
-    (fun ω => X (τ ω+t) ω)
-  convert h.stronglyMeasurable using 1
-  funext ω
-  change X (τ ω+t) ω = X (((τ ω+t:ℝ≥0):WithTop ℝ≥0).untopA) ω
-  rw [WithTop.untopA_eq_untop WithTop.coe_ne_top]
-  rfl
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.BrownianStrongMarkov
+
+namespace OAI
+
+/-! The original stopped-past filtration after a finite stopping time.
+This keeps the restart comparison on raw sigma algebras. -/
+
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+  {ℱ : Filtration ℝ≥0 mΩ} {τ : Ω → ℝ≥0}
+
+theorem stopping_time_add_nnreal
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω:WithTop ℝ≥0))) (t : ℝ≥0) :
+    IsStoppingTime ℱ (fun ω => ((τ ω+t:ℝ≥0):WithTop ℝ≥0)) := by
+  intro r
+  by_cases htr : t ≤ r
+  · have he : {ω | ((τ ω+t:ℝ≥0):WithTop ℝ≥0) ≤ r} =
+        {ω | (τ ω:WithTop ℝ≥0) ≤ ((r-t:ℝ≥0):WithTop ℝ≥0)} := by
+      ext ω
+      simp only [mem_ofPred_eq,WithTop.coe_le_coe]
+      exact (le_tsub_iff_right htr).symm
+    rw [he]
+    exact ℱ.mono (tsub_le_self : r-t ≤ r) _ (hτ (r-t))
+  · have he : {ω | ((τ ω+t:ℝ≥0):WithTop ℝ≥0) ≤ r} = ∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      intro ω hω
+      have hle : τ ω+t ≤ r := by exact_mod_cast hω
+      exact htr ((le_add_left le_rfl).trans hle)
+    rw [he]
+    exact @MeasurableSet.empty Ω (ℱ r)
+
+def shiftedStoppingFiltration
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω:WithTop ℝ≥0))) : Filtration ℝ≥0 mΩ where
+  seq t := (stopping_time_add_nnreal hτ t).measurableSpace
+  mono' := by
+    intro s t hst
+    apply IsStoppingTime.measurableSpace_mono
+      (stopping_time_add_nnreal hτ s) (stopping_time_add_nnreal hτ t)
+    intro ω
+    change ((τ ω+s:ℝ≥0):WithTop ℝ≥0) ≤ ((τ ω+t:ℝ≥0):WithTop ℝ≥0)
+    exact WithTop.coe_le_coe.mpr (add_le_add le_rfl hst)
+  le' t := (stopping_time_add_nnreal hτ t).measurableSpace_le
+
+theorem shiftedStopping_adapted
+    (hτ : IsStoppingTime ℱ (fun ω => (τ ω:WithTop ℝ≥0)))
+    (X : ℝ≥0 → Ω → ℝ) (hX : IsStronglyProgressive ℱ X) :
+    StronglyAdapted (shiftedStoppingFiltration hτ) (fun t ω => X (τ ω+t) ω) := by
+  intro t
+  have h := measurable_stoppedValue hX (stopping_time_add_nnreal hτ t)
+  change StronglyMeasurable[(stopping_time_add_nnreal hτ t).measurableSpace]
+    (fun ω => X (τ ω+t) ω)
+  convert h.stronglyMeasurable using 1
+  funext ω
+  change X (τ ω+t) ω = X (((τ ω+t:ℝ≥0):WithTop ℝ≥0).untopA) ω
+  rw [WithTop.untopA_eq_untop WithTop.coe_ne_top]
+  rfl
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerBridge.lean b/lean/OAI/Probability/SLE/Stochastic/WienerBridge.lean
index c4438268e00d800ab0a470dd2f741a150d91b952..8d5eef3ad0ce9c4fe3d8a62781ea0ece492a00fc 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerBridge.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerBridge.lean
@@ -1,76 +1,76 @@
-import OAI.Probability.SLE.Geometry.PublishedWienerSupport
-import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence
-
-namespace OAI
-
-/-! Independence of the Brownian endpoint and its pinned bridge, used in
-the finite-window Wiener support theorem. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def unitBridge {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
-    (t : unitInterval) (ω : Ω) : ℝ :=
-  B (T*⟨t,t.2.1⟩) ω-(t:ℝ)*B T ω
-
-lemma unitBridge_joint_gaussian
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) :
-    IsGaussianProcess (Sum.elim (fun _ : Unit => B T) (unitBridge B T)) P := by
-  classical
-  apply hB.isGaussianProcess.of_isGaussianProcess
-  rintro (s | t)
-  · exact ⟨{T},{toFun := fun x => x ⟨T,by simp⟩
-                map_add' := by simp
-                map_smul' := by simp},by simp⟩
-  · let q : ℝ≥0 := T*⟨t,t.2.1⟩
-    refine ⟨{q,T},
-      { toFun := fun x => x ⟨q,by simp⟩ - (t:ℝ)*x ⟨T,by simp⟩
-        map_add' := ?_
-        map_smul' := ?_ }, ?_⟩
-    · intro x y
-      simp only [Pi.add_apply]
-      ring
-    · intro c x
-      simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
-      ring
-    · intro ω
-      rfl
-
-lemma unitBridge_endpoint_covariance
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) (t : unitInterval) :
-    cov[B T,unitBridge B T t;P]=0 := by
-  have : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have ht : T*⟨t,t.2.1⟩≤T := by
-    apply NNReal.coe_le_coe.mp
-    change (T:ℝ)*(t:ℝ) ≤ T
-    nlinarith [NNReal.coe_nonneg T,t.2.2]
-  change cov[B T, fun ω => B (T*⟨t,t.2.1⟩) ω - (t:ℝ)*B T ω; P] = 0
-  rw [covariance_fun_sub_right,
-    covariance_const_mul_right,hB.covariance_eval,hB.covariance_eval,
-    min_eq_right ht,min_self]
-  · change (T:ℝ)*(t:ℝ) - (t:ℝ)*(T:ℝ) = 0
-    ring
-  · exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two
-  · exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two
-  · exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two.const_mul _
-
-lemma unitBridge_independent_endpoint
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) :
-    IndepFun (B T) (fun ω t => unitBridge B T t ω) P := by
-  have hi := (unitBridge_joint_gaussian hB T).indepFun_of_covariance_eq_zero
-    (fun _ => hB.aemeasurable T)
-    (fun t => (hB.aemeasurable _).sub ((hB.aemeasurable T).const_mul (t:ℝ)))
-    (fun _ t => unitBridge_endpoint_covariance hB T t)
-  simpa only [Function.comp_def, id_eq] using
-    hi.comp (measurable_pi_apply ()) measurable_id
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.PublishedWienerSupport
+import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence
+
+namespace OAI
+
+/-! Independence of the Brownian endpoint and its pinned bridge, used in
+the finite-window Wiener support theorem. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def unitBridge {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
+    (t : unitInterval) (ω : Ω) : ℝ :=
+  B (T*⟨t,t.2.1⟩) ω-(t:ℝ)*B T ω
+
+lemma unitBridge_joint_gaussian
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) :
+    IsGaussianProcess (Sum.elim (fun _ : Unit => B T) (unitBridge B T)) P := by
+  classical
+  apply hB.isGaussianProcess.of_isGaussianProcess
+  rintro (s | t)
+  · exact ⟨{T},{toFun := fun x => x ⟨T,by simp⟩
+                map_add' := by simp
+                map_smul' := by simp},by simp⟩
+  · let q : ℝ≥0 := T*⟨t,t.2.1⟩
+    refine ⟨{q,T},
+      { toFun := fun x => x ⟨q,by simp⟩ - (t:ℝ)*x ⟨T,by simp⟩
+        map_add' := ?_
+        map_smul' := ?_ }, ?_⟩
+    · intro x y
+      simp only [Pi.add_apply]
+      ring
+    · intro c x
+      simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
+      ring
+    · intro ω
+      rfl
+
+lemma unitBridge_endpoint_covariance
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) (t : unitInterval) :
+    cov[B T,unitBridge B T t;P]=0 := by
+  have : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have ht : T*⟨t,t.2.1⟩≤T := by
+    apply NNReal.coe_le_coe.mp
+    change (T:ℝ)*(t:ℝ) ≤ T
+    nlinarith [NNReal.coe_nonneg T,t.2.2]
+  change cov[B T, fun ω => B (T*⟨t,t.2.1⟩) ω - (t:ℝ)*B T ω; P] = 0
+  rw [covariance_fun_sub_right,
+    covariance_const_mul_right,hB.covariance_eval,hB.covariance_eval,
+    min_eq_right ht,min_self]
+  · change (T:ℝ)*(t:ℝ) - (t:ℝ)*(T:ℝ) = 0
+    ring
+  · exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two
+  · exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two
+  · exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two.const_mul _
+
+lemma unitBridge_independent_endpoint
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) :
+    IndepFun (B T) (fun ω t => unitBridge B T t ω) P := by
+  have hi := (unitBridge_joint_gaussian hB T).indepFun_of_covariance_eq_zero
+    (fun _ => hB.aemeasurable T)
+    (fun t => (hB.aemeasurable _).sub ((hB.aemeasurable T).const_mul (t:ℝ)))
+    (fun _ t => unitBridge_endpoint_covariance hB T t)
+  simpa only [Function.comp_def, id_eq] using
+    hi.comp (measurable_pi_apply ()) measurable_id
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerCoefficientLimit.lean b/lean/OAI/Probability/SLE/Stochastic/WienerCoefficientLimit.lean
index 8379fa4bee84a1c6de81acecb5be78594818454f..9cffdd255fadb1cd671df6ff4f8bf8fcb40cc358 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerCoefficientLimit.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerCoefficientLimit.lean
@@ -1,131 +1,131 @@
-import OAI.Probability.SLE.Geometry.NormalizedCoefficientLimit
-import OAI.Probability.SLE.Angle.TargetLevelMeasurability
-
-namespace OAI
-
-/-! Transfer of the exact fixed-angle coefficient limit to the Wiener
-integrals appearing in the prescribed deterministic gauge. -/
-
-noncomputable section
-open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge.Target
-
-lemma measurable_fixedLevelDensity (κ θ ℓ : ℝ) :
-    Measurable (fun b => levelDensity κ θ b ℓ) :=
-  @Measurable.comp Driver (ℝ × Driver) ℝ
-    inferInstance inferInstance inferInstance
-    (fun p => levelDensity κ p.1 p.2 ℓ) (fun b => (θ,b))
-    (measurable_levelDensity_angle κ ℓ) (measurable_const.prodMk measurable_id)
-
-lemma measurable_fixedDiscreteMass (κ θ : ℝ) (W : WienerLaw) [SFinite W] (ℓ : ℝ) (n : ℕ) :
-    Measurable (fun b => discreteMass κ θ W b ℓ n) := by
-  have ht : Measurable (fun b : Driver => levelTime κ θ b ℓ) :=
-    @Measurable.comp Driver (ℝ × Driver) ℝ≥0∞
-      inferInstance inferInstance inferInstance
-      (fun p => levelTime κ p.1 p.2 ℓ) (fun b => (θ,b))
-      (measurable_levelTime_angle κ ℓ) (measurable_const.prodMk measurable_id)
-  exact discreteMass_comp_measurable κ θ W id measurable_id ℓ ht n
-
-lemma measurable_fixedCoefficient (κ θ : ℝ) (W : WienerLaw) [SFinite W]
-    (ℓ u : ℝ) (n k : ℕ) :
-    Measurable (fun b => levelDensity κ θ b ℓ*(clip u (discreteMass κ θ W b ℓ n))^k) :=
-  (measurable_fixedLevelDensity κ θ ℓ).mul
-    (((continuous_clip u).measurable.comp (measurable_fixedDiscreteMass κ θ W ℓ n)).pow_const k)
-
-lemma levelDensity_nonneg_of_witness
-    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) (hw : IsCapacityTwoWitness U γ G F)
-    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
-    0 ≤ levelDensity κ θ b ℓ := by
-  rw [levelDensity_eq_stoppedGreen κ hκ b U hUc γ G F hw hU θ hθ ℓ hℓ]
-  exact div_nonneg (stoppedGreen_nonneg hw κ _ _)
-    (initialGreen_pos κ _ (diskMapInv_maps hθ (by simp))).le
-
-end SLEExactGauge.Target
-
-namespace SLEExactGauge.Multipoint.CanonicalMass
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  [IsProbabilityMeasure P] {κ : ℝ}
-  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
-  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
-  (C : CanonicalMass P κ B hBm γ G)
-
-theorem wiener_clipped_coefficient_limit
-    (hB : IsBrownianReal B P) (hκ : 0<κ) (hκ8 : κ<8)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ)
-    (u : ℝ) (hu : 0 ≤ u) (k : ℕ) :
-    Tendsto (fun n => ∫ b, Target.levelDensity κ θ b ℓ *
-      (Target.clip u (Target.discreteMass κ θ W b ℓ n))^k ∂W) atTop
-      (𝓝 (∫ ω, Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ *
-        (Target.clip u (massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω)
-          (C.diskTerminalScalar θ hθ ω)
-          (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ)))^k ∂P)) := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  have hLaw := Target.normalizedDriver_law κ hκ B hB
-    (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 W hW
-  have he (n : ℕ) := hLaw.integral_comp
-    (Target.measurable_fixedCoefficient κ θ W ℓ u n k).aestronglyMeasurable
-  have ht := C.normalized_clipped_coefficient_limit hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ u hu k
-  apply ht.congr'
-  filter_upwards [] with n
-  exact he n
-
-include hBm in
-theorem wiener_clipped_coefficient_bound
-    (hB : IsBrownianReal B P) (hκ : 0<κ) (hκ8 : κ<8)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
-    (F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ)
-    (u : ℝ) (hu : 0 ≤ u) (n k : ℕ) :
-    0 ≤ (∫ b, Target.levelDensity κ θ b ℓ *
-      (Target.clip u (Target.discreteMass κ θ W b ℓ n))^k ∂W) ∧
-    (∫ b, Target.levelDensity κ θ b ℓ *
-      (Target.clip u (Target.discreteMass κ θ W b ℓ n))^k ∂W) ≤ u^k := by
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  let b := Target.normalizedDriver κ B hc h0
-  let w := fun ω => Target.levelDensity κ θ (b ω) ℓ
-  let f := fun ω => (Target.clip u (Target.discreteMass κ θ W (b ω) ℓ n))^k
-  have hm (r : ℝ≥0) : Measurable (B r) := hBm.comp (measurable_const.prodMk measurable_id)
-  have hb := Target.normalizedDriver_measurable κ B hm hc h0
-  have hLaw := Target.normalizedDriver_law κ hκ B hB hm hc h0 W hW
-  have he := hLaw.integral_comp
-    (Target.measurable_fixedCoefficient κ θ W ℓ u n k).aestronglyMeasurable
-  rw [← he]
-  change 0 ≤ (∫ ω, w ω*f ω ∂P) ∧ (∫ ω, w ω*f ω ∂P) ≤ u^k
-  have hwi := Target.levelDensity_integrable_mean κ hκ hκ8 B hB b γ G F
-    (Eventually.of_forall hw) (Eventually.of_forall (Target.normalizedDriver_matches κ hκ B hc h0))
-    θ hθ ℓ hℓ
-  have hw0 (ω : Ω) : 0 ≤ w ω := Target.levelDensity_nonneg_of_witness κ hκ (b ω) _
-    ((hc ω).const_mul _) (γ ω) (G ω) (F ω) (hw ω)
-    (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ
-  have hf0 (ω : Ω) : 0 ≤ f ω := pow_nonneg (clip_nonneg hu _) k
-  have hfb (ω : Ω) : f ω ≤ u^k :=
-    pow_le_pow_left₀ (clip_nonneg hu _) (clip_le _ _) k
-  have hprod : Integrable (fun ω => w ω*f ω) P := by
-    apply (hwi.1.norm.mul_const (u^k)).mono'
-      ((Target.measurable_fixedCoefficient κ θ W ℓ u n k).comp hb).aestronglyMeasurable
-    filter_upwards [] with ω
-    change ‖w ω * f ω‖ ≤ ‖w ω‖ * u^k
-    rw [norm_mul,Real.norm_of_nonneg (hf0 ω)]
-    exact mul_le_mul_of_nonneg_left (hfb ω) (norm_nonneg _)
-  refine ⟨integral_nonneg (fun ω => mul_nonneg (hw0 ω) (hf0 ω)),?_⟩
-  calc
-    (∫ ω, w ω*f ω ∂P) ≤ ∫ ω, w ω*u^k ∂P :=
-      integral_mono hprod (hwi.1.mul_const _) (fun ω => mul_le_mul_of_nonneg_left (hfb ω) (hw0 ω))
-    _ = u^k := by rw [integral_mul_const,hwi.2,one_mul]
-
-end SLEExactGauge.Multipoint.CanonicalMass
-
-end
-
-end OAI
+import OAI.Probability.SLE.Geometry.NormalizedCoefficientLimit
+import OAI.Probability.SLE.Angle.TargetLevelMeasurability
+
+namespace OAI
+
+/-! Transfer of the exact fixed-angle coefficient limit to the Wiener
+integrals appearing in the prescribed deterministic gauge. -/
+
+noncomputable section
+open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge.Target
+
+lemma measurable_fixedLevelDensity (κ θ ℓ : ℝ) :
+    Measurable (fun b => levelDensity κ θ b ℓ) :=
+  @Measurable.comp Driver (ℝ × Driver) ℝ
+    inferInstance inferInstance inferInstance
+    (fun p => levelDensity κ p.1 p.2 ℓ) (fun b => (θ,b))
+    (measurable_levelDensity_angle κ ℓ) (measurable_const.prodMk measurable_id)
+
+lemma measurable_fixedDiscreteMass (κ θ : ℝ) (W : WienerLaw) [SFinite W] (ℓ : ℝ) (n : ℕ) :
+    Measurable (fun b => discreteMass κ θ W b ℓ n) := by
+  have ht : Measurable (fun b : Driver => levelTime κ θ b ℓ) :=
+    @Measurable.comp Driver (ℝ × Driver) ℝ≥0∞
+      inferInstance inferInstance inferInstance
+      (fun p => levelTime κ p.1 p.2 ℓ) (fun b => (θ,b))
+      (measurable_levelTime_angle κ ℓ) (measurable_const.prodMk measurable_id)
+  exact discreteMass_comp_measurable κ θ W id measurable_id ℓ ht n
+
+lemma measurable_fixedCoefficient (κ θ : ℝ) (W : WienerLaw) [SFinite W]
+    (ℓ u : ℝ) (n k : ℕ) :
+    Measurable (fun b => levelDensity κ θ b ℓ*(clip u (discreteMass κ θ W b ℓ n))^k) :=
+  (measurable_fixedLevelDensity κ θ ℓ).mul
+    (((continuous_clip u).measurable.comp (measurable_fixedDiscreteMass κ θ W ℓ n)).pow_const k)
+
+lemma levelDensity_nonneg_of_witness
+    (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) (hUc : Continuous U)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) (hw : IsCapacityTwoWitness U γ G F)
+    (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s))
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ) :
+    0 ≤ levelDensity κ θ b ℓ := by
+  rw [levelDensity_eq_stoppedGreen κ hκ b U hUc γ G F hw hU θ hθ ℓ hℓ]
+  exact div_nonneg (stoppedGreen_nonneg hw κ _ _)
+    (initialGreen_pos κ _ (diskMapInv_maps hθ (by simp))).le
+
+end SLEExactGauge.Target
+
+namespace SLEExactGauge.Multipoint.CanonicalMass
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  [IsProbabilityMeasure P] {κ : ℝ}
+  {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)}
+  {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ}
+  (C : CanonicalMass P κ B hBm γ G)
+
+theorem wiener_clipped_coefficient_limit
+    (hB : IsBrownianReal B P) (hκ : 0<κ) (hκ8 : κ<8)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ)
+    (u : ℝ) (hu : 0 ≤ u) (k : ℕ) :
+    Tendsto (fun n => ∫ b, Target.levelDensity κ θ b ℓ *
+      (Target.clip u (Target.discreteMass κ θ W b ℓ n))^k ∂W) atTop
+      (𝓝 (∫ ω, Target.levelDensity κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ *
+        (Target.clip u (massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω)
+          (C.diskTerminalScalar θ hθ ω)
+          (Target.levelTime κ θ (Target.normalizedDriver κ B hc h0 ω) ℓ)))^k ∂P)) := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  have hLaw := Target.normalizedDriver_law κ hκ B hB
+    (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 W hW
+  have he (n : ℕ) := hLaw.integral_comp
+    (Target.measurable_fixedCoefficient κ θ W ℓ u n k).aestronglyMeasurable
+  have ht := C.normalized_clipped_coefficient_limit hB hκ hκ8 hc h0 F hw W hW θ hθ ℓ hℓ u hu k
+  apply ht.congr'
+  filter_upwards [] with n
+  exact he n
+
+include hBm in
+theorem wiener_clipped_coefficient_bound
+    (hB : IsBrownianReal B P) (hκ : 0<κ) (hκ8 : κ<8)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0)
+    (F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω))
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0 ≤ ℓ)
+    (u : ℝ) (hu : 0 ≤ u) (n k : ℕ) :
+    0 ≤ (∫ b, Target.levelDensity κ θ b ℓ *
+      (Target.clip u (Target.discreteMass κ θ W b ℓ n))^k ∂W) ∧
+    (∫ b, Target.levelDensity κ θ b ℓ *
+      (Target.clip u (Target.discreteMass κ θ W b ℓ n))^k ∂W) ≤ u^k := by
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  let b := Target.normalizedDriver κ B hc h0
+  let w := fun ω => Target.levelDensity κ θ (b ω) ℓ
+  let f := fun ω => (Target.clip u (Target.discreteMass κ θ W (b ω) ℓ n))^k
+  have hm (r : ℝ≥0) : Measurable (B r) := hBm.comp (measurable_const.prodMk measurable_id)
+  have hb := Target.normalizedDriver_measurable κ B hm hc h0
+  have hLaw := Target.normalizedDriver_law κ hκ B hB hm hc h0 W hW
+  have he := hLaw.integral_comp
+    (Target.measurable_fixedCoefficient κ θ W ℓ u n k).aestronglyMeasurable
+  rw [← he]
+  change 0 ≤ (∫ ω, w ω*f ω ∂P) ∧ (∫ ω, w ω*f ω ∂P) ≤ u^k
+  have hwi := Target.levelDensity_integrable_mean κ hκ hκ8 B hB b γ G F
+    (Eventually.of_forall hw) (Eventually.of_forall (Target.normalizedDriver_matches κ hκ B hc h0))
+    θ hθ ℓ hℓ
+  have hw0 (ω : Ω) : 0 ≤ w ω := Target.levelDensity_nonneg_of_witness κ hκ (b ω) _
+    ((hc ω).const_mul _) (γ ω) (G ω) (F ω) (hw ω)
+    (Target.normalizedDriver_matches κ hκ B hc h0 ω) θ hθ ℓ hℓ
+  have hf0 (ω : Ω) : 0 ≤ f ω := pow_nonneg (clip_nonneg hu _) k
+  have hfb (ω : Ω) : f ω ≤ u^k :=
+    pow_le_pow_left₀ (clip_nonneg hu _) (clip_le _ _) k
+  have hprod : Integrable (fun ω => w ω*f ω) P := by
+    apply (hwi.1.norm.mul_const (u^k)).mono'
+      ((Target.measurable_fixedCoefficient κ θ W ℓ u n k).comp hb).aestronglyMeasurable
+    filter_upwards [] with ω
+    change ‖w ω * f ω‖ ≤ ‖w ω‖ * u^k
+    rw [norm_mul,Real.norm_of_nonneg (hf0 ω)]
+    exact mul_le_mul_of_nonneg_left (hfb ω) (norm_nonneg _)
+  refine ⟨integral_nonneg (fun ω => mul_nonneg (hw0 ω) (hf0 ω)),?_⟩
+  calc
+    (∫ ω, w ω*f ω ∂P) ≤ ∫ ω, w ω*u^k ∂P :=
+      integral_mono hprod (hwi.1.mul_const _) (fun ω => mul_le_mul_of_nonneg_left (hfb ω) (hw0 ω))
+    _ = u^k := by rw [integral_mul_const,hwi.2,one_mul]
+
+end SLEExactGauge.Multipoint.CanonicalMass
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerGrid.lean b/lean/OAI/Probability/SLE/Stochastic/WienerGrid.lean
index 06dc9db9fbfd985b6b6def4b7e48c9369fd5e1b9..08b07638c7be4d1c41b599a3368b1d36c4e7d305 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerGrid.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerGrid.lean
@@ -1,177 +1,177 @@
-import OAI.Probability.SLE.Stochastic.WienerBridge
-import Mathlib.Algebra.Order.Floor.Semiring
-
-namespace OAI
-
-/-! The uniform grid and its linear interpolation, used only to prove
-the finite-window support of the actual Wiener law. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def wienerGridTime (n k : ℕ) : ℝ≥0 := (k:ℝ≥0)/(n+1:ℝ≥0)
-
-def wienerGridCell (n : ℕ) (t : unitInterval) : ℕ :=
-  ⌊(n+1:ℝ)*(t:ℝ)⌋₊
-
-def wienerGridWeight (n : ℕ) (t : unitInterval) : ℝ :=
-  (n+1:ℝ)*(t:ℝ)-wienerGridCell n t
-
-lemma wienerGridCell_le (n : ℕ) (t : unitInterval) :
-    wienerGridCell n t ≤ n+1 := by
-  apply Nat.floor_le_of_le
-  push_cast
-  nlinarith [t.2.2]
-
-lemma wienerGridWeight_mem (n : ℕ) (t : unitInterval) :
-    wienerGridWeight n t ∈ Ico (0:ℝ) 1 := by
-  constructor
-  · exact sub_nonneg.mpr (Nat.floor_le (mul_nonneg (by positivity) t.2.1))
-  · have h := Nat.lt_floor_add_one ((n+1:ℝ)*(t:ℝ))
-    change _ < (wienerGridCell n t:ℝ)+1 at h
-    unfold wienerGridWeight
-    linarith
-
-lemma wienerGrid_bracket (n : ℕ) (t : unitInterval) :
-    (wienerGridTime n (wienerGridCell n t):ℝ) ≤ t ∧
-    (t:ℝ) ≤ wienerGridTime n (wienerGridCell n t+1) := by
-  have hN : (0:ℝ)<n+1 := by positivity
-  change ((wienerGridCell n t:ℝ)/(n+1) ≤ t) ∧
-    ((t:ℝ) ≤ ((wienerGridCell n t+1:ℕ):ℝ)/(n+1))
-  constructor
-  · rw [div_le_iff₀ hN]
-    have h := Nat.floor_le (mul_nonneg hN.le t.2.1)
-    change (wienerGridCell n t:ℝ) ≤ (n+1:ℝ)*(t:ℝ) at h
-    nlinarith
-  · rw [le_div_iff₀ hN]
-    push_cast
-    have h := Nat.lt_floor_add_one ((n+1:ℝ)*(t:ℝ))
-    change _ < (wienerGridCell n t:ℝ)+1 at h
-    nlinarith
-
-lemma wienerGrid_affine (n : ℕ) (t : unitInterval) :
-    (1-wienerGridWeight n t)*(wienerGridTime n (wienerGridCell n t):ℝ)+
-      wienerGridWeight n t*(wienerGridTime n (wienerGridCell n t+1):ℝ) = t := by
-  change (1-((n+1:ℝ)*(t:ℝ)-wienerGridCell n t))*((wienerGridCell n t:ℝ)/(n+1))+
-    ((n+1:ℝ)*(t:ℝ)-wienerGridCell n t)*(((wienerGridCell n t+1:ℕ):ℝ)/(n+1))=t
-  push_cast
-  field_simp
-  ring
-
-lemma wienerGrid_gap (n k : ℕ) :
-    (wienerGridTime n (k+1):ℝ)-(wienerGridTime n k:ℝ) = 1/(n+1:ℝ) := by
-  change ((k+1:ℕ):ℝ)/(n+1)-(k:ℝ)/(n+1)=1/(n+1:ℝ)
-  push_cast
-  ring
-
-def wienerGridInterp (n : ℕ) (f : ℝ≥0 → ℝ) (t : unitInterval) : ℝ :=
-  (1-wienerGridWeight n t)*f (wienerGridTime n (wienerGridCell n t))+
-    wienerGridWeight n t*f (wienerGridTime n (wienerGridCell n t+1))
-
-lemma wienerGridInterp_error (n : ℕ) (f : ℝ≥0 → ℝ) (t : unitInterval)
-    {ε : ℝ}
-    (ha : |f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t))| ≤ ε)
-    (hb : |f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1))| ≤ ε) :
-    |f ⟨t,t.2.1⟩-wienerGridInterp n f t| ≤ ε := by
-  have hw := wienerGridWeight_mem n t
-  have he : f ⟨t,t.2.1⟩-wienerGridInterp n f t =
-      (1-wienerGridWeight n t)*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t)))+
-      wienerGridWeight n t*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1))) := by
-    unfold wienerGridInterp
-    ring
-  rw [he]
-  calc
-    _ ≤ |(1-wienerGridWeight n t)*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t)))|+
-        |wienerGridWeight n t*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1)))| := abs_add_le _ _
-    _ = (1-wienerGridWeight n t)*|f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t))|+
-        wienerGridWeight n t*|f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1))| := by
-      rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr hw.2.le),abs_of_nonneg hw.1]
-    _ ≤ (1-wienerGridWeight n t)*ε+wienerGridWeight n t*ε :=
-      add_le_add (mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr hw.2.le))
-        (mul_le_mul_of_nonneg_left hb hw.1)
-    _ = ε := by ring
-
-lemma wienerGridTime_mono (n : ℕ) : Monotone (wienerGridTime n) := by
-  intro i j hij
-  exact div_le_div_of_nonneg_right (Nat.cast_le.mpr hij) (by positivity)
-
-lemma wienerGridTime_mem (n k : ℕ) (hk : k ≤ n+2) :
-    wienerGridTime n k ∈ Icc (0:ℝ≥0) 2 := by
-  refine ⟨zero_le,?_⟩
-  apply NNReal.coe_le_coe.mp
-  change (k:ℝ)/(n+1) ≤ 2
-  rw [div_le_iff₀ (by positivity : (0:ℝ)<n+1)]
-  have h : (k:ℝ) ≤ (n+2:ℕ) := by exact_mod_cast hk
-  push_cast at h
-  nlinarith
-
-lemma wienerGrid_dist_left (n : ℕ) (t : unitInterval) :
-    dist (⟨t,t.2.1⟩:ℝ≥0) (wienerGridTime n (wienerGridCell n t)) ≤ 1/(n+1:ℝ) := by
-  have h := wienerGrid_bracket n t
-  change |(t:ℝ)-(wienerGridTime n (wienerGridCell n t):ℝ)| ≤ _
-  rw [abs_of_nonneg (sub_nonneg.mpr h.1)]
-  have hg := wienerGrid_gap n (wienerGridCell n t)
-  linarith
-
-lemma wienerGrid_dist_right (n : ℕ) (t : unitInterval) :
-    dist (⟨t,t.2.1⟩:ℝ≥0) (wienerGridTime n (wienerGridCell n t+1)) ≤ 1/(n+1:ℝ) := by
-  have h := wienerGrid_bracket n t
-  change |(t:ℝ)-(wienerGridTime n (wienerGridCell n t+1):ℝ)| ≤ _
-  rw [abs_of_nonpos (sub_nonpos.mpr h.2)]
-  have hg := wienerGrid_gap n (wienerGridCell n t)
-  linarith
-
-lemma wienerGridInterp_uniform (f : ℝ≥0 → ℝ)
-    (hf : ContinuousOn f (Icc 0 2)) (ε : ℝ) (hε : 0 < ε) :
-    ∀ᶠ n in atTop, ∀ t : unitInterval,
-      |f ⟨t,t.2.1⟩-wienerGridInterp n f t| ≤ ε := by
-  obtain ⟨δ,hδ,hmod⟩ := Metric.uniformContinuousOn_iff.mp
-    (isCompact_Icc.uniformContinuousOn_of_continuous hf) ε hε
-  filter_upwards [(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
-    (gt_mem_nhds hδ)] with n hn t
-  have ht : (⟨t,t.2.1⟩:ℝ≥0) ∈ Icc (0:ℝ≥0) 2 := by
-    refine ⟨zero_le,?_⟩
-    apply NNReal.coe_le_coe.mp
-    change (t:ℝ) ≤ 2
-    linarith [t.2.2]
-  apply wienerGridInterp_error
-  · have h := hmod _ ht _ (wienerGridTime_mem n _ ((wienerGridCell_le n t).trans (by omega)))
-      ((wienerGrid_dist_left n t).trans_lt hn)
-    exact (by simpa only [Real.dist_eq] using h.le)
-  · have h := hmod _ ht _ (wienerGridTime_mem n _ (by have := wienerGridCell_le n t; omega))
-      ((wienerGrid_dist_right n t).trans_lt hn)
-    exact (by simpa only [Real.dist_eq] using h.le)
-
-lemma wienerGridInterp_sub (n : ℕ) (f g : ℝ≥0 → ℝ) (t : unitInterval) :
-    wienerGridInterp n f t-wienerGridInterp n g t =
-      wienerGridInterp n (fun s => f s-g s) t := by
-  unfold wienerGridInterp
-  ring
-
-lemma wienerGridInterp_difference (n : ℕ) (f g : ℝ≥0 → ℝ) (t : unitInterval)
-    {ε : ℝ}
-    (h : ∀ k ≤ n+2, |f (wienerGridTime n k)-g (wienerGridTime n k)| ≤ ε) :
-    |wienerGridInterp n f t-wienerGridInterp n g t| ≤ ε := by
-  rw [wienerGridInterp_sub]
-  have hw := wienerGridWeight_mem n t
-  have ha := h (wienerGridCell n t) ((wienerGridCell_le n t).trans (by omega))
-  have hb := h (wienerGridCell n t+1) (by have := wienerGridCell_le n t; omega)
-  unfold wienerGridInterp
-  calc
-    _ ≤ |(1-wienerGridWeight n t)*(f (wienerGridTime n (wienerGridCell n t))-g (wienerGridTime n (wienerGridCell n t)))|+
-        |wienerGridWeight n t*(f (wienerGridTime n (wienerGridCell n t+1))-g (wienerGridTime n (wienerGridCell n t+1)))| := abs_add_le _ _
-    _ = (1-wienerGridWeight n t)*|f (wienerGridTime n (wienerGridCell n t))-g (wienerGridTime n (wienerGridCell n t))|+
-        wienerGridWeight n t*|f (wienerGridTime n (wienerGridCell n t+1))-g (wienerGridTime n (wienerGridCell n t+1))| := by
-      rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr hw.2.le),abs_of_nonneg hw.1]
-    _ ≤ (1-wienerGridWeight n t)*ε+wienerGridWeight n t*ε :=
-      add_le_add (mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr hw.2.le))
-        (mul_le_mul_of_nonneg_left hb hw.1)
-    _ = ε := by ring
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.WienerBridge
+import Mathlib.Algebra.Order.Floor.Semiring
+
+namespace OAI
+
+/-! The uniform grid and its linear interpolation, used only to prove
+the finite-window support of the actual Wiener law. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def wienerGridTime (n k : ℕ) : ℝ≥0 := (k:ℝ≥0)/(n+1:ℝ≥0)
+
+def wienerGridCell (n : ℕ) (t : unitInterval) : ℕ :=
+  ⌊(n+1:ℝ)*(t:ℝ)⌋₊
+
+def wienerGridWeight (n : ℕ) (t : unitInterval) : ℝ :=
+  (n+1:ℝ)*(t:ℝ)-wienerGridCell n t
+
+lemma wienerGridCell_le (n : ℕ) (t : unitInterval) :
+    wienerGridCell n t ≤ n+1 := by
+  apply Nat.floor_le_of_le
+  push_cast
+  nlinarith [t.2.2]
+
+lemma wienerGridWeight_mem (n : ℕ) (t : unitInterval) :
+    wienerGridWeight n t ∈ Ico (0:ℝ) 1 := by
+  constructor
+  · exact sub_nonneg.mpr (Nat.floor_le (mul_nonneg (by positivity) t.2.1))
+  · have h := Nat.lt_floor_add_one ((n+1:ℝ)*(t:ℝ))
+    change _ < (wienerGridCell n t:ℝ)+1 at h
+    unfold wienerGridWeight
+    linarith
+
+lemma wienerGrid_bracket (n : ℕ) (t : unitInterval) :
+    (wienerGridTime n (wienerGridCell n t):ℝ) ≤ t ∧
+    (t:ℝ) ≤ wienerGridTime n (wienerGridCell n t+1) := by
+  have hN : (0:ℝ)<n+1 := by positivity
+  change ((wienerGridCell n t:ℝ)/(n+1) ≤ t) ∧
+    ((t:ℝ) ≤ ((wienerGridCell n t+1:ℕ):ℝ)/(n+1))
+  constructor
+  · rw [div_le_iff₀ hN]
+    have h := Nat.floor_le (mul_nonneg hN.le t.2.1)
+    change (wienerGridCell n t:ℝ) ≤ (n+1:ℝ)*(t:ℝ) at h
+    nlinarith
+  · rw [le_div_iff₀ hN]
+    push_cast
+    have h := Nat.lt_floor_add_one ((n+1:ℝ)*(t:ℝ))
+    change _ < (wienerGridCell n t:ℝ)+1 at h
+    nlinarith
+
+lemma wienerGrid_affine (n : ℕ) (t : unitInterval) :
+    (1-wienerGridWeight n t)*(wienerGridTime n (wienerGridCell n t):ℝ)+
+      wienerGridWeight n t*(wienerGridTime n (wienerGridCell n t+1):ℝ) = t := by
+  change (1-((n+1:ℝ)*(t:ℝ)-wienerGridCell n t))*((wienerGridCell n t:ℝ)/(n+1))+
+    ((n+1:ℝ)*(t:ℝ)-wienerGridCell n t)*(((wienerGridCell n t+1:ℕ):ℝ)/(n+1))=t
+  push_cast
+  field_simp
+  ring
+
+lemma wienerGrid_gap (n k : ℕ) :
+    (wienerGridTime n (k+1):ℝ)-(wienerGridTime n k:ℝ) = 1/(n+1:ℝ) := by
+  change ((k+1:ℕ):ℝ)/(n+1)-(k:ℝ)/(n+1)=1/(n+1:ℝ)
+  push_cast
+  ring
+
+def wienerGridInterp (n : ℕ) (f : ℝ≥0 → ℝ) (t : unitInterval) : ℝ :=
+  (1-wienerGridWeight n t)*f (wienerGridTime n (wienerGridCell n t))+
+    wienerGridWeight n t*f (wienerGridTime n (wienerGridCell n t+1))
+
+lemma wienerGridInterp_error (n : ℕ) (f : ℝ≥0 → ℝ) (t : unitInterval)
+    {ε : ℝ}
+    (ha : |f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t))| ≤ ε)
+    (hb : |f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1))| ≤ ε) :
+    |f ⟨t,t.2.1⟩-wienerGridInterp n f t| ≤ ε := by
+  have hw := wienerGridWeight_mem n t
+  have he : f ⟨t,t.2.1⟩-wienerGridInterp n f t =
+      (1-wienerGridWeight n t)*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t)))+
+      wienerGridWeight n t*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1))) := by
+    unfold wienerGridInterp
+    ring
+  rw [he]
+  calc
+    _ ≤ |(1-wienerGridWeight n t)*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t)))|+
+        |wienerGridWeight n t*(f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1)))| := abs_add_le _ _
+    _ = (1-wienerGridWeight n t)*|f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t))|+
+        wienerGridWeight n t*|f ⟨t,t.2.1⟩-f (wienerGridTime n (wienerGridCell n t+1))| := by
+      rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr hw.2.le),abs_of_nonneg hw.1]
+    _ ≤ (1-wienerGridWeight n t)*ε+wienerGridWeight n t*ε :=
+      add_le_add (mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr hw.2.le))
+        (mul_le_mul_of_nonneg_left hb hw.1)
+    _ = ε := by ring
+
+lemma wienerGridTime_mono (n : ℕ) : Monotone (wienerGridTime n) := by
+  intro i j hij
+  exact div_le_div_of_nonneg_right (Nat.cast_le.mpr hij) (by positivity)
+
+lemma wienerGridTime_mem (n k : ℕ) (hk : k ≤ n+2) :
+    wienerGridTime n k ∈ Icc (0:ℝ≥0) 2 := by
+  refine ⟨zero_le,?_⟩
+  apply NNReal.coe_le_coe.mp
+  change (k:ℝ)/(n+1) ≤ 2
+  rw [div_le_iff₀ (by positivity : (0:ℝ)<n+1)]
+  have h : (k:ℝ) ≤ (n+2:ℕ) := by exact_mod_cast hk
+  push_cast at h
+  nlinarith
+
+lemma wienerGrid_dist_left (n : ℕ) (t : unitInterval) :
+    dist (⟨t,t.2.1⟩:ℝ≥0) (wienerGridTime n (wienerGridCell n t)) ≤ 1/(n+1:ℝ) := by
+  have h := wienerGrid_bracket n t
+  change |(t:ℝ)-(wienerGridTime n (wienerGridCell n t):ℝ)| ≤ _
+  rw [abs_of_nonneg (sub_nonneg.mpr h.1)]
+  have hg := wienerGrid_gap n (wienerGridCell n t)
+  linarith
+
+lemma wienerGrid_dist_right (n : ℕ) (t : unitInterval) :
+    dist (⟨t,t.2.1⟩:ℝ≥0) (wienerGridTime n (wienerGridCell n t+1)) ≤ 1/(n+1:ℝ) := by
+  have h := wienerGrid_bracket n t
+  change |(t:ℝ)-(wienerGridTime n (wienerGridCell n t+1):ℝ)| ≤ _
+  rw [abs_of_nonpos (sub_nonpos.mpr h.2)]
+  have hg := wienerGrid_gap n (wienerGridCell n t)
+  linarith
+
+lemma wienerGridInterp_uniform (f : ℝ≥0 → ℝ)
+    (hf : ContinuousOn f (Icc 0 2)) (ε : ℝ) (hε : 0 < ε) :
+    ∀ᶠ n in atTop, ∀ t : unitInterval,
+      |f ⟨t,t.2.1⟩-wienerGridInterp n f t| ≤ ε := by
+  obtain ⟨δ,hδ,hmod⟩ := Metric.uniformContinuousOn_iff.mp
+    (isCompact_Icc.uniformContinuousOn_of_continuous hf) ε hε
+  filter_upwards [(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+    (gt_mem_nhds hδ)] with n hn t
+  have ht : (⟨t,t.2.1⟩:ℝ≥0) ∈ Icc (0:ℝ≥0) 2 := by
+    refine ⟨zero_le,?_⟩
+    apply NNReal.coe_le_coe.mp
+    change (t:ℝ) ≤ 2
+    linarith [t.2.2]
+  apply wienerGridInterp_error
+  · have h := hmod _ ht _ (wienerGridTime_mem n _ ((wienerGridCell_le n t).trans (by omega)))
+      ((wienerGrid_dist_left n t).trans_lt hn)
+    exact (by simpa only [Real.dist_eq] using h.le)
+  · have h := hmod _ ht _ (wienerGridTime_mem n _ (by have := wienerGridCell_le n t; omega))
+      ((wienerGrid_dist_right n t).trans_lt hn)
+    exact (by simpa only [Real.dist_eq] using h.le)
+
+lemma wienerGridInterp_sub (n : ℕ) (f g : ℝ≥0 → ℝ) (t : unitInterval) :
+    wienerGridInterp n f t-wienerGridInterp n g t =
+      wienerGridInterp n (fun s => f s-g s) t := by
+  unfold wienerGridInterp
+  ring
+
+lemma wienerGridInterp_difference (n : ℕ) (f g : ℝ≥0 → ℝ) (t : unitInterval)
+    {ε : ℝ}
+    (h : ∀ k ≤ n+2, |f (wienerGridTime n k)-g (wienerGridTime n k)| ≤ ε) :
+    |wienerGridInterp n f t-wienerGridInterp n g t| ≤ ε := by
+  rw [wienerGridInterp_sub]
+  have hw := wienerGridWeight_mem n t
+  have ha := h (wienerGridCell n t) ((wienerGridCell_le n t).trans (by omega))
+  have hb := h (wienerGridCell n t+1) (by have := wienerGridCell_le n t; omega)
+  unfold wienerGridInterp
+  calc
+    _ ≤ |(1-wienerGridWeight n t)*(f (wienerGridTime n (wienerGridCell n t))-g (wienerGridTime n (wienerGridCell n t)))|+
+        |wienerGridWeight n t*(f (wienerGridTime n (wienerGridCell n t+1))-g (wienerGridTime n (wienerGridCell n t+1)))| := abs_add_le _ _
+    _ = (1-wienerGridWeight n t)*|f (wienerGridTime n (wienerGridCell n t))-g (wienerGridTime n (wienerGridCell n t))|+
+        wienerGridWeight n t*|f (wienerGridTime n (wienerGridCell n t+1))-g (wienerGridTime n (wienerGridCell n t+1))| := by
+      rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr hw.2.le),abs_of_nonneg hw.1]
+    _ ≤ (1-wienerGridWeight n t)*ε+wienerGridWeight n t*ε :=
+      add_le_add (mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr hw.2.le))
+        (mul_le_mul_of_nonneg_left hb hw.1)
+    _ = ε := by ring
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerGridIndependence.lean b/lean/OAI/Probability/SLE/Stochastic/WienerGridIndependence.lean
index 966118fbeb237d4d38f25703ffd59aebadcde0ac..58eb67d2020c841e9730b9131f8ea4c9eaa4ff25 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerGridIndependence.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerGridIndependence.lean
@@ -1,101 +1,101 @@
-import OAI.Probability.SLE.Stochastic.WienerGrid
-
-namespace OAI
-
-/-! The Brownian interpolation residual is independent of the finite
-grid. Only covariances of the three evaluations in each residual enter. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma brownian_affine_residual_covariance
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (a b t s : ℝ≥0) (c : ℝ) (hat : a ≤ t) (htb : t ≤ b)
-    (hs : s ≤ a ∨ b ≤ s) (hc : (1-c)*(a:ℝ)+c*(b:ℝ)=t) :
-    cov[B s,fun ω => B t ω-((1-c)*B a ω+c*B b ω);P]=0 := by
-  have : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hm u := (hB.isGaussianProcess.hasGaussianLaw_eval u).memLp_two
-  have hz : MemLp (fun ω => (1-c)*B a ω+c*B b ω) 2 P :=
-    ((hm a).const_mul (1-c)).add ((hm b).const_mul c)
-  have hadd : cov[B s,fun ω => (1-c)*B a ω+c*B b ω;P] =
-      cov[B s,fun ω => (1-c)*B a ω;P]+cov[B s,fun ω => c*B b ω;P] :=
-    covariance_add_right (hm s) ((hm a).const_mul (1-c)) ((hm b).const_mul c)
-  rw [covariance_fun_sub_right (hm s) (hm t) hz,
-    hadd,
-    covariance_const_mul_right,covariance_const_mul_right,
-    hB.covariance_eval,hB.covariance_eval,hB.covariance_eval]
-  rcases hs with hs | hs
-  · rw [min_eq_left (hs.trans hat),min_eq_left hs,min_eq_left (hs.trans (hat.trans htb))]
-    ring
-  · rw [min_eq_right (htb.trans hs),min_eq_right ((hat.trans htb).trans hs),min_eq_right hs]
-    linarith
-
-def wienerGridResidual {Ω : Type*} (n : ℕ) (B : ℝ≥0 → Ω → ℝ)
-    (t : unitInterval) (ω : Ω) : ℝ :=
-  B ⟨t,t.2.1⟩ ω-wienerGridInterp n (fun s => B s ω) t
-
-lemma wienerGrid_joint_gaussian
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (n : ℕ) :
-    IsGaussianProcess (Sum.elim (fun i : Fin (n+3) => B (wienerGridTime n i))
-      (wienerGridResidual n B)) P := by
-  classical
-  apply hB.isGaussianProcess.of_isGaussianProcess
-  rintro (i | t)
-  · exact ⟨{wienerGridTime n i},
-      { toFun := fun x => x ⟨wienerGridTime n i,by simp⟩
-        map_add' := by simp
-        map_smul' := by simp },by simp⟩
-  · let a := wienerGridTime n (wienerGridCell n t)
-    let b := wienerGridTime n (wienerGridCell n t+1)
-    let u : ℝ≥0 := ⟨t,t.2.1⟩
-    let c := wienerGridWeight n t
-    refine ⟨{u,a,b},
-      { toFun := fun x => x ⟨u,by simp⟩-
-          ((1-c)*x ⟨a,by simp⟩+c*x ⟨b,by simp⟩)
-        map_add' := ?_
-        map_smul' := ?_ },?_⟩
-    · intro x y
-      simp only [Pi.add_apply]
-      ring
-    · intro d x
-      simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
-      ring
-    · intro ω
-      rfl
-
-lemma wienerGrid_residual_covariance
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (n : ℕ)
-    (i : Fin (n+3)) (t : unitInterval) :
-    cov[B (wienerGridTime n i),wienerGridResidual n B t;P]=0 := by
-  have hb := wienerGrid_bracket n t
-  have hs : wienerGridTime n i ≤ wienerGridTime n (wienerGridCell n t) ∨
-      wienerGridTime n (wienerGridCell n t+1) ≤ wienerGridTime n i := by
-    by_cases h : (i:ℕ) ≤ wienerGridCell n t
-    · exact Or.inl (wienerGridTime_mono n h)
-    · exact Or.inr (wienerGridTime_mono n (by omega))
-  exact brownian_affine_residual_covariance hB _ _ _ _ _ hb.1 hb.2 hs
-    (wienerGrid_affine n t)
-
-lemma wienerGrid_residual_independent
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (n : ℕ) :
-    IndepFun (fun ω (i : Fin (n+3)) => B (wienerGridTime n i) ω)
-      (fun ω t => wienerGridResidual n B t ω) P := by
-  apply (wienerGrid_joint_gaussian hB n).indepFun_of_covariance_eq_zero
-  · intro i
-    exact hB.aemeasurable _
-  · intro t
-    exact (hB.aemeasurable _).sub
-      (((hB.aemeasurable _).const_mul _).add ((hB.aemeasurable _).const_mul _))
-  · exact wienerGrid_residual_covariance hB n
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.WienerGrid
+
+namespace OAI
+
+/-! The Brownian interpolation residual is independent of the finite
+grid. Only covariances of the three evaluations in each residual enter. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma brownian_affine_residual_covariance
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (a b t s : ℝ≥0) (c : ℝ) (hat : a ≤ t) (htb : t ≤ b)
+    (hs : s ≤ a ∨ b ≤ s) (hc : (1-c)*(a:ℝ)+c*(b:ℝ)=t) :
+    cov[B s,fun ω => B t ω-((1-c)*B a ω+c*B b ω);P]=0 := by
+  have : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hm u := (hB.isGaussianProcess.hasGaussianLaw_eval u).memLp_two
+  have hz : MemLp (fun ω => (1-c)*B a ω+c*B b ω) 2 P :=
+    ((hm a).const_mul (1-c)).add ((hm b).const_mul c)
+  have hadd : cov[B s,fun ω => (1-c)*B a ω+c*B b ω;P] =
+      cov[B s,fun ω => (1-c)*B a ω;P]+cov[B s,fun ω => c*B b ω;P] :=
+    covariance_add_right (hm s) ((hm a).const_mul (1-c)) ((hm b).const_mul c)
+  rw [covariance_fun_sub_right (hm s) (hm t) hz,
+    hadd,
+    covariance_const_mul_right,covariance_const_mul_right,
+    hB.covariance_eval,hB.covariance_eval,hB.covariance_eval]
+  rcases hs with hs | hs
+  · rw [min_eq_left (hs.trans hat),min_eq_left hs,min_eq_left (hs.trans (hat.trans htb))]
+    ring
+  · rw [min_eq_right (htb.trans hs),min_eq_right ((hat.trans htb).trans hs),min_eq_right hs]
+    linarith
+
+def wienerGridResidual {Ω : Type*} (n : ℕ) (B : ℝ≥0 → Ω → ℝ)
+    (t : unitInterval) (ω : Ω) : ℝ :=
+  B ⟨t,t.2.1⟩ ω-wienerGridInterp n (fun s => B s ω) t
+
+lemma wienerGrid_joint_gaussian
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (n : ℕ) :
+    IsGaussianProcess (Sum.elim (fun i : Fin (n+3) => B (wienerGridTime n i))
+      (wienerGridResidual n B)) P := by
+  classical
+  apply hB.isGaussianProcess.of_isGaussianProcess
+  rintro (i | t)
+  · exact ⟨{wienerGridTime n i},
+      { toFun := fun x => x ⟨wienerGridTime n i,by simp⟩
+        map_add' := by simp
+        map_smul' := by simp },by simp⟩
+  · let a := wienerGridTime n (wienerGridCell n t)
+    let b := wienerGridTime n (wienerGridCell n t+1)
+    let u : ℝ≥0 := ⟨t,t.2.1⟩
+    let c := wienerGridWeight n t
+    refine ⟨{u,a,b},
+      { toFun := fun x => x ⟨u,by simp⟩-
+          ((1-c)*x ⟨a,by simp⟩+c*x ⟨b,by simp⟩)
+        map_add' := ?_
+        map_smul' := ?_ },?_⟩
+    · intro x y
+      simp only [Pi.add_apply]
+      ring
+    · intro d x
+      simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply]
+      ring
+    · intro ω
+      rfl
+
+lemma wienerGrid_residual_covariance
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (n : ℕ)
+    (i : Fin (n+3)) (t : unitInterval) :
+    cov[B (wienerGridTime n i),wienerGridResidual n B t;P]=0 := by
+  have hb := wienerGrid_bracket n t
+  have hs : wienerGridTime n i ≤ wienerGridTime n (wienerGridCell n t) ∨
+      wienerGridTime n (wienerGridCell n t+1) ≤ wienerGridTime n i := by
+    by_cases h : (i:ℕ) ≤ wienerGridCell n t
+    · exact Or.inl (wienerGridTime_mono n h)
+    · exact Or.inr (wienerGridTime_mono n (by omega))
+  exact brownian_affine_residual_covariance hB _ _ _ _ _ hb.1 hb.2 hs
+    (wienerGrid_affine n t)
+
+lemma wienerGrid_residual_independent
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (n : ℕ) :
+    IndepFun (fun ω (i : Fin (n+3)) => B (wienerGridTime n i) ω)
+      (fun ω t => wienerGridResidual n B t ω) P := by
+  apply (wienerGrid_joint_gaussian hB n).indepFun_of_covariance_eq_zero
+  · intro i
+    exact hB.aemeasurable _
+  · intro t
+    exact (hB.aemeasurable _).sub
+      (((hB.aemeasurable _).const_mul _).add ((hB.aemeasurable _).const_mul _))
+  · exact wienerGrid_residual_covariance hB n
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerGridSupport.lean b/lean/OAI/Probability/SLE/Stochastic/WienerGridSupport.lean
index 94bb1995052c8de058f8b7fce53c7ae2b1a3fc11..ded5dd44cacaade5aff7b6cbc4ad80693564d6c7 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerGridSupport.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerGridSupport.lean
@@ -1,92 +1,92 @@
-import OAI.Probability.SLE.Stochastic.WienerGridIndependence
-import Mathlib.MeasureTheory.Measure.OpenPos
-
-namespace OAI
-
-/-! Nondegenerate independent Gaussian increments give positive
-probability to every based finite grid tube. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-lemma gaussianReal_ball_pos (a c : ℝ) (v : ℝ≥0) (hv : v ≠ 0)
-    (r : ℝ) (hr : 0 < r) : 0 < gaussianReal a v (ball c r) := by
-  by_contra h
-  have hz : gaussianReal a v (ball c r)=0 := le_antisymm (le_of_not_gt h) bot_le
-  have hz' := gaussianReal_absolutelyContinuous' a hv hz
-  exact (measure_ball_pos volume c hr).ne' hz'
-
-lemma nat_path_error_le (x y : ℕ → ℝ) (m : ℕ) (δ : ℝ)
-    (hzero : x 0=y 0)
-    (hinc : ∀ k < m, |(x (k+1)-x k)-(y (k+1)-y k)| ≤ δ) :
-    ∀ k ≤ m, |x k-y k| ≤ (k:ℝ)*δ := by
-  intro k
-  induction k with
-  | zero => intro _; simp [hzero]
-  | succ k ih =>
-    intro hk
-    have hk' : k < m := by omega
-    calc
-      |x (k+1)-y (k+1)| = |(x k-y k)+((x (k+1)-x k)-(y (k+1)-y k))| := by congr 1; ring
-      _ ≤ |x k-y k|+|(x (k+1)-x k)-(y (k+1)-y k)| := abs_add_le _ _
-      _ ≤ (k:ℝ)*δ+δ := add_le_add (ih (by omega)) (hinc k hk')
-      _ = ((k+1:ℕ):ℝ)*δ := by push_cast; ring
-
-lemma wienerGridTime_strictMono (n : ℕ) : StrictMono (wienerGridTime n) := by
-  intro i j hij
-  apply NNReal.coe_lt_coe.mp
-  change (i:ℝ)/(n+1) < (j:ℝ)/(n+1)
-  exact (div_lt_div_iff_of_pos_right (by positivity)).mpr (by exact_mod_cast hij)
-
-theorem wienerGrid_tube_pos (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (n : ℕ) (f : ℕ → ℝ) (hf : f 0=0) (ε : ℝ) (hε : 0 < ε) :
-    0 < W {b | ∀ i : Fin (n+3), |b (wienerGridTime n i)-f i| < ε} := by
-  classical
-  have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  let m := n+2
-  let δ : ℝ := ε/(m+1:ℝ)
-  have hδ : 0 < δ := div_pos hε (by positivity)
-  let X : Fin m → Target.Driver → ℝ := fun i b =>
-    b (wienerGridTime n (i+1))-b (wienerGridTime n i)
-  let c : Fin m → ℝ := fun i => f (i+1)-f i
-  have hind : iIndepFun X W := by
-    exact hW.hasIndepIncrements m (fun i => wienerGridTime n i)
-      ((wienerGridTime_strictMono n).monotone.comp (fun _ _ h => h))
-  have hpos (i : Fin m) : 0 < W (X i ⁻¹' ball (c i) δ) := by
-    have ht : wienerGridTime n i < wienerGridTime n (i+1) :=
-      wienerGridTime_strictMono n (by omega)
-    have hv : nndist (wienerGridTime n (i+1):ℝ) (wienerGridTime n i:ℝ) ≠ 0 := by
-      intro h
-      exact (ne_of_gt (show (wienerGridTime n i:ℝ)<wienerGridTime n (i+1) from ht))
-        (eq_of_nndist_eq_zero h)
-    have hLaw := hW.hasLaw_sub (wienerGridTime n (i+1)) (wienerGridTime n i)
-    have he := hLaw.measure_eq (isOpen_ball.measurableSet : MeasurableSet (ball (c i) δ))
-    exact he ▸ gaussianReal_ball_pos 0 (c i) _ hv δ hδ
-  have hprod := hind.measure_inter_preimage_eq_mul Finset.univ
-    (sets := fun i => ball (c i) δ) (fun _ _ => isOpen_ball.measurableSet)
-  have hall : 0 < W (⋂ i : Fin m, X i ⁻¹' ball (c i) δ) := by
-    have he : W (⋂ i : Fin m, X i ⁻¹' ball (c i) δ) = ∏ i, W (X i ⁻¹' ball (c i) δ) := by
-      simpa only [Finset.mem_univ,iInter_true] using hprod
-    rw [he]
-    exact bot_lt_iff_ne_bot.mpr (Finset.prod_ne_zero_iff.mpr (fun i _ => (hpos i).ne'))
-  refine hall.trans_le (measure_mono ?_)
-  intro b hb i
-  have hb' : ∀ k < m,
-      |(b (wienerGridTime n (k+1))-b (wienerGridTime n k))-(f (k+1)-f k)| ≤ δ := by
-    intro k hk
-    have h := mem_iInter.mp hb ⟨k,hk⟩
-    exact (show |X ⟨k,hk⟩ b-c ⟨k,hk⟩| < δ from by simpa only [mem_preimage,mem_ball,Real.dist_eq] using h).le
-  have he := nat_path_error_le (fun k => b (wienerGridTime n k)) f m δ
-    (by simp [wienerGridTime,hf]) hb' i (by have := i.isLt; omega)
-  refine he.trans_lt ?_
-  calc
-    (i:ℝ)*δ ≤ (m:ℝ)*δ := mul_le_mul_of_nonneg_right (by exact_mod_cast (show (i:ℕ) ≤ m from by have := i.isLt; omega)) hδ.le
-    _ < ε := by dsimp [δ]; rw [← mul_div_assoc,div_lt_iff₀ (by positivity)]; nlinarith
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.WienerGridIndependence
+import Mathlib.MeasureTheory.Measure.OpenPos
+
+namespace OAI
+
+/-! Nondegenerate independent Gaussian increments give positive
+probability to every based finite grid tube. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+lemma gaussianReal_ball_pos (a c : ℝ) (v : ℝ≥0) (hv : v ≠ 0)
+    (r : ℝ) (hr : 0 < r) : 0 < gaussianReal a v (ball c r) := by
+  by_contra h
+  have hz : gaussianReal a v (ball c r)=0 := le_antisymm (le_of_not_gt h) bot_le
+  have hz' := gaussianReal_absolutelyContinuous' a hv hz
+  exact (measure_ball_pos volume c hr).ne' hz'
+
+lemma nat_path_error_le (x y : ℕ → ℝ) (m : ℕ) (δ : ℝ)
+    (hzero : x 0=y 0)
+    (hinc : ∀ k < m, |(x (k+1)-x k)-(y (k+1)-y k)| ≤ δ) :
+    ∀ k ≤ m, |x k-y k| ≤ (k:ℝ)*δ := by
+  intro k
+  induction k with
+  | zero => intro _; simp [hzero]
+  | succ k ih =>
+    intro hk
+    have hk' : k < m := by omega
+    calc
+      |x (k+1)-y (k+1)| = |(x k-y k)+((x (k+1)-x k)-(y (k+1)-y k))| := by congr 1; ring
+      _ ≤ |x k-y k|+|(x (k+1)-x k)-(y (k+1)-y k)| := abs_add_le _ _
+      _ ≤ (k:ℝ)*δ+δ := add_le_add (ih (by omega)) (hinc k hk')
+      _ = ((k+1:ℕ):ℝ)*δ := by push_cast; ring
+
+lemma wienerGridTime_strictMono (n : ℕ) : StrictMono (wienerGridTime n) := by
+  intro i j hij
+  apply NNReal.coe_lt_coe.mp
+  change (i:ℝ)/(n+1) < (j:ℝ)/(n+1)
+  exact (div_lt_div_iff_of_pos_right (by positivity)).mpr (by exact_mod_cast hij)
+
+theorem wienerGrid_tube_pos (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (n : ℕ) (f : ℕ → ℝ) (hf : f 0=0) (ε : ℝ) (hε : 0 < ε) :
+    0 < W {b | ∀ i : Fin (n+3), |b (wienerGridTime n i)-f i| < ε} := by
+  classical
+  have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  let m := n+2
+  let δ : ℝ := ε/(m+1:ℝ)
+  have hδ : 0 < δ := div_pos hε (by positivity)
+  let X : Fin m → Target.Driver → ℝ := fun i b =>
+    b (wienerGridTime n (i+1))-b (wienerGridTime n i)
+  let c : Fin m → ℝ := fun i => f (i+1)-f i
+  have hind : iIndepFun X W := by
+    exact hW.hasIndepIncrements m (fun i => wienerGridTime n i)
+      ((wienerGridTime_strictMono n).monotone.comp (fun _ _ h => h))
+  have hpos (i : Fin m) : 0 < W (X i ⁻¹' ball (c i) δ) := by
+    have ht : wienerGridTime n i < wienerGridTime n (i+1) :=
+      wienerGridTime_strictMono n (by omega)
+    have hv : nndist (wienerGridTime n (i+1):ℝ) (wienerGridTime n i:ℝ) ≠ 0 := by
+      intro h
+      exact (ne_of_gt (show (wienerGridTime n i:ℝ)<wienerGridTime n (i+1) from ht))
+        (eq_of_nndist_eq_zero h)
+    have hLaw := hW.hasLaw_sub (wienerGridTime n (i+1)) (wienerGridTime n i)
+    have he := hLaw.measure_eq (isOpen_ball.measurableSet : MeasurableSet (ball (c i) δ))
+    exact he ▸ gaussianReal_ball_pos 0 (c i) _ hv δ hδ
+  have hprod := hind.measure_inter_preimage_eq_mul Finset.univ
+    (sets := fun i => ball (c i) δ) (fun _ _ => isOpen_ball.measurableSet)
+  have hall : 0 < W (⋂ i : Fin m, X i ⁻¹' ball (c i) δ) := by
+    have he : W (⋂ i : Fin m, X i ⁻¹' ball (c i) δ) = ∏ i, W (X i ⁻¹' ball (c i) δ) := by
+      simpa only [Finset.mem_univ,iInter_true] using hprod
+    rw [he]
+    exact bot_lt_iff_ne_bot.mpr (Finset.prod_ne_zero_iff.mpr (fun i _ => (hpos i).ne'))
+  refine hall.trans_le (measure_mono ?_)
+  intro b hb i
+  have hb' : ∀ k < m,
+      |(b (wienerGridTime n (k+1))-b (wienerGridTime n k))-(f (k+1)-f k)| ≤ δ := by
+    intro k hk
+    have h := mem_iInter.mp hb ⟨k,hk⟩
+    exact (show |X ⟨k,hk⟩ b-c ⟨k,hk⟩| < δ from by simpa only [mem_preimage,mem_ball,Real.dist_eq] using h).le
+  have he := nat_path_error_le (fun k => b (wienerGridTime n k)) f m δ
+    (by simp [wienerGridTime,hf]) hb' i (by have := i.isLt; omega)
+  refine he.trans_lt ?_
+  calc
+    (i:ℝ)*δ ≤ (m:ℝ)*δ := mul_le_mul_of_nonneg_right (by exact_mod_cast (show (i:ℕ) ≤ m from by have := i.isLt; omega)) hδ.le
+    _ < ε := by dsimp [δ]; rw [← mul_div_assoc,div_lt_iff₀ (by positivity)]; nlinarith
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerRealSupport.lean b/lean/OAI/Probability/SLE/Stochastic/WienerRealSupport.lean
index 62255572834afeabf0d741e711e666aedc910e18..2fd80cc9fc66d31e2589531587493f11013fe1cd 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerRealSupport.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerRealSupport.lean
@@ -1,63 +1,63 @@
-import OAI.Probability.SLE.Stochastic.WienerResidualTube
-
-namespace OAI
-
-/-! Uniform support of the real Wiener law on a finite time window,
-proved from its Gaussian finite-dimensional laws and continuous paths. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-theorem wienerReal_uniform_tube_pos
-    (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (f : ℝ≥0 → ℝ) (hf : Continuous f) (hf0 : f 0=0)
-    (ε : ℝ) (hε : 0 < ε) :
-    0 < W {b | ∀ t : unitInterval, |b ⟨t,t.2.1⟩-f ⟨t,t.2.1⟩| < ε} := by
-  let η := ε/4
-  have hη : 0 < η := by dsimp [η]; positivity
-  obtain ⟨N,hN⟩ := eventually_atTop.mp (wienerGridInterp_uniform f hf.continuousOn η hη)
-  obtain ⟨n,hn,hR⟩ := exists_wienerResidualTube_pos W hW η hη N
-  let G : Set (Fin (n+3) → ℝ) :=
-    {x | ∀ i, |x i-f (wienerGridTime n i)| < η}
-  have hG : MeasurableSet G := by
-    simp only [G,Set.ofPred_forall]
-    exact MeasurableSet.iInter (fun i => measurableSet_lt
-      ((measurable_pi_apply i).sub measurable_const).abs measurable_const)
-  have hGpos : 0 < W ((fun b (i : Fin (n+3)) => b (wienerGridTime n i)) ⁻¹' G) :=
-    wienerGrid_tube_pos W hW n (fun k => f (wienerGridTime n k)) (by simpa [wienerGridTime] using hf0) η hη
-  have hind := wienerGrid_residual_independent hW n
-  have he := hind.measure_inter_preimage_eq_mul G (wienerResidualSet η) hG
-    (measurableSet_wienerResidualSet η)
-  have hpos : 0 < W (((fun b (i : Fin (n+3)) => b (wienerGridTime n i)) ⁻¹' G) ∩
-      (fun b t => wienerGridResidual n (fun s b => b s) t b) ⁻¹' wienerResidualSet η) := by
-    rw [he]
-    exact bot_lt_iff_ne_bot.mpr (mul_ne_zero hGpos.ne' hR.ne')
-  refine hpos.trans_le (measure_mono ?_)
-  intro b hb t
-  have hbG : ∀ k ≤ n+2, |b (wienerGridTime n k)-f (wienerGridTime n k)| ≤ η := by
-    intro k hk
-    exact (hb.1 (⟨k,by omega⟩ : Fin (n+3))).le
-  have hbound : ∀ t : unitInterval, |b ⟨t,t.2.1⟩-f ⟨t,t.2.1⟩| ≤ 3*η := by
-    intro u
-    refine (show Dense (range (denseSeq unitInterval)) from denseRange_denseSeq unitInterval).induction
-      (P := fun t : unitInterval => |b ⟨t,t.2.1⟩-f ⟨t,t.2.1⟩| ≤ 3*η) ?_ ?_ u
-    · rintro _ ⟨k,rfl⟩
-      let u := denseSeq unitInterval k
-      have hr : |b ⟨u,u.2.1⟩-wienerGridInterp n b u| ≤ η := hb.2 k
-      have hg := wienerGridInterp_difference n b f u hbG
-      have hf' := hN n hn u
-      have htri := abs_sub_le (b ⟨u,u.2.1⟩) (wienerGridInterp n b u) (f ⟨u,u.2.1⟩)
-      have htri' := abs_sub_le (wienerGridInterp n b u) (wienerGridInterp n f u) (f ⟨u,u.2.1⟩)
-      rw [abs_sub_comm (wienerGridInterp n f u)] at htri'
-      linarith
-    · apply isClosed_le _ continuous_const
-      fun_prop
-  exact (hbound t).trans_lt (by dsimp [η]; linarith)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.WienerResidualTube
+
+namespace OAI
+
+/-! Uniform support of the real Wiener law on a finite time window,
+proved from its Gaussian finite-dimensional laws and continuous paths. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+theorem wienerReal_uniform_tube_pos
+    (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (f : ℝ≥0 → ℝ) (hf : Continuous f) (hf0 : f 0=0)
+    (ε : ℝ) (hε : 0 < ε) :
+    0 < W {b | ∀ t : unitInterval, |b ⟨t,t.2.1⟩-f ⟨t,t.2.1⟩| < ε} := by
+  let η := ε/4
+  have hη : 0 < η := by dsimp [η]; positivity
+  obtain ⟨N,hN⟩ := eventually_atTop.mp (wienerGridInterp_uniform f hf.continuousOn η hη)
+  obtain ⟨n,hn,hR⟩ := exists_wienerResidualTube_pos W hW η hη N
+  let G : Set (Fin (n+3) → ℝ) :=
+    {x | ∀ i, |x i-f (wienerGridTime n i)| < η}
+  have hG : MeasurableSet G := by
+    simp only [G,Set.ofPred_forall]
+    exact MeasurableSet.iInter (fun i => measurableSet_lt
+      ((measurable_pi_apply i).sub measurable_const).abs measurable_const)
+  have hGpos : 0 < W ((fun b (i : Fin (n+3)) => b (wienerGridTime n i)) ⁻¹' G) :=
+    wienerGrid_tube_pos W hW n (fun k => f (wienerGridTime n k)) (by simpa [wienerGridTime] using hf0) η hη
+  have hind := wienerGrid_residual_independent hW n
+  have he := hind.measure_inter_preimage_eq_mul G (wienerResidualSet η) hG
+    (measurableSet_wienerResidualSet η)
+  have hpos : 0 < W (((fun b (i : Fin (n+3)) => b (wienerGridTime n i)) ⁻¹' G) ∩
+      (fun b t => wienerGridResidual n (fun s b => b s) t b) ⁻¹' wienerResidualSet η) := by
+    rw [he]
+    exact bot_lt_iff_ne_bot.mpr (mul_ne_zero hGpos.ne' hR.ne')
+  refine hpos.trans_le (measure_mono ?_)
+  intro b hb t
+  have hbG : ∀ k ≤ n+2, |b (wienerGridTime n k)-f (wienerGridTime n k)| ≤ η := by
+    intro k hk
+    exact (hb.1 (⟨k,by omega⟩ : Fin (n+3))).le
+  have hbound : ∀ t : unitInterval, |b ⟨t,t.2.1⟩-f ⟨t,t.2.1⟩| ≤ 3*η := by
+    intro u
+    refine (show Dense (range (denseSeq unitInterval)) from denseRange_denseSeq unitInterval).induction
+      (P := fun t : unitInterval => |b ⟨t,t.2.1⟩-f ⟨t,t.2.1⟩| ≤ 3*η) ?_ ?_ u
+    · rintro _ ⟨k,rfl⟩
+      let u := denseSeq unitInterval k
+      have hr : |b ⟨u,u.2.1⟩-wienerGridInterp n b u| ≤ η := hb.2 k
+      have hg := wienerGridInterp_difference n b f u hbG
+      have hf' := hN n hn u
+      have htri := abs_sub_le (b ⟨u,u.2.1⟩) (wienerGridInterp n b u) (f ⟨u,u.2.1⟩)
+      have htri' := abs_sub_le (wienerGridInterp n b u) (wienerGridInterp n f u) (f ⟨u,u.2.1⟩)
+      rw [abs_sub_comm (wienerGridInterp n f u)] at htri'
+      linarith
+    · apply isClosed_le _ continuous_const
+      fun_prop
+  exact (hbound t).trans_lt (by dsimp [η]; linarith)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/Stochastic/WienerResidualTube.lean b/lean/OAI/Probability/SLE/Stochastic/WienerResidualTube.lean
index 2550659d0eab77b262412cecbb715f62d72c7e5a..a0d5ef563e4c0869290cee4189369d738eeefdcb 100644
--- a/lean/OAI/Probability/SLE/Stochastic/WienerResidualTube.lean
+++ b/lean/OAI/Probability/SLE/Stochastic/WienerResidualTube.lean
@@ -1,61 +1,61 @@
-import OAI.Probability.SLE.Stochastic.WienerGridSupport
-
-namespace OAI
-
-/-! Uniform interpolation errors are eventually small on every continuous
-path. A countable dense set makes the residual event measurable. -/
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-
-namespace SLEExactGauge
-
-def wienerResidualSet (r : ℝ) : Set (unitInterval → ℝ) :=
-  {x | ∀ k : ℕ, |x (denseSeq unitInterval k)| ≤ r}
-
-lemma measurableSet_wienerResidualSet (r : ℝ) : MeasurableSet (wienerResidualSet r) := by
-  simp only [wienerResidualSet,Set.ofPred_forall]
-  exact MeasurableSet.iInter (fun k => measurableSet_le
-    (measurable_pi_apply (denseSeq unitInterval k)).abs measurable_const)
-
-def wienerResidualTube (n : ℕ) (r : ℝ) : Set Target.Driver :=
-  {b | ∀ k : ℕ, |wienerGridResidual n (fun t b => b t) (denseSeq unitInterval k) b| ≤ r}
-
-lemma measurableSet_wienerResidualTube (n : ℕ) (r : ℝ) :
-    MeasurableSet (wienerResidualTube n r) := by
-  simp only [wienerResidualTube,Set.ofPred_forall]
-  apply MeasurableSet.iInter
-  intro k
-  apply measurableSet_le _ measurable_const
-  unfold wienerGridResidual wienerGridInterp
-  exact ((continuous_eval_const _).measurable.sub
-    ((measurable_const.mul (continuous_eval_const _).measurable).add
-      (measurable_const.mul (continuous_eval_const _).measurable))).abs
-
-lemma wienerResidualTube_eventually (b : Target.Driver) (r : ℝ) (hr : 0 < r) :
-    ∀ᶠ n in atTop, b ∈ wienerResidualTube n r := by
-  filter_upwards [wienerGridInterp_uniform b (map_continuous b).continuousOn r hr] with n hn k
-  exact hn _
-
-lemma exists_wienerResidualTube_pos (W : Target.WienerLaw) (hW : Target.IsWiener W)
-    (r : ℝ) (hr : 0 < r) (N : ℕ) :
-    ∃ n, N ≤ n ∧ 0 < W (wienerResidualTube n r) := by
-  have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  by_contra! h
-  have hzero n (hn : N ≤ n) : W (wienerResidualTube n r)=0 :=
-    le_antisymm (h n hn) bot_le
-  have hae : ∀ᵐ b ∂W, ∀ n, N ≤ n → b ∉ wienerResidualTube n r := by
-    apply ae_all_iff.mpr
-    intro n
-    by_cases hn : N ≤ n
-    · exact ((measure_eq_zero_iff_ae_notMem.mp (hzero n hn))).mono (fun b hb _ => hb)
-    · exact ae_of_all _ (fun _ h => (hn h).elim)
-  obtain ⟨b,hb⟩ := hae.exists
-  obtain ⟨n,hn,hmem⟩ := ((eventually_ge_atTop N).and (wienerResidualTube_eventually b r hr)).exists
-  exact hb n hn hmem
-
-end SLEExactGauge
-
-end
-
-end OAI
+import OAI.Probability.SLE.Stochastic.WienerGridSupport
+
+namespace OAI
+
+/-! Uniform interpolation errors are eventually small on every continuous
+path. A countable dense set makes the residual event measurable. -/
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+
+namespace SLEExactGauge
+
+def wienerResidualSet (r : ℝ) : Set (unitInterval → ℝ) :=
+  {x | ∀ k : ℕ, |x (denseSeq unitInterval k)| ≤ r}
+
+lemma measurableSet_wienerResidualSet (r : ℝ) : MeasurableSet (wienerResidualSet r) := by
+  simp only [wienerResidualSet,Set.ofPred_forall]
+  exact MeasurableSet.iInter (fun k => measurableSet_le
+    (measurable_pi_apply (denseSeq unitInterval k)).abs measurable_const)
+
+def wienerResidualTube (n : ℕ) (r : ℝ) : Set Target.Driver :=
+  {b | ∀ k : ℕ, |wienerGridResidual n (fun t b => b t) (denseSeq unitInterval k) b| ≤ r}
+
+lemma measurableSet_wienerResidualTube (n : ℕ) (r : ℝ) :
+    MeasurableSet (wienerResidualTube n r) := by
+  simp only [wienerResidualTube,Set.ofPred_forall]
+  apply MeasurableSet.iInter
+  intro k
+  apply measurableSet_le _ measurable_const
+  unfold wienerGridResidual wienerGridInterp
+  exact ((continuous_eval_const _).measurable.sub
+    ((measurable_const.mul (continuous_eval_const _).measurable).add
+      (measurable_const.mul (continuous_eval_const _).measurable))).abs
+
+lemma wienerResidualTube_eventually (b : Target.Driver) (r : ℝ) (hr : 0 < r) :
+    ∀ᶠ n in atTop, b ∈ wienerResidualTube n r := by
+  filter_upwards [wienerGridInterp_uniform b (map_continuous b).continuousOn r hr] with n hn k
+  exact hn _
+
+lemma exists_wienerResidualTube_pos (W : Target.WienerLaw) (hW : Target.IsWiener W)
+    (r : ℝ) (hr : 0 < r) (N : ℕ) :
+    ∃ n, N ≤ n ∧ 0 < W (wienerResidualTube n r) := by
+  have : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  by_contra! h
+  have hzero n (hn : N ≤ n) : W (wienerResidualTube n r)=0 :=
+    le_antisymm (h n hn) bot_le
+  have hae : ∀ᵐ b ∂W, ∀ n, N ≤ n → b ∉ wienerResidualTube n r := by
+    apply ae_all_iff.mpr
+    intro n
+    by_cases hn : N ≤ n
+    · exact ((measure_eq_zero_iff_ae_notMem.mp (hzero n hn))).mono (fun b hb _ => hb)
+    · exact ae_of_all _ (fun _ h => (hn h).elim)
+  obtain ⟨b,hb⟩ := hae.exists
+  obtain ⟨n,hn,hmem⟩ := ((eventually_ge_atTop N).and (wienerResidualTube_eventually b r hr)).exists
+  exact hb n hn hmem
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StochasticFubini.lean b/lean/OAI/Probability/SLE/StochasticFubini.lean
index 5756670f422f3fd001f50897666666a3f84e107b..2459d92b24e8a8e6ad0cfb24c0c72761e6d535d4 100644
--- a/lean/OAI/Probability/SLE/StochasticFubini.lean
+++ b/lean/OAI/Probability/SLE/StochasticFubini.lean
@@ -1,731 +1,731 @@
-import Mathlib
-import OAI.Probability.SLE.ExponentialMartingales
-
-namespace OAI
-
-/-! Stochastic Fubini for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField
-theorem clipped_spatial_green_window_exponential
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ lo hi : ℝ) (hle : lo ≤ hi) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (hq : Measurable q)
-    (R : ℝ) (hR : 0 ≤ R) (T : ℝ≥0)
-    (σ τ : Z × Ω → ℝ≥0) (hσm : Measurable σ) (hτm : Measurable τ)
-    (hστ : ∀ p, σ p ≤ τ p)
-    (hσa : ∀ t : ℝ≥0, @MeasurableSet (Z × Ω)
-      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
-      {p | t < σ p})
-    (hσs : ∀ z, IsStoppingTime (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
-      (fun ω => (σ (z, ω) : WithTop ℝ≥0)))
-    (hτa : ∀ t : ℝ≥0, @MeasurableSet (Z × Ω)
-      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
-      {p | t < τ p})
-    (hτs : ∀ z, IsStoppingTime (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
-      (fun ω => (τ (z, ω) : WithTop ℝ≥0)))
-    (hτb : ∀ z ω, τ (z,ω) ≤ T)
-    (hinside : ∀ z ω t, t ≤ τ (z,ω) →
-      ‖((c (q z)).Q t ω, B t ω)‖ ≤ R ∧
-        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi)
-    (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp
-      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
-        greenLogDensity κ ((c (q z)).Q (σ (z,ω)) ω) (B (σ (z,ω)) ω) ∂ν) -
-      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
-        (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
-          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
-  classical
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
-  let ψ := spatialPartial φ 1
-  have hψc : Continuous ψ := (spatialPartial_smooth φ hφ 1).continuous
-  obtain ⟨K, hK, _⟩ := compact_smooth_bounded_lipschitz ψ
-    (spatialPartial_smooth φ hφ 1) (spatialPartial_compact φ hφc 1)
-  have hψeq (q' : ℂ × ℝ) (x : ℝ) (hb : ‖(q',x)‖ ≤ R) :
-      ψ (q',x) = greenLogCoefficient κ q' x := by
-    rw [show ψ (q',x) = deriv (fun y => φ (q',y)) x by
-      simp only [ψ, spatialPartial_eq_iteratedDeriv φ hφ, iteratedDeriv_one]]
-    rw [deriv_fiber_fderiv φ (hφ.differentiable (by simp)), hφD q' x hb,
-      ← deriv_fiber_fderiv _ ((greenLogDensity_contDiff κ).differentiable (by simp))]
-    exact (greenLogDensity_spatial_derivative κ q' x).deriv
-  let F (p : ℝ × (Z × Ω)) := ψ ((c (q p.2.1)).Q p.1.toNNReal p.2.2, B p.1.toNNReal p.2.2)
-  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable hBc hm
-  have hQm := clipped_family_joint_measurable c
-  have hFm : Measurable F := hψc.measurable.comp
-    ((hQm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
-      ((hq.comp (measurable_fst.comp measurable_snd)).prodMk (measurable_snd.comp measurable_snd)))).prodMk
-    (hBm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk (measurable_snd.comp measurable_snd))))
-  have hFc (z : Z) (ω : Ω) : Continuous (fun t : ℝ => F (t,(z,ω))) :=
-    hψc.comp ((((c (q z)).lipschitz ω).continuous.comp
-      (NNReal.continuous_coe.comp continuous_real_toNNReal)).prodMk
-      ((hBc ω).comp continuous_real_toNNReal))
-  have hFb (t : ℝ) (z : Z) (ω : Ω) : |F (t,(z,ω))| ≤ K := by
-    simpa only [Real.norm_eq_abs] using hK ((c (q z)).Q t.toNNReal ω, B t.toNNReal ω)
-  let C (n i : ℕ) (p : Z × Ω) := if σ p ≤ dyadicTime T n i ∧ dyadicTime T n i < τ p then
-    F (dyadicTime T n i,p) else 0
-  have hCa (n i : ℕ) : @Measurable (Z × Ω) ℝ
-      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)))
-      (borel ℝ) (C n i) := by
-    have hQ := clipped_family_past_measurable c (dyadicTime T n i)
-    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted (dyadicTime T n i)
-    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)
-    have hh := hψc.measurable.comp
-      ((hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)).prodMk (hBa.comp measurable_snd))
-    change Measurable (C n i)
-    dsimp only [C, F]
-    simp only [Real.toNNReal_coe]
-    have hsa : MeasurableSet {p : Z × Ω | σ p ≤ dyadicTime T n i} := by
-      simpa only [Set.compl_ofPred, not_lt] using (hσa (dyadicTime T n i)).compl
-    exact hh.ite (hsa.inter (hτa (dyadicTime T n i))) measurable_const
-  have hCm (n i : ℕ) : Measurable (C n i) :=
-    (hCa n i).mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (Filtration.le _ _))) le_rfl
-  have hCb (n i : ℕ) (z : Z) (ω : Ω) : |C n i (z,ω)| ≤ K := by
-    dsimp only [C]
-    split_ifs
-    · exact hFb _ _ _
-    · simp
-  let Y (p : Z × Ω) := greenLogDensity κ ((c (q p.1)).Q (τ p) p.2) (B (τ p) p.2) -
-    greenLogDensity κ ((c (q p.1)).Q (σ p) p.2) (B (σ p) p.2)
-  have hYm : Measurable Y :=
-    (((greenLogDensity_contDiff κ).continuous.measurable.comp
-      ((hQm.comp (hτm.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
-        (hBm.comp (hτm.prodMk measurable_snd))))).sub
-      ((greenLogDensity_contDiff κ).continuous.measurable.comp
-        ((hQm.comp (hσm.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
-          (hBm.comp (hσm.prodMk measurable_snd))))
-  have hlim (z : Z) : ∀ᵐ ω ∂P, Tendsto
-      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z,ω)) (2^n) ω)
-      atTop (𝓝 (Y (z,ω))) := by
-    have hτlim := (c (q z)).stopped_green_ito hB hBc hκ hκ8 R hR T
-      (fun ω => τ (z,ω)) (hτs z) (hτb z) (hinside z)
-    have hσlim := (c (q z)).stopped_green_ito hB hBc hκ hκ8 R hR T
-      (fun ω => σ (z,ω)) (hσs z) (fun ω => (hστ (z,ω)).trans (hτb z ω))
-      (fun ω t ht => hinside z ω t (ht.trans (hστ (z,ω))))
-    filter_upwards [hτlim, hσlim] with ω hτl hσl
-    have he (n i : ℕ) : C n i (z,ω) =
-        (if dyadicTime T n i < τ (z,ω) then
-          greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) -
-        (if dyadicTime T n i < σ (z,ω) then
-          greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) := by
-      dsimp only [C, F]
-      rw [Real.toNNReal_coe]
-      by_cases ht : dyadicTime T n i < τ (z,ω)
-      · rw [hψeq _ _ (hinside z ω _ ht.le).1]
-        by_cases hs : σ (z,ω) ≤ dyadicTime T n i
-        · simp [ht, hs, not_lt.mpr hs]
-        · simp [ht, hs, lt_of_not_ge hs]
-      · have hs : ¬ dyadicTime T n i < σ (z,ω) := fun h => ht (h.trans_le (hστ (z,ω)))
-        simp [ht, hs]
-    have hs (n : ℕ) : driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z,ω)) (2^n) ω =
-        (∑ i ∈ Finset.range (2^n),
-          (if dyadicTime T n i < τ (z,ω) then
-            greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) *
-            (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) -
-        (∑ i ∈ Finset.range (2^n),
-          (if dyadicTime T n i < σ (z,ω) then
-            greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) *
-            (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) := by
-      simp only [driverPredictableSum, he, sub_mul, Finset.sum_sub_distrib]
-    simpa only [hs, Y, sub_sub_sub_cancel_right] using hτl.sub hσl
-  let Q (ω : Ω) := ∫ t in (0 : ℝ)..(T : ℝ),
-    (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then F (t,(z,ω)) else 0 ∂ν)^2
-  have hE : ∀ᵐ ω ∂P, Tendsto (fun n => driverPredictableEnergy (dyadicTime T n)
-      (fun i ω => ∫ z, C n i (z,ω) ∂ν) (2^n) ω) atTop (𝓝 (Q ω)) := by
-    apply Eventually.of_forall
-    intro ω
-    have hfm : Measurable (fun p : ℝ × Z => F (p.1,(p.2,ω))) :=
-      hFm.comp (measurable_fst.prodMk (measurable_snd.prodMk measurable_const))
-    have hsm := hσm.comp (measurable_id.prodMk (measurable_const (a := ω)))
-    have htm := hτm.comp (measurable_id.prodMk (measurable_const (a := ω)))
-    have hh := area_window_pair_energy_tendsto (ν := ν) hfm (fun z => hFc z ω) hsm htm T
-      (fun z => hτb z ω) K K.coe_nonneg (fun t z => hFb t z ω)
-    rw [area_window_energy_identity hfm hsm htm T (fun z => hτb z ω) K K.coe_nonneg
-      (fun t z => hFb t z ω)] at hh
-    simpa only [driverPredictableEnergy, dyadicTime_step, C, areaWindowSample, Q, Function.comp_def, id_eq] using hh
-  have hh := brownian_area_exponential_limit hB hm T C hCm hCa K K.coe_nonneg hCb Y hYm hlim Q hE θ
-  have hQeq (ω : Ω) : Q ω = ∫ t in (0 : ℝ)..(T : ℝ),
-      (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
-        greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2 := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    rw [uIcc_of_le T.coe_nonneg] at ht
-    dsimp only []
-    apply congrArg (fun x : ℝ => x^2)
-    apply integral_congr_ae
-    apply Eventually.of_forall
-    intro z
-    dsimp only []
-    split_ifs with htτ
-    · have htn : t.toNNReal ≤ τ (z,ω) := Real.toNNReal_le_iff_le_coe.mpr htτ.2.le
-      dsimp only [F]
-      have he := hψeq _ _ (hinside z ω _ htn).1
-      simpa only [Real.coe_toNNReal _ ht.1] using he
-    · rfl
-  simpa only [Y, hQeq] using hh
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable BrownianField
-
- 
-
-theorem clipped_spatial_green_exponential
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (κ lo hi : ℝ) (hle : lo ≤ hi) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (hq : Measurable q)
-    (R : ℝ) (hR : 0 ≤ R) (T : ℝ≥0)
-    (τ : Z × Ω → ℝ≥0) (hτm : Measurable τ)
-    (hτa : ∀ t : ℝ≥0, @MeasurableSet (Z × Ω)
-      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
-      {p | t < τ p})
-    (hτs : ∀ z, IsStoppingTime (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
-      (fun ω => (τ (z, ω) : WithTop ℝ≥0)))
-    (hτb : ∀ z ω, τ (z,ω) ≤ T)
-    (hinside : ∀ z ω t, t ≤ τ (z,ω) →
-      ‖((c (q z)).Q t ω, B t ω)‖ ≤ R ∧
-        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi)
-    (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp
-      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
-        greenLogDensity κ (q z) 0 ∂ν) -
-      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
-        (∫ z, if t < (τ (z,ω) : ℝ) then
-          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
-  classical
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
-  let ψ := spatialPartial φ 1
-  have hψc : Continuous ψ := (spatialPartial_smooth φ hφ 1).continuous
-  obtain ⟨K, hK, _⟩ := compact_smooth_bounded_lipschitz ψ
-    (spatialPartial_smooth φ hφ 1) (spatialPartial_compact φ hφc 1)
-  have hψeq (q' : ℂ × ℝ) (x : ℝ) (hb : ‖(q',x)‖ ≤ R) :
-      ψ (q',x) = greenLogCoefficient κ q' x := by
-    rw [show ψ (q',x) = deriv (fun y => φ (q',y)) x by
-      simp only [ψ, spatialPartial_eq_iteratedDeriv φ hφ, iteratedDeriv_one]]
-    rw [deriv_fiber_fderiv φ (hφ.differentiable (by simp)), hφD q' x hb,
-      ← deriv_fiber_fderiv _ ((greenLogDensity_contDiff κ).differentiable (by simp))]
-    exact (greenLogDensity_spatial_derivative κ q' x).deriv
-  let F (p : ℝ × (Z × Ω)) := ψ ((c (q p.2.1)).Q p.1.toNNReal p.2.2, B p.1.toNNReal p.2.2)
-  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable hBc hm
-  have hQm := clipped_family_joint_measurable c
-  have hFm : Measurable F := hψc.measurable.comp
-    ((hQm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
-      ((hq.comp (measurable_fst.comp measurable_snd)).prodMk (measurable_snd.comp measurable_snd)))).prodMk
-    (hBm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk (measurable_snd.comp measurable_snd))))
-  have hFc (z : Z) (ω : Ω) : Continuous (fun t : ℝ => F (t,(z,ω))) :=
-    hψc.comp ((((c (q z)).lipschitz ω).continuous.comp
-      (NNReal.continuous_coe.comp continuous_real_toNNReal)).prodMk
-      ((hBc ω).comp continuous_real_toNNReal))
-  have hFb (t : ℝ) (z : Z) (ω : Ω) : |F (t,(z,ω))| ≤ K := by
-    simpa only [Real.norm_eq_abs] using hK ((c (q z)).Q t.toNNReal ω, B t.toNNReal ω)
-  let C (n i : ℕ) (p : Z × Ω) := if dyadicTime T n i < τ p then
-    F (dyadicTime T n i,p) else 0
-  have hCa (n i : ℕ) : @Measurable (Z × Ω) ℝ
-      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)))
-      (borel ℝ) (C n i) := by
-    have hQ := clipped_family_past_measurable c (dyadicTime T n i)
-    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted (dyadicTime T n i)
-    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)
-    have hh := hψc.measurable.comp
-      ((hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)).prodMk (hBa.comp measurable_snd))
-    change Measurable (C n i)
-    dsimp only [C, F]
-    simp only [Real.toNNReal_coe]
-    exact hh.ite (hτa (dyadicTime T n i)) measurable_const
-  have hCm (n i : ℕ) : Measurable (C n i) :=
-    (hCa n i).mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (Filtration.le _ _))) le_rfl
-  have hCb (n i : ℕ) (z : Z) (ω : Ω) : |C n i (z,ω)| ≤ K := by
-    dsimp only [C]
-    split_ifs
-    · exact hFb _ _ _
-    · simp
-  let Y (p : Z × Ω) := greenLogDensity κ ((c (q p.1)).Q (τ p) p.2) (B (τ p) p.2) -
-    greenLogDensity κ (q p.1) 0
-  have hYm : Measurable Y :=
-    (((greenLogDensity_contDiff κ).continuous.measurable.comp
-      ((hQm.comp (hτm.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
-        (hBm.comp (hτm.prodMk measurable_snd))))).sub
-      ((greenLogDensity_contDiff κ).continuous.measurable.comp ((hq.comp measurable_fst).prodMk measurable_const))
-  have hlim (z : Z) : ∀ᵐ ω ∂P, Tendsto
-      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z,ω)) (2^n) ω)
-      atTop (𝓝 (Y (z,ω))) := by
-    have hh := (c (q z)).stopped_green_ito hB hBc hκ hκ8 R hR T
-      (fun ω => τ (z,ω)) (hτs z) (hτb z) (hinside z)
-    have he (n i : ℕ) (ω : Ω) : C n i (z,ω) = if dyadicTime T n i < τ (z,ω) then
-        greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0 := by
-      dsimp only [C, F]
-      rw [Real.toNNReal_coe]
-      split_ifs with ht
-      · exact hψeq _ _ (hinside z ω _ ht.le).1
-      · rfl
-    simpa only [driverPredictableSum, he, Y] using hh
-  let Q (ω : Ω) := ∫ t in (0 : ℝ)..(T : ℝ),
-    (∫ z, if t < (τ (z,ω) : ℝ) then F (t,(z,ω)) else 0 ∂ν)^2
-  have hE : ∀ᵐ ω ∂P, Tendsto (fun n => driverPredictableEnergy (dyadicTime T n)
-      (fun i ω => ∫ z, C n i (z,ω) ∂ν) (2^n) ω) atTop (𝓝 (Q ω)) := by
-    apply Eventually.of_forall
-    intro ω
-    have hfm : Measurable (fun p : ℝ × Z => F (p.1,(p.2,ω))) :=
-      hFm.comp (measurable_fst.prodMk (measurable_snd.prodMk measurable_const))
-    have htm := hτm.comp (measurable_id.prodMk (measurable_const (a := ω)))
-    have hh := area_prefix_energy_tendsto (ν := ν) hfm (fun z => hFc z ω) htm T
-      (fun z => hτb z ω) K K.coe_nonneg (fun t z => hFb t z ω)
-    rw [area_prefix_energy_identity hfm htm T (fun z => hτb z ω) K K.coe_nonneg
-      (fun t z => hFb t z ω)] at hh
-    simpa only [driverPredictableEnergy, dyadicTime_step, C, areaPrefixSample, Q, Function.comp_def, id_eq] using hh
-  have hh := brownian_area_exponential_limit hB hm T C hCm hCa K K.coe_nonneg hCb Y hYm hlim Q hE θ
-  have hQeq (ω : Ω) : Q ω = ∫ t in (0 : ℝ)..(T : ℝ),
-      (∫ z, if t < (τ (z,ω) : ℝ) then
-        greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2 := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    rw [uIcc_of_le T.coe_nonneg] at ht
-    dsimp only []
-    apply congrArg (fun x : ℝ => x^2)
-    apply integral_congr_ae
-    apply Eventually.of_forall
-    intro z
-    dsimp only []
-    split_ifs with htτ
-    · have htn : t.toNNReal ≤ τ (z,ω) := Real.toNNReal_le_iff_le_coe.mpr htτ.le
-      dsimp only [F]
-      have he := hψeq _ _ (hinside z ω _ htn).1
-      simpa only [Real.coe_toNNReal _ ht.1] using he
-    · rfl
-  simpa only [Y, hQeq] using hh
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-def spatialFiltration {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    (f : Filtration ℝ≥0 mΩ) : Filtration ℝ≥0 (mZ.prod mΩ) where
-  seq t := mZ.prod (f t)
-  mono' _ _ h := sup_le_sup le_rfl (MeasurableSpace.comap_mono (f.mono h))
-  le' t := sup_le_sup le_rfl (MeasurableSpace.comap_mono (f.le t))
-
-lemma spatial_stopping_section {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {f : Filtration ℝ≥0 mΩ} {τ : Z × Ω → WithTop ℝ≥0}
-    (hτ : IsStoppingTime (spatialFiltration (Z := Z) f) τ) (z : Z) :
-    IsStoppingTime f (fun ω => τ (z,ω)) := by
-  intro t
-  have h := hτ t
-  let : MeasurableSpace Ω := f t
-  exact h.preimage (measurable_const.prodMk measurable_id)
-
- 
-
-structure SpatialLogLocalization
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi}
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (R : ℝ) (T : ℝ≥0) where
-  τ : Z × Ω → ℝ≥0
-  stopping : IsStoppingTime
-    (spatialFiltration (Z := Z) (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
-    (fun p => (τ p : WithTop ℝ≥0))
-  bound : ∀ p, τ p ≤ T
-  inside : ∀ z ω t, t ≤ τ (z,ω) → ‖((c (q z)).Q t ω, B t ω)‖ ≤ R ∧
-    lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi
-  boundary : ∀ z ω, τ (z,ω) < T →
-    ‖((c (q z)).Q (τ (z,ω)) ω, B (τ (z,ω)) ω)‖ = R ∨
-      ((c (q z)).Q (τ (z,ω)) ω).1.im = lo ∨ ((c (q z)).Q (τ (z,ω)) ω).1.im = hi
-
- 
-
-theorem exists_spatial_log_localization
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ lo hi : ℝ) (hle : lo ≤ hi)
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (hq : Measurable q) (R : ℝ) (T : ℝ≥0)
-    (hinit : ∀ z, ‖(q z, (0 : ℝ))‖ ≤ R ∧ lo ≤ (q z).1.im ∧ (q z).1.im ≤ hi) :
-    Nonempty (SpatialLogLocalization c q R T) := by
-  let f := spatialFiltration (Z := Z) (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
-  let D (t : ℝ≥0) (p : Z × Ω) := min (R - ‖((c (q p.1)).Q t p.2, B t p.2)‖)
-    (min (((c (q p.1)).Q t p.2).1.im - lo) (hi - ((c (q p.1)).Q t p.2).1.im))
-  have hDa : Adapted f D := by
-    intro t
-    have hQ := clipped_family_past_measurable c t
-    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted t
-    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) t
-    have hc := hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)
-    have hBb := hBa.comp (@measurable_snd Z Ω mZ _)
-    exact (measurable_const.sub (hc.prodMk hBb).norm).min
-      (((Complex.continuous_im.measurable.comp hc.fst).sub_const lo).min
-        (measurable_const.sub (Complex.continuous_im.measurable.comp hc.fst)))
-  have hDc (p : Z × Ω) : Continuous (fun t => D t p) := by
-    have hc := ((c (q p.1)).lipschitz p.2).continuous.comp NNReal.continuous_coe
-    exact (continuous_const.sub (hc.prodMk (hBc p.2)).norm).min
-      (((Complex.continuous_im.comp hc.fst).sub continuous_const).min
-        (continuous_const.sub (Complex.continuous_im.comp hc.fst)))
-  have hD0 (p : Z × Ω) : 0 ≤ D 0 p := by
-    simpa only [D, NNReal.coe_zero, (c (q p.1)).initial, hB0, le_min_iff, sub_nonneg]
-      using hinit p.1
-  obtain ⟨τ, hτ, hb, hin, hpre, hboundary⟩ := first_margin_stopping f D hDa hDc hD0 T
-  refine ⟨⟨τ, hτ, hb, ?_, ?_⟩⟩
-  · intro z ω t ht
-    simpa only [D, le_min_iff, sub_nonneg] using hin (z,ω) t ht
-  · intro z ω ht
-    have hh := hboundary (z,ω) ht
-    have hle' := hin (z,ω) (τ (z,ω)) le_rfl
-    dsimp only [D] at hh hle'
-    rw [min_eq_iff, min_eq_iff] at hh
-    rcases hh with ⟨h, _⟩ | ⟨h, _⟩
-    · exact Or.inl (sub_eq_zero.mp h).symm
-    · rcases h with ⟨h, _⟩ | ⟨h, _⟩
-      · exact Or.inr (Or.inl (sub_eq_zero.mp h))
-      · exact Or.inr (Or.inr (sub_eq_zero.mp h).symm)
-
-namespace SpatialLogLocalization
-variable {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-  {κ lo hi : ℝ} {hle : lo ≤ hi}
-  {c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q}
-  {q : Z → ℂ × ℝ} {R : ℝ} {T : ℝ≥0}
-
-lemma measurable (a : SpatialLogLocalization c q R T) : Measurable a.τ :=
-  stopping_time_measurable_nnreal a.stopping
-
-lemma exponential {P : Measure Ω} {ν : Measure Z} [IsFiniteMeasure ν]
-    (a : SpatialLogLocalization c q R T) (hq : Measurable q)
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (hκ8 : κ < 8) (hR : 0 ≤ R) (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp
-      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (a.τ (z,ω)) ω) (B (a.τ (z,ω)) ω) -
-        greenLogDensity κ (q z) 0 ∂ν) -
-      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
-        (∫ z, if t < (a.τ (z,ω) : ℝ) then
-          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
-  exact clipped_spatial_green_exponential hm hB hBc κ lo hi hle hκ hκ8 c q hq R hR T
-    a.τ a.measurable (stopping_before_measurable a.stopping)
-    (spatial_stopping_section a.stopping) (fun z ω => a.bound (z,ω)) a.inside θ
-
-end SpatialLogLocalization
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma greenLogCoefficient_logCoordinates
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (κ b : ℝ) (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t)
-    (hUb : U t = Real.sqrt κ * b) :
-    greenLogCoefficient κ (logLoewnerCoordinates G z t) b =
-      greenBrownianCoefficient κ U γ G t z := by
-  rw [greenLogCoefficient, greenLogDensity_logCoordinates hU h κ b z t hz hUb,
-    greenLogNormSq_logCoordinates κ b G z t (h.imag_positive t z hz),
-    logCoordinates_re, Complex.normSq_eq_norm_sq, greenBrownianCoefficient, hUb]
-  simp only [Complex.sub_re, Complex.ofReal_re]
-  ring
-
-lemma clipped_chart_identification
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi}
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (ω : Ω) (τ : ℝ≥0)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F)
-    (hin : ∀ t : ℝ≥0, t ≤ τ →
-      lo ≤ ((c (initialLogCoordinates z)).Q t ω).1.im ∧
-        ((c (initialLogCoordinates z)).Q t ω).1.im ≤ hi) :
-    z ∈ survivingDomain γ τ ∧ ∀ t : ℝ≥0, t ≤ τ →
-      (c (initialLogCoordinates z)).Q t ω = logLoewnerCoordinates G z t := by
-  let d := c (initialLogCoordinates z)
-  have hd (s : ℝ) (hs : s ∈ Icc 0 (τ : ℝ)) := (d.derivative ω s hs.1).mono
-    (show Icc (0 : ℝ) (τ : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-  have hb (s : ℝ) (hs : s ∈ Icc 0 (τ : ℝ)) : (d.Q s ω).1.im ∈ Icc lo hi := by
-    have hh := hin s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)
-    simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
-  obtain ⟨hl, he⟩ := clipped_log_identification ((hBc ω).const_mul _) hw z hz τ
-    lo hi hle (fun s => d.Q s ω) (d.initial ω) hd hb
-  exact ⟨hl, fun t ht => by simpa only [Real.toNNReal_coe] using he ⟨t.coe_nonneg, ht⟩⟩
-
-lemma SpatialLogLocalization.chart_identification
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi}
-    {c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q}
-    {z : Z → ℂ} {R : ℝ} {T : ℝ≥0}
-    (a : SpatialLogLocalization c (fun x => initialLogCoordinates (z x)) R T)
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (x : Z) (hz : z x ∈ upperHalfPlane) (ω : Ω)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F) :
-    z x ∈ survivingDomain γ (a.τ (x,ω)) ∧
-      ∀ t : ℝ≥0, t ≤ a.τ (x,ω) →
-        (c (initialLogCoordinates (z x))).Q t ω = logLoewnerCoordinates G (z x) t := by
-  let d := c (initialLogCoordinates (z x))
-  have hd (s : ℝ) (hs : s ∈ Icc 0 (a.τ (x,ω) : ℝ)) := (d.derivative ω s hs.1).mono
-    (show Icc (0 : ℝ) (a.τ (x,ω) : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-  have hb (s : ℝ) (hs : s ∈ Icc 0 (a.τ (x,ω) : ℝ)) : (d.Q s ω).1.im ∈ Icc lo hi := by
-    have hh := (a.inside x ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2
-    simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
-  obtain ⟨hl, he⟩ := clipped_log_identification ((hBc ω).const_mul _) hw (z x) hz (a.τ (x,ω))
-    lo hi hle (fun s => d.Q s ω) (d.initial ω) hd hb
-  exact ⟨hl, fun t ht => by simpa only [Real.toNNReal_coe] using he ⟨t.coe_nonneg, ht⟩⟩
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma stopped_difference_max {T E : Type*} [LinearOrder T] [AddGroup E]
-    (M : T → E) (s t : T) :
-    M t - M (min t s) = M (max s t) - M s := by
-  by_cases ht : t ≤ s
-  · rw [min_eq_left ht, max_eq_left ht, sub_self, sub_self]
-  · rw [min_eq_right (le_of_not_ge ht), max_eq_right (le_of_not_ge ht)]
-
- 
-
-theorem closed_stop_limit_condExp
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    (𝓕 : Filtration ℝ≥0 mΩ) (M : ℝ≥0 → Ω → ℝ) (Z W : Ω → ℝ)
-    (hZ : Integrable Z μ)
-    (hStop : ∀ (ρ : Ω → ℝ≥0) (hρ : IsStoppingTime 𝓕 (fun ω => (ρ ω : WithTop ℝ≥0))),
-      (fun ω => M (ρ ω) ω) =ᵐ[μ] μ[Z | hρ.measurableSpace])
-    (σ : ℝ≥0 → Ω → ℝ≥0)
-    (hσ : ∀ t, IsStoppingTime 𝓕 (fun ω => (σ t ω : WithTop ℝ≥0)))
-    (hσle : ∀ t ω, σ t ω ≤ t)
-    (hσmin : ∀ s t, s ≤ t → ∀ ω, σ s ω = min (σ t ω) s)
-    (K : ℝ) (hbound : ∀ t ω, ‖M t ω‖ ≤ K)
-    (hlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (σ n ω) ω) atTop (𝓝 (W ω)))
-    (s : ℝ≥0) : (fun ω => M (σ s ω) ω) =ᵐ[μ] μ[W | 𝓕 s] := by
-  have hInt (ρ : Ω → ℝ≥0) (hρ : IsStoppingTime 𝓕 (fun ω => (ρ ω : WithTop ℝ≥0))) :
-      Integrable (fun ω => M (ρ ω) ω) μ :=
-    integrable_condExp.congr (hStop ρ hρ).symm
-  have hWi : Integrable W μ := by
-    apply Integrable.of_bound (aestronglyMeasurable_of_tendsto_ae atTop
-      (fun n : ℕ => (hInt (σ n) (hσ n)).aestronglyMeasurable) hlim) K
-    filter_upwards [hlim] with ω hω
-    exact le_of_tendsto hω.norm (Eventually.of_forall fun n => hbound (σ n ω) ω)
-  have hss : (hσ s).measurableSpace ≤ 𝓕 s :=
-    (hσ s).measurableSpace_le_of_le_const (fun ω => by exact_mod_cast hσle s ω)
-  have hMs : AEStronglyMeasurable[𝓕 s] (fun ω => M (σ s ω) ω) μ :=
-    ((stronglyMeasurable_condExp.mono hss).aestronglyMeasurable).congr (hStop (σ s) (hσ s)).symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq (𝓕.le s) hWi
-    (fun _ _ _ => (hInt (σ s) (hσ s)).integrableOn) _ hMs
-  intro A hA _
-  have hI : Tendsto (fun n : ℕ => ∫ ω in A, M (σ n ω) ω ∂μ) atTop (𝓝 (∫ ω in A, W ω ∂μ)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => K)
-      (fun n => (hInt (σ n) (hσ n)).integrableOn.aestronglyMeasurable)
-      (integrable_const _) _ (ae_restrict_of_ae hlim)
-    intro n
-    exact Eventually.of_forall fun ω => hbound (σ n ω) ω
-  have he : ∀ᶠ n : ℕ in atTop, (∫ ω in A, M (σ n ω) ω ∂μ) =
-      ∫ ω in A, M (σ s ω) ω ∂μ := by
-    filter_upwards [eventually_ge_atTop (⌈(s : ℝ)⌉₊)] with n hn
-    have hsn : s ≤ (n : ℝ≥0) := by
-      exact_mod_cast (Nat.le_of_ceil_le hn)
-    let ρ (ω : Ω) := max s (σ n ω)
-    have hρ : IsStoppingTime 𝓕 (fun ω => (ρ ω : WithTop ℝ≥0)) := by
-      simpa only [ρ, WithTop.coe_max] using (isStoppingTime_const 𝓕 s).max (hσ n)
-    have hFρ : 𝓕 s ≤ hρ.measurableSpace :=
-      hρ.le_measurableSpace_of_const_le (fun ω => by exact_mod_cast (le_max_left s (σ n ω)))
-    have hFc : 𝓕 s ≤ (isStoppingTime_const 𝓕 s).measurableSpace :=
-      (isStoppingTime_const 𝓕 s).le_measurableSpace_of_const_le (fun _ => le_rfl)
-    have hsetρ : (∫ ω in A, M (ρ ω) ω ∂μ) = ∫ ω in A, Z ω ∂μ := by
-      rw [integral_congr_ae (ae_restrict_of_ae (hStop ρ hρ))]
-      have : IsFiniteMeasure (μ.trim hρ.measurableSpace_le) := isFiniteMeasure_trim _
-      exact setIntegral_condExp hρ.measurableSpace_le hZ (hFρ A hA)
-    have hsetc : (∫ ω in A, M s ω ∂μ) = ∫ ω in A, Z ω ∂μ := by
-      rw [integral_congr_ae (ae_restrict_of_ae (hStop (fun _ => s) (isStoppingTime_const 𝓕 s)))]
-      have : IsFiniteMeasure (μ.trim (isStoppingTime_const 𝓕 s).measurableSpace_le) := isFiniteMeasure_trim _
-      exact setIntegral_condExp (isStoppingTime_const 𝓕 s).measurableSpace_le hZ (hFc A hA)
-    have hid : (∫ ω in A, M (σ n ω) ω - M (σ s ω) ω ∂μ) =
-        ∫ ω in A, M (ρ ω) ω - M s ω ∂μ := by
-      apply integral_congr_ae
-      exact Eventually.of_forall fun ω => by
-        change M (σ n ω) ω - M (σ s ω) ω = M (ρ ω) ω - M s ω
-        rw [hσmin s n hsn ω]
-        exact stopped_difference_max (fun t => M t ω) s (σ n ω)
-    rw [integral_sub (hInt (σ n) (hσ n)).integrableOn (hInt (σ s) (hσ s)).integrableOn,
-      integral_sub (hInt ρ hρ).integrableOn
-        (hInt (fun _ => s) (isStoppingTime_const 𝓕 s)).integrableOn, hsetρ, hsetc, sub_self] at hid
-    exact sub_eq_zero.mp hid
-  have hc : Tendsto (fun n : ℕ => ∫ ω in A, M (σ n ω) ω ∂μ) atTop
-      (𝓝 (∫ ω in A, M (σ s ω) ω ∂μ)) := by
-    apply tendsto_const_nhds.congr'
-    filter_upwards [he] with n hn using hn.symm
-  exact (tendsto_nhds_unique hI hc).symm
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma boundedSpatialHit_le_time (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (t : ℝ≥0) :
-    boundedSpatialHit γ z r t ≤ t := by
-  apply WithTop.coe_le_coe.mp
-  rw [boundedSpatialHit_coe]
-  exact min_le_right _ _
-
-lemma boundedSpatialHit_min (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ)
-    (s t : ℝ≥0) (hst : s ≤ t) :
-    boundedSpatialHit γ z r s = min (boundedSpatialHit γ z r t) s := by
-  apply WithTop.coe_injective
-  rw [WithTop.coe_min, boundedSpatialHit_coe, boundedSpatialHit_coe, min_assoc,
-    min_eq_right (WithTop.coe_le_coe.mpr hst)]
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
-lemma spatialGreen_condExp_of_level [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
-    (u : ℝ) (hu : 0 ≤ u) (hlevel : (2 * z.im) * Real.exp (-u) < r)
-    (s : ℝ≥0) :
-    (fun ω : a.samples => greenDensity κ (fun t => Real.sqrt κ * a.B t ω)
-      (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r s) z) =ᵐ[P.comap Subtype.val]
-      (P.comap Subtype.val)[fun ω : a.samples => spatialGreenWeight κ
-        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r | a.filtration s] := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
-  obtain ⟨M, Z, hM, hZ, hMb, hZb, hClosed, hStop, heq, hNoHit⟩ :=
-    exists_physical_green_stopping_regular a.brownian hBm a.continuous_B a.zero_B
-      κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
-  let σ (t : ℝ≥0) (ω : a.samples) := boundedSpatialHit (γ ω) z r t
-  have hσ (t : ℝ≥0) : IsStoppingTime a.filtration (fun ω => (σ t ω : WithTop ℝ≥0)) :=
-    a.boundedSpatialHit_stopping z r t
-  dsimp only [filtration] at hσ
-  let W (ω : a.samples) := spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-    (γ ω) (a.G ω) z r
-  have hlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (σ n ω) ω) atTop (𝓝 (W ω)) := by
-    filter_upwards [heq, hNoHit] with ω he hn
-    simpa only [he.1, σ, W, Function.comp_apply] using
-      (a.witness ω).spatialGreen_limit ((a.continuous_B ω).const_mul _) κ z r hr hy u hlevel hn
-  have hb (t : ℝ≥0) (ω : a.samples) :
-      ‖M t ω‖ ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) := by
-    rw [Real.norm_eq_abs, abs_of_nonneg (hMb t ω).1]
-    exact (hMb t ω).2
-  have hc := closed_stop_limit_condExp a.filtration M Z W hZ hStop σ hσ
-    (fun t ω => boundedSpatialHit_le_time (γ ω) z r t)
-    (fun t v htv ω => boundedSpatialHit_min (γ ω) z r t v htv) _ hb hlim s
-  refine EventuallyEq.trans ?_ hc
-  filter_upwards [heq] with ω he
-  rw [he.1]
-  exact ((a.witness ω).physicalGreen_eq_before_spatial
-    ((a.continuous_B ω).const_mul _) κ z r hr hy u hlevel (σ s ω)
-    (boundedSpatialHit_le (γ ω) z r s)).symm
-
- 
-
-theorem spatialGreen_condExp [IsProbabilityMeasure P]
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im) (s : ℝ≥0) :
-    (fun ω : a.samples => greenDensity κ (fun t => Real.sqrt κ * a.B t ω)
-      (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r s) z) =ᵐ[P.comap Subtype.val]
-      (P.comap Subtype.val)[fun ω : a.samples => spatialGreenWeight κ
-        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r | a.filtration s] := by
-  have hz : z ∈ upperHalfPlane := hr.trans_le hy
-  let u := Real.log (4 * z.im / r)
-  have hratio : 0 < 4 * z.im / r := div_pos (mul_pos (by norm_num) hz) hr
-  have hu : 0 ≤ u := Real.log_nonneg ((le_div_iff₀ hr).mpr (by linarith))
-  have hlev : (2 * z.im) * Real.exp (-u) = r / 2 := by
-    dsimp only [u]
-    rw [Real.exp_neg, Real.exp_log hratio]
-    field_simp [(show 0 < z.im from hz).ne']
-    ring
-  exact a.spatialGreen_condExp_of_level hκ hκ8 z hz r hr hy u hu
-    (by rw [hlev]; linarith) s
-
-end RegularSLEPresentation
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.ExponentialMartingales
+
+namespace OAI
+
+/-! Stochastic Fubini for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField
+theorem clipped_spatial_green_window_exponential
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ lo hi : ℝ) (hle : lo ≤ hi) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (hq : Measurable q)
+    (R : ℝ) (hR : 0 ≤ R) (T : ℝ≥0)
+    (σ τ : Z × Ω → ℝ≥0) (hσm : Measurable σ) (hτm : Measurable τ)
+    (hστ : ∀ p, σ p ≤ τ p)
+    (hσa : ∀ t : ℝ≥0, @MeasurableSet (Z × Ω)
+      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
+      {p | t < σ p})
+    (hσs : ∀ z, IsStoppingTime (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
+      (fun ω => (σ (z, ω) : WithTop ℝ≥0)))
+    (hτa : ∀ t : ℝ≥0, @MeasurableSet (Z × Ω)
+      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
+      {p | t < τ p})
+    (hτs : ∀ z, IsStoppingTime (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
+      (fun ω => (τ (z, ω) : WithTop ℝ≥0)))
+    (hτb : ∀ z ω, τ (z,ω) ≤ T)
+    (hinside : ∀ z ω t, t ≤ τ (z,ω) →
+      ‖((c (q z)).Q t ω, B t ω)‖ ≤ R ∧
+        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi)
+    (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp
+      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
+        greenLogDensity κ ((c (q z)).Q (σ (z,ω)) ω) (B (σ (z,ω)) ω) ∂ν) -
+      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
+        (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
+          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
+  classical
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
+  let ψ := spatialPartial φ 1
+  have hψc : Continuous ψ := (spatialPartial_smooth φ hφ 1).continuous
+  obtain ⟨K, hK, _⟩ := compact_smooth_bounded_lipschitz ψ
+    (spatialPartial_smooth φ hφ 1) (spatialPartial_compact φ hφc 1)
+  have hψeq (q' : ℂ × ℝ) (x : ℝ) (hb : ‖(q',x)‖ ≤ R) :
+      ψ (q',x) = greenLogCoefficient κ q' x := by
+    rw [show ψ (q',x) = deriv (fun y => φ (q',y)) x by
+      simp only [ψ, spatialPartial_eq_iteratedDeriv φ hφ, iteratedDeriv_one]]
+    rw [deriv_fiber_fderiv φ (hφ.differentiable (by simp)), hφD q' x hb,
+      ← deriv_fiber_fderiv _ ((greenLogDensity_contDiff κ).differentiable (by simp))]
+    exact (greenLogDensity_spatial_derivative κ q' x).deriv
+  let F (p : ℝ × (Z × Ω)) := ψ ((c (q p.2.1)).Q p.1.toNNReal p.2.2, B p.1.toNNReal p.2.2)
+  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable hBc hm
+  have hQm := clipped_family_joint_measurable c
+  have hFm : Measurable F := hψc.measurable.comp
+    ((hQm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
+      ((hq.comp (measurable_fst.comp measurable_snd)).prodMk (measurable_snd.comp measurable_snd)))).prodMk
+    (hBm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk (measurable_snd.comp measurable_snd))))
+  have hFc (z : Z) (ω : Ω) : Continuous (fun t : ℝ => F (t,(z,ω))) :=
+    hψc.comp ((((c (q z)).lipschitz ω).continuous.comp
+      (NNReal.continuous_coe.comp continuous_real_toNNReal)).prodMk
+      ((hBc ω).comp continuous_real_toNNReal))
+  have hFb (t : ℝ) (z : Z) (ω : Ω) : |F (t,(z,ω))| ≤ K := by
+    simpa only [Real.norm_eq_abs] using hK ((c (q z)).Q t.toNNReal ω, B t.toNNReal ω)
+  let C (n i : ℕ) (p : Z × Ω) := if σ p ≤ dyadicTime T n i ∧ dyadicTime T n i < τ p then
+    F (dyadicTime T n i,p) else 0
+  have hCa (n i : ℕ) : @Measurable (Z × Ω) ℝ
+      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)))
+      (borel ℝ) (C n i) := by
+    have hQ := clipped_family_past_measurable c (dyadicTime T n i)
+    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted (dyadicTime T n i)
+    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)
+    have hh := hψc.measurable.comp
+      ((hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)).prodMk (hBa.comp measurable_snd))
+    change Measurable (C n i)
+    dsimp only [C, F]
+    simp only [Real.toNNReal_coe]
+    have hsa : MeasurableSet {p : Z × Ω | σ p ≤ dyadicTime T n i} := by
+      simpa only [Set.compl_ofPred, not_lt] using (hσa (dyadicTime T n i)).compl
+    exact hh.ite (hsa.inter (hτa (dyadicTime T n i))) measurable_const
+  have hCm (n i : ℕ) : Measurable (C n i) :=
+    (hCa n i).mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (Filtration.le _ _))) le_rfl
+  have hCb (n i : ℕ) (z : Z) (ω : Ω) : |C n i (z,ω)| ≤ K := by
+    dsimp only [C]
+    split_ifs
+    · exact hFb _ _ _
+    · simp
+  let Y (p : Z × Ω) := greenLogDensity κ ((c (q p.1)).Q (τ p) p.2) (B (τ p) p.2) -
+    greenLogDensity κ ((c (q p.1)).Q (σ p) p.2) (B (σ p) p.2)
+  have hYm : Measurable Y :=
+    (((greenLogDensity_contDiff κ).continuous.measurable.comp
+      ((hQm.comp (hτm.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
+        (hBm.comp (hτm.prodMk measurable_snd))))).sub
+      ((greenLogDensity_contDiff κ).continuous.measurable.comp
+        ((hQm.comp (hσm.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
+          (hBm.comp (hσm.prodMk measurable_snd))))
+  have hlim (z : Z) : ∀ᵐ ω ∂P, Tendsto
+      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z,ω)) (2^n) ω)
+      atTop (𝓝 (Y (z,ω))) := by
+    have hτlim := (c (q z)).stopped_green_ito hB hBc hκ hκ8 R hR T
+      (fun ω => τ (z,ω)) (hτs z) (hτb z) (hinside z)
+    have hσlim := (c (q z)).stopped_green_ito hB hBc hκ hκ8 R hR T
+      (fun ω => σ (z,ω)) (hσs z) (fun ω => (hστ (z,ω)).trans (hτb z ω))
+      (fun ω t ht => hinside z ω t (ht.trans (hστ (z,ω))))
+    filter_upwards [hτlim, hσlim] with ω hτl hσl
+    have he (n i : ℕ) : C n i (z,ω) =
+        (if dyadicTime T n i < τ (z,ω) then
+          greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) -
+        (if dyadicTime T n i < σ (z,ω) then
+          greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) := by
+      dsimp only [C, F]
+      rw [Real.toNNReal_coe]
+      by_cases ht : dyadicTime T n i < τ (z,ω)
+      · rw [hψeq _ _ (hinside z ω _ ht.le).1]
+        by_cases hs : σ (z,ω) ≤ dyadicTime T n i
+        · simp [ht, hs, not_lt.mpr hs]
+        · simp [ht, hs, lt_of_not_ge hs]
+      · have hs : ¬ dyadicTime T n i < σ (z,ω) := fun h => ht (h.trans_le (hστ (z,ω)))
+        simp [ht, hs]
+    have hs (n : ℕ) : driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z,ω)) (2^n) ω =
+        (∑ i ∈ Finset.range (2^n),
+          (if dyadicTime T n i < τ (z,ω) then
+            greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) *
+            (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) -
+        (∑ i ∈ Finset.range (2^n),
+          (if dyadicTime T n i < σ (z,ω) then
+            greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0) *
+            (B (dyadicTime T n (i+1)) ω - B (dyadicTime T n i) ω)) := by
+      simp only [driverPredictableSum, he, sub_mul, Finset.sum_sub_distrib]
+    simpa only [hs, Y, sub_sub_sub_cancel_right] using hτl.sub hσl
+  let Q (ω : Ω) := ∫ t in (0 : ℝ)..(T : ℝ),
+    (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then F (t,(z,ω)) else 0 ∂ν)^2
+  have hE : ∀ᵐ ω ∂P, Tendsto (fun n => driverPredictableEnergy (dyadicTime T n)
+      (fun i ω => ∫ z, C n i (z,ω) ∂ν) (2^n) ω) atTop (𝓝 (Q ω)) := by
+    apply Eventually.of_forall
+    intro ω
+    have hfm : Measurable (fun p : ℝ × Z => F (p.1,(p.2,ω))) :=
+      hFm.comp (measurable_fst.prodMk (measurable_snd.prodMk measurable_const))
+    have hsm := hσm.comp (measurable_id.prodMk (measurable_const (a := ω)))
+    have htm := hτm.comp (measurable_id.prodMk (measurable_const (a := ω)))
+    have hh := area_window_pair_energy_tendsto (ν := ν) hfm (fun z => hFc z ω) hsm htm T
+      (fun z => hτb z ω) K K.coe_nonneg (fun t z => hFb t z ω)
+    rw [area_window_energy_identity hfm hsm htm T (fun z => hτb z ω) K K.coe_nonneg
+      (fun t z => hFb t z ω)] at hh
+    simpa only [driverPredictableEnergy, dyadicTime_step, C, areaWindowSample, Q, Function.comp_def, id_eq] using hh
+  have hh := brownian_area_exponential_limit hB hm T C hCm hCa K K.coe_nonneg hCb Y hYm hlim Q hE θ
+  have hQeq (ω : Ω) : Q ω = ∫ t in (0 : ℝ)..(T : ℝ),
+      (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
+        greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2 := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    rw [uIcc_of_le T.coe_nonneg] at ht
+    dsimp only []
+    apply congrArg (fun x : ℝ => x^2)
+    apply integral_congr_ae
+    apply Eventually.of_forall
+    intro z
+    dsimp only []
+    split_ifs with htτ
+    · have htn : t.toNNReal ≤ τ (z,ω) := Real.toNNReal_le_iff_le_coe.mpr htτ.2.le
+      dsimp only [F]
+      have he := hψeq _ _ (hinside z ω _ htn).1
+      simpa only [Real.coe_toNNReal _ ht.1] using he
+    · rfl
+  simpa only [Y, hQeq] using hh
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable BrownianField
+
+ 
+
+theorem clipped_spatial_green_exponential
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (κ lo hi : ℝ) (hle : lo ≤ hi) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (hq : Measurable q)
+    (R : ℝ) (hR : 0 ≤ R) (T : ℝ≥0)
+    (τ : Z × Ω → ℝ≥0) (hτm : Measurable τ)
+    (hτa : ∀ t : ℝ≥0, @MeasurableSet (Z × Ω)
+      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))
+      {p | t < τ p})
+    (hτs : ∀ z, IsStoppingTime (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
+      (fun ω => (τ (z, ω) : WithTop ℝ≥0)))
+    (hτb : ∀ z ω, τ (z,ω) ≤ T)
+    (hinside : ∀ z ω t, t ≤ τ (z,ω) →
+      ‖((c (q z)).Q t ω, B t ω)‖ ≤ R ∧
+        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi)
+    (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp
+      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
+        greenLogDensity κ (q z) 0 ∂ν) -
+      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
+        (∫ z, if t < (τ (z,ω) : ℝ) then
+          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
+  classical
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_green_field κ R hR
+  let ψ := spatialPartial φ 1
+  have hψc : Continuous ψ := (spatialPartial_smooth φ hφ 1).continuous
+  obtain ⟨K, hK, _⟩ := compact_smooth_bounded_lipschitz ψ
+    (spatialPartial_smooth φ hφ 1) (spatialPartial_compact φ hφc 1)
+  have hψeq (q' : ℂ × ℝ) (x : ℝ) (hb : ‖(q',x)‖ ≤ R) :
+      ψ (q',x) = greenLogCoefficient κ q' x := by
+    rw [show ψ (q',x) = deriv (fun y => φ (q',y)) x by
+      simp only [ψ, spatialPartial_eq_iteratedDeriv φ hφ, iteratedDeriv_one]]
+    rw [deriv_fiber_fderiv φ (hφ.differentiable (by simp)), hφD q' x hb,
+      ← deriv_fiber_fderiv _ ((greenLogDensity_contDiff κ).differentiable (by simp))]
+    exact (greenLogDensity_spatial_derivative κ q' x).deriv
+  let F (p : ℝ × (Z × Ω)) := ψ ((c (q p.2.1)).Q p.1.toNNReal p.2.2, B p.1.toNNReal p.2.2)
+  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable hBc hm
+  have hQm := clipped_family_joint_measurable c
+  have hFm : Measurable F := hψc.measurable.comp
+    ((hQm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
+      ((hq.comp (measurable_fst.comp measurable_snd)).prodMk (measurable_snd.comp measurable_snd)))).prodMk
+    (hBm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk (measurable_snd.comp measurable_snd))))
+  have hFc (z : Z) (ω : Ω) : Continuous (fun t : ℝ => F (t,(z,ω))) :=
+    hψc.comp ((((c (q z)).lipschitz ω).continuous.comp
+      (NNReal.continuous_coe.comp continuous_real_toNNReal)).prodMk
+      ((hBc ω).comp continuous_real_toNNReal))
+  have hFb (t : ℝ) (z : Z) (ω : Ω) : |F (t,(z,ω))| ≤ K := by
+    simpa only [Real.norm_eq_abs] using hK ((c (q z)).Q t.toNNReal ω, B t.toNNReal ω)
+  let C (n i : ℕ) (p : Z × Ω) := if dyadicTime T n i < τ p then
+    F (dyadicTime T n i,p) else 0
+  have hCa (n i : ℕ) : @Measurable (Z × Ω) ℝ
+      (mZ.prod (Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)))
+      (borel ℝ) (C n i) := by
+    have hQ := clipped_family_past_measurable c (dyadicTime T n i)
+    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted (dyadicTime T n i)
+    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) (dyadicTime T n i)
+    have hh := hψc.measurable.comp
+      ((hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)).prodMk (hBa.comp measurable_snd))
+    change Measurable (C n i)
+    dsimp only [C, F]
+    simp only [Real.toNNReal_coe]
+    exact hh.ite (hτa (dyadicTime T n i)) measurable_const
+  have hCm (n i : ℕ) : Measurable (C n i) :=
+    (hCa n i).mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (Filtration.le _ _))) le_rfl
+  have hCb (n i : ℕ) (z : Z) (ω : Ω) : |C n i (z,ω)| ≤ K := by
+    dsimp only [C]
+    split_ifs
+    · exact hFb _ _ _
+    · simp
+  let Y (p : Z × Ω) := greenLogDensity κ ((c (q p.1)).Q (τ p) p.2) (B (τ p) p.2) -
+    greenLogDensity κ (q p.1) 0
+  have hYm : Measurable Y :=
+    (((greenLogDensity_contDiff κ).continuous.measurable.comp
+      ((hQm.comp (hτm.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
+        (hBm.comp (hτm.prodMk measurable_snd))))).sub
+      ((greenLogDensity_contDiff κ).continuous.measurable.comp ((hq.comp measurable_fst).prodMk measurable_const))
+  have hlim (z : Z) : ∀ᵐ ω ∂P, Tendsto
+      (fun n => driverPredictableSum B (dyadicTime T n) (fun i ω => C n i (z,ω)) (2^n) ω)
+      atTop (𝓝 (Y (z,ω))) := by
+    have hh := (c (q z)).stopped_green_ito hB hBc hκ hκ8 R hR T
+      (fun ω => τ (z,ω)) (hτs z) (hτb z) (hinside z)
+    have he (n i : ℕ) (ω : Ω) : C n i (z,ω) = if dyadicTime T n i < τ (z,ω) then
+        greenLogCoefficient κ ((c (q z)).Q (dyadicTime T n i) ω) (B (dyadicTime T n i) ω) else 0 := by
+      dsimp only [C, F]
+      rw [Real.toNNReal_coe]
+      split_ifs with ht
+      · exact hψeq _ _ (hinside z ω _ ht.le).1
+      · rfl
+    simpa only [driverPredictableSum, he, Y] using hh
+  let Q (ω : Ω) := ∫ t in (0 : ℝ)..(T : ℝ),
+    (∫ z, if t < (τ (z,ω) : ℝ) then F (t,(z,ω)) else 0 ∂ν)^2
+  have hE : ∀ᵐ ω ∂P, Tendsto (fun n => driverPredictableEnergy (dyadicTime T n)
+      (fun i ω => ∫ z, C n i (z,ω) ∂ν) (2^n) ω) atTop (𝓝 (Q ω)) := by
+    apply Eventually.of_forall
+    intro ω
+    have hfm : Measurable (fun p : ℝ × Z => F (p.1,(p.2,ω))) :=
+      hFm.comp (measurable_fst.prodMk (measurable_snd.prodMk measurable_const))
+    have htm := hτm.comp (measurable_id.prodMk (measurable_const (a := ω)))
+    have hh := area_prefix_energy_tendsto (ν := ν) hfm (fun z => hFc z ω) htm T
+      (fun z => hτb z ω) K K.coe_nonneg (fun t z => hFb t z ω)
+    rw [area_prefix_energy_identity hfm htm T (fun z => hτb z ω) K K.coe_nonneg
+      (fun t z => hFb t z ω)] at hh
+    simpa only [driverPredictableEnergy, dyadicTime_step, C, areaPrefixSample, Q, Function.comp_def, id_eq] using hh
+  have hh := brownian_area_exponential_limit hB hm T C hCm hCa K K.coe_nonneg hCb Y hYm hlim Q hE θ
+  have hQeq (ω : Ω) : Q ω = ∫ t in (0 : ℝ)..(T : ℝ),
+      (∫ z, if t < (τ (z,ω) : ℝ) then
+        greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2 := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    rw [uIcc_of_le T.coe_nonneg] at ht
+    dsimp only []
+    apply congrArg (fun x : ℝ => x^2)
+    apply integral_congr_ae
+    apply Eventually.of_forall
+    intro z
+    dsimp only []
+    split_ifs with htτ
+    · have htn : t.toNNReal ≤ τ (z,ω) := Real.toNNReal_le_iff_le_coe.mpr htτ.le
+      dsimp only [F]
+      have he := hψeq _ _ (hinside z ω _ htn).1
+      simpa only [Real.coe_toNNReal _ ht.1] using he
+    · rfl
+  simpa only [Y, hQeq] using hh
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+def spatialFiltration {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    (f : Filtration ℝ≥0 mΩ) : Filtration ℝ≥0 (mZ.prod mΩ) where
+  seq t := mZ.prod (f t)
+  mono' _ _ h := sup_le_sup le_rfl (MeasurableSpace.comap_mono (f.mono h))
+  le' t := sup_le_sup le_rfl (MeasurableSpace.comap_mono (f.le t))
+
+lemma spatial_stopping_section {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {f : Filtration ℝ≥0 mΩ} {τ : Z × Ω → WithTop ℝ≥0}
+    (hτ : IsStoppingTime (spatialFiltration (Z := Z) f) τ) (z : Z) :
+    IsStoppingTime f (fun ω => τ (z,ω)) := by
+  intro t
+  have h := hτ t
+  let : MeasurableSpace Ω := f t
+  exact h.preimage (measurable_const.prodMk measurable_id)
+
+ 
+
+structure SpatialLogLocalization
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi}
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (R : ℝ) (T : ℝ≥0) where
+  τ : Z × Ω → ℝ≥0
+  stopping : IsStoppingTime
+    (spatialFiltration (Z := Z) (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
+    (fun p => (τ p : WithTop ℝ≥0))
+  bound : ∀ p, τ p ≤ T
+  inside : ∀ z ω t, t ≤ τ (z,ω) → ‖((c (q z)).Q t ω, B t ω)‖ ≤ R ∧
+    lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi
+  boundary : ∀ z ω, τ (z,ω) < T →
+    ‖((c (q z)).Q (τ (z,ω)) ω, B (τ (z,ω)) ω)‖ = R ∨
+      ((c (q z)).Q (τ (z,ω)) ω).1.im = lo ∨ ((c (q z)).Q (τ (z,ω)) ω).1.im = hi
+
+ 
+
+theorem exists_spatial_log_localization
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ lo hi : ℝ) (hle : lo ≤ hi)
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (hq : Measurable q) (R : ℝ) (T : ℝ≥0)
+    (hinit : ∀ z, ‖(q z, (0 : ℝ))‖ ≤ R ∧ lo ≤ (q z).1.im ∧ (q z).1.im ≤ hi) :
+    Nonempty (SpatialLogLocalization c q R T) := by
+  let f := spatialFiltration (Z := Z) (Filtration.natural B (fun s => (hm s).stronglyMeasurable))
+  let D (t : ℝ≥0) (p : Z × Ω) := min (R - ‖((c (q p.1)).Q t p.2, B t p.2)‖)
+    (min (((c (q p.1)).Q t p.2).1.im - lo) (hi - ((c (q p.1)).Q t p.2).1.im))
+  have hDa : Adapted f D := by
+    intro t
+    have hQ := clipped_family_past_measurable c t
+    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted t
+    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) t
+    have hc := hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)
+    have hBb := hBa.comp (@measurable_snd Z Ω mZ _)
+    exact (measurable_const.sub (hc.prodMk hBb).norm).min
+      (((Complex.continuous_im.measurable.comp hc.fst).sub_const lo).min
+        (measurable_const.sub (Complex.continuous_im.measurable.comp hc.fst)))
+  have hDc (p : Z × Ω) : Continuous (fun t => D t p) := by
+    have hc := ((c (q p.1)).lipschitz p.2).continuous.comp NNReal.continuous_coe
+    exact (continuous_const.sub (hc.prodMk (hBc p.2)).norm).min
+      (((Complex.continuous_im.comp hc.fst).sub continuous_const).min
+        (continuous_const.sub (Complex.continuous_im.comp hc.fst)))
+  have hD0 (p : Z × Ω) : 0 ≤ D 0 p := by
+    simpa only [D, NNReal.coe_zero, (c (q p.1)).initial, hB0, le_min_iff, sub_nonneg]
+      using hinit p.1
+  obtain ⟨τ, hτ, hb, hin, hpre, hboundary⟩ := first_margin_stopping f D hDa hDc hD0 T
+  refine ⟨⟨τ, hτ, hb, ?_, ?_⟩⟩
+  · intro z ω t ht
+    simpa only [D, le_min_iff, sub_nonneg] using hin (z,ω) t ht
+  · intro z ω ht
+    have hh := hboundary (z,ω) ht
+    have hle' := hin (z,ω) (τ (z,ω)) le_rfl
+    dsimp only [D] at hh hle'
+    rw [min_eq_iff, min_eq_iff] at hh
+    rcases hh with ⟨h, _⟩ | ⟨h, _⟩
+    · exact Or.inl (sub_eq_zero.mp h).symm
+    · rcases h with ⟨h, _⟩ | ⟨h, _⟩
+      · exact Or.inr (Or.inl (sub_eq_zero.mp h))
+      · exact Or.inr (Or.inr (sub_eq_zero.mp h).symm)
+
+namespace SpatialLogLocalization
+variable {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+  {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+  {κ lo hi : ℝ} {hle : lo ≤ hi}
+  {c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q}
+  {q : Z → ℂ × ℝ} {R : ℝ} {T : ℝ≥0}
+
+lemma measurable (a : SpatialLogLocalization c q R T) : Measurable a.τ :=
+  stopping_time_measurable_nnreal a.stopping
+
+lemma exponential {P : Measure Ω} {ν : Measure Z} [IsFiniteMeasure ν]
+    (a : SpatialLogLocalization c q R T) (hq : Measurable q)
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (hκ8 : κ < 8) (hR : 0 ≤ R) (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp
+      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (a.τ (z,ω)) ω) (B (a.τ (z,ω)) ω) -
+        greenLogDensity κ (q z) 0 ∂ν) -
+      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
+        (∫ z, if t < (a.τ (z,ω) : ℝ) then
+          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
+  exact clipped_spatial_green_exponential hm hB hBc κ lo hi hle hκ hκ8 c q hq R hR T
+    a.τ a.measurable (stopping_before_measurable a.stopping)
+    (spatial_stopping_section a.stopping) (fun z ω => a.bound (z,ω)) a.inside θ
+
+end SpatialLogLocalization
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma greenLogCoefficient_logCoordinates
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (κ b : ℝ) (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t)
+    (hUb : U t = Real.sqrt κ * b) :
+    greenLogCoefficient κ (logLoewnerCoordinates G z t) b =
+      greenBrownianCoefficient κ U γ G t z := by
+  rw [greenLogCoefficient, greenLogDensity_logCoordinates hU h κ b z t hz hUb,
+    greenLogNormSq_logCoordinates κ b G z t (h.imag_positive t z hz),
+    logCoordinates_re, Complex.normSq_eq_norm_sq, greenBrownianCoefficient, hUb]
+  simp only [Complex.sub_re, Complex.ofReal_re]
+  ring
+
+lemma clipped_chart_identification
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi}
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (ω : Ω) (τ : ℝ≥0)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F)
+    (hin : ∀ t : ℝ≥0, t ≤ τ →
+      lo ≤ ((c (initialLogCoordinates z)).Q t ω).1.im ∧
+        ((c (initialLogCoordinates z)).Q t ω).1.im ≤ hi) :
+    z ∈ survivingDomain γ τ ∧ ∀ t : ℝ≥0, t ≤ τ →
+      (c (initialLogCoordinates z)).Q t ω = logLoewnerCoordinates G z t := by
+  let d := c (initialLogCoordinates z)
+  have hd (s : ℝ) (hs : s ∈ Icc 0 (τ : ℝ)) := (d.derivative ω s hs.1).mono
+    (show Icc (0 : ℝ) (τ : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+  have hb (s : ℝ) (hs : s ∈ Icc 0 (τ : ℝ)) : (d.Q s ω).1.im ∈ Icc lo hi := by
+    have hh := hin s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)
+    simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
+  obtain ⟨hl, he⟩ := clipped_log_identification ((hBc ω).const_mul _) hw z hz τ
+    lo hi hle (fun s => d.Q s ω) (d.initial ω) hd hb
+  exact ⟨hl, fun t ht => by simpa only [Real.toNNReal_coe] using he ⟨t.coe_nonneg, ht⟩⟩
+
+lemma SpatialLogLocalization.chart_identification
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi}
+    {c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q}
+    {z : Z → ℂ} {R : ℝ} {T : ℝ≥0}
+    (a : SpatialLogLocalization c (fun x => initialLogCoordinates (z x)) R T)
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (x : Z) (hz : z x ∈ upperHalfPlane) (ω : Ω)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F) :
+    z x ∈ survivingDomain γ (a.τ (x,ω)) ∧
+      ∀ t : ℝ≥0, t ≤ a.τ (x,ω) →
+        (c (initialLogCoordinates (z x))).Q t ω = logLoewnerCoordinates G (z x) t := by
+  let d := c (initialLogCoordinates (z x))
+  have hd (s : ℝ) (hs : s ∈ Icc 0 (a.τ (x,ω) : ℝ)) := (d.derivative ω s hs.1).mono
+    (show Icc (0 : ℝ) (a.τ (x,ω) : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+  have hb (s : ℝ) (hs : s ∈ Icc 0 (a.τ (x,ω) : ℝ)) : (d.Q s ω).1.im ∈ Icc lo hi := by
+    have hh := (a.inside x ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2
+    simpa only [Set.mem_Icc, Real.coe_toNNReal _ hs.1] using hh
+  obtain ⟨hl, he⟩ := clipped_log_identification ((hBc ω).const_mul _) hw (z x) hz (a.τ (x,ω))
+    lo hi hle (fun s => d.Q s ω) (d.initial ω) hd hb
+  exact ⟨hl, fun t ht => by simpa only [Real.toNNReal_coe] using he ⟨t.coe_nonneg, ht⟩⟩
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma stopped_difference_max {T E : Type*} [LinearOrder T] [AddGroup E]
+    (M : T → E) (s t : T) :
+    M t - M (min t s) = M (max s t) - M s := by
+  by_cases ht : t ≤ s
+  · rw [min_eq_left ht, max_eq_left ht, sub_self, sub_self]
+  · rw [min_eq_right (le_of_not_ge ht), max_eq_right (le_of_not_ge ht)]
+
+ 
+
+theorem closed_stop_limit_condExp
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    (𝓕 : Filtration ℝ≥0 mΩ) (M : ℝ≥0 → Ω → ℝ) (Z W : Ω → ℝ)
+    (hZ : Integrable Z μ)
+    (hStop : ∀ (ρ : Ω → ℝ≥0) (hρ : IsStoppingTime 𝓕 (fun ω => (ρ ω : WithTop ℝ≥0))),
+      (fun ω => M (ρ ω) ω) =ᵐ[μ] μ[Z | hρ.measurableSpace])
+    (σ : ℝ≥0 → Ω → ℝ≥0)
+    (hσ : ∀ t, IsStoppingTime 𝓕 (fun ω => (σ t ω : WithTop ℝ≥0)))
+    (hσle : ∀ t ω, σ t ω ≤ t)
+    (hσmin : ∀ s t, s ≤ t → ∀ ω, σ s ω = min (σ t ω) s)
+    (K : ℝ) (hbound : ∀ t ω, ‖M t ω‖ ≤ K)
+    (hlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (σ n ω) ω) atTop (𝓝 (W ω)))
+    (s : ℝ≥0) : (fun ω => M (σ s ω) ω) =ᵐ[μ] μ[W | 𝓕 s] := by
+  have hInt (ρ : Ω → ℝ≥0) (hρ : IsStoppingTime 𝓕 (fun ω => (ρ ω : WithTop ℝ≥0))) :
+      Integrable (fun ω => M (ρ ω) ω) μ :=
+    integrable_condExp.congr (hStop ρ hρ).symm
+  have hWi : Integrable W μ := by
+    apply Integrable.of_bound (aestronglyMeasurable_of_tendsto_ae atTop
+      (fun n : ℕ => (hInt (σ n) (hσ n)).aestronglyMeasurable) hlim) K
+    filter_upwards [hlim] with ω hω
+    exact le_of_tendsto hω.norm (Eventually.of_forall fun n => hbound (σ n ω) ω)
+  have hss : (hσ s).measurableSpace ≤ 𝓕 s :=
+    (hσ s).measurableSpace_le_of_le_const (fun ω => by exact_mod_cast hσle s ω)
+  have hMs : AEStronglyMeasurable[𝓕 s] (fun ω => M (σ s ω) ω) μ :=
+    ((stronglyMeasurable_condExp.mono hss).aestronglyMeasurable).congr (hStop (σ s) (hσ s)).symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq (𝓕.le s) hWi
+    (fun _ _ _ => (hInt (σ s) (hσ s)).integrableOn) _ hMs
+  intro A hA _
+  have hI : Tendsto (fun n : ℕ => ∫ ω in A, M (σ n ω) ω ∂μ) atTop (𝓝 (∫ ω in A, W ω ∂μ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => K)
+      (fun n => (hInt (σ n) (hσ n)).integrableOn.aestronglyMeasurable)
+      (integrable_const _) _ (ae_restrict_of_ae hlim)
+    intro n
+    exact Eventually.of_forall fun ω => hbound (σ n ω) ω
+  have he : ∀ᶠ n : ℕ in atTop, (∫ ω in A, M (σ n ω) ω ∂μ) =
+      ∫ ω in A, M (σ s ω) ω ∂μ := by
+    filter_upwards [eventually_ge_atTop (⌈(s : ℝ)⌉₊)] with n hn
+    have hsn : s ≤ (n : ℝ≥0) := by
+      exact_mod_cast (Nat.le_of_ceil_le hn)
+    let ρ (ω : Ω) := max s (σ n ω)
+    have hρ : IsStoppingTime 𝓕 (fun ω => (ρ ω : WithTop ℝ≥0)) := by
+      simpa only [ρ, WithTop.coe_max] using (isStoppingTime_const 𝓕 s).max (hσ n)
+    have hFρ : 𝓕 s ≤ hρ.measurableSpace :=
+      hρ.le_measurableSpace_of_const_le (fun ω => by exact_mod_cast (le_max_left s (σ n ω)))
+    have hFc : 𝓕 s ≤ (isStoppingTime_const 𝓕 s).measurableSpace :=
+      (isStoppingTime_const 𝓕 s).le_measurableSpace_of_const_le (fun _ => le_rfl)
+    have hsetρ : (∫ ω in A, M (ρ ω) ω ∂μ) = ∫ ω in A, Z ω ∂μ := by
+      rw [integral_congr_ae (ae_restrict_of_ae (hStop ρ hρ))]
+      have : IsFiniteMeasure (μ.trim hρ.measurableSpace_le) := isFiniteMeasure_trim _
+      exact setIntegral_condExp hρ.measurableSpace_le hZ (hFρ A hA)
+    have hsetc : (∫ ω in A, M s ω ∂μ) = ∫ ω in A, Z ω ∂μ := by
+      rw [integral_congr_ae (ae_restrict_of_ae (hStop (fun _ => s) (isStoppingTime_const 𝓕 s)))]
+      have : IsFiniteMeasure (μ.trim (isStoppingTime_const 𝓕 s).measurableSpace_le) := isFiniteMeasure_trim _
+      exact setIntegral_condExp (isStoppingTime_const 𝓕 s).measurableSpace_le hZ (hFc A hA)
+    have hid : (∫ ω in A, M (σ n ω) ω - M (σ s ω) ω ∂μ) =
+        ∫ ω in A, M (ρ ω) ω - M s ω ∂μ := by
+      apply integral_congr_ae
+      exact Eventually.of_forall fun ω => by
+        change M (σ n ω) ω - M (σ s ω) ω = M (ρ ω) ω - M s ω
+        rw [hσmin s n hsn ω]
+        exact stopped_difference_max (fun t => M t ω) s (σ n ω)
+    rw [integral_sub (hInt (σ n) (hσ n)).integrableOn (hInt (σ s) (hσ s)).integrableOn,
+      integral_sub (hInt ρ hρ).integrableOn
+        (hInt (fun _ => s) (isStoppingTime_const 𝓕 s)).integrableOn, hsetρ, hsetc, sub_self] at hid
+    exact sub_eq_zero.mp hid
+  have hc : Tendsto (fun n : ℕ => ∫ ω in A, M (σ n ω) ω ∂μ) atTop
+      (𝓝 (∫ ω in A, M (σ s ω) ω ∂μ)) := by
+    apply tendsto_const_nhds.congr'
+    filter_upwards [he] with n hn using hn.symm
+  exact (tendsto_nhds_unique hI hc).symm
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma boundedSpatialHit_le_time (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (t : ℝ≥0) :
+    boundedSpatialHit γ z r t ≤ t := by
+  apply WithTop.coe_le_coe.mp
+  rw [boundedSpatialHit_coe]
+  exact min_le_right _ _
+
+lemma boundedSpatialHit_min (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ)
+    (s t : ℝ≥0) (hst : s ≤ t) :
+    boundedSpatialHit γ z r s = min (boundedSpatialHit γ z r t) s := by
+  apply WithTop.coe_injective
+  rw [WithTop.coe_min, boundedSpatialHit_coe, boundedSpatialHit_coe, min_assoc,
+    min_eq_right (WithTop.coe_le_coe.mpr hst)]
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+lemma spatialGreen_condExp_of_level [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im)
+    (u : ℝ) (hu : 0 ≤ u) (hlevel : (2 * z.im) * Real.exp (-u) < r)
+    (s : ℝ≥0) :
+    (fun ω : a.samples => greenDensity κ (fun t => Real.sqrt κ * a.B t ω)
+      (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r s) z) =ᵐ[P.comap Subtype.val]
+      (P.comap Subtype.val)[fun ω : a.samples => spatialGreenWeight κ
+        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r | a.filtration s] := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
+  obtain ⟨M, Z, hM, hZ, hMb, hZb, hClosed, hStop, heq, hNoHit⟩ :=
+    exists_physical_green_stopping_regular a.brownian hBm a.continuous_B a.zero_B
+      κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
+  let σ (t : ℝ≥0) (ω : a.samples) := boundedSpatialHit (γ ω) z r t
+  have hσ (t : ℝ≥0) : IsStoppingTime a.filtration (fun ω => (σ t ω : WithTop ℝ≥0)) :=
+    a.boundedSpatialHit_stopping z r t
+  dsimp only [filtration] at hσ
+  let W (ω : a.samples) := spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+    (γ ω) (a.G ω) z r
+  have hlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (σ n ω) ω) atTop (𝓝 (W ω)) := by
+    filter_upwards [heq, hNoHit] with ω he hn
+    simpa only [he.1, σ, W, Function.comp_apply] using
+      (a.witness ω).spatialGreen_limit ((a.continuous_B ω).const_mul _) κ z r hr hy u hlevel hn
+  have hb (t : ℝ≥0) (ω : a.samples) :
+      ‖M t ω‖ ≤ Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)) := by
+    rw [Real.norm_eq_abs, abs_of_nonneg (hMb t ω).1]
+    exact (hMb t ω).2
+  have hc := closed_stop_limit_condExp a.filtration M Z W hZ hStop σ hσ
+    (fun t ω => boundedSpatialHit_le_time (γ ω) z r t)
+    (fun t v htv ω => boundedSpatialHit_min (γ ω) z r t v htv) _ hb hlim s
+  refine EventuallyEq.trans ?_ hc
+  filter_upwards [heq] with ω he
+  rw [he.1]
+  exact ((a.witness ω).physicalGreen_eq_before_spatial
+    ((a.continuous_B ω).const_mul _) κ z r hr hy u hlevel (σ s ω)
+    (boundedSpatialHit_le (γ ω) z r s)).symm
+
+ 
+
+theorem spatialGreen_condExp [IsProbabilityMeasure P]
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im) (s : ℝ≥0) :
+    (fun ω : a.samples => greenDensity κ (fun t => Real.sqrt κ * a.B t ω)
+      (γ ω) (a.G ω) (boundedSpatialHit (γ ω) z r s) z) =ᵐ[P.comap Subtype.val]
+      (P.comap Subtype.val)[fun ω : a.samples => spatialGreenWeight κ
+        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r | a.filtration s] := by
+  have hz : z ∈ upperHalfPlane := hr.trans_le hy
+  let u := Real.log (4 * z.im / r)
+  have hratio : 0 < 4 * z.im / r := div_pos (mul_pos (by norm_num) hz) hr
+  have hu : 0 ≤ u := Real.log_nonneg ((le_div_iff₀ hr).mpr (by linarith))
+  have hlev : (2 * z.im) * Real.exp (-u) = r / 2 := by
+    dsimp only [u]
+    rw [Real.exp_neg, Real.exp_log hratio]
+    field_simp [(show 0 < z.im from hz).ne']
+    ring
+  exact a.spatialGreen_condExp_of_level hκ hκ8 z hz r hr hy u hu
+    (by rw [hlev]; linarith) s
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StochasticIntegrals.lean b/lean/OAI/Probability/SLE/StochasticIntegrals.lean
index 647858af2ac23c105ebb7f6f19d093f130b868fa..82683510fff7638b6a967dd3bfe9f886daf5f351 100644
--- a/lean/OAI/Probability/SLE/StochasticIntegrals.lean
+++ b/lean/OAI/Probability/SLE/StochasticIntegrals.lean
@@ -1,692 +1,692 @@
-import Mathlib
-import OAI.Probability.SLE.SpatialMeans
-
-namespace OAI
-
-/-! Stochastic Integrals for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-namespace SLEExactGauge
-noncomputable def spatialLaterKernel (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ)
-    (r : ℝ) (z w : ℂ) : ℝ≥0∞ :=
-  if spatialHitTime γ z r = ⊤ then 0 else
-    let s := (spatialHitTime γ z r).untopD 0
-    if z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s ∧
-        spatialHitTime γ w r ≤ s then ENNReal.ofReal (halfPlaneGreen (G s z) (G s w)) else 0
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
-lemma IsCapacityTwoWitness.spatial_later_kernel_integral
-    (h : IsCapacityTwoWitness U γ G F) {r : ℝ} (hr : 0 < r) (z : ℂ) :
-    (∫⁻ w, spatialLaterKernel γ G r z w) ≤ ENNReal.ofReal (layerGreenConstant * r ^ 2) := by
-  classical
-  by_cases ht : spatialHitTime γ z r = ⊤
-  · simp only [spatialLaterKernel, ite_eq_left ht, lintegral_zero, zero_le]
-  obtain ⟨s, hs⟩ := WithTop.ne_top_iff_exists.mp ht
-  by_cases hz : z ∈ survivingDomain γ s
-  · let L := {w | w ∈ survivingDomain γ s ∧ infDist w (frontier (survivingDomain γ s)) ≤ r}
-    have hp (w : ℂ) : spatialLaterKernel γ G r z w ≤
-        L.indicator (fun w => ENNReal.ofReal (halfPlaneGreen (G s z) (G s w))) w := by
-      rw [spatialLaterKernel, ite_eq_right ht, ← hs, WithTop.untopD_coe]
-      by_cases ha : z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s ∧ spatialHitTime γ w r ≤ s
-      · rw [ite_eq_left ha]
-        obtain ⟨v, hv, he⟩ := (spatialHitTime_le_iff h.1 w r s).mp ha.2.2
-        have hw : w ∈ L := ⟨ha.2.1, (surviving_frontier_distance_le_past s v hv.2 w ha.2.1).trans_eq he⟩
-        rw [Set.indicator_of_mem hw]
-      · rw [ite_eq_right ha]
-        exact zero_le
-    calc
-      _ ≤ ∫⁻ w, L.indicator (fun w => ENNReal.ofReal (halfPlaneGreen (G s z) (G s w))) w :=
-        lintegral_mono hp
-      _ ≤ ∫⁻ w in L, ENNReal.ofReal (halfPlaneGreen (G s z) (G s w)) := lintegral_indicator_le _ _
-      _ ≤ _ := h.boundary_layer_green s hr hz
-  · have he (w : ℂ) : spatialLaterKernel γ G r z w = 0 := by
-      rw [spatialLaterKernel, ite_eq_right ht, ← hs, WithTop.untopD_coe]
-      simp only [hz, false_and, ite_false]
-    simp only [he, lintegral_zero, zero_le]
-
-lemma slab_product_zero_of_not_live_at_start
-    (κ r ε : ℝ) (z w : ℂ) (s : ℝ≥0) (hs : spatialHitTime γ z r = s)
-    (hn : ¬ (z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s)) (t : ℝ≥0) :
-    spatialSlabCoefficient κ U γ G r ε t z * spatialSlabCoefficient κ U γ G r ε t w = 0 := by
-  by_cases ha : spatialHitTime γ z r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧
-      z ∈ survivingDomain γ t
-  · by_cases hb : spatialHitTime γ w r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ w ε ∧
-        w ∈ survivingDomain γ t
-    · have hst : s ≤ t := WithTop.coe_le_coe.mp (hs ▸ ha.1)
-      exact (hn ⟨survivingDomain_antitone γ hst ha.2.2, survivingDomain_antitone γ hst hb.2.2⟩).elim
-    · simp only [spatialSlabCoefficient, ite_eq_right hb, mul_zero]
-  · simp only [spatialSlabCoefficient, ite_eq_right ha, zero_mul]
-
-lemma IsCapacityTwoWitness.slab_pair_ordered
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
-    (r : ℝ) (z w : ℂ) (hzy : ε ≤ z.im) (hwy : ε ≤ w.im) (hne : z ≠ w)
-    (horder : spatialHitTime γ w r ≤ spatialHitTime γ z r) :
-    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-      spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        spatialLaterKernel γ G r z w := by
-  classical
-  by_cases ht : spatialHitTime γ z r = ⊤
-  · have he (t : ℝ) : spatialSlabCoefficient κ U γ G r ε t.toNNReal z = 0 := by
-      have hn : ¬ (spatialHitTime γ z r ≤ t.toNNReal ∧
-          (t.toNNReal : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧ z ∈ survivingDomain γ t.toNNReal) := by
-        intro hh
-        rw [ht] at hh
-        exact WithTop.coe_ne_top (top_le_iff.mp hh.1)
-      exact ite_eq_right hn
-    simp only [he, zero_mul, abs_zero, ENNReal.ofReal_zero, lintegral_zero, zero_le]
-  obtain ⟨s, hs⟩ := WithTop.ne_top_iff_exists.mp ht
-  by_cases hl : z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s
-  · have hord : spatialHitTime γ w r ≤ s := hs ▸ horder
-    have he : spatialLaterKernel γ G r z w = ENNReal.ofReal (halfPlaneGreen (G s z) (G s w)) := by
-      rw [spatialLaterKernel, ite_eq_right ht, ← hs, WithTop.untopD_coe, ite_eq_left ⟨hl.1, hl.2, hord⟩]
-    rw [he]
-    have hb := h.slab_pair_later_start hU hκ hκ4 hε r z w hzy hwy hne s hs.symm
-    convert hb using 1
-    rw [← ENNReal.ofReal_mul (sq_nonneg _), ← ENNReal.ofReal_mul (div_nonneg (sq_nonneg _) (by norm_num))]
-    congr 1
-    ring
-  · simp only [slab_product_zero_of_not_live_at_start κ r ε z w s hs.symm hl,
-      abs_zero, ENNReal.ofReal_zero, lintegral_zero, zero_le]
-
- 
-
-lemma IsCapacityTwoWitness.slab_pair_kernel_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
-    (r : ℝ) (z w : ℂ) (hzy : ε ≤ z.im) (hwy : ε ≤ w.im) (hne : z ≠ w) :
-    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-      spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (spatialLaterKernel γ G r z w + spatialLaterKernel γ G r w z) := by
-  rcases le_total (spatialHitTime γ w r) (spatialHitTime γ z r) with ho | ho
-  · exact (h.slab_pair_ordered hU hκ hκ4 hε r z w hzy hwy hne ho).trans
-      (mul_le_mul le_rfl (le_add_right le_rfl) bot_le bot_le)
-  · have hb := h.slab_pair_ordered hU hκ hκ4 hε r w z hwy hzy hne.symm ho
-    simp only [mul_comm (spatialSlabCoefficient κ U γ G r ε _ w)] at hb
-    exact hb.trans (mul_le_mul le_rfl (le_add_left le_rfl) bot_le bot_le)
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal ComplexConjugate
-namespace SLEExactGauge
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
-lemma spatialHitTime_measurable_point (hc : Continuous γ) (r : ℝ) :
-    Measurable (fun z => spatialHitTime γ z r) := by
-  have hm := spatialHitTime_measurable (fun _ : Unit => γ) (fun _ => hc)
-    (fun _ => measurable_const) r
-  exact hm.comp (measurable_id.prodMk (measurable_const (a := ())))
-
-lemma IsCapacityTwoWitness.joint_live_measurable
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) :
-    MeasurableSet {p : ℝ≥0 × ℂ | p.2 ∈ survivingDomain γ p.1} := by
-  let c := spatialClippedFamily (fun t (_ : Unit) => U t)
-    (hU.measurable.comp measurable_fst) (fun _ => hU)
-  have hm := c.live_measurable (fun _ => γ) (fun _ => G) (fun _ => F) (fun _ => hU) (fun _ => h)
-  exact hm.preimage (measurable_fst.prodMk (measurable_snd.prodMk (measurable_const (a := ()))))
-
-lemma IsCapacityTwoWitness.killed_chart_measurable
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) :
-    Measurable (fun p : ℝ≥0 × ℂ => if p.2 ∈ survivingDomain γ p.1 then G p.1 p.2 else 0) := by
-  let c := spatialClippedFamily (fun t (_ : Unit) => U t)
-    (hU.measurable.comp measurable_fst) (fun _ => hU)
-  convert c.jointChart_measurable.comp
-    (measurable_fst.prodMk (measurable_snd.prodMk (measurable_const (a := ())))) using 1
-  funext p
-  exact (c.jointChart_eq () hU h p.1 p.2).symm
-
-lemma IsCapacityTwoWitness.spatial_later_kernel_measurable
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (r : ℝ) :
-    Measurable (fun p : ℂ × ℂ => spatialLaterKernel γ G r p.1 p.2) := by
-  classical
-  let H (p : ℝ≥0 × ℂ) := if p.2 ∈ survivingDomain γ p.1 then G p.1 p.2 else 0
-  let s (z : ℂ) := (spatialHitTime γ z r).untopD 0
-  have hm := spatialHitTime_measurable_point h.1 r
-  have hs : Measurable s := hm.untopD 0
-  have hH : Measurable H := h.killed_chart_measurable hU
-  have hz : Measurable (fun p : ℂ × ℂ => H (s p.1, p.1)) := hH.comp ((hs.comp measurable_fst).prodMk measurable_fst)
-  have hw : Measurable (fun p : ℂ × ℂ => H (s p.1, p.2)) := hH.comp ((hs.comp measurable_fst).prodMk measurable_snd)
-  have hK : Measurable (fun p : ℂ × ℂ => ENNReal.ofReal (halfPlaneGreen (H (s p.1, p.1)) (H (s p.1, p.2)))) := by
-    exact (Real.measurable_log.comp ((hz.sub (Complex.continuous_conj.measurable.comp hw)).div (hz.sub hw)).norm).ennreal_ofReal
-  have hzl : MeasurableSet {p : ℂ × ℂ | p.1 ∈ survivingDomain γ (s p.1)} :=
-    (h.joint_live_measurable hU).preimage ((hs.comp measurable_fst).prodMk measurable_fst)
-  have hwl : MeasurableSet {p : ℂ × ℂ | p.2 ∈ survivingDomain γ (s p.1)} :=
-    (h.joint_live_measurable hU).preimage ((hs.comp measurable_fst).prodMk measurable_snd)
-  have ho : MeasurableSet {p : ℂ × ℂ | spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0)} :=
-    measurableSet_le (hm.comp measurable_snd) ((hs.comp measurable_fst).withTop_coe)
-  have hnf : MeasurableSet {p : ℂ × ℂ | spatialHitTime γ p.1 r = ⊤} :=
-    measurableSet_eq_fun (hm.comp measurable_fst) measurable_const
-  have hk : Measurable (fun p : ℂ × ℂ =>
-      if p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
-        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0) then
-        ENNReal.ofReal (halfPlaneGreen (H (s p.1, p.1)) (H (s p.1, p.2))) else 0) := by
-    exact @Measurable.ite _ _ _ _ _ _
-      (fun p : ℂ × ℂ => p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
-        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0)) (fun _ => inferInstance)
-      (hzl.inter (hwl.inter ho)) hK (measurable_const (a := (0 : ℝ≥0∞)))
-  have hh : Measurable (fun p : ℂ × ℂ => if spatialHitTime γ p.1 r = ⊤ then 0 else
-      if p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
-        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0) then
-        ENNReal.ofReal (halfPlaneGreen (H (s p.1, p.1)) (H (s p.1, p.2))) else 0) := by
-    exact @Measurable.ite _ _ _ _ _ _ (fun p : ℂ × ℂ => spatialHitTime γ p.1 r = ⊤)
-      (fun _ => inferInstance) hnf
-      (measurable_const (a := (0 : ℝ≥0∞))) hk
-  convert hh using 1
-  funext p
-  dsimp only [spatialLaterKernel]
-  by_cases hn : spatialHitTime γ p.1 r = ⊤
-  · simp only [hn, ite_eq_left]
-  · rw [ite_eq_right hn, ite_eq_right hn]
-    change (if p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
-        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0) then _ else _) = _
-    split_ifs with hl
-    · simp only [H, ite_eq_left hl.1, ite_eq_left hl.2.1]
-      rfl
-    · rfl
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
-lemma IsCapacityTwoWitness.spatial_later_kernel_double
-    (h : IsCapacityTwoWitness U γ G F) {r : ℝ} (hr : 0 < r) (A : Set ℂ) :
-    (∫⁻ z in A, ∫⁻ w in A, spatialLaterKernel γ G r z w) ≤
-      ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A := by
-  calc
-    _ ≤ ∫⁻ _z in A, ENNReal.ofReal (layerGreenConstant * r ^ 2) := by
-      apply lintegral_mono
-      intro z
-      exact (lintegral_mono' Measure.restrict_le_self le_rfl).trans (h.spatial_later_kernel_integral hr z)
-    _ = _ := by rw [lintegral_const, Measure.restrict_apply_univ]
-
-lemma sym_pair_lintegral {α : Type*} [MeasurableSpace α] (μ : Measure α)
-    [SFinite μ] (K : α → α → ℝ≥0∞) (hK : Measurable (Function.uncurry K))
-    (C : ℝ≥0∞) (hC : C ≠ ∞) :
-    (∫⁻ z, ∫⁻ w, C * (K z w + K w z) ∂μ ∂μ) =
-      C * (2 * (∫⁻ z, ∫⁻ w, K z w ∂μ ∂μ)) := by
-  have hi (z : α) : (∫⁻ w, C * (K z w + K w z) ∂μ) =
-      C * ((∫⁻ w, K z w ∂μ) + (∫⁻ w, K w z ∂μ)) := by
-    rw [lintegral_const_mul' C _ hC]
-    have hm : Measurable (fun w => K z w) := hK.comp (measurable_const.prodMk measurable_id)
-    rw [lintegral_add_left hm]
-  rw [lintegral_congr hi, lintegral_const_mul' C _ hC]
-  have ho : Measurable (fun z => ∫⁻ w, K z w ∂μ) := hK.lintegral_prod_right'
-  rw [lintegral_add_left ho]
-  have hs : (∫⁻ z, ∫⁻ w, K w z ∂μ ∂μ) = (∫⁻ z, ∫⁻ w, K z w ∂μ ∂μ) :=
-    lintegral_lintegral_swap (hK.comp measurable_swap).aemeasurable
-  rw [hs, two_mul]
-
-lemma IsCapacityTwoWitness.spatial_pair_lintegral_le
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (r : ℝ)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
-    (∫⁻ z in A, ∫⁻ w in A, ∫⁻ t in Ici (0 : ℝ),
-      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-        spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
-    (∫⁻ z in A, ∫⁻ w in A,
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (spatialLaterKernel γ G r z w + spatialLaterKernel γ G r w z)) := by
-  apply lintegral_mono_ae
-  filter_upwards [ae_restrict_mem hA] with z hz
-  apply lintegral_mono_ae
-  filter_upwards [ae_restrict_mem hA, ae_restrict_of_ae ((volume : Measure ℂ).ae_ne z)] with w hw hne
-  exact h.slab_pair_kernel_bound hU hκ hκ4 hε r z w (hheight z hz) (hheight w hw) hne.symm
-
- 
-
-theorem IsCapacityTwoWitness.spatial_slab_pair_area_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
-    (∫⁻ z in A, ∫⁻ w in A, ∫⁻ t in Ici (0 : ℝ),
-      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-        spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
-  refine (h.spatial_pair_lintegral_le hU hκ hκ4 hε r A hA hheight).trans ?_
-  rw [sym_pair_lintegral (volume.restrict A) (spatialLaterKernel γ G r) (h.spatial_later_kernel_measurable hU r)
-    _ ENNReal.ofReal_ne_top]
-  exact mul_le_mul le_rfl
-    (mul_le_mul le_rfl (h.spatial_later_kernel_double hr A) bot_le bot_le) bot_le bot_le
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal ComplexConjugate
-namespace SLEExactGauge
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
-lemma IsCapacityTwoWitness.killed_green_measurable
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) {κ : ℝ} (hκ : 0 < κ) :
-    Measurable (fun p : ℝ≥0 × ℂ => greenDensity κ U γ G p.1 p.2) := by
-  let c := spatialClippedFamily (fun t (_ : Unit) => U t)
-    (hU.measurable.comp measurable_fst) (fun _ => hU)
-  let B (t : ℝ≥0) (_ : Unit) := U t / Real.sqrt κ
-  have hBm : Measurable (fun p : ℝ≥0 × Unit => B p.1 p.2) :=
-    (hU.measurable.comp measurable_fst).div_const _
-  have hm := (c.jointGreen_measurable κ B hBm).comp
-    (measurable_fst.prodMk (measurable_snd.prodMk (measurable_const (a := ()))))
-  convert hm using 1
-  funext p
-  apply (c.jointGreen_eq κ B () hU _ h p.1 p.2).symm
-  intro t
-  dsimp only [B]
-  field_simp [(Real.sqrt_pos.mpr hκ).ne']
-
-lemma IsCapacityTwoWitness.green_coefficient_measurable
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) {κ : ℝ} (hκ : 0 < κ) :
-    Measurable (fun p : ℝ≥0 × ℂ => greenBrownianCoefficient κ U γ G p.1 p.2) := by
-  classical
-  let H (p : ℝ≥0 × ℂ) := if p.2 ∈ survivingDomain γ p.1 then G p.1 p.2 else 0
-  have hZ : Measurable (fun p : ℝ≥0 × ℂ => H p - (U p.1 : ℂ)) :=
-    (h.killed_chart_measurable hU).sub (Complex.measurable_ofReal.comp (hU.measurable.comp measurable_fst))
-  have hb := (((Complex.measurable_re.comp hZ).const_mul (Real.sqrt κ * greenExponent κ)).div
-    (hZ.norm.pow_const 2)).mul (h.killed_green_measurable hU hκ)
-  convert hb using 1
-  funext p
-  by_cases hl : p.2 ∈ survivingDomain γ p.1
-  · simp only [Pi.mul_apply, Pi.div_apply, Function.comp_apply, H, ite_eq_left hl, greenBrownianCoefficient]
-  · simp only [Pi.mul_apply, Pi.div_apply, Function.comp_apply, greenBrownianCoefficient, greenDensity, ite_eq_right hl, mul_zero]
-
-lemma IsCapacityTwoWitness.spatial_slab_coefficient_measurable
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) {κ : ℝ} (hκ : 0 < κ)
-    (r ε : ℝ) :
-    Measurable (fun p : ℝ≥0 × ℂ => spatialSlabCoefficient κ U γ G r ε p.1 p.2) := by
-  classical
-  have ht : Measurable (fun p : ℝ≥0 × ℂ => (p.1 : WithTop ℝ≥0)) := measurable_fst.withTop_coe
-  have hr : Measurable (fun p : ℝ≥0 × ℂ => spatialHitTime γ p.2 r) :=
-    (spatialHitTime_measurable_point h.1 r).comp measurable_snd
-  have he : Measurable (fun p : ℝ≥0 × ℂ => spatialHitTime γ p.2 ε) :=
-    (spatialHitTime_measurable_point h.1 ε).comp measurable_snd
-  have hm : MeasurableSet {p : ℝ≥0 × ℂ | spatialHitTime γ p.2 r ≤ p.1 ∧
-      (p.1 : WithTop ℝ≥0) ≤ spatialHitTime γ p.2 ε ∧ p.2 ∈ survivingDomain γ p.1} :=
-    (measurableSet_le hr ht).inter ((measurableSet_le ht he).inter (h.joint_live_measurable hU))
-  exact @Measurable.ite _ _ _ _ _ _ (fun p : ℝ≥0 × ℂ => spatialHitTime γ p.2 r ≤ p.1 ∧
-      (p.1 : WithTop ℝ≥0) ≤ spatialHitTime γ p.2 ε ∧ p.2 ∈ survivingDomain γ p.1)
-    (fun _ => inferInstance) hm
-    (h.green_coefficient_measurable hU hκ) measurable_const
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge
-
- 
-
-lemma area_square_tonelli {T Z : Type*} [MeasurableSpace T] [MeasurableSpace Z]
-    {μ : Measure T} {ν : Measure Z} [SFinite μ] [SFinite ν]
-    {f : T × Z → ℝ≥0∞} (hf : Measurable f) :
-    (∫⁻ t, (∫⁻ z, f (t, z) ∂ν) ^ 2 ∂μ) =
-      ∫⁻ z, ∫⁻ w, ∫⁻ t, f (t, z) * f (t, w) ∂μ ∂ν ∂ν := by
-  have hft (t : T) : Measurable (fun z => f (t, z)) :=
-    hf.comp (measurable_const.prodMk measurable_id)
-  have h3 : Measurable (fun p : (T × Z) × Z => f (p.1.1, p.1.2) * f (p.1.1, p.2)) :=
-    (hf.comp measurable_fst).mul
-      (hf.comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd))
-  have hinner : Measurable (fun p : T × Z => ∫⁻ w, f (p.1, p.2) * f (p.1, w) ∂ν) :=
-    h3.lintegral_prod_right'
-  calc
-    _ = ∫⁻ t, ∫⁻ z, ∫⁻ w, f (t, z) * f (t, w) ∂ν ∂ν ∂μ := by
-      apply lintegral_congr
-      intro t
-      simp_rw [lintegral_const_mul _ (hft t)]
-      rw [lintegral_mul_const _ (hft t), pow_two]
-    _ = ∫⁻ z, ∫⁻ t, ∫⁻ w, f (t, z) * f (t, w) ∂ν ∂μ ∂ν :=
-      lintegral_lintegral_swap hinner.aemeasurable
-    _ = _ := by
-      apply lintegral_congr
-      intro z
-      apply lintegral_lintegral_swap
-      exact ((hf.comp (measurable_fst.prodMk measurable_const)).mul hf).aemeasurable
-
-lemma spatial_integral_square_le {Z : Type*} [MeasurableSpace Z] (ν : Measure Z)
-    (f : Z → ℝ) :
-    ENNReal.ofReal ((∫ z, f z ∂ν) ^ 2) ≤ (∫⁻ z, ENNReal.ofReal |f z| ∂ν) ^ 2 := by
-  have he (x : ℝ) : ‖x‖ₑ = ENNReal.ofReal |x| := by
-    rw [← ofReal_norm, Real.norm_eq_abs]
-  have hb := pow_le_pow_left₀ bot_le (enorm_integral_le_lintegral_enorm (μ := ν) f) 2
-  rw [← enorm_pow, Real.enorm_eq_ofReal (sq_nonneg _)] at hb
-  simpa only [he] using hb
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-
- 
-
-theorem IsCapacityTwoWitness.spatial_slab_absolute_bracket_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
-    (∫⁻ t in Ici (0 : ℝ), (∫⁻ z in A,
-      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|) ^ 2) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
-  have hb : Measurable (fun p : ℝ × ℂ => spatialSlabCoefficient κ U γ G r ε p.1.toNNReal p.2) := by
-    have hm : Measurable (fun p : ℝ × ℂ => (p.1.toNNReal, p.2)) :=
-      (continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd
-    exact (h.spatial_slab_coefficient_measurable hU hκ r ε).comp
-      (f := fun p : ℝ × ℂ => (p.1.toNNReal, p.2)) hm
-  rw [area_square_tonelli hb.abs.ennreal_ofReal]
-  have he (t : ℝ) (z w : ℂ) :
-      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z| *
-        ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal w| =
-      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
-        spatialSlabCoefficient κ U γ G r ε t.toNNReal w| := by
-    rw [abs_mul, ENNReal.ofReal_mul (abs_nonneg _)]
-  simp_rw [he]
-  exact h.spatial_slab_pair_area_bound hU hκ hκ4 hε hr A hA hheight
-
- 
-
-theorem IsCapacityTwoWitness.spatial_slab_bracket_bound
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
-    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal
-      ((∫ z in A, spatialSlabCoefficient κ U γ G r ε t.toNNReal z) ^ 2)) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
-  exact (lintegral_mono fun t => spatial_integral_square_le (volume.restrict A) _).trans
-    (h.spatial_slab_absolute_bracket_bound hU hκ hκ4 hε hr A hA hheight)
-
-end SLEExactGauge
-
- 
-
- 
-
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.BrownianPredictable
-
- 
-
-lemma gaussian_fourth_moment (v : ℝ≥0) :
-    (∫ x : ℝ, x ^ 4 ∂gaussianReal 0 v) = 3 * (v : ℝ) ^ 2 := by
-  let f : ℝ → ℝ := fun t => Real.exp ((v : ℝ) * t ^ 2 / 2)
-  have hf (t : ℝ) : HasDerivAt f ((v : ℝ) * t * f t) t := by
-    apply (((hasDerivAt_id t).pow 2 |>.const_mul (v : ℝ)).div_const 2).exp.congr_deriv
-    dsimp [f]
-    ring
-  have hd : deriv f = fun t => (v : ℝ) * t * f t := funext fun t => (hf t).deriv
-  have hd1 : deriv (fun t => (v : ℝ) * t * f t) =
-      fun t => ((v : ℝ) + (v : ℝ) ^ 2 * t ^ 2) * f t := by
-    funext t
-    apply HasDerivAt.deriv
-    apply (((hasDerivAt_id t).const_mul (v : ℝ)).mul (hf t)).congr_deriv
-    simp only [id_eq]
-    ring
-  have hd2 : deriv (fun t => ((v : ℝ) + (v : ℝ) ^ 2 * t ^ 2) * f t) =
-      fun t => (3 * (v : ℝ) ^ 2 * t + (v : ℝ) ^ 3 * t ^ 3) * f t := by
-    funext t
-    apply HasDerivAt.deriv
-    apply (((((hasDerivAt_id t).pow 2).const_mul ((v : ℝ) ^ 2)).const_add
-      (v : ℝ)).mul (hf t)).congr_deriv
-    simp only [id_eq, Pi.pow_apply]
-    ring
-  have hd3 : deriv (fun t => (3 * (v : ℝ) ^ 2 * t + (v : ℝ) ^ 3 * t ^ 3) * f t) =
-      fun t => (3 * (v : ℝ) ^ 2 + 6 * (v : ℝ) ^ 3 * t ^ 2 +
-        (v : ℝ) ^ 4 * t ^ 4) * f t := by
-    funext t
-    apply HasDerivAt.deriv
-    apply ((((hasDerivAt_id t).const_mul (3 * (v : ℝ) ^ 2)).add
-      (((hasDerivAt_id t).pow 3).const_mul ((v : ℝ) ^ 3))).mul (hf t)).congr_deriv
-    simp only [id_eq, Pi.pow_apply, Pi.add_apply]
-    ring
-  change (∫ x, ((fun x : ℝ => x) ^ 4) x ∂gaussianReal 0 v) = _
-  rw [← iteratedDeriv_mgf_zero (by simp), mgf_fun_id_gaussianReal]
-  simp only [zero_mul, zero_add]
-  change iteratedDeriv 4 f 0 = _
-  simp only [iteratedDeriv_succ, iteratedDeriv_zero, hd, hd1, hd2, hd3]
-  simp [f]
-
-lemma gaussian_second_moment (v : ℝ≥0) :
-    (∫ x : ℝ, x ^ 2 ∂gaussianReal 0 v) = (v : ℝ) := by
-  have hv := variance_eq_sub (memLp_id_gaussianReal (μ := 0) (v := v) 2)
-  simpa [integral_id_gaussianReal] using hv.symm
-
-lemma gaussian_square_memLp (v : ℝ≥0) :
-    MemLp (fun x : ℝ => x ^ 2) 2 (gaussianReal 0 v) := by
-  apply (memLp_two_iff_integrable_sq (by fun_prop)).2
-  have h4 := (memLp_id_gaussianReal (μ := 0) (v := v) 4).integrable_norm_pow
-    (p := 4) (by norm_num)
-  convert h4 using 1
-  funext x
-  simp only [id_eq, Real.norm_eq_abs]
-  rw [show (4 : ℕ) = 2 * 2 from rfl, pow_mul, sq_abs]
-
-lemma gaussian_square_variance (v : ℝ≥0) :
-    variance (fun x : ℝ => x ^ 2) (gaussianReal 0 v) = 2 * (v : ℝ) ^ 2 := by
-  rw [variance_eq_sub (gaussian_square_memLp v), gaussian_second_moment]
-  have heq : (fun x : ℝ => ((fun x : ℝ => x ^ 2) ^ 2) x) = fun x : ℝ => x ^ 4 := by
-    funext x
-    simp only [Pi.pow_apply]
-    ring
-  rw [show (fun x : ℝ => x ^ 2) ^ 2 = (fun x : ℝ => x ^ 4) from heq,
-    gaussian_fourth_moment]
-  ring
-
-lemma increment_hasLaw {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {s t : ℝ≥0} (hst : s ≤ t) :
-    HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t-s)) P := by
-  simpa only [add_tsub_cancel_of_le hst] using
-    (hB.toIsPreBrownianReal.shift s).hasLaw_eval (t-s)
-
-lemma increment_square_memLp {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {s t : ℝ≥0} (hst : s ≤ t) :
-    MemLp (fun ω => (B t ω - B s ω) ^ 2) 2 P := by
-  have hl := increment_hasLaw hB hst
-  exact (hl.map_eq ▸ gaussian_square_memLp (t-s)).comp_of_map hl.aemeasurable
-
-lemma increment_square_integral {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {s t : ℝ≥0} (hst : s ≤ t) :
-    (∫ ω, (B t ω - B s ω) ^ 2 ∂P) = (t-s : ℝ≥0) := by
-  have hl := increment_hasLaw hB hst
-  simpa only [Function.comp_apply, gaussian_second_moment] using
-    hl.integral_comp (show AEStronglyMeasurable (fun x : ℝ => x ^ 2)
-      (gaussianReal 0 (t-s)) from by fun_prop)
-
-lemma increment_square_variance {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {s t : ℝ≥0} (hst : s ≤ t) :
-    variance (fun ω => (B t ω - B s ω) ^ 2) P = 2 * (t-s : ℝ≥0) ^ 2 := by
-  have hl := increment_hasLaw hB hst
-  rw [← covariance_self (hl.aemeasurable.pow_const 2 : AEMeasurable (fun ω => (B t ω - B s ω) ^ 2) P)]
-  change cov[(fun x : ℝ => x ^ 2) ∘ (fun ω => B t ω - B s ω),
-    (fun x : ℝ => x ^ 2) ∘ (fun ω => B t ω - B s ω); P] = _
-  rw [hl.covariance_comp (by fun_prop) (by fun_prop), covariance_self (by fun_prop),
-    gaussian_square_variance]
-
- 
-
-abbrev pastSigma {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (s : ℝ≥0) : MeasurableSpace Ω :=
-  MeasurableSpace.comap (fun ω (t : Set.Iic s) => B t ω) inferInstance
-
-lemma brownian_indep_of_past_measurable {Ω α : Type*}
-    [MeasurableSpace Ω] [MeasurableSpace α] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (s v : ℝ≥0)
-    {X : Ω → α} (hX : Measurable[pastSigma B s] X) :
-    IndepFun X (fun ω => B (s+v) ω - B s ω) P := by
-  have hh := (hB.toIsPreBrownianReal.indepFun_shift s).symm.comp
-    measurable_id (measurable_pi_apply v)
-  change Indep (MeasurableSpace.comap X inferInstance)
-    (MeasurableSpace.comap (fun ω => B (s+v) ω - B s ω) inferInstance) P
-  exact indep_of_indep_of_le_left hh hX.comap_le
-
-lemma pastSigma_mono {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    {s t : ℝ≥0} (hst : s ≤ t) : pastSigma B s ≤ pastSigma B t := by
-  let : MeasurableSpace Ω := pastSigma B t
-  have hm : Measurable (fun ω (u : Set.Iic s) => B u ω) := by
-    apply Measurable.of_eval
-    intro u
-    exact (measurable_pi_apply (⟨u, u.2.trans hst⟩ : Set.Iic t)).comp
-      (comap_measurable (fun ω (v : Set.Iic t) => B v ω))
-  exact hm.comap_le
-
-lemma past_measurable_eval {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    {s t : ℝ≥0} (hst : s ≤ t) : Measurable[pastSigma B t] (B s) :=
-  (measurable_pi_apply (⟨s, hst⟩ : Set.Iic t)).comp
-    (comap_measurable (fun ω (v : Set.Iic t) => B v ω))
-
-lemma past_mul_feature_integral {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {s t : ℝ≥0} (hst : s ≤ t) {X : Ω → ℝ}
-    (hX : Measurable[pastSigma B s] X) (hXa : AEMeasurable X P)
-    (φ : ℝ → ℝ) (hφ : Measurable φ) :
-    (∫ ω, X ω * φ (B t ω - B s ω) ∂P) =
-      (∫ ω, X ω ∂P) * (∫ x, φ x ∂gaussianReal 0 (t-s)) := by
-  have hi := (brownian_indep_of_past_measurable hB s (t-s) hX).comp
-    measurable_id hφ
-  rw [add_tsub_cancel_of_le hst] at hi
-  have hl := increment_hasLaw hB hst
-  change IndepFun X (fun ω => φ (B t ω - B s ω)) P at hi
-  rw [hi.integral_fun_mul_eq_mul_integral hXa.aestronglyMeasurable
-      (hφ.comp_aemeasurable hl.aemeasurable).aestronglyMeasurable]
-  rw [show (∫ ω, φ (B t ω - B s ω) ∂P) = ∫ x, φ x ∂gaussianReal 0 (t-s) from
-    hl.integral_comp hφ.aestronglyMeasurable]
-
-lemma independent_product_memLp_two {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {X Y : Ω → ℝ} (hX : MemLp X 2 P) (hY : MemLp Y 2 P)
-    (hXY : IndepFun X Y P) : MemLp (fun ω => X ω * Y ω) 2 P := by
-  apply (memLp_two_iff_integrable_sq (hX.aestronglyMeasurable.mul hY.aestronglyMeasurable)).2
-  have hi := (hXY.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integrable_mul
-    hX.integrable_sq hY.integrable_sq
-  convert hi using 1
-  ext ω
-  simp only [Function.comp_apply, Pi.mul_apply, mul_pow, id_eq]
-
- 
-def predictableFeatureSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (t : ℕ → ℝ≥0) (C : ℕ → Ω → ℝ) (φ : ℕ → ℝ → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
-  ∑ i ∈ Finset.range n, C i ω * φ i (B (t (i+1)) ω - B (t i) ω)
-
-lemma predictableTerm_memLp_two {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i)) (hC2 : ∀ i, MemLp (C i) 2 P)
-    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i))
-    (hφ2 : ∀ i, MemLp (φ i) 2 (gaussianReal 0 (t (i+1)-t i))) (i : ℕ) :
-    MemLp (fun ω => C i ω * φ i (B (t (i+1)) ω - B (t i) ω)) 2 P := by
-  have ht' := ht (Nat.le_succ i)
-  have hl := increment_hasLaw hB ht'
-  have hY := (hl.map_eq ▸ hφ2 i).comp_of_map hl.aemeasurable
-  have hi := (brownian_indep_of_past_measurable hB (t i) (t (i+1)-t i) (hC i)).comp
-    measurable_id (hφ i)
-  rw [add_tsub_cancel_of_le ht'] at hi
-  exact independent_product_memLp_two (hC2 i) hY hi
-
-lemma predictableTerm_integral {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i)) (hCa : ∀ i, AEMeasurable (C i) P)
-    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i))
-    (hφ0 : ∀ i, (∫ x, φ i x ∂gaussianReal 0 (t (i+1)-t i)) = 0) (i : ℕ) :
-    (∫ ω, C i ω * φ i (B (t (i+1)) ω - B (t i) ω) ∂P) = 0 := by
-  rw [past_mul_feature_integral hB (ht (Nat.le_succ i)) (hC i) (hCa i) (φ i) (hφ i),
-    hφ0, mul_zero]
-
-lemma predictableTerm_past_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i))
-    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i)) {i j : ℕ} (hij : i+1 ≤ j) :
-    Measurable[pastSigma B (t j)]
-      (fun ω => C i ω * φ i (B (t (i+1)) ω - B (t i) ω)) := by
-  have hi : t i ≤ t j := ht (le_trans (Nat.le_succ i) hij)
-  exact ((hC i).mono (pastSigma_mono B hi) le_rfl).mul
-    ((hφ i).comp ((past_measurable_eval B (ht hij)).sub
-      (past_measurable_eval B hi)))
-
-lemma predictableTerm_covariance_zero {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
-    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i)) (hC2 : ∀ i, MemLp (C i) 2 P)
-    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i))
-    (hφ2 : ∀ i, MemLp (φ i) 2 (gaussianReal 0 (t (i+1)-t i)))
-    (hφ0 : ∀ i, (∫ x, φ i x ∂gaussianReal 0 (t (i+1)-t i)) = 0)
-    {i j : ℕ} (hij : i ≠ j) :
-    covariance (fun ω => C i ω * φ i (B (t (i+1)) ω - B (t i) ω))
-      (fun ω => C j ω * φ j (B (t (j+1)) ω - B (t j) ω)) P = 0 := by
-  wlog hlt : i < j generalizing i j
-  · rw [covariance_comm]
-    exact this hij.symm (lt_of_le_of_ne (not_lt.mp hlt) hij.symm)
-  have hAi := predictableTerm_memLp_two hB t ht C hC hC2 φ hφ hφ2 i
-  have hAj := predictableTerm_memLp_two hB t ht C hC hC2 φ hφ hφ2 j
-  rw [covariance_eq_sub hAi hAj,
-    predictableTerm_integral hB t ht C hC (fun i => (hC2 i).aemeasurable) φ hφ hφ0 i,
-    zero_mul, sub_zero]
-  have hp := (predictableTerm_past_measurable B t ht C hC φ hφ hlt).mul (hC j)
-  have hh := past_mul_feature_integral hB (ht (Nat.le_succ j)) hp
-    (hAi.aemeasurable.mul (hC2 j).aemeasurable) (φ j) (hφ j)
-  rw [hφ0, mul_zero] at hh
-  convert hh using 1
-  congr 1
-  ext ω
-  simp only [Pi.mul_apply]
-  ring
-
-end SLEExactGauge.BrownianPredictable
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.SpatialMeans
+
+namespace OAI
+
+/-! Stochastic Integrals for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+namespace SLEExactGauge
+noncomputable def spatialLaterKernel (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ)
+    (r : ℝ) (z w : ℂ) : ℝ≥0∞ :=
+  if spatialHitTime γ z r = ⊤ then 0 else
+    let s := (spatialHitTime γ z r).untopD 0
+    if z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s ∧
+        spatialHitTime γ w r ≤ s then ENNReal.ofReal (halfPlaneGreen (G s z) (G s w)) else 0
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+lemma IsCapacityTwoWitness.spatial_later_kernel_integral
+    (h : IsCapacityTwoWitness U γ G F) {r : ℝ} (hr : 0 < r) (z : ℂ) :
+    (∫⁻ w, spatialLaterKernel γ G r z w) ≤ ENNReal.ofReal (layerGreenConstant * r ^ 2) := by
+  classical
+  by_cases ht : spatialHitTime γ z r = ⊤
+  · simp only [spatialLaterKernel, ite_eq_left ht, lintegral_zero, zero_le]
+  obtain ⟨s, hs⟩ := WithTop.ne_top_iff_exists.mp ht
+  by_cases hz : z ∈ survivingDomain γ s
+  · let L := {w | w ∈ survivingDomain γ s ∧ infDist w (frontier (survivingDomain γ s)) ≤ r}
+    have hp (w : ℂ) : spatialLaterKernel γ G r z w ≤
+        L.indicator (fun w => ENNReal.ofReal (halfPlaneGreen (G s z) (G s w))) w := by
+      rw [spatialLaterKernel, ite_eq_right ht, ← hs, WithTop.untopD_coe]
+      by_cases ha : z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s ∧ spatialHitTime γ w r ≤ s
+      · rw [ite_eq_left ha]
+        obtain ⟨v, hv, he⟩ := (spatialHitTime_le_iff h.1 w r s).mp ha.2.2
+        have hw : w ∈ L := ⟨ha.2.1, (surviving_frontier_distance_le_past s v hv.2 w ha.2.1).trans_eq he⟩
+        rw [Set.indicator_of_mem hw]
+      · rw [ite_eq_right ha]
+        exact zero_le
+    calc
+      _ ≤ ∫⁻ w, L.indicator (fun w => ENNReal.ofReal (halfPlaneGreen (G s z) (G s w))) w :=
+        lintegral_mono hp
+      _ ≤ ∫⁻ w in L, ENNReal.ofReal (halfPlaneGreen (G s z) (G s w)) := lintegral_indicator_le _ _
+      _ ≤ _ := h.boundary_layer_green s hr hz
+  · have he (w : ℂ) : spatialLaterKernel γ G r z w = 0 := by
+      rw [spatialLaterKernel, ite_eq_right ht, ← hs, WithTop.untopD_coe]
+      simp only [hz, false_and, ite_false]
+    simp only [he, lintegral_zero, zero_le]
+
+lemma slab_product_zero_of_not_live_at_start
+    (κ r ε : ℝ) (z w : ℂ) (s : ℝ≥0) (hs : spatialHitTime γ z r = s)
+    (hn : ¬ (z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s)) (t : ℝ≥0) :
+    spatialSlabCoefficient κ U γ G r ε t z * spatialSlabCoefficient κ U γ G r ε t w = 0 := by
+  by_cases ha : spatialHitTime γ z r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧
+      z ∈ survivingDomain γ t
+  · by_cases hb : spatialHitTime γ w r ≤ t ∧ (t : WithTop ℝ≥0) ≤ spatialHitTime γ w ε ∧
+        w ∈ survivingDomain γ t
+    · have hst : s ≤ t := WithTop.coe_le_coe.mp (hs ▸ ha.1)
+      exact (hn ⟨survivingDomain_antitone γ hst ha.2.2, survivingDomain_antitone γ hst hb.2.2⟩).elim
+    · simp only [spatialSlabCoefficient, ite_eq_right hb, mul_zero]
+  · simp only [spatialSlabCoefficient, ite_eq_right ha, zero_mul]
+
+lemma IsCapacityTwoWitness.slab_pair_ordered
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
+    (r : ℝ) (z w : ℂ) (hzy : ε ≤ z.im) (hwy : ε ≤ w.im) (hne : z ≠ w)
+    (horder : spatialHitTime γ w r ≤ spatialHitTime γ z r) :
+    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+      spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        spatialLaterKernel γ G r z w := by
+  classical
+  by_cases ht : spatialHitTime γ z r = ⊤
+  · have he (t : ℝ) : spatialSlabCoefficient κ U γ G r ε t.toNNReal z = 0 := by
+      have hn : ¬ (spatialHitTime γ z r ≤ t.toNNReal ∧
+          (t.toNNReal : WithTop ℝ≥0) ≤ spatialHitTime γ z ε ∧ z ∈ survivingDomain γ t.toNNReal) := by
+        intro hh
+        rw [ht] at hh
+        exact WithTop.coe_ne_top (top_le_iff.mp hh.1)
+      exact ite_eq_right hn
+    simp only [he, zero_mul, abs_zero, ENNReal.ofReal_zero, lintegral_zero, zero_le]
+  obtain ⟨s, hs⟩ := WithTop.ne_top_iff_exists.mp ht
+  by_cases hl : z ∈ survivingDomain γ s ∧ w ∈ survivingDomain γ s
+  · have hord : spatialHitTime γ w r ≤ s := hs ▸ horder
+    have he : spatialLaterKernel γ G r z w = ENNReal.ofReal (halfPlaneGreen (G s z) (G s w)) := by
+      rw [spatialLaterKernel, ite_eq_right ht, ← hs, WithTop.untopD_coe, ite_eq_left ⟨hl.1, hl.2, hord⟩]
+    rw [he]
+    have hb := h.slab_pair_later_start hU hκ hκ4 hε r z w hzy hwy hne s hs.symm
+    convert hb using 1
+    rw [← ENNReal.ofReal_mul (sq_nonneg _), ← ENNReal.ofReal_mul (div_nonneg (sq_nonneg _) (by norm_num))]
+    congr 1
+    ring
+  · simp only [slab_product_zero_of_not_live_at_start κ r ε z w s hs.symm hl,
+      abs_zero, ENNReal.ofReal_zero, lintegral_zero, zero_le]
+
+ 
+
+lemma IsCapacityTwoWitness.slab_pair_kernel_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε)
+    (r : ℝ) (z w : ℂ) (hzy : ε ≤ z.im) (hwy : ε ≤ w.im) (hne : z ≠ w) :
+    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+      spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (spatialLaterKernel γ G r z w + spatialLaterKernel γ G r w z) := by
+  rcases le_total (spatialHitTime γ w r) (spatialHitTime γ z r) with ho | ho
+  · exact (h.slab_pair_ordered hU hκ hκ4 hε r z w hzy hwy hne ho).trans
+      (mul_le_mul le_rfl (le_add_right le_rfl) bot_le bot_le)
+  · have hb := h.slab_pair_ordered hU hκ hκ4 hε r w z hwy hzy hne.symm ho
+    simp only [mul_comm (spatialSlabCoefficient κ U γ G r ε _ w)] at hb
+    exact hb.trans (mul_le_mul le_rfl (le_add_left le_rfl) bot_le bot_le)
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal ComplexConjugate
+namespace SLEExactGauge
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+lemma spatialHitTime_measurable_point (hc : Continuous γ) (r : ℝ) :
+    Measurable (fun z => spatialHitTime γ z r) := by
+  have hm := spatialHitTime_measurable (fun _ : Unit => γ) (fun _ => hc)
+    (fun _ => measurable_const) r
+  exact hm.comp (measurable_id.prodMk (measurable_const (a := ())))
+
+lemma IsCapacityTwoWitness.joint_live_measurable
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) :
+    MeasurableSet {p : ℝ≥0 × ℂ | p.2 ∈ survivingDomain γ p.1} := by
+  let c := spatialClippedFamily (fun t (_ : Unit) => U t)
+    (hU.measurable.comp measurable_fst) (fun _ => hU)
+  have hm := c.live_measurable (fun _ => γ) (fun _ => G) (fun _ => F) (fun _ => hU) (fun _ => h)
+  exact hm.preimage (measurable_fst.prodMk (measurable_snd.prodMk (measurable_const (a := ()))))
+
+lemma IsCapacityTwoWitness.killed_chart_measurable
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) :
+    Measurable (fun p : ℝ≥0 × ℂ => if p.2 ∈ survivingDomain γ p.1 then G p.1 p.2 else 0) := by
+  let c := spatialClippedFamily (fun t (_ : Unit) => U t)
+    (hU.measurable.comp measurable_fst) (fun _ => hU)
+  convert c.jointChart_measurable.comp
+    (measurable_fst.prodMk (measurable_snd.prodMk (measurable_const (a := ())))) using 1
+  funext p
+  exact (c.jointChart_eq () hU h p.1 p.2).symm
+
+lemma IsCapacityTwoWitness.spatial_later_kernel_measurable
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (r : ℝ) :
+    Measurable (fun p : ℂ × ℂ => spatialLaterKernel γ G r p.1 p.2) := by
+  classical
+  let H (p : ℝ≥0 × ℂ) := if p.2 ∈ survivingDomain γ p.1 then G p.1 p.2 else 0
+  let s (z : ℂ) := (spatialHitTime γ z r).untopD 0
+  have hm := spatialHitTime_measurable_point h.1 r
+  have hs : Measurable s := hm.untopD 0
+  have hH : Measurable H := h.killed_chart_measurable hU
+  have hz : Measurable (fun p : ℂ × ℂ => H (s p.1, p.1)) := hH.comp ((hs.comp measurable_fst).prodMk measurable_fst)
+  have hw : Measurable (fun p : ℂ × ℂ => H (s p.1, p.2)) := hH.comp ((hs.comp measurable_fst).prodMk measurable_snd)
+  have hK : Measurable (fun p : ℂ × ℂ => ENNReal.ofReal (halfPlaneGreen (H (s p.1, p.1)) (H (s p.1, p.2)))) := by
+    exact (Real.measurable_log.comp ((hz.sub (Complex.continuous_conj.measurable.comp hw)).div (hz.sub hw)).norm).ennreal_ofReal
+  have hzl : MeasurableSet {p : ℂ × ℂ | p.1 ∈ survivingDomain γ (s p.1)} :=
+    (h.joint_live_measurable hU).preimage ((hs.comp measurable_fst).prodMk measurable_fst)
+  have hwl : MeasurableSet {p : ℂ × ℂ | p.2 ∈ survivingDomain γ (s p.1)} :=
+    (h.joint_live_measurable hU).preimage ((hs.comp measurable_fst).prodMk measurable_snd)
+  have ho : MeasurableSet {p : ℂ × ℂ | spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0)} :=
+    measurableSet_le (hm.comp measurable_snd) ((hs.comp measurable_fst).withTop_coe)
+  have hnf : MeasurableSet {p : ℂ × ℂ | spatialHitTime γ p.1 r = ⊤} :=
+    measurableSet_eq_fun (hm.comp measurable_fst) measurable_const
+  have hk : Measurable (fun p : ℂ × ℂ =>
+      if p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
+        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0) then
+        ENNReal.ofReal (halfPlaneGreen (H (s p.1, p.1)) (H (s p.1, p.2))) else 0) := by
+    exact @Measurable.ite _ _ _ _ _ _
+      (fun p : ℂ × ℂ => p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
+        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0)) (fun _ => inferInstance)
+      (hzl.inter (hwl.inter ho)) hK (measurable_const (a := (0 : ℝ≥0∞)))
+  have hh : Measurable (fun p : ℂ × ℂ => if spatialHitTime γ p.1 r = ⊤ then 0 else
+      if p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
+        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0) then
+        ENNReal.ofReal (halfPlaneGreen (H (s p.1, p.1)) (H (s p.1, p.2))) else 0) := by
+    exact @Measurable.ite _ _ _ _ _ _ (fun p : ℂ × ℂ => spatialHitTime γ p.1 r = ⊤)
+      (fun _ => inferInstance) hnf
+      (measurable_const (a := (0 : ℝ≥0∞))) hk
+  convert hh using 1
+  funext p
+  dsimp only [spatialLaterKernel]
+  by_cases hn : spatialHitTime γ p.1 r = ⊤
+  · simp only [hn, ite_eq_left]
+  · rw [ite_eq_right hn, ite_eq_right hn]
+    change (if p.1 ∈ survivingDomain γ (s p.1) ∧ p.2 ∈ survivingDomain γ (s p.1) ∧
+        spatialHitTime γ p.2 r ≤ (s p.1 : WithTop ℝ≥0) then _ else _) = _
+    split_ifs with hl
+    · simp only [H, ite_eq_left hl.1, ite_eq_left hl.2.1]
+      rfl
+    · rfl
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+lemma IsCapacityTwoWitness.spatial_later_kernel_double
+    (h : IsCapacityTwoWitness U γ G F) {r : ℝ} (hr : 0 < r) (A : Set ℂ) :
+    (∫⁻ z in A, ∫⁻ w in A, spatialLaterKernel γ G r z w) ≤
+      ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A := by
+  calc
+    _ ≤ ∫⁻ _z in A, ENNReal.ofReal (layerGreenConstant * r ^ 2) := by
+      apply lintegral_mono
+      intro z
+      exact (lintegral_mono' Measure.restrict_le_self le_rfl).trans (h.spatial_later_kernel_integral hr z)
+    _ = _ := by rw [lintegral_const, Measure.restrict_apply_univ]
+
+lemma sym_pair_lintegral {α : Type*} [MeasurableSpace α] (μ : Measure α)
+    [SFinite μ] (K : α → α → ℝ≥0∞) (hK : Measurable (Function.uncurry K))
+    (C : ℝ≥0∞) (hC : C ≠ ∞) :
+    (∫⁻ z, ∫⁻ w, C * (K z w + K w z) ∂μ ∂μ) =
+      C * (2 * (∫⁻ z, ∫⁻ w, K z w ∂μ ∂μ)) := by
+  have hi (z : α) : (∫⁻ w, C * (K z w + K w z) ∂μ) =
+      C * ((∫⁻ w, K z w ∂μ) + (∫⁻ w, K w z ∂μ)) := by
+    rw [lintegral_const_mul' C _ hC]
+    have hm : Measurable (fun w => K z w) := hK.comp (measurable_const.prodMk measurable_id)
+    rw [lintegral_add_left hm]
+  rw [lintegral_congr hi, lintegral_const_mul' C _ hC]
+  have ho : Measurable (fun z => ∫⁻ w, K z w ∂μ) := hK.lintegral_prod_right'
+  rw [lintegral_add_left ho]
+  have hs : (∫⁻ z, ∫⁻ w, K w z ∂μ ∂μ) = (∫⁻ z, ∫⁻ w, K z w ∂μ ∂μ) :=
+    lintegral_lintegral_swap (hK.comp measurable_swap).aemeasurable
+  rw [hs, two_mul]
+
+lemma IsCapacityTwoWitness.spatial_pair_lintegral_le
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (r : ℝ)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
+    (∫⁻ z in A, ∫⁻ w in A, ∫⁻ t in Ici (0 : ℝ),
+      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+        spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
+    (∫⁻ z in A, ∫⁻ w in A,
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (spatialLaterKernel γ G r z w + spatialLaterKernel γ G r w z)) := by
+  apply lintegral_mono_ae
+  filter_upwards [ae_restrict_mem hA] with z hz
+  apply lintegral_mono_ae
+  filter_upwards [ae_restrict_mem hA, ae_restrict_of_ae ((volume : Measure ℂ).ae_ne z)] with w hw hne
+  exact h.slab_pair_kernel_bound hU hκ hκ4 hε r z w (hheight z hz) (hheight w hw) hne.symm
+
+ 
+
+theorem IsCapacityTwoWitness.spatial_slab_pair_area_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
+    (∫⁻ z in A, ∫⁻ w in A, ∫⁻ t in Ici (0 : ℝ),
+      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+        spatialSlabCoefficient κ U γ G r ε t.toNNReal w|) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
+  refine (h.spatial_pair_lintegral_le hU hκ hκ4 hε r A hA hheight).trans ?_
+  rw [sym_pair_lintegral (volume.restrict A) (spatialLaterKernel γ G r) (h.spatial_later_kernel_measurable hU r)
+    _ ENNReal.ofReal_ne_top]
+  exact mul_le_mul le_rfl
+    (mul_le_mul le_rfl (h.spatial_later_kernel_double hr A) bot_le bot_le) bot_le bot_le
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal ComplexConjugate
+namespace SLEExactGauge
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+lemma IsCapacityTwoWitness.killed_green_measurable
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) {κ : ℝ} (hκ : 0 < κ) :
+    Measurable (fun p : ℝ≥0 × ℂ => greenDensity κ U γ G p.1 p.2) := by
+  let c := spatialClippedFamily (fun t (_ : Unit) => U t)
+    (hU.measurable.comp measurable_fst) (fun _ => hU)
+  let B (t : ℝ≥0) (_ : Unit) := U t / Real.sqrt κ
+  have hBm : Measurable (fun p : ℝ≥0 × Unit => B p.1 p.2) :=
+    (hU.measurable.comp measurable_fst).div_const _
+  have hm := (c.jointGreen_measurable κ B hBm).comp
+    (measurable_fst.prodMk (measurable_snd.prodMk (measurable_const (a := ()))))
+  convert hm using 1
+  funext p
+  apply (c.jointGreen_eq κ B () hU _ h p.1 p.2).symm
+  intro t
+  dsimp only [B]
+  field_simp [(Real.sqrt_pos.mpr hκ).ne']
+
+lemma IsCapacityTwoWitness.green_coefficient_measurable
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) {κ : ℝ} (hκ : 0 < κ) :
+    Measurable (fun p : ℝ≥0 × ℂ => greenBrownianCoefficient κ U γ G p.1 p.2) := by
+  classical
+  let H (p : ℝ≥0 × ℂ) := if p.2 ∈ survivingDomain γ p.1 then G p.1 p.2 else 0
+  have hZ : Measurable (fun p : ℝ≥0 × ℂ => H p - (U p.1 : ℂ)) :=
+    (h.killed_chart_measurable hU).sub (Complex.measurable_ofReal.comp (hU.measurable.comp measurable_fst))
+  have hb := (((Complex.measurable_re.comp hZ).const_mul (Real.sqrt κ * greenExponent κ)).div
+    (hZ.norm.pow_const 2)).mul (h.killed_green_measurable hU hκ)
+  convert hb using 1
+  funext p
+  by_cases hl : p.2 ∈ survivingDomain γ p.1
+  · simp only [Pi.mul_apply, Pi.div_apply, Function.comp_apply, H, ite_eq_left hl, greenBrownianCoefficient]
+  · simp only [Pi.mul_apply, Pi.div_apply, Function.comp_apply, greenBrownianCoefficient, greenDensity, ite_eq_right hl, mul_zero]
+
+lemma IsCapacityTwoWitness.spatial_slab_coefficient_measurable
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U) {κ : ℝ} (hκ : 0 < κ)
+    (r ε : ℝ) :
+    Measurable (fun p : ℝ≥0 × ℂ => spatialSlabCoefficient κ U γ G r ε p.1 p.2) := by
+  classical
+  have ht : Measurable (fun p : ℝ≥0 × ℂ => (p.1 : WithTop ℝ≥0)) := measurable_fst.withTop_coe
+  have hr : Measurable (fun p : ℝ≥0 × ℂ => spatialHitTime γ p.2 r) :=
+    (spatialHitTime_measurable_point h.1 r).comp measurable_snd
+  have he : Measurable (fun p : ℝ≥0 × ℂ => spatialHitTime γ p.2 ε) :=
+    (spatialHitTime_measurable_point h.1 ε).comp measurable_snd
+  have hm : MeasurableSet {p : ℝ≥0 × ℂ | spatialHitTime γ p.2 r ≤ p.1 ∧
+      (p.1 : WithTop ℝ≥0) ≤ spatialHitTime γ p.2 ε ∧ p.2 ∈ survivingDomain γ p.1} :=
+    (measurableSet_le hr ht).inter ((measurableSet_le ht he).inter (h.joint_live_measurable hU))
+  exact @Measurable.ite _ _ _ _ _ _ (fun p : ℝ≥0 × ℂ => spatialHitTime γ p.2 r ≤ p.1 ∧
+      (p.1 : WithTop ℝ≥0) ≤ spatialHitTime γ p.2 ε ∧ p.2 ∈ survivingDomain γ p.1)
+    (fun _ => inferInstance) hm
+    (h.green_coefficient_measurable hU hκ) measurable_const
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge
+
+ 
+
+lemma area_square_tonelli {T Z : Type*} [MeasurableSpace T] [MeasurableSpace Z]
+    {μ : Measure T} {ν : Measure Z} [SFinite μ] [SFinite ν]
+    {f : T × Z → ℝ≥0∞} (hf : Measurable f) :
+    (∫⁻ t, (∫⁻ z, f (t, z) ∂ν) ^ 2 ∂μ) =
+      ∫⁻ z, ∫⁻ w, ∫⁻ t, f (t, z) * f (t, w) ∂μ ∂ν ∂ν := by
+  have hft (t : T) : Measurable (fun z => f (t, z)) :=
+    hf.comp (measurable_const.prodMk measurable_id)
+  have h3 : Measurable (fun p : (T × Z) × Z => f (p.1.1, p.1.2) * f (p.1.1, p.2)) :=
+    (hf.comp measurable_fst).mul
+      (hf.comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd))
+  have hinner : Measurable (fun p : T × Z => ∫⁻ w, f (p.1, p.2) * f (p.1, w) ∂ν) :=
+    h3.lintegral_prod_right'
+  calc
+    _ = ∫⁻ t, ∫⁻ z, ∫⁻ w, f (t, z) * f (t, w) ∂ν ∂ν ∂μ := by
+      apply lintegral_congr
+      intro t
+      simp_rw [lintegral_const_mul _ (hft t)]
+      rw [lintegral_mul_const _ (hft t), pow_two]
+    _ = ∫⁻ z, ∫⁻ t, ∫⁻ w, f (t, z) * f (t, w) ∂ν ∂μ ∂ν :=
+      lintegral_lintegral_swap hinner.aemeasurable
+    _ = _ := by
+      apply lintegral_congr
+      intro z
+      apply lintegral_lintegral_swap
+      exact ((hf.comp (measurable_fst.prodMk measurable_const)).mul hf).aemeasurable
+
+lemma spatial_integral_square_le {Z : Type*} [MeasurableSpace Z] (ν : Measure Z)
+    (f : Z → ℝ) :
+    ENNReal.ofReal ((∫ z, f z ∂ν) ^ 2) ≤ (∫⁻ z, ENNReal.ofReal |f z| ∂ν) ^ 2 := by
+  have he (x : ℝ) : ‖x‖ₑ = ENNReal.ofReal |x| := by
+    rw [← ofReal_norm, Real.norm_eq_abs]
+  have hb := pow_le_pow_left₀ bot_le (enorm_integral_le_lintegral_enorm (μ := ν) f) 2
+  rw [← enorm_pow, Real.enorm_eq_ofReal (sq_nonneg _)] at hb
+  simpa only [he] using hb
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+
+ 
+
+theorem IsCapacityTwoWitness.spatial_slab_absolute_bracket_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
+    (∫⁻ t in Ici (0 : ℝ), (∫⁻ z in A,
+      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|) ^ 2) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
+  have hb : Measurable (fun p : ℝ × ℂ => spatialSlabCoefficient κ U γ G r ε p.1.toNNReal p.2) := by
+    have hm : Measurable (fun p : ℝ × ℂ => (p.1.toNNReal, p.2)) :=
+      (continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd
+    exact (h.spatial_slab_coefficient_measurable hU hκ r ε).comp
+      (f := fun p : ℝ × ℂ => (p.1.toNNReal, p.2)) hm
+  rw [area_square_tonelli hb.abs.ennreal_ofReal]
+  have he (t : ℝ) (z w : ℂ) :
+      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z| *
+        ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal w| =
+      ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z *
+        spatialSlabCoefficient κ U γ G r ε t.toNNReal w| := by
+    rw [abs_mul, ENNReal.ofReal_mul (abs_nonneg _)]
+  simp_rw [he]
+  exact h.spatial_slab_pair_area_bound hU hκ hκ4 hε hr A hA hheight
+
+ 
+
+theorem IsCapacityTwoWitness.spatial_slab_bracket_bound
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im) :
+    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal
+      ((∫ z in A, spatialSlabCoefficient κ U γ G r ε t.toNNReal z) ^ 2)) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
+  exact (lintegral_mono fun t => spatial_integral_square_le (volume.restrict A) _).trans
+    (h.spatial_slab_absolute_bracket_bound hU hκ hκ4 hε hr A hA hheight)
+
+end SLEExactGauge
+
+ 
+
+ 
+
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.BrownianPredictable
+
+ 
+
+lemma gaussian_fourth_moment (v : ℝ≥0) :
+    (∫ x : ℝ, x ^ 4 ∂gaussianReal 0 v) = 3 * (v : ℝ) ^ 2 := by
+  let f : ℝ → ℝ := fun t => Real.exp ((v : ℝ) * t ^ 2 / 2)
+  have hf (t : ℝ) : HasDerivAt f ((v : ℝ) * t * f t) t := by
+    apply (((hasDerivAt_id t).pow 2 |>.const_mul (v : ℝ)).div_const 2).exp.congr_deriv
+    dsimp [f]
+    ring
+  have hd : deriv f = fun t => (v : ℝ) * t * f t := funext fun t => (hf t).deriv
+  have hd1 : deriv (fun t => (v : ℝ) * t * f t) =
+      fun t => ((v : ℝ) + (v : ℝ) ^ 2 * t ^ 2) * f t := by
+    funext t
+    apply HasDerivAt.deriv
+    apply (((hasDerivAt_id t).const_mul (v : ℝ)).mul (hf t)).congr_deriv
+    simp only [id_eq]
+    ring
+  have hd2 : deriv (fun t => ((v : ℝ) + (v : ℝ) ^ 2 * t ^ 2) * f t) =
+      fun t => (3 * (v : ℝ) ^ 2 * t + (v : ℝ) ^ 3 * t ^ 3) * f t := by
+    funext t
+    apply HasDerivAt.deriv
+    apply (((((hasDerivAt_id t).pow 2).const_mul ((v : ℝ) ^ 2)).const_add
+      (v : ℝ)).mul (hf t)).congr_deriv
+    simp only [id_eq, Pi.pow_apply]
+    ring
+  have hd3 : deriv (fun t => (3 * (v : ℝ) ^ 2 * t + (v : ℝ) ^ 3 * t ^ 3) * f t) =
+      fun t => (3 * (v : ℝ) ^ 2 + 6 * (v : ℝ) ^ 3 * t ^ 2 +
+        (v : ℝ) ^ 4 * t ^ 4) * f t := by
+    funext t
+    apply HasDerivAt.deriv
+    apply ((((hasDerivAt_id t).const_mul (3 * (v : ℝ) ^ 2)).add
+      (((hasDerivAt_id t).pow 3).const_mul ((v : ℝ) ^ 3))).mul (hf t)).congr_deriv
+    simp only [id_eq, Pi.pow_apply, Pi.add_apply]
+    ring
+  change (∫ x, ((fun x : ℝ => x) ^ 4) x ∂gaussianReal 0 v) = _
+  rw [← iteratedDeriv_mgf_zero (by simp), mgf_fun_id_gaussianReal]
+  simp only [zero_mul, zero_add]
+  change iteratedDeriv 4 f 0 = _
+  simp only [iteratedDeriv_succ, iteratedDeriv_zero, hd, hd1, hd2, hd3]
+  simp [f]
+
+lemma gaussian_second_moment (v : ℝ≥0) :
+    (∫ x : ℝ, x ^ 2 ∂gaussianReal 0 v) = (v : ℝ) := by
+  have hv := variance_eq_sub (memLp_id_gaussianReal (μ := 0) (v := v) 2)
+  simpa [integral_id_gaussianReal] using hv.symm
+
+lemma gaussian_square_memLp (v : ℝ≥0) :
+    MemLp (fun x : ℝ => x ^ 2) 2 (gaussianReal 0 v) := by
+  apply (memLp_two_iff_integrable_sq (by fun_prop)).2
+  have h4 := (memLp_id_gaussianReal (μ := 0) (v := v) 4).integrable_norm_pow
+    (p := 4) (by norm_num)
+  convert h4 using 1
+  funext x
+  simp only [id_eq, Real.norm_eq_abs]
+  rw [show (4 : ℕ) = 2 * 2 from rfl, pow_mul, sq_abs]
+
+lemma gaussian_square_variance (v : ℝ≥0) :
+    variance (fun x : ℝ => x ^ 2) (gaussianReal 0 v) = 2 * (v : ℝ) ^ 2 := by
+  rw [variance_eq_sub (gaussian_square_memLp v), gaussian_second_moment]
+  have heq : (fun x : ℝ => ((fun x : ℝ => x ^ 2) ^ 2) x) = fun x : ℝ => x ^ 4 := by
+    funext x
+    simp only [Pi.pow_apply]
+    ring
+  rw [show (fun x : ℝ => x ^ 2) ^ 2 = (fun x : ℝ => x ^ 4) from heq,
+    gaussian_fourth_moment]
+  ring
+
+lemma increment_hasLaw {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {s t : ℝ≥0} (hst : s ≤ t) :
+    HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t-s)) P := by
+  simpa only [add_tsub_cancel_of_le hst] using
+    (hB.toIsPreBrownianReal.shift s).hasLaw_eval (t-s)
+
+lemma increment_square_memLp {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {s t : ℝ≥0} (hst : s ≤ t) :
+    MemLp (fun ω => (B t ω - B s ω) ^ 2) 2 P := by
+  have hl := increment_hasLaw hB hst
+  exact (hl.map_eq ▸ gaussian_square_memLp (t-s)).comp_of_map hl.aemeasurable
+
+lemma increment_square_integral {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {s t : ℝ≥0} (hst : s ≤ t) :
+    (∫ ω, (B t ω - B s ω) ^ 2 ∂P) = (t-s : ℝ≥0) := by
+  have hl := increment_hasLaw hB hst
+  simpa only [Function.comp_apply, gaussian_second_moment] using
+    hl.integral_comp (show AEStronglyMeasurable (fun x : ℝ => x ^ 2)
+      (gaussianReal 0 (t-s)) from by fun_prop)
+
+lemma increment_square_variance {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {s t : ℝ≥0} (hst : s ≤ t) :
+    variance (fun ω => (B t ω - B s ω) ^ 2) P = 2 * (t-s : ℝ≥0) ^ 2 := by
+  have hl := increment_hasLaw hB hst
+  rw [← covariance_self (hl.aemeasurable.pow_const 2 : AEMeasurable (fun ω => (B t ω - B s ω) ^ 2) P)]
+  change cov[(fun x : ℝ => x ^ 2) ∘ (fun ω => B t ω - B s ω),
+    (fun x : ℝ => x ^ 2) ∘ (fun ω => B t ω - B s ω); P] = _
+  rw [hl.covariance_comp (by fun_prop) (by fun_prop), covariance_self (by fun_prop),
+    gaussian_square_variance]
+
+ 
+
+abbrev pastSigma {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (s : ℝ≥0) : MeasurableSpace Ω :=
+  MeasurableSpace.comap (fun ω (t : Set.Iic s) => B t ω) inferInstance
+
+lemma brownian_indep_of_past_measurable {Ω α : Type*}
+    [MeasurableSpace Ω] [MeasurableSpace α] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (s v : ℝ≥0)
+    {X : Ω → α} (hX : Measurable[pastSigma B s] X) :
+    IndepFun X (fun ω => B (s+v) ω - B s ω) P := by
+  have hh := (hB.toIsPreBrownianReal.indepFun_shift s).symm.comp
+    measurable_id (measurable_pi_apply v)
+  change Indep (MeasurableSpace.comap X inferInstance)
+    (MeasurableSpace.comap (fun ω => B (s+v) ω - B s ω) inferInstance) P
+  exact indep_of_indep_of_le_left hh hX.comap_le
+
+lemma pastSigma_mono {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    {s t : ℝ≥0} (hst : s ≤ t) : pastSigma B s ≤ pastSigma B t := by
+  let : MeasurableSpace Ω := pastSigma B t
+  have hm : Measurable (fun ω (u : Set.Iic s) => B u ω) := by
+    apply Measurable.of_eval
+    intro u
+    exact (measurable_pi_apply (⟨u, u.2.trans hst⟩ : Set.Iic t)).comp
+      (comap_measurable (fun ω (v : Set.Iic t) => B v ω))
+  exact hm.comap_le
+
+lemma past_measurable_eval {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    {s t : ℝ≥0} (hst : s ≤ t) : Measurable[pastSigma B t] (B s) :=
+  (measurable_pi_apply (⟨s, hst⟩ : Set.Iic t)).comp
+    (comap_measurable (fun ω (v : Set.Iic t) => B v ω))
+
+lemma past_mul_feature_integral {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {s t : ℝ≥0} (hst : s ≤ t) {X : Ω → ℝ}
+    (hX : Measurable[pastSigma B s] X) (hXa : AEMeasurable X P)
+    (φ : ℝ → ℝ) (hφ : Measurable φ) :
+    (∫ ω, X ω * φ (B t ω - B s ω) ∂P) =
+      (∫ ω, X ω ∂P) * (∫ x, φ x ∂gaussianReal 0 (t-s)) := by
+  have hi := (brownian_indep_of_past_measurable hB s (t-s) hX).comp
+    measurable_id hφ
+  rw [add_tsub_cancel_of_le hst] at hi
+  have hl := increment_hasLaw hB hst
+  change IndepFun X (fun ω => φ (B t ω - B s ω)) P at hi
+  rw [hi.integral_fun_mul_eq_mul_integral hXa.aestronglyMeasurable
+      (hφ.comp_aemeasurable hl.aemeasurable).aestronglyMeasurable]
+  rw [show (∫ ω, φ (B t ω - B s ω) ∂P) = ∫ x, φ x ∂gaussianReal 0 (t-s) from
+    hl.integral_comp hφ.aestronglyMeasurable]
+
+lemma independent_product_memLp_two {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {X Y : Ω → ℝ} (hX : MemLp X 2 P) (hY : MemLp Y 2 P)
+    (hXY : IndepFun X Y P) : MemLp (fun ω => X ω * Y ω) 2 P := by
+  apply (memLp_two_iff_integrable_sq (hX.aestronglyMeasurable.mul hY.aestronglyMeasurable)).2
+  have hi := (hXY.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integrable_mul
+    hX.integrable_sq hY.integrable_sq
+  convert hi using 1
+  ext ω
+  simp only [Function.comp_apply, Pi.mul_apply, mul_pow, id_eq]
+
+ 
+def predictableFeatureSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (t : ℕ → ℝ≥0) (C : ℕ → Ω → ℝ) (φ : ℕ → ℝ → ℝ) (n : ℕ) (ω : Ω) : ℝ :=
+  ∑ i ∈ Finset.range n, C i ω * φ i (B (t (i+1)) ω - B (t i) ω)
+
+lemma predictableTerm_memLp_two {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i)) (hC2 : ∀ i, MemLp (C i) 2 P)
+    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i))
+    (hφ2 : ∀ i, MemLp (φ i) 2 (gaussianReal 0 (t (i+1)-t i))) (i : ℕ) :
+    MemLp (fun ω => C i ω * φ i (B (t (i+1)) ω - B (t i) ω)) 2 P := by
+  have ht' := ht (Nat.le_succ i)
+  have hl := increment_hasLaw hB ht'
+  have hY := (hl.map_eq ▸ hφ2 i).comp_of_map hl.aemeasurable
+  have hi := (brownian_indep_of_past_measurable hB (t i) (t (i+1)-t i) (hC i)).comp
+    measurable_id (hφ i)
+  rw [add_tsub_cancel_of_le ht'] at hi
+  exact independent_product_memLp_two (hC2 i) hY hi
+
+lemma predictableTerm_integral {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i)) (hCa : ∀ i, AEMeasurable (C i) P)
+    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i))
+    (hφ0 : ∀ i, (∫ x, φ i x ∂gaussianReal 0 (t (i+1)-t i)) = 0) (i : ℕ) :
+    (∫ ω, C i ω * φ i (B (t (i+1)) ω - B (t i) ω) ∂P) = 0 := by
+  rw [past_mul_feature_integral hB (ht (Nat.le_succ i)) (hC i) (hCa i) (φ i) (hφ i),
+    hφ0, mul_zero]
+
+lemma predictableTerm_past_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i))
+    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i)) {i j : ℕ} (hij : i+1 ≤ j) :
+    Measurable[pastSigma B (t j)]
+      (fun ω => C i ω * φ i (B (t (i+1)) ω - B (t i) ω)) := by
+  have hi : t i ≤ t j := ht (le_trans (Nat.le_succ i) hij)
+  exact ((hC i).mono (pastSigma_mono B hi) le_rfl).mul
+    ((hφ i).comp ((past_measurable_eval B (ht hij)).sub
+      (past_measurable_eval B hi)))
+
+lemma predictableTerm_covariance_zero {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (C : ℕ → Ω → ℝ)
+    (hC : ∀ i, Measurable[pastSigma B (t i)] (C i)) (hC2 : ∀ i, MemLp (C i) 2 P)
+    (φ : ℕ → ℝ → ℝ) (hφ : ∀ i, Measurable (φ i))
+    (hφ2 : ∀ i, MemLp (φ i) 2 (gaussianReal 0 (t (i+1)-t i)))
+    (hφ0 : ∀ i, (∫ x, φ i x ∂gaussianReal 0 (t (i+1)-t i)) = 0)
+    {i j : ℕ} (hij : i ≠ j) :
+    covariance (fun ω => C i ω * φ i (B (t (i+1)) ω - B (t i) ω))
+      (fun ω => C j ω * φ j (B (t (j+1)) ω - B (t j) ω)) P = 0 := by
+  wlog hlt : i < j generalizing i j
+  · rw [covariance_comm]
+    exact this hij.symm (lt_of_le_of_ne (not_lt.mp hlt) hij.symm)
+  have hAi := predictableTerm_memLp_two hB t ht C hC hC2 φ hφ hφ2 i
+  have hAj := predictableTerm_memLp_two hB t ht C hC hC2 φ hφ hφ2 j
+  rw [covariance_eq_sub hAi hAj,
+    predictableTerm_integral hB t ht C hC (fun i => (hC2 i).aemeasurable) φ hφ hφ0 i,
+    zero_mul, sub_zero]
+  have hp := (predictableTerm_past_measurable B t ht C hC φ hφ hlt).mul (hC j)
+  have hh := past_mul_feature_integral hB (ht (Nat.le_succ j)) hp
+    (hAi.aemeasurable.mul (hC2 j).aemeasurable) (φ j) (hφ j)
+  rw [hφ0, mul_zero] at hh
+  convert hh using 1
+  congr 1
+  ext ω
+  simp only [Pi.mul_apply]
+  ring
+
+end SLEExactGauge.BrownianPredictable
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StoppedFields.lean b/lean/OAI/Probability/SLE/StoppedFields.lean
index 336f22a9ab7add8b8adf9fa8d74651698f5a303e..bab6435c35b2cee51e0ab11d18f981d8c266d612 100644
--- a/lean/OAI/Probability/SLE/StoppedFields.lean
+++ b/lean/OAI/Probability/SLE/StoppedFields.lean
@@ -1,772 +1,772 @@
-import Mathlib
-import OAI.Probability.SLE.StochasticFubini
-import OAI.Probability.SLE.WindowTails
-
-namespace OAI
-
-/-! Stopped Fields for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-lemma boundedSpatialHit_antitone_radius {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) {ε r : ℝ} (hεr : ε ≤ r) (hs : r ≤ dist (γ 0) z) (T : ℝ≥0) :
-    boundedSpatialHit γ z r T ≤ boundedSpatialHit γ z ε T := by
-  apply WithTop.coe_le_coe.mp
-  rw [boundedSpatialHit_coe, boundedSpatialHit_coe]
-  exact min_le_min (spatialHitTime_antitone_radius hc z hεr hs) le_rfl
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
- 
-
-lemma joint_spatialHit_stopping {Z : Type*} [mZ : MeasurableSpace Z]
-    (z : Z → ℂ) (hz : Measurable z) (r : ℝ) :
-    IsStoppingTime (spatialFiltration (Z := Z) a.filtration)
-      (fun p : Z × a.samples => spatialHitTime (γ p.2) (z p.1) r) := by
-  let H (t : ℝ≥0) (p : Z × a.samples) := dist (γ p.2 t) (z p.1) - r
-  have ha : Adapted (spatialFiltration (Z := Z) a.filtration) H := by
-    intro t
-    have hm := a.trace_adapted t
-    let : MeasurableSpace a.samples := a.filtration t
-    exact ((hm.comp measurable_snd).dist (hz.comp measurable_fst)).sub_const r
-  have hc (p : Z × a.samples) : Continuous (fun t => H t p) :=
-    ((a.trace_continuous p.2).dist continuous_const).sub continuous_const
-  exact continuous_zero_hit_stopping (spatialFiltration (Z := Z) a.filtration) H ha hc 0
-
-lemma joint_boundedSpatialHit_stopping {Z : Type*} [mZ : MeasurableSpace Z]
-    (z : Z → ℂ) (hz : Measurable z) (r : ℝ) (T : ℝ≥0) :
-    IsStoppingTime (spatialFiltration (Z := Z) a.filtration)
-      (fun p : Z × a.samples => (boundedSpatialHit (γ p.2) (z p.1) r T : WithTop ℝ≥0)) := by
-  simpa only [boundedSpatialHit_coe] using (a.joint_spatialHit_stopping z hz r).min_const T
-
-end RegularSLEPresentation
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma clipped_spatial_window_coefficient_le
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi}
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (ω : Ω) (α S : ℝ≥0)
-    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F)
-    (hin : ∀ t : ℝ≥0, t ≤ α → lo ≤ ((c (initialLogCoordinates z)).Q t ω).1.im ∧
-        ((c (initialLogCoordinates z)).Q t ω).1.im ≤ hi)
-    (r ε : ℝ) (t : ℝ≥0) :
-    |(if min (boundedSpatialHit γ z r S) α ≤ t ∧
-        t < min (boundedSpatialHit γ z ε S) α then
-      greenLogCoefficient κ ((c (initialLogCoordinates z)).Q t ω) (B t ω) else 0)| ≤
-    |spatialSlabCoefficient κ (fun s => Real.sqrt κ * B s ω) γ G r ε t z| := by
-  split_ifs with ht
-  · have hta : t < α := ht.2.trans_le (min_le_right _ _)
-    have hte : t < boundedSpatialHit γ z ε S := ht.2.trans_le (min_le_left _ _)
-    have htS : t < S := by
-      apply WithTop.coe_lt_coe.mp
-      exact (WithTop.coe_lt_coe.mpr hte).trans_le
-        (by rw [boundedSpatialHit_coe]; exact min_le_right _ _)
-    have hsr : boundedSpatialHit γ z r S ≤ t := (min_le_iff.mp ht.1).resolve_right (not_le_of_gt hta)
-    have hr : spatialHitTime γ z r ≤ t := by
-      have hh := WithTop.coe_le_coe.mpr hsr
-      rw [boundedSpatialHit_coe] at hh
-      exact (min_le_iff.mp hh).resolve_right (not_le_of_gt (WithTop.coe_lt_coe.mpr htS))
-    have he : (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε :=
-      (WithTop.coe_le_coe.mpr hte.le).trans (boundedSpatialHit_le γ z ε S)
-    obtain ⟨hl, hq⟩ := clipped_chart_identification c hBc z hz ω α γ G F hw hin
-    have hlive := survivingDomain_antitone γ hta.le hl
-    rw [hq t hta.le, greenLogCoefficient_logCoordinates ((hBc ω).const_mul _) hw κ _ z t hlive rfl,
-      spatialSlabCoefficient, ite_eq_left ⟨hr, he, hlive⟩]
-  · simpa only [abs_zero] using abs_nonneg (spatialSlabCoefficient κ (fun s => Real.sqrt κ * B s ω) γ G r ε t z)
-
- 
-
-theorem IsCapacityTwoWitness.spatial_masked_bracket_bound
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
-    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
-    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im)
-    (b : ℝ → A → ℝ)
-    (hb : ∀ t, 0 ≤ t → ∀ z, |b t z| ≤
-      |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|) :
-    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal
-      ((∫ z, b t z ∂(volume.comap (Subtype.val : A → ℂ))) ^ 2)) ≤
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
-  apply le_trans _ (h.spatial_slab_absolute_bracket_bound hU hκ hκ4 hε hr A hA hheight)
-  apply setLIntegral_mono' measurableSet_Ici
-  intro t ht
-  apply (spatial_integral_square_le (volume.comap (Subtype.val : A → ℂ)) (b t)).trans
-  apply pow_le_pow_left₀ bot_le
-  calc
-    _ ≤ ∫⁻ z : A, ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|
-        ∂(volume.comap (Subtype.val : A → ℂ)) := lintegral_mono fun z => ENNReal.ofReal_le_ofReal (hb t ht z)
-    _ = _ := lintegral_subtype_comap (μ := (volume : Measure ℂ)) hA (fun z : ℂ => ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|)
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma clipped_family_uniform_bound
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi}
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q) :
-    ∃ L : ℝ≥0, ∀ (q : ℂ × ℝ) (ω : Ω) (t : ℝ≥0),
-      ‖(c q).Q t ω‖ ≤ ‖q‖ + L * t := by
-  obtain ⟨L, K, hb, hK⟩ := clippedLogLoewnerField_bounds lo hi hle
-  refine ⟨L, fun q ω t => ?_⟩
-  have hh := norm_image_sub_le_of_norm_deriv_le_segment'
-    (f := fun s => (c q).Q s ω) (a := (0 : ℝ)) (b := (t : ℝ))
-    (f' := fun s => clippedLogLoewnerField lo hi hle (Real.sqrt κ * B s.toNNReal ω) ((c q).Q s ω))
-    (fun s hs => ((c q).derivative ω s hs.1).mono (fun _ hx => hx.1))
-    (fun s hs => hb _ _) (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩
-  rw [(c q).initial, sub_zero] at hh
-  calc
-    _ ≤ ‖q‖ + ‖(c q).Q t ω - q‖ := by
-      simpa only [add_comm] using norm_le_insert' ((c q).Q t ω) q
-    _ ≤ _ := by linarith
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem spatial_angular_cutoff
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ lo hi : ℝ) (hle : lo ≤ hi)
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (hq : Measurable q) (hq0 : ∀ z, (q z).2 = 0)
-    (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
-    (hε0 : ∀ z, ε ≤ logAngleSq κ (q z) 0)
-    (hlo : ∀ z, lo < (q z).1.im - u / (2 * ε)) (hhi : ∀ z, (q z).1.im < hi)
-    (T : ℝ≥0) (hT : ∀ z, u < (4 * ε ^ 2 / Real.exp (2 * (q z).1.im)) * T) :
-    ∃ τ : Z × Ω → ℝ≥0,
-      IsStoppingTime (spatialFiltration (Z := Z)
-        (Filtration.natural B (fun t => (hm t).stronglyMeasurable)))
-        (fun p => (τ p : WithTop ℝ≥0)) ∧
-      (∀ z ω, τ (z,ω) < T) ∧
-      (∀ z ω t, t ≤ τ (z,ω) →
-        logRadiusClock (q z).1.im ((c (q z)).Q t ω) ≤ u ∧
-        ε ≤ logAngleSq κ ((c (q z)).Q t ω) (B t ω) ∧
-        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi) ∧
-      (∀ z ω t, t < τ (z,ω) →
-        logRadiusClock (q z).1.im ((c (q z)).Q t ω) < u ∧
-        ε < logAngleSq κ ((c (q z)).Q t ω) (B t ω) ∧
-        lo < ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im < hi) ∧
-      (∀ z ω, logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) = u ∨
-        logAngleSq κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) = ε) := by
-  let f := spatialFiltration (Z := Z) (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-  let D (t : ℝ≥0) (p : Z × Ω) :=
-    min (u - logRadiusClock (q p.1).1.im ((c (q p.1)).Q t p.2))
-      (min (logAngleSq κ ((c (q p.1)).Q t p.2) (B t p.2) - ε)
-        (min (((c (q p.1)).Q t p.2).1.im - lo) (hi - ((c (q p.1)).Q t p.2).1.im)))
-  have hDa : Adapted f D := by
-    intro t
-    have hQ := clipped_family_past_measurable c t
-    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted t
-    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) t
-    have hc := hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)
-    have hBb := hBa.comp (@measurable_snd Z Ω mZ _)
-    have hη := Complex.continuous_im.measurable.comp (hq.comp (@measurable_fst Z Ω mZ (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))).fst
-    have hclock := (hη.sub (Complex.continuous_im.measurable.comp hc.fst)).add hc.snd
-    exact (measurable_const.sub hclock).min
-      ((((continuous_logAngleSq κ).measurable.comp (hc.prodMk hBb)).sub_const ε).min
-        (((Complex.continuous_im.measurable.comp hc.fst).sub_const lo).min
-          (measurable_const.sub (Complex.continuous_im.measurable.comp hc.fst))))
-  have hDc (p : Z × Ω) : Continuous (fun t => D t p) := by
-    have hc := ((c (q p.1)).lipschitz p.2).continuous.comp NNReal.continuous_coe
-    exact (continuous_const.sub ((continuous_const.sub (Complex.continuous_im.comp hc.fst)).add hc.snd)).min
-      ((((continuous_logAngleSq κ).comp (hc.prodMk (hBc p.2))).sub continuous_const).min
-        (((Complex.continuous_im.comp hc.fst).sub continuous_const).min
-          (continuous_const.sub (Complex.continuous_im.comp hc.fst))))
-  have hD0 (p : Z × Ω) : 0 ≤ D 0 p := by
-    dsimp only [D]
-    rw [NNReal.coe_zero, (c (q p.1)).initial, hB0]
-    simp only [le_min_iff, sub_nonneg]
-    refine ⟨?_, hε0 p.1, ?_, (hhi p.1).le⟩
-    · simpa only [logRadiusClock, sub_self, zero_add, hq0] using hu
-    · have hd : 0 ≤ u / (2 * ε) := div_nonneg hu (by positivity)
-      linarith [hlo p.1]
-  obtain ⟨τ, hτ, hb, hleD, hltD, hzero⟩ := first_margin_stopping f D hDa hDc hD0 T
-  have hin (z : Z) (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ (z,ω)) :
-      logRadiusClock (q z).1.im ((c (q z)).Q t ω) ≤ u ∧
-      ε ≤ logAngleSq κ ((c (q z)).Q t ω) (B t ω) ∧
-      lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi :=
-    by simpa only [D, le_min_iff, sub_nonneg] using hleD (z,ω) t ht
-  have hg (z : Z) (ω : Ω) :
-      ((c (q z)).Q (τ (z,ω)) ω).1.im ≤ (q z).1.im ∧
-      (q z).1.im - logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) / (2 * ε) ≤ ((c (q z)).Q (τ (z,ω)) ω).1.im ∧
-      (4 * ε ^ 2 / Real.exp (2 * (q z).1.im)) * τ (z,ω) ≤ logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) := by
-    have hd (s : ℝ) (hs : s ∈ Icc 0 (τ (z,ω) : ℝ)) :
-        HasDerivWithinAt (fun t => (c (q z)).Q t ω)
-          (physicalLogLoewnerField (Real.sqrt κ * B s.toNNReal ω) ((c (q z)).Q s ω))
-          (Icc 0 (τ (z,ω) : ℝ)) s := by
-      have hst : s.toNNReal ≤ τ (z,ω) := Real.toNNReal_le_iff_le_coe.mpr hs.2
-      have hbounds := (hin z ω s.toNNReal hst).2.2
-      rw [Real.coe_toNNReal _ hs.1] at hbounds
-      have hh := ((c (q z)).derivative ω s hs.1).mono (show Icc (0 : ℝ) (τ (z,ω) : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
-      rwa [clippedLogLoewnerField_eq_physical lo hi hle _ _ ⟨hbounds.1, hbounds.2⟩] at hh
-    have ha (s : ℝ) (hs : s ∈ Icc 0 (τ (z,ω) : ℝ)) :
-        ε ≤ logAngleSq κ ((c (q z)).Q s ω) (B s.toNNReal ω) := by
-      have hh := (hin z ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.1
-      simpa only [Real.coe_toNNReal _ hs.1] using hh
-    have hq : ((c (q z)).Q 0 ω).2 = 0 := by rw [(c (q z)).initial ω, hq0 z]
-    have hh := log_loewner_height_clock_bounds (τ (z,ω)).coe_nonneg hε hq hd ha
-    simpa only [(c (q z)).initial ω] using hh
-  have hτlt (z : Z) (ω : Ω) : τ (z,ω) < T := by
-    by_contra hn
-    have he : τ (z,ω) = T := le_antisymm (hb (z,ω)) (not_lt.mp hn)
-    have hh := (hg z ω).2.2.trans (hin z ω (τ (z,ω)) le_rfl).1
-    rw [he] at hh
-    exact (not_le_of_gt (hT z)) hh
-  have he (z : Z) (ω : Ω) :
-      logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) = u ∨
-      logAngleSq κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) = ε := by
-    obtain ⟨hr, ha, hl, hh⟩ := hin z ω (τ (z,ω)) le_rfl
-    have hstrictlo : lo < ((c (q z)).Q (τ (z,ω)) ω).1.im := by
-      have hdiv := div_le_div_of_nonneg_right hr (show 0 ≤ 2 * ε by positivity)
-      linarith [(hg z ω).2.1, hlo z]
-    have hstricthi : ((c (q z)).Q (τ (z,ω)) ω).1.im < hi := (hg z ω).1.trans_lt (hhi z)
-    by_contra hn
-    push Not at hn
-    have hp : 0 < D (τ (z,ω)) (z,ω) :=
-      by simpa only [D, lt_min_iff, sub_pos] using
-        (show logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) < u ∧
-          ε < logAngleSq κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) ∧
-          lo < ((c (q z)).Q (τ (z,ω)) ω).1.im ∧
-          ((c (q z)).Q (τ (z,ω)) ω).1.im < hi from
-          ⟨lt_of_le_of_ne hr hn.1, lt_of_le_of_ne ha (Ne.symm hn.2), hstrictlo, hstricthi⟩)
-    rw [hzero (z,ω) (hτlt z ω)] at hp
-    exact lt_irrefl 0 hp
-  refine ⟨τ, hτ, hτlt, hin, ?_, he⟩
-  intro z ω t ht
-  simpa only [D, lt_min_iff, sub_pos] using hltD (z,ω) t ht
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-structure SpatialAngularFamily {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (κ : ℝ) (z : Z → ℂ) (u ε : ℝ) where
-  lo : ℝ
-  hi : ℝ
-  le : lo ≤ hi
-  c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi le q
-  T : ℝ≥0
-  τ : Z × Ω → ℝ≥0
-  stopping : IsStoppingTime (spatialFiltration (Z := Z)
-    (Filtration.natural B (fun t => (hm t).stronglyMeasurable)))
-    (fun p => (τ p : WithTop ℝ≥0))
-  bounded : ∀ p, τ p ≤ T
-  inside : ∀ x ω t, t ≤ τ (x,ω) →
-    logRadiusClock (Real.log (z x).im) ((c (initialLogCoordinates (z x))).Q t ω) ≤ u ∧
-    ε ≤ logAngleSq κ ((c (initialLogCoordinates (z x))).Q t ω) (B t ω) ∧
-    lo ≤ ((c (initialLogCoordinates (z x))).Q t ω).1.im ∧
-      ((c (initialLogCoordinates (z x))).Q t ω).1.im ≤ hi
-  before : ∀ x ω t, t < τ (x,ω) →
-    logRadiusClock (Real.log (z x).im) ((c (initialLogCoordinates (z x))).Q t ω) < u ∧
-    ε < logAngleSq κ ((c (initialLogCoordinates (z x))).Q t ω) (B t ω)
-  boundary : ∀ x ω,
-    logRadiusClock (Real.log (z x).im) ((c (initialLogCoordinates (z x))).Q (τ (x,ω)) ω) = u ∨
-    logAngleSq κ ((c (initialLogCoordinates (z x))).Q (τ (x,ω)) ω) (B (τ (x,ω)) ω) = ε
-
-noncomputable def SpatialAngularFamily.section
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ : ℝ} {z : Z → ℂ} {u ε : ℝ}
-    (a : SpatialAngularFamily B hm κ z u ε) (x : Z) :
-    AngularLocalization B hm κ (z x) u ε where
-  lo := a.lo
-  hi := a.hi
-  le := a.le
-  solution := a.c (initialLogCoordinates (z x))
-  T := a.T
-  τ := fun ω => a.τ (x,ω)
-  stopping := spatial_stopping_section a.stopping x
-  bounded := fun ω => a.bounded (x,ω)
-  inside := a.inside x
-  before := a.before x
-  boundary := a.boundary x
-
-lemma exists_spatial_angular_family
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (z : Z → ℂ) (hz : Measurable z)
-    (l h : ℝ) (hlh : l ≤ h)
-    (hη : ∀ x, l ≤ Real.log (z x).im ∧ Real.log (z x).im ≤ h)
-    (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
-    (hangle : ∀ x, ε ≤ logAngleSq κ (initialLogCoordinates (z x)) 0) :
-    Nonempty (SpatialAngularFamily B hm κ z u ε) := by
-  classical
-  let lo := l - u / (2 * ε) - 1
-  let hi := h + 1
-  have hud : 0 ≤ u / (2 * ε) := div_nonneg hu (by positivity)
-  have hle : lo ≤ hi := by dsimp [lo, hi]; linarith
-  let c (q : ℂ × ℝ) : ClippedLogSolution B hm κ lo hi hle q :=
-    (exists_clippedLogSolution B hm hBm hBc κ lo hi hle q).some
-  let v := 4 * ε ^ 2 / Real.exp (2 * h)
-  have hv : 0 < v := by dsimp [v]; positivity
-  obtain ⟨N, hN⟩ := exists_nat_gt (u / v)
-  have hT : ∀ x, u < (4 * ε ^ 2 / Real.exp (2 * (initialLogCoordinates (z x)).1.im)) * ((N : ℝ≥0) : ℝ) := by
-    intro x
-    have huv := (div_lt_iff₀ hv).mp hN
-    have he : Real.exp (2 * Real.log (z x).im) ≤ Real.exp (2 * h) :=
-      Real.exp_le_exp.mpr (by linarith [(hη x).2])
-    have hd : v ≤ 4 * ε ^ 2 / Real.exp (2 * Real.log (z x).im) :=
-      div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) he
-    rw [initialLogCoordinates_im, NNReal.coe_natCast]
-    have huv' : u < v * (N : ℝ) := by simpa only [mul_comm (N : ℝ) v] using huv
-    exact huv'.trans_le (mul_le_mul_of_nonneg_right hd (Nat.cast_nonneg N))
-  have hq : Measurable (fun x => initialLogCoordinates (z x)) :=
-    measurable_initialLogCoordinates.comp hz
-  obtain ⟨τ, hτ, hb, hin, hbefore, hboundary⟩ := spatial_angular_cutoff hm hBc hB0 κ lo hi hle c
-    (fun x => initialLogCoordinates (z x)) hq (fun x => rfl) u ε hu hε hangle
-    (fun x => by rw [initialLogCoordinates_im]; dsimp [lo]; linarith [(hη x).1])
-    (fun x => by rw [initialLogCoordinates_im]; dsimp [hi]; linarith [(hη x).2]) (N : ℝ≥0) hT
-  refine ⟨⟨lo, hi, hle, c, N, τ, hτ, fun p => (hb p.1 p.2).le, ?_, ?_, ?_⟩⟩
-  · simpa only [initialLogCoordinates_im] using hin
-  · intro x ω t ht
-    constructor
-    · simpa only [initialLogCoordinates_im] using (hbefore x ω t ht).1
-    · exact (hbefore x ω t ht).2.1
-  · simpa only [initialLogCoordinates_im] using hboundary
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma clipped_area_window_measurable
-    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    (ν : Measure Z) [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi : ℝ} {hle : lo ≤ hi}
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (hq : Measurable q)
-    (σ τ : Z × Ω → ℝ≥0) (hσ : Measurable σ) (hτ : Measurable τ) (T : ℝ≥0) :
-    Measurable (fun ω => ∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
-        greenLogDensity κ ((c (q z)).Q (σ (z,ω)) ω) (B (σ (z,ω)) ω) ∂ν) ∧
-    Measurable (fun ω => ∫ t in (0 : ℝ)..(T : ℝ),
-        (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
-          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) := by
-  classical
-  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable hBc hm
-  have hQm := clipped_family_joint_measurable c
-  have hD (η : Z × Ω → ℝ≥0) (hη : Measurable η) : Measurable (fun p : Z × Ω =>
-      greenLogDensity κ ((c (q p.1)).Q (η p) p.2) (B (η p) p.2)) :=
-    (greenLogDensity_contDiff κ).continuous.measurable.comp
-      ((hQm.comp (hη.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
-        (hBm.comp (hη.prodMk measurable_snd)))
-  refine ⟨((hD τ hτ).sub (hD σ hσ)).stronglyMeasurable.integral_prod_left'.measurable, ?_⟩
-  let f (p : ℝ × (Z × Ω)) := if (σ p.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2 : ℝ) then
-    greenLogCoefficient κ ((c (q p.2.1)).Q p.1 p.2.2) (B p.1.toNNReal p.2.2) else 0
-  have hgood : MeasurableSet {p : ℝ × (Z × Ω) | (σ p.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2 : ℝ)} :=
-    (measurableSet_le (NNReal.continuous_coe.measurable.comp (hσ.comp measurable_snd)) measurable_fst).inter
-      (measurableSet_lt measurable_fst (NNReal.continuous_coe.measurable.comp (hτ.comp measurable_snd)))
-  have hQreal : Measurable (fun p : ℝ × (Z × Ω) => (c (q p.2.1)).Q p.1.toNNReal p.2.2) :=
-    hQm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
-      ((hq.comp (measurable_fst.comp measurable_snd)).prodMk (measurable_snd.comp measurable_snd)))
-  have hcoef : Measurable (fun p : (ℂ × ℝ) × ℝ => greenLogCoefficient κ p.1 p.2) := by
-    unfold greenLogCoefficient greenLogNormSq
-    apply (greenLogDensity_contDiff κ).continuous.measurable.mul
-    fun_prop
-  have hfm' : Measurable (fun p : ℝ × (Z × Ω) =>
-      if (σ p.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2 : ℝ) then
-        greenLogCoefficient κ ((c (q p.2.1)).Q p.1.toNNReal p.2.2) (B p.1.toNNReal p.2.2) else 0) := (hcoef.comp
-    (hQreal.prodMk (hBm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
-      (measurable_snd.comp measurable_snd))))).ite hgood (measurable_const (a := (0 : ℝ)))
-  have hfm : Measurable f := by
-    convert hfm' using 1
-    ext p
-    dsimp only [f]
-    split_ifs with hp
-    · rw [Real.coe_toNNReal _ ((σ p.2).coe_nonneg.trans hp.1)]
-    · rfl
-  have hswap : Measurable (fun p : Z × (ℝ × Ω) => f (p.2.1,(p.1,p.2.2))) := hfm.comp
-    ((measurable_fst.comp measurable_snd).prodMk (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))
-  have harea : Measurable (fun p : ℝ × Ω => ∫ z, f (p.1,(z,p.2)) ∂ν) :=
-    hswap.stronglyMeasurable.integral_prod_left'.measurable
-  have hh := (harea.pow_const 2).stronglyMeasurable.integral_prod_left'
-    (μ := volume.restrict (Ioc (0 : ℝ) T))
-  simpa only [intervalIntegral.integral_of_le T.coe_nonneg, f] using hh.measurable
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma spatial_stopping_lift {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {f : Filtration ℝ≥0 mΩ} {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime f τ) :
-    IsStoppingTime (spatialFiltration (Z := Z) f) (fun p => τ p.2) := by
-  intro t
-  let : MeasurableSpace Ω := f t
-  exact (hτ t).preimage (measurable_snd : @Measurable (Z × Ω) Ω (mZ.prod (f t)) (f t) Prod.snd)
-
- 
-
-theorem exists_common_brownian_compact_stops
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hB0 : ∀ ω, B 0 ω = 0) (T : ℝ≥0) :
-    ∃ π : ℕ → Ω → ℝ≥0,
-      (∀ n, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (π n ω : WithTop ℝ≥0))) ∧
-      (∀ n ω, π n ω ≤ T) ∧
-      (∀ n ω t, t ≤ π n ω → |B t ω| ≤ (n : ℝ) + 1) ∧
-      (∀ ω, ∀ᶠ n in atTop, π n ω = T) := by
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hDa (n : ℕ) : Adapted f (fun t ω => (n : ℝ) + 1 - |B t ω|) := by
-    intro t
-    have ha := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted t
-    let : MeasurableSpace Ω := f t
-    exact measurable_const.sub ha.abs
-  have hb (n : ℕ) := first_margin_stopping f (fun t ω => (n : ℝ) + 1 - |B t ω|)
-    (hDa n) (fun ω => continuous_const.sub (hBc ω).abs)
-    (fun ω => by rw [hB0 ω, abs_zero, sub_zero]; positivity) T
-  choose π hπ hb hin hpre hat using hb
-  refine ⟨π, hπ, hb, fun n ω t ht => (sub_nonneg.mp (hin n ω t ht)), ?_⟩
-  intro ω
-  obtain ⟨N, hN⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
-  filter_upwards [eventually_ge_atTop N] with n hn
-  apply le_antisymm (hb n ω)
-  by_contra h
-  have he := hat n ω (not_le.mp h)
-  have hh := hN (π n ω) (hb n ω)
-  have hn' : (N : ℝ) ≤ n := Nat.cast_le.mpr hn
-  linarith
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem clipped_spatial_green_window_unbounded_driver
-    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
-    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
-    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hB0 : ∀ ω, B 0 ω = 0)
-    (κ lo hi : ℝ) (hle : lo ≤ hi) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
-    (q : Z → ℂ × ℝ) (hq : Measurable q)
-    (M : ℝ) (hqbound : ∀ z, ‖q z‖ ≤ M) (T : ℝ≥0)
-    (σ τ : Z × Ω → ℝ≥0)
-    (hσ : IsStoppingTime (spatialFiltration (Z := Z)
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
-      (fun p => (σ p : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime (spatialFiltration (Z := Z)
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
-      (fun p => (τ p : WithTop ℝ≥0)))
-    (hστ : ∀ p, σ p ≤ τ p) (hτb : ∀ p, τ p ≤ T)
-    (hinside : ∀ z ω t, t ≤ τ (z,ω) →
-      lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi)
-    (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp
-      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
-        greenLogDensity κ ((c (q z)).Q (σ (z,ω)) ω) (B (σ (z,ω)) ω) ∂ν) -
-      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
-        (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
-          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
-  classical
-  obtain ⟨π, hπ, hπb, hπin, hπlim⟩ := exists_common_brownian_compact_stops hm hBc hB0 T
-  obtain ⟨L, hL⟩ := clipped_family_uniform_bound c
-  let σn (n : ℕ) (p : Z × Ω) := min (σ p) (π n p.2)
-  let τn (n : ℕ) (p : Z × Ω) := min (τ p) (π n p.2)
-  have hσn (n : ℕ) : IsStoppingTime (spatialFiltration (Z := Z)
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
-      (fun p => (σn n p : WithTop ℝ≥0)) := by
-    simpa only [σn, WithTop.coe_min] using hσ.min (spatial_stopping_lift (Z := Z) (hπ n))
-  have hτn (n : ℕ) : IsStoppingTime (spatialFiltration (Z := Z)
-      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
-      (fun p => (τn n p : WithTop ℝ≥0)) := by
-    simpa only [τn, WithTop.coe_min] using hτ.min (spatial_stopping_lift (Z := Z) (hπ n))
-  have hstn (n : ℕ) (p : Z × Ω) : σn n p ≤ τn n p := min_le_min (hστ p) le_rfl
-  have htn (n : ℕ) (p : Z × Ω) : τn n p ≤ T := (min_le_left _ _).trans (hτb p)
-  let R (n : ℕ) := max M 0 + (L : ℝ) * T + ((n : ℝ) + 1)
-  have hR (n : ℕ) : 0 ≤ R n := by dsimp [R]; positivity
-  have hin (n : ℕ) (z : Z) (ω : Ω) (t : ℝ≥0) (ht : t ≤ τn n (z,ω)) :
-      ‖((c (q z)).Q t ω, B t ω)‖ ≤ R n ∧
-        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi := by
-    have htτ := ht.trans (min_le_left _ _)
-    have htb := ht.trans (htn n (z,ω))
-    refine ⟨?_, hinside z ω t htτ⟩
-    rw [Prod.norm_def, max_le_iff]
-    have hqt := hL (q z) ω t
-    have hmul : (L : ℝ) * t ≤ (L : ℝ) * T := mul_le_mul_of_nonneg_left
-      (NNReal.coe_le_coe.mpr htb) L.coe_nonneg
-    have hbm := hπin n ω t (ht.trans (min_le_right _ _))
-    have hml : M ≤ max M 0 := le_max_left _ _
-    have hmn : 0 ≤ max M 0 := le_max_right _ _
-    have hlt : 0 ≤ (L : ℝ) * T := mul_nonneg L.coe_nonneg T.coe_nonneg
-    dsimp only [R]
-    exact ⟨by linarith [hqbound z, Nat.cast_nonneg (α := ℝ) n], by simpa only [Real.norm_eq_abs] using
-      (show |B t ω| ≤ max M 0 + (L : ℝ) * T + ((n : ℝ) + 1) by linarith)⟩
-  let X (η ξ : Z × Ω → ℝ≥0) (ω : Ω) :=
-    ∫ z, greenLogDensity κ ((c (q z)).Q (ξ (z,ω)) ω) (B (ξ (z,ω)) ω) -
-      greenLogDensity κ ((c (q z)).Q (η (z,ω)) ω) (B (η (z,ω)) ω) ∂ν
-  let E (η ξ : Z × Ω → ℝ≥0) (ω : Ω) := ∫ t in (0 : ℝ)..(T : ℝ),
-    (∫ z, if (η (z,ω) : ℝ) ≤ t ∧ t < (ξ (z,ω) : ℝ) then
-      greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2
-  have hxm (n : ℕ) := clipped_area_window_measurable ν hBc c q hq (σn n) (τn n)
-    (stopping_time_measurable_nnreal (hσn n)) (stopping_time_measurable_nnreal (hτn n)) T
-  have hexp (n : ℕ) := clipped_spatial_green_window_exponential (ν := ν) hm hB hBc
-    κ lo hi hle hκ hκ8 c q hq (R n) (hR n) T (σn n) (τn n)
-      (stopping_time_measurable_nnreal (hσn n)) (stopping_time_measurable_nnreal (hτn n)) (hstn n)
-      (stopping_before_measurable (hσn n)) (spatial_stopping_section (hσn n))
-      (stopping_before_measurable (hτn n)) (spatial_stopping_section (hτn n))
-      (fun z ω => htn n (z,ω)) (hin n) θ
-  have hev (ω : Ω) : ∀ᶠ n in atTop,
-      X (σn n) (τn n) ω = X σ τ ω ∧ E (σn n) (τn n) ω = E σ τ ω := by
-    filter_upwards [hπlim ω] with n hn
-    have hs (z : Z) : σn n (z,ω) = σ (z,ω) := by
-      dsimp only [σn]; rw [hn, min_eq_left ((hστ (z,ω)).trans (hτb (z,ω))) ]
-    have ht (z : Z) : τn n (z,ω) = τ (z,ω) := by
-      dsimp only [τn]; rw [hn, min_eq_left (hτb (z,ω))]
-    simp only [X, E, hs, ht, and_self]
-  apply compensated_exponential_limit_energy
-    (fun n => X (σn n) (τn n)) (fun n => E (σn n) (τn n)) (X σ τ) (E σ τ) θ
-  · intro n
-    exact ((((hxm n).1.const_mul θ).sub (((hxm n).2.const_mul (θ^2)).div_const 2)).exp).ennreal_ofReal.aemeasurable
-  · apply Eventually.of_forall
-    intro ω
-    exact tendsto_const_nhds.congr' ((hev ω).mono fun n hn => hn.1.symm)
-  · apply Eventually.of_forall
-    intro ω
-    exact tendsto_const_nhds.congr' ((hev ω).mono fun n hn => hn.2.symm)
-  · exact hexp
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory
-open scoped ENNReal NNReal
-namespace SLEExactGauge
-
-lemma interval_square_energy_bound (f : ℝ → ℝ) (T : ℝ≥0) (V : ℝ) (hV : 0 ≤ V)
-    (hb : (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal ((f t)^2)) ≤ ENNReal.ofReal V) :
-    (∫ t in (0 : ℝ)..(T : ℝ), (f t)^2) ≤ V := by
-  rw [intervalIntegral.integral_of_le T.coe_nonneg]
-  have hnorm := enorm_integral_le_lintegral_enorm (μ := volume.restrict (Ioc (0 : ℝ) (T : ℝ)))
-    (fun t => (f t)^2)
-  simp_rw [Real.enorm_eq_ofReal (sq_nonneg _)] at hnorm
-  have hi : 0 ≤ ∫ t in Ioc (0 : ℝ) (T : ℝ), (f t)^2 := integral_nonneg (fun t => sq_nonneg _)
-  rw [Real.enorm_eq_ofReal hi] at hnorm
-  apply (ENNReal.ofReal_le_ofReal_iff hV).mp
-  apply hnorm.trans
-  apply le_trans _ hb
-  exact lintegral_mono_set (fun t ht => ht.1.le)
-
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def slabVariance (κ ε r : ℝ) (A : Set ℂ) : ℝ :=
-  (ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-    (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A))).toReal
-
-lemma slabVariance_nonneg (κ ε r : ℝ) (A : Set ℂ) : 0 ≤ slabVariance κ ε r A :=
-  ENNReal.toReal_nonneg
-
-lemma ofReal_slabVariance (κ ε r : ℝ) (A : Set ℂ) (hfin : volume A ≠ ∞) :
-    ENNReal.ofReal (slabVariance κ ε r A) =
-      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
-        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
-  apply ENNReal.ofReal_toReal
-  exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top
-    (ENNReal.mul_ne_top (by norm_num) (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hfin))
-
-namespace RegularSLEPresentation
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-
- 
-
-theorem localized_spatial_slab_exponential
-    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
-    (A : Set ℂ) (hA : MeasurableSet A)
-    [IsFiniteMeasure (volume.comap (Subtype.val : A → ℂ))]
-    (hfin : volume A ≠ ∞)
-    (ε r : ℝ) (hε : 0 < ε) (hεr : ε ≤ r)
-    (hheight : ∀ z ∈ A, r ≤ z.im)
-    (u δ : ℝ) (b : SpatialAngularFamily a.B a.measurable_B κ
-      (Subtype.val : A → ℂ) u δ)
-    (M : ℝ) (hM : ∀ z : A, ‖initialLogCoordinates z‖ ≤ M)
-    (S : ℝ≥0) (θ : ℝ) :
-    (∫⁻ ω : a.samples, ENNReal.ofReal (Real.exp
-      (θ * (∫ z : A,
-        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
-        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω
-        ∂(volume.comap (Subtype.val : A → ℂ))) -
-        θ^2 * slabVariance κ ε r A / 2)) ∂(P.comap Subtype.val)) ≤ 1 := by
-  classical
-  let ν := volume.comap (Subtype.val : A → ℂ)
-  let σ (p : A × a.samples) := min (boundedSpatialHit (γ p.2) p.1 r S) (b.τ p)
-  let τ (p : A × a.samples) := min (boundedSpatialHit (γ p.2) p.1 ε S) (b.τ p)
-  have hσ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-      (fun p => (σ p : WithTop ℝ≥0)) := by
-    simpa only [σ, WithTop.coe_min] using
-      (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe r S).min b.stopping
-  have hτ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
-      (fun p => (τ p : WithTop ℝ≥0)) := by
-    simpa only [τ, WithTop.coe_min] using
-      (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe ε S).min b.stopping
-  have hστ (p : A × a.samples) : σ p ≤ τ p := by
-    apply min_le_min _ le_rfl
-    apply boundedSpatialHit_antitone_radius (a.trace_continuous p.2) p.1 hεr
-    rw [(a.witness p.2).2.1, dist_zero_left]
-    exact (hheight p.1 p.1.2).trans ((le_abs_self p.1.val.im).trans (Complex.abs_im_le_norm p.1))
-  have htbound (p : A × a.samples) : τ p ≤ b.T := (min_le_right _ _).trans (b.bounded p)
-  have hin (z : A) (ω : a.samples) (t : ℝ≥0) (ht : t ≤ τ (z,ω)) :
-      b.lo ≤ ((b.c (initialLogCoordinates z)).Q t ω).1.im ∧
-        ((b.c (initialLogCoordinates z)).Q t ω).1.im ≤ b.hi :=
-    (b.inside z ω t (ht.trans (min_le_right _ _))).2.2
-  have hexp := clipped_spatial_green_window_unbounded_driver (ν := ν) a.measurable_B a.brownian
-    a.continuous_B a.zero_B κ b.lo b.hi b.le hκ (by linarith) b.c
-    (fun z : A => initialLogCoordinates z) (measurable_initialLogCoordinates.comp measurable_subtype_coe)
-    M hM b.T σ τ hσ hτ hστ htbound hin θ
-  let E (ω : a.samples) := ∫ t in (0 : ℝ)..(b.T : ℝ),
-    (∫ z : A, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
-      greenLogCoefficient κ ((b.c (initialLogCoordinates z)).Q t ω) (a.B t.toNNReal ω) else 0 ∂ν)^2
-  have hE (ω : a.samples) : E ω ≤ slabVariance κ ε r A := by
-    apply interval_square_energy_bound _ b.T _ (slabVariance_nonneg _ _ _ _)
-    rw [ofReal_slabVariance _ _ _ _ hfin]
-    apply (a.witness ω).spatial_masked_bracket_bound ((a.continuous_B ω).const_mul _)
-      hκ hκ4 hε (hε.trans_le hεr) A hA (fun z hz => hεr.trans (hheight z hz))
-    intro t ht z
-    have hb := clipped_spatial_window_coefficient_le b.c a.continuous_B z
-      (show z.val ∈ upperHalfPlane from hε.trans_le (hεr.trans (hheight z z.2))) ω (b.τ (z,ω)) S
-      (γ ω) (a.G ω) (a.F ω) (a.witness ω)
-      (fun s hs => (b.inside z ω s hs).2.2) r ε t.toNNReal
-    simpa only [σ, τ, ← NNReal.coe_le_coe, ← NNReal.coe_lt_coe, Real.coe_toNNReal _ ht] using hb
-  apply le_trans _ hexp
-  apply lintegral_mono
-  intro ω
-  apply ENNReal.ofReal_le_ofReal
-  apply Real.exp_le_exp.mpr
-  change θ * _ - θ^2 * slabVariance κ ε r A / 2 ≤ θ * _ - θ^2 * E ω / 2
-  have hid : (∫ z : A,
-      (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
-      (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω ∂ν) =
-      ∫ z : A, greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (τ (z,ω)) ω) (a.B (τ (z,ω)) ω) -
-        greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (σ (z,ω)) ω) (a.B (σ (z,ω)) ω) ∂ν := by
-    apply integral_congr_ae
-    exact Eventually.of_forall fun z => by simp only [AngularLocalization.physicalWeight,
-      SpatialAngularFamily.section, σ, τ, NNReal.coe_min]
-  rw [hid]
-  nlinarith [mul_le_mul_of_nonneg_left (hE ω) (sq_nonneg θ)]
-
-end RegularSLEPresentation
-end SLEExactGauge
-
- 
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-namespace SpatialAngularFamily
-variable {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ : ℝ} {z : Z → ℂ} {u δ : ℝ}
-
-end SpatialAngularFamily
-end SLEExactGauge
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.StochasticFubini
+import OAI.Probability.SLE.WindowTails
+
+namespace OAI
+
+/-! Stopped Fields for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+lemma boundedSpatialHit_antitone_radius {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) {ε r : ℝ} (hεr : ε ≤ r) (hs : r ≤ dist (γ 0) z) (T : ℝ≥0) :
+    boundedSpatialHit γ z r T ≤ boundedSpatialHit γ z ε T := by
+  apply WithTop.coe_le_coe.mp
+  rw [boundedSpatialHit_coe, boundedSpatialHit_coe]
+  exact min_le_min (spatialHitTime_antitone_radius hc z hεr hs) le_rfl
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+  {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+ 
+
+lemma joint_spatialHit_stopping {Z : Type*} [mZ : MeasurableSpace Z]
+    (z : Z → ℂ) (hz : Measurable z) (r : ℝ) :
+    IsStoppingTime (spatialFiltration (Z := Z) a.filtration)
+      (fun p : Z × a.samples => spatialHitTime (γ p.2) (z p.1) r) := by
+  let H (t : ℝ≥0) (p : Z × a.samples) := dist (γ p.2 t) (z p.1) - r
+  have ha : Adapted (spatialFiltration (Z := Z) a.filtration) H := by
+    intro t
+    have hm := a.trace_adapted t
+    let : MeasurableSpace a.samples := a.filtration t
+    exact ((hm.comp measurable_snd).dist (hz.comp measurable_fst)).sub_const r
+  have hc (p : Z × a.samples) : Continuous (fun t => H t p) :=
+    ((a.trace_continuous p.2).dist continuous_const).sub continuous_const
+  exact continuous_zero_hit_stopping (spatialFiltration (Z := Z) a.filtration) H ha hc 0
+
+lemma joint_boundedSpatialHit_stopping {Z : Type*} [mZ : MeasurableSpace Z]
+    (z : Z → ℂ) (hz : Measurable z) (r : ℝ) (T : ℝ≥0) :
+    IsStoppingTime (spatialFiltration (Z := Z) a.filtration)
+      (fun p : Z × a.samples => (boundedSpatialHit (γ p.2) (z p.1) r T : WithTop ℝ≥0)) := by
+  simpa only [boundedSpatialHit_coe] using (a.joint_spatialHit_stopping z hz r).min_const T
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma clipped_spatial_window_coefficient_le
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi}
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (ω : Ω) (α S : ℝ≥0)
+    (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) γ G F)
+    (hin : ∀ t : ℝ≥0, t ≤ α → lo ≤ ((c (initialLogCoordinates z)).Q t ω).1.im ∧
+        ((c (initialLogCoordinates z)).Q t ω).1.im ≤ hi)
+    (r ε : ℝ) (t : ℝ≥0) :
+    |(if min (boundedSpatialHit γ z r S) α ≤ t ∧
+        t < min (boundedSpatialHit γ z ε S) α then
+      greenLogCoefficient κ ((c (initialLogCoordinates z)).Q t ω) (B t ω) else 0)| ≤
+    |spatialSlabCoefficient κ (fun s => Real.sqrt κ * B s ω) γ G r ε t z| := by
+  split_ifs with ht
+  · have hta : t < α := ht.2.trans_le (min_le_right _ _)
+    have hte : t < boundedSpatialHit γ z ε S := ht.2.trans_le (min_le_left _ _)
+    have htS : t < S := by
+      apply WithTop.coe_lt_coe.mp
+      exact (WithTop.coe_lt_coe.mpr hte).trans_le
+        (by rw [boundedSpatialHit_coe]; exact min_le_right _ _)
+    have hsr : boundedSpatialHit γ z r S ≤ t := (min_le_iff.mp ht.1).resolve_right (not_le_of_gt hta)
+    have hr : spatialHitTime γ z r ≤ t := by
+      have hh := WithTop.coe_le_coe.mpr hsr
+      rw [boundedSpatialHit_coe] at hh
+      exact (min_le_iff.mp hh).resolve_right (not_le_of_gt (WithTop.coe_lt_coe.mpr htS))
+    have he : (t : WithTop ℝ≥0) ≤ spatialHitTime γ z ε :=
+      (WithTop.coe_le_coe.mpr hte.le).trans (boundedSpatialHit_le γ z ε S)
+    obtain ⟨hl, hq⟩ := clipped_chart_identification c hBc z hz ω α γ G F hw hin
+    have hlive := survivingDomain_antitone γ hta.le hl
+    rw [hq t hta.le, greenLogCoefficient_logCoordinates ((hBc ω).const_mul _) hw κ _ z t hlive rfl,
+      spatialSlabCoefficient, ite_eq_left ⟨hr, he, hlive⟩]
+  · simpa only [abs_zero] using abs_nonneg (spatialSlabCoefficient κ (fun s => Real.sqrt κ * B s ω) γ G r ε t z)
+
+ 
+
+theorem IsCapacityTwoWitness.spatial_masked_bracket_bound
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (hU : Continuous U)
+    {κ ε r : ℝ} (hκ : 0 < κ) (hκ4 : κ ≤ 4) (hε : 0 < ε) (hr : 0 < r)
+    (A : Set ℂ) (hA : MeasurableSet A) (hheight : ∀ z ∈ A, ε ≤ z.im)
+    (b : ℝ → A → ℝ)
+    (hb : ∀ t, 0 ≤ t → ∀ z, |b t z| ≤
+      |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|) :
+    (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal
+      ((∫ z, b t z ∂(volume.comap (Subtype.val : A → ℂ))) ^ 2)) ≤
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
+  apply le_trans _ (h.spatial_slab_absolute_bracket_bound hU hκ hκ4 hε hr A hA hheight)
+  apply setLIntegral_mono' measurableSet_Ici
+  intro t ht
+  apply (spatial_integral_square_le (volume.comap (Subtype.val : A → ℂ)) (b t)).trans
+  apply pow_le_pow_left₀ bot_le
+  calc
+    _ ≤ ∫⁻ z : A, ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|
+        ∂(volume.comap (Subtype.val : A → ℂ)) := lintegral_mono fun z => ENNReal.ofReal_le_ofReal (hb t ht z)
+    _ = _ := lintegral_subtype_comap (μ := (volume : Measure ℂ)) hA (fun z : ℂ => ENNReal.ofReal |spatialSlabCoefficient κ U γ G r ε t.toNNReal z|)
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma clipped_family_uniform_bound
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi}
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q) :
+    ∃ L : ℝ≥0, ∀ (q : ℂ × ℝ) (ω : Ω) (t : ℝ≥0),
+      ‖(c q).Q t ω‖ ≤ ‖q‖ + L * t := by
+  obtain ⟨L, K, hb, hK⟩ := clippedLogLoewnerField_bounds lo hi hle
+  refine ⟨L, fun q ω t => ?_⟩
+  have hh := norm_image_sub_le_of_norm_deriv_le_segment'
+    (f := fun s => (c q).Q s ω) (a := (0 : ℝ)) (b := (t : ℝ))
+    (f' := fun s => clippedLogLoewnerField lo hi hle (Real.sqrt κ * B s.toNNReal ω) ((c q).Q s ω))
+    (fun s hs => ((c q).derivative ω s hs.1).mono (fun _ hx => hx.1))
+    (fun s hs => hb _ _) (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩
+  rw [(c q).initial, sub_zero] at hh
+  calc
+    _ ≤ ‖q‖ + ‖(c q).Q t ω - q‖ := by
+      simpa only [add_comm] using norm_le_insert' ((c q).Q t ω) q
+    _ ≤ _ := by linarith
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem spatial_angular_cutoff
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ lo hi : ℝ) (hle : lo ≤ hi)
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (hq : Measurable q) (hq0 : ∀ z, (q z).2 = 0)
+    (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
+    (hε0 : ∀ z, ε ≤ logAngleSq κ (q z) 0)
+    (hlo : ∀ z, lo < (q z).1.im - u / (2 * ε)) (hhi : ∀ z, (q z).1.im < hi)
+    (T : ℝ≥0) (hT : ∀ z, u < (4 * ε ^ 2 / Real.exp (2 * (q z).1.im)) * T) :
+    ∃ τ : Z × Ω → ℝ≥0,
+      IsStoppingTime (spatialFiltration (Z := Z)
+        (Filtration.natural B (fun t => (hm t).stronglyMeasurable)))
+        (fun p => (τ p : WithTop ℝ≥0)) ∧
+      (∀ z ω, τ (z,ω) < T) ∧
+      (∀ z ω t, t ≤ τ (z,ω) →
+        logRadiusClock (q z).1.im ((c (q z)).Q t ω) ≤ u ∧
+        ε ≤ logAngleSq κ ((c (q z)).Q t ω) (B t ω) ∧
+        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi) ∧
+      (∀ z ω t, t < τ (z,ω) →
+        logRadiusClock (q z).1.im ((c (q z)).Q t ω) < u ∧
+        ε < logAngleSq κ ((c (q z)).Q t ω) (B t ω) ∧
+        lo < ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im < hi) ∧
+      (∀ z ω, logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) = u ∨
+        logAngleSq κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) = ε) := by
+  let f := spatialFiltration (Z := Z) (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+  let D (t : ℝ≥0) (p : Z × Ω) :=
+    min (u - logRadiusClock (q p.1).1.im ((c (q p.1)).Q t p.2))
+      (min (logAngleSq κ ((c (q p.1)).Q t p.2) (B t p.2) - ε)
+        (min (((c (q p.1)).Q t p.2).1.im - lo) (hi - ((c (q p.1)).Q t p.2).1.im)))
+  have hDa : Adapted f D := by
+    intro t
+    have hQ := clipped_family_past_measurable c t
+    have hBa := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted t
+    let : MeasurableSpace Ω := Filtration.natural B (fun s => (hm s).stronglyMeasurable) t
+    have hc := hQ.comp ((hq.comp measurable_fst).prodMk measurable_snd)
+    have hBb := hBa.comp (@measurable_snd Z Ω mZ _)
+    have hη := Complex.continuous_im.measurable.comp (hq.comp (@measurable_fst Z Ω mZ (Filtration.natural B (fun s => (hm s).stronglyMeasurable) t))).fst
+    have hclock := (hη.sub (Complex.continuous_im.measurable.comp hc.fst)).add hc.snd
+    exact (measurable_const.sub hclock).min
+      ((((continuous_logAngleSq κ).measurable.comp (hc.prodMk hBb)).sub_const ε).min
+        (((Complex.continuous_im.measurable.comp hc.fst).sub_const lo).min
+          (measurable_const.sub (Complex.continuous_im.measurable.comp hc.fst))))
+  have hDc (p : Z × Ω) : Continuous (fun t => D t p) := by
+    have hc := ((c (q p.1)).lipschitz p.2).continuous.comp NNReal.continuous_coe
+    exact (continuous_const.sub ((continuous_const.sub (Complex.continuous_im.comp hc.fst)).add hc.snd)).min
+      ((((continuous_logAngleSq κ).comp (hc.prodMk (hBc p.2))).sub continuous_const).min
+        (((Complex.continuous_im.comp hc.fst).sub continuous_const).min
+          (continuous_const.sub (Complex.continuous_im.comp hc.fst))))
+  have hD0 (p : Z × Ω) : 0 ≤ D 0 p := by
+    dsimp only [D]
+    rw [NNReal.coe_zero, (c (q p.1)).initial, hB0]
+    simp only [le_min_iff, sub_nonneg]
+    refine ⟨?_, hε0 p.1, ?_, (hhi p.1).le⟩
+    · simpa only [logRadiusClock, sub_self, zero_add, hq0] using hu
+    · have hd : 0 ≤ u / (2 * ε) := div_nonneg hu (by positivity)
+      linarith [hlo p.1]
+  obtain ⟨τ, hτ, hb, hleD, hltD, hzero⟩ := first_margin_stopping f D hDa hDc hD0 T
+  have hin (z : Z) (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ (z,ω)) :
+      logRadiusClock (q z).1.im ((c (q z)).Q t ω) ≤ u ∧
+      ε ≤ logAngleSq κ ((c (q z)).Q t ω) (B t ω) ∧
+      lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi :=
+    by simpa only [D, le_min_iff, sub_nonneg] using hleD (z,ω) t ht
+  have hg (z : Z) (ω : Ω) :
+      ((c (q z)).Q (τ (z,ω)) ω).1.im ≤ (q z).1.im ∧
+      (q z).1.im - logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) / (2 * ε) ≤ ((c (q z)).Q (τ (z,ω)) ω).1.im ∧
+      (4 * ε ^ 2 / Real.exp (2 * (q z).1.im)) * τ (z,ω) ≤ logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) := by
+    have hd (s : ℝ) (hs : s ∈ Icc 0 (τ (z,ω) : ℝ)) :
+        HasDerivWithinAt (fun t => (c (q z)).Q t ω)
+          (physicalLogLoewnerField (Real.sqrt κ * B s.toNNReal ω) ((c (q z)).Q s ω))
+          (Icc 0 (τ (z,ω) : ℝ)) s := by
+      have hst : s.toNNReal ≤ τ (z,ω) := Real.toNNReal_le_iff_le_coe.mpr hs.2
+      have hbounds := (hin z ω s.toNNReal hst).2.2
+      rw [Real.coe_toNNReal _ hs.1] at hbounds
+      have hh := ((c (q z)).derivative ω s hs.1).mono (show Icc (0 : ℝ) (τ (z,ω) : ℝ) ⊆ Ici 0 from fun _ hx => hx.1)
+      rwa [clippedLogLoewnerField_eq_physical lo hi hle _ _ ⟨hbounds.1, hbounds.2⟩] at hh
+    have ha (s : ℝ) (hs : s ∈ Icc 0 (τ (z,ω) : ℝ)) :
+        ε ≤ logAngleSq κ ((c (q z)).Q s ω) (B s.toNNReal ω) := by
+      have hh := (hin z ω s.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hs.2)).2.1
+      simpa only [Real.coe_toNNReal _ hs.1] using hh
+    have hq : ((c (q z)).Q 0 ω).2 = 0 := by rw [(c (q z)).initial ω, hq0 z]
+    have hh := log_loewner_height_clock_bounds (τ (z,ω)).coe_nonneg hε hq hd ha
+    simpa only [(c (q z)).initial ω] using hh
+  have hτlt (z : Z) (ω : Ω) : τ (z,ω) < T := by
+    by_contra hn
+    have he : τ (z,ω) = T := le_antisymm (hb (z,ω)) (not_lt.mp hn)
+    have hh := (hg z ω).2.2.trans (hin z ω (τ (z,ω)) le_rfl).1
+    rw [he] at hh
+    exact (not_le_of_gt (hT z)) hh
+  have he (z : Z) (ω : Ω) :
+      logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) = u ∨
+      logAngleSq κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) = ε := by
+    obtain ⟨hr, ha, hl, hh⟩ := hin z ω (τ (z,ω)) le_rfl
+    have hstrictlo : lo < ((c (q z)).Q (τ (z,ω)) ω).1.im := by
+      have hdiv := div_le_div_of_nonneg_right hr (show 0 ≤ 2 * ε by positivity)
+      linarith [(hg z ω).2.1, hlo z]
+    have hstricthi : ((c (q z)).Q (τ (z,ω)) ω).1.im < hi := (hg z ω).1.trans_lt (hhi z)
+    by_contra hn
+    push Not at hn
+    have hp : 0 < D (τ (z,ω)) (z,ω) :=
+      by simpa only [D, lt_min_iff, sub_pos] using
+        (show logRadiusClock (q z).1.im ((c (q z)).Q (τ (z,ω)) ω) < u ∧
+          ε < logAngleSq κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) ∧
+          lo < ((c (q z)).Q (τ (z,ω)) ω).1.im ∧
+          ((c (q z)).Q (τ (z,ω)) ω).1.im < hi from
+          ⟨lt_of_le_of_ne hr hn.1, lt_of_le_of_ne ha (Ne.symm hn.2), hstrictlo, hstricthi⟩)
+    rw [hzero (z,ω) (hτlt z ω)] at hp
+    exact lt_irrefl 0 hp
+  refine ⟨τ, hτ, hτlt, hin, ?_, he⟩
+  intro z ω t ht
+  simpa only [D, lt_min_iff, sub_pos] using hltD (z,ω) t ht
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+structure SpatialAngularFamily {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (κ : ℝ) (z : Z → ℂ) (u ε : ℝ) where
+  lo : ℝ
+  hi : ℝ
+  le : lo ≤ hi
+  c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi le q
+  T : ℝ≥0
+  τ : Z × Ω → ℝ≥0
+  stopping : IsStoppingTime (spatialFiltration (Z := Z)
+    (Filtration.natural B (fun t => (hm t).stronglyMeasurable)))
+    (fun p => (τ p : WithTop ℝ≥0))
+  bounded : ∀ p, τ p ≤ T
+  inside : ∀ x ω t, t ≤ τ (x,ω) →
+    logRadiusClock (Real.log (z x).im) ((c (initialLogCoordinates (z x))).Q t ω) ≤ u ∧
+    ε ≤ logAngleSq κ ((c (initialLogCoordinates (z x))).Q t ω) (B t ω) ∧
+    lo ≤ ((c (initialLogCoordinates (z x))).Q t ω).1.im ∧
+      ((c (initialLogCoordinates (z x))).Q t ω).1.im ≤ hi
+  before : ∀ x ω t, t < τ (x,ω) →
+    logRadiusClock (Real.log (z x).im) ((c (initialLogCoordinates (z x))).Q t ω) < u ∧
+    ε < logAngleSq κ ((c (initialLogCoordinates (z x))).Q t ω) (B t ω)
+  boundary : ∀ x ω,
+    logRadiusClock (Real.log (z x).im) ((c (initialLogCoordinates (z x))).Q (τ (x,ω)) ω) = u ∨
+    logAngleSq κ ((c (initialLogCoordinates (z x))).Q (τ (x,ω)) ω) (B (τ (x,ω)) ω) = ε
+
+noncomputable def SpatialAngularFamily.section
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ : ℝ} {z : Z → ℂ} {u ε : ℝ}
+    (a : SpatialAngularFamily B hm κ z u ε) (x : Z) :
+    AngularLocalization B hm κ (z x) u ε where
+  lo := a.lo
+  hi := a.hi
+  le := a.le
+  solution := a.c (initialLogCoordinates (z x))
+  T := a.T
+  τ := fun ω => a.τ (x,ω)
+  stopping := spatial_stopping_section a.stopping x
+  bounded := fun ω => a.bounded (x,ω)
+  inside := a.inside x
+  before := a.before x
+  boundary := a.boundary x
+
+lemma exists_spatial_angular_family
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (z : Z → ℂ) (hz : Measurable z)
+    (l h : ℝ) (hlh : l ≤ h)
+    (hη : ∀ x, l ≤ Real.log (z x).im ∧ Real.log (z x).im ≤ h)
+    (u ε : ℝ) (hu : 0 ≤ u) (hε : 0 < ε)
+    (hangle : ∀ x, ε ≤ logAngleSq κ (initialLogCoordinates (z x)) 0) :
+    Nonempty (SpatialAngularFamily B hm κ z u ε) := by
+  classical
+  let lo := l - u / (2 * ε) - 1
+  let hi := h + 1
+  have hud : 0 ≤ u / (2 * ε) := div_nonneg hu (by positivity)
+  have hle : lo ≤ hi := by dsimp [lo, hi]; linarith
+  let c (q : ℂ × ℝ) : ClippedLogSolution B hm κ lo hi hle q :=
+    (exists_clippedLogSolution B hm hBm hBc κ lo hi hle q).some
+  let v := 4 * ε ^ 2 / Real.exp (2 * h)
+  have hv : 0 < v := by dsimp [v]; positivity
+  obtain ⟨N, hN⟩ := exists_nat_gt (u / v)
+  have hT : ∀ x, u < (4 * ε ^ 2 / Real.exp (2 * (initialLogCoordinates (z x)).1.im)) * ((N : ℝ≥0) : ℝ) := by
+    intro x
+    have huv := (div_lt_iff₀ hv).mp hN
+    have he : Real.exp (2 * Real.log (z x).im) ≤ Real.exp (2 * h) :=
+      Real.exp_le_exp.mpr (by linarith [(hη x).2])
+    have hd : v ≤ 4 * ε ^ 2 / Real.exp (2 * Real.log (z x).im) :=
+      div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) he
+    rw [initialLogCoordinates_im, NNReal.coe_natCast]
+    have huv' : u < v * (N : ℝ) := by simpa only [mul_comm (N : ℝ) v] using huv
+    exact huv'.trans_le (mul_le_mul_of_nonneg_right hd (Nat.cast_nonneg N))
+  have hq : Measurable (fun x => initialLogCoordinates (z x)) :=
+    measurable_initialLogCoordinates.comp hz
+  obtain ⟨τ, hτ, hb, hin, hbefore, hboundary⟩ := spatial_angular_cutoff hm hBc hB0 κ lo hi hle c
+    (fun x => initialLogCoordinates (z x)) hq (fun x => rfl) u ε hu hε hangle
+    (fun x => by rw [initialLogCoordinates_im]; dsimp [lo]; linarith [(hη x).1])
+    (fun x => by rw [initialLogCoordinates_im]; dsimp [hi]; linarith [(hη x).2]) (N : ℝ≥0) hT
+  refine ⟨⟨lo, hi, hle, c, N, τ, hτ, fun p => (hb p.1 p.2).le, ?_, ?_, ?_⟩⟩
+  · simpa only [initialLogCoordinates_im] using hin
+  · intro x ω t ht
+    constructor
+    · simpa only [initialLogCoordinates_im] using (hbefore x ω t ht).1
+    · exact (hbefore x ω t ht).2.1
+  · simpa only [initialLogCoordinates_im] using hboundary
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma clipped_area_window_measurable
+    {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    (ν : Measure Z) [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi : ℝ} {hle : lo ≤ hi}
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (hq : Measurable q)
+    (σ τ : Z × Ω → ℝ≥0) (hσ : Measurable σ) (hτ : Measurable τ) (T : ℝ≥0) :
+    Measurable (fun ω => ∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
+        greenLogDensity κ ((c (q z)).Q (σ (z,ω)) ω) (B (σ (z,ω)) ω) ∂ν) ∧
+    Measurable (fun ω => ∫ t in (0 : ℝ)..(T : ℝ),
+        (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
+          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) := by
+  classical
+  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable hBc hm
+  have hQm := clipped_family_joint_measurable c
+  have hD (η : Z × Ω → ℝ≥0) (hη : Measurable η) : Measurable (fun p : Z × Ω =>
+      greenLogDensity κ ((c (q p.1)).Q (η p) p.2) (B (η p) p.2)) :=
+    (greenLogDensity_contDiff κ).continuous.measurable.comp
+      ((hQm.comp (hη.prodMk ((hq.comp measurable_fst).prodMk measurable_snd))).prodMk
+        (hBm.comp (hη.prodMk measurable_snd)))
+  refine ⟨((hD τ hτ).sub (hD σ hσ)).stronglyMeasurable.integral_prod_left'.measurable, ?_⟩
+  let f (p : ℝ × (Z × Ω)) := if (σ p.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2 : ℝ) then
+    greenLogCoefficient κ ((c (q p.2.1)).Q p.1 p.2.2) (B p.1.toNNReal p.2.2) else 0
+  have hgood : MeasurableSet {p : ℝ × (Z × Ω) | (σ p.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2 : ℝ)} :=
+    (measurableSet_le (NNReal.continuous_coe.measurable.comp (hσ.comp measurable_snd)) measurable_fst).inter
+      (measurableSet_lt measurable_fst (NNReal.continuous_coe.measurable.comp (hτ.comp measurable_snd)))
+  have hQreal : Measurable (fun p : ℝ × (Z × Ω) => (c (q p.2.1)).Q p.1.toNNReal p.2.2) :=
+    hQm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
+      ((hq.comp (measurable_fst.comp measurable_snd)).prodMk (measurable_snd.comp measurable_snd)))
+  have hcoef : Measurable (fun p : (ℂ × ℝ) × ℝ => greenLogCoefficient κ p.1 p.2) := by
+    unfold greenLogCoefficient greenLogNormSq
+    apply (greenLogDensity_contDiff κ).continuous.measurable.mul
+    fun_prop
+  have hfm' : Measurable (fun p : ℝ × (Z × Ω) =>
+      if (σ p.2 : ℝ) ≤ p.1 ∧ p.1 < (τ p.2 : ℝ) then
+        greenLogCoefficient κ ((c (q p.2.1)).Q p.1.toNNReal p.2.2) (B p.1.toNNReal p.2.2) else 0) := (hcoef.comp
+    (hQreal.prodMk (hBm.comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
+      (measurable_snd.comp measurable_snd))))).ite hgood (measurable_const (a := (0 : ℝ)))
+  have hfm : Measurable f := by
+    convert hfm' using 1
+    ext p
+    dsimp only [f]
+    split_ifs with hp
+    · rw [Real.coe_toNNReal _ ((σ p.2).coe_nonneg.trans hp.1)]
+    · rfl
+  have hswap : Measurable (fun p : Z × (ℝ × Ω) => f (p.2.1,(p.1,p.2.2))) := hfm.comp
+    ((measurable_fst.comp measurable_snd).prodMk (measurable_fst.prodMk (measurable_snd.comp measurable_snd)))
+  have harea : Measurable (fun p : ℝ × Ω => ∫ z, f (p.1,(z,p.2)) ∂ν) :=
+    hswap.stronglyMeasurable.integral_prod_left'.measurable
+  have hh := (harea.pow_const 2).stronglyMeasurable.integral_prod_left'
+    (μ := volume.restrict (Ioc (0 : ℝ) T))
+  simpa only [intervalIntegral.integral_of_le T.coe_nonneg, f] using hh.measurable
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma spatial_stopping_lift {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {f : Filtration ℝ≥0 mΩ} {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime f τ) :
+    IsStoppingTime (spatialFiltration (Z := Z) f) (fun p => τ p.2) := by
+  intro t
+  let : MeasurableSpace Ω := f t
+  exact (hτ t).preimage (measurable_snd : @Measurable (Z × Ω) Ω (mZ.prod (f t)) (f t) Prod.snd)
+
+ 
+
+theorem exists_common_brownian_compact_stops
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hB0 : ∀ ω, B 0 ω = 0) (T : ℝ≥0) :
+    ∃ π : ℕ → Ω → ℝ≥0,
+      (∀ n, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (π n ω : WithTop ℝ≥0))) ∧
+      (∀ n ω, π n ω ≤ T) ∧
+      (∀ n ω t, t ≤ π n ω → |B t ω| ≤ (n : ℝ) + 1) ∧
+      (∀ ω, ∀ᶠ n in atTop, π n ω = T) := by
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hDa (n : ℕ) : Adapted f (fun t ω => (n : ℝ) + 1 - |B t ω|) := by
+    intro t
+    have ha := (Filtration.stronglyAdapted_natural (fun s => (hm s).stronglyMeasurable)).adapted t
+    let : MeasurableSpace Ω := f t
+    exact measurable_const.sub ha.abs
+  have hb (n : ℕ) := first_margin_stopping f (fun t ω => (n : ℝ) + 1 - |B t ω|)
+    (hDa n) (fun ω => continuous_const.sub (hBc ω).abs)
+    (fun ω => by rw [hB0 ω, abs_zero, sub_zero]; positivity) T
+  choose π hπ hb hin hpre hat using hb
+  refine ⟨π, hπ, hb, fun n ω t ht => (sub_nonneg.mp (hin n ω t ht)), ?_⟩
+  intro ω
+  obtain ⟨N, hN⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
+  filter_upwards [eventually_ge_atTop N] with n hn
+  apply le_antisymm (hb n ω)
+  by_contra h
+  have he := hat n ω (not_le.mp h)
+  have hh := hN (π n ω) (hb n ω)
+  have hn' : (N : ℝ) ≤ n := Nat.cast_le.mpr hn
+  linarith
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem clipped_spatial_green_window_unbounded_driver
+    {Z Ω : Type*} [mZ : MeasurableSpace Z] [mΩ : MeasurableSpace Ω]
+    {ν : Measure Z} {P : Measure Ω} [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, Measurable (B t))
+    (hB : IsBrownianReal B P) (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hB0 : ∀ ω, B 0 ω = 0)
+    (κ lo hi : ℝ) (hle : lo ≤ hi) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (c : ∀ q : ℂ × ℝ, ClippedLogSolution B hm κ lo hi hle q)
+    (q : Z → ℂ × ℝ) (hq : Measurable q)
+    (M : ℝ) (hqbound : ∀ z, ‖q z‖ ≤ M) (T : ℝ≥0)
+    (σ τ : Z × Ω → ℝ≥0)
+    (hσ : IsStoppingTime (spatialFiltration (Z := Z)
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
+      (fun p => (σ p : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime (spatialFiltration (Z := Z)
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
+      (fun p => (τ p : WithTop ℝ≥0)))
+    (hστ : ∀ p, σ p ≤ τ p) (hτb : ∀ p, τ p ≤ T)
+    (hinside : ∀ z ω t, t ≤ τ (z,ω) →
+      lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi)
+    (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp
+      (θ * (∫ z, greenLogDensity κ ((c (q z)).Q (τ (z,ω)) ω) (B (τ (z,ω)) ω) -
+        greenLogDensity κ ((c (q z)).Q (σ (z,ω)) ω) (B (σ (z,ω)) ω) ∂ν) -
+      θ^2 * (∫ t in (0 : ℝ)..(T : ℝ),
+        (∫ z, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
+          greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2) / 2)) ∂P) ≤ 1 := by
+  classical
+  obtain ⟨π, hπ, hπb, hπin, hπlim⟩ := exists_common_brownian_compact_stops hm hBc hB0 T
+  obtain ⟨L, hL⟩ := clipped_family_uniform_bound c
+  let σn (n : ℕ) (p : Z × Ω) := min (σ p) (π n p.2)
+  let τn (n : ℕ) (p : Z × Ω) := min (τ p) (π n p.2)
+  have hσn (n : ℕ) : IsStoppingTime (spatialFiltration (Z := Z)
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
+      (fun p => (σn n p : WithTop ℝ≥0)) := by
+    simpa only [σn, WithTop.coe_min] using hσ.min (spatial_stopping_lift (Z := Z) (hπ n))
+  have hτn (n : ℕ) : IsStoppingTime (spatialFiltration (Z := Z)
+      (Filtration.natural B (fun s => (hm s).stronglyMeasurable)))
+      (fun p => (τn n p : WithTop ℝ≥0)) := by
+    simpa only [τn, WithTop.coe_min] using hτ.min (spatial_stopping_lift (Z := Z) (hπ n))
+  have hstn (n : ℕ) (p : Z × Ω) : σn n p ≤ τn n p := min_le_min (hστ p) le_rfl
+  have htn (n : ℕ) (p : Z × Ω) : τn n p ≤ T := (min_le_left _ _).trans (hτb p)
+  let R (n : ℕ) := max M 0 + (L : ℝ) * T + ((n : ℝ) + 1)
+  have hR (n : ℕ) : 0 ≤ R n := by dsimp [R]; positivity
+  have hin (n : ℕ) (z : Z) (ω : Ω) (t : ℝ≥0) (ht : t ≤ τn n (z,ω)) :
+      ‖((c (q z)).Q t ω, B t ω)‖ ≤ R n ∧
+        lo ≤ ((c (q z)).Q t ω).1.im ∧ ((c (q z)).Q t ω).1.im ≤ hi := by
+    have htτ := ht.trans (min_le_left _ _)
+    have htb := ht.trans (htn n (z,ω))
+    refine ⟨?_, hinside z ω t htτ⟩
+    rw [Prod.norm_def, max_le_iff]
+    have hqt := hL (q z) ω t
+    have hmul : (L : ℝ) * t ≤ (L : ℝ) * T := mul_le_mul_of_nonneg_left
+      (NNReal.coe_le_coe.mpr htb) L.coe_nonneg
+    have hbm := hπin n ω t (ht.trans (min_le_right _ _))
+    have hml : M ≤ max M 0 := le_max_left _ _
+    have hmn : 0 ≤ max M 0 := le_max_right _ _
+    have hlt : 0 ≤ (L : ℝ) * T := mul_nonneg L.coe_nonneg T.coe_nonneg
+    dsimp only [R]
+    exact ⟨by linarith [hqbound z, Nat.cast_nonneg (α := ℝ) n], by simpa only [Real.norm_eq_abs] using
+      (show |B t ω| ≤ max M 0 + (L : ℝ) * T + ((n : ℝ) + 1) by linarith)⟩
+  let X (η ξ : Z × Ω → ℝ≥0) (ω : Ω) :=
+    ∫ z, greenLogDensity κ ((c (q z)).Q (ξ (z,ω)) ω) (B (ξ (z,ω)) ω) -
+      greenLogDensity κ ((c (q z)).Q (η (z,ω)) ω) (B (η (z,ω)) ω) ∂ν
+  let E (η ξ : Z × Ω → ℝ≥0) (ω : Ω) := ∫ t in (0 : ℝ)..(T : ℝ),
+    (∫ z, if (η (z,ω) : ℝ) ≤ t ∧ t < (ξ (z,ω) : ℝ) then
+      greenLogCoefficient κ ((c (q z)).Q t ω) (B t.toNNReal ω) else 0 ∂ν)^2
+  have hxm (n : ℕ) := clipped_area_window_measurable ν hBc c q hq (σn n) (τn n)
+    (stopping_time_measurable_nnreal (hσn n)) (stopping_time_measurable_nnreal (hτn n)) T
+  have hexp (n : ℕ) := clipped_spatial_green_window_exponential (ν := ν) hm hB hBc
+    κ lo hi hle hκ hκ8 c q hq (R n) (hR n) T (σn n) (τn n)
+      (stopping_time_measurable_nnreal (hσn n)) (stopping_time_measurable_nnreal (hτn n)) (hstn n)
+      (stopping_before_measurable (hσn n)) (spatial_stopping_section (hσn n))
+      (stopping_before_measurable (hτn n)) (spatial_stopping_section (hτn n))
+      (fun z ω => htn n (z,ω)) (hin n) θ
+  have hev (ω : Ω) : ∀ᶠ n in atTop,
+      X (σn n) (τn n) ω = X σ τ ω ∧ E (σn n) (τn n) ω = E σ τ ω := by
+    filter_upwards [hπlim ω] with n hn
+    have hs (z : Z) : σn n (z,ω) = σ (z,ω) := by
+      dsimp only [σn]; rw [hn, min_eq_left ((hστ (z,ω)).trans (hτb (z,ω))) ]
+    have ht (z : Z) : τn n (z,ω) = τ (z,ω) := by
+      dsimp only [τn]; rw [hn, min_eq_left (hτb (z,ω))]
+    simp only [X, E, hs, ht, and_self]
+  apply compensated_exponential_limit_energy
+    (fun n => X (σn n) (τn n)) (fun n => E (σn n) (τn n)) (X σ τ) (E σ τ) θ
+  · intro n
+    exact ((((hxm n).1.const_mul θ).sub (((hxm n).2.const_mul (θ^2)).div_const 2)).exp).ennreal_ofReal.aemeasurable
+  · apply Eventually.of_forall
+    intro ω
+    exact tendsto_const_nhds.congr' ((hev ω).mono fun n hn => hn.1.symm)
+  · apply Eventually.of_forall
+    intro ω
+    exact tendsto_const_nhds.congr' ((hev ω).mono fun n hn => hn.2.symm)
+  · exact hexp
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory
+open scoped ENNReal NNReal
+namespace SLEExactGauge
+
+lemma interval_square_energy_bound (f : ℝ → ℝ) (T : ℝ≥0) (V : ℝ) (hV : 0 ≤ V)
+    (hb : (∫⁻ t in Ici (0 : ℝ), ENNReal.ofReal ((f t)^2)) ≤ ENNReal.ofReal V) :
+    (∫ t in (0 : ℝ)..(T : ℝ), (f t)^2) ≤ V := by
+  rw [intervalIntegral.integral_of_le T.coe_nonneg]
+  have hnorm := enorm_integral_le_lintegral_enorm (μ := volume.restrict (Ioc (0 : ℝ) (T : ℝ)))
+    (fun t => (f t)^2)
+  simp_rw [Real.enorm_eq_ofReal (sq_nonneg _)] at hnorm
+  have hi : 0 ≤ ∫ t in Ioc (0 : ℝ) (T : ℝ), (f t)^2 := integral_nonneg (fun t => sq_nonneg _)
+  rw [Real.enorm_eq_ofReal hi] at hnorm
+  apply (ENNReal.ofReal_le_ofReal_iff hV).mp
+  apply hnorm.trans
+  apply le_trans _ hb
+  exact lintegral_mono_set (fun t ht => ht.1.le)
+
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def slabVariance (κ ε r : ℝ) (A : Set ℂ) : ℝ :=
+  (ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+    (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A))).toReal
+
+lemma slabVariance_nonneg (κ ε r : ℝ) (A : Set ℂ) : 0 ≤ slabVariance κ ε r A :=
+  ENNReal.toReal_nonneg
+
+lemma ofReal_slabVariance (κ ε r : ℝ) (A : Set ℂ) (hfin : volume A ≠ ∞) :
+    ENNReal.ofReal (slabVariance κ ε r A) =
+      ENNReal.ofReal ((Real.sqrt κ * greenExponent κ * ε ^ (-codimension κ)) ^ 2 / 4) *
+        (2 * (ENNReal.ofReal (layerGreenConstant * r ^ 2) * volume A)) := by
+  apply ENNReal.ofReal_toReal
+  exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top
+    (ENNReal.mul_ne_top (by norm_num) (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hfin))
+
+namespace RegularSLEPresentation
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+
+ 
+
+theorem localized_spatial_slab_exponential
+    (hκ : 0 < κ) (hκ4 : κ ≤ 4)
+    (A : Set ℂ) (hA : MeasurableSet A)
+    [IsFiniteMeasure (volume.comap (Subtype.val : A → ℂ))]
+    (hfin : volume A ≠ ∞)
+    (ε r : ℝ) (hε : 0 < ε) (hεr : ε ≤ r)
+    (hheight : ∀ z ∈ A, r ≤ z.im)
+    (u δ : ℝ) (b : SpatialAngularFamily a.B a.measurable_B κ
+      (Subtype.val : A → ℂ) u δ)
+    (M : ℝ) (hM : ∀ z : A, ‖initialLogCoordinates z‖ ≤ M)
+    (S : ℝ≥0) (θ : ℝ) :
+    (∫⁻ ω : a.samples, ENNReal.ofReal (Real.exp
+      (θ * (∫ z : A,
+        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
+        (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω
+        ∂(volume.comap (Subtype.val : A → ℂ))) -
+        θ^2 * slabVariance κ ε r A / 2)) ∂(P.comap Subtype.val)) ≤ 1 := by
+  classical
+  let ν := volume.comap (Subtype.val : A → ℂ)
+  let σ (p : A × a.samples) := min (boundedSpatialHit (γ p.2) p.1 r S) (b.τ p)
+  let τ (p : A × a.samples) := min (boundedSpatialHit (γ p.2) p.1 ε S) (b.τ p)
+  have hσ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+      (fun p => (σ p : WithTop ℝ≥0)) := by
+    simpa only [σ, WithTop.coe_min] using
+      (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe r S).min b.stopping
+  have hτ : IsStoppingTime (spatialFiltration (Z := A) a.filtration)
+      (fun p => (τ p : WithTop ℝ≥0)) := by
+    simpa only [τ, WithTop.coe_min] using
+      (a.joint_boundedSpatialHit_stopping Subtype.val measurable_subtype_coe ε S).min b.stopping
+  have hστ (p : A × a.samples) : σ p ≤ τ p := by
+    apply min_le_min _ le_rfl
+    apply boundedSpatialHit_antitone_radius (a.trace_continuous p.2) p.1 hεr
+    rw [(a.witness p.2).2.1, dist_zero_left]
+    exact (hheight p.1 p.1.2).trans ((le_abs_self p.1.val.im).trans (Complex.abs_im_le_norm p.1))
+  have htbound (p : A × a.samples) : τ p ≤ b.T := (min_le_right _ _).trans (b.bounded p)
+  have hin (z : A) (ω : a.samples) (t : ℝ≥0) (ht : t ≤ τ (z,ω)) :
+      b.lo ≤ ((b.c (initialLogCoordinates z)).Q t ω).1.im ∧
+        ((b.c (initialLogCoordinates z)).Q t ω).1.im ≤ b.hi :=
+    (b.inside z ω t (ht.trans (min_le_right _ _))).2.2
+  have hexp := clipped_spatial_green_window_unbounded_driver (ν := ν) a.measurable_B a.brownian
+    a.continuous_B a.zero_B κ b.lo b.hi b.le hκ (by linarith) b.c
+    (fun z : A => initialLogCoordinates z) (measurable_initialLogCoordinates.comp measurable_subtype_coe)
+    M hM b.T σ τ hσ hτ hστ htbound hin θ
+  let E (ω : a.samples) := ∫ t in (0 : ℝ)..(b.T : ℝ),
+    (∫ z : A, if (σ (z,ω) : ℝ) ≤ t ∧ t < (τ (z,ω) : ℝ) then
+      greenLogCoefficient κ ((b.c (initialLogCoordinates z)).Q t ω) (a.B t.toNNReal ω) else 0 ∂ν)^2
+  have hE (ω : a.samples) : E ω ≤ slabVariance κ ε r A := by
+    apply interval_square_energy_bound _ b.T _ (slabVariance_nonneg _ _ _ _)
+    rw [ofReal_slabVariance _ _ _ _ hfin]
+    apply (a.witness ω).spatial_masked_bracket_bound ((a.continuous_B ω).const_mul _)
+      hκ hκ4 hε (hε.trans_le hεr) A hA (fun z hz => hεr.trans (hheight z hz))
+    intro t ht z
+    have hb := clipped_spatial_window_coefficient_le b.c a.continuous_B z
+      (show z.val ∈ upperHalfPlane from hε.trans_le (hεr.trans (hheight z z.2))) ω (b.τ (z,ω)) S
+      (γ ω) (a.G ω) (a.F ω) (a.witness ω)
+      (fun s hs => (b.inside z ω s hs).2.2) r ε t.toNNReal
+    simpa only [σ, τ, ← NNReal.coe_le_coe, ← NNReal.coe_lt_coe, Real.coe_toNNReal _ ht] using hb
+  apply le_trans _ hexp
+  apply lintegral_mono
+  intro ω
+  apply ENNReal.ofReal_le_ofReal
+  apply Real.exp_le_exp.mpr
+  change θ * _ - θ^2 * slabVariance κ ε r A / 2 ≤ θ * _ - θ^2 * E ω / 2
+  have hid : (∫ z : A,
+      (b.section z).physicalWeight (boundedSpatialHit (γ ω) z ε S) ω -
+      (b.section z).physicalWeight (boundedSpatialHit (γ ω) z r S) ω ∂ν) =
+      ∫ z : A, greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (τ (z,ω)) ω) (a.B (τ (z,ω)) ω) -
+        greenLogDensity κ ((b.c (initialLogCoordinates z)).Q (σ (z,ω)) ω) (a.B (σ (z,ω)) ω) ∂ν := by
+    apply integral_congr_ae
+    exact Eventually.of_forall fun z => by simp only [AngularLocalization.physicalWeight,
+      SpatialAngularFamily.section, σ, τ, NNReal.coe_min]
+  rw [hid]
+  nlinarith [mul_le_mul_of_nonneg_left (hE ω) (sq_nonneg θ)]
+
+end RegularSLEPresentation
+end SLEExactGauge
+
+ 
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+namespace SpatialAngularFamily
+variable {Z Ω : Type*} [MeasurableSpace Z] [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ : ℝ} {z : Z → ℂ} {u δ : ℝ}
+
+end SpatialAngularFamily
+end SLEExactGauge
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StoppedKernels.lean b/lean/OAI/Probability/SLE/StoppedKernels.lean
index 8ec793e819d213b0d8bb0a9ecf733b0e973b765d..0a1a9e1e2a378808988eab339d05f846b29af926 100644
--- a/lean/OAI/Probability/SLE/StoppedKernels.lean
+++ b/lean/OAI/Probability/SLE/StoppedKernels.lean
@@ -1,671 +1,671 @@
-import Mathlib
-import OAI.Probability.SLE.AngularLaws
-import OAI.Probability.SLE.DiskHalfPlane
-
-namespace OAI
-
-/-! Stopped Kernels for chordal SLE. -/
-
-section
-section
-section
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian RealInnerProductSpace
-namespace SLEExactGauge
-lemma diskLogRadius_laplacian {f : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) {z : ℂ}
-    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
-    (Δ (diskLogRadius f)) z = -4 / diskWeight z ^ 2 := by
-  have hh := (((hd.deriv isOpen_ball).analyticAt (isOpen_ball.mem_nhds hz))).harmonicAt_log_norm hne
-  have he := hh.2.eq_of_nhds
-  change (Δ (fun w => Real.log ‖deriv f w‖)) z = 0 at he
-  rw [show diskLogRadius f = (fun w => Real.log (diskWeight w)) +
-      (fun w => Real.log ‖deriv f w‖) from rfl,
-    (diskWeight_contDiff.contDiffAt.log (diskWeight_pos hz).ne').laplacian_add hh.1,
-    he, add_zero, log_diskWeight_laplacian hz]
-
-lemma diskLogRadius_exp {f : ℂ → ℂ} {z : ℂ}
-    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
-    Real.exp (diskLogRadius f z) = diskWeight z * ‖deriv f z‖ := by
-  simp only [diskLogRadius, Real.exp_add, Real.exp_log (diskWeight_pos hz),
-    Real.exp_log (norm_pos_iff.mpr hne)]
-
-lemma log_diskWeight_direction_bound {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (v : ℂ) :
-    diskWeight z * |fderiv ℝ (fun w => Real.log (diskWeight w)) z v| ≤ 2 * ‖v‖ := by
-  have hh := ((diskWeight_contDiff.differentiable (by norm_num) z).hasFDerivAt.log
-    (diskWeight_pos hz).ne').fderiv
-  rw [hh, diskWeight_fderiv]
-  simp only [smul_apply, smul_eq_mul, abs_mul, abs_inv, abs_of_pos (diskWeight_pos hz),
-    abs_neg, abs_of_pos (show (0 : ℝ) < 2 by norm_num)]
-  rw [← mul_assoc, mul_inv_cancel₀ (diskWeight_pos hz).ne', one_mul]
-  have hinner := abs_real_inner_le_norm z v
-  have hn : ‖z‖ ≤ 1 := (mem_ball_zero_iff.mp hz).le
-  change 2 * |inner ℝ z v| ≤ 2 * ‖v‖
-  calc
-    _ ≤ 2 * (‖z‖ * ‖v‖) := mul_le_mul_of_nonneg_left hinner (by norm_num)
-    _ ≤ 2 * ‖v‖ := by nlinarith [norm_nonneg v]
-
-lemma disk_preSchwarzian_weight {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
-    diskWeight z * ‖deriv (deriv f) z / deriv f z‖ ≤ 2560 * Real.pi + 4 := by
-  have hh := disk_preSchwarzian_bound hd hi hg hgf hz hne
-  have hb := norm_add_le
-    (((diskWeight z : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z) - 2 * star z) (2 * star z)
-  rw [sub_add_cancel, norm_mul, Complex.norm_real, Real.norm_eq_abs,
-    abs_of_pos (diskWeight_pos hz)] at hb
-  have he : ‖(2 : ℂ) * star z‖ = 2 * ‖z‖ := by simp
-  rw [he] at hb
-  have hn := mem_ball_zero_iff.mp hz
-  change ‖((diskWeight z : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z) - 2 * star z‖ ≤ _ at hh
-  linarith
-
-lemma diskLogRadius_direction_bound {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (v : ℂ) :
-    diskWeight z * |fderiv ℝ (diskLogRadius f) z v| ≤ (2560 * Real.pi + 6) * ‖v‖ := by
-  have hdd := ((hd.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hz))
-  have hn : DifferentiableAt ℝ (fun w => Real.log ‖deriv f w‖) z :=
-    ((hdd.restrictScalars ℝ).norm ℂ hne).log (norm_ne_zero_iff.mpr hne)
-  have hw : DifferentiableAt ℝ (fun w => Real.log (diskWeight w)) z :=
-    (diskWeight_contDiff.differentiable (by norm_num) z).log (diskWeight_pos hz).ne'
-  have hp := disk_preSchwarzian_weight hd hi hg hgf hz hne
-  have h1 := log_diskWeight_direction_bound hz v
-  have h2 := log_norm_direction_bound hdd hne v
-  have he : fderiv ℝ (diskLogRadius f) z =
-      fderiv ℝ (fun w => Real.log (diskWeight w)) z +
-        fderiv ℝ (fun w => Real.log ‖deriv f w‖) z := by
-    exact fderiv_fun_add hw hn
-  rw [he, add_apply]
-  have hb := mul_le_mul_of_nonneg_left (abs_add_le
-    (fderiv ℝ (fun w => Real.log (diskWeight w)) z v)
-    (fderiv ℝ (fun w => Real.log ‖deriv f w‖) z v)) (diskWeight_pos hz).le
-  have hb2 := mul_le_mul_of_nonneg_left h2 (diskWeight_pos hz).le
-  have hb3 := mul_le_mul_of_nonneg_right hp (norm_nonneg v)
-  nlinarith
-
-lemma diskLogRadius_gradient_bound {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
-    diskWeight z ^ 2 * ((fderiv ℝ (diskLogRadius f) z 1) ^ 2 +
-      (fderiv ℝ (diskLogRadius f) z I) ^ 2) ≤ 2 * (2560 * Real.pi + 6) ^ 2 := by
-  have h1 := diskLogRadius_direction_bound hd hi hg hgf hz hne 1
-  have hI := diskLogRadius_direction_bound hd hi hg hgf hz hne I
-  simp only [norm_one, norm_I, mul_one] at h1 hI
-  have hw := (diskWeight_pos hz).le
-  have hc : 0 ≤ 2560 * Real.pi + 6 := by positivity
-  have h1' := sq_le_sq₀ (mul_nonneg hw (abs_nonneg _)) hc |>.mpr h1
-  have hI' := sq_le_sq₀ (mul_nonneg hw (abs_nonneg _)) hc |>.mpr hI
-  simp only [mul_pow, sq_abs] at h1' hI'
-  nlinarith
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge
-
- 
-
-noncomputable def layerProfile (r q x : ℝ) : ℝ :=
-  r ^ 2 * Real.exp (q * (x - Real.log r)) / (1 + Real.exp (q * (x - Real.log r)))
-
-noncomputable def layerProfileDeriv (r q x : ℝ) : ℝ :=
-  q * r ^ 2 * Real.exp (q * (x - Real.log r)) /
-    (1 + Real.exp (q * (x - Real.log r))) ^ 2
-
-lemma layerProfile_hasDeriv (r q x : ℝ) :
-    HasDerivAt (layerProfile r q) (layerProfileDeriv r q x) x := by
-  have ht := (((hasDerivAt_id x).sub_const (Real.log r)).const_mul q).exp
-  have hp := ((ht.const_mul (r ^ 2)).div (ht.const_add 1)
-    (by positivity : 1 + Real.exp (q * (x - Real.log r)) ≠ 0))
-  have hne : 1 + Real.exp (q * (x - Real.log r)) ≠ 0 := by positivity
-  convert! hp using 1
-  simp only [layerProfileDeriv, id_eq, mul_one]
-  field_simp [hne]
-  ring
-
-lemma layerProfileDeriv_hasDeriv (r q x : ℝ) :
-    HasDerivAt (layerProfileDeriv r q)
-      (q * layerProfileDeriv r q x *
-        ((1 - Real.exp (q * (x - Real.log r))) /
-          (1 + Real.exp (q * (x - Real.log r))))) x := by
-  have ht := (((hasDerivAt_id x).sub_const (Real.log r)).const_mul q).exp
-  have hp := ((ht.const_mul (q * r ^ 2)).div ((ht.const_add 1).pow 2)
-    (by positivity : (1 + Real.exp (q * (x - Real.log r))) ^ 2 ≠ 0))
-  have hne : 1 + Real.exp (q * (x - Real.log r)) ≠ 0 := by positivity
-  convert! hp using 1
-  simp only [layerProfileDeriv, id_eq, Pi.pow_apply, mul_one,
-    Nat.cast_ofNat, Nat.reduceSub, pow_one]
-  field_simp [hne]
-  ring
-
-lemma layerProfile_contDiff (r q : ℝ) : ContDiff ℝ 2 (layerProfile r q) := by
-  unfold layerProfile
-  fun_prop (disch := intros; positivity)
-
-lemma layerProfile_nonneg (r q x : ℝ) : 0 ≤ layerProfile r q x := by
-  unfold layerProfile
-  positivity
-
-lemma layerProfile_le (r q x : ℝ) : layerProfile r q x ≤ r ^ 2 := by
-  unfold layerProfile
-  apply (div_le_iff₀ (by positivity)).mpr
-  have ht := Real.exp_pos (q * (x - Real.log r))
-  nlinarith [sq_nonneg r]
-
-lemma layerProfile_deriv_nonneg {r q x : ℝ} (hq : 0 ≤ q) :
-    0 ≤ deriv (layerProfile r q) x := by
-  rw [(layerProfile_hasDeriv r q x).deriv]
-  unfold layerProfileDeriv
-  positivity
-
-lemma layerProfile_second_le {r q x : ℝ} (hq : 0 ≤ q) :
-    deriv (deriv (layerProfile r q)) x ≤ q * deriv (layerProfile r q) x := by
-  have he : deriv (layerProfile r q) = layerProfileDeriv r q :=
-    funext fun x => (layerProfile_hasDeriv r q x).deriv
-  rw [he, (layerProfileDeriv_hasDeriv r q x).deriv]
-  have ht : 0 < Real.exp (q * (x - Real.log r)) := Real.exp_pos _
-  have hratio : (1 - Real.exp (q * (x - Real.log r))) /
-      (1 + Real.exp (q * (x - Real.log r))) ≤ 1 := by
-    apply (div_le_iff₀ (by positivity)).mpr
-    linarith
-  have hd : 0 ≤ layerProfileDeriv r q x := by unfold layerProfileDeriv; positivity
-  simpa using mul_le_mul_of_nonneg_left hratio (mul_nonneg hq hd)
-
-lemma layerProfile_deriv_lower {r q x : ℝ}
-    (hr : 0 < r) (hq : 0 ≤ q) (hq2 : q ≤ 2) (hx : x ≤ Real.log r) :
-    q / 4 * Real.exp x ^ 2 ≤ deriv (layerProfile r q) x := by
-  rw [(layerProfile_hasDeriv r q x).deriv]
-  let t := Real.exp (q * (x - Real.log r))
-  have ht : 0 < t := Real.exp_pos _
-  have ht1 : t ≤ 1 := by
-    apply Real.exp_le_one_iff.mpr
-    exact mul_nonpos_of_nonneg_of_nonpos hq (sub_nonpos.mpr hx)
-  have hlow : Real.exp (2 * (x - Real.log r)) ≤ t := by
-    apply Real.exp_le_exp.mpr
-    exact mul_le_mul_of_nonpos_right hq2 (sub_nonpos.mpr hx)
-  have he : Real.exp x ^ 2 = r ^ 2 * Real.exp (2 * (x - Real.log r)) := by
-    have h2 (y : ℝ) : Real.exp (2 * y) = Real.exp y ^ 2 := by simpa using Real.exp_nat_mul y 2
-    rw [mul_sub, Real.exp_sub, h2, h2, Real.exp_log hr]
-    field_simp [hr.ne']
-  have he' : Real.exp x ^ 2 ≤ r ^ 2 * t := by rw [he]; exact mul_le_mul_of_nonneg_left hlow (sq_nonneg r)
-  change q / 4 * Real.exp x ^ 2 ≤ q * r ^ 2 * t / (1 + t) ^ 2
-  apply (le_div_iff₀ (by positivity)).mpr
-  have hden : (1 + t) ^ 2 ≤ 4 := by nlinarith
-  have hh := mul_le_mul_of_nonneg_left hden (mul_nonneg (div_nonneg hq (by norm_num : (0 : ℝ) ≤ 4)) (sq_nonneg (Real.exp x)))
-  have hh' := mul_le_mul_of_nonneg_left he' hq
-  nlinarith
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-namespace SLEExactGauge
-
-noncomputable def layerExponent : ℝ := (2 * (2560 * Real.pi + 6) ^ 2 + 1)⁻¹
-
-lemma layerExponent_pos : 0 < layerExponent := by unfold layerExponent; positivity
-lemma layerExponent_le_one : layerExponent ≤ 1 := by
-  unfold layerExponent
-  apply (inv_le_one₀ (by positivity)).mpr
-  nlinarith [sq_nonneg (2560 * Real.pi + 6)]
-lemma layerExponent_gradient : layerExponent * (2 * (2560 * Real.pi + 6) ^ 2) ≤ 1 := by
-  unfold layerExponent
-  have hp : 0 < 2 * (2560 * Real.pi + 6) ^ 2 + 1 := by positivity
-  have he := inv_mul_cancel₀ hp.ne'
-  have hn : 0 ≤ (2 * (2560 * Real.pi + 6) ^ 2 + 1)⁻¹ := by positivity
-  nlinarith
-
-noncomputable def diskLayerBarrier (f : ℂ → ℂ) (r : ℝ) : ℂ → ℝ :=
-  layerProfile r layerExponent ∘ diskLogRadius f
-
-lemma diskLayerBarrier_bounds (f : ℂ → ℂ) (r : ℝ) (z : ℂ) :
-    0 ≤ diskLayerBarrier f r z ∧ diskLayerBarrier f r z ≤ r ^ 2 :=
-  ⟨layerProfile_nonneg _ _ _, layerProfile_le _ _ _⟩
-
-lemma diskLayerBarrier_contDiffAt {f : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) {z : ℂ}
-    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (r : ℝ) :
-    ContDiffAt ℝ 2 (diskLayerBarrier f r) z :=
-  (layerProfile_contDiff r layerExponent).contDiffAt.comp z (diskLogRadius_contDiffAt hd hz hne)
-
-lemma diskLayerBarrier_laplacian_le {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (r : ℝ) :
-    (Δ (diskLayerBarrier f r)) z ≤
-      -3 * deriv (layerProfile r layerExponent) (diskLogRadius f z) / diskWeight z ^ 2 := by
-  let b := deriv (layerProfile r layerExponent) (diskLogRadius f z)
-  let c := deriv (deriv (layerProfile r layerExponent)) (diskLogRadius f z)
-  let S := (fderiv ℝ (diskLogRadius f) z 1) ^ 2 + (fderiv ℝ (diskLogRadius f) z I) ^ 2
-  have hb : 0 ≤ b := layerProfile_deriv_nonneg layerExponent_pos.le
-  have hc : c ≤ layerExponent * b := layerProfile_second_le layerExponent_pos.le
-  have hS : 0 ≤ S := add_nonneg (sq_nonneg _) (sq_nonneg _)
-  have hgrad := diskLogRadius_gradient_bound hd hi hg hgf hz hne
-  change diskWeight z ^ 2 * S ≤ 2 * (2560 * Real.pi + 6) ^ 2 at hgrad
-  have hterm : c * (diskWeight z ^ 2 * S) ≤ b := calc
-    _ ≤ (layerExponent * b) * (diskWeight z ^ 2 * S) :=
-      mul_le_mul_of_nonneg_right hc (mul_nonneg (sq_nonneg _) hS)
-    _ ≤ (layerExponent * b) * (2 * (2560 * Real.pi + 6) ^ 2) :=
-      mul_le_mul_of_nonneg_left hgrad (mul_nonneg layerExponent_pos.le hb)
-    _ = b * (layerExponent * (2 * (2560 * Real.pi + 6) ^ 2)) := by ring
-    _ ≤ b * 1 := mul_le_mul_of_nonneg_left layerExponent_gradient hb
-    _ = b := mul_one b
-  have hh := plane_laplacian_chain (diskLogRadius_contDiffAt hd hz hne)
-    ((layerProfile_contDiff r layerExponent).contDiffAt)
-  rw [diskLogRadius_laplacian hd hz hne] at hh
-  change (Δ (diskLayerBarrier f r)) z = b * (-4 / diskWeight z ^ 2) + c * S at hh
-  rw [hh]
-  change b * (-4 / diskWeight z ^ 2) + c * S ≤ -3 * b / diskWeight z ^ 2
-  have hw : 0 < diskWeight z ^ 2 := sq_pos_of_pos (diskWeight_pos hz)
-  apply (le_div_iff₀ hw).mpr
-  field_simp [(diskWeight_pos hz).ne']
-  nlinarith [hterm]
-
-lemma diskLayerBarrier_superharmonic {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (r : ℝ) :
-    (Δ (diskLayerBarrier f r)) z ≤ 0 := by
-  apply (diskLayerBarrier_laplacian_le hd hi hg hgf hz hne r).trans
-  exact div_nonpos_of_nonpos_of_nonneg
-    (mul_nonpos_of_nonpos_of_nonneg (by norm_num) (layerProfile_deriv_nonneg layerExponent_pos.le))
-    (sq_nonneg _)
-
-lemma diskLayerBarrier_layer_lower {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0)
-    {r : ℝ} (hr : 0 < r) (hsmall : diskWeight z * ‖deriv f z‖ ≤ r) :
-    layerExponent / 4 * ‖deriv f z‖ ^ 2 ≤ -(Δ (diskLayerBarrier f r)) z := by
-  have hx : diskLogRadius f z ≤ Real.log r := by
-    apply Real.exp_le_exp.mp
-    rw [diskLogRadius_exp hz hne, Real.exp_log hr]
-    exact hsmall
-  have hlow := layerProfile_deriv_lower hr layerExponent_pos.le
-    (layerExponent_le_one.trans (by norm_num : (1 : ℝ) ≤ 2)) hx
-  rw [diskLogRadius_exp hz hne, mul_pow] at hlow
-  have hh := diskLayerBarrier_laplacian_le hd hi hg hgf hz hne r
-  have hb := layerProfile_deriv_nonneg (r := r) (x := diskLogRadius f z) layerExponent_pos.le
-  have hw : 0 < diskWeight z ^ 2 := sq_pos_of_pos (diskWeight_pos hz)
-  have hh' := (le_div_iff₀ hw).mp hh
-  apply (mul_le_mul_iff_right₀ hw).mp
-  nlinarith [hlow, hh']
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology Laplacian ENNReal
-namespace SLEExactGauge
-
-lemma conformal_layer_measurable {f : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (r : ℝ) :
-    MeasurableSet {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ r} := by
-  have hcont : ContinuousOn (fun z => diskWeight z * ‖deriv f z‖) (ball (0 : ℂ) 1) :=
-    diskWeight_contDiff.continuous.continuousOn.mul (hd.deriv isOpen_ball).continuousOn.norm
-  rcases continuousOn_iff_isClosed.mp hcont (Iic r) isClosed_Iic with ⟨K, hK, heK⟩
-  change MeasurableSet (ball (0 : ℂ) 1 ∩ (fun z => diskWeight z * ‖deriv f z‖) ⁻¹' Iic r)
-  rw [inter_comm, heK]
-  exact hK.measurableSet.inter measurableSet_ball
-
-lemma disk_conformal_layer_green_le {f g : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    (hne : ∀ w ∈ ball (0 : ℂ) 1, deriv f w ≠ 0)
-    {r : ℝ} (hr : 0 < r) :
-    (∫⁻ z in {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ r},
-      ENNReal.ofReal ((-Real.log ‖z‖) * ‖deriv f z‖ ^ 2)) ≤
-      ENNReal.ofReal ((8 * Real.pi / layerExponent) * r ^ 2) := by
-  let S := {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ r}
-  let V := diskLayerBarrier f r
-  have hV := disk_green_le (fun z hz => diskLayerBarrier_contDiffAt hd hz (hne z hz) r)
-    (fun z _ => (diskLayerBarrier_bounds f r z).1)
-    (fun z _ => (diskLayerBarrier_bounds f r z).2)
-    (fun z hz => diskLayerBarrier_superharmonic hd hi hg hgf hz (hne z hz) r)
-  have hS : MeasurableSet S := conformal_layer_measurable hd r
-  have hc : 0 ≤ (4 : ℝ) / layerExponent := div_nonneg (by norm_num) layerExponent_pos.le
-  calc
-    _ ≤ ∫⁻ z in S, ENNReal.ofReal ((4 / layerExponent) *
-        ((-Real.log ‖z‖) * (-(Δ V) z))) := by
-      apply setLIntegral_mono' hS
-      intro z hz
-      apply ENNReal.ofReal_le_ofReal
-      have hl := diskLayerBarrier_layer_lower hd hi hg hgf hz.1 (hne z hz.1) hr hz.2
-      have h'd : ‖deriv f z‖ ^ 2 ≤ 4 / layerExponent * (-(Δ V) z) := by
-        have he : (4 / layerExponent) * (layerExponent / 4 * ‖deriv f z‖ ^ 2) =
-            ‖deriv f z‖ ^ 2 := by field_simp [layerExponent_pos.ne']
-        rw [← he]
-        exact mul_le_mul_of_nonneg_left hl hc
-      have hlog : 0 ≤ -Real.log ‖z‖ := neg_nonneg.mpr
-        (Real.log_nonpos (norm_nonneg z) (mem_ball_zero_iff.mp hz.1).le)
-      nlinarith [mul_le_mul_of_nonneg_left h'd hlog]
-    _ ≤ ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((4 / layerExponent) *
-        ((-Real.log ‖z‖) * (-(Δ V) z))) :=
-      lintegral_mono_set (fun z hz => hz.1)
-    _ = ENNReal.ofReal (4 / layerExponent) *
-        ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((-Real.log ‖z‖) * (-(Δ V) z)) := by
-      simp only [ENNReal.ofReal_mul hc]
-      exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-    _ ≤ ENNReal.ofReal (4 / layerExponent) * ENNReal.ofReal (2 * Real.pi * r ^ 2) :=
-      mul_le_mul_right hV _
-    _ = _ := by rw [← ENNReal.ofReal_mul hc]; congr 1; ring
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge
-
-lemma complex_derivative_det (c : ℂ) :
-    (((1 : ℂ →L[ℂ] ℂ).smulRight c).restrictScalars ℝ).det = ‖c‖ ^ 2 := by
-  simp [ContinuousLinearMap.det, LinearMap.det_restrictScalars, Algebra.norm_complex_eq,
-    Complex.normSq_eq_norm_sq]
-
-lemma conformal_disk_area_change {f : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (H : ℂ → ℝ≥0∞) :
-    (∫⁻ w in f '' ball (0 : ℂ) 1, H w) =
-      ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal (‖deriv f z‖ ^ 2) * H (f z) := by
-  have hf (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) :
-      HasFDerivWithinAt f (((1 : ℂ →L[ℂ] ℂ).smulRight (deriv f z)).restrictScalars ℝ)
-        (ball (0 : ℂ) 1) z :=
-    ((hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
-  have hh := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume measurableSet_ball hf hi H
-  simpa only [complex_derivative_det, abs_pow, abs_norm] using hh
-
-lemma chart_deriv_ne_zero {D : Set ℂ} {f g : ℂ → ℂ}
-    (hD : IsOpen D) (hd : DifferentiableOn ℂ f (ball 0 1))
-    (hg : DifferentiableOn ℂ g D) (hm : MapsTo f (ball 0 1) D)
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : deriv f z ≠ 0 := by
-  have hfz := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
-  have hgz := (hg.differentiableAt (hD.mem_nhds (hm hz))).hasDerivAt
-  have heq : (g ∘ f) =ᶠ[𝓝 z] id := by
-    filter_upwards [isOpen_ball.mem_nhds hz] with w hw
-    exact hgf w hw
-  have he : deriv g (f z) * deriv f z = 1 :=
-    ((hgz.comp z hfz).congr_of_eventuallyEq heq.symm).unique (hasDerivAt_id z)
-  intro hn
-  simp only [hn, mul_zero, zero_ne_one] at he
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology
-namespace SLEExactGauge
-
- 
-
-lemma disk_radius_le_frontier_distance {D : Set ℂ} {f : ℂ → ℂ}
-    (hD : IsOpen D) (hfront : (frontier D).Nonempty)
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
-    (hImage : f '' ball (0 : ℂ) 1 = D) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) :
-    diskWeight z * ‖deriv f z‖ ≤ (256 * Real.pi) * infDist (f z) (frontier D) := by
-  have hc : 0 < 256 * Real.pi := by positivity
-  have hr : 0 < 1 - ‖z‖ := sub_pos.mpr (mem_ball_zero_iff.mp hz)
-  have hsub : ball z (1 - ‖z‖) ⊆ ball (0 : ℂ) 1 :=
-    ball_subset_ball' (by simp only [dist_zero_right]; linarith)
-  have hh : diskWeight z * ‖deriv f z‖ / (256 * Real.pi) ≤ infDist (f z) (frontier D) := by
-    apply (le_infDist hfront).mpr
-    intro a ha
-    have hnot : a ∉ f '' ball z (1 - ‖z‖) := by
-      intro hm
-      have hIn : a ∈ D := hImage ▸ image_mono hsub hm
-      exact ha.2 (by simpa only [hD.interior_eq] using hIn)
-    have hk := univalent_omitted_value_bound_ball hr (hd.mono hsub) (hi.mono hsub) a hnot
-    apply (div_le_iff₀ hc).mpr
-    rw [dist_eq_norm]
-    unfold diskWeight
-    have hn := mul_nonneg (sq_nonneg (1 - ‖z‖)) (norm_nonneg (deriv f z))
-    nlinarith [hk, hn]
-  have hb := (div_le_iff₀ hc).mp hh
-  simpa only [mul_comm] using hb
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge
-
-noncomputable def layerGreenConstant : ℝ :=
-  (8 * Real.pi / layerExponent) * (256 * Real.pi) ^ 2
-
-lemma layerGreenConstant_pos : 0 < layerGreenConstant := by
-  unfold layerGreenConstant
-  exact mul_pos (div_pos (by positivity) layerExponent_pos) (sq_pos_of_pos (by positivity))
-
-lemma centered_boundary_layer_green_le {D : Set ℂ} {f g : ℂ → ℂ}
-    (hD : IsOpen D) (hfront : (frontier D).Nonempty)
-    (hd : DifferentiableOn ℂ f (ball 0 1)) (hg : DifferentiableOn ℂ g D)
-    (hmf : MapsTo f (ball 0 1) D) (hmg : MapsTo g D (ball 0 1))
-    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
-    (hfg : ∀ w ∈ D, f (g w) = w)
-    {r : ℝ} (hr : 0 < r) :
-    (∫⁻ w in {w | w ∈ D ∧ infDist w (frontier D) ≤ r},
-      ENNReal.ofReal (-Real.log ‖g w‖)) ≤ ENNReal.ofReal (layerGreenConstant * r ^ 2) := by
-  have hImage : f '' ball (0 : ℂ) 1 = D := by
-    ext w
-    constructor
-    · rintro ⟨z, hz, rfl⟩
-      exact hmf hz
-    · intro hw
-      exact ⟨g w, hmg hw, hfg w hw⟩
-  have hi : InjOn f (ball 0 1) := by
-    intro z hz w hw he
-    simpa only [hgf z hz, hgf w hw] using congrArg g he
-  have hgrange : DifferentiableOn ℂ g (f '' ball (0 : ℂ) 1) := by rw [hImage]; exact hg
-  have hne : ∀ z ∈ ball (0 : ℂ) 1, deriv f z ≠ 0 :=
-    fun z hz => chart_deriv_ne_zero hD hd hg hmf hgf hz
-  let L := {w | w ∈ D ∧ infDist w (frontier D) ≤ r}
-  let T := {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ (256 * Real.pi) * r}
-  have hL : MeasurableSet L := hD.measurableSet.inter
-    (measurableSet_le (continuous_infDist_pt (frontier D)).measurable measurable_const)
-  have hT : MeasurableSet T := conformal_layer_measurable hd _
-  have hLD : L ⊆ D := fun w hw => hw.1
-  have hTB : T ⊆ ball (0 : ℂ) 1 := fun z hz => hz.1
-  have hscale : 0 < (256 * Real.pi) * r := by positivity
-  calc
-    _ = ∫⁻ w in D, L.indicator (fun w => ENNReal.ofReal (-Real.log ‖g w‖)) w := by
-      rw [lintegral_indicator hL, Measure.restrict_restrict hL, inter_eq_left.mpr hLD]
-    _ = ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal (‖deriv f z‖ ^ 2) *
-        L.indicator (fun w => ENNReal.ofReal (-Real.log ‖g w‖)) (f z) := by
-      rw [← hImage]
-      exact conformal_disk_area_change hd hi _
-    _ ≤ ∫⁻ z in ball (0 : ℂ) 1, T.indicator
-        (fun z => ENNReal.ofReal ((-Real.log ‖z‖) * ‖deriv f z‖ ^ 2)) z := by
-      apply setLIntegral_mono' measurableSet_ball
-      intro z hz
-      by_cases hl : f z ∈ L
-      · have ht : z ∈ T := ⟨hz, (disk_radius_le_frontier_distance hD hfront hd hi hImage hz).trans
-          (mul_le_mul_of_nonneg_left hl.2 (by positivity))⟩
-        simp only [indicator_of_mem hl, indicator_of_mem ht, hgf z hz]
-        rw [← ENNReal.ofReal_mul (sq_nonneg _)]
-        exact le_of_eq (congrArg ENNReal.ofReal (mul_comm _ _))
-      · simp only [indicator_of_notMem hl, mul_zero]
-        exact bot_le
-    _ = ∫⁻ z in T, ENNReal.ofReal ((-Real.log ‖z‖) * ‖deriv f z‖ ^ 2) := by
-      rw [lintegral_indicator hT, Measure.restrict_restrict hT, inter_eq_left.mpr hTB]
-    _ ≤ ENNReal.ofReal ((8 * Real.pi / layerExponent) * ((256 * Real.pi) * r) ^ 2) :=
-      disk_conformal_layer_green_le hd hi hgrange hgf hne hscale
-    _ = _ := by congr 1; unfold layerGreenConstant; ring
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Metric Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-noncomputable def halfPlaneToDisk (w v : ℂ) : ℂ := (v - w) / (v - star w)
-
-lemma halfPlane_denominator_ne {w v : ℂ} (hw : 0 < w.im) (hv : 0 < v.im) :
-    v - star w ≠ 0 := by
-  intro he
-  have hh := congrArg Complex.im he
-  simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hh
-  linarith
-
-lemma halfPlaneToDisk_maps {w : ℂ} (hw : 0 < w.im) :
-    MapsTo (halfPlaneToDisk w) upperHalfPlane (ball 0 1) := by
-  intro v hv
-  exact mem_ball_zero_iff.mpr (upper_mobius_norm_lt_one hv hw)
-
-lemma halfPlaneToDisk_differentiable {w : ℂ} (hw : 0 < w.im) :
-    DifferentiableOn ℂ (halfPlaneToDisk w) upperHalfPlane :=
-  (differentiableOn_id.sub_const w).div (differentiableOn_id.sub_const (star w))
-    (fun _ hv => halfPlane_denominator_ne hw hv)
-
-lemma halfPlaneToDisk_diskToHalfPlane {w ζ : ℂ} (hw : 0 < w.im)
-    (hζ : ζ ∈ ball (0 : ℂ) 1) : halfPlaneToDisk w (diskToHalfPlane w ζ) = ζ := by
-  have hn : w - star w ≠ 0 := halfPlane_denominator_ne hw hw
-  have hd := disk_denominator_ne hζ
-  have hv := halfPlane_denominator_ne hw (diskToHalfPlane_maps hw hζ)
-  unfold halfPlaneToDisk
-  apply (div_eq_iff hv).mpr
-  unfold diskToHalfPlane
-  field_simp [hd]
-  ring
-
-lemma diskToHalfPlane_deriv {w ζ : ℂ} (hζ : ζ ∈ ball (0 : ℂ) 1) :
-    HasDerivAt (diskToHalfPlane w) ((w - star w) / (1 - ζ) ^ 2) ζ := by
-  have hh := ((hasDerivAt_const ζ w).sub ((hasDerivAt_id ζ).const_mul (star w))).div
-    ((hasDerivAt_const ζ 1).sub (hasDerivAt_id ζ)) (disk_denominator_ne hζ)
-  convert hh using 1 <;> first | rfl | (dsimp; ring)
-
-lemma diskToHalfPlane_second_zero (w : ℂ) :
-    HasDerivAt (deriv (diskToHalfPlane w)) (2 * (w - star w)) 0 := by
-  have heq : deriv (diskToHalfPlane w) =ᶠ[𝓝 (0 : ℂ)]
-      (fun ζ => (w - star w) / (1 - ζ) ^ 2) := by
-    filter_upwards [isOpen_ball.mem_nhds (mem_ball_self (by norm_num : (0 : ℝ) < 1))] with ζ hζ
-    exact (diskToHalfPlane_deriv hζ).deriv
-  have hh := (hasDerivAt_const (0 : ℂ) (w - star w)).div
-    (((hasDerivAt_const (0 : ℂ) 1).sub (hasDerivAt_id 0)).pow 2) (by norm_num : ((1 : ℂ) - 0) ^ 2 ≠ 0)
-  have hh' : HasDerivAt (fun ζ : ℂ => (w - star w) / (1 - ζ) ^ 2)
-      (2 * (w - star w)) 0 := by
-    convert hh using 1 <;> first | rfl | (norm_num; ring)
-  exact hh'.congr_of_eventuallyEq heq
-
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def halfPlaneGreen (z w : ℂ) : ℝ :=
-  Real.log ‖(z - conj w) / (z - w)‖
-
-lemma hasDerivWithinAt_log_norm {f : ℝ → ℂ} {v : ℂ} {S : Set ℝ} {t : ℝ}
-    (hf : HasDerivWithinAt f v S t) (h0 : f t ≠ 0) :
-    HasDerivWithinAt (fun s => Real.log ‖f s‖) (v / f t).re S t := by
-  have hh := (hf.norm_sq.log (pow_ne_zero 2 (norm_ne_zero_iff.mpr h0))).div_const 2
-  have he : (fun s => Real.log (‖f s‖ ^ 2) / 2) = (fun s => Real.log ‖f s‖) := by
-    funext s
-    rw [Real.log_pow]
-    ring
-  rw [he] at hh
-  apply hh.congr_deriv
-  rw [Complex.div_re, Complex.sq_norm, real_inner_eq_re_inner ℂ]
-  simp only [inner, Complex.star_def, RCLike.re_to_complex, Complex.mul_re, Complex.conj_re, Complex.conj_im]
-  ring
-
-lemma hasDerivWithinAt_conj {f : ℝ → ℂ} {v : ℂ} {S : Set ℝ} {t : ℝ}
-    (hf : HasDerivWithinAt f v S t) :
-    HasDerivWithinAt (fun s => conj (f s)) (conj v) S t := by
-  exact Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt t hf
-
-lemma hasDerivWithinAt_log_norm_div {f g : ℝ → ℂ} {v w : ℂ} {S : Set ℝ} {t : ℝ}
-    (hf : HasDerivWithinAt f v S t) (hg : HasDerivWithinAt g w S t)
-    (hf0 : f t ≠ 0) (hg0 : g t ≠ 0) :
-    HasDerivWithinAt (fun s => Real.log ‖f s / g s‖)
-      (v / f t - w / g t).re S t := by
-  apply (hasDerivWithinAt_log_norm (hf.div hg hg0) (div_ne_zero hf0 hg0)).congr_deriv
-  congr 1
-  change ((v * g t - f t * w) / g t ^ 2) / (f t / g t) = v / f t - w / g t
-  field_simp
-
-lemma green_decrease_algebra (z w : ℂ) (hz : 0 < z.im) (hw : 0 < w.im) (hne : z ≠ w) :
-    ((2 / z - conj (2 / w)) / (z - conj w) - (2 / z - 2 / w) / (z - w)).re =
-      -4 * (z.im / ‖z‖ ^ 2) * (w.im / ‖w‖ ^ 2) := by
-  have hz0 : z ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hz
-  have hw0 : w ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hw
-  have hcw0 : conj w ≠ 0 := by simpa using hw0
-  have hn : z - w ≠ 0 := sub_ne_zero.mpr hne
-  have hn' : z - conj w ≠ 0 := by
-    intro hh
-    have he := congrArg Complex.im (sub_eq_zero.mp hh)
-    simp only [Complex.conj_im] at he
-    linarith
-  have h₁ : (2 / z - conj (2 / w)) / (z - conj w) = -2 / (z * conj w) := by
-    simp only [map_div₀, map_ofNat]
-    field_simp
-    ring
-  have h₂ : (2 / z - 2 / w) / (z - w) = -2 / (z * w) := by
-    field_simp
-    ring
-  rw [h₁, h₂]
-  simp only [Complex.sub_re, Complex.div_re, Complex.neg_re, Complex.neg_im,
-    Complex.mul_re, Complex.conj_re,
-    Complex.conj_im, Complex.normSq_mul, Complex.normSq_conj, Complex.sq_norm]
-  norm_num
-  ring
-
- 
-
-end SLEExactGauge
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.AngularLaws
+import OAI.Probability.SLE.DiskHalfPlane
+
+namespace OAI
+
+/-! Stopped Kernels for chordal SLE. -/
+
+section
+section
+section
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian RealInnerProductSpace
+namespace SLEExactGauge
+lemma diskLogRadius_laplacian {f : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) {z : ℂ}
+    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
+    (Δ (diskLogRadius f)) z = -4 / diskWeight z ^ 2 := by
+  have hh := (((hd.deriv isOpen_ball).analyticAt (isOpen_ball.mem_nhds hz))).harmonicAt_log_norm hne
+  have he := hh.2.eq_of_nhds
+  change (Δ (fun w => Real.log ‖deriv f w‖)) z = 0 at he
+  rw [show diskLogRadius f = (fun w => Real.log (diskWeight w)) +
+      (fun w => Real.log ‖deriv f w‖) from rfl,
+    (diskWeight_contDiff.contDiffAt.log (diskWeight_pos hz).ne').laplacian_add hh.1,
+    he, add_zero, log_diskWeight_laplacian hz]
+
+lemma diskLogRadius_exp {f : ℂ → ℂ} {z : ℂ}
+    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
+    Real.exp (diskLogRadius f z) = diskWeight z * ‖deriv f z‖ := by
+  simp only [diskLogRadius, Real.exp_add, Real.exp_log (diskWeight_pos hz),
+    Real.exp_log (norm_pos_iff.mpr hne)]
+
+lemma log_diskWeight_direction_bound {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (v : ℂ) :
+    diskWeight z * |fderiv ℝ (fun w => Real.log (diskWeight w)) z v| ≤ 2 * ‖v‖ := by
+  have hh := ((diskWeight_contDiff.differentiable (by norm_num) z).hasFDerivAt.log
+    (diskWeight_pos hz).ne').fderiv
+  rw [hh, diskWeight_fderiv]
+  simp only [smul_apply, smul_eq_mul, abs_mul, abs_inv, abs_of_pos (diskWeight_pos hz),
+    abs_neg, abs_of_pos (show (0 : ℝ) < 2 by norm_num)]
+  rw [← mul_assoc, mul_inv_cancel₀ (diskWeight_pos hz).ne', one_mul]
+  have hinner := abs_real_inner_le_norm z v
+  have hn : ‖z‖ ≤ 1 := (mem_ball_zero_iff.mp hz).le
+  change 2 * |inner ℝ z v| ≤ 2 * ‖v‖
+  calc
+    _ ≤ 2 * (‖z‖ * ‖v‖) := mul_le_mul_of_nonneg_left hinner (by norm_num)
+    _ ≤ 2 * ‖v‖ := by nlinarith [norm_nonneg v]
+
+lemma disk_preSchwarzian_weight {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
+    diskWeight z * ‖deriv (deriv f) z / deriv f z‖ ≤ 2560 * Real.pi + 4 := by
+  have hh := disk_preSchwarzian_bound hd hi hg hgf hz hne
+  have hb := norm_add_le
+    (((diskWeight z : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z) - 2 * star z) (2 * star z)
+  rw [sub_add_cancel, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_pos (diskWeight_pos hz)] at hb
+  have he : ‖(2 : ℂ) * star z‖ = 2 * ‖z‖ := by simp
+  rw [he] at hb
+  have hn := mem_ball_zero_iff.mp hz
+  change ‖((diskWeight z : ℝ) : ℂ) * (deriv (deriv f) z / deriv f z) - 2 * star z‖ ≤ _ at hh
+  linarith
+
+lemma diskLogRadius_direction_bound {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (v : ℂ) :
+    diskWeight z * |fderiv ℝ (diskLogRadius f) z v| ≤ (2560 * Real.pi + 6) * ‖v‖ := by
+  have hdd := ((hd.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hz))
+  have hn : DifferentiableAt ℝ (fun w => Real.log ‖deriv f w‖) z :=
+    ((hdd.restrictScalars ℝ).norm ℂ hne).log (norm_ne_zero_iff.mpr hne)
+  have hw : DifferentiableAt ℝ (fun w => Real.log (diskWeight w)) z :=
+    (diskWeight_contDiff.differentiable (by norm_num) z).log (diskWeight_pos hz).ne'
+  have hp := disk_preSchwarzian_weight hd hi hg hgf hz hne
+  have h1 := log_diskWeight_direction_bound hz v
+  have h2 := log_norm_direction_bound hdd hne v
+  have he : fderiv ℝ (diskLogRadius f) z =
+      fderiv ℝ (fun w => Real.log (diskWeight w)) z +
+        fderiv ℝ (fun w => Real.log ‖deriv f w‖) z := by
+    exact fderiv_fun_add hw hn
+  rw [he, add_apply]
+  have hb := mul_le_mul_of_nonneg_left (abs_add_le
+    (fderiv ℝ (fun w => Real.log (diskWeight w)) z v)
+    (fderiv ℝ (fun w => Real.log ‖deriv f w‖) z v)) (diskWeight_pos hz).le
+  have hb2 := mul_le_mul_of_nonneg_left h2 (diskWeight_pos hz).le
+  have hb3 := mul_le_mul_of_nonneg_right hp (norm_nonneg v)
+  nlinarith
+
+lemma diskLogRadius_gradient_bound {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) :
+    diskWeight z ^ 2 * ((fderiv ℝ (diskLogRadius f) z 1) ^ 2 +
+      (fderiv ℝ (diskLogRadius f) z I) ^ 2) ≤ 2 * (2560 * Real.pi + 6) ^ 2 := by
+  have h1 := diskLogRadius_direction_bound hd hi hg hgf hz hne 1
+  have hI := diskLogRadius_direction_bound hd hi hg hgf hz hne I
+  simp only [norm_one, norm_I, mul_one] at h1 hI
+  have hw := (diskWeight_pos hz).le
+  have hc : 0 ≤ 2560 * Real.pi + 6 := by positivity
+  have h1' := sq_le_sq₀ (mul_nonneg hw (abs_nonneg _)) hc |>.mpr h1
+  have hI' := sq_le_sq₀ (mul_nonneg hw (abs_nonneg _)) hc |>.mpr hI
+  simp only [mul_pow, sq_abs] at h1' hI'
+  nlinarith
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge
+
+ 
+
+noncomputable def layerProfile (r q x : ℝ) : ℝ :=
+  r ^ 2 * Real.exp (q * (x - Real.log r)) / (1 + Real.exp (q * (x - Real.log r)))
+
+noncomputable def layerProfileDeriv (r q x : ℝ) : ℝ :=
+  q * r ^ 2 * Real.exp (q * (x - Real.log r)) /
+    (1 + Real.exp (q * (x - Real.log r))) ^ 2
+
+lemma layerProfile_hasDeriv (r q x : ℝ) :
+    HasDerivAt (layerProfile r q) (layerProfileDeriv r q x) x := by
+  have ht := (((hasDerivAt_id x).sub_const (Real.log r)).const_mul q).exp
+  have hp := ((ht.const_mul (r ^ 2)).div (ht.const_add 1)
+    (by positivity : 1 + Real.exp (q * (x - Real.log r)) ≠ 0))
+  have hne : 1 + Real.exp (q * (x - Real.log r)) ≠ 0 := by positivity
+  convert! hp using 1
+  simp only [layerProfileDeriv, id_eq, mul_one]
+  field_simp [hne]
+  ring
+
+lemma layerProfileDeriv_hasDeriv (r q x : ℝ) :
+    HasDerivAt (layerProfileDeriv r q)
+      (q * layerProfileDeriv r q x *
+        ((1 - Real.exp (q * (x - Real.log r))) /
+          (1 + Real.exp (q * (x - Real.log r))))) x := by
+  have ht := (((hasDerivAt_id x).sub_const (Real.log r)).const_mul q).exp
+  have hp := ((ht.const_mul (q * r ^ 2)).div ((ht.const_add 1).pow 2)
+    (by positivity : (1 + Real.exp (q * (x - Real.log r))) ^ 2 ≠ 0))
+  have hne : 1 + Real.exp (q * (x - Real.log r)) ≠ 0 := by positivity
+  convert! hp using 1
+  simp only [layerProfileDeriv, id_eq, Pi.pow_apply, mul_one,
+    Nat.cast_ofNat, Nat.reduceSub, pow_one]
+  field_simp [hne]
+  ring
+
+lemma layerProfile_contDiff (r q : ℝ) : ContDiff ℝ 2 (layerProfile r q) := by
+  unfold layerProfile
+  fun_prop (disch := intros; positivity)
+
+lemma layerProfile_nonneg (r q x : ℝ) : 0 ≤ layerProfile r q x := by
+  unfold layerProfile
+  positivity
+
+lemma layerProfile_le (r q x : ℝ) : layerProfile r q x ≤ r ^ 2 := by
+  unfold layerProfile
+  apply (div_le_iff₀ (by positivity)).mpr
+  have ht := Real.exp_pos (q * (x - Real.log r))
+  nlinarith [sq_nonneg r]
+
+lemma layerProfile_deriv_nonneg {r q x : ℝ} (hq : 0 ≤ q) :
+    0 ≤ deriv (layerProfile r q) x := by
+  rw [(layerProfile_hasDeriv r q x).deriv]
+  unfold layerProfileDeriv
+  positivity
+
+lemma layerProfile_second_le {r q x : ℝ} (hq : 0 ≤ q) :
+    deriv (deriv (layerProfile r q)) x ≤ q * deriv (layerProfile r q) x := by
+  have he : deriv (layerProfile r q) = layerProfileDeriv r q :=
+    funext fun x => (layerProfile_hasDeriv r q x).deriv
+  rw [he, (layerProfileDeriv_hasDeriv r q x).deriv]
+  have ht : 0 < Real.exp (q * (x - Real.log r)) := Real.exp_pos _
+  have hratio : (1 - Real.exp (q * (x - Real.log r))) /
+      (1 + Real.exp (q * (x - Real.log r))) ≤ 1 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    linarith
+  have hd : 0 ≤ layerProfileDeriv r q x := by unfold layerProfileDeriv; positivity
+  simpa using mul_le_mul_of_nonneg_left hratio (mul_nonneg hq hd)
+
+lemma layerProfile_deriv_lower {r q x : ℝ}
+    (hr : 0 < r) (hq : 0 ≤ q) (hq2 : q ≤ 2) (hx : x ≤ Real.log r) :
+    q / 4 * Real.exp x ^ 2 ≤ deriv (layerProfile r q) x := by
+  rw [(layerProfile_hasDeriv r q x).deriv]
+  let t := Real.exp (q * (x - Real.log r))
+  have ht : 0 < t := Real.exp_pos _
+  have ht1 : t ≤ 1 := by
+    apply Real.exp_le_one_iff.mpr
+    exact mul_nonpos_of_nonneg_of_nonpos hq (sub_nonpos.mpr hx)
+  have hlow : Real.exp (2 * (x - Real.log r)) ≤ t := by
+    apply Real.exp_le_exp.mpr
+    exact mul_le_mul_of_nonpos_right hq2 (sub_nonpos.mpr hx)
+  have he : Real.exp x ^ 2 = r ^ 2 * Real.exp (2 * (x - Real.log r)) := by
+    have h2 (y : ℝ) : Real.exp (2 * y) = Real.exp y ^ 2 := by simpa using Real.exp_nat_mul y 2
+    rw [mul_sub, Real.exp_sub, h2, h2, Real.exp_log hr]
+    field_simp [hr.ne']
+  have he' : Real.exp x ^ 2 ≤ r ^ 2 * t := by rw [he]; exact mul_le_mul_of_nonneg_left hlow (sq_nonneg r)
+  change q / 4 * Real.exp x ^ 2 ≤ q * r ^ 2 * t / (1 + t) ^ 2
+  apply (le_div_iff₀ (by positivity)).mpr
+  have hden : (1 + t) ^ 2 ≤ 4 := by nlinarith
+  have hh := mul_le_mul_of_nonneg_left hden (mul_nonneg (div_nonneg hq (by norm_num : (0 : ℝ) ≤ 4)) (sq_nonneg (Real.exp x)))
+  have hh' := mul_le_mul_of_nonneg_left he' hq
+  nlinarith
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+namespace SLEExactGauge
+
+noncomputable def layerExponent : ℝ := (2 * (2560 * Real.pi + 6) ^ 2 + 1)⁻¹
+
+lemma layerExponent_pos : 0 < layerExponent := by unfold layerExponent; positivity
+lemma layerExponent_le_one : layerExponent ≤ 1 := by
+  unfold layerExponent
+  apply (inv_le_one₀ (by positivity)).mpr
+  nlinarith [sq_nonneg (2560 * Real.pi + 6)]
+lemma layerExponent_gradient : layerExponent * (2 * (2560 * Real.pi + 6) ^ 2) ≤ 1 := by
+  unfold layerExponent
+  have hp : 0 < 2 * (2560 * Real.pi + 6) ^ 2 + 1 := by positivity
+  have he := inv_mul_cancel₀ hp.ne'
+  have hn : 0 ≤ (2 * (2560 * Real.pi + 6) ^ 2 + 1)⁻¹ := by positivity
+  nlinarith
+
+noncomputable def diskLayerBarrier (f : ℂ → ℂ) (r : ℝ) : ℂ → ℝ :=
+  layerProfile r layerExponent ∘ diskLogRadius f
+
+lemma diskLayerBarrier_bounds (f : ℂ → ℂ) (r : ℝ) (z : ℂ) :
+    0 ≤ diskLayerBarrier f r z ∧ diskLayerBarrier f r z ≤ r ^ 2 :=
+  ⟨layerProfile_nonneg _ _ _, layerProfile_le _ _ _⟩
+
+lemma diskLayerBarrier_contDiffAt {f : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) {z : ℂ}
+    (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (r : ℝ) :
+    ContDiffAt ℝ 2 (diskLayerBarrier f r) z :=
+  (layerProfile_contDiff r layerExponent).contDiffAt.comp z (diskLogRadius_contDiffAt hd hz hne)
+
+lemma diskLayerBarrier_laplacian_le {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (r : ℝ) :
+    (Δ (diskLayerBarrier f r)) z ≤
+      -3 * deriv (layerProfile r layerExponent) (diskLogRadius f z) / diskWeight z ^ 2 := by
+  let b := deriv (layerProfile r layerExponent) (diskLogRadius f z)
+  let c := deriv (deriv (layerProfile r layerExponent)) (diskLogRadius f z)
+  let S := (fderiv ℝ (diskLogRadius f) z 1) ^ 2 + (fderiv ℝ (diskLogRadius f) z I) ^ 2
+  have hb : 0 ≤ b := layerProfile_deriv_nonneg layerExponent_pos.le
+  have hc : c ≤ layerExponent * b := layerProfile_second_le layerExponent_pos.le
+  have hS : 0 ≤ S := add_nonneg (sq_nonneg _) (sq_nonneg _)
+  have hgrad := diskLogRadius_gradient_bound hd hi hg hgf hz hne
+  change diskWeight z ^ 2 * S ≤ 2 * (2560 * Real.pi + 6) ^ 2 at hgrad
+  have hterm : c * (diskWeight z ^ 2 * S) ≤ b := calc
+    _ ≤ (layerExponent * b) * (diskWeight z ^ 2 * S) :=
+      mul_le_mul_of_nonneg_right hc (mul_nonneg (sq_nonneg _) hS)
+    _ ≤ (layerExponent * b) * (2 * (2560 * Real.pi + 6) ^ 2) :=
+      mul_le_mul_of_nonneg_left hgrad (mul_nonneg layerExponent_pos.le hb)
+    _ = b * (layerExponent * (2 * (2560 * Real.pi + 6) ^ 2)) := by ring
+    _ ≤ b * 1 := mul_le_mul_of_nonneg_left layerExponent_gradient hb
+    _ = b := mul_one b
+  have hh := plane_laplacian_chain (diskLogRadius_contDiffAt hd hz hne)
+    ((layerProfile_contDiff r layerExponent).contDiffAt)
+  rw [diskLogRadius_laplacian hd hz hne] at hh
+  change (Δ (diskLayerBarrier f r)) z = b * (-4 / diskWeight z ^ 2) + c * S at hh
+  rw [hh]
+  change b * (-4 / diskWeight z ^ 2) + c * S ≤ -3 * b / diskWeight z ^ 2
+  have hw : 0 < diskWeight z ^ 2 := sq_pos_of_pos (diskWeight_pos hz)
+  apply (le_div_iff₀ hw).mpr
+  field_simp [(diskWeight_pos hz).ne']
+  nlinarith [hterm]
+
+lemma diskLayerBarrier_superharmonic {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0) (r : ℝ) :
+    (Δ (diskLayerBarrier f r)) z ≤ 0 := by
+  apply (diskLayerBarrier_laplacian_le hd hi hg hgf hz hne r).trans
+  exact div_nonpos_of_nonpos_of_nonneg
+    (mul_nonpos_of_nonpos_of_nonneg (by norm_num) (layerProfile_deriv_nonneg layerExponent_pos.le))
+    (sq_nonneg _)
+
+lemma diskLayerBarrier_layer_lower {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (hne : deriv f z ≠ 0)
+    {r : ℝ} (hr : 0 < r) (hsmall : diskWeight z * ‖deriv f z‖ ≤ r) :
+    layerExponent / 4 * ‖deriv f z‖ ^ 2 ≤ -(Δ (diskLayerBarrier f r)) z := by
+  have hx : diskLogRadius f z ≤ Real.log r := by
+    apply Real.exp_le_exp.mp
+    rw [diskLogRadius_exp hz hne, Real.exp_log hr]
+    exact hsmall
+  have hlow := layerProfile_deriv_lower hr layerExponent_pos.le
+    (layerExponent_le_one.trans (by norm_num : (1 : ℝ) ≤ 2)) hx
+  rw [diskLogRadius_exp hz hne, mul_pow] at hlow
+  have hh := diskLayerBarrier_laplacian_le hd hi hg hgf hz hne r
+  have hb := layerProfile_deriv_nonneg (r := r) (x := diskLogRadius f z) layerExponent_pos.le
+  have hw : 0 < diskWeight z ^ 2 := sq_pos_of_pos (diskWeight_pos hz)
+  have hh' := (le_div_iff₀ hw).mp hh
+  apply (mul_le_mul_iff_right₀ hw).mp
+  nlinarith [hlow, hh']
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology Laplacian ENNReal
+namespace SLEExactGauge
+
+lemma conformal_layer_measurable {f : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (r : ℝ) :
+    MeasurableSet {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ r} := by
+  have hcont : ContinuousOn (fun z => diskWeight z * ‖deriv f z‖) (ball (0 : ℂ) 1) :=
+    diskWeight_contDiff.continuous.continuousOn.mul (hd.deriv isOpen_ball).continuousOn.norm
+  rcases continuousOn_iff_isClosed.mp hcont (Iic r) isClosed_Iic with ⟨K, hK, heK⟩
+  change MeasurableSet (ball (0 : ℂ) 1 ∩ (fun z => diskWeight z * ‖deriv f z‖) ⁻¹' Iic r)
+  rw [inter_comm, heK]
+  exact hK.measurableSet.inter measurableSet_ball
+
+lemma disk_conformal_layer_green_le {f g : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hg : DifferentiableOn ℂ g (f '' ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    (hne : ∀ w ∈ ball (0 : ℂ) 1, deriv f w ≠ 0)
+    {r : ℝ} (hr : 0 < r) :
+    (∫⁻ z in {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ r},
+      ENNReal.ofReal ((-Real.log ‖z‖) * ‖deriv f z‖ ^ 2)) ≤
+      ENNReal.ofReal ((8 * Real.pi / layerExponent) * r ^ 2) := by
+  let S := {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ r}
+  let V := diskLayerBarrier f r
+  have hV := disk_green_le (fun z hz => diskLayerBarrier_contDiffAt hd hz (hne z hz) r)
+    (fun z _ => (diskLayerBarrier_bounds f r z).1)
+    (fun z _ => (diskLayerBarrier_bounds f r z).2)
+    (fun z hz => diskLayerBarrier_superharmonic hd hi hg hgf hz (hne z hz) r)
+  have hS : MeasurableSet S := conformal_layer_measurable hd r
+  have hc : 0 ≤ (4 : ℝ) / layerExponent := div_nonneg (by norm_num) layerExponent_pos.le
+  calc
+    _ ≤ ∫⁻ z in S, ENNReal.ofReal ((4 / layerExponent) *
+        ((-Real.log ‖z‖) * (-(Δ V) z))) := by
+      apply setLIntegral_mono' hS
+      intro z hz
+      apply ENNReal.ofReal_le_ofReal
+      have hl := diskLayerBarrier_layer_lower hd hi hg hgf hz.1 (hne z hz.1) hr hz.2
+      have h'd : ‖deriv f z‖ ^ 2 ≤ 4 / layerExponent * (-(Δ V) z) := by
+        have he : (4 / layerExponent) * (layerExponent / 4 * ‖deriv f z‖ ^ 2) =
+            ‖deriv f z‖ ^ 2 := by field_simp [layerExponent_pos.ne']
+        rw [← he]
+        exact mul_le_mul_of_nonneg_left hl hc
+      have hlog : 0 ≤ -Real.log ‖z‖ := neg_nonneg.mpr
+        (Real.log_nonpos (norm_nonneg z) (mem_ball_zero_iff.mp hz.1).le)
+      nlinarith [mul_le_mul_of_nonneg_left h'd hlog]
+    _ ≤ ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((4 / layerExponent) *
+        ((-Real.log ‖z‖) * (-(Δ V) z))) :=
+      lintegral_mono_set (fun z hz => hz.1)
+    _ = ENNReal.ofReal (4 / layerExponent) *
+        ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal ((-Real.log ‖z‖) * (-(Δ V) z)) := by
+      simp only [ENNReal.ofReal_mul hc]
+      exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+    _ ≤ ENNReal.ofReal (4 / layerExponent) * ENNReal.ofReal (2 * Real.pi * r ^ 2) :=
+      mul_le_mul_right hV _
+    _ = _ := by rw [← ENNReal.ofReal_mul hc]; congr 1; ring
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge
+
+lemma complex_derivative_det (c : ℂ) :
+    (((1 : ℂ →L[ℂ] ℂ).smulRight c).restrictScalars ℝ).det = ‖c‖ ^ 2 := by
+  simp [ContinuousLinearMap.det, LinearMap.det_restrictScalars, Algebra.norm_complex_eq,
+    Complex.normSq_eq_norm_sq]
+
+lemma conformal_disk_area_change {f : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (H : ℂ → ℝ≥0∞) :
+    (∫⁻ w in f '' ball (0 : ℂ) 1, H w) =
+      ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal (‖deriv f z‖ ^ 2) * H (f z) := by
+  have hf (z : ℂ) (hz : z ∈ ball (0 : ℂ) 1) :
+      HasFDerivWithinAt f (((1 : ℂ →L[ℂ] ℂ).smulRight (deriv f z)).restrictScalars ℝ)
+        (ball (0 : ℂ) 1) z :=
+    ((hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
+  have hh := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume measurableSet_ball hf hi H
+  simpa only [complex_derivative_det, abs_pow, abs_norm] using hh
+
+lemma chart_deriv_ne_zero {D : Set ℂ} {f g : ℂ → ℂ}
+    (hD : IsOpen D) (hd : DifferentiableOn ℂ f (ball 0 1))
+    (hg : DifferentiableOn ℂ g D) (hm : MapsTo f (ball 0 1) D)
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : deriv f z ≠ 0 := by
+  have hfz := (hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt
+  have hgz := (hg.differentiableAt (hD.mem_nhds (hm hz))).hasDerivAt
+  have heq : (g ∘ f) =ᶠ[𝓝 z] id := by
+    filter_upwards [isOpen_ball.mem_nhds hz] with w hw
+    exact hgf w hw
+  have he : deriv g (f z) * deriv f z = 1 :=
+    ((hgz.comp z hfz).congr_of_eventuallyEq heq.symm).unique (hasDerivAt_id z)
+  intro hn
+  simp only [hn, mul_zero, zero_ne_one] at he
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology
+namespace SLEExactGauge
+
+ 
+
+lemma disk_radius_le_frontier_distance {D : Set ℂ} {f : ℂ → ℂ}
+    (hD : IsOpen D) (hfront : (frontier D).Nonempty)
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1))
+    (hImage : f '' ball (0 : ℂ) 1 = D) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) :
+    diskWeight z * ‖deriv f z‖ ≤ (256 * Real.pi) * infDist (f z) (frontier D) := by
+  have hc : 0 < 256 * Real.pi := by positivity
+  have hr : 0 < 1 - ‖z‖ := sub_pos.mpr (mem_ball_zero_iff.mp hz)
+  have hsub : ball z (1 - ‖z‖) ⊆ ball (0 : ℂ) 1 :=
+    ball_subset_ball' (by simp only [dist_zero_right]; linarith)
+  have hh : diskWeight z * ‖deriv f z‖ / (256 * Real.pi) ≤ infDist (f z) (frontier D) := by
+    apply (le_infDist hfront).mpr
+    intro a ha
+    have hnot : a ∉ f '' ball z (1 - ‖z‖) := by
+      intro hm
+      have hIn : a ∈ D := hImage ▸ image_mono hsub hm
+      exact ha.2 (by simpa only [hD.interior_eq] using hIn)
+    have hk := univalent_omitted_value_bound_ball hr (hd.mono hsub) (hi.mono hsub) a hnot
+    apply (div_le_iff₀ hc).mpr
+    rw [dist_eq_norm]
+    unfold diskWeight
+    have hn := mul_nonneg (sq_nonneg (1 - ‖z‖)) (norm_nonneg (deriv f z))
+    nlinarith [hk, hn]
+  have hb := (div_le_iff₀ hc).mp hh
+  simpa only [mul_comm] using hb
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge
+
+noncomputable def layerGreenConstant : ℝ :=
+  (8 * Real.pi / layerExponent) * (256 * Real.pi) ^ 2
+
+lemma layerGreenConstant_pos : 0 < layerGreenConstant := by
+  unfold layerGreenConstant
+  exact mul_pos (div_pos (by positivity) layerExponent_pos) (sq_pos_of_pos (by positivity))
+
+lemma centered_boundary_layer_green_le {D : Set ℂ} {f g : ℂ → ℂ}
+    (hD : IsOpen D) (hfront : (frontier D).Nonempty)
+    (hd : DifferentiableOn ℂ f (ball 0 1)) (hg : DifferentiableOn ℂ g D)
+    (hmf : MapsTo f (ball 0 1) D) (hmg : MapsTo g D (ball 0 1))
+    (hgf : ∀ w ∈ ball (0 : ℂ) 1, g (f w) = w)
+    (hfg : ∀ w ∈ D, f (g w) = w)
+    {r : ℝ} (hr : 0 < r) :
+    (∫⁻ w in {w | w ∈ D ∧ infDist w (frontier D) ≤ r},
+      ENNReal.ofReal (-Real.log ‖g w‖)) ≤ ENNReal.ofReal (layerGreenConstant * r ^ 2) := by
+  have hImage : f '' ball (0 : ℂ) 1 = D := by
+    ext w
+    constructor
+    · rintro ⟨z, hz, rfl⟩
+      exact hmf hz
+    · intro hw
+      exact ⟨g w, hmg hw, hfg w hw⟩
+  have hi : InjOn f (ball 0 1) := by
+    intro z hz w hw he
+    simpa only [hgf z hz, hgf w hw] using congrArg g he
+  have hgrange : DifferentiableOn ℂ g (f '' ball (0 : ℂ) 1) := by rw [hImage]; exact hg
+  have hne : ∀ z ∈ ball (0 : ℂ) 1, deriv f z ≠ 0 :=
+    fun z hz => chart_deriv_ne_zero hD hd hg hmf hgf hz
+  let L := {w | w ∈ D ∧ infDist w (frontier D) ≤ r}
+  let T := {z | z ∈ ball (0 : ℂ) 1 ∧ diskWeight z * ‖deriv f z‖ ≤ (256 * Real.pi) * r}
+  have hL : MeasurableSet L := hD.measurableSet.inter
+    (measurableSet_le (continuous_infDist_pt (frontier D)).measurable measurable_const)
+  have hT : MeasurableSet T := conformal_layer_measurable hd _
+  have hLD : L ⊆ D := fun w hw => hw.1
+  have hTB : T ⊆ ball (0 : ℂ) 1 := fun z hz => hz.1
+  have hscale : 0 < (256 * Real.pi) * r := by positivity
+  calc
+    _ = ∫⁻ w in D, L.indicator (fun w => ENNReal.ofReal (-Real.log ‖g w‖)) w := by
+      rw [lintegral_indicator hL, Measure.restrict_restrict hL, inter_eq_left.mpr hLD]
+    _ = ∫⁻ z in ball (0 : ℂ) 1, ENNReal.ofReal (‖deriv f z‖ ^ 2) *
+        L.indicator (fun w => ENNReal.ofReal (-Real.log ‖g w‖)) (f z) := by
+      rw [← hImage]
+      exact conformal_disk_area_change hd hi _
+    _ ≤ ∫⁻ z in ball (0 : ℂ) 1, T.indicator
+        (fun z => ENNReal.ofReal ((-Real.log ‖z‖) * ‖deriv f z‖ ^ 2)) z := by
+      apply setLIntegral_mono' measurableSet_ball
+      intro z hz
+      by_cases hl : f z ∈ L
+      · have ht : z ∈ T := ⟨hz, (disk_radius_le_frontier_distance hD hfront hd hi hImage hz).trans
+          (mul_le_mul_of_nonneg_left hl.2 (by positivity))⟩
+        simp only [indicator_of_mem hl, indicator_of_mem ht, hgf z hz]
+        rw [← ENNReal.ofReal_mul (sq_nonneg _)]
+        exact le_of_eq (congrArg ENNReal.ofReal (mul_comm _ _))
+      · simp only [indicator_of_notMem hl, mul_zero]
+        exact bot_le
+    _ = ∫⁻ z in T, ENNReal.ofReal ((-Real.log ‖z‖) * ‖deriv f z‖ ^ 2) := by
+      rw [lintegral_indicator hT, Measure.restrict_restrict hT, inter_eq_left.mpr hTB]
+    _ ≤ ENNReal.ofReal ((8 * Real.pi / layerExponent) * ((256 * Real.pi) * r) ^ 2) :=
+      disk_conformal_layer_green_le hd hi hgrange hgf hne hscale
+    _ = _ := by congr 1; unfold layerGreenConstant; ring
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Metric Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+noncomputable def halfPlaneToDisk (w v : ℂ) : ℂ := (v - w) / (v - star w)
+
+lemma halfPlane_denominator_ne {w v : ℂ} (hw : 0 < w.im) (hv : 0 < v.im) :
+    v - star w ≠ 0 := by
+  intro he
+  have hh := congrArg Complex.im he
+  simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hh
+  linarith
+
+lemma halfPlaneToDisk_maps {w : ℂ} (hw : 0 < w.im) :
+    MapsTo (halfPlaneToDisk w) upperHalfPlane (ball 0 1) := by
+  intro v hv
+  exact mem_ball_zero_iff.mpr (upper_mobius_norm_lt_one hv hw)
+
+lemma halfPlaneToDisk_differentiable {w : ℂ} (hw : 0 < w.im) :
+    DifferentiableOn ℂ (halfPlaneToDisk w) upperHalfPlane :=
+  (differentiableOn_id.sub_const w).div (differentiableOn_id.sub_const (star w))
+    (fun _ hv => halfPlane_denominator_ne hw hv)
+
+lemma halfPlaneToDisk_diskToHalfPlane {w ζ : ℂ} (hw : 0 < w.im)
+    (hζ : ζ ∈ ball (0 : ℂ) 1) : halfPlaneToDisk w (diskToHalfPlane w ζ) = ζ := by
+  have hn : w - star w ≠ 0 := halfPlane_denominator_ne hw hw
+  have hd := disk_denominator_ne hζ
+  have hv := halfPlane_denominator_ne hw (diskToHalfPlane_maps hw hζ)
+  unfold halfPlaneToDisk
+  apply (div_eq_iff hv).mpr
+  unfold diskToHalfPlane
+  field_simp [hd]
+  ring
+
+lemma diskToHalfPlane_deriv {w ζ : ℂ} (hζ : ζ ∈ ball (0 : ℂ) 1) :
+    HasDerivAt (diskToHalfPlane w) ((w - star w) / (1 - ζ) ^ 2) ζ := by
+  have hh := ((hasDerivAt_const ζ w).sub ((hasDerivAt_id ζ).const_mul (star w))).div
+    ((hasDerivAt_const ζ 1).sub (hasDerivAt_id ζ)) (disk_denominator_ne hζ)
+  convert hh using 1 <;> first | rfl | (dsimp; ring)
+
+lemma diskToHalfPlane_second_zero (w : ℂ) :
+    HasDerivAt (deriv (diskToHalfPlane w)) (2 * (w - star w)) 0 := by
+  have heq : deriv (diskToHalfPlane w) =ᶠ[𝓝 (0 : ℂ)]
+      (fun ζ => (w - star w) / (1 - ζ) ^ 2) := by
+    filter_upwards [isOpen_ball.mem_nhds (mem_ball_self (by norm_num : (0 : ℝ) < 1))] with ζ hζ
+    exact (diskToHalfPlane_deriv hζ).deriv
+  have hh := (hasDerivAt_const (0 : ℂ) (w - star w)).div
+    (((hasDerivAt_const (0 : ℂ) 1).sub (hasDerivAt_id 0)).pow 2) (by norm_num : ((1 : ℂ) - 0) ^ 2 ≠ 0)
+  have hh' : HasDerivAt (fun ζ : ℂ => (w - star w) / (1 - ζ) ^ 2)
+      (2 * (w - star w)) 0 := by
+    convert hh using 1 <;> first | rfl | (norm_num; ring)
+  exact hh'.congr_of_eventuallyEq heq
+
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def halfPlaneGreen (z w : ℂ) : ℝ :=
+  Real.log ‖(z - conj w) / (z - w)‖
+
+lemma hasDerivWithinAt_log_norm {f : ℝ → ℂ} {v : ℂ} {S : Set ℝ} {t : ℝ}
+    (hf : HasDerivWithinAt f v S t) (h0 : f t ≠ 0) :
+    HasDerivWithinAt (fun s => Real.log ‖f s‖) (v / f t).re S t := by
+  have hh := (hf.norm_sq.log (pow_ne_zero 2 (norm_ne_zero_iff.mpr h0))).div_const 2
+  have he : (fun s => Real.log (‖f s‖ ^ 2) / 2) = (fun s => Real.log ‖f s‖) := by
+    funext s
+    rw [Real.log_pow]
+    ring
+  rw [he] at hh
+  apply hh.congr_deriv
+  rw [Complex.div_re, Complex.sq_norm, real_inner_eq_re_inner ℂ]
+  simp only [inner, Complex.star_def, RCLike.re_to_complex, Complex.mul_re, Complex.conj_re, Complex.conj_im]
+  ring
+
+lemma hasDerivWithinAt_conj {f : ℝ → ℂ} {v : ℂ} {S : Set ℝ} {t : ℝ}
+    (hf : HasDerivWithinAt f v S t) :
+    HasDerivWithinAt (fun s => conj (f s)) (conj v) S t := by
+  exact Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt t hf
+
+lemma hasDerivWithinAt_log_norm_div {f g : ℝ → ℂ} {v w : ℂ} {S : Set ℝ} {t : ℝ}
+    (hf : HasDerivWithinAt f v S t) (hg : HasDerivWithinAt g w S t)
+    (hf0 : f t ≠ 0) (hg0 : g t ≠ 0) :
+    HasDerivWithinAt (fun s => Real.log ‖f s / g s‖)
+      (v / f t - w / g t).re S t := by
+  apply (hasDerivWithinAt_log_norm (hf.div hg hg0) (div_ne_zero hf0 hg0)).congr_deriv
+  congr 1
+  change ((v * g t - f t * w) / g t ^ 2) / (f t / g t) = v / f t - w / g t
+  field_simp
+
+lemma green_decrease_algebra (z w : ℂ) (hz : 0 < z.im) (hw : 0 < w.im) (hne : z ≠ w) :
+    ((2 / z - conj (2 / w)) / (z - conj w) - (2 / z - 2 / w) / (z - w)).re =
+      -4 * (z.im / ‖z‖ ^ 2) * (w.im / ‖w‖ ^ 2) := by
+  have hz0 : z ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hz
+  have hw0 : w ≠ 0 := fun h => by simp only [h, Complex.zero_im, lt_self_iff_false] at hw
+  have hcw0 : conj w ≠ 0 := by simpa using hw0
+  have hn : z - w ≠ 0 := sub_ne_zero.mpr hne
+  have hn' : z - conj w ≠ 0 := by
+    intro hh
+    have he := congrArg Complex.im (sub_eq_zero.mp hh)
+    simp only [Complex.conj_im] at he
+    linarith
+  have h₁ : (2 / z - conj (2 / w)) / (z - conj w) = -2 / (z * conj w) := by
+    simp only [map_div₀, map_ofNat]
+    field_simp
+    ring
+  have h₂ : (2 / z - 2 / w) / (z - w) = -2 / (z * w) := by
+    field_simp
+    ring
+  rw [h₁, h₂]
+  simp only [Complex.sub_re, Complex.div_re, Complex.neg_re, Complex.neg_im,
+    Complex.mul_re, Complex.conj_re,
+    Complex.conj_im, Complex.normSq_mul, Complex.normSq_conj, Complex.sq_norm]
+  norm_num
+  ring
+
+ 
+
+end SLEExactGauge
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StoppedMeasures.lean b/lean/OAI/Probability/SLE/StoppedMeasures.lean
index aca05511854a35d405cc1ecca2acfa2635580db3..00df20d7f62084ded135e17c46420400a789db47 100644
--- a/lean/OAI/Probability/SLE/StoppedMeasures.lean
+++ b/lean/OAI/Probability/SLE/StoppedMeasures.lean
@@ -1,718 +1,718 @@
-import Mathlib
-import OAI.Probability.SLE.BrownianStrongMarkov
-
-namespace OAI
-
-/-! Stopped Measures for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-local instance sourceRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance sourceComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance sourceRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance sourceComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-def sourceDriverPath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) : C(ℝ≥0, ℝ) :=
-  ⟨fun t => B t ω, hc ω⟩
-
-def sourceTracePath {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
-    (hc : ∀ ω, Continuous (γ ω)) (ω : Ω) : C(ℝ≥0, ℂ) := ⟨γ ω, hc ω⟩
-
-noncomputable def sourceRestartDriverPath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) (ω : Ω) : C(ℝ≥0, ℝ) :=
-  ⟨fun t => brownianRestart B τ t ω,
-    ((hc ω).comp (continuous_const.add continuous_id)).sub continuous_const⟩
-
-universe u_source_stopped
-
- 
-
-def SourceStoppedTraceLawTarget : Prop :=
-  ∀ {Ω : Type u_source_stopped} [_m : MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ), 0 < κ → κ < 8 →
-  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
-  ∀ (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)),
-  ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧
-    (∀ ω, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω) ∧
-    ∀ (τ : Ω → ℝ≥0)
-      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0))),
-      (∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
-        (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) =
-          P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧
-      (∀ᵐ ω ∂P, ∀ t : ℝ≥0,
-        Φ (sourceRestartDriverPath B hc τ ω) t ∈ upperHalfPlane →
-        γ ω (τ ω + t) = F ω (τ ω)
-          (Φ (sourceRestartDriverPath B hc τ ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ)))
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-lemma source_path_evaluation_embedding {E : Type*} [TopologicalSpace E]
-    [PolishSpace E] [MeasurableSpace E] [BorelSpace E]
-    [MeasurableSpace C(ℝ≥0, E)] [BorelSpace C(ℝ≥0, E)] [PolishSpace C(ℝ≥0, E)] :
-    MeasurableEmbedding (fun f : C(ℝ≥0, E) => fun n => f (denseSeq ℝ≥0 n)) := by
-  apply Continuous.measurableEmbedding
-  · exact continuous_pi fun n => continuous_eval_const _
-  · intro f g h
-    apply DFunLike.coe_injective
-    exact (denseRange_denseSeq ℝ≥0).equalizer f.continuous g.continuous h
-
-lemma source_path_measurable {Ω E : Type*} [MeasurableSpace Ω]
-    [TopologicalSpace E] [PolishSpace E] [MeasurableSpace E] [BorelSpace E]
-    [MeasurableSpace C(ℝ≥0, E)] [BorelSpace C(ℝ≥0, E)] [PolishSpace C(ℝ≥0, E)]
-    (f : Ω → C(ℝ≥0, E)) (hf : ∀ t, Measurable (fun ω => f ω t)) : Measurable f := by
-  apply source_path_evaluation_embedding.measurable_comp_iff.mp
-  exact Measurable.of_eval (fun _ => hf _)
-
-local instance stoppedTraceFactorRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance stoppedTraceFactorComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance stoppedTraceFactorRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance stoppedTraceFactorComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma source_trace_factor {Ω : Type*} [m : MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → C(ℝ≥0, ℂ))
-    (ha : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => γ ω t)) :
-    ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧
-      γ = Φ ∘ (fun ω => (⟨fun t => B t ω, hc ω⟩ : C(ℝ≥0, ℝ))) := by
-  let bp : Ω → C(ℝ≥0, ℝ) := fun ω => ⟨fun t => B t ω, hc ω⟩
-  have hb (t : ℝ≥0) : @Measurable Ω ℝ (MeasurableSpace.comap bp (borel _)) (borel ℝ)
-      (B t) := by
-    exact (continuous_eval_const t).measurable.comp (Measurable.of_comap_le le_rfl)
-  have hle (t : ℝ≥0) :
-      (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t ≤
-        MeasurableSpace.comap bp (borel _) := by
-    change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ _
-    exact iSup_le fun s => iSup_le fun _ => measurable_iff_comap_le.mp (hb s)
-  have hγ : @Measurable Ω C(ℝ≥0,ℂ)
-      (MeasurableSpace.comap bp (borel _)) (borel _) γ := by
-    let : MeasurableSpace Ω := MeasurableSpace.comap bp (borel _)
-    exact source_path_measurable γ (fun t => (ha t).mono (hle t) le_rfl)
-  exact hγ.exists_eq_measurable_comp
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma source_reverseField_sub_real (a u v : ℝ) (z : ℂ) :
-    reverseField a (u - v) (z - (v : ℂ)) = reverseField a u z := by
-  unfold reverseField
-  congr 2
-  apply Complex.ext <;> simp [reverseCenter]
-
- 
-
-lemma IsCapacityTwoWitness.source_shifted_reverse_identifies
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) (w : ℂ) {a : ℝ}
-    (ha : 0 < a) (haw : a ≤ w.im) (Q : ℝ → ℂ) (hQ0 : Q 0 = w)
-    (hQd : ∀ s ∈ Icc 0 (t : ℝ), HasDerivWithinAt Q
-      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (Q s))
-        (Icc 0 (t : ℝ)) s) :
-    F (T + t) (w + (U T : ℂ)) = F T (Q t + (U T : ℂ)) := by
-  have hzH : w + (U T : ℂ) ∈ upperHalfPlane := by
-    change 0 < (w + (U T : ℂ)).im
-    simpa using ha.trans_le haw
-  let z := F (T + t) (w + (U T : ℂ))
-  have hz : z ∈ survivingDomain γ (T + t) := h.2.2.2.2.2.2.1 _ hzH
-  have hzt : z ∈ survivingDomain γ T :=
-    survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le)) hz
-  have ht (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
-      (((T + t : ℝ≥0) : ℝ) - s).toNNReal = T + ((t : ℝ) - s).toNNReal := by
-    apply NNReal.coe_injective
-    rw [Real.coe_toNNReal _ (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2])]
-    simp only [NNReal.coe_add]
-    rw [Real.coe_toNNReal _ (sub_nonneg.mpr hs.2)]
-    ring
-  let R (s : ℝ) := G (((T + t : ℝ≥0) : ℝ) - s).toNNReal z - (U T : ℂ)
-  have hRd (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) : HasDerivWithinAt R
-      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (R s))
-      (Icc 0 (t : ℝ)) s := by
-    have hs' : s ∈ Icc 0 ((T + t : ℝ≥0) : ℝ) :=
-      ⟨hs.1, by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2]⟩
-    have hd := ((h.reverse_path_derivative (T+t) (w + (U T : ℂ)) ha
-      (by simpa using haw) s hs').sub_const (U T : ℂ)).mono
-      (show Icc (0 : ℝ) t ⊆ Icc 0 ((T+t : ℝ≥0) : ℝ) from
-        Icc_subset_Icc le_rfl (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg]))
-    change HasDerivWithinAt R _ _ _ at hd
-    simpa only [R, ht s hs, source_reverseField_sub_real] using hd
-  have he : EqOn Q R (Icc 0 (t : ℝ)) := by
-    apply ODE_solution_unique
-      (fun s => reverseField_lipschitz ha (U (T + ((t : ℝ) - s).toNNReal) - U T))
-      (fun s hs => (hQd s hs).continuousWithinAt)
-      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQd s ⟨hs.1, hs.2.le⟩))
-      (fun s hs => (hRd s hs).continuousWithinAt)
-      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hRd s ⟨hs.1, hs.2.le⟩))
-    simpa only [R, sub_zero, Real.toNNReal_coe, z,
-      h.2.2.2.2.2.2.2.2.1 (T+t) _ hzH, add_sub_cancel_right] using hQ0
-  have heT := he ⟨t.coe_nonneg, le_rfl⟩
-  have htime : (((T + t : ℝ≥0) : ℝ) - (t : ℝ)).toNNReal = T := by
-    simp only [NNReal.coe_add, add_sub_cancel_right, Real.toNNReal_coe]
-  have hQ : Q t + (U T : ℂ) = G T z := by
-    simpa only [R, htime, sub_add_cancel] using congrArg (fun v : ℂ => v + (U T : ℂ)) heT
-  rw [hQ, h.2.2.2.2.2.2.2.1 T z hzt]
-
-end SLEExactGauge
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance stoppedReverseFlowRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance stoppedReverseFlowRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-
-lemma source_canonical_reverse_flow (κ : ℝ) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) :
-    ∃ Q : ℝ → C(ℝ≥0, ℝ) → ℂ,
-      (∀ s, Measurable (Q s)) ∧
-      (∀ p, Q 0 p = (Real.sqrt κ * p T : ℝ) + (a : ℂ) * I) ∧
-      ∀ p s, 0 ≤ s → HasDerivWithinAt (fun r => Q r p)
-        (reverseField a (Real.sqrt κ * p ((T : ℝ) - s).toNNReal) (Q s p)) (Ici 0) s := by
-  let V (s : ℝ) (p : C(ℝ≥0, ℝ)) := Real.sqrt κ * p ((T : ℝ) - s).toNNReal
-  have hV : Continuous (fun q : ℝ × C(ℝ≥0, ℝ) => V q.1 q.2) := by
-    apply continuous_const.mul
-    exact continuous_eval.comp (continuous_snd.prodMk
-      (continuous_real_toNNReal.comp (continuous_const.sub continuous_fst)))
-  have hw : Measurable (fun p : C(ℝ≥0, ℝ) =>
-      (Real.sqrt κ * p T : ℝ) + (a : ℂ) * I) :=
-    (Complex.continuous_ofReal.comp (continuous_const.mul (continuous_eval_const T))).measurable.add_const _
-  obtain ⟨Q, hQm, hQ0, hQd⟩ := exists_measurable_reverse_flow V hV.measurable
-    (fun p => hV.comp (continuous_id.prodMk continuous_const)) _ hw ha
-  exact ⟨Q, fun s => hQm.comp (measurable_prodMk_left (x := s)), hQ0, hQd⟩
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance stoppedRestartLawRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance stoppedRestartLawComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance stoppedRestartLawRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance stoppedRestartLawComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma source_restart_continuous_driver_law
-    {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) :
-    (P.restrict A).map (sourceRestartDriverPath B hc τ) =
-      P A • P.map (sourceDriverPath B hc) := by
-  have hbm : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
-  have hrm : Measurable (sourceRestartDriverPath B hc τ) :=
-    source_path_measurable _ (fun t => (measurable_pi_apply t).comp
-      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
-  apply source_path_evaluation_embedding.map_injective
-  have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)))
-    (brownian_restart_path_law hB hm hc hτ hA)
-  have hem : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) :=
-    Measurable.of_eval (fun _ => measurable_pi_apply _)
-  rw [Measure.map_map source_path_evaluation_embedding.measurable hrm,
-    Measure.map_smul _ source_path_evaluation_embedding.measurable.aemeasurable,
-    Measure.map_map source_path_evaluation_embedding.measurable hbm]
-  rw [Measure.map_map hem (brownianRestart_measurable hm hc
-    (stopping_time_measurable_nnreal hτ)), Measure.map_smul _ hem.aemeasurable,
-    Measure.map_map hem (Measurable.of_eval hm)] at he
-  exact he
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance stoppedTraceLawRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance stoppedTraceLawComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance stoppedTraceLawRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance stoppedTraceLawComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-theorem sourceStoppedTraceLaw : SourceStoppedTraceLawTarget := by
-  intro Ω m P κ hκ hκ8 B hB hm hc γ G F hw
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let bp := sourceDriverPath B hc
-  let gp := sourceTracePath γ (fun ω => (hw ω).1)
-  have hbm : Measurable bp := source_path_measurable bp hm
-  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => gp ω t) := by
-    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω)
-      ?_ (fun ω => continuous_const.mul (hc ω)) γ G F hw
-    intro t
-    exact measurable_const.mul
-      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
-  obtain ⟨Φ, hΦm, hΦ⟩ := source_trace_factor B hm hc gp hγa
-  have hΦe (ω : Ω) : Φ (bp ω) = gp ω := (congrFun hΦ ω).symm
-  have hgm : Measurable gp := hΦ ▸ hΦm.comp hbm
-  refine ⟨Φ, hΦm, hΦe, ?_⟩
-  intro τ hτ
-  let rp := sourceRestartDriverPath B hc τ
-  have hrm : Measurable rp := source_path_measurable rp (fun t =>
-    (measurable_pi_apply t).comp (brownianRestart_measurable hm hc
-      (stopping_time_measurable_nnreal hτ)))
-  have hlaw (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
-      (P.restrict A).map rp = P A • P.map bp :=
-    source_restart_continuous_driver_law hB hm hc τ hτ hA
-  have hlaw0 : P.map rp = P.map bp := by
-    simpa only [Measure.restrict_univ, measure_univ, one_smul] using hlaw univ MeasurableSet.univ
-  constructor
-  · intro A hA
-    have he := congrArg (Measure.map Φ) (hlaw A hA)
-    rw [Measure.map_map hΦm hrm, Measure.map_smul _ hΦm.aemeasurable,
-      Measure.map_map hΦm hbm] at he
-    have hcomp : Φ ∘ bp = gp := hΦ.symm
-    simpa only [hcomp, rp, gp, Function.comp_def] using he
-  · choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) =>
-      source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity)
-    let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1)
-    have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
-    have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) :=
-      tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
-        Eventually.of_forall ha⟩
-    have hQorig (ω : Ω) (t : ℝ≥0) (n : ℕ) :
-        Q t n t (bp ω) = F ω t ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) := by
-      apply (hw ω).reverse_flow_identifies t
-        ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
-        (fun s => Q t n s (bp ω)) (hQ0 t n (bp ω))
-      intro s hs
-      exact (hQd t n (bp ω) s hs.1).mono Icc_subset_Ici_self
-    have hQlimit (ω : Ω) (t : ℝ≥0) :
-        Tendsto (fun n => Q t n t (bp ω)) atTop (𝓝 (Φ (bp ω) t)) := by
-      have ht := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy
-      simpa only [hQorig, hΦe, gp, sourceTracePath, ContinuousMap.coe_mk, Function.comp_def] using ht
-    have hset (t : ℝ≥0) : MeasurableSet
-        {p : C(ℝ≥0,ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} :=
-      measurableSet_tendsto (𝓝 (0 : ℂ)) (fun n => (hQm t n t).sub
-        ((continuous_eval_const t).measurable.comp hΦm))
-    have hQR (t : ℝ≥0) : ∀ᵐ ω ∂P,
-        Tendsto (fun n => Q t n t (rp ω)) atTop (𝓝 (Φ (rp ω) t)) := by
-      have ho : ∀ᵐ p ∂P.map bp,
-          Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by
-        apply (ae_map_iff hbm.aemeasurable (hset t)).2
-        exact Eventually.of_forall (fun ω => tendsto_sub_nhds_zero_iff.mpr (hQlimit ω t))
-      rw [← hlaw0] at ho
-      filter_upwards [(ae_map_iff hrm.aemeasurable (hset t)).1 ho] with ω hω
-      exact tendsto_sub_nhds_zero_iff.mp hω
-    have hQRall : ∀ᵐ ω ∂P, ∀ j : ℕ,
-        Tendsto (fun n => Q (denseSeq ℝ≥0 j) n (denseSeq ℝ≥0 j) (rp ω)) atTop
-          (𝓝 (Φ (rp ω) (denseSeq ℝ≥0 j))) := ae_all_iff.mpr (fun j => hQR _)
-    filter_upwards [hQRall] with ω hω
-    have hcoc (t : ℝ≥0) (n : ℕ) :
-        F ω (τ ω + t) ((Real.sqrt κ * B (τ ω + t) ω : ℝ) + (a n : ℂ) * I) =
-        F ω (τ ω) (Q t n t (rp ω) + (Real.sqrt κ * B (τ ω) ω : ℝ)) := by
-      have he := (hw ω).source_shifted_reverse_identifies (τ ω) t
-        ((Real.sqrt κ * (rp ω) t : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
-        (fun s => Q t n s (rp ω)) (hQ0 t n (rp ω))
-        (fun s hs => by
-          have hh := (hQd t n (rp ω) s hs.1).mono
-            (show Icc (0 : ℝ) t ⊆ Ici 0 from Icc_subset_Ici_self)
-          simpa only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk,
-            mul_sub] using hh)
-      convert he using 2
-      simp only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk]
-      push_cast
-      ring
-    have hdense (j : ℕ) (hj : Φ (rp ω) (denseSeq ℝ≥0 j) ∈ upperHalfPlane) :
-        γ ω (τ ω + denseSeq ℝ≥0 j) = F ω (τ ω)
-          (Φ (rp ω) (denseSeq ℝ≥0 j) + (Real.sqrt κ * B (τ ω) ω : ℝ)) := by
-      let t := denseSeq ℝ≥0 j
-      have hz : Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ) ∈ upperHalfPlane := by
-        simpa only [upperHalfPlane, mem_ofPred_eq, add_im, ofReal_im, add_zero] using hj
-      have hFt : ContinuousAt (F ω (τ ω))
-          (Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ)) :=
-        ((hw ω).2.2.2.2.1 (τ ω)).continuousOn.continuousAt
-          (isOpen_upperHalfPlane.mem_nhds hz)
-      have hlimF := hFt.tendsto.comp ((hω j).add_const ((Real.sqrt κ * B (τ ω) ω : ℝ) : ℂ))
-      have hlimγ := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 (τ ω + t)).comp hy
-      simp only [Function.comp_def, hcoc] at hlimγ
-      exact tendsto_nhds_unique hlimγ hlimF
-    let S : Set ℝ≥0 := {t | Φ (rp ω) t ∈ upperHalfPlane}
-    have hSo : IsOpen S := isOpen_upperHalfPlane.preimage (Φ (rp ω)).continuous
-    have hleft : ContinuousOn (fun t => γ ω (τ ω + t)) S :=
-      ((hw ω).1.comp (continuous_const.add continuous_id)).continuousOn
-    have hright : ContinuousOn (fun t => F ω (τ ω)
-        (Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ))) S := by
-      apply ((hw ω).2.2.2.2.1 (τ ω)).continuousOn.comp
-        ((Φ (rp ω)).continuous.add_const _).continuousOn
-      intro t ht
-      simpa only [S, upperHalfPlane, mem_ofPred_eq, add_im, ofReal_im, add_zero] using ht
-    have hEq : EqOn (fun t => γ ω (τ ω + t))
-        (fun t => F ω (τ ω) (Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ)))
-        (S ∩ range (denseSeq ℝ≥0)) := by
-      rintro t ⟨ht, j, rfl⟩
-      exact hdense j ht
-    exact hEq.of_subset_closure hleft hright inter_subset_left
-      ((denseRange_denseSeq ℝ≥0).open_subset_closure_inter hSo)
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance stoppedForwardLawRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance stoppedForwardLawComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance stoppedForwardLawRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance stoppedForwardLawComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma IsCapacityTwoWitness.source_shifted_forward_approximation
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) (w : ℂ) {a : ℝ}
-    (ha : 0 < a) (haw : a ≤ w.im) (Q : ℝ → ℂ) (hQ0 : Q 0 = w)
-    (hQd : ∀ s ∈ Icc 0 (t : ℝ), HasDerivWithinAt Q
-      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (Q s))
-        (Icc 0 (t : ℝ)) s) :
-    Q t = G T (F (T + t) (w + (U T : ℂ))) - (U T : ℂ) := by
-  have hzH : w + (U T : ℂ) ∈ upperHalfPlane := by
-    change 0 < (w + (U T : ℂ)).im
-    simpa using ha.trans_le haw
-  let z := F (T + t) (w + (U T : ℂ))
-  have hz : z ∈ survivingDomain γ (T + t) := h.2.2.2.2.2.2.1 _ hzH
-  have hzt : z ∈ survivingDomain γ T :=
-    survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le)) hz
-  have ht (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
-      (((T + t : ℝ≥0) : ℝ) - s).toNNReal = T + ((t : ℝ) - s).toNNReal := by
-    apply NNReal.coe_injective
-    rw [Real.coe_toNNReal _ (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2])]
-    simp only [NNReal.coe_add]
-    rw [Real.coe_toNNReal _ (sub_nonneg.mpr hs.2)]
-    ring
-  let R (s : ℝ) := G (((T + t : ℝ≥0) : ℝ) - s).toNNReal z - (U T : ℂ)
-  have hRd (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) : HasDerivWithinAt R
-      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (R s))
-      (Icc 0 (t : ℝ)) s := by
-    have hs' : s ∈ Icc 0 ((T + t : ℝ≥0) : ℝ) :=
-      ⟨hs.1, by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2]⟩
-    have hd := ((h.reverse_path_derivative (T+t) (w + (U T : ℂ)) ha
-      (by simpa using haw) s hs').sub_const (U T : ℂ)).mono
-      (show Icc (0 : ℝ) t ⊆ Icc 0 ((T+t : ℝ≥0) : ℝ) from
-        Icc_subset_Icc le_rfl (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg]))
-    change HasDerivWithinAt R _ _ _ at hd
-    simpa only [R, ht s hs, source_reverseField_sub_real] using hd
-  have he : EqOn Q R (Icc 0 (t : ℝ)) := by
-    apply ODE_solution_unique
-      (fun s => reverseField_lipschitz ha (U (T + ((t : ℝ) - s).toNNReal) - U T))
-      (fun s hs => (hQd s hs).continuousWithinAt)
-      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQd s ⟨hs.1, hs.2.le⟩))
-      (fun s hs => (hRd s hs).continuousWithinAt)
-      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hRd s ⟨hs.1, hs.2.le⟩))
-    simpa only [R, sub_zero, Real.toNNReal_coe, z,
-      h.2.2.2.2.2.2.2.2.1 (T+t) _ hzH, add_sub_cancel_right] using hQ0
-  have heT := he ⟨t.coe_nonneg, le_rfl⟩
-  have htime : (((T + t : ℝ≥0) : ℝ) - (t : ℝ)).toNNReal = T := by
-    simp only [NNReal.coe_add, add_sub_cancel_right, Real.toNNReal_coe]
-  have hQ : Q t + (U T : ℂ) = G T z := by
-    simpa only [R, htime, sub_add_cancel] using congrArg (fun v : ℂ => v + (U T : ℂ)) heT
-  exact eq_sub_iff_add_eq.mpr hQ
-
-universe u_source_stopped
-
-theorem sourceStoppedTraceLaw_forward :
-  ∀ {Ω : Type u_source_stopped} [_m : MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ), 0 < κ → κ < 8 →
-  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
-  ∀ (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)),
-  ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧
-    (∀ ω, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω) ∧
-    ∀ (τ : Ω → ℝ≥0)
-      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (τ ω : WithTop ℝ≥0))),
-      (∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
-        (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) =
-          P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧
-      (∀ᵐ ω ∂P, ∀ t : ℝ≥0,
-        γ ω (τ ω + t) ∈ survivingDomain (γ ω) (τ ω) →
-        Φ (sourceRestartDriverPath B hc τ ω) t =
-          G ω (τ ω) (γ ω (τ ω + t)) - (Real.sqrt κ * B (τ ω) ω : ℝ)) := by
-  intro Ω m P κ hκ hκ8 B hB hm hc γ G F hw
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let bp := sourceDriverPath B hc
-  let gp := sourceTracePath γ (fun ω => (hw ω).1)
-  have hbm : Measurable bp := source_path_measurable bp hm
-  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => gp ω t) := by
-    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω)
-      ?_ (fun ω => continuous_const.mul (hc ω)) γ G F hw
-    intro t
-    exact measurable_const.mul
-      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
-  obtain ⟨Φ, hΦm, hΦ⟩ := source_trace_factor B hm hc gp hγa
-  have hΦe (ω : Ω) : Φ (bp ω) = gp ω := (congrFun hΦ ω).symm
-  have hgm : Measurable gp := hΦ ▸ hΦm.comp hbm
-  refine ⟨Φ, hΦm, hΦe, ?_⟩
-  intro τ hτ
-  let rp := sourceRestartDriverPath B hc τ
-  have hrm : Measurable rp := source_path_measurable rp (fun t =>
-    (measurable_pi_apply t).comp (brownianRestart_measurable hm hc
-      (stopping_time_measurable_nnreal hτ)))
-  have hlaw (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
-      (P.restrict A).map rp = P A • P.map bp :=
-    source_restart_continuous_driver_law hB hm hc τ hτ hA
-  have hlaw0 : P.map rp = P.map bp := by
-    simpa only [Measure.restrict_univ, measure_univ, one_smul] using hlaw univ MeasurableSet.univ
-  constructor
-  · intro A hA
-    have he := congrArg (Measure.map Φ) (hlaw A hA)
-    rw [Measure.map_map hΦm hrm, Measure.map_smul _ hΦm.aemeasurable,
-      Measure.map_map hΦm hbm] at he
-    have hcomp : Φ ∘ bp = gp := hΦ.symm
-    simpa only [hcomp, rp, gp, Function.comp_def] using he
-  · choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) =>
-      source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity)
-    let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1)
-    have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
-    have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) :=
-      tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
-        Eventually.of_forall ha⟩
-    have hQorig (ω : Ω) (t : ℝ≥0) (n : ℕ) :
-        Q t n t (bp ω) = F ω t ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) := by
-      apply (hw ω).reverse_flow_identifies t
-        ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
-        (fun s => Q t n s (bp ω)) (hQ0 t n (bp ω))
-      intro s hs
-      exact (hQd t n (bp ω) s hs.1).mono Icc_subset_Ici_self
-    have hQlimit (ω : Ω) (t : ℝ≥0) :
-        Tendsto (fun n => Q t n t (bp ω)) atTop (𝓝 (Φ (bp ω) t)) := by
-      have ht := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy
-      simpa only [hQorig, hΦe, gp, sourceTracePath, ContinuousMap.coe_mk, Function.comp_def] using ht
-    have hset (t : ℝ≥0) : MeasurableSet
-        {p : C(ℝ≥0,ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} :=
-      measurableSet_tendsto (𝓝 (0 : ℂ)) (fun n => (hQm t n t).sub
-        ((continuous_eval_const t).measurable.comp hΦm))
-    have hQR (t : ℝ≥0) : ∀ᵐ ω ∂P,
-        Tendsto (fun n => Q t n t (rp ω)) atTop (𝓝 (Φ (rp ω) t)) := by
-      have ho : ∀ᵐ p ∂P.map bp,
-          Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by
-        apply (ae_map_iff hbm.aemeasurable (hset t)).2
-        exact Eventually.of_forall (fun ω => tendsto_sub_nhds_zero_iff.mpr (hQlimit ω t))
-      rw [← hlaw0] at ho
-      filter_upwards [(ae_map_iff hrm.aemeasurable (hset t)).1 ho] with ω hω
-      exact tendsto_sub_nhds_zero_iff.mp hω
-    have hQRall : ∀ᵐ ω ∂P, ∀ j : ℕ,
-        Tendsto (fun n => Q (denseSeq ℝ≥0 j) n (denseSeq ℝ≥0 j) (rp ω)) atTop
-          (𝓝 (Φ (rp ω) (denseSeq ℝ≥0 j))) := ae_all_iff.mpr (fun j => hQR _)
-    filter_upwards [hQRall] with ω hω
-    have hcoc (t : ℝ≥0) (n : ℕ) :
-        Q t n t (rp ω) = G ω (τ ω)
-          (F ω (τ ω + t) ((Real.sqrt κ * B (τ ω + t) ω : ℝ) + (a n : ℂ) * I)) -
-            (Real.sqrt κ * B (τ ω) ω : ℝ) := by
-      have he := (hw ω).source_shifted_forward_approximation (τ ω) t
-        ((Real.sqrt κ * (rp ω) t : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
-        (fun s => Q t n s (rp ω)) (hQ0 t n (rp ω))
-        (fun s hs => by
-          have hh := (hQd t n (rp ω) s hs.1).mono
-            (show Icc (0 : ℝ) t ⊆ Ici 0 from Icc_subset_Ici_self)
-          simpa only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk,
-            mul_sub] using hh)
-      convert he using 2
-      simp only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk]
-      push_cast
-      ring_nf
-    have hdense (j : ℕ)
-        (hj : γ ω (τ ω + denseSeq ℝ≥0 j) ∈ survivingDomain (γ ω) (τ ω)) :
-        Φ (rp ω) (denseSeq ℝ≥0 j) =
-          G ω (τ ω) (γ ω (τ ω + denseSeq ℝ≥0 j)) - (Real.sqrt κ * B (τ ω) ω : ℝ) := by
-      let t := denseSeq ℝ≥0 j
-      have hGc : ContinuousAt (G ω (τ ω)) (γ ω (τ ω + t)) :=
-        ((hw ω).2.2.2.1 (τ ω)).continuousOn.continuousAt
-          ((isOpen_survivingDomain (hw ω).1 (τ ω)).mem_nhds hj)
-      have hlimγ := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 (τ ω + t)).comp hy
-      have hlimG := (hGc.tendsto.comp hlimγ).sub_const ((Real.sqrt κ * B (τ ω) ω : ℝ) : ℂ)
-      apply tendsto_nhds_unique (hω j)
-      simpa only [hcoc, Function.comp_def] using hlimG
-    let S : Set ℝ≥0 := {t | γ ω (τ ω + t) ∈ survivingDomain (γ ω) (τ ω)}
-    have hγc : Continuous (fun t : ℝ≥0 => γ ω (τ ω + t)) :=
-      (hw ω).1.comp (continuous_const.add continuous_id)
-    have hSo : IsOpen S := (isOpen_survivingDomain (hw ω).1 (τ ω)).preimage hγc
-    have hleft : ContinuousOn (fun t : ℝ≥0 => Φ (rp ω) t) S :=
-      (Φ (rp ω)).continuous.continuousOn
-    have hright : ContinuousOn (fun t : ℝ≥0 =>
-        G ω (τ ω) (γ ω (τ ω + t)) - (Real.sqrt κ * B (τ ω) ω : ℝ)) S :=
-      (((hw ω).2.2.2.1 (τ ω)).continuousOn.comp hγc.continuousOn
-        (fun _ ht => ht)).sub continuousOn_const
-    have hEq : EqOn (fun t : ℝ≥0 => Φ (rp ω) t)
-        (fun t : ℝ≥0 => G ω (τ ω) (γ ω (τ ω + t)) - (Real.sqrt κ * B (τ ω) ω : ℝ))
-        (S ∩ range (denseSeq ℝ≥0)) := by
-      rintro t ⟨ht, j, rfl⟩
-      exact hdense j ht
-    exact hEq.of_subset_closure hleft hright inter_subset_left
-      ((denseRange_denseSeq ℝ≥0).open_subset_closure_inter hSo)
-
-end
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma source_parameter_joint_law
-    {Ω Z E : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace E]
-    {P : Measure Ω} [IsProbabilityMeasure P] {ν : Measure E} [IsProbabilityMeasure ν]
-    (hm : m ≤ mΩ)
-    {X : Ω → E} (hX : Measurable X) {Y : Ω → Z} (hY : Measurable[m] Y)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → (P.restrict A).map X = P A • ν)
-    (A : Set Ω) (hA : MeasurableSet[m] A) :
-    (P.restrict A).map (fun ω => (Y ω, X ω)) = ((P.restrict A).map Y).prod ν := by
-  have hYa : Measurable Y := hY.mono hm le_rfl
-  apply Measure.ext_prod
-  intro S T hS hT
-  have he := congrArg (fun μ : Measure E => μ T)
-    (hlaw (A ∩ Y ⁻¹' S) (hA.inter (hY hS)))
-  rw [Measure.map_apply hX hT, Measure.smul_apply, smul_eq_mul,
-    Measure.restrict_apply (hX hT)] at he
-  rw [Measure.map_apply (hYa.prodMk hX) (hS.prod hT),
-    Measure.restrict_apply ((hYa.prodMk hX) (hS.prod hT)),
-    Measure.prod_prod, Measure.map_apply hYa hS, Measure.restrict_apply (hYa hS)]
-  change P ((Y ⁻¹' S ∩ X ⁻¹' T) ∩ A) = P (Y ⁻¹' S ∩ A) * ν T
-  simpa only [inter_assoc, inter_left_comm, inter_comm] using he
-
-lemma source_parameter_condExp
-    {Ω Z E : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace E]
-    {P : Measure Ω} [IsProbabilityMeasure P] {ν : Measure E} [IsProbabilityMeasure ν]
-    (hm : m ≤ mΩ)
-    {X : Ω → E} (hX : Measurable X) {Y : Ω → Z} (hY : Measurable[m] Y)
-    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → (P.restrict A).map X = P A • ν)
-    (H : Z × E → ℝ) (hH : Measurable H) (hb : ∀ p, ‖H p‖ ≤ 1) :
-    P[(fun ω => H (Y ω, X ω)) | m] =ᵐ[P] (fun ω => ∫ e, H (Y ω, e) ∂ν) := by
-  have hYa : Measurable Y := hY.mono hm le_rfl
-  have hK : StronglyMeasurable (fun y => ∫ e, H (y, e) ∂ν) :=
-    hH.stronglyMeasurable.integral_prod_right'
-  have hKb (y : Z) : ‖∫ e, H (y, e) ∂ν‖ ≤ 1 := by
-    simpa using norm_integral_le_of_norm_le_const (μ := ν) (Eventually.of_forall (fun e => hb (y,e)))
-  have hint : Integrable (fun ω => H (Y ω, X ω)) P :=
-    (integrable_const (1 : ℝ)).mono' (hH.comp (hYa.prodMk hX)).aestronglyMeasurable
-      (Eventually.of_forall (fun ω => hb (Y ω, X ω)))
-  have hKint : Integrable (fun ω => ∫ e, H (Y ω, e) ∂ν) P :=
-    (integrable_const (1 : ℝ)).mono' (hK.comp_measurable hYa).aestronglyMeasurable
-      (Eventually.of_forall (fun ω => hKb (Y ω)))
-  symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq hm hint
-    (fun _ _ _ => hKint.integrableOn) _ (hK.comp_measurable hY).aestronglyMeasurable
-  intro A hA _
-  have hprodint : Integrable H (((P.restrict A).map Y).prod ν) :=
-    (integrable_const (1 : ℝ)).mono' hH.aestronglyMeasurable (Eventually.of_forall hb)
-  calc
-    (∫ ω in A, ∫ e, H (Y ω, e) ∂ν ∂P) =
-        ∫ y, ∫ e, H (y, e) ∂ν ∂(P.restrict A).map Y :=
-      (integral_map hYa.aemeasurable hK.aestronglyMeasurable).symm
-    _ = ∫ p, H p ∂(((P.restrict A).map Y).prod ν) := (integral_prod H hprodint).symm
-    _ = ∫ p, H p ∂(P.restrict A).map (fun ω => (Y ω, X ω)) := by
-      rw [source_parameter_joint_law hm hX hY hlaw A hA]
-    _ = ∫ ω in A, H (Y ω, X ω) ∂P :=
-      integral_map (hYa.prodMk hX).aemeasurable hH.aestronglyMeasurable
-
-end SLEExactGauge
-
-end
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
- 
-
-theorem source_killedChart_progressive
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)) (z : ℂ) :
-    IsStronglyProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun t ω => if z ∈ survivingDomain (γ ω) t then G ω t z else 0) := by
-  classical
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable hc hm
-  have hnn (n : ℕ) : -(n : ℝ) ≤ n := by linarith [Nat.cast_nonneg (α := ℝ) n]
-  let c (n : ℕ) := (exists_clippedLogSolution B hm hBm hc κ (-(n : ℝ)) n (hnn n)
-    (initialLogCoordinates z)).some
-  let v (n : ℕ) (t : ℝ≥0) (ω : Ω) : Prop :=
-    0 < z.im ∧ Real.log z.im ≤ (n : ℝ) ∧ -(n : ℝ) ≤ ((c n).Q t ω).1.im
-  let A (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℂ :=
-    if v n t ω then
-      (((c n).Q t ω).1.re : ℂ) + (Real.exp ((c n).Q t ω).1.im : ℂ) * I
-    else 0
-  have hQp (n : ℕ) : IsStronglyProgressive f (fun t : ℝ≥0 => (c n).Q t) :=
-    (show StronglyAdapted f (fun t : ℝ≥0 => (c n).Q t) from
-      fun t => ((c n).adapted t).stronglyMeasurable).isStronglyProgressive_of_continuous
-      (fun ω => ((c n).lipschitz ω).continuous.comp NNReal.continuous_coe)
-  have hAp (n : ℕ) : IsStronglyProgressive f (A n) := by
-    intro T
-    have hQ := (hQp n T).measurable
-    let : MeasurableSpace Ω := f T
-    have hv : MeasurableSet {p : Iic T × Ω | v n p.1 p.2} :=
-      (MeasurableSet.const _).inter ((MeasurableSet.const _).inter
-        (measurableSet_le measurable_const (Complex.measurable_im.comp hQ.fst)))
-    exact (Measurable.ite hv
-      ((Complex.measurable_ofReal.comp (Complex.measurable_re.comp hQ.fst)).add
-        ((Complex.measurable_ofReal.comp (Real.continuous_exp.measurable.comp
-          (Complex.measurable_im.comp hQ.fst))).mul_const I)) measurable_const).stronglyMeasurable
-  have hlim (t : ℝ≥0) (ω : Ω) : Tendsto (fun n => A n t ω) atTop
-      (𝓝 (if z ∈ survivingDomain (γ ω) t then G ω t z else 0)) := by
-    apply tendsto_nhds_of_eventually_eq
-    have hU : Continuous (fun s => Real.sqrt κ * B s ω) := (hc ω).const_mul _
-    have hd (n : ℕ) (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :=
-      ((c n).derivative ω s hs.1).mono (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self)
-    by_cases hz : z ∈ survivingDomain (γ ω) t
-    · obtain ⟨N, hN⟩ := exists_nat_ge (max (-Real.log (G ω t z).im) (Real.log z.im))
-      filter_upwards [eventually_ge_atTop N] with n hn
-      have hNn : (N : ℝ) ≤ n := by exact_mod_cast hn
-      have hlo : -(n : ℝ) ≤ Real.log (G ω t z).im := by
-        have hh := (le_max_left (-Real.log (G ω t z).im) (Real.log z.im)).trans (hN.trans hNn)
-        linarith
-      have hhi : Real.log z.im ≤ (n : ℝ) := (le_max_right _ _).trans (hN.trans hNn)
-      have he : (c n).Q t ω = logLoewnerCoordinates (G ω) z t :=
-        clipped_log_loewner_agrees hU (hw ω) t z hz _ _ _ hlo hhi
-          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) ⟨t.coe_nonneg, le_rfl⟩
-      have hv : v n t ω := ⟨hz.1.1, hhi, by simpa only [he, logCoordinates_im] using hlo⟩
-      simp only [A, ite_eq_left hz, ite_eq_left hv, he, logCoordinates_re, logCoordinates_im,
-        Real.exp_log ((hw ω).imag_positive t z hz)]
-      exact Complex.re_add_im _
-    · filter_upwards [] with n
-      have hv : ¬ v n t ω := by
-        intro hv
-        exact hz (clipped_log_identification_of_endpoint hU (hw ω) t z hv.1 _ _ _
-          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) hv.2.2 hv.2.1).1
-      simp only [A, ite_eq_right hv, ite_eq_right hz]
-  exact isStronglyProgressive_of_tendsto hAp
-    (tendsto_pi_nhds.mpr (fun t => tendsto_pi_nhds.mpr (hlim t)))
-
-end
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.BrownianStrongMarkov
+
+namespace OAI
+
+/-! Stopped Measures for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+local instance sourceRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance sourceComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance sourceRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance sourceComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+def sourceDriverPath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (ω : Ω) : C(ℝ≥0, ℝ) :=
+  ⟨fun t => B t ω, hc ω⟩
+
+def sourceTracePath {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ)
+    (hc : ∀ ω, Continuous (γ ω)) (ω : Ω) : C(ℝ≥0, ℂ) := ⟨γ ω, hc ω⟩
+
+noncomputable def sourceRestartDriverPath {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) (ω : Ω) : C(ℝ≥0, ℝ) :=
+  ⟨fun t => brownianRestart B τ t ω,
+    ((hc ω).comp (continuous_const.add continuous_id)).sub continuous_const⟩
+
+universe u_source_stopped
+
+ 
+
+def SourceStoppedTraceLawTarget : Prop :=
+  ∀ {Ω : Type u_source_stopped} [_m : MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ), 0 < κ → κ < 8 →
+  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
+  ∀ (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)),
+  ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧
+    (∀ ω, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω) ∧
+    ∀ (τ : Ω → ℝ≥0)
+      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0))),
+      (∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
+        (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) =
+          P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧
+      (∀ᵐ ω ∂P, ∀ t : ℝ≥0,
+        Φ (sourceRestartDriverPath B hc τ ω) t ∈ upperHalfPlane →
+        γ ω (τ ω + t) = F ω (τ ω)
+          (Φ (sourceRestartDriverPath B hc τ ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ)))
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+lemma source_path_evaluation_embedding {E : Type*} [TopologicalSpace E]
+    [PolishSpace E] [MeasurableSpace E] [BorelSpace E]
+    [MeasurableSpace C(ℝ≥0, E)] [BorelSpace C(ℝ≥0, E)] [PolishSpace C(ℝ≥0, E)] :
+    MeasurableEmbedding (fun f : C(ℝ≥0, E) => fun n => f (denseSeq ℝ≥0 n)) := by
+  apply Continuous.measurableEmbedding
+  · exact continuous_pi fun n => continuous_eval_const _
+  · intro f g h
+    apply DFunLike.coe_injective
+    exact (denseRange_denseSeq ℝ≥0).equalizer f.continuous g.continuous h
+
+lemma source_path_measurable {Ω E : Type*} [MeasurableSpace Ω]
+    [TopologicalSpace E] [PolishSpace E] [MeasurableSpace E] [BorelSpace E]
+    [MeasurableSpace C(ℝ≥0, E)] [BorelSpace C(ℝ≥0, E)] [PolishSpace C(ℝ≥0, E)]
+    (f : Ω → C(ℝ≥0, E)) (hf : ∀ t, Measurable (fun ω => f ω t)) : Measurable f := by
+  apply source_path_evaluation_embedding.measurable_comp_iff.mp
+  exact Measurable.of_eval (fun _ => hf _)
+
+local instance stoppedTraceFactorRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance stoppedTraceFactorComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance stoppedTraceFactorRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance stoppedTraceFactorComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma source_trace_factor {Ω : Type*} [m : MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → C(ℝ≥0, ℂ))
+    (ha : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => γ ω t)) :
+    ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧
+      γ = Φ ∘ (fun ω => (⟨fun t => B t ω, hc ω⟩ : C(ℝ≥0, ℝ))) := by
+  let bp : Ω → C(ℝ≥0, ℝ) := fun ω => ⟨fun t => B t ω, hc ω⟩
+  have hb (t : ℝ≥0) : @Measurable Ω ℝ (MeasurableSpace.comap bp (borel _)) (borel ℝ)
+      (B t) := by
+    exact (continuous_eval_const t).measurable.comp (Measurable.of_comap_le le_rfl)
+  have hle (t : ℝ≥0) :
+      (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) t ≤
+        MeasurableSpace.comap bp (borel _) := by
+    change (⨆ s ≤ t, MeasurableSpace.comap (B s) (borel ℝ)) ≤ _
+    exact iSup_le fun s => iSup_le fun _ => measurable_iff_comap_le.mp (hb s)
+  have hγ : @Measurable Ω C(ℝ≥0,ℂ)
+      (MeasurableSpace.comap bp (borel _)) (borel _) γ := by
+    let : MeasurableSpace Ω := MeasurableSpace.comap bp (borel _)
+    exact source_path_measurable γ (fun t => (ha t).mono (hle t) le_rfl)
+  exact hγ.exists_eq_measurable_comp
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma source_reverseField_sub_real (a u v : ℝ) (z : ℂ) :
+    reverseField a (u - v) (z - (v : ℂ)) = reverseField a u z := by
+  unfold reverseField
+  congr 2
+  apply Complex.ext <;> simp [reverseCenter]
+
+ 
+
+lemma IsCapacityTwoWitness.source_shifted_reverse_identifies
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) (w : ℂ) {a : ℝ}
+    (ha : 0 < a) (haw : a ≤ w.im) (Q : ℝ → ℂ) (hQ0 : Q 0 = w)
+    (hQd : ∀ s ∈ Icc 0 (t : ℝ), HasDerivWithinAt Q
+      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (Q s))
+        (Icc 0 (t : ℝ)) s) :
+    F (T + t) (w + (U T : ℂ)) = F T (Q t + (U T : ℂ)) := by
+  have hzH : w + (U T : ℂ) ∈ upperHalfPlane := by
+    change 0 < (w + (U T : ℂ)).im
+    simpa using ha.trans_le haw
+  let z := F (T + t) (w + (U T : ℂ))
+  have hz : z ∈ survivingDomain γ (T + t) := h.2.2.2.2.2.2.1 _ hzH
+  have hzt : z ∈ survivingDomain γ T :=
+    survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le)) hz
+  have ht (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
+      (((T + t : ℝ≥0) : ℝ) - s).toNNReal = T + ((t : ℝ) - s).toNNReal := by
+    apply NNReal.coe_injective
+    rw [Real.coe_toNNReal _ (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2])]
+    simp only [NNReal.coe_add]
+    rw [Real.coe_toNNReal _ (sub_nonneg.mpr hs.2)]
+    ring
+  let R (s : ℝ) := G (((T + t : ℝ≥0) : ℝ) - s).toNNReal z - (U T : ℂ)
+  have hRd (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) : HasDerivWithinAt R
+      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (R s))
+      (Icc 0 (t : ℝ)) s := by
+    have hs' : s ∈ Icc 0 ((T + t : ℝ≥0) : ℝ) :=
+      ⟨hs.1, by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2]⟩
+    have hd := ((h.reverse_path_derivative (T+t) (w + (U T : ℂ)) ha
+      (by simpa using haw) s hs').sub_const (U T : ℂ)).mono
+      (show Icc (0 : ℝ) t ⊆ Icc 0 ((T+t : ℝ≥0) : ℝ) from
+        Icc_subset_Icc le_rfl (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg]))
+    change HasDerivWithinAt R _ _ _ at hd
+    simpa only [R, ht s hs, source_reverseField_sub_real] using hd
+  have he : EqOn Q R (Icc 0 (t : ℝ)) := by
+    apply ODE_solution_unique
+      (fun s => reverseField_lipschitz ha (U (T + ((t : ℝ) - s).toNNReal) - U T))
+      (fun s hs => (hQd s hs).continuousWithinAt)
+      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQd s ⟨hs.1, hs.2.le⟩))
+      (fun s hs => (hRd s hs).continuousWithinAt)
+      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hRd s ⟨hs.1, hs.2.le⟩))
+    simpa only [R, sub_zero, Real.toNNReal_coe, z,
+      h.2.2.2.2.2.2.2.2.1 (T+t) _ hzH, add_sub_cancel_right] using hQ0
+  have heT := he ⟨t.coe_nonneg, le_rfl⟩
+  have htime : (((T + t : ℝ≥0) : ℝ) - (t : ℝ)).toNNReal = T := by
+    simp only [NNReal.coe_add, add_sub_cancel_right, Real.toNNReal_coe]
+  have hQ : Q t + (U T : ℂ) = G T z := by
+    simpa only [R, htime, sub_add_cancel] using congrArg (fun v : ℂ => v + (U T : ℂ)) heT
+  rw [hQ, h.2.2.2.2.2.2.2.1 T z hzt]
+
+end SLEExactGauge
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance stoppedReverseFlowRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance stoppedReverseFlowRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+
+lemma source_canonical_reverse_flow (κ : ℝ) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) :
+    ∃ Q : ℝ → C(ℝ≥0, ℝ) → ℂ,
+      (∀ s, Measurable (Q s)) ∧
+      (∀ p, Q 0 p = (Real.sqrt κ * p T : ℝ) + (a : ℂ) * I) ∧
+      ∀ p s, 0 ≤ s → HasDerivWithinAt (fun r => Q r p)
+        (reverseField a (Real.sqrt κ * p ((T : ℝ) - s).toNNReal) (Q s p)) (Ici 0) s := by
+  let V (s : ℝ) (p : C(ℝ≥0, ℝ)) := Real.sqrt κ * p ((T : ℝ) - s).toNNReal
+  have hV : Continuous (fun q : ℝ × C(ℝ≥0, ℝ) => V q.1 q.2) := by
+    apply continuous_const.mul
+    exact continuous_eval.comp (continuous_snd.prodMk
+      (continuous_real_toNNReal.comp (continuous_const.sub continuous_fst)))
+  have hw : Measurable (fun p : C(ℝ≥0, ℝ) =>
+      (Real.sqrt κ * p T : ℝ) + (a : ℂ) * I) :=
+    (Complex.continuous_ofReal.comp (continuous_const.mul (continuous_eval_const T))).measurable.add_const _
+  obtain ⟨Q, hQm, hQ0, hQd⟩ := exists_measurable_reverse_flow V hV.measurable
+    (fun p => hV.comp (continuous_id.prodMk continuous_const)) _ hw ha
+  exact ⟨Q, fun s => hQm.comp (measurable_prodMk_left (x := s)), hQ0, hQd⟩
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance stoppedRestartLawRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance stoppedRestartLawComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance stoppedRestartLawRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance stoppedRestartLawComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma source_restart_continuous_driver_law
+    {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    {A : Set Ω} (hA : MeasurableSet[hτ.measurableSpace] A) :
+    (P.restrict A).map (sourceRestartDriverPath B hc τ) =
+      P A • P.map (sourceDriverPath B hc) := by
+  have hbm : Measurable (sourceDriverPath B hc) := source_path_measurable _ hm
+  have hrm : Measurable (sourceRestartDriverPath B hc τ) :=
+    source_path_measurable _ (fun t => (measurable_pi_apply t).comp
+      (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ)))
+  apply source_path_evaluation_embedding.map_injective
+  have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)))
+    (brownian_restart_path_law hB hm hc hτ hA)
+  have hem : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) :=
+    Measurable.of_eval (fun _ => measurable_pi_apply _)
+  rw [Measure.map_map source_path_evaluation_embedding.measurable hrm,
+    Measure.map_smul _ source_path_evaluation_embedding.measurable.aemeasurable,
+    Measure.map_map source_path_evaluation_embedding.measurable hbm]
+  rw [Measure.map_map hem (brownianRestart_measurable hm hc
+    (stopping_time_measurable_nnreal hτ)), Measure.map_smul _ hem.aemeasurable,
+    Measure.map_map hem (Measurable.of_eval hm)] at he
+  exact he
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance stoppedTraceLawRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance stoppedTraceLawComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance stoppedTraceLawRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance stoppedTraceLawComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+theorem sourceStoppedTraceLaw : SourceStoppedTraceLawTarget := by
+  intro Ω m P κ hκ hκ8 B hB hm hc γ G F hw
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let bp := sourceDriverPath B hc
+  let gp := sourceTracePath γ (fun ω => (hw ω).1)
+  have hbm : Measurable bp := source_path_measurable bp hm
+  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => gp ω t) := by
+    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω)
+      ?_ (fun ω => continuous_const.mul (hc ω)) γ G F hw
+    intro t
+    exact measurable_const.mul
+      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
+  obtain ⟨Φ, hΦm, hΦ⟩ := source_trace_factor B hm hc gp hγa
+  have hΦe (ω : Ω) : Φ (bp ω) = gp ω := (congrFun hΦ ω).symm
+  have hgm : Measurable gp := hΦ ▸ hΦm.comp hbm
+  refine ⟨Φ, hΦm, hΦe, ?_⟩
+  intro τ hτ
+  let rp := sourceRestartDriverPath B hc τ
+  have hrm : Measurable rp := source_path_measurable rp (fun t =>
+    (measurable_pi_apply t).comp (brownianRestart_measurable hm hc
+      (stopping_time_measurable_nnreal hτ)))
+  have hlaw (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
+      (P.restrict A).map rp = P A • P.map bp :=
+    source_restart_continuous_driver_law hB hm hc τ hτ hA
+  have hlaw0 : P.map rp = P.map bp := by
+    simpa only [Measure.restrict_univ, measure_univ, one_smul] using hlaw univ MeasurableSet.univ
+  constructor
+  · intro A hA
+    have he := congrArg (Measure.map Φ) (hlaw A hA)
+    rw [Measure.map_map hΦm hrm, Measure.map_smul _ hΦm.aemeasurable,
+      Measure.map_map hΦm hbm] at he
+    have hcomp : Φ ∘ bp = gp := hΦ.symm
+    simpa only [hcomp, rp, gp, Function.comp_def] using he
+  · choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) =>
+      source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity)
+    let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1)
+    have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
+    have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) :=
+      tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
+        Eventually.of_forall ha⟩
+    have hQorig (ω : Ω) (t : ℝ≥0) (n : ℕ) :
+        Q t n t (bp ω) = F ω t ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) := by
+      apply (hw ω).reverse_flow_identifies t
+        ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
+        (fun s => Q t n s (bp ω)) (hQ0 t n (bp ω))
+      intro s hs
+      exact (hQd t n (bp ω) s hs.1).mono Icc_subset_Ici_self
+    have hQlimit (ω : Ω) (t : ℝ≥0) :
+        Tendsto (fun n => Q t n t (bp ω)) atTop (𝓝 (Φ (bp ω) t)) := by
+      have ht := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy
+      simpa only [hQorig, hΦe, gp, sourceTracePath, ContinuousMap.coe_mk, Function.comp_def] using ht
+    have hset (t : ℝ≥0) : MeasurableSet
+        {p : C(ℝ≥0,ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} :=
+      measurableSet_tendsto (𝓝 (0 : ℂ)) (fun n => (hQm t n t).sub
+        ((continuous_eval_const t).measurable.comp hΦm))
+    have hQR (t : ℝ≥0) : ∀ᵐ ω ∂P,
+        Tendsto (fun n => Q t n t (rp ω)) atTop (𝓝 (Φ (rp ω) t)) := by
+      have ho : ∀ᵐ p ∂P.map bp,
+          Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by
+        apply (ae_map_iff hbm.aemeasurable (hset t)).2
+        exact Eventually.of_forall (fun ω => tendsto_sub_nhds_zero_iff.mpr (hQlimit ω t))
+      rw [← hlaw0] at ho
+      filter_upwards [(ae_map_iff hrm.aemeasurable (hset t)).1 ho] with ω hω
+      exact tendsto_sub_nhds_zero_iff.mp hω
+    have hQRall : ∀ᵐ ω ∂P, ∀ j : ℕ,
+        Tendsto (fun n => Q (denseSeq ℝ≥0 j) n (denseSeq ℝ≥0 j) (rp ω)) atTop
+          (𝓝 (Φ (rp ω) (denseSeq ℝ≥0 j))) := ae_all_iff.mpr (fun j => hQR _)
+    filter_upwards [hQRall] with ω hω
+    have hcoc (t : ℝ≥0) (n : ℕ) :
+        F ω (τ ω + t) ((Real.sqrt κ * B (τ ω + t) ω : ℝ) + (a n : ℂ) * I) =
+        F ω (τ ω) (Q t n t (rp ω) + (Real.sqrt κ * B (τ ω) ω : ℝ)) := by
+      have he := (hw ω).source_shifted_reverse_identifies (τ ω) t
+        ((Real.sqrt κ * (rp ω) t : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
+        (fun s => Q t n s (rp ω)) (hQ0 t n (rp ω))
+        (fun s hs => by
+          have hh := (hQd t n (rp ω) s hs.1).mono
+            (show Icc (0 : ℝ) t ⊆ Ici 0 from Icc_subset_Ici_self)
+          simpa only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk,
+            mul_sub] using hh)
+      convert he using 2
+      simp only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk]
+      push_cast
+      ring
+    have hdense (j : ℕ) (hj : Φ (rp ω) (denseSeq ℝ≥0 j) ∈ upperHalfPlane) :
+        γ ω (τ ω + denseSeq ℝ≥0 j) = F ω (τ ω)
+          (Φ (rp ω) (denseSeq ℝ≥0 j) + (Real.sqrt κ * B (τ ω) ω : ℝ)) := by
+      let t := denseSeq ℝ≥0 j
+      have hz : Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ) ∈ upperHalfPlane := by
+        simpa only [upperHalfPlane, mem_ofPred_eq, add_im, ofReal_im, add_zero] using hj
+      have hFt : ContinuousAt (F ω (τ ω))
+          (Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ)) :=
+        ((hw ω).2.2.2.2.1 (τ ω)).continuousOn.continuousAt
+          (isOpen_upperHalfPlane.mem_nhds hz)
+      have hlimF := hFt.tendsto.comp ((hω j).add_const ((Real.sqrt κ * B (τ ω) ω : ℝ) : ℂ))
+      have hlimγ := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 (τ ω + t)).comp hy
+      simp only [Function.comp_def, hcoc] at hlimγ
+      exact tendsto_nhds_unique hlimγ hlimF
+    let S : Set ℝ≥0 := {t | Φ (rp ω) t ∈ upperHalfPlane}
+    have hSo : IsOpen S := isOpen_upperHalfPlane.preimage (Φ (rp ω)).continuous
+    have hleft : ContinuousOn (fun t => γ ω (τ ω + t)) S :=
+      ((hw ω).1.comp (continuous_const.add continuous_id)).continuousOn
+    have hright : ContinuousOn (fun t => F ω (τ ω)
+        (Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ))) S := by
+      apply ((hw ω).2.2.2.2.1 (τ ω)).continuousOn.comp
+        ((Φ (rp ω)).continuous.add_const _).continuousOn
+      intro t ht
+      simpa only [S, upperHalfPlane, mem_ofPred_eq, add_im, ofReal_im, add_zero] using ht
+    have hEq : EqOn (fun t => γ ω (τ ω + t))
+        (fun t => F ω (τ ω) (Φ (rp ω) t + (Real.sqrt κ * B (τ ω) ω : ℝ)))
+        (S ∩ range (denseSeq ℝ≥0)) := by
+      rintro t ⟨ht, j, rfl⟩
+      exact hdense j ht
+    exact hEq.of_subset_closure hleft hright inter_subset_left
+      ((denseRange_denseSeq ℝ≥0).open_subset_closure_inter hSo)
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance stoppedForwardLawRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance stoppedForwardLawComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance stoppedForwardLawRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance stoppedForwardLawComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma IsCapacityTwoWitness.source_shifted_forward_approximation
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T t : ℝ≥0) (w : ℂ) {a : ℝ}
+    (ha : 0 < a) (haw : a ≤ w.im) (Q : ℝ → ℂ) (hQ0 : Q 0 = w)
+    (hQd : ∀ s ∈ Icc 0 (t : ℝ), HasDerivWithinAt Q
+      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (Q s))
+        (Icc 0 (t : ℝ)) s) :
+    Q t = G T (F (T + t) (w + (U T : ℂ))) - (U T : ℂ) := by
+  have hzH : w + (U T : ℂ) ∈ upperHalfPlane := by
+    change 0 < (w + (U T : ℂ)).im
+    simpa using ha.trans_le haw
+  let z := F (T + t) (w + (U T : ℂ))
+  have hz : z ∈ survivingDomain γ (T + t) := h.2.2.2.2.2.2.1 _ hzH
+  have hzt : z ∈ survivingDomain γ T :=
+    survivingDomain_antitone γ (le_add_of_nonneg_right (show 0 ≤ t from bot_le)) hz
+  have ht (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :
+      (((T + t : ℝ≥0) : ℝ) - s).toNNReal = T + ((t : ℝ) - s).toNNReal := by
+    apply NNReal.coe_injective
+    rw [Real.coe_toNNReal _ (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2])]
+    simp only [NNReal.coe_add]
+    rw [Real.coe_toNNReal _ (sub_nonneg.mpr hs.2)]
+    ring
+  let R (s : ℝ) := G (((T + t : ℝ≥0) : ℝ) - s).toNNReal z - (U T : ℂ)
+  have hRd (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) : HasDerivWithinAt R
+      (reverseField a (U (T + ((t : ℝ) - s).toNNReal) - U T) (R s))
+      (Icc 0 (t : ℝ)) s := by
+    have hs' : s ∈ Icc 0 ((T + t : ℝ≥0) : ℝ) :=
+      ⟨hs.1, by simp only [NNReal.coe_add]; linarith [T.coe_nonneg, hs.2]⟩
+    have hd := ((h.reverse_path_derivative (T+t) (w + (U T : ℂ)) ha
+      (by simpa using haw) s hs').sub_const (U T : ℂ)).mono
+      (show Icc (0 : ℝ) t ⊆ Icc 0 ((T+t : ℝ≥0) : ℝ) from
+        Icc_subset_Icc le_rfl (by simp only [NNReal.coe_add]; linarith [T.coe_nonneg]))
+    change HasDerivWithinAt R _ _ _ at hd
+    simpa only [R, ht s hs, source_reverseField_sub_real] using hd
+  have he : EqOn Q R (Icc 0 (t : ℝ)) := by
+    apply ODE_solution_unique
+      (fun s => reverseField_lipschitz ha (U (T + ((t : ℝ) - s).toNNReal) - U T))
+      (fun s hs => (hQd s hs).continuousWithinAt)
+      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQd s ⟨hs.1, hs.2.le⟩))
+      (fun s hs => (hRd s hs).continuousWithinAt)
+      (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hRd s ⟨hs.1, hs.2.le⟩))
+    simpa only [R, sub_zero, Real.toNNReal_coe, z,
+      h.2.2.2.2.2.2.2.2.1 (T+t) _ hzH, add_sub_cancel_right] using hQ0
+  have heT := he ⟨t.coe_nonneg, le_rfl⟩
+  have htime : (((T + t : ℝ≥0) : ℝ) - (t : ℝ)).toNNReal = T := by
+    simp only [NNReal.coe_add, add_sub_cancel_right, Real.toNNReal_coe]
+  have hQ : Q t + (U T : ℂ) = G T z := by
+    simpa only [R, htime, sub_add_cancel] using congrArg (fun v : ℂ => v + (U T : ℂ)) heT
+  exact eq_sub_iff_add_eq.mpr hQ
+
+universe u_source_stopped
+
+theorem sourceStoppedTraceLaw_forward :
+  ∀ {Ω : Type u_source_stopped} [_m : MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ), 0 < κ → κ < 8 →
+  ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
+  ∀ (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)),
+  ∃ Φ : C(ℝ≥0, ℝ) → C(ℝ≥0, ℂ), Measurable Φ ∧
+    (∀ ω, Φ (sourceDriverPath B hc ω) = sourceTracePath γ (fun ω => (hw ω).1) ω) ∧
+    ∀ (τ : Ω → ℝ≥0)
+      (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (τ ω : WithTop ℝ≥0))),
+      (∀ (A : Set Ω), MeasurableSet[hτ.measurableSpace] A →
+        (P.restrict A).map (fun ω => Φ (sourceRestartDriverPath B hc τ ω)) =
+          P A • P.map (sourceTracePath γ (fun ω => (hw ω).1))) ∧
+      (∀ᵐ ω ∂P, ∀ t : ℝ≥0,
+        γ ω (τ ω + t) ∈ survivingDomain (γ ω) (τ ω) →
+        Φ (sourceRestartDriverPath B hc τ ω) t =
+          G ω (τ ω) (γ ω (τ ω + t)) - (Real.sqrt κ * B (τ ω) ω : ℝ)) := by
+  intro Ω m P κ hκ hκ8 B hB hm hc γ G F hw
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let bp := sourceDriverPath B hc
+  let gp := sourceTracePath γ (fun ω => (hw ω).1)
+  have hbm : Measurable bp := source_path_measurable bp hm
+  have hγa : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => gp ω t) := by
+    apply capacity_trace_adapted _ (fun t ω => Real.sqrt κ * B t ω)
+      ?_ (fun ω => continuous_const.mul (hc ω)) γ G F hw
+    intro t
+    exact measurable_const.mul
+      ((Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)) t).measurable
+  obtain ⟨Φ, hΦm, hΦ⟩ := source_trace_factor B hm hc gp hγa
+  have hΦe (ω : Ω) : Φ (bp ω) = gp ω := (congrFun hΦ ω).symm
+  have hgm : Measurable gp := hΦ ▸ hΦm.comp hbm
+  refine ⟨Φ, hΦm, hΦe, ?_⟩
+  intro τ hτ
+  let rp := sourceRestartDriverPath B hc τ
+  have hrm : Measurable rp := source_path_measurable rp (fun t =>
+    (measurable_pi_apply t).comp (brownianRestart_measurable hm hc
+      (stopping_time_measurable_nnreal hτ)))
+  have hlaw (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) :
+      (P.restrict A).map rp = P A • P.map bp :=
+    source_restart_continuous_driver_law hB hm hc τ hτ hA
+  have hlaw0 : P.map rp = P.map bp := by
+    simpa only [Measure.restrict_univ, measure_univ, one_smul] using hlaw univ MeasurableSet.univ
+  constructor
+  · intro A hA
+    have he := congrArg (Measure.map Φ) (hlaw A hA)
+    rw [Measure.map_map hΦm hrm, Measure.map_smul _ hΦm.aemeasurable,
+      Measure.map_map hΦm hbm] at he
+    have hcomp : Φ ∘ bp = gp := hΦ.symm
+    simpa only [hcomp, rp, gp, Function.comp_def] using he
+  · choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) =>
+      source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity)
+    let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1)
+    have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
+    have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) :=
+      tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
+        Eventually.of_forall ha⟩
+    have hQorig (ω : Ω) (t : ℝ≥0) (n : ℕ) :
+        Q t n t (bp ω) = F ω t ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) := by
+      apply (hw ω).reverse_flow_identifies t
+        ((Real.sqrt κ * B t ω : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
+        (fun s => Q t n s (bp ω)) (hQ0 t n (bp ω))
+      intro s hs
+      exact (hQd t n (bp ω) s hs.1).mono Icc_subset_Ici_self
+    have hQlimit (ω : Ω) (t : ℝ≥0) :
+        Tendsto (fun n => Q t n t (bp ω)) atTop (𝓝 (Φ (bp ω) t)) := by
+      have ht := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy
+      simpa only [hQorig, hΦe, gp, sourceTracePath, ContinuousMap.coe_mk, Function.comp_def] using ht
+    have hset (t : ℝ≥0) : MeasurableSet
+        {p : C(ℝ≥0,ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} :=
+      measurableSet_tendsto (𝓝 (0 : ℂ)) (fun n => (hQm t n t).sub
+        ((continuous_eval_const t).measurable.comp hΦm))
+    have hQR (t : ℝ≥0) : ∀ᵐ ω ∂P,
+        Tendsto (fun n => Q t n t (rp ω)) atTop (𝓝 (Φ (rp ω) t)) := by
+      have ho : ∀ᵐ p ∂P.map bp,
+          Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by
+        apply (ae_map_iff hbm.aemeasurable (hset t)).2
+        exact Eventually.of_forall (fun ω => tendsto_sub_nhds_zero_iff.mpr (hQlimit ω t))
+      rw [← hlaw0] at ho
+      filter_upwards [(ae_map_iff hrm.aemeasurable (hset t)).1 ho] with ω hω
+      exact tendsto_sub_nhds_zero_iff.mp hω
+    have hQRall : ∀ᵐ ω ∂P, ∀ j : ℕ,
+        Tendsto (fun n => Q (denseSeq ℝ≥0 j) n (denseSeq ℝ≥0 j) (rp ω)) atTop
+          (𝓝 (Φ (rp ω) (denseSeq ℝ≥0 j))) := ae_all_iff.mpr (fun j => hQR _)
+    filter_upwards [hQRall] with ω hω
+    have hcoc (t : ℝ≥0) (n : ℕ) :
+        Q t n t (rp ω) = G ω (τ ω)
+          (F ω (τ ω + t) ((Real.sqrt κ * B (τ ω + t) ω : ℝ) + (a n : ℂ) * I)) -
+            (Real.sqrt κ * B (τ ω) ω : ℝ) := by
+      have he := (hw ω).source_shifted_forward_approximation (τ ω) t
+        ((Real.sqrt κ * (rp ω) t : ℝ) + (a n : ℂ) * I) (ha n) (by simp)
+        (fun s => Q t n s (rp ω)) (hQ0 t n (rp ω))
+        (fun s hs => by
+          have hh := (hQd t n (rp ω) s hs.1).mono
+            (show Icc (0 : ℝ) t ⊆ Ici 0 from Icc_subset_Ici_self)
+          simpa only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk,
+            mul_sub] using hh)
+      convert he using 2
+      simp only [rp, sourceRestartDriverPath, brownianRestart, ContinuousMap.coe_mk]
+      push_cast
+      ring_nf
+    have hdense (j : ℕ)
+        (hj : γ ω (τ ω + denseSeq ℝ≥0 j) ∈ survivingDomain (γ ω) (τ ω)) :
+        Φ (rp ω) (denseSeq ℝ≥0 j) =
+          G ω (τ ω) (γ ω (τ ω + denseSeq ℝ≥0 j)) - (Real.sqrt κ * B (τ ω) ω : ℝ) := by
+      let t := denseSeq ℝ≥0 j
+      have hGc : ContinuousAt (G ω (τ ω)) (γ ω (τ ω + t)) :=
+        ((hw ω).2.2.2.1 (τ ω)).continuousOn.continuousAt
+          ((isOpen_survivingDomain (hw ω).1 (τ ω)).mem_nhds hj)
+      have hlimγ := ((hw ω).2.2.2.2.2.2.2.2.2.2.2.2 (τ ω + t)).comp hy
+      have hlimG := (hGc.tendsto.comp hlimγ).sub_const ((Real.sqrt κ * B (τ ω) ω : ℝ) : ℂ)
+      apply tendsto_nhds_unique (hω j)
+      simpa only [hcoc, Function.comp_def] using hlimG
+    let S : Set ℝ≥0 := {t | γ ω (τ ω + t) ∈ survivingDomain (γ ω) (τ ω)}
+    have hγc : Continuous (fun t : ℝ≥0 => γ ω (τ ω + t)) :=
+      (hw ω).1.comp (continuous_const.add continuous_id)
+    have hSo : IsOpen S := (isOpen_survivingDomain (hw ω).1 (τ ω)).preimage hγc
+    have hleft : ContinuousOn (fun t : ℝ≥0 => Φ (rp ω) t) S :=
+      (Φ (rp ω)).continuous.continuousOn
+    have hright : ContinuousOn (fun t : ℝ≥0 =>
+        G ω (τ ω) (γ ω (τ ω + t)) - (Real.sqrt κ * B (τ ω) ω : ℝ)) S :=
+      (((hw ω).2.2.2.1 (τ ω)).continuousOn.comp hγc.continuousOn
+        (fun _ ht => ht)).sub continuousOn_const
+    have hEq : EqOn (fun t : ℝ≥0 => Φ (rp ω) t)
+        (fun t : ℝ≥0 => G ω (τ ω) (γ ω (τ ω + t)) - (Real.sqrt κ * B (τ ω) ω : ℝ))
+        (S ∩ range (denseSeq ℝ≥0)) := by
+      rintro t ⟨ht, j, rfl⟩
+      exact hdense j ht
+    exact hEq.of_subset_closure hleft hright inter_subset_left
+      ((denseRange_denseSeq ℝ≥0).open_subset_closure_inter hSo)
+
+end
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma source_parameter_joint_law
+    {Ω Z E : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace E]
+    {P : Measure Ω} [IsProbabilityMeasure P] {ν : Measure E} [IsProbabilityMeasure ν]
+    (hm : m ≤ mΩ)
+    {X : Ω → E} (hX : Measurable X) {Y : Ω → Z} (hY : Measurable[m] Y)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → (P.restrict A).map X = P A • ν)
+    (A : Set Ω) (hA : MeasurableSet[m] A) :
+    (P.restrict A).map (fun ω => (Y ω, X ω)) = ((P.restrict A).map Y).prod ν := by
+  have hYa : Measurable Y := hY.mono hm le_rfl
+  apply Measure.ext_prod
+  intro S T hS hT
+  have he := congrArg (fun μ : Measure E => μ T)
+    (hlaw (A ∩ Y ⁻¹' S) (hA.inter (hY hS)))
+  rw [Measure.map_apply hX hT, Measure.smul_apply, smul_eq_mul,
+    Measure.restrict_apply (hX hT)] at he
+  rw [Measure.map_apply (hYa.prodMk hX) (hS.prod hT),
+    Measure.restrict_apply ((hYa.prodMk hX) (hS.prod hT)),
+    Measure.prod_prod, Measure.map_apply hYa hS, Measure.restrict_apply (hYa hS)]
+  change P ((Y ⁻¹' S ∩ X ⁻¹' T) ∩ A) = P (Y ⁻¹' S ∩ A) * ν T
+  simpa only [inter_assoc, inter_left_comm, inter_comm] using he
+
+lemma source_parameter_condExp
+    {Ω Z E : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace E]
+    {P : Measure Ω} [IsProbabilityMeasure P] {ν : Measure E} [IsProbabilityMeasure ν]
+    (hm : m ≤ mΩ)
+    {X : Ω → E} (hX : Measurable X) {Y : Ω → Z} (hY : Measurable[m] Y)
+    (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → (P.restrict A).map X = P A • ν)
+    (H : Z × E → ℝ) (hH : Measurable H) (hb : ∀ p, ‖H p‖ ≤ 1) :
+    P[(fun ω => H (Y ω, X ω)) | m] =ᵐ[P] (fun ω => ∫ e, H (Y ω, e) ∂ν) := by
+  have hYa : Measurable Y := hY.mono hm le_rfl
+  have hK : StronglyMeasurable (fun y => ∫ e, H (y, e) ∂ν) :=
+    hH.stronglyMeasurable.integral_prod_right'
+  have hKb (y : Z) : ‖∫ e, H (y, e) ∂ν‖ ≤ 1 := by
+    simpa using norm_integral_le_of_norm_le_const (μ := ν) (Eventually.of_forall (fun e => hb (y,e)))
+  have hint : Integrable (fun ω => H (Y ω, X ω)) P :=
+    (integrable_const (1 : ℝ)).mono' (hH.comp (hYa.prodMk hX)).aestronglyMeasurable
+      (Eventually.of_forall (fun ω => hb (Y ω, X ω)))
+  have hKint : Integrable (fun ω => ∫ e, H (Y ω, e) ∂ν) P :=
+    (integrable_const (1 : ℝ)).mono' (hK.comp_measurable hYa).aestronglyMeasurable
+      (Eventually.of_forall (fun ω => hKb (Y ω)))
+  symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq hm hint
+    (fun _ _ _ => hKint.integrableOn) _ (hK.comp_measurable hY).aestronglyMeasurable
+  intro A hA _
+  have hprodint : Integrable H (((P.restrict A).map Y).prod ν) :=
+    (integrable_const (1 : ℝ)).mono' hH.aestronglyMeasurable (Eventually.of_forall hb)
+  calc
+    (∫ ω in A, ∫ e, H (Y ω, e) ∂ν ∂P) =
+        ∫ y, ∫ e, H (y, e) ∂ν ∂(P.restrict A).map Y :=
+      (integral_map hYa.aemeasurable hK.aestronglyMeasurable).symm
+    _ = ∫ p, H p ∂(((P.restrict A).map Y).prod ν) := (integral_prod H hprodint).symm
+    _ = ∫ p, H p ∂(P.restrict A).map (fun ω => (Y ω, X ω)) := by
+      rw [source_parameter_joint_law hm hX hY hlaw A hA]
+    _ = ∫ ω in A, H (Y ω, X ω) ∂P :=
+      integral_map (hYa.prodMk hX).aemeasurable hH.aestronglyMeasurable
+
+end SLEExactGauge
+
+end
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+ 
+
+theorem source_killedChart_progressive
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)) (z : ℂ) :
+    IsStronglyProgressive (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun t ω => if z ∈ survivingDomain (γ ω) t then G ω t z else 0) := by
+  classical
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable hc hm
+  have hnn (n : ℕ) : -(n : ℝ) ≤ n := by linarith [Nat.cast_nonneg (α := ℝ) n]
+  let c (n : ℕ) := (exists_clippedLogSolution B hm hBm hc κ (-(n : ℝ)) n (hnn n)
+    (initialLogCoordinates z)).some
+  let v (n : ℕ) (t : ℝ≥0) (ω : Ω) : Prop :=
+    0 < z.im ∧ Real.log z.im ≤ (n : ℝ) ∧ -(n : ℝ) ≤ ((c n).Q t ω).1.im
+  let A (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℂ :=
+    if v n t ω then
+      (((c n).Q t ω).1.re : ℂ) + (Real.exp ((c n).Q t ω).1.im : ℂ) * I
+    else 0
+  have hQp (n : ℕ) : IsStronglyProgressive f (fun t : ℝ≥0 => (c n).Q t) :=
+    (show StronglyAdapted f (fun t : ℝ≥0 => (c n).Q t) from
+      fun t => ((c n).adapted t).stronglyMeasurable).isStronglyProgressive_of_continuous
+      (fun ω => ((c n).lipschitz ω).continuous.comp NNReal.continuous_coe)
+  have hAp (n : ℕ) : IsStronglyProgressive f (A n) := by
+    intro T
+    have hQ := (hQp n T).measurable
+    let : MeasurableSpace Ω := f T
+    have hv : MeasurableSet {p : Iic T × Ω | v n p.1 p.2} :=
+      (MeasurableSet.const _).inter ((MeasurableSet.const _).inter
+        (measurableSet_le measurable_const (Complex.measurable_im.comp hQ.fst)))
+    exact (Measurable.ite hv
+      ((Complex.measurable_ofReal.comp (Complex.measurable_re.comp hQ.fst)).add
+        ((Complex.measurable_ofReal.comp (Real.continuous_exp.measurable.comp
+          (Complex.measurable_im.comp hQ.fst))).mul_const I)) measurable_const).stronglyMeasurable
+  have hlim (t : ℝ≥0) (ω : Ω) : Tendsto (fun n => A n t ω) atTop
+      (𝓝 (if z ∈ survivingDomain (γ ω) t then G ω t z else 0)) := by
+    apply tendsto_nhds_of_eventually_eq
+    have hU : Continuous (fun s => Real.sqrt κ * B s ω) := (hc ω).const_mul _
+    have hd (n : ℕ) (s : ℝ) (hs : s ∈ Icc 0 (t : ℝ)) :=
+      ((c n).derivative ω s hs.1).mono (show Icc (0 : ℝ) (t : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self)
+    by_cases hz : z ∈ survivingDomain (γ ω) t
+    · obtain ⟨N, hN⟩ := exists_nat_ge (max (-Real.log (G ω t z).im) (Real.log z.im))
+      filter_upwards [eventually_ge_atTop N] with n hn
+      have hNn : (N : ℝ) ≤ n := by exact_mod_cast hn
+      have hlo : -(n : ℝ) ≤ Real.log (G ω t z).im := by
+        have hh := (le_max_left (-Real.log (G ω t z).im) (Real.log z.im)).trans (hN.trans hNn)
+        linarith
+      have hhi : Real.log z.im ≤ (n : ℝ) := (le_max_right _ _).trans (hN.trans hNn)
+      have he : (c n).Q t ω = logLoewnerCoordinates (G ω) z t :=
+        clipped_log_loewner_agrees hU (hw ω) t z hz _ _ _ hlo hhi
+          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) ⟨t.coe_nonneg, le_rfl⟩
+      have hv : v n t ω := ⟨hz.1.1, hhi, by simpa only [he, logCoordinates_im] using hlo⟩
+      simp only [A, ite_eq_left hz, ite_eq_left hv, he, logCoordinates_re, logCoordinates_im,
+        Real.exp_log ((hw ω).imag_positive t z hz)]
+      exact Complex.re_add_im _
+    · filter_upwards [] with n
+      have hv : ¬ v n t ω := by
+        intro hv
+        exact hz (clipped_log_identification_of_endpoint hU (hw ω) t z hv.1 _ _ _
+          (fun s => (c n).Q s ω) ((c n).initial ω) (hd n) hv.2.2 hv.2.1).1
+      simp only [A, ite_eq_right hv, ite_eq_right hz]
+  exact isStronglyProgressive_of_tendsto hAp
+    (tendsto_pi_nhds.mpr (fun t => tendsto_pi_nhds.mpr (hlim t)))
+
+end
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StoppedPower.lean b/lean/OAI/Probability/SLE/StoppedPower.lean
index 79b15e2af19f63a67420b292a84165f2e3d64819..f8f6c248378f7508580a4bf55e025aca39f6b0d5 100644
--- a/lean/OAI/Probability/SLE/StoppedPower.lean
+++ b/lean/OAI/Probability/SLE/StoppedPower.lean
@@ -1,847 +1,847 @@
-import Mathlib
-import OAI.Probability.SLE.SwallowLyapunov
-
-namespace OAI
-
-/-! Stopped Power for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-noncomputable def powerLogGenerator (κ p α : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  powerLogDensity κ p α q x *
-    ((-4 * α + κ * α * (α + 1) / 2) * (q.1.re - Real.sqrt κ * x) ^ 2 +
-      (4 * p - κ * α / 2) * Real.exp (2 * q.1.im)) /
-    (greenLogNormSq κ q x) ^ 2
-
-lemma continuous_powerLogGenerator (κ p α : ℝ) :
-    Continuous (fun q : (ℂ × ℝ) × ℝ => powerLogGenerator κ p α q.1 q.2) := by
-  have hD := (powerLogDensity_contDiff κ p α).continuous
-  have hN : Continuous (fun q : (ℂ × ℝ) × ℝ => greenLogNormSq κ q.1 q.2) := by
-    unfold greenLogNormSq
-    fun_prop
-  unfold powerLogGenerator
-  apply Continuous.div (hD.mul (by fun_prop)) (hN.pow 2)
-  intro q
-  exact pow_ne_zero 2 (greenLogNormSq_pos κ q.1 q.2).ne'
-
-lemma measurable_powerLogGenerator_comp {X : Type*} [MeasurableSpace X]
-    (κ p α : ℝ) (q : X → ℂ × ℝ) (b : X → ℝ)
-    (hq : Measurable q) (hb : Measurable b) :
-    Measurable (fun x => powerLogGenerator κ p α (q x) (b x)) := by
-  have hh := (continuous_powerLogGenerator κ p α).measurable.comp (hq.prodMk hb)
-  dsimp only [Function.comp_def] at hh
-  with_reducible exact hh
-
-lemma compact_power_drift
-    (κ p α : ℝ) (hκ : 0 ≤ κ) (lo hi : ℝ) (hle : lo ≤ hi)
-    (φ : (ℂ × ℝ) × ℝ → ℝ) (q : ℂ × ℝ) (x : ℝ)
-    (hφD : fderiv ℝ φ (q, x) =
-      fderiv ℝ (fun r : (ℂ × ℝ) × ℝ => powerLogDensity κ p α r.1 r.2) (q, x))
-    (hφ2 : iteratedDeriv 2 (fun y => φ (q, y)) x =
-      iteratedDeriv 2 (powerLogDensity κ p α q) x)
-    (hq : q.1.im ∈ Icc lo hi) :
-    fderiv ℝ φ (q, x)
-        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * x) q, 0) +
-      (1 / 2 : ℝ) * iteratedDeriv 2 (fun y => φ (q, y)) x =
-        powerLogGenerator κ p α q x := by
-  rw [hφD, hφ2, powerLogDensity_pair_directional,
-    clippedLogLoewnerField_eq_physical lo hi hle _ _ hq]
-  exact power_physical_generator κ p α hκ q x
-
- 
-
-theorem ClippedLogSolution.stopped_power_dynkin
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 ≤ κ) (p α : ℝ)
-    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
-    Integrable (fun ω => powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
-      (∫ ω, powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
-        powerLogDensity κ p α q₀ 0 +
-        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ),
-          powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_power_field κ p α R hR
-  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) :=
-    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
-  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
-    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
-      measurable_snd)).const_mul _
-  have hpair : Measurable (fun p : ℝ × Ω =>
-      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
-  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
-    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
-  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
-    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
-  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
-    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
-    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
-  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
-  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
-  have hgen (ω : Ω) (t : ℝ) (ht : t ∈ Icc 0 (τ ω : ℝ)) :
-      brownianFieldDrift B F a t ω =
-        powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω) := by
-    have ht' : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    obtain ⟨hball, hlo, hhi⟩ := hinside ω t.toNNReal ht'
-    have he : (t.toNNReal : ℝ) = t := Real.coe_toNNReal t ht.1
-    change fderiv ℝ φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
-      (V (t.toNNReal : ℝ) ω, 0) + _ = _
-    dsimp only [V]
-    rw [Real.toNNReal_coe]
-    have hh := compact_power_drift κ p α hκ lo hi hle φ _ _
-      (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
-    simpa only [F, he] using hh
-  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
-  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
-    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
-  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
-    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
-  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
-    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
-    exact Eventually.of_forall fun ω => by
-      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
-        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
-  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
-      powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω) :=
-    hφeq _ _ (hinside ω (τ ω) le_rfl).1
-  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
-  calc
-    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
-      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
-    _ = (∫ ω, F 0 ω (B 0 ω) ∂P) +
-        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a t ω) ∂P := hopt
-    _ = _ := by
-      congr 1
-      · calc
-          _ = ∫ _, powerLogDensity κ p α q₀ 0 ∂P := by
-            apply integral_congr_ae
-            filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
-            change φ (c.Q 0 ω, B 0 ω) = _
-            have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
-              simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
-            rw [hφeq _ _ hh, c.initial ω, hB0]
-          _ = _ := by simp
-      · apply integral_congr_ae
-        exact Eventually.of_forall fun ω => intervalIntegral.integral_congr
-          (fun t ht => hgen ω t (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using ht))
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma stopping_paste_after
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
-    {σ τ : Ω → ℝ≥0}
-    (hσ : IsStoppingTime f (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hle : ∀ ω, σ ω ≤ τ ω) {A : Set Ω}
-    (hA : MeasurableSet[hσ.measurableSpace] A) :
-    IsStoppingTime f (fun ω => ((if ω ∈ A then τ ω else σ ω : ℝ≥0) : WithTop ℝ≥0)) := by
-  classical
-  intro t
-  have he : {ω | ((if ω ∈ A then τ ω else σ ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
-      (A ∩ {ω | (σ ω : WithTop ℝ≥0) ≤ t} ∩ {ω | (τ ω : WithTop ℝ≥0) ≤ t}) ∪
-        (Aᶜ ∩ {ω | (σ ω : WithTop ℝ≥0) ≤ t}) := by
-    ext ω
-    simp only [mem_ofPred_eq, WithTop.coe_le_coe, mem_union, mem_inter_iff, mem_compl_iff]
-    split_ifs with h
-    · simp only [h, true_and, not_true_eq_false, false_and, or_false]
-      exact ⟨fun ht => ⟨(hle ω).trans ht, ht⟩, fun ht => ht.2⟩
-    · simp [h]
-  rw [he]
-  exact ((hA.2 t).inter (hτ t)).union ((hA.compl).2 t)
-
- 
-
-theorem ClippedLogSolution.stopped_green_bounded_clock
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (η₀ u : ℝ) (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T)
-    (hin : ∀ ω t, t ≤ σ ω → logRadiusClock η₀ (c.Q t ω) ≤ u ∧
-      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
-    Integrable (fun ω => greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) P ∧
-      (∫ ω, greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω) ∂P) =
-        greenLogDensity κ q₀ 0 := by
-  classical
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hBa : Adapted f B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  have hex (n : ℕ) : ∃ β : Ω → ℝ≥0,
-      IsStoppingTime f (fun ω => (β ω : WithTop ℝ≥0)) ∧
-      (∀ ω, β ω ≤ T) ∧
-      (∀ ω t, t ≤ β ω → |B t ω| ≤ (n : ℝ) + 1) ∧
-      (∀ ω, β ω < T → |B (β ω) ω| = (n : ℝ) + 1) := by
-    let D (t : ℝ≥0) (ω : Ω) := (n : ℝ) + 1 - |B t ω|
-    obtain ⟨β, hβ, hbβ, hβin, _, hβbd⟩ := first_margin_stopping f D
-      (fun t => by
-        let : MeasurableSpace Ω := f t
-        exact (hBa t).abs.const_sub ((n : ℝ) + 1))
-      (fun ω => continuous_const.sub (hBc ω).abs)
-      (fun ω => by simpa [D, hB0 ω] using (show 0 ≤ (n : ℝ) + 1 by positivity)) T
-    refine ⟨β, hβ, hbβ, ?_, ?_⟩
-    · intro ω t ht
-      exact sub_nonneg.mp (hβin ω t ht)
-    · intro ω ht
-      exact (sub_eq_zero.mp (hβbd ω ht)).symm
-  choose β hβ hbβ hβin hβbd using hex
-  let σn (n : ℕ) (ω : Ω) := min (σ ω) (β n ω)
-  have hsn (n : ℕ) : IsStoppingTime f (fun ω => (σn n ω : WithTop ℝ≥0)) := by
-    simpa only [σn, WithTop.coe_min] using hσ.min (hβ n)
-  have hsnb (n : ℕ) (ω : Ω) : σn n ω ≤ T := (min_le_left _ _).trans (hb ω)
-  have hsin (n : ℕ) (ω : Ω) (t : ℝ≥0) (ht : t ≤ σn n ω) :
-      logRadiusClock η₀ (c.Q t ω) ≤ u ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi :=
-    hin ω t (ht.trans (min_le_left _ _))
-  have hmean (n : ℕ) :
-      Integrable (fun ω => greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω)) P ∧
-      (∫ ω, greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω) ∂P) = greenLogDensity κ q₀ 0 := by
-    let R := max (‖q₀‖ + c.L * T) ((n : ℝ) + 1)
-    have hR : 0 ≤ R := (by positivity : 0 ≤ (n : ℝ) + 1).trans (le_max_right _ _)
-    apply c.stopped_green hB hBm hBc hκ hκ8 R hR (σn n) (hsn n) T (hsnb n)
-    intro ω t ht
-    refine ⟨?_, (hsin n ω t ht).2⟩
-    rw [Prod.norm_def, max_le_iff, Real.norm_eq_abs]
-    exact ⟨(c.norm_le ω t T (ht.trans (hsnb n ω))).trans (le_max_left _ _),
-      (hβin n ω t (ht.trans (min_le_right _ _))).trans (le_max_right _ _)⟩
-  have hstable (ω : Ω) : ∀ᶠ n : ℕ in atTop, σn n ω = σ ω := by
-    obtain ⟨N, hN⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
-    filter_upwards [eventually_ge_atTop N] with n hn
-    have hbn : β n ω = T := by
-      apply le_antisymm (hbβ n ω)
-      by_contra h
-      have hlt : β n ω < T := lt_of_not_ge h
-      have hh := hN (β n ω) (hbβ n ω)
-      rw [hβbd n ω hlt] at hh
-      have hcast : (N : ℝ) ≤ n := by exact_mod_cast hn
-      linarith
-    dsimp only [σn]
-    rw [hbn, min_eq_left (hb ω)]
-  let K := Real.exp (codimension κ * (u - Real.log 2 - η₀))
-  have hbound (n : ℕ) (ω : Ω) :
-      ‖greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω)‖ ≤ K := by
-    rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
-    exact greenLogDensity_le_radius κ η₀ u hκ hκ8 _ _ (hsin n ω _ le_rfl).1
-  have hlim (ω : Ω) : Tendsto
-      (fun n => greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω)) atTop
-      (𝓝 (greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω))) :=
-    tendsto_const_nhds.congr' ((hstable ω).mono fun n hn => by dsimp only; rw [hn])
-  have hσm := stopping_time_measurable_nnreal hσ
-  have hQm := c.measurable.comp ((measurable_coe_nnreal_real.comp hσm).prodMk measurable_id)
-  have hbm := hBm.comp (hσm.prodMk measurable_id)
-  have hmeas : Measurable (fun ω => greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) :=
-    (greenLogDensity_contDiff κ).continuous.measurable.comp (hQm.prodMk hbm)
-  have hi : Integrable (fun ω => greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) P := by
-    apply Integrable.of_bound hmeas.aestronglyMeasurable K
-    apply Eventually.of_forall
-    intro ω
-    rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
-    exact greenLogDensity_le_radius κ η₀ u hκ hκ8 _ _ (hin ω _ le_rfl).1
-  refine ⟨hi, ?_⟩
-  apply tendsto_nhds_unique (tendsto_integral_of_dominated_convergence (fun _ => K)
-    (fun n => (hmean n).1.aestronglyMeasurable) (integrable_const K)
-    (fun n => Eventually.of_forall (hbound n)) (Eventually.of_forall hlim))
-  simpa only [(hmean _).2] using
-    (tendsto_const_nhds : Tendsto (fun _ : ℕ => greenLogDensity κ q₀ 0) atTop (𝓝 _))
-
- 
-
-theorem ClippedLogSolution.stopped_green_history
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 < κ) (hκ8 : κ < 8)
-    (η₀ u : ℝ) (σ τ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hστ : ∀ ω, σ ω ≤ τ ω)
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hin : ∀ ω t, t ≤ τ ω → logRadiusClock η₀ (c.Q t ω) ≤ u ∧
-      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi)
-    (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) :
-    (∫ ω in A, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
-      ∫ ω in A, greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω) ∂P := by
-  let ρ (ω : Ω) : ℝ≥0 := if ω ∈ A then τ ω else σ ω
-  have hρ := stopping_paste_after _ hσ hτ hστ hA
-  have hρτ (ω : Ω) : ρ ω ≤ τ ω := by dsimp [ρ]; split_ifs <;> simp_all
-  obtain ⟨hiti, hitm⟩ := c.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
-    η₀ u τ hτ T hb hin
-  obtain ⟨hisi, hism⟩ := c.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
-    η₀ u σ hσ T (fun ω => (hστ ω).trans (hb ω))
-    (fun ω t ht => hin ω t (ht.trans (hστ ω)))
-  obtain ⟨hipi, hipm⟩ := c.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
-    η₀ u ρ hρ T (fun ω => (hρτ ω).trans (hb ω))
-    (fun ω t ht => hin ω t (ht.trans (hρτ ω)))
-  have hAm : MeasurableSet A := hσ.measurableSpace_le A hA
-  have hex (ω : Ω) :
-      greenLogDensity κ (c.Q (ρ ω) ω) (B (ρ ω) ω) -
-        greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω) =
-      A.indicator (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) -
-        greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) ω := by
-    by_cases hω : ω ∈ A <;> simp [ρ, hω]
-  have he : (∫ ω, A.indicator (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) -
-        greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) ω ∂P) = 0 := by
-    simp_rw [← hex]
-    rw [integral_sub hipi hisi, hipm, hism, sub_self]
-  rw [integral_indicator hAm, integral_sub hiti.integrableOn hisi.integrableOn] at he
-  exact sub_eq_zero.mp he
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma exists_brownian_compact_stops
-    {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T) :
-    ∃ σn : ℕ → Ω → ℝ≥0,
-      (∀ n, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-        (fun ω => (σn n ω : WithTop ℝ≥0))) ∧
-      (∀ n ω, σn n ω ≤ σ ω) ∧
-      (∀ n ω t, t ≤ σn n ω → |B t ω| ≤ (n : ℝ) + 1) ∧
-      (∀ ω, ∀ᶠ n : ℕ in atTop, σn n ω = σ ω) := by
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  have hBa : Adapted f B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  have hex (n : ℕ) : ∃ β : Ω → ℝ≥0,
-      IsStoppingTime f (fun ω => (β ω : WithTop ℝ≥0)) ∧
-      (∀ ω, β ω ≤ T) ∧
-      (∀ ω t, t ≤ β ω → |B t ω| ≤ (n : ℝ) + 1) ∧
-      (∀ ω, β ω < T → |B (β ω) ω| = (n : ℝ) + 1) := by
-    let D (t : ℝ≥0) (ω : Ω) := (n : ℝ) + 1 - |B t ω|
-    obtain ⟨β, hβ, hbβ, hβin, _, hβbd⟩ := first_margin_stopping f D
-      (fun t => by
-        let : MeasurableSpace Ω := f t
-        exact (hBa t).abs.const_sub ((n : ℝ) + 1))
-      (fun ω => continuous_const.sub (hBc ω).abs)
-      (fun ω => by simpa [D, hB0 ω] using (show 0 ≤ (n : ℝ) + 1 by positivity)) T
-    refine ⟨β, hβ, hbβ, ?_, ?_⟩
-    · intro ω t ht
-      exact sub_nonneg.mp (hβin ω t ht)
-    · intro ω ht
-      exact (sub_eq_zero.mp (hβbd ω ht)).symm
-  choose β hβ hbβ hβin hβbd using hex
-  let σn (n : ℕ) (ω : Ω) := min (σ ω) (β n ω)
-  refine ⟨σn, ?_, fun n ω => min_le_left _ _, ?_, ?_⟩
-  · intro n
-    simpa only [σn, WithTop.coe_min] using hσ.min (hβ n)
-  · intro n ω t ht
-    exact hβin n ω t (ht.trans (min_le_right _ _))
-  · intro ω
-    obtain ⟨N, hN⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
-    filter_upwards [eventually_ge_atTop N] with n hn
-    have hbn : β n ω = T := by
-      apply le_antisymm (hbβ n ω)
-      by_contra h
-      have hlt : β n ω < T := lt_of_not_ge h
-      have hh := hN (β n ω) (hbβ n ω)
-      rw [hβbd n ω hlt] at hh
-      have hcast : (N : ℝ) ≤ n := by exact_mod_cast hn
-      linarith
-    dsimp only [σn]
-    rw [hbn, min_eq_left (hb ω)]
-
-lemma bounded_stopped_cumulative_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (G : ℝ → Ω → ℝ) (hGm : Measurable (Function.uncurry G))
-    (σ τ : Ω → ℝ≥0) (hτm : Measurable τ) (hle : ∀ ω, τ ω ≤ σ ω)
-    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T) (C : ℝ) (hC : 0 ≤ C)
-    (hG : ∀ ω t, 0 ≤ t → t ≤ (σ ω : ℝ) → |G t ω| ≤ C) :
-    Integrable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω) P ∧
-      ∀ ω, |∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω| ≤ C * T := by
-  have hm : Measurable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω) :=
-    (cumulative_integral_measurable G hGm).comp (hτm.prodMk measurable_id)
-  have hbnd (ω : Ω) : |∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω| ≤ C * T := by
-    have hh := intervalIntegral.norm_integral_le_of_norm_le_const
-      (a := (0 : ℝ)) (b := (τ ω : ℝ)) (C := C) (f := fun t => G t ω) (fun t ht => by
-        have htt : t ∈ Ioc (0 : ℝ) (τ ω : ℝ) := by simpa only [uIoc_of_le (τ ω).coe_nonneg] using ht
-        simpa only [Real.norm_eq_abs] using hG ω t htt.1.le (htt.2.trans (by exact_mod_cast hle ω)))
-    simp only [Real.norm_eq_abs, sub_zero, abs_of_nonneg (τ ω).coe_nonneg] at hh
-    exact hh.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast (hle ω).trans (hb ω)) hC)
-  exact ⟨Integrable.of_bound hm.aestronglyMeasurable _
-    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hbnd ω), hbnd⟩
-
-lemma integral_tendsto_stopped_eval
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (F : ℝ≥0 → Ω → ℝ) (σn : ℕ → Ω → ℝ≥0) (σ : Ω → ℝ≥0)
-    (hm : ∀ n, AEStronglyMeasurable (fun ω => F (σn n ω) ω) P)
-    (K : ℝ) (hb : ∀ n ω, |F (σn n ω) ω| ≤ K)
-    (hstable : ∀ ω, ∀ᶠ n : ℕ in atTop, σn n ω = σ ω) :
-    Tendsto (fun n => ∫ ω, F (σn n ω) ω ∂P) atTop (𝓝 (∫ ω, F (σ ω) ω ∂P)) := by
-  apply tendsto_integral_of_dominated_convergence (fun _ => K) hm (integrable_const K)
-    (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hb n ω)
-  apply Eventually.of_forall
-  intro ω
-  exact tendsto_const_nhds.congr' ((hstable ω).mono fun n hn => by dsimp only; rw [hn])
-
-lemma integral_identity_of_stable_stops
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    (F H : ℝ≥0 → Ω → ℝ) (σn : ℕ → Ω → ℝ≥0) (σ : Ω → ℝ≥0)
-    (hstable : ∀ ω, ∀ᶠ n : ℕ in atTop, σn n ω = σ ω)
-    (A K C : ℝ)
-    (hmean : ∀ n, Integrable (fun ω => F (σn n ω) ω) P ∧
-      (∫ ω, F (σn n ω) ω ∂P) = A + ∫ ω, H (σn n ω) ω ∂P)
-    (hFi : Integrable (fun ω => F (σ ω) ω) P)
-    (hHim : AEStronglyMeasurable (fun ω => H (σ ω) ω) P)
-    (hHnm : ∀ n, AEStronglyMeasurable (fun ω => H (σn n ω) ω) P)
-    (hFb : ∀ n ω, |F (σn n ω) ω| ≤ K)
-    (hHb : ∀ n ω, |H (σn n ω) ω| ≤ C)
-    (hHbound : ∀ ω, |H (σ ω) ω| ≤ C) :
-    Integrable (fun ω => F (σ ω) ω) P ∧
-    Integrable (fun ω => H (σ ω) ω) P ∧
-      (∫ ω, F (σ ω) ω ∂P) = A + ∫ ω, H (σ ω) ω ∂P := by
-  refine ⟨hFi, Integrable.of_bound hHim C
-    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHbound ω), ?_⟩
-  have hlimF := integral_tendsto_stopped_eval F σn σ
-    (fun n => (hmean n).1.aestronglyMeasurable) K hFb hstable
-  have hlimH := integral_tendsto_stopped_eval H σn σ hHnm C hHb hstable
-  have he := hlimH.const_add A
-  simp_rw [← (hmean _).2] at he
-  exact tendsto_nhds_unique hlimF he
-
-lemma ClippedLogSolution.stopped_power_dynkin_compact_cap
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 ≤ κ) (p α : ℝ) (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T)
-    (hin : ∀ ω t, t ≤ σ ω → (c.Q t ω).1.im ∈ Icc lo hi)
-    (N : ℝ) (hN : 0 ≤ N) (hcap : ∀ ω t, t ≤ σ ω → |B t ω| ≤ N) :
-    Integrable (fun ω => powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)) P ∧
-      (∫ ω, powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω) ∂P) =
-        powerLogDensity κ p α q₀ 0 + ∫ ω, (∫ t in (0 : ℝ)..(σ ω : ℝ),
-          powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
-  let R := max (‖q₀‖ + c.L * T) N
-  have hR : 0 ≤ R := hN.trans (le_max_right _ _)
-  apply c.stopped_power_dynkin hB hBm hBc hκ p α R hR σ hσ T hb
-  intro ω t ht
-  refine ⟨?_, hin ω t ht⟩
-  rw [Prod.norm_def, max_le_iff, Real.norm_eq_abs]
-  exact ⟨(c.norm_le ω t T (ht.trans (hb ω))).trans (le_max_left _ _),
-    (hcap ω t ht).trans (le_max_right _ _)⟩
-
-lemma ClippedLogSolution.stopped_power_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (p α : ℝ) (σ : Ω → ℝ≥0) (hσm : Measurable σ) (K : ℝ)
-    (hval : ∀ ω, |powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)| ≤ K) :
-    Integrable (fun ω => powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)) P := by
-  have hQm := c.measurable.comp ((measurable_coe_nnreal_real.comp hσm).prodMk measurable_id)
-  have hbm := hBm.comp (hσm.prodMk measurable_id)
-  have hWm := (powerLogDensity_contDiff κ p α).continuous.measurable.comp (hQm.prodMk hbm)
-  exact Integrable.of_bound hWm.aestronglyMeasurable K
-    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hval ω)
-
-lemma ClippedLogSolution.power_generator_measurable
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (p α : ℝ) :
-    Measurable (fun q : ℝ × Ω => powerLogGenerator κ p α (c.Q q.1 q.2) (B q.1.toNNReal q.2)) := by
-  have ht : Measurable (fun q : ℝ × Ω => q.1.toNNReal) :=
-    continuous_real_toNNReal.measurable.comp measurable_fst
-  have hb : Measurable (fun q : ℝ × Ω => B q.1.toNNReal q.2) :=
-    hBm.comp (ht.prodMk measurable_snd)
-  exact measurable_powerLogGenerator_comp κ p α
-    (fun q : ℝ × Ω => c.Q q.1 q.2) (fun q : ℝ × Ω => B q.1.toNNReal q.2)
-    c.measurable hb
-
-lemma ClippedLogSolution.stopped_power_cumulative_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (p α : ℝ) (σ τ : Ω → ℝ≥0) (hτm : Measurable τ) (hleτ : ∀ ω, τ ω ≤ σ ω)
-    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T) (C : ℝ) (hC : 0 ≤ C)
-    (hdrift : ∀ ω t, t ≤ σ ω → |powerLogGenerator κ p α (c.Q t ω) (B t ω)| ≤ C) :
-    Integrable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ),
-      powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) P ∧
-      (∀ ω, |∫ t in (0 : ℝ)..(τ ω : ℝ),
-        powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)| ≤ C * T) := by
-  let G (t : ℝ) (ω : Ω) := powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)
-  have hGm : Measurable (Function.uncurry G) := c.power_generator_measurable hBm p α
-  have hGb (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (htσ : t ≤ (σ ω : ℝ)) : |G t ω| ≤ C := by
-    have hh := hdrift ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr htσ)
-    simpa only [G, Real.coe_toNNReal t ht] using hh
-  exact bounded_stopped_cumulative_integrable G hGm σ τ hτm hleτ T hb C hC hGb
-
- 
-
-theorem ClippedLogSolution.stopped_power_dynkin_bounded
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 ≤ κ) (p α : ℝ) (σ : Ω → ℝ≥0)
-    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (σ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T)
-    (hin : ∀ ω t, t ≤ σ ω → (c.Q t ω).1.im ∈ Icc lo hi)
-    (K C : ℝ) (hC : 0 ≤ C)
-    (hval : ∀ ω t, t ≤ σ ω → |powerLogDensity κ p α (c.Q t ω) (B t ω)| ≤ K)
-    (hdrift : ∀ ω t, t ≤ σ ω → |powerLogGenerator κ p α (c.Q t ω) (B t ω)| ≤ C) :
-    Integrable (fun ω => powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)) P ∧
-    Integrable (fun ω => ∫ t in (0 : ℝ)..(σ ω : ℝ),
-      powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) P ∧
-      (∫ ω, powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω) ∂P) =
-        powerLogDensity κ p α q₀ 0 +
-        ∫ ω, (∫ t in (0 : ℝ)..(σ ω : ℝ),
-          powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨σn, hsn, hsle, hsB, hstable⟩ := exists_brownian_compact_stops B hm hBc hB0 σ hσ T hb
-  have hsnb (n : ℕ) (ω : Ω) : σn n ω ≤ T := (hsle n ω).trans (hb ω)
-  let W (τ : Ω → ℝ≥0) (ω : Ω) := powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω)
-  let G (t : ℝ) (ω : Ω) := powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)
-  let H (τ : Ω → ℝ≥0) (ω : Ω) := ∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω
-  have hmean (n : ℕ) : Integrable (W (σn n)) P ∧
-      (∫ ω, W (σn n) ω ∂P) = powerLogDensity κ p α q₀ 0 + ∫ ω, H (σn n) ω ∂P := by
-    exact c.stopped_power_dynkin_compact_cap hB hBm hBc hκ p α (σn n) (hsn n)
-      T (hsnb n) (fun ω t ht => hin ω t (ht.trans (hsle n ω))) ((n : ℝ) + 1)
-      (by positivity) (hsB n)
-  have hH (τ : Ω → ℝ≥0) (hτm : Measurable τ) (hτ : ∀ ω, τ ω ≤ σ ω) :
-      Integrable (H τ) P ∧ ∀ ω, |H τ ω| ≤ C * T :=
-    c.stopped_power_cumulative_integrable hBm p α σ τ hτm hτ T hb C hC hdrift
-  have hσm := stopping_time_measurable_nnreal hσ
-  have hWi : Integrable (W σ) P := c.stopped_power_integrable hBm p α σ hσm K
-    (fun ω => hval ω (σ ω) le_rfl)
-  exact integral_identity_of_stable_stops
-    (fun t ω => powerLogDensity κ p α (c.Q t ω) (B t ω))
-    (fun t ω => ∫ u in (0 : ℝ)..(t : ℝ), G u ω) σn σ hstable
-    (powerLogDensity κ p α q₀ 0) K (C * T) hmean hWi
-    (hH σ hσm (fun _ => le_rfl)).1.aestronglyMeasurable
-    (fun n => (hH (σn n) (stopping_time_measurable_nnreal (hsn n)) (hsle n)).1.aestronglyMeasurable)
-    (fun n ω => hval ω (σn n ω) (hsle n ω))
-    (fun n => (hH (σn n) (stopping_time_measurable_nnreal (hsn n)) (hsle n)).2)
-    (hH σ hσm (fun _ => le_rfl)).2
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-
-lemma ClippedLogSolution.swallow_generator_measurable
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (β C : ℝ) :
-    Measurable (fun p : ℝ × Ω => swallowGenerator κ β C (c.Q p.1 p.2) (B p.1.toNNReal p.2)) := by
-  have hBr : Measurable (fun p : ℝ × Ω => B p.1.toNNReal p.2) :=
-    hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
-  have hp : Measurable (fun p : ℝ × Ω => (c.Q p.1 p.2, B p.1.toNNReal p.2)) :=
-    c.measurable.prodMk hBr
-  have hh := (continuous_swallowGenerator κ β C).measurable.comp hp
-  dsimp only [Function.comp_def] at hh
-  with_reducible exact hh
-
-lemma ClippedLogSolution.swallow_generator_continuous
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (β C : ℝ) (ω : Ω) :
-    Continuous (fun t : ℝ => swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)) := by
-  have hB : Continuous (fun t : ℝ => B t.toNNReal ω) :=
-    (hBc ω).comp continuous_real_toNNReal
-  have hQ : Continuous (fun t : ℝ => c.Q t ω) := (c.lipschitz ω).continuous
-  have hp : Continuous (fun t : ℝ => (c.Q t ω, B t.toNNReal ω)) := hQ.prodMk hB
-  have hh := (continuous_swallowGenerator κ β C).comp hp
-  dsimp only [Function.comp_def] at hh
-  with_reducible exact hh
-
-lemma ClippedLogSolution.swallow_cumulative_integrable
-    {P : Measure Ω} [IsFiniteMeasure P]
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (β C : ℝ)
-    (τ : Ω → ℝ≥0) (hτm : Measurable τ) (T : ℝ≥0) (hτT : ∀ ω, τ ω ≤ T)
-    (D : ℝ) (hD : 0 ≤ D)
-    (hbound : ∀ ω t, t ≤ τ ω → |swallowGenerator κ β C (c.Q t ω) (B t ω)| ≤ D) :
-    Integrable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ),
-      swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)) P := by
-  let G (t : ℝ) (ω : Ω) := swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)
-  have hGm : Measurable (Function.uncurry G) := c.swallow_generator_measurable hBm β C
-  have hGb (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (hτ : t ≤ (τ ω : ℝ)) : |G t ω| ≤ D := by
-    have hh := hbound ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hτ)
-    simpa only [G, Real.coe_toNNReal t ht] using hh
-  exact (bounded_stopped_cumulative_integrable (P := P) G hGm τ τ hτm
-    (fun _ => le_rfl) T hτT D hD hGb).1
-
-lemma bounded_nnreal_integrable {P : Measure Ω} [IsFiniteMeasure P]
-    (τ : Ω → ℝ≥0) (hτm : Measurable τ) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
-    Integrable (fun ω => (τ ω : ℝ)) P := by
-  apply Integrable.of_bound ((NNReal.continuous_coe.measurable.comp hτm).aestronglyMeasurable) T
-  exact Eventually.of_forall fun ω => by
-    change ‖(τ ω : ℝ)‖ ≤ (T : ℝ)
-    rw [Real.norm_eq_abs, abs_of_nonneg (τ ω).coe_nonneg]
-    exact NNReal.coe_le_coe.mpr (hb ω)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace RadialLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
-
-lemma exists_compact_bound (a : RadialLocalization c R T) (hκ : 0 < κ) (hR : 0 ≤ R) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ ω t, t ≤ a.τ ω → ‖(c.Q t ω, B t ω)‖ ≤ K := by
-  let M : ℝ := ‖q₀‖+c.L*T
-  have hM : 0 ≤ M := by dsimp [M]; positivity
-  refine ⟨max M ((M+R)/Real.sqrt κ), le_max_of_le_left hM, ?_⟩
-  intro ω t ht
-  have hQ : ‖c.Q t ω‖ ≤ M := c.norm_le ω t T (ht.trans (a.bounded ω))
-  have hB := norm_driver_of_radial_bound κ R M hκ hR (c.Q t ω) (B t ω) hQ (a.inside ω t ht).1
-  rw [Prod.norm_def]
-  exact max_le_max hQ (by simpa only [Real.norm_eq_abs] using hB)
-
-lemma exists_generator_bound (a : RadialLocalization c R T)
-    (hκ : 0 < κ) (hR : 0 ≤ R) (β C : ℝ) :
-    ∃ D : ℝ, 0 ≤ D ∧ ∀ ω t, t ≤ a.τ ω →
-      |swallowGenerator κ β C (c.Q t ω) (B t ω)| ≤ D := by
-  obtain ⟨K,hK,hbound⟩ := a.exists_compact_bound hκ hR
-  obtain ⟨D,hD⟩ := (isCompact_closedBall (0 : (ℂ × ℝ) × ℝ) K).exists_bound_of_continuousOn
-    (continuous_swallowGenerator κ β C).continuousOn
-  refine ⟨max D 0, le_max_right _ _, ?_⟩
-  intro ω t ht
-  have hmem : (c.Q t ω, B t ω) ∈ Metric.closedBall (0 : (ℂ × ℝ) × ℝ) K := by
-    simpa only [Metric.mem_closedBall, dist_zero_right] using hbound ω t ht
-  exact (hD _ hmem).trans (le_max_left _ _)
-
-lemma generator_cumulative_integrable {P : Measure Ω} [IsFiniteMeasure P]
-    (a : RadialLocalization c R T)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hκ : 0 < κ) (hR : 0 ≤ R) (β C : ℝ) :
-    Integrable (fun ω => ∫ t in (0 : ℝ)..(a.τ ω : ℝ),
-      swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)) P := by
-  obtain ⟨D,hD,hbound⟩ := a.exists_generator_bound hκ hR β C
-  exact c.swallow_cumulative_integrable hBm β C a.τ
-    (stopping_time_measurable_nnreal a.stopping) T a.bounded D hD hbound
-
-lemma stopped_lyapunov_bound {P : Measure Ω}
-    (a : RadialLocalization c R T)
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 4 < κ) (hR : 0 < R) (hhi : q₀.1.im ≤ hi) :
-    Integrable (fun ω => swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
-      (c.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
-    (∫ ω, swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
-      (c.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) +
-      (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) *
-        (∫ ω, (a.τ ω : ℝ) ∂P) ≤ swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q₀ 0 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hk : 0 < κ := lt_trans (by norm_num) hκ
-  let β := (κ-4)/(2*κ)
-  let C := 3*(κ-4)/8
-  let d := β*((κ-4)/4)*Real.exp ((β-2)*Real.log R)
-  let G (t : ℝ) (ω : Ω) := swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)
-  obtain ⟨K,hK,hbound⟩ := a.exists_compact_bound hk hR.le
-  have hmean := c.stopped_swallow_dynkin hB hBm hBc hk.le β C K hK a.τ a.stopping
-    T a.bounded (fun ω t ht => ⟨hbound ω t ht, (a.inside ω t ht).2,
-      (a.height_le_initial ω t ht).trans hhi⟩)
-  refine ⟨hmean.1, ?_⟩
-  have hτi : Integrable (fun ω => (a.τ ω : ℝ)) P :=
-    bounded_nnreal_integrable a.τ (stopping_time_measurable_nnreal a.stopping) T a.bounded
-  have hGi : Integrable (fun ω => ∫ t in (0 : ℝ)..(a.τ ω : ℝ), G t ω) P :=
-    a.generator_cumulative_integrable hBm hk hR.le β C
-  have hGle (ω : Ω) : (∫ t in (0 : ℝ)..(a.τ ω : ℝ), G t ω) ≤ -d*(a.τ ω : ℝ) := by
-    have hGc : Continuous (fun t : ℝ => G t ω) := c.swallow_generator_continuous hBc β C ω
-    have hh := intervalIntegral.integral_mono_on (a.τ ω).coe_nonneg
-      (hGc.intervalIntegrable _ _) (continuous_const.intervalIntegrable _ _ :
-        IntervalIntegrable (fun _ : ℝ => -d) volume 0 (a.τ ω)) (fun t ht => by
-          have hh := swallowGenerator_uniform_negative κ hκ (c.Q t.toNNReal ω)
-            (B t.toNNReal ω) R hR
-            (a.inside ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)).1
-          simpa only [G, Real.coe_toNNReal t ht.1] using hh)
-    simpa only [intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_comm] using hh
-  have htotal := integral_mono hGi (hτi.const_mul (-d)) hGle
-  rw [integral_const_mul] at htotal
-  change (∫ ω, swallowLyapunov κ β C (c.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) +
-    d*(∫ ω, (a.τ ω : ℝ) ∂P) ≤ swallowLyapunov κ β C q₀ 0
-  rw [hmean.2]
-  linarith
-
-end RadialLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem compensated_exponential_limit
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (X E : ℕ → Ω → ℝ) (Y Q : Ω → ℝ) (θ V : ℝ)
-    (hm : ∀ n, AEMeasurable (fun ω => ENNReal.ofReal
-      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))) P)
-    (hX : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
-    (hE : ∀ᵐ ω ∂P, Tendsto (fun n => E n ω) atTop (𝓝 (Q ω)))
-    (hQ : ∀ᵐ ω ∂P, Q ω ≤ V)
-    (hexp : ∀ n, (∫⁻ ω, ENNReal.ofReal
-      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2)) ∂P) ≤ 1) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) ≤ 1 := by
-  let f (n : ℕ) (ω : Ω) := ENNReal.ofReal (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))
-  have hl : ∀ᵐ ω ∂P, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) =
-      liminf (fun n => f n ω) atTop := by
-    filter_upwards [hX, hE] with ω hx he
-    exact ((ENNReal.continuous_ofReal.tendsto _).comp
-      ((Real.continuous_exp.tendsto _).comp ((hx.const_mul θ).sub ((he.const_mul (θ^2)).div_const 2)))).liminf_eq.symm
-  calc
-    _ ≤ ∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [hQ] with ω hq
-      apply ENNReal.ofReal_le_ofReal
-      apply Real.exp_le_exp.mpr
-      nlinarith [mul_le_mul_of_nonneg_left hq (sq_nonneg θ)]
-    _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := lintegral_congr_ae hl
-    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' hm
-    _ ≤ 1 := liminf_le_of_frequently_le (Frequently.of_forall hexp)
-
- 
-theorem compensated_exponential_limit_energy
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (X E : ℕ → Ω → ℝ) (Y Q : Ω → ℝ) (θ : ℝ)
-    (hm : ∀ n, AEMeasurable (fun ω => ENNReal.ofReal
-      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))) P)
-    (hX : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
-    (hE : ∀ᵐ ω ∂P, Tendsto (fun n => E n ω) atTop (𝓝 (Q ω)))
-    (hexp : ∀ n, (∫⁻ ω, ENNReal.ofReal
-      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2)) ∂P) ≤ 1) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) ∂P) ≤ 1 := by
-  let f (n : ℕ) (ω : Ω) := ENNReal.ofReal (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))
-  have hl : ∀ᵐ ω ∂P, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) =
-      liminf (fun n => f n ω) atTop := by
-    filter_upwards [hX, hE] with ω hx he
-    exact ((ENNReal.continuous_ofReal.tendsto _).comp
-      ((Real.continuous_exp.tendsto _).comp ((hx.const_mul θ).sub
-        ((he.const_mul (θ^2)).div_const 2)))).liminf_eq.symm
-  calc
-    _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := lintegral_congr_ae hl
-    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' hm
-    _ ≤ 1 := liminf_le_of_frequently_le (Frequently.of_forall hexp)
-
- 
-theorem exponential_tail_of_compensated
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {Y : Ω → ℝ} (hY : AEMeasurable Y P) {θ : ℝ} (hθ : 0 ≤ θ) (V x : ℝ)
-    (h : (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) ≤ 1) :
-    P {ω | x ≤ Y ω} ≤ ENNReal.ofReal (Real.exp (-θ * x + θ ^ 2 * V / 2)) := by
-  let a := ENNReal.ofReal (Real.exp (θ * x - θ ^ 2 * V / 2))
-  have ha : a ≠ 0 := ne_of_gt (ENNReal.ofReal_pos.mpr (Real.exp_pos _))
-  have hab : a ≠ ∞ := ENNReal.ofReal_ne_top
-  have hm := (((hY.const_mul θ).sub_const (θ ^ 2 * V / 2)).exp).ennreal_ofReal
-  have hle := meas_ge_le_lintegral_div hm ha hab
-  have hs : {ω | x ≤ Y ω} ⊆ {ω | a ≤ ENNReal.ofReal
-      (Real.exp (θ * Y ω - θ ^ 2 * V / 2))} := by
-    intro ω hω
-    exact ENNReal.ofReal_le_ofReal (Real.exp_le_exp.mpr
-      (sub_le_sub_right (mul_le_mul_of_nonneg_left hω hθ) _))
-  calc
-    _ ≤ P {ω | a ≤ ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2))} := measure_mono hs
-    _ ≤ (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) / a := hle
-    _ ≤ 1 / a := ENNReal.div_le_div_right h a
-    _ = _ := by
-      rw [one_div, ← ENNReal.ofReal_inv_of_pos (Real.exp_pos _), ← Real.exp_neg]
-      congr 2
-      ring
-
- 
-theorem subGaussian_tail_of_compensated
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {Y : Ω → ℝ} (hY : AEMeasurable Y P) {V x : ℝ} (hV : 0 < V) (hx : 0 ≤ x)
-    (h : ∀ θ : ℝ, (∫⁻ ω, ENNReal.ofReal
-      (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) ≤ 1) :
-    P {ω | x ≤ Y ω} ≤ ENNReal.ofReal (Real.exp (-x ^ 2 / (2 * V))) := by
-  have ht := exponential_tail_of_compensated hY (div_nonneg hx hV.le) V x (h (x/V))
-  convert ht using 2
-  congr 2
-  field_simp
-  ring
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.SwallowLyapunov
+
+namespace OAI
+
+/-! Stopped Power for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+noncomputable def powerLogGenerator (κ p α : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  powerLogDensity κ p α q x *
+    ((-4 * α + κ * α * (α + 1) / 2) * (q.1.re - Real.sqrt κ * x) ^ 2 +
+      (4 * p - κ * α / 2) * Real.exp (2 * q.1.im)) /
+    (greenLogNormSq κ q x) ^ 2
+
+lemma continuous_powerLogGenerator (κ p α : ℝ) :
+    Continuous (fun q : (ℂ × ℝ) × ℝ => powerLogGenerator κ p α q.1 q.2) := by
+  have hD := (powerLogDensity_contDiff κ p α).continuous
+  have hN : Continuous (fun q : (ℂ × ℝ) × ℝ => greenLogNormSq κ q.1 q.2) := by
+    unfold greenLogNormSq
+    fun_prop
+  unfold powerLogGenerator
+  apply Continuous.div (hD.mul (by fun_prop)) (hN.pow 2)
+  intro q
+  exact pow_ne_zero 2 (greenLogNormSq_pos κ q.1 q.2).ne'
+
+lemma measurable_powerLogGenerator_comp {X : Type*} [MeasurableSpace X]
+    (κ p α : ℝ) (q : X → ℂ × ℝ) (b : X → ℝ)
+    (hq : Measurable q) (hb : Measurable b) :
+    Measurable (fun x => powerLogGenerator κ p α (q x) (b x)) := by
+  have hh := (continuous_powerLogGenerator κ p α).measurable.comp (hq.prodMk hb)
+  dsimp only [Function.comp_def] at hh
+  with_reducible exact hh
+
+lemma compact_power_drift
+    (κ p α : ℝ) (hκ : 0 ≤ κ) (lo hi : ℝ) (hle : lo ≤ hi)
+    (φ : (ℂ × ℝ) × ℝ → ℝ) (q : ℂ × ℝ) (x : ℝ)
+    (hφD : fderiv ℝ φ (q, x) =
+      fderiv ℝ (fun r : (ℂ × ℝ) × ℝ => powerLogDensity κ p α r.1 r.2) (q, x))
+    (hφ2 : iteratedDeriv 2 (fun y => φ (q, y)) x =
+      iteratedDeriv 2 (powerLogDensity κ p α q) x)
+    (hq : q.1.im ∈ Icc lo hi) :
+    fderiv ℝ φ (q, x)
+        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * x) q, 0) +
+      (1 / 2 : ℝ) * iteratedDeriv 2 (fun y => φ (q, y)) x =
+        powerLogGenerator κ p α q x := by
+  rw [hφD, hφ2, powerLogDensity_pair_directional,
+    clippedLogLoewnerField_eq_physical lo hi hle _ _ hq]
+  exact power_physical_generator κ p α hκ q x
+
+ 
+
+theorem ClippedLogSolution.stopped_power_dynkin
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 ≤ κ) (p α : ℝ)
+    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
+    Integrable (fun ω => powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
+      (∫ ω, powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
+        powerLogDensity κ p α q₀ 0 +
+        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ),
+          powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_power_field κ p α R hR
+  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) :=
+    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
+  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
+    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
+      measurable_snd)).const_mul _
+  have hpair : Measurable (fun p : ℝ × Ω =>
+      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
+  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
+    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
+  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
+    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
+  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
+    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
+    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
+  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
+  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
+  have hgen (ω : Ω) (t : ℝ) (ht : t ∈ Icc 0 (τ ω : ℝ)) :
+      brownianFieldDrift B F a t ω =
+        powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω) := by
+    have ht' : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    obtain ⟨hball, hlo, hhi⟩ := hinside ω t.toNNReal ht'
+    have he : (t.toNNReal : ℝ) = t := Real.coe_toNNReal t ht.1
+    change fderiv ℝ φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
+      (V (t.toNNReal : ℝ) ω, 0) + _ = _
+    dsimp only [V]
+    rw [Real.toNNReal_coe]
+    have hh := compact_power_drift κ p α hκ lo hi hle φ _ _
+      (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
+    simpa only [F, he] using hh
+  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
+  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
+    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
+  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
+    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
+  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
+    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
+    exact Eventually.of_forall fun ω => by
+      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
+        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
+  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
+      powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω) :=
+    hφeq _ _ (hinside ω (τ ω) le_rfl).1
+  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
+  calc
+    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
+      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
+    _ = (∫ ω, F 0 ω (B 0 ω) ∂P) +
+        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a t ω) ∂P := hopt
+    _ = _ := by
+      congr 1
+      · calc
+          _ = ∫ _, powerLogDensity κ p α q₀ 0 ∂P := by
+            apply integral_congr_ae
+            filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
+            change φ (c.Q 0 ω, B 0 ω) = _
+            have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
+              simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
+            rw [hφeq _ _ hh, c.initial ω, hB0]
+          _ = _ := by simp
+      · apply integral_congr_ae
+        exact Eventually.of_forall fun ω => intervalIntegral.integral_congr
+          (fun t ht => hgen ω t (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using ht))
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma stopping_paste_after
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] (f : Filtration ℝ≥0 mΩ)
+    {σ τ : Ω → ℝ≥0}
+    (hσ : IsStoppingTime f (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime f (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hle : ∀ ω, σ ω ≤ τ ω) {A : Set Ω}
+    (hA : MeasurableSet[hσ.measurableSpace] A) :
+    IsStoppingTime f (fun ω => ((if ω ∈ A then τ ω else σ ω : ℝ≥0) : WithTop ℝ≥0)) := by
+  classical
+  intro t
+  have he : {ω | ((if ω ∈ A then τ ω else σ ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
+      (A ∩ {ω | (σ ω : WithTop ℝ≥0) ≤ t} ∩ {ω | (τ ω : WithTop ℝ≥0) ≤ t}) ∪
+        (Aᶜ ∩ {ω | (σ ω : WithTop ℝ≥0) ≤ t}) := by
+    ext ω
+    simp only [mem_ofPred_eq, WithTop.coe_le_coe, mem_union, mem_inter_iff, mem_compl_iff]
+    split_ifs with h
+    · simp only [h, true_and, not_true_eq_false, false_and, or_false]
+      exact ⟨fun ht => ⟨(hle ω).trans ht, ht⟩, fun ht => ht.2⟩
+    · simp [h]
+  rw [he]
+  exact ((hA.2 t).inter (hτ t)).union ((hA.compl).2 t)
+
+ 
+
+theorem ClippedLogSolution.stopped_green_bounded_clock
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (η₀ u : ℝ) (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T)
+    (hin : ∀ ω t, t ≤ σ ω → logRadiusClock η₀ (c.Q t ω) ≤ u ∧
+      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
+    Integrable (fun ω => greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) P ∧
+      (∫ ω, greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω) ∂P) =
+        greenLogDensity κ q₀ 0 := by
+  classical
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hBa : Adapted f B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  have hex (n : ℕ) : ∃ β : Ω → ℝ≥0,
+      IsStoppingTime f (fun ω => (β ω : WithTop ℝ≥0)) ∧
+      (∀ ω, β ω ≤ T) ∧
+      (∀ ω t, t ≤ β ω → |B t ω| ≤ (n : ℝ) + 1) ∧
+      (∀ ω, β ω < T → |B (β ω) ω| = (n : ℝ) + 1) := by
+    let D (t : ℝ≥0) (ω : Ω) := (n : ℝ) + 1 - |B t ω|
+    obtain ⟨β, hβ, hbβ, hβin, _, hβbd⟩ := first_margin_stopping f D
+      (fun t => by
+        let : MeasurableSpace Ω := f t
+        exact (hBa t).abs.const_sub ((n : ℝ) + 1))
+      (fun ω => continuous_const.sub (hBc ω).abs)
+      (fun ω => by simpa [D, hB0 ω] using (show 0 ≤ (n : ℝ) + 1 by positivity)) T
+    refine ⟨β, hβ, hbβ, ?_, ?_⟩
+    · intro ω t ht
+      exact sub_nonneg.mp (hβin ω t ht)
+    · intro ω ht
+      exact (sub_eq_zero.mp (hβbd ω ht)).symm
+  choose β hβ hbβ hβin hβbd using hex
+  let σn (n : ℕ) (ω : Ω) := min (σ ω) (β n ω)
+  have hsn (n : ℕ) : IsStoppingTime f (fun ω => (σn n ω : WithTop ℝ≥0)) := by
+    simpa only [σn, WithTop.coe_min] using hσ.min (hβ n)
+  have hsnb (n : ℕ) (ω : Ω) : σn n ω ≤ T := (min_le_left _ _).trans (hb ω)
+  have hsin (n : ℕ) (ω : Ω) (t : ℝ≥0) (ht : t ≤ σn n ω) :
+      logRadiusClock η₀ (c.Q t ω) ≤ u ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi :=
+    hin ω t (ht.trans (min_le_left _ _))
+  have hmean (n : ℕ) :
+      Integrable (fun ω => greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω)) P ∧
+      (∫ ω, greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω) ∂P) = greenLogDensity κ q₀ 0 := by
+    let R := max (‖q₀‖ + c.L * T) ((n : ℝ) + 1)
+    have hR : 0 ≤ R := (by positivity : 0 ≤ (n : ℝ) + 1).trans (le_max_right _ _)
+    apply c.stopped_green hB hBm hBc hκ hκ8 R hR (σn n) (hsn n) T (hsnb n)
+    intro ω t ht
+    refine ⟨?_, (hsin n ω t ht).2⟩
+    rw [Prod.norm_def, max_le_iff, Real.norm_eq_abs]
+    exact ⟨(c.norm_le ω t T (ht.trans (hsnb n ω))).trans (le_max_left _ _),
+      (hβin n ω t (ht.trans (min_le_right _ _))).trans (le_max_right _ _)⟩
+  have hstable (ω : Ω) : ∀ᶠ n : ℕ in atTop, σn n ω = σ ω := by
+    obtain ⟨N, hN⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
+    filter_upwards [eventually_ge_atTop N] with n hn
+    have hbn : β n ω = T := by
+      apply le_antisymm (hbβ n ω)
+      by_contra h
+      have hlt : β n ω < T := lt_of_not_ge h
+      have hh := hN (β n ω) (hbβ n ω)
+      rw [hβbd n ω hlt] at hh
+      have hcast : (N : ℝ) ≤ n := by exact_mod_cast hn
+      linarith
+    dsimp only [σn]
+    rw [hbn, min_eq_left (hb ω)]
+  let K := Real.exp (codimension κ * (u - Real.log 2 - η₀))
+  have hbound (n : ℕ) (ω : Ω) :
+      ‖greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω)‖ ≤ K := by
+    rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
+    exact greenLogDensity_le_radius κ η₀ u hκ hκ8 _ _ (hsin n ω _ le_rfl).1
+  have hlim (ω : Ω) : Tendsto
+      (fun n => greenLogDensity κ (c.Q (σn n ω) ω) (B (σn n ω) ω)) atTop
+      (𝓝 (greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω))) :=
+    tendsto_const_nhds.congr' ((hstable ω).mono fun n hn => by dsimp only; rw [hn])
+  have hσm := stopping_time_measurable_nnreal hσ
+  have hQm := c.measurable.comp ((measurable_coe_nnreal_real.comp hσm).prodMk measurable_id)
+  have hbm := hBm.comp (hσm.prodMk measurable_id)
+  have hmeas : Measurable (fun ω => greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) :=
+    (greenLogDensity_contDiff κ).continuous.measurable.comp (hQm.prodMk hbm)
+  have hi : Integrable (fun ω => greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) P := by
+    apply Integrable.of_bound hmeas.aestronglyMeasurable K
+    apply Eventually.of_forall
+    intro ω
+    rw [Real.norm_eq_abs, abs_of_nonneg (greenLogDensity_nonneg _ _ _)]
+    exact greenLogDensity_le_radius κ η₀ u hκ hκ8 _ _ (hin ω _ le_rfl).1
+  refine ⟨hi, ?_⟩
+  apply tendsto_nhds_unique (tendsto_integral_of_dominated_convergence (fun _ => K)
+    (fun n => (hmean n).1.aestronglyMeasurable) (integrable_const K)
+    (fun n => Eventually.of_forall (hbound n)) (Eventually.of_forall hlim))
+  simpa only [(hmean _).2] using
+    (tendsto_const_nhds : Tendsto (fun _ : ℕ => greenLogDensity κ q₀ 0) atTop (𝓝 _))
+
+ 
+
+theorem ClippedLogSolution.stopped_green_history
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 < κ) (hκ8 : κ < 8)
+    (η₀ u : ℝ) (σ τ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hστ : ∀ ω, σ ω ≤ τ ω)
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hin : ∀ ω t, t ≤ τ ω → logRadiusClock η₀ (c.Q t ω) ≤ u ∧
+      lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi)
+    (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) :
+    (∫ ω in A, greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
+      ∫ ω in A, greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω) ∂P := by
+  let ρ (ω : Ω) : ℝ≥0 := if ω ∈ A then τ ω else σ ω
+  have hρ := stopping_paste_after _ hσ hτ hστ hA
+  have hρτ (ω : Ω) : ρ ω ≤ τ ω := by dsimp [ρ]; split_ifs <;> simp_all
+  obtain ⟨hiti, hitm⟩ := c.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
+    η₀ u τ hτ T hb hin
+  obtain ⟨hisi, hism⟩ := c.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
+    η₀ u σ hσ T (fun ω => (hστ ω).trans (hb ω))
+    (fun ω t ht => hin ω t (ht.trans (hστ ω)))
+  obtain ⟨hipi, hipm⟩ := c.stopped_green_bounded_clock hB hBm hBc hB0 hκ hκ8
+    η₀ u ρ hρ T (fun ω => (hρτ ω).trans (hb ω))
+    (fun ω t ht => hin ω t (ht.trans (hρτ ω)))
+  have hAm : MeasurableSet A := hσ.measurableSpace_le A hA
+  have hex (ω : Ω) :
+      greenLogDensity κ (c.Q (ρ ω) ω) (B (ρ ω) ω) -
+        greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω) =
+      A.indicator (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) -
+        greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) ω := by
+    by_cases hω : ω ∈ A <;> simp [ρ, hω]
+  have he : (∫ ω, A.indicator (fun ω => greenLogDensity κ (c.Q (τ ω) ω) (B (τ ω) ω) -
+        greenLogDensity κ (c.Q (σ ω) ω) (B (σ ω) ω)) ω ∂P) = 0 := by
+    simp_rw [← hex]
+    rw [integral_sub hipi hisi, hipm, hism, sub_self]
+  rw [integral_indicator hAm, integral_sub hiti.integrableOn hisi.integrableOn] at he
+  exact sub_eq_zero.mp he
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma exists_brownian_compact_stops
+    {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T) :
+    ∃ σn : ℕ → Ω → ℝ≥0,
+      (∀ n, IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+        (fun ω => (σn n ω : WithTop ℝ≥0))) ∧
+      (∀ n ω, σn n ω ≤ σ ω) ∧
+      (∀ n ω t, t ≤ σn n ω → |B t ω| ≤ (n : ℝ) + 1) ∧
+      (∀ ω, ∀ᶠ n : ℕ in atTop, σn n ω = σ ω) := by
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  have hBa : Adapted f B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  have hex (n : ℕ) : ∃ β : Ω → ℝ≥0,
+      IsStoppingTime f (fun ω => (β ω : WithTop ℝ≥0)) ∧
+      (∀ ω, β ω ≤ T) ∧
+      (∀ ω t, t ≤ β ω → |B t ω| ≤ (n : ℝ) + 1) ∧
+      (∀ ω, β ω < T → |B (β ω) ω| = (n : ℝ) + 1) := by
+    let D (t : ℝ≥0) (ω : Ω) := (n : ℝ) + 1 - |B t ω|
+    obtain ⟨β, hβ, hbβ, hβin, _, hβbd⟩ := first_margin_stopping f D
+      (fun t => by
+        let : MeasurableSpace Ω := f t
+        exact (hBa t).abs.const_sub ((n : ℝ) + 1))
+      (fun ω => continuous_const.sub (hBc ω).abs)
+      (fun ω => by simpa [D, hB0 ω] using (show 0 ≤ (n : ℝ) + 1 by positivity)) T
+    refine ⟨β, hβ, hbβ, ?_, ?_⟩
+    · intro ω t ht
+      exact sub_nonneg.mp (hβin ω t ht)
+    · intro ω ht
+      exact (sub_eq_zero.mp (hβbd ω ht)).symm
+  choose β hβ hbβ hβin hβbd using hex
+  let σn (n : ℕ) (ω : Ω) := min (σ ω) (β n ω)
+  refine ⟨σn, ?_, fun n ω => min_le_left _ _, ?_, ?_⟩
+  · intro n
+    simpa only [σn, WithTop.coe_min] using hσ.min (hβ n)
+  · intro n ω t ht
+    exact hβin n ω t (ht.trans (min_le_right _ _))
+  · intro ω
+    obtain ⟨N, hN⟩ := continuous_path_bounded_by_nat (fun t => B t ω) (hBc ω) T
+    filter_upwards [eventually_ge_atTop N] with n hn
+    have hbn : β n ω = T := by
+      apply le_antisymm (hbβ n ω)
+      by_contra h
+      have hlt : β n ω < T := lt_of_not_ge h
+      have hh := hN (β n ω) (hbβ n ω)
+      rw [hβbd n ω hlt] at hh
+      have hcast : (N : ℝ) ≤ n := by exact_mod_cast hn
+      linarith
+    dsimp only [σn]
+    rw [hbn, min_eq_left (hb ω)]
+
+lemma bounded_stopped_cumulative_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (G : ℝ → Ω → ℝ) (hGm : Measurable (Function.uncurry G))
+    (σ τ : Ω → ℝ≥0) (hτm : Measurable τ) (hle : ∀ ω, τ ω ≤ σ ω)
+    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T) (C : ℝ) (hC : 0 ≤ C)
+    (hG : ∀ ω t, 0 ≤ t → t ≤ (σ ω : ℝ) → |G t ω| ≤ C) :
+    Integrable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω) P ∧
+      ∀ ω, |∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω| ≤ C * T := by
+  have hm : Measurable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω) :=
+    (cumulative_integral_measurable G hGm).comp (hτm.prodMk measurable_id)
+  have hbnd (ω : Ω) : |∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω| ≤ C * T := by
+    have hh := intervalIntegral.norm_integral_le_of_norm_le_const
+      (a := (0 : ℝ)) (b := (τ ω : ℝ)) (C := C) (f := fun t => G t ω) (fun t ht => by
+        have htt : t ∈ Ioc (0 : ℝ) (τ ω : ℝ) := by simpa only [uIoc_of_le (τ ω).coe_nonneg] using ht
+        simpa only [Real.norm_eq_abs] using hG ω t htt.1.le (htt.2.trans (by exact_mod_cast hle ω)))
+    simp only [Real.norm_eq_abs, sub_zero, abs_of_nonneg (τ ω).coe_nonneg] at hh
+    exact hh.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast (hle ω).trans (hb ω)) hC)
+  exact ⟨Integrable.of_bound hm.aestronglyMeasurable _
+    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hbnd ω), hbnd⟩
+
+lemma integral_tendsto_stopped_eval
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (F : ℝ≥0 → Ω → ℝ) (σn : ℕ → Ω → ℝ≥0) (σ : Ω → ℝ≥0)
+    (hm : ∀ n, AEStronglyMeasurable (fun ω => F (σn n ω) ω) P)
+    (K : ℝ) (hb : ∀ n ω, |F (σn n ω) ω| ≤ K)
+    (hstable : ∀ ω, ∀ᶠ n : ℕ in atTop, σn n ω = σ ω) :
+    Tendsto (fun n => ∫ ω, F (σn n ω) ω ∂P) atTop (𝓝 (∫ ω, F (σ ω) ω ∂P)) := by
+  apply tendsto_integral_of_dominated_convergence (fun _ => K) hm (integrable_const K)
+    (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hb n ω)
+  apply Eventually.of_forall
+  intro ω
+  exact tendsto_const_nhds.congr' ((hstable ω).mono fun n hn => by dsimp only; rw [hn])
+
+lemma integral_identity_of_stable_stops
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    (F H : ℝ≥0 → Ω → ℝ) (σn : ℕ → Ω → ℝ≥0) (σ : Ω → ℝ≥0)
+    (hstable : ∀ ω, ∀ᶠ n : ℕ in atTop, σn n ω = σ ω)
+    (A K C : ℝ)
+    (hmean : ∀ n, Integrable (fun ω => F (σn n ω) ω) P ∧
+      (∫ ω, F (σn n ω) ω ∂P) = A + ∫ ω, H (σn n ω) ω ∂P)
+    (hFi : Integrable (fun ω => F (σ ω) ω) P)
+    (hHim : AEStronglyMeasurable (fun ω => H (σ ω) ω) P)
+    (hHnm : ∀ n, AEStronglyMeasurable (fun ω => H (σn n ω) ω) P)
+    (hFb : ∀ n ω, |F (σn n ω) ω| ≤ K)
+    (hHb : ∀ n ω, |H (σn n ω) ω| ≤ C)
+    (hHbound : ∀ ω, |H (σ ω) ω| ≤ C) :
+    Integrable (fun ω => F (σ ω) ω) P ∧
+    Integrable (fun ω => H (σ ω) ω) P ∧
+      (∫ ω, F (σ ω) ω ∂P) = A + ∫ ω, H (σ ω) ω ∂P := by
+  refine ⟨hFi, Integrable.of_bound hHim C
+    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHbound ω), ?_⟩
+  have hlimF := integral_tendsto_stopped_eval F σn σ
+    (fun n => (hmean n).1.aestronglyMeasurable) K hFb hstable
+  have hlimH := integral_tendsto_stopped_eval H σn σ hHnm C hHb hstable
+  have he := hlimH.const_add A
+  simp_rw [← (hmean _).2] at he
+  exact tendsto_nhds_unique hlimF he
+
+lemma ClippedLogSolution.stopped_power_dynkin_compact_cap
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 ≤ κ) (p α : ℝ) (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T)
+    (hin : ∀ ω t, t ≤ σ ω → (c.Q t ω).1.im ∈ Icc lo hi)
+    (N : ℝ) (hN : 0 ≤ N) (hcap : ∀ ω t, t ≤ σ ω → |B t ω| ≤ N) :
+    Integrable (fun ω => powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)) P ∧
+      (∫ ω, powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω) ∂P) =
+        powerLogDensity κ p α q₀ 0 + ∫ ω, (∫ t in (0 : ℝ)..(σ ω : ℝ),
+          powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
+  let R := max (‖q₀‖ + c.L * T) N
+  have hR : 0 ≤ R := hN.trans (le_max_right _ _)
+  apply c.stopped_power_dynkin hB hBm hBc hκ p α R hR σ hσ T hb
+  intro ω t ht
+  refine ⟨?_, hin ω t ht⟩
+  rw [Prod.norm_def, max_le_iff, Real.norm_eq_abs]
+  exact ⟨(c.norm_le ω t T (ht.trans (hb ω))).trans (le_max_left _ _),
+    (hcap ω t ht).trans (le_max_right _ _)⟩
+
+lemma ClippedLogSolution.stopped_power_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (p α : ℝ) (σ : Ω → ℝ≥0) (hσm : Measurable σ) (K : ℝ)
+    (hval : ∀ ω, |powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)| ≤ K) :
+    Integrable (fun ω => powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)) P := by
+  have hQm := c.measurable.comp ((measurable_coe_nnreal_real.comp hσm).prodMk measurable_id)
+  have hbm := hBm.comp (hσm.prodMk measurable_id)
+  have hWm := (powerLogDensity_contDiff κ p α).continuous.measurable.comp (hQm.prodMk hbm)
+  exact Integrable.of_bound hWm.aestronglyMeasurable K
+    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hval ω)
+
+lemma ClippedLogSolution.power_generator_measurable
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (p α : ℝ) :
+    Measurable (fun q : ℝ × Ω => powerLogGenerator κ p α (c.Q q.1 q.2) (B q.1.toNNReal q.2)) := by
+  have ht : Measurable (fun q : ℝ × Ω => q.1.toNNReal) :=
+    continuous_real_toNNReal.measurable.comp measurable_fst
+  have hb : Measurable (fun q : ℝ × Ω => B q.1.toNNReal q.2) :=
+    hBm.comp (ht.prodMk measurable_snd)
+  exact measurable_powerLogGenerator_comp κ p α
+    (fun q : ℝ × Ω => c.Q q.1 q.2) (fun q : ℝ × Ω => B q.1.toNNReal q.2)
+    c.measurable hb
+
+lemma ClippedLogSolution.stopped_power_cumulative_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (p α : ℝ) (σ τ : Ω → ℝ≥0) (hτm : Measurable τ) (hleτ : ∀ ω, τ ω ≤ σ ω)
+    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T) (C : ℝ) (hC : 0 ≤ C)
+    (hdrift : ∀ ω t, t ≤ σ ω → |powerLogGenerator κ p α (c.Q t ω) (B t ω)| ≤ C) :
+    Integrable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ),
+      powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) P ∧
+      (∀ ω, |∫ t in (0 : ℝ)..(τ ω : ℝ),
+        powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)| ≤ C * T) := by
+  let G (t : ℝ) (ω : Ω) := powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)
+  have hGm : Measurable (Function.uncurry G) := c.power_generator_measurable hBm p α
+  have hGb (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (htσ : t ≤ (σ ω : ℝ)) : |G t ω| ≤ C := by
+    have hh := hdrift ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr htσ)
+    simpa only [G, Real.coe_toNNReal t ht] using hh
+  exact bounded_stopped_cumulative_integrable G hGm σ τ hτm hleτ T hb C hC hGb
+
+ 
+
+theorem ClippedLogSolution.stopped_power_dynkin_bounded
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 ≤ κ) (p α : ℝ) (σ : Ω → ℝ≥0)
+    (hσ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (σ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, σ ω ≤ T)
+    (hin : ∀ ω t, t ≤ σ ω → (c.Q t ω).1.im ∈ Icc lo hi)
+    (K C : ℝ) (hC : 0 ≤ C)
+    (hval : ∀ ω t, t ≤ σ ω → |powerLogDensity κ p α (c.Q t ω) (B t ω)| ≤ K)
+    (hdrift : ∀ ω t, t ≤ σ ω → |powerLogGenerator κ p α (c.Q t ω) (B t ω)| ≤ C) :
+    Integrable (fun ω => powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω)) P ∧
+    Integrable (fun ω => ∫ t in (0 : ℝ)..(σ ω : ℝ),
+      powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) P ∧
+      (∫ ω, powerLogDensity κ p α (c.Q (σ ω) ω) (B (σ ω) ω) ∂P) =
+        powerLogDensity κ p α q₀ 0 +
+        ∫ ω, (∫ t in (0 : ℝ)..(σ ω : ℝ),
+          powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨σn, hsn, hsle, hsB, hstable⟩ := exists_brownian_compact_stops B hm hBc hB0 σ hσ T hb
+  have hsnb (n : ℕ) (ω : Ω) : σn n ω ≤ T := (hsle n ω).trans (hb ω)
+  let W (τ : Ω → ℝ≥0) (ω : Ω) := powerLogDensity κ p α (c.Q (τ ω) ω) (B (τ ω) ω)
+  let G (t : ℝ) (ω : Ω) := powerLogGenerator κ p α (c.Q t ω) (B t.toNNReal ω)
+  let H (τ : Ω → ℝ≥0) (ω : Ω) := ∫ t in (0 : ℝ)..(τ ω : ℝ), G t ω
+  have hmean (n : ℕ) : Integrable (W (σn n)) P ∧
+      (∫ ω, W (σn n) ω ∂P) = powerLogDensity κ p α q₀ 0 + ∫ ω, H (σn n) ω ∂P := by
+    exact c.stopped_power_dynkin_compact_cap hB hBm hBc hκ p α (σn n) (hsn n)
+      T (hsnb n) (fun ω t ht => hin ω t (ht.trans (hsle n ω))) ((n : ℝ) + 1)
+      (by positivity) (hsB n)
+  have hH (τ : Ω → ℝ≥0) (hτm : Measurable τ) (hτ : ∀ ω, τ ω ≤ σ ω) :
+      Integrable (H τ) P ∧ ∀ ω, |H τ ω| ≤ C * T :=
+    c.stopped_power_cumulative_integrable hBm p α σ τ hτm hτ T hb C hC hdrift
+  have hσm := stopping_time_measurable_nnreal hσ
+  have hWi : Integrable (W σ) P := c.stopped_power_integrable hBm p α σ hσm K
+    (fun ω => hval ω (σ ω) le_rfl)
+  exact integral_identity_of_stable_stops
+    (fun t ω => powerLogDensity κ p α (c.Q t ω) (B t ω))
+    (fun t ω => ∫ u in (0 : ℝ)..(t : ℝ), G u ω) σn σ hstable
+    (powerLogDensity κ p α q₀ 0) K (C * T) hmean hWi
+    (hH σ hσm (fun _ => le_rfl)).1.aestronglyMeasurable
+    (fun n => (hH (σn n) (stopping_time_measurable_nnreal (hsn n)) (hsle n)).1.aestronglyMeasurable)
+    (fun n ω => hval ω (σn n ω) (hsle n ω))
+    (fun n => (hH (σn n) (stopping_time_measurable_nnreal (hsn n)) (hsle n)).2)
+    (hH σ hσm (fun _ => le_rfl)).2
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+
+lemma ClippedLogSolution.swallow_generator_measurable
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (β C : ℝ) :
+    Measurable (fun p : ℝ × Ω => swallowGenerator κ β C (c.Q p.1 p.2) (B p.1.toNNReal p.2)) := by
+  have hBr : Measurable (fun p : ℝ × Ω => B p.1.toNNReal p.2) :=
+    hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd)
+  have hp : Measurable (fun p : ℝ × Ω => (c.Q p.1 p.2, B p.1.toNNReal p.2)) :=
+    c.measurable.prodMk hBr
+  have hh := (continuous_swallowGenerator κ β C).measurable.comp hp
+  dsimp only [Function.comp_def] at hh
+  with_reducible exact hh
+
+lemma ClippedLogSolution.swallow_generator_continuous
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (β C : ℝ) (ω : Ω) :
+    Continuous (fun t : ℝ => swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)) := by
+  have hB : Continuous (fun t : ℝ => B t.toNNReal ω) :=
+    (hBc ω).comp continuous_real_toNNReal
+  have hQ : Continuous (fun t : ℝ => c.Q t ω) := (c.lipschitz ω).continuous
+  have hp : Continuous (fun t : ℝ => (c.Q t ω, B t.toNNReal ω)) := hQ.prodMk hB
+  have hh := (continuous_swallowGenerator κ β C).comp hp
+  dsimp only [Function.comp_def] at hh
+  with_reducible exact hh
+
+lemma ClippedLogSolution.swallow_cumulative_integrable
+    {P : Measure Ω} [IsFiniteMeasure P]
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (β C : ℝ)
+    (τ : Ω → ℝ≥0) (hτm : Measurable τ) (T : ℝ≥0) (hτT : ∀ ω, τ ω ≤ T)
+    (D : ℝ) (hD : 0 ≤ D)
+    (hbound : ∀ ω t, t ≤ τ ω → |swallowGenerator κ β C (c.Q t ω) (B t ω)| ≤ D) :
+    Integrable (fun ω => ∫ t in (0 : ℝ)..(τ ω : ℝ),
+      swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)) P := by
+  let G (t : ℝ) (ω : Ω) := swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)
+  have hGm : Measurable (Function.uncurry G) := c.swallow_generator_measurable hBm β C
+  have hGb (ω : Ω) (t : ℝ) (ht : 0 ≤ t) (hτ : t ≤ (τ ω : ℝ)) : |G t ω| ≤ D := by
+    have hh := hbound ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hτ)
+    simpa only [G, Real.coe_toNNReal t ht] using hh
+  exact (bounded_stopped_cumulative_integrable (P := P) G hGm τ τ hτm
+    (fun _ => le_rfl) T hτT D hD hGb).1
+
+lemma bounded_nnreal_integrable {P : Measure Ω} [IsFiniteMeasure P]
+    (τ : Ω → ℝ≥0) (hτm : Measurable τ) (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
+    Integrable (fun ω => (τ ω : ℝ)) P := by
+  apply Integrable.of_bound ((NNReal.continuous_coe.measurable.comp hτm).aestronglyMeasurable) T
+  exact Eventually.of_forall fun ω => by
+    change ‖(τ ω : ℝ)‖ ≤ (T : ℝ)
+    rw [Real.norm_eq_abs, abs_of_nonneg (τ ω).coe_nonneg]
+    exact NNReal.coe_le_coe.mpr (hb ω)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace RadialLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
+
+lemma exists_compact_bound (a : RadialLocalization c R T) (hκ : 0 < κ) (hR : 0 ≤ R) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ ω t, t ≤ a.τ ω → ‖(c.Q t ω, B t ω)‖ ≤ K := by
+  let M : ℝ := ‖q₀‖+c.L*T
+  have hM : 0 ≤ M := by dsimp [M]; positivity
+  refine ⟨max M ((M+R)/Real.sqrt κ), le_max_of_le_left hM, ?_⟩
+  intro ω t ht
+  have hQ : ‖c.Q t ω‖ ≤ M := c.norm_le ω t T (ht.trans (a.bounded ω))
+  have hB := norm_driver_of_radial_bound κ R M hκ hR (c.Q t ω) (B t ω) hQ (a.inside ω t ht).1
+  rw [Prod.norm_def]
+  exact max_le_max hQ (by simpa only [Real.norm_eq_abs] using hB)
+
+lemma exists_generator_bound (a : RadialLocalization c R T)
+    (hκ : 0 < κ) (hR : 0 ≤ R) (β C : ℝ) :
+    ∃ D : ℝ, 0 ≤ D ∧ ∀ ω t, t ≤ a.τ ω →
+      |swallowGenerator κ β C (c.Q t ω) (B t ω)| ≤ D := by
+  obtain ⟨K,hK,hbound⟩ := a.exists_compact_bound hκ hR
+  obtain ⟨D,hD⟩ := (isCompact_closedBall (0 : (ℂ × ℝ) × ℝ) K).exists_bound_of_continuousOn
+    (continuous_swallowGenerator κ β C).continuousOn
+  refine ⟨max D 0, le_max_right _ _, ?_⟩
+  intro ω t ht
+  have hmem : (c.Q t ω, B t ω) ∈ Metric.closedBall (0 : (ℂ × ℝ) × ℝ) K := by
+    simpa only [Metric.mem_closedBall, dist_zero_right] using hbound ω t ht
+  exact (hD _ hmem).trans (le_max_left _ _)
+
+lemma generator_cumulative_integrable {P : Measure Ω} [IsFiniteMeasure P]
+    (a : RadialLocalization c R T)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hκ : 0 < κ) (hR : 0 ≤ R) (β C : ℝ) :
+    Integrable (fun ω => ∫ t in (0 : ℝ)..(a.τ ω : ℝ),
+      swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)) P := by
+  obtain ⟨D,hD,hbound⟩ := a.exists_generator_bound hκ hR β C
+  exact c.swallow_cumulative_integrable hBm β C a.τ
+    (stopping_time_measurable_nnreal a.stopping) T a.bounded D hD hbound
+
+lemma stopped_lyapunov_bound {P : Measure Ω}
+    (a : RadialLocalization c R T)
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 4 < κ) (hR : 0 < R) (hhi : q₀.1.im ≤ hi) :
+    Integrable (fun ω => swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
+      (c.Q (a.τ ω) ω) (B (a.τ ω) ω)) P ∧
+    (∫ ω, swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
+      (c.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) +
+      (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) *
+        (∫ ω, (a.τ ω : ℝ) ∂P) ≤ swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q₀ 0 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hk : 0 < κ := lt_trans (by norm_num) hκ
+  let β := (κ-4)/(2*κ)
+  let C := 3*(κ-4)/8
+  let d := β*((κ-4)/4)*Real.exp ((β-2)*Real.log R)
+  let G (t : ℝ) (ω : Ω) := swallowGenerator κ β C (c.Q t ω) (B t.toNNReal ω)
+  obtain ⟨K,hK,hbound⟩ := a.exists_compact_bound hk hR.le
+  have hmean := c.stopped_swallow_dynkin hB hBm hBc hk.le β C K hK a.τ a.stopping
+    T a.bounded (fun ω t ht => ⟨hbound ω t ht, (a.inside ω t ht).2,
+      (a.height_le_initial ω t ht).trans hhi⟩)
+  refine ⟨hmean.1, ?_⟩
+  have hτi : Integrable (fun ω => (a.τ ω : ℝ)) P :=
+    bounded_nnreal_integrable a.τ (stopping_time_measurable_nnreal a.stopping) T a.bounded
+  have hGi : Integrable (fun ω => ∫ t in (0 : ℝ)..(a.τ ω : ℝ), G t ω) P :=
+    a.generator_cumulative_integrable hBm hk hR.le β C
+  have hGle (ω : Ω) : (∫ t in (0 : ℝ)..(a.τ ω : ℝ), G t ω) ≤ -d*(a.τ ω : ℝ) := by
+    have hGc : Continuous (fun t : ℝ => G t ω) := c.swallow_generator_continuous hBc β C ω
+    have hh := intervalIntegral.integral_mono_on (a.τ ω).coe_nonneg
+      (hGc.intervalIntegrable _ _) (continuous_const.intervalIntegrable _ _ :
+        IntervalIntegrable (fun _ : ℝ => -d) volume 0 (a.τ ω)) (fun t ht => by
+          have hh := swallowGenerator_uniform_negative κ hκ (c.Q t.toNNReal ω)
+            (B t.toNNReal ω) R hR
+            (a.inside ω t.toNNReal (Real.toNNReal_le_iff_le_coe.mpr ht.2)).1
+          simpa only [G, Real.coe_toNNReal t ht.1] using hh)
+    simpa only [intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_comm] using hh
+  have htotal := integral_mono hGi (hτi.const_mul (-d)) hGle
+  rw [integral_const_mul] at htotal
+  change (∫ ω, swallowLyapunov κ β C (c.Q (a.τ ω) ω) (B (a.τ ω) ω) ∂P) +
+    d*(∫ ω, (a.τ ω : ℝ) ∂P) ≤ swallowLyapunov κ β C q₀ 0
+  rw [hmean.2]
+  linarith
+
+end RadialLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem compensated_exponential_limit
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (X E : ℕ → Ω → ℝ) (Y Q : Ω → ℝ) (θ V : ℝ)
+    (hm : ∀ n, AEMeasurable (fun ω => ENNReal.ofReal
+      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))) P)
+    (hX : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
+    (hE : ∀ᵐ ω ∂P, Tendsto (fun n => E n ω) atTop (𝓝 (Q ω)))
+    (hQ : ∀ᵐ ω ∂P, Q ω ≤ V)
+    (hexp : ∀ n, (∫⁻ ω, ENNReal.ofReal
+      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2)) ∂P) ≤ 1) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) ≤ 1 := by
+  let f (n : ℕ) (ω : Ω) := ENNReal.ofReal (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))
+  have hl : ∀ᵐ ω ∂P, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) =
+      liminf (fun n => f n ω) atTop := by
+    filter_upwards [hX, hE] with ω hx he
+    exact ((ENNReal.continuous_ofReal.tendsto _).comp
+      ((Real.continuous_exp.tendsto _).comp ((hx.const_mul θ).sub ((he.const_mul (θ^2)).div_const 2)))).liminf_eq.symm
+  calc
+    _ ≤ ∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [hQ] with ω hq
+      apply ENNReal.ofReal_le_ofReal
+      apply Real.exp_le_exp.mpr
+      nlinarith [mul_le_mul_of_nonneg_left hq (sq_nonneg θ)]
+    _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := lintegral_congr_ae hl
+    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' hm
+    _ ≤ 1 := liminf_le_of_frequently_le (Frequently.of_forall hexp)
+
+ 
+theorem compensated_exponential_limit_energy
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (X E : ℕ → Ω → ℝ) (Y Q : Ω → ℝ) (θ : ℝ)
+    (hm : ∀ n, AEMeasurable (fun ω => ENNReal.ofReal
+      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))) P)
+    (hX : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
+    (hE : ∀ᵐ ω ∂P, Tendsto (fun n => E n ω) atTop (𝓝 (Q ω)))
+    (hexp : ∀ n, (∫⁻ ω, ENNReal.ofReal
+      (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2)) ∂P) ≤ 1) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) ∂P) ≤ 1 := by
+  let f (n : ℕ) (ω : Ω) := ENNReal.ofReal (Real.exp (θ * X n ω - θ ^ 2 * E n ω / 2))
+  have hl : ∀ᵐ ω ∂P, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * Q ω / 2)) =
+      liminf (fun n => f n ω) atTop := by
+    filter_upwards [hX, hE] with ω hx he
+    exact ((ENNReal.continuous_ofReal.tendsto _).comp
+      ((Real.continuous_exp.tendsto _).comp ((hx.const_mul θ).sub
+        ((he.const_mul (θ^2)).div_const 2)))).liminf_eq.symm
+  calc
+    _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := lintegral_congr_ae hl
+    _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' hm
+    _ ≤ 1 := liminf_le_of_frequently_le (Frequently.of_forall hexp)
+
+ 
+theorem exponential_tail_of_compensated
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {Y : Ω → ℝ} (hY : AEMeasurable Y P) {θ : ℝ} (hθ : 0 ≤ θ) (V x : ℝ)
+    (h : (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) ≤ 1) :
+    P {ω | x ≤ Y ω} ≤ ENNReal.ofReal (Real.exp (-θ * x + θ ^ 2 * V / 2)) := by
+  let a := ENNReal.ofReal (Real.exp (θ * x - θ ^ 2 * V / 2))
+  have ha : a ≠ 0 := ne_of_gt (ENNReal.ofReal_pos.mpr (Real.exp_pos _))
+  have hab : a ≠ ∞ := ENNReal.ofReal_ne_top
+  have hm := (((hY.const_mul θ).sub_const (θ ^ 2 * V / 2)).exp).ennreal_ofReal
+  have hle := meas_ge_le_lintegral_div hm ha hab
+  have hs : {ω | x ≤ Y ω} ⊆ {ω | a ≤ ENNReal.ofReal
+      (Real.exp (θ * Y ω - θ ^ 2 * V / 2))} := by
+    intro ω hω
+    exact ENNReal.ofReal_le_ofReal (Real.exp_le_exp.mpr
+      (sub_le_sub_right (mul_le_mul_of_nonneg_left hω hθ) _))
+  calc
+    _ ≤ P {ω | a ≤ ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2))} := measure_mono hs
+    _ ≤ (∫⁻ ω, ENNReal.ofReal (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) / a := hle
+    _ ≤ 1 / a := ENNReal.div_le_div_right h a
+    _ = _ := by
+      rw [one_div, ← ENNReal.ofReal_inv_of_pos (Real.exp_pos _), ← Real.exp_neg]
+      congr 2
+      ring
+
+ 
+theorem subGaussian_tail_of_compensated
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {Y : Ω → ℝ} (hY : AEMeasurable Y P) {V x : ℝ} (hV : 0 < V) (hx : 0 ≤ x)
+    (h : ∀ θ : ℝ, (∫⁻ ω, ENNReal.ofReal
+      (Real.exp (θ * Y ω - θ ^ 2 * V / 2)) ∂P) ≤ 1) :
+    P {ω | x ≤ Y ω} ≤ ENNReal.ofReal (Real.exp (-x ^ 2 / (2 * V))) := by
+  have ht := exponential_tail_of_compensated hY (div_nonneg hx hV.le) V x (h (x/V))
+  convert ht using 2
+  congr 2
+  field_simp
+  ring
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StripKernels.lean b/lean/OAI/Probability/SLE/StripKernels.lean
index 14645b627c20515e15851642558b30543955ab9d..51f94a27c7968fe965673f98dd59e931e17ece08 100644
--- a/lean/OAI/Probability/SLE/StripKernels.lean
+++ b/lean/OAI/Probability/SLE/StripKernels.lean
@@ -1,674 +1,674 @@
-import Mathlib
-import OAI.Probability.SLE.IntervalDensity
-
-namespace OAI
-
-/-! Strip Kernels for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology ContDiff ENNReal BigOperators NNReal
-namespace SLEExactGauge
-lemma integrable_mul_compact_test {μ : Measure ℂ} {f g : ℂ → ℝ}
-    (hf : Integrable f μ) (hg : Continuous g) (hgc : HasCompactSupport g) :
-    Integrable (fun z => f z * g z) μ := by
-  obtain ⟨C, hC⟩ := (hgc.isCompact_range hg).isBounded.exists_norm_le
-  exact hf.mul_bdd hg.aestronglyMeasurable
-    (Eventually.of_forall fun z => hC _ ⟨z,rfl⟩)
-
-lemma weak_gradient_pairing {u : ℂ → ℝ} {D : ℂ → ℂ}
-    (hu : SourceIsStripWeakGradient u D) {X Y : ℂ → ℝ}
-    (hX : ContDiff ℝ ∞ X) (hY : ContDiff ℝ ∞ Y)
-    (hcX : HasCompactSupport X) (hcY : HasCompactSupport Y)
-    (hsX : tsupport X ⊆ openUnitStrip) (hsY : tsupport Y ⊆ openUnitStrip) :
-    (∫ z in openUnitStrip, u z * ((fderiv ℝ X z) 1 + (fderiv ℝ Y z) Complex.I)) =
-      -(∫ z in openUnitStrip, (D z).re * X z + (D z).im * Y z) := by
-  have hui := hu.1.integrable (by norm_num)
-  have hDi := hu.2.1.integrable (by norm_num)
-  have hx := integrable_mul_compact_test hui
-    ((hX.continuous_fderiv (by simp)).clm_apply continuous_const) (hcX.fderiv_apply ℝ 1)
-  have hy := integrable_mul_compact_test hui
-    ((hY.continuous_fderiv (by simp)).clm_apply continuous_const) (hcY.fderiv_apply ℝ Complex.I)
-  have hdx := integrable_mul_compact_test (Complex.reCLM.integrable_comp hDi) hX.continuous hcX
-  have hdy := integrable_mul_compact_test (Complex.imCLM.integrable_comp hDi) hY.continuous hcY
-  change Integrable (fun z => (D z).re * X z) _ at hdx
-  change Integrable (fun z => (D z).im * Y z) _ at hdy
-  simp_rw [mul_add]
-  rw [integral_add hx hy, (hu.2.2 X hX hcX hsX).1, (hu.2.2 Y hY hcY hsY).2,
-    integral_add hdx hdy]
-  ring
-
-lemma complex_pairing_abs_le (D : ℂ) (X Y : ℝ) :
-    |D.re * X + D.im * Y| ≤ ‖D‖ * (|X|+|Y|) := by
-  calc
-    _ ≤ |D.re * X| + |D.im * Y| := abs_add_le _ _
-    _ ≤ ‖D‖ * |X| + ‖D‖ * |Y| := by
-      rw [abs_mul, abs_mul]
-      exact add_le_add (mul_le_mul_of_nonneg_right (Complex.abs_re_le_norm D) (abs_nonneg _))
-        (mul_le_mul_of_nonneg_right (Complex.abs_im_le_norm D) (abs_nonneg _))
-    _ = _ := by ring
-
-lemma integral_nonneg_cauchy {μ : Measure ℂ} {f g : ℂ → ℝ}
-    (hf : MemLp f 2 μ) (hg : MemLp g 2 μ)
-    (hf0 : ∀ z, 0 ≤ f z) (hg0 : ∀ z, 0 ≤ g z) :
-    (∫ z, f z * g z ∂μ) ≤
-      Real.sqrt (∫ z, (f z)^2 ∂μ) * Real.sqrt (∫ z, (g z)^2 ∂μ) := by
-  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
-    (Eventually.of_forall hf0) (Eventually.of_forall hg0)
-    (by simpa using hf) (by simpa using hg)
-  simpa only [Real.rpow_two, Real.sqrt_eq_rpow] using hh
-
-lemma integral_pairing_le {D : ℂ → ℂ}
-    (hD : MemLp D 2 (volume.restrict openUnitStrip)) {X Y : ℂ → ℝ}
-    (hX : Continuous X) (hY : Continuous Y)
-    (hcX : HasCompactSupport X) (hcY : HasCompactSupport Y) :
-    |∫ z in openUnitStrip, (D z).re * X z + (D z).im * Y z| ≤
-      Real.sqrt (stripDirichletEnergy D) *
-        Real.sqrt (∫ z in openUnitStrip, (|X z|+|Y z|)^2) := by
-  have hDi := hD.integrable (by norm_num)
-  have hdx := integrable_mul_compact_test (Complex.reCLM.integrable_comp hDi) hX hcX
-  have hdy := integrable_mul_compact_test (Complex.imCLM.integrable_comp hDi) hY hcY
-  have hK : MemLp (fun z => |X z|+|Y z|) 2 (volume.restrict openUnitStrip) :=
-    (hX.abs.add hY.abs).memLp_of_hasCompactSupport (hcX.abs.add hcY.abs)
-  calc
-    _ ≤ ∫ z in openUnitStrip, |(D z).re * X z + (D z).im * Y z| := by
-      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm
-        (fun z => (D z).re * X z + (D z).im * Y z)
-    _ ≤ ∫ z in openUnitStrip, ‖D z‖ * (|X z|+|Y z|) :=
-      integral_mono (hdx.add hdy).abs (hD.norm.integrable_mul hK)
-        (fun z => complex_pairing_abs_le ..)
-    _ ≤ _ := integral_nonneg_cauchy hD.norm hK (fun _ => norm_nonneg _)
-      (fun _ => add_nonneg (abs_nonneg _) (abs_nonneg _))
-
-noncomputable def chainDensity (t s : ℝ) (j n : ℕ) : ℂ → ℝ :=
-  rectangleDensity (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j)) n
-noncomputable def chainX (t s : ℝ) (j n : ℕ) : ℂ → ℝ :=
-  rectangleTransportX (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n
-noncomputable def chainY (t s : ℝ) (j n : ℕ) : ℂ → ℝ :=
-  rectangleTransportY (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n
-noncomputable def chainKernel (t s : ℝ) (j n : ℕ) (z : ℂ) : ℝ :=
-  |chainX t s j n z|+|chainY t s j n z|
-
-lemma chainX_smooth (t s : ℝ) (j n : ℕ) : ContDiff ℝ ∞ (chainX t s j n) :=
-  rectangleTransportX_contDiff ..
-lemma chainY_smooth (t s : ℝ) (j n : ℕ) : ContDiff ℝ ∞ (chainY t s j n) :=
-  rectangleTransportY_contDiff ..
-lemma chainX_compact {t : ℝ} (s : ℝ) (ht : 0 ≤ t) (ht1 : t ≤ 1) (j n : ℕ) :
-    HasCompactSupport (chainX t s j n) :=
-  rectangleTransportX_compact _ _ _ _ _ _ _ _
-    (scaleOrigin_step ht j) (scaleOrigin_right_step ht1 j) n
-lemma chainY_compact (t : ℝ) {s : ℝ} (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
-    HasCompactSupport (chainY t s j n) := by
-  apply rectangleTransportY_compact
-  · exact mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le
-  · have hh := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
-    nlinarith
-
-lemma chainX_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
-    tsupport (chainX t s j n) ⊆ openUnitStrip := by
-  have hh := rectangleTransportX_support (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
-    (scaleOrigin_step ht j) (scaleOrigin_right_step ht1 j)
-    (mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le)
-    (by have := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
-        nlinarith) n
-  intro z hz
-  obtain ⟨h1,h2,h3,h4⟩ := hh hz
-  have h5 := scaleOrigin_nonneg ht j
-  have h6 := scaleOrigin_add_le_one ht1 j
-  have h7 := mul_nonneg Real.pi_pos.le (scaleOrigin_nonneg hs j)
-  have h8 := mul_le_mul_of_nonneg_left (scaleOrigin_add_le_one hs1 j) Real.pi_pos.le
-  exact ⟨by linarith, by linarith, by linarith, by nlinarith⟩
-
-lemma chainY_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
-    tsupport (chainY t s j n) ⊆ openUnitStrip := by
-  have hh := rectangleTransportY_support (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
-    (mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le)
-    (by have := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
-        nlinarith) n
-  intro z hz
-  obtain ⟨h1,h2,h3,h4⟩ := hh hz
-  have h5 := scaleOrigin_nonneg ht j
-  have h6 := scaleOrigin_add_le_one ht1 j
-  have h7 := mul_nonneg Real.pi_pos.le (scaleOrigin_nonneg hs j)
-  have h8 := mul_le_mul_of_nonneg_left (scaleOrigin_add_le_one hs1 j) Real.pi_pos.le
-  exact ⟨by linarith, by linarith, by linarith, by nlinarith⟩
-
-lemma chain_divergence (t s : ℝ) (j n : ℕ) (z : ℂ) :
-    (fderiv ℝ (chainX t s j n) z) 1 + (fderiv ℝ (chainY t s j n) z) Complex.I =
-      chainDensity t s (j+1) n z - chainDensity t s j n z := by
-  unfold chainX chainY chainDensity
-  rw [rectangleTransport_divergence]
-  simp only [scaleLength_succ, mul_div_assoc]
-
-lemma chainKernel_continuous (t s : ℝ) (j n : ℕ) : Continuous (chainKernel t s j n) :=
-  (chainX_smooth t s j n).continuous.abs.add (chainY_smooth t s j n).continuous.abs
-lemma chainKernel_compact {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
-    HasCompactSupport (chainKernel t s j n) :=
-  (chainX_compact s ht ht1 j n).abs.add (chainY_compact t hs hs1 j n).abs
-lemma chainKernel_nonneg (t s : ℝ) (j n : ℕ) (z : ℂ) : 0 ≤ chainKernel t s j n z :=
-  add_nonneg (abs_nonneg _) (abs_nonneg _)
-
-lemma chainKernel_bound (t s : ℝ) (j n : ℕ) (z : ℂ) :
-    ‖chainKernel t s j n z‖ ≤ 12 * (2:ℝ)^j := by
-  rw [Real.norm_of_nonneg (chainKernel_nonneg ..)]
-  have hx := rectangleTransportX_bound (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n z
-  have hy := rectangleTransportY_bound (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n z
-  have hp : scaleLength j ≤ Real.pi*scaleLength j := by
-    nlinarith [scaleLength_pos j, Real.pi_gt_three]
-  have hx' := hx.trans (div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 8)
-    (scaleLength_pos j) hp)
-  change |chainX t s j n z| + |chainY t s j n z| ≤ _
-  calc
-    _ ≤ 8/scaleLength j + 4/scaleLength j := add_le_add hx' hy
-    _ = _ := by unfold scaleLength; field_simp; ring
-
-lemma chainKernel_integral_bound {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
-    (∫ z, ‖chainKernel t s j n z‖) ≤ 10 / (2:ℝ)^j := by
-  have hx := rectangleTransportX_integral_abs (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
-    (scaleOrigin_step ht j) (scaleOrigin_right_step ht1 j) n
-  have hy := rectangleTransportY_integral_abs (scaleOrigin t j) (Real.pi*scaleOrigin s j)
-    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
-    (mul_pos Real.pi_pos (scaleLength_pos j))
-    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
-    (mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le)
-    (by have := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
-        nlinarith) n
-  have hiX := (chainX_smooth t s j n).continuous.integrable_of_hasCompactSupport (μ := volume)
-    (chainX_compact s ht ht1 j n)
-  have hiY := (chainY_smooth t s j n).continuous.integrable_of_hasCompactSupport (μ := volume)
-    (chainY_compact t hs hs1 j n)
-  simp_rw [Real.norm_of_nonneg (chainKernel_nonneg ..)]
-  change (∫ z, |chainX t s j n z| + |chainY t s j n z|) ≤ _
-  rw [integral_add hiX.abs hiY.abs]
-  change (∫ z, |chainX t s j n z|) ≤ _ at hx
-  change (∫ z, |chainY t s j n z|) ≤ _ at hy
-  have hpi := mul_le_mul_of_nonneg_right Real.pi_lt_four.le (scaleLength_pos j).le
-  have hh : 10*scaleLength j = 10/(2:ℝ)^j := by simp [scaleLength, div_eq_mul_inv]
-  nlinarith
-
-lemma compact_sum_range {f : ℕ → ℂ → ℝ} (hf : ∀ j, HasCompactSupport (f j)) (k : ℕ) :
-    HasCompactSupport (fun z => ∑ j ∈ Finset.range k, f j z) := by
-  induction k with
-  | zero => simp [HasCompactSupport]
-  | succ k ih =>
-    simp only [Finset.sum_range_succ]
-    exact (ih.union (hf k)).of_isClosed_subset (isClosed_tsupport _) (tsupport_add _ _)
-
-lemma support_sum_range {f : ℕ → ℂ → ℝ} {S : Set ℂ}
-    (hf : ∀ j, tsupport (f j) ⊆ S) (k : ℕ) :
-    tsupport (fun z => ∑ j ∈ Finset.range k, f j z) ⊆ S := by
-  induction k with
-  | zero => simp
-  | succ k ih =>
-    simp only [Finset.sum_range_succ]
-    exact (tsupport_add _ _).trans (union_subset ih (hf k))
-
-noncomputable def aggregateX (t s : ℝ) (k n : ℕ) (z : ℂ) : ℝ :=
-  ∑ j ∈ Finset.range k, chainX t s j n z
-noncomputable def aggregateY (t s : ℝ) (k n : ℕ) (z : ℂ) : ℝ :=
-  ∑ j ∈ Finset.range k, chainY t s j n z
-
-lemma aggregateX_smooth (t s : ℝ) (k n : ℕ) : ContDiff ℝ ∞ (aggregateX t s k n) :=
-  ContDiff.sum (fun j _ => chainX_smooth t s j n)
-lemma aggregateY_smooth (t s : ℝ) (k n : ℕ) : ContDiff ℝ ∞ (aggregateY t s k n) :=
-  ContDiff.sum (fun j _ => chainY_smooth t s j n)
-lemma aggregateX_compact {t : ℝ} (s : ℝ) (ht : 0 ≤ t) (ht1 : t ≤ 1) (k n : ℕ) :
-    HasCompactSupport (aggregateX t s k n) :=
-  compact_sum_range (fun j => chainX_compact s ht ht1 j n) k
-lemma aggregateY_compact (t : ℝ) {s : ℝ} (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
-    HasCompactSupport (aggregateY t s k n) :=
-  compact_sum_range (fun j => chainY_compact t hs hs1 j n) k
-lemma aggregateX_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
-    tsupport (aggregateX t s k n) ⊆ openUnitStrip :=
-  support_sum_range (fun j => chainX_support ht ht1 hs hs1 j n) k
-lemma aggregateY_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
-    tsupport (aggregateY t s k n) ⊆ openUnitStrip :=
-  support_sum_range (fun j => chainY_support ht ht1 hs hs1 j n) k
-
-lemma aggregate_divergence (t s : ℝ) (k n : ℕ) (z : ℂ) :
-    (fderiv ℝ (aggregateX t s k n) z) 1 + (fderiv ℝ (aggregateY t s k n) z) Complex.I =
-      chainDensity t s k n z - chainDensity t s 0 n z := by
-  unfold aggregateX aggregateY
-  rw [fderiv_fun_sum (fun j _ => ((chainX_smooth t s j n).differentiable (by simp)) z),
-    fderiv_fun_sum (fun j _ => ((chainY_smooth t s j n).differentiable (by simp)) z)]
-  simp only [sum_apply, ← Finset.sum_add_distrib]
-  simp_rw [chain_divergence]
-  exact Finset.sum_range_sub (fun j => chainDensity t s j n z) k
-
-lemma aggregate_kernel_bound {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
-    (∫ z in openUnitStrip, (|aggregateX t s k n z|+|aggregateY t s k n z|)^2) ≤ 360*k := by
-  have hsum : (∫ z in openUnitStrip, ‖∑ j ∈ Finset.range k, chainKernel t s j n z‖^2) ≤
-      3*12*10*k := by
-    apply integral_sum_kernel_sq_le (fun j => chainKernel t s j n) 12 10 k
-      (by norm_num) (by norm_num)
-    · intro j _
-      exact ((chainKernel_continuous t s j n).integrable_of_hasCompactSupport
-        (chainKernel_compact ht ht1 hs hs1 j n)).integrableOn
-    · intro j _ z
-      exact chainKernel_bound t s j n z
-    · intro j _
-      refine (setIntegral_le_integral ?_ (Eventually.of_forall (fun _ => norm_nonneg _))).trans
-        (chainKernel_integral_bound ht ht1 hs hs1 j n)
-      exact ((chainKernel_continuous t s j n).integrable_of_hasCompactSupport
-        (chainKernel_compact ht ht1 hs hs1 j n)).norm
-  have hI : Integrable (fun z => (|aggregateX t s k n z|+|aggregateY t s k n z|)^2)
-      (volume.restrict openUnitStrip) := by
-    apply MemLp.integrable_sq
-    exact ((aggregateX_smooth t s k n).continuous.abs.add
-      (aggregateY_smooth t s k n).continuous.abs).memLp_of_hasCompactSupport
-      ((aggregateX_compact s ht ht1 k n).abs.add (aggregateY_compact t hs hs1 k n).abs)
-  have hJ : Integrable (fun z => ‖∑ j ∈ Finset.range k, chainKernel t s j n z‖^2)
-      (volume.restrict openUnitStrip) := by
-    have hc := continuous_finsetSum (Finset.range k) (fun j _ => chainKernel_continuous t s j n)
-    exact (memLp_two_iff_integrable_sq_norm hc.aestronglyMeasurable).1
-      (hc.memLp_of_hasCompactSupport
-        (compact_sum_range (fun j => chainKernel_compact ht ht1 hs hs1 j n) k))
-  refine (integral_mono hI hJ ?_).trans (by linarith)
-  intro z
-  have hh : |aggregateX t s k n z|+|aggregateY t s k n z| ≤
-      ∑ j ∈ Finset.range k, chainKernel t s j n z := by
-    unfold aggregateX aggregateY chainKernel
-    rw [Finset.sum_add_distrib]
-    exact add_le_add (Finset.abs_sum_le_sum_abs _ _) (Finset.abs_sum_le_sum_abs _ _)
-  apply (sq_le_sq₀ (by positivity) (norm_nonneg _)).2
-  exact hh.trans (le_abs_self _)
-
-lemma chainDensity_continuous (t s : ℝ) (j n : ℕ) : Continuous (chainDensity t s j n) :=
-  (complexTensor_contDiff (intervalDensity_smooth ..) (intervalDensity_smooth ..)).continuous
-lemma chainDensity_compact (t s : ℝ) (j n : ℕ) : HasCompactSupport (chainDensity t s j n) :=
-  complexTensor_compact (intervalDensity_compact ..) (intervalDensity_compact ..)
-
-lemma smoothed_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
-    (hu : SourceIsStripWeakGradient u D) {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
-    |(∫ z in openUnitStrip, u z * chainDensity t s k n z) -
-      (∫ z in openUnitStrip, u z * chainDensity t s 0 n z)| ≤
-      20 * Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D) := by
-  have he := weak_gradient_pairing hu (aggregateX_smooth t s k n) (aggregateY_smooth t s k n)
-    (aggregateX_compact s ht ht1 k n) (aggregateY_compact t hs hs1 k n)
-    (aggregateX_support ht ht1 hs hs1 k n) (aggregateY_support ht ht1 hs hs1 k n)
-  simp_rw [aggregate_divergence] at he
-  have hi₁ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
-    (chainDensity_continuous t s k n) (chainDensity_compact t s k n)
-  have hi₀ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
-    (chainDensity_continuous t s 0 n) (chainDensity_compact t s 0 n)
-  simp_rw [mul_sub] at he
-  rw [integral_sub hi₁ hi₀] at he
-  rw [he, abs_neg]
-  refine (integral_pairing_le hu.2.1 (aggregateX_smooth t s k n).continuous
-    (aggregateY_smooth t s k n).continuous (aggregateX_compact s ht ht1 k n)
-    (aggregateY_compact t hs hs1 k n)).trans ?_
-  have hE : 0 ≤ stripDirichletEnergy D := integral_nonneg (fun _ => sq_nonneg _)
-  have hA : 0 ≤ ∫ z in openUnitStrip, (|aggregateX t s k n z|+|aggregateY t s k n z|)^2 :=
-    integral_nonneg (fun _ => sq_nonneg _)
-  have hb := aggregate_kernel_bound ht ht1 hs hs1 k n
-  have hR : 0 ≤ ((k:ℝ)+1)*stripDirichletEnergy D := by positivity
-  have hSqE := Real.sq_sqrt hE
-  have hSqA := Real.sq_sqrt hA
-  have hSqR := Real.sq_sqrt hR
-  apply (sq_le_sq₀ (by positivity) (by positivity)).1
-  rw [mul_pow, hSqE, hSqA, mul_pow, hSqR]
-  nlinarith [mul_le_mul_of_nonneg_left hb hE]
-
-def stripRectangle (a b l : ℝ) : Set ℂ :=
-  {w | a ≤ w.re ∧ w.re ≤ a+l ∧ b ≤ w.im ∧ w.im ≤ b+Real.pi*l}
-
-lemma stripRectangle_eq_preimage (a b l : ℝ) :
-    stripRectangle a b l = Complex.measurableEquivRealProd ⁻¹'
-      (Icc a (a+l) ×ˢ Icc b (b+Real.pi*l)) := by
-  ext z
-  simp only [stripRectangle, mem_ofPred_eq, mem_preimage, mem_prod, mem_Icc,
-    Complex.measurableEquivRealProd_apply]
-  tauto
-
-lemma stripRectangle_volume (a b l : ℝ) :
-    volume (stripRectangle a b l) = ENNReal.ofReal l * ENNReal.ofReal (Real.pi*l) := by
-  rw [stripRectangle_eq_preimage,
-    Complex.volume_preserving_equiv_real_prod.measure_preimage
-      (measurableSet_Icc.prod measurableSet_Icc).nullMeasurableSet,
-    Measure.volume_eq_prod, Measure.prod_prod]
-  simp only [Real.volume_Icc, add_sub_cancel_left]
-
-lemma stripRectangle_integral_mul (a b l : ℝ) (f g : ℝ → ℝ) :
-    (∫ z in stripRectangle a b l, f z.re * g z.im) =
-      (∫ x in Icc a (a+l), f x) * (∫ y in Icc b (b+Real.pi*l), g y) := by
-  rw [stripRectangle_eq_preimage]
-  have he := (Complex.volume_preserving_equiv_real_prod.restrict_preimage
-    (s := Icc a (a+l) ×ˢ Icc b (b+Real.pi*l))
-    (measurableSet_Icc.prod measurableSet_Icc)).integral_comp
-    Complex.measurableEquivRealProd.measurableEmbedding
-    (fun p : ℝ × ℝ => f p.1 * g p.2)
-  simp only [Complex.measurableEquivRealProd_apply] at he
-  rw [he]
-  rw [Measure.volume_eq_prod, ← Measure.prod_restrict]
-  exact integral_prod_mul f g
-
-lemma stripRectangle_integral_re (a b l : ℝ) (hl : 0 ≤ l) :
-    (∫ z in stripRectangle a b l, z.re) =
-      (a*l+l^2/2)*(Real.pi*l) := by
-  have h := stripRectangle_integral_mul a b l (fun x => x) (fun _ => 1)
-  simp only [mul_one] at h
-  rw [h, integral_Icc_eq_integral_Ioc,
-    ← intervalIntegral.integral_of_le (by linarith : a ≤ a+l), integral_id]
-  simp only [integral_const, Measure.real, Measure.restrict_apply_univ, Real.volume_Icc,
-    add_sub_cancel_left, smul_eq_mul, mul_one]
-  rw [ENNReal.toReal_ofReal (mul_nonneg Real.pi_pos.le hl)]
-  ring
-
-noncomputable def spatialAverage (u : ℂ → ℝ) (Q : Set ℂ) : ℝ :=
-  (∫ z in Q, u z) / (volume Q).toReal
-
-def openStripRectangle (a b l : ℝ) : Set ℂ :=
-  {z | a < z.re ∧ z.re < a+l ∧ b < z.im ∧ z.im < b+Real.pi*l}
-
-lemma openStripRectangle_eq_preimage (a b l : ℝ) :
-    openStripRectangle a b l = Complex.measurableEquivRealProd ⁻¹'
-      (Ioo a (a+l) ×ˢ Ioo b (b+Real.pi*l)) := by
-  ext z
-  simp only [openStripRectangle, mem_ofPred_eq, mem_preimage, mem_prod, mem_Ioo,
-    Complex.measurableEquivRealProd_apply]
-  tauto
-
-lemma openStripRectangle_measurable (a b l : ℝ) : MeasurableSet (openStripRectangle a b l) := by
-  rw [openStripRectangle_eq_preimage]
-  exact (measurableSet_Ioo.prod measurableSet_Ioo).preimage
-    Complex.measurableEquivRealProd.measurable
-
-lemma openStripRectangle_ae_eq (a b l : ℝ) :
-    openStripRectangle a b l =ᵐ[volume] stripRectangle a b l := by
-  rw [openStripRectangle_eq_preimage, stripRectangle_eq_preimage]
-  apply Complex.volume_preserving_equiv_real_prod.quasiMeasurePreserving.preimage_ae_eq
-  rw [Measure.volume_eq_prod]
-  exact Measure.set_prod_ae_eq Ioo_ae_eq_Icc Ioo_ae_eq_Icc
-
-lemma rectangleDensity_nonneg (a b lx ly : ℝ) (hx : 0 < lx) (hy : 0 < ly)
-    (n : ℕ) (z : ℂ) : 0 ≤ rectangleDensity a b lx ly hx hy n z :=
-  mul_nonneg (intervalDensity_nonneg ..) (intervalDensity_nonneg ..)
-lemma rectangleDensity_bound (a b lx ly : ℝ) (hx : 0 < lx) (hy : 0 < ly)
-    (n : ℕ) (z : ℂ) : rectangleDensity a b lx ly hx hy n z ≤ 4/(lx*ly) := by
-  calc
-    _ ≤ (2/lx)*(2/ly) := mul_le_mul (intervalDensity_bound ..) (intervalDensity_bound ..)
-      (intervalDensity_nonneg ..) (by positivity)
-    _ = _ := by ring
-
-lemma rectangleDensity_tendsto (a b l : ℝ) (hl : 0 < l) (z : ℂ) :
-    Tendsto (fun n => rectangleDensity a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) n z)
-      atTop (𝓝 ((openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z)) := by
-  have hh := (intervalDensity_tendsto a l hl z.re).mul
-    (intervalDensity_tendsto b (Real.pi*l) (mul_pos Real.pi_pos hl) z.im)
-  have heq : (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z =
-      (Ioo a (a+l)).indicator (fun _ => 1/l) z.re *
-        (Ioo b (b+Real.pi*l)).indicator (fun _ => 1/(Real.pi*l)) z.im := by
-    by_cases hx : z.re ∈ Ioo a (a+l) <;> by_cases hy : z.im ∈ Ioo b (b+Real.pi*l)
-    · have hz : z ∈ openStripRectangle a b l := ⟨hx.1,hx.2,hy.1,hy.2⟩
-      simp [hz,hx,hy,one_div]
-      ring
-    · have hz : z ∉ openStripRectangle a b l := fun hz => hy ⟨hz.2.2.1,hz.2.2.2⟩
-      simp [hz,hx,hy]
-    · have hz : z ∉ openStripRectangle a b l := fun hz => hx ⟨hz.1,hz.2.1⟩
-      simp [hz,hx,hy]
-    · have hz : z ∉ openStripRectangle a b l := fun hz => hx ⟨hz.1,hz.2.1⟩
-      simp [hz,hx,hy]
-  simpa only [rectangleDensity, complexTensor, heq] using hh
-
-lemma rectangle_smoothed_mean_tendsto {u : ℂ → ℝ}
-    (hu : MemLp u 2 (volume.restrict openUnitStrip)) (a b l : ℝ) (hl : 0 < l)
-    (ha : 0 ≤ a) (ha1 : a+l ≤ 1) (hb : 0 ≤ b) (hb1 : b+Real.pi*l ≤ Real.pi) :
-    Tendsto (fun n => ∫ z in openUnitStrip,
-      u z * rectangleDensity a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) n z)
-      atTop (𝓝 (spatialAverage u (stripRectangle a b l))) := by
-  have hsub : openStripRectangle a b l ⊆ openUnitStrip := by
-    intro z hz
-    exact ⟨ha.trans_lt hz.1, hz.2.1.trans_le ha1, hb.trans_lt hz.2.2.1, hz.2.2.2.trans_le hb1⟩
-  have hint := hu.integrable (by norm_num)
-  have hlim := tendsto_integral_of_dominated_convergence
-    (μ := volume.restrict openUnitStrip)
-    (F := fun n z => u z * rectangleDensity a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) n z)
-    (f := fun z => u z * (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z)
-    (fun z => |u z| * (4/(l*(Real.pi*l))))
-    (fun n => hu.aestronglyMeasurable.mul
-      ((complexTensor_contDiff (intervalDensity_smooth ..) (intervalDensity_smooth ..)).continuous.aestronglyMeasurable))
-    (hint.abs.mul_const _) (fun n => Eventually.of_forall fun z => by
-      rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (rectangleDensity_nonneg ..)]
-      exact mul_le_mul_of_nonneg_left (rectangleDensity_bound ..) (abs_nonneg _))
-    (Eventually.of_forall fun z => (rectangleDensity_tendsto a b l hl z).const_mul (u z))
-  have heq : (∫ z in openUnitStrip,
-      u z * (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z) =
-      spatialAverage u (stripRectangle a b l) := by
-    have hfun : (fun z => u z * (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z) =
-        (openStripRectangle a b l).indicator (fun z => u z * (1/(l*(Real.pi*l)))) := by
-      funext z
-      by_cases hz : z ∈ openStripRectangle a b l <;> simp [hz]
-    rw [hfun, integral_indicator (openStripRectangle_measurable a b l),
-      Measure.restrict_restrict (openStripRectangle_measurable a b l), inter_eq_left.mpr hsub,
-      integral_mul_const, setIntegral_congr_set (openStripRectangle_ae_eq a b l)]
-    unfold spatialAverage
-    rw [stripRectangle_volume, ENNReal.toReal_mul, ENNReal.toReal_ofReal hl.le,
-      ENNReal.toReal_ofReal (mul_pos Real.pi_pos hl).le]
-    ring
-  rwa [heq] at hlim
-
-def dyadicStripRectangle (j : ℤ) (k : ℕ) (i m : Fin (2^k)) : Set ℂ :=
-  stripRectangle ((j:ℝ)+(i:ℝ)/(2:ℝ)^k)
-    (Real.pi*((m:ℝ)/(2:ℝ)^k)) (1/(2:ℝ)^k)
-
- 
-
-def SourceCriticalStripMeansTarget : Prop :=
-  ∃ K : ℝ, 0 < K ∧ ∀ (u : ℂ → ℝ) (D : ℂ → ℂ),
-    SourceIsStripWeakGradient u D → ContinuousOn u (stripRectangle 0 0 1) →
-    ∀ (k : ℕ) (i m : Fin (2^k)),
-      |spatialAverage u (dyadicStripRectangle 0 k i m) -
-        spatialAverage u (stripRectangle 0 0 1)| ≤
-      K * Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D)
-
-lemma chain_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
-    (hu : SourceIsStripWeakGradient u D) {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k : ℕ) :
-    |spatialAverage u (stripRectangle (scaleOrigin t k) (Real.pi*scaleOrigin s k) (scaleLength k)) -
-      spatialAverage u (stripRectangle 0 0 1)| ≤
-      20 * Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D) := by
-  have hlim (j : ℕ) : Tendsto (fun n => ∫ z in openUnitStrip, u z * chainDensity t s j n z)
-      atTop (𝓝 (spatialAverage u (stripRectangle (scaleOrigin t j)
-        (Real.pi*scaleOrigin s j) (scaleLength j)))) := by
-    apply rectangle_smoothed_mean_tendsto hu.1 _ _ _ (scaleLength_pos j)
-      (scaleOrigin_nonneg ht j) (scaleOrigin_add_le_one ht1 j)
-      (mul_nonneg Real.pi_pos.le (scaleOrigin_nonneg hs j))
-    have hh := mul_le_mul_of_nonneg_left (scaleOrigin_add_le_one hs1 j) Real.pi_pos.le
-    nlinarith
-  have hzero := hlim 0
-  simp only [scaleOrigin_zero, scaleLength_zero, mul_zero] at hzero
-  exact le_of_tendsto ((hlim k).sub hzero).abs
-    (Eventually.of_forall (fun n => smoothed_means_le hu ht ht1 hs hs1 k n))
-
-theorem sourceCriticalStripMeans : SourceCriticalStripMeansTarget := by
-  refine ⟨20, by norm_num, ?_⟩
-  intro u D hu _ k i m
-  by_cases hk : k = 0
-  · subst k
-    have hi : i = 0 := by apply Fin.ext; have := i.isLt; norm_num at this ⊢
-    have hm : m = 0 := by apply Fin.ext; have := m.isLt; norm_num at this ⊢
-    subst i; subst m
-    simp [dyadicStripRectangle]
-  · obtain ⟨t, ht, ht1, hit⟩ := dyadic_position_param k hk i
-    obtain ⟨s, hs, hs1, hms⟩ := dyadic_position_param k hk m
-    have hh := chain_means_le hu ht ht1 hs hs1 k
-    simpa only [dyadicStripRectangle, Int.cast_zero, zero_add, hit, hms, scaleLength] using hh
-
-lemma integral_pairing_local_le {D : ℂ → ℂ}
-    (hD : MemLp D 2 (volume.restrict openUnitStrip)) {V : Set ℂ}
-    (hV : V ⊆ openUnitStrip) {X Y : ℂ → ℝ}
-    (hX : Continuous X) (hY : Continuous Y)
-    (hcX : HasCompactSupport X) (hcY : HasCompactSupport Y)
-    (hsX : tsupport X ⊆ V) (hsY : tsupport Y ⊆ V) :
-    |∫ z in openUnitStrip, (D z).re * X z + (D z).im * Y z| ≤
-      Real.sqrt (∫ z in V, ‖D z‖^2) *
-        Real.sqrt (∫ z in V, (|X z|+|Y z|)^2) := by
-  have hm : volume.restrict V ≤ volume.restrict openUnitStrip :=
-    Measure.restrict_mono hV le_rfl
-  have hDl := hD.mono_measure hm
-  have hDi := (hD.integrable (by norm_num)).mono_measure hm
-  have hdx := integrable_mul_compact_test (Complex.reCLM.integrable_comp hDi) hX hcX
-  have hdy := integrable_mul_compact_test (Complex.imCLM.integrable_comp hDi) hY hcY
-  have hK : MemLp (fun z => |X z|+|Y z|) 2 (volume.restrict V) :=
-    (hX.abs.add hY.abs).memLp_of_hasCompactSupport (hcX.abs.add hcY.abs)
-  rw [setIntegral_eq_of_subset_of_forall_sdiff_eq_zero openUnitStrip_isOpen.measurableSet hV
-    (fun z hz => by
-      rw [image_eq_zero_of_notMem_tsupport (fun h => hz.2 (hsX h)),
-        image_eq_zero_of_notMem_tsupport (fun h => hz.2 (hsY h))]
-      ring)]
-  calc
-    _ ≤ ∫ z in V, |(D z).re * X z + (D z).im * Y z| := by
-      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm
-        (fun z => (D z).re * X z + (D z).im * Y z)
-    _ ≤ ∫ z in V, ‖D z‖ * (|X z|+|Y z|) :=
-      integral_mono (hdx.add hdy).abs (hDl.norm.integrable_mul hK)
-        (fun z => complex_pairing_abs_le ..)
-    _ ≤ _ := integral_nonneg_cauchy hDl.norm hK (fun _ => norm_nonneg _)
-      (fun _ => add_nonneg (abs_nonneg _) (abs_nonneg _))
-
-lemma rectangle_transport_L2_bound (a b l : ℝ) (hl : 0 < l) (a' b' : ℝ)
-    (ha : a ≤ a') (ha' : a'+l/2 ≤ a+l)
-    (hb : b ≤ b') (hb' : b'+(Real.pi*l)/2 ≤ b+Real.pi*l) (n : ℕ) (V : Set ℂ) :
-    (∫ z in V, (|rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z| +
-      |rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z|)^2) ≤ 120 := by
-  let X := rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
-  let Y := rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
-  have hx := (rectangleTransportX_contDiff a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n).continuous
-  have hy := (rectangleTransportY_contDiff a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n).continuous
-  have hcx := rectangleTransportX_compact a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' ha ha' n
-  have hcy := rectangleTransportY_compact a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' hb hb' n
-  have hK : Continuous (fun z => |X z|+|Y z|) := hx.abs.add hy.abs
-  have hcK : HasCompactSupport (fun z => |X z|+|Y z|) := hcx.abs.add hcy.abs
-  have hKi := hK.integrable_of_hasCompactSupport (μ := volume) hcK
-  have hKsq : Integrable (fun z => (|X z|+|Y z|)^2) volume :=
-    (hK.memLp_of_hasCompactSupport (p := 2) hcK).integrable_sq
-  have hK0 (z : ℂ) : 0 ≤ |X z|+|Y z| := by positivity
-  have hKb (z : ℂ) : |X z|+|Y z| ≤ 12/l := by
-    have hx' := rectangleTransportX_bound a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z
-    have hy' := rectangleTransportY_bound a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z
-    have hp : l ≤ Real.pi*l := by nlinarith [Real.pi_gt_three]
-    have hx'' := hx'.trans (div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 8) hl hp)
-    change |X z| ≤ _ at hx''
-    change |Y z| ≤ _ at hy'
-    calc
-      _ ≤ 8/l+4/l := add_le_add hx'' hy'
-      _ = 12/l := by ring
-  have hKI : (∫ z, |X z|+|Y z|) ≤ 10*l := by
-    rw [integral_add (hx.integrable_of_hasCompactSupport hcx).abs
-      (hy.integrable_of_hasCompactSupport hcy).abs]
-    have hx' := rectangleTransportX_integral_abs a b l (Real.pi*l) hl
-      (mul_pos Real.pi_pos hl) a' b' ha ha' n
-    have hy' := rectangleTransportY_integral_abs a b l (Real.pi*l) hl
-      (mul_pos Real.pi_pos hl) a' b' hb hb' n
-    change (∫ z, |X z|) ≤ _ at hx'
-    change (∫ z, |Y z|) ≤ _ at hy'
-    nlinarith [Real.pi_lt_four]
-  change (∫ z in V, (|X z|+|Y z|)^2) ≤ _
-  calc
-    _ ≤ ∫ z, (|X z|+|Y z|)^2 :=
-      setIntegral_le_integral hKsq (Eventually.of_forall (fun _ => sq_nonneg _))
-    _ ≤ ∫ z, (12/l)*(|X z|+|Y z|) := integral_mono hKsq (hKi.const_mul _) (fun z => by
-      nlinarith [mul_le_mul_of_nonneg_right (hKb z) (hK0 z)])
-    _ = (12/l)*∫ z, |X z|+|Y z| := integral_const_mul ..
-    _ ≤ (12/l)*(10*l) := mul_le_mul_of_nonneg_left hKI (by positivity)
-    _ = 120 := by field_simp; ring
-
-lemma openStripRectangle_subset_unit {a b l : ℝ}
-    (ha : 0 ≤ a) (ha1 : a+l ≤ 1) (hb : 0 ≤ b) (hb1 : b+Real.pi*l ≤ Real.pi) :
-    openStripRectangle a b l ⊆ openUnitStrip := by
-  intro z hz
-  exact ⟨ha.trans_lt hz.1, hz.2.1.trans_le ha1,
-    hb.trans_lt hz.2.2.1, hz.2.2.2.trans_le hb1⟩
-
-lemma local_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
-    (hu : SourceIsStripWeakGradient u D) (a b l : ℝ) (hl : 0 < l)
-    (ha : 0 ≤ a) (ha1 : a+l ≤ 1) (hb : 0 ≤ b) (hb1 : b+Real.pi*l ≤ Real.pi)
-    (a' b' : ℝ) (haa : a ≤ a') (haa' : a'+l/2 ≤ a+l)
-    (hbb : b ≤ b') (hbb' : b'+(Real.pi*l)/2 ≤ b+Real.pi*l) :
-    |spatialAverage u (stripRectangle a' b' (l/2)) -
-      spatialAverage u (stripRectangle a b l)| ≤
-      12 * Real.sqrt (∫ z in openStripRectangle a b l, ‖D z‖^2) := by
-  have hsub := openStripRectangle_subset_unit ha ha1 hb hb1
-  have hsmooth (n : ℕ) :
-      |(∫ z in openUnitStrip, u z * rectangleDensity a' b' (l/2) (Real.pi*(l/2))
-          (by positivity) (by positivity) n z) -
-        (∫ z in openUnitStrip, u z * rectangleDensity a b l (Real.pi*l)
-          hl (mul_pos Real.pi_pos hl) n z)| ≤
-      12 * Real.sqrt (∫ z in openStripRectangle a b l, ‖D z‖^2) := by
-    let X := rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
-    let Y := rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
-    have hx : ContDiff ℝ ∞ X := rectangleTransportX_contDiff ..
-    have hy : ContDiff ℝ ∞ Y := rectangleTransportY_contDiff ..
-    have hcx : HasCompactSupport X := rectangleTransportX_compact _ _ _ _ _ _ _ _ haa haa' n
-    have hcy : HasCompactSupport Y := rectangleTransportY_compact _ _ _ _ _ _ _ _ hbb hbb' n
-    have hsx : tsupport X ⊆ openStripRectangle a b l :=
-      rectangleTransportX_support _ _ _ _ _ _ _ _ haa haa' hbb hbb' n
-    have hsy : tsupport Y ⊆ openStripRectangle a b l :=
-      rectangleTransportY_support _ _ _ _ _ _ _ _ hbb hbb' n
-    have he := weak_gradient_pairing hu hx hy hcx hcy (hsx.trans hsub) (hsy.trans hsub)
-    change (∫ z in openUnitStrip, u z *
-      ((fderiv ℝ (rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n) z) 1 +
-       (fderiv ℝ (rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n) z) Complex.I)) = _ at he
-    simp_rw [rectangleTransport_divergence, show Real.pi*l/2 = Real.pi*(l/2) by ring, mul_sub] at he
-    have hi₁ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
-      (complexTensor_contDiff (intervalDensity_smooth a' (l/2) (by positivity) n)
-        (intervalDensity_smooth b' (Real.pi*(l/2)) (by positivity) n)).continuous
-      (complexTensor_compact (intervalDensity_compact ..) (intervalDensity_compact ..))
-    have hi₀ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
-      (complexTensor_contDiff (intervalDensity_smooth a l hl n)
-        (intervalDensity_smooth b (Real.pi*l) (mul_pos Real.pi_pos hl) n)).continuous
-      (complexTensor_compact (intervalDensity_compact ..) (intervalDensity_compact ..))
-    dsimp only [rectangleDensity] at he ⊢
-    rw [integral_sub hi₁ hi₀] at he
-    rw [he, abs_neg]
-    refine (integral_pairing_local_le hu.2.1 hsub hx.continuous hy.continuous hcx hcy hsx hsy).trans ?_
-    have hbnd := rectangle_transport_L2_bound a b l hl a' b' haa haa' hbb hbb' n
-      (openStripRectangle a b l)
-    change (∫ z in openStripRectangle a b l, (|X z|+|Y z|)^2) ≤ 120 at hbnd
-    have hE : 0 ≤ ∫ z in openStripRectangle a b l, ‖D z‖^2 := integral_nonneg (fun _ => sq_nonneg _)
-    have hA : 0 ≤ ∫ z in openStripRectangle a b l, (|X z|+|Y z|)^2 := integral_nonneg (fun _ => sq_nonneg _)
-    apply (sq_le_sq₀ (by positivity) (by positivity)).1
-    rw [mul_pow, Real.sq_sqrt hE, Real.sq_sqrt hA, mul_pow, Real.sq_sqrt hE]
-    nlinarith [mul_le_mul_of_nonneg_left hbnd hE]
-  have hlim₀ := rectangle_smoothed_mean_tendsto hu.1 a b l hl ha ha1 hb hb1
-  have hlim₁ := rectangle_smoothed_mean_tendsto hu.1 a' b' (l/2) (by positivity)
-    (ha.trans haa) (haa'.trans ha1) (hb.trans hbb) (by nlinarith)
-  exact le_of_tendsto (hlim₁.sub hlim₀).abs (Eventually.of_forall hsmooth)
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.IntervalDensity
+
+namespace OAI
+
+/-! Strip Kernels for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology ContDiff ENNReal BigOperators NNReal
+namespace SLEExactGauge
+lemma integrable_mul_compact_test {μ : Measure ℂ} {f g : ℂ → ℝ}
+    (hf : Integrable f μ) (hg : Continuous g) (hgc : HasCompactSupport g) :
+    Integrable (fun z => f z * g z) μ := by
+  obtain ⟨C, hC⟩ := (hgc.isCompact_range hg).isBounded.exists_norm_le
+  exact hf.mul_bdd hg.aestronglyMeasurable
+    (Eventually.of_forall fun z => hC _ ⟨z,rfl⟩)
+
+lemma weak_gradient_pairing {u : ℂ → ℝ} {D : ℂ → ℂ}
+    (hu : SourceIsStripWeakGradient u D) {X Y : ℂ → ℝ}
+    (hX : ContDiff ℝ ∞ X) (hY : ContDiff ℝ ∞ Y)
+    (hcX : HasCompactSupport X) (hcY : HasCompactSupport Y)
+    (hsX : tsupport X ⊆ openUnitStrip) (hsY : tsupport Y ⊆ openUnitStrip) :
+    (∫ z in openUnitStrip, u z * ((fderiv ℝ X z) 1 + (fderiv ℝ Y z) Complex.I)) =
+      -(∫ z in openUnitStrip, (D z).re * X z + (D z).im * Y z) := by
+  have hui := hu.1.integrable (by norm_num)
+  have hDi := hu.2.1.integrable (by norm_num)
+  have hx := integrable_mul_compact_test hui
+    ((hX.continuous_fderiv (by simp)).clm_apply continuous_const) (hcX.fderiv_apply ℝ 1)
+  have hy := integrable_mul_compact_test hui
+    ((hY.continuous_fderiv (by simp)).clm_apply continuous_const) (hcY.fderiv_apply ℝ Complex.I)
+  have hdx := integrable_mul_compact_test (Complex.reCLM.integrable_comp hDi) hX.continuous hcX
+  have hdy := integrable_mul_compact_test (Complex.imCLM.integrable_comp hDi) hY.continuous hcY
+  change Integrable (fun z => (D z).re * X z) _ at hdx
+  change Integrable (fun z => (D z).im * Y z) _ at hdy
+  simp_rw [mul_add]
+  rw [integral_add hx hy, (hu.2.2 X hX hcX hsX).1, (hu.2.2 Y hY hcY hsY).2,
+    integral_add hdx hdy]
+  ring
+
+lemma complex_pairing_abs_le (D : ℂ) (X Y : ℝ) :
+    |D.re * X + D.im * Y| ≤ ‖D‖ * (|X|+|Y|) := by
+  calc
+    _ ≤ |D.re * X| + |D.im * Y| := abs_add_le _ _
+    _ ≤ ‖D‖ * |X| + ‖D‖ * |Y| := by
+      rw [abs_mul, abs_mul]
+      exact add_le_add (mul_le_mul_of_nonneg_right (Complex.abs_re_le_norm D) (abs_nonneg _))
+        (mul_le_mul_of_nonneg_right (Complex.abs_im_le_norm D) (abs_nonneg _))
+    _ = _ := by ring
+
+lemma integral_nonneg_cauchy {μ : Measure ℂ} {f g : ℂ → ℝ}
+    (hf : MemLp f 2 μ) (hg : MemLp g 2 μ)
+    (hf0 : ∀ z, 0 ≤ f z) (hg0 : ∀ z, 0 ≤ g z) :
+    (∫ z, f z * g z ∂μ) ≤
+      Real.sqrt (∫ z, (f z)^2 ∂μ) * Real.sqrt (∫ z, (g z)^2 ∂μ) := by
+  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two
+    (Eventually.of_forall hf0) (Eventually.of_forall hg0)
+    (by simpa using hf) (by simpa using hg)
+  simpa only [Real.rpow_two, Real.sqrt_eq_rpow] using hh
+
+lemma integral_pairing_le {D : ℂ → ℂ}
+    (hD : MemLp D 2 (volume.restrict openUnitStrip)) {X Y : ℂ → ℝ}
+    (hX : Continuous X) (hY : Continuous Y)
+    (hcX : HasCompactSupport X) (hcY : HasCompactSupport Y) :
+    |∫ z in openUnitStrip, (D z).re * X z + (D z).im * Y z| ≤
+      Real.sqrt (stripDirichletEnergy D) *
+        Real.sqrt (∫ z in openUnitStrip, (|X z|+|Y z|)^2) := by
+  have hDi := hD.integrable (by norm_num)
+  have hdx := integrable_mul_compact_test (Complex.reCLM.integrable_comp hDi) hX hcX
+  have hdy := integrable_mul_compact_test (Complex.imCLM.integrable_comp hDi) hY hcY
+  have hK : MemLp (fun z => |X z|+|Y z|) 2 (volume.restrict openUnitStrip) :=
+    (hX.abs.add hY.abs).memLp_of_hasCompactSupport (hcX.abs.add hcY.abs)
+  calc
+    _ ≤ ∫ z in openUnitStrip, |(D z).re * X z + (D z).im * Y z| := by
+      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm
+        (fun z => (D z).re * X z + (D z).im * Y z)
+    _ ≤ ∫ z in openUnitStrip, ‖D z‖ * (|X z|+|Y z|) :=
+      integral_mono (hdx.add hdy).abs (hD.norm.integrable_mul hK)
+        (fun z => complex_pairing_abs_le ..)
+    _ ≤ _ := integral_nonneg_cauchy hD.norm hK (fun _ => norm_nonneg _)
+      (fun _ => add_nonneg (abs_nonneg _) (abs_nonneg _))
+
+noncomputable def chainDensity (t s : ℝ) (j n : ℕ) : ℂ → ℝ :=
+  rectangleDensity (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j)) n
+noncomputable def chainX (t s : ℝ) (j n : ℕ) : ℂ → ℝ :=
+  rectangleTransportX (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n
+noncomputable def chainY (t s : ℝ) (j n : ℕ) : ℂ → ℝ :=
+  rectangleTransportY (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n
+noncomputable def chainKernel (t s : ℝ) (j n : ℕ) (z : ℂ) : ℝ :=
+  |chainX t s j n z|+|chainY t s j n z|
+
+lemma chainX_smooth (t s : ℝ) (j n : ℕ) : ContDiff ℝ ∞ (chainX t s j n) :=
+  rectangleTransportX_contDiff ..
+lemma chainY_smooth (t s : ℝ) (j n : ℕ) : ContDiff ℝ ∞ (chainY t s j n) :=
+  rectangleTransportY_contDiff ..
+lemma chainX_compact {t : ℝ} (s : ℝ) (ht : 0 ≤ t) (ht1 : t ≤ 1) (j n : ℕ) :
+    HasCompactSupport (chainX t s j n) :=
+  rectangleTransportX_compact _ _ _ _ _ _ _ _
+    (scaleOrigin_step ht j) (scaleOrigin_right_step ht1 j) n
+lemma chainY_compact (t : ℝ) {s : ℝ} (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
+    HasCompactSupport (chainY t s j n) := by
+  apply rectangleTransportY_compact
+  · exact mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le
+  · have hh := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
+    nlinarith
+
+lemma chainX_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
+    tsupport (chainX t s j n) ⊆ openUnitStrip := by
+  have hh := rectangleTransportX_support (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
+    (scaleOrigin_step ht j) (scaleOrigin_right_step ht1 j)
+    (mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le)
+    (by have := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
+        nlinarith) n
+  intro z hz
+  obtain ⟨h1,h2,h3,h4⟩ := hh hz
+  have h5 := scaleOrigin_nonneg ht j
+  have h6 := scaleOrigin_add_le_one ht1 j
+  have h7 := mul_nonneg Real.pi_pos.le (scaleOrigin_nonneg hs j)
+  have h8 := mul_le_mul_of_nonneg_left (scaleOrigin_add_le_one hs1 j) Real.pi_pos.le
+  exact ⟨by linarith, by linarith, by linarith, by nlinarith⟩
+
+lemma chainY_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
+    tsupport (chainY t s j n) ⊆ openUnitStrip := by
+  have hh := rectangleTransportY_support (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
+    (mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le)
+    (by have := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
+        nlinarith) n
+  intro z hz
+  obtain ⟨h1,h2,h3,h4⟩ := hh hz
+  have h5 := scaleOrigin_nonneg ht j
+  have h6 := scaleOrigin_add_le_one ht1 j
+  have h7 := mul_nonneg Real.pi_pos.le (scaleOrigin_nonneg hs j)
+  have h8 := mul_le_mul_of_nonneg_left (scaleOrigin_add_le_one hs1 j) Real.pi_pos.le
+  exact ⟨by linarith, by linarith, by linarith, by nlinarith⟩
+
+lemma chain_divergence (t s : ℝ) (j n : ℕ) (z : ℂ) :
+    (fderiv ℝ (chainX t s j n) z) 1 + (fderiv ℝ (chainY t s j n) z) Complex.I =
+      chainDensity t s (j+1) n z - chainDensity t s j n z := by
+  unfold chainX chainY chainDensity
+  rw [rectangleTransport_divergence]
+  simp only [scaleLength_succ, mul_div_assoc]
+
+lemma chainKernel_continuous (t s : ℝ) (j n : ℕ) : Continuous (chainKernel t s j n) :=
+  (chainX_smooth t s j n).continuous.abs.add (chainY_smooth t s j n).continuous.abs
+lemma chainKernel_compact {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
+    HasCompactSupport (chainKernel t s j n) :=
+  (chainX_compact s ht ht1 j n).abs.add (chainY_compact t hs hs1 j n).abs
+lemma chainKernel_nonneg (t s : ℝ) (j n : ℕ) (z : ℂ) : 0 ≤ chainKernel t s j n z :=
+  add_nonneg (abs_nonneg _) (abs_nonneg _)
+
+lemma chainKernel_bound (t s : ℝ) (j n : ℕ) (z : ℂ) :
+    ‖chainKernel t s j n z‖ ≤ 12 * (2:ℝ)^j := by
+  rw [Real.norm_of_nonneg (chainKernel_nonneg ..)]
+  have hx := rectangleTransportX_bound (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n z
+  have hy := rectangleTransportY_bound (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1)) n z
+  have hp : scaleLength j ≤ Real.pi*scaleLength j := by
+    nlinarith [scaleLength_pos j, Real.pi_gt_three]
+  have hx' := hx.trans (div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 8)
+    (scaleLength_pos j) hp)
+  change |chainX t s j n z| + |chainY t s j n z| ≤ _
+  calc
+    _ ≤ 8/scaleLength j + 4/scaleLength j := add_le_add hx' hy
+    _ = _ := by unfold scaleLength; field_simp; ring
+
+lemma chainKernel_integral_bound {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (j n : ℕ) :
+    (∫ z, ‖chainKernel t s j n z‖) ≤ 10 / (2:ℝ)^j := by
+  have hx := rectangleTransportX_integral_abs (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
+    (scaleOrigin_step ht j) (scaleOrigin_right_step ht1 j) n
+  have hy := rectangleTransportY_integral_abs (scaleOrigin t j) (Real.pi*scaleOrigin s j)
+    (scaleLength j) (Real.pi*scaleLength j) (scaleLength_pos j)
+    (mul_pos Real.pi_pos (scaleLength_pos j))
+    (scaleOrigin t (j+1)) (Real.pi*scaleOrigin s (j+1))
+    (mul_le_mul_of_nonneg_left (scaleOrigin_step hs j) Real.pi_pos.le)
+    (by have := mul_le_mul_of_nonneg_left (scaleOrigin_right_step hs1 j) Real.pi_pos.le
+        nlinarith) n
+  have hiX := (chainX_smooth t s j n).continuous.integrable_of_hasCompactSupport (μ := volume)
+    (chainX_compact s ht ht1 j n)
+  have hiY := (chainY_smooth t s j n).continuous.integrable_of_hasCompactSupport (μ := volume)
+    (chainY_compact t hs hs1 j n)
+  simp_rw [Real.norm_of_nonneg (chainKernel_nonneg ..)]
+  change (∫ z, |chainX t s j n z| + |chainY t s j n z|) ≤ _
+  rw [integral_add hiX.abs hiY.abs]
+  change (∫ z, |chainX t s j n z|) ≤ _ at hx
+  change (∫ z, |chainY t s j n z|) ≤ _ at hy
+  have hpi := mul_le_mul_of_nonneg_right Real.pi_lt_four.le (scaleLength_pos j).le
+  have hh : 10*scaleLength j = 10/(2:ℝ)^j := by simp [scaleLength, div_eq_mul_inv]
+  nlinarith
+
+lemma compact_sum_range {f : ℕ → ℂ → ℝ} (hf : ∀ j, HasCompactSupport (f j)) (k : ℕ) :
+    HasCompactSupport (fun z => ∑ j ∈ Finset.range k, f j z) := by
+  induction k with
+  | zero => simp [HasCompactSupport]
+  | succ k ih =>
+    simp only [Finset.sum_range_succ]
+    exact (ih.union (hf k)).of_isClosed_subset (isClosed_tsupport _) (tsupport_add _ _)
+
+lemma support_sum_range {f : ℕ → ℂ → ℝ} {S : Set ℂ}
+    (hf : ∀ j, tsupport (f j) ⊆ S) (k : ℕ) :
+    tsupport (fun z => ∑ j ∈ Finset.range k, f j z) ⊆ S := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+    simp only [Finset.sum_range_succ]
+    exact (tsupport_add _ _).trans (union_subset ih (hf k))
+
+noncomputable def aggregateX (t s : ℝ) (k n : ℕ) (z : ℂ) : ℝ :=
+  ∑ j ∈ Finset.range k, chainX t s j n z
+noncomputable def aggregateY (t s : ℝ) (k n : ℕ) (z : ℂ) : ℝ :=
+  ∑ j ∈ Finset.range k, chainY t s j n z
+
+lemma aggregateX_smooth (t s : ℝ) (k n : ℕ) : ContDiff ℝ ∞ (aggregateX t s k n) :=
+  ContDiff.sum (fun j _ => chainX_smooth t s j n)
+lemma aggregateY_smooth (t s : ℝ) (k n : ℕ) : ContDiff ℝ ∞ (aggregateY t s k n) :=
+  ContDiff.sum (fun j _ => chainY_smooth t s j n)
+lemma aggregateX_compact {t : ℝ} (s : ℝ) (ht : 0 ≤ t) (ht1 : t ≤ 1) (k n : ℕ) :
+    HasCompactSupport (aggregateX t s k n) :=
+  compact_sum_range (fun j => chainX_compact s ht ht1 j n) k
+lemma aggregateY_compact (t : ℝ) {s : ℝ} (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
+    HasCompactSupport (aggregateY t s k n) :=
+  compact_sum_range (fun j => chainY_compact t hs hs1 j n) k
+lemma aggregateX_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
+    tsupport (aggregateX t s k n) ⊆ openUnitStrip :=
+  support_sum_range (fun j => chainX_support ht ht1 hs hs1 j n) k
+lemma aggregateY_support {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
+    tsupport (aggregateY t s k n) ⊆ openUnitStrip :=
+  support_sum_range (fun j => chainY_support ht ht1 hs hs1 j n) k
+
+lemma aggregate_divergence (t s : ℝ) (k n : ℕ) (z : ℂ) :
+    (fderiv ℝ (aggregateX t s k n) z) 1 + (fderiv ℝ (aggregateY t s k n) z) Complex.I =
+      chainDensity t s k n z - chainDensity t s 0 n z := by
+  unfold aggregateX aggregateY
+  rw [fderiv_fun_sum (fun j _ => ((chainX_smooth t s j n).differentiable (by simp)) z),
+    fderiv_fun_sum (fun j _ => ((chainY_smooth t s j n).differentiable (by simp)) z)]
+  simp only [sum_apply, ← Finset.sum_add_distrib]
+  simp_rw [chain_divergence]
+  exact Finset.sum_range_sub (fun j => chainDensity t s j n z) k
+
+lemma aggregate_kernel_bound {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
+    (∫ z in openUnitStrip, (|aggregateX t s k n z|+|aggregateY t s k n z|)^2) ≤ 360*k := by
+  have hsum : (∫ z in openUnitStrip, ‖∑ j ∈ Finset.range k, chainKernel t s j n z‖^2) ≤
+      3*12*10*k := by
+    apply integral_sum_kernel_sq_le (fun j => chainKernel t s j n) 12 10 k
+      (by norm_num) (by norm_num)
+    · intro j _
+      exact ((chainKernel_continuous t s j n).integrable_of_hasCompactSupport
+        (chainKernel_compact ht ht1 hs hs1 j n)).integrableOn
+    · intro j _ z
+      exact chainKernel_bound t s j n z
+    · intro j _
+      refine (setIntegral_le_integral ?_ (Eventually.of_forall (fun _ => norm_nonneg _))).trans
+        (chainKernel_integral_bound ht ht1 hs hs1 j n)
+      exact ((chainKernel_continuous t s j n).integrable_of_hasCompactSupport
+        (chainKernel_compact ht ht1 hs hs1 j n)).norm
+  have hI : Integrable (fun z => (|aggregateX t s k n z|+|aggregateY t s k n z|)^2)
+      (volume.restrict openUnitStrip) := by
+    apply MemLp.integrable_sq
+    exact ((aggregateX_smooth t s k n).continuous.abs.add
+      (aggregateY_smooth t s k n).continuous.abs).memLp_of_hasCompactSupport
+      ((aggregateX_compact s ht ht1 k n).abs.add (aggregateY_compact t hs hs1 k n).abs)
+  have hJ : Integrable (fun z => ‖∑ j ∈ Finset.range k, chainKernel t s j n z‖^2)
+      (volume.restrict openUnitStrip) := by
+    have hc := continuous_finsetSum (Finset.range k) (fun j _ => chainKernel_continuous t s j n)
+    exact (memLp_two_iff_integrable_sq_norm hc.aestronglyMeasurable).1
+      (hc.memLp_of_hasCompactSupport
+        (compact_sum_range (fun j => chainKernel_compact ht ht1 hs hs1 j n) k))
+  refine (integral_mono hI hJ ?_).trans (by linarith)
+  intro z
+  have hh : |aggregateX t s k n z|+|aggregateY t s k n z| ≤
+      ∑ j ∈ Finset.range k, chainKernel t s j n z := by
+    unfold aggregateX aggregateY chainKernel
+    rw [Finset.sum_add_distrib]
+    exact add_le_add (Finset.abs_sum_le_sum_abs _ _) (Finset.abs_sum_le_sum_abs _ _)
+  apply (sq_le_sq₀ (by positivity) (norm_nonneg _)).2
+  exact hh.trans (le_abs_self _)
+
+lemma chainDensity_continuous (t s : ℝ) (j n : ℕ) : Continuous (chainDensity t s j n) :=
+  (complexTensor_contDiff (intervalDensity_smooth ..) (intervalDensity_smooth ..)).continuous
+lemma chainDensity_compact (t s : ℝ) (j n : ℕ) : HasCompactSupport (chainDensity t s j n) :=
+  complexTensor_compact (intervalDensity_compact ..) (intervalDensity_compact ..)
+
+lemma smoothed_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
+    (hu : SourceIsStripWeakGradient u D) {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k n : ℕ) :
+    |(∫ z in openUnitStrip, u z * chainDensity t s k n z) -
+      (∫ z in openUnitStrip, u z * chainDensity t s 0 n z)| ≤
+      20 * Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D) := by
+  have he := weak_gradient_pairing hu (aggregateX_smooth t s k n) (aggregateY_smooth t s k n)
+    (aggregateX_compact s ht ht1 k n) (aggregateY_compact t hs hs1 k n)
+    (aggregateX_support ht ht1 hs hs1 k n) (aggregateY_support ht ht1 hs hs1 k n)
+  simp_rw [aggregate_divergence] at he
+  have hi₁ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
+    (chainDensity_continuous t s k n) (chainDensity_compact t s k n)
+  have hi₀ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
+    (chainDensity_continuous t s 0 n) (chainDensity_compact t s 0 n)
+  simp_rw [mul_sub] at he
+  rw [integral_sub hi₁ hi₀] at he
+  rw [he, abs_neg]
+  refine (integral_pairing_le hu.2.1 (aggregateX_smooth t s k n).continuous
+    (aggregateY_smooth t s k n).continuous (aggregateX_compact s ht ht1 k n)
+    (aggregateY_compact t hs hs1 k n)).trans ?_
+  have hE : 0 ≤ stripDirichletEnergy D := integral_nonneg (fun _ => sq_nonneg _)
+  have hA : 0 ≤ ∫ z in openUnitStrip, (|aggregateX t s k n z|+|aggregateY t s k n z|)^2 :=
+    integral_nonneg (fun _ => sq_nonneg _)
+  have hb := aggregate_kernel_bound ht ht1 hs hs1 k n
+  have hR : 0 ≤ ((k:ℝ)+1)*stripDirichletEnergy D := by positivity
+  have hSqE := Real.sq_sqrt hE
+  have hSqA := Real.sq_sqrt hA
+  have hSqR := Real.sq_sqrt hR
+  apply (sq_le_sq₀ (by positivity) (by positivity)).1
+  rw [mul_pow, hSqE, hSqA, mul_pow, hSqR]
+  nlinarith [mul_le_mul_of_nonneg_left hb hE]
+
+def stripRectangle (a b l : ℝ) : Set ℂ :=
+  {w | a ≤ w.re ∧ w.re ≤ a+l ∧ b ≤ w.im ∧ w.im ≤ b+Real.pi*l}
+
+lemma stripRectangle_eq_preimage (a b l : ℝ) :
+    stripRectangle a b l = Complex.measurableEquivRealProd ⁻¹'
+      (Icc a (a+l) ×ˢ Icc b (b+Real.pi*l)) := by
+  ext z
+  simp only [stripRectangle, mem_ofPred_eq, mem_preimage, mem_prod, mem_Icc,
+    Complex.measurableEquivRealProd_apply]
+  tauto
+
+lemma stripRectangle_volume (a b l : ℝ) :
+    volume (stripRectangle a b l) = ENNReal.ofReal l * ENNReal.ofReal (Real.pi*l) := by
+  rw [stripRectangle_eq_preimage,
+    Complex.volume_preserving_equiv_real_prod.measure_preimage
+      (measurableSet_Icc.prod measurableSet_Icc).nullMeasurableSet,
+    Measure.volume_eq_prod, Measure.prod_prod]
+  simp only [Real.volume_Icc, add_sub_cancel_left]
+
+lemma stripRectangle_integral_mul (a b l : ℝ) (f g : ℝ → ℝ) :
+    (∫ z in stripRectangle a b l, f z.re * g z.im) =
+      (∫ x in Icc a (a+l), f x) * (∫ y in Icc b (b+Real.pi*l), g y) := by
+  rw [stripRectangle_eq_preimage]
+  have he := (Complex.volume_preserving_equiv_real_prod.restrict_preimage
+    (s := Icc a (a+l) ×ˢ Icc b (b+Real.pi*l))
+    (measurableSet_Icc.prod measurableSet_Icc)).integral_comp
+    Complex.measurableEquivRealProd.measurableEmbedding
+    (fun p : ℝ × ℝ => f p.1 * g p.2)
+  simp only [Complex.measurableEquivRealProd_apply] at he
+  rw [he]
+  rw [Measure.volume_eq_prod, ← Measure.prod_restrict]
+  exact integral_prod_mul f g
+
+lemma stripRectangle_integral_re (a b l : ℝ) (hl : 0 ≤ l) :
+    (∫ z in stripRectangle a b l, z.re) =
+      (a*l+l^2/2)*(Real.pi*l) := by
+  have h := stripRectangle_integral_mul a b l (fun x => x) (fun _ => 1)
+  simp only [mul_one] at h
+  rw [h, integral_Icc_eq_integral_Ioc,
+    ← intervalIntegral.integral_of_le (by linarith : a ≤ a+l), integral_id]
+  simp only [integral_const, Measure.real, Measure.restrict_apply_univ, Real.volume_Icc,
+    add_sub_cancel_left, smul_eq_mul, mul_one]
+  rw [ENNReal.toReal_ofReal (mul_nonneg Real.pi_pos.le hl)]
+  ring
+
+noncomputable def spatialAverage (u : ℂ → ℝ) (Q : Set ℂ) : ℝ :=
+  (∫ z in Q, u z) / (volume Q).toReal
+
+def openStripRectangle (a b l : ℝ) : Set ℂ :=
+  {z | a < z.re ∧ z.re < a+l ∧ b < z.im ∧ z.im < b+Real.pi*l}
+
+lemma openStripRectangle_eq_preimage (a b l : ℝ) :
+    openStripRectangle a b l = Complex.measurableEquivRealProd ⁻¹'
+      (Ioo a (a+l) ×ˢ Ioo b (b+Real.pi*l)) := by
+  ext z
+  simp only [openStripRectangle, mem_ofPred_eq, mem_preimage, mem_prod, mem_Ioo,
+    Complex.measurableEquivRealProd_apply]
+  tauto
+
+lemma openStripRectangle_measurable (a b l : ℝ) : MeasurableSet (openStripRectangle a b l) := by
+  rw [openStripRectangle_eq_preimage]
+  exact (measurableSet_Ioo.prod measurableSet_Ioo).preimage
+    Complex.measurableEquivRealProd.measurable
+
+lemma openStripRectangle_ae_eq (a b l : ℝ) :
+    openStripRectangle a b l =ᵐ[volume] stripRectangle a b l := by
+  rw [openStripRectangle_eq_preimage, stripRectangle_eq_preimage]
+  apply Complex.volume_preserving_equiv_real_prod.quasiMeasurePreserving.preimage_ae_eq
+  rw [Measure.volume_eq_prod]
+  exact Measure.set_prod_ae_eq Ioo_ae_eq_Icc Ioo_ae_eq_Icc
+
+lemma rectangleDensity_nonneg (a b lx ly : ℝ) (hx : 0 < lx) (hy : 0 < ly)
+    (n : ℕ) (z : ℂ) : 0 ≤ rectangleDensity a b lx ly hx hy n z :=
+  mul_nonneg (intervalDensity_nonneg ..) (intervalDensity_nonneg ..)
+lemma rectangleDensity_bound (a b lx ly : ℝ) (hx : 0 < lx) (hy : 0 < ly)
+    (n : ℕ) (z : ℂ) : rectangleDensity a b lx ly hx hy n z ≤ 4/(lx*ly) := by
+  calc
+    _ ≤ (2/lx)*(2/ly) := mul_le_mul (intervalDensity_bound ..) (intervalDensity_bound ..)
+      (intervalDensity_nonneg ..) (by positivity)
+    _ = _ := by ring
+
+lemma rectangleDensity_tendsto (a b l : ℝ) (hl : 0 < l) (z : ℂ) :
+    Tendsto (fun n => rectangleDensity a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) n z)
+      atTop (𝓝 ((openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z)) := by
+  have hh := (intervalDensity_tendsto a l hl z.re).mul
+    (intervalDensity_tendsto b (Real.pi*l) (mul_pos Real.pi_pos hl) z.im)
+  have heq : (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z =
+      (Ioo a (a+l)).indicator (fun _ => 1/l) z.re *
+        (Ioo b (b+Real.pi*l)).indicator (fun _ => 1/(Real.pi*l)) z.im := by
+    by_cases hx : z.re ∈ Ioo a (a+l) <;> by_cases hy : z.im ∈ Ioo b (b+Real.pi*l)
+    · have hz : z ∈ openStripRectangle a b l := ⟨hx.1,hx.2,hy.1,hy.2⟩
+      simp [hz,hx,hy,one_div]
+      ring
+    · have hz : z ∉ openStripRectangle a b l := fun hz => hy ⟨hz.2.2.1,hz.2.2.2⟩
+      simp [hz,hx,hy]
+    · have hz : z ∉ openStripRectangle a b l := fun hz => hx ⟨hz.1,hz.2.1⟩
+      simp [hz,hx,hy]
+    · have hz : z ∉ openStripRectangle a b l := fun hz => hx ⟨hz.1,hz.2.1⟩
+      simp [hz,hx,hy]
+  simpa only [rectangleDensity, complexTensor, heq] using hh
+
+lemma rectangle_smoothed_mean_tendsto {u : ℂ → ℝ}
+    (hu : MemLp u 2 (volume.restrict openUnitStrip)) (a b l : ℝ) (hl : 0 < l)
+    (ha : 0 ≤ a) (ha1 : a+l ≤ 1) (hb : 0 ≤ b) (hb1 : b+Real.pi*l ≤ Real.pi) :
+    Tendsto (fun n => ∫ z in openUnitStrip,
+      u z * rectangleDensity a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) n z)
+      atTop (𝓝 (spatialAverage u (stripRectangle a b l))) := by
+  have hsub : openStripRectangle a b l ⊆ openUnitStrip := by
+    intro z hz
+    exact ⟨ha.trans_lt hz.1, hz.2.1.trans_le ha1, hb.trans_lt hz.2.2.1, hz.2.2.2.trans_le hb1⟩
+  have hint := hu.integrable (by norm_num)
+  have hlim := tendsto_integral_of_dominated_convergence
+    (μ := volume.restrict openUnitStrip)
+    (F := fun n z => u z * rectangleDensity a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) n z)
+    (f := fun z => u z * (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z)
+    (fun z => |u z| * (4/(l*(Real.pi*l))))
+    (fun n => hu.aestronglyMeasurable.mul
+      ((complexTensor_contDiff (intervalDensity_smooth ..) (intervalDensity_smooth ..)).continuous.aestronglyMeasurable))
+    (hint.abs.mul_const _) (fun n => Eventually.of_forall fun z => by
+      rw [Real.norm_eq_abs, abs_mul, abs_of_nonneg (rectangleDensity_nonneg ..)]
+      exact mul_le_mul_of_nonneg_left (rectangleDensity_bound ..) (abs_nonneg _))
+    (Eventually.of_forall fun z => (rectangleDensity_tendsto a b l hl z).const_mul (u z))
+  have heq : (∫ z in openUnitStrip,
+      u z * (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z) =
+      spatialAverage u (stripRectangle a b l) := by
+    have hfun : (fun z => u z * (openStripRectangle a b l).indicator (fun _ => 1/(l*(Real.pi*l))) z) =
+        (openStripRectangle a b l).indicator (fun z => u z * (1/(l*(Real.pi*l)))) := by
+      funext z
+      by_cases hz : z ∈ openStripRectangle a b l <;> simp [hz]
+    rw [hfun, integral_indicator (openStripRectangle_measurable a b l),
+      Measure.restrict_restrict (openStripRectangle_measurable a b l), inter_eq_left.mpr hsub,
+      integral_mul_const, setIntegral_congr_set (openStripRectangle_ae_eq a b l)]
+    unfold spatialAverage
+    rw [stripRectangle_volume, ENNReal.toReal_mul, ENNReal.toReal_ofReal hl.le,
+      ENNReal.toReal_ofReal (mul_pos Real.pi_pos hl).le]
+    ring
+  rwa [heq] at hlim
+
+def dyadicStripRectangle (j : ℤ) (k : ℕ) (i m : Fin (2^k)) : Set ℂ :=
+  stripRectangle ((j:ℝ)+(i:ℝ)/(2:ℝ)^k)
+    (Real.pi*((m:ℝ)/(2:ℝ)^k)) (1/(2:ℝ)^k)
+
+ 
+
+def SourceCriticalStripMeansTarget : Prop :=
+  ∃ K : ℝ, 0 < K ∧ ∀ (u : ℂ → ℝ) (D : ℂ → ℂ),
+    SourceIsStripWeakGradient u D → ContinuousOn u (stripRectangle 0 0 1) →
+    ∀ (k : ℕ) (i m : Fin (2^k)),
+      |spatialAverage u (dyadicStripRectangle 0 k i m) -
+        spatialAverage u (stripRectangle 0 0 1)| ≤
+      K * Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D)
+
+lemma chain_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
+    (hu : SourceIsStripWeakGradient u D) {t s : ℝ} (ht : 0 ≤ t) (ht1 : t ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) (k : ℕ) :
+    |spatialAverage u (stripRectangle (scaleOrigin t k) (Real.pi*scaleOrigin s k) (scaleLength k)) -
+      spatialAverage u (stripRectangle 0 0 1)| ≤
+      20 * Real.sqrt (((k:ℝ)+1)*stripDirichletEnergy D) := by
+  have hlim (j : ℕ) : Tendsto (fun n => ∫ z in openUnitStrip, u z * chainDensity t s j n z)
+      atTop (𝓝 (spatialAverage u (stripRectangle (scaleOrigin t j)
+        (Real.pi*scaleOrigin s j) (scaleLength j)))) := by
+    apply rectangle_smoothed_mean_tendsto hu.1 _ _ _ (scaleLength_pos j)
+      (scaleOrigin_nonneg ht j) (scaleOrigin_add_le_one ht1 j)
+      (mul_nonneg Real.pi_pos.le (scaleOrigin_nonneg hs j))
+    have hh := mul_le_mul_of_nonneg_left (scaleOrigin_add_le_one hs1 j) Real.pi_pos.le
+    nlinarith
+  have hzero := hlim 0
+  simp only [scaleOrigin_zero, scaleLength_zero, mul_zero] at hzero
+  exact le_of_tendsto ((hlim k).sub hzero).abs
+    (Eventually.of_forall (fun n => smoothed_means_le hu ht ht1 hs hs1 k n))
+
+theorem sourceCriticalStripMeans : SourceCriticalStripMeansTarget := by
+  refine ⟨20, by norm_num, ?_⟩
+  intro u D hu _ k i m
+  by_cases hk : k = 0
+  · subst k
+    have hi : i = 0 := by apply Fin.ext; have := i.isLt; norm_num at this ⊢
+    have hm : m = 0 := by apply Fin.ext; have := m.isLt; norm_num at this ⊢
+    subst i; subst m
+    simp [dyadicStripRectangle]
+  · obtain ⟨t, ht, ht1, hit⟩ := dyadic_position_param k hk i
+    obtain ⟨s, hs, hs1, hms⟩ := dyadic_position_param k hk m
+    have hh := chain_means_le hu ht ht1 hs hs1 k
+    simpa only [dyadicStripRectangle, Int.cast_zero, zero_add, hit, hms, scaleLength] using hh
+
+lemma integral_pairing_local_le {D : ℂ → ℂ}
+    (hD : MemLp D 2 (volume.restrict openUnitStrip)) {V : Set ℂ}
+    (hV : V ⊆ openUnitStrip) {X Y : ℂ → ℝ}
+    (hX : Continuous X) (hY : Continuous Y)
+    (hcX : HasCompactSupport X) (hcY : HasCompactSupport Y)
+    (hsX : tsupport X ⊆ V) (hsY : tsupport Y ⊆ V) :
+    |∫ z in openUnitStrip, (D z).re * X z + (D z).im * Y z| ≤
+      Real.sqrt (∫ z in V, ‖D z‖^2) *
+        Real.sqrt (∫ z in V, (|X z|+|Y z|)^2) := by
+  have hm : volume.restrict V ≤ volume.restrict openUnitStrip :=
+    Measure.restrict_mono hV le_rfl
+  have hDl := hD.mono_measure hm
+  have hDi := (hD.integrable (by norm_num)).mono_measure hm
+  have hdx := integrable_mul_compact_test (Complex.reCLM.integrable_comp hDi) hX hcX
+  have hdy := integrable_mul_compact_test (Complex.imCLM.integrable_comp hDi) hY hcY
+  have hK : MemLp (fun z => |X z|+|Y z|) 2 (volume.restrict V) :=
+    (hX.abs.add hY.abs).memLp_of_hasCompactSupport (hcX.abs.add hcY.abs)
+  rw [setIntegral_eq_of_subset_of_forall_sdiff_eq_zero openUnitStrip_isOpen.measurableSet hV
+    (fun z hz => by
+      rw [image_eq_zero_of_notMem_tsupport (fun h => hz.2 (hsX h)),
+        image_eq_zero_of_notMem_tsupport (fun h => hz.2 (hsY h))]
+      ring)]
+  calc
+    _ ≤ ∫ z in V, |(D z).re * X z + (D z).im * Y z| := by
+      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm
+        (fun z => (D z).re * X z + (D z).im * Y z)
+    _ ≤ ∫ z in V, ‖D z‖ * (|X z|+|Y z|) :=
+      integral_mono (hdx.add hdy).abs (hDl.norm.integrable_mul hK)
+        (fun z => complex_pairing_abs_le ..)
+    _ ≤ _ := integral_nonneg_cauchy hDl.norm hK (fun _ => norm_nonneg _)
+      (fun _ => add_nonneg (abs_nonneg _) (abs_nonneg _))
+
+lemma rectangle_transport_L2_bound (a b l : ℝ) (hl : 0 < l) (a' b' : ℝ)
+    (ha : a ≤ a') (ha' : a'+l/2 ≤ a+l)
+    (hb : b ≤ b') (hb' : b'+(Real.pi*l)/2 ≤ b+Real.pi*l) (n : ℕ) (V : Set ℂ) :
+    (∫ z in V, (|rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z| +
+      |rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z|)^2) ≤ 120 := by
+  let X := rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
+  let Y := rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
+  have hx := (rectangleTransportX_contDiff a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n).continuous
+  have hy := (rectangleTransportY_contDiff a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n).continuous
+  have hcx := rectangleTransportX_compact a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' ha ha' n
+  have hcy := rectangleTransportY_compact a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' hb hb' n
+  have hK : Continuous (fun z => |X z|+|Y z|) := hx.abs.add hy.abs
+  have hcK : HasCompactSupport (fun z => |X z|+|Y z|) := hcx.abs.add hcy.abs
+  have hKi := hK.integrable_of_hasCompactSupport (μ := volume) hcK
+  have hKsq : Integrable (fun z => (|X z|+|Y z|)^2) volume :=
+    (hK.memLp_of_hasCompactSupport (p := 2) hcK).integrable_sq
+  have hK0 (z : ℂ) : 0 ≤ |X z|+|Y z| := by positivity
+  have hKb (z : ℂ) : |X z|+|Y z| ≤ 12/l := by
+    have hx' := rectangleTransportX_bound a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z
+    have hy' := rectangleTransportY_bound a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n z
+    have hp : l ≤ Real.pi*l := by nlinarith [Real.pi_gt_three]
+    have hx'' := hx'.trans (div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 8) hl hp)
+    change |X z| ≤ _ at hx''
+    change |Y z| ≤ _ at hy'
+    calc
+      _ ≤ 8/l+4/l := add_le_add hx'' hy'
+      _ = 12/l := by ring
+  have hKI : (∫ z, |X z|+|Y z|) ≤ 10*l := by
+    rw [integral_add (hx.integrable_of_hasCompactSupport hcx).abs
+      (hy.integrable_of_hasCompactSupport hcy).abs]
+    have hx' := rectangleTransportX_integral_abs a b l (Real.pi*l) hl
+      (mul_pos Real.pi_pos hl) a' b' ha ha' n
+    have hy' := rectangleTransportY_integral_abs a b l (Real.pi*l) hl
+      (mul_pos Real.pi_pos hl) a' b' hb hb' n
+    change (∫ z, |X z|) ≤ _ at hx'
+    change (∫ z, |Y z|) ≤ _ at hy'
+    nlinarith [Real.pi_lt_four]
+  change (∫ z in V, (|X z|+|Y z|)^2) ≤ _
+  calc
+    _ ≤ ∫ z, (|X z|+|Y z|)^2 :=
+      setIntegral_le_integral hKsq (Eventually.of_forall (fun _ => sq_nonneg _))
+    _ ≤ ∫ z, (12/l)*(|X z|+|Y z|) := integral_mono hKsq (hKi.const_mul _) (fun z => by
+      nlinarith [mul_le_mul_of_nonneg_right (hKb z) (hK0 z)])
+    _ = (12/l)*∫ z, |X z|+|Y z| := integral_const_mul ..
+    _ ≤ (12/l)*(10*l) := mul_le_mul_of_nonneg_left hKI (by positivity)
+    _ = 120 := by field_simp; ring
+
+lemma openStripRectangle_subset_unit {a b l : ℝ}
+    (ha : 0 ≤ a) (ha1 : a+l ≤ 1) (hb : 0 ≤ b) (hb1 : b+Real.pi*l ≤ Real.pi) :
+    openStripRectangle a b l ⊆ openUnitStrip := by
+  intro z hz
+  exact ⟨ha.trans_lt hz.1, hz.2.1.trans_le ha1,
+    hb.trans_lt hz.2.2.1, hz.2.2.2.trans_le hb1⟩
+
+lemma local_means_le {u : ℂ → ℝ} {D : ℂ → ℂ}
+    (hu : SourceIsStripWeakGradient u D) (a b l : ℝ) (hl : 0 < l)
+    (ha : 0 ≤ a) (ha1 : a+l ≤ 1) (hb : 0 ≤ b) (hb1 : b+Real.pi*l ≤ Real.pi)
+    (a' b' : ℝ) (haa : a ≤ a') (haa' : a'+l/2 ≤ a+l)
+    (hbb : b ≤ b') (hbb' : b'+(Real.pi*l)/2 ≤ b+Real.pi*l) :
+    |spatialAverage u (stripRectangle a' b' (l/2)) -
+      spatialAverage u (stripRectangle a b l)| ≤
+      12 * Real.sqrt (∫ z in openStripRectangle a b l, ‖D z‖^2) := by
+  have hsub := openStripRectangle_subset_unit ha ha1 hb hb1
+  have hsmooth (n : ℕ) :
+      |(∫ z in openUnitStrip, u z * rectangleDensity a' b' (l/2) (Real.pi*(l/2))
+          (by positivity) (by positivity) n z) -
+        (∫ z in openUnitStrip, u z * rectangleDensity a b l (Real.pi*l)
+          hl (mul_pos Real.pi_pos hl) n z)| ≤
+      12 * Real.sqrt (∫ z in openStripRectangle a b l, ‖D z‖^2) := by
+    let X := rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
+    let Y := rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n
+    have hx : ContDiff ℝ ∞ X := rectangleTransportX_contDiff ..
+    have hy : ContDiff ℝ ∞ Y := rectangleTransportY_contDiff ..
+    have hcx : HasCompactSupport X := rectangleTransportX_compact _ _ _ _ _ _ _ _ haa haa' n
+    have hcy : HasCompactSupport Y := rectangleTransportY_compact _ _ _ _ _ _ _ _ hbb hbb' n
+    have hsx : tsupport X ⊆ openStripRectangle a b l :=
+      rectangleTransportX_support _ _ _ _ _ _ _ _ haa haa' hbb hbb' n
+    have hsy : tsupport Y ⊆ openStripRectangle a b l :=
+      rectangleTransportY_support _ _ _ _ _ _ _ _ hbb hbb' n
+    have he := weak_gradient_pairing hu hx hy hcx hcy (hsx.trans hsub) (hsy.trans hsub)
+    change (∫ z in openUnitStrip, u z *
+      ((fderiv ℝ (rectangleTransportX a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n) z) 1 +
+       (fderiv ℝ (rectangleTransportY a b l (Real.pi*l) hl (mul_pos Real.pi_pos hl) a' b' n) z) Complex.I)) = _ at he
+    simp_rw [rectangleTransport_divergence, show Real.pi*l/2 = Real.pi*(l/2) by ring, mul_sub] at he
+    have hi₁ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
+      (complexTensor_contDiff (intervalDensity_smooth a' (l/2) (by positivity) n)
+        (intervalDensity_smooth b' (Real.pi*(l/2)) (by positivity) n)).continuous
+      (complexTensor_compact (intervalDensity_compact ..) (intervalDensity_compact ..))
+    have hi₀ := integrable_mul_compact_test (hu.1.integrable (by norm_num))
+      (complexTensor_contDiff (intervalDensity_smooth a l hl n)
+        (intervalDensity_smooth b (Real.pi*l) (mul_pos Real.pi_pos hl) n)).continuous
+      (complexTensor_compact (intervalDensity_compact ..) (intervalDensity_compact ..))
+    dsimp only [rectangleDensity] at he ⊢
+    rw [integral_sub hi₁ hi₀] at he
+    rw [he, abs_neg]
+    refine (integral_pairing_local_le hu.2.1 hsub hx.continuous hy.continuous hcx hcy hsx hsy).trans ?_
+    have hbnd := rectangle_transport_L2_bound a b l hl a' b' haa haa' hbb hbb' n
+      (openStripRectangle a b l)
+    change (∫ z in openStripRectangle a b l, (|X z|+|Y z|)^2) ≤ 120 at hbnd
+    have hE : 0 ≤ ∫ z in openStripRectangle a b l, ‖D z‖^2 := integral_nonneg (fun _ => sq_nonneg _)
+    have hA : 0 ≤ ∫ z in openStripRectangle a b l, (|X z|+|Y z|)^2 := integral_nonneg (fun _ => sq_nonneg _)
+    apply (sq_le_sq₀ (by positivity) (by positivity)).1
+    rw [mul_pow, Real.sq_sqrt hE, Real.sq_sqrt hA, mul_pow, Real.sq_sqrt hE]
+    nlinarith [mul_le_mul_of_nonneg_left hbnd hE]
+  have hlim₀ := rectangle_smoothed_mean_tendsto hu.1 a b l hl ha ha1 hb hb1
+  have hlim₁ := rectangle_smoothed_mean_tendsto hu.1 a' b' (l/2) (by positivity)
+    (ha.trans haa) (haa'.trans ha1) (hb.trans hbb) (by nlinarith)
+  exact le_of_tendsto (hlim₁.sub hlim₀).abs (Eventually.of_forall hsmooth)
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/StripProbabilities.lean b/lean/OAI/Probability/SLE/StripProbabilities.lean
index a535fcaf5a418bdc2ddc3d3f7c9c6b887d018060..dd795b5175180d47e3b1838eb84fef194e681e46 100644
--- a/lean/OAI/Probability/SLE/StripProbabilities.lean
+++ b/lean/OAI/Probability/SLE/StripProbabilities.lean
@@ -1,676 +1,676 @@
-import Mathlib
-import OAI.Probability.SLE.DyadicStrips
-import OAI.Probability.SLE.NestedMeans
-
-namespace OAI
-
-/-! Strip Probabilities for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory
-open scoped Topology ContDiff ENNReal BigOperators NNReal
-namespace SLEExactGauge
-theorem comparison_path_strip_zero {η : ℝ → ℂ} {T : ℝ}
-    (hT : 0 < T) (hη : ContinuousOn η (Icc 0 T)) (hstart : η 0 = 0)
-    (hend : ‖η T‖ = 1) (hupper : ∀ t ∈ Icc 0 T, 0 ≤ (η t).im)
-    (Fbar : ℂ → OnePoint ℂ) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H)
-    (himage : ∀ t ∈ Icc 0 T, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r)
-    (v : ℝ) (hv : v ≤ 0) :
-    ∃ φ : ℝ, φ ∈ Icc 0 Real.pi ∧
-      sphericalTruncatedLog q r H (Fbar (Complex.exp ((v : ℂ)+(φ : ℂ)*Complex.I))) = 0 := by
-  have her : Real.exp v ∈ Icc ‖η 0‖ ‖η T‖ := by
-    rw [hstart,norm_zero,hend]
-    exact ⟨(Real.exp_pos v).le,by simpa using Real.exp_le_exp.mpr hv⟩
-  obtain ⟨t,ht,htr⟩ := intermediate_value_Icc hT.le hη.norm her
-  change ‖η t‖ = Real.exp v at htr
-  have hn : η t ≠ 0 := by
-    apply norm_ne_zero_iff.mp
-    rw [htr]
-    exact (Real.exp_pos v).ne'
-  let φ := (η t).arg
-  have hφ : φ ∈ Icc 0 Real.pi :=
-    ⟨Complex.arg_nonneg_iff.mpr (hupper t ht),Complex.arg_le_pi _⟩
-  have he : (v : ℂ)+(φ : ℂ)*Complex.I = Complex.log (η t) := by
-    apply Complex.ext
-    · simp [Complex.log_re,htr]
-    · simp [Complex.log_im,φ]
-  refine ⟨φ,hφ,?_⟩
-  rw [he,Complex.exp_log hn]
-  obtain ⟨w,hw,hwr⟩ := himage t ht
-  rw [hw,sphericalTruncatedLog_coe]
-  exact truncatedConformalLog_eq_zero hr hH.le hwr
-
-end SLEExactGauge
-end
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-lemma source_strip_log_locallyLipschitz {F : ℂ → ℂ} {Fbar : ℂ → OnePoint ℂ}
-    (hF : DifferentiableOn ℂ F {z | 0 < z.im})
-    (hext : ∀ z : ℂ, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
-    LocallyLipschitzOn fullOpenStrip (fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))) := by
-  let U := fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))
-  let V := truncatedConformalLog (F ∘ Complex.exp) q r H
-  have heq : EqOn U V fullOpenStrip := by
-    intro z hz
-    dsimp [U]
-    rw [hext (Complex.exp z) (exp_strip_halfplane hz),sphericalTruncatedLog_coe]
-    rfl
-  have hcomp : DifferentiableOn ℂ (F ∘ Complex.exp) fullOpenStrip :=
-    hF.comp Complex.differentiable_exp.differentiableOn exp_strip_halfplane
-  intro z hz
-  obtain ⟨K,S,hS,hzS,hK⟩ := truncated_log_lipschitz_near fullOpenStrip_isOpen hcomp hz q hr hH
-  refine ⟨K,S ∩ fullOpenStrip,mem_nhdsWithin_of_mem_nhds ((hS.inter fullOpenStrip_isOpen).mem_nhds ⟨hzS,hz⟩),?_⟩
-  intro x hx y hy
-  change edist (U x) (U y) ≤ (K : ℝ≥0∞)*edist x y
-  rw [heq hx.2,heq hy.2]
-  exact hK hx.1 hy.1
-
- 
-
-def BeforeFirstModulus (γ : ℝ≥0 → ℂ) (A : ℝ) (t : ℝ≥0) : Prop :=
-  ∀ s : ℝ≥0, s < t → ‖γ s‖ < A
-
-universe u
- 
-def SourceUniformConformalLocalizationTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 → ∃ b C c L0 : ℝ,
-    0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L0 ∧
-    ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ γ : Ω → ℝ≥0 → ℂ, IsOrdinaryChordalSLE κ γ P →
-    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
-    DifferentiableOn ℂ F {z | 0 < z.im} → InjOn F {z | 0 < z.im} →
-    ContinuousOn Fbar {z : ℂ | 0 ≤ z.im} →
-    (∀ z : ℂ, 0 < z.im → Fbar z = (F z : OnePoint ℂ)) →
-    ∀ (q : ℂ) (r : ℝ), 0 < r →
-    ∀ (η : ℝ → ℂ) (T : ℝ), 0 < T → ContinuousOn η (Icc 0 T) →
-    η 0 = 0 → ‖η T‖ = 1 → (∀ t ∈ Icc 0 T, 0 ≤ (η t).im) →
-    (∀ t ∈ Icc 0 T, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r) →
-    ∀ L : ℝ, L0 ≤ L →
-      P {xi | ∃ t : ℝ≥0, BeforeFirstModulus (γ xi) (L^b) t ∧
-        Fbar (γ xi t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (L*r)} ≤
-        ENNReal.ofReal (C*L^(-c))
-
-lemma beforeFirstModulus_norm_le {γ : ℝ≥0 → ℂ} (hc : Continuous γ) (h0 : γ 0 = 0)
-    {A : ℝ} (hA : 0 < A) {t : ℝ≥0} (ht : BeforeFirstModulus γ A t) :
-    ‖γ t‖ ≤ A := by
-  by_cases ht0 : t = 0
-  · simp [ht0,h0,hA.le]
-  have htpos : 0 < t := bot_lt_iff_ne_bot.mpr ht0
-  have hcl : t ∈ closure (Iio t : Set ℝ≥0) := by
-    rw [closure_Iio' (show (Iio t : Set ℝ≥0).Nonempty from ⟨0,htpos⟩)]
-    exact (show t ≤ t from le_rfl)
-  have hclosed : IsClosed {s : ℝ≥0 | ‖γ s‖ ≤ A} := isClosed_le hc.norm continuous_const
-  exact (closure_minimal (fun s hs => (ht s hs).le) hclosed) hcl
-
-lemma sphericalTruncatedLog_eq_height_of_notin_ball (q : ℂ) (r H : ℝ) (hr : 0 < r)
-    (x : OnePoint ℂ)
-    (hx : x ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (Real.exp H*r)) :
-    sphericalTruncatedLog q r H x = H := by
-  induction x using OnePoint.rec with
-  | infty => rfl
-  | coe z =>
-    rw [sphericalTruncatedLog_coe]
-    apply truncatedConformalLog_eq_height hr
-    apply le_of_not_gt
-    intro h
-    apply hx
-    exact ⟨z,by simpa [Metric.mem_ball,dist_eq_norm] using h,rfl⟩
-
-lemma escape_forces_strip_level {γ : ℝ≥0 → ℂ} (hc : Continuous γ) (h0 : γ 0 = 0)
-    (hupper : ∀ t, 0 ≤ (γ t).im)
-    (Fbar : ℂ → OnePoint ℂ) (q : ℂ) {r L H b : ℝ} (hr : 0 < r) (hL : 1 < L)
-    (hH : H = Real.log L)
-    (hstart : ∃ w : ℂ, Fbar 0 = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r)
-    (t : ℝ≥0) (ht : BeforeFirstModulus γ (L^b) t)
-    (hbad : Fbar (γ t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (L*r)) :
-    ∃ z : ℂ, Complex.exp z ∈ range γ ∧ 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*H ∧
-      sphericalTruncatedLog q r H (Fbar (Complex.exp z)) = H := by
-  have hLp : 0 < L := by linarith
-  have hnorm := beforeFirstModulus_norm_le hc h0 (Real.rpow_pos_of_pos hLp b) ht
-  have hn : γ t ≠ 0 := by
-    intro heq
-    rw [heq] at hbad
-    obtain ⟨w,hw,hwr⟩ := hstart
-    apply hbad
-    exact ⟨w,by rw [Metric.mem_ball,dist_eq_norm]; nlinarith,hw.symm⟩
-  refine ⟨Complex.log (γ t),by rw [Complex.exp_log hn]; exact mem_range_self t,?_,?_,?_,?_⟩
-  · simpa only [Complex.log_im] using Complex.arg_nonneg_iff.mpr (hupper t)
-  · simpa only [Complex.log_im] using Complex.arg_le_pi (γ t)
-  · rw [Complex.log_re,hH]
-    exact (Real.log_le_log (norm_pos_iff.mpr hn) hnorm).trans_eq (Real.log_rpow hLp b)
-  · rw [Complex.exp_log hn]
-    apply sphericalTruncatedLog_eq_height_of_notin_ball q r H hr
-    simpa only [hH,Real.exp_log hLp] using hbad
-
-theorem sourceUniformConformalLocalization : SourceUniformConformalLocalizationTarget := by
-  intro κ hκ hκ8
-  obtain ⟨C0,α,hC0,hα,hhit⟩ := ordinary_sle_strip_hitting_constants κ hκ hκ8
-  obtain ⟨b,C,c,H0,hb,hC,hc,hH0,hstrip⟩ := sourceStripLocalization C0 α hC0 hα
-  refine ⟨b,C,c,Real.exp (max H0 1),hb,hC,hc,?_,?_⟩
-  · exact Real.one_lt_exp_iff.mpr (lt_of_lt_of_le zero_lt_one (le_max_right _ _))
-  intro Ω mΩ P hP γ hγ F Fbar hF hi hbar hext q r hr η T hT hη hs he hu him L hL
-  let := hP
-  have hL1 : 1 < L := (Real.one_lt_exp_iff.mpr
-    (lt_of_lt_of_le zero_lt_one (le_max_right H0 1))).trans_le hL
-  have hLp : 0 < L := by linarith
-  have hlog : max H0 1 ≤ Real.log L := by
-    simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos _) hL
-  have hHH : H0 ≤ Real.log L := (le_max_left _ _).trans hlog
-  have hHp : 0 < Real.log L := hH0.trans_le hHH
-  let U := fun z : ℂ => sphericalTruncatedLog q r (Real.log L) (Fbar (Complex.exp z))
-  let X := fun xi : Ω => Complex.exp ⁻¹' range (γ xi)
-  obtain ⟨hcU,hbU,-,hgU,hEU⟩ := source_strip_log_regularity hF hi hbar hext q r (Real.log L) hr hHp
-  have hzero (v : ℝ) (hv : v ≤ 0) : ∃ y ∈ Icc (0:ℝ) Real.pi, U (v+y*Complex.I) = 0 :=
-    comparison_path_strip_zero hT hη hs he hu Fbar q r (Real.log L) hr hHp him v hv
-  have hhits (j : ℤ) (k : ℕ) (i m : Fin (2^k)) :
-      P {xi | (X xi ∩ dyadicStripRectangle j k i m).Nonempty} ≤
-        ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ))) := by
-    apply (measure_mono ?_).trans (hhit γ hγ j k i m)
-    rintro xi ⟨z,⟨t,ht⟩,hz⟩
-    exact ⟨t,z,hz,ht.symm⟩
-  have hh := hstrip (Real.log L) hHH U hcU
-    (source_strip_log_locallyLipschitz hF hext q r (Real.log L) hr hHp)
-    hbU hgU hEU hzero Ω mΩ P hP X hhits
-  have hstart : ∃ w : ℂ, Fbar 0 = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r := by
-    simpa only [hs] using him 0 ⟨le_rfl,hT.le⟩
-  obtain ⟨B,hB,htrace⟩ := hγ.2
-  have hsub : {xi | ∃ t : ℝ≥0, BeforeFirstModulus (γ xi) (L^b) t ∧
-        Fbar (γ xi t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (L*r)} ≤ᵐ[P]
-      {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*Real.log L ∧ U z = Real.log L} := by
-    filter_upwards [htrace] with xi hxi
-    obtain ⟨G,Fw,hFw⟩ := hxi
-    rintro ⟨t,ht,hbad⟩
-    exact escape_forces_strip_level hFw.1 hFw.2.1 hFw.2.2.1 Fbar q hr hL1 rfl hstart t ht hbad
-  calc
-    _ ≤ P {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*Real.log L ∧ U z = Real.log L} :=
-      measure_mono_ae hsub
-    _ ≤ ENNReal.ofReal (C*Real.exp (-c*Real.log L)) := hh
-    _ = ENNReal.ofReal (C*L^(-c)) := by
-      rw [Real.rpow_def_of_pos hLp]
-      congr 2
-      congr 1
-      ring
-
-end SLEExactGauge
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem ordinary_survival_bound_regular
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ : ℝ) (hκ : 4 < κ)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (R : ℝ) (hR : 0 < R)
-    (hN : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ R^2)
-    (T : ℝ≥0) (b : ℝ) (hb : 0 < b)
-    (hbR : b ≤ Real.exp (((κ-4)/(2*κ))*Real.log R))
-    (hbT : b ≤ (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) * T) :
-    P {ω | z ∈ survivingDomain (γ ω) T} ≤
-      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
-        (initialLogCoordinates z) 0 / b) := by
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let q₀ := initialLogCoordinates z
-  let lo (n : ℕ) : ℝ := q₀.1.im-(n:ℝ)-1
-  have hlo (n : ℕ) : lo n ≤ q₀.1.im := by dsimp [lo]; have := Nat.cast_nonneg (α := ℝ) n; linarith
-  let c (n : ℕ) := Classical.choice
-    (exists_clippedLogSolution B hm hBm hBc κ (lo n) q₀.1.im (hlo n) q₀)
-  let a (n : ℕ) : RadialLocalization (c n) R T := Classical.choice
-    (exists_radialLocalization hBc hB0 (c n) R T hN (hlo n))
-  let E (n : ℕ) : Set Ω := {ω | (a n).τ ω = T ∨
-    greenLogNormSq κ ((c n).Q ((a n).τ ω) ω) (B ((a n).τ ω) ω) = R^2}
-  have hprob (n : ℕ) : P (E n) ≤
-      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q₀ 0 / b) :=
-    (a n).exit_probability hB hBm hBc hκ hR le_rfl b hb hbR hbT
-  have hsub : {ω | z ∈ survivingDomain (γ ω) T} ≤ᵐ[P]
-      {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} := by
-    filter_upwards [hw] with ω hω
-    intro hlive
-    obtain ⟨N, hN⟩ := exists_nat_gt (q₀.1.im-Real.log (G ω T z).im)
-    filter_upwards [eventually_ge_atTop N] with n hn
-    have hlt : lo n < Real.log (G ω T z).im := by
-      have hn' : (N:ℝ) ≤ n := by exact_mod_cast hn
-      dsimp [lo]
-      linarith
-    exact (a n).alive_forces_radial_or_time_boundary hBc hz rfl le_rfl ω
-      (γ ω) (G ω) (F ω) hω hlive hlt
-  exact (measure_mono_ae hsub).trans (measure_eventually_sets_le P E _ hprob)
-
-theorem ordinary_survival_bound
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 4 < κ)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (R : ℝ) (hR : 0 < R)
-    (hN : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ R^2)
-    (T : ℝ≥0) (b : ℝ) (hb : 0 < b)
-    (hbR : b ≤ Real.exp (((κ-4)/(2*κ))*Real.log R))
-    (hbT : b ≤ (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) * T) :
-    P {ω | z ∈ survivingDomain (γ ω) T} ≤
-      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
-        (initialLogCoordinates z) 0 / b) := by
-  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
-  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
-    filter_upwards [hw,hCB] with ω hω he
-    simpa only [he] using hω
-  exact ordinary_survival_bound_regular hC hCm hCc hC0 κ hκ γ G F hwC
-    z hz R hR hN T b hb hbR hbT
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma greenLogNormSq_initial_norm (κ : ℝ) (z : ℂ) (hz : 0 < z.im) :
-    greenLogNormSq κ (initialLogCoordinates z) 0 = ‖z‖^2 := by
-  have he : Real.exp (2 * Real.log z.im) = z.im ^ 2 := by
-    rw [show 2 * Real.log z.im = Real.log z.im + Real.log z.im by ring,
-      Real.exp_add, Real.exp_log hz, pow_two]
-  rw [← Complex.normSq_eq_norm_sq]
-  simp only [greenLogNormSq, initialLogCoordinates, Complex.add_re, Complex.ofReal_re,
-      Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero,
-      sub_self, add_zero, sub_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add, he,
-      Complex.normSq_apply, pow_two]
-
-theorem ordinary_survival_radial_tail
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 4 < κ)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (z : ℂ) (hz : z ∈ upperHalfPlane) (hunit : ‖z‖ = 1)
-    (R : ℝ) (hR : 1 ≤ R) :
-    P {ω | z ∈ survivingDomain (γ ω) ((R^2).toNNReal)} ≤
-      ENNReal.ofReal ((1+3*(κ-4)/8) / min 1 (((κ-4)/(2*κ))*((κ-4)/4)) *
-        Real.exp (-((κ-4)/(2*κ))*Real.log R)) := by
-  let β := (κ-4)/(2*κ)
-  let C := 3*(κ-4)/8
-  let d := β*((κ-4)/4)
-  let m := min 1 d
-  have hk : 0 < κ := lt_trans (by norm_num) hκ
-  have hβ : 0 < β := by dsimp [β]; positivity
-  have hC : 0 ≤ C := by dsimp [C]; positivity
-  have hd : 0 < d := by dsimp [d]; positivity
-  have hm : 0 < m := lt_min (by norm_num) hd
-  have hR0 : 0 < R := lt_of_lt_of_le (by norm_num) hR
-  have hN : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ R^2 := by
-    rw [greenLogNormSq_initial_norm κ z hz, hunit]
-    nlinarith
-  have hN1 : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ (1:ℝ)^2 := by
-    rw [greenLogNormSq_initial_norm κ z hz, hunit]
-  have hF : swallowLyapunov κ β C (initialLogCoordinates z) 0 ≤ 1+C := by
-    have hr := radialLogPower_le κ β hβ.le (initialLogCoordinates z) 0 1 (by norm_num) hN1
-    have hy := heightLogPower_le κ β hβ.le (initialLogCoordinates z) 0 1 (by norm_num) hN1
-    simp only [Real.log_one, mul_zero, Real.exp_zero] at hr hy
-    exact (add_le_add hr (mul_le_mul_of_nonneg_left hy hC)).trans_eq (by ring)
-  have he : Real.exp ((β-2)*Real.log R)*R^2 = Real.exp (β*Real.log R) := by
-    have hh : R^2 = Real.exp (2*Real.log R) := by
-      rw [show 2*Real.log R = Real.log R+Real.log R by ring, Real.exp_add,Real.exp_log hR0,pow_two]
-    rw [hh,← Real.exp_add]
-    congr 1
-    ring
-  have hbR : m*Real.exp (β*Real.log R) ≤ Real.exp (β*Real.log R) := by
-    exact mul_le_of_le_one_left (Real.exp_pos _).le (min_le_left _ _)
-  have hbT : m*Real.exp (β*Real.log R) ≤ d*Real.exp ((β-2)*Real.log R)*((R^2).toNNReal:ℝ) := by
-    rw [Real.coe_toNNReal _ (sq_nonneg R),mul_assoc,he]
-    exact mul_le_mul_of_nonneg_right (min_le_right _ _) (Real.exp_pos _).le
-  have hb : 0 < m*Real.exp (β*Real.log R) := mul_pos hm (Real.exp_pos _)
-  have hp := ordinary_survival_bound hB κ hκ γ G F hw z hz R hR0 hN
-    ((R^2).toNNReal) (m*Real.exp (β*Real.log R)) hb hbR hbT
-  apply hp.trans
-  apply ENNReal.ofReal_le_ofReal
-  calc
-    swallowLyapunov κ β C (initialLogCoordinates z) 0/(m*Real.exp (β*Real.log R))
-        ≤ (1+C)/(m*Real.exp (β*Real.log R)) := div_le_div_of_nonneg_right hF hb.le
-    _ = (1+C)/m*Real.exp (-β*Real.log R) := by
-      rw [show -β*Real.log R = -(β*Real.log R) by ring, Real.exp_neg]
-      ring
-
- 
-
-theorem ordinary_survival_polynomial
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 4 < κ)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ z ∈ upperHalfPlane, ‖z‖ = 1 →
-      ∀ T : ℝ≥0, 1 ≤ T →
-      P {ω | z ∈ survivingDomain (γ ω) T} ≤ ENNReal.ofReal (C*(T:ℝ)^(-c)) := by
-  let β := (κ-4)/(2*κ)
-  let C := (1+3*(κ-4)/8) / min 1 (β*((κ-4)/4))
-  have hk : 0 < κ := lt_trans (by norm_num) hκ
-  have hβ : 0 < β := by dsimp [β]; positivity
-  have hC : 0 < C := by dsimp [C]; positivity
-  refine ⟨C,β/2,hC,by positivity,?_⟩
-  intro z hz hu T hT
-  have ht : 0 < (T:ℝ) := lt_of_lt_of_le (by norm_num) (NNReal.coe_le_coe.mpr hT)
-  have hR : 1 ≤ Real.sqrt (T:ℝ) := (Real.le_sqrt (by norm_num) T.coe_nonneg).mpr (by simpa only [one_pow, NNReal.coe_one] using NNReal.coe_le_coe.mpr hT)
-  have hp := ordinary_survival_radial_tail hB κ hκ γ G F hw z hz hu (Real.sqrt T) hR
-  rw [Real.sq_sqrt T.coe_nonneg,Real.toNNReal_coe] at hp
-  convert hp using 2
-  rw [Real.rpow_def_of_pos ht,Real.log_sqrt ht.le]
-  congr 2
-  ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma linear_subsolution_initial_nonpos (T : ℝ) (hT : 0 ≤ T)
-    (f a g : ℝ → ℝ) (ha : ContinuousOn a (Icc 0 T))
-    (hf : ∀ u ∈ Icc 0 T, HasDerivWithinAt f (g u) (Icc 0 T) u)
-    (hg : ∀ u ∈ Icc 0 T, a u*f u ≤ g u) (hfT : f T ≤ 0) : f 0 ≤ 0 := by
-  let A (u : ℝ) := a (projIcc 0 T hT u)
-  have hA : Continuous A := ha.comp_continuous
-    (continuous_subtype_val.comp continuous_projIcc) (fun u => (projIcc 0 T hT u).property)
-  have hAa (u : ℝ) (hu : u ∈ Icc 0 T) : A u = a u := by simp only [A,projIcc_of_mem hT hu]
-  let I (u : ℝ) := ∫ v in (0:ℝ)..u, A v
-  have hI (u : ℝ) : HasDerivAt I (A u) u :=
-    intervalIntegral.integral_hasDerivAt_right (hA.intervalIntegrable _ _)
-      hA.aestronglyMeasurable.stronglyMeasurableAtFilter hA.continuousAt
-  have hd (u : ℝ) (hu : u ∈ Icc 0 T) :
-      HasDerivWithinAt (fun v => f v*Real.exp (-I v))
-        ((g u-a u*f u)*Real.exp (-I u)) (Icc 0 T) u := by
-    convert (hf u hu).mul ((hI u).neg.exp.hasDerivWithinAt) using 1; try rfl
-    simp only [Pi.neg_apply]
-    rw [hAa u hu]
-    ring
-  have hmono : MonotoneOn (fun v => f v*Real.exp (-I v)) (Icc 0 T) :=
-    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
-      (fun u hu => (hd u hu).continuousWithinAt)
-      (fun u hu => (hd u (interior_subset hu)).mono interior_subset)
-      (fun u hu => mul_nonneg (sub_nonneg.mpr (hg u (interior_subset hu))) (Real.exp_pos _).le)
-  have hh := hmono (show (0:ℝ) ∈ Icc 0 T from ⟨le_rfl,hT⟩) ⟨hT,le_rfl⟩ hT
-  simp only [I,intervalIntegral.integral_same,neg_zero,Real.exp_zero,mul_one] at hh
-  exact hh.trans (mul_nonpos_of_nonpos_of_nonneg hfT (Real.exp_pos _).le)
-
-lemma IsCapacityTwoWitness.real_derivative
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
-    (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (fun v : ℝ => (G v.toNNReal z).re)
-      (2*((G u.toNNReal z).re-U u.toNNReal)/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)))
-      (Icc 0 (T:ℝ)) u := by
-  have hh := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt u (h.time_derivative T z hz u hu)
-  simpa [Function.comp_def,Complex.div_re] using hh
-
-lemma IsCapacityTwoWitness.initial_real_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
-    (hz : z ∈ survivingDomain γ T) (M : ℝ)
-    (hbound : ∀ u : ℝ≥0, u ≤ T → U u ≤ M) (hend : (G T z).re ≤ M) : z.re ≤ M := by
-  let a (u : ℝ) := 2/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))
-  have hnorm : ContinuousOn (fun u : ℝ => Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) :=
-    Complex.continuous_normSq.comp_continuousOn ((h.time_continuousOn T z hz).sub
-      (Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal)).continuousOn)
-  have hne (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
-      Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) ≠ 0 := by
-    exact (Complex.normSq_pos.mpr (h.centered_ne_zero u.toNNReal z
-      (survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu.2) hz))).ne'
-  have ha : ContinuousOn a (Icc 0 (T:ℝ)) := continuousOn_const.div hnorm hne
-  have hh := linear_subsolution_initial_nonpos T T.coe_nonneg
-    (fun u => (G u.toNNReal z).re-M) a
-    (fun u => 2*((G u.toNNReal z).re-U u.toNNReal)/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))) ha
-    (fun u hu => (h.real_derivative T z hz u hu).sub_const M)
-    (fun u hu => by
-      dsimp only [a]
-      have hb := hbound u.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hu.2)
-      have hp : 0 ≤ Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) := Complex.normSq_nonneg _
-      rw [div_mul_eq_mul_div]
-      exact div_le_div_of_nonneg_right (by linarith) hp)
-    (by simpa only [Real.toNNReal_coe] using sub_nonpos.mpr hend)
-  simpa only [Real.toNNReal_zero,h.initial_value z hz.1.1,sub_nonpos] using hh
-
-lemma IsCapacityTwoWitness.initial_real_ge
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
-    (hz : z ∈ survivingDomain γ T) (M : ℝ)
-    (hbound : ∀ u : ℝ≥0, u ≤ T → M ≤ U u) (hend : M ≤ (G T z).re) : M ≤ z.re := by
-  let a (u : ℝ) := 2/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))
-  have hnorm : ContinuousOn (fun u : ℝ => Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) :=
-    Complex.continuous_normSq.comp_continuousOn ((h.time_continuousOn T z hz).sub
-      (Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal)).continuousOn)
-  have hne (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
-      Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) ≠ 0 := by
-    exact (Complex.normSq_pos.mpr (h.centered_ne_zero u.toNNReal z
-      (survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu.2) hz))).ne'
-  have ha : ContinuousOn a (Icc 0 (T:ℝ)) := continuousOn_const.div hnorm hne
-  have hh := linear_subsolution_initial_nonpos T T.coe_nonneg
-    (fun u => M-(G u.toNNReal z).re) a
-    (fun u => -(2*((G u.toNNReal z).re-U u.toNNReal)/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)))) ha
-    (fun u hu => (h.real_derivative T z hz u hu).const_sub M)
-    (fun u hu => by
-      dsimp only [a]
-      have hb := hbound u.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hu.2)
-      have hp : 0 ≤ Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) := Complex.normSq_nonneg _
-      rw [div_mul_eq_mul_div,← neg_div]
-      exact div_le_div_of_nonneg_right (by linarith) hp)
-    (by simpa only [Real.toNNReal_coe] using sub_nonpos.mpr hend)
-  simpa only [Real.toNNReal_zero,h.initial_value z hz.1.1,sub_nonpos] using hh
-
-lemma IsCapacityTwoWitness.initial_imag_square_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
-    (hz : z ∈ survivingDomain γ T) : z.im^2 ≤ (G T z).im^2+4*T := by
-  have hd (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :=
-    ((h.imag_derivative T z hz u hu).pow 2).add ((hasDerivWithinAt_id u (Icc 0 (T:ℝ))).const_mul 4)
-  have hnonneg (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
-      0 ≤ (2:ℝ)*(G u.toNNReal z).im^(2-1)*
-        (-2*(G u.toNNReal z).im/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)))+1*4 := by
-    have hp := Complex.normSq_pos.mpr (h.centered_ne_zero u.toNNReal z
-      (survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu.2) hz))
-    have hn : (G u.toNNReal z).im^2 ≤ Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) := by
-      simp only [Complex.normSq_apply,Complex.sub_re,Complex.ofReal_re,Complex.sub_im,Complex.ofReal_im,sub_zero]
-      nlinarith [sq_nonneg ((G u.toNNReal z).re-U u.toNNReal)]
-    norm_num only [Nat.reduceSub, pow_one, one_mul]
-    field_simp
-    nlinarith
-  have hmono : MonotoneOn (fun u : ℝ => (G u.toNNReal z).im^2+4*u) (Icc 0 (T:ℝ)) :=
-    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
-      (fun u hu => (hd u hu).continuousWithinAt)
-      (fun u hu => (hd u (interior_subset hu)).mono interior_subset)
-      (fun u hu => by simpa only [Nat.cast_ofNat, mul_one, one_mul] using hnonneg u (interior_subset hu))
-  have hh := hmono (show (0:ℝ) ∈ Icc 0 (T:ℝ) from ⟨le_rfl,T.coe_nonneg⟩)
-    ⟨T.coe_nonneg,le_rfl⟩ T.coe_nonneg
-  simpa only [Real.toNNReal_zero,Real.toNNReal_coe,h.initial_value z hz.1.1,mul_zero,add_zero] using hh
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma IsCapacityTwoWitness.trace_real_bounds
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (M : ℝ)
-    (hbound : ∀ u : ℝ≥0, u ≤ T → |U u| ≤ M) : |(γ T).re| ≤ M := by
-  have hh := h
-  obtain ⟨_,_,_,_,_,_,hFmap,_,hinv,_,_,_,htip⟩ := hh
-  have hm (y : ℝ) (hy : 0 < y) :
-      |(F T ((U T:ℂ)+(y:ℂ)*Complex.I)).re| ≤ M := by
-    have hyH : (U T:ℂ)+(y:ℂ)*Complex.I ∈ upperHalfPlane := by
-      simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im,
-        Complex.mul_im, Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one,
-        mul_zero, add_zero, zero_add] using hy
-    have hz := hFmap T hyH
-    have he := hinv T _ hyH
-    apply abs_le.mpr
-    constructor
-    · apply IsCapacityTwoWitness.initial_real_ge hU h T _ hz (-M)
-      · intro u hu; exact (abs_le.mp (hbound u hu)).1
-      · rw [he]; simpa using (abs_le.mp (hbound T le_rfl)).1
-    · apply IsCapacityTwoWitness.initial_real_le hU h T _ hz M
-      · intro u hu; exact (abs_le.mp (hbound u hu)).2
-      · rw [he]; simpa using (abs_le.mp (hbound T le_rfl)).2
-  have ht := (continuous_abs.tendsto _).comp (Complex.continuous_re.tendsto _ |>.comp (htip T))
-  exact le_of_tendsto ht (eventually_nhdsWithin_of_forall fun y hy => hm y hy)
-
-lemma IsCapacityTwoWitness.trace_imag_square_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) : (γ T).im^2 ≤ 4*T := by
-  have hh := h
-  obtain ⟨_,_,_,_,_,_,hFmap,_,hinv,_,_,_,htip⟩ := hh
-  have hm (y : ℝ) (hy : 0 < y) :
-      (F T ((U T:ℂ)+(y:ℂ)*Complex.I)).im^2 ≤ y^2+4*T := by
-    have hyH : (U T:ℂ)+(y:ℂ)*Complex.I ∈ upperHalfPlane := by
-      simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im,
-        Complex.mul_im, Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one,
-        mul_zero, add_zero, zero_add] using hy
-    have hz := hFmap T hyH
-    have he := hinv T _ hyH
-    have hh := h.initial_imag_square_le T _ hz
-    rw [he] at hh
-    simpa using hh
-  have ht := (Complex.continuous_im.tendsto _ |>.comp (htip T)).pow 2
-  have hr : Tendsto (fun y : ℝ => y^2+4*(T:ℝ)) (𝓝[>] 0) (𝓝 (4*(T:ℝ))) := by
-    simpa using (((tendsto_id : Tendsto (fun y : ℝ => y) (𝓝 (0:ℝ)) (𝓝 0)).pow 2).add_const (4*(T:ℝ))).mono_left nhdsWithin_le_nhds
-  exact le_of_tendsto_of_tendsto ht hr (eventually_nhdsWithin_of_forall fun y hy => hm y hy)
-
-lemma IsCapacityTwoWitness.trace_norm_square_le
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (M : ℝ)
-    (hM : 0 ≤ M) (hbound : ∀ u : ℝ≥0, u ≤ T → |U u| ≤ M) : ‖γ T‖^2 ≤ M^2+4*T := by
-  have hr := h.trace_real_bounds hU T M hbound
-  have hi := h.trace_imag_square_le T
-  rw [Complex.sq_norm, Complex.normSq_apply]
-  have hs : (γ T).re^2 ≤ M^2 := by
-    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hM).mpr hr
-  nlinarith
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma exists_compact_square_field (R : ℝ) (hR : 0 ≤ R) :
-    ∃ φ : ℝ × ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
-      (∀ x, |x| ≤ R → φ (0,x) = x^2) ∧
-      (∀ x, |x| ≤ R → iteratedDeriv 2 (fun y => φ (0,y)) x = 2) := by
-  let b : ContDiffBump (0 : ℝ × ℝ) := ⟨R+1,R+2,by linarith,by linarith⟩
-  let φ (p : ℝ × ℝ) := b p*p.2^2
-  have hnear (x : ℝ) (hx : |x| ≤ R) : (fun y => φ (0,y)) =ᶠ[𝓝 x] (fun y => y^2) := by
-    have hball : (0,x) ∈ Metric.ball (0:ℝ × ℝ) b.rIn := by
-      simpa only [Metric.mem_ball,dist_zero_right,Prod.norm_def,norm_zero,Real.norm_eq_abs,
-        max_eq_right (abs_nonneg x)] using (show |x| < R+1 by linarith)
-    have he := (b.eventuallyEq_one_of_mem_ball hball).comp_tendsto
-      ((continuous_const.prodMk continuous_id).tendsto x)
-    filter_upwards [he] with y hy
-    change b (0,y) = 1 at hy
-    simp only [φ,hy,one_mul]
-  refine ⟨φ,b.contDiff.mul (contDiff_snd.pow 2),b.hasCompactSupport.mul_right,
-    fun x hx => (hnear x hx).eq_of_nhds,?_⟩
-  intro x hx
-  rw [(hnear x hx).iteratedDeriv_eq 2,iteratedDeriv_pow]
-  norm_num
-
-lemma brownian_stopped_square
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hin : ∀ ω t, t ≤ τ ω → |B t ω| ≤ R) :
-    Integrable (fun ω => (B (τ ω) ω)^2) P ∧
-      (∫ ω, (B (τ ω) ω)^2 ∂P) = ∫ ω, (τ ω:ℝ) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨φ,hφ,hφc,he,h2⟩ := exists_compact_square_field R hR
-  obtain ⟨K,hfield⟩ := compact_state_smooth_brownian_field B hm
-    (fun _ _ => (0:ℝ)) (fun _ _ => (0:ℝ)) 0 measurable_const (fun _ => measurable_const)
-    (fun _ => LipschitzWith.const 0) measurable_const (by simp)
-    (fun t ω ht => (hasDerivWithinAt_const t (Ici 0) (0:ℝ))) φ hφ hφc
-  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (0,x)
-  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (0,x) ((0:ℝ),0)
-  have hg (ω : Ω) (u : ℝ) (hu : u ∈ Icc 0 (τ ω:ℝ)) : brownianFieldDrift B F a u ω = 1 := by
-    have hi := hin ω u.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hu.2)
-    simp only [brownianFieldDrift,F,a,show ((0:ℝ),0) = (0:ℝ × ℝ) from rfl,map_zero,h2 _ hi]
-    norm_num
-  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hBi : Integrable (fun ω => (B (τ ω) ω)^2) P := by
-    apply Integrable.of_bound ((hBm.comp (hτm.prodMk measurable_id)).pow_const 2).aestronglyMeasurable (R^2)
-    exact Eventually.of_forall fun ω => by
-      rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
-      have hh : (B (τ ω) ω)^2 ≤ R^2 := by
-        simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hR).mpr (hin ω (τ ω) le_rfl)
-      simpa only [Function.comp_def, id_eq] using hh
-  refine ⟨hBi,?_⟩
-  have hef : (fun ω => F (τ ω) ω (B (τ ω) ω)) = fun ω => (B (τ ω) ω)^2 :=
-    funext fun ω => he _ (hin ω (τ ω) le_rfl)
-  rw [hef] at hopt
-  have h0 : (∫ ω, F 0 ω (B 0 ω) ∂P) = 0 := by
-    have he0 : (fun ω => F 0 ω (B 0 ω)) =ᵐ[P] (fun _ => (0:ℝ)) := by
-      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hω
-      change φ (0,B 0 ω) = 0
-      rw [hω,he 0 (by simpa using hR)]
-      norm_num
-    rw [integral_congr_ae he0,integral_zero]
-  rw [h0,zero_add] at hopt
-  rw [hopt]
-  apply integral_congr_ae
-  exact Eventually.of_forall fun ω => by
-    calc
-      (∫ u in (0:ℝ)..(τ ω:ℝ), brownianFieldDrift B F a u ω) = ∫ _ in (0:ℝ)..(τ ω:ℝ), (1:ℝ) := by
-        apply intervalIntegral.integral_congr
-        intro u hu
-        exact hg ω u (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using hu)
-      _ = (τ ω:ℝ) := by simp
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.DyadicStrips
+import OAI.Probability.SLE.NestedMeans
+
+namespace OAI
+
+/-! Strip Probabilities for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory
+open scoped Topology ContDiff ENNReal BigOperators NNReal
+namespace SLEExactGauge
+theorem comparison_path_strip_zero {η : ℝ → ℂ} {T : ℝ}
+    (hT : 0 < T) (hη : ContinuousOn η (Icc 0 T)) (hstart : η 0 = 0)
+    (hend : ‖η T‖ = 1) (hupper : ∀ t ∈ Icc 0 T, 0 ≤ (η t).im)
+    (Fbar : ℂ → OnePoint ℂ) (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H)
+    (himage : ∀ t ∈ Icc 0 T, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r)
+    (v : ℝ) (hv : v ≤ 0) :
+    ∃ φ : ℝ, φ ∈ Icc 0 Real.pi ∧
+      sphericalTruncatedLog q r H (Fbar (Complex.exp ((v : ℂ)+(φ : ℂ)*Complex.I))) = 0 := by
+  have her : Real.exp v ∈ Icc ‖η 0‖ ‖η T‖ := by
+    rw [hstart,norm_zero,hend]
+    exact ⟨(Real.exp_pos v).le,by simpa using Real.exp_le_exp.mpr hv⟩
+  obtain ⟨t,ht,htr⟩ := intermediate_value_Icc hT.le hη.norm her
+  change ‖η t‖ = Real.exp v at htr
+  have hn : η t ≠ 0 := by
+    apply norm_ne_zero_iff.mp
+    rw [htr]
+    exact (Real.exp_pos v).ne'
+  let φ := (η t).arg
+  have hφ : φ ∈ Icc 0 Real.pi :=
+    ⟨Complex.arg_nonneg_iff.mpr (hupper t ht),Complex.arg_le_pi _⟩
+  have he : (v : ℂ)+(φ : ℂ)*Complex.I = Complex.log (η t) := by
+    apply Complex.ext
+    · simp [Complex.log_re,htr]
+    · simp [Complex.log_im,φ]
+  refine ⟨φ,hφ,?_⟩
+  rw [he,Complex.exp_log hn]
+  obtain ⟨w,hw,hwr⟩ := himage t ht
+  rw [hw,sphericalTruncatedLog_coe]
+  exact truncatedConformalLog_eq_zero hr hH.le hwr
+
+end SLEExactGauge
+end
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+lemma source_strip_log_locallyLipschitz {F : ℂ → ℂ} {Fbar : ℂ → OnePoint ℂ}
+    (hF : DifferentiableOn ℂ F {z | 0 < z.im})
+    (hext : ∀ z : ℂ, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (q : ℂ) (r H : ℝ) (hr : 0 < r) (hH : 0 < H) :
+    LocallyLipschitzOn fullOpenStrip (fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))) := by
+  let U := fun z : ℂ => sphericalTruncatedLog q r H (Fbar (Complex.exp z))
+  let V := truncatedConformalLog (F ∘ Complex.exp) q r H
+  have heq : EqOn U V fullOpenStrip := by
+    intro z hz
+    dsimp [U]
+    rw [hext (Complex.exp z) (exp_strip_halfplane hz),sphericalTruncatedLog_coe]
+    rfl
+  have hcomp : DifferentiableOn ℂ (F ∘ Complex.exp) fullOpenStrip :=
+    hF.comp Complex.differentiable_exp.differentiableOn exp_strip_halfplane
+  intro z hz
+  obtain ⟨K,S,hS,hzS,hK⟩ := truncated_log_lipschitz_near fullOpenStrip_isOpen hcomp hz q hr hH
+  refine ⟨K,S ∩ fullOpenStrip,mem_nhdsWithin_of_mem_nhds ((hS.inter fullOpenStrip_isOpen).mem_nhds ⟨hzS,hz⟩),?_⟩
+  intro x hx y hy
+  change edist (U x) (U y) ≤ (K : ℝ≥0∞)*edist x y
+  rw [heq hx.2,heq hy.2]
+  exact hK hx.1 hy.1
+
+ 
+
+def BeforeFirstModulus (γ : ℝ≥0 → ℂ) (A : ℝ) (t : ℝ≥0) : Prop :=
+  ∀ s : ℝ≥0, s < t → ‖γ s‖ < A
+
+universe u
+ 
+def SourceUniformConformalLocalizationTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 → ∃ b C c L0 : ℝ,
+    0 < b ∧ 0 < C ∧ 0 < c ∧ 1 < L0 ∧
+    ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ γ : Ω → ℝ≥0 → ℂ, IsOrdinaryChordalSLE κ γ P →
+    ∀ (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ),
+    DifferentiableOn ℂ F {z | 0 < z.im} → InjOn F {z | 0 < z.im} →
+    ContinuousOn Fbar {z : ℂ | 0 ≤ z.im} →
+    (∀ z : ℂ, 0 < z.im → Fbar z = (F z : OnePoint ℂ)) →
+    ∀ (q : ℂ) (r : ℝ), 0 < r →
+    ∀ (η : ℝ → ℂ) (T : ℝ), 0 < T → ContinuousOn η (Icc 0 T) →
+    η 0 = 0 → ‖η T‖ = 1 → (∀ t ∈ Icc 0 T, 0 ≤ (η t).im) →
+    (∀ t ∈ Icc 0 T, ∃ w : ℂ, Fbar (η t) = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r) →
+    ∀ L : ℝ, L0 ≤ L →
+      P {xi | ∃ t : ℝ≥0, BeforeFirstModulus (γ xi) (L^b) t ∧
+        Fbar (γ xi t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (L*r)} ≤
+        ENNReal.ofReal (C*L^(-c))
+
+lemma beforeFirstModulus_norm_le {γ : ℝ≥0 → ℂ} (hc : Continuous γ) (h0 : γ 0 = 0)
+    {A : ℝ} (hA : 0 < A) {t : ℝ≥0} (ht : BeforeFirstModulus γ A t) :
+    ‖γ t‖ ≤ A := by
+  by_cases ht0 : t = 0
+  · simp [ht0,h0,hA.le]
+  have htpos : 0 < t := bot_lt_iff_ne_bot.mpr ht0
+  have hcl : t ∈ closure (Iio t : Set ℝ≥0) := by
+    rw [closure_Iio' (show (Iio t : Set ℝ≥0).Nonempty from ⟨0,htpos⟩)]
+    exact (show t ≤ t from le_rfl)
+  have hclosed : IsClosed {s : ℝ≥0 | ‖γ s‖ ≤ A} := isClosed_le hc.norm continuous_const
+  exact (closure_minimal (fun s hs => (ht s hs).le) hclosed) hcl
+
+lemma sphericalTruncatedLog_eq_height_of_notin_ball (q : ℂ) (r H : ℝ) (hr : 0 < r)
+    (x : OnePoint ℂ)
+    (hx : x ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (Real.exp H*r)) :
+    sphericalTruncatedLog q r H x = H := by
+  induction x using OnePoint.rec with
+  | infty => rfl
+  | coe z =>
+    rw [sphericalTruncatedLog_coe]
+    apply truncatedConformalLog_eq_height hr
+    apply le_of_not_gt
+    intro h
+    apply hx
+    exact ⟨z,by simpa [Metric.mem_ball,dist_eq_norm] using h,rfl⟩
+
+lemma escape_forces_strip_level {γ : ℝ≥0 → ℂ} (hc : Continuous γ) (h0 : γ 0 = 0)
+    (hupper : ∀ t, 0 ≤ (γ t).im)
+    (Fbar : ℂ → OnePoint ℂ) (q : ℂ) {r L H b : ℝ} (hr : 0 < r) (hL : 1 < L)
+    (hH : H = Real.log L)
+    (hstart : ∃ w : ℂ, Fbar 0 = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r)
+    (t : ℝ≥0) (ht : BeforeFirstModulus γ (L^b) t)
+    (hbad : Fbar (γ t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (L*r)) :
+    ∃ z : ℂ, Complex.exp z ∈ range γ ∧ 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*H ∧
+      sphericalTruncatedLog q r H (Fbar (Complex.exp z)) = H := by
+  have hLp : 0 < L := by linarith
+  have hnorm := beforeFirstModulus_norm_le hc h0 (Real.rpow_pos_of_pos hLp b) ht
+  have hn : γ t ≠ 0 := by
+    intro heq
+    rw [heq] at hbad
+    obtain ⟨w,hw,hwr⟩ := hstart
+    apply hbad
+    exact ⟨w,by rw [Metric.mem_ball,dist_eq_norm]; nlinarith,hw.symm⟩
+  refine ⟨Complex.log (γ t),by rw [Complex.exp_log hn]; exact mem_range_self t,?_,?_,?_,?_⟩
+  · simpa only [Complex.log_im] using Complex.arg_nonneg_iff.mpr (hupper t)
+  · simpa only [Complex.log_im] using Complex.arg_le_pi (γ t)
+  · rw [Complex.log_re,hH]
+    exact (Real.log_le_log (norm_pos_iff.mpr hn) hnorm).trans_eq (Real.log_rpow hLp b)
+  · rw [Complex.exp_log hn]
+    apply sphericalTruncatedLog_eq_height_of_notin_ball q r H hr
+    simpa only [hH,Real.exp_log hLp] using hbad
+
+theorem sourceUniformConformalLocalization : SourceUniformConformalLocalizationTarget := by
+  intro κ hκ hκ8
+  obtain ⟨C0,α,hC0,hα,hhit⟩ := ordinary_sle_strip_hitting_constants κ hκ hκ8
+  obtain ⟨b,C,c,H0,hb,hC,hc,hH0,hstrip⟩ := sourceStripLocalization C0 α hC0 hα
+  refine ⟨b,C,c,Real.exp (max H0 1),hb,hC,hc,?_,?_⟩
+  · exact Real.one_lt_exp_iff.mpr (lt_of_lt_of_le zero_lt_one (le_max_right _ _))
+  intro Ω mΩ P hP γ hγ F Fbar hF hi hbar hext q r hr η T hT hη hs he hu him L hL
+  let := hP
+  have hL1 : 1 < L := (Real.one_lt_exp_iff.mpr
+    (lt_of_lt_of_le zero_lt_one (le_max_right H0 1))).trans_le hL
+  have hLp : 0 < L := by linarith
+  have hlog : max H0 1 ≤ Real.log L := by
+    simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos _) hL
+  have hHH : H0 ≤ Real.log L := (le_max_left _ _).trans hlog
+  have hHp : 0 < Real.log L := hH0.trans_le hHH
+  let U := fun z : ℂ => sphericalTruncatedLog q r (Real.log L) (Fbar (Complex.exp z))
+  let X := fun xi : Ω => Complex.exp ⁻¹' range (γ xi)
+  obtain ⟨hcU,hbU,-,hgU,hEU⟩ := source_strip_log_regularity hF hi hbar hext q r (Real.log L) hr hHp
+  have hzero (v : ℝ) (hv : v ≤ 0) : ∃ y ∈ Icc (0:ℝ) Real.pi, U (v+y*Complex.I) = 0 :=
+    comparison_path_strip_zero hT hη hs he hu Fbar q r (Real.log L) hr hHp him v hv
+  have hhits (j : ℤ) (k : ℕ) (i m : Fin (2^k)) :
+      P {xi | (X xi ∩ dyadicStripRectangle j k i m).Nonempty} ≤
+        ENNReal.ofReal (C0*(2:ℝ)^(-α*(k:ℝ))) := by
+    apply (measure_mono ?_).trans (hhit γ hγ j k i m)
+    rintro xi ⟨z,⟨t,ht⟩,hz⟩
+    exact ⟨t,z,hz,ht.symm⟩
+  have hh := hstrip (Real.log L) hHH U hcU
+    (source_strip_log_locallyLipschitz hF hext q r (Real.log L) hr hHp)
+    hbU hgU hEU hzero Ω mΩ P hP X hhits
+  have hstart : ∃ w : ℂ, Fbar 0 = (w : OnePoint ℂ) ∧ ‖w-q‖ ≤ r := by
+    simpa only [hs] using him 0 ⟨le_rfl,hT.le⟩
+  obtain ⟨B,hB,htrace⟩ := hγ.2
+  have hsub : {xi | ∃ t : ℝ≥0, BeforeFirstModulus (γ xi) (L^b) t ∧
+        Fbar (γ xi t) ∉ (fun z : ℂ => (z : OnePoint ℂ)) '' Metric.ball q (L*r)} ≤ᵐ[P]
+      {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*Real.log L ∧ U z = Real.log L} := by
+    filter_upwards [htrace] with xi hxi
+    obtain ⟨G,Fw,hFw⟩ := hxi
+    rintro ⟨t,ht,hbad⟩
+    exact escape_forces_strip_level hFw.1 hFw.2.1 hFw.2.2.1 Fbar q hr hL1 rfl hstart t ht hbad
+  calc
+    _ ≤ P {xi | ∃ z ∈ X xi, 0 ≤ z.im ∧ z.im ≤ Real.pi ∧ z.re ≤ b*Real.log L ∧ U z = Real.log L} :=
+      measure_mono_ae hsub
+    _ ≤ ENNReal.ofReal (C*Real.exp (-c*Real.log L)) := hh
+    _ = ENNReal.ofReal (C*L^(-c)) := by
+      rw [Real.rpow_def_of_pos hLp]
+      congr 2
+      congr 1
+      ring
+
+end SLEExactGauge
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem ordinary_survival_bound_regular
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ : ℝ) (hκ : 4 < κ)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (R : ℝ) (hR : 0 < R)
+    (hN : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ R^2)
+    (T : ℝ≥0) (b : ℝ) (hb : 0 < b)
+    (hbR : b ≤ Real.exp (((κ-4)/(2*κ))*Real.log R))
+    (hbT : b ≤ (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) * T) :
+    P {ω | z ∈ survivingDomain (γ ω) T} ≤
+      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
+        (initialLogCoordinates z) 0 / b) := by
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let q₀ := initialLogCoordinates z
+  let lo (n : ℕ) : ℝ := q₀.1.im-(n:ℝ)-1
+  have hlo (n : ℕ) : lo n ≤ q₀.1.im := by dsimp [lo]; have := Nat.cast_nonneg (α := ℝ) n; linarith
+  let c (n : ℕ) := Classical.choice
+    (exists_clippedLogSolution B hm hBm hBc κ (lo n) q₀.1.im (hlo n) q₀)
+  let a (n : ℕ) : RadialLocalization (c n) R T := Classical.choice
+    (exists_radialLocalization hBc hB0 (c n) R T hN (hlo n))
+  let E (n : ℕ) : Set Ω := {ω | (a n).τ ω = T ∨
+    greenLogNormSq κ ((c n).Q ((a n).τ ω) ω) (B ((a n).τ ω) ω) = R^2}
+  have hprob (n : ℕ) : P (E n) ≤
+      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q₀ 0 / b) :=
+    (a n).exit_probability hB hBm hBc hκ hR le_rfl b hb hbR hbT
+  have hsub : {ω | z ∈ survivingDomain (γ ω) T} ≤ᵐ[P]
+      {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} := by
+    filter_upwards [hw] with ω hω
+    intro hlive
+    obtain ⟨N, hN⟩ := exists_nat_gt (q₀.1.im-Real.log (G ω T z).im)
+    filter_upwards [eventually_ge_atTop N] with n hn
+    have hlt : lo n < Real.log (G ω T z).im := by
+      have hn' : (N:ℝ) ≤ n := by exact_mod_cast hn
+      dsimp [lo]
+      linarith
+    exact (a n).alive_forces_radial_or_time_boundary hBc hz rfl le_rfl ω
+      (γ ω) (G ω) (F ω) hω hlive hlt
+  exact (measure_mono_ae hsub).trans (measure_eventually_sets_le P E _ hprob)
+
+theorem ordinary_survival_bound
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 4 < κ)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (R : ℝ) (hR : 0 < R)
+    (hN : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ R^2)
+    (T : ℝ≥0) (b : ℝ) (hb : 0 < b)
+    (hbR : b ≤ Real.exp (((κ-4)/(2*κ))*Real.log R))
+    (hbT : b ≤ (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) * T) :
+    P {ω | z ∈ survivingDomain (γ ω) T} ≤
+      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8)
+        (initialLogCoordinates z) 0 / b) := by
+  obtain ⟨C, hCm, hCc, hC0, hC, hCB⟩ := exists_regular_brownian_version hB
+  have hwC : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * C t ω) (γ ω) (G ω) (F ω) := by
+    filter_upwards [hw,hCB] with ω hω he
+    simpa only [he] using hω
+  exact ordinary_survival_bound_regular hC hCm hCc hC0 κ hκ γ G F hwC
+    z hz R hR hN T b hb hbR hbT
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma greenLogNormSq_initial_norm (κ : ℝ) (z : ℂ) (hz : 0 < z.im) :
+    greenLogNormSq κ (initialLogCoordinates z) 0 = ‖z‖^2 := by
+  have he : Real.exp (2 * Real.log z.im) = z.im ^ 2 := by
+    rw [show 2 * Real.log z.im = Real.log z.im + Real.log z.im by ring,
+      Real.exp_add, Real.exp_log hz, pow_two]
+  rw [← Complex.normSq_eq_norm_sq]
+  simp only [greenLogNormSq, initialLogCoordinates, Complex.add_re, Complex.ofReal_re,
+      Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero,
+      sub_self, add_zero, sub_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add, he,
+      Complex.normSq_apply, pow_two]
+
+theorem ordinary_survival_radial_tail
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 4 < κ)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (z : ℂ) (hz : z ∈ upperHalfPlane) (hunit : ‖z‖ = 1)
+    (R : ℝ) (hR : 1 ≤ R) :
+    P {ω | z ∈ survivingDomain (γ ω) ((R^2).toNNReal)} ≤
+      ENNReal.ofReal ((1+3*(κ-4)/8) / min 1 (((κ-4)/(2*κ))*((κ-4)/4)) *
+        Real.exp (-((κ-4)/(2*κ))*Real.log R)) := by
+  let β := (κ-4)/(2*κ)
+  let C := 3*(κ-4)/8
+  let d := β*((κ-4)/4)
+  let m := min 1 d
+  have hk : 0 < κ := lt_trans (by norm_num) hκ
+  have hβ : 0 < β := by dsimp [β]; positivity
+  have hC : 0 ≤ C := by dsimp [C]; positivity
+  have hd : 0 < d := by dsimp [d]; positivity
+  have hm : 0 < m := lt_min (by norm_num) hd
+  have hR0 : 0 < R := lt_of_lt_of_le (by norm_num) hR
+  have hN : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ R^2 := by
+    rw [greenLogNormSq_initial_norm κ z hz, hunit]
+    nlinarith
+  have hN1 : greenLogNormSq κ (initialLogCoordinates z) 0 ≤ (1:ℝ)^2 := by
+    rw [greenLogNormSq_initial_norm κ z hz, hunit]
+  have hF : swallowLyapunov κ β C (initialLogCoordinates z) 0 ≤ 1+C := by
+    have hr := radialLogPower_le κ β hβ.le (initialLogCoordinates z) 0 1 (by norm_num) hN1
+    have hy := heightLogPower_le κ β hβ.le (initialLogCoordinates z) 0 1 (by norm_num) hN1
+    simp only [Real.log_one, mul_zero, Real.exp_zero] at hr hy
+    exact (add_le_add hr (mul_le_mul_of_nonneg_left hy hC)).trans_eq (by ring)
+  have he : Real.exp ((β-2)*Real.log R)*R^2 = Real.exp (β*Real.log R) := by
+    have hh : R^2 = Real.exp (2*Real.log R) := by
+      rw [show 2*Real.log R = Real.log R+Real.log R by ring, Real.exp_add,Real.exp_log hR0,pow_two]
+    rw [hh,← Real.exp_add]
+    congr 1
+    ring
+  have hbR : m*Real.exp (β*Real.log R) ≤ Real.exp (β*Real.log R) := by
+    exact mul_le_of_le_one_left (Real.exp_pos _).le (min_le_left _ _)
+  have hbT : m*Real.exp (β*Real.log R) ≤ d*Real.exp ((β-2)*Real.log R)*((R^2).toNNReal:ℝ) := by
+    rw [Real.coe_toNNReal _ (sq_nonneg R),mul_assoc,he]
+    exact mul_le_mul_of_nonneg_right (min_le_right _ _) (Real.exp_pos _).le
+  have hb : 0 < m*Real.exp (β*Real.log R) := mul_pos hm (Real.exp_pos _)
+  have hp := ordinary_survival_bound hB κ hκ γ G F hw z hz R hR0 hN
+    ((R^2).toNNReal) (m*Real.exp (β*Real.log R)) hb hbR hbT
+  apply hp.trans
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    swallowLyapunov κ β C (initialLogCoordinates z) 0/(m*Real.exp (β*Real.log R))
+        ≤ (1+C)/(m*Real.exp (β*Real.log R)) := div_le_div_of_nonneg_right hF hb.le
+    _ = (1+C)/m*Real.exp (-β*Real.log R) := by
+      rw [show -β*Real.log R = -(β*Real.log R) by ring, Real.exp_neg]
+      ring
+
+ 
+
+theorem ordinary_survival_polynomial
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 4 < κ)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ z ∈ upperHalfPlane, ‖z‖ = 1 →
+      ∀ T : ℝ≥0, 1 ≤ T →
+      P {ω | z ∈ survivingDomain (γ ω) T} ≤ ENNReal.ofReal (C*(T:ℝ)^(-c)) := by
+  let β := (κ-4)/(2*κ)
+  let C := (1+3*(κ-4)/8) / min 1 (β*((κ-4)/4))
+  have hk : 0 < κ := lt_trans (by norm_num) hκ
+  have hβ : 0 < β := by dsimp [β]; positivity
+  have hC : 0 < C := by dsimp [C]; positivity
+  refine ⟨C,β/2,hC,by positivity,?_⟩
+  intro z hz hu T hT
+  have ht : 0 < (T:ℝ) := lt_of_lt_of_le (by norm_num) (NNReal.coe_le_coe.mpr hT)
+  have hR : 1 ≤ Real.sqrt (T:ℝ) := (Real.le_sqrt (by norm_num) T.coe_nonneg).mpr (by simpa only [one_pow, NNReal.coe_one] using NNReal.coe_le_coe.mpr hT)
+  have hp := ordinary_survival_radial_tail hB κ hκ γ G F hw z hz hu (Real.sqrt T) hR
+  rw [Real.sq_sqrt T.coe_nonneg,Real.toNNReal_coe] at hp
+  convert hp using 2
+  rw [Real.rpow_def_of_pos ht,Real.log_sqrt ht.le]
+  congr 2
+  ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma linear_subsolution_initial_nonpos (T : ℝ) (hT : 0 ≤ T)
+    (f a g : ℝ → ℝ) (ha : ContinuousOn a (Icc 0 T))
+    (hf : ∀ u ∈ Icc 0 T, HasDerivWithinAt f (g u) (Icc 0 T) u)
+    (hg : ∀ u ∈ Icc 0 T, a u*f u ≤ g u) (hfT : f T ≤ 0) : f 0 ≤ 0 := by
+  let A (u : ℝ) := a (projIcc 0 T hT u)
+  have hA : Continuous A := ha.comp_continuous
+    (continuous_subtype_val.comp continuous_projIcc) (fun u => (projIcc 0 T hT u).property)
+  have hAa (u : ℝ) (hu : u ∈ Icc 0 T) : A u = a u := by simp only [A,projIcc_of_mem hT hu]
+  let I (u : ℝ) := ∫ v in (0:ℝ)..u, A v
+  have hI (u : ℝ) : HasDerivAt I (A u) u :=
+    intervalIntegral.integral_hasDerivAt_right (hA.intervalIntegrable _ _)
+      hA.aestronglyMeasurable.stronglyMeasurableAtFilter hA.continuousAt
+  have hd (u : ℝ) (hu : u ∈ Icc 0 T) :
+      HasDerivWithinAt (fun v => f v*Real.exp (-I v))
+        ((g u-a u*f u)*Real.exp (-I u)) (Icc 0 T) u := by
+    convert (hf u hu).mul ((hI u).neg.exp.hasDerivWithinAt) using 1; try rfl
+    simp only [Pi.neg_apply]
+    rw [hAa u hu]
+    ring
+  have hmono : MonotoneOn (fun v => f v*Real.exp (-I v)) (Icc 0 T) :=
+    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
+      (fun u hu => (hd u hu).continuousWithinAt)
+      (fun u hu => (hd u (interior_subset hu)).mono interior_subset)
+      (fun u hu => mul_nonneg (sub_nonneg.mpr (hg u (interior_subset hu))) (Real.exp_pos _).le)
+  have hh := hmono (show (0:ℝ) ∈ Icc 0 T from ⟨le_rfl,hT⟩) ⟨hT,le_rfl⟩ hT
+  simp only [I,intervalIntegral.integral_same,neg_zero,Real.exp_zero,mul_one] at hh
+  exact hh.trans (mul_nonpos_of_nonpos_of_nonneg hfT (Real.exp_pos _).le)
+
+lemma IsCapacityTwoWitness.real_derivative
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
+    (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (fun v : ℝ => (G v.toNNReal z).re)
+      (2*((G u.toNNReal z).re-U u.toNNReal)/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)))
+      (Icc 0 (T:ℝ)) u := by
+  have hh := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt u (h.time_derivative T z hz u hu)
+  simpa [Function.comp_def,Complex.div_re] using hh
+
+lemma IsCapacityTwoWitness.initial_real_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
+    (hz : z ∈ survivingDomain γ T) (M : ℝ)
+    (hbound : ∀ u : ℝ≥0, u ≤ T → U u ≤ M) (hend : (G T z).re ≤ M) : z.re ≤ M := by
+  let a (u : ℝ) := 2/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))
+  have hnorm : ContinuousOn (fun u : ℝ => Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) :=
+    Complex.continuous_normSq.comp_continuousOn ((h.time_continuousOn T z hz).sub
+      (Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal)).continuousOn)
+  have hne (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
+      Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) ≠ 0 := by
+    exact (Complex.normSq_pos.mpr (h.centered_ne_zero u.toNNReal z
+      (survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu.2) hz))).ne'
+  have ha : ContinuousOn a (Icc 0 (T:ℝ)) := continuousOn_const.div hnorm hne
+  have hh := linear_subsolution_initial_nonpos T T.coe_nonneg
+    (fun u => (G u.toNNReal z).re-M) a
+    (fun u => 2*((G u.toNNReal z).re-U u.toNNReal)/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))) ha
+    (fun u hu => (h.real_derivative T z hz u hu).sub_const M)
+    (fun u hu => by
+      dsimp only [a]
+      have hb := hbound u.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hu.2)
+      have hp : 0 ≤ Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) := Complex.normSq_nonneg _
+      rw [div_mul_eq_mul_div]
+      exact div_le_div_of_nonneg_right (by linarith) hp)
+    (by simpa only [Real.toNNReal_coe] using sub_nonpos.mpr hend)
+  simpa only [Real.toNNReal_zero,h.initial_value z hz.1.1,sub_nonpos] using hh
+
+lemma IsCapacityTwoWitness.initial_real_ge
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
+    (hz : z ∈ survivingDomain γ T) (M : ℝ)
+    (hbound : ∀ u : ℝ≥0, u ≤ T → M ≤ U u) (hend : M ≤ (G T z).re) : M ≤ z.re := by
+  let a (u : ℝ) := 2/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))
+  have hnorm : ContinuousOn (fun u : ℝ => Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) :=
+    Complex.continuous_normSq.comp_continuousOn ((h.time_continuousOn T z hz).sub
+      (Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal)).continuousOn)
+  have hne (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
+      Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) ≠ 0 := by
+    exact (Complex.normSq_pos.mpr (h.centered_ne_zero u.toNNReal z
+      (survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu.2) hz))).ne'
+  have ha : ContinuousOn a (Icc 0 (T:ℝ)) := continuousOn_const.div hnorm hne
+  have hh := linear_subsolution_initial_nonpos T T.coe_nonneg
+    (fun u => M-(G u.toNNReal z).re) a
+    (fun u => -(2*((G u.toNNReal z).re-U u.toNNReal)/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)))) ha
+    (fun u hu => (h.real_derivative T z hz u hu).const_sub M)
+    (fun u hu => by
+      dsimp only [a]
+      have hb := hbound u.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hu.2)
+      have hp : 0 ≤ Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) := Complex.normSq_nonneg _
+      rw [div_mul_eq_mul_div,← neg_div]
+      exact div_le_div_of_nonneg_right (by linarith) hp)
+    (by simpa only [Real.toNNReal_coe] using sub_nonpos.mpr hend)
+  simpa only [Real.toNNReal_zero,h.initial_value z hz.1.1,sub_nonpos] using hh
+
+lemma IsCapacityTwoWitness.initial_imag_square_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ)
+    (hz : z ∈ survivingDomain γ T) : z.im^2 ≤ (G T z).im^2+4*T := by
+  have hd (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :=
+    ((h.imag_derivative T z hz u hu).pow 2).add ((hasDerivWithinAt_id u (Icc 0 (T:ℝ))).const_mul 4)
+  have hnonneg (u : ℝ) (hu : u ∈ Icc 0 (T:ℝ)) :
+      0 ≤ (2:ℝ)*(G u.toNNReal z).im^(2-1)*
+        (-2*(G u.toNNReal z).im/Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)))+1*4 := by
+    have hp := Complex.normSq_pos.mpr (h.centered_ne_zero u.toNNReal z
+      (survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu.2) hz))
+    have hn : (G u.toNNReal z).im^2 ≤ Complex.normSq (G u.toNNReal z-(U u.toNNReal:ℂ)) := by
+      simp only [Complex.normSq_apply,Complex.sub_re,Complex.ofReal_re,Complex.sub_im,Complex.ofReal_im,sub_zero]
+      nlinarith [sq_nonneg ((G u.toNNReal z).re-U u.toNNReal)]
+    norm_num only [Nat.reduceSub, pow_one, one_mul]
+    field_simp
+    nlinarith
+  have hmono : MonotoneOn (fun u : ℝ => (G u.toNNReal z).im^2+4*u) (Icc 0 (T:ℝ)) :=
+    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
+      (fun u hu => (hd u hu).continuousWithinAt)
+      (fun u hu => (hd u (interior_subset hu)).mono interior_subset)
+      (fun u hu => by simpa only [Nat.cast_ofNat, mul_one, one_mul] using hnonneg u (interior_subset hu))
+  have hh := hmono (show (0:ℝ) ∈ Icc 0 (T:ℝ) from ⟨le_rfl,T.coe_nonneg⟩)
+    ⟨T.coe_nonneg,le_rfl⟩ T.coe_nonneg
+  simpa only [Real.toNNReal_zero,Real.toNNReal_coe,h.initial_value z hz.1.1,mul_zero,add_zero] using hh
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma IsCapacityTwoWitness.trace_real_bounds
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (M : ℝ)
+    (hbound : ∀ u : ℝ≥0, u ≤ T → |U u| ≤ M) : |(γ T).re| ≤ M := by
+  have hh := h
+  obtain ⟨_,_,_,_,_,_,hFmap,_,hinv,_,_,_,htip⟩ := hh
+  have hm (y : ℝ) (hy : 0 < y) :
+      |(F T ((U T:ℂ)+(y:ℂ)*Complex.I)).re| ≤ M := by
+    have hyH : (U T:ℂ)+(y:ℂ)*Complex.I ∈ upperHalfPlane := by
+      simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im,
+        Complex.mul_im, Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one,
+        mul_zero, add_zero, zero_add] using hy
+    have hz := hFmap T hyH
+    have he := hinv T _ hyH
+    apply abs_le.mpr
+    constructor
+    · apply IsCapacityTwoWitness.initial_real_ge hU h T _ hz (-M)
+      · intro u hu; exact (abs_le.mp (hbound u hu)).1
+      · rw [he]; simpa using (abs_le.mp (hbound T le_rfl)).1
+    · apply IsCapacityTwoWitness.initial_real_le hU h T _ hz M
+      · intro u hu; exact (abs_le.mp (hbound u hu)).2
+      · rw [he]; simpa using (abs_le.mp (hbound T le_rfl)).2
+  have ht := (continuous_abs.tendsto _).comp (Complex.continuous_re.tendsto _ |>.comp (htip T))
+  exact le_of_tendsto ht (eventually_nhdsWithin_of_forall fun y hy => hm y hy)
+
+lemma IsCapacityTwoWitness.trace_imag_square_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) : (γ T).im^2 ≤ 4*T := by
+  have hh := h
+  obtain ⟨_,_,_,_,_,_,hFmap,_,hinv,_,_,_,htip⟩ := hh
+  have hm (y : ℝ) (hy : 0 < y) :
+      (F T ((U T:ℂ)+(y:ℂ)*Complex.I)).im^2 ≤ y^2+4*T := by
+    have hyH : (U T:ℂ)+(y:ℂ)*Complex.I ∈ upperHalfPlane := by
+      simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im,
+        Complex.mul_im, Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one,
+        mul_zero, add_zero, zero_add] using hy
+    have hz := hFmap T hyH
+    have he := hinv T _ hyH
+    have hh := h.initial_imag_square_le T _ hz
+    rw [he] at hh
+    simpa using hh
+  have ht := (Complex.continuous_im.tendsto _ |>.comp (htip T)).pow 2
+  have hr : Tendsto (fun y : ℝ => y^2+4*(T:ℝ)) (𝓝[>] 0) (𝓝 (4*(T:ℝ))) := by
+    simpa using (((tendsto_id : Tendsto (fun y : ℝ => y) (𝓝 (0:ℝ)) (𝓝 0)).pow 2).add_const (4*(T:ℝ))).mono_left nhdsWithin_le_nhds
+  exact le_of_tendsto_of_tendsto ht hr (eventually_nhdsWithin_of_forall fun y hy => hm y hy)
+
+lemma IsCapacityTwoWitness.trace_norm_square_le
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (M : ℝ)
+    (hM : 0 ≤ M) (hbound : ∀ u : ℝ≥0, u ≤ T → |U u| ≤ M) : ‖γ T‖^2 ≤ M^2+4*T := by
+  have hr := h.trace_real_bounds hU T M hbound
+  have hi := h.trace_imag_square_le T
+  rw [Complex.sq_norm, Complex.normSq_apply]
+  have hs : (γ T).re^2 ≤ M^2 := by
+    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hM).mpr hr
+  nlinarith
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma exists_compact_square_field (R : ℝ) (hR : 0 ≤ R) :
+    ∃ φ : ℝ × ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
+      (∀ x, |x| ≤ R → φ (0,x) = x^2) ∧
+      (∀ x, |x| ≤ R → iteratedDeriv 2 (fun y => φ (0,y)) x = 2) := by
+  let b : ContDiffBump (0 : ℝ × ℝ) := ⟨R+1,R+2,by linarith,by linarith⟩
+  let φ (p : ℝ × ℝ) := b p*p.2^2
+  have hnear (x : ℝ) (hx : |x| ≤ R) : (fun y => φ (0,y)) =ᶠ[𝓝 x] (fun y => y^2) := by
+    have hball : (0,x) ∈ Metric.ball (0:ℝ × ℝ) b.rIn := by
+      simpa only [Metric.mem_ball,dist_zero_right,Prod.norm_def,norm_zero,Real.norm_eq_abs,
+        max_eq_right (abs_nonneg x)] using (show |x| < R+1 by linarith)
+    have he := (b.eventuallyEq_one_of_mem_ball hball).comp_tendsto
+      ((continuous_const.prodMk continuous_id).tendsto x)
+    filter_upwards [he] with y hy
+    change b (0,y) = 1 at hy
+    simp only [φ,hy,one_mul]
+  refine ⟨φ,b.contDiff.mul (contDiff_snd.pow 2),b.hasCompactSupport.mul_right,
+    fun x hx => (hnear x hx).eq_of_nhds,?_⟩
+  intro x hx
+  rw [(hnear x hx).iteratedDeriv_eq 2,iteratedDeriv_pow]
+  norm_num
+
+lemma brownian_stopped_square
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hin : ∀ ω t, t ≤ τ ω → |B t ω| ≤ R) :
+    Integrable (fun ω => (B (τ ω) ω)^2) P ∧
+      (∫ ω, (B (τ ω) ω)^2 ∂P) = ∫ ω, (τ ω:ℝ) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨φ,hφ,hφc,he,h2⟩ := exists_compact_square_field R hR
+  obtain ⟨K,hfield⟩ := compact_state_smooth_brownian_field B hm
+    (fun _ _ => (0:ℝ)) (fun _ _ => (0:ℝ)) 0 measurable_const (fun _ => measurable_const)
+    (fun _ => LipschitzWith.const 0) measurable_const (by simp)
+    (fun t ω ht => (hasDerivWithinAt_const t (Ici 0) (0:ℝ))) φ hφ hφc
+  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (0,x)
+  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (0,x) ((0:ℝ),0)
+  have hg (ω : Ω) (u : ℝ) (hu : u ∈ Icc 0 (τ ω:ℝ)) : brownianFieldDrift B F a u ω = 1 := by
+    have hi := hin ω u.toNNReal (Real.toNNReal_le_iff_le_coe.mpr hu.2)
+    simp only [brownianFieldDrift,F,a,show ((0:ℝ),0) = (0:ℝ × ℝ) from rfl,map_zero,h2 _ hi]
+    norm_num
+  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hBi : Integrable (fun ω => (B (τ ω) ω)^2) P := by
+    apply Integrable.of_bound ((hBm.comp (hτm.prodMk measurable_id)).pow_const 2).aestronglyMeasurable (R^2)
+    exact Eventually.of_forall fun ω => by
+      rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+      have hh : (B (τ ω) ω)^2 ≤ R^2 := by
+        simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hR).mpr (hin ω (τ ω) le_rfl)
+      simpa only [Function.comp_def, id_eq] using hh
+  refine ⟨hBi,?_⟩
+  have hef : (fun ω => F (τ ω) ω (B (τ ω) ω)) = fun ω => (B (τ ω) ω)^2 :=
+    funext fun ω => he _ (hin ω (τ ω) le_rfl)
+  rw [hef] at hopt
+  have h0 : (∫ ω, F 0 ω (B 0 ω) ∂P) = 0 := by
+    have he0 : (fun ω => F 0 ω (B 0 ω)) =ᵐ[P] (fun _ => (0:ℝ)) := by
+      filter_upwards [hB.eval_zero_ae_eq_zero] with ω hω
+      change φ (0,B 0 ω) = 0
+      rw [hω,he 0 (by simpa using hR)]
+      norm_num
+    rw [integral_congr_ae he0,integral_zero]
+  rw [h0,zero_add] at hopt
+  rw [hopt]
+  apply integral_congr_ae
+  exact Eventually.of_forall fun ω => by
+    calc
+      (∫ u in (0:ℝ)..(τ ω:ℝ), brownianFieldDrift B F a u ω) = ∫ _ in (0:ℝ)..(τ ω:ℝ), (1:ℝ) := by
+        apply intervalIntegral.integral_congr
+        intro u hu
+        exact hg ω u (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using hu)
+      _ = (τ ω:ℝ) := by simp
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SwallowExit.lean b/lean/OAI/Probability/SLE/SwallowExit.lean
index d66401c4f72b6a663a5f9b113c35c27fcd46c014..30fe28610f58320cd9e68670e9a642d104d3ebb5 100644
--- a/lean/OAI/Probability/SLE/SwallowExit.lean
+++ b/lean/OAI/Probability/SLE/SwallowExit.lean
@@ -1,746 +1,746 @@
-import Mathlib
-import OAI.Probability.SLE.StoppedPower
-
-namespace OAI
-
-/-! Swallow Exit for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma nonnegative_integral_tail {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (Y : Ω → ℝ) (hYi : Integrable Y P) (hY0 : ∀ ω, 0 ≤ Y ω)
-    (M a : ℝ) (ha : 0 < a) (hmean : (∫ ω, Y ω ∂P) ≤ M) :
-    P {ω | a ≤ Y ω} ≤ ENNReal.ofReal (M/a) := by
-  have hmark := meas_ge_le_lintegral_div hYi.aemeasurable.ennreal_ofReal
-    (ne_of_gt (ENNReal.ofReal_pos.mpr ha)) ENNReal.ofReal_ne_top
-  calc
-    P {ω | a ≤ Y ω} ≤ P {ω | ENNReal.ofReal a ≤ ENNReal.ofReal (Y ω)} :=
-      measure_mono (fun ω hω => ENNReal.ofReal_le_ofReal hω)
-    _ ≤ (∫⁻ ω, ENNReal.ofReal (Y ω) ∂P) / ENNReal.ofReal a := hmark
-    _ = ENNReal.ofReal (∫ ω, Y ω ∂P) / ENNReal.ofReal a := by
-      rw [ofReal_integral_eq_lintegral_ofReal hYi (Eventually.of_forall hY0)]
-    _ ≤ ENNReal.ofReal M / ENNReal.ofReal a :=
-      ENNReal.div_le_div_right (ENNReal.ofReal_le_ofReal hmean) _
-    _ = ENNReal.ofReal (M/a) := (ENNReal.ofReal_div_of_pos ha).symm
-
-namespace RadialLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
-
-lemma exit_probability {P : Measure Ω}
-    (a : RadialLocalization c R T)
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 4 < κ) (hR : 0 < R) (hhi : q₀.1.im ≤ hi)
-    (b : ℝ) (hb : 0 < b)
-    (hbR : b ≤ Real.exp (((κ-4)/(2*κ))*Real.log R))
-    (hbT : b ≤ (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) * T) :
-    P {ω | a.τ ω = T ∨ greenLogNormSq κ (c.Q (a.τ ω) ω) (B (a.τ ω) ω) = R^2} ≤
-      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q₀ 0 / b) := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let β := (κ-4)/(2*κ)
-  let C := 3*(κ-4)/8
-  let d := β*((κ-4)/4)*Real.exp ((β-2)*Real.log R)
-  let F (ω : Ω) := swallowLyapunov κ β C (c.Q (a.τ ω) ω) (B (a.τ ω) ω)
-  let Y (ω : Ω) := F ω + d*(a.τ ω : ℝ)
-  have hk : 0 < κ := lt_trans (by norm_num) hκ
-  have hβ : 0 < β := by dsimp [β]; positivity
-  have hC : 0 ≤ C := by dsimp [C]; positivity
-  have hd : 0 ≤ d := by dsimp [d]; positivity
-  have hF0 (ω : Ω) : 0 ≤ F ω := swallowLyapunov_nonneg κ β C hC _ _
-  have hY0 (ω : Ω) : 0 ≤ Y ω := add_nonneg (hF0 ω) (mul_nonneg hd (a.τ ω).coe_nonneg)
-  have hmean := a.stopped_lyapunov_bound hB hBm hBc hκ hR hhi
-  have hτi : Integrable (fun ω => (a.τ ω : ℝ)) P :=
-    bounded_nnreal_integrable a.τ (stopping_time_measurable_nnreal a.stopping) T a.bounded
-  have hYi : Integrable Y P := hmean.1.add (hτi.const_mul d)
-  have hYmean : (∫ ω, Y ω ∂P) ≤ swallowLyapunov κ β C q₀ 0 := by
-    rw [show (∫ ω, Y ω ∂P) = (∫ ω, F ω ∂P)+d*(∫ ω, (a.τ ω : ℝ) ∂P) by
-      rw [show Y = fun ω => F ω+d*(a.τ ω : ℝ) from rfl,
-        integral_add hmean.1 (hτi.const_mul d), integral_const_mul]]
-    exact hmean.2
-  apply (measure_mono (show {ω | a.τ ω = T ∨ greenLogNormSq κ (c.Q (a.τ ω) ω) (B (a.τ ω) ω) = R^2}
-      ⊆ {ω | b ≤ Y ω} from ?_)).trans (nonnegative_integral_tail Y hYi hY0 _ b hb hYmean)
-  intro ω hω
-  rcases hω with ht | hn
-  · change b ≤ F ω+d*(a.τ ω : ℝ)
-    rw [ht]
-    exact hbT.trans (le_add_of_nonneg_left (hF0 ω))
-  · have hr := radialLogPower_at_boundary κ β (c.Q (a.τ ω) ω) (B (a.τ ω) ω) R hn
-    have hh : Real.exp (β*Real.log R) ≤ F ω := by
-      change Real.exp (β*Real.log R) ≤ radialLogPower κ β _ _ + C*heightLogPower β _
-      rw [hr]
-      exact le_add_of_nonneg_right (mul_nonneg hC (heightLogPower_pos β _).le)
-    exact (hbR.trans hh).trans (le_add_of_nonneg_right (mul_nonneg hd (a.τ ω).coe_nonneg))
-
-end RadialLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-structure BoundaryLocalization
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ a : ℝ} {q₀ : ℂ} (c : BoundaryClockSolution B hm κ a q₀)
-    (α β R H : ℝ) (T : ℝ≥0) where
-  τ : Ω → ℝ≥0
-  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun ω => (τ ω : WithTop ℝ≥0))
-  bounded : ∀ ω, τ ω ≤ T
-  inside : ∀ ω t, t ≤ τ ω → ‖(c.Q t ω,B t ω)‖ ≤ R ∧
-    3*a ≤ (c.Q t ω).re-Real.sqrt κ*B t ω ∧
-    boundaryDensity κ α β (c.Q t ω) (B t ω) ≤ H
-  before : ∀ ω t, t < τ ω → ‖(c.Q t ω,B t ω)‖ < R ∧
-    3*a < (c.Q t ω).re-Real.sqrt κ*B t ω ∧
-    boundaryDensity κ α β (c.Q t ω) (B t ω) < H
-  boundary : ∀ ω, τ ω < T → ‖(c.Q (τ ω) ω,B (τ ω) ω)‖ = R ∨
-    (c.Q (τ ω) ω).re-Real.sqrt κ*B (τ ω) ω = 3*a ∨
-    boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω) = H
-
-lemma exists_boundaryLocalization
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ a : ℝ} {q₀ : ℂ} (c : BoundaryClockSolution B hm κ a q₀)
-    (ha : 0 < a) (α β R H : ℝ) (T : ℝ≥0)
-    (hR : ‖(q₀,(0:ℝ))‖ ≤ R) (hx : 3*a ≤ q₀.re)
-    (hH : boundaryDensity κ α β q₀ 0 ≤ H) :
-    Nonempty (BoundaryLocalization c α β R H T) := by
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let V (p : ℂ × ℝ) := min (R-‖p‖) (min (p.1.re-Real.sqrt κ*p.2-3*a)
-    (H-smoothBoundaryDensity κ α β a p.1 p.2))
-  have hVc : Continuous V := by
-    exact (continuous_const.sub continuous_norm).min
-      (((Complex.continuous_re.comp continuous_fst).sub (continuous_const.mul continuous_snd)).sub
-        continuous_const |>.min (continuous_const.sub (smoothBoundaryDensity_contDiff κ α β ha).continuous))
-  let D (t : ℝ≥0) (ω : Ω) := V (c.Q t ω,B t ω)
-  have hBa : Adapted f B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  have hDa : Adapted f D := fun t => hVc.measurable.comp ((c.adapted t).prodMk (hBa t))
-  have hDc (ω : Ω) : Continuous (fun t => D t ω) :=
-    hVc.comp (((c.lipschitz ω).continuous.comp NNReal.continuous_coe).prodMk (hBc ω))
-  have hD0 (ω : Ω) : 0 ≤ D 0 ω := by
-    have he : smoothBoundaryDensity κ α β a q₀ 0 = boundaryDensity κ α β q₀ 0 :=
-      smoothBoundaryDensity_eq ha (by simpa using (show 2*a ≤ q₀.re by linarith))
-    simpa only [D,V,NNReal.coe_zero,c.initial,hB0,mul_zero,sub_zero,he,
-      le_min_iff,sub_nonneg] using And.intro hR (And.intro hx hH)
-  obtain ⟨τ,hτ,hb,hin,hpre,hbd⟩ := first_margin_stopping f D hDa hDc hD0 T
-  have hin' (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) : ‖(c.Q t ω,B t ω)‖ ≤ R ∧
-      3*a ≤ (c.Q t ω).re-Real.sqrt κ*B t ω ∧ boundaryDensity κ α β (c.Q t ω) (B t ω) ≤ H := by
-    have hh := hin ω t ht
-    simp only [D,V,le_min_iff,sub_nonneg] at hh
-    rw [smoothBoundaryDensity_eq ha (by linarith [hh.2.1])] at hh
-    exact hh
-  refine ⟨⟨τ,hτ,hb,hin',?_,?_⟩⟩
-  · intro ω t ht
-    have hh := hpre ω t ht
-    simp only [D,V,lt_min_iff,sub_pos] at hh
-    rw [smoothBoundaryDensity_eq ha (by linarith [hh.2.1])] at hh
-    exact hh
-  · intro ω ht
-    have hh := hbd ω ht
-    dsimp only [D,V] at hh
-    rw [smoothBoundaryDensity_eq ha (by linarith [(hin' ω (τ ω) le_rfl).2.1])] at hh
-    rw [min_eq_iff] at hh
-    rcases hh with ⟨hh,_⟩ | ⟨hh,_⟩
-    · exact Or.inl (sub_eq_zero.mp hh).symm
-    · rw [min_eq_iff] at hh
-      rcases hh with ⟨hh,_⟩ | ⟨hh,_⟩
-      · exact Or.inr (Or.inl (sub_eq_zero.mp hh))
-      · exact Or.inr (Or.inr (sub_eq_zero.mp hh).symm)
-
-namespace BoundaryLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ a : ℝ} {q₀ : ℂ} {c : BoundaryClockSolution B hm κ a q₀}
-    {α β R H : ℝ} {T : ℝ≥0}
-
-lemma density_crossing_probability {P : Measure Ω}
-    (l : BoundaryLocalization c α β R H T)
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (hκ : 0 < κ) (ha : 0 < a) (hβ : -α-2+κ*(β+1)/2=0)
-    (hR : 0 ≤ R) (hH : 0 < H) :
-    P {ω | H ≤ boundaryDensity κ α β (c.Q (l.τ ω) ω) (B (l.τ ω) ω)} ≤
-      ENNReal.ofReal (boundaryDensity κ α β q₀ 0/H) := by
-  obtain ⟨hi,he⟩ := c.stopped_boundary_density hB hBm hBc hκ ha α β hβ R hR
-    l.τ l.stopping T l.bounded (fun ω t ht =>
-      ⟨(l.inside ω t ht).1,by linarith [(l.inside ω t ht).2.1]⟩)
-  exact nonnegative_integral_tail _ hi (fun ω => (Real.exp_pos _).le) _ _ hH he.le
-
-end BoundaryLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-def IsPositiveBoundaryClockPath (U : ℝ≥0 → ℝ) (x : ℝ) (T : ℝ≥0) (Q : ℝ → ℂ) : Prop :=
-  Q 0 = (x:ℂ) ∧
-  (∀ t ∈ Icc 0 (T:ℝ), 0 < (Q t).re-U t.toNNReal) ∧
-  (∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Q
-    (((2/((Q t).re-U t.toNNReal):ℝ):ℂ)+
-      (((Q t).re-U t.toNNReal)⁻¹^2:ℝ)*I) (Icc 0 (T:ℝ)) t)
-
-namespace IsPositiveBoundaryClockPath
-variable {U : ℝ≥0 → ℝ} {x : ℝ} {T : ℝ≥0} {Q : ℝ → ℂ}
-
-lemma continuousOn (h : IsPositiveBoundaryClockPath U x T Q) :
-    ContinuousOn Q (Icc 0 (T:ℝ)) := fun t ht => (h.2.2 t ht).continuousWithinAt
-
-lemma positive_lower_bound (h : IsPositiveBoundaryClockPath U x T Q) (hU : Continuous U) :
-    ∃ a : ℝ, 0 < a ∧ ∀ t ∈ Icc 0 (T:ℝ), a ≤ (Q t).re-U t.toNNReal := by
-  have hc : ContinuousOn (fun t : ℝ => (Q t).re-U t.toNNReal) (Icc 0 (T:ℝ)) :=
-    (Complex.continuous_re.comp_continuousOn h.continuousOn).sub
-      ((hU.comp continuous_real_toNNReal).continuousOn)
-  obtain ⟨s,hs,hmin⟩ := (isCompact_Icc : IsCompact (Icc 0 (T:ℝ))).exists_isMinOn
-    ⟨0,le_rfl,T.coe_nonneg⟩ hc
-  exact ⟨_,h.2.1 s hs,fun t ht => hmin ht⟩
-
-lemma clock_identity (h : IsPositiveBoundaryClockPath U x T Q) (hU : Continuous U) :
-    (Q T).im = ∫ t in (0:ℝ)..(T:ℝ), ((Q t).re-U t.toNNReal)⁻¹^2 := by
-  have hc : ContinuousOn (fun t : ℝ => ((Q t).re-U t.toNNReal)⁻¹^2) (Icc 0 (T:ℝ)) :=
-    (((Complex.continuous_re.comp_continuousOn h.continuousOn).sub
-      ((hU.comp continuous_real_toNNReal).continuousOn)).inv₀ (fun t ht => (h.2.1 t ht).ne')).pow 2
-  have hd (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt (fun u => (Q u).im)
-      (((Q t).re-U t.toNNReal)⁻¹^2) (Icc 0 (T:ℝ)) t := by
-    simpa only [Function.comp_def,ContinuousLinearMap.comp_apply,Complex.imCLM_apply,Complex.add_im,
-      Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,Complex.I_im,mul_one,
-      Complex.I_re,mul_zero,add_zero,zero_add] using
-      Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (h.2.2 t ht)
-  have hdr (t : ℝ) (ht : t ∈ Ioo 0 (T:ℝ)) : HasDerivWithinAt (fun u => (Q u).im)
-      (((Q t).re-U t.toNNReal)⁻¹^2) (Ioi t) t := by
-    exact (hasDerivWithinAt_Ici_of_Icc ⟨ht.1.le,ht.2⟩
-      (hd t ⟨ht.1.le,ht.2.le⟩)).mono Ioi_subset_Ici_self
-  have he := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le T.coe_nonneg
-    (Complex.continuous_im.comp_continuousOn h.continuousOn) hdr
-    (hc.intervalIntegrable_of_Icc T.coe_nonneg)
-  dsimp only [Function.comp_def] at he
-  rw [h.1,Complex.ofReal_im,sub_zero] at he
-  exact he.symm
-
-lemma clipped_identification
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    {hm : ∀ t, Measurable (B t)} {κ a : ℝ}
-    (c : BoundaryClockSolution B hm κ a (x:ℂ)) (ha : 0 < a) (ω : Ω)
-    (h : IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q)
-    (hgap : ∀ t ∈ Icc 0 (T:ℝ), a ≤ (Q t).re-Real.sqrt κ*B t.toNNReal ω) :
-    EqOn Q (fun t => c.Q t ω) (Icc 0 (T:ℝ)) := by
-  have hd (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt Q
-      (boundaryClockField a (Real.sqrt κ*B t.toNNReal ω) (Q t)) (Icc 0 (T:ℝ)) t := by
-    simpa only [boundaryClockField,positiveBoundaryField_eq (hgap t ht),max_eq_right (hgap t ht)] using h.2.2 t ht
-  have hc (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) := (c.derivative ω t ht.1).mono
-    (show Icc (0:ℝ) (T:ℝ) ⊆ Ici 0 from fun s hs => hs.1)
-  apply ODE_solution_unique (fun t => boundaryClockField_lipschitz ha (Real.sqrt κ*B t.toNNReal ω))
-    h.continuousOn
-    (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hd t ⟨ht.1,ht.2.le⟩))
-    (c.lipschitz ω).continuous.continuousOn
-    (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hc t ⟨ht.1,ht.2.le⟩))
-  exact h.1.trans (c.initial ω).symm
-
-end IsPositiveBoundaryClockPath
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def greenDensity (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
-    (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) : ℝ := by
-  classical
-  exact if z ∈ survivingDomain γ t then
-    (fullRadius G t z) ^ (-codimension κ) *
-      (Real.sin (Complex.arg (G t z - (U t : ℂ)))) ^ greenExponent κ
-  else 0
-
-noncomputable def initialGreen (κ : ℝ) (z : ℂ) : ℝ :=
-  (2 * z.im) ^ (-codimension κ) *
-    (Real.sin (Complex.arg z)) ^ greenExponent κ
-
- 
-
-noncomputable def radiusLevelTime (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ)
-    (z : ℂ) (u : ℝ) : ℝ≥0∞ :=
-  sInf {s : ℝ≥0∞ | s < ∞ ∧ z ∈ survivingDomain γ s.toNNReal ∧
-    fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u)}
-
- 
-noncomputable def stoppedGreen (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
-    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (u : ℝ) : ℝ :=
-  if radiusLevelTime γ G z u < ∞ then
-    greenDensity κ U γ G (radiusLevelTime γ G z u).toNNReal z
-  else 0
-
-lemma logCoordinates_re (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0) :
-    (logLoewnerCoordinates G z t).1.re = (G t z).re := by
-  simp only [logLoewnerCoordinates, Real.toNNReal_coe, Complex.add_re, Complex.ofReal_re,
-    Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, zero_mul,
-    sub_self, add_zero]
-
-lemma logCoordinates_im (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0) :
-    (logLoewnerCoordinates G z t).1.im = Real.log (G t z).im := by
-  simp only [logLoewnerCoordinates, Real.toNNReal_coe, Complex.add_im, Complex.ofReal_im,
-    Complex.mul_im, Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_zero, mul_one,
-    add_zero, zero_add]
-
-lemma logCoordinates_snd (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0) :
-    (logLoewnerCoordinates G z t).2 = Real.log ‖deriv (G t) z‖ := by
-  simp only [logLoewnerCoordinates, Real.toNNReal_coe]
-
-lemma greenLogNormSq_logCoordinates (κ b : ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0)
-    (hy : 0 < (G t z).im) :
-    greenLogNormSq κ (logLoewnerCoordinates G z t) b =
-      Complex.normSq (G t z - ((Real.sqrt κ * b : ℝ) : ℂ)) := by
-  have he : Real.exp (2 * Real.log (G t z).im) = (G t z).im ^ 2 := by
-    rw [show 2 * Real.log (G t z).im = Real.log (G t z).im + Real.log (G t z).im by ring,
-      Real.exp_add, Real.exp_log hy, pow_two]
-  rw [greenLogNormSq, logCoordinates_re, logCoordinates_im, he]
-  simp only [Complex.normSq_apply, Complex.sub_re, Complex.sub_im, Complex.ofReal_re,
-    Complex.ofReal_im, sub_zero, pow_two]
-
-lemma logAngleSq_logCoordinates (κ b : ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0)
-    (hy : 0 < (G t z).im) :
-    logAngleSq κ (logLoewnerCoordinates G z t) b =
-      (Real.sin (Complex.arg (G t z - ((Real.sqrt κ * b : ℝ) : ℂ)))) ^ 2 := by
-  rw [logAngleSq, greenLogNormSq_logCoordinates κ b G z t hy, logCoordinates_im,
-    Complex.sin_arg, Complex.normSq_eq_norm_sq, div_pow]
-  congr 1
-  simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
-  rw [show 2 * Real.log (G t z).im = Real.log (G t z).im + Real.log (G t z).im by ring,
-    Real.exp_add, Real.exp_log hy, pow_two]
-
-lemma greenLogDensity_logCoordinates
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
-    (κ b : ℝ) (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t)
-    (hUb : U t = Real.sqrt κ * b) :
-    greenLogDensity κ (logLoewnerCoordinates G z t) b = greenDensity κ U γ G t z := by
-  have hy := h.imag_positive t z hz
-  have hj := h.derivative_norm_positive hU t z hz
-  have hn : 0 < ‖G t z - ((Real.sqrt κ * b : ℝ) : ℂ)‖ := by
-    apply norm_pos_iff.mpr
-    simpa only [hUb] using h.centered_ne_zero t z hz
-  have hs : 0 < Real.sin (Complex.arg (G t z - (U t : ℂ))) := by
-    rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero, hUb]
-    exact div_pos hy hn
-  have hR := h.fullRadius_positive hU t z hz
-  have hlogR : Real.log (fullRadius G t z) =
-      Real.log 2 + Real.log (G t z).im - Real.log ‖deriv (G t) z‖ := by
-    rw [fullRadius, Real.log_div (mul_ne_zero (by norm_num) hy.ne') hj.ne',
-      Real.log_mul (by norm_num) hy.ne']
-  have hlogS : Real.log (Real.sin (Complex.arg (G t z - (U t : ℂ)))) =
-      Real.log (G t z).im - Real.log ‖G t z - ((Real.sqrt κ * b : ℝ) : ℂ)‖ := by
-    rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero, hUb,
-      Real.log_div hy.ne' hn.ne']
-  rw [greenDensity, ite_eq_left hz, Real.rpow_def_of_pos hR, Real.rpow_def_of_pos hs,
-    ← Real.exp_add, hlogR, hlogS, greenLogDensity, logCoordinates_im, logCoordinates_snd,
-    greenLogNormSq_logCoordinates κ b G z t hy, Complex.normSq_eq_norm_sq, Real.log_pow]
-  congr 1
-  push_cast
-  ring
-
-lemma greenLogDensity_initial (κ : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
-    greenLogDensity κ (initialLogCoordinates z) 0 = initialGreen κ z := by
-  have hn : 0 < ‖z‖ := norm_pos_iff.mpr (by
-    have hp : 0 < z.im := hz
-    intro he; simp only [he, Complex.zero_im, lt_self_iff_false] at hp)
-  have hs : 0 < Real.sin (Complex.arg z) := by rw [Complex.sin_arg]; exact div_pos hz hn
-  have he : Real.exp (2 * Real.log z.im) = z.im ^ 2 := by
-    rw [show 2 * Real.log z.im = Real.log z.im + Real.log z.im by ring,
-      Real.exp_add, Real.exp_log hz, pow_two]
-  have hN : greenLogNormSq κ (initialLogCoordinates z) 0 = Complex.normSq z := by
-    simp only [greenLogNormSq, initialLogCoordinates, Complex.add_re, Complex.ofReal_re,
-      Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero,
-      sub_self, add_zero, sub_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add, he,
-      Complex.normSq_apply, pow_two]
-  rw [initialGreen, Real.rpow_def_of_pos (mul_pos (by norm_num) hz), Real.rpow_def_of_pos hs,
-    ← Real.exp_add, Real.log_mul (by norm_num) hz.ne', Complex.sin_arg, Real.log_div hz.ne' hn.ne']
-  rw [greenLogDensity, hN, Complex.normSq_eq_norm_sq, Real.log_pow]
-  simp only [initialLogCoordinates, Complex.add_im, Complex.ofReal_im, Complex.mul_im,
-    Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, zero_add, add_zero,
-    Nat.cast_ofNat]
-  congr 1
-  ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace RadialLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
-
-lemma alive_forces_radial_or_time_boundary (a : RadialLocalization c R T)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) {z : ℂ} (hz : z ∈ upperHalfPlane)
-    (hq : q₀ = initialLogCoordinates z) (hhi : q₀.1.im ≤ hi)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F)
-    (hlive : z ∈ survivingDomain γ T) (hlo : lo < Real.log (G T z).im) :
-    a.τ ω = T ∨ greenLogNormSq κ (c.Q (a.τ ω) ω) (B (a.τ ω) ω) = R^2 := by
-  rcases eq_or_lt_of_le (a.bounded ω) with h | h
-  · exact Or.inl h
-  rcases a.boundary ω h with hN | hh
-  · exact Or.inr hN
-  obtain ⟨_,he⟩ := a.chart_identification hBc hz hq hhi ω γ G F hw
-  have hheight := (hw.log_height_bounds T z hlive (a.τ ω)
-    ⟨(a.τ ω).coe_nonneg, NNReal.coe_le_coe.mpr (a.bounded ω)⟩).1
-  rw [he (a.τ ω) le_rfl, logCoordinates_im] at hh
-  simp only [Real.toNNReal_coe] at hheight
-  exact False.elim (not_le_of_gt hlo (hh ▸ hheight))
-
-end RadialLocalization
-
-lemma measure_eventually_sets_le {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    (E : ℕ → Set Ω) (C : ℝ≥0∞) (hC : ∀ n, P (E n) ≤ C) :
-    P {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} ≤ C := by
-  let H (n : ℕ) := ⋂ m ≥ n, E m
-  have hH : Monotone H := by
-    intro n m hnm ω hω
-    simp only [H, mem_iInter] at hω ⊢
-    exact fun k hk => hω k (hnm.trans hk)
-  have he : {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} = ⋃ n, H n := by
-    ext ω
-    simp only [mem_ofPred_eq, eventually_atTop, mem_iUnion, H, mem_iInter]
-  rw [he, hH.measure_iUnion]
-  apply iSup_le
-  intro n
-  apply (measure_mono (show H n ⊆ E n from fun ω hω => by
-    exact mem_iInter.mp (mem_iInter.mp hω n) le_rfl)).trans (hC n)
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace BoundaryLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ a x : ℝ} {c : BoundaryClockSolution B hm κ a (x:ℂ)}
-    {α β R H : ℝ} {T : ℝ≥0}
-
- 
-
-lemma true_crossing_forces_density_boundary
-    (l : BoundaryLocalization c α β R H T) (ha : 0 < a)
-    (ω : Ω) (S : ℝ≥0) (hS : S < T) (Q : ℝ → ℂ)
-    (hQ : IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x S Q)
-    (hgap : ∀ t ∈ Icc 0 (S:ℝ), 3*a < (Q t).re-Real.sqrt κ*B t.toNNReal ω)
-    (hbound : ∀ t : ℝ≥0, t ≤ S → ‖(Q t,B t ω)‖ < R)
-    (hhit : H ≤ boundaryDensity κ α β (Q S) (B S ω)) :
-    H ≤ boundaryDensity κ α β (c.Q (l.τ ω) ω) (B (l.τ ω) ω) := by
-  have he := hQ.clipped_identification c ha ω (fun t ht => by linarith [hgap t ht])
-  have hτS : l.τ ω ≤ S := by
-    by_contra h
-    have hpre := (l.before ω S (not_le.mp h)).2.2
-    have heS : Q S = c.Q S ω := he ⟨S.coe_nonneg,le_rfl⟩
-    rw [← heS] at hpre
-    exact not_lt_of_ge hhit hpre
-  have hτT : l.τ ω < T := hτS.trans_lt hS
-  have heτ : Q (l.τ ω) = c.Q (l.τ ω) ω := he ⟨(l.τ ω).coe_nonneg,NNReal.coe_le_coe.mpr hτS⟩
-  rcases l.boundary ω hτT with hnorm | hg | hf
-  · have hb := hbound (l.τ ω) hτS
-    rw [heτ,hnorm] at hb
-    exact False.elim (lt_irrefl R hb)
-  · have hp := hgap (l.τ ω) ⟨(l.τ ω).coe_nonneg,NNReal.coe_le_coe.mpr hτS⟩
-    rw [Real.toNNReal_coe,heτ,hg] at hp
-    exact False.elim (lt_irrefl _ hp)
-  · exact hf.ge
-
-end BoundaryLocalization
-end SLEExactGauge
-end
-
- 
-section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
- 
-theorem exists_measurable_brownian_version
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
-    ∃ C : ℝ≥0 → Ω → ℝ, (∀ t, Measurable (C t)) ∧ IsBrownianReal C P ∧
-      (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) := by
-  let b (n k : ℕ) : Ω → ℝ :=
-    (hB.aemeasurable (stopGridTime n k)).mk (B (stopGridTime n k))
-  have hb (n k : ℕ) : Measurable (b n k) := (hB.aemeasurable _).measurable_mk
-  have heq : ∀ᵐ ω ∂P, ∀ n k, b n k ω = B (stopGridTime n k) ω := by
-    apply ae_all_iff.mpr
-    intro n
-    apply ae_all_iff.mpr
-    intro k
-    exact (hB.aemeasurable _).ae_eq_mk.symm
-  let C (t : ℝ≥0) (ω : Ω) : ℝ :=
-    limUnder atTop (fun n => b n (stopGridIndex n t) ω)
-  have hm (t : ℝ≥0) : Measurable (C t) :=
-    (StronglyMeasurable.limUnder (fun n => (hb n _).stronglyMeasurable)).measurable
-  have hCB : ∀ᵐ ω ∂P, ∀ t, C t ω = B t ω := by
-    filter_upwards [hB.cont, heq] with ω hω heqω
-    intro t
-    change limUnder atTop (fun n => b n (stopGridIndex n t) ω) = B t ω
-    simp_rw [heqω]
-    exact (hω.continuousAt.tendsto.comp (stopGrid_tendsto t)).limUnder_eq
-  refine ⟨C, hm, ⟨?_, ?_⟩, hCB⟩
-  · exact hB.toIsPreBrownianReal.congr (fun t => hCB.mono fun ω hω => (hω t).symm)
-  · filter_upwards [hB.cont, hCB] with ω hω heqω
-    simpa only [heqω] using hω
-
- 
-
-theorem exists_continuous_measurable_brownian_version
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
-    ∃ C : ℝ≥0 → Ω → ℝ,
-      Measurable (fun q : ℝ≥0 × Ω => C q.1 q.2) ∧
-      (∀ ω, Continuous (fun t => C t ω)) ∧ IsBrownianReal C P ∧
-      (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) := by
-  classical
-  obtain ⟨V, hVm, hV, hVB⟩ := exists_measurable_brownian_version hB
-  let N := toMeasurable P {ω | ¬ Continuous (fun t => V t ω)}
-  have hNm : MeasurableSet N := measurableSet_toMeasurable _ _
-  have hN : P N = 0 := by
-    rw [show N = toMeasurable P {ω | ¬ Continuous (fun t => V t ω)} from rfl,
-      measure_toMeasurable]
-    exact ae_iff.mp hV.cont
-  let C (t : ℝ≥0) (ω : Ω) := if ω ∈ N then (0 : ℝ) else V t ω
-  have hCm (t) : Measurable (C t) := measurable_const.piecewise hNm (hVm t)
-  have hCc (ω) : Continuous (fun t => C t ω) := by
-    by_cases hω : ω ∈ N
-    · simpa only [C, ite_eq_left hω] using (continuous_const : Continuous (fun _ : ℝ≥0 => (0 : ℝ)))
-    · have hv : Continuous (fun t => V t ω) := by
-        by_contra hh
-        exact hω (subset_toMeasurable P _ hh)
-      simpa only [C, ite_eq_right hω] using hv
-  have hCV : ∀ᵐ ω ∂P, ∀ t, C t ω = V t ω := by
-    filter_upwards [show ∀ᵐ ω ∂P, ω ∉ N from (ae_iff.mpr (by simpa only [not_not, ofPred_mem_eq] using hN))] with ω hω
-    intro t
-    simp only [C, ite_eq_right hω]
-  have hCB : ∀ᵐ ω ∂P, ∀ t, C t ω = B t ω := by
-    filter_upwards [hCV, hVB] with ω hω hω'
-    exact fun t => (hω t).trans (hω' t)
-  refine ⟨C, measurable_uncurry_of_continuous_of_measurable hCc hCm, hCc, ?_, hCB⟩
-  exact ⟨hV.toIsPreBrownianReal.congr (fun t => hCV.mono fun ω hω => (hω t).symm),
-    Eventually.of_forall hCc⟩
-
- 
-
-theorem exists_regular_brownian_version
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
-    ∃ C : ℝ≥0 → Ω → ℝ,
-      Measurable (fun q : ℝ≥0 × Ω => C q.1 q.2) ∧
-      (∀ ω, Continuous (fun t => C t ω)) ∧ (∀ ω, C 0 ω = 0) ∧
-      IsBrownianReal C P ∧ (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) := by
-  obtain ⟨V, hVm, hVc, hV, hVB⟩ := exists_continuous_measurable_brownian_version hB
-  let C (t : ℝ≥0) (ω : Ω) := V t ω - V 0 ω
-  have hV0m : Measurable (V 0) := hVm.comp (measurable_const.prodMk measurable_id)
-  have hCm : Measurable (fun q : ℝ≥0 × Ω => C q.1 q.2) :=
-    hVm.sub (hV0m.comp measurable_snd)
-  have hCc (ω : Ω) : Continuous (fun t => C t ω) := (hVc ω).sub continuous_const
-  have hCV : ∀ᵐ ω ∂P, ∀ t, C t ω = V t ω := by
-    filter_upwards [hV.eval_zero_ae_eq_zero] with ω h0
-    intro t
-    simp only [C, h0, sub_zero]
-  refine ⟨C, hCm, hCc, fun ω => sub_self _, ?_, ?_⟩
-  · exact ⟨hV.toIsPreBrownianReal.congr (fun t => hCV.mono fun ω hω => (hω t).symm),
-      Eventually.of_forall hCc⟩
-  · filter_upwards [hCV, hVB] with ω hω hω'
-    exact fun t => (hω t).trans (hω' t)
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Complex
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-theorem boundary_maximal_density_regular
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    (κ α β x H : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hH : 0 < H)
-    (hβ : -α-2+κ*(β+1)/2=0)
-    (hH0 : boundaryDensity κ α β (x:ℂ) 0 ≤ H) :
-    P {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
-      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
-      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ≤
-      ENNReal.ofReal (boundaryDensity κ α β (x:ℂ) 0 / H) := by
-  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
-  let a (n : ℕ) : ℝ := (x/6)/((n:ℝ)+1)
-  have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
-  have hax (n : ℕ) : 3*a n ≤ x := by
-    have hn : 1 ≤ (n:ℝ)+1 := le_add_of_nonneg_left (Nat.cast_nonneg n)
-    have hh : a n ≤ x/6 := div_le_self (by positivity) hn
-    linarith
-  let R (n : ℕ) : ℝ := ‖((x:ℂ),(0:ℝ))‖+(n:ℝ)+1
-  have hR (n : ℕ) : 0 ≤ R n := by dsimp [R]; positivity
-  have hR0 (n : ℕ) : ‖((x:ℂ),(0:ℝ))‖ ≤ R n := by dsimp [R]; have := Nat.cast_nonneg (α := ℝ) n; linarith
-  let t (n : ℕ) : ℝ≥0 := ⟨(n:ℝ)+1,by positivity⟩
-  let c (n : ℕ) := Classical.choice (exists_boundaryClockSolution B hm hBm hBc κ (a n) (ha n) (x:ℂ))
-  let l (n : ℕ) : BoundaryLocalization (c n) α β (R n) H (t n) := Classical.choice
-    (exists_boundaryLocalization hBc hB0 (c n) (ha n) α β (R n) H (t n) (hR0 n)
-      (by simpa only [Complex.ofReal_re] using hax n) hH0)
-  let E (n : ℕ) : Set Ω := {ω | H ≤ boundaryDensity κ α β ((c n).Q ((l n).τ ω) ω) (B ((l n).τ ω) ω)}
-  have hprob (n : ℕ) : P (E n) ≤ ENNReal.ofReal (boundaryDensity κ α β (x:ℂ) 0/H) :=
-    (l n).density_crossing_probability hB hBm hBc hκ (ha n) hβ (hR n) hH
-  have hsub : {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
-      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
-      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ⊆
-      {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} := by
-    rintro ω ⟨S,Q,hQ,hhit⟩
-    obtain ⟨g,hg,hgap⟩ := hQ.positive_lower_bound ((hBc ω).const_mul _)
-    have hnc : ContinuousOn (fun u : ℝ => ‖(Q u,B u.toNNReal ω)‖) (Icc 0 (S:ℝ)) :=
-      (hQ.continuousOn.prodMk (((hBc ω).comp continuous_real_toNNReal).continuousOn)).norm
-    obtain ⟨s,hs,hmax⟩ := (isCompact_Icc : IsCompact (Icc 0 (S:ℝ))).exists_isMaxOn
-      ⟨0,le_rfl,S.coe_nonneg⟩ hnc
-    let M := ‖(Q s,B s.toNNReal ω)‖
-    obtain ⟨N,hN⟩ := exists_nat_gt (max (max (S:ℝ) M) (x/(2*g)))
-    filter_upwards [eventually_ge_atTop N] with n hn
-    have hn' : (N:ℝ) ≤ n := Nat.cast_le.mpr hn
-    have hNS : (S:ℝ) < (n:ℝ)+1 := by have := (le_max_left (S:ℝ) M).trans (le_max_left (max (S:ℝ) M) (x/(2*g))); linarith
-    have hNM : M < (n:ℝ)+1 := by have := (le_max_right (S:ℝ) M).trans (le_max_left (max (S:ℝ) M) (x/(2*g))); linarith
-    have hNg : x/(2*g) < (n:ℝ)+1 := by have := le_max_right (max (S:ℝ) M) (x/(2*g)); linarith
-    have hann : 3*a n < g := by
-      have hn0 : 0 < (n:ℝ)+1 := by positivity
-      have hh : x < ((n:ℝ)+1)*(2*g) := (div_lt_iff₀ (by positivity)).mp hNg
-      rw [show 3*a n = (x/2)/((n:ℝ)+1) by dsimp only [a]; ring]
-      apply (div_lt_iff₀ hn0).mpr
-      nlinarith
-    apply (l n).true_crossing_forces_density_boundary (ha n) ω S
-      (NNReal.coe_lt_coe.mp hNS) Q hQ
-    · intro u hu
-      exact hann.trans_le (hgap u hu)
-    · intro u hu
-      have hm := hmax ⟨u.coe_nonneg,NNReal.coe_le_coe.mpr hu⟩
-      change ‖(Q (u:ℝ),B (u:ℝ).toNNReal ω)‖ ≤ M at hm
-      rw [Real.toNNReal_coe] at hm
-      have hnorm : 0 ≤ ‖((x:ℂ),(0:ℝ))‖ := norm_nonneg _
-      change ‖(Q (u:ℝ),B u ω)‖ < ‖((x:ℂ),(0:ℝ))‖+(n:ℝ)+1
-      change ‖(Q (u:ℝ),B u ω)‖ ≤ M at hm
-      linarith
-    · exact hhit
-  exact (measure_mono hsub).trans (measure_eventually_sets_le P E _ hprob)
-
- 
-
-theorem boundary_maximal_density
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ α β x H : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hH : 0 < H)
-    (hβ : -α-2+κ*(β+1)/2=0)
-    (hH0 : boundaryDensity κ α β (x:ℂ) 0 ≤ H) :
-    P {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
-      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
-      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ≤
-      ENNReal.ofReal (boundaryDensity κ α β (x:ℂ) 0 / H) := by
-  obtain ⟨C,hCm,hCc,hC0,hC,hCB⟩ := exists_regular_brownian_version hB
-  apply (measure_mono_ae (show {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
-      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
-      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ≤ᵐ[P]
-      {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
-      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*C t ω) x T Q ∧
-      H ≤ boundaryDensity κ α β (Q T) (C T ω)} from ?_)).trans
-      (boundary_maximal_density_regular hC hCm hCc hC0 κ α β x H hκ hx hH hβ hH0)
-  filter_upwards [hCB] with ω he
-  rintro ⟨T,Q,hQ,hF⟩
-  refine ⟨T,Q,?_,?_⟩
-  · simpa only [he] using hQ
-  · simpa only [he] using hF
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory Complex
-open scoped Topology NNReal
-namespace SLEExactGauge
-
- 
-
-theorem IsCapacityTwoWitness.forward_flow_survives
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (Q : ℝ → ℂ)
-    (hpos : ∀ t ∈ Icc 0 (T:ℝ), 0 < (Q t).im)
-    (hd : ∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Q
-      (2 / (Q t - (U t.toNNReal : ℂ))) (Icc 0 (T:ℝ)) t) :
-    Q 0 ∈ survivingDomain γ T ∧ G T (Q 0) = Q T := by
-  have hc : ContinuousOn (fun t => (Q t).im) (Icc 0 (T:ℝ)) :=
-    Complex.continuous_im.comp_continuousOn (fun t ht => (hd t ht).continuousWithinAt)
-  obtain ⟨t₀,ht₀,hmin⟩ := (isCompact_Icc : IsCompact (Icc 0 (T:ℝ))).exists_isMinOn
-    ⟨0,le_rfl,T.coe_nonneg⟩ hc
-  let a := (Q t₀).im
-  have ha : 0 < a := hpos t₀ ht₀
-  have hbound (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a ≤ (Q t).im := hmin ht
-  have hmap : MapsTo (fun t : ℝ => (T:ℝ)-t) (Icc 0 (T:ℝ)) (Icc 0 (T:ℝ)) := by
-    intro t ht; constructor <;> linarith [ht.1,ht.2]
-  have hrev (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) :
-      HasDerivWithinAt (fun s : ℝ => Q ((T:ℝ)-s))
-        (reverseField a (U ((T:ℝ)-t).toNNReal) (Q ((T:ℝ)-t))) (Icc 0 (T:ℝ)) t := by
-    have he := (hd _ (hmap ht)).scomp t
-      ((hasDerivAt_id t).const_sub (T:ℝ)).hasDerivWithinAt hmap
-    simpa only [Function.comp_def,neg_smul,one_smul,reverseField,
-      reverseCenter_eq (hbound _ (hmap ht)),div_eq_mul_inv,neg_mul] using he
-  have he := h.reverse_flow_identifies T (Q T) ha
-    (hbound T ⟨T.coe_nonneg,le_rfl⟩) (fun s => Q ((T:ℝ)-s))
-    (by simp only [sub_zero]) hrev
-  simp only [sub_self] at he
-  have hlive : F T (Q T) ∈ survivingDomain γ T :=
-    h.2.2.2.2.2.2.1 T (hpos T ⟨T.coe_nonneg,le_rfl⟩)
-  rw [← he] at hlive
-  refine ⟨hlive,?_⟩
-  rw [he]
-  exact h.2.2.2.2.2.2.2.2.1 T (Q T) (hpos T ⟨T.coe_nonneg,le_rfl⟩)
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.StoppedPower
+
+namespace OAI
+
+/-! Swallow Exit for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma nonnegative_integral_tail {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (Y : Ω → ℝ) (hYi : Integrable Y P) (hY0 : ∀ ω, 0 ≤ Y ω)
+    (M a : ℝ) (ha : 0 < a) (hmean : (∫ ω, Y ω ∂P) ≤ M) :
+    P {ω | a ≤ Y ω} ≤ ENNReal.ofReal (M/a) := by
+  have hmark := meas_ge_le_lintegral_div hYi.aemeasurable.ennreal_ofReal
+    (ne_of_gt (ENNReal.ofReal_pos.mpr ha)) ENNReal.ofReal_ne_top
+  calc
+    P {ω | a ≤ Y ω} ≤ P {ω | ENNReal.ofReal a ≤ ENNReal.ofReal (Y ω)} :=
+      measure_mono (fun ω hω => ENNReal.ofReal_le_ofReal hω)
+    _ ≤ (∫⁻ ω, ENNReal.ofReal (Y ω) ∂P) / ENNReal.ofReal a := hmark
+    _ = ENNReal.ofReal (∫ ω, Y ω ∂P) / ENNReal.ofReal a := by
+      rw [ofReal_integral_eq_lintegral_ofReal hYi (Eventually.of_forall hY0)]
+    _ ≤ ENNReal.ofReal M / ENNReal.ofReal a :=
+      ENNReal.div_le_div_right (ENNReal.ofReal_le_ofReal hmean) _
+    _ = ENNReal.ofReal (M/a) := (ENNReal.ofReal_div_of_pos ha).symm
+
+namespace RadialLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
+
+lemma exit_probability {P : Measure Ω}
+    (a : RadialLocalization c R T)
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 4 < κ) (hR : 0 < R) (hhi : q₀.1.im ≤ hi)
+    (b : ℝ) (hb : 0 < b)
+    (hbR : b ≤ Real.exp (((κ-4)/(2*κ))*Real.log R))
+    (hbT : b ≤ (((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) * T) :
+    P {ω | a.τ ω = T ∨ greenLogNormSq κ (c.Q (a.τ ω) ω) (B (a.τ ω) ω) = R^2} ≤
+      ENNReal.ofReal (swallowLyapunov κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q₀ 0 / b) := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let β := (κ-4)/(2*κ)
+  let C := 3*(κ-4)/8
+  let d := β*((κ-4)/4)*Real.exp ((β-2)*Real.log R)
+  let F (ω : Ω) := swallowLyapunov κ β C (c.Q (a.τ ω) ω) (B (a.τ ω) ω)
+  let Y (ω : Ω) := F ω + d*(a.τ ω : ℝ)
+  have hk : 0 < κ := lt_trans (by norm_num) hκ
+  have hβ : 0 < β := by dsimp [β]; positivity
+  have hC : 0 ≤ C := by dsimp [C]; positivity
+  have hd : 0 ≤ d := by dsimp [d]; positivity
+  have hF0 (ω : Ω) : 0 ≤ F ω := swallowLyapunov_nonneg κ β C hC _ _
+  have hY0 (ω : Ω) : 0 ≤ Y ω := add_nonneg (hF0 ω) (mul_nonneg hd (a.τ ω).coe_nonneg)
+  have hmean := a.stopped_lyapunov_bound hB hBm hBc hκ hR hhi
+  have hτi : Integrable (fun ω => (a.τ ω : ℝ)) P :=
+    bounded_nnreal_integrable a.τ (stopping_time_measurable_nnreal a.stopping) T a.bounded
+  have hYi : Integrable Y P := hmean.1.add (hτi.const_mul d)
+  have hYmean : (∫ ω, Y ω ∂P) ≤ swallowLyapunov κ β C q₀ 0 := by
+    rw [show (∫ ω, Y ω ∂P) = (∫ ω, F ω ∂P)+d*(∫ ω, (a.τ ω : ℝ) ∂P) by
+      rw [show Y = fun ω => F ω+d*(a.τ ω : ℝ) from rfl,
+        integral_add hmean.1 (hτi.const_mul d), integral_const_mul]]
+    exact hmean.2
+  apply (measure_mono (show {ω | a.τ ω = T ∨ greenLogNormSq κ (c.Q (a.τ ω) ω) (B (a.τ ω) ω) = R^2}
+      ⊆ {ω | b ≤ Y ω} from ?_)).trans (nonnegative_integral_tail Y hYi hY0 _ b hb hYmean)
+  intro ω hω
+  rcases hω with ht | hn
+  · change b ≤ F ω+d*(a.τ ω : ℝ)
+    rw [ht]
+    exact hbT.trans (le_add_of_nonneg_left (hF0 ω))
+  · have hr := radialLogPower_at_boundary κ β (c.Q (a.τ ω) ω) (B (a.τ ω) ω) R hn
+    have hh : Real.exp (β*Real.log R) ≤ F ω := by
+      change Real.exp (β*Real.log R) ≤ radialLogPower κ β _ _ + C*heightLogPower β _
+      rw [hr]
+      exact le_add_of_nonneg_right (mul_nonneg hC (heightLogPower_pos β _).le)
+    exact (hbR.trans hh).trans (le_add_of_nonneg_right (mul_nonneg hd (a.τ ω).coe_nonneg))
+
+end RadialLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+structure BoundaryLocalization
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ a : ℝ} {q₀ : ℂ} (c : BoundaryClockSolution B hm κ a q₀)
+    (α β R H : ℝ) (T : ℝ≥0) where
+  τ : Ω → ℝ≥0
+  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun ω => (τ ω : WithTop ℝ≥0))
+  bounded : ∀ ω, τ ω ≤ T
+  inside : ∀ ω t, t ≤ τ ω → ‖(c.Q t ω,B t ω)‖ ≤ R ∧
+    3*a ≤ (c.Q t ω).re-Real.sqrt κ*B t ω ∧
+    boundaryDensity κ α β (c.Q t ω) (B t ω) ≤ H
+  before : ∀ ω t, t < τ ω → ‖(c.Q t ω,B t ω)‖ < R ∧
+    3*a < (c.Q t ω).re-Real.sqrt κ*B t ω ∧
+    boundaryDensity κ α β (c.Q t ω) (B t ω) < H
+  boundary : ∀ ω, τ ω < T → ‖(c.Q (τ ω) ω,B (τ ω) ω)‖ = R ∨
+    (c.Q (τ ω) ω).re-Real.sqrt κ*B (τ ω) ω = 3*a ∨
+    boundaryDensity κ α β (c.Q (τ ω) ω) (B (τ ω) ω) = H
+
+lemma exists_boundaryLocalization
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ a : ℝ} {q₀ : ℂ} (c : BoundaryClockSolution B hm κ a q₀)
+    (ha : 0 < a) (α β R H : ℝ) (T : ℝ≥0)
+    (hR : ‖(q₀,(0:ℝ))‖ ≤ R) (hx : 3*a ≤ q₀.re)
+    (hH : boundaryDensity κ α β q₀ 0 ≤ H) :
+    Nonempty (BoundaryLocalization c α β R H T) := by
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let V (p : ℂ × ℝ) := min (R-‖p‖) (min (p.1.re-Real.sqrt κ*p.2-3*a)
+    (H-smoothBoundaryDensity κ α β a p.1 p.2))
+  have hVc : Continuous V := by
+    exact (continuous_const.sub continuous_norm).min
+      (((Complex.continuous_re.comp continuous_fst).sub (continuous_const.mul continuous_snd)).sub
+        continuous_const |>.min (continuous_const.sub (smoothBoundaryDensity_contDiff κ α β ha).continuous))
+  let D (t : ℝ≥0) (ω : Ω) := V (c.Q t ω,B t ω)
+  have hBa : Adapted f B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  have hDa : Adapted f D := fun t => hVc.measurable.comp ((c.adapted t).prodMk (hBa t))
+  have hDc (ω : Ω) : Continuous (fun t => D t ω) :=
+    hVc.comp (((c.lipschitz ω).continuous.comp NNReal.continuous_coe).prodMk (hBc ω))
+  have hD0 (ω : Ω) : 0 ≤ D 0 ω := by
+    have he : smoothBoundaryDensity κ α β a q₀ 0 = boundaryDensity κ α β q₀ 0 :=
+      smoothBoundaryDensity_eq ha (by simpa using (show 2*a ≤ q₀.re by linarith))
+    simpa only [D,V,NNReal.coe_zero,c.initial,hB0,mul_zero,sub_zero,he,
+      le_min_iff,sub_nonneg] using And.intro hR (And.intro hx hH)
+  obtain ⟨τ,hτ,hb,hin,hpre,hbd⟩ := first_margin_stopping f D hDa hDc hD0 T
+  have hin' (ω : Ω) (t : ℝ≥0) (ht : t ≤ τ ω) : ‖(c.Q t ω,B t ω)‖ ≤ R ∧
+      3*a ≤ (c.Q t ω).re-Real.sqrt κ*B t ω ∧ boundaryDensity κ α β (c.Q t ω) (B t ω) ≤ H := by
+    have hh := hin ω t ht
+    simp only [D,V,le_min_iff,sub_nonneg] at hh
+    rw [smoothBoundaryDensity_eq ha (by linarith [hh.2.1])] at hh
+    exact hh
+  refine ⟨⟨τ,hτ,hb,hin',?_,?_⟩⟩
+  · intro ω t ht
+    have hh := hpre ω t ht
+    simp only [D,V,lt_min_iff,sub_pos] at hh
+    rw [smoothBoundaryDensity_eq ha (by linarith [hh.2.1])] at hh
+    exact hh
+  · intro ω ht
+    have hh := hbd ω ht
+    dsimp only [D,V] at hh
+    rw [smoothBoundaryDensity_eq ha (by linarith [(hin' ω (τ ω) le_rfl).2.1])] at hh
+    rw [min_eq_iff] at hh
+    rcases hh with ⟨hh,_⟩ | ⟨hh,_⟩
+    · exact Or.inl (sub_eq_zero.mp hh).symm
+    · rw [min_eq_iff] at hh
+      rcases hh with ⟨hh,_⟩ | ⟨hh,_⟩
+      · exact Or.inr (Or.inl (sub_eq_zero.mp hh))
+      · exact Or.inr (Or.inr (sub_eq_zero.mp hh).symm)
+
+namespace BoundaryLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ a : ℝ} {q₀ : ℂ} {c : BoundaryClockSolution B hm κ a q₀}
+    {α β R H : ℝ} {T : ℝ≥0}
+
+lemma density_crossing_probability {P : Measure Ω}
+    (l : BoundaryLocalization c α β R H T)
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (hκ : 0 < κ) (ha : 0 < a) (hβ : -α-2+κ*(β+1)/2=0)
+    (hR : 0 ≤ R) (hH : 0 < H) :
+    P {ω | H ≤ boundaryDensity κ α β (c.Q (l.τ ω) ω) (B (l.τ ω) ω)} ≤
+      ENNReal.ofReal (boundaryDensity κ α β q₀ 0/H) := by
+  obtain ⟨hi,he⟩ := c.stopped_boundary_density hB hBm hBc hκ ha α β hβ R hR
+    l.τ l.stopping T l.bounded (fun ω t ht =>
+      ⟨(l.inside ω t ht).1,by linarith [(l.inside ω t ht).2.1]⟩)
+  exact nonnegative_integral_tail _ hi (fun ω => (Real.exp_pos _).le) _ _ hH he.le
+
+end BoundaryLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+def IsPositiveBoundaryClockPath (U : ℝ≥0 → ℝ) (x : ℝ) (T : ℝ≥0) (Q : ℝ → ℂ) : Prop :=
+  Q 0 = (x:ℂ) ∧
+  (∀ t ∈ Icc 0 (T:ℝ), 0 < (Q t).re-U t.toNNReal) ∧
+  (∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Q
+    (((2/((Q t).re-U t.toNNReal):ℝ):ℂ)+
+      (((Q t).re-U t.toNNReal)⁻¹^2:ℝ)*I) (Icc 0 (T:ℝ)) t)
+
+namespace IsPositiveBoundaryClockPath
+variable {U : ℝ≥0 → ℝ} {x : ℝ} {T : ℝ≥0} {Q : ℝ → ℂ}
+
+lemma continuousOn (h : IsPositiveBoundaryClockPath U x T Q) :
+    ContinuousOn Q (Icc 0 (T:ℝ)) := fun t ht => (h.2.2 t ht).continuousWithinAt
+
+lemma positive_lower_bound (h : IsPositiveBoundaryClockPath U x T Q) (hU : Continuous U) :
+    ∃ a : ℝ, 0 < a ∧ ∀ t ∈ Icc 0 (T:ℝ), a ≤ (Q t).re-U t.toNNReal := by
+  have hc : ContinuousOn (fun t : ℝ => (Q t).re-U t.toNNReal) (Icc 0 (T:ℝ)) :=
+    (Complex.continuous_re.comp_continuousOn h.continuousOn).sub
+      ((hU.comp continuous_real_toNNReal).continuousOn)
+  obtain ⟨s,hs,hmin⟩ := (isCompact_Icc : IsCompact (Icc 0 (T:ℝ))).exists_isMinOn
+    ⟨0,le_rfl,T.coe_nonneg⟩ hc
+  exact ⟨_,h.2.1 s hs,fun t ht => hmin ht⟩
+
+lemma clock_identity (h : IsPositiveBoundaryClockPath U x T Q) (hU : Continuous U) :
+    (Q T).im = ∫ t in (0:ℝ)..(T:ℝ), ((Q t).re-U t.toNNReal)⁻¹^2 := by
+  have hc : ContinuousOn (fun t : ℝ => ((Q t).re-U t.toNNReal)⁻¹^2) (Icc 0 (T:ℝ)) :=
+    (((Complex.continuous_re.comp_continuousOn h.continuousOn).sub
+      ((hU.comp continuous_real_toNNReal).continuousOn)).inv₀ (fun t ht => (h.2.1 t ht).ne')).pow 2
+  have hd (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt (fun u => (Q u).im)
+      (((Q t).re-U t.toNNReal)⁻¹^2) (Icc 0 (T:ℝ)) t := by
+    simpa only [Function.comp_def,ContinuousLinearMap.comp_apply,Complex.imCLM_apply,Complex.add_im,
+      Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,Complex.I_im,mul_one,
+      Complex.I_re,mul_zero,add_zero,zero_add] using
+      Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (h.2.2 t ht)
+  have hdr (t : ℝ) (ht : t ∈ Ioo 0 (T:ℝ)) : HasDerivWithinAt (fun u => (Q u).im)
+      (((Q t).re-U t.toNNReal)⁻¹^2) (Ioi t) t := by
+    exact (hasDerivWithinAt_Ici_of_Icc ⟨ht.1.le,ht.2⟩
+      (hd t ⟨ht.1.le,ht.2.le⟩)).mono Ioi_subset_Ici_self
+  have he := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le T.coe_nonneg
+    (Complex.continuous_im.comp_continuousOn h.continuousOn) hdr
+    (hc.intervalIntegrable_of_Icc T.coe_nonneg)
+  dsimp only [Function.comp_def] at he
+  rw [h.1,Complex.ofReal_im,sub_zero] at he
+  exact he.symm
+
+lemma clipped_identification
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    {hm : ∀ t, Measurable (B t)} {κ a : ℝ}
+    (c : BoundaryClockSolution B hm κ a (x:ℂ)) (ha : 0 < a) (ω : Ω)
+    (h : IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q)
+    (hgap : ∀ t ∈ Icc 0 (T:ℝ), a ≤ (Q t).re-Real.sqrt κ*B t.toNNReal ω) :
+    EqOn Q (fun t => c.Q t ω) (Icc 0 (T:ℝ)) := by
+  have hd (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt Q
+      (boundaryClockField a (Real.sqrt κ*B t.toNNReal ω) (Q t)) (Icc 0 (T:ℝ)) t := by
+    simpa only [boundaryClockField,positiveBoundaryField_eq (hgap t ht),max_eq_right (hgap t ht)] using h.2.2 t ht
+  have hc (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) := (c.derivative ω t ht.1).mono
+    (show Icc (0:ℝ) (T:ℝ) ⊆ Ici 0 from fun s hs => hs.1)
+  apply ODE_solution_unique (fun t => boundaryClockField_lipschitz ha (Real.sqrt κ*B t.toNNReal ω))
+    h.continuousOn
+    (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hd t ⟨ht.1,ht.2.le⟩))
+    (c.lipschitz ω).continuous.continuousOn
+    (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hc t ⟨ht.1,ht.2.le⟩))
+  exact h.1.trans (c.initial ω).symm
+
+end IsPositiveBoundaryClockPath
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def greenDensity (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
+    (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) : ℝ := by
+  classical
+  exact if z ∈ survivingDomain γ t then
+    (fullRadius G t z) ^ (-codimension κ) *
+      (Real.sin (Complex.arg (G t z - (U t : ℂ)))) ^ greenExponent κ
+  else 0
+
+noncomputable def initialGreen (κ : ℝ) (z : ℂ) : ℝ :=
+  (2 * z.im) ^ (-codimension κ) *
+    (Real.sin (Complex.arg z)) ^ greenExponent κ
+
+ 
+
+noncomputable def radiusLevelTime (γ : ℝ≥0 → ℂ) (G : ℝ≥0 → ℂ → ℂ)
+    (z : ℂ) (u : ℝ) : ℝ≥0∞ :=
+  sInf {s : ℝ≥0∞ | s < ∞ ∧ z ∈ survivingDomain γ s.toNNReal ∧
+    fullRadius G s.toNNReal z = (2 * z.im) * Real.exp (-u)}
+
+ 
+noncomputable def stoppedGreen (κ : ℝ) (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ)
+    (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (u : ℝ) : ℝ :=
+  if radiusLevelTime γ G z u < ∞ then
+    greenDensity κ U γ G (radiusLevelTime γ G z u).toNNReal z
+  else 0
+
+lemma logCoordinates_re (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0) :
+    (logLoewnerCoordinates G z t).1.re = (G t z).re := by
+  simp only [logLoewnerCoordinates, Real.toNNReal_coe, Complex.add_re, Complex.ofReal_re,
+    Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero, zero_mul,
+    sub_self, add_zero]
+
+lemma logCoordinates_im (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0) :
+    (logLoewnerCoordinates G z t).1.im = Real.log (G t z).im := by
+  simp only [logLoewnerCoordinates, Real.toNNReal_coe, Complex.add_im, Complex.ofReal_im,
+    Complex.mul_im, Complex.ofReal_re, Complex.I_re, Complex.I_im, mul_zero, mul_one,
+    add_zero, zero_add]
+
+lemma logCoordinates_snd (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0) :
+    (logLoewnerCoordinates G z t).2 = Real.log ‖deriv (G t) z‖ := by
+  simp only [logLoewnerCoordinates, Real.toNNReal_coe]
+
+lemma greenLogNormSq_logCoordinates (κ b : ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0)
+    (hy : 0 < (G t z).im) :
+    greenLogNormSq κ (logLoewnerCoordinates G z t) b =
+      Complex.normSq (G t z - ((Real.sqrt κ * b : ℝ) : ℂ)) := by
+  have he : Real.exp (2 * Real.log (G t z).im) = (G t z).im ^ 2 := by
+    rw [show 2 * Real.log (G t z).im = Real.log (G t z).im + Real.log (G t z).im by ring,
+      Real.exp_add, Real.exp_log hy, pow_two]
+  rw [greenLogNormSq, logCoordinates_re, logCoordinates_im, he]
+  simp only [Complex.normSq_apply, Complex.sub_re, Complex.sub_im, Complex.ofReal_re,
+    Complex.ofReal_im, sub_zero, pow_two]
+
+lemma logAngleSq_logCoordinates (κ b : ℝ) (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ≥0)
+    (hy : 0 < (G t z).im) :
+    logAngleSq κ (logLoewnerCoordinates G z t) b =
+      (Real.sin (Complex.arg (G t z - ((Real.sqrt κ * b : ℝ) : ℂ)))) ^ 2 := by
+  rw [logAngleSq, greenLogNormSq_logCoordinates κ b G z t hy, logCoordinates_im,
+    Complex.sin_arg, Complex.normSq_eq_norm_sq, div_pow]
+  congr 1
+  simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
+  rw [show 2 * Real.log (G t z).im = Real.log (G t z).im + Real.log (G t z).im by ring,
+    Real.exp_add, Real.exp_log hy, pow_two]
+
+lemma greenLogDensity_logCoordinates
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F)
+    (κ b : ℝ) (z : ℂ) (t : ℝ≥0) (hz : z ∈ survivingDomain γ t)
+    (hUb : U t = Real.sqrt κ * b) :
+    greenLogDensity κ (logLoewnerCoordinates G z t) b = greenDensity κ U γ G t z := by
+  have hy := h.imag_positive t z hz
+  have hj := h.derivative_norm_positive hU t z hz
+  have hn : 0 < ‖G t z - ((Real.sqrt κ * b : ℝ) : ℂ)‖ := by
+    apply norm_pos_iff.mpr
+    simpa only [hUb] using h.centered_ne_zero t z hz
+  have hs : 0 < Real.sin (Complex.arg (G t z - (U t : ℂ))) := by
+    rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero, hUb]
+    exact div_pos hy hn
+  have hR := h.fullRadius_positive hU t z hz
+  have hlogR : Real.log (fullRadius G t z) =
+      Real.log 2 + Real.log (G t z).im - Real.log ‖deriv (G t) z‖ := by
+    rw [fullRadius, Real.log_div (mul_ne_zero (by norm_num) hy.ne') hj.ne',
+      Real.log_mul (by norm_num) hy.ne']
+  have hlogS : Real.log (Real.sin (Complex.arg (G t z - (U t : ℂ)))) =
+      Real.log (G t z).im - Real.log ‖G t z - ((Real.sqrt κ * b : ℝ) : ℂ)‖ := by
+    rw [Complex.sin_arg, Complex.sub_im, Complex.ofReal_im, sub_zero, hUb,
+      Real.log_div hy.ne' hn.ne']
+  rw [greenDensity, ite_eq_left hz, Real.rpow_def_of_pos hR, Real.rpow_def_of_pos hs,
+    ← Real.exp_add, hlogR, hlogS, greenLogDensity, logCoordinates_im, logCoordinates_snd,
+    greenLogNormSq_logCoordinates κ b G z t hy, Complex.normSq_eq_norm_sq, Real.log_pow]
+  congr 1
+  push_cast
+  ring
+
+lemma greenLogDensity_initial (κ : ℝ) (z : ℂ) (hz : z ∈ upperHalfPlane) :
+    greenLogDensity κ (initialLogCoordinates z) 0 = initialGreen κ z := by
+  have hn : 0 < ‖z‖ := norm_pos_iff.mpr (by
+    have hp : 0 < z.im := hz
+    intro he; simp only [he, Complex.zero_im, lt_self_iff_false] at hp)
+  have hs : 0 < Real.sin (Complex.arg z) := by rw [Complex.sin_arg]; exact div_pos hz hn
+  have he : Real.exp (2 * Real.log z.im) = z.im ^ 2 := by
+    rw [show 2 * Real.log z.im = Real.log z.im + Real.log z.im by ring,
+      Real.exp_add, Real.exp_log hz, pow_two]
+  have hN : greenLogNormSq κ (initialLogCoordinates z) 0 = Complex.normSq z := by
+    simp only [greenLogNormSq, initialLogCoordinates, Complex.add_re, Complex.ofReal_re,
+      Complex.mul_re, Complex.ofReal_im, Complex.I_re, Complex.I_im, mul_zero,
+      sub_self, add_zero, sub_zero, Complex.add_im, Complex.mul_im, mul_one, zero_add, he,
+      Complex.normSq_apply, pow_two]
+  rw [initialGreen, Real.rpow_def_of_pos (mul_pos (by norm_num) hz), Real.rpow_def_of_pos hs,
+    ← Real.exp_add, Real.log_mul (by norm_num) hz.ne', Complex.sin_arg, Real.log_div hz.ne' hn.ne']
+  rw [greenLogDensity, hN, Complex.normSq_eq_norm_sq, Real.log_pow]
+  simp only [initialLogCoordinates, Complex.add_im, Complex.ofReal_im, Complex.mul_im,
+    Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, zero_add, add_zero,
+    Nat.cast_ofNat]
+  congr 1
+  ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace RadialLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
+
+lemma alive_forces_radial_or_time_boundary (a : RadialLocalization c R T)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) {z : ℂ} (hz : z ∈ upperHalfPlane)
+    (hq : q₀ = initialLogCoordinates z) (hhi : q₀.1.im ≤ hi)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F)
+    (hlive : z ∈ survivingDomain γ T) (hlo : lo < Real.log (G T z).im) :
+    a.τ ω = T ∨ greenLogNormSq κ (c.Q (a.τ ω) ω) (B (a.τ ω) ω) = R^2 := by
+  rcases eq_or_lt_of_le (a.bounded ω) with h | h
+  · exact Or.inl h
+  rcases a.boundary ω h with hN | hh
+  · exact Or.inr hN
+  obtain ⟨_,he⟩ := a.chart_identification hBc hz hq hhi ω γ G F hw
+  have hheight := (hw.log_height_bounds T z hlive (a.τ ω)
+    ⟨(a.τ ω).coe_nonneg, NNReal.coe_le_coe.mpr (a.bounded ω)⟩).1
+  rw [he (a.τ ω) le_rfl, logCoordinates_im] at hh
+  simp only [Real.toNNReal_coe] at hheight
+  exact False.elim (not_le_of_gt hlo (hh ▸ hheight))
+
+end RadialLocalization
+
+lemma measure_eventually_sets_le {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    (E : ℕ → Set Ω) (C : ℝ≥0∞) (hC : ∀ n, P (E n) ≤ C) :
+    P {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} ≤ C := by
+  let H (n : ℕ) := ⋂ m ≥ n, E m
+  have hH : Monotone H := by
+    intro n m hnm ω hω
+    simp only [H, mem_iInter] at hω ⊢
+    exact fun k hk => hω k (hnm.trans hk)
+  have he : {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} = ⋃ n, H n := by
+    ext ω
+    simp only [mem_ofPred_eq, eventually_atTop, mem_iUnion, H, mem_iInter]
+  rw [he, hH.measure_iUnion]
+  apply iSup_le
+  intro n
+  apply (measure_mono (show H n ⊆ E n from fun ω hω => by
+    exact mem_iInter.mp (mem_iInter.mp hω n) le_rfl)).trans (hC n)
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace BoundaryLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ a x : ℝ} {c : BoundaryClockSolution B hm κ a (x:ℂ)}
+    {α β R H : ℝ} {T : ℝ≥0}
+
+ 
+
+lemma true_crossing_forces_density_boundary
+    (l : BoundaryLocalization c α β R H T) (ha : 0 < a)
+    (ω : Ω) (S : ℝ≥0) (hS : S < T) (Q : ℝ → ℂ)
+    (hQ : IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x S Q)
+    (hgap : ∀ t ∈ Icc 0 (S:ℝ), 3*a < (Q t).re-Real.sqrt κ*B t.toNNReal ω)
+    (hbound : ∀ t : ℝ≥0, t ≤ S → ‖(Q t,B t ω)‖ < R)
+    (hhit : H ≤ boundaryDensity κ α β (Q S) (B S ω)) :
+    H ≤ boundaryDensity κ α β (c.Q (l.τ ω) ω) (B (l.τ ω) ω) := by
+  have he := hQ.clipped_identification c ha ω (fun t ht => by linarith [hgap t ht])
+  have hτS : l.τ ω ≤ S := by
+    by_contra h
+    have hpre := (l.before ω S (not_le.mp h)).2.2
+    have heS : Q S = c.Q S ω := he ⟨S.coe_nonneg,le_rfl⟩
+    rw [← heS] at hpre
+    exact not_lt_of_ge hhit hpre
+  have hτT : l.τ ω < T := hτS.trans_lt hS
+  have heτ : Q (l.τ ω) = c.Q (l.τ ω) ω := he ⟨(l.τ ω).coe_nonneg,NNReal.coe_le_coe.mpr hτS⟩
+  rcases l.boundary ω hτT with hnorm | hg | hf
+  · have hb := hbound (l.τ ω) hτS
+    rw [heτ,hnorm] at hb
+    exact False.elim (lt_irrefl R hb)
+  · have hp := hgap (l.τ ω) ⟨(l.τ ω).coe_nonneg,NNReal.coe_le_coe.mpr hτS⟩
+    rw [Real.toNNReal_coe,heτ,hg] at hp
+    exact False.elim (lt_irrefl _ hp)
+  · exact hf.ge
+
+end BoundaryLocalization
+end SLEExactGauge
+end
+
+ 
+section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+ 
+theorem exists_measurable_brownian_version
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
+    ∃ C : ℝ≥0 → Ω → ℝ, (∀ t, Measurable (C t)) ∧ IsBrownianReal C P ∧
+      (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) := by
+  let b (n k : ℕ) : Ω → ℝ :=
+    (hB.aemeasurable (stopGridTime n k)).mk (B (stopGridTime n k))
+  have hb (n k : ℕ) : Measurable (b n k) := (hB.aemeasurable _).measurable_mk
+  have heq : ∀ᵐ ω ∂P, ∀ n k, b n k ω = B (stopGridTime n k) ω := by
+    apply ae_all_iff.mpr
+    intro n
+    apply ae_all_iff.mpr
+    intro k
+    exact (hB.aemeasurable _).ae_eq_mk.symm
+  let C (t : ℝ≥0) (ω : Ω) : ℝ :=
+    limUnder atTop (fun n => b n (stopGridIndex n t) ω)
+  have hm (t : ℝ≥0) : Measurable (C t) :=
+    (StronglyMeasurable.limUnder (fun n => (hb n _).stronglyMeasurable)).measurable
+  have hCB : ∀ᵐ ω ∂P, ∀ t, C t ω = B t ω := by
+    filter_upwards [hB.cont, heq] with ω hω heqω
+    intro t
+    change limUnder atTop (fun n => b n (stopGridIndex n t) ω) = B t ω
+    simp_rw [heqω]
+    exact (hω.continuousAt.tendsto.comp (stopGrid_tendsto t)).limUnder_eq
+  refine ⟨C, hm, ⟨?_, ?_⟩, hCB⟩
+  · exact hB.toIsPreBrownianReal.congr (fun t => hCB.mono fun ω hω => (hω t).symm)
+  · filter_upwards [hB.cont, hCB] with ω hω heqω
+    simpa only [heqω] using hω
+
+ 
+
+theorem exists_continuous_measurable_brownian_version
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
+    ∃ C : ℝ≥0 → Ω → ℝ,
+      Measurable (fun q : ℝ≥0 × Ω => C q.1 q.2) ∧
+      (∀ ω, Continuous (fun t => C t ω)) ∧ IsBrownianReal C P ∧
+      (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) := by
+  classical
+  obtain ⟨V, hVm, hV, hVB⟩ := exists_measurable_brownian_version hB
+  let N := toMeasurable P {ω | ¬ Continuous (fun t => V t ω)}
+  have hNm : MeasurableSet N := measurableSet_toMeasurable _ _
+  have hN : P N = 0 := by
+    rw [show N = toMeasurable P {ω | ¬ Continuous (fun t => V t ω)} from rfl,
+      measure_toMeasurable]
+    exact ae_iff.mp hV.cont
+  let C (t : ℝ≥0) (ω : Ω) := if ω ∈ N then (0 : ℝ) else V t ω
+  have hCm (t) : Measurable (C t) := measurable_const.piecewise hNm (hVm t)
+  have hCc (ω) : Continuous (fun t => C t ω) := by
+    by_cases hω : ω ∈ N
+    · simpa only [C, ite_eq_left hω] using (continuous_const : Continuous (fun _ : ℝ≥0 => (0 : ℝ)))
+    · have hv : Continuous (fun t => V t ω) := by
+        by_contra hh
+        exact hω (subset_toMeasurable P _ hh)
+      simpa only [C, ite_eq_right hω] using hv
+  have hCV : ∀ᵐ ω ∂P, ∀ t, C t ω = V t ω := by
+    filter_upwards [show ∀ᵐ ω ∂P, ω ∉ N from (ae_iff.mpr (by simpa only [not_not, ofPred_mem_eq] using hN))] with ω hω
+    intro t
+    simp only [C, ite_eq_right hω]
+  have hCB : ∀ᵐ ω ∂P, ∀ t, C t ω = B t ω := by
+    filter_upwards [hCV, hVB] with ω hω hω'
+    exact fun t => (hω t).trans (hω' t)
+  refine ⟨C, measurable_uncurry_of_continuous_of_measurable hCc hCm, hCc, ?_, hCB⟩
+  exact ⟨hV.toIsPreBrownianReal.congr (fun t => hCV.mono fun ω hω => (hω t).symm),
+    Eventually.of_forall hCc⟩
+
+ 
+
+theorem exists_regular_brownian_version
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) :
+    ∃ C : ℝ≥0 → Ω → ℝ,
+      Measurable (fun q : ℝ≥0 × Ω => C q.1 q.2) ∧
+      (∀ ω, Continuous (fun t => C t ω)) ∧ (∀ ω, C 0 ω = 0) ∧
+      IsBrownianReal C P ∧ (∀ᵐ ω ∂P, ∀ t, C t ω = B t ω) := by
+  obtain ⟨V, hVm, hVc, hV, hVB⟩ := exists_continuous_measurable_brownian_version hB
+  let C (t : ℝ≥0) (ω : Ω) := V t ω - V 0 ω
+  have hV0m : Measurable (V 0) := hVm.comp (measurable_const.prodMk measurable_id)
+  have hCm : Measurable (fun q : ℝ≥0 × Ω => C q.1 q.2) :=
+    hVm.sub (hV0m.comp measurable_snd)
+  have hCc (ω : Ω) : Continuous (fun t => C t ω) := (hVc ω).sub continuous_const
+  have hCV : ∀ᵐ ω ∂P, ∀ t, C t ω = V t ω := by
+    filter_upwards [hV.eval_zero_ae_eq_zero] with ω h0
+    intro t
+    simp only [C, h0, sub_zero]
+  refine ⟨C, hCm, hCc, fun ω => sub_self _, ?_, ?_⟩
+  · exact ⟨hV.toIsPreBrownianReal.congr (fun t => hCV.mono fun ω hω => (hω t).symm),
+      Eventually.of_forall hCc⟩
+  · filter_upwards [hCV, hVB] with ω hω hω'
+    exact fun t => (hω t).trans (hω' t)
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Complex
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+theorem boundary_maximal_density_regular
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    (κ α β x H : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hH : 0 < H)
+    (hβ : -α-2+κ*(β+1)/2=0)
+    (hH0 : boundaryDensity κ α β (x:ℂ) 0 ≤ H) :
+    P {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
+      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
+      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ≤
+      ENNReal.ofReal (boundaryDensity κ α β (x:ℂ) 0 / H) := by
+  let hm (t) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id)
+  let a (n : ℕ) : ℝ := (x/6)/((n:ℝ)+1)
+  have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity
+  have hax (n : ℕ) : 3*a n ≤ x := by
+    have hn : 1 ≤ (n:ℝ)+1 := le_add_of_nonneg_left (Nat.cast_nonneg n)
+    have hh : a n ≤ x/6 := div_le_self (by positivity) hn
+    linarith
+  let R (n : ℕ) : ℝ := ‖((x:ℂ),(0:ℝ))‖+(n:ℝ)+1
+  have hR (n : ℕ) : 0 ≤ R n := by dsimp [R]; positivity
+  have hR0 (n : ℕ) : ‖((x:ℂ),(0:ℝ))‖ ≤ R n := by dsimp [R]; have := Nat.cast_nonneg (α := ℝ) n; linarith
+  let t (n : ℕ) : ℝ≥0 := ⟨(n:ℝ)+1,by positivity⟩
+  let c (n : ℕ) := Classical.choice (exists_boundaryClockSolution B hm hBm hBc κ (a n) (ha n) (x:ℂ))
+  let l (n : ℕ) : BoundaryLocalization (c n) α β (R n) H (t n) := Classical.choice
+    (exists_boundaryLocalization hBc hB0 (c n) (ha n) α β (R n) H (t n) (hR0 n)
+      (by simpa only [Complex.ofReal_re] using hax n) hH0)
+  let E (n : ℕ) : Set Ω := {ω | H ≤ boundaryDensity κ α β ((c n).Q ((l n).τ ω) ω) (B ((l n).τ ω) ω)}
+  have hprob (n : ℕ) : P (E n) ≤ ENNReal.ofReal (boundaryDensity κ α β (x:ℂ) 0/H) :=
+    (l n).density_crossing_probability hB hBm hBc hκ (ha n) hβ (hR n) hH
+  have hsub : {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
+      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
+      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ⊆
+      {ω | ∀ᶠ n : ℕ in atTop, ω ∈ E n} := by
+    rintro ω ⟨S,Q,hQ,hhit⟩
+    obtain ⟨g,hg,hgap⟩ := hQ.positive_lower_bound ((hBc ω).const_mul _)
+    have hnc : ContinuousOn (fun u : ℝ => ‖(Q u,B u.toNNReal ω)‖) (Icc 0 (S:ℝ)) :=
+      (hQ.continuousOn.prodMk (((hBc ω).comp continuous_real_toNNReal).continuousOn)).norm
+    obtain ⟨s,hs,hmax⟩ := (isCompact_Icc : IsCompact (Icc 0 (S:ℝ))).exists_isMaxOn
+      ⟨0,le_rfl,S.coe_nonneg⟩ hnc
+    let M := ‖(Q s,B s.toNNReal ω)‖
+    obtain ⟨N,hN⟩ := exists_nat_gt (max (max (S:ℝ) M) (x/(2*g)))
+    filter_upwards [eventually_ge_atTop N] with n hn
+    have hn' : (N:ℝ) ≤ n := Nat.cast_le.mpr hn
+    have hNS : (S:ℝ) < (n:ℝ)+1 := by have := (le_max_left (S:ℝ) M).trans (le_max_left (max (S:ℝ) M) (x/(2*g))); linarith
+    have hNM : M < (n:ℝ)+1 := by have := (le_max_right (S:ℝ) M).trans (le_max_left (max (S:ℝ) M) (x/(2*g))); linarith
+    have hNg : x/(2*g) < (n:ℝ)+1 := by have := le_max_right (max (S:ℝ) M) (x/(2*g)); linarith
+    have hann : 3*a n < g := by
+      have hn0 : 0 < (n:ℝ)+1 := by positivity
+      have hh : x < ((n:ℝ)+1)*(2*g) := (div_lt_iff₀ (by positivity)).mp hNg
+      rw [show 3*a n = (x/2)/((n:ℝ)+1) by dsimp only [a]; ring]
+      apply (div_lt_iff₀ hn0).mpr
+      nlinarith
+    apply (l n).true_crossing_forces_density_boundary (ha n) ω S
+      (NNReal.coe_lt_coe.mp hNS) Q hQ
+    · intro u hu
+      exact hann.trans_le (hgap u hu)
+    · intro u hu
+      have hm := hmax ⟨u.coe_nonneg,NNReal.coe_le_coe.mpr hu⟩
+      change ‖(Q (u:ℝ),B (u:ℝ).toNNReal ω)‖ ≤ M at hm
+      rw [Real.toNNReal_coe] at hm
+      have hnorm : 0 ≤ ‖((x:ℂ),(0:ℝ))‖ := norm_nonneg _
+      change ‖(Q (u:ℝ),B u ω)‖ < ‖((x:ℂ),(0:ℝ))‖+(n:ℝ)+1
+      change ‖(Q (u:ℝ),B u ω)‖ ≤ M at hm
+      linarith
+    · exact hhit
+  exact (measure_mono hsub).trans (measure_eventually_sets_le P E _ hprob)
+
+ 
+
+theorem boundary_maximal_density
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ α β x H : ℝ) (hκ : 0 < κ) (hx : 0 < x) (hH : 0 < H)
+    (hβ : -α-2+κ*(β+1)/2=0)
+    (hH0 : boundaryDensity κ α β (x:ℂ) 0 ≤ H) :
+    P {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
+      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
+      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ≤
+      ENNReal.ofReal (boundaryDensity κ α β (x:ℂ) 0 / H) := by
+  obtain ⟨C,hCm,hCc,hC0,hC,hCB⟩ := exists_regular_brownian_version hB
+  apply (measure_mono_ae (show {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
+      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*B t ω) x T Q ∧
+      H ≤ boundaryDensity κ α β (Q T) (B T ω)} ≤ᵐ[P]
+      {ω | ∃ (T : ℝ≥0) (Q : ℝ → ℂ),
+      IsPositiveBoundaryClockPath (fun t => Real.sqrt κ*C t ω) x T Q ∧
+      H ≤ boundaryDensity κ α β (Q T) (C T ω)} from ?_)).trans
+      (boundary_maximal_density_regular hC hCm hCc hC0 κ α β x H hκ hx hH hβ hH0)
+  filter_upwards [hCB] with ω he
+  rintro ⟨T,Q,hQ,hF⟩
+  refine ⟨T,Q,?_,?_⟩
+  · simpa only [he] using hQ
+  · simpa only [he] using hF
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory Complex
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+ 
+
+theorem IsCapacityTwoWitness.forward_flow_survives
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (Q : ℝ → ℂ)
+    (hpos : ∀ t ∈ Icc 0 (T:ℝ), 0 < (Q t).im)
+    (hd : ∀ t ∈ Icc 0 (T:ℝ), HasDerivWithinAt Q
+      (2 / (Q t - (U t.toNNReal : ℂ))) (Icc 0 (T:ℝ)) t) :
+    Q 0 ∈ survivingDomain γ T ∧ G T (Q 0) = Q T := by
+  have hc : ContinuousOn (fun t => (Q t).im) (Icc 0 (T:ℝ)) :=
+    Complex.continuous_im.comp_continuousOn (fun t ht => (hd t ht).continuousWithinAt)
+  obtain ⟨t₀,ht₀,hmin⟩ := (isCompact_Icc : IsCompact (Icc 0 (T:ℝ))).exists_isMinOn
+    ⟨0,le_rfl,T.coe_nonneg⟩ hc
+  let a := (Q t₀).im
+  have ha : 0 < a := hpos t₀ ht₀
+  have hbound (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : a ≤ (Q t).im := hmin ht
+  have hmap : MapsTo (fun t : ℝ => (T:ℝ)-t) (Icc 0 (T:ℝ)) (Icc 0 (T:ℝ)) := by
+    intro t ht; constructor <;> linarith [ht.1,ht.2]
+  have hrev (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) :
+      HasDerivWithinAt (fun s : ℝ => Q ((T:ℝ)-s))
+        (reverseField a (U ((T:ℝ)-t).toNNReal) (Q ((T:ℝ)-t))) (Icc 0 (T:ℝ)) t := by
+    have he := (hd _ (hmap ht)).scomp t
+      ((hasDerivAt_id t).const_sub (T:ℝ)).hasDerivWithinAt hmap
+    simpa only [Function.comp_def,neg_smul,one_smul,reverseField,
+      reverseCenter_eq (hbound _ (hmap ht)),div_eq_mul_inv,neg_mul] using he
+  have he := h.reverse_flow_identifies T (Q T) ha
+    (hbound T ⟨T.coe_nonneg,le_rfl⟩) (fun s => Q ((T:ℝ)-s))
+    (by simp only [sub_zero]) hrev
+  simp only [sub_self] at he
+  have hlive : F T (Q T) ∈ survivingDomain γ T :=
+    h.2.2.2.2.2.2.1 T (hpos T ⟨T.coe_nonneg,le_rfl⟩)
+  rw [← he] at hlive
+  refine ⟨hlive,?_⟩
+  rw [he]
+  exact h.2.2.2.2.2.2.2.2.1 T (Q T) (hpos T ⟨T.coe_nonneg,le_rfl⟩)
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/SwallowLyapunov.lean b/lean/OAI/Probability/SLE/SwallowLyapunov.lean
index d45d0fe5d141118238fb7c2eb57dfd3511d038a2..36eaf2238e7b0c167d9400994eb69e7a5f5d5c9d 100644
--- a/lean/OAI/Probability/SLE/SwallowLyapunov.lean
+++ b/lean/OAI/Probability/SLE/SwallowLyapunov.lean
@@ -1,742 +1,742 @@
-import Mathlib
-import OAI.Probability.SLE.FiniteRadiusStop
-
-namespace OAI
-
-/-! Swallow Lyapunov for chordal SLE. -/
-
-section
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-noncomputable def radialLogPower (κ β : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  Real.exp ((β/2)*Real.log (greenLogNormSq κ q x))
-noncomputable def heightLogPower (β : ℝ) (q : ℂ × ℝ) : ℝ := Real.exp (β*q.1.im)
-noncomputable def swallowLyapunov (κ β C : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  radialLogPower κ β q x + C*heightLogPower β q
-
-lemma radialLogPower_contDiff (κ β : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => radialLogPower κ β zz.1 zz.2) := by
-  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.re) :=
-    Complex.reCLM.contDiff.comp contDiff_fst.fst
-  have him : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.im) :=
-    Complex.imCLM.contDiff.comp contDiff_fst.fst
-  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => greenLogNormSq κ zz.1 zz.2) :=
-    ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add ((contDiff_const.mul him).exp)
-  exact (contDiff_const.mul (hN.log (fun zz => (greenLogNormSq_pos κ zz.1 zz.2).ne'))).exp
-
-lemma heightLogPower_contDiff (β : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => heightLogPower β zz.1) :=
-  (contDiff_const.mul (Complex.imCLM.contDiff.comp contDiff_fst.fst)).exp
-
-lemma swallowLyapunov_contDiff (κ β C : ℝ) :
-    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => swallowLyapunov κ β C zz.1 zz.2) :=
-  (radialLogPower_contDiff κ β).add (contDiff_const.mul (heightLogPower_contDiff β))
-
-lemma radialLogPower_directional (κ β : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q v =
-      radialLogPower κ β q x * β *
-        ((q.1.re-Real.sqrt κ*x)*v.1.re + Real.exp (2*q.1.im)*v.1.im) /
-          greenLogNormSq κ q x := by
-  have hzre : HasDerivAt (fun t : ℝ => (q+t • v).1.re) v.1.re 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
-  have hzim : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
-      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
-  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q+t • v) x)
-      (2*(q.1.re-Real.sqrt κ*x)*v.1.re + 2*Real.exp (2*q.1.im)*v.1.im) 0 := by
-    convert ((hzre.sub_const (Real.sqrt κ*x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
-    simp only [zero_smul, add_zero, Nat.cast_ofNat]
-    ring
-  have hcalc := ((hN.log (by simpa only [zero_smul, add_zero] using
-    (greenLogNormSq_pos κ q x).ne')).const_mul (β/2)).exp
-  have hline : HasDerivAt (fun t : ℝ => radialLogPower κ β (q+t • v) x)
-      (radialLogPower κ β q x * β *
-        ((q.1.re-Real.sqrt κ*x)*v.1.re+Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x) 0 := by
-    convert hcalc using 1 <;> try rfl
-    simp only [zero_smul, add_zero]
-    dsimp only [radialLogPower]
-    ring
-  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => radialLogPower κ β r x) :=
-    (radialLogPower_contDiff κ β).comp (contDiff_id.prodMk contDiff_const)
-  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
-    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
-  have hfd : HasFDerivAt (fun r : ℂ × ℝ => radialLogPower κ β r x)
-      (fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q) (q+(0 : ℝ) • v) := by
-    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
-  have hchain := hfd.comp_hasDerivAt 0 haff
-  simpa only [Function.comp_def] using hchain.unique hline
-
-lemma radialLogPower_spatial_deriv (κ β : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    HasDerivAt (radialLogPower κ β q)
-      (-radialLogPower κ β q x*β*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x) x := by
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
-    simp
-  convert ((hN.log (greenLogNormSq_pos κ q x).ne').const_mul (β/2)).exp using 1 <;> try rfl
-  dsimp only [radialLogPower]
-  ring
-
-lemma radialLogPower_spatial_second (κ β : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
-    iteratedDeriv 2 (radialLogPower κ β q) x =
-      radialLogPower κ β q x*κ*β*
-        ((β-1)*(q.1.re-Real.sqrt κ*x)^2+Real.exp (2*q.1.im))/(greenLogNormSq κ q x)^2 := by
-  have hd : deriv (radialLogPower κ β q) = fun x =>
-      -radialLogPower κ β q x*β*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x :=
-    funext fun x => (radialLogPower_spatial_deriv κ β q x).deriv
-  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
-  have hN : HasDerivAt (greenLogNormSq κ q)
-      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
-    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
-    simp
-  have hs := ((((radialLogPower_spatial_deriv κ β q x).neg.mul_const β).mul_const (Real.sqrt κ)).mul hX).div hN
-    (greenLogNormSq_pos κ q x).ne'
-  rw [show (2 : ℕ) = 1+1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
-  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, Pi.neg_apply, id_eq] at hs
-  change HasDerivAt (fun y => -radialLogPower κ β q y * β * Real.sqrt κ *
-    (q.1.re-Real.sqrt κ*y)/greenLogNormSq κ q y) _ x at hs
-  rw [hs.deriv]
-  have hn := (greenLogNormSq_pos κ q x).ne'
-  dsimp only [greenLogNormSq] at hn ⊢
-  generalize radialLogPower κ β q x = f
-  generalize q.1.re-Real.sqrt κ*x = X at hn ⊢
-  generalize Real.exp (2*q.1.im) = Y at hn ⊢
-  have hk := Real.sq_sqrt hκ
-  generalize Real.sqrt κ = r at *
-  rw [← hk]
-  field_simp [hn]
-  ring
-
-lemma radialLogPower_generator (κ β : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q
-        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-      (1/2 : ℝ)*iteratedDeriv 2 (radialLogPower κ β q) x =
-      β*radialLogPower κ β q x*
-        ((2+κ*(β-1)/2)*(q.1.re-Real.sqrt κ*x)^2+(κ/2-2)*Real.exp (2*q.1.im)) /
-          (greenLogNormSq κ q x)^2 := by
-  rw [radialLogPower_directional, radialLogPower_spatial_second κ β hκ]
-  obtain ⟨hre,him,_⟩ := physicalLogLoewnerField_components κ q x
-  rw [hre,him]
-  ring
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma heightLogPower_directional (β : ℝ) (q v : ℂ × ℝ) :
-    fderiv ℝ (heightLogPower β) q v = heightLogPower β q * β * v.1.im := by
-  have hzim : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
-    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im,
-      smul_eq_mul, id_eq, one_mul] using
-      ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
-  have hline : HasDerivAt (fun t : ℝ => heightLogPower β (q+t • v))
-      (heightLogPower β q*β*v.1.im) 0 := by
-    convert (hzim.const_mul β).exp using 1 <;> try rfl
-    simp only [zero_smul, add_zero, heightLogPower]
-    ring
-  have hcd : ContDiff ℝ (⊤ : ℕ∞) (heightLogPower β) :=
-    (contDiff_const.mul (Complex.imCLM.contDiff.comp contDiff_fst)).exp
-  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
-    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
-  have hfd : HasFDerivAt (heightLogPower β) (fderiv ℝ (heightLogPower β) q)
-      (q+(0 : ℝ) • v) := by
-    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
-  have hc := hfd.comp_hasDerivAt 0 haff
-  simpa only [Function.comp_def] using hc.unique hline
-
-noncomputable def swallowGenerator (κ β C : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
-  β*radialLogPower κ β q x*
-    ((2+κ*(β-1)/2)*(q.1.re-Real.sqrt κ*x)^2+(κ/2-2)*Real.exp (2*q.1.im)) /
-      (greenLogNormSq κ q x)^2 - 2*C*β*heightLogPower β q/greenLogNormSq κ q x
-
-lemma swallowLyapunov_generator (κ β C : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun r : ℂ × ℝ => swallowLyapunov κ β C r x) q
-        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-      (1/2 : ℝ)*iteratedDeriv 2 (swallowLyapunov κ β C q) x =
-      swallowGenerator κ β C q x := by
-  have hr : DifferentiableAt ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q :=
-    (((radialLogPower_contDiff κ β).comp (contDiff_id.prodMk contDiff_const)).differentiable
-      (by simp)) q
-  have hh : DifferentiableAt ℝ (heightLogPower β) q :=
-    ((show ContDiff ℝ (⊤ : ℕ∞) (heightLogPower β) from
-      (contDiff_const.mul (Complex.imCLM.contDiff.comp contDiff_fst)).exp).differentiable (by simp)) q
-  have hd : fderiv ℝ (fun r : ℂ × ℝ => swallowLyapunov κ β C r x) q =
-      fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q +
-        C • fderiv ℝ (heightLogPower β) q := by
-    exact (hr.hasFDerivAt.add (hh.hasFDerivAt.const_mul C)).fderiv
-  have h2 : iteratedDeriv 2 (swallowLyapunov κ β C q) x =
-      iteratedDeriv 2 (radialLogPower κ β q) x := by
-    unfold swallowLyapunov
-    simp_rw [add_comm]
-    exact iteratedDeriv_const_add (by decide) _
-  rw [hd, h2, add_apply, smul_apply,
-    smul_eq_mul, heightLogPower_directional]
-  have he := radialLogPower_generator κ β hκ q x
-  obtain ⟨_, him, _⟩ := physicalLogLoewnerField_components κ q x
-  rw [him]
-  unfold swallowGenerator
-  calc
-    _ = (fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q
-        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
-        (1/2 : ℝ)*iteratedDeriv 2 (radialLogPower κ β q) x) -
-        2*C*β*heightLogPower β q/greenLogNormSq κ q x := by ring
-    _ = _ := by rw [he]
-
-lemma radialLogPower_pos (κ β : ℝ) (q : ℂ × ℝ) (x : ℝ) :
-    0 < radialLogPower κ β q x := Real.exp_pos _
-lemma heightLogPower_pos (β : ℝ) (q : ℂ × ℝ) :
-    0 < heightLogPower β q := Real.exp_pos _
-
- 
-
-lemma height_radial_ratio (κ β : ℝ) (_hβ0 : 0 ≤ β) (hβ2 : β ≤ 2)
-    (q : ℂ × ℝ) (x : ℝ) :
-    radialLogPower κ β q x * Real.exp (2*q.1.im) ≤
-      heightLogPower β q * greenLogNormSq κ q x := by
-  have hN := greenLogNormSq_pos κ q x
-  have hle : Real.exp (2*q.1.im) ≤ greenLogNormSq κ q x := by
-    unfold greenLogNormSq
-    nlinarith [sq_nonneg (q.1.re-Real.sqrt κ*x)]
-  have hlog : 2*q.1.im ≤ Real.log (greenLogNormSq κ q x) := by
-    simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos _) hle
-  unfold radialLogPower heightLogPower
-  rw [← Real.exp_add, ← Real.exp_log hN, ← Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  simp only [Real.log_exp]
-  nlinarith [mul_nonneg (sub_nonneg.mpr hβ2) (sub_nonneg.mpr hlog)]
-
-lemma swallowGenerator_negative (κ : ℝ) (hκ : 4 < κ) (q : ℂ × ℝ) (x : ℝ) :
-    swallowGenerator κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q x ≤
-      -((κ-4)/(2*κ))*((κ-4)/4)*radialLogPower κ ((κ-4)/(2*κ)) q x /
-        greenLogNormSq κ q x := by
-  let β := (κ-4)/(2*κ)
-  have hk : 0 < κ := by linarith
-  have hβ : 0 < β := by dsimp [β]; positivity
-  have hβ2 : β ≤ 2 := by dsimp [β]; apply (div_le_iff₀ (by positivity)).mpr; linarith
-  have hratio := height_radial_ratio κ β hβ.le hβ2 q x
-  have hn := greenLogNormSq_pos κ q x
-  have hc : 2+κ*(β-1)/2 = -(κ-4)/4 := by dsimp [β]; field_simp; ring
-  change swallowGenerator κ β (3*(κ-4)/8) q x ≤ -β*((κ-4)/4)*radialLogPower κ β q x /
-    greenLogNormSq κ q x
-  unfold swallowGenerator
-  rw [hc]
-  apply (mul_le_mul_iff_left₀ (sq_pos_of_pos hn)).mp
-  field_simp [hn.ne']
-  dsimp only [greenLogNormSq] at hratio ⊢
-  nlinarith [mul_nonneg (mul_nonneg hβ.le (sub_nonneg.mpr hκ.le)) (sub_nonneg.mpr hratio)]
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-structure RadialLocalization
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (R : ℝ) (T : ℝ≥0) where
-  τ : Ω → ℝ≥0
-  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-    (fun ω => (τ ω : WithTop ℝ≥0))
-  bounded : ∀ ω, τ ω ≤ T
-  inside : ∀ ω t, t ≤ τ ω → greenLogNormSq κ (c.Q t ω) (B t ω) ≤ R^2 ∧ lo ≤ (c.Q t ω).1.im
-  before : ∀ ω t, t < τ ω → greenLogNormSq κ (c.Q t ω) (B t ω) < R^2 ∧ lo < (c.Q t ω).1.im
-  boundary : ∀ ω, τ ω < T → greenLogNormSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = R^2 ∨
-    (c.Q (τ ω) ω).1.im = lo
-
-lemma greenLogNormSq_continuous (κ : ℝ) :
-    Continuous (fun v : (ℂ × ℝ) × ℝ => greenLogNormSq κ v.1 v.2) := by
-  unfold greenLogNormSq
-  fun_prop
-
-lemma exists_radialLocalization
-    {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀) (R : ℝ) (T : ℝ≥0)
-    (hN : greenLogNormSq κ q₀ 0 ≤ R^2) (hlo : lo ≤ q₀.1.im) :
-    Nonempty (RadialLocalization c R T) := by
-  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
-  let D (t : ℝ≥0) (ω : Ω) := min (R^2-greenLogNormSq κ (c.Q t ω) (B t ω)) ((c.Q t ω).1.im-lo)
-  have hBa : Adapted f B :=
-    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
-  have hDa : Adapted f D := by
-    intro t
-    have hQ := c.adapted t
-    let : MeasurableSpace Ω := f t
-    exact (measurable_const.sub ((greenLogNormSq_continuous κ).measurable.comp
-      (hQ.prodMk (hBa t)))).min ((Complex.continuous_im.measurable.comp hQ.fst).sub_const lo)
-  have hDc (ω : Ω) : Continuous (fun t => D t ω) := by
-    have hQ := (c.lipschitz ω).continuous.comp NNReal.continuous_coe
-    exact (continuous_const.sub ((greenLogNormSq_continuous κ).comp (hQ.prodMk (hBc ω)))).min
-      ((Complex.continuous_im.comp hQ.fst).sub continuous_const)
-  have hD0 (ω : Ω) : 0 ≤ D 0 ω := by
-    simpa only [D, NNReal.coe_zero, c.initial, hB0, le_min_iff, sub_nonneg] using And.intro hN hlo
-  obtain ⟨τ,hτ,hb,hin,hpre,hbd⟩ := first_margin_stopping f D hDa hDc hD0 T
-  refine ⟨⟨τ,hτ,hb,?_,?_,?_⟩⟩
-  · intro ω t ht
-    simpa only [D, le_min_iff, sub_nonneg] using hin ω t ht
-  · intro ω t ht
-    simpa only [D, lt_min_iff, sub_pos] using hpre ω t ht
-  · intro ω ht
-    have hh := hbd ω ht
-    dsimp only [D] at hh
-    rw [min_eq_iff] at hh
-    rcases hh with ⟨h,_⟩ | ⟨h,_⟩
-    · exact Or.inl (sub_eq_zero.mp h).symm
-    · exact Or.inr (sub_eq_zero.mp h)
-
-namespace RadialLocalization
-variable {Ω : Type*} [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
-lemma height_le_initial (a : RadialLocalization c R T) (ω : Ω) (t : ℝ≥0) (ht : t ≤ a.τ ω) :
-    (c.Q t ω).1.im ≤ q₀.1.im := by
-  have hd : ∀ s ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => c.Q s ω)
-      (clippedLogLoewnerField lo hi hle (Real.sqrt κ*B s.toNNReal ω) (c.Q s ω))
-      (Icc 0 (T : ℝ)) s := by
-    intro s hs
-    exact (c.derivative ω s hs.1).mono (fun s hs => hs.1)
-  have hanti : AntitoneOn (fun s : ℝ => (c.Q s ω).1.im) (Icc 0 (T : ℝ)) :=
-    clipped_log_height_antitone (U := fun u => Real.sqrt κ*B u ω)
-      (Q := fun s => c.Q s ω) T hd
-  have htT : (t : ℝ) ≤ T := NNReal.coe_le_coe.mpr (ht.trans (a.bounded ω))
-  have hh : (c.Q t ω).1.im ≤ (c.Q 0 ω).1.im :=
-    hanti ⟨le_rfl,T.coe_nonneg⟩ ⟨t.coe_nonneg,htT⟩ t.coe_nonneg
-  rw [c.initial ω] at hh
-  exact hh
-
-lemma chart_identification (a : RadialLocalization c R T)
-    (hBc : ∀ ω, Continuous (fun t => B t ω)) {z : ℂ} (hz : z ∈ upperHalfPlane)
-    (hq : q₀ = initialLogCoordinates z) (hhi : q₀.1.im ≤ hi)
-    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F) :
-    z ∈ survivingDomain γ (a.τ ω) ∧
-      ∀ t : ℝ≥0, t ≤ a.τ ω → c.Q t ω = logLoewnerCoordinates G z t := by
-  have hd (s : ℝ) (hs : s ∈ Icc 0 (a.τ ω : ℝ)) := (c.derivative ω s hs.1).mono
-    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun s hs => hs.1)
-  obtain ⟨hl,he⟩ := clipped_log_identification_of_endpoint ((hBc ω).const_mul _) hw
-    (a.τ ω) z hz lo hi hle (fun t => c.Q t ω) ((c.initial ω).trans hq) hd
-    (a.inside ω (a.τ ω) le_rfl).2
-    (by simpa [hq, initialLogCoordinates] using hhi)
-  exact ⟨hl,fun t ht => he ⟨t.coe_nonneg,by exact_mod_cast ht⟩⟩
-end RadialLocalization
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma radialLogPower_le (κ β : ℝ) (hβ : 0 ≤ β) (q : ℂ × ℝ) (x R : ℝ)
-    (_hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
-    radialLogPower κ β q x ≤ Real.exp (β*Real.log R) := by
-  have hl := Real.log_le_log (greenLogNormSq_pos κ q x) hN
-  rw [Real.log_pow] at hl
-  norm_num only [Nat.cast_ofNat] at hl
-  unfold radialLogPower
-  apply Real.exp_le_exp.mpr
-  nlinarith [mul_nonneg hβ (sub_nonneg.mpr hl)]
-
-lemma heightLogPower_le (κ β : ℝ) (hβ : 0 ≤ β) (q : ℂ × ℝ) (x R : ℝ)
-    (_hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
-    heightLogPower β q ≤ Real.exp (β*Real.log R) := by
-  have hy : Real.exp (2*q.1.im) ≤ R^2 := by
-    unfold greenLogNormSq at hN
-    nlinarith [sq_nonneg (q.1.re-Real.sqrt κ*x)]
-  have hl := Real.log_le_log (Real.exp_pos _) hy
-  rw [Real.log_exp,Real.log_pow] at hl
-  norm_num only [Nat.cast_ofNat] at hl
-  unfold heightLogPower
-  apply Real.exp_le_exp.mpr
-  nlinarith [mul_nonneg hβ (sub_nonneg.mpr hl)]
-
-lemma radialLogPower_div_lower (κ β : ℝ) (hβ : β ≤ 2) (q : ℂ × ℝ) (x R : ℝ)
-    (_hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
-    Real.exp ((β-2)*Real.log R) ≤ radialLogPower κ β q x/greenLogNormSq κ q x := by
-  have hl := Real.log_le_log (greenLogNormSq_pos κ q x) hN
-  rw [Real.log_pow] at hl
-  norm_num only [Nat.cast_ofNat] at hl
-  have he : radialLogPower κ β q x/greenLogNormSq κ q x =
-      Real.exp ((β/2-1)*Real.log (greenLogNormSq κ q x)) := by
-    unfold radialLogPower
-    conv_lhs => rhs; rw [← Real.exp_log (greenLogNormSq_pos κ q x)]
-    rw [← Real.exp_sub]
-    congr 1
-    ring
-  rw [he]
-  apply Real.exp_le_exp.mpr
-  nlinarith [mul_nonneg (sub_nonneg.mpr hβ) (sub_nonneg.mpr hl)]
-
-lemma swallowGenerator_uniform_negative (κ : ℝ) (hκ : 4 < κ) (q : ℂ × ℝ) (x R : ℝ)
-    (hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
-    swallowGenerator κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q x ≤
-      -(((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) := by
-  let β := (κ-4)/(2*κ)
-  have hk : 0 < κ := lt_trans (by norm_num) hκ
-  have hβ : 0 < β := by dsimp [β]; positivity
-  have hβ2 : β ≤ 2 := by dsimp [β]; apply (div_le_iff₀ (by positivity)).mpr; linarith
-  have hr := radialLogPower_div_lower κ β hβ2 q x R hR hN
-  have ha : 0 ≤ β*((κ-4)/4) := by positivity
-  have hm := mul_le_mul_of_nonneg_left hr ha
-  have hg := swallowGenerator_negative κ hκ q x
-  change swallowGenerator κ β (3*(κ-4)/8) q x ≤ -(β*((κ-4)/4)*Real.exp ((β-2)*Real.log R))
-  dsimp only [β] at hm ⊢
-  apply hg.trans
-  convert neg_le_neg hm using 1
-  ring
-
-lemma norm_driver_of_radial_bound (κ R K : ℝ) (hκ : 0 < κ) (hR : 0 ≤ R)
-    (q : ℂ × ℝ) (x : ℝ) (hq : ‖q‖ ≤ K) (hN : greenLogNormSq κ q x ≤ R^2) :
-    |x| ≤ (K+R)/Real.sqrt κ := by
-  have hX : |q.1.re-Real.sqrt κ*x| ≤ R := by
-    rw [abs_le]
-    have hp := Real.exp_pos (2*q.1.im)
-    unfold greenLogNormSq at hN
-    constructor <;> nlinarith
-  have hqre : |q.1.re| ≤ K := (Complex.abs_re_le_norm _).trans ((norm_fst_le q).trans hq)
-  have hx : |Real.sqrt κ*x| ≤ K+R := by
-    calc
-      _ = |q.1.re-(q.1.re-Real.sqrt κ*x)| := by congr 1; ring
-      _ ≤ |q.1.re|+|q.1.re-Real.sqrt κ*x| := abs_sub _ _
-      _ ≤ K+R := add_le_add hqre hX
-  rw [abs_mul,abs_of_pos (Real.sqrt_pos.2 hκ)] at hx
-  exact (le_div_iff₀ (Real.sqrt_pos.2 hκ)).mpr (by simpa only [mul_comm] using hx)
-
-lemma swallowLyapunov_nonneg (κ β C : ℝ) (hC : 0 ≤ C) (q : ℂ × ℝ) (x : ℝ) :
-    0 ≤ swallowLyapunov κ β C q x := by
-  exact add_nonneg (radialLogPower_pos κ β q x).le (mul_nonneg hC (heightLogPower_pos β q).le)
-
-lemma radialLogPower_at_boundary (κ β : ℝ) (q : ℂ × ℝ) (x R : ℝ)
-    (hN : greenLogNormSq κ q x = R^2) :
-    radialLogPower κ β q x = Real.exp (β*Real.log R) := by
-  unfold radialLogPower
-  rw [hN, Real.log_pow]
-  congr 1
-  norm_num only [Nat.cast_ofNat]
-  ring
-
-lemma continuous_swallowGenerator (κ β C : ℝ) :
-    Continuous (fun v : (ℂ × ℝ) × ℝ => swallowGenerator κ β C v.1 v.2) := by
-  have hN : Continuous (fun v : (ℂ × ℝ) × ℝ => greenLogNormSq κ v.1 v.2) := by
-    unfold greenLogNormSq
-    fun_prop
-  have hf := (radialLogPower_contDiff κ β).continuous
-  have hh := (heightLogPower_contDiff β).continuous
-  unfold swallowGenerator
-  apply Continuous.sub
-  · apply Continuous.div
-    · exact (continuous_const.mul hf).mul (by fun_prop)
-    · exact hN.pow 2
-    · intro v; exact pow_ne_zero 2 (greenLogNormSq_pos κ v.1 v.2).ne'
-  · exact (continuous_const.mul hh).div hN (fun v => (greenLogNormSq_pos κ v.1 v.2).ne')
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma swallowLyapunov_pair_directional (κ p a : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
-    fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => swallowLyapunov κ p a zz.1 zz.2) (q, x) (v, 0) =
-      fderiv ℝ (fun r : ℂ × ℝ => swallowLyapunov κ p a r x) q v := by
-  have hd := ((swallowLyapunov_contDiff κ p a).differentiable (by simp)) (q, x)
-  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
-  have hc := (hd.hasFDerivAt.comp q hp).fderiv
-  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) hc
-  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
-    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
-
- 
-
-lemma exists_compact_swallow_field (κ p a R : ℝ) (hR : 0 ≤ R) :
-    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
-      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = swallowLyapunov κ p a q x) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R →
-        fderiv ℝ φ (q, x) = fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => swallowLyapunov κ p a zz.1 zz.2) (q, x)) ∧
-      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
-        iteratedDeriv 2 (swallowLyapunov κ p a q) x) := by
-  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
-  let φ (zz : (ℂ × ℝ) × ℝ) : ℝ := b zz * swallowLyapunov κ p a zz.1 zz.2
-  have hnear (zz : (ℂ × ℝ) × ℝ) (hp : ‖zz‖ ≤ R) :
-      φ =ᶠ[𝓝 zz] (fun r : (ℂ × ℝ) × ℝ => swallowLyapunov κ p a r.1 r.2) := by
-    have hball : zz ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
-      simp only [Metric.mem_ball, dist_zero_right]
-      change ‖zz‖ < R + 1
-      linarith
-    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with r hr
-    simp only [φ, hr, Pi.one_apply, one_mul]
-  refine ⟨φ, b.contDiff.mul (swallowLyapunov_contDiff κ p a), b.hasCompactSupport.mul_right,
-    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
-  · intro q x h
-    exact (hnear (q, x) h).fderiv_eq
-  · intro q x h
-    have he := (hnear (q, x) h).comp_tendsto
-      ((continuous_const.prodMk continuous_id).tendsto x)
-    exact he.iteratedDeriv_eq 2
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma bounded_cumulative_grid_error
-    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
-    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hC : 0 ≤ C)
-    (hb : ∀ u ω, |G u ω| ≤ C) (τ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) :
-    |(∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), G u ω) -
-       (∫ u in (0 : ℝ)..(τ ω : ℝ), G u ω)| ≤ C * (1 / ((n : ℝ) + 1)) := by
-  have hle : (τ ω : ℝ) ≤ stopGrid n (τ ω) := (stopGrid_bounds n (τ ω)).1
-  have hs := intervalIntegral.integral_add_adjacent_intervals
-    (bounded_time_section_intervalIntegrable 0 (τ ω) (τ ω).coe_nonneg G hG C hb ω)
-    (bounded_time_section_intervalIntegrable (τ ω) (stopGrid n (τ ω)) hle G hG C hb ω)
-  rw [← hs, add_sub_cancel_left]
-  have hh := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (τ ω : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := C) (f := fun u => G u ω)
-    (fun u _ => by simpa only [Real.norm_eq_abs] using hb u ω)
-  rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hle)] at hh
-  refine hh.trans (mul_le_mul_of_nonneg_left ?_ hC)
-  have hu : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ) + 1 / ((n : ℝ) + 1) := by
-    exact_mod_cast (stopGrid_bounds n (τ ω)).2
-  linarith
-
- 
-
-theorem brownian_time_field_stopped_dynkin
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (hm : ∀ t, Measurable (B t))
-    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
-    (hF : SmoothTimeBrownianField B hm K F a)
-    (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B
-      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
-    (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P) = (∫ ω, F 0 ω (B 0 ω) ∂P) +
-      ∫ ω, (∫ u in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a u ω) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have hτm := stopping_time_measurable_nnreal hτ
-  let V (n : ℕ) (ω : Ω) := F (stopGrid n (τ ω)) ω (B (stopGrid n (τ ω)) ω)
-  let H (n : ℕ) (ω : Ω) := ∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), brownianFieldDrift B F a u ω
-  let H₀ (ω : Ω) := ∫ u in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a u ω
-  have hVm (n : ℕ) : Measurable (V n) :=
-    measurable_discrete_eval (fun k ω => F (stopGridTime n k) ω (B (stopGridTime n k) ω))
-      (fun k => hF.eval_measurable _ 0 (by decide) (hm _)) (stopGridIndex_measurable hτm n)
-  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) := hF.bound _ ω _ 0 (by decide)
-  have hVi (n : ℕ) : Integrable (V n) P :=
-    Integrable.of_bound (hVm n).aestronglyMeasurable (K : ℝ)
-      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-  have hGm := hF.generator_measurable hBm
-  have hHjoint := cumulative_integral_measurable _ hGm
-  have hHm (n : ℕ) : Measurable (H n) :=
-    measurable_discrete_eval (fun k ω => ∫ u in (0 : ℝ)..(stopGridTime n k : ℝ), brownianFieldDrift B F a u ω)
-      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
-  have hH₀m : Measurable H₀ := hHjoint.comp (hτm.prodMk measurable_id)
-  have hH₀b (ω : Ω) : |H₀ ω| ≤ 2 * (K : ℝ) * T := by
-    have hh := intervalIntegral.norm_integral_le_of_norm_le_const
-      (a := (0 : ℝ)) (b := (τ ω : ℝ)) (C := 2 * (K : ℝ))
-      (f := fun u => brownianFieldDrift B F a u ω)
-      (fun u _ => by simpa only [Real.norm_eq_abs] using hF.generator_bound u ω)
-    simp only [Real.norm_eq_abs, sub_zero, abs_of_nonneg (τ ω).coe_nonneg] at hh
-    exact hh.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hb ω) (by positivity))
-  have hH₀i : Integrable H₀ P := Integrable.of_bound hH₀m.aestronglyMeasurable _
-    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hH₀b ω)
-  have hdiff (n : ℕ) (ω : Ω) : |H n ω - H₀ ω| ≤ (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) :=
-    bounded_cumulative_grid_error _ hGm _ (by positivity) hF.generator_bound τ n ω
-  have hHb (n : ℕ) (ω : Ω) : |H n ω| ≤ 2 * (K : ℝ) * (T + 1) := by
-    have hr : 1 / ((n : ℝ) + 1) ≤ 1 := by
-      apply (div_le_one (by positivity)).mpr
-      have hn := Nat.cast_nonneg (α := ℝ) n
-      linarith
-    have hd := (hdiff n ω).trans (mul_le_mul_of_nonneg_left hr (by positivity))
-    have ht : |H n ω| ≤ |H n ω - H₀ ω| + |H₀ ω| := by
-      calc _ = |(H n ω - H₀ ω) + H₀ ω| := by rw [sub_add_cancel]
-           _ ≤ _ := abs_add_le _ _
-    nlinarith [hH₀b ω]
-  have hHi (n : ℕ) : Integrable (H n) P := Integrable.of_bound (hHm n).aestronglyMeasurable _
-    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
-  have hmean (n : ℕ) : (∫ ω, V n ω ∂P) = (∫ ω, F 0 ω (B 0 ω) ∂P) + ∫ ω, H n ω ∂P := by
-    have hh := brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
-    change (∫ ω, V n ω - H n ω ∂P) = _ at hh
-    rw [integral_sub (hVi n) (hHi n)] at hh
-    linarith
-  have hlimV : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
-      (𝓝 (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => (K : ℝ))
-      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
-    exact Eventually.of_forall fun ω => (hF.path_continuous hBc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω))
-  have hlimH : Tendsto (fun n => ∫ ω, H n ω ∂P) atTop (𝓝 (∫ ω, H₀ ω ∂P)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => 2 * (K : ℝ) * (T + 1))
-      (fun n => (hHm n).aestronglyMeasurable) (integrable_const _)
-      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
-    apply Eventually.of_forall
-    intro ω
-    apply tendsto_iff_dist_tendsto_zero.mpr
-    apply squeeze_zero (fun n => dist_nonneg) (fun n => by simpa only [Real.dist_eq] using hdiff n ω)
-    simpa only [mul_zero] using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2 * (K : ℝ))
-  have he := hlimH.const_add (∫ ω, F 0 ω (B 0 ω) ∂P)
-  simp_rw [← hmean] at he
-  exact tendsto_nhds_unique hlimV he
-
-end SLEExactGauge
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma compact_swallow_drift
-    (κ p α : ℝ) (hκ : 0 ≤ κ) (lo hi : ℝ) (hle : lo ≤ hi)
-    (φ : (ℂ × ℝ) × ℝ → ℝ) (q : ℂ × ℝ) (x : ℝ)
-    (hφD : fderiv ℝ φ (q, x) =
-      fderiv ℝ (fun r : (ℂ × ℝ) × ℝ => swallowLyapunov κ p α r.1 r.2) (q, x))
-    (hφ2 : iteratedDeriv 2 (fun y => φ (q, y)) x =
-      iteratedDeriv 2 (swallowLyapunov κ p α q) x)
-    (hq : q.1.im ∈ Icc lo hi) :
-    fderiv ℝ φ (q, x)
-        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * x) q, 0) +
-      (1 / 2 : ℝ) * iteratedDeriv 2 (fun y => φ (q, y)) x =
-        swallowGenerator κ p α q x := by
-  rw [hφD, hφ2, swallowLyapunov_pair_directional,
-    clippedLogLoewnerField_eq_physical lo hi hle _ _ hq]
-  exact swallowLyapunov_generator κ p α hκ q x
-
- 
-
-theorem ClippedLogSolution.stopped_swallow_dynkin
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
-    (hB : IsBrownianReal B P)
-    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
-    (hBc : ∀ ω, Continuous (fun t => B t ω))
-    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
-    (c : ClippedLogSolution B hm κ lo hi hle q₀)
-    (hκ : 0 ≤ κ) (p α : ℝ)
-    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => (τ ω : WithTop ℝ≥0)))
-    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
-    (hinside : ∀ ω t, t ≤ τ ω →
-      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
-    Integrable (fun ω => swallowLyapunov κ p α (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
-      (∫ ω, swallowLyapunov κ p α (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
-        swallowLyapunov κ p α q₀ 0 +
-        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ),
-          swallowGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_swallow_field κ p α R hR
-  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
-  let V (t : ℝ) (ω : Ω) :=
-    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
-  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
-    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
-      measurable_snd)).const_mul _
-  have hpair : Measurable (fun p : ℝ × Ω =>
-      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
-  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
-    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
-      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
-    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
-  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
-    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
-  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
-    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
-    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
-  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
-  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
-  have hgen (ω : Ω) (t : ℝ) (ht : t ∈ Icc 0 (τ ω : ℝ)) :
-      brownianFieldDrift B F a t ω =
-        swallowGenerator κ p α (c.Q t ω) (B t.toNNReal ω) := by
-    have ht' : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
-    obtain ⟨hball, hlo, hhi⟩ := hinside ω t.toNNReal ht'
-    have he : (t.toNNReal : ℝ) = t := Real.coe_toNNReal t ht.1
-    change fderiv ℝ φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
-      (V (t.toNNReal : ℝ) ω, 0) + _ = _
-    dsimp only [V]
-    rw [Real.toNNReal_coe]
-    have hh := compact_swallow_drift κ p α hκ lo hi hle φ _ _
-      (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
-    simpa only [F, he] using hh
-  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
-  have hτm := stopping_time_measurable_nnreal hτ
-  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
-  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
-    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
-  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
-    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
-  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
-    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
-    exact Eventually.of_forall fun ω => by
-      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
-        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
-  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
-      swallowLyapunov κ p α (c.Q (τ ω) ω) (B (τ ω) ω) :=
-    hφeq _ _ (hinside ω (τ ω) le_rfl).1
-  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
-  calc
-    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
-      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
-    _ = (∫ ω, F 0 ω (B 0 ω) ∂P) +
-        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a t ω) ∂P := hopt
-    _ = _ := by
-      congr 1
-      · calc
-          _ = ∫ _, swallowLyapunov κ p α q₀ 0 ∂P := by
-            apply integral_congr_ae
-            filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
-            change φ (c.Q 0 ω, B 0 ω) = _
-            have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
-              simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
-            rw [hφeq _ _ hh, c.initial ω, hB0]
-          _ = _ := by simp
-      · apply integral_congr_ae
-        exact Eventually.of_forall fun ω => intervalIntegral.integral_congr
-          (fun t ht => hgen ω t (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using ht))
-
-end SLEExactGauge
-end
-
- 
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.FiniteRadiusStop
+
+namespace OAI
+
+/-! Swallow Lyapunov for chordal SLE. -/
+
+section
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+noncomputable def radialLogPower (κ β : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  Real.exp ((β/2)*Real.log (greenLogNormSq κ q x))
+noncomputable def heightLogPower (β : ℝ) (q : ℂ × ℝ) : ℝ := Real.exp (β*q.1.im)
+noncomputable def swallowLyapunov (κ β C : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  radialLogPower κ β q x + C*heightLogPower β q
+
+lemma radialLogPower_contDiff (κ β : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => radialLogPower κ β zz.1 zz.2) := by
+  have hre : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.re) :=
+    Complex.reCLM.contDiff.comp contDiff_fst.fst
+  have him : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => zz.1.1.im) :=
+    Complex.imCLM.contDiff.comp contDiff_fst.fst
+  have hN : ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => greenLogNormSq κ zz.1 zz.2) :=
+    ((hre.sub (contDiff_const.mul contDiff_snd)).pow 2).add ((contDiff_const.mul him).exp)
+  exact (contDiff_const.mul (hN.log (fun zz => (greenLogNormSq_pos κ zz.1 zz.2).ne'))).exp
+
+lemma heightLogPower_contDiff (β : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => heightLogPower β zz.1) :=
+  (contDiff_const.mul (Complex.imCLM.contDiff.comp contDiff_fst.fst)).exp
+
+lemma swallowLyapunov_contDiff (κ β C : ℝ) :
+    ContDiff ℝ (⊤ : ℕ∞) (fun zz : (ℂ × ℝ) × ℝ => swallowLyapunov κ β C zz.1 zz.2) :=
+  (radialLogPower_contDiff κ β).add (contDiff_const.mul (heightLogPower_contDiff β))
+
+lemma radialLogPower_directional (κ β : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q v =
+      radialLogPower κ β q x * β *
+        ((q.1.re-Real.sqrt κ*x)*v.1.re + Real.exp (2*q.1.im)*v.1.im) /
+          greenLogNormSq κ q x := by
+  have hzre : HasDerivAt (fun t : ℝ => (q+t • v).1.re) v.1.re 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_re, Complex.smul_re, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.re).const_add q.1.re
+  have hzim : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im, smul_eq_mul,
+      id_eq, one_mul] using ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
+  have hN : HasDerivAt (fun t : ℝ => greenLogNormSq κ (q+t • v) x)
+      (2*(q.1.re-Real.sqrt κ*x)*v.1.re + 2*Real.exp (2*q.1.im)*v.1.im) 0 := by
+    convert ((hzre.sub_const (Real.sqrt κ*x)).pow 2).add ((hzim.const_mul 2).exp) using 1 <;> try rfl
+    simp only [zero_smul, add_zero, Nat.cast_ofNat]
+    ring
+  have hcalc := ((hN.log (by simpa only [zero_smul, add_zero] using
+    (greenLogNormSq_pos κ q x).ne')).const_mul (β/2)).exp
+  have hline : HasDerivAt (fun t : ℝ => radialLogPower κ β (q+t • v) x)
+      (radialLogPower κ β q x * β *
+        ((q.1.re-Real.sqrt κ*x)*v.1.re+Real.exp (2*q.1.im)*v.1.im)/greenLogNormSq κ q x) 0 := by
+    convert hcalc using 1 <;> try rfl
+    simp only [zero_smul, add_zero]
+    dsimp only [radialLogPower]
+    ring
+  have hcd : ContDiff ℝ (⊤ : ℕ∞) (fun r : ℂ × ℝ => radialLogPower κ β r x) :=
+    (radialLogPower_contDiff κ β).comp (contDiff_id.prodMk contDiff_const)
+  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
+    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
+  have hfd : HasFDerivAt (fun r : ℂ × ℝ => radialLogPower κ β r x)
+      (fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q) (q+(0 : ℝ) • v) := by
+    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
+  have hchain := hfd.comp_hasDerivAt 0 haff
+  simpa only [Function.comp_def] using hchain.unique hline
+
+lemma radialLogPower_spatial_deriv (κ β : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    HasDerivAt (radialLogPower κ β q)
+      (-radialLogPower κ β q x*β*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x) x := by
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
+    simp
+  convert ((hN.log (greenLogNormSq_pos κ q x).ne').const_mul (β/2)).exp using 1 <;> try rfl
+  dsimp only [radialLogPower]
+  ring
+
+lemma radialLogPower_spatial_second (κ β : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
+    iteratedDeriv 2 (radialLogPower κ β q) x =
+      radialLogPower κ β q x*κ*β*
+        ((β-1)*(q.1.re-Real.sqrt κ*x)^2+Real.exp (2*q.1.im))/(greenLogNormSq κ q x)^2 := by
+  have hd : deriv (radialLogPower κ β q) = fun x =>
+      -radialLogPower κ β q x*β*Real.sqrt κ*(q.1.re-Real.sqrt κ*x)/greenLogNormSq κ q x :=
+    funext fun x => (radialLogPower_spatial_deriv κ β q x).deriv
+  have hX := (hasDerivAt_const x q.1.re).sub ((hasDerivAt_id x).const_mul (Real.sqrt κ))
+  have hN : HasDerivAt (greenLogNormSq κ q)
+      (2*(q.1.re-Real.sqrt κ*x)*(-Real.sqrt κ)) x := by
+    convert (hX.pow 2).add_const (Real.exp (2*q.1.im)) using 1 <;> try rfl
+    simp
+  have hs := ((((radialLogPower_spatial_deriv κ β q x).neg.mul_const β).mul_const (Real.sqrt κ)).mul hX).div hN
+    (greenLogNormSq_pos κ q x).ne'
+  rw [show (2 : ℕ) = 1+1 from rfl, iteratedDeriv_succ, iteratedDeriv_one, hd]
+  dsimp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, Pi.neg_apply, id_eq] at hs
+  change HasDerivAt (fun y => -radialLogPower κ β q y * β * Real.sqrt κ *
+    (q.1.re-Real.sqrt κ*y)/greenLogNormSq κ q y) _ x at hs
+  rw [hs.deriv]
+  have hn := (greenLogNormSq_pos κ q x).ne'
+  dsimp only [greenLogNormSq] at hn ⊢
+  generalize radialLogPower κ β q x = f
+  generalize q.1.re-Real.sqrt κ*x = X at hn ⊢
+  generalize Real.exp (2*q.1.im) = Y at hn ⊢
+  have hk := Real.sq_sqrt hκ
+  generalize Real.sqrt κ = r at *
+  rw [← hk]
+  field_simp [hn]
+  ring
+
+lemma radialLogPower_generator (κ β : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q
+        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+      (1/2 : ℝ)*iteratedDeriv 2 (radialLogPower κ β q) x =
+      β*radialLogPower κ β q x*
+        ((2+κ*(β-1)/2)*(q.1.re-Real.sqrt κ*x)^2+(κ/2-2)*Real.exp (2*q.1.im)) /
+          (greenLogNormSq κ q x)^2 := by
+  rw [radialLogPower_directional, radialLogPower_spatial_second κ β hκ]
+  obtain ⟨hre,him,_⟩ := physicalLogLoewnerField_components κ q x
+  rw [hre,him]
+  ring
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma heightLogPower_directional (β : ℝ) (q v : ℂ × ℝ) :
+    fderiv ℝ (heightLogPower β) q v = heightLogPower β q * β * v.1.im := by
+  have hzim : HasDerivAt (fun t : ℝ => (q+t • v).1.im) v.1.im 0 := by
+    simpa only [Prod.fst_add, Prod.smul_fst, Complex.add_im, Complex.smul_im,
+      smul_eq_mul, id_eq, one_mul] using
+      ((hasDerivAt_id (0 : ℝ)).mul_const v.1.im).const_add q.1.im
+  have hline : HasDerivAt (fun t : ℝ => heightLogPower β (q+t • v))
+      (heightLogPower β q*β*v.1.im) 0 := by
+    convert (hzim.const_mul β).exp using 1 <;> try rfl
+    simp only [zero_smul, add_zero, heightLogPower]
+    ring
+  have hcd : ContDiff ℝ (⊤ : ℕ∞) (heightLogPower β) :=
+    (contDiff_const.mul (Complex.imCLM.contDiff.comp contDiff_fst)).exp
+  have haff : HasDerivAt (fun t : ℝ => q+t • v) v 0 := by
+    simpa only [id_eq, one_smul] using ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add q
+  have hfd : HasFDerivAt (heightLogPower β) (fderiv ℝ (heightLogPower β) q)
+      (q+(0 : ℝ) • v) := by
+    simpa using ((hcd.differentiable (by simp)) q).hasFDerivAt
+  have hc := hfd.comp_hasDerivAt 0 haff
+  simpa only [Function.comp_def] using hc.unique hline
+
+noncomputable def swallowGenerator (κ β C : ℝ) (q : ℂ × ℝ) (x : ℝ) : ℝ :=
+  β*radialLogPower κ β q x*
+    ((2+κ*(β-1)/2)*(q.1.re-Real.sqrt κ*x)^2+(κ/2-2)*Real.exp (2*q.1.im)) /
+      (greenLogNormSq κ q x)^2 - 2*C*β*heightLogPower β q/greenLogNormSq κ q x
+
+lemma swallowLyapunov_generator (κ β C : ℝ) (hκ : 0 ≤ κ) (q : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun r : ℂ × ℝ => swallowLyapunov κ β C r x) q
+        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+      (1/2 : ℝ)*iteratedDeriv 2 (swallowLyapunov κ β C q) x =
+      swallowGenerator κ β C q x := by
+  have hr : DifferentiableAt ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q :=
+    (((radialLogPower_contDiff κ β).comp (contDiff_id.prodMk contDiff_const)).differentiable
+      (by simp)) q
+  have hh : DifferentiableAt ℝ (heightLogPower β) q :=
+    ((show ContDiff ℝ (⊤ : ℕ∞) (heightLogPower β) from
+      (contDiff_const.mul (Complex.imCLM.contDiff.comp contDiff_fst)).exp).differentiable (by simp)) q
+  have hd : fderiv ℝ (fun r : ℂ × ℝ => swallowLyapunov κ β C r x) q =
+      fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q +
+        C • fderiv ℝ (heightLogPower β) q := by
+    exact (hr.hasFDerivAt.add (hh.hasFDerivAt.const_mul C)).fderiv
+  have h2 : iteratedDeriv 2 (swallowLyapunov κ β C q) x =
+      iteratedDeriv 2 (radialLogPower κ β q) x := by
+    unfold swallowLyapunov
+    simp_rw [add_comm]
+    exact iteratedDeriv_const_add (by decide) _
+  rw [hd, h2, add_apply, smul_apply,
+    smul_eq_mul, heightLogPower_directional]
+  have he := radialLogPower_generator κ β hκ q x
+  obtain ⟨_, him, _⟩ := physicalLogLoewnerField_components κ q x
+  rw [him]
+  unfold swallowGenerator
+  calc
+    _ = (fderiv ℝ (fun r : ℂ × ℝ => radialLogPower κ β r x) q
+        (physicalLogLoewnerField (Real.sqrt κ*x) q) +
+        (1/2 : ℝ)*iteratedDeriv 2 (radialLogPower κ β q) x) -
+        2*C*β*heightLogPower β q/greenLogNormSq κ q x := by ring
+    _ = _ := by rw [he]
+
+lemma radialLogPower_pos (κ β : ℝ) (q : ℂ × ℝ) (x : ℝ) :
+    0 < radialLogPower κ β q x := Real.exp_pos _
+lemma heightLogPower_pos (β : ℝ) (q : ℂ × ℝ) :
+    0 < heightLogPower β q := Real.exp_pos _
+
+ 
+
+lemma height_radial_ratio (κ β : ℝ) (_hβ0 : 0 ≤ β) (hβ2 : β ≤ 2)
+    (q : ℂ × ℝ) (x : ℝ) :
+    radialLogPower κ β q x * Real.exp (2*q.1.im) ≤
+      heightLogPower β q * greenLogNormSq κ q x := by
+  have hN := greenLogNormSq_pos κ q x
+  have hle : Real.exp (2*q.1.im) ≤ greenLogNormSq κ q x := by
+    unfold greenLogNormSq
+    nlinarith [sq_nonneg (q.1.re-Real.sqrt κ*x)]
+  have hlog : 2*q.1.im ≤ Real.log (greenLogNormSq κ q x) := by
+    simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos _) hle
+  unfold radialLogPower heightLogPower
+  rw [← Real.exp_add, ← Real.exp_log hN, ← Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  simp only [Real.log_exp]
+  nlinarith [mul_nonneg (sub_nonneg.mpr hβ2) (sub_nonneg.mpr hlog)]
+
+lemma swallowGenerator_negative (κ : ℝ) (hκ : 4 < κ) (q : ℂ × ℝ) (x : ℝ) :
+    swallowGenerator κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q x ≤
+      -((κ-4)/(2*κ))*((κ-4)/4)*radialLogPower κ ((κ-4)/(2*κ)) q x /
+        greenLogNormSq κ q x := by
+  let β := (κ-4)/(2*κ)
+  have hk : 0 < κ := by linarith
+  have hβ : 0 < β := by dsimp [β]; positivity
+  have hβ2 : β ≤ 2 := by dsimp [β]; apply (div_le_iff₀ (by positivity)).mpr; linarith
+  have hratio := height_radial_ratio κ β hβ.le hβ2 q x
+  have hn := greenLogNormSq_pos κ q x
+  have hc : 2+κ*(β-1)/2 = -(κ-4)/4 := by dsimp [β]; field_simp; ring
+  change swallowGenerator κ β (3*(κ-4)/8) q x ≤ -β*((κ-4)/4)*radialLogPower κ β q x /
+    greenLogNormSq κ q x
+  unfold swallowGenerator
+  rw [hc]
+  apply (mul_le_mul_iff_left₀ (sq_pos_of_pos hn)).mp
+  field_simp [hn.ne']
+  dsimp only [greenLogNormSq] at hratio ⊢
+  nlinarith [mul_nonneg (mul_nonneg hβ.le (sub_nonneg.mpr hκ.le)) (sub_nonneg.mpr hratio)]
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+structure RadialLocalization
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (R : ℝ) (T : ℝ≥0) where
+  τ : Ω → ℝ≥0
+  stopping : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+    (fun ω => (τ ω : WithTop ℝ≥0))
+  bounded : ∀ ω, τ ω ≤ T
+  inside : ∀ ω t, t ≤ τ ω → greenLogNormSq κ (c.Q t ω) (B t ω) ≤ R^2 ∧ lo ≤ (c.Q t ω).1.im
+  before : ∀ ω t, t < τ ω → greenLogNormSq κ (c.Q t ω) (B t ω) < R^2 ∧ lo < (c.Q t ω).1.im
+  boundary : ∀ ω, τ ω < T → greenLogNormSq κ (c.Q (τ ω) ω) (B (τ ω) ω) = R^2 ∨
+    (c.Q (τ ω) ω).1.im = lo
+
+lemma greenLogNormSq_continuous (κ : ℝ) :
+    Continuous (fun v : (ℂ × ℝ) × ℝ => greenLogNormSq κ v.1 v.2) := by
+  unfold greenLogNormSq
+  fun_prop
+
+lemma exists_radialLocalization
+    {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0)
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀) (R : ℝ) (T : ℝ≥0)
+    (hN : greenLogNormSq κ q₀ 0 ≤ R^2) (hlo : lo ≤ q₀.1.im) :
+    Nonempty (RadialLocalization c R T) := by
+  let f := Filtration.natural B (fun t => (hm t).stronglyMeasurable)
+  let D (t : ℝ≥0) (ω : Ω) := min (R^2-greenLogNormSq κ (c.Q t ω) (B t ω)) ((c.Q t ω).1.im-lo)
+  have hBa : Adapted f B :=
+    (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).adapted
+  have hDa : Adapted f D := by
+    intro t
+    have hQ := c.adapted t
+    let : MeasurableSpace Ω := f t
+    exact (measurable_const.sub ((greenLogNormSq_continuous κ).measurable.comp
+      (hQ.prodMk (hBa t)))).min ((Complex.continuous_im.measurable.comp hQ.fst).sub_const lo)
+  have hDc (ω : Ω) : Continuous (fun t => D t ω) := by
+    have hQ := (c.lipschitz ω).continuous.comp NNReal.continuous_coe
+    exact (continuous_const.sub ((greenLogNormSq_continuous κ).comp (hQ.prodMk (hBc ω)))).min
+      ((Complex.continuous_im.comp hQ.fst).sub continuous_const)
+  have hD0 (ω : Ω) : 0 ≤ D 0 ω := by
+    simpa only [D, NNReal.coe_zero, c.initial, hB0, le_min_iff, sub_nonneg] using And.intro hN hlo
+  obtain ⟨τ,hτ,hb,hin,hpre,hbd⟩ := first_margin_stopping f D hDa hDc hD0 T
+  refine ⟨⟨τ,hτ,hb,?_,?_,?_⟩⟩
+  · intro ω t ht
+    simpa only [D, le_min_iff, sub_nonneg] using hin ω t ht
+  · intro ω t ht
+    simpa only [D, lt_min_iff, sub_pos] using hpre ω t ht
+  · intro ω ht
+    have hh := hbd ω ht
+    dsimp only [D] at hh
+    rw [min_eq_iff] at hh
+    rcases hh with ⟨h,_⟩ | ⟨h,_⟩
+    · exact Or.inl (sub_eq_zero.mp h).symm
+    · exact Or.inr (sub_eq_zero.mp h)
+
+namespace RadialLocalization
+variable {Ω : Type*} [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    {c : ClippedLogSolution B hm κ lo hi hle q₀} {R : ℝ} {T : ℝ≥0}
+lemma height_le_initial (a : RadialLocalization c R T) (ω : Ω) (t : ℝ≥0) (ht : t ≤ a.τ ω) :
+    (c.Q t ω).1.im ≤ q₀.1.im := by
+  have hd : ∀ s ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => c.Q s ω)
+      (clippedLogLoewnerField lo hi hle (Real.sqrt κ*B s.toNNReal ω) (c.Q s ω))
+      (Icc 0 (T : ℝ)) s := by
+    intro s hs
+    exact (c.derivative ω s hs.1).mono (fun s hs => hs.1)
+  have hanti : AntitoneOn (fun s : ℝ => (c.Q s ω).1.im) (Icc 0 (T : ℝ)) :=
+    clipped_log_height_antitone (U := fun u => Real.sqrt κ*B u ω)
+      (Q := fun s => c.Q s ω) T hd
+  have htT : (t : ℝ) ≤ T := NNReal.coe_le_coe.mpr (ht.trans (a.bounded ω))
+  have hh : (c.Q t ω).1.im ≤ (c.Q 0 ω).1.im :=
+    hanti ⟨le_rfl,T.coe_nonneg⟩ ⟨t.coe_nonneg,htT⟩ t.coe_nonneg
+  rw [c.initial ω] at hh
+  exact hh
+
+lemma chart_identification (a : RadialLocalization c R T)
+    (hBc : ∀ ω, Continuous (fun t => B t ω)) {z : ℂ} (hz : z ∈ upperHalfPlane)
+    (hq : q₀ = initialLogCoordinates z) (hhi : q₀.1.im ≤ hi)
+    (ω : Ω) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) γ G F) :
+    z ∈ survivingDomain γ (a.τ ω) ∧
+      ∀ t : ℝ≥0, t ≤ a.τ ω → c.Q t ω = logLoewnerCoordinates G z t := by
+  have hd (s : ℝ) (hs : s ∈ Icc 0 (a.τ ω : ℝ)) := (c.derivative ω s hs.1).mono
+    (show Icc (0 : ℝ) (a.τ ω : ℝ) ⊆ Ici 0 from fun s hs => hs.1)
+  obtain ⟨hl,he⟩ := clipped_log_identification_of_endpoint ((hBc ω).const_mul _) hw
+    (a.τ ω) z hz lo hi hle (fun t => c.Q t ω) ((c.initial ω).trans hq) hd
+    (a.inside ω (a.τ ω) le_rfl).2
+    (by simpa [hq, initialLogCoordinates] using hhi)
+  exact ⟨hl,fun t ht => he ⟨t.coe_nonneg,by exact_mod_cast ht⟩⟩
+end RadialLocalization
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma radialLogPower_le (κ β : ℝ) (hβ : 0 ≤ β) (q : ℂ × ℝ) (x R : ℝ)
+    (_hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
+    radialLogPower κ β q x ≤ Real.exp (β*Real.log R) := by
+  have hl := Real.log_le_log (greenLogNormSq_pos κ q x) hN
+  rw [Real.log_pow] at hl
+  norm_num only [Nat.cast_ofNat] at hl
+  unfold radialLogPower
+  apply Real.exp_le_exp.mpr
+  nlinarith [mul_nonneg hβ (sub_nonneg.mpr hl)]
+
+lemma heightLogPower_le (κ β : ℝ) (hβ : 0 ≤ β) (q : ℂ × ℝ) (x R : ℝ)
+    (_hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
+    heightLogPower β q ≤ Real.exp (β*Real.log R) := by
+  have hy : Real.exp (2*q.1.im) ≤ R^2 := by
+    unfold greenLogNormSq at hN
+    nlinarith [sq_nonneg (q.1.re-Real.sqrt κ*x)]
+  have hl := Real.log_le_log (Real.exp_pos _) hy
+  rw [Real.log_exp,Real.log_pow] at hl
+  norm_num only [Nat.cast_ofNat] at hl
+  unfold heightLogPower
+  apply Real.exp_le_exp.mpr
+  nlinarith [mul_nonneg hβ (sub_nonneg.mpr hl)]
+
+lemma radialLogPower_div_lower (κ β : ℝ) (hβ : β ≤ 2) (q : ℂ × ℝ) (x R : ℝ)
+    (_hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
+    Real.exp ((β-2)*Real.log R) ≤ radialLogPower κ β q x/greenLogNormSq κ q x := by
+  have hl := Real.log_le_log (greenLogNormSq_pos κ q x) hN
+  rw [Real.log_pow] at hl
+  norm_num only [Nat.cast_ofNat] at hl
+  have he : radialLogPower κ β q x/greenLogNormSq κ q x =
+      Real.exp ((β/2-1)*Real.log (greenLogNormSq κ q x)) := by
+    unfold radialLogPower
+    conv_lhs => rhs; rw [← Real.exp_log (greenLogNormSq_pos κ q x)]
+    rw [← Real.exp_sub]
+    congr 1
+    ring
+  rw [he]
+  apply Real.exp_le_exp.mpr
+  nlinarith [mul_nonneg (sub_nonneg.mpr hβ) (sub_nonneg.mpr hl)]
+
+lemma swallowGenerator_uniform_negative (κ : ℝ) (hκ : 4 < κ) (q : ℂ × ℝ) (x R : ℝ)
+    (hR : 0 < R) (hN : greenLogNormSq κ q x ≤ R^2) :
+    swallowGenerator κ ((κ-4)/(2*κ)) (3*(κ-4)/8) q x ≤
+      -(((κ-4)/(2*κ))*((κ-4)/4)*Real.exp ((((κ-4)/(2*κ))-2)*Real.log R)) := by
+  let β := (κ-4)/(2*κ)
+  have hk : 0 < κ := lt_trans (by norm_num) hκ
+  have hβ : 0 < β := by dsimp [β]; positivity
+  have hβ2 : β ≤ 2 := by dsimp [β]; apply (div_le_iff₀ (by positivity)).mpr; linarith
+  have hr := radialLogPower_div_lower κ β hβ2 q x R hR hN
+  have ha : 0 ≤ β*((κ-4)/4) := by positivity
+  have hm := mul_le_mul_of_nonneg_left hr ha
+  have hg := swallowGenerator_negative κ hκ q x
+  change swallowGenerator κ β (3*(κ-4)/8) q x ≤ -(β*((κ-4)/4)*Real.exp ((β-2)*Real.log R))
+  dsimp only [β] at hm ⊢
+  apply hg.trans
+  convert neg_le_neg hm using 1
+  ring
+
+lemma norm_driver_of_radial_bound (κ R K : ℝ) (hκ : 0 < κ) (hR : 0 ≤ R)
+    (q : ℂ × ℝ) (x : ℝ) (hq : ‖q‖ ≤ K) (hN : greenLogNormSq κ q x ≤ R^2) :
+    |x| ≤ (K+R)/Real.sqrt κ := by
+  have hX : |q.1.re-Real.sqrt κ*x| ≤ R := by
+    rw [abs_le]
+    have hp := Real.exp_pos (2*q.1.im)
+    unfold greenLogNormSq at hN
+    constructor <;> nlinarith
+  have hqre : |q.1.re| ≤ K := (Complex.abs_re_le_norm _).trans ((norm_fst_le q).trans hq)
+  have hx : |Real.sqrt κ*x| ≤ K+R := by
+    calc
+      _ = |q.1.re-(q.1.re-Real.sqrt κ*x)| := by congr 1; ring
+      _ ≤ |q.1.re|+|q.1.re-Real.sqrt κ*x| := abs_sub _ _
+      _ ≤ K+R := add_le_add hqre hX
+  rw [abs_mul,abs_of_pos (Real.sqrt_pos.2 hκ)] at hx
+  exact (le_div_iff₀ (Real.sqrt_pos.2 hκ)).mpr (by simpa only [mul_comm] using hx)
+
+lemma swallowLyapunov_nonneg (κ β C : ℝ) (hC : 0 ≤ C) (q : ℂ × ℝ) (x : ℝ) :
+    0 ≤ swallowLyapunov κ β C q x := by
+  exact add_nonneg (radialLogPower_pos κ β q x).le (mul_nonneg hC (heightLogPower_pos β q).le)
+
+lemma radialLogPower_at_boundary (κ β : ℝ) (q : ℂ × ℝ) (x R : ℝ)
+    (hN : greenLogNormSq κ q x = R^2) :
+    radialLogPower κ β q x = Real.exp (β*Real.log R) := by
+  unfold radialLogPower
+  rw [hN, Real.log_pow]
+  congr 1
+  norm_num only [Nat.cast_ofNat]
+  ring
+
+lemma continuous_swallowGenerator (κ β C : ℝ) :
+    Continuous (fun v : (ℂ × ℝ) × ℝ => swallowGenerator κ β C v.1 v.2) := by
+  have hN : Continuous (fun v : (ℂ × ℝ) × ℝ => greenLogNormSq κ v.1 v.2) := by
+    unfold greenLogNormSq
+    fun_prop
+  have hf := (radialLogPower_contDiff κ β).continuous
+  have hh := (heightLogPower_contDiff β).continuous
+  unfold swallowGenerator
+  apply Continuous.sub
+  · apply Continuous.div
+    · exact (continuous_const.mul hf).mul (by fun_prop)
+    · exact hN.pow 2
+    · intro v; exact pow_ne_zero 2 (greenLogNormSq_pos κ v.1 v.2).ne'
+  · exact (continuous_const.mul hh).div hN (fun v => (greenLogNormSq_pos κ v.1 v.2).ne')
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma swallowLyapunov_pair_directional (κ p a : ℝ) (q v : ℂ × ℝ) (x : ℝ) :
+    fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => swallowLyapunov κ p a zz.1 zz.2) (q, x) (v, 0) =
+      fderiv ℝ (fun r : ℂ × ℝ => swallowLyapunov κ p a r x) q v := by
+  have hd := ((swallowLyapunov_contDiff κ p a).differentiable (by simp)) (q, x)
+  have hp := (hasFDerivAt_id (𝕜 := ℝ) q).prodMk (hasFDerivAt_const (𝕜 := ℝ) (E := ℂ × ℝ) x q)
+  have hc := (hd.hasFDerivAt.comp q hp).fderiv
+  have he := congrArg (fun D : (ℂ × ℝ) →L[ℝ] ℝ => D v) hc
+  simpa only [Function.comp_def, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
+    ContinuousLinearMap.id_apply, id_eq, zero_apply] using he.symm
+
+ 
+
+lemma exists_compact_swallow_field (κ p a R : ℝ) (hR : 0 ≤ R) :
+    ∃ φ : (ℂ × ℝ) × ℝ → ℝ,
+      ContDiff ℝ (⊤ : ℕ∞) φ ∧ HasCompactSupport φ ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → φ (q, x) = swallowLyapunov κ p a q x) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R →
+        fderiv ℝ φ (q, x) = fderiv ℝ (fun zz : (ℂ × ℝ) × ℝ => swallowLyapunov κ p a zz.1 zz.2) (q, x)) ∧
+      (∀ q x, ‖(q, x)‖ ≤ R → iteratedDeriv 2 (fun y => φ (q, y)) x =
+        iteratedDeriv 2 (swallowLyapunov κ p a q) x) := by
+  let b : ContDiffBump (0 : (ℂ × ℝ) × ℝ) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let φ (zz : (ℂ × ℝ) × ℝ) : ℝ := b zz * swallowLyapunov κ p a zz.1 zz.2
+  have hnear (zz : (ℂ × ℝ) × ℝ) (hp : ‖zz‖ ≤ R) :
+      φ =ᶠ[𝓝 zz] (fun r : (ℂ × ℝ) × ℝ => swallowLyapunov κ p a r.1 r.2) := by
+    have hball : zz ∈ Metric.ball (0 : (ℂ × ℝ) × ℝ) b.rIn := by
+      simp only [Metric.mem_ball, dist_zero_right]
+      change ‖zz‖ < R + 1
+      linarith
+    filter_upwards [b.eventuallyEq_one_of_mem_ball hball] with r hr
+    simp only [φ, hr, Pi.one_apply, one_mul]
+  refine ⟨φ, b.contDiff.mul (swallowLyapunov_contDiff κ p a), b.hasCompactSupport.mul_right,
+    (fun q x h => (hnear (q, x) h).eq_of_nhds), ?_, ?_⟩
+  · intro q x h
+    exact (hnear (q, x) h).fderiv_eq
+  · intro q x h
+    have he := (hnear (q, x) h).comp_tendsto
+      ((continuous_const.prodMk continuous_id).tendsto x)
+    exact he.iteratedDeriv_eq 2
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma bounded_cumulative_grid_error
+    {Ω : Type*} [MeasurableSpace Ω] (G : ℝ → Ω → ℝ)
+    (hG : Measurable (Function.uncurry G)) (C : ℝ) (hC : 0 ≤ C)
+    (hb : ∀ u ω, |G u ω| ≤ C) (τ : Ω → ℝ≥0) (n : ℕ) (ω : Ω) :
+    |(∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), G u ω) -
+       (∫ u in (0 : ℝ)..(τ ω : ℝ), G u ω)| ≤ C * (1 / ((n : ℝ) + 1)) := by
+  have hle : (τ ω : ℝ) ≤ stopGrid n (τ ω) := (stopGrid_bounds n (τ ω)).1
+  have hs := intervalIntegral.integral_add_adjacent_intervals
+    (bounded_time_section_intervalIntegrable 0 (τ ω) (τ ω).coe_nonneg G hG C hb ω)
+    (bounded_time_section_intervalIntegrable (τ ω) (stopGrid n (τ ω)) hle G hG C hb ω)
+  rw [← hs, add_sub_cancel_left]
+  have hh := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (τ ω : ℝ)) (b := (stopGrid n (τ ω) : ℝ)) (C := C) (f := fun u => G u ω)
+    (fun u _ => by simpa only [Real.norm_eq_abs] using hb u ω)
+  rw [Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr hle)] at hh
+  refine hh.trans (mul_le_mul_of_nonneg_left ?_ hC)
+  have hu : (stopGrid n (τ ω) : ℝ) ≤ (τ ω : ℝ) + 1 / ((n : ℝ) + 1) := by
+    exact_mod_cast (stopGrid_bounds n (τ ω)).2
+  linarith
+
+ 
+
+theorem brownian_time_field_stopped_dynkin
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (hm : ∀ t, Measurable (B t))
+    (hBm : Measurable (fun q : ℝ≥0 × Ω => B q.1 q.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    (K : ℝ≥0) (F a : ℝ≥0 → Ω → ℝ → ℝ)
+    (hF : SmoothTimeBrownianField B hm K F a)
+    (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B
+      (fun t => (hm t).stronglyMeasurable)) (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T) :
+    (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P) = (∫ ω, F 0 ω (B 0 ω) ∂P) +
+      ∫ ω, (∫ u in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a u ω) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have hτm := stopping_time_measurable_nnreal hτ
+  let V (n : ℕ) (ω : Ω) := F (stopGrid n (τ ω)) ω (B (stopGrid n (τ ω)) ω)
+  let H (n : ℕ) (ω : Ω) := ∫ u in (0 : ℝ)..(stopGrid n (τ ω) : ℝ), brownianFieldDrift B F a u ω
+  let H₀ (ω : Ω) := ∫ u in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a u ω
+  have hVm (n : ℕ) : Measurable (V n) :=
+    measurable_discrete_eval (fun k ω => F (stopGridTime n k) ω (B (stopGridTime n k) ω))
+      (fun k => hF.eval_measurable _ 0 (by decide) (hm _)) (stopGridIndex_measurable hτm n)
+  have hVb (n : ℕ) (ω : Ω) : |V n ω| ≤ (K : ℝ) := hF.bound _ ω _ 0 (by decide)
+  have hVi (n : ℕ) : Integrable (V n) P :=
+    Integrable.of_bound (hVm n).aestronglyMeasurable (K : ℝ)
+      (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+  have hGm := hF.generator_measurable hBm
+  have hHjoint := cumulative_integral_measurable _ hGm
+  have hHm (n : ℕ) : Measurable (H n) :=
+    measurable_discrete_eval (fun k ω => ∫ u in (0 : ℝ)..(stopGridTime n k : ℝ), brownianFieldDrift B F a u ω)
+      (fun k => hHjoint.comp measurable_prodMk_left) (stopGridIndex_measurable hτm n)
+  have hH₀m : Measurable H₀ := hHjoint.comp (hτm.prodMk measurable_id)
+  have hH₀b (ω : Ω) : |H₀ ω| ≤ 2 * (K : ℝ) * T := by
+    have hh := intervalIntegral.norm_integral_le_of_norm_le_const
+      (a := (0 : ℝ)) (b := (τ ω : ℝ)) (C := 2 * (K : ℝ))
+      (f := fun u => brownianFieldDrift B F a u ω)
+      (fun u _ => by simpa only [Real.norm_eq_abs] using hF.generator_bound u ω)
+    simp only [Real.norm_eq_abs, sub_zero, abs_of_nonneg (τ ω).coe_nonneg] at hh
+    exact hh.trans (mul_le_mul_of_nonneg_left (by exact_mod_cast hb ω) (by positivity))
+  have hH₀i : Integrable H₀ P := Integrable.of_bound hH₀m.aestronglyMeasurable _
+    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hH₀b ω)
+  have hdiff (n : ℕ) (ω : Ω) : |H n ω - H₀ ω| ≤ (2 * (K : ℝ)) * (1 / ((n : ℝ) + 1)) :=
+    bounded_cumulative_grid_error _ hGm _ (by positivity) hF.generator_bound τ n ω
+  have hHb (n : ℕ) (ω : Ω) : |H n ω| ≤ 2 * (K : ℝ) * (T + 1) := by
+    have hr : 1 / ((n : ℝ) + 1) ≤ 1 := by
+      apply (div_le_one (by positivity)).mpr
+      have hn := Nat.cast_nonneg (α := ℝ) n
+      linarith
+    have hd := (hdiff n ω).trans (mul_le_mul_of_nonneg_left hr (by positivity))
+    have ht : |H n ω| ≤ |H n ω - H₀ ω| + |H₀ ω| := by
+      calc _ = |(H n ω - H₀ ω) + H₀ ω| := by rw [sub_add_cancel]
+           _ ≤ _ := abs_add_le _ _
+    nlinarith [hH₀b ω]
+  have hHi (n : ℕ) : Integrable (H n) P := Integrable.of_bound (hHm n).aestronglyMeasurable _
+    (Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
+  have hmean (n : ℕ) : (∫ ω, V n ω ∂P) = (∫ ω, F 0 ω (B 0 ω) ∂P) + ∫ ω, H n ω ∂P := by
+    have hh := brownian_compensated_grid_integral hB hm hBm hF τ hτ T hb n
+    change (∫ ω, V n ω - H n ω ∂P) = _ at hh
+    rw [integral_sub (hVi n) (hHi n)] at hh
+    linarith
+  have hlimV : Tendsto (fun n => ∫ ω, V n ω ∂P) atTop
+      (𝓝 (∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => (K : ℝ))
+      (fun n => (hVm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hVb n ω)
+    exact Eventually.of_forall fun ω => (hF.path_continuous hBc ω).continuousAt.tendsto.comp (stopGrid_tendsto (τ ω))
+  have hlimH : Tendsto (fun n => ∫ ω, H n ω ∂P) atTop (𝓝 (∫ ω, H₀ ω ∂P)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => 2 * (K : ℝ) * (T + 1))
+      (fun n => (hHm n).aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hHb n ω)
+    apply Eventually.of_forall
+    intro ω
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun n => dist_nonneg) (fun n => by simpa only [Real.dist_eq] using hdiff n ω)
+    simpa only [mul_zero] using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2 * (K : ℝ))
+  have he := hlimH.const_add (∫ ω, F 0 ω (B 0 ω) ∂P)
+  simp_rw [← hmean] at he
+  exact tendsto_nhds_unique hlimV he
+
+end SLEExactGauge
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma compact_swallow_drift
+    (κ p α : ℝ) (hκ : 0 ≤ κ) (lo hi : ℝ) (hle : lo ≤ hi)
+    (φ : (ℂ × ℝ) × ℝ → ℝ) (q : ℂ × ℝ) (x : ℝ)
+    (hφD : fderiv ℝ φ (q, x) =
+      fderiv ℝ (fun r : (ℂ × ℝ) × ℝ => swallowLyapunov κ p α r.1 r.2) (q, x))
+    (hφ2 : iteratedDeriv 2 (fun y => φ (q, y)) x =
+      iteratedDeriv 2 (swallowLyapunov κ p α q) x)
+    (hq : q.1.im ∈ Icc lo hi) :
+    fderiv ℝ φ (q, x)
+        (clippedLogLoewnerField lo hi hle (Real.sqrt κ * x) q, 0) +
+      (1 / 2 : ℝ) * iteratedDeriv 2 (fun y => φ (q, y)) x =
+        swallowGenerator κ p α q x := by
+  rw [hφD, hφ2, swallowLyapunov_pair_directional,
+    clippedLogLoewnerField_eq_physical lo hi hle _ _ hq]
+  exact swallowLyapunov_generator κ p α hκ q x
+
+ 
+
+theorem ClippedLogSolution.stopped_swallow_dynkin
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} {hm : ∀ t, Measurable (B t)}
+    (hB : IsBrownianReal B P)
+    (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2))
+    (hBc : ∀ ω, Continuous (fun t => B t ω))
+    {κ lo hi : ℝ} {hle : lo ≤ hi} {q₀ : ℂ × ℝ}
+    (c : ClippedLogSolution B hm κ lo hi hle q₀)
+    (hκ : 0 ≤ κ) (p α : ℝ)
+    (R : ℝ) (hR : 0 ≤ R) (τ : Ω → ℝ≥0)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => (τ ω : WithTop ℝ≥0)))
+    (T : ℝ≥0) (hb : ∀ ω, τ ω ≤ T)
+    (hinside : ∀ ω t, t ≤ τ ω →
+      ‖(c.Q t ω, B t ω)‖ ≤ R ∧ lo ≤ (c.Q t ω).1.im ∧ (c.Q t ω).1.im ≤ hi) :
+    Integrable (fun ω => swallowLyapunov κ p α (c.Q (τ ω) ω) (B (τ ω) ω)) P ∧
+      (∫ ω, swallowLyapunov κ p α (c.Q (τ ω) ω) (B (τ ω) ω) ∂P) =
+        swallowLyapunov κ p α q₀ 0 +
+        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ),
+          swallowGenerator κ p α (c.Q t ω) (B t.toNNReal ω)) ∂P := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨φ, hφ, hφc, hφeq, hφD, hφ2⟩ := exists_compact_swallow_field κ p α R hR
+  obtain ⟨L₀, K₀, hL₀, _⟩ := clippedLogLoewnerField_bounds lo hi hle
+  let V (t : ℝ) (ω : Ω) :=
+    clippedLogLoewnerField lo hi hle (Real.sqrt κ * B t.toNNReal ω) (c.Q t ω)
+  have hBr : Measurable (fun p : ℝ × Ω => Real.sqrt κ * B p.1.toNNReal p.2) :=
+    (hBm.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk
+      measurable_snd)).const_mul _
+  have hpair : Measurable (fun p : ℝ × Ω =>
+      (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)) := hBr.prodMk c.measurable
+  have hVm : Measurable (fun p : ℝ × Ω => V p.1 p.2) := by
+    change Measurable ((fun p : ℝ × (ℂ × ℝ) => clippedLogLoewnerField lo hi hle p.1 p.2) ∘
+      (fun p : ℝ × Ω => (Real.sqrt κ * B p.1.toNNReal p.2, c.Q p.1 p.2)))
+    exact (continuous_clippedLogLoewnerField lo hi hle).measurable.comp hpair
+  have hVb (t : ℝ) (ω : Ω) : ‖V t ω‖ ≤ (max c.L L₀ : ℝ≥0) :=
+    (hL₀ _ _).trans (by exact_mod_cast le_max_right c.L L₀)
+  obtain ⟨K, hfield⟩ := compact_state_smooth_brownian_field B hm c.Q V (max c.L L₀)
+    c.measurable c.adapted (fun ω => (c.lipschitz ω).weaken (le_max_left _ _)) hVm hVb
+    (fun t ω ht => c.derivative ω t ht) φ hφ hφc
+  let F (t : ℝ≥0) (ω : Ω) (x : ℝ) := φ (c.Q t ω, x)
+  let a (t : ℝ≥0) (ω : Ω) (x : ℝ) := fderiv ℝ φ (c.Q t ω, x) (V t ω, 0)
+  have hgen (ω : Ω) (t : ℝ) (ht : t ∈ Icc 0 (τ ω : ℝ)) :
+      brownianFieldDrift B F a t ω =
+        swallowGenerator κ p α (c.Q t ω) (B t.toNNReal ω) := by
+    have ht' : t.toNNReal ≤ τ ω := Real.toNNReal_le_iff_le_coe.mpr ht.2
+    obtain ⟨hball, hlo, hhi⟩ := hinside ω t.toNNReal ht'
+    have he : (t.toNNReal : ℝ) = t := Real.coe_toNNReal t ht.1
+    change fderiv ℝ φ (c.Q (t.toNNReal : ℝ) ω, B t.toNNReal ω)
+      (V (t.toNNReal : ℝ) ω, 0) + _ = _
+    dsimp only [V]
+    rw [Real.toNNReal_coe]
+    have hh := compact_swallow_drift κ p α hκ lo hi hle φ _ _
+      (hφD _ _ hball) (hφ2 _ _ hball) ⟨hlo, hhi⟩
+    simpa only [F, he] using hh
+  have hopt := brownian_time_field_stopped_dynkin hB hm hBm hBc K F a hfield τ hτ T hb
+  have hτm := stopping_time_measurable_nnreal hτ
+  have hBmτ : Measurable (fun ω => B (τ ω) ω) := hBm.comp (hτm.prodMk measurable_id)
+  have hQmτ : Measurable (fun ω => c.Q (τ ω) ω) :=
+    c.measurable.comp ((NNReal.continuous_coe.measurable.comp hτm).prodMk measurable_id)
+  have hfm : Measurable (fun ω => F (τ ω) ω (B (τ ω) ω)) :=
+    hφ.continuous.measurable.comp (hQmτ.prodMk hBmτ)
+  have hfi : Integrable (fun ω => F (τ ω) ω (B (τ ω) ω)) P := by
+    apply Integrable.of_bound hfm.aestronglyMeasurable (K : ℝ)
+    exact Eventually.of_forall fun ω => by
+      simpa only [Real.norm_eq_abs, iteratedDeriv_zero] using
+        hfield.bound (τ ω) ω (B (τ ω) ω) 0 (by decide)
+  have hfeq (ω : Ω) : F (τ ω) ω (B (τ ω) ω) =
+      swallowLyapunov κ p α (c.Q (τ ω) ω) (B (τ ω) ω) :=
+    hφeq _ _ (hinside ω (τ ω) le_rfl).1
+  refine ⟨hfi.congr (Eventually.of_forall hfeq), ?_⟩
+  calc
+    _ = ∫ ω, F (τ ω) ω (B (τ ω) ω) ∂P :=
+      integral_congr_ae (Eventually.of_forall fun ω => (hfeq ω).symm)
+    _ = (∫ ω, F 0 ω (B 0 ω) ∂P) +
+        ∫ ω, (∫ t in (0 : ℝ)..(τ ω : ℝ), brownianFieldDrift B F a t ω) ∂P := hopt
+    _ = _ := by
+      congr 1
+      · calc
+          _ = ∫ _, swallowLyapunov κ p α q₀ 0 ∂P := by
+            apply integral_congr_ae
+            filter_upwards [hB.eval_zero_ae_eq_zero] with ω hB0
+            change φ (c.Q 0 ω, B 0 ω) = _
+            have hh : ‖(c.Q 0 ω, B 0 ω)‖ ≤ R := by
+              simpa only [NNReal.coe_zero] using (hinside ω 0 zero_le).1
+            rw [hφeq _ _ hh, c.initial ω, hB0]
+          _ = _ := by simp
+      · apply integral_congr_ae
+        exact Eventually.of_forall fun ω => intervalIntegral.integral_congr
+          (fun t ht => hgen ω t (by simpa only [uIcc_of_le (τ ω).coe_nonneg] using ht))
+
+end SLEExactGauge
+end
+
+ 
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/TraceMeasurability.lean b/lean/OAI/Probability/SLE/TraceMeasurability.lean
index 48fb5bf7ca1555bf7b22ba7292105917af025942..f230071b3014b596ff1a4a1ae4240aa0933de8ad 100644
--- a/lean/OAI/Probability/SLE/TraceMeasurability.lean
+++ b/lean/OAI/Probability/SLE/TraceMeasurability.lean
@@ -1,131 +1,131 @@
-import Mathlib
-import OAI.Probability.SLE.SausageSeedAll
-
-namespace OAI
-
-/-! Trace Measurability for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory
-open scoped Topology NNReal
-open Set Filter MeasureTheory Complex
-open MeasureTheory ProbabilityTheory Set Filter
-open Set Filter Metric Complex InnerProductSpace
-open scoped Topology Laplacian
-open Set Filter Metric Complex MeasureTheory
-open Set Filter Metric Complex MeasureTheory InnerProductSpace
-open scoped Topology Laplacian ENNReal
-open scoped Topology ENNReal
-open scoped Topology Laplacian RealInnerProductSpace
-open Set Filter Metric Complex
-open scoped Topology
-open Set Filter Metric
-open Set Filter MeasureTheory Metric
-open Set Filter Complex
-open scoped Topology ComplexConjugate NNReal
-open scoped Topology ENNReal NNReal ComplexConjugate
-open Set Filter Metric MeasureTheory Complex
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology ENNReal NNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open Set Filter
-open scoped Topology NNReal ENNReal ComplexConjugate
-open scoped ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-section
-open Set Filter MeasureTheory Metric TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance wholeTraceMeasurabilityPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance wholeTraceMeasurabilityPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_whole_trace_measure_measurable (h : ℝ → ℝ) (hh : IsGauge h)
-    (C : Set ℂ) (hC : IsClosed C) :
-    Measurable (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (range γ ∩ C)) := by
-  have heq : (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (range γ ∩ C)) =
-      fun γ : C(ℝ≥0,ℂ) => ⨆ n : ℕ, hausdorffGauge h (C ∩ (γ '' Icc 0 (n : ℝ≥0))) := by
-    funext γ
-    have hU : range γ ∩ C = ⋃ n : ℕ, C ∩ (γ '' Icc 0 (n : ℝ≥0)) := by
-      ext z
-      constructor
-      · rintro ⟨⟨t,rfl⟩,hc⟩
-        obtain ⟨n,hn⟩ := exists_nat_ge t
-        exact mem_iUnion.mpr ⟨n,hc,t,⟨zero_le,hn⟩,rfl⟩
-      · intro hz
-        obtain ⟨n,hc,t,ht,rfl⟩ := mem_iUnion.mp hz
-        exact ⟨mem_range_self t,hc⟩
-    rw [hU]
-    apply Monotone.measure_iUnion
-    intro i j hij
-    exact inter_subset_inter_right C (image_mono (Icc_subset_Icc_right (Nat.cast_le.mpr hij)))
-  rw [heq]
-  exact Measurable.iSup (fun n => source_compact_trace_measure_measurable h hh C hC 0 n)
-
-lemma source_whole_path_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (f : Ω → C(ℝ≥0,ℂ)) (hf : ∀ t, Measurable (fun ω => f ω t)) : Measurable f := by
-  have he : MeasurableEmbedding (fun g : C(ℝ≥0,ℂ) => fun n => g (denseSeq ℝ≥0 n)) := by
-    apply Continuous.measurableEmbedding
-    · exact continuous_pi fun n => continuous_eval_const _
-    · intro g₁ g₂ h
-      apply DFunLike.coe_injective
-      exact (denseRange_denseSeq ℝ≥0).equalizer g₁.continuous g₂.continuous h
-  apply he.measurable_comp_iff.mp
-  exact Measurable.of_eval (fun index => hf (denseSeq ℝ≥0 index))
-
-lemma source_sle_whole_trace_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
-    (hSLE : IsOrdinaryChordalSLE κ γ P) (h : ℝ → ℝ) (hh : IsGauge h)
-    (C : Set ℂ) (hC : IsClosed C) :
-    AEMeasurable (fun ω => hausdorffGauge h (wholeTrace (γ ω) ∩ C)) P := by
-  classical
-  have hcont : ∀ᵐ ω ∂P, Continuous (γ ω) := by
-    obtain ⟨B,hB,hcap⟩ := hSLE.2
-    filter_upwards [hcap] with ω hω
-    obtain ⟨G,F,hGF⟩ := hω
-    exact hGF.1
-  obtain ⟨S,hS,hSm,hSc⟩ := hcont.exists_measurable_mem
-  let f : Ω → C(ℝ≥0,ℂ) := fun ω => if hω : ω ∈ S then ⟨γ ω,hSc ω hω⟩ else 0
-  have hfm : Measurable f := by
-    apply source_whole_path_measurable
-    intro t
-    have hm : Measurable (fun ω => if ω ∈ S then γ ω t else (0 : ℂ)) :=
-      Measurable.ite hSm (hSLE.1 t) measurable_const
-    convert hm using 1
-    funext ω
-    by_cases hω : ω ∈ S <;> simp [f,hω]
-  apply ((source_whole_trace_measure_measurable h hh C hC).comp hfm).aemeasurable.congr
-  filter_upwards [hS] with ω hω
-  simp [f,hω,wholeTrace]
-end
-end SLEExactGauge
-
-namespace SLEExactGauge
-universe v_meas
- 
-def SourceWholeTraceMeasurableTarget : Prop :=
-  ∀ (Ω : Type v_meas) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ κ : ℝ, 0 < κ → κ < 8 →
-  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∀ R : ℝ, 0 < R → AEMeasurable (fun ω => hausdorffGauge h
-    (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P
-end SLEExactGauge
-namespace SLEExactGauge
-theorem sourceWholeTraceMeasurable_proved : SourceWholeTraceMeasurableTarget := by
-  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf R hR
-  exact source_sle_whole_trace_aemeasurable P κ γ hSLE h hh _ isClosed_closedBall
-end SLEExactGauge
-end
-
- 
-
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.SausageSeedAll
+
+namespace OAI
+
+/-! Trace Measurability for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory
+open scoped Topology NNReal
+open Set Filter MeasureTheory Complex
+open MeasureTheory ProbabilityTheory Set Filter
+open Set Filter Metric Complex InnerProductSpace
+open scoped Topology Laplacian
+open Set Filter Metric Complex MeasureTheory
+open Set Filter Metric Complex MeasureTheory InnerProductSpace
+open scoped Topology Laplacian ENNReal
+open scoped Topology ENNReal
+open scoped Topology Laplacian RealInnerProductSpace
+open Set Filter Metric Complex
+open scoped Topology
+open Set Filter Metric
+open Set Filter MeasureTheory Metric
+open Set Filter Complex
+open scoped Topology ComplexConjugate NNReal
+open scoped Topology ENNReal NNReal ComplexConjugate
+open Set Filter Metric MeasureTheory Complex
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology ENNReal NNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open Set Filter
+open scoped Topology NNReal ENNReal ComplexConjugate
+open scoped ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+section
+open Set Filter MeasureTheory Metric TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance wholeTraceMeasurabilityPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance wholeTraceMeasurabilityPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_whole_trace_measure_measurable (h : ℝ → ℝ) (hh : IsGauge h)
+    (C : Set ℂ) (hC : IsClosed C) :
+    Measurable (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (range γ ∩ C)) := by
+  have heq : (fun γ : C(ℝ≥0,ℂ) => hausdorffGauge h (range γ ∩ C)) =
+      fun γ : C(ℝ≥0,ℂ) => ⨆ n : ℕ, hausdorffGauge h (C ∩ (γ '' Icc 0 (n : ℝ≥0))) := by
+    funext γ
+    have hU : range γ ∩ C = ⋃ n : ℕ, C ∩ (γ '' Icc 0 (n : ℝ≥0)) := by
+      ext z
+      constructor
+      · rintro ⟨⟨t,rfl⟩,hc⟩
+        obtain ⟨n,hn⟩ := exists_nat_ge t
+        exact mem_iUnion.mpr ⟨n,hc,t,⟨zero_le,hn⟩,rfl⟩
+      · intro hz
+        obtain ⟨n,hc,t,ht,rfl⟩ := mem_iUnion.mp hz
+        exact ⟨mem_range_self t,hc⟩
+    rw [hU]
+    apply Monotone.measure_iUnion
+    intro i j hij
+    exact inter_subset_inter_right C (image_mono (Icc_subset_Icc_right (Nat.cast_le.mpr hij)))
+  rw [heq]
+  exact Measurable.iSup (fun n => source_compact_trace_measure_measurable h hh C hC 0 n)
+
+lemma source_whole_path_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (f : Ω → C(ℝ≥0,ℂ)) (hf : ∀ t, Measurable (fun ω => f ω t)) : Measurable f := by
+  have he : MeasurableEmbedding (fun g : C(ℝ≥0,ℂ) => fun n => g (denseSeq ℝ≥0 n)) := by
+    apply Continuous.measurableEmbedding
+    · exact continuous_pi fun n => continuous_eval_const _
+    · intro g₁ g₂ h
+      apply DFunLike.coe_injective
+      exact (denseRange_denseSeq ℝ≥0).equalizer g₁.continuous g₂.continuous h
+  apply he.measurable_comp_iff.mp
+  exact Measurable.of_eval (fun index => hf (denseSeq ℝ≥0 index))
+
+lemma source_sle_whole_trace_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ)
+    (hSLE : IsOrdinaryChordalSLE κ γ P) (h : ℝ → ℝ) (hh : IsGauge h)
+    (C : Set ℂ) (hC : IsClosed C) :
+    AEMeasurable (fun ω => hausdorffGauge h (wholeTrace (γ ω) ∩ C)) P := by
+  classical
+  have hcont : ∀ᵐ ω ∂P, Continuous (γ ω) := by
+    obtain ⟨B,hB,hcap⟩ := hSLE.2
+    filter_upwards [hcap] with ω hω
+    obtain ⟨G,F,hGF⟩ := hω
+    exact hGF.1
+  obtain ⟨S,hS,hSm,hSc⟩ := hcont.exists_measurable_mem
+  let f : Ω → C(ℝ≥0,ℂ) := fun ω => if hω : ω ∈ S then ⟨γ ω,hSc ω hω⟩ else 0
+  have hfm : Measurable f := by
+    apply source_whole_path_measurable
+    intro t
+    have hm : Measurable (fun ω => if ω ∈ S then γ ω t else (0 : ℂ)) :=
+      Measurable.ite hSm (hSLE.1 t) measurable_const
+    convert hm using 1
+    funext ω
+    by_cases hω : ω ∈ S <;> simp [f,hω]
+  apply ((source_whole_trace_measure_measurable h hh C hC).comp hfm).aemeasurable.congr
+  filter_upwards [hS] with ω hω
+  simp [f,hω,wholeTrace]
+end
+end SLEExactGauge
+
+namespace SLEExactGauge
+universe v_meas
+ 
+def SourceWholeTraceMeasurableTarget : Prop :=
+  ∀ (Ω : Type v_meas) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ κ : ℝ, 0 < κ → κ < 8 →
+  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∀ R : ℝ, 0 < R → AEMeasurable (fun ω => hausdorffGauge h
+    (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P
+end SLEExactGauge
+namespace SLEExactGauge
+theorem sourceWholeTraceMeasurable_proved : SourceWholeTraceMeasurableTarget := by
+  intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf R hR
+  exact source_sle_whole_trace_aemeasurable P κ γ hSLE h hh _ isClosed_closedBall
+end SLEExactGauge
+end
+
+ 
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/TracePosCombined.lean b/lean/OAI/Probability/SLE/TracePosCombined.lean
index e1d33e0f575b0c530caaad00ece544733fa5569d..5be8bb2c2b670386230560141395609cbf92864d 100644
--- a/lean/OAI/Probability/SLE/TracePosCombined.lean
+++ b/lean/OAI/Probability/SLE/TracePosCombined.lean
@@ -1,818 +1,818 @@
-import Mathlib
-import OAI.Probability.SLE.ActualRetainedKernel
-
-namespace OAI
-
-/-! Trace Pos Combined for chordal SLE. -/
-
-section
-
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_gauge_cost_nonempty (h : ℝ → ℝ) (hh : IsGauge h) (A : Set ℂ) :
-    (⨆ _ : A.Nonempty, extendedGauge h (ediam A)) = extendedGauge h (ediam A) := by
-  by_cases hA : A.Nonempty
-  · simp [hA]
-  · rw [Set.not_nonempty_iff_eq_empty.mp hA]
-    simp [extendedGauge,hh.zero]
-
-lemma source_gauge_cover_formula (h : ℝ → ℝ) (hh : IsGauge h) (K : Set ℂ) :
-    hausdorffGauge h K = ⨆ (r : ℝ≥0∞) (_ : 0 < r),
-      ⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n) (_ : ∀ n, ediam (U n) ≤ r),
-        ∑' n, extendedGauge h (ediam (U n)) := by
-  simp only [hausdorffGauge,Measure.mkMetric_apply,source_gauge_cost_nonempty h hh]
-
-lemma source_gauge_open_enlargement (h : ℝ → ℝ) (hh : IsGauge h)
-    (A : Set ℂ) (r : ℝ) (hr : (ediam A).toReal < r) (hA : ediam A ≠ ∞)
-    (ε : ℝ≥0) (hε : 0 < ε) :
-    ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ ediam U < ENNReal.ofReal r ∧
-      extendedGauge h (ediam U) ≤ extendedGauge h (ediam A) + ε := by
-  let b (n : ℕ) : ℝ := 1 / ((n:ℝ)+1)
-  have hb (n : ℕ) : 0 < b n := by dsimp [b]; positivity
-  have hb0 : Tendsto b atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-  let d := (ediam A).toReal
-  have hd : 0 ≤ d := ENNReal.toReal_nonneg
-  have hlim : Tendsto (fun n => d + 2*b n) atTop (𝓝 d) := by
-    simpa only [mul_zero,add_zero] using tendsto_const_nhds.add (hb0.const_mul 2)
-  have hhlim : Tendsto (fun n => ENNReal.ofReal (h (d+2*b n))) atTop
-      (𝓝 (extendedGauge h (ediam A))) := by
-    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-      ((hh.continuousOn d hd).tendsto.comp (tendsto_nhdsWithin_iff.mpr
-        ⟨hlim,Eventually.of_forall (fun n => show 0 ≤ d+2*b n by positivity)⟩))
-  have hcost : extendedGauge h (ediam A) < extendedGauge h (ediam A) + ε := by
-    exact ENNReal.lt_add_right (by simp [extendedGauge]) (ENNReal.coe_pos.mpr hε).ne'
-  obtain ⟨n,hn,hc⟩ := ( (hlim.eventually (gt_mem_nhds hr)).and
-    (hhlim.eventually (gt_mem_nhds hcost)) ).exists
-  let U := thickening (b n) A
-  have hUb : Bornology.IsBounded U := (isBounded_iff_ediam_ne_top.mpr hA).thickening
-  have hdiam : (ediam U).toReal ≤ d+2*b n := diam_thickening_le A (hb n).le
-  refine ⟨U,isOpen_thickening, self_subset_thickening (hb n) A,?_,?_⟩
-  · rw [←ENNReal.ofReal_toReal hUb.ediam_ne_top]
-    exact ENNReal.ofReal_lt_ofReal_iff'.mpr ⟨hdiam.trans_lt hn,by linarith⟩
-  · exact (ENNReal.ofReal_le_ofReal (hh.monotoneOn ENNReal.toReal_nonneg
-      (show 0 ≤ d+2*b n by positivity) hdiam)).trans hc.le
-
- 
-def sourceFiniteOpenCover (h : ℝ → ℝ) (r ε : ℝ≥0∞) (K : Set ℂ) : Prop :=
-  ∃ (U : ℕ → Set ℂ) (I : Finset ℕ), (∀ n ∈ I, IsOpen (U n)) ∧
-    (∀ n ∈ I, ediam (U n) ≤ r) ∧ K ⊆ ⋃ n ∈ I, U n ∧
-    (∑ n ∈ I, extendedGauge h (ediam (U n))) < ε
-
-lemma source_null_compact_finite_open_cover (h : ℝ → ℝ) (hh : IsGauge h)
-    (K : Set ℂ) (hK : IsCompact K) (hzero : hausdorffGauge h K = 0)
-    (r ε : ℝ) (hr : 0 < r) (hε : 0 < ε) :
-    sourceFiniteOpenCover h (ENNReal.ofReal r) (ENNReal.ofReal ε) K := by
-  have hsmall :
-      (⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n)
-        (_ : ∀ n, ediam (U n) ≤ ENNReal.ofReal (r/2)),
-          ∑' n, extendedGauge h (ediam (U n))) < ENNReal.ofReal (ε/2) := by
-    apply lt_of_le_of_lt (b := hausdorffGauge h K)
-    · rw [source_gauge_cover_formula h hh]
-      exact le_iSup_of_le (ENNReal.ofReal (r/2))
-        (le_iSup_of_le (ENNReal.ofReal_pos.mpr (by positivity)) le_rfl)
-    · simpa only [hzero] using ENNReal.ofReal_pos.mpr (show 0 < ε/2 by positivity)
-  obtain ⟨U,hU,hdiam,hcost⟩ := (by simpa only [iInf_lt_iff] using hsmall)
-  obtain ⟨e,he,hes⟩ := ENNReal.exists_pos_sum_of_countable
-    (ENNReal.ofReal_pos.mpr (show 0 < ε/2 by positivity)).ne' ℕ
-  have hfin (n : ℕ) : ediam (U n) ≠ ∞ :=
-    ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hdiam n)
-  have hrd (n : ℕ) : (ediam (U n)).toReal < r := by
-    have hb := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hdiam n)
-    rw [ENNReal.toReal_ofReal (by positivity)] at hb
-    linarith
-  choose V hopen hUV hVdiam hVcost using fun n =>
-    source_gauge_open_enlargement h hh (U n) r (hrd n) (hfin n) (e n) (he n)
-  obtain ⟨I,hI⟩ := hK.elim_finite_subcover V hopen (hU.trans (iUnion_mono hUV))
-  refine ⟨V,I,fun n _ => hopen n,fun n _ => (hVdiam n).le,hI,?_⟩
-  calc
-    (∑ n ∈ I, extendedGauge h (ediam (V n))) ≤ ∑' n, extendedGauge h (ediam (V n)) :=
-      ENNReal.sum_le_tsum I
-    _ ≤ ∑' n, (extendedGauge h (ediam (U n)) + e n) := ENNReal.tsum_le_tsum hVcost
-    _ = (∑' n, extendedGauge h (ediam (U n))) + ∑' n, (e n : ℝ≥0∞) := ENNReal.tsum_add
-    _ < ENNReal.ofReal (ε/2) + ENNReal.ofReal (ε/2) := ENNReal.add_lt_add hcost hes
-    _ = ENNReal.ofReal ε := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
-end
-end SLEExactGauge
-end
-
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-def sourceFineGaugeContent (h : ℝ → ℝ) (r : ℝ≥0∞) (K : Set ℂ) : ℝ≥0∞ :=
-  ⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n) (_ : ∀ n, ediam (U n) ≤ r),
-    ∑' n, extendedGauge h (ediam (U n))
-lemma source_fine_content_finite_cover (h : ℝ → ℝ) (hh : IsGauge h)
-    {K : Set ℂ} {R r ε : ℝ≥0∞} (hr : r ≤ R) (hc : sourceFiniteOpenCover h r ε K) :
-    sourceFineGaugeContent h R K < ε := by
-  classical
-  obtain ⟨U,I,hopen,hdiam,hcover,hcost⟩ := hc
-  let V (n : ℕ) : Set ℂ := if n ∈ I then U n else ∅
-  have hKV : K ⊆ ⋃ n, V n := by
-    intro z hz
-    obtain ⟨n,hn,hzn⟩ := mem_iUnion₂.mp (hcover hz)
-    exact mem_iUnion.mpr ⟨n,by simpa only [V,ite_eq_left hn] using hzn⟩
-  have hVdiam (n : ℕ) : ediam (V n) ≤ R := by
-    by_cases hn : n ∈ I
-    · simpa only [V,ite_eq_left hn] using (hdiam n hn).trans hr
-    · simp [V,hn]
-  unfold sourceFineGaugeContent
-  apply lt_of_le_of_lt (b := ∑' n, extendedGauge h (ediam (V n)))
-  · exact iInf_le_of_le V (iInf_le_of_le hKV (iInf_le_of_le hVdiam le_rfl))
-  have hsum : (∑' n, extendedGauge h (ediam (V n))) = ∑ n ∈ I, extendedGauge h (ediam (U n)) := by
-    rw [tsum_eq_sum (s := I) (fun n hn => by simp [V,hn,extendedGauge,hh.zero])]
-    exact Finset.sum_congr rfl (fun n hn => by simp only [V,ite_eq_left hn])
-  simpa only [hsum] using hcost
-
-noncomputable def sourceCoverScale (n : ℕ) : ℝ≥0∞ := ENNReal.ofReal (1/((n:ℝ)+1))
-lemma source_coverScale_pos (n : ℕ) : 0 < sourceCoverScale n := by
-  unfold sourceCoverScale
-  positivity
-lemma source_coverScale_tendsto : Tendsto sourceCoverScale atTop (𝓝 0) := by
-  unfold sourceCoverScale
-  simpa only [ENNReal.ofReal_zero] using ENNReal.tendsto_ofReal
-    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
-
-lemma source_compact_gauge_null_criterion (h : ℝ → ℝ) (hh : IsGauge h)
-    (K : Set ℂ) (hK : IsCompact K) :
-    hausdorffGauge h K = 0 ↔ ∀ n : ℕ,
-      sourceFiniteOpenCover h (sourceCoverScale n) (sourceCoverScale n) K := by
-  constructor
-  · intro hz n
-    exact source_null_compact_finite_open_cover h hh K hK hz _ _ (by positivity) (by positivity)
-  · intro hcov
-    apply le_antisymm _ zero_le
-    rw [source_gauge_cover_formula h hh]
-    refine iSup_le fun R => iSup_le fun hR => ?_
-    change sourceFineGaugeContent h R K ≤ 0
-    apply le_of_tendsto_of_tendsto tendsto_const_nhds source_coverScale_tendsto
-    filter_upwards [source_coverScale_tendsto.eventually (gt_mem_nhds hR)] with n hn
-    exact (source_fine_content_finite_cover h hh hn.le (hcov n)).le
-end
-end SLEExactGauge
-end
-
-section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance tracePosCoverComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance tracePosCoverComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_finite_open_cover_trace_isOpen (h : ℝ → ℝ) (C : Set ℂ) (hC : IsClosed C)
-    (s t : ℝ≥0) (r ε : ℝ≥0∞) :
-    IsOpen {γ : C(ℝ≥0,ℂ) | sourceFiniteOpenCover h r ε (C ∩ (γ '' Icc s t))} := by
-  classical
-  apply isOpen_iff_mem_nhds.mpr
-  intro γ hγ
-  obtain ⟨U,I,ho,hd,hcov,hcost⟩ := hγ
-  let O := ⋃ n ∈ I, U n
-  have hO : IsOpen O := isOpen_iUnion (fun n => isOpen_iUnion (fun hn => ho n hn))
-  have hmap : MapsTo γ (Icc s t) (Cᶜ ∪ O) := by
-    intro u hu
-    by_cases hc : γ u ∈ C
-    · exact Or.inr (hcov ⟨hc,⟨u,hu,rfl⟩⟩)
-    · exact Or.inl hc
-  have hev := (ContinuousMap.isOpen_setOfPred_mapsTo isCompact_Icc (hC.isOpen_compl.union hO)).mem_nhds hmap
-  filter_upwards [hev] with g hg
-  refine ⟨U,I,ho,hd,?_,hcost⟩
-  rintro z ⟨hz,u,hu,rfl⟩
-  exact (hg hu).resolve_left (fun hn => hn hz)
- 
-
-def SourceTracePositiveMeasurableTarget : Prop :=
-  ∀ (h : ℝ → ℝ), IsGauge h → ∀ (C : Set ℂ), IsClosed C → ∀ s t : ℝ≥0,
-    @MeasurableSet C(ℝ≥0, ℂ) (borel _) {γ | 0 < hausdorffGauge h (C ∩ (γ '' Set.Icc s t))}
-theorem sourceTracePositiveMeasurable_proved : SourceTracePositiveMeasurableTarget := by
-  intro h hh C hC s t
-  have hnull : MeasurableSet {γ : C(ℝ≥0,ℂ) | hausdorffGauge h (C ∩ (γ '' Icc s t)) = 0} := by
-    have heq : {γ : C(ℝ≥0,ℂ) | hausdorffGauge h (C ∩ (γ '' Icc s t)) = 0} =
-        ⋂ n : ℕ, {γ : C(ℝ≥0,ℂ) | sourceFiniteOpenCover h (sourceCoverScale n) (sourceCoverScale n)
-          (C ∩ (γ '' Icc s t))} := by
-      ext γ
-      simp only [mem_ofPred_eq,mem_iInter]
-      exact source_compact_gauge_null_criterion h hh _
-        ((isCompact_Icc.image γ.continuous).inter_left hC)
-    rw [heq]
-    exact MeasurableSet.iInter (fun n =>
-      (source_finite_open_cover_trace_isOpen h C hC s t _ _).measurableSet)
-  simpa only [compl_ofPred,←ne_eq,←pos_iff_ne_zero] using hnull.compl
-end
-end SLEExactGauge
-end
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance tracePosDilationRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance tracePosDilationComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance tracePosDilationRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance tracePosDilationComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_dilation_gauge_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    {c : ℂ} (hc : c ≠ 0) {S : Set ℂ} (hS : IsCompact S)
-    (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => c*z) '' S) := by
-  by_contra hn
-  have hz : hausdorffGauge h ((fun z => c*z) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le
-  have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
-    (D := univ) (f := fun z => c⁻¹*z) isOpen_univ (hS.image (continuous_const.mul continuous_id))
-    (subset_univ _) (by fun_prop) hz
-  have hi : (fun z => c⁻¹*z) '' ((fun z => c*z) '' S) = S := by
-    rw [←image_comp]
-    simp only [Function.comp_def,←mul_assoc,inv_mul_cancel₀ hc,one_mul,image_id']
-  change hausdorffGauge h ((fun z => c⁻¹*z) '' ((fun z => c*z) '' S)) = 0 at he
-  rw [hi] at he
-  exact hp.ne' he
-
-lemma RegularSLEPresentation.source_trace_path_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) :
-    Measurable (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) := by
-  apply source_path_measurable
-  intro t
-  exact (a.trace_adapted t).mono (a.filtration.le t) le_rfl
-
-lemma RegularSLEPresentation.source_scale_trace_law
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (c : ℝ≥0) (hc : 0 < c) :
-    (P.comap (Subtype.val : a.samples → Ω)).map
-      (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) =
-    (P.comap (Subtype.val : a.samples → Ω)).map
-      (sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
-        (a.sourceScaled c hc).trace_continuous) := by
-  exact sourceTraceLawUnique_proved κ hκ hκ8 a.B (a.sourceScaled c hc).B
-    a.brownian (a.sourceScaled c hc).brownian a.measurable_B (a.sourceScaled c hc).measurable_B
-    a.continuous_B (a.sourceScaled c hc).continuous_B _ _ a.G a.F
-    (a.sourceScaled c hc).G (a.sourceScaled c hc).F a.witness (a.sourceScaled c hc).witness
- 
-
-def SourceUniformInteriorSeedTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (h : ℝ → ℝ), IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-  ∃ p : ℝ≥0∞, 0 < p ∧ ∀ δ : ℝ≥0, 0 < δ →
-    ∃ K : Set ℂ, IsCompact K ∧ K ⊆ upperHalfPlane ∧
-      p ≤ (P.comap (Subtype.val : a.samples → Ω))
-        {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Set.Icc 0 δ))}
-theorem sourceUniformInteriorSeed_proved : SourceUniformInteriorSeedTarget := by
-  intro κ hκ hκ8 Ω m P hP γ a h hh hf
-  let : IsProbabilityMeasure P := hP
-  have hk : 0 < κ := by linarith
-  let z₀ : ℂ := 3*I
-  have hz₀ : 2 < z₀.im := by norm_num [z₀]
-  obtain ⟨T,E,hT,hE,hp,hseed⟩ :=
-    sourceInteriorRetainedTransfer_proved κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  refine ⟨Q E,hp,?_⟩
-  intro δ hδ
-  let c := NNReal.sqrt (δ/T)
-  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
-  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
-  have ht : sourceScaleTime c δ = T := by
-    unfold sourceScaleTime
-    rw [hc2]
-    field_simp
-  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
-  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
-  have hKH : K ⊆ upperHalfPlane := by
-    rintro _ ⟨z,hz,rfl⟩
-    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
-    change 0 < ((c:ℂ)*z).im
-    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos (show 0 < (c:ℝ) from hc) hz'
-  refine ⟨K,hK,hKH,?_⟩
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
-    (a.sourceScaled c hc).trace_continuous
-  let A : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))}
-  have hA : MeasurableSet A := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ
-  have hm : Measurable gp := a.source_trace_path_measurable
-  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable
-  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law hk hκ8 c hc)
-  rw [Measure.map_apply hm hA,Measure.map_apply hsm hA] at hmap
-  change Q E ≤ Q (gp ⁻¹' A)
-  rw [hmap]
-  apply measure_mono
-  intro ω hω
-  have hpos := (hseed ω hω).1
-  have heq : K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ) =
-      (fun z : ℂ => (c:ℂ)*z) '' (closedBall z₀ 1 ∩ (γ ω '' Icc 0 T)) := by
-    rw [source_scale_past c δ hc,ht, image_inter (mul_right_injective₀ (by exact_mod_cast hc.ne' : (c:ℂ) ≠ 0))]
-  change 0 < hausdorffGauge h (K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ))
-  rw [heq]
-  exact source_dilation_gauge_positive κ hk hκ8 h hh hf (by exact_mod_cast hc.ne')
-    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed) hpos
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance tracePosRestartRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance tracePosRestartComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance tracePosRestartRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-local instance tracePosRestartComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-lemma source_translation_gauge_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    (b : ℂ) {S : Set ℂ} (hS : IsCompact S)
-    (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => z+b) '' S) := by
-  by_contra hn
-  have hz : hausdorffGauge h ((fun z => z+b) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le
-  have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
-    (D := univ) (f := fun z => z-b) isOpen_univ (hS.image (continuous_id.add_const b))
-    (subset_univ _) (by fun_prop) hz
-  have hi : (fun z => z-b) '' ((fun z => z+b) '' S) = S := by
-    rw [←image_comp]
-    simp only [Function.comp_def,add_sub_cancel_right,image_id']
-  change hausdorffGauge h ((fun z => z-b) '' ((fun z => z+b) '' S)) = 0 at he
-  rw [hi] at he
-  exact hp.ne' he
-
-lemma source_restart_segment_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (s δ : ℝ≥0) (η : C(ℝ≥0,ℂ))
-    (hid : ∀ t : ℝ≥0, η t ∈ upperHalfPlane → γ (s+t) = F s (η t + (U s:ℂ)))
-    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
-    (hp : 0 < hausdorffGauge h (K ∩ (η '' Icc 0 δ))) :
-    0 < hausdorffGauge h (γ '' Icc s (s+δ)) := by
-  let L := K ∩ (η '' Icc 0 δ)
-  have hL : IsCompact L := hK.inter_right (isCompact_Icc.image η.continuous).isClosed
-  have hLH : L ⊆ upperHalfPlane := inter_subset_left.trans hKH
-  have htrans := source_translation_gauge_positive κ hκ hκ8 h hh hf (U s:ℂ) hL hp
-  have htrK : IsCompact ((fun z => z+(U s:ℂ)) '' L) := hL.image (continuous_id.add_const _)
-  have htrH : (fun z => z+(U s:ℂ)) '' L ⊆ upperHalfPlane := by
-    rintro _ ⟨z,hz,rfl⟩
-    simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,add_zero] using hLH hz
-  have hpositive := source_inverse_chart_gauge_positive κ hκ hκ8 h hh hf U γ G F hw s htrK htrH htrans
-  apply hpositive.trans_le (measure_mono ?_)
-  rintro _ ⟨_,⟨z,hz,rfl⟩,rfl⟩
-  obtain ⟨u,hu,rfl⟩ := hz.2
-  refine ⟨s+u,⟨le_add_of_nonneg_right hu.1,add_le_add le_rfl hu.2⟩,?_⟩
-  exact hid u (hKH hz.1)
-
-lemma RegularSLEPresentation.source_segment_positive_measurable_at
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h)
-    (s t : ℝ≥0) :
-    MeasurableSet[a.filtration t] {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} := by
-  let f : a.samples → C(ℝ≥0,ℂ) := fun ω =>
-    ⟨fun u => γ ω (min u t), (a.trace_continuous ω).comp (continuous_id.min continuous_const)⟩
-  have hm : @Measurable a.samples C(ℝ≥0,ℂ) (a.filtration t) (borel _) f := by
-    let : MeasurableSpace a.samples := a.filtration t
-    apply source_path_measurable
-    intro u
-    exact (a.trace_adapted (min u t)).mono (a.filtration.mono (min_le_right u t)) le_rfl
-  have heq : {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} =
-      f ⁻¹' {g : C(ℝ≥0,ℂ) | 0 < hausdorffGauge h (univ ∩ (g '' Icc s t))} := by
-    ext ω
-    simp only [mem_preimage,mem_ofPred_eq,univ_inter]
-    have hi : γ ω '' Icc s t = f ω '' Icc s t := by
-      apply image_congr
-      intro u hu
-      simp only [f,ContinuousMap.coe_mk,min_eq_left hu.2]
-    rw [hi]
-  rw [heq]
-  exact (sourceTracePositiveMeasurable_proved h hh univ isClosed_univ s t).preimage hm
-
-lemma RegularSLEPresentation.source_segment_null_measurable_at
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h) (s t : ℝ≥0) :
-    MeasurableSet[a.filtration t] {ω : a.samples | hausdorffGauge h (γ ω '' Icc s t) = 0} := by
-  simpa only [compl_ofPred,not_lt,nonpos_iff_eq_zero] using
-    (a.source_segment_positive_measurable_at h hh s t).compl
-
-lemma RegularSLEPresentation.source_uniform_null_slot_bound
-    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h)
-    (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ →
-      ∀ A : Set a.samples, MeasurableSet[a.filtration s] A →
-        (P.comap Subtype.val) (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0}) ≤
-          (1-p)*(P.comap Subtype.val) A := by
-  have hk : 0 < κ := by linarith
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨p,hp,hseed⟩ := sourceUniformInteriorSeed_proved κ hκ hκ8 Ω m P inferInstance γ a h hh hf
-  have hp1 : p ≤ 1 := by
-    obtain ⟨K,hK,hKH,hKP⟩ := hseed 1 zero_lt_one
-    exact hKP.trans (prob_le_one)
-  refine ⟨p,hp,hp1,?_⟩
-  intro s δ hδ A hA
-  obtain ⟨K,hK,hKH,hKP⟩ := hseed δ hδ
-  obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian
-    a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness
-  let τ : a.samples → ℝ≥0 := fun _ => s
-  have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s
-  have hAstop : MeasurableSet[hτ.measurableSpace] A := by
-    change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A
-    rw [IsStoppingTime.measurableSpace_const]
-    exact hA
-  obtain ⟨hlaw,hident⟩ := hrestart τ hτ
-  let rp := sourceRestartDriverPath a.B a.continuous_B τ
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
-  let C : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))}
-  let E : Set a.samples := (fun ω => Φ (rp ω)) ⁻¹' C
-  have hC : MeasurableSet C := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ
-  have hrm : Measurable rp := by
-    apply source_path_measurable
-    intro t
-    exact (a.measurable_B (s+t)).sub (a.measurable_B s)
-  have hem : Measurable (fun ω => Φ (rp ω)) := hΦm.comp hrm
-  have hEm : MeasurableSet E := hC.preimage hem
-  have hAm : MeasurableSet A := (a.filtration.le s) A hA
-  have hprob : Q (A ∩ E) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} := by
-    have he := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ C) (hlaw A hAstop)
-    rw [Measure.map_apply hem hC,Measure.smul_apply,
-      Measure.map_apply a.source_trace_path_measurable hC,
-      Measure.restrict_apply (hC.preimage hem)] at he
-    change Q (E ∩ A) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} at he
-    simpa only [inter_comm] using he
-  have hins : (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0} : Set a.samples) ≤ᵐ[Q] (A \ E : Set a.samples) := by
-    filter_upwards [hident] with ω hω
-    intro hx
-    refine ⟨hx.1,?_⟩
-    intro hsuc
-    have hpositive := source_restart_segment_positive κ hk hκ8 h hh hf
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω)
-      s δ (Φ (rp ω)) hω hK hKH hsuc
-    rw [hx.2] at hpositive
-    exact lt_irrefl _ hpositive
-  have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by
-    have he : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto
-    rw [he]
-    exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _)
-  calc Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0})
-      ≤ Q (A \ E) := measure_mono_ae hins
-    _ = Q A - Q (A ∩ E) := hdiff
-    _ ≤ Q A - Q A*p := tsub_le_tsub_left (by rw [hprob]; gcongr) _
-    _ = (1-p)*Q A := by
-      rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul,mul_comm p (Q A)]
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma RegularSLEPresentation.source_positive_fixed_interval
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h)
-    (hf : HasSmallRadiusFormula h (hFormula κ)) (s t : ℝ≥0) (hst : s < t) :
-    ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), 0 < hausdorffGauge h (γ ω '' Icc s t) := by
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_null_slot_bound hκ hκ8 h hh hf
-  let c : ℝ≥0∞ := 1-p
-  have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne'
-  have hsteps (δ : ℝ≥0) (hδ : 0 < δ) (k : ℕ) :
-      Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0} ≤ c^k := by
-    induction k with
-    | zero => simpa using (prob_le_one (μ := Q)
-        (s := {ω | hausdorffGauge h (γ ω '' Icc s (s+(0:ℝ≥0)*δ)) = 0}))
-    | succ k ih =>
-      have he : s+((k+1:ℕ):ℝ≥0)*δ = (s+(k:ℝ≥0)*δ)+δ := by push_cast; ring
-      rw [he,pow_succ,mul_comm (c^k) c]
-      let A : Set a.samples := {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0}
-      have hA : MeasurableSet[a.filtration (s+(k:ℝ≥0)*δ)] A :=
-        a.source_segment_null_measurable_at h hh s _
-      calc Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ+δ)) = 0}
-          ≤ Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc (s+(k:ℝ≥0)*δ) (s+(k:ℝ≥0)*δ+δ)) = 0}) := by
-            apply measure_mono
-            intro ω hω
-            constructor
-            · exact measure_mono_null (image_mono (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))) hω
-            · exact measure_mono_null (image_mono (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)) hω
-        _ ≤ c * Q A := hslot _ δ hδ A hA
-        _ ≤ c * c^k := by gcongr
-  have hbound : ∀ᶠ n : ℕ in atTop,
-      Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} ≤ c^n := by
-    filter_upwards [eventually_ge_atTop 1] with n hn
-    have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega)
-    let δ : ℝ≥0 := (t-s)/(n:ℝ≥0)
-    have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0
-    have he : s+(n:ℝ≥0)*δ = t := by
-      dsimp [δ]
-      rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le]
-    simpa only [he] using hsteps δ hδ n
-  have hzero : Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} = 0 :=
-    le_antisymm (ge_of_tendsto (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc) hbound) zero_le
-  apply ae_iff.mpr
-  simpa only [not_lt,nonpos_iff_eq_zero] using hzero
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma RegularSLEPresentation.source_positive_all_intervals
-    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hκ : 4 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h)
-    (hf : HasSmallRadiusFormula h (hFormula κ)) :
-    ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω),
-      ∀ s t : ℝ≥0, 0 < s → s < t → 0 < hausdorffGauge h (γ ω '' Icc s t) := by
-  have hrat : ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω),
-      ∀ r q : ℚ, (r:ℝ).toNNReal < (q:ℝ).toNNReal →
-        0 < hausdorffGauge h (γ ω '' Icc (r:ℝ).toNNReal (q:ℝ).toNNReal) := by
-    apply ae_all_iff.mpr
-    intro r
-    apply ae_all_iff.mpr
-    intro q
-    apply ae_all_iff.mpr
-    intro hrq
-    exact a.source_positive_fixed_interval hκ hκ8 h hh hf _ _ hrq
-  filter_upwards [hrat] with ω hω
-  intro s t hs hst
-  obtain ⟨r,hsr,hrt⟩ := exists_rat_btwn (show (s:ℝ) < (t:ℝ) from hst)
-  obtain ⟨q,hrq,hqt⟩ := exists_rat_btwn hrt
-  have hq : 0 < (q:ℝ) := (show 0 < (s:ℝ) from hs).trans (hsr.trans hrq)
-  have hp := hω r q ((Real.toNNReal_lt_toNNReal_iff hq).mpr hrq)
-  apply hp.trans_le (measure_mono (image_mono (Icc_subset_Icc ?_ ?_)))
-  · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hsr.le
-  · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hqt.le
- 
-
-def SourceLowerAllIntervalsTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
-    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
-      0 < hausdorffGauge h (γ ω '' Set.Icc s t)
-theorem sourceLowerAllIntervals_proved : SourceLowerAllIntervalsTarget := by
-  intro κ hκ hκ8 Ω m P hP γ hSLE h hh hf
-  let : IsProbabilityMeasure P := hP
-  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
-  have ha := a.source_positive_all_intervals hκ hκ8 h hh hf
-  rw [←a.measurePreserving_val.map_eq]
-  exact (MeasurableEmbedding.subtype_coe a.measurable_samples).ae_map_iff.mpr ha
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable def sourceSausageSet (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (e : ℝ) : Set ℂ :=
-  {z | ∃ u ∈ Set.Icc s t, dist z (γ u) < e}
-noncomputable def sourceSausageMass (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (e : ℝ) : ℝ≥0∞ :=
-  ENNReal.ofReal (e^(-codimension κ)) * MeasureTheory.volume (sourceSausageSet γ s t e)
- 
-
-def SourceSmallSausageTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : MeasureTheory.Measure Ω),
-    MeasureTheory.IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
-  ∀ s t : ℝ≥0, s < t → ∀ ε : ℝ, 0 < ε →
-    ∃ a e₀ : ℝ, 0 < a ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-      P {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal a} < ENNReal.ofReal ε
-end SLEExactGauge
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-local instance tracePosSausageComplexPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance tracePosSausageComplexPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma source_sausage_isOpen (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (e : ℝ) :
-    IsOpen (sourceSausageSet γ s t e) := by
-  have he : sourceSausageSet γ s t e = ⋃ u ∈ Icc s t, ball (γ u) e := by
-    ext z; simp [sourceSausageSet]
-  rw [he]
-  exact isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_ball
-lemma source_sausage_joint_open (s t : ℝ≥0) (e : ℝ) :
-    IsOpen {p : C(ℝ≥0,ℂ) × ℂ | p.2 ∈ sourceSausageSet p.1 s t e} := by
-  have he : {p : C(ℝ≥0,ℂ) × ℂ | p.2 ∈ sourceSausageSet p.1 s t e} =
-      ⋃ u ∈ Icc s t, {p : C(ℝ≥0,ℂ) × ℂ | dist p.2 (p.1 u) < e} := by
-    ext p; simp [sourceSausageSet]
-  rw [he]
-  apply isOpen_iUnion; intro u
-  apply isOpen_iUnion; intro hu
-  exact isOpen_lt (continuous_snd.dist ((continuous_eval_const u).comp continuous_fst)) continuous_const
-lemma source_sausage_local_volume_measurable (s t : ℝ≥0) (e : ℝ) (K : Set ℂ) :
-    Measurable (fun g : C(ℝ≥0,ℂ) => volume (K ∩ sourceSausageSet g s t e)) := by
-  have hm : Measurable (fun p : C(ℝ≥0,ℂ) × ℂ =>
-      if p.2 ∈ sourceSausageSet p.1 s t e then (1:ℝ≥0∞) else 0) :=
-    Measurable.ite (source_sausage_joint_open s t e).measurableSet measurable_const measurable_const
-  have hh := hm.lintegral_prod_right' (ν := volume.restrict K)
-  convert hh using 1
-  ext g
-  change volume (K ∩ sourceSausageSet g s t e) =
-    ∫⁻ z, (sourceSausageSet g s t e).indicator (fun _ => 1) z ∂(volume.restrict K)
-  rw [lintegral_indicator (source_sausage_isOpen g s t e).measurableSet]
-  simp [Measure.restrict_apply (source_sausage_isOpen g s t e).measurableSet,inter_comm]
-lemma source_sausage_mass_measurable (κ : ℝ) (s t : ℝ≥0) (e : ℝ) :
-    Measurable (fun g : C(ℝ≥0,ℂ) => sourceSausageMass κ g s t e) := by
-  exact measurable_const.mul (by simpa only [univ_inter] using source_sausage_local_volume_measurable s t e univ)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_second_moment_seed {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) [IsProbabilityMeasure P] (X : Ω → ℝ≥0∞) (hm : Measurable X)
-    (m M : ℝ) (hp : 0 < m) (hM : 0 < M)
-    (hfirst : ENNReal.ofReal m ≤ ∫⁻ ω, X ω ∂P)
-    (hsecond : (∫⁻ ω, (X ω)^2 ∂P) ≤ ENNReal.ofReal M) :
-    ENNReal.ofReal ((m/2)^2/M) ≤ P {ω | ENNReal.ofReal (m/2) ≤ X ω} := by
-  classical
-  let E := {ω | ENNReal.ofReal (m/2) ≤ X ω}
-  have hE : MeasurableSet E := measurableSet_le measurable_const hm
-  let Y : Ω → ℝ≥0∞ := E.indicator (fun _ => 1)
-  have hYm : Measurable Y := measurable_const.indicator hE
-  have hcs := ENNReal.lintegral_mul_le_Lp_mul_Lq P
-    (show (2:ℝ).HolderConjugate 2 by constructor <;> norm_num)
-    hm.aemeasurable hYm.aemeasurable
-  have hYsq : (∫⁻ ω, (Y ω)^(2:ℝ) ∂P) = P E := by
-    simp only [Y,Set.indicator_apply]
-    simp_rw [show ∀ (b : Prop) [Decidable b], ((if b then 1 else 0):ℝ≥0∞)^(2:ℝ) = if b then 1 else 0 by intros; split_ifs <;> norm_num]
-    simpa only [Set.indicator_apply,Pi.one_apply] using (lintegral_indicator_one (μ := P) hE)
-  rw [hYsq] at hcs
-  simp only [ENNReal.rpow_two] at hcs
-  have hcs' : (∫⁻ ω, X ω*Y ω ∂P) ≤ (ENNReal.ofReal M)^(1/(2:ℝ))*(P E)^(1/(2:ℝ)) :=
-    hcs.trans (mul_le_mul' (ENNReal.rpow_le_rpow hsecond (by norm_num)) le_rfl)
-  have hfirst' : ENNReal.ofReal m ≤ ENNReal.ofReal (m/2) +
-      (ENNReal.ofReal M)^(1/(2:ℝ))*(P E)^(1/(2:ℝ)) := by
-    apply hfirst.trans
-    calc (∫⁻ ω, X ω ∂P) ≤ ∫⁻ ω, ENNReal.ofReal (m/2)+X ω*Y ω ∂P := by
-          apply lintegral_mono
-          intro ω
-          by_cases hω : ω ∈ E
-          · simp [Y,hω]
-          · have hh : X ω ≤ ENNReal.ofReal (m/2) := (lt_of_not_ge hω).le
-            simpa [Y,hω] using hh
-      _ = ENNReal.ofReal (m/2)+(∫⁻ ω, X ω*Y ω ∂P) := by
-          rw [lintegral_add_left measurable_const]
-          simp
-      _ ≤ _ := add_le_add le_rfl hcs'
-  have hb : ENNReal.ofReal (m/2) +
-      (ENNReal.ofReal M)^(1/(2:ℝ))*(P E)^(1/(2:ℝ)) ≠ ⊤ := by finiteness
-  have hh := (ENNReal.ofReal_le_iff_le_toReal hb).mp hfirst'
-  rw [ENNReal.toReal_add (by simp) (by finiteness),ENNReal.toReal_ofReal (by positivity),
-    ENNReal.toReal_mul,←ENNReal.toReal_rpow,←ENNReal.toReal_rpow,
-    ENNReal.toReal_ofReal hM.le] at hh
-  have hrootM : (M^(1/(2:ℝ)))^2 = M := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul hM.le]; norm_num
-  have hrootP : ((P E).toReal^(1/(2:ℝ)))^2 = (P E).toReal := by
-    rw [←Real.rpow_natCast,←Real.rpow_mul ENNReal.toReal_nonneg]; norm_num
-  have hle : (m/2)^2/M ≤ (P E).toReal := by
-    apply (div_le_iff₀ hM).mpr
-    have hnon : 0 ≤ M^(1/(2:ℝ))*(P E).toReal^(1/(2:ℝ)) := by positivity
-    have hs : (m/2)^2 ≤ (M^(1/(2:ℝ))*(P E).toReal^(1/(2:ℝ)))^2 := by nlinarith
-    simpa only [mul_pow,hrootM,hrootP,mul_comm] using hs
-  exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top P E)).mpr hle
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-namespace RegularSLEPresentation
-variable {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-variable {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-omit [IsProbabilityMeasure P] in
-lemma source_finite_mass_le_sausage (hκ : 0 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 2 < z₀.im) (T : ℝ≥0) {e : ℝ} (he : 0 < e) (he1 : e ≤ 1)
-    (ω : a.samples) :
-    finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω)
-      (e/2) T (closedBall z₀ 1) ≤
-    ENNReal.ofReal ((e/2)^(-codimension κ)) *
-      volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e) := by
-  have hs : MeasurableSet (sourceSausageSet (γ ω) 0 T e) := (source_sausage_isOpen _ _ _ _).measurableSet
-  calc
-    _ ≤ ∫⁻ z in closedBall z₀ 1, (sourceSausageSet (γ ω) 0 T e).indicator
-        (fun _ => ENNReal.ofReal ((e/2)^(-codimension κ))) z := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem measurableSet_closedBall] with z hz'
-      by_cases hh : spatialHitTime (γ ω) z (e/2) ≤ T
-      · obtain ⟨u,hu,hue⟩ := (spatialHitTime_le_iff (a.trace_continuous ω) z (e/2) T).mp hh
-        have hmem : z ∈ sourceSausageSet (γ ω) 0 T e := ⟨u,hu,by rw [dist_comm,hue]; linarith⟩
-        simp only [ite_eq_left hh,Set.indicator_of_mem hmem]
-        apply ENNReal.ofReal_le_ofReal
-        exact ((a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _)
-          hκ hκ8 (by positivity) z (by have := source_unit_ball_height hz hz'; linarith)).2
-      · simp only [ite_eq_right hh,ENNReal.ofReal_zero,zero_le]
-    _ = _ := by
-      rw [lintegral_indicator hs,lintegral_const]
-      simp [Measure.restrict_apply hs,inter_comm]
-lemma source_positive_probability_initial_sausage (hκ : 4 < κ) (hκ8 : κ < 8)
-    (z₀ : ℂ) (hz : 2 < z₀.im) :
-    ∃ (T : ℝ≥0) (p b e₀ : ℝ), 0 < T ∧ 0 < p ∧ 0 < b ∧ 0 < e₀ ∧
-      ∀ e : ℝ, 0 < e → e < e₀ →
-        ENNReal.ofReal p ≤ (P.comap (Subtype.val : a.samples → Ω))
-          {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
-            volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e)} := by
-  have hk : 0 < κ := by linarith
-  let Q := P.comap (Subtype.val : a.samples → Ω)
-  obtain ⟨T,e₀,m₀,M,hT,he₀,hm₀,hM,hMom⟩ := sourceRetainedSeed_proved κ hκ hκ8 Ω m P inferInstance γ a z₀ hz
-  let C := ((1:ℝ)/2)^(-codimension κ)
-  have hC : 0 < C := Real.rpow_pos_of_pos (by norm_num) _
-  refine ⟨T,(m₀/2)^2/M,(m₀/2)/C,min 1 (2*e₀),hT,by positivity,by positivity,lt_min (by norm_num) (by positivity),?_⟩
-  intro e he hecut
-  let X := fun ω : a.samples => finiteSpatialGreenMass κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (e/2) T (closedBall z₀ 1)
-  have hXm : Measurable X := (a.sourceFiniteGreenDensity_measurable (e/2) T).lintegral_prod_right'
-  have hhalf : e/2 ≤ e₀ := by have := hecut.trans_le (min_le_right _ _); linarith
-  have hXX := hMom (e/2) (by positivity) hhalf
-  apply (source_second_moment_seed Q X hXm m₀ M hm₀ hM hXX.1 hXX.2).trans
-  apply measure_mono
-  intro ω hω
-  have hbound := a.source_finite_mass_le_sausage hk hκ8 z₀ hz T he
-    (hecut.le.trans (min_le_left _ _)) ω
-  have heq : (e/2)^(-codimension κ) = C*e^(-codimension κ) := by
-    rw [show e/2 = (1/2)*e by ring,Real.mul_rpow (by norm_num) he.le]
-  rw [heq,ENNReal.ofReal_mul hC.le,mul_assoc] at hbound
-  have hdiv : ENNReal.ofReal (m₀/2) / ENNReal.ofReal C ≤
-      ENNReal.ofReal (e^(-codimension κ))*volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e) := by
-    apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
-    simpa only [mul_comm] using (hω.trans hbound)
-  simpa only [mem_ofPred_eq,ENNReal.ofReal_div_of_pos hC] using hdiv
-end RegularSLEPresentation
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Complex Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_volume_lipschitz_image {f : ℂ → ℂ} {S : Set ℂ} {L : ℝ≥0}
-    (hf : LipschitzOnWith L f S) : volume (f '' S) ≤ (L:ℝ≥0∞)^2*volume S := by
-  have hhaar : (Measure.hausdorffMeasure (2:ℝ) : Measure ℂ).IsAddHaarMeasure := by
-    simpa only [Complex.finrank_real_complex, Nat.cast_ofNat] using
-      (inferInstance : (Measure.hausdorffMeasure (Module.finrank ℝ ℂ) : Measure ℂ).IsAddHaarMeasure)
-  let _ := hhaar
-  have he := Measure.isAddLeftInvariant_eq_smul (Measure.hausdorffMeasure (2:ℝ) : Measure ℂ) volume
-  have hc := Measure.addHaarScalarFactor_pos_of_isAddHaarMeasure
-    (Measure.hausdorffMeasure (2:ℝ) : Measure ℂ) volume
-  have hh := hf.hausdorffMeasure_image_le (d := 2) (by norm_num)
-  rw [he,Measure.smul_apply,Measure.smul_apply] at hh
-  simp only [ENNReal.smul_def,smul_eq_mul,ENNReal.rpow_two] at hh
-  rw [mul_left_comm] at hh
-  exact (ENNReal.mul_le_mul_iff_right (by exact_mod_cast hc.ne') (by simp)).mp hh
-lemma source_holomorphic_compact_lipschitz {f : ℂ → ℂ} {D K : Set ℂ}
-    (hD : IsOpen D) (hK : IsCompact K) (hKD : K ⊆ D) (hf : DifferentiableOn ℂ f D) :
-    ∃ L : ℝ≥0, LipschitzOnWith L f K := by
-  apply LocallyLipschitzOn.exists_lipschitzOnWith_of_compact hK
-  intro x hx
-  obtain ⟨L,V,hV,hLip⟩ := (hf.analyticAt (hD.mem_nhds (hKD hx))).contDiffAt.exists_lipschitzOnWith
-  exact ⟨L,V,mem_nhdsWithin_of_mem_nhds hV,hLip⟩
-lemma source_inverse_chart_compact_bilipschitz
-    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
-    (hw : IsCapacityTwoWitness U γ G F) (s : ℝ≥0)
-    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) :
-    ∃ L : ℝ≥0, 0 < L ∧ LipschitzOnWith L (F s) K ∧
-      ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ (L:ℝ)*dist (F s x) (F s y) := by
-  have hFc := hw.2.2.2.2.1 s
-  have hGc := hw.2.2.2.1 s
-  have hFD : F s '' K ⊆ survivingDomain γ s := by
-    rintro _ ⟨z,hz,rfl⟩
-    exact hw.2.2.2.2.2.2.1 s (hKH hz)
-  obtain ⟨L,hL⟩ := source_holomorphic_compact_lipschitz isOpen_upperHalfPlane hK hKH hFc
-  obtain ⟨M,hM⟩ := source_holomorphic_compact_lipschitz (isOpen_survivingDomain hw.1 s)
-    (hK.image_of_continuousOn (hFc.continuousOn.mono hKH)) hFD hGc
-  refine ⟨max 1 (max L M),lt_of_lt_of_le zero_lt_one (le_max_left _ _),hL.weaken (le_trans (le_max_left _ _) (le_max_right _ _)),?_⟩
-  intro x hx y hy
-  have hh := hM.dist_le_mul (F s x) ⟨x,hx,rfl⟩ (F s y) ⟨y,hy,rfl⟩
-  rw [hw.2.2.2.2.2.2.2.2.1 s x (hKH hx),hw.2.2.2.2.2.2.2.2.1 s y (hKH hy)] at hh
-  exact hh.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast (le_trans (le_max_right L M) (le_max_right 1 (max L M)))) dist_nonneg)
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.ActualRetainedKernel
+
+namespace OAI
+
+/-! Trace Pos Combined for chordal SLE. -/
+
+section
+
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_gauge_cost_nonempty (h : ℝ → ℝ) (hh : IsGauge h) (A : Set ℂ) :
+    (⨆ _ : A.Nonempty, extendedGauge h (ediam A)) = extendedGauge h (ediam A) := by
+  by_cases hA : A.Nonempty
+  · simp [hA]
+  · rw [Set.not_nonempty_iff_eq_empty.mp hA]
+    simp [extendedGauge,hh.zero]
+
+lemma source_gauge_cover_formula (h : ℝ → ℝ) (hh : IsGauge h) (K : Set ℂ) :
+    hausdorffGauge h K = ⨆ (r : ℝ≥0∞) (_ : 0 < r),
+      ⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n) (_ : ∀ n, ediam (U n) ≤ r),
+        ∑' n, extendedGauge h (ediam (U n)) := by
+  simp only [hausdorffGauge,Measure.mkMetric_apply,source_gauge_cost_nonempty h hh]
+
+lemma source_gauge_open_enlargement (h : ℝ → ℝ) (hh : IsGauge h)
+    (A : Set ℂ) (r : ℝ) (hr : (ediam A).toReal < r) (hA : ediam A ≠ ∞)
+    (ε : ℝ≥0) (hε : 0 < ε) :
+    ∃ U : Set ℂ, IsOpen U ∧ A ⊆ U ∧ ediam U < ENNReal.ofReal r ∧
+      extendedGauge h (ediam U) ≤ extendedGauge h (ediam A) + ε := by
+  let b (n : ℕ) : ℝ := 1 / ((n:ℝ)+1)
+  have hb (n : ℕ) : 0 < b n := by dsimp [b]; positivity
+  have hb0 : Tendsto b atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  let d := (ediam A).toReal
+  have hd : 0 ≤ d := ENNReal.toReal_nonneg
+  have hlim : Tendsto (fun n => d + 2*b n) atTop (𝓝 d) := by
+    simpa only [mul_zero,add_zero] using tendsto_const_nhds.add (hb0.const_mul 2)
+  have hhlim : Tendsto (fun n => ENNReal.ofReal (h (d+2*b n))) atTop
+      (𝓝 (extendedGauge h (ediam A))) := by
+    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+      ((hh.continuousOn d hd).tendsto.comp (tendsto_nhdsWithin_iff.mpr
+        ⟨hlim,Eventually.of_forall (fun n => show 0 ≤ d+2*b n by positivity)⟩))
+  have hcost : extendedGauge h (ediam A) < extendedGauge h (ediam A) + ε := by
+    exact ENNReal.lt_add_right (by simp [extendedGauge]) (ENNReal.coe_pos.mpr hε).ne'
+  obtain ⟨n,hn,hc⟩ := ( (hlim.eventually (gt_mem_nhds hr)).and
+    (hhlim.eventually (gt_mem_nhds hcost)) ).exists
+  let U := thickening (b n) A
+  have hUb : Bornology.IsBounded U := (isBounded_iff_ediam_ne_top.mpr hA).thickening
+  have hdiam : (ediam U).toReal ≤ d+2*b n := diam_thickening_le A (hb n).le
+  refine ⟨U,isOpen_thickening, self_subset_thickening (hb n) A,?_,?_⟩
+  · rw [←ENNReal.ofReal_toReal hUb.ediam_ne_top]
+    exact ENNReal.ofReal_lt_ofReal_iff'.mpr ⟨hdiam.trans_lt hn,by linarith⟩
+  · exact (ENNReal.ofReal_le_ofReal (hh.monotoneOn ENNReal.toReal_nonneg
+      (show 0 ≤ d+2*b n by positivity) hdiam)).trans hc.le
+
+ 
+def sourceFiniteOpenCover (h : ℝ → ℝ) (r ε : ℝ≥0∞) (K : Set ℂ) : Prop :=
+  ∃ (U : ℕ → Set ℂ) (I : Finset ℕ), (∀ n ∈ I, IsOpen (U n)) ∧
+    (∀ n ∈ I, ediam (U n) ≤ r) ∧ K ⊆ ⋃ n ∈ I, U n ∧
+    (∑ n ∈ I, extendedGauge h (ediam (U n))) < ε
+
+lemma source_null_compact_finite_open_cover (h : ℝ → ℝ) (hh : IsGauge h)
+    (K : Set ℂ) (hK : IsCompact K) (hzero : hausdorffGauge h K = 0)
+    (r ε : ℝ) (hr : 0 < r) (hε : 0 < ε) :
+    sourceFiniteOpenCover h (ENNReal.ofReal r) (ENNReal.ofReal ε) K := by
+  have hsmall :
+      (⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n)
+        (_ : ∀ n, ediam (U n) ≤ ENNReal.ofReal (r/2)),
+          ∑' n, extendedGauge h (ediam (U n))) < ENNReal.ofReal (ε/2) := by
+    apply lt_of_le_of_lt (b := hausdorffGauge h K)
+    · rw [source_gauge_cover_formula h hh]
+      exact le_iSup_of_le (ENNReal.ofReal (r/2))
+        (le_iSup_of_le (ENNReal.ofReal_pos.mpr (by positivity)) le_rfl)
+    · simpa only [hzero] using ENNReal.ofReal_pos.mpr (show 0 < ε/2 by positivity)
+  obtain ⟨U,hU,hdiam,hcost⟩ := (by simpa only [iInf_lt_iff] using hsmall)
+  obtain ⟨e,he,hes⟩ := ENNReal.exists_pos_sum_of_countable
+    (ENNReal.ofReal_pos.mpr (show 0 < ε/2 by positivity)).ne' ℕ
+  have hfin (n : ℕ) : ediam (U n) ≠ ∞ :=
+    ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hdiam n)
+  have hrd (n : ℕ) : (ediam (U n)).toReal < r := by
+    have hb := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hdiam n)
+    rw [ENNReal.toReal_ofReal (by positivity)] at hb
+    linarith
+  choose V hopen hUV hVdiam hVcost using fun n =>
+    source_gauge_open_enlargement h hh (U n) r (hrd n) (hfin n) (e n) (he n)
+  obtain ⟨I,hI⟩ := hK.elim_finite_subcover V hopen (hU.trans (iUnion_mono hUV))
+  refine ⟨V,I,fun n _ => hopen n,fun n _ => (hVdiam n).le,hI,?_⟩
+  calc
+    (∑ n ∈ I, extendedGauge h (ediam (V n))) ≤ ∑' n, extendedGauge h (ediam (V n)) :=
+      ENNReal.sum_le_tsum I
+    _ ≤ ∑' n, (extendedGauge h (ediam (U n)) + e n) := ENNReal.tsum_le_tsum hVcost
+    _ = (∑' n, extendedGauge h (ediam (U n))) + ∑' n, (e n : ℝ≥0∞) := ENNReal.tsum_add
+    _ < ENNReal.ofReal (ε/2) + ENNReal.ofReal (ε/2) := ENNReal.add_lt_add hcost hes
+    _ = ENNReal.ofReal ε := by rw [←ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring
+end
+end SLEExactGauge
+end
+
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+def sourceFineGaugeContent (h : ℝ → ℝ) (r : ℝ≥0∞) (K : Set ℂ) : ℝ≥0∞ :=
+  ⨅ (U : ℕ → Set ℂ) (_ : K ⊆ ⋃ n, U n) (_ : ∀ n, ediam (U n) ≤ r),
+    ∑' n, extendedGauge h (ediam (U n))
+lemma source_fine_content_finite_cover (h : ℝ → ℝ) (hh : IsGauge h)
+    {K : Set ℂ} {R r ε : ℝ≥0∞} (hr : r ≤ R) (hc : sourceFiniteOpenCover h r ε K) :
+    sourceFineGaugeContent h R K < ε := by
+  classical
+  obtain ⟨U,I,hopen,hdiam,hcover,hcost⟩ := hc
+  let V (n : ℕ) : Set ℂ := if n ∈ I then U n else ∅
+  have hKV : K ⊆ ⋃ n, V n := by
+    intro z hz
+    obtain ⟨n,hn,hzn⟩ := mem_iUnion₂.mp (hcover hz)
+    exact mem_iUnion.mpr ⟨n,by simpa only [V,ite_eq_left hn] using hzn⟩
+  have hVdiam (n : ℕ) : ediam (V n) ≤ R := by
+    by_cases hn : n ∈ I
+    · simpa only [V,ite_eq_left hn] using (hdiam n hn).trans hr
+    · simp [V,hn]
+  unfold sourceFineGaugeContent
+  apply lt_of_le_of_lt (b := ∑' n, extendedGauge h (ediam (V n)))
+  · exact iInf_le_of_le V (iInf_le_of_le hKV (iInf_le_of_le hVdiam le_rfl))
+  have hsum : (∑' n, extendedGauge h (ediam (V n))) = ∑ n ∈ I, extendedGauge h (ediam (U n)) := by
+    rw [tsum_eq_sum (s := I) (fun n hn => by simp [V,hn,extendedGauge,hh.zero])]
+    exact Finset.sum_congr rfl (fun n hn => by simp only [V,ite_eq_left hn])
+  simpa only [hsum] using hcost
+
+noncomputable def sourceCoverScale (n : ℕ) : ℝ≥0∞ := ENNReal.ofReal (1/((n:ℝ)+1))
+lemma source_coverScale_pos (n : ℕ) : 0 < sourceCoverScale n := by
+  unfold sourceCoverScale
+  positivity
+lemma source_coverScale_tendsto : Tendsto sourceCoverScale atTop (𝓝 0) := by
+  unfold sourceCoverScale
+  simpa only [ENNReal.ofReal_zero] using ENNReal.tendsto_ofReal
+    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
+
+lemma source_compact_gauge_null_criterion (h : ℝ → ℝ) (hh : IsGauge h)
+    (K : Set ℂ) (hK : IsCompact K) :
+    hausdorffGauge h K = 0 ↔ ∀ n : ℕ,
+      sourceFiniteOpenCover h (sourceCoverScale n) (sourceCoverScale n) K := by
+  constructor
+  · intro hz n
+    exact source_null_compact_finite_open_cover h hh K hK hz _ _ (by positivity) (by positivity)
+  · intro hcov
+    apply le_antisymm _ zero_le
+    rw [source_gauge_cover_formula h hh]
+    refine iSup_le fun R => iSup_le fun hR => ?_
+    change sourceFineGaugeContent h R K ≤ 0
+    apply le_of_tendsto_of_tendsto tendsto_const_nhds source_coverScale_tendsto
+    filter_upwards [source_coverScale_tendsto.eventually (gt_mem_nhds hR)] with n hn
+    exact (source_fine_content_finite_cover h hh hn.le (hcov n)).le
+end
+end SLEExactGauge
+end
+
+section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance tracePosCoverComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance tracePosCoverComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_finite_open_cover_trace_isOpen (h : ℝ → ℝ) (C : Set ℂ) (hC : IsClosed C)
+    (s t : ℝ≥0) (r ε : ℝ≥0∞) :
+    IsOpen {γ : C(ℝ≥0,ℂ) | sourceFiniteOpenCover h r ε (C ∩ (γ '' Icc s t))} := by
+  classical
+  apply isOpen_iff_mem_nhds.mpr
+  intro γ hγ
+  obtain ⟨U,I,ho,hd,hcov,hcost⟩ := hγ
+  let O := ⋃ n ∈ I, U n
+  have hO : IsOpen O := isOpen_iUnion (fun n => isOpen_iUnion (fun hn => ho n hn))
+  have hmap : MapsTo γ (Icc s t) (Cᶜ ∪ O) := by
+    intro u hu
+    by_cases hc : γ u ∈ C
+    · exact Or.inr (hcov ⟨hc,⟨u,hu,rfl⟩⟩)
+    · exact Or.inl hc
+  have hev := (ContinuousMap.isOpen_setOfPred_mapsTo isCompact_Icc (hC.isOpen_compl.union hO)).mem_nhds hmap
+  filter_upwards [hev] with g hg
+  refine ⟨U,I,ho,hd,?_,hcost⟩
+  rintro z ⟨hz,u,hu,rfl⟩
+  exact (hg hu).resolve_left (fun hn => hn hz)
+ 
+
+def SourceTracePositiveMeasurableTarget : Prop :=
+  ∀ (h : ℝ → ℝ), IsGauge h → ∀ (C : Set ℂ), IsClosed C → ∀ s t : ℝ≥0,
+    @MeasurableSet C(ℝ≥0, ℂ) (borel _) {γ | 0 < hausdorffGauge h (C ∩ (γ '' Set.Icc s t))}
+theorem sourceTracePositiveMeasurable_proved : SourceTracePositiveMeasurableTarget := by
+  intro h hh C hC s t
+  have hnull : MeasurableSet {γ : C(ℝ≥0,ℂ) | hausdorffGauge h (C ∩ (γ '' Icc s t)) = 0} := by
+    have heq : {γ : C(ℝ≥0,ℂ) | hausdorffGauge h (C ∩ (γ '' Icc s t)) = 0} =
+        ⋂ n : ℕ, {γ : C(ℝ≥0,ℂ) | sourceFiniteOpenCover h (sourceCoverScale n) (sourceCoverScale n)
+          (C ∩ (γ '' Icc s t))} := by
+      ext γ
+      simp only [mem_ofPred_eq,mem_iInter]
+      exact source_compact_gauge_null_criterion h hh _
+        ((isCompact_Icc.image γ.continuous).inter_left hC)
+    rw [heq]
+    exact MeasurableSet.iInter (fun n =>
+      (source_finite_open_cover_trace_isOpen h C hC s t _ _).measurableSet)
+  simpa only [compl_ofPred,←ne_eq,←pos_iff_ne_zero] using hnull.compl
+end
+end SLEExactGauge
+end
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance tracePosDilationRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance tracePosDilationComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance tracePosDilationRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance tracePosDilationComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_dilation_gauge_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    {c : ℂ} (hc : c ≠ 0) {S : Set ℂ} (hS : IsCompact S)
+    (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => c*z) '' S) := by
+  by_contra hn
+  have hz : hausdorffGauge h ((fun z => c*z) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le
+  have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
+    (D := univ) (f := fun z => c⁻¹*z) isOpen_univ (hS.image (continuous_const.mul continuous_id))
+    (subset_univ _) (by fun_prop) hz
+  have hi : (fun z => c⁻¹*z) '' ((fun z => c*z) '' S) = S := by
+    rw [←image_comp]
+    simp only [Function.comp_def,←mul_assoc,inv_mul_cancel₀ hc,one_mul,image_id']
+  change hausdorffGauge h ((fun z => c⁻¹*z) '' ((fun z => c*z) '' S)) = 0 at he
+  rw [hi] at he
+  exact hp.ne' he
+
+lemma RegularSLEPresentation.source_trace_path_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) :
+    Measurable (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) := by
+  apply source_path_measurable
+  intro t
+  exact (a.trace_adapted t).mono (a.filtration.le t) le_rfl
+
+lemma RegularSLEPresentation.source_scale_trace_law
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (c : ℝ≥0) (hc : 0 < c) :
+    (P.comap (Subtype.val : a.samples → Ω)).map
+      (sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous) =
+    (P.comap (Subtype.val : a.samples → Ω)).map
+      (sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
+        (a.sourceScaled c hc).trace_continuous) := by
+  exact sourceTraceLawUnique_proved κ hκ hκ8 a.B (a.sourceScaled c hc).B
+    a.brownian (a.sourceScaled c hc).brownian a.measurable_B (a.sourceScaled c hc).measurable_B
+    a.continuous_B (a.sourceScaled c hc).continuous_B _ _ a.G a.F
+    (a.sourceScaled c hc).G (a.sourceScaled c hc).F a.witness (a.sourceScaled c hc).witness
+ 
+
+def SourceUniformInteriorSeedTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (h : ℝ → ℝ), IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+  ∃ p : ℝ≥0∞, 0 < p ∧ ∀ δ : ℝ≥0, 0 < δ →
+    ∃ K : Set ℂ, IsCompact K ∧ K ⊆ upperHalfPlane ∧
+      p ≤ (P.comap (Subtype.val : a.samples → Ω))
+        {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Set.Icc 0 δ))}
+theorem sourceUniformInteriorSeed_proved : SourceUniformInteriorSeedTarget := by
+  intro κ hκ hκ8 Ω m P hP γ a h hh hf
+  let : IsProbabilityMeasure P := hP
+  have hk : 0 < κ := by linarith
+  let z₀ : ℂ := 3*I
+  have hz₀ : 2 < z₀.im := by norm_num [z₀]
+  obtain ⟨T,E,hT,hE,hp,hseed⟩ :=
+    sourceInteriorRetainedTransfer_proved κ hκ hκ8 Ω m P hP γ a z₀ hz₀ h hh hf
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  refine ⟨Q E,hp,?_⟩
+  intro δ hδ
+  let c := NNReal.sqrt (δ/T)
+  have hc : 0 < c := by dsimp [c]; exact NNReal.sqrt_pos.mpr (div_pos hδ hT)
+  have hc2 : c^2 = δ/T := NNReal.sq_sqrt _
+  have ht : sourceScaleTime c δ = T := by
+    unfold sourceScaleTime
+    rw [hc2]
+    field_simp
+  let K := (fun z : ℂ => (c:ℂ)*z) '' closedBall z₀ 1
+  have hK : IsCompact K := (isCompact_closedBall z₀ 1).image (continuous_const.mul continuous_id)
+  have hKH : K ⊆ upperHalfPlane := by
+    rintro _ ⟨z,hz,rfl⟩
+    have hz' : 0 < z.im := zero_lt_one.trans (source_unit_ball_height hz₀ hz)
+    change 0 < ((c:ℂ)*z).im
+    simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos (show 0 < (c:ℝ) from hc) hz'
+  refine ⟨K,hK,hKH,?_⟩
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let sp := sourceTracePath (fun ω : a.samples => sourceScaleTrace c (γ ω))
+    (a.sourceScaled c hc).trace_continuous
+  let A : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))}
+  have hA : MeasurableSet A := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ
+  have hm : Measurable gp := a.source_trace_path_measurable
+  have hsm : Measurable sp := (a.sourceScaled c hc).source_trace_path_measurable
+  have hmap := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ A) (a.source_scale_trace_law hk hκ8 c hc)
+  rw [Measure.map_apply hm hA,Measure.map_apply hsm hA] at hmap
+  change Q E ≤ Q (gp ⁻¹' A)
+  rw [hmap]
+  apply measure_mono
+  intro ω hω
+  have hpos := (hseed ω hω).1
+  have heq : K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ) =
+      (fun z : ℂ => (c:ℂ)*z) '' (closedBall z₀ 1 ∩ (γ ω '' Icc 0 T)) := by
+    rw [source_scale_past c δ hc,ht, image_inter (mul_right_injective₀ (by exact_mod_cast hc.ne' : (c:ℂ) ≠ 0))]
+  change 0 < hausdorffGauge h (K ∩ (sourceScaleTrace c (γ ω) '' Icc 0 δ))
+  rw [heq]
+  exact source_dilation_gauge_positive κ hk hκ8 h hh hf (by exact_mod_cast hc.ne')
+    ((isCompact_closedBall z₀ 1).inter_right (isCompact_Icc.image (a.trace_continuous ω)).isClosed) hpos
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance tracePosRestartRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance tracePosRestartComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance tracePosRestartRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+local instance tracePosRestartComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+lemma source_translation_gauge_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    (b : ℂ) {S : Set ℂ} (hS : IsCompact S)
+    (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => z+b) '' S) := by
+  by_contra hn
+  have hz : hausdorffGauge h ((fun z => z+b) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le
+  have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf
+    (D := univ) (f := fun z => z-b) isOpen_univ (hS.image (continuous_id.add_const b))
+    (subset_univ _) (by fun_prop) hz
+  have hi : (fun z => z-b) '' ((fun z => z+b) '' S) = S := by
+    rw [←image_comp]
+    simp only [Function.comp_def,add_sub_cancel_right,image_id']
+  change hausdorffGauge h ((fun z => z-b) '' ((fun z => z+b) '' S)) = 0 at he
+  rw [hi] at he
+  exact hp.ne' he
+
+lemma source_restart_segment_positive (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ))
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (s δ : ℝ≥0) (η : C(ℝ≥0,ℂ))
+    (hid : ∀ t : ℝ≥0, η t ∈ upperHalfPlane → γ (s+t) = F s (η t + (U s:ℂ)))
+    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane)
+    (hp : 0 < hausdorffGauge h (K ∩ (η '' Icc 0 δ))) :
+    0 < hausdorffGauge h (γ '' Icc s (s+δ)) := by
+  let L := K ∩ (η '' Icc 0 δ)
+  have hL : IsCompact L := hK.inter_right (isCompact_Icc.image η.continuous).isClosed
+  have hLH : L ⊆ upperHalfPlane := inter_subset_left.trans hKH
+  have htrans := source_translation_gauge_positive κ hκ hκ8 h hh hf (U s:ℂ) hL hp
+  have htrK : IsCompact ((fun z => z+(U s:ℂ)) '' L) := hL.image (continuous_id.add_const _)
+  have htrH : (fun z => z+(U s:ℂ)) '' L ⊆ upperHalfPlane := by
+    rintro _ ⟨z,hz,rfl⟩
+    simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,add_zero] using hLH hz
+  have hpositive := source_inverse_chart_gauge_positive κ hκ hκ8 h hh hf U γ G F hw s htrK htrH htrans
+  apply hpositive.trans_le (measure_mono ?_)
+  rintro _ ⟨_,⟨z,hz,rfl⟩,rfl⟩
+  obtain ⟨u,hu,rfl⟩ := hz.2
+  refine ⟨s+u,⟨le_add_of_nonneg_right hu.1,add_le_add le_rfl hu.2⟩,?_⟩
+  exact hid u (hKH hz.1)
+
+lemma RegularSLEPresentation.source_segment_positive_measurable_at
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h)
+    (s t : ℝ≥0) :
+    MeasurableSet[a.filtration t] {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} := by
+  let f : a.samples → C(ℝ≥0,ℂ) := fun ω =>
+    ⟨fun u => γ ω (min u t), (a.trace_continuous ω).comp (continuous_id.min continuous_const)⟩
+  have hm : @Measurable a.samples C(ℝ≥0,ℂ) (a.filtration t) (borel _) f := by
+    let : MeasurableSpace a.samples := a.filtration t
+    apply source_path_measurable
+    intro u
+    exact (a.trace_adapted (min u t)).mono (a.filtration.mono (min_le_right u t)) le_rfl
+  have heq : {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} =
+      f ⁻¹' {g : C(ℝ≥0,ℂ) | 0 < hausdorffGauge h (univ ∩ (g '' Icc s t))} := by
+    ext ω
+    simp only [mem_preimage,mem_ofPred_eq,univ_inter]
+    have hi : γ ω '' Icc s t = f ω '' Icc s t := by
+      apply image_congr
+      intro u hu
+      simp only [f,ContinuousMap.coe_mk,min_eq_left hu.2]
+    rw [hi]
+  rw [heq]
+  exact (sourceTracePositiveMeasurable_proved h hh univ isClosed_univ s t).preimage hm
+
+lemma RegularSLEPresentation.source_segment_null_measurable_at
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h) (s t : ℝ≥0) :
+    MeasurableSet[a.filtration t] {ω : a.samples | hausdorffGauge h (γ ω '' Icc s t) = 0} := by
+  simpa only [compl_ofPred,not_lt,nonpos_iff_eq_zero] using
+    (a.source_segment_positive_measurable_at h hh s t).compl
+
+lemma RegularSLEPresentation.source_uniform_null_slot_bound
+    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h)
+    (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ →
+      ∀ A : Set a.samples, MeasurableSet[a.filtration s] A →
+        (P.comap Subtype.val) (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0}) ≤
+          (1-p)*(P.comap Subtype.val) A := by
+  have hk : 0 < κ := by linarith
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨p,hp,hseed⟩ := sourceUniformInteriorSeed_proved κ hκ hκ8 Ω m P inferInstance γ a h hh hf
+  have hp1 : p ≤ 1 := by
+    obtain ⟨K,hK,hKH,hKP⟩ := hseed 1 zero_lt_one
+    exact hKP.trans (prob_le_one)
+  refine ⟨p,hp,hp1,?_⟩
+  intro s δ hδ A hA
+  obtain ⟨K,hK,hKH,hKP⟩ := hseed δ hδ
+  obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian
+    a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness
+  let τ : a.samples → ℝ≥0 := fun _ => s
+  have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s
+  have hAstop : MeasurableSet[hτ.measurableSpace] A := by
+    change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A
+    rw [IsStoppingTime.measurableSpace_const]
+    exact hA
+  obtain ⟨hlaw,hident⟩ := hrestart τ hτ
+  let rp := sourceRestartDriverPath a.B a.continuous_B τ
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous
+  let C : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))}
+  let E : Set a.samples := (fun ω => Φ (rp ω)) ⁻¹' C
+  have hC : MeasurableSet C := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ
+  have hrm : Measurable rp := by
+    apply source_path_measurable
+    intro t
+    exact (a.measurable_B (s+t)).sub (a.measurable_B s)
+  have hem : Measurable (fun ω => Φ (rp ω)) := hΦm.comp hrm
+  have hEm : MeasurableSet E := hC.preimage hem
+  have hAm : MeasurableSet A := (a.filtration.le s) A hA
+  have hprob : Q (A ∩ E) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} := by
+    have he := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ C) (hlaw A hAstop)
+    rw [Measure.map_apply hem hC,Measure.smul_apply,
+      Measure.map_apply a.source_trace_path_measurable hC,
+      Measure.restrict_apply (hC.preimage hem)] at he
+    change Q (E ∩ A) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} at he
+    simpa only [inter_comm] using he
+  have hins : (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0} : Set a.samples) ≤ᵐ[Q] (A \ E : Set a.samples) := by
+    filter_upwards [hident] with ω hω
+    intro hx
+    refine ⟨hx.1,?_⟩
+    intro hsuc
+    have hpositive := source_restart_segment_positive κ hk hκ8 h hh hf
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω)
+      s δ (Φ (rp ω)) hω hK hKH hsuc
+    rw [hx.2] at hpositive
+    exact lt_irrefl _ hpositive
+  have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by
+    have he : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto
+    rw [he]
+    exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _)
+  calc Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0})
+      ≤ Q (A \ E) := measure_mono_ae hins
+    _ = Q A - Q (A ∩ E) := hdiff
+    _ ≤ Q A - Q A*p := tsub_le_tsub_left (by rw [hprob]; gcongr) _
+    _ = (1-p)*Q A := by
+      rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul,mul_comm p (Q A)]
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma RegularSLEPresentation.source_positive_fixed_interval
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h)
+    (hf : HasSmallRadiusFormula h (hFormula κ)) (s t : ℝ≥0) (hst : s < t) :
+    ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), 0 < hausdorffGauge h (γ ω '' Icc s t) := by
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_null_slot_bound hκ hκ8 h hh hf
+  let c : ℝ≥0∞ := 1-p
+  have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne'
+  have hsteps (δ : ℝ≥0) (hδ : 0 < δ) (k : ℕ) :
+      Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0} ≤ c^k := by
+    induction k with
+    | zero => simpa using (prob_le_one (μ := Q)
+        (s := {ω | hausdorffGauge h (γ ω '' Icc s (s+(0:ℝ≥0)*δ)) = 0}))
+    | succ k ih =>
+      have he : s+((k+1:ℕ):ℝ≥0)*δ = (s+(k:ℝ≥0)*δ)+δ := by push_cast; ring
+      rw [he,pow_succ,mul_comm (c^k) c]
+      let A : Set a.samples := {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0}
+      have hA : MeasurableSet[a.filtration (s+(k:ℝ≥0)*δ)] A :=
+        a.source_segment_null_measurable_at h hh s _
+      calc Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ+δ)) = 0}
+          ≤ Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc (s+(k:ℝ≥0)*δ) (s+(k:ℝ≥0)*δ+δ)) = 0}) := by
+            apply measure_mono
+            intro ω hω
+            constructor
+            · exact measure_mono_null (image_mono (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))) hω
+            · exact measure_mono_null (image_mono (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)) hω
+        _ ≤ c * Q A := hslot _ δ hδ A hA
+        _ ≤ c * c^k := by gcongr
+  have hbound : ∀ᶠ n : ℕ in atTop,
+      Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} ≤ c^n := by
+    filter_upwards [eventually_ge_atTop 1] with n hn
+    have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega)
+    let δ : ℝ≥0 := (t-s)/(n:ℝ≥0)
+    have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0
+    have he : s+(n:ℝ≥0)*δ = t := by
+      dsimp [δ]
+      rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le]
+    simpa only [he] using hsteps δ hδ n
+  have hzero : Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} = 0 :=
+    le_antisymm (ge_of_tendsto (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc) hbound) zero_le
+  apply ae_iff.mpr
+  simpa only [not_lt,nonpos_iff_eq_zero] using hzero
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma RegularSLEPresentation.source_positive_all_intervals
+    {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hκ : 4 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h)
+    (hf : HasSmallRadiusFormula h (hFormula κ)) :
+    ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω),
+      ∀ s t : ℝ≥0, 0 < s → s < t → 0 < hausdorffGauge h (γ ω '' Icc s t) := by
+  have hrat : ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω),
+      ∀ r q : ℚ, (r:ℝ).toNNReal < (q:ℝ).toNNReal →
+        0 < hausdorffGauge h (γ ω '' Icc (r:ℝ).toNNReal (q:ℝ).toNNReal) := by
+    apply ae_all_iff.mpr
+    intro r
+    apply ae_all_iff.mpr
+    intro q
+    apply ae_all_iff.mpr
+    intro hrq
+    exact a.source_positive_fixed_interval hκ hκ8 h hh hf _ _ hrq
+  filter_upwards [hrat] with ω hω
+  intro s t hs hst
+  obtain ⟨r,hsr,hrt⟩ := exists_rat_btwn (show (s:ℝ) < (t:ℝ) from hst)
+  obtain ⟨q,hrq,hqt⟩ := exists_rat_btwn hrt
+  have hq : 0 < (q:ℝ) := (show 0 < (s:ℝ) from hs).trans (hsr.trans hrq)
+  have hp := hω r q ((Real.toNNReal_lt_toNNReal_iff hq).mpr hrq)
+  apply hp.trans_le (measure_mono (image_mono (Icc_subset_Icc ?_ ?_)))
+  · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hsr.le
+  · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hqt.le
+ 
+
+def SourceLowerAllIntervalsTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+  ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) →
+    ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t →
+      0 < hausdorffGauge h (γ ω '' Set.Icc s t)
+theorem sourceLowerAllIntervals_proved : SourceLowerAllIntervalsTarget := by
+  intro κ hκ hκ8 Ω m P hP γ hSLE h hh hf
+  let : IsProbabilityMeasure P := hP
+  obtain ⟨a⟩ := exists_regular_sle_presentation hSLE
+  have ha := a.source_positive_all_intervals hκ hκ8 h hh hf
+  rw [←a.measurePreserving_val.map_eq]
+  exact (MeasurableEmbedding.subtype_coe a.measurable_samples).ae_map_iff.mpr ha
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable def sourceSausageSet (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (e : ℝ) : Set ℂ :=
+  {z | ∃ u ∈ Set.Icc s t, dist z (γ u) < e}
+noncomputable def sourceSausageMass (κ : ℝ) (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (e : ℝ) : ℝ≥0∞ :=
+  ENNReal.ofReal (e^(-codimension κ)) * MeasureTheory.volume (sourceSausageSet γ s t e)
+ 
+
+def SourceSmallSausageTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : MeasureTheory.Measure Ω),
+    MeasureTheory.IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P →
+  ∀ s t : ℝ≥0, s < t → ∀ ε : ℝ, 0 < ε →
+    ∃ a e₀ : ℝ, 0 < a ∧ 0 < e₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+      P {ω | sourceSausageMass κ (γ ω) s t e < ENNReal.ofReal a} < ENNReal.ofReal ε
+end SLEExactGauge
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+local instance tracePosSausageComplexPathMeasurable : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance tracePosSausageComplexPathBorel : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma source_sausage_isOpen (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (e : ℝ) :
+    IsOpen (sourceSausageSet γ s t e) := by
+  have he : sourceSausageSet γ s t e = ⋃ u ∈ Icc s t, ball (γ u) e := by
+    ext z; simp [sourceSausageSet]
+  rw [he]
+  exact isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_ball
+lemma source_sausage_joint_open (s t : ℝ≥0) (e : ℝ) :
+    IsOpen {p : C(ℝ≥0,ℂ) × ℂ | p.2 ∈ sourceSausageSet p.1 s t e} := by
+  have he : {p : C(ℝ≥0,ℂ) × ℂ | p.2 ∈ sourceSausageSet p.1 s t e} =
+      ⋃ u ∈ Icc s t, {p : C(ℝ≥0,ℂ) × ℂ | dist p.2 (p.1 u) < e} := by
+    ext p; simp [sourceSausageSet]
+  rw [he]
+  apply isOpen_iUnion; intro u
+  apply isOpen_iUnion; intro hu
+  exact isOpen_lt (continuous_snd.dist ((continuous_eval_const u).comp continuous_fst)) continuous_const
+lemma source_sausage_local_volume_measurable (s t : ℝ≥0) (e : ℝ) (K : Set ℂ) :
+    Measurable (fun g : C(ℝ≥0,ℂ) => volume (K ∩ sourceSausageSet g s t e)) := by
+  have hm : Measurable (fun p : C(ℝ≥0,ℂ) × ℂ =>
+      if p.2 ∈ sourceSausageSet p.1 s t e then (1:ℝ≥0∞) else 0) :=
+    Measurable.ite (source_sausage_joint_open s t e).measurableSet measurable_const measurable_const
+  have hh := hm.lintegral_prod_right' (ν := volume.restrict K)
+  convert hh using 1
+  ext g
+  change volume (K ∩ sourceSausageSet g s t e) =
+    ∫⁻ z, (sourceSausageSet g s t e).indicator (fun _ => 1) z ∂(volume.restrict K)
+  rw [lintegral_indicator (source_sausage_isOpen g s t e).measurableSet]
+  simp [Measure.restrict_apply (source_sausage_isOpen g s t e).measurableSet,inter_comm]
+lemma source_sausage_mass_measurable (κ : ℝ) (s t : ℝ≥0) (e : ℝ) :
+    Measurable (fun g : C(ℝ≥0,ℂ) => sourceSausageMass κ g s t e) := by
+  exact measurable_const.mul (by simpa only [univ_inter] using source_sausage_local_volume_measurable s t e univ)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_second_moment_seed {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) [IsProbabilityMeasure P] (X : Ω → ℝ≥0∞) (hm : Measurable X)
+    (m M : ℝ) (hp : 0 < m) (hM : 0 < M)
+    (hfirst : ENNReal.ofReal m ≤ ∫⁻ ω, X ω ∂P)
+    (hsecond : (∫⁻ ω, (X ω)^2 ∂P) ≤ ENNReal.ofReal M) :
+    ENNReal.ofReal ((m/2)^2/M) ≤ P {ω | ENNReal.ofReal (m/2) ≤ X ω} := by
+  classical
+  let E := {ω | ENNReal.ofReal (m/2) ≤ X ω}
+  have hE : MeasurableSet E := measurableSet_le measurable_const hm
+  let Y : Ω → ℝ≥0∞ := E.indicator (fun _ => 1)
+  have hYm : Measurable Y := measurable_const.indicator hE
+  have hcs := ENNReal.lintegral_mul_le_Lp_mul_Lq P
+    (show (2:ℝ).HolderConjugate 2 by constructor <;> norm_num)
+    hm.aemeasurable hYm.aemeasurable
+  have hYsq : (∫⁻ ω, (Y ω)^(2:ℝ) ∂P) = P E := by
+    simp only [Y,Set.indicator_apply]
+    simp_rw [show ∀ (b : Prop) [Decidable b], ((if b then 1 else 0):ℝ≥0∞)^(2:ℝ) = if b then 1 else 0 by intros; split_ifs <;> norm_num]
+    simpa only [Set.indicator_apply,Pi.one_apply] using (lintegral_indicator_one (μ := P) hE)
+  rw [hYsq] at hcs
+  simp only [ENNReal.rpow_two] at hcs
+  have hcs' : (∫⁻ ω, X ω*Y ω ∂P) ≤ (ENNReal.ofReal M)^(1/(2:ℝ))*(P E)^(1/(2:ℝ)) :=
+    hcs.trans (mul_le_mul' (ENNReal.rpow_le_rpow hsecond (by norm_num)) le_rfl)
+  have hfirst' : ENNReal.ofReal m ≤ ENNReal.ofReal (m/2) +
+      (ENNReal.ofReal M)^(1/(2:ℝ))*(P E)^(1/(2:ℝ)) := by
+    apply hfirst.trans
+    calc (∫⁻ ω, X ω ∂P) ≤ ∫⁻ ω, ENNReal.ofReal (m/2)+X ω*Y ω ∂P := by
+          apply lintegral_mono
+          intro ω
+          by_cases hω : ω ∈ E
+          · simp [Y,hω]
+          · have hh : X ω ≤ ENNReal.ofReal (m/2) := (lt_of_not_ge hω).le
+            simpa [Y,hω] using hh
+      _ = ENNReal.ofReal (m/2)+(∫⁻ ω, X ω*Y ω ∂P) := by
+          rw [lintegral_add_left measurable_const]
+          simp
+      _ ≤ _ := add_le_add le_rfl hcs'
+  have hb : ENNReal.ofReal (m/2) +
+      (ENNReal.ofReal M)^(1/(2:ℝ))*(P E)^(1/(2:ℝ)) ≠ ⊤ := by finiteness
+  have hh := (ENNReal.ofReal_le_iff_le_toReal hb).mp hfirst'
+  rw [ENNReal.toReal_add (by simp) (by finiteness),ENNReal.toReal_ofReal (by positivity),
+    ENNReal.toReal_mul,←ENNReal.toReal_rpow,←ENNReal.toReal_rpow,
+    ENNReal.toReal_ofReal hM.le] at hh
+  have hrootM : (M^(1/(2:ℝ)))^2 = M := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul hM.le]; norm_num
+  have hrootP : ((P E).toReal^(1/(2:ℝ)))^2 = (P E).toReal := by
+    rw [←Real.rpow_natCast,←Real.rpow_mul ENNReal.toReal_nonneg]; norm_num
+  have hle : (m/2)^2/M ≤ (P E).toReal := by
+    apply (div_le_iff₀ hM).mpr
+    have hnon : 0 ≤ M^(1/(2:ℝ))*(P E).toReal^(1/(2:ℝ)) := by positivity
+    have hs : (m/2)^2 ≤ (M^(1/(2:ℝ))*(P E).toReal^(1/(2:ℝ)))^2 := by nlinarith
+    simpa only [mul_pow,hrootM,hrootP,mul_comm] using hs
+  exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top P E)).mpr hle
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+namespace RegularSLEPresentation
+variable {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+variable {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+omit [IsProbabilityMeasure P] in
+lemma source_finite_mass_le_sausage (hκ : 0 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 2 < z₀.im) (T : ℝ≥0) {e : ℝ} (he : 0 < e) (he1 : e ≤ 1)
+    (ω : a.samples) :
+    finiteSpatialGreenMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω)
+      (e/2) T (closedBall z₀ 1) ≤
+    ENNReal.ofReal ((e/2)^(-codimension κ)) *
+      volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e) := by
+  have hs : MeasurableSet (sourceSausageSet (γ ω) 0 T e) := (source_sausage_isOpen _ _ _ _).measurableSet
+  calc
+    _ ≤ ∫⁻ z in closedBall z₀ 1, (sourceSausageSet (γ ω) 0 T e).indicator
+        (fun _ => ENNReal.ofReal ((e/2)^(-codimension κ))) z := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem measurableSet_closedBall] with z hz'
+      by_cases hh : spatialHitTime (γ ω) z (e/2) ≤ T
+      · obtain ⟨u,hu,hue⟩ := (spatialHitTime_le_iff (a.trace_continuous ω) z (e/2) T).mp hh
+        have hmem : z ∈ sourceSausageSet (γ ω) 0 T e := ⟨u,hu,by rw [dist_comm,hue]; linarith⟩
+        simp only [ite_eq_left hh,Set.indicator_of_mem hmem]
+        apply ENNReal.ofReal_le_ofReal
+        exact ((a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _)
+          hκ hκ8 (by positivity) z (by have := source_unit_ball_height hz hz'; linarith)).2
+      · simp only [ite_eq_right hh,ENNReal.ofReal_zero,zero_le]
+    _ = _ := by
+      rw [lintegral_indicator hs,lintegral_const]
+      simp [Measure.restrict_apply hs,inter_comm]
+lemma source_positive_probability_initial_sausage (hκ : 4 < κ) (hκ8 : κ < 8)
+    (z₀ : ℂ) (hz : 2 < z₀.im) :
+    ∃ (T : ℝ≥0) (p b e₀ : ℝ), 0 < T ∧ 0 < p ∧ 0 < b ∧ 0 < e₀ ∧
+      ∀ e : ℝ, 0 < e → e < e₀ →
+        ENNReal.ofReal p ≤ (P.comap (Subtype.val : a.samples → Ω))
+          {ω | ENNReal.ofReal b ≤ ENNReal.ofReal (e^(-codimension κ))*
+            volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e)} := by
+  have hk : 0 < κ := by linarith
+  let Q := P.comap (Subtype.val : a.samples → Ω)
+  obtain ⟨T,e₀,m₀,M,hT,he₀,hm₀,hM,hMom⟩ := sourceRetainedSeed_proved κ hκ hκ8 Ω m P inferInstance γ a z₀ hz
+  let C := ((1:ℝ)/2)^(-codimension κ)
+  have hC : 0 < C := Real.rpow_pos_of_pos (by norm_num) _
+  refine ⟨T,(m₀/2)^2/M,(m₀/2)/C,min 1 (2*e₀),hT,by positivity,by positivity,lt_min (by norm_num) (by positivity),?_⟩
+  intro e he hecut
+  let X := fun ω : a.samples => finiteSpatialGreenMass κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (e/2) T (closedBall z₀ 1)
+  have hXm : Measurable X := (a.sourceFiniteGreenDensity_measurable (e/2) T).lintegral_prod_right'
+  have hhalf : e/2 ≤ e₀ := by have := hecut.trans_le (min_le_right _ _); linarith
+  have hXX := hMom (e/2) (by positivity) hhalf
+  apply (source_second_moment_seed Q X hXm m₀ M hm₀ hM hXX.1 hXX.2).trans
+  apply measure_mono
+  intro ω hω
+  have hbound := a.source_finite_mass_le_sausage hk hκ8 z₀ hz T he
+    (hecut.le.trans (min_le_left _ _)) ω
+  have heq : (e/2)^(-codimension κ) = C*e^(-codimension κ) := by
+    rw [show e/2 = (1/2)*e by ring,Real.mul_rpow (by norm_num) he.le]
+  rw [heq,ENNReal.ofReal_mul hC.le,mul_assoc] at hbound
+  have hdiv : ENNReal.ofReal (m₀/2) / ENNReal.ofReal C ≤
+      ENNReal.ofReal (e^(-codimension κ))*volume (closedBall z₀ 1 ∩ sourceSausageSet (γ ω) 0 T e) := by
+    apply (ENNReal.div_le_iff (by simpa using hC) ENNReal.ofReal_ne_top).mpr
+    simpa only [mul_comm] using (hω.trans hbound)
+  simpa only [mem_ofPred_eq,ENNReal.ofReal_div_of_pos hC] using hdiv
+end RegularSLEPresentation
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Complex Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_volume_lipschitz_image {f : ℂ → ℂ} {S : Set ℂ} {L : ℝ≥0}
+    (hf : LipschitzOnWith L f S) : volume (f '' S) ≤ (L:ℝ≥0∞)^2*volume S := by
+  have hhaar : (Measure.hausdorffMeasure (2:ℝ) : Measure ℂ).IsAddHaarMeasure := by
+    simpa only [Complex.finrank_real_complex, Nat.cast_ofNat] using
+      (inferInstance : (Measure.hausdorffMeasure (Module.finrank ℝ ℂ) : Measure ℂ).IsAddHaarMeasure)
+  let _ := hhaar
+  have he := Measure.isAddLeftInvariant_eq_smul (Measure.hausdorffMeasure (2:ℝ) : Measure ℂ) volume
+  have hc := Measure.addHaarScalarFactor_pos_of_isAddHaarMeasure
+    (Measure.hausdorffMeasure (2:ℝ) : Measure ℂ) volume
+  have hh := hf.hausdorffMeasure_image_le (d := 2) (by norm_num)
+  rw [he,Measure.smul_apply,Measure.smul_apply] at hh
+  simp only [ENNReal.smul_def,smul_eq_mul,ENNReal.rpow_two] at hh
+  rw [mul_left_comm] at hh
+  exact (ENNReal.mul_le_mul_iff_right (by exact_mod_cast hc.ne') (by simp)).mp hh
+lemma source_holomorphic_compact_lipschitz {f : ℂ → ℂ} {D K : Set ℂ}
+    (hD : IsOpen D) (hK : IsCompact K) (hKD : K ⊆ D) (hf : DifferentiableOn ℂ f D) :
+    ∃ L : ℝ≥0, LipschitzOnWith L f K := by
+  apply LocallyLipschitzOn.exists_lipschitzOnWith_of_compact hK
+  intro x hx
+  obtain ⟨L,V,hV,hLip⟩ := (hf.analyticAt (hD.mem_nhds (hKD hx))).contDiffAt.exists_lipschitzOnWith
+  exact ⟨L,V,mem_nhdsWithin_of_mem_nhds hV,hLip⟩
+lemma source_inverse_chart_compact_bilipschitz
+    (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ)
+    (hw : IsCapacityTwoWitness U γ G F) (s : ℝ≥0)
+    {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) :
+    ∃ L : ℝ≥0, 0 < L ∧ LipschitzOnWith L (F s) K ∧
+      ∀ x ∈ K, ∀ y ∈ K, dist x y ≤ (L:ℝ)*dist (F s x) (F s y) := by
+  have hFc := hw.2.2.2.2.1 s
+  have hGc := hw.2.2.2.1 s
+  have hFD : F s '' K ⊆ survivingDomain γ s := by
+    rintro _ ⟨z,hz,rfl⟩
+    exact hw.2.2.2.2.2.2.1 s (hKH hz)
+  obtain ⟨L,hL⟩ := source_holomorphic_compact_lipschitz isOpen_upperHalfPlane hK hKH hFc
+  obtain ⟨M,hM⟩ := source_holomorphic_compact_lipschitz (isOpen_survivingDomain hw.1 s)
+    (hK.image_of_continuousOn (hFc.continuousOn.mono hKH)) hFD hGc
+  refine ⟨max 1 (max L M),lt_of_lt_of_le zero_lt_one (le_max_left _ _),hL.weaken (le_trans (le_max_left _ _) (le_max_right _ _)),?_⟩
+  intro x hx y hy
+  have hh := hM.dist_le_mul (F s x) ⟨x,hx,rfl⟩ (F s y) ⟨y,hy,rfl⟩
+  rw [hw.2.2.2.2.2.2.2.2.1 s x (hKH hx),hw.2.2.2.2.2.2.2.2.1 s y (hKH hy)] at hh
+  exact hh.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast (le_trans (le_max_right L M) (le_max_right 1 (max L M)))) dist_nonneg)
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/WindowAssembly.lean b/lean/OAI/Probability/SLE/WindowAssembly.lean
index 2f87478d9a2d87fac3457747623045846a6d1511..9a8a839fae15333c264164ae6ba4253e7e458ab5 100644
--- a/lean/OAI/Probability/SLE/WindowAssembly.lean
+++ b/lean/OAI/Probability/SLE/WindowAssembly.lean
@@ -1,844 +1,844 @@
-import Mathlib
-
-namespace OAI
-
-/-! Window Assembly for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma window_prod_telescope {ι : Type*} [LinearOrder ι]
-    (s : Finset ι) (f : ι → ℝ) :
-    ∏ i ∈ s, f i = 1 + ∑ i ∈ s, (f i - 1) * ∏ j ∈ s.filter (· < i), f j := by
-  classical
-  induction s using Finset.induction_on_max with
-  | empty => simp
-  | @insert a s ha ih =>
-    have hn : a ∉ s := fun h => (lt_irrefl a) (ha a h)
-    have hfa : (insert a s).filter (· < a) = s := by
-      ext j
-      simp only [Finset.mem_filter, Finset.mem_insert]
-      constructor
-      · rintro ⟨rfl | hj, hja⟩
-        · exact (lt_irrefl _ hja).elim
-        · exact hj
-      · intro hj; exact ⟨Or.inr hj, ha j hj⟩
-    have hf (i : ι) (hi : i ∈ s) :
-        (insert a s).filter (· < i) = s.filter (· < i) := by
-      ext j
-      simp only [Finset.mem_filter, Finset.mem_insert]
-      constructor
-      · rintro ⟨rfl | hj, hji⟩
-        · exact (not_lt_of_ge (ha i hi).le hji).elim
-        · exact ⟨hj, hji⟩
-      · exact fun ⟨hj, hji⟩ => ⟨Or.inr hj, hji⟩
-    rw [Finset.prod_insert hn, Finset.sum_insert hn, hfa]
-    simp_rw [Finset.sum_congr rfl (fun i hi => congrArg (fun t => (f i-1)*∏j∈t,f j) (hf i hi))]
-    rw [ih]
-    ring
-
-lemma window_integrable_bounded {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    [IsFiniteMeasure P] {f : Ω → ℝ} (hf : Measurable f) {C : ℝ}
-    (hC : ∀ ω, ‖f ω‖ ≤ C) : Integrable f P :=
-  Integrable.mono' (integrable_const C) hf.aestronglyMeasurable (Eventually.of_forall hC)
-
-lemma window_exp_secant {K θ x : ℝ} (hK : 0 < K) (hx : |x| ≤ K) :
-    Real.exp (θ*x) ≤ Real.exp ((θ*K)^2/2) + Real.sinh (θ*K)/K*x := by
-  let a := (K-x)/(2*K)
-  let b := (K+x)/(2*K)
-  have ha : 0 ≤ a := div_nonneg (sub_nonneg.mpr (le_abs_self x |>.trans hx)) (by positivity)
-  have hb : 0 ≤ b := div_nonneg (by linarith [neg_abs_le x]) (by positivity)
-  have hab : a+b=1 := by dsimp [a,b]; field_simp; ring
-  have he := convexOn_exp.2 (x := -(θ*K)) (y := θ*K) (by trivial) (by trivial) ha hb hab
-  have harg : a * -(θ*K) + b*(θ*K) = θ*x := by dsimp [a,b]; field_simp; ring
-  simp only [smul_eq_mul, harg] at he
-  have hid : a*Real.exp (-(θ*K)) + b*Real.exp (θ*K) =
-      Real.cosh (θ*K) + Real.sinh (θ*K)/K*x := by
-    rw [Real.cosh_eq, Real.sinh_eq]
-    dsimp [a,b]
-    field_simp
-    ring
-  rw [hid] at he
-  exact he.trans (add_le_add (Real.cosh_le_exp_half_sq _) le_rfl)
-
-lemma window_weighted_cond_mean {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
-    {X W : Ω → ℝ} (hX : Integrable X P) (hWX : Integrable (W*X) P)
-    (hW : Measurable[m] W) (hW0 : ∀ ω, 0 ≤ W ω)
-    (hmean : P[X|m] ≤ᵐ[P] 0) :
-    ∫ ω, W ω*X ω ∂P ≤ 0 := by
-  have he := condExp_mul_of_stronglyMeasurable_left hW.stronglyMeasurable hWX hX
-  calc
-    _ = ∫ ω, P[W*X|m] ω ∂P := (integral_condExp hm).symm
-    _ = ∫ ω, W ω*P[X|m] ω ∂P := integral_congr_ae he
-    _ ≤ 0 := integral_nonpos_of_ae (hmean.mono fun ω h =>
-      mul_nonpos_of_nonneg_of_nonpos (hW0 ω) h)
-
-lemma window_weighted_hoeffding {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
-    {X W : Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
-    (hX : Measurable[mΩ] X) (hXb : ∀ ω, |X ω| ≤ K)
-    (hmean : P[X|m] ≤ᵐ[P] 0) (hW : Measurable[m] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω * Real.exp (θ*X ω) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp ((θ*K)^2/2) := by
-  let : MeasurableSpace Ω := mΩ
-  obtain ⟨M,hM⟩ := hWb
-  have hWa : Measurable[mΩ] W := hW.mono hm le_rfl
-  have hWi : Integrable W P := window_integrable_bounded hWa
-    (fun ω => by simpa only [Real.norm_eq_abs, abs_of_nonneg (hW0 ω)] using hM ω)
-  have hXi : Integrable X P := window_integrable_bounded hX (by simpa only [Real.norm_eq_abs] using hXb)
-  have hWXi : Integrable (fun ω => W ω*X ω) P := window_integrable_bounded (hWa.mul hX) (fun ω => by
-    rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω)]
-    exact mul_le_mul (hM ω) (hXb ω) (norm_nonneg _) (le_trans (hW0 ω) (hM ω)))
-  have hc := window_weighted_cond_mean hm hXi hWXi hW hW0 hmean
-  have hL : 0 ≤ Real.sinh (θ*K)/K := div_nonneg (by rw [Real.sinh_eq]; exact div_nonneg (sub_nonneg.mpr (Real.exp_le_exp.mpr (by nlinarith [mul_nonneg hθ hK.le]))) (by norm_num)) hK.le
-  have hEi : Integrable (fun ω => W ω*Real.exp (θ*X ω)) P :=
-    window_integrable_bounded (hWa.mul ((hX.const_mul θ).exp)) (fun ω => by
-      rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω), Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
-      exact mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-        (le_abs_self _ |>.trans (hXb ω)) hθ)) (Real.exp_pos _).le (le_trans (hW0 ω) (hM ω)))
-  calc
-    _ ≤ ∫ ω, W ω * (Real.exp ((θ*K)^2/2) + Real.sinh (θ*K)/K * X ω) ∂P := by
-      apply integral_mono hEi
-      · have hi := (hWi.mul_const (Real.exp ((θ*K)^2/2))).add
-            (hWXi.const_mul (Real.sinh (θ*K)/K))
-        convert hi using 1; first | rfl | (ext ω; simp only [Pi.add_apply, mul_add]; ring)
-      · intro ω
-        exact mul_le_mul_of_nonneg_left (window_exp_secant hK (hXb ω)) (hW0 ω)
-    _ = (∫ ω, W ω ∂P)*Real.exp ((θ*K)^2/2) +
-        Real.sinh (θ*K)/K*(∫ ω, W ω*X ω ∂P) := by
-      simp_rw [mul_add, show ∀ ω, W ω*(Real.sinh (θ*K)/K*X ω) =
-        (Real.sinh (θ*K)/K)*(W ω*X ω) from fun _ => by ring]
-      rw [integral_add (hWi.mul_const _) (hWXi.const_mul _), integral_mul_const, integral_const_mul]
-    _ ≤ _ := by linarith [mul_nonpos_of_nonneg_of_nonpos hL hc]
-
-def windowBefore {Ω ι : Type*} [LinearOrder ι]
-    (σ : ι → Ω → WithTop ℝ≥0) (i j : ι) (ω : Ω) : Prop :=
-  σ i ω < σ j ω ∨ σ i ω = σ j ω ∧ i < j
-
-lemma window_prod_before {Ω ι : Type*} [Fintype ι] [LinearOrder ι]
-    (σ : ι → Ω → WithTop ℝ≥0) (ω : Ω) (f : ι → ℝ) :
-    ∏ i, f i = 1 + ∑ i, (f i - 1) * ∏ j, if windowBefore σ j i ω then f j else 1 := by
-  classical
-  let key (i : ι) : Lex (WithTop ℝ≥0 × ι) := toLex (σ i ω, i)
-  have hinj : Function.Injective key := fun i j h => congrArg Prod.snd h
-  have hh := @window_prod_telescope ι (LinearOrder.lift' key hinj) Finset.univ f
-  simp only [Finset.prod_filter] at hh
-  change (∏ i, f i) = 1 + ∑ i, (f i-1)*∏j, if key j < key i then f j else 1 at hh
-  rw [hh]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro i hi
-  congr 1
-  apply Finset.prod_congr rfl
-  intro j hj
-  have he : key j < key i ↔ windowBefore σ j i ω := by
-    simp only [key, Prod.Lex.toLex_lt_toLex, windowBefore]
-  by_cases hb : windowBefore σ j i ω
-  · rw [ite_eq_left hb, ite_eq_left (he.mpr hb)]
-  · rw [ite_eq_right hb, ite_eq_right (mt he.mp hb)]
-
-lemma window_before_measurable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [LinearOrder ι]
-    {ℱ : Filtration ℝ≥0 mΩ} {σ : ι → Ω → WithTop ℝ≥0}
-    (hs : ∀ i, IsStoppingTime ℱ (σ i)) (i j : ι) :
-    MeasurableSet[(hs i).measurableSpace] {ω | windowBefore σ j i ω} := by
-  have hlt : MeasurableSet[(hs i).measurableSpace] {ω | σ j ω < σ i ω} := by
-    simpa only [compl_ofPred, not_le] using ((hs i).measurableSet_le_stopping_time (hs j)).compl
-  have heq : MeasurableSet[(hs i).measurableSpace] {ω | σ j ω = σ i ω} := by
-    simpa only [eq_comm] using (hs i).measurableSet_eq_stopping_time (hs j)
-  by_cases hij : j < i
-  · convert hlt.union heq using 1
-    ext ω
-    simp only [windowBefore, hij, and_true, mem_ofPred_eq, mem_union]
-  · simpa only [windowBefore, hij, and_false, or_false] using hlt
-
-lemma window_measurable_on_event {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {ℱ : Filtration ℝ≥0 mΩ} {σ τ : Ω → WithTop ℝ≥0}
-    (hs : IsStoppingTime ℱ σ) (ht : IsStoppingTime ℱ τ)
-    {f : Ω → ℝ} (hf : Measurable[ht.measurableSpace] f)
-    {E : Set Ω} (hE : MeasurableSet[hs.measurableSpace] E)
-    (hle : ∀ ω ∈ E, τ ω ≤ σ ω) (c : ℝ) :
-    Measurable[hs.measurableSpace] (fun ω => if ω ∈ E then f ω else c) := by
-  classical
-  intro B hB
-  have hEB : MeasurableSet[hs.measurableSpace] (f ⁻¹' B ∩ E) := by
-    have hh := ht.measurableSet_inter_le hs (f ⁻¹' B) (hf hB)
-    have hh' : MeasurableSet[hs.measurableSpace] (f ⁻¹' B ∩ {ω | τ ω ≤ σ ω}) :=
-      ((ht.min hs).measurableSpace_mono hs (fun _ => min_le_right _ _)) _ hh
-    convert hh'.inter hE using 1
-    ext ω
-    simp only [mem_inter_iff, mem_preimage, mem_ofPred_eq]
-    exact ⟨fun ⟨h1,h2⟩ => ⟨⟨h1,hle ω h2⟩,h2⟩, fun ⟨⟨h1,_⟩,h2⟩ => ⟨h1,h2⟩⟩
-  by_cases hc : c ∈ B
-  · convert hEB.union hE.compl using 1
-    ext ω
-    by_cases hw : ω ∈ E <;> simp [hw, hc]
-  · convert hEB using 1
-    ext ω
-    by_cases hw : ω ∈ E <;> simp [hw, hc]
-
-lemma window_integrable_nonneg {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    [IsFiniteMeasure P] {f : Ω → ℝ} (hf : Measurable f) (h0 : ∀ ω, 0 ≤ f ω)
-    {C : ℝ} (hC : ∀ ω, f ω ≤ C) : Integrable f P :=
-  window_integrable_bounded hf (fun ω => by
-    simpa only [Real.norm_eq_abs, abs_of_nonneg (h0 ω)] using hC ω)
-
-lemma window_weighted_normalized {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
-    {X W : Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
-    (hX : Measurable[mΩ] X) (hXb : ∀ ω, |X ω| ≤ K)
-    (hmean : P[X|m] ≤ᵐ[P] 0) (hW : Measurable[m] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω * (Real.exp (θ*X ω - (θ*K)^2/2)-1) ∂P) ≤ 0 := by
-  let : MeasurableSpace Ω := mΩ
-  obtain ⟨M,hM⟩ := hWb
-  have hWa : Measurable[mΩ] W := hW.mono hm le_rfl
-  have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
-  have hEi : Integrable (fun ω => W ω*Real.exp (θ*X ω)) P :=
-    window_integrable_nonneg (hWa.mul ((hX.const_mul θ).exp))
-      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le) (fun ω =>
-      mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-        (le_abs_self _ |>.trans (hXb ω)) hθ)) (Real.exp_pos _).le (le_trans (hW0 ω) (hM ω)))
-  have h := window_weighted_hoeffding hm hK hθ hX hXb hmean hW hW0 ⟨M,hM⟩
-  simp_rw [Real.exp_sub, mul_sub, mul_one, ← mul_div_assoc]
-  rw [integral_sub (hEi.div_const _) hWi, integral_div]
-  exact sub_nonpos.mpr ((div_le_iff₀ (Real.exp_pos _)).mpr h)
-
- 
-
-lemma window_finite_mgf {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
-    {ℱ : Filtration ℝ≥0 mΩ} {σ τ : ι → Ω → WithTop ℝ≥0}
-    (hs : ∀ i, IsStoppingTime ℱ (σ i)) (ht : ∀ i, IsStoppingTime ℱ (τ i))
-    (horder : ∀ i j ω, windowBefore σ j i ω → τ j ω ≤ σ i ω)
-    {X : ι → Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
-    (hX : ∀ i, Measurable[(ht i).measurableSpace] (X i))
-    (hXb : ∀ i ω, |X i ω| ≤ K)
-    (hmean : ∀ i, P[X i|(hs i).measurableSpace] ≤ᵐ[P] 0)
-    {W : Ω → ℝ} (hW : Measurable[ℱ 0] W) (hW0 : ∀ ω, 0 ≤ W ω)
-    (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω * Real.exp (θ*∑ i, X i ω) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
-  classical
-  let q := (θ*K)^2/2
-  let g (i : ι) (ω : Ω) := Real.exp (θ*X i ω-q)
-  let pre (i : ι) (ω : Ω) := ∏ j, if windowBefore σ j i ω then g j ω else 1
-  have hg (i : ι) : Measurable[(ht i).measurableSpace] (g i) :=
-    (((hX i).const_mul θ).sub_const q).exp
-  have hga (i : ι) : Measurable[mΩ] (g i) := (hg i).mono (ht i).measurableSpace_le le_rfl
-  have hg0 (i : ι) (ω : Ω) : 0 ≤ g i ω := (Real.exp_pos _).le
-  have hgB (i : ι) (ω : Ω) : g i ω ≤ Real.exp (θ*K) := by
-    apply Real.exp_le_exp.mpr
-    have := mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hXb i ω)) hθ
-    dsimp [q]
-    nlinarith [sq_nonneg (θ*K)]
-  have hp (i : ι) : Measurable[(hs i).measurableSpace] (pre i) := by
-    apply Finset.measurable_prod
-    intro j hj
-    exact window_measurable_on_event (hs i) (ht j) (hg j)
-      (window_before_measurable hs i j) (fun ω hω => horder i j ω hω) 1
-  have hp0 (i : ι) (ω : Ω) : 0 ≤ pre i ω := by
-    apply Finset.prod_nonneg
-    intro j hj
-    split_ifs <;> positivity
-  have hpB (i : ι) (ω : Ω) : pre i ω ≤ Real.exp (θ*K)^Fintype.card ι := by
-    calc
-      _ ≤ ∏ _j : ι, Real.exp (θ*K) := by
-        apply Finset.prod_le_prod₀
-        · intro j hj; split_ifs <;> positivity
-        · intro j hj; split_ifs
-          · exact hgB j ω
-          · exact Real.one_le_exp (mul_nonneg hθ hK.le)
-      _ = _ := by simp
-  have hWstart (i : ι) : Measurable[(hs i).measurableSpace] W := by
-    have hle := (isStoppingTime_const ℱ (0 : ℝ≥0)).measurableSpace_mono (hs i)
-      (fun ω => bot_le)
-    rw [IsStoppingTime.measurableSpace_const] at hle
-    exact hW.mono hle le_rfl
-  have hWa : Measurable[mΩ] W := hW.mono (ℱ.le 0) le_rfl
-  obtain ⟨M,hM⟩ := hWb
-  have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
-  have hHi (i : ι) : Integrable (fun ω => W ω*pre i ω*(g i ω-1)) P := by
-    apply window_integrable_bounded (C := (M*Real.exp (θ*K)^Fintype.card ι)*(Real.exp (θ*K)+1)) (((hWa.mul ((hp i).mono (hs i).measurableSpace_le le_rfl))).mul
-      ((hga i).sub_const 1))
-    intro ω
-    change ‖W ω*pre i ω*(g i ω-1)‖ ≤ _
-    rw [norm_mul, norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω),
-      Real.norm_eq_abs, abs_of_nonneg (hp0 i ω)]
-    apply mul_le_mul
-    · exact mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))
-    · calc
-        _ ≤ ‖g i ω‖ + ‖(1 : ℝ)‖ := norm_sub_le _ _
-        _ = g i ω + 1 := by rw [Real.norm_eq_abs, abs_of_nonneg (hg0 i ω), norm_one]
-        _ ≤ _ := add_le_add (hgB i ω) le_rfl
-    · exact norm_nonneg _
-    · exact mul_nonneg ((hW0 ω).trans (hM ω)) (by positivity)
-  have hHi0 (i : ι) : (∫ ω, W ω*pre i ω*(g i ω-1) ∂P) ≤ 0 := by
-    exact window_weighted_normalized (hs i).measurableSpace_le hK hθ
-      ((hX i).mono (ht i).measurableSpace_le le_rfl) (hXb i) (hmean i)
-      ((hWstart i).mul (hp i)) (fun ω => mul_nonneg (hW0 ω) (hp0 i ω))
-      ⟨M*Real.exp (θ*K)^Fintype.card ι, fun ω =>
-        mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))⟩
-  have hnorm : (∫ ω, W ω * ∏ i, g i ω ∂P) ≤ ∫ ω, W ω ∂P := by
-    have hid (ω : Ω) := window_prod_before σ ω (fun i => g i ω)
-    simp_rw [hid, mul_add, mul_one, Finset.mul_sum]
-    change (∫ ω, W ω + ∑ i, W ω*((g i ω-1)*pre i ω) ∂P) ≤ _
-    simp_rw [show ∀ i ω, W ω*((g i ω-1)*pre i ω) = W ω*pre i ω*(g i ω-1)
-      from fun _ _ => by ring]
-    rw [integral_add hWi (integrable_finsetSum _ (fun i hi => hHi i)), integral_finsetSum]
-    · exact add_le_of_nonpos_right (Finset.sum_nonpos fun i hi => hHi0 i)
-    · exact fun i hi => hHi i
-  have hprod (ω : Ω) : (∏ i, g i ω) =
-      Real.exp (θ*∑ i, X i ω) / Real.exp (Fintype.card ι * q) := by
-    change (∏ i, Real.exp (θ*X i ω-q)) = _
-    rw [← Real.exp_sum]
-    simp only [Finset.sum_sub_distrib, ← Finset.mul_sum, Finset.sum_const,
-      Finset.card_univ, nsmul_eq_mul]
-    exact Real.exp_sub _ _
-  simp only [hprod, ← mul_div_assoc] at hnorm
-  rw [integral_div] at hnorm
-  simpa only [q, mul_div_assoc] using (div_le_iff₀ (Real.exp_pos _)).mp hnorm
-
- 
-
-structure SourceDisjointWindows {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (D : Set ℂ) (r R K : ℝ) (γ : ℝ≥0 → Ω → ℂ)
-    (σ τ : ℂ → Ω → WithTop ℝ≥0) (Y μ : ℂ → Ω → ℝ) : Prop where
-  measurable_D : MeasurableSet D
-  bounded_D : D ⊆ ball 0 1
-  radius_pos : 0 < r
-  radius_le : r ≤ R
-  radius_small : 3 * R ≤ 1
-  bound_nonneg : 0 ≤ K
-  continuous_trace : ∀ ω, Continuous (fun t => γ t ω)
-  adapted_trace : ∀ t, Measurable[ℱ t] (γ t)
-  start : ∀ z, IsStoppingTime ℱ (σ z)
-  finish : ∀ z, IsStoppingTime ℱ (τ z)
-  time_le : ∀ z ω, σ z ω ≤ τ z ω
-  contact : ∀ z ω, σ z ω ≠ ⊤ → dist (γ ((σ z ω).untopD 0) ω) z = r
-  confined : ∀ z ω (t : ℝ≥0), σ z ω ≤ (t : WithTop ℝ≥0) → (t : WithTop ℝ≥0) < τ z ω →
-    dist (γ t ω) z < R
-  output_measurable : ∀ z, Measurable[(finish z).measurableSpace] (Y z)
-  mean_measurable : ∀ z, Measurable[(start z).measurableSpace] (μ z)
-  output_bound : ∀ z ω, 0 ≤ Y z ω ∧ Y z ω ≤ K
-  mean_nonneg : ∀ z ω, 0 ≤ μ z ω
-  mean_bounded : ∃ M : ℝ, ∀ z ω, μ z ω ≤ M
-  upper_mean : ∀ z, ∀ᵐ ω ∂P, σ z ω ≠ ⊤ → P[Y z | (start z).measurableSpace] ω ≤ μ z ω
-  no_start : ∀ z ω, σ z ω = ⊤ → Y z ω = 0 ∧ μ z ω = 0
-  no_end : ∀ z ω, τ z ω = ⊤ → Y z ω = 0
-  outside : ∀ z, z ∉ D → ∀ ω, σ z ω = ⊤ ∧ Y z ω = 0 ∧ μ z ω = 0
-  joint_output : Measurable (fun p : ℂ × Ω => Y p.1 p.2)
-  joint_mean : Measurable (fun p : ℂ × Ω => μ p.1 p.2)
-
-lemma window_cond_mean_at_infinity {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    {σ : Ω → WithTop ℝ≥0} (hs : IsStoppingTime ℱ σ)
-    {Y μ : Ω → ℝ} (hY : Integrable Y P)
-    (hmean : ∀ᵐ ω ∂P, σ ω ≠ ⊤ → P[Y|hs.measurableSpace] ω ≤ μ ω)
-    (hno : ∀ ω, σ ω = ⊤ → Y ω = 0 ∧ μ ω = 0) :
-    P[Y|hs.measurableSpace] ≤ᵐ[P] μ := by
-  have htop : IsStoppingTime ℱ (fun _ => (⊤ : WithTop ℝ≥0)) := by
-    intro t
-    simp
-  have hE : MeasurableSet[hs.measurableSpace] {ω | σ ω = ⊤} :=
-    hs.measurableSet_eq_stopping_time htop
-  have he := condExp_indicator hY hE
-  have hz : ({ω | σ ω = ⊤}.indicator Y) = 0 := by
-    ext ω
-    by_cases h : σ ω = ⊤
-    · simpa [h] using (hno ω h).1
-    · simp [h]
-  rw [hz, condExp_zero] at he
-  filter_upwards [hmean, he] with ω hm he
-  by_cases h : σ ω = ⊤
-  · have hy : P[Y|hs.measurableSpace] ω = 0 := by simpa [h] using he.symm
-    rw [hy, (hno ω h).2]
-  · exact hm h
-
-lemma window_capped_cond_mean {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
-    {Y μ : Ω → ℝ} {K : ℝ} (hK : 0 ≤ K) (hY : Measurable[mΩ] Y)
-    (hYb : ∀ ω, 0 ≤ Y ω ∧ Y ω ≤ K) (hμ : Measurable[m] μ)
-    (hμ0 : ∀ ω, 0 ≤ μ ω) (hmean : P[Y|m] ≤ᵐ[P] μ) :
-    P[(fun ω => Y ω-min (μ ω) K)|m] ≤ᵐ[P] 0 := by
-  let : MeasurableSpace Ω := mΩ
-  have hYi : Integrable Y P := window_integrable_nonneg hY (fun ω => (hYb ω).1) (fun ω => (hYb ω).2)
-  have hν : Measurable[m] (fun ω => min (μ ω) K) := hμ.min measurable_const
-  have hνi : Integrable (fun ω => min (μ ω) K) P :=
-    window_integrable_nonneg (hν.mono hm le_rfl) (fun ω => le_min (hμ0 ω) hK)
-      (fun ω => min_le_right _ _)
-  have hbound := condExp_mono hYi (integrable_const K) (Eventually.of_forall fun ω => (hYb ω).2) (m := m)
-  rw [condExp_const hm] at hbound
-  have he := condExp_sub hYi hνi m
-  rw [condExp_of_stronglyMeasurable hm hν.stronglyMeasurable hνi] at he
-  filter_upwards [he, hbound, hmean] with ω he hb hm
-  simp only [Pi.sub_apply] at he
-  have he' : P[(fun ω => Y ω-min (μ ω) K)|m] ω = P[Y|m] ω-min (μ ω) K := by
-    convert he using 1
-  rw [he']
-  exact sub_nonpos.mpr (le_min hm hb)
-
-lemma window_geometric_order {Ω ι : Type*} [MeasurableSpace Ω]
-    [LinearOrder ι] {P : Measure Ω} {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    {z : ι → ℂ} (hz : ∀ i j, i ≠ j → 2*R < dist (z i) (z j)) :
-    ∀ i j ω, windowBefore (fun i => σ (z i)) j i ω → τ (z j) ω ≤ σ (z i) ω := by
-  intro i j ω hpre
-  by_contra hn
-  have hne : σ (z i) ω ≠ ⊤ := ne_top_of_lt (lt_of_not_ge hn)
-  have hji : j ≠ i := by
-    intro he
-    subst j
-    rcases hpre with hlt | ⟨_,hlt⟩ <;> exact (lt_irrefl _ hlt)
-  have hle : σ (z j) ω ≤ σ (z i) ω := hpre.elim le_of_lt (fun h => h.1.le)
-  have hcoe : (((σ (z i) ω).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = σ (z i) ω :=
-    by
-      cases he : σ (z i) ω with
-      | top => exact (hne he).elim
-      | coe t => rfl
-  have hconf := h.confined (z j) ω ((σ (z i) ω).untopD 0) (hcoe.symm ▸ hle)
-    (hcoe.symm ▸ lt_of_not_ge hn)
-  have hcont := h.contact (z i) ω hne
-  have htri := dist_triangle (z j) (γ ((σ (z i) ω).untopD 0) ω) (z i)
-  rw [dist_comm (z j) (γ ((σ (z i) ω).untopD 0) ω), hcont] at htri
-  have := hz j i hji
-  linarith [h.radius_le]
-
-lemma window_source_finite_mgf {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
-    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
-    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    {z : ι → ℂ} (hz : ∀ i j, i ≠ j → 2*R < dist (z i) (z j))
-    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hW : Measurable[ℱ 0] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
-    (∫ ω, W ω * Real.exp (θ*∑ i, (Y (z i) ω-μ (z i) ω)) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
-  let X (i : ι) (ω : Ω) := Y (z i) ω-min (μ (z i) ω) K
-  have hYa (i : ι) : Measurable[mΩ] (Y (z i)) :=
-    (h.output_measurable (z i)).mono (h.finish (z i)).measurableSpace_le le_rfl
-  have hμa (i : ι) : Measurable[mΩ] (μ (z i)) :=
-    (h.mean_measurable (z i)).mono (h.start (z i)).measurableSpace_le le_rfl
-  have hX (i : ι) : Measurable[(h.finish (z i)).measurableSpace] (X i) :=
-    (h.output_measurable (z i)).sub (((h.mean_measurable (z i)).mono
-      ((h.start (z i)).measurableSpace_mono (h.finish (z i)) (h.time_le (z i))) le_rfl).min measurable_const)
-  have hXb (i : ι) (ω : Ω) : |X i ω| ≤ K := by
-    rw [abs_le]
-    have := h.output_bound (z i) ω
-    have := le_min (h.mean_nonneg (z i) ω) hK.le
-    have := min_le_right (μ (z i) ω) K
-    dsimp [X]
-    constructor <;> linarith
-  have hm (i : ι) : P[X i|(h.start (z i)).measurableSpace] ≤ᵐ[P] 0 :=
-    window_capped_cond_mean (h.start (z i)).measurableSpace_le hK.le (hYa i)
-      (h.output_bound (z i)) (h.mean_measurable (z i)) (h.mean_nonneg (z i))
-      (window_cond_mean_at_infinity (h.start (z i))
-        (window_integrable_nonneg (hYa i) (fun ω => (h.output_bound (z i) ω).1)
-          (fun ω => (h.output_bound (z i) ω).2)) (h.upper_mean (z i)) (h.no_start (z i)))
-  have ht := window_finite_mgf (fun i => h.start (z i)) (fun i => h.finish (z i))
-    (window_geometric_order h hz) hK hθ hX hXb hm hW hW0 hWb
-  apply le_trans _ ht
-  obtain ⟨M,hM⟩ := hWb
-  obtain ⟨L,hL⟩ := h.mean_bounded
-  have hWa : Measurable[mΩ] W := hW.mono (ℱ.le 0) le_rfl
-  have hib (f : ι → Ω → ℝ) (hf : ∀ i, Measurable[mΩ] (f i))
-      (hb : ∀ i ω, f i ω ≤ K) : Integrable (fun ω => W ω*Real.exp (θ*∑ i, f i ω)) P :=
-    window_integrable_nonneg (C := M*Real.exp (θ*(Fintype.card ι*K))) (hWa.mul ((Finset.measurable_sum _ (fun i _ => hf i)).const_mul θ).exp)
-      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
-      (fun ω => mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
-        ((Finset.sum_le_sum fun i hi => hb i ω).trans_eq (by simp)) hθ))
-        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
-  apply integral_mono
-  · exact hib (fun i ω => Y (z i) ω-μ (z i) ω) (fun i => (hYa i).sub (hμa i))
-      (fun i ω => sub_le_self _ (h.mean_nonneg (z i) ω) |>.trans (h.output_bound (z i) ω).2)
-  · exact hib X (fun i => (hX i).mono (h.finish (z i)).measurableSpace_le le_rfl)
-      (fun i ω => le_abs_self _ |>.trans (hXb i ω))
-  · intro ω
-    apply mul_le_mul_of_nonneg_left _ (hW0 ω)
-    apply Real.exp_le_exp.mpr
-    apply mul_le_mul_of_nonneg_left _ hθ
-    exact Finset.sum_le_sum fun i hi => sub_le_sub_left (min_le_left _ _) _
-
-lemma window_integral_grid_1d {f : ℝ → ℝ} (hf : Measurable f)
-    {C a : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (ha : 0 ≤ a) (n : ℕ) :
-    (∫ x in Ioc 0 a, ∑ i : Fin n, f (x + a*i - 2)) =
-      ∫ x in Ioc (-2) (a*n-2), f x := by
-  have hfi (u v : ℝ) : IntegrableOn f (Ioc u v) :=
-    window_integrable_bounded hf hC
-  have hi (i : Fin n) : IntegrableOn (fun x => f (x+a*i-2)) (Ioc 0 a) :=
-    window_integrable_bounded (hf.comp ((measurable_id.add_const _).sub_const _))
-      (fun x => hC _)
-  rw [integral_finsetSum _ (fun i _ => hi i)]
-  have hpiece (i : Fin n) : (∫ x in Ioc 0 a, f (x+a*i-2)) =
-      ∫ x in (a*(i:ℝ)-2)..(a*((i:ℝ)+1)-2), f x := by
-    rw [← intervalIntegral.integral_of_le ha]
-    have he := intervalIntegral.integral_comp_add_right f (a*i-2) (a := 0) (b := a)
-    simp only [zero_add] at he
-    have hend : a*((i:ℝ)+1)-2 = a+(a*i-2) := by ring
-    simpa only [hend, add_sub_assoc] using he
-  simp_rw [hpiece]
-  rw [Fin.sum_univ_eq_sum_range (fun k : ℕ =>
-    ∫ x in (a*(k:ℝ)-2)..(a*((k:ℝ)+1)-2), f x) n]
-  have hsum := intervalIntegral.sum_integral_adjacent_intervals
-    (a := fun k => a*k-2) (n := n) (f := f) (μ := volume)
-    (fun k hk => (intervalIntegrable_iff_integrableOn_Ioc_of_le
-      (by have : (k:ℝ) ≤ k+1 := by linarith
-          exact sub_le_sub_right (mul_le_mul_of_nonneg_left (by exact_mod_cast Nat.le_succ k) ha) 2)).mpr (hfi _ _))
-  simp only [Nat.cast_add, Nat.cast_one, Nat.cast_zero, mul_zero, zero_sub] at hsum
-  rw [hsum, intervalIntegral.integral_of_le (by nlinarith [mul_nonneg ha (Nat.cast_nonneg n)])]
-
-noncomputable def windowSquare (a b : ℝ) : Measure (ℝ × ℝ) :=
-  (volume.restrict (Ioc a b)).prod (volume.restrict (Ioc a b))
-
-instance windowSquare_finite (a b : ℝ) : IsFiniteMeasure (windowSquare a b) := by
-  dsimp [windowSquare]
-  infer_instance
-
-lemma window_grid_2d {f : ℝ × ℝ → ℝ} (hf : Measurable f)
-    {C a : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (ha : 0 ≤ a) (n : ℕ) :
-    (∫ p, ∑ ij : Fin n × Fin n,
-      f (p.1+a*ij.1-2,p.2+a*ij.2-2) ∂windowSquare 0 a) =
-      ∫ p, f p ∂windowSquare (-2) (a*n-2) := by
-  let G (x : ℝ) := ∫ y in Ioc (-2) (a*n-2), f (x,y)
-  have hG : Measurable G := (hf.stronglyMeasurable.integral_prod_right (f := fun x y => f (x,y))).measurable
-  have hGb (x : ℝ) : ‖G x‖ ≤ C*(volume.restrict (Ioc (-2) (a*n-2))).real univ :=
-    norm_integral_le_of_norm_le_const (Eventually.of_forall fun y => hC (x,y))
-  have hrow (x : ℝ) : (∫ y in Ioc 0 a, ∑ ij : Fin n × Fin n,
-      f (x+a*ij.1-2,y+a*ij.2-2)) = ∑ i : Fin n, G (x+a*i-2) := by
-    simp_rw [Fintype.sum_prod_type]
-    rw [integral_finsetSum]
-    · congr 1
-      ext i
-      exact window_integral_grid_1d (hf.comp (measurable_const.prodMk measurable_id))
-        (fun y => hC _) ha n
-    · intro i hi
-      apply window_integrable_bounded (C := n*C)
-      · exact Finset.measurable_sum _ (fun j hj => hf.comp (measurable_const.prodMk
-          ((measurable_id.add_const _).sub_const _)))
-      · intro y
-        exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun j hj => hC _).trans_eq (by simp))
-  dsimp only [windowSquare]
-  rw [integral_prod]
-  · simp_rw [hrow]
-    rw [window_integral_grid_1d hG hGb ha n]
-    exact (integral_prod f (window_integrable_bounded hf hC)).symm
-  · apply window_integrable_bounded (C := (n*n)*C)
-    · apply Finset.measurable_sum
-      intro ij hij
-      exact hf.comp (((measurable_fst.add_const _).sub_const _).prodMk
-        ((measurable_snd.add_const _).sub_const _))
-    · intro p
-      exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun ij hij => hC _).trans_eq (by simp))
-
-noncomputable def windowShiftLaw (a : ℝ) : Measure (ℝ × ℝ) :=
-  (ENNReal.ofReal (a^2))⁻¹ • windowSquare 0 a
-
-lemma windowSquare_mass {a : ℝ} (ha : 0 ≤ a) : windowSquare 0 a univ = ENNReal.ofReal (a^2) := by
-  simp [windowSquare, Measure.prod_apply, Real.volume_Ioc, sq, ← ENNReal.ofReal_mul ha]
-
-lemma windowShiftLaw_probability {a : ℝ} (ha : 0 < a) : IsProbabilityMeasure (windowShiftLaw a) := by
-  apply isProbabilityMeasure_iff.mpr
-  rw [windowShiftLaw, Measure.smul_apply, windowSquare_mass ha.le]
-  exact ENNReal.inv_mul_cancel (by positivity) (by finiteness)
-
-lemma window_rectangle_integral {f : ℂ → ℝ} (_hf : Measurable f)
-    (hz : ∀ z, z ∉ ball (0:ℂ) 1 → f z = 0) :
-    (∫ p, f (Complex.measurableEquivRealProd.symm p) ∂windowSquare (-2) 2) = ∫ z, f z := by
-  rw [windowSquare, Measure.prod_restrict]
-  rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
-  · exact (Complex.volume_preserving_equiv_real_prod.symm).integral_comp' f
-  · intro p hp
-    apply hz
-    intro hb
-    apply hp
-    have hn : ‖Complex.measurableEquivRealProd.symm p‖ < 1 := by simpa using hb
-    have hr := Complex.abs_re_le_norm (Complex.measurableEquivRealProd.symm p)
-    have hi := Complex.abs_im_le_norm (Complex.measurableEquivRealProd.symm p)
-    change |p.1| ≤ _ at hr
-    change |p.2| ≤ _ at hi
-    exact ⟨⟨by linarith [neg_abs_le p.1], by linarith [le_abs_self p.1]⟩,
-      ⟨by linarith [neg_abs_le p.2], by linarith [le_abs_self p.2]⟩⟩
-
-noncomputable def windowCenter (a : ℝ) (ξ : ℝ × ℝ) {n : ℕ} (i : Fin n × Fin n) : ℂ :=
-  Complex.measurableEquivRealProd.symm (ξ.1+a*i.1-2, ξ.2+a*i.2-2)
-
-lemma window_grid_average {f : ℂ → ℝ} (hf : Measurable f)
-    {C a : ℝ} (hC : ∀ z, ‖f z‖ ≤ C) (ha : 0 < a) {n : ℕ} (han : a*n=4)
-    (hz : ∀ z, z ∉ ball (0:ℂ) 1 → f z = 0) :
-    (∫ ξ, a^2 * ∑ i : Fin n × Fin n, f (windowCenter a ξ i) ∂windowShiftLaw a) = ∫ z, f z := by
-  rw [windowShiftLaw, integral_smul_measure, integral_const_mul]
-  have he := window_grid_2d (hf.comp Complex.measurableEquivRealProd.symm.measurable)
-    (fun p => hC _) ha.le n
-  rw [han] at he
-  norm_num only [Function.comp_apply] at he
-  dsimp only [windowCenter]
-  rw [he, window_rectangle_integral hf hz]
-  simp only [ENNReal.toReal_inv, ENNReal.toReal_ofReal (sq_nonneg a), smul_eq_mul]
-  field_simp
-
-lemma window_nat_spacing (a : ℝ) (ha : 0 ≤ a) {i j : ℕ} (hne : i ≠ j) :
-    a ≤ |a*i-a*j| := by
-  rcases lt_or_gt_of_ne hne with h | h
-  · have h' : (i:ℝ)+1 ≤ j := by exact_mod_cast h
-    have hle : a*i ≤ a*j := mul_le_mul_of_nonneg_left (by exact_mod_cast h.le) ha
-    rw [abs_of_nonpos (sub_nonpos.mpr hle)]
-    nlinarith
-  · have h' : (j:ℝ)+1 ≤ i := by exact_mod_cast h
-    have hle : a*j ≤ a*i := mul_le_mul_of_nonneg_left (by exact_mod_cast h.le) ha
-    rw [abs_of_nonneg (sub_nonneg.mpr hle)]
-    nlinarith
-
-lemma window_centers_separated {a R : ℝ} (hsep : 2*R < a) (ha : 0 ≤ a)
-    {n : ℕ} (ξ : ℝ × ℝ) (i j : Fin n × Fin n) (hne : i ≠ j) :
-    2*R < dist (windowCenter a ξ i) (windowCenter a ξ j) := by
-  rw [dist_eq_norm]
-  rcases not_and_or.mp (mt Prod.ext_iff.mpr hne) with h | h
-  · apply hsep.trans_le
-    apply le_trans (window_nat_spacing a ha (mt Fin.ext h))
-    have he : (windowCenter a ξ i - windowCenter a ξ j).re = a*(i.1:ℝ)-a*(j.1:ℝ) := by
-      dsimp [windowCenter]
-      ring
-    rw [← he]
-    exact Complex.abs_re_le_norm _
-  · apply hsep.trans_le
-    apply le_trans (window_nat_spacing a ha (mt Fin.ext h))
-    have he : (windowCenter a ξ i - windowCenter a ξ j).im = a*(i.2:ℝ)-a*(j.2:ℝ) := by
-      dsimp [windowCenter]
-      ring
-    rw [← he]
-    exact Complex.abs_im_le_norm _
-
-lemma window_choose_grid {R : ℝ} (hR : 0 < R) (hsmall : 3*R ≤ 1) :
-    ∃ (n : ℕ) (a : ℝ), 0 < a ∧ a*n=4 ∧ 2*R<a ∧ a≤4*R := by
-  let n := Nat.ceil (1/R)
-  have hl := Nat.le_ceil (1/R)
-  have hu := Nat.ceil_lt_add_one (a := 1/R) (by positivity)
-  have hn : 0 < (n:ℝ) := (by positivity : 0 < 1/R).trans_le hl
-  have hnr : 1 ≤ R*n := by
-    exact (div_le_iff₀ hR).mp hl |>.trans_eq (mul_comm _ _)
-  have hnr' : R*n < 1+R := by
-    have H := mul_lt_mul_of_pos_right hu hR
-    rw [add_mul, div_mul_cancel₀ _ hR.ne', one_mul] at H
-    simpa only [mul_comm] using H
-  refine ⟨n,4/n,by positivity,div_mul_cancel₀ 4 hn.ne',?_,?_⟩
-  · apply (lt_div_iff₀ hn).mpr
-    nlinarith
-  · apply (div_le_iff₀ hn).mpr
-    nlinarith
-lemma window_jensen_mgf {Ξ Ω : Type*} [MeasurableSpace Ξ] [MeasurableSpace Ω]
-    {Q : Measure Ξ} [IsProbabilityMeasure Q] {P : Measure Ω} [IsFiniteMeasure P]
-    {F : Ξ → Ω → ℝ} (hF : Measurable (Function.uncurry F))
-    {B θ A : ℝ} (hFb : ∀ ξ ω, ‖F ξ ω‖ ≤ B) (hθ : 0 ≤ θ)
-    {W : Ω → ℝ} (hW : Measurable W) (hW0 : ∀ ω, 0 ≤ W ω)
-    (hWb : ∃ M, ∀ ω, W ω ≤ M)
-    (hmgf : ∀ ξ, (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤ A) :
-    (∫ ω, W ω*Real.exp (θ*∫ ξ, F ξ ω ∂Q) ∂P) ≤ A := by
-  obtain ⟨M,hM⟩ := hWb
-  have hFi (ω : Ω) : Integrable (fun ξ => θ*F ξ ω) Q :=
-    window_integrable_bounded ((hF.comp (measurable_id.prodMk measurable_const)).const_mul θ)
-      (C := θ*B) (fun ξ => by rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg hθ]
-                              exact mul_le_mul_of_nonneg_left (hFb ξ ω) hθ)
-  have hEi (ω : Ω) : Integrable (fun ξ => Real.exp (θ*F ξ ω)) Q :=
-    window_integrable_nonneg ((hF.comp (measurable_id.prodMk measurable_const)).const_mul θ).exp
-      (fun ξ => (Real.exp_pos _).le) (fun ξ => Real.exp_le_exp.mpr
-        (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hFb ξ ω)) hθ))
-  have hJ (ω : Ω) : Real.exp (θ*∫ ξ, F ξ ω ∂Q) ≤ ∫ ξ, Real.exp (θ*F ξ ω) ∂Q := by
-    rw [← integral_const_mul]
-    exact convexOn_exp.map_integral_le Real.continuous_exp.continuousOn isClosed_univ
-      (Eventually.of_forall fun _ => mem_univ _) (hFi ω) (hEi ω)
-  have hZ : Measurable (fun ω => ∫ ξ, F ξ ω ∂Q) :=
-    ((hF.comp measurable_swap).stronglyMeasurable.integral_prod_right (f := fun ω ξ => F ξ ω)).measurable
-  have hZb (ω : Ω) : ‖∫ ξ, F ξ ω ∂Q‖ ≤ B := by
-    simpa using norm_integral_le_of_norm_le_const (μ := Q) (Eventually.of_forall fun ξ => hFb ξ ω)
-  have hleft : Integrable (fun ω => W ω*Real.exp (θ*∫ ξ, F ξ ω ∂Q)) P :=
-    window_integrable_nonneg (hW.mul (hZ.const_mul θ).exp)
-      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
-      (C := M*Real.exp (θ*B)) (fun ω => mul_le_mul (hM ω)
-        (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hZb ω)) hθ))
-        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
-  have hjoint : Integrable (Function.uncurry (fun ξ ω => W ω*Real.exp (θ*F ξ ω))) (Q.prod P) :=
-    window_integrable_nonneg ((hW.comp measurable_snd).mul (hF.const_mul θ).exp)
-      (fun p => mul_nonneg (hW0 p.2) (Real.exp_pos _).le)
-      (C := M*Real.exp (θ*B)) (fun p => mul_le_mul (hM p.2)
-        (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hFb p.1 p.2)) hθ))
-        (Real.exp_pos _).le ((hW0 p.2).trans (hM p.2)))
-  calc
-    _ ≤ ∫ ω, ∫ ξ, W ω*Real.exp (θ*F ξ ω) ∂Q ∂P := by
-      apply integral_mono hleft hjoint.integral_prod_right
-      intro ω
-      change W ω*Real.exp (θ*∫ ξ, F ξ ω ∂Q) ≤ ∫ ξ, W ω*Real.exp (θ*F ξ ω) ∂Q
-      rw [integral_const_mul]
-      exact mul_le_mul_of_nonneg_left (hJ ω) (hW0 ω)
-    _ = ∫ ξ, ∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P ∂Q := (integral_integral_swap hjoint).symm
-    _ ≤ ∫ _ξ, A ∂Q := integral_mono hjoint.integral_prod_left (integrable_const A) hmgf
-    _ = A := by simp
-lemma window_source_mgf_pos {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hW : Measurable[ℱ 0] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
-    (∫ ω, W ω * Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
-  obtain ⟨n,a,ha,han,hsep,haR⟩ := window_choose_grid (h.radius_pos.trans_le h.radius_le) h.radius_small
-  let : LinearOrder (Fin n × Fin n) := LinearOrder.lift' toLex toLex.injective
-  let F (ξ : ℝ × ℝ) (ω : Ω) := a^2 * ∑ i : Fin n × Fin n,
-    (Y (windowCenter a ξ i) ω - μ (windowCenter a ξ i) ω)
-  have hcenter (i : Fin n × Fin n) : Measurable (fun ξ => windowCenter a ξ i) :=
-    Complex.measurableEquivRealProd.symm.measurable.comp
-      (((measurable_fst.add_const _).sub_const _).prodMk ((measurable_snd.add_const _).sub_const _))
-  have hmeas : Measurable (Function.uncurry F) := by
-    apply Measurable.const_mul
-    apply Finset.measurable_sum
-    intro i hi
-    exact (h.joint_output.sub h.joint_mean).comp
-      (((hcenter i).comp measurable_fst).prodMk measurable_snd)
-  obtain ⟨L,hL⟩ := h.mean_bounded
-  have hnorm (z : ℂ) (ω : Ω) : ‖Y z ω-μ z ω‖ ≤ K+L := by
-    calc
-      _ ≤ ‖Y z ω‖+‖μ z ω‖ := norm_sub_le _ _
-      _ = Y z ω+μ z ω := by rw [Real.norm_eq_abs, abs_of_nonneg (h.output_bound z ω).1,
-        Real.norm_eq_abs, abs_of_nonneg (h.mean_nonneg z ω)]
-      _ ≤ _ := add_le_add (h.output_bound z ω).2 (hL z ω)
-  have hFb (ξ : ℝ × ℝ) (ω : Ω) : ‖F ξ ω‖ ≤ a^2*((n:ℝ)^2*(K+L)) := by
-    dsimp only [F]
-    rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (sq_nonneg a)]
-    apply mul_le_mul_of_nonneg_left _ (sq_nonneg a)
-    exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun i hi => hnorm _ ω).trans_eq (by simp [sq]; ring))
-  have havg (ω : Ω) : (∫ ξ, F ξ ω ∂windowShiftLaw a) = ∫ z in D, (Y z ω-μ z ω) := by
-    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by
-      obtain ⟨_,hy,hμ⟩ := h.outside z hz ω
-      simp only [hy,hμ,sub_self])]
-    exact window_grid_average ((h.joint_output.sub h.joint_mean).comp
-      (measurable_id.prodMk measurable_const)) (fun z => hnorm z ω) ha han
-      (fun z hz => by
-        obtain ⟨_,hy,hμ⟩ := h.outside z (fun hzD => hz (h.bounded_D hzD)) ω
-        change Y z ω-μ z ω=0
-        simp only [hy,hμ,sub_self])
-  let : IsProbabilityMeasure (windowShiftLaw a) := windowShiftLaw_probability ha
-  have hfinite (ξ : ℝ × ℝ) :
-      (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
-    have ht := window_source_finite_mgf h (window_centers_separated (n := n) hsep ha.le ξ)
-      (mul_nonneg hθ (sq_nonneg a)) hW hW0 hWb hK
-    have he : (Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2 ≤ 128*K^2*θ^2*R^2 := by
-      simp only [Fintype.card_prod, Fintype.card_fin, Nat.cast_mul]
-      have hs : a^2 ≤ 16*R^2 := by nlinarith [h.radius_pos.trans_le h.radius_le]
-      calc
-        _ = (a*n)^2*a^2*θ^2*K^2/2 := by ring
-        _ = 8*a^2*θ^2*K^2 := by rw [han]; ring
-        _ ≤ 8*(16*R^2)*θ^2*K^2 := by gcongr
-        _ = _ := by ring
-    have ht' : (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
-        (∫ ω, W ω ∂P) * Real.exp ((Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2) := by
-      simpa only [F, mul_assoc] using ht
-    exact ht'.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr he) (integral_nonneg hW0))
-  have ht := window_jensen_mgf (Q := windowShiftLaw a) hmeas hFb hθ (hW.mono (ℱ.le 0) le_rfl) hW0 hWb hfinite
-  simpa only [havg] using ht
-
-lemma window_source_integral_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ) :
-    Measurable (fun ω => ∫ z in D, (Y z ω-μ z ω)) :=
-  ((h.joint_output.sub h.joint_mean).comp measurable_swap).stronglyMeasurable.integral_prod_right
-    (f := fun ω z => Y z ω-μ z ω) |>.measurable
-
-lemma window_source_integral_bounded {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ) :
-    ∃ B : ℝ, ∀ ω, ‖∫ z in D, (Y z ω-μ z ω)‖ ≤ B := by
-  have hfin : volume D ≠ ∞ := (isBounded_ball.subset h.bounded_D).measure_lt_top.ne
-  let : IsFiniteMeasure (volume.restrict D) := isFiniteMeasure_restrict.mpr hfin
-  obtain ⟨M,hM⟩ := h.mean_bounded
-  refine ⟨(K+M)*(volume.restrict D).real univ, fun ω => norm_integral_le_of_norm_le_const
-    (Eventually.of_forall fun z => ?_)⟩
-  calc
-    _ ≤ ‖Y z ω‖+‖μ z ω‖ := norm_sub_le _ _
-    _ = Y z ω+μ z ω := by rw [Real.norm_eq_abs, abs_of_nonneg (h.output_bound z ω).1,
-      Real.norm_eq_abs, abs_of_nonneg (h.mean_nonneg z ω)]
-    _ ≤ _ := add_le_add (h.output_bound z ω).2 (hM z ω)
-
-lemma window_source_mgf {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
-    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hW : Measurable[ℱ 0] W)
-    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
-    (∫ ω, W ω * Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ∂P) ≤
-      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
-  rcases h.bound_nonneg.eq_or_lt with hK | hK
-  · have hz (ω : Ω) : (∫ z in D, (Y z ω-μ z ω)) ≤ 0 := by
-      apply integral_nonpos
-      intro z
-      change Y z ω-μ z ω≤0
-      have := (h.output_bound z ω).2
-      have := h.mean_nonneg z ω
-      linarith
-    have he (ω : Ω) : Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ≤ 1 :=
-      Real.exp_le_one_iff.mpr (mul_nonpos_of_nonneg_of_nonpos hθ (hz ω))
-    obtain ⟨M,hM⟩ := hWb
-    have hWa : Measurable W := hW.mono (ℱ.le 0) le_rfl
-    have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
-    have hli : Integrable (fun ω => W ω*Real.exp (θ*∫ z in D, (Y z ω-μ z ω))) P :=
-      window_integrable_nonneg (hWa.mul ((window_source_integral_measurable h).const_mul θ).exp)
-        (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
-        (fun ω => ((mul_le_mul_of_nonneg_left (he ω) (hW0 ω)).trans_eq (mul_one _)).trans (hM ω))
-    simp only [← hK, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero,
-      zero_mul, Real.exp_zero, mul_one]
-    exact integral_mono hli hWi (fun ω => (mul_le_mul_of_nonneg_left (he ω) (hW0 ω)).trans_eq (mul_one _))
-  · exact window_source_mgf_pos h hθ hW hW0 hWb hK
-lemma window_source_tail {Ω : Type*} [mΩ : MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
-    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
-    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ) {b : ℝ} (hb : 0 < b) :
-    P {ω | b < ∫ z in D, (Y z ω-μ z ω)} ≤
-      ENNReal.ofReal (Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2))) := by
-  rcases h.bound_nonneg.eq_or_lt with hK | hK
-  · simp only [← hK, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, zero_mul,
-      div_zero, Real.exp_zero, ENNReal.ofReal_one]
-    exact prob_le_one
-  · let X (ω : Ω) := ∫ z in D, (Y z ω-μ z ω)
-    let θ := b/(256*K^2*R^2)
-    have hR : 0 < R := h.radius_pos.trans_le h.radius_le
-    have hθ : 0 ≤ θ := by dsimp [θ]; positivity
-    obtain ⟨B,hB⟩ := window_source_integral_bounded h
-    have hXi : Integrable (fun ω => Real.exp (θ*X ω)) P :=
-      window_integrable_nonneg ((window_source_integral_measurable h).const_mul θ).exp
-        (fun ω => (Real.exp_pos _).le) (fun ω => Real.exp_le_exp.mpr
-          (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hB ω)) hθ))
-    have hmgf : mgf X P θ ≤ Real.exp (128*K^2*θ^2*R^2) := by
-      have H := window_source_mgf h hθ (W := fun _ => 1) measurable_const (fun _ => zero_le_one) ⟨1,fun _ => le_rfl⟩
-      simpa [mgf, X] using H
-    apply (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mp
-    rw [ENNReal.toReal_ofReal (Real.exp_pos _).le]
-    change P.real {ω | b < X ω} ≤ _
-    calc
-      _ ≤ P.real {ω | b ≤ X ω} := by
-        exact measureReal_mono (fun ω hω => (show b < X ω from hω).le) (measure_ne_top _ _)
-      _ ≤ Real.exp (-θ*b)*mgf X P θ := measure_ge_le_exp_mul_mgf b hθ hXi
-      _ ≤ Real.exp (-θ*b)*Real.exp (128*K^2*θ^2*R^2) :=
-        mul_le_mul_of_nonneg_left hmgf (Real.exp_pos _).le
-      _ = _ := by
-        rw [← Real.exp_add]
-        congr 1
-        dsimp [θ]
-        field_simp
-        ring
-
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Window Assembly for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma window_prod_telescope {ι : Type*} [LinearOrder ι]
+    (s : Finset ι) (f : ι → ℝ) :
+    ∏ i ∈ s, f i = 1 + ∑ i ∈ s, (f i - 1) * ∏ j ∈ s.filter (· < i), f j := by
+  classical
+  induction s using Finset.induction_on_max with
+  | empty => simp
+  | @insert a s ha ih =>
+    have hn : a ∉ s := fun h => (lt_irrefl a) (ha a h)
+    have hfa : (insert a s).filter (· < a) = s := by
+      ext j
+      simp only [Finset.mem_filter, Finset.mem_insert]
+      constructor
+      · rintro ⟨rfl | hj, hja⟩
+        · exact (lt_irrefl _ hja).elim
+        · exact hj
+      · intro hj; exact ⟨Or.inr hj, ha j hj⟩
+    have hf (i : ι) (hi : i ∈ s) :
+        (insert a s).filter (· < i) = s.filter (· < i) := by
+      ext j
+      simp only [Finset.mem_filter, Finset.mem_insert]
+      constructor
+      · rintro ⟨rfl | hj, hji⟩
+        · exact (not_lt_of_ge (ha i hi).le hji).elim
+        · exact ⟨hj, hji⟩
+      · exact fun ⟨hj, hji⟩ => ⟨Or.inr hj, hji⟩
+    rw [Finset.prod_insert hn, Finset.sum_insert hn, hfa]
+    simp_rw [Finset.sum_congr rfl (fun i hi => congrArg (fun t => (f i-1)*∏j∈t,f j) (hf i hi))]
+    rw [ih]
+    ring
+
+lemma window_integrable_bounded {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    [IsFiniteMeasure P] {f : Ω → ℝ} (hf : Measurable f) {C : ℝ}
+    (hC : ∀ ω, ‖f ω‖ ≤ C) : Integrable f P :=
+  Integrable.mono' (integrable_const C) hf.aestronglyMeasurable (Eventually.of_forall hC)
+
+lemma window_exp_secant {K θ x : ℝ} (hK : 0 < K) (hx : |x| ≤ K) :
+    Real.exp (θ*x) ≤ Real.exp ((θ*K)^2/2) + Real.sinh (θ*K)/K*x := by
+  let a := (K-x)/(2*K)
+  let b := (K+x)/(2*K)
+  have ha : 0 ≤ a := div_nonneg (sub_nonneg.mpr (le_abs_self x |>.trans hx)) (by positivity)
+  have hb : 0 ≤ b := div_nonneg (by linarith [neg_abs_le x]) (by positivity)
+  have hab : a+b=1 := by dsimp [a,b]; field_simp; ring
+  have he := convexOn_exp.2 (x := -(θ*K)) (y := θ*K) (by trivial) (by trivial) ha hb hab
+  have harg : a * -(θ*K) + b*(θ*K) = θ*x := by dsimp [a,b]; field_simp; ring
+  simp only [smul_eq_mul, harg] at he
+  have hid : a*Real.exp (-(θ*K)) + b*Real.exp (θ*K) =
+      Real.cosh (θ*K) + Real.sinh (θ*K)/K*x := by
+    rw [Real.cosh_eq, Real.sinh_eq]
+    dsimp [a,b]
+    field_simp
+    ring
+  rw [hid] at he
+  exact he.trans (add_le_add (Real.cosh_le_exp_half_sq _) le_rfl)
+
+lemma window_weighted_cond_mean {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
+    {X W : Ω → ℝ} (hX : Integrable X P) (hWX : Integrable (W*X) P)
+    (hW : Measurable[m] W) (hW0 : ∀ ω, 0 ≤ W ω)
+    (hmean : P[X|m] ≤ᵐ[P] 0) :
+    ∫ ω, W ω*X ω ∂P ≤ 0 := by
+  have he := condExp_mul_of_stronglyMeasurable_left hW.stronglyMeasurable hWX hX
+  calc
+    _ = ∫ ω, P[W*X|m] ω ∂P := (integral_condExp hm).symm
+    _ = ∫ ω, W ω*P[X|m] ω ∂P := integral_congr_ae he
+    _ ≤ 0 := integral_nonpos_of_ae (hmean.mono fun ω h =>
+      mul_nonpos_of_nonneg_of_nonpos (hW0 ω) h)
+
+lemma window_weighted_hoeffding {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
+    {X W : Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
+    (hX : Measurable[mΩ] X) (hXb : ∀ ω, |X ω| ≤ K)
+    (hmean : P[X|m] ≤ᵐ[P] 0) (hW : Measurable[m] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω * Real.exp (θ*X ω) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp ((θ*K)^2/2) := by
+  let : MeasurableSpace Ω := mΩ
+  obtain ⟨M,hM⟩ := hWb
+  have hWa : Measurable[mΩ] W := hW.mono hm le_rfl
+  have hWi : Integrable W P := window_integrable_bounded hWa
+    (fun ω => by simpa only [Real.norm_eq_abs, abs_of_nonneg (hW0 ω)] using hM ω)
+  have hXi : Integrable X P := window_integrable_bounded hX (by simpa only [Real.norm_eq_abs] using hXb)
+  have hWXi : Integrable (fun ω => W ω*X ω) P := window_integrable_bounded (hWa.mul hX) (fun ω => by
+    rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω)]
+    exact mul_le_mul (hM ω) (hXb ω) (norm_nonneg _) (le_trans (hW0 ω) (hM ω)))
+  have hc := window_weighted_cond_mean hm hXi hWXi hW hW0 hmean
+  have hL : 0 ≤ Real.sinh (θ*K)/K := div_nonneg (by rw [Real.sinh_eq]; exact div_nonneg (sub_nonneg.mpr (Real.exp_le_exp.mpr (by nlinarith [mul_nonneg hθ hK.le]))) (by norm_num)) hK.le
+  have hEi : Integrable (fun ω => W ω*Real.exp (θ*X ω)) P :=
+    window_integrable_bounded (hWa.mul ((hX.const_mul θ).exp)) (fun ω => by
+      rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω), Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
+      exact mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+        (le_abs_self _ |>.trans (hXb ω)) hθ)) (Real.exp_pos _).le (le_trans (hW0 ω) (hM ω)))
+  calc
+    _ ≤ ∫ ω, W ω * (Real.exp ((θ*K)^2/2) + Real.sinh (θ*K)/K * X ω) ∂P := by
+      apply integral_mono hEi
+      · have hi := (hWi.mul_const (Real.exp ((θ*K)^2/2))).add
+            (hWXi.const_mul (Real.sinh (θ*K)/K))
+        convert hi using 1; first | rfl | (ext ω; simp only [Pi.add_apply, mul_add]; ring)
+      · intro ω
+        exact mul_le_mul_of_nonneg_left (window_exp_secant hK (hXb ω)) (hW0 ω)
+    _ = (∫ ω, W ω ∂P)*Real.exp ((θ*K)^2/2) +
+        Real.sinh (θ*K)/K*(∫ ω, W ω*X ω ∂P) := by
+      simp_rw [mul_add, show ∀ ω, W ω*(Real.sinh (θ*K)/K*X ω) =
+        (Real.sinh (θ*K)/K)*(W ω*X ω) from fun _ => by ring]
+      rw [integral_add (hWi.mul_const _) (hWXi.const_mul _), integral_mul_const, integral_const_mul]
+    _ ≤ _ := by linarith [mul_nonpos_of_nonneg_of_nonpos hL hc]
+
+def windowBefore {Ω ι : Type*} [LinearOrder ι]
+    (σ : ι → Ω → WithTop ℝ≥0) (i j : ι) (ω : Ω) : Prop :=
+  σ i ω < σ j ω ∨ σ i ω = σ j ω ∧ i < j
+
+lemma window_prod_before {Ω ι : Type*} [Fintype ι] [LinearOrder ι]
+    (σ : ι → Ω → WithTop ℝ≥0) (ω : Ω) (f : ι → ℝ) :
+    ∏ i, f i = 1 + ∑ i, (f i - 1) * ∏ j, if windowBefore σ j i ω then f j else 1 := by
+  classical
+  let key (i : ι) : Lex (WithTop ℝ≥0 × ι) := toLex (σ i ω, i)
+  have hinj : Function.Injective key := fun i j h => congrArg Prod.snd h
+  have hh := @window_prod_telescope ι (LinearOrder.lift' key hinj) Finset.univ f
+  simp only [Finset.prod_filter] at hh
+  change (∏ i, f i) = 1 + ∑ i, (f i-1)*∏j, if key j < key i then f j else 1 at hh
+  rw [hh]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i hi
+  congr 1
+  apply Finset.prod_congr rfl
+  intro j hj
+  have he : key j < key i ↔ windowBefore σ j i ω := by
+    simp only [key, Prod.Lex.toLex_lt_toLex, windowBefore]
+  by_cases hb : windowBefore σ j i ω
+  · rw [ite_eq_left hb, ite_eq_left (he.mpr hb)]
+  · rw [ite_eq_right hb, ite_eq_right (mt he.mp hb)]
+
+lemma window_before_measurable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [LinearOrder ι]
+    {ℱ : Filtration ℝ≥0 mΩ} {σ : ι → Ω → WithTop ℝ≥0}
+    (hs : ∀ i, IsStoppingTime ℱ (σ i)) (i j : ι) :
+    MeasurableSet[(hs i).measurableSpace] {ω | windowBefore σ j i ω} := by
+  have hlt : MeasurableSet[(hs i).measurableSpace] {ω | σ j ω < σ i ω} := by
+    simpa only [compl_ofPred, not_le] using ((hs i).measurableSet_le_stopping_time (hs j)).compl
+  have heq : MeasurableSet[(hs i).measurableSpace] {ω | σ j ω = σ i ω} := by
+    simpa only [eq_comm] using (hs i).measurableSet_eq_stopping_time (hs j)
+  by_cases hij : j < i
+  · convert hlt.union heq using 1
+    ext ω
+    simp only [windowBefore, hij, and_true, mem_ofPred_eq, mem_union]
+  · simpa only [windowBefore, hij, and_false, or_false] using hlt
+
+lemma window_measurable_on_event {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {ℱ : Filtration ℝ≥0 mΩ} {σ τ : Ω → WithTop ℝ≥0}
+    (hs : IsStoppingTime ℱ σ) (ht : IsStoppingTime ℱ τ)
+    {f : Ω → ℝ} (hf : Measurable[ht.measurableSpace] f)
+    {E : Set Ω} (hE : MeasurableSet[hs.measurableSpace] E)
+    (hle : ∀ ω ∈ E, τ ω ≤ σ ω) (c : ℝ) :
+    Measurable[hs.measurableSpace] (fun ω => if ω ∈ E then f ω else c) := by
+  classical
+  intro B hB
+  have hEB : MeasurableSet[hs.measurableSpace] (f ⁻¹' B ∩ E) := by
+    have hh := ht.measurableSet_inter_le hs (f ⁻¹' B) (hf hB)
+    have hh' : MeasurableSet[hs.measurableSpace] (f ⁻¹' B ∩ {ω | τ ω ≤ σ ω}) :=
+      ((ht.min hs).measurableSpace_mono hs (fun _ => min_le_right _ _)) _ hh
+    convert hh'.inter hE using 1
+    ext ω
+    simp only [mem_inter_iff, mem_preimage, mem_ofPred_eq]
+    exact ⟨fun ⟨h1,h2⟩ => ⟨⟨h1,hle ω h2⟩,h2⟩, fun ⟨⟨h1,_⟩,h2⟩ => ⟨h1,h2⟩⟩
+  by_cases hc : c ∈ B
+  · convert hEB.union hE.compl using 1
+    ext ω
+    by_cases hw : ω ∈ E <;> simp [hw, hc]
+  · convert hEB using 1
+    ext ω
+    by_cases hw : ω ∈ E <;> simp [hw, hc]
+
+lemma window_integrable_nonneg {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    [IsFiniteMeasure P] {f : Ω → ℝ} (hf : Measurable f) (h0 : ∀ ω, 0 ≤ f ω)
+    {C : ℝ} (hC : ∀ ω, f ω ≤ C) : Integrable f P :=
+  window_integrable_bounded hf (fun ω => by
+    simpa only [Real.norm_eq_abs, abs_of_nonneg (h0 ω)] using hC ω)
+
+lemma window_weighted_normalized {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
+    {X W : Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
+    (hX : Measurable[mΩ] X) (hXb : ∀ ω, |X ω| ≤ K)
+    (hmean : P[X|m] ≤ᵐ[P] 0) (hW : Measurable[m] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω * (Real.exp (θ*X ω - (θ*K)^2/2)-1) ∂P) ≤ 0 := by
+  let : MeasurableSpace Ω := mΩ
+  obtain ⟨M,hM⟩ := hWb
+  have hWa : Measurable[mΩ] W := hW.mono hm le_rfl
+  have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
+  have hEi : Integrable (fun ω => W ω*Real.exp (θ*X ω)) P :=
+    window_integrable_nonneg (hWa.mul ((hX.const_mul θ).exp))
+      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le) (fun ω =>
+      mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+        (le_abs_self _ |>.trans (hXb ω)) hθ)) (Real.exp_pos _).le (le_trans (hW0 ω) (hM ω)))
+  have h := window_weighted_hoeffding hm hK hθ hX hXb hmean hW hW0 ⟨M,hM⟩
+  simp_rw [Real.exp_sub, mul_sub, mul_one, ← mul_div_assoc]
+  rw [integral_sub (hEi.div_const _) hWi, integral_div]
+  exact sub_nonpos.mpr ((div_le_iff₀ (Real.exp_pos _)).mpr h)
+
+ 
+
+lemma window_finite_mgf {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
+    {ℱ : Filtration ℝ≥0 mΩ} {σ τ : ι → Ω → WithTop ℝ≥0}
+    (hs : ∀ i, IsStoppingTime ℱ (σ i)) (ht : ∀ i, IsStoppingTime ℱ (τ i))
+    (horder : ∀ i j ω, windowBefore σ j i ω → τ j ω ≤ σ i ω)
+    {X : ι → Ω → ℝ} {K θ : ℝ} (hK : 0 < K) (hθ : 0 ≤ θ)
+    (hX : ∀ i, Measurable[(ht i).measurableSpace] (X i))
+    (hXb : ∀ i ω, |X i ω| ≤ K)
+    (hmean : ∀ i, P[X i|(hs i).measurableSpace] ≤ᵐ[P] 0)
+    {W : Ω → ℝ} (hW : Measurable[ℱ 0] W) (hW0 : ∀ ω, 0 ≤ W ω)
+    (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω * Real.exp (θ*∑ i, X i ω) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
+  classical
+  let q := (θ*K)^2/2
+  let g (i : ι) (ω : Ω) := Real.exp (θ*X i ω-q)
+  let pre (i : ι) (ω : Ω) := ∏ j, if windowBefore σ j i ω then g j ω else 1
+  have hg (i : ι) : Measurable[(ht i).measurableSpace] (g i) :=
+    (((hX i).const_mul θ).sub_const q).exp
+  have hga (i : ι) : Measurable[mΩ] (g i) := (hg i).mono (ht i).measurableSpace_le le_rfl
+  have hg0 (i : ι) (ω : Ω) : 0 ≤ g i ω := (Real.exp_pos _).le
+  have hgB (i : ι) (ω : Ω) : g i ω ≤ Real.exp (θ*K) := by
+    apply Real.exp_le_exp.mpr
+    have := mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hXb i ω)) hθ
+    dsimp [q]
+    nlinarith [sq_nonneg (θ*K)]
+  have hp (i : ι) : Measurable[(hs i).measurableSpace] (pre i) := by
+    apply Finset.measurable_prod
+    intro j hj
+    exact window_measurable_on_event (hs i) (ht j) (hg j)
+      (window_before_measurable hs i j) (fun ω hω => horder i j ω hω) 1
+  have hp0 (i : ι) (ω : Ω) : 0 ≤ pre i ω := by
+    apply Finset.prod_nonneg
+    intro j hj
+    split_ifs <;> positivity
+  have hpB (i : ι) (ω : Ω) : pre i ω ≤ Real.exp (θ*K)^Fintype.card ι := by
+    calc
+      _ ≤ ∏ _j : ι, Real.exp (θ*K) := by
+        apply Finset.prod_le_prod₀
+        · intro j hj; split_ifs <;> positivity
+        · intro j hj; split_ifs
+          · exact hgB j ω
+          · exact Real.one_le_exp (mul_nonneg hθ hK.le)
+      _ = _ := by simp
+  have hWstart (i : ι) : Measurable[(hs i).measurableSpace] W := by
+    have hle := (isStoppingTime_const ℱ (0 : ℝ≥0)).measurableSpace_mono (hs i)
+      (fun ω => bot_le)
+    rw [IsStoppingTime.measurableSpace_const] at hle
+    exact hW.mono hle le_rfl
+  have hWa : Measurable[mΩ] W := hW.mono (ℱ.le 0) le_rfl
+  obtain ⟨M,hM⟩ := hWb
+  have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
+  have hHi (i : ι) : Integrable (fun ω => W ω*pre i ω*(g i ω-1)) P := by
+    apply window_integrable_bounded (C := (M*Real.exp (θ*K)^Fintype.card ι)*(Real.exp (θ*K)+1)) (((hWa.mul ((hp i).mono (hs i).measurableSpace_le le_rfl))).mul
+      ((hga i).sub_const 1))
+    intro ω
+    change ‖W ω*pre i ω*(g i ω-1)‖ ≤ _
+    rw [norm_mul, norm_mul, Real.norm_eq_abs, abs_of_nonneg (hW0 ω),
+      Real.norm_eq_abs, abs_of_nonneg (hp0 i ω)]
+    apply mul_le_mul
+    · exact mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))
+    · calc
+        _ ≤ ‖g i ω‖ + ‖(1 : ℝ)‖ := norm_sub_le _ _
+        _ = g i ω + 1 := by rw [Real.norm_eq_abs, abs_of_nonneg (hg0 i ω), norm_one]
+        _ ≤ _ := add_le_add (hgB i ω) le_rfl
+    · exact norm_nonneg _
+    · exact mul_nonneg ((hW0 ω).trans (hM ω)) (by positivity)
+  have hHi0 (i : ι) : (∫ ω, W ω*pre i ω*(g i ω-1) ∂P) ≤ 0 := by
+    exact window_weighted_normalized (hs i).measurableSpace_le hK hθ
+      ((hX i).mono (ht i).measurableSpace_le le_rfl) (hXb i) (hmean i)
+      ((hWstart i).mul (hp i)) (fun ω => mul_nonneg (hW0 ω) (hp0 i ω))
+      ⟨M*Real.exp (θ*K)^Fintype.card ι, fun ω =>
+        mul_le_mul (hM ω) (hpB i ω) (hp0 i ω) ((hW0 ω).trans (hM ω))⟩
+  have hnorm : (∫ ω, W ω * ∏ i, g i ω ∂P) ≤ ∫ ω, W ω ∂P := by
+    have hid (ω : Ω) := window_prod_before σ ω (fun i => g i ω)
+    simp_rw [hid, mul_add, mul_one, Finset.mul_sum]
+    change (∫ ω, W ω + ∑ i, W ω*((g i ω-1)*pre i ω) ∂P) ≤ _
+    simp_rw [show ∀ i ω, W ω*((g i ω-1)*pre i ω) = W ω*pre i ω*(g i ω-1)
+      from fun _ _ => by ring]
+    rw [integral_add hWi (integrable_finsetSum _ (fun i hi => hHi i)), integral_finsetSum]
+    · exact add_le_of_nonpos_right (Finset.sum_nonpos fun i hi => hHi0 i)
+    · exact fun i hi => hHi i
+  have hprod (ω : Ω) : (∏ i, g i ω) =
+      Real.exp (θ*∑ i, X i ω) / Real.exp (Fintype.card ι * q) := by
+    change (∏ i, Real.exp (θ*X i ω-q)) = _
+    rw [← Real.exp_sum]
+    simp only [Finset.sum_sub_distrib, ← Finset.mul_sum, Finset.sum_const,
+      Finset.card_univ, nsmul_eq_mul]
+    exact Real.exp_sub _ _
+  simp only [hprod, ← mul_div_assoc] at hnorm
+  rw [integral_div] at hnorm
+  simpa only [q, mul_div_assoc] using (div_le_iff₀ (Real.exp_pos _)).mp hnorm
+
+ 
+
+structure SourceDisjointWindows {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (D : Set ℂ) (r R K : ℝ) (γ : ℝ≥0 → Ω → ℂ)
+    (σ τ : ℂ → Ω → WithTop ℝ≥0) (Y μ : ℂ → Ω → ℝ) : Prop where
+  measurable_D : MeasurableSet D
+  bounded_D : D ⊆ ball 0 1
+  radius_pos : 0 < r
+  radius_le : r ≤ R
+  radius_small : 3 * R ≤ 1
+  bound_nonneg : 0 ≤ K
+  continuous_trace : ∀ ω, Continuous (fun t => γ t ω)
+  adapted_trace : ∀ t, Measurable[ℱ t] (γ t)
+  start : ∀ z, IsStoppingTime ℱ (σ z)
+  finish : ∀ z, IsStoppingTime ℱ (τ z)
+  time_le : ∀ z ω, σ z ω ≤ τ z ω
+  contact : ∀ z ω, σ z ω ≠ ⊤ → dist (γ ((σ z ω).untopD 0) ω) z = r
+  confined : ∀ z ω (t : ℝ≥0), σ z ω ≤ (t : WithTop ℝ≥0) → (t : WithTop ℝ≥0) < τ z ω →
+    dist (γ t ω) z < R
+  output_measurable : ∀ z, Measurable[(finish z).measurableSpace] (Y z)
+  mean_measurable : ∀ z, Measurable[(start z).measurableSpace] (μ z)
+  output_bound : ∀ z ω, 0 ≤ Y z ω ∧ Y z ω ≤ K
+  mean_nonneg : ∀ z ω, 0 ≤ μ z ω
+  mean_bounded : ∃ M : ℝ, ∀ z ω, μ z ω ≤ M
+  upper_mean : ∀ z, ∀ᵐ ω ∂P, σ z ω ≠ ⊤ → P[Y z | (start z).measurableSpace] ω ≤ μ z ω
+  no_start : ∀ z ω, σ z ω = ⊤ → Y z ω = 0 ∧ μ z ω = 0
+  no_end : ∀ z ω, τ z ω = ⊤ → Y z ω = 0
+  outside : ∀ z, z ∉ D → ∀ ω, σ z ω = ⊤ ∧ Y z ω = 0 ∧ μ z ω = 0
+  joint_output : Measurable (fun p : ℂ × Ω => Y p.1 p.2)
+  joint_mean : Measurable (fun p : ℂ × Ω => μ p.1 p.2)
+
+lemma window_cond_mean_at_infinity {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    {σ : Ω → WithTop ℝ≥0} (hs : IsStoppingTime ℱ σ)
+    {Y μ : Ω → ℝ} (hY : Integrable Y P)
+    (hmean : ∀ᵐ ω ∂P, σ ω ≠ ⊤ → P[Y|hs.measurableSpace] ω ≤ μ ω)
+    (hno : ∀ ω, σ ω = ⊤ → Y ω = 0 ∧ μ ω = 0) :
+    P[Y|hs.measurableSpace] ≤ᵐ[P] μ := by
+  have htop : IsStoppingTime ℱ (fun _ => (⊤ : WithTop ℝ≥0)) := by
+    intro t
+    simp
+  have hE : MeasurableSet[hs.measurableSpace] {ω | σ ω = ⊤} :=
+    hs.measurableSet_eq_stopping_time htop
+  have he := condExp_indicator hY hE
+  have hz : ({ω | σ ω = ⊤}.indicator Y) = 0 := by
+    ext ω
+    by_cases h : σ ω = ⊤
+    · simpa [h] using (hno ω h).1
+    · simp [h]
+  rw [hz, condExp_zero] at he
+  filter_upwards [hmean, he] with ω hm he
+  by_cases h : σ ω = ⊤
+  · have hy : P[Y|hs.measurableSpace] ω = 0 := by simpa [h] using he.symm
+    rw [hy, (hno ω h).2]
+  · exact hm h
+
+lemma window_capped_cond_mean {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
+    {Y μ : Ω → ℝ} {K : ℝ} (hK : 0 ≤ K) (hY : Measurable[mΩ] Y)
+    (hYb : ∀ ω, 0 ≤ Y ω ∧ Y ω ≤ K) (hμ : Measurable[m] μ)
+    (hμ0 : ∀ ω, 0 ≤ μ ω) (hmean : P[Y|m] ≤ᵐ[P] μ) :
+    P[(fun ω => Y ω-min (μ ω) K)|m] ≤ᵐ[P] 0 := by
+  let : MeasurableSpace Ω := mΩ
+  have hYi : Integrable Y P := window_integrable_nonneg hY (fun ω => (hYb ω).1) (fun ω => (hYb ω).2)
+  have hν : Measurable[m] (fun ω => min (μ ω) K) := hμ.min measurable_const
+  have hνi : Integrable (fun ω => min (μ ω) K) P :=
+    window_integrable_nonneg (hν.mono hm le_rfl) (fun ω => le_min (hμ0 ω) hK)
+      (fun ω => min_le_right _ _)
+  have hbound := condExp_mono hYi (integrable_const K) (Eventually.of_forall fun ω => (hYb ω).2) (m := m)
+  rw [condExp_const hm] at hbound
+  have he := condExp_sub hYi hνi m
+  rw [condExp_of_stronglyMeasurable hm hν.stronglyMeasurable hνi] at he
+  filter_upwards [he, hbound, hmean] with ω he hb hm
+  simp only [Pi.sub_apply] at he
+  have he' : P[(fun ω => Y ω-min (μ ω) K)|m] ω = P[Y|m] ω-min (μ ω) K := by
+    convert he using 1
+  rw [he']
+  exact sub_nonpos.mpr (le_min hm hb)
+
+lemma window_geometric_order {Ω ι : Type*} [MeasurableSpace Ω]
+    [LinearOrder ι] {P : Measure Ω} {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    {z : ι → ℂ} (hz : ∀ i j, i ≠ j → 2*R < dist (z i) (z j)) :
+    ∀ i j ω, windowBefore (fun i => σ (z i)) j i ω → τ (z j) ω ≤ σ (z i) ω := by
+  intro i j ω hpre
+  by_contra hn
+  have hne : σ (z i) ω ≠ ⊤ := ne_top_of_lt (lt_of_not_ge hn)
+  have hji : j ≠ i := by
+    intro he
+    subst j
+    rcases hpre with hlt | ⟨_,hlt⟩ <;> exact (lt_irrefl _ hlt)
+  have hle : σ (z j) ω ≤ σ (z i) ω := hpre.elim le_of_lt (fun h => h.1.le)
+  have hcoe : (((σ (z i) ω).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = σ (z i) ω :=
+    by
+      cases he : σ (z i) ω with
+      | top => exact (hne he).elim
+      | coe t => rfl
+  have hconf := h.confined (z j) ω ((σ (z i) ω).untopD 0) (hcoe.symm ▸ hle)
+    (hcoe.symm ▸ lt_of_not_ge hn)
+  have hcont := h.contact (z i) ω hne
+  have htri := dist_triangle (z j) (γ ((σ (z i) ω).untopD 0) ω) (z i)
+  rw [dist_comm (z j) (γ ((σ (z i) ω).untopD 0) ω), hcont] at htri
+  have := hz j i hji
+  linarith [h.radius_le]
+
+lemma window_source_finite_mgf {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
+    [Fintype ι] [LinearOrder ι] {P : Measure Ω} [IsFiniteMeasure P]
+    {ℱ : Filtration ℝ≥0 mΩ} {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    {z : ι → ℂ} (hz : ∀ i j, i ≠ j → 2*R < dist (z i) (z j))
+    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hW : Measurable[ℱ 0] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
+    (∫ ω, W ω * Real.exp (θ*∑ i, (Y (z i) ω-μ (z i) ω)) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (Fintype.card ι * (θ*K)^2/2) := by
+  let X (i : ι) (ω : Ω) := Y (z i) ω-min (μ (z i) ω) K
+  have hYa (i : ι) : Measurable[mΩ] (Y (z i)) :=
+    (h.output_measurable (z i)).mono (h.finish (z i)).measurableSpace_le le_rfl
+  have hμa (i : ι) : Measurable[mΩ] (μ (z i)) :=
+    (h.mean_measurable (z i)).mono (h.start (z i)).measurableSpace_le le_rfl
+  have hX (i : ι) : Measurable[(h.finish (z i)).measurableSpace] (X i) :=
+    (h.output_measurable (z i)).sub (((h.mean_measurable (z i)).mono
+      ((h.start (z i)).measurableSpace_mono (h.finish (z i)) (h.time_le (z i))) le_rfl).min measurable_const)
+  have hXb (i : ι) (ω : Ω) : |X i ω| ≤ K := by
+    rw [abs_le]
+    have := h.output_bound (z i) ω
+    have := le_min (h.mean_nonneg (z i) ω) hK.le
+    have := min_le_right (μ (z i) ω) K
+    dsimp [X]
+    constructor <;> linarith
+  have hm (i : ι) : P[X i|(h.start (z i)).measurableSpace] ≤ᵐ[P] 0 :=
+    window_capped_cond_mean (h.start (z i)).measurableSpace_le hK.le (hYa i)
+      (h.output_bound (z i)) (h.mean_measurable (z i)) (h.mean_nonneg (z i))
+      (window_cond_mean_at_infinity (h.start (z i))
+        (window_integrable_nonneg (hYa i) (fun ω => (h.output_bound (z i) ω).1)
+          (fun ω => (h.output_bound (z i) ω).2)) (h.upper_mean (z i)) (h.no_start (z i)))
+  have ht := window_finite_mgf (fun i => h.start (z i)) (fun i => h.finish (z i))
+    (window_geometric_order h hz) hK hθ hX hXb hm hW hW0 hWb
+  apply le_trans _ ht
+  obtain ⟨M,hM⟩ := hWb
+  obtain ⟨L,hL⟩ := h.mean_bounded
+  have hWa : Measurable[mΩ] W := hW.mono (ℱ.le 0) le_rfl
+  have hib (f : ι → Ω → ℝ) (hf : ∀ i, Measurable[mΩ] (f i))
+      (hb : ∀ i ω, f i ω ≤ K) : Integrable (fun ω => W ω*Real.exp (θ*∑ i, f i ω)) P :=
+    window_integrable_nonneg (C := M*Real.exp (θ*(Fintype.card ι*K))) (hWa.mul ((Finset.measurable_sum _ (fun i _ => hf i)).const_mul θ).exp)
+      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
+      (fun ω => mul_le_mul (hM ω) (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left
+        ((Finset.sum_le_sum fun i hi => hb i ω).trans_eq (by simp)) hθ))
+        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
+  apply integral_mono
+  · exact hib (fun i ω => Y (z i) ω-μ (z i) ω) (fun i => (hYa i).sub (hμa i))
+      (fun i ω => sub_le_self _ (h.mean_nonneg (z i) ω) |>.trans (h.output_bound (z i) ω).2)
+  · exact hib X (fun i => (hX i).mono (h.finish (z i)).measurableSpace_le le_rfl)
+      (fun i ω => le_abs_self _ |>.trans (hXb i ω))
+  · intro ω
+    apply mul_le_mul_of_nonneg_left _ (hW0 ω)
+    apply Real.exp_le_exp.mpr
+    apply mul_le_mul_of_nonneg_left _ hθ
+    exact Finset.sum_le_sum fun i hi => sub_le_sub_left (min_le_left _ _) _
+
+lemma window_integral_grid_1d {f : ℝ → ℝ} (hf : Measurable f)
+    {C a : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (ha : 0 ≤ a) (n : ℕ) :
+    (∫ x in Ioc 0 a, ∑ i : Fin n, f (x + a*i - 2)) =
+      ∫ x in Ioc (-2) (a*n-2), f x := by
+  have hfi (u v : ℝ) : IntegrableOn f (Ioc u v) :=
+    window_integrable_bounded hf hC
+  have hi (i : Fin n) : IntegrableOn (fun x => f (x+a*i-2)) (Ioc 0 a) :=
+    window_integrable_bounded (hf.comp ((measurable_id.add_const _).sub_const _))
+      (fun x => hC _)
+  rw [integral_finsetSum _ (fun i _ => hi i)]
+  have hpiece (i : Fin n) : (∫ x in Ioc 0 a, f (x+a*i-2)) =
+      ∫ x in (a*(i:ℝ)-2)..(a*((i:ℝ)+1)-2), f x := by
+    rw [← intervalIntegral.integral_of_le ha]
+    have he := intervalIntegral.integral_comp_add_right f (a*i-2) (a := 0) (b := a)
+    simp only [zero_add] at he
+    have hend : a*((i:ℝ)+1)-2 = a+(a*i-2) := by ring
+    simpa only [hend, add_sub_assoc] using he
+  simp_rw [hpiece]
+  rw [Fin.sum_univ_eq_sum_range (fun k : ℕ =>
+    ∫ x in (a*(k:ℝ)-2)..(a*((k:ℝ)+1)-2), f x) n]
+  have hsum := intervalIntegral.sum_integral_adjacent_intervals
+    (a := fun k => a*k-2) (n := n) (f := f) (μ := volume)
+    (fun k hk => (intervalIntegrable_iff_integrableOn_Ioc_of_le
+      (by have : (k:ℝ) ≤ k+1 := by linarith
+          exact sub_le_sub_right (mul_le_mul_of_nonneg_left (by exact_mod_cast Nat.le_succ k) ha) 2)).mpr (hfi _ _))
+  simp only [Nat.cast_add, Nat.cast_one, Nat.cast_zero, mul_zero, zero_sub] at hsum
+  rw [hsum, intervalIntegral.integral_of_le (by nlinarith [mul_nonneg ha (Nat.cast_nonneg n)])]
+
+noncomputable def windowSquare (a b : ℝ) : Measure (ℝ × ℝ) :=
+  (volume.restrict (Ioc a b)).prod (volume.restrict (Ioc a b))
+
+instance windowSquare_finite (a b : ℝ) : IsFiniteMeasure (windowSquare a b) := by
+  dsimp [windowSquare]
+  infer_instance
+
+lemma window_grid_2d {f : ℝ × ℝ → ℝ} (hf : Measurable f)
+    {C a : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (ha : 0 ≤ a) (n : ℕ) :
+    (∫ p, ∑ ij : Fin n × Fin n,
+      f (p.1+a*ij.1-2,p.2+a*ij.2-2) ∂windowSquare 0 a) =
+      ∫ p, f p ∂windowSquare (-2) (a*n-2) := by
+  let G (x : ℝ) := ∫ y in Ioc (-2) (a*n-2), f (x,y)
+  have hG : Measurable G := (hf.stronglyMeasurable.integral_prod_right (f := fun x y => f (x,y))).measurable
+  have hGb (x : ℝ) : ‖G x‖ ≤ C*(volume.restrict (Ioc (-2) (a*n-2))).real univ :=
+    norm_integral_le_of_norm_le_const (Eventually.of_forall fun y => hC (x,y))
+  have hrow (x : ℝ) : (∫ y in Ioc 0 a, ∑ ij : Fin n × Fin n,
+      f (x+a*ij.1-2,y+a*ij.2-2)) = ∑ i : Fin n, G (x+a*i-2) := by
+    simp_rw [Fintype.sum_prod_type]
+    rw [integral_finsetSum]
+    · congr 1
+      ext i
+      exact window_integral_grid_1d (hf.comp (measurable_const.prodMk measurable_id))
+        (fun y => hC _) ha n
+    · intro i hi
+      apply window_integrable_bounded (C := n*C)
+      · exact Finset.measurable_sum _ (fun j hj => hf.comp (measurable_const.prodMk
+          ((measurable_id.add_const _).sub_const _)))
+      · intro y
+        exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun j hj => hC _).trans_eq (by simp))
+  dsimp only [windowSquare]
+  rw [integral_prod]
+  · simp_rw [hrow]
+    rw [window_integral_grid_1d hG hGb ha n]
+    exact (integral_prod f (window_integrable_bounded hf hC)).symm
+  · apply window_integrable_bounded (C := (n*n)*C)
+    · apply Finset.measurable_sum
+      intro ij hij
+      exact hf.comp (((measurable_fst.add_const _).sub_const _).prodMk
+        ((measurable_snd.add_const _).sub_const _))
+    · intro p
+      exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun ij hij => hC _).trans_eq (by simp))
+
+noncomputable def windowShiftLaw (a : ℝ) : Measure (ℝ × ℝ) :=
+  (ENNReal.ofReal (a^2))⁻¹ • windowSquare 0 a
+
+lemma windowSquare_mass {a : ℝ} (ha : 0 ≤ a) : windowSquare 0 a univ = ENNReal.ofReal (a^2) := by
+  simp [windowSquare, Measure.prod_apply, Real.volume_Ioc, sq, ← ENNReal.ofReal_mul ha]
+
+lemma windowShiftLaw_probability {a : ℝ} (ha : 0 < a) : IsProbabilityMeasure (windowShiftLaw a) := by
+  apply isProbabilityMeasure_iff.mpr
+  rw [windowShiftLaw, Measure.smul_apply, windowSquare_mass ha.le]
+  exact ENNReal.inv_mul_cancel (by positivity) (by finiteness)
+
+lemma window_rectangle_integral {f : ℂ → ℝ} (_hf : Measurable f)
+    (hz : ∀ z, z ∉ ball (0:ℂ) 1 → f z = 0) :
+    (∫ p, f (Complex.measurableEquivRealProd.symm p) ∂windowSquare (-2) 2) = ∫ z, f z := by
+  rw [windowSquare, Measure.prod_restrict]
+  rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
+  · exact (Complex.volume_preserving_equiv_real_prod.symm).integral_comp' f
+  · intro p hp
+    apply hz
+    intro hb
+    apply hp
+    have hn : ‖Complex.measurableEquivRealProd.symm p‖ < 1 := by simpa using hb
+    have hr := Complex.abs_re_le_norm (Complex.measurableEquivRealProd.symm p)
+    have hi := Complex.abs_im_le_norm (Complex.measurableEquivRealProd.symm p)
+    change |p.1| ≤ _ at hr
+    change |p.2| ≤ _ at hi
+    exact ⟨⟨by linarith [neg_abs_le p.1], by linarith [le_abs_self p.1]⟩,
+      ⟨by linarith [neg_abs_le p.2], by linarith [le_abs_self p.2]⟩⟩
+
+noncomputable def windowCenter (a : ℝ) (ξ : ℝ × ℝ) {n : ℕ} (i : Fin n × Fin n) : ℂ :=
+  Complex.measurableEquivRealProd.symm (ξ.1+a*i.1-2, ξ.2+a*i.2-2)
+
+lemma window_grid_average {f : ℂ → ℝ} (hf : Measurable f)
+    {C a : ℝ} (hC : ∀ z, ‖f z‖ ≤ C) (ha : 0 < a) {n : ℕ} (han : a*n=4)
+    (hz : ∀ z, z ∉ ball (0:ℂ) 1 → f z = 0) :
+    (∫ ξ, a^2 * ∑ i : Fin n × Fin n, f (windowCenter a ξ i) ∂windowShiftLaw a) = ∫ z, f z := by
+  rw [windowShiftLaw, integral_smul_measure, integral_const_mul]
+  have he := window_grid_2d (hf.comp Complex.measurableEquivRealProd.symm.measurable)
+    (fun p => hC _) ha.le n
+  rw [han] at he
+  norm_num only [Function.comp_apply] at he
+  dsimp only [windowCenter]
+  rw [he, window_rectangle_integral hf hz]
+  simp only [ENNReal.toReal_inv, ENNReal.toReal_ofReal (sq_nonneg a), smul_eq_mul]
+  field_simp
+
+lemma window_nat_spacing (a : ℝ) (ha : 0 ≤ a) {i j : ℕ} (hne : i ≠ j) :
+    a ≤ |a*i-a*j| := by
+  rcases lt_or_gt_of_ne hne with h | h
+  · have h' : (i:ℝ)+1 ≤ j := by exact_mod_cast h
+    have hle : a*i ≤ a*j := mul_le_mul_of_nonneg_left (by exact_mod_cast h.le) ha
+    rw [abs_of_nonpos (sub_nonpos.mpr hle)]
+    nlinarith
+  · have h' : (j:ℝ)+1 ≤ i := by exact_mod_cast h
+    have hle : a*j ≤ a*i := mul_le_mul_of_nonneg_left (by exact_mod_cast h.le) ha
+    rw [abs_of_nonneg (sub_nonneg.mpr hle)]
+    nlinarith
+
+lemma window_centers_separated {a R : ℝ} (hsep : 2*R < a) (ha : 0 ≤ a)
+    {n : ℕ} (ξ : ℝ × ℝ) (i j : Fin n × Fin n) (hne : i ≠ j) :
+    2*R < dist (windowCenter a ξ i) (windowCenter a ξ j) := by
+  rw [dist_eq_norm]
+  rcases not_and_or.mp (mt Prod.ext_iff.mpr hne) with h | h
+  · apply hsep.trans_le
+    apply le_trans (window_nat_spacing a ha (mt Fin.ext h))
+    have he : (windowCenter a ξ i - windowCenter a ξ j).re = a*(i.1:ℝ)-a*(j.1:ℝ) := by
+      dsimp [windowCenter]
+      ring
+    rw [← he]
+    exact Complex.abs_re_le_norm _
+  · apply hsep.trans_le
+    apply le_trans (window_nat_spacing a ha (mt Fin.ext h))
+    have he : (windowCenter a ξ i - windowCenter a ξ j).im = a*(i.2:ℝ)-a*(j.2:ℝ) := by
+      dsimp [windowCenter]
+      ring
+    rw [← he]
+    exact Complex.abs_im_le_norm _
+
+lemma window_choose_grid {R : ℝ} (hR : 0 < R) (hsmall : 3*R ≤ 1) :
+    ∃ (n : ℕ) (a : ℝ), 0 < a ∧ a*n=4 ∧ 2*R<a ∧ a≤4*R := by
+  let n := Nat.ceil (1/R)
+  have hl := Nat.le_ceil (1/R)
+  have hu := Nat.ceil_lt_add_one (a := 1/R) (by positivity)
+  have hn : 0 < (n:ℝ) := (by positivity : 0 < 1/R).trans_le hl
+  have hnr : 1 ≤ R*n := by
+    exact (div_le_iff₀ hR).mp hl |>.trans_eq (mul_comm _ _)
+  have hnr' : R*n < 1+R := by
+    have H := mul_lt_mul_of_pos_right hu hR
+    rw [add_mul, div_mul_cancel₀ _ hR.ne', one_mul] at H
+    simpa only [mul_comm] using H
+  refine ⟨n,4/n,by positivity,div_mul_cancel₀ 4 hn.ne',?_,?_⟩
+  · apply (lt_div_iff₀ hn).mpr
+    nlinarith
+  · apply (div_le_iff₀ hn).mpr
+    nlinarith
+lemma window_jensen_mgf {Ξ Ω : Type*} [MeasurableSpace Ξ] [MeasurableSpace Ω]
+    {Q : Measure Ξ} [IsProbabilityMeasure Q] {P : Measure Ω} [IsFiniteMeasure P]
+    {F : Ξ → Ω → ℝ} (hF : Measurable (Function.uncurry F))
+    {B θ A : ℝ} (hFb : ∀ ξ ω, ‖F ξ ω‖ ≤ B) (hθ : 0 ≤ θ)
+    {W : Ω → ℝ} (hW : Measurable W) (hW0 : ∀ ω, 0 ≤ W ω)
+    (hWb : ∃ M, ∀ ω, W ω ≤ M)
+    (hmgf : ∀ ξ, (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤ A) :
+    (∫ ω, W ω*Real.exp (θ*∫ ξ, F ξ ω ∂Q) ∂P) ≤ A := by
+  obtain ⟨M,hM⟩ := hWb
+  have hFi (ω : Ω) : Integrable (fun ξ => θ*F ξ ω) Q :=
+    window_integrable_bounded ((hF.comp (measurable_id.prodMk measurable_const)).const_mul θ)
+      (C := θ*B) (fun ξ => by rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg hθ]
+                              exact mul_le_mul_of_nonneg_left (hFb ξ ω) hθ)
+  have hEi (ω : Ω) : Integrable (fun ξ => Real.exp (θ*F ξ ω)) Q :=
+    window_integrable_nonneg ((hF.comp (measurable_id.prodMk measurable_const)).const_mul θ).exp
+      (fun ξ => (Real.exp_pos _).le) (fun ξ => Real.exp_le_exp.mpr
+        (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hFb ξ ω)) hθ))
+  have hJ (ω : Ω) : Real.exp (θ*∫ ξ, F ξ ω ∂Q) ≤ ∫ ξ, Real.exp (θ*F ξ ω) ∂Q := by
+    rw [← integral_const_mul]
+    exact convexOn_exp.map_integral_le Real.continuous_exp.continuousOn isClosed_univ
+      (Eventually.of_forall fun _ => mem_univ _) (hFi ω) (hEi ω)
+  have hZ : Measurable (fun ω => ∫ ξ, F ξ ω ∂Q) :=
+    ((hF.comp measurable_swap).stronglyMeasurable.integral_prod_right (f := fun ω ξ => F ξ ω)).measurable
+  have hZb (ω : Ω) : ‖∫ ξ, F ξ ω ∂Q‖ ≤ B := by
+    simpa using norm_integral_le_of_norm_le_const (μ := Q) (Eventually.of_forall fun ξ => hFb ξ ω)
+  have hleft : Integrable (fun ω => W ω*Real.exp (θ*∫ ξ, F ξ ω ∂Q)) P :=
+    window_integrable_nonneg (hW.mul (hZ.const_mul θ).exp)
+      (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
+      (C := M*Real.exp (θ*B)) (fun ω => mul_le_mul (hM ω)
+        (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hZb ω)) hθ))
+        (Real.exp_pos _).le ((hW0 ω).trans (hM ω)))
+  have hjoint : Integrable (Function.uncurry (fun ξ ω => W ω*Real.exp (θ*F ξ ω))) (Q.prod P) :=
+    window_integrable_nonneg ((hW.comp measurable_snd).mul (hF.const_mul θ).exp)
+      (fun p => mul_nonneg (hW0 p.2) (Real.exp_pos _).le)
+      (C := M*Real.exp (θ*B)) (fun p => mul_le_mul (hM p.2)
+        (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hFb p.1 p.2)) hθ))
+        (Real.exp_pos _).le ((hW0 p.2).trans (hM p.2)))
+  calc
+    _ ≤ ∫ ω, ∫ ξ, W ω*Real.exp (θ*F ξ ω) ∂Q ∂P := by
+      apply integral_mono hleft hjoint.integral_prod_right
+      intro ω
+      change W ω*Real.exp (θ*∫ ξ, F ξ ω ∂Q) ≤ ∫ ξ, W ω*Real.exp (θ*F ξ ω) ∂Q
+      rw [integral_const_mul]
+      exact mul_le_mul_of_nonneg_left (hJ ω) (hW0 ω)
+    _ = ∫ ξ, ∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P ∂Q := (integral_integral_swap hjoint).symm
+    _ ≤ ∫ _ξ, A ∂Q := integral_mono hjoint.integral_prod_left (integrable_const A) hmgf
+    _ = A := by simp
+lemma window_source_mgf_pos {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hW : Measurable[ℱ 0] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) (hK : 0 < K) :
+    (∫ ω, W ω * Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
+  obtain ⟨n,a,ha,han,hsep,haR⟩ := window_choose_grid (h.radius_pos.trans_le h.radius_le) h.radius_small
+  let : LinearOrder (Fin n × Fin n) := LinearOrder.lift' toLex toLex.injective
+  let F (ξ : ℝ × ℝ) (ω : Ω) := a^2 * ∑ i : Fin n × Fin n,
+    (Y (windowCenter a ξ i) ω - μ (windowCenter a ξ i) ω)
+  have hcenter (i : Fin n × Fin n) : Measurable (fun ξ => windowCenter a ξ i) :=
+    Complex.measurableEquivRealProd.symm.measurable.comp
+      (((measurable_fst.add_const _).sub_const _).prodMk ((measurable_snd.add_const _).sub_const _))
+  have hmeas : Measurable (Function.uncurry F) := by
+    apply Measurable.const_mul
+    apply Finset.measurable_sum
+    intro i hi
+    exact (h.joint_output.sub h.joint_mean).comp
+      (((hcenter i).comp measurable_fst).prodMk measurable_snd)
+  obtain ⟨L,hL⟩ := h.mean_bounded
+  have hnorm (z : ℂ) (ω : Ω) : ‖Y z ω-μ z ω‖ ≤ K+L := by
+    calc
+      _ ≤ ‖Y z ω‖+‖μ z ω‖ := norm_sub_le _ _
+      _ = Y z ω+μ z ω := by rw [Real.norm_eq_abs, abs_of_nonneg (h.output_bound z ω).1,
+        Real.norm_eq_abs, abs_of_nonneg (h.mean_nonneg z ω)]
+      _ ≤ _ := add_le_add (h.output_bound z ω).2 (hL z ω)
+  have hFb (ξ : ℝ × ℝ) (ω : Ω) : ‖F ξ ω‖ ≤ a^2*((n:ℝ)^2*(K+L)) := by
+    dsimp only [F]
+    rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (sq_nonneg a)]
+    apply mul_le_mul_of_nonneg_left _ (sq_nonneg a)
+    exact (norm_sum_le _ _).trans ((Finset.sum_le_sum fun i hi => hnorm _ ω).trans_eq (by simp [sq]; ring))
+  have havg (ω : Ω) : (∫ ξ, F ξ ω ∂windowShiftLaw a) = ∫ z in D, (Y z ω-μ z ω) := by
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by
+      obtain ⟨_,hy,hμ⟩ := h.outside z hz ω
+      simp only [hy,hμ,sub_self])]
+    exact window_grid_average ((h.joint_output.sub h.joint_mean).comp
+      (measurable_id.prodMk measurable_const)) (fun z => hnorm z ω) ha han
+      (fun z hz => by
+        obtain ⟨_,hy,hμ⟩ := h.outside z (fun hzD => hz (h.bounded_D hzD)) ω
+        change Y z ω-μ z ω=0
+        simp only [hy,hμ,sub_self])
+  let : IsProbabilityMeasure (windowShiftLaw a) := windowShiftLaw_probability ha
+  have hfinite (ξ : ℝ × ℝ) :
+      (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
+    have ht := window_source_finite_mgf h (window_centers_separated (n := n) hsep ha.le ξ)
+      (mul_nonneg hθ (sq_nonneg a)) hW hW0 hWb hK
+    have he : (Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2 ≤ 128*K^2*θ^2*R^2 := by
+      simp only [Fintype.card_prod, Fintype.card_fin, Nat.cast_mul]
+      have hs : a^2 ≤ 16*R^2 := by nlinarith [h.radius_pos.trans_le h.radius_le]
+      calc
+        _ = (a*n)^2*a^2*θ^2*K^2/2 := by ring
+        _ = 8*a^2*θ^2*K^2 := by rw [han]; ring
+        _ ≤ 8*(16*R^2)*θ^2*K^2 := by gcongr
+        _ = _ := by ring
+    have ht' : (∫ ω, W ω*Real.exp (θ*F ξ ω) ∂P) ≤
+        (∫ ω, W ω ∂P) * Real.exp ((Fintype.card (Fin n × Fin n) : ℝ)*((θ*a^2)*K)^2/2) := by
+      simpa only [F, mul_assoc] using ht
+    exact ht'.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr he) (integral_nonneg hW0))
+  have ht := window_jensen_mgf (Q := windowShiftLaw a) hmeas hFb hθ (hW.mono (ℱ.le 0) le_rfl) hW0 hWb hfinite
+  simpa only [havg] using ht
+
+lemma window_source_integral_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ) :
+    Measurable (fun ω => ∫ z in D, (Y z ω-μ z ω)) :=
+  ((h.joint_output.sub h.joint_mean).comp measurable_swap).stronglyMeasurable.integral_prod_right
+    (f := fun ω z => Y z ω-μ z ω) |>.measurable
+
+lemma window_source_integral_bounded {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ) :
+    ∃ B : ℝ, ∀ ω, ‖∫ z in D, (Y z ω-μ z ω)‖ ≤ B := by
+  have hfin : volume D ≠ ∞ := (isBounded_ball.subset h.bounded_D).measure_lt_top.ne
+  let : IsFiniteMeasure (volume.restrict D) := isFiniteMeasure_restrict.mpr hfin
+  obtain ⟨M,hM⟩ := h.mean_bounded
+  refine ⟨(K+M)*(volume.restrict D).real univ, fun ω => norm_integral_le_of_norm_le_const
+    (Eventually.of_forall fun z => ?_)⟩
+  calc
+    _ ≤ ‖Y z ω‖+‖μ z ω‖ := norm_sub_le _ _
+    _ = Y z ω+μ z ω := by rw [Real.norm_eq_abs, abs_of_nonneg (h.output_bound z ω).1,
+      Real.norm_eq_abs, abs_of_nonneg (h.mean_nonneg z ω)]
+    _ ≤ _ := add_le_add (h.output_bound z ω).2 (hM z ω)
+
+lemma window_source_mgf {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ)
+    {θ : ℝ} (hθ : 0 ≤ θ) {W : Ω → ℝ} (hW : Measurable[ℱ 0] W)
+    (hW0 : ∀ ω, 0 ≤ W ω) (hWb : ∃ M, ∀ ω, W ω ≤ M) :
+    (∫ ω, W ω * Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ∂P) ≤
+      (∫ ω, W ω ∂P) * Real.exp (128*K^2*θ^2*R^2) := by
+  rcases h.bound_nonneg.eq_or_lt with hK | hK
+  · have hz (ω : Ω) : (∫ z in D, (Y z ω-μ z ω)) ≤ 0 := by
+      apply integral_nonpos
+      intro z
+      change Y z ω-μ z ω≤0
+      have := (h.output_bound z ω).2
+      have := h.mean_nonneg z ω
+      linarith
+    have he (ω : Ω) : Real.exp (θ*∫ z in D, (Y z ω-μ z ω)) ≤ 1 :=
+      Real.exp_le_one_iff.mpr (mul_nonpos_of_nonneg_of_nonpos hθ (hz ω))
+    obtain ⟨M,hM⟩ := hWb
+    have hWa : Measurable W := hW.mono (ℱ.le 0) le_rfl
+    have hWi : Integrable W P := window_integrable_nonneg hWa hW0 hM
+    have hli : Integrable (fun ω => W ω*Real.exp (θ*∫ z in D, (Y z ω-μ z ω))) P :=
+      window_integrable_nonneg (hWa.mul ((window_source_integral_measurable h).const_mul θ).exp)
+        (fun ω => mul_nonneg (hW0 ω) (Real.exp_pos _).le)
+        (fun ω => ((mul_le_mul_of_nonneg_left (he ω) (hW0 ω)).trans_eq (mul_one _)).trans (hM ω))
+    simp only [← hK, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero,
+      zero_mul, Real.exp_zero, mul_one]
+    exact integral_mono hli hWi (fun ω => (mul_le_mul_of_nonneg_left (he ω) (hW0 ω)).trans_eq (mul_one _))
+  · exact window_source_mgf_pos h hθ hW hW0 hWb hK
+lemma window_source_tail {Ω : Type*} [mΩ : MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    {D : Set ℂ} {r R K : ℝ} {γ : ℝ≥0 → Ω → ℂ}
+    {σ τ : ℂ → Ω → WithTop ℝ≥0} {Y μ : ℂ → Ω → ℝ}
+    (h : SourceDisjointWindows P ℱ D r R K γ σ τ Y μ) {b : ℝ} (hb : 0 < b) :
+    P {ω | b < ∫ z in D, (Y z ω-μ z ω)} ≤
+      ENNReal.ofReal (Real.exp (-(1/512:ℝ)*b^2/(K^2*R^2))) := by
+  rcases h.bound_nonneg.eq_or_lt with hK | hK
+  · simp only [← hK, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, zero_mul,
+      div_zero, Real.exp_zero, ENNReal.ofReal_one]
+    exact prob_le_one
+  · let X (ω : Ω) := ∫ z in D, (Y z ω-μ z ω)
+    let θ := b/(256*K^2*R^2)
+    have hR : 0 < R := h.radius_pos.trans_le h.radius_le
+    have hθ : 0 ≤ θ := by dsimp [θ]; positivity
+    obtain ⟨B,hB⟩ := window_source_integral_bounded h
+    have hXi : Integrable (fun ω => Real.exp (θ*X ω)) P :=
+      window_integrable_nonneg ((window_source_integral_measurable h).const_mul θ).exp
+        (fun ω => (Real.exp_pos _).le) (fun ω => Real.exp_le_exp.mpr
+          (mul_le_mul_of_nonneg_left (le_abs_self _ |>.trans (hB ω)) hθ))
+    have hmgf : mgf X P θ ≤ Real.exp (128*K^2*θ^2*R^2) := by
+      have H := window_source_mgf h hθ (W := fun _ => 1) measurable_const (fun _ => zero_le_one) ⟨1,fun _ => le_rfl⟩
+      simpa [mgf, X] using H
+    apply (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mp
+    rw [ENNReal.toReal_ofReal (Real.exp_pos _).le]
+    change P.real {ω | b < X ω} ≤ _
+    calc
+      _ ≤ P.real {ω | b ≤ X ω} := by
+        exact measureReal_mono (fun ω hω => (show b < X ω from hω).le) (measure_ne_top _ _)
+      _ ≤ Real.exp (-θ*b)*mgf X P θ := measure_ge_le_exp_mul_mgf b hθ hXi
+      _ ≤ Real.exp (-θ*b)*Real.exp (128*K^2*θ^2*R^2) :=
+        mul_le_mul_of_nonneg_left hmgf (Real.exp_pos _).le
+      _ = _ := by
+        rw [← Real.exp_add]
+        congr 1
+        dsimp [θ]
+        field_simp
+        ring
+
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/WindowBGeometry.lean b/lean/OAI/Probability/SLE/WindowBGeometry.lean
index 848247f4a725fbbc02117e6c3763c6cfe0414fa0..384dee1126216418ee94cb8fca69a2d27474fbfa 100644
--- a/lean/OAI/Probability/SLE/WindowBGeometry.lean
+++ b/lean/OAI/Probability/SLE/WindowBGeometry.lean
@@ -1,220 +1,220 @@
-import Mathlib
-import OAI.Probability.SLE.WindowTails
-
-namespace OAI
-
-/-! Window BGeometry for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_witness_initial_domain
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) : survivingDomain γ 0 = upperHalfPlane := by
-  apply Set.Subset.antisymm (fun _ hz => hz.1.1)
-  intro z hz
-  have hf : F 0 z ∈ survivingDomain γ 0 := h.2.2.2.2.2.2.1 0 hz
-  have he : F 0 z = z := (h.initial_value (F 0 z) hf.1.1).symm.trans (h.2.2.2.2.2.2.2.2.1 0 z hz)
-  rwa [he] at hf
-
-lemma source_window_initial_ball
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
-    (hy : r < z.im) (ω : a.samples) : closedBall z r ⊆ survivingDomain (γ ω) 0 := by
-  rw [source_witness_initial_domain (a.witness ω)]
-  intro w hw
-  have hi := (le_abs_self (z.im-w.im)).trans (Complex.abs_im_le_norm (z-w))
-  have hd : ‖z-w‖ ≤ r := by simpa only [mem_closedBall,dist_comm,dist_eq_norm] using hw
-  change 0 < w.im
-  linarith
-
-lemma source_window_first_geometry
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
-    (hy : r < z.im) (ω : a.samples) (hfin : liveSpatialHitTime (γ ω) z r ≠ ⊤) :
-    ∃ T s : ℝ≥0, liveSpatialHitTime (γ ω) z r = T ∧ s ≤ T ∧
-      closedBall z r ⊆ survivingDomain (γ ω) s ∧ dist (γ ω T) z = r ∧
-      (∀ t ∈ Ico s T, dist (γ ω t) z ≠ r) ∧ z ∈ survivingDomain (γ ω) T := by
-  obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hfin
-  have hmem := source_liveHit_mem (a.trace_continuous ω) z r
-    (source_window_origin_bound a z r hy.le ω) T hT.symm
-  have he := source_liveTime_eq_sphere_of_finite (a.trace_continuous ω) z r
-    (source_window_origin_bound a z r hy.le ω) hfin
-  refine ⟨T,0,hT.symm,bot_le,source_window_initial_ball a z r hy ω,hmem.2,?_,hmem.1⟩
-  intro t ht hd
-  have hh := (spatialHitTime_le_iff (a.trace_continuous ω) z r t).mpr ⟨t,⟨bot_le,le_rfl⟩,hd⟩
-  rw [← he,← hT] at hh
-  exact ht.2.not_ge (WithTop.coe_le_coe.mp hh)
-
-lemma source_windowB_le_disk
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r ε R : ℝ)
-    (her : ε ≤ r) (hrR : r < R) (hy : r ≤ z.im) (ω : a.samples) :
-    sourceSpatialWindowB (γ ω) z r ε R ≤
-      (if liveSpatialHitTime (γ ω) z r ≠ ⊤ ∧ ∃ t : ℝ≥0,
-        z ∈ survivingDomain (γ ω) ((liveSpatialHitTime (γ ω) z r).untopD 0+t) ∧
-        R ≤ dist (γ ω ((liveSpatialHitTime (γ ω) z r).untopD 0+t)) z then (1:ℝ) else 0) := by
-  rcases (source_windowHB_range (γ ω) z r ε R).2 with h|h
-  · rw [h]; split_ifs <;> norm_num
-  obtain ⟨T,hσ,hz,hd,_⟩ := source_windowB_before (a.trace_continuous ω) z r ε R her hrR
-    (source_window_origin_bound a z r hy ω) h
-  have hn := ne_top_of_le_ne_top WithTop.coe_ne_top hσ
-  obtain ⟨s,hs⟩ := WithTop.ne_top_iff_exists.mp hn
-  have hst : s ≤ T := WithTop.coe_le_coe.mp (by simpa only [← hs] using hσ)
-  have he : (liveSpatialHitTime (γ ω) z r).untopD 0 + (T-s) = T := by
-    rw [← hs,WithTop.untopD_coe,add_tsub_cancel_of_le hst]
-  rw [h,ite_eq_left ⟨hn,T-s,by rwa [he],by rwa [he]⟩]
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-local instance windowBGeometryPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance windowBGeometryPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-
-lemma source_condExp_bound_supported
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} {m : MeasurableSpace Ω}
-    {f g : Ω → ℝ} {A : Set Ω} (hA : MeasurableSet[m] A)
-    (hf : Integrable f μ) (hg : Integrable g μ) (hgf : g ≤ᵐ[μ] f)
-    (hg0 : ∀ ω ∉ A, g ω = 0) (K : ℝ)
-    (hbound : ∀ᵐ ω ∂μ, ω ∈ A → μ[f | m] ω ≤ K) :
-    μ[g | m] ≤ᵐ[μ] (fun ω => K*(if ω ∈ A then 1 else 0)) := by
-  have heq : A.indicator g = g := by
-    funext ω
-    by_cases hω : ω ∈ A
-    · exact Set.indicator_of_mem hω _
-    · rw [Set.indicator_of_notMem hω,hg0 ω hω]
-  have he := condExp_indicator hg hA
-  rw [heq] at he
-  filter_upwards [he,condExp_mono (m := m) hg hf hgf,hbound] with ω he hh hb
-  by_cases hω : ω ∈ A
-  · simpa only [ite_eq_left hω,mul_one] using hh.trans (hb hω)
-  · rw [he,Set.indicator_of_notMem hω,ite_eq_right hω,mul_zero]
-
-lemma source_windowB_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r ε R : ℝ)
-    (hy : r ≤ z.im) :
-    Integrable (fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε R) (P.comap Subtype.val) := by
-  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
-  have hτ := source_window_stopping a z r ε R hy
-  have hm := (source_windowHB_stopped_measurable a z r ε R hτ).2.mono hτ.measurableSpace_le le_rfl
-  apply Integrable.of_bound hm.aestronglyMeasurable 1
-  exact Eventually.of_forall fun ω => by
-    rcases (source_windowHB_range (γ ω) z r ε R).2 with h|h <;> simp only [h,norm_zero,norm_one,zero_le_one,le_refl]
-
-lemma source_windowB_mean :
-    ∀ κ : ℝ, 4 < κ → κ < 8 → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-      (z : ℂ) (ε r L : ℝ), 0 < r → ε ≤ r → r < z.im → 1 < L →
-    ∀ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
-      (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)) | hσ.measurableSpace]
-        ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => C*L^(-c)*
-          (if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1)) := by
-  intro κ hk hk8
-  obtain ⟨C,c,hC,hc,hdisk⟩ := sourcePhysicalDisk_proved κ hk hk8
-  refine ⟨C,c,hC,hc,?_⟩
-  intro Ω m P hP γ a z ε r L hr her hy hL hσ
-  let : IsProbabilityMeasure P := hP
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  let σ := fun ω : a.samples => liveSpatialHitTime (γ ω) z r
-  let A := {ω : a.samples | σ ω ≠ ⊤}
-  have hA : MeasurableSet[hσ.measurableSpace] A :=
-    (measurableSet_eq_fun hσ.measurable measurable_const).compl
-  let E := {ω : a.samples | σ ω ≠ ⊤ ∧ ∃ t : ℝ≥0,
-    z ∈ survivingDomain (γ ω) ((σ ω).untopD 0+t) ∧
-    L*r ≤ dist (γ ω ((σ ω).untopD 0+t)) z}
-  let f := E.indicator (fun _ => (1:ℝ))
-  have hEm : MeasurableSet E := by
-    apply (hσ.measurableSpace_le _ hA).inter
-    exact source_physical_closed_exit_measurable
-      (fun ω : a.samples => ⟨γ ω,a.trace_continuous ω⟩) (source_tracePath_measurable a)
-      (fun ω => (σ ω).untopD 0) (hσ.measurable'.untopD 0) z (L*r)
-  have hf : Integrable f μ := (integrable_const (1:ℝ)).indicator hEm
-  have hf_def : f = (fun ω : a.samples => if ω ∈ E then 1 else 0) := by
-    funext ω; exact Set.indicator_apply _ _ _
-  have hgeom : ∀ ω ∈ A, ∃ T s : ℝ≥0, σ ω = T ∧ s ≤ T ∧
-      closedBall z r ⊆ survivingDomain (γ ω) s ∧ dist (γ ω T) z = r ∧
-      (∀ t ∈ Ico s T, dist (γ ω t) z ≠ r) ∧ z ∈ survivingDomain (γ ω) T :=
-    fun ω hω => source_window_first_geometry a z r hy ω hω
-  have hd := hdisk a.samples inferInstance μ inferInstance a.B a.brownian a.measurable_B
-    a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness σ hσ z r hr A hA hgeom L hL.le
-  have hb : ∀ᵐ ω ∂μ, ω ∈ A → μ[f | hσ.measurableSpace] ω ≤ C*L^(-c) := by
-    rw [hf_def]
-    convert hd using 1
-    rfl
-  have hi := source_windowB_integrable a z r ε (L*r) hy.le
-  have hgf : (fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)) ≤ᵐ[μ] f := by
-    rw [hf_def]
-    exact Eventually.of_forall fun ω => source_windowB_le_disk a z r ε (L*r) her
-      (by nlinarith) hy.le ω
-  have hh := source_condExp_bound_supported hA hf hi hgf
-    (fun ω hω => (source_windowHB_zero (a.trace_continuous ω) z r ε (L*r) (by
-      simpa only [A,σ,Set.mem_ofPred_eq,ne_eq,not_not] using hω)).2) (C*L^(-c)) hb
-  filter_upwards [hh] with ω hω
-  by_cases htop : liveSpatialHitTime (γ ω) z r = ⊤ <;>
-    simpa [A, σ, htop] using hω
-
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
- 
-
-def SourceSpatialWindowMeansTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 →
-  ∃ C Cbad c q : ℝ, 0 < C ∧ 0 < Cbad ∧ 0 < c ∧ 0 < q ∧ q < 1 ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (z : ℂ) (ε r L : ℝ), 0 < ε → ε ≤ q*r → r < z.im → 1 < L →
-  ∃ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
-    let μ := P.comap (Subtype.val : a.samples → Ω)
-    let W := fun v (ω : a.samples) => spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z v
-    let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε (L*r)
-    let B := fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)
-    μ[H | hσ.measurableSpace] ≤ᵐ[μ] (fun ω => C*ε^codimension κ*W r ω) ∧
-    μ[(fun ω => r^codimension κ*W ε ω*H ω) | hσ.measurableSpace] ≤ᵐ[μ]
-      (fun ω => r^codimension κ*W r ω) ∧
-    μ[B | hσ.measurableSpace] ≤ᵐ[μ]
-      (fun ω => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1))
-
-theorem sourceSpatialWindowMeans_proved : SourceSpatialWindowMeansTarget := by
-  intro κ hk hk8
-  have hk0 : 0 < κ := by linarith
-  obtain ⟨C,q,hC,hq,hq1,hHV⟩ := source_window_HV_mean κ hk0 hk8
-  obtain ⟨Cbad,c,hCb,hc,hB⟩ := source_windowB_mean κ hk hk8
-  refine ⟨C,Cbad,c,q,hC,hCb,hc,hq,hq1,?_⟩
-  intro Ω m P hP γ a z ε r L he heq hy hL
-  have hr : 0 < r := by
-    by_contra! hn
-    have hh := mul_nonpos_of_nonneg_of_nonpos hq.le hn
-    linarith
-  have her : ε ≤ r := heq.trans (by nlinarith)
-  obtain ⟨hs,hH,hV⟩ := hHV Ω m P hP γ a z ε r (L*r) he heq hy.le
-  exact ⟨hs,hH,hV,hB Ω m P hP γ a z ε r L hr her hy hL hs⟩
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.WindowTails
+
+namespace OAI
+
+/-! Window BGeometry for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_witness_initial_domain
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) : survivingDomain γ 0 = upperHalfPlane := by
+  apply Set.Subset.antisymm (fun _ hz => hz.1.1)
+  intro z hz
+  have hf : F 0 z ∈ survivingDomain γ 0 := h.2.2.2.2.2.2.1 0 hz
+  have he : F 0 z = z := (h.initial_value (F 0 z) hf.1.1).symm.trans (h.2.2.2.2.2.2.2.2.1 0 z hz)
+  rwa [he] at hf
+
+lemma source_window_initial_ball
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
+    (hy : r < z.im) (ω : a.samples) : closedBall z r ⊆ survivingDomain (γ ω) 0 := by
+  rw [source_witness_initial_domain (a.witness ω)]
+  intro w hw
+  have hi := (le_abs_self (z.im-w.im)).trans (Complex.abs_im_le_norm (z-w))
+  have hd : ‖z-w‖ ≤ r := by simpa only [mem_closedBall,dist_comm,dist_eq_norm] using hw
+  change 0 < w.im
+  linarith
+
+lemma source_window_first_geometry
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
+    (hy : r < z.im) (ω : a.samples) (hfin : liveSpatialHitTime (γ ω) z r ≠ ⊤) :
+    ∃ T s : ℝ≥0, liveSpatialHitTime (γ ω) z r = T ∧ s ≤ T ∧
+      closedBall z r ⊆ survivingDomain (γ ω) s ∧ dist (γ ω T) z = r ∧
+      (∀ t ∈ Ico s T, dist (γ ω t) z ≠ r) ∧ z ∈ survivingDomain (γ ω) T := by
+  obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hfin
+  have hmem := source_liveHit_mem (a.trace_continuous ω) z r
+    (source_window_origin_bound a z r hy.le ω) T hT.symm
+  have he := source_liveTime_eq_sphere_of_finite (a.trace_continuous ω) z r
+    (source_window_origin_bound a z r hy.le ω) hfin
+  refine ⟨T,0,hT.symm,bot_le,source_window_initial_ball a z r hy ω,hmem.2,?_,hmem.1⟩
+  intro t ht hd
+  have hh := (spatialHitTime_le_iff (a.trace_continuous ω) z r t).mpr ⟨t,⟨bot_le,le_rfl⟩,hd⟩
+  rw [← he,← hT] at hh
+  exact ht.2.not_ge (WithTop.coe_le_coe.mp hh)
+
+lemma source_windowB_le_disk
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r ε R : ℝ)
+    (her : ε ≤ r) (hrR : r < R) (hy : r ≤ z.im) (ω : a.samples) :
+    sourceSpatialWindowB (γ ω) z r ε R ≤
+      (if liveSpatialHitTime (γ ω) z r ≠ ⊤ ∧ ∃ t : ℝ≥0,
+        z ∈ survivingDomain (γ ω) ((liveSpatialHitTime (γ ω) z r).untopD 0+t) ∧
+        R ≤ dist (γ ω ((liveSpatialHitTime (γ ω) z r).untopD 0+t)) z then (1:ℝ) else 0) := by
+  rcases (source_windowHB_range (γ ω) z r ε R).2 with h|h
+  · rw [h]; split_ifs <;> norm_num
+  obtain ⟨T,hσ,hz,hd,_⟩ := source_windowB_before (a.trace_continuous ω) z r ε R her hrR
+    (source_window_origin_bound a z r hy ω) h
+  have hn := ne_top_of_le_ne_top WithTop.coe_ne_top hσ
+  obtain ⟨s,hs⟩ := WithTop.ne_top_iff_exists.mp hn
+  have hst : s ≤ T := WithTop.coe_le_coe.mp (by simpa only [← hs] using hσ)
+  have he : (liveSpatialHitTime (γ ω) z r).untopD 0 + (T-s) = T := by
+    rw [← hs,WithTop.untopD_coe,add_tsub_cancel_of_le hst]
+  rw [h,ite_eq_left ⟨hn,T-s,by rwa [he],by rwa [he]⟩]
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+local instance windowBGeometryPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance windowBGeometryPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+
+lemma source_condExp_bound_supported
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} {m : MeasurableSpace Ω}
+    {f g : Ω → ℝ} {A : Set Ω} (hA : MeasurableSet[m] A)
+    (hf : Integrable f μ) (hg : Integrable g μ) (hgf : g ≤ᵐ[μ] f)
+    (hg0 : ∀ ω ∉ A, g ω = 0) (K : ℝ)
+    (hbound : ∀ᵐ ω ∂μ, ω ∈ A → μ[f | m] ω ≤ K) :
+    μ[g | m] ≤ᵐ[μ] (fun ω => K*(if ω ∈ A then 1 else 0)) := by
+  have heq : A.indicator g = g := by
+    funext ω
+    by_cases hω : ω ∈ A
+    · exact Set.indicator_of_mem hω _
+    · rw [Set.indicator_of_notMem hω,hg0 ω hω]
+  have he := condExp_indicator hg hA
+  rw [heq] at he
+  filter_upwards [he,condExp_mono (m := m) hg hf hgf,hbound] with ω he hh hb
+  by_cases hω : ω ∈ A
+  · simpa only [ite_eq_left hω,mul_one] using hh.trans (hb hω)
+  · rw [he,Set.indicator_of_notMem hω,ite_eq_right hω,mul_zero]
+
+lemma source_windowB_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r ε R : ℝ)
+    (hy : r ≤ z.im) :
+    Integrable (fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε R) (P.comap Subtype.val) := by
+  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
+  have hτ := source_window_stopping a z r ε R hy
+  have hm := (source_windowHB_stopped_measurable a z r ε R hτ).2.mono hτ.measurableSpace_le le_rfl
+  apply Integrable.of_bound hm.aestronglyMeasurable 1
+  exact Eventually.of_forall fun ω => by
+    rcases (source_windowHB_range (γ ω) z r ε R).2 with h|h <;> simp only [h,norm_zero,norm_one,zero_le_one,le_refl]
+
+lemma source_windowB_mean :
+    ∀ κ : ℝ, 4 < κ → κ < 8 → ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+      (z : ℂ) (ε r L : ℝ), 0 < r → ε ≤ r → r < z.im → 1 < L →
+    ∀ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
+      (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)) | hσ.measurableSpace]
+        ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => C*L^(-c)*
+          (if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1)) := by
+  intro κ hk hk8
+  obtain ⟨C,c,hC,hc,hdisk⟩ := sourcePhysicalDisk_proved κ hk hk8
+  refine ⟨C,c,hC,hc,?_⟩
+  intro Ω m P hP γ a z ε r L hr her hy hL hσ
+  let : IsProbabilityMeasure P := hP
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  let σ := fun ω : a.samples => liveSpatialHitTime (γ ω) z r
+  let A := {ω : a.samples | σ ω ≠ ⊤}
+  have hA : MeasurableSet[hσ.measurableSpace] A :=
+    (measurableSet_eq_fun hσ.measurable measurable_const).compl
+  let E := {ω : a.samples | σ ω ≠ ⊤ ∧ ∃ t : ℝ≥0,
+    z ∈ survivingDomain (γ ω) ((σ ω).untopD 0+t) ∧
+    L*r ≤ dist (γ ω ((σ ω).untopD 0+t)) z}
+  let f := E.indicator (fun _ => (1:ℝ))
+  have hEm : MeasurableSet E := by
+    apply (hσ.measurableSpace_le _ hA).inter
+    exact source_physical_closed_exit_measurable
+      (fun ω : a.samples => ⟨γ ω,a.trace_continuous ω⟩) (source_tracePath_measurable a)
+      (fun ω => (σ ω).untopD 0) (hσ.measurable'.untopD 0) z (L*r)
+  have hf : Integrable f μ := (integrable_const (1:ℝ)).indicator hEm
+  have hf_def : f = (fun ω : a.samples => if ω ∈ E then 1 else 0) := by
+    funext ω; exact Set.indicator_apply _ _ _
+  have hgeom : ∀ ω ∈ A, ∃ T s : ℝ≥0, σ ω = T ∧ s ≤ T ∧
+      closedBall z r ⊆ survivingDomain (γ ω) s ∧ dist (γ ω T) z = r ∧
+      (∀ t ∈ Ico s T, dist (γ ω t) z ≠ r) ∧ z ∈ survivingDomain (γ ω) T :=
+    fun ω hω => source_window_first_geometry a z r hy ω hω
+  have hd := hdisk a.samples inferInstance μ inferInstance a.B a.brownian a.measurable_B
+    a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness σ hσ z r hr A hA hgeom L hL.le
+  have hb : ∀ᵐ ω ∂μ, ω ∈ A → μ[f | hσ.measurableSpace] ω ≤ C*L^(-c) := by
+    rw [hf_def]
+    convert hd using 1
+    rfl
+  have hi := source_windowB_integrable a z r ε (L*r) hy.le
+  have hgf : (fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)) ≤ᵐ[μ] f := by
+    rw [hf_def]
+    exact Eventually.of_forall fun ω => source_windowB_le_disk a z r ε (L*r) her
+      (by nlinarith) hy.le ω
+  have hh := source_condExp_bound_supported hA hf hi hgf
+    (fun ω hω => (source_windowHB_zero (a.trace_continuous ω) z r ε (L*r) (by
+      simpa only [A,σ,Set.mem_ofPred_eq,ne_eq,not_not] using hω)).2) (C*L^(-c)) hb
+  filter_upwards [hh] with ω hω
+  by_cases htop : liveSpatialHitTime (γ ω) z r = ⊤ <;>
+    simpa [A, σ, htop] using hω
+
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+ 
+
+def SourceSpatialWindowMeansTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 →
+  ∃ C Cbad c q : ℝ, 0 < C ∧ 0 < Cbad ∧ 0 < c ∧ 0 < q ∧ q < 1 ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (z : ℂ) (ε r L : ℝ), 0 < ε → ε ≤ q*r → r < z.im → 1 < L →
+  ∃ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
+    let μ := P.comap (Subtype.val : a.samples → Ω)
+    let W := fun v (ω : a.samples) => spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z v
+    let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε (L*r)
+    let B := fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε (L*r)
+    μ[H | hσ.measurableSpace] ≤ᵐ[μ] (fun ω => C*ε^codimension κ*W r ω) ∧
+    μ[(fun ω => r^codimension κ*W ε ω*H ω) | hσ.measurableSpace] ≤ᵐ[μ]
+      (fun ω => r^codimension κ*W r ω) ∧
+    μ[B | hσ.measurableSpace] ≤ᵐ[μ]
+      (fun ω => Cbad*L^(-c)*(if liveSpatialHitTime (γ ω) z r = ⊤ then 0 else 1))
+
+theorem sourceSpatialWindowMeans_proved : SourceSpatialWindowMeansTarget := by
+  intro κ hk hk8
+  have hk0 : 0 < κ := by linarith
+  obtain ⟨C,q,hC,hq,hq1,hHV⟩ := source_window_HV_mean κ hk0 hk8
+  obtain ⟨Cbad,c,hCb,hc,hB⟩ := source_windowB_mean κ hk hk8
+  refine ⟨C,Cbad,c,q,hC,hCb,hc,hq,hq1,?_⟩
+  intro Ω m P hP γ a z ε r L he heq hy hL
+  have hr : 0 < r := by
+    by_contra! hn
+    have hh := mul_nonpos_of_nonneg_of_nonpos hq.le hn
+    linarith
+  have her : ε ≤ r := heq.trans (by nlinarith)
+  obtain ⟨hs,hH,hV⟩ := hHV Ω m P hP γ a z ε r (L*r) he heq hy.le
+  exact ⟨hs,hH,hV,hB Ω m P hP γ a z ε r L hr her hy hL hs⟩
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/WindowBounds.lean b/lean/OAI/Probability/SLE/WindowBounds.lean
index 16da7de391d3429926f7de59fb9a8a92dc15d25a..71012a3c608da5ba04b0f0e948086edc5d831bab 100644
--- a/lean/OAI/Probability/SLE/WindowBounds.lean
+++ b/lean/OAI/Probability/SLE/WindowBounds.lean
@@ -1,701 +1,701 @@
-import Mathlib
-import OAI.Probability.SLE.PhysicalKernelUtilities
-import OAI.Probability.SLE.WindowGeometry
-
-namespace OAI
-
-/-! Window Bounds for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance windowBoundsPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance windowBoundsPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
-lemma source_liveTime_eq_sphere_of_finite {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) (hn : liveSpatialHitTime γ z r ≠ ⊤) :
-    liveSpatialHitTime γ z r = spatialHitTime γ z r := by
-  have he := source_liveTime_eq_killed hc z r hs
-  split_ifs at he with hl
-  · exact he
-  · exact (hn he).elim
-
-lemma source_windowV_stopped_eq {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ) (κ : ℝ) (z : ℂ) (r ε R : ℝ)
-    (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    spatialGreenWeight κ U γ G z ε * sourceSpatialWindowH γ z r ε R =
-      greenDensity κ U γ G (sourceSpatialWindowTime γ z r ε R).untopA z *
-        sourceSpatialWindowH γ z r ε R := by
-  by_cases hH : sourceSpatialWindowH γ z r ε R = 1
-  · obtain ⟨he,hn⟩ := (source_windowH_iff hc z r ε R her hs).mp hH
-    have hr := source_liveTime_eq_sphere_of_finite hc z ε (her.trans hs) hn
-    have hraw : spatialHitTime γ z ε ≠ ⊤ := hr ▸ hn
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hraw
-    rw [hH, mul_one, mul_one, he, hr, spatialGreenWeight, ite_eq_right hraw, ← ht]
-    rfl
-  · have hz : sourceSpatialWindowH γ z r ε R = 0 := by
-      unfold sourceSpatialWindowH at hH ⊢
-      split_ifs <;> simp_all
-    rw [hz,mul_zero,mul_zero]
-
-lemma source_windowV_stopped_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
-    (r ε R : ℝ) (her : ε ≤ r) (hy : r ≤ z.im)
-    (hτ : IsStoppingTime a.filtration (fun ω : a.samples => sourceSpatialWindowTime (γ ω) z r ε R)) :
-    Measurable[hτ.measurableSpace] (fun ω : a.samples =>
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z ε *
-        sourceSpatialWindowH (γ ω) z r ε R) := by
-  have hg := measurable_stoppedValue
-    (source_killedGreen_progressive a.measurable_B a.continuous_B κ
-      (fun ω : a.samples => γ ω) a.G a.F a.witness z) hτ
-  have hH := (source_windowHB_stopped_measurable a z r ε R hτ).1
-  convert hg.mul hH using 1 <;> try rfl
-  funext ω
-  apply source_windowV_stopped_eq (a.trace_continuous ω) _ _ _ _ _ _ _ her
-  rw [(a.witness ω).2.1,dist_zero_left]
-  exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm _))
-
-lemma source_liveWeight_stopped_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
-    (r : ℝ) (hy : r ≤ z.im)
-    (hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r)) :
-    Measurable[hσ.measurableSpace] (fun ω : a.samples =>
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) := by
-  have hraw : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
-  have hm := source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
-    (fun ω : a.samples => γ ω) a.G a.F a.witness z r hraw
-  apply hm.mono (hraw.measurableSpace_mono hσ ?_) le_rfl
-  intro ω
-  have hs : r ≤ dist (γ ω 0) z := by
-    rw [(a.witness ω).2.1,dist_zero_left]
-    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm _))
-  change spatialHitTime (γ ω) z r ≤ liveSpatialHitTime (γ ω) z r
-  rw [source_liveTime_eq_killed (a.trace_continuous ω) z r hs]
-  split_ifs <;> first | exact le_rfl | exact le_top
-
-lemma source_collarHB_joint
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (hD : MeasurableSet D) (r ε R : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) :
-    Measurable (fun p : ℂ × a.samples => sourceCollarH D (γ p.2) p.1 r ε R) ∧
-    Measurable (fun p : ℂ × a.samples => sourceCollarB D (γ p.2) p.1 r ε R) := by
-  let τ := fun p : ℂ × a.samples => sourceCollarFinish D (γ p.2) p.1 r ε R
-  have ht : Measurable τ := source_collarFinish_joint a D hD r ε R hy
-  have hg : Measurable (fun p : ℂ × a.samples => (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ))) :=
-    (source_tracePath_measurable a).comp measurable_snd
-  have hl := source_joint_survival_measurable
-    (fun p : ℂ × a.samples => (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ)))
-    (fun p => (τ p).untopD 0) Prod.fst hg (ht.untopD 0) measurable_fst
-  have hd : Measurable (fun p : ℂ × a.samples => dist (γ p.2 ((τ p).untopD 0)) p.1) :=
-    (continuous_eval.measurable.comp (hg.prodMk (ht.untopD 0))).dist measurable_fst
-  have hn : MeasurableSet {p | τ p ≠ ⊤} := (measurableSet_eq_fun ht measurable_const).compl
-  have hH : Measurable (fun p : ℂ × a.samples =>
-      if τ p ≠ ⊤ ∧ p.1 ∈ survivingDomain (γ p.2) ((τ p).untopD 0) ∧
-      dist (γ p.2 ((τ p).untopD 0)) p.1 ≤ ε then (1:ℝ) else 0) :=
-    Measurable.ite (hn.inter (hl.inter (measurableSet_le hd measurable_const))) measurable_const measurable_const
-  have hB : Measurable (fun p : ℂ × a.samples =>
-      if τ p ≠ ⊤ ∧ p.1 ∈ survivingDomain (γ p.2) ((τ p).untopD 0) ∧
-      R ≤ dist (γ p.2 ((τ p).untopD 0)) p.1 then (1:ℝ) else 0) :=
-    Measurable.ite (hn.inter (hl.inter (measurableSet_le measurable_const hd))) measurable_const measurable_const
-  constructor
-  · convert hH using 1
-    funext p
-    by_cases hz : p.1 ∈ D
-    · simp only [sourceCollarH,sourceSpatialWindowH,τ,sourceCollarFinish,ite_eq_left hz]
-    · simp only [sourceCollarH,τ,sourceCollarFinish,ite_eq_right hz,ne_eq,not_true_eq_false,false_and,ite_false]
-  · convert hB using 1
-    funext p
-    by_cases hz : p.1 ∈ D
-    · simp only [sourceCollarB,sourceSpatialWindowB,τ,sourceCollarFinish,ite_eq_left hz]
-    · simp only [sourceCollarB,τ,sourceCollarFinish,ite_eq_right hz,ne_eq,not_true_eq_false,false_and,ite_false]
-
-lemma source_stop_space_congr {Ω : Type*} [MeasurableSpace Ω]
-    {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} {σ τ : Ω → WithTop ℝ≥0}
-    (hσ : IsStoppingTime ℱ σ) (hτ : IsStoppingTime ℱ τ) (he : σ = τ) :
-    hσ.measurableSpace = hτ.measurableSpace := by
-  subst τ
-  rfl
-
-lemma source_collarHBV_stopped
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (r ε R : ℝ) (her : ε ≤ r) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
-    let hτ := source_collarFinish_stopping a D r ε R hy z
-    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceCollarH D (γ ω) z r ε R) ∧
-    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceCollarB D (γ ω) z r ε R) ∧
-    Measurable[hτ.measurableSpace] (fun ω : a.samples =>
-      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z ε *
-        sourceCollarH D (γ ω) z r ε R) := by
-  dsimp only
-  by_cases hz : z ∈ D
-  · have ht := source_window_stopping a z r ε R (hy z hz)
-    have hh := source_windowHB_stopped_measurable a z r ε R ht
-    have hv := source_windowV_stopped_measurable a z r ε R her (hy z hz) ht
-    have hspace := source_stop_space_congr (source_collarFinish_stopping a D r ε R hy z) ht
-      (funext fun ω => by simp only [sourceCollarFinish,ite_eq_left hz])
-    rw [hspace]
-    simpa only [sourceCollarH,sourceCollarB,ite_eq_left hz] using And.intro hh.1 (And.intro hh.2 hv)
-  · simp only [sourceCollarH,sourceCollarB,ite_eq_right hz,mul_zero]
-    exact ⟨measurable_const,measurable_const,measurable_const⟩
-
-lemma source_collarWeight_stopped
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
-    Measurable[(source_collarStart_stopping a D r hy z).measurableSpace] (fun ω : a.samples =>
-      if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r else 0) := by
-  by_cases hz : z ∈ D
-  · have hs := source_liveHit_stopping a z r (hy z hz)
-    have hw := source_liveWeight_stopped_measurable a z r (hy z hz) hs
-    have hspace := source_stop_space_congr (source_collarStart_stopping a D r hy z) hs
-      (funext fun ω => by simp only [sourceCollarStart,ite_eq_left hz])
-    rw [hspace]
-    simpa only [ite_eq_left hz] using hw
-  · simp only [ite_eq_right hz]
-    exact measurable_const
-
- theorem sourceSpatialWindowMeasurable_proved : SourceSpatialWindowMeasurableTarget := by
-  intro Ω m P κ hκ hκ8 γ a D hD ε r R hε her hrR hy
-  dsimp only
-  refine ⟨⟨source_collarStart_stopping a D r hy,
-    source_collarFinish_stopping a D r ε R hy,?_,?_⟩,?_,?_,?_,?_,?_⟩
-  · exact source_collarHBV_stopped a D r ε R her.le hy
-  · exact source_collarWeight_stopped a D r hy
-  · exact source_collarStart_joint a D hD r hy
-  · exact source_collarFinish_joint a D hD r ε R hy
-  · exact (source_collarHB_joint a D hD r ε R hy).1
-  · exact (source_collarHB_joint a D hD r ε R hy).2
-  · exact (a.spatialGreenWeight_measurable ε).mul (source_collarHB_joint a D hD r ε R hy).1
-
-end
-end SLEExactGauge
-
-end
-
- 
-section
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_liveIndicator_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (z : ℂ) (ε : ℝ) (hy : ε ≤ z.im) :
-    Measurable (fun ω : a.samples =>
-      if liveSpatialHitTime (γ ω) z ε = ⊤ then (0 : ℝ) else 1) := by
-  classical
-  have hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z ε) :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z ε
-  have hK := measurable_stoppedValue
-    (source_killedChart_progressive a.measurable_B a.continuous_B κ
-      (fun ω : a.samples => γ ω) a.G a.F a.witness z) hτ
-  have hKm : Measurable (fun ω : a.samples =>
-      (if z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z ε).untopA
-        then a.G ω (spatialHitTime (γ ω) z ε).untopA z else 0 : ℂ)) :=
-    hK.mono hτ.measurableSpace_le le_rfl
-  have htop : MeasurableSet {ω : a.samples | spatialHitTime (γ ω) z ε = ⊤} :=
-    measurableSet_eq_fun (hτ.measurable.mono hτ.measurableSpace_le le_rfl) measurable_const
-  have hl : MeasurableSet {ω : a.samples |
-      0 < (if z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z ε).untopA
-        then a.G ω (spatialHitTime (γ ω) z ε).untopA z else 0 : ℂ).im} :=
-    measurableSet_lt measurable_const (Complex.measurable_im.comp hKm)
-  have hmo : Measurable (fun ω : a.samples =>
-      if spatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else
-        if 0 < (if z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z ε).untopA
-          then a.G ω (spatialHitTime (γ ω) z ε).untopA z else 0 : ℂ).im then 1 else 0) :=
-    Measurable.ite htop measurable_const (Measurable.ite hl measurable_const measurable_const)
-  convert hmo using 1
-  funext ω
-  have hs : ε ≤ dist (γ ω 0) z := by
-    rw [(a.witness ω).2.1, dist_zero_left]
-    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  rw [liveSpatialHitTime_eq (a.trace_continuous ω) z ε hs]
-  by_cases ht : spatialHitTime (γ ω) z ε = ⊤
-  · simp only [ite_eq_left ht, ite_true]
-  · obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp ht
-    rw [ite_eq_right ht, ite_eq_right ht, ← hv]
-    change (if (if z ∈ survivingDomain (γ ω) v then (v : WithTop ℝ≥0) else ⊤) = ⊤
-      then (0:ℝ) else 1) =
-      (if 0 < (if z ∈ survivingDomain (γ ω) v then a.G ω v z else 0 : ℂ).im then 1 else 0)
-    by_cases hz : z ∈ survivingDomain (γ ω) v
-    · simp only [ite_eq_left hz, WithTop.coe_ne_top, ite_false,
-        ite_eq_left ((a.witness ω).imag_positive v z hz)]
-    · simp only [ite_eq_right hz, ite_true, Complex.zero_im, lt_self_iff_false, ite_false]
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance windowBoundsApproachPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance windowBoundsApproachPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-noncomputable def sourceApproachConstant (κ : ℝ) : ℝ :=
-  (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
-    (512 * Real.pi) ^ codimension κ
-
-lemma sourceApproachConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < sourceApproachConstant κ := by
-  exact mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num))
-    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by positivity) _)
-
-lemma source_kernel_mean_bound
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgm : Measurable (sourceTracePath γ (fun ω => (hw ω).1)))
-    (ζ : ℂ) (hζ : ζ ∈ upperHalfPlane) (ε r R : ℝ)
-    (hε : 0 < ε) (hr : 0 < r) (hy : 8 * ε ≤ r)
-    (hsmall : Real.exp 1 * (512 * Real.pi) * ε ≤ r)
-    (hR : 0 < R) (hRb : R ≤ (128 * Real.pi) * r) :
-    (∫ p, sourceApproachKernel ε r (ζ, p) ∂P.map (sourceTracePath γ (fun ω => (hw ω).1))) ≤
-      sourceApproachConstant κ * ε ^ codimension κ *
-        (R ^ (-codimension κ) * (Real.sin (Complex.arg ζ)) ^ greenExponent κ) := by
-  have hypos : 0 < ζ.im := hζ
-  let δ := 8 * ζ.im * ε / r
-  have hδ : 0 < δ := by dsimp [δ]; positivity
-  have hδy : δ ≤ ζ.im := by
-    apply (div_le_iff₀ hr).mpr
-    nlinarith [mul_le_mul_of_nonneg_left hy (show 0 ≤ ζ.im from hζ.le)]
-  have hδsmall : 1 ≤ Real.log ((2 * ζ.im) / ((128 * Real.pi) * δ)) := by
-    apply (Real.le_log_iff_exp_le (by positivity)).mpr
-    apply (le_div_iff₀ (by positivity : 0 < (128 * Real.pi) * δ)).mpr
-    calc
-      Real.exp 1 * ((128 * Real.pi) * δ) =
-          (2 * ζ.im) * (Real.exp 1 * (512 * Real.pi) * ε) / r := by dsimp [δ]; ring
-      _ ≤ (2 * ζ.im) * r / r :=
-        div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hsmall (by positivity)) hr.le
-      _ = 2 * ζ.im := mul_div_cancel_right₀ _ hr.ne'
-  have hb := euclidean_approach hB κ hκ hκ8 γ G F (Eventually.of_forall hw)
-    ζ hζ δ hδ hδy hδsmall
-  let S : Set Ω := {ω | ∃ t : ℝ≥0, dist (γ ω t) ζ < δ}
-  have hS : MeasurableSet S := by
-    have hh := measurableSet_lt (measurable_const (a := (0:ℝ)))
-      ((sourceApproachKernel_measurable ε r).comp ((measurable_const (a := ζ)).prodMk hgm))
-    convert hh using 1
-    ext ω
-    change (∃ t : ℝ≥0, dist (γ ω t) ζ < δ) ↔ 0 <
-      (if ∃ t : ℝ≥0, dist (γ ω t) ζ < δ then (1:ℝ) else 0)
-    split_ifs <;> simp_all
-  have hint : (∫ p, sourceApproachKernel ε r (ζ, p)
-      ∂P.map (sourceTracePath γ (fun ω => (hw ω).1))) = (P S).toReal := by
-    have hfun : Measurable (fun p : C(ℝ≥0, ℂ) => sourceApproachKernel ε r (ζ, p)) :=
-      (sourceApproachKernel_measurable ε r).comp (measurable_const.prodMk measurable_id)
-    calc
-      _ = ∫ ω, sourceApproachKernel ε r (ζ, sourceTracePath γ (fun ω => (hw ω).1) ω) ∂P :=
-        integral_map hgm.aemeasurable hfun.aestronglyMeasurable
-      _ = _ := ?_
-
-    change (∫ ω, S.indicator (fun _ => (1:ℝ)) ω ∂P) = _
-    simp only [integral_indicator_const _ hS, smul_eq_mul, mul_one, measureReal_def]
-  rw [hint]
-  apply hb.trans
-  have hsin : 0 ≤ Real.sin (Complex.arg ζ) := by
-    rw [Complex.sin_arg]
-    exact div_nonneg hζ.le (norm_nonneg _)
-  have hC : 0 ≤ uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ :=
-    (mul_pos (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num))
-      (Real.rpow_pos_of_pos (by positivity) _)).le
-  have hp : (4 * R / r) ^ codimension κ ≤ (512 * Real.pi) ^ codimension κ := by
-    apply Real.rpow_le_rpow (by positivity) _ (codimension_pos κ hκ8).le
-    apply (div_le_iff₀ hr).mpr
-    nlinarith [hRb]
-  have heq := source_rpow_scale (codimension κ) ζ.im ε r R hζ hε hr hR
-  dsimp only [initialGreen, δ]
-  calc
-    _ = (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
-        ((2 * ζ.im) ^ (-codimension κ) * (8 * ζ.im * ε / r) ^ codimension κ) *
-        (Real.sin (Complex.arg ζ)) ^ greenExponent κ := by ring
-    _ = (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
-        ((4 * R / r) ^ codimension κ * ε ^ codimension κ * R ^ (-codimension κ)) *
-        (Real.sin (Complex.arg ζ)) ^ greenExponent κ := by rw [heq]
-    _ ≤ (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
-        ((512 * Real.pi) ^ codimension κ * ε ^ codimension κ * R ^ (-codimension κ)) *
-        (Real.sin (Complex.arg ζ)) ^ greenExponent κ := by
-      gcongr
-    _ = _ := by dsimp [sourceApproachConstant]; ring
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-
-lemma source_liveSpatialHitTime_ne_top_iff (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
-    liveSpatialHitTime γ z r ≠ ⊤ ↔
-      ∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r := by
-  simp only [liveSpatialHitTime]
-  split_ifs with h
-  · simp only [ne_eq, WithTop.coe_ne_top, not_false_eq_true, true_iff]
-    exact h
-  · simp only [ne_eq, not_true_eq_false, false_iff]
-    exact h
-
- 
-
-lemma source_live_contact_restarts
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (r ε : ℝ)
-    (hr : 0 < r) (hε : 0 < ε) (he : ε ≤ r / 2) (hy : r ≤ z.im)
-    (τ : ℝ≥0) (hτ : spatialHitTime γ z r = τ)
-    (s : ℝ≥0) (hs : z ∈ survivingDomain γ s) (hd : dist (γ s) z ≤ ε) :
-    τ < s ∧ z ∈ survivingDomain γ τ ∧ γ s ∈ survivingDomain γ τ ∧
-      dist (G τ (γ s) - (U τ : ℂ)) (G τ z - (U τ : ℂ)) <
-        8 * (G τ z).im * ε / r := by
-  have hs0 : r ≤ dist (γ 0) z := by
-    rw [h.2.1, dist_zero_left]
-    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  have her : ε < r := by linarith
-  have hτs : τ < s := WithTop.coe_lt_coe.mp (hτ ▸
-    spatialHitTime_lt_of_dist_lt h.1 z r hs0 s (hd.trans_lt her))
-  have hl : z ∈ survivingDomain γ τ := survivingDomain_antitone γ hτs.le hs
-  have hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 τ) := by
-    intro w hw
-    refine ⟨ball_subset_upperHalfPlane hy hw, ?_⟩
-    rintro ⟨t, ht, rfl⟩
-    have hh := spatialHitTime_lt_of_dist_lt h.1 z r hs0 t (mem_ball.mp hw)
-    rw [hτ] at hh
-    exact not_lt_of_ge ht.2 (WithTop.coe_lt_coe.mp hh)
-  have hb := ball_subset_survivingDomain hr hl hfree
-  have hgs := hb (mem_ball.mpr (hd.trans_lt her))
-  refine ⟨hτs, hl, hgs, ?_⟩
-  rw [dist_sub_right]
-  have hbound := source_halfplane_map_local_bound hr hε.le he
-    ((h.2.2.2.1 τ).mono hb) (fun w hw => h.2.2.2.2.2.1 τ (hb hw)) hd
-  apply hbound.trans_lt
-  apply (div_lt_div_iff_of_pos_right hr).mpr
-  have hp : 0 < (G τ z).im * ε := mul_pos (h.2.2.2.2.2.1 τ hl) hε
-  nlinarith
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance windowBoundsLivePathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance windowBoundsLivePathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-
-lemma source_liveIndicator_zero_of_coarse_top
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (r ε : ℝ)
-    (he : ε < r) (hy : r ≤ z.im) (htop : spatialHitTime γ z r = ⊤) :
-    (if liveSpatialHitTime γ z ε = ⊤ then (0:ℝ) else 1) = 0 := by
-  have hs0 : r ≤ dist (γ 0) z := by
-    rw [h.2.1, dist_zero_left]
-    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  have hl : liveSpatialHitTime γ z ε = ⊤ := by
-    by_contra hn
-    obtain ⟨s, _, hd⟩ := (source_liveSpatialHitTime_ne_top_iff γ z ε).mp hn
-    have hh := spatialHitTime_lt_of_dist_lt h.1 z r hs0 s (hd.trans_lt he)
-    rw [htop] at hh
-    exact not_lt_of_ge le_top hh
-  simp only [ite_eq_left hl]
-
-lemma source_restarted_kernel_dominates
-    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
-    (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (r ε : ℝ)
-    (hr : 0 < r) (hε : 0 < ε) (he : ε ≤ r / 2) (hy : r ≤ z.im)
-    (τ : ℝ≥0) (hτ : spatialHitTime γ z r = τ)
-    (p : C(ℝ≥0, ℂ))
-    (hfwd : ∀ t : ℝ≥0, γ (τ + t) ∈ survivingDomain γ τ →
-      p t = G τ (γ (τ + t)) - (U τ : ℂ)) :
-    (if liveSpatialHitTime γ z ε = ⊤ then (0:ℝ) else 1) ≤
-      sourceApproachKernel ε r ((if z ∈ survivingDomain γ τ then G τ z else 0) - (U τ : ℂ), p) := by
-  by_cases ht : liveSpatialHitTime γ z ε = ⊤
-  · simp only [ite_eq_left ht, sourceApproachKernel]
-    split_ifs <;> norm_num
-  · obtain ⟨s, hs, hd⟩ := (source_liveSpatialHitTime_ne_top_iff γ z ε).mp ht
-    obtain ⟨hτs, hl, hgs, hb⟩ := source_live_contact_restarts h z r ε hr hε he hy τ hτ s hs hd
-    have heq : τ + (s - τ) = s := add_tsub_cancel_of_le hτs.le
-    have hf := hfwd (s - τ) (by simpa only [heq] using hgs)
-    simp only [heq] at hf
-    have hhit : ∃ t : ℝ≥0, dist (p t) (G τ z - (U τ : ℂ)) <
-        8 * (G τ z - (U τ : ℂ)).im * ε / r := by
-      refine ⟨s - τ, ?_⟩
-      simpa only [hf, Complex.sub_im, Complex.ofReal_im, sub_zero] using hb
-    simp only [ite_eq_right ht, ite_eq_left hl, sourceApproachKernel, ite_eq_left hhit, le_refl]
-
-lemma source_stopped_kernel_mean_bound
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
-    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
-    (hgm : Measurable (sourceTracePath γ (fun ω => (hw ω).1)))
-    {U' : ℝ≥0 → ℝ} {γ' : ℝ≥0 → ℂ} {G' F' : ℝ≥0 → ℂ → ℂ}
-    (hU : Continuous U') (hw' : IsCapacityTwoWitness U' γ' G' F')
-    (z : ℂ) (ε r : ℝ) (hε : 0 < ε) (hr : 0 < r)
-    (hy : 8 * ε ≤ r) (hsmall : Real.exp 1 * (512 * Real.pi) * ε ≤ r)
-    (τ : ℝ≥0) (hτ : spatialHitTime γ' z r = τ) :
-    (∫ p, sourceApproachKernel ε r
-      ((if z ∈ survivingDomain γ' τ then G' τ z else 0) - (U' τ : ℂ), p)
-      ∂P.map (sourceTracePath γ (fun ω => (hw ω).1))) ≤
-      sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ U' γ' G' z r := by
-  have ht : spatialHitTime γ' z r ≠ ⊤ := by rw [hτ]; exact WithTop.coe_ne_top
-  have he : (spatialHitTime γ' z r).untopD 0 = τ := by rw [hτ]; rfl
-  by_cases hz : z ∈ survivingDomain γ' τ
-  · have hζ : G' τ z - (U' τ : ℂ) ∈ upperHalfPlane := by
-      change 0 < (G' τ z - (U' τ : ℂ)).im
-      simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hw'.imag_positive τ z hz
-    have hRb : fullRadius G' τ z ≤ (128 * Real.pi) * r := by
-      have hh := hw'.radius_le_past_distance hU τ z hz τ le_rfl
-      rwa [dist_comm z, spatialHitTime_mem hw'.1 z r τ hτ] at hh
-    have hb := source_kernel_mean_bound hB κ hκ hκ8 γ G F hw hgm
-      (G' τ z - (U' τ : ℂ)) hζ ε r (fullRadius G' τ z) hε hr hy hsmall
-      (hw'.fullRadius_positive hU τ z hz) hRb
-    simpa only [ite_eq_left hz, spatialGreenWeight, ite_eq_right ht, he, greenDensity, ite_eq_left hz] using hb
-  · have hk (p : C(ℝ≥0, ℂ)) : sourceApproachKernel ε r ((0:ℂ) - (U' τ : ℂ), p) = 0 :=
-      sourceApproachKernel_zero hε.le hr.le (by simp)
-    simp only [hk, integral_zero, spatialGreenWeight, ite_eq_right ht, he,
-      greenDensity, ite_eq_right hz, mul_zero, le_refl]
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-local instance windowBoundsIntegralPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
-local instance windowBoundsIntegralPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
-local instance windowBoundsDriverPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-local instance windowBoundsDriverPathBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-
-lemma source_finite_approach_setIntegral_bound
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {γ : Ω → ℝ≥0 → ℂ}
-    (a : RegularSLEPresentation P κ γ) (z : ℂ) (ε r : ℝ)
-    (hε : 0 < ε) (hr : 0 < r) (he : ε ≤ r / 2) (hy : r ≤ z.im)
-    (h8 : 8 * ε ≤ r) (hsmall : Real.exp 1 * (512 * Real.pi) * ε ≤ r)
-    (σ : a.samples → ℝ≥0)
-    (hσ : IsStoppingTime a.filtration (fun ω => (σ ω : WithTop ℝ≥0)))
-    (A : Set a.samples) (hA : MeasurableSet[hσ.measurableSpace] A)
-    (hAσ : ∀ ω ∈ A, spatialHitTime (γ ω) z r = (σ ω : WithTop ℝ≥0)) :
-    (∫ ω in A, (if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1)
-      ∂P.comap Subtype.val) ≤
-    ∫ ω in A, sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ
-      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r ∂P.comap Subtype.val := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  let gp := sourceTracePath (fun ω : a.samples => γ ω) (fun ω => (a.witness ω).1)
-  have hgm : Measurable gp := source_path_measurable gp (fun t =>
-    (a.trace_adapted t).mono (a.filtration.le t) le_rfl)
-  let ν := μ.map gp
-  let : IsProbabilityMeasure ν := inferInstance
-  obtain ⟨Φ, hΦm, _, hΦ⟩ := sourceStoppedTraceLaw_forward κ hκ hκ8
-    a.B a.brownian a.measurable_B a.continuous_B (fun ω : a.samples => γ ω)
-    a.G a.F a.witness
-  obtain ⟨hlaw, hfwd⟩ := hΦ σ hσ
-  let rp := sourceRestartDriverPath a.B a.continuous_B σ
-  let X (ω : a.samples) := Φ (rp ω)
-  have hrm : Measurable rp := source_path_measurable rp (fun t =>
-    (measurable_pi_apply t).comp (brownianRestart_measurable a.measurable_B a.continuous_B
-      (stopping_time_measurable_nnreal hσ)))
-  have hXm : Measurable X := hΦm.comp hrm
-  let Y (ω : a.samples) : ℂ :=
-    (if z ∈ survivingDomain (γ ω) (σ ω) then a.G ω (σ ω) z else 0) -
-      (Real.sqrt κ * a.B (σ ω) ω : ℝ)
-  have hYm : Measurable[hσ.measurableSpace] Y := by
-    have hGm := measurable_stoppedValue
-      (source_killedChart_progressive a.measurable_B a.continuous_B κ
-        (fun ω : a.samples => γ ω) a.G a.F a.witness z) hσ
-    have hBm := measurable_stoppedValue
-      ((Filtration.stronglyAdapted_natural (fun t => (a.measurable_B t).stronglyMeasurable)).isStronglyProgressive_of_continuous a.continuous_B) hσ
-    exact hGm.sub (Complex.measurable_ofReal.comp (measurable_const.mul hBm))
-  let H (ω : a.samples) := sourceApproachKernel ε r (Y ω, X ω)
-  let K (ω : a.samples) := ∫ p, sourceApproachKernel ε r (Y ω, p) ∂ν
-  have hHmeas : Measurable H := (sourceApproachKernel_measurable ε r).comp
-    ((hYm.mono hσ.measurableSpace_le le_rfl).prodMk hXm)
-  have hHi : Integrable H μ :=
-    (integrable_const (1:ℝ)).mono' hHmeas.aestronglyMeasurable
-      (Eventually.of_forall (fun ω => sourceApproachKernel_norm_le ε r (Y ω, X ω)))
-  have hcond : μ[H | hσ.measurableSpace] =ᵐ[μ] K :=
-    source_parameter_condExp hσ.measurableSpace_le hXm hYm hlaw
-      (sourceApproachKernel ε r) (sourceApproachKernel_measurable ε r)
-      (sourceApproachKernel_norm_le ε r)
-  have hKi : Integrable K μ := integrable_condExp.congr hcond
-  let W (ω : a.samples) := sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ
-    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r
-  have hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
-  have hWm : Measurable W := measurable_const.mul
-    ((source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
-      (fun ω : a.samples => γ ω) a.G a.F a.witness z r hτ).mono hτ.measurableSpace_le le_rfl)
-  have hWi : Integrable W μ := by
-    apply Integrable.of_bound hWm.aestronglyMeasurable
-      (sourceApproachConstant κ * ε ^ codimension κ * r ^ (-codimension κ))
-    apply Eventually.of_forall
-    intro ω
-    have hb := (a.witness ω).spatialGreenWeight_bound
-      (continuous_const.mul (a.continuous_B ω)) hκ hκ8 hr z hy
-    have hc : 0 ≤ sourceApproachConstant κ * ε ^ codimension κ :=
-      mul_nonneg (sourceApproachConstant_pos κ hκ hκ8).le (Real.rpow_nonneg hε.le _)
-    rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hc hb.1)]
-    exact mul_le_mul_of_nonneg_left hb.2 hc
-  have hfi : Integrable (fun ω : a.samples => if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1) μ := by
-    apply Integrable.of_bound (source_liveIndicator_measurable a z ε (by linarith)).aestronglyMeasurable 1
-    apply Eventually.of_forall
-    intro ω
-    split_ifs <;> norm_num
-  have hAm : MeasurableSet A := hσ.measurableSpace_le A hA
-  calc
-    _ ≤ ∫ ω in A, H ω ∂μ := by
-      apply integral_mono_ae hfi.integrableOn hHi.integrableOn
-      filter_upwards [ae_restrict_of_ae hfwd, ae_restrict_mem hAm] with ω hf hωA
-      exact source_restarted_kernel_dominates (a.witness ω) z r ε hr hε he hy
-        (σ ω) (hAσ ω hωA) (X ω) hf
-    _ = ∫ ω in A, K ω ∂μ := by
-      have : IsFiniteMeasure (μ.trim hσ.measurableSpace_le) := isFiniteMeasure_trim _
-      rw [← integral_congr_ae (ae_restrict_of_ae hcond)]
-      exact (setIntegral_condExp hσ.measurableSpace_le hHi hA).symm
-    _ ≤ ∫ ω in A, W ω ∂μ := by
-      apply integral_mono_ae hKi.integrableOn hWi.integrableOn
-      filter_upwards [ae_restrict_mem hAm] with ω hωA
-      exact source_stopped_kernel_mean_bound a.brownian κ hκ hκ8
-        (fun ω : a.samples => γ ω) a.G a.F a.witness hgm
-        (continuous_const.mul (a.continuous_B ω)) (a.witness ω)
-        z ε r hε hr h8 hsmall (σ ω) (hAσ ω hωA)
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal
-namespace SLEExactGauge
-
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
- 
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-universe u_source_approach
- 
-
-def SourceConditionalApproachTarget : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 → ∃ C : ℝ, 0 < C ∧ ∃ q : ℝ, 0 < q ∧ q < 1 ∧
-  ∀ (Ω : Type u_source_approach) (_ : MeasurableSpace Ω) (P : Measure Ω),
-  IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (z : ℂ) (ε r : ℝ), 0 < ε → ε ≤ q * r → r ≤ z.im →
-  ∃ hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r),
-    let μ := P.comap (Subtype.val : a.samples → Ω)
-    μ[(fun ω : a.samples => if liveSpatialHitTime (γ ω) z ε = ⊤ then (0 : ℝ) else 1)
-      | hτ.measurableSpace] ≤ᵐ[μ]
-      (fun ω => C * ε ^ codimension κ * spatialGreenWeight κ
-        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r)
-
-lemma source_approach_small_scale (ε r : ℝ) (hε : 0 < ε)
-    (h : ε ≤ min (1/16) (1/(Real.exp 1 * (512 * Real.pi))) * r) :
-    0 < r ∧ ε ≤ r/2 ∧ 8*ε ≤ r ∧ Real.exp 1 * (512 * Real.pi) * ε ≤ r := by
-  have hA : 0 < Real.exp 1 * (512 * Real.pi) := by positivity
-  have hq : 0 < min (1/16:ℝ) (1/(Real.exp 1 * (512 * Real.pi))) := by positivity
-  have hr : 0 < r := by
-    by_contra! hn
-    have hh := mul_nonpos_of_nonneg_of_nonpos hq.le hn
-    linarith
-  have h16 : ε ≤ r/16 := by
-    apply h.trans
-    have hh := mul_le_mul_of_nonneg_right
-      (min_le_left (1/16:ℝ) (1/(Real.exp 1 * (512 * Real.pi)))) hr.le
-    nlinarith
-  refine ⟨hr, by linarith, by linarith, ?_⟩
-  have hqA := (le_div_iff₀ hA).mp
-    (min_le_right (1/16:ℝ) (1/(Real.exp 1 * (512 * Real.pi))))
-  have hh := mul_le_mul_of_nonneg_right hqA hr.le
-  have hhh := mul_le_mul_of_nonneg_left h hA.le
-  nlinarith only [hh, hhh]
-
- 
-
-theorem sourceConditionalApproach_proved : SourceConditionalApproachTarget := by
-  intro κ hκ hκ8
-  refine ⟨sourceApproachConstant κ, sourceApproachConstant_pos κ hκ hκ8,
-    min (1/16) (1/(Real.exp 1 * (512 * Real.pi))), by positivity, ?_, ?_⟩
-  · exact (min_le_left _ _).trans_lt (by norm_num)
-  intro Ω mΩ P hP γ a z ε r hε hεq hy
-  let : IsProbabilityMeasure P := hP
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  obtain ⟨hr, he, h8, hsmall⟩ := source_approach_small_scale ε r hε hεq
-  let τ (ω : a.samples) := spatialHitTime (γ ω) z r
-  have hτ : IsStoppingTime a.filtration τ :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
-  refine ⟨hτ, ?_⟩
-  let f (ω : a.samples) : ℝ := if liveSpatialHitTime (γ ω) z ε = ⊤ then 0 else 1
-  let W (ω : a.samples) := spatialGreenWeight κ
-    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r
-  let g (ω : a.samples) := sourceApproachConstant κ * ε ^ codimension κ * W ω
-  have hf : Integrable f μ := by
-    apply Integrable.of_bound (source_liveIndicator_measurable a z ε (by linarith)).aestronglyMeasurable 1
-    apply Eventually.of_forall
-    intro ω
-    dsimp [f]
-    split_ifs <;> norm_num
-  have hWm : Measurable[hτ.measurableSpace] W :=
-    source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
-      (fun ω : a.samples => γ ω) a.G a.F a.witness z r hτ
-  have hgm : Measurable[hτ.measurableSpace] g := measurable_const.mul hWm
-  have hg : Integrable g μ := by
-    apply Integrable.of_bound (hgm.mono hτ.measurableSpace_le le_rfl).aestronglyMeasurable
-      (sourceApproachConstant κ * ε ^ codimension κ * r ^ (-codimension κ))
-    apply Eventually.of_forall
-    intro ω
-    have hb := (a.witness ω).spatialGreenWeight_bound
-      (continuous_const.mul (a.continuous_B ω)) hκ hκ8 hr z hy
-    have hc : 0 ≤ sourceApproachConstant κ * ε ^ codimension κ :=
-      mul_nonneg (sourceApproachConstant_pos κ hκ hκ8).le (Real.rpow_nonneg hε.le _)
-    rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hc hb.1)]
-    exact mul_le_mul_of_nonneg_left hb.2 hc
-  change μ[f | hτ.measurableSpace] ≤ᵐ[μ] g
-  apply source_condExp_le_of_setIntegral_le hτ.measurableSpace_le hf hg hgm.stronglyMeasurable
-  apply source_unbounded_setIntegral_le a.filtration hτ f g hf hg
-  · intro ω ht
-    constructor
-    · exact source_liveIndicator_zero_of_coarse_top (a.witness ω) z r ε (by linarith) hy ht
-    · change sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ _ _ _ z r = 0
-      change spatialHitTime (γ ω) z r = ⊤ at ht
-      simp only [spatialGreenWeight, ite_eq_left ht, mul_zero]
-  · intro n A hA hAτ
-    exact source_finite_approach_setIntegral_bound hκ hκ8 a z ε r hε hr he hy h8 hsmall
-      (sourceStopTruncate τ n) (sourceStopTruncate_stopping a.filtration hτ n) A hA hAτ
-
-end
-end SLEExactGauge
-
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-universe u_source_spatial
- 
-
-end
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.PhysicalKernelUtilities
+import OAI.Probability.SLE.WindowGeometry
+
+namespace OAI
+
+/-! Window Bounds for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance windowBoundsPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance windowBoundsPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+lemma source_liveTime_eq_sphere_of_finite {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) (hn : liveSpatialHitTime γ z r ≠ ⊤) :
+    liveSpatialHitTime γ z r = spatialHitTime γ z r := by
+  have he := source_liveTime_eq_killed hc z r hs
+  split_ifs at he with hl
+  · exact he
+  · exact (hn he).elim
+
+lemma source_windowV_stopped_eq {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ) (κ : ℝ) (z : ℂ) (r ε R : ℝ)
+    (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    spatialGreenWeight κ U γ G z ε * sourceSpatialWindowH γ z r ε R =
+      greenDensity κ U γ G (sourceSpatialWindowTime γ z r ε R).untopA z *
+        sourceSpatialWindowH γ z r ε R := by
+  by_cases hH : sourceSpatialWindowH γ z r ε R = 1
+  · obtain ⟨he,hn⟩ := (source_windowH_iff hc z r ε R her hs).mp hH
+    have hr := source_liveTime_eq_sphere_of_finite hc z ε (her.trans hs) hn
+    have hraw : spatialHitTime γ z ε ≠ ⊤ := hr ▸ hn
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hraw
+    rw [hH, mul_one, mul_one, he, hr, spatialGreenWeight, ite_eq_right hraw, ← ht]
+    rfl
+  · have hz : sourceSpatialWindowH γ z r ε R = 0 := by
+      unfold sourceSpatialWindowH at hH ⊢
+      split_ifs <;> simp_all
+    rw [hz,mul_zero,mul_zero]
+
+lemma source_windowV_stopped_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
+    (r ε R : ℝ) (her : ε ≤ r) (hy : r ≤ z.im)
+    (hτ : IsStoppingTime a.filtration (fun ω : a.samples => sourceSpatialWindowTime (γ ω) z r ε R)) :
+    Measurable[hτ.measurableSpace] (fun ω : a.samples =>
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z ε *
+        sourceSpatialWindowH (γ ω) z r ε R) := by
+  have hg := measurable_stoppedValue
+    (source_killedGreen_progressive a.measurable_B a.continuous_B κ
+      (fun ω : a.samples => γ ω) a.G a.F a.witness z) hτ
+  have hH := (source_windowHB_stopped_measurable a z r ε R hτ).1
+  convert hg.mul hH using 1 <;> try rfl
+  funext ω
+  apply source_windowV_stopped_eq (a.trace_continuous ω) _ _ _ _ _ _ _ her
+  rw [(a.witness ω).2.1,dist_zero_left]
+  exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm _))
+
+lemma source_liveWeight_stopped_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
+    (r : ℝ) (hy : r ≤ z.im)
+    (hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r)) :
+    Measurable[hσ.measurableSpace] (fun ω : a.samples =>
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r) := by
+  have hraw : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
+  have hm := source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
+    (fun ω : a.samples => γ ω) a.G a.F a.witness z r hraw
+  apply hm.mono (hraw.measurableSpace_mono hσ ?_) le_rfl
+  intro ω
+  have hs : r ≤ dist (γ ω 0) z := by
+    rw [(a.witness ω).2.1,dist_zero_left]
+    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm _))
+  change spatialHitTime (γ ω) z r ≤ liveSpatialHitTime (γ ω) z r
+  rw [source_liveTime_eq_killed (a.trace_continuous ω) z r hs]
+  split_ifs <;> first | exact le_rfl | exact le_top
+
+lemma source_collarHB_joint
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (hD : MeasurableSet D) (r ε R : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) :
+    Measurable (fun p : ℂ × a.samples => sourceCollarH D (γ p.2) p.1 r ε R) ∧
+    Measurable (fun p : ℂ × a.samples => sourceCollarB D (γ p.2) p.1 r ε R) := by
+  let τ := fun p : ℂ × a.samples => sourceCollarFinish D (γ p.2) p.1 r ε R
+  have ht : Measurable τ := source_collarFinish_joint a D hD r ε R hy
+  have hg : Measurable (fun p : ℂ × a.samples => (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ))) :=
+    (source_tracePath_measurable a).comp measurable_snd
+  have hl := source_joint_survival_measurable
+    (fun p : ℂ × a.samples => (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ)))
+    (fun p => (τ p).untopD 0) Prod.fst hg (ht.untopD 0) measurable_fst
+  have hd : Measurable (fun p : ℂ × a.samples => dist (γ p.2 ((τ p).untopD 0)) p.1) :=
+    (continuous_eval.measurable.comp (hg.prodMk (ht.untopD 0))).dist measurable_fst
+  have hn : MeasurableSet {p | τ p ≠ ⊤} := (measurableSet_eq_fun ht measurable_const).compl
+  have hH : Measurable (fun p : ℂ × a.samples =>
+      if τ p ≠ ⊤ ∧ p.1 ∈ survivingDomain (γ p.2) ((τ p).untopD 0) ∧
+      dist (γ p.2 ((τ p).untopD 0)) p.1 ≤ ε then (1:ℝ) else 0) :=
+    Measurable.ite (hn.inter (hl.inter (measurableSet_le hd measurable_const))) measurable_const measurable_const
+  have hB : Measurable (fun p : ℂ × a.samples =>
+      if τ p ≠ ⊤ ∧ p.1 ∈ survivingDomain (γ p.2) ((τ p).untopD 0) ∧
+      R ≤ dist (γ p.2 ((τ p).untopD 0)) p.1 then (1:ℝ) else 0) :=
+    Measurable.ite (hn.inter (hl.inter (measurableSet_le measurable_const hd))) measurable_const measurable_const
+  constructor
+  · convert hH using 1
+    funext p
+    by_cases hz : p.1 ∈ D
+    · simp only [sourceCollarH,sourceSpatialWindowH,τ,sourceCollarFinish,ite_eq_left hz]
+    · simp only [sourceCollarH,τ,sourceCollarFinish,ite_eq_right hz,ne_eq,not_true_eq_false,false_and,ite_false]
+  · convert hB using 1
+    funext p
+    by_cases hz : p.1 ∈ D
+    · simp only [sourceCollarB,sourceSpatialWindowB,τ,sourceCollarFinish,ite_eq_left hz]
+    · simp only [sourceCollarB,τ,sourceCollarFinish,ite_eq_right hz,ne_eq,not_true_eq_false,false_and,ite_false]
+
+lemma source_stop_space_congr {Ω : Type*} [MeasurableSpace Ω]
+    {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} {σ τ : Ω → WithTop ℝ≥0}
+    (hσ : IsStoppingTime ℱ σ) (hτ : IsStoppingTime ℱ τ) (he : σ = τ) :
+    hσ.measurableSpace = hτ.measurableSpace := by
+  subst τ
+  rfl
+
+lemma source_collarHBV_stopped
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (r ε R : ℝ) (her : ε ≤ r) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
+    let hτ := source_collarFinish_stopping a D r ε R hy z
+    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceCollarH D (γ ω) z r ε R) ∧
+    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceCollarB D (γ ω) z r ε R) ∧
+    Measurable[hτ.measurableSpace] (fun ω : a.samples =>
+      spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z ε *
+        sourceCollarH D (γ ω) z r ε R) := by
+  dsimp only
+  by_cases hz : z ∈ D
+  · have ht := source_window_stopping a z r ε R (hy z hz)
+    have hh := source_windowHB_stopped_measurable a z r ε R ht
+    have hv := source_windowV_stopped_measurable a z r ε R her (hy z hz) ht
+    have hspace := source_stop_space_congr (source_collarFinish_stopping a D r ε R hy z) ht
+      (funext fun ω => by simp only [sourceCollarFinish,ite_eq_left hz])
+    rw [hspace]
+    simpa only [sourceCollarH,sourceCollarB,ite_eq_left hz] using And.intro hh.1 (And.intro hh.2 hv)
+  · simp only [sourceCollarH,sourceCollarB,ite_eq_right hz,mul_zero]
+    exact ⟨measurable_const,measurable_const,measurable_const⟩
+
+lemma source_collarWeight_stopped
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
+    Measurable[(source_collarStart_stopping a D r hy z).measurableSpace] (fun ω : a.samples =>
+      if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r else 0) := by
+  by_cases hz : z ∈ D
+  · have hs := source_liveHit_stopping a z r (hy z hz)
+    have hw := source_liveWeight_stopped_measurable a z r (hy z hz) hs
+    have hspace := source_stop_space_congr (source_collarStart_stopping a D r hy z) hs
+      (funext fun ω => by simp only [sourceCollarStart,ite_eq_left hz])
+    rw [hspace]
+    simpa only [ite_eq_left hz] using hw
+  · simp only [ite_eq_right hz]
+    exact measurable_const
+
+ theorem sourceSpatialWindowMeasurable_proved : SourceSpatialWindowMeasurableTarget := by
+  intro Ω m P κ hκ hκ8 γ a D hD ε r R hε her hrR hy
+  dsimp only
+  refine ⟨⟨source_collarStart_stopping a D r hy,
+    source_collarFinish_stopping a D r ε R hy,?_,?_⟩,?_,?_,?_,?_,?_⟩
+  · exact source_collarHBV_stopped a D r ε R her.le hy
+  · exact source_collarWeight_stopped a D r hy
+  · exact source_collarStart_joint a D hD r hy
+  · exact source_collarFinish_joint a D hD r ε R hy
+  · exact (source_collarHB_joint a D hD r ε R hy).1
+  · exact (source_collarHB_joint a D hD r ε R hy).2
+  · exact (a.spatialGreenWeight_measurable ε).mul (source_collarHB_joint a D hD r ε R hy).1
+
+end
+end SLEExactGauge
+
+end
+
+ 
+section
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_liveIndicator_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (z : ℂ) (ε : ℝ) (hy : ε ≤ z.im) :
+    Measurable (fun ω : a.samples =>
+      if liveSpatialHitTime (γ ω) z ε = ⊤ then (0 : ℝ) else 1) := by
+  classical
+  have hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z ε) :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z ε
+  have hK := measurable_stoppedValue
+    (source_killedChart_progressive a.measurable_B a.continuous_B κ
+      (fun ω : a.samples => γ ω) a.G a.F a.witness z) hτ
+  have hKm : Measurable (fun ω : a.samples =>
+      (if z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z ε).untopA
+        then a.G ω (spatialHitTime (γ ω) z ε).untopA z else 0 : ℂ)) :=
+    hK.mono hτ.measurableSpace_le le_rfl
+  have htop : MeasurableSet {ω : a.samples | spatialHitTime (γ ω) z ε = ⊤} :=
+    measurableSet_eq_fun (hτ.measurable.mono hτ.measurableSpace_le le_rfl) measurable_const
+  have hl : MeasurableSet {ω : a.samples |
+      0 < (if z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z ε).untopA
+        then a.G ω (spatialHitTime (γ ω) z ε).untopA z else 0 : ℂ).im} :=
+    measurableSet_lt measurable_const (Complex.measurable_im.comp hKm)
+  have hmo : Measurable (fun ω : a.samples =>
+      if spatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else
+        if 0 < (if z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z ε).untopA
+          then a.G ω (spatialHitTime (γ ω) z ε).untopA z else 0 : ℂ).im then 1 else 0) :=
+    Measurable.ite htop measurable_const (Measurable.ite hl measurable_const measurable_const)
+  convert hmo using 1
+  funext ω
+  have hs : ε ≤ dist (γ ω 0) z := by
+    rw [(a.witness ω).2.1, dist_zero_left]
+    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  rw [liveSpatialHitTime_eq (a.trace_continuous ω) z ε hs]
+  by_cases ht : spatialHitTime (γ ω) z ε = ⊤
+  · simp only [ite_eq_left ht, ite_true]
+  · obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp ht
+    rw [ite_eq_right ht, ite_eq_right ht, ← hv]
+    change (if (if z ∈ survivingDomain (γ ω) v then (v : WithTop ℝ≥0) else ⊤) = ⊤
+      then (0:ℝ) else 1) =
+      (if 0 < (if z ∈ survivingDomain (γ ω) v then a.G ω v z else 0 : ℂ).im then 1 else 0)
+    by_cases hz : z ∈ survivingDomain (γ ω) v
+    · simp only [ite_eq_left hz, WithTop.coe_ne_top, ite_false,
+        ite_eq_left ((a.witness ω).imag_positive v z hz)]
+    · simp only [ite_eq_right hz, ite_true, Complex.zero_im, lt_self_iff_false, ite_false]
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance windowBoundsApproachPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance windowBoundsApproachPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+noncomputable def sourceApproachConstant (κ : ℝ) : ℝ :=
+  (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
+    (512 * Real.pi) ^ codimension κ
+
+lemma sourceApproachConstant_pos (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < sourceApproachConstant κ := by
+  exact mul_pos (mul_pos (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num))
+    (Real.rpow_pos_of_pos (by positivity) _)) (Real.rpow_pos_of_pos (by positivity) _)
+
+lemma source_kernel_mean_bound
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgm : Measurable (sourceTracePath γ (fun ω => (hw ω).1)))
+    (ζ : ℂ) (hζ : ζ ∈ upperHalfPlane) (ε r R : ℝ)
+    (hε : 0 < ε) (hr : 0 < r) (hy : 8 * ε ≤ r)
+    (hsmall : Real.exp 1 * (512 * Real.pi) * ε ≤ r)
+    (hR : 0 < R) (hRb : R ≤ (128 * Real.pi) * r) :
+    (∫ p, sourceApproachKernel ε r (ζ, p) ∂P.map (sourceTracePath γ (fun ω => (hw ω).1))) ≤
+      sourceApproachConstant κ * ε ^ codimension κ *
+        (R ^ (-codimension κ) * (Real.sin (Complex.arg ζ)) ^ greenExponent κ) := by
+  have hypos : 0 < ζ.im := hζ
+  let δ := 8 * ζ.im * ε / r
+  have hδ : 0 < δ := by dsimp [δ]; positivity
+  have hδy : δ ≤ ζ.im := by
+    apply (div_le_iff₀ hr).mpr
+    nlinarith [mul_le_mul_of_nonneg_left hy (show 0 ≤ ζ.im from hζ.le)]
+  have hδsmall : 1 ≤ Real.log ((2 * ζ.im) / ((128 * Real.pi) * δ)) := by
+    apply (Real.le_log_iff_exp_le (by positivity)).mpr
+    apply (le_div_iff₀ (by positivity : 0 < (128 * Real.pi) * δ)).mpr
+    calc
+      Real.exp 1 * ((128 * Real.pi) * δ) =
+          (2 * ζ.im) * (Real.exp 1 * (512 * Real.pi) * ε) / r := by dsimp [δ]; ring
+      _ ≤ (2 * ζ.im) * r / r :=
+        div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hsmall (by positivity)) hr.le
+      _ = 2 * ζ.im := mul_div_cancel_right₀ _ hr.ne'
+  have hb := euclidean_approach hB κ hκ hκ8 γ G F (Eventually.of_forall hw)
+    ζ hζ δ hδ hδy hδsmall
+  let S : Set Ω := {ω | ∃ t : ℝ≥0, dist (γ ω t) ζ < δ}
+  have hS : MeasurableSet S := by
+    have hh := measurableSet_lt (measurable_const (a := (0:ℝ)))
+      ((sourceApproachKernel_measurable ε r).comp ((measurable_const (a := ζ)).prodMk hgm))
+    convert hh using 1
+    ext ω
+    change (∃ t : ℝ≥0, dist (γ ω t) ζ < δ) ↔ 0 <
+      (if ∃ t : ℝ≥0, dist (γ ω t) ζ < δ then (1:ℝ) else 0)
+    split_ifs <;> simp_all
+  have hint : (∫ p, sourceApproachKernel ε r (ζ, p)
+      ∂P.map (sourceTracePath γ (fun ω => (hw ω).1))) = (P S).toReal := by
+    have hfun : Measurable (fun p : C(ℝ≥0, ℂ) => sourceApproachKernel ε r (ζ, p)) :=
+      (sourceApproachKernel_measurable ε r).comp (measurable_const.prodMk measurable_id)
+    calc
+      _ = ∫ ω, sourceApproachKernel ε r (ζ, sourceTracePath γ (fun ω => (hw ω).1) ω) ∂P :=
+        integral_map hgm.aemeasurable hfun.aestronglyMeasurable
+      _ = _ := ?_
+
+    change (∫ ω, S.indicator (fun _ => (1:ℝ)) ω ∂P) = _
+    simp only [integral_indicator_const _ hS, smul_eq_mul, mul_one, measureReal_def]
+  rw [hint]
+  apply hb.trans
+  have hsin : 0 ≤ Real.sin (Complex.arg ζ) := by
+    rw [Complex.sin_arg]
+    exact div_nonneg hζ.le (norm_nonneg _)
+  have hC : 0 ≤ uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ :=
+    (mul_pos (uniformInverseAngleConstant_pos κ 1 hκ hκ8 (by norm_num))
+      (Real.rpow_pos_of_pos (by positivity) _)).le
+  have hp : (4 * R / r) ^ codimension κ ≤ (512 * Real.pi) ^ codimension κ := by
+    apply Real.rpow_le_rpow (by positivity) _ (codimension_pos κ hκ8).le
+    apply (div_le_iff₀ hr).mpr
+    nlinarith [hRb]
+  have heq := source_rpow_scale (codimension κ) ζ.im ε r R hζ hε hr hR
+  dsimp only [initialGreen, δ]
+  calc
+    _ = (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
+        ((2 * ζ.im) ^ (-codimension κ) * (8 * ζ.im * ε / r) ^ codimension κ) *
+        (Real.sin (Complex.arg ζ)) ^ greenExponent κ := by ring
+    _ = (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
+        ((4 * R / r) ^ codimension κ * ε ^ codimension κ * R ^ (-codimension κ)) *
+        (Real.sin (Complex.arg ζ)) ^ greenExponent κ := by rw [heq]
+    _ ≤ (uniformInverseAngleConstant κ 1 * (128 * Real.pi) ^ codimension κ) *
+        ((512 * Real.pi) ^ codimension κ * ε ^ codimension κ * R ^ (-codimension κ)) *
+        (Real.sin (Complex.arg ζ)) ^ greenExponent κ := by
+      gcongr
+    _ = _ := by dsimp [sourceApproachConstant]; ring
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+
+lemma source_liveSpatialHitTime_ne_top_iff (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) :
+    liveSpatialHitTime γ z r ≠ ⊤ ↔
+      ∃ t : ℝ≥0, z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ r := by
+  simp only [liveSpatialHitTime]
+  split_ifs with h
+  · simp only [ne_eq, WithTop.coe_ne_top, not_false_eq_true, true_iff]
+    exact h
+  · simp only [ne_eq, not_true_eq_false, false_iff]
+    exact h
+
+ 
+
+lemma source_live_contact_restarts
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (r ε : ℝ)
+    (hr : 0 < r) (hε : 0 < ε) (he : ε ≤ r / 2) (hy : r ≤ z.im)
+    (τ : ℝ≥0) (hτ : spatialHitTime γ z r = τ)
+    (s : ℝ≥0) (hs : z ∈ survivingDomain γ s) (hd : dist (γ s) z ≤ ε) :
+    τ < s ∧ z ∈ survivingDomain γ τ ∧ γ s ∈ survivingDomain γ τ ∧
+      dist (G τ (γ s) - (U τ : ℂ)) (G τ z - (U τ : ℂ)) <
+        8 * (G τ z).im * ε / r := by
+  have hs0 : r ≤ dist (γ 0) z := by
+    rw [h.2.1, dist_zero_left]
+    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  have her : ε < r := by linarith
+  have hτs : τ < s := WithTop.coe_lt_coe.mp (hτ ▸
+    spatialHitTime_lt_of_dist_lt h.1 z r hs0 s (hd.trans_lt her))
+  have hl : z ∈ survivingDomain γ τ := survivingDomain_antitone γ hτs.le hs
+  have hfree : ball z r ⊆ upperHalfPlane \ (γ '' Icc 0 τ) := by
+    intro w hw
+    refine ⟨ball_subset_upperHalfPlane hy hw, ?_⟩
+    rintro ⟨t, ht, rfl⟩
+    have hh := spatialHitTime_lt_of_dist_lt h.1 z r hs0 t (mem_ball.mp hw)
+    rw [hτ] at hh
+    exact not_lt_of_ge ht.2 (WithTop.coe_lt_coe.mp hh)
+  have hb := ball_subset_survivingDomain hr hl hfree
+  have hgs := hb (mem_ball.mpr (hd.trans_lt her))
+  refine ⟨hτs, hl, hgs, ?_⟩
+  rw [dist_sub_right]
+  have hbound := source_halfplane_map_local_bound hr hε.le he
+    ((h.2.2.2.1 τ).mono hb) (fun w hw => h.2.2.2.2.2.1 τ (hb hw)) hd
+  apply hbound.trans_lt
+  apply (div_lt_div_iff_of_pos_right hr).mpr
+  have hp : 0 < (G τ z).im * ε := mul_pos (h.2.2.2.2.2.1 τ hl) hε
+  nlinarith
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance windowBoundsLivePathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance windowBoundsLivePathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+
+lemma source_liveIndicator_zero_of_coarse_top
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (r ε : ℝ)
+    (he : ε < r) (hy : r ≤ z.im) (htop : spatialHitTime γ z r = ⊤) :
+    (if liveSpatialHitTime γ z ε = ⊤ then (0:ℝ) else 1) = 0 := by
+  have hs0 : r ≤ dist (γ 0) z := by
+    rw [h.2.1, dist_zero_left]
+    exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  have hl : liveSpatialHitTime γ z ε = ⊤ := by
+    by_contra hn
+    obtain ⟨s, _, hd⟩ := (source_liveSpatialHitTime_ne_top_iff γ z ε).mp hn
+    have hh := spatialHitTime_lt_of_dist_lt h.1 z r hs0 s (hd.trans_lt he)
+    rw [htop] at hh
+    exact not_lt_of_ge le_top hh
+  simp only [ite_eq_left hl]
+
+lemma source_restarted_kernel_dominates
+    {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ}
+    (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (r ε : ℝ)
+    (hr : 0 < r) (hε : 0 < ε) (he : ε ≤ r / 2) (hy : r ≤ z.im)
+    (τ : ℝ≥0) (hτ : spatialHitTime γ z r = τ)
+    (p : C(ℝ≥0, ℂ))
+    (hfwd : ∀ t : ℝ≥0, γ (τ + t) ∈ survivingDomain γ τ →
+      p t = G τ (γ (τ + t)) - (U τ : ℂ)) :
+    (if liveSpatialHitTime γ z ε = ⊤ then (0:ℝ) else 1) ≤
+      sourceApproachKernel ε r ((if z ∈ survivingDomain γ τ then G τ z else 0) - (U τ : ℂ), p) := by
+  by_cases ht : liveSpatialHitTime γ z ε = ⊤
+  · simp only [ite_eq_left ht, sourceApproachKernel]
+    split_ifs <;> norm_num
+  · obtain ⟨s, hs, hd⟩ := (source_liveSpatialHitTime_ne_top_iff γ z ε).mp ht
+    obtain ⟨hτs, hl, hgs, hb⟩ := source_live_contact_restarts h z r ε hr hε he hy τ hτ s hs hd
+    have heq : τ + (s - τ) = s := add_tsub_cancel_of_le hτs.le
+    have hf := hfwd (s - τ) (by simpa only [heq] using hgs)
+    simp only [heq] at hf
+    have hhit : ∃ t : ℝ≥0, dist (p t) (G τ z - (U τ : ℂ)) <
+        8 * (G τ z - (U τ : ℂ)).im * ε / r := by
+      refine ⟨s - τ, ?_⟩
+      simpa only [hf, Complex.sub_im, Complex.ofReal_im, sub_zero] using hb
+    simp only [ite_eq_right ht, ite_eq_left hl, sourceApproachKernel, ite_eq_left hhit, le_refl]
+
+lemma source_stopped_kernel_mean_bound
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8)
+    (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω))
+    (hgm : Measurable (sourceTracePath γ (fun ω => (hw ω).1)))
+    {U' : ℝ≥0 → ℝ} {γ' : ℝ≥0 → ℂ} {G' F' : ℝ≥0 → ℂ → ℂ}
+    (hU : Continuous U') (hw' : IsCapacityTwoWitness U' γ' G' F')
+    (z : ℂ) (ε r : ℝ) (hε : 0 < ε) (hr : 0 < r)
+    (hy : 8 * ε ≤ r) (hsmall : Real.exp 1 * (512 * Real.pi) * ε ≤ r)
+    (τ : ℝ≥0) (hτ : spatialHitTime γ' z r = τ) :
+    (∫ p, sourceApproachKernel ε r
+      ((if z ∈ survivingDomain γ' τ then G' τ z else 0) - (U' τ : ℂ), p)
+      ∂P.map (sourceTracePath γ (fun ω => (hw ω).1))) ≤
+      sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ U' γ' G' z r := by
+  have ht : spatialHitTime γ' z r ≠ ⊤ := by rw [hτ]; exact WithTop.coe_ne_top
+  have he : (spatialHitTime γ' z r).untopD 0 = τ := by rw [hτ]; rfl
+  by_cases hz : z ∈ survivingDomain γ' τ
+  · have hζ : G' τ z - (U' τ : ℂ) ∈ upperHalfPlane := by
+      change 0 < (G' τ z - (U' τ : ℂ)).im
+      simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hw'.imag_positive τ z hz
+    have hRb : fullRadius G' τ z ≤ (128 * Real.pi) * r := by
+      have hh := hw'.radius_le_past_distance hU τ z hz τ le_rfl
+      rwa [dist_comm z, spatialHitTime_mem hw'.1 z r τ hτ] at hh
+    have hb := source_kernel_mean_bound hB κ hκ hκ8 γ G F hw hgm
+      (G' τ z - (U' τ : ℂ)) hζ ε r (fullRadius G' τ z) hε hr hy hsmall
+      (hw'.fullRadius_positive hU τ z hz) hRb
+    simpa only [ite_eq_left hz, spatialGreenWeight, ite_eq_right ht, he, greenDensity, ite_eq_left hz] using hb
+  · have hk (p : C(ℝ≥0, ℂ)) : sourceApproachKernel ε r ((0:ℂ) - (U' τ : ℂ), p) = 0 :=
+      sourceApproachKernel_zero hε.le hr.le (by simp)
+    simp only [hk, integral_zero, spatialGreenWeight, ite_eq_right ht, he,
+      greenDensity, ite_eq_right hz, mul_zero, le_refl]
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+local instance windowBoundsIntegralPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℂ) := borel _
+local instance windowBoundsIntegralPathBorelSpace : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩
+local instance windowBoundsDriverPathMeasurableSpace : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+local instance windowBoundsDriverPathBorelSpace : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+
+lemma source_finite_approach_setIntegral_bound
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {γ : Ω → ℝ≥0 → ℂ}
+    (a : RegularSLEPresentation P κ γ) (z : ℂ) (ε r : ℝ)
+    (hε : 0 < ε) (hr : 0 < r) (he : ε ≤ r / 2) (hy : r ≤ z.im)
+    (h8 : 8 * ε ≤ r) (hsmall : Real.exp 1 * (512 * Real.pi) * ε ≤ r)
+    (σ : a.samples → ℝ≥0)
+    (hσ : IsStoppingTime a.filtration (fun ω => (σ ω : WithTop ℝ≥0)))
+    (A : Set a.samples) (hA : MeasurableSet[hσ.measurableSpace] A)
+    (hAσ : ∀ ω ∈ A, spatialHitTime (γ ω) z r = (σ ω : WithTop ℝ≥0)) :
+    (∫ ω in A, (if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1)
+      ∂P.comap Subtype.val) ≤
+    ∫ ω in A, sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ
+      (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r ∂P.comap Subtype.val := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  let gp := sourceTracePath (fun ω : a.samples => γ ω) (fun ω => (a.witness ω).1)
+  have hgm : Measurable gp := source_path_measurable gp (fun t =>
+    (a.trace_adapted t).mono (a.filtration.le t) le_rfl)
+  let ν := μ.map gp
+  let : IsProbabilityMeasure ν := inferInstance
+  obtain ⟨Φ, hΦm, _, hΦ⟩ := sourceStoppedTraceLaw_forward κ hκ hκ8
+    a.B a.brownian a.measurable_B a.continuous_B (fun ω : a.samples => γ ω)
+    a.G a.F a.witness
+  obtain ⟨hlaw, hfwd⟩ := hΦ σ hσ
+  let rp := sourceRestartDriverPath a.B a.continuous_B σ
+  let X (ω : a.samples) := Φ (rp ω)
+  have hrm : Measurable rp := source_path_measurable rp (fun t =>
+    (measurable_pi_apply t).comp (brownianRestart_measurable a.measurable_B a.continuous_B
+      (stopping_time_measurable_nnreal hσ)))
+  have hXm : Measurable X := hΦm.comp hrm
+  let Y (ω : a.samples) : ℂ :=
+    (if z ∈ survivingDomain (γ ω) (σ ω) then a.G ω (σ ω) z else 0) -
+      (Real.sqrt κ * a.B (σ ω) ω : ℝ)
+  have hYm : Measurable[hσ.measurableSpace] Y := by
+    have hGm := measurable_stoppedValue
+      (source_killedChart_progressive a.measurable_B a.continuous_B κ
+        (fun ω : a.samples => γ ω) a.G a.F a.witness z) hσ
+    have hBm := measurable_stoppedValue
+      ((Filtration.stronglyAdapted_natural (fun t => (a.measurable_B t).stronglyMeasurable)).isStronglyProgressive_of_continuous a.continuous_B) hσ
+    exact hGm.sub (Complex.measurable_ofReal.comp (measurable_const.mul hBm))
+  let H (ω : a.samples) := sourceApproachKernel ε r (Y ω, X ω)
+  let K (ω : a.samples) := ∫ p, sourceApproachKernel ε r (Y ω, p) ∂ν
+  have hHmeas : Measurable H := (sourceApproachKernel_measurable ε r).comp
+    ((hYm.mono hσ.measurableSpace_le le_rfl).prodMk hXm)
+  have hHi : Integrable H μ :=
+    (integrable_const (1:ℝ)).mono' hHmeas.aestronglyMeasurable
+      (Eventually.of_forall (fun ω => sourceApproachKernel_norm_le ε r (Y ω, X ω)))
+  have hcond : μ[H | hσ.measurableSpace] =ᵐ[μ] K :=
+    source_parameter_condExp hσ.measurableSpace_le hXm hYm hlaw
+      (sourceApproachKernel ε r) (sourceApproachKernel_measurable ε r)
+      (sourceApproachKernel_norm_le ε r)
+  have hKi : Integrable K μ := integrable_condExp.congr hcond
+  let W (ω : a.samples) := sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ
+    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r
+  have hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
+  have hWm : Measurable W := measurable_const.mul
+    ((source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
+      (fun ω : a.samples => γ ω) a.G a.F a.witness z r hτ).mono hτ.measurableSpace_le le_rfl)
+  have hWi : Integrable W μ := by
+    apply Integrable.of_bound hWm.aestronglyMeasurable
+      (sourceApproachConstant κ * ε ^ codimension κ * r ^ (-codimension κ))
+    apply Eventually.of_forall
+    intro ω
+    have hb := (a.witness ω).spatialGreenWeight_bound
+      (continuous_const.mul (a.continuous_B ω)) hκ hκ8 hr z hy
+    have hc : 0 ≤ sourceApproachConstant κ * ε ^ codimension κ :=
+      mul_nonneg (sourceApproachConstant_pos κ hκ hκ8).le (Real.rpow_nonneg hε.le _)
+    rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hc hb.1)]
+    exact mul_le_mul_of_nonneg_left hb.2 hc
+  have hfi : Integrable (fun ω : a.samples => if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1) μ := by
+    apply Integrable.of_bound (source_liveIndicator_measurable a z ε (by linarith)).aestronglyMeasurable 1
+    apply Eventually.of_forall
+    intro ω
+    split_ifs <;> norm_num
+  have hAm : MeasurableSet A := hσ.measurableSpace_le A hA
+  calc
+    _ ≤ ∫ ω in A, H ω ∂μ := by
+      apply integral_mono_ae hfi.integrableOn hHi.integrableOn
+      filter_upwards [ae_restrict_of_ae hfwd, ae_restrict_mem hAm] with ω hf hωA
+      exact source_restarted_kernel_dominates (a.witness ω) z r ε hr hε he hy
+        (σ ω) (hAσ ω hωA) (X ω) hf
+    _ = ∫ ω in A, K ω ∂μ := by
+      have : IsFiniteMeasure (μ.trim hσ.measurableSpace_le) := isFiniteMeasure_trim _
+      rw [← integral_congr_ae (ae_restrict_of_ae hcond)]
+      exact (setIntegral_condExp hσ.measurableSpace_le hHi hA).symm
+    _ ≤ ∫ ω in A, W ω ∂μ := by
+      apply integral_mono_ae hKi.integrableOn hWi.integrableOn
+      filter_upwards [ae_restrict_mem hAm] with ω hωA
+      exact source_stopped_kernel_mean_bound a.brownian κ hκ hκ8
+        (fun ω : a.samples => γ ω) a.G a.F a.witness hgm
+        (continuous_const.mul (a.continuous_B ω)) (a.witness ω)
+        z ε r hε hr h8 hsmall (σ ω) (hAσ ω hωA)
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal
+namespace SLEExactGauge
+
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+ 
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+universe u_source_approach
+ 
+
+def SourceConditionalApproachTarget : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 → ∃ C : ℝ, 0 < C ∧ ∃ q : ℝ, 0 < q ∧ q < 1 ∧
+  ∀ (Ω : Type u_source_approach) (_ : MeasurableSpace Ω) (P : Measure Ω),
+  IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (z : ℂ) (ε r : ℝ), 0 < ε → ε ≤ q * r → r ≤ z.im →
+  ∃ hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r),
+    let μ := P.comap (Subtype.val : a.samples → Ω)
+    μ[(fun ω : a.samples => if liveSpatialHitTime (γ ω) z ε = ⊤ then (0 : ℝ) else 1)
+      | hτ.measurableSpace] ≤ᵐ[μ]
+      (fun ω => C * ε ^ codimension κ * spatialGreenWeight κ
+        (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r)
+
+lemma source_approach_small_scale (ε r : ℝ) (hε : 0 < ε)
+    (h : ε ≤ min (1/16) (1/(Real.exp 1 * (512 * Real.pi))) * r) :
+    0 < r ∧ ε ≤ r/2 ∧ 8*ε ≤ r ∧ Real.exp 1 * (512 * Real.pi) * ε ≤ r := by
+  have hA : 0 < Real.exp 1 * (512 * Real.pi) := by positivity
+  have hq : 0 < min (1/16:ℝ) (1/(Real.exp 1 * (512 * Real.pi))) := by positivity
+  have hr : 0 < r := by
+    by_contra! hn
+    have hh := mul_nonpos_of_nonneg_of_nonpos hq.le hn
+    linarith
+  have h16 : ε ≤ r/16 := by
+    apply h.trans
+    have hh := mul_le_mul_of_nonneg_right
+      (min_le_left (1/16:ℝ) (1/(Real.exp 1 * (512 * Real.pi)))) hr.le
+    nlinarith
+  refine ⟨hr, by linarith, by linarith, ?_⟩
+  have hqA := (le_div_iff₀ hA).mp
+    (min_le_right (1/16:ℝ) (1/(Real.exp 1 * (512 * Real.pi))))
+  have hh := mul_le_mul_of_nonneg_right hqA hr.le
+  have hhh := mul_le_mul_of_nonneg_left h hA.le
+  nlinarith only [hh, hhh]
+
+ 
+
+theorem sourceConditionalApproach_proved : SourceConditionalApproachTarget := by
+  intro κ hκ hκ8
+  refine ⟨sourceApproachConstant κ, sourceApproachConstant_pos κ hκ hκ8,
+    min (1/16) (1/(Real.exp 1 * (512 * Real.pi))), by positivity, ?_, ?_⟩
+  · exact (min_le_left _ _).trans_lt (by norm_num)
+  intro Ω mΩ P hP γ a z ε r hε hεq hy
+  let : IsProbabilityMeasure P := hP
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  obtain ⟨hr, he, h8, hsmall⟩ := source_approach_small_scale ε r hε hεq
+  let τ (ω : a.samples) := spatialHitTime (γ ω) z r
+  have hτ : IsStoppingTime a.filtration τ :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
+  refine ⟨hτ, ?_⟩
+  let f (ω : a.samples) : ℝ := if liveSpatialHitTime (γ ω) z ε = ⊤ then 0 else 1
+  let W (ω : a.samples) := spatialGreenWeight κ
+    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z r
+  let g (ω : a.samples) := sourceApproachConstant κ * ε ^ codimension κ * W ω
+  have hf : Integrable f μ := by
+    apply Integrable.of_bound (source_liveIndicator_measurable a z ε (by linarith)).aestronglyMeasurable 1
+    apply Eventually.of_forall
+    intro ω
+    dsimp [f]
+    split_ifs <;> norm_num
+  have hWm : Measurable[hτ.measurableSpace] W :=
+    source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
+      (fun ω : a.samples => γ ω) a.G a.F a.witness z r hτ
+  have hgm : Measurable[hτ.measurableSpace] g := measurable_const.mul hWm
+  have hg : Integrable g μ := by
+    apply Integrable.of_bound (hgm.mono hτ.measurableSpace_le le_rfl).aestronglyMeasurable
+      (sourceApproachConstant κ * ε ^ codimension κ * r ^ (-codimension κ))
+    apply Eventually.of_forall
+    intro ω
+    have hb := (a.witness ω).spatialGreenWeight_bound
+      (continuous_const.mul (a.continuous_B ω)) hκ hκ8 hr z hy
+    have hc : 0 ≤ sourceApproachConstant κ * ε ^ codimension κ :=
+      mul_nonneg (sourceApproachConstant_pos κ hκ hκ8).le (Real.rpow_nonneg hε.le _)
+    rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg hc hb.1)]
+    exact mul_le_mul_of_nonneg_left hb.2 hc
+  change μ[f | hτ.measurableSpace] ≤ᵐ[μ] g
+  apply source_condExp_le_of_setIntegral_le hτ.measurableSpace_le hf hg hgm.stronglyMeasurable
+  apply source_unbounded_setIntegral_le a.filtration hτ f g hf hg
+  · intro ω ht
+    constructor
+    · exact source_liveIndicator_zero_of_coarse_top (a.witness ω) z r ε (by linarith) hy ht
+    · change sourceApproachConstant κ * ε ^ codimension κ * spatialGreenWeight κ _ _ _ z r = 0
+      change spatialHitTime (γ ω) z r = ⊤ at ht
+      simp only [spatialGreenWeight, ite_eq_left ht, mul_zero]
+  · intro n A hA hAτ
+    exact source_finite_approach_setIntegral_bound hκ hκ8 a z ε r hε hr he hy h8 hsmall
+      (sourceStopTruncate τ n) (sourceStopTruncate_stopping a.filtration hτ n) A hA hAτ
+
+end
+end SLEExactGauge
+
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+universe u_source_spatial
+ 
+
+end
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/WindowGeometry.lean b/lean/OAI/Probability/SLE/WindowGeometry.lean
index a2e1596646696f94c23edb09003aa95a60670e46..3b1cd09d83716372fb5002ecb17b5f8d8b9553f6 100644
--- a/lean/OAI/Probability/SLE/WindowGeometry.lean
+++ b/lean/OAI/Probability/SLE/WindowGeometry.lean
@@ -1,684 +1,684 @@
-import Mathlib
-import OAI.Probability.SLE.ChartMeasurability
-import OAI.Probability.SLE.SpatialMeans
-import OAI.Probability.SLE.StoppedMeasures
-
-namespace OAI
-
-/-! Window Geometry for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-def sourceSpatialWindowSet (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : Set ℝ≥0 :=
-  {t | liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0) ∧
-    (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t)}
-noncomputable def sourceSpatialWindowTime (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : WithTop ℝ≥0 :=
-  if (sourceSpatialWindowSet γ z r ε R).Nonempty then
-    ↑(sInf (sourceSpatialWindowSet γ z r ε R)) else ⊤
-noncomputable def sourceSpatialWindowH (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
-  if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
-    z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
-    dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z ≤ ε then 1 else 0
-noncomputable def sourceSpatialWindowB (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
-  if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
-    z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
-    R ≤ dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z then 1 else 0
-
- 
-
-def SourceSpatialWindowGeometryTarget : Prop :=
-  ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
-  Continuous U → IsCapacityTwoWitness U γ G F →
-  ∀ (κ : ℝ), 0 < κ → κ < 8 → ∀ (z : ℂ) (ε r R : ℝ),
-  0 < ε → ε < r → r < R → r ≤ z.im →
-  let σ := liveSpatialHitTime γ z r
-  let τ := sourceSpatialWindowTime γ z r ε R
-  let H := sourceSpatialWindowH γ z r ε R
-  let B := sourceSpatialWindowB γ z r ε R
-  let W := spatialGreenWeight κ U γ G z ε
-  σ ≤ τ ∧
-  (∀ t : ℝ≥0, σ ≤ (t : WithTop ℝ≥0) → (t : WithTop ℝ≥0) < τ →
-    z ∈ survivingDomain γ t ∧ ε < dist (γ t) z ∧ dist (γ t) z < R) ∧
-  (τ = ⊤ → H = 0 ∧ B = 0) ∧
-  (σ = ⊤ → τ = ⊤) ∧
-  0 ≤ H ∧ H ≤ 1 ∧ 0 ≤ B ∧ B ≤ 1 ∧
-  H ≤ (if liveSpatialHitTime γ z ε = ⊤ then (0 : ℝ) else 1) ∧
-  (B = 1 → ∃ t : ℝ≥0, σ ≤ (t : WithTop ℝ≥0) ∧
-    z ∈ survivingDomain γ t ∧ R ≤ dist (γ t) z ∧
-    (t : WithTop ℝ≥0) < liveSpatialHitTime γ z ε) ∧
-  (if liveSpatialHitTime γ z ε = ⊤ then (0 : ℝ) else 1) ≤ H+B ∧
-  W ≤ W*H + ε^(-codimension κ)*B
-
-lemma source_liveHit_mem {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) (t : ℝ≥0)
-    (ht : liveSpatialHitTime γ z r = t) :
-    z ∈ survivingDomain γ t ∧ dist (γ t) z = r := by
-  rw [liveSpatialHitTime_eq hc z r hs] at ht
-  split_ifs at ht with hn hl
-  · exact (WithTop.coe_ne_top ht.symm).elim
-  · constructor
-    · simpa only [ht, WithTop.untopD_coe] using hl
-    · exact spatialHitTime_mem hc z r t ht
-  · exact (WithTop.coe_ne_top ht.symm).elim
-
-lemma source_liveHit_le (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (t : ℝ≥0)
-    (hl : z ∈ survivingDomain γ t) (hd : dist (γ t) z ≤ r) :
-    liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0) := by
-  unfold liveSpatialHitTime
-  rw [ite_eq_left ⟨t, hl, hd⟩]
-  exact WithTop.coe_le_coe.mpr (csInf_le ⟨0, fun _ _ => bot_le⟩ ⟨hl, hd⟩)
-
-lemma source_liveHit_antitone {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) {ε r : ℝ} (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    liveSpatialHitTime γ z r ≤ liveSpatialHitTime γ z ε := by
-  by_cases hn : liveSpatialHitTime γ z ε = ⊤
-  · rw [hn]; exact le_top
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-  have hmem := source_liveHit_mem hc z ε (her.trans hs) t ht.symm
-  rw [← ht]
-  exact source_liveHit_le γ z r t hmem.1 (hmem.2.le.trans her)
-
-lemma source_windowSet_closed {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) : IsClosed (sourceSpatialWindowSet γ z r ε R) := by
-  have hl : IsOpen {t : ℝ≥0 | z ∈ survivingDomain γ t} := by
-    let gp : C(ℝ≥0,ℂ) := ⟨γ,hc⟩
-    exact source_survival_open.preimage
-      ((continuous_const.prodMk continuous_id).prodMk continuous_const :
-        Continuous (fun t : ℝ≥0 => ((gp,t),z)))
-  exact (isClosed_le continuous_const (WithTop.continuous_coe : Continuous ((↑) : ℝ≥0 → WithTop ℝ≥0))).inter
-    ((isClosed_le (hc.dist continuous_const) continuous_const).union
-      ((isClosed_le continuous_const (hc.dist continuous_const)).union hl.isClosed_compl))
-
-lemma source_window_le_of_mem (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ)
-    (t : ℝ≥0) (ht : t ∈ sourceSpatialWindowSet γ z r ε R) :
-    sourceSpatialWindowTime γ z r ε R ≤ (t : WithTop ℝ≥0) := by
-  unfold sourceSpatialWindowTime
-  rw [ite_eq_left ⟨t,ht⟩]
-  exact WithTop.coe_le_coe.mpr (csInf_le ⟨0,fun _ _ => bot_le⟩ ht)
-
-lemma source_window_mem {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (t : ℝ≥0)
-    (ht : sourceSpatialWindowTime γ z r ε R = t) :
-    t ∈ sourceSpatialWindowSet γ z r ε R := by
-  unfold sourceSpatialWindowTime at ht
-  split_ifs at ht with hn
-  · have he := WithTop.coe_injective ht
-    rw [← he]
-    exact (source_windowSet_closed hc z r ε R).csInf_mem hn ⟨0,fun _ _ => bot_le⟩
-  · exact (WithTop.coe_ne_top ht.symm).elim
-
-lemma source_window_start_le {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) :
-    liveSpatialHitTime γ z r ≤ sourceSpatialWindowTime γ z r ε R := by
-  by_cases hn : sourceSpatialWindowTime γ z r ε R = ⊤
-  · rw [hn]; exact le_top
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-  rw [← ht]
-  exact (source_window_mem hc z r ε R t ht.symm).1
-
-lemma source_before_window (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ)
-    (t : ℝ≥0) (hs : liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0))
-    (ht : (t : WithTop ℝ≥0) < sourceSpatialWindowTime γ z r ε R) :
-    z ∈ survivingDomain γ t ∧ ε < dist (γ t) z ∧ dist (γ t) z < R := by
-  have hh : ¬ (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t) := by
-    intro hh
-    exact not_le_of_gt ht (source_window_le_of_mem γ z r ε R t ⟨hs,hh⟩)
-  push Not at hh
-  exact ⟨hh.2.2,hh.1,hh.2.1⟩
-
-lemma source_window_le_fine {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    sourceSpatialWindowTime γ z r ε R ≤ liveSpatialHitTime γ z ε := by
-  by_cases hn : liveSpatialHitTime γ z ε = ⊤
-  · rw [hn]; exact le_top
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-  have hm := source_liveHit_mem hc z ε (her.trans hs) t ht.symm
-  rw [← ht]
-  exact source_window_le_of_mem γ z r ε R t
-    ⟨by simpa only [← ht] using source_liveHit_antitone hc z her hs, Or.inl hm.2.le⟩
-
-lemma source_windowH_iff {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    sourceSpatialWindowH γ z r ε R = 1 ↔
-      sourceSpatialWindowTime γ z r ε R = liveSpatialHitTime γ z ε ∧
-      liveSpatialHitTime γ z ε ≠ ⊤ := by
-  constructor
-  · intro hH
-    have hh : sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
-        z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
-        dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z ≤ ε := by
-      simpa only [sourceSpatialWindowH, ite_eq_left_iff, zero_ne_one, imp_false, not_not] using hH
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hh.1
-    have hm : z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ ε := by
-      simpa only [← ht, WithTop.untopD_coe] using hh.2
-    have hf := source_liveHit_le γ z ε t hm.1 hm.2
-    rw [ht] at hf
-    have he := le_antisymm (source_window_le_fine hc z r ε R her hs) hf
-    exact ⟨he, he ▸ hh.1⟩
-  · rintro ⟨he,hn⟩
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-    have hm := source_liveHit_mem hc z ε (her.trans hs) t ht.symm
-    simp only [sourceSpatialWindowH, he, ← ht, WithTop.coe_ne_top, ne_eq, not_false_eq_true,
-      WithTop.untopD_coe, hm.1, hm.2.le, and_self, ite_true]
-
-lemma source_windowB_before {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hrR : r < R)
-    (hs : r ≤ dist (γ 0) z) (hB : sourceSpatialWindowB γ z r ε R = 1) :
-    ∃ t : ℝ≥0, liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0) ∧
-      z ∈ survivingDomain γ t ∧ R ≤ dist (γ t) z ∧
-      (t : WithTop ℝ≥0) < liveSpatialHitTime γ z ε := by
-  have hh : sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
-        z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
-        R ≤ dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z := by
-    simpa only [sourceSpatialWindowB, ite_eq_left_iff, zero_ne_one, imp_false, not_not] using hB
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hh.1
-  have hm : z ∈ survivingDomain γ t ∧ R ≤ dist (γ t) z := by
-    simpa only [← ht, WithTop.untopD_coe] using hh.2
-  refine ⟨t,(source_window_mem hc z r ε R t ht.symm).1,hm.1,hm.2,?_⟩
-  apply lt_of_le_of_ne (by simpa only [← ht] using source_window_le_fine hc z r ε R her hs)
-  intro he
-  have hf := (source_liveHit_mem hc z ε (her.trans hs) t he.symm).2
-  linarith
-
-lemma source_window_missed {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    (if liveSpatialHitTime γ z ε = ⊤ then (0 : ℝ) else 1) ≤
-      sourceSpatialWindowH γ z r ε R + sourceSpatialWindowB γ z r ε R := by
-  by_cases hn : liveSpatialHitTime γ z ε = ⊤
-  · simp only [ite_eq_left hn]
-    unfold sourceSpatialWindowH sourceSpatialWindowB
-    split_ifs <;> norm_num
-  rw [ite_eq_right hn]
-  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-  have hle := source_window_le_fine hc z r ε R her hs
-  have hwn : sourceSpatialWindowTime γ z r ε R ≠ ⊤ :=
-    ne_top_of_le_ne_top hn hle
-  obtain ⟨s,hs'⟩ := WithTop.ne_top_iff_exists.mp hwn
-  have hst : s ≤ t := WithTop.coe_le_coe.mp (by simpa only [← ht,← hs'] using hle)
-  have hl : z ∈ survivingDomain γ s := survivingDomain_antitone γ hst
-    (source_liveHit_mem hc z ε (her.trans hs) t ht.symm).1
-  have hm := (source_window_mem hc z r ε R s hs'.symm).2
-  rcases hm with hd | hd | hd
-  · have hH : sourceSpatialWindowH γ z r ε R = 1 := by
-      simp only [sourceSpatialWindowH, ← hs', WithTop.coe_ne_top, ne_eq, not_false_eq_true,
-        WithTop.untopD_coe, hl, hd, and_self, ite_true]
-    rw [hH]; unfold sourceSpatialWindowB; split_ifs <;> norm_num
-  · have hB : sourceSpatialWindowB γ z r ε R = 1 := by
-      simp only [sourceSpatialWindowB, ← hs', WithTop.coe_ne_top, ne_eq, not_false_eq_true,
-        WithTop.untopD_coe, hl, hd, and_self, ite_true]
-    rw [hB]; unfold sourceSpatialWindowH; split_ifs <;> norm_num
-  · exact (hd hl).elim
-
-theorem sourceSpatialWindowGeometry_proved : SourceSpatialWindowGeometryTarget := by
-  intro U γ G F hU hw κ hκ hκ8 z ε r R he her hrR hry
-  have hs : r ≤ dist (γ 0) z := by
-    rw [hw.2.1, dist_zero_left]
-    exact hry.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  dsimp only
-  have hH0 : 0 ≤ sourceSpatialWindowH γ z r ε R := by
-    unfold sourceSpatialWindowH; split_ifs <;> norm_num
-  have hH1 : sourceSpatialWindowH γ z r ε R ≤ 1 := by
-    unfold sourceSpatialWindowH; split_ifs <;> norm_num
-  have hB0 : 0 ≤ sourceSpatialWindowB γ z r ε R := by
-    unfold sourceSpatialWindowB; split_ifs <;> norm_num
-  have hB1 : sourceSpatialWindowB γ z r ε R ≤ 1 := by
-    unfold sourceSpatialWindowB; split_ifs <;> norm_num
-  refine ⟨source_window_start_le hw.1 z r ε R,
-    source_before_window γ z r ε R, ?_, ?_, hH0,hH1,hB0,hB1,?_,
-    source_windowB_before hw.1 z r ε R her.le hrR hs,
-    source_window_missed hw.1 z r ε R her.le hs, ?_⟩
-  · intro ht
-    simp only [sourceSpatialWindowH, sourceSpatialWindowB, ht, ne_eq, not_true_eq_false,
-      false_and, ite_false, and_self]
-  · intro ht
-    exact top_le_iff.mp (by simpa only [ht] using source_window_start_le hw.1 z r ε R)
-  · by_cases hn : liveSpatialHitTime γ z ε = ⊤
-    · rw [ite_eq_left hn]
-      have hh : sourceSpatialWindowH γ z r ε R ≠ 1 := by
-        intro hh
-        exact ((source_windowH_iff hw.1 z r ε R her.le hs).mp hh).2 hn
-      unfold sourceSpatialWindowH at hh ⊢
-      split_ifs at hh ⊢ <;> norm_num at *
-    · rw [ite_eq_right hn]; exact hH1
-  · have hW := hw.spatialGreenWeight_bound hU hκ hκ8 he z (her.le.trans hry)
-    have hR : 0 ≤ ε ^ (-codimension κ) := Real.rpow_nonneg he.le _
-    by_cases hh : sourceSpatialWindowH γ z r ε R = 1
-    · rw [hh,mul_one]; exact le_add_of_nonneg_right (mul_nonneg hR hB0)
-    have hh0 : sourceSpatialWindowH γ z r ε R = 0 := by
-      unfold sourceSpatialWindowH at hh ⊢
-      split_ifs at hh ⊢ <;> norm_num at *
-    rw [hh0,mul_zero,zero_add]
-    by_cases hn : liveSpatialHitTime γ z ε = ⊤
-    · rw [spatialGreenWeight_eq_live hw.1 κ U G z ε (her.le.trans hs), ite_eq_left hn]
-      exact mul_nonneg hR hB0
-    · have hm := source_window_missed hw.1 z r ε R her.le hs
-      rw [ite_eq_right hn,hh0,zero_add] at hm
-      have hb := le_antisymm hB1 hm
-      rw [hb,mul_one]
-      exact hW.2
-
-end
-end SLEExactGauge
-
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-local instance windowGeometryPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
-local instance windowGeometryPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
- 
-noncomputable def sourceCollarStart (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) : WithTop ℝ≥0 :=
-  if z ∈ D then liveSpatialHitTime γ z r else ⊤
-noncomputable def sourceCollarFinish (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : WithTop ℝ≥0 :=
-  if z ∈ D then sourceSpatialWindowTime γ z r ε R else ⊤
-noncomputable def sourceCollarH (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
-  if z ∈ D then sourceSpatialWindowH γ z r ε R else 0
-noncomputable def sourceCollarB (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
-  if z ∈ D then sourceSpatialWindowB γ z r ε R else 0
- 
-
-def SourceSpatialWindowMeasurableTarget : Prop :=
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω) (κ : ℝ),
-  0 < κ → κ < 8 → ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (D : Set ℂ), MeasurableSet D → ∀ ε r R : ℝ,
-  0 < ε → ε < r → r < R → (∀ z ∈ D, r ≤ z.im) →
-  let σ := fun z (ω : a.samples) => sourceCollarStart D (γ ω) z r
-  let τ := fun z (ω : a.samples) => sourceCollarFinish D (γ ω) z r ε R
-  let H := fun z (ω : a.samples) => sourceCollarH D (γ ω) z r ε R
-  let B := fun z (ω : a.samples) => sourceCollarB D (γ ω) z r ε R
-  let V := fun z (ω : a.samples) => spatialGreenWeight κ
-    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z ε * H z ω
-  (∃ (hσ : ∀ z, IsStoppingTime a.filtration (σ z))
-    (hτ : ∀ z, IsStoppingTime a.filtration (τ z)),
-    (∀ z, Measurable[(hτ z).measurableSpace] (H z) ∧
-      Measurable[(hτ z).measurableSpace] (B z) ∧
-      Measurable[(hτ z).measurableSpace] (V z)) ∧
-    (∀ z, Measurable[(hσ z).measurableSpace] (fun ω =>
-      if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-        (γ ω) (a.G ω) z r else 0))) ∧
-  Measurable (fun p : ℂ × a.samples => σ p.1 p.2) ∧
-  Measurable (fun p : ℂ × a.samples => τ p.1 p.2) ∧
-  Measurable (fun p : ℂ × a.samples => H p.1 p.2) ∧
-  Measurable (fun p : ℂ × a.samples => B p.1 p.2) ∧
-  Measurable (fun p : ℂ × a.samples => V p.1 p.2)
-
-lemma source_window_le_iff {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (T : ℝ≥0) :
-    sourceSpatialWindowTime γ z r ε R ≤ (T : WithTop ℝ≥0) ↔
-      ∃ t ∈ Icc (0 : ℝ≥0) T, t ∈ sourceSpatialWindowSet γ z r ε R := by
-  constructor
-  · intro hh
-    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp (ne_top_of_le_ne_top WithTop.coe_ne_top hh)
-    exact ⟨t,⟨bot_le, WithTop.coe_le_coe.mp (by simpa only [← ht] using hh)⟩,
-      source_window_mem hc z r ε R t ht.symm⟩
-  · rintro ⟨t,ht,hmem⟩
-    exact (source_window_le_of_mem γ z r ε R t hmem).trans (WithTop.coe_le_coe.mpr ht.2)
-
-lemma source_liveTime_eq_killed {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) :
-    liveSpatialHitTime γ z r =
-      if z ∈ survivingDomain γ (spatialHitTime γ z r).untopA
-      then spatialHitTime γ z r else ⊤ := by
-  rw [liveSpatialHitTime_eq hc z r hs]
-  by_cases hn : spatialHitTime γ z r = ⊤
-  · simp only [hn, ite_self]
-  · obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
-    rw [ite_eq_right hn, ← ht]
-    rfl
-
-lemma source_live_stopped_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
-    {τ : a.samples → WithTop ℝ≥0} (hτ : IsStoppingTime a.filtration τ) :
-    MeasurableSet[hτ.measurableSpace] {ω | z ∈ survivingDomain (γ ω) (τ ω).untopA} := by
-  have hm := measurable_stoppedValue
-    (source_killedChart_progressive a.measurable_B a.continuous_B κ
-      (fun ω : a.samples => γ ω) a.G a.F a.witness z) hτ
-  have he : {ω : a.samples | z ∈ survivingDomain (γ ω) (τ ω).untopA} =
-      {ω : a.samples | 0 < (stoppedValue (fun t (ω : a.samples) =>
-        if z ∈ survivingDomain (γ ω) t then a.G ω t z else 0) τ ω).im} := by
-    ext ω
-    change (z ∈ survivingDomain (γ ω) (τ ω).untopA) ↔
-      0 < (if z ∈ survivingDomain (γ ω) (τ ω).untopA then a.G ω (τ ω).untopA z else 0).im
-    by_cases hl : z ∈ survivingDomain (γ ω) (τ ω).untopA
-    · simp only [hl, true_iff]
-      exact (a.witness ω).2.2.2.2.2.1 _ hl
-    · simp [hl]
-  rw [he]
-  exact measurableSet_lt measurable_const (Complex.measurable_im.comp hm)
-
-lemma source_liveHit_stopping
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
-    (r : ℝ) (hy : r ≤ z.im) :
-    IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r) := by
-  have hσ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
-  have hA := source_live_stopped_measurable a z hσ
-  intro T
-  have he : {ω : a.samples | liveSpatialHitTime (γ ω) z r ≤ (T : WithTop ℝ≥0)} =
-      {ω : a.samples | z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z r).untopA} ∩
-      {ω : a.samples | spatialHitTime (γ ω) z r ≤ (T : WithTop ℝ≥0)} := by
-    ext ω
-    have hs : r ≤ dist (γ ω 0) z := by
-      rw [(a.witness ω).2.1, dist_zero_left]
-      exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-    simp only [Set.mem_ofPred_eq, Set.mem_inter_iff]
-    rw [source_liveTime_eq_killed (a.trace_continuous ω) z r hs]
-    by_cases hl : z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z r).untopA
-    · simp only [hl, true_and, ite_true]
-    · simp only [hl, false_and, iff_false, ite_false]
-      exact not_le_of_gt (WithTop.coe_lt_top _)
-  rw [he]
-  exact hA.2 T
-
- 
-lemma source_windowKernel_closed (T : ℝ≥0) (ε R : ℝ) :
-    IsClosed {p : (ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ | ∃ t ∈ Icc (0 : ℝ≥0) T,
-      p.1.1 ≤ t ∧ (dist (p.1.2 t) p.2 ≤ ε ∨ R ≤ dist (p.1.2 t) p.2 ∨
-        p.2 ∉ survivingDomain p.1.2 t)} := by
-  let A : Set (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) :=
-    {p | p.1.1.1 ≤ p.2.1 ∧ (dist (p.1.1.2 p.2.1) p.1.2 ≤ ε ∨
-      R ≤ dist (p.1.1.2 p.2.1) p.1.2 ∨ p.1.2 ∉ survivingDomain p.1.1.2 p.2.1)}
-  have ht : Continuous (fun p : (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) => p.2.1) :=
-    continuous_subtype_val.comp continuous_snd
-  have hv : Continuous (fun p : (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) =>
-      p.1.1.2 p.2.1) := continuous_eval.comp (continuous_fst.fst.snd.prodMk ht)
-  have hl : IsOpen {p : (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) |
-      p.1.2 ∈ survivingDomain p.1.1.2 p.2.1} := source_survival_open.preimage
-      ((continuous_fst.fst.snd.prodMk ht).prodMk continuous_fst.snd)
-  have hA : IsClosed A := (isClosed_le continuous_fst.fst.fst ht).inter
-    ((isClosed_le (hv.dist continuous_fst.snd) continuous_const).union
-      ((isClosed_le continuous_const (hv.dist continuous_fst.snd)).union hl.isClosed_compl))
-  have hp := isClosedMap_fst_of_compactSpace A hA
-  convert hp using 1
-  ext p
-  simp only [A, Set.mem_ofPred_eq, Set.mem_image, Prod.exists, Subtype.exists,
-    exists_and_right, exists_eq_right]
-  aesop
-
-lemma source_min_time_coe (σ : WithTop ℝ≥0) (T : ℝ≥0) :
-    (((min σ (T : WithTop ℝ≥0)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = min σ T := by
-  have hn : min σ (T : WithTop ℝ≥0) ≠ ⊤ := ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_right _ _)
-  obtain ⟨s,hs⟩ := WithTop.ne_top_iff_exists.mp hn
-  rw [← hs, WithTop.untopD_coe]
-
-lemma source_window_kernel_iff {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (T : ℝ≥0) :
-    sourceSpatialWindowTime γ z r ε R ≤ (T : WithTop ℝ≥0) ↔
-    liveSpatialHitTime γ z r ≤ (T : WithTop ℝ≥0) ∧
-    ∃ t ∈ Icc (0 : ℝ≥0) T, (min (liveSpatialHitTime γ z r) T).untopD 0 ≤ t ∧
-      (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t) := by
-  rw [source_window_le_iff hc z r ε R T]
-  constructor
-  · rintro ⟨t,ht,hs,he⟩
-    have hT := hs.trans (WithTop.coe_le_coe.mpr ht.2)
-    refine ⟨hT,t,ht,?_,he⟩
-    apply WithTop.coe_le_coe.mp
-    rw [source_min_time_coe, min_eq_left hT]
-    exact hs
-  · rintro ⟨hT,t,ht,hs,he⟩
-    refine ⟨t,ht,?_,he⟩
-    have hs' := WithTop.coe_le_coe.mpr hs
-    rw [source_min_time_coe, min_eq_left hT] at hs'
-    exact hs'
-
-def sourceCapPath (γ : C(ℝ≥0,ℂ)) (T : ℝ≥0) : C(ℝ≥0,ℂ) :=
-  ⟨fun t => γ (min t T), γ.continuous.comp (continuous_id.min continuous_const)⟩
-
-lemma source_cap_domain (γ : C(ℝ≥0,ℂ)) (T t : ℝ≥0) (ht : t ≤ T) :
-    survivingDomain (sourceCapPath γ T) t = survivingDomain γ t := by
-  have he : sourceCapPath γ T '' Icc 0 t = γ '' Icc 0 t := by
-    apply Set.image_congr
-    intro s hs
-    change γ (min s T) = γ s
-    rw [min_eq_left (hs.2.trans ht)]
-  simp only [survivingDomain, he]
-
-lemma source_window_kernel_cap (γ : C(ℝ≥0,ℂ)) (z : ℂ) (ε R : ℝ) (T b : ℝ≥0) :
-    (∃ t ∈ Icc (0 : ℝ≥0) T, b ≤ t ∧
-      (dist (sourceCapPath γ T t) z ≤ ε ∨ R ≤ dist (sourceCapPath γ T t) z ∨
-        z ∉ survivingDomain (sourceCapPath γ T) t)) ↔
-    (∃ t ∈ Icc (0 : ℝ≥0) T, b ≤ t ∧
-      (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t)) := by
-  apply exists_congr
-  intro t
-  apply and_congr_right
-  intro ht
-  rw [source_cap_domain γ T t ht.2]
-  have he : sourceCapPath γ T t = γ t := by dsimp [sourceCapPath]; rw [min_eq_left ht.2]
-  rw [he]
-
-lemma source_tracePath_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) :
-    Measurable (fun ω : a.samples => (⟨γ ω,a.trace_continuous ω⟩ : C(ℝ≥0,ℂ))) := by
-  apply source_path_measurable
-  intro t
-  exact (a.trace_adapted t).mono (a.filtration.le t) le_rfl
-
-lemma source_capPath_adapted
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (T : ℝ≥0) :
-    Measurable[a.filtration T] (fun ω : a.samples =>
-      sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T) := by
-  refine @source_path_measurable a.samples ℂ (a.filtration T) _ _ _ _ _ _ _ _ ?_
-  intro t
-  exact (a.trace_adapted (min t T)).mono (a.filtration.mono (min_le_right _ _)) le_rfl
-
-lemma source_window_stopping
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
-    (r ε R : ℝ) (hy : r ≤ z.im) :
-    IsStoppingTime a.filtration (fun ω : a.samples => sourceSpatialWindowTime (γ ω) z r ε R) := by
-  let σ := fun ω : a.samples => liveSpatialHitTime (γ ω) z r
-  have hσ : IsStoppingTime a.filtration σ := source_liveHit_stopping a z r hy
-  intro T
-  have hm : Measurable[a.filtration T] (fun ω : a.samples => (min (σ ω) T).untopD 0) :=
-    ((hσ.min_const T).measurable_of_le (fun _ => min_le_right _ _)).untopD 0
-  have hp : Measurable[a.filtration T] (fun ω : a.samples =>
-      (((min (σ ω) T).untopD 0, sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T),z)) :=
-    (hm.prodMk (source_capPath_adapted a T)).prodMk measurable_const
-  have hK := (source_windowKernel_closed T ε R).measurableSet.preimage hp
-  have he : {ω : a.samples | sourceSpatialWindowTime (γ ω) z r ε R ≤ (T : WithTop ℝ≥0)} =
-      {ω : a.samples | σ ω ≤ (T : WithTop ℝ≥0)} ∩
-      {ω : a.samples | ∃ t ∈ Icc (0 : ℝ≥0) T, (min (σ ω) T).untopD 0 ≤ t ∧
-        (dist (sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T t) z ≤ ε ∨
-         R ≤ dist (sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T t) z ∨
-         z ∉ survivingDomain (sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T) t)} := by
-    ext ω
-    simp only [Set.mem_ofPred_eq, Set.mem_inter_iff]
-    rw [source_window_kernel_iff (a.trace_continuous ω) z r ε R T,
-      source_window_kernel_cap]
-    rfl
-  rw [he]
-  exact (hσ T).inter hK
-
-lemma source_collarStart_stopping
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
-    IsStoppingTime a.filtration (fun ω : a.samples => sourceCollarStart D (γ ω) z r) := by
-  by_cases hz : z ∈ D
-  · simpa only [sourceCollarStart, ite_eq_left hz] using source_liveHit_stopping a z r (hy z hz)
-  · simpa only [sourceCollarStart, ite_eq_right hz] using (show IsStoppingTime a.filtration (fun _ => (⊤ : WithTop ℝ≥0)) from fun _ => MeasurableSet.const _)
-
-lemma source_collarFinish_stopping
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (r ε R : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
-    IsStoppingTime a.filtration (fun ω : a.samples => sourceCollarFinish D (γ ω) z r ε R) := by
-  by_cases hz : z ∈ D
-  · simpa only [sourceCollarFinish, ite_eq_left hz] using source_window_stopping a z r ε R (hy z hz)
-  · simpa only [sourceCollarFinish, ite_eq_right hz] using (show IsStoppingTime a.filtration (fun _ => (⊤ : WithTop ℝ≥0)) from fun _ => MeasurableSet.const _)
-
-lemma source_joint_survival_measurable
-    {X : Type*} [MeasurableSpace X] (g : X → C(ℝ≥0,ℂ)) (t : X → ℝ≥0) (z : X → ℂ)
-    (hg : Measurable g) (ht : Measurable t) (hz : Measurable z) :
-    MeasurableSet {x | z x ∈ survivingDomain (g x) (t x)} :=
-  source_survival_open.measurableSet.preimage ((hg.prodMk ht).prodMk hz)
-
-lemma source_collarStart_joint
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (hD : MeasurableSet D) (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) :
-    Measurable (fun p : ℂ × a.samples => sourceCollarStart D (γ p.2) p.1 r) := by
-  have hhit := spatialHitTime_measurable (fun ω : a.samples => γ ω) a.trace_continuous
-    (fun t => (a.trace_adapted t).mono (a.filtration.le t) le_rfl) r
-  have hl := source_joint_survival_measurable
-    (fun p : ℂ × a.samples => (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ)))
-    (fun p => (spatialHitTime (γ p.2) p.1 r).untopA) Prod.fst
-    ((source_tracePath_measurable a).comp measurable_snd)
-    (WithTop.measurable_untopA.comp hhit) measurable_fst
-  have hm : Measurable (fun p : ℂ × a.samples =>
-      if p.1 ∈ D then (if p.1 ∈ survivingDomain (γ p.2) (spatialHitTime (γ p.2) p.1 r).untopA
-      then spatialHitTime (γ p.2) p.1 r else ⊤) else ⊤) :=
-    Measurable.ite (hD.preimage measurable_fst)
-    (Measurable.ite hl hhit (measurable_const (a := (⊤ : WithTop ℝ≥0))))
-    (measurable_const (a := (⊤ : WithTop ℝ≥0)))
-  convert hm using 1
-  funext p
-  by_cases hz : p.1 ∈ D
-  · simp only [sourceCollarStart, ite_eq_left hz]
-    apply source_liveTime_eq_killed (a.trace_continuous p.2)
-    rw [(a.witness p.2).2.1, dist_zero_left]
-    exact (hy p.1 hz).trans ((le_abs_self p.1.im).trans (Complex.abs_im_le_norm _))
-  · simp only [sourceCollarStart, ite_eq_right hz]
-
-lemma source_collarFinish_joint
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
-    (hD : MeasurableSet D) (r ε R : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) :
-    Measurable (fun p : ℂ × a.samples => sourceCollarFinish D (γ p.2) p.1 r ε R) := by
-  have hσ := source_collarStart_joint a D hD r hy
-  apply measurable_of_Iic
-  intro b
-  by_cases hb : b = ⊤
-  · subst b
-    simp only [preimage, mem_Iic, le_top, ofPred_true]
-    exact MeasurableSet.univ
-  obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hb
-  subst b
-  have hp : Measurable (fun p : ℂ × a.samples =>
-      (((min (sourceCollarStart D (γ p.2) p.1 r) T).untopD 0,
-      (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ))),p.1)) :=
-    (((hσ.min measurable_const).untopD 0).prodMk
-      ((source_tracePath_measurable a).comp measurable_snd)).prodMk measurable_fst
-  have hK := (source_windowKernel_closed T ε R).measurableSet.preimage hp
-  have he : (fun p : ℂ × a.samples => sourceCollarFinish D (γ p.2) p.1 r ε R) ⁻¹' Iic (T : WithTop ℝ≥0) =
-      (Prod.fst ⁻¹' D) ∩ {p : ℂ × a.samples | sourceCollarStart D (γ p.2) p.1 r ≤ (T : WithTop ℝ≥0)} ∩
-      {p : ℂ × a.samples | ∃ t ∈ Icc (0 : ℝ≥0) T,
-      (min (sourceCollarStart D (γ p.2) p.1 r) T).untopD 0 ≤ t ∧
-      (dist (γ p.2 t) p.1 ≤ ε ∨ R ≤ dist (γ p.2 t) p.1 ∨ p.1 ∉ survivingDomain (γ p.2) t)} := by
-    ext p
-    simp only [mem_preimage, mem_Iic, mem_inter_iff, mem_ofPred_eq]
-    by_cases hz : p.1 ∈ D
-    · simp only [hz, true_and, sourceCollarStart, sourceCollarFinish]
-      exact source_window_kernel_iff (a.trace_continuous p.2) p.1 r ε R T
-    · simp only [hz, false_and, sourceCollarFinish, iff_false]
-      exact not_le_of_gt (WithTop.coe_lt_top _)
-  rw [he]
-  exact ((hD.preimage measurable_fst).inter (measurableSet_le hσ measurable_const)).inter hK
-
-lemma source_spatialWeight_stopping_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
-    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
-    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
-    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
-      (γ ω) (G ω) (F ω)) (z : ℂ) (r : ℝ)
-    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
-      (fun ω => spatialHitTime (γ ω) z r)) :
-    Measurable[hτ.measurableSpace] (fun ω => spatialGreenWeight κ
-      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z r) := by
-  have hmG := measurable_stoppedValue (source_killedGreen_progressive hm hc κ γ G F hw z) hτ
-  have he : MeasurableSet[hτ.measurableSpace] {ω | spatialHitTime (γ ω) z r = ⊤} :=
-    measurableSet_eq_fun hτ.measurable measurable_const
-  have hmo : Measurable[hτ.measurableSpace] (fun ω =>
-      if spatialHitTime (γ ω) z r = ⊤ then (0:ℝ) else
-        greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω)
-          (spatialHitTime (γ ω) z r).untopA z) :=
-    Measurable.ite he measurable_const hmG
-  convert hmo using 1
-  funext ω
-  by_cases ht : spatialHitTime (γ ω) z r = ⊤
-  · simp only [spatialGreenWeight, ite_eq_left ht]
-  · obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp ht
-    simp only [spatialGreenWeight, ite_eq_right ht]
-    rw [← hv]
-    rfl
-
-lemma source_windowH_untopA (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) :
-    sourceSpatialWindowH γ z r ε R =
-    if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
-      z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopA) ∧
-      dist (γ ((sourceSpatialWindowTime γ z r ε R).untopA)) z ≤ ε then 1 else 0 := by
-  by_cases ht : sourceSpatialWindowTime γ z r ε R = ⊤
-  · simp only [sourceSpatialWindowH, ht, ne_eq, not_true_eq_false, false_and, ite_false]
-  · obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp ht
-    unfold sourceSpatialWindowH
-    rw [← ht]
-    rfl
-
-lemma source_windowB_untopA (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) :
-    sourceSpatialWindowB γ z r ε R =
-    if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
-      z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopA) ∧
-      R ≤ dist (γ ((sourceSpatialWindowTime γ z r ε R).untopA)) z then 1 else 0 := by
-  by_cases ht : sourceSpatialWindowTime γ z r ε R = ⊤
-  · simp only [sourceSpatialWindowB, ht, ne_eq, not_true_eq_false, false_and, ite_false]
-  · obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp ht
-    unfold sourceSpatialWindowB
-    rw [← ht]
-    rfl
-
-lemma source_windowHB_stopped_measurable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
-    (r ε R : ℝ)
-    (hτ : IsStoppingTime a.filtration (fun ω : a.samples => sourceSpatialWindowTime (γ ω) z r ε R)) :
-    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R) ∧
-    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε R) := by
-  have htp := (show StronglyAdapted a.filtration (fun t (ω : a.samples) => γ ω t) from
-    fun t => (a.trace_adapted t).stronglyMeasurable).isStronglyProgressive_of_continuous a.trace_continuous
-  have hv := measurable_stoppedValue htp hτ
-  let : MeasurableSpace a.samples := hτ.measurableSpace
-  have hl := source_live_stopped_measurable a z hτ
-  have hn : MeasurableSet[hτ.measurableSpace] {ω : a.samples | sourceSpatialWindowTime (γ ω) z r ε R ≠ ⊤} :=
-    (measurableSet_eq_fun hτ.measurable measurable_const).compl
-  have hd : Measurable[hτ.measurableSpace] (fun ω : a.samples =>
-      dist (γ ω (sourceSpatialWindowTime (γ ω) z r ε R).untopA) z) := hv.dist measurable_const
-  have hH : Measurable[hτ.measurableSpace] (fun ω : a.samples =>
-      if sourceSpatialWindowTime (γ ω) z r ε R ≠ ⊤ ∧
-      z ∈ survivingDomain (γ ω) (sourceSpatialWindowTime (γ ω) z r ε R).untopA ∧
-      dist (γ ω (sourceSpatialWindowTime (γ ω) z r ε R).untopA) z ≤ ε then (1:ℝ) else 0) :=
-    Measurable.ite (hn.inter (hl.inter (measurableSet_le hd measurable_const)))
-    (measurable_const (a := (1:ℝ))) (measurable_const (a := (0:ℝ)))
-  have hB : Measurable[hτ.measurableSpace] (fun ω : a.samples =>
-      if sourceSpatialWindowTime (γ ω) z r ε R ≠ ⊤ ∧
-      z ∈ survivingDomain (γ ω) (sourceSpatialWindowTime (γ ω) z r ε R).untopA ∧
-      R ≤ dist (γ ω (sourceSpatialWindowTime (γ ω) z r ε R).untopA) z then (1:ℝ) else 0) :=
-    Measurable.ite (hn.inter (hl.inter (measurableSet_le measurable_const hd)))
-    (measurable_const (a := (1:ℝ))) (measurable_const (a := (0:ℝ)))
-  constructor
-  · convert hH using 1
-    funext ω
-    exact source_windowH_untopA (γ ω) z r ε R
-  · convert hB using 1
-    funext ω
-    exact source_windowB_untopA (γ ω) z r ε R
-
-end
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.ChartMeasurability
+import OAI.Probability.SLE.SpatialMeans
+import OAI.Probability.SLE.StoppedMeasures
+
+namespace OAI
+
+/-! Window Geometry for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+def sourceSpatialWindowSet (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : Set ℝ≥0 :=
+  {t | liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0) ∧
+    (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t)}
+noncomputable def sourceSpatialWindowTime (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : WithTop ℝ≥0 :=
+  if (sourceSpatialWindowSet γ z r ε R).Nonempty then
+    ↑(sInf (sourceSpatialWindowSet γ z r ε R)) else ⊤
+noncomputable def sourceSpatialWindowH (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
+  if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
+    z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
+    dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z ≤ ε then 1 else 0
+noncomputable def sourceSpatialWindowB (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
+  if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
+    z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
+    R ≤ dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z then 1 else 0
+
+ 
+
+def SourceSpatialWindowGeometryTarget : Prop :=
+  ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ),
+  Continuous U → IsCapacityTwoWitness U γ G F →
+  ∀ (κ : ℝ), 0 < κ → κ < 8 → ∀ (z : ℂ) (ε r R : ℝ),
+  0 < ε → ε < r → r < R → r ≤ z.im →
+  let σ := liveSpatialHitTime γ z r
+  let τ := sourceSpatialWindowTime γ z r ε R
+  let H := sourceSpatialWindowH γ z r ε R
+  let B := sourceSpatialWindowB γ z r ε R
+  let W := spatialGreenWeight κ U γ G z ε
+  σ ≤ τ ∧
+  (∀ t : ℝ≥0, σ ≤ (t : WithTop ℝ≥0) → (t : WithTop ℝ≥0) < τ →
+    z ∈ survivingDomain γ t ∧ ε < dist (γ t) z ∧ dist (γ t) z < R) ∧
+  (τ = ⊤ → H = 0 ∧ B = 0) ∧
+  (σ = ⊤ → τ = ⊤) ∧
+  0 ≤ H ∧ H ≤ 1 ∧ 0 ≤ B ∧ B ≤ 1 ∧
+  H ≤ (if liveSpatialHitTime γ z ε = ⊤ then (0 : ℝ) else 1) ∧
+  (B = 1 → ∃ t : ℝ≥0, σ ≤ (t : WithTop ℝ≥0) ∧
+    z ∈ survivingDomain γ t ∧ R ≤ dist (γ t) z ∧
+    (t : WithTop ℝ≥0) < liveSpatialHitTime γ z ε) ∧
+  (if liveSpatialHitTime γ z ε = ⊤ then (0 : ℝ) else 1) ≤ H+B ∧
+  W ≤ W*H + ε^(-codimension κ)*B
+
+lemma source_liveHit_mem {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) (t : ℝ≥0)
+    (ht : liveSpatialHitTime γ z r = t) :
+    z ∈ survivingDomain γ t ∧ dist (γ t) z = r := by
+  rw [liveSpatialHitTime_eq hc z r hs] at ht
+  split_ifs at ht with hn hl
+  · exact (WithTop.coe_ne_top ht.symm).elim
+  · constructor
+    · simpa only [ht, WithTop.untopD_coe] using hl
+    · exact spatialHitTime_mem hc z r t ht
+  · exact (WithTop.coe_ne_top ht.symm).elim
+
+lemma source_liveHit_le (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) (t : ℝ≥0)
+    (hl : z ∈ survivingDomain γ t) (hd : dist (γ t) z ≤ r) :
+    liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0) := by
+  unfold liveSpatialHitTime
+  rw [ite_eq_left ⟨t, hl, hd⟩]
+  exact WithTop.coe_le_coe.mpr (csInf_le ⟨0, fun _ _ => bot_le⟩ ⟨hl, hd⟩)
+
+lemma source_liveHit_antitone {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) {ε r : ℝ} (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    liveSpatialHitTime γ z r ≤ liveSpatialHitTime γ z ε := by
+  by_cases hn : liveSpatialHitTime γ z ε = ⊤
+  · rw [hn]; exact le_top
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+  have hmem := source_liveHit_mem hc z ε (her.trans hs) t ht.symm
+  rw [← ht]
+  exact source_liveHit_le γ z r t hmem.1 (hmem.2.le.trans her)
+
+lemma source_windowSet_closed {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) : IsClosed (sourceSpatialWindowSet γ z r ε R) := by
+  have hl : IsOpen {t : ℝ≥0 | z ∈ survivingDomain γ t} := by
+    let gp : C(ℝ≥0,ℂ) := ⟨γ,hc⟩
+    exact source_survival_open.preimage
+      ((continuous_const.prodMk continuous_id).prodMk continuous_const :
+        Continuous (fun t : ℝ≥0 => ((gp,t),z)))
+  exact (isClosed_le continuous_const (WithTop.continuous_coe : Continuous ((↑) : ℝ≥0 → WithTop ℝ≥0))).inter
+    ((isClosed_le (hc.dist continuous_const) continuous_const).union
+      ((isClosed_le continuous_const (hc.dist continuous_const)).union hl.isClosed_compl))
+
+lemma source_window_le_of_mem (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ)
+    (t : ℝ≥0) (ht : t ∈ sourceSpatialWindowSet γ z r ε R) :
+    sourceSpatialWindowTime γ z r ε R ≤ (t : WithTop ℝ≥0) := by
+  unfold sourceSpatialWindowTime
+  rw [ite_eq_left ⟨t,ht⟩]
+  exact WithTop.coe_le_coe.mpr (csInf_le ⟨0,fun _ _ => bot_le⟩ ht)
+
+lemma source_window_mem {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (t : ℝ≥0)
+    (ht : sourceSpatialWindowTime γ z r ε R = t) :
+    t ∈ sourceSpatialWindowSet γ z r ε R := by
+  unfold sourceSpatialWindowTime at ht
+  split_ifs at ht with hn
+  · have he := WithTop.coe_injective ht
+    rw [← he]
+    exact (source_windowSet_closed hc z r ε R).csInf_mem hn ⟨0,fun _ _ => bot_le⟩
+  · exact (WithTop.coe_ne_top ht.symm).elim
+
+lemma source_window_start_le {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) :
+    liveSpatialHitTime γ z r ≤ sourceSpatialWindowTime γ z r ε R := by
+  by_cases hn : sourceSpatialWindowTime γ z r ε R = ⊤
+  · rw [hn]; exact le_top
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+  rw [← ht]
+  exact (source_window_mem hc z r ε R t ht.symm).1
+
+lemma source_before_window (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ)
+    (t : ℝ≥0) (hs : liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0))
+    (ht : (t : WithTop ℝ≥0) < sourceSpatialWindowTime γ z r ε R) :
+    z ∈ survivingDomain γ t ∧ ε < dist (γ t) z ∧ dist (γ t) z < R := by
+  have hh : ¬ (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t) := by
+    intro hh
+    exact not_le_of_gt ht (source_window_le_of_mem γ z r ε R t ⟨hs,hh⟩)
+  push Not at hh
+  exact ⟨hh.2.2,hh.1,hh.2.1⟩
+
+lemma source_window_le_fine {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    sourceSpatialWindowTime γ z r ε R ≤ liveSpatialHitTime γ z ε := by
+  by_cases hn : liveSpatialHitTime γ z ε = ⊤
+  · rw [hn]; exact le_top
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+  have hm := source_liveHit_mem hc z ε (her.trans hs) t ht.symm
+  rw [← ht]
+  exact source_window_le_of_mem γ z r ε R t
+    ⟨by simpa only [← ht] using source_liveHit_antitone hc z her hs, Or.inl hm.2.le⟩
+
+lemma source_windowH_iff {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    sourceSpatialWindowH γ z r ε R = 1 ↔
+      sourceSpatialWindowTime γ z r ε R = liveSpatialHitTime γ z ε ∧
+      liveSpatialHitTime γ z ε ≠ ⊤ := by
+  constructor
+  · intro hH
+    have hh : sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
+        z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
+        dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z ≤ ε := by
+      simpa only [sourceSpatialWindowH, ite_eq_left_iff, zero_ne_one, imp_false, not_not] using hH
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hh.1
+    have hm : z ∈ survivingDomain γ t ∧ dist (γ t) z ≤ ε := by
+      simpa only [← ht, WithTop.untopD_coe] using hh.2
+    have hf := source_liveHit_le γ z ε t hm.1 hm.2
+    rw [ht] at hf
+    have he := le_antisymm (source_window_le_fine hc z r ε R her hs) hf
+    exact ⟨he, he ▸ hh.1⟩
+  · rintro ⟨he,hn⟩
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+    have hm := source_liveHit_mem hc z ε (her.trans hs) t ht.symm
+    simp only [sourceSpatialWindowH, he, ← ht, WithTop.coe_ne_top, ne_eq, not_false_eq_true,
+      WithTop.untopD_coe, hm.1, hm.2.le, and_self, ite_true]
+
+lemma source_windowB_before {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hrR : r < R)
+    (hs : r ≤ dist (γ 0) z) (hB : sourceSpatialWindowB γ z r ε R = 1) :
+    ∃ t : ℝ≥0, liveSpatialHitTime γ z r ≤ (t : WithTop ℝ≥0) ∧
+      z ∈ survivingDomain γ t ∧ R ≤ dist (γ t) z ∧
+      (t : WithTop ℝ≥0) < liveSpatialHitTime γ z ε := by
+  have hh : sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
+        z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopD 0) ∧
+        R ≤ dist (γ ((sourceSpatialWindowTime γ z r ε R).untopD 0)) z := by
+    simpa only [sourceSpatialWindowB, ite_eq_left_iff, zero_ne_one, imp_false, not_not] using hB
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hh.1
+  have hm : z ∈ survivingDomain γ t ∧ R ≤ dist (γ t) z := by
+    simpa only [← ht, WithTop.untopD_coe] using hh.2
+  refine ⟨t,(source_window_mem hc z r ε R t ht.symm).1,hm.1,hm.2,?_⟩
+  apply lt_of_le_of_ne (by simpa only [← ht] using source_window_le_fine hc z r ε R her hs)
+  intro he
+  have hf := (source_liveHit_mem hc z ε (her.trans hs) t he.symm).2
+  linarith
+
+lemma source_window_missed {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    (if liveSpatialHitTime γ z ε = ⊤ then (0 : ℝ) else 1) ≤
+      sourceSpatialWindowH γ z r ε R + sourceSpatialWindowB γ z r ε R := by
+  by_cases hn : liveSpatialHitTime γ z ε = ⊤
+  · simp only [ite_eq_left hn]
+    unfold sourceSpatialWindowH sourceSpatialWindowB
+    split_ifs <;> norm_num
+  rw [ite_eq_right hn]
+  obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+  have hle := source_window_le_fine hc z r ε R her hs
+  have hwn : sourceSpatialWindowTime γ z r ε R ≠ ⊤ :=
+    ne_top_of_le_ne_top hn hle
+  obtain ⟨s,hs'⟩ := WithTop.ne_top_iff_exists.mp hwn
+  have hst : s ≤ t := WithTop.coe_le_coe.mp (by simpa only [← ht,← hs'] using hle)
+  have hl : z ∈ survivingDomain γ s := survivingDomain_antitone γ hst
+    (source_liveHit_mem hc z ε (her.trans hs) t ht.symm).1
+  have hm := (source_window_mem hc z r ε R s hs'.symm).2
+  rcases hm with hd | hd | hd
+  · have hH : sourceSpatialWindowH γ z r ε R = 1 := by
+      simp only [sourceSpatialWindowH, ← hs', WithTop.coe_ne_top, ne_eq, not_false_eq_true,
+        WithTop.untopD_coe, hl, hd, and_self, ite_true]
+    rw [hH]; unfold sourceSpatialWindowB; split_ifs <;> norm_num
+  · have hB : sourceSpatialWindowB γ z r ε R = 1 := by
+      simp only [sourceSpatialWindowB, ← hs', WithTop.coe_ne_top, ne_eq, not_false_eq_true,
+        WithTop.untopD_coe, hl, hd, and_self, ite_true]
+    rw [hB]; unfold sourceSpatialWindowH; split_ifs <;> norm_num
+  · exact (hd hl).elim
+
+theorem sourceSpatialWindowGeometry_proved : SourceSpatialWindowGeometryTarget := by
+  intro U γ G F hU hw κ hκ hκ8 z ε r R he her hrR hry
+  have hs : r ≤ dist (γ 0) z := by
+    rw [hw.2.1, dist_zero_left]
+    exact hry.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  dsimp only
+  have hH0 : 0 ≤ sourceSpatialWindowH γ z r ε R := by
+    unfold sourceSpatialWindowH; split_ifs <;> norm_num
+  have hH1 : sourceSpatialWindowH γ z r ε R ≤ 1 := by
+    unfold sourceSpatialWindowH; split_ifs <;> norm_num
+  have hB0 : 0 ≤ sourceSpatialWindowB γ z r ε R := by
+    unfold sourceSpatialWindowB; split_ifs <;> norm_num
+  have hB1 : sourceSpatialWindowB γ z r ε R ≤ 1 := by
+    unfold sourceSpatialWindowB; split_ifs <;> norm_num
+  refine ⟨source_window_start_le hw.1 z r ε R,
+    source_before_window γ z r ε R, ?_, ?_, hH0,hH1,hB0,hB1,?_,
+    source_windowB_before hw.1 z r ε R her.le hrR hs,
+    source_window_missed hw.1 z r ε R her.le hs, ?_⟩
+  · intro ht
+    simp only [sourceSpatialWindowH, sourceSpatialWindowB, ht, ne_eq, not_true_eq_false,
+      false_and, ite_false, and_self]
+  · intro ht
+    exact top_le_iff.mp (by simpa only [ht] using source_window_start_le hw.1 z r ε R)
+  · by_cases hn : liveSpatialHitTime γ z ε = ⊤
+    · rw [ite_eq_left hn]
+      have hh : sourceSpatialWindowH γ z r ε R ≠ 1 := by
+        intro hh
+        exact ((source_windowH_iff hw.1 z r ε R her.le hs).mp hh).2 hn
+      unfold sourceSpatialWindowH at hh ⊢
+      split_ifs at hh ⊢ <;> norm_num at *
+    · rw [ite_eq_right hn]; exact hH1
+  · have hW := hw.spatialGreenWeight_bound hU hκ hκ8 he z (her.le.trans hry)
+    have hR : 0 ≤ ε ^ (-codimension κ) := Real.rpow_nonneg he.le _
+    by_cases hh : sourceSpatialWindowH γ z r ε R = 1
+    · rw [hh,mul_one]; exact le_add_of_nonneg_right (mul_nonneg hR hB0)
+    have hh0 : sourceSpatialWindowH γ z r ε R = 0 := by
+      unfold sourceSpatialWindowH at hh ⊢
+      split_ifs at hh ⊢ <;> norm_num at *
+    rw [hh0,mul_zero,zero_add]
+    by_cases hn : liveSpatialHitTime γ z ε = ⊤
+    · rw [spatialGreenWeight_eq_live hw.1 κ U G z ε (her.le.trans hs), ite_eq_left hn]
+      exact mul_nonneg hR hB0
+    · have hm := source_window_missed hw.1 z r ε R her.le hs
+      rw [ite_eq_right hn,hh0,zero_add] at hm
+      have hb := le_antisymm hB1 hm
+      rw [hb,mul_one]
+      exact hW.2
+
+end
+end SLEExactGauge
+
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+local instance windowGeometryPathMeasurableSpace : MeasurableSpace C(ℝ≥0,ℂ) := borel _
+local instance windowGeometryPathBorelSpace : BorelSpace C(ℝ≥0,ℂ) := ⟨rfl⟩
+ 
+noncomputable def sourceCollarStart (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r : ℝ) : WithTop ℝ≥0 :=
+  if z ∈ D then liveSpatialHitTime γ z r else ⊤
+noncomputable def sourceCollarFinish (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : WithTop ℝ≥0 :=
+  if z ∈ D then sourceSpatialWindowTime γ z r ε R else ⊤
+noncomputable def sourceCollarH (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
+  if z ∈ D then sourceSpatialWindowH γ z r ε R else 0
+noncomputable def sourceCollarB (D : Set ℂ) (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) : ℝ :=
+  if z ∈ D then sourceSpatialWindowB γ z r ε R else 0
+ 
+
+def SourceSpatialWindowMeasurableTarget : Prop :=
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω) (κ : ℝ),
+  0 < κ → κ < 8 → ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (D : Set ℂ), MeasurableSet D → ∀ ε r R : ℝ,
+  0 < ε → ε < r → r < R → (∀ z ∈ D, r ≤ z.im) →
+  let σ := fun z (ω : a.samples) => sourceCollarStart D (γ ω) z r
+  let τ := fun z (ω : a.samples) => sourceCollarFinish D (γ ω) z r ε R
+  let H := fun z (ω : a.samples) => sourceCollarH D (γ ω) z r ε R
+  let B := fun z (ω : a.samples) => sourceCollarB D (γ ω) z r ε R
+  let V := fun z (ω : a.samples) => spatialGreenWeight κ
+    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z ε * H z ω
+  (∃ (hσ : ∀ z, IsStoppingTime a.filtration (σ z))
+    (hτ : ∀ z, IsStoppingTime a.filtration (τ z)),
+    (∀ z, Measurable[(hτ z).measurableSpace] (H z) ∧
+      Measurable[(hτ z).measurableSpace] (B z) ∧
+      Measurable[(hτ z).measurableSpace] (V z)) ∧
+    (∀ z, Measurable[(hσ z).measurableSpace] (fun ω =>
+      if z ∈ D then spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+        (γ ω) (a.G ω) z r else 0))) ∧
+  Measurable (fun p : ℂ × a.samples => σ p.1 p.2) ∧
+  Measurable (fun p : ℂ × a.samples => τ p.1 p.2) ∧
+  Measurable (fun p : ℂ × a.samples => H p.1 p.2) ∧
+  Measurable (fun p : ℂ × a.samples => B p.1 p.2) ∧
+  Measurable (fun p : ℂ × a.samples => V p.1 p.2)
+
+lemma source_window_le_iff {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (T : ℝ≥0) :
+    sourceSpatialWindowTime γ z r ε R ≤ (T : WithTop ℝ≥0) ↔
+      ∃ t ∈ Icc (0 : ℝ≥0) T, t ∈ sourceSpatialWindowSet γ z r ε R := by
+  constructor
+  · intro hh
+    obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp (ne_top_of_le_ne_top WithTop.coe_ne_top hh)
+    exact ⟨t,⟨bot_le, WithTop.coe_le_coe.mp (by simpa only [← ht] using hh)⟩,
+      source_window_mem hc z r ε R t ht.symm⟩
+  · rintro ⟨t,ht,hmem⟩
+    exact (source_window_le_of_mem γ z r ε R t hmem).trans (WithTop.coe_le_coe.mpr ht.2)
+
+lemma source_liveTime_eq_killed {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r : ℝ) (hs : r ≤ dist (γ 0) z) :
+    liveSpatialHitTime γ z r =
+      if z ∈ survivingDomain γ (spatialHitTime γ z r).untopA
+      then spatialHitTime γ z r else ⊤ := by
+  rw [liveSpatialHitTime_eq hc z r hs]
+  by_cases hn : spatialHitTime γ z r = ⊤
+  · simp only [hn, ite_self]
+  · obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp hn
+    rw [ite_eq_right hn, ← ht]
+    rfl
+
+lemma source_live_stopped_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
+    {τ : a.samples → WithTop ℝ≥0} (hτ : IsStoppingTime a.filtration τ) :
+    MeasurableSet[hτ.measurableSpace] {ω | z ∈ survivingDomain (γ ω) (τ ω).untopA} := by
+  have hm := measurable_stoppedValue
+    (source_killedChart_progressive a.measurable_B a.continuous_B κ
+      (fun ω : a.samples => γ ω) a.G a.F a.witness z) hτ
+  have he : {ω : a.samples | z ∈ survivingDomain (γ ω) (τ ω).untopA} =
+      {ω : a.samples | 0 < (stoppedValue (fun t (ω : a.samples) =>
+        if z ∈ survivingDomain (γ ω) t then a.G ω t z else 0) τ ω).im} := by
+    ext ω
+    change (z ∈ survivingDomain (γ ω) (τ ω).untopA) ↔
+      0 < (if z ∈ survivingDomain (γ ω) (τ ω).untopA then a.G ω (τ ω).untopA z else 0).im
+    by_cases hl : z ∈ survivingDomain (γ ω) (τ ω).untopA
+    · simp only [hl, true_iff]
+      exact (a.witness ω).2.2.2.2.2.1 _ hl
+    · simp [hl]
+  rw [he]
+  exact measurableSet_lt measurable_const (Complex.measurable_im.comp hm)
+
+lemma source_liveHit_stopping
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
+    (r : ℝ) (hy : r ≤ z.im) :
+    IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r) := by
+  have hσ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r) :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
+  have hA := source_live_stopped_measurable a z hσ
+  intro T
+  have he : {ω : a.samples | liveSpatialHitTime (γ ω) z r ≤ (T : WithTop ℝ≥0)} =
+      {ω : a.samples | z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z r).untopA} ∩
+      {ω : a.samples | spatialHitTime (γ ω) z r ≤ (T : WithTop ℝ≥0)} := by
+    ext ω
+    have hs : r ≤ dist (γ ω 0) z := by
+      rw [(a.witness ω).2.1, dist_zero_left]
+      exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+    simp only [Set.mem_ofPred_eq, Set.mem_inter_iff]
+    rw [source_liveTime_eq_killed (a.trace_continuous ω) z r hs]
+    by_cases hl : z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z r).untopA
+    · simp only [hl, true_and, ite_true]
+    · simp only [hl, false_and, iff_false, ite_false]
+      exact not_le_of_gt (WithTop.coe_lt_top _)
+  rw [he]
+  exact hA.2 T
+
+ 
+lemma source_windowKernel_closed (T : ℝ≥0) (ε R : ℝ) :
+    IsClosed {p : (ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ | ∃ t ∈ Icc (0 : ℝ≥0) T,
+      p.1.1 ≤ t ∧ (dist (p.1.2 t) p.2 ≤ ε ∨ R ≤ dist (p.1.2 t) p.2 ∨
+        p.2 ∉ survivingDomain p.1.2 t)} := by
+  let A : Set (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) :=
+    {p | p.1.1.1 ≤ p.2.1 ∧ (dist (p.1.1.2 p.2.1) p.1.2 ≤ ε ∨
+      R ≤ dist (p.1.1.2 p.2.1) p.1.2 ∨ p.1.2 ∉ survivingDomain p.1.1.2 p.2.1)}
+  have ht : Continuous (fun p : (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) => p.2.1) :=
+    continuous_subtype_val.comp continuous_snd
+  have hv : Continuous (fun p : (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) =>
+      p.1.1.2 p.2.1) := continuous_eval.comp (continuous_fst.fst.snd.prodMk ht)
+  have hl : IsOpen {p : (((ℝ≥0 × C(ℝ≥0,ℂ)) × ℂ) × Icc (0 : ℝ≥0) T) |
+      p.1.2 ∈ survivingDomain p.1.1.2 p.2.1} := source_survival_open.preimage
+      ((continuous_fst.fst.snd.prodMk ht).prodMk continuous_fst.snd)
+  have hA : IsClosed A := (isClosed_le continuous_fst.fst.fst ht).inter
+    ((isClosed_le (hv.dist continuous_fst.snd) continuous_const).union
+      ((isClosed_le continuous_const (hv.dist continuous_fst.snd)).union hl.isClosed_compl))
+  have hp := isClosedMap_fst_of_compactSpace A hA
+  convert hp using 1
+  ext p
+  simp only [A, Set.mem_ofPred_eq, Set.mem_image, Prod.exists, Subtype.exists,
+    exists_and_right, exists_eq_right]
+  aesop
+
+lemma source_min_time_coe (σ : WithTop ℝ≥0) (T : ℝ≥0) :
+    (((min σ (T : WithTop ℝ≥0)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) = min σ T := by
+  have hn : min σ (T : WithTop ℝ≥0) ≠ ⊤ := ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_right _ _)
+  obtain ⟨s,hs⟩ := WithTop.ne_top_iff_exists.mp hn
+  rw [← hs, WithTop.untopD_coe]
+
+lemma source_window_kernel_iff {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (T : ℝ≥0) :
+    sourceSpatialWindowTime γ z r ε R ≤ (T : WithTop ℝ≥0) ↔
+    liveSpatialHitTime γ z r ≤ (T : WithTop ℝ≥0) ∧
+    ∃ t ∈ Icc (0 : ℝ≥0) T, (min (liveSpatialHitTime γ z r) T).untopD 0 ≤ t ∧
+      (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t) := by
+  rw [source_window_le_iff hc z r ε R T]
+  constructor
+  · rintro ⟨t,ht,hs,he⟩
+    have hT := hs.trans (WithTop.coe_le_coe.mpr ht.2)
+    refine ⟨hT,t,ht,?_,he⟩
+    apply WithTop.coe_le_coe.mp
+    rw [source_min_time_coe, min_eq_left hT]
+    exact hs
+  · rintro ⟨hT,t,ht,hs,he⟩
+    refine ⟨t,ht,?_,he⟩
+    have hs' := WithTop.coe_le_coe.mpr hs
+    rw [source_min_time_coe, min_eq_left hT] at hs'
+    exact hs'
+
+def sourceCapPath (γ : C(ℝ≥0,ℂ)) (T : ℝ≥0) : C(ℝ≥0,ℂ) :=
+  ⟨fun t => γ (min t T), γ.continuous.comp (continuous_id.min continuous_const)⟩
+
+lemma source_cap_domain (γ : C(ℝ≥0,ℂ)) (T t : ℝ≥0) (ht : t ≤ T) :
+    survivingDomain (sourceCapPath γ T) t = survivingDomain γ t := by
+  have he : sourceCapPath γ T '' Icc 0 t = γ '' Icc 0 t := by
+    apply Set.image_congr
+    intro s hs
+    change γ (min s T) = γ s
+    rw [min_eq_left (hs.2.trans ht)]
+  simp only [survivingDomain, he]
+
+lemma source_window_kernel_cap (γ : C(ℝ≥0,ℂ)) (z : ℂ) (ε R : ℝ) (T b : ℝ≥0) :
+    (∃ t ∈ Icc (0 : ℝ≥0) T, b ≤ t ∧
+      (dist (sourceCapPath γ T t) z ≤ ε ∨ R ≤ dist (sourceCapPath γ T t) z ∨
+        z ∉ survivingDomain (sourceCapPath γ T) t)) ↔
+    (∃ t ∈ Icc (0 : ℝ≥0) T, b ≤ t ∧
+      (dist (γ t) z ≤ ε ∨ R ≤ dist (γ t) z ∨ z ∉ survivingDomain γ t)) := by
+  apply exists_congr
+  intro t
+  apply and_congr_right
+  intro ht
+  rw [source_cap_domain γ T t ht.2]
+  have he : sourceCapPath γ T t = γ t := by dsimp [sourceCapPath]; rw [min_eq_left ht.2]
+  rw [he]
+
+lemma source_tracePath_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) :
+    Measurable (fun ω : a.samples => (⟨γ ω,a.trace_continuous ω⟩ : C(ℝ≥0,ℂ))) := by
+  apply source_path_measurable
+  intro t
+  exact (a.trace_adapted t).mono (a.filtration.le t) le_rfl
+
+lemma source_capPath_adapted
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (T : ℝ≥0) :
+    Measurable[a.filtration T] (fun ω : a.samples =>
+      sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T) := by
+  refine @source_path_measurable a.samples ℂ (a.filtration T) _ _ _ _ _ _ _ _ ?_
+  intro t
+  exact (a.trace_adapted (min t T)).mono (a.filtration.mono (min_le_right _ _)) le_rfl
+
+lemma source_window_stopping
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
+    (r ε R : ℝ) (hy : r ≤ z.im) :
+    IsStoppingTime a.filtration (fun ω : a.samples => sourceSpatialWindowTime (γ ω) z r ε R) := by
+  let σ := fun ω : a.samples => liveSpatialHitTime (γ ω) z r
+  have hσ : IsStoppingTime a.filtration σ := source_liveHit_stopping a z r hy
+  intro T
+  have hm : Measurable[a.filtration T] (fun ω : a.samples => (min (σ ω) T).untopD 0) :=
+    ((hσ.min_const T).measurable_of_le (fun _ => min_le_right _ _)).untopD 0
+  have hp : Measurable[a.filtration T] (fun ω : a.samples =>
+      (((min (σ ω) T).untopD 0, sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T),z)) :=
+    (hm.prodMk (source_capPath_adapted a T)).prodMk measurable_const
+  have hK := (source_windowKernel_closed T ε R).measurableSet.preimage hp
+  have he : {ω : a.samples | sourceSpatialWindowTime (γ ω) z r ε R ≤ (T : WithTop ℝ≥0)} =
+      {ω : a.samples | σ ω ≤ (T : WithTop ℝ≥0)} ∩
+      {ω : a.samples | ∃ t ∈ Icc (0 : ℝ≥0) T, (min (σ ω) T).untopD 0 ≤ t ∧
+        (dist (sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T t) z ≤ ε ∨
+         R ≤ dist (sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T t) z ∨
+         z ∉ survivingDomain (sourceCapPath ⟨γ ω,a.trace_continuous ω⟩ T) t)} := by
+    ext ω
+    simp only [Set.mem_ofPred_eq, Set.mem_inter_iff]
+    rw [source_window_kernel_iff (a.trace_continuous ω) z r ε R T,
+      source_window_kernel_cap]
+    rfl
+  rw [he]
+  exact (hσ T).inter hK
+
+lemma source_collarStart_stopping
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
+    IsStoppingTime a.filtration (fun ω : a.samples => sourceCollarStart D (γ ω) z r) := by
+  by_cases hz : z ∈ D
+  · simpa only [sourceCollarStart, ite_eq_left hz] using source_liveHit_stopping a z r (hy z hz)
+  · simpa only [sourceCollarStart, ite_eq_right hz] using (show IsStoppingTime a.filtration (fun _ => (⊤ : WithTop ℝ≥0)) from fun _ => MeasurableSet.const _)
+
+lemma source_collarFinish_stopping
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (r ε R : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) (z : ℂ) :
+    IsStoppingTime a.filtration (fun ω : a.samples => sourceCollarFinish D (γ ω) z r ε R) := by
+  by_cases hz : z ∈ D
+  · simpa only [sourceCollarFinish, ite_eq_left hz] using source_window_stopping a z r ε R (hy z hz)
+  · simpa only [sourceCollarFinish, ite_eq_right hz] using (show IsStoppingTime a.filtration (fun _ => (⊤ : WithTop ℝ≥0)) from fun _ => MeasurableSet.const _)
+
+lemma source_joint_survival_measurable
+    {X : Type*} [MeasurableSpace X] (g : X → C(ℝ≥0,ℂ)) (t : X → ℝ≥0) (z : X → ℂ)
+    (hg : Measurable g) (ht : Measurable t) (hz : Measurable z) :
+    MeasurableSet {x | z x ∈ survivingDomain (g x) (t x)} :=
+  source_survival_open.measurableSet.preimage ((hg.prodMk ht).prodMk hz)
+
+lemma source_collarStart_joint
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (hD : MeasurableSet D) (r : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) :
+    Measurable (fun p : ℂ × a.samples => sourceCollarStart D (γ p.2) p.1 r) := by
+  have hhit := spatialHitTime_measurable (fun ω : a.samples => γ ω) a.trace_continuous
+    (fun t => (a.trace_adapted t).mono (a.filtration.le t) le_rfl) r
+  have hl := source_joint_survival_measurable
+    (fun p : ℂ × a.samples => (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ)))
+    (fun p => (spatialHitTime (γ p.2) p.1 r).untopA) Prod.fst
+    ((source_tracePath_measurable a).comp measurable_snd)
+    (WithTop.measurable_untopA.comp hhit) measurable_fst
+  have hm : Measurable (fun p : ℂ × a.samples =>
+      if p.1 ∈ D then (if p.1 ∈ survivingDomain (γ p.2) (spatialHitTime (γ p.2) p.1 r).untopA
+      then spatialHitTime (γ p.2) p.1 r else ⊤) else ⊤) :=
+    Measurable.ite (hD.preimage measurable_fst)
+    (Measurable.ite hl hhit (measurable_const (a := (⊤ : WithTop ℝ≥0))))
+    (measurable_const (a := (⊤ : WithTop ℝ≥0)))
+  convert hm using 1
+  funext p
+  by_cases hz : p.1 ∈ D
+  · simp only [sourceCollarStart, ite_eq_left hz]
+    apply source_liveTime_eq_killed (a.trace_continuous p.2)
+    rw [(a.witness p.2).2.1, dist_zero_left]
+    exact (hy p.1 hz).trans ((le_abs_self p.1.im).trans (Complex.abs_im_le_norm _))
+  · simp only [sourceCollarStart, ite_eq_right hz]
+
+lemma source_collarFinish_joint
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (D : Set ℂ)
+    (hD : MeasurableSet D) (r ε R : ℝ) (hy : ∀ z ∈ D, r ≤ z.im) :
+    Measurable (fun p : ℂ × a.samples => sourceCollarFinish D (γ p.2) p.1 r ε R) := by
+  have hσ := source_collarStart_joint a D hD r hy
+  apply measurable_of_Iic
+  intro b
+  by_cases hb : b = ⊤
+  · subst b
+    simp only [preimage, mem_Iic, le_top, ofPred_true]
+    exact MeasurableSet.univ
+  obtain ⟨T,hT⟩ := WithTop.ne_top_iff_exists.mp hb
+  subst b
+  have hp : Measurable (fun p : ℂ × a.samples =>
+      (((min (sourceCollarStart D (γ p.2) p.1 r) T).untopD 0,
+      (⟨γ p.2,a.trace_continuous p.2⟩ : C(ℝ≥0,ℂ))),p.1)) :=
+    (((hσ.min measurable_const).untopD 0).prodMk
+      ((source_tracePath_measurable a).comp measurable_snd)).prodMk measurable_fst
+  have hK := (source_windowKernel_closed T ε R).measurableSet.preimage hp
+  have he : (fun p : ℂ × a.samples => sourceCollarFinish D (γ p.2) p.1 r ε R) ⁻¹' Iic (T : WithTop ℝ≥0) =
+      (Prod.fst ⁻¹' D) ∩ {p : ℂ × a.samples | sourceCollarStart D (γ p.2) p.1 r ≤ (T : WithTop ℝ≥0)} ∩
+      {p : ℂ × a.samples | ∃ t ∈ Icc (0 : ℝ≥0) T,
+      (min (sourceCollarStart D (γ p.2) p.1 r) T).untopD 0 ≤ t ∧
+      (dist (γ p.2 t) p.1 ≤ ε ∨ R ≤ dist (γ p.2 t) p.1 ∨ p.1 ∉ survivingDomain (γ p.2) t)} := by
+    ext p
+    simp only [mem_preimage, mem_Iic, mem_inter_iff, mem_ofPred_eq]
+    by_cases hz : p.1 ∈ D
+    · simp only [hz, true_and, sourceCollarStart, sourceCollarFinish]
+      exact source_window_kernel_iff (a.trace_continuous p.2) p.1 r ε R T
+    · simp only [hz, false_and, sourceCollarFinish, iff_false]
+      exact not_le_of_gt (WithTop.coe_lt_top _)
+  rw [he]
+  exact ((hD.preimage measurable_fst).inter (measurableSet_le hσ measurable_const)).inter hK
+
+lemma source_spatialWeight_stopping_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ}
+    (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω))
+    (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ)
+    (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω)
+      (γ ω) (G ω) (F ω)) (z : ℂ) (r : ℝ)
+    (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable))
+      (fun ω => spatialHitTime (γ ω) z r)) :
+    Measurable[hτ.measurableSpace] (fun ω => spatialGreenWeight κ
+      (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) z r) := by
+  have hmG := measurable_stoppedValue (source_killedGreen_progressive hm hc κ γ G F hw z) hτ
+  have he : MeasurableSet[hτ.measurableSpace] {ω | spatialHitTime (γ ω) z r = ⊤} :=
+    measurableSet_eq_fun hτ.measurable measurable_const
+  have hmo : Measurable[hτ.measurableSpace] (fun ω =>
+      if spatialHitTime (γ ω) z r = ⊤ then (0:ℝ) else
+        greenDensity κ (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω)
+          (spatialHitTime (γ ω) z r).untopA z) :=
+    Measurable.ite he measurable_const hmG
+  convert hmo using 1
+  funext ω
+  by_cases ht : spatialHitTime (γ ω) z r = ⊤
+  · simp only [spatialGreenWeight, ite_eq_left ht]
+  · obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp ht
+    simp only [spatialGreenWeight, ite_eq_right ht]
+    rw [← hv]
+    rfl
+
+lemma source_windowH_untopA (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) :
+    sourceSpatialWindowH γ z r ε R =
+    if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
+      z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopA) ∧
+      dist (γ ((sourceSpatialWindowTime γ z r ε R).untopA)) z ≤ ε then 1 else 0 := by
+  by_cases ht : sourceSpatialWindowTime γ z r ε R = ⊤
+  · simp only [sourceSpatialWindowH, ht, ne_eq, not_true_eq_false, false_and, ite_false]
+  · obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp ht
+    unfold sourceSpatialWindowH
+    rw [← ht]
+    rfl
+
+lemma source_windowB_untopA (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) :
+    sourceSpatialWindowB γ z r ε R =
+    if sourceSpatialWindowTime γ z r ε R ≠ ⊤ ∧
+      z ∈ survivingDomain γ ((sourceSpatialWindowTime γ z r ε R).untopA) ∧
+      R ≤ dist (γ ((sourceSpatialWindowTime γ z r ε R).untopA)) z then 1 else 0 := by
+  by_cases ht : sourceSpatialWindowTime γ z r ε R = ⊤
+  · simp only [sourceSpatialWindowB, ht, ne_eq, not_true_eq_false, false_and, ite_false]
+  · obtain ⟨t,ht⟩ := WithTop.ne_top_iff_exists.mp ht
+    unfold sourceSpatialWindowB
+    rw [← ht]
+    rfl
+
+lemma source_windowHB_stopped_measurable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ)
+    (r ε R : ℝ)
+    (hτ : IsStoppingTime a.filtration (fun ω : a.samples => sourceSpatialWindowTime (γ ω) z r ε R)) :
+    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R) ∧
+    Measurable[hτ.measurableSpace] (fun ω : a.samples => sourceSpatialWindowB (γ ω) z r ε R) := by
+  have htp := (show StronglyAdapted a.filtration (fun t (ω : a.samples) => γ ω t) from
+    fun t => (a.trace_adapted t).stronglyMeasurable).isStronglyProgressive_of_continuous a.trace_continuous
+  have hv := measurable_stoppedValue htp hτ
+  let : MeasurableSpace a.samples := hτ.measurableSpace
+  have hl := source_live_stopped_measurable a z hτ
+  have hn : MeasurableSet[hτ.measurableSpace] {ω : a.samples | sourceSpatialWindowTime (γ ω) z r ε R ≠ ⊤} :=
+    (measurableSet_eq_fun hτ.measurable measurable_const).compl
+  have hd : Measurable[hτ.measurableSpace] (fun ω : a.samples =>
+      dist (γ ω (sourceSpatialWindowTime (γ ω) z r ε R).untopA) z) := hv.dist measurable_const
+  have hH : Measurable[hτ.measurableSpace] (fun ω : a.samples =>
+      if sourceSpatialWindowTime (γ ω) z r ε R ≠ ⊤ ∧
+      z ∈ survivingDomain (γ ω) (sourceSpatialWindowTime (γ ω) z r ε R).untopA ∧
+      dist (γ ω (sourceSpatialWindowTime (γ ω) z r ε R).untopA) z ≤ ε then (1:ℝ) else 0) :=
+    Measurable.ite (hn.inter (hl.inter (measurableSet_le hd measurable_const)))
+    (measurable_const (a := (1:ℝ))) (measurable_const (a := (0:ℝ)))
+  have hB : Measurable[hτ.measurableSpace] (fun ω : a.samples =>
+      if sourceSpatialWindowTime (γ ω) z r ε R ≠ ⊤ ∧
+      z ∈ survivingDomain (γ ω) (sourceSpatialWindowTime (γ ω) z r ε R).untopA ∧
+      R ≤ dist (γ ω (sourceSpatialWindowTime (γ ω) z r ε R).untopA) z then (1:ℝ) else 0) :=
+    Measurable.ite (hn.inter (hl.inter (measurableSet_le measurable_const hd)))
+    (measurable_const (a := (1:ℝ))) (measurable_const (a := (0:ℝ)))
+  constructor
+  · convert hH using 1
+    funext ω
+    exact source_windowH_untopA (γ ω) z r ε R
+  · convert hB using 1
+    funext ω
+    exact source_windowB_untopA (γ ω) z r ε R
+
+end
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/WindowPropagation.lean b/lean/OAI/Probability/SLE/WindowPropagation.lean
index 6af95ebfac13f542d29150f0aaeeea316885357b..b75bbdc46f5bfa08f3accbb8ebb853e0f4173255 100644
--- a/lean/OAI/Probability/SLE/WindowPropagation.lean
+++ b/lean/OAI/Probability/SLE/WindowPropagation.lean
@@ -1,739 +1,739 @@
-import Mathlib
-import OAI.Probability.SLE.PhysicalMeshPast
-
-namespace OAI
-
-/-! Window Propagation for chordal SLE. -/
-
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
- 
-
-lemma geometric_over_quartic_ge_linear {β : ℝ} (hβ : 1 < β) :
-    ∃ k : ℝ, 0 < k ∧ ∀ n : ℕ, k*(n+1:ℝ) ≤ β^n/(n+1:ℝ)^4 := by
-  have hβ0 : 0 < β := by linarith
-  have hir : 0 < β⁻¹ := inv_pos.mpr hβ0
-  have hi1 : β⁻¹ < 1 := (inv_lt_one₀ hβ0).mpr hβ
-  have hnorm : ‖β⁻¹‖ < 1 := by simpa only [Real.norm_eq_abs,abs_of_pos hir] using hi1
-  have hs := summable_pow_mul_geometric_of_norm_lt_one 5 hnorm
-  have hss := ((summable_nat_add_iff 1).mpr hs).div_const (β⁻¹)
-  have hseq : Summable (fun n : ℕ => (n+1:ℝ)^5/β^n) := by
-    apply hss.congr
-    intro n
-    simp only [Nat.cast_add,Nat.cast_one,pow_succ,inv_pow]
-    field_simp [ne_of_gt hβ0]
-  let T : ℝ := ∑' n : ℕ, (n+1:ℝ)^5/β^n
-  have hn (n : ℕ) : (n+1:ℝ)^5/β^n ≤ T :=
-    hseq.le_tsum n (fun j _ => by positivity)
-  have hT : 0 < T := by have := hn 0; norm_num at this; linarith
-  refine ⟨T⁻¹,inv_pos.mpr hT,fun n => ?_⟩
-  have hn0 : 0 < (n+1:ℝ) := by positivity
-  have hp0 : 0 < β^n := pow_pos hβ0 _
-  have hn' : (n+1:ℝ)^5 ≤ T*β^n := (div_le_iff₀ hp0).mp (hn n)
-  apply (le_div_iff₀ (pow_pos hn0 4)).mpr
-  apply (mul_le_mul_iff_right₀ hT).mp
-  calc
-    T * (T⁻¹ * (n+1:ℝ) * (n+1:ℝ)^4) = (n+1:ℝ)^5 := by field_simp
-    _ ≤ T * β^n := hn'
-
-lemma all_levels_exceptional_sum {b β : ℝ} (hb : 0 < b) (hβ : 1 < β) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ D T : ℝ, 0 ≤ D → 1 ≤ T →
-      (∑' n : ℕ, ENNReal.ofReal (D*Real.exp (-b*T*(β^n/(n+1:ℝ)^4)))) ≤
-        ENNReal.ofReal (D*C*Real.exp (-c*T)) := by
-  obtain ⟨k,hk,hkn⟩ := geometric_over_quartic_ge_linear hβ
-  let c := b*k
-  have hc : 0 < c := mul_pos hb hk
-  let r := Real.exp (-c)
-  have hr : 0 < r := Real.exp_pos _
-  have hr1 : r < 1 := (Real.exp_lt_one_iff).mpr (neg_neg_of_pos hc)
-  refine ⟨(1-r)⁻¹,c,inv_pos.mpr (sub_pos.mpr hr1),hc,?_⟩
-  intro D T hD hT
-  have hT0 : 0 ≤ T := by linarith
-  have hn (n : ℕ) : D*Real.exp (-b*T*(β^n/(n+1:ℝ)^4)) ≤
-      (D*Real.exp (-c*T))*r^n := by
-    have hbound := mul_le_mul_of_nonneg_left (hkn n) (mul_nonneg hb.le hT0)
-    have hneg : -b*T*(β^n/(n+1:ℝ)^4) ≤ -c*T + (n:ℝ)*(-c) := by
-      have hh : c*(T-1)*(n:ℝ) ≥ 0 := by positivity
-      dsimp [c] at *
-      nlinarith
-    calc
-      _ ≤ D*Real.exp (-c*T + (n:ℝ)*(-c)) := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hneg) hD
-      _ = _ := by rw [Real.exp_add,Real.exp_nat_mul]; ring
-  have hsum := (hasSum_geometric_of_lt_one hr.le hr1).mul_left (D*Real.exp (-c*T))
-  calc
-    _ ≤ ∑' n : ℕ, ENNReal.ofReal ((D*Real.exp (-c*T))*r^n) :=
-      ENNReal.tsum_le_tsum (fun n => ENNReal.ofReal_le_ofReal (hn n))
-    _ = ENNReal.ofReal ((D*Real.exp (-c*T))*(1-r)⁻¹) := by
-      rw [← ENNReal.ofReal_tsum_of_nonneg (fun n => by positivity) hsum.summable,hsum.tsum_eq]
-    _ = _ := by congr 1; ring
-
- 
-lemma reciprocal_square_partial_sum_strong (n : ℕ) :
-    (∑ j ∈ Finset.range (n+1), 1/(j+1:ℝ)^2) ≤ 2 - 1/(n+1:ℝ) := by
-  induction n with
-  | zero => norm_num
-  | succ n ih =>
-    rw [Finset.sum_range_succ]
-    have hn : 0 < (n+1:ℝ) := by positivity
-    have hn2 : 0 < (n+2:ℝ) := by positivity
-    have hp : 1/(n+2:ℝ)^2 ≤ 1/(n+1:ℝ) - 1/(n+2:ℝ) := by
-      calc
-        _ ≤ 1/((n+1:ℝ)*(n+2:ℝ)) := by
-          apply one_div_le_one_div_of_le (mul_pos hn hn2)
-          nlinarith
-        _ = _ := by field_simp; ring
-    simp only [Nat.cast_add,Nat.cast_one] at *
-    rw [show (n:ℝ)+1+1 = n+2 by ring]
-    linarith
-
-lemma reciprocal_square_partial_sum_le_two (n : ℕ) :
-    (∑ j ∈ Finset.range n, 1/(j+1:ℝ)^2) ≤ 2 := by
-  cases n with
-  | zero => norm_num
-  | succ n =>
-    have := reciprocal_square_partial_sum_strong n
-    have : 0 ≤ 1/(n+1:ℝ) := by positivity
-    linarith [reciprocal_square_partial_sum_strong n]
-
-lemma discrete_gronwall_exp (U α e : ℕ → ℝ) (hU : 0 ≤ U 0)
-    (hα : ∀ n, 0 ≤ α n) (he : ∀ n, 0 ≤ e n)
-    (hrec : ∀ n, U (n+1) ≤ (1+α n)*U n+e n) :
-    ∀ n, U n ≤ (U 0 + ∑ j ∈ Finset.range n, e j) *
-      Real.exp (∑ j ∈ Finset.range n, α j) := by
-  intro n
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ,Finset.sum_range_succ]
-    have hV : 0 ≤ U 0 + ∑ j ∈ Finset.range n, e j :=
-      add_nonneg hU (Finset.sum_nonneg (fun j _ => he j))
-    have hsum : 0 ≤ (∑ j ∈ Finset.range n, α j) + α n :=
-      add_nonneg (Finset.sum_nonneg (fun j _ => hα j)) (hα n)
-    have hExp : 1 ≤ Real.exp ((∑ j ∈ Finset.range n, α j)+α n) :=
-      Real.one_le_exp_iff.mpr hsum
-    have hfac : 1+α n ≤ Real.exp (α n) := by linarith [Real.add_one_le_exp (α n)]
-    calc
-      U (n+1) ≤ (1+α n)*U n+e n := hrec n
-      _ ≤ (1+α n)*((U 0+∑ j ∈ Finset.range n,e j)*
-          Real.exp (∑ j ∈ Finset.range n,α j))+e n :=
-        by
-          gcongr
-          · linarith [hα n]
-      _ ≤ (U 0+∑ j ∈ Finset.range n,e j)*
-          Real.exp ((∑ j ∈ Finset.range n,α j)+α n) +
-          e n*Real.exp ((∑ j ∈ Finset.range n,α j)+α n) := by
-        apply add_le_add
-        · rw [Real.exp_add]
-          nlinarith [mul_le_mul_of_nonneg_right hfac
-            (mul_nonneg hV (Real.exp_pos (∑ j ∈ Finset.range n,α j)).le)]
-        · exact le_mul_of_one_le_right (he n) hExp
-      _ = _ := by ring
-
-lemma two_recurrences_normalized (S A α e : ℕ → ℝ) (C₁ K : ℝ)
-    (hC : 0 ≤ C₁) (hK : 1 ≤ K) (hCK : C₁ ≤ K)
-    (hnonneg : ∀ n, 0 ≤ S n ∧ 0 ≤ A n)
-    (hα : ∀ n, 0 ≤ α n) (he : ∀ n, 0 ≤ e n)
-    (hrec : ∀ n, S (n+1) ≤ S n+α n*A n+e n ∧
-      A (n+1) ≤ C₁*S n+α n*A n+e n) :
-    ∀ n, max (S (n+1)) (A (n+1)/K) ≤
-      (1+K*α n)*max (S n) (A n/K)+e n := by
-  have hK0 : 0 < K := by linarith
-  intro n
-  let U := max (S n) (A n/K)
-  have hS : S n ≤ U := le_max_left _ _
-  have hA : A n ≤ K*U := by
-    have := (div_le_iff₀ hK0).mp (le_max_right (S n) (A n/K))
-    simpa only [mul_comm] using this
-  have hU : 0 ≤ U := (hnonneg n).1.trans hS
-  apply max_le
-  · calc
-      S (n+1) ≤ S n+α n*A n+e n := (hrec n).1
-      _ ≤ U+α n*(K*U)+e n := by gcongr; exact hα n
-      _ = _ := by ring
-  · apply (div_le_iff₀ hK0).mpr
-    calc
-      A (n+1) ≤ C₁*S n+α n*A n+e n := (hrec n).2
-      _ ≤ K*U+K*(K*α n*U)+K*e n := by
-        apply add_le_add
-        · apply add_le_add
-          · exact (mul_le_mul_of_nonneg_left hS hC).trans
-              (mul_le_mul_of_nonneg_right hCK hU)
-          · calc
-              α n*A n ≤ α n*(K*U) := mul_le_mul_of_nonneg_left hA (hα n)
-              _ = K*α n*U := by ring
-              _ ≤ K*(K*α n*U) := le_mul_of_one_le_left (mul_nonneg (mul_nonneg hK0.le (hα n)) hU) hK
-        · exact le_mul_of_one_le_left (he n) hK
-      _ = _ := by ring
-
-lemma window_recurrences_uniform (C₀ C₁ E θ a d : ℝ)
-    (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) (hE : 0 ≤ E)
-    (hθ : 0 ≤ θ) (hθ1 : θ < 1) (ha : 0 ≤ a) (ha1 : a ≤ 1/2)
-    (hd : 0 < d) (S A : ℕ → ℝ)
-    (hnonneg : ∀ n, 0 ≤ S n ∧ 0 ≤ A n)
-    (hini : S 0 ≤ C₀*d ∧ A 0 ≤ C₀*d)
-    (hrec : ∀ n, S (n+1) ≤ S n+(a*θ^n)*A n+E*d/(n+1:ℝ)^2 ∧
-      A (n+1) ≤ C₁*S n+(a*θ^n)*A n+E*d/(n+1:ℝ)^2) :
-    ∀ n, S n ≤ ((C₀+2*E)*Real.exp ((max 1 (2*C₁))/2/(1-θ)))*d := by
-  let K := max 1 (2*C₁)
-  have hK : 1 ≤ K := le_max_left _ _
-  have hK0 : 0 < K := by linarith
-  have hCK : C₁ ≤ K := (by linarith : C₁ ≤ 2*C₁).trans (le_max_right _ _)
-  let U := fun n => max (S n) (A n/K)
-  let α := fun n : ℕ => a*θ^n
-  let e := fun n : ℕ => E*d/(n+1:ℝ)^2
-  have hα (n : ℕ) : 0 ≤ α n := mul_nonneg ha (pow_nonneg hθ _)
-  have he (n : ℕ) : 0 ≤ e n := by dsimp [e]; positivity
-  have hU0 : 0 ≤ U 0 := (hnonneg 0).1.trans (le_max_left _ _)
-  have hrecU := two_recurrences_normalized S A α e C₁ K hC₁ hK hCK hnonneg hα he hrec
-  have hG := discrete_gronwall_exp U (fun n => K*α n) e hU0
-    (fun n => mul_nonneg hK0.le (hα n)) he hrecU
-  have hIni : U 0 ≤ C₀*d := by
-    apply max_le hini.1
-    apply (div_le_iff₀ hK0).mpr
-    exact hini.2.trans (le_mul_of_one_le_right (mul_nonneg hC₀ hd.le) hK)
-  have hGeom := hasSum_geometric_of_lt_one hθ hθ1
-  intro n
-  have hsum_geom : (∑ j ∈ Finset.range n, θ^j) ≤ (1-θ)⁻¹ := by
-    rw [← hGeom.tsum_eq]
-    exact hGeom.summable.sum_le_tsum _ (fun j _ => pow_nonneg hθ j)
-  have he_sum : (∑ j ∈ Finset.range n, e j) ≤ 2*E*d := by
-    calc
-      _ = (E*d)*(∑ j ∈ Finset.range n, 1/(j+1:ℝ)^2) := by
-        rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j hj; dsimp [e]; ring
-      _ ≤ (E*d)*2 := mul_le_mul_of_nonneg_left
-        (reciprocal_square_partial_sum_le_two n) (mul_nonneg hE hd.le)
-      _ = _ := by ring
-  have hα_sum : (∑ j ∈ Finset.range n, K*α j) ≤ K/2/(1-θ) := by
-    calc
-      _ = (K*a)*(∑ j ∈ Finset.range n, θ^j) := by
-        rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j hj; dsimp [α]; ring
-      _ ≤ (K*a)*(1-θ)⁻¹ := mul_le_mul_of_nonneg_left hsum_geom (mul_nonneg hK0.le ha)
-      _ ≤ (K*(1/2))*(1-θ)⁻¹ := mul_le_mul_of_nonneg_right
-        (mul_le_mul_of_nonneg_left ha1 hK0.le) (inv_nonneg.mpr (sub_pos.mpr hθ1).le)
-      _ = _ := by ring
-  calc
-    S n ≤ U n := le_max_left _ _
-    _ ≤ (U 0+∑ j ∈ Finset.range n,e j)*Real.exp (∑ j ∈ Finset.range n,K*α j) := hG n
-    _ ≤ (C₀*d+2*E*d)*Real.exp (K/2/(1-θ)) := by
-      apply mul_le_mul (add_le_add hIni he_sum) (Real.exp_le_exp.mpr hα_sum)
-      · exact (Real.exp_pos _).le
-      · positivity
-    _ = _ := by dsimp [K]; ring
-
-universe u
- 
-
-def SourceWindowPropagationTarget : Prop :=
-  ∀ p C₀ C₁ E D b θ β : ℝ,
-    0 < p → p < 1 → 0 ≤ C₀ → 0 ≤ C₁ → 0 ≤ E → 0 ≤ D → 0 < b →
-    0 ≤ θ → θ < 1 → 1 < β →
-    ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
-      ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-      ∀ δ a : ℝ, 0 < δ → δ ≤ 1 → 0 ≤ a → a ≤ 1/2 →
-      ∀ (S A : ℕ → Ω → ℝ) (B : ℕ → Set Ω),
-      (∀ n ω, 0 ≤ S n ω ∧ 0 ≤ A n ω) →
-      (∀ ω, S 0 ω ≤ C₀*δ^(-p) ∧ A 0 ω ≤ C₀*δ^(-p)) →
-      (∀ n, P (B n) ≤ ENNReal.ofReal
-        (D*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4)))) →
-      (∀ n ω, ω ∉ B n →
-        S (n+1) ω ≤ S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2 ∧
-        A (n+1) ω ≤ C₁*S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2) →
-      P {ω | ∃ n, M*δ^(-p) < S n ω} ≤ ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ)))
-
-theorem sourceWindowPropagation : SourceWindowPropagationTarget := by
-  intro p C₀ C₁ E D b θ β hp hp1 hC₀ hC₁ hE hD hb hθ hθ1 hβ
-  obtain ⟨C,c,hC,hc,hseries⟩ := all_levels_exceptional_sum hb hβ
-  let M₀ := (C₀+2*E)*Real.exp ((max 1 (2*C₁))/2/(1-θ))
-  have hM₀ : 0 ≤ M₀ := mul_nonneg (by linarith) (Real.exp_pos _).le
-  refine ⟨M₀+1,D*C+1,c,by linarith,by positivity,hc,?_⟩
-  intro Ω m P hP δ a hδ hδ1 ha ha1 S A B hnonneg hini hfail hrec
-  have hd : 0 < δ^(-p) := Real.rpow_pos_of_pos hδ _
-  have hT : 1 ≤ δ^(-2:ℝ) := by
-    rw [Real.rpow_neg hδ.le,Real.rpow_two]
-    exact one_le_inv₀ (sq_pos_of_pos hδ) |>.mpr (by nlinarith)
-  have hsubset : {ω | ∃ n, (M₀+1)*δ^(-p) < S n ω} ⊆ ⋃ n, B n := by
-    intro ω hω
-    by_contra hnot
-    have hgood (n : ℕ) : ω ∉ B n := by
-      intro hn
-      exact hnot (mem_iUnion.mpr ⟨n,hn⟩)
-    have hbound := window_recurrences_uniform C₀ C₁ E θ a (δ^(-p))
-      hC₀ hC₁ hE hθ hθ1 ha ha1 hd (fun n => S n ω) (fun n => A n ω)
-      (fun n => hnonneg n ω) (hini ω) (fun n => hrec n ω (hgood n))
-    obtain ⟨n,hn⟩ := hω
-    have hn' : S n ω ≤ M₀*δ^(-p) := hbound n
-    nlinarith
-  calc
-    _ ≤ P (⋃ n, B n) := measure_mono hsubset
-    _ ≤ ∑' n, P (B n) := measure_iUnion_le _
-    _ ≤ ∑' n : ℕ, ENNReal.ofReal
-        (D*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4))) := ENNReal.tsum_le_tsum hfail
-    _ ≤ ENNReal.ofReal (D*C*Real.exp (-c*δ^(-2:ℝ))) := hseries D _ hD hT
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (by nlinarith [Real.exp_pos (-c*δ^(-2:ℝ))])
-
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma source_mesh_finite_measure_propagation (p C₀ C₁ E b θ β : ℝ)
-    (hp : 0 < p) (hp1 : p < 1) (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) (hE : 0 ≤ E)
-    (hb : 0 < b) (hθ : 0 ≤ θ) (hθ1 : θ < 1) (hβ : 1 < β) :
-    ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
-      ∀ (Ω : Type) (_ : MeasurableSpace Ω) (ν : Measure Ω) (v δ a : ℝ),
-      0 ≤ v → 0 < δ → δ ≤ 1 → 0 ≤ a → a ≤ 1/2 →
-      ∀ (S A : ℕ → Ω → ℝ) (B : ℕ → Set Ω),
-      (∀ n ω, 0 ≤ S n ω ∧ 0 ≤ A n ω) →
-      (∀ ω, S 0 ω ≤ C₀*δ^(-p) ∧ A 0 ω ≤ C₀*δ^(-p)) →
-      (∀ n, ν (B n) ≤ ENNReal.ofReal
-        (3*v*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4)))) →
-      (∀ n ω, ω ∉ B n →
-        S (n+1) ω ≤ S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2 ∧
-        A (n+1) ω ≤ C₁*S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2) →
-      ν {ω | ∃ n, M*δ^(-p) < S n ω} ≤ ENNReal.ofReal (v*C*Real.exp (-c*δ^(-2:ℝ))) := by
-  obtain ⟨C,c,hC,hc,hseries⟩ := all_levels_exceptional_sum hb hβ
-  let M₀ := (C₀+2*E)*Real.exp ((max 1 (2*C₁))/2/(1-θ))
-  have hM₀ : 0 ≤ M₀ := mul_nonneg (by linarith) (Real.exp_pos _).le
-  refine ⟨M₀+1,3*C,c,by linarith,by positivity,hc,?_⟩
-  intro Ω m ν v δ a hv hδ hδ1 ha ha1 S A B hnonneg hini hfail hrec
-  have hd : 0 < δ^(-p) := Real.rpow_pos_of_pos hδ _
-  have hT : 1 ≤ δ^(-2:ℝ) := by
-    rw [Real.rpow_neg hδ.le,Real.rpow_two]
-    exact one_le_inv₀ (sq_pos_of_pos hδ) |>.mpr (by nlinarith)
-  have hsubset : {ω | ∃ n, (M₀+1)*δ^(-p) < S n ω} ⊆ ⋃ n, B n := by
-    intro ω hω
-    by_contra hnot
-    have hgood (n : ℕ) : ω ∉ B n := fun hn => hnot (mem_iUnion.mpr ⟨n,hn⟩)
-    have hbound := window_recurrences_uniform C₀ C₁ E θ a (δ^(-p))
-      hC₀ hC₁ hE hθ hθ1 ha ha1 hd (fun n => S n ω) (fun n => A n ω)
-      (fun n => hnonneg n ω) (hini ω) (fun n => hrec n ω (hgood n))
-    obtain ⟨n,hn⟩ := hω
-    have hn' : S n ω ≤ M₀*δ^(-p) := hbound n
-    nlinarith
-  calc
-    _ ≤ ν (⋃ n, B n) := measure_mono hsubset
-    _ ≤ ∑' n, ν (B n) := measure_iUnion_le _
-    _ ≤ ∑' n : ℕ, ENNReal.ofReal
-        (3*v*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4))) := ENNReal.tsum_le_tsum hfail
-    _ ≤ ENNReal.ofReal (3*v*C*Real.exp (-c*δ^(-2:ℝ))) := hseries (3*v) _ (by positivity) hT
-    _ = _ := by congr 1; ring
-
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_mesh_uniform_past_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8)
-    (m : SourceMeshParameters κ) :
-    ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-      (c₀ : ℂ) (r δ : ℝ), 0 < r → m.q*δ ≤ r → r ≤ δ → δ ≤ 1 → 3*m.L*δ ≤ 1 →
-      (hy : ∀ z ∈ ball c₀ 1, δ < z.im) →
-      ∀ (E : Set a.samples), MeasurableSet E →
-      (∀ (v : ℝ) (_hv : 0 < v) (hvd : v ≤ δ) (z : ℂ),
-        MeasurableSet[(source_collarStart_stopping a (ball c₀ 1) v
-          (fun z hz => hvd.trans (hy z hz).le) z).measurableSpace] E) →
-      ((P.comap Subtype.val).restrict E) {ω | ∃ n : ℕ, M*δ^(-codimension κ) <
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^n) (ball c₀ 1)} ≤
-        ENNReal.ofReal ((P.comap Subtype.val).real E*C*Real.exp (-c*δ^(-2:ℝ))) := by
-  let p := codimension κ
-  let K := m.q^(-p)
-  let b := 1/(512*K^2*m.L^2)
-  let θ := m.u^(-m.c)
-  let β := (m.q^p/(m.u*m.q))^2
-  have hk : 0 < κ := by linarith
-  have hp : 0 < p := by dsimp [p,codimension,dimension]; linarith
-  have hp1 : p < 1 := by dsimp [p,codimension,dimension]; linarith
-  have hK : 0 < K := Real.rpow_pos_of_pos m.hq _
-  have hL : 0 < m.L := zero_lt_one.trans m.hL
-  have hu : 0 < m.u := zero_lt_one.trans m.hu
-  have hb : 0 < b := by dsimp [b]; positivity
-  have hθ : 0 < θ := Real.rpow_pos_of_pos hu _
-  have hθ1 : θ < 1 := Real.rpow_lt_one_of_one_lt_of_neg m.hu (neg_neg_of_pos m.hc)
-  have hβ : 1 < β := by dsimp [β]; nlinarith [m.hratio]
-  obtain ⟨M,C,c,hM,hC,hc,hprop⟩ := source_mesh_finite_measure_propagation p
-    ((volume (ball (0:ℂ) 1)).toReal*K) m.C (2*K) b θ β hp hp1 (by positivity) m.hC.le
-    (by positivity) hb hθ.le hθ1 hβ
-  refine ⟨M,C,c,hM,hC,hc,?_⟩
-  intro Ω ms P hP γ a c₀ r δ hr hqr hrδ hδ1 hR hy E hE hEs
-  let : IsProbabilityMeasure P := hP
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  have hδ : 0 < δ := hr.trans_le hrδ
-  let S (n : ℕ) (ω : a.samples) := spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
-    (γ ω) (a.G ω) (r*m.q^n) (ball c₀ 1)
-  let A (n : ℕ) (ω : a.samples) := sourceMeshCount p (ball c₀ 1) (γ ω) (r*m.q^n)
-  have hrn (n : ℕ) : 0 < r*m.q^n := mul_pos hr (pow_pos m.hq _)
-  have hyr (n : ℕ) (z : ℂ) (hz : z ∈ ball c₀ 1) : r*m.q^n ≤ z.im :=
-    (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n).trans (hy z hz).le
-  have hfin : volume (ball c₀ 1) ≠ ∞ := by finiteness
-  have hnonneg (n : ℕ) (ω : a.samples) : 0 ≤ S n ω ∧ 0 ≤ A n ω :=
-    ⟨(a.spatialRealMass_bound hk hκ8 _ (hrn n) _ measurableSet_ball hfin (hyr n) ω).1,
-      (source_mesh_count_bound a _ measurableSet_ball hfin p _ (hrn n) (hyr n) ω).1⟩
-  have hini (ω : a.samples) : S 0 ω ≤ ((volume (ball (0:ℂ) 1)).toReal*K)*δ^(-p) ∧
-      A 0 ω ≤ ((volume (ball (0:ℂ) 1)).toReal*K)*δ^(-p) := by
-    have hs := (a.spatialRealMass_bound hk hκ8 r hr _ measurableSet_ball hfin
-      (fun z hz => hrδ.trans (hy z hz).le) ω).2
-    have ha := (source_mesh_count_bound a _ measurableSet_ball hfin p r hr
-      (fun z hz => hrδ.trans (hy z hz).le) ω).2
-    have hbound : (volume (ball c₀ 1)).toReal*r^(-p) ≤ ((volume (ball (0:ℂ) 1)).toReal*K)*δ^(-p) := by
-      rw [source_mesh_ball_volume c₀]
-      calc
-        _ ≤ (volume (ball (0:ℂ) 1)).toReal*(m.q^(-p)*δ^(-p)) :=
-          mul_le_mul_of_nonneg_left (source_mesh_initial_scale hp m.hq hδ hqr) ENNReal.toReal_nonneg
-        _ = _ := by ring
-    constructor
-    · simpa only [S,pow_zero,mul_one] using hs.trans hbound
-    · change (r*m.q^0)^(-p)*(∫ z in ball c₀ 1, if sourceCollarStart (ball c₀ 1) (γ ω) z (r*m.q^0) = ⊤ then (0:ℝ) else 1) ≤ _
-      rw [pow_zero,mul_one]
-      exact ha.trans hbound
-  have hstep (n : ℕ) := source_mesh_grid_step a hκ hκ8 m (ball c₀ 1) measurableSet_ball c₀
-    subset_rfl r δ hr hrδ hR hy n E hE
-    (hEs _ (hrn n) (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n))
-  choose F hFm hFt hFg using hstep
-  exact hprop a.samples inferInstance (μ.restrict E) (μ.real E) δ (m.Cbad*m.L^(-m.c)*K)
-    measureReal_nonneg hδ hδ1 (mul_nonneg (mul_nonneg m.hCb.le (Real.rpow_nonneg hL.le _)) hK.le) m.halpha S A F hnonneg hini hFt hFg
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_conditional_indicator_bound_on
-    {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] (hm : m ≤ mΩ) (B F : Set Ω)
-    (hB : MeasurableSet[m] B) (hF : MeasurableSet F) (G : ℝ) (hG : 0 ≤ G)
-    (ht : ∀ A : Set Ω, MeasurableSet[m] A →
-      (μ.restrict (A∩B)) F ≤ ENNReal.ofReal (μ.real (A∩B)*G)) :
-    ∀ᵐ ω ∂μ, ω ∈ B → μ[(fun ω => if ω ∈ F then (1:ℝ) else 0) | m] ω ≤ G := by
-  let f : Ω → ℝ := fun ω => if ω ∈ F then 1 else 0
-  have hf : Integrable f μ := by
-    exact (integrable_const (1:ℝ)).indicator hF
-  have hBm : MeasurableSet B := hm _ hB
-  have hfg : μ[B.indicator f | m] ≤ᵐ[μ] B.indicator (fun _ => G) := by
-    apply source_condExp_le_of_setIntegral_le hm (hf.indicator hBm)
-      ((integrable_const _).indicator hBm) (stronglyMeasurable_const.indicator hB)
-    intro A hA
-    rw [setIntegral_indicator hBm,setIntegral_indicator hBm]
-    change (∫ ω in A∩B, F.indicator (fun _ => (1:ℝ)) ω ∂μ) ≤ _
-    have he : (∫ ω in A∩B, F.indicator (fun _ => (1:ℝ)) ω ∂μ) = (μ.restrict (A∩B)).real F := by
-      exact integral_indicator_one hF
-    rw [he]
-    have hh := (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity : 0 ≤ μ.real (A∩B)*G)).mp (ht A hA)
-    simpa only [integral_const,Measure.restrict_apply_univ,smul_eq_mul,measureReal_def] using hh
-  filter_upwards [hfg,condExp_indicator hf hB] with ω hg he
-  intro hω
-  rw [he,Set.indicator_of_mem hω,Set.indicator_of_mem hω] at hg
-  exact hg
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
- 
-
-def SourcePhysicalMeshTailTarget : Prop :=
-  ∀ κ : ℝ, 4 < κ → κ < 8 → ∀ η : ℝ, 0 < η →
-  ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < η ∧
-  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (ρ : a.samples → WithTop ℝ≥0) (hρ : IsStoppingTime a.filtration ρ)
-    (z₀ : ℂ) (r_f δ : ℝ), 0 < r_f → r_f ≤ δ → δ ≤ δ₀ →
-  let μ := P.comap (Subtype.val : a.samples → Ω)
-  let E := {ω : a.samples | C*δ^(-codimension κ) < spatialRealMass κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r_f (Metric.ball z₀ 1)}
-  ∀ᵐ ω ∂μ, ρ ω ≠ ⊤ →
-    Metric.closedBall z₀ (1+η) ⊆ survivingDomain (γ ω) ((ρ ω).untopD 0) →
-    μ[(fun ω => if ω ∈ E then (1:ℝ) else 0) | hρ.measurableSpace] ω ≤
-      C*Real.exp (-c*δ^(-2:ℝ))
-
-theorem sourcePhysicalMeshTail_proved : SourcePhysicalMeshTailTarget := by
-  intro κ hκ hκ8 η hη
-  obtain ⟨m⟩ := source_mesh_parameters κ hκ hκ8
-  obtain ⟨M,C,c,hM,hC,hc,htail⟩ := source_mesh_uniform_past_tail κ hκ hκ8 m
-  obtain ⟨δ₀,hδ₀,hδη,hδ1,hδR⟩ := source_mesh_choose_delta η m.L hη (zero_lt_one.trans m.hL)
-  let CT := max M C
-  have hCT : 0 < CT := hM.trans_le (le_max_left _ _)
-  refine ⟨CT,c,δ₀,hCT,hc,hδ₀,by linarith,?_⟩
-  intro Ω ms P hP γ a ρ hρ z₀ rf δ hf hfδ hδsmall μ F
-  let : IsProbabilityMeasure P := hP
-  have hδ : 0 < δ := hf.trans_le hfδ
-  have hδh : δ < η/2 := hδsmall.trans_lt hδη
-  by_cases hz : 1+η ≤ z₀.im
-  · obtain ⟨r,n,hr,hqr,hrδ,hfr⟩ := source_mesh_select_start m.hq m.hq1 hf hfδ
-    have hy (z : ℂ) (hz' : z ∈ ball z₀ 1) : δ < z.im :=
-      source_mesh_height hz hz' (by linarith)
-    let τ (ω : a.samples) := spatialHitTime (γ ω) z₀ (1+η/2)
-    have hτ : IsStoppingTime a.filtration τ := source_mesh_center_stopping a z₀ η
-    let B := {ω : a.samples | ρ ω ≤ τ ω}
-    have hB : MeasurableSet[hρ.measurableSpace] B := hρ.measurableSet_le_stopping_time hτ
-    have hF : MeasurableSet F := measurableSet_lt measurable_const (a.spatialRealMass_measurable (ball z₀ 1) rf)
-    have hbound : ∀ᵐ ω ∂μ, ω ∈ B →
-        μ[(fun ω => if ω ∈ F then (1:ℝ) else 0) | hρ.measurableSpace] ω ≤ CT*Real.exp (-c*δ^(-2:ℝ)) := by
-      apply source_conditional_indicator_bound_on hρ.measurableSpace_le B F hB hF _ (by positivity)
-      intro A hA
-      have hE : MeasurableSet (A∩B) := hρ.measurableSpace_le _ (hA.inter hB)
-      have hEτ : MeasurableSet[hτ.measurableSpace] (A∩B) :=
-        ((hρ.min hτ).measurableSpace_mono hτ (fun ω => min_le_right _ _)) _ (hρ.measurableSet_inter_le hτ A hA)
-      have hEs (v : ℝ) (hv : 0 < v) (hvd : v ≤ δ) (z : ℂ) :
-          MeasurableSet[(source_collarStart_stopping a (ball z₀ 1) v
-            (fun z hz => hvd.trans (hy z hz).le) z).measurableSpace] (A∩B) := by
-        apply (hτ.measurableSpace_mono (source_collarStart_stopping a (ball z₀ 1) v
-          (fun z hz => hvd.trans (hy z hz).le) z) _ ) _ hEτ
-        intro ω
-        exact source_mesh_center_before a (ball z₀ 1) z₀ subset_rfl η v hη
-          (hvd.trans_lt hδh) hz (fun z hz => hvd.trans (hy z hz).le) z ω
-      have hR : 3*m.L*δ ≤ 1 := (mul_le_mul_of_nonneg_left hδsmall (mul_nonneg (by norm_num) (zero_lt_one.trans m.hL).le)).trans hδR
-      have ht := htail Ω ms P hP γ a z₀ r δ hr hqr hrδ (hδsmall.trans hδ1) hR hy (A∩B) hE hEs
-      have hsub : F ⊆ {ω : a.samples | ∃ k : ℕ, M*δ^(-codimension κ) <
-          spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^k) (ball z₀ 1)} := by
-        intro ω hω
-        refine ⟨n,?_⟩
-        have hh : M*δ^(-codimension κ) ≤ CT*δ^(-codimension κ) :=
-          mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg hδ.le _)
-        change CT*δ^(-codimension κ) < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1) at hω
-        rw [hfr] at hω
-        exact hh.trans_lt hω
-      calc
-        _ ≤ ((P.comap Subtype.val).restrict (A∩B)) {ω : a.samples | ∃ k : ℕ, M*δ^(-codimension κ) <
-            spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^k) (ball z₀ 1)} := measure_mono hsub
-        _ ≤ _ := ht.trans (ENNReal.ofReal_le_ofReal (by
-          have hh := mul_le_mul_of_nonneg_left (le_max_right M C) (measureReal_nonneg (μ := μ) (s := A∩B))
-          have hh' := mul_le_mul_of_nonneg_right hh (Real.exp_pos (-c*δ^(-2:ℝ))).le
-          simpa only [mul_assoc] using hh'))
-    filter_upwards [hbound] with ω hω
-    intro hρfin hfree
-    apply hω
-    change ρ ω ≤ spatialHitTime (γ ω) z₀ (1+η/2)
-    obtain ⟨s,hs⟩ := WithTop.ne_top_iff_exists.mp hρfin
-    rw [←hs] at hfree ⊢
-    simp only [WithTop.untopD_coe] at hfree
-    exact source_mesh_free_before_center (a.trace_continuous ω) s z₀ η hη hfree
-  · exact Eventually.of_forall fun ω _ hfree => False.elim (hz (source_mesh_free_im hη hfree).le)
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-lemma source_indicator_integral_tail {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
-    {μ : Measure Ω} [IsProbabilityMeasure μ]
-    (hm : m ≤ mΩ) (E : Set Ω) (hE : MeasurableSet E) (K : ℝ) (hK : 0 ≤ K)
-    (ht : μ[(fun ω => if ω ∈ E then (1:ℝ) else 0) | m] ≤ᵐ[μ] fun _ => K) :
-    μ E ≤ ENNReal.ofReal K := by
-  have hi : Integrable (fun ω => if ω ∈ E then (1:ℝ) else 0) μ :=
-    (integrable_const (1:ℝ)).indicator hE
-  have hh := integral_mono_ae integrable_condExp (integrable_const K) ht
-  rw [integral_condExp hm] at hh
-  have he : (∫ ω, (if ω ∈ E then (1:ℝ) else 0) ∂μ) = μ.real E := integral_indicator_one hE
-  rw [he] at hh
-  simp only [integral_const,measureReal_def,measure_univ,ENNReal.toReal_one,one_smul] at hh
-  exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) hK).mpr hh
-
-lemma source_initial_unit_mesh_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
-    ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < 1 ∧
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
-    ∀ (rf δ : ℝ), 0 < rf → rf ≤ δ → δ ≤ δ₀ →
-      (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) <
-        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
-        ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
-  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := sourcePhysicalMeshTail_proved κ hκ hκ8 1 (by norm_num)
-  refine ⟨C,c,δ₀,hC,hc,hδ,hδ1,?_⟩
-  intro Ω m P hP γ a z₀ hz rf δ hf hfd hd
-  let : IsProbabilityMeasure P := hP
-  let ρ : a.samples → WithTop ℝ≥0 := fun _ => 0
-  have hρ : IsStoppingTime a.filtration ρ := isStoppingTime_const a.filtration 0
-  have hb := ht Ω m P hP γ a ρ hρ z₀ rf δ hf hfd hd
-  apply source_indicator_integral_tail hρ.measurableSpace_le _
-    (measurableSet_lt measurable_const (a.spatialRealMass_measurable _ _)) _ (by positivity)
-  filter_upwards [hb] with ω hω
-  apply hω (by simp [ρ])
-  change closedBall z₀ (1+1) ⊆ survivingDomain (γ ω) 0
-  rw [source_witness_initial_domain (a.witness ω)]
-  intro z hz'
-  have hdist : abs (z.im-z₀.im) ≤ dist z z₀ := by
-    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
-  have hh : dist z z₀ ≤ 1+1 := hz'
-  have ha := neg_abs_le (z.im-z₀.im)
-  change 0 < z.im
-  linarith
-
-lemma source_mesh_tail_to_linear {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (X : Ω → ℝ) (p C c δ₀ v rf : ℝ) (hp : 0 < p) (hp1 : p ≤ 1)
-    (hC : 0 < C) (hc : 0 < c) (hδ : 0 < δ₀) (hv : 0 ≤ v) (_hf : 0 < rf)
-    (hbound : ∀ ω, X ω ≤ v*rf^(-p))
-    (ht : ∀ δ : ℝ, rf ≤ δ → δ ≤ δ₀ → μ {ω | C*δ^(-p) < X ω} ≤
-      ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ)))) :
-    ∀ t : ℝ, 1 ≤ t →
-      μ {ω | max C v*δ₀^(-p)*t < X ω} ≤
-        ENNReal.ofReal (C*Real.exp (-(c*δ₀^(-2:ℝ))*t)) := by
-  intro t ht1
-  have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht1
-  let δ := δ₀/t
-  have hδp : 0 < δ := div_pos hδ ht0
-  have hδd : δ ≤ δ₀ := by dsimp [δ]; exact div_le_self hδ.le ht1
-  have he : δ^(-p) = δ₀^(-p)*t^p := by
-    dsimp [δ]
-    rw [Real.div_rpow hδ.le ht0.le,Real.rpow_neg ht0.le,div_inv_eq_mul]
-  have htp : t^p ≤ t := by
-    simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le ht1 hp1
-  have hM : 0 < max C v := hC.trans_le (le_max_left _ _)
-  have hn (a : ℝ) (ha : a ≤ max C v) (ha0 : 0 ≤ a) :
-      a*δ^(-p) ≤ max C v*δ₀^(-p)*t := by
-    rw [he,←mul_assoc]
-    exact mul_le_mul (mul_le_mul_of_nonneg_right ha (Real.rpow_nonneg hδ.le _))
-      htp (Real.rpow_nonneg ht0.le _) (by positivity)
-  by_cases hfd : rf ≤ δ
-  · apply (measure_mono (show {ω | max C v*δ₀^(-p)*t < X ω} ⊆
-        {ω | C*δ^(-p) < X ω} from fun ω hω => (hn C (le_max_left _ _) hC.le).trans_lt hω)).trans
-    apply (ht δ hfd hδd).trans
-    apply ENNReal.ofReal_le_ofReal
-    apply mul_le_mul_of_nonneg_left _ hC.le
-    apply Real.exp_le_exp.mpr
-    have hex : δ^(-2:ℝ) = δ₀^(-2:ℝ)*t^2 := by
-      dsimp [δ]
-      rw [Real.div_rpow hδ.le ht0.le,Real.rpow_neg ht0.le,div_inv_eq_mul,Real.rpow_two]
-    rw [hex]
-    have hsq : t ≤ t^2 := by nlinarith
-    nlinarith [mul_nonneg (mul_nonneg hc.le (Real.rpow_nonneg hδ.le (-2))) (sub_nonneg.mpr hsq)]
-  · have hx (ω : Ω) : X ω ≤ max C v*δ₀^(-p)*t := by
-      calc
-        X ω ≤ v*rf^(-p) := hbound ω
-        _ ≤ v*δ^(-p) := mul_le_mul_of_nonneg_left
-          (Real.rpow_le_rpow_of_nonpos hδp (le_of_not_ge hfd) (by linarith)) hv
-        _ ≤ _ := hn v (le_max_right _ _) hv
-    have he : {ω | max C v*δ₀^(-p)*t < X ω} = ∅ := by
-      ext ω; simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]; exact not_lt_of_ge (hx ω)
-    rw [he,measure_empty]
-    exact bot_le
-end
-end SLEExactGauge
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-lemma source_linear_tail_all {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    [IsProbabilityMeasure μ] (X : Ω → ℝ) (M C b : ℝ)
-    (hM : 0 < M) (_hC : 0 < C) (hb : 0 < b)
-    (ht : ∀ t : ℝ, 1 ≤ t → μ {ω | M*t < X ω} ≤ ENNReal.ofReal (C*Real.exp (-b*t))) :
-    ∀ x : ℝ, 0 < x → μ {ω | x < X ω} ≤
-      ENNReal.ofReal ((max 1 C*Real.exp b)*Real.exp (-(b/M)*x)) := by
-  intro x hx
-  by_cases hxM : M ≤ x
-  · have hh := ht (x/M) ((one_le_div hM).mpr hxM)
-    rw [mul_div_cancel₀ _ hM.ne'] at hh
-    apply hh.trans
-    apply ENNReal.ofReal_le_ofReal
-    have hm : C ≤ max 1 C*Real.exp b := by
-      calc
-        C ≤ max 1 C := le_max_right _ _
-        _ ≤ _ := le_mul_of_one_le_right (by positivity) ((Real.one_le_exp_iff).mpr hb.le)
-    convert mul_le_mul_of_nonneg_right hm (Real.exp_pos (-(b/M)*x)).le using 1
-    congr 2
-    ring
-  · apply (prob_le_one : μ {ω | x < X ω} ≤ 1).trans
-    apply (ENNReal.one_le_ofReal).mpr
-    rw [mul_assoc,←Real.exp_add]
-    have he : 1 ≤ Real.exp (b+-(b/M)*x) := by
-      apply Real.one_le_exp_iff.mpr
-      have hh : (b/M)*x ≤ b := by
-        calc
-          (b/M)*x ≤ (b/M)*M := mul_le_mul_of_nonneg_left (le_of_not_ge hxM) (by positivity)
-          _ = b := div_mul_cancel₀ _ hM.ne'
-      linarith
-    calc
-      1 ≤ max 1 C := le_max_left _ _
-      _ ≤ _ := le_mul_of_one_le_right (by positivity) he
-
-lemma source_exponential_second_moment {Ω : Type*} [MeasurableSpace Ω]
-    (μ : Measure Ω) (X : Ω → ℝ) (hXm : Measurable X) (hX : ∀ ω, 0 ≤ X ω)
-    (A b : ℝ) (hA : 0 ≤ A) (hb : 0 < b)
-    (ht : ∀ t : ℝ, 0 < t → μ {ω | t < X ω} ≤ ENNReal.ofReal (A*Real.exp (-b*t))) :
-    (∫⁻ ω, ENNReal.ofReal (X ω)^2 ∂μ) ≤ ENNReal.ofReal (8*A/b^2) := by
-  have h0 : ∀ x : ℝ, (∫ t in (0:ℝ)..x, (2:ℝ)*t) = x^2 := by
-    intro x
-    rw [intervalIntegral.integral_const_mul,integral_id]
-    ring
-  have he := lintegral_comp_eq_lintegral_meas_lt_mul μ (Eventually.of_forall hX)
-    hXm.aemeasurable (g := fun t : ℝ => 2*t)
-    (fun t ht => (continuous_const.mul continuous_id).intervalIntegrable 0 t)
-    (ae_restrict_mem measurableSet_Ioi |>.mono fun t ht => mul_nonneg (by norm_num) (show 0 < t from ht).le)
-  simp only [h0,ENNReal.ofReal_pow (hX _)] at he
-  rw [he]
-  have hbi : IntegrableOn (fun t : ℝ => Real.exp (-(b/2)*t)) (Ioi 0) :=
-    integrableOn_exp_mul_Ioi (by linarith : -(b/2) < 0) 0
-  calc
-    _ ≤ ∫⁻ t in Ioi (0:ℝ), ENNReal.ofReal ((4*A/b)*Real.exp (-(b/2)*t)) := by
-      apply setLIntegral_mono' measurableSet_Ioi
-      intro t ht0
-      have htpos : 0 < t := ht0
-      calc
-        _ ≤ ENNReal.ofReal (A*Real.exp (-b*t))*ENNReal.ofReal (2*t) :=
-          by gcongr; exact ht t htpos
-        _ = ENNReal.ofReal ((A*Real.exp (-b*t))*(2*t)) := (ENNReal.ofReal_mul (by positivity)).symm
-        _ ≤ _ := by
-          apply ENNReal.ofReal_le_ofReal
-          have hte : (b/2)*t ≤ Real.exp ((b/2)*t) := (Real.add_one_le_exp _).trans' (by linarith)
-          have hmul := mul_le_mul_of_nonneg_left hte (Real.exp_pos (-(b/2)*t)).le
-          have hev : Real.exp (-(b/2)*t)*Real.exp ((b/2)*t) = 1 := by
-            rw [←Real.exp_add]; ring_nf; exact Real.exp_zero
-          rw [hev] at hmul
-          have hex : Real.exp (-b*t) = Real.exp (-(b/2)*t)*Real.exp (-(b/2)*t) := by
-            rw [←Real.exp_add]; congr 1; ring
-          rw [hex]
-          have htb : t*Real.exp (-(b/2)*t) ≤ 2/b := by
-            apply (le_div_iff₀ hb).mpr
-            nlinarith
-          calc
-            A*(Real.exp (-(b/2)*t)*Real.exp (-(b/2)*t))*(2*t) =
-                (2*A*Real.exp (-(b/2)*t))*(t*Real.exp (-(b/2)*t)) := by ring
-            _ ≤ (2*A*Real.exp (-(b/2)*t))*(2/b) :=
-              mul_le_mul_of_nonneg_left htb (by positivity)
-            _ = (4*A/b)*Real.exp (-(b/2)*t) := by ring
-    _ = ENNReal.ofReal (8*A/b^2) := by
-      rw [←ofReal_integral_eq_lintegral_ofReal (hbi.const_mul (4*A/b))
-        (Eventually.of_forall (fun t => by positivity))]
-      rw [integral_const_mul,integral_exp_mul_Ioi (by linarith : -(b/2) < 0)]
-      simp only [mul_zero,Real.exp_zero]
-      congr 1
-      field_simp
-      ring
-end
-end SLEExactGauge
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.PhysicalMeshPast
+
+namespace OAI
+
+/-! Window Propagation for chordal SLE. -/
+
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+ 
+
+lemma geometric_over_quartic_ge_linear {β : ℝ} (hβ : 1 < β) :
+    ∃ k : ℝ, 0 < k ∧ ∀ n : ℕ, k*(n+1:ℝ) ≤ β^n/(n+1:ℝ)^4 := by
+  have hβ0 : 0 < β := by linarith
+  have hir : 0 < β⁻¹ := inv_pos.mpr hβ0
+  have hi1 : β⁻¹ < 1 := (inv_lt_one₀ hβ0).mpr hβ
+  have hnorm : ‖β⁻¹‖ < 1 := by simpa only [Real.norm_eq_abs,abs_of_pos hir] using hi1
+  have hs := summable_pow_mul_geometric_of_norm_lt_one 5 hnorm
+  have hss := ((summable_nat_add_iff 1).mpr hs).div_const (β⁻¹)
+  have hseq : Summable (fun n : ℕ => (n+1:ℝ)^5/β^n) := by
+    apply hss.congr
+    intro n
+    simp only [Nat.cast_add,Nat.cast_one,pow_succ,inv_pow]
+    field_simp [ne_of_gt hβ0]
+  let T : ℝ := ∑' n : ℕ, (n+1:ℝ)^5/β^n
+  have hn (n : ℕ) : (n+1:ℝ)^5/β^n ≤ T :=
+    hseq.le_tsum n (fun j _ => by positivity)
+  have hT : 0 < T := by have := hn 0; norm_num at this; linarith
+  refine ⟨T⁻¹,inv_pos.mpr hT,fun n => ?_⟩
+  have hn0 : 0 < (n+1:ℝ) := by positivity
+  have hp0 : 0 < β^n := pow_pos hβ0 _
+  have hn' : (n+1:ℝ)^5 ≤ T*β^n := (div_le_iff₀ hp0).mp (hn n)
+  apply (le_div_iff₀ (pow_pos hn0 4)).mpr
+  apply (mul_le_mul_iff_right₀ hT).mp
+  calc
+    T * (T⁻¹ * (n+1:ℝ) * (n+1:ℝ)^4) = (n+1:ℝ)^5 := by field_simp
+    _ ≤ T * β^n := hn'
+
+lemma all_levels_exceptional_sum {b β : ℝ} (hb : 0 < b) (hβ : 1 < β) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ D T : ℝ, 0 ≤ D → 1 ≤ T →
+      (∑' n : ℕ, ENNReal.ofReal (D*Real.exp (-b*T*(β^n/(n+1:ℝ)^4)))) ≤
+        ENNReal.ofReal (D*C*Real.exp (-c*T)) := by
+  obtain ⟨k,hk,hkn⟩ := geometric_over_quartic_ge_linear hβ
+  let c := b*k
+  have hc : 0 < c := mul_pos hb hk
+  let r := Real.exp (-c)
+  have hr : 0 < r := Real.exp_pos _
+  have hr1 : r < 1 := (Real.exp_lt_one_iff).mpr (neg_neg_of_pos hc)
+  refine ⟨(1-r)⁻¹,c,inv_pos.mpr (sub_pos.mpr hr1),hc,?_⟩
+  intro D T hD hT
+  have hT0 : 0 ≤ T := by linarith
+  have hn (n : ℕ) : D*Real.exp (-b*T*(β^n/(n+1:ℝ)^4)) ≤
+      (D*Real.exp (-c*T))*r^n := by
+    have hbound := mul_le_mul_of_nonneg_left (hkn n) (mul_nonneg hb.le hT0)
+    have hneg : -b*T*(β^n/(n+1:ℝ)^4) ≤ -c*T + (n:ℝ)*(-c) := by
+      have hh : c*(T-1)*(n:ℝ) ≥ 0 := by positivity
+      dsimp [c] at *
+      nlinarith
+    calc
+      _ ≤ D*Real.exp (-c*T + (n:ℝ)*(-c)) := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hneg) hD
+      _ = _ := by rw [Real.exp_add,Real.exp_nat_mul]; ring
+  have hsum := (hasSum_geometric_of_lt_one hr.le hr1).mul_left (D*Real.exp (-c*T))
+  calc
+    _ ≤ ∑' n : ℕ, ENNReal.ofReal ((D*Real.exp (-c*T))*r^n) :=
+      ENNReal.tsum_le_tsum (fun n => ENNReal.ofReal_le_ofReal (hn n))
+    _ = ENNReal.ofReal ((D*Real.exp (-c*T))*(1-r)⁻¹) := by
+      rw [← ENNReal.ofReal_tsum_of_nonneg (fun n => by positivity) hsum.summable,hsum.tsum_eq]
+    _ = _ := by congr 1; ring
+
+ 
+lemma reciprocal_square_partial_sum_strong (n : ℕ) :
+    (∑ j ∈ Finset.range (n+1), 1/(j+1:ℝ)^2) ≤ 2 - 1/(n+1:ℝ) := by
+  induction n with
+  | zero => norm_num
+  | succ n ih =>
+    rw [Finset.sum_range_succ]
+    have hn : 0 < (n+1:ℝ) := by positivity
+    have hn2 : 0 < (n+2:ℝ) := by positivity
+    have hp : 1/(n+2:ℝ)^2 ≤ 1/(n+1:ℝ) - 1/(n+2:ℝ) := by
+      calc
+        _ ≤ 1/((n+1:ℝ)*(n+2:ℝ)) := by
+          apply one_div_le_one_div_of_le (mul_pos hn hn2)
+          nlinarith
+        _ = _ := by field_simp; ring
+    simp only [Nat.cast_add,Nat.cast_one] at *
+    rw [show (n:ℝ)+1+1 = n+2 by ring]
+    linarith
+
+lemma reciprocal_square_partial_sum_le_two (n : ℕ) :
+    (∑ j ∈ Finset.range n, 1/(j+1:ℝ)^2) ≤ 2 := by
+  cases n with
+  | zero => norm_num
+  | succ n =>
+    have := reciprocal_square_partial_sum_strong n
+    have : 0 ≤ 1/(n+1:ℝ) := by positivity
+    linarith [reciprocal_square_partial_sum_strong n]
+
+lemma discrete_gronwall_exp (U α e : ℕ → ℝ) (hU : 0 ≤ U 0)
+    (hα : ∀ n, 0 ≤ α n) (he : ∀ n, 0 ≤ e n)
+    (hrec : ∀ n, U (n+1) ≤ (1+α n)*U n+e n) :
+    ∀ n, U n ≤ (U 0 + ∑ j ∈ Finset.range n, e j) *
+      Real.exp (∑ j ∈ Finset.range n, α j) := by
+  intro n
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ,Finset.sum_range_succ]
+    have hV : 0 ≤ U 0 + ∑ j ∈ Finset.range n, e j :=
+      add_nonneg hU (Finset.sum_nonneg (fun j _ => he j))
+    have hsum : 0 ≤ (∑ j ∈ Finset.range n, α j) + α n :=
+      add_nonneg (Finset.sum_nonneg (fun j _ => hα j)) (hα n)
+    have hExp : 1 ≤ Real.exp ((∑ j ∈ Finset.range n, α j)+α n) :=
+      Real.one_le_exp_iff.mpr hsum
+    have hfac : 1+α n ≤ Real.exp (α n) := by linarith [Real.add_one_le_exp (α n)]
+    calc
+      U (n+1) ≤ (1+α n)*U n+e n := hrec n
+      _ ≤ (1+α n)*((U 0+∑ j ∈ Finset.range n,e j)*
+          Real.exp (∑ j ∈ Finset.range n,α j))+e n :=
+        by
+          gcongr
+          · linarith [hα n]
+      _ ≤ (U 0+∑ j ∈ Finset.range n,e j)*
+          Real.exp ((∑ j ∈ Finset.range n,α j)+α n) +
+          e n*Real.exp ((∑ j ∈ Finset.range n,α j)+α n) := by
+        apply add_le_add
+        · rw [Real.exp_add]
+          nlinarith [mul_le_mul_of_nonneg_right hfac
+            (mul_nonneg hV (Real.exp_pos (∑ j ∈ Finset.range n,α j)).le)]
+        · exact le_mul_of_one_le_right (he n) hExp
+      _ = _ := by ring
+
+lemma two_recurrences_normalized (S A α e : ℕ → ℝ) (C₁ K : ℝ)
+    (hC : 0 ≤ C₁) (hK : 1 ≤ K) (hCK : C₁ ≤ K)
+    (hnonneg : ∀ n, 0 ≤ S n ∧ 0 ≤ A n)
+    (hα : ∀ n, 0 ≤ α n) (he : ∀ n, 0 ≤ e n)
+    (hrec : ∀ n, S (n+1) ≤ S n+α n*A n+e n ∧
+      A (n+1) ≤ C₁*S n+α n*A n+e n) :
+    ∀ n, max (S (n+1)) (A (n+1)/K) ≤
+      (1+K*α n)*max (S n) (A n/K)+e n := by
+  have hK0 : 0 < K := by linarith
+  intro n
+  let U := max (S n) (A n/K)
+  have hS : S n ≤ U := le_max_left _ _
+  have hA : A n ≤ K*U := by
+    have := (div_le_iff₀ hK0).mp (le_max_right (S n) (A n/K))
+    simpa only [mul_comm] using this
+  have hU : 0 ≤ U := (hnonneg n).1.trans hS
+  apply max_le
+  · calc
+      S (n+1) ≤ S n+α n*A n+e n := (hrec n).1
+      _ ≤ U+α n*(K*U)+e n := by gcongr; exact hα n
+      _ = _ := by ring
+  · apply (div_le_iff₀ hK0).mpr
+    calc
+      A (n+1) ≤ C₁*S n+α n*A n+e n := (hrec n).2
+      _ ≤ K*U+K*(K*α n*U)+K*e n := by
+        apply add_le_add
+        · apply add_le_add
+          · exact (mul_le_mul_of_nonneg_left hS hC).trans
+              (mul_le_mul_of_nonneg_right hCK hU)
+          · calc
+              α n*A n ≤ α n*(K*U) := mul_le_mul_of_nonneg_left hA (hα n)
+              _ = K*α n*U := by ring
+              _ ≤ K*(K*α n*U) := le_mul_of_one_le_left (mul_nonneg (mul_nonneg hK0.le (hα n)) hU) hK
+        · exact le_mul_of_one_le_left (he n) hK
+      _ = _ := by ring
+
+lemma window_recurrences_uniform (C₀ C₁ E θ a d : ℝ)
+    (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) (hE : 0 ≤ E)
+    (hθ : 0 ≤ θ) (hθ1 : θ < 1) (ha : 0 ≤ a) (ha1 : a ≤ 1/2)
+    (hd : 0 < d) (S A : ℕ → ℝ)
+    (hnonneg : ∀ n, 0 ≤ S n ∧ 0 ≤ A n)
+    (hini : S 0 ≤ C₀*d ∧ A 0 ≤ C₀*d)
+    (hrec : ∀ n, S (n+1) ≤ S n+(a*θ^n)*A n+E*d/(n+1:ℝ)^2 ∧
+      A (n+1) ≤ C₁*S n+(a*θ^n)*A n+E*d/(n+1:ℝ)^2) :
+    ∀ n, S n ≤ ((C₀+2*E)*Real.exp ((max 1 (2*C₁))/2/(1-θ)))*d := by
+  let K := max 1 (2*C₁)
+  have hK : 1 ≤ K := le_max_left _ _
+  have hK0 : 0 < K := by linarith
+  have hCK : C₁ ≤ K := (by linarith : C₁ ≤ 2*C₁).trans (le_max_right _ _)
+  let U := fun n => max (S n) (A n/K)
+  let α := fun n : ℕ => a*θ^n
+  let e := fun n : ℕ => E*d/(n+1:ℝ)^2
+  have hα (n : ℕ) : 0 ≤ α n := mul_nonneg ha (pow_nonneg hθ _)
+  have he (n : ℕ) : 0 ≤ e n := by dsimp [e]; positivity
+  have hU0 : 0 ≤ U 0 := (hnonneg 0).1.trans (le_max_left _ _)
+  have hrecU := two_recurrences_normalized S A α e C₁ K hC₁ hK hCK hnonneg hα he hrec
+  have hG := discrete_gronwall_exp U (fun n => K*α n) e hU0
+    (fun n => mul_nonneg hK0.le (hα n)) he hrecU
+  have hIni : U 0 ≤ C₀*d := by
+    apply max_le hini.1
+    apply (div_le_iff₀ hK0).mpr
+    exact hini.2.trans (le_mul_of_one_le_right (mul_nonneg hC₀ hd.le) hK)
+  have hGeom := hasSum_geometric_of_lt_one hθ hθ1
+  intro n
+  have hsum_geom : (∑ j ∈ Finset.range n, θ^j) ≤ (1-θ)⁻¹ := by
+    rw [← hGeom.tsum_eq]
+    exact hGeom.summable.sum_le_tsum _ (fun j _ => pow_nonneg hθ j)
+  have he_sum : (∑ j ∈ Finset.range n, e j) ≤ 2*E*d := by
+    calc
+      _ = (E*d)*(∑ j ∈ Finset.range n, 1/(j+1:ℝ)^2) := by
+        rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j hj; dsimp [e]; ring
+      _ ≤ (E*d)*2 := mul_le_mul_of_nonneg_left
+        (reciprocal_square_partial_sum_le_two n) (mul_nonneg hE hd.le)
+      _ = _ := by ring
+  have hα_sum : (∑ j ∈ Finset.range n, K*α j) ≤ K/2/(1-θ) := by
+    calc
+      _ = (K*a)*(∑ j ∈ Finset.range n, θ^j) := by
+        rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j hj; dsimp [α]; ring
+      _ ≤ (K*a)*(1-θ)⁻¹ := mul_le_mul_of_nonneg_left hsum_geom (mul_nonneg hK0.le ha)
+      _ ≤ (K*(1/2))*(1-θ)⁻¹ := mul_le_mul_of_nonneg_right
+        (mul_le_mul_of_nonneg_left ha1 hK0.le) (inv_nonneg.mpr (sub_pos.mpr hθ1).le)
+      _ = _ := by ring
+  calc
+    S n ≤ U n := le_max_left _ _
+    _ ≤ (U 0+∑ j ∈ Finset.range n,e j)*Real.exp (∑ j ∈ Finset.range n,K*α j) := hG n
+    _ ≤ (C₀*d+2*E*d)*Real.exp (K/2/(1-θ)) := by
+      apply mul_le_mul (add_le_add hIni he_sum) (Real.exp_le_exp.mpr hα_sum)
+      · exact (Real.exp_pos _).le
+      · positivity
+    _ = _ := by dsimp [K]; ring
+
+universe u
+ 
+
+def SourceWindowPropagationTarget : Prop :=
+  ∀ p C₀ C₁ E D b θ β : ℝ,
+    0 < p → p < 1 → 0 ≤ C₀ → 0 ≤ C₁ → 0 ≤ E → 0 ≤ D → 0 < b →
+    0 ≤ θ → θ < 1 → 1 < β →
+    ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
+      ∀ (Ω : Type u) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+      ∀ δ a : ℝ, 0 < δ → δ ≤ 1 → 0 ≤ a → a ≤ 1/2 →
+      ∀ (S A : ℕ → Ω → ℝ) (B : ℕ → Set Ω),
+      (∀ n ω, 0 ≤ S n ω ∧ 0 ≤ A n ω) →
+      (∀ ω, S 0 ω ≤ C₀*δ^(-p) ∧ A 0 ω ≤ C₀*δ^(-p)) →
+      (∀ n, P (B n) ≤ ENNReal.ofReal
+        (D*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4)))) →
+      (∀ n ω, ω ∉ B n →
+        S (n+1) ω ≤ S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2 ∧
+        A (n+1) ω ≤ C₁*S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2) →
+      P {ω | ∃ n, M*δ^(-p) < S n ω} ≤ ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ)))
+
+theorem sourceWindowPropagation : SourceWindowPropagationTarget := by
+  intro p C₀ C₁ E D b θ β hp hp1 hC₀ hC₁ hE hD hb hθ hθ1 hβ
+  obtain ⟨C,c,hC,hc,hseries⟩ := all_levels_exceptional_sum hb hβ
+  let M₀ := (C₀+2*E)*Real.exp ((max 1 (2*C₁))/2/(1-θ))
+  have hM₀ : 0 ≤ M₀ := mul_nonneg (by linarith) (Real.exp_pos _).le
+  refine ⟨M₀+1,D*C+1,c,by linarith,by positivity,hc,?_⟩
+  intro Ω m P hP δ a hδ hδ1 ha ha1 S A B hnonneg hini hfail hrec
+  have hd : 0 < δ^(-p) := Real.rpow_pos_of_pos hδ _
+  have hT : 1 ≤ δ^(-2:ℝ) := by
+    rw [Real.rpow_neg hδ.le,Real.rpow_two]
+    exact one_le_inv₀ (sq_pos_of_pos hδ) |>.mpr (by nlinarith)
+  have hsubset : {ω | ∃ n, (M₀+1)*δ^(-p) < S n ω} ⊆ ⋃ n, B n := by
+    intro ω hω
+    by_contra hnot
+    have hgood (n : ℕ) : ω ∉ B n := by
+      intro hn
+      exact hnot (mem_iUnion.mpr ⟨n,hn⟩)
+    have hbound := window_recurrences_uniform C₀ C₁ E θ a (δ^(-p))
+      hC₀ hC₁ hE hθ hθ1 ha ha1 hd (fun n => S n ω) (fun n => A n ω)
+      (fun n => hnonneg n ω) (hini ω) (fun n => hrec n ω (hgood n))
+    obtain ⟨n,hn⟩ := hω
+    have hn' : S n ω ≤ M₀*δ^(-p) := hbound n
+    nlinarith
+  calc
+    _ ≤ P (⋃ n, B n) := measure_mono hsubset
+    _ ≤ ∑' n, P (B n) := measure_iUnion_le _
+    _ ≤ ∑' n : ℕ, ENNReal.ofReal
+        (D*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4))) := ENNReal.tsum_le_tsum hfail
+    _ ≤ ENNReal.ofReal (D*C*Real.exp (-c*δ^(-2:ℝ))) := hseries D _ hD hT
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (by nlinarith [Real.exp_pos (-c*δ^(-2:ℝ))])
+
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma source_mesh_finite_measure_propagation (p C₀ C₁ E b θ β : ℝ)
+    (hp : 0 < p) (hp1 : p < 1) (hC₀ : 0 ≤ C₀) (hC₁ : 0 ≤ C₁) (hE : 0 ≤ E)
+    (hb : 0 < b) (hθ : 0 ≤ θ) (hθ1 : θ < 1) (hβ : 1 < β) :
+    ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
+      ∀ (Ω : Type) (_ : MeasurableSpace Ω) (ν : Measure Ω) (v δ a : ℝ),
+      0 ≤ v → 0 < δ → δ ≤ 1 → 0 ≤ a → a ≤ 1/2 →
+      ∀ (S A : ℕ → Ω → ℝ) (B : ℕ → Set Ω),
+      (∀ n ω, 0 ≤ S n ω ∧ 0 ≤ A n ω) →
+      (∀ ω, S 0 ω ≤ C₀*δ^(-p) ∧ A 0 ω ≤ C₀*δ^(-p)) →
+      (∀ n, ν (B n) ≤ ENNReal.ofReal
+        (3*v*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4)))) →
+      (∀ n ω, ω ∉ B n →
+        S (n+1) ω ≤ S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2 ∧
+        A (n+1) ω ≤ C₁*S n ω + (a*θ^n)*A n ω + E*δ^(-p)/(n+1:ℝ)^2) →
+      ν {ω | ∃ n, M*δ^(-p) < S n ω} ≤ ENNReal.ofReal (v*C*Real.exp (-c*δ^(-2:ℝ))) := by
+  obtain ⟨C,c,hC,hc,hseries⟩ := all_levels_exceptional_sum hb hβ
+  let M₀ := (C₀+2*E)*Real.exp ((max 1 (2*C₁))/2/(1-θ))
+  have hM₀ : 0 ≤ M₀ := mul_nonneg (by linarith) (Real.exp_pos _).le
+  refine ⟨M₀+1,3*C,c,by linarith,by positivity,hc,?_⟩
+  intro Ω m ν v δ a hv hδ hδ1 ha ha1 S A B hnonneg hini hfail hrec
+  have hd : 0 < δ^(-p) := Real.rpow_pos_of_pos hδ _
+  have hT : 1 ≤ δ^(-2:ℝ) := by
+    rw [Real.rpow_neg hδ.le,Real.rpow_two]
+    exact one_le_inv₀ (sq_pos_of_pos hδ) |>.mpr (by nlinarith)
+  have hsubset : {ω | ∃ n, (M₀+1)*δ^(-p) < S n ω} ⊆ ⋃ n, B n := by
+    intro ω hω
+    by_contra hnot
+    have hgood (n : ℕ) : ω ∉ B n := fun hn => hnot (mem_iUnion.mpr ⟨n,hn⟩)
+    have hbound := window_recurrences_uniform C₀ C₁ E θ a (δ^(-p))
+      hC₀ hC₁ hE hθ hθ1 ha ha1 hd (fun n => S n ω) (fun n => A n ω)
+      (fun n => hnonneg n ω) (hini ω) (fun n => hrec n ω (hgood n))
+    obtain ⟨n,hn⟩ := hω
+    have hn' : S n ω ≤ M₀*δ^(-p) := hbound n
+    nlinarith
+  calc
+    _ ≤ ν (⋃ n, B n) := measure_mono hsubset
+    _ ≤ ∑' n, ν (B n) := measure_iUnion_le _
+    _ ≤ ∑' n : ℕ, ENNReal.ofReal
+        (3*v*Real.exp (-b*δ^(-2:ℝ)*(β^n/(n+1:ℝ)^4))) := ENNReal.tsum_le_tsum hfail
+    _ ≤ ENNReal.ofReal (3*v*C*Real.exp (-c*δ^(-2:ℝ))) := hseries (3*v) _ (by positivity) hT
+    _ = _ := by congr 1; ring
+
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_mesh_uniform_past_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8)
+    (m : SourceMeshParameters κ) :
+    ∃ M C c : ℝ, 0 < M ∧ 0 < C ∧ 0 < c ∧
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+      (c₀ : ℂ) (r δ : ℝ), 0 < r → m.q*δ ≤ r → r ≤ δ → δ ≤ 1 → 3*m.L*δ ≤ 1 →
+      (hy : ∀ z ∈ ball c₀ 1, δ < z.im) →
+      ∀ (E : Set a.samples), MeasurableSet E →
+      (∀ (v : ℝ) (_hv : 0 < v) (hvd : v ≤ δ) (z : ℂ),
+        MeasurableSet[(source_collarStart_stopping a (ball c₀ 1) v
+          (fun z hz => hvd.trans (hy z hz).le) z).measurableSpace] E) →
+      ((P.comap Subtype.val).restrict E) {ω | ∃ n : ℕ, M*δ^(-codimension κ) <
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^n) (ball c₀ 1)} ≤
+        ENNReal.ofReal ((P.comap Subtype.val).real E*C*Real.exp (-c*δ^(-2:ℝ))) := by
+  let p := codimension κ
+  let K := m.q^(-p)
+  let b := 1/(512*K^2*m.L^2)
+  let θ := m.u^(-m.c)
+  let β := (m.q^p/(m.u*m.q))^2
+  have hk : 0 < κ := by linarith
+  have hp : 0 < p := by dsimp [p,codimension,dimension]; linarith
+  have hp1 : p < 1 := by dsimp [p,codimension,dimension]; linarith
+  have hK : 0 < K := Real.rpow_pos_of_pos m.hq _
+  have hL : 0 < m.L := zero_lt_one.trans m.hL
+  have hu : 0 < m.u := zero_lt_one.trans m.hu
+  have hb : 0 < b := by dsimp [b]; positivity
+  have hθ : 0 < θ := Real.rpow_pos_of_pos hu _
+  have hθ1 : θ < 1 := Real.rpow_lt_one_of_one_lt_of_neg m.hu (neg_neg_of_pos m.hc)
+  have hβ : 1 < β := by dsimp [β]; nlinarith [m.hratio]
+  obtain ⟨M,C,c,hM,hC,hc,hprop⟩ := source_mesh_finite_measure_propagation p
+    ((volume (ball (0:ℂ) 1)).toReal*K) m.C (2*K) b θ β hp hp1 (by positivity) m.hC.le
+    (by positivity) hb hθ.le hθ1 hβ
+  refine ⟨M,C,c,hM,hC,hc,?_⟩
+  intro Ω ms P hP γ a c₀ r δ hr hqr hrδ hδ1 hR hy E hE hEs
+  let : IsProbabilityMeasure P := hP
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  have hδ : 0 < δ := hr.trans_le hrδ
+  let S (n : ℕ) (ω : a.samples) := spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω)
+    (γ ω) (a.G ω) (r*m.q^n) (ball c₀ 1)
+  let A (n : ℕ) (ω : a.samples) := sourceMeshCount p (ball c₀ 1) (γ ω) (r*m.q^n)
+  have hrn (n : ℕ) : 0 < r*m.q^n := mul_pos hr (pow_pos m.hq _)
+  have hyr (n : ℕ) (z : ℂ) (hz : z ∈ ball c₀ 1) : r*m.q^n ≤ z.im :=
+    (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n).trans (hy z hz).le
+  have hfin : volume (ball c₀ 1) ≠ ∞ := by finiteness
+  have hnonneg (n : ℕ) (ω : a.samples) : 0 ≤ S n ω ∧ 0 ≤ A n ω :=
+    ⟨(a.spatialRealMass_bound hk hκ8 _ (hrn n) _ measurableSet_ball hfin (hyr n) ω).1,
+      (source_mesh_count_bound a _ measurableSet_ball hfin p _ (hrn n) (hyr n) ω).1⟩
+  have hini (ω : a.samples) : S 0 ω ≤ ((volume (ball (0:ℂ) 1)).toReal*K)*δ^(-p) ∧
+      A 0 ω ≤ ((volume (ball (0:ℂ) 1)).toReal*K)*δ^(-p) := by
+    have hs := (a.spatialRealMass_bound hk hκ8 r hr _ measurableSet_ball hfin
+      (fun z hz => hrδ.trans (hy z hz).le) ω).2
+    have ha := (source_mesh_count_bound a _ measurableSet_ball hfin p r hr
+      (fun z hz => hrδ.trans (hy z hz).le) ω).2
+    have hbound : (volume (ball c₀ 1)).toReal*r^(-p) ≤ ((volume (ball (0:ℂ) 1)).toReal*K)*δ^(-p) := by
+      rw [source_mesh_ball_volume c₀]
+      calc
+        _ ≤ (volume (ball (0:ℂ) 1)).toReal*(m.q^(-p)*δ^(-p)) :=
+          mul_le_mul_of_nonneg_left (source_mesh_initial_scale hp m.hq hδ hqr) ENNReal.toReal_nonneg
+        _ = _ := by ring
+    constructor
+    · simpa only [S,pow_zero,mul_one] using hs.trans hbound
+    · change (r*m.q^0)^(-p)*(∫ z in ball c₀ 1, if sourceCollarStart (ball c₀ 1) (γ ω) z (r*m.q^0) = ⊤ then (0:ℝ) else 1) ≤ _
+      rw [pow_zero,mul_one]
+      exact ha.trans hbound
+  have hstep (n : ℕ) := source_mesh_grid_step a hκ hκ8 m (ball c₀ 1) measurableSet_ball c₀
+    subset_rfl r δ hr hrδ hR hy n E hE
+    (hEs _ (hrn n) (source_mesh_grid_radius_le m.hq m.hq1 hr hrδ n))
+  choose F hFm hFt hFg using hstep
+  exact hprop a.samples inferInstance (μ.restrict E) (μ.real E) δ (m.Cbad*m.L^(-m.c)*K)
+    measureReal_nonneg hδ hδ1 (mul_nonneg (mul_nonneg m.hCb.le (Real.rpow_nonneg hL.le _)) hK.le) m.halpha S A F hnonneg hini hFt hFg
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_conditional_indicator_bound_on
+    {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] (hm : m ≤ mΩ) (B F : Set Ω)
+    (hB : MeasurableSet[m] B) (hF : MeasurableSet F) (G : ℝ) (hG : 0 ≤ G)
+    (ht : ∀ A : Set Ω, MeasurableSet[m] A →
+      (μ.restrict (A∩B)) F ≤ ENNReal.ofReal (μ.real (A∩B)*G)) :
+    ∀ᵐ ω ∂μ, ω ∈ B → μ[(fun ω => if ω ∈ F then (1:ℝ) else 0) | m] ω ≤ G := by
+  let f : Ω → ℝ := fun ω => if ω ∈ F then 1 else 0
+  have hf : Integrable f μ := by
+    exact (integrable_const (1:ℝ)).indicator hF
+  have hBm : MeasurableSet B := hm _ hB
+  have hfg : μ[B.indicator f | m] ≤ᵐ[μ] B.indicator (fun _ => G) := by
+    apply source_condExp_le_of_setIntegral_le hm (hf.indicator hBm)
+      ((integrable_const _).indicator hBm) (stronglyMeasurable_const.indicator hB)
+    intro A hA
+    rw [setIntegral_indicator hBm,setIntegral_indicator hBm]
+    change (∫ ω in A∩B, F.indicator (fun _ => (1:ℝ)) ω ∂μ) ≤ _
+    have he : (∫ ω in A∩B, F.indicator (fun _ => (1:ℝ)) ω ∂μ) = (μ.restrict (A∩B)).real F := by
+      exact integral_indicator_one hF
+    rw [he]
+    have hh := (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) (by positivity : 0 ≤ μ.real (A∩B)*G)).mp (ht A hA)
+    simpa only [integral_const,Measure.restrict_apply_univ,smul_eq_mul,measureReal_def] using hh
+  filter_upwards [hfg,condExp_indicator hf hB] with ω hg he
+  intro hω
+  rw [he,Set.indicator_of_mem hω,Set.indicator_of_mem hω] at hg
+  exact hg
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+ 
+
+def SourcePhysicalMeshTailTarget : Prop :=
+  ∀ κ : ℝ, 4 < κ → κ < 8 → ∀ η : ℝ, 0 < η →
+  ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < η ∧
+  ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (ρ : a.samples → WithTop ℝ≥0) (hρ : IsStoppingTime a.filtration ρ)
+    (z₀ : ℂ) (r_f δ : ℝ), 0 < r_f → r_f ≤ δ → δ ≤ δ₀ →
+  let μ := P.comap (Subtype.val : a.samples → Ω)
+  let E := {ω : a.samples | C*δ^(-codimension κ) < spatialRealMass κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) r_f (Metric.ball z₀ 1)}
+  ∀ᵐ ω ∂μ, ρ ω ≠ ⊤ →
+    Metric.closedBall z₀ (1+η) ⊆ survivingDomain (γ ω) ((ρ ω).untopD 0) →
+    μ[(fun ω => if ω ∈ E then (1:ℝ) else 0) | hρ.measurableSpace] ω ≤
+      C*Real.exp (-c*δ^(-2:ℝ))
+
+theorem sourcePhysicalMeshTail_proved : SourcePhysicalMeshTailTarget := by
+  intro κ hκ hκ8 η hη
+  obtain ⟨m⟩ := source_mesh_parameters κ hκ hκ8
+  obtain ⟨M,C,c,hM,hC,hc,htail⟩ := source_mesh_uniform_past_tail κ hκ hκ8 m
+  obtain ⟨δ₀,hδ₀,hδη,hδ1,hδR⟩ := source_mesh_choose_delta η m.L hη (zero_lt_one.trans m.hL)
+  let CT := max M C
+  have hCT : 0 < CT := hM.trans_le (le_max_left _ _)
+  refine ⟨CT,c,δ₀,hCT,hc,hδ₀,by linarith,?_⟩
+  intro Ω ms P hP γ a ρ hρ z₀ rf δ hf hfδ hδsmall μ F
+  let : IsProbabilityMeasure P := hP
+  have hδ : 0 < δ := hf.trans_le hfδ
+  have hδh : δ < η/2 := hδsmall.trans_lt hδη
+  by_cases hz : 1+η ≤ z₀.im
+  · obtain ⟨r,n,hr,hqr,hrδ,hfr⟩ := source_mesh_select_start m.hq m.hq1 hf hfδ
+    have hy (z : ℂ) (hz' : z ∈ ball z₀ 1) : δ < z.im :=
+      source_mesh_height hz hz' (by linarith)
+    let τ (ω : a.samples) := spatialHitTime (γ ω) z₀ (1+η/2)
+    have hτ : IsStoppingTime a.filtration τ := source_mesh_center_stopping a z₀ η
+    let B := {ω : a.samples | ρ ω ≤ τ ω}
+    have hB : MeasurableSet[hρ.measurableSpace] B := hρ.measurableSet_le_stopping_time hτ
+    have hF : MeasurableSet F := measurableSet_lt measurable_const (a.spatialRealMass_measurable (ball z₀ 1) rf)
+    have hbound : ∀ᵐ ω ∂μ, ω ∈ B →
+        μ[(fun ω => if ω ∈ F then (1:ℝ) else 0) | hρ.measurableSpace] ω ≤ CT*Real.exp (-c*δ^(-2:ℝ)) := by
+      apply source_conditional_indicator_bound_on hρ.measurableSpace_le B F hB hF _ (by positivity)
+      intro A hA
+      have hE : MeasurableSet (A∩B) := hρ.measurableSpace_le _ (hA.inter hB)
+      have hEτ : MeasurableSet[hτ.measurableSpace] (A∩B) :=
+        ((hρ.min hτ).measurableSpace_mono hτ (fun ω => min_le_right _ _)) _ (hρ.measurableSet_inter_le hτ A hA)
+      have hEs (v : ℝ) (hv : 0 < v) (hvd : v ≤ δ) (z : ℂ) :
+          MeasurableSet[(source_collarStart_stopping a (ball z₀ 1) v
+            (fun z hz => hvd.trans (hy z hz).le) z).measurableSpace] (A∩B) := by
+        apply (hτ.measurableSpace_mono (source_collarStart_stopping a (ball z₀ 1) v
+          (fun z hz => hvd.trans (hy z hz).le) z) _ ) _ hEτ
+        intro ω
+        exact source_mesh_center_before a (ball z₀ 1) z₀ subset_rfl η v hη
+          (hvd.trans_lt hδh) hz (fun z hz => hvd.trans (hy z hz).le) z ω
+      have hR : 3*m.L*δ ≤ 1 := (mul_le_mul_of_nonneg_left hδsmall (mul_nonneg (by norm_num) (zero_lt_one.trans m.hL).le)).trans hδR
+      have ht := htail Ω ms P hP γ a z₀ r δ hr hqr hrδ (hδsmall.trans hδ1) hR hy (A∩B) hE hEs
+      have hsub : F ⊆ {ω : a.samples | ∃ k : ℕ, M*δ^(-codimension κ) <
+          spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^k) (ball z₀ 1)} := by
+        intro ω hω
+        refine ⟨n,?_⟩
+        have hh : M*δ^(-codimension κ) ≤ CT*δ^(-codimension κ) :=
+          mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.rpow_nonneg hδ.le _)
+        change CT*δ^(-codimension κ) < spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1) at hω
+        rw [hfr] at hω
+        exact hh.trans_lt hω
+      calc
+        _ ≤ ((P.comap Subtype.val).restrict (A∩B)) {ω : a.samples | ∃ k : ℕ, M*δ^(-codimension κ) <
+            spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (r*m.q^k) (ball z₀ 1)} := measure_mono hsub
+        _ ≤ _ := ht.trans (ENNReal.ofReal_le_ofReal (by
+          have hh := mul_le_mul_of_nonneg_left (le_max_right M C) (measureReal_nonneg (μ := μ) (s := A∩B))
+          have hh' := mul_le_mul_of_nonneg_right hh (Real.exp_pos (-c*δ^(-2:ℝ))).le
+          simpa only [mul_assoc] using hh'))
+    filter_upwards [hbound] with ω hω
+    intro hρfin hfree
+    apply hω
+    change ρ ω ≤ spatialHitTime (γ ω) z₀ (1+η/2)
+    obtain ⟨s,hs⟩ := WithTop.ne_top_iff_exists.mp hρfin
+    rw [←hs] at hfree ⊢
+    simp only [WithTop.untopD_coe] at hfree
+    exact source_mesh_free_before_center (a.trace_continuous ω) s z₀ η hη hfree
+  · exact Eventually.of_forall fun ω _ hfree => False.elim (hz (source_mesh_free_im hη hfree).le)
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+lemma source_indicator_integral_tail {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
+    {μ : Measure Ω} [IsProbabilityMeasure μ]
+    (hm : m ≤ mΩ) (E : Set Ω) (hE : MeasurableSet E) (K : ℝ) (hK : 0 ≤ K)
+    (ht : μ[(fun ω => if ω ∈ E then (1:ℝ) else 0) | m] ≤ᵐ[μ] fun _ => K) :
+    μ E ≤ ENNReal.ofReal K := by
+  have hi : Integrable (fun ω => if ω ∈ E then (1:ℝ) else 0) μ :=
+    (integrable_const (1:ℝ)).indicator hE
+  have hh := integral_mono_ae integrable_condExp (integrable_const K) ht
+  rw [integral_condExp hm] at hh
+  have he : (∫ ω, (if ω ∈ E then (1:ℝ) else 0) ∂μ) = μ.real E := integral_indicator_one hE
+  rw [he] at hh
+  simp only [integral_const,measureReal_def,measure_univ,ENNReal.toReal_one,one_smul] at hh
+  exact (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top _ _) hK).mpr hh
+
+lemma source_initial_unit_mesh_tail (κ : ℝ) (hκ : 4 < κ) (hκ8 : κ < 8) :
+    ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧ δ₀ < 1 ∧
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ) (z₀ : ℂ), 2 < z₀.im →
+    ∀ (rf δ : ℝ), 0 < rf → rf ≤ δ → δ ≤ δ₀ →
+      (P.comap Subtype.val) {ω : a.samples | C*δ^(-codimension κ) <
+        spatialRealMass κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) rf (ball z₀ 1)} ≤
+        ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
+  obtain ⟨C,c,δ₀,hC,hc,hδ,hδ1,ht⟩ := sourcePhysicalMeshTail_proved κ hκ hκ8 1 (by norm_num)
+  refine ⟨C,c,δ₀,hC,hc,hδ,hδ1,?_⟩
+  intro Ω m P hP γ a z₀ hz rf δ hf hfd hd
+  let : IsProbabilityMeasure P := hP
+  let ρ : a.samples → WithTop ℝ≥0 := fun _ => 0
+  have hρ : IsStoppingTime a.filtration ρ := isStoppingTime_const a.filtration 0
+  have hb := ht Ω m P hP γ a ρ hρ z₀ rf δ hf hfd hd
+  apply source_indicator_integral_tail hρ.measurableSpace_le _
+    (measurableSet_lt measurable_const (a.spatialRealMass_measurable _ _)) _ (by positivity)
+  filter_upwards [hb] with ω hω
+  apply hω (by simp [ρ])
+  change closedBall z₀ (1+1) ⊆ survivingDomain (γ ω) 0
+  rw [source_witness_initial_domain (a.witness ω)]
+  intro z hz'
+  have hdist : abs (z.im-z₀.im) ≤ dist z z₀ := by
+    simpa only [dist_eq_norm,Complex.sub_im] using Complex.abs_im_le_norm (z-z₀)
+  have hh : dist z z₀ ≤ 1+1 := hz'
+  have ha := neg_abs_le (z.im-z₀.im)
+  change 0 < z.im
+  linarith
+
+lemma source_mesh_tail_to_linear {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (X : Ω → ℝ) (p C c δ₀ v rf : ℝ) (hp : 0 < p) (hp1 : p ≤ 1)
+    (hC : 0 < C) (hc : 0 < c) (hδ : 0 < δ₀) (hv : 0 ≤ v) (_hf : 0 < rf)
+    (hbound : ∀ ω, X ω ≤ v*rf^(-p))
+    (ht : ∀ δ : ℝ, rf ≤ δ → δ ≤ δ₀ → μ {ω | C*δ^(-p) < X ω} ≤
+      ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ)))) :
+    ∀ t : ℝ, 1 ≤ t →
+      μ {ω | max C v*δ₀^(-p)*t < X ω} ≤
+        ENNReal.ofReal (C*Real.exp (-(c*δ₀^(-2:ℝ))*t)) := by
+  intro t ht1
+  have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht1
+  let δ := δ₀/t
+  have hδp : 0 < δ := div_pos hδ ht0
+  have hδd : δ ≤ δ₀ := by dsimp [δ]; exact div_le_self hδ.le ht1
+  have he : δ^(-p) = δ₀^(-p)*t^p := by
+    dsimp [δ]
+    rw [Real.div_rpow hδ.le ht0.le,Real.rpow_neg ht0.le,div_inv_eq_mul]
+  have htp : t^p ≤ t := by
+    simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le ht1 hp1
+  have hM : 0 < max C v := hC.trans_le (le_max_left _ _)
+  have hn (a : ℝ) (ha : a ≤ max C v) (ha0 : 0 ≤ a) :
+      a*δ^(-p) ≤ max C v*δ₀^(-p)*t := by
+    rw [he,←mul_assoc]
+    exact mul_le_mul (mul_le_mul_of_nonneg_right ha (Real.rpow_nonneg hδ.le _))
+      htp (Real.rpow_nonneg ht0.le _) (by positivity)
+  by_cases hfd : rf ≤ δ
+  · apply (measure_mono (show {ω | max C v*δ₀^(-p)*t < X ω} ⊆
+        {ω | C*δ^(-p) < X ω} from fun ω hω => (hn C (le_max_left _ _) hC.le).trans_lt hω)).trans
+    apply (ht δ hfd hδd).trans
+    apply ENNReal.ofReal_le_ofReal
+    apply mul_le_mul_of_nonneg_left _ hC.le
+    apply Real.exp_le_exp.mpr
+    have hex : δ^(-2:ℝ) = δ₀^(-2:ℝ)*t^2 := by
+      dsimp [δ]
+      rw [Real.div_rpow hδ.le ht0.le,Real.rpow_neg ht0.le,div_inv_eq_mul,Real.rpow_two]
+    rw [hex]
+    have hsq : t ≤ t^2 := by nlinarith
+    nlinarith [mul_nonneg (mul_nonneg hc.le (Real.rpow_nonneg hδ.le (-2))) (sub_nonneg.mpr hsq)]
+  · have hx (ω : Ω) : X ω ≤ max C v*δ₀^(-p)*t := by
+      calc
+        X ω ≤ v*rf^(-p) := hbound ω
+        _ ≤ v*δ^(-p) := mul_le_mul_of_nonneg_left
+          (Real.rpow_le_rpow_of_nonpos hδp (le_of_not_ge hfd) (by linarith)) hv
+        _ ≤ _ := hn v (le_max_right _ _) hv
+    have he : {ω | max C v*δ₀^(-p)*t < X ω} = ∅ := by
+      ext ω; simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]; exact not_lt_of_ge (hx ω)
+    rw [he,measure_empty]
+    exact bot_le
+end
+end SLEExactGauge
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+lemma source_linear_tail_all {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+    [IsProbabilityMeasure μ] (X : Ω → ℝ) (M C b : ℝ)
+    (hM : 0 < M) (_hC : 0 < C) (hb : 0 < b)
+    (ht : ∀ t : ℝ, 1 ≤ t → μ {ω | M*t < X ω} ≤ ENNReal.ofReal (C*Real.exp (-b*t))) :
+    ∀ x : ℝ, 0 < x → μ {ω | x < X ω} ≤
+      ENNReal.ofReal ((max 1 C*Real.exp b)*Real.exp (-(b/M)*x)) := by
+  intro x hx
+  by_cases hxM : M ≤ x
+  · have hh := ht (x/M) ((one_le_div hM).mpr hxM)
+    rw [mul_div_cancel₀ _ hM.ne'] at hh
+    apply hh.trans
+    apply ENNReal.ofReal_le_ofReal
+    have hm : C ≤ max 1 C*Real.exp b := by
+      calc
+        C ≤ max 1 C := le_max_right _ _
+        _ ≤ _ := le_mul_of_one_le_right (by positivity) ((Real.one_le_exp_iff).mpr hb.le)
+    convert mul_le_mul_of_nonneg_right hm (Real.exp_pos (-(b/M)*x)).le using 1
+    congr 2
+    ring
+  · apply (prob_le_one : μ {ω | x < X ω} ≤ 1).trans
+    apply (ENNReal.one_le_ofReal).mpr
+    rw [mul_assoc,←Real.exp_add]
+    have he : 1 ≤ Real.exp (b+-(b/M)*x) := by
+      apply Real.one_le_exp_iff.mpr
+      have hh : (b/M)*x ≤ b := by
+        calc
+          (b/M)*x ≤ (b/M)*M := mul_le_mul_of_nonneg_left (le_of_not_ge hxM) (by positivity)
+          _ = b := div_mul_cancel₀ _ hM.ne'
+      linarith
+    calc
+      1 ≤ max 1 C := le_max_left _ _
+      _ ≤ _ := le_mul_of_one_le_right (by positivity) he
+
+lemma source_exponential_second_moment {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Measure Ω) (X : Ω → ℝ) (hXm : Measurable X) (hX : ∀ ω, 0 ≤ X ω)
+    (A b : ℝ) (hA : 0 ≤ A) (hb : 0 < b)
+    (ht : ∀ t : ℝ, 0 < t → μ {ω | t < X ω} ≤ ENNReal.ofReal (A*Real.exp (-b*t))) :
+    (∫⁻ ω, ENNReal.ofReal (X ω)^2 ∂μ) ≤ ENNReal.ofReal (8*A/b^2) := by
+  have h0 : ∀ x : ℝ, (∫ t in (0:ℝ)..x, (2:ℝ)*t) = x^2 := by
+    intro x
+    rw [intervalIntegral.integral_const_mul,integral_id]
+    ring
+  have he := lintegral_comp_eq_lintegral_meas_lt_mul μ (Eventually.of_forall hX)
+    hXm.aemeasurable (g := fun t : ℝ => 2*t)
+    (fun t ht => (continuous_const.mul continuous_id).intervalIntegrable 0 t)
+    (ae_restrict_mem measurableSet_Ioi |>.mono fun t ht => mul_nonneg (by norm_num) (show 0 < t from ht).le)
+  simp only [h0,ENNReal.ofReal_pow (hX _)] at he
+  rw [he]
+  have hbi : IntegrableOn (fun t : ℝ => Real.exp (-(b/2)*t)) (Ioi 0) :=
+    integrableOn_exp_mul_Ioi (by linarith : -(b/2) < 0) 0
+  calc
+    _ ≤ ∫⁻ t in Ioi (0:ℝ), ENNReal.ofReal ((4*A/b)*Real.exp (-(b/2)*t)) := by
+      apply setLIntegral_mono' measurableSet_Ioi
+      intro t ht0
+      have htpos : 0 < t := ht0
+      calc
+        _ ≤ ENNReal.ofReal (A*Real.exp (-b*t))*ENNReal.ofReal (2*t) :=
+          by gcongr; exact ht t htpos
+        _ = ENNReal.ofReal ((A*Real.exp (-b*t))*(2*t)) := (ENNReal.ofReal_mul (by positivity)).symm
+        _ ≤ _ := by
+          apply ENNReal.ofReal_le_ofReal
+          have hte : (b/2)*t ≤ Real.exp ((b/2)*t) := (Real.add_one_le_exp _).trans' (by linarith)
+          have hmul := mul_le_mul_of_nonneg_left hte (Real.exp_pos (-(b/2)*t)).le
+          have hev : Real.exp (-(b/2)*t)*Real.exp ((b/2)*t) = 1 := by
+            rw [←Real.exp_add]; ring_nf; exact Real.exp_zero
+          rw [hev] at hmul
+          have hex : Real.exp (-b*t) = Real.exp (-(b/2)*t)*Real.exp (-(b/2)*t) := by
+            rw [←Real.exp_add]; congr 1; ring
+          rw [hex]
+          have htb : t*Real.exp (-(b/2)*t) ≤ 2/b := by
+            apply (le_div_iff₀ hb).mpr
+            nlinarith
+          calc
+            A*(Real.exp (-(b/2)*t)*Real.exp (-(b/2)*t))*(2*t) =
+                (2*A*Real.exp (-(b/2)*t))*(t*Real.exp (-(b/2)*t)) := by ring
+            _ ≤ (2*A*Real.exp (-(b/2)*t))*(2/b) :=
+              mul_le_mul_of_nonneg_left htb (by positivity)
+            _ = (4*A/b)*Real.exp (-(b/2)*t) := by ring
+    _ = ENNReal.ofReal (8*A/b^2) := by
+      rw [←ofReal_integral_eq_lintegral_ofReal (hbi.const_mul (4*A/b))
+        (Eventually.of_forall (fun t => by positivity))]
+      rw [integral_const_mul,integral_exp_mul_Ioi (by linarith : -(b/2) < 0)]
+      simp only [mul_zero,Real.exp_zero]
+      congr 1
+      field_simp
+      ring
+end
+end SLEExactGauge
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLE/WindowTails.lean b/lean/OAI/Probability/SLE/WindowTails.lean
index 62cc35aa799b271ff88718b61fe7a45a28182e73..6d73a133f57c67da7b64ee0195a64c18ff84e6af 100644
--- a/lean/OAI/Probability/SLE/WindowTails.lean
+++ b/lean/OAI/Probability/SLE/WindowTails.lean
@@ -1,539 +1,539 @@
-import Mathlib
-import OAI.Probability.SLE.WindowBounds
-
-namespace OAI
-
-/-! Window Tails for chordal SLE. -/
-
-section
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped ENNReal
-open Set Filter MeasureTheory ProbabilityTheory Metric Classical
-namespace SLEExactGauge
-noncomputable section
-universe u_source_spatial
-def SourceSpatialConditionalMeanTarget : Prop :=
-  ∀ (Ω : Type u_source_spatial) (_ : MeasurableSpace Ω) (P : Measure Ω),
-  IsProbabilityMeasure P → ∀ (κ : ℝ), 0 < κ → κ < 8 →
-  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-    (z : ℂ) (ε r : ℝ), 0 < ε → ε ≤ r → r ≤ z.im →
-  ∃ hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r),
-    let μ := P.comap (Subtype.val : a.samples → Ω)
-    let W := fun v ω => spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
-      (γ ω) (a.G ω) z v
-    μ[W ε | hτ.measurableSpace] =ᵐ[μ] W r ∧
-    ∀ Y : a.samples → ℝ, Integrable Y μ → Y ≤ᵐ[μ] W ε →
-      μ[Y | hτ.measurableSpace] ≤ᵐ[μ] W r
-
-lemma spatialHitTime_antitone_radius {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) {ε r : ℝ} (hεr : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    spatialHitTime γ z r ≤ spatialHitTime γ z ε := by
-  by_cases he : spatialHitTime γ z ε = ⊤
-  · rw [he]; exact le_top
-  obtain ⟨t, ht⟩ := WithTop.ne_top_iff_exists.mp he
-  rw [← ht]
-  have hd := spatialHitTime_mem hc z ε t ht.symm
-  exact spatialHitTime_le_of_dist_le hc z r hs t (hd.le.trans hεr)
-
-lemma source_nested_stops_condExp
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    (ℱ : Filtration ℝ≥0 mΩ) (τ σ : Ω → WithTop ℝ≥0)
-    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) (hτσ : τ ≤ σ)
-    (M : ℝ≥0 → Ω → ℝ) (Z V W : Ω → ℝ) (hZ : Integrable Z μ)
-    (hStop : ∀ (ρ : Ω → ℝ≥0) (hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0))),
-      (fun ω => M (ρ ω) ω) =ᵐ[μ] μ[Z | hρ.measurableSpace])
-    (K : ℝ) (hbound : ∀ t ω, ‖M t ω‖ ≤ K)
-    (hVlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate τ n ω) ω) atTop (𝓝 (V ω)))
-    (hWlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate σ n ω) ω) atTop (𝓝 (W ω)))
-    (hzero : ∀ ω, τ ω = ⊤ → V ω = 0 ∧ W ω = 0)
-    (hterm : ∀ᵐ ω ∂μ, ∀ t : ℝ≥0, τ ω ≤ (t : WithTop ℝ≥0) → M (sourceStopTruncate τ t ω) ω = V ω) :
-    μ[W | hτ.measurableSpace] =ᵐ[μ] V := by
-  let τn := sourceStopTruncate τ
-  let σn := sourceStopTruncate σ
-  have hτn (t : ℝ≥0) := sourceStopTruncate_stopping ℱ hτ t
-  have hσn (t : ℝ≥0) := sourceStopTruncate_stopping ℱ hσ t
-  have hVi (t : ℝ≥0) : Integrable (fun ω => M (τn t ω) ω) μ :=
-    integrable_condExp.congr (hStop (τn t) (hτn t)).symm
-  have hWi (t : ℝ≥0) : Integrable (fun ω => M (σn t ω) ω) μ :=
-    integrable_condExp.congr (hStop (σn t) (hσn t)).symm
-  have hVm (n : ℕ) : StronglyMeasurable[hτ.measurableSpace]
-      (μ[Z | (hτn n).measurableSpace]) := by
-    apply stronglyMeasurable_condExp.mono
-    apply IsStoppingTime.measurableSpace_mono
-    intro ω
-    dsimp only
-    rw [sourceStopTruncate_coe]
-    exact min_le_left _ _
-  have hVmeas : AEStronglyMeasurable[hτ.measurableSpace] V μ := by
-    refine ⟨fun ω => limUnder atTop (fun n : ℕ => μ[Z | (hτn n).measurableSpace] ω),
-      (by
-        let : MeasurableSpace Ω := hτ.measurableSpace
-        exact StronglyMeasurable.limUnder hVm), ?_⟩
-    filter_upwards [hVlim, ae_all_iff.mpr (fun n : ℕ => hStop (τn n) (hτn n))] with ω hω he
-    have hf : (fun n : ℕ => μ[Z | (hτn n).measurableSpace] ω) =
-        (fun n : ℕ => M (τn n ω) ω) := funext fun n => (he n).symm
-    rw [hf]
-    exact hω.limUnder_eq.symm
-  have hVint : Integrable V μ := by
-    apply Integrable.of_bound (hVmeas.mono hτ.measurableSpace_le) K
-    filter_upwards [hVlim] with ω hω
-    exact le_of_tendsto hω.norm (Eventually.of_forall fun n => hbound _ _)
-  have hWint : Integrable W μ := by
-    apply Integrable.of_bound (aestronglyMeasurable_of_tendsto_ae atTop
-      (fun n : ℕ => (hWi n).aestronglyMeasurable) hWlim) K
-    filter_upwards [hWlim] with ω hω
-    exact le_of_tendsto hω.norm (Eventually.of_forall fun n => hbound _ _)
-  apply EventuallyEq.symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq hτ.measurableSpace_le hWint
-    (fun _ _ _ => hVint.integrableOn) _ hVmeas
-  intro A hA _
-  have hAm (m : ℕ) : MeasurableSet[(hτn m).measurableSpace]
-      (A ∩ {ω | τ ω ≤ (m : ℝ≥0)}) := by
-    have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ (m : ℝ≥0)) A hA
-    convert hh using 1
-    · congr 1
-      funext ω
-      exact sourceStopTruncate_coe τ m ω
-  have hset (m : ℕ) : (∫ ω in A ∩ {ω | τ ω ≤ (m : ℝ≥0)}, V ω ∂μ) =
-      ∫ ω in A ∩ {ω | τ ω ≤ (m : ℝ≥0)}, W ω ∂μ := by
-    let E := A ∩ {ω | τ ω ≤ (m : ℝ≥0)}
-    have hI : Tendsto (fun n : ℕ => ∫ ω in E, M (σn n ω) ω ∂μ) atTop
-        (𝓝 (∫ ω in E, W ω ∂μ)) := by
-      apply tendsto_integral_of_dominated_convergence (fun _ => K)
-        (fun n => (hWi n).integrableOn.aestronglyMeasurable)
-        (integrable_const _) _ (ae_restrict_of_ae hWlim)
-      intro n
-      exact Eventually.of_forall fun ω => hbound _ _
-    have he : ∀ᶠ n : ℕ in atTop, (∫ ω in E, M (σn n ω) ω ∂μ) =
-        ∫ ω in E, V ω ∂μ := by
-      filter_upwards [eventually_ge_atTop m] with n hn
-      have hmn : (m : ℝ≥0) ≤ (n : ℝ≥0) := by exact_mod_cast hn
-      have hle : (hτn m).measurableSpace ≤ (hσn n).measurableSpace := by
-        apply IsStoppingTime.measurableSpace_mono
-        intro ω
-        dsimp only
-        rw [sourceStopTruncate_coe, sourceStopTruncate_coe]
-        exact min_le_min (hτσ ω) (WithTop.coe_le_coe.mpr hmn)
-      have h1 : (∫ ω in E, M (σn n ω) ω ∂μ) = ∫ ω in E, Z ω ∂μ := by
-        rw [integral_congr_ae (ae_restrict_of_ae (hStop (σn n) (hσn n)))]
-        have : IsFiniteMeasure (μ.trim (hσn n).measurableSpace_le) := isFiniteMeasure_trim _
-        exact setIntegral_condExp (hσn n).measurableSpace_le hZ (hle E (hAm m))
-      have h2 : (∫ ω in E, M (τn m ω) ω ∂μ) = ∫ ω in E, Z ω ∂μ := by
-        rw [integral_congr_ae (ae_restrict_of_ae (hStop (τn m) (hτn m)))]
-        have : IsFiniteMeasure (μ.trim (hτn m).measurableSpace_le) := isFiniteMeasure_trim _
-        exact setIntegral_condExp (hτn m).measurableSpace_le hZ (hAm m)
-      rw [h1, ← h2]
-      apply integral_congr_ae
-      filter_upwards [ae_restrict_of_ae hterm, ae_restrict_mem
-        ((hτn m).measurableSpace_le E (hAm m))] with ω hω hE
-      exact hω m hE.2
-    have hc : Tendsto (fun n : ℕ => ∫ ω in E, M (σn n ω) ω ∂μ) atTop
-        (𝓝 (∫ ω in E, V ω ∂μ)) := tendsto_const_nhds.congr' (he.mono fun n hn => hn.symm)
-    exact (tendsto_nhds_unique hI hc).symm
-  have hlim (f : Ω → ℝ) (hf : Integrable f μ) (hz : ∀ ω, τ ω = ⊤ → f ω = 0) :
-      Tendsto (fun n : ℕ => ∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, f ω ∂μ) atTop
-        (𝓝 (∫ ω in A, f ω ∂μ)) := by
-    have he (n : ℕ) : (∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, f ω ∂μ) =
-        ∫ ω in A, {ω | τ ω ≤ (n : ℝ≥0)}.indicator f ω ∂μ :=
-      (setIntegral_indicator (μ := μ) (s := A) (f := f)
-        ((ℱ.le (n : ℝ≥0)) _ (hτ n))).symm
-    simp_rw [he]
-    apply tendsto_integral_of_dominated_convergence (fun ω => ‖f ω‖)
-      (fun n => (hf.indicator ((ℱ.le (n : ℝ≥0)) _ (hτ n))).integrableOn.aestronglyMeasurable)
-      hf.norm.integrableOn
-    · intro n
-      exact Eventually.of_forall fun ω => norm_indicator_le_norm_self _ _
-    · apply Eventually.of_forall
-      intro ω
-      apply (sourceStop_indicator_tendsto (τ ω) (f ω) (hz ω)).congr'
-      apply Eventually.of_forall
-      intro n
-      classical
-      by_cases hn : τ ω ≤ (n : WithTop ℝ≥0) <;> simp [Set.indicator, hn]
-  have h1 := hlim V hVint (fun ω hω => (hzero ω hω).1)
-  have h2 := hlim W hWint (fun ω hω => (hzero ω hω).2)
-  exact tendsto_nhds_unique (h1.congr' (Eventually.of_forall hset)) h2
-
- 
-
-theorem sourceSpatialConditionalMean_proved : SourceSpatialConditionalMeanTarget := by
-  intro Ω mΩ P hP κ hκ hκ8 γ a z ε r hε hεr hry
-  let : IsProbabilityMeasure P := hP
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  let τ (ω : a.samples) := spatialHitTime (γ ω) z r
-  let σ (ω : a.samples) := spatialHitTime (γ ω) z ε
-  have hτ : IsStoppingTime a.filtration τ :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
-  have hσ : IsStoppingTime a.filtration σ :=
-    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z ε
-  refine ⟨hτ, ?_⟩
-  let W (v : ℝ) (ω : a.samples) := spatialGreenWeight κ
-    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z v
-  change μ[W ε | hτ.measurableSpace] =ᵐ[μ] W r ∧
-    ∀ Y : a.samples → ℝ, Integrable Y μ → Y ≤ᵐ[μ] W ε →
-      μ[Y | hτ.measurableSpace] ≤ᵐ[μ] W r
-  have hr : 0 < r := hε.trans_le hεr
-  have hz : z ∈ upperHalfPlane := hr.trans_le hry
-  have hεy : ε ≤ z.im := hεr.trans hry
-  have hτσ : τ ≤ σ := by
-    intro ω
-    apply spatialHitTime_antitone_radius (a.trace_continuous ω) z hεr
-    rw [(a.witness ω).2.1, dist_zero_left]
-    exact hry.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-  let u := Real.log (4 * z.im / ε)
-  have hratio : 0 < 4 * z.im / ε := div_pos (mul_pos (by norm_num) hz) hε
-  have hu : 0 ≤ u := Real.log_nonneg ((le_div_iff₀ hε).mpr (by linarith))
-  have hlev : (2 * z.im) * Real.exp (-u) = ε / 2 := by
-    dsimp only [u]
-    rw [Real.exp_neg, Real.exp_log hratio]
-    field_simp [(show 0 < z.im from hz).ne']
-    ring
-  have hlevelε : (2 * z.im) * Real.exp (-u) < ε := by rw [hlev]; linarith
-  have hlevelr := hlevelε.trans_le hεr
-  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
-    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
-  obtain ⟨M, Z, hM, hZ, hMb, hZb, hClosed, hStop, heq, hNoHit⟩ :=
-    exists_physical_green_stopping_regular a.brownian hBm a.continuous_B a.zero_B
-      κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
-  have hVlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate τ n ω) ω)
-      atTop (𝓝 (W r ω)) := by
-    filter_upwards [heq, hNoHit] with ω he hn
-    simpa only [he.1, τ, W, sourceStopTruncate, boundedSpatialHit, Function.comp_apply] using
-      (a.witness ω).spatialGreen_limit ((a.continuous_B ω).const_mul _)
-        κ z r hr hry u hlevelr hn
-  have hWlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate σ n ω) ω)
-      atTop (𝓝 (W ε ω)) := by
-    filter_upwards [heq, hNoHit] with ω he hn
-    simpa only [he.1, σ, W, sourceStopTruncate, boundedSpatialHit, Function.comp_apply] using
-      (a.witness ω).spatialGreen_limit ((a.continuous_B ω).const_mul _)
-        κ z ε hε hεy u hlevelε hn
-  have hzero : ∀ ω, τ ω = ⊤ → W r ω = 0 ∧ W ε ω = 0 := by
-    intro ω hω
-    have hσtop : σ ω = ⊤ := top_unique (hω ▸ hτσ ω)
-    simp only [τ] at hω
-    simp only [σ] at hσtop
-    simp [W, spatialGreenWeight, hω, hσtop]
-  have hterm : ∀ᵐ ω ∂μ, ∀ t : ℝ≥0, τ ω ≤ (t : WithTop ℝ≥0) →
-      M (sourceStopTruncate τ t ω) ω = W r ω := by
-    filter_upwards [heq] with ω he
-    intro t ht
-    have hn : spatialHitTime (γ ω) z r ≠ ⊤ := ne_top_of_le_ne_top WithTop.coe_ne_top ht
-    have htrunc : sourceStopTruncate τ t ω = (spatialHitTime (γ ω) z r).untopD 0 := by
-      exact congrArg (fun v : WithTop ℝ≥0 => v.untopD 0) (min_eq_left ht)
-    have hle : (((spatialHitTime (γ ω) z r).untopD 0 : ℝ≥0) : WithTop ℝ≥0) ≤
-        spatialHitTime (γ ω) z r := by
-      obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp hn
-      rw [← hv, WithTop.untopD_coe]
-    change M (sourceStopTruncate τ t ω) ω =
-      if spatialHitTime (γ ω) z r = ⊤ then 0 else
-        greenDensity κ (fun s => Real.sqrt κ * a.B s ω) (γ ω) (a.G ω)
-          ((spatialHitTime (γ ω) z r).untopD 0) z
-    rw [ite_eq_right hn, htrunc, he.1]
-    exact (a.witness ω).physicalGreen_eq_before_spatial ((a.continuous_B ω).const_mul _)
-      κ z r hr hry u hlevelr _ hle
-  have hcond := source_nested_stops_condExp a.filtration τ σ hτ hσ hτσ M Z (W r) (W ε)
-    hZ hStop (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
-    (fun t ω => by rw [Real.norm_eq_abs, abs_of_nonneg (hMb t ω).1]; exact (hMb t ω).2)
-    hVlim hWlim hzero hterm
-  refine ⟨hcond, ?_⟩
-  intro Y hY hYW
-  have hWi := (a.spatialGreen_expectation hκ hκ8 z ε hε hεy).1
-  have hle := condExp_mono (m := hτ.measurableSpace) hY hWi hYW
-  filter_upwards [hcond, hle] with ω hω hi
-  exact hi.trans_eq hω
-
-end
-end SLEExactGauge
-
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
- 
-
-lemma source_condExp_le_changed_stop
-    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
-    (ℱ : Filtration ℝ≥0 mΩ) {σ τ : Ω → WithTop ℝ≥0}
-    (hσ : IsStoppingTime ℱ σ) (hτ : IsStoppingTime ℱ τ)
-    (hle : σ ≤ τ) (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A)
-    (heq : ∀ ω ∈ A, τ ω = σ ω)
-    {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
-    (hgm : StronglyMeasurable[hσ.measurableSpace] g)
-    (hf0 : ∀ ω ∉ A, f ω = 0) (hg0 : ∀ ω ∉ A, g ω = 0)
-    (hbound : μ[f | hσ.measurableSpace] ≤ᵐ[μ] g) :
-    μ[f | hτ.measurableSpace] ≤ᵐ[μ] g := by
-  have hm := hσ.measurableSpace_mono hτ hle
-  apply source_condExp_le_of_setIntegral_le hτ.measurableSpace_le hf hg (hgm.mono hm)
-  intro E hE
-  have hEA : MeasurableSet[hσ.measurableSpace] (E ∩ A) := by
-    have hh := hτ.measurableSet_inter_le hσ E hE
-    have hraw : MeasurableSet[hσ.measurableSpace] (E ∩ {ω | τ ω ≤ σ ω}) :=
-      (IsStoppingTime.measurableSet_min_iff hτ hσ _).mp hh |>.2
-    have hsub : A ⊆ {ω | τ ω ≤ σ ω} := fun ω hω => (heq ω hω).le
-    convert hraw.inter hA using 1
-    exact (Set.inter_assoc _ _ _ |>.trans (congrArg (Set.inter E) (Set.inter_eq_right.mpr hsub))).symm
-  have hfi : A.indicator f = f := by
-    funext ω
-    by_cases hω : ω ∈ A
-    · exact Set.indicator_of_mem hω _
-    · rw [Set.indicator_of_notMem hω, hf0 ω hω]
-  have hgi : A.indicator g = g := by
-    funext ω
-    by_cases hω : ω ∈ A
-    · exact Set.indicator_of_mem hω _
-    · rw [Set.indicator_of_notMem hω, hg0 ω hω]
-  have hAm := hσ.measurableSpace_le _ hA
-  rw [← hfi, setIntegral_indicator hAm, ← hgi, setIntegral_indicator hAm]
-  rw [← setIntegral_condExp hσ.measurableSpace_le hf hEA]
-  exact setIntegral_mono_ae integrable_condExp.integrableOn hg.integrableOn hbound
-
-lemma source_windowHB_range (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) :
-    (sourceSpatialWindowH γ z r ε R = 0 ∨ sourceSpatialWindowH γ z r ε R = 1) ∧
-    (sourceSpatialWindowB γ z r ε R = 0 ∨ sourceSpatialWindowB γ z r ε R = 1) := by
-  constructor <;> (first | unfold sourceSpatialWindowH | unfold sourceSpatialWindowB) <;>
-    split_ifs <;> simp
-
-lemma source_windowHB_zero {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (hs : liveSpatialHitTime γ z r = ⊤) :
-    sourceSpatialWindowH γ z r ε R = 0 ∧ sourceSpatialWindowB γ z r ε R = 0 := by
-  have ht : sourceSpatialWindowTime γ z r ε R = ⊤ :=
-    top_unique (by simpa only [hs] using source_window_start_le hc z r ε R)
-  simp only [sourceSpatialWindowH,sourceSpatialWindowB,ht,ne_eq,not_true_eq_false,false_and,ite_false,and_self]
-
-lemma source_windowH_le_indicator {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
-    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
-    sourceSpatialWindowH γ z r ε R ≤
-      (if liveSpatialHitTime γ z ε = ⊤ then (0:ℝ) else 1) := by
-  rcases (source_windowHB_range γ z r ε R).1 with h | h
-  · rw [h]; split_ifs <;> norm_num
-  · rw [h, ite_eq_right ((source_windowH_iff hc z r ε R her hs).mp h).2]
-
-end
-end SLEExactGauge
-
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_window_origin_bound
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
-    (hy : r ≤ z.im) (ω : a.samples) : r ≤ dist (γ ω 0) z := by
-  rw [(a.witness ω).2.1,dist_zero_left]
-  exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
-
-lemma source_window_change_live_mean
-    {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (z : ℂ) (r : ℝ) (hy : r ≤ z.im)
-    (hraw : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r))
-    (hs : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r))
-    (f g : a.samples → ℝ)
-    (hf : Integrable f (P.comap Subtype.val)) (hg : Integrable g (P.comap Subtype.val))
-    (hgm : StronglyMeasurable[hraw.measurableSpace] g)
-    (hf0 : ∀ ω : a.samples, liveSpatialHitTime (γ ω) z r = ⊤ → f ω = 0)
-    (hg0 : ∀ ω : a.samples, liveSpatialHitTime (γ ω) z r = ⊤ → g ω = 0)
-    (hbound : (P.comap Subtype.val)[f | hraw.measurableSpace] ≤ᵐ[P.comap Subtype.val] g) :
-    (P.comap Subtype.val)[f | hs.measurableSpace] ≤ᵐ[P.comap Subtype.val] g := by
-  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
-  let A := {ω : a.samples | z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z r).untopA}
-  have hA : MeasurableSet[hraw.measurableSpace] A := source_live_stopped_measurable a z hraw
-  have he (ω : a.samples) : liveSpatialHitTime (γ ω) z r =
-      if ω ∈ A then spatialHitTime (γ ω) z r else ⊤ :=
-    source_liveTime_eq_killed (a.trace_continuous ω) z r (source_window_origin_bound a z r hy ω)
-  have hle : (fun ω : a.samples => spatialHitTime (γ ω) z r) ≤
-      (fun ω : a.samples => liveSpatialHitTime (γ ω) z r) := by
-    intro ω; dsimp only; rw [he]; split_ifs <;> first | exact le_rfl | exact le_top
-  apply source_condExp_le_changed_stop a.filtration hraw hs hle A hA
-    (fun ω hω => by rw [he,ite_eq_left hω]) hf hg hgm
-  · intro ω hω; exact hf0 ω (by rw [he,ite_eq_right hω])
-  · intro ω hω; exact hg0 ω (by rw [he,ite_eq_right hω])
-  · exact hbound
-
-lemma source_window_weight_zero
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
-    (hy : r ≤ z.im) (ω : a.samples) (h : liveSpatialHitTime (γ ω) z r = ⊤) :
-    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r = 0 := by
-  rw [spatialGreenWeight_eq_live (a.trace_continuous ω) κ _ _ z r
-    (source_window_origin_bound a z r hy ω),ite_eq_left h]
-
-lemma source_windowH_integrable
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r ε R : ℝ)
-    (hy : r ≤ z.im) :
-    Integrable (fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R) (P.comap Subtype.val) := by
-  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
-  have hτ := source_window_stopping a z r ε R hy
-  have hm := (source_windowHB_stopped_measurable a z r ε R hτ).1.mono hτ.measurableSpace_le le_rfl
-  apply Integrable.of_bound hm.aestronglyMeasurable 1
-  exact Eventually.of_forall fun ω => by
-    rcases (source_windowHB_range (γ ω) z r ε R).1 with h|h <;> simp only [h,norm_zero,norm_one,zero_le_one,le_refl]
-
-lemma source_windowV_integrable_le
-    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
-    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (hk : 0 < κ) (hk8 : κ < 8)
-    (z : ℂ) (r ε R : ℝ) (he : 0 < ε) (her : ε ≤ r) (hy : r ≤ z.im) :
-    let W := fun ω : a.samples => spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z ε
-    let V := fun ω : a.samples => W ω*sourceSpatialWindowH (γ ω) z r ε R
-    Integrable V (P.comap Subtype.val) ∧ V ≤ᵐ[P.comap Subtype.val] W := by
-  dsimp only
-  let W := fun ω : a.samples => spatialGreenWeight κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z ε
-  let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R
-  have hτ := source_window_stopping a z r ε R hy
-  have hHm : Measurable H :=
-    (source_windowHB_stopped_measurable a z r ε R hτ).1.mono hτ.measurableSpace_le le_rfl
-  have hVm : Measurable (fun ω => W ω * H ω) :=
-    (source_windowV_stopped_measurable a z r ε R her hy hτ).mono hτ.measurableSpace_le le_rfl
-  have hWi := (a.spatialGreen_expectation hk hk8 z ε he (her.trans hy)).1
-  have hW0 (ω : a.samples) : 0 ≤ W ω :=
-    ((a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hk hk8 he z (her.trans hy)).1
-  constructor
-  · apply Integrable.mono' hWi hVm.aestronglyMeasurable
-    filter_upwards [] with ω
-    change ‖W ω * H ω‖ ≤ W ω
-    dsimp only [H]
-    rcases (source_windowHB_range (γ ω) z r ε R).1 with h|h
-    · simp only [h,mul_zero,norm_zero]; exact hW0 ω
-    · simp only [h,mul_one,Real.norm_eq_abs,abs_of_nonneg (hW0 ω),le_refl]
-  · filter_upwards [] with ω
-    rcases (source_windowHB_range (γ ω) z r ε R).1 with h|h
-    · simpa only [h,mul_zero] using hW0 ω
-    · simp only [h,mul_one,le_refl]
-
-end
-end SLEExactGauge
-
-end
-
- 
-section
-open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-noncomputable section
-attribute [local instance] Classical.propDecidable
-
-lemma source_windowH_mean_of_raw
-    {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hk : 0 < κ) (hk8 : κ < 8) (z : ℂ) (ε r R C : ℝ)
-    (he : 0 < ε) (her : ε ≤ r) (hy : r ≤ z.im)
-    (hraw : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r))
-    (hs : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r))
-    (hbound : (P.comap Subtype.val)[(fun ω : a.samples =>
-      if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1) | hraw.measurableSpace]
-      ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => C*ε^codimension κ*
-        spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) :
-    (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R) | hs.measurableSpace]
-      ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => C*ε^codimension κ*
-        spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r) := by
-  let μ : Measure a.samples := P.comap Subtype.val
-  let : IsProbabilityMeasure μ := a.probability
-  have hi := source_windowH_integrable a z r ε R hy
-  have hWi := (a.spatialGreen_expectation hk hk8 z r (he.trans_le her) hy).1
-  have hWm := source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
-    (fun ω : a.samples => γ ω) a.G a.F a.witness z r hraw
-  have hfi : Integrable (fun ω : a.samples => if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1) μ := by
-    apply Integrable.of_bound (source_liveIndicator_measurable a z ε (her.trans hy)).aestronglyMeasurable 1
-    exact Eventually.of_forall fun ω => by split_ifs <;> norm_num
-  have hcomp := condExp_mono (m := hraw.measurableSpace) hi hfi
-    (Eventually.of_forall fun ω => source_windowH_le_indicator (a.trace_continuous ω) z r ε R her
-      (source_window_origin_bound a z r hy ω))
-  apply source_window_change_live_mean a z r hy hraw hs _ _ hi (hWi.const_mul _)
-      (measurable_const.mul hWm).stronglyMeasurable
-  · intro ω hω; exact (source_windowHB_zero (a.trace_continuous ω) z r ε R hω).1
-  · intro ω hω; rw [source_window_weight_zero a z r hy ω hω,mul_zero]
-  · exact hcomp.trans hbound
-
-lemma source_windowV_mean
-    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
-    (hk : 0 < κ) (hk8 : κ < 8) (z : ℂ) (ε r R : ℝ)
-    (he : 0 < ε) (her : ε ≤ r) (hy : r ≤ z.im)
-    (hs : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r)) :
-    let W := fun v (ω : a.samples) => spatialGreenWeight κ
-      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
-    (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*W ε ω*
-      sourceSpatialWindowH (γ ω) z r ε R) | hs.measurableSpace]
-      ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => r^codimension κ*W r ω) := by
-  dsimp only
-  let μ : Measure a.samples := P.comap Subtype.val
-  let W := fun v (ω : a.samples) => spatialGreenWeight κ
-    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
-  let V := fun ω : a.samples => W ε ω*sourceSpatialWindowH (γ ω) z r ε R
-  have hi := source_windowV_integrable_le a hk hk8 z r ε R he her hy
-  have hWi := (a.spatialGreen_expectation hk hk8 z r (he.trans_le her) hy).1
-  obtain ⟨hraw,_,hvbound⟩ := sourceSpatialConditionalMean_proved Ω m P inferInstance κ hk hk8 γ a z ε r he her hy
-  have hWm := source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
-    (fun ω : a.samples => γ ω) a.G a.F a.witness z r hraw
-  have hv : μ[V | hs.measurableSpace] ≤ᵐ[μ] W r := by
-    apply source_window_change_live_mean a z r hy hraw hs V (W r) hi.1 hWi hWm.stronglyMeasurable
-    · intro ω hω; dsimp only [V]; rw [(source_windowHB_zero (a.trace_continuous ω) z r ε R hω).1,mul_zero]
-    · exact source_window_weight_zero a z r hy
-    · exact hvbound V hi.1 hi.2
-  change μ[(fun ω => r^codimension κ*W ε ω*sourceSpatialWindowH (γ ω) z r ε R) |
-    hs.measurableSpace] ≤ᵐ[μ] (fun ω => r^codimension κ*W r ω)
-  have heqv : (fun ω => r^codimension κ*W ε ω*sourceSpatialWindowH (γ ω) z r ε R) =
-      (fun ω => r^codimension κ*V ω) := by funext ω; dsimp only [V]; ring
-  rw [heqv]
-  filter_upwards [condExp_smul (μ := μ) (r^codimension κ) V hs.measurableSpace, hv] with ω hc hv
-  change μ[(fun ω => r^codimension κ*V ω) | hs.measurableSpace] ω =
-    r^codimension κ*μ[V | hs.measurableSpace] ω at hc
-  rw [hc]
-  exact mul_le_mul_of_nonneg_left hv (Real.rpow_nonneg (he.trans_le her).le _)
-
-lemma source_window_HV_mean :
-    ∀ κ : ℝ, 0 < κ → κ < 8 → ∃ C q : ℝ, 0 < C ∧ 0 < q ∧ q < 1 ∧
-    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
-    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
-      (z : ℂ) (ε r R : ℝ), 0 < ε → ε ≤ q*r → r ≤ z.im →
-    ∃ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
-      let μ := P.comap (Subtype.val : a.samples → Ω)
-      let W := fun v (ω : a.samples) => spatialGreenWeight κ
-        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
-      let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R
-      μ[H | hσ.measurableSpace] ≤ᵐ[μ] (fun ω => C*ε^codimension κ*W r ω) ∧
-      μ[(fun ω => r^codimension κ*W ε ω*H ω) | hσ.measurableSpace] ≤ᵐ[μ]
-        (fun ω => r^codimension κ*W r ω) := by
-  intro κ hk hk8
-  obtain ⟨C,hC,q,hq,hq1,happ⟩ := sourceConditionalApproach_proved κ hk hk8
-  refine ⟨C,q,hC,hq,hq1,?_⟩
-  intro Ω m P hP γ a z ε r R he heq hy
-  let : IsProbabilityMeasure P := hP
-  have hr : 0 < r := by
-    by_contra! hn
-    have hh := mul_nonpos_of_nonneg_of_nonpos hq.le hn
-    linarith
-  have her : ε < r := heq.trans_lt (by nlinarith)
-  have hs := source_liveHit_stopping a z r hy
-  obtain ⟨hraw,hbound⟩ := happ Ω m P hP γ a z ε r he heq hy
-  exact ⟨hs,source_windowH_mean_of_raw a hk hk8 z ε r R C he her.le hy hraw hs hbound,
-    source_windowV_mean a hk hk8 z ε r R he her.le hy hs⟩
-
-end
-end SLEExactGauge
-
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.WindowBounds
+
+namespace OAI
+
+/-! Window Tails for chordal SLE. -/
+
+section
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical TopologicalSpace
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped ENNReal
+open Set Filter MeasureTheory ProbabilityTheory Metric Classical
+namespace SLEExactGauge
+noncomputable section
+universe u_source_spatial
+def SourceSpatialConditionalMeanTarget : Prop :=
+  ∀ (Ω : Type u_source_spatial) (_ : MeasurableSpace Ω) (P : Measure Ω),
+  IsProbabilityMeasure P → ∀ (κ : ℝ), 0 < κ → κ < 8 →
+  ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+    (z : ℂ) (ε r : ℝ), 0 < ε → ε ≤ r → r ≤ z.im →
+  ∃ hτ : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r),
+    let μ := P.comap (Subtype.val : a.samples → Ω)
+    let W := fun v ω => spatialGreenWeight κ (fun t => Real.sqrt κ * a.B t ω)
+      (γ ω) (a.G ω) z v
+    μ[W ε | hτ.measurableSpace] =ᵐ[μ] W r ∧
+    ∀ Y : a.samples → ℝ, Integrable Y μ → Y ≤ᵐ[μ] W ε →
+      μ[Y | hτ.measurableSpace] ≤ᵐ[μ] W r
+
+lemma spatialHitTime_antitone_radius {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) {ε r : ℝ} (hεr : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    spatialHitTime γ z r ≤ spatialHitTime γ z ε := by
+  by_cases he : spatialHitTime γ z ε = ⊤
+  · rw [he]; exact le_top
+  obtain ⟨t, ht⟩ := WithTop.ne_top_iff_exists.mp he
+  rw [← ht]
+  have hd := spatialHitTime_mem hc z ε t ht.symm
+  exact spatialHitTime_le_of_dist_le hc z r hs t (hd.le.trans hεr)
+
+lemma source_nested_stops_condExp
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    (ℱ : Filtration ℝ≥0 mΩ) (τ σ : Ω → WithTop ℝ≥0)
+    (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) (hτσ : τ ≤ σ)
+    (M : ℝ≥0 → Ω → ℝ) (Z V W : Ω → ℝ) (hZ : Integrable Z μ)
+    (hStop : ∀ (ρ : Ω → ℝ≥0) (hρ : IsStoppingTime ℱ (fun ω => (ρ ω : WithTop ℝ≥0))),
+      (fun ω => M (ρ ω) ω) =ᵐ[μ] μ[Z | hρ.measurableSpace])
+    (K : ℝ) (hbound : ∀ t ω, ‖M t ω‖ ≤ K)
+    (hVlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate τ n ω) ω) atTop (𝓝 (V ω)))
+    (hWlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate σ n ω) ω) atTop (𝓝 (W ω)))
+    (hzero : ∀ ω, τ ω = ⊤ → V ω = 0 ∧ W ω = 0)
+    (hterm : ∀ᵐ ω ∂μ, ∀ t : ℝ≥0, τ ω ≤ (t : WithTop ℝ≥0) → M (sourceStopTruncate τ t ω) ω = V ω) :
+    μ[W | hτ.measurableSpace] =ᵐ[μ] V := by
+  let τn := sourceStopTruncate τ
+  let σn := sourceStopTruncate σ
+  have hτn (t : ℝ≥0) := sourceStopTruncate_stopping ℱ hτ t
+  have hσn (t : ℝ≥0) := sourceStopTruncate_stopping ℱ hσ t
+  have hVi (t : ℝ≥0) : Integrable (fun ω => M (τn t ω) ω) μ :=
+    integrable_condExp.congr (hStop (τn t) (hτn t)).symm
+  have hWi (t : ℝ≥0) : Integrable (fun ω => M (σn t ω) ω) μ :=
+    integrable_condExp.congr (hStop (σn t) (hσn t)).symm
+  have hVm (n : ℕ) : StronglyMeasurable[hτ.measurableSpace]
+      (μ[Z | (hτn n).measurableSpace]) := by
+    apply stronglyMeasurable_condExp.mono
+    apply IsStoppingTime.measurableSpace_mono
+    intro ω
+    dsimp only
+    rw [sourceStopTruncate_coe]
+    exact min_le_left _ _
+  have hVmeas : AEStronglyMeasurable[hτ.measurableSpace] V μ := by
+    refine ⟨fun ω => limUnder atTop (fun n : ℕ => μ[Z | (hτn n).measurableSpace] ω),
+      (by
+        let : MeasurableSpace Ω := hτ.measurableSpace
+        exact StronglyMeasurable.limUnder hVm), ?_⟩
+    filter_upwards [hVlim, ae_all_iff.mpr (fun n : ℕ => hStop (τn n) (hτn n))] with ω hω he
+    have hf : (fun n : ℕ => μ[Z | (hτn n).measurableSpace] ω) =
+        (fun n : ℕ => M (τn n ω) ω) := funext fun n => (he n).symm
+    rw [hf]
+    exact hω.limUnder_eq.symm
+  have hVint : Integrable V μ := by
+    apply Integrable.of_bound (hVmeas.mono hτ.measurableSpace_le) K
+    filter_upwards [hVlim] with ω hω
+    exact le_of_tendsto hω.norm (Eventually.of_forall fun n => hbound _ _)
+  have hWint : Integrable W μ := by
+    apply Integrable.of_bound (aestronglyMeasurable_of_tendsto_ae atTop
+      (fun n : ℕ => (hWi n).aestronglyMeasurable) hWlim) K
+    filter_upwards [hWlim] with ω hω
+    exact le_of_tendsto hω.norm (Eventually.of_forall fun n => hbound _ _)
+  apply EventuallyEq.symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq hτ.measurableSpace_le hWint
+    (fun _ _ _ => hVint.integrableOn) _ hVmeas
+  intro A hA _
+  have hAm (m : ℕ) : MeasurableSet[(hτn m).measurableSpace]
+      (A ∩ {ω | τ ω ≤ (m : ℝ≥0)}) := by
+    have hh := hτ.measurableSet_inter_le (isStoppingTime_const ℱ (m : ℝ≥0)) A hA
+    convert hh using 1
+    · congr 1
+      funext ω
+      exact sourceStopTruncate_coe τ m ω
+  have hset (m : ℕ) : (∫ ω in A ∩ {ω | τ ω ≤ (m : ℝ≥0)}, V ω ∂μ) =
+      ∫ ω in A ∩ {ω | τ ω ≤ (m : ℝ≥0)}, W ω ∂μ := by
+    let E := A ∩ {ω | τ ω ≤ (m : ℝ≥0)}
+    have hI : Tendsto (fun n : ℕ => ∫ ω in E, M (σn n ω) ω ∂μ) atTop
+        (𝓝 (∫ ω in E, W ω ∂μ)) := by
+      apply tendsto_integral_of_dominated_convergence (fun _ => K)
+        (fun n => (hWi n).integrableOn.aestronglyMeasurable)
+        (integrable_const _) _ (ae_restrict_of_ae hWlim)
+      intro n
+      exact Eventually.of_forall fun ω => hbound _ _
+    have he : ∀ᶠ n : ℕ in atTop, (∫ ω in E, M (σn n ω) ω ∂μ) =
+        ∫ ω in E, V ω ∂μ := by
+      filter_upwards [eventually_ge_atTop m] with n hn
+      have hmn : (m : ℝ≥0) ≤ (n : ℝ≥0) := by exact_mod_cast hn
+      have hle : (hτn m).measurableSpace ≤ (hσn n).measurableSpace := by
+        apply IsStoppingTime.measurableSpace_mono
+        intro ω
+        dsimp only
+        rw [sourceStopTruncate_coe, sourceStopTruncate_coe]
+        exact min_le_min (hτσ ω) (WithTop.coe_le_coe.mpr hmn)
+      have h1 : (∫ ω in E, M (σn n ω) ω ∂μ) = ∫ ω in E, Z ω ∂μ := by
+        rw [integral_congr_ae (ae_restrict_of_ae (hStop (σn n) (hσn n)))]
+        have : IsFiniteMeasure (μ.trim (hσn n).measurableSpace_le) := isFiniteMeasure_trim _
+        exact setIntegral_condExp (hσn n).measurableSpace_le hZ (hle E (hAm m))
+      have h2 : (∫ ω in E, M (τn m ω) ω ∂μ) = ∫ ω in E, Z ω ∂μ := by
+        rw [integral_congr_ae (ae_restrict_of_ae (hStop (τn m) (hτn m)))]
+        have : IsFiniteMeasure (μ.trim (hτn m).measurableSpace_le) := isFiniteMeasure_trim _
+        exact setIntegral_condExp (hτn m).measurableSpace_le hZ (hAm m)
+      rw [h1, ← h2]
+      apply integral_congr_ae
+      filter_upwards [ae_restrict_of_ae hterm, ae_restrict_mem
+        ((hτn m).measurableSpace_le E (hAm m))] with ω hω hE
+      exact hω m hE.2
+    have hc : Tendsto (fun n : ℕ => ∫ ω in E, M (σn n ω) ω ∂μ) atTop
+        (𝓝 (∫ ω in E, V ω ∂μ)) := tendsto_const_nhds.congr' (he.mono fun n hn => hn.symm)
+    exact (tendsto_nhds_unique hI hc).symm
+  have hlim (f : Ω → ℝ) (hf : Integrable f μ) (hz : ∀ ω, τ ω = ⊤ → f ω = 0) :
+      Tendsto (fun n : ℕ => ∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, f ω ∂μ) atTop
+        (𝓝 (∫ ω in A, f ω ∂μ)) := by
+    have he (n : ℕ) : (∫ ω in A ∩ {ω | τ ω ≤ (n : ℝ≥0)}, f ω ∂μ) =
+        ∫ ω in A, {ω | τ ω ≤ (n : ℝ≥0)}.indicator f ω ∂μ :=
+      (setIntegral_indicator (μ := μ) (s := A) (f := f)
+        ((ℱ.le (n : ℝ≥0)) _ (hτ n))).symm
+    simp_rw [he]
+    apply tendsto_integral_of_dominated_convergence (fun ω => ‖f ω‖)
+      (fun n => (hf.indicator ((ℱ.le (n : ℝ≥0)) _ (hτ n))).integrableOn.aestronglyMeasurable)
+      hf.norm.integrableOn
+    · intro n
+      exact Eventually.of_forall fun ω => norm_indicator_le_norm_self _ _
+    · apply Eventually.of_forall
+      intro ω
+      apply (sourceStop_indicator_tendsto (τ ω) (f ω) (hz ω)).congr'
+      apply Eventually.of_forall
+      intro n
+      classical
+      by_cases hn : τ ω ≤ (n : WithTop ℝ≥0) <;> simp [Set.indicator, hn]
+  have h1 := hlim V hVint (fun ω hω => (hzero ω hω).1)
+  have h2 := hlim W hWint (fun ω hω => (hzero ω hω).2)
+  exact tendsto_nhds_unique (h1.congr' (Eventually.of_forall hset)) h2
+
+ 
+
+theorem sourceSpatialConditionalMean_proved : SourceSpatialConditionalMeanTarget := by
+  intro Ω mΩ P hP κ hκ hκ8 γ a z ε r hε hεr hry
+  let : IsProbabilityMeasure P := hP
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  let τ (ω : a.samples) := spatialHitTime (γ ω) z r
+  let σ (ω : a.samples) := spatialHitTime (γ ω) z ε
+  have hτ : IsStoppingTime a.filtration τ :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z r
+  have hσ : IsStoppingTime a.filtration σ :=
+    a.diskHitAfter_stopping (fun _ => 0) (isStoppingTime_const a.filtration 0) z ε
+  refine ⟨hτ, ?_⟩
+  let W (v : ℝ) (ω : a.samples) := spatialGreenWeight κ
+    (fun t => Real.sqrt κ * a.B t ω) (γ ω) (a.G ω) z v
+  change μ[W ε | hτ.measurableSpace] =ᵐ[μ] W r ∧
+    ∀ Y : a.samples → ℝ, Integrable Y μ → Y ≤ᵐ[μ] W ε →
+      μ[Y | hτ.measurableSpace] ≤ᵐ[μ] W r
+  have hr : 0 < r := hε.trans_le hεr
+  have hz : z ∈ upperHalfPlane := hr.trans_le hry
+  have hεy : ε ≤ z.im := hεr.trans hry
+  have hτσ : τ ≤ σ := by
+    intro ω
+    apply spatialHitTime_antitone_radius (a.trace_continuous ω) z hεr
+    rw [(a.witness ω).2.1, dist_zero_left]
+    exact hry.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+  let u := Real.log (4 * z.im / ε)
+  have hratio : 0 < 4 * z.im / ε := div_pos (mul_pos (by norm_num) hz) hε
+  have hu : 0 ≤ u := Real.log_nonneg ((le_div_iff₀ hε).mpr (by linarith))
+  have hlev : (2 * z.im) * Real.exp (-u) = ε / 2 := by
+    dsimp only [u]
+    rw [Real.exp_neg, Real.exp_log hratio]
+    field_simp [(show 0 < z.im from hz).ne']
+    ring
+  have hlevelε : (2 * z.im) * Real.exp (-u) < ε := by rw [hlev]; linarith
+  have hlevelr := hlevelε.trans_le hεr
+  have hBm : Measurable (fun p : ℝ≥0 × a.samples => a.B p.1 p.2) :=
+    measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B
+  obtain ⟨M, Z, hM, hZ, hMb, hZb, hClosed, hStop, heq, hNoHit⟩ :=
+    exists_physical_green_stopping_regular a.brownian hBm a.continuous_B a.zero_B
+      κ hκ hκ8 (γ ∘ Subtype.val) a.G a.F (Eventually.of_forall a.witness) z hz u hu
+  have hVlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate τ n ω) ω)
+      atTop (𝓝 (W r ω)) := by
+    filter_upwards [heq, hNoHit] with ω he hn
+    simpa only [he.1, τ, W, sourceStopTruncate, boundedSpatialHit, Function.comp_apply] using
+      (a.witness ω).spatialGreen_limit ((a.continuous_B ω).const_mul _)
+        κ z r hr hry u hlevelr hn
+  have hWlim : ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => M (sourceStopTruncate σ n ω) ω)
+      atTop (𝓝 (W ε ω)) := by
+    filter_upwards [heq, hNoHit] with ω he hn
+    simpa only [he.1, σ, W, sourceStopTruncate, boundedSpatialHit, Function.comp_apply] using
+      (a.witness ω).spatialGreen_limit ((a.continuous_B ω).const_mul _)
+        κ z ε hε hεy u hlevelε hn
+  have hzero : ∀ ω, τ ω = ⊤ → W r ω = 0 ∧ W ε ω = 0 := by
+    intro ω hω
+    have hσtop : σ ω = ⊤ := top_unique (hω ▸ hτσ ω)
+    simp only [τ] at hω
+    simp only [σ] at hσtop
+    simp [W, spatialGreenWeight, hω, hσtop]
+  have hterm : ∀ᵐ ω ∂μ, ∀ t : ℝ≥0, τ ω ≤ (t : WithTop ℝ≥0) →
+      M (sourceStopTruncate τ t ω) ω = W r ω := by
+    filter_upwards [heq] with ω he
+    intro t ht
+    have hn : spatialHitTime (γ ω) z r ≠ ⊤ := ne_top_of_le_ne_top WithTop.coe_ne_top ht
+    have htrunc : sourceStopTruncate τ t ω = (spatialHitTime (γ ω) z r).untopD 0 := by
+      exact congrArg (fun v : WithTop ℝ≥0 => v.untopD 0) (min_eq_left ht)
+    have hle : (((spatialHitTime (γ ω) z r).untopD 0 : ℝ≥0) : WithTop ℝ≥0) ≤
+        spatialHitTime (γ ω) z r := by
+      obtain ⟨v, hv⟩ := WithTop.ne_top_iff_exists.mp hn
+      rw [← hv, WithTop.untopD_coe]
+    change M (sourceStopTruncate τ t ω) ω =
+      if spatialHitTime (γ ω) z r = ⊤ then 0 else
+        greenDensity κ (fun s => Real.sqrt κ * a.B s ω) (γ ω) (a.G ω)
+          ((spatialHitTime (γ ω) z r).untopD 0) z
+    rw [ite_eq_right hn, htrunc, he.1]
+    exact (a.witness ω).physicalGreen_eq_before_spatial ((a.continuous_B ω).const_mul _)
+      κ z r hr hry u hlevelr _ hle
+  have hcond := source_nested_stops_condExp a.filtration τ σ hτ hσ hτσ M Z (W r) (W ε)
+    hZ hStop (Real.exp (codimension κ * (u - Real.log 2 - Real.log z.im)))
+    (fun t ω => by rw [Real.norm_eq_abs, abs_of_nonneg (hMb t ω).1]; exact (hMb t ω).2)
+    hVlim hWlim hzero hterm
+  refine ⟨hcond, ?_⟩
+  intro Y hY hYW
+  have hWi := (a.spatialGreen_expectation hκ hκ8 z ε hε hεy).1
+  have hle := condExp_mono (m := hτ.measurableSpace) hY hWi hYW
+  filter_upwards [hcond, hle] with ω hω hi
+  exact hi.trans_eq hω
+
+end
+end SLEExactGauge
+
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+ 
+
+lemma source_condExp_le_changed_stop
+    {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
+    (ℱ : Filtration ℝ≥0 mΩ) {σ τ : Ω → WithTop ℝ≥0}
+    (hσ : IsStoppingTime ℱ σ) (hτ : IsStoppingTime ℱ τ)
+    (hle : σ ≤ τ) (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A)
+    (heq : ∀ ω ∈ A, τ ω = σ ω)
+    {f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
+    (hgm : StronglyMeasurable[hσ.measurableSpace] g)
+    (hf0 : ∀ ω ∉ A, f ω = 0) (hg0 : ∀ ω ∉ A, g ω = 0)
+    (hbound : μ[f | hσ.measurableSpace] ≤ᵐ[μ] g) :
+    μ[f | hτ.measurableSpace] ≤ᵐ[μ] g := by
+  have hm := hσ.measurableSpace_mono hτ hle
+  apply source_condExp_le_of_setIntegral_le hτ.measurableSpace_le hf hg (hgm.mono hm)
+  intro E hE
+  have hEA : MeasurableSet[hσ.measurableSpace] (E ∩ A) := by
+    have hh := hτ.measurableSet_inter_le hσ E hE
+    have hraw : MeasurableSet[hσ.measurableSpace] (E ∩ {ω | τ ω ≤ σ ω}) :=
+      (IsStoppingTime.measurableSet_min_iff hτ hσ _).mp hh |>.2
+    have hsub : A ⊆ {ω | τ ω ≤ σ ω} := fun ω hω => (heq ω hω).le
+    convert hraw.inter hA using 1
+    exact (Set.inter_assoc _ _ _ |>.trans (congrArg (Set.inter E) (Set.inter_eq_right.mpr hsub))).symm
+  have hfi : A.indicator f = f := by
+    funext ω
+    by_cases hω : ω ∈ A
+    · exact Set.indicator_of_mem hω _
+    · rw [Set.indicator_of_notMem hω, hf0 ω hω]
+  have hgi : A.indicator g = g := by
+    funext ω
+    by_cases hω : ω ∈ A
+    · exact Set.indicator_of_mem hω _
+    · rw [Set.indicator_of_notMem hω, hg0 ω hω]
+  have hAm := hσ.measurableSpace_le _ hA
+  rw [← hfi, setIntegral_indicator hAm, ← hgi, setIntegral_indicator hAm]
+  rw [← setIntegral_condExp hσ.measurableSpace_le hf hEA]
+  exact setIntegral_mono_ae integrable_condExp.integrableOn hg.integrableOn hbound
+
+lemma source_windowHB_range (γ : ℝ≥0 → ℂ) (z : ℂ) (r ε R : ℝ) :
+    (sourceSpatialWindowH γ z r ε R = 0 ∨ sourceSpatialWindowH γ z r ε R = 1) ∧
+    (sourceSpatialWindowB γ z r ε R = 0 ∨ sourceSpatialWindowB γ z r ε R = 1) := by
+  constructor <;> (first | unfold sourceSpatialWindowH | unfold sourceSpatialWindowB) <;>
+    split_ifs <;> simp
+
+lemma source_windowHB_zero {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (hs : liveSpatialHitTime γ z r = ⊤) :
+    sourceSpatialWindowH γ z r ε R = 0 ∧ sourceSpatialWindowB γ z r ε R = 0 := by
+  have ht : sourceSpatialWindowTime γ z r ε R = ⊤ :=
+    top_unique (by simpa only [hs] using source_window_start_le hc z r ε R)
+  simp only [sourceSpatialWindowH,sourceSpatialWindowB,ht,ne_eq,not_true_eq_false,false_and,ite_false,and_self]
+
+lemma source_windowH_le_indicator {γ : ℝ≥0 → ℂ} (hc : Continuous γ)
+    (z : ℂ) (r ε R : ℝ) (her : ε ≤ r) (hs : r ≤ dist (γ 0) z) :
+    sourceSpatialWindowH γ z r ε R ≤
+      (if liveSpatialHitTime γ z ε = ⊤ then (0:ℝ) else 1) := by
+  rcases (source_windowHB_range γ z r ε R).1 with h | h
+  · rw [h]; split_ifs <;> norm_num
+  · rw [h, ite_eq_right ((source_windowH_iff hc z r ε R her hs).mp h).2]
+
+end
+end SLEExactGauge
+
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_window_origin_bound
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
+    (hy : r ≤ z.im) (ω : a.samples) : r ≤ dist (γ ω 0) z := by
+  rw [(a.witness ω).2.1,dist_zero_left]
+  exact hy.trans ((le_abs_self z.im).trans (Complex.abs_im_le_norm z))
+
+lemma source_window_change_live_mean
+    {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (z : ℂ) (r : ℝ) (hy : r ≤ z.im)
+    (hraw : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r))
+    (hs : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r))
+    (f g : a.samples → ℝ)
+    (hf : Integrable f (P.comap Subtype.val)) (hg : Integrable g (P.comap Subtype.val))
+    (hgm : StronglyMeasurable[hraw.measurableSpace] g)
+    (hf0 : ∀ ω : a.samples, liveSpatialHitTime (γ ω) z r = ⊤ → f ω = 0)
+    (hg0 : ∀ ω : a.samples, liveSpatialHitTime (γ ω) z r = ⊤ → g ω = 0)
+    (hbound : (P.comap Subtype.val)[f | hraw.measurableSpace] ≤ᵐ[P.comap Subtype.val] g) :
+    (P.comap Subtype.val)[f | hs.measurableSpace] ≤ᵐ[P.comap Subtype.val] g := by
+  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
+  let A := {ω : a.samples | z ∈ survivingDomain (γ ω) (spatialHitTime (γ ω) z r).untopA}
+  have hA : MeasurableSet[hraw.measurableSpace] A := source_live_stopped_measurable a z hraw
+  have he (ω : a.samples) : liveSpatialHitTime (γ ω) z r =
+      if ω ∈ A then spatialHitTime (γ ω) z r else ⊤ :=
+    source_liveTime_eq_killed (a.trace_continuous ω) z r (source_window_origin_bound a z r hy ω)
+  have hle : (fun ω : a.samples => spatialHitTime (γ ω) z r) ≤
+      (fun ω : a.samples => liveSpatialHitTime (γ ω) z r) := by
+    intro ω; dsimp only; rw [he]; split_ifs <;> first | exact le_rfl | exact le_top
+  apply source_condExp_le_changed_stop a.filtration hraw hs hle A hA
+    (fun ω hω => by rw [he,ite_eq_left hω]) hf hg hgm
+  · intro ω hω; exact hf0 ω (by rw [he,ite_eq_right hω])
+  · intro ω hω; exact hg0 ω (by rw [he,ite_eq_right hω])
+  · exact hbound
+
+lemma source_window_weight_zero
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r : ℝ)
+    (hy : r ≤ z.im) (ω : a.samples) (h : liveSpatialHitTime (γ ω) z r = ⊤) :
+    spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r = 0 := by
+  rw [spatialGreenWeight_eq_live (a.trace_continuous ω) κ _ _ z r
+    (source_window_origin_bound a z r hy ω),ite_eq_left h]
+
+lemma source_windowH_integrable
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (z : ℂ) (r ε R : ℝ)
+    (hy : r ≤ z.im) :
+    Integrable (fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R) (P.comap Subtype.val) := by
+  let : IsProbabilityMeasure (P.comap (Subtype.val : a.samples → Ω)) := a.probability
+  have hτ := source_window_stopping a z r ε R hy
+  have hm := (source_windowHB_stopped_measurable a z r ε R hτ).1.mono hτ.measurableSpace_le le_rfl
+  apply Integrable.of_bound hm.aestronglyMeasurable 1
+  exact Eventually.of_forall fun ω => by
+    rcases (source_windowHB_range (γ ω) z r ε R).1 with h|h <;> simp only [h,norm_zero,norm_one,zero_le_one,le_refl]
+
+lemma source_windowV_integrable_le
+    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ}
+    {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (hk : 0 < κ) (hk8 : κ < 8)
+    (z : ℂ) (r ε R : ℝ) (he : 0 < ε) (her : ε ≤ r) (hy : r ≤ z.im) :
+    let W := fun ω : a.samples => spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z ε
+    let V := fun ω : a.samples => W ω*sourceSpatialWindowH (γ ω) z r ε R
+    Integrable V (P.comap Subtype.val) ∧ V ≤ᵐ[P.comap Subtype.val] W := by
+  dsimp only
+  let W := fun ω : a.samples => spatialGreenWeight κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z ε
+  let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R
+  have hτ := source_window_stopping a z r ε R hy
+  have hHm : Measurable H :=
+    (source_windowHB_stopped_measurable a z r ε R hτ).1.mono hτ.measurableSpace_le le_rfl
+  have hVm : Measurable (fun ω => W ω * H ω) :=
+    (source_windowV_stopped_measurable a z r ε R her hy hτ).mono hτ.measurableSpace_le le_rfl
+  have hWi := (a.spatialGreen_expectation hk hk8 z ε he (her.trans hy)).1
+  have hW0 (ω : a.samples) : 0 ≤ W ω :=
+    ((a.witness ω).spatialGreenWeight_bound ((a.continuous_B ω).const_mul _) hk hk8 he z (her.trans hy)).1
+  constructor
+  · apply Integrable.mono' hWi hVm.aestronglyMeasurable
+    filter_upwards [] with ω
+    change ‖W ω * H ω‖ ≤ W ω
+    dsimp only [H]
+    rcases (source_windowHB_range (γ ω) z r ε R).1 with h|h
+    · simp only [h,mul_zero,norm_zero]; exact hW0 ω
+    · simp only [h,mul_one,Real.norm_eq_abs,abs_of_nonneg (hW0 ω),le_refl]
+  · filter_upwards [] with ω
+    rcases (source_windowHB_range (γ ω) z r ε R).1 with h|h
+    · simpa only [h,mul_zero] using hW0 ω
+    · simp only [h,mul_one,le_refl]
+
+end
+end SLEExactGauge
+
+end
+
+ 
+section
+open Set Filter MeasureTheory ProbabilityTheory Metric Complex Classical
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+noncomputable section
+attribute [local instance] Classical.propDecidable
+
+lemma source_windowH_mean_of_raw
+    {Ω : Type*} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hk : 0 < κ) (hk8 : κ < 8) (z : ℂ) (ε r R C : ℝ)
+    (he : 0 < ε) (her : ε ≤ r) (hy : r ≤ z.im)
+    (hraw : IsStoppingTime a.filtration (fun ω : a.samples => spatialHitTime (γ ω) z r))
+    (hs : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r))
+    (hbound : (P.comap Subtype.val)[(fun ω : a.samples =>
+      if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1) | hraw.measurableSpace]
+      ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => C*ε^codimension κ*
+        spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r)) :
+    (P.comap Subtype.val)[(fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R) | hs.measurableSpace]
+      ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => C*ε^codimension κ*
+        spatialGreenWeight κ (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z r) := by
+  let μ : Measure a.samples := P.comap Subtype.val
+  let : IsProbabilityMeasure μ := a.probability
+  have hi := source_windowH_integrable a z r ε R hy
+  have hWi := (a.spatialGreen_expectation hk hk8 z r (he.trans_le her) hy).1
+  have hWm := source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
+    (fun ω : a.samples => γ ω) a.G a.F a.witness z r hraw
+  have hfi : Integrable (fun ω : a.samples => if liveSpatialHitTime (γ ω) z ε = ⊤ then (0:ℝ) else 1) μ := by
+    apply Integrable.of_bound (source_liveIndicator_measurable a z ε (her.trans hy)).aestronglyMeasurable 1
+    exact Eventually.of_forall fun ω => by split_ifs <;> norm_num
+  have hcomp := condExp_mono (m := hraw.measurableSpace) hi hfi
+    (Eventually.of_forall fun ω => source_windowH_le_indicator (a.trace_continuous ω) z r ε R her
+      (source_window_origin_bound a z r hy ω))
+  apply source_window_change_live_mean a z r hy hraw hs _ _ hi (hWi.const_mul _)
+      (measurable_const.mul hWm).stronglyMeasurable
+  · intro ω hω; exact (source_windowHB_zero (a.trace_continuous ω) z r ε R hω).1
+  · intro ω hω; rw [source_window_weight_zero a z r hy ω hω,mul_zero]
+  · exact hcomp.trans hbound
+
+lemma source_windowV_mean
+    {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ)
+    (hk : 0 < κ) (hk8 : κ < 8) (z : ℂ) (ε r R : ℝ)
+    (he : 0 < ε) (her : ε ≤ r) (hy : r ≤ z.im)
+    (hs : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r)) :
+    let W := fun v (ω : a.samples) => spatialGreenWeight κ
+      (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
+    (P.comap Subtype.val)[(fun ω : a.samples => r^codimension κ*W ε ω*
+      sourceSpatialWindowH (γ ω) z r ε R) | hs.measurableSpace]
+      ≤ᵐ[P.comap Subtype.val] (fun ω : a.samples => r^codimension κ*W r ω) := by
+  dsimp only
+  let μ : Measure a.samples := P.comap Subtype.val
+  let W := fun v (ω : a.samples) => spatialGreenWeight κ
+    (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
+  let V := fun ω : a.samples => W ε ω*sourceSpatialWindowH (γ ω) z r ε R
+  have hi := source_windowV_integrable_le a hk hk8 z r ε R he her hy
+  have hWi := (a.spatialGreen_expectation hk hk8 z r (he.trans_le her) hy).1
+  obtain ⟨hraw,_,hvbound⟩ := sourceSpatialConditionalMean_proved Ω m P inferInstance κ hk hk8 γ a z ε r he her hy
+  have hWm := source_spatialWeight_stopping_measurable a.measurable_B a.continuous_B κ
+    (fun ω : a.samples => γ ω) a.G a.F a.witness z r hraw
+  have hv : μ[V | hs.measurableSpace] ≤ᵐ[μ] W r := by
+    apply source_window_change_live_mean a z r hy hraw hs V (W r) hi.1 hWi hWm.stronglyMeasurable
+    · intro ω hω; dsimp only [V]; rw [(source_windowHB_zero (a.trace_continuous ω) z r ε R hω).1,mul_zero]
+    · exact source_window_weight_zero a z r hy
+    · exact hvbound V hi.1 hi.2
+  change μ[(fun ω => r^codimension κ*W ε ω*sourceSpatialWindowH (γ ω) z r ε R) |
+    hs.measurableSpace] ≤ᵐ[μ] (fun ω => r^codimension κ*W r ω)
+  have heqv : (fun ω => r^codimension κ*W ε ω*sourceSpatialWindowH (γ ω) z r ε R) =
+      (fun ω => r^codimension κ*V ω) := by funext ω; dsimp only [V]; ring
+  rw [heqv]
+  filter_upwards [condExp_smul (μ := μ) (r^codimension κ) V hs.measurableSpace, hv] with ω hc hv
+  change μ[(fun ω => r^codimension κ*V ω) | hs.measurableSpace] ω =
+    r^codimension κ*μ[V | hs.measurableSpace] ω at hc
+  rw [hc]
+  exact mul_le_mul_of_nonneg_left hv (Real.rpow_nonneg (he.trans_le her).le _)
+
+lemma source_window_HV_mean :
+    ∀ κ : ℝ, 0 < κ → κ < 8 → ∃ C q : ℝ, 0 < C ∧ 0 < q ∧ q < 1 ∧
+    ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P →
+    ∀ (γ : Ω → ℝ≥0 → ℂ) (a : RegularSLEPresentation P κ γ)
+      (z : ℂ) (ε r R : ℝ), 0 < ε → ε ≤ q*r → r ≤ z.im →
+    ∃ hσ : IsStoppingTime a.filtration (fun ω : a.samples => liveSpatialHitTime (γ ω) z r),
+      let μ := P.comap (Subtype.val : a.samples → Ω)
+      let W := fun v (ω : a.samples) => spatialGreenWeight κ
+        (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) z v
+      let H := fun ω : a.samples => sourceSpatialWindowH (γ ω) z r ε R
+      μ[H | hσ.measurableSpace] ≤ᵐ[μ] (fun ω => C*ε^codimension κ*W r ω) ∧
+      μ[(fun ω => r^codimension κ*W ε ω*H ω) | hσ.measurableSpace] ≤ᵐ[μ]
+        (fun ω => r^codimension κ*W r ω) := by
+  intro κ hk hk8
+  obtain ⟨C,hC,q,hq,hq1,happ⟩ := sourceConditionalApproach_proved κ hk hk8
+  refine ⟨C,q,hC,hq,hq1,?_⟩
+  intro Ω m P hP γ a z ε r R he heq hy
+  let : IsProbabilityMeasure P := hP
+  have hr : 0 < r := by
+    by_contra! hn
+    have hh := mul_nonpos_of_nonneg_of_nonpos hq.le hn
+    linarith
+  have her : ε < r := heq.trans_lt (by nlinarith)
+  have hs := source_liveHit_stopping a z r hy
+  obtain ⟨hraw,hbound⟩ := happ Ω m P hP γ a z ε r he heq hy
+  exact ⟨hs,source_windowH_mean_of_raw a hk hk8 z ε r R C he her.le hy hraw hs hbound,
+    source_windowV_mean a hk hk8 z ε r R he her.le hy hs⟩
+
+end
+end SLEExactGauge
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/AdaptedFailure.lean b/lean/OAI/Probability/SLEGauge/AdaptedFailure.lean
index a1c7589680e06708f0f5b1f9d2289390ea62dd0d..ddf3394d654c8ee3cb3329302a5a51bb6989f135 100644
--- a/lean/OAI/Probability/SLEGauge/AdaptedFailure.lean
+++ b/lean/OAI/Probability/SLEGauge/AdaptedFailure.lean
@@ -1,574 +1,574 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.SpatialPastTail
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.AdaptedFailure
-
-variable {Ω : Type*} [m₀ : MeasurableSpace Ω]
-
- 
-def failed (success : ℕ→Set Ω) : ℕ→Set Ω
-  | 0 => univ
-  | n+1 => failed success n \ success n
-
-omit m₀ in
-lemma failed_iff [MeasurableSpace Ω] (success : ℕ→Set Ω) (n : ℕ) (ω : Ω) :
-    ω∈failed success n ↔ ∀i<n,ω∉success i := by
-  induction n with
-  | zero => simp [failed]
-  | succ n ih =>
-    simp only [failed,Set.mem_sdiff,ih,Nat.lt_succ_iff_lt_or_eq]
-    constructor
-    · rintro ⟨h,h'⟩ i (hi | rfl)
-      · exact h i hi
-      · exact h'
-    · intro h
-      exact ⟨fun i hi=>h i (Or.inl hi),h n (Or.inr rfl)⟩
-
-lemma failed_measurable (ℱ : Filtration ℕ m₀)
-    (success : ℕ→Set Ω) (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (n : ℕ) :
-    MeasurableSet[ℱ n] (failed success n) := by
-  induction n with
-  | zero => exact MeasurableSet.univ
-  | succ n ih => exact (ℱ.mono (Nat.le_succ n) _ ih).diff (hs n)
-
- 
-
-theorem failure_product (P : Measure Ω) [IsProbabilityMeasure P]
-    (ℱ : Filtration ℕ m₀) (success : ℕ→Set Ω)
-    (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (p : ℕ→ℝ)
-    (hp1 : ∀i,p i≤1)
-    (htest : ∀i,∀A : Set Ω,MeasurableSet[ℱ i] A →
-      p i*P.real A≤P.real (A∩success i)) (n : ℕ) :
-    P.real (failed success n)≤∏i∈Finset.range n,(1-p i) := by
-  induction n with
-  | zero => simp [failed]
-  | succ n ih =>
-    have hs' : MeasurableSet (success n) := ℱ.le (n+1) _ (hs n)
-    have hsplit := measureReal_inter_add_sdiff (μ := P) (s := failed success n) hs'
-    have hstep := htest n (failed success n) (failed_measurable ℱ success hs n)
-    calc
-      P.real (failed success (n+1))≤(1-p n)*P.real (failed success n) := by
-        change P.real (failed success n \ success n)≤_
-        nlinarith
-      _ ≤ (1-p n)*(∏i∈Finset.range n,(1-p i)) :=
-        mul_le_mul_of_nonneg_left ih (sub_nonneg.mpr (hp1 n))
-      _ = _ := by rw [Finset.prod_range_succ]; ring
-
-lemma product_le_exp (p : ℕ→ℝ) (hp1 : ∀i,p i≤1) (n : ℕ) :
-    (∏i∈Finset.range n,(1-p i))≤Real.exp (-(∑i∈Finset.range n,p i)) := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    calc
-      _ = (∏i∈Finset.range n,(1-p i))*(1-p n) := Finset.prod_range_succ _ _
-      _ ≤ Real.exp (-(∑i∈Finset.range n,p i))*Real.exp (-p n) :=
-        mul_le_mul ih (Real.one_sub_le_exp_neg _) (sub_nonneg.mpr (hp1 n)) (Real.exp_pos _).le
-      _ = _ := by rw [←Real.exp_add,Finset.sum_range_succ,neg_add_rev]; congr 1; ring
-
- 
-
-lemma tested_of_conditional (P : Measure Ω) [IsProbabilityMeasure P]
-    {m : MeasurableSpace Ω} (hm : m ≤ m₀)
-    (S : Set Ω) (hS : MeasurableSet[m₀] S) (p : ℝ)
-    (hp : ∀ᵐ ω∂P,p≤P[S.indicator (fun _=> (1:ℝ)) | m] ω)
-    (A : Set Ω) (hA : MeasurableSet[m] A) :
-    p*P.real A≤P.real (A∩S) := by
-  have hInt : Integrable (S.indicator (fun _ : Ω=>(1:ℝ))) P :=
-    (integrable_const 1).indicator hS
-  calc
-    p*P.real A = ∫ω in A,p∂P := by simp only [setIntegral_const,smul_eq_mul]; ring
-    _ ≤ ∫ω in A,P[S.indicator (fun _=>(1:ℝ)) | m] ω∂P :=
-      integral_mono_ae (integrable_const _) integrable_condExp.integrableOn
-        (ae_restrict_of_ae hp)
-    _ = ∫ω in A,S.indicator (fun _=>(1:ℝ)) ω∂P := setIntegral_condExp hm hInt hA
-    _ = P.real (A∩S) := by rw [setIntegral_indicator hS]; simp
-
- 
-
-theorem conditional_failure (P : Measure Ω) [IsProbabilityMeasure P]
-    (ℱ : Filtration ℕ m₀) (success : ℕ→Set Ω)
-    (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (p : ℕ→ℝ)
-    (hp1 : ∀i,p i≤1)
-    (hp : ∀i,∀ᵐ ω∂P,p i≤P[(success i).indicator (fun _=>(1:ℝ)) | ℱ i] ω)
-    (n : ℕ) (B : ℝ) (hB : B≤∑i∈Finset.range n,p i) :
-    P.real (failed success n)≤Real.exp (-B) := by
-  refine (failure_product P ℱ success hs p hp1 ?_ n).trans
-    ((product_le_exp p hp1 n).trans (Real.exp_le_exp.mpr (neg_le_neg hB)))
-  intro i A hA
-  exact tested_of_conditional P (ℱ.le i) (success i) (ℱ.le (i+1) _ (hs i))
-    (p i) (hp i) A hA
-
- 
-
-lemma weighted_past_bound (P : Measure Ω) [IsFiniteMeasure P]
-    {m : MeasurableSpace Ω} (hm : m ≤ m₀)
-    (f : Ω→ℝ) (hf : Integrable f P) (K : ℝ)
-    (hb : ∀ᵐ ω∂P,P[f | m] ω≤K) (A : Set Ω) (hA : MeasurableSet[m] A) :
-    (∫ω in A,f ω∂P)≤K*P.real A := by
-  calc
-    (∫ω in A,f ω∂P) = ∫ω in A,P[f | m] ω∂P := (setIntegral_condExp hm hf hA).symm
-    _ ≤ ∫ω in A,K∂P := integral_mono_ae integrable_condExp.integrableOn
-      (integrable_const _) (ae_restrict_of_ae hb)
-    _ = K*P.real A := by simp only [setIntegral_const,smul_eq_mul]; ring
-
- 
-
-theorem weighted_conditional_failure (P : Measure Ω) [IsProbabilityMeasure P]
-    (ℱ : Filtration ℕ m₀) (success : ℕ→Set Ω)
-    (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (p : ℕ→ℝ)
-    (hp1 : ∀i,p i≤1)
-    (hp : ∀i,∀ᵐ ω∂P,p i≤P[(success i).indicator (fun _=>(1:ℝ)) | ℱ i] ω)
-    (n : ℕ) (B : ℝ) (hB : B≤∑i∈Finset.range n,p i)
-    (f : Ω→ℝ) (hf : Integrable f P) (K : ℝ) (hK : 0≤K)
-    (hb : ∀ᵐ ω∂P,P[f | ℱ n] ω≤K) :
-    (∫ω in failed success n,f ω∂P)≤K*Real.exp (-B) :=
-  (weighted_past_bound P (ℱ.le n) f hf K hb (failed success n)
-    (failed_measurable ℱ success hs n)).trans
-    (mul_le_mul_of_nonneg_left (conditional_failure P ℱ success hs p hp1 hp n B hB) hK)
-
-end SLEExactGauge.AdaptedFailure
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart CompactSequence MomentBatches AdaptedFailure
-
- 
-
-structure RadialTestSchedule (a y : ℝ) where
-  start : ℕ → ℝ
-  boundary : ℕ → ℝ
-  level : ℕ
-  threshold : ℕ → ℝ
-  boundary_pos : ∀i,0 < boundary i
-  boundary_le : ∀i,boundary i ≤ y
-  start_pos : ∀i,0 < start i
-  start_le : ∀i,start i ≤ boundary i
-  next : ∀i,boundary (i+1)=start i*Real.exp (-2*a*level)
-  threshold_nonneg : ∀i,0 ≤ threshold i
-  gap : ∀i,∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧ start i=boundary i*Real.exp (-2*a*s)
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4 < a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-def testSuccess (z : ℂ) (D : RadialTestSchedule a z.im) (i : ℕ) : Set GDriver :=
-  {b | ENNReal.ofReal (D.threshold i) ≤ M.stoppedTest z (D.start i) D.level b}
-
-def testFailure (z : ℂ) (D : RadialTestSchedule a z.im) (n : ℕ) : Set GDriver :=
-  failed (M.testSuccess z D) n
-
-lemma testFailure_raw (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ) :
-    MeasurableSet[(radialTime_stopping a (by linarith) z hz (D.boundary n)
-      (D.boundary_pos n) (D.boundary_le n)).measurableSpace] (M.testFailure z D n) := by
-  induction n with
-  | zero => exact MeasurableSet.univ
-  | succ n ih =>
-    have hs := M.stoppedTest_measurable z hz (D.start n) (D.start_pos n)
-      ((D.start_le n).trans (D.boundary_le n)) D.level
-    have hn : D.boundary (n+1) ≤ D.boundary n := by
-      rw [D.next]
-      exact (testEndpoint_le (ha := ha) _ (D.start_pos n).le _).trans (D.start_le n)
-    have hmono := (radialTime_stopping a (by linarith) z hz (D.boundary n)
-      (D.boundary_pos n) (D.boundary_le n)).measurableSpace_mono
-      (radialTime_stopping a (by linarith) z hz (D.boundary (n+1))
-        (D.boundary_pos _) (D.boundary_le _))
-      (radialTime_antitone a (by linarith) z hz _ _ (D.boundary_pos _) hn (D.boundary_le n))
-    apply (hmono _ ih).diff
-    unfold testSuccess
-    have he := measurableSet_le (f := fun _=>ENNReal.ofReal (D.threshold n)) measurable_const hs
-    exact ⟨he.1,fun t=>by simpa only [D.next] using he.2 t⟩
-
- 
-
-theorem testFailure_product [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ) :
-    (radialLaw W a z (D.boundary n)).real (M.testFailure z D n) ≤
-      ∏i∈Finset.range n,(1-M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2) := by
-  let := M.probability
-  have hP (i : ℕ) : IsProbabilityMeasure (radialLaw W a z (D.boundary i)) :=
-    radialLaw_probability W hW a ha z hz _ (D.boundary_pos _) (D.boundary_le _)
-  induction n with
-  | zero =>
-    let := hP 0
-    simp [testFailure,failed]
-  | succ n ih =>
-    let := hP (n+1)
-    have hraw := M.testFailure_raw z hz D n
-    obtain ⟨s,hs,hgap⟩ := D.gap n
-    have hl := M.stopped_success_lower hW z hz (D.boundary n) (D.start n)
-      (D.start_pos n) (D.start_le n) (D.boundary_le n) s hs hgap D.level
-      (D.threshold n) (D.threshold_nonneg n) (M.testFailure z D n) hraw
-    rw [←D.next] at hl
-    have hlr := ENNReal.toReal_mono (measure_ne_top _ _) hl
-    simp only [ENNReal.toReal_mul,ENNReal.toReal_div,ENNReal.toReal_one,ENNReal.toReal_ofNat] at hlr
-    change 1/2*M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}*
-      (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n) ≤
-      (radialLaw W a z (D.boundary (n+1))).real
-        (M.testFailure z D n∩M.testSuccess z D n) at hlr
-    have hm : MeasurableSet (M.testSuccess z D n) := by
-      have hT := radialTime_stopping a (by linarith) z hz _
-        (testEndpoint_pos (a := a) _ (D.start_pos n) D.level)
-        ((testEndpoint_le (ha := ha) _ (D.start_pos n).le _).trans
-          ((D.start_le n).trans (D.boundary_le n)))
-      exact hT.measurableSpace_le _ (measurableSet_le measurable_const
-        (M.stoppedTest_measurable z hz (D.start n) (D.start_pos n)
-          ((D.start_le n).trans (D.boundary_le n)) D.level))
-    have hsplit := measureReal_inter_add_sdiff (μ := radialLaw W a z (D.boundary (n+1)))
-      (s := M.testFailure z D n) hm
-    have hn : D.boundary (n+1) ≤ D.boundary n := by
-      rw [D.next]
-      exact (testEndpoint_le (ha := ha) _ (D.start_pos n).le _).trans (D.start_le n)
-    have he := congrArg ENNReal.toReal (radialLaw_consistent_le W hW a ha z hz
-      (D.boundary n) (D.boundary (n+1)) (D.boundary_pos _) hn (D.boundary_le n)
-      (M.testFailure z D n) hraw)
-    change (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n)=
-      (radialLaw W a z (D.boundary n)).real (M.testFailure z D n) at he
-    have hp : M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x} ≤ 1 := measureReal_le_one
-    calc
-      _ ≤ (1-M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}/2)*
-          (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n) := by
-        change (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n \ M.testSuccess z D n) ≤ _
-        nlinarith
-      _ = (1-M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}/2)*
-          (radialLaw W a z (D.boundary n)).real (M.testFailure z D n) := by rw [he]
-      _ ≤ (1-M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}/2)*
-          (∏i∈Finset.range n,(1-M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2)) :=
-        mul_le_mul_of_nonneg_left ih (by linarith)
-      _ = _ := by rw [Finset.prod_range_succ]; ring
-
-lemma testFailure_exponential [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ)
-    (B : ℝ) (hB : B ≤ ∑i∈Finset.range n,M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2) :
-    radialLaw W a z (D.boundary n) (M.testFailure z D n) ≤ ENNReal.ofReal (Real.exp (-B)) := by
-  let := M.probability
-  have := radialLaw_probability W hW a ha z hz (D.boundary n) (D.boundary_pos _) (D.boundary_le _)
-  have hp (i : ℕ) : M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2 ≤ 1 := by
-    have : M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x} ≤ 1 := measureReal_le_one
-    linarith
-  have hh := (M.testFailure_product hW z hz D n).trans
-    ((product_le_exp _ hp n).trans (Real.exp_le_exp.mpr (neg_le_neg hB)))
-  simpa only [Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using ENNReal.ofReal_le_ofReal hh
-
- 
-theorem testFailure_chordal [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ)
-    (ρ : ℝ) (hρ : 0 < ρ) (hρσ : ρ ≤ D.boundary n)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=D.boundary n*Real.exp (-2*a*s))
-    (B : ℝ) (hB : B ≤ ∑i∈Finset.range n,M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2) :
-    W (M.testFailure z D n∩{b | radialTime a b.1 z ρ≠∞}) ≤
-      Jacobi.visitConstant a*ENNReal.ofReal ((ρ/z.im)^(alpha a))*ENNReal.ofReal (Real.exp (-B)) := by
-  have hr := M.testFailure_raw z hz D n
-  have hh := radial_change_law_past_bound W hW a ha z hz (D.boundary n) ρ hρ hρσ
-    (D.boundary_le n) s hs hgap (M.testFailure z D n) hr
-  rw [radialLaw_consistent_le W hW a ha z hz _ ρ hρ hρσ (D.boundary_le n) _ hr] at hh
-  exact hh.trans (mul_le_mul le_rfl (M.testFailure_exponential hW z hz D n B hB) zero_le zero_le)
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator LoewnerRestart
-
- 
-def batchRadius (a : ℝ) (l n i : ℕ) : ℝ :=
-  Real.exp (-(n:ℝ)^2-2*a*(i:ℝ)*((l:ℝ)+(n:ℝ)))
-
-lemma batchRadius_pos (a : ℝ) (l n i : ℕ) : 0 < batchRadius a l n i := Real.exp_pos _
-
-lemma batchRadius_zero (a : ℝ) (l n : ℕ) : batchRadius a l n 0=Real.exp (-(n:ℝ)^2) := by
-  simp [batchRadius]
-
-lemma batchRadius_succ (a : ℝ) (l n i : ℕ) :
-    batchRadius a l n (i+1)=batchRadius a l n i*Real.exp (-2*a*l)*Real.exp (-2*a*n) := by
-  simp only [batchRadius,←Real.exp_add,Nat.cast_add,Nat.cast_one]
-  congr 1
-  ring
-
-lemma batchRadius_antitone (a : ℝ) (ha : 0 ≤ a) (l n : ℕ) : Antitone (batchRadius a l n) := by
-  intro i j hij
-  apply Real.exp_le_exp.mpr
-  have hj : (i:ℝ) ≤ j := by exact_mod_cast hij
-  have hm : 0 ≤ 2*a*((l:ℝ)+(n:ℝ)) := by positivity
-  nlinarith [mul_le_mul_of_nonneg_left hj hm]
-
-lemma mixingTime_nonneg (a : ℝ) : 0 ≤ Jacobi.mixingTime (4*a) := by
-  unfold Jacobi.mixingTime Jacobi.majorantBase
-  positivity
-
-def BatchReady (a : ℝ) (n : ℕ) : Prop :=
-  3 ≤ n ∧ Jacobi.mixingTime (4*a) ≤ (n:ℝ)-Real.log 2/(2*a) ∧
-  400*Real.exp (-(n:ℝ)^2) ≤ Real.exp (-(n:ℝ))*Real.exp (-2*a*Jacobi.mixingTime (4*a))
-
-lemma eventually_batchReady (a : ℝ) (_ha : 0 < a) : ∀ᶠ n in atTop,BatchReady a n := by
-  have hR (x : ℝ) : ∀ᶠ n : ℕ in atTop,x ≤ (n:ℝ) :=
-    (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop x)
-  filter_upwards [eventually_ge_atTop 3,hR (Jacobi.mixingTime (4*a)+Real.log 2/(2*a)),
-    hR (Real.log 400+2*a*Jacobi.mixingTime (4*a))] with n hn hmix hinit
-  refine ⟨hn,by linarith,?_⟩
-  have hnR : (3:ℝ) ≤ n := by exact_mod_cast hn
-  have hsq : (n:ℝ) ≤ (n:ℝ)^2-(n:ℝ) := by nlinarith
-  rw [show (400:ℝ)=Real.exp (Real.log 400) by rw [Real.exp_log (by norm_num)],
-    ←Real.exp_add,←Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  linarith
-
-lemma BatchReady.mixing {a : ℝ} (ha : 0<a) {n : ℕ} (h : BatchReady a n) :
-    Jacobi.mixingTime (4*a) ≤ (n:ℝ) := by
-  have : 0 ≤ Real.log 2/(2*a) := div_nonneg (Real.log_nonneg (by norm_num)) (by positivity)
-  linarith [h.2.1]
-
- 
-
-def batchBoundary (a y : ℝ) (l n : ℕ) : ℕ→ℝ
-  | 0 => y
-  | i+1 => 400*batchRadius a l n i*Real.exp (-2*a*l)
-
-lemma batchBoundary_pos (a y : ℝ) (hy : 0<y) (l n i : ℕ) : 0<batchBoundary a y l n i := by
-  cases i with
-  | zero => exact hy
-  | succ i => dsimp [batchBoundary]; have := batchRadius_pos a l n i; positivity
-
-lemma batchBoundary_start (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n : ℕ)
-    (h : BatchReady a n) (hyn : Real.exp (-(n:ℝ)) ≤ y) (i : ℕ) :
-    400*batchRadius a l n i ≤ batchBoundary a y l n i ∧
-      ∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧
-        400*batchRadius a l n i=batchBoundary a y l n i*Real.exp (-2*a*s) := by
-  cases i with
-  | zero =>
-    apply clock_gap_of_small a ha _ (mixingTime_nonneg a) y _ hy
-      (by have := batchRadius_pos a l n 0; positivity)
-    rw [batchRadius_zero]
-    exact h.2.2.trans (mul_le_mul_of_nonneg_right hyn (Real.exp_pos _).le)
-  | succ i =>
-    have hg : 400*batchRadius a l n (i+1)=batchBoundary a y l n (i+1)*Real.exp (-2*a*n) := by
-      rw [batchRadius_succ,batchBoundary]; ring
-    refine ⟨?_,(n:ℝ),h.mixing ha,hg⟩
-    rw [hg]
-    exact mul_le_of_le_one_right (batchBoundary_pos a y hy l n _).le
-      (Real.exp_le_one_iff.mpr (by have : (0:ℝ) ≤ n := Nat.cast_nonneg n; nlinarith))
-
-lemma batchBoundary_le (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n : ℕ)
-    (h : BatchReady a n) (hyn : Real.exp (-(n:ℝ)) ≤ y) (i : ℕ) :
-    batchBoundary a y l n i ≤ y := by
-  induction i with
-  | zero => exact le_rfl
-  | succ i ih =>
-    change 400*batchRadius a l n i*Real.exp (-2*a*l) ≤ y
-    have hstart := (batchBoundary_start a y ha hy l n h hyn i).1.trans ih
-    refine (mul_le_of_le_one_right (by have := batchRadius_pos a l n i; positivity) ?_).trans hstart
-    exact Real.exp_le_one_iff.mpr (by have : (0:ℝ) ≤ l := Nat.cast_nonneg l; nlinarith)
-
-def batchSchedule (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n : ℕ)
-    (h : BatchReady a n) (hyn : Real.exp (-(n:ℝ)) ≤ y) (b : ℕ→ℝ) (hb : ∀i,0 ≤ b i) :
-    RadialTestSchedule a y where
-  start i := 400*batchRadius a l n i
-  boundary := batchBoundary a y l n
-  level := l
-  threshold := b
-  boundary_pos := batchBoundary_pos a y hy l n
-  boundary_le := batchBoundary_le a y ha hy l n h hyn
-  start_pos i := by have := batchRadius_pos a l n i; positivity
-  start_le i := (batchBoundary_start a y ha hy l n h hyn i).1
-  next _ := rfl
-  threshold_nonneg := hb
-  gap i := (batchBoundary_start a y ha hy l n h hyn i).2
-
-lemma batch_final_gap (a y : ℝ) (ha : 0<a) (l n J : ℕ) (hJ : 0<J) :
-    800*batchRadius a l n J=batchBoundary a y l n J*
-      Real.exp (-2*a*((n:ℝ)-Real.log 2/(2*a))) := by
-  obtain ⟨i,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hJ.ne'
-  rw [batchRadius_succ,batchBoundary,
-    show -2*a*((n:ℝ)-Real.log 2/(2*a))= -2*a*n+Real.log 2 by field_simp; ring,
-    Real.exp_add,Real.exp_log (by norm_num : (0:ℝ)<2)]
-  ring
-
-lemma batch_final_le (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n J : ℕ) (hJ : 0<J)
-    (h : BatchReady a n) : 800*batchRadius a l n J ≤ batchBoundary a y l n J := by
-  rw [batch_final_gap a y ha l n J hJ]
-  apply mul_le_of_le_one_right (batchBoundary_pos a y hy l n J).le
-  apply Real.exp_le_one_iff.mpr
-  have ht := (mixingTime_nonneg a).trans h.2.1
-  nlinarith
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart CompactSequence MomentBatches
-
-lemma sum_getD_eq_map_sum (L : List ℝ) (f : ℝ→ℝ) :
-    (∑i∈Finset.range L.length,f (L.getD i 1))=(L.map f).sum := by
-  rw [←Fin.sum_univ_eq_sum_range]
-  simp only [List.getD_eq_getElem _ _ (Fin.isLt _)]
-  rw [←List.sum_ofFn]
-  change (List.ofFn (f ∘ (fun i : Fin L.length=>L[i.1]))).sum = _
-  rw [←List.map_ofFn,List.ofFn_getElem]
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4 < a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-def batchLevel (n : ℕ) : ℕ := momentLevel M.law sequenceMassReal sequenceLevelReal n
-
-def testCount (n : ℕ) : ℕ := (noncleanup (moment M.law sequenceMassReal) n).length
-
-def testThreshold (n i : ℕ) : ℝ := (noncleanup (moment M.law sequenceMassReal) n).getD i 1
-
-def finalRadius (n : ℕ) : ℝ := batchRadius a (M.batchLevel n) n (M.testCount n)
-
-lemma finalRadius_pos (n : ℕ) : 0<M.finalRadius n := batchRadius_pos ..
-
-lemma finalRadius_le (n : ℕ) : M.finalRadius n ≤ Real.exp (-(n:ℝ)^2) := by
-  rw [←batchRadius_zero a (M.batchLevel n) n]
-  exact batchRadius_antitone a (by linarith) _ _ (Nat.zero_le _)
-
-lemma testThreshold_pos (n i : ℕ) : 0 < M.testThreshold n i := by
-  classical
-  unfold testThreshold
-  by_cases hi : i < (noncleanup (moment M.law sequenceMassReal) n).length
-  · rw [List.getD_eq_getElem _ _ hi]
-    have hm := List.getElem_mem hi
-    obtain ⟨k,hk,h⟩ := List.mem_flatMap.mp hm
-    obtain ⟨_,he⟩ := List.mem_replicate.mp h
-    rw [he]
-    positivity
-  · rw [List.getD_eq_default _ _ (Nat.le_of_not_gt hi)]
-    norm_num
-
- 
-
-def testIndex (n : ℕ) (hn : 0<n) (i : ℕ) (hi : i<M.testCount n) : M.Entries :=
-  ⟨⟨n,hn⟩,⟨i,by
-    change i<(batch (moment M.law sequenceMassReal) n).length
-    change i < (noncleanup (moment M.law sequenceMassReal) n).length at hi
-    simpa only [batch,List.length_append,List.length_singleton] using Nat.lt_succ_of_lt hi⟩⟩
-
-lemma testIndex_threshold (n : ℕ) (hn : 0<n) (i : ℕ) (hi : i<M.testCount n) :
-    M.entryThreshold (M.testIndex n hn i hi)=M.testThreshold n i := by
-  simp only [entryThreshold,threshold,testIndex,batch,List.getElem_append_left (by exact hi),
-    testThreshold,List.getD_eq_getElem _ _ (by exact hi)]
-
-lemma testIndex_radius (n : ℕ) (hn : 0<n) (i : ℕ) (hi : i<M.testCount n) :
-    M.entryRadius (M.testIndex n hn i hi)=batchRadius a (M.batchLevel n) n i := rfl
-
-lemma cleanup_radius (n : ℕ) (hn : 0<n) :
-    M.entryRadius (cleanupIndex (moment M.law sequenceMassReal) n hn)=M.finalRadius n := rfl
-
-lemma batch_success (n : ℕ) (hn : 3≤n) :
-    (2:ℝ)^n/8 ≤ ∑i∈Finset.range (M.testCount n),
-      M.law.real {x | M.testThreshold n i ≤ sequenceLevelReal (M.batchLevel n) x} := by
-  let := M.probability
-  have hnp : 0<n := by omega
-  have hfin := finite_multiplicity_support hnp (M.moments_positive 1 le_rfl)
-    (M.moments_positive n hnp) (M.moments_positive (n+1) (by omega))
-    (mean_pow_le_moment M.law sequenceMassReal (Eventually.of_forall sequenceMassReal_nonneg)
-      (by simpa only [pow_one] using M.integrable_moments 1) n (M.integrable_moments n))
-  change _ ≤ ∑i∈Finset.range (noncleanup (moment M.law sequenceMassReal) n).length,
-    (fun b=>M.law.real {x | b ≤ sequenceLevelReal (M.batchLevel n) x})
-      ((noncleanup (moment M.law sequenceMassReal) n).getD i 1)
-  rw [sum_getD_eq_map_sum _ (fun b=>M.law.real {x | b ≤ sequenceLevelReal (M.batchLevel n) x}),
-    noncleanup_sum,sum_eq_tsum_multiplicity hfin]
-  exact M.success n hn
-
-lemma testCount_pos (n : ℕ) (hn : 3≤n) : 0<M.testCount n := by
-  have hs := M.batch_success n hn
-  by_contra! hh
-  have he := Nat.eq_zero_of_le_zero hh
-  rw [he] at hs
-  simp only [Finset.range_zero,Finset.sum_empty] at hs
-  have : (0:ℝ)<2^n/8 := by positivity
-  linarith
-
-lemma batch_half_success (n : ℕ) (hn : 3≤n) :
-    (2:ℝ)^n/16 ≤ ∑i∈Finset.range (M.testCount n),
-      M.law.real {x | M.testThreshold n i ≤ sequenceLevelReal (M.batchLevel n) x}/2 := by
-  rw [←Finset.sum_div]
-  linarith [M.batch_success n hn]
-
-def batchFailures (z : ℂ) (n : ℕ) : Set GDriver :=
-  {b | ∀i<M.testCount n,¬(ENNReal.ofReal (M.testThreshold n i) ≤
-    M.stoppedTest z (400*batchRadius a (M.batchLevel n) n i) (M.batchLevel n) b)}
-
-lemma batchFailures_eq_testFailure (z : ℂ) (hz : 0<z.im) (n : ℕ)
-    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
-    M.batchFailures z n=M.testFailure z
-      (batchSchedule a z.im (by linarith) hz (M.batchLevel n) n hr hy
-        (M.testThreshold n) (fun i=>(M.testThreshold_pos n i).le)) (M.testCount n) := by
-  ext b
-  simp only [batchFailures,mem_ofPred_eq,testFailure,AdaptedFailure.failed_iff]
-  rfl
-
-lemma batch_failures_measurable (z : ℂ) (hz : 0<z.im) (n : ℕ)
-    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
-    MeasurableSet (M.batchFailures z n) := by
-  rw [M.batchFailures_eq_testFailure z hz n hr hy]
-  exact (radialTime_stopping a (by linarith) z hz _
-    (batchBoundary_pos a z.im hz _ _ _) (batchBoundary_le a z.im (by linarith) hz _ _ hr hy _)).measurableSpace_le _
-    (M.testFailure_raw z hz _ _)
-
- 
-
-theorem batch_failure_chordal [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0<z.im) (n : ℕ) (hr : BatchReady a n)
-    (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
-    W (M.batchFailures z n∩{b | radialTime a b.1 z (800*M.finalRadius n)≠∞}) ≤
-      Jacobi.visitConstant a*ENNReal.ofReal ((800*M.finalRadius n/z.im)^(alpha a))*
-        ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
-  rw [M.batchFailures_eq_testFailure z hz n hr hy]
-  apply M.testFailure_chordal hW z hz _ _ _
-    (by have := M.finalRadius_pos n; positivity)
-    (batch_final_le a z.im (by linarith) hz _ _ _ (M.testCount_pos n hr.1) hr)
-    ((n:ℝ)-Real.log 2/(2*a)) hr.2.1
-    (batch_final_gap a z.im (by linarith) _ _ _ (M.testCount_pos n hr.1))
-    ((2:ℝ)^n/16)
-  exact M.batch_half_success n hr.1
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.SpatialPastTail
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.AdaptedFailure
+
+variable {Ω : Type*} [m₀ : MeasurableSpace Ω]
+
+ 
+def failed (success : ℕ→Set Ω) : ℕ→Set Ω
+  | 0 => univ
+  | n+1 => failed success n \ success n
+
+omit m₀ in
+lemma failed_iff [MeasurableSpace Ω] (success : ℕ→Set Ω) (n : ℕ) (ω : Ω) :
+    ω∈failed success n ↔ ∀i<n,ω∉success i := by
+  induction n with
+  | zero => simp [failed]
+  | succ n ih =>
+    simp only [failed,Set.mem_sdiff,ih,Nat.lt_succ_iff_lt_or_eq]
+    constructor
+    · rintro ⟨h,h'⟩ i (hi | rfl)
+      · exact h i hi
+      · exact h'
+    · intro h
+      exact ⟨fun i hi=>h i (Or.inl hi),h n (Or.inr rfl)⟩
+
+lemma failed_measurable (ℱ : Filtration ℕ m₀)
+    (success : ℕ→Set Ω) (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (n : ℕ) :
+    MeasurableSet[ℱ n] (failed success n) := by
+  induction n with
+  | zero => exact MeasurableSet.univ
+  | succ n ih => exact (ℱ.mono (Nat.le_succ n) _ ih).diff (hs n)
+
+ 
+
+theorem failure_product (P : Measure Ω) [IsProbabilityMeasure P]
+    (ℱ : Filtration ℕ m₀) (success : ℕ→Set Ω)
+    (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (p : ℕ→ℝ)
+    (hp1 : ∀i,p i≤1)
+    (htest : ∀i,∀A : Set Ω,MeasurableSet[ℱ i] A →
+      p i*P.real A≤P.real (A∩success i)) (n : ℕ) :
+    P.real (failed success n)≤∏i∈Finset.range n,(1-p i) := by
+  induction n with
+  | zero => simp [failed]
+  | succ n ih =>
+    have hs' : MeasurableSet (success n) := ℱ.le (n+1) _ (hs n)
+    have hsplit := measureReal_inter_add_sdiff (μ := P) (s := failed success n) hs'
+    have hstep := htest n (failed success n) (failed_measurable ℱ success hs n)
+    calc
+      P.real (failed success (n+1))≤(1-p n)*P.real (failed success n) := by
+        change P.real (failed success n \ success n)≤_
+        nlinarith
+      _ ≤ (1-p n)*(∏i∈Finset.range n,(1-p i)) :=
+        mul_le_mul_of_nonneg_left ih (sub_nonneg.mpr (hp1 n))
+      _ = _ := by rw [Finset.prod_range_succ]; ring
+
+lemma product_le_exp (p : ℕ→ℝ) (hp1 : ∀i,p i≤1) (n : ℕ) :
+    (∏i∈Finset.range n,(1-p i))≤Real.exp (-(∑i∈Finset.range n,p i)) := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    calc
+      _ = (∏i∈Finset.range n,(1-p i))*(1-p n) := Finset.prod_range_succ _ _
+      _ ≤ Real.exp (-(∑i∈Finset.range n,p i))*Real.exp (-p n) :=
+        mul_le_mul ih (Real.one_sub_le_exp_neg _) (sub_nonneg.mpr (hp1 n)) (Real.exp_pos _).le
+      _ = _ := by rw [←Real.exp_add,Finset.sum_range_succ,neg_add_rev]; congr 1; ring
+
+ 
+
+lemma tested_of_conditional (P : Measure Ω) [IsProbabilityMeasure P]
+    {m : MeasurableSpace Ω} (hm : m ≤ m₀)
+    (S : Set Ω) (hS : MeasurableSet[m₀] S) (p : ℝ)
+    (hp : ∀ᵐ ω∂P,p≤P[S.indicator (fun _=> (1:ℝ)) | m] ω)
+    (A : Set Ω) (hA : MeasurableSet[m] A) :
+    p*P.real A≤P.real (A∩S) := by
+  have hInt : Integrable (S.indicator (fun _ : Ω=>(1:ℝ))) P :=
+    (integrable_const 1).indicator hS
+  calc
+    p*P.real A = ∫ω in A,p∂P := by simp only [setIntegral_const,smul_eq_mul]; ring
+    _ ≤ ∫ω in A,P[S.indicator (fun _=>(1:ℝ)) | m] ω∂P :=
+      integral_mono_ae (integrable_const _) integrable_condExp.integrableOn
+        (ae_restrict_of_ae hp)
+    _ = ∫ω in A,S.indicator (fun _=>(1:ℝ)) ω∂P := setIntegral_condExp hm hInt hA
+    _ = P.real (A∩S) := by rw [setIntegral_indicator hS]; simp
+
+ 
+
+theorem conditional_failure (P : Measure Ω) [IsProbabilityMeasure P]
+    (ℱ : Filtration ℕ m₀) (success : ℕ→Set Ω)
+    (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (p : ℕ→ℝ)
+    (hp1 : ∀i,p i≤1)
+    (hp : ∀i,∀ᵐ ω∂P,p i≤P[(success i).indicator (fun _=>(1:ℝ)) | ℱ i] ω)
+    (n : ℕ) (B : ℝ) (hB : B≤∑i∈Finset.range n,p i) :
+    P.real (failed success n)≤Real.exp (-B) := by
+  refine (failure_product P ℱ success hs p hp1 ?_ n).trans
+    ((product_le_exp p hp1 n).trans (Real.exp_le_exp.mpr (neg_le_neg hB)))
+  intro i A hA
+  exact tested_of_conditional P (ℱ.le i) (success i) (ℱ.le (i+1) _ (hs i))
+    (p i) (hp i) A hA
+
+ 
+
+lemma weighted_past_bound (P : Measure Ω) [IsFiniteMeasure P]
+    {m : MeasurableSpace Ω} (hm : m ≤ m₀)
+    (f : Ω→ℝ) (hf : Integrable f P) (K : ℝ)
+    (hb : ∀ᵐ ω∂P,P[f | m] ω≤K) (A : Set Ω) (hA : MeasurableSet[m] A) :
+    (∫ω in A,f ω∂P)≤K*P.real A := by
+  calc
+    (∫ω in A,f ω∂P) = ∫ω in A,P[f | m] ω∂P := (setIntegral_condExp hm hf hA).symm
+    _ ≤ ∫ω in A,K∂P := integral_mono_ae integrable_condExp.integrableOn
+      (integrable_const _) (ae_restrict_of_ae hb)
+    _ = K*P.real A := by simp only [setIntegral_const,smul_eq_mul]; ring
+
+ 
+
+theorem weighted_conditional_failure (P : Measure Ω) [IsProbabilityMeasure P]
+    (ℱ : Filtration ℕ m₀) (success : ℕ→Set Ω)
+    (hs : ∀i,MeasurableSet[ℱ (i+1)] (success i)) (p : ℕ→ℝ)
+    (hp1 : ∀i,p i≤1)
+    (hp : ∀i,∀ᵐ ω∂P,p i≤P[(success i).indicator (fun _=>(1:ℝ)) | ℱ i] ω)
+    (n : ℕ) (B : ℝ) (hB : B≤∑i∈Finset.range n,p i)
+    (f : Ω→ℝ) (hf : Integrable f P) (K : ℝ) (hK : 0≤K)
+    (hb : ∀ᵐ ω∂P,P[f | ℱ n] ω≤K) :
+    (∫ω in failed success n,f ω∂P)≤K*Real.exp (-B) :=
+  (weighted_past_bound P (ℱ.le n) f hf K hb (failed success n)
+    (failed_measurable ℱ success hs n)).trans
+    (mul_le_mul_of_nonneg_left (conditional_failure P ℱ success hs p hp1 hp n B hB) hK)
+
+end SLEExactGauge.AdaptedFailure
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart CompactSequence MomentBatches AdaptedFailure
+
+ 
+
+structure RadialTestSchedule (a y : ℝ) where
+  start : ℕ → ℝ
+  boundary : ℕ → ℝ
+  level : ℕ
+  threshold : ℕ → ℝ
+  boundary_pos : ∀i,0 < boundary i
+  boundary_le : ∀i,boundary i ≤ y
+  start_pos : ∀i,0 < start i
+  start_le : ∀i,start i ≤ boundary i
+  next : ∀i,boundary (i+1)=start i*Real.exp (-2*a*level)
+  threshold_nonneg : ∀i,0 ≤ threshold i
+  gap : ∀i,∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧ start i=boundary i*Real.exp (-2*a*s)
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4 < a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+def testSuccess (z : ℂ) (D : RadialTestSchedule a z.im) (i : ℕ) : Set GDriver :=
+  {b | ENNReal.ofReal (D.threshold i) ≤ M.stoppedTest z (D.start i) D.level b}
+
+def testFailure (z : ℂ) (D : RadialTestSchedule a z.im) (n : ℕ) : Set GDriver :=
+  failed (M.testSuccess z D) n
+
+lemma testFailure_raw (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ) :
+    MeasurableSet[(radialTime_stopping a (by linarith) z hz (D.boundary n)
+      (D.boundary_pos n) (D.boundary_le n)).measurableSpace] (M.testFailure z D n) := by
+  induction n with
+  | zero => exact MeasurableSet.univ
+  | succ n ih =>
+    have hs := M.stoppedTest_measurable z hz (D.start n) (D.start_pos n)
+      ((D.start_le n).trans (D.boundary_le n)) D.level
+    have hn : D.boundary (n+1) ≤ D.boundary n := by
+      rw [D.next]
+      exact (testEndpoint_le (ha := ha) _ (D.start_pos n).le _).trans (D.start_le n)
+    have hmono := (radialTime_stopping a (by linarith) z hz (D.boundary n)
+      (D.boundary_pos n) (D.boundary_le n)).measurableSpace_mono
+      (radialTime_stopping a (by linarith) z hz (D.boundary (n+1))
+        (D.boundary_pos _) (D.boundary_le _))
+      (radialTime_antitone a (by linarith) z hz _ _ (D.boundary_pos _) hn (D.boundary_le n))
+    apply (hmono _ ih).diff
+    unfold testSuccess
+    have he := measurableSet_le (f := fun _=>ENNReal.ofReal (D.threshold n)) measurable_const hs
+    exact ⟨he.1,fun t=>by simpa only [D.next] using he.2 t⟩
+
+ 
+
+theorem testFailure_product [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ) :
+    (radialLaw W a z (D.boundary n)).real (M.testFailure z D n) ≤
+      ∏i∈Finset.range n,(1-M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2) := by
+  let := M.probability
+  have hP (i : ℕ) : IsProbabilityMeasure (radialLaw W a z (D.boundary i)) :=
+    radialLaw_probability W hW a ha z hz _ (D.boundary_pos _) (D.boundary_le _)
+  induction n with
+  | zero =>
+    let := hP 0
+    simp [testFailure,failed]
+  | succ n ih =>
+    let := hP (n+1)
+    have hraw := M.testFailure_raw z hz D n
+    obtain ⟨s,hs,hgap⟩ := D.gap n
+    have hl := M.stopped_success_lower hW z hz (D.boundary n) (D.start n)
+      (D.start_pos n) (D.start_le n) (D.boundary_le n) s hs hgap D.level
+      (D.threshold n) (D.threshold_nonneg n) (M.testFailure z D n) hraw
+    rw [←D.next] at hl
+    have hlr := ENNReal.toReal_mono (measure_ne_top _ _) hl
+    simp only [ENNReal.toReal_mul,ENNReal.toReal_div,ENNReal.toReal_one,ENNReal.toReal_ofNat] at hlr
+    change 1/2*M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}*
+      (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n) ≤
+      (radialLaw W a z (D.boundary (n+1))).real
+        (M.testFailure z D n∩M.testSuccess z D n) at hlr
+    have hm : MeasurableSet (M.testSuccess z D n) := by
+      have hT := radialTime_stopping a (by linarith) z hz _
+        (testEndpoint_pos (a := a) _ (D.start_pos n) D.level)
+        ((testEndpoint_le (ha := ha) _ (D.start_pos n).le _).trans
+          ((D.start_le n).trans (D.boundary_le n)))
+      exact hT.measurableSpace_le _ (measurableSet_le measurable_const
+        (M.stoppedTest_measurable z hz (D.start n) (D.start_pos n)
+          ((D.start_le n).trans (D.boundary_le n)) D.level))
+    have hsplit := measureReal_inter_add_sdiff (μ := radialLaw W a z (D.boundary (n+1)))
+      (s := M.testFailure z D n) hm
+    have hn : D.boundary (n+1) ≤ D.boundary n := by
+      rw [D.next]
+      exact (testEndpoint_le (ha := ha) _ (D.start_pos n).le _).trans (D.start_le n)
+    have he := congrArg ENNReal.toReal (radialLaw_consistent_le W hW a ha z hz
+      (D.boundary n) (D.boundary (n+1)) (D.boundary_pos _) hn (D.boundary_le n)
+      (M.testFailure z D n) hraw)
+    change (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n)=
+      (radialLaw W a z (D.boundary n)).real (M.testFailure z D n) at he
+    have hp : M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x} ≤ 1 := measureReal_le_one
+    calc
+      _ ≤ (1-M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}/2)*
+          (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n) := by
+        change (radialLaw W a z (D.boundary (n+1))).real (M.testFailure z D n \ M.testSuccess z D n) ≤ _
+        nlinarith
+      _ = (1-M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}/2)*
+          (radialLaw W a z (D.boundary n)).real (M.testFailure z D n) := by rw [he]
+      _ ≤ (1-M.law.real {x | D.threshold n ≤ sequenceLevelReal D.level x}/2)*
+          (∏i∈Finset.range n,(1-M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2)) :=
+        mul_le_mul_of_nonneg_left ih (by linarith)
+      _ = _ := by rw [Finset.prod_range_succ]; ring
+
+lemma testFailure_exponential [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ)
+    (B : ℝ) (hB : B ≤ ∑i∈Finset.range n,M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2) :
+    radialLaw W a z (D.boundary n) (M.testFailure z D n) ≤ ENNReal.ofReal (Real.exp (-B)) := by
+  let := M.probability
+  have := radialLaw_probability W hW a ha z hz (D.boundary n) (D.boundary_pos _) (D.boundary_le _)
+  have hp (i : ℕ) : M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2 ≤ 1 := by
+    have : M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x} ≤ 1 := measureReal_le_one
+    linarith
+  have hh := (M.testFailure_product hW z hz D n).trans
+    ((product_le_exp _ hp n).trans (Real.exp_le_exp.mpr (neg_le_neg hB)))
+  simpa only [Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using ENNReal.ofReal_le_ofReal hh
+
+ 
+theorem testFailure_chordal [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0 < z.im) (D : RadialTestSchedule a z.im) (n : ℕ)
+    (ρ : ℝ) (hρ : 0 < ρ) (hρσ : ρ ≤ D.boundary n)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=D.boundary n*Real.exp (-2*a*s))
+    (B : ℝ) (hB : B ≤ ∑i∈Finset.range n,M.law.real {x | D.threshold i ≤ sequenceLevelReal D.level x}/2) :
+    W (M.testFailure z D n∩{b | radialTime a b.1 z ρ≠∞}) ≤
+      Jacobi.visitConstant a*ENNReal.ofReal ((ρ/z.im)^(alpha a))*ENNReal.ofReal (Real.exp (-B)) := by
+  have hr := M.testFailure_raw z hz D n
+  have hh := radial_change_law_past_bound W hW a ha z hz (D.boundary n) ρ hρ hρσ
+    (D.boundary_le n) s hs hgap (M.testFailure z D n) hr
+  rw [radialLaw_consistent_le W hW a ha z hz _ ρ hρ hρσ (D.boundary_le n) _ hr] at hh
+  exact hh.trans (mul_le_mul le_rfl (M.testFailure_exponential hW z hz D n B hB) zero_le zero_le)
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator LoewnerRestart
+
+ 
+def batchRadius (a : ℝ) (l n i : ℕ) : ℝ :=
+  Real.exp (-(n:ℝ)^2-2*a*(i:ℝ)*((l:ℝ)+(n:ℝ)))
+
+lemma batchRadius_pos (a : ℝ) (l n i : ℕ) : 0 < batchRadius a l n i := Real.exp_pos _
+
+lemma batchRadius_zero (a : ℝ) (l n : ℕ) : batchRadius a l n 0=Real.exp (-(n:ℝ)^2) := by
+  simp [batchRadius]
+
+lemma batchRadius_succ (a : ℝ) (l n i : ℕ) :
+    batchRadius a l n (i+1)=batchRadius a l n i*Real.exp (-2*a*l)*Real.exp (-2*a*n) := by
+  simp only [batchRadius,←Real.exp_add,Nat.cast_add,Nat.cast_one]
+  congr 1
+  ring
+
+lemma batchRadius_antitone (a : ℝ) (ha : 0 ≤ a) (l n : ℕ) : Antitone (batchRadius a l n) := by
+  intro i j hij
+  apply Real.exp_le_exp.mpr
+  have hj : (i:ℝ) ≤ j := by exact_mod_cast hij
+  have hm : 0 ≤ 2*a*((l:ℝ)+(n:ℝ)) := by positivity
+  nlinarith [mul_le_mul_of_nonneg_left hj hm]
+
+lemma mixingTime_nonneg (a : ℝ) : 0 ≤ Jacobi.mixingTime (4*a) := by
+  unfold Jacobi.mixingTime Jacobi.majorantBase
+  positivity
+
+def BatchReady (a : ℝ) (n : ℕ) : Prop :=
+  3 ≤ n ∧ Jacobi.mixingTime (4*a) ≤ (n:ℝ)-Real.log 2/(2*a) ∧
+  400*Real.exp (-(n:ℝ)^2) ≤ Real.exp (-(n:ℝ))*Real.exp (-2*a*Jacobi.mixingTime (4*a))
+
+lemma eventually_batchReady (a : ℝ) (_ha : 0 < a) : ∀ᶠ n in atTop,BatchReady a n := by
+  have hR (x : ℝ) : ∀ᶠ n : ℕ in atTop,x ≤ (n:ℝ) :=
+    (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_ge_atTop x)
+  filter_upwards [eventually_ge_atTop 3,hR (Jacobi.mixingTime (4*a)+Real.log 2/(2*a)),
+    hR (Real.log 400+2*a*Jacobi.mixingTime (4*a))] with n hn hmix hinit
+  refine ⟨hn,by linarith,?_⟩
+  have hnR : (3:ℝ) ≤ n := by exact_mod_cast hn
+  have hsq : (n:ℝ) ≤ (n:ℝ)^2-(n:ℝ) := by nlinarith
+  rw [show (400:ℝ)=Real.exp (Real.log 400) by rw [Real.exp_log (by norm_num)],
+    ←Real.exp_add,←Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  linarith
+
+lemma BatchReady.mixing {a : ℝ} (ha : 0<a) {n : ℕ} (h : BatchReady a n) :
+    Jacobi.mixingTime (4*a) ≤ (n:ℝ) := by
+  have : 0 ≤ Real.log 2/(2*a) := div_nonneg (Real.log_nonneg (by norm_num)) (by positivity)
+  linarith [h.2.1]
+
+ 
+
+def batchBoundary (a y : ℝ) (l n : ℕ) : ℕ→ℝ
+  | 0 => y
+  | i+1 => 400*batchRadius a l n i*Real.exp (-2*a*l)
+
+lemma batchBoundary_pos (a y : ℝ) (hy : 0<y) (l n i : ℕ) : 0<batchBoundary a y l n i := by
+  cases i with
+  | zero => exact hy
+  | succ i => dsimp [batchBoundary]; have := batchRadius_pos a l n i; positivity
+
+lemma batchBoundary_start (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n : ℕ)
+    (h : BatchReady a n) (hyn : Real.exp (-(n:ℝ)) ≤ y) (i : ℕ) :
+    400*batchRadius a l n i ≤ batchBoundary a y l n i ∧
+      ∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧
+        400*batchRadius a l n i=batchBoundary a y l n i*Real.exp (-2*a*s) := by
+  cases i with
+  | zero =>
+    apply clock_gap_of_small a ha _ (mixingTime_nonneg a) y _ hy
+      (by have := batchRadius_pos a l n 0; positivity)
+    rw [batchRadius_zero]
+    exact h.2.2.trans (mul_le_mul_of_nonneg_right hyn (Real.exp_pos _).le)
+  | succ i =>
+    have hg : 400*batchRadius a l n (i+1)=batchBoundary a y l n (i+1)*Real.exp (-2*a*n) := by
+      rw [batchRadius_succ,batchBoundary]; ring
+    refine ⟨?_,(n:ℝ),h.mixing ha,hg⟩
+    rw [hg]
+    exact mul_le_of_le_one_right (batchBoundary_pos a y hy l n _).le
+      (Real.exp_le_one_iff.mpr (by have : (0:ℝ) ≤ n := Nat.cast_nonneg n; nlinarith))
+
+lemma batchBoundary_le (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n : ℕ)
+    (h : BatchReady a n) (hyn : Real.exp (-(n:ℝ)) ≤ y) (i : ℕ) :
+    batchBoundary a y l n i ≤ y := by
+  induction i with
+  | zero => exact le_rfl
+  | succ i ih =>
+    change 400*batchRadius a l n i*Real.exp (-2*a*l) ≤ y
+    have hstart := (batchBoundary_start a y ha hy l n h hyn i).1.trans ih
+    refine (mul_le_of_le_one_right (by have := batchRadius_pos a l n i; positivity) ?_).trans hstart
+    exact Real.exp_le_one_iff.mpr (by have : (0:ℝ) ≤ l := Nat.cast_nonneg l; nlinarith)
+
+def batchSchedule (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n : ℕ)
+    (h : BatchReady a n) (hyn : Real.exp (-(n:ℝ)) ≤ y) (b : ℕ→ℝ) (hb : ∀i,0 ≤ b i) :
+    RadialTestSchedule a y where
+  start i := 400*batchRadius a l n i
+  boundary := batchBoundary a y l n
+  level := l
+  threshold := b
+  boundary_pos := batchBoundary_pos a y hy l n
+  boundary_le := batchBoundary_le a y ha hy l n h hyn
+  start_pos i := by have := batchRadius_pos a l n i; positivity
+  start_le i := (batchBoundary_start a y ha hy l n h hyn i).1
+  next _ := rfl
+  threshold_nonneg := hb
+  gap i := (batchBoundary_start a y ha hy l n h hyn i).2
+
+lemma batch_final_gap (a y : ℝ) (ha : 0<a) (l n J : ℕ) (hJ : 0<J) :
+    800*batchRadius a l n J=batchBoundary a y l n J*
+      Real.exp (-2*a*((n:ℝ)-Real.log 2/(2*a))) := by
+  obtain ⟨i,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hJ.ne'
+  rw [batchRadius_succ,batchBoundary,
+    show -2*a*((n:ℝ)-Real.log 2/(2*a))= -2*a*n+Real.log 2 by field_simp; ring,
+    Real.exp_add,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  ring
+
+lemma batch_final_le (a y : ℝ) (ha : 0<a) (hy : 0<y) (l n J : ℕ) (hJ : 0<J)
+    (h : BatchReady a n) : 800*batchRadius a l n J ≤ batchBoundary a y l n J := by
+  rw [batch_final_gap a y ha l n J hJ]
+  apply mul_le_of_le_one_right (batchBoundary_pos a y hy l n J).le
+  apply Real.exp_le_one_iff.mpr
+  have ht := (mixingTime_nonneg a).trans h.2.1
+  nlinarith
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart CompactSequence MomentBatches
+
+lemma sum_getD_eq_map_sum (L : List ℝ) (f : ℝ→ℝ) :
+    (∑i∈Finset.range L.length,f (L.getD i 1))=(L.map f).sum := by
+  rw [←Fin.sum_univ_eq_sum_range]
+  simp only [List.getD_eq_getElem _ _ (Fin.isLt _)]
+  rw [←List.sum_ofFn]
+  change (List.ofFn (f ∘ (fun i : Fin L.length=>L[i.1]))).sum = _
+  rw [←List.map_ofFn,List.ofFn_getElem]
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4 < a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+def batchLevel (n : ℕ) : ℕ := momentLevel M.law sequenceMassReal sequenceLevelReal n
+
+def testCount (n : ℕ) : ℕ := (noncleanup (moment M.law sequenceMassReal) n).length
+
+def testThreshold (n i : ℕ) : ℝ := (noncleanup (moment M.law sequenceMassReal) n).getD i 1
+
+def finalRadius (n : ℕ) : ℝ := batchRadius a (M.batchLevel n) n (M.testCount n)
+
+lemma finalRadius_pos (n : ℕ) : 0<M.finalRadius n := batchRadius_pos ..
+
+lemma finalRadius_le (n : ℕ) : M.finalRadius n ≤ Real.exp (-(n:ℝ)^2) := by
+  rw [←batchRadius_zero a (M.batchLevel n) n]
+  exact batchRadius_antitone a (by linarith) _ _ (Nat.zero_le _)
+
+lemma testThreshold_pos (n i : ℕ) : 0 < M.testThreshold n i := by
+  classical
+  unfold testThreshold
+  by_cases hi : i < (noncleanup (moment M.law sequenceMassReal) n).length
+  · rw [List.getD_eq_getElem _ _ hi]
+    have hm := List.getElem_mem hi
+    obtain ⟨k,hk,h⟩ := List.mem_flatMap.mp hm
+    obtain ⟨_,he⟩ := List.mem_replicate.mp h
+    rw [he]
+    positivity
+  · rw [List.getD_eq_default _ _ (Nat.le_of_not_gt hi)]
+    norm_num
+
+ 
+
+def testIndex (n : ℕ) (hn : 0<n) (i : ℕ) (hi : i<M.testCount n) : M.Entries :=
+  ⟨⟨n,hn⟩,⟨i,by
+    change i<(batch (moment M.law sequenceMassReal) n).length
+    change i < (noncleanup (moment M.law sequenceMassReal) n).length at hi
+    simpa only [batch,List.length_append,List.length_singleton] using Nat.lt_succ_of_lt hi⟩⟩
+
+lemma testIndex_threshold (n : ℕ) (hn : 0<n) (i : ℕ) (hi : i<M.testCount n) :
+    M.entryThreshold (M.testIndex n hn i hi)=M.testThreshold n i := by
+  simp only [entryThreshold,threshold,testIndex,batch,List.getElem_append_left (by exact hi),
+    testThreshold,List.getD_eq_getElem _ _ (by exact hi)]
+
+lemma testIndex_radius (n : ℕ) (hn : 0<n) (i : ℕ) (hi : i<M.testCount n) :
+    M.entryRadius (M.testIndex n hn i hi)=batchRadius a (M.batchLevel n) n i := rfl
+
+lemma cleanup_radius (n : ℕ) (hn : 0<n) :
+    M.entryRadius (cleanupIndex (moment M.law sequenceMassReal) n hn)=M.finalRadius n := rfl
+
+lemma batch_success (n : ℕ) (hn : 3≤n) :
+    (2:ℝ)^n/8 ≤ ∑i∈Finset.range (M.testCount n),
+      M.law.real {x | M.testThreshold n i ≤ sequenceLevelReal (M.batchLevel n) x} := by
+  let := M.probability
+  have hnp : 0<n := by omega
+  have hfin := finite_multiplicity_support hnp (M.moments_positive 1 le_rfl)
+    (M.moments_positive n hnp) (M.moments_positive (n+1) (by omega))
+    (mean_pow_le_moment M.law sequenceMassReal (Eventually.of_forall sequenceMassReal_nonneg)
+      (by simpa only [pow_one] using M.integrable_moments 1) n (M.integrable_moments n))
+  change _ ≤ ∑i∈Finset.range (noncleanup (moment M.law sequenceMassReal) n).length,
+    (fun b=>M.law.real {x | b ≤ sequenceLevelReal (M.batchLevel n) x})
+      ((noncleanup (moment M.law sequenceMassReal) n).getD i 1)
+  rw [sum_getD_eq_map_sum _ (fun b=>M.law.real {x | b ≤ sequenceLevelReal (M.batchLevel n) x}),
+    noncleanup_sum,sum_eq_tsum_multiplicity hfin]
+  exact M.success n hn
+
+lemma testCount_pos (n : ℕ) (hn : 3≤n) : 0<M.testCount n := by
+  have hs := M.batch_success n hn
+  by_contra! hh
+  have he := Nat.eq_zero_of_le_zero hh
+  rw [he] at hs
+  simp only [Finset.range_zero,Finset.sum_empty] at hs
+  have : (0:ℝ)<2^n/8 := by positivity
+  linarith
+
+lemma batch_half_success (n : ℕ) (hn : 3≤n) :
+    (2:ℝ)^n/16 ≤ ∑i∈Finset.range (M.testCount n),
+      M.law.real {x | M.testThreshold n i ≤ sequenceLevelReal (M.batchLevel n) x}/2 := by
+  rw [←Finset.sum_div]
+  linarith [M.batch_success n hn]
+
+def batchFailures (z : ℂ) (n : ℕ) : Set GDriver :=
+  {b | ∀i<M.testCount n,¬(ENNReal.ofReal (M.testThreshold n i) ≤
+    M.stoppedTest z (400*batchRadius a (M.batchLevel n) n i) (M.batchLevel n) b)}
+
+lemma batchFailures_eq_testFailure (z : ℂ) (hz : 0<z.im) (n : ℕ)
+    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
+    M.batchFailures z n=M.testFailure z
+      (batchSchedule a z.im (by linarith) hz (M.batchLevel n) n hr hy
+        (M.testThreshold n) (fun i=>(M.testThreshold_pos n i).le)) (M.testCount n) := by
+  ext b
+  simp only [batchFailures,mem_ofPred_eq,testFailure,AdaptedFailure.failed_iff]
+  rfl
+
+lemma batch_failures_measurable (z : ℂ) (hz : 0<z.im) (n : ℕ)
+    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
+    MeasurableSet (M.batchFailures z n) := by
+  rw [M.batchFailures_eq_testFailure z hz n hr hy]
+  exact (radialTime_stopping a (by linarith) z hz _
+    (batchBoundary_pos a z.im hz _ _ _) (batchBoundary_le a z.im (by linarith) hz _ _ hr hy _)).measurableSpace_le _
+    (M.testFailure_raw z hz _ _)
+
+ 
+
+theorem batch_failure_chordal [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0<z.im) (n : ℕ) (hr : BatchReady a n)
+    (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
+    W (M.batchFailures z n∩{b | radialTime a b.1 z (800*M.finalRadius n)≠∞}) ≤
+      Jacobi.visitConstant a*ENNReal.ofReal ((800*M.finalRadius n/z.im)^(alpha a))*
+        ENNReal.ofReal (Real.exp (-((2:ℝ)^n/16))) := by
+  rw [M.batchFailures_eq_testFailure z hz n hr hy]
+  apply M.testFailure_chordal hW z hz _ _ _
+    (by have := M.finalRadius_pos n; positivity)
+    (batch_final_le a z.im (by linarith) hz _ _ _ (M.testCount_pos n hr.1) hr)
+    ((n:ℝ)-Real.log 2/(2*a)) hr.2.1
+    (batch_final_gap a z.im (by linarith) _ _ _ (M.testCount_pos n hr.1))
+    ((2:ℝ)^n/16)
+  exact M.batch_half_success n hr.1
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/AngleGenerators.lean b/lean/OAI/Probability/SLEGauge/AngleGenerators.lean
index 329239067fac3b2a5d5650bbcb19a6048f38ec29..abb6d93cf34e01286051408a1fac0a35ab27287c 100644
--- a/lean/OAI/Probability/SLEGauge/AngleGenerators.lean
+++ b/lean/OAI/Probability/SLEGauge/AngleGenerators.lean
@@ -1,451 +1,451 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialStopping
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-namespace ExactSLEGreen
-lemma physicalTheta_second_noise (v : State) (hv : 0<Y v) :
-    fderiv ℝ (fun w => fderiv ℝ physicalTheta w noise) v noise = 2*X v*Y v/(Q v)^2 := by
-  have he : (fun w => fderiv ℝ physicalTheta w noise) =ᶠ[𝓝 v] physicalV := by
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hv)] with w hw
-    rw [(hasFDerivAt_physicalTheta w hw).fderiv,thetaDeriv_noise]
-  rw [he.fderiv_eq]
-  have hd : HasFDerivAt physicalV (velocityDeriv v) v := by
-    have hQinv := (hasDerivAt_inv (Q_pos hv).ne').comp_hasFDerivAt v (hasFDerivAt_Q v)
-    convert Y.hasFDerivAt.mul hQinv using 1 <;> try rfl
-    apply ContinuousLinearMap.ext
-    intro w
-    simp only [velocityDeriv,add_apply,smul_apply,
-      sub_apply,smul_eq_mul,Function.comp_apply]
-    have hq := (Q_pos hv).ne'
-    field_simp
-    ring
-  rw [hd.fderiv]
-  simp [velocityDeriv,noise,X,Y]
-  ring
-
-lemma regX_drift (a : ℝ) (v : State) : X (physicalDrift a v) = a*X v/Q v := by
-  simp [physicalDrift,X_apply,Complex.div_re,Q_eq]
-lemma regY_drift (a : ℝ) (v : State) : Y (physicalDrift a v) = -a*Y v/Q v := by
-  simp [physicalDrift,Y_apply,Complex.div_im,Q_eq]; ring
-lemma regJ_drift (a : ℝ) (v : State) :
-    J (physicalDrift a v) = -a*((X v)^2-(Y v)^2)/(Q v)^2 := by
-  simp only [physicalDrift,J_apply,Complex.div_re,Complex.neg_re,Complex.ofReal_re,
-    Complex.neg_im,Complex.ofReal_im,neg_zero,zero_mul,map_pow,← Q_eq]
-  simp only [pow_two,Complex.mul_re,X_apply,Y_apply,zero_div,add_zero]
-
-lemma theta_generator (a : ℝ) (v : State) (hv : 0<Y v) :
-    (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ physicalTheta w noise) v noise +
-      fderiv ℝ physicalTheta v (physicalDrift a v) =
-        (1-2*a)*Real.cot (physicalTheta v)*(physicalV v)^2 := by
-  have hy := hv.ne'
-  have hq := (Q_pos hv).ne'
-  rw [physicalTheta_second_noise v hv,(hasFDerivAt_physicalTheta v hv).fderiv]
-  simp only [thetaDeriv,add_apply,smul_apply,
-    smul_eq_mul,regX_drift,regY_drift,physicalTheta_cot,physicalV]
-  field_simp
-  ring
-
-noncomputable def physicalClock (a cap : ℝ) (v : State) : ℝ :=
-  (Real.log (cap/2)+J v-Real.log (Y v))/(2*a)
-noncomputable def clockDeriv (a : ℝ) (v : State) : State →L[ℝ] ℝ :=
-  (1/(2*a)) • (J-(1/Y v) • Y)
-lemma hasFDerivAt_physicalClock (a cap : ℝ) (v : State) (hv : 0<Y v) :
-    HasFDerivAt (physicalClock a cap) (clockDeriv a v) v := by
-  convert ((J.hasFDerivAt.const_add (Real.log (cap/2))).sub (Y.hasFDerivAt.log hv.ne')).const_mul (1/(2*a)) using 1 <;> try rfl
-  · ext w
-    simp only [physicalClock,Pi.sub_apply]
-    ring
-  · simp only [clockDeriv,one_div]
-lemma clockDeriv_noise (a : ℝ) (v : State) : clockDeriv a v noise = 0 := by
-  simp [clockDeriv,noise,J,Y]
-lemma physicalClock_drift (a cap : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
-    fderiv ℝ (physicalClock a cap) v (physicalDrift a v) = (physicalV v)^2 := by
-  have hy := hv.ne'
-  have hq := (Q_pos hv).ne'
-  rw [(hasFDerivAt_physicalClock a cap v hv).fderiv]
-  simp only [clockDeriv,sub_apply,smul_apply,smul_eq_mul,
-    regY_drift,regJ_drift,physicalV]
-  field_simp
-  dsimp only [Q]
-  ring
-lemma physicalClock_second_noise (a cap : ℝ) (v : State) (hv : 0<Y v) :
-    fderiv ℝ (fun w => fderiv ℝ (physicalClock a cap) w noise) v noise = 0 := by
-  have he : (fun w => fderiv ℝ (physicalClock a cap) w noise) =ᶠ[𝓝 v] (fun _ => (0:ℝ)) := by
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hv)] with w hw
-    rw [(hasFDerivAt_physicalClock a cap w hw).fderiv,clockDeriv_noise]
-  rw [he.fderiv_eq,(hasFDerivAt_const (0:ℝ) v).fderiv]
-  rfl
-lemma physicalTheta_smooth (v : State) (hv : 0<Y v) : ContDiffAt ℝ 2 physicalTheta v := by
-  exact contDiffAt_const.sub ((X.contDiff.contDiffAt.div Y.contDiff.contDiffAt hv.ne').arctan)
-lemma physicalClock_smooth (a cap : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ 2 (physicalClock a cap) v := by
-  exact ((contDiffAt_const.add J.contDiff.contDiffAt).sub
-    (Y.contDiff.contDiffAt.log hv.ne')).div_const (2*a)
-end ExactSLEGreen
-
-open Filter Set
-open scoped Topology
-namespace ExactSLECalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-noncomputable def N (f : E → ℝ) (e : E) (x : E) : ℝ := fderiv ℝ f x e
-noncomputable def L (f : E → ℝ) (A e : E) (x : E) : ℝ :=
-  (1/2:ℝ) * N (N f e) e x + fderiv ℝ f x A
-lemma N_differentiableAt {f : E → ℝ} {x e : E} (hf : ContDiffAt ℝ 2 f x) :
-    DifferentiableAt ℝ (N f e) x :=
-  ((hf.fderiv_right (m := 1) (by norm_num)).clm_apply contDiffAt_const).differentiableAt_one
-lemma N_mul {f g : E → ℝ} {x e : E}
-    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
-    N (fun y => f y*g y) e x = f x*N g e x+g x*N f e x := by
-  change (fderiv ℝ (f*g) x) e = _
-  rw [(hf.hasFDerivAt.mul hg.hasFDerivAt).fderiv]
-  simp only [add_apply,smul_apply,smul_eq_mul,N]
-lemma L_mul {f g : E → ℝ} {x A e : E}
-    (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) :
-    L (fun y => f y*g y) A e x = f x*L g A e x+g x*L f A e x+N f e x*N g e x := by
-  have hfd := hf.differentiableAt (by norm_num)
-  have hgd := hg.differentiableAt (by norm_num)
-  have heq : N (fun y => f y*g y) e =ᶠ[𝓝 x] (fun y => f y*N g e y+g y*N f e y) := by
-    filter_upwards [hf.eventually (by norm_num),hg.eventually (by norm_num)] with y hfy hgy
-    exact N_mul (hfy.differentiableAt (by norm_num)) (hgy.differentiableAt (by norm_num))
-  have hd := (hfd.hasFDerivAt.mul (N_differentiableAt (e := e) hg).hasFDerivAt).add
-    (hgd.hasFDerivAt.mul (N_differentiableAt (e := e) hf).hasFDerivAt)
-  have hd' := hd.fderiv
-  have hp := (hfd.hasFDerivAt.mul hgd.hasFDerivAt).fderiv
-  change fderiv ℝ (fun y => f y*N g e y+g y*N f e y) x = _ at hd'
-  change fderiv ℝ (fun y => f y*g y) x = _ at hp
-  unfold L
-  rw [N,heq.fderiv_eq,hd',hp]
-  simp only [add_apply,smul_apply,smul_eq_mul,N]
-  ring
-lemma N_comp {f : E → ℝ} {h : ℝ → ℝ} {x e : E}
-    (hf : DifferentiableAt ℝ f x) (hh : DifferentiableAt ℝ h (f x)) :
-    N (h ∘ f) e x = deriv h (f x)*N f e x := by
-  rw [N,(hh.hasDerivAt.comp_hasFDerivAt x hf.hasFDerivAt).fderiv]
-  simp only [smul_apply,smul_eq_mul,N]
-lemma L_comp {f : E → ℝ} {h : ℝ → ℝ} {x A e : E}
-    (hf : ContDiffAt ℝ 2 f x) (hh : ContDiffAt ℝ 2 h (f x)) :
-    L (h ∘ f) A e x = deriv h (f x)*L f A e x+
-      (1/2:ℝ)*deriv (deriv h) (f x)*(N f e x)^2 := by
-  have hfd := hf.differentiableAt (by norm_num)
-  have hhd := hh.differentiableAt (by norm_num)
-  have hdd : DifferentiableAt ℝ (deriv h) (f x) := by
-    convert N_differentiableAt (e := (1:ℝ)) hh using 1
-    rfl
-  have heq : N (h ∘ f) e =ᶠ[𝓝 x] (fun y => deriv h (f y)*N f e y) := by
-    filter_upwards [hf.eventually (by norm_num),
-      hfd.continuousAt.preimage_mem_nhds (hh.eventually (by norm_num))] with y hfy hhy
-    change ContDiffAt ℝ 2 h (f y) at hhy
-    exact N_comp (hfy.differentiableAt (by norm_num)) (hhy.differentiableAt (by norm_num))
-  have hd := (hdd.hasDerivAt.comp_hasFDerivAt x hfd.hasFDerivAt).mul
-    (N_differentiableAt (e := e) hf).hasFDerivAt
-  have hd' := hd.fderiv
-  change fderiv ℝ (fun y => deriv h (f y)*N f e y) x = _ at hd'
-  unfold L
-  rw [N,heq.fderiv_eq,hd',(hhd.hasDerivAt.comp_hasFDerivAt x hfd.hasFDerivAt).fderiv]
-  simp only [add_apply,smul_apply,smul_eq_mul,N,Function.comp_apply]
-  ring
-end ExactSLECalculus
-
-namespace ExactSLECalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-@[simp] lemma N_const (c : ℝ) (e x : E) : N (fun _ => c) e x = 0 := by
-  rw [N,(hasFDerivAt_const c x).fderiv]; rfl
-@[simp] lemma L_const (c : ℝ) (A e x : E) : L (fun _ => c) A e x = 0 := by
-  have heq : N (fun _ : E => c) e = (fun _ => (0:ℝ)) := funext (N_const c e)
-  simp only [L,heq,N,(hasFDerivAt_const c x).fderiv,(hasFDerivAt_const (0:ℝ) x).fderiv,
-    zero_apply,mul_zero,add_zero]
-lemma N_add {f g : E → ℝ} {x e : E}
-    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
-    N (fun y => f y+g y) e x = N f e x+N g e x := by
-  have h := (hf.hasFDerivAt.add hg.hasFDerivAt).fderiv
-  change fderiv ℝ (fun y => f y+g y) x = _ at h
-  simp only [N,h,add_apply]
-lemma L_add {f g : E → ℝ} {x A e : E}
-    (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) :
-    L (fun y => f y+g y) A e x = L f A e x+L g A e x := by
-  have heq : N (fun y => f y+g y) e =ᶠ[𝓝 x] (fun y => N f e y+N g e y) := by
-    filter_upwards [hf.eventually (by norm_num),hg.eventually (by norm_num)] with y hfy hgy
-    exact N_add (hfy.differentiableAt (by norm_num)) (hgy.differentiableAt (by norm_num))
-  have h := ((hf.differentiableAt (by norm_num)).hasFDerivAt.add
-    (hg.differentiableAt (by norm_num)).hasFDerivAt).fderiv
-  have hd := ((N_differentiableAt (e := e) hf).hasFDerivAt.add
-    (N_differentiableAt (e := e) hg).hasFDerivAt).fderiv
-  change fderiv ℝ (fun y => f y+g y) x = _ at h
-  change fderiv ℝ (fun y => N f e y+N g e y) x = _ at hd
-  unfold L
-  rw [N,heq.fderiv_eq,hd,h]
-  simp only [add_apply,N]
-  ring
-lemma N_const_mul {f : E → ℝ} {x e : E} (hf : DifferentiableAt ℝ f x) (c : ℝ) :
-    N (fun y => c*f y) e x = c*N f e x := by
-  simpa only [N_const,zero_mul,mul_zero,add_zero] using N_mul (differentiableAt_const c) hf (e := e)
-lemma L_const_mul {f : E → ℝ} {x A e : E} (hf : ContDiffAt ℝ 2 f x) (c : ℝ) :
-    L (fun y => c*f y) A e x = c*L f A e x := by
-  simpa only [N_const,L_const,mul_zero,zero_mul,add_zero] using
-    L_mul (contDiffAt_const (c := c)) hf (A := A) (e := e)
-end ExactSLECalculus
-
- 
-
-namespace ExactSLEFiniteTilt
-noncomputable def sinePower (r : ℝ) (θ : ℝ) : ℝ := Real.sin θ ^ r
-noncomputable def angleGenerator (β : ℝ) (f : ℝ → ℝ) (θ : ℝ) : ℝ :=
-  (1 / 2 : ℝ) * deriv (deriv f) θ + β * Real.cot θ * deriv f θ
-
-open Filter
-open scoped Topology
-
-theorem sinePower_hasDerivAt (r θ : ℝ) (hθ : 0 < Real.sin θ) :
-    HasDerivAt (sinePower r) (Real.cos θ * r * Real.sin θ ^ (r - 1)) θ := by
-  exact (Real.hasDerivAt_sin θ).rpow_const (Or.inl hθ.ne')
-
-theorem sinePower_secondDeriv (r θ : ℝ) (hθ : 0 < Real.sin θ) :
-    deriv (deriv (sinePower r)) θ =
-      -Real.sin θ * r * Real.sin θ ^ (r - 1) +
-      (Real.cos θ * r) * (Real.cos θ * (r - 1) * Real.sin θ ^ (r - 2)) := by
-  have he : deriv (sinePower r) =ᶠ[𝓝 θ]
-      (fun x => Real.cos x * r * Real.sin x ^ (r - 1)) := by
-    have hp : ∀ᶠ x in 𝓝 θ, 0 < Real.sin x :=
-      (isOpen_lt continuous_const Real.continuous_sin).mem_nhds hθ
-    filter_upwards [hp] with x hx
-    exact (sinePower_hasDerivAt r x hx).deriv
-  rw [he.deriv_eq]
-  have hd := ((Real.hasDerivAt_cos θ).mul_const r).mul
-    ((Real.hasDerivAt_sin θ).rpow_const (p := r - 1) (Or.inl hθ.ne'))
-  convert hd.deriv using 1
-  congr 2
-  ring_nf
-
-theorem angleGenerator_sinePower (β r θ : ℝ) (hθ : 0 < Real.sin θ) :
-    angleGenerator β (sinePower r) θ = sinePower r θ *
-      (-r / 2 + r * (r - 1 + 2 * β) / 2 * Real.cot θ ^ 2) := by
-  unfold angleGenerator
-  rw [sinePower_secondDeriv r θ hθ, (sinePower_hasDerivAt r θ hθ).deriv]
-  dsimp [sinePower]
-  rw [Real.rpow_sub hθ, Real.rpow_sub hθ, Real.rpow_one, Real.rpow_two,
-    Real.cot_eq_cos_div_sin]
-  have htrig := Real.sin_sq_add_cos_sq θ
-  field_simp
-  nlinarith [sq_nonneg (Real.sin θ), sq_nonneg (Real.cos θ)]
-
-theorem sine_generator (a r θ : ℝ) (hθ : θ ∈ Set.Ioo 0 Real.pi) :
-    angleGenerator (1 - 2 * a) (sinePower r) θ / sinePower r θ =
-      -r / 2 + r * (r - (4 * a - 1)) / 2 * Real.cot θ ^ 2 := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  rw [angleGenerator_sinePower _ _ _ hs]
-  have hp : sinePower r θ ≠ 0 := (Real.rpow_pos_of_pos hs r).ne'
-  field_simp
-  ring
-
-theorem green_angle_eigenfunction (a θ : ℝ) (hθ : θ ∈ Set.Ioo 0 Real.pi) :
-    angleGenerator (1 - 2 * a) (sinePower (4 * a - 1)) θ =
-      -(4 * a - 1) / 2 * sinePower (4 * a - 1) θ := by
-  rw [angleGenerator_sinePower _ _ _ (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)]
-  ring
-
-theorem tilted_lyapunov (a q θ : ℝ) (_ha : 1 / 4 < a)
-    (hq : 0 < q) (hqp : q < 4 * a - 1) (hθ : θ ∈ Set.Ioo 0 Real.pi) :
-    angleGenerator (2 * a) (sinePower (-q)) θ =
-      sinePower (-q) θ * (q / 2 - q * (4 * a - 1 - q) / 2 * Real.cot θ ^ 2) ∧
-    angleGenerator (2 * a) (sinePower (-q)) θ ≤ q / 2 * sinePower (-q) θ := by
-  have he : angleGenerator (2 * a) (sinePower (-q)) θ =
-      sinePower (-q) θ * (q / 2 - q * (4 * a - 1 - q) / 2 * Real.cot θ ^ 2) := by
-    rw [angleGenerator_sinePower _ _ _ (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)]
-    ring
-  refine ⟨he, ?_⟩
-  rw [he]
-  have hf : 0 < sinePower (-q) θ :=
-    Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2) _
-  have hc : 0 ≤ q * (4 * a - 1 - q) / 2 * Real.cot θ ^ 2 := by positivity
-  nlinarith
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt
-noncomputable def clockExp (a cap lam : ℝ) (v : State) : ℝ :=
-  Real.exp (lam*physicalClock a cap v)
-noncomputable def angleLift (g : ℝ → ℝ) (v : State) : ℝ := g (physicalTheta v)
-lemma clock_N_zero (a cap : ℝ) (v : State) (hv : 0<Y v) :
-    N (physicalClock a cap) noise v = 0 := by
-  rw [N,(hasFDerivAt_physicalClock a cap v hv).fderiv,clockDeriv_noise]
-lemma clock_L (a cap : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
-    L (physicalClock a cap) (physicalDrift a v) noise v = (physicalV v)^2 := by
-  change (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ (physicalClock a cap) w noise) v noise +
-    fderiv ℝ (physicalClock a cap) v (physicalDrift a v) = _
-  rw [physicalClock_second_noise a cap v hv,physicalClock_drift a cap ha v hv]
-  ring
-lemma theta_N (v : State) (hv : 0<Y v) : N physicalTheta noise v = physicalV v := by
-  rw [N,(hasFDerivAt_physicalTheta v hv).fderiv,thetaDeriv_noise]
-lemma theta_L (a : ℝ) (v : State) (hv : 0<Y v) :
-    L physicalTheta (physicalDrift a v) noise v =
-      (1-2*a)*Real.cot (physicalTheta v)*(physicalV v)^2 := theta_generator a v hv
-lemma clockExp_smooth (a cap lam : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ 2 (clockExp a cap lam) v :=
-  (contDiffAt_const.mul (physicalClock_smooth a cap v hv)).exp
-lemma clockExp_N (a cap lam : ℝ) (v : State) (hv : 0<Y v) :
-    N (clockExp a cap lam) noise v = 0 := by
-  have he : clockExp a cap lam = Real.exp ∘ (fun w => lam*physicalClock a cap w) := rfl
-  rw [he,N_comp ((contDiffAt_const.mul (physicalClock_smooth a cap v hv)).differentiableAt (by norm_num))
-    Real.differentiable_exp.differentiableAt]
-  rw [N_const_mul ((physicalClock_smooth a cap v hv).differentiableAt (by norm_num)),clock_N_zero a cap v hv]
-  ring
-lemma clockExp_L (a cap lam : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
-    L (clockExp a cap lam) (physicalDrift a v) noise v =
-      clockExp a cap lam v*lam*(physicalV v)^2 := by
-  have he : clockExp a cap lam = Real.exp ∘ (fun w => lam*physicalClock a cap w) := rfl
-  rw [he,L_comp (contDiffAt_const.mul (physicalClock_smooth a cap v hv)) Real.contDiff_exp.contDiffAt,
-    L_const_mul (physicalClock_smooth a cap v hv),clock_L a cap ha v hv,
-    N_const_mul ((physicalClock_smooth a cap v hv).differentiableAt (by norm_num)),clock_N_zero a cap v hv]
-  simp only [Real.deriv_exp]
-  change _ = Real.exp (lam*physicalClock a cap v)*lam*(physicalV v)^2
-  ring
-lemma angleLift_smooth (g : ℝ → ℝ) (v : State) (hv : 0<Y v)
-    (hg : ContDiffAt ℝ 2 g (physicalTheta v)) : ContDiffAt ℝ 2 (angleLift g) v :=
-  hg.comp v (physicalTheta_smooth v hv)
-lemma angleLift_N (g : ℝ → ℝ) (v : State) (hv : 0<Y v)
-    (hg : DifferentiableAt ℝ g (physicalTheta v)) :
-    N (angleLift g) noise v = deriv g (physicalTheta v)*physicalV v := by
-  change N (g ∘ physicalTheta) noise v = _
-  rw [N_comp ((physicalTheta_smooth v hv).differentiableAt (by norm_num)) hg,theta_N v hv]
-lemma angleLift_L (g : ℝ → ℝ) (a : ℝ) (v : State) (hv : 0<Y v)
-    (hg : ContDiffAt ℝ 2 g (physicalTheta v)) :
-    L (angleLift g) (physicalDrift a v) noise v =
-      angleGenerator (1-2*a) g (physicalTheta v)*(physicalV v)^2 := by
-  change L (g ∘ physicalTheta) (physicalDrift a v) noise v = _
-  rw [L_comp (physicalTheta_smooth v hv) hg,theta_L a v hv,theta_N v hv]
-  dsimp [angleGenerator]
-  ring
-lemma exponential_angle_generator (g : ℝ → ℝ) (a cap lam : ℝ) (ha : a≠0)
-    (v : State) (hv : 0<Y v) (hg : ContDiffAt ℝ 2 g (physicalTheta v)) :
-    L (fun w => clockExp a cap lam w*angleLift g w) (physicalDrift a v) noise v =
-      clockExp a cap lam v*(lam*g (physicalTheta v)+angleGenerator (1-2*a) g (physicalTheta v))*(physicalV v)^2 := by
-  rw [L_mul (clockExp_smooth a cap lam v hv) (angleLift_smooth g v hv hg),
-    clockExp_L a cap lam ha v hv,clockExp_N a cap lam v hv,angleLift_L g a v hv hg]
-  dsimp only [angleLift]
-  ring
-lemma sinePower_smooth (r θ : ℝ) (hθ : 0<Real.sin θ) : ContDiffAt ℝ 2 (sinePower r) θ :=
-  Real.contDiff_sin.contDiffAt.rpow_const_of_ne hθ.ne'
-end ExactSLEGreen
-
-namespace ExactSLEFiniteTilt
-noncomputable def regP (a : ℝ) : ℝ := 4*a-1
-noncomputable def regQ (a : ℝ) : ℝ := min 1 (regP a)/2
-noncomputable def regEta (a : ℝ) : ℝ := (1-regQ a)/(2-regQ a)
-noncomputable def regM (a : ℝ) : ℝ := (1+regEta a)/(1-2*regEta a)
-noncomputable def regBeta (a : ℝ) : ℝ := min 1 (a/(4*regM a))
-noncomputable def regK (a : ℝ) : ℝ := regQ a*(regP a-regQ a)/2
-noncomputable def regA (a : ℝ) : ℝ := regP a/2+(regP a-regQ a)*(1+regQ a)/2+2
-noncomputable def regH (a : ℝ) : ℝ := (regK a/(2*regA a))^(1/(2-regQ a))
-noncomputable def regC (a : ℝ) : ℝ := regA a * regH a ^ (-regP a)
-noncomputable def regFactor (a δ θ : ℝ) : ℝ :=
-  sinePower (regP a) θ + δ*(1+sinePower (regQ a) θ)
-noncomputable def regD (a δ θ : ℝ) : ℝ := δ/regFactor a δ θ * Real.sin θ^(regQ a-2)
-noncomputable def regNormalizedDrift (a δ σ θ : ℝ) : ℝ :=
-  (angleGenerator (1-2*a) (regFactor a δ) θ + regP a/2 * regFactor a δ θ +
-    σ*δ*(1+sinePower (regQ a) θ))/regFactor a δ θ
-end ExactSLEFiniteTilt
-
-namespace ExactSLEFiniteTilt
-
-lemma regP_pos {a : ℝ} (ha : 1/4 < a) : 0 < regP a := by dsimp [regP]; linarith
-lemma regQ_pos {a : ℝ} (ha : 1/4 < a) : 0 < regQ a := by
-  dsimp [regQ]; exact div_pos (lt_min (by norm_num) (regP_pos ha)) (by norm_num)
-lemma regQ_lt_one {a : ℝ} (_ha : 1/4 < a) : regQ a < 1 := by
-  have hh := min_le_left (1 : ℝ) (regP a)
-  dsimp [regQ]; linarith
-lemma regQ_lt_p {a : ℝ} (ha : 1/4 < a) : regQ a < regP a := by
-  have hh := min_le_right (1 : ℝ) (regP a)
-  have hp := regP_pos ha
-  dsimp [regQ]; linarith
-lemma regEta_pos {a : ℝ} (ha : 1/4 < a) : 0 < regEta a := by
-  have hq := regQ_lt_one ha
-  exact div_pos (by linarith) (by linarith)
-lemma regEta_lt_half {a : ℝ} (ha : 1/4 < a) : regEta a < 1/2 := by
-  have hq := regQ_pos ha
-  have hq1 := regQ_lt_one ha
-  dsimp [regEta]
-  apply (div_lt_iff₀ (by linarith)).mpr
-  linarith
-lemma regEta_identity {a : ℝ} (ha : 1/4 < a) : regEta a*(regQ a-2) = regQ a-1 := by
-  have hq := regQ_lt_one ha
-  dsimp [regEta]
-  field_simp [show 2-regQ a ≠ 0 by linarith]
-  ring
-lemma regM_pos {a : ℝ} (ha : 1/4 < a) : 0 < regM a := by
-  have hη := regEta_pos ha
-  have hη1 := regEta_lt_half ha
-  exact div_pos (by linarith) (by linarith)
-lemma regM_eq {a : ℝ} (ha : 1/4 < a) : regM a = 3/regQ a-2 := by
-  have hq := regQ_pos ha
-  have hq1 := regQ_lt_one ha
-  have hη1 := regEta_lt_half ha
-  dsimp only [regM]
-  apply (div_eq_iff (show 1-2*regEta a ≠ 0 by linarith)).mpr
-  dsimp only [regEta]
-  field_simp [show 2-regQ a ≠ 0 by linarith, hq.ne']
-  ring
-lemma regBeta_pos {a : ℝ} (ha : 1/4 < a) : 0 < regBeta a := by
-  exact lt_min (by norm_num) (div_pos (by linarith) (by have := regM_pos ha; positivity))
-lemma regBeta_le_one (a : ℝ) : regBeta a ≤ 1 := min_le_left _ _
-lemma regBeta_discount {a : ℝ} (ha : 1/4 < a) : 2*regM a*regBeta a < a := by
-  have hM := regM_pos ha
-  have hb := min_le_right (1 : ℝ) (a/(4*regM a))
-  change regBeta a ≤ a/(4*regM a) at hb
-  have hm := (le_div_iff₀ (by positivity : 0 < 4*regM a)).mp hb
-  nlinarith
-lemma regK_pos {a : ℝ} (ha : 1/4 < a) : 0 < regK a := by
-  exact div_pos (mul_pos (regQ_pos ha) (sub_pos.mpr (regQ_lt_p ha))) (by norm_num)
-lemma regA_pos {a : ℝ} (ha : 1/4 < a) : 0 < regA a := by
-  have hp := regP_pos ha
-  have hq := regQ_pos ha
-  have hpq := sub_pos.mpr (regQ_lt_p ha)
-  dsimp [regA]; positivity
-lemma reg_base_lt_one {a : ℝ} (ha : 1/4 < a) : regK a/(2*regA a) < 1 := by
-  have hp := regP_pos ha
-  have hq := regQ_pos ha
-  have hpq := sub_pos.mpr (regQ_lt_p ha)
-  have hA := regA_pos ha
-  apply (div_lt_one (by positivity)).mpr
-  have hprod := mul_pos hpq (show 0 < 1+regQ a/2 by linarith)
-  dsimp [regK,regA]
-  nlinarith
-lemma regH_pos {a : ℝ} (ha : 1/4 < a) : 0 < regH a := by
-  exact Real.rpow_pos_of_pos (div_pos (regK_pos ha) (by have := regA_pos ha; positivity)) _
-lemma regH_lt_one {a : ℝ} (ha : 1/4 < a) : regH a < 1 := by
-  have hq1 := regQ_lt_one ha
-  exact Real.rpow_lt_one (by have := regK_pos ha; have := regA_pos ha; positivity)
-    (reg_base_lt_one ha) (div_pos zero_lt_one (by linarith))
-lemma regH_power {a : ℝ} (ha : 1/4 < a) : regH a ^ (regQ a-2) = 2*regA a/regK a := by
-  have hq1 := regQ_lt_one ha
-  have hk := regK_pos ha
-  have hA := regA_pos ha
-  dsimp [regH]
-  rw [←Real.rpow_mul (by positivity : 0 ≤ regK a/(2*regA a))]
-  have he : 1/(2-regQ a)*(regQ a-2) = -1 := by field_simp [show 2-regQ a ≠ 0 by linarith]; ring
-  rw [he,Real.rpow_neg_one,inv_div]
-lemma regC_pos {a : ℝ} (ha : 1/4 < a) : 0 < regC a :=
-  mul_pos (regA_pos ha) (Real.rpow_pos_of_pos (regH_pos ha) _)
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLEFiniteTilt
-open Filter
-end ExactSLEFiniteTilt
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialStopping
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+namespace ExactSLEGreen
+lemma physicalTheta_second_noise (v : State) (hv : 0<Y v) :
+    fderiv ℝ (fun w => fderiv ℝ physicalTheta w noise) v noise = 2*X v*Y v/(Q v)^2 := by
+  have he : (fun w => fderiv ℝ physicalTheta w noise) =ᶠ[𝓝 v] physicalV := by
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hv)] with w hw
+    rw [(hasFDerivAt_physicalTheta w hw).fderiv,thetaDeriv_noise]
+  rw [he.fderiv_eq]
+  have hd : HasFDerivAt physicalV (velocityDeriv v) v := by
+    have hQinv := (hasDerivAt_inv (Q_pos hv).ne').comp_hasFDerivAt v (hasFDerivAt_Q v)
+    convert Y.hasFDerivAt.mul hQinv using 1 <;> try rfl
+    apply ContinuousLinearMap.ext
+    intro w
+    simp only [velocityDeriv,add_apply,smul_apply,
+      sub_apply,smul_eq_mul,Function.comp_apply]
+    have hq := (Q_pos hv).ne'
+    field_simp
+    ring
+  rw [hd.fderiv]
+  simp [velocityDeriv,noise,X,Y]
+  ring
+
+lemma regX_drift (a : ℝ) (v : State) : X (physicalDrift a v) = a*X v/Q v := by
+  simp [physicalDrift,X_apply,Complex.div_re,Q_eq]
+lemma regY_drift (a : ℝ) (v : State) : Y (physicalDrift a v) = -a*Y v/Q v := by
+  simp [physicalDrift,Y_apply,Complex.div_im,Q_eq]; ring
+lemma regJ_drift (a : ℝ) (v : State) :
+    J (physicalDrift a v) = -a*((X v)^2-(Y v)^2)/(Q v)^2 := by
+  simp only [physicalDrift,J_apply,Complex.div_re,Complex.neg_re,Complex.ofReal_re,
+    Complex.neg_im,Complex.ofReal_im,neg_zero,zero_mul,map_pow,← Q_eq]
+  simp only [pow_two,Complex.mul_re,X_apply,Y_apply,zero_div,add_zero]
+
+lemma theta_generator (a : ℝ) (v : State) (hv : 0<Y v) :
+    (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ physicalTheta w noise) v noise +
+      fderiv ℝ physicalTheta v (physicalDrift a v) =
+        (1-2*a)*Real.cot (physicalTheta v)*(physicalV v)^2 := by
+  have hy := hv.ne'
+  have hq := (Q_pos hv).ne'
+  rw [physicalTheta_second_noise v hv,(hasFDerivAt_physicalTheta v hv).fderiv]
+  simp only [thetaDeriv,add_apply,smul_apply,
+    smul_eq_mul,regX_drift,regY_drift,physicalTheta_cot,physicalV]
+  field_simp
+  ring
+
+noncomputable def physicalClock (a cap : ℝ) (v : State) : ℝ :=
+  (Real.log (cap/2)+J v-Real.log (Y v))/(2*a)
+noncomputable def clockDeriv (a : ℝ) (v : State) : State →L[ℝ] ℝ :=
+  (1/(2*a)) • (J-(1/Y v) • Y)
+lemma hasFDerivAt_physicalClock (a cap : ℝ) (v : State) (hv : 0<Y v) :
+    HasFDerivAt (physicalClock a cap) (clockDeriv a v) v := by
+  convert ((J.hasFDerivAt.const_add (Real.log (cap/2))).sub (Y.hasFDerivAt.log hv.ne')).const_mul (1/(2*a)) using 1 <;> try rfl
+  · ext w
+    simp only [physicalClock,Pi.sub_apply]
+    ring
+  · simp only [clockDeriv,one_div]
+lemma clockDeriv_noise (a : ℝ) (v : State) : clockDeriv a v noise = 0 := by
+  simp [clockDeriv,noise,J,Y]
+lemma physicalClock_drift (a cap : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
+    fderiv ℝ (physicalClock a cap) v (physicalDrift a v) = (physicalV v)^2 := by
+  have hy := hv.ne'
+  have hq := (Q_pos hv).ne'
+  rw [(hasFDerivAt_physicalClock a cap v hv).fderiv]
+  simp only [clockDeriv,sub_apply,smul_apply,smul_eq_mul,
+    regY_drift,regJ_drift,physicalV]
+  field_simp
+  dsimp only [Q]
+  ring
+lemma physicalClock_second_noise (a cap : ℝ) (v : State) (hv : 0<Y v) :
+    fderiv ℝ (fun w => fderiv ℝ (physicalClock a cap) w noise) v noise = 0 := by
+  have he : (fun w => fderiv ℝ (physicalClock a cap) w noise) =ᶠ[𝓝 v] (fun _ => (0:ℝ)) := by
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hv)] with w hw
+    rw [(hasFDerivAt_physicalClock a cap w hw).fderiv,clockDeriv_noise]
+  rw [he.fderiv_eq,(hasFDerivAt_const (0:ℝ) v).fderiv]
+  rfl
+lemma physicalTheta_smooth (v : State) (hv : 0<Y v) : ContDiffAt ℝ 2 physicalTheta v := by
+  exact contDiffAt_const.sub ((X.contDiff.contDiffAt.div Y.contDiff.contDiffAt hv.ne').arctan)
+lemma physicalClock_smooth (a cap : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ 2 (physicalClock a cap) v := by
+  exact ((contDiffAt_const.add J.contDiff.contDiffAt).sub
+    (Y.contDiff.contDiffAt.log hv.ne')).div_const (2*a)
+end ExactSLEGreen
+
+open Filter Set
+open scoped Topology
+namespace ExactSLECalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+noncomputable def N (f : E → ℝ) (e : E) (x : E) : ℝ := fderiv ℝ f x e
+noncomputable def L (f : E → ℝ) (A e : E) (x : E) : ℝ :=
+  (1/2:ℝ) * N (N f e) e x + fderiv ℝ f x A
+lemma N_differentiableAt {f : E → ℝ} {x e : E} (hf : ContDiffAt ℝ 2 f x) :
+    DifferentiableAt ℝ (N f e) x :=
+  ((hf.fderiv_right (m := 1) (by norm_num)).clm_apply contDiffAt_const).differentiableAt_one
+lemma N_mul {f g : E → ℝ} {x e : E}
+    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
+    N (fun y => f y*g y) e x = f x*N g e x+g x*N f e x := by
+  change (fderiv ℝ (f*g) x) e = _
+  rw [(hf.hasFDerivAt.mul hg.hasFDerivAt).fderiv]
+  simp only [add_apply,smul_apply,smul_eq_mul,N]
+lemma L_mul {f g : E → ℝ} {x A e : E}
+    (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) :
+    L (fun y => f y*g y) A e x = f x*L g A e x+g x*L f A e x+N f e x*N g e x := by
+  have hfd := hf.differentiableAt (by norm_num)
+  have hgd := hg.differentiableAt (by norm_num)
+  have heq : N (fun y => f y*g y) e =ᶠ[𝓝 x] (fun y => f y*N g e y+g y*N f e y) := by
+    filter_upwards [hf.eventually (by norm_num),hg.eventually (by norm_num)] with y hfy hgy
+    exact N_mul (hfy.differentiableAt (by norm_num)) (hgy.differentiableAt (by norm_num))
+  have hd := (hfd.hasFDerivAt.mul (N_differentiableAt (e := e) hg).hasFDerivAt).add
+    (hgd.hasFDerivAt.mul (N_differentiableAt (e := e) hf).hasFDerivAt)
+  have hd' := hd.fderiv
+  have hp := (hfd.hasFDerivAt.mul hgd.hasFDerivAt).fderiv
+  change fderiv ℝ (fun y => f y*N g e y+g y*N f e y) x = _ at hd'
+  change fderiv ℝ (fun y => f y*g y) x = _ at hp
+  unfold L
+  rw [N,heq.fderiv_eq,hd',hp]
+  simp only [add_apply,smul_apply,smul_eq_mul,N]
+  ring
+lemma N_comp {f : E → ℝ} {h : ℝ → ℝ} {x e : E}
+    (hf : DifferentiableAt ℝ f x) (hh : DifferentiableAt ℝ h (f x)) :
+    N (h ∘ f) e x = deriv h (f x)*N f e x := by
+  rw [N,(hh.hasDerivAt.comp_hasFDerivAt x hf.hasFDerivAt).fderiv]
+  simp only [smul_apply,smul_eq_mul,N]
+lemma L_comp {f : E → ℝ} {h : ℝ → ℝ} {x A e : E}
+    (hf : ContDiffAt ℝ 2 f x) (hh : ContDiffAt ℝ 2 h (f x)) :
+    L (h ∘ f) A e x = deriv h (f x)*L f A e x+
+      (1/2:ℝ)*deriv (deriv h) (f x)*(N f e x)^2 := by
+  have hfd := hf.differentiableAt (by norm_num)
+  have hhd := hh.differentiableAt (by norm_num)
+  have hdd : DifferentiableAt ℝ (deriv h) (f x) := by
+    convert N_differentiableAt (e := (1:ℝ)) hh using 1
+    rfl
+  have heq : N (h ∘ f) e =ᶠ[𝓝 x] (fun y => deriv h (f y)*N f e y) := by
+    filter_upwards [hf.eventually (by norm_num),
+      hfd.continuousAt.preimage_mem_nhds (hh.eventually (by norm_num))] with y hfy hhy
+    change ContDiffAt ℝ 2 h (f y) at hhy
+    exact N_comp (hfy.differentiableAt (by norm_num)) (hhy.differentiableAt (by norm_num))
+  have hd := (hdd.hasDerivAt.comp_hasFDerivAt x hfd.hasFDerivAt).mul
+    (N_differentiableAt (e := e) hf).hasFDerivAt
+  have hd' := hd.fderiv
+  change fderiv ℝ (fun y => deriv h (f y)*N f e y) x = _ at hd'
+  unfold L
+  rw [N,heq.fderiv_eq,hd',(hhd.hasDerivAt.comp_hasFDerivAt x hfd.hasFDerivAt).fderiv]
+  simp only [add_apply,smul_apply,smul_eq_mul,N,Function.comp_apply]
+  ring
+end ExactSLECalculus
+
+namespace ExactSLECalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+@[simp] lemma N_const (c : ℝ) (e x : E) : N (fun _ => c) e x = 0 := by
+  rw [N,(hasFDerivAt_const c x).fderiv]; rfl
+@[simp] lemma L_const (c : ℝ) (A e x : E) : L (fun _ => c) A e x = 0 := by
+  have heq : N (fun _ : E => c) e = (fun _ => (0:ℝ)) := funext (N_const c e)
+  simp only [L,heq,N,(hasFDerivAt_const c x).fderiv,(hasFDerivAt_const (0:ℝ) x).fderiv,
+    zero_apply,mul_zero,add_zero]
+lemma N_add {f g : E → ℝ} {x e : E}
+    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
+    N (fun y => f y+g y) e x = N f e x+N g e x := by
+  have h := (hf.hasFDerivAt.add hg.hasFDerivAt).fderiv
+  change fderiv ℝ (fun y => f y+g y) x = _ at h
+  simp only [N,h,add_apply]
+lemma L_add {f g : E → ℝ} {x A e : E}
+    (hf : ContDiffAt ℝ 2 f x) (hg : ContDiffAt ℝ 2 g x) :
+    L (fun y => f y+g y) A e x = L f A e x+L g A e x := by
+  have heq : N (fun y => f y+g y) e =ᶠ[𝓝 x] (fun y => N f e y+N g e y) := by
+    filter_upwards [hf.eventually (by norm_num),hg.eventually (by norm_num)] with y hfy hgy
+    exact N_add (hfy.differentiableAt (by norm_num)) (hgy.differentiableAt (by norm_num))
+  have h := ((hf.differentiableAt (by norm_num)).hasFDerivAt.add
+    (hg.differentiableAt (by norm_num)).hasFDerivAt).fderiv
+  have hd := ((N_differentiableAt (e := e) hf).hasFDerivAt.add
+    (N_differentiableAt (e := e) hg).hasFDerivAt).fderiv
+  change fderiv ℝ (fun y => f y+g y) x = _ at h
+  change fderiv ℝ (fun y => N f e y+N g e y) x = _ at hd
+  unfold L
+  rw [N,heq.fderiv_eq,hd,h]
+  simp only [add_apply,N]
+  ring
+lemma N_const_mul {f : E → ℝ} {x e : E} (hf : DifferentiableAt ℝ f x) (c : ℝ) :
+    N (fun y => c*f y) e x = c*N f e x := by
+  simpa only [N_const,zero_mul,mul_zero,add_zero] using N_mul (differentiableAt_const c) hf (e := e)
+lemma L_const_mul {f : E → ℝ} {x A e : E} (hf : ContDiffAt ℝ 2 f x) (c : ℝ) :
+    L (fun y => c*f y) A e x = c*L f A e x := by
+  simpa only [N_const,L_const,mul_zero,zero_mul,add_zero] using
+    L_mul (contDiffAt_const (c := c)) hf (A := A) (e := e)
+end ExactSLECalculus
+
+ 
+
+namespace ExactSLEFiniteTilt
+noncomputable def sinePower (r : ℝ) (θ : ℝ) : ℝ := Real.sin θ ^ r
+noncomputable def angleGenerator (β : ℝ) (f : ℝ → ℝ) (θ : ℝ) : ℝ :=
+  (1 / 2 : ℝ) * deriv (deriv f) θ + β * Real.cot θ * deriv f θ
+
+open Filter
+open scoped Topology
+
+theorem sinePower_hasDerivAt (r θ : ℝ) (hθ : 0 < Real.sin θ) :
+    HasDerivAt (sinePower r) (Real.cos θ * r * Real.sin θ ^ (r - 1)) θ := by
+  exact (Real.hasDerivAt_sin θ).rpow_const (Or.inl hθ.ne')
+
+theorem sinePower_secondDeriv (r θ : ℝ) (hθ : 0 < Real.sin θ) :
+    deriv (deriv (sinePower r)) θ =
+      -Real.sin θ * r * Real.sin θ ^ (r - 1) +
+      (Real.cos θ * r) * (Real.cos θ * (r - 1) * Real.sin θ ^ (r - 2)) := by
+  have he : deriv (sinePower r) =ᶠ[𝓝 θ]
+      (fun x => Real.cos x * r * Real.sin x ^ (r - 1)) := by
+    have hp : ∀ᶠ x in 𝓝 θ, 0 < Real.sin x :=
+      (isOpen_lt continuous_const Real.continuous_sin).mem_nhds hθ
+    filter_upwards [hp] with x hx
+    exact (sinePower_hasDerivAt r x hx).deriv
+  rw [he.deriv_eq]
+  have hd := ((Real.hasDerivAt_cos θ).mul_const r).mul
+    ((Real.hasDerivAt_sin θ).rpow_const (p := r - 1) (Or.inl hθ.ne'))
+  convert hd.deriv using 1
+  congr 2
+  ring_nf
+
+theorem angleGenerator_sinePower (β r θ : ℝ) (hθ : 0 < Real.sin θ) :
+    angleGenerator β (sinePower r) θ = sinePower r θ *
+      (-r / 2 + r * (r - 1 + 2 * β) / 2 * Real.cot θ ^ 2) := by
+  unfold angleGenerator
+  rw [sinePower_secondDeriv r θ hθ, (sinePower_hasDerivAt r θ hθ).deriv]
+  dsimp [sinePower]
+  rw [Real.rpow_sub hθ, Real.rpow_sub hθ, Real.rpow_one, Real.rpow_two,
+    Real.cot_eq_cos_div_sin]
+  have htrig := Real.sin_sq_add_cos_sq θ
+  field_simp
+  nlinarith [sq_nonneg (Real.sin θ), sq_nonneg (Real.cos θ)]
+
+theorem sine_generator (a r θ : ℝ) (hθ : θ ∈ Set.Ioo 0 Real.pi) :
+    angleGenerator (1 - 2 * a) (sinePower r) θ / sinePower r θ =
+      -r / 2 + r * (r - (4 * a - 1)) / 2 * Real.cot θ ^ 2 := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  rw [angleGenerator_sinePower _ _ _ hs]
+  have hp : sinePower r θ ≠ 0 := (Real.rpow_pos_of_pos hs r).ne'
+  field_simp
+  ring
+
+theorem green_angle_eigenfunction (a θ : ℝ) (hθ : θ ∈ Set.Ioo 0 Real.pi) :
+    angleGenerator (1 - 2 * a) (sinePower (4 * a - 1)) θ =
+      -(4 * a - 1) / 2 * sinePower (4 * a - 1) θ := by
+  rw [angleGenerator_sinePower _ _ _ (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)]
+  ring
+
+theorem tilted_lyapunov (a q θ : ℝ) (_ha : 1 / 4 < a)
+    (hq : 0 < q) (hqp : q < 4 * a - 1) (hθ : θ ∈ Set.Ioo 0 Real.pi) :
+    angleGenerator (2 * a) (sinePower (-q)) θ =
+      sinePower (-q) θ * (q / 2 - q * (4 * a - 1 - q) / 2 * Real.cot θ ^ 2) ∧
+    angleGenerator (2 * a) (sinePower (-q)) θ ≤ q / 2 * sinePower (-q) θ := by
+  have he : angleGenerator (2 * a) (sinePower (-q)) θ =
+      sinePower (-q) θ * (q / 2 - q * (4 * a - 1 - q) / 2 * Real.cot θ ^ 2) := by
+    rw [angleGenerator_sinePower _ _ _ (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)]
+    ring
+  refine ⟨he, ?_⟩
+  rw [he]
+  have hf : 0 < sinePower (-q) θ :=
+    Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2) _
+  have hc : 0 ≤ q * (4 * a - 1 - q) / 2 * Real.cot θ ^ 2 := by positivity
+  nlinarith
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt
+noncomputable def clockExp (a cap lam : ℝ) (v : State) : ℝ :=
+  Real.exp (lam*physicalClock a cap v)
+noncomputable def angleLift (g : ℝ → ℝ) (v : State) : ℝ := g (physicalTheta v)
+lemma clock_N_zero (a cap : ℝ) (v : State) (hv : 0<Y v) :
+    N (physicalClock a cap) noise v = 0 := by
+  rw [N,(hasFDerivAt_physicalClock a cap v hv).fderiv,clockDeriv_noise]
+lemma clock_L (a cap : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
+    L (physicalClock a cap) (physicalDrift a v) noise v = (physicalV v)^2 := by
+  change (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ (physicalClock a cap) w noise) v noise +
+    fderiv ℝ (physicalClock a cap) v (physicalDrift a v) = _
+  rw [physicalClock_second_noise a cap v hv,physicalClock_drift a cap ha v hv]
+  ring
+lemma theta_N (v : State) (hv : 0<Y v) : N physicalTheta noise v = physicalV v := by
+  rw [N,(hasFDerivAt_physicalTheta v hv).fderiv,thetaDeriv_noise]
+lemma theta_L (a : ℝ) (v : State) (hv : 0<Y v) :
+    L physicalTheta (physicalDrift a v) noise v =
+      (1-2*a)*Real.cot (physicalTheta v)*(physicalV v)^2 := theta_generator a v hv
+lemma clockExp_smooth (a cap lam : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ 2 (clockExp a cap lam) v :=
+  (contDiffAt_const.mul (physicalClock_smooth a cap v hv)).exp
+lemma clockExp_N (a cap lam : ℝ) (v : State) (hv : 0<Y v) :
+    N (clockExp a cap lam) noise v = 0 := by
+  have he : clockExp a cap lam = Real.exp ∘ (fun w => lam*physicalClock a cap w) := rfl
+  rw [he,N_comp ((contDiffAt_const.mul (physicalClock_smooth a cap v hv)).differentiableAt (by norm_num))
+    Real.differentiable_exp.differentiableAt]
+  rw [N_const_mul ((physicalClock_smooth a cap v hv).differentiableAt (by norm_num)),clock_N_zero a cap v hv]
+  ring
+lemma clockExp_L (a cap lam : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
+    L (clockExp a cap lam) (physicalDrift a v) noise v =
+      clockExp a cap lam v*lam*(physicalV v)^2 := by
+  have he : clockExp a cap lam = Real.exp ∘ (fun w => lam*physicalClock a cap w) := rfl
+  rw [he,L_comp (contDiffAt_const.mul (physicalClock_smooth a cap v hv)) Real.contDiff_exp.contDiffAt,
+    L_const_mul (physicalClock_smooth a cap v hv),clock_L a cap ha v hv,
+    N_const_mul ((physicalClock_smooth a cap v hv).differentiableAt (by norm_num)),clock_N_zero a cap v hv]
+  simp only [Real.deriv_exp]
+  change _ = Real.exp (lam*physicalClock a cap v)*lam*(physicalV v)^2
+  ring
+lemma angleLift_smooth (g : ℝ → ℝ) (v : State) (hv : 0<Y v)
+    (hg : ContDiffAt ℝ 2 g (physicalTheta v)) : ContDiffAt ℝ 2 (angleLift g) v :=
+  hg.comp v (physicalTheta_smooth v hv)
+lemma angleLift_N (g : ℝ → ℝ) (v : State) (hv : 0<Y v)
+    (hg : DifferentiableAt ℝ g (physicalTheta v)) :
+    N (angleLift g) noise v = deriv g (physicalTheta v)*physicalV v := by
+  change N (g ∘ physicalTheta) noise v = _
+  rw [N_comp ((physicalTheta_smooth v hv).differentiableAt (by norm_num)) hg,theta_N v hv]
+lemma angleLift_L (g : ℝ → ℝ) (a : ℝ) (v : State) (hv : 0<Y v)
+    (hg : ContDiffAt ℝ 2 g (physicalTheta v)) :
+    L (angleLift g) (physicalDrift a v) noise v =
+      angleGenerator (1-2*a) g (physicalTheta v)*(physicalV v)^2 := by
+  change L (g ∘ physicalTheta) (physicalDrift a v) noise v = _
+  rw [L_comp (physicalTheta_smooth v hv) hg,theta_L a v hv,theta_N v hv]
+  dsimp [angleGenerator]
+  ring
+lemma exponential_angle_generator (g : ℝ → ℝ) (a cap lam : ℝ) (ha : a≠0)
+    (v : State) (hv : 0<Y v) (hg : ContDiffAt ℝ 2 g (physicalTheta v)) :
+    L (fun w => clockExp a cap lam w*angleLift g w) (physicalDrift a v) noise v =
+      clockExp a cap lam v*(lam*g (physicalTheta v)+angleGenerator (1-2*a) g (physicalTheta v))*(physicalV v)^2 := by
+  rw [L_mul (clockExp_smooth a cap lam v hv) (angleLift_smooth g v hv hg),
+    clockExp_L a cap lam ha v hv,clockExp_N a cap lam v hv,angleLift_L g a v hv hg]
+  dsimp only [angleLift]
+  ring
+lemma sinePower_smooth (r θ : ℝ) (hθ : 0<Real.sin θ) : ContDiffAt ℝ 2 (sinePower r) θ :=
+  Real.contDiff_sin.contDiffAt.rpow_const_of_ne hθ.ne'
+end ExactSLEGreen
+
+namespace ExactSLEFiniteTilt
+noncomputable def regP (a : ℝ) : ℝ := 4*a-1
+noncomputable def regQ (a : ℝ) : ℝ := min 1 (regP a)/2
+noncomputable def regEta (a : ℝ) : ℝ := (1-regQ a)/(2-regQ a)
+noncomputable def regM (a : ℝ) : ℝ := (1+regEta a)/(1-2*regEta a)
+noncomputable def regBeta (a : ℝ) : ℝ := min 1 (a/(4*regM a))
+noncomputable def regK (a : ℝ) : ℝ := regQ a*(regP a-regQ a)/2
+noncomputable def regA (a : ℝ) : ℝ := regP a/2+(regP a-regQ a)*(1+regQ a)/2+2
+noncomputable def regH (a : ℝ) : ℝ := (regK a/(2*regA a))^(1/(2-regQ a))
+noncomputable def regC (a : ℝ) : ℝ := regA a * regH a ^ (-regP a)
+noncomputable def regFactor (a δ θ : ℝ) : ℝ :=
+  sinePower (regP a) θ + δ*(1+sinePower (regQ a) θ)
+noncomputable def regD (a δ θ : ℝ) : ℝ := δ/regFactor a δ θ * Real.sin θ^(regQ a-2)
+noncomputable def regNormalizedDrift (a δ σ θ : ℝ) : ℝ :=
+  (angleGenerator (1-2*a) (regFactor a δ) θ + regP a/2 * regFactor a δ θ +
+    σ*δ*(1+sinePower (regQ a) θ))/regFactor a δ θ
+end ExactSLEFiniteTilt
+
+namespace ExactSLEFiniteTilt
+
+lemma regP_pos {a : ℝ} (ha : 1/4 < a) : 0 < regP a := by dsimp [regP]; linarith
+lemma regQ_pos {a : ℝ} (ha : 1/4 < a) : 0 < regQ a := by
+  dsimp [regQ]; exact div_pos (lt_min (by norm_num) (regP_pos ha)) (by norm_num)
+lemma regQ_lt_one {a : ℝ} (_ha : 1/4 < a) : regQ a < 1 := by
+  have hh := min_le_left (1 : ℝ) (regP a)
+  dsimp [regQ]; linarith
+lemma regQ_lt_p {a : ℝ} (ha : 1/4 < a) : regQ a < regP a := by
+  have hh := min_le_right (1 : ℝ) (regP a)
+  have hp := regP_pos ha
+  dsimp [regQ]; linarith
+lemma regEta_pos {a : ℝ} (ha : 1/4 < a) : 0 < regEta a := by
+  have hq := regQ_lt_one ha
+  exact div_pos (by linarith) (by linarith)
+lemma regEta_lt_half {a : ℝ} (ha : 1/4 < a) : regEta a < 1/2 := by
+  have hq := regQ_pos ha
+  have hq1 := regQ_lt_one ha
+  dsimp [regEta]
+  apply (div_lt_iff₀ (by linarith)).mpr
+  linarith
+lemma regEta_identity {a : ℝ} (ha : 1/4 < a) : regEta a*(regQ a-2) = regQ a-1 := by
+  have hq := regQ_lt_one ha
+  dsimp [regEta]
+  field_simp [show 2-regQ a ≠ 0 by linarith]
+  ring
+lemma regM_pos {a : ℝ} (ha : 1/4 < a) : 0 < regM a := by
+  have hη := regEta_pos ha
+  have hη1 := regEta_lt_half ha
+  exact div_pos (by linarith) (by linarith)
+lemma regM_eq {a : ℝ} (ha : 1/4 < a) : regM a = 3/regQ a-2 := by
+  have hq := regQ_pos ha
+  have hq1 := regQ_lt_one ha
+  have hη1 := regEta_lt_half ha
+  dsimp only [regM]
+  apply (div_eq_iff (show 1-2*regEta a ≠ 0 by linarith)).mpr
+  dsimp only [regEta]
+  field_simp [show 2-regQ a ≠ 0 by linarith, hq.ne']
+  ring
+lemma regBeta_pos {a : ℝ} (ha : 1/4 < a) : 0 < regBeta a := by
+  exact lt_min (by norm_num) (div_pos (by linarith) (by have := regM_pos ha; positivity))
+lemma regBeta_le_one (a : ℝ) : regBeta a ≤ 1 := min_le_left _ _
+lemma regBeta_discount {a : ℝ} (ha : 1/4 < a) : 2*regM a*regBeta a < a := by
+  have hM := regM_pos ha
+  have hb := min_le_right (1 : ℝ) (a/(4*regM a))
+  change regBeta a ≤ a/(4*regM a) at hb
+  have hm := (le_div_iff₀ (by positivity : 0 < 4*regM a)).mp hb
+  nlinarith
+lemma regK_pos {a : ℝ} (ha : 1/4 < a) : 0 < regK a := by
+  exact div_pos (mul_pos (regQ_pos ha) (sub_pos.mpr (regQ_lt_p ha))) (by norm_num)
+lemma regA_pos {a : ℝ} (ha : 1/4 < a) : 0 < regA a := by
+  have hp := regP_pos ha
+  have hq := regQ_pos ha
+  have hpq := sub_pos.mpr (regQ_lt_p ha)
+  dsimp [regA]; positivity
+lemma reg_base_lt_one {a : ℝ} (ha : 1/4 < a) : regK a/(2*regA a) < 1 := by
+  have hp := regP_pos ha
+  have hq := regQ_pos ha
+  have hpq := sub_pos.mpr (regQ_lt_p ha)
+  have hA := regA_pos ha
+  apply (div_lt_one (by positivity)).mpr
+  have hprod := mul_pos hpq (show 0 < 1+regQ a/2 by linarith)
+  dsimp [regK,regA]
+  nlinarith
+lemma regH_pos {a : ℝ} (ha : 1/4 < a) : 0 < regH a := by
+  exact Real.rpow_pos_of_pos (div_pos (regK_pos ha) (by have := regA_pos ha; positivity)) _
+lemma regH_lt_one {a : ℝ} (ha : 1/4 < a) : regH a < 1 := by
+  have hq1 := regQ_lt_one ha
+  exact Real.rpow_lt_one (by have := regK_pos ha; have := regA_pos ha; positivity)
+    (reg_base_lt_one ha) (div_pos zero_lt_one (by linarith))
+lemma regH_power {a : ℝ} (ha : 1/4 < a) : regH a ^ (regQ a-2) = 2*regA a/regK a := by
+  have hq1 := regQ_lt_one ha
+  have hk := regK_pos ha
+  have hA := regA_pos ha
+  dsimp [regH]
+  rw [←Real.rpow_mul (by positivity : 0 ≤ regK a/(2*regA a))]
+  have he : 1/(2-regQ a)*(regQ a-2) = -1 := by field_simp [show 2-regQ a ≠ 0 by linarith]; ring
+  rw [he,Real.rpow_neg_one,inv_div]
+lemma regC_pos {a : ℝ} (ha : 1/4 < a) : 0 < regC a :=
+  mul_pos (regA_pos ha) (Real.rpow_pos_of_pos (regH_pos ha) _)
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLEFiniteTilt
+open Filter
+end ExactSLEFiniteTilt
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/AngleLyapunov.lean b/lean/OAI/Probability/SLEGauge/AngleLyapunov.lean
index d14f14fb74dba018fecb43c1224a57ea1e8d8cd1..f25db9b2cffbde11ddcca41bf88dc0abcb5de573 100644
--- a/lean/OAI/Probability/SLEGauge/AngleLyapunov.lean
+++ b/lean/OAI/Probability/SLEGauge/AngleLyapunov.lean
@@ -1,523 +1,523 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RawBrownianRestart
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace ExactSLEFiniteTilt
-
- 
-lemma exists_tilted_coercivity {a q : ℝ} (ha : 1/4<a) (hq : 0<q)
-    (hqp : q<4*a-1) :
-    ∃ C : ℝ, 0<C ∧ ∀ θ∈Ioo 0 Real.pi,
-      angleGenerator (2*a) (sinePower (-q)) θ + sinePower (-q) θ ≤ C := by
-  let A := q/2+1
-  let K := q*(4*a-1-q)/2
-  have hA : 0<A := by dsimp [A]; linarith
-  have hK : 0<K := by dsimp [K]; exact div_pos (mul_pos hq (by linarith)) (by norm_num)
-  let δ := Real.sqrt (K/(A+K))
-  have hδ : 0<δ := Real.sqrt_pos.mpr (div_pos hK (add_pos hA hK))
-  have hδ2 : δ^2=K/(A+K) := Real.sq_sqrt (by positivity)
-  have hδK : (A+K)*δ^2=K := by rw [hδ2]; field_simp
-  refine ⟨A*δ^(-q), mul_pos hA (Real.rpow_pos_of_pos hδ _), ?_⟩
-  intro θ hθ
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  have hF : 0 < sinePower (-q) θ := Real.rpow_pos_of_pos hs _
-  rw [(tilted_lyapunov a q θ ha hq hqp hθ).1]
-  change sinePower (-q) θ * (q/2-K*Real.cot θ^2)+sinePower (-q) θ≤_
-  by_cases hx : δ≤Real.sin θ
-  · have hle : sinePower (-q) θ ≤ δ^(-q) :=
-      Real.rpow_le_rpow_of_nonpos hδ hx (by linarith)
-    have hn : 0 ≤ sinePower (-q) θ * (K*Real.cot θ^2) :=
-      mul_nonneg hF.le (mul_nonneg hK.le (sq_nonneg _))
-    have hh := mul_le_mul_of_nonneg_left hle hA.le
-    dsimp only [A] at hh ⊢
-    nlinarith only [hh,hn]
-  · have hx' : Real.sin θ ≤ δ := (lt_of_not_ge hx).le
-    have hx2 : Real.sin θ^2≤δ^2 := pow_le_pow_left₀ hs.le hx' 2
-    have hsq := Real.sin_sq_add_cos_sq θ
-    have hk : A*(Real.sin θ)^2 ≤ K*(Real.cos θ)^2 := by
-      have hh := mul_le_mul_of_nonneg_left hx2 (add_pos hA hK).le
-      rw [hδK] at hh
-      nlinarith only [hh, congrArg (fun x => K*x) hsq]
-    have hcot : A ≤ K*Real.cot θ^2 := by
-      rw [Real.cot_eq_cos_div_sin, div_pow, ← mul_div_assoc]
-      exact (le_div_iff₀ (sq_pos_of_pos hs)).mpr hk
-    have hh := mul_le_mul_of_nonneg_left hcot hF.le
-    have hc : 0≤A*δ^(-q) := by positivity
-    dsimp only [A] at hh
-    nlinarith only [hh,hc]
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt
-
-lemma physicalTheta_smooth_all (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) physicalTheta v :=
-  contDiffAt_const.sub ((X.contDiff.contDiffAt.div Y.contDiff.contDiffAt hv.ne').arctan)
-
-lemma physicalClock_smooth_all (a cap : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalClock a cap) v :=
-  ((contDiffAt_const.add J.contDiff.contDiffAt).sub
-    (Y.contDiff.contDiffAt.log hv.ne')).div_const (2*a)
-
-lemma clockExp_smooth_all (a cap lam : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (clockExp a cap lam) v :=
-  (contDiffAt_const.mul (physicalClock_smooth_all a cap v hv)).exp
-
-lemma angleLift_sinePower_smooth_all (q : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (angleLift (sinePower q)) v := by
-  have hp := physicalTheta_mem v
-  exact ((physicalTheta_smooth_all v hv).sin).rpow_const_of_ne
-    (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2).ne'
-
-lemma green_exponential_angle_generator (g : ℝ → ℝ) (a cap lam : ℝ) (ha : a≠0)
-    (v : State) (hv : 0<Y v) (hg : ContDiffAt ℝ 2 g (physicalTheta v)) :
-    L (fun w => physicalGreen a w * (clockExp a cap lam w*angleLift g w))
-      (physicalDrift a v) noise v =
-      physicalGreen a v * clockExp a cap lam v *
-        (lam*g (physicalTheta v)+angleGenerator (2*a) g (physicalTheta v))*(physicalV v)^2 := by
-  rw [green_product_generator a ha _ v hv ((clockExp_smooth a cap lam v hv).mul
-    (angleLift_smooth g v hv hg))]
-  rw [centeredGenerator,exponential_angle_generator g a cap lam ha v hv hg,
-    N_mul ((clockExp_smooth a cap lam v hv).differentiableAt (by norm_num))
-      ((angleLift_smooth g v hv hg).differentiableAt (by norm_num)),
-    clockExp_N a cap lam v hv, angleLift_N g v hv (hg.differentiableAt (by norm_num))]
-  simp only [angleGenerator,physicalTheta_cot,physicalV,noiseLog,p]
-  field_simp [(Q_pos hv).ne']
-  ring
-
- 
-
-def angleBudgetFactor (a q cap s C : ℝ) (v : State) : ℝ :=
-  clockExp a cap 1 v * (angleLift (sinePower (-q)) v - C) + C*Real.exp s
-
-lemma angleBudgetFactor_smooth (a q cap s C : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (angleBudgetFactor a q cap s C) v :=
-  ((clockExp_smooth_all a cap 1 v hv).mul
-    ((angleLift_sinePower_smooth_all (-q) v hv).sub contDiffAt_const)).add contDiffAt_const
-
-lemma angleBudgetFactor_nonneg (a q cap s C : ℝ) (hC : 0≤C)
-    (v : State) (hc : physicalClock a cap v ≤ s) :
-    0≤angleBudgetFactor a q cap s C v := by
-  have hp := physicalTheta_mem v
-  have hF : 0<angleLift (sinePower (-q)) v :=
-    Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2) _
-  have hE : 0<clockExp a cap 1 v := Real.exp_pos _
-  have hEs : clockExp a cap 1 v≤Real.exp s := by
-    exact Real.exp_le_exp.mpr (by simpa only [one_mul] using hc)
-  have hh := mul_nonneg hC (sub_nonneg.mpr hEs)
-  have hh' := mul_pos hE hF
-  dsimp only [angleBudgetFactor]
-  nlinarith only [hh,hh']
-
-lemma angleBudgetFactor_terminal (a q cap s C : ℝ) (v : State)
-    (hc : physicalClock a cap v=s) :
-    angleBudgetFactor a q cap s C v = Real.exp s * angleLift (sinePower (-q)) v := by
-  unfold angleBudgetFactor clockExp
-  rw [hc,one_mul]
-  ring
-
-lemma green_angleBudget_generator (a q cap s C : ℝ) (ha : a≠0)
-    (v : State) (hv : 0<Y v) :
-    L (fun w => physicalGreen a w * angleBudgetFactor a q cap s C w)
-      (physicalDrift a v) noise v =
-      physicalGreen a v * clockExp a cap 1 v *
-        (sinePower (-q) (physicalTheta v) +
-          angleGenerator (2*a) (sinePower (-q)) (physicalTheta v) - C)*(physicalV v)^2 := by
-  have hp := physicalTheta_mem v
-  have hs := sinePower_smooth (-q) (physicalTheta v)
-    (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2)
-  have hG : ContDiffAt ℝ 2 (physicalGreen a) v :=
-    (contDiffAt_physicalGreen a v hv).of_le (by norm_cast)
-  have he : (fun w => physicalGreen a w * angleBudgetFactor a q cap s C w) =
-      (fun w => physicalGreen a w * (clockExp a cap 1 w * angleLift (sinePower (-q)) w) +
-        (-C)*(physicalGreen a w * (clockExp a cap 1 w * angleLift (fun _ => (1:ℝ)) w)) +
-        (C*Real.exp s)*physicalGreen a w) := by
-    funext w
-    dsimp only [angleBudgetFactor,angleLift]
-    ring
-  have hz : L (physicalGreen a) (physicalDrift a v) noise v = 0 :=
-    green_generator_zero a ha v hv
-  rw [he,L_add ((hG.mul ((clockExp_smooth a cap 1 v hv).mul (angleLift_smooth _ v hv hs))).add
-      (contDiffAt_const.mul (hG.mul ((clockExp_smooth a cap 1 v hv).mul
-        (angleLift_smooth _ v hv contDiffAt_const))))) (contDiffAt_const.mul hG),
-    L_add (hG.mul ((clockExp_smooth a cap 1 v hv).mul (angleLift_smooth _ v hv hs)))
-      (contDiffAt_const.mul (hG.mul ((clockExp_smooth a cap 1 v hv).mul
-        (angleLift_smooth _ v hv contDiffAt_const)))),
-    L_const_mul (hG.mul ((clockExp_smooth a cap 1 v hv).mul
-      (angleLift_smooth _ v hv contDiffAt_const))),L_const_mul hG,
-    green_exponential_angle_generator _ a cap 1 ha v hv hs,
-    green_exponential_angle_generator _ a cap 1 ha v hv contDiffAt_const,
-    hz]
-  have h1 : deriv (fun _ : ℝ => (1:ℝ)) = fun _ => 0 := funext fun _ => deriv_const _ _
-  simp only [angleGenerator,h1,deriv_const]
-  ring
-
-lemma green_angleBudget_generator_nonpos (a q cap s C : ℝ) (ha : a≠0)
-    (hC : ∀ θ∈Ioo 0 Real.pi,
-      angleGenerator (2*a) (sinePower (-q)) θ+sinePower (-q) θ≤C)
-    (v : State) (hv : 0<Y v) :
-    L (fun w => physicalGreen a w * angleBudgetFactor a q cap s C w)
-      (physicalDrift a v) noise v ≤0 := by
-  rw [green_angleBudget_generator a q cap s C ha v hv]
-  have hh := hC (physicalTheta v) (physicalTheta_mem v)
-  apply mul_nonpos_of_nonpos_of_nonneg _ (sq_nonneg _)
-  exact mul_nonpos_of_nonneg_of_nonpos (mul_nonneg (Real.exp_pos _).le (Real.exp_pos _).le)
-    (by linarith)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem compact_test_stopped_dynkin
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
-    (T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hTB : ∀ b, T b ≤ B) :
-    Integrable (fun b => f (X (T b) b)) W ∧
-    Integrable (fun b => ∫ u in (0 : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) W ∧
-    (∫ b, f (X (T b) b) ∂W) = (∫ b, f (X 0 b) ∂W) +
-      ∫ b, (∫ u in (0 : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) ∂W := by
-  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
-    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
-  have hM : Martingale M rawFiltration W :=
-    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
-  have hgc (b : Driver) : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
-    (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
-  have hmc (b : Driver) : Continuous (fun t => M t b) :=
-    (hf.continuous.comp (hXc b)).sub
-      ((intervalIntegral.continuous_primitive (fun s t => (hgc b).intervalIntegrable s t) 0).comp NNReal.continuous_coe)
-  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
-  have hmb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
-    calc
-      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
-      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
-        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
-          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
-            (fun u _ => hgb (X u.toNNReal b)))
-      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
-  have hFa : StronglyAdapted rawFiltration (fun t b => f (X t b)) := fun t =>
-    (hf.continuous.measurable.comp (hXm t)).stronglyMeasurable
-  have hFm : Measurable (fun b => f (X (T b) b)) :=
-    (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
-      (fun b => hf.continuous.comp (hXc b))) hT).mono hT.measurableSpace_le le_rfl
-  have hFi : Integrable (fun b => f (X (T b) b)) W :=
-    Integrable.of_bound hFm.aestronglyMeasurable C (Eventually.of_forall fun b => hfb _)
-  have hMi : Integrable (fun b => M (T b) b) W :=
-    Integrable.of_bound (ExactSLEOptional.stopped_measurable rawFiltration hM hmc T hT).aestronglyMeasurable
-      (C+D*(B+1)) (Eventually.of_forall fun b => hmb _ ((hTB b).trans (le_add_of_nonneg_right zero_le_one)) b)
-  have hi : Integrable (fun b => ∫ u in (0 : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) W := by
-    convert hFi.sub hMi using 1
-    funext b
-    dsimp [M]
-    ring
-  have heq := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hmc B
-    (C+D*(B+1)) hmb (fun _ => 0) T (isStoppingTime_const _ _) hT (fun _ => bot_le) hTB univ MeasurableSet.univ
-  simp only [setIntegral_univ, M, NNReal.coe_zero, intervalIntegral.integral_same, sub_zero] at heq
-  rw [integral_sub hFi hi] at heq
-  exact ⟨hFi, hi, by linarith⟩
-
-end ExactSLEGenerator
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEGenerator
-
-noncomputable def lifetimeCutoff (a ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
-  greenCutoff a ε N v * lifetimeTest (lifetimeExponent a) v
-
-lemma lifetimeCutoff_smooth (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (lifetimeCutoff a ε N) := by
-  rw [contDiff_iff_contDiffAt]
-  intro v
-  by_cases hv : 0 < Y v
-  · exact (greenCutoff_smooth a ε hε N).contDiffAt.mul (lifetimeTest_smooth _ v hv)
-  · apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
-    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
-    simp only [lifetimeCutoff, greenCutoff, heightTurn_zero hε (le_of_lt hw),
-      mul_zero, zero_mul]
-
-lemma lifetimeCutoff_compact (a ε : ℝ) (N : ℝ≥0) :
-    HasCompactSupport (lifetimeCutoff a ε N) := (greenCutoff_compact a ε N).mul_right
-
-lemma lifetimeCutoff_eq_near (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0)
-    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
-    lifetimeCutoff a ε N =ᶠ[𝓝 v]
-      fun w => physicalGreen a w * lifetimeTest (lifetimeExponent a) w :=
-  (greenCutoff_eq_near a ε hε N v hv hy).mul (EventuallyEq.refl _ _)
-
-noncomputable def lifetimeCost (a : ℝ) (v : State) : ℝ :=
-  lifetimeDrift a * physicalGreen a v * Q v ^ (lifetimeExponent a / 2 - 1)
-
-lemma lifetimeCost_pos {a : ℝ} (ha : 1/4 < a) {v : State} (hv : 0 < Y v) :
-    0 < lifetimeCost a v :=
-  mul_pos (mul_pos (lifetimeDrift_pos ha) (Real.exp_pos _)) (Real.rpow_pos_of_pos (Q_pos hv) _)
-
-lemma lifetimeCost_continuousAt (a : ℝ) (v : State) (hv : 0 < Y v) :
-    ContinuousAt (lifetimeCost a) v := by
-  have hQ : ContDiffAt ℝ 1 Q v :=
-    ((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt
-  exact (continuousAt_const.mul (contDiffAt_physicalGreen a v hv).continuousAt).mul
-    ((hQ.rpow_const_of_ne (Q_pos hv).ne').continuousAt)
-
-lemma lifetimeCutoff_generator_le (a ε : ℝ) (ha : 1/4 < a) (hε : 0 < ε) (N : ℝ≥0)
-    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
-    generator (lifetimeCutoff a ε N) (cutoffDrift a ε) noise v ≤ -lifetimeCost a v := by
-  have he := lifetimeCutoff_eq_near a ε hε N v hv hy
-  have hd : (fun w => fderiv ℝ (lifetimeCutoff a ε N) w noise) =ᶠ[𝓝 v]
-      (fun w => fderiv ℝ (fun u => physicalGreen a u * lifetimeTest (lifetimeExponent a) u) w noise) :=
-    he.fderiv.mono (fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw)
-  change (1/2:ℝ) * fderiv ℝ (fun w => fderiv ℝ (lifetimeCutoff a ε N) w noise) v noise +
-    fderiv ℝ (lifetimeCutoff a ε N) v (cutoffDrift a ε v) ≤ _
-  rw [hd.fderiv_eq, he.fderiv_eq, cutoffDrift_eq_physical a ε v hy]
-  convert green_lifetime_generator_le a ha v (hε.trans_le hy) using 1 <;> first | rfl | (dsimp [lifetimeCost]; ring)
-
- 
-
-theorem local_lifetime_budget (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (a ε : ℝ) (ha : 1/4 < a) (hε : 0 < ε) (z : ℂ) (N B : ℝ≥0)
-    (T : Driver → ℝ≥0)
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hTB : ∀ b, T b ≤ B)
-    (hsafe : ∀ b t, t ≤ T b → ε ≤ Y (localState a ε hε z t b) ∧
-      ‖localState a ε hε z t b‖ ≤ N) :
-    Integrable (fun b => physicalGreen a (localState a ε hε z (T b) b) *
-      lifetimeTest (lifetimeExponent a) (localState a ε hε z (T b) b)) W ∧
-    (∫ b, physicalGreen a (localState a ε hε z (T b) b) *
-      lifetimeTest (lifetimeExponent a) (localState a ε hε z (T b) b) ∂W) ≤
-      physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) ∧
-    (∫⁻ b, ∫⁻ u in Ioc (0 : ℝ) (T b : ℝ),
-      ENNReal.ofReal (lifetimeCost a (localState a ε hε z u.toNNReal b)) ∂volume ∂W) ≤
-      ENNReal.ofReal (physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0)) := by
-  let V := localState a ε hε z
-  let f := lifetimeCutoff a ε N
-  let g := generator f (cutoffDrift a ε) noise
-  have hf := lifetimeCutoff_smooth a ε hε N
-  have hfc := lifetimeCutoff_compact a ε N
-  have hgc : Continuous g := generator_continuous f hf _ (cutoffDrift_lipschitz a ε hε).continuous noise
-  have hVc (b : Driver) : Continuous (fun u : ℝ => V u.toNNReal b) :=
-    (localState_continuous a ε hε z b).comp continuous_real_toNNReal
-  have he (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) :
-      f (V t b) = physicalGreen a (V t b) * lifetimeTest (lifetimeExponent a) (V t b) :=
-    (lifetimeCutoff_eq_near a ε hε N _ (hsafe b t ht).2 (hsafe b t ht).1).eq_of_nhds
-  have hfpos (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) : 0 ≤ f (V t b) := by
-    rw [he b t ht]
-    exact mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg _ _ (hε.le.trans (hsafe b t ht).1))
-  have hgen (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) :
-      g (V t b) ≤ -lifetimeCost a (V t b) :=
-    lifetimeCutoff_generator_le a ε ha hε N _ (hsafe b t ht).2 (hsafe b t ht).1
-  have hpos (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) : 0 < lifetimeCost a (V t b) :=
-    lifetimeCost_pos ha (hε.trans_le (hsafe b t ht).1)
-  have hDy := compact_test_stopped_dynkin W hW V
-    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
-    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
-    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
-    (by simp [noise, Prod.norm_def]) (localState_eq a ε hε z) f hf hfc T B hT hTB
-  let H : Driver → ℝ := fun b => -(∫ u in (0 : ℝ)..(T b : ℝ), g (V u.toNNReal b))
-  have hHi : Integrable H W := hDy.2.1.neg
-  have hHnonneg (b : Driver) : 0 ≤ H b := by
-    dsimp [H]
-    apply neg_nonneg.mpr
-    rw [intervalIntegral.integral_of_le (T b).coe_nonneg]
-    apply setIntegral_nonpos measurableSet_Ioc
-    intro u hu
-    have ht : u.toNNReal ≤ T b := by simpa using Real.toNNReal_le_toNNReal hu.2
-    exact (hgen b u.toNNReal ht).trans (neg_nonpos.mpr (hpos b u.toNNReal ht).le)
-  have hHbound : (∫ b, H b ∂W) ≤ physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) := by
-    have hinit (b : Driver) : f (V 0 b) = physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) := by
-      rw [he b 0 (bot_le)]
-      simp only [V,localState_zero]
-    have hd := hDy.2.2
-    simp_rw [hinit] at hd
-    simp only [integral_const, probReal_univ, one_smul] at hd
-    have hn : 0 ≤ ∫ b, f (V (T b) b) ∂W := integral_nonneg (fun b => hfpos b (T b) le_rfl)
-    change (∫ b, -(∫ u in (0 : ℝ)..(T b : ℝ), g (V u.toNNReal b)) ∂W) ≤ _
-    rw [integral_neg]
-    linarith
-  refine ⟨?_, ?_, ?_⟩
-  · simpa only [he _ _ le_rfl] using hDy.1
-  · have hn : (∫ b, H b ∂W) ≥ 0 := integral_nonneg hHnonneg
-    have hd := hDy.2.2
-    have hinit (b : Driver) : f (V 0 b) = physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) := by
-      rw [he b 0 (bot_le)]
-      simp only [V,localState_zero]
-    simp_rw [hinit, he _ _ le_rfl] at hd
-    simp only [integral_const, probReal_univ, one_smul] at hd
-    dsimp only [H] at hn
-    rw [integral_neg] at hn
-    change (∫ b, physicalGreen a (V (T b) b) * lifetimeTest (lifetimeExponent a) (V (T b) b) ∂W) ≤ _
-    linarith
-  · suffices hp : ∀ b, (∫⁻ u in Ioc (0 : ℝ) (T b : ℝ), ENNReal.ofReal (lifetimeCost a (V u.toNNReal b))) ≤ ENNReal.ofReal (H b) by
-      calc
-        _ ≤ ∫⁻ b, ENNReal.ofReal (H b) ∂W := lintegral_mono hp
-        _ = ENNReal.ofReal (∫ b, H b ∂W) := (ofReal_integral_eq_lintegral_ofReal hHi (Eventually.of_forall hHnonneg)).symm
-        _ ≤ _ := ENNReal.ofReal_le_ofReal hHbound
-    intro b
-    have ht (u : ℝ) (hu : u ∈ Icc 0 (T b : ℝ)) : u.toNNReal ≤ T b := by
-      simpa using Real.toNNReal_le_toNNReal hu.2
-    have hcostc : ContinuousOn (fun u : ℝ => lifetimeCost a (V u.toNNReal b)) (Icc 0 (T b : ℝ)) := by
-      intro u hu
-      exact ((lifetimeCost_continuousAt a _ (hε.trans_le (hsafe b u.toNNReal (ht u hu)).1)).comp (f := fun u : ℝ => V u.toNNReal b) (hVc b).continuousAt).continuousWithinAt
-    have hci : IntervalIntegrable (fun u : ℝ => lifetimeCost a (V u.toNNReal b)) volume 0 (T b) :=
-      hcostc.intervalIntegrable_of_Icc (T b).coe_nonneg
-    have hgi : IntervalIntegrable (fun u : ℝ => g (V u.toNNReal b)) volume 0 (T b) := (hgc.comp (hVc b)).intervalIntegrable 0 (T b)
-    have hcb : (∫ u in (0 : ℝ)..(T b : ℝ), lifetimeCost a (V u.toNNReal b)) ≤ H b := by
-      dsimp only [H]
-      rw [← intervalIntegral.integral_neg]
-      apply intervalIntegral.integral_mono_on (T b).coe_nonneg hci hgi.neg
-      intro u hu
-      dsimp only [Pi.neg_apply]
-      linarith only [hgen b u.toNNReal (ht u hu)]
-    rw [intervalIntegral.integral_of_le (T b).coe_nonneg] at hcb
-    have hn : 0 ≤ᵐ[volume.restrict (Ioc (0 : ℝ) (T b : ℝ))] (fun u => lifetimeCost a (V u.toNNReal b)) := by
-      filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
-      exact (hpos b u.toNNReal (ht u ⟨hu.1.le,hu.2⟩)).le
-    rw [← ofReal_integral_eq_lintegral_ofReal hci.1 hn]
-    exact ENNReal.ofReal_le_ofReal hcb
-
-end ExactSLEGreen
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
-
-def angleBudgetCutoff (a q cap s C ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
-  greenCutoff a ε N v * angleBudgetFactor a q cap s C v
-
-lemma angleBudgetCutoff_smooth (a q cap s C ε : ℝ) (hε : 0<ε) (N : ℝ≥0) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (angleBudgetCutoff a q cap s C ε N) := by
-  rw [contDiff_iff_contDiffAt]
-  intro v
-  by_cases hv : 0<Y v
-  · exact (greenCutoff_smooth a ε hε N).contDiffAt.mul
-      (angleBudgetFactor_smooth a q cap s C v hv)
-  · apply (contDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq
-    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
-    simp only [angleBudgetCutoff, greenCutoff, heightTurn_zero hε (le_of_lt hw),
-      mul_zero, zero_mul]
-
-lemma angleBudgetCutoff_compact (a q cap s C ε : ℝ) (N : ℝ≥0) :
-    HasCompactSupport (angleBudgetCutoff a q cap s C ε N) :=
-  (greenCutoff_compact a ε N).mul_right
-
-lemma angleBudgetCutoff_eq_near (a q cap s C ε : ℝ) (hε : 0<ε) (N : ℝ≥0)
-    (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
-    angleBudgetCutoff a q cap s C ε N =ᶠ[𝓝 v]
-      fun w => physicalGreen a w * angleBudgetFactor a q cap s C w :=
-  (greenCutoff_eq_near a ε hε N v hv hy).mul (EventuallyEq.refl _ _)
-
-lemma angleBudgetCutoff_generator_nonpos (a q cap s C ε : ℝ) (ha : a≠0)
-    (hC : ∀ θ∈Ioo 0 Real.pi,
-      angleGenerator (2*a) (sinePower (-q)) θ + sinePower (-q) θ ≤ C)
-    (hε : 0<ε) (N : ℝ≥0) (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
-    generator (angleBudgetCutoff a q cap s C ε N) (cutoffDrift a ε) noise v ≤ 0 := by
-  have he := angleBudgetCutoff_eq_near a q cap s C ε hε N v hv hy
-  have hd : (fun w => fderiv ℝ (angleBudgetCutoff a q cap s C ε N) w noise) =ᶠ[𝓝 v]
-      (fun w => fderiv ℝ (fun u => physicalGreen a u * angleBudgetFactor a q cap s C u) w noise) :=
-    he.fderiv.mono (fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw)
-  change (1/2:ℝ) * fderiv ℝ (fun w => fderiv ℝ (angleBudgetCutoff a q cap s C ε N) w noise) v noise +
-    fderiv ℝ (angleBudgetCutoff a q cap s C ε N) v (cutoffDrift a ε v) ≤ 0
-  rw [hd.fderiv_eq, he.fderiv_eq, cutoffDrift_eq_physical a ε v hy]
-  exact green_angleBudget_generator_nonpos a q cap s C ha hC v (hε.trans_le hy)
-
- 
-
-theorem local_angle_budget (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (a q cap s C ε : ℝ) (ha : a≠0)
-    (hC : ∀ θ∈Ioo 0 Real.pi,
-      angleGenerator (2*a) (sinePower (-q)) θ + sinePower (-q) θ ≤ C)
-    (hε : 0<ε) (z : ℂ) (N B : ℝ≥0) (T : Driver → ℝ≥0)
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hTB : ∀ b, T b≤B)
-    (hsafe : ∀ b t, t≤T b → ε≤Y (localState a ε hε z t b) ∧
-      ‖localState a ε hε z t b‖≤N) :
-    Integrable (fun b => physicalGreen a (localState a ε hε z (T b) b) *
-      angleBudgetFactor a q cap s C (localState a ε hε z (T b) b)) W ∧
-    (∫ b, physicalGreen a (localState a ε hε z (T b) b) *
-      angleBudgetFactor a q cap s C (localState a ε hε z (T b) b) ∂W) ≤
-      physicalGreen a (z,0) * angleBudgetFactor a q cap s C (z,0) := by
-  let V := localState a ε hε z
-  let f := angleBudgetCutoff a q cap s C ε N
-  let g := generator f (cutoffDrift a ε) noise
-  have hf := angleBudgetCutoff_smooth a q cap s C ε hε N
-  have hfc := angleBudgetCutoff_compact a q cap s C ε N
-  have he (b : Driver) (t : ℝ≥0) (ht : t≤T b) :
-      f (V t b) = physicalGreen a (V t b) * angleBudgetFactor a q cap s C (V t b) :=
-    (angleBudgetCutoff_eq_near a q cap s C ε hε N _ (hsafe b t ht).2
-      (hsafe b t ht).1).eq_of_nhds
-  have hgen (b : Driver) (t : ℝ≥0) (ht : t≤T b) : g (V t b)≤0 :=
-    angleBudgetCutoff_generator_nonpos a q cap s C ε ha hC hε N _
-      (hsafe b t ht).2 (hsafe b t ht).1
-  have hDy := compact_test_stopped_dynkin W hW V
-    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
-    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
-    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
-    (by simp [noise, Prod.norm_def]) (localState_eq a ε hε z) f hf hfc T B hT hTB
-  have hneg : (∫ b, (∫ u in (0:ℝ)..(T b : ℝ), g (V u.toNNReal b)) ∂W) ≤ 0 := by
-    apply integral_nonpos
-    intro b
-    change (∫ u in (0:ℝ)..(T b : ℝ), g (V u.toNNReal b)) ≤ 0
-    rw [intervalIntegral.integral_of_le (T b).coe_nonneg]
-    apply setIntegral_nonpos measurableSet_Ioc
-    intro u hu
-    exact hgen b u.toNNReal (by simpa using Real.toNNReal_le_toNNReal hu.2)
-  have hinit (b : Driver) : f (V 0 b) = physicalGreen a (z,0) * angleBudgetFactor a q cap s C (z,0) := by
-    rw [he b 0 bot_le]
-    simp only [V,localState_zero]
-  refine ⟨by simpa only [he _ _ le_rfl] using hDy.1, ?_⟩
-  have hd := hDy.2.2
-  simp_rw [hinit,he _ _ le_rfl] at hd
-  simp only [integral_const,probReal_univ,one_smul] at hd
-  change (∫ b, physicalGreen a (V (T b) b)*angleBudgetFactor a q cap s C (V (T b) b) ∂W)≤_
-  linarith
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RawBrownianRestart
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace ExactSLEFiniteTilt
+
+ 
+lemma exists_tilted_coercivity {a q : ℝ} (ha : 1/4<a) (hq : 0<q)
+    (hqp : q<4*a-1) :
+    ∃ C : ℝ, 0<C ∧ ∀ θ∈Ioo 0 Real.pi,
+      angleGenerator (2*a) (sinePower (-q)) θ + sinePower (-q) θ ≤ C := by
+  let A := q/2+1
+  let K := q*(4*a-1-q)/2
+  have hA : 0<A := by dsimp [A]; linarith
+  have hK : 0<K := by dsimp [K]; exact div_pos (mul_pos hq (by linarith)) (by norm_num)
+  let δ := Real.sqrt (K/(A+K))
+  have hδ : 0<δ := Real.sqrt_pos.mpr (div_pos hK (add_pos hA hK))
+  have hδ2 : δ^2=K/(A+K) := Real.sq_sqrt (by positivity)
+  have hδK : (A+K)*δ^2=K := by rw [hδ2]; field_simp
+  refine ⟨A*δ^(-q), mul_pos hA (Real.rpow_pos_of_pos hδ _), ?_⟩
+  intro θ hθ
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  have hF : 0 < sinePower (-q) θ := Real.rpow_pos_of_pos hs _
+  rw [(tilted_lyapunov a q θ ha hq hqp hθ).1]
+  change sinePower (-q) θ * (q/2-K*Real.cot θ^2)+sinePower (-q) θ≤_
+  by_cases hx : δ≤Real.sin θ
+  · have hle : sinePower (-q) θ ≤ δ^(-q) :=
+      Real.rpow_le_rpow_of_nonpos hδ hx (by linarith)
+    have hn : 0 ≤ sinePower (-q) θ * (K*Real.cot θ^2) :=
+      mul_nonneg hF.le (mul_nonneg hK.le (sq_nonneg _))
+    have hh := mul_le_mul_of_nonneg_left hle hA.le
+    dsimp only [A] at hh ⊢
+    nlinarith only [hh,hn]
+  · have hx' : Real.sin θ ≤ δ := (lt_of_not_ge hx).le
+    have hx2 : Real.sin θ^2≤δ^2 := pow_le_pow_left₀ hs.le hx' 2
+    have hsq := Real.sin_sq_add_cos_sq θ
+    have hk : A*(Real.sin θ)^2 ≤ K*(Real.cos θ)^2 := by
+      have hh := mul_le_mul_of_nonneg_left hx2 (add_pos hA hK).le
+      rw [hδK] at hh
+      nlinarith only [hh, congrArg (fun x => K*x) hsq]
+    have hcot : A ≤ K*Real.cot θ^2 := by
+      rw [Real.cot_eq_cos_div_sin, div_pow, ← mul_div_assoc]
+      exact (le_div_iff₀ (sq_pos_of_pos hs)).mpr hk
+    have hh := mul_le_mul_of_nonneg_left hcot hF.le
+    have hc : 0≤A*δ^(-q) := by positivity
+    dsimp only [A] at hh
+    nlinarith only [hh,hc]
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt
+
+lemma physicalTheta_smooth_all (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) physicalTheta v :=
+  contDiffAt_const.sub ((X.contDiff.contDiffAt.div Y.contDiff.contDiffAt hv.ne').arctan)
+
+lemma physicalClock_smooth_all (a cap : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalClock a cap) v :=
+  ((contDiffAt_const.add J.contDiff.contDiffAt).sub
+    (Y.contDiff.contDiffAt.log hv.ne')).div_const (2*a)
+
+lemma clockExp_smooth_all (a cap lam : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (clockExp a cap lam) v :=
+  (contDiffAt_const.mul (physicalClock_smooth_all a cap v hv)).exp
+
+lemma angleLift_sinePower_smooth_all (q : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (angleLift (sinePower q)) v := by
+  have hp := physicalTheta_mem v
+  exact ((physicalTheta_smooth_all v hv).sin).rpow_const_of_ne
+    (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2).ne'
+
+lemma green_exponential_angle_generator (g : ℝ → ℝ) (a cap lam : ℝ) (ha : a≠0)
+    (v : State) (hv : 0<Y v) (hg : ContDiffAt ℝ 2 g (physicalTheta v)) :
+    L (fun w => physicalGreen a w * (clockExp a cap lam w*angleLift g w))
+      (physicalDrift a v) noise v =
+      physicalGreen a v * clockExp a cap lam v *
+        (lam*g (physicalTheta v)+angleGenerator (2*a) g (physicalTheta v))*(physicalV v)^2 := by
+  rw [green_product_generator a ha _ v hv ((clockExp_smooth a cap lam v hv).mul
+    (angleLift_smooth g v hv hg))]
+  rw [centeredGenerator,exponential_angle_generator g a cap lam ha v hv hg,
+    N_mul ((clockExp_smooth a cap lam v hv).differentiableAt (by norm_num))
+      ((angleLift_smooth g v hv hg).differentiableAt (by norm_num)),
+    clockExp_N a cap lam v hv, angleLift_N g v hv (hg.differentiableAt (by norm_num))]
+  simp only [angleGenerator,physicalTheta_cot,physicalV,noiseLog,p]
+  field_simp [(Q_pos hv).ne']
+  ring
+
+ 
+
+def angleBudgetFactor (a q cap s C : ℝ) (v : State) : ℝ :=
+  clockExp a cap 1 v * (angleLift (sinePower (-q)) v - C) + C*Real.exp s
+
+lemma angleBudgetFactor_smooth (a q cap s C : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (angleBudgetFactor a q cap s C) v :=
+  ((clockExp_smooth_all a cap 1 v hv).mul
+    ((angleLift_sinePower_smooth_all (-q) v hv).sub contDiffAt_const)).add contDiffAt_const
+
+lemma angleBudgetFactor_nonneg (a q cap s C : ℝ) (hC : 0≤C)
+    (v : State) (hc : physicalClock a cap v ≤ s) :
+    0≤angleBudgetFactor a q cap s C v := by
+  have hp := physicalTheta_mem v
+  have hF : 0<angleLift (sinePower (-q)) v :=
+    Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2) _
+  have hE : 0<clockExp a cap 1 v := Real.exp_pos _
+  have hEs : clockExp a cap 1 v≤Real.exp s := by
+    exact Real.exp_le_exp.mpr (by simpa only [one_mul] using hc)
+  have hh := mul_nonneg hC (sub_nonneg.mpr hEs)
+  have hh' := mul_pos hE hF
+  dsimp only [angleBudgetFactor]
+  nlinarith only [hh,hh']
+
+lemma angleBudgetFactor_terminal (a q cap s C : ℝ) (v : State)
+    (hc : physicalClock a cap v=s) :
+    angleBudgetFactor a q cap s C v = Real.exp s * angleLift (sinePower (-q)) v := by
+  unfold angleBudgetFactor clockExp
+  rw [hc,one_mul]
+  ring
+
+lemma green_angleBudget_generator (a q cap s C : ℝ) (ha : a≠0)
+    (v : State) (hv : 0<Y v) :
+    L (fun w => physicalGreen a w * angleBudgetFactor a q cap s C w)
+      (physicalDrift a v) noise v =
+      physicalGreen a v * clockExp a cap 1 v *
+        (sinePower (-q) (physicalTheta v) +
+          angleGenerator (2*a) (sinePower (-q)) (physicalTheta v) - C)*(physicalV v)^2 := by
+  have hp := physicalTheta_mem v
+  have hs := sinePower_smooth (-q) (physicalTheta v)
+    (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2)
+  have hG : ContDiffAt ℝ 2 (physicalGreen a) v :=
+    (contDiffAt_physicalGreen a v hv).of_le (by norm_cast)
+  have he : (fun w => physicalGreen a w * angleBudgetFactor a q cap s C w) =
+      (fun w => physicalGreen a w * (clockExp a cap 1 w * angleLift (sinePower (-q)) w) +
+        (-C)*(physicalGreen a w * (clockExp a cap 1 w * angleLift (fun _ => (1:ℝ)) w)) +
+        (C*Real.exp s)*physicalGreen a w) := by
+    funext w
+    dsimp only [angleBudgetFactor,angleLift]
+    ring
+  have hz : L (physicalGreen a) (physicalDrift a v) noise v = 0 :=
+    green_generator_zero a ha v hv
+  rw [he,L_add ((hG.mul ((clockExp_smooth a cap 1 v hv).mul (angleLift_smooth _ v hv hs))).add
+      (contDiffAt_const.mul (hG.mul ((clockExp_smooth a cap 1 v hv).mul
+        (angleLift_smooth _ v hv contDiffAt_const))))) (contDiffAt_const.mul hG),
+    L_add (hG.mul ((clockExp_smooth a cap 1 v hv).mul (angleLift_smooth _ v hv hs)))
+      (contDiffAt_const.mul (hG.mul ((clockExp_smooth a cap 1 v hv).mul
+        (angleLift_smooth _ v hv contDiffAt_const)))),
+    L_const_mul (hG.mul ((clockExp_smooth a cap 1 v hv).mul
+      (angleLift_smooth _ v hv contDiffAt_const))),L_const_mul hG,
+    green_exponential_angle_generator _ a cap 1 ha v hv hs,
+    green_exponential_angle_generator _ a cap 1 ha v hv contDiffAt_const,
+    hz]
+  have h1 : deriv (fun _ : ℝ => (1:ℝ)) = fun _ => 0 := funext fun _ => deriv_const _ _
+  simp only [angleGenerator,h1,deriv_const]
+  ring
+
+lemma green_angleBudget_generator_nonpos (a q cap s C : ℝ) (ha : a≠0)
+    (hC : ∀ θ∈Ioo 0 Real.pi,
+      angleGenerator (2*a) (sinePower (-q)) θ+sinePower (-q) θ≤C)
+    (v : State) (hv : 0<Y v) :
+    L (fun w => physicalGreen a w * angleBudgetFactor a q cap s C w)
+      (physicalDrift a v) noise v ≤0 := by
+  rw [green_angleBudget_generator a q cap s C ha v hv]
+  have hh := hC (physicalTheta v) (physicalTheta_mem v)
+  apply mul_nonpos_of_nonpos_of_nonneg _ (sq_nonneg _)
+  exact mul_nonpos_of_nonneg_of_nonpos (mul_nonneg (Real.exp_pos _).le (Real.exp_pos _).le)
+    (by linarith)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem compact_test_stopped_dynkin
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
+    (T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hTB : ∀ b, T b ≤ B) :
+    Integrable (fun b => f (X (T b) b)) W ∧
+    Integrable (fun b => ∫ u in (0 : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) W ∧
+    (∫ b, f (X (T b) b) ∂W) = (∫ b, f (X 0 b) ∂W) +
+      ∫ b, (∫ u in (0 : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) ∂W := by
+  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
+    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
+  have hM : Martingale M rawFiltration W :=
+    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
+  have hgc (b : Driver) : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
+    (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
+  have hmc (b : Driver) : Continuous (fun t => M t b) :=
+    (hf.continuous.comp (hXc b)).sub
+      ((intervalIntegral.continuous_primitive (fun s t => (hgc b).intervalIntegrable s t) 0).comp NNReal.continuous_coe)
+  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
+  have hmb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
+    calc
+      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
+      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
+        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
+          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
+            (fun u _ => hgb (X u.toNNReal b)))
+      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
+  have hFa : StronglyAdapted rawFiltration (fun t b => f (X t b)) := fun t =>
+    (hf.continuous.measurable.comp (hXm t)).stronglyMeasurable
+  have hFm : Measurable (fun b => f (X (T b) b)) :=
+    (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
+      (fun b => hf.continuous.comp (hXc b))) hT).mono hT.measurableSpace_le le_rfl
+  have hFi : Integrable (fun b => f (X (T b) b)) W :=
+    Integrable.of_bound hFm.aestronglyMeasurable C (Eventually.of_forall fun b => hfb _)
+  have hMi : Integrable (fun b => M (T b) b) W :=
+    Integrable.of_bound (ExactSLEOptional.stopped_measurable rawFiltration hM hmc T hT).aestronglyMeasurable
+      (C+D*(B+1)) (Eventually.of_forall fun b => hmb _ ((hTB b).trans (le_add_of_nonneg_right zero_le_one)) b)
+  have hi : Integrable (fun b => ∫ u in (0 : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) W := by
+    convert hFi.sub hMi using 1
+    funext b
+    dsimp [M]
+    ring
+  have heq := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hmc B
+    (C+D*(B+1)) hmb (fun _ => 0) T (isStoppingTime_const _ _) hT (fun _ => bot_le) hTB univ MeasurableSet.univ
+  simp only [setIntegral_univ, M, NNReal.coe_zero, intervalIntegral.integral_same, sub_zero] at heq
+  rw [integral_sub hFi hi] at heq
+  exact ⟨hFi, hi, by linarith⟩
+
+end ExactSLEGenerator
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEGenerator
+
+noncomputable def lifetimeCutoff (a ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
+  greenCutoff a ε N v * lifetimeTest (lifetimeExponent a) v
+
+lemma lifetimeCutoff_smooth (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (lifetimeCutoff a ε N) := by
+  rw [contDiff_iff_contDiffAt]
+  intro v
+  by_cases hv : 0 < Y v
+  · exact (greenCutoff_smooth a ε hε N).contDiffAt.mul (lifetimeTest_smooth _ v hv)
+  · apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
+    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
+    simp only [lifetimeCutoff, greenCutoff, heightTurn_zero hε (le_of_lt hw),
+      mul_zero, zero_mul]
+
+lemma lifetimeCutoff_compact (a ε : ℝ) (N : ℝ≥0) :
+    HasCompactSupport (lifetimeCutoff a ε N) := (greenCutoff_compact a ε N).mul_right
+
+lemma lifetimeCutoff_eq_near (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0)
+    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
+    lifetimeCutoff a ε N =ᶠ[𝓝 v]
+      fun w => physicalGreen a w * lifetimeTest (lifetimeExponent a) w :=
+  (greenCutoff_eq_near a ε hε N v hv hy).mul (EventuallyEq.refl _ _)
+
+noncomputable def lifetimeCost (a : ℝ) (v : State) : ℝ :=
+  lifetimeDrift a * physicalGreen a v * Q v ^ (lifetimeExponent a / 2 - 1)
+
+lemma lifetimeCost_pos {a : ℝ} (ha : 1/4 < a) {v : State} (hv : 0 < Y v) :
+    0 < lifetimeCost a v :=
+  mul_pos (mul_pos (lifetimeDrift_pos ha) (Real.exp_pos _)) (Real.rpow_pos_of_pos (Q_pos hv) _)
+
+lemma lifetimeCost_continuousAt (a : ℝ) (v : State) (hv : 0 < Y v) :
+    ContinuousAt (lifetimeCost a) v := by
+  have hQ : ContDiffAt ℝ 1 Q v :=
+    ((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt
+  exact (continuousAt_const.mul (contDiffAt_physicalGreen a v hv).continuousAt).mul
+    ((hQ.rpow_const_of_ne (Q_pos hv).ne').continuousAt)
+
+lemma lifetimeCutoff_generator_le (a ε : ℝ) (ha : 1/4 < a) (hε : 0 < ε) (N : ℝ≥0)
+    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
+    generator (lifetimeCutoff a ε N) (cutoffDrift a ε) noise v ≤ -lifetimeCost a v := by
+  have he := lifetimeCutoff_eq_near a ε hε N v hv hy
+  have hd : (fun w => fderiv ℝ (lifetimeCutoff a ε N) w noise) =ᶠ[𝓝 v]
+      (fun w => fderiv ℝ (fun u => physicalGreen a u * lifetimeTest (lifetimeExponent a) u) w noise) :=
+    he.fderiv.mono (fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw)
+  change (1/2:ℝ) * fderiv ℝ (fun w => fderiv ℝ (lifetimeCutoff a ε N) w noise) v noise +
+    fderiv ℝ (lifetimeCutoff a ε N) v (cutoffDrift a ε v) ≤ _
+  rw [hd.fderiv_eq, he.fderiv_eq, cutoffDrift_eq_physical a ε v hy]
+  convert green_lifetime_generator_le a ha v (hε.trans_le hy) using 1 <;> first | rfl | (dsimp [lifetimeCost]; ring)
+
+ 
+
+theorem local_lifetime_budget (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (a ε : ℝ) (ha : 1/4 < a) (hε : 0 < ε) (z : ℂ) (N B : ℝ≥0)
+    (T : Driver → ℝ≥0)
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hTB : ∀ b, T b ≤ B)
+    (hsafe : ∀ b t, t ≤ T b → ε ≤ Y (localState a ε hε z t b) ∧
+      ‖localState a ε hε z t b‖ ≤ N) :
+    Integrable (fun b => physicalGreen a (localState a ε hε z (T b) b) *
+      lifetimeTest (lifetimeExponent a) (localState a ε hε z (T b) b)) W ∧
+    (∫ b, physicalGreen a (localState a ε hε z (T b) b) *
+      lifetimeTest (lifetimeExponent a) (localState a ε hε z (T b) b) ∂W) ≤
+      physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) ∧
+    (∫⁻ b, ∫⁻ u in Ioc (0 : ℝ) (T b : ℝ),
+      ENNReal.ofReal (lifetimeCost a (localState a ε hε z u.toNNReal b)) ∂volume ∂W) ≤
+      ENNReal.ofReal (physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0)) := by
+  let V := localState a ε hε z
+  let f := lifetimeCutoff a ε N
+  let g := generator f (cutoffDrift a ε) noise
+  have hf := lifetimeCutoff_smooth a ε hε N
+  have hfc := lifetimeCutoff_compact a ε N
+  have hgc : Continuous g := generator_continuous f hf _ (cutoffDrift_lipschitz a ε hε).continuous noise
+  have hVc (b : Driver) : Continuous (fun u : ℝ => V u.toNNReal b) :=
+    (localState_continuous a ε hε z b).comp continuous_real_toNNReal
+  have he (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) :
+      f (V t b) = physicalGreen a (V t b) * lifetimeTest (lifetimeExponent a) (V t b) :=
+    (lifetimeCutoff_eq_near a ε hε N _ (hsafe b t ht).2 (hsafe b t ht).1).eq_of_nhds
+  have hfpos (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) : 0 ≤ f (V t b) := by
+    rw [he b t ht]
+    exact mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg _ _ (hε.le.trans (hsafe b t ht).1))
+  have hgen (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) :
+      g (V t b) ≤ -lifetimeCost a (V t b) :=
+    lifetimeCutoff_generator_le a ε ha hε N _ (hsafe b t ht).2 (hsafe b t ht).1
+  have hpos (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) : 0 < lifetimeCost a (V t b) :=
+    lifetimeCost_pos ha (hε.trans_le (hsafe b t ht).1)
+  have hDy := compact_test_stopped_dynkin W hW V
+    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
+    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
+    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
+    (by simp [noise, Prod.norm_def]) (localState_eq a ε hε z) f hf hfc T B hT hTB
+  let H : Driver → ℝ := fun b => -(∫ u in (0 : ℝ)..(T b : ℝ), g (V u.toNNReal b))
+  have hHi : Integrable H W := hDy.2.1.neg
+  have hHnonneg (b : Driver) : 0 ≤ H b := by
+    dsimp [H]
+    apply neg_nonneg.mpr
+    rw [intervalIntegral.integral_of_le (T b).coe_nonneg]
+    apply setIntegral_nonpos measurableSet_Ioc
+    intro u hu
+    have ht : u.toNNReal ≤ T b := by simpa using Real.toNNReal_le_toNNReal hu.2
+    exact (hgen b u.toNNReal ht).trans (neg_nonpos.mpr (hpos b u.toNNReal ht).le)
+  have hHbound : (∫ b, H b ∂W) ≤ physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) := by
+    have hinit (b : Driver) : f (V 0 b) = physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) := by
+      rw [he b 0 (bot_le)]
+      simp only [V,localState_zero]
+    have hd := hDy.2.2
+    simp_rw [hinit] at hd
+    simp only [integral_const, probReal_univ, one_smul] at hd
+    have hn : 0 ≤ ∫ b, f (V (T b) b) ∂W := integral_nonneg (fun b => hfpos b (T b) le_rfl)
+    change (∫ b, -(∫ u in (0 : ℝ)..(T b : ℝ), g (V u.toNNReal b)) ∂W) ≤ _
+    rw [integral_neg]
+    linarith
+  refine ⟨?_, ?_, ?_⟩
+  · simpa only [he _ _ le_rfl] using hDy.1
+  · have hn : (∫ b, H b ∂W) ≥ 0 := integral_nonneg hHnonneg
+    have hd := hDy.2.2
+    have hinit (b : Driver) : f (V 0 b) = physicalGreen a (z,0) * lifetimeTest (lifetimeExponent a) (z,0) := by
+      rw [he b 0 (bot_le)]
+      simp only [V,localState_zero]
+    simp_rw [hinit, he _ _ le_rfl] at hd
+    simp only [integral_const, probReal_univ, one_smul] at hd
+    dsimp only [H] at hn
+    rw [integral_neg] at hn
+    change (∫ b, physicalGreen a (V (T b) b) * lifetimeTest (lifetimeExponent a) (V (T b) b) ∂W) ≤ _
+    linarith
+  · suffices hp : ∀ b, (∫⁻ u in Ioc (0 : ℝ) (T b : ℝ), ENNReal.ofReal (lifetimeCost a (V u.toNNReal b))) ≤ ENNReal.ofReal (H b) by
+      calc
+        _ ≤ ∫⁻ b, ENNReal.ofReal (H b) ∂W := lintegral_mono hp
+        _ = ENNReal.ofReal (∫ b, H b ∂W) := (ofReal_integral_eq_lintegral_ofReal hHi (Eventually.of_forall hHnonneg)).symm
+        _ ≤ _ := ENNReal.ofReal_le_ofReal hHbound
+    intro b
+    have ht (u : ℝ) (hu : u ∈ Icc 0 (T b : ℝ)) : u.toNNReal ≤ T b := by
+      simpa using Real.toNNReal_le_toNNReal hu.2
+    have hcostc : ContinuousOn (fun u : ℝ => lifetimeCost a (V u.toNNReal b)) (Icc 0 (T b : ℝ)) := by
+      intro u hu
+      exact ((lifetimeCost_continuousAt a _ (hε.trans_le (hsafe b u.toNNReal (ht u hu)).1)).comp (f := fun u : ℝ => V u.toNNReal b) (hVc b).continuousAt).continuousWithinAt
+    have hci : IntervalIntegrable (fun u : ℝ => lifetimeCost a (V u.toNNReal b)) volume 0 (T b) :=
+      hcostc.intervalIntegrable_of_Icc (T b).coe_nonneg
+    have hgi : IntervalIntegrable (fun u : ℝ => g (V u.toNNReal b)) volume 0 (T b) := (hgc.comp (hVc b)).intervalIntegrable 0 (T b)
+    have hcb : (∫ u in (0 : ℝ)..(T b : ℝ), lifetimeCost a (V u.toNNReal b)) ≤ H b := by
+      dsimp only [H]
+      rw [← intervalIntegral.integral_neg]
+      apply intervalIntegral.integral_mono_on (T b).coe_nonneg hci hgi.neg
+      intro u hu
+      dsimp only [Pi.neg_apply]
+      linarith only [hgen b u.toNNReal (ht u hu)]
+    rw [intervalIntegral.integral_of_le (T b).coe_nonneg] at hcb
+    have hn : 0 ≤ᵐ[volume.restrict (Ioc (0 : ℝ) (T b : ℝ))] (fun u => lifetimeCost a (V u.toNNReal b)) := by
+      filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
+      exact (hpos b u.toNNReal (ht u ⟨hu.1.le,hu.2⟩)).le
+    rw [← ofReal_integral_eq_lintegral_ofReal hci.1 hn]
+    exact ENNReal.ofReal_le_ofReal hcb
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
+
+def angleBudgetCutoff (a q cap s C ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
+  greenCutoff a ε N v * angleBudgetFactor a q cap s C v
+
+lemma angleBudgetCutoff_smooth (a q cap s C ε : ℝ) (hε : 0<ε) (N : ℝ≥0) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (angleBudgetCutoff a q cap s C ε N) := by
+  rw [contDiff_iff_contDiffAt]
+  intro v
+  by_cases hv : 0<Y v
+  · exact (greenCutoff_smooth a ε hε N).contDiffAt.mul
+      (angleBudgetFactor_smooth a q cap s C v hv)
+  · apply (contDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq
+    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
+    simp only [angleBudgetCutoff, greenCutoff, heightTurn_zero hε (le_of_lt hw),
+      mul_zero, zero_mul]
+
+lemma angleBudgetCutoff_compact (a q cap s C ε : ℝ) (N : ℝ≥0) :
+    HasCompactSupport (angleBudgetCutoff a q cap s C ε N) :=
+  (greenCutoff_compact a ε N).mul_right
+
+lemma angleBudgetCutoff_eq_near (a q cap s C ε : ℝ) (hε : 0<ε) (N : ℝ≥0)
+    (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
+    angleBudgetCutoff a q cap s C ε N =ᶠ[𝓝 v]
+      fun w => physicalGreen a w * angleBudgetFactor a q cap s C w :=
+  (greenCutoff_eq_near a ε hε N v hv hy).mul (EventuallyEq.refl _ _)
+
+lemma angleBudgetCutoff_generator_nonpos (a q cap s C ε : ℝ) (ha : a≠0)
+    (hC : ∀ θ∈Ioo 0 Real.pi,
+      angleGenerator (2*a) (sinePower (-q)) θ + sinePower (-q) θ ≤ C)
+    (hε : 0<ε) (N : ℝ≥0) (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
+    generator (angleBudgetCutoff a q cap s C ε N) (cutoffDrift a ε) noise v ≤ 0 := by
+  have he := angleBudgetCutoff_eq_near a q cap s C ε hε N v hv hy
+  have hd : (fun w => fderiv ℝ (angleBudgetCutoff a q cap s C ε N) w noise) =ᶠ[𝓝 v]
+      (fun w => fderiv ℝ (fun u => physicalGreen a u * angleBudgetFactor a q cap s C u) w noise) :=
+    he.fderiv.mono (fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw)
+  change (1/2:ℝ) * fderiv ℝ (fun w => fderiv ℝ (angleBudgetCutoff a q cap s C ε N) w noise) v noise +
+    fderiv ℝ (angleBudgetCutoff a q cap s C ε N) v (cutoffDrift a ε v) ≤ 0
+  rw [hd.fderiv_eq, he.fderiv_eq, cutoffDrift_eq_physical a ε v hy]
+  exact green_angleBudget_generator_nonpos a q cap s C ha hC v (hε.trans_le hy)
+
+ 
+
+theorem local_angle_budget (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (a q cap s C ε : ℝ) (ha : a≠0)
+    (hC : ∀ θ∈Ioo 0 Real.pi,
+      angleGenerator (2*a) (sinePower (-q)) θ + sinePower (-q) θ ≤ C)
+    (hε : 0<ε) (z : ℂ) (N B : ℝ≥0) (T : Driver → ℝ≥0)
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hTB : ∀ b, T b≤B)
+    (hsafe : ∀ b t, t≤T b → ε≤Y (localState a ε hε z t b) ∧
+      ‖localState a ε hε z t b‖≤N) :
+    Integrable (fun b => physicalGreen a (localState a ε hε z (T b) b) *
+      angleBudgetFactor a q cap s C (localState a ε hε z (T b) b)) W ∧
+    (∫ b, physicalGreen a (localState a ε hε z (T b) b) *
+      angleBudgetFactor a q cap s C (localState a ε hε z (T b) b) ∂W) ≤
+      physicalGreen a (z,0) * angleBudgetFactor a q cap s C (z,0) := by
+  let V := localState a ε hε z
+  let f := angleBudgetCutoff a q cap s C ε N
+  let g := generator f (cutoffDrift a ε) noise
+  have hf := angleBudgetCutoff_smooth a q cap s C ε hε N
+  have hfc := angleBudgetCutoff_compact a q cap s C ε N
+  have he (b : Driver) (t : ℝ≥0) (ht : t≤T b) :
+      f (V t b) = physicalGreen a (V t b) * angleBudgetFactor a q cap s C (V t b) :=
+    (angleBudgetCutoff_eq_near a q cap s C ε hε N _ (hsafe b t ht).2
+      (hsafe b t ht).1).eq_of_nhds
+  have hgen (b : Driver) (t : ℝ≥0) (ht : t≤T b) : g (V t b)≤0 :=
+    angleBudgetCutoff_generator_nonpos a q cap s C ε ha hC hε N _
+      (hsafe b t ht).2 (hsafe b t ht).1
+  have hDy := compact_test_stopped_dynkin W hW V
+    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
+    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
+    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
+    (by simp [noise, Prod.norm_def]) (localState_eq a ε hε z) f hf hfc T B hT hTB
+  have hneg : (∫ b, (∫ u in (0:ℝ)..(T b : ℝ), g (V u.toNNReal b)) ∂W) ≤ 0 := by
+    apply integral_nonpos
+    intro b
+    change (∫ u in (0:ℝ)..(T b : ℝ), g (V u.toNNReal b)) ≤ 0
+    rw [intervalIntegral.integral_of_le (T b).coe_nonneg]
+    apply setIntegral_nonpos measurableSet_Ioc
+    intro u hu
+    exact hgen b u.toNNReal (by simpa using Real.toNNReal_le_toNNReal hu.2)
+  have hinit (b : Driver) : f (V 0 b) = physicalGreen a (z,0) * angleBudgetFactor a q cap s C (z,0) := by
+    rw [he b 0 bot_le]
+    simp only [V,localState_zero]
+  refine ⟨by simpa only [he _ _ le_rfl] using hDy.1, ?_⟩
+  have hd := hDy.2.2
+  simp_rw [hinit,he _ _ le_rfl] at hd
+  simp only [integral_const,probReal_univ,one_smul] at hd
+  change (∫ b, physicalGreen a (V (T b) b)*angleBudgetFactor a q cap s C (V (T b) b) ∂W)≤_
+  linarith
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/AngleMeasure.lean b/lean/OAI/Probability/SLEGauge/AngleMeasure.lean
index a3e5d2c9e8daee3d26850dc966d050e66a0f8eb6..9795bf65d2e2cc564649107bee07919d8e85733c 100644
--- a/lean/OAI/Probability/SLEGauge/AngleMeasure.lean
+++ b/lean/OAI/Probability/SLEGauge/AngleMeasure.lean
@@ -1,573 +1,573 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ConformalFlow
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen
-
-abbrev Angle := Ioo (0:ℝ) Real.pi
-
-def angleArea : Measure Angle := volume.comap (Subtype.val : Angle→ℝ)
-
-def angleWeight (a : ℝ) (θ : Angle) : ℝ≥0∞ := ENNReal.ofReal ((Real.sin θ)^(4*a))
-
-def angleNormalization (a : ℝ) : ℝ≥0∞ := ∫⁻θ,angleWeight a θ∂angleArea
-
- 
-
-def angleLaw (a : ℝ) : Measure Angle :=
-  (angleNormalization a)⁻¹ • angleArea.withDensity (angleWeight a)
-
-lemma angleArea_univ : angleArea univ=ENNReal.ofReal Real.pi := by
-  rw [angleArea,MeasurableEmbedding.comap_apply
-    (MeasurableEmbedding.subtype_coe measurableSet_Ioo) _ univ]
-  rw [show (Subtype.val : Angle→ℝ) '' univ=Ioo (0:ℝ) Real.pi by ext x; simp,
-    Real.volume_Ioo,sub_zero]
-
-instance : IsFiniteMeasure angleArea := ⟨by rw [angleArea_univ]; exact ENNReal.ofReal_lt_top⟩
-
-lemma angleArea_ne_zero : angleArea univ≠0 := by
-  rw [angleArea_univ]
-  exact (ENNReal.ofReal_pos.mpr Real.pi_pos).ne'
-
-lemma angleWeight_measurable (a : ℝ) : Measurable (angleWeight a) := by
-  unfold angleWeight
-  fun_prop
-
-lemma angleWeight_pos (a : ℝ) (θ : Angle) : 0<angleWeight a θ :=
-  ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) _)
-
-lemma angleWeight_le_one (a : ℝ) (ha : 0≤a) (θ : Angle) : angleWeight a θ≤1 := by
-  rw [←ENNReal.ofReal_one]
-  exact ENNReal.ofReal_le_ofReal (Real.rpow_le_one
-    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le (Real.sin_le_one θ) (by positivity))
-
-lemma angleNormalization_pos (a : ℝ) : 0<angleNormalization a := by
-  rw [angleNormalization,lintegral_pos_iff_support (angleWeight_measurable a)]
-  have hs : Function.support (angleWeight a)=univ := by
-    ext θ
-    simp [Function.mem_support,(angleWeight_pos a θ).ne']
-  rw [hs]
-  exact bot_lt_iff_ne_bot.mpr angleArea_ne_zero
-
-lemma angleNormalization_lt_top (a : ℝ) (ha : 0≤a) : angleNormalization a<∞ := by
-  calc
-    _ ≤ ∫⁻_ : Angle,(1:ℝ≥0∞)∂angleArea := lintegral_mono (angleWeight_le_one a ha)
-    _ < ∞ := by simp [angleArea_univ]
-
-lemma angleLaw_probability (a : ℝ) (ha : 0≤a) : IsProbabilityMeasure (angleLaw a) := by
-  rw [isProbabilityMeasure_iff]
-  rw [angleLaw,Measure.smul_apply,smul_eq_mul,withDensity_apply _ MeasurableSet.univ,
-    Measure.restrict_univ]
-  exact ENNReal.inv_mul_cancel (angleNormalization_pos a).ne' (angleNormalization_lt_top a ha).ne
-
-lemma angleLaw_integral (a : ℝ) (f : Angle→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻θ,f θ∂angleLaw a)=(angleNormalization a)⁻¹*
-      ∫⁻θ,angleWeight a θ*f θ∂angleArea := by
-  rw [angleLaw,lintegral_smul_measure,smul_eq_mul,
-    lintegral_withDensity_eq_lintegral_mul _ (angleWeight_measurable a) hf]
-  rfl
-
-lemma angleWeight_cancel (a : ℝ) (θ : Angle) :
-    angleWeight a θ*ENNReal.ofReal ((Real.sin θ)^(-p a))=
-      ENNReal.ofReal (Real.sin θ) := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
-  rw [angleWeight,←ENNReal.ofReal_mul (Real.rpow_nonneg hs.le _),←Real.rpow_add hs]
-  simp [p]
-
- 
-lemma angle_inverse_power_finite (a : ℝ) :
-    (∫⁻θ : Angle,ENNReal.ofReal ((Real.sin θ)^(-p a))∂angleLaw a)<∞ := by
-  rw [angleLaw_integral a _ (by fun_prop)]
-  simp_rw [angleWeight_cancel]
-  apply ENNReal.mul_lt_top
-  · exact ENNReal.inv_lt_top.mpr (angleNormalization_pos a)
-  · calc
-      _ ≤ ∫⁻_ : Angle,(1:ℝ≥0∞)∂angleArea := by
-        apply lintegral_mono
-        intro θ
-        rw [←ENNReal.ofReal_one]
-        exact ENNReal.ofReal_le_ofReal (Real.sin_le_one θ)
-      _ < ∞ := by simp [angleArea_univ]
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Polynomial
-
-namespace SLEExactGauge.Jacobi
-
- 
-def operator (nu : ℝ) (p : ℝ[X]) : ℝ[X] :=
-  (1-X^2)*p.derivative.derivative - C (1+nu)*X*p.derivative
-
- 
-def raise (nu n : ℝ) (p : ℝ[X]) : ℝ[X] :=
-  C (1/(2*n+nu)) * (C (n+nu)*X*p - (1-X^2)*p.derivative)
-
-lemma raise_derivative (nu n : ℝ) (p : ℝ[X])
-    (hp : operator nu p = -C (n*(n+nu))*p) :
-    (raise nu n p).derivative =
-      C (1/(2*n+nu))*C (n+1)*(C (n+nu)*p+X*p.derivative) := by
-  dsimp only [raise,operator] at *
-  simp only [derivative_mul,derivative_sub,derivative_C,derivative_X,derivative_one,
-    derivative_pow,zero_mul,zero_sub,zero_add]
-  simp only [map_mul,map_add,map_one,Nat.cast_ofNat,map_ofNat] at hp ⊢
-  linear_combination -C (1/(2*n+nu)) * hp
-
-lemma raise_eigen (nu n : ℝ) (p : ℝ[X])
-    (hp : operator nu p = -C (n*(n+nu))*p) :
-    operator nu (raise nu n p) = -C ((n+1)*(n+1+nu)) * raise nu n p := by
-  have hd := raise_derivative nu n p hp
-  have hd2 := congrArg Polynomial.derivative hd
-  simp only [derivative_mul,derivative_add,derivative_C,derivative_X,
-    zero_mul,mul_zero,zero_add,one_mul,add_zero] at hd2
-  rw [operator,hd2,hd]
-  dsimp only [raise,operator] at *
-  simp only [map_mul,map_add,map_one] at hp ⊢
-  linear_combination C (1/(2*n+nu))*(C n+1)*X * hp
-
-lemma raise_lowering (nu n : ℝ) (p : ℝ[X])
-    (hp : operator nu p = -C (n*(n+nu))*p) :
-    (1-X^2)*(raise nu n p).derivative + C (n+1)*X*raise nu n p =
-      C ((n+1)*(n+nu)/(2*n+nu))*p := by
-  rw [raise_derivative nu n p hp]
-  simp only [raise,map_mul,map_add,map_one,div_eq_mul_inv]
-  ring
-
- 
-def poly (nu : ℝ) : ℕ → ℝ[X]
-  | 0 => 1
-  | n+1 => raise nu n (poly nu n)
-
-@[simp] lemma poly_zero (nu : ℝ) : poly nu 0=1 := rfl
-@[simp] lemma poly_succ (nu : ℝ) (n : ℕ) :
-    poly nu (n+1)=raise nu n (poly nu n) := rfl
-
-lemma poly_eigen (nu : ℝ) (n : ℕ) :
-    operator nu (poly nu n) = -C ((n:ℝ)*(n+nu))*poly nu n := by
-  induction n with
-  | zero => simp [operator]
-  | succ n ih => simpa only [poly_succ,Nat.cast_add,Nat.cast_one] using
-      raise_eigen nu n (poly nu n) ih
-
-lemma poly_lowering (nu : ℝ) (n : ℕ) :
-    (1-X^2)*(poly nu (n+1)).derivative + C ((n:ℝ)+1)*X*poly nu (n+1) =
-      C (((n:ℝ)+1)*(n+nu)/(2*n+nu))*poly nu n :=
-  raise_lowering nu n (poly nu n) (poly_eigen nu n)
-
-lemma raise_eq_X_sub (nu n : ℝ) (p : ℝ[X]) (h : 2*n+nu≠0) :
-    raise nu n p = X*p - C (1/(2*n+nu))*((1-X^2)*p.derivative+C n*X*p) := by
-  have hc : C (1/(2*n+nu))*(2*C n+C nu)=(1:ℝ[X]) := by
-    have he : (2*C n+C nu : ℝ[X])=C (2*n+nu) := by simp only [map_add,map_mul,map_ofNat]
-    rw [he,←map_mul,one_div_mul_cancel h,map_one]
-  simp only [raise,map_add]
-  linear_combination X*p*hc
-
-lemma poly_one (nu : ℝ) (hnu : nu≠0) : poly nu 1=X := by
-  rw [show 1=0+1 from rfl,poly_succ,poly_zero,Nat.cast_zero,raise_eq_X_sub nu 0 1 (by simpa using hnu)]
-  simp
-
- 
-def beta (nu : ℝ) (n : ℕ) : ℝ :=
-  ((n:ℝ)+1)*(n+nu)/((2*n+nu+2)*(2*n+nu))
-
-lemma poly_recurrence (nu : ℝ) (hnu : 0<nu) (n : ℕ) :
-    poly nu (n+2)=X*poly nu (n+1)-C (beta nu n)*poly nu n := by
-  have hd : 2*((n:ℝ)+1)+nu≠0 := ne_of_gt (by positivity)
-  change raise nu ((n+1:ℕ):ℝ) (poly nu (n+1))=_
-  rw [Nat.cast_add,Nat.cast_one,raise_eq_X_sub nu ((n:ℝ)+1) _ hd,poly_lowering]
-  congr 1
-  rw [←mul_assoc,←map_mul]
-  congr 2
-  unfold beta
-  field_simp
-  ring
-
-lemma beta_pos (nu : ℝ) (hnu : 0<nu) (n : ℕ) : 0<beta nu n := by
-  unfold beta
-  positivity
-
-lemma beta_le_one (nu : ℝ) (hnu : 0<nu) (n : ℕ) : beta nu n≤1 := by
-  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
-  unfold beta
-  apply (div_le_one (by positivity)).mpr
-  nlinarith [sq_nonneg ((n:ℝ)),sq_nonneg nu,mul_nonneg hn hnu.le]
-
-lemma beta_lower (nu : ℝ) (hnu : 1≤nu) (n : ℕ) :
-    1/(nu+2)^2≤beta nu n := by
-  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
-  have hnu0 : 0<nu := lt_of_lt_of_le zero_lt_one hnu
-  have h1 : 0<(nu+2)^2 := by positivity
-  have h2 : 0<(2*(n:ℝ)+nu+2)*(2*n+nu) := by positivity
-  have hden : (2*(n:ℝ)+nu+2)*(2*n+nu)≤(nu+2)^2*((n:ℝ)+1)^2 := by
-    have hb1 : 2*(n:ℝ)+nu+2≤(nu+2)*((n:ℝ)+1) := by
-      nlinarith [mul_nonneg hnu0.le hn]
-    have hb2 : 2*(n:ℝ)+nu≤(nu+2)*((n:ℝ)+1) := by linarith
-    have hh := mul_le_mul hb1 hb2 (by positivity : 0≤2*(n:ℝ)+nu)
-      (by positivity : 0≤(nu+2)*((n:ℝ)+1))
-    nlinarith only [hh]
-  have hnum : ((n:ℝ)+1)^2≤((n:ℝ)+1)*(n+nu) := by nlinarith
-  rw [beta,div_le_div_iff₀ h1 h2]
-  have hm := mul_le_mul_of_nonneg_left hnum h1.le
-  nlinarith only [hden,hm]
-
-lemma poly_eval_bound (nu : ℝ) (hnu : 0<nu) (n : ℕ) (x : ℝ) (hx : |x|≤1) :
-    |(poly nu n).eval x|≤2^n := by
-  induction n using Nat.twoStepInduction with
-  | zero => simp
-  | one =>
-    rw [poly_one nu hnu.ne',eval_X,pow_one]
-    exact hx.trans (by norm_num)
-  | more n hn hn1 =>
-    rw [poly_recurrence nu hnu n,eval_sub,eval_mul,eval_X,eval_mul,eval_C]
-    calc
-      _ ≤ |x*(poly nu (n+1)).eval x|+|beta nu n*(poly nu n).eval x| := abs_sub _ _
-      _ = |x| * |(poly nu (n+1)).eval x|+beta nu n * |(poly nu n).eval x| := by
-        rw [abs_mul,abs_mul,abs_of_pos (beta_pos nu hnu n)]
-      _ ≤ 2^(n+1)+2^n := add_le_add
-        ((mul_le_mul hx hn1 (abs_nonneg _) zero_le_one).trans_eq (one_mul _))
-        ((mul_le_mul (beta_le_one nu hnu n) hn (abs_nonneg _) zero_le_one).trans_eq (one_mul _))
-      _ ≤ 2^(n+2) := by
-        simp only [pow_add,pow_one]
-        norm_num
-        nlinarith [show (0:ℝ)≤2^n by positivity]
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open Polynomial MeasureTheory Set
-open scoped Interval
-
-namespace SLEExactGauge.Jacobi
-
-def weight (nu x : ℝ) : ℝ := (1-x^2)^((nu-1)/2)
-def flux (nu x : ℝ) : ℝ := (1-x^2)^((nu+1)/2)
-
-def inner (nu : ℝ) (p q : ℝ[X]) : ℝ :=
-  ∫ x in (-1:ℝ)..1, weight nu x * p.eval x * q.eval x
-
-lemma continuous_weight (nu : ℝ) (hnu : 1≤nu) : Continuous (weight nu) :=
-  (continuous_const.sub (continuous_id.pow 2)).rpow_const (fun _ => Or.inr (by linarith))
-
-lemma continuous_flux (nu : ℝ) (hnu : 1≤nu) : Continuous (flux nu) :=
-  (continuous_const.sub (continuous_id.pow 2)).rpow_const (fun _ => Or.inr (by linarith))
-
-lemma flux_eq (nu : ℝ) (hnu : 1<nu) (x : ℝ) (hx : x∈Icc (-1:ℝ) 1) :
-    flux nu x = (1-x^2)*weight nu x := by
-  have hx0 : 0≤1-x^2 := by nlinarith [mul_nonneg (sub_nonneg.mpr hx.2) (show 0≤1+x by linarith [hx.1])]
-  have he : (nu+1)/2=(nu-1)/2+1 := by ring
-  rw [flux,he,Real.rpow_add' hx0 (by linarith : (nu-1)/2+1≠0),Real.rpow_one,weight]
-  ring
-
-lemma hasDerivAt_flux (nu : ℝ) (hnu : 1≤nu) (x : ℝ) :
-    HasDerivAt (flux nu) (-(nu+1)*x*weight nu x) x := by
-  have hd := ((hasDerivAt_const x (1:ℝ)).sub ((hasDerivAt_id x).pow 2)).rpow_const
-    (Or.inr (by linarith : 1≤(nu+1)/2))
-  convert hd using 1
-  · rfl
-  · dsimp [weight]
-    rw [show (nu+1)/2-1=(nu-1)/2 by ring]
-    ring
-
-lemma inner_comm (nu : ℝ) (p q : ℝ[X]) : inner nu p q=inner nu q p := by
-  apply intervalIntegral.integral_congr
-  intro x hx
-  ring
-
-lemma inner_integrable (nu : ℝ) (hnu : 1≤nu) (p q : ℝ[X]) :
-    IntervalIntegrable (fun x => weight nu x*p.eval x*q.eval x) volume (-1) 1 :=
-  (((continuous_weight nu hnu).mul p.continuous).mul q.continuous).intervalIntegrable _ _
-
-lemma inner_add_left (nu : ℝ) (hnu : 1≤nu) (p q r : ℝ[X]) :
-    inner nu (p+q) r=inner nu p r+inner nu q r := by
-  simp only [inner,eval_add,mul_add,add_mul]
-  exact intervalIntegral.integral_add (inner_integrable nu hnu p r) (inner_integrable nu hnu q r)
-
-lemma inner_sub_left (nu : ℝ) (hnu : 1≤nu) (p q r : ℝ[X]) :
-    inner nu (p-q) r=inner nu p r-inner nu q r := by
-  simp only [inner,eval_sub,mul_sub,sub_mul]
-  exact intervalIntegral.integral_sub (inner_integrable nu hnu p r) (inner_integrable nu hnu q r)
-
-lemma inner_C_mul_left (nu c : ℝ) (p q : ℝ[X]) :
-    inner nu (C c*p) q=c*inner nu p q := by
-  simp only [inner,eval_mul,eval_C]
-  simp_rw [show ∀ x, weight nu x*(c*p.eval x)*q.eval x=c*(weight nu x*p.eval x*q.eval x) by intro x; ring]
-  exact intervalIntegral.integral_const_mul _ _
-
-lemma inner_neg_left (nu : ℝ) (p q : ℝ[X]) :
-    inner nu (-p) q= -inner nu p q := by
-  simp [inner]
-
-lemma inner_X (nu : ℝ) (p q : ℝ[X]) : inner nu (X*p) q=inner nu p (X*q) := by
-  apply intervalIntegral.integral_congr
-  intro x hx
-  simp only [eval_mul,eval_X]
-  ring
-
- 
-lemma operator_symmetric (nu : ℝ) (hnu : 1<nu) (p q : ℝ[X]) :
-    inner nu p (operator nu q)=inner nu (operator nu p) q := by
-  let f (x : ℝ) := flux nu x * (p.eval x*q.derivative.eval x-p.derivative.eval x*q.eval x)
-  let g (x : ℝ) := weight nu x*p.eval x*(operator nu q).eval x-
-    weight nu x*(operator nu p).eval x*q.eval x
-  have hd (x : ℝ) (hx : x∈uIcc (-1:ℝ) 1) : HasDerivAt f (g x) x := by
-    have hxx : x∈Icc (-1:ℝ) 1 := by simpa only [uIcc_of_le (by norm_num : (-1:ℝ)≤1)] using hx
-    have hh := (hasDerivAt_flux nu hnu.le x).mul
-      (((p.hasDerivAt x).mul (q.derivative.hasDerivAt x)).sub
-        ((p.derivative.hasDerivAt x).mul (q.hasDerivAt x)))
-    convert hh using 1
-    try rfl
-    all_goals simp only [g,operator,Pi.mul_apply,Pi.sub_apply,eval_sub,eval_mul,eval_pow,eval_X,eval_one,eval_C,flux_eq nu hnu x hxx]
-    all_goals ring
-  have hi : IntervalIntegrable g volume (-1) 1 :=
-    (inner_integrable nu hnu.le p (operator nu q)).sub
-      (inner_integrable nu hnu.le (operator nu p) q)
-  have hh := intervalIntegral.integral_eq_sub_of_hasDerivAt hd hi
-  have hz : flux nu 1=0 ∧ flux nu (-1)=0 := by
-    simp [flux,Real.zero_rpow (by linarith : (nu+1)/2≠0)]
-  have hg : (∫ x in (-1:ℝ)..1, g x) = inner nu p (operator nu q)-inner nu (operator nu p) q :=
-    intervalIntegral.integral_sub (inner_integrable nu hnu.le p (operator nu q))
-      (inner_integrable nu hnu.le (operator nu p) q)
-  rw [hg] at hh
-  simpa [f,hz.1,hz.2,sub_eq_zero] using hh
-
-lemma eigenvalue_strictMono (nu : ℝ) (hnu : 0<nu) :
-    StrictMono (fun n : ℕ => (n:ℝ)*(n+nu)) := by
-  intro n m hnm
-  have hnm' : (n:ℝ) < m := Nat.cast_lt.mpr hnm
-  nlinarith [Nat.cast_nonneg (α := ℝ) n,Nat.cast_nonneg (α := ℝ) m]
-
- 
-lemma poly_orthogonal (nu : ℝ) (hnu : 1<nu) (n m : ℕ) (hnm : n≠m) :
-    inner nu (poly nu n) (poly nu m)=0 := by
-  have hs := operator_symmetric nu hnu (poly nu n) (poly nu m)
-  simp only [poly_eigen,neg_mul] at hs
-  rw [inner_comm nu (poly nu n),inner_neg_left,inner_C_mul_left,
-    inner_neg_left,inner_C_mul_left,inner_comm nu (poly nu m)] at hs
-  have he := (eigenvalue_strictMono nu (by linarith)).injective.ne hnm
-  have hp : ((n:ℝ)*(n+nu)-(m:ℝ)*(m+nu))*inner nu (poly nu n) (poly nu m)=0 := by linarith
-  exact (mul_eq_zero.mp hp).resolve_left (sub_ne_zero.mpr he)
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open Polynomial MeasureTheory Set
-open scoped Interval
-
-namespace SLEExactGauge.Jacobi
-
-lemma poly_monic_degree (nu : ℝ) (hnu : 0<nu) (n : ℕ) :
-    (poly nu n).Monic ∧ (poly nu n).degree=n := by
-  induction n using Nat.twoStepInduction with
-  | zero => simp only [poly_zero,Nat.cast_zero,monic_one,degree_one,and_self]
-  | one => rw [poly_one nu hnu.ne']; exact ⟨monic_X,degree_X⟩
-  | more n hn hn1 =>
-    have hd : (C (beta nu n)*poly nu n).degree < (X*poly nu (n+1)).degree := by
-      rw [degree_C_mul (beta_pos nu hnu n).ne',degree_mul,degree_X,hn.2,hn1.2]
-      norm_cast
-      omega
-    rw [poly_recurrence nu hnu n]
-    refine ⟨(monic_X.mul hn1.1).sub_of_left hd, ?_⟩
-    rw [degree_sub_eq_left_of_degree_lt hd,degree_mul,degree_X,hn1.2]
-    norm_cast
-    omega
-
- 
-def basis (nu : ℝ) (hnu : 0<nu) : Module.Basis ℕ ℝ ℝ[X] :=
-  let seq : Polynomial.Sequence ℝ := ⟨poly nu, fun n => (poly_monic_degree nu hnu n).2⟩
-  seq.basis (fun n => by
-    change IsUnit (poly nu n).leadingCoeff
-    rw [(poly_monic_degree nu hnu n).1.leadingCoeff]
-    exact isUnit_one)
-
-@[simp] lemma basis_apply (nu : ℝ) (hnu : 0<nu) (n : ℕ) : basis nu hnu n=poly nu n := by
-  simp only [basis,Polynomial.Sequence.basis_eq_self]
-
-def normSq (nu : ℝ) (n : ℕ) : ℝ := inner nu (poly nu n) (poly nu n)
-
-lemma normSq_zero_pos (nu : ℝ) (hnu : 1≤nu) : 0<normSq nu 0 := by
-  change 0<∫ x in (-1:ℝ)..1, weight nu x*(poly nu 0).eval x*(poly nu 0).eval x
-  simp only [poly_zero,eval_one,mul_one]
-  apply intervalIntegral.integral_pos (by norm_num) (continuous_weight nu hnu).continuousOn
-  · intro x hx
-    apply Real.rpow_nonneg
-    nlinarith [mul_nonneg (sub_nonneg.mpr hx.2) (show 0≤1+x by linarith [hx.1])]
-  · refine ⟨0,by norm_num, ?_⟩
-    simp [weight]
-
-lemma normSq_succ (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
-    normSq nu (n+1)=beta nu n*normSq nu n := by
-  have hnu0 : 0<nu := by linarith
-  have hr (j : ℕ) : X*poly nu (j+1)=poly nu (j+2)+C (beta nu j)*poly nu j := by
-    rw [poly_recurrence nu hnu0 j]
-    ring
-  have hl : inner nu (X*poly nu (n+1)) (poly nu n)=beta nu n*normSq nu n := by
-    rw [hr,inner_add_left nu hnu.le,inner_C_mul_left,
-      poly_orthogonal nu hnu (n+2) n (by omega),zero_add]
-    rfl
-  rw [inner_X,inner_comm] at hl
-  rw [←hl]
-  cases n with
-  | zero =>
-    change inner nu (poly nu 1) (poly nu 1)=_
-    rw [poly_zero,mul_one,poly_one nu hnu0.ne']
-  | succ n =>
-    rw [hr n,inner_add_left nu hnu.le,inner_C_mul_left,
-      poly_orthogonal nu hnu n (n+1+1) (by omega),mul_zero,add_zero]
-    rfl
-
-lemma normSq_pos (nu : ℝ) (hnu : 1<nu) (n : ℕ) : 0<normSq nu n := by
-  induction n with
-  | zero => exact normSq_zero_pos nu hnu.le
-  | succ n hn =>
-    rw [normSq_succ nu hnu n]
-    exact mul_pos (beta_pos nu (by linarith) n) hn
-
-lemma normSq_lower (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
-    normSq nu 0 / ((nu+2)^2)^n ≤ normSq nu n := by
-  induction n with
-  | zero => simp
-  | succ n hn =>
-    rw [normSq_succ nu hnu,pow_succ]
-    have hb := beta_lower nu hnu.le n
-    have hz := normSq_zero_pos nu hnu.le
-    have hh := mul_le_mul hb hn (by positivity : 0≤normSq nu 0/((nu+2)^2)^n)
-      (beta_pos nu (by linarith) n).le
-    calc
-      _ = (1/(nu+2)^2)*(normSq nu 0/((nu+2)^2)^n) := by
-        simp only [div_eq_mul_inv,mul_inv_rev]
-        ring
-      _ ≤ _ := hh
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open Polynomial MeasureTheory Set
-open scoped ENNReal Interval Topology
-namespace SLEExactGauge.Jacobi
-
-abbrev Interval := Icc (-1:ℝ) 1
-
-def baseMeasure : Measure Interval := volume.comap Subtype.val
-
-instance base_finite : IsFiniteMeasure baseMeasure := by
-  constructor
-  rw [baseMeasure,(MeasurableEmbedding.subtype_coe measurableSet_Icc).comap_apply]
-  simp only [image_univ,Subtype.range_coe_subtype,ofPred_mem_eq]
-  exact isCompact_Icc.measure_lt_top
-
-lemma weight_nonneg (nu : ℝ) (x : Interval) : 0≤weight nu x := by
-  apply Real.rpow_nonneg
-  have hx := x.2
-  nlinarith [mul_nonneg (sub_nonneg.mpr hx.2) (show (0:ℝ)≤1+x.1 by linarith [hx.1])]
-
-lemma continuous_weight_interval (nu : ℝ) (hnu : 1≤nu) :
-    Continuous (fun x : Interval => weight nu x) := (continuous_weight nu hnu).comp continuous_subtype_val
-
-lemma integrable_weight_interval (nu : ℝ) (hnu : 1≤nu) :
-    Integrable (fun x : Interval => weight nu x) baseMeasure := by
-  exact (continuous_weight_interval nu hnu).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
-
-lemma integral_base (f : ℝ → ℝ) :
-    (∫ x : Interval, f x ∂baseMeasure)=∫ x in (-1:ℝ)..1, f x := by
-  rw [baseMeasure,integral_subtype_comap measurableSet_Icc,integral_Icc_eq_integral_Ioc,
-    intervalIntegral.integral_of_le (by norm_num : (-1:ℝ)≤1)]
-
-lemma integral_weight_base (nu : ℝ) :
-    (∫ x : Interval, weight nu x ∂baseMeasure)=normSq nu 0 := by
-  rw [integral_base]
-  simp only [normSq,inner,poly_zero,eval_one,mul_one]
-
-def rawMeasure (nu : ℝ) : Measure Interval :=
-  baseMeasure.withDensity (fun x => ENNReal.ofReal (weight nu x))
-
-def pi (nu : ℝ) : Measure Interval := (ENNReal.ofReal (normSq nu 0))⁻¹ • rawMeasure nu
-
-lemma rawMeasure_univ (nu : ℝ) (hnu : 1≤nu) :
-    rawMeasure nu univ=ENNReal.ofReal (normSq nu 0) := by
-  rw [rawMeasure,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
-  rw [← ofReal_integral_eq_lintegral_ofReal (integrable_weight_interval nu hnu)
-    (Filter.Eventually.of_forall (weight_nonneg nu)),integral_weight_base]
-
-lemma pi_probability (nu : ℝ) (hnu : 1≤nu) : IsProbabilityMeasure (pi nu) := by
-  constructor
-  rw [pi,Measure.smul_apply,rawMeasure_univ nu hnu,smul_eq_mul]
-  exact ENNReal.inv_mul_cancel ((ENNReal.ofReal_pos.mpr (normSq_zero_pos nu hnu)).ne') ENNReal.ofReal_ne_top
-
-lemma integral_pi_mul (nu : ℝ) (hnu : 1≤nu) (p q : ℝ[X]) :
-    (∫ x : Interval, p.eval (x:ℝ)*q.eval (x:ℝ) ∂pi nu)=inner nu p q/normSq nu 0 := by
-  rw [pi,integral_smul_measure,rawMeasure,integral_withDensity_eq_integral_toReal_smul
-    ((continuous_weight_interval nu hnu).measurable.ennreal_ofReal)
-    (Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
-  simp_rw [ENNReal.toReal_ofReal (weight_nonneg nu _),smul_eq_mul]
-  rw [ENNReal.toReal_inv,ENNReal.toReal_ofReal (normSq_zero_pos nu hnu).le]
-  have hi : (∫ x : Interval, weight nu x*(p.eval (x:ℝ)*q.eval (x:ℝ)) ∂baseMeasure)=inner nu p q := by
-    simp_rw [←mul_assoc]
-    exact integral_base (fun x : ℝ => weight nu x*p.eval x*q.eval x)
-  rw [hi]
-  ring
-
-lemma integral_pi_poly (nu : ℝ) (hnu : 1<nu) (n m : ℕ) :
-    (∫ x : Interval, (poly nu n).eval (x:ℝ) * (poly nu m).eval (x:ℝ) ∂pi nu)=
-      if n=m then normSq nu n/normSq nu 0 else 0 := by
-  rw [integral_pi_mul nu hnu.le]
-  split_ifs with hnm
-  · subst m; rfl
-  · rw [poly_orthogonal nu hnu n m hnm,zero_div]
-
-lemma poly_interval_integrable (nu : ℝ) (hnu : 1<nu) (p : ℝ[X]) :
-    Integrable (fun x : Interval => p.eval (x:ℝ)) (pi nu) := by
-  let := pi_probability nu hnu.le
-  exact (p.continuous.comp continuous_subtype_val).integrable_of_hasCompactSupport
-    (HasCompactSupport.of_compactSpace _)
-
-lemma integral_pi_poly_single (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
-    (∫ x : Interval, (poly nu n).eval (x:ℝ) ∂pi nu)=if n=0 then 1 else 0 := by
-  have hh := integral_pi_poly nu hnu n 0
-  simp only [poly_zero,eval_one,mul_one] at hh
-  rw [hh]
-  split_ifs with hn
-  · subst n; exact div_self (normSq_zero_pos nu hnu.le).ne'
-  · rfl
-
-end SLEExactGauge.Jacobi
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ConformalFlow
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen
+
+abbrev Angle := Ioo (0:ℝ) Real.pi
+
+def angleArea : Measure Angle := volume.comap (Subtype.val : Angle→ℝ)
+
+def angleWeight (a : ℝ) (θ : Angle) : ℝ≥0∞ := ENNReal.ofReal ((Real.sin θ)^(4*a))
+
+def angleNormalization (a : ℝ) : ℝ≥0∞ := ∫⁻θ,angleWeight a θ∂angleArea
+
+ 
+
+def angleLaw (a : ℝ) : Measure Angle :=
+  (angleNormalization a)⁻¹ • angleArea.withDensity (angleWeight a)
+
+lemma angleArea_univ : angleArea univ=ENNReal.ofReal Real.pi := by
+  rw [angleArea,MeasurableEmbedding.comap_apply
+    (MeasurableEmbedding.subtype_coe measurableSet_Ioo) _ univ]
+  rw [show (Subtype.val : Angle→ℝ) '' univ=Ioo (0:ℝ) Real.pi by ext x; simp,
+    Real.volume_Ioo,sub_zero]
+
+instance : IsFiniteMeasure angleArea := ⟨by rw [angleArea_univ]; exact ENNReal.ofReal_lt_top⟩
+
+lemma angleArea_ne_zero : angleArea univ≠0 := by
+  rw [angleArea_univ]
+  exact (ENNReal.ofReal_pos.mpr Real.pi_pos).ne'
+
+lemma angleWeight_measurable (a : ℝ) : Measurable (angleWeight a) := by
+  unfold angleWeight
+  fun_prop
+
+lemma angleWeight_pos (a : ℝ) (θ : Angle) : 0<angleWeight a θ :=
+  ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) _)
+
+lemma angleWeight_le_one (a : ℝ) (ha : 0≤a) (θ : Angle) : angleWeight a θ≤1 := by
+  rw [←ENNReal.ofReal_one]
+  exact ENNReal.ofReal_le_ofReal (Real.rpow_le_one
+    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le (Real.sin_le_one θ) (by positivity))
+
+lemma angleNormalization_pos (a : ℝ) : 0<angleNormalization a := by
+  rw [angleNormalization,lintegral_pos_iff_support (angleWeight_measurable a)]
+  have hs : Function.support (angleWeight a)=univ := by
+    ext θ
+    simp [Function.mem_support,(angleWeight_pos a θ).ne']
+  rw [hs]
+  exact bot_lt_iff_ne_bot.mpr angleArea_ne_zero
+
+lemma angleNormalization_lt_top (a : ℝ) (ha : 0≤a) : angleNormalization a<∞ := by
+  calc
+    _ ≤ ∫⁻_ : Angle,(1:ℝ≥0∞)∂angleArea := lintegral_mono (angleWeight_le_one a ha)
+    _ < ∞ := by simp [angleArea_univ]
+
+lemma angleLaw_probability (a : ℝ) (ha : 0≤a) : IsProbabilityMeasure (angleLaw a) := by
+  rw [isProbabilityMeasure_iff]
+  rw [angleLaw,Measure.smul_apply,smul_eq_mul,withDensity_apply _ MeasurableSet.univ,
+    Measure.restrict_univ]
+  exact ENNReal.inv_mul_cancel (angleNormalization_pos a).ne' (angleNormalization_lt_top a ha).ne
+
+lemma angleLaw_integral (a : ℝ) (f : Angle→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻θ,f θ∂angleLaw a)=(angleNormalization a)⁻¹*
+      ∫⁻θ,angleWeight a θ*f θ∂angleArea := by
+  rw [angleLaw,lintegral_smul_measure,smul_eq_mul,
+    lintegral_withDensity_eq_lintegral_mul _ (angleWeight_measurable a) hf]
+  rfl
+
+lemma angleWeight_cancel (a : ℝ) (θ : Angle) :
+    angleWeight a θ*ENNReal.ofReal ((Real.sin θ)^(-p a))=
+      ENNReal.ofReal (Real.sin θ) := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
+  rw [angleWeight,←ENNReal.ofReal_mul (Real.rpow_nonneg hs.le _),←Real.rpow_add hs]
+  simp [p]
+
+ 
+lemma angle_inverse_power_finite (a : ℝ) :
+    (∫⁻θ : Angle,ENNReal.ofReal ((Real.sin θ)^(-p a))∂angleLaw a)<∞ := by
+  rw [angleLaw_integral a _ (by fun_prop)]
+  simp_rw [angleWeight_cancel]
+  apply ENNReal.mul_lt_top
+  · exact ENNReal.inv_lt_top.mpr (angleNormalization_pos a)
+  · calc
+      _ ≤ ∫⁻_ : Angle,(1:ℝ≥0∞)∂angleArea := by
+        apply lintegral_mono
+        intro θ
+        rw [←ENNReal.ofReal_one]
+        exact ENNReal.ofReal_le_ofReal (Real.sin_le_one θ)
+      _ < ∞ := by simp [angleArea_univ]
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Polynomial
+
+namespace SLEExactGauge.Jacobi
+
+ 
+def operator (nu : ℝ) (p : ℝ[X]) : ℝ[X] :=
+  (1-X^2)*p.derivative.derivative - C (1+nu)*X*p.derivative
+
+ 
+def raise (nu n : ℝ) (p : ℝ[X]) : ℝ[X] :=
+  C (1/(2*n+nu)) * (C (n+nu)*X*p - (1-X^2)*p.derivative)
+
+lemma raise_derivative (nu n : ℝ) (p : ℝ[X])
+    (hp : operator nu p = -C (n*(n+nu))*p) :
+    (raise nu n p).derivative =
+      C (1/(2*n+nu))*C (n+1)*(C (n+nu)*p+X*p.derivative) := by
+  dsimp only [raise,operator] at *
+  simp only [derivative_mul,derivative_sub,derivative_C,derivative_X,derivative_one,
+    derivative_pow,zero_mul,zero_sub,zero_add]
+  simp only [map_mul,map_add,map_one,Nat.cast_ofNat,map_ofNat] at hp ⊢
+  linear_combination -C (1/(2*n+nu)) * hp
+
+lemma raise_eigen (nu n : ℝ) (p : ℝ[X])
+    (hp : operator nu p = -C (n*(n+nu))*p) :
+    operator nu (raise nu n p) = -C ((n+1)*(n+1+nu)) * raise nu n p := by
+  have hd := raise_derivative nu n p hp
+  have hd2 := congrArg Polynomial.derivative hd
+  simp only [derivative_mul,derivative_add,derivative_C,derivative_X,
+    zero_mul,mul_zero,zero_add,one_mul,add_zero] at hd2
+  rw [operator,hd2,hd]
+  dsimp only [raise,operator] at *
+  simp only [map_mul,map_add,map_one] at hp ⊢
+  linear_combination C (1/(2*n+nu))*(C n+1)*X * hp
+
+lemma raise_lowering (nu n : ℝ) (p : ℝ[X])
+    (hp : operator nu p = -C (n*(n+nu))*p) :
+    (1-X^2)*(raise nu n p).derivative + C (n+1)*X*raise nu n p =
+      C ((n+1)*(n+nu)/(2*n+nu))*p := by
+  rw [raise_derivative nu n p hp]
+  simp only [raise,map_mul,map_add,map_one,div_eq_mul_inv]
+  ring
+
+ 
+def poly (nu : ℝ) : ℕ → ℝ[X]
+  | 0 => 1
+  | n+1 => raise nu n (poly nu n)
+
+@[simp] lemma poly_zero (nu : ℝ) : poly nu 0=1 := rfl
+@[simp] lemma poly_succ (nu : ℝ) (n : ℕ) :
+    poly nu (n+1)=raise nu n (poly nu n) := rfl
+
+lemma poly_eigen (nu : ℝ) (n : ℕ) :
+    operator nu (poly nu n) = -C ((n:ℝ)*(n+nu))*poly nu n := by
+  induction n with
+  | zero => simp [operator]
+  | succ n ih => simpa only [poly_succ,Nat.cast_add,Nat.cast_one] using
+      raise_eigen nu n (poly nu n) ih
+
+lemma poly_lowering (nu : ℝ) (n : ℕ) :
+    (1-X^2)*(poly nu (n+1)).derivative + C ((n:ℝ)+1)*X*poly nu (n+1) =
+      C (((n:ℝ)+1)*(n+nu)/(2*n+nu))*poly nu n :=
+  raise_lowering nu n (poly nu n) (poly_eigen nu n)
+
+lemma raise_eq_X_sub (nu n : ℝ) (p : ℝ[X]) (h : 2*n+nu≠0) :
+    raise nu n p = X*p - C (1/(2*n+nu))*((1-X^2)*p.derivative+C n*X*p) := by
+  have hc : C (1/(2*n+nu))*(2*C n+C nu)=(1:ℝ[X]) := by
+    have he : (2*C n+C nu : ℝ[X])=C (2*n+nu) := by simp only [map_add,map_mul,map_ofNat]
+    rw [he,←map_mul,one_div_mul_cancel h,map_one]
+  simp only [raise,map_add]
+  linear_combination X*p*hc
+
+lemma poly_one (nu : ℝ) (hnu : nu≠0) : poly nu 1=X := by
+  rw [show 1=0+1 from rfl,poly_succ,poly_zero,Nat.cast_zero,raise_eq_X_sub nu 0 1 (by simpa using hnu)]
+  simp
+
+ 
+def beta (nu : ℝ) (n : ℕ) : ℝ :=
+  ((n:ℝ)+1)*(n+nu)/((2*n+nu+2)*(2*n+nu))
+
+lemma poly_recurrence (nu : ℝ) (hnu : 0<nu) (n : ℕ) :
+    poly nu (n+2)=X*poly nu (n+1)-C (beta nu n)*poly nu n := by
+  have hd : 2*((n:ℝ)+1)+nu≠0 := ne_of_gt (by positivity)
+  change raise nu ((n+1:ℕ):ℝ) (poly nu (n+1))=_
+  rw [Nat.cast_add,Nat.cast_one,raise_eq_X_sub nu ((n:ℝ)+1) _ hd,poly_lowering]
+  congr 1
+  rw [←mul_assoc,←map_mul]
+  congr 2
+  unfold beta
+  field_simp
+  ring
+
+lemma beta_pos (nu : ℝ) (hnu : 0<nu) (n : ℕ) : 0<beta nu n := by
+  unfold beta
+  positivity
+
+lemma beta_le_one (nu : ℝ) (hnu : 0<nu) (n : ℕ) : beta nu n≤1 := by
+  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
+  unfold beta
+  apply (div_le_one (by positivity)).mpr
+  nlinarith [sq_nonneg ((n:ℝ)),sq_nonneg nu,mul_nonneg hn hnu.le]
+
+lemma beta_lower (nu : ℝ) (hnu : 1≤nu) (n : ℕ) :
+    1/(nu+2)^2≤beta nu n := by
+  have hn : (0:ℝ)≤n := Nat.cast_nonneg n
+  have hnu0 : 0<nu := lt_of_lt_of_le zero_lt_one hnu
+  have h1 : 0<(nu+2)^2 := by positivity
+  have h2 : 0<(2*(n:ℝ)+nu+2)*(2*n+nu) := by positivity
+  have hden : (2*(n:ℝ)+nu+2)*(2*n+nu)≤(nu+2)^2*((n:ℝ)+1)^2 := by
+    have hb1 : 2*(n:ℝ)+nu+2≤(nu+2)*((n:ℝ)+1) := by
+      nlinarith [mul_nonneg hnu0.le hn]
+    have hb2 : 2*(n:ℝ)+nu≤(nu+2)*((n:ℝ)+1) := by linarith
+    have hh := mul_le_mul hb1 hb2 (by positivity : 0≤2*(n:ℝ)+nu)
+      (by positivity : 0≤(nu+2)*((n:ℝ)+1))
+    nlinarith only [hh]
+  have hnum : ((n:ℝ)+1)^2≤((n:ℝ)+1)*(n+nu) := by nlinarith
+  rw [beta,div_le_div_iff₀ h1 h2]
+  have hm := mul_le_mul_of_nonneg_left hnum h1.le
+  nlinarith only [hden,hm]
+
+lemma poly_eval_bound (nu : ℝ) (hnu : 0<nu) (n : ℕ) (x : ℝ) (hx : |x|≤1) :
+    |(poly nu n).eval x|≤2^n := by
+  induction n using Nat.twoStepInduction with
+  | zero => simp
+  | one =>
+    rw [poly_one nu hnu.ne',eval_X,pow_one]
+    exact hx.trans (by norm_num)
+  | more n hn hn1 =>
+    rw [poly_recurrence nu hnu n,eval_sub,eval_mul,eval_X,eval_mul,eval_C]
+    calc
+      _ ≤ |x*(poly nu (n+1)).eval x|+|beta nu n*(poly nu n).eval x| := abs_sub _ _
+      _ = |x| * |(poly nu (n+1)).eval x|+beta nu n * |(poly nu n).eval x| := by
+        rw [abs_mul,abs_mul,abs_of_pos (beta_pos nu hnu n)]
+      _ ≤ 2^(n+1)+2^n := add_le_add
+        ((mul_le_mul hx hn1 (abs_nonneg _) zero_le_one).trans_eq (one_mul _))
+        ((mul_le_mul (beta_le_one nu hnu n) hn (abs_nonneg _) zero_le_one).trans_eq (one_mul _))
+      _ ≤ 2^(n+2) := by
+        simp only [pow_add,pow_one]
+        norm_num
+        nlinarith [show (0:ℝ)≤2^n by positivity]
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open Polynomial MeasureTheory Set
+open scoped Interval
+
+namespace SLEExactGauge.Jacobi
+
+def weight (nu x : ℝ) : ℝ := (1-x^2)^((nu-1)/2)
+def flux (nu x : ℝ) : ℝ := (1-x^2)^((nu+1)/2)
+
+def inner (nu : ℝ) (p q : ℝ[X]) : ℝ :=
+  ∫ x in (-1:ℝ)..1, weight nu x * p.eval x * q.eval x
+
+lemma continuous_weight (nu : ℝ) (hnu : 1≤nu) : Continuous (weight nu) :=
+  (continuous_const.sub (continuous_id.pow 2)).rpow_const (fun _ => Or.inr (by linarith))
+
+lemma continuous_flux (nu : ℝ) (hnu : 1≤nu) : Continuous (flux nu) :=
+  (continuous_const.sub (continuous_id.pow 2)).rpow_const (fun _ => Or.inr (by linarith))
+
+lemma flux_eq (nu : ℝ) (hnu : 1<nu) (x : ℝ) (hx : x∈Icc (-1:ℝ) 1) :
+    flux nu x = (1-x^2)*weight nu x := by
+  have hx0 : 0≤1-x^2 := by nlinarith [mul_nonneg (sub_nonneg.mpr hx.2) (show 0≤1+x by linarith [hx.1])]
+  have he : (nu+1)/2=(nu-1)/2+1 := by ring
+  rw [flux,he,Real.rpow_add' hx0 (by linarith : (nu-1)/2+1≠0),Real.rpow_one,weight]
+  ring
+
+lemma hasDerivAt_flux (nu : ℝ) (hnu : 1≤nu) (x : ℝ) :
+    HasDerivAt (flux nu) (-(nu+1)*x*weight nu x) x := by
+  have hd := ((hasDerivAt_const x (1:ℝ)).sub ((hasDerivAt_id x).pow 2)).rpow_const
+    (Or.inr (by linarith : 1≤(nu+1)/2))
+  convert hd using 1
+  · rfl
+  · dsimp [weight]
+    rw [show (nu+1)/2-1=(nu-1)/2 by ring]
+    ring
+
+lemma inner_comm (nu : ℝ) (p q : ℝ[X]) : inner nu p q=inner nu q p := by
+  apply intervalIntegral.integral_congr
+  intro x hx
+  ring
+
+lemma inner_integrable (nu : ℝ) (hnu : 1≤nu) (p q : ℝ[X]) :
+    IntervalIntegrable (fun x => weight nu x*p.eval x*q.eval x) volume (-1) 1 :=
+  (((continuous_weight nu hnu).mul p.continuous).mul q.continuous).intervalIntegrable _ _
+
+lemma inner_add_left (nu : ℝ) (hnu : 1≤nu) (p q r : ℝ[X]) :
+    inner nu (p+q) r=inner nu p r+inner nu q r := by
+  simp only [inner,eval_add,mul_add,add_mul]
+  exact intervalIntegral.integral_add (inner_integrable nu hnu p r) (inner_integrable nu hnu q r)
+
+lemma inner_sub_left (nu : ℝ) (hnu : 1≤nu) (p q r : ℝ[X]) :
+    inner nu (p-q) r=inner nu p r-inner nu q r := by
+  simp only [inner,eval_sub,mul_sub,sub_mul]
+  exact intervalIntegral.integral_sub (inner_integrable nu hnu p r) (inner_integrable nu hnu q r)
+
+lemma inner_C_mul_left (nu c : ℝ) (p q : ℝ[X]) :
+    inner nu (C c*p) q=c*inner nu p q := by
+  simp only [inner,eval_mul,eval_C]
+  simp_rw [show ∀ x, weight nu x*(c*p.eval x)*q.eval x=c*(weight nu x*p.eval x*q.eval x) by intro x; ring]
+  exact intervalIntegral.integral_const_mul _ _
+
+lemma inner_neg_left (nu : ℝ) (p q : ℝ[X]) :
+    inner nu (-p) q= -inner nu p q := by
+  simp [inner]
+
+lemma inner_X (nu : ℝ) (p q : ℝ[X]) : inner nu (X*p) q=inner nu p (X*q) := by
+  apply intervalIntegral.integral_congr
+  intro x hx
+  simp only [eval_mul,eval_X]
+  ring
+
+ 
+lemma operator_symmetric (nu : ℝ) (hnu : 1<nu) (p q : ℝ[X]) :
+    inner nu p (operator nu q)=inner nu (operator nu p) q := by
+  let f (x : ℝ) := flux nu x * (p.eval x*q.derivative.eval x-p.derivative.eval x*q.eval x)
+  let g (x : ℝ) := weight nu x*p.eval x*(operator nu q).eval x-
+    weight nu x*(operator nu p).eval x*q.eval x
+  have hd (x : ℝ) (hx : x∈uIcc (-1:ℝ) 1) : HasDerivAt f (g x) x := by
+    have hxx : x∈Icc (-1:ℝ) 1 := by simpa only [uIcc_of_le (by norm_num : (-1:ℝ)≤1)] using hx
+    have hh := (hasDerivAt_flux nu hnu.le x).mul
+      (((p.hasDerivAt x).mul (q.derivative.hasDerivAt x)).sub
+        ((p.derivative.hasDerivAt x).mul (q.hasDerivAt x)))
+    convert hh using 1
+    try rfl
+    all_goals simp only [g,operator,Pi.mul_apply,Pi.sub_apply,eval_sub,eval_mul,eval_pow,eval_X,eval_one,eval_C,flux_eq nu hnu x hxx]
+    all_goals ring
+  have hi : IntervalIntegrable g volume (-1) 1 :=
+    (inner_integrable nu hnu.le p (operator nu q)).sub
+      (inner_integrable nu hnu.le (operator nu p) q)
+  have hh := intervalIntegral.integral_eq_sub_of_hasDerivAt hd hi
+  have hz : flux nu 1=0 ∧ flux nu (-1)=0 := by
+    simp [flux,Real.zero_rpow (by linarith : (nu+1)/2≠0)]
+  have hg : (∫ x in (-1:ℝ)..1, g x) = inner nu p (operator nu q)-inner nu (operator nu p) q :=
+    intervalIntegral.integral_sub (inner_integrable nu hnu.le p (operator nu q))
+      (inner_integrable nu hnu.le (operator nu p) q)
+  rw [hg] at hh
+  simpa [f,hz.1,hz.2,sub_eq_zero] using hh
+
+lemma eigenvalue_strictMono (nu : ℝ) (hnu : 0<nu) :
+    StrictMono (fun n : ℕ => (n:ℝ)*(n+nu)) := by
+  intro n m hnm
+  have hnm' : (n:ℝ) < m := Nat.cast_lt.mpr hnm
+  nlinarith [Nat.cast_nonneg (α := ℝ) n,Nat.cast_nonneg (α := ℝ) m]
+
+ 
+lemma poly_orthogonal (nu : ℝ) (hnu : 1<nu) (n m : ℕ) (hnm : n≠m) :
+    inner nu (poly nu n) (poly nu m)=0 := by
+  have hs := operator_symmetric nu hnu (poly nu n) (poly nu m)
+  simp only [poly_eigen,neg_mul] at hs
+  rw [inner_comm nu (poly nu n),inner_neg_left,inner_C_mul_left,
+    inner_neg_left,inner_C_mul_left,inner_comm nu (poly nu m)] at hs
+  have he := (eigenvalue_strictMono nu (by linarith)).injective.ne hnm
+  have hp : ((n:ℝ)*(n+nu)-(m:ℝ)*(m+nu))*inner nu (poly nu n) (poly nu m)=0 := by linarith
+  exact (mul_eq_zero.mp hp).resolve_left (sub_ne_zero.mpr he)
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open Polynomial MeasureTheory Set
+open scoped Interval
+
+namespace SLEExactGauge.Jacobi
+
+lemma poly_monic_degree (nu : ℝ) (hnu : 0<nu) (n : ℕ) :
+    (poly nu n).Monic ∧ (poly nu n).degree=n := by
+  induction n using Nat.twoStepInduction with
+  | zero => simp only [poly_zero,Nat.cast_zero,monic_one,degree_one,and_self]
+  | one => rw [poly_one nu hnu.ne']; exact ⟨monic_X,degree_X⟩
+  | more n hn hn1 =>
+    have hd : (C (beta nu n)*poly nu n).degree < (X*poly nu (n+1)).degree := by
+      rw [degree_C_mul (beta_pos nu hnu n).ne',degree_mul,degree_X,hn.2,hn1.2]
+      norm_cast
+      omega
+    rw [poly_recurrence nu hnu n]
+    refine ⟨(monic_X.mul hn1.1).sub_of_left hd, ?_⟩
+    rw [degree_sub_eq_left_of_degree_lt hd,degree_mul,degree_X,hn1.2]
+    norm_cast
+    omega
+
+ 
+def basis (nu : ℝ) (hnu : 0<nu) : Module.Basis ℕ ℝ ℝ[X] :=
+  let seq : Polynomial.Sequence ℝ := ⟨poly nu, fun n => (poly_monic_degree nu hnu n).2⟩
+  seq.basis (fun n => by
+    change IsUnit (poly nu n).leadingCoeff
+    rw [(poly_monic_degree nu hnu n).1.leadingCoeff]
+    exact isUnit_one)
+
+@[simp] lemma basis_apply (nu : ℝ) (hnu : 0<nu) (n : ℕ) : basis nu hnu n=poly nu n := by
+  simp only [basis,Polynomial.Sequence.basis_eq_self]
+
+def normSq (nu : ℝ) (n : ℕ) : ℝ := inner nu (poly nu n) (poly nu n)
+
+lemma normSq_zero_pos (nu : ℝ) (hnu : 1≤nu) : 0<normSq nu 0 := by
+  change 0<∫ x in (-1:ℝ)..1, weight nu x*(poly nu 0).eval x*(poly nu 0).eval x
+  simp only [poly_zero,eval_one,mul_one]
+  apply intervalIntegral.integral_pos (by norm_num) (continuous_weight nu hnu).continuousOn
+  · intro x hx
+    apply Real.rpow_nonneg
+    nlinarith [mul_nonneg (sub_nonneg.mpr hx.2) (show 0≤1+x by linarith [hx.1])]
+  · refine ⟨0,by norm_num, ?_⟩
+    simp [weight]
+
+lemma normSq_succ (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
+    normSq nu (n+1)=beta nu n*normSq nu n := by
+  have hnu0 : 0<nu := by linarith
+  have hr (j : ℕ) : X*poly nu (j+1)=poly nu (j+2)+C (beta nu j)*poly nu j := by
+    rw [poly_recurrence nu hnu0 j]
+    ring
+  have hl : inner nu (X*poly nu (n+1)) (poly nu n)=beta nu n*normSq nu n := by
+    rw [hr,inner_add_left nu hnu.le,inner_C_mul_left,
+      poly_orthogonal nu hnu (n+2) n (by omega),zero_add]
+    rfl
+  rw [inner_X,inner_comm] at hl
+  rw [←hl]
+  cases n with
+  | zero =>
+    change inner nu (poly nu 1) (poly nu 1)=_
+    rw [poly_zero,mul_one,poly_one nu hnu0.ne']
+  | succ n =>
+    rw [hr n,inner_add_left nu hnu.le,inner_C_mul_left,
+      poly_orthogonal nu hnu n (n+1+1) (by omega),mul_zero,add_zero]
+    rfl
+
+lemma normSq_pos (nu : ℝ) (hnu : 1<nu) (n : ℕ) : 0<normSq nu n := by
+  induction n with
+  | zero => exact normSq_zero_pos nu hnu.le
+  | succ n hn =>
+    rw [normSq_succ nu hnu n]
+    exact mul_pos (beta_pos nu (by linarith) n) hn
+
+lemma normSq_lower (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
+    normSq nu 0 / ((nu+2)^2)^n ≤ normSq nu n := by
+  induction n with
+  | zero => simp
+  | succ n hn =>
+    rw [normSq_succ nu hnu,pow_succ]
+    have hb := beta_lower nu hnu.le n
+    have hz := normSq_zero_pos nu hnu.le
+    have hh := mul_le_mul hb hn (by positivity : 0≤normSq nu 0/((nu+2)^2)^n)
+      (beta_pos nu (by linarith) n).le
+    calc
+      _ = (1/(nu+2)^2)*(normSq nu 0/((nu+2)^2)^n) := by
+        simp only [div_eq_mul_inv,mul_inv_rev]
+        ring
+      _ ≤ _ := hh
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open Polynomial MeasureTheory Set
+open scoped ENNReal Interval Topology
+namespace SLEExactGauge.Jacobi
+
+abbrev Interval := Icc (-1:ℝ) 1
+
+def baseMeasure : Measure Interval := volume.comap Subtype.val
+
+instance base_finite : IsFiniteMeasure baseMeasure := by
+  constructor
+  rw [baseMeasure,(MeasurableEmbedding.subtype_coe measurableSet_Icc).comap_apply]
+  simp only [image_univ,Subtype.range_coe_subtype,ofPred_mem_eq]
+  exact isCompact_Icc.measure_lt_top
+
+lemma weight_nonneg (nu : ℝ) (x : Interval) : 0≤weight nu x := by
+  apply Real.rpow_nonneg
+  have hx := x.2
+  nlinarith [mul_nonneg (sub_nonneg.mpr hx.2) (show (0:ℝ)≤1+x.1 by linarith [hx.1])]
+
+lemma continuous_weight_interval (nu : ℝ) (hnu : 1≤nu) :
+    Continuous (fun x : Interval => weight nu x) := (continuous_weight nu hnu).comp continuous_subtype_val
+
+lemma integrable_weight_interval (nu : ℝ) (hnu : 1≤nu) :
+    Integrable (fun x : Interval => weight nu x) baseMeasure := by
+  exact (continuous_weight_interval nu hnu).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+
+lemma integral_base (f : ℝ → ℝ) :
+    (∫ x : Interval, f x ∂baseMeasure)=∫ x in (-1:ℝ)..1, f x := by
+  rw [baseMeasure,integral_subtype_comap measurableSet_Icc,integral_Icc_eq_integral_Ioc,
+    intervalIntegral.integral_of_le (by norm_num : (-1:ℝ)≤1)]
+
+lemma integral_weight_base (nu : ℝ) :
+    (∫ x : Interval, weight nu x ∂baseMeasure)=normSq nu 0 := by
+  rw [integral_base]
+  simp only [normSq,inner,poly_zero,eval_one,mul_one]
+
+def rawMeasure (nu : ℝ) : Measure Interval :=
+  baseMeasure.withDensity (fun x => ENNReal.ofReal (weight nu x))
+
+def pi (nu : ℝ) : Measure Interval := (ENNReal.ofReal (normSq nu 0))⁻¹ • rawMeasure nu
+
+lemma rawMeasure_univ (nu : ℝ) (hnu : 1≤nu) :
+    rawMeasure nu univ=ENNReal.ofReal (normSq nu 0) := by
+  rw [rawMeasure,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
+  rw [← ofReal_integral_eq_lintegral_ofReal (integrable_weight_interval nu hnu)
+    (Filter.Eventually.of_forall (weight_nonneg nu)),integral_weight_base]
+
+lemma pi_probability (nu : ℝ) (hnu : 1≤nu) : IsProbabilityMeasure (pi nu) := by
+  constructor
+  rw [pi,Measure.smul_apply,rawMeasure_univ nu hnu,smul_eq_mul]
+  exact ENNReal.inv_mul_cancel ((ENNReal.ofReal_pos.mpr (normSq_zero_pos nu hnu)).ne') ENNReal.ofReal_ne_top
+
+lemma integral_pi_mul (nu : ℝ) (hnu : 1≤nu) (p q : ℝ[X]) :
+    (∫ x : Interval, p.eval (x:ℝ)*q.eval (x:ℝ) ∂pi nu)=inner nu p q/normSq nu 0 := by
+  rw [pi,integral_smul_measure,rawMeasure,integral_withDensity_eq_integral_toReal_smul
+    ((continuous_weight_interval nu hnu).measurable.ennreal_ofReal)
+    (Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
+  simp_rw [ENNReal.toReal_ofReal (weight_nonneg nu _),smul_eq_mul]
+  rw [ENNReal.toReal_inv,ENNReal.toReal_ofReal (normSq_zero_pos nu hnu).le]
+  have hi : (∫ x : Interval, weight nu x*(p.eval (x:ℝ)*q.eval (x:ℝ)) ∂baseMeasure)=inner nu p q := by
+    simp_rw [←mul_assoc]
+    exact integral_base (fun x : ℝ => weight nu x*p.eval x*q.eval x)
+  rw [hi]
+  ring
+
+lemma integral_pi_poly (nu : ℝ) (hnu : 1<nu) (n m : ℕ) :
+    (∫ x : Interval, (poly nu n).eval (x:ℝ) * (poly nu m).eval (x:ℝ) ∂pi nu)=
+      if n=m then normSq nu n/normSq nu 0 else 0 := by
+  rw [integral_pi_mul nu hnu.le]
+  split_ifs with hnm
+  · subst m; rfl
+  · rw [poly_orthogonal nu hnu n m hnm,zero_div]
+
+lemma poly_interval_integrable (nu : ℝ) (hnu : 1<nu) (p : ℝ[X]) :
+    Integrable (fun x : Interval => p.eval (x:ℝ)) (pi nu) := by
+  let := pi_probability nu hnu.le
+  exact (p.continuous.comp continuous_subtype_val).integrable_of_hasCompactSupport
+    (HasCompactSupport.of_compactSpace _)
+
+lemma integral_pi_poly_single (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
+    (∫ x : Interval, (poly nu n).eval (x:ℝ) ∂pi nu)=if n=0 then 1 else 0 := by
+  have hh := integral_pi_poly nu hnu n 0
+  simp only [poly_zero,eval_one,mul_one] at hh
+  rw [hh]
+  split_ifs with hn
+  · subst n; exact div_self (normSq_zero_pos nu hnu.le).ne'
+  · rfl
+
+end SLEExactGauge.Jacobi
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/AngularControlMass.lean b/lean/OAI/Probability/SLEGauge/AngularControlMass.lean
index 18a987807714752fef23935d89ec08a1ad450c78..3628aac1ab366a4fcfa032a991e0a0e848b0e377 100644
--- a/lean/OAI/Probability/SLEGauge/AngularControlMass.lean
+++ b/lean/OAI/Probability/SLEGauge/AngularControlMass.lean
@@ -1,489 +1,489 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.GlobalAngleBudget
-import OAI.Probability.SLEGauge.FutureSupport
-import OAI.Probability.SLEGauge.FiniteProductLimit
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-
- 
-
-def angularDensity (a q : ℝ) (j : ℕ) (b : ExactSLEGenerator.Driver) (z : UpperPlane) : ℝ≥0∞ :=
-  ENNReal.ofReal (Real.exp (-alpha a*Real.log 2)*
-    radialTestWeight a (angleLift (sinePower (-q))) b.1 z.1
-      (levelRadius a (2*z.1.im) (j+1)))
-
-def angularCompactMass (a q : ℝ) (j : ℕ) (K : Set UpperPlane)
-    (b : ExactSLEGenerator.Driver) : ℝ≥0∞ := ∫⁻z in K, angularDensity a q j b z ∂upperArea
-
-lemma angularDensity_measurable (a q : ℝ) (ha : 1/4<a) (j : ℕ) :
-    Measurable (Function.uncurry (angularDensity a q j)) := by
-  have he (z : UpperPlane) : levelRadius a (2*z.1.im) (j+1)=z.1.im*Real.exp (-2*a*(j+1)) := by
-    simp [levelRadius]
-  unfold angularDensity Function.uncurry
-  simp only [he]
-  exact (measurable_const.mul (joint_radialTestWeight_measurable a ha _ (Real.exp_pos _)
-      (level_step_le_one a ha j) _ (angleLift_sinePower_measurable (-q)))).ennreal_ofReal
-
-lemma angularCompactMass_measurable (a q : ℝ) (ha : 1/4<a) (j : ℕ) (K : Set UpperPlane) :
-    Measurable (angularCompactMass a q j K) := (angularDensity_measurable a q ha j).lintegral_prod_right
-
-lemma angularInitial_continuous (a q C : ℝ) : Continuous (fun z : UpperPlane =>
-    physicalGreen a (z.1,0)*(angleLift (sinePower (-q)) (z.1,0)+C)) := by
-  apply continuous_iff_continuousAt.mpr
-  intro z
-  have hsin := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem (z.1,0)).1
-    (physicalTheta_mem (z.1,0)).2
-  have hi : Continuous (fun w : UpperPlane => (w.1,(0:ℝ))) :=
-    continuous_subtype_val.prodMk continuous_const
-  exact ((contDiffAt_physicalGreen a (z.1,0) z.2).continuousAt.mul
-    (((angleLift_smooth (sinePower (-q)) _ z.2 (sinePower_smooth (-q) _ hsin)).continuousAt).add_const C)).comp
-      (f := fun w : UpperPlane => (w.1,(0:ℝ))) hi.continuousAt
-
- 
-theorem angularCompactMass_uniform_mean (a q : ℝ) (ha : 1/4<a) (hq : 0<q)
-    (hqp : q<4*a-1) (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (K : Set UpperPlane) (hK : IsCompact K) :
-    ∃ B : ℝ≥0∞, B<∞ ∧ ∀ j, (∫⁻b, angularCompactMass a q j K b ∂W)≤B := by
-  obtain ⟨C,hC,hbound⟩ := exists_finite_level_angle_bound a q ha hq hqp
-  let A : UpperPlane→ℝ := fun z => physicalGreen a (z.1,0)*
-    (angleLift (sinePower (-q)) (z.1,0)+C)
-  have hA : Continuous A := angularInitial_continuous a q C
-  let R := Real.exp (-alpha a*Real.log 2)
-  let B : ℝ≥0∞ := ∫⁻z in K, ENNReal.ofReal (R*A z) ∂upperArea
-  have hB : B<∞ := (ContinuousOn.integrableOn_compact (μ := upperArea) hK
-    (show ContinuousOn (fun z => R*A z) K from (continuous_const.mul hA).continuousOn)).lintegral_lt_top
-  refine ⟨B,hB,?_⟩
-  intro j
-  have hm := angularDensity_measurable a q ha j
-  simp only [angularCompactMass]
-  rw [lintegral_lintegral_swap hm.aemeasurable]
-  apply setLIntegral_mono' hK.measurableSet
-  intro z _
-  let f : ExactSLEGenerator.Driver→ℝ := fun b =>
-    radialTestWeight a (angleLift (sinePower (-q))) b.1 z.1
-      (levelRadius a (2*z.1.im) (j+1))
-  have hf : Measurable f := by
-    have hh := (joint_radialTestWeight_measurable a ha _ (Real.exp_pos _)
-      (level_step_le_one a ha j) _ (angleLift_sinePower_measurable (-q))).comp
-      (measurable_id.prodMk (measurable_const (a := z)))
-    simpa [f,levelRadius,Function.comp_def] using hh
-  simp_rw [angularDensity,ENNReal.ofReal_mul (Real.exp_pos _).le]
-  rw [lintegral_const_mul _ hf.ennreal_ofReal]
-  exact (mul_le_mul' le_rfl (hbound W hW z.1 z.2 (j+1) (by positivity))).trans_eq
-    (ENNReal.ofReal_mul (Real.exp_pos _).le).symm
-
- 
-def convexAngularMass (a q : ℝ) (s : Finset ℕ) (c : ℕ→ℝ)
-    (K : Set UpperPlane) (b : ExactSLEGenerator.Driver) : ℝ≥0∞ :=
-  ∑j∈s, ENNReal.ofReal (c j)*angularCompactMass a q j K b
-
-lemma convexAngularMass_measurable (a q : ℝ) (ha : 1/4<a) (s : Finset ℕ)
-    (c : ℕ→ℝ) (K : Set UpperPlane) : Measurable (convexAngularMass a q s c K) := by
-  apply Finset.measurable_sum
-  intro j _
-  exact measurable_const.mul (angularCompactMass_measurable a q ha j K)
-
-lemma convexAngularMass_mean_bound (a q : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) (K : Set UpperPlane) (B : ℝ≥0∞)
-    (hB : ∀ j, (∫⁻b, angularCompactMass a q j K b ∂W)≤B)
-    (s : Finset ℕ) (c : ℕ→ℝ) (hc : ∀ j∈s,0≤c j) (hs : ∑ j∈s,c j=1) :
-    (∫⁻b, convexAngularMass a q s c K b ∂W)≤B := by
-  have hc' : ∑j∈s,ENNReal.ofReal (c j)=1 := by
-    rw [←ENNReal.ofReal_sum_of_nonneg hc,hs,ENNReal.ofReal_one]
-  simp only [convexAngularMass]
-  rw [lintegral_finsetSum s (f := fun j b => ENNReal.ofReal (c j)*angularCompactMass a q j K b) (fun j _ =>
-    measurable_const.mul (angularCompactMass_measurable a q ha j K))]
-  simp_rw [lintegral_const_mul _ (angularCompactMass_measurable a q ha _ K)]
-  exact (Finset.sum_le_sum (fun j _ => mul_le_mul' le_rfl (hB j))).trans_eq (by
-    rw [←Finset.sum_mul,hc',one_mul])
-
- 
-
-theorem convexAngularMass_liminf_finite (a q : ℝ) (ha : 1/4<a) (hq : 0<q)
-    (hqp : q<4*a-1) (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (K : ℕ→Set UpperPlane) (hK : ∀n,IsCompact (K n)) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀ k j,j∈s k→0≤c k j) (hs : ∀k,∑j∈s k,c k j=1) :
-    ∀ᵐ b ∂W, ∀ n, liminf (fun k => convexAngularMass a q (s k) (c k) (K n) b) atTop<∞ := by
-  apply ae_all_iff.mpr
-  intro n
-  obtain ⟨B,hB,hBj⟩ := angularCompactMass_uniform_mean a q ha hq hqp W hW (K n) (hK n)
-  apply ae_lt_top (Measurable.liminf (fun k =>
-    convexAngularMass_measurable a q ha (s k) (c k) (K n)))
-  apply ne_top_of_lt
-  apply lt_of_le_of_lt (lintegral_liminf_le' (fun k =>
-    (convexAngularMass_measurable a q ha (s k) (c k) (K n)).aemeasurable))
-  exact (liminf_le_of_frequently_le (Eventually.of_forall (fun k =>
-    convexAngularMass_mean_bound a q ha W (K n) B hBj (s k) (c k) (hc k) (hs k))).frequently).trans_lt hB
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_raw_mass_endpoint (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
-        (∀ i j, t i≠∞ → t j=∞ →
-          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) := by
-  obtain ⟨s,c,hc,hs,hsum,hμ,hmeas,hprog,hmom,hL1,hmean,hcond,hH,hconv⟩ :=
-    actual_raw_mass_extraction a ha W hW t
-  let μ := recoveredMass a ha s c hc
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) :=
-    (hμ v b).1
-  let K : ℝ≥0∞ → Kernel ExactSLEGenerator.Driver UpperPlane := fun v => ⟨μ v,hmeas v⟩
-  have hint (i : ℕ) (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂μ (t i) b) W :=
-    memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using hmom i f 1)
-  have hfut (i j : ℕ) : ∀ᵐ b ∂W, t i≠∞ → t j=∞ →
-      (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0 := by
-    by_cases hi : t i=∞
-    · exact Eventually.of_forall (fun _ h => False.elim (h hi))
-    by_cases hj : t j=∞
-    · have hz := conditional_future_support a ha W hW (t i).toNNReal (K (t i)) (K (t j))
-        (hint i) (hint j) (by
-          filter_upwards [hconv] with b hb
-          exact hb.1 i j (by rw [hj]; exact le_top))
-        (hcond i j (t i).toNNReal (ENNReal.coe_toNNReal hi).symm hj)
-      filter_upwards [hz] with b hb
-      exact fun _ _ => hb
-    · exact Eventually.of_forall (fun _ _ h => False.elim (hj h))
-  refine ⟨s,c,hc,hs,hsum,hμ,recoveredMass_zero a ha s c hc,hmeas,hprog,hmom,?_,hL1,?_,hH,?_⟩
-  · intro i A hA m
-    exact compact_mass_memLp (K (t i)) W (m : ℝ≥0∞) (fun f => hmom i f m) A hA
-  · intro i hi
-    exact initialGreenMeasure_eq_average a (K (t i)) W (hint i) (hmean i hi)
-  · filter_upwards [hconv,ae_all_iff.mpr (fun i => ae_all_iff.mpr (hfut i))] with b hb hf
-    exact ⟨hb.1,hb.2,hf⟩
-
-end ExactSLEGreen
-end
- 
-
- 
-section
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEKernel SLEExactGauge.MassVersions
-
- 
-
-def localPairMeasure (α d C : ℝ) (φ : C_c(UpperPlane,ℝ≥0)) : Measure (UpperPlane × UpperPlane) :=
-  (upperArea.prod upperArea).withDensity (fun p =>
-    (φ p.1:ℝ≥0∞)*(φ p.2:ℝ≥0∞)*(ENNReal.ofReal C*pairKernel α d p))
-
-lemma localPairMeasure_finite (α d C : ℝ) (hα : 0<α) (hα1 : α<1) (hd : 0<d)
-    (φ : C_c(UpperPlane,ℝ≥0)) : IsFiniteMeasure (localPairMeasure α d C φ) := by
-  let K := tsupport φ
-  let B : ℝ≥0∞ := ENNReal.ofReal ‖φ.toReal.toBoundedContinuousFunction‖
-  have hB (x : UpperPlane) : (φ x:ℝ≥0∞) ≤ B := by
-    have h := φ.toReal.toBoundedContinuousFunction.norm_coe_le_norm x
-    have h' : (φ x:ℝ)≤‖φ.toReal.toBoundedContinuousFunction‖ := by
-      simpa only [CompactlySupportedContinuousMap.toBoundedContinuousFunction_apply,
-        CompactlySupportedContinuousMap.toReal_apply,Real.norm_of_nonneg (φ x).coe_nonneg] using h
-    simpa only [ENNReal.ofReal_coe_nnreal] using ENNReal.ofReal_le_ofReal h'
-  have hK : IsCompact K := φ.hasCompactSupport
-  apply isFiniteMeasure_withDensity
-  apply ne_of_lt
-  calc
-    _ ≤ ∫⁻p, (K ×ˢ K).indicator (fun p => (B*B*ENNReal.ofReal C)*pairKernel α d p) p
-        ∂upperArea.prod upperArea := by
-      apply lintegral_mono
-      intro p
-      by_cases hx : p.1∈K
-      · by_cases hy : p.2∈K
-        · rw [indicator_of_mem (show p∈K ×ˢ K from ⟨hx,hy⟩)]
-          calc
-            _ ≤ (B*B)*(ENNReal.ofReal C*pairKernel α d p) :=
-              mul_le_mul' (mul_le_mul' (hB p.1) (hB p.2)) le_rfl
-            _ = _ := by ac_rfl
-        · simp [K, image_eq_zero_of_notMem_tsupport hy, hy]
-      · simp [K, image_eq_zero_of_notMem_tsupport hx, hx]
-    _ = (B*B*ENNReal.ofReal C)*(∫⁻p in K ×ˢ K, pairKernel α d p ∂upperArea.prod upperArea) := by
-      rw [lintegral_indicator (hK.measurableSet.prod hK.measurableSet),
-        lintegral_const_mul _ (pairKernel_measurable α d hα.le)]
-    _ < ∞ := ENNReal.mul_lt_top (by dsimp [B]; finiteness)
-      (pairKernel_integrable_on_compact α d hα hα1 hd K hK)
-
-lemma localPairMeasure_diagonal (α d C : ℝ) (φ : C_c(UpperPlane,ℝ≥0)) :
-    localPairMeasure α d C φ (diagonal UpperPlane) = 0 :=
-  diagonal_null_of_ac_prod upperArea _ (withDensity_absolutelyContinuous _ _)
-
- 
-
-theorem localized_actualConvexMass_pair_bound (a : ℝ) (ha : 1/4<a) :
-    ∃ C : ℝ,0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ (φ : C_c(UpperPlane,ℝ≥0)) (d : ℝ),0<d → (∀ z∈tsupport φ,d≤2*z.1.im) →
-    ∀ (s : Finset ℕ) (c : ℕ → ℝ), (∀ j∈s,0≤c j) → (∑j∈s,c j=1) →
-    ∀ t : ℝ≥0∞,
-      W.bind (fun b => ((actualConvexMass a s c t b).withDensity (fun z => φ z)).prod
-        ((actualConvexMass a s c t b).withDensity (fun z => φ z))) ≤
-          localPairMeasure (alpha a) d C φ := by
-  obtain ⟨C,hC,hb⟩ := actualConvexDensity_pair_bound a ha
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW φ d hd hdφ s c hc hsum t
-  let ρ := actualConvexDensity a s c t
-  have hρ : Measurable (Function.uncurry ρ) := actualConvexDensity_measurable a ha s c t
-  have hφ : Measurable (fun z => (φ z:ℝ≥0∞)) := φ.continuous.measurable.coe_nnreal_ennreal
-  have he (b : ExactSLEGenerator.Driver) :
-      (actualConvexMass a s c t b).withDensity (fun z => φ z) =
-        upperArea.withDensity (fun z => ρ b z*(φ z:ℝ≥0∞)) := by
-    rw [actualConvexMass_density a ha s c t b]
-    exact (withDensity_mul upperArea (hρ.comp (measurable_const.prodMk measurable_id)) hφ).symm
-  simp_rw [he]
-  exact average_localized_density_prod_le W upperArea ρ hρ φ
-    (fun p => ENNReal.ofReal C*pairKernel (alpha a) d p)
-    (fun x hx y hy => hb W hW s c hc hsum t x y d hd (hdφ x hx) (hdφ y hy))
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
-open MeasureTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [BorelSpace X]
-
- 
-
-theorem ae_atomless_of_localizations (W : Measure Ω) (ν : Ω → Measure X)
-    (h : ∀ φ : C_c(X,ℝ≥0),∀ᵐ ω ∂W,
-      NullSingletonClass ((ν ω).withDensity (fun x => φ x))) :
-    ∀ᵐ ω ∂W,NullSingletonClass (ν ω) := by
-  let K := CompactExhaustion.choice X
-  have hcut (n : ℕ) : ∃ φ : C_c(X,ℝ≥0),∀ x∈K n,φ x=1 := by
-    obtain ⟨g,hg1,-,hgc,-⟩ := exists_continuous_one_zero_of_isCompact
-      (K.isCompact n) isClosed_empty (disjoint_empty (K n))
-    let f : C_c(X,ℝ) := ⟨g,hgc⟩
-    refine ⟨f.nnrealPart,?_⟩
-    intro x hx
-    rw [CompactlySupportedContinuousMap.nnrealPart_apply]
-    change (g x).toNNReal=1
-    simp only [hg1 hx,Pi.one_apply,Real.toNNReal_one]
-  choose φ hφ using hcut
-  filter_upwards [ae_all_iff.mpr (fun n => h (φ n))] with ω hω
-  constructor
-  intro x
-  obtain ⟨n,hn⟩ := K.exists_mem x
-  let := hω n
-  have hz : ((ν ω).withDensity (fun z => φ n z)) {x}=0 := measure_singleton x
-  rw [withDensity_apply _ (measurableSet_singleton x),lintegral_singleton,hφ n x hn] at hz
-  simpa only [ENNReal.coe_one,one_mul] using hz
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEKernel SLEExactGauge.MassVersions
-
- 
-
-theorem actual_limit_pair_bound (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-    (hc : ∀ k j,j∈s k → 0≤c k j) (hsum : ∀ k,∑j∈s k,c k j=1)
-    (t : ℝ≥0∞) (ν : ExactSLEGenerator.Driver → Measure UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (ν b)] (hν : Measurable ν)
-    (hconv : ∀ᵐ b ∂W,∀ f : C_c(UpperPlane,ℝ),
-      Tendsto (fun k => ∑j∈s k,c k j * ∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂ν b)))
-    (φ : C_c(UpperPlane,ℝ≥0)) :
-    ∃ (Q : Measure (UpperPlane × UpperPlane)), IsFiniteMeasure Q ∧
-      Q (diagonal UpperPlane)=0 ∧
-      W.bind (fun b => (ν b |>.withDensity (fun z => φ z)).prod
-        (ν b |>.withDensity (fun z => φ z))) ≤ Q := by
-  obtain ⟨C,hC,hb⟩ := localized_actualConvexMass_pair_bound a ha
-  obtain ⟨d,hd,hdφ⟩ := upper_compact_lower (tsupport φ) φ.hasCompactSupport
-  let Q := localPairMeasure (alpha a) d C φ
-  have hα1 : alpha a<1 := by
-    have ha0 : 0<a := by linarith
-    dsimp [alpha]
-    have h : (0:ℝ)<1/(4*a) := by positivity
-    linarith
-  have hQ : IsFiniteMeasure Q := localPairMeasure_finite _ _ _ (alpha_pos ha) hα1 hd φ
-  let := hQ
-  let (k : ℕ) (b : ExactSLEGenerator.Driver) :
-      IsFiniteMeasureOnCompacts (actualConvexMass a (s k) (c k) t b) :=
-    actualConvexMass_finiteOnCompacts a ha (s k) (c k) t b
-  let μk := fun k b => localizedFinite (actualConvexMass a (s k) (c k) t b) φ
-  let μ := fun b => localizedFinite (ν b) φ
-  have hμk (k) : Measurable (μk k) := measurable_localizedFinite _
-    (actualConvexMass_measurable a ha (s k) (c k) t) φ
-  have hμ : Measurable μ := measurable_localizedFinite ν hν φ
-  have hl : ∀ᵐ b ∂W,Tendsto (fun k => μk k b) atTop (𝓝 (μ b)) := by
-    filter_upwards [hconv] with b hb'
-    apply localizedFinite_tendsto
-    intro f
-    simpa only [actualConvexMass_integral a ha _ _ (hc _) t b f] using hb' f
-  refine ⟨Q,hQ,localPairMeasure_diagonal _ _ _ φ,?_⟩
-  exact average_prod_le_of_weak_limit W μk μ hμk hμ hl Q
-    (fun k => hb W hW φ d hd hdφ (s k) (c k) (hc k) (hsum k) t)
-
- 
-
-theorem actual_limit_localized_atomless (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-    (hc : ∀ k j,j∈s k → 0≤c k j) (hsum : ∀ k,∑j∈s k,c k j=1)
-    (t : ℝ≥0∞) (ν : ExactSLEGenerator.Driver → Measure UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (ν b)] (hν : Measurable ν)
-    (hconv : ∀ᵐ b ∂W,∀ f : C_c(UpperPlane,ℝ),
-      Tendsto (fun k => ∑j∈s k,c k j * ∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂ν b)))
-    (φ : C_c(UpperPlane,ℝ≥0)) :
-    ∀ᵐ b ∂W,NullSingletonClass ((ν b).withDensity (fun z => φ z)) := by
-  obtain ⟨Q,hQ,hd,hb⟩ := actual_limit_pair_bound a ha W hW s c hc hsum t ν hν hconv φ
-  apply ae_nullSingleton_of_average_prod_diagonal W
-    (fun b => localizedFinite (ν b) φ) (measurable_localizedFinite ν hν φ)
-  apply le_antisymm _ bot_le
-  exact (hb (diagonal UpperPlane)).trans_eq hd
-
- 
-
-theorem actual_limit_atomless (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-    (hc : ∀ k j,j∈s k → 0≤c k j) (hsum : ∀ k,∑j∈s k,c k j=1)
-    (t : ℝ≥0∞) (ν : ExactSLEGenerator.Driver → Measure UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (ν b)] (hν : Measurable ν)
-    (hconv : ∀ᵐ b ∂W,∀ f : C_c(UpperPlane,ℝ),
-      Tendsto (fun k => ∑j∈s k,c k j * ∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂ν b))) :
-    ∀ᵐ b ∂W,NullSingletonClass (ν b) :=
-  ae_atomless_of_localizations W ν
-    (actual_limit_localized_atomless a ha W hW s c hc hsum t ν hν hconv)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_raw_atomless_mass_endpoint (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
-        (∀ i j, t i≠∞ → t j=∞ →
-          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
-        (∀ i, NullSingletonClass (μ (t i) b)) := by
-  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
-    actual_raw_mass_endpoint a ha W hW t
-  let μ := recoveredMass a ha s c hc
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) :=
-    (hμ v b).1
-  have hnon (i : ℕ) : ∀ᵐ b ∂W,NullSingletonClass (μ (t i) b) := by
-    apply actual_limit_atomless a ha W hW s c hc hsum (t i) (μ (t i)) (hmeas (t i))
-    filter_upwards [hconv] with b hb
-    exact hb.2.1 i
-  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
-  filter_upwards [hconv,ae_all_iff.mpr hnon] with b hb hn
-  exact ⟨hb.1,hb.2.1,hb.2.2,hn⟩
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.GlobalAngleBudget
+import OAI.Probability.SLEGauge.FutureSupport
+import OAI.Probability.SLEGauge.FiniteProductLimit
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+
+ 
+
+def angularDensity (a q : ℝ) (j : ℕ) (b : ExactSLEGenerator.Driver) (z : UpperPlane) : ℝ≥0∞ :=
+  ENNReal.ofReal (Real.exp (-alpha a*Real.log 2)*
+    radialTestWeight a (angleLift (sinePower (-q))) b.1 z.1
+      (levelRadius a (2*z.1.im) (j+1)))
+
+def angularCompactMass (a q : ℝ) (j : ℕ) (K : Set UpperPlane)
+    (b : ExactSLEGenerator.Driver) : ℝ≥0∞ := ∫⁻z in K, angularDensity a q j b z ∂upperArea
+
+lemma angularDensity_measurable (a q : ℝ) (ha : 1/4<a) (j : ℕ) :
+    Measurable (Function.uncurry (angularDensity a q j)) := by
+  have he (z : UpperPlane) : levelRadius a (2*z.1.im) (j+1)=z.1.im*Real.exp (-2*a*(j+1)) := by
+    simp [levelRadius]
+  unfold angularDensity Function.uncurry
+  simp only [he]
+  exact (measurable_const.mul (joint_radialTestWeight_measurable a ha _ (Real.exp_pos _)
+      (level_step_le_one a ha j) _ (angleLift_sinePower_measurable (-q)))).ennreal_ofReal
+
+lemma angularCompactMass_measurable (a q : ℝ) (ha : 1/4<a) (j : ℕ) (K : Set UpperPlane) :
+    Measurable (angularCompactMass a q j K) := (angularDensity_measurable a q ha j).lintegral_prod_right
+
+lemma angularInitial_continuous (a q C : ℝ) : Continuous (fun z : UpperPlane =>
+    physicalGreen a (z.1,0)*(angleLift (sinePower (-q)) (z.1,0)+C)) := by
+  apply continuous_iff_continuousAt.mpr
+  intro z
+  have hsin := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem (z.1,0)).1
+    (physicalTheta_mem (z.1,0)).2
+  have hi : Continuous (fun w : UpperPlane => (w.1,(0:ℝ))) :=
+    continuous_subtype_val.prodMk continuous_const
+  exact ((contDiffAt_physicalGreen a (z.1,0) z.2).continuousAt.mul
+    (((angleLift_smooth (sinePower (-q)) _ z.2 (sinePower_smooth (-q) _ hsin)).continuousAt).add_const C)).comp
+      (f := fun w : UpperPlane => (w.1,(0:ℝ))) hi.continuousAt
+
+ 
+theorem angularCompactMass_uniform_mean (a q : ℝ) (ha : 1/4<a) (hq : 0<q)
+    (hqp : q<4*a-1) (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (K : Set UpperPlane) (hK : IsCompact K) :
+    ∃ B : ℝ≥0∞, B<∞ ∧ ∀ j, (∫⁻b, angularCompactMass a q j K b ∂W)≤B := by
+  obtain ⟨C,hC,hbound⟩ := exists_finite_level_angle_bound a q ha hq hqp
+  let A : UpperPlane→ℝ := fun z => physicalGreen a (z.1,0)*
+    (angleLift (sinePower (-q)) (z.1,0)+C)
+  have hA : Continuous A := angularInitial_continuous a q C
+  let R := Real.exp (-alpha a*Real.log 2)
+  let B : ℝ≥0∞ := ∫⁻z in K, ENNReal.ofReal (R*A z) ∂upperArea
+  have hB : B<∞ := (ContinuousOn.integrableOn_compact (μ := upperArea) hK
+    (show ContinuousOn (fun z => R*A z) K from (continuous_const.mul hA).continuousOn)).lintegral_lt_top
+  refine ⟨B,hB,?_⟩
+  intro j
+  have hm := angularDensity_measurable a q ha j
+  simp only [angularCompactMass]
+  rw [lintegral_lintegral_swap hm.aemeasurable]
+  apply setLIntegral_mono' hK.measurableSet
+  intro z _
+  let f : ExactSLEGenerator.Driver→ℝ := fun b =>
+    radialTestWeight a (angleLift (sinePower (-q))) b.1 z.1
+      (levelRadius a (2*z.1.im) (j+1))
+  have hf : Measurable f := by
+    have hh := (joint_radialTestWeight_measurable a ha _ (Real.exp_pos _)
+      (level_step_le_one a ha j) _ (angleLift_sinePower_measurable (-q))).comp
+      (measurable_id.prodMk (measurable_const (a := z)))
+    simpa [f,levelRadius,Function.comp_def] using hh
+  simp_rw [angularDensity,ENNReal.ofReal_mul (Real.exp_pos _).le]
+  rw [lintegral_const_mul _ hf.ennreal_ofReal]
+  exact (mul_le_mul' le_rfl (hbound W hW z.1 z.2 (j+1) (by positivity))).trans_eq
+    (ENNReal.ofReal_mul (Real.exp_pos _).le).symm
+
+ 
+def convexAngularMass (a q : ℝ) (s : Finset ℕ) (c : ℕ→ℝ)
+    (K : Set UpperPlane) (b : ExactSLEGenerator.Driver) : ℝ≥0∞ :=
+  ∑j∈s, ENNReal.ofReal (c j)*angularCompactMass a q j K b
+
+lemma convexAngularMass_measurable (a q : ℝ) (ha : 1/4<a) (s : Finset ℕ)
+    (c : ℕ→ℝ) (K : Set UpperPlane) : Measurable (convexAngularMass a q s c K) := by
+  apply Finset.measurable_sum
+  intro j _
+  exact measurable_const.mul (angularCompactMass_measurable a q ha j K)
+
+lemma convexAngularMass_mean_bound (a q : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) (K : Set UpperPlane) (B : ℝ≥0∞)
+    (hB : ∀ j, (∫⁻b, angularCompactMass a q j K b ∂W)≤B)
+    (s : Finset ℕ) (c : ℕ→ℝ) (hc : ∀ j∈s,0≤c j) (hs : ∑ j∈s,c j=1) :
+    (∫⁻b, convexAngularMass a q s c K b ∂W)≤B := by
+  have hc' : ∑j∈s,ENNReal.ofReal (c j)=1 := by
+    rw [←ENNReal.ofReal_sum_of_nonneg hc,hs,ENNReal.ofReal_one]
+  simp only [convexAngularMass]
+  rw [lintegral_finsetSum s (f := fun j b => ENNReal.ofReal (c j)*angularCompactMass a q j K b) (fun j _ =>
+    measurable_const.mul (angularCompactMass_measurable a q ha j K))]
+  simp_rw [lintegral_const_mul _ (angularCompactMass_measurable a q ha _ K)]
+  exact (Finset.sum_le_sum (fun j _ => mul_le_mul' le_rfl (hB j))).trans_eq (by
+    rw [←Finset.sum_mul,hc',one_mul])
+
+ 
+
+theorem convexAngularMass_liminf_finite (a q : ℝ) (ha : 1/4<a) (hq : 0<q)
+    (hqp : q<4*a-1) (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (K : ℕ→Set UpperPlane) (hK : ∀n,IsCompact (K n)) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀ k j,j∈s k→0≤c k j) (hs : ∀k,∑j∈s k,c k j=1) :
+    ∀ᵐ b ∂W, ∀ n, liminf (fun k => convexAngularMass a q (s k) (c k) (K n) b) atTop<∞ := by
+  apply ae_all_iff.mpr
+  intro n
+  obtain ⟨B,hB,hBj⟩ := angularCompactMass_uniform_mean a q ha hq hqp W hW (K n) (hK n)
+  apply ae_lt_top (Measurable.liminf (fun k =>
+    convexAngularMass_measurable a q ha (s k) (c k) (K n)))
+  apply ne_top_of_lt
+  apply lt_of_le_of_lt (lintegral_liminf_le' (fun k =>
+    (convexAngularMass_measurable a q ha (s k) (c k) (K n)).aemeasurable))
+  exact (liminf_le_of_frequently_le (Eventually.of_forall (fun k =>
+    convexAngularMass_mean_bound a q ha W (K n) B hBj (s k) (c k) (hc k) (hs k))).frequently).trans_lt hB
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_raw_mass_endpoint (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
+        (∀ i j, t i≠∞ → t j=∞ →
+          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) := by
+  obtain ⟨s,c,hc,hs,hsum,hμ,hmeas,hprog,hmom,hL1,hmean,hcond,hH,hconv⟩ :=
+    actual_raw_mass_extraction a ha W hW t
+  let μ := recoveredMass a ha s c hc
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) :=
+    (hμ v b).1
+  let K : ℝ≥0∞ → Kernel ExactSLEGenerator.Driver UpperPlane := fun v => ⟨μ v,hmeas v⟩
+  have hint (i : ℕ) (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂μ (t i) b) W :=
+    memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using hmom i f 1)
+  have hfut (i j : ℕ) : ∀ᵐ b ∂W, t i≠∞ → t j=∞ →
+      (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0 := by
+    by_cases hi : t i=∞
+    · exact Eventually.of_forall (fun _ h => False.elim (h hi))
+    by_cases hj : t j=∞
+    · have hz := conditional_future_support a ha W hW (t i).toNNReal (K (t i)) (K (t j))
+        (hint i) (hint j) (by
+          filter_upwards [hconv] with b hb
+          exact hb.1 i j (by rw [hj]; exact le_top))
+        (hcond i j (t i).toNNReal (ENNReal.coe_toNNReal hi).symm hj)
+      filter_upwards [hz] with b hb
+      exact fun _ _ => hb
+    · exact Eventually.of_forall (fun _ _ h => False.elim (hj h))
+  refine ⟨s,c,hc,hs,hsum,hμ,recoveredMass_zero a ha s c hc,hmeas,hprog,hmom,?_,hL1,?_,hH,?_⟩
+  · intro i A hA m
+    exact compact_mass_memLp (K (t i)) W (m : ℝ≥0∞) (fun f => hmom i f m) A hA
+  · intro i hi
+    exact initialGreenMeasure_eq_average a (K (t i)) W (hint i) (hmean i hi)
+  · filter_upwards [hconv,ae_all_iff.mpr (fun i => ae_all_iff.mpr (hfut i))] with b hb hf
+    exact ⟨hb.1,hb.2,hf⟩
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEKernel SLEExactGauge.MassVersions
+
+ 
+
+def localPairMeasure (α d C : ℝ) (φ : C_c(UpperPlane,ℝ≥0)) : Measure (UpperPlane × UpperPlane) :=
+  (upperArea.prod upperArea).withDensity (fun p =>
+    (φ p.1:ℝ≥0∞)*(φ p.2:ℝ≥0∞)*(ENNReal.ofReal C*pairKernel α d p))
+
+lemma localPairMeasure_finite (α d C : ℝ) (hα : 0<α) (hα1 : α<1) (hd : 0<d)
+    (φ : C_c(UpperPlane,ℝ≥0)) : IsFiniteMeasure (localPairMeasure α d C φ) := by
+  let K := tsupport φ
+  let B : ℝ≥0∞ := ENNReal.ofReal ‖φ.toReal.toBoundedContinuousFunction‖
+  have hB (x : UpperPlane) : (φ x:ℝ≥0∞) ≤ B := by
+    have h := φ.toReal.toBoundedContinuousFunction.norm_coe_le_norm x
+    have h' : (φ x:ℝ)≤‖φ.toReal.toBoundedContinuousFunction‖ := by
+      simpa only [CompactlySupportedContinuousMap.toBoundedContinuousFunction_apply,
+        CompactlySupportedContinuousMap.toReal_apply,Real.norm_of_nonneg (φ x).coe_nonneg] using h
+    simpa only [ENNReal.ofReal_coe_nnreal] using ENNReal.ofReal_le_ofReal h'
+  have hK : IsCompact K := φ.hasCompactSupport
+  apply isFiniteMeasure_withDensity
+  apply ne_of_lt
+  calc
+    _ ≤ ∫⁻p, (K ×ˢ K).indicator (fun p => (B*B*ENNReal.ofReal C)*pairKernel α d p) p
+        ∂upperArea.prod upperArea := by
+      apply lintegral_mono
+      intro p
+      by_cases hx : p.1∈K
+      · by_cases hy : p.2∈K
+        · rw [indicator_of_mem (show p∈K ×ˢ K from ⟨hx,hy⟩)]
+          calc
+            _ ≤ (B*B)*(ENNReal.ofReal C*pairKernel α d p) :=
+              mul_le_mul' (mul_le_mul' (hB p.1) (hB p.2)) le_rfl
+            _ = _ := by ac_rfl
+        · simp [K, image_eq_zero_of_notMem_tsupport hy, hy]
+      · simp [K, image_eq_zero_of_notMem_tsupport hx, hx]
+    _ = (B*B*ENNReal.ofReal C)*(∫⁻p in K ×ˢ K, pairKernel α d p ∂upperArea.prod upperArea) := by
+      rw [lintegral_indicator (hK.measurableSet.prod hK.measurableSet),
+        lintegral_const_mul _ (pairKernel_measurable α d hα.le)]
+    _ < ∞ := ENNReal.mul_lt_top (by dsimp [B]; finiteness)
+      (pairKernel_integrable_on_compact α d hα hα1 hd K hK)
+
+lemma localPairMeasure_diagonal (α d C : ℝ) (φ : C_c(UpperPlane,ℝ≥0)) :
+    localPairMeasure α d C φ (diagonal UpperPlane) = 0 :=
+  diagonal_null_of_ac_prod upperArea _ (withDensity_absolutelyContinuous _ _)
+
+ 
+
+theorem localized_actualConvexMass_pair_bound (a : ℝ) (ha : 1/4<a) :
+    ∃ C : ℝ,0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ (φ : C_c(UpperPlane,ℝ≥0)) (d : ℝ),0<d → (∀ z∈tsupport φ,d≤2*z.1.im) →
+    ∀ (s : Finset ℕ) (c : ℕ → ℝ), (∀ j∈s,0≤c j) → (∑j∈s,c j=1) →
+    ∀ t : ℝ≥0∞,
+      W.bind (fun b => ((actualConvexMass a s c t b).withDensity (fun z => φ z)).prod
+        ((actualConvexMass a s c t b).withDensity (fun z => φ z))) ≤
+          localPairMeasure (alpha a) d C φ := by
+  obtain ⟨C,hC,hb⟩ := actualConvexDensity_pair_bound a ha
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW φ d hd hdφ s c hc hsum t
+  let ρ := actualConvexDensity a s c t
+  have hρ : Measurable (Function.uncurry ρ) := actualConvexDensity_measurable a ha s c t
+  have hφ : Measurable (fun z => (φ z:ℝ≥0∞)) := φ.continuous.measurable.coe_nnreal_ennreal
+  have he (b : ExactSLEGenerator.Driver) :
+      (actualConvexMass a s c t b).withDensity (fun z => φ z) =
+        upperArea.withDensity (fun z => ρ b z*(φ z:ℝ≥0∞)) := by
+    rw [actualConvexMass_density a ha s c t b]
+    exact (withDensity_mul upperArea (hρ.comp (measurable_const.prodMk measurable_id)) hφ).symm
+  simp_rw [he]
+  exact average_localized_density_prod_le W upperArea ρ hρ φ
+    (fun p => ENNReal.ofReal C*pairKernel (alpha a) d p)
+    (fun x hx y hy => hb W hW s c hc hsum t x y d hd (hdφ x hx) (hdφ y hy))
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+open MeasureTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [BorelSpace X]
+
+ 
+
+theorem ae_atomless_of_localizations (W : Measure Ω) (ν : Ω → Measure X)
+    (h : ∀ φ : C_c(X,ℝ≥0),∀ᵐ ω ∂W,
+      NullSingletonClass ((ν ω).withDensity (fun x => φ x))) :
+    ∀ᵐ ω ∂W,NullSingletonClass (ν ω) := by
+  let K := CompactExhaustion.choice X
+  have hcut (n : ℕ) : ∃ φ : C_c(X,ℝ≥0),∀ x∈K n,φ x=1 := by
+    obtain ⟨g,hg1,-,hgc,-⟩ := exists_continuous_one_zero_of_isCompact
+      (K.isCompact n) isClosed_empty (disjoint_empty (K n))
+    let f : C_c(X,ℝ) := ⟨g,hgc⟩
+    refine ⟨f.nnrealPart,?_⟩
+    intro x hx
+    rw [CompactlySupportedContinuousMap.nnrealPart_apply]
+    change (g x).toNNReal=1
+    simp only [hg1 hx,Pi.one_apply,Real.toNNReal_one]
+  choose φ hφ using hcut
+  filter_upwards [ae_all_iff.mpr (fun n => h (φ n))] with ω hω
+  constructor
+  intro x
+  obtain ⟨n,hn⟩ := K.exists_mem x
+  let := hω n
+  have hz : ((ν ω).withDensity (fun z => φ n z)) {x}=0 := measure_singleton x
+  rw [withDensity_apply _ (measurableSet_singleton x),lintegral_singleton,hφ n x hn] at hz
+  simpa only [ENNReal.coe_one,one_mul] using hz
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEKernel SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_limit_pair_bound (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+    (hc : ∀ k j,j∈s k → 0≤c k j) (hsum : ∀ k,∑j∈s k,c k j=1)
+    (t : ℝ≥0∞) (ν : ExactSLEGenerator.Driver → Measure UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (ν b)] (hν : Measurable ν)
+    (hconv : ∀ᵐ b ∂W,∀ f : C_c(UpperPlane,ℝ),
+      Tendsto (fun k => ∑j∈s k,c k j * ∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂ν b)))
+    (φ : C_c(UpperPlane,ℝ≥0)) :
+    ∃ (Q : Measure (UpperPlane × UpperPlane)), IsFiniteMeasure Q ∧
+      Q (diagonal UpperPlane)=0 ∧
+      W.bind (fun b => (ν b |>.withDensity (fun z => φ z)).prod
+        (ν b |>.withDensity (fun z => φ z))) ≤ Q := by
+  obtain ⟨C,hC,hb⟩ := localized_actualConvexMass_pair_bound a ha
+  obtain ⟨d,hd,hdφ⟩ := upper_compact_lower (tsupport φ) φ.hasCompactSupport
+  let Q := localPairMeasure (alpha a) d C φ
+  have hα1 : alpha a<1 := by
+    have ha0 : 0<a := by linarith
+    dsimp [alpha]
+    have h : (0:ℝ)<1/(4*a) := by positivity
+    linarith
+  have hQ : IsFiniteMeasure Q := localPairMeasure_finite _ _ _ (alpha_pos ha) hα1 hd φ
+  let := hQ
+  let (k : ℕ) (b : ExactSLEGenerator.Driver) :
+      IsFiniteMeasureOnCompacts (actualConvexMass a (s k) (c k) t b) :=
+    actualConvexMass_finiteOnCompacts a ha (s k) (c k) t b
+  let μk := fun k b => localizedFinite (actualConvexMass a (s k) (c k) t b) φ
+  let μ := fun b => localizedFinite (ν b) φ
+  have hμk (k) : Measurable (μk k) := measurable_localizedFinite _
+    (actualConvexMass_measurable a ha (s k) (c k) t) φ
+  have hμ : Measurable μ := measurable_localizedFinite ν hν φ
+  have hl : ∀ᵐ b ∂W,Tendsto (fun k => μk k b) atTop (𝓝 (μ b)) := by
+    filter_upwards [hconv] with b hb'
+    apply localizedFinite_tendsto
+    intro f
+    simpa only [actualConvexMass_integral a ha _ _ (hc _) t b f] using hb' f
+  refine ⟨Q,hQ,localPairMeasure_diagonal _ _ _ φ,?_⟩
+  exact average_prod_le_of_weak_limit W μk μ hμk hμ hl Q
+    (fun k => hb W hW φ d hd hdφ (s k) (c k) (hc k) (hsum k) t)
+
+ 
+
+theorem actual_limit_localized_atomless (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+    (hc : ∀ k j,j∈s k → 0≤c k j) (hsum : ∀ k,∑j∈s k,c k j=1)
+    (t : ℝ≥0∞) (ν : ExactSLEGenerator.Driver → Measure UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (ν b)] (hν : Measurable ν)
+    (hconv : ∀ᵐ b ∂W,∀ f : C_c(UpperPlane,ℝ),
+      Tendsto (fun k => ∑j∈s k,c k j * ∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂ν b)))
+    (φ : C_c(UpperPlane,ℝ≥0)) :
+    ∀ᵐ b ∂W,NullSingletonClass ((ν b).withDensity (fun z => φ z)) := by
+  obtain ⟨Q,hQ,hd,hb⟩ := actual_limit_pair_bound a ha W hW s c hc hsum t ν hν hconv φ
+  apply ae_nullSingleton_of_average_prod_diagonal W
+    (fun b => localizedFinite (ν b) φ) (measurable_localizedFinite ν hν φ)
+  apply le_antisymm _ bot_le
+  exact (hb (diagonal UpperPlane)).trans_eq hd
+
+ 
+
+theorem actual_limit_atomless (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+    (hc : ∀ k j,j∈s k → 0≤c k j) (hsum : ∀ k,∑j∈s k,c k j=1)
+    (t : ℝ≥0∞) (ν : ExactSLEGenerator.Driver → Measure UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (ν b)] (hν : Measurable ν)
+    (hconv : ∀ᵐ b ∂W,∀ f : C_c(UpperPlane,ℝ),
+      Tendsto (fun k => ∑j∈s k,c k j * ∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂ν b))) :
+    ∀ᵐ b ∂W,NullSingletonClass (ν b) :=
+  ae_atomless_of_localizations W ν
+    (actual_limit_localized_atomless a ha W hW s c hc hsum t ν hν hconv)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_raw_atomless_mass_endpoint (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
+        (∀ i j, t i≠∞ → t j=∞ →
+          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
+        (∀ i, NullSingletonClass (μ (t i) b)) := by
+  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
+    actual_raw_mass_endpoint a ha W hW t
+  let μ := recoveredMass a ha s c hc
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) :=
+    (hμ v b).1
+  have hnon (i : ℕ) : ∀ᵐ b ∂W,NullSingletonClass (μ (t i) b) := by
+    apply actual_limit_atomless a ha W hW s c hc hsum (t i) (μ (t i)) (hmeas (t i))
+    filter_upwards [hconv] with b hb
+    exact hb.2.1 i
+  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
+  filter_upwards [hconv,ae_all_iff.mpr hnon] with b hb hn
+  exact ⟨hb.1,hb.2.1,hb.2.2,hn⟩
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/BrownianGerm.lean b/lean/OAI/Probability/SLEGauge/BrownianGerm.lean
index 29fc213854821be2a4dd45a7a6cdf69bd7e1ab40..eb751c956fe9483770ae0d6a27e2d82565b45be0 100644
--- a/lean/OAI/Probability/SLEGauge/BrownianGerm.lean
+++ b/lean/OAI/Probability/SLEGauge/BrownianGerm.lean
@@ -1,559 +1,559 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.FutureSupport
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.BrownianGerm
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-
- 
-@[instance_reducible] def future (B : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) : MeasurableSpace Ω :=
-  MeasurableSpace.comap (fun ω u => B (t+u) ω - B t ω) inferInstance
-
-omit mΩ in
-lemma measurable_shift (B : ℝ≥0 → Ω → ℝ) (s u : ℝ≥0) :
-    Measurable[future B s] (fun ω => B (s+u) ω-B s ω) :=
-  (measurable_pi_apply u).comp
-    (comap_measurable (fun ω v => B (s+v) ω-B s ω))
-
-lemma future_le (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (t : ℝ≥0) :
-    future B t ≤ mΩ :=
-  (Measurable.of_eval fun offset => (hm (t + offset)).sub (hm t)).comap_le
-
-omit mΩ in
-lemma future_antitone (B : ℝ≥0 → Ω → ℝ) : Antitone (future B) := by
-  intro s t hst
-  suffices Measurable[future B s] (fun ω u => B (t+u) ω - B t ω) from
-    this.comap_le
-  let : MeasurableSpace Ω := future B s
-  refine Measurable.of_eval fun u => ?_
-  have h := (measurable_shift B s (t-s+u)).sub (measurable_shift B s (t-s))
-  convert h using 1; ext ω
-  simp only [← add_assoc, add_tsub_cancel_of_le hst, Pi.sub_apply]
-  ring
-
- 
-
-omit mΩ in
-lemma measurable_eval_iSup_future (B : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0) (t : ℝ≥0) :
-    Measurable[⨆ s : {s : ℝ≥0 // 0<s}, future B s.1] (B t) := by
-  let ε : ℕ → ℝ≥0 := fun n => ((n:ℝ≥0)+1)⁻¹
-  have hep (n : ℕ) : 0<ε n := by dsimp [ε]; positivity
-  have he : Tendsto ε atTop (𝓝 0) := by
-    simpa [ε,one_div] using (tendsto_one_div_add_atTop_nhds_zero_nat :
-      Tendsto (fun n : ℕ => 1/((n:ℝ≥0)+1)) atTop (𝓝 0))
-  let : MeasurableSpace Ω := ⨆ s : {s : ℝ≥0 // 0<s}, future B s.1
-  apply measurable_of_tendsto_metrizable
-    (f := fun n ω => B (ε n+t) ω-B (ε n) ω)
-  · intro n
-    exact (measurable_shift B (ε n) t).mono (le_iSup_of_le ⟨ε n,hep n⟩ le_rfl) le_rfl
-  · apply tendsto_pi_nhds.mpr
-    intro ω
-    have h := ((hc ω).tendsto (0+t)).comp (he.add_const t)
-    have h' := ((hc ω).tendsto 0).comp he
-    simpa [hz ω] using h.sub h'
-
- 
-
-theorem measure_zero_or_one (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, StronglyMeasurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
-    (P : Measure Ω) (hB : IsPreBrownianReal B P)
-    {E : Set Ω}
-    (hE : ∀ t : ℝ≥0, 0<t → MeasurableSet[Filtration.natural B hm t] E) :
-    P E=0 ∨ P E=1 := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  let g : MeasurableSpace Ω := ⨅ t : {t : ℝ≥0 // 0<t}, Filtration.natural B hm t.1
-  let f : {t : ℝ≥0 // 0<t} → MeasurableSpace Ω := fun t => future B t.1
-  have hg1 : g ≤ Filtration.natural B hm 1 :=
-    iInf_le (fun t : {t : ℝ≥0 // 0<t} => Filtration.natural B hm t.1) ⟨1,by norm_num⟩
-  have hg : g ≤ mΩ := hg1.trans ((Filtration.natural B hm).le 1)
-  have hi (t : {t : ℝ≥0 // 0<t}) : Indep (f t) g P := by
-    have h := (IndepFun_iff_Indep _ _ _).mp (hB.indepFun_shift t.1)
-    rw [← Filtration.natural_eq_comap B hm t.1] at h
-    exact indep_of_indep_of_le_right h (iInf_le _ t)
-  have hd : Directed (· ≤ ·) f := by
-    intro s t
-    refine ⟨⟨min s.1 t.1,lt_min s.2 t.2⟩, ?_, ?_⟩
-    · exact future_antitone B (min_le_left _ _)
-    · exact future_antitone B (min_le_right _ _)
-  have hind : Indep (⨆ t, f t) g P :=
-    indep_iSup_of_directed_le hi (fun t => @future_le Ω mΩ B (fun u => (hm u).measurable) t.1) hg hd
-  have hgE : MeasurableSet[g] E := by
-    exact (MeasurableSpace.measurableSet_iInf).mpr fun t => hE t.1 t.2
-  have hng : Filtration.natural B hm 1 ≤ ⨆ t, f t := by
-    rw [Filtration.natural_eq_comap]
-    let : MeasurableSpace Ω := ⨆ t, f t
-    exact (Measurable.of_eval fun t : Iic (1:ℝ≥0) =>
-      measurable_eval_iSup_future B hc hz t.1).comap_le
-  exact measure_eq_zero_or_one_of_indepSet_self
-    (hind.indepSet_of_measurableSet (hng _ (hE 1 (by norm_num))) hgE)
-
- 
-
-lemma measurableSet_limsup_germ
-    (ℱ : Filtration ℝ≥0 mΩ) (τ : ℕ → ℝ≥0) (hτ : Tendsto τ atTop (𝓝 0))
-    (E : ℕ → Set Ω) (hE : ∀ n, MeasurableSet[ℱ (τ n)] (E n))
-    (t : ℝ≥0) (ht : 0<t) : MeasurableSet[ℱ t] (limsup E atTop) := by
-  classical
-  have hev : ∀ᶠ n in atTop, τ n≤t :=
-    (hτ.eventually (gt_mem_nhds ht)).mono (fun _ h => h.le)
-  have heq : E =ᶠ[atTop] (fun n => if τ n≤t then E n else ∅) :=
-    hev.mono fun n hn => by simp [hn]
-  rw [limsup_congr heq]
-  let : MeasurableSpace Ω := ℱ t
-  apply MeasurableSet.measurableSet_limsup
-  intro n
-  split_ifs with h
-  · exact ℱ.mono h _ (hE n)
-  · exact MeasurableSet.empty
-
- 
-theorem measure_zero_or_one_limsup (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, StronglyMeasurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
-    (P : Measure Ω) (hB : IsPreBrownianReal B P)
-    (τ : ℕ → ℝ≥0) (hτ : Tendsto τ atTop (𝓝 0))
-    (E : ℕ → Set Ω) (hE : ∀ n, MeasurableSet[Filtration.natural B hm (τ n)] (E n)) :
-    P (limsup E atTop)=0 ∨ P (limsup E atTop)=1 :=
-  measure_zero_or_one B hm hc hz P hB
-    (measurableSet_limsup_germ (Filtration.natural B hm) τ hτ E hE)
-
-end SLEExactGauge.BrownianGerm
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.GermPositivity
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-
-omit mΩ in
-lemma mem_limsup_iff_mem_iInter {E : ℕ → Set Ω} {ω : Ω}
-    (hE : Antitone (fun n => ω ∈ E n)) :
-    ω ∈ limsup E atTop ↔ ω ∈ ⋂ n, E n := by
-  rw [mem_limsup_iff_frequently_mem,mem_iInter]
-  constructor
-  · intro h n
-    obtain ⟨m,hm,hmem⟩ := frequently_atTop.mp h n
-    exact hE hm hmem
-  · exact fun h => Filter.Frequently.of_forall h
-
- 
-
-lemma limsup_ae_eq_iInter {P : Measure Ω} {E : ℕ → Set Ω}
-    (hE : ∀ᵐ ω ∂P, Antitone (fun n => ω ∈ E n)) :
-    ∀ᵐ ω ∂P, ω ∈ limsup E atTop ↔ ω ∈ ⋂ n, E n := by
-  exact hE.mono fun _ h => mem_limsup_iff_mem_iInter h
-
- 
-
-theorem ae_all_mem (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, StronglyMeasurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
-    (P : Measure Ω) (hB : IsPreBrownianReal B P)
-    (τ : ℕ → ℝ≥0) (hτ : Tendsto τ atTop (𝓝 0))
-    (E : ℕ → Set Ω) (hE : ∀ n, MeasurableSet[Filtration.natural B hm (τ n)] (E n))
-    (hanti : ∀ᵐ ω ∂P, Antitone (fun n => ω ∈ E n))
-    {p : ℝ≥0∞} (hp : 0<p) (hlower : ∀ n, p≤P (E n)) :
-    ∀ᵐ ω ∂P, ∀ n, ω ∈ E n := by
-  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
-  have he (n : ℕ) : MeasurableSet (E n) :=
-    (Filtration.natural B hm).le (τ n) _ (hE n)
-  have hmI : MeasurableSet (⋂ n, E n) := MeasurableSet.iInter he
-  have hprob : p≤P (⋂ n, E n) := by
-    rw [measure_iInter_of_ae_antitone hanti (fun n => (he n).nullMeasurableSet)
-      ⟨0,measure_ne_top _ _⟩]
-    exact le_iInf hlower
-  have hEq : P (limsup E atTop)=P (⋂ n, E n) :=
-    measure_congr (eventuallyEqSet_iff.mpr (limsup_ae_eq_iInter hanti))
-  have hpos : 0<P (limsup E atTop) := by rw [hEq]; exact hp.trans_le hprob
-  have hz1 := BrownianGerm.measure_zero_or_one_limsup B hm hc hz P hB τ hτ E hE
-  have hone : P (limsup E atTop)=1 := hz1.resolve_left (ne_of_gt hpos)
-  have hi : ∀ᵐ ω ∂P, ω ∈ ⋂ n, E n := by
-    apply (ae_iff_measure_eq hmI.nullMeasurableSet).mpr
-    rw [measure_univ]
-    convert! hEq.symm.trans hone using 1
-  simpa only [mem_iInter] using hi
-
- 
-def smallTime (n : ℕ) : ℝ≥0 := ((n:ℝ≥0)+1)⁻¹
-
-lemma smallTime_pos (n : ℕ) : 0<smallTime n := by dsimp [smallTime]; positivity
-
-lemma smallTime_antitone : Antitone smallTime := by
-  intro n m hnm
-  dsimp [smallTime]
-  apply inv_anti₀ (by positivity)
-  exact_mod_cast Nat.add_le_add_right hnm 1
-
-lemma tendsto_smallTime : Tendsto smallTime atTop (𝓝 0) := by
-  change Tendsto (fun n : ℕ => ((n:ℝ≥0)+1)⁻¹) atTop (𝓝 0)
-  simpa only [one_div] using (tendsto_one_div_add_atTop_nhds_zero_nat :
-    Tendsto (fun n : ℕ => 1/((n:ℝ≥0)+1)) atTop (𝓝 0))
-
- 
-
-theorem ae_positive_all_times (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, StronglyMeasurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
-    (P : Measure Ω) (hB : IsPreBrownianReal B P)
-    (A : ℝ≥0 → Ω → ℝ≥0∞)
-    (hAm : ∀ t, Measurable[Filtration.natural B hm t] (A t))
-    (hAo : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
-    {p : ℝ≥0∞} (hp : 0<p)
-    (hlower : ∀ t : ℝ≥0, 0<t → p≤P {ω | 0<A t ω}) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0<t → 0<A t ω := by
-  let E (n : ℕ) : Set Ω := {ω | 0<A (smallTime n) ω}
-  have hEm (n : ℕ) : MeasurableSet[Filtration.natural B hm (smallTime n)] (E n) :=
-    measurableSet_lt measurable_const (hAm _)
-  have hant : ∀ᵐ ω ∂P, Antitone (fun n => ω∈E n) := by
-    filter_upwards [hAo] with ω hω n m hnm hmem
-    exact hmem.trans_le (hω (smallTime_antitone hnm))
-  have hs := ae_all_mem B hm hc hz P hB smallTime tendsto_smallTime E hEm hant hp
-    (fun n => hlower _ (smallTime_pos n))
-  filter_upwards [hs,hAo] with ω hsω hoω t ht
-  obtain ⟨n,hn⟩ := (tendsto_smallTime.eventually (gt_mem_nhds ht)).exists
-  exact (hsω n).trans_le (hoω hn.le)
-
- 
-
-theorem ae_positive_all_times_of_same_probability (B : ℝ≥0 → Ω → ℝ)
-    (hm : ∀ t, StronglyMeasurable (B t))
-    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
-    (P : Measure Ω) (hB : IsPreBrownianReal B P)
-    (A : ℝ≥0 → Ω → ℝ≥0∞)
-    (hAm : ∀ t, Measurable[Filtration.natural B hm t] (A t))
-    (hAo : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
-    (hpos : ∃ u : ℝ≥0, 0<u ∧ 0<P {ω | 0<A u ω})
-    (heq : ∀ s t : ℝ≥0, 0<s → 0<t → P {ω | 0<A s ω}=P {ω | 0<A t ω}) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0<t → 0<A t ω := by
-  obtain ⟨u,hu,hp⟩ := hpos
-  exact ae_positive_all_times B hm hc hz P hB A hAm hAo hp
-    (fun t ht => (heq u t hu ht).le)
-
-end SLEExactGauge.GermPositivity
-end
-end
- 
-
- 
-section
- 
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma physicalGreen_split_bound (a : ℝ) (ha : 1/4 < a) (R : ℝ) (hR : 0 < R)
-    (v : State) (hv : 0 < Y v) :
-    physicalGreen a v ≤ (physicalGreen a v * lifetimeTest (lifetimeExponent a) v) /
-      R ^ (lifetimeExponent a / 2) +
-      (R / (lifetimeDrift a * R ^ (lifetimeExponent a / 2))) * lifetimeCost a v := by
-  let q := lifetimeExponent a
-  let c := lifetimeDrift a
-  have hq : 0 < q := lifetimeExponent_pos ha
-  have hc : 0 < c := lifetimeDrift_pos ha
-  have hp : 0 < R ^ (q/2) := Real.rpow_pos_of_pos hR _
-  have hG : 0 < physicalGreen a v := Real.exp_pos _
-  have hF : 0 ≤ physicalGreen a v * lifetimeTest q v :=
-    mul_nonneg hG.le (lifetimeTest_nonneg q v hv.le)
-  have hC : 0 ≤ R / (c * R^(q/2)) * lifetimeCost a v := by
-    exact mul_nonneg (by positivity) (lifetimeCost_pos ha hv).le
-  change physicalGreen a v ≤ _ / R^(q/2) + (R/(c*R^(q/2))) * lifetimeCost a v
-  by_cases hs : R ≤ Q v
-  · have ht : R^(q/2) ≤ lifetimeTest q v :=
-      (Real.rpow_le_rpow hR.le hs (by positivity)).trans (lifetimeTest_coercive q v hv.le)
-    have h := mul_le_mul_of_nonneg_left ht hG.le
-    exact ((le_div_iff₀ hp).mpr h).trans (le_add_of_nonneg_right hC)
-  · have heq : R^(q/2-1) = R^(q/2)/R := by
-      rw [Real.rpow_sub hR,Real.rpow_one]
-    have he : R^(q/2)/R ≤ Q v^(q/2-1) := by
-      rw [← heq]
-      apply Real.rpow_le_rpow_of_nonpos (Q_pos hv) (le_of_not_ge hs)
-      have hq1 := lifetimeExponent_le_one a
-      dsimp only [q]
-      linarith
-    have hprod := mul_le_mul_of_nonneg_left he
-      (show 0 ≤ R / (c * R^(q/2)) * (c * physicalGreen a v) by positivity)
-    have hcancel : R / (c * R^(q/2)) * (c * physicalGreen a v) * (R^(q/2)/R) =
-        physicalGreen a v := by field_simp
-    rw [hcancel] at hprod
-    have hb : physicalGreen a v ≤ (R/(c*R^(q/2))) * lifetimeCost a v := by
-      convert hprod using 1; first | rfl | (dsimp [lifetimeCost,c,q]; ring)
-    exact hb.trans (le_add_of_nonneg_left (div_nonneg hF hp.le))
-
-lemma killedGreen_split_bound (a : ℝ) (ha : 1/4 < a) (R : ℝ) (hR : 0 < R)
-    (z : UpperPlane) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    killedObservable a z (physicalGreen a) t b ≤
-      killedObservable a z (fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v)
-        t b / R^(lifetimeExponent a/2) +
-      (R / (lifetimeDrift a * R^(lifetimeExponent a/2))) *
-        killedObservable a z (lifetimeCost a) t b := by
-  unfold killedObservable
-  split_ifs with ht
-  · apply physicalGreen_split_bound a ha R hR
-    obtain ⟨f,hf⟩ := ht
-    have he : loewnerMap a b.1 t z.1 = f t := by
-      simpa only [Real.toNNReal_coe] using
-        loewnerMap_eq_solution_time hf (right_mem_Icc.mpr t.coe_nonneg)
-    simpa only [actualState,Y_apply,he,Complex.sub_im,Complex.ofReal_im,sub_zero] using
-      (hf.2 (t : ℝ) (right_mem_Icc.mpr t.coe_nonneg)).1
-  · simp only [zero_div,mul_zero,add_zero,le_refl]
-
-lemma killedObservable_real_joint_measurable (a : ℝ) (z : UpperPlane)
-    (F : State → ℝ) (hF : Measurable F) :
-    Measurable (fun p : ℝ × ExactSLEGenerator.Driver =>
-      ENNReal.ofReal (killedObservable a z F p.1.toNNReal p.2)) :=
-  ((killedObservable_joint_measurable a z F hF).comp
-    ((measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd)).ennreal_ofReal
-
-lemma killedGreen_timeIntegral_bound (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (R : ℝ) (hR : 0 < R) (T : ℝ≥0) :
-    (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b,
-      ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ∂W) ≤
-    ENNReal.ofReal (((T : ℝ) / R^(lifetimeExponent a/2) +
-      R / (lifetimeDrift a * R^(lifetimeExponent a/2))) *
-      (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0))) := by
-  let q := lifetimeExponent a
-  let c := lifetimeDrift a
-  let M := physicalGreen a (z.1,0) * lifetimeTest q (z.1,0)
-  let A : ℝ := 1 / R^(q/2)
-  let B : ℝ := R / (c*R^(q/2))
-  let F : ℝ → ExactSLEGenerator.Driver → ℝ≥0∞ := fun u b => ENNReal.ofReal
-    (killedObservable a z (fun v => physicalGreen a v * lifetimeTest q v) u.toNNReal b)
-  let C : ℝ → ExactSLEGenerator.Driver → ℝ≥0∞ := fun u b => ENNReal.ofReal
-    (killedObservable a z (lifetimeCost a) u.toNNReal b)
-  have hM : 0 ≤ M := mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg q _ z.2.le)
-  have hc : 0 < c := lifetimeDrift_pos ha
-  have hA : 0 ≤ A := by dsimp [A]; positivity
-  have hB : 0 ≤ B := by dsimp [B]; positivity
-  have hFm : Measurable (Function.uncurry F) :=
-    killedObservable_real_joint_measurable a z _ (greenLifetime_measurable a)
-  have hCm : Measurable (Function.uncurry C) :=
-    killedObservable_real_joint_measurable a z _ (lifetimeCost_measurable a)
-  have hFu (u : ℝ) : Measurable (F u) :=
-    hFm.comp (measurable_const.prodMk measurable_id)
-  have hCu (u : ℝ) : Measurable (C u) :=
-    hCm.comp (measurable_const.prodMk measurable_id)
-  have hFim : Measurable (fun u => ∫⁻ b, F u b ∂W) := hFm.lintegral_prod_right
-  have hCim : Measurable (fun u => ∫⁻ b, C u b ∂W) := hCm.lintegral_prod_right
-  have hFub (u : ℝ) : (∫⁻ b, F u b ∂W) ≤ ENNReal.ofReal M :=
-    global_lifetime_test_budget a ha W hW z u.toNNReal
-  have hFint : (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, F u b ∂W) ≤
-      ENNReal.ofReal M * ENNReal.ofReal (T : ℝ) := by
-    calc
-      _ ≤ ∫⁻ _u in Ioc (0 : ℝ) (T : ℝ), ENNReal.ofReal M := lintegral_mono hFub
-      _ = _ := by simp only [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioc,sub_zero]
-  have hCint : (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, C u b ∂W) ≤ ENNReal.ofReal M := by
-    rw [lintegral_lintegral_swap hCm.aemeasurable]
-    apply le_trans (lintegral_mono (fun b => lintegral_mono_set Ioc_subset_Ioi_self))
-    exact global_lifetime_occupation_budget a ha W hW z
-  have hsplit (u : ℝ) (b : ExactSLEGenerator.Driver) :
-      ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ≤
-        ENNReal.ofReal A * F u b + ENNReal.ofReal B * C u b := by
-    have hFn := killedObservable_nonneg a z
-      (fun v => physicalGreen a v * lifetimeTest q v)
-      (fun v hv => mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg q v hv.le)) u.toNNReal b
-    have hCn := killedObservable_nonneg a z (lifetimeCost a)
-      (fun v hv => (lifetimeCost_pos ha hv).le) u.toNNReal b
-    have h := ENNReal.ofReal_le_ofReal (killedGreen_split_bound a ha R hR z u.toNNReal b)
-    have he : killedObservable a z
-        (fun v => physicalGreen a v * lifetimeTest q v) u.toNNReal b / R^(q/2) =
-        A * killedObservable a z
-          (fun v => physicalGreen a v * lifetimeTest q v) u.toNNReal b := by dsimp [A]; ring
-    change _ ≤ ENNReal.ofReal (_ / R^(q/2) + B * _) at h
-    rw [he, ENNReal.ofReal_add (mul_nonneg hA hFn) (mul_nonneg hB hCn),
-      ENNReal.ofReal_mul hA, ENNReal.ofReal_mul hB] at h
-    exact h
-  have hsplitInt (u : ℝ) :
-      (∫⁻ b, ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ∂W) ≤
-        ENNReal.ofReal A * (∫⁻ b, F u b ∂W) + ENNReal.ofReal B * (∫⁻ b, C u b ∂W) := by
-    calc
-      _ ≤ ∫⁻ b, ENNReal.ofReal A * F u b + ENNReal.ofReal B * C u b ∂W :=
-        lintegral_mono (hsplit u)
-      _ = _ := by rw [lintegral_add_left ((hFu u).const_mul (ENNReal.ofReal A)),
-          lintegral_const_mul _ (hFu u), lintegral_const_mul _ (hCu u)]
-  calc
-    _ ≤ ∫⁻ u in Ioc (0 : ℝ) (T : ℝ),
-        ENNReal.ofReal A * (∫⁻ b, F u b ∂W) + ENNReal.ofReal B * (∫⁻ b, C u b ∂W) :=
-      lintegral_mono hsplitInt
-    _ = ENNReal.ofReal A * (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, F u b ∂W) +
-        ENNReal.ofReal B * (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, C u b ∂W) := by
-      rw [lintegral_add_left (hFim.const_mul (ENNReal.ofReal A)),
-        lintegral_const_mul _ hFim, lintegral_const_mul _ hCim]
-    _ ≤ ENNReal.ofReal A * (ENNReal.ofReal M * ENNReal.ofReal (T : ℝ)) +
-        ENNReal.ofReal B * ENNReal.ofReal M :=
-      add_le_add (mul_le_mul_right hFint _) (mul_le_mul_right hCint _)
-    _ = _ := by
-      rw [← ENNReal.ofReal_mul hM, ← ENNReal.ofReal_mul hA, ← ENNReal.ofReal_mul hB,
-        ← ENNReal.ofReal_add (mul_nonneg hA (mul_nonneg hM T.coe_nonneg)) (mul_nonneg hB hM)]
-      congr 1
-      dsimp [A,B,M,q,c]
-      ring
-
- 
-
-theorem killedGreen_polynomial_decay (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (T : ℝ≥0) (hT : 0 < T) :
-    (∫ b, killedObservable a z (physicalGreen a) T b ∂W) ≤
-      (1 + 1 / lifetimeDrift a) *
-        (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) /
-        (T : ℝ)^(lifetimeExponent a/2) := by
-  have ht : 0 < (T : ℝ) := hT
-  have htime := killedGreen_timeIntegral_bound a ha W hW z T ht T
-  have hlower : ENNReal.ofReal (T : ℝ) *
-      (∫⁻ b, ENNReal.ofReal (killedObservable a z (physicalGreen a) T b) ∂W) ≤
-      ∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b,
-        ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ∂W := by
-    calc
-      _ = ∫⁻ _u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b,
-          ENNReal.ofReal (killedObservable a z (physicalGreen a) T b) ∂W := by
-        simp only [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioc,sub_zero,mul_comm]
-      _ ≤ _ := by
-        apply lintegral_mono_ae
-        filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
-        apply killedGreen_lintegral_antitone a ha W hW z
-        simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hu.2
-  have HB := hlower.trans htime
-  rw [← ofReal_integral_eq_lintegral_ofReal (killedGreen_integrable a ha W hW z T)
-      (Eventually.of_forall (killedGreen_nonneg a z T)),
-    ← ENNReal.ofReal_mul ht.le] at HB
-  have hp : 0 < (T : ℝ)^(lifetimeExponent a/2) := Real.rpow_pos_of_pos ht _
-  have hc : 0 < lifetimeDrift a := lifetimeDrift_pos ha
-  have hM : 0 ≤ physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0) :=
-    mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg _ _ z.2.le)
-  have HR := (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp HB
-  have he : ((T : ℝ)/(T : ℝ)^(lifetimeExponent a/2) +
-      (T : ℝ)/(lifetimeDrift a*(T : ℝ)^(lifetimeExponent a/2))) *
-      (physicalGreen a (z.1,0)*lifetimeTest (lifetimeExponent a) (z.1,0)) =
-      (T : ℝ) * ((1+1/lifetimeDrift a) *
-        (physicalGreen a (z.1,0)*lifetimeTest (lifetimeExponent a) (z.1,0)) /
-        (T : ℝ)^(lifetimeExponent a/2)) := by ring
-  rw [he] at HR
-  exact (mul_le_mul_iff_right₀ ht).mp HR
-
- 
-
-theorem killedGreen_mean_tendsto_zero (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    Tendsto (fun T : ℝ≥0 => ∫ b, killedObservable a z (physicalGreen a) T b ∂W)
-      atTop (𝓝 0) := by
-  apply squeeze_zero'
-    (Eventually.of_forall (fun T => integral_nonneg (killedGreen_nonneg a z T)))
-  · filter_upwards [eventually_gt_atTop (0 : ℝ≥0)] with T hT
-    exact killedGreen_polynomial_decay a ha W hW z T hT
-  · exact tendsto_const_nhds.div_atTop
-      ((tendsto_rpow_atTop (by linarith [lifetimeExponent_pos ha] : 0 < lifetimeExponent a/2)).comp
-        (NNReal.tendsto_coe_atTop.mpr tendsto_id))
-
-theorem survivingGreen_mean_tendsto_zero (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    Tendsto (fun T : ℝ≥0 => ∫ b, survivingGreen a T b z.1 ∂W) atTop (𝓝 0) := by
-  simpa only [survivingGreen_eq_killed,integral_const_mul,mul_zero] using
-    (killedGreen_mean_tendsto_zero a ha W hW z).const_mul (Real.exp (-alpha a*Real.log 2))
-
-theorem survivingGreen_lintegral_tendsto_zero (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    Tendsto (fun T : ℝ≥0 => ∫⁻ b, ENNReal.ofReal (survivingGreen a T b z.1) ∂W)
-      atTop (𝓝 0) := by
-  have he (T : ℝ≥0) : ENNReal.ofReal (∫ b, survivingGreen a T b z.1 ∂W) =
-      ∫⁻ b, ENNReal.ofReal (survivingGreen a T b z.1) ∂W :=
-    ofReal_integral_eq_lintegral_ofReal ((actual_surviving_green a ha T).2.2 W hW z.1 z.2).1
-      (Eventually.of_forall (fun b => by
-        change 0 ≤ survivingGreen a T b z.1
-        rw [survivingGreen_eq_killed]
-        exact mul_nonneg (Real.exp_pos _).le (killedGreen_nonneg a z T b)))
-  simpa only [he,ENNReal.ofReal_zero] using
-    ENNReal.tendsto_ofReal (survivingGreen_mean_tendsto_zero a ha W hW z)
-
-end ExactSLEGreen
-end
- 
-
- 
-section
- 
-
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma survivingGreen_lintegral_le_initial (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (z : UpperPlane) :
-    (∫⁻b, ENNReal.ofReal (survivingGreen a t b z.1) ∂W) ≤
-      ENNReal.ofReal (initialGreen a z) := by
-  obtain ⟨hi,hb⟩ := (actual_surviving_green a ha t).2.2 W hW z.1 z.2
-  rw [← ofReal_integral_eq_lintegral_ofReal hi
-    (Eventually.of_forall (fun b => survivingGreen_nonneg a t b z.1))]
-  exact ENNReal.ofReal_le_ofReal hb
-
- 
-
-theorem survivingGreen_compact_mean_tendsto_zero (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (K : Set UpperPlane) (hK : IsCompact K) :
-    Tendsto (fun n : ℕ => ∫⁻b, ∫⁻z in K,
-      ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W) atTop (𝓝 0) := by
-  have hm (n : ℕ) : Measurable (fun z : UpperPlane =>
-      ∫⁻b, ENNReal.ofReal (survivingGreen a n b z.1) ∂W) :=
-    (survivingGreen_joint_measurable a ha n).ennreal_ofReal.lintegral_prod_left
-  have hb (n : ℕ) : (fun z : UpperPlane =>
-        ∫⁻b, ENNReal.ofReal (survivingGreen a n b z.1) ∂W) ≤ᵐ[upperArea.restrict K]
-      (fun z => ENNReal.ofReal (initialGreen a z)) :=
-    Eventually.of_forall (survivingGreen_lintegral_le_initial a ha W hW n)
-  have hf : (∫⁻z in K, ENNReal.ofReal (initialGreen a z) ∂upperArea) ≠ ∞ := by
-    rw [← withDensity_apply _ hK.measurableSet]
-    change initialGreenMeasure a K ≠ ∞
-    exact hK.measure_lt_top.ne
-  have hlim : ∀ᵐ z ∂upperArea.restrict K,
-      Tendsto (fun n : ℕ => ∫⁻b, ENNReal.ofReal (survivingGreen a n b z.1) ∂W)
-        atTop (𝓝 (0 : ℝ≥0∞)) := Eventually.of_forall (fun z =>
-    (survivingGreen_lintegral_tendsto_zero a ha W hW z).comp tendsto_natCast_atTop_atTop)
-  have h := tendsto_lintegral_of_dominated_convergence
-    (fun z => ENNReal.ofReal (initialGreen a z)) hm hb hf hlim
-  simp only [lintegral_zero] at h
-  convert h using 1
-  funext n
-  exact lintegral_lintegral_swap
-    (survivingGreen_joint_measurable a ha n).ennreal_ofReal.aemeasurable
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.FutureSupport
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.BrownianGerm
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+
+ 
+@[instance_reducible] def future (B : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) : MeasurableSpace Ω :=
+  MeasurableSpace.comap (fun ω u => B (t+u) ω - B t ω) inferInstance
+
+omit mΩ in
+lemma measurable_shift (B : ℝ≥0 → Ω → ℝ) (s u : ℝ≥0) :
+    Measurable[future B s] (fun ω => B (s+u) ω-B s ω) :=
+  (measurable_pi_apply u).comp
+    (comap_measurable (fun ω v => B (s+v) ω-B s ω))
+
+lemma future_le (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) (t : ℝ≥0) :
+    future B t ≤ mΩ :=
+  (Measurable.of_eval fun offset => (hm (t + offset)).sub (hm t)).comap_le
+
+omit mΩ in
+lemma future_antitone (B : ℝ≥0 → Ω → ℝ) : Antitone (future B) := by
+  intro s t hst
+  suffices Measurable[future B s] (fun ω u => B (t+u) ω - B t ω) from
+    this.comap_le
+  let : MeasurableSpace Ω := future B s
+  refine Measurable.of_eval fun u => ?_
+  have h := (measurable_shift B s (t-s+u)).sub (measurable_shift B s (t-s))
+  convert h using 1; ext ω
+  simp only [← add_assoc, add_tsub_cancel_of_le hst, Pi.sub_apply]
+  ring
+
+ 
+
+omit mΩ in
+lemma measurable_eval_iSup_future (B : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0) (t : ℝ≥0) :
+    Measurable[⨆ s : {s : ℝ≥0 // 0<s}, future B s.1] (B t) := by
+  let ε : ℕ → ℝ≥0 := fun n => ((n:ℝ≥0)+1)⁻¹
+  have hep (n : ℕ) : 0<ε n := by dsimp [ε]; positivity
+  have he : Tendsto ε atTop (𝓝 0) := by
+    simpa [ε,one_div] using (tendsto_one_div_add_atTop_nhds_zero_nat :
+      Tendsto (fun n : ℕ => 1/((n:ℝ≥0)+1)) atTop (𝓝 0))
+  let : MeasurableSpace Ω := ⨆ s : {s : ℝ≥0 // 0<s}, future B s.1
+  apply measurable_of_tendsto_metrizable
+    (f := fun n ω => B (ε n+t) ω-B (ε n) ω)
+  · intro n
+    exact (measurable_shift B (ε n) t).mono (le_iSup_of_le ⟨ε n,hep n⟩ le_rfl) le_rfl
+  · apply tendsto_pi_nhds.mpr
+    intro ω
+    have h := ((hc ω).tendsto (0+t)).comp (he.add_const t)
+    have h' := ((hc ω).tendsto 0).comp he
+    simpa [hz ω] using h.sub h'
+
+ 
+
+theorem measure_zero_or_one (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, StronglyMeasurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
+    (P : Measure Ω) (hB : IsPreBrownianReal B P)
+    {E : Set Ω}
+    (hE : ∀ t : ℝ≥0, 0<t → MeasurableSet[Filtration.natural B hm t] E) :
+    P E=0 ∨ P E=1 := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  let g : MeasurableSpace Ω := ⨅ t : {t : ℝ≥0 // 0<t}, Filtration.natural B hm t.1
+  let f : {t : ℝ≥0 // 0<t} → MeasurableSpace Ω := fun t => future B t.1
+  have hg1 : g ≤ Filtration.natural B hm 1 :=
+    iInf_le (fun t : {t : ℝ≥0 // 0<t} => Filtration.natural B hm t.1) ⟨1,by norm_num⟩
+  have hg : g ≤ mΩ := hg1.trans ((Filtration.natural B hm).le 1)
+  have hi (t : {t : ℝ≥0 // 0<t}) : Indep (f t) g P := by
+    have h := (IndepFun_iff_Indep _ _ _).mp (hB.indepFun_shift t.1)
+    rw [← Filtration.natural_eq_comap B hm t.1] at h
+    exact indep_of_indep_of_le_right h (iInf_le _ t)
+  have hd : Directed (· ≤ ·) f := by
+    intro s t
+    refine ⟨⟨min s.1 t.1,lt_min s.2 t.2⟩, ?_, ?_⟩
+    · exact future_antitone B (min_le_left _ _)
+    · exact future_antitone B (min_le_right _ _)
+  have hind : Indep (⨆ t, f t) g P :=
+    indep_iSup_of_directed_le hi (fun t => @future_le Ω mΩ B (fun u => (hm u).measurable) t.1) hg hd
+  have hgE : MeasurableSet[g] E := by
+    exact (MeasurableSpace.measurableSet_iInf).mpr fun t => hE t.1 t.2
+  have hng : Filtration.natural B hm 1 ≤ ⨆ t, f t := by
+    rw [Filtration.natural_eq_comap]
+    let : MeasurableSpace Ω := ⨆ t, f t
+    exact (Measurable.of_eval fun t : Iic (1:ℝ≥0) =>
+      measurable_eval_iSup_future B hc hz t.1).comap_le
+  exact measure_eq_zero_or_one_of_indepSet_self
+    (hind.indepSet_of_measurableSet (hng _ (hE 1 (by norm_num))) hgE)
+
+ 
+
+lemma measurableSet_limsup_germ
+    (ℱ : Filtration ℝ≥0 mΩ) (τ : ℕ → ℝ≥0) (hτ : Tendsto τ atTop (𝓝 0))
+    (E : ℕ → Set Ω) (hE : ∀ n, MeasurableSet[ℱ (τ n)] (E n))
+    (t : ℝ≥0) (ht : 0<t) : MeasurableSet[ℱ t] (limsup E atTop) := by
+  classical
+  have hev : ∀ᶠ n in atTop, τ n≤t :=
+    (hτ.eventually (gt_mem_nhds ht)).mono (fun _ h => h.le)
+  have heq : E =ᶠ[atTop] (fun n => if τ n≤t then E n else ∅) :=
+    hev.mono fun n hn => by simp [hn]
+  rw [limsup_congr heq]
+  let : MeasurableSpace Ω := ℱ t
+  apply MeasurableSet.measurableSet_limsup
+  intro n
+  split_ifs with h
+  · exact ℱ.mono h _ (hE n)
+  · exact MeasurableSet.empty
+
+ 
+theorem measure_zero_or_one_limsup (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, StronglyMeasurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
+    (P : Measure Ω) (hB : IsPreBrownianReal B P)
+    (τ : ℕ → ℝ≥0) (hτ : Tendsto τ atTop (𝓝 0))
+    (E : ℕ → Set Ω) (hE : ∀ n, MeasurableSet[Filtration.natural B hm (τ n)] (E n)) :
+    P (limsup E atTop)=0 ∨ P (limsup E atTop)=1 :=
+  measure_zero_or_one B hm hc hz P hB
+    (measurableSet_limsup_germ (Filtration.natural B hm) τ hτ E hE)
+
+end SLEExactGauge.BrownianGerm
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.GermPositivity
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+
+omit mΩ in
+lemma mem_limsup_iff_mem_iInter {E : ℕ → Set Ω} {ω : Ω}
+    (hE : Antitone (fun n => ω ∈ E n)) :
+    ω ∈ limsup E atTop ↔ ω ∈ ⋂ n, E n := by
+  rw [mem_limsup_iff_frequently_mem,mem_iInter]
+  constructor
+  · intro h n
+    obtain ⟨m,hm,hmem⟩ := frequently_atTop.mp h n
+    exact hE hm hmem
+  · exact fun h => Filter.Frequently.of_forall h
+
+ 
+
+lemma limsup_ae_eq_iInter {P : Measure Ω} {E : ℕ → Set Ω}
+    (hE : ∀ᵐ ω ∂P, Antitone (fun n => ω ∈ E n)) :
+    ∀ᵐ ω ∂P, ω ∈ limsup E atTop ↔ ω ∈ ⋂ n, E n := by
+  exact hE.mono fun _ h => mem_limsup_iff_mem_iInter h
+
+ 
+
+theorem ae_all_mem (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, StronglyMeasurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
+    (P : Measure Ω) (hB : IsPreBrownianReal B P)
+    (τ : ℕ → ℝ≥0) (hτ : Tendsto τ atTop (𝓝 0))
+    (E : ℕ → Set Ω) (hE : ∀ n, MeasurableSet[Filtration.natural B hm (τ n)] (E n))
+    (hanti : ∀ᵐ ω ∂P, Antitone (fun n => ω ∈ E n))
+    {p : ℝ≥0∞} (hp : 0<p) (hlower : ∀ n, p≤P (E n)) :
+    ∀ᵐ ω ∂P, ∀ n, ω ∈ E n := by
+  let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
+  have he (n : ℕ) : MeasurableSet (E n) :=
+    (Filtration.natural B hm).le (τ n) _ (hE n)
+  have hmI : MeasurableSet (⋂ n, E n) := MeasurableSet.iInter he
+  have hprob : p≤P (⋂ n, E n) := by
+    rw [measure_iInter_of_ae_antitone hanti (fun n => (he n).nullMeasurableSet)
+      ⟨0,measure_ne_top _ _⟩]
+    exact le_iInf hlower
+  have hEq : P (limsup E atTop)=P (⋂ n, E n) :=
+    measure_congr (eventuallyEqSet_iff.mpr (limsup_ae_eq_iInter hanti))
+  have hpos : 0<P (limsup E atTop) := by rw [hEq]; exact hp.trans_le hprob
+  have hz1 := BrownianGerm.measure_zero_or_one_limsup B hm hc hz P hB τ hτ E hE
+  have hone : P (limsup E atTop)=1 := hz1.resolve_left (ne_of_gt hpos)
+  have hi : ∀ᵐ ω ∂P, ω ∈ ⋂ n, E n := by
+    apply (ae_iff_measure_eq hmI.nullMeasurableSet).mpr
+    rw [measure_univ]
+    convert! hEq.symm.trans hone using 1
+  simpa only [mem_iInter] using hi
+
+ 
+def smallTime (n : ℕ) : ℝ≥0 := ((n:ℝ≥0)+1)⁻¹
+
+lemma smallTime_pos (n : ℕ) : 0<smallTime n := by dsimp [smallTime]; positivity
+
+lemma smallTime_antitone : Antitone smallTime := by
+  intro n m hnm
+  dsimp [smallTime]
+  apply inv_anti₀ (by positivity)
+  exact_mod_cast Nat.add_le_add_right hnm 1
+
+lemma tendsto_smallTime : Tendsto smallTime atTop (𝓝 0) := by
+  change Tendsto (fun n : ℕ => ((n:ℝ≥0)+1)⁻¹) atTop (𝓝 0)
+  simpa only [one_div] using (tendsto_one_div_add_atTop_nhds_zero_nat :
+    Tendsto (fun n : ℕ => 1/((n:ℝ≥0)+1)) atTop (𝓝 0))
+
+ 
+
+theorem ae_positive_all_times (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, StronglyMeasurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
+    (P : Measure Ω) (hB : IsPreBrownianReal B P)
+    (A : ℝ≥0 → Ω → ℝ≥0∞)
+    (hAm : ∀ t, Measurable[Filtration.natural B hm t] (A t))
+    (hAo : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
+    {p : ℝ≥0∞} (hp : 0<p)
+    (hlower : ∀ t : ℝ≥0, 0<t → p≤P {ω | 0<A t ω}) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0<t → 0<A t ω := by
+  let E (n : ℕ) : Set Ω := {ω | 0<A (smallTime n) ω}
+  have hEm (n : ℕ) : MeasurableSet[Filtration.natural B hm (smallTime n)] (E n) :=
+    measurableSet_lt measurable_const (hAm _)
+  have hant : ∀ᵐ ω ∂P, Antitone (fun n => ω∈E n) := by
+    filter_upwards [hAo] with ω hω n m hnm hmem
+    exact hmem.trans_le (hω (smallTime_antitone hnm))
+  have hs := ae_all_mem B hm hc hz P hB smallTime tendsto_smallTime E hEm hant hp
+    (fun n => hlower _ (smallTime_pos n))
+  filter_upwards [hs,hAo] with ω hsω hoω t ht
+  obtain ⟨n,hn⟩ := (tendsto_smallTime.eventually (gt_mem_nhds ht)).exists
+  exact (hsω n).trans_le (hoω hn.le)
+
+ 
+
+theorem ae_positive_all_times_of_same_probability (B : ℝ≥0 → Ω → ℝ)
+    (hm : ∀ t, StronglyMeasurable (B t))
+    (hc : ∀ ω, Continuous (fun t => B t ω)) (hz : ∀ ω, B 0 ω=0)
+    (P : Measure Ω) (hB : IsPreBrownianReal B P)
+    (A : ℝ≥0 → Ω → ℝ≥0∞)
+    (hAm : ∀ t, Measurable[Filtration.natural B hm t] (A t))
+    (hAo : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
+    (hpos : ∃ u : ℝ≥0, 0<u ∧ 0<P {ω | 0<A u ω})
+    (heq : ∀ s t : ℝ≥0, 0<s → 0<t → P {ω | 0<A s ω}=P {ω | 0<A t ω}) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0<t → 0<A t ω := by
+  obtain ⟨u,hu,hp⟩ := hpos
+  exact ae_positive_all_times B hm hc hz P hB A hAm hAo hp
+    (fun t ht => (heq u t hu ht).le)
+
+end SLEExactGauge.GermPositivity
+end
+end
+ 
+
+ 
+section
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma physicalGreen_split_bound (a : ℝ) (ha : 1/4 < a) (R : ℝ) (hR : 0 < R)
+    (v : State) (hv : 0 < Y v) :
+    physicalGreen a v ≤ (physicalGreen a v * lifetimeTest (lifetimeExponent a) v) /
+      R ^ (lifetimeExponent a / 2) +
+      (R / (lifetimeDrift a * R ^ (lifetimeExponent a / 2))) * lifetimeCost a v := by
+  let q := lifetimeExponent a
+  let c := lifetimeDrift a
+  have hq : 0 < q := lifetimeExponent_pos ha
+  have hc : 0 < c := lifetimeDrift_pos ha
+  have hp : 0 < R ^ (q/2) := Real.rpow_pos_of_pos hR _
+  have hG : 0 < physicalGreen a v := Real.exp_pos _
+  have hF : 0 ≤ physicalGreen a v * lifetimeTest q v :=
+    mul_nonneg hG.le (lifetimeTest_nonneg q v hv.le)
+  have hC : 0 ≤ R / (c * R^(q/2)) * lifetimeCost a v := by
+    exact mul_nonneg (by positivity) (lifetimeCost_pos ha hv).le
+  change physicalGreen a v ≤ _ / R^(q/2) + (R/(c*R^(q/2))) * lifetimeCost a v
+  by_cases hs : R ≤ Q v
+  · have ht : R^(q/2) ≤ lifetimeTest q v :=
+      (Real.rpow_le_rpow hR.le hs (by positivity)).trans (lifetimeTest_coercive q v hv.le)
+    have h := mul_le_mul_of_nonneg_left ht hG.le
+    exact ((le_div_iff₀ hp).mpr h).trans (le_add_of_nonneg_right hC)
+  · have heq : R^(q/2-1) = R^(q/2)/R := by
+      rw [Real.rpow_sub hR,Real.rpow_one]
+    have he : R^(q/2)/R ≤ Q v^(q/2-1) := by
+      rw [← heq]
+      apply Real.rpow_le_rpow_of_nonpos (Q_pos hv) (le_of_not_ge hs)
+      have hq1 := lifetimeExponent_le_one a
+      dsimp only [q]
+      linarith
+    have hprod := mul_le_mul_of_nonneg_left he
+      (show 0 ≤ R / (c * R^(q/2)) * (c * physicalGreen a v) by positivity)
+    have hcancel : R / (c * R^(q/2)) * (c * physicalGreen a v) * (R^(q/2)/R) =
+        physicalGreen a v := by field_simp
+    rw [hcancel] at hprod
+    have hb : physicalGreen a v ≤ (R/(c*R^(q/2))) * lifetimeCost a v := by
+      convert hprod using 1; first | rfl | (dsimp [lifetimeCost,c,q]; ring)
+    exact hb.trans (le_add_of_nonneg_left (div_nonneg hF hp.le))
+
+lemma killedGreen_split_bound (a : ℝ) (ha : 1/4 < a) (R : ℝ) (hR : 0 < R)
+    (z : UpperPlane) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    killedObservable a z (physicalGreen a) t b ≤
+      killedObservable a z (fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v)
+        t b / R^(lifetimeExponent a/2) +
+      (R / (lifetimeDrift a * R^(lifetimeExponent a/2))) *
+        killedObservable a z (lifetimeCost a) t b := by
+  unfold killedObservable
+  split_ifs with ht
+  · apply physicalGreen_split_bound a ha R hR
+    obtain ⟨f,hf⟩ := ht
+    have he : loewnerMap a b.1 t z.1 = f t := by
+      simpa only [Real.toNNReal_coe] using
+        loewnerMap_eq_solution_time hf (right_mem_Icc.mpr t.coe_nonneg)
+    simpa only [actualState,Y_apply,he,Complex.sub_im,Complex.ofReal_im,sub_zero] using
+      (hf.2 (t : ℝ) (right_mem_Icc.mpr t.coe_nonneg)).1
+  · simp only [zero_div,mul_zero,add_zero,le_refl]
+
+lemma killedObservable_real_joint_measurable (a : ℝ) (z : UpperPlane)
+    (F : State → ℝ) (hF : Measurable F) :
+    Measurable (fun p : ℝ × ExactSLEGenerator.Driver =>
+      ENNReal.ofReal (killedObservable a z F p.1.toNNReal p.2)) :=
+  ((killedObservable_joint_measurable a z F hF).comp
+    ((measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd)).ennreal_ofReal
+
+lemma killedGreen_timeIntegral_bound (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (R : ℝ) (hR : 0 < R) (T : ℝ≥0) :
+    (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b,
+      ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ∂W) ≤
+    ENNReal.ofReal (((T : ℝ) / R^(lifetimeExponent a/2) +
+      R / (lifetimeDrift a * R^(lifetimeExponent a/2))) *
+      (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0))) := by
+  let q := lifetimeExponent a
+  let c := lifetimeDrift a
+  let M := physicalGreen a (z.1,0) * lifetimeTest q (z.1,0)
+  let A : ℝ := 1 / R^(q/2)
+  let B : ℝ := R / (c*R^(q/2))
+  let F : ℝ → ExactSLEGenerator.Driver → ℝ≥0∞ := fun u b => ENNReal.ofReal
+    (killedObservable a z (fun v => physicalGreen a v * lifetimeTest q v) u.toNNReal b)
+  let C : ℝ → ExactSLEGenerator.Driver → ℝ≥0∞ := fun u b => ENNReal.ofReal
+    (killedObservable a z (lifetimeCost a) u.toNNReal b)
+  have hM : 0 ≤ M := mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg q _ z.2.le)
+  have hc : 0 < c := lifetimeDrift_pos ha
+  have hA : 0 ≤ A := by dsimp [A]; positivity
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hFm : Measurable (Function.uncurry F) :=
+    killedObservable_real_joint_measurable a z _ (greenLifetime_measurable a)
+  have hCm : Measurable (Function.uncurry C) :=
+    killedObservable_real_joint_measurable a z _ (lifetimeCost_measurable a)
+  have hFu (u : ℝ) : Measurable (F u) :=
+    hFm.comp (measurable_const.prodMk measurable_id)
+  have hCu (u : ℝ) : Measurable (C u) :=
+    hCm.comp (measurable_const.prodMk measurable_id)
+  have hFim : Measurable (fun u => ∫⁻ b, F u b ∂W) := hFm.lintegral_prod_right
+  have hCim : Measurable (fun u => ∫⁻ b, C u b ∂W) := hCm.lintegral_prod_right
+  have hFub (u : ℝ) : (∫⁻ b, F u b ∂W) ≤ ENNReal.ofReal M :=
+    global_lifetime_test_budget a ha W hW z u.toNNReal
+  have hFint : (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, F u b ∂W) ≤
+      ENNReal.ofReal M * ENNReal.ofReal (T : ℝ) := by
+    calc
+      _ ≤ ∫⁻ _u in Ioc (0 : ℝ) (T : ℝ), ENNReal.ofReal M := lintegral_mono hFub
+      _ = _ := by simp only [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioc,sub_zero]
+  have hCint : (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, C u b ∂W) ≤ ENNReal.ofReal M := by
+    rw [lintegral_lintegral_swap hCm.aemeasurable]
+    apply le_trans (lintegral_mono (fun b => lintegral_mono_set Ioc_subset_Ioi_self))
+    exact global_lifetime_occupation_budget a ha W hW z
+  have hsplit (u : ℝ) (b : ExactSLEGenerator.Driver) :
+      ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ≤
+        ENNReal.ofReal A * F u b + ENNReal.ofReal B * C u b := by
+    have hFn := killedObservable_nonneg a z
+      (fun v => physicalGreen a v * lifetimeTest q v)
+      (fun v hv => mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg q v hv.le)) u.toNNReal b
+    have hCn := killedObservable_nonneg a z (lifetimeCost a)
+      (fun v hv => (lifetimeCost_pos ha hv).le) u.toNNReal b
+    have h := ENNReal.ofReal_le_ofReal (killedGreen_split_bound a ha R hR z u.toNNReal b)
+    have he : killedObservable a z
+        (fun v => physicalGreen a v * lifetimeTest q v) u.toNNReal b / R^(q/2) =
+        A * killedObservable a z
+          (fun v => physicalGreen a v * lifetimeTest q v) u.toNNReal b := by dsimp [A]; ring
+    change _ ≤ ENNReal.ofReal (_ / R^(q/2) + B * _) at h
+    rw [he, ENNReal.ofReal_add (mul_nonneg hA hFn) (mul_nonneg hB hCn),
+      ENNReal.ofReal_mul hA, ENNReal.ofReal_mul hB] at h
+    exact h
+  have hsplitInt (u : ℝ) :
+      (∫⁻ b, ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ∂W) ≤
+        ENNReal.ofReal A * (∫⁻ b, F u b ∂W) + ENNReal.ofReal B * (∫⁻ b, C u b ∂W) := by
+    calc
+      _ ≤ ∫⁻ b, ENNReal.ofReal A * F u b + ENNReal.ofReal B * C u b ∂W :=
+        lintegral_mono (hsplit u)
+      _ = _ := by rw [lintegral_add_left ((hFu u).const_mul (ENNReal.ofReal A)),
+          lintegral_const_mul _ (hFu u), lintegral_const_mul _ (hCu u)]
+  calc
+    _ ≤ ∫⁻ u in Ioc (0 : ℝ) (T : ℝ),
+        ENNReal.ofReal A * (∫⁻ b, F u b ∂W) + ENNReal.ofReal B * (∫⁻ b, C u b ∂W) :=
+      lintegral_mono hsplitInt
+    _ = ENNReal.ofReal A * (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, F u b ∂W) +
+        ENNReal.ofReal B * (∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b, C u b ∂W) := by
+      rw [lintegral_add_left (hFim.const_mul (ENNReal.ofReal A)),
+        lintegral_const_mul _ hFim, lintegral_const_mul _ hCim]
+    _ ≤ ENNReal.ofReal A * (ENNReal.ofReal M * ENNReal.ofReal (T : ℝ)) +
+        ENNReal.ofReal B * ENNReal.ofReal M :=
+      add_le_add (mul_le_mul_right hFint _) (mul_le_mul_right hCint _)
+    _ = _ := by
+      rw [← ENNReal.ofReal_mul hM, ← ENNReal.ofReal_mul hA, ← ENNReal.ofReal_mul hB,
+        ← ENNReal.ofReal_add (mul_nonneg hA (mul_nonneg hM T.coe_nonneg)) (mul_nonneg hB hM)]
+      congr 1
+      dsimp [A,B,M,q,c]
+      ring
+
+ 
+
+theorem killedGreen_polynomial_decay (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (T : ℝ≥0) (hT : 0 < T) :
+    (∫ b, killedObservable a z (physicalGreen a) T b ∂W) ≤
+      (1 + 1 / lifetimeDrift a) *
+        (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) /
+        (T : ℝ)^(lifetimeExponent a/2) := by
+  have ht : 0 < (T : ℝ) := hT
+  have htime := killedGreen_timeIntegral_bound a ha W hW z T ht T
+  have hlower : ENNReal.ofReal (T : ℝ) *
+      (∫⁻ b, ENNReal.ofReal (killedObservable a z (physicalGreen a) T b) ∂W) ≤
+      ∫⁻ u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b,
+        ENNReal.ofReal (killedObservable a z (physicalGreen a) u.toNNReal b) ∂W := by
+    calc
+      _ = ∫⁻ _u in Ioc (0 : ℝ) (T : ℝ), ∫⁻ b,
+          ENNReal.ofReal (killedObservable a z (physicalGreen a) T b) ∂W := by
+        simp only [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioc,sub_zero,mul_comm]
+      _ ≤ _ := by
+        apply lintegral_mono_ae
+        filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
+        apply killedGreen_lintegral_antitone a ha W hW z
+        simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hu.2
+  have HB := hlower.trans htime
+  rw [← ofReal_integral_eq_lintegral_ofReal (killedGreen_integrable a ha W hW z T)
+      (Eventually.of_forall (killedGreen_nonneg a z T)),
+    ← ENNReal.ofReal_mul ht.le] at HB
+  have hp : 0 < (T : ℝ)^(lifetimeExponent a/2) := Real.rpow_pos_of_pos ht _
+  have hc : 0 < lifetimeDrift a := lifetimeDrift_pos ha
+  have hM : 0 ≤ physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0) :=
+    mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg _ _ z.2.le)
+  have HR := (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp HB
+  have he : ((T : ℝ)/(T : ℝ)^(lifetimeExponent a/2) +
+      (T : ℝ)/(lifetimeDrift a*(T : ℝ)^(lifetimeExponent a/2))) *
+      (physicalGreen a (z.1,0)*lifetimeTest (lifetimeExponent a) (z.1,0)) =
+      (T : ℝ) * ((1+1/lifetimeDrift a) *
+        (physicalGreen a (z.1,0)*lifetimeTest (lifetimeExponent a) (z.1,0)) /
+        (T : ℝ)^(lifetimeExponent a/2)) := by ring
+  rw [he] at HR
+  exact (mul_le_mul_iff_right₀ ht).mp HR
+
+ 
+
+theorem killedGreen_mean_tendsto_zero (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    Tendsto (fun T : ℝ≥0 => ∫ b, killedObservable a z (physicalGreen a) T b ∂W)
+      atTop (𝓝 0) := by
+  apply squeeze_zero'
+    (Eventually.of_forall (fun T => integral_nonneg (killedGreen_nonneg a z T)))
+  · filter_upwards [eventually_gt_atTop (0 : ℝ≥0)] with T hT
+    exact killedGreen_polynomial_decay a ha W hW z T hT
+  · exact tendsto_const_nhds.div_atTop
+      ((tendsto_rpow_atTop (by linarith [lifetimeExponent_pos ha] : 0 < lifetimeExponent a/2)).comp
+        (NNReal.tendsto_coe_atTop.mpr tendsto_id))
+
+theorem survivingGreen_mean_tendsto_zero (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    Tendsto (fun T : ℝ≥0 => ∫ b, survivingGreen a T b z.1 ∂W) atTop (𝓝 0) := by
+  simpa only [survivingGreen_eq_killed,integral_const_mul,mul_zero] using
+    (killedGreen_mean_tendsto_zero a ha W hW z).const_mul (Real.exp (-alpha a*Real.log 2))
+
+theorem survivingGreen_lintegral_tendsto_zero (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    Tendsto (fun T : ℝ≥0 => ∫⁻ b, ENNReal.ofReal (survivingGreen a T b z.1) ∂W)
+      atTop (𝓝 0) := by
+  have he (T : ℝ≥0) : ENNReal.ofReal (∫ b, survivingGreen a T b z.1 ∂W) =
+      ∫⁻ b, ENNReal.ofReal (survivingGreen a T b z.1) ∂W :=
+    ofReal_integral_eq_lintegral_ofReal ((actual_surviving_green a ha T).2.2 W hW z.1 z.2).1
+      (Eventually.of_forall (fun b => by
+        change 0 ≤ survivingGreen a T b z.1
+        rw [survivingGreen_eq_killed]
+        exact mul_nonneg (Real.exp_pos _).le (killedGreen_nonneg a z T b)))
+  simpa only [he,ENNReal.ofReal_zero] using
+    ENNReal.tendsto_ofReal (survivingGreen_mean_tendsto_zero a ha W hW z)
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma survivingGreen_lintegral_le_initial (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (z : UpperPlane) :
+    (∫⁻b, ENNReal.ofReal (survivingGreen a t b z.1) ∂W) ≤
+      ENNReal.ofReal (initialGreen a z) := by
+  obtain ⟨hi,hb⟩ := (actual_surviving_green a ha t).2.2 W hW z.1 z.2
+  rw [← ofReal_integral_eq_lintegral_ofReal hi
+    (Eventually.of_forall (fun b => survivingGreen_nonneg a t b z.1))]
+  exact ENNReal.ofReal_le_ofReal hb
+
+ 
+
+theorem survivingGreen_compact_mean_tendsto_zero (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (K : Set UpperPlane) (hK : IsCompact K) :
+    Tendsto (fun n : ℕ => ∫⁻b, ∫⁻z in K,
+      ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W) atTop (𝓝 0) := by
+  have hm (n : ℕ) : Measurable (fun z : UpperPlane =>
+      ∫⁻b, ENNReal.ofReal (survivingGreen a n b z.1) ∂W) :=
+    (survivingGreen_joint_measurable a ha n).ennreal_ofReal.lintegral_prod_left
+  have hb (n : ℕ) : (fun z : UpperPlane =>
+        ∫⁻b, ENNReal.ofReal (survivingGreen a n b z.1) ∂W) ≤ᵐ[upperArea.restrict K]
+      (fun z => ENNReal.ofReal (initialGreen a z)) :=
+    Eventually.of_forall (survivingGreen_lintegral_le_initial a ha W hW n)
+  have hf : (∫⁻z in K, ENNReal.ofReal (initialGreen a z) ∂upperArea) ≠ ∞ := by
+    rw [← withDensity_apply _ hK.measurableSet]
+    change initialGreenMeasure a K ≠ ∞
+    exact hK.measure_lt_top.ne
+  have hlim : ∀ᵐ z ∂upperArea.restrict K,
+      Tendsto (fun n : ℕ => ∫⁻b, ENNReal.ofReal (survivingGreen a n b z.1) ∂W)
+        atTop (𝓝 (0 : ℝ≥0∞)) := Eventually.of_forall (fun z =>
+    (survivingGreen_lintegral_tendsto_zero a ha W hW z).comp tendsto_natCast_atTop_atTop)
+  have h := tendsto_lintegral_of_dominated_convergence
+    (fun z => ENNReal.ofReal (initialGreen a z)) hm hb hf hlim
+  simp only [lintegral_zero] at h
+  convert h using 1
+  funext n
+  exact lintegral_lintegral_swap
+    (survivingGreen_joint_measurable a ha n).ennreal_ofReal.aemeasurable
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/BrownianMartingales.lean b/lean/OAI/Probability/SLEGauge/BrownianMartingales.lean
index a86f6c8c95b80f5a8a3295a92190c5ee2b346600..6883b2283d3e51dfb268590e5fe8a33d12a5e621 100644
--- a/lean/OAI/Probability/SLEGauge/BrownianMartingales.lean
+++ b/lean/OAI/Probability/SLEGauge/BrownianMartingales.lean
@@ -1,395 +1,395 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.FlowDerivatives
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-namespace ExactSLEGenerator
-open Set
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-theorem compact_smooth_brownian_martingale
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f) :
-    Martingale (fun t b => f (X t b) - ∫ u in (0 : ℝ)..(t : ℝ),
-      ((1/2 : ℝ) * directional 2 f e (X u.toNNReal b) +
-        fderiv ℝ f (X u.toNNReal b) (A (X u.toNNReal b)))) rawFiltration W := by
-  obtain ⟨C,M,hC,hM,hLip,hThird,hfb,hDb,hQb⟩ := compact_smooth_test_bounds f hf hfc e
-  have hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x+r • e)) 0) := by
-    simp_rw [iteratedDeriv_line f hf e,zero_smul,add_zero]
-    exact (directional_contDiff f hf e 2).continuous
-  have hf3 : ContDiff ℝ 3 f := hf.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)
-  have hSmooth (x : E) : ContDiff ℝ 3 (fun w : ℝ => f (x+w • e)) :=
-    hf3.comp (contDiff_const.add (contDiff_id.smul contDiff_const))
-  have h := additive_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f
-    (fderiv ℝ f) (fun x => (hf.differentiable (by simp) x).hasFDerivAt)
-    (hf.continuous_fderiv (by simp)) C M hC hM hLip hSmooth hThird hQc hfb hDb hQb
-  simpa only [iteratedDeriv_line f hf e,zero_smul,add_zero] using h
-
-end ExactSLEGenerator
-
- 
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped ENNReal NNReal Topology
-namespace ExactSLEOptional
-variable {Ω : Type*} {m : MeasurableSpace Ω} (ℱ : Filtration ℝ≥0 m)
-noncomputable def roundUp (N : ℕ) (s : ℝ≥0) : ℝ≥0 := (⌈(N : ℝ≥0) * s⌉₊ : ℝ≥0) / N
-
-theorem le_roundUp {N : ℕ} (hN : 0 < N) (s : ℝ≥0) : s ≤ roundUp N s := by
-  rw [roundUp, le_div_iff₀ (by exact_mod_cast hN)]
-  simpa [mul_comm] using Nat.le_ceil ((N : ℝ≥0) * s)
-
-theorem roundUp_lt {N : ℕ} (hN : 0 < N) (s : ℝ≥0) :
-    roundUp N s < s + (N : ℝ≥0)⁻¹ := by
-  rw [roundUp, div_lt_iff₀ (by exact_mod_cast hN)]
-  have h := Nat.ceil_lt_add_one (a := (N : ℝ≥0) * s) bot_le
-  have hn : (N : ℝ≥0) ≠ 0 := by exact_mod_cast hN.ne'
-  simpa [add_mul, mul_add, hn, mul_comm] using h
-
-theorem roundUp_le_iff {N : ℕ} (hN : 0 < N) (s t : ℝ≥0) :
-    roundUp N s ≤ t ↔ s ≤ (⌊(N : ℝ≥0) * t⌋₊ : ℝ≥0) / N := by
-  have hn : (0 : ℝ≥0) < N := by exact_mod_cast hN
-  rw [roundUp, div_le_iff₀ hn, le_div_iff₀ hn, mul_comm t, mul_comm s]
-  rw [← Nat.le_floor_iff bot_le, Nat.ceil_le]
-
-theorem roundUp_stopping (S : Ω → ℝ≥0)
-    (hS : IsStoppingTime ℱ (fun b => (S b : WithTop ℝ≥0)))
-    {N : ℕ} (hN : 0 < N) :
-    IsStoppingTime ℱ (fun b => (roundUp N (S b) : WithTop ℝ≥0)) := by
-  intro t
-  have ht : (⌊(N : ℝ≥0) * t⌋₊ : ℝ≥0) / N ≤ t := by
-    rw [div_le_iff₀ (by exact_mod_cast hN), mul_comm t]
-    exact Nat.floor_le bot_le
-  have he : {b : Ω | (roundUp N (S b) : WithTop ℝ≥0) ≤ t} =
-      {b : Ω | (S b : WithTop ℝ≥0) ≤ (((⌊(N : ℝ≥0) * t⌋₊ : ℝ≥0) / N : ℝ≥0) : WithTop ℝ≥0)} := by
-    ext b
-    exact_mod_cast roundUp_le_iff hN (S b) t
-  rw [he]
-  exact ℱ.mono ht _ (hS _)
-
-lemma roundUp_tendsto (s : ℝ≥0) :
-    Tendsto (fun n : ℕ => roundUp (n + 1) s) atTop (𝓝 s) := by
-  have hi : Tendsto (fun n : ℕ => ((n + 1 : ℕ) : ℝ≥0)⁻¹) atTop (𝓝 0) :=
-    tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop.comp (tendsto_add_atTop_nat 1))
-  have hr : Tendsto (fun n : ℕ => s + ((n + 1 : ℕ) : ℝ≥0)⁻¹) atTop (𝓝 s) := by
-    simpa using tendsto_const_nhds.add hi
-  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hr
-    (fun n => le_roundUp (Nat.succ_pos n) s) (fun n => (roundUp_lt (Nat.succ_pos n) s).le)
-
-lemma roundUp_le_add_one {N : ℕ} (hN : 0 < N) (s : ℝ≥0) :
-    roundUp N s ≤ s+1 := by
-  exact (roundUp_lt hN s).le.trans (add_le_add le_rfl
-    ((inv_le_one₀ (by exact_mod_cast hN : (0 : ℝ≥0) < (N : ℝ≥0))).mpr
-      (by exact_mod_cast Nat.succ_le_iff.mpr hN)))
-
-lemma roundUp_countable_range (S : Ω → ℝ≥0) (N : ℕ) :
-    (Set.range (fun ω => (roundUp N (S ω) : WithTop ℝ≥0))).Countable := by
-  apply (Set.countable_range (fun k : ℕ => (((k : ℝ≥0)/N : ℝ≥0) : WithTop ℝ≥0))).mono
-  rintro _ ⟨ω,rfl⟩
-  exact ⟨⌈(N : ℝ≥0) * S ω⌉₊,rfl⟩
-
-variable {μ : Measure Ω} [IsFiniteMeasure μ]
-variable {M : ℝ≥0 → Ω → ℝ}
-
-omit [IsFiniteMeasure μ] in
-lemma stopped_measurable (hM : Martingale M ℱ μ) (hc : ∀ ω, Continuous (fun t => M t ω))
-    (S : Ω → ℝ≥0) (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0))) :
-    Measurable (fun ω => M (S ω) ω) := by
-  exact
-    (measurable_stoppedValue (hM.stronglyAdapted.isStronglyProgressive_of_continuous hc) hS).mono hS.measurableSpace_le le_rfl
-
- 
-
-lemma setIntegral_stopped_eq_terminal (hM : Martingale M ℱ μ)
-    (hc : ∀ ω, Continuous (fun t => M t ω)) (K : ℝ≥0) (C : ℝ)
-    (hb : ∀ t ≤ K+1, ∀ ω, ‖M t ω‖ ≤ C)
-    (S : Ω → ℝ≥0) (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0)))
-    (hSK : ∀ ω, S ω ≤ K) (A : Set Ω) (hA : MeasurableSet[hS.measurableSpace] A) :
-    ∫ ω in A, M (S ω) ω ∂μ = ∫ ω in A, M (K+1) ω ∂μ := by
-  have heq (n : ℕ) : ∫ ω in A, M (roundUp (n+1) (S ω)) ω ∂μ =
-      ∫ ω in A, M (K+1) ω ∂μ := by
-    let Sn := fun ω => (roundUp (n+1) (S ω) : WithTop ℝ≥0)
-    have hSn : IsStoppingTime ℱ Sn := roundUp_stopping ℱ S hS (Nat.succ_pos n)
-    have hbnd : ∀ ω, Sn ω ≤ (K+1 : ℝ≥0) := by
-      intro ω
-      exact WithTop.coe_le_coe.mpr ((roundUp_le_add_one (Nat.succ_pos n) (S ω)).trans (add_le_add (hSK ω) le_rfl))
-    have hm := hM.stoppedValue_ae_eq_condExp_of_le_const_of_countable_range
-      hSn hbnd (roundUp_countable_range S (n+1))
-    have hASn : MeasurableSet[hSn.measurableSpace] A :=
-      hS.measurableSpace_mono hSn (fun ω => WithTop.coe_le_coe.mpr (le_roundUp (Nat.succ_pos n) (S ω))) _ hA
-    calc
-      _ = ∫ ω in A, stoppedValue M Sn ω ∂μ := by
-        congr 1
-      _ = ∫ ω in A, μ[M (K+1) | hSn.measurableSpace] ω ∂μ :=
-        integral_congr_ae (ae_restrict_of_ae hm)
-      _ = _ := setIntegral_condExp hSn.measurableSpace_le (hM.integrable _) hASn
-  have ht : Tendsto (fun n : ℕ => ∫ ω in A, M (roundUp (n+1) (S ω)) ω ∂μ)
-      atTop (𝓝 (∫ ω in A, M (S ω) ω ∂μ)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => C)
-    · intro n
-      exact (stopped_measurable ℱ hM hc _ (roundUp_stopping ℱ S hS (Nat.succ_pos n))).aestronglyMeasurable
-    · exact integrable_const C
-    · intro n
-      exact Filter.Eventually.of_forall fun ω => hb _
-        ((roundUp_le_add_one (Nat.succ_pos n) (S ω)).trans (add_le_add (hSK ω) le_rfl)) ω
-    · exact Filter.Eventually.of_forall fun ω => (hc ω).tendsto (S ω) |>.comp (roundUp_tendsto (S ω))
-  exact tendsto_nhds_unique ht (by simp_rw [heq]; exact tendsto_const_nhds)
-
-lemma setIntegral_optional_sampling (hM : Martingale M ℱ μ)
-    (hc : ∀ ω, Continuous (fun t => M t ω)) (K : ℝ≥0) (C : ℝ)
-    (hb : ∀ t ≤ K+1, ∀ ω, ‖M t ω‖ ≤ C)
-    (S T : Ω → ℝ≥0)
-    (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0)))
-    (hT : IsStoppingTime ℱ (fun ω => (T ω : WithTop ℝ≥0)))
-    (hST : ∀ ω, S ω ≤ T ω) (hTK : ∀ ω, T ω ≤ K)
-    (A : Set Ω) (hA : MeasurableSet[hS.measurableSpace] A) :
-    ∫ ω in A, M (S ω) ω ∂μ = ∫ ω in A, M (T ω) ω ∂μ := by
-  rw [setIntegral_stopped_eq_terminal ℱ hM hc K C hb S hS (fun ω => (hST ω).trans (hTK ω)) A hA,
-    setIntegral_stopped_eq_terminal ℱ hM hc K C hb T hT hTK A
-      (hS.measurableSpace_mono hT (fun ω => WithTop.coe_le_coe.mpr (hST ω)) _ hA)]
-
-end ExactSLEOptional
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
-noncomputable def generator (f : E → ℝ) (A : E → E) (e : E) (x : E) : ℝ :=
-  (1/2 : ℝ) * directional 2 f e x + fderiv ℝ f x (A x)
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma generator_continuous (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
-    (A : E → E) (hA : Continuous A) (e : E) : Continuous (generator f A e) :=
-  (continuous_const.mul (directional_contDiff f hf e 2).continuous).add
-    ((hf.continuous_fderiv (by simp)).clm_apply hA)
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma compact_generator_bounds (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
-    (hfc : HasCompactSupport f) (A : E → E) (K : ℝ) (hK : 0 ≤ K)
-    (hAb : ∀ x, ‖A x‖ ≤ K) (e : E) :
-    ∃ M C : ℝ, 0 ≤ M ∧ 0 ≤ C ∧ (∀ x, ‖f x‖ ≤ M) ∧
-      (∀ x, ‖generator f A e x‖ ≤ C) := by
-  obtain ⟨D,M,hD,hM,hLip,hThird,hfb,hDb,hQb⟩ := compact_smooth_test_bounds f hf hfc e
-  refine ⟨M,M/2+M*K,hM,by positivity,?_,?_⟩
-  · intro x
-    exact hfb x
-  · intro x
-    have hq : ‖directional 2 f e x‖ ≤ M := by
-      simpa only [iteratedDeriv_line f hf e,zero_smul,add_zero,Real.norm_eq_abs] using hQb x
-    have hv : ‖fderiv ℝ f x (A x)‖ ≤ M*K :=
-      ((fderiv ℝ f x).le_opNorm _).trans (mul_le_mul (hDb x) (hAb x) (norm_nonneg _) hM)
-    calc
-      _ ≤ ‖(1/2 : ℝ)*directional 2 f e x‖ + ‖fderiv ℝ f x (A x)‖ := norm_add_le _ _
-      _ ≤ (1/2 : ℝ)*M+M*K := add_le_add (by
-        rw [norm_mul,Real.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/2)]
-        exact mul_le_mul_of_nonneg_left hq (by norm_num)) hv
-      _ = _ := by ring
-
- 
-
-theorem compact_test_optional_sampling
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
-    (S T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
-    (hzero : ∀ b t, t ≤ T b → generator f A e (X t b) = 0)
-    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    ∫ b in U, f (X (S b) b) ∂W = ∫ b in U, f (X (T b) b) ∂W := by
-  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
-    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
-  have hM : Martingale M rawFiltration W :=
-    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
-  have hc (b : Driver) : Continuous (fun t => M t b) := by
-    have hg : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
-      (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
-    exact (hf.continuous.comp (hXc b)).sub
-      ((intervalIntegral.continuous_primitive (fun s t => hg.intervalIntegrable s t) 0).comp NNReal.continuous_coe)
-  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
-  have hb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
-    calc
-      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
-      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
-        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
-          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
-            (fun u _ => hgb (X u.toNNReal b)))
-      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
-  have heq (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) : M t b = f (X t b) := by
-    change f (X t b) - _ = _
-    suffices (∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)) = 0 by rw [this,sub_zero]
-    apply Eq.trans _ (intervalIntegral.integral_zero (a := (0 : ℝ)) (b := (t : ℝ)) (μ := volume) (E := ℝ))
-    apply intervalIntegral.integral_congr
-    intro u hu
-    apply hzero b u.toNNReal
-    have hu' : u ≤ (t : ℝ) := (by simpa only [uIcc_of_le t.coe_nonneg] using hu : u ∈ Icc 0 (t : ℝ)).2
-    exact (Real.toNNReal_le_toNNReal hu').trans (by simpa using ht)
-  have ho := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hc B
-    (C+D*(B+1)) hb S T hS hT hST hTB U hU
-  simpa only [heq _ _ (hST _),heq _ _ le_rfl] using ho
-
-end ExactSLEGenerator
-
-namespace ExactSLEMain
-
-abbrev Driver := {b : C(ℝ≥0, ℝ) // b 0 = 0}
-instance : MeasurableSpace Driver := borel Driver
-
-def odeUntil (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ≥0) (f : ℝ → ℂ) : Prop :=
-  f 0 = z ∧ ∀ u ∈ Set.Icc 0 (t : ℝ),
-    0 < (f u).im ∧ HasDerivWithinAt f ((A : ℂ) / (f u - (b u.toNNReal : ℂ)))
-      (Set.Icc 0 (t : ℝ)) u
-
-def alive (A : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) : Prop :=
-  ∃ f : ℝ → ℂ, odeUntil A b z t f
-
-noncomputable def loewnerMap (A : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) : ℂ := by
-  classical
-  exact if h : alive A b t z then (Classical.choose h) t else 0
-
-theorem odeUntil_continuousOn {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z t f) : ContinuousOn f (Set.Icc 0 (t : ℝ)) := by
-  exact HasDerivWithinAt.continuousOn (fun u hu => (hf.2 u hu).2)
-
-theorem odeUntil_height_lower {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z t f) :
-    ∃ ε : ℝ, 0 < ε ∧ ∀ u ∈ Set.Icc 0 (t : ℝ), ε ≤ (f u).im := by
-  have hnon : (Set.Icc 0 (t : ℝ)).Nonempty := ⟨0, by simp⟩
-  obtain ⟨u, hu, hmin⟩ := isCompact_Icc.exists_isMinOn hnon
-    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf))
-  exact ⟨(f u).im, (hf.2 u hu).1, hmin⟩
-
-theorem loewner_vector_difference (A β ε : ℝ) (hε : 0 < ε) (v w : ℂ)
-    (hv : ε ≤ v.im) (hw : ε ≤ w.im) :
-    ‖(A : ℂ) / (v - (β : ℂ)) - (A : ℂ) / (w - (β : ℂ))‖ ≤
-      (|A| / ε ^ 2) * ‖v - w‖ := by
-  have hv0 : 0 < v.im := hε.trans_le hv
-  have hw0 : 0 < w.im := hε.trans_le hw
-  have hnv : ε ≤ ‖v - (β : ℂ)‖ := by
-    calc
-      ε ≤ v.im := hv
-      _ = (v - (β : ℂ)).im := by simp
-      _ ≤ _ := Complex.im_le_norm _
-  have hnw : ε ≤ ‖w - (β : ℂ)‖ := by
-    calc
-      ε ≤ w.im := hw
-      _ = (w - (β : ℂ)).im := by simp
-      _ ≤ _ := Complex.im_le_norm _
-  have hvz : v - (β : ℂ) ≠ 0 := norm_pos_iff.mp (hε.trans_le hnv)
-  have hwz : w - (β : ℂ) ≠ 0 := norm_pos_iff.mp (hε.trans_le hnw)
-  have heq : (A : ℂ) / (v - (β : ℂ)) - (A : ℂ) / (w - (β : ℂ)) =
-      (-(A : ℂ)) * (v - w) / ((v - (β : ℂ)) * (w - (β : ℂ))) := by
-    field_simp
-    ring
-  rw [heq, norm_div, norm_mul, norm_mul, norm_neg, Complex.norm_real, Real.norm_eq_abs]
-  calc
-    |A| * ‖v - w‖ / (‖v - (β : ℂ)‖ * ‖w - (β : ℂ)‖) ≤ |A| * ‖v - w‖ / ε ^ 2 := by
-      apply div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hε)
-      simpa [pow_two] using mul_le_mul hnv hnw hε.le (norm_nonneg _)
-    _ = _ := by ring
-
-theorem odeUntil_unique (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ≥0)
-    {f g : ℝ → ℂ} (hf : odeUntil A b z t f) (hg : odeUntil A b z t g) :
-    Set.EqOn f g (Set.Icc 0 (t : ℝ)) := by
-  obtain ⟨εf, hεf, hflower⟩ := odeUntil_height_lower hf
-  obtain ⟨εg, hεg, hglower⟩ := odeUntil_height_lower hg
-  let ε := min εf εg
-  have hε : 0 < ε := lt_min hεf hεg
-  have hderiv : ∀ u ∈ Set.Ico 0 (t : ℝ), HasDerivWithinAt (fun u => f u - g u)
-      ((A : ℂ) / (f u - (b u.toNNReal : ℂ)) - (A : ℂ) / (g u - (b u.toNNReal : ℂ)))
-      (Set.Ici u) u := by
-    intro u hu
-    exact ((hf.2 u (Set.mem_Icc_of_Ico hu)).2.sub (hg.2 u (Set.mem_Icc_of_Ico hu)).2).mono_of_mem_nhdsWithin
-      (Icc_mem_nhdsGE_of_mem hu)
-  have hzero : ∀ u ∈ Set.Icc 0 (t : ℝ), f u - g u = 0 := by
-    apply eq_zero_of_abs_deriv_le_mul_abs_self_of_eq_zero_right
-      ((odeUntil_continuousOn hf).sub (odeUntil_continuousOn hg)) hderiv
-      (show f 0 - g 0 = 0 by rw [hf.1, hg.1, sub_self])
-    intro u hu
-    exact loewner_vector_difference A (b u.toNNReal) ε hε (f u) (g u)
-      ((min_le_left _ _).trans (hflower u (Set.mem_Icc_of_Ico hu)))
-      ((min_le_right _ _).trans (hglower u (Set.mem_Icc_of_Ico hu)))
-  exact fun u hu => sub_eq_zero.mp (hzero u hu)
-
-theorem odeUntil_mono {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z t f) (hst : s ≤ t) : odeUntil A b z s f := by
-  refine ⟨hf.1, fun u hu => ?_⟩
-  have hsub : Set.Icc 0 (s : ℝ) ⊆ Set.Icc 0 (t : ℝ) :=
-    Set.Icc_subset_Icc_right (show (s : ℝ) ≤ t from hst)
-  exact ⟨(hf.2 u (hsub hu)).1, (hf.2 u (hsub hu)).2.mono hsub⟩
-
-theorem alive_mono {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
-    (h : alive A b t z) (hst : s ≤ t) : alive A b s z := by
-  obtain ⟨f, hf⟩ := h
-  exact ⟨f, odeUntil_mono hf hst⟩
-
-theorem loewnerMap_eq {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z t f) : loewnerMap A b t z = f t := by
-  have he : alive A b t z := ⟨f, hf⟩
-  rw [loewnerMap, dite_eq_left he]
-  exact odeUntil_unique A b z t (Classical.choose_spec he) hf (by simp)
-
-def shiftDriver (b : ℝ≥0 → ℝ) (s : ℝ≥0) : ℝ≥0 → ℝ := fun u => b (s + u) - b s
-
-theorem odeUntil_shift {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z (s + t) f) :
-    odeUntil A (shiftDriver b s) (f s - (b s : ℂ)) t
-      (fun u => f (u + s) - (b s : ℂ)) := by
-  refine ⟨by simp, fun u hu => ?_⟩
-  have hmaps : Set.MapsTo (fun u : ℝ => u + s) (Set.Icc 0 (t : ℝ)) (Set.Icc 0 ((s + t : ℝ≥0) : ℝ)) := by
-    intro v hv
-    constructor <;> simp only [NNReal.coe_add] <;> linarith [s.coe_nonneg, hv.1, hv.2]
-  have hnn : (u + (s : ℝ)).toNNReal = s + u.toNNReal := by
-    apply NNReal.coe_injective
-    simp only [NNReal.coe_add, Real.coe_toNNReal _ (add_nonneg hu.1 s.coe_nonneg),
-      Real.coe_toNNReal _ hu.1]
-    ring
-  have hder := ((hf.2 (u + s) (hmaps hu)).2.scomp u
-    ((hasDerivAt_id u).add_const (s : ℝ)).hasDerivWithinAt hmaps).sub_const (b s : ℂ)
-  constructor
-  · simpa using (hf.2 (u + s) (hmaps hu)).1
-  · simpa [shiftDriver, hnn, sub_sub_sub_cancel_right] using hder
-
-theorem loewnerMap_restart (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (s t : ℝ≥0)
-    (h : alive A b (s + t) z) :
-    alive A (shiftDriver b s) t (loewnerMap A b s z - (b s : ℂ)) ∧
-    loewnerMap A (shiftDriver b s) t (loewnerMap A b s z - (b s : ℂ)) =
-      loewnerMap A b (s + t) z - (b s : ℂ) := by
-  obtain ⟨f, hf⟩ := h
-  have hs : odeUntil A b z s f := odeUntil_mono hf (by exact le_add_of_nonneg_right (show 0 ≤ t from bot_le))
-  have hfs := loewnerMap_eq hs
-  rw [hfs]
-  constructor
-  · exact ⟨_, odeUntil_shift hf⟩
-  · rw [loewnerMap_eq (odeUntil_shift hf), loewnerMap_eq hf]
-    simp [add_comm]
-
-end ExactSLEMain
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.FlowDerivatives
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+namespace ExactSLEGenerator
+open Set
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+theorem compact_smooth_brownian_martingale
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f) :
+    Martingale (fun t b => f (X t b) - ∫ u in (0 : ℝ)..(t : ℝ),
+      ((1/2 : ℝ) * directional 2 f e (X u.toNNReal b) +
+        fderiv ℝ f (X u.toNNReal b) (A (X u.toNNReal b)))) rawFiltration W := by
+  obtain ⟨C,M,hC,hM,hLip,hThird,hfb,hDb,hQb⟩ := compact_smooth_test_bounds f hf hfc e
+  have hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x+r • e)) 0) := by
+    simp_rw [iteratedDeriv_line f hf e,zero_smul,add_zero]
+    exact (directional_contDiff f hf e 2).continuous
+  have hf3 : ContDiff ℝ 3 f := hf.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)
+  have hSmooth (x : E) : ContDiff ℝ 3 (fun w : ℝ => f (x+w • e)) :=
+    hf3.comp (contDiff_const.add (contDiff_id.smul contDiff_const))
+  have h := additive_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f
+    (fderiv ℝ f) (fun x => (hf.differentiable (by simp) x).hasFDerivAt)
+    (hf.continuous_fderiv (by simp)) C M hC hM hLip hSmooth hThird hQc hfb hDb hQb
+  simpa only [iteratedDeriv_line f hf e,zero_smul,add_zero] using h
+
+end ExactSLEGenerator
+
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped ENNReal NNReal Topology
+namespace ExactSLEOptional
+variable {Ω : Type*} {m : MeasurableSpace Ω} (ℱ : Filtration ℝ≥0 m)
+noncomputable def roundUp (N : ℕ) (s : ℝ≥0) : ℝ≥0 := (⌈(N : ℝ≥0) * s⌉₊ : ℝ≥0) / N
+
+theorem le_roundUp {N : ℕ} (hN : 0 < N) (s : ℝ≥0) : s ≤ roundUp N s := by
+  rw [roundUp, le_div_iff₀ (by exact_mod_cast hN)]
+  simpa [mul_comm] using Nat.le_ceil ((N : ℝ≥0) * s)
+
+theorem roundUp_lt {N : ℕ} (hN : 0 < N) (s : ℝ≥0) :
+    roundUp N s < s + (N : ℝ≥0)⁻¹ := by
+  rw [roundUp, div_lt_iff₀ (by exact_mod_cast hN)]
+  have h := Nat.ceil_lt_add_one (a := (N : ℝ≥0) * s) bot_le
+  have hn : (N : ℝ≥0) ≠ 0 := by exact_mod_cast hN.ne'
+  simpa [add_mul, mul_add, hn, mul_comm] using h
+
+theorem roundUp_le_iff {N : ℕ} (hN : 0 < N) (s t : ℝ≥0) :
+    roundUp N s ≤ t ↔ s ≤ (⌊(N : ℝ≥0) * t⌋₊ : ℝ≥0) / N := by
+  have hn : (0 : ℝ≥0) < N := by exact_mod_cast hN
+  rw [roundUp, div_le_iff₀ hn, le_div_iff₀ hn, mul_comm t, mul_comm s]
+  rw [← Nat.le_floor_iff bot_le, Nat.ceil_le]
+
+theorem roundUp_stopping (S : Ω → ℝ≥0)
+    (hS : IsStoppingTime ℱ (fun b => (S b : WithTop ℝ≥0)))
+    {N : ℕ} (hN : 0 < N) :
+    IsStoppingTime ℱ (fun b => (roundUp N (S b) : WithTop ℝ≥0)) := by
+  intro t
+  have ht : (⌊(N : ℝ≥0) * t⌋₊ : ℝ≥0) / N ≤ t := by
+    rw [div_le_iff₀ (by exact_mod_cast hN), mul_comm t]
+    exact Nat.floor_le bot_le
+  have he : {b : Ω | (roundUp N (S b) : WithTop ℝ≥0) ≤ t} =
+      {b : Ω | (S b : WithTop ℝ≥0) ≤ (((⌊(N : ℝ≥0) * t⌋₊ : ℝ≥0) / N : ℝ≥0) : WithTop ℝ≥0)} := by
+    ext b
+    exact_mod_cast roundUp_le_iff hN (S b) t
+  rw [he]
+  exact ℱ.mono ht _ (hS _)
+
+lemma roundUp_tendsto (s : ℝ≥0) :
+    Tendsto (fun n : ℕ => roundUp (n + 1) s) atTop (𝓝 s) := by
+  have hi : Tendsto (fun n : ℕ => ((n + 1 : ℕ) : ℝ≥0)⁻¹) atTop (𝓝 0) :=
+    tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop.comp (tendsto_add_atTop_nat 1))
+  have hr : Tendsto (fun n : ℕ => s + ((n + 1 : ℕ) : ℝ≥0)⁻¹) atTop (𝓝 s) := by
+    simpa using tendsto_const_nhds.add hi
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hr
+    (fun n => le_roundUp (Nat.succ_pos n) s) (fun n => (roundUp_lt (Nat.succ_pos n) s).le)
+
+lemma roundUp_le_add_one {N : ℕ} (hN : 0 < N) (s : ℝ≥0) :
+    roundUp N s ≤ s+1 := by
+  exact (roundUp_lt hN s).le.trans (add_le_add le_rfl
+    ((inv_le_one₀ (by exact_mod_cast hN : (0 : ℝ≥0) < (N : ℝ≥0))).mpr
+      (by exact_mod_cast Nat.succ_le_iff.mpr hN)))
+
+lemma roundUp_countable_range (S : Ω → ℝ≥0) (N : ℕ) :
+    (Set.range (fun ω => (roundUp N (S ω) : WithTop ℝ≥0))).Countable := by
+  apply (Set.countable_range (fun k : ℕ => (((k : ℝ≥0)/N : ℝ≥0) : WithTop ℝ≥0))).mono
+  rintro _ ⟨ω,rfl⟩
+  exact ⟨⌈(N : ℝ≥0) * S ω⌉₊,rfl⟩
+
+variable {μ : Measure Ω} [IsFiniteMeasure μ]
+variable {M : ℝ≥0 → Ω → ℝ}
+
+omit [IsFiniteMeasure μ] in
+lemma stopped_measurable (hM : Martingale M ℱ μ) (hc : ∀ ω, Continuous (fun t => M t ω))
+    (S : Ω → ℝ≥0) (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0))) :
+    Measurable (fun ω => M (S ω) ω) := by
+  exact
+    (measurable_stoppedValue (hM.stronglyAdapted.isStronglyProgressive_of_continuous hc) hS).mono hS.measurableSpace_le le_rfl
+
+ 
+
+lemma setIntegral_stopped_eq_terminal (hM : Martingale M ℱ μ)
+    (hc : ∀ ω, Continuous (fun t => M t ω)) (K : ℝ≥0) (C : ℝ)
+    (hb : ∀ t ≤ K+1, ∀ ω, ‖M t ω‖ ≤ C)
+    (S : Ω → ℝ≥0) (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0)))
+    (hSK : ∀ ω, S ω ≤ K) (A : Set Ω) (hA : MeasurableSet[hS.measurableSpace] A) :
+    ∫ ω in A, M (S ω) ω ∂μ = ∫ ω in A, M (K+1) ω ∂μ := by
+  have heq (n : ℕ) : ∫ ω in A, M (roundUp (n+1) (S ω)) ω ∂μ =
+      ∫ ω in A, M (K+1) ω ∂μ := by
+    let Sn := fun ω => (roundUp (n+1) (S ω) : WithTop ℝ≥0)
+    have hSn : IsStoppingTime ℱ Sn := roundUp_stopping ℱ S hS (Nat.succ_pos n)
+    have hbnd : ∀ ω, Sn ω ≤ (K+1 : ℝ≥0) := by
+      intro ω
+      exact WithTop.coe_le_coe.mpr ((roundUp_le_add_one (Nat.succ_pos n) (S ω)).trans (add_le_add (hSK ω) le_rfl))
+    have hm := hM.stoppedValue_ae_eq_condExp_of_le_const_of_countable_range
+      hSn hbnd (roundUp_countable_range S (n+1))
+    have hASn : MeasurableSet[hSn.measurableSpace] A :=
+      hS.measurableSpace_mono hSn (fun ω => WithTop.coe_le_coe.mpr (le_roundUp (Nat.succ_pos n) (S ω))) _ hA
+    calc
+      _ = ∫ ω in A, stoppedValue M Sn ω ∂μ := by
+        congr 1
+      _ = ∫ ω in A, μ[M (K+1) | hSn.measurableSpace] ω ∂μ :=
+        integral_congr_ae (ae_restrict_of_ae hm)
+      _ = _ := setIntegral_condExp hSn.measurableSpace_le (hM.integrable _) hASn
+  have ht : Tendsto (fun n : ℕ => ∫ ω in A, M (roundUp (n+1) (S ω)) ω ∂μ)
+      atTop (𝓝 (∫ ω in A, M (S ω) ω ∂μ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => C)
+    · intro n
+      exact (stopped_measurable ℱ hM hc _ (roundUp_stopping ℱ S hS (Nat.succ_pos n))).aestronglyMeasurable
+    · exact integrable_const C
+    · intro n
+      exact Filter.Eventually.of_forall fun ω => hb _
+        ((roundUp_le_add_one (Nat.succ_pos n) (S ω)).trans (add_le_add (hSK ω) le_rfl)) ω
+    · exact Filter.Eventually.of_forall fun ω => (hc ω).tendsto (S ω) |>.comp (roundUp_tendsto (S ω))
+  exact tendsto_nhds_unique ht (by simp_rw [heq]; exact tendsto_const_nhds)
+
+lemma setIntegral_optional_sampling (hM : Martingale M ℱ μ)
+    (hc : ∀ ω, Continuous (fun t => M t ω)) (K : ℝ≥0) (C : ℝ)
+    (hb : ∀ t ≤ K+1, ∀ ω, ‖M t ω‖ ≤ C)
+    (S T : Ω → ℝ≥0)
+    (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0)))
+    (hT : IsStoppingTime ℱ (fun ω => (T ω : WithTop ℝ≥0)))
+    (hST : ∀ ω, S ω ≤ T ω) (hTK : ∀ ω, T ω ≤ K)
+    (A : Set Ω) (hA : MeasurableSet[hS.measurableSpace] A) :
+    ∫ ω in A, M (S ω) ω ∂μ = ∫ ω in A, M (T ω) ω ∂μ := by
+  rw [setIntegral_stopped_eq_terminal ℱ hM hc K C hb S hS (fun ω => (hST ω).trans (hTK ω)) A hA,
+    setIntegral_stopped_eq_terminal ℱ hM hc K C hb T hT hTK A
+      (hS.measurableSpace_mono hT (fun ω => WithTop.coe_le_coe.mpr (hST ω)) _ hA)]
+
+end ExactSLEOptional
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+noncomputable def generator (f : E → ℝ) (A : E → E) (e : E) (x : E) : ℝ :=
+  (1/2 : ℝ) * directional 2 f e x + fderiv ℝ f x (A x)
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma generator_continuous (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
+    (A : E → E) (hA : Continuous A) (e : E) : Continuous (generator f A e) :=
+  (continuous_const.mul (directional_contDiff f hf e 2).continuous).add
+    ((hf.continuous_fderiv (by simp)).clm_apply hA)
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma compact_generator_bounds (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
+    (hfc : HasCompactSupport f) (A : E → E) (K : ℝ) (hK : 0 ≤ K)
+    (hAb : ∀ x, ‖A x‖ ≤ K) (e : E) :
+    ∃ M C : ℝ, 0 ≤ M ∧ 0 ≤ C ∧ (∀ x, ‖f x‖ ≤ M) ∧
+      (∀ x, ‖generator f A e x‖ ≤ C) := by
+  obtain ⟨D,M,hD,hM,hLip,hThird,hfb,hDb,hQb⟩ := compact_smooth_test_bounds f hf hfc e
+  refine ⟨M,M/2+M*K,hM,by positivity,?_,?_⟩
+  · intro x
+    exact hfb x
+  · intro x
+    have hq : ‖directional 2 f e x‖ ≤ M := by
+      simpa only [iteratedDeriv_line f hf e,zero_smul,add_zero,Real.norm_eq_abs] using hQb x
+    have hv : ‖fderiv ℝ f x (A x)‖ ≤ M*K :=
+      ((fderiv ℝ f x).le_opNorm _).trans (mul_le_mul (hDb x) (hAb x) (norm_nonneg _) hM)
+    calc
+      _ ≤ ‖(1/2 : ℝ)*directional 2 f e x‖ + ‖fderiv ℝ f x (A x)‖ := norm_add_le _ _
+      _ ≤ (1/2 : ℝ)*M+M*K := add_le_add (by
+        rw [norm_mul,Real.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/2)]
+        exact mul_le_mul_of_nonneg_left hq (by norm_num)) hv
+      _ = _ := by ring
+
+ 
+
+theorem compact_test_optional_sampling
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
+    (S T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
+    (hzero : ∀ b t, t ≤ T b → generator f A e (X t b) = 0)
+    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    ∫ b in U, f (X (S b) b) ∂W = ∫ b in U, f (X (T b) b) ∂W := by
+  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
+    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
+  have hM : Martingale M rawFiltration W :=
+    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
+  have hc (b : Driver) : Continuous (fun t => M t b) := by
+    have hg : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
+      (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
+    exact (hf.continuous.comp (hXc b)).sub
+      ((intervalIntegral.continuous_primitive (fun s t => hg.intervalIntegrable s t) 0).comp NNReal.continuous_coe)
+  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
+  have hb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
+    calc
+      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
+      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
+        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
+          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
+            (fun u _ => hgb (X u.toNNReal b)))
+      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
+  have heq (b : Driver) (t : ℝ≥0) (ht : t ≤ T b) : M t b = f (X t b) := by
+    change f (X t b) - _ = _
+    suffices (∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)) = 0 by rw [this,sub_zero]
+    apply Eq.trans _ (intervalIntegral.integral_zero (a := (0 : ℝ)) (b := (t : ℝ)) (μ := volume) (E := ℝ))
+    apply intervalIntegral.integral_congr
+    intro u hu
+    apply hzero b u.toNNReal
+    have hu' : u ≤ (t : ℝ) := (by simpa only [uIcc_of_le t.coe_nonneg] using hu : u ∈ Icc 0 (t : ℝ)).2
+    exact (Real.toNNReal_le_toNNReal hu').trans (by simpa using ht)
+  have ho := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hc B
+    (C+D*(B+1)) hb S T hS hT hST hTB U hU
+  simpa only [heq _ _ (hST _),heq _ _ le_rfl] using ho
+
+end ExactSLEGenerator
+
+namespace ExactSLEMain
+
+abbrev Driver := {b : C(ℝ≥0, ℝ) // b 0 = 0}
+instance : MeasurableSpace Driver := borel Driver
+
+def odeUntil (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ≥0) (f : ℝ → ℂ) : Prop :=
+  f 0 = z ∧ ∀ u ∈ Set.Icc 0 (t : ℝ),
+    0 < (f u).im ∧ HasDerivWithinAt f ((A : ℂ) / (f u - (b u.toNNReal : ℂ)))
+      (Set.Icc 0 (t : ℝ)) u
+
+def alive (A : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) : Prop :=
+  ∃ f : ℝ → ℂ, odeUntil A b z t f
+
+noncomputable def loewnerMap (A : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) : ℂ := by
+  classical
+  exact if h : alive A b t z then (Classical.choose h) t else 0
+
+theorem odeUntil_continuousOn {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z t f) : ContinuousOn f (Set.Icc 0 (t : ℝ)) := by
+  exact HasDerivWithinAt.continuousOn (fun u hu => (hf.2 u hu).2)
+
+theorem odeUntil_height_lower {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z t f) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ u ∈ Set.Icc 0 (t : ℝ), ε ≤ (f u).im := by
+  have hnon : (Set.Icc 0 (t : ℝ)).Nonempty := ⟨0, by simp⟩
+  obtain ⟨u, hu, hmin⟩ := isCompact_Icc.exists_isMinOn hnon
+    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf))
+  exact ⟨(f u).im, (hf.2 u hu).1, hmin⟩
+
+theorem loewner_vector_difference (A β ε : ℝ) (hε : 0 < ε) (v w : ℂ)
+    (hv : ε ≤ v.im) (hw : ε ≤ w.im) :
+    ‖(A : ℂ) / (v - (β : ℂ)) - (A : ℂ) / (w - (β : ℂ))‖ ≤
+      (|A| / ε ^ 2) * ‖v - w‖ := by
+  have hv0 : 0 < v.im := hε.trans_le hv
+  have hw0 : 0 < w.im := hε.trans_le hw
+  have hnv : ε ≤ ‖v - (β : ℂ)‖ := by
+    calc
+      ε ≤ v.im := hv
+      _ = (v - (β : ℂ)).im := by simp
+      _ ≤ _ := Complex.im_le_norm _
+  have hnw : ε ≤ ‖w - (β : ℂ)‖ := by
+    calc
+      ε ≤ w.im := hw
+      _ = (w - (β : ℂ)).im := by simp
+      _ ≤ _ := Complex.im_le_norm _
+  have hvz : v - (β : ℂ) ≠ 0 := norm_pos_iff.mp (hε.trans_le hnv)
+  have hwz : w - (β : ℂ) ≠ 0 := norm_pos_iff.mp (hε.trans_le hnw)
+  have heq : (A : ℂ) / (v - (β : ℂ)) - (A : ℂ) / (w - (β : ℂ)) =
+      (-(A : ℂ)) * (v - w) / ((v - (β : ℂ)) * (w - (β : ℂ))) := by
+    field_simp
+    ring
+  rw [heq, norm_div, norm_mul, norm_mul, norm_neg, Complex.norm_real, Real.norm_eq_abs]
+  calc
+    |A| * ‖v - w‖ / (‖v - (β : ℂ)‖ * ‖w - (β : ℂ)‖) ≤ |A| * ‖v - w‖ / ε ^ 2 := by
+      apply div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hε)
+      simpa [pow_two] using mul_le_mul hnv hnw hε.le (norm_nonneg _)
+    _ = _ := by ring
+
+theorem odeUntil_unique (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ≥0)
+    {f g : ℝ → ℂ} (hf : odeUntil A b z t f) (hg : odeUntil A b z t g) :
+    Set.EqOn f g (Set.Icc 0 (t : ℝ)) := by
+  obtain ⟨εf, hεf, hflower⟩ := odeUntil_height_lower hf
+  obtain ⟨εg, hεg, hglower⟩ := odeUntil_height_lower hg
+  let ε := min εf εg
+  have hε : 0 < ε := lt_min hεf hεg
+  have hderiv : ∀ u ∈ Set.Ico 0 (t : ℝ), HasDerivWithinAt (fun u => f u - g u)
+      ((A : ℂ) / (f u - (b u.toNNReal : ℂ)) - (A : ℂ) / (g u - (b u.toNNReal : ℂ)))
+      (Set.Ici u) u := by
+    intro u hu
+    exact ((hf.2 u (Set.mem_Icc_of_Ico hu)).2.sub (hg.2 u (Set.mem_Icc_of_Ico hu)).2).mono_of_mem_nhdsWithin
+      (Icc_mem_nhdsGE_of_mem hu)
+  have hzero : ∀ u ∈ Set.Icc 0 (t : ℝ), f u - g u = 0 := by
+    apply eq_zero_of_abs_deriv_le_mul_abs_self_of_eq_zero_right
+      ((odeUntil_continuousOn hf).sub (odeUntil_continuousOn hg)) hderiv
+      (show f 0 - g 0 = 0 by rw [hf.1, hg.1, sub_self])
+    intro u hu
+    exact loewner_vector_difference A (b u.toNNReal) ε hε (f u) (g u)
+      ((min_le_left _ _).trans (hflower u (Set.mem_Icc_of_Ico hu)))
+      ((min_le_right _ _).trans (hglower u (Set.mem_Icc_of_Ico hu)))
+  exact fun u hu => sub_eq_zero.mp (hzero u hu)
+
+theorem odeUntil_mono {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z t f) (hst : s ≤ t) : odeUntil A b z s f := by
+  refine ⟨hf.1, fun u hu => ?_⟩
+  have hsub : Set.Icc 0 (s : ℝ) ⊆ Set.Icc 0 (t : ℝ) :=
+    Set.Icc_subset_Icc_right (show (s : ℝ) ≤ t from hst)
+  exact ⟨(hf.2 u (hsub hu)).1, (hf.2 u (hsub hu)).2.mono hsub⟩
+
+theorem alive_mono {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
+    (h : alive A b t z) (hst : s ≤ t) : alive A b s z := by
+  obtain ⟨f, hf⟩ := h
+  exact ⟨f, odeUntil_mono hf hst⟩
+
+theorem loewnerMap_eq {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z t f) : loewnerMap A b t z = f t := by
+  have he : alive A b t z := ⟨f, hf⟩
+  rw [loewnerMap, dite_eq_left he]
+  exact odeUntil_unique A b z t (Classical.choose_spec he) hf (by simp)
+
+def shiftDriver (b : ℝ≥0 → ℝ) (s : ℝ≥0) : ℝ≥0 → ℝ := fun u => b (s + u) - b s
+
+theorem odeUntil_shift {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z (s + t) f) :
+    odeUntil A (shiftDriver b s) (f s - (b s : ℂ)) t
+      (fun u => f (u + s) - (b s : ℂ)) := by
+  refine ⟨by simp, fun u hu => ?_⟩
+  have hmaps : Set.MapsTo (fun u : ℝ => u + s) (Set.Icc 0 (t : ℝ)) (Set.Icc 0 ((s + t : ℝ≥0) : ℝ)) := by
+    intro v hv
+    constructor <;> simp only [NNReal.coe_add] <;> linarith [s.coe_nonneg, hv.1, hv.2]
+  have hnn : (u + (s : ℝ)).toNNReal = s + u.toNNReal := by
+    apply NNReal.coe_injective
+    simp only [NNReal.coe_add, Real.coe_toNNReal _ (add_nonneg hu.1 s.coe_nonneg),
+      Real.coe_toNNReal _ hu.1]
+    ring
+  have hder := ((hf.2 (u + s) (hmaps hu)).2.scomp u
+    ((hasDerivAt_id u).add_const (s : ℝ)).hasDerivWithinAt hmaps).sub_const (b s : ℂ)
+  constructor
+  · simpa using (hf.2 (u + s) (hmaps hu)).1
+  · simpa [shiftDriver, hnn, sub_sub_sub_cancel_right] using hder
+
+theorem loewnerMap_restart (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (s t : ℝ≥0)
+    (h : alive A b (s + t) z) :
+    alive A (shiftDriver b s) t (loewnerMap A b s z - (b s : ℂ)) ∧
+    loewnerMap A (shiftDriver b s) t (loewnerMap A b s z - (b s : ℂ)) =
+      loewnerMap A b (s + t) z - (b s : ℂ) := by
+  obtain ⟨f, hf⟩ := h
+  have hs : odeUntil A b z s f := odeUntil_mono hf (by exact le_add_of_nonneg_right (show 0 ≤ t from bot_le))
+  have hfs := loewnerMap_eq hs
+  rw [hfs]
+  constructor
+  · exact ⟨_, odeUntil_shift hf⟩
+  · rw [loewnerMap_eq (odeUntil_shift hf), loewnerMap_eq hf]
+    simp [add_comm]
+
+end ExactSLEMain
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/BrownianNodes.lean b/lean/OAI/Probability/SLEGauge/BrownianNodes.lean
index e14e922232c574be5beec29f28f8a875efbaf82d..1c17d5650cf9ef07b9b2853b774ac4cd901d2c82 100644
--- a/lean/OAI/Probability/SLEGauge/BrownianNodes.lean
+++ b/lean/OAI/Probability/SLEGauge/BrownianNodes.lean
@@ -1,464 +1,464 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.InverseRegularity
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.BrownianCalculus
-open scoped ContinuousMapZero
-variable {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-include hW
-lemma increment_indep_history (s h : ℝ≥0) : IndepFun (increment s h) (history s) W := by
-  have hi := (hW.toIsPreBrownianReal.indepFun_shift s).comp
-    (measurable_pi_apply h) measurable_id
-  exact hi
-
-lemma increment_indep_past_function (s h : ℝ≥0) {F : (Iic s → ℝ) → ℝ}
-    (hF : Measurable F) : IndepFun (increment s h) (F ∘ history s) W := by
-  exact (increment_indep_history hW s h).comp measurable_id hF
-
-lemma predictable_increment_centered (s h : ℝ≥0) {F : (Iic s → ℝ) → ℝ}
-    (hF : Measurable F) :
-    (∫ b, F (history s b) * increment s h b ∂W) = 0 := by
-  have hi := (increment_indep_past_function hW s h hF).symm
-  have hh := hi.integral_mul_eq_mul_integral
-    (hF.comp (measurable_history s)).aestronglyMeasurable
-    (measurable_increment s h).aestronglyMeasurable
-  change (∫ b, F (history s b) * increment s h b ∂W) =
-    (∫ b, F (history s b) ∂W) * (∫ b, increment s h b ∂W) at hh
-  rw [hh, increment_mean hW, mul_zero]
-
-lemma increment_pow_integrable (s h : ℝ≥0) (n : ℕ) :
-    Integrable (fun b => (increment s h b)^n) W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hm := (increment_law hW s h).hasGaussianLaw.memLp (p := (n : ℝ≥0∞)) (by simp)
-  apply (integrable_norm_iff ((measurable_increment s h).pow_const n).aestronglyMeasurable).mp
-  simpa only [norm_pow] using hm.integrable_norm_pow'
-
-lemma predictable_innovation_centered (s h : ℝ≥0) {F : (Iic s → ℝ) → ℝ}
-    (hF : Measurable F) {J : ℝ → ℝ} (hJ : Measurable J)
-    (hmean : (∫ x, J x ∂gaussianReal 0 h) = 0) :
-    (∫ b, F (history s b) * J (increment s h b) ∂W) = 0 := by
-  have hi := (increment_indep_history hW s h).comp hJ hF
-  have hh := hi.symm.integral_mul_eq_mul_integral
-    (hF.comp (measurable_history s)).aestronglyMeasurable
-    (hJ.comp (measurable_increment s h)).aestronglyMeasurable
-  change (∫ b, F (history s b) * J (increment s h b) ∂W) =
-    (∫ b, F (history s b) ∂W) * (∫ b, J (increment s h b) ∂W) at hh
-  have hm : (∫ b, J (increment s h b) ∂W) = 0 := by
-    exact ((increment_law hW s h).integral_comp hJ.aestronglyMeasurable).trans hmean
-  rw [hh, hm, mul_zero]
-
-lemma predictable_compensated_square_centered (s h : ℝ≥0)
-    {F : (Iic s → ℝ) → ℝ} (hF : Measurable F) :
-    (∫ b, F (history s b) * ((increment s h b)^2 - (h:ℝ)) ∂W) = 0 := by
-  apply predictable_innovation_centered hW s h hF (J := fun x : ℝ => x^2-(h:ℝ)) (by fun_prop)
-  have hi : Integrable (fun x : ℝ => x^2) (gaussianReal 0 h) := by
-    simpa only [id_eq] using (memLp_id_gaussianReal (μ := 0) (v := h) 2).integrable_sq
-  rw [integral_sub hi (integrable_const (h:ℝ)), integral_sq_gaussian]
-  simp
-
-lemma increment_abs_cube_integrable (s h : ℝ≥0) :
-    Integrable (fun b => |increment s h b|^3) W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hm := (increment_law hW s h).hasGaussianLaw.memLp (p := (3 : ℝ≥0∞)) (by simp)
-  simpa only [Real.norm_eq_abs] using hm.integrable_norm_pow' (p := 3)
-
-omit hW in
-lemma abs_cube_bound {h x : ℝ} (hh : 0 < h) :
-    |x|^3 ≤ (h*x^2 + x^4) / (2*Real.sqrt h) := by
-  have hr := Real.sqrt_pos.mpr hh
-  have hs := Real.sq_sqrt hh.le
-  apply (le_div_iff₀ (by positivity : 0 < 2*Real.sqrt h)).mpr
-  have hx4 : |x|^4 = x^4 := by
-    calc |x|^4 = (|x|^2)^2 := by ring
-         _ = (x^2)^2 := by rw [sq_abs]
-         _ = x^4 := by ring
-  have he : h*x^2 + x^4 - |x|^3*(2*Real.sqrt h) =
-      (|x|^2 - Real.sqrt h*|x|)^2 := by
-    calc
-      _ = (Real.sqrt h)^2*|x|^2 + |x|^4 - |x|^3*(2*Real.sqrt h) := by
-        rw [hs, sq_abs, hx4]
-      _ = _ := by ring
-  linarith [sq_nonneg (|x|^2-Real.sqrt h*|x|)]
-
-lemma increment_abs_cube_le (s h : ℝ≥0) :
-    (∫ b, |increment s h b|^3 ∂W) ≤ 2*(h:ℝ)*Real.sqrt h := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  by_cases hh : h = 0
-  · subst h
-    simp [increment]
-  have hp : 0 < (h:ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr hh)
-  have hr := Real.sqrt_pos.mpr hp
-  have hi2 := increment_pow_integrable hW s h 2
-  have hi4 := increment_pow_integrable hW s h 4
-  have hm := integral_mono (increment_abs_cube_integrable hW s h)
-    (((hi2.const_mul (h:ℝ)).add hi4).div_const (2*Real.sqrt h))
-    (fun b => abs_cube_bound (x := increment s h b) hp)
-  have he : (∫ b, ((h:ℝ)*(increment s h b)^2+(increment s h b)^4) /
-      (2*Real.sqrt h) ∂W) = 2*(h:ℝ)*Real.sqrt h := by
-    rw [integral_div, integral_add (hi2.const_mul _) hi4, integral_const_mul,
-      increment_sq hW, increment_fourth hW]
-    apply (div_eq_iff (by positivity : 2*Real.sqrt (h:ℝ) ≠ 0)).mpr
-    nlinarith [Real.sq_sqrt hp.le]
-  exact hm.trans_eq he
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
-lemma taylor_second_bound {f : ℝ → ℝ} (hf : ContDiff ℝ 3 f)
-    {C : ℝ} (hC : ∀ z, |iteratedDeriv 3 f z| ≤ C) (x y : ℝ) :
-    |f y - f x - deriv f x*(y-x) - deriv (deriv f) x*(y-x)^2/2| ≤
-      C/6 * |y-x|^3 := by
-  by_cases heq : x = y
-  · subst y
-    simp
-  have hc : ContDiffOn ℝ (2+1 : ℕ) f (uIcc x y) := hf.contDiffOn
-  obtain ⟨z, hz, he⟩ := taylor_mean_remainder_lagrange_iteratedDeriv
-    (n := 2) heq hc
-  have hi (k : ℕ) (hk : k ≤ 2) :
-      iteratedDerivWithin k f (uIcc x y) x = iteratedDeriv k f x := by
-    apply iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_uIcc heq)
-    · exact (hf.of_le (by norm_cast; omega)).contDiffAt
-    · exact left_mem_uIcc
-  have ht : taylorWithinEval f 2 (uIcc x y) x y =
-      f x + deriv f x*(y-x) + deriv (deriv f) x*(y-x)^2/2 := by
-    simp only [taylor_within_apply, Finset.sum_range_succ, Finset.sum_range_zero,
-      zero_add, hi 0 (by omega), hi 1 (by omega), hi 2 (by omega)]
-    norm_num [iteratedDeriv_succ, iteratedDeriv_zero]
-    ring
-  have he' : f y - f x - deriv f x*(y-x) - deriv (deriv f) x*(y-x)^2/2 =
-      iteratedDeriv 3 f z * (y-x)^3 / 6 := by
-    rw [ht] at he
-    norm_num at he
-    linarith
-  rw [he', abs_div, abs_mul, abs_pow]
-  norm_num
-  calc |iteratedDeriv 3 f z| * |y-x|^3 / 6 ≤ C * |y-x|^3 / 6 := by gcongr; exact hC z
-       _ = _ := by ring
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
- 
-
-lemma expected_taylor_step {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s h : ℝ≥0) (f : (Iic s → ℝ) → ℝ → ℝ)
-    (hsm : ∀ p, ContDiff ℝ 3 (f p))
-    (hm1 : Measurable (fun p => deriv (f p) (p ⟨s, by simp⟩)))
-    (hm2 : Measurable (fun p => deriv (deriv (f p)) (p ⟨s, by simp⟩)))
-    {C M₁ M₂ : ℝ} (hC : 0 ≤ C)
-    (hb1 : ∀ p, |deriv (f p) (p ⟨s, by simp⟩)| ≤ M₁)
-    (hb2 : ∀ p, |deriv (deriv (f p)) (p ⟨s, by simp⟩)| ≤ M₂)
-    (hb3 : ∀ p x, |iteratedDeriv 3 (f p) x| ≤ C)
-    (hi : Integrable (fun b => f (history s b) (b (s+h)) - f (history s b) (b s)) W) :
-    |(∫ b, f (history s b) (b (s+h)) - f (history s b) (b s) ∂W) -
-      (h:ℝ)/2 * (∫ b, deriv (deriv (f (history s b))) (b s) ∂W)| ≤
-      C/3 * (h:ℝ) * Real.sqrt h := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let A₁ := fun p : Iic s → ℝ => deriv (f p) (p ⟨s, by simp⟩)
-  let A₂ := fun p : Iic s → ℝ => deriv (deriv (f p)) (p ⟨s, by simp⟩)
-  have ha₁ : AEStronglyMeasurable (fun b => A₁ (history s b)) W :=
-    (hm1.comp (measurable_history s)).aestronglyMeasurable
-  have ha₂ : AEStronglyMeasurable (fun b => A₂ (history s b)) W :=
-    (hm2.comp (measurable_history s)).aestronglyMeasurable
-  have i₁ : Integrable (fun b => A₁ (history s b)*increment s h b) W := by
-    have hg : Integrable (increment s h) W := by
-      simpa only [pow_one] using increment_pow_integrable hW s h 1
-    exact hg.bdd_mul ha₁ (Filter.Eventually.of_forall fun b => by
-      simpa only [Real.norm_eq_abs] using hb1 (history s b))
-  have i₂ : Integrable (fun b => A₂ (history s b)*(increment s h b)^2) W :=
-    (increment_pow_integrable hW s h 2).bdd_mul ha₂
-      (Filter.Eventually.of_forall fun b => by
-        simpa only [Real.norm_eq_abs] using hb2 (history s b))
-  have ia₂ : Integrable (fun b => A₂ (history s b)) W :=
-    (integrable_const M₂).mono' ha₂ (Filter.Eventually.of_forall fun b => by
-      simpa only [Real.norm_eq_abs] using hb2 (history s b))
-  have e₂ : (∫ b, A₂ (history s b)*(increment s h b)^2 ∂W) =
-      (h:ℝ)*(∫ b, A₂ (history s b) ∂W) := by
-    have hh := predictable_compensated_square_centered hW s h hm2
-    change (∫ b, A₂ (history s b)*((increment s h b)^2-(h:ℝ)) ∂W) = 0 at hh
-    simp_rw [mul_sub] at hh
-    rw [integral_sub i₂ (ia₂.mul_const (h:ℝ)), integral_mul_const] at hh
-    linarith
-  let R := fun b => f (history s b) (b (s+h)) - f (history s b) (b s) -
-    A₁ (history s b)*increment s h b - A₂ (history s b)*(increment s h b)^2/2
-  have he : (∫ b, R b ∂W) =
-      (∫ b, f (history s b) (b (s+h)) - f (history s b) (b s) ∂W) -
-        (h:ℝ)/2*(∫ b, deriv (deriv (f (history s b))) (b s) ∂W) := by
-    dsimp only [R]
-    have hs1 := integral_sub hi i₁
-    have hs2 := integral_sub (hi.sub i₁) (i₂.div_const 2)
-    simp only [Pi.sub_apply] at hs1 hs2
-    rw [hs2, hs1, predictable_increment_centered hW s h hm1, integral_div, e₂]
-    dsimp only [A₂, history]
-    ring
-  rw [← he]
-  have hb (b : Driver) : ‖R b‖ ≤ C/6 * |increment s h b|^3 := by
-    simpa only [R, A₁, A₂, history, increment, Real.norm_eq_abs] using
-      taylor_second_bound (hsm (history s b)) (hb3 (history s b)) (b s) (b (s+h))
-  have hiR := (increment_abs_cube_integrable hW s h).const_mul (C/6)
-  have hbR := norm_integral_le_of_norm_le hiR (Filter.Eventually.of_forall hb)
-  rw [Real.norm_eq_abs, integral_const_mul] at hbR
-  calc
-    _ ≤ C/6 * (∫ b, |increment s h b|^3 ∂W) := hbR
-    _ ≤ C/6 * (2*(h:ℝ)*Real.sqrt h) := by
-      gcongr
-      exact increment_abs_cube_le hW s h
-    _ = _ := by ring
-
-end SLEExactGauge.BrownianCalculus
-
- 
-
-noncomputable section
-open Set
-namespace SLEExactGauge.BrownianCalculus
-
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-
-lemma state_taylor_bound {F : E → ℝ} {D : E → E →L[ℝ] ℝ}
-    (hD : ∀ x, HasFDerivAt F (D x) x) {C : ℝ} (hC : 0 ≤ C)
-    (hLip : ∀ x y, ‖D x-D y‖ ≤ C*‖x-y‖) (x v : E) :
-    |F (x+v)-F x-D x v| ≤ C*‖v‖^2 := by
-  let f : ℝ → ℝ := fun t => F (x+t • v)-t*(D x v)
-  have hd (t : ℝ) : HasDerivAt f ((D (x+t • v)-D x) v) t := by
-    have hh := (hD (x+t • v)).comp_hasDerivAt t
-      (((hasDerivAt_id t).smul_const v).const_add x)
-    have hk := hh.sub ((hasDerivAt_id t).mul_const (D x v))
-    convert! hk using 1
-    simp
-  have hb (t : ℝ) (ht : t ∈ Icc 0 1) : ‖(D (x+t • v)-D x) v‖ ≤ C*‖v‖^2 := by
-    calc _ ≤ ‖D (x+t • v)-D x‖*‖v‖ := ContinuousLinearMap.le_opNorm _ _
-      _ ≤ (C*‖(x+t • v)-x‖)*‖v‖ :=
-        mul_le_mul_of_nonneg_right (hLip _ _) (norm_nonneg _)
-      _ = C*(t*‖v‖)*‖v‖ := by
-        rw [add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg ht.1]
-      _ = t*(C*‖v‖^2) := by ring
-      _ ≤ 1*(C*‖v‖^2) := mul_le_mul_of_nonneg_right ht.2 (mul_nonneg hC (sq_nonneg _))
-      _ = _ := one_mul _
-  have hh := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
-    (fun t (_ : t ∈ Icc (0:ℝ) 1) => (hd t).hasDerivWithinAt) hb
-    (convex_Icc (0:ℝ) 1) (by simp : (0:ℝ) ∈ Icc 0 1) (by simp : (1:ℝ) ∈ Icc 0 1)
-  simp only [f, one_smul, zero_smul, add_zero, one_mul, zero_mul, sub_zero,
-    Real.norm_eq_abs, abs_one, mul_one] at hh
-  convert! hh using 1
-  congr 1
-  ring
-
-lemma state_euler_remainder {F : E → ℝ} {D : E → E →L[ℝ] ℝ}
-    (hD : ∀ x, HasFDerivAt F (D x) x) {C : ℝ} (hC : 0 ≤ C)
-    (hLip : ∀ x y, ‖D x-D y‖ ≤ C*‖x-y‖) (x v : E) {M h : ℝ}
-    (hh : 0 ≤ h) (hv : ‖v‖ ≤ M) :
-    F (x+h • v)-F x ≤ h*(D x v) + C*M^2*h^2 := by
-  have hb := (le_abs_self _).trans (state_taylor_bound hD hC hLip x (h • v))
-  rw [map_smul, smul_eq_mul, norm_smul, Real.norm_eq_abs, abs_of_nonneg hh] at hb
-  have hM : 0 ≤ M := (norm_nonneg v).trans hv
-  have hn : ‖v‖^2 ≤ M^2 := by nlinarith [norm_nonneg v]
-  have hh' := mul_le_mul_of_nonneg_left hn (mul_nonneg hC (sq_nonneg h))
-  nlinarith
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem expected_euler_step {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s h : ℝ≥0) (F : E → ℝ → ℝ) (D : E → ℝ → E →L[ℝ] ℝ)
-    (X V : (Iic s → ℝ) → E)
-    (mF : Measurable (Function.uncurry F))
-    (mB1 : Measurable (fun q : E × ℝ => deriv (F q.1) q.2))
-    (mB2 : Measurable (fun q : E × ℝ => deriv (deriv (F q.1)) q.2))
-    (mX : Measurable X) (mV : Measurable V)
-    (hD : ∀ x b, HasFDerivAt (fun y => F y b) (D x b) x)
-    (hsm : ∀ x, ContDiff ℝ 3 (F x))
-    {C₀ C₁ C₂ C₃ Cx Ccross M : ℝ}
-    (hC₃ : 0 ≤ C₃) (hCx : 0 ≤ Cx) (hCcross : 0 ≤ Ccross)
-    (hF : ∀ x b, |F x b| ≤ C₀)
-    (hB1 : ∀ x b, |deriv (F x) b| ≤ C₁)
-    (hB2 : ∀ x b, |deriv (deriv (F x)) b| ≤ C₂)
-    (hB3 : ∀ x b, |iteratedDeriv 3 (F x) b| ≤ C₃)
-    (hLipD : ∀ x y b, ‖D x b-D y b‖ ≤ Cx*‖x-y‖)
-    (hLipB2 : ∀ x y b, |deriv (deriv (F x)) b-deriv (deriv (F y)) b| ≤ Ccross*‖x-y‖)
-    (hV : ∀ p, ‖V p‖ ≤ M)
-    (hgen : ∀ p, D (X p) (p ⟨s,by simp⟩) (V p) +
-      deriv (deriv (F (X p))) (p ⟨s,by simp⟩)/2 ≤ 0) :
-    (∫ b, F (X (history s b)+(h:ℝ) • V (history s b)) (b (s+h)) -
-      F (X (history s b)) (b s) ∂W) ≤
-        (Cx*M^2+Ccross*M/2)*(h:ℝ)^2 + C₃/3*(h:ℝ)*Real.sqrt h := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let Y := fun p => X p+(h:ℝ) • V p
-  have mY : Measurable Y := mX.add (mV.const_smul (h:ℝ))
-  let f := fun p v => F (Y p) v
-  let old := fun b : Driver => F (X (history s b)) (b s)
-  let step := fun b : Driver => F (Y (history s b)) (b s)
-  let next := fun b : Driver => F (Y (history s b)) (b (s+h))
-  have mOld : Measurable old := mF.comp ((mX.comp (measurable_history s)).prodMk
-    ((continuous_eval_const s).measurable))
-  have mStep : Measurable step := mF.comp ((mY.comp (measurable_history s)).prodMk
-    ((continuous_eval_const s).measurable))
-  have mNext : Measurable next := mF.comp ((mY.comp (measurable_history s)).prodMk
-    ((continuous_eval_const (s+h)).measurable))
-  have iOld : Integrable old W := (integrable_const C₀).mono' mOld.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hF _ _)
-  have iStep : Integrable step W := (integrable_const C₀).mono' mStep.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hF _ _)
-  have iNext : Integrable next W := (integrable_const C₀).mono' mNext.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hF _ _)
-  let Q := fun b : Driver => deriv (deriv (F (Y (history s b)))) (b s)
-  have mQ : Measurable Q := mB2.comp ((mY.comp (measurable_history s)).prodMk
-    ((continuous_eval_const s).measurable))
-  have iQ : Integrable Q W := (integrable_const C₂).mono' mQ.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hB2 _ _)
-  have hm1 : Measurable (fun p => deriv (f p) (p ⟨s,by simp⟩)) :=
-    mB1.comp (mY.prodMk (measurable_pi_apply _))
-  have hm2 : Measurable (fun p => deriv (deriv (f p)) (p ⟨s,by simp⟩)) :=
-    mB2.comp (mY.prodMk (measurable_pi_apply _))
-  have ht := expected_taylor_step hW s h f (fun p => hsm (Y p)) hm1 hm2 hC₃
-    (fun p => hB1 _ _) (fun p => hB2 _ _) (fun p v => hB3 _ _) (iNext.sub iStep)
-  change |(∫ b, next b-step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W)| ≤ _ at ht
-  have hp (b : Driver) : step b-old b+(h:ℝ)/2*Q b ≤
-      (Cx*M^2+Ccross*M/2)*(h:ℝ)^2 := by
-    let p := history s b
-    have hs : p ⟨s,by simp⟩ = b s := rfl
-    have hy : ‖Y p-X p‖ ≤ (h:ℝ)*M := by
-      simp only [Y, add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg h.coe_nonneg]
-      exact mul_le_mul_of_nonneg_left (hV p) h.coe_nonneg
-    have hd := state_euler_remainder (fun x => hD x (b s)) hCx
-      (fun x y => hLipD x y (b s)) (X p) (V p) h.coe_nonneg (hV p)
-    have hb := (le_abs_self _).trans (hLipB2 (Y p) (X p) (b s))
-    have hcross := hb.trans (mul_le_mul_of_nonneg_left hy hCcross)
-    have hg := hgen p
-    rw [hs] at hg
-    have hg' := mul_nonpos_of_nonneg_of_nonpos h.coe_nonneg hg
-    have hc' := mul_le_mul_of_nonneg_left hcross (div_nonneg h.coe_nonneg (by norm_num : (0:ℝ) ≤ 2))
-    dsimp only [step, old, Q]
-    change F (Y p) (b s)-F (X p) (b s)+(h:ℝ)/2*deriv (deriv (F (Y p))) (b s) ≤ _
-    dsimp only [Y] at hd ⊢
-    nlinarith
-  have hib : (∫ b, step b-old b+(h:ℝ)/2*Q b ∂W) ≤
-      (Cx*M^2+Ccross*M/2)*(h:ℝ)^2 := by
-    have hh := integral_mono_ae ((iStep.sub iOld).add (iQ.const_mul ((h:ℝ)/2)))
-      (integrable_const ((Cx*M^2+Ccross*M/2)*(h:ℝ)^2)) (Filter.Eventually.of_forall hp)
-    simpa only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul, Pi.add_apply, Pi.sub_apply] using hh
-  change (∫ b, (step-old) b+((h:ℝ)/2*Q b) ∂W) ≤ _ at hib
-  rw [integral_add (iStep.sub iOld) (iQ.const_mul ((h:ℝ)/2)), integral_const_mul] at hib
-  change (∫ b, step b-old b ∂W)+(h:ℝ)/2*(∫ b, Q b ∂W) ≤ _ at hib
-  rw [integral_sub iStep iOld] at hib
-  rw [integral_sub iNext iStep] at ht
-  change (∫ b, next b-old b ∂W) ≤ _
-  rw [integral_sub iNext iOld]
-  linarith [le_abs_self ((∫ b, next b ∂W)-(∫ b, step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W))]
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-structure SmoothTest (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [MeasurableSpace E] where
-  F : E → ℝ → ℝ
-  D : E → ℝ → E →L[ℝ] ℝ
-  C₀ : ℝ
-  C₁ : ℝ
-  C₂ : ℝ
-  C₃ : ℝ
-  Cx : ℝ
-  Ccross : ℝ
-  hC₃ : 0 ≤ C₃
-  hCx : 0 ≤ Cx
-  hCcross : 0 ≤ Ccross
-  cF : Continuous (Function.uncurry F)
-  mF : Measurable (Function.uncurry F)
-  mB1 : Measurable (fun q : E × ℝ => deriv (F q.1) q.2)
-  mB2 : Measurable (fun q : E × ℝ => deriv (deriv (F q.1)) q.2)
-  hD : ∀ x b, HasFDerivAt (fun y => F y b) (D x b) x
-  hsm : ∀ x, ContDiff ℝ 3 (F x)
-  hF : ∀ x b, |F x b| ≤ C₀
-  hB1 : ∀ x b, |deriv (F x) b| ≤ C₁
-  hB2 : ∀ x b, |deriv (deriv (F x)) b| ≤ C₂
-  hB3 : ∀ x b, |iteratedDeriv 3 (F x) b| ≤ C₃
-  hLipD : ∀ x y b, ‖D x b-D y b‖ ≤ Cx*‖x-y‖
-  hLipB2 : ∀ x y b, |deriv (deriv (F x)) b-deriv (deriv (F y)) b| ≤ Ccross*‖x-y‖
-
-def extendHistory (s : ℝ≥0) (p : Iic s → ℝ) (t : ℝ≥0) : ℝ :=
-  if ht : t ≤ s then p ⟨t,ht⟩ else 0
-
-lemma measurable_extendHistory (s : ℝ≥0) : Measurable (extendHistory s) := by
-  apply Measurable.of_eval
-  intro t
-  by_cases ht : t ≤ s
-  · simpa only [extendHistory, dite_eq_left ht] using (measurable_pi_apply (⟨t,ht⟩ : Iic s))
-  · simp only [extendHistory, dite_eq_right ht]
-    exact measurable_const
-
- 
-def stochasticNodes (v : E → ℝ → E) (h : ℝ≥0) (x : E) : ℕ → (ℝ≥0 → ℝ) → E
-  | 0, _ => x
-  | k+1, b => stochasticNodes v h x k b + (h:ℝ) •
-      v (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h))
-
-lemma measurable_stochasticNodes (v : E → ℝ → E)
-    (hv : Measurable (Function.uncurry v)) (h : ℝ≥0) (x : E) (k : ℕ) :
-    Measurable (stochasticNodes v h x k) := by
-  induction k with
-  | zero => exact measurable_const
-  | succ k ih => exact ih.add ((hv.comp (ih.prodMk (measurable_pi_apply _))).const_smul (h:ℝ))
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma stochasticNodes_congr (v : E → ℝ → E) (h : ℝ≥0) (x : E) (k : ℕ)
-    {b c : ℝ≥0 → ℝ} (he : ∀ t ≤ (k:ℝ≥0)*h, b t = c t) :
-    stochasticNodes v h x k b = stochasticNodes v h x k c := by
-  induction k with
-  | zero => rfl
-  | succ k ih =>
-    have hle : (k:ℝ≥0)*h ≤ ((k+1:ℕ):ℝ≥0)*h :=
-      mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ k) h.coe_nonneg
-    have hn := ih (fun t ht => he t (ht.trans hle))
-    simp only [stochasticNodes, hn, he _ hle]
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma stochasticNodes_history (v : E → ℝ → E) (h : ℝ≥0) (x : E) (k : ℕ)
-    (b : Driver) :
-    stochasticNodes v h x k (extendHistory ((k:ℝ≥0)*h) (history ((k:ℝ≥0)*h) b)) =
-      stochasticNodes v h x k b := by
-  apply stochasticNodes_congr
-  intro t ht
-  simp only [extendHistory, dite_eq_left ht, history]
-
-end SLEExactGauge.BrownianCalculus
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.InverseRegularity
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.BrownianCalculus
+open scoped ContinuousMapZero
+variable {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+include hW
+lemma increment_indep_history (s h : ℝ≥0) : IndepFun (increment s h) (history s) W := by
+  have hi := (hW.toIsPreBrownianReal.indepFun_shift s).comp
+    (measurable_pi_apply h) measurable_id
+  exact hi
+
+lemma increment_indep_past_function (s h : ℝ≥0) {F : (Iic s → ℝ) → ℝ}
+    (hF : Measurable F) : IndepFun (increment s h) (F ∘ history s) W := by
+  exact (increment_indep_history hW s h).comp measurable_id hF
+
+lemma predictable_increment_centered (s h : ℝ≥0) {F : (Iic s → ℝ) → ℝ}
+    (hF : Measurable F) :
+    (∫ b, F (history s b) * increment s h b ∂W) = 0 := by
+  have hi := (increment_indep_past_function hW s h hF).symm
+  have hh := hi.integral_mul_eq_mul_integral
+    (hF.comp (measurable_history s)).aestronglyMeasurable
+    (measurable_increment s h).aestronglyMeasurable
+  change (∫ b, F (history s b) * increment s h b ∂W) =
+    (∫ b, F (history s b) ∂W) * (∫ b, increment s h b ∂W) at hh
+  rw [hh, increment_mean hW, mul_zero]
+
+lemma increment_pow_integrable (s h : ℝ≥0) (n : ℕ) :
+    Integrable (fun b => (increment s h b)^n) W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hm := (increment_law hW s h).hasGaussianLaw.memLp (p := (n : ℝ≥0∞)) (by simp)
+  apply (integrable_norm_iff ((measurable_increment s h).pow_const n).aestronglyMeasurable).mp
+  simpa only [norm_pow] using hm.integrable_norm_pow'
+
+lemma predictable_innovation_centered (s h : ℝ≥0) {F : (Iic s → ℝ) → ℝ}
+    (hF : Measurable F) {J : ℝ → ℝ} (hJ : Measurable J)
+    (hmean : (∫ x, J x ∂gaussianReal 0 h) = 0) :
+    (∫ b, F (history s b) * J (increment s h b) ∂W) = 0 := by
+  have hi := (increment_indep_history hW s h).comp hJ hF
+  have hh := hi.symm.integral_mul_eq_mul_integral
+    (hF.comp (measurable_history s)).aestronglyMeasurable
+    (hJ.comp (measurable_increment s h)).aestronglyMeasurable
+  change (∫ b, F (history s b) * J (increment s h b) ∂W) =
+    (∫ b, F (history s b) ∂W) * (∫ b, J (increment s h b) ∂W) at hh
+  have hm : (∫ b, J (increment s h b) ∂W) = 0 := by
+    exact ((increment_law hW s h).integral_comp hJ.aestronglyMeasurable).trans hmean
+  rw [hh, hm, mul_zero]
+
+lemma predictable_compensated_square_centered (s h : ℝ≥0)
+    {F : (Iic s → ℝ) → ℝ} (hF : Measurable F) :
+    (∫ b, F (history s b) * ((increment s h b)^2 - (h:ℝ)) ∂W) = 0 := by
+  apply predictable_innovation_centered hW s h hF (J := fun x : ℝ => x^2-(h:ℝ)) (by fun_prop)
+  have hi : Integrable (fun x : ℝ => x^2) (gaussianReal 0 h) := by
+    simpa only [id_eq] using (memLp_id_gaussianReal (μ := 0) (v := h) 2).integrable_sq
+  rw [integral_sub hi (integrable_const (h:ℝ)), integral_sq_gaussian]
+  simp
+
+lemma increment_abs_cube_integrable (s h : ℝ≥0) :
+    Integrable (fun b => |increment s h b|^3) W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hm := (increment_law hW s h).hasGaussianLaw.memLp (p := (3 : ℝ≥0∞)) (by simp)
+  simpa only [Real.norm_eq_abs] using hm.integrable_norm_pow' (p := 3)
+
+omit hW in
+lemma abs_cube_bound {h x : ℝ} (hh : 0 < h) :
+    |x|^3 ≤ (h*x^2 + x^4) / (2*Real.sqrt h) := by
+  have hr := Real.sqrt_pos.mpr hh
+  have hs := Real.sq_sqrt hh.le
+  apply (le_div_iff₀ (by positivity : 0 < 2*Real.sqrt h)).mpr
+  have hx4 : |x|^4 = x^4 := by
+    calc |x|^4 = (|x|^2)^2 := by ring
+         _ = (x^2)^2 := by rw [sq_abs]
+         _ = x^4 := by ring
+  have he : h*x^2 + x^4 - |x|^3*(2*Real.sqrt h) =
+      (|x|^2 - Real.sqrt h*|x|)^2 := by
+    calc
+      _ = (Real.sqrt h)^2*|x|^2 + |x|^4 - |x|^3*(2*Real.sqrt h) := by
+        rw [hs, sq_abs, hx4]
+      _ = _ := by ring
+  linarith [sq_nonneg (|x|^2-Real.sqrt h*|x|)]
+
+lemma increment_abs_cube_le (s h : ℝ≥0) :
+    (∫ b, |increment s h b|^3 ∂W) ≤ 2*(h:ℝ)*Real.sqrt h := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  by_cases hh : h = 0
+  · subst h
+    simp [increment]
+  have hp : 0 < (h:ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr hh)
+  have hr := Real.sqrt_pos.mpr hp
+  have hi2 := increment_pow_integrable hW s h 2
+  have hi4 := increment_pow_integrable hW s h 4
+  have hm := integral_mono (increment_abs_cube_integrable hW s h)
+    (((hi2.const_mul (h:ℝ)).add hi4).div_const (2*Real.sqrt h))
+    (fun b => abs_cube_bound (x := increment s h b) hp)
+  have he : (∫ b, ((h:ℝ)*(increment s h b)^2+(increment s h b)^4) /
+      (2*Real.sqrt h) ∂W) = 2*(h:ℝ)*Real.sqrt h := by
+    rw [integral_div, integral_add (hi2.const_mul _) hi4, integral_const_mul,
+      increment_sq hW, increment_fourth hW]
+    apply (div_eq_iff (by positivity : 2*Real.sqrt (h:ℝ) ≠ 0)).mpr
+    nlinarith [Real.sq_sqrt hp.le]
+  exact hm.trans_eq he
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+lemma taylor_second_bound {f : ℝ → ℝ} (hf : ContDiff ℝ 3 f)
+    {C : ℝ} (hC : ∀ z, |iteratedDeriv 3 f z| ≤ C) (x y : ℝ) :
+    |f y - f x - deriv f x*(y-x) - deriv (deriv f) x*(y-x)^2/2| ≤
+      C/6 * |y-x|^3 := by
+  by_cases heq : x = y
+  · subst y
+    simp
+  have hc : ContDiffOn ℝ (2+1 : ℕ) f (uIcc x y) := hf.contDiffOn
+  obtain ⟨z, hz, he⟩ := taylor_mean_remainder_lagrange_iteratedDeriv
+    (n := 2) heq hc
+  have hi (k : ℕ) (hk : k ≤ 2) :
+      iteratedDerivWithin k f (uIcc x y) x = iteratedDeriv k f x := by
+    apply iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_uIcc heq)
+    · exact (hf.of_le (by norm_cast; omega)).contDiffAt
+    · exact left_mem_uIcc
+  have ht : taylorWithinEval f 2 (uIcc x y) x y =
+      f x + deriv f x*(y-x) + deriv (deriv f) x*(y-x)^2/2 := by
+    simp only [taylor_within_apply, Finset.sum_range_succ, Finset.sum_range_zero,
+      zero_add, hi 0 (by omega), hi 1 (by omega), hi 2 (by omega)]
+    norm_num [iteratedDeriv_succ, iteratedDeriv_zero]
+    ring
+  have he' : f y - f x - deriv f x*(y-x) - deriv (deriv f) x*(y-x)^2/2 =
+      iteratedDeriv 3 f z * (y-x)^3 / 6 := by
+    rw [ht] at he
+    norm_num at he
+    linarith
+  rw [he', abs_div, abs_mul, abs_pow]
+  norm_num
+  calc |iteratedDeriv 3 f z| * |y-x|^3 / 6 ≤ C * |y-x|^3 / 6 := by gcongr; exact hC z
+       _ = _ := by ring
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+ 
+
+lemma expected_taylor_step {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s h : ℝ≥0) (f : (Iic s → ℝ) → ℝ → ℝ)
+    (hsm : ∀ p, ContDiff ℝ 3 (f p))
+    (hm1 : Measurable (fun p => deriv (f p) (p ⟨s, by simp⟩)))
+    (hm2 : Measurable (fun p => deriv (deriv (f p)) (p ⟨s, by simp⟩)))
+    {C M₁ M₂ : ℝ} (hC : 0 ≤ C)
+    (hb1 : ∀ p, |deriv (f p) (p ⟨s, by simp⟩)| ≤ M₁)
+    (hb2 : ∀ p, |deriv (deriv (f p)) (p ⟨s, by simp⟩)| ≤ M₂)
+    (hb3 : ∀ p x, |iteratedDeriv 3 (f p) x| ≤ C)
+    (hi : Integrable (fun b => f (history s b) (b (s+h)) - f (history s b) (b s)) W) :
+    |(∫ b, f (history s b) (b (s+h)) - f (history s b) (b s) ∂W) -
+      (h:ℝ)/2 * (∫ b, deriv (deriv (f (history s b))) (b s) ∂W)| ≤
+      C/3 * (h:ℝ) * Real.sqrt h := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let A₁ := fun p : Iic s → ℝ => deriv (f p) (p ⟨s, by simp⟩)
+  let A₂ := fun p : Iic s → ℝ => deriv (deriv (f p)) (p ⟨s, by simp⟩)
+  have ha₁ : AEStronglyMeasurable (fun b => A₁ (history s b)) W :=
+    (hm1.comp (measurable_history s)).aestronglyMeasurable
+  have ha₂ : AEStronglyMeasurable (fun b => A₂ (history s b)) W :=
+    (hm2.comp (measurable_history s)).aestronglyMeasurable
+  have i₁ : Integrable (fun b => A₁ (history s b)*increment s h b) W := by
+    have hg : Integrable (increment s h) W := by
+      simpa only [pow_one] using increment_pow_integrable hW s h 1
+    exact hg.bdd_mul ha₁ (Filter.Eventually.of_forall fun b => by
+      simpa only [Real.norm_eq_abs] using hb1 (history s b))
+  have i₂ : Integrable (fun b => A₂ (history s b)*(increment s h b)^2) W :=
+    (increment_pow_integrable hW s h 2).bdd_mul ha₂
+      (Filter.Eventually.of_forall fun b => by
+        simpa only [Real.norm_eq_abs] using hb2 (history s b))
+  have ia₂ : Integrable (fun b => A₂ (history s b)) W :=
+    (integrable_const M₂).mono' ha₂ (Filter.Eventually.of_forall fun b => by
+      simpa only [Real.norm_eq_abs] using hb2 (history s b))
+  have e₂ : (∫ b, A₂ (history s b)*(increment s h b)^2 ∂W) =
+      (h:ℝ)*(∫ b, A₂ (history s b) ∂W) := by
+    have hh := predictable_compensated_square_centered hW s h hm2
+    change (∫ b, A₂ (history s b)*((increment s h b)^2-(h:ℝ)) ∂W) = 0 at hh
+    simp_rw [mul_sub] at hh
+    rw [integral_sub i₂ (ia₂.mul_const (h:ℝ)), integral_mul_const] at hh
+    linarith
+  let R := fun b => f (history s b) (b (s+h)) - f (history s b) (b s) -
+    A₁ (history s b)*increment s h b - A₂ (history s b)*(increment s h b)^2/2
+  have he : (∫ b, R b ∂W) =
+      (∫ b, f (history s b) (b (s+h)) - f (history s b) (b s) ∂W) -
+        (h:ℝ)/2*(∫ b, deriv (deriv (f (history s b))) (b s) ∂W) := by
+    dsimp only [R]
+    have hs1 := integral_sub hi i₁
+    have hs2 := integral_sub (hi.sub i₁) (i₂.div_const 2)
+    simp only [Pi.sub_apply] at hs1 hs2
+    rw [hs2, hs1, predictable_increment_centered hW s h hm1, integral_div, e₂]
+    dsimp only [A₂, history]
+    ring
+  rw [← he]
+  have hb (b : Driver) : ‖R b‖ ≤ C/6 * |increment s h b|^3 := by
+    simpa only [R, A₁, A₂, history, increment, Real.norm_eq_abs] using
+      taylor_second_bound (hsm (history s b)) (hb3 (history s b)) (b s) (b (s+h))
+  have hiR := (increment_abs_cube_integrable hW s h).const_mul (C/6)
+  have hbR := norm_integral_le_of_norm_le hiR (Filter.Eventually.of_forall hb)
+  rw [Real.norm_eq_abs, integral_const_mul] at hbR
+  calc
+    _ ≤ C/6 * (∫ b, |increment s h b|^3 ∂W) := hbR
+    _ ≤ C/6 * (2*(h:ℝ)*Real.sqrt h) := by
+      gcongr
+      exact increment_abs_cube_le hW s h
+    _ = _ := by ring
+
+end SLEExactGauge.BrownianCalculus
+
+ 
+
+noncomputable section
+open Set
+namespace SLEExactGauge.BrownianCalculus
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma state_taylor_bound {F : E → ℝ} {D : E → E →L[ℝ] ℝ}
+    (hD : ∀ x, HasFDerivAt F (D x) x) {C : ℝ} (hC : 0 ≤ C)
+    (hLip : ∀ x y, ‖D x-D y‖ ≤ C*‖x-y‖) (x v : E) :
+    |F (x+v)-F x-D x v| ≤ C*‖v‖^2 := by
+  let f : ℝ → ℝ := fun t => F (x+t • v)-t*(D x v)
+  have hd (t : ℝ) : HasDerivAt f ((D (x+t • v)-D x) v) t := by
+    have hh := (hD (x+t • v)).comp_hasDerivAt t
+      (((hasDerivAt_id t).smul_const v).const_add x)
+    have hk := hh.sub ((hasDerivAt_id t).mul_const (D x v))
+    convert! hk using 1
+    simp
+  have hb (t : ℝ) (ht : t ∈ Icc 0 1) : ‖(D (x+t • v)-D x) v‖ ≤ C*‖v‖^2 := by
+    calc _ ≤ ‖D (x+t • v)-D x‖*‖v‖ := ContinuousLinearMap.le_opNorm _ _
+      _ ≤ (C*‖(x+t • v)-x‖)*‖v‖ :=
+        mul_le_mul_of_nonneg_right (hLip _ _) (norm_nonneg _)
+      _ = C*(t*‖v‖)*‖v‖ := by
+        rw [add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg ht.1]
+      _ = t*(C*‖v‖^2) := by ring
+      _ ≤ 1*(C*‖v‖^2) := mul_le_mul_of_nonneg_right ht.2 (mul_nonneg hC (sq_nonneg _))
+      _ = _ := one_mul _
+  have hh := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
+    (fun t (_ : t ∈ Icc (0:ℝ) 1) => (hd t).hasDerivWithinAt) hb
+    (convex_Icc (0:ℝ) 1) (by simp : (0:ℝ) ∈ Icc 0 1) (by simp : (1:ℝ) ∈ Icc 0 1)
+  simp only [f, one_smul, zero_smul, add_zero, one_mul, zero_mul, sub_zero,
+    Real.norm_eq_abs, abs_one, mul_one] at hh
+  convert! hh using 1
+  congr 1
+  ring
+
+lemma state_euler_remainder {F : E → ℝ} {D : E → E →L[ℝ] ℝ}
+    (hD : ∀ x, HasFDerivAt F (D x) x) {C : ℝ} (hC : 0 ≤ C)
+    (hLip : ∀ x y, ‖D x-D y‖ ≤ C*‖x-y‖) (x v : E) {M h : ℝ}
+    (hh : 0 ≤ h) (hv : ‖v‖ ≤ M) :
+    F (x+h • v)-F x ≤ h*(D x v) + C*M^2*h^2 := by
+  have hb := (le_abs_self _).trans (state_taylor_bound hD hC hLip x (h • v))
+  rw [map_smul, smul_eq_mul, norm_smul, Real.norm_eq_abs, abs_of_nonneg hh] at hb
+  have hM : 0 ≤ M := (norm_nonneg v).trans hv
+  have hn : ‖v‖^2 ≤ M^2 := by nlinarith [norm_nonneg v]
+  have hh' := mul_le_mul_of_nonneg_left hn (mul_nonneg hC (sq_nonneg h))
+  nlinarith
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem expected_euler_step {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s h : ℝ≥0) (F : E → ℝ → ℝ) (D : E → ℝ → E →L[ℝ] ℝ)
+    (X V : (Iic s → ℝ) → E)
+    (mF : Measurable (Function.uncurry F))
+    (mB1 : Measurable (fun q : E × ℝ => deriv (F q.1) q.2))
+    (mB2 : Measurable (fun q : E × ℝ => deriv (deriv (F q.1)) q.2))
+    (mX : Measurable X) (mV : Measurable V)
+    (hD : ∀ x b, HasFDerivAt (fun y => F y b) (D x b) x)
+    (hsm : ∀ x, ContDiff ℝ 3 (F x))
+    {C₀ C₁ C₂ C₃ Cx Ccross M : ℝ}
+    (hC₃ : 0 ≤ C₃) (hCx : 0 ≤ Cx) (hCcross : 0 ≤ Ccross)
+    (hF : ∀ x b, |F x b| ≤ C₀)
+    (hB1 : ∀ x b, |deriv (F x) b| ≤ C₁)
+    (hB2 : ∀ x b, |deriv (deriv (F x)) b| ≤ C₂)
+    (hB3 : ∀ x b, |iteratedDeriv 3 (F x) b| ≤ C₃)
+    (hLipD : ∀ x y b, ‖D x b-D y b‖ ≤ Cx*‖x-y‖)
+    (hLipB2 : ∀ x y b, |deriv (deriv (F x)) b-deriv (deriv (F y)) b| ≤ Ccross*‖x-y‖)
+    (hV : ∀ p, ‖V p‖ ≤ M)
+    (hgen : ∀ p, D (X p) (p ⟨s,by simp⟩) (V p) +
+      deriv (deriv (F (X p))) (p ⟨s,by simp⟩)/2 ≤ 0) :
+    (∫ b, F (X (history s b)+(h:ℝ) • V (history s b)) (b (s+h)) -
+      F (X (history s b)) (b s) ∂W) ≤
+        (Cx*M^2+Ccross*M/2)*(h:ℝ)^2 + C₃/3*(h:ℝ)*Real.sqrt h := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let Y := fun p => X p+(h:ℝ) • V p
+  have mY : Measurable Y := mX.add (mV.const_smul (h:ℝ))
+  let f := fun p v => F (Y p) v
+  let old := fun b : Driver => F (X (history s b)) (b s)
+  let step := fun b : Driver => F (Y (history s b)) (b s)
+  let next := fun b : Driver => F (Y (history s b)) (b (s+h))
+  have mOld : Measurable old := mF.comp ((mX.comp (measurable_history s)).prodMk
+    ((continuous_eval_const s).measurable))
+  have mStep : Measurable step := mF.comp ((mY.comp (measurable_history s)).prodMk
+    ((continuous_eval_const s).measurable))
+  have mNext : Measurable next := mF.comp ((mY.comp (measurable_history s)).prodMk
+    ((continuous_eval_const (s+h)).measurable))
+  have iOld : Integrable old W := (integrable_const C₀).mono' mOld.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hF _ _)
+  have iStep : Integrable step W := (integrable_const C₀).mono' mStep.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hF _ _)
+  have iNext : Integrable next W := (integrable_const C₀).mono' mNext.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hF _ _)
+  let Q := fun b : Driver => deriv (deriv (F (Y (history s b)))) (b s)
+  have mQ : Measurable Q := mB2.comp ((mY.comp (measurable_history s)).prodMk
+    ((continuous_eval_const s).measurable))
+  have iQ : Integrable Q W := (integrable_const C₂).mono' mQ.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hB2 _ _)
+  have hm1 : Measurable (fun p => deriv (f p) (p ⟨s,by simp⟩)) :=
+    mB1.comp (mY.prodMk (measurable_pi_apply _))
+  have hm2 : Measurable (fun p => deriv (deriv (f p)) (p ⟨s,by simp⟩)) :=
+    mB2.comp (mY.prodMk (measurable_pi_apply _))
+  have ht := expected_taylor_step hW s h f (fun p => hsm (Y p)) hm1 hm2 hC₃
+    (fun p => hB1 _ _) (fun p => hB2 _ _) (fun p v => hB3 _ _) (iNext.sub iStep)
+  change |(∫ b, next b-step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W)| ≤ _ at ht
+  have hp (b : Driver) : step b-old b+(h:ℝ)/2*Q b ≤
+      (Cx*M^2+Ccross*M/2)*(h:ℝ)^2 := by
+    let p := history s b
+    have hs : p ⟨s,by simp⟩ = b s := rfl
+    have hy : ‖Y p-X p‖ ≤ (h:ℝ)*M := by
+      simp only [Y, add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg h.coe_nonneg]
+      exact mul_le_mul_of_nonneg_left (hV p) h.coe_nonneg
+    have hd := state_euler_remainder (fun x => hD x (b s)) hCx
+      (fun x y => hLipD x y (b s)) (X p) (V p) h.coe_nonneg (hV p)
+    have hb := (le_abs_self _).trans (hLipB2 (Y p) (X p) (b s))
+    have hcross := hb.trans (mul_le_mul_of_nonneg_left hy hCcross)
+    have hg := hgen p
+    rw [hs] at hg
+    have hg' := mul_nonpos_of_nonneg_of_nonpos h.coe_nonneg hg
+    have hc' := mul_le_mul_of_nonneg_left hcross (div_nonneg h.coe_nonneg (by norm_num : (0:ℝ) ≤ 2))
+    dsimp only [step, old, Q]
+    change F (Y p) (b s)-F (X p) (b s)+(h:ℝ)/2*deriv (deriv (F (Y p))) (b s) ≤ _
+    dsimp only [Y] at hd ⊢
+    nlinarith
+  have hib : (∫ b, step b-old b+(h:ℝ)/2*Q b ∂W) ≤
+      (Cx*M^2+Ccross*M/2)*(h:ℝ)^2 := by
+    have hh := integral_mono_ae ((iStep.sub iOld).add (iQ.const_mul ((h:ℝ)/2)))
+      (integrable_const ((Cx*M^2+Ccross*M/2)*(h:ℝ)^2)) (Filter.Eventually.of_forall hp)
+    simpa only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul, Pi.add_apply, Pi.sub_apply] using hh
+  change (∫ b, (step-old) b+((h:ℝ)/2*Q b) ∂W) ≤ _ at hib
+  rw [integral_add (iStep.sub iOld) (iQ.const_mul ((h:ℝ)/2)), integral_const_mul] at hib
+  change (∫ b, step b-old b ∂W)+(h:ℝ)/2*(∫ b, Q b ∂W) ≤ _ at hib
+  rw [integral_sub iStep iOld] at hib
+  rw [integral_sub iNext iStep] at ht
+  change (∫ b, next b-old b ∂W) ≤ _
+  rw [integral_sub iNext iOld]
+  linarith [le_abs_self ((∫ b, next b ∂W)-(∫ b, step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W))]
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+structure SmoothTest (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [MeasurableSpace E] where
+  F : E → ℝ → ℝ
+  D : E → ℝ → E →L[ℝ] ℝ
+  C₀ : ℝ
+  C₁ : ℝ
+  C₂ : ℝ
+  C₃ : ℝ
+  Cx : ℝ
+  Ccross : ℝ
+  hC₃ : 0 ≤ C₃
+  hCx : 0 ≤ Cx
+  hCcross : 0 ≤ Ccross
+  cF : Continuous (Function.uncurry F)
+  mF : Measurable (Function.uncurry F)
+  mB1 : Measurable (fun q : E × ℝ => deriv (F q.1) q.2)
+  mB2 : Measurable (fun q : E × ℝ => deriv (deriv (F q.1)) q.2)
+  hD : ∀ x b, HasFDerivAt (fun y => F y b) (D x b) x
+  hsm : ∀ x, ContDiff ℝ 3 (F x)
+  hF : ∀ x b, |F x b| ≤ C₀
+  hB1 : ∀ x b, |deriv (F x) b| ≤ C₁
+  hB2 : ∀ x b, |deriv (deriv (F x)) b| ≤ C₂
+  hB3 : ∀ x b, |iteratedDeriv 3 (F x) b| ≤ C₃
+  hLipD : ∀ x y b, ‖D x b-D y b‖ ≤ Cx*‖x-y‖
+  hLipB2 : ∀ x y b, |deriv (deriv (F x)) b-deriv (deriv (F y)) b| ≤ Ccross*‖x-y‖
+
+def extendHistory (s : ℝ≥0) (p : Iic s → ℝ) (t : ℝ≥0) : ℝ :=
+  if ht : t ≤ s then p ⟨t,ht⟩ else 0
+
+lemma measurable_extendHistory (s : ℝ≥0) : Measurable (extendHistory s) := by
+  apply Measurable.of_eval
+  intro t
+  by_cases ht : t ≤ s
+  · simpa only [extendHistory, dite_eq_left ht] using (measurable_pi_apply (⟨t,ht⟩ : Iic s))
+  · simp only [extendHistory, dite_eq_right ht]
+    exact measurable_const
+
+ 
+def stochasticNodes (v : E → ℝ → E) (h : ℝ≥0) (x : E) : ℕ → (ℝ≥0 → ℝ) → E
+  | 0, _ => x
+  | k+1, b => stochasticNodes v h x k b + (h:ℝ) •
+      v (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h))
+
+lemma measurable_stochasticNodes (v : E → ℝ → E)
+    (hv : Measurable (Function.uncurry v)) (h : ℝ≥0) (x : E) (k : ℕ) :
+    Measurable (stochasticNodes v h x k) := by
+  induction k with
+  | zero => exact measurable_const
+  | succ k ih => exact ih.add ((hv.comp (ih.prodMk (measurable_pi_apply _))).const_smul (h:ℝ))
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma stochasticNodes_congr (v : E → ℝ → E) (h : ℝ≥0) (x : E) (k : ℕ)
+    {b c : ℝ≥0 → ℝ} (he : ∀ t ≤ (k:ℝ≥0)*h, b t = c t) :
+    stochasticNodes v h x k b = stochasticNodes v h x k c := by
+  induction k with
+  | zero => rfl
+  | succ k ih =>
+    have hle : (k:ℝ≥0)*h ≤ ((k+1:ℕ):ℝ≥0)*h :=
+      mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ k) h.coe_nonneg
+    have hn := ih (fun t ht => he t (ht.trans hle))
+    simp only [stochasticNodes, hn, he _ hle]
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma stochasticNodes_history (v : E → ℝ → E) (h : ℝ≥0) (x : E) (k : ℕ)
+    (b : Driver) :
+    stochasticNodes v h x k (extendHistory ((k:ℝ≥0)*h) (history ((k:ℝ≥0)*h) b)) =
+      stochasticNodes v h x k b := by
+  apply stochasticNodes_congr
+  intro t ht
+  simp only [extendHistory, dite_eq_left ht, history]
+
+end SLEExactGauge.BrownianCalculus
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/BrownianRestart.lean b/lean/OAI/Probability/SLEGauge/BrownianRestart.lean
index 11afb3edf92f48656ac400dc90d1450880e9ddeb..087d7e66959039246bf2b7d6f501930beeeee23b 100644
--- a/lean/OAI/Probability/SLEGauge/BrownianRestart.lean
+++ b/lean/OAI/Probability/SLEGauge/BrownianRestart.lean
@@ -1,394 +1,394 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.BrownianShifts
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.StoppingRounding
-
-def round (n : ℕ) (t : ℝ≥0) : ℝ≥0 := (⌈((n+1:ℕ):ℝ≥0)*t⌉₊ : ℝ≥0)/(n+1:ℕ)
-
-theorem le_round (n : ℕ) (t : ℝ≥0) : t≤round n t := by
-  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
-  rw [round,le_div_iff₀ hd]
-  simpa only [mul_comm] using Nat.le_ceil (((n+1:ℕ):ℝ≥0)*t)
-
-theorem round_le (n : ℕ) (t : ℝ≥0) : round n t≤t+((n+1:ℕ):ℝ≥0)⁻¹ := by
-  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
-  rw [round,div_le_iff₀ hd,add_mul,inv_mul_cancel₀ hd.ne']
-  exact (Nat.ceil_lt_add_one (bot_le (a:=((n+1:ℕ):ℝ≥0)*t))).le.trans_eq (by rw [mul_comm])
-
-theorem round_le_iff (n : ℕ) (s t : ℝ≥0) :
-    round n s≤t ↔ s≤(⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ) := by
-  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
-  rw [round,div_le_iff₀ hd,←Nat.le_floor_iff (bot_le (a:=t*((n+1:ℕ):ℝ≥0))),
-    Nat.ceil_le,le_div_iff₀ hd]
-  simp only [mul_comm]
-
-theorem floor_time_le (n : ℕ) (t : ℝ≥0) :
-    (⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ)≤t := by
-  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
-  rw [div_le_iff₀ hd]
-  simpa only [mul_comm] using Nat.floor_le (bot_le (a:=((n+1:ℕ):ℝ≥0)*t))
-
-theorem countable_range (n : ℕ) : (Set.range (round n)).Countable := by
-  apply (Set.countable_range (fun k : ℕ => (k:ℝ≥0)/(n+1:ℕ))).mono
-  rintro _ ⟨t,rfl⟩
-  exact ⟨_,rfl⟩
-
-theorem tendsto_round (t : ℝ≥0) : Tendsto (fun n => round n t) atTop (𝓝 t) := by
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _
-    (fun n => le_round n t) (fun n => round_le n t)
-  have ht : Tendsto (fun n : ℕ => ((n+1:ℕ):ℝ≥0)⁻¹) atTop (𝓝 0) := by
-    have h := tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R:=ℝ≥0) |>.comp
-      (tendsto_add_atTop_nat 1))
-    exact h
-  simpa only [add_zero] using tendsto_const_nhds.add ht
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ}
-    {S : Ω→ℝ≥0}
-
-theorem isStoppingTime (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0))) (n : ℕ) :
-    IsStoppingTime ℱ (fun ω => (round n (S ω) : WithTop ℝ≥0)) := by
-  intro t
-  have h := hS ((⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ))
-  have he : {ω | (round n (S ω) : WithTop ℝ≥0)≤t} =
-      {ω | (S ω : WithTop ℝ≥0)≤((⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ) : ℝ≥0)} := by
-    ext ω
-    exact (WithTop.coe_le_coe.trans (round_le_iff n (S ω) t)).trans WithTop.coe_le_coe.symm
-  rw [he]
-  exact ℱ.mono (floor_time_le n t) _ h
-
-end SLEExactGauge.StoppingRounding
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology Function
-open scoped NNReal ENNReal ContinuousMapZero BoundedContinuousFunction
-namespace SLEExactGauge.BrownianRestart
-
-def filtration : Filtration ℝ≥0 instMeasurableSpaceDriver :=
-  Filtration.natural (fun t (b : Driver) => b t)
-    (fun t => (continuous_eval_const t).stronglyMeasurable)
-
-lemma continuous_shift_joint : Continuous (fun p : ℝ≥0×Driver => shift p.1 p.2) := by
-  apply ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff.mpr
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  change Continuous (fun p : (ℝ≥0×Driver)×ℝ≥0 => p.1.2 (p.1.1+p.2)-p.1.2 p.1.1)
-  exact (continuous_eval.comp ((continuous_snd.comp continuous_fst).prodMk
-    ((continuous_fst.comp continuous_fst).add continuous_snd))).sub
-    (continuous_eval.comp ((continuous_snd.comp continuous_fst).prodMk
-      (continuous_fst.comp continuous_fst)))
-
-lemma indep_filtration_shift {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W) (t : ℝ≥0) :
-    Indep (filtration t) (MeasurableSpace.comap (shift t) inferInstance) W := by
-  rw [filtration, Filtration.natural_eq_comap]
-  change IndepFun (fun b : Driver => fun s : Iic t => b s) (shift t) W
-  exact (indep_shift_history hW t).symm
-
-lemma integral_shift {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W) (t : ℝ≥0) (f : Driver →ᵇ ℝ) :
-    (∫b,f (shift t b)∂W)=∫b,f b∂W := by
-  rw [← integral_map (measurable_shift t).aemeasurable f.continuous.aestronglyMeasurable,
-    shift_preserves_law hW t]
-
-lemma setIntegral_shift {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W) (t : ℝ≥0)
-    {A : Set Driver} (hA : MeasurableSet[filtration t] A) (f : Driver →ᵇ ℝ) :
-    (∫b in A,f (shift t b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
-  rw [(indep_filtration_shift hW t).setIntegral_eq_mul (filtration.le t)
-    (measurable_shift t).aemeasurable hA f.continuous.aestronglyMeasurable,
-    integral_shift hW t f]
-  simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul]
-
-lemma measurable_finite_stop {T : Driver→ℝ≥0}
-    (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) : Measurable T := by
-  apply measurable_of_Iic
-  intro t
-  convert filtration.le t _ (hT t) using 1
-  ext b
-  change T b ≤ t ↔ (T b : WithTop ℝ≥0) ≤ (t : WithTop ℝ≥0)
-  exact WithTop.coe_le_coe.symm
-
-lemma measurable_shift_finite_stop {T : Driver→ℝ≥0}
-    (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) :
-    Measurable (fun b => shift (T b) b) :=
-  continuous_shift_joint.measurable.comp ((measurable_finite_stop hT).prodMk measurable_id)
-
-lemma integrable_bcf_comp {W : Measure Driver} [IsFiniteMeasure W]
-    (f : Driver →ᵇ ℝ) {Y : Driver→Driver} (hY : Measurable Y) :
-    Integrable (fun b => f (Y b)) W := by
-  apply Integrable.mono' (integrable_const ‖f‖)
-    (f.continuous.measurable.comp hY).aestronglyMeasurable
-  exact Eventually.of_forall (fun b => f.norm_coe_le_norm (Y b))
-
- 
-theorem setIntegral_shift_countable {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0)))
-    (hc : (Set.range T).Countable) {A : Set Driver}
-    (hA : MeasurableSet[hT.measurableSpace] A) (f : Driver →ᵇ ℝ) :
-    (∫b in A,f (shift (T b) b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let : Countable (Set.range T) := hc.to_subtype
-  let U : Set.range T→Set Driver := fun t => A∩{b | T b=t.1}
-  have hm (t : Set.range T) : MeasurableSet[filtration t.1] (U t) := by
-    have h := hA.inter (hT.measurableSet_eq' t.1)
-    have h' := (hT.measurableSet_inter_eq_iff A t.1).mp h
-    simpa only [U,WithTop.coe_eq_coe] using h'
-  have hm' (t : Set.range T) : MeasurableSet (U t) := filtration.le t.1 _ (hm t)
-  have hd : Pairwise (Disjoint on U) := by
-    intro t u htu
-    apply disjoint_left.mpr
-    intro b hb hc
-    exact htu (Subtype.ext (hb.2.symm.trans hc.2))
-  have hU : (⋃t,U t)=A := by
-    ext b
-    simp only [mem_iUnion,U,mem_inter_iff,mem_ofPred_eq]
-    constructor
-    · rintro ⟨t,h,_⟩; exact h
-    · intro hb; exact ⟨⟨T b,mem_range_self b⟩,hb,rfl⟩
-  have hi := integrable_bcf_comp f (measurable_shift_finite_stop hT) (W:=W)
-  calc
-    (∫b in A,f (shift (T b) b)∂W) = ∑'t,∫b in U t,f (shift (T b) b)∂W := by
-      rw [←hU, integral_iUnion hm' hd hi.integrableOn]
-    _ = ∑'t,∫_b in U t,(∫b,f b∂W)∂W := by
-      apply tsum_congr
-      intro t
-      rw [setIntegral_congr_fun (hm' t) (fun b hb => by rw [hb.2])]
-      exact setIntegral_shift hW t.1 (hm t) f
-    _ = ∫_b in A,(∫b,f b∂W)∂W := by
-      rw [←hU,integral_iUnion hm' hd (integrable_const _).integrableOn]
-
- 
-
-theorem setIntegral_shift_finite {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0)))
-    {A : Set Driver} (hA : MeasurableSet[hT.measurableSpace] A) (f : Driver →ᵇ ℝ) :
-    (∫b in A,f (shift (T b) b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let Tn := fun n b => StoppingRounding.round n (T b)
-  have hn (n : ℕ) : IsStoppingTime filtration (fun b => (Tn n b : WithTop ℝ≥0)) :=
-    StoppingRounding.isStoppingTime hT n
-  have hAn (n : ℕ) : MeasurableSet[(hn n).measurableSpace] A :=
-    (hT.measurableSpace_mono (hn n) (fun b => WithTop.coe_le_coe.mpr
-      (StoppingRounding.le_round n (T b)))) _ hA
-  have he (n : ℕ) : (∫b in A,f (shift (Tn n b) b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
-    apply setIntegral_shift_countable hW (hn n) _ (hAn n) f
-    exact (StoppingRounding.countable_range n).mono (Set.range_comp_subset_range T (StoppingRounding.round n))
-  have ht : Tendsto (fun n => ∫b in A,f (shift (Tn n b) b)∂W) atTop
-      (𝓝 (∫b in A,f (shift (T b) b)∂W)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ => ‖f‖)
-      (fun n => (f.continuous.measurable.comp (measurable_shift_finite_stop (hn n))).aestronglyMeasurable)
-      (integrable_const _) (fun n => Eventually.of_forall (fun b => f.norm_coe_le_norm _))
-    apply Eventually.of_forall
-    intro b
-    exact f.continuous.continuousAt.tendsto.comp
-      (continuous_shift_joint.continuousAt.tendsto.comp
-        ((StoppingRounding.tendsto_round (T b)).prodMk_nhds tendsto_const_nhds))
-  simp only [he] at ht
-  exact (tendsto_nhds_unique tendsto_const_nhds ht).symm
-
- 
-theorem map_shift_finite {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) :
-    W.map (fun b => shift (T b) b)=W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro f
-  rw [integral_map (measurable_shift_finite_stop hT).aemeasurable
-    f.continuous.aestronglyMeasurable]
-  simpa using setIntegral_shift_finite hW hT MeasurableSet.univ f
-
- 
-theorem indep_shift_finite {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) :
-    Indep hT.measurableSpace (MeasurableSpace.comap (fun b => shift (T b) b) inferInstance) W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  apply indep_comap_of_bcf hT.measurableSpace_le (measurable_shift_finite_stop hT).aemeasurable
-  intro A hA f
-  rw [setIntegral_shift_finite hW hT hA f]
-  rw [←integral_map (measurable_shift_finite_stop hT).aemeasurable
-    f.continuous.aestronglyMeasurable,map_shift_finite hW hT]
-  simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul]
-
-end SLEExactGauge.BrownianRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology Function
-open scoped NNReal ENNReal ContinuousMapZero BoundedContinuousFunction
-namespace SLEExactGauge.BrownianRestart
-
- 
-
-def shiftAt (S : Driver → ℝ≥0∞) (b : Driver) := shift (S b).toNNReal b
-
-def truncateStop (S : Driver → ℝ≥0∞) (t : ℝ≥0) (b : Driver) : ℝ≥0 :=
-  (min (S b) (t : ℝ≥0∞)).toNNReal
-
-lemma coe_truncateStop (S : Driver → ℝ≥0∞) (t : ℝ≥0) :
-    (fun b => (truncateStop S t b : WithTop ℝ≥0)) = fun b => min (S b) (t : WithTop ℝ≥0) := by
-  funext b
-  simp only [truncateStop, ENNReal.some_eq_coe']
-  exact ENNReal.coe_toNNReal (ne_top_of_le_ne_top ENNReal.coe_ne_top (min_le_right _ _))
-
-lemma isStoppingTime_truncateStop {S : Driver → ℝ≥0∞}
-    (hS : IsStoppingTime filtration S) (t : ℝ≥0) :
-    IsStoppingTime filtration (fun b => (truncateStop S t b : WithTop ℝ≥0)) := by
-  rw [coe_truncateStop]
-  exact hS.min_const t
-
-lemma measurable_shiftAt {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S) :
-    Measurable (shiftAt S) := by
-  exact continuous_shift_joint.measurable.comp
-    ((ENNReal.measurable_toNNReal.comp hS.measurable').prodMk measurable_id)
-
-lemma finiteStop_iUnion_le (S : Driver → ℝ≥0∞) :
-    (⋃ n : ℕ, {b | S b ≤ (n : ℝ≥0∞)}) = {b | S b ≠ ∞} := by
-  ext b
-  simp only [mem_iUnion, mem_ofPred_eq]
-  constructor
-  · rintro ⟨n,hn⟩
-    exact ne_top_of_le_ne_top (by simp) hn
-  · intro hb
-    exact (ENNReal.exists_nat_gt hb).imp (fun _ h => h.le)
-
-lemma setIntegral_shiftAt_le {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
-    {A : Set Driver} (hA : MeasurableSet[hS.measurableSpace] A)
-    (t : ℝ≥0) (f : Driver →ᵇ ℝ) :
-    (∫b in A ∩ {b | S b ≤ (t : ℝ≥0∞)}, f (shiftAt S b) ∂W) =
-      ∫_b in A ∩ {b | S b ≤ (t : ℝ≥0∞)}, (∫b, f b ∂W) ∂W := by
-  have he := coe_truncateStop S t
-  have hA' : MeasurableSet[(isStoppingTime_truncateStop hS t).measurableSpace]
-      (A ∩ {b | S b ≤ (t : ℝ≥0∞)}) := by
-    have hspace : (isStoppingTime_truncateStop hS t).measurableSpace =
-        (hS.min_const t).measurableSpace := by
-      congr 1
-    rw [hspace]
-    convert (hS.measurableSet_inter_le_const_iff A t).mp
-      (hA.inter (hS.measurableSet_le' t)) using 1
-    simp only [ENNReal.some_eq_coe']
-    rfl
-  have h := setIntegral_shift_finite hW (isStoppingTime_truncateStop hS t) hA' f
-  rw [←h]
-  apply setIntegral_congr_fun
-  · exact hS.measurableSpace_le _ (hA.inter (measurableSet_le (show Measurable[hS.measurableSpace] S from hS.measurable) measurable_const))
-  · intro b hb
-    simp only [shiftAt, truncateStop, min_eq_left (show S b ≤ (t : ℝ≥0∞) from hb.2)]
-
- 
-
-theorem setIntegral_shiftAt_finite {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
-    {A : Set Driver} (hA : MeasurableSet[hS.measurableSpace] A) (f : Driver →ᵇ ℝ) :
-    (∫b in A ∩ {b | S b ≠ ∞}, f (shiftAt S b) ∂W) =
-      ∫_b in A ∩ {b | S b ≠ ∞}, (∫b, f b ∂W) ∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let U : ℕ → Set Driver := fun n => A ∩ {b | S b ≤ (n : ℝ≥0∞)}
-  have hmeas n : MeasurableSet (U n) :=
-    (hS.measurableSpace_le _ hA).inter (measurableSet_le (show Measurable S from hS.measurable') measurable_const)
-  have hmono : Monotone U := by
-    intro n m hnm b hb
-    exact ⟨hb.1, (show S b ≤ (n : ℝ≥0∞) from hb.2).trans (Nat.cast_le.mpr hnm)⟩
-  have hU : (⋃ n, U n) = A ∩ {b | S b ≠ ∞} := by
-    simp only [U, ←inter_iUnion, finiteStop_iUnion_le]
-  have h₁ := tendsto_setIntegral_of_monotone hmeas hmono
-    (integrable_bcf_comp f (measurable_shiftAt hS) (W:=W)).integrableOn
-  have h₂ := tendsto_setIntegral_of_monotone hmeas hmono
-    (integrable_const (∫b, f b ∂W) (μ:=W)).integrableOn
-  have he (n : ℕ) : (∫b in U n, f (shiftAt S b) ∂W) =
-      ∫_b in U n, (∫b, f b ∂W) ∂W := by
-    simpa only [U, ENNReal.coe_natCast] using setIntegral_shiftAt_le hW hS hA (n:ℝ≥0) f
-  simp only [hU, he] at h₁ h₂
-  exact tendsto_nhds_unique h₁ h₂
-
- 
-
-theorem map_shiftAt_restrict {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
-    {A : Set Driver} (hA : MeasurableSet[hS.measurableSpace] A) :
-    (W.restrict (A ∩ {b | S b ≠ ∞})).map (shiftAt S) =
-      W (A ∩ {b | S b ≠ ∞}) • W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have : IsFiniteMeasure (W (A ∩ {b | S b ≠ ∞}) • W) :=
-    W.smul_finite (measure_ne_top W _)
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro f
-  rw [integral_map (measurable_shiftAt hS).aemeasurable f.continuous.aestronglyMeasurable,
-    setIntegral_shiftAt_finite hW hS hA f, integral_smul_measure]
-  simp only [integral_const, measureReal_def, Measure.restrict_apply_univ]
-
- 
-
-theorem map_data_shiftAt {X : Type*} [MeasurableSpace X] {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
-    {D : Driver → X} (hD : Measurable[hS.measurableSpace] D) :
-    (W.restrict {b | S b ≠ ∞}).map (fun b => (D b,shiftAt S b)) =
-      ((W.restrict {b | S b ≠ ∞}).map D).prod W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
-  have hY := measurable_shiftAt hS
-  apply Measure.ext_prod
-  intro A B hA hB
-  rw [Measure.map_apply (hD'.prodMk hY) (hA.prod hB), Measure.prod_prod,
-    Measure.map_apply hD' hA, Measure.restrict_apply (hD' hA)]
-  have he := congrArg (fun μ : Measure Driver => μ B)
-    (map_shiftAt_restrict hW hS (hD hA))
-  rw [Measure.map_apply hY hB, Measure.restrict_apply (hY hB), Measure.smul_apply,
-    smul_eq_mul] at he
-  rw [Measure.restrict_apply ((hD'.prodMk hY) (hA.prod hB))]
-  convert he using 1
-  congr 1
-  ext b
-  simp only [mem_inter_iff, mem_preimage, mem_prod, mem_ofPred_eq]
-  tauto
-
- 
-
-theorem lintegral_data_shiftAt {X : Type*} [MeasurableSpace X] {W : Measure Driver}
-    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
-    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
-    {D : Driver → X} (hD : Measurable[hS.measurableSpace] D)
-    {F : X × Driver → ℝ≥0∞} (hF : Measurable F) :
-    (∫⁻b in {b | S b ≠ ∞}, F (D b,shiftAt S b) ∂W) =
-      ∫⁻b in {b | S b ≠ ∞}, ∫⁻c, F (D b,c) ∂W ∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
-  rw [←lintegral_map hF (hD'.prodMk (measurable_shiftAt hS)),map_data_shiftAt hW hS hD,
-    lintegral_prod _ hF.aemeasurable, lintegral_map hF.lintegral_prod_right' hD']
-
-end SLEExactGauge.BrownianRestart
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.BrownianShifts
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.StoppingRounding
+
+def round (n : ℕ) (t : ℝ≥0) : ℝ≥0 := (⌈((n+1:ℕ):ℝ≥0)*t⌉₊ : ℝ≥0)/(n+1:ℕ)
+
+theorem le_round (n : ℕ) (t : ℝ≥0) : t≤round n t := by
+  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
+  rw [round,le_div_iff₀ hd]
+  simpa only [mul_comm] using Nat.le_ceil (((n+1:ℕ):ℝ≥0)*t)
+
+theorem round_le (n : ℕ) (t : ℝ≥0) : round n t≤t+((n+1:ℕ):ℝ≥0)⁻¹ := by
+  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
+  rw [round,div_le_iff₀ hd,add_mul,inv_mul_cancel₀ hd.ne']
+  exact (Nat.ceil_lt_add_one (bot_le (a:=((n+1:ℕ):ℝ≥0)*t))).le.trans_eq (by rw [mul_comm])
+
+theorem round_le_iff (n : ℕ) (s t : ℝ≥0) :
+    round n s≤t ↔ s≤(⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ) := by
+  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
+  rw [round,div_le_iff₀ hd,←Nat.le_floor_iff (bot_le (a:=t*((n+1:ℕ):ℝ≥0))),
+    Nat.ceil_le,le_div_iff₀ hd]
+  simp only [mul_comm]
+
+theorem floor_time_le (n : ℕ) (t : ℝ≥0) :
+    (⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ)≤t := by
+  have hd : (0:ℝ≥0)<(n+1:ℕ) := by positivity
+  rw [div_le_iff₀ hd]
+  simpa only [mul_comm] using Nat.floor_le (bot_le (a:=((n+1:ℕ):ℝ≥0)*t))
+
+theorem countable_range (n : ℕ) : (Set.range (round n)).Countable := by
+  apply (Set.countable_range (fun k : ℕ => (k:ℝ≥0)/(n+1:ℕ))).mono
+  rintro _ ⟨t,rfl⟩
+  exact ⟨_,rfl⟩
+
+theorem tendsto_round (t : ℝ≥0) : Tendsto (fun n => round n t) atTop (𝓝 t) := by
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _
+    (fun n => le_round n t) (fun n => round_le n t)
+  have ht : Tendsto (fun n : ℕ => ((n+1:ℕ):ℝ≥0)⁻¹) atTop (𝓝 0) := by
+    have h := tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R:=ℝ≥0) |>.comp
+      (tendsto_add_atTop_nat 1))
+    exact h
+  simpa only [add_zero] using tendsto_const_nhds.add ht
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ}
+    {S : Ω→ℝ≥0}
+
+theorem isStoppingTime (hS : IsStoppingTime ℱ (fun ω => (S ω : WithTop ℝ≥0))) (n : ℕ) :
+    IsStoppingTime ℱ (fun ω => (round n (S ω) : WithTop ℝ≥0)) := by
+  intro t
+  have h := hS ((⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ))
+  have he : {ω | (round n (S ω) : WithTop ℝ≥0)≤t} =
+      {ω | (S ω : WithTop ℝ≥0)≤((⌊((n+1:ℕ):ℝ≥0)*t⌋₊ : ℝ≥0)/(n+1:ℕ) : ℝ≥0)} := by
+    ext ω
+    exact (WithTop.coe_le_coe.trans (round_le_iff n (S ω) t)).trans WithTop.coe_le_coe.symm
+  rw [he]
+  exact ℱ.mono (floor_time_le n t) _ h
+
+end SLEExactGauge.StoppingRounding
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology Function
+open scoped NNReal ENNReal ContinuousMapZero BoundedContinuousFunction
+namespace SLEExactGauge.BrownianRestart
+
+def filtration : Filtration ℝ≥0 instMeasurableSpaceDriver :=
+  Filtration.natural (fun t (b : Driver) => b t)
+    (fun t => (continuous_eval_const t).stronglyMeasurable)
+
+lemma continuous_shift_joint : Continuous (fun p : ℝ≥0×Driver => shift p.1 p.2) := by
+  apply ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff.mpr
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  change Continuous (fun p : (ℝ≥0×Driver)×ℝ≥0 => p.1.2 (p.1.1+p.2)-p.1.2 p.1.1)
+  exact (continuous_eval.comp ((continuous_snd.comp continuous_fst).prodMk
+    ((continuous_fst.comp continuous_fst).add continuous_snd))).sub
+    (continuous_eval.comp ((continuous_snd.comp continuous_fst).prodMk
+      (continuous_fst.comp continuous_fst)))
+
+lemma indep_filtration_shift {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W) (t : ℝ≥0) :
+    Indep (filtration t) (MeasurableSpace.comap (shift t) inferInstance) W := by
+  rw [filtration, Filtration.natural_eq_comap]
+  change IndepFun (fun b : Driver => fun s : Iic t => b s) (shift t) W
+  exact (indep_shift_history hW t).symm
+
+lemma integral_shift {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W) (t : ℝ≥0) (f : Driver →ᵇ ℝ) :
+    (∫b,f (shift t b)∂W)=∫b,f b∂W := by
+  rw [← integral_map (measurable_shift t).aemeasurable f.continuous.aestronglyMeasurable,
+    shift_preserves_law hW t]
+
+lemma setIntegral_shift {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W) (t : ℝ≥0)
+    {A : Set Driver} (hA : MeasurableSet[filtration t] A) (f : Driver →ᵇ ℝ) :
+    (∫b in A,f (shift t b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
+  rw [(indep_filtration_shift hW t).setIntegral_eq_mul (filtration.le t)
+    (measurable_shift t).aemeasurable hA f.continuous.aestronglyMeasurable,
+    integral_shift hW t f]
+  simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul]
+
+lemma measurable_finite_stop {T : Driver→ℝ≥0}
+    (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) : Measurable T := by
+  apply measurable_of_Iic
+  intro t
+  convert filtration.le t _ (hT t) using 1
+  ext b
+  change T b ≤ t ↔ (T b : WithTop ℝ≥0) ≤ (t : WithTop ℝ≥0)
+  exact WithTop.coe_le_coe.symm
+
+lemma measurable_shift_finite_stop {T : Driver→ℝ≥0}
+    (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) :
+    Measurable (fun b => shift (T b) b) :=
+  continuous_shift_joint.measurable.comp ((measurable_finite_stop hT).prodMk measurable_id)
+
+lemma integrable_bcf_comp {W : Measure Driver} [IsFiniteMeasure W]
+    (f : Driver →ᵇ ℝ) {Y : Driver→Driver} (hY : Measurable Y) :
+    Integrable (fun b => f (Y b)) W := by
+  apply Integrable.mono' (integrable_const ‖f‖)
+    (f.continuous.measurable.comp hY).aestronglyMeasurable
+  exact Eventually.of_forall (fun b => f.norm_coe_le_norm (Y b))
+
+ 
+theorem setIntegral_shift_countable {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0)))
+    (hc : (Set.range T).Countable) {A : Set Driver}
+    (hA : MeasurableSet[hT.measurableSpace] A) (f : Driver →ᵇ ℝ) :
+    (∫b in A,f (shift (T b) b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let : Countable (Set.range T) := hc.to_subtype
+  let U : Set.range T→Set Driver := fun t => A∩{b | T b=t.1}
+  have hm (t : Set.range T) : MeasurableSet[filtration t.1] (U t) := by
+    have h := hA.inter (hT.measurableSet_eq' t.1)
+    have h' := (hT.measurableSet_inter_eq_iff A t.1).mp h
+    simpa only [U,WithTop.coe_eq_coe] using h'
+  have hm' (t : Set.range T) : MeasurableSet (U t) := filtration.le t.1 _ (hm t)
+  have hd : Pairwise (Disjoint on U) := by
+    intro t u htu
+    apply disjoint_left.mpr
+    intro b hb hc
+    exact htu (Subtype.ext (hb.2.symm.trans hc.2))
+  have hU : (⋃t,U t)=A := by
+    ext b
+    simp only [mem_iUnion,U,mem_inter_iff,mem_ofPred_eq]
+    constructor
+    · rintro ⟨t,h,_⟩; exact h
+    · intro hb; exact ⟨⟨T b,mem_range_self b⟩,hb,rfl⟩
+  have hi := integrable_bcf_comp f (measurable_shift_finite_stop hT) (W:=W)
+  calc
+    (∫b in A,f (shift (T b) b)∂W) = ∑'t,∫b in U t,f (shift (T b) b)∂W := by
+      rw [←hU, integral_iUnion hm' hd hi.integrableOn]
+    _ = ∑'t,∫_b in U t,(∫b,f b∂W)∂W := by
+      apply tsum_congr
+      intro t
+      rw [setIntegral_congr_fun (hm' t) (fun b hb => by rw [hb.2])]
+      exact setIntegral_shift hW t.1 (hm t) f
+    _ = ∫_b in A,(∫b,f b∂W)∂W := by
+      rw [←hU,integral_iUnion hm' hd (integrable_const _).integrableOn]
+
+ 
+
+theorem setIntegral_shift_finite {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0)))
+    {A : Set Driver} (hA : MeasurableSet[hT.measurableSpace] A) (f : Driver →ᵇ ℝ) :
+    (∫b in A,f (shift (T b) b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let Tn := fun n b => StoppingRounding.round n (T b)
+  have hn (n : ℕ) : IsStoppingTime filtration (fun b => (Tn n b : WithTop ℝ≥0)) :=
+    StoppingRounding.isStoppingTime hT n
+  have hAn (n : ℕ) : MeasurableSet[(hn n).measurableSpace] A :=
+    (hT.measurableSpace_mono (hn n) (fun b => WithTop.coe_le_coe.mpr
+      (StoppingRounding.le_round n (T b)))) _ hA
+  have he (n : ℕ) : (∫b in A,f (shift (Tn n b) b)∂W)=∫_b in A,(∫b,f b∂W)∂W := by
+    apply setIntegral_shift_countable hW (hn n) _ (hAn n) f
+    exact (StoppingRounding.countable_range n).mono (Set.range_comp_subset_range T (StoppingRounding.round n))
+  have ht : Tendsto (fun n => ∫b in A,f (shift (Tn n b) b)∂W) atTop
+      (𝓝 (∫b in A,f (shift (T b) b)∂W)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ => ‖f‖)
+      (fun n => (f.continuous.measurable.comp (measurable_shift_finite_stop (hn n))).aestronglyMeasurable)
+      (integrable_const _) (fun n => Eventually.of_forall (fun b => f.norm_coe_le_norm _))
+    apply Eventually.of_forall
+    intro b
+    exact f.continuous.continuousAt.tendsto.comp
+      (continuous_shift_joint.continuousAt.tendsto.comp
+        ((StoppingRounding.tendsto_round (T b)).prodMk_nhds tendsto_const_nhds))
+  simp only [he] at ht
+  exact (tendsto_nhds_unique tendsto_const_nhds ht).symm
+
+ 
+theorem map_shift_finite {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) :
+    W.map (fun b => shift (T b) b)=W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro f
+  rw [integral_map (measurable_shift_finite_stop hT).aemeasurable
+    f.continuous.aestronglyMeasurable]
+  simpa using setIntegral_shift_finite hW hT MeasurableSet.univ f
+
+ 
+theorem indep_shift_finite {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {T : Driver→ℝ≥0} (hT : IsStoppingTime filtration (fun b => (T b : WithTop ℝ≥0))) :
+    Indep hT.measurableSpace (MeasurableSpace.comap (fun b => shift (T b) b) inferInstance) W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  apply indep_comap_of_bcf hT.measurableSpace_le (measurable_shift_finite_stop hT).aemeasurable
+  intro A hA f
+  rw [setIntegral_shift_finite hW hT hA f]
+  rw [←integral_map (measurable_shift_finite_stop hT).aemeasurable
+    f.continuous.aestronglyMeasurable,map_shift_finite hW hT]
+  simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul]
+
+end SLEExactGauge.BrownianRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology Function
+open scoped NNReal ENNReal ContinuousMapZero BoundedContinuousFunction
+namespace SLEExactGauge.BrownianRestart
+
+ 
+
+def shiftAt (S : Driver → ℝ≥0∞) (b : Driver) := shift (S b).toNNReal b
+
+def truncateStop (S : Driver → ℝ≥0∞) (t : ℝ≥0) (b : Driver) : ℝ≥0 :=
+  (min (S b) (t : ℝ≥0∞)).toNNReal
+
+lemma coe_truncateStop (S : Driver → ℝ≥0∞) (t : ℝ≥0) :
+    (fun b => (truncateStop S t b : WithTop ℝ≥0)) = fun b => min (S b) (t : WithTop ℝ≥0) := by
+  funext b
+  simp only [truncateStop, ENNReal.some_eq_coe']
+  exact ENNReal.coe_toNNReal (ne_top_of_le_ne_top ENNReal.coe_ne_top (min_le_right _ _))
+
+lemma isStoppingTime_truncateStop {S : Driver → ℝ≥0∞}
+    (hS : IsStoppingTime filtration S) (t : ℝ≥0) :
+    IsStoppingTime filtration (fun b => (truncateStop S t b : WithTop ℝ≥0)) := by
+  rw [coe_truncateStop]
+  exact hS.min_const t
+
+lemma measurable_shiftAt {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S) :
+    Measurable (shiftAt S) := by
+  exact continuous_shift_joint.measurable.comp
+    ((ENNReal.measurable_toNNReal.comp hS.measurable').prodMk measurable_id)
+
+lemma finiteStop_iUnion_le (S : Driver → ℝ≥0∞) :
+    (⋃ n : ℕ, {b | S b ≤ (n : ℝ≥0∞)}) = {b | S b ≠ ∞} := by
+  ext b
+  simp only [mem_iUnion, mem_ofPred_eq]
+  constructor
+  · rintro ⟨n,hn⟩
+    exact ne_top_of_le_ne_top (by simp) hn
+  · intro hb
+    exact (ENNReal.exists_nat_gt hb).imp (fun _ h => h.le)
+
+lemma setIntegral_shiftAt_le {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
+    {A : Set Driver} (hA : MeasurableSet[hS.measurableSpace] A)
+    (t : ℝ≥0) (f : Driver →ᵇ ℝ) :
+    (∫b in A ∩ {b | S b ≤ (t : ℝ≥0∞)}, f (shiftAt S b) ∂W) =
+      ∫_b in A ∩ {b | S b ≤ (t : ℝ≥0∞)}, (∫b, f b ∂W) ∂W := by
+  have he := coe_truncateStop S t
+  have hA' : MeasurableSet[(isStoppingTime_truncateStop hS t).measurableSpace]
+      (A ∩ {b | S b ≤ (t : ℝ≥0∞)}) := by
+    have hspace : (isStoppingTime_truncateStop hS t).measurableSpace =
+        (hS.min_const t).measurableSpace := by
+      congr 1
+    rw [hspace]
+    convert (hS.measurableSet_inter_le_const_iff A t).mp
+      (hA.inter (hS.measurableSet_le' t)) using 1
+    simp only [ENNReal.some_eq_coe']
+    rfl
+  have h := setIntegral_shift_finite hW (isStoppingTime_truncateStop hS t) hA' f
+  rw [←h]
+  apply setIntegral_congr_fun
+  · exact hS.measurableSpace_le _ (hA.inter (measurableSet_le (show Measurable[hS.measurableSpace] S from hS.measurable) measurable_const))
+  · intro b hb
+    simp only [shiftAt, truncateStop, min_eq_left (show S b ≤ (t : ℝ≥0∞) from hb.2)]
+
+ 
+
+theorem setIntegral_shiftAt_finite {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
+    {A : Set Driver} (hA : MeasurableSet[hS.measurableSpace] A) (f : Driver →ᵇ ℝ) :
+    (∫b in A ∩ {b | S b ≠ ∞}, f (shiftAt S b) ∂W) =
+      ∫_b in A ∩ {b | S b ≠ ∞}, (∫b, f b ∂W) ∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let U : ℕ → Set Driver := fun n => A ∩ {b | S b ≤ (n : ℝ≥0∞)}
+  have hmeas n : MeasurableSet (U n) :=
+    (hS.measurableSpace_le _ hA).inter (measurableSet_le (show Measurable S from hS.measurable') measurable_const)
+  have hmono : Monotone U := by
+    intro n m hnm b hb
+    exact ⟨hb.1, (show S b ≤ (n : ℝ≥0∞) from hb.2).trans (Nat.cast_le.mpr hnm)⟩
+  have hU : (⋃ n, U n) = A ∩ {b | S b ≠ ∞} := by
+    simp only [U, ←inter_iUnion, finiteStop_iUnion_le]
+  have h₁ := tendsto_setIntegral_of_monotone hmeas hmono
+    (integrable_bcf_comp f (measurable_shiftAt hS) (W:=W)).integrableOn
+  have h₂ := tendsto_setIntegral_of_monotone hmeas hmono
+    (integrable_const (∫b, f b ∂W) (μ:=W)).integrableOn
+  have he (n : ℕ) : (∫b in U n, f (shiftAt S b) ∂W) =
+      ∫_b in U n, (∫b, f b ∂W) ∂W := by
+    simpa only [U, ENNReal.coe_natCast] using setIntegral_shiftAt_le hW hS hA (n:ℝ≥0) f
+  simp only [hU, he] at h₁ h₂
+  exact tendsto_nhds_unique h₁ h₂
+
+ 
+
+theorem map_shiftAt_restrict {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
+    {A : Set Driver} (hA : MeasurableSet[hS.measurableSpace] A) :
+    (W.restrict (A ∩ {b | S b ≠ ∞})).map (shiftAt S) =
+      W (A ∩ {b | S b ≠ ∞}) • W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have : IsFiniteMeasure (W (A ∩ {b | S b ≠ ∞}) • W) :=
+    W.smul_finite (measure_ne_top W _)
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro f
+  rw [integral_map (measurable_shiftAt hS).aemeasurable f.continuous.aestronglyMeasurable,
+    setIntegral_shiftAt_finite hW hS hA f, integral_smul_measure]
+  simp only [integral_const, measureReal_def, Measure.restrict_apply_univ]
+
+ 
+
+theorem map_data_shiftAt {X : Type*} [MeasurableSpace X] {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
+    {D : Driver → X} (hD : Measurable[hS.measurableSpace] D) :
+    (W.restrict {b | S b ≠ ∞}).map (fun b => (D b,shiftAt S b)) =
+      ((W.restrict {b | S b ≠ ∞}).map D).prod W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
+  have hY := measurable_shiftAt hS
+  apply Measure.ext_prod
+  intro A B hA hB
+  rw [Measure.map_apply (hD'.prodMk hY) (hA.prod hB), Measure.prod_prod,
+    Measure.map_apply hD' hA, Measure.restrict_apply (hD' hA)]
+  have he := congrArg (fun μ : Measure Driver => μ B)
+    (map_shiftAt_restrict hW hS (hD hA))
+  rw [Measure.map_apply hY hB, Measure.restrict_apply (hY hB), Measure.smul_apply,
+    smul_eq_mul] at he
+  rw [Measure.restrict_apply ((hD'.prodMk hY) (hA.prod hB))]
+  convert he using 1
+  congr 1
+  ext b
+  simp only [mem_inter_iff, mem_preimage, mem_prod, mem_ofPred_eq]
+  tauto
+
+ 
+
+theorem lintegral_data_shiftAt {X : Type*} [MeasurableSpace X] {W : Measure Driver}
+    (hW : IsBrownianReal (fun t (b : Driver) => b t) W)
+    {S : Driver → ℝ≥0∞} (hS : IsStoppingTime filtration S)
+    {D : Driver → X} (hD : Measurable[hS.measurableSpace] D)
+    {F : X × Driver → ℝ≥0∞} (hF : Measurable F) :
+    (∫⁻b in {b | S b ≠ ∞}, F (D b,shiftAt S b) ∂W) =
+      ∫⁻b in {b | S b ≠ ∞}, ∫⁻c, F (D b,c) ∂W ∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
+  rw [←lintegral_map hF (hD'.prodMk (measurable_shiftAt hS)),map_data_shiftAt hW hS hD,
+    lintegral_prod _ hF.aemeasurable, lintegral_map hF.lintegral_prod_right' hD']
+
+end SLEExactGauge.BrownianRestart
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/BrownianShifts.lean b/lean/OAI/Probability/SLEGauge/BrownianShifts.lean
index 69f90b767fcea9f705a4b9a85c6be55a3823d420..e65e948ac2fde395da1162a0c20f3bbcc91eb79b 100644
--- a/lean/OAI/Probability/SLEGauge/BrownianShifts.lean
+++ b/lean/OAI/Probability/SLEGauge/BrownianShifts.lean
@@ -1,481 +1,481 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ReverseGenerators
-import OAI.Probability.SLEGauge.ReverseMoments
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-noncomputable section
-namespace SLEExactGauge.BrownianRestart
-def shift (t : ℝ≥0) (b : Driver) : Driver :=
-  ⟨⟨fun s => b (t+s)-b t, (map_continuous b).comp (continuous_const.add continuous_id) |>.sub continuous_const⟩,
-    by simp⟩
-
-@[simp] lemma shift_apply (t : ℝ≥0) (b : Driver) (s : ℝ≥0) :
-    shift t b s = b (t+s)-b t := rfl
-
-lemma measurable_shift (t : ℝ≥0) : Measurable (shift t) := by
-  apply code_embedding.measurable_comp_iff.mp
-  exact Measurable.of_eval fun _ =>
-    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
-
-lemma indep_shift_history {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
-    IndepFun (shift t) (fun b : Driver => fun s : Iic t => b s) W := by
-  have hi := (hW.toIsPreBrownianReal.indepFun_shift t).comp
-    (Measurable.of_eval (f := fun path : ℝ≥0 → ℝ => fun time : ℚ => path (Real.toNNReal (time:ℝ)))
-      (fun _ => measurable_pi_apply _)) measurable_id
-  have he : MeasurableSpace.comap (code ∘ shift t) inferInstance =
-      MeasurableSpace.comap (shift t) inferInstance := by
-    rw [← MeasurableSpace.comap_comp,code_embedding.comap_eq]
-  change Indep (MeasurableSpace.comap (code ∘ shift t) inferInstance)
-    (MeasurableSpace.comap (fun b : Driver => fun s : Iic t => b s) inferInstance) W at hi
-  rw [he] at hi
-  exact hi
-
-lemma evaluation_law_unique {W V : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W)
-    (hV : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) V) : W = V := by
-  let ev : Driver → (ℝ≥0 → ℝ) := fun b => b
-  have mev : Measurable ev := Measurable.of_eval fun time => (continuous_eval_const time).measurable
-  have hw : IsProjectiveLimit (W.map ev) BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map I.measurable_restrict mev]
-    exact (hW.hasLaw I).map_eq
-  have hv : IsProjectiveLimit (V.map ev) BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map I.measurable_restrict mev]
-    exact (hV.hasLaw I).map_eq
-  have he := hw.unique hv
-  have mr : Measurable (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ))) :=
-    Measurable.of_eval fun _ => measurable_pi_apply _
-  have hc := congrArg (Measure.map (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ)))) he
-  rw [Measure.map_map mr mev, Measure.map_map mr mev] at hc
-  exact code_embedding.map_injective hc
-
-lemma brownian_shift_map {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
-    IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) (W.map (shift t)) where
-  hasLaw I := by
-    refine ⟨(Measurable.of_eval fun _ => (continuous_eval_const _).measurable).aemeasurable, ?_⟩
-    have mm : Measurable (fun b : Driver => I.restrict (fun s => b s)) :=
-      Measurable.of_eval fun time => (continuous_eval_const (time:ℝ≥0)).measurable
-    rw [Measure.map_map mm (measurable_shift t)]
-    exact ((hW.shift t).hasLaw I).map_eq
-  cont := Filter.Eventually.of_forall fun b => map_continuous b
-
-lemma shift_preserves_law {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
-    W.map (shift t) = W := evaluation_law_unique (brownian_shift_map hW t) hW
-
-end SLEExactGauge.BrownianRestart
-
-namespace SLEExactGauge.BrownianRestart
-
- 
-def past (t : ℝ≥0) (b : Driver) : Driver :=
-  ⟨⟨fun s => b (min t s), (map_continuous b).comp (continuous_const.min continuous_id)⟩,
-    by simp⟩
-
-@[simp] lemma prefix_apply (t : ℝ≥0) (b : Driver) (s : ℝ≥0) :
-    past t b s = b (min t s) := rfl
-
-lemma measurable_prefix (t : ℝ≥0) : Measurable (past t) := by
-  apply code_embedding.measurable_comp_iff.mp
-  exact Measurable.of_eval fun _ => (continuous_eval_const _).measurable
-
- 
-def join (t : ℝ≥0) (b c : Driver) : Driver where
-  toFun s := if s ≤ t then b s else b t + c (s-t)
-  continuous_toFun := by
-    apply Continuous.if_le (map_continuous b)
-      (continuous_const.add ((map_continuous c).comp (continuous_id.sub continuous_const)))
-      continuous_id continuous_const
-    intro s hs
-    change s = t at hs
-    subst s
-    simp
-  map_zero' := by simp
-
-@[simp] lemma join_apply (t : ℝ≥0) (b c : Driver) (s : ℝ≥0) :
-    join t b c s = if s ≤ t then b s else b t + c (s-t) := rfl
-
-lemma measurable_join (t : ℝ≥0) : Measurable (fun q : Driver × Driver => join t q.1 q.2) := by
-  apply code_embedding.measurable_comp_iff.mp
-  apply Measurable.of_eval
-  intro q
-  change Measurable (fun v : Driver × Driver => if Real.toNNReal (q:ℝ) ≤ t then
-    v.1 (Real.toNNReal (q:ℝ)) else v.1 t + v.2 (Real.toNNReal (q:ℝ)-t))
-  split_ifs
-  · exact (continuous_eval_const _).measurable.comp measurable_fst
-  · exact ((continuous_eval_const _).measurable.comp measurable_fst).add
-      ((continuous_eval_const _).measurable.comp measurable_snd)
-
-lemma join_prefix (t : ℝ≥0) (b c : Driver) : join t (past t b) c = join t b c := by
-  ext s
-  simp only [join_apply,prefix_apply,min_self]
-  split_ifs with hs
-  · rw [min_eq_right hs]
-  · rfl
-
-lemma join_reconstruct (t : ℝ≥0) (b : Driver) : join t (past t b) (shift t b) = b := by
-  rw [join_prefix]
-  ext s
-  simp only [join_apply,shift_apply]
-  split_ifs with hs
-  · rfl
-  · rw [add_tsub_cancel_of_le (le_of_not_ge hs)]
-    ring
-
-lemma indep_shift_prefix {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
-    IndepFun (shift t) (past t) W := by
-  have hi := (indep_shift_history hW t).comp measurable_id
-    (Measurable.of_eval (f := fun path : Iic t → ℝ =>
-      fun time : ℚ => path ⟨min t (Real.toNNReal (time:ℝ)),by simpa only [Set.mem_Iic] using (min_le_left t (Real.toNNReal (time:ℝ)))⟩)
-        (fun q : ℚ => measurable_pi_apply (⟨min t (Real.toNNReal (q:ℝ)),by simpa only [Set.mem_Iic] using (min_le_left t (Real.toNNReal (q:ℝ)))⟩ : Iic t)))
-  have he : MeasurableSpace.comap (code ∘ past t) inferInstance =
-      MeasurableSpace.comap (past t) inferInstance := by
-    rw [← MeasurableSpace.comap_comp,code_embedding.comap_eq]
-  change Indep (MeasurableSpace.comap (shift t) inferInstance)
-    (MeasurableSpace.comap (code ∘ past t) inferInstance) W at hi
-  rw [he] at hi
-  exact hi
-
- 
-
-lemma split_map {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
-    W.map (fun b => (past t b,shift t b)) = (W.map (past t)).prod W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  rw [(indep_shift_prefix hW t).symm.map_prod_eq_prod_map_map
-    (measurable_prefix t).aemeasurable (measurable_shift t).aemeasurable,
-    shift_preserves_law hW t]
-
-lemma lintegral_split {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0)
-    {F : Driver × Driver → ℝ≥0∞} (hF : Measurable F) :
-    (∫⁻ b, F (past t b, shift t b) ∂W) =
-      ∫⁻ b, ∫⁻ c, F (past t b,c) ∂W ∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  rw [← lintegral_map hF ((measurable_prefix t).prodMk (measurable_shift t)),split_map hW t]
-  rw [lintegral_prod _ hF.aemeasurable]
-  rw [lintegral_map hF.lintegral_prod_right' (measurable_prefix t)]
-
- 
-
-theorem conditional_join_kernel {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0)
-    {F H : Driver → ℝ≥0∞} (hF : Measurable F) (hH : Measurable H) :
-    (∫⁻ b, H (past t b) * F b ∂W) =
-      ∫⁻ b, H (past t b) * (∫⁻ c, F (join t b c) ∂W) ∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hm := (hH.comp measurable_fst).mul (hF.comp (measurable_join t))
-  have he := lintegral_split hW t hm
-  change (∫⁻ b, H (past t b) * F (join t (past t b) (shift t b)) ∂W) =
-    ∫⁻ b, ∫⁻ c, H (past t b) * F (join t (past t b) c) ∂W ∂W at he
-  simp only [join_reconstruct] at he
-  simp only [join_prefix] at he
-  rw [he]
-  apply lintegral_congr
-  intro b
-  exact lintegral_const_mul _ ((hF.comp (measurable_join t)).comp
-    (measurable_const.prodMk measurable_id))
-
-end SLEExactGauge.BrownianRestart
-
-namespace SLEExactGauge.BrownianRestart
-
- 
-
-def reverse (T : ℝ≥0) (b : Driver) : Driver :=
-  ⟨⟨fun s => b (max T s) - b (T-s),
-    ((map_continuous b).comp (continuous_const.max continuous_id)).sub
-      ((map_continuous b).comp (continuous_const.sub continuous_id))⟩, by simp⟩
-
-@[simp] lemma reverse_apply (T : ℝ≥0) (b : Driver) (s : ℝ≥0) :
-    reverse T b s = b (max T s)-b (T-s) := rfl
-
-lemma measurable_reverse (T : ℝ≥0) : Measurable (reverse T) := by
-  apply code_embedding.measurable_comp_iff.mp
-  exact Measurable.of_eval fun _ =>
-    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
-
-lemma reverse_brownian {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (T : ℝ≥0) :
-    IsBrownianReal (fun s b => reverse T b s) W := by
-  have hB := hW.toIsPreBrownianReal
-  have hG : IsGaussianProcess (fun s b => reverse T b s) W := by
-    apply hB.isGaussianProcess.of_isGaussianProcess
-    intro s
-    exact ⟨{max T s,T-s},
-      { toFun x := x ⟨max T s,by simp⟩ - x ⟨T-s,by simp⟩
-        map_add' x y := by simp; abel
-        map_smul' c x := by simp; ring },by simp⟩
-  refine ⟨hG.isPreBrownianReal_of_covariance ?_ ?_,?_⟩
-  · intro t
-    simp only [reverse_apply]
-    rw [integral_sub (hB.integrable_eval _) (hB.integrable_eval _),
-      hB.integral_eval,hB.integral_eval,sub_self]
-  · intro s t hst
-    let := hG.isProbabilityMeasure
-    simp only [reverse_apply]
-    rw [covariance_fun_sub_left, covariance_fun_sub_right, covariance_fun_sub_right]
-    any_goals exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two
-    any_goals exact hB.isGaussianProcess.hasGaussianLaw_sub.memLp_two
-    simp only [hB.covariance_fun_eval]
-    rw [min_eq_left (max_le_max_left T hst),
-      min_eq_right ((tsub_le_self).trans (le_max_left T s)),
-      min_eq_left ((tsub_le_self).trans (le_max_left T t)),
-      min_eq_right (tsub_le_tsub_left hst T)]
-    by_cases hs : s ≤ T
-    · rw [max_eq_left hs,NNReal.coe_sub hs]
-      ring
-    · rw [max_eq_right (le_of_not_ge hs),tsub_eq_zero_of_le (le_of_not_ge hs)]
-      simp
-  · exact Filter.Eventually.of_forall fun b => map_continuous (reverse T b)
-
-lemma reverse_preserves_law {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (T : ℝ≥0) :
-    W.map (reverse T) = W := by
-  apply evaluation_law_unique _ hW
-  refine ⟨⟨?_⟩, Filter.Eventually.of_forall fun b => map_continuous b⟩
-  intro I
-  refine ⟨(Measurable.of_eval fun _ => (continuous_eval_const _).measurable).aemeasurable,?_⟩
-  have mm : Measurable (fun b : Driver => I.restrict (fun s => b s)) :=
-    Measurable.of_eval fun time => (continuous_eval_const (time:ℝ≥0)).measurable
-  rw [Measure.map_map mm (measurable_reverse T)]
-  exact ((reverse_brownian hW T).hasLaw I).map_eq
-
-lemma reverse_before (T s : ℝ≥0) (hs : s ≤ T) (b : Driver) :
-    reverse T b s = b T - b (T-s) := by simp [max_eq_left hs]
-
-lemma reverse_after (T s : ℝ≥0) (hs : T ≤ s) (b : Driver) :
-    reverse T b s = b s := by simp [max_eq_right hs,tsub_eq_zero_of_le hs]
-
-end SLEExactGauge.BrownianRestart
-
-namespace SLEExactGauge.BrownianRestart
-
-def backward (T : ℝ≥0) (b : Driver) : Driver := -reverse T b
-
-@[simp] lemma backward_apply (T : ℝ≥0) (b : Driver) (s : ℝ≥0) :
-    backward T b s = -(b (max T s)-b (T-s)) := rfl
-
-lemma measurable_backward (T : ℝ≥0) : Measurable (backward T) := by
-  apply code_embedding.measurable_comp_iff.mp
-  exact Measurable.of_eval fun _ =>
-    ((continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable).neg
-
-lemma backward_preserves_law {W : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (T : ℝ≥0) :
-    W.map (backward T) = W := by
-  apply evaluation_law_unique _ hW
-  refine ⟨⟨?_⟩, Filter.Eventually.of_forall fun b => map_continuous b⟩
-  intro I
-  refine ⟨(Measurable.of_eval fun _ => (continuous_eval_const _).measurable).aemeasurable,?_⟩
-  have mm : Measurable (fun b : Driver => I.restrict (fun s => b s)) :=
-    Measurable.of_eval fun time => (continuous_eval_const (time:ℝ≥0)).measurable
-  rw [Measure.map_map mm (measurable_backward T)]
-  exact ((reverse_brownian hW T).neg.hasLaw I).map_eq
-
-lemma backward_before {T s : ℝ} (hs : s ∈ Icc 0 T) (b : Driver) :
-    backward T.toNNReal b s.toNNReal = b (T-s).toNNReal-b T.toNNReal := by
-  have hT : 0 ≤ T := hs.1.trans hs.2
-  have hst : s.toNNReal ≤ T.toNNReal := Real.toNNReal_mono hs.2
-  have hsub : T.toNNReal-s.toNNReal = (T-s).toNNReal := by
-    apply NNReal.eq
-    simp only [NNReal.coe_sub hst,Real.coe_toNNReal',max_eq_left hT,max_eq_left hs.1,
-      max_eq_left (sub_nonneg.mpr hs.2)]
-  simp only [backward_apply,max_eq_left hst,hsub]
-  ring
-end SLEExactGauge.BrownianRestart
-
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative
-
-def reversedState (κ y T : ℝ) (b : Driver) : ℝ → ReverseDerivative.State :=
-  ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y
-    (BrownianRestart.backward T.toNNReal b)
-
-def reversedCurve (κ y T : ℝ) (b : Driver) (s : ℝ) : ℂ :=
-  (reversedState κ y T b s).2.1
-
-lemma reverse_reparam_solution {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
-    (b : Driver) : PositiveSolution κ b
-      (reversedCurve κ y T b T+driverValue b T) T
-      (fun t => reversedCurve κ y T b (T-t)+driverValue b T) := by
-  let β := BrownianRestart.backward T.toNNReal b
-  let g := reversedState κ y T b
-  let G := reversedCurve κ y T b
-  have hq : 0 < y/4 := by positivity
-  have ha : 0 ≤ a κ := (div_pos (by norm_num) hκ).le
-  have hg := ReverseDerivative.reverseFlow_solution hq (a κ) (T+y^2) y β β.continuous hT
-  have hga := ReverseDerivative.reverseFlow_actual hq ha (by linarith : y/4 ≤ y)
-    (T+y^2) β β.continuous hT
-  have hgc : ContinuousOn G (Icc 0 T) := fun t ht => (hg.2 t ht).continuousWithinAt.snd.fst
-  have hheight (t : ℝ) (ht : t ∈ Icc 0 T) : y ≤ (G t).im :=
-    ReverseDerivative.reverseFlow_height hq ha hy.le (T+y^2) β β.continuous hT t ht
-  have hrev (t : ℝ) (ht : t ∈ Icc 0 T) : T-t ∈ Icc 0 T := by constructor <;> linarith [ht.1,ht.2]
-  refine ⟨hT,(hgc.comp (continuous_const.sub continuous_id).continuousOn hrev).add continuousOn_const,?_,?_,?_⟩
-  · simp
-  · intro t ht
-    simpa using hy.trans_le (hheight (T-t) (hrev t ht))
-  · intro t ht
-    let projG : ReverseDerivative.State →L[ℝ] ℂ :=
-      (ContinuousLinearMap.fst ℝ ℂ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℂ × ℝ))
-    have hd := projG.hasFDerivAt.comp_hasDerivWithinAt (T-t) (hga (T-t) (hrev t ht))
-    change HasDerivWithinAt G (-(a κ:ℂ)/(G (T-t)-β (T-t).toNNReal)) (Icc 0 T) (T-t) at hd
-    have hβ : β (T-t).toNNReal = driverValue b t-driverValue b T := by
-      have hh := BrownianRestart.backward_before (hrev t ht) b
-      simpa only [sub_sub_cancel,driverValue] using hh
-    have hden : G (T-t)-(β (T-t).toNNReal:ℂ) =
-        G (T-t)+(driverValue b T:ℂ)-driverValue b t := by rw [hβ,Complex.ofReal_sub]; ring
-    rw [hden] at hd
-    have hh := (hd.scomp t (((hasDerivAt_const t T).sub (hasDerivAt_id t)).hasDerivWithinAt) hrev).add_const
-      (driverValue b T:ℂ)
-    convert! hh using 1; first | rfl | (simp only [one_smul,zero_sub,neg_smul,neg_div,neg_neg]; rfl)
-
-lemma inverseFlow_eq_reversedCurve {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
-    (b : Driver) : inverseFlow κ b T (driverValue b T+y*Complex.I) =
-      reversedCurve κ y T b T+driverValue b T := by
-  have hf := reverse_reparam_solution hκ hy hT b
-  have he := hf.flow_eq
-  have hq : 0 < y/4 := by positivity
-  have hg := ReverseDerivative.reverseFlow_solution hq (a κ) (T+y^2) y
-    (BrownianRestart.backward T.toNNReal b) (BrownianRestart.backward T.toNNReal b).continuous hT
-  have hG0 : reversedCurve κ y T b 0 = (y:ℂ)*Complex.I := by
-    change (ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y _ 0).2.1 = _
-    rw [hg.1]
-    apply Complex.ext <;> simp [ReverseDerivative.initial]
-  rw [sub_self,hG0] at he
-  have he' : flow κ b (reversedCurve κ y T b T+driverValue b T) T =
-      driverValue b T+y*Complex.I := he.trans (add_comm _ _)
-  rw [← he',inverseFlow_flow hT hf.surviving]
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.ReverseDerivative
-open MeasureTheory BrownianCalculus
-
-lemma reverseFlow_log_integral {a q y T : ℝ} (hq : 0 < q) (ha : 0 ≤ a)
-    (hy : q ≤ y) (hT : 0 ≤ T) (U : ℝ) (b : Driver) :
-    (reverseFlow a q U y b T).2.2 =
-      ∫ s in (0:ℝ)..T, a*((((reverseFlow a q U y b s).2.1-(b s.toNNReal:ℂ))⁻¹)^2).re := by
-  let g := reverseFlow a q U y b
-  have hg := reverseFlow_solution hq a U y b b.continuous hT
-  have hd := reverseFlow_actual hq ha hy U b b.continuous hT
-  have hc : ContinuousOn g (Icc 0 T) := fun t ht => (hg.2 t ht).continuousWithinAt
-  have hz (t : ℝ) (ht : t ∈ Icc 0 T) : (g t).2.1-(b t.toNNReal:ℂ) ≠ 0 := by
-    have hh := reverseFlow_height hq ha (hq.le.trans hy) U b b.continuous hT t ht
-    intro he
-    have hi := congrArg Complex.im he
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-    change (reverseFlow a q U y b t).2.1.im = 0 at hi
-    linarith
-  have hm : ContinuousOn (fun t => (g t).2.1-(b t.toNNReal:ℂ)) (Icc 0 T) :=
-    hc.snd.fst.sub ((Complex.continuous_ofReal.comp (b.continuous.comp continuous_real_toNNReal)).continuousOn)
-  have hr : ContinuousOn (fun t => a*((((g t).2.1-(b t.toNNReal:ℂ))⁻¹)^2).re) (Icc 0 T) := by
-    exact continuousOn_const.mul (Complex.continuous_re.comp_continuousOn ((hm.inv₀ hz).pow 2))
-  let projL : State →L[ℝ] ℝ :=
-    (ContinuousLinearMap.snd ℝ ℂ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℂ × ℝ))
-  have hL (s : ℝ) (hs : s ∈ Icc 0 T) : HasDerivWithinAt (fun t => (g t).2.2)
-      (a*((((g s).2.1-(b s.toNNReal:ℂ))⁻¹)^2).re) (Icc 0 T) s :=
-    projL.hasFDerivAt.comp_hasDerivWithinAt s (hd s hs)
-  have he := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le hT hc.snd.snd
-    (fun s hs => ((hL s ⟨hs.1.le,hs.2.le⟩).hasDerivAt (Icc_mem_nhds hs.1 hs.2)).hasDerivWithinAt)
-    (hr.intervalIntegrable_of_Icc hT)
-  change (∫ s in (0:ℝ)..T, a*((((g s).2.1-(b s.toNNReal:ℂ))⁻¹)^2).re) =
-    (reverseFlow a q U y b T).2.2-(reverseFlow a q U y b 0).2.2 at he
-  rw [hg.1] at he
-  simpa only [initial,sub_zero] using he.symm
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative
-
-def reverseCoeff (κ y T : ℝ) (b : Driver) (s : ℝ) : ℂ :=
-  (a κ:ℂ)*((reversedCurve κ y T b s-
-    (BrownianRestart.backward T.toNNReal b s.toNNReal:ℂ))⁻¹)^2
-
-lemma continuousOn_reverseCoeff {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
-    (b : Driver) : ContinuousOn (reverseCoeff κ y T b) (Icc 0 T) := by
-  let β := BrownianRestart.backward T.toNNReal b
-  have hq : 0 < y/4 := by positivity
-  have ha : 0 ≤ a κ := (div_pos (by norm_num) hκ).le
-  have hg := ReverseDerivative.reverseFlow_solution hq (a κ) (T+y^2) y β β.continuous hT
-  have hc : ContinuousOn (reversedCurve κ y T b) (Icc 0 T) := fun t ht => (hg.2 t ht).continuousWithinAt.snd.fst
-  have hz (t : ℝ) (ht : t ∈ Icc 0 T) : reversedCurve κ y T b t-(β t.toNNReal:ℂ) ≠ 0 := by
-    have hh := ReverseDerivative.reverseFlow_height hq ha hy.le (T+y^2) β β.continuous hT t ht
-    intro he
-    have hi := congrArg Complex.im he
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-    change (ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y β t).2.1.im = 0 at hi
-    linarith
-  exact continuousOn_const.mul
-    (((hc.sub ((Complex.continuous_ofReal.comp (β.continuous.comp continuous_real_toNNReal)).continuousOn)).inv₀ hz).pow 2)
-
-lemma reversed_log_eq_integral {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
-    (b : Driver) : (reversedState κ y T b T).2.2 =
-      (∫ s in (0:ℝ)..T, reverseCoeff κ y T b s).re := by
-  have hq : 0 < y/4 := by positivity
-  have hh := ReverseDerivative.reverseFlow_log_integral hq
-    (by exact (div_pos (by norm_num) hκ).le : 0 ≤ a κ) (by linarith : y/4 ≤ y)
-    hT (T+y^2) (BrownianRestart.backward T.toNNReal b)
-  change (reversedState κ y T b T).2.2 = Complex.reCLM (∫ s in (0:ℝ)..T, reverseCoeff κ y T b s)
-  rw [← Complex.reCLM.intervalIntegral_comp_comm ((continuousOn_reverseCoeff hκ hy hT b).intervalIntegrable_of_Icc hT)]
-  simpa only [Complex.reCLM_apply,reverseCoeff,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,
-    reversedState,reversedCurve] using hh
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ReverseGenerators
+import OAI.Probability.SLEGauge.ReverseMoments
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+noncomputable section
+namespace SLEExactGauge.BrownianRestart
+def shift (t : ℝ≥0) (b : Driver) : Driver :=
+  ⟨⟨fun s => b (t+s)-b t, (map_continuous b).comp (continuous_const.add continuous_id) |>.sub continuous_const⟩,
+    by simp⟩
+
+@[simp] lemma shift_apply (t : ℝ≥0) (b : Driver) (s : ℝ≥0) :
+    shift t b s = b (t+s)-b t := rfl
+
+lemma measurable_shift (t : ℝ≥0) : Measurable (shift t) := by
+  apply code_embedding.measurable_comp_iff.mp
+  exact Measurable.of_eval fun _ =>
+    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
+
+lemma indep_shift_history {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
+    IndepFun (shift t) (fun b : Driver => fun s : Iic t => b s) W := by
+  have hi := (hW.toIsPreBrownianReal.indepFun_shift t).comp
+    (Measurable.of_eval (f := fun path : ℝ≥0 → ℝ => fun time : ℚ => path (Real.toNNReal (time:ℝ)))
+      (fun _ => measurable_pi_apply _)) measurable_id
+  have he : MeasurableSpace.comap (code ∘ shift t) inferInstance =
+      MeasurableSpace.comap (shift t) inferInstance := by
+    rw [← MeasurableSpace.comap_comp,code_embedding.comap_eq]
+  change Indep (MeasurableSpace.comap (code ∘ shift t) inferInstance)
+    (MeasurableSpace.comap (fun b : Driver => fun s : Iic t => b s) inferInstance) W at hi
+  rw [he] at hi
+  exact hi
+
+lemma evaluation_law_unique {W V : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W)
+    (hV : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) V) : W = V := by
+  let ev : Driver → (ℝ≥0 → ℝ) := fun b => b
+  have mev : Measurable ev := Measurable.of_eval fun time => (continuous_eval_const time).measurable
+  have hw : IsProjectiveLimit (W.map ev) BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map I.measurable_restrict mev]
+    exact (hW.hasLaw I).map_eq
+  have hv : IsProjectiveLimit (V.map ev) BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map I.measurable_restrict mev]
+    exact (hV.hasLaw I).map_eq
+  have he := hw.unique hv
+  have mr : Measurable (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ))) :=
+    Measurable.of_eval fun _ => measurable_pi_apply _
+  have hc := congrArg (Measure.map (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ)))) he
+  rw [Measure.map_map mr mev, Measure.map_map mr mev] at hc
+  exact code_embedding.map_injective hc
+
+lemma brownian_shift_map {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
+    IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) (W.map (shift t)) where
+  hasLaw I := by
+    refine ⟨(Measurable.of_eval fun _ => (continuous_eval_const _).measurable).aemeasurable, ?_⟩
+    have mm : Measurable (fun b : Driver => I.restrict (fun s => b s)) :=
+      Measurable.of_eval fun time => (continuous_eval_const (time:ℝ≥0)).measurable
+    rw [Measure.map_map mm (measurable_shift t)]
+    exact ((hW.shift t).hasLaw I).map_eq
+  cont := Filter.Eventually.of_forall fun b => map_continuous b
+
+lemma shift_preserves_law {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
+    W.map (shift t) = W := evaluation_law_unique (brownian_shift_map hW t) hW
+
+end SLEExactGauge.BrownianRestart
+
+namespace SLEExactGauge.BrownianRestart
+
+ 
+def past (t : ℝ≥0) (b : Driver) : Driver :=
+  ⟨⟨fun s => b (min t s), (map_continuous b).comp (continuous_const.min continuous_id)⟩,
+    by simp⟩
+
+@[simp] lemma prefix_apply (t : ℝ≥0) (b : Driver) (s : ℝ≥0) :
+    past t b s = b (min t s) := rfl
+
+lemma measurable_prefix (t : ℝ≥0) : Measurable (past t) := by
+  apply code_embedding.measurable_comp_iff.mp
+  exact Measurable.of_eval fun _ => (continuous_eval_const _).measurable
+
+ 
+def join (t : ℝ≥0) (b c : Driver) : Driver where
+  toFun s := if s ≤ t then b s else b t + c (s-t)
+  continuous_toFun := by
+    apply Continuous.if_le (map_continuous b)
+      (continuous_const.add ((map_continuous c).comp (continuous_id.sub continuous_const)))
+      continuous_id continuous_const
+    intro s hs
+    change s = t at hs
+    subst s
+    simp
+  map_zero' := by simp
+
+@[simp] lemma join_apply (t : ℝ≥0) (b c : Driver) (s : ℝ≥0) :
+    join t b c s = if s ≤ t then b s else b t + c (s-t) := rfl
+
+lemma measurable_join (t : ℝ≥0) : Measurable (fun q : Driver × Driver => join t q.1 q.2) := by
+  apply code_embedding.measurable_comp_iff.mp
+  apply Measurable.of_eval
+  intro q
+  change Measurable (fun v : Driver × Driver => if Real.toNNReal (q:ℝ) ≤ t then
+    v.1 (Real.toNNReal (q:ℝ)) else v.1 t + v.2 (Real.toNNReal (q:ℝ)-t))
+  split_ifs
+  · exact (continuous_eval_const _).measurable.comp measurable_fst
+  · exact ((continuous_eval_const _).measurable.comp measurable_fst).add
+      ((continuous_eval_const _).measurable.comp measurable_snd)
+
+lemma join_prefix (t : ℝ≥0) (b c : Driver) : join t (past t b) c = join t b c := by
+  ext s
+  simp only [join_apply,prefix_apply,min_self]
+  split_ifs with hs
+  · rw [min_eq_right hs]
+  · rfl
+
+lemma join_reconstruct (t : ℝ≥0) (b : Driver) : join t (past t b) (shift t b) = b := by
+  rw [join_prefix]
+  ext s
+  simp only [join_apply,shift_apply]
+  split_ifs with hs
+  · rfl
+  · rw [add_tsub_cancel_of_le (le_of_not_ge hs)]
+    ring
+
+lemma indep_shift_prefix {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
+    IndepFun (shift t) (past t) W := by
+  have hi := (indep_shift_history hW t).comp measurable_id
+    (Measurable.of_eval (f := fun path : Iic t → ℝ =>
+      fun time : ℚ => path ⟨min t (Real.toNNReal (time:ℝ)),by simpa only [Set.mem_Iic] using (min_le_left t (Real.toNNReal (time:ℝ)))⟩)
+        (fun q : ℚ => measurable_pi_apply (⟨min t (Real.toNNReal (q:ℝ)),by simpa only [Set.mem_Iic] using (min_le_left t (Real.toNNReal (q:ℝ)))⟩ : Iic t)))
+  have he : MeasurableSpace.comap (code ∘ past t) inferInstance =
+      MeasurableSpace.comap (past t) inferInstance := by
+    rw [← MeasurableSpace.comap_comp,code_embedding.comap_eq]
+  change Indep (MeasurableSpace.comap (shift t) inferInstance)
+    (MeasurableSpace.comap (code ∘ past t) inferInstance) W at hi
+  rw [he] at hi
+  exact hi
+
+ 
+
+lemma split_map {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0) :
+    W.map (fun b => (past t b,shift t b)) = (W.map (past t)).prod W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  rw [(indep_shift_prefix hW t).symm.map_prod_eq_prod_map_map
+    (measurable_prefix t).aemeasurable (measurable_shift t).aemeasurable,
+    shift_preserves_law hW t]
+
+lemma lintegral_split {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0)
+    {F : Driver × Driver → ℝ≥0∞} (hF : Measurable F) :
+    (∫⁻ b, F (past t b, shift t b) ∂W) =
+      ∫⁻ b, ∫⁻ c, F (past t b,c) ∂W ∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  rw [← lintegral_map hF ((measurable_prefix t).prodMk (measurable_shift t)),split_map hW t]
+  rw [lintegral_prod _ hF.aemeasurable]
+  rw [lintegral_map hF.lintegral_prod_right' (measurable_prefix t)]
+
+ 
+
+theorem conditional_join_kernel {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (t : ℝ≥0)
+    {F H : Driver → ℝ≥0∞} (hF : Measurable F) (hH : Measurable H) :
+    (∫⁻ b, H (past t b) * F b ∂W) =
+      ∫⁻ b, H (past t b) * (∫⁻ c, F (join t b c) ∂W) ∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hm := (hH.comp measurable_fst).mul (hF.comp (measurable_join t))
+  have he := lintegral_split hW t hm
+  change (∫⁻ b, H (past t b) * F (join t (past t b) (shift t b)) ∂W) =
+    ∫⁻ b, ∫⁻ c, H (past t b) * F (join t (past t b) c) ∂W ∂W at he
+  simp only [join_reconstruct] at he
+  simp only [join_prefix] at he
+  rw [he]
+  apply lintegral_congr
+  intro b
+  exact lintegral_const_mul _ ((hF.comp (measurable_join t)).comp
+    (measurable_const.prodMk measurable_id))
+
+end SLEExactGauge.BrownianRestart
+
+namespace SLEExactGauge.BrownianRestart
+
+ 
+
+def reverse (T : ℝ≥0) (b : Driver) : Driver :=
+  ⟨⟨fun s => b (max T s) - b (T-s),
+    ((map_continuous b).comp (continuous_const.max continuous_id)).sub
+      ((map_continuous b).comp (continuous_const.sub continuous_id))⟩, by simp⟩
+
+@[simp] lemma reverse_apply (T : ℝ≥0) (b : Driver) (s : ℝ≥0) :
+    reverse T b s = b (max T s)-b (T-s) := rfl
+
+lemma measurable_reverse (T : ℝ≥0) : Measurable (reverse T) := by
+  apply code_embedding.measurable_comp_iff.mp
+  exact Measurable.of_eval fun _ =>
+    (continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable
+
+lemma reverse_brownian {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (T : ℝ≥0) :
+    IsBrownianReal (fun s b => reverse T b s) W := by
+  have hB := hW.toIsPreBrownianReal
+  have hG : IsGaussianProcess (fun s b => reverse T b s) W := by
+    apply hB.isGaussianProcess.of_isGaussianProcess
+    intro s
+    exact ⟨{max T s,T-s},
+      { toFun x := x ⟨max T s,by simp⟩ - x ⟨T-s,by simp⟩
+        map_add' x y := by simp; abel
+        map_smul' c x := by simp; ring },by simp⟩
+  refine ⟨hG.isPreBrownianReal_of_covariance ?_ ?_,?_⟩
+  · intro t
+    simp only [reverse_apply]
+    rw [integral_sub (hB.integrable_eval _) (hB.integrable_eval _),
+      hB.integral_eval,hB.integral_eval,sub_self]
+  · intro s t hst
+    let := hG.isProbabilityMeasure
+    simp only [reverse_apply]
+    rw [covariance_fun_sub_left, covariance_fun_sub_right, covariance_fun_sub_right]
+    any_goals exact (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two
+    any_goals exact hB.isGaussianProcess.hasGaussianLaw_sub.memLp_two
+    simp only [hB.covariance_fun_eval]
+    rw [min_eq_left (max_le_max_left T hst),
+      min_eq_right ((tsub_le_self).trans (le_max_left T s)),
+      min_eq_left ((tsub_le_self).trans (le_max_left T t)),
+      min_eq_right (tsub_le_tsub_left hst T)]
+    by_cases hs : s ≤ T
+    · rw [max_eq_left hs,NNReal.coe_sub hs]
+      ring
+    · rw [max_eq_right (le_of_not_ge hs),tsub_eq_zero_of_le (le_of_not_ge hs)]
+      simp
+  · exact Filter.Eventually.of_forall fun b => map_continuous (reverse T b)
+
+lemma reverse_preserves_law {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (T : ℝ≥0) :
+    W.map (reverse T) = W := by
+  apply evaluation_law_unique _ hW
+  refine ⟨⟨?_⟩, Filter.Eventually.of_forall fun b => map_continuous b⟩
+  intro I
+  refine ⟨(Measurable.of_eval fun _ => (continuous_eval_const _).measurable).aemeasurable,?_⟩
+  have mm : Measurable (fun b : Driver => I.restrict (fun s => b s)) :=
+    Measurable.of_eval fun time => (continuous_eval_const (time:ℝ≥0)).measurable
+  rw [Measure.map_map mm (measurable_reverse T)]
+  exact ((reverse_brownian hW T).hasLaw I).map_eq
+
+lemma reverse_before (T s : ℝ≥0) (hs : s ≤ T) (b : Driver) :
+    reverse T b s = b T - b (T-s) := by simp [max_eq_left hs]
+
+lemma reverse_after (T s : ℝ≥0) (hs : T ≤ s) (b : Driver) :
+    reverse T b s = b s := by simp [max_eq_right hs,tsub_eq_zero_of_le hs]
+
+end SLEExactGauge.BrownianRestart
+
+namespace SLEExactGauge.BrownianRestart
+
+def backward (T : ℝ≥0) (b : Driver) : Driver := -reverse T b
+
+@[simp] lemma backward_apply (T : ℝ≥0) (b : Driver) (s : ℝ≥0) :
+    backward T b s = -(b (max T s)-b (T-s)) := rfl
+
+lemma measurable_backward (T : ℝ≥0) : Measurable (backward T) := by
+  apply code_embedding.measurable_comp_iff.mp
+  exact Measurable.of_eval fun _ =>
+    ((continuous_eval_const _).measurable.sub (continuous_eval_const _).measurable).neg
+
+lemma backward_preserves_law {W : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W) (T : ℝ≥0) :
+    W.map (backward T) = W := by
+  apply evaluation_law_unique _ hW
+  refine ⟨⟨?_⟩, Filter.Eventually.of_forall fun b => map_continuous b⟩
+  intro I
+  refine ⟨(Measurable.of_eval fun _ => (continuous_eval_const _).measurable).aemeasurable,?_⟩
+  have mm : Measurable (fun b : Driver => I.restrict (fun s => b s)) :=
+    Measurable.of_eval fun time => (continuous_eval_const (time:ℝ≥0)).measurable
+  rw [Measure.map_map mm (measurable_backward T)]
+  exact ((reverse_brownian hW T).neg.hasLaw I).map_eq
+
+lemma backward_before {T s : ℝ} (hs : s ∈ Icc 0 T) (b : Driver) :
+    backward T.toNNReal b s.toNNReal = b (T-s).toNNReal-b T.toNNReal := by
+  have hT : 0 ≤ T := hs.1.trans hs.2
+  have hst : s.toNNReal ≤ T.toNNReal := Real.toNNReal_mono hs.2
+  have hsub : T.toNNReal-s.toNNReal = (T-s).toNNReal := by
+    apply NNReal.eq
+    simp only [NNReal.coe_sub hst,Real.coe_toNNReal',max_eq_left hT,max_eq_left hs.1,
+      max_eq_left (sub_nonneg.mpr hs.2)]
+  simp only [backward_apply,max_eq_left hst,hsub]
+  ring
+end SLEExactGauge.BrownianRestart
+
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative
+
+def reversedState (κ y T : ℝ) (b : Driver) : ℝ → ReverseDerivative.State :=
+  ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y
+    (BrownianRestart.backward T.toNNReal b)
+
+def reversedCurve (κ y T : ℝ) (b : Driver) (s : ℝ) : ℂ :=
+  (reversedState κ y T b s).2.1
+
+lemma reverse_reparam_solution {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
+    (b : Driver) : PositiveSolution κ b
+      (reversedCurve κ y T b T+driverValue b T) T
+      (fun t => reversedCurve κ y T b (T-t)+driverValue b T) := by
+  let β := BrownianRestart.backward T.toNNReal b
+  let g := reversedState κ y T b
+  let G := reversedCurve κ y T b
+  have hq : 0 < y/4 := by positivity
+  have ha : 0 ≤ a κ := (div_pos (by norm_num) hκ).le
+  have hg := ReverseDerivative.reverseFlow_solution hq (a κ) (T+y^2) y β β.continuous hT
+  have hga := ReverseDerivative.reverseFlow_actual hq ha (by linarith : y/4 ≤ y)
+    (T+y^2) β β.continuous hT
+  have hgc : ContinuousOn G (Icc 0 T) := fun t ht => (hg.2 t ht).continuousWithinAt.snd.fst
+  have hheight (t : ℝ) (ht : t ∈ Icc 0 T) : y ≤ (G t).im :=
+    ReverseDerivative.reverseFlow_height hq ha hy.le (T+y^2) β β.continuous hT t ht
+  have hrev (t : ℝ) (ht : t ∈ Icc 0 T) : T-t ∈ Icc 0 T := by constructor <;> linarith [ht.1,ht.2]
+  refine ⟨hT,(hgc.comp (continuous_const.sub continuous_id).continuousOn hrev).add continuousOn_const,?_,?_,?_⟩
+  · simp
+  · intro t ht
+    simpa using hy.trans_le (hheight (T-t) (hrev t ht))
+  · intro t ht
+    let projG : ReverseDerivative.State →L[ℝ] ℂ :=
+      (ContinuousLinearMap.fst ℝ ℂ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℂ × ℝ))
+    have hd := projG.hasFDerivAt.comp_hasDerivWithinAt (T-t) (hga (T-t) (hrev t ht))
+    change HasDerivWithinAt G (-(a κ:ℂ)/(G (T-t)-β (T-t).toNNReal)) (Icc 0 T) (T-t) at hd
+    have hβ : β (T-t).toNNReal = driverValue b t-driverValue b T := by
+      have hh := BrownianRestart.backward_before (hrev t ht) b
+      simpa only [sub_sub_cancel,driverValue] using hh
+    have hden : G (T-t)-(β (T-t).toNNReal:ℂ) =
+        G (T-t)+(driverValue b T:ℂ)-driverValue b t := by rw [hβ,Complex.ofReal_sub]; ring
+    rw [hden] at hd
+    have hh := (hd.scomp t (((hasDerivAt_const t T).sub (hasDerivAt_id t)).hasDerivWithinAt) hrev).add_const
+      (driverValue b T:ℂ)
+    convert! hh using 1; first | rfl | (simp only [one_smul,zero_sub,neg_smul,neg_div,neg_neg]; rfl)
+
+lemma inverseFlow_eq_reversedCurve {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
+    (b : Driver) : inverseFlow κ b T (driverValue b T+y*Complex.I) =
+      reversedCurve κ y T b T+driverValue b T := by
+  have hf := reverse_reparam_solution hκ hy hT b
+  have he := hf.flow_eq
+  have hq : 0 < y/4 := by positivity
+  have hg := ReverseDerivative.reverseFlow_solution hq (a κ) (T+y^2) y
+    (BrownianRestart.backward T.toNNReal b) (BrownianRestart.backward T.toNNReal b).continuous hT
+  have hG0 : reversedCurve κ y T b 0 = (y:ℂ)*Complex.I := by
+    change (ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y _ 0).2.1 = _
+    rw [hg.1]
+    apply Complex.ext <;> simp [ReverseDerivative.initial]
+  rw [sub_self,hG0] at he
+  have he' : flow κ b (reversedCurve κ y T b T+driverValue b T) T =
+      driverValue b T+y*Complex.I := he.trans (add_comm _ _)
+  rw [← he',inverseFlow_flow hT hf.surviving]
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.ReverseDerivative
+open MeasureTheory BrownianCalculus
+
+lemma reverseFlow_log_integral {a q y T : ℝ} (hq : 0 < q) (ha : 0 ≤ a)
+    (hy : q ≤ y) (hT : 0 ≤ T) (U : ℝ) (b : Driver) :
+    (reverseFlow a q U y b T).2.2 =
+      ∫ s in (0:ℝ)..T, a*((((reverseFlow a q U y b s).2.1-(b s.toNNReal:ℂ))⁻¹)^2).re := by
+  let g := reverseFlow a q U y b
+  have hg := reverseFlow_solution hq a U y b b.continuous hT
+  have hd := reverseFlow_actual hq ha hy U b b.continuous hT
+  have hc : ContinuousOn g (Icc 0 T) := fun t ht => (hg.2 t ht).continuousWithinAt
+  have hz (t : ℝ) (ht : t ∈ Icc 0 T) : (g t).2.1-(b t.toNNReal:ℂ) ≠ 0 := by
+    have hh := reverseFlow_height hq ha (hq.le.trans hy) U b b.continuous hT t ht
+    intro he
+    have hi := congrArg Complex.im he
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+    change (reverseFlow a q U y b t).2.1.im = 0 at hi
+    linarith
+  have hm : ContinuousOn (fun t => (g t).2.1-(b t.toNNReal:ℂ)) (Icc 0 T) :=
+    hc.snd.fst.sub ((Complex.continuous_ofReal.comp (b.continuous.comp continuous_real_toNNReal)).continuousOn)
+  have hr : ContinuousOn (fun t => a*((((g t).2.1-(b t.toNNReal:ℂ))⁻¹)^2).re) (Icc 0 T) := by
+    exact continuousOn_const.mul (Complex.continuous_re.comp_continuousOn ((hm.inv₀ hz).pow 2))
+  let projL : State →L[ℝ] ℝ :=
+    (ContinuousLinearMap.snd ℝ ℂ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℂ × ℝ))
+  have hL (s : ℝ) (hs : s ∈ Icc 0 T) : HasDerivWithinAt (fun t => (g t).2.2)
+      (a*((((g s).2.1-(b s.toNNReal:ℂ))⁻¹)^2).re) (Icc 0 T) s :=
+    projL.hasFDerivAt.comp_hasDerivWithinAt s (hd s hs)
+  have he := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le hT hc.snd.snd
+    (fun s hs => ((hL s ⟨hs.1.le,hs.2.le⟩).hasDerivAt (Icc_mem_nhds hs.1 hs.2)).hasDerivWithinAt)
+    (hr.intervalIntegrable_of_Icc hT)
+  change (∫ s in (0:ℝ)..T, a*((((g s).2.1-(b s.toNNReal:ℂ))⁻¹)^2).re) =
+    (reverseFlow a q U y b T).2.2-(reverseFlow a q U y b 0).2.2 at he
+  rw [hg.1] at he
+  simpa only [initial,sub_zero] using he.symm
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative
+
+def reverseCoeff (κ y T : ℝ) (b : Driver) (s : ℝ) : ℂ :=
+  (a κ:ℂ)*((reversedCurve κ y T b s-
+    (BrownianRestart.backward T.toNNReal b s.toNNReal:ℂ))⁻¹)^2
+
+lemma continuousOn_reverseCoeff {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
+    (b : Driver) : ContinuousOn (reverseCoeff κ y T b) (Icc 0 T) := by
+  let β := BrownianRestart.backward T.toNNReal b
+  have hq : 0 < y/4 := by positivity
+  have ha : 0 ≤ a κ := (div_pos (by norm_num) hκ).le
+  have hg := ReverseDerivative.reverseFlow_solution hq (a κ) (T+y^2) y β β.continuous hT
+  have hc : ContinuousOn (reversedCurve κ y T b) (Icc 0 T) := fun t ht => (hg.2 t ht).continuousWithinAt.snd.fst
+  have hz (t : ℝ) (ht : t ∈ Icc 0 T) : reversedCurve κ y T b t-(β t.toNNReal:ℂ) ≠ 0 := by
+    have hh := ReverseDerivative.reverseFlow_height hq ha hy.le (T+y^2) β β.continuous hT t ht
+    intro he
+    have hi := congrArg Complex.im he
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+    change (ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y β t).2.1.im = 0 at hi
+    linarith
+  exact continuousOn_const.mul
+    (((hc.sub ((Complex.continuous_ofReal.comp (β.continuous.comp continuous_real_toNNReal)).continuousOn)).inv₀ hz).pow 2)
+
+lemma reversed_log_eq_integral {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
+    (b : Driver) : (reversedState κ y T b T).2.2 =
+      (∫ s in (0:ℝ)..T, reverseCoeff κ y T b s).re := by
+  have hq : 0 < y/4 := by positivity
+  have hh := ReverseDerivative.reverseFlow_log_integral hq
+    (by exact (div_pos (by norm_num) hκ).le : 0 ≤ a κ) (by linarith : y/4 ≤ y)
+    hT (T+y^2) (BrownianRestart.backward T.toNNReal b)
+  change (reversedState κ y T b T).2.2 = Complex.reCLM (∫ s in (0:ℝ)..T, reverseCoeff κ y T b s)
+  rw [← Complex.reCLM.intervalIntegral_comp_comm ((continuousOn_reverseCoeff hκ hy hT b).intervalIntegrable_of_Icc hT)]
+  simpa only [Complex.reCLM_apply,reverseCoeff,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,
+    reversedState,reversedCurve] using hh
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CandidateCover.lean b/lean/OAI/Probability/SLEGauge/CandidateCover.lean
index 2d758661fde9d1ced091f4482e94402afa4eb377..5f456308b184f377b61504cc1766b0c1f1a3d864 100644
--- a/lean/OAI/Probability/SLEGauge/CandidateCover.lean
+++ b/lean/OAI/Probability/SLEGauge/CandidateCover.lean
@@ -1,550 +1,550 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.OptionalPastPalm
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped ENNReal NNReal Topology Classical
-namespace SLEExactGauge.CandidateCover
-open Deterministic
-
-lemma closedBall_subset_triple_of_inter {X : Type*} [PseudoMetricSpace X]
-    {x y : X} {r s : ℝ} (hrs : r ≤ s)
-    (hi : ¬Disjoint (closedBall x r) (closedBall y s)) :
-    closedBall x r ⊆ closedBall y (3*s) := by
-  obtain ⟨z,hzx,hzy⟩ := Set.not_disjoint_iff.mp hi
-  apply closedBall_subset_closedBall'
-  have hd : dist x y ≤ r+s :=
-    (dist_triangle_left x y z).trans (add_le_add hzx hzy)
-  linarith
-
- 
-
-theorem finite_disjoint_triples {X ι : Type*} [PseudoMetricSpace X]
-    (s : Finset ι) (x : ι→X) (r : ι→ℝ) (hr : ∀i∈s,0 ≤ r i) :
-    ∃t : Finset ι,t ⊆ s ∧
-      ((t : Set ι).PairwiseDisjoint (fun i=>closedBall (x i) (r i))) ∧
-      ∀i∈s,∃j∈t,closedBall (x i) (r i) ⊆ closedBall (x j) (3*r j) := by
-  revert hr
-  refine Finset.strongInductionOn s ?_
-  intro s ih hr
-  by_cases hs : s.Nonempty
-  · obtain ⟨a,ha,hmax⟩ := s.exists_max_image r hs
-    let s' := s.filter (fun i=>Disjoint (closedBall (x i) (r i)) (closedBall (x a) (r a)))
-    have hstrict : s' ⊂ s := by
-      apply Finset.filter_ssubset.mpr
-      refine ⟨a,ha,?_⟩
-      intro hd
-      have hx : x a∈closedBall (x a) (r a) := mem_closedBall_self (hr a ha)
-      exact Set.disjoint_left.mp hd hx hx
-    obtain ⟨t,ht,hd,hcover⟩ := ih s' hstrict (fun i hi=>hr i (Finset.mem_filter.mp hi).1)
-    refine ⟨insert a t,?_,?_,?_⟩
-    · intro i hi
-      rcases Finset.mem_insert.mp hi with hia | hit
-      · simpa only [hia] using ha
-      · exact (Finset.mem_filter.mp (ht hit)).1
-    · rw [Finset.coe_insert,Set.pairwiseDisjoint_insert]
-      refine ⟨hd,?_⟩
-      intro j hj _
-      exact (Finset.mem_filter.mp (ht hj)).2.symm
-    · intro i hi
-      by_cases hi' : i∈s'
-      · obtain ⟨j,hj,hij⟩ := hcover i hi'
-        exact ⟨j,Finset.mem_insert_of_mem hj,hij⟩
-      · have hn : ¬Disjoint (closedBall (x i) (r i)) (closedBall (x a) (r a)) := by
-          intro hdi
-          exact hi' (Finset.mem_filter.mpr ⟨hi,hdi⟩)
-        exact ⟨a,Finset.mem_insert_self _ _,closedBall_subset_triple_of_inter (hmax i hi) hn⟩
-  · have he : s=∅ := Finset.not_nonempty_iff_eq_empty.mp hs
-    subst s
-    exact ⟨∅,by simp,by simp,by simp⟩
-
-lemma ediam_closedBall_le_of_nonneg {X : Type*} [PseudoMetricSpace X]
-    (x : X) (r : ℝ) : ediam (closedBall x r) ≤ ENNReal.ofReal (2*r) := by
-  apply ediam_le_of_forall_dist_le
-  intro y hy z hz
-  change dist y x ≤ r at hy
-  change dist z x ≤ r at hz
-  exact (dist_triangle_right y z x).trans (by linarith : dist y x+dist z x ≤ 2*r)
-
-lemma diameterCost_le_radius {h : ℝ→ℝ} (hh : Target.IsHausdorffGauge h)
-    {D : ℝ≥0∞} {R : ℝ} (hR : 0 ≤ R) (hD : D ≤ ENNReal.ofReal R) :
-    diameterCost h D ≤ ENNReal.ofReal (h R) := by
-  have hDfin : D≠∞ := (hD.trans_lt ENNReal.ofReal_lt_top).ne
-  rw [diameterCost,ite_eq_right hDfin]
-  apply ENNReal.ofReal_le_ofReal
-  apply hh.2.1 ENNReal.toReal_nonneg hR
-  simpa only [ENNReal.toReal_ofReal hR] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hD
-
-lemma disjoint_charge {X ι : Type*} [MeasurableSpace X]
-    (μ : Measure X) (t : Finset ι) (A : ι→Set X)
-    (hm : ∀i∈t,MeasurableSet (A i)) (hd : (t : Set ι).PairwiseDisjoint A)
-    (B : Set X) (hB : ∀i∈t,A i ⊆ B) (cost : ι→ℝ≥0∞) (C : ℝ≥0∞)
-    (hcost : ∀i∈t,cost i ≤ C*μ (A i)) :
-    (∑i∈t,cost i) ≤ C*μ B := by
-  calc
-    (∑i∈t,cost i) ≤ ∑i∈t,C*μ (A i) := Finset.sum_le_sum hcost
-    _ = C*(∑i∈t,μ (A i)) := (Finset.mul_sum t _ C).symm
-    _ = C*μ (⋃i∈t,A i) := by rw [measure_biUnion_finset hd hm]
-    _ ≤ C*μ B := mul_le_mul le_rfl (measure_mono (iUnion₂_subset hB)) zero_le zero_le
-
- 
-
-theorem finite_candidate_charge {X ι : Type*} [PseudoMetricSpace X]
-    [MeasurableSpace X] [BorelSpace X]
-    (μ : Measure X) (s : Finset ι) (x : ι→X) (r : ι→ℝ)
-    (hr : ∀i∈s,0 ≤ r i) (H B : Set X) (hH : MeasurableSet H)
-    (hB : ∀i∈s,closedBall (x i) (r i)∩H ⊆ B)
-    (cost : ι→ℝ≥0∞) (C : ℝ≥0∞)
-    (hc : ∀i∈s,cost i ≤ C*μ (closedBall (x i) (r i)∩H)) :
-    ∃t : Finset ι,t ⊆ s ∧
-      ((t : Set ι).PairwiseDisjoint (fun i=>closedBall (x i) (r i))) ∧
-      (∀i∈s,∃j∈t,closedBall (x i) (r i) ⊆ closedBall (x j) (3*r j)) ∧
-      (∑i∈t,cost i) ≤ C*μ B := by
-  obtain ⟨t,ht,hd,hcover⟩ := finite_disjoint_triples s x r hr
-  refine ⟨t,ht,hd,hcover,?_⟩
-  apply disjoint_charge μ t (fun i=>closedBall (x i) (r i)∩H)
-    (fun _ _=>measurableSet_closedBall.inter hH)
-    (hd.mono (fun _=>inter_subset_left)) B (fun i hi=>hB i (ht hi)) cost C
-  exact fun i hi=>hc i (ht hi)
-
-lemma quadratic_multiple_le {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι→ℝ)
-    (hb : ∀i,0 ≤ b i) (hρ : ∀i,0<ρ i) (C : ℝ) (hC : 1 ≤ C) (i : ι) :
-    QuadraticGauge.gauge d b ρ (C*ρ i) ≤ C^2*b i*(ρ i)^d := by
-  calc
-    _ ≤ QuadraticGauge.entry d (b i) (ρ i) (C*ρ i) :=
-      QuadraticGauge.gauge_le_entry hb hρ _ i
-    _ = _ := by
-      unfold QuadraticGauge.entry
-      rw [mul_div_cancel_right₀ C (hρ i).ne',max_eq_right (by nlinarith : 1 ≤ C^2)]
-      ring
-
- 
-
-lemma quadratic_candidate_cost {ι : Type*} [Nonempty ι]
-    (d : ℝ) (b ρ : ι→ℝ) (hb : ∀i,0 ≤ b i) (hρ : ∀i,0<ρ i)
-    (hh : Target.IsHausdorffGauge (QuadraticGauge.gauge d b ρ))
-    (μ : Measure ℂ) (H : Set ℂ) (z : ℂ) (i : ι) (A c : ℝ)
-    (hA : 1 ≤ A) (hc : 0<c)
-    (hqual : ENNReal.ofReal (c*b i*(ρ i)^d) ≤ μ (closedBall z (A*ρ i)∩H)) :
-    diameterCost (QuadraticGauge.gauge d b ρ) (ediam (closedBall z (3*(A*ρ i)))) ≤
-      ENNReal.ofReal ((6*A)^2/c)*μ (closedBall z (A*ρ i)∩H) := by
-  have hmul : (6*A)^2/c*(c*b i*(ρ i)^d)=(6*A)^2*b i*(ρ i)^d := by
-    field_simp
-  have hscale : 1 ≤ 6*A := by linarith
-  have hR : 0 ≤ 6*A*ρ i := by have := hρ i; positivity
-  calc
-    _ ≤ ENNReal.ofReal (QuadraticGauge.gauge d b ρ ((6*A)*ρ i)) :=
-      diameterCost_le_radius hh hR (by
-        convert ediam_closedBall_le_of_nonneg z (3*(A*ρ i)) using 1; ring_nf)
-    _ ≤ ENNReal.ofReal ((6*A)^2*b i*(ρ i)^d) :=
-      ENNReal.ofReal_le_ofReal (quadratic_multiple_le d b ρ hb hρ _ hscale i)
-    _ = ENNReal.ofReal ((6*A)^2/c)*ENNReal.ofReal (c*b i*(ρ i)^d) := by
-      rw [←ENNReal.ofReal_mul (by positivity : 0 ≤ (6*A)^2/c),hmul]
-    _ ≤ _ := mul_le_mul le_rfl hqual zero_le zero_le
-
- 
-
-theorem finite_quadratic_candidates {ι η : Type*} [Nonempty ι]
-    (d : ℝ) (b ρ : ι→ℝ) (hb : ∀i,0 ≤ b i) (hρ : ∀i,0<ρ i)
-    (hh : Target.IsHausdorffGauge (QuadraticGauge.gauge d b ρ))
-    (μ : Measure ℂ) (s : Finset η) (x : η→ℂ) (entry : η→ι)
-    (H B : Set ℂ) (hH : MeasurableSet H) (A c : ℝ) (hA : 1 ≤ A) (hc : 0<c)
-    (hB : ∀j∈s,closedBall (x j) (A*ρ (entry j))∩H ⊆ B)
-    (hqual : ∀j∈s,ENNReal.ofReal (c*b (entry j)*(ρ (entry j))^d) ≤
-      μ (closedBall (x j) (A*ρ (entry j))∩H)) :
-    ∃t : Finset η,t ⊆ s ∧
-      ((t : Set η).PairwiseDisjoint (fun j=>closedBall (x j) (A*ρ (entry j)))) ∧
-      (∀j∈s,∃k∈t,closedBall (x j) (A*ρ (entry j)) ⊆
-        closedBall (x k) (3*(A*ρ (entry k)))) ∧
-      (∑j∈t,diameterCost (QuadraticGauge.gauge d b ρ)
-        (ediam (closedBall (x j) (3*(A*ρ (entry j)))))) ≤
-        ENNReal.ofReal ((6*A)^2/c)*μ B := by
-  apply finite_candidate_charge μ s x (fun j=>A*ρ (entry j))
-    (fun j _=>mul_nonneg (by linarith) (hρ _).le) H B hH hB
-  exact fun j hj=>quadratic_candidate_cost d b ρ hb hρ hh μ H (x j) (entry j) A c hA hc (hqual j hj)
-
-end SLEExactGauge.CandidateCover
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.BoundaryGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen
-
- 
-lemma height_power_lintegral (α H : ℝ) (hα : α<1) (hH : 0≤H) :
-    (∫⁻y in Ioc (0:ℝ) H,ENNReal.ofReal (y^(-α)))=
-      ENNReal.ofReal (H^(1-α)/(1-α)) := by
-  have hI : IntegrableOn (fun y : ℝ=>y^(-α)) (Ioc 0 H) volume := by
-    simpa only [uIoc_of_le hH] using
-      (intervalIntegral.intervalIntegrable_rpow' (a:=0) (b:=H) (r:=-α) (by linarith)).def'
-  have hn : 0≤ᵐ[volume.restrict (Ioc (0:ℝ) H)] (fun y : ℝ=>y^(-α)) := by
-    filter_upwards [ae_restrict_mem measurableSet_Ioc] with y hy
-    exact Real.rpow_nonneg hy.1.le _
-  rw [←ofReal_integral_eq_lintegral_ofReal hI hn,
-    ←intervalIntegral.integral_of_le hH,integral_rpow (Or.inl (by linarith : -1< -α))]
-  rw [show -α+1=1-α by ring,Real.zero_rpow (by linarith : 1-α≠0),sub_zero]
-
- 
-def rectangle (l u H : ℝ) : Set UpperPlane :=
-  {z | z.1.re∈Icc l u ∧ z.1.im≤H}
-
-lemma rectangle_measurable (l u H : ℝ) : MeasurableSet (rectangle l u H) := by
-  exact ((Complex.measurable_re.comp measurable_subtype_coe) measurableSet_Icc).inter
-    (measurableSet_le (Complex.measurable_im.comp measurable_subtype_coe) measurable_const)
-
-lemma image_rectangle (l u H : ℝ) :
-    Subtype.val '' rectangle l u H=
-      Complex.measurableEquivRealProd ⁻¹' (Icc l u ×ˢ Ioc 0 H) := by
-  ext z
-  constructor
-  · rintro ⟨w,hw,rfl⟩
-    exact ⟨hw.1,w.2,hw.2⟩
-  · rintro ⟨hzx,hzy⟩
-    exact ⟨⟨z,hzy.1⟩,⟨hzx,hzy.2⟩,rfl⟩
-
-lemma rectangle_height_integral (α l u H : ℝ) (hα : α<1) (hH : 0≤H) :
-    (∫⁻z in rectangle l u H,ENNReal.ofReal (z.1.im^(-α))∂upperArea)=
-      ENNReal.ofReal (u-l)*ENNReal.ofReal (H^(1-α)/(1-α)) := by
-  have hs := setLIntegral_subtype (μ:=(volume : Measure ℂ)) upperPlane_open.measurableSet
-    (rectangle l u H) (fun z=>ENNReal.ofReal (z.im^(-α)))
-  change (∫⁻z in rectangle l u H,ENNReal.ofReal (z.1.im^(-α))∂upperArea)=_ at hs
-  rw [hs]
-  change (∫⁻z in (Subtype.val : UpperPlane → ℂ) '' rectangle l u H,
-    ENNReal.ofReal (z.im^(-α))∂volume)=_
-  rw [image_rectangle l u H]
-  have ht := Complex.volume_preserving_equiv_real_prod.setLIntegral_comp_preimage_emb
-    Complex.measurableEquivRealProd.measurableEmbedding
-    (fun z : ℝ×ℝ=>ENNReal.ofReal (z.2^(-α))) (Icc l u ×ˢ Ioc 0 H)
-  change (∫⁻z in Complex.measurableEquivRealProd ⁻¹' (Icc l u ×ˢ Ioc 0 H),
-    ENNReal.ofReal (z.im^(-α))∂volume)=_ at ht
-  rw [ht]
-  change (∫⁻z in Icc l u ×ˢ Ioc (0:ℝ) H, ENNReal.ofReal (z.2^(-α))
-    ∂(volume : Measure ℝ).prod volume)=_
-  rw [←Measure.prod_restrict,lintegral_prod _ (by fun_prop)]
-  dsimp only
-  rw [height_power_lintegral α H hα hH,lintegral_const,Measure.restrict_apply_univ,
-    Real.volume_Icc,mul_comm]
-
-lemma initialGreen_height_bound (a : ℝ) (ha : 1/4<a) (z : UpperPlane) :
-    initialGreen a z ≤ Real.exp (-alpha a*Real.log 2)*z.1.im^(-alpha a) := by
-  have hn : 0<‖z.1‖ := z.2.trans_le (Complex.im_le_norm z.1)
-  have hs : z.1.im/‖z.1‖≤1 := (div_le_one hn).mpr (Complex.im_le_norm z.1)
-  have hpow : (z.1.im/‖z.1‖)^(p a)≤1 :=
-    Real.rpow_le_one (div_nonneg z.2.le hn.le) hs (by unfold p; linarith)
-  rw [initialGreen,physicalGreen_eq_radius_sine a (z.1,0) z.2]
-  have he : physicalRadius (z.1,(0:ℝ))=z.1.im := by simp [physicalRadius,Y,J]
-  rw [he]
-  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-  exact mul_le_of_le_one_right (Real.rpow_nonneg z.2.le _) hpow
-
- 
-theorem initialGreen_rectangle_bound (a : ℝ) (ha : 1/4<a)
-    (l u H : ℝ) (hH : 0≤H) :
-    initialGreenMeasure a (rectangle l u H) ≤
-      ENNReal.ofReal (Real.exp (-alpha a*Real.log 2))*
-      ENNReal.ofReal (u-l)*ENNReal.ofReal (H^(1-alpha a)/(1-alpha a)) := by
-  have hα : alpha a<1 := by
-    unfold alpha
-    have h : 0<1/(4*a) := div_pos zero_lt_one (by linarith)
-    linarith
-  rw [initialGreenMeasure,withDensity_apply _ (rectangle_measurable l u H)]
-  calc
-    _ ≤ ∫⁻z in rectangle l u H,
-        ENNReal.ofReal (Real.exp (-alpha a*Real.log 2))*
-        ENNReal.ofReal (z.1.im^(-alpha a))∂upperArea := by
-      apply lintegral_mono
-      intro z
-      dsimp only
-      rw [←ENNReal.ofReal_mul (Real.exp_pos _).le]
-      exact ENNReal.ofReal_le_ofReal (initialGreen_height_bound a ha z)
-    _ = _ := by
-      rw [lintegral_const_mul _ (by fun_prop),
-        rectangle_height_integral (alpha a) l u H hα hH,mul_assoc]
-
-lemma initialGreen_rectangle_finite (a : ℝ) (ha : 1/4<a)
-    (l u H : ℝ) (hH : 0≤H) : initialGreenMeasure a (rectangle l u H)<∞ :=
-  (initialGreen_rectangle_bound a ha l u H hH).trans_lt
-    (ENNReal.mul_lt_top (ENNReal.mul_lt_top ENNReal.ofReal_lt_top ENNReal.ofReal_lt_top)
-      ENNReal.ofReal_lt_top)
-
-open FlowIdentification TraceMassContinuity
-
- 
-
-theorem canonical_rectangle_mean {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver}
-    {μ : ℝ≥0∞→GDriver→Measure UpperPlane} (D : CanonicalMassData κ a ha W μ)
-    (l u H : ℝ) (hH : 0≤H) :
-    (∫⁻b,μ ∞ b (rectangle l u H)∂W)=initialGreenMeasure a (rectangle l u H) ∧
-    (∫⁻b,μ ∞ b (rectangle l u H)∂W)<∞ ∧
-    ∀ᵐ b∂W,μ ∞ b (rectangle l u H)<∞ := by
-  have hmeas : Measurable (fun b=>μ ∞ b (rectangle l u H)) :=
-    (Measure.measurable_measure.mp D.terminal_measurable) _ (rectangle_measurable l u H)
-  have he : (∫⁻b,μ ∞ b (rectangle l u H)∂W)=initialGreenMeasure a (rectangle l u H) := by
-    rw [←Measure.bind_apply (rectangle_measurable l u H) D.terminal_measurable.aemeasurable,D.mean]
-  have hf := initialGreen_rectangle_finite a ha l u H hH
-  exact ⟨he,he.trans_lt hf,ae_lt_top hmeas (he.trans_lt hf).ne⟩
-
-end SLEExactGauge.BoundaryGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.BoxGrid
-
-def box (m : ℝ) : Set ℂ := {z | z.re∈Icc (-m) m ∧ z.im∈Icc 0 m}
-
-def region (m : ℝ) : Set ℂ := {z | z.re∈Icc (-(m+2)) (m+2) ∧ z.im≤ m+2}
-
-lemma region_measurable (m : ℝ) : MeasurableSet (region m) :=
-  (Complex.measurable_re measurableSet_Icc).inter
-    (measurableSet_le Complex.measurable_im measurable_const)
-
-lemma preimage_region (m : ℝ) :
-    (Subtype.val : ExactSLEGreen.UpperPlane→ℂ) ⁻¹' region m =
-      BoundaryGreen.rectangle (-(m+2)) (m+2) (m+2) := rfl
-
-def columns (m r : ℝ) : ℕ := ⌊2*m/r⌋₊+1
-
-def rows (m r : ℝ) : ℕ := ⌊m/r⌋₊+1
-
-abbrev Cells (m r : ℝ) := Fin (columns m r) × Fin (rows m r)
-
-def center (m r : ℝ) (i j : ℕ) : ℂ :=
-  ⟨-m+((i:ℝ)+1/2)*r,((j:ℝ)+1/2)*r⟩
-
-lemma center_im_pos (m r : ℝ) (hr : 0<r) (i j : ℕ) : 0<(center m r i j).im := by
-  dsimp [center]
-  positivity
-
-lemma columns_width (m r : ℝ) (hm : 0≤ m) (hr : 0<r) :
-    (columns m r:ℝ)*r ≤ 2*m+r := by
-  have h := Nat.floor_le (div_nonneg (by positivity : 0≤2*m) hr.le)
-  have hh := (le_div_iff₀ hr).mp h
-  simp only [columns,Nat.cast_add,Nat.cast_one]
-  nlinarith
-
-lemma rows_height (m r : ℝ) (hm : 0≤ m) (hr : 0<r) :
-    (rows m r:ℝ)*r ≤ m+r := by
-  have h := Nat.floor_le (div_nonneg hm hr.le)
-  have hh := (le_div_iff₀ hr).mp h
-  simp only [rows,Nat.cast_add,Nat.cast_one]
-  nlinarith
-
-lemma center_bounds (m r : ℝ) (hm : 0≤ m) (hr : 0<r) (k : Cells m r) :
-    -m≤(center m r k.1 k.2).re ∧ (center m r k.1 k.2).re≤ m+r ∧
-      0<(center m r k.1 k.2).im ∧ (center m r k.1 k.2).im≤ m+r := by
-  have hi : (k.1.1:ℝ)+1≤columns m r := by exact_mod_cast k.1.2
-  have hj : (k.2.1:ℝ)+1≤rows m r := by exact_mod_cast k.2.2
-  have hx := columns_width m r hm hr
-  have hy := rows_height m r hm hr
-  have hip := mul_le_mul_of_nonneg_right hi hr.le
-  have hjp := mul_le_mul_of_nonneg_right hj hr.le
-  have hn := Nat.cast_nonneg (α:=ℝ) k.1.1
-  refine ⟨?_,?_,center_im_pos m r hr _ _,?_⟩ <;> dsimp [center] <;> nlinarith
-
-lemma covered (m r : ℝ) (_hm : 0≤ m) (hr : 0<r) :
-    box m ⊆ ⋃k : Cells m r,closedBall (center m r k.1 k.2) r := by
-  intro z hz
-  let i : ℕ := ⌊(z.re+m)/r⌋₊
-  let j : ℕ := ⌊z.im/r⌋₊
-  have hix : i<columns m r := Nat.lt_succ_of_le (Nat.floor_mono
-    ((div_le_div_iff_of_pos_right hr).mpr (by linarith [hz.1.2])))
-  have hjy : j<rows m r := Nat.lt_succ_of_le (Nat.floor_mono
-    ((div_le_div_iff_of_pos_right hr).mpr hz.2.2))
-  refine mem_iUnion.mpr ⟨(⟨i,hix⟩,⟨j,hjy⟩),?_⟩
-  have hi0 := (le_div_iff₀ hr).mp (Nat.floor_le (div_nonneg (show 0≤z.re+m by linarith [hz.1.1]) hr.le))
-  have hi1 := (div_lt_iff₀ hr).mp (Nat.lt_floor_add_one ((z.re+m)/r))
-  have hj0 := (le_div_iff₀ hr).mp (Nat.floor_le (div_nonneg hz.2.1 hr.le))
-  have hj1 := (div_lt_iff₀ hr).mp (Nat.lt_floor_add_one (z.im/r))
-  change dist z (center m r i j) ≤ r
-  rw [dist_eq_norm]
-  apply (Complex.norm_le_abs_re_add_abs_im _).trans
-  have hx : |z.re-(-m+((i:ℝ)+1/2)*r)|≤r/2 := by
-    apply abs_le.mpr
-    dsimp [i]
-    constructor <;> linarith
-  have hy : |z.im-((j:ℝ)+1/2)*r|≤r/2 := by
-    apply abs_le.mpr
-    dsimp [j]
-    constructor <;> linarith
-  simpa only [Complex.sub_re,Complex.sub_im,center] using add_le_add hx hy |>.trans_eq (by ring)
-
-lemma ball_subset_region (m r R : ℝ) (hm : 0≤ m) (hr : 0<r) (hr1 : r≤1)
-    (hR : R≤1) (k : Cells m r) :
-    closedBall (center m r k.1 k.2) R ⊆ region m := by
-  intro z hz
-  have hd : ‖z-center m r k.1 k.2‖≤R := by simpa only [mem_closedBall,dist_eq_norm] using hz
-  have hx := (Complex.abs_re_le_norm (z-center m r k.1 k.2)).trans hd
-  have hy := (Complex.abs_im_le_norm (z-center m r k.1 k.2)).trans hd
-  have h := center_bounds m r hm hr k
-  simp only [Complex.sub_re,Complex.sub_im,abs_le] at hx hy
-  exact ⟨⟨by linarith [h.1],by linarith [h.2.1]⟩,by linarith [h.2.2.2]⟩
-
-end SLEExactGauge.BoxGrid
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Finset intervalIntegral
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.BoundaryRows
-
- 
-def rowConstant (α : ℝ) : ℝ := (1/2:ℝ)^(-α)+1/(1-α)
-
-lemma rowConstant_pos {α : ℝ} (hα : α<1) : 0<rowConstant α := by
-  unfold rowConstant
-  exact add_pos (Real.rpow_pos_of_pos (by norm_num) _) (div_pos zero_lt_one (by linarith))
-
- 
-
-theorem half_integer_sum (α : ℝ) (hα : 0≤α) (hα1 : α<1) (N : ℕ) :
-    (∑j∈Finset.range N,((j:ℝ)+1/2)^(-α))≤rowConstant α*(N:ℝ)^(1-α) := by
-  rcases N with _ | N
-  · simp [Real.zero_rpow (by linarith : 1-α≠0)]
-  have hn : (0:ℝ)≤N := Nat.cast_nonneg N
-  have hb : 0<1-α := by linarith
-  have hanti : AntitoneOn (fun x : ℝ=>x^(-α)) (Icc (1/2) (1/2+(N:ℝ))) := by
-    intro x hx y hy hxy
-    exact Real.rpow_le_rpow_of_nonpos (by linarith [hx.1] : 0<x) hxy (neg_nonpos.mpr hα)
-  have hsum := hanti.sum_le_integral
-  have hi : (∫x in (1/2:ℝ)..1/2+N,x^(-α))=
-      (((N:ℝ)+1/2)^(1-α)-(1/2:ℝ)^(1-α))/(1-α) := by
-    rw [integral_rpow (Or.inl (by linarith : -1 < -α))]
-    rw [show -α+1=1-α by ring,add_comm (1/2:ℝ) (N:ℝ)]
-  have hpow : ((N:ℝ)+1/2)^(1-α)≤((N:ℝ)+1)^(1-α) :=
-    Real.rpow_le_rpow (by positivity) (by linarith) hb.le
-  have hpone : 1≤((N:ℝ)+1)^(1-α) := Real.one_le_rpow (by linarith) hb.le
-  rw [Finset.sum_range_succ']
-  have he : (∑j∈Finset.range N,((↑(j+1):ℝ)+1/2)^(-α))≤
-      (((N:ℝ)+1/2)^(1-α)-(1/2:ℝ)^(1-α))/(1-α) := by
-    convert hsum.trans_eq hi using 1; congr 1
-    ext j
-    congr 1
-    ring
-  simp only [Nat.cast_zero,zero_add,Nat.cast_succ]
-  simp only [Nat.cast_add,Nat.cast_one] at he
-  calc
-    _ ≤ (((N:ℝ)+1/2)^(1-α)-(1/2:ℝ)^(1-α))/(1-α)+(1/2:ℝ)^(-α) := add_le_add he le_rfl
-    _ ≤ ((N:ℝ)+1)^(1-α)/(1-α)+(1/2:ℝ)^(-α) := by
-      apply add_le_add_left
-      exact (div_le_div_iff_of_pos_right hb).mpr
-        ((sub_le_self _ (Real.rpow_nonneg (by norm_num) _)).trans hpow)
-    _ ≤ ((N:ℝ)+1)^(1-α)/(1-α)+(1/2:ℝ)^(-α)*((N:ℝ)+1)^(1-α) :=
-      add_le_add le_rfl (le_mul_of_one_le_right (Real.rpow_nonneg (by norm_num : (0:ℝ)≤1/2) (-α)) hpone)
-    _ = _ := by unfold rowConstant; ring
-
-lemma scaled_half_integer_sum (α : ℝ) (hα : 0≤α) (hα1 : α<1)
-    (r : ℝ) (hr : 0<r) (N : ℕ) :
-    (∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))≤
-      rowConstant α/r*((N:ℝ)*r)^(1-α) := by
-  have hsplit : (∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))=
-      r^(-α)*(∑j∈Finset.range N,((j:ℝ)+1/2)^(-α)) := by
-    rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [Real.mul_rpow (by positivity) hr.le,mul_comm]
-  have hscale : r^(-α)=r^(1-α)/r := by
-    rw [show 1-α=(-α)+1 by ring,Real.rpow_add hr,Real.rpow_one,mul_div_cancel_right₀ _ hr.ne']
-  calc
-    _ = r^(-α)*(∑j∈Finset.range N,((j:ℝ)+1/2)^(-α)) := hsplit
-    _ ≤ r^(-α)*(rowConstant α*(N:ℝ)^(1-α)) :=
-      mul_le_mul_of_nonneg_left (half_integer_sum α hα hα1 N) (Real.rpow_nonneg hr.le _)
-    _ = _ := by rw [hscale,Real.mul_rpow (Nat.cast_nonneg N) hr.le]; ring
-
- 
-
-theorem rows_below_height (α : ℝ) (hα : 0≤α) (hα1 : α<1)
-    (r : ℝ) (hr : 0<r) (N : ℕ) (Y : ℝ) (hY : (N:ℝ)*r≤Y) :
-    (∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))≤rowConstant α/r*Y^(1-α) := by
-  refine (scaled_half_integer_sum α hα hα1 r hr N).trans ?_
-  exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (by positivity) hY (by linarith))
-    (div_nonneg (rowConstant_pos hα1).le hr.le)
-
- 
-
-theorem truncated_rows (α : ℝ) (hα : 0≤α) (hα1 : α<1)
-    (r : ℝ) (hr : 0<r) (N : ℕ) (ε : ℝ) (hε : 0≤ε) :
-    (∑j∈(Finset.range N).filter (fun j : ℕ=>((j:ℝ)+1/2)*r<ε),
-      (((j:ℝ)+1/2)*r)^(-α))≤rowConstant α/r*(ε+r)^(1-α) := by
-  have hsub : (Finset.range N).filter (fun j : ℕ=>((j:ℝ)+1/2)*r<ε) ⊆
-      Finset.range ⌈ε/r⌉₊ := by
-    intro j hj
-    rcases Finset.mem_filter.mp hj with ⟨_,hj⟩
-    apply Finset.mem_range.mpr
-    apply Nat.lt_ceil.mpr
-    apply (lt_div_iff₀ hr).mpr
-    nlinarith
-  have htop : (⌈ε/r⌉₊:ℝ)*r≤ε+r := by
-    have he := (mul_lt_mul_of_pos_right (Nat.ceil_lt_add_one (div_nonneg hε hr.le)) hr).le
-    rwa [add_mul,div_mul_cancel₀ _ hr.ne',one_mul] at he
-  exact (Finset.sum_le_sum_of_subset_of_nonneg hsub
-    (fun j _ _=>Real.rpow_nonneg (by positivity) _)).trans
-    (rows_below_height α hα hα1 r hr _ _ htop)
-
- 
-
-theorem rectangle_rows (α : ℝ) (hα : 0≤α) (hα1 : α<1)
-    (r : ℝ) (hr : 0<r) (M N : ℕ) (X Y : ℝ)
-    (hX : (M:ℝ)*r≤X) (hY : (N:ℝ)*r≤Y) :
-    (∑_i∈Finset.range M,∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))≤
-      rowConstant α*X*Y^(1-α)/r^2 := by
-  have hy : 0≤Y := (mul_nonneg (Nat.cast_nonneg N) hr.le).trans hY
-  simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-  calc
-    _ ≤ (M:ℝ)*(rowConstant α/r*Y^(1-α)) :=
-      mul_le_mul_of_nonneg_left (rows_below_height α hα hα1 r hr N Y hY) (Nat.cast_nonneg M)
-    _ ≤ (X/r)*(rowConstant α/r*Y^(1-α)) :=
-      mul_le_mul_of_nonneg_right ((le_div_iff₀ hr).mpr hX)
-        (mul_nonneg (div_nonneg (rowConstant_pos hα1).le hr.le) (Real.rpow_nonneg hy _))
-    _ = _ := by ring
-
- 
-theorem rectangle_truncated_rows (α : ℝ) (hα : 0≤α) (hα1 : α<1)
-    (r : ℝ) (hr : 0<r) (M N : ℕ) (X ε : ℝ)
-    (hX : (M:ℝ)*r≤X) (hε : 0≤ε) :
-    (∑_i∈Finset.range M,
-      ∑j∈(Finset.range N).filter (fun j : ℕ=>((j:ℝ)+1/2)*r<ε),
-        (((j:ℝ)+1/2)*r)^(-α))≤rowConstant α*X*(ε+r)^(1-α)/r^2 := by
-  simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
-  calc
-    _ ≤ (M:ℝ)*(rowConstant α/r*(ε+r)^(1-α)) :=
-      mul_le_mul_of_nonneg_left (truncated_rows α hα hα1 r hr N ε hε) (Nat.cast_nonneg M)
-    _ ≤ (X/r)*(rowConstant α/r*(ε+r)^(1-α)) :=
-      mul_le_mul_of_nonneg_right ((le_div_iff₀ hr).mpr hX)
-        (mul_nonneg (div_nonneg (rowConstant_pos hα1).le hr.le) (Real.rpow_nonneg (by linarith) _))
-    _ = _ := by ring
-
-end SLEExactGauge.BoundaryRows
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.OptionalPastPalm
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped ENNReal NNReal Topology Classical
+namespace SLEExactGauge.CandidateCover
+open Deterministic
+
+lemma closedBall_subset_triple_of_inter {X : Type*} [PseudoMetricSpace X]
+    {x y : X} {r s : ℝ} (hrs : r ≤ s)
+    (hi : ¬Disjoint (closedBall x r) (closedBall y s)) :
+    closedBall x r ⊆ closedBall y (3*s) := by
+  obtain ⟨z,hzx,hzy⟩ := Set.not_disjoint_iff.mp hi
+  apply closedBall_subset_closedBall'
+  have hd : dist x y ≤ r+s :=
+    (dist_triangle_left x y z).trans (add_le_add hzx hzy)
+  linarith
+
+ 
+
+theorem finite_disjoint_triples {X ι : Type*} [PseudoMetricSpace X]
+    (s : Finset ι) (x : ι→X) (r : ι→ℝ) (hr : ∀i∈s,0 ≤ r i) :
+    ∃t : Finset ι,t ⊆ s ∧
+      ((t : Set ι).PairwiseDisjoint (fun i=>closedBall (x i) (r i))) ∧
+      ∀i∈s,∃j∈t,closedBall (x i) (r i) ⊆ closedBall (x j) (3*r j) := by
+  revert hr
+  refine Finset.strongInductionOn s ?_
+  intro s ih hr
+  by_cases hs : s.Nonempty
+  · obtain ⟨a,ha,hmax⟩ := s.exists_max_image r hs
+    let s' := s.filter (fun i=>Disjoint (closedBall (x i) (r i)) (closedBall (x a) (r a)))
+    have hstrict : s' ⊂ s := by
+      apply Finset.filter_ssubset.mpr
+      refine ⟨a,ha,?_⟩
+      intro hd
+      have hx : x a∈closedBall (x a) (r a) := mem_closedBall_self (hr a ha)
+      exact Set.disjoint_left.mp hd hx hx
+    obtain ⟨t,ht,hd,hcover⟩ := ih s' hstrict (fun i hi=>hr i (Finset.mem_filter.mp hi).1)
+    refine ⟨insert a t,?_,?_,?_⟩
+    · intro i hi
+      rcases Finset.mem_insert.mp hi with hia | hit
+      · simpa only [hia] using ha
+      · exact (Finset.mem_filter.mp (ht hit)).1
+    · rw [Finset.coe_insert,Set.pairwiseDisjoint_insert]
+      refine ⟨hd,?_⟩
+      intro j hj _
+      exact (Finset.mem_filter.mp (ht hj)).2.symm
+    · intro i hi
+      by_cases hi' : i∈s'
+      · obtain ⟨j,hj,hij⟩ := hcover i hi'
+        exact ⟨j,Finset.mem_insert_of_mem hj,hij⟩
+      · have hn : ¬Disjoint (closedBall (x i) (r i)) (closedBall (x a) (r a)) := by
+          intro hdi
+          exact hi' (Finset.mem_filter.mpr ⟨hi,hdi⟩)
+        exact ⟨a,Finset.mem_insert_self _ _,closedBall_subset_triple_of_inter (hmax i hi) hn⟩
+  · have he : s=∅ := Finset.not_nonempty_iff_eq_empty.mp hs
+    subst s
+    exact ⟨∅,by simp,by simp,by simp⟩
+
+lemma ediam_closedBall_le_of_nonneg {X : Type*} [PseudoMetricSpace X]
+    (x : X) (r : ℝ) : ediam (closedBall x r) ≤ ENNReal.ofReal (2*r) := by
+  apply ediam_le_of_forall_dist_le
+  intro y hy z hz
+  change dist y x ≤ r at hy
+  change dist z x ≤ r at hz
+  exact (dist_triangle_right y z x).trans (by linarith : dist y x+dist z x ≤ 2*r)
+
+lemma diameterCost_le_radius {h : ℝ→ℝ} (hh : Target.IsHausdorffGauge h)
+    {D : ℝ≥0∞} {R : ℝ} (hR : 0 ≤ R) (hD : D ≤ ENNReal.ofReal R) :
+    diameterCost h D ≤ ENNReal.ofReal (h R) := by
+  have hDfin : D≠∞ := (hD.trans_lt ENNReal.ofReal_lt_top).ne
+  rw [diameterCost,ite_eq_right hDfin]
+  apply ENNReal.ofReal_le_ofReal
+  apply hh.2.1 ENNReal.toReal_nonneg hR
+  simpa only [ENNReal.toReal_ofReal hR] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hD
+
+lemma disjoint_charge {X ι : Type*} [MeasurableSpace X]
+    (μ : Measure X) (t : Finset ι) (A : ι→Set X)
+    (hm : ∀i∈t,MeasurableSet (A i)) (hd : (t : Set ι).PairwiseDisjoint A)
+    (B : Set X) (hB : ∀i∈t,A i ⊆ B) (cost : ι→ℝ≥0∞) (C : ℝ≥0∞)
+    (hcost : ∀i∈t,cost i ≤ C*μ (A i)) :
+    (∑i∈t,cost i) ≤ C*μ B := by
+  calc
+    (∑i∈t,cost i) ≤ ∑i∈t,C*μ (A i) := Finset.sum_le_sum hcost
+    _ = C*(∑i∈t,μ (A i)) := (Finset.mul_sum t _ C).symm
+    _ = C*μ (⋃i∈t,A i) := by rw [measure_biUnion_finset hd hm]
+    _ ≤ C*μ B := mul_le_mul le_rfl (measure_mono (iUnion₂_subset hB)) zero_le zero_le
+
+ 
+
+theorem finite_candidate_charge {X ι : Type*} [PseudoMetricSpace X]
+    [MeasurableSpace X] [BorelSpace X]
+    (μ : Measure X) (s : Finset ι) (x : ι→X) (r : ι→ℝ)
+    (hr : ∀i∈s,0 ≤ r i) (H B : Set X) (hH : MeasurableSet H)
+    (hB : ∀i∈s,closedBall (x i) (r i)∩H ⊆ B)
+    (cost : ι→ℝ≥0∞) (C : ℝ≥0∞)
+    (hc : ∀i∈s,cost i ≤ C*μ (closedBall (x i) (r i)∩H)) :
+    ∃t : Finset ι,t ⊆ s ∧
+      ((t : Set ι).PairwiseDisjoint (fun i=>closedBall (x i) (r i))) ∧
+      (∀i∈s,∃j∈t,closedBall (x i) (r i) ⊆ closedBall (x j) (3*r j)) ∧
+      (∑i∈t,cost i) ≤ C*μ B := by
+  obtain ⟨t,ht,hd,hcover⟩ := finite_disjoint_triples s x r hr
+  refine ⟨t,ht,hd,hcover,?_⟩
+  apply disjoint_charge μ t (fun i=>closedBall (x i) (r i)∩H)
+    (fun _ _=>measurableSet_closedBall.inter hH)
+    (hd.mono (fun _=>inter_subset_left)) B (fun i hi=>hB i (ht hi)) cost C
+  exact fun i hi=>hc i (ht hi)
+
+lemma quadratic_multiple_le {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι→ℝ)
+    (hb : ∀i,0 ≤ b i) (hρ : ∀i,0<ρ i) (C : ℝ) (hC : 1 ≤ C) (i : ι) :
+    QuadraticGauge.gauge d b ρ (C*ρ i) ≤ C^2*b i*(ρ i)^d := by
+  calc
+    _ ≤ QuadraticGauge.entry d (b i) (ρ i) (C*ρ i) :=
+      QuadraticGauge.gauge_le_entry hb hρ _ i
+    _ = _ := by
+      unfold QuadraticGauge.entry
+      rw [mul_div_cancel_right₀ C (hρ i).ne',max_eq_right (by nlinarith : 1 ≤ C^2)]
+      ring
+
+ 
+
+lemma quadratic_candidate_cost {ι : Type*} [Nonempty ι]
+    (d : ℝ) (b ρ : ι→ℝ) (hb : ∀i,0 ≤ b i) (hρ : ∀i,0<ρ i)
+    (hh : Target.IsHausdorffGauge (QuadraticGauge.gauge d b ρ))
+    (μ : Measure ℂ) (H : Set ℂ) (z : ℂ) (i : ι) (A c : ℝ)
+    (hA : 1 ≤ A) (hc : 0<c)
+    (hqual : ENNReal.ofReal (c*b i*(ρ i)^d) ≤ μ (closedBall z (A*ρ i)∩H)) :
+    diameterCost (QuadraticGauge.gauge d b ρ) (ediam (closedBall z (3*(A*ρ i)))) ≤
+      ENNReal.ofReal ((6*A)^2/c)*μ (closedBall z (A*ρ i)∩H) := by
+  have hmul : (6*A)^2/c*(c*b i*(ρ i)^d)=(6*A)^2*b i*(ρ i)^d := by
+    field_simp
+  have hscale : 1 ≤ 6*A := by linarith
+  have hR : 0 ≤ 6*A*ρ i := by have := hρ i; positivity
+  calc
+    _ ≤ ENNReal.ofReal (QuadraticGauge.gauge d b ρ ((6*A)*ρ i)) :=
+      diameterCost_le_radius hh hR (by
+        convert ediam_closedBall_le_of_nonneg z (3*(A*ρ i)) using 1; ring_nf)
+    _ ≤ ENNReal.ofReal ((6*A)^2*b i*(ρ i)^d) :=
+      ENNReal.ofReal_le_ofReal (quadratic_multiple_le d b ρ hb hρ _ hscale i)
+    _ = ENNReal.ofReal ((6*A)^2/c)*ENNReal.ofReal (c*b i*(ρ i)^d) := by
+      rw [←ENNReal.ofReal_mul (by positivity : 0 ≤ (6*A)^2/c),hmul]
+    _ ≤ _ := mul_le_mul le_rfl hqual zero_le zero_le
+
+ 
+
+theorem finite_quadratic_candidates {ι η : Type*} [Nonempty ι]
+    (d : ℝ) (b ρ : ι→ℝ) (hb : ∀i,0 ≤ b i) (hρ : ∀i,0<ρ i)
+    (hh : Target.IsHausdorffGauge (QuadraticGauge.gauge d b ρ))
+    (μ : Measure ℂ) (s : Finset η) (x : η→ℂ) (entry : η→ι)
+    (H B : Set ℂ) (hH : MeasurableSet H) (A c : ℝ) (hA : 1 ≤ A) (hc : 0<c)
+    (hB : ∀j∈s,closedBall (x j) (A*ρ (entry j))∩H ⊆ B)
+    (hqual : ∀j∈s,ENNReal.ofReal (c*b (entry j)*(ρ (entry j))^d) ≤
+      μ (closedBall (x j) (A*ρ (entry j))∩H)) :
+    ∃t : Finset η,t ⊆ s ∧
+      ((t : Set η).PairwiseDisjoint (fun j=>closedBall (x j) (A*ρ (entry j)))) ∧
+      (∀j∈s,∃k∈t,closedBall (x j) (A*ρ (entry j)) ⊆
+        closedBall (x k) (3*(A*ρ (entry k)))) ∧
+      (∑j∈t,diameterCost (QuadraticGauge.gauge d b ρ)
+        (ediam (closedBall (x j) (3*(A*ρ (entry j)))))) ≤
+        ENNReal.ofReal ((6*A)^2/c)*μ B := by
+  apply finite_candidate_charge μ s x (fun j=>A*ρ (entry j))
+    (fun j _=>mul_nonneg (by linarith) (hρ _).le) H B hH hB
+  exact fun j hj=>quadratic_candidate_cost d b ρ hb hρ hh μ H (x j) (entry j) A c hA hc (hqual j hj)
+
+end SLEExactGauge.CandidateCover
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.BoundaryGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen
+
+ 
+lemma height_power_lintegral (α H : ℝ) (hα : α<1) (hH : 0≤H) :
+    (∫⁻y in Ioc (0:ℝ) H,ENNReal.ofReal (y^(-α)))=
+      ENNReal.ofReal (H^(1-α)/(1-α)) := by
+  have hI : IntegrableOn (fun y : ℝ=>y^(-α)) (Ioc 0 H) volume := by
+    simpa only [uIoc_of_le hH] using
+      (intervalIntegral.intervalIntegrable_rpow' (a:=0) (b:=H) (r:=-α) (by linarith)).def'
+  have hn : 0≤ᵐ[volume.restrict (Ioc (0:ℝ) H)] (fun y : ℝ=>y^(-α)) := by
+    filter_upwards [ae_restrict_mem measurableSet_Ioc] with y hy
+    exact Real.rpow_nonneg hy.1.le _
+  rw [←ofReal_integral_eq_lintegral_ofReal hI hn,
+    ←intervalIntegral.integral_of_le hH,integral_rpow (Or.inl (by linarith : -1< -α))]
+  rw [show -α+1=1-α by ring,Real.zero_rpow (by linarith : 1-α≠0),sub_zero]
+
+ 
+def rectangle (l u H : ℝ) : Set UpperPlane :=
+  {z | z.1.re∈Icc l u ∧ z.1.im≤H}
+
+lemma rectangle_measurable (l u H : ℝ) : MeasurableSet (rectangle l u H) := by
+  exact ((Complex.measurable_re.comp measurable_subtype_coe) measurableSet_Icc).inter
+    (measurableSet_le (Complex.measurable_im.comp measurable_subtype_coe) measurable_const)
+
+lemma image_rectangle (l u H : ℝ) :
+    Subtype.val '' rectangle l u H=
+      Complex.measurableEquivRealProd ⁻¹' (Icc l u ×ˢ Ioc 0 H) := by
+  ext z
+  constructor
+  · rintro ⟨w,hw,rfl⟩
+    exact ⟨hw.1,w.2,hw.2⟩
+  · rintro ⟨hzx,hzy⟩
+    exact ⟨⟨z,hzy.1⟩,⟨hzx,hzy.2⟩,rfl⟩
+
+lemma rectangle_height_integral (α l u H : ℝ) (hα : α<1) (hH : 0≤H) :
+    (∫⁻z in rectangle l u H,ENNReal.ofReal (z.1.im^(-α))∂upperArea)=
+      ENNReal.ofReal (u-l)*ENNReal.ofReal (H^(1-α)/(1-α)) := by
+  have hs := setLIntegral_subtype (μ:=(volume : Measure ℂ)) upperPlane_open.measurableSet
+    (rectangle l u H) (fun z=>ENNReal.ofReal (z.im^(-α)))
+  change (∫⁻z in rectangle l u H,ENNReal.ofReal (z.1.im^(-α))∂upperArea)=_ at hs
+  rw [hs]
+  change (∫⁻z in (Subtype.val : UpperPlane → ℂ) '' rectangle l u H,
+    ENNReal.ofReal (z.im^(-α))∂volume)=_
+  rw [image_rectangle l u H]
+  have ht := Complex.volume_preserving_equiv_real_prod.setLIntegral_comp_preimage_emb
+    Complex.measurableEquivRealProd.measurableEmbedding
+    (fun z : ℝ×ℝ=>ENNReal.ofReal (z.2^(-α))) (Icc l u ×ˢ Ioc 0 H)
+  change (∫⁻z in Complex.measurableEquivRealProd ⁻¹' (Icc l u ×ˢ Ioc 0 H),
+    ENNReal.ofReal (z.im^(-α))∂volume)=_ at ht
+  rw [ht]
+  change (∫⁻z in Icc l u ×ˢ Ioc (0:ℝ) H, ENNReal.ofReal (z.2^(-α))
+    ∂(volume : Measure ℝ).prod volume)=_
+  rw [←Measure.prod_restrict,lintegral_prod _ (by fun_prop)]
+  dsimp only
+  rw [height_power_lintegral α H hα hH,lintegral_const,Measure.restrict_apply_univ,
+    Real.volume_Icc,mul_comm]
+
+lemma initialGreen_height_bound (a : ℝ) (ha : 1/4<a) (z : UpperPlane) :
+    initialGreen a z ≤ Real.exp (-alpha a*Real.log 2)*z.1.im^(-alpha a) := by
+  have hn : 0<‖z.1‖ := z.2.trans_le (Complex.im_le_norm z.1)
+  have hs : z.1.im/‖z.1‖≤1 := (div_le_one hn).mpr (Complex.im_le_norm z.1)
+  have hpow : (z.1.im/‖z.1‖)^(p a)≤1 :=
+    Real.rpow_le_one (div_nonneg z.2.le hn.le) hs (by unfold p; linarith)
+  rw [initialGreen,physicalGreen_eq_radius_sine a (z.1,0) z.2]
+  have he : physicalRadius (z.1,(0:ℝ))=z.1.im := by simp [physicalRadius,Y,J]
+  rw [he]
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+  exact mul_le_of_le_one_right (Real.rpow_nonneg z.2.le _) hpow
+
+ 
+theorem initialGreen_rectangle_bound (a : ℝ) (ha : 1/4<a)
+    (l u H : ℝ) (hH : 0≤H) :
+    initialGreenMeasure a (rectangle l u H) ≤
+      ENNReal.ofReal (Real.exp (-alpha a*Real.log 2))*
+      ENNReal.ofReal (u-l)*ENNReal.ofReal (H^(1-alpha a)/(1-alpha a)) := by
+  have hα : alpha a<1 := by
+    unfold alpha
+    have h : 0<1/(4*a) := div_pos zero_lt_one (by linarith)
+    linarith
+  rw [initialGreenMeasure,withDensity_apply _ (rectangle_measurable l u H)]
+  calc
+    _ ≤ ∫⁻z in rectangle l u H,
+        ENNReal.ofReal (Real.exp (-alpha a*Real.log 2))*
+        ENNReal.ofReal (z.1.im^(-alpha a))∂upperArea := by
+      apply lintegral_mono
+      intro z
+      dsimp only
+      rw [←ENNReal.ofReal_mul (Real.exp_pos _).le]
+      exact ENNReal.ofReal_le_ofReal (initialGreen_height_bound a ha z)
+    _ = _ := by
+      rw [lintegral_const_mul _ (by fun_prop),
+        rectangle_height_integral (alpha a) l u H hα hH,mul_assoc]
+
+lemma initialGreen_rectangle_finite (a : ℝ) (ha : 1/4<a)
+    (l u H : ℝ) (hH : 0≤H) : initialGreenMeasure a (rectangle l u H)<∞ :=
+  (initialGreen_rectangle_bound a ha l u H hH).trans_lt
+    (ENNReal.mul_lt_top (ENNReal.mul_lt_top ENNReal.ofReal_lt_top ENNReal.ofReal_lt_top)
+      ENNReal.ofReal_lt_top)
+
+open FlowIdentification TraceMassContinuity
+
+ 
+
+theorem canonical_rectangle_mean {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver}
+    {μ : ℝ≥0∞→GDriver→Measure UpperPlane} (D : CanonicalMassData κ a ha W μ)
+    (l u H : ℝ) (hH : 0≤H) :
+    (∫⁻b,μ ∞ b (rectangle l u H)∂W)=initialGreenMeasure a (rectangle l u H) ∧
+    (∫⁻b,μ ∞ b (rectangle l u H)∂W)<∞ ∧
+    ∀ᵐ b∂W,μ ∞ b (rectangle l u H)<∞ := by
+  have hmeas : Measurable (fun b=>μ ∞ b (rectangle l u H)) :=
+    (Measure.measurable_measure.mp D.terminal_measurable) _ (rectangle_measurable l u H)
+  have he : (∫⁻b,μ ∞ b (rectangle l u H)∂W)=initialGreenMeasure a (rectangle l u H) := by
+    rw [←Measure.bind_apply (rectangle_measurable l u H) D.terminal_measurable.aemeasurable,D.mean]
+  have hf := initialGreen_rectangle_finite a ha l u H hH
+  exact ⟨he,he.trans_lt hf,ae_lt_top hmeas (he.trans_lt hf).ne⟩
+
+end SLEExactGauge.BoundaryGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.BoxGrid
+
+def box (m : ℝ) : Set ℂ := {z | z.re∈Icc (-m) m ∧ z.im∈Icc 0 m}
+
+def region (m : ℝ) : Set ℂ := {z | z.re∈Icc (-(m+2)) (m+2) ∧ z.im≤ m+2}
+
+lemma region_measurable (m : ℝ) : MeasurableSet (region m) :=
+  (Complex.measurable_re measurableSet_Icc).inter
+    (measurableSet_le Complex.measurable_im measurable_const)
+
+lemma preimage_region (m : ℝ) :
+    (Subtype.val : ExactSLEGreen.UpperPlane→ℂ) ⁻¹' region m =
+      BoundaryGreen.rectangle (-(m+2)) (m+2) (m+2) := rfl
+
+def columns (m r : ℝ) : ℕ := ⌊2*m/r⌋₊+1
+
+def rows (m r : ℝ) : ℕ := ⌊m/r⌋₊+1
+
+abbrev Cells (m r : ℝ) := Fin (columns m r) × Fin (rows m r)
+
+def center (m r : ℝ) (i j : ℕ) : ℂ :=
+  ⟨-m+((i:ℝ)+1/2)*r,((j:ℝ)+1/2)*r⟩
+
+lemma center_im_pos (m r : ℝ) (hr : 0<r) (i j : ℕ) : 0<(center m r i j).im := by
+  dsimp [center]
+  positivity
+
+lemma columns_width (m r : ℝ) (hm : 0≤ m) (hr : 0<r) :
+    (columns m r:ℝ)*r ≤ 2*m+r := by
+  have h := Nat.floor_le (div_nonneg (by positivity : 0≤2*m) hr.le)
+  have hh := (le_div_iff₀ hr).mp h
+  simp only [columns,Nat.cast_add,Nat.cast_one]
+  nlinarith
+
+lemma rows_height (m r : ℝ) (hm : 0≤ m) (hr : 0<r) :
+    (rows m r:ℝ)*r ≤ m+r := by
+  have h := Nat.floor_le (div_nonneg hm hr.le)
+  have hh := (le_div_iff₀ hr).mp h
+  simp only [rows,Nat.cast_add,Nat.cast_one]
+  nlinarith
+
+lemma center_bounds (m r : ℝ) (hm : 0≤ m) (hr : 0<r) (k : Cells m r) :
+    -m≤(center m r k.1 k.2).re ∧ (center m r k.1 k.2).re≤ m+r ∧
+      0<(center m r k.1 k.2).im ∧ (center m r k.1 k.2).im≤ m+r := by
+  have hi : (k.1.1:ℝ)+1≤columns m r := by exact_mod_cast k.1.2
+  have hj : (k.2.1:ℝ)+1≤rows m r := by exact_mod_cast k.2.2
+  have hx := columns_width m r hm hr
+  have hy := rows_height m r hm hr
+  have hip := mul_le_mul_of_nonneg_right hi hr.le
+  have hjp := mul_le_mul_of_nonneg_right hj hr.le
+  have hn := Nat.cast_nonneg (α:=ℝ) k.1.1
+  refine ⟨?_,?_,center_im_pos m r hr _ _,?_⟩ <;> dsimp [center] <;> nlinarith
+
+lemma covered (m r : ℝ) (_hm : 0≤ m) (hr : 0<r) :
+    box m ⊆ ⋃k : Cells m r,closedBall (center m r k.1 k.2) r := by
+  intro z hz
+  let i : ℕ := ⌊(z.re+m)/r⌋₊
+  let j : ℕ := ⌊z.im/r⌋₊
+  have hix : i<columns m r := Nat.lt_succ_of_le (Nat.floor_mono
+    ((div_le_div_iff_of_pos_right hr).mpr (by linarith [hz.1.2])))
+  have hjy : j<rows m r := Nat.lt_succ_of_le (Nat.floor_mono
+    ((div_le_div_iff_of_pos_right hr).mpr hz.2.2))
+  refine mem_iUnion.mpr ⟨(⟨i,hix⟩,⟨j,hjy⟩),?_⟩
+  have hi0 := (le_div_iff₀ hr).mp (Nat.floor_le (div_nonneg (show 0≤z.re+m by linarith [hz.1.1]) hr.le))
+  have hi1 := (div_lt_iff₀ hr).mp (Nat.lt_floor_add_one ((z.re+m)/r))
+  have hj0 := (le_div_iff₀ hr).mp (Nat.floor_le (div_nonneg hz.2.1 hr.le))
+  have hj1 := (div_lt_iff₀ hr).mp (Nat.lt_floor_add_one (z.im/r))
+  change dist z (center m r i j) ≤ r
+  rw [dist_eq_norm]
+  apply (Complex.norm_le_abs_re_add_abs_im _).trans
+  have hx : |z.re-(-m+((i:ℝ)+1/2)*r)|≤r/2 := by
+    apply abs_le.mpr
+    dsimp [i]
+    constructor <;> linarith
+  have hy : |z.im-((j:ℝ)+1/2)*r|≤r/2 := by
+    apply abs_le.mpr
+    dsimp [j]
+    constructor <;> linarith
+  simpa only [Complex.sub_re,Complex.sub_im,center] using add_le_add hx hy |>.trans_eq (by ring)
+
+lemma ball_subset_region (m r R : ℝ) (hm : 0≤ m) (hr : 0<r) (hr1 : r≤1)
+    (hR : R≤1) (k : Cells m r) :
+    closedBall (center m r k.1 k.2) R ⊆ region m := by
+  intro z hz
+  have hd : ‖z-center m r k.1 k.2‖≤R := by simpa only [mem_closedBall,dist_eq_norm] using hz
+  have hx := (Complex.abs_re_le_norm (z-center m r k.1 k.2)).trans hd
+  have hy := (Complex.abs_im_le_norm (z-center m r k.1 k.2)).trans hd
+  have h := center_bounds m r hm hr k
+  simp only [Complex.sub_re,Complex.sub_im,abs_le] at hx hy
+  exact ⟨⟨by linarith [h.1],by linarith [h.2.1]⟩,by linarith [h.2.2.2]⟩
+
+end SLEExactGauge.BoxGrid
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Finset intervalIntegral
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.BoundaryRows
+
+ 
+def rowConstant (α : ℝ) : ℝ := (1/2:ℝ)^(-α)+1/(1-α)
+
+lemma rowConstant_pos {α : ℝ} (hα : α<1) : 0<rowConstant α := by
+  unfold rowConstant
+  exact add_pos (Real.rpow_pos_of_pos (by norm_num) _) (div_pos zero_lt_one (by linarith))
+
+ 
+
+theorem half_integer_sum (α : ℝ) (hα : 0≤α) (hα1 : α<1) (N : ℕ) :
+    (∑j∈Finset.range N,((j:ℝ)+1/2)^(-α))≤rowConstant α*(N:ℝ)^(1-α) := by
+  rcases N with _ | N
+  · simp [Real.zero_rpow (by linarith : 1-α≠0)]
+  have hn : (0:ℝ)≤N := Nat.cast_nonneg N
+  have hb : 0<1-α := by linarith
+  have hanti : AntitoneOn (fun x : ℝ=>x^(-α)) (Icc (1/2) (1/2+(N:ℝ))) := by
+    intro x hx y hy hxy
+    exact Real.rpow_le_rpow_of_nonpos (by linarith [hx.1] : 0<x) hxy (neg_nonpos.mpr hα)
+  have hsum := hanti.sum_le_integral
+  have hi : (∫x in (1/2:ℝ)..1/2+N,x^(-α))=
+      (((N:ℝ)+1/2)^(1-α)-(1/2:ℝ)^(1-α))/(1-α) := by
+    rw [integral_rpow (Or.inl (by linarith : -1 < -α))]
+    rw [show -α+1=1-α by ring,add_comm (1/2:ℝ) (N:ℝ)]
+  have hpow : ((N:ℝ)+1/2)^(1-α)≤((N:ℝ)+1)^(1-α) :=
+    Real.rpow_le_rpow (by positivity) (by linarith) hb.le
+  have hpone : 1≤((N:ℝ)+1)^(1-α) := Real.one_le_rpow (by linarith) hb.le
+  rw [Finset.sum_range_succ']
+  have he : (∑j∈Finset.range N,((↑(j+1):ℝ)+1/2)^(-α))≤
+      (((N:ℝ)+1/2)^(1-α)-(1/2:ℝ)^(1-α))/(1-α) := by
+    convert hsum.trans_eq hi using 1; congr 1
+    ext j
+    congr 1
+    ring
+  simp only [Nat.cast_zero,zero_add,Nat.cast_succ]
+  simp only [Nat.cast_add,Nat.cast_one] at he
+  calc
+    _ ≤ (((N:ℝ)+1/2)^(1-α)-(1/2:ℝ)^(1-α))/(1-α)+(1/2:ℝ)^(-α) := add_le_add he le_rfl
+    _ ≤ ((N:ℝ)+1)^(1-α)/(1-α)+(1/2:ℝ)^(-α) := by
+      apply add_le_add_left
+      exact (div_le_div_iff_of_pos_right hb).mpr
+        ((sub_le_self _ (Real.rpow_nonneg (by norm_num) _)).trans hpow)
+    _ ≤ ((N:ℝ)+1)^(1-α)/(1-α)+(1/2:ℝ)^(-α)*((N:ℝ)+1)^(1-α) :=
+      add_le_add le_rfl (le_mul_of_one_le_right (Real.rpow_nonneg (by norm_num : (0:ℝ)≤1/2) (-α)) hpone)
+    _ = _ := by unfold rowConstant; ring
+
+lemma scaled_half_integer_sum (α : ℝ) (hα : 0≤α) (hα1 : α<1)
+    (r : ℝ) (hr : 0<r) (N : ℕ) :
+    (∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))≤
+      rowConstant α/r*((N:ℝ)*r)^(1-α) := by
+  have hsplit : (∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))=
+      r^(-α)*(∑j∈Finset.range N,((j:ℝ)+1/2)^(-α)) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [Real.mul_rpow (by positivity) hr.le,mul_comm]
+  have hscale : r^(-α)=r^(1-α)/r := by
+    rw [show 1-α=(-α)+1 by ring,Real.rpow_add hr,Real.rpow_one,mul_div_cancel_right₀ _ hr.ne']
+  calc
+    _ = r^(-α)*(∑j∈Finset.range N,((j:ℝ)+1/2)^(-α)) := hsplit
+    _ ≤ r^(-α)*(rowConstant α*(N:ℝ)^(1-α)) :=
+      mul_le_mul_of_nonneg_left (half_integer_sum α hα hα1 N) (Real.rpow_nonneg hr.le _)
+    _ = _ := by rw [hscale,Real.mul_rpow (Nat.cast_nonneg N) hr.le]; ring
+
+ 
+
+theorem rows_below_height (α : ℝ) (hα : 0≤α) (hα1 : α<1)
+    (r : ℝ) (hr : 0<r) (N : ℕ) (Y : ℝ) (hY : (N:ℝ)*r≤Y) :
+    (∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))≤rowConstant α/r*Y^(1-α) := by
+  refine (scaled_half_integer_sum α hα hα1 r hr N).trans ?_
+  exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (by positivity) hY (by linarith))
+    (div_nonneg (rowConstant_pos hα1).le hr.le)
+
+ 
+
+theorem truncated_rows (α : ℝ) (hα : 0≤α) (hα1 : α<1)
+    (r : ℝ) (hr : 0<r) (N : ℕ) (ε : ℝ) (hε : 0≤ε) :
+    (∑j∈(Finset.range N).filter (fun j : ℕ=>((j:ℝ)+1/2)*r<ε),
+      (((j:ℝ)+1/2)*r)^(-α))≤rowConstant α/r*(ε+r)^(1-α) := by
+  have hsub : (Finset.range N).filter (fun j : ℕ=>((j:ℝ)+1/2)*r<ε) ⊆
+      Finset.range ⌈ε/r⌉₊ := by
+    intro j hj
+    rcases Finset.mem_filter.mp hj with ⟨_,hj⟩
+    apply Finset.mem_range.mpr
+    apply Nat.lt_ceil.mpr
+    apply (lt_div_iff₀ hr).mpr
+    nlinarith
+  have htop : (⌈ε/r⌉₊:ℝ)*r≤ε+r := by
+    have he := (mul_lt_mul_of_pos_right (Nat.ceil_lt_add_one (div_nonneg hε hr.le)) hr).le
+    rwa [add_mul,div_mul_cancel₀ _ hr.ne',one_mul] at he
+  exact (Finset.sum_le_sum_of_subset_of_nonneg hsub
+    (fun j _ _=>Real.rpow_nonneg (by positivity) _)).trans
+    (rows_below_height α hα hα1 r hr _ _ htop)
+
+ 
+
+theorem rectangle_rows (α : ℝ) (hα : 0≤α) (hα1 : α<1)
+    (r : ℝ) (hr : 0<r) (M N : ℕ) (X Y : ℝ)
+    (hX : (M:ℝ)*r≤X) (hY : (N:ℝ)*r≤Y) :
+    (∑_i∈Finset.range M,∑j∈Finset.range N,(((j:ℝ)+1/2)*r)^(-α))≤
+      rowConstant α*X*Y^(1-α)/r^2 := by
+  have hy : 0≤Y := (mul_nonneg (Nat.cast_nonneg N) hr.le).trans hY
+  simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+  calc
+    _ ≤ (M:ℝ)*(rowConstant α/r*Y^(1-α)) :=
+      mul_le_mul_of_nonneg_left (rows_below_height α hα hα1 r hr N Y hY) (Nat.cast_nonneg M)
+    _ ≤ (X/r)*(rowConstant α/r*Y^(1-α)) :=
+      mul_le_mul_of_nonneg_right ((le_div_iff₀ hr).mpr hX)
+        (mul_nonneg (div_nonneg (rowConstant_pos hα1).le hr.le) (Real.rpow_nonneg hy _))
+    _ = _ := by ring
+
+ 
+theorem rectangle_truncated_rows (α : ℝ) (hα : 0≤α) (hα1 : α<1)
+    (r : ℝ) (hr : 0<r) (M N : ℕ) (X ε : ℝ)
+    (hX : (M:ℝ)*r≤X) (hε : 0≤ε) :
+    (∑_i∈Finset.range M,
+      ∑j∈(Finset.range N).filter (fun j : ℕ=>((j:ℝ)+1/2)*r<ε),
+        (((j:ℝ)+1/2)*r)^(-α))≤rowConstant α*X*(ε+r)^(1-α)/r^2 := by
+  simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]
+  calc
+    _ ≤ (M:ℝ)*(rowConstant α/r*(ε+r)^(1-α)) :=
+      mul_le_mul_of_nonneg_left (truncated_rows α hα hα1 r hr N ε hε) (Nat.cast_nonneg M)
+    _ ≤ (X/r)*(rowConstant α/r*(ε+r)^(1-α)) :=
+      mul_le_mul_of_nonneg_right ((le_div_iff₀ hr).mpr hX)
+        (mul_nonneg (div_nonneg (rowConstant_pos hα1).le hr.le) (Real.rpow_nonneg (by linarith) _))
+    _ = _ := by ring
+
+end SLEExactGauge.BoundaryRows
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CenteredSequenceLaw.lean b/lean/OAI/Probability/SLEGauge/CenteredSequenceLaw.lean
index dfa8e74e05002b7617bd34b318104f0f8ee0cac3..57fbd0bca18920050f2d69a2e0902010d1eee4c9 100644
--- a/lean/OAI/Probability/SLEGauge/CenteredSequenceLaw.lean
+++ b/lean/OAI/Probability/SLEGauge/CenteredSequenceLaw.lean
@@ -1,518 +1,518 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.NormalizedRadialTests
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
-
-lemma centeredMass_raw_measurable_of_le (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) {i l : ℕ} (hil : i≤l)
-    (hμ : Measurable[(centeredTime_stopping a ha θ i).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 i) b)) :
-    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>centeredMass a μ i (θ,b)) :=
-  (centeredMass_raw_measurable a ha μ θ i hμ).mono
-    ((centeredTime_stopping a ha θ i).measurableSpace_mono (centeredTime_stopping a ha θ l)
-      (fun b=>centeredTime_monotone a ha θ b hil)) le_rfl
-
-lemma centered_vector_consistent (W : Measure ExactSLEGenerator.Driver)
-    [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b))
-    {l k : ℕ} (hlk : l≤k) :
-    (finiteAngleLaw W a k).map (fun p=>fun i : Fin (l+1)=>centeredMass a μ i p)=
-      (finiteAngleLaw W a l).map (fun p=>fun i : Fin (l+1)=>centeredMass a μ i p) := by
-  have hm : Measurable (fun p=>fun i : Fin (l+1)=>centeredMass a μ i p) :=
-    Measurable.of_eval (fun index=>centeredMass_measurable a ha μ hμ index)
-  apply Measure.ext
-  intro A hA
-  rw [Measure.map_apply hm hA,Measure.map_apply hm hA]
-  simp only [←lintegral_indicator_one (hm hA)]
-  apply finiteAngleLaw_lintegral_consistent W hW a ha hlk
-  · exact measurable_const.indicator (hm hA)
-  · intro θ
-    let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
-    have hv : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-        (fun b=>fun i : Fin (l+1)=>centeredMass a μ i (θ,b)) :=
-      Measurable.of_eval (fun index=>centeredMass_raw_measurable_of_le a ha μ θ
-        (by omega) (hμraw index θ))
-    exact measurable_const.indicator (hv hA)
-
- 
-
-theorem exists_centered_sequence_law (W : Measure ExactSLEGenerator.Driver)
-    [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b))
-    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) :
-    ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
-      (∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
-        (fun p=>fun i : Fin (n+1)=>centeredMass a μ i p)) ∧
-      (∀ᵐ x∂Q,Monotone x) := by
-  have (n : ℕ) : IsProbabilityMeasure (finiteAngleLaw W a n) :=
-    finiteAngleLaw_probability W hW a ha n
-  obtain ⟨Q,hQ,hQlaw⟩ := exists_sequence_law (finiteAngleLaw W a)
-    (centeredMass a μ) (centeredMass_measurable a ha μ hμ)
-    (fun _ _ hlk=>centered_vector_consistent W hW a ha μ hμ hμraw hlk)
-  refine ⟨Q,hQ,hQlaw,?_⟩
-  have hX (l : ℕ) : ∀ᵐ p∂finiteAngleLaw W a l,Monotone (fun n=>centeredMass a μ n p) := by
-    have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
-      Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
-    filter_upwards [(finiteAngleLaw_absolutelyContinuous W a l).ae_le hb] with p hp
-    exact centeredMass_monotone a ha μ p.2 hp p.1
-  have hs (n : ℕ) : ∀ᵐ x∂Q,x n≤x (n+1) := by
-    let i : Fin (n+1+1) := ⟨n,by omega⟩
-    let j : Fin (n+1+1) := ⟨n+1,by omega⟩
-    have hm : MeasurableSet {x : Fin (n+1+1)→ℝ≥0∞ | x i≤x j} :=
-      measurableSet_le (measurable_pi_apply _) (measurable_pi_apply _)
-    change ∀ᵐ x∂Q,(initialVector (n+1) x) i≤(initialVector (n+1) x) j
-    rw [←ae_map_iff (measurable_initialVector (n+1)).aemeasurable hm,hQlaw]
-    apply (ae_map_iff (Measurable.of_eval
-      (fun index : Fin (n+1+1)=>centeredMass_measurable a ha μ hμ index)).aemeasurable hm).mpr
-    filter_upwards [hX (n+1)] with p hp
-    exact hp (by simp only [i,j]; omega)
-  filter_upwards [ae_all_iff.mpr hs] with x hx
-  exact monotone_nat_of_le_succ hx
-
- 
-
-def diskSequenceMass (x : ℕ→ℝ≥0∞) : ℝ≥0∞ := ⨆n,x n
-
-lemma diskSequenceMass_measurable : Measurable diskSequenceMass :=
-  Measurable.iSup (fun _=>measurable_pi_apply _)
-
-lemma diskSequenceMass_tendsto {x : ℕ→ℝ≥0∞} (hx : Monotone x) :
-    Tendsto x atTop (𝓝 (diskSequenceMass x)) := tendsto_atTop_iSup hx
-
- 
-
-theorem diskSequence_moment (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (Q : Measure (ℕ→ℝ≥0∞))
-    (hQlaw : ∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
-      (fun p=>fun i : Fin (n+1)=>centeredMass a μ i p))
-    (hQmono : ∀ᵐ x∂Q,Monotone x) (m : ℕ) :
-    (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,finiteMoment W a μ m l := by
-  simp only [diskSequenceMass,ENNReal.iSup_pow]
-  rw [lintegral_iSup' (fun _=>(measurable_pi_apply _).pow_const m |>.aemeasurable)
-    (hQmono.mono (fun _ hx _ _ hij=>pow_le_pow_left' (hx hij) m))]
-  congr with l
-  exact lintegral_coordinate Q (finiteAngleLaw W a l) (centeredMass a μ)
-    (centeredMass_measurable a ha μ hμ) l (hQlaw l) (fun x=>x^m)
-    (measurable_id.pow_const m)
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-open TraceMassIntegration TraceMassContinuity CompactSequence
-
- 
-
-theorem actual_centered_finite_moments (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
-      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      (∀v b,IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) ∧
-      (∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
-        Measurable[hS.measurableSpace] (fun b=>μ (S b) b)) ∧
-      (∀ᵐ b∂W,TraceMassFacts κ b (fun v=>μ v (driverHomeomorph b)) ∧
-        (∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),Tendsto
-          (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-             (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
-          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
-        (∀u v : ℝ≥0,u<v →
-          0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) univ)) ∧
-      ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
-        (∀n,Q.map (initialVector n)=(finiteAngleLaw (W.map driverHomeomorph) (Target.a κ) n).map
-          (fun p=>fun i : Fin (n+1)=>centeredMass (Target.a κ) μ i p)) ∧
-        (∀ᵐ x∂Q,Monotone x ∧ Tendsto x atTop (𝓝 (diskSequenceMass x)) ∧ diskSequenceMass x<∞) ∧
-        (∀ m : ℕ, 1 ≤ m →
-          (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ ∧
-          (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,finiteMoment (W.map driverHomeomorph) (Target.a κ) μ m l ∧
-          Tendsto (finiteMoment (W.map driverHomeomorph) (Target.a κ) μ m) atTop
-            (𝓝 (∫⁻x,(diskSequenceMass x)^m∂Q))) := by
-  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
-    actual_canonical_trace_mass κ hκ hκ8 W hW
-  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  let V := W.map driverHomeomorph
-  have hBW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) V := brownian_map_driver W hW
-  have : IsProbabilityMeasure V := hBW.isGaussianProcess.isProbabilityMeasure
-  have (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (μ v b) := (hreg v b).1
-  have hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) :=
-    recoveredMass_joint_measurable (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  have hr (l : ℕ) (θ : Angle) :
-      Measurable[(centeredTime_stopping (Target.a κ) (a_gt_quarter hκ hκ8) θ l).measurableSpace]
-        (fun b=>μ (levelTime (Target.a κ) θ.1 b.1 l) b) := (hstop _ _).1
-  have hp : ∀ᵐ b∂V,TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b) := by
-    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-    filter_upwards [hpath] with b hb
-    change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
-      (fun v=>μ v (driverHomeomorph b))
-    simpa only [Homeomorph.symm_apply_apply] using hb.1
-  have hv : ∀ᵐ b∂V,∀f : C_c(UpperPlane,ℝ),Tendsto
-      (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) ∞ b) atTop (𝓝 (∫z,f z∂μ ∞ b)) := by
-    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-    exact hpath.mono (fun b hb=>hb.2.1 ∞)
-  have hmono : ∀ᵐ b∂V,Monotone (fun v=>μ v b) := hp.mono (fun b hb=>hb.monotone)
-  obtain ⟨Q,hQ,hlaw,hQmono⟩ := exists_centered_sequence_law V hBW (Target.a κ)
-    (a_gt_quarter hκ hκ8) μ hμ hr hmono
-  have he (m : ℕ) : (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,finiteMoment V (Target.a κ) μ m l :=
-    diskSequence_moment V (Target.a κ) (a_gt_quarter hκ hκ8) μ hμ Q hlaw hQmono m
-  have hf (m : ℕ) (hm : 1 ≤ m) : (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ := by
-    obtain ⟨D,hD,hDb⟩ := common_limit_mixed_moment_bound (Target.a κ) (a_gt_quarter hκ hκ8) m hm
-    rw [he]
-    apply lt_of_le_of_lt (iSup_le (fun l=> ?_)) hD
-    exact hDb V hBW s c hc hsum μ (fun v b=>(hreg v b).1) hμ
-      (hmono.mono (fun b hb v=>hb le_top)) hv l
-  have hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞ := ae_lt_top (diskSequenceMass_measurable)
-    (show (∫⁻x,diskSequenceMass x∂Q)≠∞ by simpa only [pow_one] using (hf 1 le_rfl).ne)
-  refine ⟨s,c,hc,hs,hsum,hreg,hμ,fun S hS=>(hstop S hS).1,?_,Q,hQ,hlaw,?_,?_⟩
-  · filter_upwards [hpath] with b hb
-    exact ⟨hb.1,hb.2.1,fun u v huv=>(hb.2.2 u v huv).1⟩
-  · filter_upwards [hQmono,hfin] with x hx hxf
-    exact ⟨hx,diskSequenceMass_tendsto hx,hxf⟩
-  · intro m hm
-    refine ⟨hf m hm,he m,?_⟩
-    rw [he]
-    exact tendsto_atTop_iSup (finiteMoment_monotone V hBW (Target.a κ)
-      (a_gt_quarter hκ hκ8) μ hμ hr hmono m)
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-open TraceMassIntegration TraceMassContinuity
-
-lemma diskMap_norm_lt_one (θ : Angle) (z : UpperPlane) : ‖diskMap θ.1 z.1‖<1 := by
-  let e := upperAffine (Real.cos θ.1) (Real.sin θ.1)
-    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
-  have he := diskMap_affine θ.1 θ.2.1 θ.2.2 (e.symm z).1
-  rw [←upperAffine_apply,Homeomorph.apply_symm_apply] at he
-  rw [he]
-  have h := LiteralTrace.halfPlane_disk_norm_lt_one (e.symm z).2
-    (show 0<Complex.I.im by norm_num)
-  simpa only [map_mul,Complex.star_def,Complex.conj_I,sub_neg_eq_add] using h
-
-lemma diskWeight_pos (a : ℝ) (θ : Angle) (z : UpperPlane) :
-    0<diskWeight a θ.1 z := by
-  apply Real.rpow_pos_of_pos
-  apply norm_pos_iff.mpr
-  rw [disk_deriv θ.1 θ.2.1 θ.2.2 z]
-  apply div_ne_zero
-  · exact mul_ne_zero (Complex.ofReal_ne_zero.mpr (mul_pos (by norm_num)
-      (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)).ne') Complex.I_ne_zero
-  · exact pow_ne_zero _ (disk_denominator_ne θ.1 θ.2.1 θ.2.2 z)
-
-lemma diskMass_unit_pos (a : ℝ) (θ : Angle) (ν : Measure UpperPlane)
-    (hν : 0<ν univ) : 0<diskMass a θ.1 ν (Metric.ball (0:ℂ) 1) := by
-  have he : (fun z : UpperPlane=>diskMap θ.1 z.1) ⁻¹' Metric.ball (0:ℂ) 1=univ := by
-    ext z
-    simp only [mem_preimage,Metric.mem_ball,dist_zero_right,mem_univ,iff_true]
-    exact diskMap_norm_lt_one θ z
-  rw [diskMass,Measure.map_apply (diskMap_measurable θ.1) Metric.isOpen_ball.measurableSet,
-    he,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,
-    lintegral_pos_iff_support ((diskWeight_measurable a θ.1 θ.2.1 θ.2.2).ennreal_ofReal)]
-  have hs : Function.support (fun z : UpperPlane=>ENNReal.ofReal (diskWeight a θ.1 z))=univ := by
-    ext z
-    simp only [Function.mem_support,mem_univ,iff_true]
-    exact (ENNReal.ofReal_pos.mpr (diskWeight_pos a θ z)).ne'
-  rw [hs]
-  exact hν
-
- 
-
-def subdiskMass (a : ℝ) (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (r : ℝ) (l : ℕ) (p : Angle×GDriver) : ℝ≥0∞ :=
-  diskMass a p.1.1 (μ (levelTime a p.1.1 p.2.1 l) p.2) (Metric.closedBall (0:ℂ) r)
-
-lemma subdiskMass_quarter (a : ℝ) (μ : ℝ≥0∞→GDriver→Measure UpperPlane) :
-    subdiskMass a μ (1/4)=centeredMass a μ := rfl
-
-lemma subdiskMass_measurable (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (r : ℝ) (l : ℕ) :
-    Measurable (subdiskMass a μ r l) := by
-  exact (Measure.measurable_coe Metric.isClosed_closedBall.measurableSet).comp
-    (diskMass_joint_measurable a Prod.fst measurable_fst
-      (fun p : Angle×GDriver=>μ (levelTime a p.1.1 p.2.1 l) p.2)
-      (hμ.comp ((centeredTime_measurable a ha l).prodMk measurable_snd)))
-
-lemma centeredTime_pos (a : ℝ) (ha : 0<a) (θ : Angle) (b : GDriver)
-    (l : ℕ) (hl : 0<l) : 0<levelTime a θ.1 b.1 l := by
-  rw [levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous]
-  apply bot_lt_iff_ne_bot.mpr
-  have hz : 0<(fromDisk θ.1 0).im := by
-    rw [fromDisk_zero_im]
-    exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
-  simpa only [ENNReal.bot_eq_zero,fromDisk_zero_im] using radialTime_ne_zero a ha (Real.exp (-2*a*l))
-    (Real.exp_pos _) (Real.exp_lt_one_iff.mpr (by
-      have hl' : (0:ℝ)<l := Nat.cast_pos.mpr hl
-      nlinarith)) b (fromDisk θ.1 0) hz
-
- 
-
-def subdiskRadius (n : ℕ) : ℝ := 1-1/(n+2:ℝ)
-
-lemma subdiskRadius_pos (n : ℕ) : 0<subdiskRadius n := by
-  unfold subdiskRadius
-  have hn : (1:ℝ)<n+2 := by linarith [Nat.cast_nonneg (α := ℝ) n]
-  have hn0 : (0:ℝ)<n+2 := by positivity
-  exact sub_pos.mpr ((div_lt_one hn0).mpr hn)
-
-lemma subdiskRadius_lt_one (n : ℕ) : subdiskRadius n<1 := by
-  unfold subdiskRadius
-  have : (0:ℝ)<1/(n+2:ℝ) := by positivity
-  linarith
-
-lemma unitBall_eq_iUnion_subdisk : Metric.ball (0:ℂ) 1=
-    ⋃n : ℕ,Metric.closedBall (0:ℂ) (subdiskRadius n) := by
-  ext z
-  simp only [Metric.mem_ball,Metric.mem_closedBall,dist_zero_right,mem_iUnion]
-  constructor
-  · intro hz
-    obtain ⟨n,hn⟩ := exists_nat_one_div_lt (sub_pos.mpr hz)
-    refine ⟨n,?_⟩
-    have hle : (1:ℝ)/(n+2)≤1/(n+1) :=
-      one_div_le_one_div_of_le (by positivity) (by linarith)
-    dsimp [subdiskRadius]
-    linarith
-  · rintro ⟨n,hn⟩
-    exact hn.trans_lt (subdiskRadius_lt_one n)
-
- 
-
-theorem exists_subdisk_positive_mean (W : Measure GDriver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hpos : ∀ᵐ b∂W,∀v : ℝ≥0∞,0<v → 0<μ v b univ) :
-    ∃n : ℕ,0<∫⁻p,subdiskMass a μ (subdiskRadius n) 1 p∂finiteAngleLaw W a 1 := by
-  have : IsProbabilityMeasure (finiteAngleLaw W a 1) := finiteAngleLaw_probability W hW a ha 1
-  have hp : ∀ᵐ p∂finiteAngleLaw W a 1,
-      0<diskMass a p.1.1 (μ (levelTime a p.1.1 p.2.1 1) p.2) (Metric.ball (0:ℂ) 1) := by
-    have hb : ∀ᵐ p∂(angleLaw a).prod W,∀v : ℝ≥0∞,0<v → 0<μ v p.2 univ :=
-      Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hpos
-    filter_upwards [(finiteAngleLaw_absolutelyContinuous W a 1).ae_le hb] with p hp
-    exact diskMass_unit_pos a p.1 _ (hp _ (centeredTime_pos a (by linarith) p.1 p.2 1 (by omega)))
-  by_contra! hn
-  have hzero (n : ℕ) : ∀ᵐ p∂finiteAngleLaw W a 1,subdiskMass a μ (subdiskRadius n) 1 p=0 :=
-    (lintegral_eq_zero_iff (subdiskMass_measurable a ha μ hμ (subdiskRadius n) 1)).mp
-      (le_antisymm (hn n) bot_le)
-  have hf : ∀ᵐ p∂finiteAngleLaw W a 1,False := by
-    filter_upwards [hp,ae_all_iff.mpr hzero] with p hp hz
-    rw [unitBall_eq_iUnion_subdisk] at hp
-    have hnull := measure_iUnion_null (fun n=>hz n)
-    exact hp.ne' hnull
-  simpa using ae_iff.mp hf
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions KernelSimilarity
-
-lemma cayley_subdisk_bounds (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
-    (hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤r) :
-    dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im := by
-  have hn : 1≤‖z.1+Complex.I‖ := by
-    apply le_trans _ (Complex.im_le_norm _)
-    simp only [Complex.add_im,Complex.I_im]
-    linarith [z.2]
-  have hp : 0<‖z.1+Complex.I‖ := lt_of_lt_of_le (by norm_num) hn
-  have hz' : ‖z.1-Complex.I‖≤r*‖z.1+Complex.I‖ := by
-    rw [norm_div] at hz
-    exact (div_le_iff₀ hp).mp hz
-  have htri : ‖z.1+Complex.I‖≤‖z.1-Complex.I‖+2 := by
-    calc
-      _ = ‖(z.1-Complex.I)+(Complex.I+Complex.I)‖ := by congr 1; ring
-      _ ≤ ‖z.1-Complex.I‖+‖Complex.I+Complex.I‖ := norm_add_le _ _
-      _ ≤ ‖z.1-Complex.I‖+2 := by
-        have := norm_add_le Complex.I Complex.I
-        simp only [Complex.norm_I] at this
-        linarith
-  constructor
-  · rw [dist_eq_norm,le_div_iff₀ (sub_pos.mpr hr1)]
-    nlinarith [mul_le_mul_of_nonneg_left htri hr]
-  · have h2 : ‖z.1-Complex.I‖^2≤r^2*‖z.1+Complex.I‖^2 := by
-      simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _) hz' 2
-    have he : ‖z.1+Complex.I‖^2-‖z.1-Complex.I‖^2=4*z.1.im := by
-      simp only [Complex.sq_norm,Complex.normSq_apply,Complex.add_re,Complex.add_im,
-        Complex.sub_re,Complex.sub_im,Complex.I_re,Complex.I_im,add_zero,sub_zero]
-      ring
-    have hrr : 0≤1-r^2 := by nlinarith
-    have hsq : 1≤‖z.1+Complex.I‖^2 := by nlinarith
-    nlinarith [mul_le_mul_of_nonneg_left hsq hrr]
-
-lemma subdisk_region_compact (r : ℝ) (hr : 0≤r) (hr1 : r<1) :
-    IsCompact {z : UpperPlane | dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im} := by
-  have hη : 0<(1-r^2)/4 := by nlinarith
-  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
-  have he : (Subtype.val : UpperPlane→ℂ) ''
-      {z | dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im}=
-      Metric.closedBall Complex.I (2*r/(1-r)) ∩ {z : ℂ | (1-r^2)/4≤z.im} := by
-    ext z
-    constructor
-    · rintro ⟨w,hw,rfl⟩; exact hw
-    · rintro ⟨hz,hi⟩
-      exact ⟨⟨z,hη.trans_le hi⟩,⟨hz,hi⟩,rfl⟩
-  rw [he]
-  exact (isCompact_closedBall _ _).inter_right (isClosed_le continuous_const Complex.continuous_im)
-
-lemma subdisk_cutoff_exists (r : ℝ) (hr : 0≤r) (hr1 : r<1) :
-    ∃f : C_c(UpperPlane,ℝ),
-      (∀z,0≤f z ∧ f z≤1) ∧
-      (∀z : UpperPlane, ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤r → f z=1) ∧
-      (∀z∈tsupport f,(1-r^2)/8<z.1.im) := by
-  have hη : 0<(1-r^2)/4 := by nlinarith
-  obtain ⟨f,hf,hfc,hfs,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen
-    (subdisk_region_compact r hr hr1)
-    (isOpen_lt continuous_const (Complex.continuous_im.comp continuous_subtype_val))
-    (show {z : UpperPlane | dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im} ⊆
-        {z : UpperPlane | (1-r^2)/8<z.1.im} from fun z hz=>by dsimp; linarith [hz.2])
-  exact ⟨⟨f,hfc⟩,h01,fun z hz=>hf (cayley_subdisk_bounds r hr hr1 z hz),fun z hz=>hfs hz⟩
-
-def subdiskCutoff (r : ℝ) (hr : 0≤r) (hr1 : r<1) : C_c(UpperPlane,ℝ) :=
-  Classical.choose (subdisk_cutoff_exists r hr hr1)
-
-lemma subdiskCutoff_nonneg (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane) :
-    0≤ subdiskCutoff r hr hr1 z :=
-  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).1 z |>.1
-lemma subdiskCutoff_le_one (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane) :
-    subdiskCutoff r hr hr1 z≤1 :=
-  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).1 z |>.2
-lemma subdiskCutoff_one (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
-    (hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤r) : subdiskCutoff r hr hr1 z=1 :=
-  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).2.1 z hz
-lemma subdiskCutoff_support (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
-    (hz : z∈tsupport (subdiskCutoff r hr hr1)) : (1-r^2)/8<z.1.im :=
-  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).2.2 z hz
-
-def subdiskTest (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1) : C_c(UpperPlane,ℝ) :=
-  (2^(2-alpha a):ℝ) • normalizedTest a (Real.cos θ.1) (Real.sin θ.1)
-    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) (subdiskCutoff r hr hr1)
-
-lemma subdiskTest_nonneg (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1)
-    (z : UpperPlane) : 0≤ subdiskTest a θ r hr hr1 z :=
-  mul_nonneg (Real.rpow_nonneg (by norm_num) _)
-    (normalizedTest_nonneg a _ _ _ _ (subdiskCutoff_nonneg r hr hr1) z)
-
-lemma subdiskTest_bound (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1)
-    (z : UpperPlane) : ‖subdiskTest a θ r hr hr1 z‖≤2^(2-alpha a)*(Real.sin θ.1)^(alpha a-2) := by
-  change ‖(2^(2-alpha a):ℝ)*normalizedTest a _ _ _ _ z‖≤_
-  rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (by norm_num) _)]
-  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
-  simpa only [one_mul] using normalizedTest_bound a _ _ _ (subdiskCutoff r hr hr1) (B:=1)
-    (fun z=>by rw [Real.norm_eq_abs,abs_of_nonneg (subdiskCutoff_nonneg r hr hr1 z)]
-               exact subdiskCutoff_le_one r hr hr1 z) z
-
-lemma subdiskTest_support (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1) :
-    Subtype.val '' tsupport (subdiskTest a θ r hr hr1) ⊆
-      affine (Real.cos θ.1:ℂ) (Real.sin θ.1) (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) ''
-        (Subtype.val '' tsupport (subdiskCutoff r hr hr1)) := by
-  apply (image_mono (tsupport_smul_subset_right (fun _ : UpperPlane => (2^(2-alpha a):ℝ))
-    (normalizedTest a (Real.cos θ.1) (Real.sin θ.1) (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
-      (subdiskCutoff r hr hr1)))).trans
-  exact normalizedTest_support a _ _ _ _
-
-lemma diskWeight_le_subdiskTest (a : ℝ) (ha : 1/4<a) (θ : Angle)
-    (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
-    (hz : diskMap θ.1 z.1∈Metric.closedBall (0:ℂ) r) :
-    diskWeight a θ.1 z≤ subdiskTest a θ r hr hr1 z := by
-  have hy := Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
-  have he : ‖((((upperAffine (Real.cos θ.1) (Real.sin θ.1) hy).symm z).1)-Complex.I)/
-      ((((upperAffine (Real.cos θ.1) (Real.sin θ.1) hy).symm z).1)+Complex.I)‖≤r := by
-    have h := diskMap_affine θ.1 θ.2.1 θ.2.2
-      (((upperAffine (Real.cos θ.1) (Real.sin θ.1) hy).symm z).1)
-    rw [←upperAffine_apply,Homeomorph.apply_symm_apply] at h
-    simpa only [Metric.mem_closedBall,dist_zero_right,h] using hz
-  change ‖deriv (diskMap θ.1) z.1‖^(2-alpha a)≤
-    2^(2-alpha a)*((Real.sin θ.1)^(alpha a-2)*subdiskCutoff r hr hr1 _)
-  rw [subdiskCutoff_one r hr hr1 _ he,mul_one]
-  calc
-    _ ≤ (2/(Real.sin θ.1))^(2-alpha a) := Real.rpow_le_rpow (norm_nonneg _)
-      (disk_deriv_bound θ.1 θ.2.1 θ.2.2 z) (dimension_pos a ha).le
-    _ = _ := by
-      rw [div_eq_mul_inv,Real.mul_rpow (by norm_num : (0:ℝ)≤2) (inv_nonneg.mpr hy.le),
-        Real.inv_rpow hy.le,←Real.rpow_neg hy.le,neg_sub]
-
-lemma subdiskMass_le_enorm (a : ℝ) (ha : 1/4<a) (θ : Angle)
-    (r : ℝ) (hr : 0≤r) (hr1 : r<1)
-    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] :
-    diskMass a θ.1 ν (Metric.closedBall (0:ℂ) r)≤‖∫z,subdiskTest a θ r hr hr1 z∂ν‖ₑ := by
-  apply le_trans _ (Real.ofReal_le_enorm _)
-  rw [diskMass,Measure.map_apply (diskMap_measurable θ.1) Metric.isClosed_closedBall.measurableSet,
-    withDensity_apply _ ((diskMap_measurable θ.1) Metric.isClosed_closedBall.measurableSet),
-    ofReal_integral_eq_lintegral_ofReal (subdiskTest a θ r hr hr1).integrable
-      (Eventually.of_forall (subdiskTest_nonneg a θ r hr hr1)),
-    ←lintegral_indicator ((diskMap_measurable θ.1) Metric.isClosed_closedBall.measurableSet)]
-  apply lintegral_mono
-  intro z
-  by_cases hz : diskMap θ.1 z.1∈Metric.closedBall (0:ℂ) r
-  · simp only [Set.indicator, Set.mem_preimage, hz, ite_true]
-    exact ENNReal.ofReal_le_ofReal (diskWeight_le_subdiskTest a ha θ r hr hr1 z hz)
-  · simp only [Set.indicator, Set.mem_preimage, hz, ite_false]; positivity
-
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.NormalizedRadialTests
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
+
+lemma centeredMass_raw_measurable_of_le (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) {i l : ℕ} (hil : i≤l)
+    (hμ : Measurable[(centeredTime_stopping a ha θ i).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 i) b)) :
+    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>centeredMass a μ i (θ,b)) :=
+  (centeredMass_raw_measurable a ha μ θ i hμ).mono
+    ((centeredTime_stopping a ha θ i).measurableSpace_mono (centeredTime_stopping a ha θ l)
+      (fun b=>centeredTime_monotone a ha θ b hil)) le_rfl
+
+lemma centered_vector_consistent (W : Measure ExactSLEGenerator.Driver)
+    [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b))
+    {l k : ℕ} (hlk : l≤k) :
+    (finiteAngleLaw W a k).map (fun p=>fun i : Fin (l+1)=>centeredMass a μ i p)=
+      (finiteAngleLaw W a l).map (fun p=>fun i : Fin (l+1)=>centeredMass a μ i p) := by
+  have hm : Measurable (fun p=>fun i : Fin (l+1)=>centeredMass a μ i p) :=
+    Measurable.of_eval (fun index=>centeredMass_measurable a ha μ hμ index)
+  apply Measure.ext
+  intro A hA
+  rw [Measure.map_apply hm hA,Measure.map_apply hm hA]
+  simp only [←lintegral_indicator_one (hm hA)]
+  apply finiteAngleLaw_lintegral_consistent W hW a ha hlk
+  · exact measurable_const.indicator (hm hA)
+  · intro θ
+    let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
+    have hv : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+        (fun b=>fun i : Fin (l+1)=>centeredMass a μ i (θ,b)) :=
+      Measurable.of_eval (fun index=>centeredMass_raw_measurable_of_le a ha μ θ
+        (by omega) (hμraw index θ))
+    exact measurable_const.indicator (hv hA)
+
+ 
+
+theorem exists_centered_sequence_law (W : Measure ExactSLEGenerator.Driver)
+    [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b))
+    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) :
+    ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
+      (∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
+        (fun p=>fun i : Fin (n+1)=>centeredMass a μ i p)) ∧
+      (∀ᵐ x∂Q,Monotone x) := by
+  have (n : ℕ) : IsProbabilityMeasure (finiteAngleLaw W a n) :=
+    finiteAngleLaw_probability W hW a ha n
+  obtain ⟨Q,hQ,hQlaw⟩ := exists_sequence_law (finiteAngleLaw W a)
+    (centeredMass a μ) (centeredMass_measurable a ha μ hμ)
+    (fun _ _ hlk=>centered_vector_consistent W hW a ha μ hμ hμraw hlk)
+  refine ⟨Q,hQ,hQlaw,?_⟩
+  have hX (l : ℕ) : ∀ᵐ p∂finiteAngleLaw W a l,Monotone (fun n=>centeredMass a μ n p) := by
+    have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
+      Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
+    filter_upwards [(finiteAngleLaw_absolutelyContinuous W a l).ae_le hb] with p hp
+    exact centeredMass_monotone a ha μ p.2 hp p.1
+  have hs (n : ℕ) : ∀ᵐ x∂Q,x n≤x (n+1) := by
+    let i : Fin (n+1+1) := ⟨n,by omega⟩
+    let j : Fin (n+1+1) := ⟨n+1,by omega⟩
+    have hm : MeasurableSet {x : Fin (n+1+1)→ℝ≥0∞ | x i≤x j} :=
+      measurableSet_le (measurable_pi_apply _) (measurable_pi_apply _)
+    change ∀ᵐ x∂Q,(initialVector (n+1) x) i≤(initialVector (n+1) x) j
+    rw [←ae_map_iff (measurable_initialVector (n+1)).aemeasurable hm,hQlaw]
+    apply (ae_map_iff (Measurable.of_eval
+      (fun index : Fin (n+1+1)=>centeredMass_measurable a ha μ hμ index)).aemeasurable hm).mpr
+    filter_upwards [hX (n+1)] with p hp
+    exact hp (by simp only [i,j]; omega)
+  filter_upwards [ae_all_iff.mpr hs] with x hx
+  exact monotone_nat_of_le_succ hx
+
+ 
+
+def diskSequenceMass (x : ℕ→ℝ≥0∞) : ℝ≥0∞ := ⨆n,x n
+
+lemma diskSequenceMass_measurable : Measurable diskSequenceMass :=
+  Measurable.iSup (fun _=>measurable_pi_apply _)
+
+lemma diskSequenceMass_tendsto {x : ℕ→ℝ≥0∞} (hx : Monotone x) :
+    Tendsto x atTop (𝓝 (diskSequenceMass x)) := tendsto_atTop_iSup hx
+
+ 
+
+theorem diskSequence_moment (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (Q : Measure (ℕ→ℝ≥0∞))
+    (hQlaw : ∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
+      (fun p=>fun i : Fin (n+1)=>centeredMass a μ i p))
+    (hQmono : ∀ᵐ x∂Q,Monotone x) (m : ℕ) :
+    (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,finiteMoment W a μ m l := by
+  simp only [diskSequenceMass,ENNReal.iSup_pow]
+  rw [lintegral_iSup' (fun _=>(measurable_pi_apply _).pow_const m |>.aemeasurable)
+    (hQmono.mono (fun _ hx _ _ hij=>pow_le_pow_left' (hx hij) m))]
+  congr with l
+  exact lintegral_coordinate Q (finiteAngleLaw W a l) (centeredMass a μ)
+    (centeredMass_measurable a ha μ hμ) l (hQlaw l) (fun x=>x^m)
+    (measurable_id.pow_const m)
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+open TraceMassIntegration TraceMassContinuity CompactSequence
+
+ 
+
+theorem actual_centered_finite_moments (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
+      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      (∀v b,IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) ∧
+      (∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
+        Measurable[hS.measurableSpace] (fun b=>μ (S b) b)) ∧
+      (∀ᵐ b∂W,TraceMassFacts κ b (fun v=>μ v (driverHomeomorph b)) ∧
+        (∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),Tendsto
+          (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+             (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
+          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
+        (∀u v : ℝ≥0,u<v →
+          0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) univ)) ∧
+      ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
+        (∀n,Q.map (initialVector n)=(finiteAngleLaw (W.map driverHomeomorph) (Target.a κ) n).map
+          (fun p=>fun i : Fin (n+1)=>centeredMass (Target.a κ) μ i p)) ∧
+        (∀ᵐ x∂Q,Monotone x ∧ Tendsto x atTop (𝓝 (diskSequenceMass x)) ∧ diskSequenceMass x<∞) ∧
+        (∀ m : ℕ, 1 ≤ m →
+          (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ ∧
+          (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,finiteMoment (W.map driverHomeomorph) (Target.a κ) μ m l ∧
+          Tendsto (finiteMoment (W.map driverHomeomorph) (Target.a κ) μ m) atTop
+            (𝓝 (∫⁻x,(diskSequenceMass x)^m∂Q))) := by
+  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
+    actual_canonical_trace_mass κ hκ hκ8 W hW
+  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  let V := W.map driverHomeomorph
+  have hBW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) V := brownian_map_driver W hW
+  have : IsProbabilityMeasure V := hBW.isGaussianProcess.isProbabilityMeasure
+  have (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (μ v b) := (hreg v b).1
+  have hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) :=
+    recoveredMass_joint_measurable (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  have hr (l : ℕ) (θ : Angle) :
+      Measurable[(centeredTime_stopping (Target.a κ) (a_gt_quarter hκ hκ8) θ l).measurableSpace]
+        (fun b=>μ (levelTime (Target.a κ) θ.1 b.1 l) b) := (hstop _ _).1
+  have hp : ∀ᵐ b∂V,TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b) := by
+    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+    filter_upwards [hpath] with b hb
+    change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
+      (fun v=>μ v (driverHomeomorph b))
+    simpa only [Homeomorph.symm_apply_apply] using hb.1
+  have hv : ∀ᵐ b∂V,∀f : C_c(UpperPlane,ℝ),Tendsto
+      (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) ∞ b) atTop (𝓝 (∫z,f z∂μ ∞ b)) := by
+    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+    exact hpath.mono (fun b hb=>hb.2.1 ∞)
+  have hmono : ∀ᵐ b∂V,Monotone (fun v=>μ v b) := hp.mono (fun b hb=>hb.monotone)
+  obtain ⟨Q,hQ,hlaw,hQmono⟩ := exists_centered_sequence_law V hBW (Target.a κ)
+    (a_gt_quarter hκ hκ8) μ hμ hr hmono
+  have he (m : ℕ) : (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,finiteMoment V (Target.a κ) μ m l :=
+    diskSequence_moment V (Target.a κ) (a_gt_quarter hκ hκ8) μ hμ Q hlaw hQmono m
+  have hf (m : ℕ) (hm : 1 ≤ m) : (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ := by
+    obtain ⟨D,hD,hDb⟩ := common_limit_mixed_moment_bound (Target.a κ) (a_gt_quarter hκ hκ8) m hm
+    rw [he]
+    apply lt_of_le_of_lt (iSup_le (fun l=> ?_)) hD
+    exact hDb V hBW s c hc hsum μ (fun v b=>(hreg v b).1) hμ
+      (hmono.mono (fun b hb v=>hb le_top)) hv l
+  have hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞ := ae_lt_top (diskSequenceMass_measurable)
+    (show (∫⁻x,diskSequenceMass x∂Q)≠∞ by simpa only [pow_one] using (hf 1 le_rfl).ne)
+  refine ⟨s,c,hc,hs,hsum,hreg,hμ,fun S hS=>(hstop S hS).1,?_,Q,hQ,hlaw,?_,?_⟩
+  · filter_upwards [hpath] with b hb
+    exact ⟨hb.1,hb.2.1,fun u v huv=>(hb.2.2 u v huv).1⟩
+  · filter_upwards [hQmono,hfin] with x hx hxf
+    exact ⟨hx,diskSequenceMass_tendsto hx,hxf⟩
+  · intro m hm
+    refine ⟨hf m hm,he m,?_⟩
+    rw [he]
+    exact tendsto_atTop_iSup (finiteMoment_monotone V hBW (Target.a κ)
+      (a_gt_quarter hκ hκ8) μ hμ hr hmono m)
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+open TraceMassIntegration TraceMassContinuity
+
+lemma diskMap_norm_lt_one (θ : Angle) (z : UpperPlane) : ‖diskMap θ.1 z.1‖<1 := by
+  let e := upperAffine (Real.cos θ.1) (Real.sin θ.1)
+    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
+  have he := diskMap_affine θ.1 θ.2.1 θ.2.2 (e.symm z).1
+  rw [←upperAffine_apply,Homeomorph.apply_symm_apply] at he
+  rw [he]
+  have h := LiteralTrace.halfPlane_disk_norm_lt_one (e.symm z).2
+    (show 0<Complex.I.im by norm_num)
+  simpa only [map_mul,Complex.star_def,Complex.conj_I,sub_neg_eq_add] using h
+
+lemma diskWeight_pos (a : ℝ) (θ : Angle) (z : UpperPlane) :
+    0<diskWeight a θ.1 z := by
+  apply Real.rpow_pos_of_pos
+  apply norm_pos_iff.mpr
+  rw [disk_deriv θ.1 θ.2.1 θ.2.2 z]
+  apply div_ne_zero
+  · exact mul_ne_zero (Complex.ofReal_ne_zero.mpr (mul_pos (by norm_num)
+      (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)).ne') Complex.I_ne_zero
+  · exact pow_ne_zero _ (disk_denominator_ne θ.1 θ.2.1 θ.2.2 z)
+
+lemma diskMass_unit_pos (a : ℝ) (θ : Angle) (ν : Measure UpperPlane)
+    (hν : 0<ν univ) : 0<diskMass a θ.1 ν (Metric.ball (0:ℂ) 1) := by
+  have he : (fun z : UpperPlane=>diskMap θ.1 z.1) ⁻¹' Metric.ball (0:ℂ) 1=univ := by
+    ext z
+    simp only [mem_preimage,Metric.mem_ball,dist_zero_right,mem_univ,iff_true]
+    exact diskMap_norm_lt_one θ z
+  rw [diskMass,Measure.map_apply (diskMap_measurable θ.1) Metric.isOpen_ball.measurableSet,
+    he,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,
+    lintegral_pos_iff_support ((diskWeight_measurable a θ.1 θ.2.1 θ.2.2).ennreal_ofReal)]
+  have hs : Function.support (fun z : UpperPlane=>ENNReal.ofReal (diskWeight a θ.1 z))=univ := by
+    ext z
+    simp only [Function.mem_support,mem_univ,iff_true]
+    exact (ENNReal.ofReal_pos.mpr (diskWeight_pos a θ z)).ne'
+  rw [hs]
+  exact hν
+
+ 
+
+def subdiskMass (a : ℝ) (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (r : ℝ) (l : ℕ) (p : Angle×GDriver) : ℝ≥0∞ :=
+  diskMass a p.1.1 (μ (levelTime a p.1.1 p.2.1 l) p.2) (Metric.closedBall (0:ℂ) r)
+
+lemma subdiskMass_quarter (a : ℝ) (μ : ℝ≥0∞→GDriver→Measure UpperPlane) :
+    subdiskMass a μ (1/4)=centeredMass a μ := rfl
+
+lemma subdiskMass_measurable (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (r : ℝ) (l : ℕ) :
+    Measurable (subdiskMass a μ r l) := by
+  exact (Measure.measurable_coe Metric.isClosed_closedBall.measurableSet).comp
+    (diskMass_joint_measurable a Prod.fst measurable_fst
+      (fun p : Angle×GDriver=>μ (levelTime a p.1.1 p.2.1 l) p.2)
+      (hμ.comp ((centeredTime_measurable a ha l).prodMk measurable_snd)))
+
+lemma centeredTime_pos (a : ℝ) (ha : 0<a) (θ : Angle) (b : GDriver)
+    (l : ℕ) (hl : 0<l) : 0<levelTime a θ.1 b.1 l := by
+  rw [levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous]
+  apply bot_lt_iff_ne_bot.mpr
+  have hz : 0<(fromDisk θ.1 0).im := by
+    rw [fromDisk_zero_im]
+    exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
+  simpa only [ENNReal.bot_eq_zero,fromDisk_zero_im] using radialTime_ne_zero a ha (Real.exp (-2*a*l))
+    (Real.exp_pos _) (Real.exp_lt_one_iff.mpr (by
+      have hl' : (0:ℝ)<l := Nat.cast_pos.mpr hl
+      nlinarith)) b (fromDisk θ.1 0) hz
+
+ 
+
+def subdiskRadius (n : ℕ) : ℝ := 1-1/(n+2:ℝ)
+
+lemma subdiskRadius_pos (n : ℕ) : 0<subdiskRadius n := by
+  unfold subdiskRadius
+  have hn : (1:ℝ)<n+2 := by linarith [Nat.cast_nonneg (α := ℝ) n]
+  have hn0 : (0:ℝ)<n+2 := by positivity
+  exact sub_pos.mpr ((div_lt_one hn0).mpr hn)
+
+lemma subdiskRadius_lt_one (n : ℕ) : subdiskRadius n<1 := by
+  unfold subdiskRadius
+  have : (0:ℝ)<1/(n+2:ℝ) := by positivity
+  linarith
+
+lemma unitBall_eq_iUnion_subdisk : Metric.ball (0:ℂ) 1=
+    ⋃n : ℕ,Metric.closedBall (0:ℂ) (subdiskRadius n) := by
+  ext z
+  simp only [Metric.mem_ball,Metric.mem_closedBall,dist_zero_right,mem_iUnion]
+  constructor
+  · intro hz
+    obtain ⟨n,hn⟩ := exists_nat_one_div_lt (sub_pos.mpr hz)
+    refine ⟨n,?_⟩
+    have hle : (1:ℝ)/(n+2)≤1/(n+1) :=
+      one_div_le_one_div_of_le (by positivity) (by linarith)
+    dsimp [subdiskRadius]
+    linarith
+  · rintro ⟨n,hn⟩
+    exact hn.trans_lt (subdiskRadius_lt_one n)
+
+ 
+
+theorem exists_subdisk_positive_mean (W : Measure GDriver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hpos : ∀ᵐ b∂W,∀v : ℝ≥0∞,0<v → 0<μ v b univ) :
+    ∃n : ℕ,0<∫⁻p,subdiskMass a μ (subdiskRadius n) 1 p∂finiteAngleLaw W a 1 := by
+  have : IsProbabilityMeasure (finiteAngleLaw W a 1) := finiteAngleLaw_probability W hW a ha 1
+  have hp : ∀ᵐ p∂finiteAngleLaw W a 1,
+      0<diskMass a p.1.1 (μ (levelTime a p.1.1 p.2.1 1) p.2) (Metric.ball (0:ℂ) 1) := by
+    have hb : ∀ᵐ p∂(angleLaw a).prod W,∀v : ℝ≥0∞,0<v → 0<μ v p.2 univ :=
+      Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hpos
+    filter_upwards [(finiteAngleLaw_absolutelyContinuous W a 1).ae_le hb] with p hp
+    exact diskMass_unit_pos a p.1 _ (hp _ (centeredTime_pos a (by linarith) p.1 p.2 1 (by omega)))
+  by_contra! hn
+  have hzero (n : ℕ) : ∀ᵐ p∂finiteAngleLaw W a 1,subdiskMass a μ (subdiskRadius n) 1 p=0 :=
+    (lintegral_eq_zero_iff (subdiskMass_measurable a ha μ hμ (subdiskRadius n) 1)).mp
+      (le_antisymm (hn n) bot_le)
+  have hf : ∀ᵐ p∂finiteAngleLaw W a 1,False := by
+    filter_upwards [hp,ae_all_iff.mpr hzero] with p hp hz
+    rw [unitBall_eq_iUnion_subdisk] at hp
+    have hnull := measure_iUnion_null (fun n=>hz n)
+    exact hp.ne' hnull
+  simpa using ae_iff.mp hf
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions KernelSimilarity
+
+lemma cayley_subdisk_bounds (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
+    (hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤r) :
+    dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im := by
+  have hn : 1≤‖z.1+Complex.I‖ := by
+    apply le_trans _ (Complex.im_le_norm _)
+    simp only [Complex.add_im,Complex.I_im]
+    linarith [z.2]
+  have hp : 0<‖z.1+Complex.I‖ := lt_of_lt_of_le (by norm_num) hn
+  have hz' : ‖z.1-Complex.I‖≤r*‖z.1+Complex.I‖ := by
+    rw [norm_div] at hz
+    exact (div_le_iff₀ hp).mp hz
+  have htri : ‖z.1+Complex.I‖≤‖z.1-Complex.I‖+2 := by
+    calc
+      _ = ‖(z.1-Complex.I)+(Complex.I+Complex.I)‖ := by congr 1; ring
+      _ ≤ ‖z.1-Complex.I‖+‖Complex.I+Complex.I‖ := norm_add_le _ _
+      _ ≤ ‖z.1-Complex.I‖+2 := by
+        have := norm_add_le Complex.I Complex.I
+        simp only [Complex.norm_I] at this
+        linarith
+  constructor
+  · rw [dist_eq_norm,le_div_iff₀ (sub_pos.mpr hr1)]
+    nlinarith [mul_le_mul_of_nonneg_left htri hr]
+  · have h2 : ‖z.1-Complex.I‖^2≤r^2*‖z.1+Complex.I‖^2 := by
+      simpa only [mul_pow] using pow_le_pow_left₀ (norm_nonneg _) hz' 2
+    have he : ‖z.1+Complex.I‖^2-‖z.1-Complex.I‖^2=4*z.1.im := by
+      simp only [Complex.sq_norm,Complex.normSq_apply,Complex.add_re,Complex.add_im,
+        Complex.sub_re,Complex.sub_im,Complex.I_re,Complex.I_im,add_zero,sub_zero]
+      ring
+    have hrr : 0≤1-r^2 := by nlinarith
+    have hsq : 1≤‖z.1+Complex.I‖^2 := by nlinarith
+    nlinarith [mul_le_mul_of_nonneg_left hsq hrr]
+
+lemma subdisk_region_compact (r : ℝ) (hr : 0≤r) (hr1 : r<1) :
+    IsCompact {z : UpperPlane | dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im} := by
+  have hη : 0<(1-r^2)/4 := by nlinarith
+  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
+  have he : (Subtype.val : UpperPlane→ℂ) ''
+      {z | dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im}=
+      Metric.closedBall Complex.I (2*r/(1-r)) ∩ {z : ℂ | (1-r^2)/4≤z.im} := by
+    ext z
+    constructor
+    · rintro ⟨w,hw,rfl⟩; exact hw
+    · rintro ⟨hz,hi⟩
+      exact ⟨⟨z,hη.trans_le hi⟩,⟨hz,hi⟩,rfl⟩
+  rw [he]
+  exact (isCompact_closedBall _ _).inter_right (isClosed_le continuous_const Complex.continuous_im)
+
+lemma subdisk_cutoff_exists (r : ℝ) (hr : 0≤r) (hr1 : r<1) :
+    ∃f : C_c(UpperPlane,ℝ),
+      (∀z,0≤f z ∧ f z≤1) ∧
+      (∀z : UpperPlane, ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤r → f z=1) ∧
+      (∀z∈tsupport f,(1-r^2)/8<z.1.im) := by
+  have hη : 0<(1-r^2)/4 := by nlinarith
+  obtain ⟨f,hf,hfc,hfs,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen
+    (subdisk_region_compact r hr hr1)
+    (isOpen_lt continuous_const (Complex.continuous_im.comp continuous_subtype_val))
+    (show {z : UpperPlane | dist z.1 Complex.I≤2*r/(1-r) ∧ (1-r^2)/4≤z.1.im} ⊆
+        {z : UpperPlane | (1-r^2)/8<z.1.im} from fun z hz=>by dsimp; linarith [hz.2])
+  exact ⟨⟨f,hfc⟩,h01,fun z hz=>hf (cayley_subdisk_bounds r hr hr1 z hz),fun z hz=>hfs hz⟩
+
+def subdiskCutoff (r : ℝ) (hr : 0≤r) (hr1 : r<1) : C_c(UpperPlane,ℝ) :=
+  Classical.choose (subdisk_cutoff_exists r hr hr1)
+
+lemma subdiskCutoff_nonneg (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane) :
+    0≤ subdiskCutoff r hr hr1 z :=
+  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).1 z |>.1
+lemma subdiskCutoff_le_one (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane) :
+    subdiskCutoff r hr hr1 z≤1 :=
+  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).1 z |>.2
+lemma subdiskCutoff_one (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
+    (hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤r) : subdiskCutoff r hr hr1 z=1 :=
+  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).2.1 z hz
+lemma subdiskCutoff_support (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
+    (hz : z∈tsupport (subdiskCutoff r hr hr1)) : (1-r^2)/8<z.1.im :=
+  (Classical.choose_spec (subdisk_cutoff_exists r hr hr1)).2.2 z hz
+
+def subdiskTest (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1) : C_c(UpperPlane,ℝ) :=
+  (2^(2-alpha a):ℝ) • normalizedTest a (Real.cos θ.1) (Real.sin θ.1)
+    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) (subdiskCutoff r hr hr1)
+
+lemma subdiskTest_nonneg (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1)
+    (z : UpperPlane) : 0≤ subdiskTest a θ r hr hr1 z :=
+  mul_nonneg (Real.rpow_nonneg (by norm_num) _)
+    (normalizedTest_nonneg a _ _ _ _ (subdiskCutoff_nonneg r hr hr1) z)
+
+lemma subdiskTest_bound (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1)
+    (z : UpperPlane) : ‖subdiskTest a θ r hr hr1 z‖≤2^(2-alpha a)*(Real.sin θ.1)^(alpha a-2) := by
+  change ‖(2^(2-alpha a):ℝ)*normalizedTest a _ _ _ _ z‖≤_
+  rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (by norm_num) _)]
+  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
+  simpa only [one_mul] using normalizedTest_bound a _ _ _ (subdiskCutoff r hr hr1) (B:=1)
+    (fun z=>by rw [Real.norm_eq_abs,abs_of_nonneg (subdiskCutoff_nonneg r hr hr1 z)]
+               exact subdiskCutoff_le_one r hr hr1 z) z
+
+lemma subdiskTest_support (a : ℝ) (θ : Angle) (r : ℝ) (hr : 0≤r) (hr1 : r<1) :
+    Subtype.val '' tsupport (subdiskTest a θ r hr hr1) ⊆
+      affine (Real.cos θ.1:ℂ) (Real.sin θ.1) (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) ''
+        (Subtype.val '' tsupport (subdiskCutoff r hr hr1)) := by
+  apply (image_mono (tsupport_smul_subset_right (fun _ : UpperPlane => (2^(2-alpha a):ℝ))
+    (normalizedTest a (Real.cos θ.1) (Real.sin θ.1) (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
+      (subdiskCutoff r hr hr1)))).trans
+  exact normalizedTest_support a _ _ _ _
+
+lemma diskWeight_le_subdiskTest (a : ℝ) (ha : 1/4<a) (θ : Angle)
+    (r : ℝ) (hr : 0≤r) (hr1 : r<1) (z : UpperPlane)
+    (hz : diskMap θ.1 z.1∈Metric.closedBall (0:ℂ) r) :
+    diskWeight a θ.1 z≤ subdiskTest a θ r hr hr1 z := by
+  have hy := Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
+  have he : ‖((((upperAffine (Real.cos θ.1) (Real.sin θ.1) hy).symm z).1)-Complex.I)/
+      ((((upperAffine (Real.cos θ.1) (Real.sin θ.1) hy).symm z).1)+Complex.I)‖≤r := by
+    have h := diskMap_affine θ.1 θ.2.1 θ.2.2
+      (((upperAffine (Real.cos θ.1) (Real.sin θ.1) hy).symm z).1)
+    rw [←upperAffine_apply,Homeomorph.apply_symm_apply] at h
+    simpa only [Metric.mem_closedBall,dist_zero_right,h] using hz
+  change ‖deriv (diskMap θ.1) z.1‖^(2-alpha a)≤
+    2^(2-alpha a)*((Real.sin θ.1)^(alpha a-2)*subdiskCutoff r hr hr1 _)
+  rw [subdiskCutoff_one r hr hr1 _ he,mul_one]
+  calc
+    _ ≤ (2/(Real.sin θ.1))^(2-alpha a) := Real.rpow_le_rpow (norm_nonneg _)
+      (disk_deriv_bound θ.1 θ.2.1 θ.2.2 z) (dimension_pos a ha).le
+    _ = _ := by
+      rw [div_eq_mul_inv,Real.mul_rpow (by norm_num : (0:ℝ)≤2) (inv_nonneg.mpr hy.le),
+        Real.inv_rpow hy.le,←Real.rpow_neg hy.le,neg_sub]
+
+lemma subdiskMass_le_enorm (a : ℝ) (ha : 1/4<a) (θ : Angle)
+    (r : ℝ) (hr : 0≤r) (hr1 : r<1)
+    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] :
+    diskMass a θ.1 ν (Metric.closedBall (0:ℂ) r)≤‖∫z,subdiskTest a θ r hr hr1 z∂ν‖ₑ := by
+  apply le_trans _ (Real.ofReal_le_enorm _)
+  rw [diskMass,Measure.map_apply (diskMap_measurable θ.1) Metric.isClosed_closedBall.measurableSet,
+    withDensity_apply _ ((diskMap_measurable θ.1) Metric.isClosed_closedBall.measurableSet),
+    ofReal_integral_eq_lintegral_ofReal (subdiskTest a θ r hr hr1).integrable
+      (Eventually.of_forall (subdiskTest_nonneg a θ r hr hr1)),
+    ←lintegral_indicator ((diskMap_measurable θ.1) Metric.isClosed_closedBall.measurableSet)]
+  apply lintegral_mono
+  intro z
+  by_cases hz : diskMap θ.1 z.1∈Metric.closedBall (0:ℂ) r
+  · simp only [Set.indicator, Set.mem_preimage, hz, ite_true]
+    exact ENNReal.ofReal_le_ofReal (diskWeight_le_subdiskTest a ha θ r hr hr1 z hz)
+  · simp only [Set.indicator, Set.mem_preimage, hz, ite_false]; positivity
+
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CleanupCost.lean b/lean/OAI/Probability/SLEGauge/CleanupCost.lean
index 5644a2e8e94613ac3ca96d5b4de5422ed5ee5247..17fbffa9e087bd41d757b3504fbe502359020fe1 100644
--- a/lean/OAI/Probability/SLEGauge/CleanupCost.lean
+++ b/lean/OAI/Probability/SLEGauge/CleanupCost.lean
@@ -1,599 +1,599 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ModelCandidates
-import OAI.Probability.SLEGauge.CandidateCover
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Finset
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CleanupCost
-open BoundaryRows
-
-lemma nat_mul_exp_neg_linear (c : ℝ) (hc : 0<c) :
-    Tendsto (fun n : ℕ=>(n:ℝ)*Real.exp (-c*n)) atTop (𝓝 0) := by
-  have ht : Tendsto (fun n : ℕ=>c*(n:ℝ)) atTop atTop :=
-    (tendsto_const_mul_atTop_of_pos hc).mpr tendsto_natCast_atTop_atTop
-  have hh := ((Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero 1).comp ht).const_mul c⁻¹
-  simp only [Function.comp_def,pow_one,mul_zero] at hh
-  convert hh using 1
-  funext n
-  rw [neg_mul,←mul_assoc,←mul_assoc,inv_mul_cancel₀ hc.ne',one_mul]
-
-lemma nat_mul_exp_neg_two_pow :
-    Tendsto (fun n : ℕ=>(n:ℝ)*Real.exp (-((2:ℝ)^n)/16)) atTop (𝓝 0) := by
-  apply squeeze_zero (fun n=>mul_nonneg (Nat.cast_nonneg n) (Real.exp_pos _).le)
-    (g:=fun n : ℕ=>(n:ℝ)*Real.exp (-(1/16:ℝ)*n))
-  · intro n
-    have hn : (n:ℝ)≤(2:ℝ)^n := by exact_mod_cast (Nat.lt_two_pow_self (n:=n)).le
-    apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
-    exact Real.exp_le_exp.mpr (by linarith)
-  · exact nat_mul_exp_neg_linear (1/16) (by norm_num)
-
-lemma nat_mul_boundary_power {α : ℝ} (hα : α<1) (r : ℕ→ℝ)
-    (hr : ∀n,0≤r n) (hsmall : ∀ᶠ n in atTop,r n≤Real.exp (-(n:ℝ)^2)) :
-    Tendsto (fun n : ℕ=>(n:ℝ)*(Real.exp (-(n:ℝ))+r n)^(1-α)) atTop (𝓝 0) := by
-  have hβ : 0<1-α := by linarith
-  have hg : Tendsto (fun n : ℕ=>(2:ℝ)^(1-α)*((n:ℝ)*Real.exp (-(1-α)*n))) atTop (𝓝 0) := by
-    simpa only [mul_zero] using (nat_mul_exp_neg_linear (1-α) hβ).const_mul ((2:ℝ)^(1-α))
-  apply squeeze_zero' (Eventually.of_forall (fun n=>mul_nonneg (Nat.cast_nonneg n)
-    (Real.rpow_nonneg (add_nonneg (Real.exp_pos _).le (hr n)) _))) _ hg
-  filter_upwards [hsmall,eventually_ge_atTop 1] with n hn hn1
-  have hn1' : (1:ℝ)≤n := by exact_mod_cast hn1
-  have hh : -(n:ℝ)^2≤-(n:ℝ) := by nlinarith
-  have he : Real.exp (-(n:ℝ))+r n≤2*Real.exp (-(n:ℝ)) := by
-    have hh' := Real.exp_le_exp.mpr hh
-    linarith
-  calc
-    _ ≤ (n:ℝ)*(2*Real.exp (-(n:ℝ)))^(1-α) :=
-      mul_le_mul_of_nonneg_left
-        (Real.rpow_le_rpow (add_nonneg (Real.exp_pos _).le (hr n)) he hβ.le) (Nat.cast_nonneg n)
-    _ = _ := by
-      rw [Real.mul_rpow (by norm_num : (0:ℝ)≤2) (Real.exp_pos _).le,
-        Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-      rw [show -(n:ℝ)*(1-α)=-(1-α)*(n:ℝ) by ring]
-      ring
-
- 
-
-def envelope (α : ℝ) (r : ℕ→ℝ) (n : ℕ) : ℝ :=
-  (n:ℝ)*Real.exp (-((2:ℝ)^n)/16)+(n:ℝ)*(Real.exp (-(n:ℝ))+r n)^(1-α)
-
-lemma envelope_tendsto {α : ℝ} (hα : α<1) (r : ℕ→ℝ)
-    (hr : ∀n,0≤r n) (hsmall : ∀ᶠ n in atTop,r n≤Real.exp (-(n:ℝ)^2)) :
-    Tendsto (envelope α r) atTop (𝓝 0) := by
-  change Tendsto (fun n : ℕ=>(n:ℝ)*Real.exp (-((2:ℝ)^n)/16) +
-    (n:ℝ)*(Real.exp (-(n:ℝ))+r n)^(1-α)) atTop (𝓝 0)
-  simpa only [zero_add] using
-    nat_mul_exp_neg_two_pow.add (nat_mul_boundary_power hα r hr hsmall)
-
-lemma envelope_uniform_bound {α : ℝ} (hα : α<1) (r : ℕ→ℝ)
-    (hr : ∀n,0≤r n) (hsmall : ∀ᶠ n in atTop,r n≤Real.exp (-(n:ℝ)^2)) :
-    ∃C : ℝ,0<C ∧ ∀n,envelope α r n≤C := by
-  obtain ⟨C,hC,hbound⟩ := (Metric.isBounded_range_of_tendsto (envelope α r)
-    (envelope_tendsto hα r hr hsmall)).exists_pos_norm_le
-  exact ⟨C,hC,fun n=>(le_abs_self _).trans (hbound _ (mem_range_self n))⟩
-
- 
-
-lemma two_region_cell_sum (α r : ℝ) (hα : 0≤α) (hα1 : α<1) (hr : 0<r)
-    (M N : ℕ) (X Y ε A q : ℝ)
-    (hX : (M:ℝ)*r≤X) (hY : (N:ℝ)*r≤Y)
-    (hε : 0≤ε) (hA : 0≤A) (hq : 0≤q) (P : ℕ→ℕ→ℝ)
-    (hnear : ∀i∈Finset.range M,∀j∈Finset.range N,
-      ((j:ℝ)+1/2)*r<ε → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α))
-    (hfar : ∀i∈Finset.range M,∀j∈Finset.range N,
-      ε≤((j:ℝ)+1/2)*r → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α)*q) :
-    (∑i∈Finset.range M,∑j∈Finset.range N,P i j)≤
-      A*r^α*(rowConstant α*X/r^2)*(q*Y^(1-α)+(ε+r)^(1-α)) := by
-  classical
-  let y : ℕ→ℝ := fun j=>((j:ℝ)+1/2)*r
-  have hn (j : ℕ) : 0≤A*r^α*(y j)^(-α) := by dsimp [y]; positivity
-  have hb (i : ℕ) (hi : i∈Finset.range M) (j : ℕ) (hj : j∈Finset.range N) :
-      P i j≤A*r^α*((y j)^(-α)*q+(if y j<ε then (y j)^(-α) else 0)) := by
-    by_cases hy : y j<ε
-    · rw [ite_eq_left hy,mul_add]
-      exact (hnear i hi j hj hy).trans (le_add_of_nonneg_left (by positivity))
-    · rw [ite_eq_right hy,add_zero,←mul_assoc]
-      exact hfar i hi j hj (le_of_not_gt hy)
-  have hsplit : (∑i∈Finset.range M,∑j∈Finset.range N,
-        A*r^α*((y j)^(-α)*q+(if y j<ε then (y j)^(-α) else 0)))=
-      A*r^α*(q*(∑i∈Finset.range M,∑j∈Finset.range N,(y j)^(-α))+
-        (∑i∈Finset.range M,∑j∈(Finset.range N).filter (fun j=>y j<ε),(y j)^(-α))) := by
-    simp_rw [Finset.sum_filter,mul_add,Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul]
-    ring
-  have hh1 := rectangle_rows α hα hα1 r hr M N X Y hX hY
-  have hh2 := rectangle_truncated_rows α hα hα1 r hr M N X ε hX hε
-  calc
-    _ ≤ ∑i∈Finset.range M,∑j∈Finset.range N,
-        A*r^α*((y j)^(-α)*q+(if y j<ε then (y j)^(-α) else 0)) := by
-      exact Finset.sum_le_sum (fun i hi=>Finset.sum_le_sum (hb i hi))
-    _ = _ := hsplit
-    _ ≤ A*r^α*(q*(rowConstant α*X*Y^(1-α)/r^2)+
-        rowConstant α*X*(ε+r)^(1-α)/r^2) :=
-      mul_le_mul_of_nonneg_left (add_le_add (mul_le_mul_of_nonneg_left hh1 hq) hh2)
-        (mul_nonneg hA (Real.rpow_nonneg hr.le _))
-    _ = _ := by ring
-
- 
-
-theorem cleanup_sum (d α r : ℝ) (hα : 0≤α) (hα1 : α<1) (hr : 0<r) (hdα : d+α=2)
-    (M N n : ℕ) (X Y ε A q : ℝ)
-    (hX : (M:ℝ)*r≤X) (hY : (N:ℝ)*r≤Y)
-    (hε : 0≤ε) (hA : 0≤A) (hq : 0≤q) (P : ℕ→ℕ→ℝ)
-    (hnear : ∀i∈Finset.range M,∀j∈Finset.range N,
-      ((j:ℝ)+1/2)*r<ε → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α))
-    (hfar : ∀i∈Finset.range M,∀j∈Finset.range N,
-      ε≤((j:ℝ)+1/2)*r → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α)*q) :
-    (∑i∈Finset.range M,∑j∈Finset.range N,2*(n:ℝ)*r^d*P i j)≤
-      2*(n:ℝ)*A*rowConstant α*X*(q*Y^(1-α)+(ε+r)^(1-α)) := by
-  simp_rw [←Finset.mul_sum]
-  have hh := mul_le_mul_of_nonneg_left
-    (two_region_cell_sum α r hα hα1 hr M N X Y ε A q hX hY hε hA hq P hnear hfar)
-    (show 0≤2*(n:ℝ)*r^d by positivity)
-  refine hh.trans_eq ?_
-  have hcancel : r^d*r^α/r^2=1 := by
-    rw [←Real.rpow_add hr,hdα,Real.rpow_two,div_self (pow_ne_zero 2 hr.ne')]
-  calc
-    _ = 2*(n:ℝ)*A*rowConstant α*X*(q*Y^(1-α)+(ε+r)^(1-α))*(r^d*r^α/r^2) := by ring
-    _ = _ := by rw [hcancel,mul_one]
-
-end SLEExactGauge.CleanupCost
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology Classical
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart CompactSequence MomentBatches BoxGrid
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-def cleanupCost (m : ℝ) (n : ℕ) (b : GDriver) : ℝ≥0∞ :=
-  ∑k : Cells m (M.finalRadius n),
-    (M.cleanupEvent (center m (M.finalRadius n) k.1 k.2) n).indicator
-      (fun _=>ENNReal.ofReal (4*(n:ℝ)*(M.finalRadius n)^(2-alpha a))) b
-
-lemma cleanupCost_measurable (m : ℝ) (n : ℕ) : Measurable (M.cleanupCost m n) := by
-  apply Finset.measurable_sum
-  intro k hk
-  exact measurable_const.indicator (M.cleanupEvent_measurable _
-    (center_im_pos m _ (M.finalRadius_pos n) _ _) n)
-
-lemma cleanupCost_integral [IsProbabilityMeasure W] (m : ℝ) (n : ℕ) :
-    (∫⁻b,M.cleanupCost m n b∂W)=ENNReal.ofReal
-      (∑i∈Finset.range (columns m (M.finalRadius n)),
-       ∑j∈Finset.range (rows m (M.finalRadius n)),
-        4*(n:ℝ)*(M.finalRadius n)^(2-alpha a)*
-          W.real (M.cleanupEvent (center m (M.finalRadius n) i j) n)) := by
-  have hr := M.finalRadius_pos n
-  simp only [cleanupCost]
-  rw [lintegral_finsetSum]
-  · simp_rw [lintegral_indicator_const (M.cleanupEvent_measurable _
-      (center_im_pos m _ hr _ _) n)]
-    rw [Fintype.sum_prod_type]
-
-    simp_rw [←Fin.sum_univ_eq_sum_range]
-    rw [ENNReal.ofReal_sum_of_nonneg (fun i _=>Finset.sum_nonneg (fun j _=>by positivity))]
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [ENNReal.ofReal_sum_of_nonneg (fun j _=>by positivity)]
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [ENNReal.ofReal_mul (show 0≤4*(n:ℝ)*(M.finalRadius n)^(2-alpha a) by positivity),Measure.real,
-      ENNReal.ofReal_toReal (measure_ne_top _ _)]
-  · intro k hk
-    exact measurable_const.indicator (M.cleanupEvent_measurable _
-      (center_im_pos m _ hr _ _) n)
-
- 
-lemma cleanupCost_integral_bound [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (m : ℝ) (hm : 0≤ m) (n : ℕ) (hr : BatchReady a n)
-    (hr1 : M.finalRadius n≤1) :
-    (∫⁻b,M.cleanupCost m n b∂W) ≤ ENNReal.ofReal
-      (4*(n:ℝ)*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1)*
-        (Real.exp (-((2:ℝ)^n/16))*(m+1)^(1-alpha a)+
-          (Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a))) := by
-  rw [M.cleanupCost_integral]
-  apply ENNReal.ofReal_le_ofReal
-  have hsum := CleanupCost.cleanup_sum (2-alpha a) (alpha a) (M.finalRadius n)
-    (alpha_pos ha).le (alpha_lt_one (ha:=ha)) (M.finalRadius_pos n) (by ring)
-    (columns m (M.finalRadius n)) (rows m (M.finalRadius n)) n
-    (2*m+1) (m+1) (Real.exp (-(n:ℝ))) M.cellConstant (Real.exp (-((2:ℝ)^n/16)))
-    ((columns_width m _ hm (M.finalRadius_pos n)).trans (by linarith))
-    ((rows_height m _ hm (M.finalRadius_pos n)).trans (by linarith))
-    (Real.exp_pos _).le M.cellConstant_pos.le (Real.exp_pos _).le
-    (fun i j=>W.real (M.cleanupEvent (center m (M.finalRadius n) i j) n))
-    (fun i _ j _ _=>M.cleanup_near hW _ (center_im_pos m _ (M.finalRadius_pos n) i j) n)
-    (fun i _ j _ hy=>M.cleanup_far hW _ (center_im_pos m _ (M.finalRadius_pos n) i j) n hr hy)
-  have hh := mul_le_mul_of_nonneg_left hsum (show (0:ℝ)≤2 by norm_num)
-  simp_rw [Finset.mul_sum] at hh
-  calc
-    _ = ∑i∈Finset.range (columns m (M.finalRadius n)),
-       ∑j∈Finset.range (rows m (M.finalRadius n)),
-       2*(2*(n:ℝ)*(M.finalRadius n)^(2-alpha a)*W.real
-         (M.cleanupEvent (center m (M.finalRadius n) i j) n)) := by
-      apply Finset.sum_congr rfl
-      intro i hi
-      apply Finset.sum_congr rfl
-      intro j hj
-      ring
-    _ ≤ _ := hh.trans_eq (by ring)
-
-lemma cleanupCost_uniform_bound [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (m : ℝ) (hm : 0≤ m) :
-    ∃B : ℝ≥0∞,B<∞ ∧ ∀ᶠ n in atTop,(∫⁻b,M.cleanupCost m n b∂W)≤B := by
-  obtain ⟨C,hC,hbound⟩ := CleanupCost.envelope_uniform_bound (alpha_lt_one (ha:=ha))
-    M.finalRadius (fun n=>(M.finalRadius_pos n).le) (Eventually.of_forall M.finalRadius_le)
-  let D := 4*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1)*max 1 ((m+1)^(1-alpha a))
-  refine ⟨ENNReal.ofReal (D*C),ENNReal.ofReal_lt_top,?_⟩
-  filter_upwards [eventually_batchReady a (by linarith)] with n hn
-  have hs : M.finalRadius n≤1 := (M.finalRadius_le n).trans
-    (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
-  apply (M.cleanupCost_integral_bound hW m hm n hn hs).trans
-  apply ENNReal.ofReal_le_ofReal
-  have hrow := BoundaryRows.rowConstant_pos (alpha_lt_one (ha:=ha))
-  have hD : 0≤D := by
-    dsimp [D]
-    have := M.cellConstant_pos
-    positivity
-  have henv : 4*(n:ℝ)*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1)*
-        (Real.exp (-((2:ℝ)^n/16))*(m+1)^(1-alpha a)+
-          (Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a)) ≤
-      D*CleanupCost.envelope (alpha a) M.finalRadius n := by
-    have hy : (m+1)^(1-alpha a) ≤ max 1 ((m+1)^(1-alpha a)) := le_max_right ..
-    have hu : (Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a) ≤
-        max 1 ((m+1)^(1-alpha a))*(Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a) := by
-      exact le_mul_of_one_le_left (Real.rpow_nonneg
-        (add_nonneg (Real.exp_pos _).le (M.finalRadius_pos n).le) _) (le_max_left ..)
-    have hh := mul_le_mul_of_nonneg_left
-      (add_le_add (mul_le_mul_of_nonneg_left hy (Real.exp_pos (-((2:ℝ)^n/16))).le) hu)
-      (show 0≤4*(n:ℝ)*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1) by
-        have := M.cellConstant_pos; positivity)
-    exact hh.trans_eq (by dsimp [D,CleanupCost.envelope]; rw [neg_div]; ring)
-  exact henv.trans (mul_le_mul_of_nonneg_left (hbound n) hD)
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology Classical
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart CompactSequence MomentBatches BoxGrid Deterministic CandidateCover
-
-def allFrontiers (a : ℝ) (b : GDriver) : Set ℂ :=
-  ⋃T : ℝ≥0,frontier {z | alive a b.1 T z}
-
-lemma exp_neg_nat_sq_tendsto :
-    Tendsto (fun n : ℕ=>Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
-  have hs : Tendsto (fun n : ℕ=>(n:ℝ)^2) atTop atTop :=
-    (tendsto_pow_atTop (by norm_num : (2:ℕ)≠0)).comp tendsto_natCast_atTop_atTop
-  exact Real.tendsto_exp_atBot.comp (tendsto_neg_atTop_atBot.comp hs)
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-abbrev GridCells (m : ℝ) (n : ℕ) := Cells m (M.finalRadius n)
-abbrev CandidateIndices (m : ℝ) (n : ℕ) := M.GridCells m n × Fin (M.testCount n)
-abbrev CoverIndices (m : ℝ) (n : ℕ) := M.CandidateIndices m n ⊕ M.GridCells m n
-
-def gridCenter (m : ℝ) (n : ℕ) (k : M.GridCells m n) : ℂ :=
-  center m (M.finalRadius n) k.1 k.2
-
-def costBound (m : ℝ) (n : ℕ) (b : GDriver) : ℝ≥0∞ :=
-  ENNReal.ofReal ((6*M.candidateScale)^2/M.candidateLower)*M.spatialMass b (region m)+
-    M.cleanupCost m n b
-
-lemma costBound_measurable (m : ℝ) (n : ℕ) : Measurable (M.costBound m n) :=
-  (measurable_const.mul (M.spatialMass_measurable _ (region_measurable m))).add
-    (M.cleanupCost_measurable m n)
-
-lemma costBound_uniform [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (m : ℝ) (hm : 0≤ m) :
-    ∃B : ℝ≥0∞,B<∞ ∧ ∀ᶠ n in atTop,(∫⁻b,M.costBound m n b∂W)≤B := by
-  obtain ⟨C,hC,hbound⟩ := M.cleanupCost_uniform_bound hW m hm
-  have hmean : (∫⁻b,M.spatialMass b (region m)∂W)<∞ := by
-    simp_rw [M.spatialMass_apply _ _ (region_measurable m),preimage_region]
-    exact (BoundaryGreen.canonical_rectangle_mean M.canonical _ _ _ (by linarith)).2.1
-  refine ⟨ENNReal.ofReal ((6*M.candidateScale)^2/M.candidateLower)*
-    (∫⁻b,M.spatialMass b (region m)∂W)+C,
-    ENNReal.add_lt_top.mpr ⟨ENNReal.mul_lt_top ENNReal.ofReal_lt_top hmean,hC⟩,?_⟩
-  filter_upwards [hbound] with n hn
-  simp only [costBound]
-  have hf : Measurable (fun b=>ENNReal.ofReal ((6*M.candidateScale)^2/M.candidateLower)*
-      M.spatialMass b (region m)) := measurable_const.mul (M.spatialMass_measurable _ (region_measurable m))
-  rw [lintegral_add_left hf,
-    lintegral_const_mul _ (M.spatialMass_measurable _ (region_measurable m))]
-  exact add_le_add le_rfl hn
-
-lemma finalRadius_le_one (n : ℕ) : M.finalRadius n≤1 :=
-  (M.finalRadius_le n).trans (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
-
-lemma gauge_admissible_actual : Target.IsHausdorffGauge (M.gauge (2-alpha a)) :=
-  M.gauge_admissible (by linarith [alpha_lt_one (ha:=ha)]) (by linarith [alpha_pos ha])
-
-lemma cleanup_ball_cost (n : ℕ) (hn : 0<n) (z : ℂ) :
-    diameterCost (M.gauge (2-alpha a)) (ediam (closedBall z (M.finalRadius n))) ≤
-      ENNReal.ofReal (4*(n:ℝ)*(M.finalRadius n)^(2-alpha a)) := by
-  apply (diameterCost_le_radius M.gauge_admissible_actual
-    (by have := M.finalRadius_pos n; positivity)
-    (ediam_closedBall_le_of_nonneg z (M.finalRadius n))).trans
-  apply ENNReal.ofReal_le_ofReal
-  rw [M.gauge_eq_quadratic]
-  have h := quadratic_multiple_le (2-alpha a) M.entryThreshold M.entryRadius
-    (fun i=>(M.entryThreshold_pos i).le) M.entryRadius_pos 2 (by norm_num)
-    (cleanupIndex (moment M.law sequenceMassReal) n hn)
-  rw [M.cleanup_radius n hn,show M.entryThreshold (cleanupIndex (moment M.law sequenceMassReal) n hn)=
-    (n:ℝ) from cleanup_threshold ..] at h
-  norm_num only [show (2:ℝ)^2=4 by norm_num] at h
-  exact h
-
- 
-
-lemma finite_cover (m : ℝ) (hm : 0≤ m) (n : ℕ) (hn : 0<n)
-    (hsmall : M.candidateScale*Real.exp (-(n:ℝ)^2)≤1) :
-    ∀ᵐ b∂W,∃U : M.CoverIndices m n→Set ℂ,
-      (∀i,ediam (U i)≤ENNReal.ofReal (6*M.candidateScale*Real.exp (-(n:ℝ)^2))) ∧
-      allFrontiers a b∩box m ⊆ ⋃i,U i ∧
-      (∑i,diameterCost (M.gauge (2-alpha a)) (ediam (U i)))≤M.costBound m n b := by
-  have hcover : ∀ᵐ b∂W,∀k : M.GridCells m n,∀T : ℝ≥0,∀v∈closedBall (M.gridCenter m n k) (M.finalRadius n),
-      v∈frontier {z | alive a b.1 T z} →
-        (∃i<M.testCount n,M.candidate n i (M.gridCenter m n k) b) ∨
-          b∈M.cleanupEvent (M.gridCenter m n k) n :=
-    ae_all_iff.mpr (fun k=>M.cleanup_covers _
-      (center_im_pos m _ (M.finalRadius_pos n) _ _) n)
-  filter_upwards [hcover] with b hb
-  let s : Finset (M.CandidateIndices m n) := Finset.univ.filter
-    (fun j=>M.candidate n j.2 (M.gridCenter m n j.1) b)
-  let e : M.CandidateIndices m n→M.Entries := fun j=>M.testIndex n hn j.2 j.2.2
-  have her (j : M.CandidateIndices m n) : M.entryRadius (e j)=batchRadius a (M.batchLevel n) n j.2 := rfl
-  have het (j : M.CandidateIndices m n) : M.entryThreshold (e j)=M.testThreshold n j.2 :=
-    M.testIndex_threshold n hn j.2 j.2.2
-  have hrbound (j : M.CandidateIndices m n) : M.entryRadius (e j)≤Real.exp (-(n:ℝ)^2) := by
-    rw [her,←batchRadius_zero a (M.batchLevel n) n]
-    exact batchRadius_antitone a (by linarith) _ _ (Nat.zero_le _)
-  have hrball (j : M.CandidateIndices m n) : M.candidateScale*M.entryRadius (e j)≤1 :=
-    (mul_le_mul_of_nonneg_left (hrbound j) M.candidateScale_pos.le).trans hsmall
-  have hqual (j : M.CandidateIndices m n) (hj : j∈s) :
-      ENNReal.ofReal (M.candidateLower*M.entryThreshold (e j)*(M.entryRadius (e j))^(2-alpha a)) ≤
-        M.spatialMass b (closedBall (M.gridCenter m n j.1) (M.candidateScale*M.entryRadius (e j))∩univ) := by
-    rw [inter_univ,her,het]
-    exact (Finset.mem_filter.mp hj).2
-  obtain ⟨t,ht,hd,hcov,hcost⟩ := finite_quadratic_candidates (2-alpha a) M.entryThreshold M.entryRadius
-    (fun j=>(M.entryThreshold_pos j).le) M.entryRadius_pos M.gauge_admissible_actual
-    (M.spatialMass b) s (fun j=>M.gridCenter m n j.1) e univ (region m) MeasurableSet.univ
-    M.candidateScale M.candidateLower M.candidateScale_one M.candidateLower_pos
-    (fun j _=>(inter_subset_left.trans (ball_subset_region m _ _ hm (M.finalRadius_pos n)
-      (M.finalRadius_le_one n) (hrball j) j.1))) hqual
-  let U : M.CoverIndices m n→Set ℂ := Sum.elim
-    (fun j=>if j∈t then closedBall (M.gridCenter m n j.1) (3*(M.candidateScale*M.entryRadius (e j))) else ∅)
-    (fun k=>if b∈M.cleanupEvent (M.gridCenter m n k) n then
-      closedBall (M.gridCenter m n k) (M.finalRadius n) else ∅)
-  have hδ : 0≤6*M.candidateScale*Real.exp (-(n:ℝ)^2) := by have := M.candidateScale_pos; positivity
-  refine ⟨U,?_,?_,?_⟩
-  · intro i
-    cases i with
-    | inl j =>
-      dsimp [U]
-      split_ifs with hj
-      · apply (ediam_closedBall_le_of_nonneg _ _).trans
-        apply ENNReal.ofReal_le_ofReal
-        nlinarith [mul_le_mul_of_nonneg_left (hrbound j) M.candidateScale_pos.le]
-      · simp
-    | inr k =>
-      dsimp [U]
-      split_ifs
-      · apply (ediam_closedBall_le_of_nonneg _ _).trans
-        apply ENNReal.ofReal_le_ofReal
-        have he := Real.exp_pos (-(n:ℝ)^2)
-        nlinarith [M.finalRadius_le n,M.candidateScale_one]
-      · simp
-  · intro v hv
-    obtain ⟨T,hT⟩ := mem_iUnion.mp hv.1
-    obtain ⟨k,hk⟩ := mem_iUnion.mp (covered m _ hm (M.finalRadius_pos n) hv.2)
-    rcases hb k T v hk hT with hc | hc
-    · obtain ⟨i,hi,hc⟩ := hc
-      let j : M.CandidateIndices m n := (k,⟨i,hi⟩)
-      have hjs : j∈s := Finset.mem_filter.mpr ⟨Finset.mem_univ _,hc⟩
-      obtain ⟨q,hqt,hq⟩ := hcov j hjs
-      refine mem_iUnion.mpr ⟨Sum.inl q,?_⟩
-      change v∈if q∈t then _ else ∅
-      rw [ite_eq_left hqt]
-      apply hq
-      apply closedBall_subset_closedBall (ε₁:=M.finalRadius n) (ε₂:=M.candidateScale*M.entryRadius (e j)) _ hk
-      have hri : M.finalRadius n≤M.entryRadius (e j) :=
-        batchRadius_antitone a (by linarith) _ _ hi.le
-      exact hri.trans (le_mul_of_one_le_left (M.entryRadius_pos _).le M.candidateScale_one)
-    · refine mem_iUnion.mpr ⟨Sum.inr k,?_⟩
-      change v∈if b∈M.cleanupEvent (M.gridCenter m n k) n then _ else ∅
-      rw [ite_eq_left hc]
-      exact hk
-  · rw [Fintype.sum_sum_type]
-    unfold costBound
-    apply add_le_add
-    · change (∑j,diameterCost (M.gauge (2-alpha a)) (ediam
-        (if j∈t then closedBall (M.gridCenter m n j.1) (3*(M.candidateScale*M.entryRadius (e j))) else ∅))) ≤ _
-      simp only [apply_ite,ediam_empty,diameterCost_zero M.gauge_admissible_actual.2.2.1,
-        ←Finset.sum_filter]
-      simpa only [Finset.filter_mem_eq_inter,Finset.univ_inter,M.gauge_eq_quadratic] using hcost
-    · change (∑k,diameterCost (M.gauge (2-alpha a)) (ediam
-        (if b∈M.cleanupEvent (M.gridCenter m n k) n then closedBall (M.gridCenter m n k) (M.finalRadius n) else ∅))) ≤ _
-      unfold cleanupCost
-      apply Finset.sum_le_sum
-      intro k hk
-      change diameterCost (M.gauge (2-alpha a)) (ediam (if b∈M.cleanupEvent (M.gridCenter m n k) n then
-        closedBall (M.gridCenter m n k) (M.finalRadius n) else ∅)) ≤
-        (M.cleanupEvent (M.gridCenter m n k) n).indicator
-          (fun _=>ENNReal.ofReal (4*(n:ℝ)*(M.finalRadius n)^(2-alpha a))) b
-      by_cases hc : b∈M.cleanupEvent (M.gridCenter m n k) n
-      · simp only [ite_eq_left hc,indicator_of_mem hc]
-        exact M.cleanup_ball_cost n hn _
-      · simp only [ite_eq_right hc,ediam_empty,diameterCost_zero M.gauge_admissible_actual.2.2.1]
-        exact bot_le
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart CompactSequence MomentBatches BoxGrid Deterministic
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
- 
-
-theorem frontier_box_finite [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (m : ℝ) (hm : 0≤ m) :
-    ∀ᵐ b∂W,Target.hausdorff (M.gauge (2-alpha a)) (allFrontiers a b∩box m)<∞ := by
-  obtain ⟨B,hB,hbound⟩ := M.costBound_uniform hW m hm
-  have hfin := (CoverCost.liminf_cost_bound W (M.costBound m)
-    (M.costBound_measurable m) B hB hbound).2.2
-  have hc : ∀ᵐ b∂W,∀n : ℕ,0<n → M.candidateScale*Real.exp (-(n:ℝ)^2)≤1 →
-      ∃U : M.CoverIndices m n→Set ℂ,
-        (∀i,ediam (U i)≤ENNReal.ofReal (6*M.candidateScale*Real.exp (-(n:ℝ)^2))) ∧
-        allFrontiers a b∩box m⊆⋃i,U i ∧
-        (∑i,diameterCost (M.gauge (2-alpha a)) (ediam (U i)))≤M.costBound m n b := by
-    apply ae_all_iff.mpr
-    intro n
-    by_cases hn : 0<n
-    · by_cases hs : M.candidateScale*Real.exp (-(n:ℝ)^2)≤1
-      · exact (M.finite_cover m hm n hn hs).mono (fun b hb _ _=>hb)
-      · exact Eventually.of_forall (fun _ _ h=>(hs h).elim)
-    · exact Eventually.of_forall (fun _ h _=>(hn h).elim)
-  have he : Tendsto (fun n : ℕ=>M.candidateScale*Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
-    simpa only [mul_zero] using exp_neg_nat_sq_tendsto.const_mul M.candidateScale
-  have hsmall : ∀ᶠ n : ℕ in atTop,M.candidateScale*Real.exp (-(n:ℝ)^2)≤1 :=
-    (he.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1))).mono (fun _ h=>h.le)
-  have hδ : Tendsto (fun n : ℕ=>ENNReal.ofReal (6*M.candidateScale*Real.exp (-(n:ℝ)^2))) atTop (𝓝 0) := by
-    have hh : Tendsto (fun n : ℕ=>6*M.candidateScale*Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using exp_neg_nat_sq_tendsto.const_mul (6*M.candidateScale)
-    simpa only [ENNReal.ofReal_zero] using ENNReal.tendsto_ofReal hh
-  filter_upwards [hc,hfin] with b hb hf
-  apply lt_of_le_of_lt _ hf
-  apply CoverCost.hausdorff_le_liminf_of_exists
-    (ι:=fun n=>M.CoverIndices m n) _ _ _ hδ _
-  filter_upwards [hsmall,eventually_gt_atTop (0:ℕ)] with n hs hn
-  obtain ⟨U,hd,hcov,hcost⟩ := hb n hn hs
-  exact ⟨U,hd,hcov,by simpa only [tsum_fintype] using hcost⟩
-
- 
-theorem frontier_boxes_finite [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W) :
-    ∀ᵐ b∂W,∀m : ℕ,Target.hausdorff (M.gauge (2-alpha a)) (allFrontiers a b∩box m)<∞ :=
-  ae_all_iff.mpr (fun m=>M.frontier_box_finite hW m (Nat.cast_nonneg m))
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification TraceMassIntegration
-
- 
-def dimension (κ : ℝ) : ℝ := 1+κ/8
-
-lemma parameterDimension_pos {κ : ℝ} (hκ : 0<κ) : 0<dimension κ := by
-  unfold dimension
-  linarith
-
-lemma dimension_lt_two {κ : ℝ} (hκ8 : κ<8) : dimension κ<2 := by
-  unfold dimension
-  linarith
-
- 
-
-theorem exists_parameter_moment_gauge (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) :
-    ∃h : ℝ→ℝ, Target.IsHausdorffGauge h ∧
-      (∀C r : ℝ, 1≤C → 0≤r → h (C*r)≤C^2*h r) ∧
-      ∀W : Target.WienerLaw, Target.IsWiener W →
-        ∃M : CenteredMomentModel κ (Target.a κ) (a_gt_quarter hκ hκ8)
-          (W.map driverHomeomorph), h=M.gauge (dimension κ) := by
-  classical
-  by_cases h : ∃W : Target.WienerLaw, Target.IsWiener W
-  · obtain ⟨W,hW⟩ := h
-    obtain ⟨M⟩ := actual_centered_moment_model κ hκ hκ8 W hW
-    refine ⟨M.gauge (dimension κ),
-      M.gauge_admissible (parameterDimension_pos hκ) (dimension_lt_two hκ8).le,
-      M.gauge_doubling (parameterDimension_pos hκ) (dimension_lt_two hκ8).le,?_⟩
-    intro V hV
-    have e : V=W := BrownianLaw.evaluation_law_unique hV hW
-    subst V
-    exact ⟨M,rfl⟩
-  · refine ⟨id,⟨continuous_id.continuousOn,monotone_id.monotoneOn _,rfl,fun _ hr=>hr⟩,?_,?_⟩
-    · intro C r hC hr
-      change C*r ≤ C^2*r
-      exact mul_le_mul_of_nonneg_right (by nlinarith : C≤C^2) hr
-    · intro W hW
-      exact (h ⟨W,hW⟩).elim
-
- 
-def parameterMomentGauge (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) : ℝ→ℝ :=
-  Classical.choose (exists_parameter_moment_gauge κ hκ hκ8)
-
-lemma parameterMomentGauge_admissible (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) :
-    Target.IsHausdorffGauge (parameterMomentGauge κ hκ hκ8) :=
-  (Classical.choose_spec (exists_parameter_moment_gauge κ hκ hκ8)).1
-
-lemma parameterMomentGauge_doubling (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (C r : ℝ) (hC : 1≤C) (hr : 0≤r) :
-    parameterMomentGauge κ hκ hκ8 (C*r) ≤ C^2*parameterMomentGauge κ hκ hκ8 r :=
-  (Classical.choose_spec (exists_parameter_moment_gauge κ hκ hκ8)).2.1 C r hC hr
-
-lemma parameterMomentGauge_model (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Target.WienerLaw) (hW : Target.IsWiener W) :
-    ∃M : CenteredMomentModel κ (Target.a κ) (a_gt_quarter hκ hκ8)
-      (W.map driverHomeomorph), parameterMomentGauge κ hκ hκ8=M.gauge (dimension κ) :=
-  (Classical.choose_spec (exists_parameter_moment_gauge κ hκ hκ8)).2.2 W hW
-
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ModelCandidates
+import OAI.Probability.SLEGauge.CandidateCover
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Finset
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CleanupCost
+open BoundaryRows
+
+lemma nat_mul_exp_neg_linear (c : ℝ) (hc : 0<c) :
+    Tendsto (fun n : ℕ=>(n:ℝ)*Real.exp (-c*n)) atTop (𝓝 0) := by
+  have ht : Tendsto (fun n : ℕ=>c*(n:ℝ)) atTop atTop :=
+    (tendsto_const_mul_atTop_of_pos hc).mpr tendsto_natCast_atTop_atTop
+  have hh := ((Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero 1).comp ht).const_mul c⁻¹
+  simp only [Function.comp_def,pow_one,mul_zero] at hh
+  convert hh using 1
+  funext n
+  rw [neg_mul,←mul_assoc,←mul_assoc,inv_mul_cancel₀ hc.ne',one_mul]
+
+lemma nat_mul_exp_neg_two_pow :
+    Tendsto (fun n : ℕ=>(n:ℝ)*Real.exp (-((2:ℝ)^n)/16)) atTop (𝓝 0) := by
+  apply squeeze_zero (fun n=>mul_nonneg (Nat.cast_nonneg n) (Real.exp_pos _).le)
+    (g:=fun n : ℕ=>(n:ℝ)*Real.exp (-(1/16:ℝ)*n))
+  · intro n
+    have hn : (n:ℝ)≤(2:ℝ)^n := by exact_mod_cast (Nat.lt_two_pow_self (n:=n)).le
+    apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg n)
+    exact Real.exp_le_exp.mpr (by linarith)
+  · exact nat_mul_exp_neg_linear (1/16) (by norm_num)
+
+lemma nat_mul_boundary_power {α : ℝ} (hα : α<1) (r : ℕ→ℝ)
+    (hr : ∀n,0≤r n) (hsmall : ∀ᶠ n in atTop,r n≤Real.exp (-(n:ℝ)^2)) :
+    Tendsto (fun n : ℕ=>(n:ℝ)*(Real.exp (-(n:ℝ))+r n)^(1-α)) atTop (𝓝 0) := by
+  have hβ : 0<1-α := by linarith
+  have hg : Tendsto (fun n : ℕ=>(2:ℝ)^(1-α)*((n:ℝ)*Real.exp (-(1-α)*n))) atTop (𝓝 0) := by
+    simpa only [mul_zero] using (nat_mul_exp_neg_linear (1-α) hβ).const_mul ((2:ℝ)^(1-α))
+  apply squeeze_zero' (Eventually.of_forall (fun n=>mul_nonneg (Nat.cast_nonneg n)
+    (Real.rpow_nonneg (add_nonneg (Real.exp_pos _).le (hr n)) _))) _ hg
+  filter_upwards [hsmall,eventually_ge_atTop 1] with n hn hn1
+  have hn1' : (1:ℝ)≤n := by exact_mod_cast hn1
+  have hh : -(n:ℝ)^2≤-(n:ℝ) := by nlinarith
+  have he : Real.exp (-(n:ℝ))+r n≤2*Real.exp (-(n:ℝ)) := by
+    have hh' := Real.exp_le_exp.mpr hh
+    linarith
+  calc
+    _ ≤ (n:ℝ)*(2*Real.exp (-(n:ℝ)))^(1-α) :=
+      mul_le_mul_of_nonneg_left
+        (Real.rpow_le_rpow (add_nonneg (Real.exp_pos _).le (hr n)) he hβ.le) (Nat.cast_nonneg n)
+    _ = _ := by
+      rw [Real.mul_rpow (by norm_num : (0:ℝ)≤2) (Real.exp_pos _).le,
+        Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+      rw [show -(n:ℝ)*(1-α)=-(1-α)*(n:ℝ) by ring]
+      ring
+
+ 
+
+def envelope (α : ℝ) (r : ℕ→ℝ) (n : ℕ) : ℝ :=
+  (n:ℝ)*Real.exp (-((2:ℝ)^n)/16)+(n:ℝ)*(Real.exp (-(n:ℝ))+r n)^(1-α)
+
+lemma envelope_tendsto {α : ℝ} (hα : α<1) (r : ℕ→ℝ)
+    (hr : ∀n,0≤r n) (hsmall : ∀ᶠ n in atTop,r n≤Real.exp (-(n:ℝ)^2)) :
+    Tendsto (envelope α r) atTop (𝓝 0) := by
+  change Tendsto (fun n : ℕ=>(n:ℝ)*Real.exp (-((2:ℝ)^n)/16) +
+    (n:ℝ)*(Real.exp (-(n:ℝ))+r n)^(1-α)) atTop (𝓝 0)
+  simpa only [zero_add] using
+    nat_mul_exp_neg_two_pow.add (nat_mul_boundary_power hα r hr hsmall)
+
+lemma envelope_uniform_bound {α : ℝ} (hα : α<1) (r : ℕ→ℝ)
+    (hr : ∀n,0≤r n) (hsmall : ∀ᶠ n in atTop,r n≤Real.exp (-(n:ℝ)^2)) :
+    ∃C : ℝ,0<C ∧ ∀n,envelope α r n≤C := by
+  obtain ⟨C,hC,hbound⟩ := (Metric.isBounded_range_of_tendsto (envelope α r)
+    (envelope_tendsto hα r hr hsmall)).exists_pos_norm_le
+  exact ⟨C,hC,fun n=>(le_abs_self _).trans (hbound _ (mem_range_self n))⟩
+
+ 
+
+lemma two_region_cell_sum (α r : ℝ) (hα : 0≤α) (hα1 : α<1) (hr : 0<r)
+    (M N : ℕ) (X Y ε A q : ℝ)
+    (hX : (M:ℝ)*r≤X) (hY : (N:ℝ)*r≤Y)
+    (hε : 0≤ε) (hA : 0≤A) (hq : 0≤q) (P : ℕ→ℕ→ℝ)
+    (hnear : ∀i∈Finset.range M,∀j∈Finset.range N,
+      ((j:ℝ)+1/2)*r<ε → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α))
+    (hfar : ∀i∈Finset.range M,∀j∈Finset.range N,
+      ε≤((j:ℝ)+1/2)*r → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α)*q) :
+    (∑i∈Finset.range M,∑j∈Finset.range N,P i j)≤
+      A*r^α*(rowConstant α*X/r^2)*(q*Y^(1-α)+(ε+r)^(1-α)) := by
+  classical
+  let y : ℕ→ℝ := fun j=>((j:ℝ)+1/2)*r
+  have hn (j : ℕ) : 0≤A*r^α*(y j)^(-α) := by dsimp [y]; positivity
+  have hb (i : ℕ) (hi : i∈Finset.range M) (j : ℕ) (hj : j∈Finset.range N) :
+      P i j≤A*r^α*((y j)^(-α)*q+(if y j<ε then (y j)^(-α) else 0)) := by
+    by_cases hy : y j<ε
+    · rw [ite_eq_left hy,mul_add]
+      exact (hnear i hi j hj hy).trans (le_add_of_nonneg_left (by positivity))
+    · rw [ite_eq_right hy,add_zero,←mul_assoc]
+      exact hfar i hi j hj (le_of_not_gt hy)
+  have hsplit : (∑i∈Finset.range M,∑j∈Finset.range N,
+        A*r^α*((y j)^(-α)*q+(if y j<ε then (y j)^(-α) else 0)))=
+      A*r^α*(q*(∑i∈Finset.range M,∑j∈Finset.range N,(y j)^(-α))+
+        (∑i∈Finset.range M,∑j∈(Finset.range N).filter (fun j=>y j<ε),(y j)^(-α))) := by
+    simp_rw [Finset.sum_filter,mul_add,Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul]
+    ring
+  have hh1 := rectangle_rows α hα hα1 r hr M N X Y hX hY
+  have hh2 := rectangle_truncated_rows α hα hα1 r hr M N X ε hX hε
+  calc
+    _ ≤ ∑i∈Finset.range M,∑j∈Finset.range N,
+        A*r^α*((y j)^(-α)*q+(if y j<ε then (y j)^(-α) else 0)) := by
+      exact Finset.sum_le_sum (fun i hi=>Finset.sum_le_sum (hb i hi))
+    _ = _ := hsplit
+    _ ≤ A*r^α*(q*(rowConstant α*X*Y^(1-α)/r^2)+
+        rowConstant α*X*(ε+r)^(1-α)/r^2) :=
+      mul_le_mul_of_nonneg_left (add_le_add (mul_le_mul_of_nonneg_left hh1 hq) hh2)
+        (mul_nonneg hA (Real.rpow_nonneg hr.le _))
+    _ = _ := by ring
+
+ 
+
+theorem cleanup_sum (d α r : ℝ) (hα : 0≤α) (hα1 : α<1) (hr : 0<r) (hdα : d+α=2)
+    (M N n : ℕ) (X Y ε A q : ℝ)
+    (hX : (M:ℝ)*r≤X) (hY : (N:ℝ)*r≤Y)
+    (hε : 0≤ε) (hA : 0≤A) (hq : 0≤q) (P : ℕ→ℕ→ℝ)
+    (hnear : ∀i∈Finset.range M,∀j∈Finset.range N,
+      ((j:ℝ)+1/2)*r<ε → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α))
+    (hfar : ∀i∈Finset.range M,∀j∈Finset.range N,
+      ε≤((j:ℝ)+1/2)*r → P i j≤A*r^α*(((j:ℝ)+1/2)*r)^(-α)*q) :
+    (∑i∈Finset.range M,∑j∈Finset.range N,2*(n:ℝ)*r^d*P i j)≤
+      2*(n:ℝ)*A*rowConstant α*X*(q*Y^(1-α)+(ε+r)^(1-α)) := by
+  simp_rw [←Finset.mul_sum]
+  have hh := mul_le_mul_of_nonneg_left
+    (two_region_cell_sum α r hα hα1 hr M N X Y ε A q hX hY hε hA hq P hnear hfar)
+    (show 0≤2*(n:ℝ)*r^d by positivity)
+  refine hh.trans_eq ?_
+  have hcancel : r^d*r^α/r^2=1 := by
+    rw [←Real.rpow_add hr,hdα,Real.rpow_two,div_self (pow_ne_zero 2 hr.ne')]
+  calc
+    _ = 2*(n:ℝ)*A*rowConstant α*X*(q*Y^(1-α)+(ε+r)^(1-α))*(r^d*r^α/r^2) := by ring
+    _ = _ := by rw [hcancel,mul_one]
+
+end SLEExactGauge.CleanupCost
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology Classical
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart CompactSequence MomentBatches BoxGrid
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+def cleanupCost (m : ℝ) (n : ℕ) (b : GDriver) : ℝ≥0∞ :=
+  ∑k : Cells m (M.finalRadius n),
+    (M.cleanupEvent (center m (M.finalRadius n) k.1 k.2) n).indicator
+      (fun _=>ENNReal.ofReal (4*(n:ℝ)*(M.finalRadius n)^(2-alpha a))) b
+
+lemma cleanupCost_measurable (m : ℝ) (n : ℕ) : Measurable (M.cleanupCost m n) := by
+  apply Finset.measurable_sum
+  intro k hk
+  exact measurable_const.indicator (M.cleanupEvent_measurable _
+    (center_im_pos m _ (M.finalRadius_pos n) _ _) n)
+
+lemma cleanupCost_integral [IsProbabilityMeasure W] (m : ℝ) (n : ℕ) :
+    (∫⁻b,M.cleanupCost m n b∂W)=ENNReal.ofReal
+      (∑i∈Finset.range (columns m (M.finalRadius n)),
+       ∑j∈Finset.range (rows m (M.finalRadius n)),
+        4*(n:ℝ)*(M.finalRadius n)^(2-alpha a)*
+          W.real (M.cleanupEvent (center m (M.finalRadius n) i j) n)) := by
+  have hr := M.finalRadius_pos n
+  simp only [cleanupCost]
+  rw [lintegral_finsetSum]
+  · simp_rw [lintegral_indicator_const (M.cleanupEvent_measurable _
+      (center_im_pos m _ hr _ _) n)]
+    rw [Fintype.sum_prod_type]
+
+    simp_rw [←Fin.sum_univ_eq_sum_range]
+    rw [ENNReal.ofReal_sum_of_nonneg (fun i _=>Finset.sum_nonneg (fun j _=>by positivity))]
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [ENNReal.ofReal_sum_of_nonneg (fun j _=>by positivity)]
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [ENNReal.ofReal_mul (show 0≤4*(n:ℝ)*(M.finalRadius n)^(2-alpha a) by positivity),Measure.real,
+      ENNReal.ofReal_toReal (measure_ne_top _ _)]
+  · intro k hk
+    exact measurable_const.indicator (M.cleanupEvent_measurable _
+      (center_im_pos m _ hr _ _) n)
+
+ 
+lemma cleanupCost_integral_bound [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (m : ℝ) (hm : 0≤ m) (n : ℕ) (hr : BatchReady a n)
+    (hr1 : M.finalRadius n≤1) :
+    (∫⁻b,M.cleanupCost m n b∂W) ≤ ENNReal.ofReal
+      (4*(n:ℝ)*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1)*
+        (Real.exp (-((2:ℝ)^n/16))*(m+1)^(1-alpha a)+
+          (Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a))) := by
+  rw [M.cleanupCost_integral]
+  apply ENNReal.ofReal_le_ofReal
+  have hsum := CleanupCost.cleanup_sum (2-alpha a) (alpha a) (M.finalRadius n)
+    (alpha_pos ha).le (alpha_lt_one (ha:=ha)) (M.finalRadius_pos n) (by ring)
+    (columns m (M.finalRadius n)) (rows m (M.finalRadius n)) n
+    (2*m+1) (m+1) (Real.exp (-(n:ℝ))) M.cellConstant (Real.exp (-((2:ℝ)^n/16)))
+    ((columns_width m _ hm (M.finalRadius_pos n)).trans (by linarith))
+    ((rows_height m _ hm (M.finalRadius_pos n)).trans (by linarith))
+    (Real.exp_pos _).le M.cellConstant_pos.le (Real.exp_pos _).le
+    (fun i j=>W.real (M.cleanupEvent (center m (M.finalRadius n) i j) n))
+    (fun i _ j _ _=>M.cleanup_near hW _ (center_im_pos m _ (M.finalRadius_pos n) i j) n)
+    (fun i _ j _ hy=>M.cleanup_far hW _ (center_im_pos m _ (M.finalRadius_pos n) i j) n hr hy)
+  have hh := mul_le_mul_of_nonneg_left hsum (show (0:ℝ)≤2 by norm_num)
+  simp_rw [Finset.mul_sum] at hh
+  calc
+    _ = ∑i∈Finset.range (columns m (M.finalRadius n)),
+       ∑j∈Finset.range (rows m (M.finalRadius n)),
+       2*(2*(n:ℝ)*(M.finalRadius n)^(2-alpha a)*W.real
+         (M.cleanupEvent (center m (M.finalRadius n) i j) n)) := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      apply Finset.sum_congr rfl
+      intro j hj
+      ring
+    _ ≤ _ := hh.trans_eq (by ring)
+
+lemma cleanupCost_uniform_bound [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (m : ℝ) (hm : 0≤ m) :
+    ∃B : ℝ≥0∞,B<∞ ∧ ∀ᶠ n in atTop,(∫⁻b,M.cleanupCost m n b∂W)≤B := by
+  obtain ⟨C,hC,hbound⟩ := CleanupCost.envelope_uniform_bound (alpha_lt_one (ha:=ha))
+    M.finalRadius (fun n=>(M.finalRadius_pos n).le) (Eventually.of_forall M.finalRadius_le)
+  let D := 4*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1)*max 1 ((m+1)^(1-alpha a))
+  refine ⟨ENNReal.ofReal (D*C),ENNReal.ofReal_lt_top,?_⟩
+  filter_upwards [eventually_batchReady a (by linarith)] with n hn
+  have hs : M.finalRadius n≤1 := (M.finalRadius_le n).trans
+    (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
+  apply (M.cleanupCost_integral_bound hW m hm n hn hs).trans
+  apply ENNReal.ofReal_le_ofReal
+  have hrow := BoundaryRows.rowConstant_pos (alpha_lt_one (ha:=ha))
+  have hD : 0≤D := by
+    dsimp [D]
+    have := M.cellConstant_pos
+    positivity
+  have henv : 4*(n:ℝ)*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1)*
+        (Real.exp (-((2:ℝ)^n/16))*(m+1)^(1-alpha a)+
+          (Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a)) ≤
+      D*CleanupCost.envelope (alpha a) M.finalRadius n := by
+    have hy : (m+1)^(1-alpha a) ≤ max 1 ((m+1)^(1-alpha a)) := le_max_right ..
+    have hu : (Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a) ≤
+        max 1 ((m+1)^(1-alpha a))*(Real.exp (-(n:ℝ))+M.finalRadius n)^(1-alpha a) := by
+      exact le_mul_of_one_le_left (Real.rpow_nonneg
+        (add_nonneg (Real.exp_pos _).le (M.finalRadius_pos n).le) _) (le_max_left ..)
+    have hh := mul_le_mul_of_nonneg_left
+      (add_le_add (mul_le_mul_of_nonneg_left hy (Real.exp_pos (-((2:ℝ)^n/16))).le) hu)
+      (show 0≤4*(n:ℝ)*M.cellConstant*BoundaryRows.rowConstant (alpha a)*(2*m+1) by
+        have := M.cellConstant_pos; positivity)
+    exact hh.trans_eq (by dsimp [D,CleanupCost.envelope]; rw [neg_div]; ring)
+  exact henv.trans (mul_le_mul_of_nonneg_left (hbound n) hD)
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology Classical
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart CompactSequence MomentBatches BoxGrid Deterministic CandidateCover
+
+def allFrontiers (a : ℝ) (b : GDriver) : Set ℂ :=
+  ⋃T : ℝ≥0,frontier {z | alive a b.1 T z}
+
+lemma exp_neg_nat_sq_tendsto :
+    Tendsto (fun n : ℕ=>Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
+  have hs : Tendsto (fun n : ℕ=>(n:ℝ)^2) atTop atTop :=
+    (tendsto_pow_atTop (by norm_num : (2:ℕ)≠0)).comp tendsto_natCast_atTop_atTop
+  exact Real.tendsto_exp_atBot.comp (tendsto_neg_atTop_atBot.comp hs)
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+abbrev GridCells (m : ℝ) (n : ℕ) := Cells m (M.finalRadius n)
+abbrev CandidateIndices (m : ℝ) (n : ℕ) := M.GridCells m n × Fin (M.testCount n)
+abbrev CoverIndices (m : ℝ) (n : ℕ) := M.CandidateIndices m n ⊕ M.GridCells m n
+
+def gridCenter (m : ℝ) (n : ℕ) (k : M.GridCells m n) : ℂ :=
+  center m (M.finalRadius n) k.1 k.2
+
+def costBound (m : ℝ) (n : ℕ) (b : GDriver) : ℝ≥0∞ :=
+  ENNReal.ofReal ((6*M.candidateScale)^2/M.candidateLower)*M.spatialMass b (region m)+
+    M.cleanupCost m n b
+
+lemma costBound_measurable (m : ℝ) (n : ℕ) : Measurable (M.costBound m n) :=
+  (measurable_const.mul (M.spatialMass_measurable _ (region_measurable m))).add
+    (M.cleanupCost_measurable m n)
+
+lemma costBound_uniform [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (m : ℝ) (hm : 0≤ m) :
+    ∃B : ℝ≥0∞,B<∞ ∧ ∀ᶠ n in atTop,(∫⁻b,M.costBound m n b∂W)≤B := by
+  obtain ⟨C,hC,hbound⟩ := M.cleanupCost_uniform_bound hW m hm
+  have hmean : (∫⁻b,M.spatialMass b (region m)∂W)<∞ := by
+    simp_rw [M.spatialMass_apply _ _ (region_measurable m),preimage_region]
+    exact (BoundaryGreen.canonical_rectangle_mean M.canonical _ _ _ (by linarith)).2.1
+  refine ⟨ENNReal.ofReal ((6*M.candidateScale)^2/M.candidateLower)*
+    (∫⁻b,M.spatialMass b (region m)∂W)+C,
+    ENNReal.add_lt_top.mpr ⟨ENNReal.mul_lt_top ENNReal.ofReal_lt_top hmean,hC⟩,?_⟩
+  filter_upwards [hbound] with n hn
+  simp only [costBound]
+  have hf : Measurable (fun b=>ENNReal.ofReal ((6*M.candidateScale)^2/M.candidateLower)*
+      M.spatialMass b (region m)) := measurable_const.mul (M.spatialMass_measurable _ (region_measurable m))
+  rw [lintegral_add_left hf,
+    lintegral_const_mul _ (M.spatialMass_measurable _ (region_measurable m))]
+  exact add_le_add le_rfl hn
+
+lemma finalRadius_le_one (n : ℕ) : M.finalRadius n≤1 :=
+  (M.finalRadius_le n).trans (Real.exp_le_one_iff.mpr (neg_nonpos.mpr (sq_nonneg _)))
+
+lemma gauge_admissible_actual : Target.IsHausdorffGauge (M.gauge (2-alpha a)) :=
+  M.gauge_admissible (by linarith [alpha_lt_one (ha:=ha)]) (by linarith [alpha_pos ha])
+
+lemma cleanup_ball_cost (n : ℕ) (hn : 0<n) (z : ℂ) :
+    diameterCost (M.gauge (2-alpha a)) (ediam (closedBall z (M.finalRadius n))) ≤
+      ENNReal.ofReal (4*(n:ℝ)*(M.finalRadius n)^(2-alpha a)) := by
+  apply (diameterCost_le_radius M.gauge_admissible_actual
+    (by have := M.finalRadius_pos n; positivity)
+    (ediam_closedBall_le_of_nonneg z (M.finalRadius n))).trans
+  apply ENNReal.ofReal_le_ofReal
+  rw [M.gauge_eq_quadratic]
+  have h := quadratic_multiple_le (2-alpha a) M.entryThreshold M.entryRadius
+    (fun i=>(M.entryThreshold_pos i).le) M.entryRadius_pos 2 (by norm_num)
+    (cleanupIndex (moment M.law sequenceMassReal) n hn)
+  rw [M.cleanup_radius n hn,show M.entryThreshold (cleanupIndex (moment M.law sequenceMassReal) n hn)=
+    (n:ℝ) from cleanup_threshold ..] at h
+  norm_num only [show (2:ℝ)^2=4 by norm_num] at h
+  exact h
+
+ 
+
+lemma finite_cover (m : ℝ) (hm : 0≤ m) (n : ℕ) (hn : 0<n)
+    (hsmall : M.candidateScale*Real.exp (-(n:ℝ)^2)≤1) :
+    ∀ᵐ b∂W,∃U : M.CoverIndices m n→Set ℂ,
+      (∀i,ediam (U i)≤ENNReal.ofReal (6*M.candidateScale*Real.exp (-(n:ℝ)^2))) ∧
+      allFrontiers a b∩box m ⊆ ⋃i,U i ∧
+      (∑i,diameterCost (M.gauge (2-alpha a)) (ediam (U i)))≤M.costBound m n b := by
+  have hcover : ∀ᵐ b∂W,∀k : M.GridCells m n,∀T : ℝ≥0,∀v∈closedBall (M.gridCenter m n k) (M.finalRadius n),
+      v∈frontier {z | alive a b.1 T z} →
+        (∃i<M.testCount n,M.candidate n i (M.gridCenter m n k) b) ∨
+          b∈M.cleanupEvent (M.gridCenter m n k) n :=
+    ae_all_iff.mpr (fun k=>M.cleanup_covers _
+      (center_im_pos m _ (M.finalRadius_pos n) _ _) n)
+  filter_upwards [hcover] with b hb
+  let s : Finset (M.CandidateIndices m n) := Finset.univ.filter
+    (fun j=>M.candidate n j.2 (M.gridCenter m n j.1) b)
+  let e : M.CandidateIndices m n→M.Entries := fun j=>M.testIndex n hn j.2 j.2.2
+  have her (j : M.CandidateIndices m n) : M.entryRadius (e j)=batchRadius a (M.batchLevel n) n j.2 := rfl
+  have het (j : M.CandidateIndices m n) : M.entryThreshold (e j)=M.testThreshold n j.2 :=
+    M.testIndex_threshold n hn j.2 j.2.2
+  have hrbound (j : M.CandidateIndices m n) : M.entryRadius (e j)≤Real.exp (-(n:ℝ)^2) := by
+    rw [her,←batchRadius_zero a (M.batchLevel n) n]
+    exact batchRadius_antitone a (by linarith) _ _ (Nat.zero_le _)
+  have hrball (j : M.CandidateIndices m n) : M.candidateScale*M.entryRadius (e j)≤1 :=
+    (mul_le_mul_of_nonneg_left (hrbound j) M.candidateScale_pos.le).trans hsmall
+  have hqual (j : M.CandidateIndices m n) (hj : j∈s) :
+      ENNReal.ofReal (M.candidateLower*M.entryThreshold (e j)*(M.entryRadius (e j))^(2-alpha a)) ≤
+        M.spatialMass b (closedBall (M.gridCenter m n j.1) (M.candidateScale*M.entryRadius (e j))∩univ) := by
+    rw [inter_univ,her,het]
+    exact (Finset.mem_filter.mp hj).2
+  obtain ⟨t,ht,hd,hcov,hcost⟩ := finite_quadratic_candidates (2-alpha a) M.entryThreshold M.entryRadius
+    (fun j=>(M.entryThreshold_pos j).le) M.entryRadius_pos M.gauge_admissible_actual
+    (M.spatialMass b) s (fun j=>M.gridCenter m n j.1) e univ (region m) MeasurableSet.univ
+    M.candidateScale M.candidateLower M.candidateScale_one M.candidateLower_pos
+    (fun j _=>(inter_subset_left.trans (ball_subset_region m _ _ hm (M.finalRadius_pos n)
+      (M.finalRadius_le_one n) (hrball j) j.1))) hqual
+  let U : M.CoverIndices m n→Set ℂ := Sum.elim
+    (fun j=>if j∈t then closedBall (M.gridCenter m n j.1) (3*(M.candidateScale*M.entryRadius (e j))) else ∅)
+    (fun k=>if b∈M.cleanupEvent (M.gridCenter m n k) n then
+      closedBall (M.gridCenter m n k) (M.finalRadius n) else ∅)
+  have hδ : 0≤6*M.candidateScale*Real.exp (-(n:ℝ)^2) := by have := M.candidateScale_pos; positivity
+  refine ⟨U,?_,?_,?_⟩
+  · intro i
+    cases i with
+    | inl j =>
+      dsimp [U]
+      split_ifs with hj
+      · apply (ediam_closedBall_le_of_nonneg _ _).trans
+        apply ENNReal.ofReal_le_ofReal
+        nlinarith [mul_le_mul_of_nonneg_left (hrbound j) M.candidateScale_pos.le]
+      · simp
+    | inr k =>
+      dsimp [U]
+      split_ifs
+      · apply (ediam_closedBall_le_of_nonneg _ _).trans
+        apply ENNReal.ofReal_le_ofReal
+        have he := Real.exp_pos (-(n:ℝ)^2)
+        nlinarith [M.finalRadius_le n,M.candidateScale_one]
+      · simp
+  · intro v hv
+    obtain ⟨T,hT⟩ := mem_iUnion.mp hv.1
+    obtain ⟨k,hk⟩ := mem_iUnion.mp (covered m _ hm (M.finalRadius_pos n) hv.2)
+    rcases hb k T v hk hT with hc | hc
+    · obtain ⟨i,hi,hc⟩ := hc
+      let j : M.CandidateIndices m n := (k,⟨i,hi⟩)
+      have hjs : j∈s := Finset.mem_filter.mpr ⟨Finset.mem_univ _,hc⟩
+      obtain ⟨q,hqt,hq⟩ := hcov j hjs
+      refine mem_iUnion.mpr ⟨Sum.inl q,?_⟩
+      change v∈if q∈t then _ else ∅
+      rw [ite_eq_left hqt]
+      apply hq
+      apply closedBall_subset_closedBall (ε₁:=M.finalRadius n) (ε₂:=M.candidateScale*M.entryRadius (e j)) _ hk
+      have hri : M.finalRadius n≤M.entryRadius (e j) :=
+        batchRadius_antitone a (by linarith) _ _ hi.le
+      exact hri.trans (le_mul_of_one_le_left (M.entryRadius_pos _).le M.candidateScale_one)
+    · refine mem_iUnion.mpr ⟨Sum.inr k,?_⟩
+      change v∈if b∈M.cleanupEvent (M.gridCenter m n k) n then _ else ∅
+      rw [ite_eq_left hc]
+      exact hk
+  · rw [Fintype.sum_sum_type]
+    unfold costBound
+    apply add_le_add
+    · change (∑j,diameterCost (M.gauge (2-alpha a)) (ediam
+        (if j∈t then closedBall (M.gridCenter m n j.1) (3*(M.candidateScale*M.entryRadius (e j))) else ∅))) ≤ _
+      simp only [apply_ite,ediam_empty,diameterCost_zero M.gauge_admissible_actual.2.2.1,
+        ←Finset.sum_filter]
+      simpa only [Finset.filter_mem_eq_inter,Finset.univ_inter,M.gauge_eq_quadratic] using hcost
+    · change (∑k,diameterCost (M.gauge (2-alpha a)) (ediam
+        (if b∈M.cleanupEvent (M.gridCenter m n k) n then closedBall (M.gridCenter m n k) (M.finalRadius n) else ∅))) ≤ _
+      unfold cleanupCost
+      apply Finset.sum_le_sum
+      intro k hk
+      change diameterCost (M.gauge (2-alpha a)) (ediam (if b∈M.cleanupEvent (M.gridCenter m n k) n then
+        closedBall (M.gridCenter m n k) (M.finalRadius n) else ∅)) ≤
+        (M.cleanupEvent (M.gridCenter m n k) n).indicator
+          (fun _=>ENNReal.ofReal (4*(n:ℝ)*(M.finalRadius n)^(2-alpha a))) b
+      by_cases hc : b∈M.cleanupEvent (M.gridCenter m n k) n
+      · simp only [ite_eq_left hc,indicator_of_mem hc]
+        exact M.cleanup_ball_cost n hn _
+      · simp only [ite_eq_right hc,ediam_empty,diameterCost_zero M.gauge_admissible_actual.2.2.1]
+        exact bot_le
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart CompactSequence MomentBatches BoxGrid Deterministic
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+ 
+
+theorem frontier_box_finite [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (m : ℝ) (hm : 0≤ m) :
+    ∀ᵐ b∂W,Target.hausdorff (M.gauge (2-alpha a)) (allFrontiers a b∩box m)<∞ := by
+  obtain ⟨B,hB,hbound⟩ := M.costBound_uniform hW m hm
+  have hfin := (CoverCost.liminf_cost_bound W (M.costBound m)
+    (M.costBound_measurable m) B hB hbound).2.2
+  have hc : ∀ᵐ b∂W,∀n : ℕ,0<n → M.candidateScale*Real.exp (-(n:ℝ)^2)≤1 →
+      ∃U : M.CoverIndices m n→Set ℂ,
+        (∀i,ediam (U i)≤ENNReal.ofReal (6*M.candidateScale*Real.exp (-(n:ℝ)^2))) ∧
+        allFrontiers a b∩box m⊆⋃i,U i ∧
+        (∑i,diameterCost (M.gauge (2-alpha a)) (ediam (U i)))≤M.costBound m n b := by
+    apply ae_all_iff.mpr
+    intro n
+    by_cases hn : 0<n
+    · by_cases hs : M.candidateScale*Real.exp (-(n:ℝ)^2)≤1
+      · exact (M.finite_cover m hm n hn hs).mono (fun b hb _ _=>hb)
+      · exact Eventually.of_forall (fun _ _ h=>(hs h).elim)
+    · exact Eventually.of_forall (fun _ h _=>(hn h).elim)
+  have he : Tendsto (fun n : ℕ=>M.candidateScale*Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
+    simpa only [mul_zero] using exp_neg_nat_sq_tendsto.const_mul M.candidateScale
+  have hsmall : ∀ᶠ n : ℕ in atTop,M.candidateScale*Real.exp (-(n:ℝ)^2)≤1 :=
+    (he.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1))).mono (fun _ h=>h.le)
+  have hδ : Tendsto (fun n : ℕ=>ENNReal.ofReal (6*M.candidateScale*Real.exp (-(n:ℝ)^2))) atTop (𝓝 0) := by
+    have hh : Tendsto (fun n : ℕ=>6*M.candidateScale*Real.exp (-(n:ℝ)^2)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using exp_neg_nat_sq_tendsto.const_mul (6*M.candidateScale)
+    simpa only [ENNReal.ofReal_zero] using ENNReal.tendsto_ofReal hh
+  filter_upwards [hc,hfin] with b hb hf
+  apply lt_of_le_of_lt _ hf
+  apply CoverCost.hausdorff_le_liminf_of_exists
+    (ι:=fun n=>M.CoverIndices m n) _ _ _ hδ _
+  filter_upwards [hsmall,eventually_gt_atTop (0:ℕ)] with n hs hn
+  obtain ⟨U,hd,hcov,hcost⟩ := hb n hn hs
+  exact ⟨U,hd,hcov,by simpa only [tsum_fintype] using hcost⟩
+
+ 
+theorem frontier_boxes_finite [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W) :
+    ∀ᵐ b∂W,∀m : ℕ,Target.hausdorff (M.gauge (2-alpha a)) (allFrontiers a b∩box m)<∞ :=
+  ae_all_iff.mpr (fun m=>M.frontier_box_finite hW m (Nat.cast_nonneg m))
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification TraceMassIntegration
+
+ 
+def dimension (κ : ℝ) : ℝ := 1+κ/8
+
+lemma parameterDimension_pos {κ : ℝ} (hκ : 0<κ) : 0<dimension κ := by
+  unfold dimension
+  linarith
+
+lemma dimension_lt_two {κ : ℝ} (hκ8 : κ<8) : dimension κ<2 := by
+  unfold dimension
+  linarith
+
+ 
+
+theorem exists_parameter_moment_gauge (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) :
+    ∃h : ℝ→ℝ, Target.IsHausdorffGauge h ∧
+      (∀C r : ℝ, 1≤C → 0≤r → h (C*r)≤C^2*h r) ∧
+      ∀W : Target.WienerLaw, Target.IsWiener W →
+        ∃M : CenteredMomentModel κ (Target.a κ) (a_gt_quarter hκ hκ8)
+          (W.map driverHomeomorph), h=M.gauge (dimension κ) := by
+  classical
+  by_cases h : ∃W : Target.WienerLaw, Target.IsWiener W
+  · obtain ⟨W,hW⟩ := h
+    obtain ⟨M⟩ := actual_centered_moment_model κ hκ hκ8 W hW
+    refine ⟨M.gauge (dimension κ),
+      M.gauge_admissible (parameterDimension_pos hκ) (dimension_lt_two hκ8).le,
+      M.gauge_doubling (parameterDimension_pos hκ) (dimension_lt_two hκ8).le,?_⟩
+    intro V hV
+    have e : V=W := BrownianLaw.evaluation_law_unique hV hW
+    subst V
+    exact ⟨M,rfl⟩
+  · refine ⟨id,⟨continuous_id.continuousOn,monotone_id.monotoneOn _,rfl,fun _ hr=>hr⟩,?_,?_⟩
+    · intro C r hC hr
+      change C*r ≤ C^2*r
+      exact mul_le_mul_of_nonneg_right (by nlinarith : C≤C^2) hr
+    · intro W hW
+      exact (h ⟨W,hW⟩).elim
+
+ 
+def parameterMomentGauge (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) : ℝ→ℝ :=
+  Classical.choose (exists_parameter_moment_gauge κ hκ hκ8)
+
+lemma parameterMomentGauge_admissible (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) :
+    Target.IsHausdorffGauge (parameterMomentGauge κ hκ hκ8) :=
+  (Classical.choose_spec (exists_parameter_moment_gauge κ hκ hκ8)).1
+
+lemma parameterMomentGauge_doubling (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (C r : ℝ) (hC : 1≤C) (hr : 0≤r) :
+    parameterMomentGauge κ hκ hκ8 (C*r) ≤ C^2*parameterMomentGauge κ hκ hκ8 r :=
+  (Classical.choose_spec (exists_parameter_moment_gauge κ hκ hκ8)).2.1 C r hC hr
+
+lemma parameterMomentGauge_model (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Target.WienerLaw) (hW : Target.IsWiener W) :
+    ∃M : CenteredMomentModel κ (Target.a κ) (a_gt_quarter hκ hκ8)
+      (W.map driverHomeomorph), parameterMomentGauge κ hκ hκ8=M.gauge (dimension κ) :=
+  (Classical.choose_spec (exists_parameter_moment_gauge κ hκ hκ8)).2.2 W hW
+
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CommonStoppedMass.lean b/lean/OAI/Probability/SLEGauge/CommonStoppedMass.lean
index f5923bcd980c3fa601d41bc3bc7bab8a58c56b4b..1a249398a398105bd882e25239f1511132ae69b5 100644
--- a/lean/OAI/Probability/SLEGauge/CommonStoppedMass.lean
+++ b/lean/OAI/Probability/SLEGauge/CommonStoppedMass.lean
@@ -1,570 +1,570 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ConformalRestartMap
-import OAI.Probability.SLEGauge.StoppedMassMeasurability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
-
- 
-
-theorem recoveredMass_all_stopping (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂recoveredMass a ha s c hc v b))) :
-    let μ := recoveredMass a ha s c hc
-    ∀ (S : ExactSLEGenerator.Driver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
-      Measurable[hS.measurableSpace] (fun b => μ (S b) b) ∧
-      (∀f : C_c(UpperPlane,ℝ), ∀m : ℕ,
-        MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) W) ∧
-      (∀K : Set UpperPlane, IsCompact K → ∀m : ℕ,
-        MemLp (fun b => (μ (S b) b K).toReal) (m:ℝ≥0∞) W) ∧
-      (∀f : C_c(UpperPlane,ℝ), Tendsto (fun k => eLpNorm
-        (fun b => (∑j∈s k,c k j*∫z,f z∂actualUpperMass a
-          (Real.exp (-2*a*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b)) 1 W)
-        atTop (𝓝 0)) ∧
-      (∀H : ExactSLEGenerator.Driver × UpperPlane→ℝ≥0∞,
-        Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=(∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
-          ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W) := by
-  intro μ S hS
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) :=
-    radonLimit_finite _
-  have hmeas := recoveredMass_stopping_raw_measurable a ha s c hc S hS
-  let K : Kernel ExactSLEGenerator.Driver UpperPlane :=
-    ⟨fun b => μ (S b) b,hmeas.mono hS.measurableSpace_le le_rfl⟩
-  let ν : Kernel ExactSLEGenerator.Driver UpperPlane :=
-    ⟨μ ∞,recoveredMass_measurable a ha s c hc ∞⟩
-  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (K b) := inferInstanceAs
-    (IsFiniteMeasureOnCompacts (μ (S b) b))
-  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (ν b) := inferInstanceAs
-    (IsFiniteMeasureOnCompacts (μ ∞ b))
-  have hcv (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S b) b)
-      atTop (𝓝 (∫z,f z∂μ (S b) b)) := hconv.mono (fun b hb => hb (S b) f)
-  have hct (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
-      atTop (𝓝 (∫z,f z∂ν b)) := hconv.mono (fun b hb => hb ∞ f)
-  have hmom (f : C_c(UpperPlane,ℝ)) (m : ℕ) :
-      MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) W :=
-    actual_stopped_radon_limit_memLp a ha W hW S hS s c hc hsum _ f (hcv f) m
-  have hint (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂K b) W :=
-    memLp_one_iff_integrable.mp (by simpa only [K,Kernel.coe_mk,Nat.cast_one] using hmom f 1)
-  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (∞:ℝ≥0∞)) :=
-    fun t => by simp
-  have hintop (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂ν b) W :=
-    memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using
-      actual_stopped_radon_limit_memLp a ha W hW _ htop s c hc hsum _ f (hct f) 1)
-  have hcond (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
-      (hU : MeasurableSet[hS.measurableSpace] U) :
-      (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂K b)∂W)=
-        ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea)∂W :=
-    actual_stopped_limit_compact_conditional a ha W hW S hS s c hs hc hsum
-      K ν f (hcv f) (hct f) U hU
-  refine ⟨hmeas,hmom,?_,?_,?_⟩
-  · intro A hA m
-    exact compact_mass_memLp K W (m:ℝ≥0∞) (fun f => hmom f m) A hA
-  · intro f
-    exact actual_stopped_radon_limit_L1 a ha W hW S hS _ (fun _ => Real.exp_pos _)
-      (level_step_le_one a ha) s c hc hsum _ f (hcv f)
-  · intro H hH
-    exact stopped_compact_conditional_to_random_tests a ha W hW S hS K ν hint hintop hcond H hH
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceRestriction
-
-variable {X Y : Type*} [MeasurableSpace X] [MetricSpace Y]
-  [MeasurableSpace Y] [BorelSpace Y]
-
-lemma firstVisit_finite_of_exhaustion (ν : Measure X)
-    (ι : X→Y) (hι : Measurable ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
-    (hlim : ∀A : Set X, MeasurableSet A → Tendsto
-      (fun n : ℕ => (ν.restrict (visited ι γ n)) A) atTop (𝓝 (ν A))) :
-    ∀ᵐ x ∂ν, firstVisit ι γ x<∞ := by
-  let V := ι ⁻¹' range γ
-  have hV : MeasurableSet V := by
-    rw [show V=⋃n : ℕ,visited ι γ n from (iUnion_visited_nat ι γ).symm]
-    exact MeasurableSet.iUnion (fun n => measurableSet_visited ι hι γ hγ n)
-  have hz (n : ℕ) : (ν.restrict (visited ι γ n)) Vᶜ=0 := by
-    rw [Measure.restrict_apply hV.compl]
-    have he : Vᶜ∩visited ι γ n=∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      rintro x ⟨hn,u,hu,he⟩
-      exact hn ⟨u,he⟩
-    rw [he,measure_empty]
-  have hnull : ν Vᶜ=0 := by
-    have h := hlim Vᶜ hV.compl
-    simp_rw [hz] at h
-    exact tendsto_nhds_unique h tendsto_const_nhds
-  filter_upwards [measure_eq_zero_iff_ae_notMem.mp hnull] with x hx
-  apply (firstVisit_lt_top_iff ι γ x).mpr
-  exact not_not.mp hx
-
- 
-omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
-lemma firstVisit_fiber_subset [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y]
-    (ι : X→Y) (γ : ℝ≥0→Y) (hγ : Continuous γ)
-    (t : ℝ≥0) : {x | firstVisit ι γ x=(t : ℝ≥0∞)}⊆ι ⁻¹' {γ t} := by
-  rintro x hx
-  change firstVisit ι γ x=(t : ℝ≥0∞) at hx
-  by_cases h : ({u : ℝ≥0 | γ u=ι x}).Nonempty
-  · have hclosed : IsClosed {u : ℝ≥0 | γ u=ι x} := isClosed_eq hγ continuous_const
-    have hmem := hclosed.csInf_mem h ⟨0,fun _ _ => bot_le⟩
-    have he : sInf {u : ℝ≥0 | γ u=ι x}=t := by
-      apply ENNReal.coe_injective
-      simpa only [firstVisit,dite_eq_left h] using hx
-    rw [he] at hmem
-    exact hmem.symm
-  · simp only [firstVisit,dite_eq_right h] at hx
-    exact False.elim (ENNReal.coe_ne_top hx.symm)
-
-lemma firstVisit_fiber_null (ν : Measure X) [NullSingletonClass ν]
-    (ι : X→Y) (hinj : Function.Injective ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
-    (t : ℝ≥0) : ν {x | firstVisit ι γ x=(t : ℝ≥0∞)}=0 :=
-  measure_mono_null (firstVisit_fiber_subset ι γ hγ t)
-    (preimage_singleton_null_of_injective ν ι hinj (γ t))
-
- 
-
-lemma restrict_strictFirstVisit (ν : Measure X) [NullSingletonClass ν]
-    (ι : X→Y) (hinj : Function.Injective ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
-    (t : ℝ≥0) :
-    ν.restrict {x | firstVisit ι γ x<(t : ℝ≥0∞)}=ν.restrict (visited ι γ t) := by
-  apply Measure.restrict_congr_set
-  filter_upwards [measure_eq_zero_iff_ae_notMem.mp
-    (firstVisit_fiber_null ν ι hinj γ hγ t)] with x hx
-  apply propext
-  rw [←firstVisit_le_iff ι γ hγ x t,lt_iff_le_and_ne]
-  exact and_iff_left hx
-
-end SLEExactGauge.TraceRestriction
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceMassContinuity
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen TraceRestriction
-
- 
-
-structure TraceMassFacts (κ : ℝ) (b : TDriver) (μ : ℝ≥0∞→Measure UpperPlane) : Prop where
-  continuous_trace : ContinuousOn (Target.trace κ b) (Ici 0)
-  atomless : NullSingletonClass (μ ∞)
-  restriction : ∀t : ℝ≥0, μ t=(μ ∞).restrict
-    (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t)
-  monotone : Monotone μ
-  increment_support : ∀s t : ℝ≥0, s≤t → (μ t-μ s)
-    {z : UpperPlane | z.1∉internalTrace κ b '' Icc s t}=0
-  continuous_tests : ∀f : C_c(UpperPlane,ℝ), Continuous (fun t : ℝ≥0 => ∫z,f z∂μ t)
-  local_variation : ∀K : Set UpperPlane, IsCompact K → ∀t : ℝ≥0, Tendsto
-    (fun u : ℝ≥0 => ((μ u-μ t)+(μ t-μ u)) K) (𝓝 t) (𝓝 0)
-  exhaustion : ∀A : Set UpperPlane, MeasurableSet A → Tendsto
-    (fun n : ℕ => μ (n:ℝ≥0) A) atTop (𝓝 (μ ∞ A))
-  remaining_decay : ∀K : Set UpperPlane, IsCompact K → Tendsto
-    (fun n : ℕ => (μ ∞-μ (n:ℝ≥0)) K) atTop (𝓝 0)
-  firstVisit_measurable : Measurable (firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b))
-  firstVisit_finite : ∀ᵐ z ∂μ ∞, firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b) z<∞
-  firstVisit_identity : ∀t : ℝ≥0, μ t=(μ ∞).restrict
-    {z | firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b) z≤(t:ℝ≥0∞)}
-  strict_firstVisit_identity : ∀t : ℝ≥0, μ t=(μ ∞).restrict
-    {z | firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b) z<(t:ℝ≥0∞)}
-  initial_positive : ∀t : ℝ≥0∞, 0<t → 0<μ t univ
-  initial_positive_compact : ∀t : ℝ≥0∞, 0<t → ∃K : Set UpperPlane, IsCompact K ∧
-    0<μ t K ∧ μ t K<∞
-
-lemma traceMassFacts_of_restrictions (κ : ℝ) (hκ : 0<κ) (b : TDriver)
-    (μ : ℝ≥0∞→Measure UpperPlane) [∀v,IsFiniteMeasureOnCompacts (μ v)]
-    (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
-    (hatom : NullSingletonClass (μ ∞))
-    (hres : ∀t : ℝ≥0, μ t=(μ ∞).restrict
-      (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t))
-    (hexh : ∀A : Set UpperPlane, MeasurableSet A → Tendsto
-      (fun n : ℕ => μ (n:ℝ≥0) A) atTop (𝓝 (μ ∞ A)))
-    (hrem : ∀K : Set UpperPlane, IsCompact K → Tendsto
-      (fun n : ℕ => (μ ∞-μ (n:ℝ≥0)) K) atTop (𝓝 0))
-    (hpos : ∀t : ℝ≥0∞, 0<t → 0<μ t univ)
-    (hposK : ∀t : ℝ≥0∞, 0<t → ∃K : Set UpperPlane, IsCompact K ∧ 0<μ t K ∧ μ t K<∞) :
-    TraceMassFacts κ b μ := by
-  let := hatom
-  have hγ := continuous_internalTrace hκ b hcont
-  refine ⟨hcont,hatom,hres,?_,?_,?_,?_,hexh,hrem,?_,?_,?_,?_,hpos,hposK⟩
-  · intro u v huv
-    induction v using ENNReal.recTopCoe with
-    | top =>
-      induction u using ENNReal.recTopCoe with
-      | top => exact le_rfl
-      | coe u => rw [hres u]; exact Measure.restrict_le_self
-    | coe v =>
-      have hu : u≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top huv
-      rw [←ENNReal.coe_toNNReal hu,hres,hres]
-      exact Measure.restrict_mono (visited_mono _ _ (by
-        apply ENNReal.coe_le_coe.mp
-        simpa only [ENNReal.coe_toNNReal hu] using huv)) le_rfl
-  · intro s t hst
-    rw [hres,hres]
-    exact increment_carried_by_segment (μ ∞) _ measurable_subtype_coe _ hγ hst
-  · intro f
-    simp_rw [hres]
-    exact continuous_integral_visited (μ ∞) _ measurable_subtype_coe Subtype.val_injective
-      _ hγ f f.integrable
-  · intro K hK t
-    simp_rw [hres]
-    exact local_variation_visited (μ ∞) hκ b hcont K hK.measurableSet hK.measure_lt_top.ne t
-  · exact measurable_firstVisit _ measurable_subtype_coe _ hγ
-  · apply firstVisit_finite_of_exhaustion (μ ∞) _ measurable_subtype_coe _ hγ
-    intro A hA
-    simpa only [←hres] using hexh A hA
-  · intro t
-    rw [hres]
-    congr 1
-    ext z
-    exact (firstVisit_le_iff _ _ hγ z t).symm
-  · intro t
-    rw [hres,restrict_strictFirstVisit (μ ∞) _ Subtype.val_injective _ hγ]
-
- 
-
-lemma TraceMassFacts.capacity_increment_support {κ : ℝ} (hκ : 0<κ) {b : TDriver}
-    {μ : ℝ≥0∞→Measure UpperPlane} (hμ : TraceMassFacts κ b μ)
-    (s t : ℝ) (hs : 0 ≤ s) (hst : s ≤ t) :
-    (μ ((κ*t).toNNReal : ℝ≥0∞)-μ ((κ*s).toNNReal : ℝ≥0∞))
-      {z : UpperPlane | z.1∉Target.trace κ b '' Icc s t}=0 := by
-  have hst' : (κ*s).toNNReal≤(κ*t).toNNReal :=
-    Real.toNNReal_mono (mul_le_mul_of_nonneg_left hst hκ.le)
-  have h := hμ.increment_support _ _ hst'
-  have hs0 : 0≤κ*s := mul_nonneg hκ.le hs
-  have ht0 : 0≤κ*t := mul_nonneg hκ.le (hs.trans hst)
-  simpa only [internalTrace_image hκ,Real.coe_toNNReal _ hs0,Real.coe_toNNReal _ ht0,
-    mul_div_cancel_left₀ _ hκ.ne'] using h
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceMassContinuity
-open GermPositivity
-
- 
-
-def massTimeList (n : ℕ) : ℝ≥0∞ :=
-  if n%4=0 then ∞ else if n%4=1 then ((n/4 : ℕ) : ℝ≥0)
-    else if n%4=2 then smallTime (n/4)
-      else TopologicalSpace.denseSeq ℝ≥0 (n/4)
-
-@[simp] lemma massTimeList_top : massTimeList 0=∞ := by simp [massTimeList]
-
-@[simp] lemma massTimeList_nat (n : ℕ) : massTimeList (4*n+1)=(n : ℝ≥0) := by
-  simp [massTimeList,Nat.mul_add_div]
-
-@[simp] lemma massTimeList_small (n : ℕ) : massTimeList (4*n+2)=(smallTime n : ℝ≥0∞) := by
-  simp [massTimeList,Nat.mul_add_div]
-
-@[simp] lemma massTimeList_dense (n : ℕ) :
-    massTimeList (4*n+3)=(TopologicalSpace.denseSeq ℝ≥0 n : ℝ≥0∞) := by
-  simp [massTimeList,Nat.mul_add_div]
-
-lemma massTimeList_is_dense : Dense {u : ℝ≥0 | ∃i,massTimeList i=u} := by
-  apply Dense.mono (s₁ := range (TopologicalSpace.denseSeq ℝ≥0)) _
-    (TopologicalSpace.denseRange_denseSeq ℝ≥0)
-  rintro u ⟨n,rfl⟩
-  exact ⟨4*n+3,massTimeList_dense n⟩
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceMassContinuity
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions TraceMassIntegration
-
- 
-
-theorem actual_common_trace_mass (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
-      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      (∀v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀b, μ 0 b=0) ∧ (∀v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      μ ∞ ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ) ∧
-      (∀ (S : ExactSLEGenerator.Driver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
-        Measurable[hS.measurableSpace] (fun b => μ (S b) b) ∧
-        (∀f : C_c(UpperPlane,ℝ), ∀m : ℕ,
-          MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
-        (∀K : Set UpperPlane, IsCompact K → ∀m : ℕ,
-          MemLp (fun b => (μ (S b) b K).toReal) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
-        (∀f : C_c(UpperPlane,ℝ), Tendsto (fun k => eLpNorm
-          (fun b => (∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-            (Real.exp (-2*(Target.a κ)*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b))
-              1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
-        (∀H : ExactSLEGenerator.Driver × UpperPlane→ℝ≥0∞,
-          Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
-          (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂(W.map driverHomeomorph))=
-            (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂(W.map driverHomeomorph))+
-            ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen (Target.a κ) (S b) b z.1)
-              ∂upperArea∂(W.map driverHomeomorph))) ∧
-      ∀ᵐ b ∂W, TraceMassFacts κ b (fun v => μ v (driverHomeomorph b)) ∧
-        (∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-          (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-            (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
-          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) := by
-  have hBW := brownian_map_driver W hW
-  have : IsProbabilityMeasure (W.map driverHomeomorph) := hBW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hpath⟩ :=
-    actual_raw_mass_trace κ hκ hκ8 W hW massTimeList massTimeList_is_dense
-      (fun n => 4*n+1) massTimeList_nat (fun n => 4*n+2) massTimeList_small 0 massTimeList_top
-  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  have hcv : ∀ᵐ b ∂(W.map driverHomeomorph), ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ),
-      Tendsto (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b) atTop (𝓝 (∫z,f z∂μ v b)) := by
-    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-    filter_upwards [hpath] with b hb
-    exact hb.2.2.2.1
-  refine ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,?_,?_,?_⟩
-  · simpa only [massTimeList_top] using hmean 0 massTimeList_top
-  · exact recoveredMass_all_stopping (Target.a κ) (a_gt_quarter hκ hκ8)
-      (W.map driverHomeomorph) hBW s c hc hs hsum hcv
-  · filter_upwards [hpath] with b hb
-    rcases hb with ⟨hcont,hatom,hres,hconv,hexh,hrem,hpos,hposK⟩
-    let (v : ℝ≥0∞) : IsFiniteMeasureOnCompacts (μ v (driverHomeomorph b)) := (hreg _ _).1
-    exact ⟨traceMassFacts_of_restrictions κ hκ b (fun v => μ v (driverHomeomorph b))
-      hcont hatom hres hexh hrem hpos hposK,hconv⟩
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGenerator
-
- 
-def concatenate (s : ℝ≥0) (b c : GDriver) : GDriver :=
-  ⟨⟨fun u => b.1 (min u s)+c.1 (u-s),
-    (b.1.continuous.comp (continuous_id.min continuous_const)).add
-      (c.1.continuous.comp (continuous_id.sub continuous_const))⟩,
-    by
-      change b.1 (min 0 s)+c.1 (0-s)=0
-      simp only [min_eq_left (show 0 ≤ s from bot_le),zero_tsub,b.2,c.2,add_zero]⟩
-
-@[simp] lemma concatenate_apply (s u : ℝ≥0) (b c : GDriver) :
-    (concatenate s b c).1 u=b.1 (min u s)+c.1 (u-s) := rfl
-
-lemma continuous_concatenate : Continuous (fun p : (ℝ≥0×GDriver)×GDriver =>
-    concatenate p.1.1 p.1.2 p.2) := by
-  apply Continuous.subtype_mk
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  exact (continuous_eval.comp
-    ((continuous_subtype_val.comp continuous_fst.fst.snd).prodMk
-      (continuous_snd.min continuous_fst.fst.fst))).add
-    (continuous_eval.comp
-      ((continuous_subtype_val.comp continuous_fst.snd).prodMk
-        (continuous_snd.sub continuous_fst.fst.fst)))
-
-lemma concatenate_before (s : ℝ≥0) (b c : GDriver) (u : ℝ≥0) (hu : u ≤ s) :
-    (concatenate s b c).1 u=b.1 u := by
-  simp only [concatenate_apply,min_eq_left hu,tsub_eq_zero_of_le hu,c.2,add_zero]
-
-lemma concatenate_reconstruct (s : ℝ≥0) (b : GDriver) :
-    concatenate s (pastDriver s b) (restart s b)=b := by
-  apply Subtype.ext
-  ext u
-  by_cases hu : u ≤ s
-  · rw [concatenate_before _ _ _ _ hu,pastDriver_eq _ _ _ hu]
-  · have hsu : s ≤ u := le_of_not_ge hu
-    simp only [concatenate_apply,min_eq_right hsu,restart_apply,pastDriver_eq s b s le_rfl,
-      add_tsub_cancel_of_le hsu]
-    ring
-
- 
-lemma raw_event_congr {u : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[rawFiltration u] A) {b c : GDriver}
-    (h : ∀v ≤ u,b.1 v=c.1 v) : b∈A ↔ c∈A := by
-  rw [rawFiltration,Filtration.natural_eq_comap,MeasurableSpace.measurableSet_comap] at hA
-  obtain ⟨B,hB,rfl⟩ := hA
-  have he : (fun v : Iic u => b.1 v)=(fun v : Iic u => c.1 v) :=
-    funext (fun v => h v v.2)
-  change (fun v : Iic u => b.1 v)∈B ↔ (fun v : Iic u => c.1 v)∈B
-  rw [he]
-
- 
-lemma raw_stop_congr {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {b c : GDriver} {s : ℝ≥0} (hb : S b=s) (h : ∀u ≤ s,b.1 u=c.1 u) : S c=s := by
-  have hc : S c ≤ s := (raw_event_congr (hS s) h).mp (by
-    change S b ≤ (s:ℝ≥0∞)
-    rw [hb])
-  have hcf : S c≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hc
-  have hct : (S c).toNNReal ≤ s := ENNReal.coe_le_coe.mp (by
-    simpa only [ENNReal.coe_toNNReal hcf] using hc)
-  have hbc : S b ≤ S c := by
-    have hh := (raw_event_congr (hS (S c).toNNReal)
-      (fun u hu => h u (hu.trans hct))).mpr
-      (show S c ≤ ((S c).toNNReal:ℝ≥0∞) by rw [ENNReal.coe_toNNReal hcf])
-    change S b ≤ ((S c).toNNReal:ℝ≥0∞) at hh
-    simpa only [ENNReal.coe_toNNReal hcf] using hh
-  exact le_antisymm hc (hb ▸ hbc)
-
- 
-def shiftStop (S : GDriver→ℝ≥0∞) (t : ℝ≥0) : GDriver→ℝ≥0∞ := fun b => S b+t
-
-lemma shiftStop_isStoppingTime {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    (t : ℝ≥0) : IsStoppingTime rawFiltration (shiftStop S t) := by
-  intro u
-  change MeasurableSet[rawFiltration u] {b | shiftStop S t b ≤ (u:ℝ≥0∞)}
-  by_cases htu : t ≤ u
-  · have he : {b : GDriver | shiftStop S t b ≤ u}={b | S b ≤ ((u-t:ℝ≥0):ℝ≥0∞)} := by
-      ext b
-      simp only [shiftStop,mem_ofPred_eq,ENNReal.coe_sub]
-      exact (ENNReal.le_sub_iff_add_le_right ENNReal.coe_ne_top
-        (ENNReal.coe_le_coe.mpr htu)).symm
-    rw [he]
-    exact (rawFiltration.mono tsub_le_self) _ (hS (u-t))
-  · have he : {b : GDriver | shiftStop S t b ≤ u}=∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      intro b hb
-      exact htu (ENNReal.coe_le_coe.mp ((le_add_left le_rfl).trans hb))
-    rw [he]
-    exact @MeasurableSet.empty GDriver (rawFiltration u)
-
-lemma measurableSpace_le_shiftStop {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) (t : ℝ≥0) :
-    hS.measurableSpace ≤ (shiftStop_isStoppingTime hS t).measurableSpace :=
-  hS.measurableSpace_mono (shiftStop_isStoppingTime hS t) (fun b => show S b ≤ S b+(t:ℝ≥0∞) from le_self_add)
-
- 
-lemma raw_stopped_event_congr {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A)
-    {b c : GDriver} {u : ℝ≥0} (hb : S b ≤ u) (h : ∀v ≤ u,b.1 v=c.1 v) : b∈A ↔ c∈A := by
-  have hc : S c ≤ u := (raw_event_congr (hS u) h).mp hb
-  have hh := raw_event_congr (hA.2 u) h
-  change (b∈A ∧ S b ≤ (u:ℝ≥0∞)) ↔ (c∈A ∧ S c ≤ (u:ℝ≥0∞)) at hh
-  simpa only [hb,hc,and_true] using hh
-
- 
-
-def concatenatedEvent (S : GDriver→ℝ≥0∞) (A : Set GDriver) :
-    Set ((ℝ≥0×GDriver)×GDriver) :=
-  {p | S (concatenate p.1.1 p.1.2 p.2) ≤ (p.1.1:ℝ≥0∞) ∧
-    concatenate p.1.1 p.1.2 p.2∈A}
-
-lemma concatenatedEvent_measurable {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) {A : Set GDriver} (hA : MeasurableSet A) :
-    MeasurableSet (concatenatedEvent S A) := by
-  have : PolishSpace GDriver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
-  have hc : Measurable (fun p : (ℝ≥0×GDriver)×GDriver => concatenate p.1.1 p.1.2 p.2) :=
-    continuous_concatenate.measurable
-  have ht : Measurable (fun p : (ℝ≥0×GDriver)×GDriver => (p.1.1:ℝ≥0∞)) :=
-    ENNReal.continuous_coe.measurable.comp measurable_fst.fst
-  exact (measurableSet_le (hS.measurable'.comp hc) ht).inter (hc hA)
-
-lemma concatenatedEvent_past {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) {t : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
-    (s : ℝ≥0) (b c : GDriver) :
-    ((s,b),c)∈concatenatedEvent S A ↔ ((s,b),pastDriver t c)∈concatenatedEvent S A := by
-  have hsmall : ∀u ≤ s,(concatenate s b c).1 u=(concatenate s b (pastDriver t c)).1 u := by
-    intro u hu
-    rw [concatenate_before _ _ _ _ hu,concatenate_before _ _ _ _ hu]
-  have hlong : ∀u ≤ s+t,(concatenate s b c).1 u=(concatenate s b (pastDriver t c)).1 u := by
-    intro u hu
-    simp only [concatenate_apply]
-    rw [pastDriver_eq t c (u-s) (tsub_le_iff_right.mpr (by simpa only [add_comm] using hu))]
-  have hg := raw_event_congr (hS s) hsmall
-  change (S (concatenate s b c) ≤ (s:ℝ≥0∞) ∧ concatenate s b c∈A) ↔
-    (S (concatenate s b (pastDriver t c)) ≤ (s:ℝ≥0∞) ∧
-      concatenate s b (pastDriver t c)∈A)
-  change (S (concatenate s b c) ≤ (s:ℝ≥0∞)) ↔
-    (S (concatenate s b (pastDriver t c)) ≤ (s:ℝ≥0∞)) at hg
-  rw [←hg]
-  apply and_congr_right
-  intro h
-  exact raw_stopped_event_congr (shiftStop_isStoppingTime hS t) hA
-    (by
-      change S (concatenate s b c)+(t:ℝ≥0∞) ≤ ((s+t:ℝ≥0):ℝ≥0∞)
-      rw [ENNReal.coe_add]
-      exact add_le_add h le_rfl) hlong
-
-lemma concatenatedEvent_raw_joint {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) {t : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A) :
-    @MeasurableSet ((ℝ≥0×GDriver)×GDriver)
-      ((inferInstance : MeasurableSpace (ℝ≥0×GDriver)).prod (rawFiltration t))
-      (concatenatedEvent S A) := by
-  have hM : @Measurable ((ℝ≥0×GDriver)×GDriver) ((ℝ≥0×GDriver)×GDriver)
-      ((inferInstance : MeasurableSpace (ℝ≥0×GDriver)).prod (rawFiltration t)) inferInstance
-      (fun p => (p.1,pastDriver t p.2)) :=
-    measurable_fst.prodMk ((pastDriver_raw_measurable t).comp measurable_snd)
-  have hh := hM (concatenatedEvent_measurable hS
-    ((shiftStop_isStoppingTime hS t).measurableSpace_le A hA))
-  convert hh using 1
-  ext p
-  exact concatenatedEvent_past hS hA p.1.1 p.1.2 p.2
-
-lemma concatenatedEvent_stoppedData {S : GDriver→ℝ≥0∞} {A : Set GDriver}
-    {b : GDriver} (hb : S b≠∞) :
-    (stoppedData S b,restartAt S b)∈concatenatedEvent S A ↔ b∈A := by
-  simp only [concatenatedEvent,stoppedData,restartAt,mem_ofPred_eq,concatenate_reconstruct,
-    ENNReal.coe_toNNReal hb,le_refl,true_and]
-
-end SLEExactGauge.LoewnerRestart
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ConformalRestartMap
+import OAI.Probability.SLEGauge.StoppedMassMeasurability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
+
+ 
+
+theorem recoveredMass_all_stopping (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂recoveredMass a ha s c hc v b))) :
+    let μ := recoveredMass a ha s c hc
+    ∀ (S : ExactSLEGenerator.Driver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
+      Measurable[hS.measurableSpace] (fun b => μ (S b) b) ∧
+      (∀f : C_c(UpperPlane,ℝ), ∀m : ℕ,
+        MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) W) ∧
+      (∀K : Set UpperPlane, IsCompact K → ∀m : ℕ,
+        MemLp (fun b => (μ (S b) b K).toReal) (m:ℝ≥0∞) W) ∧
+      (∀f : C_c(UpperPlane,ℝ), Tendsto (fun k => eLpNorm
+        (fun b => (∑j∈s k,c k j*∫z,f z∂actualUpperMass a
+          (Real.exp (-2*a*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b)) 1 W)
+        atTop (𝓝 0)) ∧
+      (∀H : ExactSLEGenerator.Driver × UpperPlane→ℝ≥0∞,
+        Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=(∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
+          ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W) := by
+  intro μ S hS
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) :=
+    radonLimit_finite _
+  have hmeas := recoveredMass_stopping_raw_measurable a ha s c hc S hS
+  let K : Kernel ExactSLEGenerator.Driver UpperPlane :=
+    ⟨fun b => μ (S b) b,hmeas.mono hS.measurableSpace_le le_rfl⟩
+  let ν : Kernel ExactSLEGenerator.Driver UpperPlane :=
+    ⟨μ ∞,recoveredMass_measurable a ha s c hc ∞⟩
+  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (K b) := inferInstanceAs
+    (IsFiniteMeasureOnCompacts (μ (S b) b))
+  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (ν b) := inferInstanceAs
+    (IsFiniteMeasureOnCompacts (μ ∞ b))
+  have hcv (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S b) b)
+      atTop (𝓝 (∫z,f z∂μ (S b) b)) := hconv.mono (fun b hb => hb (S b) f)
+  have hct (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
+      atTop (𝓝 (∫z,f z∂ν b)) := hconv.mono (fun b hb => hb ∞ f)
+  have hmom (f : C_c(UpperPlane,ℝ)) (m : ℕ) :
+      MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) W :=
+    actual_stopped_radon_limit_memLp a ha W hW S hS s c hc hsum _ f (hcv f) m
+  have hint (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂K b) W :=
+    memLp_one_iff_integrable.mp (by simpa only [K,Kernel.coe_mk,Nat.cast_one] using hmom f 1)
+  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (∞:ℝ≥0∞)) :=
+    fun t => by simp
+  have hintop (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂ν b) W :=
+    memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using
+      actual_stopped_radon_limit_memLp a ha W hW _ htop s c hc hsum _ f (hct f) 1)
+  have hcond (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
+      (hU : MeasurableSet[hS.measurableSpace] U) :
+      (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂K b)∂W)=
+        ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea)∂W :=
+    actual_stopped_limit_compact_conditional a ha W hW S hS s c hs hc hsum
+      K ν f (hcv f) (hct f) U hU
+  refine ⟨hmeas,hmom,?_,?_,?_⟩
+  · intro A hA m
+    exact compact_mass_memLp K W (m:ℝ≥0∞) (fun f => hmom f m) A hA
+  · intro f
+    exact actual_stopped_radon_limit_L1 a ha W hW S hS _ (fun _ => Real.exp_pos _)
+      (level_step_le_one a ha) s c hc hsum _ f (hcv f)
+  · intro H hH
+    exact stopped_compact_conditional_to_random_tests a ha W hW S hS K ν hint hintop hcond H hH
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceRestriction
+
+variable {X Y : Type*} [MeasurableSpace X] [MetricSpace Y]
+  [MeasurableSpace Y] [BorelSpace Y]
+
+lemma firstVisit_finite_of_exhaustion (ν : Measure X)
+    (ι : X→Y) (hι : Measurable ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
+    (hlim : ∀A : Set X, MeasurableSet A → Tendsto
+      (fun n : ℕ => (ν.restrict (visited ι γ n)) A) atTop (𝓝 (ν A))) :
+    ∀ᵐ x ∂ν, firstVisit ι γ x<∞ := by
+  let V := ι ⁻¹' range γ
+  have hV : MeasurableSet V := by
+    rw [show V=⋃n : ℕ,visited ι γ n from (iUnion_visited_nat ι γ).symm]
+    exact MeasurableSet.iUnion (fun n => measurableSet_visited ι hι γ hγ n)
+  have hz (n : ℕ) : (ν.restrict (visited ι γ n)) Vᶜ=0 := by
+    rw [Measure.restrict_apply hV.compl]
+    have he : Vᶜ∩visited ι γ n=∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      rintro x ⟨hn,u,hu,he⟩
+      exact hn ⟨u,he⟩
+    rw [he,measure_empty]
+  have hnull : ν Vᶜ=0 := by
+    have h := hlim Vᶜ hV.compl
+    simp_rw [hz] at h
+    exact tendsto_nhds_unique h tendsto_const_nhds
+  filter_upwards [measure_eq_zero_iff_ae_notMem.mp hnull] with x hx
+  apply (firstVisit_lt_top_iff ι γ x).mpr
+  exact not_not.mp hx
+
+ 
+omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
+lemma firstVisit_fiber_subset [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y]
+    (ι : X→Y) (γ : ℝ≥0→Y) (hγ : Continuous γ)
+    (t : ℝ≥0) : {x | firstVisit ι γ x=(t : ℝ≥0∞)}⊆ι ⁻¹' {γ t} := by
+  rintro x hx
+  change firstVisit ι γ x=(t : ℝ≥0∞) at hx
+  by_cases h : ({u : ℝ≥0 | γ u=ι x}).Nonempty
+  · have hclosed : IsClosed {u : ℝ≥0 | γ u=ι x} := isClosed_eq hγ continuous_const
+    have hmem := hclosed.csInf_mem h ⟨0,fun _ _ => bot_le⟩
+    have he : sInf {u : ℝ≥0 | γ u=ι x}=t := by
+      apply ENNReal.coe_injective
+      simpa only [firstVisit,dite_eq_left h] using hx
+    rw [he] at hmem
+    exact hmem.symm
+  · simp only [firstVisit,dite_eq_right h] at hx
+    exact False.elim (ENNReal.coe_ne_top hx.symm)
+
+lemma firstVisit_fiber_null (ν : Measure X) [NullSingletonClass ν]
+    (ι : X→Y) (hinj : Function.Injective ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
+    (t : ℝ≥0) : ν {x | firstVisit ι γ x=(t : ℝ≥0∞)}=0 :=
+  measure_mono_null (firstVisit_fiber_subset ι γ hγ t)
+    (preimage_singleton_null_of_injective ν ι hinj (γ t))
+
+ 
+
+lemma restrict_strictFirstVisit (ν : Measure X) [NullSingletonClass ν]
+    (ι : X→Y) (hinj : Function.Injective ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
+    (t : ℝ≥0) :
+    ν.restrict {x | firstVisit ι γ x<(t : ℝ≥0∞)}=ν.restrict (visited ι γ t) := by
+  apply Measure.restrict_congr_set
+  filter_upwards [measure_eq_zero_iff_ae_notMem.mp
+    (firstVisit_fiber_null ν ι hinj γ hγ t)] with x hx
+  apply propext
+  rw [←firstVisit_le_iff ι γ hγ x t,lt_iff_le_and_ne]
+  exact and_iff_left hx
+
+end SLEExactGauge.TraceRestriction
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceMassContinuity
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen TraceRestriction
+
+ 
+
+structure TraceMassFacts (κ : ℝ) (b : TDriver) (μ : ℝ≥0∞→Measure UpperPlane) : Prop where
+  continuous_trace : ContinuousOn (Target.trace κ b) (Ici 0)
+  atomless : NullSingletonClass (μ ∞)
+  restriction : ∀t : ℝ≥0, μ t=(μ ∞).restrict
+    (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t)
+  monotone : Monotone μ
+  increment_support : ∀s t : ℝ≥0, s≤t → (μ t-μ s)
+    {z : UpperPlane | z.1∉internalTrace κ b '' Icc s t}=0
+  continuous_tests : ∀f : C_c(UpperPlane,ℝ), Continuous (fun t : ℝ≥0 => ∫z,f z∂μ t)
+  local_variation : ∀K : Set UpperPlane, IsCompact K → ∀t : ℝ≥0, Tendsto
+    (fun u : ℝ≥0 => ((μ u-μ t)+(μ t-μ u)) K) (𝓝 t) (𝓝 0)
+  exhaustion : ∀A : Set UpperPlane, MeasurableSet A → Tendsto
+    (fun n : ℕ => μ (n:ℝ≥0) A) atTop (𝓝 (μ ∞ A))
+  remaining_decay : ∀K : Set UpperPlane, IsCompact K → Tendsto
+    (fun n : ℕ => (μ ∞-μ (n:ℝ≥0)) K) atTop (𝓝 0)
+  firstVisit_measurable : Measurable (firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b))
+  firstVisit_finite : ∀ᵐ z ∂μ ∞, firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b) z<∞
+  firstVisit_identity : ∀t : ℝ≥0, μ t=(μ ∞).restrict
+    {z | firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b) z≤(t:ℝ≥0∞)}
+  strict_firstVisit_identity : ∀t : ℝ≥0, μ t=(μ ∞).restrict
+    {z | firstVisit (fun z : UpperPlane => z.1) (internalTrace κ b) z<(t:ℝ≥0∞)}
+  initial_positive : ∀t : ℝ≥0∞, 0<t → 0<μ t univ
+  initial_positive_compact : ∀t : ℝ≥0∞, 0<t → ∃K : Set UpperPlane, IsCompact K ∧
+    0<μ t K ∧ μ t K<∞
+
+lemma traceMassFacts_of_restrictions (κ : ℝ) (hκ : 0<κ) (b : TDriver)
+    (μ : ℝ≥0∞→Measure UpperPlane) [∀v,IsFiniteMeasureOnCompacts (μ v)]
+    (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
+    (hatom : NullSingletonClass (μ ∞))
+    (hres : ∀t : ℝ≥0, μ t=(μ ∞).restrict
+      (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t))
+    (hexh : ∀A : Set UpperPlane, MeasurableSet A → Tendsto
+      (fun n : ℕ => μ (n:ℝ≥0) A) atTop (𝓝 (μ ∞ A)))
+    (hrem : ∀K : Set UpperPlane, IsCompact K → Tendsto
+      (fun n : ℕ => (μ ∞-μ (n:ℝ≥0)) K) atTop (𝓝 0))
+    (hpos : ∀t : ℝ≥0∞, 0<t → 0<μ t univ)
+    (hposK : ∀t : ℝ≥0∞, 0<t → ∃K : Set UpperPlane, IsCompact K ∧ 0<μ t K ∧ μ t K<∞) :
+    TraceMassFacts κ b μ := by
+  let := hatom
+  have hγ := continuous_internalTrace hκ b hcont
+  refine ⟨hcont,hatom,hres,?_,?_,?_,?_,hexh,hrem,?_,?_,?_,?_,hpos,hposK⟩
+  · intro u v huv
+    induction v using ENNReal.recTopCoe with
+    | top =>
+      induction u using ENNReal.recTopCoe with
+      | top => exact le_rfl
+      | coe u => rw [hres u]; exact Measure.restrict_le_self
+    | coe v =>
+      have hu : u≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top huv
+      rw [←ENNReal.coe_toNNReal hu,hres,hres]
+      exact Measure.restrict_mono (visited_mono _ _ (by
+        apply ENNReal.coe_le_coe.mp
+        simpa only [ENNReal.coe_toNNReal hu] using huv)) le_rfl
+  · intro s t hst
+    rw [hres,hres]
+    exact increment_carried_by_segment (μ ∞) _ measurable_subtype_coe _ hγ hst
+  · intro f
+    simp_rw [hres]
+    exact continuous_integral_visited (μ ∞) _ measurable_subtype_coe Subtype.val_injective
+      _ hγ f f.integrable
+  · intro K hK t
+    simp_rw [hres]
+    exact local_variation_visited (μ ∞) hκ b hcont K hK.measurableSet hK.measure_lt_top.ne t
+  · exact measurable_firstVisit _ measurable_subtype_coe _ hγ
+  · apply firstVisit_finite_of_exhaustion (μ ∞) _ measurable_subtype_coe _ hγ
+    intro A hA
+    simpa only [←hres] using hexh A hA
+  · intro t
+    rw [hres]
+    congr 1
+    ext z
+    exact (firstVisit_le_iff _ _ hγ z t).symm
+  · intro t
+    rw [hres,restrict_strictFirstVisit (μ ∞) _ Subtype.val_injective _ hγ]
+
+ 
+
+lemma TraceMassFacts.capacity_increment_support {κ : ℝ} (hκ : 0<κ) {b : TDriver}
+    {μ : ℝ≥0∞→Measure UpperPlane} (hμ : TraceMassFacts κ b μ)
+    (s t : ℝ) (hs : 0 ≤ s) (hst : s ≤ t) :
+    (μ ((κ*t).toNNReal : ℝ≥0∞)-μ ((κ*s).toNNReal : ℝ≥0∞))
+      {z : UpperPlane | z.1∉Target.trace κ b '' Icc s t}=0 := by
+  have hst' : (κ*s).toNNReal≤(κ*t).toNNReal :=
+    Real.toNNReal_mono (mul_le_mul_of_nonneg_left hst hκ.le)
+  have h := hμ.increment_support _ _ hst'
+  have hs0 : 0≤κ*s := mul_nonneg hκ.le hs
+  have ht0 : 0≤κ*t := mul_nonneg hκ.le (hs.trans hst)
+  simpa only [internalTrace_image hκ,Real.coe_toNNReal _ hs0,Real.coe_toNNReal _ ht0,
+    mul_div_cancel_left₀ _ hκ.ne'] using h
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceMassContinuity
+open GermPositivity
+
+ 
+
+def massTimeList (n : ℕ) : ℝ≥0∞ :=
+  if n%4=0 then ∞ else if n%4=1 then ((n/4 : ℕ) : ℝ≥0)
+    else if n%4=2 then smallTime (n/4)
+      else TopologicalSpace.denseSeq ℝ≥0 (n/4)
+
+@[simp] lemma massTimeList_top : massTimeList 0=∞ := by simp [massTimeList]
+
+@[simp] lemma massTimeList_nat (n : ℕ) : massTimeList (4*n+1)=(n : ℝ≥0) := by
+  simp [massTimeList,Nat.mul_add_div]
+
+@[simp] lemma massTimeList_small (n : ℕ) : massTimeList (4*n+2)=(smallTime n : ℝ≥0∞) := by
+  simp [massTimeList,Nat.mul_add_div]
+
+@[simp] lemma massTimeList_dense (n : ℕ) :
+    massTimeList (4*n+3)=(TopologicalSpace.denseSeq ℝ≥0 n : ℝ≥0∞) := by
+  simp [massTimeList,Nat.mul_add_div]
+
+lemma massTimeList_is_dense : Dense {u : ℝ≥0 | ∃i,massTimeList i=u} := by
+  apply Dense.mono (s₁ := range (TopologicalSpace.denseSeq ℝ≥0)) _
+    (TopologicalSpace.denseRange_denseSeq ℝ≥0)
+  rintro u ⟨n,rfl⟩
+  exact ⟨4*n+3,massTimeList_dense n⟩
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceMassContinuity
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions TraceMassIntegration
+
+ 
+
+theorem actual_common_trace_mass (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
+      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      (∀v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀b, μ 0 b=0) ∧ (∀v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      μ ∞ ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ) ∧
+      (∀ (S : ExactSLEGenerator.Driver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
+        Measurable[hS.measurableSpace] (fun b => μ (S b) b) ∧
+        (∀f : C_c(UpperPlane,ℝ), ∀m : ℕ,
+          MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
+        (∀K : Set UpperPlane, IsCompact K → ∀m : ℕ,
+          MemLp (fun b => (μ (S b) b K).toReal) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
+        (∀f : C_c(UpperPlane,ℝ), Tendsto (fun k => eLpNorm
+          (fun b => (∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+            (Real.exp (-2*(Target.a κ)*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b))
+              1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
+        (∀H : ExactSLEGenerator.Driver × UpperPlane→ℝ≥0∞,
+          Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
+          (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂(W.map driverHomeomorph))=
+            (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂(W.map driverHomeomorph))+
+            ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen (Target.a κ) (S b) b z.1)
+              ∂upperArea∂(W.map driverHomeomorph))) ∧
+      ∀ᵐ b ∂W, TraceMassFacts κ b (fun v => μ v (driverHomeomorph b)) ∧
+        (∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+          (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+            (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
+          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) := by
+  have hBW := brownian_map_driver W hW
+  have : IsProbabilityMeasure (W.map driverHomeomorph) := hBW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hpath⟩ :=
+    actual_raw_mass_trace κ hκ hκ8 W hW massTimeList massTimeList_is_dense
+      (fun n => 4*n+1) massTimeList_nat (fun n => 4*n+2) massTimeList_small 0 massTimeList_top
+  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  have hcv : ∀ᵐ b ∂(W.map driverHomeomorph), ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ),
+      Tendsto (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b) atTop (𝓝 (∫z,f z∂μ v b)) := by
+    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+    filter_upwards [hpath] with b hb
+    exact hb.2.2.2.1
+  refine ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,?_,?_,?_⟩
+  · simpa only [massTimeList_top] using hmean 0 massTimeList_top
+  · exact recoveredMass_all_stopping (Target.a κ) (a_gt_quarter hκ hκ8)
+      (W.map driverHomeomorph) hBW s c hc hs hsum hcv
+  · filter_upwards [hpath] with b hb
+    rcases hb with ⟨hcont,hatom,hres,hconv,hexh,hrem,hpos,hposK⟩
+    let (v : ℝ≥0∞) : IsFiniteMeasureOnCompacts (μ v (driverHomeomorph b)) := (hreg _ _).1
+    exact ⟨traceMassFacts_of_restrictions κ hκ b (fun v => μ v (driverHomeomorph b))
+      hcont hatom hres hexh hrem hpos hposK,hconv⟩
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGenerator
+
+ 
+def concatenate (s : ℝ≥0) (b c : GDriver) : GDriver :=
+  ⟨⟨fun u => b.1 (min u s)+c.1 (u-s),
+    (b.1.continuous.comp (continuous_id.min continuous_const)).add
+      (c.1.continuous.comp (continuous_id.sub continuous_const))⟩,
+    by
+      change b.1 (min 0 s)+c.1 (0-s)=0
+      simp only [min_eq_left (show 0 ≤ s from bot_le),zero_tsub,b.2,c.2,add_zero]⟩
+
+@[simp] lemma concatenate_apply (s u : ℝ≥0) (b c : GDriver) :
+    (concatenate s b c).1 u=b.1 (min u s)+c.1 (u-s) := rfl
+
+lemma continuous_concatenate : Continuous (fun p : (ℝ≥0×GDriver)×GDriver =>
+    concatenate p.1.1 p.1.2 p.2) := by
+  apply Continuous.subtype_mk
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  exact (continuous_eval.comp
+    ((continuous_subtype_val.comp continuous_fst.fst.snd).prodMk
+      (continuous_snd.min continuous_fst.fst.fst))).add
+    (continuous_eval.comp
+      ((continuous_subtype_val.comp continuous_fst.snd).prodMk
+        (continuous_snd.sub continuous_fst.fst.fst)))
+
+lemma concatenate_before (s : ℝ≥0) (b c : GDriver) (u : ℝ≥0) (hu : u ≤ s) :
+    (concatenate s b c).1 u=b.1 u := by
+  simp only [concatenate_apply,min_eq_left hu,tsub_eq_zero_of_le hu,c.2,add_zero]
+
+lemma concatenate_reconstruct (s : ℝ≥0) (b : GDriver) :
+    concatenate s (pastDriver s b) (restart s b)=b := by
+  apply Subtype.ext
+  ext u
+  by_cases hu : u ≤ s
+  · rw [concatenate_before _ _ _ _ hu,pastDriver_eq _ _ _ hu]
+  · have hsu : s ≤ u := le_of_not_ge hu
+    simp only [concatenate_apply,min_eq_right hsu,restart_apply,pastDriver_eq s b s le_rfl,
+      add_tsub_cancel_of_le hsu]
+    ring
+
+ 
+lemma raw_event_congr {u : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[rawFiltration u] A) {b c : GDriver}
+    (h : ∀v ≤ u,b.1 v=c.1 v) : b∈A ↔ c∈A := by
+  rw [rawFiltration,Filtration.natural_eq_comap,MeasurableSpace.measurableSet_comap] at hA
+  obtain ⟨B,hB,rfl⟩ := hA
+  have he : (fun v : Iic u => b.1 v)=(fun v : Iic u => c.1 v) :=
+    funext (fun v => h v v.2)
+  change (fun v : Iic u => b.1 v)∈B ↔ (fun v : Iic u => c.1 v)∈B
+  rw [he]
+
+ 
+lemma raw_stop_congr {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {b c : GDriver} {s : ℝ≥0} (hb : S b=s) (h : ∀u ≤ s,b.1 u=c.1 u) : S c=s := by
+  have hc : S c ≤ s := (raw_event_congr (hS s) h).mp (by
+    change S b ≤ (s:ℝ≥0∞)
+    rw [hb])
+  have hcf : S c≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hc
+  have hct : (S c).toNNReal ≤ s := ENNReal.coe_le_coe.mp (by
+    simpa only [ENNReal.coe_toNNReal hcf] using hc)
+  have hbc : S b ≤ S c := by
+    have hh := (raw_event_congr (hS (S c).toNNReal)
+      (fun u hu => h u (hu.trans hct))).mpr
+      (show S c ≤ ((S c).toNNReal:ℝ≥0∞) by rw [ENNReal.coe_toNNReal hcf])
+    change S b ≤ ((S c).toNNReal:ℝ≥0∞) at hh
+    simpa only [ENNReal.coe_toNNReal hcf] using hh
+  exact le_antisymm hc (hb ▸ hbc)
+
+ 
+def shiftStop (S : GDriver→ℝ≥0∞) (t : ℝ≥0) : GDriver→ℝ≥0∞ := fun b => S b+t
+
+lemma shiftStop_isStoppingTime {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    (t : ℝ≥0) : IsStoppingTime rawFiltration (shiftStop S t) := by
+  intro u
+  change MeasurableSet[rawFiltration u] {b | shiftStop S t b ≤ (u:ℝ≥0∞)}
+  by_cases htu : t ≤ u
+  · have he : {b : GDriver | shiftStop S t b ≤ u}={b | S b ≤ ((u-t:ℝ≥0):ℝ≥0∞)} := by
+      ext b
+      simp only [shiftStop,mem_ofPred_eq,ENNReal.coe_sub]
+      exact (ENNReal.le_sub_iff_add_le_right ENNReal.coe_ne_top
+        (ENNReal.coe_le_coe.mpr htu)).symm
+    rw [he]
+    exact (rawFiltration.mono tsub_le_self) _ (hS (u-t))
+  · have he : {b : GDriver | shiftStop S t b ≤ u}=∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      intro b hb
+      exact htu (ENNReal.coe_le_coe.mp ((le_add_left le_rfl).trans hb))
+    rw [he]
+    exact @MeasurableSet.empty GDriver (rawFiltration u)
+
+lemma measurableSpace_le_shiftStop {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) (t : ℝ≥0) :
+    hS.measurableSpace ≤ (shiftStop_isStoppingTime hS t).measurableSpace :=
+  hS.measurableSpace_mono (shiftStop_isStoppingTime hS t) (fun b => show S b ≤ S b+(t:ℝ≥0∞) from le_self_add)
+
+ 
+lemma raw_stopped_event_congr {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A)
+    {b c : GDriver} {u : ℝ≥0} (hb : S b ≤ u) (h : ∀v ≤ u,b.1 v=c.1 v) : b∈A ↔ c∈A := by
+  have hc : S c ≤ u := (raw_event_congr (hS u) h).mp hb
+  have hh := raw_event_congr (hA.2 u) h
+  change (b∈A ∧ S b ≤ (u:ℝ≥0∞)) ↔ (c∈A ∧ S c ≤ (u:ℝ≥0∞)) at hh
+  simpa only [hb,hc,and_true] using hh
+
+ 
+
+def concatenatedEvent (S : GDriver→ℝ≥0∞) (A : Set GDriver) :
+    Set ((ℝ≥0×GDriver)×GDriver) :=
+  {p | S (concatenate p.1.1 p.1.2 p.2) ≤ (p.1.1:ℝ≥0∞) ∧
+    concatenate p.1.1 p.1.2 p.2∈A}
+
+lemma concatenatedEvent_measurable {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) {A : Set GDriver} (hA : MeasurableSet A) :
+    MeasurableSet (concatenatedEvent S A) := by
+  have : PolishSpace GDriver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
+  have hc : Measurable (fun p : (ℝ≥0×GDriver)×GDriver => concatenate p.1.1 p.1.2 p.2) :=
+    continuous_concatenate.measurable
+  have ht : Measurable (fun p : (ℝ≥0×GDriver)×GDriver => (p.1.1:ℝ≥0∞)) :=
+    ENNReal.continuous_coe.measurable.comp measurable_fst.fst
+  exact (measurableSet_le (hS.measurable'.comp hc) ht).inter (hc hA)
+
+lemma concatenatedEvent_past {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) {t : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
+    (s : ℝ≥0) (b c : GDriver) :
+    ((s,b),c)∈concatenatedEvent S A ↔ ((s,b),pastDriver t c)∈concatenatedEvent S A := by
+  have hsmall : ∀u ≤ s,(concatenate s b c).1 u=(concatenate s b (pastDriver t c)).1 u := by
+    intro u hu
+    rw [concatenate_before _ _ _ _ hu,concatenate_before _ _ _ _ hu]
+  have hlong : ∀u ≤ s+t,(concatenate s b c).1 u=(concatenate s b (pastDriver t c)).1 u := by
+    intro u hu
+    simp only [concatenate_apply]
+    rw [pastDriver_eq t c (u-s) (tsub_le_iff_right.mpr (by simpa only [add_comm] using hu))]
+  have hg := raw_event_congr (hS s) hsmall
+  change (S (concatenate s b c) ≤ (s:ℝ≥0∞) ∧ concatenate s b c∈A) ↔
+    (S (concatenate s b (pastDriver t c)) ≤ (s:ℝ≥0∞) ∧
+      concatenate s b (pastDriver t c)∈A)
+  change (S (concatenate s b c) ≤ (s:ℝ≥0∞)) ↔
+    (S (concatenate s b (pastDriver t c)) ≤ (s:ℝ≥0∞)) at hg
+  rw [←hg]
+  apply and_congr_right
+  intro h
+  exact raw_stopped_event_congr (shiftStop_isStoppingTime hS t) hA
+    (by
+      change S (concatenate s b c)+(t:ℝ≥0∞) ≤ ((s+t:ℝ≥0):ℝ≥0∞)
+      rw [ENNReal.coe_add]
+      exact add_le_add h le_rfl) hlong
+
+lemma concatenatedEvent_raw_joint {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) {t : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A) :
+    @MeasurableSet ((ℝ≥0×GDriver)×GDriver)
+      ((inferInstance : MeasurableSpace (ℝ≥0×GDriver)).prod (rawFiltration t))
+      (concatenatedEvent S A) := by
+  have hM : @Measurable ((ℝ≥0×GDriver)×GDriver) ((ℝ≥0×GDriver)×GDriver)
+      ((inferInstance : MeasurableSpace (ℝ≥0×GDriver)).prod (rawFiltration t)) inferInstance
+      (fun p => (p.1,pastDriver t p.2)) :=
+    measurable_fst.prodMk ((pastDriver_raw_measurable t).comp measurable_snd)
+  have hh := hM (concatenatedEvent_measurable hS
+    ((shiftStop_isStoppingTime hS t).measurableSpace_le A hA))
+  convert hh using 1
+  ext p
+  exact concatenatedEvent_past hS hA p.1.1 p.1.2 p.2
+
+lemma concatenatedEvent_stoppedData {S : GDriver→ℝ≥0∞} {A : Set GDriver}
+    {b : GDriver} (hb : S b≠∞) :
+    (stoppedData S b,restartAt S b)∈concatenatedEvent S A ↔ b∈A := by
+  simp only [concatenatedEvent,stoppedData,restartAt,mem_ofPred_eq,concatenate_reconstruct,
+    ENNReal.coe_toNNReal hb,le_refl,true_and]
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ConformalFlow.lean b/lean/OAI/Probability/SLEGauge/ConformalFlow.lean
index fc880db4f3f0cd1446b5cdc47d5148bcfa33e7f5..d84550e27f05c96a2cfb8c1d130068c30fae1b8c 100644
--- a/lean/OAI/Probability/SLEGauge/ConformalFlow.lean
+++ b/lean/OAI/Probability/SLEGauge/ConformalFlow.lean
@@ -1,450 +1,450 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.LoewnerSolutions
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-namespace ExactSLEMain
-open Set Metric
-theorem hasDerivAt_loewnerMap {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) :
-    HasDerivAt (loewnerMap A b T)
-      (Complex.exp (∫ u in (0 : ℝ)..(T : ℝ),
-        -(A : ℂ) / (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2)) z := by
-  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hz
-  have hzball : z ∈ ball z δ := mem_ball_self hδ
-  let q : ℂ → ℝ → ℂ := fun w u => -(A : ℂ) /
-    (heightClamp ε (F u.toNNReal w - (b u.toNNReal : ℂ)) *
-     heightClamp ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
-  have hqc (w : ℂ) : Continuous (q w) := continuous_const.div
-    (((heightClamp_lipschitz ε).continuous.comp
-      (((hFc w).comp continuous_real_toNNReal).sub
-        (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))).mul
-     ((heightClamp_lipschitz ε).continuous.comp
-      (((hFc z).comp continuous_real_toNNReal).sub
-        (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))))
-    (fun u => mul_ne_zero (heightClamp_ne_zero hε _) (heightClamp_ne_zero hε _))
-  have hqw (u : ℝ) : Continuous (fun w => q w u) := continuous_const.div
-    ((((heightClamp_lipschitz ε).continuous.comp
-      ((cutoff_loewner_continuous_start hε hb hFc hFe u.toNNReal).sub continuous_const))).mul
-      continuous_const)
-    (fun w => mul_ne_zero (heightClamp_ne_zero hε _) (heightClamp_ne_zero hε _))
-  have hIc : Continuous (fun w => ∫ u in (0 : ℝ)..(T : ℝ), q w u) :=
-    intervalIntegral.continuous_of_dominated_interval
-      (fun w => (hqc w).aestronglyMeasurable)
-      (fun w => ae_of_all volume (fun u hu => invprod_heightClamp_bound A ε hε _ _))
-      (show IntervalIntegrable (fun _ : ℝ => |A|/ε^2) volume 0 T from intervalIntegrable_const)
-      (ae_of_all volume (fun u hu => hqw u))
-  have hqe (w : ℂ) (hw : w ∈ ball z δ) (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      q w u = -(A : ℂ) /
-        ((F u.toNNReal w - (b u.toNNReal : ℂ)) * (F u.toNNReal z - (b u.toNNReal : ℂ))) := by
-    have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    have hc₁ : heightClamp ε (F u.toNNReal w - (b u.toNNReal : ℂ)) =
-        F u.toNNReal w - (b u.toNNReal : ℂ) := heightClamp_eq (by simpa using (hIm w hw _ huT).le)
-    have hc₂ : heightClamp ε (F u.toNNReal z - (b u.toNNReal : ℂ)) =
-        F u.toNNReal z - (b u.toNNReal : ℂ) := heightClamp_eq (by simpa using (hIm z hzball _ huT).le)
-    simp only [q,hc₁,hc₂]
-  have hIe (w : ℂ) (hw : w ∈ ball z δ) :
-      (∫ u in (0 : ℝ)..(T : ℝ), q w u) =
-      ∫ u in (0 : ℝ)..(T : ℝ), -(A : ℂ) /
-        ((F u.toNNReal w - (b u.toNNReal : ℂ)) * (F u.toNNReal z - (b u.toNNReal : ℂ))) := by
-    apply intervalIntegral.integral_congr
-    intro u hu
-    exact hqe w hw u (by simpa only [uIcc_of_le T.coe_nonneg] using hu)
-  have hIz : (∫ u in (0 : ℝ)..(T : ℝ), q z u) =
-      ∫ u in (0 : ℝ)..(T : ℝ),
-        -(A : ℂ) / (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2 := by
-    rw [hIe z hzball]
-    apply intervalIntegral.integral_congr
-    intro u hu
-    have hu' : u ∈ Icc (0 : ℝ) T := by simpa only [uIcc_of_le T.coe_nonneg] using hu
-    have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu'.2).trans_eq (by simp)
-    dsimp only
-    rw [hmap z hzball _ huT,pow_two]
-  rw [← hIz,hasDerivAt_iff_tendsto_slope]
-  have ht := (Complex.continuous_exp.comp hIc).continuousAt.tendsto.mono_left
-    (nhdsWithin_le_nhds : 𝓝[≠] z ≤ 𝓝 z)
-  apply ht.congr'
-  filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds
-      (show ∀ᶠ w in 𝓝 z, w ∈ ball z δ from Metric.ball_mem_nhds z hδ),
-    self_mem_nhdsWithin] with w hw hwn
-  have hwz : w ≠ z := by simpa only [mem_compl_iff,mem_singleton_iff] using hwn
-  have hd := loewner_solution_difference hb (hode w hw) (hode z hzball)
-  simp only [Real.toNNReal_coe] at hd
-  rw [← hIe w hw] at hd
-  rw [slope,vsub_eq_sub,smul_eq_mul,hmap w hw T le_rfl,hmap z hzball T le_rfl,hd]
-  rw [← mul_assoc,inv_mul_cancel₀ (sub_ne_zero.mpr hwz),one_mul]
-  rfl
-
-lemma deriv_loewnerMap_ne_zero {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) : deriv (loewnerMap A b T) z ≠ 0 := by
-  rw [(hasDerivAt_loewnerMap hb hz).deriv]
-  exact Complex.exp_ne_zero _
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set
-
-lemma loewnerMap_eq_solution_time {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {T : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z T f) {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) :
-    loewnerMap A b u.toNNReal z = f u := by
-  have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-  simpa only [Real.coe_toNNReal _ hu.1] using loewnerMap_eq (odeUntil_mono hf huT)
-
-lemma odeUntil_center_ne_zero {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {T : ℝ≥0}
-    {f : ℝ → ℂ} (hf : odeUntil A b z T f) {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) :
-    f u - (b u.toNNReal : ℂ) ≠ 0 := by
-  intro he
-  have hi := congrArg Complex.im he
-  have hp := (hf.2 u hu).1
-  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-  linarith
-
-lemma odeUntil_center_continuousOn {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil A b z T f) :
-    ContinuousOn (fun u : ℝ => f u - (b u.toNNReal : ℂ)) (Icc (0 : ℝ) T) :=
-  (odeUntil_continuousOn hf).sub
-    (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)).continuousOn
-
-lemma loewner_derivative_integrable {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) :
-    IntervalIntegrable (fun u : ℝ => -(A : ℂ) /
-      (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2) volume 0 T := by
-  obtain ⟨f,hf⟩ := hz
-  have hc : ContinuousOn (fun u : ℝ => -(A : ℂ) / (f u - (b u.toNNReal : ℂ))^2)
-      (Icc (0 : ℝ) T) := continuousOn_const.div
-    ((odeUntil_center_continuousOn hb hf).pow 2)
-    (fun u hu => pow_ne_zero 2 (odeUntil_center_ne_zero hf hu))
-  apply (hc.intervalIntegrable_of_Icc T.coe_nonneg).congr
-  intro u hu
-  dsimp only
-  rw [loewnerMap_eq_solution_time hf (by
-    simpa only [uIcc_of_le T.coe_nonneg] using (uIoc_subset_uIcc hu))]
-
- 
-
-theorem log_norm_deriv_loewnerMap {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) :
-    Real.log ‖deriv (loewnerMap A b T) z‖ =
-      ∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ) /
-        (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2).re := by
-  rw [(hasDerivAt_loewnerMap hb hz).deriv,Complex.norm_exp,Real.log_exp]
-  exact (Complex.reCLM.intervalIntegral_comp_comm (loewner_derivative_integrable hb hz)).symm
-
-lemma loewner_radius_drift (A : ℝ) (z : ℂ) (hz : z ≠ 0) :
-    -A / Complex.normSq z - (-(A : ℂ)/z^2).re =
-      -2*A*(z.im/Complex.normSq z)^2 := by
-  have hn : Complex.normSq z ≠ 0 := (Complex.normSq_pos.mpr hz).ne'
-  simp only [Complex.div_re,Complex.neg_re,Complex.ofReal_re,Complex.neg_im,
-    Complex.ofReal_im,neg_zero,zero_mul,zero_div,add_zero,map_pow]
-  field_simp
-  simp only [pow_two,Complex.mul_re,Complex.normSq_apply]
-  ring
-
-lemma real_scalar_linear_solution_on (H c : ℝ → ℝ) (T : ℝ≥0)
-    (hc : ContinuousOn c (Icc (0 : ℝ) T))
-    (hH : ∀ u ∈ Icc (0 : ℝ) T, HasDerivWithinAt H (c u * H u) (Icc (0 : ℝ) T) u) :
-    H T = H 0 * Real.exp (∫ u in (0 : ℝ)..(T : ℝ), c u) := by
-  have hc' : ContinuousOn (fun u => (c u : ℂ)) (Icc (0 : ℝ) T) :=
-    Complex.continuous_ofReal.comp_continuousOn hc
-  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt (fun u => (H u : ℂ)) ((c u : ℂ)*(H u : ℂ)) (Icc (0 : ℝ) T) u := by
-    simpa only [Complex.ofReal_mul] using (hH u hu).ofReal_comp
-  have he := scalar_linear_solution_on (fun u => (H u : ℂ)) (fun u => (c u : ℂ)) T hc' hd
-  have hi := Complex.ofRealCLM.intervalIntegral_comp_comm (μ := volume) (hc.intervalIntegrable_of_Icc T.coe_nonneg)
-  change (∫ u in (0 : ℝ)..(T : ℝ), (c u : ℂ)) = (∫ u in (0 : ℝ)..(T : ℝ), c u) at hi
-  rw [hi,← Complex.ofReal_exp,← Complex.ofReal_mul] at he
-  exact Complex.ofReal_injective he
-
- 
-theorem loewner_height_formula {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil A b z T f) :
-    (f T).im = z.im * Real.exp (∫ u in (0 : ℝ)..(T : ℝ),
-      -A / Complex.normSq (f u - (b u.toNNReal : ℂ))) := by
-  have hc : ContinuousOn (fun u : ℝ => -A / Complex.normSq (f u - (b u.toNNReal : ℂ)))
-      (Icc (0 : ℝ) T) := continuousOn_const.div
-    (Complex.continuous_normSq.comp_continuousOn (odeUntil_center_continuousOn hb hf))
-    (fun u hu => (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf hu)).ne')
-  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt (fun u => (f u).im)
-        ((-A / Complex.normSq (f u - (b u.toNNReal : ℂ))) * (f u).im) (Icc (0 : ℝ) T) u := by
-    have h := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u (hf.2 u hu).2
-    convert h using 1 <;> try rfl
-    simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
-      zero_mul,zero_div,zero_sub,Complex.sub_im,sub_zero]
-    ring
-  simpa only [hf.1] using real_scalar_linear_solution_on (fun u => (f u).im) _ T hc hd
-
- 
-
-theorem loewner_radius_ratio {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hz : alive A b T z) :
-    (loewnerMap A b T z).im / ‖deriv (loewnerMap A b T) z‖ =
-      z.im * Real.exp (-2*A * (∫ u in (0 : ℝ)..(T : ℝ),
-        ((loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ)).im /
-          Complex.normSq (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ)))^2)) := by
-  obtain ⟨f,hf⟩ := hz
-  have hz : alive A b T z := ⟨f,hf⟩
-  let Z : ℝ → ℂ := fun u => f u - (b u.toNNReal : ℂ)
-  have hZc : ContinuousOn Z (Icc (0 : ℝ) T) := odeUntil_center_continuousOn hb hf
-  have hZn (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : Z u ≠ 0 := odeUntil_center_ne_zero hf hu
-  have hJc : ContinuousOn (fun u => (-(A : ℂ)/(Z u)^2).re) (Icc (0 : ℝ) T) :=
-    Complex.continuous_re.comp_continuousOn
-      (continuousOn_const.div (hZc.pow 2) (fun u hu => pow_ne_zero 2 (hZn u hu)))
-  have hYc : ContinuousOn (fun u => -A/Complex.normSq (Z u)) (Icc (0 : ℝ) T) :=
-    continuousOn_const.div (Complex.continuous_normSq.comp_continuousOn hZc)
-      (fun u hu => (Complex.normSq_pos.mpr (hZn u hu)).ne')
-  have hJe : (∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/
-      (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2).re) =
-      ∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/(Z u)^2).re := by
-    apply intervalIntegral.integral_congr
-    intro u hu
-    dsimp only [Z]
-    rw [loewnerMap_eq_solution_time hf (by simpa only [uIcc_of_le T.coe_nonneg] using hu)]
-  have hi : (∫ u in (0 : ℝ)..(T : ℝ), -A/Complex.normSq (Z u)) -
-      (∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/(Z u)^2).re) =
-      -2*A*(∫ u in (0 : ℝ)..(T : ℝ), ((Z u).im/Complex.normSq (Z u))^2) := by
-    rw [← intervalIntegral.integral_sub (hYc.intervalIntegrable_of_Icc T.coe_nonneg)
-      (hJc.intervalIntegrable_of_Icc T.coe_nonneg),← intervalIntegral.integral_const_mul]
-    apply intervalIntegral.integral_congr
-    intro u hu
-    exact loewner_radius_drift A (Z u) (hZn u (by simpa only [uIcc_of_le T.coe_nonneg] using hu))
-  have hd : ‖deriv (loewnerMap A b T) z‖ =
-      Real.exp (∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/(Z u)^2).re) := by
-    rw [← hJe, ← log_norm_deriv_loewnerMap hb hz,Real.exp_log (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb hz))]
-  rw [loewnerMap_eq hf,loewner_height_formula hb hf,hd,mul_div_assoc,← Real.exp_sub]
-  change z.im * Real.exp ((∫ u in (0 : ℝ)..(T : ℝ), -A/Complex.normSq (Z u)) - _) = _
-  rw [hi]
-  congr 3
-  apply intervalIntegral.integral_congr
-  intro u hu
-  dsimp only [Z]
-  rw [loewnerMap_eq_solution_time hf (by simpa only [uIcc_of_le T.coe_nonneg] using hu)]
-
-end ExactSLEMain
-
-namespace ExactSLEGreen
-abbrev State := ℂ × ℝ
-noncomputable def X : State →L[ℝ] ℝ := Complex.reCLM.comp (ContinuousLinearMap.fst ℝ ℂ ℝ)
-noncomputable def Y : State →L[ℝ] ℝ := Complex.imCLM.comp (ContinuousLinearMap.fst ℝ ℂ ℝ)
-noncomputable def J : State →L[ℝ] ℝ := ContinuousLinearMap.snd ℝ ℂ ℝ
-noncomputable def Q (v : State) : ℝ := (X v)^2 + (Y v)^2
-noncomputable def p (a : ℝ) : ℝ := 4*a-1
-noncomputable def alpha (a : ℝ) : ℝ := 1-1/(4*a)
-noncomputable def logGreen (a : ℝ) (v : State) : ℝ :=
-  alpha a * J v + (p a-alpha a)*Real.log (Y v) - (p a/2)*Real.log (Q v)
-noncomputable def physicalGreen (a : ℝ) (v : State) : ℝ := Real.exp (logGreen a v)
-noncomputable def logGreenDeriv (a : ℝ) (v : State) : State →L[ℝ] ℝ :=
-  alpha a • J + ((p a-alpha a)/(Y v)) • Y - ((p a/2)/(Q v)) •
-    ((2*X v) • X + (2*Y v) • Y)
-noncomputable def physicalDrift (a : ℝ) (v : State) : State :=
-  ((a : ℂ)/v.1, (-(a : ℂ)/v.1^2).re)
-def noise : State := (-1,0)
-
-@[simp] lemma X_apply (v : State) : X v = v.1.re := rfl
-@[simp] lemma Y_apply (v : State) : Y v = v.1.im := rfl
-@[simp] lemma J_apply (v : State) : J v = v.2 := rfl
-lemma Q_eq (v : State) : Q v = Complex.normSq v.1 := by
-  simp only [Q,X_apply,Y_apply,Complex.normSq_apply,pow_two]
-lemma Q_pos {v : State} (hv : 0 < Y v) : 0 < Q v := by
-  change 0 < X v ^ 2 + Y v ^ 2
-  nlinarith [sq_nonneg (X v),sq_pos_of_pos hv]
-
-lemma hasFDerivAt_Q (v : State) :
-    HasFDerivAt Q ((2*X v) • X + (2*Y v) • Y) v := by
-  have hX : HasFDerivAt (fun v => X v) X v := X.hasFDerivAt
-  have hY : HasFDerivAt (fun v => Y v) Y v := Y.hasFDerivAt
-  convert (hX.pow 2).add (hY.pow 2) using 1 <;> try rfl
-  simp only [Nat.reduceSub,pow_one,nsmul_eq_mul,Nat.cast_ofNat]
-
-lemma hasFDerivAt_logGreen (a : ℝ) (v : State) (hv : 0 < Y v) :
-    HasFDerivAt (logGreen a) (logGreenDeriv a v) v := by
-  have hY := Y.hasFDerivAt.log hv.ne'
-  have hQ := (hasFDerivAt_Q v).log (Q_pos hv).ne'
-  convert ((J.hasFDerivAt.const_mul (alpha a)).add (hY.const_mul (p a-alpha a))).sub
-    (hQ.const_mul (p a/2)) using 1 <;> try rfl
-  simp only [logGreenDeriv,smul_smul,div_eq_mul_inv]
-
-lemma hasFDerivAt_physicalGreen (a : ℝ) (v : State) (hv : 0 < Y v) :
-    HasFDerivAt (physicalGreen a) (physicalGreen a v • logGreenDeriv a v) v :=
-  (hasFDerivAt_logGreen a v hv).exp
-
-lemma contDiffAt_physicalGreen (a : ℝ) (v : State) (hv : 0 < Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalGreen a) v := by
-  have hQc : ContDiff ℝ (↑(⊤ : ℕ∞)) Q := (X.contDiff.pow 2).add (Y.contDiff.pow 2)
-  exact ((((contDiffAt_const (c := alpha a)).mul (J.contDiff.contDiffAt)).add
-    ((contDiffAt_const (c := p a-alpha a)).mul (Y.contDiff.contDiffAt.log hv.ne'))).sub
-    ((contDiffAt_const (c := p a/2)).mul (hQc.contDiffAt.log (Q_pos hv).ne'))).exp
-
-lemma logGreenDeriv_noise (a : ℝ) (v : State) :
-    logGreenDeriv a v noise = p a * X v / Q v := by
-  simp only [logGreenDeriv,add_apply,sub_apply,
-    smul_apply,smul_eq_mul,noise,X_apply,Y_apply,J_apply,
-    Complex.neg_re,Complex.one_re,Complex.neg_im,Complex.one_im,neg_zero,mul_zero,
-    mul_neg,mul_one,add_zero,zero_sub]
-  ring
-
-lemma logGreenDeriv_drift (a : ℝ) (v : State) (hv : 0 < Y v) :
-    logGreenDeriv a v (physicalDrift a v) =
-      -a*(alpha a+p a)*(X v^2-Y v^2)/(Q v)^2 - a*(p a-alpha a)/(Q v) := by
-  have hy : Y v ≠ 0 := hv.ne'
-  have hq : Q v ≠ 0 := (Q_pos hv).ne'
-  simp only [logGreenDeriv,add_apply,sub_apply,
-    smul_apply,smul_eq_mul,physicalDrift,X_apply,Y_apply,J_apply,
-    Complex.div_re,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
-    Complex.neg_re,Complex.neg_im,neg_zero,zero_mul,zero_div,add_zero,zero_sub,map_pow]
-  simp only [← Q_eq,pow_two,Complex.mul_re]
-  change alpha a * (-a * (X v*X v-Y v*Y v)/(Q v*Q v)) +
-    ((p a-alpha a)/Y v)*(- (a*Y v/Q v)) - ((p a/2)/Q v)*
-      (2*X v*(a*X v/Q v) + 2*Y v*(- (a*Y v/Q v))) = _
-  field_simp
-  simp only [X_apply,Y_apply]
-  ring
-
-noncomputable def noiseLog (a : ℝ) (v : State) : ℝ := p a * X v / Q v
-noncomputable def noiseLogDeriv (a : ℝ) (v : State) : State →L[ℝ] ℝ :=
-  (p a / Q v) • X - (p a * X v / Q v^2) • ((2*X v) • X + (2*Y v) • Y)
-
-lemma hasFDerivAt_noiseLog (a : ℝ) (v : State) (hv : 0 < Y v) :
-    HasFDerivAt (noiseLog a) (noiseLogDeriv a v) v := by
-  have hQinv := (hasDerivAt_inv (Q_pos hv).ne').comp_hasFDerivAt v (hasFDerivAt_Q v)
-  convert (X.hasFDerivAt.const_mul (p a)).mul hQinv using 1 <;> try rfl
-  apply ContinuousLinearMap.ext
-  intro w
-  simp only [noiseLogDeriv,smul_apply,sub_apply,
-    add_apply, smul_eq_mul,smul_smul,Function.comp_apply]
-  ring
-
-lemma noiseLogDeriv_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
-    noiseLogDeriv a v noise = p a * (X v^2-Y v^2)/(Q v)^2 := by
-  have hq := (Q_pos hv).ne'
-  simp only [noiseLogDeriv,sub_apply,smul_apply,add_apply,
-    smul_eq_mul,noise,X_apply,Y_apply,Complex.neg_re,Complex.one_re,
-    Complex.neg_im,Complex.one_im,neg_zero,mul_zero,mul_neg,mul_one,add_zero]
-  field_simp
-  unfold Q
-  simp only [X_apply,Y_apply]
-  ring
-
-lemma green_fderiv_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
-    fderiv ℝ (physicalGreen a) v noise = physicalGreen a v * noiseLog a v := by
-  rw [(hasFDerivAt_physicalGreen a v hv).fderiv]
-  simp only [smul_apply,smul_eq_mul,logGreenDeriv_noise,noiseLog]
-
-lemma hasFDerivAt_green_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
-    HasFDerivAt (fun w => fderiv ℝ (physicalGreen a) w noise)
-      ((noiseLog a v) • (physicalGreen a v • logGreenDeriv a v) +
-        (physicalGreen a v) • noiseLogDeriv a v) v := by
-  have h := (hasFDerivAt_physicalGreen a v hv).mul (hasFDerivAt_noiseLog a v hv)
-  rw [add_comm]
-  apply h.congr_of_eventuallyEq
-  filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hv)] with w hw
-  exact green_fderiv_noise a w hw
-
-lemma green_second_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
-    fderiv ℝ (fun w => fderiv ℝ (physicalGreen a) w noise) v noise =
-      physicalGreen a v * ((p a)^2 * (X v)^2 + p a * (X v^2-Y v^2)) / (Q v)^2 := by
-  rw [(hasFDerivAt_green_noise a v hv).fderiv]
-  simp only [add_apply,smul_apply,smul_eq_mul,
-    logGreenDeriv_noise, noiseLogDeriv_noise a v hv,noiseLog]
-  ring
-
- 
-lemma green_generator_zero (a : ℝ) (ha : a ≠ 0) (v : State) (hv : 0 < Y v) :
-    (1/2 : ℝ) * fderiv ℝ (fun w => fderiv ℝ (physicalGreen a) w noise) v noise +
-      fderiv ℝ (physicalGreen a) v (physicalDrift a v) = 0 := by
-  rw [green_second_noise a v hv,(hasFDerivAt_physicalGreen a v hv).fderiv]
-  simp only [smul_apply,smul_eq_mul,logGreenDeriv_drift a v hv]
-  have hq := (Q_pos hv).ne'
-  unfold alpha p
-  field_simp
-  unfold Q
-  ring
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open Metric
-
-def spaceBump (N : ℝ≥0) : ContDiffBump (0 : State) where
-  rIn := N+1
-  rOut := N+2
-  rIn_pos := by positivity
-  rIn_lt_rOut := by linarith
-
-noncomputable def heightTurn (ε : ℝ) (v : State) : ℝ :=
-  Real.smoothTransition (4 * Y v / ε - 1)
-
-noncomputable def greenCutoff (a ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
-  spaceBump N v * heightTurn ε v * physicalGreen a v
-
-lemma heightTurn_smooth (ε : ℝ) : ContDiff ℝ (↑(⊤ : ℕ∞)) (heightTurn ε) := by
-  exact Real.smoothTransition.contDiff.comp
-    (((contDiff_const.mul Y.contDiff).div_const ε).sub contDiff_const)
-
-lemma heightTurn_zero {ε : ℝ} (hε : 0 < ε) {v : State} (hv : Y v ≤ ε/4) :
-    heightTurn ε v = 0 := by
-  apply Real.smoothTransition.zero_of_nonpos
-  rw [sub_nonpos,div_le_iff₀ hε]
-  linarith
-
-lemma heightTurn_one {ε : ℝ} (hε : 0 < ε) {v : State} (hv : ε/2 ≤ Y v) :
-    heightTurn ε v = 1 := by
-  apply Real.smoothTransition.one_of_one_le
-  rw [le_sub_iff_add_le,le_div_iff₀ hε]
-  linarith
-
-lemma greenCutoff_smooth (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (greenCutoff a ε N) := by
-  rw [contDiff_iff_contDiffAt]
-  intro v
-  by_cases hv : 0 < Y v
-  · exact (((spaceBump N).contDiffAt.mul (heightTurn_smooth ε).contDiffAt).mul
-      (contDiffAt_physicalGreen a v hv))
-  · apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
-    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
-    simp only [greenCutoff,heightTurn_zero hε (le_of_lt hw),mul_zero,zero_mul]
-
-lemma greenCutoff_compact (a ε : ℝ) (N : ℝ≥0) : HasCompactSupport (greenCutoff a ε N) :=
-  ((spaceBump N).hasCompactSupport.mul_right).mul_right
-
-lemma greenCutoff_eq_near (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0)
-    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
-    greenCutoff a ε N =ᶠ[𝓝 v] physicalGreen a := by
-  have hv' : v ∈ ball (0 : State) (spaceBump N).rIn := by
-    simp only [mem_ball,dist_zero_right,spaceBump]
-    linarith
-  have hy' : ε/2 < Y v := lt_of_lt_of_le (by linarith) hy
-  filter_upwards [(spaceBump N).eventuallyEq_one_of_mem_ball hv',
-    Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy')] with w hw hyw
-  simp only [greenCutoff,hw,heightTurn_one hε (le_of_lt hyw),Pi.one_apply,one_mul]
-
-lemma greenCutoff_generator_zero (a ε : ℝ) (ha : a ≠ 0) (hε : 0 < ε) (N : ℝ≥0)
-    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
-    (1/2 : ℝ) * fderiv ℝ (fun w => fderiv ℝ (greenCutoff a ε N) w noise) v noise +
-      fderiv ℝ (greenCutoff a ε N) v (physicalDrift a v) = 0 := by
-  have he := greenCutoff_eq_near a ε hε N v hv hy
-  have hd : (fun w => fderiv ℝ (greenCutoff a ε N) w noise) =ᶠ[𝓝 v]
-      (fun w => fderiv ℝ (physicalGreen a) w noise) :=
-    he.fderiv.mono fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw
-  rw [hd.fderiv_eq,he.fderiv_eq]
-  exact green_generator_zero a ha v (hε.trans_le hy)
-
-end ExactSLEGreen
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.LoewnerSolutions
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+namespace ExactSLEMain
+open Set Metric
+theorem hasDerivAt_loewnerMap {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) :
+    HasDerivAt (loewnerMap A b T)
+      (Complex.exp (∫ u in (0 : ℝ)..(T : ℝ),
+        -(A : ℂ) / (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2)) z := by
+  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hz
+  have hzball : z ∈ ball z δ := mem_ball_self hδ
+  let q : ℂ → ℝ → ℂ := fun w u => -(A : ℂ) /
+    (heightClamp ε (F u.toNNReal w - (b u.toNNReal : ℂ)) *
+     heightClamp ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
+  have hqc (w : ℂ) : Continuous (q w) := continuous_const.div
+    (((heightClamp_lipschitz ε).continuous.comp
+      (((hFc w).comp continuous_real_toNNReal).sub
+        (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))).mul
+     ((heightClamp_lipschitz ε).continuous.comp
+      (((hFc z).comp continuous_real_toNNReal).sub
+        (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))))
+    (fun u => mul_ne_zero (heightClamp_ne_zero hε _) (heightClamp_ne_zero hε _))
+  have hqw (u : ℝ) : Continuous (fun w => q w u) := continuous_const.div
+    ((((heightClamp_lipschitz ε).continuous.comp
+      ((cutoff_loewner_continuous_start hε hb hFc hFe u.toNNReal).sub continuous_const))).mul
+      continuous_const)
+    (fun w => mul_ne_zero (heightClamp_ne_zero hε _) (heightClamp_ne_zero hε _))
+  have hIc : Continuous (fun w => ∫ u in (0 : ℝ)..(T : ℝ), q w u) :=
+    intervalIntegral.continuous_of_dominated_interval
+      (fun w => (hqc w).aestronglyMeasurable)
+      (fun w => ae_of_all volume (fun u hu => invprod_heightClamp_bound A ε hε _ _))
+      (show IntervalIntegrable (fun _ : ℝ => |A|/ε^2) volume 0 T from intervalIntegrable_const)
+      (ae_of_all volume (fun u hu => hqw u))
+  have hqe (w : ℂ) (hw : w ∈ ball z δ) (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      q w u = -(A : ℂ) /
+        ((F u.toNNReal w - (b u.toNNReal : ℂ)) * (F u.toNNReal z - (b u.toNNReal : ℂ))) := by
+    have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    have hc₁ : heightClamp ε (F u.toNNReal w - (b u.toNNReal : ℂ)) =
+        F u.toNNReal w - (b u.toNNReal : ℂ) := heightClamp_eq (by simpa using (hIm w hw _ huT).le)
+    have hc₂ : heightClamp ε (F u.toNNReal z - (b u.toNNReal : ℂ)) =
+        F u.toNNReal z - (b u.toNNReal : ℂ) := heightClamp_eq (by simpa using (hIm z hzball _ huT).le)
+    simp only [q,hc₁,hc₂]
+  have hIe (w : ℂ) (hw : w ∈ ball z δ) :
+      (∫ u in (0 : ℝ)..(T : ℝ), q w u) =
+      ∫ u in (0 : ℝ)..(T : ℝ), -(A : ℂ) /
+        ((F u.toNNReal w - (b u.toNNReal : ℂ)) * (F u.toNNReal z - (b u.toNNReal : ℂ))) := by
+    apply intervalIntegral.integral_congr
+    intro u hu
+    exact hqe w hw u (by simpa only [uIcc_of_le T.coe_nonneg] using hu)
+  have hIz : (∫ u in (0 : ℝ)..(T : ℝ), q z u) =
+      ∫ u in (0 : ℝ)..(T : ℝ),
+        -(A : ℂ) / (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2 := by
+    rw [hIe z hzball]
+    apply intervalIntegral.integral_congr
+    intro u hu
+    have hu' : u ∈ Icc (0 : ℝ) T := by simpa only [uIcc_of_le T.coe_nonneg] using hu
+    have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu'.2).trans_eq (by simp)
+    dsimp only
+    rw [hmap z hzball _ huT,pow_two]
+  rw [← hIz,hasDerivAt_iff_tendsto_slope]
+  have ht := (Complex.continuous_exp.comp hIc).continuousAt.tendsto.mono_left
+    (nhdsWithin_le_nhds : 𝓝[≠] z ≤ 𝓝 z)
+  apply ht.congr'
+  filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds
+      (show ∀ᶠ w in 𝓝 z, w ∈ ball z δ from Metric.ball_mem_nhds z hδ),
+    self_mem_nhdsWithin] with w hw hwn
+  have hwz : w ≠ z := by simpa only [mem_compl_iff,mem_singleton_iff] using hwn
+  have hd := loewner_solution_difference hb (hode w hw) (hode z hzball)
+  simp only [Real.toNNReal_coe] at hd
+  rw [← hIe w hw] at hd
+  rw [slope,vsub_eq_sub,smul_eq_mul,hmap w hw T le_rfl,hmap z hzball T le_rfl,hd]
+  rw [← mul_assoc,inv_mul_cancel₀ (sub_ne_zero.mpr hwz),one_mul]
+  rfl
+
+lemma deriv_loewnerMap_ne_zero {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) : deriv (loewnerMap A b T) z ≠ 0 := by
+  rw [(hasDerivAt_loewnerMap hb hz).deriv]
+  exact Complex.exp_ne_zero _
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set
+
+lemma loewnerMap_eq_solution_time {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {T : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z T f) {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) :
+    loewnerMap A b u.toNNReal z = f u := by
+  have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+  simpa only [Real.coe_toNNReal _ hu.1] using loewnerMap_eq (odeUntil_mono hf huT)
+
+lemma odeUntil_center_ne_zero {A : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {T : ℝ≥0}
+    {f : ℝ → ℂ} (hf : odeUntil A b z T f) {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) :
+    f u - (b u.toNNReal : ℂ) ≠ 0 := by
+  intro he
+  have hi := congrArg Complex.im he
+  have hp := (hf.2 u hu).1
+  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+  linarith
+
+lemma odeUntil_center_continuousOn {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil A b z T f) :
+    ContinuousOn (fun u : ℝ => f u - (b u.toNNReal : ℂ)) (Icc (0 : ℝ) T) :=
+  (odeUntil_continuousOn hf).sub
+    (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)).continuousOn
+
+lemma loewner_derivative_integrable {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) :
+    IntervalIntegrable (fun u : ℝ => -(A : ℂ) /
+      (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2) volume 0 T := by
+  obtain ⟨f,hf⟩ := hz
+  have hc : ContinuousOn (fun u : ℝ => -(A : ℂ) / (f u - (b u.toNNReal : ℂ))^2)
+      (Icc (0 : ℝ) T) := continuousOn_const.div
+    ((odeUntil_center_continuousOn hb hf).pow 2)
+    (fun u hu => pow_ne_zero 2 (odeUntil_center_ne_zero hf hu))
+  apply (hc.intervalIntegrable_of_Icc T.coe_nonneg).congr
+  intro u hu
+  dsimp only
+  rw [loewnerMap_eq_solution_time hf (by
+    simpa only [uIcc_of_le T.coe_nonneg] using (uIoc_subset_uIcc hu))]
+
+ 
+
+theorem log_norm_deriv_loewnerMap {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) :
+    Real.log ‖deriv (loewnerMap A b T) z‖ =
+      ∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ) /
+        (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2).re := by
+  rw [(hasDerivAt_loewnerMap hb hz).deriv,Complex.norm_exp,Real.log_exp]
+  exact (Complex.reCLM.intervalIntegral_comp_comm (loewner_derivative_integrable hb hz)).symm
+
+lemma loewner_radius_drift (A : ℝ) (z : ℂ) (hz : z ≠ 0) :
+    -A / Complex.normSq z - (-(A : ℂ)/z^2).re =
+      -2*A*(z.im/Complex.normSq z)^2 := by
+  have hn : Complex.normSq z ≠ 0 := (Complex.normSq_pos.mpr hz).ne'
+  simp only [Complex.div_re,Complex.neg_re,Complex.ofReal_re,Complex.neg_im,
+    Complex.ofReal_im,neg_zero,zero_mul,zero_div,add_zero,map_pow]
+  field_simp
+  simp only [pow_two,Complex.mul_re,Complex.normSq_apply]
+  ring
+
+lemma real_scalar_linear_solution_on (H c : ℝ → ℝ) (T : ℝ≥0)
+    (hc : ContinuousOn c (Icc (0 : ℝ) T))
+    (hH : ∀ u ∈ Icc (0 : ℝ) T, HasDerivWithinAt H (c u * H u) (Icc (0 : ℝ) T) u) :
+    H T = H 0 * Real.exp (∫ u in (0 : ℝ)..(T : ℝ), c u) := by
+  have hc' : ContinuousOn (fun u => (c u : ℂ)) (Icc (0 : ℝ) T) :=
+    Complex.continuous_ofReal.comp_continuousOn hc
+  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt (fun u => (H u : ℂ)) ((c u : ℂ)*(H u : ℂ)) (Icc (0 : ℝ) T) u := by
+    simpa only [Complex.ofReal_mul] using (hH u hu).ofReal_comp
+  have he := scalar_linear_solution_on (fun u => (H u : ℂ)) (fun u => (c u : ℂ)) T hc' hd
+  have hi := Complex.ofRealCLM.intervalIntegral_comp_comm (μ := volume) (hc.intervalIntegrable_of_Icc T.coe_nonneg)
+  change (∫ u in (0 : ℝ)..(T : ℝ), (c u : ℂ)) = (∫ u in (0 : ℝ)..(T : ℝ), c u) at hi
+  rw [hi,← Complex.ofReal_exp,← Complex.ofReal_mul] at he
+  exact Complex.ofReal_injective he
+
+ 
+theorem loewner_height_formula {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil A b z T f) :
+    (f T).im = z.im * Real.exp (∫ u in (0 : ℝ)..(T : ℝ),
+      -A / Complex.normSq (f u - (b u.toNNReal : ℂ))) := by
+  have hc : ContinuousOn (fun u : ℝ => -A / Complex.normSq (f u - (b u.toNNReal : ℂ)))
+      (Icc (0 : ℝ) T) := continuousOn_const.div
+    (Complex.continuous_normSq.comp_continuousOn (odeUntil_center_continuousOn hb hf))
+    (fun u hu => (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf hu)).ne')
+  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt (fun u => (f u).im)
+        ((-A / Complex.normSq (f u - (b u.toNNReal : ℂ))) * (f u).im) (Icc (0 : ℝ) T) u := by
+    have h := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u (hf.2 u hu).2
+    convert h using 1 <;> try rfl
+    simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
+      zero_mul,zero_div,zero_sub,Complex.sub_im,sub_zero]
+    ring
+  simpa only [hf.1] using real_scalar_linear_solution_on (fun u => (f u).im) _ T hc hd
+
+ 
+
+theorem loewner_radius_ratio {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hz : alive A b T z) :
+    (loewnerMap A b T z).im / ‖deriv (loewnerMap A b T) z‖ =
+      z.im * Real.exp (-2*A * (∫ u in (0 : ℝ)..(T : ℝ),
+        ((loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ)).im /
+          Complex.normSq (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ)))^2)) := by
+  obtain ⟨f,hf⟩ := hz
+  have hz : alive A b T z := ⟨f,hf⟩
+  let Z : ℝ → ℂ := fun u => f u - (b u.toNNReal : ℂ)
+  have hZc : ContinuousOn Z (Icc (0 : ℝ) T) := odeUntil_center_continuousOn hb hf
+  have hZn (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : Z u ≠ 0 := odeUntil_center_ne_zero hf hu
+  have hJc : ContinuousOn (fun u => (-(A : ℂ)/(Z u)^2).re) (Icc (0 : ℝ) T) :=
+    Complex.continuous_re.comp_continuousOn
+      (continuousOn_const.div (hZc.pow 2) (fun u hu => pow_ne_zero 2 (hZn u hu)))
+  have hYc : ContinuousOn (fun u => -A/Complex.normSq (Z u)) (Icc (0 : ℝ) T) :=
+    continuousOn_const.div (Complex.continuous_normSq.comp_continuousOn hZc)
+      (fun u hu => (Complex.normSq_pos.mpr (hZn u hu)).ne')
+  have hJe : (∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/
+      (loewnerMap A b u.toNNReal z - (b u.toNNReal : ℂ))^2).re) =
+      ∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/(Z u)^2).re := by
+    apply intervalIntegral.integral_congr
+    intro u hu
+    dsimp only [Z]
+    rw [loewnerMap_eq_solution_time hf (by simpa only [uIcc_of_le T.coe_nonneg] using hu)]
+  have hi : (∫ u in (0 : ℝ)..(T : ℝ), -A/Complex.normSq (Z u)) -
+      (∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/(Z u)^2).re) =
+      -2*A*(∫ u in (0 : ℝ)..(T : ℝ), ((Z u).im/Complex.normSq (Z u))^2) := by
+    rw [← intervalIntegral.integral_sub (hYc.intervalIntegrable_of_Icc T.coe_nonneg)
+      (hJc.intervalIntegrable_of_Icc T.coe_nonneg),← intervalIntegral.integral_const_mul]
+    apply intervalIntegral.integral_congr
+    intro u hu
+    exact loewner_radius_drift A (Z u) (hZn u (by simpa only [uIcc_of_le T.coe_nonneg] using hu))
+  have hd : ‖deriv (loewnerMap A b T) z‖ =
+      Real.exp (∫ u in (0 : ℝ)..(T : ℝ), (-(A : ℂ)/(Z u)^2).re) := by
+    rw [← hJe, ← log_norm_deriv_loewnerMap hb hz,Real.exp_log (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb hz))]
+  rw [loewnerMap_eq hf,loewner_height_formula hb hf,hd,mul_div_assoc,← Real.exp_sub]
+  change z.im * Real.exp ((∫ u in (0 : ℝ)..(T : ℝ), -A/Complex.normSq (Z u)) - _) = _
+  rw [hi]
+  congr 3
+  apply intervalIntegral.integral_congr
+  intro u hu
+  dsimp only [Z]
+  rw [loewnerMap_eq_solution_time hf (by simpa only [uIcc_of_le T.coe_nonneg] using hu)]
+
+end ExactSLEMain
+
+namespace ExactSLEGreen
+abbrev State := ℂ × ℝ
+noncomputable def X : State →L[ℝ] ℝ := Complex.reCLM.comp (ContinuousLinearMap.fst ℝ ℂ ℝ)
+noncomputable def Y : State →L[ℝ] ℝ := Complex.imCLM.comp (ContinuousLinearMap.fst ℝ ℂ ℝ)
+noncomputable def J : State →L[ℝ] ℝ := ContinuousLinearMap.snd ℝ ℂ ℝ
+noncomputable def Q (v : State) : ℝ := (X v)^2 + (Y v)^2
+noncomputable def p (a : ℝ) : ℝ := 4*a-1
+noncomputable def alpha (a : ℝ) : ℝ := 1-1/(4*a)
+noncomputable def logGreen (a : ℝ) (v : State) : ℝ :=
+  alpha a * J v + (p a-alpha a)*Real.log (Y v) - (p a/2)*Real.log (Q v)
+noncomputable def physicalGreen (a : ℝ) (v : State) : ℝ := Real.exp (logGreen a v)
+noncomputable def logGreenDeriv (a : ℝ) (v : State) : State →L[ℝ] ℝ :=
+  alpha a • J + ((p a-alpha a)/(Y v)) • Y - ((p a/2)/(Q v)) •
+    ((2*X v) • X + (2*Y v) • Y)
+noncomputable def physicalDrift (a : ℝ) (v : State) : State :=
+  ((a : ℂ)/v.1, (-(a : ℂ)/v.1^2).re)
+def noise : State := (-1,0)
+
+@[simp] lemma X_apply (v : State) : X v = v.1.re := rfl
+@[simp] lemma Y_apply (v : State) : Y v = v.1.im := rfl
+@[simp] lemma J_apply (v : State) : J v = v.2 := rfl
+lemma Q_eq (v : State) : Q v = Complex.normSq v.1 := by
+  simp only [Q,X_apply,Y_apply,Complex.normSq_apply,pow_two]
+lemma Q_pos {v : State} (hv : 0 < Y v) : 0 < Q v := by
+  change 0 < X v ^ 2 + Y v ^ 2
+  nlinarith [sq_nonneg (X v),sq_pos_of_pos hv]
+
+lemma hasFDerivAt_Q (v : State) :
+    HasFDerivAt Q ((2*X v) • X + (2*Y v) • Y) v := by
+  have hX : HasFDerivAt (fun v => X v) X v := X.hasFDerivAt
+  have hY : HasFDerivAt (fun v => Y v) Y v := Y.hasFDerivAt
+  convert (hX.pow 2).add (hY.pow 2) using 1 <;> try rfl
+  simp only [Nat.reduceSub,pow_one,nsmul_eq_mul,Nat.cast_ofNat]
+
+lemma hasFDerivAt_logGreen (a : ℝ) (v : State) (hv : 0 < Y v) :
+    HasFDerivAt (logGreen a) (logGreenDeriv a v) v := by
+  have hY := Y.hasFDerivAt.log hv.ne'
+  have hQ := (hasFDerivAt_Q v).log (Q_pos hv).ne'
+  convert ((J.hasFDerivAt.const_mul (alpha a)).add (hY.const_mul (p a-alpha a))).sub
+    (hQ.const_mul (p a/2)) using 1 <;> try rfl
+  simp only [logGreenDeriv,smul_smul,div_eq_mul_inv]
+
+lemma hasFDerivAt_physicalGreen (a : ℝ) (v : State) (hv : 0 < Y v) :
+    HasFDerivAt (physicalGreen a) (physicalGreen a v • logGreenDeriv a v) v :=
+  (hasFDerivAt_logGreen a v hv).exp
+
+lemma contDiffAt_physicalGreen (a : ℝ) (v : State) (hv : 0 < Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalGreen a) v := by
+  have hQc : ContDiff ℝ (↑(⊤ : ℕ∞)) Q := (X.contDiff.pow 2).add (Y.contDiff.pow 2)
+  exact ((((contDiffAt_const (c := alpha a)).mul (J.contDiff.contDiffAt)).add
+    ((contDiffAt_const (c := p a-alpha a)).mul (Y.contDiff.contDiffAt.log hv.ne'))).sub
+    ((contDiffAt_const (c := p a/2)).mul (hQc.contDiffAt.log (Q_pos hv).ne'))).exp
+
+lemma logGreenDeriv_noise (a : ℝ) (v : State) :
+    logGreenDeriv a v noise = p a * X v / Q v := by
+  simp only [logGreenDeriv,add_apply,sub_apply,
+    smul_apply,smul_eq_mul,noise,X_apply,Y_apply,J_apply,
+    Complex.neg_re,Complex.one_re,Complex.neg_im,Complex.one_im,neg_zero,mul_zero,
+    mul_neg,mul_one,add_zero,zero_sub]
+  ring
+
+lemma logGreenDeriv_drift (a : ℝ) (v : State) (hv : 0 < Y v) :
+    logGreenDeriv a v (physicalDrift a v) =
+      -a*(alpha a+p a)*(X v^2-Y v^2)/(Q v)^2 - a*(p a-alpha a)/(Q v) := by
+  have hy : Y v ≠ 0 := hv.ne'
+  have hq : Q v ≠ 0 := (Q_pos hv).ne'
+  simp only [logGreenDeriv,add_apply,sub_apply,
+    smul_apply,smul_eq_mul,physicalDrift,X_apply,Y_apply,J_apply,
+    Complex.div_re,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
+    Complex.neg_re,Complex.neg_im,neg_zero,zero_mul,zero_div,add_zero,zero_sub,map_pow]
+  simp only [← Q_eq,pow_two,Complex.mul_re]
+  change alpha a * (-a * (X v*X v-Y v*Y v)/(Q v*Q v)) +
+    ((p a-alpha a)/Y v)*(- (a*Y v/Q v)) - ((p a/2)/Q v)*
+      (2*X v*(a*X v/Q v) + 2*Y v*(- (a*Y v/Q v))) = _
+  field_simp
+  simp only [X_apply,Y_apply]
+  ring
+
+noncomputable def noiseLog (a : ℝ) (v : State) : ℝ := p a * X v / Q v
+noncomputable def noiseLogDeriv (a : ℝ) (v : State) : State →L[ℝ] ℝ :=
+  (p a / Q v) • X - (p a * X v / Q v^2) • ((2*X v) • X + (2*Y v) • Y)
+
+lemma hasFDerivAt_noiseLog (a : ℝ) (v : State) (hv : 0 < Y v) :
+    HasFDerivAt (noiseLog a) (noiseLogDeriv a v) v := by
+  have hQinv := (hasDerivAt_inv (Q_pos hv).ne').comp_hasFDerivAt v (hasFDerivAt_Q v)
+  convert (X.hasFDerivAt.const_mul (p a)).mul hQinv using 1 <;> try rfl
+  apply ContinuousLinearMap.ext
+  intro w
+  simp only [noiseLogDeriv,smul_apply,sub_apply,
+    add_apply, smul_eq_mul,smul_smul,Function.comp_apply]
+  ring
+
+lemma noiseLogDeriv_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
+    noiseLogDeriv a v noise = p a * (X v^2-Y v^2)/(Q v)^2 := by
+  have hq := (Q_pos hv).ne'
+  simp only [noiseLogDeriv,sub_apply,smul_apply,add_apply,
+    smul_eq_mul,noise,X_apply,Y_apply,Complex.neg_re,Complex.one_re,
+    Complex.neg_im,Complex.one_im,neg_zero,mul_zero,mul_neg,mul_one,add_zero]
+  field_simp
+  unfold Q
+  simp only [X_apply,Y_apply]
+  ring
+
+lemma green_fderiv_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
+    fderiv ℝ (physicalGreen a) v noise = physicalGreen a v * noiseLog a v := by
+  rw [(hasFDerivAt_physicalGreen a v hv).fderiv]
+  simp only [smul_apply,smul_eq_mul,logGreenDeriv_noise,noiseLog]
+
+lemma hasFDerivAt_green_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
+    HasFDerivAt (fun w => fderiv ℝ (physicalGreen a) w noise)
+      ((noiseLog a v) • (physicalGreen a v • logGreenDeriv a v) +
+        (physicalGreen a v) • noiseLogDeriv a v) v := by
+  have h := (hasFDerivAt_physicalGreen a v hv).mul (hasFDerivAt_noiseLog a v hv)
+  rw [add_comm]
+  apply h.congr_of_eventuallyEq
+  filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hv)] with w hw
+  exact green_fderiv_noise a w hw
+
+lemma green_second_noise (a : ℝ) (v : State) (hv : 0 < Y v) :
+    fderiv ℝ (fun w => fderiv ℝ (physicalGreen a) w noise) v noise =
+      physicalGreen a v * ((p a)^2 * (X v)^2 + p a * (X v^2-Y v^2)) / (Q v)^2 := by
+  rw [(hasFDerivAt_green_noise a v hv).fderiv]
+  simp only [add_apply,smul_apply,smul_eq_mul,
+    logGreenDeriv_noise, noiseLogDeriv_noise a v hv,noiseLog]
+  ring
+
+ 
+lemma green_generator_zero (a : ℝ) (ha : a ≠ 0) (v : State) (hv : 0 < Y v) :
+    (1/2 : ℝ) * fderiv ℝ (fun w => fderiv ℝ (physicalGreen a) w noise) v noise +
+      fderiv ℝ (physicalGreen a) v (physicalDrift a v) = 0 := by
+  rw [green_second_noise a v hv,(hasFDerivAt_physicalGreen a v hv).fderiv]
+  simp only [smul_apply,smul_eq_mul,logGreenDeriv_drift a v hv]
+  have hq := (Q_pos hv).ne'
+  unfold alpha p
+  field_simp
+  unfold Q
+  ring
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open Metric
+
+def spaceBump (N : ℝ≥0) : ContDiffBump (0 : State) where
+  rIn := N+1
+  rOut := N+2
+  rIn_pos := by positivity
+  rIn_lt_rOut := by linarith
+
+noncomputable def heightTurn (ε : ℝ) (v : State) : ℝ :=
+  Real.smoothTransition (4 * Y v / ε - 1)
+
+noncomputable def greenCutoff (a ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
+  spaceBump N v * heightTurn ε v * physicalGreen a v
+
+lemma heightTurn_smooth (ε : ℝ) : ContDiff ℝ (↑(⊤ : ℕ∞)) (heightTurn ε) := by
+  exact Real.smoothTransition.contDiff.comp
+    (((contDiff_const.mul Y.contDiff).div_const ε).sub contDiff_const)
+
+lemma heightTurn_zero {ε : ℝ} (hε : 0 < ε) {v : State} (hv : Y v ≤ ε/4) :
+    heightTurn ε v = 0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  rw [sub_nonpos,div_le_iff₀ hε]
+  linarith
+
+lemma heightTurn_one {ε : ℝ} (hε : 0 < ε) {v : State} (hv : ε/2 ≤ Y v) :
+    heightTurn ε v = 1 := by
+  apply Real.smoothTransition.one_of_one_le
+  rw [le_sub_iff_add_le,le_div_iff₀ hε]
+  linarith
+
+lemma greenCutoff_smooth (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (greenCutoff a ε N) := by
+  rw [contDiff_iff_contDiffAt]
+  intro v
+  by_cases hv : 0 < Y v
+  · exact (((spaceBump N).contDiffAt.mul (heightTurn_smooth ε).contDiffAt).mul
+      (contDiffAt_physicalGreen a v hv))
+  · apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
+    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
+    simp only [greenCutoff,heightTurn_zero hε (le_of_lt hw),mul_zero,zero_mul]
+
+lemma greenCutoff_compact (a ε : ℝ) (N : ℝ≥0) : HasCompactSupport (greenCutoff a ε N) :=
+  ((spaceBump N).hasCompactSupport.mul_right).mul_right
+
+lemma greenCutoff_eq_near (a ε : ℝ) (hε : 0 < ε) (N : ℝ≥0)
+    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
+    greenCutoff a ε N =ᶠ[𝓝 v] physicalGreen a := by
+  have hv' : v ∈ ball (0 : State) (spaceBump N).rIn := by
+    simp only [mem_ball,dist_zero_right,spaceBump]
+    linarith
+  have hy' : ε/2 < Y v := lt_of_lt_of_le (by linarith) hy
+  filter_upwards [(spaceBump N).eventuallyEq_one_of_mem_ball hv',
+    Y.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy')] with w hw hyw
+  simp only [greenCutoff,hw,heightTurn_one hε (le_of_lt hyw),Pi.one_apply,one_mul]
+
+lemma greenCutoff_generator_zero (a ε : ℝ) (ha : a ≠ 0) (hε : 0 < ε) (N : ℝ≥0)
+    (v : State) (hv : ‖v‖ ≤ N) (hy : ε ≤ Y v) :
+    (1/2 : ℝ) * fderiv ℝ (fun w => fderiv ℝ (greenCutoff a ε N) w noise) v noise +
+      fderiv ℝ (greenCutoff a ε N) v (physicalDrift a v) = 0 := by
+  have he := greenCutoff_eq_near a ε hε N v hv hy
+  have hd : (fun w => fderiv ℝ (greenCutoff a ε N) w noise) =ᶠ[𝓝 v]
+      (fun w => fderiv ℝ (physicalGreen a) w noise) :=
+    he.fderiv.mono fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw
+  rw [hd.fderiv_eq,he.fderiv_eq]
+  exact green_generator_zero a ha v (hε.trans_le hy)
+
+end ExactSLEGreen
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ConformalRestartMap.lean b/lean/OAI/Probability/SLEGauge/ConformalRestartMap.lean
index 43ddcbc81ff3ccfbaefb4445540bae244cb748cc..659f103a039f607fa297c438485689948e05f84b 100644
--- a/lean/OAI/Probability/SLEGauge/ConformalRestartMap.lean
+++ b/lean/OAI/Probability/SLEGauge/ConformalRestartMap.lean
@@ -1,550 +1,550 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.OptionalGreen
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen
-
-lemma literal_parameter (a : ℝ) : Target.a (2/a)=a := div_div_cancel₀ (by norm_num : (2:ℝ)≠0)
-
-lemma literal_flow (a : ℝ) (b : GDriver) (z : ℂ) (t : ℝ≥0) :
-    LiteralTrace.flow (2/a) (driverHomeomorph.symm b) z t=loewnerMap a b.1 t z := by
-  rw [←flow_eq_literal,flow_eq_loewnerMap,literal_parameter,driverHomeomorph.apply_symm_apply]
-
-lemma literal_surviving (a : ℝ) (b : GDriver) (z : ℂ) (t : ℝ≥0) :
-    LiteralTrace.Surviving (2/a) (driverHomeomorph.symm b) z t ↔ alive a b.1 t z := by
-  rw [←surviving_eq,surviving_iff_alive,literal_parameter,driverHomeomorph.apply_symm_apply]
-
- 
-def inverseMap (a : ℝ) (b : GDriver) (s : ℝ≥0) (w : ℂ) : ℂ :=
-  LiteralTrace.inverseFlow (2/a) (driverHomeomorph.symm b) s (w+(b.1 s:ℂ))
-
-lemma inverseMap_alive {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) : alive a b.1 s (inverseMap a b s w) := by
-  apply (literal_surviving a b _ s).mp
-  apply LiteralTrace.inverseFlow_surviving (by positivity) _ s.coe_nonneg
-  simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using hw
-
-lemma alive_initial_im {a : ℝ} {b : ℝ≥0→ℝ} {s : ℝ≥0} {z : ℂ}
-    (hz : alive a b s z) : 0<z.im := by
-  obtain ⟨f,hf⟩ := hz
-  simpa only [hf.1] using (hf.2 0 ⟨le_rfl,s.coe_nonneg⟩).1
-
-lemma loewnerMap_im_pos {a : ℝ} {b : ℝ≥0→ℝ} {s : ℝ≥0} {z : ℂ}
-    (hz : alive a b s z) : 0<(loewnerMap a b s z).im := by
-  obtain ⟨f,hf⟩ := hz
-  rw [loewnerMap_eq hf]
-  exact (hf.2 s ⟨s.coe_nonneg,le_rfl⟩).1
-
-lemma centered_inverseMap {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    loewnerMap a b.1 s (inverseMap a b s w)-(b.1 s:ℂ)=w := by
-  rw [←literal_flow,inverseMap,LiteralTrace.flow_inverseFlow (by positivity) _ s.coe_nonneg
-    (by simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using hw),add_sub_cancel_right]
-
-lemma inverseMap_centered (a : ℝ) (b : GDriver) (s : ℝ≥0) {z : ℂ}
-    (hz : alive a b.1 s z) :
-    inverseMap a b s (loewnerMap a b.1 s z-(b.1 s:ℂ))=z := by
-  rw [inverseMap,sub_add_cancel,←literal_flow]
-  exact LiteralTrace.inverseFlow_flow s.coe_nonneg ((literal_surviving a b z s).mpr hz)
-
-lemma inverseMap_hasStrictDerivAt {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    HasStrictDerivAt (inverseMap a b s)
-      (deriv (loewnerMap a b.1 s) (inverseMap a b s w))⁻¹ w := by
-  have h := (LiteralTrace.inverseFlow_hasStrictDerivAt (by positivity : 0<2/a)
-    (driverHomeomorph.symm b) s.coe_nonneg
-    (show 0<(w+(b.1 s:ℂ)).im by simpa using hw)).scomp w
-      ((hasStrictDerivAt_id w).add_const (b.1 s:ℂ))
-  simp only [literal_flow, Function.comp_def, id_eq, one_smul] at h
-  convert! h using 1
-
-lemma deriv_inverseMap {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    deriv (inverseMap a b s) w=(deriv (loewnerMap a b.1 s) (inverseMap a b s w))⁻¹ :=
-  (inverseMap_hasStrictDerivAt ha b s hw).hasDerivAt.deriv
-
-lemma deriv_inverseMap_ne_zero {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) : deriv (inverseMap a b s) w≠0 := by
-  rw [deriv_inverseMap ha b s hw]
-  exact inv_ne_zero (deriv_loewnerMap_ne_zero b.1.continuous (inverseMap_alive ha b s hw))
-
- 
-def inverseUpper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) : UpperPlane :=
-  ⟨inverseMap a b s w.1,alive_initial_im (inverseMap_alive ha b s w.2)⟩
-
-lemma continuous_inverseUpper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
-    Continuous (inverseUpper a ha b s) := by
-  apply Continuous.subtype_mk
-  apply continuous_iff_continuousAt.mpr
-  intro w
-  exact (inverseMap_hasStrictDerivAt ha b s w.2).hasDerivAt.continuousAt.comp
-    continuous_subtype_val.continuousAt
-
-lemma inverseUpper_injective (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
-    Function.Injective (inverseUpper a ha b s) := by
-  intro u v h
-  apply Subtype.ext
-  have hh := congrArg (fun z : UpperPlane => loewnerMap a b.1 s z.1-(b.1 s:ℂ)) h
-  simpa only [inverseUpper,centered_inverseMap ha b s u.2,centered_inverseMap ha b s v.2] using hh
-
- 
-def restartHomeomorph (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
-    UpperPlane ≃ₜ {z : UpperPlane // alive a b.1 s z.1} where
-  toFun w := ⟨inverseUpper a ha b s w,inverseMap_alive ha b s w.2⟩
-  invFun z := ⟨loewnerMap a b.1 s z.1.1-(b.1 s:ℂ),by
-    simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using loewnerMap_im_pos z.2⟩
-  left_inv w := by
-    apply Subtype.ext
-    exact centered_inverseMap ha b s w.2
-  right_inv z := by
-    apply Subtype.ext
-    apply Subtype.ext
-    exact inverseMap_centered a b s z.2
-  continuous_toFun := (continuous_inverseUpper a ha b s).subtype_mk _
-  continuous_invFun := by
-    apply Continuous.subtype_mk
-    apply continuous_iff_continuousAt.mpr
-    intro z
-    have hc : ContinuousAt (fun z : {z : UpperPlane // alive a b.1 s z.1} => z.1.1) z :=
-      (continuous_subtype_val.comp continuous_subtype_val).continuousAt
-    exact ((hasDerivAt_loewnerMap b.1.continuous z.2).continuousAt.comp_of_eq hc rfl).sub continuousAt_const
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen
-
-abbrev AliveData (a : ℝ) := {p : ℝ≥0×(GDriver×UpperPlane) //
-  alive a p.2.1.1 p.1 p.2.2.1}
-
-def centeredData (a : ℝ) (p : AliveData a) : ℝ≥0×(GDriver×UpperPlane) :=
-  (p.1.1,(p.1.2.1,⟨loewnerMap a p.1.2.1.1 p.1.1 p.1.2.2.1-(p.1.2.1.1 p.1.1:ℂ),by
-    simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using loewnerMap_im_pos p.2⟩))
-
-lemma measurable_centeredData (a : ℝ) (ha : 1/4<a) : Measurable (centeredData a) := by
-  have hv : Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) => (p.2.1.1 p.1:ℂ)) :=
-    (Complex.continuous_ofReal.comp (continuous_subtype_val.comp (continuous_fst.comp
-      continuous_snd) |>.eval continuous_fst)).measurable
-  exact (measurable_fst.comp measurable_subtype_coe).prodMk
-    ((measurable_fst.comp (measurable_snd.comp measurable_subtype_coe)).prodMk
-      (((loewnerMap_time_joint_measurable a ha).comp measurable_subtype_coe).sub
-        (hv.comp measurable_subtype_coe) |>.subtype_mk))
-
-lemma centeredData_injective (a : ℝ) : Function.Injective (centeredData a) := by
-  rintro ⟨⟨s,b,z⟩,hz⟩ ⟨⟨t,c,w⟩,hw⟩ h
-  have ht : s=t := congrArg Prod.fst h
-  have hc : b=c := congrArg (fun p => p.2.1) h
-  subst t; subst c
-  have hz' : loewnerMap a b.1 s z.1-(b.1 s:ℂ)=loewnerMap a b.1 s w.1-(b.1 s:ℂ) :=
-    congrArg (fun p => p.2.2.1) h
-  have hzw := congrArg (inverseMap a b s) hz'
-  rw [inverseMap_centered a b s hz,inverseMap_centered a b s hw] at hzw
-  have : z=w := Subtype.ext hzw
-  subst w
-  rfl
-
- 
-lemma inverseUpper_joint_measurable (a : ℝ) (ha : 1/4<a) :
-    Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) =>
-      inverseUpper a (lt_trans (by norm_num) ha) p.2.1 p.1 p.2.2) := by
-  have ha0 : 0<a := lt_trans (by norm_num) ha
-  have : PolishSpace GDriver :=
-    (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
-  have : PolishSpace UpperPlane := upperPlane_open.polishSpace
-  have : StandardBorelSpace (AliveData a) := (measurableSet_alive_joint a ha).standardBorel
-  let inv : (ℝ≥0×(GDriver×UpperPlane))→AliveData a := fun p =>
-    ⟨(p.1,(p.2.1,inverseUpper a ha0 p.2.1 p.1 p.2.2)),inverseMap_alive ha0 _ _ p.2.2.2⟩
-  have hi : Measurable inv := by
-    apply ((measurable_centeredData a ha).measurableEmbedding (centeredData_injective a)).measurable_comp_iff.mp
-    have he : centeredData a ∘ inv=id := by
-      funext p
-      apply Prod.ext
-      · rfl
-      apply Prod.ext
-      · rfl
-      apply Subtype.ext
-      exact centered_inverseMap ha0 _ _ p.2.2.2
-    rw [he]
-    exact measurable_id
-  exact (measurable_subtype_coe.comp hi).snd.snd
-
-lemma norm_deriv_inverseMap_eq_exp (a : ℝ) (ha : 0<a) (b : GDriver)
-    (s : ℝ≥0) (w : UpperPlane) :
-    ‖deriv (inverseMap a b s) w.1‖=
-      Real.exp (-(killedState a s b (inverseUpper a ha b s w).1).2) := by
-  simp only [inverseUpper]
-  rw [deriv_inverseMap ha b s w.2,norm_inv,killedState,
-    ite_eq_left (inverseMap_alive ha b s w.2),actualState,Real.exp_neg,Real.exp_log]
-  exact norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous (inverseMap_alive ha b s w.2))
-
- 
-lemma norm_deriv_inverseMap_joint_measurable (a : ℝ) (ha : 1/4<a) :
-    Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) =>
-      ‖deriv (inverseMap a p.2.1 p.1) p.2.2.1‖) := by
-  simp_rw [norm_deriv_inverseMap_eq_exp a (lt_trans (by norm_num) ha)]
-  exact (((killedState_time_joint_measurable a ha).comp
-    (measurable_fst.prodMk (measurable_snd.fst.prodMk
-      (inverseUpper_joint_measurable a ha)))).snd.neg).exp
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.Conformal
-
-def realDerivative (v : ℂ) : ℂ→L[ℝ]ℂ :=
-  (ContinuousLinearMap.toSpanSingleton ℂ v).restrictScalars ℝ
-
-lemma det_realDerivative (v : ℂ) : (realDerivative v).det=‖v‖^2 := by
-  simp [realDerivative,ContinuousLinearMap.det,LinearMap.det_restrictScalars,
-    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
-
-lemma lintegral_image_complex {f : ℂ→ℂ} {s : Set ℂ} (hs : MeasurableSet s)
-    (hd : ∀z∈s,DifferentiableAt ℂ f z) (hi : InjOn f s) (g : ℂ→ℝ≥0∞) :
-    (∫⁻z in f '' s,g z∂volume)=
-      ∫⁻w in s,ENNReal.ofReal (‖deriv f w‖^2)*g (f w)∂volume := by
-  have hf (z : ℂ) (hz : z∈s) : HasFDerivWithinAt f (realDerivative (deriv f z)) s z :=
-    ((hd z hz).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
-  simpa only [det_realDerivative,abs_pow, abs_norm] using
-    lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hs hf hi g
-
- 
-
-lemma map_withDensity_composed {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
-    (μ : Measure X) {e : X→Y} (he : Measurable e) {f : Y→ℝ≥0∞} (hf : Measurable f) :
-    (μ.withDensity (fun x => f (e x))).map e=(μ.map e).withDensity f := by
-  ext A hA
-  rw [Measure.map_apply he hA,withDensity_apply _ (he hA),
-    withDensity_apply _ hA,Measure.restrict_map he hA,lintegral_map hf he]
-
-end SLEExactGauge.Conformal
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen
-
- 
-
-def upperExtend (f : UpperPlane→ℝ≥0∞) (z : ℂ) : ℝ≥0∞ :=
-  if hz : 0<z.im then f ⟨z,hz⟩ else 0
-
-@[simp] lemma upperExtend_coe (f : UpperPlane→ℝ≥0∞) (z : UpperPlane) :
-    upperExtend f z.1=f z := dite_eq_left z.2
-
-lemma image_inverseMap_upper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
-    inverseMap a b s '' {w : ℂ | 0<w.im}=
-      Subtype.val '' {z : UpperPlane | alive a b.1 s z.1} := by
-  ext z
-  constructor
-  · rintro ⟨w,hw,rfl⟩
-    exact ⟨inverseUpper a ha b s ⟨w,hw⟩,inverseMap_alive ha b s hw,rfl⟩
-  · rintro ⟨z,hz,rfl⟩
-    refine ⟨loewnerMap a b.1 s z.1-(b.1 s:ℂ),?_,inverseMap_centered a b s hz⟩
-    simpa only [mem_ofPred_eq,Complex.sub_im,Complex.ofReal_im,sub_zero] using
-      loewnerMap_im_pos hz
-
-lemma inverseMap_injOn_upper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
-    InjOn (inverseMap a b s) {w : ℂ | 0<w.im} := by
-  intro u hu v hv huv
-  have h := congrArg (fun z => loewnerMap a b.1 s z-(b.1 s:ℂ)) huv
-  simpa only [centered_inverseMap ha b s hu,centered_inverseMap ha b s hv] using h
-
- 
-
-theorem inverseUpper_lintegral_area (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    (f : UpperPlane→ℝ≥0∞) :
-    (∫⁻w : UpperPlane, ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
-      f (inverseUpper a ha b s w) ∂upperArea)=
-    ∫⁻z in {z : UpperPlane | alive a b.1 s z.1},f z∂upperArea := by
-  have h := Conformal.lintegral_image_complex upperPlane_open.measurableSet
-    (fun w hw => (inverseMap_hasStrictDerivAt ha b s hw).hasDerivAt.differentiableAt)
-    (inverseMap_injOn_upper a ha b s) (upperExtend f)
-  rw [image_inverseMap_upper a ha b s] at h
-  have hleft := setLIntegral_subtype (μ:=(volume : Measure ℂ))
-    upperPlane_open.measurableSet {z : UpperPlane | alive a b.1 s z.1} (upperExtend f)
-  have hright := lintegral_subtype_comap (μ:=(volume : Measure ℂ))
-    upperPlane_open.measurableSet (fun w =>
-      ENNReal.ofReal (‖deriv (inverseMap a b s) w‖^2)*upperExtend f (inverseMap a b s w))
-  have hh := hright.trans (h.symm.trans hleft.symm)
-  have he (w : UpperPlane) : upperExtend f (inverseMap a b s w.1)=
-      f (inverseUpper a ha b s w) := dite_eq_left (alive_initial_im (inverseMap_alive ha b s w.2))
-  simp only [upperExtend_coe, he] at hh
-  convert! hh using 1
-
- 
-theorem map_inverseUpper_area (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
-    (upperArea.withDensity (fun w : UpperPlane =>
-       ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2))).map
-        (inverseUpper a (lt_trans (by norm_num) ha) b s)=
-      upperArea.restrict {z : UpperPlane | alive a b.1 s z.1} := by
-  let ha0 : 0<a := lt_trans (by norm_num) ha
-  let J := inverseUpper a ha0 b s
-  have hJ : Measurable J := (continuous_inverseUpper a ha0 b s).measurable
-  have hsurv : MeasurableSet {z : UpperPlane | alive a b.1 s z.1} :=
-    (measurableSet_alive_joint a ha).preimage
-      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
-  ext A hA
-  rw [Measure.map_apply hJ hA,withDensity_apply _ (hJ hA),
-    Measure.restrict_apply hA]
-  have h := inverseUpper_lintegral_area a ha0 b s (A.indicator (fun _ =>1))
-  have hi : (fun w : UpperPlane => ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
-      A.indicator (fun _ =>1) (J w))=
-      (J ⁻¹' A).indicator (fun w => ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)) := by
-    funext w
-    by_cases hw : J w ∈ A <;> simp [hw]
-  change (∫⁻w : UpperPlane, ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
-      A.indicator (fun _ =>1) (J w) ∂upperArea)=_ at h
-  rw [hi,lintegral_indicator (hJ hA),lintegral_indicator hA,
-    setLIntegral_one,Measure.restrict_apply hA] at h
-  exact h
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen
-
- 
-def restartWeight (a : ℝ) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) : ℝ≥0∞ :=
-  ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^(2-alpha a))
-
-lemma restartWeight_pos (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) :
-    0<restartWeight a b s w := ENNReal.ofReal_pos.mpr
-      (Real.rpow_pos_of_pos (norm_pos_iff.mpr (deriv_inverseMap_ne_zero ha b s w.2)) _)
-
-lemma restartWeight_ne_top (a : ℝ) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) :
-    restartWeight a b s w≠∞ := ENNReal.ofReal_ne_top
-
-lemma restartWeight_joint_measurable (a : ℝ) (ha : 1/4<a) :
-    Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) => restartWeight a p.2.1 p.1 p.2.2) :=
-  ((norm_deriv_inverseMap_joint_measurable a ha).pow_const _).ennreal_ofReal
-
-lemma restartWeight_measurable (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
-    Measurable (restartWeight a b s) := by
-  have hin : Measurable (fun w : UpperPlane => (s,(b,w))) :=
-    measurable_const.prodMk (measurable_const.prodMk measurable_id)
-  have hh := (restartWeight_joint_measurable a ha).comp hin
-  change Measurable (fun w : UpperPlane => restartWeight a b s w) at hh
-  exact hh
-
-lemma green_jacobian_restart (a : ℝ) (ha : 0<a) (b : GDriver) (s t : ℝ≥0)
-    (w : UpperPlane) :
-    ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
-      ENNReal.ofReal (survivingGreen a (s+t) b (inverseUpper a ha b s w).1)=
-      restartWeight a b s w*ENNReal.ofReal (survivingGreen a t (restart s b) w.1) := by
-  have hi := inverseMap_alive ha b s w.2
-  have hp : 0<‖deriv (inverseMap a b s) w.1‖ :=
-    norm_pos_iff.mpr (deriv_inverseMap_ne_zero ha b s w.2)
-  have hr : ‖deriv (loewnerMap a b.1 s) (inverseMap a b s w.1)‖ =
-      ‖deriv (inverseMap a b s) w.1‖⁻¹ := by
-    rw [deriv_inverseMap ha b s w.2,norm_inv,inv_inv]
-  change ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
-      ENNReal.ofReal (survivingGreen a (s+t) b (inverseMap a b s w.1))=_
-  rw [survivingGreen_restart a b s t hi,centered_inverseMap ha b s w.2,hr,
-    ENNReal.ofReal_mul (Real.rpow_nonneg (inv_nonneg.mpr hp.le) _),←mul_assoc,
-    ←ENNReal.ofReal_mul (sq_nonneg _)]
-  congr 1
-  rw [←Real.rpow_two,←Real.rpow_neg_eq_inv_rpow,←Real.rpow_add hp,←sub_eq_add_neg]
-  rfl
-
- 
-
-theorem inverseUpper_lintegral_green (a : ℝ) (ha : 0<a) (b : GDriver) (s t : ℝ≥0)
-    (f : UpperPlane→ℝ≥0∞) :
-    (∫⁻w : UpperPlane,restartWeight a b s w*ENNReal.ofReal
-      (survivingGreen a t (restart s b) w.1)*f (inverseUpper a ha b s w)∂upperArea)=
-    ∫⁻z in {z : UpperPlane | alive a b.1 s z.1},
-      ENNReal.ofReal (survivingGreen a (s+t) b z.1)*f z∂upperArea := by
-  have h := inverseUpper_lintegral_area a ha b s
-    (fun z => ENNReal.ofReal (survivingGreen a (s+t) b z.1)*f z)
-  simpa only [←mul_assoc,green_jacobian_restart a ha b s t] using h
-
- 
-
-def restartPush (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) (μ : Measure UpperPlane) :
-    Measure UpperPlane := (μ.withDensity (restartWeight a b s)).map (inverseUpper a ha b s)
-
-lemma restartPush_lintegral (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (μ : Measure UpperPlane) {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    (∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b s μ)=
-      ∫⁻w,restartWeight a b s w*f (inverseUpper a (lt_trans (by norm_num) ha) b s w)∂μ := by
-  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
-  have hF : Measurable (fun w : UpperPlane =>
-      f (inverseUpper a (lt_trans (by norm_num) ha) b s w)) := hf.comp hJ
-  rw [restartPush,lintegral_map hf hJ]
-  exact lintegral_withDensity_eq_lintegral_mul μ (restartWeight_measurable a ha b s) hF
-
- 
-
-theorem restartPush_survivingKernel (a : ℝ) (ha : 1/4<a) (b : GDriver) (s t : ℝ≥0) :
-    restartPush a (lt_trans (by norm_num) ha) b s (survivingKernel a ha t (restart s b))=
-      (survivingKernel a ha (s+t) b).restrict {z : UpperPlane | alive a b.1 s z.1} := by
-  let ha0 : 0<a := lt_trans (by norm_num) ha
-  let J := inverseUpper a ha0 b s
-  have hJ : Measurable J := (continuous_inverseUpper a ha0 b s).measurable
-  have hsurv : MeasurableSet {z : UpperPlane | alive a b.1 s z.1} :=
-    (measurableSet_alive_joint a ha).preimage
-      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
-  ext A hA
-  have h := inverseUpper_lintegral_green a ha0 b s t (A.indicator (fun _ =>1))
-  have heL := restartPush_lintegral a ha b s (survivingKernel a ha t (restart s b))
-    (measurable_const.indicator hA : Measurable (A.indicator (fun _ =>(1:ℝ≥0∞))))
-  rw [lintegral_indicator hA,setLIntegral_one] at heL
-  have hF : Measurable (fun w : UpperPlane => restartWeight a b s w*
-      A.indicator (fun _ =>(1:ℝ≥0∞)) (inverseUpper a ha0 b s w)) :=
-    (restartWeight_measurable a ha b s).mul ((measurable_const.indicator hA).comp hJ)
-  rw [survivingKernel_apply,lintegral_withDensity_eq_lintegral_mul upperArea
-    (survivingGreen_spatial_measurable a ha t (restart s b)).ennreal_ofReal hF] at heL
-  have heR : (∫⁻z in {z : UpperPlane | alive a b.1 s z.1},
-      ENNReal.ofReal (survivingGreen a (s+t) b z.1)*A.indicator (fun _ =>1) z∂upperArea)=
-      ((survivingKernel a ha (s+t) b).restrict {z : UpperPlane | alive a b.1 s z.1}) A := by
-    rw [survivingKernel_apply,Measure.restrict_apply hA,withDensity_apply _ (hA.inter hsurv)]
-    calc
-      _ = ∫⁻z in {z : UpperPlane | alive a b.1 s z.1},
-          A.indicator (fun z => ENNReal.ofReal (survivingGreen a (s+t) b z.1)) z∂upperArea := by
-        congr 1
-        funext z
-        by_cases hz : z∈A <;> simp [hz]
-      _ = _ := by rw [lintegral_indicator hA,Measure.restrict_restrict hA]
-  rw [←heR,←h]
-  rw [survivingKernel_apply]
-  simpa only [Pi.mul_apply,mul_left_comm,mul_assoc] using heL
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-
-lemma inverseMap_driver_congr (a : ℝ) (ha : 0<a) {b c : GDriver} {s : ℝ≥0}
-    (h : ∀ u ≤ s,b.1 u=c.1 u) {w : ℂ} (hw : 0<w.im) :
-    inverseMap a b s w=inverseMap a c s w := by
-  have hi := inverseMap_alive ha b s hw
-  have hc := (alive_driver_congr h).mp hi
-  have he : loewnerMap a c.1 s (inverseMap a b s w)-(c.1 s:ℂ)=w := by
-    rw [←loewnerMap_driver_congr h,←h s le_rfl]
-    exact centered_inverseMap ha b s hw
-  have hj := inverseMap_centered a c s hc
-  rw [he] at hj
-  exact hj.symm
-
-lemma inverseUpper_driver_congr (a : ℝ) (ha : 0<a) {b c : GDriver} {s : ℝ≥0}
-    (h : ∀ u ≤ s,b.1 u=c.1 u) : inverseUpper a ha b s=inverseUpper a ha c s := by
-  funext w
-  apply Subtype.ext
-  exact inverseMap_driver_congr a ha h w.2
-
-lemma restartWeight_driver_congr (a : ℝ) (ha : 0<a) {b c : GDriver} {s : ℝ≥0}
-    (h : ∀ u ≤ s,b.1 u=c.1 u) : restartWeight a b s=restartWeight a c s := by
-  funext w
-  simp only [restartWeight,deriv_inverseMap ha _ _ w.2,
-    inverseMap_driver_congr a ha h w.2,loewnerMap_driver_congr h]
-
- 
-
-def stoppedData (S : GDriver→ℝ≥0∞) (b : GDriver) : ℝ≥0×GDriver :=
-  ((S b).toNNReal,pastDriver (S b).toNNReal b)
-
-lemma stoppedData_raw_measurable {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (stoppedData S) :=
-  (ENNReal.measurable_toNNReal.comp hS.measurable).prodMk
-    (pastDriver_stopping_measurable S hS)
-
-lemma inverseUpper_stoppedData (a : ℝ) (ha : 0<a) (S : GDriver→ℝ≥0∞)
-    (b : GDriver) : inverseUpper a ha (stoppedData S b).2 (stoppedData S b).1=
-      inverseUpper a ha b (S b).toNNReal :=
-  inverseUpper_driver_congr a ha (pastDriver_eq _ b)
-
-lemma restartWeight_stoppedData (a : ℝ) (ha : 0<a) (S : GDriver→ℝ≥0∞)
-    (b : GDriver) : restartWeight a (stoppedData S b).2 (stoppedData S b).1=
-      restartWeight a b (S b).toNNReal :=
-  restartWeight_driver_congr a ha (pastDriver_eq _ b)
-
-lemma inverseUpper_stopped_raw_joint (a : ℝ) (ha : 1/4<a)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    @Measurable (GDriver×UpperPlane) UpperPlane
-      (hS.measurableSpace.prod inferInstance) inferInstance
-      (fun p => inverseUpper a (lt_trans (by norm_num) ha) p.1 (S p.1).toNNReal p.2) := by
-  have hD := stoppedData_raw_measurable hS
-  have hin : @Measurable (GDriver×UpperPlane) (ℝ≥0×(GDriver×UpperPlane))
-      (hS.measurableSpace.prod inferInstance) inferInstance (fun p =>
-      ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) :=
-    (hD.fst.comp measurable_fst).prodMk ((hD.snd.comp measurable_fst).prodMk measurable_snd)
-  have hh := (inverseUpper_joint_measurable a ha).comp hin
-  simpa only [Function.comp_def,inverseUpper_stoppedData] using hh
-
-lemma restartWeight_stopped_raw_joint (a : ℝ) (ha : 1/4<a)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    @Measurable (GDriver×UpperPlane) ℝ≥0∞
-      (hS.measurableSpace.prod inferInstance) inferInstance
-      (fun p => restartWeight a p.1 (S p.1).toNNReal p.2) := by
-  have hD := stoppedData_raw_measurable hS
-  have hin : @Measurable (GDriver×UpperPlane) (ℝ≥0×(GDriver×UpperPlane))
-      (hS.measurableSpace.prod inferInstance) inferInstance (fun p =>
-      ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) :=
-    (hD.fst.comp measurable_fst).prodMk ((hD.snd.comp measurable_fst).prodMk measurable_snd)
-  have hh := (restartWeight_joint_measurable a ha).comp hin
-  simpa only [Function.comp_def,restartWeight_stoppedData a (lt_trans (by norm_num) ha)] using hh
-
-end SLEExactGauge.LoewnerRestart
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.OptionalGreen
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen
+
+lemma literal_parameter (a : ℝ) : Target.a (2/a)=a := div_div_cancel₀ (by norm_num : (2:ℝ)≠0)
+
+lemma literal_flow (a : ℝ) (b : GDriver) (z : ℂ) (t : ℝ≥0) :
+    LiteralTrace.flow (2/a) (driverHomeomorph.symm b) z t=loewnerMap a b.1 t z := by
+  rw [←flow_eq_literal,flow_eq_loewnerMap,literal_parameter,driverHomeomorph.apply_symm_apply]
+
+lemma literal_surviving (a : ℝ) (b : GDriver) (z : ℂ) (t : ℝ≥0) :
+    LiteralTrace.Surviving (2/a) (driverHomeomorph.symm b) z t ↔ alive a b.1 t z := by
+  rw [←surviving_eq,surviving_iff_alive,literal_parameter,driverHomeomorph.apply_symm_apply]
+
+ 
+def inverseMap (a : ℝ) (b : GDriver) (s : ℝ≥0) (w : ℂ) : ℂ :=
+  LiteralTrace.inverseFlow (2/a) (driverHomeomorph.symm b) s (w+(b.1 s:ℂ))
+
+lemma inverseMap_alive {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) : alive a b.1 s (inverseMap a b s w) := by
+  apply (literal_surviving a b _ s).mp
+  apply LiteralTrace.inverseFlow_surviving (by positivity) _ s.coe_nonneg
+  simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using hw
+
+lemma alive_initial_im {a : ℝ} {b : ℝ≥0→ℝ} {s : ℝ≥0} {z : ℂ}
+    (hz : alive a b s z) : 0<z.im := by
+  obtain ⟨f,hf⟩ := hz
+  simpa only [hf.1] using (hf.2 0 ⟨le_rfl,s.coe_nonneg⟩).1
+
+lemma loewnerMap_im_pos {a : ℝ} {b : ℝ≥0→ℝ} {s : ℝ≥0} {z : ℂ}
+    (hz : alive a b s z) : 0<(loewnerMap a b s z).im := by
+  obtain ⟨f,hf⟩ := hz
+  rw [loewnerMap_eq hf]
+  exact (hf.2 s ⟨s.coe_nonneg,le_rfl⟩).1
+
+lemma centered_inverseMap {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    loewnerMap a b.1 s (inverseMap a b s w)-(b.1 s:ℂ)=w := by
+  rw [←literal_flow,inverseMap,LiteralTrace.flow_inverseFlow (by positivity) _ s.coe_nonneg
+    (by simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using hw),add_sub_cancel_right]
+
+lemma inverseMap_centered (a : ℝ) (b : GDriver) (s : ℝ≥0) {z : ℂ}
+    (hz : alive a b.1 s z) :
+    inverseMap a b s (loewnerMap a b.1 s z-(b.1 s:ℂ))=z := by
+  rw [inverseMap,sub_add_cancel,←literal_flow]
+  exact LiteralTrace.inverseFlow_flow s.coe_nonneg ((literal_surviving a b z s).mpr hz)
+
+lemma inverseMap_hasStrictDerivAt {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    HasStrictDerivAt (inverseMap a b s)
+      (deriv (loewnerMap a b.1 s) (inverseMap a b s w))⁻¹ w := by
+  have h := (LiteralTrace.inverseFlow_hasStrictDerivAt (by positivity : 0<2/a)
+    (driverHomeomorph.symm b) s.coe_nonneg
+    (show 0<(w+(b.1 s:ℂ)).im by simpa using hw)).scomp w
+      ((hasStrictDerivAt_id w).add_const (b.1 s:ℂ))
+  simp only [literal_flow, Function.comp_def, id_eq, one_smul] at h
+  convert! h using 1
+
+lemma deriv_inverseMap {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    deriv (inverseMap a b s) w=(deriv (loewnerMap a b.1 s) (inverseMap a b s w))⁻¹ :=
+  (inverseMap_hasStrictDerivAt ha b s hw).hasDerivAt.deriv
+
+lemma deriv_inverseMap_ne_zero {a : ℝ} (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) : deriv (inverseMap a b s) w≠0 := by
+  rw [deriv_inverseMap ha b s hw]
+  exact inv_ne_zero (deriv_loewnerMap_ne_zero b.1.continuous (inverseMap_alive ha b s hw))
+
+ 
+def inverseUpper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) : UpperPlane :=
+  ⟨inverseMap a b s w.1,alive_initial_im (inverseMap_alive ha b s w.2)⟩
+
+lemma continuous_inverseUpper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
+    Continuous (inverseUpper a ha b s) := by
+  apply Continuous.subtype_mk
+  apply continuous_iff_continuousAt.mpr
+  intro w
+  exact (inverseMap_hasStrictDerivAt ha b s w.2).hasDerivAt.continuousAt.comp
+    continuous_subtype_val.continuousAt
+
+lemma inverseUpper_injective (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
+    Function.Injective (inverseUpper a ha b s) := by
+  intro u v h
+  apply Subtype.ext
+  have hh := congrArg (fun z : UpperPlane => loewnerMap a b.1 s z.1-(b.1 s:ℂ)) h
+  simpa only [inverseUpper,centered_inverseMap ha b s u.2,centered_inverseMap ha b s v.2] using hh
+
+ 
+def restartHomeomorph (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
+    UpperPlane ≃ₜ {z : UpperPlane // alive a b.1 s z.1} where
+  toFun w := ⟨inverseUpper a ha b s w,inverseMap_alive ha b s w.2⟩
+  invFun z := ⟨loewnerMap a b.1 s z.1.1-(b.1 s:ℂ),by
+    simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using loewnerMap_im_pos z.2⟩
+  left_inv w := by
+    apply Subtype.ext
+    exact centered_inverseMap ha b s w.2
+  right_inv z := by
+    apply Subtype.ext
+    apply Subtype.ext
+    exact inverseMap_centered a b s z.2
+  continuous_toFun := (continuous_inverseUpper a ha b s).subtype_mk _
+  continuous_invFun := by
+    apply Continuous.subtype_mk
+    apply continuous_iff_continuousAt.mpr
+    intro z
+    have hc : ContinuousAt (fun z : {z : UpperPlane // alive a b.1 s z.1} => z.1.1) z :=
+      (continuous_subtype_val.comp continuous_subtype_val).continuousAt
+    exact ((hasDerivAt_loewnerMap b.1.continuous z.2).continuousAt.comp_of_eq hc rfl).sub continuousAt_const
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen
+
+abbrev AliveData (a : ℝ) := {p : ℝ≥0×(GDriver×UpperPlane) //
+  alive a p.2.1.1 p.1 p.2.2.1}
+
+def centeredData (a : ℝ) (p : AliveData a) : ℝ≥0×(GDriver×UpperPlane) :=
+  (p.1.1,(p.1.2.1,⟨loewnerMap a p.1.2.1.1 p.1.1 p.1.2.2.1-(p.1.2.1.1 p.1.1:ℂ),by
+    simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using loewnerMap_im_pos p.2⟩))
+
+lemma measurable_centeredData (a : ℝ) (ha : 1/4<a) : Measurable (centeredData a) := by
+  have hv : Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) => (p.2.1.1 p.1:ℂ)) :=
+    (Complex.continuous_ofReal.comp (continuous_subtype_val.comp (continuous_fst.comp
+      continuous_snd) |>.eval continuous_fst)).measurable
+  exact (measurable_fst.comp measurable_subtype_coe).prodMk
+    ((measurable_fst.comp (measurable_snd.comp measurable_subtype_coe)).prodMk
+      (((loewnerMap_time_joint_measurable a ha).comp measurable_subtype_coe).sub
+        (hv.comp measurable_subtype_coe) |>.subtype_mk))
+
+lemma centeredData_injective (a : ℝ) : Function.Injective (centeredData a) := by
+  rintro ⟨⟨s,b,z⟩,hz⟩ ⟨⟨t,c,w⟩,hw⟩ h
+  have ht : s=t := congrArg Prod.fst h
+  have hc : b=c := congrArg (fun p => p.2.1) h
+  subst t; subst c
+  have hz' : loewnerMap a b.1 s z.1-(b.1 s:ℂ)=loewnerMap a b.1 s w.1-(b.1 s:ℂ) :=
+    congrArg (fun p => p.2.2.1) h
+  have hzw := congrArg (inverseMap a b s) hz'
+  rw [inverseMap_centered a b s hz,inverseMap_centered a b s hw] at hzw
+  have : z=w := Subtype.ext hzw
+  subst w
+  rfl
+
+ 
+lemma inverseUpper_joint_measurable (a : ℝ) (ha : 1/4<a) :
+    Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) =>
+      inverseUpper a (lt_trans (by norm_num) ha) p.2.1 p.1 p.2.2) := by
+  have ha0 : 0<a := lt_trans (by norm_num) ha
+  have : PolishSpace GDriver :=
+    (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
+  have : PolishSpace UpperPlane := upperPlane_open.polishSpace
+  have : StandardBorelSpace (AliveData a) := (measurableSet_alive_joint a ha).standardBorel
+  let inv : (ℝ≥0×(GDriver×UpperPlane))→AliveData a := fun p =>
+    ⟨(p.1,(p.2.1,inverseUpper a ha0 p.2.1 p.1 p.2.2)),inverseMap_alive ha0 _ _ p.2.2.2⟩
+  have hi : Measurable inv := by
+    apply ((measurable_centeredData a ha).measurableEmbedding (centeredData_injective a)).measurable_comp_iff.mp
+    have he : centeredData a ∘ inv=id := by
+      funext p
+      apply Prod.ext
+      · rfl
+      apply Prod.ext
+      · rfl
+      apply Subtype.ext
+      exact centered_inverseMap ha0 _ _ p.2.2.2
+    rw [he]
+    exact measurable_id
+  exact (measurable_subtype_coe.comp hi).snd.snd
+
+lemma norm_deriv_inverseMap_eq_exp (a : ℝ) (ha : 0<a) (b : GDriver)
+    (s : ℝ≥0) (w : UpperPlane) :
+    ‖deriv (inverseMap a b s) w.1‖=
+      Real.exp (-(killedState a s b (inverseUpper a ha b s w).1).2) := by
+  simp only [inverseUpper]
+  rw [deriv_inverseMap ha b s w.2,norm_inv,killedState,
+    ite_eq_left (inverseMap_alive ha b s w.2),actualState,Real.exp_neg,Real.exp_log]
+  exact norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous (inverseMap_alive ha b s w.2))
+
+ 
+lemma norm_deriv_inverseMap_joint_measurable (a : ℝ) (ha : 1/4<a) :
+    Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) =>
+      ‖deriv (inverseMap a p.2.1 p.1) p.2.2.1‖) := by
+  simp_rw [norm_deriv_inverseMap_eq_exp a (lt_trans (by norm_num) ha)]
+  exact (((killedState_time_joint_measurable a ha).comp
+    (measurable_fst.prodMk (measurable_snd.fst.prodMk
+      (inverseUpper_joint_measurable a ha)))).snd.neg).exp
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.Conformal
+
+def realDerivative (v : ℂ) : ℂ→L[ℝ]ℂ :=
+  (ContinuousLinearMap.toSpanSingleton ℂ v).restrictScalars ℝ
+
+lemma det_realDerivative (v : ℂ) : (realDerivative v).det=‖v‖^2 := by
+  simp [realDerivative,ContinuousLinearMap.det,LinearMap.det_restrictScalars,
+    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
+
+lemma lintegral_image_complex {f : ℂ→ℂ} {s : Set ℂ} (hs : MeasurableSet s)
+    (hd : ∀z∈s,DifferentiableAt ℂ f z) (hi : InjOn f s) (g : ℂ→ℝ≥0∞) :
+    (∫⁻z in f '' s,g z∂volume)=
+      ∫⁻w in s,ENNReal.ofReal (‖deriv f w‖^2)*g (f w)∂volume := by
+  have hf (z : ℂ) (hz : z∈s) : HasFDerivWithinAt f (realDerivative (deriv f z)) s z :=
+    ((hd z hz).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt
+  simpa only [det_realDerivative,abs_pow, abs_norm] using
+    lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hs hf hi g
+
+ 
+
+lemma map_withDensity_composed {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    (μ : Measure X) {e : X→Y} (he : Measurable e) {f : Y→ℝ≥0∞} (hf : Measurable f) :
+    (μ.withDensity (fun x => f (e x))).map e=(μ.map e).withDensity f := by
+  ext A hA
+  rw [Measure.map_apply he hA,withDensity_apply _ (he hA),
+    withDensity_apply _ hA,Measure.restrict_map he hA,lintegral_map hf he]
+
+end SLEExactGauge.Conformal
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen
+
+ 
+
+def upperExtend (f : UpperPlane→ℝ≥0∞) (z : ℂ) : ℝ≥0∞ :=
+  if hz : 0<z.im then f ⟨z,hz⟩ else 0
+
+@[simp] lemma upperExtend_coe (f : UpperPlane→ℝ≥0∞) (z : UpperPlane) :
+    upperExtend f z.1=f z := dite_eq_left z.2
+
+lemma image_inverseMap_upper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
+    inverseMap a b s '' {w : ℂ | 0<w.im}=
+      Subtype.val '' {z : UpperPlane | alive a b.1 s z.1} := by
+  ext z
+  constructor
+  · rintro ⟨w,hw,rfl⟩
+    exact ⟨inverseUpper a ha b s ⟨w,hw⟩,inverseMap_alive ha b s hw,rfl⟩
+  · rintro ⟨z,hz,rfl⟩
+    refine ⟨loewnerMap a b.1 s z.1-(b.1 s:ℂ),?_,inverseMap_centered a b s hz⟩
+    simpa only [mem_ofPred_eq,Complex.sub_im,Complex.ofReal_im,sub_zero] using
+      loewnerMap_im_pos hz
+
+lemma inverseMap_injOn_upper (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) :
+    InjOn (inverseMap a b s) {w : ℂ | 0<w.im} := by
+  intro u hu v hv huv
+  have h := congrArg (fun z => loewnerMap a b.1 s z-(b.1 s:ℂ)) huv
+  simpa only [centered_inverseMap ha b s hu,centered_inverseMap ha b s hv] using h
+
+ 
+
+theorem inverseUpper_lintegral_area (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    (f : UpperPlane→ℝ≥0∞) :
+    (∫⁻w : UpperPlane, ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
+      f (inverseUpper a ha b s w) ∂upperArea)=
+    ∫⁻z in {z : UpperPlane | alive a b.1 s z.1},f z∂upperArea := by
+  have h := Conformal.lintegral_image_complex upperPlane_open.measurableSet
+    (fun w hw => (inverseMap_hasStrictDerivAt ha b s hw).hasDerivAt.differentiableAt)
+    (inverseMap_injOn_upper a ha b s) (upperExtend f)
+  rw [image_inverseMap_upper a ha b s] at h
+  have hleft := setLIntegral_subtype (μ:=(volume : Measure ℂ))
+    upperPlane_open.measurableSet {z : UpperPlane | alive a b.1 s z.1} (upperExtend f)
+  have hright := lintegral_subtype_comap (μ:=(volume : Measure ℂ))
+    upperPlane_open.measurableSet (fun w =>
+      ENNReal.ofReal (‖deriv (inverseMap a b s) w‖^2)*upperExtend f (inverseMap a b s w))
+  have hh := hright.trans (h.symm.trans hleft.symm)
+  have he (w : UpperPlane) : upperExtend f (inverseMap a b s w.1)=
+      f (inverseUpper a ha b s w) := dite_eq_left (alive_initial_im (inverseMap_alive ha b s w.2))
+  simp only [upperExtend_coe, he] at hh
+  convert! hh using 1
+
+ 
+theorem map_inverseUpper_area (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
+    (upperArea.withDensity (fun w : UpperPlane =>
+       ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2))).map
+        (inverseUpper a (lt_trans (by norm_num) ha) b s)=
+      upperArea.restrict {z : UpperPlane | alive a b.1 s z.1} := by
+  let ha0 : 0<a := lt_trans (by norm_num) ha
+  let J := inverseUpper a ha0 b s
+  have hJ : Measurable J := (continuous_inverseUpper a ha0 b s).measurable
+  have hsurv : MeasurableSet {z : UpperPlane | alive a b.1 s z.1} :=
+    (measurableSet_alive_joint a ha).preimage
+      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
+  ext A hA
+  rw [Measure.map_apply hJ hA,withDensity_apply _ (hJ hA),
+    Measure.restrict_apply hA]
+  have h := inverseUpper_lintegral_area a ha0 b s (A.indicator (fun _ =>1))
+  have hi : (fun w : UpperPlane => ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
+      A.indicator (fun _ =>1) (J w))=
+      (J ⁻¹' A).indicator (fun w => ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)) := by
+    funext w
+    by_cases hw : J w ∈ A <;> simp [hw]
+  change (∫⁻w : UpperPlane, ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
+      A.indicator (fun _ =>1) (J w) ∂upperArea)=_ at h
+  rw [hi,lintegral_indicator (hJ hA),lintegral_indicator hA,
+    setLIntegral_one,Measure.restrict_apply hA] at h
+  exact h
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen
+
+ 
+def restartWeight (a : ℝ) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) : ℝ≥0∞ :=
+  ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^(2-alpha a))
+
+lemma restartWeight_pos (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) :
+    0<restartWeight a b s w := ENNReal.ofReal_pos.mpr
+      (Real.rpow_pos_of_pos (norm_pos_iff.mpr (deriv_inverseMap_ne_zero ha b s w.2)) _)
+
+lemma restartWeight_ne_top (a : ℝ) (b : GDriver) (s : ℝ≥0) (w : UpperPlane) :
+    restartWeight a b s w≠∞ := ENNReal.ofReal_ne_top
+
+lemma restartWeight_joint_measurable (a : ℝ) (ha : 1/4<a) :
+    Measurable (fun p : ℝ≥0×(GDriver×UpperPlane) => restartWeight a p.2.1 p.1 p.2.2) :=
+  ((norm_deriv_inverseMap_joint_measurable a ha).pow_const _).ennreal_ofReal
+
+lemma restartWeight_measurable (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
+    Measurable (restartWeight a b s) := by
+  have hin : Measurable (fun w : UpperPlane => (s,(b,w))) :=
+    measurable_const.prodMk (measurable_const.prodMk measurable_id)
+  have hh := (restartWeight_joint_measurable a ha).comp hin
+  change Measurable (fun w : UpperPlane => restartWeight a b s w) at hh
+  exact hh
+
+lemma green_jacobian_restart (a : ℝ) (ha : 0<a) (b : GDriver) (s t : ℝ≥0)
+    (w : UpperPlane) :
+    ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
+      ENNReal.ofReal (survivingGreen a (s+t) b (inverseUpper a ha b s w).1)=
+      restartWeight a b s w*ENNReal.ofReal (survivingGreen a t (restart s b) w.1) := by
+  have hi := inverseMap_alive ha b s w.2
+  have hp : 0<‖deriv (inverseMap a b s) w.1‖ :=
+    norm_pos_iff.mpr (deriv_inverseMap_ne_zero ha b s w.2)
+  have hr : ‖deriv (loewnerMap a b.1 s) (inverseMap a b s w.1)‖ =
+      ‖deriv (inverseMap a b s) w.1‖⁻¹ := by
+    rw [deriv_inverseMap ha b s w.2,norm_inv,inv_inv]
+  change ENNReal.ofReal (‖deriv (inverseMap a b s) w.1‖^2)*
+      ENNReal.ofReal (survivingGreen a (s+t) b (inverseMap a b s w.1))=_
+  rw [survivingGreen_restart a b s t hi,centered_inverseMap ha b s w.2,hr,
+    ENNReal.ofReal_mul (Real.rpow_nonneg (inv_nonneg.mpr hp.le) _),←mul_assoc,
+    ←ENNReal.ofReal_mul (sq_nonneg _)]
+  congr 1
+  rw [←Real.rpow_two,←Real.rpow_neg_eq_inv_rpow,←Real.rpow_add hp,←sub_eq_add_neg]
+  rfl
+
+ 
+
+theorem inverseUpper_lintegral_green (a : ℝ) (ha : 0<a) (b : GDriver) (s t : ℝ≥0)
+    (f : UpperPlane→ℝ≥0∞) :
+    (∫⁻w : UpperPlane,restartWeight a b s w*ENNReal.ofReal
+      (survivingGreen a t (restart s b) w.1)*f (inverseUpper a ha b s w)∂upperArea)=
+    ∫⁻z in {z : UpperPlane | alive a b.1 s z.1},
+      ENNReal.ofReal (survivingGreen a (s+t) b z.1)*f z∂upperArea := by
+  have h := inverseUpper_lintegral_area a ha b s
+    (fun z => ENNReal.ofReal (survivingGreen a (s+t) b z.1)*f z)
+  simpa only [←mul_assoc,green_jacobian_restart a ha b s t] using h
+
+ 
+
+def restartPush (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0) (μ : Measure UpperPlane) :
+    Measure UpperPlane := (μ.withDensity (restartWeight a b s)).map (inverseUpper a ha b s)
+
+lemma restartPush_lintegral (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (μ : Measure UpperPlane) {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    (∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b s μ)=
+      ∫⁻w,restartWeight a b s w*f (inverseUpper a (lt_trans (by norm_num) ha) b s w)∂μ := by
+  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
+  have hF : Measurable (fun w : UpperPlane =>
+      f (inverseUpper a (lt_trans (by norm_num) ha) b s w)) := hf.comp hJ
+  rw [restartPush,lintegral_map hf hJ]
+  exact lintegral_withDensity_eq_lintegral_mul μ (restartWeight_measurable a ha b s) hF
+
+ 
+
+theorem restartPush_survivingKernel (a : ℝ) (ha : 1/4<a) (b : GDriver) (s t : ℝ≥0) :
+    restartPush a (lt_trans (by norm_num) ha) b s (survivingKernel a ha t (restart s b))=
+      (survivingKernel a ha (s+t) b).restrict {z : UpperPlane | alive a b.1 s z.1} := by
+  let ha0 : 0<a := lt_trans (by norm_num) ha
+  let J := inverseUpper a ha0 b s
+  have hJ : Measurable J := (continuous_inverseUpper a ha0 b s).measurable
+  have hsurv : MeasurableSet {z : UpperPlane | alive a b.1 s z.1} :=
+    (measurableSet_alive_joint a ha).preimage
+      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
+  ext A hA
+  have h := inverseUpper_lintegral_green a ha0 b s t (A.indicator (fun _ =>1))
+  have heL := restartPush_lintegral a ha b s (survivingKernel a ha t (restart s b))
+    (measurable_const.indicator hA : Measurable (A.indicator (fun _ =>(1:ℝ≥0∞))))
+  rw [lintegral_indicator hA,setLIntegral_one] at heL
+  have hF : Measurable (fun w : UpperPlane => restartWeight a b s w*
+      A.indicator (fun _ =>(1:ℝ≥0∞)) (inverseUpper a ha0 b s w)) :=
+    (restartWeight_measurable a ha b s).mul ((measurable_const.indicator hA).comp hJ)
+  rw [survivingKernel_apply,lintegral_withDensity_eq_lintegral_mul upperArea
+    (survivingGreen_spatial_measurable a ha t (restart s b)).ennreal_ofReal hF] at heL
+  have heR : (∫⁻z in {z : UpperPlane | alive a b.1 s z.1},
+      ENNReal.ofReal (survivingGreen a (s+t) b z.1)*A.indicator (fun _ =>1) z∂upperArea)=
+      ((survivingKernel a ha (s+t) b).restrict {z : UpperPlane | alive a b.1 s z.1}) A := by
+    rw [survivingKernel_apply,Measure.restrict_apply hA,withDensity_apply _ (hA.inter hsurv)]
+    calc
+      _ = ∫⁻z in {z : UpperPlane | alive a b.1 s z.1},
+          A.indicator (fun z => ENNReal.ofReal (survivingGreen a (s+t) b z.1)) z∂upperArea := by
+        congr 1
+        funext z
+        by_cases hz : z∈A <;> simp [hz]
+      _ = _ := by rw [lintegral_indicator hA,Measure.restrict_restrict hA]
+  rw [←heR,←h]
+  rw [survivingKernel_apply]
+  simpa only [Pi.mul_apply,mul_left_comm,mul_assoc] using heL
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+
+lemma inverseMap_driver_congr (a : ℝ) (ha : 0<a) {b c : GDriver} {s : ℝ≥0}
+    (h : ∀ u ≤ s,b.1 u=c.1 u) {w : ℂ} (hw : 0<w.im) :
+    inverseMap a b s w=inverseMap a c s w := by
+  have hi := inverseMap_alive ha b s hw
+  have hc := (alive_driver_congr h).mp hi
+  have he : loewnerMap a c.1 s (inverseMap a b s w)-(c.1 s:ℂ)=w := by
+    rw [←loewnerMap_driver_congr h,←h s le_rfl]
+    exact centered_inverseMap ha b s hw
+  have hj := inverseMap_centered a c s hc
+  rw [he] at hj
+  exact hj.symm
+
+lemma inverseUpper_driver_congr (a : ℝ) (ha : 0<a) {b c : GDriver} {s : ℝ≥0}
+    (h : ∀ u ≤ s,b.1 u=c.1 u) : inverseUpper a ha b s=inverseUpper a ha c s := by
+  funext w
+  apply Subtype.ext
+  exact inverseMap_driver_congr a ha h w.2
+
+lemma restartWeight_driver_congr (a : ℝ) (ha : 0<a) {b c : GDriver} {s : ℝ≥0}
+    (h : ∀ u ≤ s,b.1 u=c.1 u) : restartWeight a b s=restartWeight a c s := by
+  funext w
+  simp only [restartWeight,deriv_inverseMap ha _ _ w.2,
+    inverseMap_driver_congr a ha h w.2,loewnerMap_driver_congr h]
+
+ 
+
+def stoppedData (S : GDriver→ℝ≥0∞) (b : GDriver) : ℝ≥0×GDriver :=
+  ((S b).toNNReal,pastDriver (S b).toNNReal b)
+
+lemma stoppedData_raw_measurable {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (stoppedData S) :=
+  (ENNReal.measurable_toNNReal.comp hS.measurable).prodMk
+    (pastDriver_stopping_measurable S hS)
+
+lemma inverseUpper_stoppedData (a : ℝ) (ha : 0<a) (S : GDriver→ℝ≥0∞)
+    (b : GDriver) : inverseUpper a ha (stoppedData S b).2 (stoppedData S b).1=
+      inverseUpper a ha b (S b).toNNReal :=
+  inverseUpper_driver_congr a ha (pastDriver_eq _ b)
+
+lemma restartWeight_stoppedData (a : ℝ) (ha : 0<a) (S : GDriver→ℝ≥0∞)
+    (b : GDriver) : restartWeight a (stoppedData S b).2 (stoppedData S b).1=
+      restartWeight a b (S b).toNNReal :=
+  restartWeight_driver_congr a ha (pastDriver_eq _ b)
+
+lemma inverseUpper_stopped_raw_joint (a : ℝ) (ha : 1/4<a)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    @Measurable (GDriver×UpperPlane) UpperPlane
+      (hS.measurableSpace.prod inferInstance) inferInstance
+      (fun p => inverseUpper a (lt_trans (by norm_num) ha) p.1 (S p.1).toNNReal p.2) := by
+  have hD := stoppedData_raw_measurable hS
+  have hin : @Measurable (GDriver×UpperPlane) (ℝ≥0×(GDriver×UpperPlane))
+      (hS.measurableSpace.prod inferInstance) inferInstance (fun p =>
+      ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) :=
+    (hD.fst.comp measurable_fst).prodMk ((hD.snd.comp measurable_fst).prodMk measurable_snd)
+  have hh := (inverseUpper_joint_measurable a ha).comp hin
+  simpa only [Function.comp_def,inverseUpper_stoppedData] using hh
+
+lemma restartWeight_stopped_raw_joint (a : ℝ) (ha : 1/4<a)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    @Measurable (GDriver×UpperPlane) ℝ≥0∞
+      (hS.measurableSpace.prod inferInstance) inferInstance
+      (fun p => restartWeight a p.1 (S p.1).toNNReal p.2) := by
+  have hD := stoppedData_raw_measurable hS
+  have hin : @Measurable (GDriver×UpperPlane) (ℝ≥0×(GDriver×UpperPlane))
+      (hS.measurableSpace.prod inferInstance) inferInstance (fun p =>
+      ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) :=
+    (hD.fst.comp measurable_fst).prodMk ((hD.snd.comp measurable_fst).prodMk measurable_snd)
+  have hh := (restartWeight_joint_measurable a ha).comp hin
+  simpa only [Function.comp_def,restartWeight_stoppedData a (lt_trans (by norm_num) ha)] using hh
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CutoffDrift.lean b/lean/OAI/Probability/SLEGauge/CutoffDrift.lean
index 2f2526fb4c74a8586332fba75210e7291e15534f..695360eb29f69cad675fe046652a6749197d4804 100644
--- a/lean/OAI/Probability/SLEGauge/CutoffDrift.lean
+++ b/lean/OAI/Probability/SLEGauge/CutoffDrift.lean
@@ -1,445 +1,445 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ConformalFlow
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-namespace ExactSLEGreen
-open ExactSLEMain
-noncomputable def cutoffDrift (a ε : ℝ) (v : State) : State :=
-  (cutoffVector a ε v.1, (-(a : ℂ) * (cutoffVector 1 ε v.1)^2).re)
-
-lemma cutoffDrift_second_bound (a ε : ℝ) (hε : 0 < ε) (z : ℂ) :
-    ‖(-(a : ℂ) * (cutoffVector 1 ε z)^2).re‖ ≤ |a|/ε^2 := by
-  calc
-    _ ≤ ‖-(a : ℂ) * (cutoffVector 1 ε z)^2‖ := Complex.abs_re_le_norm _
-    _ = |a| * ‖cutoffVector 1 ε z‖^2 := by simp only [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs,norm_pow]
-    _ ≤ |a| * (1/ε)^2 := mul_le_mul_of_nonneg_left
-      (pow_le_pow_left₀ (norm_nonneg _) (by simpa using cutoffVector_bound 1 ε hε z) 2) (abs_nonneg _)
-    _ = _ := by ring
-
-lemma cutoffDrift_second_lipschitz (a ε : ℝ) (hε : 0 < ε) :
-    LipschitzWith ⟨4*|a|/ε^3,by positivity⟩
-      (fun z : ℂ => (-(a : ℂ) * (cutoffVector 1 ε z)^2).re) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  rw [Real.dist_eq,dist_eq_norm,← Complex.sub_re]
-  have hb (u : ℂ) : ‖cutoffVector 1 ε u‖ ≤ 1/ε := by simpa using cutoffVector_bound 1 ε hε u
-  have hd : ‖cutoffVector 1 ε z - cutoffVector 1 ε w‖ ≤ (2/ε^2)*‖z-w‖ := by
-    have h := (cutoffVector_lipschitz 1 ε hε).dist_le_mul z w
-    simp only [dist_eq_norm] at h
-    change ‖cutoffVector 1 ε z - cutoffVector 1 ε w‖ ≤ (2*|1|/ε^2)*‖z-w‖ at h
-    simpa only [abs_one,mul_one] using h
-  have he : -(a : ℂ) * (cutoffVector 1 ε z)^2 - -(a : ℂ) * (cutoffVector 1 ε w)^2 =
-      -(a : ℂ) * (cutoffVector 1 ε z - cutoffVector 1 ε w) *
-        (cutoffVector 1 ε z + cutoffVector 1 ε w) := by ring
-  calc
-    _ ≤ ‖-(a : ℂ) * (cutoffVector 1 ε z)^2 - -(a : ℂ) * (cutoffVector 1 ε w)^2‖ := Complex.abs_re_le_norm _
-    _ = |a| * ‖cutoffVector 1 ε z - cutoffVector 1 ε w‖ *
-        ‖cutoffVector 1 ε z + cutoffVector 1 ε w‖ := by rw [he]; simp only [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs]
-    _ ≤ (|a| * ((2/ε^2)*‖z-w‖)) * (1/ε+1/ε) :=
-      mul_le_mul (mul_le_mul_of_nonneg_left hd (abs_nonneg _))
-        ((norm_add_le _ _).trans (add_le_add (hb z) (hb w))) (norm_nonneg _) (by positivity)
-    _ = _ := by change _ = (4*|a|/ε^3)*‖z-w‖; field_simp; ring
-
-lemma cutoffDrift_bound (a ε : ℝ) (hε : 0 < ε) (v : State) :
-    ‖cutoffDrift a ε v‖ ≤ max (|a|/ε) (|a|/ε^2) := by
-  exact max_le_max (cutoffVector_bound a ε hε v.1) (cutoffDrift_second_bound a ε hε v.1)
-
-lemma cutoffDrift_lipschitz (a ε : ℝ) (hε : 0 < ε) :
-    LipschitzWith (max ⟨2*|a|/ε^2,by positivity⟩ ⟨4*|a|/ε^3,by positivity⟩)
-      (cutoffDrift a ε) := by
-  convert ((cutoffVector_lipschitz a ε hε).prodMk (cutoffDrift_second_lipschitz a ε hε)).comp LipschitzWith.prod_fst using 1 <;> first | rfl | simp only [mul_one]
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEMain
-
-lemma cutoffDrift_eq_physical (a ε : ℝ) (v : State) (hy : ε ≤ v.1.im) :
-    cutoffDrift a ε v = ((a : ℂ)/v.1, (-(a : ℂ)/v.1^2).re) := by
-  simp only [cutoffDrift,cutoffVector,heightClamp_eq hy,Complex.ofReal_one,one_div]
-  congr 1
-  rw [inv_pow,div_eq_mul_inv]
-
- 
-
-theorem state_eq_loewner_on_safe
-    (a ε : ℝ) (hε : 0 < ε) (b : ℝ≥0 → ℝ) (hb : Continuous b) (_hb0 : b 0 = 0)
-    (z : ℂ) (V : ℝ≥0 → State) (hVc : Continuous V)
-    (hVe : ∀ t, V t = (z,0) + b t • ((-1 : ℂ), (0 : ℝ)) +
-      ∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (V u.toNNReal))
-    (T : ℝ≥0) (hy : ∀ t ≤ T, ε ≤ (V t).1.im) :
-    alive a b T z ∧
-      (∀ t ≤ T, loewnerMap a b t z - (b t : ℂ) = (V t).1) ∧
-      Real.log ‖deriv (loewnerMap a b T) z‖ = (V T).2 := by
-  have hAc := (cutoffDrift_lipschitz a ε hε).continuous
-  have hIc (t : ℝ≥0) : IntervalIntegrable
-      (fun u : ℝ => cutoffDrift a ε (V u.toNNReal)) volume 0 t :=
-    (hAc.comp (hVc.comp continuous_real_toNNReal)).intervalIntegrable 0 t
-  have hfst (t : ℝ≥0) : (V t).1 = z - (b t : ℂ) +
-      ∫ u in (0 : ℝ)..(t : ℝ), cutoffVector a ε (V u.toNNReal).1 := by
-    have h := congrArg Prod.fst (hVe t)
-    have hi := (ContinuousLinearMap.fst ℝ ℂ ℝ).intervalIntegral_comp_comm (hIc t)
-    simp only [Prod.fst_add] at h
-    change (V t).1 = z + b t • (-1 : ℂ) +
-      (ContinuousLinearMap.fst ℝ ℂ ℝ) (∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (V u.toNNReal)) at h
-    rw [← hi] at h
-    simpa only [ContinuousLinearMap.coe_fst',Prod.fst_add,Prod.fst_smul,Complex.real_smul,Complex.ofReal_neg,
-      mul_neg,mul_one,sub_eq_add_neg,cutoffDrift] using h
-  have hsnd (t : ℝ≥0) : (V t).2 =
-      ∫ u in (0 : ℝ)..(t : ℝ), (-(a : ℂ)*(cutoffVector 1 ε (V u.toNNReal).1)^2).re := by
-    have h := congrArg Prod.snd (hVe t)
-    have hi := (ContinuousLinearMap.snd ℝ ℂ ℝ).intervalIntegral_comp_comm (hIc t)
-    simp only [Prod.snd_add] at h
-    change (V t).2 = (0 : ℝ) + b t • (0 : ℝ) +
-      (ContinuousLinearMap.snd ℝ ℂ ℝ) (∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (V u.toNNReal)) at h
-    rw [← hi] at h
-    simpa only [ContinuousLinearMap.coe_snd',Prod.snd_add,Prod.snd_smul,smul_zero,add_zero,zero_add,cutoffDrift] using h
-  let G : ℝ≥0 → ℂ := fun t => (V t).1 + (b t : ℂ)
-  have hGc : Continuous G := hVc.fst.add (Complex.continuous_ofReal.comp hb)
-  have hGe (t : ℝ≥0) : G t = z + ∫ u in (0 : ℝ)..(t : ℝ),
-      cutoffVector a ε (G u.toNNReal + -(b u.toNNReal : ℂ)) := by
-    simp only [G,add_neg_cancel_right]
-    rw [hfst t]
-    abel
-  have hG0 : G 0 = z := by simpa using hGe 0
-  have hode : odeUntil a b z T (fun u : ℝ => G u.toNNReal) := by
-    refine ⟨by simpa using hG0,?_⟩
-    intro u hu
-    have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    have huy := hy _ hut
-    have hd := ExactSLEFlow.integral_flow_hasDerivWithinAt
-      (cutoffVector_lipschitz a ε hε).continuous (Complex.continuous_ofReal.comp hb).neg
-      hGc hGe T u hu
-    have hGim : (G u.toNNReal).im = (V u.toNNReal).1.im := by simp [G]
-    refine ⟨by rw [hGim]; exact hε.trans_le huy,?_⟩
-    simpa only [Function.comp_def,Pi.neg_apply,cutoffVector,G,add_neg_cancel_right,heightClamp_eq huy,sub_eq_add_neg] using hd
-  have hlo (t : ℝ≥0) (ht : t ≤ T) : loewnerMap a b t z - (b t : ℂ) = (V t).1 := by
-    rw [loewnerMap_eq (odeUntil_mono hode ht)]
-    simp only [Real.toNNReal_coe,G,add_sub_cancel_right]
-  refine ⟨⟨_,hode⟩,hlo,?_⟩
-  rw [log_norm_deriv_loewnerMap hb ⟨_,hode⟩, hsnd T]
-  apply intervalIntegral.integral_congr
-  intro u hu
-  have hu' : u ∈ Icc (0 : ℝ) T := by simpa only [uIcc_of_le T.coe_nonneg] using hu
-  have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu'.2).trans_eq (by simp)
-  dsimp only
-  rw [hlo _ hut,cutoffVector,heightClamp_eq (hy _ hut)]
-  simp only [Complex.ofReal_one,inv_pow,div_eq_mul_inv,one_mul]
-
-end ExactSLEGreen
-
-namespace ExactSLEHitting
-variable {Ω : Type*} {m : MeasurableSpace Ω}
-variable (ℱ : Filtration ℝ≥0 m) (f : ℝ≥0 → Ω → ℝ)
-
-lemma firstContact_mem (hc : ∀ ω, Continuous (fun t => f t ω)) (ω : Ω)
-    (hex : ∃ t, f t ω ≤ 0) :
-    f (hittingAfter f (Iic 0) 0 ω).untopA ω ≤ 0 := by
-  have he : ∃ t : ℝ≥0, 0 ≤ t ∧ f t ω ∈ Iic 0 := by
-    obtain ⟨t,ht⟩ := hex
-    exact ⟨t,bot_le,ht⟩
-  rw [hittingAfter,ite_eq_left he]
-  change f (sInf {t : ℝ≥0 | 0 ≤ t ∧ f t ω ≤ 0}) ω ≤ 0
-  have hcl : IsClosed {t : ℝ≥0 | 0 ≤ t ∧ f t ω ≤ 0} :=
-    isClosed_Ici.inter (isClosed_le (hc ω) continuous_const)
-  exact (hcl.csInf_mem he ⟨0,fun _ _ => bot_le⟩).2
-
-lemma firstContact_le_iff (hc : ∀ ω, Continuous (fun t => f t ω))
-    (ω : Ω) (t : ℝ≥0) :
-    hittingAfter f (Iic 0) 0 ω ≤ t ↔ ∃ u ≤ t, f u ω ≤ 0 := by
-  constructor
-  · intro h
-    have hex : ∃ u, f u ω ≤ 0 := by
-      by_contra hn
-      have hz : hittingAfter f (Iic 0) 0 ω = ⊤ := by
-        rw [hittingAfter_eq_top_iff]
-        intro j _ hj
-        exact hn ⟨j,hj⟩
-      rw [hz] at h
-      exact WithTop.not_top_le_coe t h
-    refine ⟨(hittingAfter f (Iic 0) 0 ω).untopA,?_, firstContact_mem f hc ω hex⟩
-    exact WithTop.untopA_le h
-  · rintro ⟨u,hu,hfu⟩
-    exact (hittingAfter_le_of_mem (by exact bot_le) hfu).trans (WithTop.coe_le_coe.mpr hu)
-
-lemma measurable_has_contact (hc : ∀ ω, Continuous (fun t => f t ω))
-    (hm : ∀ t, Measurable[ℱ t] (f t)) (t : ℝ≥0) :
-    MeasurableSet[ℱ t] {ω | ∃ u ≤ t, f u ω ≤ 0} := by
-  let g : Icc (0 : ℝ≥0) t → Ω → ℝ≥0∞ := fun u ω => ENNReal.ofReal (f u ω)
-  have hg : @Measurable Ω ℝ≥0∞ (ℱ t) _ (⨅ u, g u) := by
-    apply measurable_iInf_of_upperSemicontinuous
-    · intro u
-      exact ((hm u).mono (ℱ.mono u.2.2) le_rfl).ennreal_ofReal
-    · intro ω
-      exact (ENNReal.continuous_ofReal.comp ((hc ω).comp continuous_subtype_val)).upperSemicontinuous
-  have he : {ω | ∃ u ≤ t, f u ω ≤ 0} = {ω | (⨅ u, g u) ω = 0} := by
-    ext ω
-    simp only [mem_ofPred_eq,iInf_apply]
-    constructor
-    · rintro ⟨u,hu,hfu⟩
-      apply le_antisymm _ bot_le
-      exact (iInf_le (fun u => g u ω) ⟨u,bot_le,hu⟩).trans (by change ENNReal.ofReal (f u ω) ≤ 0; exact le_of_eq (ENNReal.ofReal_eq_zero.mpr hfu))
-    · intro h
-      obtain ⟨u,hu,hmin⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr (show (0 : ℝ≥0) ≤ t from bot_le)) (hc ω).continuousOn
-      refine ⟨u,hu.2,?_⟩
-      have hl : ENNReal.ofReal (f u ω) ≤ ⨅ v, g v ω := by
-        apply le_iInf
-        intro v
-        exact ENNReal.ofReal_le_ofReal (hmin v.2)
-      rw [h] at hl
-      exact ENNReal.ofReal_eq_zero.mp (le_antisymm hl bot_le)
-  rw [he]
-  exact measurableSet_eq_fun hg measurable_const
-
-lemma firstContact_stopping (hc : ∀ ω, Continuous (fun t => f t ω))
-    (hm : ∀ t, Measurable[ℱ t] (f t)) :
-    IsStoppingTime ℱ (hittingAfter f (Iic 0) 0) := by
-  intro t
-  have he : {ω | hittingAfter f (Iic 0) 0 ω ≤ t} = {ω | ∃ u ≤ t, f u ω ≤ 0} := by
-    ext ω
-    exact firstContact_le_iff f hc ω t
-  rw [he]
-  exact measurable_has_contact ℱ f hc hm t
-
-end ExactSLEHitting
-
-namespace ExactSLEHitting
-variable {Ω : Type*} {m : MeasurableSpace Ω}
-variable (ℱ : Filtration ℝ≥0 m) (f : ℝ≥0 → Ω → ℝ)
-
-lemma boundedContact_le_iff (hc : ∀ ω, Continuous (fun t => f t ω))
-    (ω : Ω) (B t : ℝ≥0) (ht : t < B) :
-    hittingBtwn f (Iic 0) 0 B ω ≤ t ↔ ∃ u ≤ t, f u ω ≤ 0 := by
-  constructor
-  · intro h
-    have hex : ∃ u ∈ Icc (0 : ℝ≥0) B, f u ω ∈ Iic 0 := by
-      by_contra hn
-      have he : hittingBtwn f (Iic 0) 0 B ω = B := by simp only [hittingBtwn,ite_eq_right hn]
-      rw [he] at h
-      exact not_le.mpr ht h
-    have hcl : IsClosed (Icc (0 : ℝ≥0) B ∩ {u | f u ω ∈ Iic 0}) :=
-      isClosed_Icc.inter (isClosed_Iic.preimage (hc ω))
-    have hmem : f (hittingBtwn f (Iic 0) 0 B ω) ω ≤ 0 := by
-      rw [hittingBtwn,ite_eq_left hex]
-      exact (hcl.csInf_mem hex bddBelow_Icc.inter_of_left).2
-    exact ⟨_,h,hmem⟩
-  · rintro ⟨u,hu,hfu⟩
-    exact (hittingBtwn_le_of_mem bot_le (hu.trans ht.le) hfu).trans hu
-
-lemma boundedContact_stopping (hc : ∀ ω, Continuous (fun t => f t ω))
-    (hm : ∀ t, Measurable[ℱ t] (f t)) (B : ℝ≥0) :
-    IsStoppingTime ℱ (fun ω => ((hittingBtwn f (Iic (0 : ℝ)) (0 : ℝ≥0) B ω : ℝ≥0) : WithTop ℝ≥0)) := by
-  intro t
-  by_cases ht : t < B
-  · have he : {ω | ((hittingBtwn f (Iic (0 : ℝ)) (0 : ℝ≥0) B ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
-        {ω | ∃ u ≤ t, f u ω ≤ 0} := by
-      ext ω
-      simp only [mem_ofPred_eq,WithTop.coe_le_coe]
-      exact boundedContact_le_iff f hc ω B t ht
-    rw [he]
-    exact measurable_has_contact ℱ f hc hm t
-  · have he : {ω | ((hittingBtwn f (Iic (0 : ℝ)) (0 : ℝ≥0) B ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} = univ := by
-      apply eq_univ_of_forall
-      intro ω
-      exact WithTop.coe_le_coe.mpr ((hittingBtwn_le ω).trans (le_of_not_gt ht))
-    rw [he]
-    exact MeasurableSet.univ
-
-lemma nonneg_before_boundedContact (hc : ∀ ω, Continuous (fun t => f t ω))
-    (h0 : ∀ ω, 0 ≤ f 0 ω) (B : ℝ≥0) (ω : Ω) (t : ℝ≥0)
-    (ht : t ≤ hittingBtwn f (Iic 0) 0 B ω) : 0 ≤ f t ω := by
-  by_cases hz : t = 0
-  · simpa only [hz] using h0 ω
-  · have ht0 : 0 < t := bot_lt_iff_ne_bot.mpr hz
-    have : NeBot (𝓝[<] t) := nhdsWithin_Iio_neBot' ⟨0,ht0⟩ le_rfl
-    apply ge_of_tendsto ((hc ω).continuousAt.tendsto.mono_left (show 𝓝[<] t ≤ 𝓝 t from inf_le_left))
-    filter_upwards [self_mem_nhdsWithin] with u hu
-    have hn : f u ω ∉ Iic 0 := notMem_of_lt_hittingBtwn (hu.trans_le ht) bot_le
-    exact le_of_lt (lt_of_not_ge hn)
-
-lemma boundedContact_eq_zero (hc : ∀ ω, Continuous (fun t => f t ω))
-    (h0 : ∀ ω, 0 ≤ f 0 ω) (B : ℝ≥0) (ω : Ω)
-    (ht : hittingBtwn f (Iic 0) 0 B ω < B) :
-    f (hittingBtwn f (Iic 0) 0 B ω) ω = 0 := by
-  apply le_antisymm _ (nonneg_before_boundedContact f hc h0 B ω _ le_rfl)
-  obtain ⟨u,hu,hfu⟩ := (boundedContact_le_iff f hc ω B _ ht).mp le_rfl
-  have he : hittingBtwn f (Iic 0) 0 B ω = u :=
-    le_antisymm (hittingBtwn_le_of_mem bot_le (hu.trans ht.le) hfu) hu
-  simpa only [he] using hfu
-
-end ExactSLEHitting
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-noncomputable def physicalRadius (v : State) : ℝ := Y v * Real.exp (-J v)
-lemma physicalRadius_continuous : Continuous physicalRadius :=
-  Y.continuous.mul (Real.continuous_exp.comp J.continuous.neg)
-
-lemma alpha_pos {a : ℝ} (ha : 1/4 < a) : 0 < alpha a := by
-  have ha0 : 0 < a := by linarith
-  unfold alpha
-  rw [sub_pos,div_lt_one (by positivity)]
-  linarith
-
-lemma physicalGreen_bound (a ρ : ℝ) (ha : 1/4 < a) (hρ : 0 < ρ)
-    (v : State) (hy : 0 < Y v) (hr : ρ ≤ physicalRadius v) :
-    physicalGreen a v ≤ Real.exp (-alpha a * Real.log ρ) := by
-  have hq := Q_pos hy
-  have hlq : 2*Real.log (Y v) ≤ Real.log (Q v) := by
-    have he := Real.log_pow (Y v) 2
-    norm_num only [Nat.cast_ofNat] at he
-    rw [← he]
-    apply Real.log_le_log (by positivity)
-    dsimp only [Q]
-    nlinarith [sq_nonneg (X v)]
-  have hlr : Real.log ρ ≤ Real.log (Y v) - J v := by
-    have h := Real.log_le_log hρ hr
-    simpa only [physicalRadius,Real.log_mul hy.ne' (Real.exp_ne_zero _),Real.log_exp,sub_eq_add_neg] using h
-  have hp : 0 ≤ p a / 2 := by unfold p; linarith
-  have ha' := alpha_pos ha
-  apply Real.exp_le_exp.mpr
-  unfold logGreen
-  nlinarith [mul_le_mul_of_nonneg_left hlq hp,mul_le_mul_of_nonneg_left hlr ha'.le]
-
-lemma physicalGreen_eq_radius_sine (a : ℝ) (v : State) (hy : 0 < Y v) :
-    physicalGreen a v = (physicalRadius v) ^ (-alpha a) * (Y v / ‖v.1‖) ^ (p a) := by
-  have hzn : 0 < ‖v.1‖ := lt_of_lt_of_le hy (Complex.im_le_norm _)
-  have hr : 0 < physicalRadius v := mul_pos hy (Real.exp_pos _)
-  have hqlog : Real.log (Q v) = 2*Real.log ‖v.1‖ := by
-    rw [Q_eq,Complex.normSq_eq_norm_sq,Real.log_pow]
-    norm_num
-  rw [Real.rpow_def_of_pos hr,Real.rpow_def_of_pos (div_pos hy hzn),← Real.exp_add]
-  rw [physicalRadius,Real.log_mul hy.ne' (Real.exp_ne_zero _),Real.log_exp,
-    Real.log_div hy.ne' hzn.ne']
-  simp only [physicalGreen,logGreen] at *
-  rw [hqlog]
-  congr 1
-  ring
-
- 
-lemma exists_localState (a ε : ℝ) (hε : 0 < ε) (z : ℂ) :
-    ∃ V : ℝ≥0 → ExactSLEGenerator.Driver → State,
-      (∀ b, Continuous (fun t => V t b)) ∧
-      (∀ t, Measurable[rawFiltration t] (V t)) ∧
-      (∀ b, V 0 b = (z,0)) ∧
-      (∀ (s h : ℝ≥0) b, V (s+h) b = V s b + increment s h b • noise +
-        ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), cutoffDrift a ε (V t.toNNReal b)) := by
-  exact exists_adapted_additive_brownian (cutoffDrift a ε)
-    ⟨max (|a|/ε) (|a|/ε^2),by positivity⟩ _
-    (cutoffDrift_lipschitz a ε hε) (cutoffDrift_bound a ε hε) (z,0) noise
-
-noncomputable def localState (a ε : ℝ) (hε : 0 < ε) (z : ℂ) : ℝ≥0 → ExactSLEGenerator.Driver → State :=
-  Classical.choose (exists_localState a ε hε z)
-lemma localState_continuous (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (b : ExactSLEGenerator.Driver) :
-    Continuous (fun t => localState a ε hε z t b) :=
-  (Classical.choose_spec (exists_localState a ε hε z)).1 b
-lemma localState_adapted (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (t : ℝ≥0) :
-    Measurable[rawFiltration t] (localState a ε hε z t) :=
-  (Classical.choose_spec (exists_localState a ε hε z)).2.1 t
-lemma localState_zero (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (b : ExactSLEGenerator.Driver) :
-    localState a ε hε z 0 b = (z,0) :=
-  (Classical.choose_spec (exists_localState a ε hε z)).2.2.1 b
-lemma localState_eq (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (s h : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    localState a ε hε z (s+h) b = localState a ε hε z s b + increment s h b • noise +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), cutoffDrift a ε (localState a ε hε z t.toNNReal b) :=
-  (Classical.choose_spec (exists_localState a ε hε z)).2.2.2 s h b
-lemma localState_eq_zero (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    localState a ε hε z t b = (z,0) + b.1 t • noise +
-      ∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (localState a ε hε z u.toNNReal b) := by
-  simpa only [zero_add,localState_zero,increment,b.2,sub_zero,NNReal.coe_zero]
-    using localState_eq a ε hε z 0 t b
-
- 
-
-theorem localGreen_optional (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a ε : ℝ) (ha : a ≠ 0) (hε : 0 < ε) (z : ℂ) (N B : ℝ≥0)
-    (S T : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
-    (hsafe : ∀ b t, t ≤ T b → ε ≤ Y (localState a ε hε z t b) ∧
-      ‖localState a ε hε z t b‖ ≤ N)
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    ∫ b in U, physicalGreen a (localState a ε hε z (S b) b) ∂W =
-      ∫ b in U, physicalGreen a (localState a ε hε z (T b) b) ∂W := by
-  have hf := greenCutoff_smooth a ε hε N
-  have hG (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ T b) :
-      generator (greenCutoff a ε N) (cutoffDrift a ε) noise (localState a ε hε z t b) = 0 := by
-    have hv := hsafe b t ht
-    have hg := greenCutoff_generator_zero a ε ha hε N (localState a ε hε z t b) hv.2 hv.1
-    change (1/2 : ℝ) * fderiv ℝ (fun w => fderiv ℝ (greenCutoff a ε N) w noise)
-      (localState a ε hε z t b) noise + fderiv ℝ (greenCutoff a ε N)
-      (localState a ε hε z t b) (cutoffDrift a ε (localState a ε hε z t b)) = 0
-    rw [cutoffDrift_eq_physical a ε _ hv.1]
-    exact hg
-  have ho := compact_test_optional_sampling W hW (localState a ε hε z)
-    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
-    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
-    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
-    (by simp [noise,Prod.norm_def]) (localState_eq a ε hε z)
-    (greenCutoff a ε N) hf (greenCutoff_compact a ε N) S T B hS hT hST hTB hG U hU
-  have he (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ T b) :
-      greenCutoff a ε N (localState a ε hε z t b) = physicalGreen a (localState a ε hε z t b) :=
-    (greenCutoff_eq_near a ε hε N _ (hsafe b t ht).2 (hsafe b t ht).1).eq_of_nhds
-  simpa only [he _ _ (hST _),he _ _ le_rfl] using ho
-
-end ExactSLEGreen
-
-namespace ExactSLEBarrier
-
- 
-
-structure Evolution (a : ℝ) (T : ℝ≥0) (y j q : ℝ → ℝ) : Prop where
-  yc : ContinuousOn y (Icc 0 (T : ℝ))
-  jc : ContinuousOn j (Icc 0 (T : ℝ))
-  yp : ∀ t ∈ Icc 0 (T : ℝ), 0 < y t
-  qp : ∀ t ∈ Icc 0 (T : ℝ), 0 < q t
-  yd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt y (-a*y t/q t) (Icc 0 (T : ℝ)) t
-  jd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt j
-    (-a*(q t-2*(y t)^2)/(q t)^2) (Icc 0 (T : ℝ)) t
-
-variable {a : ℝ} {T : ℝ≥0} {y j q : ℝ → ℝ}
-
-lemma Evolution.logHeight_deriv (h : Evolution a T y j q) {t : ℝ}
-    (ht : t ∈ Icc 0 (T : ℝ)) :
-    HasDerivWithinAt (fun t => Real.log (y t)) (-a/q t) (Icc 0 (T : ℝ)) t := by
-  convert (h.yd t ht).log (h.yp t ht).ne' using 1
-  field_simp [(h.yp t ht).ne']
-
-lemma Evolution.logRadius_deriv (h : Evolution a T y j q) {t : ℝ}
-    (ht : t ∈ Icc 0 (T : ℝ)) :
-    HasDerivWithinAt (fun t => Real.log (y t)-j t)
-      (-2*a*(y t)^2/(q t)^2) (Icc 0 (T : ℝ)) t := by
-  convert (h.logHeight_deriv ht).sub (h.jd t ht) using 1
-  try rfl
-  field_simp [(h.qp t ht).ne']
-  ring
-
-lemma Evolution.height_le (h : Evolution a T y j q) (ha : 0 ≤ a)
-    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : y t ≤ y 0 := by
-  have hm : AntitoneOn y (Icc 0 (T : ℝ)) :=
-    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) h.yc
-      (fun u hu => (h.yd u (interior_subset hu)).mono interior_subset)
-      (fun u hu => div_nonpos_of_nonpos_of_nonneg (by nlinarith [h.yp u (interior_subset hu)])
-        (h.qp u (interior_subset hu)).le)
-  exact hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
-
-end ExactSLEBarrier
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ConformalFlow
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+namespace ExactSLEGreen
+open ExactSLEMain
+noncomputable def cutoffDrift (a ε : ℝ) (v : State) : State :=
+  (cutoffVector a ε v.1, (-(a : ℂ) * (cutoffVector 1 ε v.1)^2).re)
+
+lemma cutoffDrift_second_bound (a ε : ℝ) (hε : 0 < ε) (z : ℂ) :
+    ‖(-(a : ℂ) * (cutoffVector 1 ε z)^2).re‖ ≤ |a|/ε^2 := by
+  calc
+    _ ≤ ‖-(a : ℂ) * (cutoffVector 1 ε z)^2‖ := Complex.abs_re_le_norm _
+    _ = |a| * ‖cutoffVector 1 ε z‖^2 := by simp only [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs,norm_pow]
+    _ ≤ |a| * (1/ε)^2 := mul_le_mul_of_nonneg_left
+      (pow_le_pow_left₀ (norm_nonneg _) (by simpa using cutoffVector_bound 1 ε hε z) 2) (abs_nonneg _)
+    _ = _ := by ring
+
+lemma cutoffDrift_second_lipschitz (a ε : ℝ) (hε : 0 < ε) :
+    LipschitzWith ⟨4*|a|/ε^3,by positivity⟩
+      (fun z : ℂ => (-(a : ℂ) * (cutoffVector 1 ε z)^2).re) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  rw [Real.dist_eq,dist_eq_norm,← Complex.sub_re]
+  have hb (u : ℂ) : ‖cutoffVector 1 ε u‖ ≤ 1/ε := by simpa using cutoffVector_bound 1 ε hε u
+  have hd : ‖cutoffVector 1 ε z - cutoffVector 1 ε w‖ ≤ (2/ε^2)*‖z-w‖ := by
+    have h := (cutoffVector_lipschitz 1 ε hε).dist_le_mul z w
+    simp only [dist_eq_norm] at h
+    change ‖cutoffVector 1 ε z - cutoffVector 1 ε w‖ ≤ (2*|1|/ε^2)*‖z-w‖ at h
+    simpa only [abs_one,mul_one] using h
+  have he : -(a : ℂ) * (cutoffVector 1 ε z)^2 - -(a : ℂ) * (cutoffVector 1 ε w)^2 =
+      -(a : ℂ) * (cutoffVector 1 ε z - cutoffVector 1 ε w) *
+        (cutoffVector 1 ε z + cutoffVector 1 ε w) := by ring
+  calc
+    _ ≤ ‖-(a : ℂ) * (cutoffVector 1 ε z)^2 - -(a : ℂ) * (cutoffVector 1 ε w)^2‖ := Complex.abs_re_le_norm _
+    _ = |a| * ‖cutoffVector 1 ε z - cutoffVector 1 ε w‖ *
+        ‖cutoffVector 1 ε z + cutoffVector 1 ε w‖ := by rw [he]; simp only [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs]
+    _ ≤ (|a| * ((2/ε^2)*‖z-w‖)) * (1/ε+1/ε) :=
+      mul_le_mul (mul_le_mul_of_nonneg_left hd (abs_nonneg _))
+        ((norm_add_le _ _).trans (add_le_add (hb z) (hb w))) (norm_nonneg _) (by positivity)
+    _ = _ := by change _ = (4*|a|/ε^3)*‖z-w‖; field_simp; ring
+
+lemma cutoffDrift_bound (a ε : ℝ) (hε : 0 < ε) (v : State) :
+    ‖cutoffDrift a ε v‖ ≤ max (|a|/ε) (|a|/ε^2) := by
+  exact max_le_max (cutoffVector_bound a ε hε v.1) (cutoffDrift_second_bound a ε hε v.1)
+
+lemma cutoffDrift_lipschitz (a ε : ℝ) (hε : 0 < ε) :
+    LipschitzWith (max ⟨2*|a|/ε^2,by positivity⟩ ⟨4*|a|/ε^3,by positivity⟩)
+      (cutoffDrift a ε) := by
+  convert ((cutoffVector_lipschitz a ε hε).prodMk (cutoffDrift_second_lipschitz a ε hε)).comp LipschitzWith.prod_fst using 1 <;> first | rfl | simp only [mul_one]
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEMain
+
+lemma cutoffDrift_eq_physical (a ε : ℝ) (v : State) (hy : ε ≤ v.1.im) :
+    cutoffDrift a ε v = ((a : ℂ)/v.1, (-(a : ℂ)/v.1^2).re) := by
+  simp only [cutoffDrift,cutoffVector,heightClamp_eq hy,Complex.ofReal_one,one_div]
+  congr 1
+  rw [inv_pow,div_eq_mul_inv]
+
+ 
+
+theorem state_eq_loewner_on_safe
+    (a ε : ℝ) (hε : 0 < ε) (b : ℝ≥0 → ℝ) (hb : Continuous b) (_hb0 : b 0 = 0)
+    (z : ℂ) (V : ℝ≥0 → State) (hVc : Continuous V)
+    (hVe : ∀ t, V t = (z,0) + b t • ((-1 : ℂ), (0 : ℝ)) +
+      ∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (V u.toNNReal))
+    (T : ℝ≥0) (hy : ∀ t ≤ T, ε ≤ (V t).1.im) :
+    alive a b T z ∧
+      (∀ t ≤ T, loewnerMap a b t z - (b t : ℂ) = (V t).1) ∧
+      Real.log ‖deriv (loewnerMap a b T) z‖ = (V T).2 := by
+  have hAc := (cutoffDrift_lipschitz a ε hε).continuous
+  have hIc (t : ℝ≥0) : IntervalIntegrable
+      (fun u : ℝ => cutoffDrift a ε (V u.toNNReal)) volume 0 t :=
+    (hAc.comp (hVc.comp continuous_real_toNNReal)).intervalIntegrable 0 t
+  have hfst (t : ℝ≥0) : (V t).1 = z - (b t : ℂ) +
+      ∫ u in (0 : ℝ)..(t : ℝ), cutoffVector a ε (V u.toNNReal).1 := by
+    have h := congrArg Prod.fst (hVe t)
+    have hi := (ContinuousLinearMap.fst ℝ ℂ ℝ).intervalIntegral_comp_comm (hIc t)
+    simp only [Prod.fst_add] at h
+    change (V t).1 = z + b t • (-1 : ℂ) +
+      (ContinuousLinearMap.fst ℝ ℂ ℝ) (∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (V u.toNNReal)) at h
+    rw [← hi] at h
+    simpa only [ContinuousLinearMap.coe_fst',Prod.fst_add,Prod.fst_smul,Complex.real_smul,Complex.ofReal_neg,
+      mul_neg,mul_one,sub_eq_add_neg,cutoffDrift] using h
+  have hsnd (t : ℝ≥0) : (V t).2 =
+      ∫ u in (0 : ℝ)..(t : ℝ), (-(a : ℂ)*(cutoffVector 1 ε (V u.toNNReal).1)^2).re := by
+    have h := congrArg Prod.snd (hVe t)
+    have hi := (ContinuousLinearMap.snd ℝ ℂ ℝ).intervalIntegral_comp_comm (hIc t)
+    simp only [Prod.snd_add] at h
+    change (V t).2 = (0 : ℝ) + b t • (0 : ℝ) +
+      (ContinuousLinearMap.snd ℝ ℂ ℝ) (∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (V u.toNNReal)) at h
+    rw [← hi] at h
+    simpa only [ContinuousLinearMap.coe_snd',Prod.snd_add,Prod.snd_smul,smul_zero,add_zero,zero_add,cutoffDrift] using h
+  let G : ℝ≥0 → ℂ := fun t => (V t).1 + (b t : ℂ)
+  have hGc : Continuous G := hVc.fst.add (Complex.continuous_ofReal.comp hb)
+  have hGe (t : ℝ≥0) : G t = z + ∫ u in (0 : ℝ)..(t : ℝ),
+      cutoffVector a ε (G u.toNNReal + -(b u.toNNReal : ℂ)) := by
+    simp only [G,add_neg_cancel_right]
+    rw [hfst t]
+    abel
+  have hG0 : G 0 = z := by simpa using hGe 0
+  have hode : odeUntil a b z T (fun u : ℝ => G u.toNNReal) := by
+    refine ⟨by simpa using hG0,?_⟩
+    intro u hu
+    have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    have huy := hy _ hut
+    have hd := ExactSLEFlow.integral_flow_hasDerivWithinAt
+      (cutoffVector_lipschitz a ε hε).continuous (Complex.continuous_ofReal.comp hb).neg
+      hGc hGe T u hu
+    have hGim : (G u.toNNReal).im = (V u.toNNReal).1.im := by simp [G]
+    refine ⟨by rw [hGim]; exact hε.trans_le huy,?_⟩
+    simpa only [Function.comp_def,Pi.neg_apply,cutoffVector,G,add_neg_cancel_right,heightClamp_eq huy,sub_eq_add_neg] using hd
+  have hlo (t : ℝ≥0) (ht : t ≤ T) : loewnerMap a b t z - (b t : ℂ) = (V t).1 := by
+    rw [loewnerMap_eq (odeUntil_mono hode ht)]
+    simp only [Real.toNNReal_coe,G,add_sub_cancel_right]
+  refine ⟨⟨_,hode⟩,hlo,?_⟩
+  rw [log_norm_deriv_loewnerMap hb ⟨_,hode⟩, hsnd T]
+  apply intervalIntegral.integral_congr
+  intro u hu
+  have hu' : u ∈ Icc (0 : ℝ) T := by simpa only [uIcc_of_le T.coe_nonneg] using hu
+  have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu'.2).trans_eq (by simp)
+  dsimp only
+  rw [hlo _ hut,cutoffVector,heightClamp_eq (hy _ hut)]
+  simp only [Complex.ofReal_one,inv_pow,div_eq_mul_inv,one_mul]
+
+end ExactSLEGreen
+
+namespace ExactSLEHitting
+variable {Ω : Type*} {m : MeasurableSpace Ω}
+variable (ℱ : Filtration ℝ≥0 m) (f : ℝ≥0 → Ω → ℝ)
+
+lemma firstContact_mem (hc : ∀ ω, Continuous (fun t => f t ω)) (ω : Ω)
+    (hex : ∃ t, f t ω ≤ 0) :
+    f (hittingAfter f (Iic 0) 0 ω).untopA ω ≤ 0 := by
+  have he : ∃ t : ℝ≥0, 0 ≤ t ∧ f t ω ∈ Iic 0 := by
+    obtain ⟨t,ht⟩ := hex
+    exact ⟨t,bot_le,ht⟩
+  rw [hittingAfter,ite_eq_left he]
+  change f (sInf {t : ℝ≥0 | 0 ≤ t ∧ f t ω ≤ 0}) ω ≤ 0
+  have hcl : IsClosed {t : ℝ≥0 | 0 ≤ t ∧ f t ω ≤ 0} :=
+    isClosed_Ici.inter (isClosed_le (hc ω) continuous_const)
+  exact (hcl.csInf_mem he ⟨0,fun _ _ => bot_le⟩).2
+
+lemma firstContact_le_iff (hc : ∀ ω, Continuous (fun t => f t ω))
+    (ω : Ω) (t : ℝ≥0) :
+    hittingAfter f (Iic 0) 0 ω ≤ t ↔ ∃ u ≤ t, f u ω ≤ 0 := by
+  constructor
+  · intro h
+    have hex : ∃ u, f u ω ≤ 0 := by
+      by_contra hn
+      have hz : hittingAfter f (Iic 0) 0 ω = ⊤ := by
+        rw [hittingAfter_eq_top_iff]
+        intro j _ hj
+        exact hn ⟨j,hj⟩
+      rw [hz] at h
+      exact WithTop.not_top_le_coe t h
+    refine ⟨(hittingAfter f (Iic 0) 0 ω).untopA,?_, firstContact_mem f hc ω hex⟩
+    exact WithTop.untopA_le h
+  · rintro ⟨u,hu,hfu⟩
+    exact (hittingAfter_le_of_mem (by exact bot_le) hfu).trans (WithTop.coe_le_coe.mpr hu)
+
+lemma measurable_has_contact (hc : ∀ ω, Continuous (fun t => f t ω))
+    (hm : ∀ t, Measurable[ℱ t] (f t)) (t : ℝ≥0) :
+    MeasurableSet[ℱ t] {ω | ∃ u ≤ t, f u ω ≤ 0} := by
+  let g : Icc (0 : ℝ≥0) t → Ω → ℝ≥0∞ := fun u ω => ENNReal.ofReal (f u ω)
+  have hg : @Measurable Ω ℝ≥0∞ (ℱ t) _ (⨅ u, g u) := by
+    apply measurable_iInf_of_upperSemicontinuous
+    · intro u
+      exact ((hm u).mono (ℱ.mono u.2.2) le_rfl).ennreal_ofReal
+    · intro ω
+      exact (ENNReal.continuous_ofReal.comp ((hc ω).comp continuous_subtype_val)).upperSemicontinuous
+  have he : {ω | ∃ u ≤ t, f u ω ≤ 0} = {ω | (⨅ u, g u) ω = 0} := by
+    ext ω
+    simp only [mem_ofPred_eq,iInf_apply]
+    constructor
+    · rintro ⟨u,hu,hfu⟩
+      apply le_antisymm _ bot_le
+      exact (iInf_le (fun u => g u ω) ⟨u,bot_le,hu⟩).trans (by change ENNReal.ofReal (f u ω) ≤ 0; exact le_of_eq (ENNReal.ofReal_eq_zero.mpr hfu))
+    · intro h
+      obtain ⟨u,hu,hmin⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr (show (0 : ℝ≥0) ≤ t from bot_le)) (hc ω).continuousOn
+      refine ⟨u,hu.2,?_⟩
+      have hl : ENNReal.ofReal (f u ω) ≤ ⨅ v, g v ω := by
+        apply le_iInf
+        intro v
+        exact ENNReal.ofReal_le_ofReal (hmin v.2)
+      rw [h] at hl
+      exact ENNReal.ofReal_eq_zero.mp (le_antisymm hl bot_le)
+  rw [he]
+  exact measurableSet_eq_fun hg measurable_const
+
+lemma firstContact_stopping (hc : ∀ ω, Continuous (fun t => f t ω))
+    (hm : ∀ t, Measurable[ℱ t] (f t)) :
+    IsStoppingTime ℱ (hittingAfter f (Iic 0) 0) := by
+  intro t
+  have he : {ω | hittingAfter f (Iic 0) 0 ω ≤ t} = {ω | ∃ u ≤ t, f u ω ≤ 0} := by
+    ext ω
+    exact firstContact_le_iff f hc ω t
+  rw [he]
+  exact measurable_has_contact ℱ f hc hm t
+
+end ExactSLEHitting
+
+namespace ExactSLEHitting
+variable {Ω : Type*} {m : MeasurableSpace Ω}
+variable (ℱ : Filtration ℝ≥0 m) (f : ℝ≥0 → Ω → ℝ)
+
+lemma boundedContact_le_iff (hc : ∀ ω, Continuous (fun t => f t ω))
+    (ω : Ω) (B t : ℝ≥0) (ht : t < B) :
+    hittingBtwn f (Iic 0) 0 B ω ≤ t ↔ ∃ u ≤ t, f u ω ≤ 0 := by
+  constructor
+  · intro h
+    have hex : ∃ u ∈ Icc (0 : ℝ≥0) B, f u ω ∈ Iic 0 := by
+      by_contra hn
+      have he : hittingBtwn f (Iic 0) 0 B ω = B := by simp only [hittingBtwn,ite_eq_right hn]
+      rw [he] at h
+      exact not_le.mpr ht h
+    have hcl : IsClosed (Icc (0 : ℝ≥0) B ∩ {u | f u ω ∈ Iic 0}) :=
+      isClosed_Icc.inter (isClosed_Iic.preimage (hc ω))
+    have hmem : f (hittingBtwn f (Iic 0) 0 B ω) ω ≤ 0 := by
+      rw [hittingBtwn,ite_eq_left hex]
+      exact (hcl.csInf_mem hex bddBelow_Icc.inter_of_left).2
+    exact ⟨_,h,hmem⟩
+  · rintro ⟨u,hu,hfu⟩
+    exact (hittingBtwn_le_of_mem bot_le (hu.trans ht.le) hfu).trans hu
+
+lemma boundedContact_stopping (hc : ∀ ω, Continuous (fun t => f t ω))
+    (hm : ∀ t, Measurable[ℱ t] (f t)) (B : ℝ≥0) :
+    IsStoppingTime ℱ (fun ω => ((hittingBtwn f (Iic (0 : ℝ)) (0 : ℝ≥0) B ω : ℝ≥0) : WithTop ℝ≥0)) := by
+  intro t
+  by_cases ht : t < B
+  · have he : {ω | ((hittingBtwn f (Iic (0 : ℝ)) (0 : ℝ≥0) B ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} =
+        {ω | ∃ u ≤ t, f u ω ≤ 0} := by
+      ext ω
+      simp only [mem_ofPred_eq,WithTop.coe_le_coe]
+      exact boundedContact_le_iff f hc ω B t ht
+    rw [he]
+    exact measurable_has_contact ℱ f hc hm t
+  · have he : {ω | ((hittingBtwn f (Iic (0 : ℝ)) (0 : ℝ≥0) B ω : ℝ≥0) : WithTop ℝ≥0) ≤ t} = univ := by
+      apply eq_univ_of_forall
+      intro ω
+      exact WithTop.coe_le_coe.mpr ((hittingBtwn_le ω).trans (le_of_not_gt ht))
+    rw [he]
+    exact MeasurableSet.univ
+
+lemma nonneg_before_boundedContact (hc : ∀ ω, Continuous (fun t => f t ω))
+    (h0 : ∀ ω, 0 ≤ f 0 ω) (B : ℝ≥0) (ω : Ω) (t : ℝ≥0)
+    (ht : t ≤ hittingBtwn f (Iic 0) 0 B ω) : 0 ≤ f t ω := by
+  by_cases hz : t = 0
+  · simpa only [hz] using h0 ω
+  · have ht0 : 0 < t := bot_lt_iff_ne_bot.mpr hz
+    have : NeBot (𝓝[<] t) := nhdsWithin_Iio_neBot' ⟨0,ht0⟩ le_rfl
+    apply ge_of_tendsto ((hc ω).continuousAt.tendsto.mono_left (show 𝓝[<] t ≤ 𝓝 t from inf_le_left))
+    filter_upwards [self_mem_nhdsWithin] with u hu
+    have hn : f u ω ∉ Iic 0 := notMem_of_lt_hittingBtwn (hu.trans_le ht) bot_le
+    exact le_of_lt (lt_of_not_ge hn)
+
+lemma boundedContact_eq_zero (hc : ∀ ω, Continuous (fun t => f t ω))
+    (h0 : ∀ ω, 0 ≤ f 0 ω) (B : ℝ≥0) (ω : Ω)
+    (ht : hittingBtwn f (Iic 0) 0 B ω < B) :
+    f (hittingBtwn f (Iic 0) 0 B ω) ω = 0 := by
+  apply le_antisymm _ (nonneg_before_boundedContact f hc h0 B ω _ le_rfl)
+  obtain ⟨u,hu,hfu⟩ := (boundedContact_le_iff f hc ω B _ ht).mp le_rfl
+  have he : hittingBtwn f (Iic 0) 0 B ω = u :=
+    le_antisymm (hittingBtwn_le_of_mem bot_le (hu.trans ht.le) hfu) hu
+  simpa only [he] using hfu
+
+end ExactSLEHitting
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+noncomputable def physicalRadius (v : State) : ℝ := Y v * Real.exp (-J v)
+lemma physicalRadius_continuous : Continuous physicalRadius :=
+  Y.continuous.mul (Real.continuous_exp.comp J.continuous.neg)
+
+lemma alpha_pos {a : ℝ} (ha : 1/4 < a) : 0 < alpha a := by
+  have ha0 : 0 < a := by linarith
+  unfold alpha
+  rw [sub_pos,div_lt_one (by positivity)]
+  linarith
+
+lemma physicalGreen_bound (a ρ : ℝ) (ha : 1/4 < a) (hρ : 0 < ρ)
+    (v : State) (hy : 0 < Y v) (hr : ρ ≤ physicalRadius v) :
+    physicalGreen a v ≤ Real.exp (-alpha a * Real.log ρ) := by
+  have hq := Q_pos hy
+  have hlq : 2*Real.log (Y v) ≤ Real.log (Q v) := by
+    have he := Real.log_pow (Y v) 2
+    norm_num only [Nat.cast_ofNat] at he
+    rw [← he]
+    apply Real.log_le_log (by positivity)
+    dsimp only [Q]
+    nlinarith [sq_nonneg (X v)]
+  have hlr : Real.log ρ ≤ Real.log (Y v) - J v := by
+    have h := Real.log_le_log hρ hr
+    simpa only [physicalRadius,Real.log_mul hy.ne' (Real.exp_ne_zero _),Real.log_exp,sub_eq_add_neg] using h
+  have hp : 0 ≤ p a / 2 := by unfold p; linarith
+  have ha' := alpha_pos ha
+  apply Real.exp_le_exp.mpr
+  unfold logGreen
+  nlinarith [mul_le_mul_of_nonneg_left hlq hp,mul_le_mul_of_nonneg_left hlr ha'.le]
+
+lemma physicalGreen_eq_radius_sine (a : ℝ) (v : State) (hy : 0 < Y v) :
+    physicalGreen a v = (physicalRadius v) ^ (-alpha a) * (Y v / ‖v.1‖) ^ (p a) := by
+  have hzn : 0 < ‖v.1‖ := lt_of_lt_of_le hy (Complex.im_le_norm _)
+  have hr : 0 < physicalRadius v := mul_pos hy (Real.exp_pos _)
+  have hqlog : Real.log (Q v) = 2*Real.log ‖v.1‖ := by
+    rw [Q_eq,Complex.normSq_eq_norm_sq,Real.log_pow]
+    norm_num
+  rw [Real.rpow_def_of_pos hr,Real.rpow_def_of_pos (div_pos hy hzn),← Real.exp_add]
+  rw [physicalRadius,Real.log_mul hy.ne' (Real.exp_ne_zero _),Real.log_exp,
+    Real.log_div hy.ne' hzn.ne']
+  simp only [physicalGreen,logGreen] at *
+  rw [hqlog]
+  congr 1
+  ring
+
+ 
+lemma exists_localState (a ε : ℝ) (hε : 0 < ε) (z : ℂ) :
+    ∃ V : ℝ≥0 → ExactSLEGenerator.Driver → State,
+      (∀ b, Continuous (fun t => V t b)) ∧
+      (∀ t, Measurable[rawFiltration t] (V t)) ∧
+      (∀ b, V 0 b = (z,0)) ∧
+      (∀ (s h : ℝ≥0) b, V (s+h) b = V s b + increment s h b • noise +
+        ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), cutoffDrift a ε (V t.toNNReal b)) := by
+  exact exists_adapted_additive_brownian (cutoffDrift a ε)
+    ⟨max (|a|/ε) (|a|/ε^2),by positivity⟩ _
+    (cutoffDrift_lipschitz a ε hε) (cutoffDrift_bound a ε hε) (z,0) noise
+
+noncomputable def localState (a ε : ℝ) (hε : 0 < ε) (z : ℂ) : ℝ≥0 → ExactSLEGenerator.Driver → State :=
+  Classical.choose (exists_localState a ε hε z)
+lemma localState_continuous (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (b : ExactSLEGenerator.Driver) :
+    Continuous (fun t => localState a ε hε z t b) :=
+  (Classical.choose_spec (exists_localState a ε hε z)).1 b
+lemma localState_adapted (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (t : ℝ≥0) :
+    Measurable[rawFiltration t] (localState a ε hε z t) :=
+  (Classical.choose_spec (exists_localState a ε hε z)).2.1 t
+lemma localState_zero (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (b : ExactSLEGenerator.Driver) :
+    localState a ε hε z 0 b = (z,0) :=
+  (Classical.choose_spec (exists_localState a ε hε z)).2.2.1 b
+lemma localState_eq (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (s h : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    localState a ε hε z (s+h) b = localState a ε hε z s b + increment s h b • noise +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), cutoffDrift a ε (localState a ε hε z t.toNNReal b) :=
+  (Classical.choose_spec (exists_localState a ε hε z)).2.2.2 s h b
+lemma localState_eq_zero (a ε : ℝ) (hε : 0 < ε) (z : ℂ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    localState a ε hε z t b = (z,0) + b.1 t • noise +
+      ∫ u in (0 : ℝ)..(t : ℝ), cutoffDrift a ε (localState a ε hε z u.toNNReal b) := by
+  simpa only [zero_add,localState_zero,increment,b.2,sub_zero,NNReal.coe_zero]
+    using localState_eq a ε hε z 0 t b
+
+ 
+
+theorem localGreen_optional (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a ε : ℝ) (ha : a ≠ 0) (hε : 0 < ε) (z : ℂ) (N B : ℝ≥0)
+    (S T : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
+    (hsafe : ∀ b t, t ≤ T b → ε ≤ Y (localState a ε hε z t b) ∧
+      ‖localState a ε hε z t b‖ ≤ N)
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    ∫ b in U, physicalGreen a (localState a ε hε z (S b) b) ∂W =
+      ∫ b in U, physicalGreen a (localState a ε hε z (T b) b) ∂W := by
+  have hf := greenCutoff_smooth a ε hε N
+  have hG (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ T b) :
+      generator (greenCutoff a ε N) (cutoffDrift a ε) noise (localState a ε hε z t b) = 0 := by
+    have hv := hsafe b t ht
+    have hg := greenCutoff_generator_zero a ε ha hε N (localState a ε hε z t b) hv.2 hv.1
+    change (1/2 : ℝ) * fderiv ℝ (fun w => fderiv ℝ (greenCutoff a ε N) w noise)
+      (localState a ε hε z t b) noise + fderiv ℝ (greenCutoff a ε N)
+      (localState a ε hε z t b) (cutoffDrift a ε (localState a ε hε z t b)) = 0
+    rw [cutoffDrift_eq_physical a ε _ hv.1]
+    exact hg
+  have ho := compact_test_optional_sampling W hW (localState a ε hε z)
+    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
+    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
+    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
+    (by simp [noise,Prod.norm_def]) (localState_eq a ε hε z)
+    (greenCutoff a ε N) hf (greenCutoff_compact a ε N) S T B hS hT hST hTB hG U hU
+  have he (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ T b) :
+      greenCutoff a ε N (localState a ε hε z t b) = physicalGreen a (localState a ε hε z t b) :=
+    (greenCutoff_eq_near a ε hε N _ (hsafe b t ht).2 (hsafe b t ht).1).eq_of_nhds
+  simpa only [he _ _ (hST _),he _ _ le_rfl] using ho
+
+end ExactSLEGreen
+
+namespace ExactSLEBarrier
+
+ 
+
+structure Evolution (a : ℝ) (T : ℝ≥0) (y j q : ℝ → ℝ) : Prop where
+  yc : ContinuousOn y (Icc 0 (T : ℝ))
+  jc : ContinuousOn j (Icc 0 (T : ℝ))
+  yp : ∀ t ∈ Icc 0 (T : ℝ), 0 < y t
+  qp : ∀ t ∈ Icc 0 (T : ℝ), 0 < q t
+  yd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt y (-a*y t/q t) (Icc 0 (T : ℝ)) t
+  jd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt j
+    (-a*(q t-2*(y t)^2)/(q t)^2) (Icc 0 (T : ℝ)) t
+
+variable {a : ℝ} {T : ℝ≥0} {y j q : ℝ → ℝ}
+
+lemma Evolution.logHeight_deriv (h : Evolution a T y j q) {t : ℝ}
+    (ht : t ∈ Icc 0 (T : ℝ)) :
+    HasDerivWithinAt (fun t => Real.log (y t)) (-a/q t) (Icc 0 (T : ℝ)) t := by
+  convert (h.yd t ht).log (h.yp t ht).ne' using 1
+  field_simp [(h.yp t ht).ne']
+
+lemma Evolution.logRadius_deriv (h : Evolution a T y j q) {t : ℝ}
+    (ht : t ∈ Icc 0 (T : ℝ)) :
+    HasDerivWithinAt (fun t => Real.log (y t)-j t)
+      (-2*a*(y t)^2/(q t)^2) (Icc 0 (T : ℝ)) t := by
+  convert (h.logHeight_deriv ht).sub (h.jd t ht) using 1
+  try rfl
+  field_simp [(h.qp t ht).ne']
+  ring
+
+lemma Evolution.height_le (h : Evolution a T y j q) (ha : 0 ≤ a)
+    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : y t ≤ y 0 := by
+  have hm : AntitoneOn y (Icc 0 (T : ℝ)) :=
+    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) h.yc
+      (fun u hu => (h.yd u (interior_subset hu)).mono interior_subset)
+      (fun u hu => div_nonpos_of_nonpos_of_nonneg (by nlinarith [h.yp u (interior_subset hu)])
+        (h.qp u (interior_subset hu)).le)
+  exact hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
+
+end ExactSLEBarrier
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CutoffExhaustion.lean b/lean/OAI/Probability/SLEGauge/CutoffExhaustion.lean
index e708c94aeae60a230d09942b006e9f7eb80c0cbe..c97059c1bae5ee98b6c18b81f2a10cc93c454c9e 100644
--- a/lean/OAI/Probability/SLEGauge/CutoffExhaustion.lean
+++ b/lean/OAI/Probability/SLEGauge/CutoffExhaustion.lean
@@ -1,464 +1,464 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadiusComparison
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEStops
-open scoped Classical
-lemma unfinishedCutoff_exhausts {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
-    (b : ExactSLEGenerator.Driver) (hb : ∀ i, radialTime a b.1 (z i) (ρ i)<∞) :
-    ∀ᶠ n in Filter.atTop, ∀ i, radialTime a b.1 (z i) (ρ i) <
-      (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞) := by
-  have he : ∀ᶠ n in atTop, ∀ i, (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=
-      radialTime a b.1 (z i) (ρ i) := by
-    rw [Filter.eventually_all]
-    intro i
-    exact radialStop_eventually_eq a ha (z i) (hz i) (ρ i) (hρ i) (hr i) b (hb i).ne
-  have hn : ∀ᶠ n : ℕ in atTop, ∀ i, radialTime a b.1 (z i) (ρ i)<(n:ℝ≥0∞) := by
-    rw [Filter.eventually_all]
-    intro i
-    obtain ⟨N,hN⟩ := ENNReal.exists_nat_gt (hb i).ne
-    filter_upwards [Filter.eventually_ge_atTop N] with n hn
-    exact hN.trans_le (by exact_mod_cast hn)
-  filter_upwards [he,hn] with n he hn i
-  rw [unfinishedCutoff_coe,unfinishedCutoffTop]
-  have hi : (Finset.univ.inf fun j =>
-      if (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞)=radialTime a b.1 (z j) (ρ j)
-      then ∞ else (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞))=∞ := by
-    apply top_unique
-    apply Finset.le_inf
-    intro j hj
-    rw [ite_eq_left (he j)]
-  rw [hi,min_top_right]
-  exact hn i
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
- 
-
-theorem actual_unfinished_cutoff {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im) :
-    (∀ n, IsStoppingTime rawFiltration
-      (fun b => (unfinishedCutoff a ha z hz ρ hρ hr n b : WithTop ℝ≥0))) ∧
-    (∀ n b t i, t≤unfinishedCutoff a ha z hz ρ hρ hr n b →
-      (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i) → alive a b.1 t (z i)) ∧
-    (∀ b, (∀ i, radialTime a b.1 (z i) (ρ i)<∞) →
-      ∀ᶠ n in Filter.atTop, ∀ i, radialTime a b.1 (z i) (ρ i) <
-        (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)) := by
-  refine ⟨unfinishedCutoff_stopping a ha z hz ρ hρ hr,?_,unfinishedCutoff_exhausts a ha z hz ρ hρ hr⟩
-  intro n b t i ht hti
-  exact radialStop_actual a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b t
-    (unfinishedCutoff_before_individual a ha z hz ρ hρ hr n b t i ht hti)
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEFlow
-open Set Filter
-lemma localState_noiseFree_deriv (a ε : ℝ) (hε : 0<ε) (z : ℂ)
-    (b : ExactSLEGenerator.Driver) (T : ℝ≥0) (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
-    HasDerivWithinAt (fun v : ℝ => localState a ε hε z v.toNNReal b-b.1 v.toNNReal • noise)
-      (cutoffDrift a ε (localState a ε hε z u.toNNReal b)) (Icc (0:ℝ) T) u := by
-  have hc : Continuous (fun v : ℝ => cutoffDrift a ε (localState a ε hε z v.toNNReal b)) :=
-    (cutoffDrift_lipschitz a ε hε).continuous.comp
-      ((localState_continuous a ε hε z b).comp continuous_real_toNNReal)
-  have hd := (intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 u)
-    hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt).const_add ((z,0):State)
-  apply hd.hasDerivWithinAt.congr_of_mem _ hu
-  intro v hv
-  rw [localState_eq_zero,Real.coe_toNNReal _ hv.1]
-  abel
-lemma localState_lower_cutoff (a η ε : ℝ) (hη : 0<η) (hε : 0<ε) (hηε : η≤ε)
-    (z : ℂ) (b : ExactSLEGenerator.Driver) (T : ℝ≥0)
-    (hy : ∀ t≤T, ε≤Y (localState a ε hε z t b)) (t : ℝ≥0) (ht : t≤T) :
-    localState a η hη z t b=localState a ε hε z t b := by
-  let f : ℝ → State := fun u => localState a η hη z u.toNNReal b-b.1 u.toNNReal • noise
-  let g : ℝ → State := fun u => localState a ε hε z u.toNNReal b-b.1 u.toNNReal • noise
-  let u : ℝ → State := fun v => b.1 v.toNNReal • noise
-  have hf : Continuous f := ((localState_continuous a η hη z b).comp continuous_real_toNNReal).sub
-    ((b.1.continuous.comp continuous_real_toNNReal).smul continuous_const)
-  have hg : Continuous g := ((localState_continuous a ε hε z b).comp continuous_real_toNNReal).sub
-    ((b.1.continuous.comp continuous_real_toNNReal).smul continuous_const)
-  have hfd (v : ℝ) (hv : v∈Icc (0:ℝ) T) :
-      HasDerivWithinAt f (cutoffDrift a η (f v+u v)) (Icc (0:ℝ) T) v := by
-    simpa only [f,u,sub_add_cancel] using localState_noiseFree_deriv a η hη z b T v hv
-  have hgd (v : ℝ) (hv : v∈Icc (0:ℝ) T) :
-      HasDerivWithinAt g (cutoffDrift a η (g v+u v)) (Icc (0:ℝ) T) v := by
-    have hvT : v.toNNReal≤T := (Real.toNNReal_le_toNNReal hv.2).trans_eq (by simp)
-    have he : cutoffDrift a ε (localState a ε hε z v.toNNReal b)=
-        cutoffDrift a η (localState a ε hε z v.toNNReal b) := by
-      rw [cutoffDrift_eq_physical a ε _ (hy _ hvT),cutoffDrift_eq_physical a η _ (hηε.trans (hy _ hvT))]
-    simpa only [g,u,sub_add_cancel,←he] using localState_noiseFree_deriv a ε hε z b T v hv
-  have he := additive_flow_unique (cutoffDrift_lipschitz a η hη) hf.continuousOn hg.continuousOn
-    hfd hgd (by simp [f,g,localState_zero]) (show (t:ℝ)∈Icc (0:ℝ) T from ⟨t.coe_nonneg,ht⟩)
-  simpa only [f,g,Real.toNNReal_coe,sub_left_inj] using he
-lemma localState_noise_bounds (a η : ℝ) (hη : 0<η) (z : ℂ)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    ‖localState a η hη z t b‖≤‖z‖+|b.1 t|+max (|a|/η) (|a|/η^2)*t ∧
-    |b.1 t|≤‖localState a η hη z t b‖+‖z‖+max (|a|/η) (|a|/η^2)*t := by
-  let K := max (|a|/η) (|a|/η^2)
-  let I : State := ∫ v in (0:ℝ)..(t:ℝ), cutoffDrift a η (localState a η hη z v.toNNReal b)
-  have hI : ‖I‖≤K*t := by
-    simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
-      intervalIntegral.norm_integral_le_of_norm_le_const (a:=(0:ℝ)) (b:=(t:ℝ))
-        (C:=K) (fun v _ => cutoffDrift_bound a η hη _)
-  have hn : ‖noise‖=1 := by simp [noise,Prod.norm_def]
-  have hnoise : ‖b.1 t • noise‖=|b.1 t| := by rw [norm_smul,hn,mul_one,Real.norm_eq_abs]
-  have hz : ‖((z,0):State)‖=‖z‖ := by simp [Prod.norm_def]
-  have he : localState a η hη z t b=(z,0)+b.1 t • noise+I := localState_eq_zero a η hη z t b
-  constructor
-  · rw [he]
-    calc
-      _ ≤ ‖((z,0):State)+b.1 t • noise‖+‖I‖ := norm_add_le _ _
-      _ ≤ ‖((z,0):State)‖+‖b.1 t • noise‖+K*t := add_le_add (norm_add_le _ _) hI
-      _ = _ := by rw [hz,hnoise]
-  · have he' : b.1 t • noise=localState a η hη z t b-(z,0)-I := by rw [he]; abel
-    rw [←hnoise,he']
-    calc
-      _ ≤ ‖localState a η hη z t b-(z,0)‖+‖I‖ := norm_sub_le _ _
-      _ ≤ ‖localState a η hη z t b‖+‖((z,0):State)‖+K*t := add_le_add (norm_sub_le _ _) hI
-      _ = _ := by rw [hz]
-lemma finite_pos_lower {ι : Type*} (s : Finset ι) (f : ι → ℝ) (hf : ∀ i∈s,0<f i) :
-    ∃ η : ℝ, 0<η ∧ ∀ i∈s, η≤f i := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => exact ⟨1,by norm_num,by simp⟩
-  | @insert i s hi ih =>
-    obtain ⟨η,hη,hηs⟩ := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
-    refine ⟨min (f i) η,lt_min (hf i (Finset.mem_insert_self _ _)) hη,?_⟩
-    intro j hj
-    rcases Finset.mem_insert.mp hj with rfl|hj
-    · exact min_le_left _ _
-    · exact (min_le_right _ _).trans (hηs j hj)
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
- 
-
-theorem actual_common_cutoff_safe {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im) (n : ℕ) :
-    ∃ η : ℝ, ∃ hη : 0<η, ∃ M : ℝ≥0, ∀ (b : ExactSLEGenerator.Driver) (t : ℝ≥0),
-      t≤unfinishedCutoff a ha z hz ρ hρ hr n b →
-      (∃ i, (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i)) →
-      ‖vectorState a η hη z t b‖≤M ∧
-      ∀ i, (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i) →
-        η≤Y (vectorState a η hη z t b i) ∧
-        vectorState a η hη z t b i=actualState a b.1 t (z i) := by
-  let L := fun i => radialLocalization a ha (z i) (hz i) (ρ i) (angleThreshold (z i) n)
-    (hρ i) (hr i) (angleThreshold_pos (z i) (hz i) n) (angleThreshold_initial (z i) (hz i) n)
-  obtain ⟨η,hη,hηi⟩ := finite_pos_lower Finset.univ (fun i => (L i).ε) (fun i _ => (L i).hε)
-  let N : ℝ≥0 := Finset.univ.sup (fun i => (L i).N)
-  let K : ℝ := max (|a|/η) (|a|/η^2)
-  have hK : 0≤K := le_trans (by positivity : 0≤|a|/η) (le_max_left _ _)
-  let M : ℝ≥0 := ⟨N+2*‖z‖+2*K*n,by positivity⟩
-  refine ⟨η,hη,M,?_⟩
-  intro b t ht hex
-  have htN : (t:ℝ)≤n := by
-    have hcut : (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)≤(n:ℝ≥0∞) := by
-      rw [unfinishedCutoff_coe]
-      exact unfinishedCutoffTop_le a ha z hz ρ hρ hr n b
-    have h : (t:ℝ≥0∞)≤(n:ℝ≥0∞) := (ENNReal.coe_le_coe.mpr ht).trans hcut
-    exact_mod_cast h
-  have hstate (i : Fin m) (hi : (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i)) :
-      η≤Y (vectorState a η hη z t b i) ∧
-      vectorState a η hη z t b i=actualState a b.1 t (z i) ∧
-      ‖vectorState a η hη z t b i‖≤(L i).N := by
-    have hti := unfinishedCutoff_before_individual a ha z hz ρ hρ hr n b t i ht hi
-    have he : localState a η hη (z i) t b=localState a (L i).ε (L i).hε (z i) t b :=
-      localState_lower_cutoff a η (L i).ε hη (L i).hε (hηi i (Finset.mem_univ i)) (z i) b
-        (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b)
-        (fun u hu => (regionStop_safe a (L i).ε (L i).hε (z i) (L i).N (ρ i)
-          (angleThreshold (z i) n) (L i).B (L i).initial b u hu).1) t hti
-    have hs := regionStop_safe a (L i).ε (L i).hε (z i) (L i).N (ρ i)
-      (angleThreshold (z i) n) (L i).B (L i).initial b t hti
-    have ha' := regionStop_actualState a (L i).ε (L i).hε (z i) (L i).N (ρ i)
-      (angleThreshold (z i) n) (L i).B (L i).initial b t hti
-    change η≤Y (localState a η hη (z i) t b) ∧
-      localState a η hη (z i) t b=actualState a b.1 t (z i) ∧ _
-    rw [he]
-    exact ⟨(hηi i (Finset.mem_univ i)).trans hs.1,ha'.2.symm,by simpa only [vectorState,he] using hs.2.1⟩
-  obtain ⟨j,hj⟩ := hex
-  have hjN : ((L j).N:ℝ)≤N := by exact_mod_cast Finset.le_sup (f:=fun i => (L i).N) (Finset.mem_univ j)
-  have hbn : |b.1 t|≤N+‖z‖+K*n := by
-    have h := (localState_noise_bounds a η hη (z j) b t).2
-    have hs := (hstate j hj).2.2.trans hjN
-    have hzj := norm_le_pi_norm z j
-    have hkt := mul_le_mul_of_nonneg_left htN hK
-    change ‖localState a η hη (z j) t b‖≤(N:ℝ) at hs
-    dsimp only [K] at *
-    linarith
-  constructor
-  · apply (pi_norm_le_iff_of_nonneg M.coe_nonneg).mpr
-    intro i
-    have h := (localState_noise_bounds a η hη (z i) b t).1
-    have hzj := norm_le_pi_norm z i
-    have hkt := mul_le_mul_of_nonneg_left htN hK
-    change ‖localState a η hη (z i) t b‖≤N+2*‖z‖+2*K*n
-    dsimp only [K] at *
-    linarith
-  · intro i hi
-    exact ⟨(hstate i hi).1,(hstate i hi).2.1⟩
-end ExactSLEGreen
-namespace ExactSLEStops
-open Set Filter
-open scoped Classical
-variable {Ω : Type*} [MeasurableSpace Ω]
-noncomputable def clip (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) : ℝ≥0 :=
-  (min (τ ω) (B ω : ℝ≥0∞)).toNNReal
-omit [MeasurableSpace Ω] in
-lemma clip_coe (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) :
-    (clip τ B ω : ℝ≥0∞)=min (τ ω) (B ω : ℝ≥0∞) :=
-  ENNReal.coe_toNNReal (ne_top_of_le_ne_top (by simp) (min_le_right _ _))
-omit [MeasurableSpace Ω] in
-lemma clip_le (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) : clip τ B ω≤B ω := by
-  apply ENNReal.coe_le_coe.mp
-  rw [clip_coe]
-  exact min_le_right _ _
-omit [MeasurableSpace Ω] in
-lemma le_clip_iff (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) (t : ℝ≥0) :
-    t≤clip τ B ω ↔ (t:ℝ≥0∞)≤τ ω ∧ t≤B ω := by
-  rw [←ENNReal.coe_le_coe,clip_coe,le_min_iff,ENNReal.coe_le_coe]
-omit [MeasurableSpace Ω] in
-lemma clip_le_time (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) :
-    (clip τ B ω : ℝ≥0∞)≤τ ω := by rw [clip_coe]; exact min_le_left _ _
-lemma clip_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (hτ : IsStoppingTime F τ)
-    (hB : IsStoppingTime F (fun ω => (B ω : WithTop ℝ≥0))) :
-    IsStoppingTime F (fun ω => (clip τ B ω : WithTop ℝ≥0)) := by
-  have h := hτ.min hB
-  intro t
-  have ht := h t
-  change MeasurableSet[F t] {ω | min (τ ω) (B ω : ℝ≥0∞)≤(t:ℝ≥0∞)} at ht
-  change MeasurableSet[F t] {ω | (clip τ B ω : ℝ≥0∞)≤(t:ℝ≥0∞)}
-  simpa only [clip_coe] using ht
- 
-noncomputable def phase (A B C : ℝ≥0∞) (t : ℝ≥0) : Fin 4 :=
-  if C≤(t:ℝ≥0∞) then 3 else if B≤(t:ℝ≥0∞) then 2 else if A≤(t:ℝ≥0∞) then 1 else 0
-lemma phase_measurable (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (A B C : Ω → ℝ≥0∞) (S : Ω → ℝ≥0)
-    (hA : IsStoppingTime F A) (hB : IsStoppingTime F B) (hC : IsStoppingTime F C)
-    (hS : IsStoppingTime F (fun ω => (S ω : WithTop ℝ≥0))) :
-    Measurable[hS.measurableSpace] (fun ω => phase (A ω) (B ω) (C ω) (S ω)) := by
-  have hA' : MeasurableSet[hS.measurableSpace] {ω | A ω≤(S ω : ℝ≥0∞)} := hA.measurableSet_stopping_time_le hS
-  have hB' : MeasurableSet[hS.measurableSpace] {ω | B ω≤(S ω : ℝ≥0∞)} := hB.measurableSet_stopping_time_le hS
-  have hC' : MeasurableSet[hS.measurableSpace] {ω | C ω≤(S ω : ℝ≥0∞)} := hC.measurableSet_stopping_time_le hS
-  exact measurable_const.ite hC' (measurable_const.ite hB' (measurable_const.ite hA' measurable_const))
-lemma phase_one_iff (A B C : ℝ≥0∞) (t : ℝ≥0) (_hAB : A≤B) (hBC : B≤C) :
-    phase A B C t=1 ↔ A≤(t:ℝ≥0∞) ∧ (t:ℝ≥0∞)<B := by
-  unfold phase
-  split_ifs with hC hB hA <;> simp_all
-  order
-lemma phase_two_iff (A B C : ℝ≥0∞) (t : ℝ≥0) (_hAB : A≤B) (_hBC : B≤C) :
-    phase A B C t=2 ↔ B≤(t:ℝ≥0∞) ∧ (t:ℝ≥0∞)<C := by
-  unfold phase
-  split_ifs with hC hB hA <;> simp_all
-lemma phase_zero_iff (A B C : ℝ≥0∞) (t : ℝ≥0) (hAB : A≤B) (hBC : B≤C) :
-    phase A B C t=0 ↔ (t:ℝ≥0∞)<A := by
-  unfold phase
-  split_ifs with hC hB hA <;> simp_all <;> order
-lemma phase_three_iff (A B C : ℝ≥0∞) (t : ℝ≥0) :
-    phase A B C t=3 ↔ C≤(t:ℝ≥0∞) := by
-  unfold phase
-  split_ifs <;> simp_all
- 
-omit [MeasurableSpace Ω] in
-lemma no_actual_change_between (l : List (Ω → ℝ≥0)) (B : Ω → ℝ≥0) (τ : Ω → ℝ≥0∞)
-    (hτ : clip τ B∈l) (ω : Ω) (k : ℕ) (hk : k+1<(sort l).length)
-    (hB : (sort l)[k+1] ω≤B ω) :
-    τ ω≤((sort l)[k] ω : ℝ≥0∞) ∨ ((sort l)[k+1] ω : ℝ≥0∞)≤τ ω := by
-  rcases no_change_between l ω k hk (clip τ B) hτ with h|h
-  · by_cases ht : τ ω≤((sort l)[k] ω : ℝ≥0∞)
-    · exact Or.inl ht
-    · right
-      have hs : B ω≤(sort l)[k] ω := by
-        have h' := ENNReal.coe_le_coe.mpr h
-        rw [clip_coe,min_le_iff] at h'
-        exact ENNReal.coe_le_coe.mp (h'.resolve_left ht)
-      exact (ENNReal.coe_le_coe.mpr (hB.trans hs)).trans (not_le.mp ht).le
-  · exact Or.inr ((le_clip_iff τ B ω _).mp h).1
-end ExactSLEStops
-
- 
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt Set Filter
-open scoped ENNReal NNReal
-lemma radialTime_antitone (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ σ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im) (b : ExactSLEGenerator.Driver) :
-    radialTime a b.1 z σ≤radialTime a b.1 z ρ := by
-  rcases hρσ.eq_or_lt with rfl|h
-  · exact le_rfl
-  · by_cases ht : radialTime a b.1 z ρ=∞
-    · rw [ht]; exact le_top
-    · exact (radialTime_strict_of_finite a ha z hz ρ σ hρ h hσ b ht).le
-lemma actual_radius_before_target (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (ht : alive a b.1 t z) (htr : (t:ℝ≥0∞)≤radialTime a b.1 z ρ) :
-    ρ≤physicalRadius (actualState a b.1 t z) := by
-  rcases htr.eq_or_lt with he|he
-  · have hn : radialTime a b.1 z ρ≠∞ := by rw [←he]; simp
-    have h := (radialTime_attained a ha z hz ρ hρ hr b hn).2
-    rw [←he,ENNReal.toNNReal_coe] at h
-    exact h.ge
-  · obtain ⟨n,hn⟩ := (radialStop_after_survival a ha z hz ρ hρ hr b t ht he).exists
-    let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-      (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-    have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t hn.le
-    have heq := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t hn.le
-    rw [heq.2]
-    exact hs.2.2.1
-lemma actual_physicalClock_le (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap ρ : ℝ) (hcap : 0<cap) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (ht : alive a b.1 t z) (htr : (t:ℝ≥0∞)≤radialTime a b.1 z ρ) :
-    physicalClock a cap (actualState a b.1 t z)≤Real.log (cap/(2*ρ))/(2*a) := by
-  have hy : 0<Y (actualState a b.1 t z) := by
-    obtain ⟨f,hf⟩ := ht
-    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq hf]
-      using (hf.2 t ⟨t.coe_nonneg,le_rfl⟩).1
-  have hR := actual_radius_before_target a ha z hz ρ hρ hr b t ht htr
-  have hR0 : 0<physicalRadius (actualState a b.1 t z) := hρ.trans_le hR
-  rw [physicalClock_radius a cap hcap _ hy]
-  apply (div_le_div_iff_of_pos_right (by positivity : 0<2*a)).mpr
-  apply Real.log_le_log (by positivity)
-  exact div_le_div_of_nonneg_left hcap.le (by positivity) (by linarith)
-lemma actual_physicalClock_at (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap ρ : ℝ) (hcap : 0<cap) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver)
-    (hf : radialTime a b.1 z ρ≠∞) :
-    physicalClock a cap (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z)=
-      Real.log (cap/(2*ρ))/(2*a) := by
-  have h := radialTime_attained a ha z hz ρ hρ hr b hf
-  have hy : 0<Y (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z) := by
-    obtain ⟨f,hf⟩ := h.1
-    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq hf]
-      using (hf.2 _ ⟨(radialTime a b.1 z ρ).toNNReal.coe_nonneg,le_rfl⟩).1
-  rw [physicalClock_radius a cap hcap _ hy,h.2]
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt Set Filter
-open scoped ENNReal NNReal
-noncomputable def levelRadius (a cap u : ℝ) : ℝ := cap/2*Real.exp (-2*a*u)
-noncomputable def levelTime (a : ℝ) (b : ExactSLEGenerator.Driver) (z : ℂ) (cap u : ℝ) : ℝ≥0∞ :=
-  radialTime a b.1 z (levelRadius a cap u)
-lemma levelRadius_pos (a cap u : ℝ) (hc : 0<cap) : 0<levelRadius a cap u := by
-  unfold levelRadius; positivity
-lemma levelRadius_le {a cap u : ℝ} (ha : 0<a) (hc : 0<cap) (hu : 0≤u) :
-    levelRadius a cap u≤cap/2 := by
-  have h := Real.exp_le_one_iff.mpr (show -2*a*u≤0 by nlinarith)
-  simpa only [levelRadius,mul_one] using mul_le_mul_of_nonneg_left h (show 0≤cap/2 by positivity)
-lemma levelRadius_antitone {a cap : ℝ} (ha : 0<a) (hc : 0<cap) :
-    Antitone (levelRadius a cap) := by
-  intro u v huv
-  apply mul_le_mul_of_nonneg_left _ (show 0≤cap/2 by positivity)
-  exact Real.exp_le_exp.mpr (by nlinarith)
-lemma levelTime_mono (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (b : ExactSLEGenerator.Driver)
-    (u v : ℝ) (hu : 0≤u) (huv : u≤v) :
-    levelTime a b z cap u≤levelTime a b z cap v := by
-  apply radialTime_antitone a ha z hz _ _ (levelRadius_pos a cap v hc)
-    (levelRadius_antitone ha hc huv)
-  exact (levelRadius_le ha hc hu).trans (by linarith)
-lemma levelRadius_clock {a cap u : ℝ} (ha : a≠0) (hc : 0<cap) :
-    Real.log (cap/(2*levelRadius a cap u))/(2*a)=u := by
-  have h : cap/(2*levelRadius a cap u)=Real.exp (2*a*u) := by
-    unfold levelRadius
-    rw [show 2*(cap/2*Real.exp (-2*a*u))=cap*Real.exp (-(2*a*u)) by ring_nf,
-      Real.exp_neg,div_mul_eq_div_div,div_self hc.ne',one_div,inv_inv]
-  rw [h,Real.log_exp]
-  field_simp
-lemma levelTime_clock (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
-    (hf : levelTime a b z cap u≠∞) :
-    physicalClock a cap (actualState a b.1 (levelTime a b z cap u).toNNReal z)=u := by
-  unfold levelTime
-  rw [actual_physicalClock_at a ha z hz cap _ hc (levelRadius_pos a cap u hc)
-    ((levelRadius_le ha hc hu).trans (by linarith)) b hf,levelRadius_clock ha.ne' hc]
-lemma levelTime_fullRadius (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (b : ExactSLEGenerator.Driver)
-    (hf : levelTime a b z cap 0≠∞) :
-    fullLoewnerRadius a b.1 z (levelTime a b z cap 0).toNNReal=cap := by
-  have h := radialTime_attained a ha z hz _ (levelRadius_pos a cap 0 hc)
-    ((levelRadius_le ha hc le_rfl).trans (by linarith)) b hf
-  unfold levelTime
-  rw [←actual_radius_eq_full b.1.continuous h.1,h.2]
-  simp [levelRadius]
-  ring
-lemma actual_physicalClock_mono (a : ℝ) (ha : 0<a) (cap : ℝ) (_hc : 0<cap)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (S T : ℝ≥0) (hST : S≤T) (hT : alive a b.1 T z) :
-    physicalClock a cap (actualState a b.1 S z)≤physicalClock a cap (actualState a b.1 T z) := by
-  obtain ⟨f,hf⟩ := hT
-  have hS : alive a b.1 S z := ⟨f,odeUntil_mono hf hST⟩
-  have hy : 0<Y (actualState a b.1 S z) := by
-    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq (odeUntil_mono hf hST)]
-      using (hf.2 S ⟨S.coe_nonneg,by exact_mod_cast hST⟩).1
-  have hR : 0<fullLoewnerRadius a b.1 z S := by
-    rw [←actual_radius_eq_full b.1.continuous hS]
-    unfold physicalRadius
-    positivity
-  have heT := actual_clock_eq ha.ne' hR b.1.continuous hf hST le_rfl rfl
-  have heS := actual_clock_eq ha.ne' hR b.1.continuous hf hST hST rfl
-  have hS0 : activeODEClock b.1 T S f S=0 := by simp [activeODEClock]
-  have hu := activeODEClock_nonneg (b:=b.1) (T:=T) (σ:=S) (f:=f) (show (S:ℝ)≤T from hST)
-  rw [hS0] at heS
-  have he : physicalClock a cap (actualState a b.1 T z)-physicalClock a cap (actualState a b.1 S z)=
-      physicalClock a (fullLoewnerRadius a b.1 z S) (actualState a b.1 T z)-
-      physicalClock a (fullLoewnerRadius a b.1 z S) (actualState a b.1 S z) := by
-    unfold physicalClock
-    ring
-  rw [heT,heS,sub_zero] at he
-  linarith
-lemma levelTime_clock_le (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (ht : alive a b.1 t z) (htt : (t:ℝ≥0∞)≤levelTime a b z cap u) :
-    physicalClock a cap (actualState a b.1 t z)≤u := by
-  have h := actual_physicalClock_le a ha z hz cap _ hc (levelRadius_pos a cap u hc)
-    ((levelRadius_le ha hc hu).trans (by linarith)) b t ht htt
-  rwa [levelRadius_clock ha.ne' hc] at h
-lemma levelTime_clock_ge (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (ht : alive a b.1 t z) (htt : levelTime a b z cap u≤(t:ℝ≥0∞)) :
-    u≤physicalClock a cap (actualState a b.1 t z) := by
-  have hf : levelTime a b z cap u≠∞ := ne_top_of_le_ne_top (by simp) htt
-  have hs : (levelTime a b z cap u).toNNReal≤t := by
-    exact ENNReal.coe_le_coe.mp (ENNReal.coe_toNNReal_le_self.trans htt)
-  rw [←levelTime_clock a ha z hz cap u hc hr hu b hf]
-  exact actual_physicalClock_mono a ha cap hc b z _ t hs ht
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-open scoped Classical
-abbrev PhaseVector (m : ℕ) := Fin m → Fin 4
-noncomputable def phaseActive {m : ℕ} (p : PhaseVector m) : Finset (Fin m) :=
-  Finset.univ.filter fun i => p i=1 ∨ p i=2
-noncomputable def phaseSigma {m : ℕ} (a : ℝ) (p : PhaseVector m) (i : Fin m) : ℝ :=
-  if p i=2 then regBeta a else -regBeta a
-noncomputable def phaseOffset {m : ℕ} (a : ℝ) (U : Fin m → ℝ) (p : PhaseVector m) (i : Fin m) : ℝ :=
-  if p i=2 then -regBeta a*U i else 0
-noncomputable def phaseConstant {m : ℕ} (a : ℝ) (U : Fin m → ℝ) (p : PhaseVector m) : ℝ :=
-  ∏ i, if p i=3 then Real.exp (regP a*U i/2)/3 else 1
-noncomputable def phaseProduct {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ)
-    (p : PhaseVector m) (v : Fin m → State) : ℝ :=
-  phaseConstant a U p*regularizedProduct (phaseActive p) a cap (phaseSigma a p) (phaseOffset a U p) v
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadiusComparison
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEStops
+open scoped Classical
+lemma unfinishedCutoff_exhausts {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
+    (b : ExactSLEGenerator.Driver) (hb : ∀ i, radialTime a b.1 (z i) (ρ i)<∞) :
+    ∀ᶠ n in Filter.atTop, ∀ i, radialTime a b.1 (z i) (ρ i) <
+      (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞) := by
+  have he : ∀ᶠ n in atTop, ∀ i, (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=
+      radialTime a b.1 (z i) (ρ i) := by
+    rw [Filter.eventually_all]
+    intro i
+    exact radialStop_eventually_eq a ha (z i) (hz i) (ρ i) (hρ i) (hr i) b (hb i).ne
+  have hn : ∀ᶠ n : ℕ in atTop, ∀ i, radialTime a b.1 (z i) (ρ i)<(n:ℝ≥0∞) := by
+    rw [Filter.eventually_all]
+    intro i
+    obtain ⟨N,hN⟩ := ENNReal.exists_nat_gt (hb i).ne
+    filter_upwards [Filter.eventually_ge_atTop N] with n hn
+    exact hN.trans_le (by exact_mod_cast hn)
+  filter_upwards [he,hn] with n he hn i
+  rw [unfinishedCutoff_coe,unfinishedCutoffTop]
+  have hi : (Finset.univ.inf fun j =>
+      if (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞)=radialTime a b.1 (z j) (ρ j)
+      then ∞ else (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞))=∞ := by
+    apply top_unique
+    apply Finset.le_inf
+    intro j hj
+    rw [ite_eq_left (he j)]
+  rw [hi,min_top_right]
+  exact hn i
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+ 
+
+theorem actual_unfinished_cutoff {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im) :
+    (∀ n, IsStoppingTime rawFiltration
+      (fun b => (unfinishedCutoff a ha z hz ρ hρ hr n b : WithTop ℝ≥0))) ∧
+    (∀ n b t i, t≤unfinishedCutoff a ha z hz ρ hρ hr n b →
+      (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i) → alive a b.1 t (z i)) ∧
+    (∀ b, (∀ i, radialTime a b.1 (z i) (ρ i)<∞) →
+      ∀ᶠ n in Filter.atTop, ∀ i, radialTime a b.1 (z i) (ρ i) <
+        (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)) := by
+  refine ⟨unfinishedCutoff_stopping a ha z hz ρ hρ hr,?_,unfinishedCutoff_exhausts a ha z hz ρ hρ hr⟩
+  intro n b t i ht hti
+  exact radialStop_actual a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b t
+    (unfinishedCutoff_before_individual a ha z hz ρ hρ hr n b t i ht hti)
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEFlow
+open Set Filter
+lemma localState_noiseFree_deriv (a ε : ℝ) (hε : 0<ε) (z : ℂ)
+    (b : ExactSLEGenerator.Driver) (T : ℝ≥0) (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
+    HasDerivWithinAt (fun v : ℝ => localState a ε hε z v.toNNReal b-b.1 v.toNNReal • noise)
+      (cutoffDrift a ε (localState a ε hε z u.toNNReal b)) (Icc (0:ℝ) T) u := by
+  have hc : Continuous (fun v : ℝ => cutoffDrift a ε (localState a ε hε z v.toNNReal b)) :=
+    (cutoffDrift_lipschitz a ε hε).continuous.comp
+      ((localState_continuous a ε hε z b).comp continuous_real_toNNReal)
+  have hd := (intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 u)
+    hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt).const_add ((z,0):State)
+  apply hd.hasDerivWithinAt.congr_of_mem _ hu
+  intro v hv
+  rw [localState_eq_zero,Real.coe_toNNReal _ hv.1]
+  abel
+lemma localState_lower_cutoff (a η ε : ℝ) (hη : 0<η) (hε : 0<ε) (hηε : η≤ε)
+    (z : ℂ) (b : ExactSLEGenerator.Driver) (T : ℝ≥0)
+    (hy : ∀ t≤T, ε≤Y (localState a ε hε z t b)) (t : ℝ≥0) (ht : t≤T) :
+    localState a η hη z t b=localState a ε hε z t b := by
+  let f : ℝ → State := fun u => localState a η hη z u.toNNReal b-b.1 u.toNNReal • noise
+  let g : ℝ → State := fun u => localState a ε hε z u.toNNReal b-b.1 u.toNNReal • noise
+  let u : ℝ → State := fun v => b.1 v.toNNReal • noise
+  have hf : Continuous f := ((localState_continuous a η hη z b).comp continuous_real_toNNReal).sub
+    ((b.1.continuous.comp continuous_real_toNNReal).smul continuous_const)
+  have hg : Continuous g := ((localState_continuous a ε hε z b).comp continuous_real_toNNReal).sub
+    ((b.1.continuous.comp continuous_real_toNNReal).smul continuous_const)
+  have hfd (v : ℝ) (hv : v∈Icc (0:ℝ) T) :
+      HasDerivWithinAt f (cutoffDrift a η (f v+u v)) (Icc (0:ℝ) T) v := by
+    simpa only [f,u,sub_add_cancel] using localState_noiseFree_deriv a η hη z b T v hv
+  have hgd (v : ℝ) (hv : v∈Icc (0:ℝ) T) :
+      HasDerivWithinAt g (cutoffDrift a η (g v+u v)) (Icc (0:ℝ) T) v := by
+    have hvT : v.toNNReal≤T := (Real.toNNReal_le_toNNReal hv.2).trans_eq (by simp)
+    have he : cutoffDrift a ε (localState a ε hε z v.toNNReal b)=
+        cutoffDrift a η (localState a ε hε z v.toNNReal b) := by
+      rw [cutoffDrift_eq_physical a ε _ (hy _ hvT),cutoffDrift_eq_physical a η _ (hηε.trans (hy _ hvT))]
+    simpa only [g,u,sub_add_cancel,←he] using localState_noiseFree_deriv a ε hε z b T v hv
+  have he := additive_flow_unique (cutoffDrift_lipschitz a η hη) hf.continuousOn hg.continuousOn
+    hfd hgd (by simp [f,g,localState_zero]) (show (t:ℝ)∈Icc (0:ℝ) T from ⟨t.coe_nonneg,ht⟩)
+  simpa only [f,g,Real.toNNReal_coe,sub_left_inj] using he
+lemma localState_noise_bounds (a η : ℝ) (hη : 0<η) (z : ℂ)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    ‖localState a η hη z t b‖≤‖z‖+|b.1 t|+max (|a|/η) (|a|/η^2)*t ∧
+    |b.1 t|≤‖localState a η hη z t b‖+‖z‖+max (|a|/η) (|a|/η^2)*t := by
+  let K := max (|a|/η) (|a|/η^2)
+  let I : State := ∫ v in (0:ℝ)..(t:ℝ), cutoffDrift a η (localState a η hη z v.toNNReal b)
+  have hI : ‖I‖≤K*t := by
+    simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
+      intervalIntegral.norm_integral_le_of_norm_le_const (a:=(0:ℝ)) (b:=(t:ℝ))
+        (C:=K) (fun v _ => cutoffDrift_bound a η hη _)
+  have hn : ‖noise‖=1 := by simp [noise,Prod.norm_def]
+  have hnoise : ‖b.1 t • noise‖=|b.1 t| := by rw [norm_smul,hn,mul_one,Real.norm_eq_abs]
+  have hz : ‖((z,0):State)‖=‖z‖ := by simp [Prod.norm_def]
+  have he : localState a η hη z t b=(z,0)+b.1 t • noise+I := localState_eq_zero a η hη z t b
+  constructor
+  · rw [he]
+    calc
+      _ ≤ ‖((z,0):State)+b.1 t • noise‖+‖I‖ := norm_add_le _ _
+      _ ≤ ‖((z,0):State)‖+‖b.1 t • noise‖+K*t := add_le_add (norm_add_le _ _) hI
+      _ = _ := by rw [hz,hnoise]
+  · have he' : b.1 t • noise=localState a η hη z t b-(z,0)-I := by rw [he]; abel
+    rw [←hnoise,he']
+    calc
+      _ ≤ ‖localState a η hη z t b-(z,0)‖+‖I‖ := norm_sub_le _ _
+      _ ≤ ‖localState a η hη z t b‖+‖((z,0):State)‖+K*t := add_le_add (norm_sub_le _ _) hI
+      _ = _ := by rw [hz]
+lemma finite_pos_lower {ι : Type*} (s : Finset ι) (f : ι → ℝ) (hf : ∀ i∈s,0<f i) :
+    ∃ η : ℝ, 0<η ∧ ∀ i∈s, η≤f i := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => exact ⟨1,by norm_num,by simp⟩
+  | @insert i s hi ih =>
+    obtain ⟨η,hη,hηs⟩ := ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))
+    refine ⟨min (f i) η,lt_min (hf i (Finset.mem_insert_self _ _)) hη,?_⟩
+    intro j hj
+    rcases Finset.mem_insert.mp hj with rfl|hj
+    · exact min_le_left _ _
+    · exact (min_le_right _ _).trans (hηs j hj)
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+ 
+
+theorem actual_common_cutoff_safe {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im) (n : ℕ) :
+    ∃ η : ℝ, ∃ hη : 0<η, ∃ M : ℝ≥0, ∀ (b : ExactSLEGenerator.Driver) (t : ℝ≥0),
+      t≤unfinishedCutoff a ha z hz ρ hρ hr n b →
+      (∃ i, (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i)) →
+      ‖vectorState a η hη z t b‖≤M ∧
+      ∀ i, (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i) →
+        η≤Y (vectorState a η hη z t b i) ∧
+        vectorState a η hη z t b i=actualState a b.1 t (z i) := by
+  let L := fun i => radialLocalization a ha (z i) (hz i) (ρ i) (angleThreshold (z i) n)
+    (hρ i) (hr i) (angleThreshold_pos (z i) (hz i) n) (angleThreshold_initial (z i) (hz i) n)
+  obtain ⟨η,hη,hηi⟩ := finite_pos_lower Finset.univ (fun i => (L i).ε) (fun i _ => (L i).hε)
+  let N : ℝ≥0 := Finset.univ.sup (fun i => (L i).N)
+  let K : ℝ := max (|a|/η) (|a|/η^2)
+  have hK : 0≤K := le_trans (by positivity : 0≤|a|/η) (le_max_left _ _)
+  let M : ℝ≥0 := ⟨N+2*‖z‖+2*K*n,by positivity⟩
+  refine ⟨η,hη,M,?_⟩
+  intro b t ht hex
+  have htN : (t:ℝ)≤n := by
+    have hcut : (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)≤(n:ℝ≥0∞) := by
+      rw [unfinishedCutoff_coe]
+      exact unfinishedCutoffTop_le a ha z hz ρ hρ hr n b
+    have h : (t:ℝ≥0∞)≤(n:ℝ≥0∞) := (ENNReal.coe_le_coe.mpr ht).trans hcut
+    exact_mod_cast h
+  have hstate (i : Fin m) (hi : (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i)) :
+      η≤Y (vectorState a η hη z t b i) ∧
+      vectorState a η hη z t b i=actualState a b.1 t (z i) ∧
+      ‖vectorState a η hη z t b i‖≤(L i).N := by
+    have hti := unfinishedCutoff_before_individual a ha z hz ρ hρ hr n b t i ht hi
+    have he : localState a η hη (z i) t b=localState a (L i).ε (L i).hε (z i) t b :=
+      localState_lower_cutoff a η (L i).ε hη (L i).hε (hηi i (Finset.mem_univ i)) (z i) b
+        (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b)
+        (fun u hu => (regionStop_safe a (L i).ε (L i).hε (z i) (L i).N (ρ i)
+          (angleThreshold (z i) n) (L i).B (L i).initial b u hu).1) t hti
+    have hs := regionStop_safe a (L i).ε (L i).hε (z i) (L i).N (ρ i)
+      (angleThreshold (z i) n) (L i).B (L i).initial b t hti
+    have ha' := regionStop_actualState a (L i).ε (L i).hε (z i) (L i).N (ρ i)
+      (angleThreshold (z i) n) (L i).B (L i).initial b t hti
+    change η≤Y (localState a η hη (z i) t b) ∧
+      localState a η hη (z i) t b=actualState a b.1 t (z i) ∧ _
+    rw [he]
+    exact ⟨(hηi i (Finset.mem_univ i)).trans hs.1,ha'.2.symm,by simpa only [vectorState,he] using hs.2.1⟩
+  obtain ⟨j,hj⟩ := hex
+  have hjN : ((L j).N:ℝ)≤N := by exact_mod_cast Finset.le_sup (f:=fun i => (L i).N) (Finset.mem_univ j)
+  have hbn : |b.1 t|≤N+‖z‖+K*n := by
+    have h := (localState_noise_bounds a η hη (z j) b t).2
+    have hs := (hstate j hj).2.2.trans hjN
+    have hzj := norm_le_pi_norm z j
+    have hkt := mul_le_mul_of_nonneg_left htN hK
+    change ‖localState a η hη (z j) t b‖≤(N:ℝ) at hs
+    dsimp only [K] at *
+    linarith
+  constructor
+  · apply (pi_norm_le_iff_of_nonneg M.coe_nonneg).mpr
+    intro i
+    have h := (localState_noise_bounds a η hη (z i) b t).1
+    have hzj := norm_le_pi_norm z i
+    have hkt := mul_le_mul_of_nonneg_left htN hK
+    change ‖localState a η hη (z i) t b‖≤N+2*‖z‖+2*K*n
+    dsimp only [K] at *
+    linarith
+  · intro i hi
+    exact ⟨(hstate i hi).1,(hstate i hi).2.1⟩
+end ExactSLEGreen
+namespace ExactSLEStops
+open Set Filter
+open scoped Classical
+variable {Ω : Type*} [MeasurableSpace Ω]
+noncomputable def clip (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) : ℝ≥0 :=
+  (min (τ ω) (B ω : ℝ≥0∞)).toNNReal
+omit [MeasurableSpace Ω] in
+lemma clip_coe (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) :
+    (clip τ B ω : ℝ≥0∞)=min (τ ω) (B ω : ℝ≥0∞) :=
+  ENNReal.coe_toNNReal (ne_top_of_le_ne_top (by simp) (min_le_right _ _))
+omit [MeasurableSpace Ω] in
+lemma clip_le (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) : clip τ B ω≤B ω := by
+  apply ENNReal.coe_le_coe.mp
+  rw [clip_coe]
+  exact min_le_right _ _
+omit [MeasurableSpace Ω] in
+lemma le_clip_iff (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) (t : ℝ≥0) :
+    t≤clip τ B ω ↔ (t:ℝ≥0∞)≤τ ω ∧ t≤B ω := by
+  rw [←ENNReal.coe_le_coe,clip_coe,le_min_iff,ENNReal.coe_le_coe]
+omit [MeasurableSpace Ω] in
+lemma clip_le_time (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (ω : Ω) :
+    (clip τ B ω : ℝ≥0∞)≤τ ω := by rw [clip_coe]; exact min_le_left _ _
+lemma clip_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (τ : Ω → ℝ≥0∞) (B : Ω → ℝ≥0) (hτ : IsStoppingTime F τ)
+    (hB : IsStoppingTime F (fun ω => (B ω : WithTop ℝ≥0))) :
+    IsStoppingTime F (fun ω => (clip τ B ω : WithTop ℝ≥0)) := by
+  have h := hτ.min hB
+  intro t
+  have ht := h t
+  change MeasurableSet[F t] {ω | min (τ ω) (B ω : ℝ≥0∞)≤(t:ℝ≥0∞)} at ht
+  change MeasurableSet[F t] {ω | (clip τ B ω : ℝ≥0∞)≤(t:ℝ≥0∞)}
+  simpa only [clip_coe] using ht
+ 
+noncomputable def phase (A B C : ℝ≥0∞) (t : ℝ≥0) : Fin 4 :=
+  if C≤(t:ℝ≥0∞) then 3 else if B≤(t:ℝ≥0∞) then 2 else if A≤(t:ℝ≥0∞) then 1 else 0
+lemma phase_measurable (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (A B C : Ω → ℝ≥0∞) (S : Ω → ℝ≥0)
+    (hA : IsStoppingTime F A) (hB : IsStoppingTime F B) (hC : IsStoppingTime F C)
+    (hS : IsStoppingTime F (fun ω => (S ω : WithTop ℝ≥0))) :
+    Measurable[hS.measurableSpace] (fun ω => phase (A ω) (B ω) (C ω) (S ω)) := by
+  have hA' : MeasurableSet[hS.measurableSpace] {ω | A ω≤(S ω : ℝ≥0∞)} := hA.measurableSet_stopping_time_le hS
+  have hB' : MeasurableSet[hS.measurableSpace] {ω | B ω≤(S ω : ℝ≥0∞)} := hB.measurableSet_stopping_time_le hS
+  have hC' : MeasurableSet[hS.measurableSpace] {ω | C ω≤(S ω : ℝ≥0∞)} := hC.measurableSet_stopping_time_le hS
+  exact measurable_const.ite hC' (measurable_const.ite hB' (measurable_const.ite hA' measurable_const))
+lemma phase_one_iff (A B C : ℝ≥0∞) (t : ℝ≥0) (_hAB : A≤B) (hBC : B≤C) :
+    phase A B C t=1 ↔ A≤(t:ℝ≥0∞) ∧ (t:ℝ≥0∞)<B := by
+  unfold phase
+  split_ifs with hC hB hA <;> simp_all
+  order
+lemma phase_two_iff (A B C : ℝ≥0∞) (t : ℝ≥0) (_hAB : A≤B) (_hBC : B≤C) :
+    phase A B C t=2 ↔ B≤(t:ℝ≥0∞) ∧ (t:ℝ≥0∞)<C := by
+  unfold phase
+  split_ifs with hC hB hA <;> simp_all
+lemma phase_zero_iff (A B C : ℝ≥0∞) (t : ℝ≥0) (hAB : A≤B) (hBC : B≤C) :
+    phase A B C t=0 ↔ (t:ℝ≥0∞)<A := by
+  unfold phase
+  split_ifs with hC hB hA <;> simp_all <;> order
+lemma phase_three_iff (A B C : ℝ≥0∞) (t : ℝ≥0) :
+    phase A B C t=3 ↔ C≤(t:ℝ≥0∞) := by
+  unfold phase
+  split_ifs <;> simp_all
+ 
+omit [MeasurableSpace Ω] in
+lemma no_actual_change_between (l : List (Ω → ℝ≥0)) (B : Ω → ℝ≥0) (τ : Ω → ℝ≥0∞)
+    (hτ : clip τ B∈l) (ω : Ω) (k : ℕ) (hk : k+1<(sort l).length)
+    (hB : (sort l)[k+1] ω≤B ω) :
+    τ ω≤((sort l)[k] ω : ℝ≥0∞) ∨ ((sort l)[k+1] ω : ℝ≥0∞)≤τ ω := by
+  rcases no_change_between l ω k hk (clip τ B) hτ with h|h
+  · by_cases ht : τ ω≤((sort l)[k] ω : ℝ≥0∞)
+    · exact Or.inl ht
+    · right
+      have hs : B ω≤(sort l)[k] ω := by
+        have h' := ENNReal.coe_le_coe.mpr h
+        rw [clip_coe,min_le_iff] at h'
+        exact ENNReal.coe_le_coe.mp (h'.resolve_left ht)
+      exact (ENNReal.coe_le_coe.mpr (hB.trans hs)).trans (not_le.mp ht).le
+  · exact Or.inr ((le_clip_iff τ B ω _).mp h).1
+end ExactSLEStops
+
+ 
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt Set Filter
+open scoped ENNReal NNReal
+lemma radialTime_antitone (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ σ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im) (b : ExactSLEGenerator.Driver) :
+    radialTime a b.1 z σ≤radialTime a b.1 z ρ := by
+  rcases hρσ.eq_or_lt with rfl|h
+  · exact le_rfl
+  · by_cases ht : radialTime a b.1 z ρ=∞
+    · rw [ht]; exact le_top
+    · exact (radialTime_strict_of_finite a ha z hz ρ σ hρ h hσ b ht).le
+lemma actual_radius_before_target (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (ht : alive a b.1 t z) (htr : (t:ℝ≥0∞)≤radialTime a b.1 z ρ) :
+    ρ≤physicalRadius (actualState a b.1 t z) := by
+  rcases htr.eq_or_lt with he|he
+  · have hn : radialTime a b.1 z ρ≠∞ := by rw [←he]; simp
+    have h := (radialTime_attained a ha z hz ρ hρ hr b hn).2
+    rw [←he,ENNReal.toNNReal_coe] at h
+    exact h.ge
+  · obtain ⟨n,hn⟩ := (radialStop_after_survival a ha z hz ρ hρ hr b t ht he).exists
+    let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+      (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+    have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t hn.le
+    have heq := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t hn.le
+    rw [heq.2]
+    exact hs.2.2.1
+lemma actual_physicalClock_le (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap ρ : ℝ) (hcap : 0<cap) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (ht : alive a b.1 t z) (htr : (t:ℝ≥0∞)≤radialTime a b.1 z ρ) :
+    physicalClock a cap (actualState a b.1 t z)≤Real.log (cap/(2*ρ))/(2*a) := by
+  have hy : 0<Y (actualState a b.1 t z) := by
+    obtain ⟨f,hf⟩ := ht
+    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq hf]
+      using (hf.2 t ⟨t.coe_nonneg,le_rfl⟩).1
+  have hR := actual_radius_before_target a ha z hz ρ hρ hr b t ht htr
+  have hR0 : 0<physicalRadius (actualState a b.1 t z) := hρ.trans_le hR
+  rw [physicalClock_radius a cap hcap _ hy]
+  apply (div_le_div_iff_of_pos_right (by positivity : 0<2*a)).mpr
+  apply Real.log_le_log (by positivity)
+  exact div_le_div_of_nonneg_left hcap.le (by positivity) (by linarith)
+lemma actual_physicalClock_at (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap ρ : ℝ) (hcap : 0<cap) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver)
+    (hf : radialTime a b.1 z ρ≠∞) :
+    physicalClock a cap (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z)=
+      Real.log (cap/(2*ρ))/(2*a) := by
+  have h := radialTime_attained a ha z hz ρ hρ hr b hf
+  have hy : 0<Y (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z) := by
+    obtain ⟨f,hf⟩ := h.1
+    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq hf]
+      using (hf.2 _ ⟨(radialTime a b.1 z ρ).toNNReal.coe_nonneg,le_rfl⟩).1
+  rw [physicalClock_radius a cap hcap _ hy,h.2]
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt Set Filter
+open scoped ENNReal NNReal
+noncomputable def levelRadius (a cap u : ℝ) : ℝ := cap/2*Real.exp (-2*a*u)
+noncomputable def levelTime (a : ℝ) (b : ExactSLEGenerator.Driver) (z : ℂ) (cap u : ℝ) : ℝ≥0∞ :=
+  radialTime a b.1 z (levelRadius a cap u)
+lemma levelRadius_pos (a cap u : ℝ) (hc : 0<cap) : 0<levelRadius a cap u := by
+  unfold levelRadius; positivity
+lemma levelRadius_le {a cap u : ℝ} (ha : 0<a) (hc : 0<cap) (hu : 0≤u) :
+    levelRadius a cap u≤cap/2 := by
+  have h := Real.exp_le_one_iff.mpr (show -2*a*u≤0 by nlinarith)
+  simpa only [levelRadius,mul_one] using mul_le_mul_of_nonneg_left h (show 0≤cap/2 by positivity)
+lemma levelRadius_antitone {a cap : ℝ} (ha : 0<a) (hc : 0<cap) :
+    Antitone (levelRadius a cap) := by
+  intro u v huv
+  apply mul_le_mul_of_nonneg_left _ (show 0≤cap/2 by positivity)
+  exact Real.exp_le_exp.mpr (by nlinarith)
+lemma levelTime_mono (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (b : ExactSLEGenerator.Driver)
+    (u v : ℝ) (hu : 0≤u) (huv : u≤v) :
+    levelTime a b z cap u≤levelTime a b z cap v := by
+  apply radialTime_antitone a ha z hz _ _ (levelRadius_pos a cap v hc)
+    (levelRadius_antitone ha hc huv)
+  exact (levelRadius_le ha hc hu).trans (by linarith)
+lemma levelRadius_clock {a cap u : ℝ} (ha : a≠0) (hc : 0<cap) :
+    Real.log (cap/(2*levelRadius a cap u))/(2*a)=u := by
+  have h : cap/(2*levelRadius a cap u)=Real.exp (2*a*u) := by
+    unfold levelRadius
+    rw [show 2*(cap/2*Real.exp (-2*a*u))=cap*Real.exp (-(2*a*u)) by ring_nf,
+      Real.exp_neg,div_mul_eq_div_div,div_self hc.ne',one_div,inv_inv]
+  rw [h,Real.log_exp]
+  field_simp
+lemma levelTime_clock (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
+    (hf : levelTime a b z cap u≠∞) :
+    physicalClock a cap (actualState a b.1 (levelTime a b z cap u).toNNReal z)=u := by
+  unfold levelTime
+  rw [actual_physicalClock_at a ha z hz cap _ hc (levelRadius_pos a cap u hc)
+    ((levelRadius_le ha hc hu).trans (by linarith)) b hf,levelRadius_clock ha.ne' hc]
+lemma levelTime_fullRadius (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (b : ExactSLEGenerator.Driver)
+    (hf : levelTime a b z cap 0≠∞) :
+    fullLoewnerRadius a b.1 z (levelTime a b z cap 0).toNNReal=cap := by
+  have h := radialTime_attained a ha z hz _ (levelRadius_pos a cap 0 hc)
+    ((levelRadius_le ha hc le_rfl).trans (by linarith)) b hf
+  unfold levelTime
+  rw [←actual_radius_eq_full b.1.continuous h.1,h.2]
+  simp [levelRadius]
+  ring
+lemma actual_physicalClock_mono (a : ℝ) (ha : 0<a) (cap : ℝ) (_hc : 0<cap)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (S T : ℝ≥0) (hST : S≤T) (hT : alive a b.1 T z) :
+    physicalClock a cap (actualState a b.1 S z)≤physicalClock a cap (actualState a b.1 T z) := by
+  obtain ⟨f,hf⟩ := hT
+  have hS : alive a b.1 S z := ⟨f,odeUntil_mono hf hST⟩
+  have hy : 0<Y (actualState a b.1 S z) := by
+    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq (odeUntil_mono hf hST)]
+      using (hf.2 S ⟨S.coe_nonneg,by exact_mod_cast hST⟩).1
+  have hR : 0<fullLoewnerRadius a b.1 z S := by
+    rw [←actual_radius_eq_full b.1.continuous hS]
+    unfold physicalRadius
+    positivity
+  have heT := actual_clock_eq ha.ne' hR b.1.continuous hf hST le_rfl rfl
+  have heS := actual_clock_eq ha.ne' hR b.1.continuous hf hST hST rfl
+  have hS0 : activeODEClock b.1 T S f S=0 := by simp [activeODEClock]
+  have hu := activeODEClock_nonneg (b:=b.1) (T:=T) (σ:=S) (f:=f) (show (S:ℝ)≤T from hST)
+  rw [hS0] at heS
+  have he : physicalClock a cap (actualState a b.1 T z)-physicalClock a cap (actualState a b.1 S z)=
+      physicalClock a (fullLoewnerRadius a b.1 z S) (actualState a b.1 T z)-
+      physicalClock a (fullLoewnerRadius a b.1 z S) (actualState a b.1 S z) := by
+    unfold physicalClock
+    ring
+  rw [heT,heS,sub_zero] at he
+  linarith
+lemma levelTime_clock_le (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (ht : alive a b.1 t z) (htt : (t:ℝ≥0∞)≤levelTime a b z cap u) :
+    physicalClock a cap (actualState a b.1 t z)≤u := by
+  have h := actual_physicalClock_le a ha z hz cap _ hc (levelRadius_pos a cap u hc)
+    ((levelRadius_le ha hc hu).trans (by linarith)) b t ht htt
+  rwa [levelRadius_clock ha.ne' hc] at h
+lemma levelTime_clock_ge (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (ht : alive a b.1 t z) (htt : levelTime a b z cap u≤(t:ℝ≥0∞)) :
+    u≤physicalClock a cap (actualState a b.1 t z) := by
+  have hf : levelTime a b z cap u≠∞ := ne_top_of_le_ne_top (by simp) htt
+  have hs : (levelTime a b z cap u).toNNReal≤t := by
+    exact ENNReal.coe_le_coe.mp (ENNReal.coe_toNNReal_le_self.trans htt)
+  rw [←levelTime_clock a ha z hz cap u hc hr hu b hf]
+  exact actual_physicalClock_mono a ha cap hc b z _ t hs ht
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+open scoped Classical
+abbrev PhaseVector (m : ℕ) := Fin m → Fin 4
+noncomputable def phaseActive {m : ℕ} (p : PhaseVector m) : Finset (Fin m) :=
+  Finset.univ.filter fun i => p i=1 ∨ p i=2
+noncomputable def phaseSigma {m : ℕ} (a : ℝ) (p : PhaseVector m) (i : Fin m) : ℝ :=
+  if p i=2 then regBeta a else -regBeta a
+noncomputable def phaseOffset {m : ℕ} (a : ℝ) (U : Fin m → ℝ) (p : PhaseVector m) (i : Fin m) : ℝ :=
+  if p i=2 then -regBeta a*U i else 0
+noncomputable def phaseConstant {m : ℕ} (a : ℝ) (U : Fin m → ℝ) (p : PhaseVector m) : ℝ :=
+  ∏ i, if p i=3 then Real.exp (regP a*U i/2)/3 else 1
+noncomputable def phaseProduct {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ)
+    (p : PhaseVector m) (v : Fin m → State) : ℝ :=
+  phaseConstant a U p*regularizedProduct (phaseActive p) a cap (phaseSigma a p) (phaseOffset a U p) v
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/CutoffFlow.lean b/lean/OAI/Probability/SLEGauge/CutoffFlow.lean
index 5eaf92adfe8802e9be67090fdad4c3b27342a67c..2dc67a755555e375cd4068cdebc91905b0d16205 100644
--- a/lean/OAI/Probability/SLEGauge/CutoffFlow.lean
+++ b/lean/OAI/Probability/SLEGauge/CutoffFlow.lean
@@ -1,420 +1,420 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.LiteralTrace
-
-open scoped ContinuousMapZero
-abbrev Driver := C(ℝ≥0, ℝ)₀
-instance : MeasurableSpace Driver := borel Driver
-instance : BorelSpace Driver := ⟨rfl⟩
-abbrev WienerLaw := Measure Driver
-
-def IsWiener (W : WienerLaw) : Prop :=
-  ProbabilityTheory.IsBrownianReal (fun t b => b t) W
-
-def a (κ : ℝ) := 2 / κ
-def d (κ : ℝ) := 1 + 1 / (4 * a κ)
-def α (κ : ℝ) := 2 - d κ
-def p (κ : ℝ) := 4 * a κ - 1
-
-def driverValue (b : Driver) (t : ℝ) : ℝ := b t.toNNReal
-
-def diskMap (θ : ℝ) (z : ℂ) : ℂ :=
-  (z - Complex.exp (θ * Complex.I)) / (z - Complex.exp (-θ * Complex.I))
-
-def diskMapInv (θ : ℝ) (x : ℂ) : ℂ :=
-  (Complex.exp (θ * Complex.I) - x * Complex.exp (-θ * Complex.I)) / (1 - x)
-
- 
-def cutoffField (κ : ℝ) (b : Driver) (j : ℕ) (t : ℝ) (z : ℂ) : ℂ :=
-  (a κ : ℂ) * star (z - driverValue b t) /
-    (max (‖z - driverValue b t‖ ^ 2) ((j : ℝ) ^ (-(4 : ℤ))) : ℝ)
-
-def mesh (N : ℕ) : ℝ := (2 : ℝ) ^ (-(N : ℤ))
-
-def eulerSteps (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) : ℕ → ℂ
-  | 0 => z
-  | k + 1 => eulerSteps κ b j N z k +
-      mesh N * cutoffField κ b j ((k : ℝ) * mesh N) (eulerSteps κ b j N z k)
-
-def eulerFlow (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
-  let k := ⌊t / mesh N⌋₊
-  let w :=  t / mesh N - (k : ℝ)
-  (1 - w : ℝ) * eulerSteps κ b j N z k + w * eulerSteps κ b j N z (k + 1)
-
-def cutoffFlow (κ : ℝ) (b : Driver) (j : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
-  limUnder atTop (fun N : ℕ => eulerFlow κ b j N z t)
-
-def Surviving (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Prop :=
-  ∃ j : ℕ, 1 ≤ j ∧ (j : ℝ)⁻¹ <
-    sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 t)
-
-instance (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Decidable (Surviving κ b z t) :=
-  Classical.propDecidable _
-
-def flow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
-  if h : Surviving κ b z t then cutoffFlow κ b (Nat.find h) z t else 0
-
-def centeredFlow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
-  flow κ b z t - driverValue b t
-
- 
-def clock (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  ∫ u in Icc (0 : ℝ) t,
-    (flow κ b (diskMapInv θ x) u).im ^ 2 /
-      ‖centeredFlow κ b (diskMapInv θ x) u‖ ^ 4
-
-def initialRadius (θ : ℝ) (x : ℂ) : ℝ :=
-  2 * (diskMapInv θ x).im * ‖deriv (diskMap θ) (diskMapInv θ x)‖
-
-def radius (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  initialRadius θ x * Real.exp (-2 * a κ * clock κ θ b x t)
-
-def angle (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  Complex.arg (centeredFlow κ b (diskMapInv θ x) t)
-
-def green (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
-  if Surviving κ b (diskMapInv θ x) t then
-    radius κ θ b x t ^ (-α κ) * Real.sin (angle κ θ b x t) ^ p κ
-  else 0
-
- 
-def levelTime (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ≥0∞ :=
-  sInf {t | ∃ q : ℚ, 0 < q ∧ t = ENNReal.ofReal (q : ℝ) ∧
-    Surviving κ b (diskMapInv θ 0) q ∧ ℓ < clock κ θ b 0 q}
-
-def diskK : Set ℂ := closedBall 0 (1 / 4)
-
-def potential (κ θ : ℝ) (b : Driver) (t : ℝ) : ℝ≥0∞ :=
-  ∫⁻ x in diskK, ENNReal.ofReal (green κ θ b x t)
-
- 
-
-def concat (b c : Driver) (t : ℝ) : Driver where
-  toFun u := if u ≤ t.toNNReal then b u else b t.toNNReal + c (u - t.toNNReal)
-  continuous_toFun := by
-    apply Continuous.if_le (map_continuous b)
-      (continuous_const.add ((map_continuous c).comp (continuous_id.sub continuous_const)))
-      continuous_id continuous_const
-    intro u hu
-    change u = t.toNNReal at hu
-    subst u
-    simp
-  map_zero' := by simp
-
- 
-def potentialDifference (κ θ : ℝ) (W : WienerLaw) (b : Driver) (h i : ℕ) : ℝ :=
-  let t := (i : ℝ) * mesh h
-  let v := potential κ θ b t
-  let w := ∫⁻ c, potential κ θ (concat b c t) (((i : ℝ) + 1) * mesh h) ∂W
-  if v < ∞ ∧ w < ∞ then v.toReal - w.toReal else 0
-
-def discreteMass (κ θ : ℝ) (W : WienerLaw) (b : Driver) (ℓ : ℝ) (h : ℕ) : ℝ :=
-  ∑ i ∈ Finset.range (h * 2 ^ h),
-    if ENNReal.ofReal (((i : ℝ) + 1) * mesh h) ≤ levelTime κ θ b ℓ
-    then potentialDifference κ θ W b h i else 0
-
-def levelDensity (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ :=
-  let τ := levelTime κ θ b ℓ
-  if τ < ∞ then Real.exp (2 * a κ * α κ * ℓ) *
-    Real.sin (angle κ θ b 0 τ.toReal) ^ p κ / Real.sin θ ^ p κ else 0
-
- 
-def angleLaw (κ : ℝ) : Measure ℝ :=
-  let unnormalized := (volume.restrict (Ioo (0 : ℝ) Real.pi)).withDensity
-    (fun θ => ENNReal.ofReal (Real.sin θ ^ (4 * a κ)))
-  (unnormalized Set.univ)⁻¹ • unnormalized
-
-def clip (u x : ℝ) : ℝ := min (max x 0) u
-
- 
-
-def finiteMoment (κ : ℝ) (W : WienerLaw) (n : ℕ) (ℓ : ℝ) : ℝ :=
-  limUnder atTop (fun u : ℕ => limUnder atTop (fun h : ℕ =>
-    ∫ θ, ∫ b, levelDensity κ θ b ℓ *
-      (clip u (discreteMass κ θ W b ℓ h)) ^ n ∂W ∂angleLaw κ))
-
-def moment (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
-  limUnder atTop (fun ℓ : ℕ => finiteMoment κ W n ℓ)
-
- 
-def batchLevel (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℕ :=
-  1 + Nat.card {j : ℕ // 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4}
-
-def threshold (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
-  4 * moment κ W (n + 1) / moment κ W n
-
-def multiplicity (κ : ℝ) (W : WienerLaw) (n : ℕ) (k : ℤ) : ℕ :=
-  if (2 : ℝ) ^ k ≤ threshold κ W n then
-    ⌊(4 : ℝ) ^ n * ((2 : ℝ) ^ k) ^ n / moment κ W n⌋₊ else 0
-
- 
-
-abbrev Entry := ℕ × Option (ℤ × ℕ)
-
-def ValidEntry (κ : ℝ) (W : WienerLaw) (e : Entry) : Prop :=
-  1 ≤ e.1 ∧ match e.2 with
-    | none => True
-    | some (k,j) => j < multiplicity κ W e.1 k
-
-def entryValue (e : Entry) : ℝ :=
-  match e.2 with
-    | none => e.1
-    | some (k,_) => (2 : ℝ) ^ k
-
-def entryPosition (κ : ℝ) (W : WienerLaw) (e : Entry) : ℕ :=
-  match e.2 with
-    | none => ∑' k : ℤ, multiplicity κ W e.1 k
-    | some (k,j) => (∑' l : ℤ, if l < k then multiplicity κ W e.1 l else 0) + j
-
-def entryRadius (κ : ℝ) (W : WienerLaw) (e : Entry) : ℝ :=
-  Real.exp (-(e.1 : ℝ) ^ 2 - 2 * a κ * (entryPosition κ W e : ℝ) *
-    ((batchLevel κ W e.1 : ℝ) + (e.1 : ℝ)))
-
-def entryCost (κ : ℝ) (W : WienerLaw) (e : Entry) (r : ℝ) : ℝ :=
-  entryValue e * entryRadius κ W e ^ d κ * max 1 ((r / entryRadius κ W e) ^ 2)
-
- 
-def gauge (κ : ℝ) (W : WienerLaw) (r : ℝ) : ℝ :=
-  if r ≤ 0 then 0 else sInf {v | ∃ e : Entry, ValidEntry κ W e ∧ v = entryCost κ W e r}
-
-def IsHausdorffGauge (h : ℝ → ℝ) : Prop :=
-  ContinuousOn h (Ici 0) ∧ MonotoneOn h (Ici 0) ∧ h 0 = 0 ∧ ∀ r > 0, 0 < h r
-
- 
-def hausdorff (h : ℝ → ℝ) : OuterMeasure ℂ :=
-  OuterMeasure.mkMetric (fun r => if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal))
-
-def inverseFlow (κ : ℝ) (b : Driver) (t : ℝ) (w : ℂ) : ℂ :=
-  @ite ℂ (∃ z : ℂ, Surviving κ b z t ∧ flow κ b z t = w) (Classical.propDecidable _)
-    (Classical.epsilon (fun z : ℂ => Surviving κ b z t ∧ flow κ b z t = w)) 0
-
- 
-def trace (κ : ℝ) (b : Driver) (t : ℝ) : ℂ :=
-  limUnder (nhdsWithin 0 (Ioi 0)) (fun ε : ℝ =>
-    inverseFlow κ b (κ * t) (driverValue b (κ * t) + ε * Complex.I))
-
- 
-
-def MainObligation : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 → ∀ W : WienerLaw, IsWiener W →
-    IsHausdorffGauge (gauge κ W) ∧
-    ∀ᵐ b ∂W, ∀ s t : ℝ, 0 < s → s < t →
-      0 < hausdorff (gauge κ W) ((trace κ b) '' Icc s t) ∧
-      hausdorff (gauge κ W) ((trace κ b) '' Icc s t) < ∞
-
-end SLEExactGauge.LiteralTrace
-
- 
-
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.Cutoff
-
-def denominator (q : ℝ) (z : ℂ) : ℝ := max (‖z‖^2) (q^2)
-
-def inverse (q : ℝ) (z : ℂ) : ℂ := star z / (denominator q z : ℂ)
-
-lemma denominator_pos {q : ℝ} (hq : 0 < q) (z : ℂ) : 0 < denominator q z :=
-  lt_of_lt_of_le (sq_pos_of_pos hq) (le_max_right _ _)
-
-lemma q_mul_norm_le_denominator {q : ℝ} (hq : 0 ≤ q) (z : ℂ) :
-    q * ‖z‖ ≤ denominator q z := by
-  by_cases he : ‖z‖ ≤ q
-  · exact (mul_le_mul_of_nonneg_left he hq).trans (by
-      change q*q ≤ max (‖z‖^2) (q^2)
-      rw [← sq]
-      exact le_max_right _ _)
-  · exact (mul_le_mul_of_nonneg_right (le_of_not_ge he) (norm_nonneg _)).trans (by
-      change ‖z‖*‖z‖ ≤ max (‖z‖^2) (q^2)
-      rw [← sq]
-      exact le_max_left _ _)
-
-lemma norm_inverse_le {q : ℝ} (hq : 0 < q) (z : ℂ) :
-    ‖inverse q z‖ ≤ q⁻¹ := by
-  rw [inverse, norm_div, norm_star, Complex.norm_real,
-    Real.norm_eq_abs, abs_of_pos (denominator_pos hq z)]
-  apply (div_le_iff₀ (denominator_pos hq z)).mpr
-  have hh := q_mul_norm_le_denominator hq.le z
-  have he : q⁻¹ * denominator q z = denominator q z / q := by ring
-  rw [he]
-  exact (le_div_iff₀ hq).mpr (by simpa only [mul_comm] using hh)
-
-lemma denominator_sub_le {q : ℝ} {z w : ℂ} (hzw : ‖z‖ ≤ ‖w‖) :
-    0 ≤ denominator q w - denominator q z ∧
-    denominator q w - denominator q z ≤ (‖w‖+‖z‖)*‖w-z‖ := by
-  have hn : ‖z‖^2 ≤ ‖w‖^2 := sq_le_sq₀ (norm_nonneg _) (norm_nonneg _) |>.mpr hzw
-  have hden : denominator q z ≤ denominator q w := max_le_max_right _ hn
-  refine ⟨sub_nonneg.mpr hden, ?_⟩
-  have hd : max (‖w‖^2) (q^2) - max (‖z‖^2) (q^2) ≤ ‖w‖^2 - ‖z‖^2 := by
-    by_cases hz : q^2 ≤ ‖z‖^2
-    · rw [max_eq_left (hz.trans hn), max_eq_left hz]
-    · rw [max_eq_right (le_of_not_ge hz)]
-      by_cases hw : q^2 ≤ ‖w‖^2
-      · rw [max_eq_left hw]
-        linarith
-      · rw [max_eq_right (le_of_not_ge hw)]
-        linarith
-  refine hd.trans ?_
-  calc ‖w‖^2 - ‖z‖^2 = (‖w‖+‖z‖)*(‖w‖-‖z‖) := by ring
-       _ ≤ _ := mul_le_mul_of_nonneg_left (norm_sub_norm_le w z)
-         (add_nonneg (norm_nonneg _) (norm_nonneg _))
-
-lemma inverse_sub_bound_ordered {q : ℝ} (hq : 0 < q) (z w : ℂ)
-    (hzw : ‖z‖ ≤ ‖w‖) :
-    ‖inverse q z - inverse q w‖ ≤ (3/q^2)*‖z-w‖ := by
-  let dz := denominator q z
-  let dw := denominator q w
-  have hz : 0 < dz := denominator_pos hq z
-  have hw : 0 < dw := denominator_pos hq w
-  have hzne : (dz : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt hz
-  have hwne : (dw : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt hw
-  have he : inverse q z - inverse q w =
-      (star z - star w)/(dw:ℂ) + star z * ((dw:ℂ)-(dz:ℂ))/((dz:ℂ)*(dw:ℂ)) := by
-    dsimp [inverse]
-    change star z/(dz:ℂ)-star w/(dw:ℂ) = _
-    change star z/(dz:ℂ)-star w/(dw:ℂ) =
-      (star z-star w)/(dw:ℂ)+star z*((dw:ℂ)-(dz:ℂ))/((dz:ℂ)*(dw:ℂ))
-    field_simp [hzne, hwne]
-    ring
-  have hd := denominator_sub_le (q := q) hzw
-  have hdz : q^2 ≤ dz := le_max_right _ _
-  have hdw : q^2 ≤ dw := le_max_right _ _
-  have hqw := q_mul_norm_le_denominator hq.le w
-  have htwo : ‖z‖*(dw-dz) ≤ 2 * dw * ‖z-w‖ := by
-    have hm := mul_le_mul_of_nonneg_left hd.2 (norm_nonneg z)
-    rw [norm_sub_rev w z] at hm
-    have hn : ‖z‖*(‖w‖+‖z‖) ≤ 2*dw := by
-      have hww : ‖w‖^2 ≤ dw := le_max_left _ _
-      nlinarith [norm_nonneg z, norm_nonneg w]
-    calc ‖z‖*(dw-dz) ≤ ‖z‖*((‖w‖+‖z‖)*‖z-w‖) := hm
-         _ = (‖z‖*(‖w‖+‖z‖))*‖z-w‖ := by ring
-         _ ≤ _ := mul_le_mul_of_nonneg_right hn (norm_nonneg _)
-  have hb₁ : ‖star z - star w‖ / dw ≤ ‖z-w‖ / q^2 := by
-    rw [← star_sub, norm_star]
-    exact div_le_div_of_nonneg_left (norm_nonneg _) (sq_pos_of_pos hq) hdw
-  have hb₂ : ‖z‖*(dw-dz)/(dz*dw) ≤ 2*‖z-w‖/(q^2) := by
-    calc
-      _ ≤ 2*dw*‖z-w‖/(dz*dw) := div_le_div_of_nonneg_right htwo (mul_nonneg hz.le hw.le)
-      _ = 2*‖z-w‖/dz := by field_simp
-      _ ≤ _ := div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hq) hdz
-  rw [he]
-  calc
-    _ ≤ ‖(star z-star w)/(dw:ℂ)‖ +
-        ‖star z*((dw:ℂ)-(dz:ℂ))/((dz:ℂ)*(dw:ℂ))‖ := norm_add_le _ _
-    _ = ‖star z-star w‖/dw + ‖z‖*(dw-dz)/(dz*dw) := by
-      rw [norm_div, norm_div, norm_mul, norm_mul, norm_star,
-        ← Complex.ofReal_sub, Complex.norm_real, Complex.norm_real,
-        Complex.norm_real, Real.norm_eq_abs, Real.norm_eq_abs, Real.norm_eq_abs,
-        abs_of_pos hw, abs_of_pos hz, abs_of_nonneg hd.1]
-    _ ≤ ‖z-w‖/q^2 + 2*‖z-w‖/q^2 := add_le_add hb₁ hb₂
-    _ = _ := by ring
-
-lemma inverse_sub_bound {q : ℝ} (hq : 0 < q) (z w : ℂ) :
-    ‖inverse q z - inverse q w‖ ≤ (3/q^2)*‖z-w‖ := by
-  rcases le_total ‖z‖ ‖w‖ with h | h
-  · exact inverse_sub_bound_ordered hq z w h
-  · simpa only [norm_sub_rev] using inverse_sub_bound_ordered hq w z h
-
-lemma lipschitz_inverse {q : ℝ} (hq : 0 < q) :
-    LipschitzWith ⟨3/q^2, by positivity⟩ (inverse q) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  change dist (inverse q z) (inverse q w) ≤ (3/q^2)*dist z w
-  simpa only [dist_eq_norm] using inverse_sub_bound hq z w
-
- 
-def field (a q : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ) : ℂ :=
-  (a : ℂ) * inverse q (z - (b t : ℂ))
-
-lemma norm_field_le {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ) :
-    ‖field a q b t z‖ ≤ |a| / q := by
-  rw [field, norm_mul, Complex.norm_real, Real.norm_eq_abs]
-  calc |a| * ‖inverse q (z - (b t : ℂ))‖ ≤ |a| * q⁻¹ :=
-          mul_le_mul_of_nonneg_left (norm_inverse_le hq _) (abs_nonneg a)
-       _ = _ := div_eq_mul_inv _ _ |>.symm
-
-lemma field_sub_le {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ) (t : ℝ) (z w : ℂ) :
-    ‖field a q b t z - field a q b t w‖ ≤ (|a| *3/q^2)*‖z-w‖ := by
-  rw [field, field, ← mul_sub, norm_mul, Complex.norm_real, Real.norm_eq_abs]
-  have hh := mul_le_mul_of_nonneg_left
-    (inverse_sub_bound hq (z-(b t:ℂ)) (w-(b t:ℂ))) (abs_nonneg a)
-  simpa only [sub_sub_sub_cancel_right, mul_div_assoc, mul_assoc] using hh
-
-lemma continuous_field_time {q : ℝ} (hq : 0 < q) (a : ℝ) {b : ℝ → ℝ}
-    (hb : Continuous b) (z : ℂ) : Continuous (fun t => field a q b t z) := by
-  exact continuous_const.mul ((lipschitz_inverse hq).continuous.comp
-    (continuous_const.sub (Complex.continuous_ofReal.comp hb)))
-
-lemma field_joint_sub_le {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ)
-    (s t : ℝ) (z w : ℂ) :
-    ‖field a q b s z - field a q b t w‖ ≤
-      (|a| *3/q^2)*(‖z-w‖ + |b s-b t|) := by
-  rw [field, field, ← mul_sub, norm_mul, Complex.norm_real, Real.norm_eq_abs]
-  have hdiff : (z-(b s:ℂ))-(w-(b t:ℂ)) = (z-w)-((b s:ℂ)-(b t:ℂ)) := by ring
-  calc
-    _ ≤ |a| * ((3/q^2)*‖(z-(b s:ℂ))-(w-(b t:ℂ))‖) :=
-      mul_le_mul_of_nonneg_left (inverse_sub_bound hq _ _) (abs_nonneg _)
-    _ = (|a| *3/q^2)*‖(z-w)-((b s:ℂ)-(b t:ℂ))‖ := by rw [hdiff]; ring
-    _ ≤ (|a| *3/q^2)*(‖z-w‖ + ‖(b s:ℂ)-(b t:ℂ)‖) :=
-      mul_le_mul_of_nonneg_left (norm_sub_le _ _) (by positivity)
-    _ = _ := by rw [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs]
-
- 
-
-theorem exists_solution_on_compact {q : ℝ} (hq : 0 < q) (a : ℝ) {b : ℝ → ℝ}
-    (hb : Continuous b) (z : ℂ) {T : ℝ} (hT : 0 ≤ T) :
-    ∃ g : ℝ → ℂ, g 0 = z ∧ ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt g (field a q b t (g t)) (Icc 0 T) t := by
-  let M : ℝ≥0 := ⟨|a|/q, by positivity⟩
-  let K : ℝ≥0 := ⟨|a| *3/q^2, by positivity⟩
-  let R : ℝ≥0 := ⟨(|a|/q)*T, mul_nonneg (by positivity) hT⟩
-  have hp : IsPicardLindelof (field a q b) (tmin := 0) (tmax := T)
-      ⟨0, ⟨le_rfl, hT⟩⟩ z R 0 M K := by
-    constructor
-    · intro t ht
-      apply LipschitzWith.lipschitzOnWith
-      apply LipschitzWith.of_dist_le_mul
-      intro w v
-      change dist (field a q b t w) (field a q b t v) ≤ (|a| *3/q^2)*dist w v
-      simpa only [dist_eq_norm] using field_sub_le hq a b t w v
-    · intro w hw
-      exact (continuous_field_time hq a hb w).continuousOn
-    · intro t ht w hw
-      exact norm_field_le hq a b t w
-    · change (|a|/q)*max (T-0) (0-0) ≤ (|a|/q)*T-0
-      simp only [sub_zero, sub_self, max_eq_left hT, le_refl]
-  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
-
-end SLEExactGauge.Cutoff
-
- 
-
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.Euler
-
-def nodes (f : ℝ → ℂ → ℂ) (h : ℝ) (z : ℂ) : ℕ → ℂ
-  | 0 => z
-  | k+1 => nodes f h z k + h • f ((k:ℝ)*h) (nodes f h z k)
-
-end SLEExactGauge.Euler
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.LiteralTrace
+
+open scoped ContinuousMapZero
+abbrev Driver := C(ℝ≥0, ℝ)₀
+instance : MeasurableSpace Driver := borel Driver
+instance : BorelSpace Driver := ⟨rfl⟩
+abbrev WienerLaw := Measure Driver
+
+def IsWiener (W : WienerLaw) : Prop :=
+  ProbabilityTheory.IsBrownianReal (fun t b => b t) W
+
+def a (κ : ℝ) := 2 / κ
+def d (κ : ℝ) := 1 + 1 / (4 * a κ)
+def α (κ : ℝ) := 2 - d κ
+def p (κ : ℝ) := 4 * a κ - 1
+
+def driverValue (b : Driver) (t : ℝ) : ℝ := b t.toNNReal
+
+def diskMap (θ : ℝ) (z : ℂ) : ℂ :=
+  (z - Complex.exp (θ * Complex.I)) / (z - Complex.exp (-θ * Complex.I))
+
+def diskMapInv (θ : ℝ) (x : ℂ) : ℂ :=
+  (Complex.exp (θ * Complex.I) - x * Complex.exp (-θ * Complex.I)) / (1 - x)
+
+ 
+def cutoffField (κ : ℝ) (b : Driver) (j : ℕ) (t : ℝ) (z : ℂ) : ℂ :=
+  (a κ : ℂ) * star (z - driverValue b t) /
+    (max (‖z - driverValue b t‖ ^ 2) ((j : ℝ) ^ (-(4 : ℤ))) : ℝ)
+
+def mesh (N : ℕ) : ℝ := (2 : ℝ) ^ (-(N : ℤ))
+
+def eulerSteps (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) : ℕ → ℂ
+  | 0 => z
+  | k + 1 => eulerSteps κ b j N z k +
+      mesh N * cutoffField κ b j ((k : ℝ) * mesh N) (eulerSteps κ b j N z k)
+
+def eulerFlow (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
+  let k := ⌊t / mesh N⌋₊
+  let w :=  t / mesh N - (k : ℝ)
+  (1 - w : ℝ) * eulerSteps κ b j N z k + w * eulerSteps κ b j N z (k + 1)
+
+def cutoffFlow (κ : ℝ) (b : Driver) (j : ℕ) (z : ℂ) (t : ℝ) : ℂ :=
+  limUnder atTop (fun N : ℕ => eulerFlow κ b j N z t)
+
+def Surviving (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Prop :=
+  ∃ j : ℕ, 1 ≤ j ∧ (j : ℝ)⁻¹ <
+    sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 t)
+
+instance (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : Decidable (Surviving κ b z t) :=
+  Classical.propDecidable _
+
+def flow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
+  if h : Surviving κ b z t then cutoffFlow κ b (Nat.find h) z t else 0
+
+def centeredFlow (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℂ :=
+  flow κ b z t - driverValue b t
+
+ 
+def clock (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  ∫ u in Icc (0 : ℝ) t,
+    (flow κ b (diskMapInv θ x) u).im ^ 2 /
+      ‖centeredFlow κ b (diskMapInv θ x) u‖ ^ 4
+
+def initialRadius (θ : ℝ) (x : ℂ) : ℝ :=
+  2 * (diskMapInv θ x).im * ‖deriv (diskMap θ) (diskMapInv θ x)‖
+
+def radius (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  initialRadius θ x * Real.exp (-2 * a κ * clock κ θ b x t)
+
+def angle (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  Complex.arg (centeredFlow κ b (diskMapInv θ x) t)
+
+def green (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) : ℝ :=
+  if Surviving κ b (diskMapInv θ x) t then
+    radius κ θ b x t ^ (-α κ) * Real.sin (angle κ θ b x t) ^ p κ
+  else 0
+
+ 
+def levelTime (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ≥0∞ :=
+  sInf {t | ∃ q : ℚ, 0 < q ∧ t = ENNReal.ofReal (q : ℝ) ∧
+    Surviving κ b (diskMapInv θ 0) q ∧ ℓ < clock κ θ b 0 q}
+
+def diskK : Set ℂ := closedBall 0 (1 / 4)
+
+def potential (κ θ : ℝ) (b : Driver) (t : ℝ) : ℝ≥0∞ :=
+  ∫⁻ x in diskK, ENNReal.ofReal (green κ θ b x t)
+
+ 
+
+def concat (b c : Driver) (t : ℝ) : Driver where
+  toFun u := if u ≤ t.toNNReal then b u else b t.toNNReal + c (u - t.toNNReal)
+  continuous_toFun := by
+    apply Continuous.if_le (map_continuous b)
+      (continuous_const.add ((map_continuous c).comp (continuous_id.sub continuous_const)))
+      continuous_id continuous_const
+    intro u hu
+    change u = t.toNNReal at hu
+    subst u
+    simp
+  map_zero' := by simp
+
+ 
+def potentialDifference (κ θ : ℝ) (W : WienerLaw) (b : Driver) (h i : ℕ) : ℝ :=
+  let t := (i : ℝ) * mesh h
+  let v := potential κ θ b t
+  let w := ∫⁻ c, potential κ θ (concat b c t) (((i : ℝ) + 1) * mesh h) ∂W
+  if v < ∞ ∧ w < ∞ then v.toReal - w.toReal else 0
+
+def discreteMass (κ θ : ℝ) (W : WienerLaw) (b : Driver) (ℓ : ℝ) (h : ℕ) : ℝ :=
+  ∑ i ∈ Finset.range (h * 2 ^ h),
+    if ENNReal.ofReal (((i : ℝ) + 1) * mesh h) ≤ levelTime κ θ b ℓ
+    then potentialDifference κ θ W b h i else 0
+
+def levelDensity (κ θ : ℝ) (b : Driver) (ℓ : ℝ) : ℝ :=
+  let τ := levelTime κ θ b ℓ
+  if τ < ∞ then Real.exp (2 * a κ * α κ * ℓ) *
+    Real.sin (angle κ θ b 0 τ.toReal) ^ p κ / Real.sin θ ^ p κ else 0
+
+ 
+def angleLaw (κ : ℝ) : Measure ℝ :=
+  let unnormalized := (volume.restrict (Ioo (0 : ℝ) Real.pi)).withDensity
+    (fun θ => ENNReal.ofReal (Real.sin θ ^ (4 * a κ)))
+  (unnormalized Set.univ)⁻¹ • unnormalized
+
+def clip (u x : ℝ) : ℝ := min (max x 0) u
+
+ 
+
+def finiteMoment (κ : ℝ) (W : WienerLaw) (n : ℕ) (ℓ : ℝ) : ℝ :=
+  limUnder atTop (fun u : ℕ => limUnder atTop (fun h : ℕ =>
+    ∫ θ, ∫ b, levelDensity κ θ b ℓ *
+      (clip u (discreteMass κ θ W b ℓ h)) ^ n ∂W ∂angleLaw κ))
+
+def moment (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
+  limUnder atTop (fun ℓ : ℕ => finiteMoment κ W n ℓ)
+
+ 
+def batchLevel (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℕ :=
+  1 + Nat.card {j : ℕ // 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4}
+
+def threshold (κ : ℝ) (W : WienerLaw) (n : ℕ) : ℝ :=
+  4 * moment κ W (n + 1) / moment κ W n
+
+def multiplicity (κ : ℝ) (W : WienerLaw) (n : ℕ) (k : ℤ) : ℕ :=
+  if (2 : ℝ) ^ k ≤ threshold κ W n then
+    ⌊(4 : ℝ) ^ n * ((2 : ℝ) ^ k) ^ n / moment κ W n⌋₊ else 0
+
+ 
+
+abbrev Entry := ℕ × Option (ℤ × ℕ)
+
+def ValidEntry (κ : ℝ) (W : WienerLaw) (e : Entry) : Prop :=
+  1 ≤ e.1 ∧ match e.2 with
+    | none => True
+    | some (k,j) => j < multiplicity κ W e.1 k
+
+def entryValue (e : Entry) : ℝ :=
+  match e.2 with
+    | none => e.1
+    | some (k,_) => (2 : ℝ) ^ k
+
+def entryPosition (κ : ℝ) (W : WienerLaw) (e : Entry) : ℕ :=
+  match e.2 with
+    | none => ∑' k : ℤ, multiplicity κ W e.1 k
+    | some (k,j) => (∑' l : ℤ, if l < k then multiplicity κ W e.1 l else 0) + j
+
+def entryRadius (κ : ℝ) (W : WienerLaw) (e : Entry) : ℝ :=
+  Real.exp (-(e.1 : ℝ) ^ 2 - 2 * a κ * (entryPosition κ W e : ℝ) *
+    ((batchLevel κ W e.1 : ℝ) + (e.1 : ℝ)))
+
+def entryCost (κ : ℝ) (W : WienerLaw) (e : Entry) (r : ℝ) : ℝ :=
+  entryValue e * entryRadius κ W e ^ d κ * max 1 ((r / entryRadius κ W e) ^ 2)
+
+ 
+def gauge (κ : ℝ) (W : WienerLaw) (r : ℝ) : ℝ :=
+  if r ≤ 0 then 0 else sInf {v | ∃ e : Entry, ValidEntry κ W e ∧ v = entryCost κ W e r}
+
+def IsHausdorffGauge (h : ℝ → ℝ) : Prop :=
+  ContinuousOn h (Ici 0) ∧ MonotoneOn h (Ici 0) ∧ h 0 = 0 ∧ ∀ r > 0, 0 < h r
+
+ 
+def hausdorff (h : ℝ → ℝ) : OuterMeasure ℂ :=
+  OuterMeasure.mkMetric (fun r => if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal))
+
+def inverseFlow (κ : ℝ) (b : Driver) (t : ℝ) (w : ℂ) : ℂ :=
+  @ite ℂ (∃ z : ℂ, Surviving κ b z t ∧ flow κ b z t = w) (Classical.propDecidable _)
+    (Classical.epsilon (fun z : ℂ => Surviving κ b z t ∧ flow κ b z t = w)) 0
+
+ 
+def trace (κ : ℝ) (b : Driver) (t : ℝ) : ℂ :=
+  limUnder (nhdsWithin 0 (Ioi 0)) (fun ε : ℝ =>
+    inverseFlow κ b (κ * t) (driverValue b (κ * t) + ε * Complex.I))
+
+ 
+
+def MainObligation : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 → ∀ W : WienerLaw, IsWiener W →
+    IsHausdorffGauge (gauge κ W) ∧
+    ∀ᵐ b ∂W, ∀ s t : ℝ, 0 < s → s < t →
+      0 < hausdorff (gauge κ W) ((trace κ b) '' Icc s t) ∧
+      hausdorff (gauge κ W) ((trace κ b) '' Icc s t) < ∞
+
+end SLEExactGauge.LiteralTrace
+
+ 
+
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.Cutoff
+
+def denominator (q : ℝ) (z : ℂ) : ℝ := max (‖z‖^2) (q^2)
+
+def inverse (q : ℝ) (z : ℂ) : ℂ := star z / (denominator q z : ℂ)
+
+lemma denominator_pos {q : ℝ} (hq : 0 < q) (z : ℂ) : 0 < denominator q z :=
+  lt_of_lt_of_le (sq_pos_of_pos hq) (le_max_right _ _)
+
+lemma q_mul_norm_le_denominator {q : ℝ} (hq : 0 ≤ q) (z : ℂ) :
+    q * ‖z‖ ≤ denominator q z := by
+  by_cases he : ‖z‖ ≤ q
+  · exact (mul_le_mul_of_nonneg_left he hq).trans (by
+      change q*q ≤ max (‖z‖^2) (q^2)
+      rw [← sq]
+      exact le_max_right _ _)
+  · exact (mul_le_mul_of_nonneg_right (le_of_not_ge he) (norm_nonneg _)).trans (by
+      change ‖z‖*‖z‖ ≤ max (‖z‖^2) (q^2)
+      rw [← sq]
+      exact le_max_left _ _)
+
+lemma norm_inverse_le {q : ℝ} (hq : 0 < q) (z : ℂ) :
+    ‖inverse q z‖ ≤ q⁻¹ := by
+  rw [inverse, norm_div, norm_star, Complex.norm_real,
+    Real.norm_eq_abs, abs_of_pos (denominator_pos hq z)]
+  apply (div_le_iff₀ (denominator_pos hq z)).mpr
+  have hh := q_mul_norm_le_denominator hq.le z
+  have he : q⁻¹ * denominator q z = denominator q z / q := by ring
+  rw [he]
+  exact (le_div_iff₀ hq).mpr (by simpa only [mul_comm] using hh)
+
+lemma denominator_sub_le {q : ℝ} {z w : ℂ} (hzw : ‖z‖ ≤ ‖w‖) :
+    0 ≤ denominator q w - denominator q z ∧
+    denominator q w - denominator q z ≤ (‖w‖+‖z‖)*‖w-z‖ := by
+  have hn : ‖z‖^2 ≤ ‖w‖^2 := sq_le_sq₀ (norm_nonneg _) (norm_nonneg _) |>.mpr hzw
+  have hden : denominator q z ≤ denominator q w := max_le_max_right _ hn
+  refine ⟨sub_nonneg.mpr hden, ?_⟩
+  have hd : max (‖w‖^2) (q^2) - max (‖z‖^2) (q^2) ≤ ‖w‖^2 - ‖z‖^2 := by
+    by_cases hz : q^2 ≤ ‖z‖^2
+    · rw [max_eq_left (hz.trans hn), max_eq_left hz]
+    · rw [max_eq_right (le_of_not_ge hz)]
+      by_cases hw : q^2 ≤ ‖w‖^2
+      · rw [max_eq_left hw]
+        linarith
+      · rw [max_eq_right (le_of_not_ge hw)]
+        linarith
+  refine hd.trans ?_
+  calc ‖w‖^2 - ‖z‖^2 = (‖w‖+‖z‖)*(‖w‖-‖z‖) := by ring
+       _ ≤ _ := mul_le_mul_of_nonneg_left (norm_sub_norm_le w z)
+         (add_nonneg (norm_nonneg _) (norm_nonneg _))
+
+lemma inverse_sub_bound_ordered {q : ℝ} (hq : 0 < q) (z w : ℂ)
+    (hzw : ‖z‖ ≤ ‖w‖) :
+    ‖inverse q z - inverse q w‖ ≤ (3/q^2)*‖z-w‖ := by
+  let dz := denominator q z
+  let dw := denominator q w
+  have hz : 0 < dz := denominator_pos hq z
+  have hw : 0 < dw := denominator_pos hq w
+  have hzne : (dz : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt hz
+  have hwne : (dw : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt hw
+  have he : inverse q z - inverse q w =
+      (star z - star w)/(dw:ℂ) + star z * ((dw:ℂ)-(dz:ℂ))/((dz:ℂ)*(dw:ℂ)) := by
+    dsimp [inverse]
+    change star z/(dz:ℂ)-star w/(dw:ℂ) = _
+    change star z/(dz:ℂ)-star w/(dw:ℂ) =
+      (star z-star w)/(dw:ℂ)+star z*((dw:ℂ)-(dz:ℂ))/((dz:ℂ)*(dw:ℂ))
+    field_simp [hzne, hwne]
+    ring
+  have hd := denominator_sub_le (q := q) hzw
+  have hdz : q^2 ≤ dz := le_max_right _ _
+  have hdw : q^2 ≤ dw := le_max_right _ _
+  have hqw := q_mul_norm_le_denominator hq.le w
+  have htwo : ‖z‖*(dw-dz) ≤ 2 * dw * ‖z-w‖ := by
+    have hm := mul_le_mul_of_nonneg_left hd.2 (norm_nonneg z)
+    rw [norm_sub_rev w z] at hm
+    have hn : ‖z‖*(‖w‖+‖z‖) ≤ 2*dw := by
+      have hww : ‖w‖^2 ≤ dw := le_max_left _ _
+      nlinarith [norm_nonneg z, norm_nonneg w]
+    calc ‖z‖*(dw-dz) ≤ ‖z‖*((‖w‖+‖z‖)*‖z-w‖) := hm
+         _ = (‖z‖*(‖w‖+‖z‖))*‖z-w‖ := by ring
+         _ ≤ _ := mul_le_mul_of_nonneg_right hn (norm_nonneg _)
+  have hb₁ : ‖star z - star w‖ / dw ≤ ‖z-w‖ / q^2 := by
+    rw [← star_sub, norm_star]
+    exact div_le_div_of_nonneg_left (norm_nonneg _) (sq_pos_of_pos hq) hdw
+  have hb₂ : ‖z‖*(dw-dz)/(dz*dw) ≤ 2*‖z-w‖/(q^2) := by
+    calc
+      _ ≤ 2*dw*‖z-w‖/(dz*dw) := div_le_div_of_nonneg_right htwo (mul_nonneg hz.le hw.le)
+      _ = 2*‖z-w‖/dz := by field_simp
+      _ ≤ _ := div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hq) hdz
+  rw [he]
+  calc
+    _ ≤ ‖(star z-star w)/(dw:ℂ)‖ +
+        ‖star z*((dw:ℂ)-(dz:ℂ))/((dz:ℂ)*(dw:ℂ))‖ := norm_add_le _ _
+    _ = ‖star z-star w‖/dw + ‖z‖*(dw-dz)/(dz*dw) := by
+      rw [norm_div, norm_div, norm_mul, norm_mul, norm_star,
+        ← Complex.ofReal_sub, Complex.norm_real, Complex.norm_real,
+        Complex.norm_real, Real.norm_eq_abs, Real.norm_eq_abs, Real.norm_eq_abs,
+        abs_of_pos hw, abs_of_pos hz, abs_of_nonneg hd.1]
+    _ ≤ ‖z-w‖/q^2 + 2*‖z-w‖/q^2 := add_le_add hb₁ hb₂
+    _ = _ := by ring
+
+lemma inverse_sub_bound {q : ℝ} (hq : 0 < q) (z w : ℂ) :
+    ‖inverse q z - inverse q w‖ ≤ (3/q^2)*‖z-w‖ := by
+  rcases le_total ‖z‖ ‖w‖ with h | h
+  · exact inverse_sub_bound_ordered hq z w h
+  · simpa only [norm_sub_rev] using inverse_sub_bound_ordered hq w z h
+
+lemma lipschitz_inverse {q : ℝ} (hq : 0 < q) :
+    LipschitzWith ⟨3/q^2, by positivity⟩ (inverse q) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  change dist (inverse q z) (inverse q w) ≤ (3/q^2)*dist z w
+  simpa only [dist_eq_norm] using inverse_sub_bound hq z w
+
+ 
+def field (a q : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ) : ℂ :=
+  (a : ℂ) * inverse q (z - (b t : ℂ))
+
+lemma norm_field_le {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ) :
+    ‖field a q b t z‖ ≤ |a| / q := by
+  rw [field, norm_mul, Complex.norm_real, Real.norm_eq_abs]
+  calc |a| * ‖inverse q (z - (b t : ℂ))‖ ≤ |a| * q⁻¹ :=
+          mul_le_mul_of_nonneg_left (norm_inverse_le hq _) (abs_nonneg a)
+       _ = _ := div_eq_mul_inv _ _ |>.symm
+
+lemma field_sub_le {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ) (t : ℝ) (z w : ℂ) :
+    ‖field a q b t z - field a q b t w‖ ≤ (|a| *3/q^2)*‖z-w‖ := by
+  rw [field, field, ← mul_sub, norm_mul, Complex.norm_real, Real.norm_eq_abs]
+  have hh := mul_le_mul_of_nonneg_left
+    (inverse_sub_bound hq (z-(b t:ℂ)) (w-(b t:ℂ))) (abs_nonneg a)
+  simpa only [sub_sub_sub_cancel_right, mul_div_assoc, mul_assoc] using hh
+
+lemma continuous_field_time {q : ℝ} (hq : 0 < q) (a : ℝ) {b : ℝ → ℝ}
+    (hb : Continuous b) (z : ℂ) : Continuous (fun t => field a q b t z) := by
+  exact continuous_const.mul ((lipschitz_inverse hq).continuous.comp
+    (continuous_const.sub (Complex.continuous_ofReal.comp hb)))
+
+lemma field_joint_sub_le {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ)
+    (s t : ℝ) (z w : ℂ) :
+    ‖field a q b s z - field a q b t w‖ ≤
+      (|a| *3/q^2)*(‖z-w‖ + |b s-b t|) := by
+  rw [field, field, ← mul_sub, norm_mul, Complex.norm_real, Real.norm_eq_abs]
+  have hdiff : (z-(b s:ℂ))-(w-(b t:ℂ)) = (z-w)-((b s:ℂ)-(b t:ℂ)) := by ring
+  calc
+    _ ≤ |a| * ((3/q^2)*‖(z-(b s:ℂ))-(w-(b t:ℂ))‖) :=
+      mul_le_mul_of_nonneg_left (inverse_sub_bound hq _ _) (abs_nonneg _)
+    _ = (|a| *3/q^2)*‖(z-w)-((b s:ℂ)-(b t:ℂ))‖ := by rw [hdiff]; ring
+    _ ≤ (|a| *3/q^2)*(‖z-w‖ + ‖(b s:ℂ)-(b t:ℂ)‖) :=
+      mul_le_mul_of_nonneg_left (norm_sub_le _ _) (by positivity)
+    _ = _ := by rw [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs]
+
+ 
+
+theorem exists_solution_on_compact {q : ℝ} (hq : 0 < q) (a : ℝ) {b : ℝ → ℝ}
+    (hb : Continuous b) (z : ℂ) {T : ℝ} (hT : 0 ≤ T) :
+    ∃ g : ℝ → ℂ, g 0 = z ∧ ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt g (field a q b t (g t)) (Icc 0 T) t := by
+  let M : ℝ≥0 := ⟨|a|/q, by positivity⟩
+  let K : ℝ≥0 := ⟨|a| *3/q^2, by positivity⟩
+  let R : ℝ≥0 := ⟨(|a|/q)*T, mul_nonneg (by positivity) hT⟩
+  have hp : IsPicardLindelof (field a q b) (tmin := 0) (tmax := T)
+      ⟨0, ⟨le_rfl, hT⟩⟩ z R 0 M K := by
+    constructor
+    · intro t ht
+      apply LipschitzWith.lipschitzOnWith
+      apply LipschitzWith.of_dist_le_mul
+      intro w v
+      change dist (field a q b t w) (field a q b t v) ≤ (|a| *3/q^2)*dist w v
+      simpa only [dist_eq_norm] using field_sub_le hq a b t w v
+    · intro w hw
+      exact (continuous_field_time hq a hb w).continuousOn
+    · intro t ht w hw
+      exact norm_field_le hq a b t w
+    · change (|a|/q)*max (T-0) (0-0) ≤ (|a|/q)*T-0
+      simp only [sub_zero, sub_self, max_eq_left hT, le_refl]
+  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
+
+end SLEExactGauge.Cutoff
+
+ 
+
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.Euler
+
+def nodes (f : ℝ → ℂ → ℂ) (h : ℝ) (z : ℂ) : ℕ → ℂ
+  | 0 => z
+  | k+1 => nodes f h z k + h • f ((k:ℝ)*h) (nodes f h z k)
+
+end SLEExactGauge.Euler
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/DynkinIdentity.lean b/lean/OAI/Probability/SLEGauge/DynkinIdentity.lean
index bf132294fa735115d8e65ff5185bda4c62cd3bba..e375c074b49b2722bb1ebabb23419ca3428befc1 100644
--- a/lean/OAI/Probability/SLEGauge/DynkinIdentity.lean
+++ b/lean/OAI/Probability/SLEGauge/DynkinIdentity.lean
@@ -1,442 +1,442 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RiemannIntegrals
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-namespace ExactSLEGenerator
-variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace Ω]
-variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-theorem additive_brownian_dynkin (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (D : E → E →L[ℝ] ℝ) (hD : ∀ x, HasFDerivAt f (D x) x)
-    (hDc : Continuous D)
-    (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M)
-    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
-    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
-    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
-    (hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0))
-    (hfb : ∀ x, |f x| ≤ M) (hDb : ∀ x, ‖D x‖ ≤ M)
-    (hQb : ∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0| ≤ M)
-    (T : ℝ≥0) :
-    (∫ b, f (X T b) ∂W) - (∫ b, f (X 0 b) ∂W) =
-      (1/2 : ℝ) * (∫ b, (∫ t in 0..(T : ℝ),
-        iteratedDeriv 2 (fun r : ℝ => f (X t.toNNReal b + r • e)) 0) ∂W) +
-      (∫ b, (∫ t in 0..(T : ℝ), D (X t.toNNReal b) (A (X t.toNNReal b))) ∂W) := by
-  have hfc : Continuous f := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt)
-  let X' : ℝ → Driver → E := fun t b => X t.toNNReal b
-  have hX'c (b : Driver) : Continuous (fun t => X' t b) :=
-    (hXc b).comp continuous_real_toNNReal
-  have hX'm (t : ℝ) : Measurable (X' t) := (hXm t.toNNReal).mono (rawFiltration.le _) le_rfl
-  have hXs (t : ℝ≥0) (b : Driver) : X' t b = X t b := by simp [X']
-  let Q : E → ℝ := fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0
-  have hZc : Continuous (fun x => D x e) := hDc.clm_apply continuous_const
-  have hZi (t : ℝ) : Integrable (fun b => D (X' t b) e) W := by
-    apply Integrable.of_bound (hZc.measurable.comp (hX'm t)).aestronglyMeasurable M
-    refine ae_of_all W (fun b => ?_)
-    calc
-      _ ≤ ‖D (X' t b)‖ * ‖e‖ := (D _).le_opNorm _
-      _ ≤ M * 1 := mul_le_mul (hDb _) he (norm_nonneg _) hM
-      _ = M := mul_one _
-  have hVm (u v : ℝ) : AEStronglyMeasurable
-      (fun b => D (X' u b) (∫ t in u..v, A (X' t b))) W := by
-    have hd : AEStronglyMeasurable (fun b => D (X' u b)) W :=
-      hDc.comp_aestronglyMeasurable (hX'm u).aestronglyMeasurable
-    have hi : StronglyMeasurable (fun b => ∫ t in u..v, A (X' t b)) :=
-      stronglyMeasurable_intervalIntegral_of_continuous (fun t b => A (X' t b))
-        (fun b => hAc.comp (hX'c b)) (fun t => hAc.measurable.comp (hX'm t)) u v
-    exact (continuous_fst.clm_apply continuous_snd).comp_aestronglyMeasurable
-      (hd.prodMk hi.aestronglyMeasurable)
-  have hTaylor (s h : ℝ≥0) (b : Driver) :
-      |f (X' ((s+h : ℝ≥0) : ℝ) b) - f (X' s b) - D (X' s b) e * increment s h b -
-        Q (X' s b) * increment s h b ^ 2 / 2 -
-          D (X' s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤
-      C * (K*h*(|increment s h b|+K*h)+|increment s h b|^3) := by
-    have hv : ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤ K * (h : ℝ) := by
-      have hb := intervalIntegral.norm_integral_le_of_norm_le_const
-        (a := (s : ℝ)) (b := ((s+h : ℝ≥0) : ℝ)) (fun t ht => hAb (X' t b))
-      simpa only [NNReal.coe_add, add_sub_cancel_left, abs_of_nonneg h.coe_nonneg] using hb
-    have ht := mixed_taylor_remainder f D hD e he C hC hLip hSmooth hThird (X s b)
-      (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)) (increment s h b)
-    rw [← hXeq s h b] at ht
-    simp only [hXs]
-    change |f (X (s+h) b) - f (X s b) - D (X s b) e * increment s h b -
-      Q (X s b) * increment s h b ^ 2 / 2 - D (X s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤ _ at ht
-    have hs : |increment s h b| + ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤
-        |increment s h b| + K*(h : ℝ) := by linarith
-    have hm := mul_le_mul hv hs
-      (add_nonneg (abs_nonneg (increment s h b)) (norm_nonneg _))
-      (mul_nonneg hK h.coe_nonneg)
-    exact ht.trans (mul_le_mul_of_nonneg_left (add_le_add hm (le_refl _)) hC)
-  have hg := dynkin_of_mixed_taylor W hW 0
-    (fun t b => f (X' t b)) (fun t b => D (X' t b) e) (fun t b => Q (X' t b))
-    (fun t b => D (X' t b)) (fun t b => A (X' t b))
-    (fun t => Integrable.of_bound (hfc.measurable.comp (hX'm t)).aestronglyMeasurable M
-      (ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hfb (X' t b))))
-    (fun t => by
-      have ht : rawFiltration (0+t) = rawFiltration t := congrArg rawFiltration (zero_add t)
-      rw [ht]
-      simpa only [hXs, Function.comp_def] using hZc.measurable.comp (hXm t))
-    (fun t => by
-      have ht : rawFiltration (0+t) = rawFiltration t := congrArg rawFiltration (zero_add t)
-      rw [ht]
-      simpa only [hXs, Function.comp_def, Q] using hQc.measurable.comp (hXm t))
-    (fun t => hZi t) (fun b => hQc.comp (hX'c b))
-    (fun t => (hQc.measurable.comp (hX'm t)).aestronglyMeasurable)
-    (fun b => hDc.comp (hX'c b)) (fun b => hAc.comp (hX'c b)) hVm C M K hC hM hK
-    (fun b t => hQb _) (fun b t => hDb _) (fun b t => hAb _)
-    (fun s h b => by simpa only [NNReal.coe_add, zero_add] using hTaylor s h b) T
-  simpa only [hXs, X', Q, Real.toNNReal_zero] using hg
-
-end ExactSLEGenerator
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
- 
-
-theorem weighted_dynkin_of_mixed_taylor (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (start : ℝ≥0) (F Z Y : ℝ → Driver → ℝ) (D : ℝ → Driver → E →L[ℝ] ℝ) (A : ℝ → Driver → E)
-    (hF : ∀ t : ℝ≥0, Integrable (F t) W)
-    (hZm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Z t))
-    (hYm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Y t))
-    (hZi : ∀ t : ℝ≥0, Integrable (Z t) W)
-    (hYc : ∀ b, Continuous (fun t => Y t b))
-    (hYm' : ∀ t, AEStronglyMeasurable (Y t) W)
-    (hDc : ∀ b, Continuous (fun t => D t b))
-    (hAc : ∀ b, Continuous (fun t => A t b))
-    (hVm : ∀ u v, AEStronglyMeasurable (fun b => D u b (∫ t in u..v, A t b)) W)
-    (C M K : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M) (hK : 0 ≤ K)
-    (hYb : ∀ b t, |Y t b| ≤ M) (hDb : ∀ b t, ‖D t b‖ ≤ M)
-    (hAb : ∀ b t, ‖A t b‖ ≤ K)
-    (hTaylor : ∀ (s h : ℝ≥0) b,
-      |F (s+h) b - F s b - Z s b * increment (start+s) h b -
-        Y s b * increment (start+s) h b ^ 2 / 2 - D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b)| ≤
-      C * (K * h * (|increment (start+s) h b| + K * h) + |increment (start+s) h b| ^ 3))
-    (J : Driver → ℝ) (hJm : Measurable[rawFiltration start] J) (hJb : ∀ b, |J b| ≤ 1)
-    (T : ℝ≥0) :
-    (∫ b, J b * F T b ∂W) - (∫ b, J b * F 0 b ∂W) =
-      (1/2 : ℝ) * (∫ b, J b * (∫ t in 0..(T : ℝ), Y t b) ∂W) +
-      (∫ b, J b * (∫ t in 0..(T : ℝ), D t b (A t b)) ∂W) := by
-  have hJ : AEStronglyMeasurable J W := (hJm.mono (rawFiltration.le _) le_rfl).aestronglyMeasurable
-  have hJbound : ∀ᵐ b ∂W, ‖J b‖ ≤ 1 := ae_of_all W (fun b => by simpa using hJb b)
-  have hJmt (t : ℝ≥0) : Measurable[rawFiltration (start+t)] J :=
-    hJm.mono (rawFiltration.mono (le_self_add)) le_rfl
-  have h := dynkin_of_mixed_taylor W hW start
-    (fun t b => J b * F t b) (fun t b => J b * Z t b) (fun t b => J b * Y t b)
-    (fun t b => J b • D t b) A
-    (fun t => (hF t).bdd_mul hJ hJbound)
-    (fun t => (hJmt t).mul (hZm t)) (fun t => (hJmt t).mul (hYm t))
-    (fun t => (hZi t).bdd_mul hJ hJbound)
-    (fun b => continuous_const.mul (hYc b))
-    (fun t => hJ.mul (hYm' t))
-    (fun b => (continuous_const : Continuous (fun _ : ℝ => J b)).smul (hDc b)) hAc
-    (fun u v => hJ.mul (hVm u v))
-    C M K hC hM hK
-    (fun b t => by
-      rw [abs_mul]
-      exact (mul_le_mul (hJb b) (hYb b t) (abs_nonneg _) (by norm_num)).trans (by simp))
-    (fun b t => by
-      rw [norm_smul, Real.norm_eq_abs]
-      exact (mul_le_mul (hJb b) (hDb b t) (norm_nonneg _) (by norm_num)).trans (by simp))
-    hAb
-    (fun s h b => by
-      simp only [smul_apply, smul_eq_mul]
-      have he : J b * F (s+h) b - J b * F s b - J b * Z s b * increment (start+s) h b -
-          J b * Y s b * increment (start+s) h b ^ 2 / 2 -
-          J b * D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b) =
-          J b * (F (s+h) b - F s b - Z s b * increment (start+s) h b -
-          Y s b * increment (start+s) h b ^ 2 / 2 - D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b)) := by ring
-      rw [he, abs_mul]
-      exact (mul_le_mul_of_nonneg_right (hJb b) (abs_nonneg _)).trans (by simpa only [one_mul] using hTaylor s h b)) T
-  simpa only [smul_apply, smul_eq_mul, intervalIntegral.integral_const_mul] using h
-
-end ExactSLEGenerator
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-theorem additive_brownian_weighted_dynkin (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (start : ℝ≥0) (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration (start+t)] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment (start+s) h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (D : E → E →L[ℝ] ℝ) (hD : ∀ x, HasFDerivAt f (D x) x)
-    (hDc : Continuous D)
-    (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M)
-    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
-    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
-    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
-    (hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0))
-    (hfb : ∀ x, |f x| ≤ M) (hDb : ∀ x, ‖D x‖ ≤ M)
-    (hQb : ∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0| ≤ M)
-    (J : Driver → ℝ) (hJm : Measurable[rawFiltration start] J) (hJb : ∀ b, |J b| ≤ 1)
-    (T : ℝ≥0) :
-    (∫ b, J b * f (X T b) ∂W) - (∫ b, J b * f (X 0 b) ∂W) =
-      (1/2 : ℝ) * (∫ b, J b * (∫ t in 0..(T : ℝ),
-        iteratedDeriv 2 (fun r : ℝ => f (X t.toNNReal b + r • e)) 0) ∂W) +
-      (∫ b, J b * (∫ t in 0..(T : ℝ), D (X t.toNNReal b) (A (X t.toNNReal b))) ∂W) := by
-  have hfc : Continuous f := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt)
-  let X' : ℝ → Driver → E := fun t b => X t.toNNReal b
-  have hX'c (b : Driver) : Continuous (fun t => X' t b) :=
-    (hXc b).comp continuous_real_toNNReal
-  have hX'm (t : ℝ) : Measurable (X' t) := (hXm t.toNNReal).mono (rawFiltration.le _) le_rfl
-  have hXs (t : ℝ≥0) (b : Driver) : X' t b = X t b := by simp [X']
-  let Q : E → ℝ := fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0
-  have hZc : Continuous (fun x => D x e) := hDc.clm_apply continuous_const
-  have hZi (t : ℝ) : Integrable (fun b => D (X' t b) e) W := by
-    apply Integrable.of_bound (hZc.measurable.comp (hX'm t)).aestronglyMeasurable M
-    refine ae_of_all W (fun b => ?_)
-    calc
-      _ ≤ ‖D (X' t b)‖ * ‖e‖ := (D _).le_opNorm _
-      _ ≤ M * 1 := mul_le_mul (hDb _) he (norm_nonneg _) hM
-      _ = M := mul_one _
-  have hVm (u v : ℝ) : AEStronglyMeasurable
-      (fun b => D (X' u b) (∫ t in u..v, A (X' t b))) W := by
-    have hd : AEStronglyMeasurable (fun b => D (X' u b)) W :=
-      hDc.comp_aestronglyMeasurable (hX'm u).aestronglyMeasurable
-    have hi : StronglyMeasurable (fun b => ∫ t in u..v, A (X' t b)) :=
-      stronglyMeasurable_intervalIntegral_of_continuous (fun t b => A (X' t b))
-        (fun b => hAc.comp (hX'c b)) (fun t => hAc.measurable.comp (hX'm t)) u v
-    exact (continuous_fst.clm_apply continuous_snd).comp_aestronglyMeasurable
-      (hd.prodMk hi.aestronglyMeasurable)
-  have hTaylor (s h : ℝ≥0) (b : Driver) :
-      |f (X' ((s+h : ℝ≥0) : ℝ) b) - f (X' s b) - D (X' s b) e * increment (start+s) h b -
-        Q (X' s b) * increment (start+s) h b ^ 2 / 2 -
-          D (X' s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤
-      C * (K*h*(|increment (start+s) h b|+K*h)+|increment (start+s) h b|^3) := by
-    have hv : ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤ K * (h : ℝ) := by
-      have hb := intervalIntegral.norm_integral_le_of_norm_le_const
-        (a := (s : ℝ)) (b := ((s+h : ℝ≥0) : ℝ)) (fun t ht => hAb (X' t b))
-      simpa only [NNReal.coe_add, add_sub_cancel_left, abs_of_nonneg h.coe_nonneg] using hb
-    have ht := mixed_taylor_remainder f D hD e he C hC hLip hSmooth hThird (X s b)
-      (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)) (increment (start+s) h b)
-    rw [← hXeq s h b] at ht
-    simp only [hXs]
-    change |f (X (s+h) b) - f (X s b) - D (X s b) e * increment (start+s) h b -
-      Q (X s b) * increment (start+s) h b ^ 2 / 2 - D (X s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤ _ at ht
-    have hs : |increment (start+s) h b| + ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤
-        |increment (start+s) h b| + K*(h : ℝ) := by linarith
-    have hm := mul_le_mul hv hs
-      (add_nonneg (abs_nonneg (increment (start+s) h b)) (norm_nonneg _))
-      (mul_nonneg hK h.coe_nonneg)
-    exact ht.trans (mul_le_mul_of_nonneg_left (add_le_add hm (le_refl _)) hC)
-  have hg := weighted_dynkin_of_mixed_taylor W hW start
-    (fun t b => f (X' t b)) (fun t b => D (X' t b) e) (fun t b => Q (X' t b))
-    (fun t b => D (X' t b)) (fun t b => A (X' t b))
-    (fun t => Integrable.of_bound (hfc.measurable.comp (hX'm t)).aestronglyMeasurable M
-      (ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hfb (X' t b))))
-    (fun t => by
-      simpa only [hXs, Function.comp_def] using hZc.measurable.comp (hXm t))
-    (fun t => by
-      simpa only [hXs, Function.comp_def, Q] using hQc.measurable.comp (hXm t))
-    (fun t => hZi t) (fun b => hQc.comp (hX'c b))
-    (fun t => (hQc.measurable.comp (hX'm t)).aestronglyMeasurable)
-    (fun b => hDc.comp (hX'c b)) (fun b => hAc.comp (hX'c b)) hVm C M K hC hM hK
-    (fun b t => hQb _) (fun b t => hDb _) (fun b t => hAb _)
-    (fun s h b => by simpa only [NNReal.coe_add] using hTaylor s h b) J hJm hJb T
-  simpa only [hXs, X', Q, Real.toNNReal_zero] using hg
-
-end ExactSLEGenerator
- 
-
-open MeasureTheory Filter Set Metric Function ODE
-open scoped NNReal Topology
-namespace ExactSLEFlow
-variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [MeasurableSpace Ω]
-
-omit [CompleteSpace E] in
-lemma stronglyMeasurable_intervalIntegral_of_continuous
-    (A : ℝ → Ω → E) (hc : ∀ b, Continuous (fun t => A t b))
-    (hm : ∀ t, Measurable (A t)) (u v : ℝ) :
-    StronglyMeasurable (fun b => ∫ t in u..v, A t b) := by
-  have hj : StronglyMeasurable (Function.uncurry A) :=
-    (measurable_uncurry_of_continuous_of_measurable hc hm).stronglyMeasurable
-  by_cases huv : u ≤ v
-  · simp_rw [intervalIntegral.integral_of_le huv]
-    exact hj.integral_prod_left
-  · simp_rw [intervalIntegral.integral_of_ge (le_of_not_ge huv)]
-    exact hj.integral_prod_left.neg
-
-omit [NormedSpace ℝ E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E]
-  [SecondCountableTopology E] in
-lemma bounded_forcing_picard (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A)
-    (hb : ∀ x, ‖A x‖ ≤ L) (u : ℝ → E) (hu : Continuous u) (x : E) (T : ℝ≥0) :
-    IsPicardLindelof (fun t y => A (y + u t))
-      (⟨0, by simp⟩ : Icc (0 : ℝ) T) x (L*T) 0 L K := by
-  constructor
-  · intro t ht
-    simpa only [mul_one, Function.comp_def, NNReal.coe_mul] using
-      (hA.comp (isometry_add_right (u t)).lipschitzWith).lipschitzOnWith
-        (s := closedBall x (L*T))
-  · intro y hy
-    exact (hA.continuous.comp (continuous_const.add hu)).continuousOn
-  · intro t ht y hy
-    exact hb _
-  · simp
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma picard_iterates_tendsto {f : ℝ → E → E} {tmin tmax : ℝ}
-    {t₀ : Icc tmin tmax} {x : E} {a L K : ℝ≥0}
-    (hf : IsPicardLindelof f t₀ x a 0 L K)
-    (α : ODE.FunSpace t₀ x 0 L)
-    (hα : IsFixedPt (ODE.FunSpace.next hf (mem_closedBall_self (by simp))) α)
-    (β : ODE.FunSpace t₀ x 0 L) (t : ℝ) :
-    Tendsto (fun n : ℕ => ((ODE.FunSpace.next hf (mem_closedBall_self (by simp)))^[n] β).compProj t)
-      atTop (𝓝 (α.compProj t)) := by
-  let t' : Icc tmin tmax := projIcc tmin tmax (le_trans t₀.2.1 t₀.2.2) t
-  rw [tendsto_iff_dist_tendsto_zero]
-  have hb (n : ℕ) := ODE.FunSpace.dist_iterate_next_apply_le hf
-    (mem_closedBall_self (by simp)) β α n t'
-  have he : ∀ n : ℕ, ((ODE.FunSpace.next hf (mem_closedBall_self (by simp)))^[n] α) = α :=
-    fun n => hα.iterate n
-  simp only [he] at hb
-  have ht := (FloorSemiring.tendsto_pow_div_factorial_atTop
-    ((K : ℝ) * |(t' : ℝ) - (t₀ : ℝ)|)).mul_const (dist β α)
-  simp only [zero_mul] at ht
-  exact squeeze_zero (fun n => dist_nonneg)
-    (fun n => hb n) ht
-
- 
-
-theorem measurable_additive_flow_on_interval
-    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
-    (u : ℝ → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
-    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) (T : ℝ≥0) :
-    ∃ Y : ℝ → Ω → E,
-      (∀ ω, Continuous (fun t => Y t ω)) ∧
-      (∀ t, Measurable (Y t)) ∧
-      (∀ ω t, t ∈ Icc (0 : ℝ) T →
-        Y t ω = x ω + ∫ v in 0..t, A (Y v ω + u v ω)) ∧
-      (∀ ω t, t ∈ Icc (0 : ℝ) T →
-        HasDerivWithinAt (fun v => Y v ω) (A (Y t ω + u t ω)) (Icc (0 : ℝ) T) t) := by
-  let t₀ : Icc (0 : ℝ) T := ⟨0, by simp⟩
-  have hf (ω : Ω) : IsPicardLindelof (fun t y => A (y + u t ω))
-      t₀ (x ω) (L*T) 0 L K := bounded_forcing_picard A L K hA hb _ (huc ω) _ _
-  have hfix (ω : Ω) := ODE.FunSpace.exists_isFixedPt_next (hf ω)
-    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
-  choose α hα using hfix
-  let Y : ℝ → Ω → E := fun t ω => (α ω).compProj t
-  let β : (ω : Ω) → ODE.FunSpace t₀ (x ω) 0 L := fun ω => default
-  let N (ω : Ω) := ODE.FunSpace.next (hf ω)
-    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
-  have hm (n : ℕ) (t : ℝ) : Measurable (fun ω => ((N ω)^[n] (β ω)).compProj t) := by
-    induction n generalizing t with
-    | zero => change Measurable x; exact hxm
-    | succ n ih =>
-      have hc (ω : Ω) : Continuous (fun v => A (((N ω)^[n] (β ω)).compProj v + u v ω)) :=
-        hA.continuous.comp ((ODE.FunSpace.continuous_compProj _).add (huc ω))
-      have hi := stronglyMeasurable_intervalIntegral_of_continuous
-        (fun v ω => A (((N ω)^[n] (β ω)).compProj v + u v ω)) hc
-        (fun v => hA.continuous.measurable.comp ((ih v).add (hum v)))
-        0 (projIcc (0 : ℝ) T T.coe_nonneg t)
-      simp only [iterate_succ_apply']
-      change Measurable (fun ω => x ω + ∫ v in (0 : ℝ)..
-        (projIcc (0 : ℝ) T T.coe_nonneg t : ℝ),
-          A (((N ω)^[n] (β ω)).compProj v + u v ω))
-      exact hxm.add hi.measurable
-  refine ⟨Y, fun ω => (α ω).continuous_compProj, fun t => ?_, ?_, ?_⟩
-  · exact measurable_of_tendsto_metrizable (fun n => hm n t)
-      (tendsto_pi_nhds.mpr (fun ω => picard_iterates_tendsto (hf ω) (α ω) (hα ω) (β ω) t))
-  · intro ω t ht
-    have he := (ODE.FunSpace.isFixedPt_next_iff (hf ω)
-      (mem_closedBall_self (by simp))).mp (hα ω) ⟨t, ht⟩
-    change (α ω).compProj t = x ω + ∫ v in 0..t, A ((α ω).compProj v + u v ω)
-    rw [ODE.FunSpace.compProj_of_mem ht]
-    exact he
-  · intro ω t ht
-    apply ODE.hasDerivWithinAt_picard_Icc t₀.2 (hf ω).continuousOn_uncurry
-      (α ω).continuous_compProj.continuousOn
-      (fun _ _ => (α ω).compProj_mem_closedBall (hf ω).mul_max_le) (x ω) ht |>.congr_of_mem _ ht
-    intro v hv
-    change (α ω).compProj v = _
-    nth_rw 1 [← hα ω]
-    rw [ODE.FunSpace.compProj_of_mem hv]
-    rfl
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma additive_flow_unique {A : E → E} {K : ℝ≥0} (hA : LipschitzWith K A)
-    {u : ℝ → E} {f g : ℝ → E} {T : ℝ≥0}
-    (hfc : ContinuousOn f (Icc 0 (T : ℝ))) (hgc : ContinuousOn g (Icc 0 (T : ℝ)))
-    (hfd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt f (A (f t + u t)) (Icc 0 (T : ℝ)) t)
-    (hgd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt g (A (g t + u t)) (Icc 0 (T : ℝ)) t)
-    (hinit : f 0 = g 0) : EqOn f g (Icc 0 (T : ℝ)) := by
-  apply ODE_solution_unique (K := K) (v := fun t y => A (y + u t))
-    (fun t => by simpa only [mul_one, Function.comp_def] using hA.comp (isometry_add_right (u t)).lipschitzWith)
-    hfc (fun t ht => (hfd t (mem_Icc_of_Ico ht)).mono_of_mem_nhdsWithin
-      (Icc_mem_nhdsGE_of_mem ht))
-    hgc (fun t ht => (hgd t (mem_Icc_of_Ico ht)).mono_of_mem_nhdsWithin
-      (Icc_mem_nhdsGE_of_mem ht)) hinit
-
- 
-
-theorem measurable_additive_flow
-    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
-    (u : ℝ≥0 → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
-    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) :
-    ∃ Y : ℝ≥0 → Ω → E,
-      (∀ ω, Continuous (fun t => Y t ω)) ∧
-      (∀ t, Measurable (Y t)) ∧
-      (∀ ω t, Y t ω = x ω + ∫ v in (0 : ℝ)..(t : ℝ),
-        A (Y v.toNNReal ω + u v.toNNReal ω)) := by
-  have hex (T : ℝ≥0) := measurable_additive_flow_on_interval A L K hA hb
-    (fun t ω => u t.toNNReal ω) (fun ω => (huc ω).comp continuous_real_toNNReal)
-    (fun t => hum _) x hxm T
-  choose F hFc hFm hFe hFd using hex
-  have hF0 (T : ℝ≥0) (ω : Ω) : F T 0 ω = x ω := by
-    simpa using hFe T ω 0 (by simp)
-  have hcompat {S T : ℝ≥0} (hST : S ≤ T) (ω : Ω) (v : ℝ) (hv : v ∈ Icc 0 (S : ℝ)) :
-      F S v ω = F T v ω := by
-    have hsub : Icc (0 : ℝ) S ⊆ Icc (0 : ℝ) T := Icc_subset_Icc_right hST
-    apply additive_flow_unique hA (hFc S ω).continuousOn (hFc T ω).continuousOn
-      (hFd S ω) (fun t ht => (hFd T ω t (hsub ht)).mono hsub)
-      (by rw [hF0, hF0]) hv
-  let Y : ℝ≥0 → Ω → E := fun t ω => F t t ω
-  have hYF {T t : ℝ≥0} (ht : t ≤ T) (ω : Ω) : Y t ω = F T t ω :=
-    hcompat ht ω t (by simp)
-  refine ⟨Y, ?_, fun t => hFm t t, ?_⟩
-  · intro ω
-    apply continuous_iff_continuousAt.mpr
-    intro t
-    have hc : ContinuousAt (fun v : ℝ≥0 => F (t+1) v ω) t :=
-      ((hFc (t+1) ω).comp continuous_subtype_val).continuousAt
-    apply hc.congr
-    filter_upwards [eventually_lt_nhds (show t < t+1 by simp)] with v hv
-    exact (hYF hv.le ω).symm
-  · intro ω t
-    rw [hYF le_rfl ω, hFe t ω t (by simp)]
-    congr 1
-    apply intervalIntegral.integral_congr
-    intro v hv
-    have hv' : v ∈ Icc (0 : ℝ) t := by simpa [uIcc_of_le t.coe_nonneg] using hv
-    have hvt : v.toNNReal ≤ t := (Real.toNNReal_le_toNNReal hv'.2).trans_eq (by simp)
-    dsimp only
-    rw [hYF hvt ω, Real.coe_toNNReal _ hv'.1]
-
-end ExactSLEFlow
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RiemannIntegrals
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+namespace ExactSLEGenerator
+variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace Ω]
+variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+theorem additive_brownian_dynkin (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (D : E → E →L[ℝ] ℝ) (hD : ∀ x, HasFDerivAt f (D x) x)
+    (hDc : Continuous D)
+    (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M)
+    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
+    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
+    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
+    (hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0))
+    (hfb : ∀ x, |f x| ≤ M) (hDb : ∀ x, ‖D x‖ ≤ M)
+    (hQb : ∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0| ≤ M)
+    (T : ℝ≥0) :
+    (∫ b, f (X T b) ∂W) - (∫ b, f (X 0 b) ∂W) =
+      (1/2 : ℝ) * (∫ b, (∫ t in 0..(T : ℝ),
+        iteratedDeriv 2 (fun r : ℝ => f (X t.toNNReal b + r • e)) 0) ∂W) +
+      (∫ b, (∫ t in 0..(T : ℝ), D (X t.toNNReal b) (A (X t.toNNReal b))) ∂W) := by
+  have hfc : Continuous f := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt)
+  let X' : ℝ → Driver → E := fun t b => X t.toNNReal b
+  have hX'c (b : Driver) : Continuous (fun t => X' t b) :=
+    (hXc b).comp continuous_real_toNNReal
+  have hX'm (t : ℝ) : Measurable (X' t) := (hXm t.toNNReal).mono (rawFiltration.le _) le_rfl
+  have hXs (t : ℝ≥0) (b : Driver) : X' t b = X t b := by simp [X']
+  let Q : E → ℝ := fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0
+  have hZc : Continuous (fun x => D x e) := hDc.clm_apply continuous_const
+  have hZi (t : ℝ) : Integrable (fun b => D (X' t b) e) W := by
+    apply Integrable.of_bound (hZc.measurable.comp (hX'm t)).aestronglyMeasurable M
+    refine ae_of_all W (fun b => ?_)
+    calc
+      _ ≤ ‖D (X' t b)‖ * ‖e‖ := (D _).le_opNorm _
+      _ ≤ M * 1 := mul_le_mul (hDb _) he (norm_nonneg _) hM
+      _ = M := mul_one _
+  have hVm (u v : ℝ) : AEStronglyMeasurable
+      (fun b => D (X' u b) (∫ t in u..v, A (X' t b))) W := by
+    have hd : AEStronglyMeasurable (fun b => D (X' u b)) W :=
+      hDc.comp_aestronglyMeasurable (hX'm u).aestronglyMeasurable
+    have hi : StronglyMeasurable (fun b => ∫ t in u..v, A (X' t b)) :=
+      stronglyMeasurable_intervalIntegral_of_continuous (fun t b => A (X' t b))
+        (fun b => hAc.comp (hX'c b)) (fun t => hAc.measurable.comp (hX'm t)) u v
+    exact (continuous_fst.clm_apply continuous_snd).comp_aestronglyMeasurable
+      (hd.prodMk hi.aestronglyMeasurable)
+  have hTaylor (s h : ℝ≥0) (b : Driver) :
+      |f (X' ((s+h : ℝ≥0) : ℝ) b) - f (X' s b) - D (X' s b) e * increment s h b -
+        Q (X' s b) * increment s h b ^ 2 / 2 -
+          D (X' s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤
+      C * (K*h*(|increment s h b|+K*h)+|increment s h b|^3) := by
+    have hv : ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤ K * (h : ℝ) := by
+      have hb := intervalIntegral.norm_integral_le_of_norm_le_const
+        (a := (s : ℝ)) (b := ((s+h : ℝ≥0) : ℝ)) (fun t ht => hAb (X' t b))
+      simpa only [NNReal.coe_add, add_sub_cancel_left, abs_of_nonneg h.coe_nonneg] using hb
+    have ht := mixed_taylor_remainder f D hD e he C hC hLip hSmooth hThird (X s b)
+      (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)) (increment s h b)
+    rw [← hXeq s h b] at ht
+    simp only [hXs]
+    change |f (X (s+h) b) - f (X s b) - D (X s b) e * increment s h b -
+      Q (X s b) * increment s h b ^ 2 / 2 - D (X s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤ _ at ht
+    have hs : |increment s h b| + ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤
+        |increment s h b| + K*(h : ℝ) := by linarith
+    have hm := mul_le_mul hv hs
+      (add_nonneg (abs_nonneg (increment s h b)) (norm_nonneg _))
+      (mul_nonneg hK h.coe_nonneg)
+    exact ht.trans (mul_le_mul_of_nonneg_left (add_le_add hm (le_refl _)) hC)
+  have hg := dynkin_of_mixed_taylor W hW 0
+    (fun t b => f (X' t b)) (fun t b => D (X' t b) e) (fun t b => Q (X' t b))
+    (fun t b => D (X' t b)) (fun t b => A (X' t b))
+    (fun t => Integrable.of_bound (hfc.measurable.comp (hX'm t)).aestronglyMeasurable M
+      (ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hfb (X' t b))))
+    (fun t => by
+      have ht : rawFiltration (0+t) = rawFiltration t := congrArg rawFiltration (zero_add t)
+      rw [ht]
+      simpa only [hXs, Function.comp_def] using hZc.measurable.comp (hXm t))
+    (fun t => by
+      have ht : rawFiltration (0+t) = rawFiltration t := congrArg rawFiltration (zero_add t)
+      rw [ht]
+      simpa only [hXs, Function.comp_def, Q] using hQc.measurable.comp (hXm t))
+    (fun t => hZi t) (fun b => hQc.comp (hX'c b))
+    (fun t => (hQc.measurable.comp (hX'm t)).aestronglyMeasurable)
+    (fun b => hDc.comp (hX'c b)) (fun b => hAc.comp (hX'c b)) hVm C M K hC hM hK
+    (fun b t => hQb _) (fun b t => hDb _) (fun b t => hAb _)
+    (fun s h b => by simpa only [NNReal.coe_add, zero_add] using hTaylor s h b) T
+  simpa only [hXs, X', Q, Real.toNNReal_zero] using hg
+
+end ExactSLEGenerator
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ 
+
+theorem weighted_dynkin_of_mixed_taylor (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (start : ℝ≥0) (F Z Y : ℝ → Driver → ℝ) (D : ℝ → Driver → E →L[ℝ] ℝ) (A : ℝ → Driver → E)
+    (hF : ∀ t : ℝ≥0, Integrable (F t) W)
+    (hZm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Z t))
+    (hYm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Y t))
+    (hZi : ∀ t : ℝ≥0, Integrable (Z t) W)
+    (hYc : ∀ b, Continuous (fun t => Y t b))
+    (hYm' : ∀ t, AEStronglyMeasurable (Y t) W)
+    (hDc : ∀ b, Continuous (fun t => D t b))
+    (hAc : ∀ b, Continuous (fun t => A t b))
+    (hVm : ∀ u v, AEStronglyMeasurable (fun b => D u b (∫ t in u..v, A t b)) W)
+    (C M K : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M) (hK : 0 ≤ K)
+    (hYb : ∀ b t, |Y t b| ≤ M) (hDb : ∀ b t, ‖D t b‖ ≤ M)
+    (hAb : ∀ b t, ‖A t b‖ ≤ K)
+    (hTaylor : ∀ (s h : ℝ≥0) b,
+      |F (s+h) b - F s b - Z s b * increment (start+s) h b -
+        Y s b * increment (start+s) h b ^ 2 / 2 - D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b)| ≤
+      C * (K * h * (|increment (start+s) h b| + K * h) + |increment (start+s) h b| ^ 3))
+    (J : Driver → ℝ) (hJm : Measurable[rawFiltration start] J) (hJb : ∀ b, |J b| ≤ 1)
+    (T : ℝ≥0) :
+    (∫ b, J b * F T b ∂W) - (∫ b, J b * F 0 b ∂W) =
+      (1/2 : ℝ) * (∫ b, J b * (∫ t in 0..(T : ℝ), Y t b) ∂W) +
+      (∫ b, J b * (∫ t in 0..(T : ℝ), D t b (A t b)) ∂W) := by
+  have hJ : AEStronglyMeasurable J W := (hJm.mono (rawFiltration.le _) le_rfl).aestronglyMeasurable
+  have hJbound : ∀ᵐ b ∂W, ‖J b‖ ≤ 1 := ae_of_all W (fun b => by simpa using hJb b)
+  have hJmt (t : ℝ≥0) : Measurable[rawFiltration (start+t)] J :=
+    hJm.mono (rawFiltration.mono (le_self_add)) le_rfl
+  have h := dynkin_of_mixed_taylor W hW start
+    (fun t b => J b * F t b) (fun t b => J b * Z t b) (fun t b => J b * Y t b)
+    (fun t b => J b • D t b) A
+    (fun t => (hF t).bdd_mul hJ hJbound)
+    (fun t => (hJmt t).mul (hZm t)) (fun t => (hJmt t).mul (hYm t))
+    (fun t => (hZi t).bdd_mul hJ hJbound)
+    (fun b => continuous_const.mul (hYc b))
+    (fun t => hJ.mul (hYm' t))
+    (fun b => (continuous_const : Continuous (fun _ : ℝ => J b)).smul (hDc b)) hAc
+    (fun u v => hJ.mul (hVm u v))
+    C M K hC hM hK
+    (fun b t => by
+      rw [abs_mul]
+      exact (mul_le_mul (hJb b) (hYb b t) (abs_nonneg _) (by norm_num)).trans (by simp))
+    (fun b t => by
+      rw [norm_smul, Real.norm_eq_abs]
+      exact (mul_le_mul (hJb b) (hDb b t) (norm_nonneg _) (by norm_num)).trans (by simp))
+    hAb
+    (fun s h b => by
+      simp only [smul_apply, smul_eq_mul]
+      have he : J b * F (s+h) b - J b * F s b - J b * Z s b * increment (start+s) h b -
+          J b * Y s b * increment (start+s) h b ^ 2 / 2 -
+          J b * D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b) =
+          J b * (F (s+h) b - F s b - Z s b * increment (start+s) h b -
+          Y s b * increment (start+s) h b ^ 2 / 2 - D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b)) := by ring
+      rw [he, abs_mul]
+      exact (mul_le_mul_of_nonneg_right (hJb b) (abs_nonneg _)).trans (by simpa only [one_mul] using hTaylor s h b)) T
+  simpa only [smul_apply, smul_eq_mul, intervalIntegral.integral_const_mul] using h
+
+end ExactSLEGenerator
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+theorem additive_brownian_weighted_dynkin (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (start : ℝ≥0) (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration (start+t)] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment (start+s) h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (D : E → E →L[ℝ] ℝ) (hD : ∀ x, HasFDerivAt f (D x) x)
+    (hDc : Continuous D)
+    (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M)
+    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
+    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
+    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
+    (hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0))
+    (hfb : ∀ x, |f x| ≤ M) (hDb : ∀ x, ‖D x‖ ≤ M)
+    (hQb : ∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0| ≤ M)
+    (J : Driver → ℝ) (hJm : Measurable[rawFiltration start] J) (hJb : ∀ b, |J b| ≤ 1)
+    (T : ℝ≥0) :
+    (∫ b, J b * f (X T b) ∂W) - (∫ b, J b * f (X 0 b) ∂W) =
+      (1/2 : ℝ) * (∫ b, J b * (∫ t in 0..(T : ℝ),
+        iteratedDeriv 2 (fun r : ℝ => f (X t.toNNReal b + r • e)) 0) ∂W) +
+      (∫ b, J b * (∫ t in 0..(T : ℝ), D (X t.toNNReal b) (A (X t.toNNReal b))) ∂W) := by
+  have hfc : Continuous f := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt)
+  let X' : ℝ → Driver → E := fun t b => X t.toNNReal b
+  have hX'c (b : Driver) : Continuous (fun t => X' t b) :=
+    (hXc b).comp continuous_real_toNNReal
+  have hX'm (t : ℝ) : Measurable (X' t) := (hXm t.toNNReal).mono (rawFiltration.le _) le_rfl
+  have hXs (t : ℝ≥0) (b : Driver) : X' t b = X t b := by simp [X']
+  let Q : E → ℝ := fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0
+  have hZc : Continuous (fun x => D x e) := hDc.clm_apply continuous_const
+  have hZi (t : ℝ) : Integrable (fun b => D (X' t b) e) W := by
+    apply Integrable.of_bound (hZc.measurable.comp (hX'm t)).aestronglyMeasurable M
+    refine ae_of_all W (fun b => ?_)
+    calc
+      _ ≤ ‖D (X' t b)‖ * ‖e‖ := (D _).le_opNorm _
+      _ ≤ M * 1 := mul_le_mul (hDb _) he (norm_nonneg _) hM
+      _ = M := mul_one _
+  have hVm (u v : ℝ) : AEStronglyMeasurable
+      (fun b => D (X' u b) (∫ t in u..v, A (X' t b))) W := by
+    have hd : AEStronglyMeasurable (fun b => D (X' u b)) W :=
+      hDc.comp_aestronglyMeasurable (hX'm u).aestronglyMeasurable
+    have hi : StronglyMeasurable (fun b => ∫ t in u..v, A (X' t b)) :=
+      stronglyMeasurable_intervalIntegral_of_continuous (fun t b => A (X' t b))
+        (fun b => hAc.comp (hX'c b)) (fun t => hAc.measurable.comp (hX'm t)) u v
+    exact (continuous_fst.clm_apply continuous_snd).comp_aestronglyMeasurable
+      (hd.prodMk hi.aestronglyMeasurable)
+  have hTaylor (s h : ℝ≥0) (b : Driver) :
+      |f (X' ((s+h : ℝ≥0) : ℝ) b) - f (X' s b) - D (X' s b) e * increment (start+s) h b -
+        Q (X' s b) * increment (start+s) h b ^ 2 / 2 -
+          D (X' s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤
+      C * (K*h*(|increment (start+s) h b|+K*h)+|increment (start+s) h b|^3) := by
+    have hv : ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤ K * (h : ℝ) := by
+      have hb := intervalIntegral.norm_integral_le_of_norm_le_const
+        (a := (s : ℝ)) (b := ((s+h : ℝ≥0) : ℝ)) (fun t ht => hAb (X' t b))
+      simpa only [NNReal.coe_add, add_sub_cancel_left, abs_of_nonneg h.coe_nonneg] using hb
+    have ht := mixed_taylor_remainder f D hD e he C hC hLip hSmooth hThird (X s b)
+      (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)) (increment (start+s) h b)
+    rw [← hXeq s h b] at ht
+    simp only [hXs]
+    change |f (X (s+h) b) - f (X s b) - D (X s b) e * increment (start+s) h b -
+      Q (X s b) * increment (start+s) h b ^ 2 / 2 - D (X s b) (∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b))| ≤ _ at ht
+    have hs : |increment (start+s) h b| + ‖∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X' t b)‖ ≤
+        |increment (start+s) h b| + K*(h : ℝ) := by linarith
+    have hm := mul_le_mul hv hs
+      (add_nonneg (abs_nonneg (increment (start+s) h b)) (norm_nonneg _))
+      (mul_nonneg hK h.coe_nonneg)
+    exact ht.trans (mul_le_mul_of_nonneg_left (add_le_add hm (le_refl _)) hC)
+  have hg := weighted_dynkin_of_mixed_taylor W hW start
+    (fun t b => f (X' t b)) (fun t b => D (X' t b) e) (fun t b => Q (X' t b))
+    (fun t b => D (X' t b)) (fun t b => A (X' t b))
+    (fun t => Integrable.of_bound (hfc.measurable.comp (hX'm t)).aestronglyMeasurable M
+      (ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hfb (X' t b))))
+    (fun t => by
+      simpa only [hXs, Function.comp_def] using hZc.measurable.comp (hXm t))
+    (fun t => by
+      simpa only [hXs, Function.comp_def, Q] using hQc.measurable.comp (hXm t))
+    (fun t => hZi t) (fun b => hQc.comp (hX'c b))
+    (fun t => (hQc.measurable.comp (hX'm t)).aestronglyMeasurable)
+    (fun b => hDc.comp (hX'c b)) (fun b => hAc.comp (hX'c b)) hVm C M K hC hM hK
+    (fun b t => hQb _) (fun b t => hDb _) (fun b t => hAb _)
+    (fun s h b => by simpa only [NNReal.coe_add] using hTaylor s h b) J hJm hJb T
+  simpa only [hXs, X', Q, Real.toNNReal_zero] using hg
+
+end ExactSLEGenerator
+ 
+
+open MeasureTheory Filter Set Metric Function ODE
+open scoped NNReal Topology
+namespace ExactSLEFlow
+variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [MeasurableSpace Ω]
+
+omit [CompleteSpace E] in
+lemma stronglyMeasurable_intervalIntegral_of_continuous
+    (A : ℝ → Ω → E) (hc : ∀ b, Continuous (fun t => A t b))
+    (hm : ∀ t, Measurable (A t)) (u v : ℝ) :
+    StronglyMeasurable (fun b => ∫ t in u..v, A t b) := by
+  have hj : StronglyMeasurable (Function.uncurry A) :=
+    (measurable_uncurry_of_continuous_of_measurable hc hm).stronglyMeasurable
+  by_cases huv : u ≤ v
+  · simp_rw [intervalIntegral.integral_of_le huv]
+    exact hj.integral_prod_left
+  · simp_rw [intervalIntegral.integral_of_ge (le_of_not_ge huv)]
+    exact hj.integral_prod_left.neg
+
+omit [NormedSpace ℝ E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E]
+  [SecondCountableTopology E] in
+lemma bounded_forcing_picard (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A)
+    (hb : ∀ x, ‖A x‖ ≤ L) (u : ℝ → E) (hu : Continuous u) (x : E) (T : ℝ≥0) :
+    IsPicardLindelof (fun t y => A (y + u t))
+      (⟨0, by simp⟩ : Icc (0 : ℝ) T) x (L*T) 0 L K := by
+  constructor
+  · intro t ht
+    simpa only [mul_one, Function.comp_def, NNReal.coe_mul] using
+      (hA.comp (isometry_add_right (u t)).lipschitzWith).lipschitzOnWith
+        (s := closedBall x (L*T))
+  · intro y hy
+    exact (hA.continuous.comp (continuous_const.add hu)).continuousOn
+  · intro t ht y hy
+    exact hb _
+  · simp
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma picard_iterates_tendsto {f : ℝ → E → E} {tmin tmax : ℝ}
+    {t₀ : Icc tmin tmax} {x : E} {a L K : ℝ≥0}
+    (hf : IsPicardLindelof f t₀ x a 0 L K)
+    (α : ODE.FunSpace t₀ x 0 L)
+    (hα : IsFixedPt (ODE.FunSpace.next hf (mem_closedBall_self (by simp))) α)
+    (β : ODE.FunSpace t₀ x 0 L) (t : ℝ) :
+    Tendsto (fun n : ℕ => ((ODE.FunSpace.next hf (mem_closedBall_self (by simp)))^[n] β).compProj t)
+      atTop (𝓝 (α.compProj t)) := by
+  let t' : Icc tmin tmax := projIcc tmin tmax (le_trans t₀.2.1 t₀.2.2) t
+  rw [tendsto_iff_dist_tendsto_zero]
+  have hb (n : ℕ) := ODE.FunSpace.dist_iterate_next_apply_le hf
+    (mem_closedBall_self (by simp)) β α n t'
+  have he : ∀ n : ℕ, ((ODE.FunSpace.next hf (mem_closedBall_self (by simp)))^[n] α) = α :=
+    fun n => hα.iterate n
+  simp only [he] at hb
+  have ht := (FloorSemiring.tendsto_pow_div_factorial_atTop
+    ((K : ℝ) * |(t' : ℝ) - (t₀ : ℝ)|)).mul_const (dist β α)
+  simp only [zero_mul] at ht
+  exact squeeze_zero (fun n => dist_nonneg)
+    (fun n => hb n) ht
+
+ 
+
+theorem measurable_additive_flow_on_interval
+    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
+    (u : ℝ → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
+    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) (T : ℝ≥0) :
+    ∃ Y : ℝ → Ω → E,
+      (∀ ω, Continuous (fun t => Y t ω)) ∧
+      (∀ t, Measurable (Y t)) ∧
+      (∀ ω t, t ∈ Icc (0 : ℝ) T →
+        Y t ω = x ω + ∫ v in 0..t, A (Y v ω + u v ω)) ∧
+      (∀ ω t, t ∈ Icc (0 : ℝ) T →
+        HasDerivWithinAt (fun v => Y v ω) (A (Y t ω + u t ω)) (Icc (0 : ℝ) T) t) := by
+  let t₀ : Icc (0 : ℝ) T := ⟨0, by simp⟩
+  have hf (ω : Ω) : IsPicardLindelof (fun t y => A (y + u t ω))
+      t₀ (x ω) (L*T) 0 L K := bounded_forcing_picard A L K hA hb _ (huc ω) _ _
+  have hfix (ω : Ω) := ODE.FunSpace.exists_isFixedPt_next (hf ω)
+    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
+  choose α hα using hfix
+  let Y : ℝ → Ω → E := fun t ω => (α ω).compProj t
+  let β : (ω : Ω) → ODE.FunSpace t₀ (x ω) 0 L := fun ω => default
+  let N (ω : Ω) := ODE.FunSpace.next (hf ω)
+    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
+  have hm (n : ℕ) (t : ℝ) : Measurable (fun ω => ((N ω)^[n] (β ω)).compProj t) := by
+    induction n generalizing t with
+    | zero => change Measurable x; exact hxm
+    | succ n ih =>
+      have hc (ω : Ω) : Continuous (fun v => A (((N ω)^[n] (β ω)).compProj v + u v ω)) :=
+        hA.continuous.comp ((ODE.FunSpace.continuous_compProj _).add (huc ω))
+      have hi := stronglyMeasurable_intervalIntegral_of_continuous
+        (fun v ω => A (((N ω)^[n] (β ω)).compProj v + u v ω)) hc
+        (fun v => hA.continuous.measurable.comp ((ih v).add (hum v)))
+        0 (projIcc (0 : ℝ) T T.coe_nonneg t)
+      simp only [iterate_succ_apply']
+      change Measurable (fun ω => x ω + ∫ v in (0 : ℝ)..
+        (projIcc (0 : ℝ) T T.coe_nonneg t : ℝ),
+          A (((N ω)^[n] (β ω)).compProj v + u v ω))
+      exact hxm.add hi.measurable
+  refine ⟨Y, fun ω => (α ω).continuous_compProj, fun t => ?_, ?_, ?_⟩
+  · exact measurable_of_tendsto_metrizable (fun n => hm n t)
+      (tendsto_pi_nhds.mpr (fun ω => picard_iterates_tendsto (hf ω) (α ω) (hα ω) (β ω) t))
+  · intro ω t ht
+    have he := (ODE.FunSpace.isFixedPt_next_iff (hf ω)
+      (mem_closedBall_self (by simp))).mp (hα ω) ⟨t, ht⟩
+    change (α ω).compProj t = x ω + ∫ v in 0..t, A ((α ω).compProj v + u v ω)
+    rw [ODE.FunSpace.compProj_of_mem ht]
+    exact he
+  · intro ω t ht
+    apply ODE.hasDerivWithinAt_picard_Icc t₀.2 (hf ω).continuousOn_uncurry
+      (α ω).continuous_compProj.continuousOn
+      (fun _ _ => (α ω).compProj_mem_closedBall (hf ω).mul_max_le) (x ω) ht |>.congr_of_mem _ ht
+    intro v hv
+    change (α ω).compProj v = _
+    nth_rw 1 [← hα ω]
+    rw [ODE.FunSpace.compProj_of_mem hv]
+    rfl
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma additive_flow_unique {A : E → E} {K : ℝ≥0} (hA : LipschitzWith K A)
+    {u : ℝ → E} {f g : ℝ → E} {T : ℝ≥0}
+    (hfc : ContinuousOn f (Icc 0 (T : ℝ))) (hgc : ContinuousOn g (Icc 0 (T : ℝ)))
+    (hfd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt f (A (f t + u t)) (Icc 0 (T : ℝ)) t)
+    (hgd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt g (A (g t + u t)) (Icc 0 (T : ℝ)) t)
+    (hinit : f 0 = g 0) : EqOn f g (Icc 0 (T : ℝ)) := by
+  apply ODE_solution_unique (K := K) (v := fun t y => A (y + u t))
+    (fun t => by simpa only [mul_one, Function.comp_def] using hA.comp (isometry_add_right (u t)).lipschitzWith)
+    hfc (fun t ht => (hfd t (mem_Icc_of_Ico ht)).mono_of_mem_nhdsWithin
+      (Icc_mem_nhdsGE_of_mem ht))
+    hgc (fun t ht => (hgd t (mem_Icc_of_Ico ht)).mono_of_mem_nhdsWithin
+      (Icc_mem_nhdsGE_of_mem ht)) hinit
+
+ 
+
+theorem measurable_additive_flow
+    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
+    (u : ℝ≥0 → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
+    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) :
+    ∃ Y : ℝ≥0 → Ω → E,
+      (∀ ω, Continuous (fun t => Y t ω)) ∧
+      (∀ t, Measurable (Y t)) ∧
+      (∀ ω t, Y t ω = x ω + ∫ v in (0 : ℝ)..(t : ℝ),
+        A (Y v.toNNReal ω + u v.toNNReal ω)) := by
+  have hex (T : ℝ≥0) := measurable_additive_flow_on_interval A L K hA hb
+    (fun t ω => u t.toNNReal ω) (fun ω => (huc ω).comp continuous_real_toNNReal)
+    (fun t => hum _) x hxm T
+  choose F hFc hFm hFe hFd using hex
+  have hF0 (T : ℝ≥0) (ω : Ω) : F T 0 ω = x ω := by
+    simpa using hFe T ω 0 (by simp)
+  have hcompat {S T : ℝ≥0} (hST : S ≤ T) (ω : Ω) (v : ℝ) (hv : v ∈ Icc 0 (S : ℝ)) :
+      F S v ω = F T v ω := by
+    have hsub : Icc (0 : ℝ) S ⊆ Icc (0 : ℝ) T := Icc_subset_Icc_right hST
+    apply additive_flow_unique hA (hFc S ω).continuousOn (hFc T ω).continuousOn
+      (hFd S ω) (fun t ht => (hFd T ω t (hsub ht)).mono hsub)
+      (by rw [hF0, hF0]) hv
+  let Y : ℝ≥0 → Ω → E := fun t ω => F t t ω
+  have hYF {T t : ℝ≥0} (ht : t ≤ T) (ω : Ω) : Y t ω = F T t ω :=
+    hcompat ht ω t (by simp)
+  refine ⟨Y, ?_, fun t => hFm t t, ?_⟩
+  · intro ω
+    apply continuous_iff_continuousAt.mpr
+    intro t
+    have hc : ContinuousAt (fun v : ℝ≥0 => F (t+1) v ω) t :=
+      ((hFc (t+1) ω).comp continuous_subtype_val).continuousAt
+    apply hc.congr
+    filter_upwards [eventually_lt_nhds (show t < t+1 by simp)] with v hv
+    exact (hYF hv.le ω).symm
+  · intro ω t
+    rw [hYF le_rfl ω, hFe t ω t (by simp)]
+    congr 1
+    apply intervalIntegral.integral_congr
+    intro v hv
+    have hv' : v ∈ Icc (0 : ℝ) t := by simpa [uIcc_of_le t.coe_nonneg] using hv
+    have hvt : v.toNNReal ≤ t := (Real.toNNReal_le_toNNReal hv'.2).trans_eq (by simp)
+    dsimp only
+    rw [hYF hvt ω, Real.coe_toNNReal _ hv'.1]
+
+end ExactSLEFlow
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/EulerConvergence.lean b/lean/OAI/Probability/SLEGauge/EulerConvergence.lean
index 13a56e5929850a008e5da0027778e32f1e35e718..512ef2a75516f6570f95f3a6e71763b308105659 100644
--- a/lean/OAI/Probability/SLEGauge/EulerConvergence.lean
+++ b/lean/OAI/Probability/SLEGauge/EulerConvergence.lean
@@ -1,467 +1,467 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StoppedEstimates
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-namespace SLEExactGauge
-namespace Parameter
-theorem weighted_young_pair (P : Parameter) {C ε t X Y Z : ℝ}
-    (hC : 0 < C) (hε : 0 < ε) (ht : 0 < t)
-    (hX : 0 ≤ X) (hY : 0 ≤ Y) (hZ : 0 ≤ Z) :
-    C * t ^ (-1 - P.η) * X ^ P.η * Y ^ P.η * Z ^ (1 - 2 * P.η) ≤
-      ε * (X + Y) + P.youngConstant C ε * t ^ (-P.M) * Z := by
-  let r := 1 - 2 * P.η
-  have hr : 0 < r := by dsimp [r]; linarith [P.η_lt_half]
-  have hη := P.η_pos
-  have hηε : 0 < P.η / ε := div_pos hη hε
-  have hεη : 0 < ε / P.η := div_pos hε hη
-  let B := C ^ (1 / r) * (P.η / ε) ^ (2 * P.η / r) * t ^ (-P.M)
-  have hB : 0 < B := by dsimp [B]; positivity
-  have heM : (-P.M) * r = -1 - P.η := by
-    have hM : P.M * r = 1 + P.η := by
-      dsimp [M, r]
-      exact div_mul_cancel₀ _ (by linarith [P.η_lt_half])
-    nlinarith
-  have hBr : B ^ r = C * (P.η / ε) ^ (2 * P.η) * t ^ (-1 - P.η) := by
-    dsimp only [B]
-    rw [Real.mul_rpow (by positivity : 0 ≤ C ^ (1 / r) *
-        (P.η / ε) ^ (2 * P.η / r)) (by positivity),
-      Real.mul_rpow (by positivity : 0 ≤ C ^ (1 / r)) (by positivity),
-      ← Real.rpow_mul hC.le, ← Real.rpow_mul hηε.le,
-      ← Real.rpow_mul ht.le, heM]
-    rw [div_mul_cancel₀ _ hr.ne', div_mul_cancel₀ _ hr.ne', Real.rpow_one]
-  have hinv : (ε / P.η) ^ P.η * (ε / P.η) ^ P.η *
-      (P.η / ε) ^ (2 * P.η) = 1 := by
-    rw [← Real.rpow_add hεη, show P.η + P.η = 2 * P.η by ring,
-      ← Real.mul_rpow hεη.le hηε.le]
-    have hprod : ε / P.η * (P.η / ε) = 1 := by field_simp
-    rw [hprod, Real.one_rpow]
-  have hamgm := Real.geom_mean_le_arith_mean3_weighted hη.le hη.le hr.le
-    (mul_nonneg hεη.le hX) (mul_nonneg hεη.le hY) (mul_nonneg hB.le hZ)
-    (show P.η + P.η + r = 1 by dsimp [r]; ring)
-  have hleft : (ε / P.η * X) ^ P.η * (ε / P.η * Y) ^ P.η * (B * Z) ^ r =
-      C * t ^ (-1 - P.η) * X ^ P.η * Y ^ P.η * Z ^ r := by
-    rw [Real.mul_rpow hεη.le hX, Real.mul_rpow hεη.le hY,
-      Real.mul_rpow hB.le hZ, hBr]
-    calc
-      _ = ((ε / P.η) ^ P.η * (ε / P.η) ^ P.η * (P.η / ε) ^ (2 * P.η)) *
-        (C * t ^ (-1 - P.η) * X ^ P.η * Y ^ P.η * Z ^ r) := by ring
-      _ = _ := by rw [hinv, one_mul]
-  have hright : P.η * (ε / P.η * X) + P.η * (ε / P.η * Y) + r * (B * Z) =
-      ε * (X + Y) + P.youngConstant C ε * t ^ (-P.M) * Z := by
-    dsimp only [B, youngConstant]
-    change P.η * (ε / P.η * X) + P.η * (ε / P.η * Y) +
-      r * (C ^ (1 / r) * (P.η / ε) ^ (2 * P.η / r) * t ^ (-P.M) * Z) =
-      ε * (X + Y) + r * C ^ (1 / r) * (P.η / ε) ^ (2 * P.η / r) * t ^ (-P.M) * Z
-    field_simp
-  rw [hleft, hright] at hamgm
-  exact hamgm
-
-end Parameter
-end SLEExactGauge
-
-open Filter Topology
-
-namespace SLEExactGauge.Parameter
-
-lemma hasDerivAt_factor_first (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
-    HasDerivAt (fun t : ℝ => P.angularDerivative δ (Real.sin t) (Real.cos t))
-      (P.p*(P.p-1)*Real.cos θ^2*Real.sin θ^(P.p-2)-P.p*Real.sin θ^P.p+
-       δ*(P.q*(P.q-1)*Real.cos θ^2*Real.sin θ^(P.q-2)-P.q*Real.sin θ^P.q)) θ := by
-  convert! (hasDerivAt_sinePower_first P.p θ hs).add
-    ((hasDerivAt_sinePower_first P.q θ hs).const_mul δ) using 1
-  ext t
-  unfold angularDerivative
-  simp only [Pi.add_apply]
-  ring
-
-lemma deriv_factor (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
-    deriv (fun t : ℝ => P.f δ (Real.sin t)) θ =
-      P.angularDerivative δ (Real.sin θ) (Real.cos θ) :=
-  (P.hasDerivAt_factor hs).deriv
-
-lemma factor_deriv_differentiable (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
-    HasDerivAt (deriv (fun t : ℝ => P.f δ (Real.sin t)))
-      (P.p*(P.p-1)*Real.cos θ^2*Real.sin θ^(P.p-2)-P.p*Real.sin θ^P.p+
-       δ*(P.q*(P.q-1)*Real.cos θ^2*Real.sin θ^(P.q-2)-P.q*Real.sin θ^P.q)) θ := by
-  have he : deriv (fun t : ℝ => P.f δ (Real.sin t)) =ᶠ[nhds θ]
-      (fun t : ℝ => P.angularDerivative δ (Real.sin t) (Real.cos t)) := by
-    filter_upwards [Real.continuous_sin.continuousAt.eventually (Ioi_mem_nhds hs)] with t ht
-    exact P.deriv_factor ht
-  exact (P.hasDerivAt_factor_first hs).congr_of_eventuallyEq he
-
-lemma angleGenerator_factor (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
-    P.angleGenerator (fun t : ℝ => P.f δ (Real.sin t)) θ =
-      P.angleGenerator (fun t : ℝ => Real.sin t^P.p) θ+
-      δ*P.angleGenerator (fun t : ℝ => Real.sin t^P.q) θ := by
-  unfold angleGenerator
-  rw [(P.factor_deriv_differentiable hs).deriv,P.deriv_factor hs,
-    secondDeriv_sinePower P.p θ hs,secondDeriv_sinePower P.q θ hs,
-    deriv_sinePower P.p θ hs,deriv_sinePower P.q θ hs]
-  unfold angularDerivative
-  ring
-
- 
-
-lemma shifted_sine_generator (P : Parameter) (r θ : ℝ) (hs : 0 < Real.sin θ) :
-    P.angleGenerator (fun t : ℝ => Real.sin t^r) θ+P.p/2*Real.sin θ^r =
-      (P.p-r)*(r+1)/2*Real.sin θ^r-r*(P.p-r)/2*Real.sin θ^(r-2) := by
-  have hp1 : Real.sin θ^(r-1) = Real.sin θ^r/Real.sin θ := by
-    rw [Real.rpow_sub hs,Real.rpow_one]
-  have hp2 : Real.sin θ^(r-2) = Real.sin θ^r/Real.sin θ^2 := by
-    rw [Real.rpow_sub hs,Real.rpow_two]
-  have hcos : Real.cos θ^2 = 1-Real.sin θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
-  unfold angleGenerator
-  rw [secondDeriv_sinePower r θ hs,deriv_sinePower r θ hs,hp1,hp2]
-  unfold p
-  field_simp
-  rw [hcos]
-  ring
-
- 
-
-theorem exact_factor_drift (P : Parameter) {δ θ : ℝ} (_hδ : 0 < δ)
-    (hs : 0 < Real.sin θ) (σ : ℝ) :
-    (P.angleGenerator (fun t : ℝ => P.f δ (Real.sin t)) θ+
-      P.p/2*P.f δ (Real.sin θ)+σ*δ*(1+Real.sin θ^P.q))/P.f δ (Real.sin θ) =
-      P.drift δ (Real.sin θ) σ := by
-  rw [P.angleGenerator_factor hs]
-  have hp := P.shifted_sine_generator P.p θ hs
-  have hq := P.shifted_sine_generator P.q θ hs
-  simp only [sub_self,zero_mul,mul_zero,zero_div] at hp
-  unfold drift f k at *
-  linear_combination (norm := ring_nf) hp / (Real.sin θ ^ P.p+δ*(1+Real.sin θ^P.q)) +
-    δ / (Real.sin θ ^ P.p+δ*(1+Real.sin θ^P.q)) * hq
-
-end SLEExactGauge.Parameter
-
- 
-
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-namespace SLEExactGauge.AngleCalculus
-
- 
-def angleD (z : ℂ) : ℂ →L[ℝ] ℝ :=
-  Complex.imCLM.comp (z⁻¹ • (1 : ℂ →L[ℝ] ℂ))
-
-lemma hasFDerivAt_arg {z : ℂ} (hz : z ∈ Complex.slitPlane) :
-    HasFDerivAt Complex.arg (angleD z) z := by
-  simpa only [angleD,Function.comp_def,Complex.imCLM_apply,Complex.log_im] using
-    Complex.imCLM.hasFDerivAt.comp z (Complex.hasStrictFDerivAt_log_real hz).hasFDerivAt
-
-lemma angleD_apply (z w : ℂ) : angleD z w = (w/z).im := by
-  simp [angleD,div_eq_mul_inv,mul_comm]
-
- 
-def V (z : ℂ) : ℝ := z.im / Complex.normSq z
-
-lemma driver_angle {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
-    HasDerivAt (fun u : ℝ => Complex.arg (z-u)) (V (z-b)) b := by
-  have hz : z-(b:ℂ) ∈ Complex.slitPlane := Or.inr (by simpa using ne_of_gt hy)
-  have hd := (hasFDerivAt_arg hz).comp_hasDerivAt b
-    ((hasDerivAt_const b z).sub Complex.ofRealCLM.hasDerivAt)
-  convert! hd using 1
-  rw [angleD_apply]
-  simp [V,Complex.div_im,Complex.normSq_apply]
-  ring
-
-lemma hasDerivAt_driver_V (z : ℂ) (b : ℝ) (hy : 0 < z.im) :
-    HasDerivAt (fun u : ℝ => V (z-u))
-      (2*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2) b := by
-  have hz : z-(b:ℂ) ≠ 0 := by
-    intro hh
-    have he := congrArg Complex.im hh
-    simp at he
-    linarith
-  have hn : Complex.normSq (z-(b:ℂ)) ≠ 0 := (Complex.normSq_pos.mpr hz).ne'
-  have hd : HasDerivAt (fun u : ℝ => (z.re-u)^2+z.im^2) (-2*(z.re-b)) b := by
-    convert! (((hasDerivAt_const b z.re).sub (hasDerivAt_id b)).pow 2).add_const (z.im^2) using 1
-    simp only [Pi.sub_apply,id_eq]
-    ring
-  have he := (hasDerivAt_const b z.im).div hd (by simpa [Complex.normSq_apply,pow_two] using hn)
-  convert! he using 1
-  · ext u
-    simp [V,Complex.normSq_apply,pow_two]
-  · simp [Complex.normSq_apply]
-    ring
-
-lemma deriv_driver_angle {z : ℂ} (hy : 0 < z.im) :
-    deriv (fun u : ℝ => Complex.arg (z-u)) = fun u : ℝ => V (z-u) := by
-  funext u
-  exact (driver_angle hy u).deriv
-
-lemma secondDeriv_driver_angle {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
-    deriv (deriv (fun u : ℝ => Complex.arg (z-u))) b =
-      2*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2 := by
-  rw [deriv_driver_angle hy]
-  exact (hasDerivAt_driver_V z b hy).deriv
-
-lemma loewner_angle {g : ℝ → ℂ} {a b t : ℝ}
-    (hg : HasDerivAt g ((a:ℂ)/(g t-b)) t) (hy : 0 < (g t).im) :
-    HasDerivAt (fun u => Complex.arg (g u-b))
-      (-2*a*(g t-b).re*(g t-b).im/(Complex.normSq (g t-b))^2) t := by
-  have hz : g t-(b:ℂ) ∈ Complex.slitPlane := Or.inr (by simpa using ne_of_gt hy)
-  convert! (hasFDerivAt_arg hz).comp_hasDerivAt t (hg.sub_const (b:ℂ)) using 1
-  rw [angleD_apply]
-  simp [Complex.div_im,Complex.div_re]
-  ring
-
- 
-
-lemma angle_generator_coefficients (a : ℝ) {z : ℂ} (hy : 0 < z.im) :
-    (-2*a*z.re*z.im/(Complex.normSq z)^2)+
-      (2*z.re*z.im/(Complex.normSq z)^2)/2 =
-      (1-2*a)*(Real.cos z.arg/Real.sin z.arg)*(V z)^2 := by
-  have hz : z ≠ 0 := by intro h; simp [h] at hy
-  have hn : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz
-  have hN : Complex.normSq z ≠ 0 := (Complex.normSq_pos.mpr hz).ne'
-  rw [Complex.sin_arg,Complex.cos_arg hz,V]
-  field_simp
-  ring
-
-end SLEExactGauge.AngleCalculus
-
-namespace SLEExactGauge.AngleCalculus
-
-lemma sin_angle_pos {z : ℂ} (hy : 0 < z.im) : 0 < Real.sin z.arg := by
-  rw [Complex.sin_arg]
-  apply div_pos hy
-  apply norm_pos_iff.mpr
-  intro h
-  simp [h] at hy
-
- 
-
-def physicalGenerator (a : ℝ) (φ : ℝ → ℝ) (z : ℂ) (b : ℝ) : ℝ :=
-  fderiv ℝ (fun w : ℂ => φ (Complex.arg (w-b))) z ((a:ℂ)/(z-b))+
-    deriv (deriv (fun c : ℝ => φ (Complex.arg (z-c)))) b/2
-
-lemma driver_comp_first (φ : ℝ → ℝ)
-    (hφ : ∀ θ, 0 < Real.sin θ → HasDerivAt φ (deriv φ θ) θ)
-    {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
-    HasDerivAt (fun c : ℝ => φ (Complex.arg (z-c)))
-      (deriv φ (Complex.arg (z-b))*V (z-b)) b :=
-  (hφ _ (sin_angle_pos (by simpa using hy))).comp b (driver_angle hy b)
-
-lemma driver_comp_second (φ : ℝ → ℝ)
-    (hφ : ∀ θ, 0 < Real.sin θ → HasDerivAt φ (deriv φ θ) θ)
-    (hφ' : ∀ θ, 0 < Real.sin θ → HasDerivAt (deriv φ) (deriv (deriv φ) θ) θ)
-    {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
-    deriv (deriv (fun c : ℝ => φ (Complex.arg (z-c)))) b =
-      deriv (deriv φ) (Complex.arg (z-b))*(V (z-b))^2+
-      deriv φ (Complex.arg (z-b))*(2*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2) := by
-  have he : deriv (fun c : ℝ => φ (Complex.arg (z-c))) =
-      fun c : ℝ => deriv φ (Complex.arg (z-c))*V (z-c) := by
-    funext c
-    exact (driver_comp_first φ hφ hy c).deriv
-  rw [he]
-  have hd := ((hφ' _ (sin_angle_pos (by simpa using hy))).comp b
-    (driver_angle hy b)).mul (hasDerivAt_driver_V z b hy)
-  convert! hd.deriv using 1
-  simp only [Function.comp_def]
-  ring
-
-lemma physical_generator_angle (a : ℝ) (φ : ℝ → ℝ)
-    (hφ : ∀ θ, 0 < Real.sin θ → HasDerivAt φ (deriv φ θ) θ)
-    (hφ' : ∀ θ, 0 < Real.sin θ → HasDerivAt (deriv φ) (deriv (deriv φ) θ) θ)
-    {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
-    physicalGenerator a φ z b = (V (z-b))^2 *
-      (deriv (deriv φ) (z-b).arg/2+(1-2*a)*
-        (Real.cos (z-b).arg/Real.sin (z-b).arg)*deriv φ (z-b).arg) := by
-  have hy' : 0 < (z-(b:ℂ)).im := by simpa using hy
-  have hz : z-(b:ℂ) ∈ Complex.slitPlane := Or.inr hy'.ne'
-  have hd := (hφ _ (sin_angle_pos hy')).hasFDerivAt.comp z
-    ((hasFDerivAt_arg hz).comp z ((hasFDerivAt_id z).sub_const (b:ℂ)))
-  have hv : angleD (z-b) ((a:ℂ)/(z-b)) =
-      -2*a*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2 := by
-    rw [angleD_apply]
-    simp [Complex.div_im,Complex.div_re]
-    ring
-  simp only [Function.comp_def,id_eq] at hd
-  unfold physicalGenerator
-  rw [hd.fderiv,driver_comp_second φ hφ hφ' hy b]
-  simp only [ContinuousLinearMap.comp_apply,
-    ContinuousLinearMap.id_apply,ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul]
-  rw [hv]
-  have hc := angle_generator_coefficients a hy'
-  linear_combination deriv φ (z-b).arg * hc
-
-end SLEExactGauge.AngleCalculus
-
-noncomputable section
-open scoped Topology
-namespace SLEExactGauge.Calculus
-
-lemma fderiv_separable (e δ : ℝ → ℝ) (φ ψ : ℂ → ℝ) (u : ℝ) (z : ℂ)
-    (e' δ' : ℝ) (Dφ Dψ : ℂ →L[ℝ] ℝ)
-    (he : HasDerivAt e e' u) (hδ : HasDerivAt δ δ' u)
-    (hφ : HasFDerivAt φ Dφ z) (hψ : HasFDerivAt ψ Dψ z) (v : ℝ) (w : ℂ) :
-    fderiv ℝ (fun x : ℝ × ℂ => e x.1*(φ x.2+δ x.1*ψ x.2)) (u,z) (v,w) =
-      e'*v*(φ z+δ u*ψ z)+e u*(Dφ w+δ'*v*ψ z+δ u*Dψ w) := by
-  have hfst : HasFDerivAt (fun x : ℝ × ℂ => x.1) (ContinuousLinearMap.fst ℝ ℝ ℂ) (u,z) := hasFDerivAt_fst
-  have hsnd : HasFDerivAt (fun x : ℝ × ℂ => x.2) (ContinuousLinearMap.snd ℝ ℝ ℂ) (u,z) := hasFDerivAt_snd
-  have heF := he.hasFDerivAt.comp (u,z) hfst
-  have hδF := hδ.hasFDerivAt.comp (u,z) hfst
-  have hφF := hφ.comp (u,z) hsnd
-  have hψF := hψ.comp (u,z) hsnd
-  have hd := heF.mul (hφF.add (hδF.mul hψF))
-  simp only [Function.comp_def,Pi.mul_apply,Pi.add_apply] at hd
-  change HasFDerivAt (fun x : ℝ × ℂ => e x.1*(φ x.2+δ x.1*ψ x.2)) _ (u,z) at hd
-  rw [hd.fderiv]
-  simp only [add_apply,ContinuousLinearMap.comp_apply,
-    smul_apply,
-    ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul]
-  change e u * (Dφ w + (δ u * Dψ w + ψ z * (v * δ'))) + (φ z + δ u * ψ z) * (v * e') = _
-  ring
-
-lemma secondDeriv_linear (C δ : ℝ) (φ ψ : ℝ → ℝ)
-    (hφ : ∀ x, DifferentiableAt ℝ φ x) (hψ : ∀ x, DifferentiableAt ℝ ψ x)
-    (hφ' : ∀ x, DifferentiableAt ℝ (deriv φ) x)
-    (hψ' : ∀ x, DifferentiableAt ℝ (deriv ψ) x) (b : ℝ) :
-    deriv (deriv (fun x => C*(φ x+δ*(1+ψ x)))) b =
-      C*(deriv (deriv φ) b+δ*deriv (deriv ψ) b) := by
-  have he : deriv (fun x => C*(φ x+δ*(1+ψ x))) =
-      fun x => C*(deriv φ x+δ*deriv ψ x) := by
-    funext x
-    simpa using ((hφ x).hasDerivAt.add (((hψ x).hasDerivAt.const_add 1).const_mul δ)).const_mul C |>.deriv
-  rw [he]
-  simpa using (((hφ' b).hasDerivAt.add ((hψ' b).hasDerivAt.const_mul δ)).const_mul C).deriv
-
-end SLEExactGauge.Calculus
-
-namespace SLEExactGauge.Parameter
-open AngleCalculus Calculus
-
- 
-
-def physicalFactor (P : Parameter) (δ : ℝ → ℝ) (x : ℝ × ℂ) (b : ℝ) : ℝ :=
-  Real.exp (P.p*x.1/2)/3 * P.f (δ x.1) (Real.sin (x.2-b).arg)
-
-lemma physical_factor_state_derivative (P : Parameter) (δ : ℝ → ℝ)
-    (u σ b v : ℝ) (z w : ℂ) (hy : 0 < z.im)
-    (hδ : HasDerivAt δ (σ*δ u) u) :
-    fderiv ℝ (fun x : ℝ × ℂ => P.physicalFactor δ x b) (u,z) (v,w) =
-      Real.exp (P.p*u/2)/3 *
-        (fderiv ℝ (fun z : ℂ => P.f (δ u) (Real.sin (z-b).arg)) z w+
-          (P.p/2*P.f (δ u) (Real.sin (z-b).arg)+σ*δ u*(1+Real.sin (z-b).arg^P.q))*v) := by
-  let e : ℝ → ℝ := fun r => Real.exp (P.p*r/2)/3
-  let φ : ℂ → ℝ := fun w => Real.sin (w-b).arg^P.p
-  let ψ : ℂ → ℝ := fun w => 1+Real.sin (w-b).arg^P.q
-  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
-  have ha : HasFDerivAt (fun w : ℂ => (w-b).arg) (angleD (z-b)) z := by
-    convert! (hasFDerivAt_arg (show z-(b:ℂ) ∈ Complex.slitPlane from Or.inr (by simpa using hy.ne'))).comp z
-      ((hasFDerivAt_id z).sub_const (b:ℂ)) using 1
-  have hφ := (hasDerivAt_sinePower P.p (z-b).arg hs).hasFDerivAt.comp z ha
-  have hψ := ((hasDerivAt_sinePower P.q (z-b).arg hs).const_add 1).hasFDerivAt.comp z ha
-  change HasFDerivAt φ _ z at hφ
-  change HasFDerivAt ψ _ z at hψ
-  have he : HasDerivAt e (P.p/2*e u) u := by
-    convert! (((hasDerivAt_id u).const_mul P.p).div_const 2).exp.div_const 3 using 1
-    dsimp [e]
-    ring
-  have hd := fderiv_separable e δ φ ψ u z (P.p/2*e u) (σ*δ u) _ _ he hδ hφ hψ v w
-  have hc := hφ.add (hψ.const_mul (δ u))
-  change HasFDerivAt (fun z : ℂ => P.f (δ u) (Real.sin (z-b).arg)) _ z at hc
-  change fderiv ℝ (fun x : ℝ × ℂ => e x.1*(φ x.2+δ x.1*ψ x.2)) (u,z) (v,w) = _
-  rw [hd,hc.fderiv]
-  simp only [add_apply,smul_apply]
-  change (P.p/2*e u)*v*(φ z+δ u*ψ z)+e u*(_+σ*δ u*v*ψ z+δ u*_) =
-    e u*(_+δ u*_+(P.p/2*(φ z+δ u*ψ z)+σ*δ u*ψ z)*v)
-  ring
-
- 
-
-theorem physical_factor_generator (P : Parameter) (δ : ℝ → ℝ) (u σ b : ℝ)
-    (z : ℂ) (hy : 0 < z.im) (hδ0 : 0 < δ u)
-    (hδ : HasDerivAt δ (σ*δ u) u) :
-    fderiv ℝ (fun x : ℝ × ℂ => P.physicalFactor δ x b) (u,z)
-      ((V (z-b))^2,(P.a:ℂ)/(z-b))+
-      deriv (deriv (P.physicalFactor δ (u,z))) b/2 =
-      P.physicalFactor δ (u,z) b*(V (z-b))^2*
-        P.drift (δ u) (Real.sin (z-b).arg) σ := by
-  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
-  have hphy := physical_generator_angle P.a (fun θ => P.f (δ u) (Real.sin θ))
-    (fun θ hθ => (P.hasDerivAt_factor hθ).differentiableAt.hasDerivAt)
-    (fun θ hθ => (P.factor_deriv_differentiable hθ).differentiableAt.hasDerivAt) hy b
-  change physicalGenerator P.a (fun θ => P.f (δ u) (Real.sin θ)) z b =
-    (V (z-b))^2*P.angleGenerator (fun θ => P.f (δ u) (Real.sin θ)) (z-b).arg at hphy
-  have hsecond : deriv (deriv (P.physicalFactor δ (u,z))) b =
-      Real.exp (P.p*u/2)/3*deriv (deriv (fun c : ℝ => P.f (δ u) (Real.sin (z-c).arg))) b := by
-    change deriv (deriv (fun c : ℝ => Real.exp (P.p*u/2)/3*P.f (δ u) (Real.sin (z-c).arg))) b = _
-    rw [deriv_const_mul_field',deriv_const_mul_field]
-  rw [P.physical_factor_state_derivative δ u σ b _ z _ hy hδ,hsecond]
-  have heq := (div_eq_iff (P.f_pos hδ0 hs).ne').mp (P.exact_factor_drift hδ0 hs σ)
-  unfold physicalGenerator at hphy
-  unfold physicalFactor
-  linear_combination (Real.exp (P.p*u/2)/3)*hphy +
-    (Real.exp (P.p*u/2)/3*(V (z-b))^2)*heq
-
-end SLEExactGauge.Parameter
-
-namespace SLEExactGauge.Parameter
-open AngleCalculus Calculus
-
-open scoped ContDiff
-lemma contDiffAt_arg_real {z : ℂ} (hz : z ∈ Complex.slitPlane) (n : ℕ∞ω) :
-    ContDiffAt ℝ n Complex.arg z := by
-  have h := Complex.imCLM.contDiff.contDiffAt.comp z
-    ((Complex.contDiffAt_log (n := n) hz).restrict_scalars ℝ)
-  change ContDiffAt ℝ n (fun x => (Complex.log x).im) z at h
-  simpa only [Complex.log_im] using h
-
-lemma contDiffAt_physical_factor (P : Parameter) (δ : ℝ → ℝ)
-    (u b : ℝ) (z : ℂ) (hy : 0 < z.im) (n : ℕ∞ω)
-    (hδ : ContDiffAt ℝ n δ u) :
-    ContDiffAt ℝ n (fun x : ℝ × ℂ => P.physicalFactor δ x b) (u,z) := by
-  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
-  have ha := (contDiffAt_arg_real (show z-(b:ℂ) ∈ Complex.slitPlane
-      from Or.inr (by simpa using hy.ne')) n).comp (u,z)
-    (contDiffAt_snd.sub contDiffAt_const)
-  have hsin := ha.sin
-  have hp := hsin.rpow_const_of_ne (p := P.p) hs.ne'
-  have hq := hsin.rpow_const_of_ne (p := P.q) hs.ne'
-  have he : ContDiffAt ℝ n (fun x : ℝ × ℂ => Real.exp (P.p*x.1/2)/3) (u,z) := by fun_prop
-  exact he.mul (hp.add ((hδ.comp (u,z) contDiffAt_fst).mul (contDiffAt_const.add hq)))
-
-end SLEExactGauge.Parameter
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-end SLEExactGauge.BrownianCalculus
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StoppedEstimates
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+namespace SLEExactGauge
+namespace Parameter
+theorem weighted_young_pair (P : Parameter) {C ε t X Y Z : ℝ}
+    (hC : 0 < C) (hε : 0 < ε) (ht : 0 < t)
+    (hX : 0 ≤ X) (hY : 0 ≤ Y) (hZ : 0 ≤ Z) :
+    C * t ^ (-1 - P.η) * X ^ P.η * Y ^ P.η * Z ^ (1 - 2 * P.η) ≤
+      ε * (X + Y) + P.youngConstant C ε * t ^ (-P.M) * Z := by
+  let r := 1 - 2 * P.η
+  have hr : 0 < r := by dsimp [r]; linarith [P.η_lt_half]
+  have hη := P.η_pos
+  have hηε : 0 < P.η / ε := div_pos hη hε
+  have hεη : 0 < ε / P.η := div_pos hε hη
+  let B := C ^ (1 / r) * (P.η / ε) ^ (2 * P.η / r) * t ^ (-P.M)
+  have hB : 0 < B := by dsimp [B]; positivity
+  have heM : (-P.M) * r = -1 - P.η := by
+    have hM : P.M * r = 1 + P.η := by
+      dsimp [M, r]
+      exact div_mul_cancel₀ _ (by linarith [P.η_lt_half])
+    nlinarith
+  have hBr : B ^ r = C * (P.η / ε) ^ (2 * P.η) * t ^ (-1 - P.η) := by
+    dsimp only [B]
+    rw [Real.mul_rpow (by positivity : 0 ≤ C ^ (1 / r) *
+        (P.η / ε) ^ (2 * P.η / r)) (by positivity),
+      Real.mul_rpow (by positivity : 0 ≤ C ^ (1 / r)) (by positivity),
+      ← Real.rpow_mul hC.le, ← Real.rpow_mul hηε.le,
+      ← Real.rpow_mul ht.le, heM]
+    rw [div_mul_cancel₀ _ hr.ne', div_mul_cancel₀ _ hr.ne', Real.rpow_one]
+  have hinv : (ε / P.η) ^ P.η * (ε / P.η) ^ P.η *
+      (P.η / ε) ^ (2 * P.η) = 1 := by
+    rw [← Real.rpow_add hεη, show P.η + P.η = 2 * P.η by ring,
+      ← Real.mul_rpow hεη.le hηε.le]
+    have hprod : ε / P.η * (P.η / ε) = 1 := by field_simp
+    rw [hprod, Real.one_rpow]
+  have hamgm := Real.geom_mean_le_arith_mean3_weighted hη.le hη.le hr.le
+    (mul_nonneg hεη.le hX) (mul_nonneg hεη.le hY) (mul_nonneg hB.le hZ)
+    (show P.η + P.η + r = 1 by dsimp [r]; ring)
+  have hleft : (ε / P.η * X) ^ P.η * (ε / P.η * Y) ^ P.η * (B * Z) ^ r =
+      C * t ^ (-1 - P.η) * X ^ P.η * Y ^ P.η * Z ^ r := by
+    rw [Real.mul_rpow hεη.le hX, Real.mul_rpow hεη.le hY,
+      Real.mul_rpow hB.le hZ, hBr]
+    calc
+      _ = ((ε / P.η) ^ P.η * (ε / P.η) ^ P.η * (P.η / ε) ^ (2 * P.η)) *
+        (C * t ^ (-1 - P.η) * X ^ P.η * Y ^ P.η * Z ^ r) := by ring
+      _ = _ := by rw [hinv, one_mul]
+  have hright : P.η * (ε / P.η * X) + P.η * (ε / P.η * Y) + r * (B * Z) =
+      ε * (X + Y) + P.youngConstant C ε * t ^ (-P.M) * Z := by
+    dsimp only [B, youngConstant]
+    change P.η * (ε / P.η * X) + P.η * (ε / P.η * Y) +
+      r * (C ^ (1 / r) * (P.η / ε) ^ (2 * P.η / r) * t ^ (-P.M) * Z) =
+      ε * (X + Y) + r * C ^ (1 / r) * (P.η / ε) ^ (2 * P.η / r) * t ^ (-P.M) * Z
+    field_simp
+  rw [hleft, hright] at hamgm
+  exact hamgm
+
+end Parameter
+end SLEExactGauge
+
+open Filter Topology
+
+namespace SLEExactGauge.Parameter
+
+lemma hasDerivAt_factor_first (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
+    HasDerivAt (fun t : ℝ => P.angularDerivative δ (Real.sin t) (Real.cos t))
+      (P.p*(P.p-1)*Real.cos θ^2*Real.sin θ^(P.p-2)-P.p*Real.sin θ^P.p+
+       δ*(P.q*(P.q-1)*Real.cos θ^2*Real.sin θ^(P.q-2)-P.q*Real.sin θ^P.q)) θ := by
+  convert! (hasDerivAt_sinePower_first P.p θ hs).add
+    ((hasDerivAt_sinePower_first P.q θ hs).const_mul δ) using 1
+  ext t
+  unfold angularDerivative
+  simp only [Pi.add_apply]
+  ring
+
+lemma deriv_factor (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
+    deriv (fun t : ℝ => P.f δ (Real.sin t)) θ =
+      P.angularDerivative δ (Real.sin θ) (Real.cos θ) :=
+  (P.hasDerivAt_factor hs).deriv
+
+lemma factor_deriv_differentiable (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
+    HasDerivAt (deriv (fun t : ℝ => P.f δ (Real.sin t)))
+      (P.p*(P.p-1)*Real.cos θ^2*Real.sin θ^(P.p-2)-P.p*Real.sin θ^P.p+
+       δ*(P.q*(P.q-1)*Real.cos θ^2*Real.sin θ^(P.q-2)-P.q*Real.sin θ^P.q)) θ := by
+  have he : deriv (fun t : ℝ => P.f δ (Real.sin t)) =ᶠ[nhds θ]
+      (fun t : ℝ => P.angularDerivative δ (Real.sin t) (Real.cos t)) := by
+    filter_upwards [Real.continuous_sin.continuousAt.eventually (Ioi_mem_nhds hs)] with t ht
+    exact P.deriv_factor ht
+  exact (P.hasDerivAt_factor_first hs).congr_of_eventuallyEq he
+
+lemma angleGenerator_factor (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
+    P.angleGenerator (fun t : ℝ => P.f δ (Real.sin t)) θ =
+      P.angleGenerator (fun t : ℝ => Real.sin t^P.p) θ+
+      δ*P.angleGenerator (fun t : ℝ => Real.sin t^P.q) θ := by
+  unfold angleGenerator
+  rw [(P.factor_deriv_differentiable hs).deriv,P.deriv_factor hs,
+    secondDeriv_sinePower P.p θ hs,secondDeriv_sinePower P.q θ hs,
+    deriv_sinePower P.p θ hs,deriv_sinePower P.q θ hs]
+  unfold angularDerivative
+  ring
+
+ 
+
+lemma shifted_sine_generator (P : Parameter) (r θ : ℝ) (hs : 0 < Real.sin θ) :
+    P.angleGenerator (fun t : ℝ => Real.sin t^r) θ+P.p/2*Real.sin θ^r =
+      (P.p-r)*(r+1)/2*Real.sin θ^r-r*(P.p-r)/2*Real.sin θ^(r-2) := by
+  have hp1 : Real.sin θ^(r-1) = Real.sin θ^r/Real.sin θ := by
+    rw [Real.rpow_sub hs,Real.rpow_one]
+  have hp2 : Real.sin θ^(r-2) = Real.sin θ^r/Real.sin θ^2 := by
+    rw [Real.rpow_sub hs,Real.rpow_two]
+  have hcos : Real.cos θ^2 = 1-Real.sin θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
+  unfold angleGenerator
+  rw [secondDeriv_sinePower r θ hs,deriv_sinePower r θ hs,hp1,hp2]
+  unfold p
+  field_simp
+  rw [hcos]
+  ring
+
+ 
+
+theorem exact_factor_drift (P : Parameter) {δ θ : ℝ} (_hδ : 0 < δ)
+    (hs : 0 < Real.sin θ) (σ : ℝ) :
+    (P.angleGenerator (fun t : ℝ => P.f δ (Real.sin t)) θ+
+      P.p/2*P.f δ (Real.sin θ)+σ*δ*(1+Real.sin θ^P.q))/P.f δ (Real.sin θ) =
+      P.drift δ (Real.sin θ) σ := by
+  rw [P.angleGenerator_factor hs]
+  have hp := P.shifted_sine_generator P.p θ hs
+  have hq := P.shifted_sine_generator P.q θ hs
+  simp only [sub_self,zero_mul,mul_zero,zero_div] at hp
+  unfold drift f k at *
+  linear_combination (norm := ring_nf) hp / (Real.sin θ ^ P.p+δ*(1+Real.sin θ^P.q)) +
+    δ / (Real.sin θ ^ P.p+δ*(1+Real.sin θ^P.q)) * hq
+
+end SLEExactGauge.Parameter
+
+ 
+
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+namespace SLEExactGauge.AngleCalculus
+
+ 
+def angleD (z : ℂ) : ℂ →L[ℝ] ℝ :=
+  Complex.imCLM.comp (z⁻¹ • (1 : ℂ →L[ℝ] ℂ))
+
+lemma hasFDerivAt_arg {z : ℂ} (hz : z ∈ Complex.slitPlane) :
+    HasFDerivAt Complex.arg (angleD z) z := by
+  simpa only [angleD,Function.comp_def,Complex.imCLM_apply,Complex.log_im] using
+    Complex.imCLM.hasFDerivAt.comp z (Complex.hasStrictFDerivAt_log_real hz).hasFDerivAt
+
+lemma angleD_apply (z w : ℂ) : angleD z w = (w/z).im := by
+  simp [angleD,div_eq_mul_inv,mul_comm]
+
+ 
+def V (z : ℂ) : ℝ := z.im / Complex.normSq z
+
+lemma driver_angle {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
+    HasDerivAt (fun u : ℝ => Complex.arg (z-u)) (V (z-b)) b := by
+  have hz : z-(b:ℂ) ∈ Complex.slitPlane := Or.inr (by simpa using ne_of_gt hy)
+  have hd := (hasFDerivAt_arg hz).comp_hasDerivAt b
+    ((hasDerivAt_const b z).sub Complex.ofRealCLM.hasDerivAt)
+  convert! hd using 1
+  rw [angleD_apply]
+  simp [V,Complex.div_im,Complex.normSq_apply]
+  ring
+
+lemma hasDerivAt_driver_V (z : ℂ) (b : ℝ) (hy : 0 < z.im) :
+    HasDerivAt (fun u : ℝ => V (z-u))
+      (2*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2) b := by
+  have hz : z-(b:ℂ) ≠ 0 := by
+    intro hh
+    have he := congrArg Complex.im hh
+    simp at he
+    linarith
+  have hn : Complex.normSq (z-(b:ℂ)) ≠ 0 := (Complex.normSq_pos.mpr hz).ne'
+  have hd : HasDerivAt (fun u : ℝ => (z.re-u)^2+z.im^2) (-2*(z.re-b)) b := by
+    convert! (((hasDerivAt_const b z.re).sub (hasDerivAt_id b)).pow 2).add_const (z.im^2) using 1
+    simp only [Pi.sub_apply,id_eq]
+    ring
+  have he := (hasDerivAt_const b z.im).div hd (by simpa [Complex.normSq_apply,pow_two] using hn)
+  convert! he using 1
+  · ext u
+    simp [V,Complex.normSq_apply,pow_two]
+  · simp [Complex.normSq_apply]
+    ring
+
+lemma deriv_driver_angle {z : ℂ} (hy : 0 < z.im) :
+    deriv (fun u : ℝ => Complex.arg (z-u)) = fun u : ℝ => V (z-u) := by
+  funext u
+  exact (driver_angle hy u).deriv
+
+lemma secondDeriv_driver_angle {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
+    deriv (deriv (fun u : ℝ => Complex.arg (z-u))) b =
+      2*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2 := by
+  rw [deriv_driver_angle hy]
+  exact (hasDerivAt_driver_V z b hy).deriv
+
+lemma loewner_angle {g : ℝ → ℂ} {a b t : ℝ}
+    (hg : HasDerivAt g ((a:ℂ)/(g t-b)) t) (hy : 0 < (g t).im) :
+    HasDerivAt (fun u => Complex.arg (g u-b))
+      (-2*a*(g t-b).re*(g t-b).im/(Complex.normSq (g t-b))^2) t := by
+  have hz : g t-(b:ℂ) ∈ Complex.slitPlane := Or.inr (by simpa using ne_of_gt hy)
+  convert! (hasFDerivAt_arg hz).comp_hasDerivAt t (hg.sub_const (b:ℂ)) using 1
+  rw [angleD_apply]
+  simp [Complex.div_im,Complex.div_re]
+  ring
+
+ 
+
+lemma angle_generator_coefficients (a : ℝ) {z : ℂ} (hy : 0 < z.im) :
+    (-2*a*z.re*z.im/(Complex.normSq z)^2)+
+      (2*z.re*z.im/(Complex.normSq z)^2)/2 =
+      (1-2*a)*(Real.cos z.arg/Real.sin z.arg)*(V z)^2 := by
+  have hz : z ≠ 0 := by intro h; simp [h] at hy
+  have hn : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz
+  have hN : Complex.normSq z ≠ 0 := (Complex.normSq_pos.mpr hz).ne'
+  rw [Complex.sin_arg,Complex.cos_arg hz,V]
+  field_simp
+  ring
+
+end SLEExactGauge.AngleCalculus
+
+namespace SLEExactGauge.AngleCalculus
+
+lemma sin_angle_pos {z : ℂ} (hy : 0 < z.im) : 0 < Real.sin z.arg := by
+  rw [Complex.sin_arg]
+  apply div_pos hy
+  apply norm_pos_iff.mpr
+  intro h
+  simp [h] at hy
+
+ 
+
+def physicalGenerator (a : ℝ) (φ : ℝ → ℝ) (z : ℂ) (b : ℝ) : ℝ :=
+  fderiv ℝ (fun w : ℂ => φ (Complex.arg (w-b))) z ((a:ℂ)/(z-b))+
+    deriv (deriv (fun c : ℝ => φ (Complex.arg (z-c)))) b/2
+
+lemma driver_comp_first (φ : ℝ → ℝ)
+    (hφ : ∀ θ, 0 < Real.sin θ → HasDerivAt φ (deriv φ θ) θ)
+    {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
+    HasDerivAt (fun c : ℝ => φ (Complex.arg (z-c)))
+      (deriv φ (Complex.arg (z-b))*V (z-b)) b :=
+  (hφ _ (sin_angle_pos (by simpa using hy))).comp b (driver_angle hy b)
+
+lemma driver_comp_second (φ : ℝ → ℝ)
+    (hφ : ∀ θ, 0 < Real.sin θ → HasDerivAt φ (deriv φ θ) θ)
+    (hφ' : ∀ θ, 0 < Real.sin θ → HasDerivAt (deriv φ) (deriv (deriv φ) θ) θ)
+    {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
+    deriv (deriv (fun c : ℝ => φ (Complex.arg (z-c)))) b =
+      deriv (deriv φ) (Complex.arg (z-b))*(V (z-b))^2+
+      deriv φ (Complex.arg (z-b))*(2*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2) := by
+  have he : deriv (fun c : ℝ => φ (Complex.arg (z-c))) =
+      fun c : ℝ => deriv φ (Complex.arg (z-c))*V (z-c) := by
+    funext c
+    exact (driver_comp_first φ hφ hy c).deriv
+  rw [he]
+  have hd := ((hφ' _ (sin_angle_pos (by simpa using hy))).comp b
+    (driver_angle hy b)).mul (hasDerivAt_driver_V z b hy)
+  convert! hd.deriv using 1
+  simp only [Function.comp_def]
+  ring
+
+lemma physical_generator_angle (a : ℝ) (φ : ℝ → ℝ)
+    (hφ : ∀ θ, 0 < Real.sin θ → HasDerivAt φ (deriv φ θ) θ)
+    (hφ' : ∀ θ, 0 < Real.sin θ → HasDerivAt (deriv φ) (deriv (deriv φ) θ) θ)
+    {z : ℂ} (hy : 0 < z.im) (b : ℝ) :
+    physicalGenerator a φ z b = (V (z-b))^2 *
+      (deriv (deriv φ) (z-b).arg/2+(1-2*a)*
+        (Real.cos (z-b).arg/Real.sin (z-b).arg)*deriv φ (z-b).arg) := by
+  have hy' : 0 < (z-(b:ℂ)).im := by simpa using hy
+  have hz : z-(b:ℂ) ∈ Complex.slitPlane := Or.inr hy'.ne'
+  have hd := (hφ _ (sin_angle_pos hy')).hasFDerivAt.comp z
+    ((hasFDerivAt_arg hz).comp z ((hasFDerivAt_id z).sub_const (b:ℂ)))
+  have hv : angleD (z-b) ((a:ℂ)/(z-b)) =
+      -2*a*(z-b).re*(z-b).im/(Complex.normSq (z-b))^2 := by
+    rw [angleD_apply]
+    simp [Complex.div_im,Complex.div_re]
+    ring
+  simp only [Function.comp_def,id_eq] at hd
+  unfold physicalGenerator
+  rw [hd.fderiv,driver_comp_second φ hφ hφ' hy b]
+  simp only [ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply,ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul]
+  rw [hv]
+  have hc := angle_generator_coefficients a hy'
+  linear_combination deriv φ (z-b).arg * hc
+
+end SLEExactGauge.AngleCalculus
+
+noncomputable section
+open scoped Topology
+namespace SLEExactGauge.Calculus
+
+lemma fderiv_separable (e δ : ℝ → ℝ) (φ ψ : ℂ → ℝ) (u : ℝ) (z : ℂ)
+    (e' δ' : ℝ) (Dφ Dψ : ℂ →L[ℝ] ℝ)
+    (he : HasDerivAt e e' u) (hδ : HasDerivAt δ δ' u)
+    (hφ : HasFDerivAt φ Dφ z) (hψ : HasFDerivAt ψ Dψ z) (v : ℝ) (w : ℂ) :
+    fderiv ℝ (fun x : ℝ × ℂ => e x.1*(φ x.2+δ x.1*ψ x.2)) (u,z) (v,w) =
+      e'*v*(φ z+δ u*ψ z)+e u*(Dφ w+δ'*v*ψ z+δ u*Dψ w) := by
+  have hfst : HasFDerivAt (fun x : ℝ × ℂ => x.1) (ContinuousLinearMap.fst ℝ ℝ ℂ) (u,z) := hasFDerivAt_fst
+  have hsnd : HasFDerivAt (fun x : ℝ × ℂ => x.2) (ContinuousLinearMap.snd ℝ ℝ ℂ) (u,z) := hasFDerivAt_snd
+  have heF := he.hasFDerivAt.comp (u,z) hfst
+  have hδF := hδ.hasFDerivAt.comp (u,z) hfst
+  have hφF := hφ.comp (u,z) hsnd
+  have hψF := hψ.comp (u,z) hsnd
+  have hd := heF.mul (hφF.add (hδF.mul hψF))
+  simp only [Function.comp_def,Pi.mul_apply,Pi.add_apply] at hd
+  change HasFDerivAt (fun x : ℝ × ℂ => e x.1*(φ x.2+δ x.1*ψ x.2)) _ (u,z) at hd
+  rw [hd.fderiv]
+  simp only [add_apply,ContinuousLinearMap.comp_apply,
+    smul_apply,
+    ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul]
+  change e u * (Dφ w + (δ u * Dψ w + ψ z * (v * δ'))) + (φ z + δ u * ψ z) * (v * e') = _
+  ring
+
+lemma secondDeriv_linear (C δ : ℝ) (φ ψ : ℝ → ℝ)
+    (hφ : ∀ x, DifferentiableAt ℝ φ x) (hψ : ∀ x, DifferentiableAt ℝ ψ x)
+    (hφ' : ∀ x, DifferentiableAt ℝ (deriv φ) x)
+    (hψ' : ∀ x, DifferentiableAt ℝ (deriv ψ) x) (b : ℝ) :
+    deriv (deriv (fun x => C*(φ x+δ*(1+ψ x)))) b =
+      C*(deriv (deriv φ) b+δ*deriv (deriv ψ) b) := by
+  have he : deriv (fun x => C*(φ x+δ*(1+ψ x))) =
+      fun x => C*(deriv φ x+δ*deriv ψ x) := by
+    funext x
+    simpa using ((hφ x).hasDerivAt.add (((hψ x).hasDerivAt.const_add 1).const_mul δ)).const_mul C |>.deriv
+  rw [he]
+  simpa using (((hφ' b).hasDerivAt.add ((hψ' b).hasDerivAt.const_mul δ)).const_mul C).deriv
+
+end SLEExactGauge.Calculus
+
+namespace SLEExactGauge.Parameter
+open AngleCalculus Calculus
+
+ 
+
+def physicalFactor (P : Parameter) (δ : ℝ → ℝ) (x : ℝ × ℂ) (b : ℝ) : ℝ :=
+  Real.exp (P.p*x.1/2)/3 * P.f (δ x.1) (Real.sin (x.2-b).arg)
+
+lemma physical_factor_state_derivative (P : Parameter) (δ : ℝ → ℝ)
+    (u σ b v : ℝ) (z w : ℂ) (hy : 0 < z.im)
+    (hδ : HasDerivAt δ (σ*δ u) u) :
+    fderiv ℝ (fun x : ℝ × ℂ => P.physicalFactor δ x b) (u,z) (v,w) =
+      Real.exp (P.p*u/2)/3 *
+        (fderiv ℝ (fun z : ℂ => P.f (δ u) (Real.sin (z-b).arg)) z w+
+          (P.p/2*P.f (δ u) (Real.sin (z-b).arg)+σ*δ u*(1+Real.sin (z-b).arg^P.q))*v) := by
+  let e : ℝ → ℝ := fun r => Real.exp (P.p*r/2)/3
+  let φ : ℂ → ℝ := fun w => Real.sin (w-b).arg^P.p
+  let ψ : ℂ → ℝ := fun w => 1+Real.sin (w-b).arg^P.q
+  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
+  have ha : HasFDerivAt (fun w : ℂ => (w-b).arg) (angleD (z-b)) z := by
+    convert! (hasFDerivAt_arg (show z-(b:ℂ) ∈ Complex.slitPlane from Or.inr (by simpa using hy.ne'))).comp z
+      ((hasFDerivAt_id z).sub_const (b:ℂ)) using 1
+  have hφ := (hasDerivAt_sinePower P.p (z-b).arg hs).hasFDerivAt.comp z ha
+  have hψ := ((hasDerivAt_sinePower P.q (z-b).arg hs).const_add 1).hasFDerivAt.comp z ha
+  change HasFDerivAt φ _ z at hφ
+  change HasFDerivAt ψ _ z at hψ
+  have he : HasDerivAt e (P.p/2*e u) u := by
+    convert! (((hasDerivAt_id u).const_mul P.p).div_const 2).exp.div_const 3 using 1
+    dsimp [e]
+    ring
+  have hd := fderiv_separable e δ φ ψ u z (P.p/2*e u) (σ*δ u) _ _ he hδ hφ hψ v w
+  have hc := hφ.add (hψ.const_mul (δ u))
+  change HasFDerivAt (fun z : ℂ => P.f (δ u) (Real.sin (z-b).arg)) _ z at hc
+  change fderiv ℝ (fun x : ℝ × ℂ => e x.1*(φ x.2+δ x.1*ψ x.2)) (u,z) (v,w) = _
+  rw [hd,hc.fderiv]
+  simp only [add_apply,smul_apply]
+  change (P.p/2*e u)*v*(φ z+δ u*ψ z)+e u*(_+σ*δ u*v*ψ z+δ u*_) =
+    e u*(_+δ u*_+(P.p/2*(φ z+δ u*ψ z)+σ*δ u*ψ z)*v)
+  ring
+
+ 
+
+theorem physical_factor_generator (P : Parameter) (δ : ℝ → ℝ) (u σ b : ℝ)
+    (z : ℂ) (hy : 0 < z.im) (hδ0 : 0 < δ u)
+    (hδ : HasDerivAt δ (σ*δ u) u) :
+    fderiv ℝ (fun x : ℝ × ℂ => P.physicalFactor δ x b) (u,z)
+      ((V (z-b))^2,(P.a:ℂ)/(z-b))+
+      deriv (deriv (P.physicalFactor δ (u,z))) b/2 =
+      P.physicalFactor δ (u,z) b*(V (z-b))^2*
+        P.drift (δ u) (Real.sin (z-b).arg) σ := by
+  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
+  have hphy := physical_generator_angle P.a (fun θ => P.f (δ u) (Real.sin θ))
+    (fun θ hθ => (P.hasDerivAt_factor hθ).differentiableAt.hasDerivAt)
+    (fun θ hθ => (P.factor_deriv_differentiable hθ).differentiableAt.hasDerivAt) hy b
+  change physicalGenerator P.a (fun θ => P.f (δ u) (Real.sin θ)) z b =
+    (V (z-b))^2*P.angleGenerator (fun θ => P.f (δ u) (Real.sin θ)) (z-b).arg at hphy
+  have hsecond : deriv (deriv (P.physicalFactor δ (u,z))) b =
+      Real.exp (P.p*u/2)/3*deriv (deriv (fun c : ℝ => P.f (δ u) (Real.sin (z-c).arg))) b := by
+    change deriv (deriv (fun c : ℝ => Real.exp (P.p*u/2)/3*P.f (δ u) (Real.sin (z-c).arg))) b = _
+    rw [deriv_const_mul_field',deriv_const_mul_field]
+  rw [P.physical_factor_state_derivative δ u σ b _ z _ hy hδ,hsecond]
+  have heq := (div_eq_iff (P.f_pos hδ0 hs).ne').mp (P.exact_factor_drift hδ0 hs σ)
+  unfold physicalGenerator at hphy
+  unfold physicalFactor
+  linear_combination (Real.exp (P.p*u/2)/3)*hphy +
+    (Real.exp (P.p*u/2)/3*(V (z-b))^2)*heq
+
+end SLEExactGauge.Parameter
+
+namespace SLEExactGauge.Parameter
+open AngleCalculus Calculus
+
+open scoped ContDiff
+lemma contDiffAt_arg_real {z : ℂ} (hz : z ∈ Complex.slitPlane) (n : ℕ∞ω) :
+    ContDiffAt ℝ n Complex.arg z := by
+  have h := Complex.imCLM.contDiff.contDiffAt.comp z
+    ((Complex.contDiffAt_log (n := n) hz).restrict_scalars ℝ)
+  change ContDiffAt ℝ n (fun x => (Complex.log x).im) z at h
+  simpa only [Complex.log_im] using h
+
+lemma contDiffAt_physical_factor (P : Parameter) (δ : ℝ → ℝ)
+    (u b : ℝ) (z : ℂ) (hy : 0 < z.im) (n : ℕ∞ω)
+    (hδ : ContDiffAt ℝ n δ u) :
+    ContDiffAt ℝ n (fun x : ℝ × ℂ => P.physicalFactor δ x b) (u,z) := by
+  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
+  have ha := (contDiffAt_arg_real (show z-(b:ℂ) ∈ Complex.slitPlane
+      from Or.inr (by simpa using hy.ne')) n).comp (u,z)
+    (contDiffAt_snd.sub contDiffAt_const)
+  have hsin := ha.sin
+  have hp := hsin.rpow_const_of_ne (p := P.p) hs.ne'
+  have hq := hsin.rpow_const_of_ne (p := P.q) hs.ne'
+  have he : ContDiffAt ℝ n (fun x : ℝ × ℂ => Real.exp (P.p*x.1/2)/3) (u,z) := by fun_prop
+  exact he.mul (hp.add ((hδ.comp (u,z) contDiffAt_fst).mul (contDiffAt_const.add hq)))
+
+end SLEExactGauge.Parameter
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+end SLEExactGauge.BrownianCalculus
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/EulerFlow.lean b/lean/OAI/Probability/SLEGauge/EulerFlow.lean
index 48b2faa7c04ee546e694744701cbb14be56c12ea..a56ad63b30ecb6b3cb3adcc58c47fdf16c92eadf 100644
--- a/lean/OAI/Probability/SLEGauge/EulerFlow.lean
+++ b/lean/OAI/Probability/SLEGauge/EulerFlow.lean
@@ -1,451 +1,451 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.CutoffFlow
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.Euler
-def interpolate (f : ℝ → ℂ → ℂ) (h : ℝ) (z : ℂ) (t : ℝ) : ℂ :=
-  let k := ⌊t/h⌋₊
-  let w := t/h-(k:ℝ)
-  (1-w) • nodes f h z k + w • nodes f h z (k+1)
-
-lemma finite_gronwall {e : ℕ → ℝ} {L h ε : ℝ} (hL : 0 ≤ L) (hh : 0 ≤ h)
-    (hε : 0 ≤ ε) (he0 : e 0 ≤ 0) {N : ℕ}
-    (hrec : ∀ k < N, e (k+1) ≤ (1+L*h)*e k + ε*h) :
-    e N ≤ ε*((N:ℝ)*h)*Real.exp (L*((N:ℝ)*h)) := by
-  have hstep : ∀ k ≤ N, e k ≤ ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h)) := by
-    intro k hk
-    induction k with
-    | zero => simpa using he0
-    | succ k ih =>
-      have hi := ih (by omega)
-      have hk' : k < N := by omega
-      have he : 1+L*h ≤ Real.exp (L*h) := by
-        simpa only [add_comm] using Real.add_one_le_exp (L*h)
-      have he1 : 1 ≤ Real.exp (L*(((k+1:ℕ):ℝ)*h)) :=
-        Real.one_le_exp (by positivity)
-      calc e (k+1)
-          ≤ (1+L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) + ε*h :=
-            (hrec k hk').trans (add_le_add (mul_le_mul_of_nonneg_left hi
-              (show 0 ≤ 1+L*h from by positivity)) le_rfl)
-        _ ≤ Real.exp (L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) +
-            (ε*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
-          exact add_le_add (mul_le_mul_of_nonneg_right he (by positivity))
-            (le_mul_of_one_le_right (mul_nonneg hε hh) he1)
-        _ = ε*(((k+1:ℕ):ℝ)*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
-          have hex : Real.exp (L*(((k+1:ℕ):ℝ)*h)) =
-              Real.exp (L*h)*Real.exp (L*((k:ℝ)*h)) := by
-            rw [← Real.exp_add]
-            congr 1
-            push_cast
-            ring
-          rw [hex]
-          push_cast
-          ring
-  exact hstep N le_rfl
-
-lemma nodes_error {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L h ε T : ℝ}
-    (hL : 0 ≤ L) (hh : 0 ≤ h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
-    (hlip : ∀ t ∈ Icc 0 T, ∀ x y, ‖f t x-f t y‖ ≤ L*‖x-y‖)
-    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
-      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
-    (N : ℕ) (hN : (N:ℝ)*h ≤ T) :
-    ‖nodes f h z N - g ((N:ℝ)*h)‖ ≤ ε*T*Real.exp (L*T) := by
-  have hrec (k : ℕ) (hk : k < N) :
-      ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖ ≤
-        (1+L*h)*‖nodes f h z k-g ((k:ℝ)*h)‖ + ε*h := by
-    have hkT : ((k+1:ℕ):ℝ)*h ≤ T :=
-      (mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.succ_le_of_lt hk) hh).trans hN
-    have hk0 : 0 ≤ (k:ℝ)*h := by positivity
-    have hkT' : (k:ℝ)*h ≤ T := (by push_cast; nlinarith : (k:ℝ)*h ≤ ((k+1:ℕ):ℝ)*h).trans hkT
-    let r := g (((k+1:ℕ):ℝ)*h)-g ((k:ℝ)*h)-h • f ((k:ℝ)*h) (g ((k:ℝ)*h))
-    have he : nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h) =
-        (nodes f h z k-g ((k:ℝ)*h)) +
-          h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h))) - r := by
-      simp only [nodes, r, smul_sub]
-      abel
-    rw [he]
-    calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
-          ‖h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h)))‖ + ‖r‖ :=
-          (norm_sub_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl)
-      _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
-          h*(L*‖nodes f h z k-g ((k:ℝ)*h)‖) + ε*h := by
-        rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hh]
-        exact add_le_add (add_le_add le_rfl (mul_le_mul_of_nonneg_left
-          (hlip _ ⟨hk0,hkT'⟩ _ _) hh)) (hres k hkT)
-      _ = _ := by ring
-  have hb := finite_gronwall (e := fun k => ‖nodes f h z k-g ((k:ℝ)*h)‖)
-    hL hh hε (by simp [nodes, hg0]) hrec
-  have hT : 0 ≤ T := (mul_nonneg (Nat.cast_nonneg _) hh).trans hN
-  exact hb.trans (by gcongr)
-
-lemma residual_bound {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {T s h ε : ℝ}
-    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
-    (hs : 0 ≤ s) (hh : 0 ≤ h) (hsh : s+h ≤ T)
-    (hosc : ∀ t ∈ Icc s (s+h), ‖f t (g t)-f s (g s)‖ ≤ ε) :
-    ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
-  have hsub : Icc s (s+h) ⊆ Icc 0 T := fun t ht => ⟨hs.trans ht.1, ht.2.trans hsh⟩
-  have hd (t : ℝ) (ht : t ∈ Icc s (s+h)) :
-      HasDerivWithinAt (fun u => g u-u • f s (g s))
-        (f t (g t)-f s (g s)) (Icc s (s+h)) t := by
-    convert! ((hder t (hsub ht)).mono hsub).sub
-      (((hasDerivAt_id t).smul_const (f s (g s))).hasDerivWithinAt) using 1
-    simp only [one_smul]
-  have hb := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd hosc
-    (convex_Icc s (s+h)) (left_mem_Icc.mpr (by linarith))
-    (right_mem_Icc.mpr (by linarith))
-  have he : (g (s+h)-(s+h) • f s (g s))-(g s-s • f s (g s)) =
-      g (s+h)-g s-h • f s (g s) := by rw [add_smul]; abel
-  simpa only [he, add_sub_cancel_left, Real.norm_eq_abs, abs_of_nonneg hh] using hb
-
-lemma exists_residual_bound {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {T : ℝ}
-    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
-    (hcf : ContinuousOn (fun t => f t (g t)) (Icc 0 T)) {ε : ℝ} (hε : 0 < ε) :
-    ∃ δ > 0, ∀ h : ℝ, 0 ≤ h → h < δ → ∀ s : ℝ, 0 ≤ s → s+h ≤ T →
-      ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
-  have hu := isCompact_Icc.uniformContinuousOn_of_continuous hcf
-  obtain ⟨δ,hδ,hb⟩ := Metric.uniformContinuousOn_iff.mp hu ε hε
-  refine ⟨δ,hδ, fun h hh hhδ s hs hsT => residual_bound hder hs hh hsT ?_⟩
-  intro u hu
-  apply le_of_lt
-  have huT : u ∈ Icc 0 T := ⟨hs.trans hu.1, hu.2.trans hsT⟩
-  have hsT' : s ∈ Icc 0 T := ⟨hs, by linarith⟩
-  have hd : dist u s < δ := by
-    rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.mpr hu.1)]
-    linarith [hu.2]
-  simpa only [dist_eq_norm] using hb u huT s hsT' hd
-
-lemma norm_interpolate_sub_le {x y z : ℂ} {w E : ℝ}
-    (hw : 0 ≤ w) (hw1 : w ≤ 1) (hx : ‖x-z‖ ≤ E) (hy : ‖y-z‖ ≤ E) :
-    ‖(1-w) • x+w • y-z‖ ≤ E := by
-  have he : (1-w) • x+w • y-z = (1-w) • (x-z)+w • (y-z) := by
-    simp only [sub_smul, one_smul, smul_sub]
-    abel
-  rw [he]
-  calc _ ≤ ‖(1-w) • (x-z)‖+‖w • (y-z)‖ := norm_add_le _ _
-    _ = (1-w)*‖x-z‖+w*‖y-z‖ := by
-      rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs,
-        abs_of_nonneg (sub_nonneg.mpr hw1), abs_of_nonneg hw]
-    _ ≤ (1-w)*E+w*E := add_le_add (mul_le_mul_of_nonneg_left hx (sub_nonneg.mpr hw1))
-      (mul_le_mul_of_nonneg_left hy hw)
-    _ = E := by ring
-
-lemma interpolate_error {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L M h ε T t : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hh : 0 < h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
-      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
-    (hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v|)
-    (ht : 0 ≤ t) (htT : t+h ≤ T) :
-    ‖interpolate f h z t-g t‖ ≤ ε*T*Real.exp (L*T)+M*h := by
-  let k := ⌊t/h⌋₊
-  have hf0 : (k:ℝ) ≤ t/h := Nat.floor_le (div_nonneg ht hh.le)
-  have hf1 : t/h < (k:ℝ)+1 := Nat.lt_floor_add_one _
-  have hk0 : 0 ≤ (k:ℝ)*h := by positivity
-  have hkt : (k:ℝ)*h ≤ t := (le_div_iff₀ hh).mp hf0
-  have htks : t < ((k+1:ℕ):ℝ)*h := by
-    push_cast
-    exact (div_lt_iff₀ hh).mp hf1
-  have hksT : ((k+1:ℕ):ℝ)*h ≤ T := by push_cast; nlinarith
-  have hkT : (k:ℝ)*h ≤ T := hkt.trans (by linarith)
-  have hw : 0 ≤ t/h-(k:ℝ) := by linarith
-  have hw1 : t/h-(k:ℝ) ≤ 1 := by linarith
-  apply norm_interpolate_sub_le hw hw1
-  · calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖+‖g ((k:ℝ)*h)-g t‖ :=
-        norm_sub_le_norm_sub_add_norm_sub _ _ _
-      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
-        (nodes_error hL hh.le hε hg0 hlip hres k hkT)
-        ((hpath _ ⟨hk0,hkT⟩ _ ⟨ht,by linarith⟩).trans (by
-          rw [abs_of_nonpos (sub_nonpos.mpr hkt)]
-          apply mul_le_mul_of_nonneg_left _ hM
-          push_cast at htks
-          linarith))
-  · calc _ ≤ ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖+
-          ‖g (((k+1:ℕ):ℝ)*h)-g t‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
-      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
-        (nodes_error hL hh.le hε hg0 hlip hres (k+1) hksT)
-        ((hpath _ ⟨by positivity,hksT⟩ _ ⟨ht,by linarith⟩).trans (by
-          rw [abs_of_nonneg (by linarith : 0 ≤ ((k+1:ℕ):ℝ)*h-t)]
-          apply mul_le_mul_of_nonneg_left _ hM
-          push_cast
-          linarith))
-
-lemma converge {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L M T t : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
-    (ht : 0 ≤ t) (htT : t < T)
-    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
-    Tendsto (fun n => interpolate f (h n) z t) atTop (𝓝 (g t)) := by
-  have hT : 0 < T := lt_of_le_of_lt ht htT
-  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
-    intro u hu v hv
-    simpa only [Real.norm_eq_abs] using
-      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
-  rw [Metric.tendsto_nhds]
-  intro ε hε
-  let E := T*Real.exp (L*T)
-  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
-  let η := ε/(2*E)
-  have hη : 0 < η := div_pos hε (by positivity)
-  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
-  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
-  have hev2 : ∀ᶠ n in atTop, h n < T-t :=
-    (tendsto_order.1 hh0).2 (T-t) (sub_pos.mpr htT)
-  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
-    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
-  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
-  rw [dist_eq_norm]
-  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
-    (fun k hk => by
-      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
-      rw [hc] at hk ⊢
-      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n) (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
-    hpath ht (by linarith : t+h n ≤ T)
-  apply lt_of_le_of_lt hb
-  have hηE : η*T*Real.exp (L*T) = ε/2 := by
-    dsimp [η, E]
-    field_simp
-  have hsmall : M*h n < ε/2 := by
-    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
-    nlinarith [hh n]
-  rw [hηE]
-  linarith
-
-lemma converge_uniformOn {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L M T U : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
-    (hU : 0 ≤ U) (hUT : U < T)
-    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
-    TendstoUniformlyOn (fun n => interpolate f (h n) z) g atTop (Icc 0 U) := by
-  have hT : 0 < T := lt_of_le_of_lt hU hUT
-  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
-    intro u hu v hv
-    simpa only [Real.norm_eq_abs] using
-      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  let E := T*Real.exp (L*T)
-  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
-  let η := ε/(2*E)
-  have hη : 0 < η := div_pos hε (by positivity)
-  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
-  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
-  have hev2 : ∀ᶠ n in atTop, h n < T-U :=
-    (tendsto_order.1 hh0).2 (T-U) (sub_pos.mpr hUT)
-  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
-    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
-  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
-  intro t ht
-  rw [dist_comm, dist_eq_norm]
-  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
-    (fun k hk => by
-      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
-      rw [hc] at hk ⊢
-      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n)
-        (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
-    hpath ht.1 (by linarith [ht.2] : t+h n ≤ T)
-  apply lt_of_le_of_lt hb
-  have hηE : η*T*Real.exp (L*T) = ε/2 := by
-    dsimp [η, E]
-    field_simp
-  have hsmall : M*h n < ε/2 := by
-    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
-    nlinarith [hh n]
-  rw [hηE]
-  linarith
-
-end SLEExactGauge.Euler
-
-namespace SLEExactGauge.LiteralTrace
-
-lemma continuous_driverValue (b : Driver) : Continuous (driverValue b) :=
-  (map_continuous b).comp continuous_real_toNNReal
-
-lemma mesh_pos (n : ℕ) : 0 < mesh n := zpow_pos (by norm_num) _
-
-lemma mesh_tendsto_zero : Tendsto mesh atTop (𝓝 0) := by
-  have he : mesh = fun n : ℕ => ((2:ℝ)⁻¹)^n := by
-    funext n
-    simp only [mesh, zpow_neg, zpow_natCast, inv_pow]
-  rw [he]
-  exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-
-lemma cutoffField_eq (κ : ℝ) (b : Driver) (j : ℕ) :
-    cutoffField κ b j = Cutoff.field (a κ) ((j:ℝ)^(-(2:ℤ))) (driverValue b) := by
-  funext t z
-  have hp : ((j:ℝ)^(-(2:ℤ)))^2 = (j:ℝ)^(-(4:ℤ)) := by
-    rw [← zpow_natCast, ← zpow_mul]
-    norm_num
-  simp only [cutoffField, Cutoff.field, Cutoff.inverse, Cutoff.denominator, hp, mul_div_assoc]
-
-lemma eulerSteps_eq_nodes (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) :
-    eulerSteps κ b j N z = Euler.nodes (cutoffField κ b j) (mesh N) z := by
-  funext k
-  induction k with
-  | zero => rfl
-  | succ k ih => simp only [eulerSteps, Euler.nodes, ih, Complex.real_smul]
-
-lemma eulerFlow_eq_interpolate (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) :
-    eulerFlow κ b j N z = Euler.interpolate (cutoffField κ b j) (mesh N) z := by
-  funext t
-  simp only [eulerFlow, Euler.interpolate, eulerSteps_eq_nodes, Complex.real_smul]
-
- 
-
-theorem cutoffFlow_solution (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ)
-    {T : ℝ} (hT : 0 ≤ T) :
-    TendstoUniformlyOn (fun N => eulerFlow κ b j N z) (cutoffFlow κ b j z) atTop (Icc 0 T) ∧
-    cutoffFlow κ b j z 0 = z ∧
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (cutoffFlow κ b j z)
-      (cutoffField κ b j t (cutoffFlow κ b j z t)) (Icc 0 T) t := by
-  let q : ℝ := (j:ℝ)^(-(2:ℤ))
-  have hq : 0 < q := zpow_pos (by exact_mod_cast hj) _
-  obtain ⟨g,hg0,hg⟩ := Cutoff.exists_solution_on_compact hq (a κ) (continuous_driverValue b) z
-    (show 0 ≤ T+1 by linarith)
-  have hg' : ∀ t ∈ Icc 0 (T+1),
-      HasDerivWithinAt g (cutoffField κ b j t (g t)) (Icc 0 (T+1)) t := by
-    simpa only [cutoffField_eq] using hg
-  have hgc : ContinuousOn g (Icc 0 (T+1)) := fun t ht => (hg' t ht).continuousWithinAt
-  have hfc : ContinuousOn (fun t => cutoffField κ b j t (g t)) (Icc 0 (T+1)) := by
-    simp only [cutoffField_eq, Cutoff.field]
-    exact continuousOn_const.mul ((Cutoff.lipschitz_inverse hq).continuous.comp_continuousOn
-      (hgc.sub ((Complex.continuous_ofReal.comp (continuous_driverValue b)).continuousOn)))
-  have hconv : TendstoUniformlyOn (fun N => eulerFlow κ b j N z) g atTop (Icc 0 T) := by
-    simp only [eulerFlow_eq_interpolate]
-    apply Euler.converge_uniformOn (L := |a κ| * 3/q^2) (M := |a κ|/q)
-      (by positivity) (by positivity) hg0 _ hg' hfc _ hT (by linarith)
-      mesh_pos mesh_tendsto_zero
-    · intro t ht x y
-      rw [cutoffField_eq]
-      exact Cutoff.field_sub_le hq _ _ _ _ _
-    · intro t ht
-      rw [cutoffField_eq]
-      exact Cutoff.norm_field_le hq _ _ _ _
-  have heq : ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t = g t := by
-    intro t ht
-    exact (hconv.tendsto_at ht).limUnder_eq
-  refine ⟨?_, (heq 0 ⟨le_rfl,hT⟩).trans hg0, ?_⟩
-  · exact hconv.congr_right (fun t ht => (heq t ht).symm)
-  · intro t ht
-    rw [heq t ht]
-    exact ((hg' t ⟨ht.1,by linarith [ht.2]⟩).mono
-      (show Icc 0 T ⊆ Icc 0 (T+1) from fun u hu => ⟨hu.1,by linarith [hu.2]⟩)).congr heq (heq t ht)
-
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-
- 
-def PositiveSolution (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (f : ℝ → ℂ) : Prop :=
-  0 ≤ T ∧ ContinuousOn f (Icc 0 T) ∧ f 0 = z ∧
-  (∀ t ∈ Icc 0 T, 0 < (f t).im) ∧
-  ∀ t ∈ Icc 0 T, HasDerivWithinAt f ((a κ : ℂ)/(f t-driverValue b t)) (Icc 0 T) t
-
-lemma cutoffField_of_height (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
-    {t : ℝ} {z : ℂ} (hz : (j:ℝ)⁻¹ ≤ z.im) :
-    cutoffField κ b j t z = (a κ : ℂ)/(z-driverValue b t) := by
-  have hjpos : (0:ℝ) < j := by exact_mod_cast (Nat.zero_lt_of_lt hj)
-  have hjone : (1:ℝ) ≤ j := by exact_mod_cast hj
-  have hi : (j:ℝ)⁻¹ ≤ 1 := (inv_le_one₀ hjpos).mpr hjone
-  have hin : 0 ≤ (j:ℝ)⁻¹ := (inv_pos.mpr hjpos).le
-  have hn : (j:ℝ)⁻¹ ≤ ‖z-driverValue b t‖ :=
-    hz.trans (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using Complex.im_le_norm (z-driverValue b t))
-  have hsq : (j:ℝ)⁻¹^2 ≤ (j:ℝ)⁻¹ := by nlinarith
-  have hpow : (j:ℝ)^(-(4:ℤ)) ≤ ‖z-driverValue b t‖^2 := by
-    norm_num only [zpow_neg,zpow_ofNat]
-    rw [← inv_pow]
-    calc
-      (j:ℝ)⁻¹^4 = ((j:ℝ)⁻¹^2)^2 := by ring
-      _ ≤ ((j:ℝ)⁻¹)^2 := pow_le_pow_left₀ (sq_nonneg _) hsq 2
-      _ ≤ ‖z-driverValue b t‖^2 := pow_le_pow_left₀ hin hn 2
-  unfold cutoffField
-  simp only [max_eq_left hpow,div_eq_mul_inv]
-  rw [Complex.inv_def (z-driverValue b t)]
-  simp only [Complex.normSq_eq_norm_sq,Complex.ofReal_inv,starRingEnd_apply,mul_assoc]
-
-lemma cutoffFlow_continuous (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j) (z : ℂ)
-    {T : ℝ} (hT : 0 ≤ T) : ContinuousOn (cutoffFlow κ b j z) (Icc 0 T) :=
-  fun t ht => ((cutoffFlow_solution κ b j hj z hT).2.2 t ht).continuousWithinAt
-
-lemma cutoff_solution_unique (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
-    {z : ℂ} {T : ℝ} (hT : 0 ≤ T) {f : ℝ → ℂ}
-    (hc : ContinuousOn f (Icc 0 T)) (h0 : f 0 = z)
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt f (cutoffField κ b j t (f t)) (Icc 0 T) t) :
-    EqOn (cutoffFlow κ b j z) f (Icc 0 T) := by
-  let q : ℝ := (j:ℝ)^(-(2:ℤ))
-  have hq : 0 < q := zpow_pos (by exact_mod_cast (Nat.zero_lt_of_lt hj)) _
-  let K : ℝ≥0 := ⟨|a κ| * 3/q^2,by positivity⟩
-  apply ODE_solution_unique (v := cutoffField κ b j) (K := K)
-  · intro t
-    apply LipschitzWith.of_dist_le_mul
-    intro z w
-    simp only [dist_eq_norm,cutoffField_eq]
-    exact Cutoff.field_sub_le hq _ _ _ _ _
-  · exact cutoffFlow_continuous κ b hj z hT
-  · intro t ht
-    exact ((cutoffFlow_solution κ b j hj z hT).2.2 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
-      (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1))
-  · exact hc
-  · intro t ht
-    exact (hd t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
-      (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1))
-  · rw [(cutoffFlow_solution κ b j hj z hT).2.1,h0]
-
-lemma PositiveSolution.cutoff_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {j : ℕ} (hj : 1 ≤ j)
-    (hgap : ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ ≤ (f t).im) :
-    EqOn (cutoffFlow κ b j z) f (Icc 0 T) := by
-  apply cutoff_solution_unique κ b hj hf.1 hf.2.1 hf.2.2.1
-  intro t ht
-  rw [cutoffField_of_height κ b hj (hgap t ht)]
-  exact hf.2.2.2.2 t ht
-
-lemma PositiveSolution.exists_cutoff {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) :
-    ∃ j : ℕ, 1 ≤ j ∧ ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ < (f t).im := by
-  obtain ⟨t0,ht0,hmin⟩ := isCompact_Icc.exists_isMinOn ⟨0,le_rfl,hf.1⟩
-    (Complex.continuous_im.comp_continuousOn hf.2.1)
-  have hy := hf.2.2.2.1 t0 ht0
-  have ht : Tendsto (fun n : ℕ => (n:ℝ)⁻¹) atTop (𝓝 0) :=
-    tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R := ℝ))
-  obtain ⟨j,hj,hj'⟩ := ((eventually_ge_atTop 1).and (ht.eventually (eventually_lt_nhds hy))).exists
-  exact ⟨j,hj,fun t ht => hj'.trans_le (hmin ht)⟩
-
-lemma PositiveSolution.unique {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f g : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) (hg : PositiveSolution κ b z T g) : EqOn f g (Icc 0 T) := by
-  obtain ⟨j,hj,hjf⟩ := hf.exists_cutoff
-  obtain ⟨k,hk,hkg⟩ := hg.exists_cutoff
-  let n := max j k
-  have hn : 1 ≤ n := hj.trans (le_max_left _ _)
-  have hinj : (n:ℝ)⁻¹ ≤ (j:ℝ)⁻¹ := inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hj)) (by exact_mod_cast le_max_left j k)
-  have hink : (n:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ := inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hk)) (by exact_mod_cast le_max_right j k)
-  have he := hf.cutoff_eq hn (fun t ht => hinj.trans (hjf t ht).le)
-  have hg' := hg.cutoff_eq hn (fun t ht => hink.trans (hkg t ht).le)
-  exact fun t ht => (he ht).symm.trans (hg' ht)
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.CutoffFlow
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.Euler
+def interpolate (f : ℝ → ℂ → ℂ) (h : ℝ) (z : ℂ) (t : ℝ) : ℂ :=
+  let k := ⌊t/h⌋₊
+  let w := t/h-(k:ℝ)
+  (1-w) • nodes f h z k + w • nodes f h z (k+1)
+
+lemma finite_gronwall {e : ℕ → ℝ} {L h ε : ℝ} (hL : 0 ≤ L) (hh : 0 ≤ h)
+    (hε : 0 ≤ ε) (he0 : e 0 ≤ 0) {N : ℕ}
+    (hrec : ∀ k < N, e (k+1) ≤ (1+L*h)*e k + ε*h) :
+    e N ≤ ε*((N:ℝ)*h)*Real.exp (L*((N:ℝ)*h)) := by
+  have hstep : ∀ k ≤ N, e k ≤ ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h)) := by
+    intro k hk
+    induction k with
+    | zero => simpa using he0
+    | succ k ih =>
+      have hi := ih (by omega)
+      have hk' : k < N := by omega
+      have he : 1+L*h ≤ Real.exp (L*h) := by
+        simpa only [add_comm] using Real.add_one_le_exp (L*h)
+      have he1 : 1 ≤ Real.exp (L*(((k+1:ℕ):ℝ)*h)) :=
+        Real.one_le_exp (by positivity)
+      calc e (k+1)
+          ≤ (1+L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) + ε*h :=
+            (hrec k hk').trans (add_le_add (mul_le_mul_of_nonneg_left hi
+              (show 0 ≤ 1+L*h from by positivity)) le_rfl)
+        _ ≤ Real.exp (L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) +
+            (ε*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
+          exact add_le_add (mul_le_mul_of_nonneg_right he (by positivity))
+            (le_mul_of_one_le_right (mul_nonneg hε hh) he1)
+        _ = ε*(((k+1:ℕ):ℝ)*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
+          have hex : Real.exp (L*(((k+1:ℕ):ℝ)*h)) =
+              Real.exp (L*h)*Real.exp (L*((k:ℝ)*h)) := by
+            rw [← Real.exp_add]
+            congr 1
+            push_cast
+            ring
+          rw [hex]
+          push_cast
+          ring
+  exact hstep N le_rfl
+
+lemma nodes_error {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L h ε T : ℝ}
+    (hL : 0 ≤ L) (hh : 0 ≤ h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
+    (hlip : ∀ t ∈ Icc 0 T, ∀ x y, ‖f t x-f t y‖ ≤ L*‖x-y‖)
+    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
+      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
+    (N : ℕ) (hN : (N:ℝ)*h ≤ T) :
+    ‖nodes f h z N - g ((N:ℝ)*h)‖ ≤ ε*T*Real.exp (L*T) := by
+  have hrec (k : ℕ) (hk : k < N) :
+      ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖ ≤
+        (1+L*h)*‖nodes f h z k-g ((k:ℝ)*h)‖ + ε*h := by
+    have hkT : ((k+1:ℕ):ℝ)*h ≤ T :=
+      (mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.succ_le_of_lt hk) hh).trans hN
+    have hk0 : 0 ≤ (k:ℝ)*h := by positivity
+    have hkT' : (k:ℝ)*h ≤ T := (by push_cast; nlinarith : (k:ℝ)*h ≤ ((k+1:ℕ):ℝ)*h).trans hkT
+    let r := g (((k+1:ℕ):ℝ)*h)-g ((k:ℝ)*h)-h • f ((k:ℝ)*h) (g ((k:ℝ)*h))
+    have he : nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h) =
+        (nodes f h z k-g ((k:ℝ)*h)) +
+          h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h))) - r := by
+      simp only [nodes, r, smul_sub]
+      abel
+    rw [he]
+    calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
+          ‖h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h)))‖ + ‖r‖ :=
+          (norm_sub_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl)
+      _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
+          h*(L*‖nodes f h z k-g ((k:ℝ)*h)‖) + ε*h := by
+        rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hh]
+        exact add_le_add (add_le_add le_rfl (mul_le_mul_of_nonneg_left
+          (hlip _ ⟨hk0,hkT'⟩ _ _) hh)) (hres k hkT)
+      _ = _ := by ring
+  have hb := finite_gronwall (e := fun k => ‖nodes f h z k-g ((k:ℝ)*h)‖)
+    hL hh hε (by simp [nodes, hg0]) hrec
+  have hT : 0 ≤ T := (mul_nonneg (Nat.cast_nonneg _) hh).trans hN
+  exact hb.trans (by gcongr)
+
+lemma residual_bound {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {T s h ε : ℝ}
+    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
+    (hs : 0 ≤ s) (hh : 0 ≤ h) (hsh : s+h ≤ T)
+    (hosc : ∀ t ∈ Icc s (s+h), ‖f t (g t)-f s (g s)‖ ≤ ε) :
+    ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
+  have hsub : Icc s (s+h) ⊆ Icc 0 T := fun t ht => ⟨hs.trans ht.1, ht.2.trans hsh⟩
+  have hd (t : ℝ) (ht : t ∈ Icc s (s+h)) :
+      HasDerivWithinAt (fun u => g u-u • f s (g s))
+        (f t (g t)-f s (g s)) (Icc s (s+h)) t := by
+    convert! ((hder t (hsub ht)).mono hsub).sub
+      (((hasDerivAt_id t).smul_const (f s (g s))).hasDerivWithinAt) using 1
+    simp only [one_smul]
+  have hb := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd hosc
+    (convex_Icc s (s+h)) (left_mem_Icc.mpr (by linarith))
+    (right_mem_Icc.mpr (by linarith))
+  have he : (g (s+h)-(s+h) • f s (g s))-(g s-s • f s (g s)) =
+      g (s+h)-g s-h • f s (g s) := by rw [add_smul]; abel
+  simpa only [he, add_sub_cancel_left, Real.norm_eq_abs, abs_of_nonneg hh] using hb
+
+lemma exists_residual_bound {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {T : ℝ}
+    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
+    (hcf : ContinuousOn (fun t => f t (g t)) (Icc 0 T)) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, ∀ h : ℝ, 0 ≤ h → h < δ → ∀ s : ℝ, 0 ≤ s → s+h ≤ T →
+      ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
+  have hu := isCompact_Icc.uniformContinuousOn_of_continuous hcf
+  obtain ⟨δ,hδ,hb⟩ := Metric.uniformContinuousOn_iff.mp hu ε hε
+  refine ⟨δ,hδ, fun h hh hhδ s hs hsT => residual_bound hder hs hh hsT ?_⟩
+  intro u hu
+  apply le_of_lt
+  have huT : u ∈ Icc 0 T := ⟨hs.trans hu.1, hu.2.trans hsT⟩
+  have hsT' : s ∈ Icc 0 T := ⟨hs, by linarith⟩
+  have hd : dist u s < δ := by
+    rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.mpr hu.1)]
+    linarith [hu.2]
+  simpa only [dist_eq_norm] using hb u huT s hsT' hd
+
+lemma norm_interpolate_sub_le {x y z : ℂ} {w E : ℝ}
+    (hw : 0 ≤ w) (hw1 : w ≤ 1) (hx : ‖x-z‖ ≤ E) (hy : ‖y-z‖ ≤ E) :
+    ‖(1-w) • x+w • y-z‖ ≤ E := by
+  have he : (1-w) • x+w • y-z = (1-w) • (x-z)+w • (y-z) := by
+    simp only [sub_smul, one_smul, smul_sub]
+    abel
+  rw [he]
+  calc _ ≤ ‖(1-w) • (x-z)‖+‖w • (y-z)‖ := norm_add_le _ _
+    _ = (1-w)*‖x-z‖+w*‖y-z‖ := by
+      rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs,
+        abs_of_nonneg (sub_nonneg.mpr hw1), abs_of_nonneg hw]
+    _ ≤ (1-w)*E+w*E := add_le_add (mul_le_mul_of_nonneg_left hx (sub_nonneg.mpr hw1))
+      (mul_le_mul_of_nonneg_left hy hw)
+    _ = E := by ring
+
+lemma interpolate_error {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L M h ε T t : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hh : 0 < h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
+      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
+    (hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v|)
+    (ht : 0 ≤ t) (htT : t+h ≤ T) :
+    ‖interpolate f h z t-g t‖ ≤ ε*T*Real.exp (L*T)+M*h := by
+  let k := ⌊t/h⌋₊
+  have hf0 : (k:ℝ) ≤ t/h := Nat.floor_le (div_nonneg ht hh.le)
+  have hf1 : t/h < (k:ℝ)+1 := Nat.lt_floor_add_one _
+  have hk0 : 0 ≤ (k:ℝ)*h := by positivity
+  have hkt : (k:ℝ)*h ≤ t := (le_div_iff₀ hh).mp hf0
+  have htks : t < ((k+1:ℕ):ℝ)*h := by
+    push_cast
+    exact (div_lt_iff₀ hh).mp hf1
+  have hksT : ((k+1:ℕ):ℝ)*h ≤ T := by push_cast; nlinarith
+  have hkT : (k:ℝ)*h ≤ T := hkt.trans (by linarith)
+  have hw : 0 ≤ t/h-(k:ℝ) := by linarith
+  have hw1 : t/h-(k:ℝ) ≤ 1 := by linarith
+  apply norm_interpolate_sub_le hw hw1
+  · calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖+‖g ((k:ℝ)*h)-g t‖ :=
+        norm_sub_le_norm_sub_add_norm_sub _ _ _
+      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
+        (nodes_error hL hh.le hε hg0 hlip hres k hkT)
+        ((hpath _ ⟨hk0,hkT⟩ _ ⟨ht,by linarith⟩).trans (by
+          rw [abs_of_nonpos (sub_nonpos.mpr hkt)]
+          apply mul_le_mul_of_nonneg_left _ hM
+          push_cast at htks
+          linarith))
+  · calc _ ≤ ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖+
+          ‖g (((k+1:ℕ):ℝ)*h)-g t‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
+      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
+        (nodes_error hL hh.le hε hg0 hlip hres (k+1) hksT)
+        ((hpath _ ⟨by positivity,hksT⟩ _ ⟨ht,by linarith⟩).trans (by
+          rw [abs_of_nonneg (by linarith : 0 ≤ ((k+1:ℕ):ℝ)*h-t)]
+          apply mul_le_mul_of_nonneg_left _ hM
+          push_cast
+          linarith))
+
+lemma converge {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L M T t : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
+    (ht : 0 ≤ t) (htT : t < T)
+    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
+    Tendsto (fun n => interpolate f (h n) z t) atTop (𝓝 (g t)) := by
+  have hT : 0 < T := lt_of_le_of_lt ht htT
+  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
+    intro u hu v hv
+    simpa only [Real.norm_eq_abs] using
+      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
+  rw [Metric.tendsto_nhds]
+  intro ε hε
+  let E := T*Real.exp (L*T)
+  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
+  let η := ε/(2*E)
+  have hη : 0 < η := div_pos hε (by positivity)
+  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
+  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
+  have hev2 : ∀ᶠ n in atTop, h n < T-t :=
+    (tendsto_order.1 hh0).2 (T-t) (sub_pos.mpr htT)
+  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
+    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
+  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
+  rw [dist_eq_norm]
+  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
+    (fun k hk => by
+      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
+      rw [hc] at hk ⊢
+      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n) (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
+    hpath ht (by linarith : t+h n ≤ T)
+  apply lt_of_le_of_lt hb
+  have hηE : η*T*Real.exp (L*T) = ε/2 := by
+    dsimp [η, E]
+    field_simp
+  have hsmall : M*h n < ε/2 := by
+    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
+    nlinarith [hh n]
+  rw [hηE]
+  linarith
+
+lemma converge_uniformOn {f : ℝ → ℂ → ℂ} {g : ℝ → ℂ} {z : ℂ} {L M T U : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
+    (hU : 0 ≤ U) (hUT : U < T)
+    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
+    TendstoUniformlyOn (fun n => interpolate f (h n) z) g atTop (Icc 0 U) := by
+  have hT : 0 < T := lt_of_le_of_lt hU hUT
+  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
+    intro u hu v hv
+    simpa only [Real.norm_eq_abs] using
+      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  let E := T*Real.exp (L*T)
+  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
+  let η := ε/(2*E)
+  have hη : 0 < η := div_pos hε (by positivity)
+  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
+  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
+  have hev2 : ∀ᶠ n in atTop, h n < T-U :=
+    (tendsto_order.1 hh0).2 (T-U) (sub_pos.mpr hUT)
+  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
+    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
+  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
+  intro t ht
+  rw [dist_comm, dist_eq_norm]
+  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
+    (fun k hk => by
+      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
+      rw [hc] at hk ⊢
+      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n)
+        (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
+    hpath ht.1 (by linarith [ht.2] : t+h n ≤ T)
+  apply lt_of_le_of_lt hb
+  have hηE : η*T*Real.exp (L*T) = ε/2 := by
+    dsimp [η, E]
+    field_simp
+  have hsmall : M*h n < ε/2 := by
+    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
+    nlinarith [hh n]
+  rw [hηE]
+  linarith
+
+end SLEExactGauge.Euler
+
+namespace SLEExactGauge.LiteralTrace
+
+lemma continuous_driverValue (b : Driver) : Continuous (driverValue b) :=
+  (map_continuous b).comp continuous_real_toNNReal
+
+lemma mesh_pos (n : ℕ) : 0 < mesh n := zpow_pos (by norm_num) _
+
+lemma mesh_tendsto_zero : Tendsto mesh atTop (𝓝 0) := by
+  have he : mesh = fun n : ℕ => ((2:ℝ)⁻¹)^n := by
+    funext n
+    simp only [mesh, zpow_neg, zpow_natCast, inv_pow]
+  rw [he]
+  exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+
+lemma cutoffField_eq (κ : ℝ) (b : Driver) (j : ℕ) :
+    cutoffField κ b j = Cutoff.field (a κ) ((j:ℝ)^(-(2:ℤ))) (driverValue b) := by
+  funext t z
+  have hp : ((j:ℝ)^(-(2:ℤ)))^2 = (j:ℝ)^(-(4:ℤ)) := by
+    rw [← zpow_natCast, ← zpow_mul]
+    norm_num
+  simp only [cutoffField, Cutoff.field, Cutoff.inverse, Cutoff.denominator, hp, mul_div_assoc]
+
+lemma eulerSteps_eq_nodes (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) :
+    eulerSteps κ b j N z = Euler.nodes (cutoffField κ b j) (mesh N) z := by
+  funext k
+  induction k with
+  | zero => rfl
+  | succ k ih => simp only [eulerSteps, Euler.nodes, ih, Complex.real_smul]
+
+lemma eulerFlow_eq_interpolate (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) :
+    eulerFlow κ b j N z = Euler.interpolate (cutoffField κ b j) (mesh N) z := by
+  funext t
+  simp only [eulerFlow, Euler.interpolate, eulerSteps_eq_nodes, Complex.real_smul]
+
+ 
+
+theorem cutoffFlow_solution (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ)
+    {T : ℝ} (hT : 0 ≤ T) :
+    TendstoUniformlyOn (fun N => eulerFlow κ b j N z) (cutoffFlow κ b j z) atTop (Icc 0 T) ∧
+    cutoffFlow κ b j z 0 = z ∧
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (cutoffFlow κ b j z)
+      (cutoffField κ b j t (cutoffFlow κ b j z t)) (Icc 0 T) t := by
+  let q : ℝ := (j:ℝ)^(-(2:ℤ))
+  have hq : 0 < q := zpow_pos (by exact_mod_cast hj) _
+  obtain ⟨g,hg0,hg⟩ := Cutoff.exists_solution_on_compact hq (a κ) (continuous_driverValue b) z
+    (show 0 ≤ T+1 by linarith)
+  have hg' : ∀ t ∈ Icc 0 (T+1),
+      HasDerivWithinAt g (cutoffField κ b j t (g t)) (Icc 0 (T+1)) t := by
+    simpa only [cutoffField_eq] using hg
+  have hgc : ContinuousOn g (Icc 0 (T+1)) := fun t ht => (hg' t ht).continuousWithinAt
+  have hfc : ContinuousOn (fun t => cutoffField κ b j t (g t)) (Icc 0 (T+1)) := by
+    simp only [cutoffField_eq, Cutoff.field]
+    exact continuousOn_const.mul ((Cutoff.lipschitz_inverse hq).continuous.comp_continuousOn
+      (hgc.sub ((Complex.continuous_ofReal.comp (continuous_driverValue b)).continuousOn)))
+  have hconv : TendstoUniformlyOn (fun N => eulerFlow κ b j N z) g atTop (Icc 0 T) := by
+    simp only [eulerFlow_eq_interpolate]
+    apply Euler.converge_uniformOn (L := |a κ| * 3/q^2) (M := |a κ|/q)
+      (by positivity) (by positivity) hg0 _ hg' hfc _ hT (by linarith)
+      mesh_pos mesh_tendsto_zero
+    · intro t ht x y
+      rw [cutoffField_eq]
+      exact Cutoff.field_sub_le hq _ _ _ _ _
+    · intro t ht
+      rw [cutoffField_eq]
+      exact Cutoff.norm_field_le hq _ _ _ _
+  have heq : ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t = g t := by
+    intro t ht
+    exact (hconv.tendsto_at ht).limUnder_eq
+  refine ⟨?_, (heq 0 ⟨le_rfl,hT⟩).trans hg0, ?_⟩
+  · exact hconv.congr_right (fun t ht => (heq t ht).symm)
+  · intro t ht
+    rw [heq t ht]
+    exact ((hg' t ⟨ht.1,by linarith [ht.2]⟩).mono
+      (show Icc 0 T ⊆ Icc 0 (T+1) from fun u hu => ⟨hu.1,by linarith [hu.2]⟩)).congr heq (heq t ht)
+
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+
+ 
+def PositiveSolution (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (f : ℝ → ℂ) : Prop :=
+  0 ≤ T ∧ ContinuousOn f (Icc 0 T) ∧ f 0 = z ∧
+  (∀ t ∈ Icc 0 T, 0 < (f t).im) ∧
+  ∀ t ∈ Icc 0 T, HasDerivWithinAt f ((a κ : ℂ)/(f t-driverValue b t)) (Icc 0 T) t
+
+lemma cutoffField_of_height (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
+    {t : ℝ} {z : ℂ} (hz : (j:ℝ)⁻¹ ≤ z.im) :
+    cutoffField κ b j t z = (a κ : ℂ)/(z-driverValue b t) := by
+  have hjpos : (0:ℝ) < j := by exact_mod_cast (Nat.zero_lt_of_lt hj)
+  have hjone : (1:ℝ) ≤ j := by exact_mod_cast hj
+  have hi : (j:ℝ)⁻¹ ≤ 1 := (inv_le_one₀ hjpos).mpr hjone
+  have hin : 0 ≤ (j:ℝ)⁻¹ := (inv_pos.mpr hjpos).le
+  have hn : (j:ℝ)⁻¹ ≤ ‖z-driverValue b t‖ :=
+    hz.trans (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using Complex.im_le_norm (z-driverValue b t))
+  have hsq : (j:ℝ)⁻¹^2 ≤ (j:ℝ)⁻¹ := by nlinarith
+  have hpow : (j:ℝ)^(-(4:ℤ)) ≤ ‖z-driverValue b t‖^2 := by
+    norm_num only [zpow_neg,zpow_ofNat]
+    rw [← inv_pow]
+    calc
+      (j:ℝ)⁻¹^4 = ((j:ℝ)⁻¹^2)^2 := by ring
+      _ ≤ ((j:ℝ)⁻¹)^2 := pow_le_pow_left₀ (sq_nonneg _) hsq 2
+      _ ≤ ‖z-driverValue b t‖^2 := pow_le_pow_left₀ hin hn 2
+  unfold cutoffField
+  simp only [max_eq_left hpow,div_eq_mul_inv]
+  rw [Complex.inv_def (z-driverValue b t)]
+  simp only [Complex.normSq_eq_norm_sq,Complex.ofReal_inv,starRingEnd_apply,mul_assoc]
+
+lemma cutoffFlow_continuous (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j) (z : ℂ)
+    {T : ℝ} (hT : 0 ≤ T) : ContinuousOn (cutoffFlow κ b j z) (Icc 0 T) :=
+  fun t ht => ((cutoffFlow_solution κ b j hj z hT).2.2 t ht).continuousWithinAt
+
+lemma cutoff_solution_unique (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
+    {z : ℂ} {T : ℝ} (hT : 0 ≤ T) {f : ℝ → ℂ}
+    (hc : ContinuousOn f (Icc 0 T)) (h0 : f 0 = z)
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt f (cutoffField κ b j t (f t)) (Icc 0 T) t) :
+    EqOn (cutoffFlow κ b j z) f (Icc 0 T) := by
+  let q : ℝ := (j:ℝ)^(-(2:ℤ))
+  have hq : 0 < q := zpow_pos (by exact_mod_cast (Nat.zero_lt_of_lt hj)) _
+  let K : ℝ≥0 := ⟨|a κ| * 3/q^2,by positivity⟩
+  apply ODE_solution_unique (v := cutoffField κ b j) (K := K)
+  · intro t
+    apply LipschitzWith.of_dist_le_mul
+    intro z w
+    simp only [dist_eq_norm,cutoffField_eq]
+    exact Cutoff.field_sub_le hq _ _ _ _ _
+  · exact cutoffFlow_continuous κ b hj z hT
+  · intro t ht
+    exact ((cutoffFlow_solution κ b j hj z hT).2.2 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
+      (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1))
+  · exact hc
+  · intro t ht
+    exact (hd t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
+      (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1))
+  · rw [(cutoffFlow_solution κ b j hj z hT).2.1,h0]
+
+lemma PositiveSolution.cutoff_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {j : ℕ} (hj : 1 ≤ j)
+    (hgap : ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ ≤ (f t).im) :
+    EqOn (cutoffFlow κ b j z) f (Icc 0 T) := by
+  apply cutoff_solution_unique κ b hj hf.1 hf.2.1 hf.2.2.1
+  intro t ht
+  rw [cutoffField_of_height κ b hj (hgap t ht)]
+  exact hf.2.2.2.2 t ht
+
+lemma PositiveSolution.exists_cutoff {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) :
+    ∃ j : ℕ, 1 ≤ j ∧ ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ < (f t).im := by
+  obtain ⟨t0,ht0,hmin⟩ := isCompact_Icc.exists_isMinOn ⟨0,le_rfl,hf.1⟩
+    (Complex.continuous_im.comp_continuousOn hf.2.1)
+  have hy := hf.2.2.2.1 t0 ht0
+  have ht : Tendsto (fun n : ℕ => (n:ℝ)⁻¹) atTop (𝓝 0) :=
+    tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R := ℝ))
+  obtain ⟨j,hj,hj'⟩ := ((eventually_ge_atTop 1).and (ht.eventually (eventually_lt_nhds hy))).exists
+  exact ⟨j,hj,fun t ht => hj'.trans_le (hmin ht)⟩
+
+lemma PositiveSolution.unique {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f g : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) (hg : PositiveSolution κ b z T g) : EqOn f g (Icc 0 T) := by
+  obtain ⟨j,hj,hjf⟩ := hf.exists_cutoff
+  obtain ⟨k,hk,hkg⟩ := hg.exists_cutoff
+  let n := max j k
+  have hn : 1 ≤ n := hj.trans (le_max_left _ _)
+  have hinj : (n:ℝ)⁻¹ ≤ (j:ℝ)⁻¹ := inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hj)) (by exact_mod_cast le_max_left j k)
+  have hink : (n:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ := inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hk)) (by exact_mod_cast le_max_right j k)
+  have he := hf.cutoff_eq hn (fun t ht => hinj.trans (hjf t ht).le)
+  have hg' := hg.cutoff_eq hn (fun t ht => hink.trans (hkg t ht).le)
+  exact fun t ht => (he ht).symm.trans (hg' ht)
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/EulerMeasurability.lean b/lean/OAI/Probability/SLEGauge/EulerMeasurability.lean
index 8a60e8c4ed65ddd6c8f1e2f0a80deed7b4dddd92..b2007d30e7566a87ed8e90f47203412f71cc6bb8 100644
--- a/lean/OAI/Probability/SLEGauge/EulerMeasurability.lean
+++ b/lean/OAI/Probability/SLEGauge/EulerMeasurability.lean
@@ -1,438 +1,438 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.BrownianNodes
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-namespace SLEExactGauge.BrownianCalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-lemma measurable_nodes_driver (v : E → ℝ → E)
-    (hv : Measurable (Function.uncurry v)) (h : ℝ≥0) (x : E) (k : ℕ) :
-    Measurable (fun b : Driver => stochasticNodes v h x k b) :=
-  (measurable_stochasticNodes v hv h x k).comp
-    (Measurable.of_eval fun time => (continuous_eval_const time).measurable)
-
-lemma expected_nodes_step {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E)
-    (mv : Measurable (Function.uncurry v)) (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
-    (hgen : ∀ x b, A.D x b (v x b)+deriv (deriv (A.F x)) b/2 ≤ 0)
-    (h : ℝ≥0) (x : E) (k : ℕ) :
-    (∫ b : Driver, A.F (stochasticNodes v h x (k+1) b) (b (((k+1:ℕ):ℝ≥0)*h))-
-      A.F (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h)) ∂W) ≤
-      (A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2 + A.C₃/3*(h:ℝ)*Real.sqrt h := by
-  let s := (k:ℝ≥0)*h
-  let X := fun p : Iic s → ℝ => stochasticNodes v h x k (extendHistory s p)
-  let V := fun p : Iic s → ℝ => v (X p) (p ⟨s,by simp⟩)
-  have mX : Measurable X := (measurable_stochasticNodes v mv h x k).comp
-    (measurable_extendHistory s)
-  have mV : Measurable V := mv.comp (mX.prodMk (measurable_pi_apply _))
-  have he := expected_euler_step hW s h A.F A.D X V A.mF A.mB1 A.mB2 mX mV
-    A.hD A.hsm A.hC₃ A.hCx A.hCcross A.hF A.hB1 A.hB2 A.hB3 A.hLipD A.hLipB2
-    (fun p => hv _ _) (fun p => hgen _ _)
-  have hn (b : Driver) : X (history s b) = stochasticNodes v h x k b :=
-    stochasticNodes_history v h x k b
-  have hs : ((k+1:ℕ):ℝ≥0)*h = s+h := by dsimp [s]; push_cast; ring
-  simpa only [X, V, hn, history, stochasticNodes, hs] using he
-
- 
-
-theorem expected_nodes {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E)
-    (mv : Measurable (Function.uncurry v)) (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
-    (hgen : ∀ x b, A.D x b (v x b)+deriv (deriv (A.F x)) b/2 ≤ 0)
-    (h : ℝ≥0) (x : E) (k : ℕ) :
-    (∫ b : Driver, A.F (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h)) ∂W) ≤
-      A.F x 0+(k:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2 + A.C₃/3*(h:ℝ)*Real.sqrt h) := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hi (k : ℕ) : Integrable (fun b : Driver =>
-      A.F (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h))) W := by
-    apply (integrable_const A.C₀).mono'
-    · exact (A.mF.comp ((measurable_nodes_driver v mv h x k).prodMk
-        (continuous_eval_const _).measurable)).aestronglyMeasurable
-    · exact Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using A.hF _ _
-  induction k with
-  | zero => simp [stochasticNodes, map_zero]
-  | succ k ih =>
-    have he := expected_nodes_step hW A v mv M hv hgen h x k
-    rw [integral_sub (hi (k+1)) (hi k)] at he
-    push_cast at he ⊢
-    linarith
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem expected_nodes_limit {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E)
-    (mv : Measurable (Function.uncurry v)) (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
-    (hgen : ∀ x b, A.D x b (v x b)+deriv (deriv (A.F x)) b/2 ≤ 0)
-    (h : ℕ → ℝ≥0) (k : ℕ → ℕ) (x : E) (t : ℝ≥0) (X : Driver → E)
-    (hh : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0))
-    (htime : Tendsto (fun n => (k n:ℝ≥0)*h n) atTop (𝓝 t))
-    (hX : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n => stochasticNodes v (h n) x (k n) b) atTop (𝓝 (X b))) :
-    (∫ b, A.F (X b) (b t) ∂W) ≤ A.F x 0 := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hc : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n =>
-      A.F (stochasticNodes v (h n) x (k n) b) (b ((k n:ℝ≥0)*h n)))
-      atTop (𝓝 (A.F (X b) (b t))) := by
-    filter_upwards [hX] with b hb
-    exact A.cF.continuousAt.tendsto.comp (hb.prodMk_nhds (b.continuous.continuousAt.tendsto.comp htime))
-  have hi : Tendsto (fun n => ∫ b : Driver,
-      A.F (stochasticNodes v (h n) x (k n) b) (b ((k n:ℝ≥0)*h n)) ∂W)
-      atTop (𝓝 (∫ b, A.F (X b) (b t) ∂W)) :=
-    tendsto_integral_of_dominated_convergence (fun _ : Driver => A.C₀)
-    (fun n => (A.mF.comp ((measurable_nodes_driver v mv (h n) x (k n)).prodMk
-      (continuous_eval_const _).measurable)).aestronglyMeasurable)
-    (integrable_const A.C₀)
-    (fun n => Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using A.hF _ _) hc
-  have hk : Tendsto (fun n => (k n:ℝ)*(h n:ℝ)) atTop (𝓝 (t:ℝ)) := by
-    simpa only [Function.comp_def, NNReal.coe_mul, NNReal.coe_natCast] using
-      NNReal.continuous_coe.continuousAt.tendsto.comp htime
-  have hs : Tendsto (fun n => Real.sqrt (h n:ℝ)) atTop (𝓝 0) := by
-    simpa only [Function.comp_def, Real.sqrt_zero] using (Real.continuous_sqrt.tendsto 0).comp hh
-  have he : Tendsto (fun n => (k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2 +
-      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 0) := by
-    have hh' := hk.mul (((tendsto_const_nhds (x := A.Cx*M^2+A.Ccross*M/2)).mul hh).add
-      ((tendsto_const_nhds (x := A.C₃/3)).mul hs))
-    convert! hh' using 1
-    · ext n
-      ring
-    · simp
-  have hb : Tendsto (fun n => A.F x 0+(k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2 +
-      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 (A.F x 0)) := by
-    simpa using tendsto_const_nhds.add he
-  exact le_of_tendsto_of_tendsto hi hb
-    (Filter.Eventually.of_forall fun n => expected_nodes hW A v mv M hv hgen (h n) x (k n))
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
- 
-
-lemma expected_shifted_taylor {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s h : ℝ≥0) (F : (Iic s → ℝ) → ℝ → ℝ) (B : (Iic s → ℝ) → ℝ)
-    (hsm : ∀ p, ContDiff ℝ 3 (F p))
-    (hm1 : Measurable (fun p => deriv (F p) (B p)))
-    (hm2 : Measurable (fun p => deriv (deriv (F p)) (B p)))
-    {C M₁ M₂ : ℝ} (hC : 0 ≤ C)
-    (hb1 : ∀ p, |deriv (F p) (B p)| ≤ M₁)
-    (hb2 : ∀ p, |deriv (deriv (F p)) (B p)| ≤ M₂)
-    (hb3 : ∀ p x, |iteratedDeriv 3 (F p) x| ≤ C)
-    (hi : Integrable (fun b => F (history s b) (B (history s b)+increment s h b) -
-      F (history s b) (B (history s b))) W) :
-    |(∫ b, F (history s b) (B (history s b)+increment s h b) -
-      F (history s b) (B (history s b)) ∂W) -
-      (h:ℝ)/2 * (∫ b, deriv (deriv (F (history s b))) (B (history s b)) ∂W)| ≤
-      C/3 * (h:ℝ) * Real.sqrt h := by
-  let c := fun p : Iic s → ℝ => B p-p ⟨s,by simp⟩
-  let f := fun p u => F p (u+c p)
-  have hbase (p : Iic s → ℝ) : p ⟨s,by simp⟩+c p = B p := by dsimp [c]; ring
-  have hd1 (p : Iic s → ℝ) : deriv (f p) = fun u => deriv (F p) (u+c p) := by
-    ext u
-    exact deriv_comp_add_const (F p) (c p) u
-  have hd2 (p : Iic s → ℝ) : deriv (deriv (f p)) = fun u => deriv (deriv (F p)) (u+c p) := by
-    rw [hd1]
-    ext u
-    exact deriv_comp_add_const (deriv (F p)) (c p) u
-  have hsmooth (p : Iic s → ℝ) : ContDiff ℝ 3 (f p) :=
-    (hsm p).comp (contDiff_id.add contDiff_const)
-  have hm1' : Measurable (fun p => deriv (f p) (p ⟨s,by simp⟩)) := by
-    simpa only [hd1, hbase] using hm1
-  have hm2' : Measurable (fun p => deriv (deriv (f p)) (p ⟨s,by simp⟩)) := by
-    simpa only [hd2, hbase] using hm2
-  have hf (b : Driver) : f (history s b) (b (s+h))-f (history s b) (b s) =
-      F (history s b) (B (history s b)+increment s h b)-F (history s b) (B (history s b)) := by
-    dsimp [f,c,history,increment]
-    congr 2 <;> ring
-  have hi' : Integrable (fun b => f (history s b) (b (s+h))-f (history s b) (b s)) W := by
-    simpa only [hf] using hi
-  have he := expected_taylor_step hW s h f hsmooth hm1' hm2' hC
-    (fun p => by simpa only [hd1,hbase] using hb1 p)
-    (fun p => by simpa only [hd2,hbase] using hb2 p)
-    (fun p u => by simpa only [f, iteratedDeriv_comp_add_const, Function.comp_apply] using hb3 p (u+c p)) hi'
-  simpa only [hf,hd2,show ∀ b : Driver, b s+c (history s b) = B (history s b) from
-    fun b => hbase (history s b)] using he
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-
- 
-
-lemma expected_shift_with_drift {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s h : ℝ≥0) (f : (Iic s → ℝ) → ℝ → ℝ) (B old : (Iic s → ℝ) → ℝ)
-    (mf : Measurable (Function.uncurry f)) (mB : Measurable B) (mOld : Measurable old)
-    (hsm : ∀ p, ContDiff ℝ 3 (f p))
-    (hm1 : Measurable (fun p => deriv (f p) (B p)))
-    (hm2 : Measurable (fun p => deriv (deriv (f p)) (B p)))
-    {C₀ C₁ C₂ C₃ R : ℝ} (hC₃ : 0 ≤ C₃)
-    (hb0 : ∀ p u, |f p u| ≤ C₀) (hOld : ∀ p, |old p| ≤ C₀)
-    (hb1 : ∀ p, |deriv (f p) (B p)| ≤ C₁)
-    (hb2 : ∀ p, |deriv (deriv (f p)) (B p)| ≤ C₂)
-    (hb3 : ∀ p u, |iteratedDeriv 3 (f p) u| ≤ C₃)
-    (hstep : ∀ p, f p (B p)-old p+(h:ℝ)/2*deriv (deriv (f p)) (B p) ≤ R) :
-    (∫ b, f (history s b) (B (history s b)+increment s h b)-old (history s b) ∂W) ≤
-      R + C₃/3*(h:ℝ)*Real.sqrt h := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let step := fun b : Driver => f (history s b) (B (history s b))
-  let next := fun b : Driver => f (history s b) (B (history s b)+increment s h b)
-  let prev := fun b : Driver => old (history s b)
-  let Q := fun b : Driver => deriv (deriv (f (history s b))) (B (history s b))
-  have ms : Measurable step := mf.comp ((measurable_history s).prodMk (mB.comp (measurable_history s)))
-  have mn : Measurable next := mf.comp ((measurable_history s).prodMk
-    ((mB.comp (measurable_history s)).add (measurable_increment s h)))
-  have mp : Measurable prev := mOld.comp (measurable_history s)
-  have mq : Measurable Q := hm2.comp (measurable_history s)
-  have is : Integrable step W := (integrable_const C₀).mono' ms.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb0 _ _)
-  have inx : Integrable next W := (integrable_const C₀).mono' mn.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb0 _ _)
-  have ip : Integrable prev W := (integrable_const C₀).mono' mp.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hOld _)
-  have iq : Integrable Q W := (integrable_const C₂).mono' mq.aestronglyMeasurable
-    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb2 _)
-  have ht := expected_shifted_taylor hW s h f B hsm hm1 hm2 hC₃ hb1 hb2 hb3 (inx.sub is)
-  change |(∫ b, next b-step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W)| ≤ _ at ht
-  have hi := integral_mono_ae ((is.sub ip).add (iq.const_mul ((h:ℝ)/2)))
-    (integrable_const R) (Filter.Eventually.of_forall fun b => hstep (history s b))
-  have hi' : (∫ b, step b-prev b+(h:ℝ)/2*Q b ∂W) ≤ R := by
-    simpa only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul,
-      Pi.add_apply, Pi.sub_apply] using hi
-  change (∫ b, (step-prev) b+(h:ℝ)/2*Q b ∂W) ≤ R at hi'
-  rw [integral_add (is.sub ip) (iq.const_mul ((h:ℝ)/2)), integral_const_mul] at hi'
-  change (∫ b, step b-prev b ∂W)+(h:ℝ)/2*(∫ b, Q b ∂W) ≤ R at hi'
-  rw [integral_sub is ip] at hi'
-  rw [integral_sub inx is] at ht
-  change (∫ b, next b-prev b ∂W) ≤ _
-  rw [integral_sub inx ip]
-  linarith [le_abs_self ((∫ b, next b ∂W)-(∫ b, step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W))]
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
-omit [BorelSpace E] [SecondCountableTopology E] in
-lemma SmoothTest.euler_generator_error (A : SmoothTest E) (x v : E) (b : ℝ)
-    {h M : ℝ} (hh : 0 ≤ h) (hv : ‖v‖ ≤ M)
-    (hg : A.D x b v+deriv (deriv (A.F x)) b/2 ≤ 0) :
-    A.F (x+h • v) b-A.F x b+h/2*deriv (deriv (A.F (x+h • v))) b ≤
-      (A.Cx*M^2+A.Ccross*M/2)*h^2 := by
-  have hy : ‖x+h • v-x‖ ≤ h*M := by
-    simp only [add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg hh]
-    exact mul_le_mul_of_nonneg_left hv hh
-  have hd := state_euler_remainder (fun x => A.hD x b) A.hCx
-    (fun x y => A.hLipD x y b) x v hh hv
-  have hb := (le_abs_self _).trans (A.hLipB2 (x+h • v) x b)
-  have hc := hb.trans (mul_le_mul_of_nonneg_left hy A.hCcross)
-  have hg' := mul_nonpos_of_nonneg_of_nonpos hh hg
-  have hc' := mul_le_mul_of_nonneg_left hc (div_nonneg hh (by norm_num : (0:ℝ) ≤ 2))
-  nlinarith
-
- 
-
-lemma expected_active_euler_step {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s h : ℝ≥0) (A : SmoothTest E)
-    (X V : (Iic s → ℝ) → E) (B : (Iic s → ℝ) → ℝ)
-    (mX : Measurable X) (mV : Measurable V) (mB : Measurable B)
-    (S : Set (Iic s → ℝ)) (hS : MeasurableSet S)
-    {M : ℝ} (hV : ∀ p, ‖V p‖ ≤ M)
-    (hgen : ∀ p ∈ S, A.D (X p) (B p) (V p)+deriv (deriv (A.F (X p))) (B p)/2 ≤ 0) :
-    (∫ b, (if history s b ∈ S then
-      A.F (X (history s b)+(h:ℝ) • V (history s b)) (B (history s b)+increment s h b)
-      else A.F (X (history s b)) (B (history s b)))-
-      A.F (X (history s b)) (B (history s b)) ∂W) ≤
-      (A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2+A.C₃/3*(h:ℝ)*Real.sqrt h := by
-  classical
-  let Y := fun p => X p+(h:ℝ) • V p
-  let old := fun p => A.F (X p) (B p)
-  let f := fun p u => if p ∈ S then A.F (Y p) u else old p
-  have mY : Measurable Y := mX.add (mV.const_smul (h:ℝ))
-  have mOld : Measurable old := A.mF.comp (mX.prodMk mB)
-  have mf : Measurable (Function.uncurry f) :=
-    (A.mF.comp ((mY.comp measurable_fst).prodMk measurable_snd)).ite
-      (hS.preimage measurable_fst) (mOld.comp measurable_fst)
-  have hd1 (p : Iic s → ℝ) : deriv (f p) = fun u => if p ∈ S then deriv (A.F (Y p)) u else 0 := by
-    by_cases hp : p ∈ S <;> simp [f,hp]
-  have hd2 (p : Iic s → ℝ) : deriv (deriv (f p)) = fun u =>
-      if p ∈ S then deriv (deriv (A.F (Y p))) u else 0 := by
-    rw [hd1]
-    by_cases hp : p ∈ S <;> simp [hp]
-  have hm1 : Measurable (fun p => deriv (f p) (B p)) := by
-    simp only [hd1]
-    exact (A.mB1.comp (mY.prodMk mB)).ite hS measurable_const
-  have hm2 : Measurable (fun p => deriv (deriv (f p)) (B p)) := by
-    simp only [hd2]
-    exact (A.mB2.comp (mY.prodMk mB)).ite hS measurable_const
-  have hC₁ : 0 ≤ A.C₁ := (abs_nonneg _).trans (A.hB1 0 0)
-  have hC₂ : 0 ≤ A.C₂ := (abs_nonneg _).trans (A.hB2 0 0)
-  have hM : 0 ≤ M := (norm_nonneg _).trans (hV (fun _ => 0))
-  refine expected_shift_with_drift (C₀ := A.C₀) (C₁ := A.C₁) (C₂ := A.C₂)
-    hW s h f B old mf mB mOld ?_ hm1 hm2 A.hC₃ ?_ ?_ ?_ ?_ ?_ ?_
-  · intro p
-    by_cases hp : p ∈ S
-    · simpa only [f,ite_eq_left hp] using A.hsm (Y p)
-    · simpa only [f,ite_eq_right hp] using (contDiff_const (c := old p) : ContDiff ℝ 3 _)
-  · intro p u
-    by_cases hp : p ∈ S
-    · simpa only [f,ite_eq_left hp] using A.hF (Y p) u
-    · simpa only [f,ite_eq_right hp,old] using A.hF (X p) (B p)
-  · intro p
-    exact A.hF _ _
-  · intro p
-    by_cases hp : p ∈ S
-    · simpa only [hd1,ite_eq_left hp] using A.hB1 (Y p) (B p)
-    · simpa only [hd1,ite_eq_right hp,abs_zero] using hC₁
-  · intro p
-    by_cases hp : p ∈ S
-    · simpa only [hd2,ite_eq_left hp] using A.hB2 (Y p) (B p)
-    · simpa only [hd2,ite_eq_right hp,abs_zero] using hC₂
-  · intro p u
-    by_cases hp : p ∈ S
-    · simpa only [f,ite_eq_left hp] using A.hB3 (Y p) u
-    · simpa [f,hp,iteratedDeriv_const] using A.hC₃
-  · intro p
-    rw [hd2]
-    by_cases hp : p ∈ S
-    · simpa only [f,ite_eq_left hp,old,Y] using A.euler_generator_error (X p) (V p) (B p)
-        h.coe_nonneg (hV p) (hgen p hp)
-    · simp only [f,ite_eq_right hp,sub_self,mul_zero,add_zero]
-      exact mul_nonneg (add_nonneg (mul_nonneg A.hCx (sq_nonneg M))
-        (div_nonneg (mul_nonneg A.hCcross hM) (by norm_num))) (sq_nonneg _)
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-def frozenNodes (v : E → ℝ → E) (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
-    (h : ℝ≥0) (q : E × ℝ) : ℕ → (ℝ≥0 → ℝ) → E × ℝ
-  | 0, _ => q
-  | k+1, b =>
-    let qk := frozenNodes v U h q k b
-    if (b,qk) ∈ U k then
-      (qk.1+(h:ℝ) • v qk.1 qk.2, qk.2+(b (((k:ℝ≥0)+1)*h)-b ((k:ℝ≥0)*h)))
-    else qk
-
-lemma measurable_frozenNodes (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mU : ∀ k, MeasurableSet (U k))
-    (h : ℝ≥0) (q : E × ℝ) (k : ℕ) : Measurable (frozenNodes v U h q k) := by
-  induction k with
-  | zero => exact measurable_const
-  | succ k ih =>
-    exact ((ih.fst.add ((mv.comp ih).const_smul (h:ℝ))).prodMk
-      (ih.snd.add ((measurable_pi_apply _).sub (measurable_pi_apply _)))).ite
-      ((mU k).preimage (measurable_id.prodMk ih)) ih
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma frozenNodes_congr (v : E → ℝ → E) (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
-    (h : ℝ≥0) (q : E × ℝ)
-    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
-      ((b,x) ∈ U k ↔ (c,x) ∈ U k)) (k : ℕ)
-    {b c : ℝ≥0 → ℝ} (he : ∀ t ≤ (k:ℝ≥0)*h, b t = c t) :
-    frozenNodes v U h q k b = frozenNodes v U h q k c := by
-  induction k with
-  | zero => rfl
-  | succ k ih =>
-    have hle : (k:ℝ≥0)*h ≤ ((k+1:ℕ):ℝ≥0)*h :=
-      mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ k) h.coe_nonneg
-    have he' : ∀ t ≤ (k:ℝ≥0)*h, b t = c t := fun t ht => he t (ht.trans hle)
-    have hn := ih he'
-    have hu := hU k b c he' (frozenNodes v U h q k c)
-    have hb := he (((k:ℝ≥0)+1)*h) (by push_cast; exact le_rfl)
-    simp only [frozenNodes, hn, hu, hb, he _ hle]
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma frozenNodes_history (v : E → ℝ → E) (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
-    (h : ℝ≥0) (q : E × ℝ)
-    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
-      ((b,x) ∈ U k ↔ (c,x) ∈ U k)) (k : ℕ) (b : Driver) :
-    frozenNodes v U h q k (extendHistory ((k:ℝ≥0)*h) (history ((k:ℝ≥0)*h) b)) =
-      frozenNodes v U h q k b := by
-  apply frozenNodes_congr v U h q hU
-  intro t ht
-  simp only [extendHistory,dite_eq_left ht,history]
-
-lemma expected_frozen_step {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mU : ∀ k, MeasurableSet (U k))
-    (h : ℝ≥0) (q : E × ℝ)
-    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
-      ((b,x) ∈ U k ↔ (c,x) ∈ U k))
-    {M : ℝ} (hv : ∀ x b, ‖v x b‖ ≤ M)
-    (hg : ∀ k b x, (b,x) ∈ U k → A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
-    (k : ℕ) :
-    (∫ b : Driver, A.F (frozenNodes v U h q (k+1) b).1 (frozenNodes v U h q (k+1) b).2-
-      A.F (frozenNodes v U h q k b).1 (frozenNodes v U h q k b).2 ∂W) ≤
-      (A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2+A.C₃/3*(h:ℝ)*Real.sqrt h := by
-  let s := (k:ℝ≥0)*h
-  let X := fun p : Iic s → ℝ => frozenNodes v U h q k (extendHistory s p)
-  have mX : Measurable X := (measurable_frozenNodes v mv U mU h q k).comp (measurable_extendHistory s)
-  let S := {p : Iic s → ℝ | (extendHistory s p,X p) ∈ U k}
-  have mS : MeasurableSet S := (mU k).preimage ((measurable_extendHistory s).prodMk mX)
-  have he := expected_active_euler_step hW s h A (fun p => (X p).1)
-    (fun p => v (X p).1 (X p).2) (fun p => (X p).2) mX.fst (mv.comp mX) mX.snd S mS
-    (fun p => hv _ _) (fun p hp => hg k (extendHistory s p) (X p) hp)
-  have hx (b : Driver) : X (history s b) = frozenNodes v U h q k b :=
-    frozenNodes_history v U h q hU k b
-  have hs (b : Driver) : history s b ∈ S ↔ ((b : ℝ≥0 → ℝ),frozenNodes v U h q k b) ∈ U k := by
-    change (extendHistory s (history s b),X (history s b)) ∈ U k ↔ _
-    rw [hx]
-    apply hU k _ _
-    intro t ht
-    change t ≤ s at ht
-    simp only [extendHistory,dite_eq_left ht,history]
-  have ht : s+h = ((k:ℝ≥0)+1)*h := by dsimp [s]; ring
-  convert! he using 1
-  congr 1
-  funext b
-  rw [hx,hs]
-  simp only [frozenNodes,increment,ht,s]
-  split_ifs <;> rfl
-
-end SLEExactGauge.BrownianCalculus
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.BrownianNodes
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+namespace SLEExactGauge.BrownianCalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+lemma measurable_nodes_driver (v : E → ℝ → E)
+    (hv : Measurable (Function.uncurry v)) (h : ℝ≥0) (x : E) (k : ℕ) :
+    Measurable (fun b : Driver => stochasticNodes v h x k b) :=
+  (measurable_stochasticNodes v hv h x k).comp
+    (Measurable.of_eval fun time => (continuous_eval_const time).measurable)
+
+lemma expected_nodes_step {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E)
+    (mv : Measurable (Function.uncurry v)) (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
+    (hgen : ∀ x b, A.D x b (v x b)+deriv (deriv (A.F x)) b/2 ≤ 0)
+    (h : ℝ≥0) (x : E) (k : ℕ) :
+    (∫ b : Driver, A.F (stochasticNodes v h x (k+1) b) (b (((k+1:ℕ):ℝ≥0)*h))-
+      A.F (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h)) ∂W) ≤
+      (A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2 + A.C₃/3*(h:ℝ)*Real.sqrt h := by
+  let s := (k:ℝ≥0)*h
+  let X := fun p : Iic s → ℝ => stochasticNodes v h x k (extendHistory s p)
+  let V := fun p : Iic s → ℝ => v (X p) (p ⟨s,by simp⟩)
+  have mX : Measurable X := (measurable_stochasticNodes v mv h x k).comp
+    (measurable_extendHistory s)
+  have mV : Measurable V := mv.comp (mX.prodMk (measurable_pi_apply _))
+  have he := expected_euler_step hW s h A.F A.D X V A.mF A.mB1 A.mB2 mX mV
+    A.hD A.hsm A.hC₃ A.hCx A.hCcross A.hF A.hB1 A.hB2 A.hB3 A.hLipD A.hLipB2
+    (fun p => hv _ _) (fun p => hgen _ _)
+  have hn (b : Driver) : X (history s b) = stochasticNodes v h x k b :=
+    stochasticNodes_history v h x k b
+  have hs : ((k+1:ℕ):ℝ≥0)*h = s+h := by dsimp [s]; push_cast; ring
+  simpa only [X, V, hn, history, stochasticNodes, hs] using he
+
+ 
+
+theorem expected_nodes {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E)
+    (mv : Measurable (Function.uncurry v)) (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
+    (hgen : ∀ x b, A.D x b (v x b)+deriv (deriv (A.F x)) b/2 ≤ 0)
+    (h : ℝ≥0) (x : E) (k : ℕ) :
+    (∫ b : Driver, A.F (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h)) ∂W) ≤
+      A.F x 0+(k:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2 + A.C₃/3*(h:ℝ)*Real.sqrt h) := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hi (k : ℕ) : Integrable (fun b : Driver =>
+      A.F (stochasticNodes v h x k b) (b ((k:ℝ≥0)*h))) W := by
+    apply (integrable_const A.C₀).mono'
+    · exact (A.mF.comp ((measurable_nodes_driver v mv h x k).prodMk
+        (continuous_eval_const _).measurable)).aestronglyMeasurable
+    · exact Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using A.hF _ _
+  induction k with
+  | zero => simp [stochasticNodes, map_zero]
+  | succ k ih =>
+    have he := expected_nodes_step hW A v mv M hv hgen h x k
+    rw [integral_sub (hi (k+1)) (hi k)] at he
+    push_cast at he ⊢
+    linarith
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem expected_nodes_limit {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E)
+    (mv : Measurable (Function.uncurry v)) (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
+    (hgen : ∀ x b, A.D x b (v x b)+deriv (deriv (A.F x)) b/2 ≤ 0)
+    (h : ℕ → ℝ≥0) (k : ℕ → ℕ) (x : E) (t : ℝ≥0) (X : Driver → E)
+    (hh : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0))
+    (htime : Tendsto (fun n => (k n:ℝ≥0)*h n) atTop (𝓝 t))
+    (hX : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n => stochasticNodes v (h n) x (k n) b) atTop (𝓝 (X b))) :
+    (∫ b, A.F (X b) (b t) ∂W) ≤ A.F x 0 := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hc : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n =>
+      A.F (stochasticNodes v (h n) x (k n) b) (b ((k n:ℝ≥0)*h n)))
+      atTop (𝓝 (A.F (X b) (b t))) := by
+    filter_upwards [hX] with b hb
+    exact A.cF.continuousAt.tendsto.comp (hb.prodMk_nhds (b.continuous.continuousAt.tendsto.comp htime))
+  have hi : Tendsto (fun n => ∫ b : Driver,
+      A.F (stochasticNodes v (h n) x (k n) b) (b ((k n:ℝ≥0)*h n)) ∂W)
+      atTop (𝓝 (∫ b, A.F (X b) (b t) ∂W)) :=
+    tendsto_integral_of_dominated_convergence (fun _ : Driver => A.C₀)
+    (fun n => (A.mF.comp ((measurable_nodes_driver v mv (h n) x (k n)).prodMk
+      (continuous_eval_const _).measurable)).aestronglyMeasurable)
+    (integrable_const A.C₀)
+    (fun n => Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using A.hF _ _) hc
+  have hk : Tendsto (fun n => (k n:ℝ)*(h n:ℝ)) atTop (𝓝 (t:ℝ)) := by
+    simpa only [Function.comp_def, NNReal.coe_mul, NNReal.coe_natCast] using
+      NNReal.continuous_coe.continuousAt.tendsto.comp htime
+  have hs : Tendsto (fun n => Real.sqrt (h n:ℝ)) atTop (𝓝 0) := by
+    simpa only [Function.comp_def, Real.sqrt_zero] using (Real.continuous_sqrt.tendsto 0).comp hh
+  have he : Tendsto (fun n => (k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2 +
+      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 0) := by
+    have hh' := hk.mul (((tendsto_const_nhds (x := A.Cx*M^2+A.Ccross*M/2)).mul hh).add
+      ((tendsto_const_nhds (x := A.C₃/3)).mul hs))
+    convert! hh' using 1
+    · ext n
+      ring
+    · simp
+  have hb : Tendsto (fun n => A.F x 0+(k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2 +
+      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 (A.F x 0)) := by
+    simpa using tendsto_const_nhds.add he
+  exact le_of_tendsto_of_tendsto hi hb
+    (Filter.Eventually.of_forall fun n => expected_nodes hW A v mv M hv hgen (h n) x (k n))
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+ 
+
+lemma expected_shifted_taylor {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s h : ℝ≥0) (F : (Iic s → ℝ) → ℝ → ℝ) (B : (Iic s → ℝ) → ℝ)
+    (hsm : ∀ p, ContDiff ℝ 3 (F p))
+    (hm1 : Measurable (fun p => deriv (F p) (B p)))
+    (hm2 : Measurable (fun p => deriv (deriv (F p)) (B p)))
+    {C M₁ M₂ : ℝ} (hC : 0 ≤ C)
+    (hb1 : ∀ p, |deriv (F p) (B p)| ≤ M₁)
+    (hb2 : ∀ p, |deriv (deriv (F p)) (B p)| ≤ M₂)
+    (hb3 : ∀ p x, |iteratedDeriv 3 (F p) x| ≤ C)
+    (hi : Integrable (fun b => F (history s b) (B (history s b)+increment s h b) -
+      F (history s b) (B (history s b))) W) :
+    |(∫ b, F (history s b) (B (history s b)+increment s h b) -
+      F (history s b) (B (history s b)) ∂W) -
+      (h:ℝ)/2 * (∫ b, deriv (deriv (F (history s b))) (B (history s b)) ∂W)| ≤
+      C/3 * (h:ℝ) * Real.sqrt h := by
+  let c := fun p : Iic s → ℝ => B p-p ⟨s,by simp⟩
+  let f := fun p u => F p (u+c p)
+  have hbase (p : Iic s → ℝ) : p ⟨s,by simp⟩+c p = B p := by dsimp [c]; ring
+  have hd1 (p : Iic s → ℝ) : deriv (f p) = fun u => deriv (F p) (u+c p) := by
+    ext u
+    exact deriv_comp_add_const (F p) (c p) u
+  have hd2 (p : Iic s → ℝ) : deriv (deriv (f p)) = fun u => deriv (deriv (F p)) (u+c p) := by
+    rw [hd1]
+    ext u
+    exact deriv_comp_add_const (deriv (F p)) (c p) u
+  have hsmooth (p : Iic s → ℝ) : ContDiff ℝ 3 (f p) :=
+    (hsm p).comp (contDiff_id.add contDiff_const)
+  have hm1' : Measurable (fun p => deriv (f p) (p ⟨s,by simp⟩)) := by
+    simpa only [hd1, hbase] using hm1
+  have hm2' : Measurable (fun p => deriv (deriv (f p)) (p ⟨s,by simp⟩)) := by
+    simpa only [hd2, hbase] using hm2
+  have hf (b : Driver) : f (history s b) (b (s+h))-f (history s b) (b s) =
+      F (history s b) (B (history s b)+increment s h b)-F (history s b) (B (history s b)) := by
+    dsimp [f,c,history,increment]
+    congr 2 <;> ring
+  have hi' : Integrable (fun b => f (history s b) (b (s+h))-f (history s b) (b s)) W := by
+    simpa only [hf] using hi
+  have he := expected_taylor_step hW s h f hsmooth hm1' hm2' hC
+    (fun p => by simpa only [hd1,hbase] using hb1 p)
+    (fun p => by simpa only [hd2,hbase] using hb2 p)
+    (fun p u => by simpa only [f, iteratedDeriv_comp_add_const, Function.comp_apply] using hb3 p (u+c p)) hi'
+  simpa only [hf,hd2,show ∀ b : Driver, b s+c (history s b) = B (history s b) from
+    fun b => hbase (history s b)] using he
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+
+ 
+
+lemma expected_shift_with_drift {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s h : ℝ≥0) (f : (Iic s → ℝ) → ℝ → ℝ) (B old : (Iic s → ℝ) → ℝ)
+    (mf : Measurable (Function.uncurry f)) (mB : Measurable B) (mOld : Measurable old)
+    (hsm : ∀ p, ContDiff ℝ 3 (f p))
+    (hm1 : Measurable (fun p => deriv (f p) (B p)))
+    (hm2 : Measurable (fun p => deriv (deriv (f p)) (B p)))
+    {C₀ C₁ C₂ C₃ R : ℝ} (hC₃ : 0 ≤ C₃)
+    (hb0 : ∀ p u, |f p u| ≤ C₀) (hOld : ∀ p, |old p| ≤ C₀)
+    (hb1 : ∀ p, |deriv (f p) (B p)| ≤ C₁)
+    (hb2 : ∀ p, |deriv (deriv (f p)) (B p)| ≤ C₂)
+    (hb3 : ∀ p u, |iteratedDeriv 3 (f p) u| ≤ C₃)
+    (hstep : ∀ p, f p (B p)-old p+(h:ℝ)/2*deriv (deriv (f p)) (B p) ≤ R) :
+    (∫ b, f (history s b) (B (history s b)+increment s h b)-old (history s b) ∂W) ≤
+      R + C₃/3*(h:ℝ)*Real.sqrt h := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let step := fun b : Driver => f (history s b) (B (history s b))
+  let next := fun b : Driver => f (history s b) (B (history s b)+increment s h b)
+  let prev := fun b : Driver => old (history s b)
+  let Q := fun b : Driver => deriv (deriv (f (history s b))) (B (history s b))
+  have ms : Measurable step := mf.comp ((measurable_history s).prodMk (mB.comp (measurable_history s)))
+  have mn : Measurable next := mf.comp ((measurable_history s).prodMk
+    ((mB.comp (measurable_history s)).add (measurable_increment s h)))
+  have mp : Measurable prev := mOld.comp (measurable_history s)
+  have mq : Measurable Q := hm2.comp (measurable_history s)
+  have is : Integrable step W := (integrable_const C₀).mono' ms.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb0 _ _)
+  have inx : Integrable next W := (integrable_const C₀).mono' mn.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb0 _ _)
+  have ip : Integrable prev W := (integrable_const C₀).mono' mp.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hOld _)
+  have iq : Integrable Q W := (integrable_const C₂).mono' mq.aestronglyMeasurable
+    (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using hb2 _)
+  have ht := expected_shifted_taylor hW s h f B hsm hm1 hm2 hC₃ hb1 hb2 hb3 (inx.sub is)
+  change |(∫ b, next b-step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W)| ≤ _ at ht
+  have hi := integral_mono_ae ((is.sub ip).add (iq.const_mul ((h:ℝ)/2)))
+    (integrable_const R) (Filter.Eventually.of_forall fun b => hstep (history s b))
+  have hi' : (∫ b, step b-prev b+(h:ℝ)/2*Q b ∂W) ≤ R := by
+    simpa only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul,
+      Pi.add_apply, Pi.sub_apply] using hi
+  change (∫ b, (step-prev) b+(h:ℝ)/2*Q b ∂W) ≤ R at hi'
+  rw [integral_add (is.sub ip) (iq.const_mul ((h:ℝ)/2)), integral_const_mul] at hi'
+  change (∫ b, step b-prev b ∂W)+(h:ℝ)/2*(∫ b, Q b ∂W) ≤ R at hi'
+  rw [integral_sub is ip] at hi'
+  rw [integral_sub inx is] at ht
+  change (∫ b, next b-prev b ∂W) ≤ _
+  rw [integral_sub inx ip]
+  linarith [le_abs_self ((∫ b, next b ∂W)-(∫ b, step b ∂W)-(h:ℝ)/2*(∫ b, Q b ∂W))]
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+omit [BorelSpace E] [SecondCountableTopology E] in
+lemma SmoothTest.euler_generator_error (A : SmoothTest E) (x v : E) (b : ℝ)
+    {h M : ℝ} (hh : 0 ≤ h) (hv : ‖v‖ ≤ M)
+    (hg : A.D x b v+deriv (deriv (A.F x)) b/2 ≤ 0) :
+    A.F (x+h • v) b-A.F x b+h/2*deriv (deriv (A.F (x+h • v))) b ≤
+      (A.Cx*M^2+A.Ccross*M/2)*h^2 := by
+  have hy : ‖x+h • v-x‖ ≤ h*M := by
+    simp only [add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg hh]
+    exact mul_le_mul_of_nonneg_left hv hh
+  have hd := state_euler_remainder (fun x => A.hD x b) A.hCx
+    (fun x y => A.hLipD x y b) x v hh hv
+  have hb := (le_abs_self _).trans (A.hLipB2 (x+h • v) x b)
+  have hc := hb.trans (mul_le_mul_of_nonneg_left hy A.hCcross)
+  have hg' := mul_nonpos_of_nonneg_of_nonpos hh hg
+  have hc' := mul_le_mul_of_nonneg_left hc (div_nonneg hh (by norm_num : (0:ℝ) ≤ 2))
+  nlinarith
+
+ 
+
+lemma expected_active_euler_step {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s h : ℝ≥0) (A : SmoothTest E)
+    (X V : (Iic s → ℝ) → E) (B : (Iic s → ℝ) → ℝ)
+    (mX : Measurable X) (mV : Measurable V) (mB : Measurable B)
+    (S : Set (Iic s → ℝ)) (hS : MeasurableSet S)
+    {M : ℝ} (hV : ∀ p, ‖V p‖ ≤ M)
+    (hgen : ∀ p ∈ S, A.D (X p) (B p) (V p)+deriv (deriv (A.F (X p))) (B p)/2 ≤ 0) :
+    (∫ b, (if history s b ∈ S then
+      A.F (X (history s b)+(h:ℝ) • V (history s b)) (B (history s b)+increment s h b)
+      else A.F (X (history s b)) (B (history s b)))-
+      A.F (X (history s b)) (B (history s b)) ∂W) ≤
+      (A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2+A.C₃/3*(h:ℝ)*Real.sqrt h := by
+  classical
+  let Y := fun p => X p+(h:ℝ) • V p
+  let old := fun p => A.F (X p) (B p)
+  let f := fun p u => if p ∈ S then A.F (Y p) u else old p
+  have mY : Measurable Y := mX.add (mV.const_smul (h:ℝ))
+  have mOld : Measurable old := A.mF.comp (mX.prodMk mB)
+  have mf : Measurable (Function.uncurry f) :=
+    (A.mF.comp ((mY.comp measurable_fst).prodMk measurable_snd)).ite
+      (hS.preimage measurable_fst) (mOld.comp measurable_fst)
+  have hd1 (p : Iic s → ℝ) : deriv (f p) = fun u => if p ∈ S then deriv (A.F (Y p)) u else 0 := by
+    by_cases hp : p ∈ S <;> simp [f,hp]
+  have hd2 (p : Iic s → ℝ) : deriv (deriv (f p)) = fun u =>
+      if p ∈ S then deriv (deriv (A.F (Y p))) u else 0 := by
+    rw [hd1]
+    by_cases hp : p ∈ S <;> simp [hp]
+  have hm1 : Measurable (fun p => deriv (f p) (B p)) := by
+    simp only [hd1]
+    exact (A.mB1.comp (mY.prodMk mB)).ite hS measurable_const
+  have hm2 : Measurable (fun p => deriv (deriv (f p)) (B p)) := by
+    simp only [hd2]
+    exact (A.mB2.comp (mY.prodMk mB)).ite hS measurable_const
+  have hC₁ : 0 ≤ A.C₁ := (abs_nonneg _).trans (A.hB1 0 0)
+  have hC₂ : 0 ≤ A.C₂ := (abs_nonneg _).trans (A.hB2 0 0)
+  have hM : 0 ≤ M := (norm_nonneg _).trans (hV (fun _ => 0))
+  refine expected_shift_with_drift (C₀ := A.C₀) (C₁ := A.C₁) (C₂ := A.C₂)
+    hW s h f B old mf mB mOld ?_ hm1 hm2 A.hC₃ ?_ ?_ ?_ ?_ ?_ ?_
+  · intro p
+    by_cases hp : p ∈ S
+    · simpa only [f,ite_eq_left hp] using A.hsm (Y p)
+    · simpa only [f,ite_eq_right hp] using (contDiff_const (c := old p) : ContDiff ℝ 3 _)
+  · intro p u
+    by_cases hp : p ∈ S
+    · simpa only [f,ite_eq_left hp] using A.hF (Y p) u
+    · simpa only [f,ite_eq_right hp,old] using A.hF (X p) (B p)
+  · intro p
+    exact A.hF _ _
+  · intro p
+    by_cases hp : p ∈ S
+    · simpa only [hd1,ite_eq_left hp] using A.hB1 (Y p) (B p)
+    · simpa only [hd1,ite_eq_right hp,abs_zero] using hC₁
+  · intro p
+    by_cases hp : p ∈ S
+    · simpa only [hd2,ite_eq_left hp] using A.hB2 (Y p) (B p)
+    · simpa only [hd2,ite_eq_right hp,abs_zero] using hC₂
+  · intro p u
+    by_cases hp : p ∈ S
+    · simpa only [f,ite_eq_left hp] using A.hB3 (Y p) u
+    · simpa [f,hp,iteratedDeriv_const] using A.hC₃
+  · intro p
+    rw [hd2]
+    by_cases hp : p ∈ S
+    · simpa only [f,ite_eq_left hp,old,Y] using A.euler_generator_error (X p) (V p) (B p)
+        h.coe_nonneg (hV p) (hgen p hp)
+    · simp only [f,ite_eq_right hp,sub_self,mul_zero,add_zero]
+      exact mul_nonneg (add_nonneg (mul_nonneg A.hCx (sq_nonneg M))
+        (div_nonneg (mul_nonneg A.hCcross hM) (by norm_num))) (sq_nonneg _)
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+def frozenNodes (v : E → ℝ → E) (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
+    (h : ℝ≥0) (q : E × ℝ) : ℕ → (ℝ≥0 → ℝ) → E × ℝ
+  | 0, _ => q
+  | k+1, b =>
+    let qk := frozenNodes v U h q k b
+    if (b,qk) ∈ U k then
+      (qk.1+(h:ℝ) • v qk.1 qk.2, qk.2+(b (((k:ℝ≥0)+1)*h)-b ((k:ℝ≥0)*h)))
+    else qk
+
+lemma measurable_frozenNodes (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mU : ∀ k, MeasurableSet (U k))
+    (h : ℝ≥0) (q : E × ℝ) (k : ℕ) : Measurable (frozenNodes v U h q k) := by
+  induction k with
+  | zero => exact measurable_const
+  | succ k ih =>
+    exact ((ih.fst.add ((mv.comp ih).const_smul (h:ℝ))).prodMk
+      (ih.snd.add ((measurable_pi_apply _).sub (measurable_pi_apply _)))).ite
+      ((mU k).preimage (measurable_id.prodMk ih)) ih
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma frozenNodes_congr (v : E → ℝ → E) (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
+    (h : ℝ≥0) (q : E × ℝ)
+    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
+      ((b,x) ∈ U k ↔ (c,x) ∈ U k)) (k : ℕ)
+    {b c : ℝ≥0 → ℝ} (he : ∀ t ≤ (k:ℝ≥0)*h, b t = c t) :
+    frozenNodes v U h q k b = frozenNodes v U h q k c := by
+  induction k with
+  | zero => rfl
+  | succ k ih =>
+    have hle : (k:ℝ≥0)*h ≤ ((k+1:ℕ):ℝ≥0)*h :=
+      mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ k) h.coe_nonneg
+    have he' : ∀ t ≤ (k:ℝ≥0)*h, b t = c t := fun t ht => he t (ht.trans hle)
+    have hn := ih he'
+    have hu := hU k b c he' (frozenNodes v U h q k c)
+    have hb := he (((k:ℝ≥0)+1)*h) (by push_cast; exact le_rfl)
+    simp only [frozenNodes, hn, hu, hb, he _ hle]
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma frozenNodes_history (v : E → ℝ → E) (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
+    (h : ℝ≥0) (q : E × ℝ)
+    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
+      ((b,x) ∈ U k ↔ (c,x) ∈ U k)) (k : ℕ) (b : Driver) :
+    frozenNodes v U h q k (extendHistory ((k:ℝ≥0)*h) (history ((k:ℝ≥0)*h) b)) =
+      frozenNodes v U h q k b := by
+  apply frozenNodes_congr v U h q hU
+  intro t ht
+  simp only [extendHistory,dite_eq_left ht,history]
+
+lemma expected_frozen_step {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mU : ∀ k, MeasurableSet (U k))
+    (h : ℝ≥0) (q : E × ℝ)
+    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
+      ((b,x) ∈ U k ↔ (c,x) ∈ U k))
+    {M : ℝ} (hv : ∀ x b, ‖v x b‖ ≤ M)
+    (hg : ∀ k b x, (b,x) ∈ U k → A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
+    (k : ℕ) :
+    (∫ b : Driver, A.F (frozenNodes v U h q (k+1) b).1 (frozenNodes v U h q (k+1) b).2-
+      A.F (frozenNodes v U h q k b).1 (frozenNodes v U h q k b).2 ∂W) ≤
+      (A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2+A.C₃/3*(h:ℝ)*Real.sqrt h := by
+  let s := (k:ℝ≥0)*h
+  let X := fun p : Iic s → ℝ => frozenNodes v U h q k (extendHistory s p)
+  have mX : Measurable X := (measurable_frozenNodes v mv U mU h q k).comp (measurable_extendHistory s)
+  let S := {p : Iic s → ℝ | (extendHistory s p,X p) ∈ U k}
+  have mS : MeasurableSet S := (mU k).preimage ((measurable_extendHistory s).prodMk mX)
+  have he := expected_active_euler_step hW s h A (fun p => (X p).1)
+    (fun p => v (X p).1 (X p).2) (fun p => (X p).2) mX.fst (mv.comp mX) mX.snd S mS
+    (fun p => hv _ _) (fun p hp => hg k (extendHistory s p) (X p) hp)
+  have hx (b : Driver) : X (history s b) = frozenNodes v U h q k b :=
+    frozenNodes_history v U h q hU k b
+  have hs (b : Driver) : history s b ∈ S ↔ ((b : ℝ≥0 → ℝ),frozenNodes v U h q k b) ∈ U k := by
+    change (extendHistory s (history s b),X (history s b)) ∈ U k ↔ _
+    rw [hx]
+    apply hU k _ _
+    intro t ht
+    change t ≤ s at ht
+    simp only [extendHistory,dite_eq_left ht,history]
+  have ht : s+h = ((k:ℝ≥0)+1)*h := by dsimp [s]; ring
+  convert! he using 1
+  congr 1
+  funext b
+  rw [hx,hs]
+  simp only [frozenNodes,increment,ht,s]
+  split_ifs <;> rfl
+
+end SLEExactGauge.BrownianCalculus
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/EulerMoments.lean b/lean/OAI/Probability/SLEGauge/EulerMoments.lean
index de66e475bcc1863bccb0ef3255543897ef04f926..6ac64362cf302d5fcb547019f1b04425e219fb01 100644
--- a/lean/OAI/Probability/SLEGauge/EulerMoments.lean
+++ b/lean/OAI/Probability/SLEGauge/EulerMoments.lean
@@ -1,432 +1,432 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.FrozenExpectations
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.EulerGeneral
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-omit [CompleteSpace E] in
-lemma interpolate_at_node (f : ℝ → E → E) {h : ℝ} (hh : 0 < h) (z : E) (k : ℕ) :
-    interpolate f h z ((k:ℝ)*h) = nodes f h z k := by
-  simp [interpolate, mul_div_cancel_right₀ _ hh.ne']
-
- 
-
-omit [CompleteSpace E] in
-lemma nodes_converge_moving {f : ℝ → E → E} {g : ℝ → E} {z : E} {L M T U s : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
-    (hU : 0 ≤ U) (hUT : U < T) (hs : s ∈ Icc 0 U)
-    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0))
-    (k : ℕ → ℕ) (hk : ∀ᶠ n in atTop, (k n:ℝ)*h n ≤ U)
-    (hks : Tendsto (fun n => (k n:ℝ)*h n) atTop (𝓝 s)) :
-    Tendsto (fun n => nodes f (h n) z (k n)) atTop (𝓝 (g s)) := by
-  have hu := converge_uniformOn hL hM hg0 hlip hder hcf hbound hU hUT hh hh0
-  have hgs : ContinuousWithinAt g (Icc 0 U) s :=
-    (hder s ⟨hs.1, hs.2.trans hUT.le⟩).continuousWithinAt.mono (Icc_subset_Icc_right hUT.le)
-  have hs' : Tendsto (fun n => (k n:ℝ)*h n) atTop (𝓝[Icc 0 U] s) := by
-    rw [tendsto_nhdsWithin_iff]
-    exact ⟨hks, hk.mono fun n hn => ⟨mul_nonneg (Nat.cast_nonneg _) (hh n).le,hn⟩⟩
-  simpa only [interpolate_at_node _ (hh _) _ _] using hu.tendsto_comp hgs hs'
-
-end SLEExactGauge.EulerGeneral
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-
-lemma frozenNodes_eq_min (v : E → ℝ → E) (f : ℝ → E → E)
-    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (h : ℝ≥0) (z : E) (b : ℝ≥0 → ℝ)
-    (hf : ∀ (k : ℕ) x, f ((k:ℝ)*h) x = v x (b ((k:ℝ≥0)*h))) (K : ℕ)
-    (ha : ∀ k < K, (b,(EulerGeneral.nodes f h z k,b ((k:ℝ≥0)*h))) ∈ U k)
-    (hi : ∀ k ≥ K, ∀ x, (b,x) ∉ U k) (k : ℕ) :
-    frozenNodes v U h (z,b 0) k b =
-      (EulerGeneral.nodes f h z (min k K),b (((min k K:ℕ):ℝ≥0)*h)) := by
-  induction k with
-  | zero => simp [frozenNodes,EulerGeneral.nodes]
-  | succ k ih =>
-    by_cases hk : k < K
-    · have hk1 : k+1 ≤ K := hk
-      rw [frozenNodes,ih]
-      simp only [min_eq_left hk.le, min_eq_left hk1, ha k hk, ite_eq_left, EulerGeneral.nodes, hf]
-      congr 1
-      simp only [Nat.cast_add,Nat.cast_one]
-      ring
-    · have hk' : K ≤ k := Nat.le_of_not_gt hk
-      have hk1 : K ≤ k+1 := hk'.trans (Nat.le_succ _)
-      rw [frozenNodes,ih]
-      simp only [min_eq_right hk',min_eq_right hk1,hi k hk',ite_false]
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.EulerGeneral
-
-lemma ceil_mesh_times_converge {h : ℕ → ℝ} (hh : ∀ n, 0 < h n)
-    (hh0 : Tendsto h atTop (𝓝 0)) {τ : ℝ} (hτ : 0 ≤ τ) :
-    Tendsto (fun n => (⌈τ/h n⌉₊:ℝ)*h n) atTop (𝓝 τ) := by
-  have hb (n : ℕ) : 0 ≤ (⌈τ/h n⌉₊:ℝ)*h n-τ ∧ (⌈τ/h n⌉₊:ℝ)*h n-τ ≤ h n := by
-    have hl := Nat.le_ceil (τ/h n)
-    have hu := Nat.ceil_lt_add_one (div_nonneg hτ (hh n).le)
-    have hl' := (div_le_iff₀ (hh n)).mp hl
-    have hu' := (mul_lt_mul_of_pos_right hu (hh n))
-    rw [add_mul,div_mul_cancel₀ _ (hh n).ne',one_mul] at hu'
-    constructor <;> linarith
-  have hz : Tendsto (fun n => (⌈τ/h n⌉₊:ℝ)*h n-τ) atTop (𝓝 0) :=
-    squeeze_zero (fun n => (hb n).1) (fun n => (hb n).2) hh0
-  simpa using hz.add_const τ
-
-lemma effective_stop_times_converge {h : ℕ → ℝ} (hh : ∀ n, 0 < h n)
-    (hh0 : Tendsto h atTop (𝓝 0)) {τ t : ℝ} (hτ : 0 ≤ τ) (ht : 0 ≤ t) :
-    Tendsto (fun n => ((min ⌊t/h n⌋₊ ⌈τ/h n⌉₊:ℕ):ℝ)*h n) atTop (𝓝 (min t τ)) := by
-  have h1 := mesh_times_converge hh hh0 ht
-  have h2 := ceil_mesh_times_converge hh hh0 hτ
-  simpa only [Nat.cast_min, min_mul_of_nonneg _ _ (hh _).le] using h1.min h2
-
-end SLEExactGauge.EulerGeneral
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
- 
-
-omit [CompleteSpace E] in
-lemma ordinary_nodes_eventually_inside (f : ℝ → E → E) (g : ℝ → E) (z : E)
-    (b : ℝ≥0 → ℝ) (D : Set (E × ℝ)) {τ T L M ε : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
-    (hτ : 0 ≤ τ) (hτT : τ < T) (hε : 0 < ε)
-    (hD : ∀ u ∈ Icc 0 τ, ∀ x, ‖x-g u‖ < ε → (x,b u.toNNReal) ∈ D)
-    {h : ℕ → ℝ≥0} (hh : ∀ n, 0 < h n) (hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0)) :
-    ∀ᶠ n in atTop, ∀ k : ℕ, (k:ℝ)*(h n:ℝ) < τ →
-      (EulerGeneral.nodes f (h n) z k,b ((k:ℝ≥0)*h n)) ∈ D := by
-  have hu := EulerGeneral.converge_uniformOn hL hM hg0 hlip hder hcf hbound hτ hτT
-    (fun n => show 0 < (h n:ℝ) from hh n) hh0
-  have he := (Metric.tendstoUniformlyOn_iff.mp hu) ε hε
-  filter_upwards [he] with n hn
-  intro k hk
-  have htime : (k:ℝ)*(h n:ℝ) ∈ Icc 0 τ := ⟨by positivity,hk.le⟩
-  have hn' := hn _ htime
-  rw [EulerGeneral.interpolate_at_node _ (show 0 < (h n:ℝ) from hh n) _ _] at hn'
-  have hb : ((k:ℝ)*(h n:ℝ)).toNNReal = (k:ℝ≥0)*h n := by
-    apply NNReal.coe_injective
-    simp [Real.toNNReal_of_nonneg (show 0 ≤ (k:ℝ)*(h n:ℝ) by positivity)]
-  rw [← hb]
-  apply hD _ htime _
-  rw [dist_comm] at hn'
-  simpa only [dist_eq_norm] using hn'
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
- 
-
-omit [CompleteSpace E] in
-theorem frozen_nodes_path_limit (v : E → ℝ → E) (g : ℝ → E) (z : E)
-    (b : ℝ≥0 → ℝ) (cb : Continuous b) (D : Set (E × ℝ))
-    (U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) {τ T t L M ε : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖v x (b u.toNNReal)-v y (b u.toNNReal)‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (v (g u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => v (g u) (b u.toNNReal)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖v (g u) (b u.toNNReal)‖ ≤ M)
-    (hτ : 0 ≤ τ) (hτT : τ < T) (ht : 0 ≤ t) (hε : 0 < ε)
-    (hD : ∀ u ∈ Icc 0 τ, ∀ x, ‖x-g u‖ < ε → (x,b u.toNNReal) ∈ D)
-    {h : ℕ → ℝ≥0} (hh : ∀ n, 0 < h n) (hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0))
-    (hU : ∀ n k x, (b,x) ∈ U n k ↔ (k:ℝ)*(h n:ℝ) < τ ∧ x ∈ D) :
-    Tendsto (fun n => frozenNodes v (U n) (h n) (z,b 0) ⌊t/(h n:ℝ)⌋₊ b)
-      atTop (𝓝 (g (min t τ),b (min t τ).toNNReal)) := by
-  let f : ℝ → E → E := fun u x => v x (b u.toNNReal)
-  let K : ℕ → ℕ := fun n => ⌈τ/(h n:ℝ)⌉₊
-  let k : ℕ → ℕ := fun n => min ⌊t/(h n:ℝ)⌋₊ (K n)
-  have hn := ordinary_nodes_eventually_inside f g z b D hL hM hg0 hlip hder hcf hbound
-    hτ hτT hε hD hh hh0
-  have he : ∀ᶠ n in atTop, frozenNodes v (U n) (h n) (z,b 0) ⌊t/(h n:ℝ)⌋₊ b =
-      (EulerGeneral.nodes f (h n) z (k n),b ((k n:ℝ≥0)*h n)) := by
-    filter_upwards [hn] with n hn
-    refine frozenNodes_eq_min v f (U n) (h n) z b ?_ (K n) ?_ ?_ _
-    · intro j x
-      dsimp [f]
-      congr 2
-      apply NNReal.coe_injective
-      simp [Real.toNNReal_of_nonneg (show 0 ≤ (j:ℝ)*(h n:ℝ) by positivity)]
-    · intro j hj
-      have htj : (j:ℝ)*(h n:ℝ) < τ :=
-        (lt_div_iff₀ (show 0 < (h n:ℝ) from hh n)).mp (Nat.lt_ceil.mp hj)
-      exact (hU n j _).mpr ⟨htj,hn j htj⟩
-    · intro j hj x hx
-      have htj : τ ≤ (j:ℝ)*(h n:ℝ) :=
-        (div_le_iff₀ (show 0 < (h n:ℝ) from hh n)).mp (Nat.ceil_le.mp hj)
-      exact (not_lt_of_ge htj) ((hU n j x).mp hx).1
-  have hk0 : Tendsto (fun n => (k n:ℝ)*(h n:ℝ)) atTop (𝓝 (min t τ)) :=
-    EulerGeneral.effective_stop_times_converge (fun n => hh n) hh0 hτ ht
-  let S : ℝ := (τ+T)/2
-  have hsS : min t τ < S := lt_of_le_of_lt (min_le_right _ _) (by dsimp [S]; linarith)
-  have hkS : ∀ᶠ n in atTop, (k n:ℝ)*(h n:ℝ) ≤ S :=
-    (hk0.eventually (gt_mem_nhds hsS)).mono fun _ hn => hn.le
-  have hg := EulerGeneral.nodes_converge_moving hL hM hg0 hlip hder hcf hbound
-    (show 0 ≤ S by dsimp [S]; linarith) (show S < T by dsimp [S]; linarith)
-    (show min t τ ∈ Icc 0 S from ⟨le_min ht hτ,hsS.le⟩)
-    (fun n => show 0 < (h n:ℝ) from hh n) hh0 k hkS hk0
-  have hb0 : Tendsto (fun n => (k n:ℝ≥0)*h n) atTop (𝓝 (min t τ).toNNReal) := by
-    convert! continuous_real_toNNReal.continuousAt.tendsto.comp hk0 using 1
-    funext n
-    apply NNReal.coe_injective
-    simp [Real.toNNReal_of_nonneg (show 0 ≤ (k n:ℝ)*(h n:ℝ) by positivity)]
-  exact (hg.prodMk_nhds (cb.continuousAt.tendsto.comp hb0)).congr' (Filter.EventuallyEq.symm he)
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-omit [CompleteSpace E] in
-theorem expected_stopped_smooth {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hg0 : ∀ b, g b 0 = z)
-    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
-    (τ : (ℝ≥0 → ℝ) → ℝ) (mτ : Measurable τ)
-    (hτpast : ∀ (s : ℝ≥0) b c, (∀ u ≤ s, b u = c u) → ((s:ℝ) < τ b ↔ (s:ℝ) < τ c))
-    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
-    (D : Set (E × ℝ)) (mD : MeasurableSet D)
-    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ D)
-    (hgen : ∀ x ∈ D, A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
-    (ht : 0 ≤ t) :
-    (∫ b : Driver, A.F (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ A.F z 0 := by
-  let h : ℕ → ℝ≥0 := fun n => (1/2)^n
-  have hh : ∀ n, 0 < h n := fun n => pow_pos (by norm_num) n
-  have hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0) := by
-    simp only [h,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat]
-    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-  let U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)) :=
-    fun n k => {q | (k:ℝ)*(h n:ℝ) < τ q.1 ∧ q.2 ∈ D}
-  have mU : ∀ n k, MeasurableSet (U n k) := by
-    intro n k
-    exact (measurableSet_lt measurable_const (mτ.comp measurable_fst)).inter
-      (mD.preimage measurable_snd)
-  have hu : ∀ n (k : ℕ) b c, (∀ u ≤ (k:ℝ≥0)*h n, b u = c u) → ∀ x,
-      ((b,x) ∈ U n k ↔ (c,x) ∈ U n k) := by
-    intro n k b c hbc x
-    change ((k:ℝ)*(h n:ℝ) < τ b ∧ x ∈ D ↔ (k:ℝ)*(h n:ℝ) < τ c ∧ x ∈ D)
-    rw [show (k:ℝ)*(h n:ℝ) = (((k:ℝ≥0)*h n):ℝ) by simp]
-    exact and_congr_left fun _ => hτpast _ b c hbc
-  apply expected_frozen_limit hW A v mv U mU h (fun n => ⌊t/(h n:ℝ)⌋₊) (z,0) hu M hv
-    (fun _ _ _ x hx => hgen x hx.2)
-    (fun b => (g b (min t (τ b)), b (min t (τ b)).toNNReal)) t hh0
-    (EulerGeneral.mesh_times_converge (fun n => hh n) hh0 ht)
-  filter_upwards [] with b
-  obtain ⟨ε,hε,hDb⟩ := hD b
-  have hp := frozen_nodes_path_limit v (g b) z b b.continuous D U hL hM (hg0 b)
-    (fun _ _ => hlip _) (hder b) (hcf b) (fun u _ => hv _ _) (hτ b).1 (hτ b).2 ht
-    hε hDb hh hh0 (fun _ _ _ => Iff.rfl)
-  simpa only [map_zero] using hp
-
-end SLEExactGauge.BrownianCalculus
-
-open scoped ContDiff Manifold
-namespace SLEExactGauge.BrownianCalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-def driverJets (F : E × ℝ → ℝ) : ℕ → E × ℝ → ℝ
-  | 0 => F
-  | n+1 => fun q => fderiv ℝ (driverJets F n) q (0,1)
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma driverJets_smooth {F : E × ℝ → ℝ} (hF : ContDiff ℝ ∞ F) (n : ℕ) :
-    ContDiff ℝ ∞ (driverJets F n) := by
-  induction n with
-  | zero => exact hF
-  | succ n ih =>
-    exact (contDiff_infty_iff_fderiv.mp ih).2.clm_apply contDiff_const
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma driverJets_compact {F : E × ℝ → ℝ} (hF : HasCompactSupport F) (n : ℕ) :
-    HasCompactSupport (driverJets F n) := by
-  induction n with
-  | zero => exact hF
-  | succ n ih => exact ih.fderiv_apply ℝ (0,1)
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma driverJets_restriction {F : E × ℝ → ℝ} (hF : ContDiff ℝ ∞ F) (n : ℕ)
-    (x : E) (b : ℝ) : driverJets F n (x,b) = iteratedDeriv n (fun c => F (x,c)) b := by
-  induction n generalizing b with
-  | zero => rfl
-  | succ n ih =>
-    have he : iteratedDeriv n (fun c => F (x,c)) = fun c => driverJets F n (x,c) := by
-      funext c
-      exact (ih c).symm
-    rw [iteratedDeriv_succ,he]
-    have hd := ((driverJets_smooth hF n).differentiable (by simp) (x,b)).hasFDerivAt.comp_hasDerivAt b
-      ((hasDerivAt_const b x).prodMk (hasDerivAt_id b))
-    exact hd.deriv.symm
-
- 
-
-omit [SecondCountableTopology E] in
-theorem exists_smoothTest_of_compact (F : E × ℝ → ℝ)
-    (hF : ContDiff ℝ ∞ F) (hK : HasCompactSupport F) :
-    ∃ A : SmoothTest E, A.F = fun x b => F (x,b) := by
-  let D : E × ℝ → E →L[ℝ] ℝ := fun q =>
-    (fderiv ℝ F q).comp (ContinuousLinearMap.inl ℝ E ℝ)
-  have hD : ContDiff ℝ ∞ D :=
-    (contDiff_infty_iff_fderiv.mp hF).2.clm_comp contDiff_const
-  have hDK : HasCompactSupport D := by
-    exact (hK.fderiv ℝ).comp_left (g := fun A : E × ℝ →L[ℝ] ℝ => A.comp (ContinuousLinearMap.inl ℝ E ℝ)) (show (0 : E × ℝ →L[ℝ] ℝ).comp (ContinuousLinearMap.inl ℝ E ℝ) = 0 from rfl)
-  obtain ⟨C₀,hC₀⟩ := hK.exists_bound_of_continuous hF.continuous
-  obtain ⟨C₁,hC₁⟩ := (driverJets_compact hK 1).exists_bound_of_continuous
-    (driverJets_smooth hF 1).continuous
-  obtain ⟨C₂,hC₂⟩ := (driverJets_compact hK 2).exists_bound_of_continuous
-    (driverJets_smooth hF 2).continuous
-  obtain ⟨C₃,hC₃⟩ := (driverJets_compact hK 3).exists_bound_of_continuous
-    (driverJets_smooth hF 3).continuous
-  obtain ⟨Cx,hCx⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hDK hD (by simp)
-  obtain ⟨Cc,hCc⟩ := ContDiff.lipschitzWith_of_hasCompactSupport (driverJets_compact hK 2)
-    (driverJets_smooth hF 2) (by simp)
-  have hj1 (x : E) (b : ℝ) : driverJets F 1 (x,b) = deriv (fun c => F (x,c)) b := by
-    simpa using driverJets_restriction hF 1 x b
-  have hj2 (x : E) (b : ℝ) : driverJets F 2 (x,b) = deriv (deriv (fun c => F (x,c))) b := by
-    simpa only [iteratedDeriv_succ,iteratedDeriv_zero] using driverJets_restriction hF 2 x b
-  refine ⟨{
-    F := fun x b => F (x,b)
-    D := fun x b => D (x,b)
-    C₀ := C₀
-    C₁ := C₁
-    C₂ := C₂
-    C₃ := max C₃ 0
-    Cx := Cx
-    Ccross := Cc
-    hC₃ := le_max_right _ _
-    hCx := Cx.coe_nonneg
-    hCcross := Cc.coe_nonneg
-    cF := hF.continuous
-    mF := hF.continuous.measurable
-    mB1 := ?_
-    mB2 := ?_
-    hD := ?_
-    hsm := ?_
-    hF := ?_
-    hB1 := ?_
-    hB2 := ?_
-    hB3 := ?_
-    hLipD := ?_
-    hLipB2 := ?_ }, rfl⟩
-  · convert! (driverJets_smooth hF 1).continuous.measurable using 1
-    funext q
-    exact (hj1 q.1 q.2).symm
-  · convert! (driverJets_smooth hF 2).continuous.measurable using 1
-    funext q
-    exact (hj2 q.1 q.2).symm
-  · intro x b
-    have hi : HasFDerivAt (fun y : E => (y,b)) (ContinuousLinearMap.inl ℝ E ℝ) x := by
-      convert! (hasFDerivAt_id x).prodMk (hasFDerivAt_const b x) using 1
-    exact ((hF.differentiable (by simp) (x,b)).hasFDerivAt).comp x hi
-  · intro x
-    exact (hF.comp (contDiff_const.prodMk contDiff_id)).of_le (WithTop.coe_le_coe.mpr le_top)
-  · intro x b
-    simpa only [Real.norm_eq_abs] using hC₀ (x,b)
-  · intro x b
-    simpa only [hj1,Real.norm_eq_abs] using hC₁ (x,b)
-  · intro x b
-    simpa only [hj2,Real.norm_eq_abs] using hC₂ (x,b)
-  · intro x b
-    rw [← driverJets_restriction hF 3 x b,← Real.norm_eq_abs]
-    exact (hC₃ (x,b)).trans (le_max_left _ _)
-  · intro x y b
-    simpa [Prod.norm_def] using hCx.norm_sub_le (x,b) (y,b)
-  · intro x y b
-    simpa [hj2,Prod.norm_def,Real.norm_eq_abs] using hCc.norm_sub_le (x,b) (y,b)
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.Localization
-variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
-
- 
-
-theorem compact_extension {K O : Set E} (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O)
-    (f : E → F) (hf : ContDiffOn ℝ ∞ f O) :
-    ∃ g : E → F, ContDiff ℝ ∞ g ∧ HasCompactSupport g ∧ ∀ x ∈ K, g =ᶠ[𝓝 x] f := by
-  obtain ⟨L,hL,hKL,hLO⟩ := exists_compact_between hK hO hKO
-  obtain ⟨φ,hφ1,hφ0,hφbound⟩ :=
-    exists_contMDiffMap_one_nhds_of_subset_interior (𝓘(ℝ,E)) hK.isClosed hKL (n := ⊤)
-  have hφsmooth : ContDiff ℝ ∞ (φ : E → ℝ) := contMDiff_iff_contDiff.mp φ.contMDiff
-  have hsupp : tsupport (φ : E → ℝ) ⊆ L := by
-    apply closure_minimal _ hL.isClosed
-    intro x hx
-    by_contra hn
-    exact hx (hφ0 x hn)
-  have hφK : HasCompactSupport (φ : E → ℝ) := hL.of_isClosed_subset isClosed_closure hsupp
-  refine ⟨fun x => φ x • f x,?_,hφK.smul_right,?_⟩
-  · apply contDiff_iff_contDiffAt.mpr
-    intro x
-    by_cases hx : x ∈ tsupport (φ : E → ℝ)
-    · exact hφsmooth.contDiffAt.smul ((hf x (hLO (hsupp hx))).contDiffAt (hO.mem_nhds (hLO (hsupp hx))))
-    · have he : (φ : E → ℝ) =ᶠ[𝓝 x] 0 := by
-        exact (notMem_tsupport_iff_eventuallyEq.mp hx)
-      apply (contDiffAt_const (c := (0 : F))).congr_of_eventuallyEq
-      filter_upwards [he] with y hy
-      simp [hy]
-  · intro x hx
-    have hnear : ∀ᶠ y in 𝓝 x, φ y = 1 := hφ1.filter_mono (nhds_le_nhdsSet hx)
-    filter_upwards [hnear] with y hy
-    simp [hy]
-
-end SLEExactGauge.Localization
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
-
-end SLEExactGauge.BrownianCalculus
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.FrozenExpectations
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.EulerGeneral
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+omit [CompleteSpace E] in
+lemma interpolate_at_node (f : ℝ → E → E) {h : ℝ} (hh : 0 < h) (z : E) (k : ℕ) :
+    interpolate f h z ((k:ℝ)*h) = nodes f h z k := by
+  simp [interpolate, mul_div_cancel_right₀ _ hh.ne']
+
+ 
+
+omit [CompleteSpace E] in
+lemma nodes_converge_moving {f : ℝ → E → E} {g : ℝ → E} {z : E} {L M T U s : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
+    (hU : 0 ≤ U) (hUT : U < T) (hs : s ∈ Icc 0 U)
+    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0))
+    (k : ℕ → ℕ) (hk : ∀ᶠ n in atTop, (k n:ℝ)*h n ≤ U)
+    (hks : Tendsto (fun n => (k n:ℝ)*h n) atTop (𝓝 s)) :
+    Tendsto (fun n => nodes f (h n) z (k n)) atTop (𝓝 (g s)) := by
+  have hu := converge_uniformOn hL hM hg0 hlip hder hcf hbound hU hUT hh hh0
+  have hgs : ContinuousWithinAt g (Icc 0 U) s :=
+    (hder s ⟨hs.1, hs.2.trans hUT.le⟩).continuousWithinAt.mono (Icc_subset_Icc_right hUT.le)
+  have hs' : Tendsto (fun n => (k n:ℝ)*h n) atTop (𝓝[Icc 0 U] s) := by
+    rw [tendsto_nhdsWithin_iff]
+    exact ⟨hks, hk.mono fun n hn => ⟨mul_nonneg (Nat.cast_nonneg _) (hh n).le,hn⟩⟩
+  simpa only [interpolate_at_node _ (hh _) _ _] using hu.tendsto_comp hgs hs'
+
+end SLEExactGauge.EulerGeneral
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma frozenNodes_eq_min (v : E → ℝ → E) (f : ℝ → E → E)
+    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (h : ℝ≥0) (z : E) (b : ℝ≥0 → ℝ)
+    (hf : ∀ (k : ℕ) x, f ((k:ℝ)*h) x = v x (b ((k:ℝ≥0)*h))) (K : ℕ)
+    (ha : ∀ k < K, (b,(EulerGeneral.nodes f h z k,b ((k:ℝ≥0)*h))) ∈ U k)
+    (hi : ∀ k ≥ K, ∀ x, (b,x) ∉ U k) (k : ℕ) :
+    frozenNodes v U h (z,b 0) k b =
+      (EulerGeneral.nodes f h z (min k K),b (((min k K:ℕ):ℝ≥0)*h)) := by
+  induction k with
+  | zero => simp [frozenNodes,EulerGeneral.nodes]
+  | succ k ih =>
+    by_cases hk : k < K
+    · have hk1 : k+1 ≤ K := hk
+      rw [frozenNodes,ih]
+      simp only [min_eq_left hk.le, min_eq_left hk1, ha k hk, ite_eq_left, EulerGeneral.nodes, hf]
+      congr 1
+      simp only [Nat.cast_add,Nat.cast_one]
+      ring
+    · have hk' : K ≤ k := Nat.le_of_not_gt hk
+      have hk1 : K ≤ k+1 := hk'.trans (Nat.le_succ _)
+      rw [frozenNodes,ih]
+      simp only [min_eq_right hk',min_eq_right hk1,hi k hk',ite_false]
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.EulerGeneral
+
+lemma ceil_mesh_times_converge {h : ℕ → ℝ} (hh : ∀ n, 0 < h n)
+    (hh0 : Tendsto h atTop (𝓝 0)) {τ : ℝ} (hτ : 0 ≤ τ) :
+    Tendsto (fun n => (⌈τ/h n⌉₊:ℝ)*h n) atTop (𝓝 τ) := by
+  have hb (n : ℕ) : 0 ≤ (⌈τ/h n⌉₊:ℝ)*h n-τ ∧ (⌈τ/h n⌉₊:ℝ)*h n-τ ≤ h n := by
+    have hl := Nat.le_ceil (τ/h n)
+    have hu := Nat.ceil_lt_add_one (div_nonneg hτ (hh n).le)
+    have hl' := (div_le_iff₀ (hh n)).mp hl
+    have hu' := (mul_lt_mul_of_pos_right hu (hh n))
+    rw [add_mul,div_mul_cancel₀ _ (hh n).ne',one_mul] at hu'
+    constructor <;> linarith
+  have hz : Tendsto (fun n => (⌈τ/h n⌉₊:ℝ)*h n-τ) atTop (𝓝 0) :=
+    squeeze_zero (fun n => (hb n).1) (fun n => (hb n).2) hh0
+  simpa using hz.add_const τ
+
+lemma effective_stop_times_converge {h : ℕ → ℝ} (hh : ∀ n, 0 < h n)
+    (hh0 : Tendsto h atTop (𝓝 0)) {τ t : ℝ} (hτ : 0 ≤ τ) (ht : 0 ≤ t) :
+    Tendsto (fun n => ((min ⌊t/h n⌋₊ ⌈τ/h n⌉₊:ℕ):ℝ)*h n) atTop (𝓝 (min t τ)) := by
+  have h1 := mesh_times_converge hh hh0 ht
+  have h2 := ceil_mesh_times_converge hh hh0 hτ
+  simpa only [Nat.cast_min, min_mul_of_nonneg _ _ (hh _).le] using h1.min h2
+
+end SLEExactGauge.EulerGeneral
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ 
+
+omit [CompleteSpace E] in
+lemma ordinary_nodes_eventually_inside (f : ℝ → E → E) (g : ℝ → E) (z : E)
+    (b : ℝ≥0 → ℝ) (D : Set (E × ℝ)) {τ T L M ε : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
+    (hτ : 0 ≤ τ) (hτT : τ < T) (hε : 0 < ε)
+    (hD : ∀ u ∈ Icc 0 τ, ∀ x, ‖x-g u‖ < ε → (x,b u.toNNReal) ∈ D)
+    {h : ℕ → ℝ≥0} (hh : ∀ n, 0 < h n) (hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0)) :
+    ∀ᶠ n in atTop, ∀ k : ℕ, (k:ℝ)*(h n:ℝ) < τ →
+      (EulerGeneral.nodes f (h n) z k,b ((k:ℝ≥0)*h n)) ∈ D := by
+  have hu := EulerGeneral.converge_uniformOn hL hM hg0 hlip hder hcf hbound hτ hτT
+    (fun n => show 0 < (h n:ℝ) from hh n) hh0
+  have he := (Metric.tendstoUniformlyOn_iff.mp hu) ε hε
+  filter_upwards [he] with n hn
+  intro k hk
+  have htime : (k:ℝ)*(h n:ℝ) ∈ Icc 0 τ := ⟨by positivity,hk.le⟩
+  have hn' := hn _ htime
+  rw [EulerGeneral.interpolate_at_node _ (show 0 < (h n:ℝ) from hh n) _ _] at hn'
+  have hb : ((k:ℝ)*(h n:ℝ)).toNNReal = (k:ℝ≥0)*h n := by
+    apply NNReal.coe_injective
+    simp [Real.toNNReal_of_nonneg (show 0 ≤ (k:ℝ)*(h n:ℝ) by positivity)]
+  rw [← hb]
+  apply hD _ htime _
+  rw [dist_comm] at hn'
+  simpa only [dist_eq_norm] using hn'
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ 
+
+omit [CompleteSpace E] in
+theorem frozen_nodes_path_limit (v : E → ℝ → E) (g : ℝ → E) (z : E)
+    (b : ℝ≥0 → ℝ) (cb : Continuous b) (D : Set (E × ℝ))
+    (U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) {τ T t L M ε : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖v x (b u.toNNReal)-v y (b u.toNNReal)‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (v (g u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => v (g u) (b u.toNNReal)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖v (g u) (b u.toNNReal)‖ ≤ M)
+    (hτ : 0 ≤ τ) (hτT : τ < T) (ht : 0 ≤ t) (hε : 0 < ε)
+    (hD : ∀ u ∈ Icc 0 τ, ∀ x, ‖x-g u‖ < ε → (x,b u.toNNReal) ∈ D)
+    {h : ℕ → ℝ≥0} (hh : ∀ n, 0 < h n) (hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0))
+    (hU : ∀ n k x, (b,x) ∈ U n k ↔ (k:ℝ)*(h n:ℝ) < τ ∧ x ∈ D) :
+    Tendsto (fun n => frozenNodes v (U n) (h n) (z,b 0) ⌊t/(h n:ℝ)⌋₊ b)
+      atTop (𝓝 (g (min t τ),b (min t τ).toNNReal)) := by
+  let f : ℝ → E → E := fun u x => v x (b u.toNNReal)
+  let K : ℕ → ℕ := fun n => ⌈τ/(h n:ℝ)⌉₊
+  let k : ℕ → ℕ := fun n => min ⌊t/(h n:ℝ)⌋₊ (K n)
+  have hn := ordinary_nodes_eventually_inside f g z b D hL hM hg0 hlip hder hcf hbound
+    hτ hτT hε hD hh hh0
+  have he : ∀ᶠ n in atTop, frozenNodes v (U n) (h n) (z,b 0) ⌊t/(h n:ℝ)⌋₊ b =
+      (EulerGeneral.nodes f (h n) z (k n),b ((k n:ℝ≥0)*h n)) := by
+    filter_upwards [hn] with n hn
+    refine frozenNodes_eq_min v f (U n) (h n) z b ?_ (K n) ?_ ?_ _
+    · intro j x
+      dsimp [f]
+      congr 2
+      apply NNReal.coe_injective
+      simp [Real.toNNReal_of_nonneg (show 0 ≤ (j:ℝ)*(h n:ℝ) by positivity)]
+    · intro j hj
+      have htj : (j:ℝ)*(h n:ℝ) < τ :=
+        (lt_div_iff₀ (show 0 < (h n:ℝ) from hh n)).mp (Nat.lt_ceil.mp hj)
+      exact (hU n j _).mpr ⟨htj,hn j htj⟩
+    · intro j hj x hx
+      have htj : τ ≤ (j:ℝ)*(h n:ℝ) :=
+        (div_le_iff₀ (show 0 < (h n:ℝ) from hh n)).mp (Nat.ceil_le.mp hj)
+      exact (not_lt_of_ge htj) ((hU n j x).mp hx).1
+  have hk0 : Tendsto (fun n => (k n:ℝ)*(h n:ℝ)) atTop (𝓝 (min t τ)) :=
+    EulerGeneral.effective_stop_times_converge (fun n => hh n) hh0 hτ ht
+  let S : ℝ := (τ+T)/2
+  have hsS : min t τ < S := lt_of_le_of_lt (min_le_right _ _) (by dsimp [S]; linarith)
+  have hkS : ∀ᶠ n in atTop, (k n:ℝ)*(h n:ℝ) ≤ S :=
+    (hk0.eventually (gt_mem_nhds hsS)).mono fun _ hn => hn.le
+  have hg := EulerGeneral.nodes_converge_moving hL hM hg0 hlip hder hcf hbound
+    (show 0 ≤ S by dsimp [S]; linarith) (show S < T by dsimp [S]; linarith)
+    (show min t τ ∈ Icc 0 S from ⟨le_min ht hτ,hsS.le⟩)
+    (fun n => show 0 < (h n:ℝ) from hh n) hh0 k hkS hk0
+  have hb0 : Tendsto (fun n => (k n:ℝ≥0)*h n) atTop (𝓝 (min t τ).toNNReal) := by
+    convert! continuous_real_toNNReal.continuousAt.tendsto.comp hk0 using 1
+    funext n
+    apply NNReal.coe_injective
+    simp [Real.toNNReal_of_nonneg (show 0 ≤ (k n:ℝ)*(h n:ℝ) by positivity)]
+  exact (hg.prodMk_nhds (cb.continuousAt.tendsto.comp hb0)).congr' (Filter.EventuallyEq.symm he)
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+omit [CompleteSpace E] in
+theorem expected_stopped_smooth {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hg0 : ∀ b, g b 0 = z)
+    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
+    (τ : (ℝ≥0 → ℝ) → ℝ) (mτ : Measurable τ)
+    (hτpast : ∀ (s : ℝ≥0) b c, (∀ u ≤ s, b u = c u) → ((s:ℝ) < τ b ↔ (s:ℝ) < τ c))
+    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
+    (D : Set (E × ℝ)) (mD : MeasurableSet D)
+    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ D)
+    (hgen : ∀ x ∈ D, A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
+    (ht : 0 ≤ t) :
+    (∫ b : Driver, A.F (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ A.F z 0 := by
+  let h : ℕ → ℝ≥0 := fun n => (1/2)^n
+  have hh : ∀ n, 0 < h n := fun n => pow_pos (by norm_num) n
+  have hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0) := by
+    simp only [h,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat]
+    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+  let U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)) :=
+    fun n k => {q | (k:ℝ)*(h n:ℝ) < τ q.1 ∧ q.2 ∈ D}
+  have mU : ∀ n k, MeasurableSet (U n k) := by
+    intro n k
+    exact (measurableSet_lt measurable_const (mτ.comp measurable_fst)).inter
+      (mD.preimage measurable_snd)
+  have hu : ∀ n (k : ℕ) b c, (∀ u ≤ (k:ℝ≥0)*h n, b u = c u) → ∀ x,
+      ((b,x) ∈ U n k ↔ (c,x) ∈ U n k) := by
+    intro n k b c hbc x
+    change ((k:ℝ)*(h n:ℝ) < τ b ∧ x ∈ D ↔ (k:ℝ)*(h n:ℝ) < τ c ∧ x ∈ D)
+    rw [show (k:ℝ)*(h n:ℝ) = (((k:ℝ≥0)*h n):ℝ) by simp]
+    exact and_congr_left fun _ => hτpast _ b c hbc
+  apply expected_frozen_limit hW A v mv U mU h (fun n => ⌊t/(h n:ℝ)⌋₊) (z,0) hu M hv
+    (fun _ _ _ x hx => hgen x hx.2)
+    (fun b => (g b (min t (τ b)), b (min t (τ b)).toNNReal)) t hh0
+    (EulerGeneral.mesh_times_converge (fun n => hh n) hh0 ht)
+  filter_upwards [] with b
+  obtain ⟨ε,hε,hDb⟩ := hD b
+  have hp := frozen_nodes_path_limit v (g b) z b b.continuous D U hL hM (hg0 b)
+    (fun _ _ => hlip _) (hder b) (hcf b) (fun u _ => hv _ _) (hτ b).1 (hτ b).2 ht
+    hε hDb hh hh0 (fun _ _ _ => Iff.rfl)
+  simpa only [map_zero] using hp
+
+end SLEExactGauge.BrownianCalculus
+
+open scoped ContDiff Manifold
+namespace SLEExactGauge.BrownianCalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+def driverJets (F : E × ℝ → ℝ) : ℕ → E × ℝ → ℝ
+  | 0 => F
+  | n+1 => fun q => fderiv ℝ (driverJets F n) q (0,1)
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma driverJets_smooth {F : E × ℝ → ℝ} (hF : ContDiff ℝ ∞ F) (n : ℕ) :
+    ContDiff ℝ ∞ (driverJets F n) := by
+  induction n with
+  | zero => exact hF
+  | succ n ih =>
+    exact (contDiff_infty_iff_fderiv.mp ih).2.clm_apply contDiff_const
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma driverJets_compact {F : E × ℝ → ℝ} (hF : HasCompactSupport F) (n : ℕ) :
+    HasCompactSupport (driverJets F n) := by
+  induction n with
+  | zero => exact hF
+  | succ n ih => exact ih.fderiv_apply ℝ (0,1)
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma driverJets_restriction {F : E × ℝ → ℝ} (hF : ContDiff ℝ ∞ F) (n : ℕ)
+    (x : E) (b : ℝ) : driverJets F n (x,b) = iteratedDeriv n (fun c => F (x,c)) b := by
+  induction n generalizing b with
+  | zero => rfl
+  | succ n ih =>
+    have he : iteratedDeriv n (fun c => F (x,c)) = fun c => driverJets F n (x,c) := by
+      funext c
+      exact (ih c).symm
+    rw [iteratedDeriv_succ,he]
+    have hd := ((driverJets_smooth hF n).differentiable (by simp) (x,b)).hasFDerivAt.comp_hasDerivAt b
+      ((hasDerivAt_const b x).prodMk (hasDerivAt_id b))
+    exact hd.deriv.symm
+
+ 
+
+omit [SecondCountableTopology E] in
+theorem exists_smoothTest_of_compact (F : E × ℝ → ℝ)
+    (hF : ContDiff ℝ ∞ F) (hK : HasCompactSupport F) :
+    ∃ A : SmoothTest E, A.F = fun x b => F (x,b) := by
+  let D : E × ℝ → E →L[ℝ] ℝ := fun q =>
+    (fderiv ℝ F q).comp (ContinuousLinearMap.inl ℝ E ℝ)
+  have hD : ContDiff ℝ ∞ D :=
+    (contDiff_infty_iff_fderiv.mp hF).2.clm_comp contDiff_const
+  have hDK : HasCompactSupport D := by
+    exact (hK.fderiv ℝ).comp_left (g := fun A : E × ℝ →L[ℝ] ℝ => A.comp (ContinuousLinearMap.inl ℝ E ℝ)) (show (0 : E × ℝ →L[ℝ] ℝ).comp (ContinuousLinearMap.inl ℝ E ℝ) = 0 from rfl)
+  obtain ⟨C₀,hC₀⟩ := hK.exists_bound_of_continuous hF.continuous
+  obtain ⟨C₁,hC₁⟩ := (driverJets_compact hK 1).exists_bound_of_continuous
+    (driverJets_smooth hF 1).continuous
+  obtain ⟨C₂,hC₂⟩ := (driverJets_compact hK 2).exists_bound_of_continuous
+    (driverJets_smooth hF 2).continuous
+  obtain ⟨C₃,hC₃⟩ := (driverJets_compact hK 3).exists_bound_of_continuous
+    (driverJets_smooth hF 3).continuous
+  obtain ⟨Cx,hCx⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hDK hD (by simp)
+  obtain ⟨Cc,hCc⟩ := ContDiff.lipschitzWith_of_hasCompactSupport (driverJets_compact hK 2)
+    (driverJets_smooth hF 2) (by simp)
+  have hj1 (x : E) (b : ℝ) : driverJets F 1 (x,b) = deriv (fun c => F (x,c)) b := by
+    simpa using driverJets_restriction hF 1 x b
+  have hj2 (x : E) (b : ℝ) : driverJets F 2 (x,b) = deriv (deriv (fun c => F (x,c))) b := by
+    simpa only [iteratedDeriv_succ,iteratedDeriv_zero] using driverJets_restriction hF 2 x b
+  refine ⟨{
+    F := fun x b => F (x,b)
+    D := fun x b => D (x,b)
+    C₀ := C₀
+    C₁ := C₁
+    C₂ := C₂
+    C₃ := max C₃ 0
+    Cx := Cx
+    Ccross := Cc
+    hC₃ := le_max_right _ _
+    hCx := Cx.coe_nonneg
+    hCcross := Cc.coe_nonneg
+    cF := hF.continuous
+    mF := hF.continuous.measurable
+    mB1 := ?_
+    mB2 := ?_
+    hD := ?_
+    hsm := ?_
+    hF := ?_
+    hB1 := ?_
+    hB2 := ?_
+    hB3 := ?_
+    hLipD := ?_
+    hLipB2 := ?_ }, rfl⟩
+  · convert! (driverJets_smooth hF 1).continuous.measurable using 1
+    funext q
+    exact (hj1 q.1 q.2).symm
+  · convert! (driverJets_smooth hF 2).continuous.measurable using 1
+    funext q
+    exact (hj2 q.1 q.2).symm
+  · intro x b
+    have hi : HasFDerivAt (fun y : E => (y,b)) (ContinuousLinearMap.inl ℝ E ℝ) x := by
+      convert! (hasFDerivAt_id x).prodMk (hasFDerivAt_const b x) using 1
+    exact ((hF.differentiable (by simp) (x,b)).hasFDerivAt).comp x hi
+  · intro x
+    exact (hF.comp (contDiff_const.prodMk contDiff_id)).of_le (WithTop.coe_le_coe.mpr le_top)
+  · intro x b
+    simpa only [Real.norm_eq_abs] using hC₀ (x,b)
+  · intro x b
+    simpa only [hj1,Real.norm_eq_abs] using hC₁ (x,b)
+  · intro x b
+    simpa only [hj2,Real.norm_eq_abs] using hC₂ (x,b)
+  · intro x b
+    rw [← driverJets_restriction hF 3 x b,← Real.norm_eq_abs]
+    exact (hC₃ (x,b)).trans (le_max_left _ _)
+  · intro x y b
+    simpa [Prod.norm_def] using hCx.norm_sub_le (x,b) (y,b)
+  · intro x y b
+    simpa [hj2,Prod.norm_def,Real.norm_eq_abs] using hCc.norm_sub_le (x,b) (y,b)
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.Localization
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+ 
+
+theorem compact_extension {K O : Set E} (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O)
+    (f : E → F) (hf : ContDiffOn ℝ ∞ f O) :
+    ∃ g : E → F, ContDiff ℝ ∞ g ∧ HasCompactSupport g ∧ ∀ x ∈ K, g =ᶠ[𝓝 x] f := by
+  obtain ⟨L,hL,hKL,hLO⟩ := exists_compact_between hK hO hKO
+  obtain ⟨φ,hφ1,hφ0,hφbound⟩ :=
+    exists_contMDiffMap_one_nhds_of_subset_interior (𝓘(ℝ,E)) hK.isClosed hKL (n := ⊤)
+  have hφsmooth : ContDiff ℝ ∞ (φ : E → ℝ) := contMDiff_iff_contDiff.mp φ.contMDiff
+  have hsupp : tsupport (φ : E → ℝ) ⊆ L := by
+    apply closure_minimal _ hL.isClosed
+    intro x hx
+    by_contra hn
+    exact hx (hφ0 x hn)
+  have hφK : HasCompactSupport (φ : E → ℝ) := hL.of_isClosed_subset isClosed_closure hsupp
+  refine ⟨fun x => φ x • f x,?_,hφK.smul_right,?_⟩
+  · apply contDiff_iff_contDiffAt.mpr
+    intro x
+    by_cases hx : x ∈ tsupport (φ : E → ℝ)
+    · exact hφsmooth.contDiffAt.smul ((hf x (hLO (hsupp hx))).contDiffAt (hO.mem_nhds (hLO (hsupp hx))))
+    · have he : (φ : E → ℝ) =ᶠ[𝓝 x] 0 := by
+        exact (notMem_tsupport_iff_eventuallyEq.mp hx)
+      apply (contDiffAt_const (c := (0 : F))).congr_of_eventuallyEq
+      filter_upwards [he] with y hy
+      simp [hy]
+  · intro x hx
+    have hnear : ∀ᶠ y in 𝓝 x, φ y = 1 := hφ1.filter_mono (nhds_le_nhdsSet hx)
+    filter_upwards [hnear] with y hy
+    simp [hy]
+
+end SLEExactGauge.Localization
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+end SLEExactGauge.BrownianCalculus
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ExactGaugeEndpoint.lean b/lean/OAI/Probability/SLEGauge/ExactGaugeEndpoint.lean
index dc4d452338e73d234e26681c1b9f023948d966a0..61ccf86f99d44e264cca6e251c22fabf713cbb09 100644
--- a/lean/OAI/Probability/SLEGauge/ExactGaugeEndpoint.lean
+++ b/lean/OAI/Probability/SLEGauge/ExactGaugeEndpoint.lean
@@ -1,83 +1,83 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.CleanupCost
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart CenteredMoments BoxGrid
-
-lemma allFrontiers_im_nonneg (a : ℝ) (b : GDriver) {z : ℂ}
-    (hz : z∈allFrontiers a b) : 0≤z.im := by
-  obtain ⟨T,hT⟩ := mem_iUnion.mp hz
-  exact closure_minimal (fun w hw=>(alive_initial_im hw).le)
-    (isClosed_le continuous_const Complex.continuous_im) (frontier_subset_closure hT)
-
-lemma compact_subset_integer_box (K : Set ℂ) (hK : IsCompact K)
-    (hH : ∀z∈K,0≤z.im) : ∃m : ℕ,K⊆box m := by
-  obtain ⟨R,hR⟩ := (Metric.isBounded_iff_subset_closedBall (0:ℂ)).mp hK.isBounded
-  obtain ⟨m,hm⟩ := exists_nat_gt R
-  refine ⟨m,fun z hz=>?_⟩
-  have hn : ‖z‖≤(m:ℝ) := (by simpa only [mem_closedBall,dist_zero_right] using hR hz : ‖z‖≤R).trans hm.le
-  have hx := abs_le.mp ((Complex.abs_re_le_norm z).trans hn)
-  have hy := (le_abs_self z.im).trans ((Complex.abs_im_le_norm z).trans hn)
-  exact ⟨hx,⟨hH z hz,hy⟩⟩
-
-lemma capacity_dimension_eq (κ : ℝ) (_hκ : 0<κ) :
-    dimension κ = 2-alpha (Target.a κ) := by
-  unfold dimension alpha Target.a
-  field_simp
-  ring
-
-namespace Target
-
- 
-
-theorem exactHausdorffGauge :
-    ∀ κ : ℝ, 0 < κ → κ < 8 →
-      ∃ h : ℝ → ℝ, IsHausdorffGauge h ∧
-        ∀ W : WienerLaw, IsWiener W →
-          ∀ᵐ b ∂W, ∀ s t : ℝ, 0 < s → s < t →
-            0 < hausdorff h ((trace κ b) '' Icc s t) ∧
-            hausdorff h ((trace κ b) '' Icc s t) < ∞ := by
-  intro κ hκ hκ8
-  refine ⟨parameterMomentGauge κ hκ hκ8,parameterMomentGauge_admissible κ hκ hκ8,?_⟩
-  intro W hW
-  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-  let : IsProbabilityMeasure (W.map driverHomeomorph) := inferInstance
-  obtain ⟨M,hM⟩ := parameterMomentGauge_model κ hκ hκ8 W hW
-  rw [hM,capacity_dimension_eq κ hκ]
-  have hB := brownian_map_driver W hW
-  have hl := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp (M.hausdorff_positive hκ hB)
-  have hu := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp (M.frontier_boxes_finite hB)
-  filter_upwards [hl,hu,trace_regular_actual_domains hκ hκ8 hW] with b hl hu hreg
-  intro s t hs hst
-  refine ⟨?_,?_⟩
-  · simpa only [show driverMeasurableEquiv b=driverHomeomorph b from rfl,
-      Homeomorph.symm_apply_apply] using hl s t hs hst
-  · let K := trace κ b '' Icc s t
-    have hK : IsCompact K := isCompact_Icc.image_of_continuousOn
-      (hreg.1.mono (fun u hu=>hs.le.trans hu.1))
-    have hKF : K⊆allFrontiers (a κ) (driverHomeomorph b) := by
-      rintro z ⟨u,hu,rfl⟩
-      exact mem_iUnion.mpr ⟨(κ*u).toNNReal,hreg.2.1 u (hs.le.trans hu.1)⟩
-    obtain ⟨m,hm⟩ := compact_subset_integer_box K hK
-      (fun z hz=>allFrontiers_im_nonneg _ _ (hKF hz))
-    exact (hausdorff (M.gauge (2-alpha (a κ)))).mono
-      (subset_inter hKF hm) |>.trans_lt (hu m)
-
-end Target
-end SLEExactGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.CleanupCost
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart CenteredMoments BoxGrid
+
+lemma allFrontiers_im_nonneg (a : ℝ) (b : GDriver) {z : ℂ}
+    (hz : z∈allFrontiers a b) : 0≤z.im := by
+  obtain ⟨T,hT⟩ := mem_iUnion.mp hz
+  exact closure_minimal (fun w hw=>(alive_initial_im hw).le)
+    (isClosed_le continuous_const Complex.continuous_im) (frontier_subset_closure hT)
+
+lemma compact_subset_integer_box (K : Set ℂ) (hK : IsCompact K)
+    (hH : ∀z∈K,0≤z.im) : ∃m : ℕ,K⊆box m := by
+  obtain ⟨R,hR⟩ := (Metric.isBounded_iff_subset_closedBall (0:ℂ)).mp hK.isBounded
+  obtain ⟨m,hm⟩ := exists_nat_gt R
+  refine ⟨m,fun z hz=>?_⟩
+  have hn : ‖z‖≤(m:ℝ) := (by simpa only [mem_closedBall,dist_zero_right] using hR hz : ‖z‖≤R).trans hm.le
+  have hx := abs_le.mp ((Complex.abs_re_le_norm z).trans hn)
+  have hy := (le_abs_self z.im).trans ((Complex.abs_im_le_norm z).trans hn)
+  exact ⟨hx,⟨hH z hz,hy⟩⟩
+
+lemma capacity_dimension_eq (κ : ℝ) (_hκ : 0<κ) :
+    dimension κ = 2-alpha (Target.a κ) := by
+  unfold dimension alpha Target.a
+  field_simp
+  ring
+
+namespace Target
+
+ 
+
+theorem exactHausdorffGauge :
+    ∀ κ : ℝ, 0 < κ → κ < 8 →
+      ∃ h : ℝ → ℝ, IsHausdorffGauge h ∧
+        ∀ W : WienerLaw, IsWiener W →
+          ∀ᵐ b ∂W, ∀ s t : ℝ, 0 < s → s < t →
+            0 < hausdorff h ((trace κ b) '' Icc s t) ∧
+            hausdorff h ((trace κ b) '' Icc s t) < ∞ := by
+  intro κ hκ hκ8
+  refine ⟨parameterMomentGauge κ hκ hκ8,parameterMomentGauge_admissible κ hκ hκ8,?_⟩
+  intro W hW
+  let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+  let : IsProbabilityMeasure (W.map driverHomeomorph) := inferInstance
+  obtain ⟨M,hM⟩ := parameterMomentGauge_model κ hκ hκ8 W hW
+  rw [hM,capacity_dimension_eq κ hκ]
+  have hB := brownian_map_driver W hW
+  have hl := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp (M.hausdorff_positive hκ hB)
+  have hu := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp (M.frontier_boxes_finite hB)
+  filter_upwards [hl,hu,trace_regular_actual_domains hκ hκ8 hW] with b hl hu hreg
+  intro s t hs hst
+  refine ⟨?_,?_⟩
+  · simpa only [show driverMeasurableEquiv b=driverHomeomorph b from rfl,
+      Homeomorph.symm_apply_apply] using hl s t hs hst
+  · let K := trace κ b '' Icc s t
+    have hK : IsCompact K := isCompact_Icc.image_of_continuousOn
+      (hreg.1.mono (fun u hu=>hs.le.trans hu.1))
+    have hKF : K⊆allFrontiers (a κ) (driverHomeomorph b) := by
+      rintro z ⟨u,hu,rfl⟩
+      exact mem_iUnion.mpr ⟨(κ*u).toNNReal,hreg.2.1 u (hs.le.trans hu.1)⟩
+    obtain ⟨m,hm⟩ := compact_subset_integer_box K hK
+      (fun z hz=>allFrontiers_im_nonneg _ _ (hKF hz))
+    exact (hausdorff (M.gauge (2-alpha (a κ)))).mono
+      (subset_inter hKF hm) |>.trans_lt (hu m)
+
+end Target
+end SLEExactGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/FiniteAngleEigen.lean b/lean/OAI/Probability/SLEGauge/FiniteAngleEigen.lean
index f2d162b567268c2537ff68aad86bc7939f4614b2..3916bd872222c1e69702eb66e46f53b687f4aad9 100644
--- a/lean/OAI/Probability/SLEGauge/FiniteAngleEigen.lean
+++ b/lean/OAI/Probability/SLEGauge/FiniteAngleEigen.lean
@@ -1,535 +1,535 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.GlobalAngleBudget
-import OAI.Probability.SLEGauge.JacobiKernel
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
-
-def angleEigenCutoff (g : ℝ → ℝ) (a cap lam ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
-  greenCutoff a ε N v * (clockExp a cap lam v * angleLift g v)
-
-lemma angleEigenCutoff_smooth (g : ℝ → ℝ)
-    (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ (↑(⊤ : ℕ∞)) g θ)
-    (a cap lam ε : ℝ) (hε : 0<ε) (N : ℝ≥0) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (angleEigenCutoff g a cap lam ε N) := by
-  rw [contDiff_iff_contDiffAt]
-  intro v
-  by_cases hv : 0<Y v
-  · exact (greenCutoff_smooth a ε hε N).contDiffAt.mul
-      ((clockExp_smooth_all a cap lam v hv).mul
-        ((hg _ (physicalTheta_mem v)).comp v (physicalTheta_smooth_all v hv)))
-  · apply (contDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq
-    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
-    simp only [angleEigenCutoff,greenCutoff,heightTurn_zero hε (le_of_lt hw),
-      mul_zero,zero_mul]
-
-lemma angleEigenCutoff_compact (g : ℝ → ℝ) (a cap lam ε : ℝ) (N : ℝ≥0) :
-    HasCompactSupport (angleEigenCutoff g a cap lam ε N) :=
-  (greenCutoff_compact a ε N).mul_right
-
-lemma angleEigenCutoff_eq_near (g : ℝ → ℝ) (a cap lam ε : ℝ) (hε : 0<ε)
-    (N : ℝ≥0) (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
-    angleEigenCutoff g a cap lam ε N =ᶠ[𝓝 v]
-      fun w => physicalGreen a w * (clockExp a cap lam w * angleLift g w) :=
-  (greenCutoff_eq_near a ε hε N v hv hy).mul (EventuallyEq.refl _ _)
-
-lemma angleEigenCutoff_generator (g : ℝ → ℝ)
-    (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ 2 g θ)
-    (a cap lam ε : ℝ) (ha : a≠0)
-    (heigen : ∀ θ ∈ Ioo 0 Real.pi, angleGenerator (2*a) g θ = -lam*g θ)
-    (hε : 0<ε) (N : ℝ≥0) (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
-    generator (angleEigenCutoff g a cap lam ε N) (cutoffDrift a ε) noise v = 0 := by
-  have he := angleEigenCutoff_eq_near g a cap lam ε hε N v hv hy
-  have hd : (fun w => fderiv ℝ (angleEigenCutoff g a cap lam ε N) w noise) =ᶠ[𝓝 v]
-      (fun w => fderiv ℝ (fun u => physicalGreen a u *
-        (clockExp a cap lam u * angleLift g u)) w noise) :=
-    he.fderiv.mono (fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw)
-  change (1/2:ℝ) * fderiv ℝ (fun w => fderiv ℝ
-    (angleEigenCutoff g a cap lam ε N) w noise) v noise +
-    fderiv ℝ (angleEigenCutoff g a cap lam ε N) v (cutoffDrift a ε v) = 0
-  rw [hd.fderiv_eq,he.fderiv_eq,cutoffDrift_eq_physical a ε v hy]
-  change L (fun w => physicalGreen a w * (clockExp a cap lam w * angleLift g w))
-    (physicalDrift a v) noise v = 0
-  rw [green_exponential_angle_generator g a cap lam ha v (hε.trans_le hy)
-    (hg _ (physicalTheta_mem v)),heigen _ (physicalTheta_mem v)]
-  ring
-
- 
-theorem local_angle_eigen (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (g : ℝ → ℝ) (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ (↑(⊤ : ℕ∞)) g θ)
-    (a cap lam ε : ℝ) (ha : a≠0)
-    (heigen : ∀ θ ∈ Ioo 0 Real.pi, angleGenerator (2*a) g θ = -lam*g θ)
-    (hε : 0<ε) (z : ℂ) (N B : ℝ≥0) (T : ExactSLEGenerator.Driver → ℝ≥0)
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hTB : ∀ b, T b≤B)
-    (hsafe : ∀ b t, t≤T b → ε≤Y (localState a ε hε z t b) ∧
-      ‖localState a ε hε z t b‖≤N) :
-    Integrable (fun b => physicalGreen a (localState a ε hε z (T b) b) *
-      (clockExp a cap lam (localState a ε hε z (T b) b) *
-        angleLift g (localState a ε hε z (T b) b))) W ∧
-    (∫ b, physicalGreen a (localState a ε hε z (T b) b) *
-      (clockExp a cap lam (localState a ε hε z (T b) b) *
-        angleLift g (localState a ε hε z (T b) b)) ∂W) =
-      physicalGreen a (z,0) * (clockExp a cap lam (z,0) * angleLift g (z,0)) := by
-  let V := localState a ε hε z
-  let f := angleEigenCutoff g a cap lam ε N
-  let D := generator f (cutoffDrift a ε) noise
-  have hf := angleEigenCutoff_smooth g hg a cap lam ε hε N
-  have hfc := angleEigenCutoff_compact g a cap lam ε N
-  have he (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t≤T b) :
-      f (V t b) = physicalGreen a (V t b) * (clockExp a cap lam (V t b)*angleLift g (V t b)) :=
-    (angleEigenCutoff_eq_near g a cap lam ε hε N _ (hsafe b t ht).2
-      (hsafe b t ht).1).eq_of_nhds
-  have hgen (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t≤T b) : D (V t b)=0 :=
-    angleEigenCutoff_generator g (fun θ hθ => (hg θ hθ).of_le (by norm_cast))
-      a cap lam ε ha heigen hε N _ (hsafe b t ht).2 (hsafe b t ht).1
-  have hDy := compact_test_stopped_dynkin W hW V
-    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
-    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
-    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
-    (by simp [noise,Prod.norm_def]) (localState_eq a ε hε z) f hf hfc T B hT hTB
-  have hzero (b : ExactSLEGenerator.Driver) : (∫ u in (0:ℝ)..(T b : ℝ), generator f (cutoffDrift a ε) noise (V u.toNNReal b)) = 0 := by
-    rw [intervalIntegral.integral_of_le (T b).coe_nonneg]
-    apply setIntegral_eq_zero_of_forall_eq_zero
-    intro u hu
-    exact hgen b u.toNNReal (by simpa using Real.toNNReal_le_toNNReal hu.2)
-  have hinit (b : ExactSLEGenerator.Driver) : f (V 0 b) =
-      physicalGreen a (z,0)*(clockExp a cap lam (z,0)*angleLift g (z,0)) := by
-    rw [he b 0 bot_le]
-    simp only [V,localState_zero]
-  refine ⟨by simpa only [he _ _ le_rfl] using hDy.1, ?_⟩
-  have hd := hDy.2.2
-  simp_rw [hinit,he _ _ le_rfl,hzero] at hd
-  simpa only [integral_const,probReal_univ,one_smul,integral_zero,add_zero] using hd
-
- 
-
-theorem finite_level_angle_eigen (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (g : ℝ → ℝ) (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ (↑(⊤ : ℕ∞)) g θ)
-    (C : ℝ) (hC : 0≤C) (hbound : ∀ θ ∈ Ioo 0 Real.pi, ‖g θ‖≤C)
-    (a lam : ℝ) (ha : 1/4<a) (hlam : 0≤lam)
-    (heigen : ∀ θ ∈ Ioo 0 Real.pi, angleGenerator (2*a) g θ = -lam*g θ)
-    (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0≤ s) :
-    Integrable (fun b : ExactSLEGenerator.Driver => radialTestWeight a (angleLift g) b.1 z
-      (levelRadius a (2*z.im) s)) W ∧
-    (∫ b, radialTestWeight a (angleLift g) b.1 z
-      (levelRadius a (2*z.im) s) ∂W) =
-      physicalGreen a (z,0) * Real.exp (-lam*s) * g (physicalTheta (z,0)) := by
-  have ha0 : 0<a := by linarith
-  let cap := 2*z.im
-  have hcap : 0<cap := by dsimp [cap]; positivity
-  let ρ := levelRadius a cap s
-  have hρ : 0<ρ := levelRadius_pos a cap s hcap
-  have hr : ρ≤z.im := by
-    simpa only [cap,mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0)] using
-      (levelRadius_le ha0 hcap hs)
-  let L (n : ℕ) := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T (n : ℕ) := radialStop a ha0 z hz ρ hρ hr n
-  let V (n : ℕ) (b : ExactSLEGenerator.Driver) := localState a (L n).ε (L n).hε z (T n b) b
-  let f (n : ℕ) (b : ExactSLEGenerator.Driver) := physicalGreen a (V n b)*
-    (clockExp a cap lam (V n b)*angleLift g (V n b))
-  let F (b : ExactSLEGenerator.Driver) := Real.exp (lam*s)*radialTestWeight a (angleLift g) b.1 z ρ
-  have hsafe (n : ℕ) (b : ExactSLEGenerator.Driver) (u : ℝ≥0) (hu : u≤T n b) :
-      (L n).ε≤Y (localState a (L n).ε (L n).hε z u b) ∧
-        ‖localState a (L n).ε (L n).hε z u b‖≤(L n).N := by
-    have h := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b u hu
-    exact ⟨h.1,h.2.1⟩
-  have hact (n : ℕ) (b : ExactSLEGenerator.Driver) : alive a b.1 (T n b) z ∧
-      actualState a b.1 (T n b) z = V n b :=
-    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b _ le_rfl
-  have hc (n : ℕ) (b : ExactSLEGenerator.Driver) : physicalClock a cap (V n b)≤ s := by
-    rw [←(hact n b).2]
-    have h := actual_physicalClock_le a ha0 z hz cap ρ hcap hρ hr b (T n b)
-      (hact n b).1 (radialStop_le_radialTime a ha0 z hz ρ hρ hr n b)
-    simpa only [ρ,levelRadius_clock ha0.ne' hcap] using h
-  have hfi (n : ℕ) : Integrable (f n) W ∧
-      (∫ b, f n b ∂W)=physicalGreen a (z,0)*angleLift g (z,0) := by
-    have h := local_angle_eigen W hW g hg a cap lam (L n).ε ha0.ne' heigen (L n).hε
-      z (L n).N (L n).B (T n) (radialStop_stopping a ha0 z hz ρ hρ hr n)
-      (regionStop_le a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B)
-      (hsafe n)
-    refine ⟨h.1, ?_⟩
-    have hi : clockExp a cap lam (z,0) = 1 := by
-      simp only [cap,clockExp,physicalClock_initial,mul_zero,Real.exp_zero]
-    simpa only [hi,one_mul] using h.2
-  have hfactor (n : ℕ) (b : ExactSLEGenerator.Driver) :
-      ‖clockExp a cap lam (V n b)*angleLift g (V n b)‖ ≤ Real.exp (lam*s)*C := by
-    rw [norm_mul,Real.norm_eq_abs,abs_of_pos (show 0 < clockExp a cap lam (V n b) from Real.exp_pos _)]
-    exact mul_le_mul (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (hc n b) hlam))
-      (hbound _ (physicalTheta_mem _)) (norm_nonneg _) (Real.exp_pos _).le
-  have hb (n : ℕ) (b : ExactSLEGenerator.Driver) :
-      ‖f n b‖ ≤ radialApproximation a ha0 z hz ρ hρ hr n b*(Real.exp (lam*s)*C) := by
-    change ‖physicalGreen a (V n b)*(clockExp a cap lam (V n b)*angleLift g (V n b))‖≤_
-    rw [norm_mul,Real.norm_eq_abs,abs_of_pos (show 0 < physicalGreen a (V n b) from Real.exp_pos _)]
-    exact mul_le_mul_of_nonneg_left (hfactor n b) (Real.exp_pos _).le
-  have hlim (b : ExactSLEGenerator.Driver) : Tendsto (fun n => f n b) atTop (𝓝 (F b)) := by
-    by_cases ht : radialTime a b.1 z ρ=∞
-    · have hF : F b=0 := by simp only [F,radialTestWeight,ite_eq_left ht,mul_zero]
-      rw [hF]
-      have hG := radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
-      have hG0 : radialWeight a b.1 z ρ=0 := by simp only [radialWeight,ite_eq_left ht]
-      rw [hG0] at hG
-      apply tendsto_zero_iff_norm_tendsto_zero.mpr
-      apply squeeze_zero (fun n => norm_nonneg _) (fun n => hb n b)
-      simpa only [zero_mul] using hG.mul_const (Real.exp (lam*s)*C)
-    · apply tendsto_const_nhds.congr'
-      filter_upwards [radialStop_eventually_eq a ha0 z hz ρ hρ hr b ht] with n hn
-      have he : T n b=(radialTime a b.1 z ρ).toNNReal := by
-        have hh := congrArg ENNReal.toNNReal hn
-        simpa only [ENNReal.toNNReal_coe] using hh
-      have hv : V n b=actualState a b.1 (radialTime a b.1 z ρ).toNNReal z := by
-        rw [←(hact n b).2,he]
-      have hclock : physicalClock a cap (V n b)=s := by
-        rw [hv,actual_physicalClock_at a ha0 z hz cap ρ hcap hρ hr b ht]
-        exact levelRadius_clock ha0.ne' hcap
-      dsimp only [f,F]
-      rw [clockExp,hclock,radialTestWeight,ite_eq_right ht,hv]
-      ring
-  let bound := Real.exp (-alpha a*Real.log ρ)*(Real.exp (lam*s)*C)
-  have hbnd (n : ℕ) : ∀ᵐ b ∂W, ‖f n b‖≤bound := Eventually.of_forall fun b =>
-    (hb n b).trans (mul_le_mul_of_nonneg_right
-      (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).2
-      (mul_nonneg (Real.exp_pos _).le hC))
-  have hFm : AEStronglyMeasurable F W := aestronglyMeasurable_of_tendsto_ae
-    atTop (fun n => (hfi n).1.aestronglyMeasurable)
-    (Eventually.of_forall hlim)
-  have hFi : Integrable F W := Integrable.of_bound hFm bound (by
-    filter_upwards [] with b
-    exact le_of_tendsto (hlim b).norm (Eventually.of_forall fun n =>
-      (hb n b).trans (mul_le_mul_of_nonneg_right
-        (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).2
-        (mul_nonneg (Real.exp_pos _).le hC))))
-  have hexp : Real.exp (-lam*s)*Real.exp (lam*s)=1 := by
-    rw [←Real.exp_add]
-    convert Real.exp_zero using 1
-    ring_nf
-  have hconst : (fun b : ExactSLEGenerator.Driver => radialTestWeight a (angleLift g) b.1 z ρ) =
-      fun b => Real.exp (-lam*s)*F b := by
-    funext b
-    dsimp only [F]
-    rw [←mul_assoc,hexp,one_mul]
-  have hint : (∫ b, F b ∂W)=physicalGreen a (z,0)*angleLift g (z,0) := by
-    have hconv := tendsto_integral_of_dominated_convergence (fun _ : ExactSLEGenerator.Driver => bound)
-      (fun n => (hfi n).1.aestronglyMeasurable) (integrable_const bound) hbnd
-      (Eventually.of_forall hlim)
-    apply tendsto_nhds_unique hconv
-    simpa only [(hfi _).2] using
-      (tendsto_const_nhds : Tendsto (fun _ : ℕ => physicalGreen a (z,0)*angleLift g (z,0))
-        atTop (𝓝 (physicalGreen a (z,0)*angleLift g (z,0))))
-  change Integrable (fun b : ExactSLEGenerator.Driver => radialTestWeight a (angleLift g) b.1 z ρ) W ∧ _
-  rw [hconst]
-  refine ⟨hFi.const_mul _, ?_⟩
-  rw [integral_const_mul,hint]
-  dsimp only [angleLift]
-  ring
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Polynomial MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.Jacobi
-open ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
-
-def anglePoly (nu : ℝ) (n : ℕ) (θ : ℝ) : ℝ := (poly nu n).eval (Real.cos θ)
-
-lemma eval_cos_deriv (p : ℝ[X]) (θ : ℝ) :
-    deriv (fun θ => p.eval (Real.cos θ)) θ = p.derivative.eval (Real.cos θ)*(-Real.sin θ) :=
-  ((p.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)).deriv
-
-lemma eval_cos_second (p : ℝ[X]) (θ : ℝ) :
-    deriv (deriv (fun θ => p.eval (Real.cos θ))) θ =
-      p.derivative.derivative.eval (Real.cos θ)*Real.sin θ^2
-      -p.derivative.eval (Real.cos θ)*Real.cos θ := by
-  have he : deriv (fun θ => p.eval (Real.cos θ)) =
-      fun θ => p.derivative.eval (Real.cos θ)*(-Real.sin θ) := funext (eval_cos_deriv p)
-  rw [he]
-  have hd := (((p.derivative.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)).mul
-    (Real.hasDerivAt_sin θ).neg).deriv
-  change deriv (fun θ => p.derivative.eval (Real.cos θ)*(-Real.sin θ)) θ = _ at hd
-  rw [hd]
-  dsimp only [Function.comp_apply, Pi.neg_apply]
-  ring
-
-lemma angleGenerator_poly (p : ℝ[X]) (nu : ℝ) (θ : ℝ) (hθ : 0<Real.sin θ) :
-    angleGenerator (nu/2) (fun θ => p.eval (Real.cos θ)) θ =
-      (operator nu p).eval (Real.cos θ)/2 := by
-  rw [angleGenerator,eval_cos_second,eval_cos_deriv]
-  simp only [operator,eval_sub,eval_mul,eval_pow,eval_one,eval_X,eval_C,Real.cot_eq_cos_div_sin]
-  have hid := Real.sin_sq_add_cos_sq θ
-  field_simp
-  linear_combination p.derivative.derivative.eval (Real.cos θ) * hid
-
-lemma anglePoly_smooth (nu : ℝ) (n : ℕ) : ContDiff ℝ (↑(⊤ : ℕ∞)) (anglePoly nu n) := by
-  change ContDiff ℝ _ ((fun x : ℝ => (poly nu n).eval x) ∘ Real.cos)
-  have hp (p : ℝ[X]) : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x : ℝ => p.eval x) := by
-    induction p using Polynomial.induction_on' with
-    | add p q hp hq => simpa only [eval_add] using hp.add hq
-    | monomial n c =>
-      simp only [eval_monomial]
-      exact (contDiff_const (c:=c)).mul (contDiff_id.pow n)
-  exact (hp _).comp Real.contDiff_cos
-
-lemma anglePoly_bound (nu : ℝ) (hnu : 0<nu) (n : ℕ) (θ : ℝ) :
-    ‖anglePoly nu n θ‖≤(2:ℝ)^n :=
-  poly_eval_bound nu hnu n (Real.cos θ) (Real.abs_cos_le_one θ)
-
-lemma anglePoly_eigen (a : ℝ) (n : ℕ) (θ : ℝ) (hθ : θ∈Ioo 0 Real.pi) :
-    angleGenerator (2*a) (anglePoly (4*a) n) θ = -eigenvalue (4*a) n*anglePoly (4*a) n θ := by
-  rw [show 2*a=(4*a)/2 by ring]
-  unfold anglePoly
-  rw [angleGenerator_poly _ _ _ (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2),poly_eigen]
-  simp only [eval_mul,eval_neg,eval_C,eigenvalue]
-  ring
-
-theorem finite_angle_poly (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0 ≤ s) (n : ℕ) :
-    Integrable (fun b : ExactSLEGenerator.Driver =>
-      radialTestWeight a (angleLift (anglePoly (4*a) n)) b.1 z (levelRadius a (2*z.im) s)) W ∧
-    (∫ b, radialTestWeight a (angleLift (anglePoly (4*a) n)) b.1 z (levelRadius a (2*z.im) s) ∂W)=
-      physicalGreen a (z,0)*Real.exp (-s*eigenvalue (4*a) n)*
-        (poly (4*a) n).eval (Real.cos (physicalTheta (z,0))) := by
-  simpa only [mul_comm (eigenvalue (4*a) n) s,neg_mul,anglePoly] using
-    finite_level_angle_eigen W hW (anglePoly (4*a) n)
-      (fun θ _ => (anglePoly_smooth (4*a) n).contDiffAt) (2^n) (by positivity)
-      (fun θ _ => anglePoly_bound (4*a) (by linarith) n θ) a (eigenvalue (4*a) n) ha
-      (eigenvalue_nonneg (4*a) (by linarith) n) (anglePoly_eigen a n) z hz s hs
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Polynomial MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.Jacobi
-open ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
-
-lemma radialState_measurable (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
-    Measurable (fun b : ExactSLEGenerator.Driver => radialState a b.1 z ρ) := by
-  have he : z.im*(ρ/z.im)=ρ := mul_div_cancel₀ _ hz.ne'
-  simpa only [Function.comp_def, id_eq, he] using (joint_radialState_measurable a ha (ρ/z.im) (div_pos hρ hz)
-    ((div_le_one hz).mpr hr)).comp
-      (measurable_id.prodMk (measurable_const (a:= (⟨z,hz⟩ : UpperPlane))))
-
-lemma radialWeight_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
-    Measurable (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ) := by
-  have he : z.im*(ρ/z.im)=ρ := mul_div_cancel₀ _ hz.ne'
-  simpa only [Function.comp_def, id_eq, he] using (joint_radialWeight_measurable a ha (ρ/z.im) (div_pos hρ hz)
-    ((div_le_one hz).mpr hr)).comp
-      (measurable_id.prodMk (measurable_const (a:= (⟨z,hz⟩ : UpperPlane))))
-
-def cosState (v : State) : Interval :=
-  ⟨Real.cos (physicalTheta v),Real.neg_one_le_cos _,Real.cos_le_one _⟩
-
-lemma cosState_measurable : Measurable cosState := by
-  unfold cosState physicalTheta
-  fun_prop
-
-def finiteCos (a : ℝ) (z : ℂ) (ρ : ℝ) (b : ExactSLEGenerator.Driver) : Interval :=
-  cosState (radialState a b.1 z ρ)
-
-lemma finiteCos_measurable (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) : Measurable (finiteCos a z ρ) :=
-  cosState_measurable.comp (radialState_measurable a ha z hz ρ hρ hr)
-
- 
-def finiteCosLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (z : ℂ) (s : ℝ) : Measure Interval :=
-  (radialLaw W a z (levelRadius a (2*z.im) s)).map (finiteCos a z (levelRadius a (2*z.im) s))
-
-lemma levelRadius_initial_le (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (s : ℝ) (hs : 0 ≤ s) : levelRadius a (2*z.im) s≤z.im := by
-  simpa only [mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0)] using
-    levelRadius_le ha (by positivity : 0<2*z.im) hs
-
-lemma finiteCosLaw_probability (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0 ≤ s) :
-    IsProbabilityMeasure (finiteCosLaw W a z s) := by
-  have hρ := levelRadius_pos a (2*z.im) s (by positivity)
-  have hr := levelRadius_initial_le a (by linarith) z hz s hs
-  have := radialLaw_probability W hW a ha z hz _ hρ hr
-  exact (Measure.isProbabilityMeasure_map_iff
-    (finiteCos_measurable a (by linarith) z hz _ hρ hr).aemeasurable).mpr inferInstance
-
-lemma finiteCosLaw_poly_integral (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0 ≤ s) (n : ℕ) :
-    (∫ w : Interval,(poly (4*a) n).eval (w:ℝ) ∂finiteCosLaw W a z s)=
-      Real.exp (-s*eigenvalue (4*a) n)*(poly (4*a) n).eval (cosState (z,0):ℝ) := by
-  have hρ := levelRadius_pos a (2*z.im) s (by positivity)
-  have hr := levelRadius_initial_le a (by linarith) z hz s hs
-  let ρ := levelRadius a (2*z.im) s
-  have hG : 0<physicalGreen a (z,0) := Real.exp_pos _
-  have hf : AEStronglyMeasurable (fun w : Interval => (poly (4*a) n).eval (w:ℝ)) (finiteCosLaw W a z s) :=
-    ((poly (4*a) n).continuous.comp continuous_subtype_val).aestronglyMeasurable
-  rw [finiteCosLaw,integral_map (finiteCos_measurable a (by linarith) z hz ρ hρ hr).aemeasurable hf]
-  rw [radialLaw,integral_withDensity_eq_integral_toReal_smul
-    ((radialWeight_measurable a ha z hz ρ hρ hr).div_const _).ennreal_ofReal
-    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
-  have hn (b : ExactSLEGenerator.Driver) : 0≤radialWeight a b.1 z ρ/physicalGreen a (z,0) :=
-    div_nonneg (radialWeight_bound a ha z hz ρ hρ hr b).1 hG.le
-  simp_rw [ENNReal.toReal_ofReal (hn _),smul_eq_mul]
-  have he (b : ExactSLEGenerator.Driver) :
-      (radialWeight a b.1 z ρ/physicalGreen a (z,0))*(poly (4*a) n).eval (finiteCos a z ρ b:ℝ)=
-        radialTestWeight a (angleLift (anglePoly (4*a) n)) b.1 z ρ/physicalGreen a (z,0) := by
-    rw [radialTestWeight_eq_mul]
-    dsimp only [finiteCos,cosState,angleLift,anglePoly]
-    ring
-  simp_rw [he]
-  rw [integral_div,(finite_angle_poly W hW a ha z hz s hs n).2]
-  dsimp only [cosState]
-  field_simp
-
- 
-
-theorem finiteCosLaw_eq_transition (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a) ≤ s) :
-    finiteCosLaw W a z s=transition (4*a) s (cosState (z,0)) := by
-  have hs0 : 0 ≤ s := le_trans (by unfold mixingTime majorantBase; positivity) hs
-  have := finiteCosLaw_probability W hW a ha z hz s hs0
-  exact transition_identified (4*a) (by linarith) s hs _ _
-    (finiteCosLaw_poly_integral W hW a ha z hz s hs0)
-
- 
-theorem finiteCosLaw_integral_bounds (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a) ≤ s)
-    (f : Interval→ℝ≥0∞) (hf : Measurable f) :
-    (1/2:ℝ≥0∞)*(∫⁻w,f w∂pi (4*a))≤∫⁻w,f w∂finiteCosLaw W a z s ∧
-      (∫⁻w,f w∂finiteCosLaw W a z s)≤2*(∫⁻w,f w∂pi (4*a)) := by
-  rw [finiteCosLaw_eq_transition W hW a ha z hz s hs]
-  exact transition_lintegral_bounds (4*a) (by linarith) s hs _ f hf
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Polynomial MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.Jacobi
-open CenteredMoments ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
-
-def arccosAngle (x : Interval) : Angle :=
-  if h : Real.arccos (x:ℝ) ∈ Ioo 0 Real.pi then ⟨Real.arccos (x:ℝ),h⟩
-  else ⟨Real.pi/2,by constructor <;> linarith [Real.pi_pos]⟩
-
-lemma arccosAngle_measurable : Measurable arccosAngle := by
-  suffices Measurable (fun x : Interval => (arccosAngle x : ℝ)) from this.subtype_mk
-  simp only [arccosAngle,apply_dite,dite_eq_ite]
-  exact (Real.continuous_arccos.measurable.comp measurable_subtype_coe).ite
-    ((Real.continuous_arccos.measurable.comp measurable_subtype_coe) measurableSet_Ioo)
-    measurable_const
-
-@[simp] lemma arccosAngle_angleCos (θ : Angle) : arccosAngle (angleCos θ)=θ := by
-  have hc : Real.arccos (Real.cos (θ:ℝ))=(θ:ℝ) := Real.arccos_cos θ.2.1.le θ.2.2.le
-  have hh : Real.arccos (angleCos θ : ℝ)∈Ioo 0 Real.pi := by
-    change Real.arccos (Real.cos (θ:ℝ))∈Ioo 0 Real.pi
-    rw [hc]
-    exact θ.2
-  apply Subtype.ext
-  simp only [arccosAngle,dite_eq_left hh]
-  exact hc
-
-def angleState (v : State) : Angle := ⟨physicalTheta v,physicalTheta_mem v⟩
-
-lemma angleState_measurable : Measurable angleState := by
-  unfold angleState physicalTheta
-  fun_prop
-
-lemma arccosAngle_cosState (v : State) : arccosAngle (cosState v)=angleState v :=
-  arccosAngle_angleCos (angleState v)
-
-def finiteAngle (a : ℝ) (z : ℂ) (ρ : ℝ) (b : ExactSLEGenerator.Driver) : Angle :=
-  angleState (radialState a b.1 z ρ)
-
-lemma finiteAngle_measurable (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) : Measurable (finiteAngle a z ρ) :=
-  angleState_measurable.comp (radialState_measurable a ha z hz ρ hρ hr)
-
-def finiteAngleLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (z : ℂ) (s : ℝ) : Measure Angle :=
-  (radialLaw W a z (levelRadius a (2*z.im) s)).map (finiteAngle a z (levelRadius a (2*z.im) s))
-
-lemma finiteCosLaw_map_arccos (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0≤ s) :
-    (finiteCosLaw W a z s).map arccosAngle=finiteAngleLaw W a z s := by
-  have hρ := levelRadius_pos a (2*z.im) s (by positivity : 0<2*z.im)
-  have hr := levelRadius_initial_le a ha z hz s hs
-  rw [finiteCosLaw,Measure.map_map arccosAngle_measurable
-    (finiteCos_measurable a ha z hz _ hρ hr)]
-  congr 1
-  funext b
-  exact arccosAngle_cosState _
-
-lemma pi_map_arccos (a : ℝ) (ha : 1<4*a) : (pi (4*a)).map arccosAngle=angleLaw a := by
-  rw [←map_angleLaw_eq_pi a ha,Measure.map_map arccosAngle_measurable angleCos_measurable]
-  simp only [Function.comp_def,arccosAngle_angleCos,Measure.map_id']
-
- 
-
-theorem finiteAngleLaw_integral_bounds (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a)≤ s)
-    (f : Angle→ℝ≥0∞) (hf : Measurable f) :
-    (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a)≤∫⁻θ,f θ∂finiteAngleLaw W a z s ∧
-      (∫⁻θ,f θ∂finiteAngleLaw W a z s)≤2*(∫⁻θ,f θ∂angleLaw a) := by
-  have hs0 : 0≤ s := le_trans (by unfold mixingTime majorantBase; positivity) hs
-  rw [←pi_map_arccos a (by linarith),←finiteCosLaw_map_arccos W a (by linarith) z hz s hs0]
-  simp only [lintegral_map hf arccosAngle_measurable]
-  exact finiteCosLaw_integral_bounds W hW a ha z hz s hs (f ∘ arccosAngle)
-    (hf.comp arccosAngle_measurable)
-
- 
-
-theorem finiteAngleLaw_inverse_power_finite (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a)≤ s) :
-    (∫⁻θ : Angle,ENNReal.ofReal ((Real.sin θ)^(-p a))∂finiteAngleLaw W a z s)<∞ := by
-  apply lt_of_le_of_lt (finiteAngleLaw_integral_bounds W hW a ha z hz s hs _ (by fun_prop)).2
-  exact ENNReal.mul_lt_top (by norm_num) (angle_inverse_power_finite a)
-
-end SLEExactGauge.Jacobi
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.GlobalAngleBudget
+import OAI.Probability.SLEGauge.JacobiKernel
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
+
+def angleEigenCutoff (g : ℝ → ℝ) (a cap lam ε : ℝ) (N : ℝ≥0) (v : State) : ℝ :=
+  greenCutoff a ε N v * (clockExp a cap lam v * angleLift g v)
+
+lemma angleEigenCutoff_smooth (g : ℝ → ℝ)
+    (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ (↑(⊤ : ℕ∞)) g θ)
+    (a cap lam ε : ℝ) (hε : 0<ε) (N : ℝ≥0) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (angleEigenCutoff g a cap lam ε N) := by
+  rw [contDiff_iff_contDiffAt]
+  intro v
+  by_cases hv : 0<Y v
+  · exact (greenCutoff_smooth a ε hε N).contDiffAt.mul
+      ((clockExp_smooth_all a cap lam v hv).mul
+        ((hg _ (physicalTheta_mem v)).comp v (physicalTheta_smooth_all v hv)))
+  · apply (contDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq
+    have hv' : Y v < ε/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
+    simp only [angleEigenCutoff,greenCutoff,heightTurn_zero hε (le_of_lt hw),
+      mul_zero,zero_mul]
+
+lemma angleEigenCutoff_compact (g : ℝ → ℝ) (a cap lam ε : ℝ) (N : ℝ≥0) :
+    HasCompactSupport (angleEigenCutoff g a cap lam ε N) :=
+  (greenCutoff_compact a ε N).mul_right
+
+lemma angleEigenCutoff_eq_near (g : ℝ → ℝ) (a cap lam ε : ℝ) (hε : 0<ε)
+    (N : ℝ≥0) (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
+    angleEigenCutoff g a cap lam ε N =ᶠ[𝓝 v]
+      fun w => physicalGreen a w * (clockExp a cap lam w * angleLift g w) :=
+  (greenCutoff_eq_near a ε hε N v hv hy).mul (EventuallyEq.refl _ _)
+
+lemma angleEigenCutoff_generator (g : ℝ → ℝ)
+    (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ 2 g θ)
+    (a cap lam ε : ℝ) (ha : a≠0)
+    (heigen : ∀ θ ∈ Ioo 0 Real.pi, angleGenerator (2*a) g θ = -lam*g θ)
+    (hε : 0<ε) (N : ℝ≥0) (v : State) (hv : ‖v‖≤N) (hy : ε≤Y v) :
+    generator (angleEigenCutoff g a cap lam ε N) (cutoffDrift a ε) noise v = 0 := by
+  have he := angleEigenCutoff_eq_near g a cap lam ε hε N v hv hy
+  have hd : (fun w => fderiv ℝ (angleEigenCutoff g a cap lam ε N) w noise) =ᶠ[𝓝 v]
+      (fun w => fderiv ℝ (fun u => physicalGreen a u *
+        (clockExp a cap lam u * angleLift g u)) w noise) :=
+    he.fderiv.mono (fun w hw => congrArg (fun f : State →L[ℝ] ℝ => f noise) hw)
+  change (1/2:ℝ) * fderiv ℝ (fun w => fderiv ℝ
+    (angleEigenCutoff g a cap lam ε N) w noise) v noise +
+    fderiv ℝ (angleEigenCutoff g a cap lam ε N) v (cutoffDrift a ε v) = 0
+  rw [hd.fderiv_eq,he.fderiv_eq,cutoffDrift_eq_physical a ε v hy]
+  change L (fun w => physicalGreen a w * (clockExp a cap lam w * angleLift g w))
+    (physicalDrift a v) noise v = 0
+  rw [green_exponential_angle_generator g a cap lam ha v (hε.trans_le hy)
+    (hg _ (physicalTheta_mem v)),heigen _ (physicalTheta_mem v)]
+  ring
+
+ 
+theorem local_angle_eigen (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (g : ℝ → ℝ) (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ (↑(⊤ : ℕ∞)) g θ)
+    (a cap lam ε : ℝ) (ha : a≠0)
+    (heigen : ∀ θ ∈ Ioo 0 Real.pi, angleGenerator (2*a) g θ = -lam*g θ)
+    (hε : 0<ε) (z : ℂ) (N B : ℝ≥0) (T : ExactSLEGenerator.Driver → ℝ≥0)
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hTB : ∀ b, T b≤B)
+    (hsafe : ∀ b t, t≤T b → ε≤Y (localState a ε hε z t b) ∧
+      ‖localState a ε hε z t b‖≤N) :
+    Integrable (fun b => physicalGreen a (localState a ε hε z (T b) b) *
+      (clockExp a cap lam (localState a ε hε z (T b) b) *
+        angleLift g (localState a ε hε z (T b) b))) W ∧
+    (∫ b, physicalGreen a (localState a ε hε z (T b) b) *
+      (clockExp a cap lam (localState a ε hε z (T b) b) *
+        angleLift g (localState a ε hε z (T b) b)) ∂W) =
+      physicalGreen a (z,0) * (clockExp a cap lam (z,0) * angleLift g (z,0)) := by
+  let V := localState a ε hε z
+  let f := angleEigenCutoff g a cap lam ε N
+  let D := generator f (cutoffDrift a ε) noise
+  have hf := angleEigenCutoff_smooth g hg a cap lam ε hε N
+  have hfc := angleEigenCutoff_compact g a cap lam ε N
+  have he (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t≤T b) :
+      f (V t b) = physicalGreen a (V t b) * (clockExp a cap lam (V t b)*angleLift g (V t b)) :=
+    (angleEigenCutoff_eq_near g a cap lam ε hε N _ (hsafe b t ht).2
+      (hsafe b t ht).1).eq_of_nhds
+  have hgen (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t≤T b) : D (V t b)=0 :=
+    angleEigenCutoff_generator g (fun θ hθ => (hg θ hθ).of_le (by norm_cast))
+      a cap lam ε ha heigen hε N _ (hsafe b t ht).2 (hsafe b t ht).1
+  have hDy := compact_test_stopped_dynkin W hW V
+    (localState_continuous a ε hε z) (localState_adapted a ε hε z)
+    (cutoffDrift a ε) (cutoffDrift_lipschitz a ε hε).continuous
+    (max (|a|/ε) (|a|/ε^2)) (by positivity) (cutoffDrift_bound a ε hε) noise
+    (by simp [noise,Prod.norm_def]) (localState_eq a ε hε z) f hf hfc T B hT hTB
+  have hzero (b : ExactSLEGenerator.Driver) : (∫ u in (0:ℝ)..(T b : ℝ), generator f (cutoffDrift a ε) noise (V u.toNNReal b)) = 0 := by
+    rw [intervalIntegral.integral_of_le (T b).coe_nonneg]
+    apply setIntegral_eq_zero_of_forall_eq_zero
+    intro u hu
+    exact hgen b u.toNNReal (by simpa using Real.toNNReal_le_toNNReal hu.2)
+  have hinit (b : ExactSLEGenerator.Driver) : f (V 0 b) =
+      physicalGreen a (z,0)*(clockExp a cap lam (z,0)*angleLift g (z,0)) := by
+    rw [he b 0 bot_le]
+    simp only [V,localState_zero]
+  refine ⟨by simpa only [he _ _ le_rfl] using hDy.1, ?_⟩
+  have hd := hDy.2.2
+  simp_rw [hinit,he _ _ le_rfl,hzero] at hd
+  simpa only [integral_const,probReal_univ,one_smul,integral_zero,add_zero] using hd
+
+ 
+
+theorem finite_level_angle_eigen (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (g : ℝ → ℝ) (hg : ∀ θ ∈ Ioo 0 Real.pi, ContDiffAt ℝ (↑(⊤ : ℕ∞)) g θ)
+    (C : ℝ) (hC : 0≤C) (hbound : ∀ θ ∈ Ioo 0 Real.pi, ‖g θ‖≤C)
+    (a lam : ℝ) (ha : 1/4<a) (hlam : 0≤lam)
+    (heigen : ∀ θ ∈ Ioo 0 Real.pi, angleGenerator (2*a) g θ = -lam*g θ)
+    (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0≤ s) :
+    Integrable (fun b : ExactSLEGenerator.Driver => radialTestWeight a (angleLift g) b.1 z
+      (levelRadius a (2*z.im) s)) W ∧
+    (∫ b, radialTestWeight a (angleLift g) b.1 z
+      (levelRadius a (2*z.im) s) ∂W) =
+      physicalGreen a (z,0) * Real.exp (-lam*s) * g (physicalTheta (z,0)) := by
+  have ha0 : 0<a := by linarith
+  let cap := 2*z.im
+  have hcap : 0<cap := by dsimp [cap]; positivity
+  let ρ := levelRadius a cap s
+  have hρ : 0<ρ := levelRadius_pos a cap s hcap
+  have hr : ρ≤z.im := by
+    simpa only [cap,mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0)] using
+      (levelRadius_le ha0 hcap hs)
+  let L (n : ℕ) := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T (n : ℕ) := radialStop a ha0 z hz ρ hρ hr n
+  let V (n : ℕ) (b : ExactSLEGenerator.Driver) := localState a (L n).ε (L n).hε z (T n b) b
+  let f (n : ℕ) (b : ExactSLEGenerator.Driver) := physicalGreen a (V n b)*
+    (clockExp a cap lam (V n b)*angleLift g (V n b))
+  let F (b : ExactSLEGenerator.Driver) := Real.exp (lam*s)*radialTestWeight a (angleLift g) b.1 z ρ
+  have hsafe (n : ℕ) (b : ExactSLEGenerator.Driver) (u : ℝ≥0) (hu : u≤T n b) :
+      (L n).ε≤Y (localState a (L n).ε (L n).hε z u b) ∧
+        ‖localState a (L n).ε (L n).hε z u b‖≤(L n).N := by
+    have h := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b u hu
+    exact ⟨h.1,h.2.1⟩
+  have hact (n : ℕ) (b : ExactSLEGenerator.Driver) : alive a b.1 (T n b) z ∧
+      actualState a b.1 (T n b) z = V n b :=
+    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b _ le_rfl
+  have hc (n : ℕ) (b : ExactSLEGenerator.Driver) : physicalClock a cap (V n b)≤ s := by
+    rw [←(hact n b).2]
+    have h := actual_physicalClock_le a ha0 z hz cap ρ hcap hρ hr b (T n b)
+      (hact n b).1 (radialStop_le_radialTime a ha0 z hz ρ hρ hr n b)
+    simpa only [ρ,levelRadius_clock ha0.ne' hcap] using h
+  have hfi (n : ℕ) : Integrable (f n) W ∧
+      (∫ b, f n b ∂W)=physicalGreen a (z,0)*angleLift g (z,0) := by
+    have h := local_angle_eigen W hW g hg a cap lam (L n).ε ha0.ne' heigen (L n).hε
+      z (L n).N (L n).B (T n) (radialStop_stopping a ha0 z hz ρ hρ hr n)
+      (regionStop_le a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B)
+      (hsafe n)
+    refine ⟨h.1, ?_⟩
+    have hi : clockExp a cap lam (z,0) = 1 := by
+      simp only [cap,clockExp,physicalClock_initial,mul_zero,Real.exp_zero]
+    simpa only [hi,one_mul] using h.2
+  have hfactor (n : ℕ) (b : ExactSLEGenerator.Driver) :
+      ‖clockExp a cap lam (V n b)*angleLift g (V n b)‖ ≤ Real.exp (lam*s)*C := by
+    rw [norm_mul,Real.norm_eq_abs,abs_of_pos (show 0 < clockExp a cap lam (V n b) from Real.exp_pos _)]
+    exact mul_le_mul (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (hc n b) hlam))
+      (hbound _ (physicalTheta_mem _)) (norm_nonneg _) (Real.exp_pos _).le
+  have hb (n : ℕ) (b : ExactSLEGenerator.Driver) :
+      ‖f n b‖ ≤ radialApproximation a ha0 z hz ρ hρ hr n b*(Real.exp (lam*s)*C) := by
+    change ‖physicalGreen a (V n b)*(clockExp a cap lam (V n b)*angleLift g (V n b))‖≤_
+    rw [norm_mul,Real.norm_eq_abs,abs_of_pos (show 0 < physicalGreen a (V n b) from Real.exp_pos _)]
+    exact mul_le_mul_of_nonneg_left (hfactor n b) (Real.exp_pos _).le
+  have hlim (b : ExactSLEGenerator.Driver) : Tendsto (fun n => f n b) atTop (𝓝 (F b)) := by
+    by_cases ht : radialTime a b.1 z ρ=∞
+    · have hF : F b=0 := by simp only [F,radialTestWeight,ite_eq_left ht,mul_zero]
+      rw [hF]
+      have hG := radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
+      have hG0 : radialWeight a b.1 z ρ=0 := by simp only [radialWeight,ite_eq_left ht]
+      rw [hG0] at hG
+      apply tendsto_zero_iff_norm_tendsto_zero.mpr
+      apply squeeze_zero (fun n => norm_nonneg _) (fun n => hb n b)
+      simpa only [zero_mul] using hG.mul_const (Real.exp (lam*s)*C)
+    · apply tendsto_const_nhds.congr'
+      filter_upwards [radialStop_eventually_eq a ha0 z hz ρ hρ hr b ht] with n hn
+      have he : T n b=(radialTime a b.1 z ρ).toNNReal := by
+        have hh := congrArg ENNReal.toNNReal hn
+        simpa only [ENNReal.toNNReal_coe] using hh
+      have hv : V n b=actualState a b.1 (radialTime a b.1 z ρ).toNNReal z := by
+        rw [←(hact n b).2,he]
+      have hclock : physicalClock a cap (V n b)=s := by
+        rw [hv,actual_physicalClock_at a ha0 z hz cap ρ hcap hρ hr b ht]
+        exact levelRadius_clock ha0.ne' hcap
+      dsimp only [f,F]
+      rw [clockExp,hclock,radialTestWeight,ite_eq_right ht,hv]
+      ring
+  let bound := Real.exp (-alpha a*Real.log ρ)*(Real.exp (lam*s)*C)
+  have hbnd (n : ℕ) : ∀ᵐ b ∂W, ‖f n b‖≤bound := Eventually.of_forall fun b =>
+    (hb n b).trans (mul_le_mul_of_nonneg_right
+      (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).2
+      (mul_nonneg (Real.exp_pos _).le hC))
+  have hFm : AEStronglyMeasurable F W := aestronglyMeasurable_of_tendsto_ae
+    atTop (fun n => (hfi n).1.aestronglyMeasurable)
+    (Eventually.of_forall hlim)
+  have hFi : Integrable F W := Integrable.of_bound hFm bound (by
+    filter_upwards [] with b
+    exact le_of_tendsto (hlim b).norm (Eventually.of_forall fun n =>
+      (hb n b).trans (mul_le_mul_of_nonneg_right
+        (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).2
+        (mul_nonneg (Real.exp_pos _).le hC))))
+  have hexp : Real.exp (-lam*s)*Real.exp (lam*s)=1 := by
+    rw [←Real.exp_add]
+    convert Real.exp_zero using 1
+    ring_nf
+  have hconst : (fun b : ExactSLEGenerator.Driver => radialTestWeight a (angleLift g) b.1 z ρ) =
+      fun b => Real.exp (-lam*s)*F b := by
+    funext b
+    dsimp only [F]
+    rw [←mul_assoc,hexp,one_mul]
+  have hint : (∫ b, F b ∂W)=physicalGreen a (z,0)*angleLift g (z,0) := by
+    have hconv := tendsto_integral_of_dominated_convergence (fun _ : ExactSLEGenerator.Driver => bound)
+      (fun n => (hfi n).1.aestronglyMeasurable) (integrable_const bound) hbnd
+      (Eventually.of_forall hlim)
+    apply tendsto_nhds_unique hconv
+    simpa only [(hfi _).2] using
+      (tendsto_const_nhds : Tendsto (fun _ : ℕ => physicalGreen a (z,0)*angleLift g (z,0))
+        atTop (𝓝 (physicalGreen a (z,0)*angleLift g (z,0))))
+  change Integrable (fun b : ExactSLEGenerator.Driver => radialTestWeight a (angleLift g) b.1 z ρ) W ∧ _
+  rw [hconst]
+  refine ⟨hFi.const_mul _, ?_⟩
+  rw [integral_const_mul,hint]
+  dsimp only [angleLift]
+  ring
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Polynomial MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.Jacobi
+open ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
+
+def anglePoly (nu : ℝ) (n : ℕ) (θ : ℝ) : ℝ := (poly nu n).eval (Real.cos θ)
+
+lemma eval_cos_deriv (p : ℝ[X]) (θ : ℝ) :
+    deriv (fun θ => p.eval (Real.cos θ)) θ = p.derivative.eval (Real.cos θ)*(-Real.sin θ) :=
+  ((p.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)).deriv
+
+lemma eval_cos_second (p : ℝ[X]) (θ : ℝ) :
+    deriv (deriv (fun θ => p.eval (Real.cos θ))) θ =
+      p.derivative.derivative.eval (Real.cos θ)*Real.sin θ^2
+      -p.derivative.eval (Real.cos θ)*Real.cos θ := by
+  have he : deriv (fun θ => p.eval (Real.cos θ)) =
+      fun θ => p.derivative.eval (Real.cos θ)*(-Real.sin θ) := funext (eval_cos_deriv p)
+  rw [he]
+  have hd := (((p.derivative.hasDerivAt (Real.cos θ)).comp θ (Real.hasDerivAt_cos θ)).mul
+    (Real.hasDerivAt_sin θ).neg).deriv
+  change deriv (fun θ => p.derivative.eval (Real.cos θ)*(-Real.sin θ)) θ = _ at hd
+  rw [hd]
+  dsimp only [Function.comp_apply, Pi.neg_apply]
+  ring
+
+lemma angleGenerator_poly (p : ℝ[X]) (nu : ℝ) (θ : ℝ) (hθ : 0<Real.sin θ) :
+    angleGenerator (nu/2) (fun θ => p.eval (Real.cos θ)) θ =
+      (operator nu p).eval (Real.cos θ)/2 := by
+  rw [angleGenerator,eval_cos_second,eval_cos_deriv]
+  simp only [operator,eval_sub,eval_mul,eval_pow,eval_one,eval_X,eval_C,Real.cot_eq_cos_div_sin]
+  have hid := Real.sin_sq_add_cos_sq θ
+  field_simp
+  linear_combination p.derivative.derivative.eval (Real.cos θ) * hid
+
+lemma anglePoly_smooth (nu : ℝ) (n : ℕ) : ContDiff ℝ (↑(⊤ : ℕ∞)) (anglePoly nu n) := by
+  change ContDiff ℝ _ ((fun x : ℝ => (poly nu n).eval x) ∘ Real.cos)
+  have hp (p : ℝ[X]) : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x : ℝ => p.eval x) := by
+    induction p using Polynomial.induction_on' with
+    | add p q hp hq => simpa only [eval_add] using hp.add hq
+    | monomial n c =>
+      simp only [eval_monomial]
+      exact (contDiff_const (c:=c)).mul (contDiff_id.pow n)
+  exact (hp _).comp Real.contDiff_cos
+
+lemma anglePoly_bound (nu : ℝ) (hnu : 0<nu) (n : ℕ) (θ : ℝ) :
+    ‖anglePoly nu n θ‖≤(2:ℝ)^n :=
+  poly_eval_bound nu hnu n (Real.cos θ) (Real.abs_cos_le_one θ)
+
+lemma anglePoly_eigen (a : ℝ) (n : ℕ) (θ : ℝ) (hθ : θ∈Ioo 0 Real.pi) :
+    angleGenerator (2*a) (anglePoly (4*a) n) θ = -eigenvalue (4*a) n*anglePoly (4*a) n θ := by
+  rw [show 2*a=(4*a)/2 by ring]
+  unfold anglePoly
+  rw [angleGenerator_poly _ _ _ (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2),poly_eigen]
+  simp only [eval_mul,eval_neg,eval_C,eigenvalue]
+  ring
+
+theorem finite_angle_poly (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0 ≤ s) (n : ℕ) :
+    Integrable (fun b : ExactSLEGenerator.Driver =>
+      radialTestWeight a (angleLift (anglePoly (4*a) n)) b.1 z (levelRadius a (2*z.im) s)) W ∧
+    (∫ b, radialTestWeight a (angleLift (anglePoly (4*a) n)) b.1 z (levelRadius a (2*z.im) s) ∂W)=
+      physicalGreen a (z,0)*Real.exp (-s*eigenvalue (4*a) n)*
+        (poly (4*a) n).eval (Real.cos (physicalTheta (z,0))) := by
+  simpa only [mul_comm (eigenvalue (4*a) n) s,neg_mul,anglePoly] using
+    finite_level_angle_eigen W hW (anglePoly (4*a) n)
+      (fun θ _ => (anglePoly_smooth (4*a) n).contDiffAt) (2^n) (by positivity)
+      (fun θ _ => anglePoly_bound (4*a) (by linarith) n θ) a (eigenvalue (4*a) n) ha
+      (eigenvalue_nonneg (4*a) (by linarith) n) (anglePoly_eigen a n) z hz s hs
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Polynomial MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.Jacobi
+open ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
+
+lemma radialState_measurable (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
+    Measurable (fun b : ExactSLEGenerator.Driver => radialState a b.1 z ρ) := by
+  have he : z.im*(ρ/z.im)=ρ := mul_div_cancel₀ _ hz.ne'
+  simpa only [Function.comp_def, id_eq, he] using (joint_radialState_measurable a ha (ρ/z.im) (div_pos hρ hz)
+    ((div_le_one hz).mpr hr)).comp
+      (measurable_id.prodMk (measurable_const (a:= (⟨z,hz⟩ : UpperPlane))))
+
+lemma radialWeight_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
+    Measurable (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ) := by
+  have he : z.im*(ρ/z.im)=ρ := mul_div_cancel₀ _ hz.ne'
+  simpa only [Function.comp_def, id_eq, he] using (joint_radialWeight_measurable a ha (ρ/z.im) (div_pos hρ hz)
+    ((div_le_one hz).mpr hr)).comp
+      (measurable_id.prodMk (measurable_const (a:= (⟨z,hz⟩ : UpperPlane))))
+
+def cosState (v : State) : Interval :=
+  ⟨Real.cos (physicalTheta v),Real.neg_one_le_cos _,Real.cos_le_one _⟩
+
+lemma cosState_measurable : Measurable cosState := by
+  unfold cosState physicalTheta
+  fun_prop
+
+def finiteCos (a : ℝ) (z : ℂ) (ρ : ℝ) (b : ExactSLEGenerator.Driver) : Interval :=
+  cosState (radialState a b.1 z ρ)
+
+lemma finiteCos_measurable (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) : Measurable (finiteCos a z ρ) :=
+  cosState_measurable.comp (radialState_measurable a ha z hz ρ hρ hr)
+
+ 
+def finiteCosLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (z : ℂ) (s : ℝ) : Measure Interval :=
+  (radialLaw W a z (levelRadius a (2*z.im) s)).map (finiteCos a z (levelRadius a (2*z.im) s))
+
+lemma levelRadius_initial_le (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (s : ℝ) (hs : 0 ≤ s) : levelRadius a (2*z.im) s≤z.im := by
+  simpa only [mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0)] using
+    levelRadius_le ha (by positivity : 0<2*z.im) hs
+
+lemma finiteCosLaw_probability (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0 ≤ s) :
+    IsProbabilityMeasure (finiteCosLaw W a z s) := by
+  have hρ := levelRadius_pos a (2*z.im) s (by positivity)
+  have hr := levelRadius_initial_le a (by linarith) z hz s hs
+  have := radialLaw_probability W hW a ha z hz _ hρ hr
+  exact (Measure.isProbabilityMeasure_map_iff
+    (finiteCos_measurable a (by linarith) z hz _ hρ hr).aemeasurable).mpr inferInstance
+
+lemma finiteCosLaw_poly_integral (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0 ≤ s) (n : ℕ) :
+    (∫ w : Interval,(poly (4*a) n).eval (w:ℝ) ∂finiteCosLaw W a z s)=
+      Real.exp (-s*eigenvalue (4*a) n)*(poly (4*a) n).eval (cosState (z,0):ℝ) := by
+  have hρ := levelRadius_pos a (2*z.im) s (by positivity)
+  have hr := levelRadius_initial_le a (by linarith) z hz s hs
+  let ρ := levelRadius a (2*z.im) s
+  have hG : 0<physicalGreen a (z,0) := Real.exp_pos _
+  have hf : AEStronglyMeasurable (fun w : Interval => (poly (4*a) n).eval (w:ℝ)) (finiteCosLaw W a z s) :=
+    ((poly (4*a) n).continuous.comp continuous_subtype_val).aestronglyMeasurable
+  rw [finiteCosLaw,integral_map (finiteCos_measurable a (by linarith) z hz ρ hρ hr).aemeasurable hf]
+  rw [radialLaw,integral_withDensity_eq_integral_toReal_smul
+    ((radialWeight_measurable a ha z hz ρ hρ hr).div_const _).ennreal_ofReal
+    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  have hn (b : ExactSLEGenerator.Driver) : 0≤radialWeight a b.1 z ρ/physicalGreen a (z,0) :=
+    div_nonneg (radialWeight_bound a ha z hz ρ hρ hr b).1 hG.le
+  simp_rw [ENNReal.toReal_ofReal (hn _),smul_eq_mul]
+  have he (b : ExactSLEGenerator.Driver) :
+      (radialWeight a b.1 z ρ/physicalGreen a (z,0))*(poly (4*a) n).eval (finiteCos a z ρ b:ℝ)=
+        radialTestWeight a (angleLift (anglePoly (4*a) n)) b.1 z ρ/physicalGreen a (z,0) := by
+    rw [radialTestWeight_eq_mul]
+    dsimp only [finiteCos,cosState,angleLift,anglePoly]
+    ring
+  simp_rw [he]
+  rw [integral_div,(finite_angle_poly W hW a ha z hz s hs n).2]
+  dsimp only [cosState]
+  field_simp
+
+ 
+
+theorem finiteCosLaw_eq_transition (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a) ≤ s) :
+    finiteCosLaw W a z s=transition (4*a) s (cosState (z,0)) := by
+  have hs0 : 0 ≤ s := le_trans (by unfold mixingTime majorantBase; positivity) hs
+  have := finiteCosLaw_probability W hW a ha z hz s hs0
+  exact transition_identified (4*a) (by linarith) s hs _ _
+    (finiteCosLaw_poly_integral W hW a ha z hz s hs0)
+
+ 
+theorem finiteCosLaw_integral_bounds (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a) ≤ s)
+    (f : Interval→ℝ≥0∞) (hf : Measurable f) :
+    (1/2:ℝ≥0∞)*(∫⁻w,f w∂pi (4*a))≤∫⁻w,f w∂finiteCosLaw W a z s ∧
+      (∫⁻w,f w∂finiteCosLaw W a z s)≤2*(∫⁻w,f w∂pi (4*a)) := by
+  rw [finiteCosLaw_eq_transition W hW a ha z hz s hs]
+  exact transition_lintegral_bounds (4*a) (by linarith) s hs _ f hf
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Polynomial MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.Jacobi
+open CenteredMoments ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
+
+def arccosAngle (x : Interval) : Angle :=
+  if h : Real.arccos (x:ℝ) ∈ Ioo 0 Real.pi then ⟨Real.arccos (x:ℝ),h⟩
+  else ⟨Real.pi/2,by constructor <;> linarith [Real.pi_pos]⟩
+
+lemma arccosAngle_measurable : Measurable arccosAngle := by
+  suffices Measurable (fun x : Interval => (arccosAngle x : ℝ)) from this.subtype_mk
+  simp only [arccosAngle,apply_dite,dite_eq_ite]
+  exact (Real.continuous_arccos.measurable.comp measurable_subtype_coe).ite
+    ((Real.continuous_arccos.measurable.comp measurable_subtype_coe) measurableSet_Ioo)
+    measurable_const
+
+@[simp] lemma arccosAngle_angleCos (θ : Angle) : arccosAngle (angleCos θ)=θ := by
+  have hc : Real.arccos (Real.cos (θ:ℝ))=(θ:ℝ) := Real.arccos_cos θ.2.1.le θ.2.2.le
+  have hh : Real.arccos (angleCos θ : ℝ)∈Ioo 0 Real.pi := by
+    change Real.arccos (Real.cos (θ:ℝ))∈Ioo 0 Real.pi
+    rw [hc]
+    exact θ.2
+  apply Subtype.ext
+  simp only [arccosAngle,dite_eq_left hh]
+  exact hc
+
+def angleState (v : State) : Angle := ⟨physicalTheta v,physicalTheta_mem v⟩
+
+lemma angleState_measurable : Measurable angleState := by
+  unfold angleState physicalTheta
+  fun_prop
+
+lemma arccosAngle_cosState (v : State) : arccosAngle (cosState v)=angleState v :=
+  arccosAngle_angleCos (angleState v)
+
+def finiteAngle (a : ℝ) (z : ℂ) (ρ : ℝ) (b : ExactSLEGenerator.Driver) : Angle :=
+  angleState (radialState a b.1 z ρ)
+
+lemma finiteAngle_measurable (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) : Measurable (finiteAngle a z ρ) :=
+  angleState_measurable.comp (radialState_measurable a ha z hz ρ hρ hr)
+
+def finiteAngleLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (z : ℂ) (s : ℝ) : Measure Angle :=
+  (radialLaw W a z (levelRadius a (2*z.im) s)).map (finiteAngle a z (levelRadius a (2*z.im) s))
+
+lemma finiteCosLaw_map_arccos (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0≤ s) :
+    (finiteCosLaw W a z s).map arccosAngle=finiteAngleLaw W a z s := by
+  have hρ := levelRadius_pos a (2*z.im) s (by positivity : 0<2*z.im)
+  have hr := levelRadius_initial_le a ha z hz s hs
+  rw [finiteCosLaw,Measure.map_map arccosAngle_measurable
+    (finiteCos_measurable a ha z hz _ hρ hr)]
+  congr 1
+  funext b
+  exact arccosAngle_cosState _
+
+lemma pi_map_arccos (a : ℝ) (ha : 1<4*a) : (pi (4*a)).map arccosAngle=angleLaw a := by
+  rw [←map_angleLaw_eq_pi a ha,Measure.map_map arccosAngle_measurable angleCos_measurable]
+  simp only [Function.comp_def,arccosAngle_angleCos,Measure.map_id']
+
+ 
+
+theorem finiteAngleLaw_integral_bounds (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a)≤ s)
+    (f : Angle→ℝ≥0∞) (hf : Measurable f) :
+    (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a)≤∫⁻θ,f θ∂finiteAngleLaw W a z s ∧
+      (∫⁻θ,f θ∂finiteAngleLaw W a z s)≤2*(∫⁻θ,f θ∂angleLaw a) := by
+  have hs0 : 0≤ s := le_trans (by unfold mixingTime majorantBase; positivity) hs
+  rw [←pi_map_arccos a (by linarith),←finiteCosLaw_map_arccos W a (by linarith) z hz s hs0]
+  simp only [lintegral_map hf arccosAngle_measurable]
+  exact finiteCosLaw_integral_bounds W hW a ha z hz s hs (f ∘ arccosAngle)
+    (hf.comp arccosAngle_measurable)
+
+ 
+
+theorem finiteAngleLaw_inverse_power_finite (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : mixingTime (4*a)≤ s) :
+    (∫⁻θ : Angle,ENNReal.ofReal ((Real.sin θ)^(-p a))∂finiteAngleLaw W a z s)<∞ := by
+  apply lt_of_le_of_lt (finiteAngleLaw_integral_bounds W hW a ha z hz s hs _ (by fun_prop)).2
+  exact ENNReal.mul_lt_top (by norm_num) (angle_inverse_power_finite a)
+
+end SLEExactGauge.Jacobi
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/FiniteProductLimit.lean b/lean/OAI/Probability/SLEGauge/FiniteProductLimit.lean
index d90dc85a4cc0a8afa8d5e309ae0de18f82fa1601..6c1cca06f59b6b19afc997a5111199b4d9939490 100644
--- a/lean/OAI/Probability/SLEGauge/FiniteProductLimit.lean
+++ b/lean/OAI/Probability/SLEGauge/FiniteProductLimit.lean
@@ -1,571 +1,571 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.MassVersionsCore
-import OAI.Probability.SLEGauge.RadonKernels
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory Filter Set Topology TopologicalSpace
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge.MassVersions
-
- 
-
-theorem tendsto_finiteMeasure_prod
-    {X Y I : Type*} [MeasurableSpace X] [MeasurableSpace Y]
-    [TopologicalSpace X] [TopologicalSpace Y]
-    [SecondCountableTopology X] [SecondCountableTopology Y]
-    [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y]
-    [OpensMeasurableSpace X] [OpensMeasurableSpace Y]
-    {l : Filter I} {μ : I → FiniteMeasure X} {ν : I → FiniteMeasure Y}
-    {μ₀ : FiniteMeasure X} {ν₀ : FiniteMeasure Y}
-    (hμ : Tendsto μ l (𝓝 μ₀)) (hν : Tendsto ν l (𝓝 ν₀)) :
-    Tendsto (fun i => (μ i).prod (ν i)) l (𝓝 (μ₀.prod ν₀)) := by
-  have hm : Tendsto (fun i => ((μ i).prod (ν i)).mass) l
-      (𝓝 (μ₀.mass * ν₀.mass)) := by
-    simp only [FiniteMeasure.mass_prod]
-    exact hμ.mass.mul hν.mass
-  by_cases hμ0 : μ₀ = 0
-  · simp only [hμ0, FiniteMeasure.zero_mass, zero_mul] at hm
-    simpa only [hμ0, FiniteMeasure.zero_prod] using
-      FiniteMeasure.tendsto_zero_of_tendsto_zero_mass hm
-  by_cases hν0 : ν₀ = 0
-  · simp only [hν0, FiniteMeasure.zero_mass, mul_zero] at hm
-    simpa only [hν0, FiniteMeasure.prod_zero] using
-      FiniteMeasure.tendsto_zero_of_tendsto_zero_mass hm
-  have : Nonempty X := by
-    by_contra h
-    have : IsEmpty X := not_nonempty_iff.mp h
-    exact hμ0 (Subtype.ext (Measure.eq_zero_of_isEmpty _))
-  have : Nonempty Y := by
-    by_contra h
-    have : IsEmpty Y := not_nonempty_iff.mp h
-    exact hν0 (Subtype.ext (Measure.eq_zero_of_isEmpty _))
-  have hp := ProbabilityMeasure.continuous_prod.tendsto
-    (μ₀.normalize, ν₀.normalize) |>.comp
-      ((μ₀.tendsto_normalize_of_tendsto hμ hμ0).prodMk_nhds
-        (ν₀.tendsto_normalize_of_tendsto hν hν0))
-  have he (m : FiniteMeasure X) (n : FiniteMeasure Y) :
-      m.prod n = (m.mass * n.mass) •
-        (m.normalize.prod n.normalize).toFiniteMeasure := by
-    apply Subtype.ext
-    change m.toMeasure.prod n.toMeasure =
-      (((m.mass * n.mass) • (m.normalize.prod n.normalize).toFiniteMeasure).toMeasure)
-    rw [FiniteMeasure.toMeasure_smul]
-    change m.toMeasure.prod n.toMeasure =
-      (m.mass * n.mass) • (m.normalize.toMeasure.prod n.normalize.toMeasure)
-    have hm' := congrArg FiniteMeasure.toMeasure m.self_eq_mass_smul_normalize
-    have hn' := congrArg FiniteMeasure.toMeasure n.self_eq_mass_smul_normalize
-    simp only [FiniteMeasure.toMeasure_smul,
-      ProbabilityMeasure.toMeasure_comp_toFiniteMeasure_eq_toMeasure] at hm' hn'
-    calc
-      _ = (m.mass • m.normalize.toMeasure).prod (n.mass • n.normalize.toMeasure) :=
-        congrArg₂ Measure.prod hm' hn'
-      _ = _ := by rw [Measure.prod_smul_left, Measure.prod_smul_right, smul_smul]
-  simp_rw [he]
-  exact (hμ.mass.mul hν.mass).smul
-    (ProbabilityMeasure.toFiniteMeasure_continuous.tendsto _ |>.comp hp)
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge.MassVersions
-
-variable {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
-
-noncomputable def localizedFinite (μ : Measure X) [IsFiniteMeasureOnCompacts μ]
-    (φ : C_c(X,ℝ≥0)) : FiniteMeasure X :=
-  ⟨μ.withDensity (fun x => φ x), isFiniteMeasure_withDensity (by
-    rw [lintegral_coe_eq_integral _ (by
-      have h : Integrable (fun x => φ.toReal x) μ := φ.toReal.integrable
-      simpa only [CompactlySupportedContinuousMap.toReal_apply] using h)]
-    exact ENNReal.ofReal_ne_top)⟩
-
-lemma localizedFinite_integral (μ : Measure X) [IsFiniteMeasureOnCompacts μ]
-    (φ : C_c(X,ℝ≥0)) (f : X → ℝ) :
-    (∫x, f x ∂localizedFinite μ φ) = ∫x, (φ x:ℝ)*f x ∂μ := by
-  exact integral_withDensity_eq_integral_smul φ.continuous.measurable f
-
- 
-
-theorem localizedFinite_tendsto {I : Type*} {l : Filter I}
-    (μ : I → Measure X) (μ₀ : Measure X)
-    [∀ i, IsFiniteMeasureOnCompacts (μ i)] [IsFiniteMeasureOnCompacts μ₀]
-    (h : ∀ f : C_c(X,ℝ), Tendsto (fun i => ∫x,f x∂μ i) l (𝓝 (∫x,f x∂μ₀)))
-    (φ : C_c(X,ℝ≥0)) :
-    Tendsto (fun i => localizedFinite (μ i) φ) l (𝓝 (localizedFinite μ₀ φ)) := by
-  apply FiniteMeasure.tendsto_iff_forall_integral_tendsto.mpr
-  intro f
-  let F : C_c(X,ℝ) := f • φ.toReal
-  simpa only [localizedFinite_integral, F, CompactlySupportedContinuousMap.smulc_apply,
-    smul_eq_mul, CompactlySupportedContinuousMap.toReal_apply, mul_comm] using h F
-
- 
-theorem measure_le_of_nonnegative_compact_tests
-    [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-    (μ ν : Measure X) [IsFiniteMeasureOnCompacts μ] [ν.OuterRegular]
-    (h : ∀ f : C_c(X,ℝ), (∀ x, 0≤f x) →
-      ENNReal.ofReal (∫x,f x∂μ) ≤ ∫⁻x, ENNReal.ofReal (f x)∂ν) : μ≤ν := by
-  have ho (U : Set X) (hU : IsOpen U) : μ U≤ν U := by
-    obtain ⟨f,hf,hf0,he⟩ := open_countable_tests U hU
-    rw [he μ inferInstance]
-    refine iSup_le (fun n => (h (f n) (fun x => (hf n x).1)).trans ?_)
-    calc
-      _ ≤ ∫⁻x, U.indicator (fun _ => (1:ℝ≥0∞)) x∂ν := by
-        apply lintegral_mono
-        intro x
-        by_cases hx : x∈U
-        · simp only [indicator_of_mem hx]
-          exact (ENNReal.ofReal_le_ofReal (hf n x).2).trans_eq (by simp)
-        · simp [hx, hf0 n hx]
-      _ = ν U := by simp [hU.measurableSet]
-  intro S
-  rw [S.measure_eq_iInf_isOpen ν]
-  exact le_iInf (fun U => le_iInf (fun hSU => le_iInf (fun hU =>
-    (measure_mono hSU).trans (ho U hU))))
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [BorelSpace X]
-
- 
-
-theorem bind_le_of_weak_limit (W : Measure Ω)
-    (μ : ℕ → Ω → FiniteMeasure X) (ν : Ω → FiniteMeasure X)
-    (hμ : ∀ n, Measurable (μ n)) (hν : Measurable ν)
-    (hconv : ∀ᵐ ω ∂W, Tendsto (fun n => μ n ω) atTop (𝓝 (ν ω)))
-    (Q : Measure X) [IsFiniteMeasure Q] [Q.OuterRegular]
-    (hb : ∀ n, W.bind (fun ω => (μ n ω).toMeasure) ≤ Q) :
-    W.bind (fun ω => (ν ω).toMeasure) ≤ Q := by
-  have hm (n : ℕ) : Measurable (fun ω => (μ n ω).toMeasure) :=
-    measurable_subtype_coe.comp (hμ n)
-  have hn : Measurable (fun ω => (ν ω).toMeasure) := measurable_subtype_coe.comp hν
-  have ht (f : BoundedContinuousFunction X ℝ≥0) :
-      (∫⁻x, (f x : ℝ≥0∞) ∂W.bind (fun ω => (ν ω).toMeasure)) ≤ ∫⁻x, (f x : ℝ≥0∞) ∂Q := by
-    have hf : Measurable (fun x => (f x : ℝ≥0∞)) := f.continuous.measurable.coe_nnreal_ennreal
-    rw [Measure.lintegral_bind hn.aemeasurable hf.aemeasurable]
-    calc
-      _ = ∫⁻ω, liminf (fun n => ∫⁻x, (f x : ℝ≥0∞) ∂(μ n ω)) atTop ∂W := by
-        apply lintegral_congr_ae
-        filter_upwards [hconv] with ω hω
-        exact (FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp hω f).liminf_eq.symm
-      _ ≤ liminf (fun n => ∫⁻ω, ∫⁻x, (f x : ℝ≥0∞) ∂(μ n ω) ∂W) atTop :=
-        lintegral_liminf_le (fun n => (Measure.measurable_lintegral hf).comp (hm n))
-      _ ≤ ∫⁻x, (f x : ℝ≥0∞) ∂Q := by
-        have hh (n : ℕ) : (∫⁻ω, ∫⁻x, (f x : ℝ≥0∞) ∂(μ n ω) ∂W) ≤ ∫⁻x, (f x : ℝ≥0∞) ∂Q := by
-          rw [← Measure.lintegral_bind (hm n).aemeasurable hf.aemeasurable]
-          exact lintegral_mono' (hb n) le_rfl
-        simpa only [liminf_const] using liminf_le_liminf (f := atTop) (Eventually.of_forall hh)
-  have havg : IsFiniteMeasure (W.bind (fun ω => (ν ω).toMeasure)) := by
-    constructor
-    have h := ht (BoundedContinuousFunction.const X 1)
-    simp only [BoundedContinuousFunction.const_apply, ENNReal.coe_one, lintegral_one] at h
-    exact h.trans_lt (measure_lt_top Q univ)
-  apply measure_le_of_nonnegative_compact_tests
-  intro f hf
-  rw [ofReal_integral_eq_lintegral_ofReal f.integrable (Eventually.of_forall hf)]
-  exact ht f.nnrealPart.toBoundedContinuousFunction
-
- 
-
-theorem average_prod_le_of_weak_limit
-    [TopologicalSpace.PseudoMetrizableSpace X]
-    (W : Measure Ω) (μ : ℕ → Ω → FiniteMeasure X) (ν : Ω → FiniteMeasure X)
-    (hμ : ∀ n, Measurable (μ n)) (hν : Measurable ν)
-    (hconv : ∀ᵐ ω ∂W, Tendsto (fun n => μ n ω) atTop (𝓝 (ν ω)))
-    (Q : Measure (X × X)) [IsFiniteMeasure Q] [Q.OuterRegular]
-    (hb : ∀ n, W.bind (fun ω => (μ n ω).toMeasure.prod (μ n ω).toMeasure) ≤ Q) :
-    W.bind (fun ω => (ν ω).toMeasure.prod (ν ω).toMeasure) ≤ Q := by
-  apply bind_le_of_weak_limit W
-    (fun n ω => (μ n ω).prod (μ n ω)) (fun ω => (ν ω).prod (ν ω))
-    (fun n => (FiniteMeasure.measurable_fun_prod.comp ((hμ n).prodMk (hμ n))).subtype_mk)
-    ((FiniteMeasure.measurable_fun_prod.comp (hν.prodMk hν)).subtype_mk) _ Q hb
-  filter_upwards [hconv] with ω hω
-  exact tendsto_finiteMeasure_prod hω hω
-
- 
-
-omit [LocallyCompactSpace X] in
-theorem ae_nullSingleton_of_average_prod_diagonal [LocallyCompactSpace X]
-    (W : Measure Ω) (ν : Ω → FiniteMeasure X) (hν : Measurable ν)
-    (hdiag : W.bind (fun ω => (ν ω).toMeasure.prod (ν ω).toMeasure) (diagonal X) = 0) :
-    ∀ᵐ ω ∂W, NullSingletonClass (ν ω).toMeasure := by
-  have hm : Measurable (fun ω => (ν ω).toMeasure.prod (ν ω).toMeasure) :=
-    FiniteMeasure.measurable_fun_prod.comp (hν.prodMk hν)
-  have hd : MeasurableSet (diagonal X) := isClosed_diagonal.measurableSet
-  rw [Measure.bind_apply hd hm.aemeasurable] at hdiag
-  have hae := (lintegral_eq_zero_iff' ((Measure.measurable_coe hd).comp hm).aemeasurable).mp hdiag
-  filter_upwards [hae] with ω hω
-  constructor
-  intro x
-  apply mul_self_eq_zero.mp
-  apply le_antisymm _ bot_le
-  have hsub : ({x} : Set X) ×ˢ {x} ⊆ diagonal X := by
-    rintro p ⟨hp,hq⟩
-    simp only [mem_singleton_iff] at hp hq
-    exact hp.trans hq.symm
-  simpa only [Measure.prod_prod, Pi.zero_apply, bot_eq_zero] using
-    (measure_mono (μ := (ν ω).toMeasure.prod (ν ω).toMeasure) hsub).trans_eq hω
-
- 
-
-omit [LocallyCompactSpace X] in
-lemma diagonal_null_of_ac_prod [LocallyCompactSpace X]
-    (V : Measure X) [SFinite V] [NullSingletonClass V]
-    (Q : Measure (X × X)) (hQ : Q ≪ V.prod V) : Q (diagonal X) = 0 := by
-  apply hQ
-  rw [Measure.prod_apply isClosed_diagonal.measurableSet]
-  have he (x : X) : Prod.mk x ⁻¹' diagonal X = {x} := by ext y; simp [diagonal, eq_comm]
-  simp_rw [he, measure_singleton, lintegral_zero]
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEKernel
-
- 
-
-theorem actual_mixed_density_product_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
-    ∃ C : ℝ, 0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ (x : Fin m → UpperPlane) (c : ℝ), 0<c → (∀ i, c≤2*(x i).1.im) →
-    ∀ q : Fin m → ℝ, (∀ i, 0<q i) → (∀ i, q i≤1) → ∀ t : Fin m → ℝ≥0∞,
-      (∫⁻ b, ∏ i, timeMassDensity a (q i) (t i) b (x i).1 ∂W) ≤
-        ENNReal.ofReal C * kernel (alpha a) c (fun i => (x i).1) := by
-  obtain ⟨C,hC,hCb⟩ := actual_density_product_bound a ha m
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW x c hc hcx q hq hq1 t
-  let z : Fin m → ℂ := fun i => (x i).1
-  have hz (i) : 0<(z i).im := (x i).2
-  apply (lintegral_mono (fun b => Finset.prod_le_prod
-    (fun i _ => timeMassDensity_le a (q i) (t i) b (z i)))).trans
-  by_cases hinj : Function.Injective z
-  · let cap := fun i => cappedDistance c z i
-    let target := fun i => 2*(z i).im*q i
-    have hcap (i) : 0<cap i := cappedDistance_pos hc z hinj i
-    have ht (i) : 0<target i ∧ target i≤2*(z i).im := by
-      refine ⟨mul_pos (mul_pos (by norm_num) (hz i)) (hq i),?_⟩
-      dsimp [target]
-      exact mul_le_of_le_one_right (mul_nonneg (by norm_num) (hz i).le) (hq1 i)
-    have hh := hCb W hW z hinj hz cap target
-      (fun i => ⟨hcap i,(cappedDistance_le c z i).trans (hcx i)⟩)
-      (fun i j hij => by simpa [dist_eq_norm] using cappedDistance_le_dist c z i j hij.symm) ht
-    have he (b : ExactSLEGenerator.Driver) (i) : massDensity a (q i) b (z i)=
-        ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i)) := by
-      simp [massDensity,hz i,target]
-    simp_rw [he]
-    have hk : ENNReal.ofReal (C*∏ i, cap i^(-alpha a))=ENNReal.ofReal C*kernel (alpha a) c z := by
-      rw [ENNReal.ofReal_mul hC.le,ENNReal.ofReal_prod_of_nonneg
-        (fun i _ => Real.rpow_nonneg (hcap i).le _)]
-      congr 1
-      unfold kernel
-      apply Finset.prod_congr rfl
-      intro i _
-      exact (ENNReal.ofReal_rpow_of_pos (hcap i)).symm
-    exact hh.trans_eq hk
-  · rw [kernel_top_of_not_injective (alpha a) c (alpha_pos ha) hc z hinj,
-      ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr hC))]
-    exact le_top
-
- 
-def pairKernel (α c : ℝ) (p : UpperPlane × UpperPlane) : ℝ≥0∞ :=
-  kernel α c (fun i => (MeasurableEquiv.finTwoArrow.symm p i).1)
-
-lemma pairKernel_measurable (α c : ℝ) (hα : 0≤α) : Measurable (pairKernel α c) :=
-  (measurable_kernel hα c).comp (by
-    apply Measurable.of_eval
-    intro i
-    exact measurable_subtype_coe.comp
-      ((measurable_pi_apply i).comp MeasurableEquiv.finTwoArrow.symm.measurable))
-
-lemma pairKernel_integrable_on_compact (α c : ℝ) (hα : 0<α) (hα1 : α<1)
-    (hc : 0<c) (K : Set UpperPlane) (hK : IsCompact K) :
-    (∫⁻ p in K ×ˢ K, pairKernel α c p ∂upperArea.prod upperArea) < ∞ := by
-  let F : ℂ × ℂ → ℝ≥0∞ := fun p => kernel α c (MeasurableEquiv.finTwoArrow.symm p)
-  have hF : Measurable F := (measurable_kernel hα.le c).comp
-    MeasurableEquiv.finTwoArrow.symm.measurable
-  have he (x y : UpperPlane) : pairKernel α c (x,y)=F (x.1,y.1) := by
-    unfold pairKernel F
-    congr 1
-    funext i
-    fin_cases i <;> rfl
-  rw [←Measure.prod_restrict,lintegral_prod _ (pairKernel_measurable α c hα.le).aemeasurable]
-  simp_rw [he]
-  have hsub (x : ℂ) : (∫⁻ y in K, F (x,y.1) ∂upperArea) =
-      ∫⁻ y in Subtype.val '' K, F (x,y) :=
-    setLIntegral_subtype upperPlane_open.measurableSet K (fun y => F (x,y))
-  simp_rw [hsub]
-  rw [show (∫⁻ x in K, ∫⁻ y in Subtype.val '' K, F (x.1,y) ∂volume ∂upperArea) =
-      ∫⁻ x in Subtype.val '' K, ∫⁻ y in Subtype.val '' K, F (x,y) ∂volume ∂volume from
-        setLIntegral_subtype (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet K
-          (fun x => ∫⁻y in Subtype.val '' K, F (x,y))]
-  rw [←lintegral_prod _ hF.aemeasurable,
-    (measurePreserving_finTwoArrow (volume.restrict (Subtype.val '' K))).symm.lintegral_comp
-      (measurable_kernel hα.le c)]
-  exact capped_distance_kernel_integral α c hα hα1 hc 2 (Subtype.val '' K)
-    (hK.image continuous_subtype_val).measurableSet (hK.image continuous_subtype_val).isBounded
-
-instance upperArea_atomless : NullSingletonClass upperArea := by
-  constructor
-  intro z
-  rw [upperArea,upper_embedding.comap_apply]
-  simp
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-
- 
-
-lemma average_density_prod_le (W : Measure Ω) [SFinite W]
-    (V : Measure X) [SFinite V] (ρ : Ω → X → ℝ≥0∞)
-    (hρ : Measurable (Function.uncurry ρ)) (q : X × X → ℝ≥0∞)
-    (hb : ∀ x y, (∫⁻ω, ρ ω x * ρ ω y ∂W) ≤ q (x,y)) :
-    W.bind (fun ω => (V.withDensity (ρ ω)).prod (V.withDensity (ρ ω))) ≤
-      (V.prod V).withDensity q := by
-  have hr : Measurable (fun p : Ω × (X × X) => ρ p.1 p.2.1 * ρ p.1 p.2.2) :=
-    (hρ.comp (measurable_fst.prodMk measurable_snd.fst)).mul
-      (hρ.comp (measurable_fst.prodMk measurable_snd.snd))
-  have he (ω : Ω) : (V.withDensity (ρ ω)).prod (V.withDensity (ρ ω)) =
-      (V.prod V).withDensity (fun p => ρ ω p.1 * ρ ω p.2) :=
-    prod_withDensity (hρ.comp (measurable_const.prodMk measurable_id))
-      (hρ.comp (measurable_const.prodMk measurable_id))
-  simp_rw [he]
-  apply Measure.le_iff.mpr
-  intro S hS
-  rw [Measure.bind_apply hS (measurable_withDensity hr).aemeasurable]
-  simp_rw [withDensity_apply _ hS]
-  rw [lintegral_lintegral_swap hr.aemeasurable]
-  exact lintegral_mono (fun p => hb p.1 p.2)
-
- 
-lemma average_localized_density_prod_le [TopologicalSpace X] [BorelSpace X]
-    (W : Measure Ω) [SFinite W] (V : Measure X) [SFinite V]
-    (ρ : Ω → X → ℝ≥0∞) (hρ : Measurable (Function.uncurry ρ))
-    (φ : C_c(X,ℝ≥0)) (q : X × X → ℝ≥0∞)
-    (hb : ∀ x∈tsupport φ,∀ y∈tsupport φ,(∫⁻ω,ρ ω x*ρ ω y∂W)≤q (x,y)) :
-    W.bind (fun ω => (V.withDensity (fun x => ρ ω x*(φ x:ℝ≥0∞))).prod
-      (V.withDensity (fun x => ρ ω x*(φ x:ℝ≥0∞)))) ≤
-        (V.prod V).withDensity (fun p => (φ p.1:ℝ≥0∞)*(φ p.2:ℝ≥0∞)*q p) := by
-  have hφ : Measurable (fun x => (φ x:ℝ≥0∞)) := φ.continuous.measurable.coe_nnreal_ennreal
-  refine average_density_prod_le W V (fun ω x => ρ ω x*(φ x:ℝ≥0∞))
-    (hρ.mul (hφ.comp measurable_snd)) _ ?_
-  intro x y
-  by_cases hx : x∈tsupport φ
-  · by_cases hy : y∈tsupport φ
-    · have hm : Measurable (fun ω => ρ ω x*ρ ω y) :=
-        (hρ.comp (measurable_id.prodMk measurable_const)).mul
-          (hρ.comp (measurable_id.prodMk measurable_const))
-      have he (ω : Ω) : (ρ ω x*(φ x:ℝ≥0∞))*(ρ ω y*(φ y:ℝ≥0∞)) =
-          ((φ x:ℝ≥0∞)*(φ y:ℝ≥0∞))*(ρ ω x*ρ ω y) := by ac_rfl
-      simp_rw [he]
-      rw [lintegral_const_mul _ hm]
-      exact mul_le_mul' le_rfl (hb x hx y hy)
-    · simp [image_eq_zero_of_notMem_tsupport hy]
-  · simp [image_eq_zero_of_notMem_tsupport hx]
-
- 
-lemma convex_density_moment_le (W : Measure Ω) {ι : Type*}
-    (s : Finset ι) (c : ι → ℝ≥0∞) (hc : ∑i∈s,c i=1)
-    (f g : ι → Ω → ℝ≥0∞) (hf : ∀ i∈s,Measurable (f i))
-    (hg : ∀ i∈s,Measurable (g i)) (Q : ℝ≥0∞)
-    (hb : ∀ i∈s,∀ j∈s,(∫⁻ω,f i ω*g j ω∂W)≤Q) :
-    (∫⁻ω,(∑i∈s,c i*f i ω)*(∑i∈s,c i*g i ω)∂W) ≤ Q := by
-  have he (ω : Ω) : (∑i∈s,c i*f i ω)*(∑i∈s,c i*g i ω) =
-      ∑i∈s,∑j∈s, (c i*c j)*(f i ω*g j ω) := by
-    rw [Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro i _
-    rw [Finset.mul_sum]
-    apply Finset.sum_congr rfl
-    intro j _
-    ac_rfl
-  simp_rw [he]
-  rw [lintegral_finsetSum s (f := fun i ω => ∑j∈s, (c i*c j)*(f i ω*g j ω)) (fun i hi =>
-    Finset.measurable_sum s (fun j hj => measurable_const.mul ((hf i hi).mul (hg j hj))))]
-  calc
-    _ = ∑i∈s,∑j∈s,(c i*c j)*(∫⁻ω,f i ω*g j ω∂W) := by
-      apply Finset.sum_congr rfl
-      intro i hi
-      rw [lintegral_finsetSum s (f := fun j ω => (c i*c j)*(f i ω*g j ω)) (fun j hj => measurable_const.mul ((hf i hi).mul (hg j hj)))]
-      apply Finset.sum_congr rfl
-      intro j hj
-      exact lintegral_const_mul _ ((hf i hi).mul (hg j hj))
-    _ ≤ ∑i∈s,∑j∈s,(c i*c j)*Q := Finset.sum_le_sum (fun i hi =>
-      Finset.sum_le_sum (fun j hj => mul_le_mul' le_rfl (hb i hi j hj)))
-    _ = Q := by simp only [←Finset.sum_mul,←Finset.mul_sum,hc,mul_one,one_mul]
-
-lemma measurable_localizedFinite [TopologicalSpace X] [BorelSpace X]
-    (μ : Ω → Measure X) [∀ ω, IsFiniteMeasureOnCompacts (μ ω)]
-    (hμ : Measurable μ) (φ : C_c(X,ℝ≥0)) :
-    Measurable (fun ω => localizedFinite (μ ω) φ) := by
-  apply Measurable.subtype_mk
-  apply Measure.measurable_of_measurable_coe
-  intro S hS
-  simp only [withDensity_apply _ hS]
-  simp_rw [←lintegral_indicator hS]
-  exact (Measure.measurable_lintegral
-    (φ.continuous.measurable.coe_nnreal_ennreal.indicator hS)).comp hμ
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEKernel SLEExactGauge.MassVersions
-
-def actualConvexMass (a : ℝ) (s : Finset ℕ) (c : ℕ → ℝ)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : Measure UpperPlane :=
-  ∑j∈s, ENNReal.ofReal (c j) • actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b
-
-def actualConvexDensity (a : ℝ) (s : Finset ℕ) (c : ℕ → ℝ)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : UpperPlane) : ℝ≥0∞ :=
-  ∑j∈s, ENNReal.ofReal (c j) * timeMassDensity a (Real.exp (-2*a*((j:ℝ)+1))) t b z.1
-
-lemma actualConvexDensity_measurable (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
-    (c : ℕ → ℝ) (t : ℝ≥0∞) :
-    Measurable (Function.uncurry (actualConvexDensity a s c t)) := by
-  apply Finset.measurable_sum
-  intro j _
-  exact measurable_const.mul ((timeMassDensity_measurable a ha _ (Real.exp_pos _)
-    (level_step_le_one a ha j) t).comp
-      (measurable_fst.prodMk (measurable_subtype_coe.comp measurable_snd)))
-
-lemma actualConvexMass_density (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
-    (c : ℕ → ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    actualConvexMass a s c t b = upperArea.withDensity (actualConvexDensity a s c t b) := by
-  apply Measure.ext
-  intro S hS
-  simp only [actualConvexMass,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul,
-    actualUpperMass,withDensity_apply _ hS,actualConvexDensity]
-  have hm (j : ℕ) : Measurable (fun z : UpperPlane =>
-      timeMassDensity a (Real.exp (-2*a*((j:ℝ)+1))) t b z.1) :=
-    (timeMassDensity_measurable a ha _ (Real.exp_pos _) (level_step_le_one a ha j) t).comp
-      (measurable_const.prodMk measurable_subtype_coe)
-  rw [lintegral_finsetSum s (f := fun j (z : UpperPlane) => ENNReal.ofReal (c j) *
-    timeMassDensity a (Real.exp (-2*a*((j:ℝ)+1))) t b z.1)
-    (fun j _ => measurable_const.mul (hm j))]
-  exact Finset.sum_congr rfl (fun j _ => (lintegral_const_mul _ (hm j)).symm)
-
-lemma actualConvexMass_measurable (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
-    (c : ℕ → ℝ) (t : ℝ≥0∞) : Measurable (actualConvexMass a s c t) := by
-  simp_rw [show actualConvexMass a s c t =
-    fun b => upperArea.withDensity (actualConvexDensity a s c t b) from
-      funext (actualConvexMass_density a ha s c t)]
-  exact measurable_withDensity (actualConvexDensity_measurable a ha s c t)
-
-lemma actualConvexMass_finiteOnCompacts (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
-    (c : ℕ → ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    IsFiniteMeasureOnCompacts (actualConvexMass a s c t b) := by
-  constructor
-  intro K hK
-  rw [actualConvexMass,Measure.finsetSum_apply]
-  apply ENNReal.sum_lt_top.mpr
-  intro j _
-  have := actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _)
-    (level_step_le_one a ha j) t b
-  exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top hK.measure_lt_top
-
-lemma actualConvexMass_integral (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
-    (c : ℕ → ℝ) (hc : ∀ j∈s,0≤c j) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
-    (∫z,f z∂actualConvexMass a s c t b) =
-      ∑j∈s,c j*(∫z,f z∂actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b) := by
-  let (j : ℕ) := actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _)
-    (level_step_le_one a ha j) t b
-  rw [actualConvexMass,integral_finsetSum_measure (fun j _ =>
-    f.integrable.smul_measure ENNReal.ofReal_ne_top)]
-  apply Finset.sum_congr rfl
-  intro j hj
-  rw [integral_smul_measure, ENNReal.toReal_ofReal (hc j hj),smul_eq_mul]
-
-lemma actual_pair_density_bound (a : ℝ) (ha : 1/4<a) :
-    ∃ C : ℝ,0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ (x y : UpperPlane) (c : ℝ),0<c → c≤2*x.1.im → c≤2*y.1.im →
-    ∀ (q r : ℝ),0<q → q≤1 → 0<r → r≤1 → ∀ (u v : ℝ≥0∞),
-      (∫⁻b,timeMassDensity a q u b x.1*timeMassDensity a r v b y.1∂W) ≤
-        ENNReal.ofReal C * pairKernel (alpha a) c (x,y) := by
-  obtain ⟨C,hC,hb⟩ := actual_mixed_density_product_bound a ha 2
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW x y c hc hx hy q r hq hq1 hr hr1 u v
-  have h := hb W hW (MeasurableEquiv.finTwoArrow.symm (x,y)) c hc
-    (by intro i; fin_cases i <;> assumption) ![q,r]
-    (by intro i; fin_cases i <;> assumption)
-    (by intro i; fin_cases i <;> assumption) ![u,v]
-  simpa only [Fin.prod_univ_two,Matrix.cons_val_zero,Matrix.cons_val_one,
-    pairKernel,MeasurableEquiv.finTwoArrow_symm_apply,Fin.cons_zero,Fin.cons_one] using h
-
- 
-
-theorem actualConvexDensity_pair_bound (a : ℝ) (ha : 1/4<a) :
-    ∃ C : ℝ,0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ (s : Finset ℕ) (c : ℕ → ℝ), (∀ j∈s,0≤c j) → (∑j∈s,c j=1) →
-    ∀ (t : ℝ≥0∞) (x y : UpperPlane) (d : ℝ),0<d → d≤2*x.1.im → d≤2*y.1.im →
-      (∫⁻b,actualConvexDensity a s c t b x*actualConvexDensity a s c t b y∂W) ≤
-        ENNReal.ofReal C * pairKernel (alpha a) d (x,y) := by
-  obtain ⟨C,hC,hb⟩ := actual_pair_density_bound a ha
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW s c hc hsum t x y d hd hx hy
-  have hc' : ∑ j∈s, ENNReal.ofReal (c j)=1 := by
-    rw [←ENNReal.ofReal_sum_of_nonneg hc,hsum,ENNReal.ofReal_one]
-  apply convex_density_moment_le W s (fun j => ENNReal.ofReal (c j)) hc'
-  · intro j _
-    exact (timeMassDensity_measurable a ha _ (Real.exp_pos _)
-      (level_step_le_one a ha j) t).comp (measurable_id.prodMk measurable_const)
-  · intro j _
-    exact (timeMassDensity_measurable a ha _ (Real.exp_pos _)
-      (level_step_le_one a ha j) t).comp (measurable_id.prodMk measurable_const)
-  · intro i _ j _
-    exact hb W hW x y d hd hx hy _ _ (Real.exp_pos _) (level_step_le_one a ha i)
-      (Real.exp_pos _) (level_step_le_one a ha j) t t
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.MassVersionsCore
+import OAI.Probability.SLEGauge.RadonKernels
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory Filter Set Topology TopologicalSpace
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge.MassVersions
+
+ 
+
+theorem tendsto_finiteMeasure_prod
+    {X Y I : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    [TopologicalSpace X] [TopologicalSpace Y]
+    [SecondCountableTopology X] [SecondCountableTopology Y]
+    [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y]
+    [OpensMeasurableSpace X] [OpensMeasurableSpace Y]
+    {l : Filter I} {μ : I → FiniteMeasure X} {ν : I → FiniteMeasure Y}
+    {μ₀ : FiniteMeasure X} {ν₀ : FiniteMeasure Y}
+    (hμ : Tendsto μ l (𝓝 μ₀)) (hν : Tendsto ν l (𝓝 ν₀)) :
+    Tendsto (fun i => (μ i).prod (ν i)) l (𝓝 (μ₀.prod ν₀)) := by
+  have hm : Tendsto (fun i => ((μ i).prod (ν i)).mass) l
+      (𝓝 (μ₀.mass * ν₀.mass)) := by
+    simp only [FiniteMeasure.mass_prod]
+    exact hμ.mass.mul hν.mass
+  by_cases hμ0 : μ₀ = 0
+  · simp only [hμ0, FiniteMeasure.zero_mass, zero_mul] at hm
+    simpa only [hμ0, FiniteMeasure.zero_prod] using
+      FiniteMeasure.tendsto_zero_of_tendsto_zero_mass hm
+  by_cases hν0 : ν₀ = 0
+  · simp only [hν0, FiniteMeasure.zero_mass, mul_zero] at hm
+    simpa only [hν0, FiniteMeasure.prod_zero] using
+      FiniteMeasure.tendsto_zero_of_tendsto_zero_mass hm
+  have : Nonempty X := by
+    by_contra h
+    have : IsEmpty X := not_nonempty_iff.mp h
+    exact hμ0 (Subtype.ext (Measure.eq_zero_of_isEmpty _))
+  have : Nonempty Y := by
+    by_contra h
+    have : IsEmpty Y := not_nonempty_iff.mp h
+    exact hν0 (Subtype.ext (Measure.eq_zero_of_isEmpty _))
+  have hp := ProbabilityMeasure.continuous_prod.tendsto
+    (μ₀.normalize, ν₀.normalize) |>.comp
+      ((μ₀.tendsto_normalize_of_tendsto hμ hμ0).prodMk_nhds
+        (ν₀.tendsto_normalize_of_tendsto hν hν0))
+  have he (m : FiniteMeasure X) (n : FiniteMeasure Y) :
+      m.prod n = (m.mass * n.mass) •
+        (m.normalize.prod n.normalize).toFiniteMeasure := by
+    apply Subtype.ext
+    change m.toMeasure.prod n.toMeasure =
+      (((m.mass * n.mass) • (m.normalize.prod n.normalize).toFiniteMeasure).toMeasure)
+    rw [FiniteMeasure.toMeasure_smul]
+    change m.toMeasure.prod n.toMeasure =
+      (m.mass * n.mass) • (m.normalize.toMeasure.prod n.normalize.toMeasure)
+    have hm' := congrArg FiniteMeasure.toMeasure m.self_eq_mass_smul_normalize
+    have hn' := congrArg FiniteMeasure.toMeasure n.self_eq_mass_smul_normalize
+    simp only [FiniteMeasure.toMeasure_smul,
+      ProbabilityMeasure.toMeasure_comp_toFiniteMeasure_eq_toMeasure] at hm' hn'
+    calc
+      _ = (m.mass • m.normalize.toMeasure).prod (n.mass • n.normalize.toMeasure) :=
+        congrArg₂ Measure.prod hm' hn'
+      _ = _ := by rw [Measure.prod_smul_left, Measure.prod_smul_right, smul_smul]
+  simp_rw [he]
+  exact (hμ.mass.mul hν.mass).smul
+    (ProbabilityMeasure.toFiniteMeasure_continuous.tendsto _ |>.comp hp)
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge.MassVersions
+
+variable {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
+
+noncomputable def localizedFinite (μ : Measure X) [IsFiniteMeasureOnCompacts μ]
+    (φ : C_c(X,ℝ≥0)) : FiniteMeasure X :=
+  ⟨μ.withDensity (fun x => φ x), isFiniteMeasure_withDensity (by
+    rw [lintegral_coe_eq_integral _ (by
+      have h : Integrable (fun x => φ.toReal x) μ := φ.toReal.integrable
+      simpa only [CompactlySupportedContinuousMap.toReal_apply] using h)]
+    exact ENNReal.ofReal_ne_top)⟩
+
+lemma localizedFinite_integral (μ : Measure X) [IsFiniteMeasureOnCompacts μ]
+    (φ : C_c(X,ℝ≥0)) (f : X → ℝ) :
+    (∫x, f x ∂localizedFinite μ φ) = ∫x, (φ x:ℝ)*f x ∂μ := by
+  exact integral_withDensity_eq_integral_smul φ.continuous.measurable f
+
+ 
+
+theorem localizedFinite_tendsto {I : Type*} {l : Filter I}
+    (μ : I → Measure X) (μ₀ : Measure X)
+    [∀ i, IsFiniteMeasureOnCompacts (μ i)] [IsFiniteMeasureOnCompacts μ₀]
+    (h : ∀ f : C_c(X,ℝ), Tendsto (fun i => ∫x,f x∂μ i) l (𝓝 (∫x,f x∂μ₀)))
+    (φ : C_c(X,ℝ≥0)) :
+    Tendsto (fun i => localizedFinite (μ i) φ) l (𝓝 (localizedFinite μ₀ φ)) := by
+  apply FiniteMeasure.tendsto_iff_forall_integral_tendsto.mpr
+  intro f
+  let F : C_c(X,ℝ) := f • φ.toReal
+  simpa only [localizedFinite_integral, F, CompactlySupportedContinuousMap.smulc_apply,
+    smul_eq_mul, CompactlySupportedContinuousMap.toReal_apply, mul_comm] using h F
+
+ 
+theorem measure_le_of_nonnegative_compact_tests
+    [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+    (μ ν : Measure X) [IsFiniteMeasureOnCompacts μ] [ν.OuterRegular]
+    (h : ∀ f : C_c(X,ℝ), (∀ x, 0≤f x) →
+      ENNReal.ofReal (∫x,f x∂μ) ≤ ∫⁻x, ENNReal.ofReal (f x)∂ν) : μ≤ν := by
+  have ho (U : Set X) (hU : IsOpen U) : μ U≤ν U := by
+    obtain ⟨f,hf,hf0,he⟩ := open_countable_tests U hU
+    rw [he μ inferInstance]
+    refine iSup_le (fun n => (h (f n) (fun x => (hf n x).1)).trans ?_)
+    calc
+      _ ≤ ∫⁻x, U.indicator (fun _ => (1:ℝ≥0∞)) x∂ν := by
+        apply lintegral_mono
+        intro x
+        by_cases hx : x∈U
+        · simp only [indicator_of_mem hx]
+          exact (ENNReal.ofReal_le_ofReal (hf n x).2).trans_eq (by simp)
+        · simp [hx, hf0 n hx]
+      _ = ν U := by simp [hU.measurableSet]
+  intro S
+  rw [S.measure_eq_iInf_isOpen ν]
+  exact le_iInf (fun U => le_iInf (fun hSU => le_iInf (fun hU =>
+    (measure_mono hSU).trans (ho U hU))))
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [BorelSpace X]
+
+ 
+
+theorem bind_le_of_weak_limit (W : Measure Ω)
+    (μ : ℕ → Ω → FiniteMeasure X) (ν : Ω → FiniteMeasure X)
+    (hμ : ∀ n, Measurable (μ n)) (hν : Measurable ν)
+    (hconv : ∀ᵐ ω ∂W, Tendsto (fun n => μ n ω) atTop (𝓝 (ν ω)))
+    (Q : Measure X) [IsFiniteMeasure Q] [Q.OuterRegular]
+    (hb : ∀ n, W.bind (fun ω => (μ n ω).toMeasure) ≤ Q) :
+    W.bind (fun ω => (ν ω).toMeasure) ≤ Q := by
+  have hm (n : ℕ) : Measurable (fun ω => (μ n ω).toMeasure) :=
+    measurable_subtype_coe.comp (hμ n)
+  have hn : Measurable (fun ω => (ν ω).toMeasure) := measurable_subtype_coe.comp hν
+  have ht (f : BoundedContinuousFunction X ℝ≥0) :
+      (∫⁻x, (f x : ℝ≥0∞) ∂W.bind (fun ω => (ν ω).toMeasure)) ≤ ∫⁻x, (f x : ℝ≥0∞) ∂Q := by
+    have hf : Measurable (fun x => (f x : ℝ≥0∞)) := f.continuous.measurable.coe_nnreal_ennreal
+    rw [Measure.lintegral_bind hn.aemeasurable hf.aemeasurable]
+    calc
+      _ = ∫⁻ω, liminf (fun n => ∫⁻x, (f x : ℝ≥0∞) ∂(μ n ω)) atTop ∂W := by
+        apply lintegral_congr_ae
+        filter_upwards [hconv] with ω hω
+        exact (FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp hω f).liminf_eq.symm
+      _ ≤ liminf (fun n => ∫⁻ω, ∫⁻x, (f x : ℝ≥0∞) ∂(μ n ω) ∂W) atTop :=
+        lintegral_liminf_le (fun n => (Measure.measurable_lintegral hf).comp (hm n))
+      _ ≤ ∫⁻x, (f x : ℝ≥0∞) ∂Q := by
+        have hh (n : ℕ) : (∫⁻ω, ∫⁻x, (f x : ℝ≥0∞) ∂(μ n ω) ∂W) ≤ ∫⁻x, (f x : ℝ≥0∞) ∂Q := by
+          rw [← Measure.lintegral_bind (hm n).aemeasurable hf.aemeasurable]
+          exact lintegral_mono' (hb n) le_rfl
+        simpa only [liminf_const] using liminf_le_liminf (f := atTop) (Eventually.of_forall hh)
+  have havg : IsFiniteMeasure (W.bind (fun ω => (ν ω).toMeasure)) := by
+    constructor
+    have h := ht (BoundedContinuousFunction.const X 1)
+    simp only [BoundedContinuousFunction.const_apply, ENNReal.coe_one, lintegral_one] at h
+    exact h.trans_lt (measure_lt_top Q univ)
+  apply measure_le_of_nonnegative_compact_tests
+  intro f hf
+  rw [ofReal_integral_eq_lintegral_ofReal f.integrable (Eventually.of_forall hf)]
+  exact ht f.nnrealPart.toBoundedContinuousFunction
+
+ 
+
+theorem average_prod_le_of_weak_limit
+    [TopologicalSpace.PseudoMetrizableSpace X]
+    (W : Measure Ω) (μ : ℕ → Ω → FiniteMeasure X) (ν : Ω → FiniteMeasure X)
+    (hμ : ∀ n, Measurable (μ n)) (hν : Measurable ν)
+    (hconv : ∀ᵐ ω ∂W, Tendsto (fun n => μ n ω) atTop (𝓝 (ν ω)))
+    (Q : Measure (X × X)) [IsFiniteMeasure Q] [Q.OuterRegular]
+    (hb : ∀ n, W.bind (fun ω => (μ n ω).toMeasure.prod (μ n ω).toMeasure) ≤ Q) :
+    W.bind (fun ω => (ν ω).toMeasure.prod (ν ω).toMeasure) ≤ Q := by
+  apply bind_le_of_weak_limit W
+    (fun n ω => (μ n ω).prod (μ n ω)) (fun ω => (ν ω).prod (ν ω))
+    (fun n => (FiniteMeasure.measurable_fun_prod.comp ((hμ n).prodMk (hμ n))).subtype_mk)
+    ((FiniteMeasure.measurable_fun_prod.comp (hν.prodMk hν)).subtype_mk) _ Q hb
+  filter_upwards [hconv] with ω hω
+  exact tendsto_finiteMeasure_prod hω hω
+
+ 
+
+omit [LocallyCompactSpace X] in
+theorem ae_nullSingleton_of_average_prod_diagonal [LocallyCompactSpace X]
+    (W : Measure Ω) (ν : Ω → FiniteMeasure X) (hν : Measurable ν)
+    (hdiag : W.bind (fun ω => (ν ω).toMeasure.prod (ν ω).toMeasure) (diagonal X) = 0) :
+    ∀ᵐ ω ∂W, NullSingletonClass (ν ω).toMeasure := by
+  have hm : Measurable (fun ω => (ν ω).toMeasure.prod (ν ω).toMeasure) :=
+    FiniteMeasure.measurable_fun_prod.comp (hν.prodMk hν)
+  have hd : MeasurableSet (diagonal X) := isClosed_diagonal.measurableSet
+  rw [Measure.bind_apply hd hm.aemeasurable] at hdiag
+  have hae := (lintegral_eq_zero_iff' ((Measure.measurable_coe hd).comp hm).aemeasurable).mp hdiag
+  filter_upwards [hae] with ω hω
+  constructor
+  intro x
+  apply mul_self_eq_zero.mp
+  apply le_antisymm _ bot_le
+  have hsub : ({x} : Set X) ×ˢ {x} ⊆ diagonal X := by
+    rintro p ⟨hp,hq⟩
+    simp only [mem_singleton_iff] at hp hq
+    exact hp.trans hq.symm
+  simpa only [Measure.prod_prod, Pi.zero_apply, bot_eq_zero] using
+    (measure_mono (μ := (ν ω).toMeasure.prod (ν ω).toMeasure) hsub).trans_eq hω
+
+ 
+
+omit [LocallyCompactSpace X] in
+lemma diagonal_null_of_ac_prod [LocallyCompactSpace X]
+    (V : Measure X) [SFinite V] [NullSingletonClass V]
+    (Q : Measure (X × X)) (hQ : Q ≪ V.prod V) : Q (diagonal X) = 0 := by
+  apply hQ
+  rw [Measure.prod_apply isClosed_diagonal.measurableSet]
+  have he (x : X) : Prod.mk x ⁻¹' diagonal X = {x} := by ext y; simp [diagonal, eq_comm]
+  simp_rw [he, measure_singleton, lintegral_zero]
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEKernel
+
+ 
+
+theorem actual_mixed_density_product_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
+    ∃ C : ℝ, 0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ (x : Fin m → UpperPlane) (c : ℝ), 0<c → (∀ i, c≤2*(x i).1.im) →
+    ∀ q : Fin m → ℝ, (∀ i, 0<q i) → (∀ i, q i≤1) → ∀ t : Fin m → ℝ≥0∞,
+      (∫⁻ b, ∏ i, timeMassDensity a (q i) (t i) b (x i).1 ∂W) ≤
+        ENNReal.ofReal C * kernel (alpha a) c (fun i => (x i).1) := by
+  obtain ⟨C,hC,hCb⟩ := actual_density_product_bound a ha m
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW x c hc hcx q hq hq1 t
+  let z : Fin m → ℂ := fun i => (x i).1
+  have hz (i) : 0<(z i).im := (x i).2
+  apply (lintegral_mono (fun b => Finset.prod_le_prod
+    (fun i _ => timeMassDensity_le a (q i) (t i) b (z i)))).trans
+  by_cases hinj : Function.Injective z
+  · let cap := fun i => cappedDistance c z i
+    let target := fun i => 2*(z i).im*q i
+    have hcap (i) : 0<cap i := cappedDistance_pos hc z hinj i
+    have ht (i) : 0<target i ∧ target i≤2*(z i).im := by
+      refine ⟨mul_pos (mul_pos (by norm_num) (hz i)) (hq i),?_⟩
+      dsimp [target]
+      exact mul_le_of_le_one_right (mul_nonneg (by norm_num) (hz i).le) (hq1 i)
+    have hh := hCb W hW z hinj hz cap target
+      (fun i => ⟨hcap i,(cappedDistance_le c z i).trans (hcx i)⟩)
+      (fun i j hij => by simpa [dist_eq_norm] using cappedDistance_le_dist c z i j hij.symm) ht
+    have he (b : ExactSLEGenerator.Driver) (i) : massDensity a (q i) b (z i)=
+        ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i)) := by
+      simp [massDensity,hz i,target]
+    simp_rw [he]
+    have hk : ENNReal.ofReal (C*∏ i, cap i^(-alpha a))=ENNReal.ofReal C*kernel (alpha a) c z := by
+      rw [ENNReal.ofReal_mul hC.le,ENNReal.ofReal_prod_of_nonneg
+        (fun i _ => Real.rpow_nonneg (hcap i).le _)]
+      congr 1
+      unfold kernel
+      apply Finset.prod_congr rfl
+      intro i _
+      exact (ENNReal.ofReal_rpow_of_pos (hcap i)).symm
+    exact hh.trans_eq hk
+  · rw [kernel_top_of_not_injective (alpha a) c (alpha_pos ha) hc z hinj,
+      ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr hC))]
+    exact le_top
+
+ 
+def pairKernel (α c : ℝ) (p : UpperPlane × UpperPlane) : ℝ≥0∞ :=
+  kernel α c (fun i => (MeasurableEquiv.finTwoArrow.symm p i).1)
+
+lemma pairKernel_measurable (α c : ℝ) (hα : 0≤α) : Measurable (pairKernel α c) :=
+  (measurable_kernel hα c).comp (by
+    apply Measurable.of_eval
+    intro i
+    exact measurable_subtype_coe.comp
+      ((measurable_pi_apply i).comp MeasurableEquiv.finTwoArrow.symm.measurable))
+
+lemma pairKernel_integrable_on_compact (α c : ℝ) (hα : 0<α) (hα1 : α<1)
+    (hc : 0<c) (K : Set UpperPlane) (hK : IsCompact K) :
+    (∫⁻ p in K ×ˢ K, pairKernel α c p ∂upperArea.prod upperArea) < ∞ := by
+  let F : ℂ × ℂ → ℝ≥0∞ := fun p => kernel α c (MeasurableEquiv.finTwoArrow.symm p)
+  have hF : Measurable F := (measurable_kernel hα.le c).comp
+    MeasurableEquiv.finTwoArrow.symm.measurable
+  have he (x y : UpperPlane) : pairKernel α c (x,y)=F (x.1,y.1) := by
+    unfold pairKernel F
+    congr 1
+    funext i
+    fin_cases i <;> rfl
+  rw [←Measure.prod_restrict,lintegral_prod _ (pairKernel_measurable α c hα.le).aemeasurable]
+  simp_rw [he]
+  have hsub (x : ℂ) : (∫⁻ y in K, F (x,y.1) ∂upperArea) =
+      ∫⁻ y in Subtype.val '' K, F (x,y) :=
+    setLIntegral_subtype upperPlane_open.measurableSet K (fun y => F (x,y))
+  simp_rw [hsub]
+  rw [show (∫⁻ x in K, ∫⁻ y in Subtype.val '' K, F (x.1,y) ∂volume ∂upperArea) =
+      ∫⁻ x in Subtype.val '' K, ∫⁻ y in Subtype.val '' K, F (x,y) ∂volume ∂volume from
+        setLIntegral_subtype (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet K
+          (fun x => ∫⁻y in Subtype.val '' K, F (x,y))]
+  rw [←lintegral_prod _ hF.aemeasurable,
+    (measurePreserving_finTwoArrow (volume.restrict (Subtype.val '' K))).symm.lintegral_comp
+      (measurable_kernel hα.le c)]
+  exact capped_distance_kernel_integral α c hα hα1 hc 2 (Subtype.val '' K)
+    (hK.image continuous_subtype_val).measurableSet (hK.image continuous_subtype_val).isBounded
+
+instance upperArea_atomless : NullSingletonClass upperArea := by
+  constructor
+  intro z
+  rw [upperArea,upper_embedding.comap_apply]
+  simp
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+
+ 
+
+lemma average_density_prod_le (W : Measure Ω) [SFinite W]
+    (V : Measure X) [SFinite V] (ρ : Ω → X → ℝ≥0∞)
+    (hρ : Measurable (Function.uncurry ρ)) (q : X × X → ℝ≥0∞)
+    (hb : ∀ x y, (∫⁻ω, ρ ω x * ρ ω y ∂W) ≤ q (x,y)) :
+    W.bind (fun ω => (V.withDensity (ρ ω)).prod (V.withDensity (ρ ω))) ≤
+      (V.prod V).withDensity q := by
+  have hr : Measurable (fun p : Ω × (X × X) => ρ p.1 p.2.1 * ρ p.1 p.2.2) :=
+    (hρ.comp (measurable_fst.prodMk measurable_snd.fst)).mul
+      (hρ.comp (measurable_fst.prodMk measurable_snd.snd))
+  have he (ω : Ω) : (V.withDensity (ρ ω)).prod (V.withDensity (ρ ω)) =
+      (V.prod V).withDensity (fun p => ρ ω p.1 * ρ ω p.2) :=
+    prod_withDensity (hρ.comp (measurable_const.prodMk measurable_id))
+      (hρ.comp (measurable_const.prodMk measurable_id))
+  simp_rw [he]
+  apply Measure.le_iff.mpr
+  intro S hS
+  rw [Measure.bind_apply hS (measurable_withDensity hr).aemeasurable]
+  simp_rw [withDensity_apply _ hS]
+  rw [lintegral_lintegral_swap hr.aemeasurable]
+  exact lintegral_mono (fun p => hb p.1 p.2)
+
+ 
+lemma average_localized_density_prod_le [TopologicalSpace X] [BorelSpace X]
+    (W : Measure Ω) [SFinite W] (V : Measure X) [SFinite V]
+    (ρ : Ω → X → ℝ≥0∞) (hρ : Measurable (Function.uncurry ρ))
+    (φ : C_c(X,ℝ≥0)) (q : X × X → ℝ≥0∞)
+    (hb : ∀ x∈tsupport φ,∀ y∈tsupport φ,(∫⁻ω,ρ ω x*ρ ω y∂W)≤q (x,y)) :
+    W.bind (fun ω => (V.withDensity (fun x => ρ ω x*(φ x:ℝ≥0∞))).prod
+      (V.withDensity (fun x => ρ ω x*(φ x:ℝ≥0∞)))) ≤
+        (V.prod V).withDensity (fun p => (φ p.1:ℝ≥0∞)*(φ p.2:ℝ≥0∞)*q p) := by
+  have hφ : Measurable (fun x => (φ x:ℝ≥0∞)) := φ.continuous.measurable.coe_nnreal_ennreal
+  refine average_density_prod_le W V (fun ω x => ρ ω x*(φ x:ℝ≥0∞))
+    (hρ.mul (hφ.comp measurable_snd)) _ ?_
+  intro x y
+  by_cases hx : x∈tsupport φ
+  · by_cases hy : y∈tsupport φ
+    · have hm : Measurable (fun ω => ρ ω x*ρ ω y) :=
+        (hρ.comp (measurable_id.prodMk measurable_const)).mul
+          (hρ.comp (measurable_id.prodMk measurable_const))
+      have he (ω : Ω) : (ρ ω x*(φ x:ℝ≥0∞))*(ρ ω y*(φ y:ℝ≥0∞)) =
+          ((φ x:ℝ≥0∞)*(φ y:ℝ≥0∞))*(ρ ω x*ρ ω y) := by ac_rfl
+      simp_rw [he]
+      rw [lintegral_const_mul _ hm]
+      exact mul_le_mul' le_rfl (hb x hx y hy)
+    · simp [image_eq_zero_of_notMem_tsupport hy]
+  · simp [image_eq_zero_of_notMem_tsupport hx]
+
+ 
+lemma convex_density_moment_le (W : Measure Ω) {ι : Type*}
+    (s : Finset ι) (c : ι → ℝ≥0∞) (hc : ∑i∈s,c i=1)
+    (f g : ι → Ω → ℝ≥0∞) (hf : ∀ i∈s,Measurable (f i))
+    (hg : ∀ i∈s,Measurable (g i)) (Q : ℝ≥0∞)
+    (hb : ∀ i∈s,∀ j∈s,(∫⁻ω,f i ω*g j ω∂W)≤Q) :
+    (∫⁻ω,(∑i∈s,c i*f i ω)*(∑i∈s,c i*g i ω)∂W) ≤ Q := by
+  have he (ω : Ω) : (∑i∈s,c i*f i ω)*(∑i∈s,c i*g i ω) =
+      ∑i∈s,∑j∈s, (c i*c j)*(f i ω*g j ω) := by
+    rw [Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro i _
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    ac_rfl
+  simp_rw [he]
+  rw [lintegral_finsetSum s (f := fun i ω => ∑j∈s, (c i*c j)*(f i ω*g j ω)) (fun i hi =>
+    Finset.measurable_sum s (fun j hj => measurable_const.mul ((hf i hi).mul (hg j hj))))]
+  calc
+    _ = ∑i∈s,∑j∈s,(c i*c j)*(∫⁻ω,f i ω*g j ω∂W) := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      rw [lintegral_finsetSum s (f := fun j ω => (c i*c j)*(f i ω*g j ω)) (fun j hj => measurable_const.mul ((hf i hi).mul (hg j hj)))]
+      apply Finset.sum_congr rfl
+      intro j hj
+      exact lintegral_const_mul _ ((hf i hi).mul (hg j hj))
+    _ ≤ ∑i∈s,∑j∈s,(c i*c j)*Q := Finset.sum_le_sum (fun i hi =>
+      Finset.sum_le_sum (fun j hj => mul_le_mul' le_rfl (hb i hi j hj)))
+    _ = Q := by simp only [←Finset.sum_mul,←Finset.mul_sum,hc,mul_one,one_mul]
+
+lemma measurable_localizedFinite [TopologicalSpace X] [BorelSpace X]
+    (μ : Ω → Measure X) [∀ ω, IsFiniteMeasureOnCompacts (μ ω)]
+    (hμ : Measurable μ) (φ : C_c(X,ℝ≥0)) :
+    Measurable (fun ω => localizedFinite (μ ω) φ) := by
+  apply Measurable.subtype_mk
+  apply Measure.measurable_of_measurable_coe
+  intro S hS
+  simp only [withDensity_apply _ hS]
+  simp_rw [←lintegral_indicator hS]
+  exact (Measure.measurable_lintegral
+    (φ.continuous.measurable.coe_nnreal_ennreal.indicator hS)).comp hμ
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEKernel SLEExactGauge.MassVersions
+
+def actualConvexMass (a : ℝ) (s : Finset ℕ) (c : ℕ → ℝ)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : Measure UpperPlane :=
+  ∑j∈s, ENNReal.ofReal (c j) • actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b
+
+def actualConvexDensity (a : ℝ) (s : Finset ℕ) (c : ℕ → ℝ)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : UpperPlane) : ℝ≥0∞ :=
+  ∑j∈s, ENNReal.ofReal (c j) * timeMassDensity a (Real.exp (-2*a*((j:ℝ)+1))) t b z.1
+
+lemma actualConvexDensity_measurable (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
+    (c : ℕ → ℝ) (t : ℝ≥0∞) :
+    Measurable (Function.uncurry (actualConvexDensity a s c t)) := by
+  apply Finset.measurable_sum
+  intro j _
+  exact measurable_const.mul ((timeMassDensity_measurable a ha _ (Real.exp_pos _)
+    (level_step_le_one a ha j) t).comp
+      (measurable_fst.prodMk (measurable_subtype_coe.comp measurable_snd)))
+
+lemma actualConvexMass_density (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
+    (c : ℕ → ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    actualConvexMass a s c t b = upperArea.withDensity (actualConvexDensity a s c t b) := by
+  apply Measure.ext
+  intro S hS
+  simp only [actualConvexMass,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul,
+    actualUpperMass,withDensity_apply _ hS,actualConvexDensity]
+  have hm (j : ℕ) : Measurable (fun z : UpperPlane =>
+      timeMassDensity a (Real.exp (-2*a*((j:ℝ)+1))) t b z.1) :=
+    (timeMassDensity_measurable a ha _ (Real.exp_pos _) (level_step_le_one a ha j) t).comp
+      (measurable_const.prodMk measurable_subtype_coe)
+  rw [lintegral_finsetSum s (f := fun j (z : UpperPlane) => ENNReal.ofReal (c j) *
+    timeMassDensity a (Real.exp (-2*a*((j:ℝ)+1))) t b z.1)
+    (fun j _ => measurable_const.mul (hm j))]
+  exact Finset.sum_congr rfl (fun j _ => (lintegral_const_mul _ (hm j)).symm)
+
+lemma actualConvexMass_measurable (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
+    (c : ℕ → ℝ) (t : ℝ≥0∞) : Measurable (actualConvexMass a s c t) := by
+  simp_rw [show actualConvexMass a s c t =
+    fun b => upperArea.withDensity (actualConvexDensity a s c t b) from
+      funext (actualConvexMass_density a ha s c t)]
+  exact measurable_withDensity (actualConvexDensity_measurable a ha s c t)
+
+lemma actualConvexMass_finiteOnCompacts (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
+    (c : ℕ → ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    IsFiniteMeasureOnCompacts (actualConvexMass a s c t b) := by
+  constructor
+  intro K hK
+  rw [actualConvexMass,Measure.finsetSum_apply]
+  apply ENNReal.sum_lt_top.mpr
+  intro j _
+  have := actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _)
+    (level_step_le_one a ha j) t b
+  exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top hK.measure_lt_top
+
+lemma actualConvexMass_integral (a : ℝ) (ha : 1/4<a) (s : Finset ℕ)
+    (c : ℕ → ℝ) (hc : ∀ j∈s,0≤c j) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
+    (∫z,f z∂actualConvexMass a s c t b) =
+      ∑j∈s,c j*(∫z,f z∂actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b) := by
+  let (j : ℕ) := actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _)
+    (level_step_le_one a ha j) t b
+  rw [actualConvexMass,integral_finsetSum_measure (fun j _ =>
+    f.integrable.smul_measure ENNReal.ofReal_ne_top)]
+  apply Finset.sum_congr rfl
+  intro j hj
+  rw [integral_smul_measure, ENNReal.toReal_ofReal (hc j hj),smul_eq_mul]
+
+lemma actual_pair_density_bound (a : ℝ) (ha : 1/4<a) :
+    ∃ C : ℝ,0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ (x y : UpperPlane) (c : ℝ),0<c → c≤2*x.1.im → c≤2*y.1.im →
+    ∀ (q r : ℝ),0<q → q≤1 → 0<r → r≤1 → ∀ (u v : ℝ≥0∞),
+      (∫⁻b,timeMassDensity a q u b x.1*timeMassDensity a r v b y.1∂W) ≤
+        ENNReal.ofReal C * pairKernel (alpha a) c (x,y) := by
+  obtain ⟨C,hC,hb⟩ := actual_mixed_density_product_bound a ha 2
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW x y c hc hx hy q r hq hq1 hr hr1 u v
+  have h := hb W hW (MeasurableEquiv.finTwoArrow.symm (x,y)) c hc
+    (by intro i; fin_cases i <;> assumption) ![q,r]
+    (by intro i; fin_cases i <;> assumption)
+    (by intro i; fin_cases i <;> assumption) ![u,v]
+  simpa only [Fin.prod_univ_two,Matrix.cons_val_zero,Matrix.cons_val_one,
+    pairKernel,MeasurableEquiv.finTwoArrow_symm_apply,Fin.cons_zero,Fin.cons_one] using h
+
+ 
+
+theorem actualConvexDensity_pair_bound (a : ℝ) (ha : 1/4<a) :
+    ∃ C : ℝ,0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ (s : Finset ℕ) (c : ℕ → ℝ), (∀ j∈s,0≤c j) → (∑j∈s,c j=1) →
+    ∀ (t : ℝ≥0∞) (x y : UpperPlane) (d : ℝ),0<d → d≤2*x.1.im → d≤2*y.1.im →
+      (∫⁻b,actualConvexDensity a s c t b x*actualConvexDensity a s c t b y∂W) ≤
+        ENNReal.ofReal C * pairKernel (alpha a) d (x,y) := by
+  obtain ⟨C,hC,hb⟩ := actual_pair_density_bound a ha
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW s c hc hsum t x y d hd hx hy
+  have hc' : ∑ j∈s, ENNReal.ofReal (c j)=1 := by
+    rw [←ENNReal.ofReal_sum_of_nonneg hc,hsum,ENNReal.ofReal_one]
+  apply convex_density_moment_le W s (fun j => ENNReal.ofReal (c j)) hc'
+  · intro j _
+    exact (timeMassDensity_measurable a ha _ (Real.exp_pos _)
+      (level_step_le_one a ha j) t).comp (measurable_id.prodMk measurable_const)
+  · intro j _
+    exact (timeMassDensity_measurable a ha _ (Real.exp_pos _)
+      (level_step_le_one a ha j) t).comp (measurable_id.prodMk measurable_const)
+  · intro i _ j _
+    exact hb W hW x y d hd hx hy _ _ (Real.exp_pos _) (level_step_le_one a ha i)
+      (Real.exp_pos _) (level_step_le_one a ha j) t t
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/FiniteVariationMartingale.lean b/lean/OAI/Probability/SLEGauge/FiniteVariationMartingale.lean
index 45713fe853eaa792a3d2b75fdda321341372b496..ab6f985a4f71ff7966c549677e136743ede67905 100644
--- a/lean/OAI/Probability/SLEGauge/FiniteVariationMartingale.lean
+++ b/lean/OAI/Probability/SLEGauge/FiniteVariationMartingale.lean
@@ -1,536 +1,536 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.PostStopPotential
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.FiniteVariationMartingale
-open MeshEnergy TaylorVariation
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-variable {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-variable {M : ℝ≥0 → Ω → ℝ}
-
- 
-
-lemma integral_weighted_increment (hM : Martingale M ℱ P)
-    {s t : ℝ≥0} (hst : s≤t) {H : Ω → ℝ}
-    (hm : StronglyMeasurable[ℱ s] H) {L : ℝ}
-    (hb : ∀ᵐ ω ∂P, ‖H ω‖≤L) :
-    ∫ ω, H ω*(M t ω-M s ω) ∂P = 0 := by
-  have hi := (hM.integrable t).sub (hM.integrable s)
-  have hprod : Integrable (H*(M t-M s)) P :=
-    hi.bdd_mul (hm.mono (ℱ.le s)).aestronglyMeasurable hb
-  have hzero : P[M t-M s | ℱ s] =ᵐ[P] 0 := by
-    have hc := condExp_sub (hM.integrable t) (hM.integrable s) (ℱ s)
-    have hs := condExp_of_stronglyMeasurable (ℱ.le s) (hM.stronglyMeasurable s)
-      (hM.integrable s)
-    filter_upwards [hc,hM.condExp_ae_eq hst] with ω he ht
-    simp only [Pi.sub_apply,hs,ht,sub_self] at he
-    exact he
-  have hp := condExp_mul_of_stronglyMeasurable_left hm hprod hi
-  calc
-    _ = ∫ ω, P[H*(M t-M s) | ℱ s] ω ∂P := (integral_condExp (ℱ.le s)).symm
-    _ = ∫ ω, (0:ℝ) ∂P := by
-      apply integral_congr_ae
-      filter_upwards [hp,hzero] with ω he hz
-      simpa [hz] using he
-    _ = 0 := integral_zero _ _
-
-omit [IsFiniteMeasure P] in
-lemma integrable_detector [IsFiniteMeasure P] (hM : Martingale M ℱ P) (t : ℝ≥0) :
-    Integrable (fun ω => detector (M t ω)) P :=
-  (hM.integrable t).norm.mono'
-    (lipschitz_detector.continuous.comp_aestronglyMeasurable
-      (hM.integrable t).aestronglyMeasurable)
-    (ae_of_all _ fun ω => abs_detector_le (M t ω))
-
- 
-
-theorem ae_eq_zero_at (hM : Martingale M ℱ P)
-    (hz : M 0 =ᵐ[P] 0) (C : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
-    (hc : ∀ᵐ ω ∂P, ContinuousOn (fun t => C t ω) (Icc 0 T))
-    (hcontrol : ∀ᵐ ω ∂P, ∀ s t, s≤t → t≤T → |M t ω-M s ω|≤C t ω-C s ω)
-    (hint : Integrable (fun ω => C T ω-C 0 ω) P) : M T =ᵐ[P] 0 := by
-  let H (n i : ℕ) (ω : Ω) := detectorDeriv (M (grid T n i) ω)
-  have hm (n i : ℕ) : StronglyMeasurable[ℱ (grid T n i)] (H n i) :=
-    lipschitz_detectorDeriv.continuous.comp_stronglyMeasurable (hM.stronglyMeasurable _)
-  have hb (n i : ℕ) : ∀ᵐ ω ∂P, ‖H n i ω‖≤1 :=
-    ae_of_all _ fun ω => abs_detectorDeriv_le _
-  have hi (n i : ℕ) : Integrable
-      (fun ω => H n i ω * (M (grid T n (i+1)) ω-M (grid T n i) ω)) P :=
-    ((hM.integrable _).sub (hM.integrable _)).bdd_mul
-      ((hm n i).mono (ℱ.le _)).aestronglyMeasurable (hb n i)
-  have hzero (n i : ℕ) : (∫ ω, H n i ω *
-      (M (grid T n (i+1)) ω-M (grid T n i) ω) ∂P)=0 :=
-    integral_weighted_increment hM (grid_mono T n (Nat.le_succ i)) (hm n i) (hb n i)
-  have hsint (n : ℕ) : Integrable (fun ω => ∑ i ∈ Finset.range (n+1), H n i ω *
-      (M (grid T n (i+1)) ω-M (grid T n i) ω)) P :=
-    integrable_finsetSum _ fun i _ => hi n i
-  have hsz (n : ℕ) : (∫ ω, ∑ i ∈ Finset.range (n+1), H n i ω *
-      (M (grid T n (i+1)) ω-M (grid T n i) ω) ∂P)=0 := by
-    rw [integral_finsetSum _ (fun i _ => hi n i)]
-    simp only [hzero,Finset.sum_const_zero]
-  let F (n : ℕ) (ω : Ω) := error detector detectorDeriv (fun t => M t ω) T n
-  have hFi (n : ℕ) : Integrable (F n) P :=
-    ((integrable_detector hM T).sub (integrable_detector hM 0)).sub (hsint n)
-  have hz0 : (∫ ω, detector (M 0 ω) ∂P)=0 := by
-    calc
-      _ = ∫ ω, (0:ℝ) ∂P := integral_congr_ae (hz.mono fun ω he => by simp [he])
-      _ = 0 := integral_zero _ _
-  have hFint (n : ℕ) : (∫ ω, F n ω ∂P) = ∫ ω, detector (M T ω) ∂P := by
-    change (∫ ω, detector (M T ω)-detector (M 0 ω)-
-      ∑ i ∈ Finset.range (n+1), H n i ω *
-        (M (grid T n (i+1)) ω-M (grid T n i) ω) ∂P)=_
-    have hh := integral_sub ((integrable_detector hM T).sub (integrable_detector hM 0)) (hsint n)
-    simp only [Pi.sub_apply] at hh
-    rw [hh, integral_sub (integrable_detector hM T) (integrable_detector hM 0),
-      hz0,hsz,sub_zero,sub_zero]
-  have hFb (n : ℕ) : ∀ᵐ ω ∂P, ‖F n ω‖≤2*(C T ω-C 0 ω) := by
-    filter_upwards [hcontrol] with ω hω
-    simpa [F] using abs_error_le_variation detector detectorDeriv hasDerivAt_detector
-      (L := 1) abs_detectorDeriv_le (fun t => M t ω) (fun t => C t ω) T hω n
-  have hFl : ∀ᵐ ω ∂P, Tendsto (fun n => F n ω) atTop (𝓝 0) := by
-    filter_upwards [hc,hcontrol] with ω hcω hω
-    exact tendsto_error_zero detector detectorDeriv hasDerivAt_detector
-      lipschitz_detectorDeriv (fun t => M t ω) (fun t => C t ω) T hcω hω
-  have hlim : Tendsto (fun n => ∫ ω, F n ω ∂P) atTop (𝓝 0) := by
-    simpa only [integral_zero] using tendsto_integral_of_dominated_convergence
-      (fun ω => 2*(C T ω-C 0 ω)) (fun n => (hFi n).aestronglyMeasurable)
-      (hint.const_mul 2) hFb hFl
-  simp only [hFint] at hlim
-  have he : (∫ ω, detector (M T ω) ∂P)=0 :=
-    tendsto_nhds_unique tendsto_const_nhds hlim
-  have hdet := (integral_eq_zero_iff_of_nonneg
-    (fun ω => detector_nonneg (M T ω)) (integrable_detector hM T)).mp he
-  filter_upwards [hdet] with ω hω
-  exact (detector_eq_zero_iff _).mp hω
-
- 
-theorem ae_all_eq_zero (hM : Martingale M ℱ P) (hz : M 0 =ᵐ[P] 0)
-    (hmc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω))
-    (C : ℝ≥0 → Ω → ℝ) (hc : ∀ᵐ ω ∂P, Continuous (fun t => C t ω))
-    (hcontrol : ∀ᵐ ω ∂P, ∀ s t, s≤t → |M t ω-M s ω|≤C t ω-C s ω)
-    (hint : ∀ T, Integrable (fun ω => C T ω-C 0 ω) P) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, M t ω=0 := by
-  obtain ⟨D,hD,hd⟩ := TopologicalSpace.exists_countable_dense ℝ≥0
-  have heq : ∀ᵐ ω ∂P, ∀ t∈D, M t ω=0 := by
-    apply (ae_ball_iff hD).mpr
-    intro t _
-    exact ae_eq_zero_at hM hz C t (hc.mono fun _ h => h.continuousOn)
-      (hcontrol.mono fun _ h s u hsu _ => h s u hsu) (hint t)
-  filter_upwards [heq,hmc] with ω hω hmω
-  have he : (fun t => M t ω)=(fun _ => (0:ℝ)) :=
-    Continuous.ext_on hd hmω continuous_const hω
-  exact fun t => congrFun he t
-
-end SLEExactGauge.FiniteVariationMartingale
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CompensatorUniqueness
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-variable {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-variable {A B : ℝ≥0 → Ω → ℝ} {TA TB : Ω → ℝ}
-
- 
-
-omit [IsFiniteMeasure P] in
-lemma integrable_at [IsFiniteMeasure P] (hm : StronglyAdapted ℱ A)
-    (hz : A 0 =ᵐ[P] 0) (hmono : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
-    (hdom : ∀ᵐ ω ∂P, ∀ t, A t ω ≤ TA ω) (hi : Integrable TA P) (t : ℝ≥0) :
-    Integrable (A t) P := by
-  apply hi.norm.mono' ((hm t).mono (ℱ.le t)).aestronglyMeasurable
-  filter_upwards [hz,hmono,hdom] with ω hzω hmω hdω
-  have hnonneg : 0 ≤ A t ω := by simpa only [hzω,Pi.zero_apply] using hmω (show 0 ≤ t from bot_le)
-  simpa only [Real.norm_eq_abs,abs_of_nonneg hnonneg] using
-    (hdω t).trans (le_abs_self (TA ω))
-
- 
-
-lemma martingale_sub (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B)
-    (hAi : ∀ t, Integrable (A t) P) (hBi : ∀ t, Integrable (B t) P)
-    (hTA : Integrable TA P) (hTB : Integrable TB P)
-    (hpot : ∀ t, P[TA - A t | ℱ t] =ᵐ[P] P[TB - B t | ℱ t]) :
-    Martingale (A-B) ℱ P := by
-  apply (martingale_condExp (TA-TB) ℱ P).congr (hAm.sub hBm)
-  intro t
-  have ha := condExp_of_stronglyMeasurable (ℱ.le t) (hAm t) (hAi t)
-  have hb := condExp_of_stronglyMeasurable (ℱ.le t) (hBm t) (hBi t)
-  filter_upwards [condExp_sub hTA hTB (ℱ t),condExp_sub hTA (hAi t) (ℱ t),
-    condExp_sub hTB (hBi t) (ℱ t),hpot t] with ω hab hae hbe hp
-  simp only [Pi.sub_apply,ha,hb] at hab hae hbe ⊢
-  linarith
-
- 
-
-theorem ae_all_eq (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B)
-    (hAz : A 0 =ᵐ[P] 0) (hBz : B 0 =ᵐ[P] 0)
-    (hAc : ∀ᵐ ω ∂P, Continuous (fun t => A t ω))
-    (hBc : ∀ᵐ ω ∂P, Continuous (fun t => B t ω))
-    (hAo : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
-    (hBo : ∀ᵐ ω ∂P, Monotone (fun t => B t ω))
-    (hAd : ∀ᵐ ω ∂P, ∀ t, A t ω ≤ TA ω)
-    (hBd : ∀ᵐ ω ∂P, ∀ t, B t ω ≤ TB ω)
-    (hTA : Integrable TA P) (hTB : Integrable TB P)
-    (hpot : ∀ t, P[TA-A t | ℱ t] =ᵐ[P] P[TB-B t | ℱ t]) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, A t ω=B t ω := by
-  have hAi := integrable_at hAm hAz hAo hAd hTA
-  have hBi := integrable_at hBm hBz hBo hBd hTB
-  have hM := martingale_sub hAm hBm hAi hBi hTA hTB hpot
-  have hz : (A-B) 0 =ᵐ[P] 0 := by
-    filter_upwards [hAz,hBz] with ω ha hb
-    simp only [Pi.sub_apply,ha,hb,sub_self,Pi.zero_apply]
-  have hmc : ∀ᵐ ω ∂P, Continuous (fun t => (A-B) t ω) := by
-    filter_upwards [hAc,hBc] with ω ha hb
-    exact ha.sub hb
-  have hc : ∀ᵐ ω ∂P, Continuous (fun t => (A+B) t ω) := by
-    filter_upwards [hAc,hBc] with ω ha hb
-    exact ha.add hb
-  have hcontrol : ∀ᵐ ω ∂P, ∀ s t, s≤t →
-      |(A-B) t ω-(A-B) s ω|≤(A+B) t ω-(A+B) s ω := by
-    filter_upwards [hAo,hBo] with ω ha hb s t hst
-    have hat := ha hst
-    have hbt := hb hst
-    simp only [Pi.sub_apply,Pi.add_apply]
-    exact abs_le.mpr ⟨by linarith,by linarith⟩
-  have hint (T : ℝ≥0) : Integrable (fun ω => (A+B) T ω-(A+B) 0 ω) P :=
-    ((hAi T).add (hBi T)).sub ((hAi 0).add (hBi 0))
-  have he := FiniteVariationMartingale.ae_all_eq_zero hM hz hmc (A+B) hc hcontrol hint
-  filter_upwards [he] with ω hω t
-  exact sub_eq_zero.mp (hω t)
-
- 
-omit [IsFiniteMeasure P] in
-theorem ae_terminal_eq_of_ae_all_eq [IsFiniteMeasure P]
-    (hab : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, A t ω=B t ω)
-    (hAl : ∀ᵐ ω ∂P, Tendsto (fun t => A t ω) atTop (𝓝 (TA ω)))
-    (hBl : ∀ᵐ ω ∂P, Tendsto (fun t => B t ω) atTop (𝓝 (TB ω))) :
-    TA =ᵐ[P] TB := by
-  filter_upwards [hab,hAl,hBl] with ω he ha hb
-  simp only [he] at ha
-  exact tendsto_nhds_unique ha hb
-
-end SLEExactGauge.CompensatorUniqueness
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CompensatorUniqueness
-open MassVersions
-variable {Ω X : Type*} [mΩ : MeasurableSpace Ω]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-    (μ ν : ℝ≥0 → Ω → Measure X) (Tμ Tν : Ω → Measure X)
-    [∀ t ω, (μ t ω).Regular] [∀ t ω, (ν t ω).Regular]
-    [∀ ω, (Tμ ω).Regular] [∀ ω, (Tν ω).Regular]
-
- 
-theorem ae_all_measure_eq
-    (hμm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂μ t ω))
-    (hνm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂ν t ω))
-    (hμ0 : μ 0 =ᵐ[P] 0) (hν0 : ν 0 =ᵐ[P] 0)
-    (hμc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂μ t ω))
-    (hνc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂ν t ω))
-    (hμo : ∀ᵐ ω ∂P, Monotone (fun t => μ t ω))
-    (hνo : ∀ᵐ ω ∂P, Monotone (fun t => ν t ω))
-    (hμd : ∀ᵐ ω ∂P, ∀ t, μ t ω≤Tμ ω)
-    (hνd : ∀ᵐ ω ∂P, ∀ t, ν t ω≤Tν ω)
-    (hμi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tμ ω) P)
-    (hνi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tν ω) P)
-    (hpot : ∀ f : C_c(X,ℝ), (∀ x, 0≤f x) → ∀ t,
-      P[(fun ω => (∫x,f x∂Tμ ω)-(∫x,f x∂μ t ω)) | ℱ t] =ᵐ[P]
-        P[(fun ω => (∫x,f x∂Tν ω)-(∫x,f x∂ν t ω)) | ℱ t]) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω := by
-  have hpos (f : C_c(X,ℝ)) (hf : ∀ x,0≤f x) :
-      ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (∫x,f x∂μ t ω)=∫x,f x∂ν t ω := by
-    have hμz : (fun ω => ∫x,f x∂μ 0 ω) =ᵐ[P] 0 := by
-      filter_upwards [hμ0] with ω hω
-      simp only [hω,Pi.zero_apply,integral_zero_measure]
-    have hνz : (fun ω => ∫x,f x∂ν 0 ω) =ᵐ[P] 0 := by
-      filter_upwards [hν0] with ω hω
-      simp only [hω,Pi.zero_apply,integral_zero_measure]
-    have hμo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂μ t ω) := by
-      filter_upwards [hμo] with ω hω s t hst
-      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
-    have hνo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂ν t ω) := by
-      filter_upwards [hνo] with ω hω s t hst
-      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
-    have hμd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂μ t ω)≤∫x,f x∂Tμ ω := by
-      filter_upwards [hμd] with ω hω t
-      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
-    have hνd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂ν t ω)≤∫x,f x∂Tν ω := by
-      filter_upwards [hνd] with ω hω t
-      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
-    exact ae_all_eq (hμm f) (hνm f) hμz hνz (hμc f) (hνc f)
-      hμo' hνo' hμd' hνd' (hμi f) (hνi f) (hpot f hf)
-  apply ae_all_measure_eq_of_test_eq P μ ν
-  intro i
-  let f : C_c(X,ℝ) := determiningTests i
-  filter_upwards [hpos (positiveTest f) (positiveTest_nonneg f),
-    hpos (positiveTest (-f)) (positiveTest_nonneg (-f))] with ω hp hn t
-  calc
-    (∫x,f x∂μ t ω) = (∫x,positiveTest f x∂μ t ω)-(∫x,positiveTest (-f) x∂μ t ω) :=
-      (integral_positiveTest_sub f _).symm
-    _ = (∫x,positiveTest f x∂ν t ω)-(∫x,positiveTest (-f) x∂ν t ω) := by rw [hp t,hn t]
-    _ = (∫x,f x∂ν t ω) := integral_positiveTest_sub f _
-
- 
-omit [IsFiniteMeasure P] [∀ time sample, (μ time sample).Regular]
-    [∀ time sample, (ν time sample).Regular] in
-theorem ae_terminal_measure_eq [IsFiniteMeasure P]
-    [∀ time sample, (μ time sample).Regular] [∀ time sample, (ν time sample).Regular]
-    (he : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω)
-    (hμl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
-      Tendsto (fun t : ℝ≥0 => ∫x,f x∂μ t ω) atTop (𝓝 (∫x,f x∂Tμ ω)))
-    (hνl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
-      Tendsto (fun t : ℝ≥0 => ∫x,f x∂ν t ω) atTop (𝓝 (∫x,f x∂Tν ω))) :
-    Tμ =ᵐ[P] Tν := by
-  have hi (i : ℕ) : (fun ω => ∫x,determiningTests i x∂Tμ ω) =ᵐ[P]
-      (fun ω => ∫x,determiningTests i x∂Tν ω) := by
-    filter_upwards [he,hμl (determiningTests i),hνl (determiningTests i)] with ω hω hμω hνω
-    simp only [hω] at hμω
-    exact tendsto_nhds_unique hμω hνω
-  filter_upwards [ae_all_iff.mpr hi] with ω hω
-  exact measure_eq_of_determiningTests _ _ hω
-
- 
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X]
-    [IsFiniteMeasure P] [∀ time sample, (μ time sample).Regular]
-    [∀ time sample, (ν time sample).Regular] in
-theorem ae_all_random_weight_eq [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [BorelSpace X] [IsFiniteMeasure P]
-    [∀ time sample, (μ time sample).Regular] [∀ time sample, (ν time sample).Regular]
-    (he : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω) :
-    ∀ᵐ ω ∂P, ∀ (t : Ω → ℝ≥0) (w : Ω → X → ℝ≥0∞),
-      (∫⁻x,w ω x∂μ (t ω) ω)=∫⁻x,w ω x∂ν (t ω) ω := by
-  filter_upwards [he] with ω hω t w
-  rw [hω (t ω)]
-
-end SLEExactGauge.CompensatorUniqueness
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CompensatorUniqueness
-open MassVersions
-variable {Ω X : Type*} [mΩ : MeasurableSpace Ω]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
-
- 
-
-theorem ae_all_measure_eq_of_ae_regular
-    (μ ν : ℝ≥0 → Ω → Measure X) (Tμ Tν : Ω → Measure X)
-    (hμr : ∀ᵐ ω ∂P, ∀t, (μ t ω).Regular)
-    (hνr : ∀ᵐ ω ∂P, ∀t, (ν t ω).Regular)
-    (hTμr : ∀ᵐ ω ∂P, (Tμ ω).Regular)
-    (hTνr : ∀ᵐ ω ∂P, (Tν ω).Regular)
-    (hμm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂μ t ω))
-    (hνm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂ν t ω))
-    (hμ0 : μ 0 =ᵐ[P] 0) (hν0 : ν 0 =ᵐ[P] 0)
-    (hμc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂μ t ω))
-    (hνc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂ν t ω))
-    (hμo : ∀ᵐ ω ∂P, Monotone (fun t => μ t ω))
-    (hνo : ∀ᵐ ω ∂P, Monotone (fun t => ν t ω))
-    (hμd : ∀ᵐ ω ∂P, ∀ t, μ t ω≤Tμ ω)
-    (hνd : ∀ᵐ ω ∂P, ∀ t, ν t ω≤Tν ω)
-    (hμi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tμ ω) P)
-    (hνi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tν ω) P)
-    (hpot : ∀ f : C_c(X,ℝ), (∀ x, 0≤f x) → ∀ t,
-      P[(fun ω => (∫x,f x∂Tμ ω)-(∫x,f x∂μ t ω)) | ℱ t] =ᵐ[P]
-        P[(fun ω => (∫x,f x∂Tν ω)-(∫x,f x∂ν t ω)) | ℱ t]) :
-    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω := by
-  have hpos (f : C_c(X,ℝ)) (hf : ∀ x,0≤f x) :
-      ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (∫x,f x∂μ t ω)=∫x,f x∂ν t ω := by
-    have hμz : (fun ω => ∫x,f x∂μ 0 ω) =ᵐ[P] 0 := by
-      filter_upwards [hμ0] with ω hω
-      simp only [hω,Pi.zero_apply,integral_zero_measure]
-    have hνz : (fun ω => ∫x,f x∂ν 0 ω) =ᵐ[P] 0 := by
-      filter_upwards [hν0] with ω hω
-      simp only [hω,Pi.zero_apply,integral_zero_measure]
-    have hμo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂μ t ω) := by
-      filter_upwards [hμo,hμr] with ω hω hr s t hst
-      let := hr t
-      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
-    have hνo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂ν t ω) := by
-      filter_upwards [hνo,hνr] with ω hω hr s t hst
-      let := hr t
-      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
-    have hμd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂μ t ω)≤∫x,f x∂Tμ ω := by
-      filter_upwards [hμd,hTμr] with ω hω hr t
-      let := hr
-      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
-    have hνd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂ν t ω)≤∫x,f x∂Tν ω := by
-      filter_upwards [hνd,hTνr] with ω hω hr t
-      let := hr
-      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
-    exact ae_all_eq (hμm f) (hνm f) hμz hνz (hμc f) (hνc f)
-      hμo' hνo' hμd' hνd' (hμi f) (hνi f) (hpot f hf)
-  have he (i : ℕ) : ∀ᵐ ω ∂P, ∀t,
-      (∫x,determiningTests i x∂μ t ω)=∫x,determiningTests i x∂ν t ω := by
-    let f : C_c(X,ℝ) := determiningTests i
-    filter_upwards [hμr,hνr,hpos (positiveTest f) (positiveTest_nonneg f),
-      hpos (positiveTest (-f)) (positiveTest_nonneg (-f))] with ω hr hs hp hn t
-    let := hr t
-    let := hs t
-    calc
-      (∫x,f x∂μ t ω) = (∫x,positiveTest f x∂μ t ω)-(∫x,positiveTest (-f) x∂μ t ω) :=
-        (integral_positiveTest_sub f _).symm
-      _ = (∫x,positiveTest f x∂ν t ω)-(∫x,positiveTest (-f) x∂ν t ω) := by rw [hp t,hn t]
-      _ = (∫x,f x∂ν t ω) := integral_positiveTest_sub f _
-  filter_upwards [ae_all_iff.mpr he,hμr,hνr] with ω hω hr hs t
-  let := hr t
-  let := hs t
-  exact measure_eq_of_determiningTests _ _ (fun i => hω i t)
-
- 
-omit [IsFiniteMeasure P] in
-theorem ae_terminal_measure_eq_of_ae_regular [IsFiniteMeasure P]
-    (μ ν : ℝ≥0 → Ω → Measure X) (Tμ Tν : Ω → Measure X)
-    (hTμr : ∀ᵐ ω ∂P, (Tμ ω).Regular)
-    (hTνr : ∀ᵐ ω ∂P, (Tν ω).Regular)
-    (he : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω)
-    (hμl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
-      Tendsto (fun t : ℝ≥0 => ∫x,f x∂μ t ω) atTop (𝓝 (∫x,f x∂Tμ ω)))
-    (hνl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
-      Tendsto (fun t : ℝ≥0 => ∫x,f x∂ν t ω) atTop (𝓝 (∫x,f x∂Tν ω))) :
-    Tμ =ᵐ[P] Tν := by
-  have hi (i : ℕ) : (fun ω => ∫x,determiningTests i x∂Tμ ω) =ᵐ[P]
-      (fun ω => ∫x,determiningTests i x∂Tν ω) := by
-    filter_upwards [he,hμl (determiningTests i),hνl (determiningTests i)] with ω hω hμω hνω
-    simp only [hω] at hμω
-    exact tendsto_nhds_unique hμω hνω
-  filter_upwards [ae_all_iff.mpr hi,hTμr,hTνr] with ω hω hr hs
-  let := hr
-  let := hs
-  exact measure_eq_of_determiningTests _ _ hω
-
-end SLEExactGauge.CompensatorUniqueness
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-
- 
-
-lemma integrable_test_of_ae_measure_le {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (W : Measure Ω) (M T : Ω→Measure X) {f : X→ℝ} (hf : ∀x,0≤f x)
-    (hm : AEStronglyMeasurable (fun ω => ∫x,f x∂M ω) W)
-    (hl : ∀ᵐ ω ∂W, Integrable f (T ω))
-    (hi : Integrable (fun ω => ∫x,f x∂T ω) W)
-    (hd : ∀ᵐ ω ∂W, M ω≤T ω) : Integrable (fun ω => ∫x,f x∂M ω) W := by
-  apply hi.norm.mono' hm
-  filter_upwards [hl,hd] with ω hω hdω
-  simp only [Real.norm_eq_abs]
-  rw [abs_of_nonneg (integral_nonneg (μ:=M ω) hf),abs_of_nonneg (integral_nonneg (μ:=T ω) hf)]
-  exact integral_mono_measure hdω (Eventually.of_forall hf) hω
-
-lemma double_lintegral_ofReal_eq {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (W : Measure Ω) (M : Ω→Measure X) {f : X→ℝ} (hf : ∀x,0≤f x)
-    (hlocal : ∀ᵐ ω ∂W, Integrable f (M ω))
-    (hmean : Integrable (fun ω => ∫x,f x∂M ω) W) (A : Set Ω) :
-    (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂M ω∂W)=
-      ENNReal.ofReal (∫ω in A,∫x,f x∂M ω∂W) := by
-  calc
-    _ = ∫⁻ω in A,ENNReal.ofReal (∫x,f x∂M ω)∂W := by
-      apply lintegral_congr_ae
-      filter_upwards [ae_restrict_of_ae hlocal] with ω hω
-      exact (ofReal_integral_eq_lintegral_ofReal hω (Eventually.of_forall hf)).symm
-    _ = _ := (ofReal_integral_eq_lintegral_ofReal hmean.restrict
-      (Eventually.of_forall fun ω => integral_nonneg hf)).symm
-
- 
-
-lemma condExp_sub_eq_of_lintegral_potential {Ω X : Type*} [mΩ : MeasurableSpace Ω]
-    [MeasurableSpace X] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
-    (W : @Measure Ω mΩ) [IsFiniteMeasure W]
-    (K L TK TL : Ω→Measure X) {f : X→ℝ} (hf : ∀x,0≤f x)
-    (hKi : Integrable (fun ω => ∫x,f x∂K ω) W)
-    (hLi : Integrable (fun ω => ∫x,f x∂L ω) W)
-    (hTKi : Integrable (fun ω => ∫x,f x∂TK ω) W)
-    (hTLi : Integrable (fun ω => ∫x,f x∂TL ω) W)
-    (hKl : ∀ᵐ ω ∂W, Integrable f (K ω))
-    (hLl : ∀ᵐ ω ∂W, Integrable f (L ω))
-    (hTKl : ∀ᵐ ω ∂W, Integrable f (TK ω))
-    (hTLl : ∀ᵐ ω ∂W, Integrable f (TL ω))
-    (V : Set Ω→ℝ≥0∞)
-    (hK : ∀A,MeasurableSet[m] A →
-      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TK ω∂W)=
-        (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂K ω∂W)+V A)
-    (hL : ∀A,MeasurableSet[m] A →
-      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TL ω∂W)=
-        (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂L ω∂W)+V A) :
-    W[(fun ω => (∫x,f x∂TK ω)-(∫x,f x∂K ω)) | m] =ᵐ[W]
-      W[(fun ω => (∫x,f x∂TL ω)-(∫x,f x∂L ω)) | m] := by
-  apply ae_eq_condExp_of_forall_setIntegral_eq hm (hTLi.sub hLi)
-    (fun _ _ _ => integrable_condExp.integrableOn) _
-    stronglyMeasurable_condExp.aestronglyMeasurable
-  intro A hA _
-  have hk : Integrable (fun ω => (∫x,f x∂TK ω)-(∫x,f x∂K ω)) W := hTKi.sub hKi
-  rw [setIntegral_condExp hm hk hA]
-  simp only [Pi.sub_apply]
-  rw [integral_sub hTKi.restrict hKi.restrict,integral_sub hTLi.restrict hLi.restrict]
-  have he : (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TK ω∂W)+
-      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂L ω∂W)=
-      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TL ω∂W)+
-      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂K ω∂W) := by
-    rw [hK A hA,hL A hA]
-    ac_rfl
-  rw [@double_lintegral_ofReal_eq Ω X mΩ _ W TK f hf hTKl hTKi,
-    @double_lintegral_ofReal_eq Ω X mΩ _ W L f hf hLl hLi,
-    @double_lintegral_ofReal_eq Ω X mΩ _ W TL f hf hTLl hTLi,
-    @double_lintegral_ofReal_eq Ω X mΩ _ W K f hf hKl hKi] at he
-  have hh := congrArg ENNReal.toReal he
-  have hn (M : Ω→Measure X) : 0≤∫ω in A,∫x,f x∂M ω∂W :=
-    integral_nonneg (fun _ => integral_nonneg hf)
-  simp only [ENNReal.toReal_add ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top,
-    ENNReal.toReal_ofReal (hn TK),ENNReal.toReal_ofReal (hn L),
-    ENNReal.toReal_ofReal (hn TL),ENNReal.toReal_ofReal (hn K)] at hh
-  linarith
-
-end SLEExactGauge.MassVersions
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.PostStopPotential
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.FiniteVariationMartingale
+open MeshEnergy TaylorVariation
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+variable {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+variable {M : ℝ≥0 → Ω → ℝ}
+
+ 
+
+lemma integral_weighted_increment (hM : Martingale M ℱ P)
+    {s t : ℝ≥0} (hst : s≤t) {H : Ω → ℝ}
+    (hm : StronglyMeasurable[ℱ s] H) {L : ℝ}
+    (hb : ∀ᵐ ω ∂P, ‖H ω‖≤L) :
+    ∫ ω, H ω*(M t ω-M s ω) ∂P = 0 := by
+  have hi := (hM.integrable t).sub (hM.integrable s)
+  have hprod : Integrable (H*(M t-M s)) P :=
+    hi.bdd_mul (hm.mono (ℱ.le s)).aestronglyMeasurable hb
+  have hzero : P[M t-M s | ℱ s] =ᵐ[P] 0 := by
+    have hc := condExp_sub (hM.integrable t) (hM.integrable s) (ℱ s)
+    have hs := condExp_of_stronglyMeasurable (ℱ.le s) (hM.stronglyMeasurable s)
+      (hM.integrable s)
+    filter_upwards [hc,hM.condExp_ae_eq hst] with ω he ht
+    simp only [Pi.sub_apply,hs,ht,sub_self] at he
+    exact he
+  have hp := condExp_mul_of_stronglyMeasurable_left hm hprod hi
+  calc
+    _ = ∫ ω, P[H*(M t-M s) | ℱ s] ω ∂P := (integral_condExp (ℱ.le s)).symm
+    _ = ∫ ω, (0:ℝ) ∂P := by
+      apply integral_congr_ae
+      filter_upwards [hp,hzero] with ω he hz
+      simpa [hz] using he
+    _ = 0 := integral_zero _ _
+
+omit [IsFiniteMeasure P] in
+lemma integrable_detector [IsFiniteMeasure P] (hM : Martingale M ℱ P) (t : ℝ≥0) :
+    Integrable (fun ω => detector (M t ω)) P :=
+  (hM.integrable t).norm.mono'
+    (lipschitz_detector.continuous.comp_aestronglyMeasurable
+      (hM.integrable t).aestronglyMeasurable)
+    (ae_of_all _ fun ω => abs_detector_le (M t ω))
+
+ 
+
+theorem ae_eq_zero_at (hM : Martingale M ℱ P)
+    (hz : M 0 =ᵐ[P] 0) (C : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
+    (hc : ∀ᵐ ω ∂P, ContinuousOn (fun t => C t ω) (Icc 0 T))
+    (hcontrol : ∀ᵐ ω ∂P, ∀ s t, s≤t → t≤T → |M t ω-M s ω|≤C t ω-C s ω)
+    (hint : Integrable (fun ω => C T ω-C 0 ω) P) : M T =ᵐ[P] 0 := by
+  let H (n i : ℕ) (ω : Ω) := detectorDeriv (M (grid T n i) ω)
+  have hm (n i : ℕ) : StronglyMeasurable[ℱ (grid T n i)] (H n i) :=
+    lipschitz_detectorDeriv.continuous.comp_stronglyMeasurable (hM.stronglyMeasurable _)
+  have hb (n i : ℕ) : ∀ᵐ ω ∂P, ‖H n i ω‖≤1 :=
+    ae_of_all _ fun ω => abs_detectorDeriv_le _
+  have hi (n i : ℕ) : Integrable
+      (fun ω => H n i ω * (M (grid T n (i+1)) ω-M (grid T n i) ω)) P :=
+    ((hM.integrable _).sub (hM.integrable _)).bdd_mul
+      ((hm n i).mono (ℱ.le _)).aestronglyMeasurable (hb n i)
+  have hzero (n i : ℕ) : (∫ ω, H n i ω *
+      (M (grid T n (i+1)) ω-M (grid T n i) ω) ∂P)=0 :=
+    integral_weighted_increment hM (grid_mono T n (Nat.le_succ i)) (hm n i) (hb n i)
+  have hsint (n : ℕ) : Integrable (fun ω => ∑ i ∈ Finset.range (n+1), H n i ω *
+      (M (grid T n (i+1)) ω-M (grid T n i) ω)) P :=
+    integrable_finsetSum _ fun i _ => hi n i
+  have hsz (n : ℕ) : (∫ ω, ∑ i ∈ Finset.range (n+1), H n i ω *
+      (M (grid T n (i+1)) ω-M (grid T n i) ω) ∂P)=0 := by
+    rw [integral_finsetSum _ (fun i _ => hi n i)]
+    simp only [hzero,Finset.sum_const_zero]
+  let F (n : ℕ) (ω : Ω) := error detector detectorDeriv (fun t => M t ω) T n
+  have hFi (n : ℕ) : Integrable (F n) P :=
+    ((integrable_detector hM T).sub (integrable_detector hM 0)).sub (hsint n)
+  have hz0 : (∫ ω, detector (M 0 ω) ∂P)=0 := by
+    calc
+      _ = ∫ ω, (0:ℝ) ∂P := integral_congr_ae (hz.mono fun ω he => by simp [he])
+      _ = 0 := integral_zero _ _
+  have hFint (n : ℕ) : (∫ ω, F n ω ∂P) = ∫ ω, detector (M T ω) ∂P := by
+    change (∫ ω, detector (M T ω)-detector (M 0 ω)-
+      ∑ i ∈ Finset.range (n+1), H n i ω *
+        (M (grid T n (i+1)) ω-M (grid T n i) ω) ∂P)=_
+    have hh := integral_sub ((integrable_detector hM T).sub (integrable_detector hM 0)) (hsint n)
+    simp only [Pi.sub_apply] at hh
+    rw [hh, integral_sub (integrable_detector hM T) (integrable_detector hM 0),
+      hz0,hsz,sub_zero,sub_zero]
+  have hFb (n : ℕ) : ∀ᵐ ω ∂P, ‖F n ω‖≤2*(C T ω-C 0 ω) := by
+    filter_upwards [hcontrol] with ω hω
+    simpa [F] using abs_error_le_variation detector detectorDeriv hasDerivAt_detector
+      (L := 1) abs_detectorDeriv_le (fun t => M t ω) (fun t => C t ω) T hω n
+  have hFl : ∀ᵐ ω ∂P, Tendsto (fun n => F n ω) atTop (𝓝 0) := by
+    filter_upwards [hc,hcontrol] with ω hcω hω
+    exact tendsto_error_zero detector detectorDeriv hasDerivAt_detector
+      lipschitz_detectorDeriv (fun t => M t ω) (fun t => C t ω) T hcω hω
+  have hlim : Tendsto (fun n => ∫ ω, F n ω ∂P) atTop (𝓝 0) := by
+    simpa only [integral_zero] using tendsto_integral_of_dominated_convergence
+      (fun ω => 2*(C T ω-C 0 ω)) (fun n => (hFi n).aestronglyMeasurable)
+      (hint.const_mul 2) hFb hFl
+  simp only [hFint] at hlim
+  have he : (∫ ω, detector (M T ω) ∂P)=0 :=
+    tendsto_nhds_unique tendsto_const_nhds hlim
+  have hdet := (integral_eq_zero_iff_of_nonneg
+    (fun ω => detector_nonneg (M T ω)) (integrable_detector hM T)).mp he
+  filter_upwards [hdet] with ω hω
+  exact (detector_eq_zero_iff _).mp hω
+
+ 
+theorem ae_all_eq_zero (hM : Martingale M ℱ P) (hz : M 0 =ᵐ[P] 0)
+    (hmc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω))
+    (C : ℝ≥0 → Ω → ℝ) (hc : ∀ᵐ ω ∂P, Continuous (fun t => C t ω))
+    (hcontrol : ∀ᵐ ω ∂P, ∀ s t, s≤t → |M t ω-M s ω|≤C t ω-C s ω)
+    (hint : ∀ T, Integrable (fun ω => C T ω-C 0 ω) P) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, M t ω=0 := by
+  obtain ⟨D,hD,hd⟩ := TopologicalSpace.exists_countable_dense ℝ≥0
+  have heq : ∀ᵐ ω ∂P, ∀ t∈D, M t ω=0 := by
+    apply (ae_ball_iff hD).mpr
+    intro t _
+    exact ae_eq_zero_at hM hz C t (hc.mono fun _ h => h.continuousOn)
+      (hcontrol.mono fun _ h s u hsu _ => h s u hsu) (hint t)
+  filter_upwards [heq,hmc] with ω hω hmω
+  have he : (fun t => M t ω)=(fun _ => (0:ℝ)) :=
+    Continuous.ext_on hd hmω continuous_const hω
+  exact fun t => congrFun he t
+
+end SLEExactGauge.FiniteVariationMartingale
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CompensatorUniqueness
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+variable {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+variable {A B : ℝ≥0 → Ω → ℝ} {TA TB : Ω → ℝ}
+
+ 
+
+omit [IsFiniteMeasure P] in
+lemma integrable_at [IsFiniteMeasure P] (hm : StronglyAdapted ℱ A)
+    (hz : A 0 =ᵐ[P] 0) (hmono : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
+    (hdom : ∀ᵐ ω ∂P, ∀ t, A t ω ≤ TA ω) (hi : Integrable TA P) (t : ℝ≥0) :
+    Integrable (A t) P := by
+  apply hi.norm.mono' ((hm t).mono (ℱ.le t)).aestronglyMeasurable
+  filter_upwards [hz,hmono,hdom] with ω hzω hmω hdω
+  have hnonneg : 0 ≤ A t ω := by simpa only [hzω,Pi.zero_apply] using hmω (show 0 ≤ t from bot_le)
+  simpa only [Real.norm_eq_abs,abs_of_nonneg hnonneg] using
+    (hdω t).trans (le_abs_self (TA ω))
+
+ 
+
+lemma martingale_sub (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B)
+    (hAi : ∀ t, Integrable (A t) P) (hBi : ∀ t, Integrable (B t) P)
+    (hTA : Integrable TA P) (hTB : Integrable TB P)
+    (hpot : ∀ t, P[TA - A t | ℱ t] =ᵐ[P] P[TB - B t | ℱ t]) :
+    Martingale (A-B) ℱ P := by
+  apply (martingale_condExp (TA-TB) ℱ P).congr (hAm.sub hBm)
+  intro t
+  have ha := condExp_of_stronglyMeasurable (ℱ.le t) (hAm t) (hAi t)
+  have hb := condExp_of_stronglyMeasurable (ℱ.le t) (hBm t) (hBi t)
+  filter_upwards [condExp_sub hTA hTB (ℱ t),condExp_sub hTA (hAi t) (ℱ t),
+    condExp_sub hTB (hBi t) (ℱ t),hpot t] with ω hab hae hbe hp
+  simp only [Pi.sub_apply,ha,hb] at hab hae hbe ⊢
+  linarith
+
+ 
+
+theorem ae_all_eq (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B)
+    (hAz : A 0 =ᵐ[P] 0) (hBz : B 0 =ᵐ[P] 0)
+    (hAc : ∀ᵐ ω ∂P, Continuous (fun t => A t ω))
+    (hBc : ∀ᵐ ω ∂P, Continuous (fun t => B t ω))
+    (hAo : ∀ᵐ ω ∂P, Monotone (fun t => A t ω))
+    (hBo : ∀ᵐ ω ∂P, Monotone (fun t => B t ω))
+    (hAd : ∀ᵐ ω ∂P, ∀ t, A t ω ≤ TA ω)
+    (hBd : ∀ᵐ ω ∂P, ∀ t, B t ω ≤ TB ω)
+    (hTA : Integrable TA P) (hTB : Integrable TB P)
+    (hpot : ∀ t, P[TA-A t | ℱ t] =ᵐ[P] P[TB-B t | ℱ t]) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, A t ω=B t ω := by
+  have hAi := integrable_at hAm hAz hAo hAd hTA
+  have hBi := integrable_at hBm hBz hBo hBd hTB
+  have hM := martingale_sub hAm hBm hAi hBi hTA hTB hpot
+  have hz : (A-B) 0 =ᵐ[P] 0 := by
+    filter_upwards [hAz,hBz] with ω ha hb
+    simp only [Pi.sub_apply,ha,hb,sub_self,Pi.zero_apply]
+  have hmc : ∀ᵐ ω ∂P, Continuous (fun t => (A-B) t ω) := by
+    filter_upwards [hAc,hBc] with ω ha hb
+    exact ha.sub hb
+  have hc : ∀ᵐ ω ∂P, Continuous (fun t => (A+B) t ω) := by
+    filter_upwards [hAc,hBc] with ω ha hb
+    exact ha.add hb
+  have hcontrol : ∀ᵐ ω ∂P, ∀ s t, s≤t →
+      |(A-B) t ω-(A-B) s ω|≤(A+B) t ω-(A+B) s ω := by
+    filter_upwards [hAo,hBo] with ω ha hb s t hst
+    have hat := ha hst
+    have hbt := hb hst
+    simp only [Pi.sub_apply,Pi.add_apply]
+    exact abs_le.mpr ⟨by linarith,by linarith⟩
+  have hint (T : ℝ≥0) : Integrable (fun ω => (A+B) T ω-(A+B) 0 ω) P :=
+    ((hAi T).add (hBi T)).sub ((hAi 0).add (hBi 0))
+  have he := FiniteVariationMartingale.ae_all_eq_zero hM hz hmc (A+B) hc hcontrol hint
+  filter_upwards [he] with ω hω t
+  exact sub_eq_zero.mp (hω t)
+
+ 
+omit [IsFiniteMeasure P] in
+theorem ae_terminal_eq_of_ae_all_eq [IsFiniteMeasure P]
+    (hab : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, A t ω=B t ω)
+    (hAl : ∀ᵐ ω ∂P, Tendsto (fun t => A t ω) atTop (𝓝 (TA ω)))
+    (hBl : ∀ᵐ ω ∂P, Tendsto (fun t => B t ω) atTop (𝓝 (TB ω))) :
+    TA =ᵐ[P] TB := by
+  filter_upwards [hab,hAl,hBl] with ω he ha hb
+  simp only [he] at ha
+  exact tendsto_nhds_unique ha hb
+
+end SLEExactGauge.CompensatorUniqueness
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CompensatorUniqueness
+open MassVersions
+variable {Ω X : Type*} [mΩ : MeasurableSpace Ω]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+    (μ ν : ℝ≥0 → Ω → Measure X) (Tμ Tν : Ω → Measure X)
+    [∀ t ω, (μ t ω).Regular] [∀ t ω, (ν t ω).Regular]
+    [∀ ω, (Tμ ω).Regular] [∀ ω, (Tν ω).Regular]
+
+ 
+theorem ae_all_measure_eq
+    (hμm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂μ t ω))
+    (hνm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂ν t ω))
+    (hμ0 : μ 0 =ᵐ[P] 0) (hν0 : ν 0 =ᵐ[P] 0)
+    (hμc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂μ t ω))
+    (hνc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂ν t ω))
+    (hμo : ∀ᵐ ω ∂P, Monotone (fun t => μ t ω))
+    (hνo : ∀ᵐ ω ∂P, Monotone (fun t => ν t ω))
+    (hμd : ∀ᵐ ω ∂P, ∀ t, μ t ω≤Tμ ω)
+    (hνd : ∀ᵐ ω ∂P, ∀ t, ν t ω≤Tν ω)
+    (hμi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tμ ω) P)
+    (hνi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tν ω) P)
+    (hpot : ∀ f : C_c(X,ℝ), (∀ x, 0≤f x) → ∀ t,
+      P[(fun ω => (∫x,f x∂Tμ ω)-(∫x,f x∂μ t ω)) | ℱ t] =ᵐ[P]
+        P[(fun ω => (∫x,f x∂Tν ω)-(∫x,f x∂ν t ω)) | ℱ t]) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω := by
+  have hpos (f : C_c(X,ℝ)) (hf : ∀ x,0≤f x) :
+      ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (∫x,f x∂μ t ω)=∫x,f x∂ν t ω := by
+    have hμz : (fun ω => ∫x,f x∂μ 0 ω) =ᵐ[P] 0 := by
+      filter_upwards [hμ0] with ω hω
+      simp only [hω,Pi.zero_apply,integral_zero_measure]
+    have hνz : (fun ω => ∫x,f x∂ν 0 ω) =ᵐ[P] 0 := by
+      filter_upwards [hν0] with ω hω
+      simp only [hω,Pi.zero_apply,integral_zero_measure]
+    have hμo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂μ t ω) := by
+      filter_upwards [hμo] with ω hω s t hst
+      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
+    have hνo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂ν t ω) := by
+      filter_upwards [hνo] with ω hω s t hst
+      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
+    have hμd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂μ t ω)≤∫x,f x∂Tμ ω := by
+      filter_upwards [hμd] with ω hω t
+      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
+    have hνd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂ν t ω)≤∫x,f x∂Tν ω := by
+      filter_upwards [hνd] with ω hω t
+      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
+    exact ae_all_eq (hμm f) (hνm f) hμz hνz (hμc f) (hνc f)
+      hμo' hνo' hμd' hνd' (hμi f) (hνi f) (hpot f hf)
+  apply ae_all_measure_eq_of_test_eq P μ ν
+  intro i
+  let f : C_c(X,ℝ) := determiningTests i
+  filter_upwards [hpos (positiveTest f) (positiveTest_nonneg f),
+    hpos (positiveTest (-f)) (positiveTest_nonneg (-f))] with ω hp hn t
+  calc
+    (∫x,f x∂μ t ω) = (∫x,positiveTest f x∂μ t ω)-(∫x,positiveTest (-f) x∂μ t ω) :=
+      (integral_positiveTest_sub f _).symm
+    _ = (∫x,positiveTest f x∂ν t ω)-(∫x,positiveTest (-f) x∂ν t ω) := by rw [hp t,hn t]
+    _ = (∫x,f x∂ν t ω) := integral_positiveTest_sub f _
+
+ 
+omit [IsFiniteMeasure P] [∀ time sample, (μ time sample).Regular]
+    [∀ time sample, (ν time sample).Regular] in
+theorem ae_terminal_measure_eq [IsFiniteMeasure P]
+    [∀ time sample, (μ time sample).Regular] [∀ time sample, (ν time sample).Regular]
+    (he : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω)
+    (hμl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
+      Tendsto (fun t : ℝ≥0 => ∫x,f x∂μ t ω) atTop (𝓝 (∫x,f x∂Tμ ω)))
+    (hνl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
+      Tendsto (fun t : ℝ≥0 => ∫x,f x∂ν t ω) atTop (𝓝 (∫x,f x∂Tν ω))) :
+    Tμ =ᵐ[P] Tν := by
+  have hi (i : ℕ) : (fun ω => ∫x,determiningTests i x∂Tμ ω) =ᵐ[P]
+      (fun ω => ∫x,determiningTests i x∂Tν ω) := by
+    filter_upwards [he,hμl (determiningTests i),hνl (determiningTests i)] with ω hω hμω hνω
+    simp only [hω] at hμω
+    exact tendsto_nhds_unique hμω hνω
+  filter_upwards [ae_all_iff.mpr hi] with ω hω
+  exact measure_eq_of_determiningTests _ _ hω
+
+ 
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X]
+    [IsFiniteMeasure P] [∀ time sample, (μ time sample).Regular]
+    [∀ time sample, (ν time sample).Regular] in
+theorem ae_all_random_weight_eq [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [BorelSpace X] [IsFiniteMeasure P]
+    [∀ time sample, (μ time sample).Regular] [∀ time sample, (ν time sample).Regular]
+    (he : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω) :
+    ∀ᵐ ω ∂P, ∀ (t : Ω → ℝ≥0) (w : Ω → X → ℝ≥0∞),
+      (∫⁻x,w ω x∂μ (t ω) ω)=∫⁻x,w ω x∂ν (t ω) ω := by
+  filter_upwards [he] with ω hω t w
+  rw [hω (t ω)]
+
+end SLEExactGauge.CompensatorUniqueness
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CompensatorUniqueness
+open MassVersions
+variable {Ω X : Type*} [mΩ : MeasurableSpace Ω]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    {P : Measure Ω} [IsFiniteMeasure P] {ℱ : Filtration ℝ≥0 mΩ}
+
+ 
+
+theorem ae_all_measure_eq_of_ae_regular
+    (μ ν : ℝ≥0 → Ω → Measure X) (Tμ Tν : Ω → Measure X)
+    (hμr : ∀ᵐ ω ∂P, ∀t, (μ t ω).Regular)
+    (hνr : ∀ᵐ ω ∂P, ∀t, (ν t ω).Regular)
+    (hTμr : ∀ᵐ ω ∂P, (Tμ ω).Regular)
+    (hTνr : ∀ᵐ ω ∂P, (Tν ω).Regular)
+    (hμm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂μ t ω))
+    (hνm : ∀ f : C_c(X,ℝ), StronglyAdapted ℱ (fun t ω => ∫x,f x∂ν t ω))
+    (hμ0 : μ 0 =ᵐ[P] 0) (hν0 : ν 0 =ᵐ[P] 0)
+    (hμc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂μ t ω))
+    (hνc : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P, Continuous (fun t => ∫x,f x∂ν t ω))
+    (hμo : ∀ᵐ ω ∂P, Monotone (fun t => μ t ω))
+    (hνo : ∀ᵐ ω ∂P, Monotone (fun t => ν t ω))
+    (hμd : ∀ᵐ ω ∂P, ∀ t, μ t ω≤Tμ ω)
+    (hνd : ∀ᵐ ω ∂P, ∀ t, ν t ω≤Tν ω)
+    (hμi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tμ ω) P)
+    (hνi : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x,f x∂Tν ω) P)
+    (hpot : ∀ f : C_c(X,ℝ), (∀ x, 0≤f x) → ∀ t,
+      P[(fun ω => (∫x,f x∂Tμ ω)-(∫x,f x∂μ t ω)) | ℱ t] =ᵐ[P]
+        P[(fun ω => (∫x,f x∂Tν ω)-(∫x,f x∂ν t ω)) | ℱ t]) :
+    ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω := by
+  have hpos (f : C_c(X,ℝ)) (hf : ∀ x,0≤f x) :
+      ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (∫x,f x∂μ t ω)=∫x,f x∂ν t ω := by
+    have hμz : (fun ω => ∫x,f x∂μ 0 ω) =ᵐ[P] 0 := by
+      filter_upwards [hμ0] with ω hω
+      simp only [hω,Pi.zero_apply,integral_zero_measure]
+    have hνz : (fun ω => ∫x,f x∂ν 0 ω) =ᵐ[P] 0 := by
+      filter_upwards [hν0] with ω hω
+      simp only [hω,Pi.zero_apply,integral_zero_measure]
+    have hμo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂μ t ω) := by
+      filter_upwards [hμo,hμr] with ω hω hr s t hst
+      let := hr t
+      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
+    have hνo' : ∀ᵐ ω ∂P, Monotone (fun t => ∫x,f x∂ν t ω) := by
+      filter_upwards [hνo,hνr] with ω hω hr s t hst
+      let := hr t
+      exact integral_mono_measure (hω hst) (Eventually.of_forall hf) f.integrable
+    have hμd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂μ t ω)≤∫x,f x∂Tμ ω := by
+      filter_upwards [hμd,hTμr] with ω hω hr t
+      let := hr
+      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
+    have hνd' : ∀ᵐ ω ∂P, ∀ t, (∫x,f x∂ν t ω)≤∫x,f x∂Tν ω := by
+      filter_upwards [hνd,hTνr] with ω hω hr t
+      let := hr
+      exact integral_mono_measure (hω t) (Eventually.of_forall hf) f.integrable
+    exact ae_all_eq (hμm f) (hνm f) hμz hνz (hμc f) (hνc f)
+      hμo' hνo' hμd' hνd' (hμi f) (hνi f) (hpot f hf)
+  have he (i : ℕ) : ∀ᵐ ω ∂P, ∀t,
+      (∫x,determiningTests i x∂μ t ω)=∫x,determiningTests i x∂ν t ω := by
+    let f : C_c(X,ℝ) := determiningTests i
+    filter_upwards [hμr,hνr,hpos (positiveTest f) (positiveTest_nonneg f),
+      hpos (positiveTest (-f)) (positiveTest_nonneg (-f))] with ω hr hs hp hn t
+    let := hr t
+    let := hs t
+    calc
+      (∫x,f x∂μ t ω) = (∫x,positiveTest f x∂μ t ω)-(∫x,positiveTest (-f) x∂μ t ω) :=
+        (integral_positiveTest_sub f _).symm
+      _ = (∫x,positiveTest f x∂ν t ω)-(∫x,positiveTest (-f) x∂ν t ω) := by rw [hp t,hn t]
+      _ = (∫x,f x∂ν t ω) := integral_positiveTest_sub f _
+  filter_upwards [ae_all_iff.mpr he,hμr,hνr] with ω hω hr hs t
+  let := hr t
+  let := hs t
+  exact measure_eq_of_determiningTests _ _ (fun i => hω i t)
+
+ 
+omit [IsFiniteMeasure P] in
+theorem ae_terminal_measure_eq_of_ae_regular [IsFiniteMeasure P]
+    (μ ν : ℝ≥0 → Ω → Measure X) (Tμ Tν : Ω → Measure X)
+    (hTμr : ∀ᵐ ω ∂P, (Tμ ω).Regular)
+    (hTνr : ∀ᵐ ω ∂P, (Tν ω).Regular)
+    (he : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, μ t ω=ν t ω)
+    (hμl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
+      Tendsto (fun t : ℝ≥0 => ∫x,f x∂μ t ω) atTop (𝓝 (∫x,f x∂Tμ ω)))
+    (hνl : ∀ f : C_c(X,ℝ), ∀ᵐ ω ∂P,
+      Tendsto (fun t : ℝ≥0 => ∫x,f x∂ν t ω) atTop (𝓝 (∫x,f x∂Tν ω))) :
+    Tμ =ᵐ[P] Tν := by
+  have hi (i : ℕ) : (fun ω => ∫x,determiningTests i x∂Tμ ω) =ᵐ[P]
+      (fun ω => ∫x,determiningTests i x∂Tν ω) := by
+    filter_upwards [he,hμl (determiningTests i),hνl (determiningTests i)] with ω hω hμω hνω
+    simp only [hω] at hμω
+    exact tendsto_nhds_unique hμω hνω
+  filter_upwards [ae_all_iff.mpr hi,hTμr,hTνr] with ω hω hr hs
+  let := hr
+  let := hs
+  exact measure_eq_of_determiningTests _ _ hω
+
+end SLEExactGauge.CompensatorUniqueness
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+
+ 
+
+lemma integrable_test_of_ae_measure_le {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (W : Measure Ω) (M T : Ω→Measure X) {f : X→ℝ} (hf : ∀x,0≤f x)
+    (hm : AEStronglyMeasurable (fun ω => ∫x,f x∂M ω) W)
+    (hl : ∀ᵐ ω ∂W, Integrable f (T ω))
+    (hi : Integrable (fun ω => ∫x,f x∂T ω) W)
+    (hd : ∀ᵐ ω ∂W, M ω≤T ω) : Integrable (fun ω => ∫x,f x∂M ω) W := by
+  apply hi.norm.mono' hm
+  filter_upwards [hl,hd] with ω hω hdω
+  simp only [Real.norm_eq_abs]
+  rw [abs_of_nonneg (integral_nonneg (μ:=M ω) hf),abs_of_nonneg (integral_nonneg (μ:=T ω) hf)]
+  exact integral_mono_measure hdω (Eventually.of_forall hf) hω
+
+lemma double_lintegral_ofReal_eq {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (W : Measure Ω) (M : Ω→Measure X) {f : X→ℝ} (hf : ∀x,0≤f x)
+    (hlocal : ∀ᵐ ω ∂W, Integrable f (M ω))
+    (hmean : Integrable (fun ω => ∫x,f x∂M ω) W) (A : Set Ω) :
+    (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂M ω∂W)=
+      ENNReal.ofReal (∫ω in A,∫x,f x∂M ω∂W) := by
+  calc
+    _ = ∫⁻ω in A,ENNReal.ofReal (∫x,f x∂M ω)∂W := by
+      apply lintegral_congr_ae
+      filter_upwards [ae_restrict_of_ae hlocal] with ω hω
+      exact (ofReal_integral_eq_lintegral_ofReal hω (Eventually.of_forall hf)).symm
+    _ = _ := (ofReal_integral_eq_lintegral_ofReal hmean.restrict
+      (Eventually.of_forall fun ω => integral_nonneg hf)).symm
+
+ 
+
+lemma condExp_sub_eq_of_lintegral_potential {Ω X : Type*} [mΩ : MeasurableSpace Ω]
+    [MeasurableSpace X] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
+    (W : @Measure Ω mΩ) [IsFiniteMeasure W]
+    (K L TK TL : Ω→Measure X) {f : X→ℝ} (hf : ∀x,0≤f x)
+    (hKi : Integrable (fun ω => ∫x,f x∂K ω) W)
+    (hLi : Integrable (fun ω => ∫x,f x∂L ω) W)
+    (hTKi : Integrable (fun ω => ∫x,f x∂TK ω) W)
+    (hTLi : Integrable (fun ω => ∫x,f x∂TL ω) W)
+    (hKl : ∀ᵐ ω ∂W, Integrable f (K ω))
+    (hLl : ∀ᵐ ω ∂W, Integrable f (L ω))
+    (hTKl : ∀ᵐ ω ∂W, Integrable f (TK ω))
+    (hTLl : ∀ᵐ ω ∂W, Integrable f (TL ω))
+    (V : Set Ω→ℝ≥0∞)
+    (hK : ∀A,MeasurableSet[m] A →
+      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TK ω∂W)=
+        (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂K ω∂W)+V A)
+    (hL : ∀A,MeasurableSet[m] A →
+      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TL ω∂W)=
+        (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂L ω∂W)+V A) :
+    W[(fun ω => (∫x,f x∂TK ω)-(∫x,f x∂K ω)) | m] =ᵐ[W]
+      W[(fun ω => (∫x,f x∂TL ω)-(∫x,f x∂L ω)) | m] := by
+  apply ae_eq_condExp_of_forall_setIntegral_eq hm (hTLi.sub hLi)
+    (fun _ _ _ => integrable_condExp.integrableOn) _
+    stronglyMeasurable_condExp.aestronglyMeasurable
+  intro A hA _
+  have hk : Integrable (fun ω => (∫x,f x∂TK ω)-(∫x,f x∂K ω)) W := hTKi.sub hKi
+  rw [setIntegral_condExp hm hk hA]
+  simp only [Pi.sub_apply]
+  rw [integral_sub hTKi.restrict hKi.restrict,integral_sub hTLi.restrict hLi.restrict]
+  have he : (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TK ω∂W)+
+      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂L ω∂W)=
+      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂TL ω∂W)+
+      (∫⁻ω in A,∫⁻x,ENNReal.ofReal (f x)∂K ω∂W) := by
+    rw [hK A hA,hL A hA]
+    ac_rfl
+  rw [@double_lintegral_ofReal_eq Ω X mΩ _ W TK f hf hTKl hTKi,
+    @double_lintegral_ofReal_eq Ω X mΩ _ W L f hf hLl hLi,
+    @double_lintegral_ofReal_eq Ω X mΩ _ W TL f hf hTLl hTLi,
+    @double_lintegral_ofReal_eq Ω X mΩ _ W K f hf hKl hKi] at he
+  have hh := congrArg ENNReal.toReal he
+  have hn (M : Ω→Measure X) : 0≤∫ω in A,∫x,f x∂M ω∂W :=
+    integral_nonneg (fun _ => integral_nonneg hf)
+  simp only [ENNReal.toReal_add ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top,
+    ENNReal.toReal_ofReal (hn TK),ENNReal.toReal_ofReal (hn L),
+    ENNReal.toReal_ofReal (hn TL),ENNReal.toReal_ofReal (hn K)] at hh
+  linarith
+
+end SLEExactGauge.MassVersions
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/FlowDerivatives.lean b/lean/OAI/Probability/SLEGauge/FlowDerivatives.lean
index 2d45b0a01da02f0ffeb4332440cf73554ddd72fd..8f518829a06aafcaa01ebcbfa2ec5bfa810a27e1 100644
--- a/lean/OAI/Probability/SLEGauge/FlowDerivatives.lean
+++ b/lean/OAI/Probability/SLEGauge/FlowDerivatives.lean
@@ -1,461 +1,461 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.DynkinIdentity
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-namespace ExactSLEFlow
-variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [MeasurableSpace Ω]
-omit [MeasurableSpace E] [BorelSpace E] in
-lemma integral_flow_hasDerivWithinAt
-    {A : E → E} (hA : Continuous A) {u f : ℝ≥0 → E}
-    (hu : Continuous u) (hf : Continuous f) {x : E}
-    (heq : ∀ t, f t = x + ∫ v in (0 : ℝ)..(t : ℝ), A (f v.toNNReal + u v.toNNReal))
-    (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) T) :
-    HasDerivWithinAt (fun v : ℝ => f v.toNNReal) (A (f t.toNNReal + u t.toNNReal))
-      (Icc (0 : ℝ) T) t := by
-  have hc : Continuous (fun v : ℝ => A (f v.toNNReal + u v.toNNReal)) :=
-    hA.comp ((hf.comp continuous_real_toNNReal).add (hu.comp continuous_real_toNNReal))
-  have hd := (intervalIntegral.integral_hasDerivAt_right
-    (hc.intervalIntegrable 0 t) hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt).const_add x
-  apply hd.hasDerivWithinAt.congr_of_mem _ ht
-  intro v hv
-  simpa only [Real.coe_toNNReal _ hv.1] using heq v.toNNReal
-
-omit [MeasurableSpace E] [BorelSpace E] in
-lemma integral_flow_unique
-    {A : E → E} {K : ℝ≥0} (hA : LipschitzWith K A) {u v f g : ℝ≥0 → E}
-    (hu : Continuous u) (hv : Continuous v) (hf : Continuous f) (hg : Continuous g) {x : E}
-    (hfe : ∀ t, f t = x + ∫ w in (0 : ℝ)..(t : ℝ), A (f w.toNNReal + u w.toNNReal))
-    (hge : ∀ t, g t = x + ∫ w in (0 : ℝ)..(t : ℝ), A (g w.toNNReal + v w.toNNReal))
-    {T : ℝ≥0} (huv : ∀ t ≤ T, u t = v t) {t : ℝ≥0} (ht : t ≤ T) : f t = g t := by
-  have hfd := integral_flow_hasDerivWithinAt hA.continuous hu hf hfe T
-  have hgd : ∀ w ∈ Icc (0 : ℝ) T, HasDerivWithinAt (fun w : ℝ => g w.toNNReal)
-      (A (g w.toNNReal + u w.toNNReal)) (Icc (0 : ℝ) T) w := by
-    intro w hw
-    rw [huv w.toNNReal ((Real.toNNReal_le_toNNReal hw.2).trans_eq (by simp))]
-    exact integral_flow_hasDerivWithinAt hA.continuous hv hg hge T w hw
-  have hf0 : f 0 = x := by simpa using hfe 0
-  have hg0 : g 0 = x := by simpa using hge 0
-  have h0 : f 0 = g 0 := hf0.trans hg0.symm
-  have he := additive_flow_unique hA
-    ((hf.comp continuous_real_toNNReal).continuousOn)
-    ((hg.comp continuous_real_toNNReal).continuousOn) hfd hgd (by simpa using h0)
-    (show (t : ℝ) ∈ Icc (0 : ℝ) T from ⟨t.coe_nonneg, ht⟩)
-  simpa using he
-
- 
-
-theorem additive_flow_adapted
-    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
-    (u : ℝ≥0 → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
-    (x : Ω → E) (Y : ℝ≥0 → Ω → E) (hYc : ∀ ω, Continuous (fun t => Y t ω))
-    (hYe : ∀ ω t, Y t ω = x ω + ∫ v in (0 : ℝ)..(t : ℝ),
-      A (Y v.toNNReal ω + u v.toNNReal ω))
-    (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (hum : ∀ t, Measurable[F t] (u t)) (hxm : Measurable[F 0] x) :
-    ∀ t, Measurable[F t] (Y t) := by
-  intro T
-  let : MeasurableSpace Ω := F T
-  let uT : ℝ≥0 → Ω → E := fun t ω => u (min t T) ω
-  have huTc (ω : Ω) : Continuous (fun t => uT t ω) :=
-    (huc ω).comp (continuous_id.min continuous_const)
-  have huTm (t : ℝ≥0) : Measurable (uT t) :=
-    (hum (min t T)).mono (F.mono (min_le_right _ _)) le_rfl
-  have hx : Measurable x := hxm.mono (F.mono bot_le) le_rfl
-  obtain ⟨Z, hZc, hZm, hZe⟩ := measurable_additive_flow A L K hA hb uT huTc huTm x hx
-  have he : Y T = Z T := by
-    funext ω
-    exact integral_flow_unique hA (huc ω) (huTc ω) (hYc ω) (hZc ω) (hYe ω) (hZe ω)
-      (T := T) (fun t ht => by simp [uT, min_eq_left ht]) le_rfl
-  rw [he]
-  exact hZm T
-
-end ExactSLEFlow
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem exists_adapted_additive_brownian
-    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
-    (x e : E) :
-    ∃ X : ℝ≥0 → Driver → E,
-      (∀ b, Continuous (fun t => X t b)) ∧
-      (∀ t, Measurable[rawFiltration t] (X t)) ∧
-      (∀ b, X 0 b = x) ∧
-      (∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-        ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b)) := by
-  let u : ℝ≥0 → Driver → E := fun t b => b.1 t • e
-  have huc (b : Driver) : Continuous (fun t => u t b) := b.1.continuous.smul continuous_const
-  have hum (t : ℝ≥0) : Measurable (u t) := (driver_eval_measurable t).measurable.smul_const e
-  have humF (t : ℝ≥0) : Measurable[rawFiltration t] (u t) := by
-    have h := Filtration.stronglyAdapted_natural driver_eval_measurable t
-    exact h.measurable.smul_const e
-  obtain ⟨Y, hYc, hYm, hYe⟩ := ExactSLEFlow.measurable_additive_flow A L K hA hb
-    u huc hum (fun _ => x) measurable_const
-  have hYF := ExactSLEFlow.additive_flow_adapted A L K hA hb u huc (fun _ => x)
-    Y hYc hYe rawFiltration humF measurable_const
-  let X : ℝ≥0 → Driver → E := fun t b => Y t b + u t b
-  refine ⟨X, fun b => (hYc b).add (huc b), fun t => (hYF t).add (humF t), ?_, ?_⟩
-  · intro b
-    have h0 := hYe b 0
-    simp only [NNReal.coe_zero, intervalIntegral.integral_same, add_zero] at h0
-    simp [X, u, b.2, h0]
-  · intro s h b
-    have hc : Continuous (fun t : ℝ => A (X t.toNNReal b)) :=
-      hA.continuous.comp (((hYc b).add (huc b)).comp continuous_real_toNNReal)
-    have hi := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
-      (hc.intervalIntegrable (0 : ℝ) s) (hc.intervalIntegrable (s : ℝ) (s+h : ℝ≥0))
-    have hYs := hYe b s
-    have hYt := hYe b (s+h)
-    change Y (s+h) b + b.1 (s+h) • e =
-      Y s b + b.1 s • e + (b.1 (s+h) - b.1 s) • e + ∫ t in (s : ℝ)..(s+h : ℝ≥0), A (X t.toNNReal b)
-    change Y s b = x + ∫ t in (0 : ℝ)..(s : ℝ), A (X t.toNNReal b) at hYs
-    change Y (s+h) b = x + ∫ t in (0 : ℝ)..(s+h : ℝ≥0), A (X t.toNNReal b) at hYt
-    rw [hYs, hYt, ← hi, sub_smul]
-    abel
-
- 
-omit [CompleteSpace E] in
-lemma adapted_intervalIntegral {Ω : Type*} [MeasurableSpace Ω]
-    (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›) (g : ℝ≥0 → Ω → E)
-    (hc : ∀ ω, Continuous (fun t => g t ω)) (hm : ∀ t, Measurable[F t] (g t))
-    (T : ℝ≥0) :
-    Measurable[F T] (fun ω => ∫ t in (0 : ℝ)..(T : ℝ), g t.toNNReal ω) := by
-  let : MeasurableSpace Ω := F T
-  have hc' (ω : Ω) : Continuous (fun t : ℝ => g (min t.toNNReal T) ω) :=
-    (hc ω).comp (continuous_real_toNNReal.min continuous_const)
-  have hm' (t : ℝ) : Measurable (g (min t.toNNReal T)) :=
-    (hm _).mono (F.mono (min_le_right _ _)) le_rfl
-  have hme := stronglyMeasurable_intervalIntegral_of_continuous
-    (fun t ω => g (min t.toNNReal T) ω) hc' hm' 0 T
-  have he : (fun ω => ∫ t in (0 : ℝ)..(T : ℝ), g t.toNNReal ω) =
-      (fun ω => ∫ t in (0 : ℝ)..(T : ℝ), g (min t.toNNReal T) ω) := by
-    funext ω
-    apply intervalIntegral.integral_congr
-    intro t ht
-    have ht' : t ≤ (T : ℝ) := (by simpa only [uIcc_of_le T.coe_nonneg] using ht : t ∈ Set.Icc 0 (T : ℝ)).2
-    change g t.toNNReal ω = g (min t.toNNReal T) ω
-    have htT : t.toNNReal ≤ T := (Real.toNNReal_le_toNNReal ht').trans_eq (by simp)
-    rw [min_eq_left htT]
-  rw [he]
-  exact hme.measurable
-
-end ExactSLEGenerator
-
-namespace ExactSLEGenerator
-
- 
-
-theorem martingale_compensated_of_weighted
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (F G : ℝ≥0 → Driver → ℝ)
-    (hFm : ∀ t, Measurable[rawFiltration t] (F t))
-    (hFi : ∀ t, Integrable (F t) W)
-    (hGc : ∀ b, Continuous (fun t => G t b))
-    (hGm : ∀ t, Measurable[rawFiltration t] (G t))
-    (C : ℝ) (hGb : ∀ b t, |G t b| ≤ C)
-    (hw : ∀ (s h : ℝ≥0) (J : Driver → ℝ), Measurable[rawFiltration s] J →
-      (∀ b, |J b| ≤ 1) →
-      (∫ b, J b * F (s+h) b ∂W) - (∫ b, J b * F s b ∂W) =
-        ∫ b, J b * (∫ u in (s : ℝ)..(s+h : ℝ≥0), G u.toNNReal b) ∂W) :
-    Martingale (fun t b => F t b - ∫ u in (0 : ℝ)..(t : ℝ), G u.toNNReal b)
-      rawFiltration W := by
-  let I : ℝ≥0 → Driver → ℝ := fun t b => ∫ u in (0 : ℝ)..(t : ℝ), G u.toNNReal b
-  have hIm (t : ℝ≥0) : Measurable[rawFiltration t] (I t) :=
-    adapted_intervalIntegral rawFiltration G hGc hGm t
-  have hGg (t : ℝ≥0) : Measurable (G t) := (hGm t).mono (rawFiltration.le t) le_rfl
-  have hseg (s t : ℝ) : Integrable (fun b => ∫ u in s..t, G u.toNNReal b) W := by
-    apply Integrable.of_bound
-      (stronglyMeasurable_intervalIntegral_of_continuous (fun u b => G u.toNNReal b)
-        (fun b => (hGc b).comp continuous_real_toNNReal) (fun u => hGg _) s t).aestronglyMeasurable
-      (C * |t-s|)
-    exact ae_of_all W (fun b => intervalIntegral.norm_integral_le_of_norm_le_const
-      (fun u hu => by simpa only [Real.norm_eq_abs] using hGb b u.toNNReal))
-  have hIi (t : ℝ≥0) : Integrable (I t) W := hseg 0 t
-  have hMa (t : ℝ≥0) : StronglyMeasurable[rawFiltration t] (fun b => F t b - I t b) :=
-    ((hFm t).sub (hIm t)).stronglyMeasurable
-  have hMi (t : ℝ≥0) : Integrable (fun b => F t b - I t b) W := (hFi t).sub (hIi t)
-  refine ⟨hMa, ?_⟩
-  intro s t hst
-  symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq (rawFiltration.le s) (hMi t)
-    (fun E hE hEfin => (hMi s).integrableOn) _ (hMa s).aestronglyMeasurable
-  intro E hE hEfin
-  have hEe : MeasurableSet E := rawFiltration.le s _ hE
-  let J : Driver → ℝ := E.indicator (fun _ => 1)
-  have hJm : Measurable[rawFiltration s] J := measurable_const.indicator hE
-  have hJb (b : Driver) : |J b| ≤ 1 := by
-    by_cases hb : b ∈ E <;> simp [J, hb]
-  have hJF (H : Driver → ℝ) : (fun b => J b * H b) = E.indicator H := by
-    funext b
-    by_cases hb : b ∈ E <;> simp [J, hb]
-  have he := hw s (t-s) J hJm hJb
-  rw [add_tsub_cancel_of_le hst] at he
-  simp only [hJF, integral_indicator hEe] at he
-  have hIadd (b : Driver) : I t b = I s b + ∫ u in (s : ℝ)..(t : ℝ), G u.toNNReal b := by
-    have hc := (hGc b).comp continuous_real_toNNReal
-    exact (intervalIntegral.integral_add_adjacent_intervals (μ := volume)
-      (hc.intervalIntegrable (0 : ℝ) s) (hc.intervalIntegrable (s : ℝ) t)).symm
-  have hIset : (∫ b in E, I t b ∂W) = (∫ b in E, I s b ∂W) +
-      (∫ b in E, (∫ u in (s : ℝ)..(t : ℝ), G u.toNNReal b) ∂W) := by
-    simp_rw [hIadd]
-    exact integral_add (hIi s).integrableOn (hseg s t).integrableOn
-  change (∫ b in E, F s b - I s b ∂W) = ∫ b in E, F t b - I t b ∂W
-  rw [integral_sub (hFi s).integrableOn (hIi s).integrableOn,
-    integral_sub (hFi t).integrableOn (hIi t).integrableOn, hIset]
-  linarith
-
-end ExactSLEGenerator
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma intervalIntegral_nnreal_shift (H : ℝ≥0 → E) (start s t : ℝ≥0) :
-    (∫ u in (s : ℝ)..(t : ℝ), H (start+u.toNNReal)) =
-      ∫ u in (start+s : ℝ≥0)..(start+t : ℝ≥0), H u.toNNReal := by
-  calc
-    _ = ∫ u in (s : ℝ)..(t : ℝ), H (((start : ℝ)+u).toNNReal) := by
-      apply intervalIntegral.integral_congr
-      intro u hu
-      have hu0 : 0 ≤ u := le_trans (le_min s.coe_nonneg t.coe_nonneg) hu.1
-      change H (start+u.toNNReal) = H (((start : ℝ)+u).toNNReal)
-      congr 1
-      ext
-      simp only [NNReal.coe_add, Real.coe_toNNReal _ hu0,
-        Real.coe_toNNReal _ (add_nonneg start.coe_nonneg hu0)]
-    _ = _ := by
-      simpa only [NNReal.coe_add] using
-        (intervalIntegral.integral_comp_add_left (fun u : ℝ => H u.toNNReal) (start : ℝ))
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma shift_additive_equation
-    (X : ℝ≥0 → Driver → E) (A : E → E) (e : E)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (start s h : ℝ≥0) (b : Driver) :
-    X (start+(s+h)) b = X (start+s) b + increment (start+s) h b • e +
-      ∫ t in (s : ℝ)..(s+h : ℝ≥0), A (X (start+t.toNNReal) b) := by
-  rw [intervalIntegral_nnreal_shift (fun v => A (X v b)) start s (s+h), ← add_assoc]
-  exact hXeq (start+s) h b
-
-lemma integrable_intervalIntegral_of_bound
-    (W : Measure Driver) [IsProbabilityMeasure W] (G : ℝ≥0 → Driver → ℝ)
-    (hc : ∀ b, Continuous (fun t => G t b)) (hm : ∀ t, Measurable (G t))
-    (C : ℝ) (hb : ∀ b t, |G t b| ≤ C) (s t : ℝ) :
-    Integrable (fun b => ∫ u in s..t, G u.toNNReal b) W := by
-  apply Integrable.of_bound
-    (stronglyMeasurable_intervalIntegral_of_continuous (fun u b => G u.toNNReal b)
-      (fun b => (hc b).comp continuous_real_toNNReal) (fun u => hm _) s t).aestronglyMeasurable
-    (C * |t-s|)
-  exact ae_of_all W (fun b => intervalIntegral.norm_integral_le_of_norm_le_const
-    (fun u hu => by simpa only [Real.norm_eq_abs] using hb b u.toNNReal))
-
- 
-
-theorem additive_brownian_martingale (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (D : E → E →L[ℝ] ℝ) (hD : ∀ x, HasFDerivAt f (D x) x)
-    (hDc : Continuous D)
-    (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M)
-    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
-    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
-    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
-    (hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0))
-    (hfb : ∀ x, |f x| ≤ M) (hDb : ∀ x, ‖D x‖ ≤ M)
-    (hQb : ∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0| ≤ M) :
-    Martingale (fun t b => f (X t b) - ∫ u in (0 : ℝ)..(t : ℝ),
-      ((1/2 : ℝ) * iteratedDeriv 2 (fun r : ℝ => f (X u.toNNReal b + r • e)) 0 +
-        D (X u.toNNReal b) (A (X u.toNNReal b)))) rawFiltration W := by
-  let Q : E → ℝ := fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0
-  let V : E → ℝ := fun x => D x (A x)
-  let G : E → ℝ := fun x => (1/2 : ℝ)*Q x + V x
-  have hVc : Continuous V := hDc.clm_apply hAc
-  have hGc : Continuous G := (continuous_const.mul hQc).add hVc
-  have hVb (x : E) : |V x| ≤ M*K := by
-    calc
-      _ = ‖D x (A x)‖ := (Real.norm_eq_abs _).symm
-      _ ≤ ‖D x‖ * ‖A x‖ := (D x).le_opNorm _
-      _ ≤ M*K := mul_le_mul (hDb x) (hAb x) (norm_nonneg _) hM
-  have hGb (x : E) : |G x| ≤ M/2 + M*K := by
-    calc
-      _ ≤ |(1/2 : ℝ)*Q x| + |V x| := abs_add_le _ _
-      _ ≤ (1/2 : ℝ)*M + M*K := add_le_add (by
-        rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/2)]
-        exact mul_le_mul_of_nonneg_left (hQb x) (by norm_num)) (hVb x)
-      _ = _ := by ring
-  have hfc : Continuous f := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt)
-  have hXg (t : ℝ≥0) : Measurable (X t) := (hXm t).mono (rawFiltration.le t) le_rfl
-  apply martingale_compensated_of_weighted W (fun t b => f (X t b))
-    (fun t b => G (X t b))
-    (fun t => hfc.measurable.comp (hXm t))
-    (fun t => Integrable.of_bound (hfc.measurable.comp (hXg t)).aestronglyMeasurable M
-      (ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hfb (X t b))))
-    (fun b => hGc.comp (hXc b))
-    (fun t => hGc.measurable.comp (hXm t)) (M/2+M*K) (fun b t => hGb _) 
-  intro s h J hJm hJb
-  have hc (b : Driver) : Continuous (fun t => X (s+t) b) :=
-    (hXc b).comp (continuous_const.add continuous_id)
-  have hdyn := additive_brownian_weighted_dynkin W hW s (fun t b => X (s+t) b)
-    hc (fun t => hXm (s+t)) A hAc K hK hAb e he (shift_additive_equation X A e hXeq s)
-    f D hD hDc C M hC hM hLip hSmooth hThird hQc hfb hDb hQb J hJm hJb h
-  have hiQ (b : Driver) : (∫ u in (0 : ℝ)..(h : ℝ), Q (X (s+u.toNNReal) b)) =
-      ∫ u in (s : ℝ)..(s+h : ℝ≥0), Q (X u.toNNReal b) := by
-    simpa only [NNReal.coe_zero, add_zero] using
-      (intervalIntegral_nnreal_shift (fun t => Q (X t b)) s 0 h)
-  have hiV (b : Driver) : (∫ u in (0 : ℝ)..(h : ℝ), V (X (s+u.toNNReal) b)) =
-      ∫ u in (s : ℝ)..(s+h : ℝ≥0), V (X u.toNNReal b) := by
-    simpa only [NNReal.coe_zero, add_zero] using
-      (intervalIntegral_nnreal_shift (fun t => V (X t b)) s 0 h)
-  change (∫ b, J b * f (X (s+h) b) ∂W) - (∫ b, J b * f (X (s+0) b) ∂W) =
-    (1/2 : ℝ) * (∫ b, J b * (∫ u in (0 : ℝ)..(h : ℝ), Q (X (s+u.toNNReal) b)) ∂W) +
-    (∫ b, J b * (∫ u in (0 : ℝ)..(h : ℝ), V (X (s+u.toNNReal) b)) ∂W) at hdyn
-  simp_rw [add_zero, hiQ, hiV] at hdyn
-  rw [hdyn]
-  have hJ : AEStronglyMeasurable J W := (hJm.mono (rawFiltration.le _) le_rfl).aestronglyMeasurable
-  have hJi : ∀ᵐ b ∂W, ‖J b‖ ≤ 1 := ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hJb b)
-  have hQi := integrable_intervalIntegral_of_bound W (fun t b => Q (X t b))
-    (fun b => hQc.comp (hXc b)) (fun t => hQc.measurable.comp (hXg t)) M
-    (fun b t => hQb _) s (s+h : ℝ≥0)
-  have hVi := integrable_intervalIntegral_of_bound W (fun t b => V (X t b))
-    (fun b => hVc.comp (hXc b)) (fun t => hVc.measurable.comp (hXg t)) (M*K)
-    (fun b t => hVb _) s (s+h : ℝ≥0)
-  have hiG (b : Driver) : (∫ u in (s : ℝ)..(s+h : ℝ≥0), G (X u.toNNReal b)) =
-      (1/2 : ℝ)*(∫ u in (s : ℝ)..(s+h : ℝ≥0), Q (X u.toNNReal b)) +
-      (∫ u in (s : ℝ)..(s+h : ℝ≥0), V (X u.toNNReal b)) := by
-    have hqc : Continuous (fun u : ℝ => Q (X u.toNNReal b)) :=
-      hQc.comp ((hXc b).comp continuous_real_toNNReal)
-    have hvc : Continuous (fun u : ℝ => V (X u.toNNReal b)) :=
-      hVc.comp ((hXc b).comp continuous_real_toNNReal)
-    rw [show (fun u : ℝ => G (X u.toNNReal b)) =
-      (fun u : ℝ => (1/2 : ℝ)*Q (X u.toNNReal b) + V (X u.toNNReal b)) from rfl]
-    rw [intervalIntegral.integral_add ((hqc.intervalIntegrable _ _).const_mul _) (hvc.intervalIntegrable _ _),
-      intervalIntegral.integral_const_mul]
-  simp_rw [hiG, mul_add, ← mul_assoc, mul_comm _ (1/2 : ℝ), mul_assoc]
-  rw [integral_add ((hQi.bdd_mul hJ hJi).const_mul _) (hVi.bdd_mul hJ hJi), integral_const_mul]
-
-end ExactSLEGenerator
-
-namespace ExactSLEGenerator
-open Set
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
-noncomputable def directional (n : ℕ) (f : E → ℝ) (e : E) : E → ℝ :=
-  Nat.rec f (fun _ g x => fderiv ℝ g x e) n
-
-omit [CompleteSpace E] in
-lemma directional_contDiff (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (e : E) (n : ℕ) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (directional n f e) := by
-  induction n with
-  | zero => exact hf
-  | succ n ih =>
-    exact (ih.fderiv_right (by simp)).clm_apply contDiff_const
-
-omit [CompleteSpace E] in
-lemma directional_compact (f : E → ℝ) (hf : HasCompactSupport f) (e : E) (n : ℕ) :
-    HasCompactSupport (directional n f e) := by
-  induction n with
-  | zero => exact hf
-  | succ n ih => exact ih.fderiv_apply ℝ e
-
-omit [CompleteSpace E] in
-lemma iteratedDeriv_line (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (e x : E) (n : ℕ) :
-    iteratedDeriv n (fun r : ℝ => f (x + r • e)) = fun r => directional n f e (x+r • e) := by
-  induction n with
-  | zero => rfl
-  | succ n ih =>
-    rw [iteratedDeriv_succ,ih]
-    funext r
-    have hd : HasDerivAt (fun r : ℝ => x+r • e) e r := by
-      convert (hasDerivAt_const r x).add ((hasDerivAt_id r).smul_const e) using 1 <;> first | rfl | simp only [zero_add,one_smul]
-    exact (((directional_contDiff f hf e n).differentiable (by simp) (x+r • e)).hasFDerivAt.comp_hasDerivAt r hd).deriv
-
-omit [CompleteSpace E] in
-lemma compact_smooth_fderiv_bound (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
-    (hfc : HasCompactSupport f) :
-    ∃ C : ℝ, ∀ x y, ‖fderiv ℝ f y - fderiv ℝ f x‖ ≤ C * ‖y-x‖ := by
-  have hd : ContDiff ℝ (↑(⊤ : ℕ∞)) (fderiv ℝ f) := hf.fderiv_right (by simp)
-  have hdc : HasCompactSupport (fderiv ℝ f) := hfc.fderiv ℝ
-  have hddc : HasCompactSupport (fderiv ℝ (fderiv ℝ f)) := hdc.fderiv ℝ
-  have hdd : Continuous (fderiv ℝ (fderiv ℝ f)) := hd.continuous_fderiv (by simp)
-  have hbnd : ∃ C : ℝ, ∀ x : E, ‖fderiv ℝ (fderiv ℝ f) x‖ ≤ C := HasCompactSupport.exists_bound_of_continuous (f := fderiv ℝ (fderiv ℝ f)) hddc hdd
-  obtain ⟨C,hC⟩ := hbnd
-  refine ⟨C, fun x y => ?_⟩
-  exact Convex.norm_image_sub_le_of_norm_fderiv_le
-    (fun u _ => hd.differentiable (by simp) u) (fun u _ => hC u)
-    convex_univ (mem_univ x) (mem_univ y)
-
-omit [CompleteSpace E] in
-lemma compact_smooth_line_bound (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
-    (hfc : HasCompactSupport f) (e : E) (n : ℕ) :
-    ∃ C : ℝ, ∀ x w, |iteratedDeriv n (fun r : ℝ => f (x+r • e)) w| ≤ C := by
-  obtain ⟨C,hC⟩ := (directional_compact f hfc e n).exists_bound_of_continuous
-    (directional_contDiff f hf e n).continuous
-  refine ⟨C, fun x w => ?_⟩
-  rw [iteratedDeriv_line f hf e x n]
-  exact (show |directional n f e (x+w • e)| ≤ C by
-    simpa only [Real.norm_eq_abs] using hC (x+w • e))
-
-omit [CompleteSpace E] in
-lemma compact_smooth_test_bounds (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
-    (hfc : HasCompactSupport f) (e : E) :
-    ∃ C M : ℝ, 0 ≤ C ∧ 0 ≤ M ∧
-      (∀ x y, ‖fderiv ℝ f y - fderiv ℝ f x‖ ≤ C * ‖y-x‖) ∧
-      (∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x+r • e)) w| ≤ C) ∧
-      (∀ x, |f x| ≤ M) ∧ (∀ x, ‖fderiv ℝ f x‖ ≤ M) ∧
-      (∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x+r • e)) 0| ≤ M) := by
-  obtain ⟨C₀,hC₀⟩ := compact_smooth_fderiv_bound f hf hfc
-  obtain ⟨C₁,hC₁⟩ := compact_smooth_line_bound f hf hfc e 3
-  obtain ⟨M₀,hM₀⟩ := hfc.exists_bound_of_continuous hf.continuous
-  have hdc : HasCompactSupport (fderiv ℝ f) := hfc.fderiv ℝ
-  obtain ⟨M₁,hM₁⟩ := hdc.exists_bound_of_continuous (hf.continuous_fderiv (by simp))
-  obtain ⟨M₂,hM₂⟩ := compact_smooth_line_bound f hf hfc e 2
-  let C := max 0 (max C₀ C₁)
-  let M := max 0 (max M₀ (max M₁ M₂))
-  have hC₀' : C₀ ≤ C := (le_max_left _ _).trans (le_max_right _ _)
-  have hC₁' : C₁ ≤ C := (le_max_right _ _).trans (le_max_right _ _)
-  have hM₀' : M₀ ≤ M := (le_max_left _ _).trans (le_max_right _ _)
-  have hM₁' : M₁ ≤ M := (le_max_left _ _).trans ((le_max_right _ _).trans (le_max_right _ _))
-  have hM₂' : M₂ ≤ M := (le_max_right _ _).trans ((le_max_right _ _).trans (le_max_right _ _))
-  refine ⟨C,M,le_max_left _ _,le_max_left _ _,?_,?_,?_,?_,?_⟩
-  · intro x y
-    exact (hC₀ x y).trans (mul_le_mul_of_nonneg_right hC₀' (norm_nonneg _))
-  · intro x w
-    exact (hC₁ x w).trans hC₁'
-  · intro x
-    exact (show |f x| ≤ M₀ by simpa only [Real.norm_eq_abs] using hM₀ x).trans hM₀'
-  · intro x
-    exact (hM₁ x).trans hM₁'
-  · intro x
-    exact (hM₂ x 0).trans hM₂'
-
-variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-end ExactSLEGenerator
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.DynkinIdentity
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+namespace ExactSLEFlow
+variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [MeasurableSpace Ω]
+omit [MeasurableSpace E] [BorelSpace E] in
+lemma integral_flow_hasDerivWithinAt
+    {A : E → E} (hA : Continuous A) {u f : ℝ≥0 → E}
+    (hu : Continuous u) (hf : Continuous f) {x : E}
+    (heq : ∀ t, f t = x + ∫ v in (0 : ℝ)..(t : ℝ), A (f v.toNNReal + u v.toNNReal))
+    (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) T) :
+    HasDerivWithinAt (fun v : ℝ => f v.toNNReal) (A (f t.toNNReal + u t.toNNReal))
+      (Icc (0 : ℝ) T) t := by
+  have hc : Continuous (fun v : ℝ => A (f v.toNNReal + u v.toNNReal)) :=
+    hA.comp ((hf.comp continuous_real_toNNReal).add (hu.comp continuous_real_toNNReal))
+  have hd := (intervalIntegral.integral_hasDerivAt_right
+    (hc.intervalIntegrable 0 t) hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt).const_add x
+  apply hd.hasDerivWithinAt.congr_of_mem _ ht
+  intro v hv
+  simpa only [Real.coe_toNNReal _ hv.1] using heq v.toNNReal
+
+omit [MeasurableSpace E] [BorelSpace E] in
+lemma integral_flow_unique
+    {A : E → E} {K : ℝ≥0} (hA : LipschitzWith K A) {u v f g : ℝ≥0 → E}
+    (hu : Continuous u) (hv : Continuous v) (hf : Continuous f) (hg : Continuous g) {x : E}
+    (hfe : ∀ t, f t = x + ∫ w in (0 : ℝ)..(t : ℝ), A (f w.toNNReal + u w.toNNReal))
+    (hge : ∀ t, g t = x + ∫ w in (0 : ℝ)..(t : ℝ), A (g w.toNNReal + v w.toNNReal))
+    {T : ℝ≥0} (huv : ∀ t ≤ T, u t = v t) {t : ℝ≥0} (ht : t ≤ T) : f t = g t := by
+  have hfd := integral_flow_hasDerivWithinAt hA.continuous hu hf hfe T
+  have hgd : ∀ w ∈ Icc (0 : ℝ) T, HasDerivWithinAt (fun w : ℝ => g w.toNNReal)
+      (A (g w.toNNReal + u w.toNNReal)) (Icc (0 : ℝ) T) w := by
+    intro w hw
+    rw [huv w.toNNReal ((Real.toNNReal_le_toNNReal hw.2).trans_eq (by simp))]
+    exact integral_flow_hasDerivWithinAt hA.continuous hv hg hge T w hw
+  have hf0 : f 0 = x := by simpa using hfe 0
+  have hg0 : g 0 = x := by simpa using hge 0
+  have h0 : f 0 = g 0 := hf0.trans hg0.symm
+  have he := additive_flow_unique hA
+    ((hf.comp continuous_real_toNNReal).continuousOn)
+    ((hg.comp continuous_real_toNNReal).continuousOn) hfd hgd (by simpa using h0)
+    (show (t : ℝ) ∈ Icc (0 : ℝ) T from ⟨t.coe_nonneg, ht⟩)
+  simpa using he
+
+ 
+
+theorem additive_flow_adapted
+    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
+    (u : ℝ≥0 → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
+    (x : Ω → E) (Y : ℝ≥0 → Ω → E) (hYc : ∀ ω, Continuous (fun t => Y t ω))
+    (hYe : ∀ ω t, Y t ω = x ω + ∫ v in (0 : ℝ)..(t : ℝ),
+      A (Y v.toNNReal ω + u v.toNNReal ω))
+    (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (hum : ∀ t, Measurable[F t] (u t)) (hxm : Measurable[F 0] x) :
+    ∀ t, Measurable[F t] (Y t) := by
+  intro T
+  let : MeasurableSpace Ω := F T
+  let uT : ℝ≥0 → Ω → E := fun t ω => u (min t T) ω
+  have huTc (ω : Ω) : Continuous (fun t => uT t ω) :=
+    (huc ω).comp (continuous_id.min continuous_const)
+  have huTm (t : ℝ≥0) : Measurable (uT t) :=
+    (hum (min t T)).mono (F.mono (min_le_right _ _)) le_rfl
+  have hx : Measurable x := hxm.mono (F.mono bot_le) le_rfl
+  obtain ⟨Z, hZc, hZm, hZe⟩ := measurable_additive_flow A L K hA hb uT huTc huTm x hx
+  have he : Y T = Z T := by
+    funext ω
+    exact integral_flow_unique hA (huc ω) (huTc ω) (hYc ω) (hZc ω) (hYe ω) (hZe ω)
+      (T := T) (fun t ht => by simp [uT, min_eq_left ht]) le_rfl
+  rw [he]
+  exact hZm T
+
+end ExactSLEFlow
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem exists_adapted_additive_brownian
+    (A : E → E) (L K : ℝ≥0) (hA : LipschitzWith K A) (hb : ∀ x, ‖A x‖ ≤ L)
+    (x e : E) :
+    ∃ X : ℝ≥0 → Driver → E,
+      (∀ b, Continuous (fun t => X t b)) ∧
+      (∀ t, Measurable[rawFiltration t] (X t)) ∧
+      (∀ b, X 0 b = x) ∧
+      (∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+        ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b)) := by
+  let u : ℝ≥0 → Driver → E := fun t b => b.1 t • e
+  have huc (b : Driver) : Continuous (fun t => u t b) := b.1.continuous.smul continuous_const
+  have hum (t : ℝ≥0) : Measurable (u t) := (driver_eval_measurable t).measurable.smul_const e
+  have humF (t : ℝ≥0) : Measurable[rawFiltration t] (u t) := by
+    have h := Filtration.stronglyAdapted_natural driver_eval_measurable t
+    exact h.measurable.smul_const e
+  obtain ⟨Y, hYc, hYm, hYe⟩ := ExactSLEFlow.measurable_additive_flow A L K hA hb
+    u huc hum (fun _ => x) measurable_const
+  have hYF := ExactSLEFlow.additive_flow_adapted A L K hA hb u huc (fun _ => x)
+    Y hYc hYe rawFiltration humF measurable_const
+  let X : ℝ≥0 → Driver → E := fun t b => Y t b + u t b
+  refine ⟨X, fun b => (hYc b).add (huc b), fun t => (hYF t).add (humF t), ?_, ?_⟩
+  · intro b
+    have h0 := hYe b 0
+    simp only [NNReal.coe_zero, intervalIntegral.integral_same, add_zero] at h0
+    simp [X, u, b.2, h0]
+  · intro s h b
+    have hc : Continuous (fun t : ℝ => A (X t.toNNReal b)) :=
+      hA.continuous.comp (((hYc b).add (huc b)).comp continuous_real_toNNReal)
+    have hi := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
+      (hc.intervalIntegrable (0 : ℝ) s) (hc.intervalIntegrable (s : ℝ) (s+h : ℝ≥0))
+    have hYs := hYe b s
+    have hYt := hYe b (s+h)
+    change Y (s+h) b + b.1 (s+h) • e =
+      Y s b + b.1 s • e + (b.1 (s+h) - b.1 s) • e + ∫ t in (s : ℝ)..(s+h : ℝ≥0), A (X t.toNNReal b)
+    change Y s b = x + ∫ t in (0 : ℝ)..(s : ℝ), A (X t.toNNReal b) at hYs
+    change Y (s+h) b = x + ∫ t in (0 : ℝ)..(s+h : ℝ≥0), A (X t.toNNReal b) at hYt
+    rw [hYs, hYt, ← hi, sub_smul]
+    abel
+
+ 
+omit [CompleteSpace E] in
+lemma adapted_intervalIntegral {Ω : Type*} [MeasurableSpace Ω]
+    (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›) (g : ℝ≥0 → Ω → E)
+    (hc : ∀ ω, Continuous (fun t => g t ω)) (hm : ∀ t, Measurable[F t] (g t))
+    (T : ℝ≥0) :
+    Measurable[F T] (fun ω => ∫ t in (0 : ℝ)..(T : ℝ), g t.toNNReal ω) := by
+  let : MeasurableSpace Ω := F T
+  have hc' (ω : Ω) : Continuous (fun t : ℝ => g (min t.toNNReal T) ω) :=
+    (hc ω).comp (continuous_real_toNNReal.min continuous_const)
+  have hm' (t : ℝ) : Measurable (g (min t.toNNReal T)) :=
+    (hm _).mono (F.mono (min_le_right _ _)) le_rfl
+  have hme := stronglyMeasurable_intervalIntegral_of_continuous
+    (fun t ω => g (min t.toNNReal T) ω) hc' hm' 0 T
+  have he : (fun ω => ∫ t in (0 : ℝ)..(T : ℝ), g t.toNNReal ω) =
+      (fun ω => ∫ t in (0 : ℝ)..(T : ℝ), g (min t.toNNReal T) ω) := by
+    funext ω
+    apply intervalIntegral.integral_congr
+    intro t ht
+    have ht' : t ≤ (T : ℝ) := (by simpa only [uIcc_of_le T.coe_nonneg] using ht : t ∈ Set.Icc 0 (T : ℝ)).2
+    change g t.toNNReal ω = g (min t.toNNReal T) ω
+    have htT : t.toNNReal ≤ T := (Real.toNNReal_le_toNNReal ht').trans_eq (by simp)
+    rw [min_eq_left htT]
+  rw [he]
+  exact hme.measurable
+
+end ExactSLEGenerator
+
+namespace ExactSLEGenerator
+
+ 
+
+theorem martingale_compensated_of_weighted
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (F G : ℝ≥0 → Driver → ℝ)
+    (hFm : ∀ t, Measurable[rawFiltration t] (F t))
+    (hFi : ∀ t, Integrable (F t) W)
+    (hGc : ∀ b, Continuous (fun t => G t b))
+    (hGm : ∀ t, Measurable[rawFiltration t] (G t))
+    (C : ℝ) (hGb : ∀ b t, |G t b| ≤ C)
+    (hw : ∀ (s h : ℝ≥0) (J : Driver → ℝ), Measurable[rawFiltration s] J →
+      (∀ b, |J b| ≤ 1) →
+      (∫ b, J b * F (s+h) b ∂W) - (∫ b, J b * F s b ∂W) =
+        ∫ b, J b * (∫ u in (s : ℝ)..(s+h : ℝ≥0), G u.toNNReal b) ∂W) :
+    Martingale (fun t b => F t b - ∫ u in (0 : ℝ)..(t : ℝ), G u.toNNReal b)
+      rawFiltration W := by
+  let I : ℝ≥0 → Driver → ℝ := fun t b => ∫ u in (0 : ℝ)..(t : ℝ), G u.toNNReal b
+  have hIm (t : ℝ≥0) : Measurable[rawFiltration t] (I t) :=
+    adapted_intervalIntegral rawFiltration G hGc hGm t
+  have hGg (t : ℝ≥0) : Measurable (G t) := (hGm t).mono (rawFiltration.le t) le_rfl
+  have hseg (s t : ℝ) : Integrable (fun b => ∫ u in s..t, G u.toNNReal b) W := by
+    apply Integrable.of_bound
+      (stronglyMeasurable_intervalIntegral_of_continuous (fun u b => G u.toNNReal b)
+        (fun b => (hGc b).comp continuous_real_toNNReal) (fun u => hGg _) s t).aestronglyMeasurable
+      (C * |t-s|)
+    exact ae_of_all W (fun b => intervalIntegral.norm_integral_le_of_norm_le_const
+      (fun u hu => by simpa only [Real.norm_eq_abs] using hGb b u.toNNReal))
+  have hIi (t : ℝ≥0) : Integrable (I t) W := hseg 0 t
+  have hMa (t : ℝ≥0) : StronglyMeasurable[rawFiltration t] (fun b => F t b - I t b) :=
+    ((hFm t).sub (hIm t)).stronglyMeasurable
+  have hMi (t : ℝ≥0) : Integrable (fun b => F t b - I t b) W := (hFi t).sub (hIi t)
+  refine ⟨hMa, ?_⟩
+  intro s t hst
+  symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq (rawFiltration.le s) (hMi t)
+    (fun E hE hEfin => (hMi s).integrableOn) _ (hMa s).aestronglyMeasurable
+  intro E hE hEfin
+  have hEe : MeasurableSet E := rawFiltration.le s _ hE
+  let J : Driver → ℝ := E.indicator (fun _ => 1)
+  have hJm : Measurable[rawFiltration s] J := measurable_const.indicator hE
+  have hJb (b : Driver) : |J b| ≤ 1 := by
+    by_cases hb : b ∈ E <;> simp [J, hb]
+  have hJF (H : Driver → ℝ) : (fun b => J b * H b) = E.indicator H := by
+    funext b
+    by_cases hb : b ∈ E <;> simp [J, hb]
+  have he := hw s (t-s) J hJm hJb
+  rw [add_tsub_cancel_of_le hst] at he
+  simp only [hJF, integral_indicator hEe] at he
+  have hIadd (b : Driver) : I t b = I s b + ∫ u in (s : ℝ)..(t : ℝ), G u.toNNReal b := by
+    have hc := (hGc b).comp continuous_real_toNNReal
+    exact (intervalIntegral.integral_add_adjacent_intervals (μ := volume)
+      (hc.intervalIntegrable (0 : ℝ) s) (hc.intervalIntegrable (s : ℝ) t)).symm
+  have hIset : (∫ b in E, I t b ∂W) = (∫ b in E, I s b ∂W) +
+      (∫ b in E, (∫ u in (s : ℝ)..(t : ℝ), G u.toNNReal b) ∂W) := by
+    simp_rw [hIadd]
+    exact integral_add (hIi s).integrableOn (hseg s t).integrableOn
+  change (∫ b in E, F s b - I s b ∂W) = ∫ b in E, F t b - I t b ∂W
+  rw [integral_sub (hFi s).integrableOn (hIi s).integrableOn,
+    integral_sub (hFi t).integrableOn (hIi t).integrableOn, hIset]
+  linarith
+
+end ExactSLEGenerator
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma intervalIntegral_nnreal_shift (H : ℝ≥0 → E) (start s t : ℝ≥0) :
+    (∫ u in (s : ℝ)..(t : ℝ), H (start+u.toNNReal)) =
+      ∫ u in (start+s : ℝ≥0)..(start+t : ℝ≥0), H u.toNNReal := by
+  calc
+    _ = ∫ u in (s : ℝ)..(t : ℝ), H (((start : ℝ)+u).toNNReal) := by
+      apply intervalIntegral.integral_congr
+      intro u hu
+      have hu0 : 0 ≤ u := le_trans (le_min s.coe_nonneg t.coe_nonneg) hu.1
+      change H (start+u.toNNReal) = H (((start : ℝ)+u).toNNReal)
+      congr 1
+      ext
+      simp only [NNReal.coe_add, Real.coe_toNNReal _ hu0,
+        Real.coe_toNNReal _ (add_nonneg start.coe_nonneg hu0)]
+    _ = _ := by
+      simpa only [NNReal.coe_add] using
+        (intervalIntegral.integral_comp_add_left (fun u : ℝ => H u.toNNReal) (start : ℝ))
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma shift_additive_equation
+    (X : ℝ≥0 → Driver → E) (A : E → E) (e : E)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (start s h : ℝ≥0) (b : Driver) :
+    X (start+(s+h)) b = X (start+s) b + increment (start+s) h b • e +
+      ∫ t in (s : ℝ)..(s+h : ℝ≥0), A (X (start+t.toNNReal) b) := by
+  rw [intervalIntegral_nnreal_shift (fun v => A (X v b)) start s (s+h), ← add_assoc]
+  exact hXeq (start+s) h b
+
+lemma integrable_intervalIntegral_of_bound
+    (W : Measure Driver) [IsProbabilityMeasure W] (G : ℝ≥0 → Driver → ℝ)
+    (hc : ∀ b, Continuous (fun t => G t b)) (hm : ∀ t, Measurable (G t))
+    (C : ℝ) (hb : ∀ b t, |G t b| ≤ C) (s t : ℝ) :
+    Integrable (fun b => ∫ u in s..t, G u.toNNReal b) W := by
+  apply Integrable.of_bound
+    (stronglyMeasurable_intervalIntegral_of_continuous (fun u b => G u.toNNReal b)
+      (fun b => (hc b).comp continuous_real_toNNReal) (fun u => hm _) s t).aestronglyMeasurable
+    (C * |t-s|)
+  exact ae_of_all W (fun b => intervalIntegral.norm_integral_le_of_norm_le_const
+    (fun u hu => by simpa only [Real.norm_eq_abs] using hb b u.toNNReal))
+
+ 
+
+theorem additive_brownian_martingale (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (D : E → E →L[ℝ] ℝ) (hD : ∀ x, HasFDerivAt f (D x) x)
+    (hDc : Continuous D)
+    (C M : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M)
+    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
+    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
+    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
+    (hQc : Continuous (fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0))
+    (hfb : ∀ x, |f x| ≤ M) (hDb : ∀ x, ‖D x‖ ≤ M)
+    (hQb : ∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0| ≤ M) :
+    Martingale (fun t b => f (X t b) - ∫ u in (0 : ℝ)..(t : ℝ),
+      ((1/2 : ℝ) * iteratedDeriv 2 (fun r : ℝ => f (X u.toNNReal b + r • e)) 0 +
+        D (X u.toNNReal b) (A (X u.toNNReal b)))) rawFiltration W := by
+  let Q : E → ℝ := fun x => iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0
+  let V : E → ℝ := fun x => D x (A x)
+  let G : E → ℝ := fun x => (1/2 : ℝ)*Q x + V x
+  have hVc : Continuous V := hDc.clm_apply hAc
+  have hGc : Continuous G := (continuous_const.mul hQc).add hVc
+  have hVb (x : E) : |V x| ≤ M*K := by
+    calc
+      _ = ‖D x (A x)‖ := (Real.norm_eq_abs _).symm
+      _ ≤ ‖D x‖ * ‖A x‖ := (D x).le_opNorm _
+      _ ≤ M*K := mul_le_mul (hDb x) (hAb x) (norm_nonneg _) hM
+  have hGb (x : E) : |G x| ≤ M/2 + M*K := by
+    calc
+      _ ≤ |(1/2 : ℝ)*Q x| + |V x| := abs_add_le _ _
+      _ ≤ (1/2 : ℝ)*M + M*K := add_le_add (by
+        rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/2)]
+        exact mul_le_mul_of_nonneg_left (hQb x) (by norm_num)) (hVb x)
+      _ = _ := by ring
+  have hfc : Continuous f := continuous_iff_continuousAt.mpr (fun x => (hD x).continuousAt)
+  have hXg (t : ℝ≥0) : Measurable (X t) := (hXm t).mono (rawFiltration.le t) le_rfl
+  apply martingale_compensated_of_weighted W (fun t b => f (X t b))
+    (fun t b => G (X t b))
+    (fun t => hfc.measurable.comp (hXm t))
+    (fun t => Integrable.of_bound (hfc.measurable.comp (hXg t)).aestronglyMeasurable M
+      (ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hfb (X t b))))
+    (fun b => hGc.comp (hXc b))
+    (fun t => hGc.measurable.comp (hXm t)) (M/2+M*K) (fun b t => hGb _) 
+  intro s h J hJm hJb
+  have hc (b : Driver) : Continuous (fun t => X (s+t) b) :=
+    (hXc b).comp (continuous_const.add continuous_id)
+  have hdyn := additive_brownian_weighted_dynkin W hW s (fun t b => X (s+t) b)
+    hc (fun t => hXm (s+t)) A hAc K hK hAb e he (shift_additive_equation X A e hXeq s)
+    f D hD hDc C M hC hM hLip hSmooth hThird hQc hfb hDb hQb J hJm hJb h
+  have hiQ (b : Driver) : (∫ u in (0 : ℝ)..(h : ℝ), Q (X (s+u.toNNReal) b)) =
+      ∫ u in (s : ℝ)..(s+h : ℝ≥0), Q (X u.toNNReal b) := by
+    simpa only [NNReal.coe_zero, add_zero] using
+      (intervalIntegral_nnreal_shift (fun t => Q (X t b)) s 0 h)
+  have hiV (b : Driver) : (∫ u in (0 : ℝ)..(h : ℝ), V (X (s+u.toNNReal) b)) =
+      ∫ u in (s : ℝ)..(s+h : ℝ≥0), V (X u.toNNReal b) := by
+    simpa only [NNReal.coe_zero, add_zero] using
+      (intervalIntegral_nnreal_shift (fun t => V (X t b)) s 0 h)
+  change (∫ b, J b * f (X (s+h) b) ∂W) - (∫ b, J b * f (X (s+0) b) ∂W) =
+    (1/2 : ℝ) * (∫ b, J b * (∫ u in (0 : ℝ)..(h : ℝ), Q (X (s+u.toNNReal) b)) ∂W) +
+    (∫ b, J b * (∫ u in (0 : ℝ)..(h : ℝ), V (X (s+u.toNNReal) b)) ∂W) at hdyn
+  simp_rw [add_zero, hiQ, hiV] at hdyn
+  rw [hdyn]
+  have hJ : AEStronglyMeasurable J W := (hJm.mono (rawFiltration.le _) le_rfl).aestronglyMeasurable
+  have hJi : ∀ᵐ b ∂W, ‖J b‖ ≤ 1 := ae_of_all W (fun b => by simpa only [Real.norm_eq_abs] using hJb b)
+  have hQi := integrable_intervalIntegral_of_bound W (fun t b => Q (X t b))
+    (fun b => hQc.comp (hXc b)) (fun t => hQc.measurable.comp (hXg t)) M
+    (fun b t => hQb _) s (s+h : ℝ≥0)
+  have hVi := integrable_intervalIntegral_of_bound W (fun t b => V (X t b))
+    (fun b => hVc.comp (hXc b)) (fun t => hVc.measurable.comp (hXg t)) (M*K)
+    (fun b t => hVb _) s (s+h : ℝ≥0)
+  have hiG (b : Driver) : (∫ u in (s : ℝ)..(s+h : ℝ≥0), G (X u.toNNReal b)) =
+      (1/2 : ℝ)*(∫ u in (s : ℝ)..(s+h : ℝ≥0), Q (X u.toNNReal b)) +
+      (∫ u in (s : ℝ)..(s+h : ℝ≥0), V (X u.toNNReal b)) := by
+    have hqc : Continuous (fun u : ℝ => Q (X u.toNNReal b)) :=
+      hQc.comp ((hXc b).comp continuous_real_toNNReal)
+    have hvc : Continuous (fun u : ℝ => V (X u.toNNReal b)) :=
+      hVc.comp ((hXc b).comp continuous_real_toNNReal)
+    rw [show (fun u : ℝ => G (X u.toNNReal b)) =
+      (fun u : ℝ => (1/2 : ℝ)*Q (X u.toNNReal b) + V (X u.toNNReal b)) from rfl]
+    rw [intervalIntegral.integral_add ((hqc.intervalIntegrable _ _).const_mul _) (hvc.intervalIntegrable _ _),
+      intervalIntegral.integral_const_mul]
+  simp_rw [hiG, mul_add, ← mul_assoc, mul_comm _ (1/2 : ℝ), mul_assoc]
+  rw [integral_add ((hQi.bdd_mul hJ hJi).const_mul _) (hVi.bdd_mul hJ hJi), integral_const_mul]
+
+end ExactSLEGenerator
+
+namespace ExactSLEGenerator
+open Set
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+noncomputable def directional (n : ℕ) (f : E → ℝ) (e : E) : E → ℝ :=
+  Nat.rec f (fun _ g x => fderiv ℝ g x e) n
+
+omit [CompleteSpace E] in
+lemma directional_contDiff (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (e : E) (n : ℕ) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (directional n f e) := by
+  induction n with
+  | zero => exact hf
+  | succ n ih =>
+    exact (ih.fderiv_right (by simp)).clm_apply contDiff_const
+
+omit [CompleteSpace E] in
+lemma directional_compact (f : E → ℝ) (hf : HasCompactSupport f) (e : E) (n : ℕ) :
+    HasCompactSupport (directional n f e) := by
+  induction n with
+  | zero => exact hf
+  | succ n ih => exact ih.fderiv_apply ℝ e
+
+omit [CompleteSpace E] in
+lemma iteratedDeriv_line (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (e x : E) (n : ℕ) :
+    iteratedDeriv n (fun r : ℝ => f (x + r • e)) = fun r => directional n f e (x+r • e) := by
+  induction n with
+  | zero => rfl
+  | succ n ih =>
+    rw [iteratedDeriv_succ,ih]
+    funext r
+    have hd : HasDerivAt (fun r : ℝ => x+r • e) e r := by
+      convert (hasDerivAt_const r x).add ((hasDerivAt_id r).smul_const e) using 1 <;> first | rfl | simp only [zero_add,one_smul]
+    exact (((directional_contDiff f hf e n).differentiable (by simp) (x+r • e)).hasFDerivAt.comp_hasDerivAt r hd).deriv
+
+omit [CompleteSpace E] in
+lemma compact_smooth_fderiv_bound (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
+    (hfc : HasCompactSupport f) :
+    ∃ C : ℝ, ∀ x y, ‖fderiv ℝ f y - fderiv ℝ f x‖ ≤ C * ‖y-x‖ := by
+  have hd : ContDiff ℝ (↑(⊤ : ℕ∞)) (fderiv ℝ f) := hf.fderiv_right (by simp)
+  have hdc : HasCompactSupport (fderiv ℝ f) := hfc.fderiv ℝ
+  have hddc : HasCompactSupport (fderiv ℝ (fderiv ℝ f)) := hdc.fderiv ℝ
+  have hdd : Continuous (fderiv ℝ (fderiv ℝ f)) := hd.continuous_fderiv (by simp)
+  have hbnd : ∃ C : ℝ, ∀ x : E, ‖fderiv ℝ (fderiv ℝ f) x‖ ≤ C := HasCompactSupport.exists_bound_of_continuous (f := fderiv ℝ (fderiv ℝ f)) hddc hdd
+  obtain ⟨C,hC⟩ := hbnd
+  refine ⟨C, fun x y => ?_⟩
+  exact Convex.norm_image_sub_le_of_norm_fderiv_le
+    (fun u _ => hd.differentiable (by simp) u) (fun u _ => hC u)
+    convex_univ (mem_univ x) (mem_univ y)
+
+omit [CompleteSpace E] in
+lemma compact_smooth_line_bound (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
+    (hfc : HasCompactSupport f) (e : E) (n : ℕ) :
+    ∃ C : ℝ, ∀ x w, |iteratedDeriv n (fun r : ℝ => f (x+r • e)) w| ≤ C := by
+  obtain ⟨C,hC⟩ := (directional_compact f hfc e n).exists_bound_of_continuous
+    (directional_contDiff f hf e n).continuous
+  refine ⟨C, fun x w => ?_⟩
+  rw [iteratedDeriv_line f hf e x n]
+  exact (show |directional n f e (x+w • e)| ≤ C by
+    simpa only [Real.norm_eq_abs] using hC (x+w • e))
+
+omit [CompleteSpace E] in
+lemma compact_smooth_test_bounds (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f)
+    (hfc : HasCompactSupport f) (e : E) :
+    ∃ C M : ℝ, 0 ≤ C ∧ 0 ≤ M ∧
+      (∀ x y, ‖fderiv ℝ f y - fderiv ℝ f x‖ ≤ C * ‖y-x‖) ∧
+      (∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x+r • e)) w| ≤ C) ∧
+      (∀ x, |f x| ≤ M) ∧ (∀ x, ‖fderiv ℝ f x‖ ≤ M) ∧
+      (∀ x, |iteratedDeriv 2 (fun r : ℝ => f (x+r • e)) 0| ≤ M) := by
+  obtain ⟨C₀,hC₀⟩ := compact_smooth_fderiv_bound f hf hfc
+  obtain ⟨C₁,hC₁⟩ := compact_smooth_line_bound f hf hfc e 3
+  obtain ⟨M₀,hM₀⟩ := hfc.exists_bound_of_continuous hf.continuous
+  have hdc : HasCompactSupport (fderiv ℝ f) := hfc.fderiv ℝ
+  obtain ⟨M₁,hM₁⟩ := hdc.exists_bound_of_continuous (hf.continuous_fderiv (by simp))
+  obtain ⟨M₂,hM₂⟩ := compact_smooth_line_bound f hf hfc e 2
+  let C := max 0 (max C₀ C₁)
+  let M := max 0 (max M₀ (max M₁ M₂))
+  have hC₀' : C₀ ≤ C := (le_max_left _ _).trans (le_max_right _ _)
+  have hC₁' : C₁ ≤ C := (le_max_right _ _).trans (le_max_right _ _)
+  have hM₀' : M₀ ≤ M := (le_max_left _ _).trans (le_max_right _ _)
+  have hM₁' : M₁ ≤ M := (le_max_left _ _).trans ((le_max_right _ _).trans (le_max_right _ _))
+  have hM₂' : M₂ ≤ M := (le_max_right _ _).trans ((le_max_right _ _).trans (le_max_right _ _))
+  refine ⟨C,M,le_max_left _ _,le_max_left _ _,?_,?_,?_,?_,?_⟩
+  · intro x y
+    exact (hC₀ x y).trans (mul_le_mul_of_nonneg_right hC₀' (norm_nonneg _))
+  · intro x w
+    exact (hC₁ x w).trans hC₁'
+  · intro x
+    exact (show |f x| ≤ M₀ by simpa only [Real.norm_eq_abs] using hM₀ x).trans hM₀'
+  · intro x
+    exact (hM₁ x).trans hM₁'
+  · intro x
+    exact (hM₂ x 0).trans hM₂'
+
+variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+end ExactSLEGenerator
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/FrozenExpectations.lean b/lean/OAI/Probability/SLEGauge/FrozenExpectations.lean
index b9c766fe28f621b0a448b4a18d42758a070fdede..a929c227e10f222ce42b4f8d208d20840534dc27 100644
--- a/lean/OAI/Probability/SLEGauge/FrozenExpectations.lean
+++ b/lean/OAI/Probability/SLEGauge/FrozenExpectations.lean
@@ -1,465 +1,465 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.EulerMeasurability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-lemma expected_frozen_nodes {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mU : ∀ k, MeasurableSet (U k))
-    (h : ℝ≥0) (q : E × ℝ)
-    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
-      ((b,x) ∈ U k ↔ (c,x) ∈ U k))
-    {M : ℝ} (hv : ∀ x b, ‖v x b‖ ≤ M)
-    (hg : ∀ k b x, (b,x) ∈ U k → A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
-    (k : ℕ) :
-    (∫ b : Driver, A.F (frozenNodes v U h q k b).1 (frozenNodes v U h q k b).2 ∂W) ≤
-      A.F q.1 q.2+(k:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2+A.C₃/3*(h:ℝ)*Real.sqrt h) := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hi (i : ℕ) : Integrable (fun b : Driver =>
-      A.F (frozenNodes v U h q i b).1 (frozenNodes v U h q i b).2) W := by
-    have hm := (measurable_frozenNodes v mv U mU h q i).comp
-      (Measurable.of_eval fun time => (show Continuous (fun driver : Driver => driver time) from continuous_eval_const time).measurable)
-    exact (integrable_const A.C₀).mono' (A.mF.comp hm).aestronglyMeasurable
-      (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using A.hF _ _)
-  induction k with
-  | zero => simp [frozenNodes,Measure.real]
-  | succ k ih =>
-    have he := expected_frozen_step hW A v mv U mU h q hU hv hg k
-    rw [integral_sub (hi (k+1)) (hi k)] at he
-    push_cast
-    linarith
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem expected_frozen_limit {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
-    (mU : ∀ n k, MeasurableSet (U n k)) (h : ℕ → ℝ≥0) (k : ℕ → ℕ) (q : E × ℝ)
-    (hU : ∀ n (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h n, b t = c t) → ∀ x,
-      ((b,x) ∈ U n k ↔ (c,x) ∈ U n k))
-    (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
-    (hg : ∀ n k b x, (b,x) ∈ U n k →
-      A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
-    (X : Driver → E × ℝ) (t : ℝ)
-    (hh : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0))
-    (ht : Tendsto (fun n => (k n:ℝ)*(h n:ℝ)) atTop (𝓝 t))
-    (hX : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n => frozenNodes v (U n) (h n) q (k n) b)
-      atTop (𝓝 (X b))) :
-    (∫ b, A.F (X b).1 (X b).2 ∂W) ≤ A.F q.1 q.2 := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hc : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n =>
-      A.F (frozenNodes v (U n) (h n) q (k n) b).1 (frozenNodes v (U n) (h n) q (k n) b).2)
-      atTop (𝓝 (A.F (X b).1 (X b).2)) := by
-    filter_upwards [hX] with b hb
-    exact A.cF.continuousAt.tendsto.comp hb
-  have hi : Tendsto (fun n => ∫ b : Driver,
-      A.F (frozenNodes v (U n) (h n) q (k n) b).1 (frozenNodes v (U n) (h n) q (k n) b).2 ∂W)
-      atTop (𝓝 (∫ b, A.F (X b).1 (X b).2 ∂W)) := by
-    apply tendsto_integral_of_dominated_convergence (fun _ : Driver => A.C₀)
-    · intro n
-      have hm : Measurable (fun b : Driver => (b : ℝ≥0 → ℝ)) :=
-        Measurable.of_eval fun time => (continuous_eval_const time).measurable
-      exact (A.mF.comp ((measurable_frozenNodes v mv (U n) (mU n) (h n) q (k n)).comp hm)).aestronglyMeasurable
-    · exact integrable_const A.C₀
-    · intro n
-      exact Filter.Eventually.of_forall fun b => by
-        simpa only [Real.norm_eq_abs] using A.hF _ _
-    · exact hc
-  have hs : Tendsto (fun n => Real.sqrt (h n:ℝ)) atTop (𝓝 0) := by
-    simpa only [Function.comp_def,Real.sqrt_zero] using (Real.continuous_sqrt.tendsto 0).comp hh
-  have he : Tendsto (fun n => (k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2+
-      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 0) := by
-    have hh' := ht.mul (((tendsto_const_nhds (x := A.Cx*M^2+A.Ccross*M/2)).mul hh).add
-      ((tendsto_const_nhds (x := A.C₃/3)).mul hs))
-    convert! hh' using 1
-    · ext n; ring
-    · simp
-  have hb : Tendsto (fun n => A.F q.1 q.2+(k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2+
-      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 (A.F q.1 q.2)) := by
-    simpa using tendsto_const_nhds.add he
-  exact le_of_tendsto_of_tendsto hi hb (Filter.Eventually.of_forall fun n =>
-    expected_frozen_nodes hW A v mv (U n) (mU n) (h n) q (hU n) hv (hg n) (k n))
-
-end SLEExactGauge.BrownianCalculus
-
- 
-
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.EulerGeneral
-
-variable {Vspace : Type*} [NormedAddCommGroup Vspace] [NormedSpace ℝ Vspace] [CompleteSpace Vspace]
-
-def nodes (f : ℝ → Vspace → Vspace) (h : ℝ) (z : Vspace) : ℕ → Vspace
-  | 0 => z
-  | k+1 => nodes f h z k + h • f ((k:ℝ)*h) (nodes f h z k)
-
-def interpolate (f : ℝ → Vspace → Vspace) (h : ℝ) (z : Vspace) (t : ℝ) : Vspace :=
-  let k := ⌊t/h⌋₊
-  let w := t/h-(k:ℝ)
-  (1-w) • nodes f h z k + w • nodes f h z (k+1)
-
-lemma finite_gronwall {e : ℕ → ℝ} {L h ε : ℝ} (hL : 0 ≤ L) (hh : 0 ≤ h)
-    (hε : 0 ≤ ε) (he0 : e 0 ≤ 0) {N : ℕ}
-    (hrec : ∀ k < N, e (k+1) ≤ (1+L*h)*e k + ε*h) :
-    e N ≤ ε*((N:ℝ)*h)*Real.exp (L*((N:ℝ)*h)) := by
-  have hstep : ∀ k ≤ N, e k ≤ ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h)) := by
-    intro k hk
-    induction k with
-    | zero => simpa using he0
-    | succ k ih =>
-      have hi := ih (by omega)
-      have hk' : k < N := by omega
-      have he : 1+L*h ≤ Real.exp (L*h) := by
-        simpa only [add_comm] using Real.add_one_le_exp (L*h)
-      have he1 : 1 ≤ Real.exp (L*(((k+1:ℕ):ℝ)*h)) :=
-        Real.one_le_exp (by positivity)
-      calc e (k+1)
-          ≤ (1+L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) + ε*h :=
-            (hrec k hk').trans (add_le_add (mul_le_mul_of_nonneg_left hi
-              (show 0 ≤ 1+L*h from by positivity)) le_rfl)
-        _ ≤ Real.exp (L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) +
-            (ε*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
-          exact add_le_add (mul_le_mul_of_nonneg_right he (by positivity))
-            (le_mul_of_one_le_right (mul_nonneg hε hh) he1)
-        _ = ε*(((k+1:ℕ):ℝ)*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
-          have hex : Real.exp (L*(((k+1:ℕ):ℝ)*h)) =
-              Real.exp (L*h)*Real.exp (L*((k:ℝ)*h)) := by
-            rw [← Real.exp_add]
-            congr 1
-            push_cast
-            ring
-          rw [hex]
-          push_cast
-          ring
-  exact hstep N le_rfl
-
-omit [CompleteSpace Vspace] in
-lemma nodes_error {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L h ε T : ℝ}
-    (hL : 0 ≤ L) (hh : 0 ≤ h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
-    (hlip : ∀ t ∈ Icc 0 T, ∀ x y, ‖f t x-f t y‖ ≤ L*‖x-y‖)
-    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
-      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
-    (N : ℕ) (hN : (N:ℝ)*h ≤ T) :
-    ‖nodes f h z N - g ((N:ℝ)*h)‖ ≤ ε*T*Real.exp (L*T) := by
-  have hrec (k : ℕ) (hk : k < N) :
-      ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖ ≤
-        (1+L*h)*‖nodes f h z k-g ((k:ℝ)*h)‖ + ε*h := by
-    have hkT : ((k+1:ℕ):ℝ)*h ≤ T :=
-      (mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.succ_le_of_lt hk) hh).trans hN
-    have hk0 : 0 ≤ (k:ℝ)*h := by positivity
-    have hkT' : (k:ℝ)*h ≤ T := (by push_cast; nlinarith : (k:ℝ)*h ≤ ((k+1:ℕ):ℝ)*h).trans hkT
-    let r := g (((k+1:ℕ):ℝ)*h)-g ((k:ℝ)*h)-h • f ((k:ℝ)*h) (g ((k:ℝ)*h))
-    have he : nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h) =
-        (nodes f h z k-g ((k:ℝ)*h)) +
-          h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h))) - r := by
-      simp only [nodes, r, smul_sub]
-      abel
-    rw [he]
-    calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
-          ‖h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h)))‖ + ‖r‖ :=
-          (norm_sub_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl)
-      _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
-          h*(L*‖nodes f h z k-g ((k:ℝ)*h)‖) + ε*h := by
-        rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hh]
-        exact add_le_add (add_le_add le_rfl (mul_le_mul_of_nonneg_left
-          (hlip _ ⟨hk0,hkT'⟩ _ _) hh)) (hres k hkT)
-      _ = _ := by ring
-  have hb := finite_gronwall (e := fun k => ‖nodes f h z k-g ((k:ℝ)*h)‖)
-    hL hh hε (by simp [nodes, hg0]) hrec
-  have hT : 0 ≤ T := (mul_nonneg (Nat.cast_nonneg _) hh).trans hN
-  exact hb.trans (by gcongr)
-
-omit [CompleteSpace Vspace] in
-lemma residual_bound {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {T s h ε : ℝ}
-    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
-    (hs : 0 ≤ s) (hh : 0 ≤ h) (hsh : s+h ≤ T)
-    (hosc : ∀ t ∈ Icc s (s+h), ‖f t (g t)-f s (g s)‖ ≤ ε) :
-    ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
-  have hsub : Icc s (s+h) ⊆ Icc 0 T := fun t ht => ⟨hs.trans ht.1, ht.2.trans hsh⟩
-  have hd (t : ℝ) (ht : t ∈ Icc s (s+h)) :
-      HasDerivWithinAt (fun u => g u-u • f s (g s))
-        (f t (g t)-f s (g s)) (Icc s (s+h)) t := by
-    convert! ((hder t (hsub ht)).mono hsub).sub
-      (((hasDerivAt_id t).smul_const (f s (g s))).hasDerivWithinAt) using 1
-    simp only [one_smul]
-  have hb := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd hosc
-    (convex_Icc s (s+h)) (left_mem_Icc.mpr (by linarith))
-    (right_mem_Icc.mpr (by linarith))
-  have he : (g (s+h)-(s+h) • f s (g s))-(g s-s • f s (g s)) =
-      g (s+h)-g s-h • f s (g s) := by rw [add_smul]; abel
-  simpa only [he, add_sub_cancel_left, Real.norm_eq_abs, abs_of_nonneg hh] using hb
-
-omit [CompleteSpace Vspace] in
-lemma exists_residual_bound {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {T : ℝ}
-    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
-    (hcf : ContinuousOn (fun t => f t (g t)) (Icc 0 T)) {ε : ℝ} (hε : 0 < ε) :
-    ∃ δ > 0, ∀ h : ℝ, 0 ≤ h → h < δ → ∀ s : ℝ, 0 ≤ s → s+h ≤ T →
-      ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
-  have hu := isCompact_Icc.uniformContinuousOn_of_continuous hcf
-  obtain ⟨δ,hδ,hb⟩ := Metric.uniformContinuousOn_iff.mp hu ε hε
-  refine ⟨δ,hδ, fun h hh hhδ s hs hsT => residual_bound hder hs hh hsT ?_⟩
-  intro u hu
-  apply le_of_lt
-  have huT : u ∈ Icc 0 T := ⟨hs.trans hu.1, hu.2.trans hsT⟩
-  have hsT' : s ∈ Icc 0 T := ⟨hs, by linarith⟩
-  have hd : dist u s < δ := by
-    rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.mpr hu.1)]
-    linarith [hu.2]
-  simpa only [dist_eq_norm] using hb u huT s hsT' hd
-
-omit [CompleteSpace Vspace] in
-lemma norm_interpolate_sub_le {x y z : Vspace} {w E : ℝ}
-    (hw : 0 ≤ w) (hw1 : w ≤ 1) (hx : ‖x-z‖ ≤ E) (hy : ‖y-z‖ ≤ E) :
-    ‖(1-w) • x+w • y-z‖ ≤ E := by
-  have he : (1-w) • x+w • y-z = (1-w) • (x-z)+w • (y-z) := by
-    simp only [sub_smul, one_smul, smul_sub]
-    abel
-  rw [he]
-  calc _ ≤ ‖(1-w) • (x-z)‖+‖w • (y-z)‖ := norm_add_le _ _
-    _ = (1-w)*‖x-z‖+w*‖y-z‖ := by
-      rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs,
-        abs_of_nonneg (sub_nonneg.mpr hw1), abs_of_nonneg hw]
-    _ ≤ (1-w)*E+w*E := add_le_add (mul_le_mul_of_nonneg_left hx (sub_nonneg.mpr hw1))
-      (mul_le_mul_of_nonneg_left hy hw)
-    _ = E := by ring
-
-omit [CompleteSpace Vspace] in
-lemma interpolate_error {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M h ε T t : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hh : 0 < h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
-      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
-    (hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v|)
-    (ht : 0 ≤ t) (htT : t+h ≤ T) :
-    ‖interpolate f h z t-g t‖ ≤ ε*T*Real.exp (L*T)+M*h := by
-  let k := ⌊t/h⌋₊
-  have hf0 : (k:ℝ) ≤ t/h := Nat.floor_le (div_nonneg ht hh.le)
-  have hf1 : t/h < (k:ℝ)+1 := Nat.lt_floor_add_one _
-  have hk0 : 0 ≤ (k:ℝ)*h := by positivity
-  have hkt : (k:ℝ)*h ≤ t := (le_div_iff₀ hh).mp hf0
-  have htks : t < ((k+1:ℕ):ℝ)*h := by
-    push_cast
-    exact (div_lt_iff₀ hh).mp hf1
-  have hksT : ((k+1:ℕ):ℝ)*h ≤ T := by push_cast; nlinarith
-  have hkT : (k:ℝ)*h ≤ T := hkt.trans (by linarith)
-  have hw : 0 ≤ t/h-(k:ℝ) := by linarith
-  have hw1 : t/h-(k:ℝ) ≤ 1 := by linarith
-  apply norm_interpolate_sub_le hw hw1
-  · calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖+‖g ((k:ℝ)*h)-g t‖ :=
-        norm_sub_le_norm_sub_add_norm_sub _ _ _
-      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
-        (nodes_error hL hh.le hε hg0 hlip hres k hkT)
-        ((hpath _ ⟨hk0,hkT⟩ _ ⟨ht,by linarith⟩).trans (by
-          rw [abs_of_nonpos (sub_nonpos.mpr hkt)]
-          apply mul_le_mul_of_nonneg_left _ hM
-          push_cast at htks
-          linarith))
-  · calc _ ≤ ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖+
-          ‖g (((k+1:ℕ):ℝ)*h)-g t‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
-      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
-        (nodes_error hL hh.le hε hg0 hlip hres (k+1) hksT)
-        ((hpath _ ⟨by positivity,hksT⟩ _ ⟨ht,by linarith⟩).trans (by
-          rw [abs_of_nonneg (by linarith : 0 ≤ ((k+1:ℕ):ℝ)*h-t)]
-          apply mul_le_mul_of_nonneg_left _ hM
-          push_cast
-          linarith))
-
-omit [CompleteSpace Vspace] in
-lemma converge {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M T t : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
-    (ht : 0 ≤ t) (htT : t < T)
-    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
-    Tendsto (fun n => interpolate f (h n) z t) atTop (𝓝 (g t)) := by
-  have hT : 0 < T := lt_of_le_of_lt ht htT
-  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
-    intro u hu v hv
-    simpa only [Real.norm_eq_abs] using
-      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
-  rw [Metric.tendsto_nhds]
-  intro ε hε
-  let E := T*Real.exp (L*T)
-  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
-  let η := ε/(2*E)
-  have hη : 0 < η := div_pos hε (by positivity)
-  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
-  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
-  have hev2 : ∀ᶠ n in atTop, h n < T-t :=
-    (tendsto_order.1 hh0).2 (T-t) (sub_pos.mpr htT)
-  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
-    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
-  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
-  rw [dist_eq_norm]
-  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
-    (fun k hk => by
-      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
-      rw [hc] at hk ⊢
-      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n) (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
-    hpath ht (by linarith : t+h n ≤ T)
-  apply lt_of_le_of_lt hb
-  have hηE : η*T*Real.exp (L*T) = ε/2 := by
-    dsimp [η, E]
-    field_simp
-  have hsmall : M*h n < ε/2 := by
-    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
-    nlinarith [hh n]
-  rw [hηE]
-  linarith
-
-omit [CompleteSpace Vspace] in
-lemma converge_uniformOn {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M T U : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
-    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
-    (hU : 0 ≤ U) (hUT : U < T)
-    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
-    TendstoUniformlyOn (fun n => interpolate f (h n) z) g atTop (Icc 0 U) := by
-  have hT : 0 < T := lt_of_le_of_lt hU hUT
-  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
-    intro u hu v hv
-    simpa only [Real.norm_eq_abs] using
-      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  let E := T*Real.exp (L*T)
-  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
-  let η := ε/(2*E)
-  have hη : 0 < η := div_pos hε (by positivity)
-  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
-  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
-  have hev2 : ∀ᶠ n in atTop, h n < T-U :=
-    (tendsto_order.1 hh0).2 (T-U) (sub_pos.mpr hUT)
-  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
-    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
-  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
-  intro t ht
-  rw [dist_comm, dist_eq_norm]
-  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
-    (fun k hk => by
-      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
-      rw [hc] at hk ⊢
-      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n)
-        (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
-    hpath ht.1 (by linarith [ht.2] : t+h n ≤ T)
-  apply lt_of_le_of_lt hb
-  have hηE : η*T*Real.exp (L*T) = ε/2 := by
-    dsimp [η, E]
-    field_simp
-  have hsmall : M*h n < ε/2 := by
-    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
-    nlinarith [hh n]
-  rw [hηE]
-  linarith
-
-end SLEExactGauge.EulerGeneral
-
-namespace SLEExactGauge.EulerGeneral
-variable {Vspace : Type*} [NormedAddCommGroup Vspace] [NormedSpace ℝ Vspace] [CompleteSpace Vspace]
-
-omit [CompleteSpace Vspace] in
-lemma norm_nodes_sub_interpolate {f : ℝ → Vspace → Vspace} {h t M : ℝ} {z : Vspace}
-    (hh : 0 < h) (ht : 0 ≤ t) (hM : 0 ≤ M)
-    (hf : ∀ u ≥ 0, ∀ w, ‖f u w‖ ≤ M) :
-    ‖nodes f h z ⌊t/h⌋₊ - interpolate f h z t‖ ≤ h*M := by
-  let k := ⌊t/h⌋₊
-  let w := t/h-(k:ℝ)
-  have hfloor : (k:ℝ) ≤ t/h := Nat.floor_le (div_nonneg ht hh.le)
-  have hlt : t/h < (k:ℝ)+1 := Nat.lt_floor_add_one (t/h)
-  have hw : 0 ≤ w := sub_nonneg.mpr hfloor
-  have hw1 : w ≤ 1 := by dsimp [w]; linarith
-  have he : nodes f h z k - interpolate f h z t =
-      -(w*h) • f ((k:ℝ)*h) (nodes f h z k) := by
-    dsimp only [interpolate]
-    change nodes f h z k- ((1-w) • nodes f h z k+w • nodes f h z (k+1)) = _
-    rw [nodes, smul_add, sub_smul, one_smul, smul_smul]
-    module
-  change ‖nodes f h z k - interpolate f h z t‖ ≤ _
-  rw [he, norm_smul, Real.norm_eq_abs, abs_neg, abs_of_nonneg (mul_nonneg hw hh.le)]
-  calc w*h*‖f ((k:ℝ)*h) (nodes f h z k)‖
-      ≤ w*h*M := mul_le_mul_of_nonneg_left (hf _ (by positivity) _) (by positivity)
-    _ ≤ h*M := by nlinarith [mul_nonneg hh.le hM]
-
-omit [CompleteSpace Vspace] in
-lemma nodes_converge {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M T t : ℝ}
-    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
-    (hlip : ∀ u ∈ Set.Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
-    (hder : ∀ u ∈ Set.Icc 0 T, HasDerivWithinAt g (f u (g u)) (Set.Icc 0 T) u)
-    (hcf : ContinuousOn (fun u => f u (g u)) (Set.Icc 0 T))
-    (hbound : ∀ u ≥ 0, ∀ w, ‖f u w‖ ≤ M)
-    (ht : 0 ≤ t) (htT : t < T)
-    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
-    Tendsto (fun n => nodes f (h n) z ⌊t/h n⌋₊) atTop (𝓝 (g t)) := by
-  have hi := converge hL hM hg0 hlip hder hcf (fun u hu => hbound u hu.1 _) ht htT hh hh0
-  have hd : Tendsto (fun n => nodes f (h n) z ⌊t/h n⌋₊-interpolate f (h n) z t)
-      atTop (𝓝 0) := by
-    apply squeeze_zero_norm (fun n => norm_nodes_sub_interpolate (hh n) ht hM hbound)
-    simpa using hh0.mul_const M
-  have he := hd.add hi
-  simpa only [sub_add_cancel, zero_add] using he
-
-lemma mesh_times_converge {h : ℕ → ℝ} (hh : ∀ n, 0 < h n)
-    (hh0 : Tendsto h atTop (𝓝 0)) {t : ℝ} (ht : 0 ≤ t) :
-    Tendsto (fun n => (⌊t/h n⌋₊:ℝ)*h n) atTop (𝓝 t) := by
-  have hb (n : ℕ) : 0 ≤ t-(⌊t/h n⌋₊:ℝ)*h n ∧ t-(⌊t/h n⌋₊:ℝ)*h n ≤ h n := by
-    have hl := Nat.floor_le (div_nonneg ht (hh n).le)
-    have hu := Nat.lt_floor_add_one (t/h n)
-    have hl' := (le_div_iff₀ (hh n)).mp hl
-    have hu' := (div_lt_iff₀ (hh n)).mp hu
-    constructor <;> linarith
-  have hz : Tendsto (fun n => t-(⌊t/h n⌋₊:ℝ)*h n) atTop (𝓝 0) :=
-    squeeze_zero (fun n => (hb n).1) (fun n => (hb n).2) hh0
-  have he := (tendsto_const_nhds (x := t) (f := atTop)).sub hz
-  simpa only [sub_sub_cancel, sub_zero] using he
-
-end SLEExactGauge.EulerGeneral
-
-namespace SLEExactGauge.EulerGeneral
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
-end SLEExactGauge.EulerGeneral
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.EulerMeasurability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+lemma expected_frozen_nodes {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (U : ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mU : ∀ k, MeasurableSet (U k))
+    (h : ℝ≥0) (q : E × ℝ)
+    (hU : ∀ (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h, b t = c t) → ∀ x,
+      ((b,x) ∈ U k ↔ (c,x) ∈ U k))
+    {M : ℝ} (hv : ∀ x b, ‖v x b‖ ≤ M)
+    (hg : ∀ k b x, (b,x) ∈ U k → A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
+    (k : ℕ) :
+    (∫ b : Driver, A.F (frozenNodes v U h q k b).1 (frozenNodes v U h q k b).2 ∂W) ≤
+      A.F q.1 q.2+(k:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h:ℝ)^2+A.C₃/3*(h:ℝ)*Real.sqrt h) := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hi (i : ℕ) : Integrable (fun b : Driver =>
+      A.F (frozenNodes v U h q i b).1 (frozenNodes v U h q i b).2) W := by
+    have hm := (measurable_frozenNodes v mv U mU h q i).comp
+      (Measurable.of_eval fun time => (show Continuous (fun driver : Driver => driver time) from continuous_eval_const time).measurable)
+    exact (integrable_const A.C₀).mono' (A.mF.comp hm).aestronglyMeasurable
+      (Filter.Eventually.of_forall fun b => by simpa only [Real.norm_eq_abs] using A.hF _ _)
+  induction k with
+  | zero => simp [frozenNodes,Measure.real]
+  | succ k ih =>
+    have he := expected_frozen_step hW A v mv U mU h q hU hv hg k
+    rw [integral_sub (hi (k+1)) (hi k)] at he
+    push_cast
+    linarith
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem expected_frozen_limit {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)))
+    (mU : ∀ n k, MeasurableSet (U n k)) (h : ℕ → ℝ≥0) (k : ℕ → ℕ) (q : E × ℝ)
+    (hU : ∀ n (k : ℕ) b c, (∀ t ≤ (k:ℝ≥0)*h n, b t = c t) → ∀ x,
+      ((b,x) ∈ U n k ↔ (c,x) ∈ U n k))
+    (M : ℝ) (hv : ∀ x b, ‖v x b‖ ≤ M)
+    (hg : ∀ n k b x, (b,x) ∈ U n k →
+      A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
+    (X : Driver → E × ℝ) (t : ℝ)
+    (hh : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0))
+    (ht : Tendsto (fun n => (k n:ℝ)*(h n:ℝ)) atTop (𝓝 t))
+    (hX : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n => frozenNodes v (U n) (h n) q (k n) b)
+      atTop (𝓝 (X b))) :
+    (∫ b, A.F (X b).1 (X b).2 ∂W) ≤ A.F q.1 q.2 := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hc : ∀ᵐ (b : Driver) ∂W, Tendsto (fun n =>
+      A.F (frozenNodes v (U n) (h n) q (k n) b).1 (frozenNodes v (U n) (h n) q (k n) b).2)
+      atTop (𝓝 (A.F (X b).1 (X b).2)) := by
+    filter_upwards [hX] with b hb
+    exact A.cF.continuousAt.tendsto.comp hb
+  have hi : Tendsto (fun n => ∫ b : Driver,
+      A.F (frozenNodes v (U n) (h n) q (k n) b).1 (frozenNodes v (U n) (h n) q (k n) b).2 ∂W)
+      atTop (𝓝 (∫ b, A.F (X b).1 (X b).2 ∂W)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ : Driver => A.C₀)
+    · intro n
+      have hm : Measurable (fun b : Driver => (b : ℝ≥0 → ℝ)) :=
+        Measurable.of_eval fun time => (continuous_eval_const time).measurable
+      exact (A.mF.comp ((measurable_frozenNodes v mv (U n) (mU n) (h n) q (k n)).comp hm)).aestronglyMeasurable
+    · exact integrable_const A.C₀
+    · intro n
+      exact Filter.Eventually.of_forall fun b => by
+        simpa only [Real.norm_eq_abs] using A.hF _ _
+    · exact hc
+  have hs : Tendsto (fun n => Real.sqrt (h n:ℝ)) atTop (𝓝 0) := by
+    simpa only [Function.comp_def,Real.sqrt_zero] using (Real.continuous_sqrt.tendsto 0).comp hh
+  have he : Tendsto (fun n => (k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2+
+      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 0) := by
+    have hh' := ht.mul (((tendsto_const_nhds (x := A.Cx*M^2+A.Ccross*M/2)).mul hh).add
+      ((tendsto_const_nhds (x := A.C₃/3)).mul hs))
+    convert! hh' using 1
+    · ext n; ring
+    · simp
+  have hb : Tendsto (fun n => A.F q.1 q.2+(k n:ℝ)*((A.Cx*M^2+A.Ccross*M/2)*(h n:ℝ)^2+
+      A.C₃/3*(h n:ℝ)*Real.sqrt (h n))) atTop (𝓝 (A.F q.1 q.2)) := by
+    simpa using tendsto_const_nhds.add he
+  exact le_of_tendsto_of_tendsto hi hb (Filter.Eventually.of_forall fun n =>
+    expected_frozen_nodes hW A v mv (U n) (mU n) (h n) q (hU n) hv (hg n) (k n))
+
+end SLEExactGauge.BrownianCalculus
+
+ 
+
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.EulerGeneral
+
+variable {Vspace : Type*} [NormedAddCommGroup Vspace] [NormedSpace ℝ Vspace] [CompleteSpace Vspace]
+
+def nodes (f : ℝ → Vspace → Vspace) (h : ℝ) (z : Vspace) : ℕ → Vspace
+  | 0 => z
+  | k+1 => nodes f h z k + h • f ((k:ℝ)*h) (nodes f h z k)
+
+def interpolate (f : ℝ → Vspace → Vspace) (h : ℝ) (z : Vspace) (t : ℝ) : Vspace :=
+  let k := ⌊t/h⌋₊
+  let w := t/h-(k:ℝ)
+  (1-w) • nodes f h z k + w • nodes f h z (k+1)
+
+lemma finite_gronwall {e : ℕ → ℝ} {L h ε : ℝ} (hL : 0 ≤ L) (hh : 0 ≤ h)
+    (hε : 0 ≤ ε) (he0 : e 0 ≤ 0) {N : ℕ}
+    (hrec : ∀ k < N, e (k+1) ≤ (1+L*h)*e k + ε*h) :
+    e N ≤ ε*((N:ℝ)*h)*Real.exp (L*((N:ℝ)*h)) := by
+  have hstep : ∀ k ≤ N, e k ≤ ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h)) := by
+    intro k hk
+    induction k with
+    | zero => simpa using he0
+    | succ k ih =>
+      have hi := ih (by omega)
+      have hk' : k < N := by omega
+      have he : 1+L*h ≤ Real.exp (L*h) := by
+        simpa only [add_comm] using Real.add_one_le_exp (L*h)
+      have he1 : 1 ≤ Real.exp (L*(((k+1:ℕ):ℝ)*h)) :=
+        Real.one_le_exp (by positivity)
+      calc e (k+1)
+          ≤ (1+L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) + ε*h :=
+            (hrec k hk').trans (add_le_add (mul_le_mul_of_nonneg_left hi
+              (show 0 ≤ 1+L*h from by positivity)) le_rfl)
+        _ ≤ Real.exp (L*h)*(ε*((k:ℝ)*h)*Real.exp (L*((k:ℝ)*h))) +
+            (ε*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
+          exact add_le_add (mul_le_mul_of_nonneg_right he (by positivity))
+            (le_mul_of_one_le_right (mul_nonneg hε hh) he1)
+        _ = ε*(((k+1:ℕ):ℝ)*h)*Real.exp (L*(((k+1:ℕ):ℝ)*h)) := by
+          have hex : Real.exp (L*(((k+1:ℕ):ℝ)*h)) =
+              Real.exp (L*h)*Real.exp (L*((k:ℝ)*h)) := by
+            rw [← Real.exp_add]
+            congr 1
+            push_cast
+            ring
+          rw [hex]
+          push_cast
+          ring
+  exact hstep N le_rfl
+
+omit [CompleteSpace Vspace] in
+lemma nodes_error {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L h ε T : ℝ}
+    (hL : 0 ≤ L) (hh : 0 ≤ h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
+    (hlip : ∀ t ∈ Icc 0 T, ∀ x y, ‖f t x-f t y‖ ≤ L*‖x-y‖)
+    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
+      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
+    (N : ℕ) (hN : (N:ℝ)*h ≤ T) :
+    ‖nodes f h z N - g ((N:ℝ)*h)‖ ≤ ε*T*Real.exp (L*T) := by
+  have hrec (k : ℕ) (hk : k < N) :
+      ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖ ≤
+        (1+L*h)*‖nodes f h z k-g ((k:ℝ)*h)‖ + ε*h := by
+    have hkT : ((k+1:ℕ):ℝ)*h ≤ T :=
+      (mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.succ_le_of_lt hk) hh).trans hN
+    have hk0 : 0 ≤ (k:ℝ)*h := by positivity
+    have hkT' : (k:ℝ)*h ≤ T := (by push_cast; nlinarith : (k:ℝ)*h ≤ ((k+1:ℕ):ℝ)*h).trans hkT
+    let r := g (((k+1:ℕ):ℝ)*h)-g ((k:ℝ)*h)-h • f ((k:ℝ)*h) (g ((k:ℝ)*h))
+    have he : nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h) =
+        (nodes f h z k-g ((k:ℝ)*h)) +
+          h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h))) - r := by
+      simp only [nodes, r, smul_sub]
+      abel
+    rw [he]
+    calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
+          ‖h • (f ((k:ℝ)*h) (nodes f h z k)-f ((k:ℝ)*h) (g ((k:ℝ)*h)))‖ + ‖r‖ :=
+          (norm_sub_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl)
+      _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖ +
+          h*(L*‖nodes f h z k-g ((k:ℝ)*h)‖) + ε*h := by
+        rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hh]
+        exact add_le_add (add_le_add le_rfl (mul_le_mul_of_nonneg_left
+          (hlip _ ⟨hk0,hkT'⟩ _ _) hh)) (hres k hkT)
+      _ = _ := by ring
+  have hb := finite_gronwall (e := fun k => ‖nodes f h z k-g ((k:ℝ)*h)‖)
+    hL hh hε (by simp [nodes, hg0]) hrec
+  have hT : 0 ≤ T := (mul_nonneg (Nat.cast_nonneg _) hh).trans hN
+  exact hb.trans (by gcongr)
+
+omit [CompleteSpace Vspace] in
+lemma residual_bound {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {T s h ε : ℝ}
+    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
+    (hs : 0 ≤ s) (hh : 0 ≤ h) (hsh : s+h ≤ T)
+    (hosc : ∀ t ∈ Icc s (s+h), ‖f t (g t)-f s (g s)‖ ≤ ε) :
+    ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
+  have hsub : Icc s (s+h) ⊆ Icc 0 T := fun t ht => ⟨hs.trans ht.1, ht.2.trans hsh⟩
+  have hd (t : ℝ) (ht : t ∈ Icc s (s+h)) :
+      HasDerivWithinAt (fun u => g u-u • f s (g s))
+        (f t (g t)-f s (g s)) (Icc s (s+h)) t := by
+    convert! ((hder t (hsub ht)).mono hsub).sub
+      (((hasDerivAt_id t).smul_const (f s (g s))).hasDerivWithinAt) using 1
+    simp only [one_smul]
+  have hb := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd hosc
+    (convex_Icc s (s+h)) (left_mem_Icc.mpr (by linarith))
+    (right_mem_Icc.mpr (by linarith))
+  have he : (g (s+h)-(s+h) • f s (g s))-(g s-s • f s (g s)) =
+      g (s+h)-g s-h • f s (g s) := by rw [add_smul]; abel
+  simpa only [he, add_sub_cancel_left, Real.norm_eq_abs, abs_of_nonneg hh] using hb
+
+omit [CompleteSpace Vspace] in
+lemma exists_residual_bound {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {T : ℝ}
+    (hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt g (f t (g t)) (Icc 0 T) t)
+    (hcf : ContinuousOn (fun t => f t (g t)) (Icc 0 T)) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, ∀ h : ℝ, 0 ≤ h → h < δ → ∀ s : ℝ, 0 ≤ s → s+h ≤ T →
+      ‖g (s+h)-g s-h • f s (g s)‖ ≤ ε*h := by
+  have hu := isCompact_Icc.uniformContinuousOn_of_continuous hcf
+  obtain ⟨δ,hδ,hb⟩ := Metric.uniformContinuousOn_iff.mp hu ε hε
+  refine ⟨δ,hδ, fun h hh hhδ s hs hsT => residual_bound hder hs hh hsT ?_⟩
+  intro u hu
+  apply le_of_lt
+  have huT : u ∈ Icc 0 T := ⟨hs.trans hu.1, hu.2.trans hsT⟩
+  have hsT' : s ∈ Icc 0 T := ⟨hs, by linarith⟩
+  have hd : dist u s < δ := by
+    rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.mpr hu.1)]
+    linarith [hu.2]
+  simpa only [dist_eq_norm] using hb u huT s hsT' hd
+
+omit [CompleteSpace Vspace] in
+lemma norm_interpolate_sub_le {x y z : Vspace} {w E : ℝ}
+    (hw : 0 ≤ w) (hw1 : w ≤ 1) (hx : ‖x-z‖ ≤ E) (hy : ‖y-z‖ ≤ E) :
+    ‖(1-w) • x+w • y-z‖ ≤ E := by
+  have he : (1-w) • x+w • y-z = (1-w) • (x-z)+w • (y-z) := by
+    simp only [sub_smul, one_smul, smul_sub]
+    abel
+  rw [he]
+  calc _ ≤ ‖(1-w) • (x-z)‖+‖w • (y-z)‖ := norm_add_le _ _
+    _ = (1-w)*‖x-z‖+w*‖y-z‖ := by
+      rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs,
+        abs_of_nonneg (sub_nonneg.mpr hw1), abs_of_nonneg hw]
+    _ ≤ (1-w)*E+w*E := add_le_add (mul_le_mul_of_nonneg_left hx (sub_nonneg.mpr hw1))
+      (mul_le_mul_of_nonneg_left hy hw)
+    _ = E := by ring
+
+omit [CompleteSpace Vspace] in
+lemma interpolate_error {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M h ε T t : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hh : 0 < h) (hε : 0 ≤ ε) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hres : ∀ k : ℕ, ((k+1:ℕ):ℝ)*h ≤ T →
+      ‖g (((k+1:ℕ):ℝ)*h) - g ((k:ℝ)*h) - h • f ((k:ℝ)*h) (g ((k:ℝ)*h))‖ ≤ ε*h)
+    (hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v|)
+    (ht : 0 ≤ t) (htT : t+h ≤ T) :
+    ‖interpolate f h z t-g t‖ ≤ ε*T*Real.exp (L*T)+M*h := by
+  let k := ⌊t/h⌋₊
+  have hf0 : (k:ℝ) ≤ t/h := Nat.floor_le (div_nonneg ht hh.le)
+  have hf1 : t/h < (k:ℝ)+1 := Nat.lt_floor_add_one _
+  have hk0 : 0 ≤ (k:ℝ)*h := by positivity
+  have hkt : (k:ℝ)*h ≤ t := (le_div_iff₀ hh).mp hf0
+  have htks : t < ((k+1:ℕ):ℝ)*h := by
+    push_cast
+    exact (div_lt_iff₀ hh).mp hf1
+  have hksT : ((k+1:ℕ):ℝ)*h ≤ T := by push_cast; nlinarith
+  have hkT : (k:ℝ)*h ≤ T := hkt.trans (by linarith)
+  have hw : 0 ≤ t/h-(k:ℝ) := by linarith
+  have hw1 : t/h-(k:ℝ) ≤ 1 := by linarith
+  apply norm_interpolate_sub_le hw hw1
+  · calc _ ≤ ‖nodes f h z k-g ((k:ℝ)*h)‖+‖g ((k:ℝ)*h)-g t‖ :=
+        norm_sub_le_norm_sub_add_norm_sub _ _ _
+      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
+        (nodes_error hL hh.le hε hg0 hlip hres k hkT)
+        ((hpath _ ⟨hk0,hkT⟩ _ ⟨ht,by linarith⟩).trans (by
+          rw [abs_of_nonpos (sub_nonpos.mpr hkt)]
+          apply mul_le_mul_of_nonneg_left _ hM
+          push_cast at htks
+          linarith))
+  · calc _ ≤ ‖nodes f h z (k+1)-g (((k+1:ℕ):ℝ)*h)‖+
+          ‖g (((k+1:ℕ):ℝ)*h)-g t‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
+      _ ≤ ε*T*Real.exp (L*T)+M*h := add_le_add
+        (nodes_error hL hh.le hε hg0 hlip hres (k+1) hksT)
+        ((hpath _ ⟨by positivity,hksT⟩ _ ⟨ht,by linarith⟩).trans (by
+          rw [abs_of_nonneg (by linarith : 0 ≤ ((k+1:ℕ):ℝ)*h-t)]
+          apply mul_le_mul_of_nonneg_left _ hM
+          push_cast
+          linarith))
+
+omit [CompleteSpace Vspace] in
+lemma converge {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M T t : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
+    (ht : 0 ≤ t) (htT : t < T)
+    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
+    Tendsto (fun n => interpolate f (h n) z t) atTop (𝓝 (g t)) := by
+  have hT : 0 < T := lt_of_le_of_lt ht htT
+  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
+    intro u hu v hv
+    simpa only [Real.norm_eq_abs] using
+      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
+  rw [Metric.tendsto_nhds]
+  intro ε hε
+  let E := T*Real.exp (L*T)
+  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
+  let η := ε/(2*E)
+  have hη : 0 < η := div_pos hε (by positivity)
+  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
+  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
+  have hev2 : ∀ᶠ n in atTop, h n < T-t :=
+    (tendsto_order.1 hh0).2 (T-t) (sub_pos.mpr htT)
+  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
+    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
+  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
+  rw [dist_eq_norm]
+  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
+    (fun k hk => by
+      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
+      rw [hc] at hk ⊢
+      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n) (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
+    hpath ht (by linarith : t+h n ≤ T)
+  apply lt_of_le_of_lt hb
+  have hηE : η*T*Real.exp (L*T) = ε/2 := by
+    dsimp [η, E]
+    field_simp
+  have hsmall : M*h n < ε/2 := by
+    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
+    nlinarith [hh n]
+  rw [hηE]
+  linarith
+
+omit [CompleteSpace Vspace] in
+lemma converge_uniformOn {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M T U : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Icc 0 T, HasDerivWithinAt g (f u (g u)) (Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Icc 0 T))
+    (hbound : ∀ u ∈ Icc 0 T, ‖f u (g u)‖ ≤ M)
+    (hU : 0 ≤ U) (hUT : U < T)
+    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
+    TendstoUniformlyOn (fun n => interpolate f (h n) z) g atTop (Icc 0 U) := by
+  have hT : 0 < T := lt_of_le_of_lt hU hUT
+  have hpath : ∀ u ∈ Icc 0 T, ∀ v ∈ Icc 0 T, ‖g u-g v‖ ≤ M*|u-v| := by
+    intro u hu v hv
+    simpa only [Real.norm_eq_abs] using
+      Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hder hbound (convex_Icc 0 T) hv hu
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  let E := T*Real.exp (L*T)
+  have hE : 0 < E := mul_pos hT (Real.exp_pos _)
+  let η := ε/(2*E)
+  have hη : 0 < η := div_pos hε (by positivity)
+  obtain ⟨δ,hδ,hres⟩ := exists_residual_bound hder hcf hη
+  have hev1 : ∀ᶠ n in atTop, h n < δ := (tendsto_order.1 hh0).2 δ hδ
+  have hev2 : ∀ᶠ n in atTop, h n < T-U :=
+    (tendsto_order.1 hh0).2 (T-U) (sub_pos.mpr hUT)
+  have hev3 : ∀ᶠ n in atTop, h n < ε/(2*(M+1)) :=
+    (tendsto_order.1 hh0).2 _ (div_pos hε (by positivity))
+  filter_upwards [hev1,hev2,hev3] with n hnδ hnT hnε
+  intro t ht
+  rw [dist_comm, dist_eq_norm]
+  have hb := interpolate_error hL hM (hh n) hη.le hg0 hlip
+    (fun k hk => by
+      have hc : ((k+1:ℕ):ℝ)*h n = (k:ℝ)*h n+h n := by push_cast; ring
+      rw [hc] at hk ⊢
+      exact hres (h n) (hh n).le hnδ ((k:ℝ)*h n)
+        (mul_nonneg (Nat.cast_nonneg _) (hh n).le) hk)
+    hpath ht.1 (by linarith [ht.2] : t+h n ≤ T)
+  apply lt_of_le_of_lt hb
+  have hηE : η*T*Real.exp (L*T) = ε/2 := by
+    dsimp [η, E]
+    field_simp
+  have hsmall : M*h n < ε/2 := by
+    have hhε := (lt_div_iff₀ (by positivity : 0 < 2*(M+1))).mp hnε
+    nlinarith [hh n]
+  rw [hηE]
+  linarith
+
+end SLEExactGauge.EulerGeneral
+
+namespace SLEExactGauge.EulerGeneral
+variable {Vspace : Type*} [NormedAddCommGroup Vspace] [NormedSpace ℝ Vspace] [CompleteSpace Vspace]
+
+omit [CompleteSpace Vspace] in
+lemma norm_nodes_sub_interpolate {f : ℝ → Vspace → Vspace} {h t M : ℝ} {z : Vspace}
+    (hh : 0 < h) (ht : 0 ≤ t) (hM : 0 ≤ M)
+    (hf : ∀ u ≥ 0, ∀ w, ‖f u w‖ ≤ M) :
+    ‖nodes f h z ⌊t/h⌋₊ - interpolate f h z t‖ ≤ h*M := by
+  let k := ⌊t/h⌋₊
+  let w := t/h-(k:ℝ)
+  have hfloor : (k:ℝ) ≤ t/h := Nat.floor_le (div_nonneg ht hh.le)
+  have hlt : t/h < (k:ℝ)+1 := Nat.lt_floor_add_one (t/h)
+  have hw : 0 ≤ w := sub_nonneg.mpr hfloor
+  have hw1 : w ≤ 1 := by dsimp [w]; linarith
+  have he : nodes f h z k - interpolate f h z t =
+      -(w*h) • f ((k:ℝ)*h) (nodes f h z k) := by
+    dsimp only [interpolate]
+    change nodes f h z k- ((1-w) • nodes f h z k+w • nodes f h z (k+1)) = _
+    rw [nodes, smul_add, sub_smul, one_smul, smul_smul]
+    module
+  change ‖nodes f h z k - interpolate f h z t‖ ≤ _
+  rw [he, norm_smul, Real.norm_eq_abs, abs_neg, abs_of_nonneg (mul_nonneg hw hh.le)]
+  calc w*h*‖f ((k:ℝ)*h) (nodes f h z k)‖
+      ≤ w*h*M := mul_le_mul_of_nonneg_left (hf _ (by positivity) _) (by positivity)
+    _ ≤ h*M := by nlinarith [mul_nonneg hh.le hM]
+
+omit [CompleteSpace Vspace] in
+lemma nodes_converge {f : ℝ → Vspace → Vspace} {g : ℝ → Vspace} {z : Vspace} {L M T t : ℝ}
+    (hL : 0 ≤ L) (hM : 0 ≤ M) (hg0 : g 0 = z)
+    (hlip : ∀ u ∈ Set.Icc 0 T, ∀ x y, ‖f u x-f u y‖ ≤ L*‖x-y‖)
+    (hder : ∀ u ∈ Set.Icc 0 T, HasDerivWithinAt g (f u (g u)) (Set.Icc 0 T) u)
+    (hcf : ContinuousOn (fun u => f u (g u)) (Set.Icc 0 T))
+    (hbound : ∀ u ≥ 0, ∀ w, ‖f u w‖ ≤ M)
+    (ht : 0 ≤ t) (htT : t < T)
+    {h : ℕ → ℝ} (hh : ∀ n, 0 < h n) (hh0 : Tendsto h atTop (𝓝 0)) :
+    Tendsto (fun n => nodes f (h n) z ⌊t/h n⌋₊) atTop (𝓝 (g t)) := by
+  have hi := converge hL hM hg0 hlip hder hcf (fun u hu => hbound u hu.1 _) ht htT hh hh0
+  have hd : Tendsto (fun n => nodes f (h n) z ⌊t/h n⌋₊-interpolate f (h n) z t)
+      atTop (𝓝 0) := by
+    apply squeeze_zero_norm (fun n => norm_nodes_sub_interpolate (hh n) ht hM hbound)
+    simpa using hh0.mul_const M
+  have he := hd.add hi
+  simpa only [sub_add_cancel, zero_add] using he
+
+lemma mesh_times_converge {h : ℕ → ℝ} (hh : ∀ n, 0 < h n)
+    (hh0 : Tendsto h atTop (𝓝 0)) {t : ℝ} (ht : 0 ≤ t) :
+    Tendsto (fun n => (⌊t/h n⌋₊:ℝ)*h n) atTop (𝓝 t) := by
+  have hb (n : ℕ) : 0 ≤ t-(⌊t/h n⌋₊:ℝ)*h n ∧ t-(⌊t/h n⌋₊:ℝ)*h n ≤ h n := by
+    have hl := Nat.floor_le (div_nonneg ht (hh n).le)
+    have hu := Nat.lt_floor_add_one (t/h n)
+    have hl' := (le_div_iff₀ (hh n)).mp hl
+    have hu' := (div_lt_iff₀ (hh n)).mp hu
+    constructor <;> linarith
+  have hz : Tendsto (fun n => t-(⌊t/h n⌋₊:ℝ)*h n) atTop (𝓝 0) :=
+    squeeze_zero (fun n => (hb n).1) (fun n => (hb n).2) hh0
+  have he := (tendsto_const_nhds (x := t) (f := atTop)).sub hz
+  simpa only [sub_sub_cancel, sub_zero] using he
+
+end SLEExactGauge.EulerGeneral
+
+namespace SLEExactGauge.EulerGeneral
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+end SLEExactGauge.EulerGeneral
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/FutureSupport.lean b/lean/OAI/Probability/SLEGauge/FutureSupport.lean
index 45a3b629eb94fa5031f1d157dba100f93aaf3880..716887ecd4bc59a70802be5393fe20babf8921e9 100644
--- a/lean/OAI/Probability/SLEGauge/FutureSupport.lean
+++ b/lean/OAI/Probability/SLEGauge/FutureSupport.lean
@@ -1,416 +1,416 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialRestart
-import OAI.Probability.SLEGauge.RadonKernels
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [BorelSpace X]
-
- 
-
-theorem random_set_sub_null_of_compact_means (μ ν : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (μ ω)] [∀ ω, IsFiniteMeasureOnCompacts (ν ω)]
-    (W : Measure Ω) [IsFiniteMeasureOnCompacts (ν ∘ₘ W)]
-    (D : Set (Ω × X)) (hD : MeasurableSet D)
-    (hle : ∀ᵐ ω ∂W, μ ω≤ν ω)
-    (he : ∀ K : Set X, IsCompact K →
-      (∫⁻ω, ν ω ({x | (ω,x)∈D}∩K) ∂W) = ∫⁻ω, μ ω ({x | (ω,x)∈D}∩K) ∂W) :
-    ∀ᵐ ω ∂W, (ν ω-μ ω) {x | (ω,x)∈D}=0 := by
-  have := isSFiniteKernel_of_locally_finite μ
-  have := isSFiniteKernel_of_locally_finite ν
-  let K := CompactExhaustion.choice X
-  have hfinite (n : ℕ) : (∫⁻ω, ν ω ({x | (ω,x)∈D}∩K n) ∂W)<∞ :=
-    lt_of_le_of_lt (lintegral_mono (fun ω => measure_mono inter_subset_right)) (by
-      rw [← Measure.bind_apply (K.isCompact n).measurableSet ν.aemeasurable]
-      exact (K.isCompact n).measure_lt_top)
-  have heq (n : ℕ) : (fun ω => μ ω ({x | (ω,x)∈D}∩K n)) =ᵐ[W]
-      (fun ω => ν ω ({x | (ω,x)∈D}∩K n)) := by
-    apply ae_eq_of_ae_le_of_lintegral_le
-    · filter_upwards [hle] with ω hω
-      exact hω _
-    · rw [← he (K n) (K.isCompact n)]
-      exact (hfinite n).ne
-    · exact (Kernel.measurable_kernel_prodMk_left
-        (hD.inter (measurable_snd (K.isCompact n).measurableSet))).aemeasurable
-    · exact (he (K n) (K.isCompact n)).le
-  filter_upwards [hle,ae_all_iff.mpr heq] with ω hω heω
-  have hsec : MeasurableSet {x | (ω,x)∈D} := measurable_prodMk_left hD
-  have hnull (n : ℕ) : (ν ω-μ ω) ({x | (ω,x)∈D}∩K n)=0 := by
-    have : IsFiniteMeasure ((μ ω).restrict (K n)) :=
-      ⟨by simpa only [Measure.restrict_apply_univ] using (K.isCompact n).measure_lt_top (μ := μ ω)⟩
-    rw [← Measure.restrict_apply hsec,
-      Measure.restrict_sub_eq_restrict_sub_restrict (K.isCompact n).measurableSet,
-      Measure.sub_apply hsec (Measure.restrict_mono Subset.rfl hω),
-      Measure.restrict_apply hsec,Measure.restrict_apply hsec,heω n,tsub_self]
-  have heU : (⋃ n, {x | (ω,x)∈D}∩K n)={x | (ω,x)∈D} := by
-    rw [← inter_iUnion,K.iUnion_eq,inter_univ]
-  rw [← heU]
-  exact measure_iUnion_null hnull
-
-end SLEExactGauge.MassVersions
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma survivingGreen_eq_zero_iff (a : ℝ) (t : ℝ≥0)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    survivingGreen a t b z=0 ↔ ¬alive a b.1 t z := by
-  unfold survivingGreen
-  split_ifs with h <;> simp [h,physicalGreen,Real.exp_ne_zero]
-
- 
-
-lemma deadSet_raw_measurable (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
-    MeasurableSet[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)]
-      {p : ExactSLEGenerator.Driver × UpperPlane | ¬alive a p.1.1 t p.2.1} := by
-  have h := (actual_surviving_green a ha t).1 (measurableSet_singleton (0 : ℝ))
-  change MeasurableSet[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)]
-    {p : ExactSLEGenerator.Driver × UpperPlane | survivingGreen a t p.1 p.2.1=0} at h
-  simpa only [survivingGreen_eq_zero_iff] using h
-
- 
-
-theorem conditional_future_support (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
-    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂μ b) W)
-    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂ν b) W)
-    (hle : ∀ᵐ b ∂W, μ b≤ν b)
-    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
-      MeasurableSet[rawFiltration t] U →
-      (∫b in U, (∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
-        ∫b in U, (∫z,f z*survivingGreen a t b z.1 ∂upperArea) ∂W) :
-    ∀ᵐ b ∂W, (ν b-μ b) {z : UpperPlane | ¬alive a b.1 t z.1}=0 := by
-  let D : Set (ExactSLEGenerator.Driver × UpperPlane) :=
-    {p | ¬alive a p.1.1 t p.2.1}
-  have hD := deadSet_raw_measurable a ha t
-  have hD₀ : MeasurableSet D := prod_sigma_le (rawFiltration.le t) _ _ hD
-  have := average_finiteOnCompacts ν W hν
-  apply random_set_sub_null_of_compact_means μ ν W D hD₀ hle
-  intro K hK
-  let E : Set (ExactSLEGenerator.Driver × UpperPlane) := D∩Prod.snd ⁻¹' K
-  have hE : MeasurableSet[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] E :=
-    hD.inter (measurable_snd hK.measurableSet)
-  have hE₀ : MeasurableSet E := prod_sigma_le (rawFiltration.le t) _ _ hE
-  let H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞ := E.indicator (fun _ => 1)
-  have hH : Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H :=
-    measurable_const.indicator hE
-  have hid := compact_conditional_to_random_tests a ha W hW t μ ν hμ hν he H hH
-  have hev (ρ : Measure UpperPlane) (b : ExactSLEGenerator.Driver) :
-      (∫⁻z, H (b,z) ∂ρ)=ρ ({z | (b,z)∈D}∩K) := by
-    have hh := lintegral_indicator_const_comp (μ:=ρ)
-      ((measurable_const (a:=b)).prodMk measurable_id) hE₀ (1 : ℝ≥0∞)
-    have hs : (fun z : UpperPlane => (b,id z)) ⁻¹' E = {z | (b,z)∈D}∩K := by
-      ext z
-      rfl
-    rw [hs, one_mul] at hh
-    exact hh
-  have hz (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-      H (b,z)*ENNReal.ofReal (survivingGreen a t b z.1)=0 := by
-    by_cases h : (b,z)∈E
-    · rw [(survivingGreen_eq_zero_iff a t b z.1).mpr h.1,ENNReal.ofReal_zero,mul_zero]
-    · simp only [H,indicator_of_notMem h,zero_mul]
-  simpa only [hev,hz,lintegral_zero,add_zero] using hid
-
-end ExactSLEGreen
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-lemma radialTime_ne_zero (a : ℝ) (ha : 0<a) (q : ℝ) (hq : 0<q) (hq1 : q<1)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im) :
-    radialTime a b.1 z (z.im*q) ≠ 0 := by
-  intro he
-  have hh := (radialTime_attained a ha z hz (z.im*q) (mul_pos hz hq)
-    (mul_le_of_le_one_right hz.le hq1.le) b (by rw [he]; exact ENNReal.zero_ne_top)).2
-  rw [he,ENNReal.toNNReal_zero,actualState_zero ha b hz] at hh
-  simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one] at hh
-  nlinarith
-
-lemma actualUpperMass_zero (a : ℝ) (ha : 0<a) (q : ℝ) (hq : 0<q) (hq1 : q<1)
-    (b : ExactSLEGenerator.Driver) : actualUpperMass a q 0 b=0 := by
-  have hz (z : UpperPlane) : timeMassDensity a q 0 b z.1=0 := by
-    simp only [timeMassDensity,le_zero_iff,ite_eq_right (radialTime_ne_zero a ha q hq hq1 b z.1 z.2)]
-  ext A hA
-  simp [actualUpperMass,hz]
-
-lemma level_step_lt_one (a : ℝ) (ha : 1/4<a) (j : ℕ) :
-    Real.exp (-2*a*((j:ℝ)+1))<1 := by
-  apply Real.exp_lt_one_iff.mpr
-  have ha0 : 0<a := by linarith
-  have hj : (0:ℝ)≤j := Nat.cast_nonneg _
-  nlinarith
-
- 
-
-theorem recoveredMass_zero (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
-    (b : ExactSLEGenerator.Driver) : recoveredMass a ha s c hc 0 b=0 := by
-  apply radonLimit_eq
-  intro f
-  simp only [actualMassFunctionals_apply, actualUpperMass_zero a (by linarith)
-    _ (Real.exp_pos _) (level_step_lt_one a ha _),integral_zero_measure,mul_zero,Finset.sum_const_zero]
-  exact tendsto_const_nhds
-
-end ExactSLEGreen
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] [BorelSpace X]
-
- 
-
-theorem compact_mass_memLp (μ : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (μ ω)] (W : Measure Ω) (p : ℝ≥0∞)
-    (hp : ∀ f : C_c(X,ℝ), MemLp (fun ω => ∫x, f x ∂μ ω) p W)
-    (K : Set X) (hK : IsCompact K) :
-    MemLp (fun ω => (μ ω K).toReal) p W := by
-  obtain ⟨g,hg1,-,hgc,hg01⟩ := exists_continuous_one_zero_of_isCompact
-    hK isClosed_empty (disjoint_empty K)
-  let f : C_c(X,ℝ) := ⟨g,hgc⟩
-  apply (hp f).mono'
-    ((μ.measurable_coe hK.measurableSet).ennreal_toReal.aestronglyMeasurable)
-  apply Eventually.of_forall
-  intro ω
-  rw [Real.norm_of_nonneg ENNReal.toReal_nonneg]
-  have hpos : 0≤∫x,f x∂μ ω := integral_nonneg (fun x => (hg01 x).1)
-  exact ENNReal.toReal_le_of_le_ofReal hpos
-    (f.integrable.measure_le_integral (Eventually.of_forall (fun x => (hg01 x).1))
-      (fun x hx => le_of_eq ((hg1 hx).symm)))
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-noncomputable def initialGreen (a : ℝ) (z : UpperPlane) : ℝ :=
-  Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)
-
-lemma initialGreen_pos (a : ℝ) (z : UpperPlane) : 0 < initialGreen a z :=
-  mul_pos (Real.exp_pos _) (Real.exp_pos _)
-
-lemma initialGreen_continuous (a : ℝ) : Continuous (initialGreen a) := by
-  have hc : Continuous (fun z : UpperPlane => (z.1,(0:ℝ))) :=
-    continuous_subtype_val.prodMk continuous_const
-  have hg : Continuous (fun z : UpperPlane => physicalGreen a (z.1,0)) := by
-    apply continuous_iff_continuousAt.mpr
-    intro z
-    exact (hasFDerivAt_physicalGreen a (z.1,0) z.2).continuousAt.comp (f := fun z : UpperPlane => (z.1,(0:ℝ))) hc.continuousAt
-  exact continuous_const.mul hg
-
- 
-noncomputable def initialGreenMeasure (a : ℝ) : Measure UpperPlane :=
-  upperArea.withDensity (fun z => ENNReal.ofReal (initialGreen a z))
-
-lemma initialGreenMeasure_compact_integrable (a : ℝ) (f : C_c(UpperPlane,ℝ)) :
-    Integrable f (initialGreenMeasure a) := by
-  rw [initialGreenMeasure,integrable_withDensity_iff_integrable_smul'
-    (initialGreen_continuous a).measurable.ennreal_ofReal
-    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
-  simp only [ENNReal.toReal_ofReal (initialGreen_pos a _).le,smul_eq_mul]
-  let g : C_c(UpperPlane,ℝ) := ⟨⟨fun z => initialGreen a z*f z,
-    (initialGreen_continuous a).mul f.continuous⟩, f.hasCompactSupport.mul_left⟩
-  exact g.integrable
-
-instance initialGreenMeasure_finite (a : ℝ) : IsFiniteMeasureOnCompacts (initialGreenMeasure a) :=
-  finiteOnCompacts_of_integrable_tests _ (initialGreenMeasure_compact_integrable a)
-
-lemma initialGreenMeasure_compact_eval (a : ℝ) (f : C_c(UpperPlane,ℝ)) :
-    (∫z,f z∂initialGreenMeasure a) = ∫z, f z*initialGreen a z∂upperArea := by
-  rw [initialGreenMeasure,integral_withDensity_eq_integral_toReal_smul]
-  · simp only [ENNReal.toReal_ofReal (initialGreen_pos a _).le,smul_eq_mul,mul_comm]
-  · exact (initialGreen_continuous a).measurable.ennreal_ofReal
-  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
-
- 
-
-theorem initialGreenMeasure_eq_average (a : ℝ)
-    (μ : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (μ b)] (W : Measure ExactSLEGenerator.Driver)
-    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂μ b) W)
-    (he : ∀ f : C_c(UpperPlane,ℝ), (∫b, (∫z,f z∂μ b) ∂W)=∫z, f z*initialGreen a z∂upperArea) :
-    μ ∘ₘ W=initialGreenMeasure a := by
-  have := average_finiteOnCompacts μ W hμ
-  apply Measure.ext_of_integral_eq_on_compactlySupported
-  intro f
-  rw [average_compact_eval μ W hμ,initialGreenMeasure_compact_eval,he]
-
- 
-
-theorem terminal_null_of_area_null (a : ℝ)
-    (μ : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (μ b)] (W : Measure ExactSLEGenerator.Driver)
-    (he : μ ∘ₘ W=initialGreenMeasure a) (A : Set UpperPlane)
-    (hA : MeasurableSet A) (hz : upperArea A=0) : ∀ᵐ b ∂W, μ b A=0 := by
-  apply (lintegral_eq_zero_iff (μ.measurable_coe hA)).mp
-  rw [← Measure.bind_apply hA μ.aemeasurable,he]
-  exact withDensity_absolutelyContinuous _ _ hz
-
-end ExactSLEGreen
-end
- 
-
- 
-section
- 
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma remainingMassDensity_mean_antitone (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
-    Antitone (fun t : ℝ≥0 => ∫ b, remainingMassDensity a q t b z.1 ∂W) := by
-  let ρ := z.1.im * q
-  have hρ : 0 < ρ := mul_pos z.2 hq
-  have hr : ρ ≤ z.1.im := mul_le_of_le_one_right z.2.le hq1
-  let K : ℝ≥0 → ExactSLEGenerator.Driver → ℝ := fun t b =>
-    if (t : ℝ≥0∞) < radialTime a b.1 z.1 ρ then radialWeight a b.1 z.1 ρ else 0
-  have hK (t : ℝ≥0) : Integrable (K t) W := by
-    have hm : MeasurableSet {b : ExactSLEGenerator.Driver |
-        (t : ℝ≥0∞) < radialTime a b.1 z.1 ρ} := by
-      have hle : MeasurableSet {b : ExactSLEGenerator.Driver |
-          radialTime a b.1 z.1 ρ ≤ (t : ℝ≥0∞)} := rawFiltration.le t _
-        ((radialTime_stopping a (by linarith) z.1 z.2 ρ hρ hr).measurableSet_le t)
-      simpa only [compl_ofPred, not_le] using hle.compl
-    apply ((radialWeight_mean W hW a ha z.1 z.2 ρ hρ hr).1.indicator hm).congr
-    filter_upwards [] with b
-    simp only [K, indicator_apply, mem_ofPred_eq]
-  have hmean (t : ℝ≥0) : (∫b, remainingMassDensity a q t b z.1 ∂W) =
-      Real.exp (-alpha a*Real.log 2) * ∫b, K t b ∂W := by
-    have H := (radial_conditional_weight W hW a ha z.1 z.2 ρ hρ hr
-      (fun _ => t) (isStoppingTime_const rawFiltration t) univ MeasurableSet.univ).2
-    simp only [setIntegral_univ] at H
-    simp only [remainingMassDensity,ite_eq_left z.2,integral_const_mul]
-    rw [← H]
-  intro s t hst
-  dsimp only
-  rw [hmean,hmean]
-  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-  apply integral_mono (hK t) (hK s)
-  intro b
-  have hn := (radialWeight_bound a ha z.1 z.2 ρ hρ hr b).1
-  have hst' : (s : ℝ≥0∞) ≤ t := ENNReal.coe_le_coe.mpr hst
-  dsimp only [K]
-  split_ifs with ht hs hs
-  · exact le_rfl
-  · exact False.elim (hs (hst'.trans_lt ht))
-  · exact hn
-  · exact le_rfl
-
-lemma survivingGreen_mean_antitone (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    Antitone (fun t : ℝ≥0 => ∫b, survivingGreen a t b z.1 ∂W) := by
-  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
-  have hq (n : ℕ) : 0 < q n ∧ q n ≤ 1 := by
-    constructor
-    · exact div_pos zero_lt_one (by positivity)
-    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
-  have hq0 : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-  have hlim (t : ℝ≥0) :
-      Tendsto (fun n => ∫b, remainingMassDensity a (q n) t b z.1 ∂W) atTop
-        (𝓝 (∫b, survivingGreen a t b z.1 ∂W)) := by
-    apply tendsto_integral_of_dominated_convergence (fun b => survivingGreen a t b z.1)
-    · intro n
-      exact (remainingMassDensity_mean_bound a ha W hW (q n) (hq n).1 (hq n).2 t z.1 z.2).1.aestronglyMeasurable
-    · exact (actual_surviving_green a ha t).2.2 W hW z.1 z.2 |>.1
-    · intro n
-      filter_upwards [] with b
-      rw [Real.norm_of_nonneg (remainingMassDensity_nonneg a (q n) t b z.1)]
-      exact remainingMassDensity_le_surviving a (q n) t b z.1 z.2
-    · exact Eventually.of_forall (fun b => remaining_to_surviving a ha t q hq hq0 b z.1 z.2)
-  intro s t hst
-  exact le_of_tendsto_of_tendsto' (hlim t) (hlim s) (fun n =>
-    remainingMassDensity_mean_antitone a ha W hW z (q n) (hq n).1 (hq n).2 hst)
-
-lemma survivingGreen_eq_killed (a : ℝ) (t : ℝ≥0)
-    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-    survivingGreen a t b z.1 = Real.exp (-alpha a * Real.log 2) *
-      killedObservable a z (physicalGreen a) t b := by
-  unfold survivingGreen killedObservable
-  split_ifs <;> simp only [mul_zero]
-
-lemma physicalGreen_measurable (a : ℝ) : Measurable (physicalGreen a) := by
-  unfold physicalGreen logGreen Q X Y J
-  fun_prop
-
-lemma killedGreen_nonneg (a : ℝ) (z : UpperPlane) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    0 ≤ killedObservable a z (physicalGreen a) t b :=
-  killedObservable_nonneg a z (physicalGreen a) (fun _ _ => (Real.exp_pos _).le) t b
-
-lemma killedGreen_integrable (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (t : ℝ≥0) :
-    Integrable (killedObservable a z (physicalGreen a) t) W := by
-  have h := (actual_surviving_green a ha t).2.2 W hW z.1 z.2 |>.1
-  simpa only [survivingGreen_eq_killed,integrable_const_mul_iff (isUnit_iff_ne_zero.mpr (Real.exp_ne_zero _))] using h
-
-lemma killedGreen_mean_antitone (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    Antitone (fun t : ℝ≥0 => ∫b, killedObservable a z (physicalGreen a) t b ∂W) := by
-  intro s t hst
-  have h := survivingGreen_mean_antitone a ha W hW z hst
-  simpa only [survivingGreen_eq_killed,integral_const_mul,
-    mul_le_mul_iff_right₀ (Real.exp_pos _)] using h
-
-lemma killedGreen_lintegral_antitone (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    Antitone (fun t : ℝ≥0 => ∫⁻b, ENNReal.ofReal
-      (killedObservable a z (physicalGreen a) t b) ∂W) := by
-  intro s t hst
-  dsimp only
-  rw [← ofReal_integral_eq_lintegral_ofReal (killedGreen_integrable a ha W hW z t)
-    (Eventually.of_forall (killedGreen_nonneg a z t)),
-    ← ofReal_integral_eq_lintegral_ofReal (killedGreen_integrable a ha W hW z s)
-    (Eventually.of_forall (killedGreen_nonneg a z s))]
-  exact ENNReal.ofReal_le_ofReal (killedGreen_mean_antitone a ha W hW z hst)
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialRestart
+import OAI.Probability.SLEGauge.RadonKernels
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [BorelSpace X]
+
+ 
+
+theorem random_set_sub_null_of_compact_means (μ ν : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (μ ω)] [∀ ω, IsFiniteMeasureOnCompacts (ν ω)]
+    (W : Measure Ω) [IsFiniteMeasureOnCompacts (ν ∘ₘ W)]
+    (D : Set (Ω × X)) (hD : MeasurableSet D)
+    (hle : ∀ᵐ ω ∂W, μ ω≤ν ω)
+    (he : ∀ K : Set X, IsCompact K →
+      (∫⁻ω, ν ω ({x | (ω,x)∈D}∩K) ∂W) = ∫⁻ω, μ ω ({x | (ω,x)∈D}∩K) ∂W) :
+    ∀ᵐ ω ∂W, (ν ω-μ ω) {x | (ω,x)∈D}=0 := by
+  have := isSFiniteKernel_of_locally_finite μ
+  have := isSFiniteKernel_of_locally_finite ν
+  let K := CompactExhaustion.choice X
+  have hfinite (n : ℕ) : (∫⁻ω, ν ω ({x | (ω,x)∈D}∩K n) ∂W)<∞ :=
+    lt_of_le_of_lt (lintegral_mono (fun ω => measure_mono inter_subset_right)) (by
+      rw [← Measure.bind_apply (K.isCompact n).measurableSet ν.aemeasurable]
+      exact (K.isCompact n).measure_lt_top)
+  have heq (n : ℕ) : (fun ω => μ ω ({x | (ω,x)∈D}∩K n)) =ᵐ[W]
+      (fun ω => ν ω ({x | (ω,x)∈D}∩K n)) := by
+    apply ae_eq_of_ae_le_of_lintegral_le
+    · filter_upwards [hle] with ω hω
+      exact hω _
+    · rw [← he (K n) (K.isCompact n)]
+      exact (hfinite n).ne
+    · exact (Kernel.measurable_kernel_prodMk_left
+        (hD.inter (measurable_snd (K.isCompact n).measurableSet))).aemeasurable
+    · exact (he (K n) (K.isCompact n)).le
+  filter_upwards [hle,ae_all_iff.mpr heq] with ω hω heω
+  have hsec : MeasurableSet {x | (ω,x)∈D} := measurable_prodMk_left hD
+  have hnull (n : ℕ) : (ν ω-μ ω) ({x | (ω,x)∈D}∩K n)=0 := by
+    have : IsFiniteMeasure ((μ ω).restrict (K n)) :=
+      ⟨by simpa only [Measure.restrict_apply_univ] using (K.isCompact n).measure_lt_top (μ := μ ω)⟩
+    rw [← Measure.restrict_apply hsec,
+      Measure.restrict_sub_eq_restrict_sub_restrict (K.isCompact n).measurableSet,
+      Measure.sub_apply hsec (Measure.restrict_mono Subset.rfl hω),
+      Measure.restrict_apply hsec,Measure.restrict_apply hsec,heω n,tsub_self]
+  have heU : (⋃ n, {x | (ω,x)∈D}∩K n)={x | (ω,x)∈D} := by
+    rw [← inter_iUnion,K.iUnion_eq,inter_univ]
+  rw [← heU]
+  exact measure_iUnion_null hnull
+
+end SLEExactGauge.MassVersions
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma survivingGreen_eq_zero_iff (a : ℝ) (t : ℝ≥0)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    survivingGreen a t b z=0 ↔ ¬alive a b.1 t z := by
+  unfold survivingGreen
+  split_ifs with h <;> simp [h,physicalGreen,Real.exp_ne_zero]
+
+ 
+
+lemma deadSet_raw_measurable (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
+    MeasurableSet[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)]
+      {p : ExactSLEGenerator.Driver × UpperPlane | ¬alive a p.1.1 t p.2.1} := by
+  have h := (actual_surviving_green a ha t).1 (measurableSet_singleton (0 : ℝ))
+  change MeasurableSet[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)]
+    {p : ExactSLEGenerator.Driver × UpperPlane | survivingGreen a t p.1 p.2.1=0} at h
+  simpa only [survivingGreen_eq_zero_iff] using h
+
+ 
+
+theorem conditional_future_support (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
+    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂μ b) W)
+    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂ν b) W)
+    (hle : ∀ᵐ b ∂W, μ b≤ν b)
+    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
+      MeasurableSet[rawFiltration t] U →
+      (∫b in U, (∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
+        ∫b in U, (∫z,f z*survivingGreen a t b z.1 ∂upperArea) ∂W) :
+    ∀ᵐ b ∂W, (ν b-μ b) {z : UpperPlane | ¬alive a b.1 t z.1}=0 := by
+  let D : Set (ExactSLEGenerator.Driver × UpperPlane) :=
+    {p | ¬alive a p.1.1 t p.2.1}
+  have hD := deadSet_raw_measurable a ha t
+  have hD₀ : MeasurableSet D := prod_sigma_le (rawFiltration.le t) _ _ hD
+  have := average_finiteOnCompacts ν W hν
+  apply random_set_sub_null_of_compact_means μ ν W D hD₀ hle
+  intro K hK
+  let E : Set (ExactSLEGenerator.Driver × UpperPlane) := D∩Prod.snd ⁻¹' K
+  have hE : MeasurableSet[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] E :=
+    hD.inter (measurable_snd hK.measurableSet)
+  have hE₀ : MeasurableSet E := prod_sigma_le (rawFiltration.le t) _ _ hE
+  let H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞ := E.indicator (fun _ => 1)
+  have hH : Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H :=
+    measurable_const.indicator hE
+  have hid := compact_conditional_to_random_tests a ha W hW t μ ν hμ hν he H hH
+  have hev (ρ : Measure UpperPlane) (b : ExactSLEGenerator.Driver) :
+      (∫⁻z, H (b,z) ∂ρ)=ρ ({z | (b,z)∈D}∩K) := by
+    have hh := lintegral_indicator_const_comp (μ:=ρ)
+      ((measurable_const (a:=b)).prodMk measurable_id) hE₀ (1 : ℝ≥0∞)
+    have hs : (fun z : UpperPlane => (b,id z)) ⁻¹' E = {z | (b,z)∈D}∩K := by
+      ext z
+      rfl
+    rw [hs, one_mul] at hh
+    exact hh
+  have hz (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+      H (b,z)*ENNReal.ofReal (survivingGreen a t b z.1)=0 := by
+    by_cases h : (b,z)∈E
+    · rw [(survivingGreen_eq_zero_iff a t b z.1).mpr h.1,ENNReal.ofReal_zero,mul_zero]
+    · simp only [H,indicator_of_notMem h,zero_mul]
+  simpa only [hev,hz,lintegral_zero,add_zero] using hid
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+lemma radialTime_ne_zero (a : ℝ) (ha : 0<a) (q : ℝ) (hq : 0<q) (hq1 : q<1)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im) :
+    radialTime a b.1 z (z.im*q) ≠ 0 := by
+  intro he
+  have hh := (radialTime_attained a ha z hz (z.im*q) (mul_pos hz hq)
+    (mul_le_of_le_one_right hz.le hq1.le) b (by rw [he]; exact ENNReal.zero_ne_top)).2
+  rw [he,ENNReal.toNNReal_zero,actualState_zero ha b hz] at hh
+  simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one] at hh
+  nlinarith
+
+lemma actualUpperMass_zero (a : ℝ) (ha : 0<a) (q : ℝ) (hq : 0<q) (hq1 : q<1)
+    (b : ExactSLEGenerator.Driver) : actualUpperMass a q 0 b=0 := by
+  have hz (z : UpperPlane) : timeMassDensity a q 0 b z.1=0 := by
+    simp only [timeMassDensity,le_zero_iff,ite_eq_right (radialTime_ne_zero a ha q hq hq1 b z.1 z.2)]
+  ext A hA
+  simp [actualUpperMass,hz]
+
+lemma level_step_lt_one (a : ℝ) (ha : 1/4<a) (j : ℕ) :
+    Real.exp (-2*a*((j:ℝ)+1))<1 := by
+  apply Real.exp_lt_one_iff.mpr
+  have ha0 : 0<a := by linarith
+  have hj : (0:ℝ)≤j := Nat.cast_nonneg _
+  nlinarith
+
+ 
+
+theorem recoveredMass_zero (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
+    (b : ExactSLEGenerator.Driver) : recoveredMass a ha s c hc 0 b=0 := by
+  apply radonLimit_eq
+  intro f
+  simp only [actualMassFunctionals_apply, actualUpperMass_zero a (by linarith)
+    _ (Real.exp_pos _) (level_step_lt_one a ha _),integral_zero_measure,mul_zero,Finset.sum_const_zero]
+  exact tendsto_const_nhds
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] [BorelSpace X]
+
+ 
+
+theorem compact_mass_memLp (μ : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (μ ω)] (W : Measure Ω) (p : ℝ≥0∞)
+    (hp : ∀ f : C_c(X,ℝ), MemLp (fun ω => ∫x, f x ∂μ ω) p W)
+    (K : Set X) (hK : IsCompact K) :
+    MemLp (fun ω => (μ ω K).toReal) p W := by
+  obtain ⟨g,hg1,-,hgc,hg01⟩ := exists_continuous_one_zero_of_isCompact
+    hK isClosed_empty (disjoint_empty K)
+  let f : C_c(X,ℝ) := ⟨g,hgc⟩
+  apply (hp f).mono'
+    ((μ.measurable_coe hK.measurableSet).ennreal_toReal.aestronglyMeasurable)
+  apply Eventually.of_forall
+  intro ω
+  rw [Real.norm_of_nonneg ENNReal.toReal_nonneg]
+  have hpos : 0≤∫x,f x∂μ ω := integral_nonneg (fun x => (hg01 x).1)
+  exact ENNReal.toReal_le_of_le_ofReal hpos
+    (f.integrable.measure_le_integral (Eventually.of_forall (fun x => (hg01 x).1))
+      (fun x hx => le_of_eq ((hg1 hx).symm)))
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+noncomputable def initialGreen (a : ℝ) (z : UpperPlane) : ℝ :=
+  Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)
+
+lemma initialGreen_pos (a : ℝ) (z : UpperPlane) : 0 < initialGreen a z :=
+  mul_pos (Real.exp_pos _) (Real.exp_pos _)
+
+lemma initialGreen_continuous (a : ℝ) : Continuous (initialGreen a) := by
+  have hc : Continuous (fun z : UpperPlane => (z.1,(0:ℝ))) :=
+    continuous_subtype_val.prodMk continuous_const
+  have hg : Continuous (fun z : UpperPlane => physicalGreen a (z.1,0)) := by
+    apply continuous_iff_continuousAt.mpr
+    intro z
+    exact (hasFDerivAt_physicalGreen a (z.1,0) z.2).continuousAt.comp (f := fun z : UpperPlane => (z.1,(0:ℝ))) hc.continuousAt
+  exact continuous_const.mul hg
+
+ 
+noncomputable def initialGreenMeasure (a : ℝ) : Measure UpperPlane :=
+  upperArea.withDensity (fun z => ENNReal.ofReal (initialGreen a z))
+
+lemma initialGreenMeasure_compact_integrable (a : ℝ) (f : C_c(UpperPlane,ℝ)) :
+    Integrable f (initialGreenMeasure a) := by
+  rw [initialGreenMeasure,integrable_withDensity_iff_integrable_smul'
+    (initialGreen_continuous a).measurable.ennreal_ofReal
+    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  simp only [ENNReal.toReal_ofReal (initialGreen_pos a _).le,smul_eq_mul]
+  let g : C_c(UpperPlane,ℝ) := ⟨⟨fun z => initialGreen a z*f z,
+    (initialGreen_continuous a).mul f.continuous⟩, f.hasCompactSupport.mul_left⟩
+  exact g.integrable
+
+instance initialGreenMeasure_finite (a : ℝ) : IsFiniteMeasureOnCompacts (initialGreenMeasure a) :=
+  finiteOnCompacts_of_integrable_tests _ (initialGreenMeasure_compact_integrable a)
+
+lemma initialGreenMeasure_compact_eval (a : ℝ) (f : C_c(UpperPlane,ℝ)) :
+    (∫z,f z∂initialGreenMeasure a) = ∫z, f z*initialGreen a z∂upperArea := by
+  rw [initialGreenMeasure,integral_withDensity_eq_integral_toReal_smul]
+  · simp only [ENNReal.toReal_ofReal (initialGreen_pos a _).le,smul_eq_mul,mul_comm]
+  · exact (initialGreen_continuous a).measurable.ennreal_ofReal
+  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
+
+ 
+
+theorem initialGreenMeasure_eq_average (a : ℝ)
+    (μ : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (μ b)] (W : Measure ExactSLEGenerator.Driver)
+    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂μ b) W)
+    (he : ∀ f : C_c(UpperPlane,ℝ), (∫b, (∫z,f z∂μ b) ∂W)=∫z, f z*initialGreen a z∂upperArea) :
+    μ ∘ₘ W=initialGreenMeasure a := by
+  have := average_finiteOnCompacts μ W hμ
+  apply Measure.ext_of_integral_eq_on_compactlySupported
+  intro f
+  rw [average_compact_eval μ W hμ,initialGreenMeasure_compact_eval,he]
+
+ 
+
+theorem terminal_null_of_area_null (a : ℝ)
+    (μ : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (μ b)] (W : Measure ExactSLEGenerator.Driver)
+    (he : μ ∘ₘ W=initialGreenMeasure a) (A : Set UpperPlane)
+    (hA : MeasurableSet A) (hz : upperArea A=0) : ∀ᵐ b ∂W, μ b A=0 := by
+  apply (lintegral_eq_zero_iff (μ.measurable_coe hA)).mp
+  rw [← Measure.bind_apply hA μ.aemeasurable,he]
+  exact withDensity_absolutelyContinuous _ _ hz
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma remainingMassDensity_mean_antitone (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
+    Antitone (fun t : ℝ≥0 => ∫ b, remainingMassDensity a q t b z.1 ∂W) := by
+  let ρ := z.1.im * q
+  have hρ : 0 < ρ := mul_pos z.2 hq
+  have hr : ρ ≤ z.1.im := mul_le_of_le_one_right z.2.le hq1
+  let K : ℝ≥0 → ExactSLEGenerator.Driver → ℝ := fun t b =>
+    if (t : ℝ≥0∞) < radialTime a b.1 z.1 ρ then radialWeight a b.1 z.1 ρ else 0
+  have hK (t : ℝ≥0) : Integrable (K t) W := by
+    have hm : MeasurableSet {b : ExactSLEGenerator.Driver |
+        (t : ℝ≥0∞) < radialTime a b.1 z.1 ρ} := by
+      have hle : MeasurableSet {b : ExactSLEGenerator.Driver |
+          radialTime a b.1 z.1 ρ ≤ (t : ℝ≥0∞)} := rawFiltration.le t _
+        ((radialTime_stopping a (by linarith) z.1 z.2 ρ hρ hr).measurableSet_le t)
+      simpa only [compl_ofPred, not_le] using hle.compl
+    apply ((radialWeight_mean W hW a ha z.1 z.2 ρ hρ hr).1.indicator hm).congr
+    filter_upwards [] with b
+    simp only [K, indicator_apply, mem_ofPred_eq]
+  have hmean (t : ℝ≥0) : (∫b, remainingMassDensity a q t b z.1 ∂W) =
+      Real.exp (-alpha a*Real.log 2) * ∫b, K t b ∂W := by
+    have H := (radial_conditional_weight W hW a ha z.1 z.2 ρ hρ hr
+      (fun _ => t) (isStoppingTime_const rawFiltration t) univ MeasurableSet.univ).2
+    simp only [setIntegral_univ] at H
+    simp only [remainingMassDensity,ite_eq_left z.2,integral_const_mul]
+    rw [← H]
+  intro s t hst
+  dsimp only
+  rw [hmean,hmean]
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+  apply integral_mono (hK t) (hK s)
+  intro b
+  have hn := (radialWeight_bound a ha z.1 z.2 ρ hρ hr b).1
+  have hst' : (s : ℝ≥0∞) ≤ t := ENNReal.coe_le_coe.mpr hst
+  dsimp only [K]
+  split_ifs with ht hs hs
+  · exact le_rfl
+  · exact False.elim (hs (hst'.trans_lt ht))
+  · exact hn
+  · exact le_rfl
+
+lemma survivingGreen_mean_antitone (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    Antitone (fun t : ℝ≥0 => ∫b, survivingGreen a t b z.1 ∂W) := by
+  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
+  have hq (n : ℕ) : 0 < q n ∧ q n ≤ 1 := by
+    constructor
+    · exact div_pos zero_lt_one (by positivity)
+    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
+  have hq0 : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  have hlim (t : ℝ≥0) :
+      Tendsto (fun n => ∫b, remainingMassDensity a (q n) t b z.1 ∂W) atTop
+        (𝓝 (∫b, survivingGreen a t b z.1 ∂W)) := by
+    apply tendsto_integral_of_dominated_convergence (fun b => survivingGreen a t b z.1)
+    · intro n
+      exact (remainingMassDensity_mean_bound a ha W hW (q n) (hq n).1 (hq n).2 t z.1 z.2).1.aestronglyMeasurable
+    · exact (actual_surviving_green a ha t).2.2 W hW z.1 z.2 |>.1
+    · intro n
+      filter_upwards [] with b
+      rw [Real.norm_of_nonneg (remainingMassDensity_nonneg a (q n) t b z.1)]
+      exact remainingMassDensity_le_surviving a (q n) t b z.1 z.2
+    · exact Eventually.of_forall (fun b => remaining_to_surviving a ha t q hq hq0 b z.1 z.2)
+  intro s t hst
+  exact le_of_tendsto_of_tendsto' (hlim t) (hlim s) (fun n =>
+    remainingMassDensity_mean_antitone a ha W hW z (q n) (hq n).1 (hq n).2 hst)
+
+lemma survivingGreen_eq_killed (a : ℝ) (t : ℝ≥0)
+    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+    survivingGreen a t b z.1 = Real.exp (-alpha a * Real.log 2) *
+      killedObservable a z (physicalGreen a) t b := by
+  unfold survivingGreen killedObservable
+  split_ifs <;> simp only [mul_zero]
+
+lemma physicalGreen_measurable (a : ℝ) : Measurable (physicalGreen a) := by
+  unfold physicalGreen logGreen Q X Y J
+  fun_prop
+
+lemma killedGreen_nonneg (a : ℝ) (z : UpperPlane) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    0 ≤ killedObservable a z (physicalGreen a) t b :=
+  killedObservable_nonneg a z (physicalGreen a) (fun _ _ => (Real.exp_pos _).le) t b
+
+lemma killedGreen_integrable (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (t : ℝ≥0) :
+    Integrable (killedObservable a z (physicalGreen a) t) W := by
+  have h := (actual_surviving_green a ha t).2.2 W hW z.1 z.2 |>.1
+  simpa only [survivingGreen_eq_killed,integrable_const_mul_iff (isUnit_iff_ne_zero.mpr (Real.exp_ne_zero _))] using h
+
+lemma killedGreen_mean_antitone (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    Antitone (fun t : ℝ≥0 => ∫b, killedObservable a z (physicalGreen a) t b ∂W) := by
+  intro s t hst
+  have h := survivingGreen_mean_antitone a ha W hW z hst
+  simpa only [survivingGreen_eq_killed,integral_const_mul,
+    mul_le_mul_iff_right₀ (Real.exp_pos _)] using h
+
+lemma killedGreen_lintegral_antitone (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    Antitone (fun t : ℝ≥0 => ∫⁻b, ENNReal.ofReal
+      (killedObservable a z (physicalGreen a) t b) ∂W) := by
+  intro s t hst
+  dsimp only
+  rw [← ofReal_integral_eq_lintegral_ofReal (killedGreen_integrable a ha W hW z t)
+    (Eventually.of_forall (killedGreen_nonneg a z t)),
+    ← ofReal_integral_eq_lintegral_ofReal (killedGreen_integrable a ha W hW z s)
+    (Eventually.of_forall (killedGreen_nonneg a z s))]
+  exact ENNReal.ofReal_le_ofReal (killedGreen_mean_antitone a ha W hW z hst)
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/GaugeBounds.lean b/lean/OAI/Probability/SLEGauge/GaugeBounds.lean
index 3f5b898d65e1f746c57b39b484bee1884489f01c..d8d44a8614d088545ad948bfaa53ea728be30332 100644
--- a/lean/OAI/Probability/SLEGauge/GaugeBounds.lean
+++ b/lean/OAI/Probability/SLEGauge/GaugeBounds.lean
@@ -1,439 +1,439 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-
-namespace SLEExactGauge.Target
-
-open scoped ContinuousMapZero
-
-end SLEExactGauge.Target
-
- 
-
- 
-
-namespace SLEExactGauge.BrownianLaw
-open SLEExactGauge.Target
-open ProbabilityTheory Function
-open scoped ContinuousMapZero
-
-instance : PolishSpace Driver := ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace
-
-def code (b : Driver) : ℚ → ℝ := fun q => b (Real.toNNReal (q:ℝ))
-
-lemma measurable_code : Measurable code :=
-  Measurable.of_eval fun _ => (continuous_eval_const _).measurable
-
-lemma dense_rational_time : DenseRange (fun q : ℚ => Real.toNNReal (q:ℝ)) := by
-  have hd : Surjective Real.toNNReal := fun x => ⟨(x:ℝ),Real.toNNReal_coe⟩
-  exact hd.denseRange.comp Rat.denseRange_cast continuous_real_toNNReal
-
-lemma code_injective : Injective code := by
-  intro b c h
-  apply ContinuousMapZero.ext
-  have he := dense_rational_time.equalizer (map_continuous b) (map_continuous c) h
-  exact congrFun he
-
-lemma code_embedding : MeasurableEmbedding code := measurable_code.measurableEmbedding code_injective
-
-lemma evaluation_law_unique {W V : Measure Driver}
-    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W)
-    (hV : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) V) : W = V := by
-  let ev : Driver → (ℝ≥0 → ℝ) := fun b => b
-  have mev : Measurable ev := Measurable.of_eval fun time => (continuous_eval_const time).measurable
-  have hw : IsProjectiveLimit (W.map ev) BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map I.measurable_restrict mev]
-    exact (hW.hasLaw I).map_eq
-  have hv : IsProjectiveLimit (V.map ev) BrownianReal.projectiveFamily := by
-    intro I
-    rw [Measure.map_map I.measurable_restrict mev]
-    exact (hV.hasLaw I).map_eq
-  have he := hw.unique hv
-  have mr : Measurable (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ))) :=
-    Measurable.of_eval fun _ => measurable_pi_apply _
-  have hc := congrArg (Measure.map (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ)))) he
-  rw [Measure.map_map mr mev, Measure.map_map mr mev] at hc
-  exact code_embedding.map_injective hc
-
-end SLEExactGauge.BrownianLaw
-
-namespace SLEExactGauge.Deterministic
-
-open SLEExactGauge.Target
-
- 
-
-def diameterCost (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=
-  if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal)
-
-lemma diameterCost_zero {h : ℝ → ℝ} (h0 : h 0 = 0) :
-    diameterCost h 0 = 0 := by
-  simp [diameterCost, h0]
-
-lemma gauge_nonneg {h : ℝ → ℝ} (hh : IsHausdorffGauge h)
-    {r : ℝ} (hr : 0 ≤ r) : 0 ≤ h r := by
-  rw [← hh.2.2.1]
-  exact hh.2.1 (by simp) hr hr
-
- 
-
-theorem hausdorff_countable_covers {h : ℝ → ℝ} (h0 : h 0 = 0) (E : Set ℂ) :
-    hausdorff h E =
-      ⨆ (δ : ℝ≥0∞) (_ : 0 < δ),
-        ⨅ (U : ℕ → Set ℂ) (_ : E ⊆ ⋃ n, U n) (_ : ∀ n, Metric.ediam (U n) ≤ δ),
-          ∑' n, diameterCost h (Metric.ediam (U n)) := by
-  change OuterMeasure.mkMetric (diameterCost h) E = _
-  rw [OuterMeasure.coe_mkMetric, Measure.mkMetric_apply]
-  congr 1 with δ
-  congr 1 with hδ
-  congr 1 with U
-  congr 1 with hEU
-  congr 1 with hU
-  apply tsum_congr
-  intro n
-  by_cases hn : (U n).Nonempty
-  · simp [hn]
-  · have hempty : U n = ∅ := Set.not_nonempty_iff_eq_empty.mp hn
-    simp [hempty, diameterCost_zero h0]
-
- 
-theorem hausdorff_tendsto_content (h : ℝ → ℝ) (E : Set ℂ) :
-    Tendsto
-      (fun δ : ℝ≥0∞ =>
-        OuterMeasure.mkMetric'.pre (fun U : Set ℂ => diameterCost h (Metric.ediam U)) δ E)
-      (𝓝[>] 0) (𝓝 (hausdorff h E)) :=
-  OuterMeasure.mkMetric'.tendsto_pre _ _
-
- 
-
-theorem mass_bound {h : ℝ → ℝ} {ν : OuterMeasure ℂ} {δ c : ℝ≥0∞}
-    (hδ : 0 < δ) (hc0 : c ≠ 0) (hc : c ≠ ∞)
-    (hν : ∀ U : Set ℂ, Metric.ediam U ≤ δ →
-      ν U ≤ c * diameterCost h (Metric.ediam U)) :
-    ν ≤ c • hausdorff h := by
-  have hm : ν ≤ OuterMeasure.mkMetric (c • diameterCost h) :=
-    OuterMeasure.le_mkMetric _ ν δ hδ hν
-  change ν ≤ c • OuterMeasure.mkMetric (diameterCost h)
-  simpa only [OuterMeasure.mkMetric_smul _ hc hc0] using hm
-
- 
-
-theorem hausdorff_pos_of_mass_bound {h : ℝ → ℝ} {ν : OuterMeasure ℂ}
-    {δ c : ℝ≥0∞} {E : Set ℂ}
-    (hδ : 0 < δ) (hc0 : c ≠ 0) (hc : c ≠ ∞)
-    (hν : ∀ U : Set ℂ, Metric.ediam U ≤ δ →
-      ν U ≤ c * diameterCost h (Metric.ediam U))
-    (hE : 0 < ν E) : 0 < hausdorff h E := by
-  have H := mass_bound hδ hc0 hc hν E
-  by_contra hn
-  have hz : hausdorff h E = 0 := le_antisymm (not_lt.mp hn) zero_le
-  have hle : ν E ≤ 0 := by simpa [smul_apply, hz] using H
-  exact hE.not_ge hle
-
- 
-
-theorem hausdorff_le_liminf_cost {ι : ℕ → Type*} [∀ n, Countable (ι n)]
-    (h : ℝ → ℝ) (E : Set ℂ) (r : ℕ → ℝ≥0∞)
-    (hr : Tendsto r atTop (𝓝 0)) (U : ∀ n, ι n → Set ℂ)
-    (hdiam : ∀ᶠ n in atTop, ∀ i, Metric.ediam (U n i) ≤ r n)
-    (hcover : ∀ᶠ n in atTop, E ⊆ ⋃ i, U n i) :
-    hausdorff h E ≤ liminf (fun n => ∑' i, diameterCost h (Metric.ediam (U n i))) atTop := by
-  change OuterMeasure.mkMetric (diameterCost h) E ≤ _
-  rw [OuterMeasure.coe_mkMetric]
-  exact Measure.mkMetric_le_liminf_tsum E r hr U hdiam hcover _
-
-section Intervals
-
-variable {X Ω : Type*}
-
- 
-
-theorem all_real_intervals_of_rational (ν : OuterMeasure X) (γ : ℝ → X)
-    (H : ∀ p q : ℚ, 0 < p → p < q →
-      0 < ν (γ '' Icc (p : ℝ) (q : ℝ)) ∧
-        ν (γ '' Icc (p : ℝ) (q : ℝ)) < ∞) :
-    ∀ s t : ℝ, 0 < s → s < t →
-      0 < ν (γ '' Icc s t) ∧ ν (γ '' Icc s t) < ∞ := by
-  intro s t hs hst
-  obtain ⟨p, hsp, hpt⟩ := exists_rat_btwn hst
-  obtain ⟨q, hpq, hqt⟩ := exists_rat_btwn hpt
-  have hp0 : (0 : ℚ) < p := by exact_mod_cast hs.trans hsp
-  have hpq' : p < q := by exact_mod_cast hpq
-  have hinner : γ '' Icc (p : ℝ) (q : ℝ) ⊆ γ '' Icc s t :=
-    image_mono (Icc_subset_Icc hsp.le hqt.le)
-  have hpos := (H p q hp0 hpq').1.trans_le (ν.mono hinner)
-  obtain ⟨u, hu0, hus⟩ := exists_rat_btwn hs
-  obtain ⟨v, htv, _⟩ := exists_rat_btwn (lt_add_one t)
-  have hu0' : (0 : ℚ) < u := by exact_mod_cast hu0
-  have huv : u < v := by exact_mod_cast hus.trans (hst.trans htv)
-  have houter : γ '' Icc s t ⊆ γ '' Icc (u : ℝ) (v : ℝ) :=
-    image_mono (Icc_subset_Icc hus.le htv.le)
-  exact ⟨hpos, (ν.mono houter).trans_lt (H u v hu0' huv).2⟩
-
- 
-
-theorem ae_all_real_intervals_iff_rational [MeasurableSpace Ω]
-    (P : Measure Ω) (ν : OuterMeasure X) (γ : Ω → ℝ → X) :
-    (∀ᵐ ω ∂P, ∀ s t : ℝ, 0 < s → s < t →
-      0 < ν (γ ω '' Icc s t) ∧ ν (γ ω '' Icc s t) < ∞) ↔
-    (∀ p q : ℚ, 0 < p → p < q →
-      ∀ᵐ ω ∂P, 0 < ν (γ ω '' Icc (p : ℝ) (q : ℝ)) ∧
-        ν (γ ω '' Icc (p : ℝ) (q : ℝ)) < ∞) := by
-  constructor
-  · intro H p q hp hpq
-    filter_upwards [H] with ω hω
-    exact hω p q (by exact_mod_cast hp) (by exact_mod_cast hpq)
-  · intro H
-    have H' : ∀ᵐ ω ∂P, ∀ p q : ℚ, 0 < p → p < q →
-        0 < ν (γ ω '' Icc (p : ℝ) (q : ℝ)) ∧
-          ν (γ ω '' Icc (p : ℝ) (q : ℝ)) < ∞ := by
-      simpa only [ae_all_iff] using H
-    filter_upwards [H'] with ω hω
-    exact all_real_intervals_of_rational ν (γ ω) hω
-
-end Intervals
-end SLEExactGauge.Deterministic
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Topology
-namespace SLEExactGauge.QuadraticGauge
-
- 
-
-def entry (d b ρ r : ℝ) : ℝ := b * ρ ^ d * max 1 ((r / ρ)^2)
-
- 
-def gauge {ι : Type*} (d : ℝ) (b ρ : ι → ℝ) (r : ℝ) : ℝ :=
-  ⨅ i, entry d (b i) (ρ i) r
-
-lemma entry_nonneg {d b ρ r : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) :
-    0 ≤ entry d b ρ r := by
-  unfold entry
-  positivity
-
-lemma base_le_entry {d b ρ r : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) :
-    b * ρ^d ≤ entry d b ρ r := by
-  unfold entry
-  exact le_mul_of_one_le_right (mul_nonneg hb (Real.rpow_nonneg hρ.le _)) (le_max_left _ _)
-
-lemma entry_eq_base {d b ρ r : ℝ} (hρ : 0 < ρ) (hr : 0 ≤ r) (hrρ : r ≤ ρ) :
-    entry d b ρ r = b * ρ^d := by
-  have hq : 0 ≤ r/ρ := div_nonneg hr hρ.le
-  have hq1 : r/ρ ≤ 1 := (div_le_one hρ).mpr hrρ
-  have hs : (r/ρ)^2 ≤ 1 := by nlinarith
-  simp only [entry,max_eq_left hs,mul_one]
-
-lemma entry_mono {d b ρ : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) :
-    MonotoneOn (entry d b ρ) (Ici 0) := by
-  intro r hr s hs hrs
-  apply mul_le_mul_of_nonneg_left _ (mul_nonneg hb (Real.rpow_nonneg hρ.le _))
-  apply max_le_max le_rfl
-  exact pow_le_pow_left₀ (div_nonneg hr hρ.le) (div_le_div_of_nonneg_right hrs hρ.le) 2
-
-lemma entry_scale {d b ρ C r : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) (hC : 1 ≤ C) :
-    entry d b ρ (C*r) ≤ C^2 * entry d b ρ r := by
-  have hC2 : 1 ≤ C^2 := by nlinarith
-  have ht : max 1 ((C*r/ρ)^2) ≤ C^2 * max 1 ((r/ρ)^2) := by
-    apply max_le
-    · exact hC2.trans (le_mul_of_one_le_right (sq_nonneg C) (le_max_left _ _))
-    · calc
-        (C*r/ρ)^2 = C^2 * (r/ρ)^2 := by ring
-        _ ≤ C^2 * max 1 ((r/ρ)^2) := mul_le_mul_of_nonneg_left (le_max_right _ _) (sq_nonneg C)
-  have hm := mul_le_mul_of_nonneg_left ht (mul_nonneg hb (Real.rpow_nonneg hρ.le d))
-  dsimp only [entry]
-  nlinarith only [hm]
-
-lemma power_le_entry {d b ρ r c : ℝ} (hd : 0 ≤ d) (hd2 : d ≤ 2)
-    (hc : 0 ≤ c) (hcb : c ≤ b) (hρ : 0 < ρ) (hr : 0 < r) :
-    c*r^d ≤ entry d b ρ r := by
-  have hb : 0 ≤ b := hc.trans hcb
-  by_cases hrρ : r ≤ ρ
-  · calc
-      c*r^d ≤ b*ρ^d := mul_le_mul hcb (Real.rpow_le_rpow hr.le hrρ hd)
-        (Real.rpow_nonneg hr.le _) hb
-      _ ≤ entry d b ρ r := base_le_entry hb hρ
-  · have hρr : ρ ≤ r := (lt_of_not_ge hrρ).le
-    have he : (r/ρ)^d ≤ (r/ρ)^2 := by
-      simpa only [Real.rpow_two] using
-        Real.rpow_le_rpow_of_exponent_le ((one_le_div hρ).mpr hρr) hd2
-    have hh : r^d ≤ ρ^d * (r/ρ)^2 := by
-      rw [Real.div_rpow hr.le hρ.le] at he
-      have hρd : 0 < ρ^d := Real.rpow_pos_of_pos hρ _
-      nlinarith [(div_le_iff₀ hρd).mp he]
-    calc
-      c*r^d ≤ b*(ρ^d * (r/ρ)^2) := mul_le_mul hcb hh (Real.rpow_nonneg hr.le _) hb
-      _ ≤ entry d b ρ r := by
-        unfold entry
-        nlinarith only [mul_le_mul_of_nonneg_left (le_max_right 1 ((r/ρ)^2))
-          (mul_nonneg hb (Real.rpow_nonneg hρ.le d))]
-
-section Envelope
-variable {ι : Type*} [Nonempty ι] {d : ℝ} {b ρ : ι → ℝ}
-variable (hb : ∀ i, 0 ≤ b i) (hρ : ∀ i, 0 < ρ i)
-include hb hρ
-
-omit [Nonempty ι] in
-lemma entry_bddBelow (r : ℝ) : BddBelow (range (fun i => entry d (b i) (ρ i) r)) := by
-  refine ⟨0,?_⟩
-  rintro _ ⟨i,rfl⟩
-  exact entry_nonneg (hb i) (hρ i)
-
-lemma gauge_nonneg (r : ℝ) : 0 ≤ gauge d b ρ r :=
-  le_ciInf (fun i => entry_nonneg (hb i) (hρ i))
-
-omit [Nonempty ι] in
-lemma gauge_le_entry (r : ℝ) (i : ι) : gauge d b ρ r ≤ entry d (b i) (ρ i) r :=
-  ciInf_le (entry_bddBelow hb hρ r) i
-
-omit [Nonempty ι] in
-lemma gauge_mono : MonotoneOn (gauge d b ρ) (Ici 0) := by
-  intro r hr s hs hrs
-  exact ciInf_mono (entry_bddBelow hb hρ r) (fun i => entry_mono (hb i) (hρ i) hr hs hrs)
-
-lemma gauge_scale {C r : ℝ} (hC : 1 ≤ C) :
-    gauge d b ρ (C*r) ≤ C^2 * gauge d b ρ r := by
-  have hCp : 0 < C^2 := sq_pos_of_pos (lt_of_lt_of_le zero_lt_one hC)
-  rw [mul_comm (C^2)]
-  apply (div_le_iff₀ hCp).mp
-  apply le_ciInf
-  intro i
-  apply (div_le_iff₀ hCp).mpr
-  exact (gauge_le_entry hb hρ (C*r) i).trans (by
-    simpa only [mul_comm] using entry_scale (hb i) (hρ i) hC)
-
-lemma gauge_comparison {r s : ℝ} (hr : 0 < r) (hs : 0 < s) :
-    gauge d b ρ r ≤ max 1 ((r/s)^2) * gauge d b ρ s := by
-  by_cases hrs : r ≤ s
-  · have hh : (r/s)^2 ≤ 1 := by
-      have hq : 0 ≤ r/s := div_nonneg hr.le hs.le
-      have hq1 : r/s ≤ 1 := (div_le_one hs).mpr hrs
-      nlinarith
-    rw [max_eq_left hh,one_mul]
-    exact gauge_mono hb hρ hr.le hs.le hrs
-  · have hq : 1 ≤ r/s := (one_le_div hs).mpr (le_of_not_ge hrs)
-    have hq2 : 1 ≤ (r/s)^2 := by nlinarith
-    rw [max_eq_right hq2]
-    convert gauge_scale (d := d) hb hρ (r := s) hq using 1
-    field_simp
-
-lemma gauge_continuousAt {r : ℝ} (hr : 0 < r) : ContinuousAt (gauge d b ρ) r := by
-  have hu : ContinuousAt (fun s : ℝ => max 1 ((s/r)^2) * gauge d b ρ r) r := by
-    fun_prop
-  have hl : ContinuousAt (fun s : ℝ => gauge d b ρ r / max 1 ((r/s)^2)) r := by
-    apply ContinuousAt.div continuousAt_const
-    · apply ContinuousAt.sup continuousAt_const
-      exact ((continuousAt_const.div continuousAt_id hr.ne').pow 2)
-    · exact ne_of_gt (lt_of_lt_of_le zero_lt_one (le_max_left _ _))
-  have hu' : Tendsto (fun s : ℝ => max 1 ((s/r)^2) * gauge d b ρ r) (𝓝 r)
-      (𝓝 (gauge d b ρ r)) := by simpa [hr.ne'] using hu.tendsto
-  have hl' : Tendsto (fun s : ℝ => gauge d b ρ r / max 1 ((r/s)^2)) (𝓝 r)
-      (𝓝 (gauge d b ρ r)) := by simpa [hr.ne'] using hl.tendsto
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hl' hu'
-  · filter_upwards [Ioi_mem_nhds hr] with s hs
-    exact (div_le_iff₀ (lt_of_lt_of_le zero_lt_one (le_max_left _ _))).mpr
-      (by simpa only [mul_comm] using gauge_comparison hb hρ hr hs)
-  · filter_upwards [Ioi_mem_nhds hr] with s hs
-    exact gauge_comparison hb hρ hs hr
-
-lemma gauge_zero (hsmall : ∀ ε : ℝ, 0 < ε → ∃ i, b i*(ρ i)^d < ε) :
-    gauge d b ρ 0 = 0 := by
-  apply le_antisymm _ (gauge_nonneg hb hρ 0)
-  by_contra! hp
-  obtain ⟨i,hi⟩ := hsmall _ hp
-  have he := gauge_le_entry (d := d) hb hρ 0 i
-  rw [entry_eq_base (hρ i) le_rfl (hρ i).le] at he
-  exact (not_lt_of_ge he) hi
-
-lemma gauge_continuousWithinAt_zero
-    (hsmall : ∀ ε : ℝ, 0 < ε → ∃ i, b i*(ρ i)^d < ε) :
-    ContinuousWithinAt (gauge d b ρ) (Ici 0) 0 := by
-  rw [Metric.continuousWithinAt_iff]
-  intro ε hε
-  obtain ⟨i,hi⟩ := hsmall ε hε
-  refine ⟨ρ i,hρ i,?_⟩
-  intro r hr hrd
-  change 0 ≤ r at hr
-  have hrρ : r ≤ ρ i := by
-    exact le_of_lt (by simpa only [Real.dist_eq, sub_zero,abs_of_nonneg hr] using hrd)
-  rw [gauge_zero hb hρ hsmall,Real.dist_eq,sub_zero,abs_of_nonneg (gauge_nonneg hb hρ r)]
-  exact (gauge_le_entry hb hρ r i).trans_lt (by
-    rwa [entry_eq_base (hρ i) hr hrρ])
-
- 
-
-theorem isHausdorffGauge (hd : 0 ≤ d) (hd2 : d ≤ 2)
-    {c : ℝ} (hc : 0 < c) (hcb : ∀ i, c ≤ b i)
-    (hsmall : ∀ ε : ℝ, 0 < ε → ∃ i, b i*(ρ i)^d < ε) :
-    Target.IsHausdorffGauge (gauge d b ρ) := by
-  refine ⟨?_,gauge_mono hb hρ,gauge_zero hb hρ hsmall,?_⟩
-  · intro r hr
-    rcases (show 0 ≤ r from hr).eq_or_lt with he | hp
-    · subst r
-      exact gauge_continuousWithinAt_zero hb hρ hsmall
-    · exact (gauge_continuousAt hb hρ hp).continuousWithinAt
-  · intro r hr
-    exact (mul_pos hc (Real.rpow_pos_of_pos hr d)).trans_le
-      (le_ciInf (fun i => power_le_entry hd hd2 hc.le (hcb i) (hρ i) hr))
-
-end Envelope
-
- 
-
-theorem finite_exclusions {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι → ℝ)
-    (hb : ∀ i, 0 < b i) (hρ : ∀ i, 0 < ρ i) (E : Finset ι)
-    (F : ℝ → ℝ) (M C : ℝ)
-    (hF : ∀ r, 0 < r → 0 ≤ F r ∧ F r ≤ M)
-    (hbound : ∀ i, i ∉ E → ∀ r, 0 < r → F r ≤ C * entry d (b i) (ρ i) r) :
-    ∃ C' : ℝ, 0 < C' ∧ ∀ r, 0 < r → F r ≤ C' * gauge d b ρ r := by
-  classical
-  let A : ι → ℝ := fun i => b i * (ρ i)^d
-  have hA (i : ι) : 0 < A i := mul_pos (hb i) (Real.rpow_pos_of_pos (hρ i) d)
-  let D : ℝ := max (max 1 C) (∑ i ∈ E, max (M/A i) 0)
-  have h1 : 1 ≤ D := (le_max_left 1 C).trans (le_max_left _ _)
-  have hD : 0 < D := zero_lt_one.trans_le h1
-  have hCD : C ≤ D := (le_max_right 1 C).trans (le_max_left _ _)
-  refine ⟨D,hD,?_⟩
-  intro r hr
-  have hi (i : ι) : F r ≤ D * entry d (b i) (ρ i) r := by
-    by_cases hie : i ∈ E
-    · have hm : M/A i ≤ D := calc
-        M/A i ≤ max (M/A i) 0 := le_max_left _ _
-        _ ≤ ∑ j ∈ E, max (M/A j) 0 :=
-          Finset.single_le_sum (f := fun j => max (M/A j) 0) (fun j _ => le_max_right _ _) hie
-        _ ≤ D := le_max_right _ _
-      calc
-        F r ≤ M := (hF r hr).2
-        _ ≤ D * A i := (div_le_iff₀ (hA i)).mp hm
-        _ ≤ D * entry d (b i) (ρ i) r :=
-          mul_le_mul_of_nonneg_left (base_le_entry (hb i).le (hρ i)) hD.le
-    · exact (hbound i hie r hr).trans
-        (mul_le_mul_of_nonneg_right hCD (entry_nonneg (hb i).le (hρ i)))
-  rw [mul_comm D]
-  apply (div_le_iff₀ hD).mp
-  apply le_ciInf
-  intro i
-  exact (div_le_iff₀ hD).mpr (by simpa only [mul_comm] using hi i)
-
-end SLEExactGauge.QuadraticGauge
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+
+namespace SLEExactGauge.Target
+
+open scoped ContinuousMapZero
+
+end SLEExactGauge.Target
+
+ 
+
+ 
+
+namespace SLEExactGauge.BrownianLaw
+open SLEExactGauge.Target
+open ProbabilityTheory Function
+open scoped ContinuousMapZero
+
+instance : PolishSpace Driver := ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace
+
+def code (b : Driver) : ℚ → ℝ := fun q => b (Real.toNNReal (q:ℝ))
+
+lemma measurable_code : Measurable code :=
+  Measurable.of_eval fun _ => (continuous_eval_const _).measurable
+
+lemma dense_rational_time : DenseRange (fun q : ℚ => Real.toNNReal (q:ℝ)) := by
+  have hd : Surjective Real.toNNReal := fun x => ⟨(x:ℝ),Real.toNNReal_coe⟩
+  exact hd.denseRange.comp Rat.denseRange_cast continuous_real_toNNReal
+
+lemma code_injective : Injective code := by
+  intro b c h
+  apply ContinuousMapZero.ext
+  have he := dense_rational_time.equalizer (map_continuous b) (map_continuous c) h
+  exact congrFun he
+
+lemma code_embedding : MeasurableEmbedding code := measurable_code.measurableEmbedding code_injective
+
+lemma evaluation_law_unique {W V : Measure Driver}
+    (hW : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) W)
+    (hV : IsBrownianReal (fun (s : ℝ≥0) (b : Driver) => b s) V) : W = V := by
+  let ev : Driver → (ℝ≥0 → ℝ) := fun b => b
+  have mev : Measurable ev := Measurable.of_eval fun time => (continuous_eval_const time).measurable
+  have hw : IsProjectiveLimit (W.map ev) BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map I.measurable_restrict mev]
+    exact (hW.hasLaw I).map_eq
+  have hv : IsProjectiveLimit (V.map ev) BrownianReal.projectiveFamily := by
+    intro I
+    rw [Measure.map_map I.measurable_restrict mev]
+    exact (hV.hasLaw I).map_eq
+  have he := hw.unique hv
+  have mr : Measurable (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ))) :=
+    Measurable.of_eval fun _ => measurable_pi_apply _
+  have hc := congrArg (Measure.map (fun f : ℝ≥0 → ℝ => fun q : ℚ => f (Real.toNNReal (q:ℝ)))) he
+  rw [Measure.map_map mr mev, Measure.map_map mr mev] at hc
+  exact code_embedding.map_injective hc
+
+end SLEExactGauge.BrownianLaw
+
+namespace SLEExactGauge.Deterministic
+
+open SLEExactGauge.Target
+
+ 
+
+def diameterCost (h : ℝ → ℝ) (r : ℝ≥0∞) : ℝ≥0∞ :=
+  if r = ∞ then ∞ else ENNReal.ofReal (h r.toReal)
+
+lemma diameterCost_zero {h : ℝ → ℝ} (h0 : h 0 = 0) :
+    diameterCost h 0 = 0 := by
+  simp [diameterCost, h0]
+
+lemma gauge_nonneg {h : ℝ → ℝ} (hh : IsHausdorffGauge h)
+    {r : ℝ} (hr : 0 ≤ r) : 0 ≤ h r := by
+  rw [← hh.2.2.1]
+  exact hh.2.1 (by simp) hr hr
+
+ 
+
+theorem hausdorff_countable_covers {h : ℝ → ℝ} (h0 : h 0 = 0) (E : Set ℂ) :
+    hausdorff h E =
+      ⨆ (δ : ℝ≥0∞) (_ : 0 < δ),
+        ⨅ (U : ℕ → Set ℂ) (_ : E ⊆ ⋃ n, U n) (_ : ∀ n, Metric.ediam (U n) ≤ δ),
+          ∑' n, diameterCost h (Metric.ediam (U n)) := by
+  change OuterMeasure.mkMetric (diameterCost h) E = _
+  rw [OuterMeasure.coe_mkMetric, Measure.mkMetric_apply]
+  congr 1 with δ
+  congr 1 with hδ
+  congr 1 with U
+  congr 1 with hEU
+  congr 1 with hU
+  apply tsum_congr
+  intro n
+  by_cases hn : (U n).Nonempty
+  · simp [hn]
+  · have hempty : U n = ∅ := Set.not_nonempty_iff_eq_empty.mp hn
+    simp [hempty, diameterCost_zero h0]
+
+ 
+theorem hausdorff_tendsto_content (h : ℝ → ℝ) (E : Set ℂ) :
+    Tendsto
+      (fun δ : ℝ≥0∞ =>
+        OuterMeasure.mkMetric'.pre (fun U : Set ℂ => diameterCost h (Metric.ediam U)) δ E)
+      (𝓝[>] 0) (𝓝 (hausdorff h E)) :=
+  OuterMeasure.mkMetric'.tendsto_pre _ _
+
+ 
+
+theorem mass_bound {h : ℝ → ℝ} {ν : OuterMeasure ℂ} {δ c : ℝ≥0∞}
+    (hδ : 0 < δ) (hc0 : c ≠ 0) (hc : c ≠ ∞)
+    (hν : ∀ U : Set ℂ, Metric.ediam U ≤ δ →
+      ν U ≤ c * diameterCost h (Metric.ediam U)) :
+    ν ≤ c • hausdorff h := by
+  have hm : ν ≤ OuterMeasure.mkMetric (c • diameterCost h) :=
+    OuterMeasure.le_mkMetric _ ν δ hδ hν
+  change ν ≤ c • OuterMeasure.mkMetric (diameterCost h)
+  simpa only [OuterMeasure.mkMetric_smul _ hc hc0] using hm
+
+ 
+
+theorem hausdorff_pos_of_mass_bound {h : ℝ → ℝ} {ν : OuterMeasure ℂ}
+    {δ c : ℝ≥0∞} {E : Set ℂ}
+    (hδ : 0 < δ) (hc0 : c ≠ 0) (hc : c ≠ ∞)
+    (hν : ∀ U : Set ℂ, Metric.ediam U ≤ δ →
+      ν U ≤ c * diameterCost h (Metric.ediam U))
+    (hE : 0 < ν E) : 0 < hausdorff h E := by
+  have H := mass_bound hδ hc0 hc hν E
+  by_contra hn
+  have hz : hausdorff h E = 0 := le_antisymm (not_lt.mp hn) zero_le
+  have hle : ν E ≤ 0 := by simpa [smul_apply, hz] using H
+  exact hE.not_ge hle
+
+ 
+
+theorem hausdorff_le_liminf_cost {ι : ℕ → Type*} [∀ n, Countable (ι n)]
+    (h : ℝ → ℝ) (E : Set ℂ) (r : ℕ → ℝ≥0∞)
+    (hr : Tendsto r atTop (𝓝 0)) (U : ∀ n, ι n → Set ℂ)
+    (hdiam : ∀ᶠ n in atTop, ∀ i, Metric.ediam (U n i) ≤ r n)
+    (hcover : ∀ᶠ n in atTop, E ⊆ ⋃ i, U n i) :
+    hausdorff h E ≤ liminf (fun n => ∑' i, diameterCost h (Metric.ediam (U n i))) atTop := by
+  change OuterMeasure.mkMetric (diameterCost h) E ≤ _
+  rw [OuterMeasure.coe_mkMetric]
+  exact Measure.mkMetric_le_liminf_tsum E r hr U hdiam hcover _
+
+section Intervals
+
+variable {X Ω : Type*}
+
+ 
+
+theorem all_real_intervals_of_rational (ν : OuterMeasure X) (γ : ℝ → X)
+    (H : ∀ p q : ℚ, 0 < p → p < q →
+      0 < ν (γ '' Icc (p : ℝ) (q : ℝ)) ∧
+        ν (γ '' Icc (p : ℝ) (q : ℝ)) < ∞) :
+    ∀ s t : ℝ, 0 < s → s < t →
+      0 < ν (γ '' Icc s t) ∧ ν (γ '' Icc s t) < ∞ := by
+  intro s t hs hst
+  obtain ⟨p, hsp, hpt⟩ := exists_rat_btwn hst
+  obtain ⟨q, hpq, hqt⟩ := exists_rat_btwn hpt
+  have hp0 : (0 : ℚ) < p := by exact_mod_cast hs.trans hsp
+  have hpq' : p < q := by exact_mod_cast hpq
+  have hinner : γ '' Icc (p : ℝ) (q : ℝ) ⊆ γ '' Icc s t :=
+    image_mono (Icc_subset_Icc hsp.le hqt.le)
+  have hpos := (H p q hp0 hpq').1.trans_le (ν.mono hinner)
+  obtain ⟨u, hu0, hus⟩ := exists_rat_btwn hs
+  obtain ⟨v, htv, _⟩ := exists_rat_btwn (lt_add_one t)
+  have hu0' : (0 : ℚ) < u := by exact_mod_cast hu0
+  have huv : u < v := by exact_mod_cast hus.trans (hst.trans htv)
+  have houter : γ '' Icc s t ⊆ γ '' Icc (u : ℝ) (v : ℝ) :=
+    image_mono (Icc_subset_Icc hus.le htv.le)
+  exact ⟨hpos, (ν.mono houter).trans_lt (H u v hu0' huv).2⟩
+
+ 
+
+theorem ae_all_real_intervals_iff_rational [MeasurableSpace Ω]
+    (P : Measure Ω) (ν : OuterMeasure X) (γ : Ω → ℝ → X) :
+    (∀ᵐ ω ∂P, ∀ s t : ℝ, 0 < s → s < t →
+      0 < ν (γ ω '' Icc s t) ∧ ν (γ ω '' Icc s t) < ∞) ↔
+    (∀ p q : ℚ, 0 < p → p < q →
+      ∀ᵐ ω ∂P, 0 < ν (γ ω '' Icc (p : ℝ) (q : ℝ)) ∧
+        ν (γ ω '' Icc (p : ℝ) (q : ℝ)) < ∞) := by
+  constructor
+  · intro H p q hp hpq
+    filter_upwards [H] with ω hω
+    exact hω p q (by exact_mod_cast hp) (by exact_mod_cast hpq)
+  · intro H
+    have H' : ∀ᵐ ω ∂P, ∀ p q : ℚ, 0 < p → p < q →
+        0 < ν (γ ω '' Icc (p : ℝ) (q : ℝ)) ∧
+          ν (γ ω '' Icc (p : ℝ) (q : ℝ)) < ∞ := by
+      simpa only [ae_all_iff] using H
+    filter_upwards [H'] with ω hω
+    exact all_real_intervals_of_rational ν (γ ω) hω
+
+end Intervals
+end SLEExactGauge.Deterministic
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Topology
+namespace SLEExactGauge.QuadraticGauge
+
+ 
+
+def entry (d b ρ r : ℝ) : ℝ := b * ρ ^ d * max 1 ((r / ρ)^2)
+
+ 
+def gauge {ι : Type*} (d : ℝ) (b ρ : ι → ℝ) (r : ℝ) : ℝ :=
+  ⨅ i, entry d (b i) (ρ i) r
+
+lemma entry_nonneg {d b ρ r : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) :
+    0 ≤ entry d b ρ r := by
+  unfold entry
+  positivity
+
+lemma base_le_entry {d b ρ r : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) :
+    b * ρ^d ≤ entry d b ρ r := by
+  unfold entry
+  exact le_mul_of_one_le_right (mul_nonneg hb (Real.rpow_nonneg hρ.le _)) (le_max_left _ _)
+
+lemma entry_eq_base {d b ρ r : ℝ} (hρ : 0 < ρ) (hr : 0 ≤ r) (hrρ : r ≤ ρ) :
+    entry d b ρ r = b * ρ^d := by
+  have hq : 0 ≤ r/ρ := div_nonneg hr hρ.le
+  have hq1 : r/ρ ≤ 1 := (div_le_one hρ).mpr hrρ
+  have hs : (r/ρ)^2 ≤ 1 := by nlinarith
+  simp only [entry,max_eq_left hs,mul_one]
+
+lemma entry_mono {d b ρ : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) :
+    MonotoneOn (entry d b ρ) (Ici 0) := by
+  intro r hr s hs hrs
+  apply mul_le_mul_of_nonneg_left _ (mul_nonneg hb (Real.rpow_nonneg hρ.le _))
+  apply max_le_max le_rfl
+  exact pow_le_pow_left₀ (div_nonneg hr hρ.le) (div_le_div_of_nonneg_right hrs hρ.le) 2
+
+lemma entry_scale {d b ρ C r : ℝ} (hb : 0 ≤ b) (hρ : 0 < ρ) (hC : 1 ≤ C) :
+    entry d b ρ (C*r) ≤ C^2 * entry d b ρ r := by
+  have hC2 : 1 ≤ C^2 := by nlinarith
+  have ht : max 1 ((C*r/ρ)^2) ≤ C^2 * max 1 ((r/ρ)^2) := by
+    apply max_le
+    · exact hC2.trans (le_mul_of_one_le_right (sq_nonneg C) (le_max_left _ _))
+    · calc
+        (C*r/ρ)^2 = C^2 * (r/ρ)^2 := by ring
+        _ ≤ C^2 * max 1 ((r/ρ)^2) := mul_le_mul_of_nonneg_left (le_max_right _ _) (sq_nonneg C)
+  have hm := mul_le_mul_of_nonneg_left ht (mul_nonneg hb (Real.rpow_nonneg hρ.le d))
+  dsimp only [entry]
+  nlinarith only [hm]
+
+lemma power_le_entry {d b ρ r c : ℝ} (hd : 0 ≤ d) (hd2 : d ≤ 2)
+    (hc : 0 ≤ c) (hcb : c ≤ b) (hρ : 0 < ρ) (hr : 0 < r) :
+    c*r^d ≤ entry d b ρ r := by
+  have hb : 0 ≤ b := hc.trans hcb
+  by_cases hrρ : r ≤ ρ
+  · calc
+      c*r^d ≤ b*ρ^d := mul_le_mul hcb (Real.rpow_le_rpow hr.le hrρ hd)
+        (Real.rpow_nonneg hr.le _) hb
+      _ ≤ entry d b ρ r := base_le_entry hb hρ
+  · have hρr : ρ ≤ r := (lt_of_not_ge hrρ).le
+    have he : (r/ρ)^d ≤ (r/ρ)^2 := by
+      simpa only [Real.rpow_two] using
+        Real.rpow_le_rpow_of_exponent_le ((one_le_div hρ).mpr hρr) hd2
+    have hh : r^d ≤ ρ^d * (r/ρ)^2 := by
+      rw [Real.div_rpow hr.le hρ.le] at he
+      have hρd : 0 < ρ^d := Real.rpow_pos_of_pos hρ _
+      nlinarith [(div_le_iff₀ hρd).mp he]
+    calc
+      c*r^d ≤ b*(ρ^d * (r/ρ)^2) := mul_le_mul hcb hh (Real.rpow_nonneg hr.le _) hb
+      _ ≤ entry d b ρ r := by
+        unfold entry
+        nlinarith only [mul_le_mul_of_nonneg_left (le_max_right 1 ((r/ρ)^2))
+          (mul_nonneg hb (Real.rpow_nonneg hρ.le d))]
+
+section Envelope
+variable {ι : Type*} [Nonempty ι] {d : ℝ} {b ρ : ι → ℝ}
+variable (hb : ∀ i, 0 ≤ b i) (hρ : ∀ i, 0 < ρ i)
+include hb hρ
+
+omit [Nonempty ι] in
+lemma entry_bddBelow (r : ℝ) : BddBelow (range (fun i => entry d (b i) (ρ i) r)) := by
+  refine ⟨0,?_⟩
+  rintro _ ⟨i,rfl⟩
+  exact entry_nonneg (hb i) (hρ i)
+
+lemma gauge_nonneg (r : ℝ) : 0 ≤ gauge d b ρ r :=
+  le_ciInf (fun i => entry_nonneg (hb i) (hρ i))
+
+omit [Nonempty ι] in
+lemma gauge_le_entry (r : ℝ) (i : ι) : gauge d b ρ r ≤ entry d (b i) (ρ i) r :=
+  ciInf_le (entry_bddBelow hb hρ r) i
+
+omit [Nonempty ι] in
+lemma gauge_mono : MonotoneOn (gauge d b ρ) (Ici 0) := by
+  intro r hr s hs hrs
+  exact ciInf_mono (entry_bddBelow hb hρ r) (fun i => entry_mono (hb i) (hρ i) hr hs hrs)
+
+lemma gauge_scale {C r : ℝ} (hC : 1 ≤ C) :
+    gauge d b ρ (C*r) ≤ C^2 * gauge d b ρ r := by
+  have hCp : 0 < C^2 := sq_pos_of_pos (lt_of_lt_of_le zero_lt_one hC)
+  rw [mul_comm (C^2)]
+  apply (div_le_iff₀ hCp).mp
+  apply le_ciInf
+  intro i
+  apply (div_le_iff₀ hCp).mpr
+  exact (gauge_le_entry hb hρ (C*r) i).trans (by
+    simpa only [mul_comm] using entry_scale (hb i) (hρ i) hC)
+
+lemma gauge_comparison {r s : ℝ} (hr : 0 < r) (hs : 0 < s) :
+    gauge d b ρ r ≤ max 1 ((r/s)^2) * gauge d b ρ s := by
+  by_cases hrs : r ≤ s
+  · have hh : (r/s)^2 ≤ 1 := by
+      have hq : 0 ≤ r/s := div_nonneg hr.le hs.le
+      have hq1 : r/s ≤ 1 := (div_le_one hs).mpr hrs
+      nlinarith
+    rw [max_eq_left hh,one_mul]
+    exact gauge_mono hb hρ hr.le hs.le hrs
+  · have hq : 1 ≤ r/s := (one_le_div hs).mpr (le_of_not_ge hrs)
+    have hq2 : 1 ≤ (r/s)^2 := by nlinarith
+    rw [max_eq_right hq2]
+    convert gauge_scale (d := d) hb hρ (r := s) hq using 1
+    field_simp
+
+lemma gauge_continuousAt {r : ℝ} (hr : 0 < r) : ContinuousAt (gauge d b ρ) r := by
+  have hu : ContinuousAt (fun s : ℝ => max 1 ((s/r)^2) * gauge d b ρ r) r := by
+    fun_prop
+  have hl : ContinuousAt (fun s : ℝ => gauge d b ρ r / max 1 ((r/s)^2)) r := by
+    apply ContinuousAt.div continuousAt_const
+    · apply ContinuousAt.sup continuousAt_const
+      exact ((continuousAt_const.div continuousAt_id hr.ne').pow 2)
+    · exact ne_of_gt (lt_of_lt_of_le zero_lt_one (le_max_left _ _))
+  have hu' : Tendsto (fun s : ℝ => max 1 ((s/r)^2) * gauge d b ρ r) (𝓝 r)
+      (𝓝 (gauge d b ρ r)) := by simpa [hr.ne'] using hu.tendsto
+  have hl' : Tendsto (fun s : ℝ => gauge d b ρ r / max 1 ((r/s)^2)) (𝓝 r)
+      (𝓝 (gauge d b ρ r)) := by simpa [hr.ne'] using hl.tendsto
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hl' hu'
+  · filter_upwards [Ioi_mem_nhds hr] with s hs
+    exact (div_le_iff₀ (lt_of_lt_of_le zero_lt_one (le_max_left _ _))).mpr
+      (by simpa only [mul_comm] using gauge_comparison hb hρ hr hs)
+  · filter_upwards [Ioi_mem_nhds hr] with s hs
+    exact gauge_comparison hb hρ hs hr
+
+lemma gauge_zero (hsmall : ∀ ε : ℝ, 0 < ε → ∃ i, b i*(ρ i)^d < ε) :
+    gauge d b ρ 0 = 0 := by
+  apply le_antisymm _ (gauge_nonneg hb hρ 0)
+  by_contra! hp
+  obtain ⟨i,hi⟩ := hsmall _ hp
+  have he := gauge_le_entry (d := d) hb hρ 0 i
+  rw [entry_eq_base (hρ i) le_rfl (hρ i).le] at he
+  exact (not_lt_of_ge he) hi
+
+lemma gauge_continuousWithinAt_zero
+    (hsmall : ∀ ε : ℝ, 0 < ε → ∃ i, b i*(ρ i)^d < ε) :
+    ContinuousWithinAt (gauge d b ρ) (Ici 0) 0 := by
+  rw [Metric.continuousWithinAt_iff]
+  intro ε hε
+  obtain ⟨i,hi⟩ := hsmall ε hε
+  refine ⟨ρ i,hρ i,?_⟩
+  intro r hr hrd
+  change 0 ≤ r at hr
+  have hrρ : r ≤ ρ i := by
+    exact le_of_lt (by simpa only [Real.dist_eq, sub_zero,abs_of_nonneg hr] using hrd)
+  rw [gauge_zero hb hρ hsmall,Real.dist_eq,sub_zero,abs_of_nonneg (gauge_nonneg hb hρ r)]
+  exact (gauge_le_entry hb hρ r i).trans_lt (by
+    rwa [entry_eq_base (hρ i) hr hrρ])
+
+ 
+
+theorem isHausdorffGauge (hd : 0 ≤ d) (hd2 : d ≤ 2)
+    {c : ℝ} (hc : 0 < c) (hcb : ∀ i, c ≤ b i)
+    (hsmall : ∀ ε : ℝ, 0 < ε → ∃ i, b i*(ρ i)^d < ε) :
+    Target.IsHausdorffGauge (gauge d b ρ) := by
+  refine ⟨?_,gauge_mono hb hρ,gauge_zero hb hρ hsmall,?_⟩
+  · intro r hr
+    rcases (show 0 ≤ r from hr).eq_or_lt with he | hp
+    · subst r
+      exact gauge_continuousWithinAt_zero hb hρ hsmall
+    · exact (gauge_continuousAt hb hρ hp).continuousWithinAt
+  · intro r hr
+    exact (mul_pos hc (Real.rpow_pos_of_pos hr d)).trans_le
+      (le_ciInf (fun i => power_le_entry hd hd2 hc.le (hcb i) (hρ i) hr))
+
+end Envelope
+
+ 
+
+theorem finite_exclusions {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι → ℝ)
+    (hb : ∀ i, 0 < b i) (hρ : ∀ i, 0 < ρ i) (E : Finset ι)
+    (F : ℝ → ℝ) (M C : ℝ)
+    (hF : ∀ r, 0 < r → 0 ≤ F r ∧ F r ≤ M)
+    (hbound : ∀ i, i ∉ E → ∀ r, 0 < r → F r ≤ C * entry d (b i) (ρ i) r) :
+    ∃ C' : ℝ, 0 < C' ∧ ∀ r, 0 < r → F r ≤ C' * gauge d b ρ r := by
+  classical
+  let A : ι → ℝ := fun i => b i * (ρ i)^d
+  have hA (i : ι) : 0 < A i := mul_pos (hb i) (Real.rpow_pos_of_pos (hρ i) d)
+  let D : ℝ := max (max 1 C) (∑ i ∈ E, max (M/A i) 0)
+  have h1 : 1 ≤ D := (le_max_left 1 C).trans (le_max_left _ _)
+  have hD : 0 < D := zero_lt_one.trans_le h1
+  have hCD : C ≤ D := (le_max_right 1 C).trans (le_max_left _ _)
+  refine ⟨D,hD,?_⟩
+  intro r hr
+  have hi (i : ι) : F r ≤ D * entry d (b i) (ρ i) r := by
+    by_cases hie : i ∈ E
+    · have hm : M/A i ≤ D := calc
+        M/A i ≤ max (M/A i) 0 := le_max_left _ _
+        _ ≤ ∑ j ∈ E, max (M/A j) 0 :=
+          Finset.single_le_sum (f := fun j => max (M/A j) 0) (fun j _ => le_max_right _ _) hie
+        _ ≤ D := le_max_right _ _
+      calc
+        F r ≤ M := (hF r hr).2
+        _ ≤ D * A i := (div_le_iff₀ (hA i)).mp hm
+        _ ≤ D * entry d (b i) (ρ i) r :=
+          mul_le_mul_of_nonneg_left (base_le_entry (hb i).le (hρ i)) hD.le
+    · exact (hbound i hie r hr).trans
+        (mul_le_mul_of_nonneg_right hCD (entry_nonneg (hb i).le (hρ i)))
+  rw [mul_comm D]
+  apply (div_le_iff₀ hD).mp
+  apply le_ciInf
+  intro i
+  exact (div_le_iff₀ hD).mpr (by simpa only [mul_comm] using hi i)
+
+end SLEExactGauge.QuadraticGauge
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/GlobalAngleBudget.lean b/lean/OAI/Probability/SLEGauge/GlobalAngleBudget.lean
index 6e03b971e938031e724dc3fd08508ab7dca8b6e9..653342d319cfe735570df6893c532cd65514874d 100644
--- a/lean/OAI/Probability/SLEGauge/GlobalAngleBudget.lean
+++ b/lean/OAI/Probability/SLEGauge/GlobalAngleBudget.lean
@@ -1,512 +1,512 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.AngleLyapunov
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-
-def radialTestWeight (a : ℝ) (F : State→ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) : ℝ :=
-  if radialTime a b z ρ=∞ then 0 else
-    let v := actualState a b (radialTime a b z ρ).toNNReal z
-    physicalGreen a v * F v
-
-lemma radialTestWeight_nonneg (a : ℝ) (F : State→ℝ) (hF : ∀v, 0≤F v)
-    (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) : 0≤radialTestWeight a F b z ρ := by
-  unfold radialTestWeight
-  split_ifs
-  · exact le_rfl
-  · exact mul_nonneg (Real.exp_pos _).le (hF _)
-
-lemma sinePower_angleLift_pos (q : ℝ) (v : State) : 0<angleLift (sinePower q) v :=
-  Real.rpow_pos_of_pos
-    (Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2) _
-
-lemma physicalClock_initial (a : ℝ) (z : ℂ) : physicalClock a (2*z.im) (z,0)=0 := by
-  simp only [physicalClock,J_apply,Y_apply]
-  rw [show 2*z.im/2=z.im by ring]
-  simp
-
-lemma angleBudgetFactor_initial_bound (a q s C : ℝ) (hs : 0≤ s) (hC : 0≤C)
-    (z : ℂ) :
-    Real.exp (-s) * angleBudgetFactor a q (2*z.im) s C (z,0) ≤
-      angleLift (sinePower (-q)) (z,0)+C := by
-  have hE := (Real.exp_pos (-s)).le
-  have hE1 : Real.exp (-s)≤1 := Real.exp_le_one_iff.mpr (by linarith)
-  have hF := (sinePower_angleLift_pos (-q) (z,0)).le
-  have he : Real.exp (-s)*Real.exp s=1 := by rw [←Real.exp_add]; simp
-  have hprod := mul_le_mul_of_nonneg_right hE1 hF
-  have hCprod := mul_nonneg hE hC
-  unfold angleBudgetFactor clockExp
-  rw [physicalClock_initial a z, mul_zero,Real.exp_zero,one_mul]
-  nlinarith only [he,hprod,hCprod,hC]
-
- 
-
-theorem finite_level_angle_budget (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a q C : ℝ) (ha : 0<a) (hC0 : 0≤C)
-    (hC : ∀ θ∈Ioo 0 Real.pi,
-      angleGenerator (2*a) (sinePower (-q)) θ+sinePower (-q) θ≤C)
-    (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0≤ s) :
-    (∫⁻ b, ENNReal.ofReal (radialTestWeight a (angleLift (sinePower (-q))) b.1 z
-      (levelRadius a (2*z.im) s)) ∂W) ≤
-      ENNReal.ofReal (physicalGreen a (z,0)*(angleLift (sinePower (-q)) (z,0)+C)) := by
-  let cap := 2*z.im
-  have hcap : 0<cap := by dsimp [cap]; positivity
-  let ρ := levelRadius a cap s
-  have hρ : 0<ρ := levelRadius_pos a cap s hcap
-  have hr : ρ≤z.im := by
-    simpa only [cap,mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0)] using
-      (levelRadius_le ha hcap hs)
-  let L (n : ℕ) := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T (n : ℕ) := radialStop a ha z hz ρ hρ hr n
-  let V (n : ℕ) (b : ExactSLEGenerator.Driver) := localState a (L n).ε (L n).hε z (T n b) b
-  let f (n : ℕ) (b : ExactSLEGenerator.Driver) := Real.exp (-s) *
-    (physicalGreen a (V n b)*angleBudgetFactor a q cap s C (V n b))
-  let A := physicalGreen a (z,0)*(angleLift (sinePower (-q)) (z,0)+C)
-  have hsafe (n : ℕ) (b : ExactSLEGenerator.Driver) (u : ℝ≥0) (hu : u≤T n b) :
-      (L n).ε≤Y (localState a (L n).ε (L n).hε z u b) ∧
-        ‖localState a (L n).ε (L n).hε z u b‖≤(L n).N := by
-    have h := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b u hu
-    exact ⟨h.1,h.2.1⟩
-  have hact (n : ℕ) (b : ExactSLEGenerator.Driver) : alive a b.1 (T n b) z ∧
-      actualState a b.1 (T n b) z = V n b :=
-    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b _ le_rfl
-  have hc (n : ℕ) (b : ExactSLEGenerator.Driver) : physicalClock a cap (V n b) ≤ s := by
-    rw [←(hact n b).2]
-    have h := actual_physicalClock_le a ha z hz cap ρ hcap hρ hr b (T n b)
-      (hact n b).1 (radialStop_le_radialTime a ha z hz ρ hρ hr n b)
-    simpa only [ρ,levelRadius_clock ha.ne' hcap] using h
-  have hp (n : ℕ) (b : ExactSLEGenerator.Driver) : 0≤f n b :=
-    mul_nonneg (Real.exp_pos _).le (mul_nonneg (Real.exp_pos _).le
-      (angleBudgetFactor_nonneg a q cap s C hC0 _ (hc n b)))
-  have hb (n : ℕ) : Integrable (f n) W ∧ (∫ b, f n b ∂W)≤A := by
-    have hh := local_angle_budget W hW a q cap s C (L n).ε ha.ne' hC (L n).hε
-      z (L n).N (L n).B (T n) (radialStop_stopping a ha z hz ρ hρ hr n)
-      (regionStop_le a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B)
-      (hsafe n)
-    refine ⟨hh.1.const_mul _, ?_⟩
-    change (∫ b, Real.exp (-s)*(physicalGreen a (V n b)*angleBudgetFactor a q cap s C (V n b)) ∂W)≤A
-    rw [integral_const_mul]
-    calc
-      _ ≤ Real.exp (-s)*(physicalGreen a (z,0)*angleBudgetFactor a q cap s C (z,0)) :=
-        mul_le_mul_of_nonneg_left hh.2 (Real.exp_pos _).le
-      _ = physicalGreen a (z,0)*(Real.exp (-s)*angleBudgetFactor a q cap s C (z,0)) := by ring
-      _ ≤ A := mul_le_mul_of_nonneg_left
-        (angleBudgetFactor_initial_bound a q s C hs hC0 z) (Real.exp_pos _).le
-  have hbf (n : ℕ) : (∫⁻ b, ENNReal.ofReal (f n b) ∂W)≤ENNReal.ofReal A := by
-    rw [←ofReal_integral_eq_lintegral_ofReal (hb n).1 (Eventually.of_forall (hp n))]
-    exact ENNReal.ofReal_le_ofReal (hb n).2
-  have hlim (b : ExactSLEGenerator.Driver) :
-      ENNReal.ofReal (radialTestWeight a (angleLift (sinePower (-q))) b.1 z ρ) ≤
-        liminf (fun n => ENNReal.ofReal (f n b)) atTop := by
-    by_cases ht : radialTime a b.1 z ρ=∞
-    · simp only [radialTestWeight,ite_eq_left ht,ENNReal.ofReal_zero]
-      exact bot_le
-    · apply le_liminf_of_le (by isBoundedDefault)
-      filter_upwards [radialStop_eventually_eq a ha z hz ρ hρ hr b ht] with n hn
-      have he : T n b=(radialTime a b.1 z ρ).toNNReal := by
-        have hh := congrArg ENNReal.toNNReal hn
-        simpa only [ENNReal.toNNReal_coe] using hh
-      have hv : V n b=actualState a b.1 (radialTime a b.1 z ρ).toNNReal z := by
-        rw [←(hact n b).2,he]
-      have htC : physicalClock a cap (V n b)=s := by
-        rw [hv, actual_physicalClock_at a ha z hz cap ρ hcap hρ hr b ht]
-        exact levelRadius_clock ha.ne' hcap
-      have hexp : Real.exp (-s)*Real.exp s=1 := by rw [←Real.exp_add]; simp
-      apply le_of_eq
-      congr 1
-      simp only [radialTestWeight,ite_eq_right ht,f,angleBudgetFactor_terminal a q cap s C _ htC]
-      rw [hv]
-      nlinarith only [congrArg (fun x => x*(physicalGreen a
-        (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z)*
-          angleLift (sinePower (-q)) (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z))) hexp]
-  exact (lintegral_mono hlim).trans ((lintegral_liminf_le' (fun n =>
-    (hb n).1.aestronglyMeasurable.aemeasurable.ennreal_ofReal)).trans
-      (liminf_le_of_frequently_le (Eventually.of_forall hbf).frequently))
-
- 
-
-theorem exists_finite_level_angle_bound (a q : ℝ) (ha : 1/4<a) (hq : 0<q)
-    (hqp : q<4*a-1) : ∃ C : ℝ, 0<C ∧
-    ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-      ∀ (z : ℂ), 0<z.im → ∀ s : ℝ, 0≤ s →
-        (∫⁻ b, ENNReal.ofReal (radialTestWeight a (angleLift (sinePower (-q))) b.1 z
-          (levelRadius a (2*z.im) s)) ∂W) ≤
-          ENNReal.ofReal (physicalGreen a (z,0)*(angleLift (sinePower (-q)) (z,0)+C)) := by
-  obtain ⟨C,hC0,hC⟩ := exists_tilted_coercivity ha hq hqp
-  refine ⟨C,hC0,?_⟩
-  intro W hprob hW z hz s hs
-  exact finite_level_angle_budget W hW a q C (by linarith) hC0.le hC z hz s hs
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-
-lemma localizationStop_eventually_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver)
-    (L : ∀n, RadialLocalization a z ρ (angleThreshold z n))
-    (hfin : radialTime a b.1 z ρ≠∞) :
-    ∀ᶠ n in atTop,
-      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b : ℝ≥0∞)=
-        radialTime a b.1 z ρ := by
-  let t := (radialTime a b.1 z ρ).toNNReal
-  have ht := radialTime_attained a ha z hz ρ hρ hr b hfin
-  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht.1
-  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
-  let T := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
-  let V := localState a (L n).ε (L n).hε z T b
-  have hTt : T≤t := regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ
-    (angleThreshold z n) (L n).B (L n).initial b t ht.2
-  have he := regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-    (L n).B (L n).initial b T le_rfl
-  rcases regionStop_true_exit a (L n).ε ha (L n).hε z (L n).N ρ (angleThreshold z n) hρ
-    (angleThreshold_pos z hz n) (L n).B (L n).initial (L n).small (L n).large (L n).time b with hR|hS
-  · exact (regionStop_eq_radialTime a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b hR).symm
-  · have hs := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b T le_rfl
-    have hnp : 0<‖V.1‖ := ((L n).hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
-    have hd : Y V/‖V.1‖=angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-    have hη' := hηs T hTt
-    rw [he.2,hd] at hη'
-    exact False.elim (not_le.mpr hn hη')
-
- 
-def radialState (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) : State :=
-  if radialTime a b z ρ=∞ then 0 else
-    actualState a b (radialTime a b z ρ).toNNReal z
-
-lemma localizedState_eventually_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver)
-    (L : ∀n, RadialLocalization a z ρ (angleThreshold z n)) :
-    ∀ᶠ n in atTop,
-      (let V := localState a (L n).ε (L n).hε z
-        (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b
-       if physicalRadius V=ρ then V else 0)=radialState a b.1 z ρ := by
-  let T (n : ℕ) := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
-  let V (n : ℕ) := localState a (L n).ε (L n).hε z (T n) b
-  have he (n : ℕ) : actualState a b.1 (T n) z=V n :=
-    (regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b _ le_rfl).2
-  by_cases hf : radialTime a b.1 z ρ=∞
-  · apply Eventually.of_forall
-    intro n
-    have hn : physicalRadius (V n)≠ρ := by
-      intro hn
-      have hh := regionStop_eq_radialTime a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-        (L n).B (L n).initial b hn
-      exact ENNReal.coe_ne_top ((hf.symm.trans hh).symm)
-    exact (ite_eq_right hn).trans (by simp [radialState,hf])
-  · filter_upwards [localizationStop_eventually_eq a ha z hz ρ hρ hr b L hf] with n hn
-    have ht : T n=(radialTime a b.1 z ρ).toNNReal := by
-      have hh := congrArg ENNReal.toNNReal hn
-      simpa only [ENNReal.toNNReal_coe] using hh
-    have hv : V n=actualState a b.1 (radialTime a b.1 z ρ).toNNReal z := by rw [←he n,ht]
-    have hh : physicalRadius (V n)=ρ := by
-      rw [hv]; exact (radialTime_attained a ha z hz ρ hρ hr b hf).2
-    exact (ite_eq_left hh).trans (by simpa only [radialState,ite_eq_right hf] using hv)
-
- 
-
-theorem joint_radialState_measurable (a : ℝ) (ha : 0<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) :
-    Measurable (fun p : ExactSLEGenerator.Driver×UpperPlane =>
-      radialState a p.1.1 p.2.1 (p.2.1.im*q)) := by
-  let Ω := ExactSLEGenerator.Driver×UpperPlane
-  let z : Ω→ℂ := fun p => p.2.1
-  let b : Ω→ExactSLEGenerator.Driver := Prod.fst
-  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
-  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
-  have hbM : Measurable b := measurable_fst
-  let ρ : Ω→ℝ := fun ω => (z ω).im*q
-  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
-  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
-  let δ : ℕ→Ω→ℝ := fun n ω => angleThreshold (z ω) n
-  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha (z ω) (hz ω) (ρ ω) (δ n ω)
-    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
-  have hρM : Measurable ρ := hzM.im.mul_const q
-  have hδM (n : ℕ) : Measurable (δ n) := by
-    dsimp [δ,angleThreshold]
-    exact (hzM.im.div hzM.norm).div_const _
-  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
-    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
-    unfold heightFloor
-    fun_prop
-  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
-    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1,_⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    unfold logCeiling heightFloor
-    fun_prop
-  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
-    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1,_⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    fun_prop
-  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
-    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
-  have hTM (n : ℕ) : Measurable (T n) :=
-    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _
-      (hεM n) hzM hbM (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
-  let V (n : ℕ) (ω : Ω) := localState a (L n ω).ε (L n ω).hε (z ω) (T n ω) (b ω)
-  have hVM (n : ℕ) : Measurable (V n) :=
-    (localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε)
-      z b (hεM n) hzM hbM).comp ((hTM n).prodMk measurable_id)
-  apply measurable_of_tendsto_metrizable
-    (f := fun n ω => if physicalRadius (V n ω)=ρ ω then V n ω else (0:State))
-  · intro n
-    exact Measurable.ite (measurableSet_eq_fun
-      (physicalRadius_continuous.measurable.comp (hVM n)) hρM) (hVM n) measurable_const
-  · apply tendsto_pi_nhds.mpr
-    intro ω
-    apply tendsto_const_nhds.congr'
-    exact (localizedState_eventually_eq a ha (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω) (b ω)
-      (fun n => L n ω)).mono (fun n hn => hn.symm)
-
-lemma radialTestWeight_eq_mul (a : ℝ) (F : State→ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) :
-    radialTestWeight a F b z ρ=radialWeight a b z ρ*F (radialState a b z ρ) := by
-  unfold radialTestWeight radialWeight radialState
-  split_ifs <;> simp
-
-lemma joint_radialTestWeight_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (F : State→ℝ) (hF : Measurable F) :
-    Measurable (fun p : ExactSLEGenerator.Driver×UpperPlane =>
-      radialTestWeight a F p.1.1 p.2.1 (p.2.1.im*q)) := by
-  simp_rw [radialTestWeight_eq_mul]
-  exact (joint_radialWeight_measurable a ha q hq hq1).mul
-    (hF.comp (joint_radialState_measurable a (by linarith) q hq hq1))
-
-lemma angleLift_sinePower_measurable (q : ℝ) : Measurable (angleLift (sinePower q)) := by
-  unfold angleLift sinePower physicalTheta
-  fun_prop
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open ExactSLEMain ExactSLEGreen
-
-def scaleState (c : ℝ≥0) (v : State) : State := (((c:ℝ):ℂ)⁻¹*v.1,v.2)
-
-lemma scaleState_Y (c : ℝ≥0) (v : State) :
-    Y (scaleState c v)=(c:ℝ)⁻¹*Y v := by
-  simp [scaleState,←Complex.ofReal_inv,Complex.mul_im]
-
-lemma scaleState_X (c : ℝ≥0) (v : State) :
-    X (scaleState c v)=(c:ℝ)⁻¹*X v := by
-  simp [scaleState,←Complex.ofReal_inv,Complex.mul_re]
-
-lemma scaleState_Q (c : ℝ≥0) (v : State) :
-    Q (scaleState c v)=((c:ℝ)⁻¹)^2*Q v := by
-  simp only [Q,scaleState_X,scaleState_Y]
-  ring
-
-lemma scaleState_J (c : ℝ≥0) (v : State) : J (scaleState c v)=J v := rfl
-
-lemma scaleState_radius (c : ℝ≥0) (v : State) :
-    physicalRadius (scaleState c v)=(c:ℝ)⁻¹*physicalRadius v := by
-  simp only [physicalRadius,scaleState_Y,scaleState_J]
-  ring
-
-lemma scaleState_logGreen (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    (v : State) (hv : 0<Y v) :
-    logGreen a (scaleState c v)=alpha a*Real.log c+logGreen a v := by
-  have hcR : 0<(c:ℝ) := hc
-  have hn : (c:ℝ)≠0 := hcR.ne'
-  have hq := (Q_pos hv).ne'
-  simp only [logGreen,scaleState_Y,scaleState_Q,scaleState_J,
-    Real.log_mul (inv_ne_zero hn) hv.ne',
-    Real.log_mul (pow_ne_zero 2 (inv_ne_zero hn)) hq,
-    Real.log_pow,Real.log_inv]
-  ring
-
-lemma scaleState_green (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    (v : State) (hv : 0<Y v) :
-    physicalGreen a (scaleState c v)=(c:ℝ)^(alpha a)*physicalGreen a v := by
-  rw [physicalGreen,scaleState_logGreen a hc v hv,Real.exp_add,
-    Real.rpow_def_of_pos (show 0<(c:ℝ) from hc)]
-  rw [mul_comm (Real.log c)]
-  rfl
-
-lemma physicalRadius_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (t : ℝ≥0) (z : ℂ)
-    (hz : alive a (scaled c b) t z) :
-    physicalRadius (actualState a (scaled c b) t z) =
-      (c:ℝ)⁻¹*physicalRadius (actualState a b (c^2*t) (((c:ℝ):ℂ)*z)) := by
-  rw [actualState_scaled a hc hb t z hz]
-  exact scaleState_radius c _
-
-lemma image_radialHits (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
-    (fun u : ℝ≥0∞ => (c:ℝ≥0∞)^2*u) ''
-      {u : ℝ≥0∞ | ∃ t : ℝ≥0, u=t ∧ alive a (scaled c b) t z ∧
-        physicalRadius (actualState a (scaled c b) t z)=ρ} =
-      {u : ℝ≥0∞ | ∃ t : ℝ≥0, u=t ∧ alive a b t (((c:ℝ):ℂ)*z) ∧
-        physicalRadius (actualState a b t (((c:ℝ):ℂ)*z))=(c:ℝ)*ρ} := by
-  have hn : (c:ℝ)≠0 := NNReal.coe_ne_zero.mpr hc.ne'
-  ext u
-  constructor
-  · rintro ⟨w,⟨t,rfl,ht,hr⟩,rfl⟩
-    refine ⟨c^2*t,by simp, (alive_scaled_iff a hc b t z).mp ht,?_⟩
-    rw [physicalRadius_scaled a hc hb t z ht] at hr
-    calc
-      _ = (c:ℝ)*((c:ℝ)⁻¹*physicalRadius (actualState a b (c^2*t) (((c:ℝ):ℂ)*z))) := by simp [hn]
-      _ = _ := by rw [hr]
-  · rintro ⟨t,rfl,ht,hr⟩
-    let v : ℝ≥0 := (c^2)⁻¹*t
-    have hvt : c^2*v=t := by simp [v,hc.ne']
-    have hv : alive a (scaled c b) v z := by
-      rw [alive_scaled_iff a hc b v z,hvt]
-      exact ht
-    refine ⟨v,⟨v,rfl,hv,?_⟩,?_⟩
-    · rw [physicalRadius_scaled a hc hb v z hv,hvt,hr]
-      simp [hn]
-    · simp only [←ENNReal.coe_pow,←ENNReal.coe_mul,hvt]
-
- 
-lemma radialTime_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
-    (c:ℝ≥0∞)^2*radialTime a (scaled c b) z ρ =
-      radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) := by
-  let e := ENNReal.mulLeftOrderIso ((c:ℝ≥0∞)^2)
-    (ENNReal.isUnit_iff.mpr ⟨pow_ne_zero _ (by exact_mod_cast hc.ne'),
-      ENNReal.pow_ne_top ENNReal.coe_ne_top⟩)
-  change e (sInf _) = sInf _
-  rw [e.map_sInf,←sInf_image]
-  congr 1
-  exact image_radialHits a hc hb z ρ
-
-lemma radialTime_scaled_eq_top_iff (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
-    radialTime a (scaled c b) z ρ = ∞ ↔
-      radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) = ∞ := by
-  rw [←radialTime_scaled a hc hb z ρ]
-  simp [ENNReal.mul_eq_top,
-    pow_ne_zero 2 (show (c:ℝ≥0∞)≠0 by exact_mod_cast hc.ne')]
-
-end SLEExactGauge.LoewnerScaling
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open ExactSLEMain ExactSLEGreen
-
-lemma Y_actual_pos {a : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
-    (ht : alive a b t z) : 0 < Y (actualState a b t z) := by
-  obtain ⟨f,hf⟩ := ht
-  simpa only [Y_apply,actualState,Complex.sub_im,Complex.ofReal_im,sub_zero,
-    loewnerMap_eq hf] using (hf.2 t ⟨t.coe_nonneg,le_rfl⟩).1
-
-lemma radialTime_scaled_toNNReal (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
-    (radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ)).toNNReal =
-      c^2*(radialTime a (scaled c b) z ρ).toNNReal := by
-  rw [←radialTime_scaled a hc hb z ρ,ENNReal.toNNReal_mul]
-  simp
-
-lemma radialTime_scaled_le_iff (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) (t : ℝ≥0∞) :
-    radialTime a (scaled c b) z ρ ≤ t ↔
-      radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) ≤ (c:ℝ≥0∞)^2*t := by
-  rw [←radialTime_scaled a hc hb z ρ]
-  exact (ENNReal.mul_le_mul_iff_right
-    (pow_ne_zero _ (by exact_mod_cast hc.ne')) (ENNReal.pow_ne_top ENNReal.coe_ne_top)).symm
-
-lemma radialWeight_scaled (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hρz : ρ≤z.im) :
-    radialWeight a (scaleDriver c b).1 z ρ =
-      (c:ℝ)^(alpha a)*radialWeight a b.1 (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) := by
-  change radialWeight a (scaled c b.1) z ρ = _
-  have ht := radialTime_scaled_eq_top_iff a hc b.1.continuous z ρ
-  by_cases hs : radialTime a (scaled c b.1) z ρ=∞
-  · simp only [radialWeight,ht.mp hs,ite_true,mul_zero]
-    exact ite_eq_left hs
-  · have hl := (radialTime_attained a ha z hz ρ hρ hρz (scaleDriver c b) hs).1
-    change alive a (scaled c b.1) (radialTime a (scaled c b.1) z ρ).toNNReal z at hl
-    have hl' := (alive_scaled_iff a hc b.1 _ z).mp hl
-    rw [radialWeight,ite_eq_right hs,radialWeight,ite_eq_right (ht.not.mp hs),
-      radialTime_scaled_toNNReal a hc b.1.continuous z ρ,
-      actualState_scaled a hc b.1.continuous _ z hl]
-    exact scaleState_green a hc _ (Y_actual_pos hl')
-
-lemma massDensity_scaled_upper (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (b : ExactSLEGenerator.Driver)
-    (z : ℂ) (hz : 0<z.im) :
-    massDensity a q (scaleDriver c b) z = ENNReal.ofReal ((c:ℝ)^(alpha a))*
-      massDensity a q b (((c:ℝ):ℂ)*z) := by
-  have hz' : 0<(((c:ℝ):ℂ)*z).im := by
-    simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero]
-      using mul_pos (show 0<(c:ℝ) from hc) hz
-  simp only [massDensity,ite_eq_left hz,ite_eq_left hz',fullRadialDensity]
-  have he : 2*z.im*q/2=z.im*q := by ring
-  have he' : 2*(((c:ℝ):ℂ)*z).im*q/2=(c:ℝ)*(z.im*q) := by
-    simp only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero]
-    ring
-  have hcp : 0≤(c:ℝ)^(alpha a) := (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _).le
-  rw [he,he',radialWeight_scaled a ha hc b z hz (z.im*q) (mul_pos hz hq)
-    (mul_le_of_le_one_right hz.le hq1),←ENNReal.ofReal_mul hcp]
-  congr 1
-  ring
-
-lemma timeMassDensity_scaled_upper (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver)
-    (z : ℂ) (hz : 0<z.im) :
-    timeMassDensity a q t (scaleDriver c b) z = ENNReal.ofReal ((c:ℝ)^(alpha a))*
-      timeMassDensity a q ((c:ℝ≥0∞)^2*t) b (((c:ℝ):ℂ)*z) := by
-  have he : (((c:ℝ):ℂ)*z).im*q=(c:ℝ)*(z.im*q) := by
-    simp only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero]
-    ring
-  simp only [timeMassDensity,he]
-  change (if radialTime a (scaled c b.1) z (z.im*q)≤t then _ else _) = _
-  simp only [radialTime_scaled_le_iff a hc b.1.continuous z (z.im*q) t,
-    massDensity_scaled_upper a ha hc q hq hq1 b z hz]
-  split_ifs <;> simp
-
-end SLEExactGauge.LoewnerScaling
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.AngleLyapunov
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+
+def radialTestWeight (a : ℝ) (F : State→ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) : ℝ :=
+  if radialTime a b z ρ=∞ then 0 else
+    let v := actualState a b (radialTime a b z ρ).toNNReal z
+    physicalGreen a v * F v
+
+lemma radialTestWeight_nonneg (a : ℝ) (F : State→ℝ) (hF : ∀v, 0≤F v)
+    (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) : 0≤radialTestWeight a F b z ρ := by
+  unfold radialTestWeight
+  split_ifs
+  · exact le_rfl
+  · exact mul_nonneg (Real.exp_pos _).le (hF _)
+
+lemma sinePower_angleLift_pos (q : ℝ) (v : State) : 0<angleLift (sinePower q) v :=
+  Real.rpow_pos_of_pos
+    (Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2) _
+
+lemma physicalClock_initial (a : ℝ) (z : ℂ) : physicalClock a (2*z.im) (z,0)=0 := by
+  simp only [physicalClock,J_apply,Y_apply]
+  rw [show 2*z.im/2=z.im by ring]
+  simp
+
+lemma angleBudgetFactor_initial_bound (a q s C : ℝ) (hs : 0≤ s) (hC : 0≤C)
+    (z : ℂ) :
+    Real.exp (-s) * angleBudgetFactor a q (2*z.im) s C (z,0) ≤
+      angleLift (sinePower (-q)) (z,0)+C := by
+  have hE := (Real.exp_pos (-s)).le
+  have hE1 : Real.exp (-s)≤1 := Real.exp_le_one_iff.mpr (by linarith)
+  have hF := (sinePower_angleLift_pos (-q) (z,0)).le
+  have he : Real.exp (-s)*Real.exp s=1 := by rw [←Real.exp_add]; simp
+  have hprod := mul_le_mul_of_nonneg_right hE1 hF
+  have hCprod := mul_nonneg hE hC
+  unfold angleBudgetFactor clockExp
+  rw [physicalClock_initial a z, mul_zero,Real.exp_zero,one_mul]
+  nlinarith only [he,hprod,hCprod,hC]
+
+ 
+
+theorem finite_level_angle_budget (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a q C : ℝ) (ha : 0<a) (hC0 : 0≤C)
+    (hC : ∀ θ∈Ioo 0 Real.pi,
+      angleGenerator (2*a) (sinePower (-q)) θ+sinePower (-q) θ≤C)
+    (z : ℂ) (hz : 0<z.im) (s : ℝ) (hs : 0≤ s) :
+    (∫⁻ b, ENNReal.ofReal (radialTestWeight a (angleLift (sinePower (-q))) b.1 z
+      (levelRadius a (2*z.im) s)) ∂W) ≤
+      ENNReal.ofReal (physicalGreen a (z,0)*(angleLift (sinePower (-q)) (z,0)+C)) := by
+  let cap := 2*z.im
+  have hcap : 0<cap := by dsimp [cap]; positivity
+  let ρ := levelRadius a cap s
+  have hρ : 0<ρ := levelRadius_pos a cap s hcap
+  have hr : ρ≤z.im := by
+    simpa only [cap,mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0)] using
+      (levelRadius_le ha hcap hs)
+  let L (n : ℕ) := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T (n : ℕ) := radialStop a ha z hz ρ hρ hr n
+  let V (n : ℕ) (b : ExactSLEGenerator.Driver) := localState a (L n).ε (L n).hε z (T n b) b
+  let f (n : ℕ) (b : ExactSLEGenerator.Driver) := Real.exp (-s) *
+    (physicalGreen a (V n b)*angleBudgetFactor a q cap s C (V n b))
+  let A := physicalGreen a (z,0)*(angleLift (sinePower (-q)) (z,0)+C)
+  have hsafe (n : ℕ) (b : ExactSLEGenerator.Driver) (u : ℝ≥0) (hu : u≤T n b) :
+      (L n).ε≤Y (localState a (L n).ε (L n).hε z u b) ∧
+        ‖localState a (L n).ε (L n).hε z u b‖≤(L n).N := by
+    have h := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b u hu
+    exact ⟨h.1,h.2.1⟩
+  have hact (n : ℕ) (b : ExactSLEGenerator.Driver) : alive a b.1 (T n b) z ∧
+      actualState a b.1 (T n b) z = V n b :=
+    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b _ le_rfl
+  have hc (n : ℕ) (b : ExactSLEGenerator.Driver) : physicalClock a cap (V n b) ≤ s := by
+    rw [←(hact n b).2]
+    have h := actual_physicalClock_le a ha z hz cap ρ hcap hρ hr b (T n b)
+      (hact n b).1 (radialStop_le_radialTime a ha z hz ρ hρ hr n b)
+    simpa only [ρ,levelRadius_clock ha.ne' hcap] using h
+  have hp (n : ℕ) (b : ExactSLEGenerator.Driver) : 0≤f n b :=
+    mul_nonneg (Real.exp_pos _).le (mul_nonneg (Real.exp_pos _).le
+      (angleBudgetFactor_nonneg a q cap s C hC0 _ (hc n b)))
+  have hb (n : ℕ) : Integrable (f n) W ∧ (∫ b, f n b ∂W)≤A := by
+    have hh := local_angle_budget W hW a q cap s C (L n).ε ha.ne' hC (L n).hε
+      z (L n).N (L n).B (T n) (radialStop_stopping a ha z hz ρ hρ hr n)
+      (regionStop_le a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B)
+      (hsafe n)
+    refine ⟨hh.1.const_mul _, ?_⟩
+    change (∫ b, Real.exp (-s)*(physicalGreen a (V n b)*angleBudgetFactor a q cap s C (V n b)) ∂W)≤A
+    rw [integral_const_mul]
+    calc
+      _ ≤ Real.exp (-s)*(physicalGreen a (z,0)*angleBudgetFactor a q cap s C (z,0)) :=
+        mul_le_mul_of_nonneg_left hh.2 (Real.exp_pos _).le
+      _ = physicalGreen a (z,0)*(Real.exp (-s)*angleBudgetFactor a q cap s C (z,0)) := by ring
+      _ ≤ A := mul_le_mul_of_nonneg_left
+        (angleBudgetFactor_initial_bound a q s C hs hC0 z) (Real.exp_pos _).le
+  have hbf (n : ℕ) : (∫⁻ b, ENNReal.ofReal (f n b) ∂W)≤ENNReal.ofReal A := by
+    rw [←ofReal_integral_eq_lintegral_ofReal (hb n).1 (Eventually.of_forall (hp n))]
+    exact ENNReal.ofReal_le_ofReal (hb n).2
+  have hlim (b : ExactSLEGenerator.Driver) :
+      ENNReal.ofReal (radialTestWeight a (angleLift (sinePower (-q))) b.1 z ρ) ≤
+        liminf (fun n => ENNReal.ofReal (f n b)) atTop := by
+    by_cases ht : radialTime a b.1 z ρ=∞
+    · simp only [radialTestWeight,ite_eq_left ht,ENNReal.ofReal_zero]
+      exact bot_le
+    · apply le_liminf_of_le (by isBoundedDefault)
+      filter_upwards [radialStop_eventually_eq a ha z hz ρ hρ hr b ht] with n hn
+      have he : T n b=(radialTime a b.1 z ρ).toNNReal := by
+        have hh := congrArg ENNReal.toNNReal hn
+        simpa only [ENNReal.toNNReal_coe] using hh
+      have hv : V n b=actualState a b.1 (radialTime a b.1 z ρ).toNNReal z := by
+        rw [←(hact n b).2,he]
+      have htC : physicalClock a cap (V n b)=s := by
+        rw [hv, actual_physicalClock_at a ha z hz cap ρ hcap hρ hr b ht]
+        exact levelRadius_clock ha.ne' hcap
+      have hexp : Real.exp (-s)*Real.exp s=1 := by rw [←Real.exp_add]; simp
+      apply le_of_eq
+      congr 1
+      simp only [radialTestWeight,ite_eq_right ht,f,angleBudgetFactor_terminal a q cap s C _ htC]
+      rw [hv]
+      nlinarith only [congrArg (fun x => x*(physicalGreen a
+        (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z)*
+          angleLift (sinePower (-q)) (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z))) hexp]
+  exact (lintegral_mono hlim).trans ((lintegral_liminf_le' (fun n =>
+    (hb n).1.aestronglyMeasurable.aemeasurable.ennreal_ofReal)).trans
+      (liminf_le_of_frequently_le (Eventually.of_forall hbf).frequently))
+
+ 
+
+theorem exists_finite_level_angle_bound (a q : ℝ) (ha : 1/4<a) (hq : 0<q)
+    (hqp : q<4*a-1) : ∃ C : ℝ, 0<C ∧
+    ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+      ∀ (z : ℂ), 0<z.im → ∀ s : ℝ, 0≤ s →
+        (∫⁻ b, ENNReal.ofReal (radialTestWeight a (angleLift (sinePower (-q))) b.1 z
+          (levelRadius a (2*z.im) s)) ∂W) ≤
+          ENNReal.ofReal (physicalGreen a (z,0)*(angleLift (sinePower (-q)) (z,0)+C)) := by
+  obtain ⟨C,hC0,hC⟩ := exists_tilted_coercivity ha hq hqp
+  refine ⟨C,hC0,?_⟩
+  intro W hprob hW z hz s hs
+  exact finite_level_angle_budget W hW a q C (by linarith) hC0.le hC z hz s hs
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+
+lemma localizationStop_eventually_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver)
+    (L : ∀n, RadialLocalization a z ρ (angleThreshold z n))
+    (hfin : radialTime a b.1 z ρ≠∞) :
+    ∀ᶠ n in atTop,
+      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b : ℝ≥0∞)=
+        radialTime a b.1 z ρ := by
+  let t := (radialTime a b.1 z ρ).toNNReal
+  have ht := radialTime_attained a ha z hz ρ hρ hr b hfin
+  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht.1
+  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
+  let T := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
+  let V := localState a (L n).ε (L n).hε z T b
+  have hTt : T≤t := regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ
+    (angleThreshold z n) (L n).B (L n).initial b t ht.2
+  have he := regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+    (L n).B (L n).initial b T le_rfl
+  rcases regionStop_true_exit a (L n).ε ha (L n).hε z (L n).N ρ (angleThreshold z n) hρ
+    (angleThreshold_pos z hz n) (L n).B (L n).initial (L n).small (L n).large (L n).time b with hR|hS
+  · exact (regionStop_eq_radialTime a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b hR).symm
+  · have hs := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b T le_rfl
+    have hnp : 0<‖V.1‖ := ((L n).hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
+    have hd : Y V/‖V.1‖=angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+    have hη' := hηs T hTt
+    rw [he.2,hd] at hη'
+    exact False.elim (not_le.mpr hn hη')
+
+ 
+def radialState (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) : State :=
+  if radialTime a b z ρ=∞ then 0 else
+    actualState a b (radialTime a b z ρ).toNNReal z
+
+lemma localizedState_eventually_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b : ExactSLEGenerator.Driver)
+    (L : ∀n, RadialLocalization a z ρ (angleThreshold z n)) :
+    ∀ᶠ n in atTop,
+      (let V := localState a (L n).ε (L n).hε z
+        (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b
+       if physicalRadius V=ρ then V else 0)=radialState a b.1 z ρ := by
+  let T (n : ℕ) := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
+  let V (n : ℕ) := localState a (L n).ε (L n).hε z (T n) b
+  have he (n : ℕ) : actualState a b.1 (T n) z=V n :=
+    (regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b _ le_rfl).2
+  by_cases hf : radialTime a b.1 z ρ=∞
+  · apply Eventually.of_forall
+    intro n
+    have hn : physicalRadius (V n)≠ρ := by
+      intro hn
+      have hh := regionStop_eq_radialTime a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+        (L n).B (L n).initial b hn
+      exact ENNReal.coe_ne_top ((hf.symm.trans hh).symm)
+    exact (ite_eq_right hn).trans (by simp [radialState,hf])
+  · filter_upwards [localizationStop_eventually_eq a ha z hz ρ hρ hr b L hf] with n hn
+    have ht : T n=(radialTime a b.1 z ρ).toNNReal := by
+      have hh := congrArg ENNReal.toNNReal hn
+      simpa only [ENNReal.toNNReal_coe] using hh
+    have hv : V n=actualState a b.1 (radialTime a b.1 z ρ).toNNReal z := by rw [←he n,ht]
+    have hh : physicalRadius (V n)=ρ := by
+      rw [hv]; exact (radialTime_attained a ha z hz ρ hρ hr b hf).2
+    exact (ite_eq_left hh).trans (by simpa only [radialState,ite_eq_right hf] using hv)
+
+ 
+
+theorem joint_radialState_measurable (a : ℝ) (ha : 0<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) :
+    Measurable (fun p : ExactSLEGenerator.Driver×UpperPlane =>
+      radialState a p.1.1 p.2.1 (p.2.1.im*q)) := by
+  let Ω := ExactSLEGenerator.Driver×UpperPlane
+  let z : Ω→ℂ := fun p => p.2.1
+  let b : Ω→ExactSLEGenerator.Driver := Prod.fst
+  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
+  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
+  have hbM : Measurable b := measurable_fst
+  let ρ : Ω→ℝ := fun ω => (z ω).im*q
+  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
+  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
+  let δ : ℕ→Ω→ℝ := fun n ω => angleThreshold (z ω) n
+  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha (z ω) (hz ω) (ρ ω) (δ n ω)
+    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
+  have hρM : Measurable ρ := hzM.im.mul_const q
+  have hδM (n : ℕ) : Measurable (δ n) := by
+    dsimp [δ,angleThreshold]
+    exact (hzM.im.div hzM.norm).div_const _
+  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
+    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
+    unfold heightFloor
+    fun_prop
+  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
+    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1,_⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    unfold logCeiling heightFloor
+    fun_prop
+  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
+    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1,_⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    fun_prop
+  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
+    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
+  have hTM (n : ℕ) : Measurable (T n) :=
+    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _
+      (hεM n) hzM hbM (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
+  let V (n : ℕ) (ω : Ω) := localState a (L n ω).ε (L n ω).hε (z ω) (T n ω) (b ω)
+  have hVM (n : ℕ) : Measurable (V n) :=
+    (localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε)
+      z b (hεM n) hzM hbM).comp ((hTM n).prodMk measurable_id)
+  apply measurable_of_tendsto_metrizable
+    (f := fun n ω => if physicalRadius (V n ω)=ρ ω then V n ω else (0:State))
+  · intro n
+    exact Measurable.ite (measurableSet_eq_fun
+      (physicalRadius_continuous.measurable.comp (hVM n)) hρM) (hVM n) measurable_const
+  · apply tendsto_pi_nhds.mpr
+    intro ω
+    apply tendsto_const_nhds.congr'
+    exact (localizedState_eventually_eq a ha (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω) (b ω)
+      (fun n => L n ω)).mono (fun n hn => hn.symm)
+
+lemma radialTestWeight_eq_mul (a : ℝ) (F : State→ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (ρ : ℝ) :
+    radialTestWeight a F b z ρ=radialWeight a b z ρ*F (radialState a b z ρ) := by
+  unfold radialTestWeight radialWeight radialState
+  split_ifs <;> simp
+
+lemma joint_radialTestWeight_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (F : State→ℝ) (hF : Measurable F) :
+    Measurable (fun p : ExactSLEGenerator.Driver×UpperPlane =>
+      radialTestWeight a F p.1.1 p.2.1 (p.2.1.im*q)) := by
+  simp_rw [radialTestWeight_eq_mul]
+  exact (joint_radialWeight_measurable a ha q hq hq1).mul
+    (hF.comp (joint_radialState_measurable a (by linarith) q hq hq1))
+
+lemma angleLift_sinePower_measurable (q : ℝ) : Measurable (angleLift (sinePower q)) := by
+  unfold angleLift sinePower physicalTheta
+  fun_prop
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open ExactSLEMain ExactSLEGreen
+
+def scaleState (c : ℝ≥0) (v : State) : State := (((c:ℝ):ℂ)⁻¹*v.1,v.2)
+
+lemma scaleState_Y (c : ℝ≥0) (v : State) :
+    Y (scaleState c v)=(c:ℝ)⁻¹*Y v := by
+  simp [scaleState,←Complex.ofReal_inv,Complex.mul_im]
+
+lemma scaleState_X (c : ℝ≥0) (v : State) :
+    X (scaleState c v)=(c:ℝ)⁻¹*X v := by
+  simp [scaleState,←Complex.ofReal_inv,Complex.mul_re]
+
+lemma scaleState_Q (c : ℝ≥0) (v : State) :
+    Q (scaleState c v)=((c:ℝ)⁻¹)^2*Q v := by
+  simp only [Q,scaleState_X,scaleState_Y]
+  ring
+
+lemma scaleState_J (c : ℝ≥0) (v : State) : J (scaleState c v)=J v := rfl
+
+lemma scaleState_radius (c : ℝ≥0) (v : State) :
+    physicalRadius (scaleState c v)=(c:ℝ)⁻¹*physicalRadius v := by
+  simp only [physicalRadius,scaleState_Y,scaleState_J]
+  ring
+
+lemma scaleState_logGreen (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    (v : State) (hv : 0<Y v) :
+    logGreen a (scaleState c v)=alpha a*Real.log c+logGreen a v := by
+  have hcR : 0<(c:ℝ) := hc
+  have hn : (c:ℝ)≠0 := hcR.ne'
+  have hq := (Q_pos hv).ne'
+  simp only [logGreen,scaleState_Y,scaleState_Q,scaleState_J,
+    Real.log_mul (inv_ne_zero hn) hv.ne',
+    Real.log_mul (pow_ne_zero 2 (inv_ne_zero hn)) hq,
+    Real.log_pow,Real.log_inv]
+  ring
+
+lemma scaleState_green (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    (v : State) (hv : 0<Y v) :
+    physicalGreen a (scaleState c v)=(c:ℝ)^(alpha a)*physicalGreen a v := by
+  rw [physicalGreen,scaleState_logGreen a hc v hv,Real.exp_add,
+    Real.rpow_def_of_pos (show 0<(c:ℝ) from hc)]
+  rw [mul_comm (Real.log c)]
+  rfl
+
+lemma physicalRadius_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (t : ℝ≥0) (z : ℂ)
+    (hz : alive a (scaled c b) t z) :
+    physicalRadius (actualState a (scaled c b) t z) =
+      (c:ℝ)⁻¹*physicalRadius (actualState a b (c^2*t) (((c:ℝ):ℂ)*z)) := by
+  rw [actualState_scaled a hc hb t z hz]
+  exact scaleState_radius c _
+
+lemma image_radialHits (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
+    (fun u : ℝ≥0∞ => (c:ℝ≥0∞)^2*u) ''
+      {u : ℝ≥0∞ | ∃ t : ℝ≥0, u=t ∧ alive a (scaled c b) t z ∧
+        physicalRadius (actualState a (scaled c b) t z)=ρ} =
+      {u : ℝ≥0∞ | ∃ t : ℝ≥0, u=t ∧ alive a b t (((c:ℝ):ℂ)*z) ∧
+        physicalRadius (actualState a b t (((c:ℝ):ℂ)*z))=(c:ℝ)*ρ} := by
+  have hn : (c:ℝ)≠0 := NNReal.coe_ne_zero.mpr hc.ne'
+  ext u
+  constructor
+  · rintro ⟨w,⟨t,rfl,ht,hr⟩,rfl⟩
+    refine ⟨c^2*t,by simp, (alive_scaled_iff a hc b t z).mp ht,?_⟩
+    rw [physicalRadius_scaled a hc hb t z ht] at hr
+    calc
+      _ = (c:ℝ)*((c:ℝ)⁻¹*physicalRadius (actualState a b (c^2*t) (((c:ℝ):ℂ)*z))) := by simp [hn]
+      _ = _ := by rw [hr]
+  · rintro ⟨t,rfl,ht,hr⟩
+    let v : ℝ≥0 := (c^2)⁻¹*t
+    have hvt : c^2*v=t := by simp [v,hc.ne']
+    have hv : alive a (scaled c b) v z := by
+      rw [alive_scaled_iff a hc b v z,hvt]
+      exact ht
+    refine ⟨v,⟨v,rfl,hv,?_⟩,?_⟩
+    · rw [physicalRadius_scaled a hc hb v z hv,hvt,hr]
+      simp [hn]
+    · simp only [←ENNReal.coe_pow,←ENNReal.coe_mul,hvt]
+
+ 
+lemma radialTime_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
+    (c:ℝ≥0∞)^2*radialTime a (scaled c b) z ρ =
+      radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) := by
+  let e := ENNReal.mulLeftOrderIso ((c:ℝ≥0∞)^2)
+    (ENNReal.isUnit_iff.mpr ⟨pow_ne_zero _ (by exact_mod_cast hc.ne'),
+      ENNReal.pow_ne_top ENNReal.coe_ne_top⟩)
+  change e (sInf _) = sInf _
+  rw [e.map_sInf,←sInf_image]
+  congr 1
+  exact image_radialHits a hc hb z ρ
+
+lemma radialTime_scaled_eq_top_iff (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
+    radialTime a (scaled c b) z ρ = ∞ ↔
+      radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) = ∞ := by
+  rw [←radialTime_scaled a hc hb z ρ]
+  simp [ENNReal.mul_eq_top,
+    pow_ne_zero 2 (show (c:ℝ≥0∞)≠0 by exact_mod_cast hc.ne')]
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open ExactSLEMain ExactSLEGreen
+
+lemma Y_actual_pos {a : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
+    (ht : alive a b t z) : 0 < Y (actualState a b t z) := by
+  obtain ⟨f,hf⟩ := ht
+  simpa only [Y_apply,actualState,Complex.sub_im,Complex.ofReal_im,sub_zero,
+    loewnerMap_eq hf] using (hf.2 t ⟨t.coe_nonneg,le_rfl⟩).1
+
+lemma radialTime_scaled_toNNReal (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) :
+    (radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ)).toNNReal =
+      c^2*(radialTime a (scaled c b) z ρ).toNNReal := by
+  rw [←radialTime_scaled a hc hb z ρ,ENNReal.toNNReal_mul]
+  simp
+
+lemma radialTime_scaled_le_iff (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (z : ℂ) (ρ : ℝ) (t : ℝ≥0∞) :
+    radialTime a (scaled c b) z ρ ≤ t ↔
+      radialTime a b (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) ≤ (c:ℝ≥0∞)^2*t := by
+  rw [←radialTime_scaled a hc hb z ρ]
+  exact (ENNReal.mul_le_mul_iff_right
+    (pow_ne_zero _ (by exact_mod_cast hc.ne')) (ENNReal.pow_ne_top ENNReal.coe_ne_top)).symm
+
+lemma radialWeight_scaled (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hρz : ρ≤z.im) :
+    radialWeight a (scaleDriver c b).1 z ρ =
+      (c:ℝ)^(alpha a)*radialWeight a b.1 (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) := by
+  change radialWeight a (scaled c b.1) z ρ = _
+  have ht := radialTime_scaled_eq_top_iff a hc b.1.continuous z ρ
+  by_cases hs : radialTime a (scaled c b.1) z ρ=∞
+  · simp only [radialWeight,ht.mp hs,ite_true,mul_zero]
+    exact ite_eq_left hs
+  · have hl := (radialTime_attained a ha z hz ρ hρ hρz (scaleDriver c b) hs).1
+    change alive a (scaled c b.1) (radialTime a (scaled c b.1) z ρ).toNNReal z at hl
+    have hl' := (alive_scaled_iff a hc b.1 _ z).mp hl
+    rw [radialWeight,ite_eq_right hs,radialWeight,ite_eq_right (ht.not.mp hs),
+      radialTime_scaled_toNNReal a hc b.1.continuous z ρ,
+      actualState_scaled a hc b.1.continuous _ z hl]
+    exact scaleState_green a hc _ (Y_actual_pos hl')
+
+lemma massDensity_scaled_upper (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (b : ExactSLEGenerator.Driver)
+    (z : ℂ) (hz : 0<z.im) :
+    massDensity a q (scaleDriver c b) z = ENNReal.ofReal ((c:ℝ)^(alpha a))*
+      massDensity a q b (((c:ℝ):ℂ)*z) := by
+  have hz' : 0<(((c:ℝ):ℂ)*z).im := by
+    simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero]
+      using mul_pos (show 0<(c:ℝ) from hc) hz
+  simp only [massDensity,ite_eq_left hz,ite_eq_left hz',fullRadialDensity]
+  have he : 2*z.im*q/2=z.im*q := by ring
+  have he' : 2*(((c:ℝ):ℂ)*z).im*q/2=(c:ℝ)*(z.im*q) := by
+    simp only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero]
+    ring
+  have hcp : 0≤(c:ℝ)^(alpha a) := (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _).le
+  rw [he,he',radialWeight_scaled a ha hc b z hz (z.im*q) (mul_pos hz hq)
+    (mul_le_of_le_one_right hz.le hq1),←ENNReal.ofReal_mul hcp]
+  congr 1
+  ring
+
+lemma timeMassDensity_scaled_upper (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver)
+    (z : ℂ) (hz : 0<z.im) :
+    timeMassDensity a q t (scaleDriver c b) z = ENNReal.ofReal ((c:ℝ)^(alpha a))*
+      timeMassDensity a q ((c:ℝ≥0∞)^2*t) b (((c:ℝ):ℂ)*z) := by
+  have he : (((c:ℝ):ℂ)*z).im*q=(c:ℝ)*(z.im*q) := by
+    simp only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero]
+    ring
+  simp only [timeMassDensity,he]
+  change (if radialTime a (scaled c b.1) z (z.im*q)≤t then _ else _) = _
+  simp only [radialTime_scaled_le_iff a hc b.1.continuous z (z.im*q) t,
+    massDensity_scaled_upper a ha hc q hq hq1 b z hz]
+  split_ifs <;> simp
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/GreenIdentities.lean b/lean/OAI/Probability/SLEGauge/GreenIdentities.lean
index 7ce384e42392c99b5c75027930667242f60b741f..5edd72808d45bad4126fbe375ab0da349a816712 100644
--- a/lean/OAI/Probability/SLEGauge/GreenIdentities.lean
+++ b/lean/OAI/Probability/SLEGauge/GreenIdentities.lean
@@ -1,458 +1,458 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.SurvivalLocalization
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-open Filter Topology
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter ExactSLEGenerator ExactSLEMain
-open scoped Classical NNReal ENNReal Topology
-theorem actual_surviving_green (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
-    (letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-     Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => survivingGreen a t p.1 p.2.1)) ∧
-    (∀ (q : ℕ → ℝ), (∀n,0<q n ∧ q n≤1) → Tendsto q atTop (𝓝 0) →
-      ∀ (b : ExactSLEGenerator.Driver) (z : ℂ), 0<z.im →
-      Tendsto (fun n => remainingMassDensity a (q n) t b z) atTop (𝓝 (survivingGreen a t b z))) ∧
-    ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-      ∀ (z : ℂ), 0<z.im → Integrable (fun b => survivingGreen a t b z) W ∧
-      (∫b, survivingGreen a t b z ∂W) ≤ Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)
- := by
-  let q : ℕ → ℝ := fun n => 1/((n:ℝ)+1)
-  have hq (n : ℕ) : 0<q n ∧ q n≤1 := by
-    dsimp [q]
-    constructor
-    · positivity
-    · apply (div_le_one (by positivity)).mpr
-      linarith [Nat.cast_nonneg (α:=ℝ) n]
-  have hlim : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-  have hm : letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-      Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => survivingGreen a t p.1 p.2.1) := by
-    let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-    apply measurable_of_tendsto_metrizable (f:=fun n p => remainingMassDensity a (q n) t p.1 p.2.1)
-    · intro n
-      have H := (radialRemaining_raw_joint a ha (q n) (hq n).1 (hq n).2 t).const_mul
-        (Real.exp (-alpha a*Real.log 2))
-      convert H using 1
-      funext p
-      simp only [remainingMassDensity,ite_eq_left p.2.2]
-    · exact tendsto_pi_nhds.mpr (fun p => remaining_to_surviving a ha t q hq hlim p.1 p.2.1 p.2.2)
-  refine ⟨hm,fun q hq hlim => remaining_to_surviving a ha t q hq hlim,?_⟩
-  intro W inst hW z hz
-  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun n b => remainingMassDensity a (q n) t b z
-  let G : ExactSLEGenerator.Driver → ℝ := fun b => survivingGreen a t b z
-  let C : ℝ := Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)
-  have hC : 0≤C := by unfold C physicalGreen; positivity
-  have hFi (n : ℕ) := (remainingMassDensity_mean_bound a ha W hW (q n) (hq n).1 (hq n).2 t z hz).1
-  have hFm (n : ℕ) : AEStronglyMeasurable (F n) W := (hFi n).aestronglyMeasurable
-  have hconv : ∀ᵐ b ∂W,Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
-    Eventually.of_forall (fun b => remaining_to_surviving a ha t q hq hlim b z hz)
-  have hbound (n : ℕ) : eLpNorm (F n) 1 W≤ENNReal.ofReal C := by
-    rw [eLpNorm_one_eq_lintegral_enorm (hFm n),←ofReal_integral_norm_eq_lintegral_enorm (hFi n)]
-    have he : (fun b => ‖F n b‖) = F n := by
-      funext b
-      exact Real.norm_of_nonneg (remainingMassDensity_nonneg a (q n) t b z)
-    rw [he]
-    exact ENNReal.ofReal_le_ofReal
-      (remainingMassDensity_mean_bound a ha W hW (q n) (hq n).1 (hq n).2 t z hz).2
-  have hGm := aestronglyMeasurable_of_tendsto_ae atTop hFm hconv
-  have hGb := Lp.eLpNorm_le_of_ae_tendsto (Eventually.of_forall hbound) hFm hGm hconv
-  have hGi : Integrable G W := memLp_one_iff_integrable.mp (hGb.trans_lt ENNReal.ofReal_lt_top)
-  refine ⟨hGi,?_⟩
-  rw [eLpNorm_one_eq_lintegral_enorm hGm,←ofReal_integral_norm_eq_lintegral_enorm hGi] at hGb
-  have he : (fun b => ‖G b‖) = G := by
-    funext b
-    exact Real.norm_of_nonneg (survivingGreen_nonneg a t b z)
-  rw [he] at hGb
-  exact (ENNReal.ofReal_le_ofReal_iff hC).mp hGb
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
-open scoped Classical NNReal ENNReal Topology
-lemma remainingMassDensity_le_surviving (a q : ℝ) (t : ℝ≥0)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im) :
-    remainingMassDensity a q t b z ≤ survivingGreen a t b z := by
-  unfold remainingMassDensity radialRemaining survivingGreen
-  simp only [ite_eq_left hz]
-  by_cases ht : alive a b.1 t z
-  · simp only [ht,true_and,ite_true]
-    split_ifs
-    · exact le_rfl
-    · simpa only [mul_zero] using (show 0≤Real.exp (-alpha a*Real.log 2)*physicalGreen a (actualState a b.1 t z) from by unfold physicalGreen; positivity)
-  · simp only [ht,false_and,ite_false,mul_zero,le_refl]
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
-open scoped NNReal ENNReal Topology
- 
-
-theorem actual_remaining_spatial_L1 (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (K : Set UpperPlane) (hK : MeasurableSet K)
-    (hKfin : upperArea K<∞) (c : ℝ) (hc : 0<c) (hcK : ∀ z∈K,c≤z.1.im)
-    (f : UpperPlane → ℝ) (hf : Measurable f) (B : ℝ) (hB : 0≤B) (hfB : ∀z∈K,‖f z‖≤B)
-    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
-    Integrable (fun p : UpperPlane × ExactSLEGenerator.Driver => f p.1*survivingGreen a t p.2 p.1.1)
-      ((upperArea.restrict K).prod W) ∧
-    Tendsto (fun n => ∫p : UpperPlane × ExactSLEGenerator.Driver,
-      ‖f p.1*remainingMassDensity a (q n) t p.2 p.1.1-f p.1*survivingGreen a t p.2 p.1.1‖
-      ∂((upperArea.restrict K).prod W)) atTop (𝓝 0)
- := by
-  let ν := upperArea.restrict K
-  have : IsFiniteMeasure ν := ⟨by simpa [ν] using hKfin⟩
-  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*survivingGreen a t p.2 p.1.1
-  let H : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*remainingMassDensity a (q n) t p.2 p.1.1
-  have hHm (n : ℕ) : Measurable (H n) :=
-    (hf.comp measurable_fst).mul ((remainingMassDensity_measurable a ha (q n) (hq n).1 (hq n).2 t).comp
-      (measurable_snd.prodMk (measurable_subtype_coe.comp measurable_fst)))
-  have hconv (p : UpperPlane × ExactSLEGenerator.Driver) : Tendsto (fun n => H n p) atTop (𝓝 (F p)) :=
-    (remaining_to_surviving a ha t q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)
-  have hFm : Measurable F := measurable_of_tendsto_metrizable hHm (tendsto_pi_nhds.mpr hconv)
-  have hi (z : UpperPlane) : Integrable (fun b => F (z,b)) W :=
-    ((actual_surviving_green a ha t).2.2 W hW z.1 z.2).1.const_mul (f z)
-  have he (z : UpperPlane) : (∫b,‖F (z,b)‖∂W) = ‖f z‖*(∫b,survivingGreen a t b z.1∂W) := by
-    simp only [F,norm_mul,Real.norm_of_nonneg (survivingGreen_nonneg _ _ _ _),integral_const_mul]
-  let C := Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log c)
-  have hC : 0≤C := by dsimp [C]; positivity
-  have hb (z : UpperPlane) (hz : z∈K) : (∫b,‖F (z,b)‖∂W)≤B*C := by
-    rw [he]
-    have hG := ((actual_surviving_green a ha t).2.2 W hW z.1 z.2).2
-    have hgreen : physicalGreen a (z.1,0)≤Real.exp (-alpha a*Real.log c) := by
-      apply physicalGreen_bound a c ha hc _ z.2
-      simpa [physicalRadius,Y] using hcK z hz
-    have hpos : 0≤(∫b,survivingGreen a t b z.1∂W) := integral_nonneg (fun b => survivingGreen_nonneg _ _ _ _)
-    exact mul_le_mul (hfB z hz) (hG.trans (mul_le_mul_of_nonneg_left hgreen (Real.exp_pos _).le))
-      hpos hB
-  have hFi : Integrable F (ν.prod W) := by
-    apply (integrable_prod_iff hFm.aestronglyMeasurable).mpr
-    refine ⟨Eventually.of_forall hi,?_⟩
-    apply Integrable.of_bound (hFm.aestronglyMeasurable.norm.integral_prod_right') (B*C)
-    filter_upwards [ae_restrict_mem hK] with z hz
-    rw [Real.norm_of_nonneg (integral_nonneg (fun b => norm_nonneg (F (z,b))))]
-    exact hb z hz
-  refine ⟨hFi,?_⟩
-  have hD : Integrable (fun p => 2*‖F p‖) (ν.prod W) := hFi.norm.const_mul 2
-  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖H n p-F p‖≤2*‖F p‖ := by
-    have hle : ‖H n p‖≤‖F p‖ := by
-      dsimp only [H,F]
-      rw [norm_mul,norm_mul,Real.norm_of_nonneg (remainingMassDensity_nonneg _ _ _ _ _),
-        Real.norm_of_nonneg (survivingGreen_nonneg _ _ _ _)]
-      exact mul_le_mul_of_nonneg_left (remainingMassDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
-    calc ‖H n p-F p‖≤‖H n p‖+‖F p‖ := norm_sub_le _ _
-      _≤2*‖F p‖ := by linarith
-  have hlim' := tendsto_integral_of_dominated_convergence (fun p => 2*‖F p‖)
-    (fun n => ((hHm n).sub hFm).norm.aestronglyMeasurable) hD
-    (fun n => Eventually.of_forall (fun p => by simpa using hbound n p))
-    (Eventually.of_forall (fun p => by simpa using ((hconv p).sub_const (F p)).norm))
-  simpa using hlim'
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped Classical NNReal ENNReal
-lemma actualMassEval_raw_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0) (K : Set ℂ) (f : ℂ → ℝ) (hf : Measurable f) :
-    Measurable[rawFiltration t] (actualMassEval a q t K f) := by
-  have hD := timeMassDensity_raw_joint a ha q hq hq1 t
-  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ => f p.2*(timeMassDensity a q t p.1 p.2).toReal) :=
-    (hf.comp measurable_snd).mul hD.ennreal_toReal
-  exact hm.stronglyMeasurable.integral_prod_right'.measurable
-lemma actualUpperMass_eval_raw_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
-    Measurable[rawFiltration t] (fun b => ∫z,f z∂actualUpperMass a q t b) := by
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  exact actualMassEval_raw_measurable a ha q hq hq1 t _ _ (upperExtension_measurable f)
-lemma raw_ae_measurable_limit {Ω : Type*} {m m₀ : MeasurableSpace Ω} (W : Measure[m₀] Ω)
-    (F : ℕ → Ω → ℝ) (g : Ω → ℝ) (hF : ∀ k, Measurable[m] (F k))
-    (h : ∀ᵐ b ∂W, Tendsto (fun k => F k b) atTop (𝓝 (g b))) :
-    AEStronglyMeasurable[m] g W := by
-  refine ⟨fun b => limsup (fun k => F k b) atTop, (Measurable.limsup hF).stronglyMeasurable,?_⟩
-  filter_upwards [h] with b hb
-  exact hb.limsup_eq.symm
- 
-
-theorem actual_radon_raw_evaluations (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, coeff k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
-          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
-      (∀ i (t₀ : ℝ≥0), t i=t₀ → ∀ f : C_c(UpperPlane,ℝ),
-        AEStronglyMeasurable[rawFiltration t₀] (fun b => ∫z,f z∂μ b i) W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i)))
- := by
-  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,h⟩ := actual_radon_terminal_mean a ha W hW t
-  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,?_,h⟩
-  intro i t₀ hi f
-  apply raw_ae_measurable_limit W
-    (fun k b => ∑ j∈s k,c k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-    (fun b => ∫z,f z∂μ b i)
-  · intro k
-    rw [hi]
-    apply Finset.measurable_sum
-    intro j hj
-    have hq1 : Real.exp (-2*a*(j+1))≤1 := by
-      apply Real.exp_le_one_iff.mpr
-      have ha0 : 0<a := by linarith
-      have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-      nlinarith
-    exact measurable_const.mul (actualUpperMass_eval_raw_measurable a ha _ (Real.exp_pos _) hq1 t₀ f)
-  · filter_upwards [h] with b hb
-    exact hb.2 i f
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped Classical NNReal ENNReal Topology
-lemma actualUpperMass_finite_kernel (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0) (f : C_c(UpperPlane,ℝ))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[rawFiltration t] U) :
-    (∫b in U, (∫z,f z∂actualUpperMass a q ∞ b)-(∫z,f z∂actualUpperMass a q t b) ∂W) =
-      ∫b in U, (∫z : UpperPlane,f z*remainingMassDensity a q t b z.1∂upperArea) ∂W := by
-  let K := Subtype.val '' tsupport f
-  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
-  obtain ⟨c,hc,hcK⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  rw [actual_finite_conditional_mass a ha W hW q hq hq1 t K hK.measurableSet hK.isBounded c hc
-    (by rintro z ⟨w,hw,rfl⟩; exact hcK w hw) (upperExtension f)
-    (upperExtension_measurable f) (upperExtension_bounded f) U hU]
-  apply integral_congr_ae
-  filter_upwards [] with b
-  simp_rw [←upperExtension_apply f]
-  trans ∫z in {z : ℂ | 0<z.im},upperExtension f z*remainingMassDensity a q t b z
-  · rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
-    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
-    · rw [upperExtension_zero f z (by rintro ⟨w,hw,rfl⟩; exact hz w.2),zero_mul]
-    · rw [upperExtension_zero f z hz,zero_mul]
-  · exact (integral_subtype_comap (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet
-      (fun z : ℂ => upperExtension f z*remainingMassDensity a q t b z)).symm
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped Classical NNReal ENNReal Topology
-lemma actualUpper_remaining_integral_tendsto (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
-    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
-    Tendsto (fun n => ∫b in U, (∫z : UpperPlane,f z*remainingMassDensity a (q n) t b z.1∂upperArea) ∂W)
-      atTop (𝓝 (∫b in U, (∫z : UpperPlane,f z*survivingGreen a t b z.1∂upperArea) ∂W)) := by
-  let K := tsupport f
-  have hK : IsCompact K := f.hasCompactSupport
-  have hfin : upperArea K<∞ := by
-    rw [upperArea,upper_embedding.comap_apply]
-    exact (hK.image continuous_subtype_val).measure_lt_top
-  have : IsFiniteMeasure (upperArea.restrict K) := ⟨by simpa using hfin⟩
-  obtain ⟨c,hc,hcK⟩ := upper_compact_lower K hK
-  obtain ⟨B,hB,hBf⟩ := upperExtension_bounded f
-  have H := actual_remaining_spatial_L1 a ha W hW t K hK.measurableSet hfin (c/2) (by positivity)
-    (fun z hz => by have := hcK z hz; linarith) f f.continuous.measurable B hB
-    (fun z hz => by simpa only [upperExtension_apply] using hBf z.1 ⟨z,hz,rfl⟩) q hq hlim
-  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*survivingGreen a t p.2 p.1.1
-  let R : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*remainingMassDensity a (q n) t p.2 p.1.1
-  let μ := (upperArea.restrict K).prod (W.restrict U)
-  have hFi : Integrable F μ := H.1.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self)
-  have hRm (n : ℕ) : Measurable (R n) :=
-    (f.continuous.measurable.comp measurable_fst).mul
-      ((remainingMassDensity_measurable a ha (q n) (hq n).1 (hq n).2 t).comp
-        (measurable_snd.prodMk (measurable_subtype_coe.comp measurable_fst)))
-  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖R n p‖≤‖F p‖ := by
-    dsimp only [R,F]
-    rw [norm_mul,norm_mul,Real.norm_of_nonneg (remainingMassDensity_nonneg _ _ _ _ _),
-      Real.norm_of_nonneg (survivingGreen_nonneg _ _ _ _)]
-    exact mul_le_mul_of_nonneg_left (remainingMassDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
-  have hRi (n : ℕ) : Integrable (R n) μ := hFi.norm.mono' (hRm n).aestronglyMeasurable
-    (Eventually.of_forall (hbound n))
-  have ht := tendsto_integral_of_dominated_convergence (fun p => ‖F p‖)
-    (fun n => (hRm n).aestronglyMeasurable) hFi.norm (fun n => Eventually.of_forall (hbound n))
-    (Eventually.of_forall (fun p => (remaining_to_surviving a ha t q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)))
-  have he (g : UpperPlane → ℝ) : (∫z in K,f z*g z∂upperArea) = ∫z,f z*g z∂upperArea :=
-    setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by rw [image_eq_zero_of_notMem_tsupport hz,zero_mul])
-  have hRint (n : ℕ) : (∫p,R n p∂μ) = ∫b in U,(∫z : UpperPlane,f z*remainingMassDensity a (q n) t b z.1∂upperArea)∂W := by
-    rw [integral_prod_symm (R n) (hRi n)]
-    apply integral_congr_ae
-    filter_upwards [] with b
-    exact he _
-  have hFint : (∫p,F p∂μ) = ∫b in U,(∫z : UpperPlane,f z*survivingGreen a t b z.1∂upperArea)∂W := by
-    rw [integral_prod_symm F hFi]
-    apply integral_congr_ae
-    filter_upwards [] with b
-    exact he _
-  change Tendsto (fun n => ∫p,R n p∂μ) atTop (𝓝 (∫p,F p∂μ)) at ht
-  simpa only [hRint,hFint] using ht
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped Classical NNReal ENNReal Topology
-lemma setIntegral_tendsto_of_L1 {Ω : Type*} [MeasurableSpace Ω] (W : Measure Ω)
-    (F : ℕ → Ω → ℝ) (g : Ω → ℝ) (hF : ∀n,Integrable (F n) W) (hg : Integrable g W)
-    (hlim : Tendsto (fun n => eLpNorm (fun b => F n b-g b) 1 W) atTop (𝓝 0)) (U : Set Ω) :
-    Tendsto (fun n => ∫b in U,F n b∂W) atTop (𝓝 (∫b in U,g b∂W)) := by
-  have hsmall : Tendsto (fun n => eLpNorm (fun b => F n b-g b) 1 (W.restrict U)) atTop (𝓝 0) :=
-    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
-      (Eventually.of_forall (fun _ => bot_le))
-      (Eventually.of_forall (fun n => eLpNorm_mono_measure _ Measure.restrict_le_self))
-  exact tendsto_integral_of_L1 g hg.integrableOn.aestronglyMeasurable
-    (Eventually.of_forall (fun n => (hF n).integrableOn))
-    (hsmall.congr fun index => eLpNorm_one_eq_lintegral_enorm
-      ((hF index).integrableOn.aestronglyMeasurable.sub hg.integrableOn.aestronglyMeasurable))
-lemma scalar_convex_tail_limit (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-    (hs : ∀k j,j∈s k→k≤j) (hc : ∀k j,j∈s k→0≤c k j) (hsum : ∀k,∑j∈s k,c k j=1)
-    (x : ℕ → ℝ) (L : ℝ) (hx : Tendsto x atTop (𝓝 L)) :
-    Tendsto (fun k => ∑j∈s k,c k j*x j) atTop (𝓝 L) := by
-  rw [Metric.tendsto_atTop] at hx ⊢
-  intro ε hε
-  obtain ⟨N,hN⟩ := hx (ε/2) (by positivity)
-  refine ⟨N,fun k hk => ?_⟩
-  rw [Real.dist_eq]
-  have he : (∑j∈s k,c k j*x j)-L = ∑j∈s k,c k j*(x j-L) := by
-    simp only [mul_sub,Finset.sum_sub_distrib,←Finset.sum_mul,hsum k,one_mul]
-  rw [he]
-  calc
-    |∑j∈s k,c k j*(x j-L)| ≤ ∑j∈s k,|c k j*(x j-L)| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑j∈s k,c k j*(ε/2) := by
-      apply Finset.sum_le_sum
-      intro j hj
-      rw [abs_mul,abs_of_nonneg (hc k j hj)]
-      exact mul_le_mul_of_nonneg_left (by simpa [Real.dist_eq] using (hN j (hk.trans (hs k j hj))).le) (hc k j hj)
-    _ = ε/2 := by rw [←Finset.sum_mul,hsum k,one_mul]
-    _ < ε := by linarith
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped Classical NNReal ENNReal
-theorem actual_radon_compact_conditional (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, coeff k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
-          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
-      (∀ i (t₀ : ℝ≥0), t i=t₀ → ∀ f : C_c(UpperPlane,ℝ),
-        AEStronglyMeasurable[rawFiltration t₀] (fun b => ∫z,f z∂μ b i) W) ∧
-      (∀ i j (t₀ : ℝ≥0), t i=t₀ → t j=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        ∀ U : Set ExactSLEGenerator.Driver, MeasurableSet[rawFiltration t₀] U →
-        (∫b in U, (∫z,f z∂μ b j)-(∫z,f z∂μ b i) ∂W) =
-          ∫b in U, (∫z : UpperPlane,f z*survivingGreen a t₀ b z.1 ∂upperArea) ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i)))
- := by
-  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,hraw,h⟩ := actual_radon_raw_evaluations a ha W hW t
-  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,hraw,?_,h⟩
-  intro i j t₀ hi hj f U hU
-  let q : ℕ → ℝ := fun n => Real.exp (-2*a*((n:ℝ)+1))
-  have hq (n : ℕ) : 0<q n ∧ q n≤1 := by
-    refine ⟨Real.exp_pos _,Real.exp_le_one_iff.mpr ?_⟩
-    have hn : (0:ℝ)≤n := Nat.cast_nonneg _
-    have ha0 : 0<a := by linarith
-    nlinarith
-  have hqlim : Tendsto q atTop (𝓝 0) := by
-    apply Real.tendsto_exp_atBot.comp
-    apply Tendsto.const_mul_atTop_of_neg (by linarith : -2*a<0)
-    apply Filter.tendsto_atTop.mpr
-    intro B
-    filter_upwards [(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop).eventually (eventually_ge_atTop B)] with n hn
-    linarith
-  let F : ℕ → ℕ → ExactSLEGenerator.Driver → ℝ := fun l k b =>
-    ∑ n∈s k,c k n*∫z,f z∂actualUpperMass a (q n) (t l) b
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW 1 (by norm_num) f
-  have hEval (n : ℕ) (v : ℝ≥0∞) : Integrable (fun b => ∫z,f z∂actualUpperMass a (q n) v b) W :=
-    (hCb (q n) (hq n).1 (hq n).2 v).1.integrable (by norm_num)
-  have hFi (l k : ℕ) : Integrable (F l k) W :=
-    integrable_finsetSum (s k) (fun n hn => (hEval n (t l)).const_mul (c k n))
-  have hμi (l : ℕ) : Integrable (fun b => ∫z,f z∂μ b l) W :=
-    (hmom l f 1).integrable (by norm_num)
-  have hlim (l : ℕ) := setIntegral_tendsto_of_L1 W (F l) (fun b => ∫z,f z∂μ b l)
-    (hFi l) (hμi l) (hL1 l f) U
-  let R : ℕ → ℝ := fun n => ∫b in U,(∫z : UpperPlane,f z*remainingMassDensity a (q n) t₀ b z.1∂upperArea)∂W
-  let L : ℝ := ∫b in U,(∫z : UpperPlane,f z*survivingGreen a t₀ b z.1∂upperArea)∂W
-  have hRlim : Tendsto R atTop (𝓝 L) := actualUpper_remaining_integral_tendsto a ha W hW t₀ f U q hq hqlim
-  have hcombo := scalar_convex_tail_limit s c hs hc hsum R L hRlim
-  have he (k : ℕ) : (∫b in U,F j k b∂W)-(∫b in U,F i k b∂W) = ∑n∈s k,c k n*R n := by
-    dsimp only [F]
-    rw [integral_finsetSum _ (fun n hn => ((hEval n (t j)).const_mul (c k n)).integrableOn),
-      integral_finsetSum _ (fun n hn => ((hEval n (t i)).const_mul (c k n)).integrableOn)]
-    simp only [integral_const_mul]
-    rw [←Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl
-    intro n hn
-    rw [←mul_sub,←integral_sub (hEval n (t j)).integrableOn (hEval n (t i)).integrableOn]
-    rw [hj,hi,actualUpperMass_finite_kernel a ha W hW (q n) (hq n).1 (hq n).2 t₀ f U hU]
-  rw [integral_sub (hμi j).integrableOn (hμi i).integrableOn]
-  exact tendsto_nhds_unique ((hlim j).sub (hlim i)) (by simpa only [he] using hcombo)
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.SurvivalLocalization
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+open Filter Topology
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter ExactSLEGenerator ExactSLEMain
+open scoped Classical NNReal ENNReal Topology
+theorem actual_surviving_green (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
+    (letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+     Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => survivingGreen a t p.1 p.2.1)) ∧
+    (∀ (q : ℕ → ℝ), (∀n,0<q n ∧ q n≤1) → Tendsto q atTop (𝓝 0) →
+      ∀ (b : ExactSLEGenerator.Driver) (z : ℂ), 0<z.im →
+      Tendsto (fun n => remainingMassDensity a (q n) t b z) atTop (𝓝 (survivingGreen a t b z))) ∧
+    ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+      ∀ (z : ℂ), 0<z.im → Integrable (fun b => survivingGreen a t b z) W ∧
+      (∫b, survivingGreen a t b z ∂W) ≤ Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)
+ := by
+  let q : ℕ → ℝ := fun n => 1/((n:ℝ)+1)
+  have hq (n : ℕ) : 0<q n ∧ q n≤1 := by
+    dsimp [q]
+    constructor
+    · positivity
+    · apply (div_le_one (by positivity)).mpr
+      linarith [Nat.cast_nonneg (α:=ℝ) n]
+  have hlim : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  have hm : letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+      Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => survivingGreen a t p.1 p.2.1) := by
+    let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+    apply measurable_of_tendsto_metrizable (f:=fun n p => remainingMassDensity a (q n) t p.1 p.2.1)
+    · intro n
+      have H := (radialRemaining_raw_joint a ha (q n) (hq n).1 (hq n).2 t).const_mul
+        (Real.exp (-alpha a*Real.log 2))
+      convert H using 1
+      funext p
+      simp only [remainingMassDensity,ite_eq_left p.2.2]
+    · exact tendsto_pi_nhds.mpr (fun p => remaining_to_surviving a ha t q hq hlim p.1 p.2.1 p.2.2)
+  refine ⟨hm,fun q hq hlim => remaining_to_surviving a ha t q hq hlim,?_⟩
+  intro W inst hW z hz
+  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun n b => remainingMassDensity a (q n) t b z
+  let G : ExactSLEGenerator.Driver → ℝ := fun b => survivingGreen a t b z
+  let C : ℝ := Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)
+  have hC : 0≤C := by unfold C physicalGreen; positivity
+  have hFi (n : ℕ) := (remainingMassDensity_mean_bound a ha W hW (q n) (hq n).1 (hq n).2 t z hz).1
+  have hFm (n : ℕ) : AEStronglyMeasurable (F n) W := (hFi n).aestronglyMeasurable
+  have hconv : ∀ᵐ b ∂W,Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
+    Eventually.of_forall (fun b => remaining_to_surviving a ha t q hq hlim b z hz)
+  have hbound (n : ℕ) : eLpNorm (F n) 1 W≤ENNReal.ofReal C := by
+    rw [eLpNorm_one_eq_lintegral_enorm (hFm n),←ofReal_integral_norm_eq_lintegral_enorm (hFi n)]
+    have he : (fun b => ‖F n b‖) = F n := by
+      funext b
+      exact Real.norm_of_nonneg (remainingMassDensity_nonneg a (q n) t b z)
+    rw [he]
+    exact ENNReal.ofReal_le_ofReal
+      (remainingMassDensity_mean_bound a ha W hW (q n) (hq n).1 (hq n).2 t z hz).2
+  have hGm := aestronglyMeasurable_of_tendsto_ae atTop hFm hconv
+  have hGb := Lp.eLpNorm_le_of_ae_tendsto (Eventually.of_forall hbound) hFm hGm hconv
+  have hGi : Integrable G W := memLp_one_iff_integrable.mp (hGb.trans_lt ENNReal.ofReal_lt_top)
+  refine ⟨hGi,?_⟩
+  rw [eLpNorm_one_eq_lintegral_enorm hGm,←ofReal_integral_norm_eq_lintegral_enorm hGi] at hGb
+  have he : (fun b => ‖G b‖) = G := by
+    funext b
+    exact Real.norm_of_nonneg (survivingGreen_nonneg a t b z)
+  rw [he] at hGb
+  exact (ENNReal.ofReal_le_ofReal_iff hC).mp hGb
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
+open scoped Classical NNReal ENNReal Topology
+lemma remainingMassDensity_le_surviving (a q : ℝ) (t : ℝ≥0)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im) :
+    remainingMassDensity a q t b z ≤ survivingGreen a t b z := by
+  unfold remainingMassDensity radialRemaining survivingGreen
+  simp only [ite_eq_left hz]
+  by_cases ht : alive a b.1 t z
+  · simp only [ht,true_and,ite_true]
+    split_ifs
+    · exact le_rfl
+    · simpa only [mul_zero] using (show 0≤Real.exp (-alpha a*Real.log 2)*physicalGreen a (actualState a b.1 t z) from by unfold physicalGreen; positivity)
+  · simp only [ht,false_and,ite_false,mul_zero,le_refl]
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
+open scoped NNReal ENNReal Topology
+ 
+
+theorem actual_remaining_spatial_L1 (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (K : Set UpperPlane) (hK : MeasurableSet K)
+    (hKfin : upperArea K<∞) (c : ℝ) (hc : 0<c) (hcK : ∀ z∈K,c≤z.1.im)
+    (f : UpperPlane → ℝ) (hf : Measurable f) (B : ℝ) (hB : 0≤B) (hfB : ∀z∈K,‖f z‖≤B)
+    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
+    Integrable (fun p : UpperPlane × ExactSLEGenerator.Driver => f p.1*survivingGreen a t p.2 p.1.1)
+      ((upperArea.restrict K).prod W) ∧
+    Tendsto (fun n => ∫p : UpperPlane × ExactSLEGenerator.Driver,
+      ‖f p.1*remainingMassDensity a (q n) t p.2 p.1.1-f p.1*survivingGreen a t p.2 p.1.1‖
+      ∂((upperArea.restrict K).prod W)) atTop (𝓝 0)
+ := by
+  let ν := upperArea.restrict K
+  have : IsFiniteMeasure ν := ⟨by simpa [ν] using hKfin⟩
+  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*survivingGreen a t p.2 p.1.1
+  let H : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*remainingMassDensity a (q n) t p.2 p.1.1
+  have hHm (n : ℕ) : Measurable (H n) :=
+    (hf.comp measurable_fst).mul ((remainingMassDensity_measurable a ha (q n) (hq n).1 (hq n).2 t).comp
+      (measurable_snd.prodMk (measurable_subtype_coe.comp measurable_fst)))
+  have hconv (p : UpperPlane × ExactSLEGenerator.Driver) : Tendsto (fun n => H n p) atTop (𝓝 (F p)) :=
+    (remaining_to_surviving a ha t q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)
+  have hFm : Measurable F := measurable_of_tendsto_metrizable hHm (tendsto_pi_nhds.mpr hconv)
+  have hi (z : UpperPlane) : Integrable (fun b => F (z,b)) W :=
+    ((actual_surviving_green a ha t).2.2 W hW z.1 z.2).1.const_mul (f z)
+  have he (z : UpperPlane) : (∫b,‖F (z,b)‖∂W) = ‖f z‖*(∫b,survivingGreen a t b z.1∂W) := by
+    simp only [F,norm_mul,Real.norm_of_nonneg (survivingGreen_nonneg _ _ _ _),integral_const_mul]
+  let C := Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log c)
+  have hC : 0≤C := by dsimp [C]; positivity
+  have hb (z : UpperPlane) (hz : z∈K) : (∫b,‖F (z,b)‖∂W)≤B*C := by
+    rw [he]
+    have hG := ((actual_surviving_green a ha t).2.2 W hW z.1 z.2).2
+    have hgreen : physicalGreen a (z.1,0)≤Real.exp (-alpha a*Real.log c) := by
+      apply physicalGreen_bound a c ha hc _ z.2
+      simpa [physicalRadius,Y] using hcK z hz
+    have hpos : 0≤(∫b,survivingGreen a t b z.1∂W) := integral_nonneg (fun b => survivingGreen_nonneg _ _ _ _)
+    exact mul_le_mul (hfB z hz) (hG.trans (mul_le_mul_of_nonneg_left hgreen (Real.exp_pos _).le))
+      hpos hB
+  have hFi : Integrable F (ν.prod W) := by
+    apply (integrable_prod_iff hFm.aestronglyMeasurable).mpr
+    refine ⟨Eventually.of_forall hi,?_⟩
+    apply Integrable.of_bound (hFm.aestronglyMeasurable.norm.integral_prod_right') (B*C)
+    filter_upwards [ae_restrict_mem hK] with z hz
+    rw [Real.norm_of_nonneg (integral_nonneg (fun b => norm_nonneg (F (z,b))))]
+    exact hb z hz
+  refine ⟨hFi,?_⟩
+  have hD : Integrable (fun p => 2*‖F p‖) (ν.prod W) := hFi.norm.const_mul 2
+  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖H n p-F p‖≤2*‖F p‖ := by
+    have hle : ‖H n p‖≤‖F p‖ := by
+      dsimp only [H,F]
+      rw [norm_mul,norm_mul,Real.norm_of_nonneg (remainingMassDensity_nonneg _ _ _ _ _),
+        Real.norm_of_nonneg (survivingGreen_nonneg _ _ _ _)]
+      exact mul_le_mul_of_nonneg_left (remainingMassDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
+    calc ‖H n p-F p‖≤‖H n p‖+‖F p‖ := norm_sub_le _ _
+      _≤2*‖F p‖ := by linarith
+  have hlim' := tendsto_integral_of_dominated_convergence (fun p => 2*‖F p‖)
+    (fun n => ((hHm n).sub hFm).norm.aestronglyMeasurable) hD
+    (fun n => Eventually.of_forall (fun p => by simpa using hbound n p))
+    (Eventually.of_forall (fun p => by simpa using ((hconv p).sub_const (F p)).norm))
+  simpa using hlim'
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped Classical NNReal ENNReal
+lemma actualMassEval_raw_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0) (K : Set ℂ) (f : ℂ → ℝ) (hf : Measurable f) :
+    Measurable[rawFiltration t] (actualMassEval a q t K f) := by
+  have hD := timeMassDensity_raw_joint a ha q hq hq1 t
+  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ => f p.2*(timeMassDensity a q t p.1 p.2).toReal) :=
+    (hf.comp measurable_snd).mul hD.ennreal_toReal
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+lemma actualUpperMass_eval_raw_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
+    Measurable[rawFiltration t] (fun b => ∫z,f z∂actualUpperMass a q t b) := by
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  exact actualMassEval_raw_measurable a ha q hq hq1 t _ _ (upperExtension_measurable f)
+lemma raw_ae_measurable_limit {Ω : Type*} {m m₀ : MeasurableSpace Ω} (W : Measure[m₀] Ω)
+    (F : ℕ → Ω → ℝ) (g : Ω → ℝ) (hF : ∀ k, Measurable[m] (F k))
+    (h : ∀ᵐ b ∂W, Tendsto (fun k => F k b) atTop (𝓝 (g b))) :
+    AEStronglyMeasurable[m] g W := by
+  refine ⟨fun b => limsup (fun k => F k b) atTop, (Measurable.limsup hF).stronglyMeasurable,?_⟩
+  filter_upwards [h] with b hb
+  exact hb.limsup_eq.symm
+ 
+
+theorem actual_radon_raw_evaluations (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, coeff k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
+          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
+      (∀ i (t₀ : ℝ≥0), t i=t₀ → ∀ f : C_c(UpperPlane,ℝ),
+        AEStronglyMeasurable[rawFiltration t₀] (fun b => ∫z,f z∂μ b i) W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i)))
+ := by
+  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,h⟩ := actual_radon_terminal_mean a ha W hW t
+  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,?_,h⟩
+  intro i t₀ hi f
+  apply raw_ae_measurable_limit W
+    (fun k b => ∑ j∈s k,c k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+    (fun b => ∫z,f z∂μ b i)
+  · intro k
+    rw [hi]
+    apply Finset.measurable_sum
+    intro j hj
+    have hq1 : Real.exp (-2*a*(j+1))≤1 := by
+      apply Real.exp_le_one_iff.mpr
+      have ha0 : 0<a := by linarith
+      have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+      nlinarith
+    exact measurable_const.mul (actualUpperMass_eval_raw_measurable a ha _ (Real.exp_pos _) hq1 t₀ f)
+  · filter_upwards [h] with b hb
+    exact hb.2 i f
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped Classical NNReal ENNReal Topology
+lemma actualUpperMass_finite_kernel (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0) (f : C_c(UpperPlane,ℝ))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[rawFiltration t] U) :
+    (∫b in U, (∫z,f z∂actualUpperMass a q ∞ b)-(∫z,f z∂actualUpperMass a q t b) ∂W) =
+      ∫b in U, (∫z : UpperPlane,f z*remainingMassDensity a q t b z.1∂upperArea) ∂W := by
+  let K := Subtype.val '' tsupport f
+  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
+  obtain ⟨c,hc,hcK⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  rw [actual_finite_conditional_mass a ha W hW q hq hq1 t K hK.measurableSet hK.isBounded c hc
+    (by rintro z ⟨w,hw,rfl⟩; exact hcK w hw) (upperExtension f)
+    (upperExtension_measurable f) (upperExtension_bounded f) U hU]
+  apply integral_congr_ae
+  filter_upwards [] with b
+  simp_rw [←upperExtension_apply f]
+  trans ∫z in {z : ℂ | 0<z.im},upperExtension f z*remainingMassDensity a q t b z
+  · rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
+    · rw [upperExtension_zero f z (by rintro ⟨w,hw,rfl⟩; exact hz w.2),zero_mul]
+    · rw [upperExtension_zero f z hz,zero_mul]
+  · exact (integral_subtype_comap (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet
+      (fun z : ℂ => upperExtension f z*remainingMassDensity a q t b z)).symm
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped Classical NNReal ENNReal Topology
+lemma actualUpper_remaining_integral_tendsto (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
+    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
+    Tendsto (fun n => ∫b in U, (∫z : UpperPlane,f z*remainingMassDensity a (q n) t b z.1∂upperArea) ∂W)
+      atTop (𝓝 (∫b in U, (∫z : UpperPlane,f z*survivingGreen a t b z.1∂upperArea) ∂W)) := by
+  let K := tsupport f
+  have hK : IsCompact K := f.hasCompactSupport
+  have hfin : upperArea K<∞ := by
+    rw [upperArea,upper_embedding.comap_apply]
+    exact (hK.image continuous_subtype_val).measure_lt_top
+  have : IsFiniteMeasure (upperArea.restrict K) := ⟨by simpa using hfin⟩
+  obtain ⟨c,hc,hcK⟩ := upper_compact_lower K hK
+  obtain ⟨B,hB,hBf⟩ := upperExtension_bounded f
+  have H := actual_remaining_spatial_L1 a ha W hW t K hK.measurableSet hfin (c/2) (by positivity)
+    (fun z hz => by have := hcK z hz; linarith) f f.continuous.measurable B hB
+    (fun z hz => by simpa only [upperExtension_apply] using hBf z.1 ⟨z,hz,rfl⟩) q hq hlim
+  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*survivingGreen a t p.2 p.1.1
+  let R : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*remainingMassDensity a (q n) t p.2 p.1.1
+  let μ := (upperArea.restrict K).prod (W.restrict U)
+  have hFi : Integrable F μ := H.1.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self)
+  have hRm (n : ℕ) : Measurable (R n) :=
+    (f.continuous.measurable.comp measurable_fst).mul
+      ((remainingMassDensity_measurable a ha (q n) (hq n).1 (hq n).2 t).comp
+        (measurable_snd.prodMk (measurable_subtype_coe.comp measurable_fst)))
+  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖R n p‖≤‖F p‖ := by
+    dsimp only [R,F]
+    rw [norm_mul,norm_mul,Real.norm_of_nonneg (remainingMassDensity_nonneg _ _ _ _ _),
+      Real.norm_of_nonneg (survivingGreen_nonneg _ _ _ _)]
+    exact mul_le_mul_of_nonneg_left (remainingMassDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
+  have hRi (n : ℕ) : Integrable (R n) μ := hFi.norm.mono' (hRm n).aestronglyMeasurable
+    (Eventually.of_forall (hbound n))
+  have ht := tendsto_integral_of_dominated_convergence (fun p => ‖F p‖)
+    (fun n => (hRm n).aestronglyMeasurable) hFi.norm (fun n => Eventually.of_forall (hbound n))
+    (Eventually.of_forall (fun p => (remaining_to_surviving a ha t q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)))
+  have he (g : UpperPlane → ℝ) : (∫z in K,f z*g z∂upperArea) = ∫z,f z*g z∂upperArea :=
+    setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by rw [image_eq_zero_of_notMem_tsupport hz,zero_mul])
+  have hRint (n : ℕ) : (∫p,R n p∂μ) = ∫b in U,(∫z : UpperPlane,f z*remainingMassDensity a (q n) t b z.1∂upperArea)∂W := by
+    rw [integral_prod_symm (R n) (hRi n)]
+    apply integral_congr_ae
+    filter_upwards [] with b
+    exact he _
+  have hFint : (∫p,F p∂μ) = ∫b in U,(∫z : UpperPlane,f z*survivingGreen a t b z.1∂upperArea)∂W := by
+    rw [integral_prod_symm F hFi]
+    apply integral_congr_ae
+    filter_upwards [] with b
+    exact he _
+  change Tendsto (fun n => ∫p,R n p∂μ) atTop (𝓝 (∫p,F p∂μ)) at ht
+  simpa only [hRint,hFint] using ht
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped Classical NNReal ENNReal Topology
+lemma setIntegral_tendsto_of_L1 {Ω : Type*} [MeasurableSpace Ω] (W : Measure Ω)
+    (F : ℕ → Ω → ℝ) (g : Ω → ℝ) (hF : ∀n,Integrable (F n) W) (hg : Integrable g W)
+    (hlim : Tendsto (fun n => eLpNorm (fun b => F n b-g b) 1 W) atTop (𝓝 0)) (U : Set Ω) :
+    Tendsto (fun n => ∫b in U,F n b∂W) atTop (𝓝 (∫b in U,g b∂W)) := by
+  have hsmall : Tendsto (fun n => eLpNorm (fun b => F n b-g b) 1 (W.restrict U)) atTop (𝓝 0) :=
+    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
+      (Eventually.of_forall (fun _ => bot_le))
+      (Eventually.of_forall (fun n => eLpNorm_mono_measure _ Measure.restrict_le_self))
+  exact tendsto_integral_of_L1 g hg.integrableOn.aestronglyMeasurable
+    (Eventually.of_forall (fun n => (hF n).integrableOn))
+    (hsmall.congr fun index => eLpNorm_one_eq_lintegral_enorm
+      ((hF index).integrableOn.aestronglyMeasurable.sub hg.integrableOn.aestronglyMeasurable))
+lemma scalar_convex_tail_limit (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+    (hs : ∀k j,j∈s k→k≤j) (hc : ∀k j,j∈s k→0≤c k j) (hsum : ∀k,∑j∈s k,c k j=1)
+    (x : ℕ → ℝ) (L : ℝ) (hx : Tendsto x atTop (𝓝 L)) :
+    Tendsto (fun k => ∑j∈s k,c k j*x j) atTop (𝓝 L) := by
+  rw [Metric.tendsto_atTop] at hx ⊢
+  intro ε hε
+  obtain ⟨N,hN⟩ := hx (ε/2) (by positivity)
+  refine ⟨N,fun k hk => ?_⟩
+  rw [Real.dist_eq]
+  have he : (∑j∈s k,c k j*x j)-L = ∑j∈s k,c k j*(x j-L) := by
+    simp only [mul_sub,Finset.sum_sub_distrib,←Finset.sum_mul,hsum k,one_mul]
+  rw [he]
+  calc
+    |∑j∈s k,c k j*(x j-L)| ≤ ∑j∈s k,|c k j*(x j-L)| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑j∈s k,c k j*(ε/2) := by
+      apply Finset.sum_le_sum
+      intro j hj
+      rw [abs_mul,abs_of_nonneg (hc k j hj)]
+      exact mul_le_mul_of_nonneg_left (by simpa [Real.dist_eq] using (hN j (hk.trans (hs k j hj))).le) (hc k j hj)
+    _ = ε/2 := by rw [←Finset.sum_mul,hsum k,one_mul]
+    _ < ε := by linarith
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped Classical NNReal ENNReal
+theorem actual_radon_compact_conditional (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, coeff k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
+          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
+      (∀ i (t₀ : ℝ≥0), t i=t₀ → ∀ f : C_c(UpperPlane,ℝ),
+        AEStronglyMeasurable[rawFiltration t₀] (fun b => ∫z,f z∂μ b i) W) ∧
+      (∀ i j (t₀ : ℝ≥0), t i=t₀ → t j=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        ∀ U : Set ExactSLEGenerator.Driver, MeasurableSet[rawFiltration t₀] U →
+        (∫b in U, (∫z,f z∂μ b j)-(∫z,f z∂μ b i) ∂W) =
+          ∫b in U, (∫z : UpperPlane,f z*survivingGreen a t₀ b z.1 ∂upperArea) ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i)))
+ := by
+  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,hraw,h⟩ := actual_radon_raw_evaluations a ha W hW t
+  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,hmean,hraw,?_,h⟩
+  intro i j t₀ hi hj f U hU
+  let q : ℕ → ℝ := fun n => Real.exp (-2*a*((n:ℝ)+1))
+  have hq (n : ℕ) : 0<q n ∧ q n≤1 := by
+    refine ⟨Real.exp_pos _,Real.exp_le_one_iff.mpr ?_⟩
+    have hn : (0:ℝ)≤n := Nat.cast_nonneg _
+    have ha0 : 0<a := by linarith
+    nlinarith
+  have hqlim : Tendsto q atTop (𝓝 0) := by
+    apply Real.tendsto_exp_atBot.comp
+    apply Tendsto.const_mul_atTop_of_neg (by linarith : -2*a<0)
+    apply Filter.tendsto_atTop.mpr
+    intro B
+    filter_upwards [(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop).eventually (eventually_ge_atTop B)] with n hn
+    linarith
+  let F : ℕ → ℕ → ExactSLEGenerator.Driver → ℝ := fun l k b =>
+    ∑ n∈s k,c k n*∫z,f z∂actualUpperMass a (q n) (t l) b
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW 1 (by norm_num) f
+  have hEval (n : ℕ) (v : ℝ≥0∞) : Integrable (fun b => ∫z,f z∂actualUpperMass a (q n) v b) W :=
+    (hCb (q n) (hq n).1 (hq n).2 v).1.integrable (by norm_num)
+  have hFi (l k : ℕ) : Integrable (F l k) W :=
+    integrable_finsetSum (s k) (fun n hn => (hEval n (t l)).const_mul (c k n))
+  have hμi (l : ℕ) : Integrable (fun b => ∫z,f z∂μ b l) W :=
+    (hmom l f 1).integrable (by norm_num)
+  have hlim (l : ℕ) := setIntegral_tendsto_of_L1 W (F l) (fun b => ∫z,f z∂μ b l)
+    (hFi l) (hμi l) (hL1 l f) U
+  let R : ℕ → ℝ := fun n => ∫b in U,(∫z : UpperPlane,f z*remainingMassDensity a (q n) t₀ b z.1∂upperArea)∂W
+  let L : ℝ := ∫b in U,(∫z : UpperPlane,f z*survivingGreen a t₀ b z.1∂upperArea)∂W
+  have hRlim : Tendsto R atTop (𝓝 L) := actualUpper_remaining_integral_tendsto a ha W hW t₀ f U q hq hqlim
+  have hcombo := scalar_convex_tail_limit s c hs hc hsum R L hRlim
+  have he (k : ℕ) : (∫b in U,F j k b∂W)-(∫b in U,F i k b∂W) = ∑n∈s k,c k n*R n := by
+    dsimp only [F]
+    rw [integral_finsetSum _ (fun n hn => ((hEval n (t j)).const_mul (c k n)).integrableOn),
+      integral_finsetSum _ (fun n hn => ((hEval n (t i)).const_mul (c k n)).integrableOn)]
+    simp only [integral_const_mul]
+    rw [←Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro n hn
+    rw [←mul_sub,←integral_sub (hEval n (t j)).integrableOn (hEval n (t i)).integrableOn]
+    rw [hj,hi,actualUpperMass_finite_kernel a ha W hW (q n) (hq n).1 (hq n).2 t₀ f U hU]
+  rw [integral_sub (hμi j).integrableOn (hμi i).integrableOn]
+  exact tendsto_nhds_unique ((hlim j).sub (hlim i)) (by simpa only [he] using hcombo)
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/HalfPlaneDistortion.lean b/lean/OAI/Probability/SLEGauge/HalfPlaneDistortion.lean
index 085d74e1ac637de278e3f1d6595f29668109e358..d07da252573fe70deac2ad89577ca99e059cf851 100644
--- a/lean/OAI/Probability/SLEGauge/HalfPlaneDistortion.lean
+++ b/lean/OAI/Probability/SLEGauge/HalfPlaneDistortion.lean
@@ -1,505 +1,505 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ModelFiniteTests
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain LoewnerRestart FlowIdentification
-
-lemma lineMap_im_lower {z w : ℂ} {y : ℝ} (hz : y≤z.im) (hw : y≤w.im)
-    {t : ℝ} (ht : t∈Icc 0 1) : y≤(AffineMap.lineMap z w t).im := by
-  rw [AffineMap.lineMap_apply]
-  simp only [vadd_eq_add,vsub_eq_sub,Complex.sub_im,Complex.add_im,Complex.real_smul,Complex.mul_im,Complex.ofReal_re,
-    Complex.ofReal_im,zero_mul,add_zero]
-  nlinarith [mul_nonneg (sub_nonneg.mpr ht.2) (sub_nonneg.mpr hz),
-    mul_nonneg ht.1 (sub_nonneg.mpr hw)]
-
-lemma lineMap_hasDerivAt_complex_comp {f : ℂ→ℂ} {f' z w : ℂ} {t : ℝ}
-    (hd : HasDerivAt f f' (AffineMap.lineMap z w t)) :
-    HasDerivAt (fun t : ℝ=>f (AffineMap.lineMap z w t)) (f'*(w-z)) t := by
-  have hh := (hd.hasFDerivAt.restrictScalars ℝ).comp_hasDerivAt t
-    (AffineMap.hasDerivAt_lineMap (𝕜 := ℝ) (a := z) (b := w) (x := t))
-  convert! hh using 1
-  change f'*(w-z)=(w-z)*f'
-  ring
-
- 
-lemma norm_exp_bound_on_halfPlane_segment {f : ℂ→ℂ} {M y : ℝ}
-    (hy : 0<y) (hM : 0≤M)
-    (hd : ∀v : ℂ,0<v.im → DifferentiableAt ℂ f v)
-    (hb : ∀v : ℂ,0<v.im → ‖deriv f v‖≤(M/v.im)*‖f v‖)
-    {z w : ℂ} (hz : y≤z.im) (hw : y≤w.im) :
-    ‖f w‖≤‖f z‖*Real.exp ((M/y)*‖w-z‖) := by
-  let K : ℝ := (M/y)*‖w-z‖
-  let g : ℝ→ℂ := fun t=>f (AffineMap.lineMap z w t)
-  let dg : ℝ→ℂ := fun t=>deriv f (AffineMap.lineMap z w t)*(w-z)
-  have hi (t : ℝ) (ht : t∈Icc 0 1) : 0<(AffineMap.lineMap z w t).im :=
-    hy.trans_le (lineMap_im_lower hz hw ht)
-  have hg (t : ℝ) (ht : t∈Icc 0 1) : HasDerivAt g (dg t) t :=
-    lineMap_hasDerivAt_complex_comp (hd _ (hi t ht)).hasDerivAt
-  have hc : ContinuousOn g (Icc 0 1) := fun t ht=>(hg t ht).continuousAt.continuousWithinAt
-  have hdb (t : ℝ) (ht : t∈Ico 0 1) : ‖dg t‖≤K*‖g t‖+0 := by
-    dsimp only [dg,g,K]
-    rw [norm_mul,add_zero]
-    calc
-      _ ≤ ((M/(AffineMap.lineMap z w t).im)*‖f (AffineMap.lineMap z w t)‖)*‖w-z‖ :=
-        mul_le_mul_of_nonneg_right (hb _ (hi t (Ico_subset_Icc_self ht))) (norm_nonneg _)
-      _ ≤ ((M/y)*‖f (AffineMap.lineMap z w t)‖)*‖w-z‖ := by
-        gcongr
-        exact lineMap_im_lower hz hw (Ico_subset_Icc_self ht)
-      _ = _ := by ring
-  have hh := norm_le_gronwallBound_of_norm_deriv_right_le hc
-    (fun t ht=>(hg t (Ico_subset_Icc_self ht)).hasDerivWithinAt)
-    (le_rfl : ‖g 0‖≤‖g 0‖) hdb 1 (by norm_num : (1:ℝ)∈Icc 0 1)
-  simpa only [g,AffineMap.lineMap_apply_zero,AffineMap.lineMap_apply_one,sub_zero,
-    gronwallBound_ε0,mul_one] using hh
-
- 
-theorem inverseMap_deriv_segment_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z w : ℂ} {y : ℝ} (hy : 0<y) (hz : y≤z.im) (hw : y≤w.im) :
-    ‖deriv (inverseMap a b T) w‖ ≤ ‖deriv (inverseMap a b T) z‖*
-      Real.exp ((3/y)*‖w-z‖) :=
-  norm_exp_bound_on_halfPlane_segment hy (by norm_num)
-    (fun v hv=>(hasDerivAt_deriv_inverseMap ha b T hv).differentiableAt)
-    (fun v hv=>inverseMap_preSchwarzian_bound ha b T hv) hz hw
-
- 
-
-theorem inverseMap_deriv_two_sided {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z w : ℂ} {y L : ℝ} (hy : 0<y) (hz : y≤z.im) (hw : y≤w.im)
-    (hd : ‖w-z‖≤L) :
-    ‖deriv (inverseMap a b T) z‖*Real.exp (-(3/y)*L)≤‖deriv (inverseMap a b T) w‖ ∧
-    ‖deriv (inverseMap a b T) w‖≤‖deriv (inverseMap a b T) z‖*Real.exp ((3/y)*L) := by
-  have hK : 0≤3/y := by positivity
-  have up := (inverseMap_deriv_segment_bound ha b T hy hz hw).trans
-    (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hd hK)) (norm_nonneg _))
-  have rev := (inverseMap_deriv_segment_bound ha b T hy hw hz).trans
-    (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr
-      (mul_le_mul_of_nonneg_left (by simpa only [norm_sub_rev] using hd) hK)) (norm_nonneg _))
-  refine ⟨?_,up⟩
-  have hh := mul_le_mul_of_nonneg_right rev (Real.exp_pos (-(3/y)*L)).le
-  rwa [mul_assoc,←Real.exp_add,show (3/y)*L+(-(3/y)*L)=0 by ring,Real.exp_zero,mul_one] at hh
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain LoewnerRestart FlowIdentification
-
-lemma im_lower_of_mem_closedBall {w z : ℂ} {r : ℝ} (hw : w∈closedBall z r) :
-    z.im-r≤w.im := by
-  have hh := (Complex.im_le_norm (z-w)).trans (by
-    simpa only [norm_sub_rev,←dist_eq_norm] using mem_closedBall.mp hw)
-  rw [Complex.sub_im] at hh
-  linarith
-
-lemma smallBall_im_half {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
-    z.im/2≤w.im := by
-  have := im_lower_of_mem_closedBall hw
-  linarith
-
-lemma inverseMap_deriv_smallBall_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
-    ‖deriv (inverseMap a b T) w‖≤3*‖deriv (inverseMap a b T) z‖ := by
-  have h := inverseMap_deriv_segment_bound ha b T (by positivity : 0<z.im/2)
-    (by linarith : z.im/2≤z.im) (smallBall_im_half hz hw)
-  have hk : (3/(z.im/2))*‖w-z‖≤1 := by
-    have hh : ‖w-z‖≤z.im/100 := by simpa only [mem_closedBall,dist_eq_norm] using hw
-    apply le_trans (mul_le_mul_of_nonneg_left hh (by positivity))
-    field_simp
-    linarith
-  exact h.trans ((mul_le_mul_of_nonneg_left
-    ((Real.exp_le_exp.mpr hk).trans Real.exp_one_lt_three.le) (norm_nonneg _)).trans_eq (mul_comm _ _))
-
-lemma inverseMap_second_deriv_smallBall_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
-    ‖deriv (deriv (inverseMap a b T)) w‖≤(18/z.im)*‖deriv (inverseMap a b T) z‖ := by
-  have hi := smallBall_im_half hz hw
-  have hw0 : 0<w.im := lt_of_lt_of_le (by positivity : 0<z.im/2) hi
-  apply (inverseMap_preSchwarzian_bound ha b T hw0).trans
-  calc
-    _ ≤ (3/(z.im/2))*(3*‖deriv (inverseMap a b T) z‖) := by
-      gcongr
-      exact inverseMap_deriv_smallBall_bound ha b T hz hw
-    _ = _ := by ring
-
-lemma inverseMap_deriv_close_on_smallBall {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
-    ‖deriv (inverseMap a b T) w-deriv (inverseMap a b T) z‖≤‖deriv (inverseMap a b T) z‖/2 := by
-  have hd (v : ℂ) (hv : v∈closedBall z (z.im/100)) :
-      HasFDerivWithinAt (deriv (inverseMap a b T))
-        (ContinuousLinearMap.toSpanSingleton ℂ (deriv (deriv (inverseMap a b T)) v))
-        (closedBall z (z.im/100)) v := by
-    have hh := hasDerivAt_deriv_inverseMap ha b T
-      (lt_of_lt_of_le (by positivity : 0<z.im/2) (smallBall_im_half hz hv))
-    rw [←hh.deriv] at hh
-    exact hh.hasFDerivAt.hasFDerivWithinAt
-  have hh := (convex_closedBall z (z.im/100)).norm_image_sub_le_of_norm_hasFDerivWithin_le hd
-    (fun v hv=>by simpa only [ContinuousLinearMap.norm_toSpanSingleton] using
-      inverseMap_second_deriv_smallBall_bound ha b T hz hv)
-    (mem_closedBall_self (by positivity)) hw
-  have hn : ‖w-z‖≤z.im/100 := by simpa only [mem_closedBall,dist_eq_norm] using hw
-  apply hh.trans
-  calc
-    _ ≤ ((18/z.im)*‖deriv (inverseMap a b T) z‖)*(z.im/100) := by gcongr
-    _ = (18/100:ℝ)*‖deriv (inverseMap a b T) z‖ := by field_simp
-    _ ≤ ‖deriv (inverseMap a b T) z‖/2 := by nlinarith [norm_nonneg (deriv (inverseMap a b T) z)]
-
-lemma inverseMap_approximates_smallBall {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z : ℂ} (hz : 0<z.im) :
-    ApproximatesLinearOn (inverseMap a b T)
-      (ContinuousLinearMap.toSpanSingleton ℂ (deriv (inverseMap a b T) z))
-      (closedBall z (z.im/100)) (‖deriv (inverseMap a b T) z‖₊/2) := by
-  let f := inverseMap a b T
-  let D := deriv f z
-  have hd (v : ℂ) (hv : v∈closedBall z (z.im/100)) :
-      HasFDerivWithinAt (fun w=>f w-D*w)
-        (ContinuousLinearMap.toSpanSingleton ℂ (deriv f v-D)) (closedBall z (z.im/100)) v := by
-    have hh := (inverseMap_hasStrictDerivAt ha b T
-      (lt_of_lt_of_le (by positivity : 0<z.im/2) (smallBall_im_half hz hv))).hasDerivAt
-    rw [←hh.deriv] at hh
-    have h := hh.sub ((hasDerivAt_id v).const_mul D)
-    simp only [mul_one] at h
-    exact h.hasFDerivAt.hasFDerivWithinAt
-  intro x hx y hy
-  have hh := (convex_closedBall z (z.im/100)).norm_image_sub_le_of_norm_hasFDerivWithin_le hd
-    (fun v hv=>by simpa only [ContinuousLinearMap.norm_toSpanSingleton] using
-      inverseMap_deriv_close_on_smallBall ha b T hz hv) hy hx
-  convert! hh using 1
-  · congr 1
-    change f x-f y-(x-y)*D=f x-D*x-(f y-D*y)
-    ring
-
- 
-def spanRightInverse (D : ℂ) (hD : D≠0) :
-    (ContinuousLinearMap.toSpanSingleton ℂ D).NonlinearRightInverse where
-  toFun w := w/D
-  nnnorm := ‖D‖₊⁻¹
-  bound' w := by simp only [norm_div,NNReal.coe_inv,coe_nnnorm]; exact le_of_eq (by ring)
-  right_inv' w := by change (w/D)*D=w; exact div_mul_cancel₀ w hD
-
- 
-
-theorem inverseMap_inner_ball {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {z : ℂ} (hz : 0<z.im) :
-    closedBall (inverseMap a b T z) (z.im*‖deriv (inverseMap a b T) z‖/200) ⊆
-      inverseMap a b T '' closedBall z (z.im/100) := by
-  have hh := (inverseMap_approximates_smallBall ha b T hz).surjOn_closedBall_of_nonlinearRightInverse
-    (spanRightInverse _ (deriv_inverseMap_ne_zero ha b T hz))
-    (by positivity : 0≤z.im/100) (Subset.rfl)
-  have he : (((((spanRightInverse (deriv (inverseMap a b T) z)
-      (deriv_inverseMap_ne_zero ha b T hz)).nnnorm:ℝ))⁻¹-
-        (‖deriv (inverseMap a b T) z‖₊/2:ℝ≥0))*(z.im/100))=
-      z.im*‖deriv (inverseMap a b T) z‖/200 := by
-    simp only [spanRightInverse,NNReal.coe_inv,coe_nnnorm,inv_inv,NNReal.coe_div,NNReal.coe_ofNat]
-    ring
-  rw [he] at hh
-  exact hh
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain LoewnerRestart FlowIdentification
-open CenteredMoments KernelSimilarity ExactSLEGreen
-
- 
-def diskAt (w v : ℂ) : ℂ := (v-w)/(v-star w)
-
-def centeredRegion (w : ℂ) (q : ℝ) : Set ℂ := {v | 0<v.im ∧ ‖diskAt w v‖≤q}
-
-def subdiskReach (q : ℝ) : ℝ := 2*q/(1-q)
-def subdiskHeight (q : ℝ) : ℝ := (1-q^2)/4
-def subdiskDistortion (q : ℝ) : ℝ := Real.exp (3*subdiskReach q/subdiskHeight q)
-
-lemma subdiskReach_nonneg {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 0≤ subdiskReach q := by
-  unfold subdiskReach
-  positivity
-lemma subdiskHeight_pos {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 0<subdiskHeight q := by
-  unfold subdiskHeight
-  nlinarith
-lemma subdiskHeight_le_one (q : ℝ) : subdiskHeight q≤1 := by
-  unfold subdiskHeight
-  nlinarith [sq_nonneg q]
-lemma subdiskDistortion_pos (q : ℝ) : 0<subdiskDistortion q := Real.exp_pos _
-lemma subdiskDistortion_ge_one {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 1≤ subdiskDistortion q := by
-  apply Real.one_le_exp_iff.mpr
-  exact div_nonneg (mul_nonneg (by norm_num) (subdiskReach_nonneg hq hq1))
-    (subdiskHeight_pos hq hq1).le
-
-lemma diskAt_denominator_ne {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) : v-star w≠0 := by
-  intro he
-  have hh := congrArg Complex.im he
-  simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,Complex.zero_im,sub_neg_eq_add] at hh
-  linarith
-
-lemma diskAt_affine {w : ℂ} (hw : 0<w.im) (v : ℂ) :
-    diskAt w (affine (w.re:ℂ) w.im hw v)=(v-Complex.I)/(v+Complex.I) := by
-  have hy : (w.im:ℂ)≠0 := Complex.ofReal_ne_zero.mpr hw.ne'
-  have hwc : w=(w.re:ℂ)+(w.im:ℂ)*Complex.I := (Complex.re_add_im w).symm
-  have hconj : star w=(w.re:ℂ)-(w.im:ℂ)*Complex.I := by
-    apply Complex.ext <;> simp
-  rw [diskAt,affine_apply]
-  rw [show (w.re:ℂ)+(w.im:ℂ)*v-w=(w.im:ℂ)*(v-Complex.I) by
-      apply Complex.ext <;> simp; ring,
-    hconj,show (w.re:ℂ)+(w.im:ℂ)*v-((w.re:ℂ)-(w.im:ℂ)*Complex.I)=
-      (w.im:ℂ)*(v+Complex.I) by ring,mul_div_mul_left _ _ hy]
-
-lemma centeredRegion_bounds {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
-    (hv : v∈centeredRegion w q) :
-    ‖v-w‖≤w.im*subdiskReach q ∧ w.im*subdiskHeight q≤v.im := by
-  let z : UpperPlane := (upperAffine w.re w.im hw).symm ⟨v,hv.1⟩
-  have he : affine (w.re:ℂ) w.im hw z.1=v := by
-    exact congrArg Subtype.val ((upperAffine w.re w.im hw).apply_symm_apply ⟨v,hv.1⟩)
-  have hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤q := by
-    rw [←diskAt_affine hw z.1,he]
-    exact hv.2
-  have hb := cayley_subdisk_bounds q hq hq1 z hz
-  have hn : ‖v-w‖=w.im*‖z.1-Complex.I‖ := by
-    rw [←he,affine_apply,show (w.re:ℂ)+(w.im:ℂ)*z.1-w=(w.im:ℂ)*(z.1-Complex.I) by
-      apply Complex.ext <;> simp; ring,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hw]
-  constructor
-  · rw [hn]
-    exact mul_le_mul_of_nonneg_left (by simpa only [dist_eq_norm,subdiskReach] using hb.1) hw.le
-  · rw [←he,affine_apply]
-    simp only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
-      zero_mul,zero_add,add_zero]
-    exact mul_le_mul_of_nonneg_left hb.2 hw.le
-
-lemma smallBall_subset_centeredRegion {w : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 1/2≤q) :
-    closedBall w (w.im/100) ⊆ centeredRegion w q := by
-  intro v hv
-  have hi := smallBall_im_half hw hv
-  have hv0 : 0<v.im := lt_of_lt_of_le (by positivity : 0<w.im/2) hi
-  refine ⟨hv0,?_⟩
-  have hn : ‖v-w‖≤w.im/100 := by simpa only [mem_closedBall,dist_eq_norm] using hv
-  have hden : w.im≤‖v-star w‖ := by
-    apply le_trans _ (Complex.im_le_norm _)
-    simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,sub_neg_eq_add]
-    linarith
-  change ‖(v-w)/(v-star w)‖≤q
-  rw [norm_div]
-  exact ((div_le_div₀ (by positivity : 0≤w.im/100) hn hw hden).trans (by
-    calc
-      (w.im/100)/w.im=(1/100:ℝ) := by field_simp
-      _ ≤ q := by linarith))
-
-lemma centeredRegion_deriv_bounds {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
-    (hv : v∈centeredRegion w q) :
-    ‖deriv (inverseMap a b T) w‖/subdiskDistortion q≤‖deriv (inverseMap a b T) v‖ ∧
-    ‖deriv (inverseMap a b T) v‖≤‖deriv (inverseMap a b T) w‖*subdiskDistortion q := by
-  have hb := centeredRegion_bounds hw hq hq1 hv
-  have hh := inverseMap_deriv_two_sided ha b T (mul_pos hw (subdiskHeight_pos hq hq1))
-    (mul_le_of_le_one_right hw.le (subdiskHeight_le_one q)) hb.2 hb.1
-  have he : (3/(w.im*subdiskHeight q))*(w.im*subdiskReach q)=3*subdiskReach q/subdiskHeight q := by
-    field_simp
-  rw [show -(3/(w.im*subdiskHeight q))*(w.im*subdiskReach q)=
-      -((3/(w.im*subdiskHeight q))*(w.im*subdiskReach q)) by ring,he,Real.exp_neg] at hh
-  simpa only [subdiskDistortion,div_eq_mul_inv] using hh
-
- 
-lemma centeredRegion_image_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
-    (hv : v∈centeredRegion w q) :
-    ‖inverseMap a b T v-inverseMap a b T w‖≤
-      subdiskReach q*subdiskDistortion q*(w.im*‖deriv (inverseMap a b T) w‖) := by
-  have hb := centeredRegion_bounds hw hq hq1 hv
-  let y : ℝ := w.im*subdiskHeight q
-  have hy : 0<y := mul_pos hw (subdiskHeight_pos hq hq1)
-  have hw' : y≤w.im := mul_le_of_le_one_right hw.le (subdiskHeight_le_one q)
-  have hxy (t : ℝ) (ht : t∈Icc 0 1) : y≤(AffineMap.lineMap w v t).im :=
-    lineMap_im_lower hw' hb.2 ht
-  have hdist (t : ℝ) (ht : t∈Icc 0 1) : ‖AffineMap.lineMap w v t-w‖≤w.im*subdiskReach q := by
-    rw [AffineMap.lineMap_apply]
-    simp only [vadd_eq_add,vsub_eq_sub,add_sub_cancel_right,norm_smul,Real.norm_eq_abs,abs_of_nonneg ht.1]
-    exact (mul_le_of_le_one_left (norm_nonneg _) ht.2).trans hb.1
-  have he : (3/y)*(w.im*subdiskReach q)=3*subdiskReach q/subdiskHeight q := by
-    dsimp [y]; field_simp
-  have hd (t : ℝ) (ht : t∈Icc 0 1) :
-      HasDerivAt (fun t : ℝ=>inverseMap a b T (AffineMap.lineMap w v t))
-        (deriv (inverseMap a b T) (AffineMap.lineMap w v t)*(v-w)) t := by
-    apply lineMap_hasDerivAt_complex_comp
-    exact ((inverseMap_hasStrictDerivAt ha b T (hy.trans_le (hxy t ht))).hasDerivAt).differentiableAt.hasDerivAt
-  have hbound (t : ℝ) (ht : t∈Icc 0 1) :
-      ‖deriv (inverseMap a b T) (AffineMap.lineMap w v t)*(v-w)‖≤
-      (‖deriv (inverseMap a b T) w‖*subdiskDistortion q)*‖v-w‖ := by
-    rw [norm_mul]
-    refine mul_le_mul_of_nonneg_right ?_ (norm_nonneg _)
-    have hh := (inverseMap_deriv_two_sided ha b T hy hw' (hxy t ht) (hdist t ht)).2
-    rwa [he] at hh
-  have hh := (convex_Icc (0:ℝ) 1).norm_image_sub_le_of_norm_hasDerivWithin_le
-    (fun t ht=>(hd t ht).hasDerivWithinAt) hbound
-    (by norm_num : (0:ℝ)∈Icc 0 1) (by norm_num : (1:ℝ)∈Icc 0 1)
-  simp only [AffineMap.lineMap_apply_zero,AffineMap.lineMap_apply_one,sub_zero,norm_one,mul_one] at hh
-  exact hh.trans (by
-    calc
-      _ ≤ (‖deriv (inverseMap a b T) w‖*subdiskDistortion q)*(w.im*subdiskReach q) := mul_le_mul_of_nonneg_left hb.1
-        (mul_nonneg (norm_nonneg _) (subdiskDistortion_pos q).le)
-      _ = _ := by ring)
-
- 
-
-theorem inverseMap_centered_localization {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 1/2≤q) (hq1 : q<1) :
-    closedBall (inverseMap a b T w) (w.im*‖deriv (inverseMap a b T) w‖/200) ⊆
-      inverseMap a b T '' centeredRegion w q ∧
-    inverseMap a b T '' centeredRegion w q ⊆ closedBall (inverseMap a b T w)
-      (subdiskReach q*subdiskDistortion q*(w.im*‖deriv (inverseMap a b T) w‖)) := by
-  refine ⟨(inverseMap_inner_ball ha b T hw).trans
-    (image_mono (smallBall_subset_centeredRegion hw hq)),?_⟩
-  rintro _ ⟨v,hv,rfl⟩
-  simpa only [mem_closedBall,dist_eq_norm] using centeredRegion_image_bound ha b T hw (by linarith) hq1 hv
-
-lemma diskAt_hasDerivAt {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) :
-    HasDerivAt (diskAt w) (((2*w.im:ℝ):ℂ)*Complex.I/(v-star w)^2) v := by
-  have hh := ((hasDerivAt_id v).sub_const w).div
-    ((hasDerivAt_id v).sub_const (star w)) (diskAt_denominator_ne hw hv)
-  convert! hh using 1
-  simp only [id_eq,one_mul,mul_one]
-  congr 1
-  apply Complex.ext <;> simp; ring
-
-lemma diskAt_deriv_norm {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) :
-    ‖deriv (diskAt w) v‖=2*w.im/‖v-star w‖^2 := by
-  rw [(diskAt_hasDerivAt hw hv).deriv,norm_div,norm_mul,norm_pow,
-    Complex.norm_real,Real.norm_eq_abs,abs_of_pos (mul_pos (by norm_num) hw),
-    Complex.norm_I,mul_one]
-
-lemma diskAt_deriv_bound {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
-    (hv : v∈centeredRegion w q) :
-    2/((subdiskReach q+2)^2*w.im) ≤ ‖deriv (diskAt w) v‖ ∧
-    ‖deriv (diskAt w) v‖ ≤ 2/w.im := by
-  rw [diskAt_deriv_norm hw hv.1]
-  have hb := centeredRegion_bounds hw hq hq1 hv
-  have hn : w.im≤‖v-star w‖ := by
-    apply le_trans _ (Complex.im_le_norm _)
-    simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,sub_neg_eq_add]
-    linarith [hv.1]
-  have hu : ‖v-star w‖≤w.im*(subdiskReach q+2) := by
-    calc
-      _ = ‖(v-w)+(w-star w)‖ := by congr 1; ring
-      _ ≤ ‖v-w‖+‖w-star w‖ := norm_add_le _ _
-      _ = ‖v-w‖+2*w.im := by
-        rw [show w-star w=((2*w.im:ℝ):ℂ)*Complex.I by
-          apply Complex.ext <;> simp; ring,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-          abs_of_pos (mul_pos (by norm_num) hw),Complex.norm_I,mul_one]
-      _ ≤ _ := by linarith [hb.1]
-  have hq' : 0<subdiskReach q+2 := by linarith [subdiskReach_nonneg hq hq1]
-  constructor
-  · calc
-      _ = 2*w.im/(w.im*(subdiskReach q+2))^2 := by field_simp
-      _ ≤ _ := div_le_div_of_nonneg_left (by positivity)
-        (sq_pos_of_pos (hw.trans_le hn)) (pow_le_pow_left₀ (norm_nonneg _) hu 2)
-  · calc
-      _ ≤ 2*w.im/w.im^2 := div_le_div_of_nonneg_left (by positivity)
-        (sq_pos_of_pos hw) (pow_le_pow_left₀ hw.le hn 2)
-      _ = _ := by field_simp
-
- 
-
-theorem centered_disk_deriv_ratio {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
-    (hv : v∈centeredRegion w q) :
-    (w.im*‖deriv (inverseMap a b T) w‖)/(2*subdiskDistortion q) ≤
-        ‖deriv (inverseMap a b T) v‖/‖deriv (diskAt w) v‖ ∧
-    ‖deriv (inverseMap a b T) v‖/‖deriv (diskAt w) v‖ ≤
-        (subdiskDistortion q*(subdiskReach q+2)^2/2)*(w.im*‖deriv (inverseMap a b T) w‖) := by
-  have hf := centeredRegion_deriv_bounds ha b T hw hq hq1 hv
-  have hg := diskAt_deriv_bound hw hq hq1 hv
-  have hq' : 0<subdiskReach q+2 := by linarith [subdiskReach_nonneg hq hq1]
-  have hlow : 0<2/((subdiskReach q+2)^2*w.im) := by positivity
-  have hnorm : 0<‖deriv (diskAt w) v‖ := hlow.trans_le hg.1
-  constructor
-  · calc
-      _ = (‖deriv (inverseMap a b T) w‖/subdiskDistortion q)/(2/w.im) := by field_simp
-      _ ≤ _ := div_le_div₀ (norm_nonneg _) hf.1 hnorm hg.2
-  · calc
-      _ ≤ (‖deriv (inverseMap a b T) w‖*subdiskDistortion q)/(2/((subdiskReach q+2)^2*w.im)) :=
-        div_le_div₀ (mul_nonneg (norm_nonneg _) (subdiskDistortion_pos q).le) hf.2 hlow hg.1
-      _ = _ := by field_simp
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain ExactSLEGreen LoewnerRestart FlowIdentification
-
-lemma centered_im_pos {a : ℝ} {b : GDriver} {t : ℝ≥0} {z : ℂ}
-    (hz : alive a b.1 t z) : 0<(loewnerMap a b.1 t z-(b.1 t:ℂ)).im := by
-  simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using loewnerMap_im_pos hz
-
-lemma inverseMap_spatial_radius {a : ℝ} (ha : 0<a) (b : GDriver) (t : ℝ≥0)
-    {z : ℂ} (hz : alive a b.1 t z) :
-    (loewnerMap a b.1 t z-(b.1 t:ℂ)).im *
-        ‖deriv (inverseMap a b t) (loewnerMap a b.1 t z-(b.1 t:ℂ))‖ =
-      physicalRadius (actualState a b.1 t z) := by
-  rw [deriv_inverseMap ha b t (centered_im_pos hz),inverseMap_centered a b t hz,
-    norm_inv]
-  have hd := norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hz)
-  simp only [physicalRadius,actualState,Y_apply,J_apply,Real.exp_neg,Real.exp_log hd]
-
- 
-
-theorem radius_ball_subset_alive {a : ℝ} (ha : 0<a) (b : GDriver) (t : ℝ≥0)
-    {z : ℂ} (hz : alive a b.1 t z) :
-    closedBall z (physicalRadius (actualState a b.1 t z)/200) ⊆ {v | alive a b.1 t v} := by
-  have hh := inverseMap_inner_ball ha b t (centered_im_pos hz)
-  rw [inverseMap_spatial_radius ha b t hz,inverseMap_centered a b t hz] at hh
-  intro v hv
-  obtain ⟨w,hw,rfl⟩ := hh hv
-  exact inverseMap_alive ha b t (lt_of_lt_of_le
-    (div_pos (centered_im_pos hz) (by norm_num))
-    (smallBall_im_half (centered_im_pos hz) hw))
-
- 
-theorem centered_localization_at_alive {a : ℝ} (ha : 0<a) (b : GDriver) (t : ℝ≥0)
-    {z : ℂ} (hz : alive a b.1 t z) {q : ℝ} (hq : 1/2≤q) (hq1 : q<1) :
-    closedBall z (physicalRadius (actualState a b.1 t z)/200) ⊆
-        inverseMap a b t '' centeredRegion (loewnerMap a b.1 t z-(b.1 t:ℂ)) q ∧
-    inverseMap a b t '' centeredRegion (loewnerMap a b.1 t z-(b.1 t:ℂ)) q ⊆
-      closedBall z (subdiskReach q*subdiskDistortion q*physicalRadius (actualState a b.1 t z)) := by
-  have hh := inverseMap_centered_localization ha b t (centered_im_pos hz) hq hq1
-  rwa [inverseMap_spatial_radius ha b t hz,inverseMap_centered a b t hz] at hh
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ModelFiniteTests
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain LoewnerRestart FlowIdentification
+
+lemma lineMap_im_lower {z w : ℂ} {y : ℝ} (hz : y≤z.im) (hw : y≤w.im)
+    {t : ℝ} (ht : t∈Icc 0 1) : y≤(AffineMap.lineMap z w t).im := by
+  rw [AffineMap.lineMap_apply]
+  simp only [vadd_eq_add,vsub_eq_sub,Complex.sub_im,Complex.add_im,Complex.real_smul,Complex.mul_im,Complex.ofReal_re,
+    Complex.ofReal_im,zero_mul,add_zero]
+  nlinarith [mul_nonneg (sub_nonneg.mpr ht.2) (sub_nonneg.mpr hz),
+    mul_nonneg ht.1 (sub_nonneg.mpr hw)]
+
+lemma lineMap_hasDerivAt_complex_comp {f : ℂ→ℂ} {f' z w : ℂ} {t : ℝ}
+    (hd : HasDerivAt f f' (AffineMap.lineMap z w t)) :
+    HasDerivAt (fun t : ℝ=>f (AffineMap.lineMap z w t)) (f'*(w-z)) t := by
+  have hh := (hd.hasFDerivAt.restrictScalars ℝ).comp_hasDerivAt t
+    (AffineMap.hasDerivAt_lineMap (𝕜 := ℝ) (a := z) (b := w) (x := t))
+  convert! hh using 1
+  change f'*(w-z)=(w-z)*f'
+  ring
+
+ 
+lemma norm_exp_bound_on_halfPlane_segment {f : ℂ→ℂ} {M y : ℝ}
+    (hy : 0<y) (hM : 0≤M)
+    (hd : ∀v : ℂ,0<v.im → DifferentiableAt ℂ f v)
+    (hb : ∀v : ℂ,0<v.im → ‖deriv f v‖≤(M/v.im)*‖f v‖)
+    {z w : ℂ} (hz : y≤z.im) (hw : y≤w.im) :
+    ‖f w‖≤‖f z‖*Real.exp ((M/y)*‖w-z‖) := by
+  let K : ℝ := (M/y)*‖w-z‖
+  let g : ℝ→ℂ := fun t=>f (AffineMap.lineMap z w t)
+  let dg : ℝ→ℂ := fun t=>deriv f (AffineMap.lineMap z w t)*(w-z)
+  have hi (t : ℝ) (ht : t∈Icc 0 1) : 0<(AffineMap.lineMap z w t).im :=
+    hy.trans_le (lineMap_im_lower hz hw ht)
+  have hg (t : ℝ) (ht : t∈Icc 0 1) : HasDerivAt g (dg t) t :=
+    lineMap_hasDerivAt_complex_comp (hd _ (hi t ht)).hasDerivAt
+  have hc : ContinuousOn g (Icc 0 1) := fun t ht=>(hg t ht).continuousAt.continuousWithinAt
+  have hdb (t : ℝ) (ht : t∈Ico 0 1) : ‖dg t‖≤K*‖g t‖+0 := by
+    dsimp only [dg,g,K]
+    rw [norm_mul,add_zero]
+    calc
+      _ ≤ ((M/(AffineMap.lineMap z w t).im)*‖f (AffineMap.lineMap z w t)‖)*‖w-z‖ :=
+        mul_le_mul_of_nonneg_right (hb _ (hi t (Ico_subset_Icc_self ht))) (norm_nonneg _)
+      _ ≤ ((M/y)*‖f (AffineMap.lineMap z w t)‖)*‖w-z‖ := by
+        gcongr
+        exact lineMap_im_lower hz hw (Ico_subset_Icc_self ht)
+      _ = _ := by ring
+  have hh := norm_le_gronwallBound_of_norm_deriv_right_le hc
+    (fun t ht=>(hg t (Ico_subset_Icc_self ht)).hasDerivWithinAt)
+    (le_rfl : ‖g 0‖≤‖g 0‖) hdb 1 (by norm_num : (1:ℝ)∈Icc 0 1)
+  simpa only [g,AffineMap.lineMap_apply_zero,AffineMap.lineMap_apply_one,sub_zero,
+    gronwallBound_ε0,mul_one] using hh
+
+ 
+theorem inverseMap_deriv_segment_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z w : ℂ} {y : ℝ} (hy : 0<y) (hz : y≤z.im) (hw : y≤w.im) :
+    ‖deriv (inverseMap a b T) w‖ ≤ ‖deriv (inverseMap a b T) z‖*
+      Real.exp ((3/y)*‖w-z‖) :=
+  norm_exp_bound_on_halfPlane_segment hy (by norm_num)
+    (fun v hv=>(hasDerivAt_deriv_inverseMap ha b T hv).differentiableAt)
+    (fun v hv=>inverseMap_preSchwarzian_bound ha b T hv) hz hw
+
+ 
+
+theorem inverseMap_deriv_two_sided {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z w : ℂ} {y L : ℝ} (hy : 0<y) (hz : y≤z.im) (hw : y≤w.im)
+    (hd : ‖w-z‖≤L) :
+    ‖deriv (inverseMap a b T) z‖*Real.exp (-(3/y)*L)≤‖deriv (inverseMap a b T) w‖ ∧
+    ‖deriv (inverseMap a b T) w‖≤‖deriv (inverseMap a b T) z‖*Real.exp ((3/y)*L) := by
+  have hK : 0≤3/y := by positivity
+  have up := (inverseMap_deriv_segment_bound ha b T hy hz hw).trans
+    (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hd hK)) (norm_nonneg _))
+  have rev := (inverseMap_deriv_segment_bound ha b T hy hw hz).trans
+    (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr
+      (mul_le_mul_of_nonneg_left (by simpa only [norm_sub_rev] using hd) hK)) (norm_nonneg _))
+  refine ⟨?_,up⟩
+  have hh := mul_le_mul_of_nonneg_right rev (Real.exp_pos (-(3/y)*L)).le
+  rwa [mul_assoc,←Real.exp_add,show (3/y)*L+(-(3/y)*L)=0 by ring,Real.exp_zero,mul_one] at hh
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain LoewnerRestart FlowIdentification
+
+lemma im_lower_of_mem_closedBall {w z : ℂ} {r : ℝ} (hw : w∈closedBall z r) :
+    z.im-r≤w.im := by
+  have hh := (Complex.im_le_norm (z-w)).trans (by
+    simpa only [norm_sub_rev,←dist_eq_norm] using mem_closedBall.mp hw)
+  rw [Complex.sub_im] at hh
+  linarith
+
+lemma smallBall_im_half {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
+    z.im/2≤w.im := by
+  have := im_lower_of_mem_closedBall hw
+  linarith
+
+lemma inverseMap_deriv_smallBall_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
+    ‖deriv (inverseMap a b T) w‖≤3*‖deriv (inverseMap a b T) z‖ := by
+  have h := inverseMap_deriv_segment_bound ha b T (by positivity : 0<z.im/2)
+    (by linarith : z.im/2≤z.im) (smallBall_im_half hz hw)
+  have hk : (3/(z.im/2))*‖w-z‖≤1 := by
+    have hh : ‖w-z‖≤z.im/100 := by simpa only [mem_closedBall,dist_eq_norm] using hw
+    apply le_trans (mul_le_mul_of_nonneg_left hh (by positivity))
+    field_simp
+    linarith
+  exact h.trans ((mul_le_mul_of_nonneg_left
+    ((Real.exp_le_exp.mpr hk).trans Real.exp_one_lt_three.le) (norm_nonneg _)).trans_eq (mul_comm _ _))
+
+lemma inverseMap_second_deriv_smallBall_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
+    ‖deriv (deriv (inverseMap a b T)) w‖≤(18/z.im)*‖deriv (inverseMap a b T) z‖ := by
+  have hi := smallBall_im_half hz hw
+  have hw0 : 0<w.im := lt_of_lt_of_le (by positivity : 0<z.im/2) hi
+  apply (inverseMap_preSchwarzian_bound ha b T hw0).trans
+  calc
+    _ ≤ (3/(z.im/2))*(3*‖deriv (inverseMap a b T) z‖) := by
+      gcongr
+      exact inverseMap_deriv_smallBall_bound ha b T hz hw
+    _ = _ := by ring
+
+lemma inverseMap_deriv_close_on_smallBall {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z w : ℂ} (hz : 0<z.im) (hw : w∈closedBall z (z.im/100)) :
+    ‖deriv (inverseMap a b T) w-deriv (inverseMap a b T) z‖≤‖deriv (inverseMap a b T) z‖/2 := by
+  have hd (v : ℂ) (hv : v∈closedBall z (z.im/100)) :
+      HasFDerivWithinAt (deriv (inverseMap a b T))
+        (ContinuousLinearMap.toSpanSingleton ℂ (deriv (deriv (inverseMap a b T)) v))
+        (closedBall z (z.im/100)) v := by
+    have hh := hasDerivAt_deriv_inverseMap ha b T
+      (lt_of_lt_of_le (by positivity : 0<z.im/2) (smallBall_im_half hz hv))
+    rw [←hh.deriv] at hh
+    exact hh.hasFDerivAt.hasFDerivWithinAt
+  have hh := (convex_closedBall z (z.im/100)).norm_image_sub_le_of_norm_hasFDerivWithin_le hd
+    (fun v hv=>by simpa only [ContinuousLinearMap.norm_toSpanSingleton] using
+      inverseMap_second_deriv_smallBall_bound ha b T hz hv)
+    (mem_closedBall_self (by positivity)) hw
+  have hn : ‖w-z‖≤z.im/100 := by simpa only [mem_closedBall,dist_eq_norm] using hw
+  apply hh.trans
+  calc
+    _ ≤ ((18/z.im)*‖deriv (inverseMap a b T) z‖)*(z.im/100) := by gcongr
+    _ = (18/100:ℝ)*‖deriv (inverseMap a b T) z‖ := by field_simp
+    _ ≤ ‖deriv (inverseMap a b T) z‖/2 := by nlinarith [norm_nonneg (deriv (inverseMap a b T) z)]
+
+lemma inverseMap_approximates_smallBall {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z : ℂ} (hz : 0<z.im) :
+    ApproximatesLinearOn (inverseMap a b T)
+      (ContinuousLinearMap.toSpanSingleton ℂ (deriv (inverseMap a b T) z))
+      (closedBall z (z.im/100)) (‖deriv (inverseMap a b T) z‖₊/2) := by
+  let f := inverseMap a b T
+  let D := deriv f z
+  have hd (v : ℂ) (hv : v∈closedBall z (z.im/100)) :
+      HasFDerivWithinAt (fun w=>f w-D*w)
+        (ContinuousLinearMap.toSpanSingleton ℂ (deriv f v-D)) (closedBall z (z.im/100)) v := by
+    have hh := (inverseMap_hasStrictDerivAt ha b T
+      (lt_of_lt_of_le (by positivity : 0<z.im/2) (smallBall_im_half hz hv))).hasDerivAt
+    rw [←hh.deriv] at hh
+    have h := hh.sub ((hasDerivAt_id v).const_mul D)
+    simp only [mul_one] at h
+    exact h.hasFDerivAt.hasFDerivWithinAt
+  intro x hx y hy
+  have hh := (convex_closedBall z (z.im/100)).norm_image_sub_le_of_norm_hasFDerivWithin_le hd
+    (fun v hv=>by simpa only [ContinuousLinearMap.norm_toSpanSingleton] using
+      inverseMap_deriv_close_on_smallBall ha b T hz hv) hy hx
+  convert! hh using 1
+  · congr 1
+    change f x-f y-(x-y)*D=f x-D*x-(f y-D*y)
+    ring
+
+ 
+def spanRightInverse (D : ℂ) (hD : D≠0) :
+    (ContinuousLinearMap.toSpanSingleton ℂ D).NonlinearRightInverse where
+  toFun w := w/D
+  nnnorm := ‖D‖₊⁻¹
+  bound' w := by simp only [norm_div,NNReal.coe_inv,coe_nnnorm]; exact le_of_eq (by ring)
+  right_inv' w := by change (w/D)*D=w; exact div_mul_cancel₀ w hD
+
+ 
+
+theorem inverseMap_inner_ball {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {z : ℂ} (hz : 0<z.im) :
+    closedBall (inverseMap a b T z) (z.im*‖deriv (inverseMap a b T) z‖/200) ⊆
+      inverseMap a b T '' closedBall z (z.im/100) := by
+  have hh := (inverseMap_approximates_smallBall ha b T hz).surjOn_closedBall_of_nonlinearRightInverse
+    (spanRightInverse _ (deriv_inverseMap_ne_zero ha b T hz))
+    (by positivity : 0≤z.im/100) (Subset.rfl)
+  have he : (((((spanRightInverse (deriv (inverseMap a b T) z)
+      (deriv_inverseMap_ne_zero ha b T hz)).nnnorm:ℝ))⁻¹-
+        (‖deriv (inverseMap a b T) z‖₊/2:ℝ≥0))*(z.im/100))=
+      z.im*‖deriv (inverseMap a b T) z‖/200 := by
+    simp only [spanRightInverse,NNReal.coe_inv,coe_nnnorm,inv_inv,NNReal.coe_div,NNReal.coe_ofNat]
+    ring
+  rw [he] at hh
+  exact hh
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain LoewnerRestart FlowIdentification
+open CenteredMoments KernelSimilarity ExactSLEGreen
+
+ 
+def diskAt (w v : ℂ) : ℂ := (v-w)/(v-star w)
+
+def centeredRegion (w : ℂ) (q : ℝ) : Set ℂ := {v | 0<v.im ∧ ‖diskAt w v‖≤q}
+
+def subdiskReach (q : ℝ) : ℝ := 2*q/(1-q)
+def subdiskHeight (q : ℝ) : ℝ := (1-q^2)/4
+def subdiskDistortion (q : ℝ) : ℝ := Real.exp (3*subdiskReach q/subdiskHeight q)
+
+lemma subdiskReach_nonneg {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 0≤ subdiskReach q := by
+  unfold subdiskReach
+  positivity
+lemma subdiskHeight_pos {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 0<subdiskHeight q := by
+  unfold subdiskHeight
+  nlinarith
+lemma subdiskHeight_le_one (q : ℝ) : subdiskHeight q≤1 := by
+  unfold subdiskHeight
+  nlinarith [sq_nonneg q]
+lemma subdiskDistortion_pos (q : ℝ) : 0<subdiskDistortion q := Real.exp_pos _
+lemma subdiskDistortion_ge_one {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 1≤ subdiskDistortion q := by
+  apply Real.one_le_exp_iff.mpr
+  exact div_nonneg (mul_nonneg (by norm_num) (subdiskReach_nonneg hq hq1))
+    (subdiskHeight_pos hq hq1).le
+
+lemma diskAt_denominator_ne {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) : v-star w≠0 := by
+  intro he
+  have hh := congrArg Complex.im he
+  simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,Complex.zero_im,sub_neg_eq_add] at hh
+  linarith
+
+lemma diskAt_affine {w : ℂ} (hw : 0<w.im) (v : ℂ) :
+    diskAt w (affine (w.re:ℂ) w.im hw v)=(v-Complex.I)/(v+Complex.I) := by
+  have hy : (w.im:ℂ)≠0 := Complex.ofReal_ne_zero.mpr hw.ne'
+  have hwc : w=(w.re:ℂ)+(w.im:ℂ)*Complex.I := (Complex.re_add_im w).symm
+  have hconj : star w=(w.re:ℂ)-(w.im:ℂ)*Complex.I := by
+    apply Complex.ext <;> simp
+  rw [diskAt,affine_apply]
+  rw [show (w.re:ℂ)+(w.im:ℂ)*v-w=(w.im:ℂ)*(v-Complex.I) by
+      apply Complex.ext <;> simp; ring,
+    hconj,show (w.re:ℂ)+(w.im:ℂ)*v-((w.re:ℂ)-(w.im:ℂ)*Complex.I)=
+      (w.im:ℂ)*(v+Complex.I) by ring,mul_div_mul_left _ _ hy]
+
+lemma centeredRegion_bounds {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
+    (hv : v∈centeredRegion w q) :
+    ‖v-w‖≤w.im*subdiskReach q ∧ w.im*subdiskHeight q≤v.im := by
+  let z : UpperPlane := (upperAffine w.re w.im hw).symm ⟨v,hv.1⟩
+  have he : affine (w.re:ℂ) w.im hw z.1=v := by
+    exact congrArg Subtype.val ((upperAffine w.re w.im hw).apply_symm_apply ⟨v,hv.1⟩)
+  have hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤q := by
+    rw [←diskAt_affine hw z.1,he]
+    exact hv.2
+  have hb := cayley_subdisk_bounds q hq hq1 z hz
+  have hn : ‖v-w‖=w.im*‖z.1-Complex.I‖ := by
+    rw [←he,affine_apply,show (w.re:ℂ)+(w.im:ℂ)*z.1-w=(w.im:ℂ)*(z.1-Complex.I) by
+      apply Complex.ext <;> simp; ring,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hw]
+  constructor
+  · rw [hn]
+    exact mul_le_mul_of_nonneg_left (by simpa only [dist_eq_norm,subdiskReach] using hb.1) hw.le
+  · rw [←he,affine_apply]
+    simp only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
+      zero_mul,zero_add,add_zero]
+    exact mul_le_mul_of_nonneg_left hb.2 hw.le
+
+lemma smallBall_subset_centeredRegion {w : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 1/2≤q) :
+    closedBall w (w.im/100) ⊆ centeredRegion w q := by
+  intro v hv
+  have hi := smallBall_im_half hw hv
+  have hv0 : 0<v.im := lt_of_lt_of_le (by positivity : 0<w.im/2) hi
+  refine ⟨hv0,?_⟩
+  have hn : ‖v-w‖≤w.im/100 := by simpa only [mem_closedBall,dist_eq_norm] using hv
+  have hden : w.im≤‖v-star w‖ := by
+    apply le_trans _ (Complex.im_le_norm _)
+    simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,sub_neg_eq_add]
+    linarith
+  change ‖(v-w)/(v-star w)‖≤q
+  rw [norm_div]
+  exact ((div_le_div₀ (by positivity : 0≤w.im/100) hn hw hden).trans (by
+    calc
+      (w.im/100)/w.im=(1/100:ℝ) := by field_simp
+      _ ≤ q := by linarith))
+
+lemma centeredRegion_deriv_bounds {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
+    (hv : v∈centeredRegion w q) :
+    ‖deriv (inverseMap a b T) w‖/subdiskDistortion q≤‖deriv (inverseMap a b T) v‖ ∧
+    ‖deriv (inverseMap a b T) v‖≤‖deriv (inverseMap a b T) w‖*subdiskDistortion q := by
+  have hb := centeredRegion_bounds hw hq hq1 hv
+  have hh := inverseMap_deriv_two_sided ha b T (mul_pos hw (subdiskHeight_pos hq hq1))
+    (mul_le_of_le_one_right hw.le (subdiskHeight_le_one q)) hb.2 hb.1
+  have he : (3/(w.im*subdiskHeight q))*(w.im*subdiskReach q)=3*subdiskReach q/subdiskHeight q := by
+    field_simp
+  rw [show -(3/(w.im*subdiskHeight q))*(w.im*subdiskReach q)=
+      -((3/(w.im*subdiskHeight q))*(w.im*subdiskReach q)) by ring,he,Real.exp_neg] at hh
+  simpa only [subdiskDistortion,div_eq_mul_inv] using hh
+
+ 
+lemma centeredRegion_image_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
+    (hv : v∈centeredRegion w q) :
+    ‖inverseMap a b T v-inverseMap a b T w‖≤
+      subdiskReach q*subdiskDistortion q*(w.im*‖deriv (inverseMap a b T) w‖) := by
+  have hb := centeredRegion_bounds hw hq hq1 hv
+  let y : ℝ := w.im*subdiskHeight q
+  have hy : 0<y := mul_pos hw (subdiskHeight_pos hq hq1)
+  have hw' : y≤w.im := mul_le_of_le_one_right hw.le (subdiskHeight_le_one q)
+  have hxy (t : ℝ) (ht : t∈Icc 0 1) : y≤(AffineMap.lineMap w v t).im :=
+    lineMap_im_lower hw' hb.2 ht
+  have hdist (t : ℝ) (ht : t∈Icc 0 1) : ‖AffineMap.lineMap w v t-w‖≤w.im*subdiskReach q := by
+    rw [AffineMap.lineMap_apply]
+    simp only [vadd_eq_add,vsub_eq_sub,add_sub_cancel_right,norm_smul,Real.norm_eq_abs,abs_of_nonneg ht.1]
+    exact (mul_le_of_le_one_left (norm_nonneg _) ht.2).trans hb.1
+  have he : (3/y)*(w.im*subdiskReach q)=3*subdiskReach q/subdiskHeight q := by
+    dsimp [y]; field_simp
+  have hd (t : ℝ) (ht : t∈Icc 0 1) :
+      HasDerivAt (fun t : ℝ=>inverseMap a b T (AffineMap.lineMap w v t))
+        (deriv (inverseMap a b T) (AffineMap.lineMap w v t)*(v-w)) t := by
+    apply lineMap_hasDerivAt_complex_comp
+    exact ((inverseMap_hasStrictDerivAt ha b T (hy.trans_le (hxy t ht))).hasDerivAt).differentiableAt.hasDerivAt
+  have hbound (t : ℝ) (ht : t∈Icc 0 1) :
+      ‖deriv (inverseMap a b T) (AffineMap.lineMap w v t)*(v-w)‖≤
+      (‖deriv (inverseMap a b T) w‖*subdiskDistortion q)*‖v-w‖ := by
+    rw [norm_mul]
+    refine mul_le_mul_of_nonneg_right ?_ (norm_nonneg _)
+    have hh := (inverseMap_deriv_two_sided ha b T hy hw' (hxy t ht) (hdist t ht)).2
+    rwa [he] at hh
+  have hh := (convex_Icc (0:ℝ) 1).norm_image_sub_le_of_norm_hasDerivWithin_le
+    (fun t ht=>(hd t ht).hasDerivWithinAt) hbound
+    (by norm_num : (0:ℝ)∈Icc 0 1) (by norm_num : (1:ℝ)∈Icc 0 1)
+  simp only [AffineMap.lineMap_apply_zero,AffineMap.lineMap_apply_one,sub_zero,norm_one,mul_one] at hh
+  exact hh.trans (by
+    calc
+      _ ≤ (‖deriv (inverseMap a b T) w‖*subdiskDistortion q)*(w.im*subdiskReach q) := mul_le_mul_of_nonneg_left hb.1
+        (mul_nonneg (norm_nonneg _) (subdiskDistortion_pos q).le)
+      _ = _ := by ring)
+
+ 
+
+theorem inverseMap_centered_localization {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 1/2≤q) (hq1 : q<1) :
+    closedBall (inverseMap a b T w) (w.im*‖deriv (inverseMap a b T) w‖/200) ⊆
+      inverseMap a b T '' centeredRegion w q ∧
+    inverseMap a b T '' centeredRegion w q ⊆ closedBall (inverseMap a b T w)
+      (subdiskReach q*subdiskDistortion q*(w.im*‖deriv (inverseMap a b T) w‖)) := by
+  refine ⟨(inverseMap_inner_ball ha b T hw).trans
+    (image_mono (smallBall_subset_centeredRegion hw hq)),?_⟩
+  rintro _ ⟨v,hv,rfl⟩
+  simpa only [mem_closedBall,dist_eq_norm] using centeredRegion_image_bound ha b T hw (by linarith) hq1 hv
+
+lemma diskAt_hasDerivAt {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) :
+    HasDerivAt (diskAt w) (((2*w.im:ℝ):ℂ)*Complex.I/(v-star w)^2) v := by
+  have hh := ((hasDerivAt_id v).sub_const w).div
+    ((hasDerivAt_id v).sub_const (star w)) (diskAt_denominator_ne hw hv)
+  convert! hh using 1
+  simp only [id_eq,one_mul,mul_one]
+  congr 1
+  apply Complex.ext <;> simp; ring
+
+lemma diskAt_deriv_norm {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) :
+    ‖deriv (diskAt w) v‖=2*w.im/‖v-star w‖^2 := by
+  rw [(diskAt_hasDerivAt hw hv).deriv,norm_div,norm_mul,norm_pow,
+    Complex.norm_real,Real.norm_eq_abs,abs_of_pos (mul_pos (by norm_num) hw),
+    Complex.norm_I,mul_one]
+
+lemma diskAt_deriv_bound {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
+    (hv : v∈centeredRegion w q) :
+    2/((subdiskReach q+2)^2*w.im) ≤ ‖deriv (diskAt w) v‖ ∧
+    ‖deriv (diskAt w) v‖ ≤ 2/w.im := by
+  rw [diskAt_deriv_norm hw hv.1]
+  have hb := centeredRegion_bounds hw hq hq1 hv
+  have hn : w.im≤‖v-star w‖ := by
+    apply le_trans _ (Complex.im_le_norm _)
+    simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,sub_neg_eq_add]
+    linarith [hv.1]
+  have hu : ‖v-star w‖≤w.im*(subdiskReach q+2) := by
+    calc
+      _ = ‖(v-w)+(w-star w)‖ := by congr 1; ring
+      _ ≤ ‖v-w‖+‖w-star w‖ := norm_add_le _ _
+      _ = ‖v-w‖+2*w.im := by
+        rw [show w-star w=((2*w.im:ℝ):ℂ)*Complex.I by
+          apply Complex.ext <;> simp; ring,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+          abs_of_pos (mul_pos (by norm_num) hw),Complex.norm_I,mul_one]
+      _ ≤ _ := by linarith [hb.1]
+  have hq' : 0<subdiskReach q+2 := by linarith [subdiskReach_nonneg hq hq1]
+  constructor
+  · calc
+      _ = 2*w.im/(w.im*(subdiskReach q+2))^2 := by field_simp
+      _ ≤ _ := div_le_div_of_nonneg_left (by positivity)
+        (sq_pos_of_pos (hw.trans_le hn)) (pow_le_pow_left₀ (norm_nonneg _) hu 2)
+  · calc
+      _ ≤ 2*w.im/w.im^2 := div_le_div_of_nonneg_left (by positivity)
+        (sq_pos_of_pos hw) (pow_le_pow_left₀ hw.le hn 2)
+      _ = _ := by field_simp
+
+ 
+
+theorem centered_disk_deriv_ratio {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w v : ℂ} (hw : 0<w.im) {q : ℝ} (hq : 0≤q) (hq1 : q<1)
+    (hv : v∈centeredRegion w q) :
+    (w.im*‖deriv (inverseMap a b T) w‖)/(2*subdiskDistortion q) ≤
+        ‖deriv (inverseMap a b T) v‖/‖deriv (diskAt w) v‖ ∧
+    ‖deriv (inverseMap a b T) v‖/‖deriv (diskAt w) v‖ ≤
+        (subdiskDistortion q*(subdiskReach q+2)^2/2)*(w.im*‖deriv (inverseMap a b T) w‖) := by
+  have hf := centeredRegion_deriv_bounds ha b T hw hq hq1 hv
+  have hg := diskAt_deriv_bound hw hq hq1 hv
+  have hq' : 0<subdiskReach q+2 := by linarith [subdiskReach_nonneg hq hq1]
+  have hlow : 0<2/((subdiskReach q+2)^2*w.im) := by positivity
+  have hnorm : 0<‖deriv (diskAt w) v‖ := hlow.trans_le hg.1
+  constructor
+  · calc
+      _ = (‖deriv (inverseMap a b T) w‖/subdiskDistortion q)/(2/w.im) := by field_simp
+      _ ≤ _ := div_le_div₀ (norm_nonneg _) hf.1 hnorm hg.2
+  · calc
+      _ ≤ (‖deriv (inverseMap a b T) w‖*subdiskDistortion q)/(2/((subdiskReach q+2)^2*w.im)) :=
+        div_le_div₀ (mul_nonneg (norm_nonneg _) (subdiskDistortion_pos q).le) hf.2 hlow hg.1
+      _ = _ := by field_simp
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain ExactSLEGreen LoewnerRestart FlowIdentification
+
+lemma centered_im_pos {a : ℝ} {b : GDriver} {t : ℝ≥0} {z : ℂ}
+    (hz : alive a b.1 t z) : 0<(loewnerMap a b.1 t z-(b.1 t:ℂ)).im := by
+  simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using loewnerMap_im_pos hz
+
+lemma inverseMap_spatial_radius {a : ℝ} (ha : 0<a) (b : GDriver) (t : ℝ≥0)
+    {z : ℂ} (hz : alive a b.1 t z) :
+    (loewnerMap a b.1 t z-(b.1 t:ℂ)).im *
+        ‖deriv (inverseMap a b t) (loewnerMap a b.1 t z-(b.1 t:ℂ))‖ =
+      physicalRadius (actualState a b.1 t z) := by
+  rw [deriv_inverseMap ha b t (centered_im_pos hz),inverseMap_centered a b t hz,
+    norm_inv]
+  have hd := norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hz)
+  simp only [physicalRadius,actualState,Y_apply,J_apply,Real.exp_neg,Real.exp_log hd]
+
+ 
+
+theorem radius_ball_subset_alive {a : ℝ} (ha : 0<a) (b : GDriver) (t : ℝ≥0)
+    {z : ℂ} (hz : alive a b.1 t z) :
+    closedBall z (physicalRadius (actualState a b.1 t z)/200) ⊆ {v | alive a b.1 t v} := by
+  have hh := inverseMap_inner_ball ha b t (centered_im_pos hz)
+  rw [inverseMap_spatial_radius ha b t hz,inverseMap_centered a b t hz] at hh
+  intro v hv
+  obtain ⟨w,hw,rfl⟩ := hh hv
+  exact inverseMap_alive ha b t (lt_of_lt_of_le
+    (div_pos (centered_im_pos hz) (by norm_num))
+    (smallBall_im_half (centered_im_pos hz) hw))
+
+ 
+theorem centered_localization_at_alive {a : ℝ} (ha : 0<a) (b : GDriver) (t : ℝ≥0)
+    {z : ℂ} (hz : alive a b.1 t z) {q : ℝ} (hq : 1/2≤q) (hq1 : q<1) :
+    closedBall z (physicalRadius (actualState a b.1 t z)/200) ⊆
+        inverseMap a b t '' centeredRegion (loewnerMap a b.1 t z-(b.1 t:ℂ)) q ∧
+    inverseMap a b t '' centeredRegion (loewnerMap a b.1 t z-(b.1 t:ℂ)) q ⊆
+      closedBall z (subdiskReach q*subdiskDistortion q*physicalRadius (actualState a b.1 t z)) := by
+  have hh := inverseMap_centered_localization ha b t (centered_im_pos hz) hq hq1
+  rwa [inverseMap_spatial_radius ha b t hz,inverseMap_centered a b t hz] at hh
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/HeightRadius.lean b/lean/OAI/Probability/SLEGauge/HeightRadius.lean
index 0481af23bd573d5e200e7e1a2405cb1a35554ddd..27fc605f8567ff00137a1e5d3ae89f44082ff6ca 100644
--- a/lean/OAI/Probability/SLEGauge/HeightRadius.lean
+++ b/lean/OAI/Probability/SLEGauge/HeightRadius.lean
@@ -1,605 +1,605 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialRestart
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace ExactSLEBarrier
-variable {a : ℝ} {T : ℝ≥0} {y j q : ℝ → ℝ}
-
-lemma Evolution.logHeight_add_deriv (h : Evolution a T y j q) {t : ℝ}
-    (ht : t ∈ Icc 0 (T : ℝ)) :
-    HasDerivWithinAt (fun t => Real.log (y t)+j t)
-      (-2*a*(q t-(y t)^2)/(q t)^2) (Icc 0 (T : ℝ)) t := by
-  convert (h.logHeight_deriv ht).add (h.jd t ht) using 1; try rfl
-  field_simp [(h.qp t ht).ne']
-  ring
-
-lemma Evolution.logHeight_add_le (h : Evolution a T y j q) (ha : 0≤a)
-    (hq : ∀ u∈Icc 0 (T : ℝ), (y u)^2≤q u)
-    (t : ℝ) (ht : t∈Icc 0 (T : ℝ)) :
-    Real.log (y t)+j t≤Real.log (y 0)+j 0 := by
-  have hc : ContinuousOn (fun t => Real.log (y t)+j t) (Icc 0 (T : ℝ)) :=
-    (h.yc.log (fun u hu => (h.yp u hu).ne')).add h.jc
-  have hm : AntitoneOn (fun t => Real.log (y t)+j t) (Icc 0 (T : ℝ)) :=
-    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) hc
-      (fun u hu => (h.logHeight_add_deriv (interior_subset hu)).mono interior_subset)
-      (fun u hu => div_nonpos_of_nonpos_of_nonneg
-        (mul_nonpos_of_nonpos_of_nonneg (by linarith)
-          (sub_nonneg.mpr (hq u (interior_subset hu)))) (sq_nonneg _))
-  exact hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
-
-lemma Evolution.height_sq_le_radius (h : Evolution a T y j q) (ha : 0≤a)
-    (hq : ∀ u∈Icc 0 (T : ℝ), (y u)^2≤q u) (hj0 : j 0=0)
-    (t : ℝ) (ht : t∈Icc 0 (T : ℝ)) :
-    (y t)^2≤y 0*(y t*Real.exp (-j t)) := by
-  have hl := h.logHeight_add_le ha hq t ht
-  rw [hj0,add_zero] at hl
-  have he : y t≤y 0*Real.exp (-j t) := by
-    have hh := Real.exp_le_exp.mpr (show Real.log (y t)≤Real.log (y 0)+ -j t by linarith)
-    simpa only [Real.exp_add,Real.exp_log (h.yp t ht),
-      Real.exp_log (h.yp 0 ⟨le_rfl,T.coe_nonneg⟩)] using hh
-  have hh := mul_le_mul_of_nonneg_left he (h.yp t ht).le
-  nlinarith only [hh]
-
-lemma Evolution.height_sq_lower (h : Evolution a T y j q) (ha : 0≤a)
-    (hq : ∀ u∈Icc 0 (T : ℝ), (y u)^2≤q u)
-    (t : ℝ) (ht : t∈Icc 0 (T : ℝ)) :
-    (y 0)^2≤(y t)^2+2*a*t := by
-  let F : ℝ→ℝ := fun u => (y u)^2+2*a*u
-  have hc : ContinuousOn F (Icc 0 (T : ℝ)) :=
-    (h.yc.pow 2).add (continuousOn_const.mul continuousOn_id)
-  have hd (u : ℝ) (hu : u∈Icc 0 (T : ℝ)) :
-      HasDerivWithinAt F (2*a*(1-(y u)^2/q u)) (Icc 0 (T : ℝ)) u := by
-    convert ((h.yd u hu).pow 2).add ((hasDerivWithinAt_id u _).const_mul (2*a)) using 1 <;> try rfl
-    simp only [Nat.reduceSub,pow_one,Nat.cast_ofNat]
-    ring
-  have hm : MonotoneOn F (Icc 0 (T : ℝ)) :=
-    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 (T : ℝ)) hc
-      (fun u hu => (hd u (interior_subset hu)).mono interior_subset)
-      (fun u hu => mul_nonneg (by positivity) (sub_nonneg.mpr
-        ((div_le_one (h.qp u (interior_subset hu))).mpr (hq u (interior_subset hu)))))
-  simpa only [F,mul_zero,add_zero] using hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
-
-end ExactSLEBarrier
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator
-
- 
-
-lemma actualState_evolution (a : ℝ) (b : ExactSLEGenerator.Driver) (z : UpperPlane)
-    (T : ℝ≥0) (hT : alive a b.1 T z.1) :
-    ExactSLEBarrier.Evolution a T
-      (fun u => Y (actualState a b.1 u.toNNReal z.1))
-      (fun u => J (actualState a b.1 u.toNNReal z.1))
-      (fun u => Q (actualState a b.1 u.toNNReal z.1)) := by
-  obtain ⟨n,hn⟩ := (exhaustionStop_eventually a z b T hT).exists
-  let ε := exhaustionHeight z n
-  let V := localState a ε (exhaustionHeight_pos z n) z.1
-  have hsafe (u : ℝ≥0) (hu : u≤T) : ε≤Y (V u b) :=
-    (regionStop_safe a ε (exhaustionHeight_pos z n) z.1 (exhaustionNorm z n) 0 0 n
-      (exhaustionRegion_initial z n) b u (hu.trans hn)).1
-  have he (u : ℝ) (hu : u∈Icc 0 (T : ℝ)) :
-      V u.toNNReal b=actualState a b.1 u.toNNReal z.1 :=
-    (exhaustionStop_agrees a z n b u.toNNReal
-      ((show u.toNNReal≤T by simpa using Real.toNNReal_le_toNNReal hu.2).trans hn)).2.symm
-  dsimp only [V] at he
-  have h := localState_evolution a ε (exhaustionHeight_pos z n) z.1 b T hsafe
-  refine ⟨?_,?_,?_,?_,?_,?_⟩
-  · exact h.yc.congr (fun u hu => congrArg Y (he u hu).symm)
-  · exact h.jc.congr (fun u hu => congrArg J (he u hu).symm)
-  · intro u hu; simpa only [←he u hu] using h.yp u hu
-  · intro u hu; simpa only [←he u hu] using h.qp u hu
-  · intro u hu
-    have hh := (h.yd u hu).congr_of_mem (fun v hv => congrArg Y (he v hv).symm) hu
-    simpa only [he u hu] using hh
-  · intro u hu
-    have hh := (h.jd u hu).congr_of_mem (fun v hv => congrArg J (he v hv).symm) hu
-    simpa only [he u hu] using hh
-
-lemma actual_height_sq_le_radius (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
-    (z : UpperPlane) (t : ℝ≥0) (ht : alive a b.1 t z.1) :
-    (Y (actualState a b.1 t z.1))^2≤z.1.im*physicalRadius (actualState a b.1 t z.1) := by
-  have h := (actualState_evolution a b z t ht).height_sq_le_radius ha.le
-    (fun u _ => by dsimp [Q]; nlinarith only [sq_nonneg (X (actualState a b.1 u.toNNReal z.1))])
-    (by simp [actualState_zero ha b z.2]) (t : ℝ) ⟨t.coe_nonneg,le_rfl⟩
-  simpa only [Real.toNNReal_coe,Real.toNNReal_zero,actualState_zero ha b z.2,Y_apply,
-    physicalRadius] using h
-
-lemma actual_height_sq_lower (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
-    (z : UpperPlane) (t : ℝ≥0) (ht : alive a b.1 t z.1) :
-    z.1.im^2≤(Y (actualState a b.1 t z.1))^2+2*a*t := by
-  have h := (actualState_evolution a b z t ht).height_sq_lower ha.le
-    (fun u _ => by dsimp [Q]; nlinarith only [sq_nonneg (X (actualState a b.1 u.toNNReal z.1))])
-    (t : ℝ) ⟨t.coe_nonneg,le_rfl⟩
-  simpa only [Real.toNNReal_coe,Real.toNNReal_zero,actualState_zero ha b z.2,Y_apply] using h
-
- 
-
-lemma level_height_bound (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
-    (z : UpperPlane) (s : ℝ) (hs : 0≤ s)
-    (hfin : radialTime a b.1 z.1 (levelRadius a (2*z.1.im) s)≠∞) :
-    Y (actualState a b.1 (radialTime a b.1 z.1 (levelRadius a (2*z.1.im) s)).toNNReal z.1)≤
-      z.1.im*Real.exp (-a*s) := by
-  have hcap : 0<2*z.1.im := mul_pos (by norm_num) z.2
-  have hr : levelRadius a (2*z.1.im) s≤z.1.im := by
-    simpa using levelRadius_le ha hcap hs
-  have ht := radialTime_attained a ha z.1 z.2 _ (levelRadius_pos _ _ _ hcap) hr b hfin
-  have hh := actual_height_sq_le_radius a ha b z _ ht.1
-  rw [ht.2] at hh
-  have hexp : Real.exp (-2*a*s)=(Real.exp (-a*s))^2 := by
-    rw [←Real.exp_nat_mul]; congr 1; ring
-  conv_rhs at hh =>
-    simp only [levelRadius,mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0),hexp]
-  have hpos : 0<z.1.im*Real.exp (-a*s) := mul_pos z.2 (Real.exp_pos _)
-  nlinarith only [hh,hpos]
-
- 
-
-lemma radialTime_half_lower (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
-    (z : UpperPlane) (r : ℝ) (hr : 0<r) (hr2 : r≤z.1.im/2)
-    (hfin : radialTime a b.1 z.1 r≠∞) :
-    z.1.im^2/(4*a)≤((radialTime a b.1 z.1 r).toNNReal : ℝ) := by
-  have ht := radialTime_attained a ha z.1 z.2 r hr (by linarith [z.2]) b hfin
-  have hl := actual_height_sq_lower a ha b z _ ht.1
-  have hu := actual_height_sq_le_radius a ha b z _ ht.1
-  rw [ht.2] at hu
-  have hh := mul_le_mul_of_nonneg_left hr2 z.2.le
-  apply (div_le_iff₀ (by positivity : 0<4*a)).mpr
-  nlinarith only [hl,hu,hh]
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Metric Set Filter Topology
-open scoped NNReal BoundedContinuousFunction
-namespace SLEExactGauge.AnalyticCompactness
-
-lemma lipschitzOn_subball {f : ℂ → ℂ} {c : ℂ} {r R M : ℝ}
-    (hr : 0 ≤ r) (hrR : r < R)
-    (hf : DifferentiableOn ℂ f (ball c R))
-    (hM : ∀ z ∈ ball c R, ‖f z‖ ≤ M) :
-    LipschitzOnWith (2*M/(R-r)).toNNReal f (closedBall c r) := by
-  have hsub : closedBall c r ⊆ ball c R := closedBall_subset_ball hrR
-  have hM0 : 0 ≤ M := (norm_nonneg _).trans (hM c (mem_ball_self (hr.trans_lt hrR)))
-  apply (convex_closedBall c r).lipschitzOnWith_of_nnnorm_deriv_le
-  · intro z hz
-    exact (hf z (hsub hz)).differentiableAt (isOpen_ball.mem_nhds (hsub hz))
-  · intro z hz
-    have hb : ball z (R-r) ⊆ ball c R := by
-      intro w hw
-      exact (dist_triangle w z c).trans_lt (by
-        have hw' : dist w z < R-r := hw
-        have hz' : dist z c ≤ r := hz
-        linarith)
-    have hh := Complex.norm_deriv_le_div_of_mapsTo_ball (hf.mono hb)
-      (R₂ := 2*M) (by
-        intro w hw
-        change dist (f w) (f z) ≤ 2*M
-        rw [dist_eq_norm]
-        exact (norm_sub_le _ _).trans (by linarith [hM w (hb hw),hM z (hsub hz)]))
-      (sub_pos.mpr hrR)
-    exact_mod_cast (hh.trans (Real.le_coe_toNNReal _))
-
- 
-
-theorem exists_subseq_uniform {F : ℕ → ℂ → ℂ} {c : ℂ} {r R M : ℝ}
-    (hr : 0 ≤ r) (hrR : r < R)
-    (hF : ∀ n, DifferentiableOn ℂ (F n) (ball c R))
-    (hM : ∀ n z, z ∈ ball c R → ‖F n z‖ ≤ M) :
-    ∃ g : ℂ → ℂ, ∃ φ : ℕ → ℕ, StrictMono φ ∧
-      TendstoUniformlyOn (fun n => F (φ n)) g atTop (closedBall c r) ∧
-      DifferentiableOn ℂ g (ball c r) := by
-  classical
-  let K := closedBall c r
-  have : CompactSpace K := isCompact_iff_compactSpace.mp (isCompact_closedBall c r)
-  have hl (n : ℕ) := lipschitzOn_subball hr hrR (hF n) (hM n)
-  let f : ℕ → K →ᵇ ℂ := fun n => BoundedContinuousFunction.mkOfCompact
-    ⟨fun z => F n z, (hl n).continuousOn.domRestrict⟩
-  have hlf (n : ℕ) : LipschitzWith (2*M/(R-r)).toNNReal (f n) :=
-    lipschitzOnWith_iff_restrict.mp (hl n)
-  have heq : Equicontinuous ((↑) : range f → K → ℂ) := by
-    apply UniformEquicontinuous.equicontinuous
-    apply LipschitzWith.uniformEquicontinuous _ (2*M/(R-r)).toNNReal
-    rintro ⟨g,⟨n,rfl⟩⟩
-    exact hlf n
-  have hc := BoundedContinuousFunction.arzela_ascoli (closedBall (0:ℂ) M)
-    (isCompact_closedBall _ _) (range f) (by
-      rintro g z ⟨n,rfl⟩
-      change dist (F n z) 0 ≤ M
-      simpa only [dist_zero_right] using
-        hM n z (closedBall_subset_ball hrR z.2)) heq
-  obtain ⟨g,-,φ,hφ,hg⟩ := hc.tendsto_subseq (fun n => subset_closure (mem_range_self n))
-  let g' : ℂ → ℂ := fun z => if hz : z ∈ K then g ⟨z,hz⟩ else 0
-  have he (z : ℂ) (hz : z ∈ K) : g' z = g ⟨z,hz⟩ := dite_eq_left hz
-  have hu : TendstoUniformlyOn (fun n => F (φ n)) g' atTop K := by
-    rw [Metric.tendstoUniformlyOn_iff]
-    intro ε hε
-    have hh := Metric.tendstoUniformly_iff.mp
-      (BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hg) ε hε
-    filter_upwards [hh] with n hn z hz
-    rw [he z hz]
-    exact hn ⟨z,hz⟩
-  refine ⟨g',φ,hφ,hu,?_⟩
-  exact (hu.mono ball_subset_closedBall).tendstoLocallyUniformlyOn.differentiableOn
-    (Eventually.of_forall (fun n => (hF (φ n)).mono (ball_subset_ball hrR.le))) isOpen_ball
-
- 
-
-theorem uniform_zero_of_accumulation {F : ℕ → ℂ → ℂ} {c : ℂ} {r R M : ℝ}
-    (hr : 0 < r) (hrR : r < R)
-    (hF : ∀ n, DifferentiableOn ℂ (F n) (ball c R))
-    (hM : ∀ n z, z ∈ ball c R → ‖F n z‖ ≤ M)
-    (z : ℕ → ℂ) (hz : ∀ j, z j ∈ closedBall c r)
-    (hzc : ∀ j, z j ≠ c) (hzlim : Tendsto z atTop (𝓝 c))
-    (hFz : ∀ j, Tendsto (fun n => F n (z j)) atTop (𝓝 0)) :
-    TendstoUniformlyOn F (fun _ => 0) atTop (closedBall c r) := by
-  classical
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  by_contra hbad
-  have hb : ∃ᶠ n in atTop, ∃ w ∈ closedBall c r, ε ≤ ‖F n w‖ := by
-    simpa only [not_forall,exists_prop,not_lt,dist_zero_left] using (show ∃ᶠ n in atTop,
-      ¬∀ w ∈ closedBall c r, dist (0:ℂ) (F n w)<ε from Filter.not_eventually.mp hbad)
-  obtain ⟨ψ,hψ,hψbad⟩ := subseq_forall_of_frequently (x := fun n : ℕ => n)
-    (p := fun n => ∃ w ∈ closedBall c r, ε ≤ ‖F n w‖) tendsto_id hb
-  let s : ℝ := (r+R)/2
-  have hrs : r<s := by dsimp [s]; linarith
-  have hsR : s<R := by dsimp [s]; linarith
-  have hs : 0<s := hr.trans hrs
-  obtain ⟨g,φ,hφ,hg,hgd⟩ := exists_subseq_uniform hs.le hsR
-    (fun n => hF (ψ n)) (fun n => hM (ψ n))
-  have hgzero (j : ℕ) : g (z j)=0 := by
-    have hpt := hg.tendsto_at (closedBall_subset_closedBall hrs.le (hz j))
-    exact tendsto_nhds_unique hpt ((hFz j).comp (hψ.comp hφ.tendsto_atTop))
-  have hcl : c∈closure ({w : ℂ | g w=0} \ {c}) := by
-    apply mem_closure_of_tendsto hzlim
-    exact Eventually.of_forall (fun j => ⟨hgzero j,hzc j⟩)
-  have heq : EqOn g (fun _ => 0) (ball c s) :=
-    hgd.analyticOnNhd isOpen_ball |>.eqOn_of_preconnected_of_mem_closure
-      analyticOnNhd_const (convex_ball c s).isPreconnected (mem_ball_self hs) hcl
-  have hsmall := Metric.tendstoUniformlyOn_iff.mp hg ε hε
-  obtain ⟨n,hn⟩ := hsmall.exists
-  obtain ⟨w,hw,hwbad⟩ := hψbad (φ n)
-  have hh := hn w (closedBall_subset_closedBall hrs.le hw)
-  rw [heq (closedBall_subset_ball hrs hw),dist_zero_left] at hh
-  exact (not_lt_of_ge hwbad) hh
-
- 
-
-theorem uniform_zero_on_horizontal {F : ℕ → ℂ → ℂ} {C M : ℝ} (hC : 0 ≤ C)
-    (hF : ∀ n, DifferentiableOn ℂ (F n) {z : ℂ | 0 < z.im})
-    (hM : ∀ n z, z ∈ ball (((C^2+2 : ℝ) : ℂ)*Complex.I) (C^2+2-1/4) →
-      ‖F n z‖ ≤ M)
-    (hv : ∀ y : ℝ, 0 < y → Tendsto (fun n => F n ((y:ℂ)*Complex.I)) atTop (𝓝 0)) :
-    TendstoUniformlyOn F (fun _ => 0) atTop {z : ℂ | z.im=1 ∧ |z.re|≤C} := by
-  let A : ℝ := C^2+2
-  let c : ℂ := (A:ℂ)*Complex.I
-  let r : ℝ := A-1/2
-  let R : ℝ := A-1/4
-  have hA : 2 ≤ A := by dsimp [A]; nlinarith [sq_nonneg C]
-  have hr : 0 < r := by dsimp [r]; linarith
-  have hrR : r < R := by dsimp [r,R]; linarith
-  have hcim : c.im=A := by simp [c]
-  have hcre : c.re=0 := by simp [c]
-  have hball : ball c R ⊆ {z : ℂ | 0 < z.im} := by
-    intro z hz
-    have hh := Complex.im_le_norm (c-z)
-    rw [Complex.sub_im,hcim,norm_sub_rev,← dist_eq_norm] at hh
-    have hz' : dist z c < A-1/4 := hz
-    change 0 < z.im
-    linarith
-  let z : ℕ → ℂ := fun j => c + ((1/((j:ℝ)+2):ℝ):ℂ)*Complex.I
-  have hz (j : ℕ) : z j∈closedBall c r := by
-    have hj : 0 < 1/((j:ℝ)+2) := by positivity
-    have hj1 : 1/((j:ℝ)+2) ≤ 1/2 := by
-      apply one_div_le_one_div_of_le (by norm_num)
-      linarith [Nat.cast_nonneg (α := ℝ) j]
-    rw [mem_closedBall,dist_eq_norm]
-    simp only [z,add_sub_cancel_left,norm_mul,Complex.norm_real,
-      Real.norm_eq_abs,abs_of_pos hj,Complex.norm_I,mul_one]
-    dsimp [r]
-    linarith
-  have hzc (j : ℕ) : z j≠c := by
-    intro h
-    have hh := congrArg Complex.im h
-    simp only [z,Complex.add_im,Complex.mul_I_im,Complex.ofReal_re] at hh
-    have hp : 0 < 1/((j:ℝ)+2) := by positivity
-    linarith
-  have hzlim : Tendsto z atTop (𝓝 c) := by
-    have hh : Tendsto (fun j : ℕ => 1/((j:ℝ)+2)) atTop (𝓝 0) :=
-      tendsto_const_nhds.div_atTop (tendsto_atTop_add_const_right atTop 2 tendsto_natCast_atTop_atTop)
-    have h' := (Complex.continuous_ofReal.tendsto 0 |>.comp hh).mul_const Complex.I |>.const_add c
-    simpa only [z,Function.comp_apply,Complex.ofReal_zero,zero_mul,add_zero] using h'
-  have hFz (j : ℕ) : Tendsto (fun n => F n (z j)) atTop (𝓝 0) := by
-    have he : z j = (((A+1/((j:ℝ)+2):ℝ):ℂ)*Complex.I) := by
-      dsimp [z,c]
-      push_cast
-      ring
-    rw [he]
-    apply hv
-    have hp : 0 < 1/((j:ℝ)+2) := by positivity
-    linarith
-  apply (uniform_zero_of_accumulation hr hrR (fun n => (hF n).mono hball)
-    hM z hz hzc hzlim hFz).mono
-  intro w hw
-  rw [mem_closedBall,dist_eq_norm]
-  have hsq : ‖w-c‖^2=w.re^2+(1-A)^2 := by
-    rw [← Complex.normSq_eq_norm_sq,Complex.normSq_apply]
-    simp only [Complex.sub_re,Complex.sub_im,hcim,hcre,sub_zero,hw.1]
-    ring
-  have hsq2 : w.re^2 ≤ C^2 := by
-    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg w.re) hC).mpr hw.2
-  have hbound : ‖w-c‖^2 ≤ r^2 := by
-    rw [hsq]
-    dsimp [r,A] at *
-    nlinarith
-  exact (sq_le_sq₀ (norm_nonneg _) hr.le).mp hbound
-
-end SLEExactGauge.AnalyticCompactness
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Metric Set Filter Topology
-namespace SLEExactGauge.AngularLimits
-open AnalyticCompactness
-
- 
-
-theorem uniform_angular_of_radial {X : Type*} (f : X → ℂ → ℂ) (g : X → ℂ)
-    (S : Set X) {M : ℝ}
-    (hf : ∀ x∈S, DifferentiableOn ℂ (f x) {z : ℂ | 0<z.im})
-    (hM : ∀ x∈S, ∀ z : ℂ, 0<z.im → ‖z‖≤1 → ‖f x z-g x‖≤M)
-    (hr : TendstoUniformlyOn (fun y : ℝ => fun x => f x ((y:ℂ)*Complex.I))
-      g (𝓝[>] 0) S) {C : ℝ} (hC : 0≤C) :
-    TendstoUniformlyOn
-      (fun y : ℝ => fun p : X×ℝ => f p.1 ((y:ℂ)*((p.2:ℂ)+Complex.I)))
-      (fun p => g p.1) (𝓝[>] 0) (S ×ˢ Icc (-C) C) := by
-  classical
-  apply tendstoUniformlyOn_of_seq_tendstoUniformlyOn
-  intro y hy
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  by_contra hbad
-  let A : ℝ := C^2+2
-  have hA : 0<A := by dsimp [A]; positivity
-  have hb : ∃ᶠ n in atTop, ∃ p∈S ×ˢ Icc (-C) C,
-      ε≤‖g p.1-f p.1 ((y n:ℂ)*((p.2:ℂ)+Complex.I))‖ := by
-    simpa only [not_forall,exists_prop,not_lt,dist_eq_norm] using not_eventually.mp hbad
-  have he : ∀ᶠ n in atTop, 0<y n ∧ y n≤1/(2*A) := by
-    filter_upwards [hy.eventually self_mem_nhdsWithin,
-      (tendsto_nhds_of_tendsto_nhdsWithin hy).eventually
-        (Iio_mem_nhds (by positivity : (0:ℝ)<1/(2*A)))] with n hn hn'
-    exact ⟨hn,hn'.le⟩
-  have hb' := hb.and_eventually he
-  obtain ⟨ψ,hψ,hψbad⟩ := subseq_forall_of_frequently
-    (x := fun n : ℕ => n)
-    (p := fun n => (∃ p∈S ×ˢ Icc (-C) C,
-      ε≤‖g p.1-f p.1 ((y n:ℂ)*((p.2:ℂ)+Complex.I))‖) ∧
-        0<y n ∧ y n≤1/(2*A)) tendsto_id hb'
-  choose p hp hpb using (fun n => (hψbad n).1)
-  have hyp (n : ℕ) : 0<y (ψ n) := (hψbad n).2.1
-  have hyb (n : ℕ) : y (ψ n)≤1/(2*A) := (hψbad n).2.2
-  let F : ℕ → ℂ → ℂ := fun n z => g (p n).1-f (p n).1 ((y (ψ n):ℂ)*z)
-  have hF (n : ℕ) : DifferentiableOn ℂ (F n) {z : ℂ | 0<z.im} := by
-    apply (differentiableOn_const _).sub
-    apply (hf (p n).1 (hp n).1).comp
-      ((differentiable_const _).mul differentiable_id).differentiableOn
-    intro z hz
-    change 0<((y (ψ n):ℂ)*z).im
-    simpa only [Complex.mul_im,Complex.ofReal_im,Complex.ofReal_re,zero_mul,add_zero] using
-      mul_pos (hyp n) hz
-  have hFM (n : ℕ) (z : ℂ)
-      (hz : z∈ball ((A:ℂ)*Complex.I) (A-1/4)) : ‖F n z‖≤M := by
-    have hzi : 0<z.im := by
-      have hh := Complex.im_le_norm ((A:ℂ)*Complex.I-z)
-      simp only [Complex.sub_im,Complex.mul_I_im,Complex.ofReal_re] at hh
-      rw [norm_sub_rev,← dist_eq_norm] at hh
-      have hz' : dist z ((A:ℂ)*Complex.I)<A-1/4 := hz
-      linarith
-    have hzn : ‖z‖≤2*A := by
-      have hnorm := norm_add_le (z-(A:ℂ)*Complex.I) ((A:ℂ)*Complex.I)
-      rw [sub_add_cancel,← dist_eq_norm] at hnorm
-      simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hA,
-        Complex.norm_I,mul_one] at hnorm
-      have hz' : dist z ((A:ℂ)*Complex.I)<A-1/4 := hz
-      linarith
-    have hpim : 0<((y (ψ n):ℂ)*z).im := by
-      simpa only [Complex.mul_im,Complex.ofReal_im,Complex.ofReal_re,zero_mul,add_zero] using
-        mul_pos (hyp n) hzi
-    have hpnom : ‖(y (ψ n):ℂ)*z‖≤1 := by
-      rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (hyp n)]
-      calc
-        _ ≤ (1/(2*A))*(2*A) := mul_le_mul (hyb n) hzn (norm_nonneg _) (by positivity)
-        _ = 1 := one_div_mul_cancel (by positivity)
-    dsimp [F]
-    rw [norm_sub_rev]
-    exact hM (p n).1 (hp n).1 _ hpim hpnom
-  have hFv (v : ℝ) (hv : 0<v) :
-      Tendsto (fun n => F n ((v:ℂ)*Complex.I)) atTop (𝓝 0) := by
-    have hy0 := tendsto_nhds_of_tendsto_nhdsWithin (hy.comp hψ)
-    have hz : Tendsto (fun n => y (ψ n)*v) atTop (𝓝[>] 0) := by
-      apply tendsto_nhdsWithin_iff.mpr
-      refine ⟨?_,Eventually.of_forall (fun n => mul_pos (hyp n) hv)⟩
-      simpa only [Function.comp_apply,zero_mul] using hy0.mul_const v
-    have hu := hr.seq_tendstoUniformlyOn _ hz
-    rw [Metric.tendsto_nhds]
-    intro η hη
-    filter_upwards [Metric.tendstoUniformlyOn_iff.mp hu η hη] with n hn
-    have hh := hn (p n).1 (hp n).1
-    simpa only [F,dist_zero_right,dist_eq_norm,Complex.ofReal_mul,mul_assoc,sub_zero] using hh
-  have hu := uniform_zero_on_horizontal hC hF hFM hFv
-  have hs := Metric.tendstoUniformlyOn_iff.mp hu ε hε
-  obtain ⟨n,hn⟩ := hs.exists
-  have hseg : ((p n).2:ℂ)+Complex.I ∈ {z : ℂ | z.im=1 ∧ |z.re|≤C} := by
-    simpa only [mem_ofPred_eq,Complex.add_im,Complex.add_re,Complex.ofReal_im,Complex.ofReal_re,
-      Complex.I_im,Complex.I_re,zero_add,add_zero] using
-      (show (1:ℝ)=1 ∧ |(p n).2|≤C from ⟨rfl,abs_le.mpr (hp n).2⟩)
-  have hh := hn _ hseg
-  rw [dist_zero_left] at hh
-  exact (not_lt_of_ge (hpb n)) hh
-
-end SLEExactGauge.AngularLimits
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Metric Set Filter Topology MeasureTheory
-namespace SLEExactGauge.TraceChaining
-
- 
-
-theorem radial_uniform_limit_nhds_of_bands {X E : Type*} [MetricSpace E]
-    [CompleteSpace E] (f : X → ℝ → E) (g : X → E) (S : Set X)
-    {r : ℝ} (hr : 0<r) (hr1 : r<1) {e : ℕ → ℝ} (he : Summable e)
-    (hb : ∀ᶠ n in atTop, ∀ t∈S, ∀ y : ℝ, r^(n+1)≤y → y≤r^n →
-      dist (f t y) (f t (r^n))≤e n)
-    (hg : ∀ t∈S, Tendsto (f t) (𝓝[>] 0) (𝓝 (g t))) :
-    TendstoUniformlyOn (fun y t => f t y) g (𝓝[>] 0) S := by
-  have hu := radial_uniform_limit_of_bands f g S hr hr1 he hb hg
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  have hε2 : 0<ε/2 := by linarith
-  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp hb
-  obtain ⟨n₁,hn₁⟩ := eventually_atTop.mp
-    (he.tendsto_atTop_zero.eventually (Iio_mem_nhds hε2))
-  obtain ⟨n₂,hn₂⟩ := eventually_atTop.mp
-    (Metric.tendstoUniformlyOn_iff.mp hu (ε/2) hε2)
-  let n₃ := max n₀ (max n₁ n₂)
-  have hv : 0<r^(n₃+1) := pow_pos hr _
-  filter_upwards [self_mem_nhdsWithin,
-    (show ∀ᶠ y : ℝ in 𝓝[>] 0, y<r^(n₃+1) from
-      nhdsWithin_le_nhds (Iio_mem_nhds hv))] with y hy hyb
-  intro t ht
-  have hy1 : y≤1 := hyb.le.trans (pow_le_one₀ hr.le hr1.le)
-  obtain ⟨n,hnlow,hnup⟩ := exists_nat_pow_near_of_lt_one hy hy1 hr hr1
-  have hn3 : n₃≤n := by
-    by_contra hh
-    have hnlt : n+1≤n₃+1 := by omega
-    have hlo := pow_le_pow_of_le_one hr.le hr1.le hnlt
-    linarith
-  have hbnd := hn₀ n (le_trans (le_max_left _ _) hn3) t ht y hnlow.le hnup
-  have hen := hn₁ n (le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hn3)
-  have hgn := hn₂ n (le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hn3) t ht
-  calc
-    _ ≤ dist (g t) (f t (r^n))+dist (f t (r^n)) (f t y) := dist_triangle _ _ _
-    _ < ε := by rw [dist_comm (f t (r^n))]; linarith
-
-end SLEExactGauge.TraceChaining
-namespace SLEExactGauge.LiteralTrace
-open TraceChaining AngularLimits
-
- 
-
-theorem radial_uniform_all_windows {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W, ∀ m H : ℕ,
-      TendstoUniformlyOn (fun y t => radialValue κ b t y)
-        (fun t => trace κ b (t/κ)) (𝓝[>] 0) (Icc (1/((m:ℝ)+1)) H) := by
-  obtain ⟨N,hN,hbands⟩ := radial_uniform_bands hκ hκ8 hW
-  filter_upwards [hbands,trace_limit_all_positive hκ hκ8 hW] with b hb hlim
-  intro m H
-  apply radial_uniform_limit_nhds_of_bands (fun t => radialValue κ b t)
-    (fun t => trace κ b (t/κ)) _ (traceRatio_pos N) (traceRatio_lt_one hN)
-    (traceError_summable hκ hκ8 hN H)
-  · filter_upwards [hb m H] with n hn t ht y hy hy'
-    simpa only [dist_eq_norm] using hn t ht.1 ht.2 y hy hy'
-  · intro t ht
-    have ht0 : 0<t := (by positivity : (0:ℝ)<1/((m:ℝ)+1)).trans_le ht.1
-    have hh := hlim (t/κ) (div_pos ht0 hκ)
-    simpa only [mul_div_cancel₀ _ hκ.ne'] using hh
-
-lemma inverseFlow_shift_bound {κ t T R : ℝ} (hκ : 0<κ) (ht : 0≤t)
-    (htT : t≤T) (b : Driver) (hB : ∀ u∈Icc 0 T, |driverValue b u|≤R)
-    {z : ℂ} (hzi : 0<z.im) (hzn : ‖z‖≤1) :
-    ‖inverseFlow κ b t (driverValue b t+z)‖≤R+3+2*a κ*T := by
-  have hw : 0<((driverValue b t:ℂ)+z).im := by simpa using hzi
-  have hp := surviving_flow_positiveSolution ht (inverseFlow_surviving hκ b ht hw)
-  have hre : |(inverseFlow κ b t (driverValue b t+z)).re|≤R+1 := by
-    apply hp.initial_abs_re_le hκ ht
-    · intro u hu
-      exact (hB u ⟨hu.1,hu.2.trans htT⟩).trans (by linarith)
-    · rw [flow_inverseFlow hκ b ht hw]
-      simp only [Complex.add_re,Complex.ofReal_re]
-      exact (abs_add_le _ _).trans
-        (add_le_add (hB t ⟨ht,htT⟩) ((Complex.abs_re_le_norm z).trans hzn))
-  have hi : (inverseFlow κ b t (driverValue b t+z)).im≤1+2*a κ*T+1 :=
-    inverseFlow_im_le hκ ht htT b hw (by simpa using (Complex.im_le_norm z).trans hzn)
-  have hip := inverseFlow_im_pos hκ ht b hw
-  apply (Complex.norm_le_abs_re_add_abs_im _).trans
-  rw [abs_of_pos hip]
-  linarith
-
- 
-
-theorem angular_uniform_all_windows {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W, ∀ m H : ℕ, ∀ C : ℝ, 0≤C →
-      TendstoUniformlyOn
-        (fun y : ℝ => fun p : ℝ×ℝ => inverseFlow κ b p.1
-          (driverValue b p.1+(y:ℂ)*((p.2:ℂ)+Complex.I)))
-        (fun p => trace κ b (p.1/κ)) (𝓝[>] 0)
-        (Icc (1/((m:ℝ)+1)) H ×ˢ Icc (-C) C) := by
-  filter_upwards [radial_uniform_all_windows hκ hκ8 hW] with b hb
-  intro m H C hC
-  have hc : Continuous (driverValue b) := (map_continuous b).comp continuous_real_toNNReal
-  obtain ⟨R,hR,hRB⟩ := (isCompact_Icc.image hc).isBounded.exists_pos_norm_le
-  have hB : ∀ u∈Icc (0:ℝ) H, |driverValue b u|≤R := by
-    intro u hu
-    simpa only [Real.norm_eq_abs] using hRB _ (mem_image_of_mem _ hu)
-  let B : ℝ := R+3+2*a κ*H
-  let S := Icc (1/((m:ℝ)+1)) (H:ℝ)
-  have ht0 (t : ℝ) (ht : t∈S) : 0≤t :=
-    (by positivity : (0:ℝ)≤1/((m:ℝ)+1)).trans ht.1
-  have hg (t : ℝ) (ht : t∈S) : ‖trace κ b (t/κ)‖≤B := by
-    apply le_of_tendsto ((hb m H).tendsto_at ht).norm
-    filter_upwards [self_mem_nhdsWithin,
-      (show ∀ᶠ y : ℝ in 𝓝[>] 0, y<1 from nhdsWithin_le_nhds (Iio_mem_nhds zero_lt_one))]
-      with y hy hy1
-    exact inverseFlow_shift_bound hκ (ht0 t ht) ht.2 b hB
-      (by simpa using hy) (by simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs,
-        abs_of_pos (show 0<y from hy),Complex.norm_I,mul_one]; exact hy1.le)
-  apply uniform_angular_of_radial (fun t z => inverseFlow κ b t (driverValue b t+z))
-    (fun t => trace κ b (t/κ)) S (M := 2*B) _ _ (hb m H) hC
-  · intro t ht z hz
-    apply ((inverseFlow_hasStrictDerivAt hκ b (ht0 t ht) (by simpa using hz)).hasDerivAt.comp z
-      ((hasDerivAt_id z).const_add (driverValue b t:ℂ))).differentiableAt.differentiableWithinAt
-  · intro t ht z hzi hzn
-    exact (norm_sub_le _ _).trans (by
-      have hh := inverseFlow_shift_bound hκ (ht0 t ht) ht.2 b hB hzi hzn
-      have hh' := hg t ht
-      change ‖inverseFlow κ b t (driverValue b t+z)‖≤B at hh
-      linarith)
-
-end SLEExactGauge.LiteralTrace
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialRestart
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace ExactSLEBarrier
+variable {a : ℝ} {T : ℝ≥0} {y j q : ℝ → ℝ}
+
+lemma Evolution.logHeight_add_deriv (h : Evolution a T y j q) {t : ℝ}
+    (ht : t ∈ Icc 0 (T : ℝ)) :
+    HasDerivWithinAt (fun t => Real.log (y t)+j t)
+      (-2*a*(q t-(y t)^2)/(q t)^2) (Icc 0 (T : ℝ)) t := by
+  convert (h.logHeight_deriv ht).add (h.jd t ht) using 1; try rfl
+  field_simp [(h.qp t ht).ne']
+  ring
+
+lemma Evolution.logHeight_add_le (h : Evolution a T y j q) (ha : 0≤a)
+    (hq : ∀ u∈Icc 0 (T : ℝ), (y u)^2≤q u)
+    (t : ℝ) (ht : t∈Icc 0 (T : ℝ)) :
+    Real.log (y t)+j t≤Real.log (y 0)+j 0 := by
+  have hc : ContinuousOn (fun t => Real.log (y t)+j t) (Icc 0 (T : ℝ)) :=
+    (h.yc.log (fun u hu => (h.yp u hu).ne')).add h.jc
+  have hm : AntitoneOn (fun t => Real.log (y t)+j t) (Icc 0 (T : ℝ)) :=
+    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) hc
+      (fun u hu => (h.logHeight_add_deriv (interior_subset hu)).mono interior_subset)
+      (fun u hu => div_nonpos_of_nonpos_of_nonneg
+        (mul_nonpos_of_nonpos_of_nonneg (by linarith)
+          (sub_nonneg.mpr (hq u (interior_subset hu)))) (sq_nonneg _))
+  exact hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
+
+lemma Evolution.height_sq_le_radius (h : Evolution a T y j q) (ha : 0≤a)
+    (hq : ∀ u∈Icc 0 (T : ℝ), (y u)^2≤q u) (hj0 : j 0=0)
+    (t : ℝ) (ht : t∈Icc 0 (T : ℝ)) :
+    (y t)^2≤y 0*(y t*Real.exp (-j t)) := by
+  have hl := h.logHeight_add_le ha hq t ht
+  rw [hj0,add_zero] at hl
+  have he : y t≤y 0*Real.exp (-j t) := by
+    have hh := Real.exp_le_exp.mpr (show Real.log (y t)≤Real.log (y 0)+ -j t by linarith)
+    simpa only [Real.exp_add,Real.exp_log (h.yp t ht),
+      Real.exp_log (h.yp 0 ⟨le_rfl,T.coe_nonneg⟩)] using hh
+  have hh := mul_le_mul_of_nonneg_left he (h.yp t ht).le
+  nlinarith only [hh]
+
+lemma Evolution.height_sq_lower (h : Evolution a T y j q) (ha : 0≤a)
+    (hq : ∀ u∈Icc 0 (T : ℝ), (y u)^2≤q u)
+    (t : ℝ) (ht : t∈Icc 0 (T : ℝ)) :
+    (y 0)^2≤(y t)^2+2*a*t := by
+  let F : ℝ→ℝ := fun u => (y u)^2+2*a*u
+  have hc : ContinuousOn F (Icc 0 (T : ℝ)) :=
+    (h.yc.pow 2).add (continuousOn_const.mul continuousOn_id)
+  have hd (u : ℝ) (hu : u∈Icc 0 (T : ℝ)) :
+      HasDerivWithinAt F (2*a*(1-(y u)^2/q u)) (Icc 0 (T : ℝ)) u := by
+    convert ((h.yd u hu).pow 2).add ((hasDerivWithinAt_id u _).const_mul (2*a)) using 1 <;> try rfl
+    simp only [Nat.reduceSub,pow_one,Nat.cast_ofNat]
+    ring
+  have hm : MonotoneOn F (Icc 0 (T : ℝ)) :=
+    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 (T : ℝ)) hc
+      (fun u hu => (hd u (interior_subset hu)).mono interior_subset)
+      (fun u hu => mul_nonneg (by positivity) (sub_nonneg.mpr
+        ((div_le_one (h.qp u (interior_subset hu))).mpr (hq u (interior_subset hu)))))
+  simpa only [F,mul_zero,add_zero] using hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
+
+end ExactSLEBarrier
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator
+
+ 
+
+lemma actualState_evolution (a : ℝ) (b : ExactSLEGenerator.Driver) (z : UpperPlane)
+    (T : ℝ≥0) (hT : alive a b.1 T z.1) :
+    ExactSLEBarrier.Evolution a T
+      (fun u => Y (actualState a b.1 u.toNNReal z.1))
+      (fun u => J (actualState a b.1 u.toNNReal z.1))
+      (fun u => Q (actualState a b.1 u.toNNReal z.1)) := by
+  obtain ⟨n,hn⟩ := (exhaustionStop_eventually a z b T hT).exists
+  let ε := exhaustionHeight z n
+  let V := localState a ε (exhaustionHeight_pos z n) z.1
+  have hsafe (u : ℝ≥0) (hu : u≤T) : ε≤Y (V u b) :=
+    (regionStop_safe a ε (exhaustionHeight_pos z n) z.1 (exhaustionNorm z n) 0 0 n
+      (exhaustionRegion_initial z n) b u (hu.trans hn)).1
+  have he (u : ℝ) (hu : u∈Icc 0 (T : ℝ)) :
+      V u.toNNReal b=actualState a b.1 u.toNNReal z.1 :=
+    (exhaustionStop_agrees a z n b u.toNNReal
+      ((show u.toNNReal≤T by simpa using Real.toNNReal_le_toNNReal hu.2).trans hn)).2.symm
+  dsimp only [V] at he
+  have h := localState_evolution a ε (exhaustionHeight_pos z n) z.1 b T hsafe
+  refine ⟨?_,?_,?_,?_,?_,?_⟩
+  · exact h.yc.congr (fun u hu => congrArg Y (he u hu).symm)
+  · exact h.jc.congr (fun u hu => congrArg J (he u hu).symm)
+  · intro u hu; simpa only [←he u hu] using h.yp u hu
+  · intro u hu; simpa only [←he u hu] using h.qp u hu
+  · intro u hu
+    have hh := (h.yd u hu).congr_of_mem (fun v hv => congrArg Y (he v hv).symm) hu
+    simpa only [he u hu] using hh
+  · intro u hu
+    have hh := (h.jd u hu).congr_of_mem (fun v hv => congrArg J (he v hv).symm) hu
+    simpa only [he u hu] using hh
+
+lemma actual_height_sq_le_radius (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
+    (z : UpperPlane) (t : ℝ≥0) (ht : alive a b.1 t z.1) :
+    (Y (actualState a b.1 t z.1))^2≤z.1.im*physicalRadius (actualState a b.1 t z.1) := by
+  have h := (actualState_evolution a b z t ht).height_sq_le_radius ha.le
+    (fun u _ => by dsimp [Q]; nlinarith only [sq_nonneg (X (actualState a b.1 u.toNNReal z.1))])
+    (by simp [actualState_zero ha b z.2]) (t : ℝ) ⟨t.coe_nonneg,le_rfl⟩
+  simpa only [Real.toNNReal_coe,Real.toNNReal_zero,actualState_zero ha b z.2,Y_apply,
+    physicalRadius] using h
+
+lemma actual_height_sq_lower (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
+    (z : UpperPlane) (t : ℝ≥0) (ht : alive a b.1 t z.1) :
+    z.1.im^2≤(Y (actualState a b.1 t z.1))^2+2*a*t := by
+  have h := (actualState_evolution a b z t ht).height_sq_lower ha.le
+    (fun u _ => by dsimp [Q]; nlinarith only [sq_nonneg (X (actualState a b.1 u.toNNReal z.1))])
+    (t : ℝ) ⟨t.coe_nonneg,le_rfl⟩
+  simpa only [Real.toNNReal_coe,Real.toNNReal_zero,actualState_zero ha b z.2,Y_apply] using h
+
+ 
+
+lemma level_height_bound (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
+    (z : UpperPlane) (s : ℝ) (hs : 0≤ s)
+    (hfin : radialTime a b.1 z.1 (levelRadius a (2*z.1.im) s)≠∞) :
+    Y (actualState a b.1 (radialTime a b.1 z.1 (levelRadius a (2*z.1.im) s)).toNNReal z.1)≤
+      z.1.im*Real.exp (-a*s) := by
+  have hcap : 0<2*z.1.im := mul_pos (by norm_num) z.2
+  have hr : levelRadius a (2*z.1.im) s≤z.1.im := by
+    simpa using levelRadius_le ha hcap hs
+  have ht := radialTime_attained a ha z.1 z.2 _ (levelRadius_pos _ _ _ hcap) hr b hfin
+  have hh := actual_height_sq_le_radius a ha b z _ ht.1
+  rw [ht.2] at hh
+  have hexp : Real.exp (-2*a*s)=(Real.exp (-a*s))^2 := by
+    rw [←Real.exp_nat_mul]; congr 1; ring
+  conv_rhs at hh =>
+    simp only [levelRadius,mul_div_cancel_left₀ _ (by norm_num : (2:ℝ)≠0),hexp]
+  have hpos : 0<z.1.im*Real.exp (-a*s) := mul_pos z.2 (Real.exp_pos _)
+  nlinarith only [hh,hpos]
+
+ 
+
+lemma radialTime_half_lower (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
+    (z : UpperPlane) (r : ℝ) (hr : 0<r) (hr2 : r≤z.1.im/2)
+    (hfin : radialTime a b.1 z.1 r≠∞) :
+    z.1.im^2/(4*a)≤((radialTime a b.1 z.1 r).toNNReal : ℝ) := by
+  have ht := radialTime_attained a ha z.1 z.2 r hr (by linarith [z.2]) b hfin
+  have hl := actual_height_sq_lower a ha b z _ ht.1
+  have hu := actual_height_sq_le_radius a ha b z _ ht.1
+  rw [ht.2] at hu
+  have hh := mul_le_mul_of_nonneg_left hr2 z.2.le
+  apply (div_le_iff₀ (by positivity : 0<4*a)).mpr
+  nlinarith only [hl,hu,hh]
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Metric Set Filter Topology
+open scoped NNReal BoundedContinuousFunction
+namespace SLEExactGauge.AnalyticCompactness
+
+lemma lipschitzOn_subball {f : ℂ → ℂ} {c : ℂ} {r R M : ℝ}
+    (hr : 0 ≤ r) (hrR : r < R)
+    (hf : DifferentiableOn ℂ f (ball c R))
+    (hM : ∀ z ∈ ball c R, ‖f z‖ ≤ M) :
+    LipschitzOnWith (2*M/(R-r)).toNNReal f (closedBall c r) := by
+  have hsub : closedBall c r ⊆ ball c R := closedBall_subset_ball hrR
+  have hM0 : 0 ≤ M := (norm_nonneg _).trans (hM c (mem_ball_self (hr.trans_lt hrR)))
+  apply (convex_closedBall c r).lipschitzOnWith_of_nnnorm_deriv_le
+  · intro z hz
+    exact (hf z (hsub hz)).differentiableAt (isOpen_ball.mem_nhds (hsub hz))
+  · intro z hz
+    have hb : ball z (R-r) ⊆ ball c R := by
+      intro w hw
+      exact (dist_triangle w z c).trans_lt (by
+        have hw' : dist w z < R-r := hw
+        have hz' : dist z c ≤ r := hz
+        linarith)
+    have hh := Complex.norm_deriv_le_div_of_mapsTo_ball (hf.mono hb)
+      (R₂ := 2*M) (by
+        intro w hw
+        change dist (f w) (f z) ≤ 2*M
+        rw [dist_eq_norm]
+        exact (norm_sub_le _ _).trans (by linarith [hM w (hb hw),hM z (hsub hz)]))
+      (sub_pos.mpr hrR)
+    exact_mod_cast (hh.trans (Real.le_coe_toNNReal _))
+
+ 
+
+theorem exists_subseq_uniform {F : ℕ → ℂ → ℂ} {c : ℂ} {r R M : ℝ}
+    (hr : 0 ≤ r) (hrR : r < R)
+    (hF : ∀ n, DifferentiableOn ℂ (F n) (ball c R))
+    (hM : ∀ n z, z ∈ ball c R → ‖F n z‖ ≤ M) :
+    ∃ g : ℂ → ℂ, ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      TendstoUniformlyOn (fun n => F (φ n)) g atTop (closedBall c r) ∧
+      DifferentiableOn ℂ g (ball c r) := by
+  classical
+  let K := closedBall c r
+  have : CompactSpace K := isCompact_iff_compactSpace.mp (isCompact_closedBall c r)
+  have hl (n : ℕ) := lipschitzOn_subball hr hrR (hF n) (hM n)
+  let f : ℕ → K →ᵇ ℂ := fun n => BoundedContinuousFunction.mkOfCompact
+    ⟨fun z => F n z, (hl n).continuousOn.domRestrict⟩
+  have hlf (n : ℕ) : LipschitzWith (2*M/(R-r)).toNNReal (f n) :=
+    lipschitzOnWith_iff_restrict.mp (hl n)
+  have heq : Equicontinuous ((↑) : range f → K → ℂ) := by
+    apply UniformEquicontinuous.equicontinuous
+    apply LipschitzWith.uniformEquicontinuous _ (2*M/(R-r)).toNNReal
+    rintro ⟨g,⟨n,rfl⟩⟩
+    exact hlf n
+  have hc := BoundedContinuousFunction.arzela_ascoli (closedBall (0:ℂ) M)
+    (isCompact_closedBall _ _) (range f) (by
+      rintro g z ⟨n,rfl⟩
+      change dist (F n z) 0 ≤ M
+      simpa only [dist_zero_right] using
+        hM n z (closedBall_subset_ball hrR z.2)) heq
+  obtain ⟨g,-,φ,hφ,hg⟩ := hc.tendsto_subseq (fun n => subset_closure (mem_range_self n))
+  let g' : ℂ → ℂ := fun z => if hz : z ∈ K then g ⟨z,hz⟩ else 0
+  have he (z : ℂ) (hz : z ∈ K) : g' z = g ⟨z,hz⟩ := dite_eq_left hz
+  have hu : TendstoUniformlyOn (fun n => F (φ n)) g' atTop K := by
+    rw [Metric.tendstoUniformlyOn_iff]
+    intro ε hε
+    have hh := Metric.tendstoUniformly_iff.mp
+      (BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hg) ε hε
+    filter_upwards [hh] with n hn z hz
+    rw [he z hz]
+    exact hn ⟨z,hz⟩
+  refine ⟨g',φ,hφ,hu,?_⟩
+  exact (hu.mono ball_subset_closedBall).tendstoLocallyUniformlyOn.differentiableOn
+    (Eventually.of_forall (fun n => (hF (φ n)).mono (ball_subset_ball hrR.le))) isOpen_ball
+
+ 
+
+theorem uniform_zero_of_accumulation {F : ℕ → ℂ → ℂ} {c : ℂ} {r R M : ℝ}
+    (hr : 0 < r) (hrR : r < R)
+    (hF : ∀ n, DifferentiableOn ℂ (F n) (ball c R))
+    (hM : ∀ n z, z ∈ ball c R → ‖F n z‖ ≤ M)
+    (z : ℕ → ℂ) (hz : ∀ j, z j ∈ closedBall c r)
+    (hzc : ∀ j, z j ≠ c) (hzlim : Tendsto z atTop (𝓝 c))
+    (hFz : ∀ j, Tendsto (fun n => F n (z j)) atTop (𝓝 0)) :
+    TendstoUniformlyOn F (fun _ => 0) atTop (closedBall c r) := by
+  classical
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  by_contra hbad
+  have hb : ∃ᶠ n in atTop, ∃ w ∈ closedBall c r, ε ≤ ‖F n w‖ := by
+    simpa only [not_forall,exists_prop,not_lt,dist_zero_left] using (show ∃ᶠ n in atTop,
+      ¬∀ w ∈ closedBall c r, dist (0:ℂ) (F n w)<ε from Filter.not_eventually.mp hbad)
+  obtain ⟨ψ,hψ,hψbad⟩ := subseq_forall_of_frequently (x := fun n : ℕ => n)
+    (p := fun n => ∃ w ∈ closedBall c r, ε ≤ ‖F n w‖) tendsto_id hb
+  let s : ℝ := (r+R)/2
+  have hrs : r<s := by dsimp [s]; linarith
+  have hsR : s<R := by dsimp [s]; linarith
+  have hs : 0<s := hr.trans hrs
+  obtain ⟨g,φ,hφ,hg,hgd⟩ := exists_subseq_uniform hs.le hsR
+    (fun n => hF (ψ n)) (fun n => hM (ψ n))
+  have hgzero (j : ℕ) : g (z j)=0 := by
+    have hpt := hg.tendsto_at (closedBall_subset_closedBall hrs.le (hz j))
+    exact tendsto_nhds_unique hpt ((hFz j).comp (hψ.comp hφ.tendsto_atTop))
+  have hcl : c∈closure ({w : ℂ | g w=0} \ {c}) := by
+    apply mem_closure_of_tendsto hzlim
+    exact Eventually.of_forall (fun j => ⟨hgzero j,hzc j⟩)
+  have heq : EqOn g (fun _ => 0) (ball c s) :=
+    hgd.analyticOnNhd isOpen_ball |>.eqOn_of_preconnected_of_mem_closure
+      analyticOnNhd_const (convex_ball c s).isPreconnected (mem_ball_self hs) hcl
+  have hsmall := Metric.tendstoUniformlyOn_iff.mp hg ε hε
+  obtain ⟨n,hn⟩ := hsmall.exists
+  obtain ⟨w,hw,hwbad⟩ := hψbad (φ n)
+  have hh := hn w (closedBall_subset_closedBall hrs.le hw)
+  rw [heq (closedBall_subset_ball hrs hw),dist_zero_left] at hh
+  exact (not_lt_of_ge hwbad) hh
+
+ 
+
+theorem uniform_zero_on_horizontal {F : ℕ → ℂ → ℂ} {C M : ℝ} (hC : 0 ≤ C)
+    (hF : ∀ n, DifferentiableOn ℂ (F n) {z : ℂ | 0 < z.im})
+    (hM : ∀ n z, z ∈ ball (((C^2+2 : ℝ) : ℂ)*Complex.I) (C^2+2-1/4) →
+      ‖F n z‖ ≤ M)
+    (hv : ∀ y : ℝ, 0 < y → Tendsto (fun n => F n ((y:ℂ)*Complex.I)) atTop (𝓝 0)) :
+    TendstoUniformlyOn F (fun _ => 0) atTop {z : ℂ | z.im=1 ∧ |z.re|≤C} := by
+  let A : ℝ := C^2+2
+  let c : ℂ := (A:ℂ)*Complex.I
+  let r : ℝ := A-1/2
+  let R : ℝ := A-1/4
+  have hA : 2 ≤ A := by dsimp [A]; nlinarith [sq_nonneg C]
+  have hr : 0 < r := by dsimp [r]; linarith
+  have hrR : r < R := by dsimp [r,R]; linarith
+  have hcim : c.im=A := by simp [c]
+  have hcre : c.re=0 := by simp [c]
+  have hball : ball c R ⊆ {z : ℂ | 0 < z.im} := by
+    intro z hz
+    have hh := Complex.im_le_norm (c-z)
+    rw [Complex.sub_im,hcim,norm_sub_rev,← dist_eq_norm] at hh
+    have hz' : dist z c < A-1/4 := hz
+    change 0 < z.im
+    linarith
+  let z : ℕ → ℂ := fun j => c + ((1/((j:ℝ)+2):ℝ):ℂ)*Complex.I
+  have hz (j : ℕ) : z j∈closedBall c r := by
+    have hj : 0 < 1/((j:ℝ)+2) := by positivity
+    have hj1 : 1/((j:ℝ)+2) ≤ 1/2 := by
+      apply one_div_le_one_div_of_le (by norm_num)
+      linarith [Nat.cast_nonneg (α := ℝ) j]
+    rw [mem_closedBall,dist_eq_norm]
+    simp only [z,add_sub_cancel_left,norm_mul,Complex.norm_real,
+      Real.norm_eq_abs,abs_of_pos hj,Complex.norm_I,mul_one]
+    dsimp [r]
+    linarith
+  have hzc (j : ℕ) : z j≠c := by
+    intro h
+    have hh := congrArg Complex.im h
+    simp only [z,Complex.add_im,Complex.mul_I_im,Complex.ofReal_re] at hh
+    have hp : 0 < 1/((j:ℝ)+2) := by positivity
+    linarith
+  have hzlim : Tendsto z atTop (𝓝 c) := by
+    have hh : Tendsto (fun j : ℕ => 1/((j:ℝ)+2)) atTop (𝓝 0) :=
+      tendsto_const_nhds.div_atTop (tendsto_atTop_add_const_right atTop 2 tendsto_natCast_atTop_atTop)
+    have h' := (Complex.continuous_ofReal.tendsto 0 |>.comp hh).mul_const Complex.I |>.const_add c
+    simpa only [z,Function.comp_apply,Complex.ofReal_zero,zero_mul,add_zero] using h'
+  have hFz (j : ℕ) : Tendsto (fun n => F n (z j)) atTop (𝓝 0) := by
+    have he : z j = (((A+1/((j:ℝ)+2):ℝ):ℂ)*Complex.I) := by
+      dsimp [z,c]
+      push_cast
+      ring
+    rw [he]
+    apply hv
+    have hp : 0 < 1/((j:ℝ)+2) := by positivity
+    linarith
+  apply (uniform_zero_of_accumulation hr hrR (fun n => (hF n).mono hball)
+    hM z hz hzc hzlim hFz).mono
+  intro w hw
+  rw [mem_closedBall,dist_eq_norm]
+  have hsq : ‖w-c‖^2=w.re^2+(1-A)^2 := by
+    rw [← Complex.normSq_eq_norm_sq,Complex.normSq_apply]
+    simp only [Complex.sub_re,Complex.sub_im,hcim,hcre,sub_zero,hw.1]
+    ring
+  have hsq2 : w.re^2 ≤ C^2 := by
+    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg w.re) hC).mpr hw.2
+  have hbound : ‖w-c‖^2 ≤ r^2 := by
+    rw [hsq]
+    dsimp [r,A] at *
+    nlinarith
+  exact (sq_le_sq₀ (norm_nonneg _) hr.le).mp hbound
+
+end SLEExactGauge.AnalyticCompactness
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Metric Set Filter Topology
+namespace SLEExactGauge.AngularLimits
+open AnalyticCompactness
+
+ 
+
+theorem uniform_angular_of_radial {X : Type*} (f : X → ℂ → ℂ) (g : X → ℂ)
+    (S : Set X) {M : ℝ}
+    (hf : ∀ x∈S, DifferentiableOn ℂ (f x) {z : ℂ | 0<z.im})
+    (hM : ∀ x∈S, ∀ z : ℂ, 0<z.im → ‖z‖≤1 → ‖f x z-g x‖≤M)
+    (hr : TendstoUniformlyOn (fun y : ℝ => fun x => f x ((y:ℂ)*Complex.I))
+      g (𝓝[>] 0) S) {C : ℝ} (hC : 0≤C) :
+    TendstoUniformlyOn
+      (fun y : ℝ => fun p : X×ℝ => f p.1 ((y:ℂ)*((p.2:ℂ)+Complex.I)))
+      (fun p => g p.1) (𝓝[>] 0) (S ×ˢ Icc (-C) C) := by
+  classical
+  apply tendstoUniformlyOn_of_seq_tendstoUniformlyOn
+  intro y hy
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  by_contra hbad
+  let A : ℝ := C^2+2
+  have hA : 0<A := by dsimp [A]; positivity
+  have hb : ∃ᶠ n in atTop, ∃ p∈S ×ˢ Icc (-C) C,
+      ε≤‖g p.1-f p.1 ((y n:ℂ)*((p.2:ℂ)+Complex.I))‖ := by
+    simpa only [not_forall,exists_prop,not_lt,dist_eq_norm] using not_eventually.mp hbad
+  have he : ∀ᶠ n in atTop, 0<y n ∧ y n≤1/(2*A) := by
+    filter_upwards [hy.eventually self_mem_nhdsWithin,
+      (tendsto_nhds_of_tendsto_nhdsWithin hy).eventually
+        (Iio_mem_nhds (by positivity : (0:ℝ)<1/(2*A)))] with n hn hn'
+    exact ⟨hn,hn'.le⟩
+  have hb' := hb.and_eventually he
+  obtain ⟨ψ,hψ,hψbad⟩ := subseq_forall_of_frequently
+    (x := fun n : ℕ => n)
+    (p := fun n => (∃ p∈S ×ˢ Icc (-C) C,
+      ε≤‖g p.1-f p.1 ((y n:ℂ)*((p.2:ℂ)+Complex.I))‖) ∧
+        0<y n ∧ y n≤1/(2*A)) tendsto_id hb'
+  choose p hp hpb using (fun n => (hψbad n).1)
+  have hyp (n : ℕ) : 0<y (ψ n) := (hψbad n).2.1
+  have hyb (n : ℕ) : y (ψ n)≤1/(2*A) := (hψbad n).2.2
+  let F : ℕ → ℂ → ℂ := fun n z => g (p n).1-f (p n).1 ((y (ψ n):ℂ)*z)
+  have hF (n : ℕ) : DifferentiableOn ℂ (F n) {z : ℂ | 0<z.im} := by
+    apply (differentiableOn_const _).sub
+    apply (hf (p n).1 (hp n).1).comp
+      ((differentiable_const _).mul differentiable_id).differentiableOn
+    intro z hz
+    change 0<((y (ψ n):ℂ)*z).im
+    simpa only [Complex.mul_im,Complex.ofReal_im,Complex.ofReal_re,zero_mul,add_zero] using
+      mul_pos (hyp n) hz
+  have hFM (n : ℕ) (z : ℂ)
+      (hz : z∈ball ((A:ℂ)*Complex.I) (A-1/4)) : ‖F n z‖≤M := by
+    have hzi : 0<z.im := by
+      have hh := Complex.im_le_norm ((A:ℂ)*Complex.I-z)
+      simp only [Complex.sub_im,Complex.mul_I_im,Complex.ofReal_re] at hh
+      rw [norm_sub_rev,← dist_eq_norm] at hh
+      have hz' : dist z ((A:ℂ)*Complex.I)<A-1/4 := hz
+      linarith
+    have hzn : ‖z‖≤2*A := by
+      have hnorm := norm_add_le (z-(A:ℂ)*Complex.I) ((A:ℂ)*Complex.I)
+      rw [sub_add_cancel,← dist_eq_norm] at hnorm
+      simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hA,
+        Complex.norm_I,mul_one] at hnorm
+      have hz' : dist z ((A:ℂ)*Complex.I)<A-1/4 := hz
+      linarith
+    have hpim : 0<((y (ψ n):ℂ)*z).im := by
+      simpa only [Complex.mul_im,Complex.ofReal_im,Complex.ofReal_re,zero_mul,add_zero] using
+        mul_pos (hyp n) hzi
+    have hpnom : ‖(y (ψ n):ℂ)*z‖≤1 := by
+      rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (hyp n)]
+      calc
+        _ ≤ (1/(2*A))*(2*A) := mul_le_mul (hyb n) hzn (norm_nonneg _) (by positivity)
+        _ = 1 := one_div_mul_cancel (by positivity)
+    dsimp [F]
+    rw [norm_sub_rev]
+    exact hM (p n).1 (hp n).1 _ hpim hpnom
+  have hFv (v : ℝ) (hv : 0<v) :
+      Tendsto (fun n => F n ((v:ℂ)*Complex.I)) atTop (𝓝 0) := by
+    have hy0 := tendsto_nhds_of_tendsto_nhdsWithin (hy.comp hψ)
+    have hz : Tendsto (fun n => y (ψ n)*v) atTop (𝓝[>] 0) := by
+      apply tendsto_nhdsWithin_iff.mpr
+      refine ⟨?_,Eventually.of_forall (fun n => mul_pos (hyp n) hv)⟩
+      simpa only [Function.comp_apply,zero_mul] using hy0.mul_const v
+    have hu := hr.seq_tendstoUniformlyOn _ hz
+    rw [Metric.tendsto_nhds]
+    intro η hη
+    filter_upwards [Metric.tendstoUniformlyOn_iff.mp hu η hη] with n hn
+    have hh := hn (p n).1 (hp n).1
+    simpa only [F,dist_zero_right,dist_eq_norm,Complex.ofReal_mul,mul_assoc,sub_zero] using hh
+  have hu := uniform_zero_on_horizontal hC hF hFM hFv
+  have hs := Metric.tendstoUniformlyOn_iff.mp hu ε hε
+  obtain ⟨n,hn⟩ := hs.exists
+  have hseg : ((p n).2:ℂ)+Complex.I ∈ {z : ℂ | z.im=1 ∧ |z.re|≤C} := by
+    simpa only [mem_ofPred_eq,Complex.add_im,Complex.add_re,Complex.ofReal_im,Complex.ofReal_re,
+      Complex.I_im,Complex.I_re,zero_add,add_zero] using
+      (show (1:ℝ)=1 ∧ |(p n).2|≤C from ⟨rfl,abs_le.mpr (hp n).2⟩)
+  have hh := hn _ hseg
+  rw [dist_zero_left] at hh
+  exact (not_lt_of_ge (hpb n)) hh
+
+end SLEExactGauge.AngularLimits
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Metric Set Filter Topology MeasureTheory
+namespace SLEExactGauge.TraceChaining
+
+ 
+
+theorem radial_uniform_limit_nhds_of_bands {X E : Type*} [MetricSpace E]
+    [CompleteSpace E] (f : X → ℝ → E) (g : X → E) (S : Set X)
+    {r : ℝ} (hr : 0<r) (hr1 : r<1) {e : ℕ → ℝ} (he : Summable e)
+    (hb : ∀ᶠ n in atTop, ∀ t∈S, ∀ y : ℝ, r^(n+1)≤y → y≤r^n →
+      dist (f t y) (f t (r^n))≤e n)
+    (hg : ∀ t∈S, Tendsto (f t) (𝓝[>] 0) (𝓝 (g t))) :
+    TendstoUniformlyOn (fun y t => f t y) g (𝓝[>] 0) S := by
+  have hu := radial_uniform_limit_of_bands f g S hr hr1 he hb hg
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  have hε2 : 0<ε/2 := by linarith
+  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp hb
+  obtain ⟨n₁,hn₁⟩ := eventually_atTop.mp
+    (he.tendsto_atTop_zero.eventually (Iio_mem_nhds hε2))
+  obtain ⟨n₂,hn₂⟩ := eventually_atTop.mp
+    (Metric.tendstoUniformlyOn_iff.mp hu (ε/2) hε2)
+  let n₃ := max n₀ (max n₁ n₂)
+  have hv : 0<r^(n₃+1) := pow_pos hr _
+  filter_upwards [self_mem_nhdsWithin,
+    (show ∀ᶠ y : ℝ in 𝓝[>] 0, y<r^(n₃+1) from
+      nhdsWithin_le_nhds (Iio_mem_nhds hv))] with y hy hyb
+  intro t ht
+  have hy1 : y≤1 := hyb.le.trans (pow_le_one₀ hr.le hr1.le)
+  obtain ⟨n,hnlow,hnup⟩ := exists_nat_pow_near_of_lt_one hy hy1 hr hr1
+  have hn3 : n₃≤n := by
+    by_contra hh
+    have hnlt : n+1≤n₃+1 := by omega
+    have hlo := pow_le_pow_of_le_one hr.le hr1.le hnlt
+    linarith
+  have hbnd := hn₀ n (le_trans (le_max_left _ _) hn3) t ht y hnlow.le hnup
+  have hen := hn₁ n (le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hn3)
+  have hgn := hn₂ n (le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hn3) t ht
+  calc
+    _ ≤ dist (g t) (f t (r^n))+dist (f t (r^n)) (f t y) := dist_triangle _ _ _
+    _ < ε := by rw [dist_comm (f t (r^n))]; linarith
+
+end SLEExactGauge.TraceChaining
+namespace SLEExactGauge.LiteralTrace
+open TraceChaining AngularLimits
+
+ 
+
+theorem radial_uniform_all_windows {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W, ∀ m H : ℕ,
+      TendstoUniformlyOn (fun y t => radialValue κ b t y)
+        (fun t => trace κ b (t/κ)) (𝓝[>] 0) (Icc (1/((m:ℝ)+1)) H) := by
+  obtain ⟨N,hN,hbands⟩ := radial_uniform_bands hκ hκ8 hW
+  filter_upwards [hbands,trace_limit_all_positive hκ hκ8 hW] with b hb hlim
+  intro m H
+  apply radial_uniform_limit_nhds_of_bands (fun t => radialValue κ b t)
+    (fun t => trace κ b (t/κ)) _ (traceRatio_pos N) (traceRatio_lt_one hN)
+    (traceError_summable hκ hκ8 hN H)
+  · filter_upwards [hb m H] with n hn t ht y hy hy'
+    simpa only [dist_eq_norm] using hn t ht.1 ht.2 y hy hy'
+  · intro t ht
+    have ht0 : 0<t := (by positivity : (0:ℝ)<1/((m:ℝ)+1)).trans_le ht.1
+    have hh := hlim (t/κ) (div_pos ht0 hκ)
+    simpa only [mul_div_cancel₀ _ hκ.ne'] using hh
+
+lemma inverseFlow_shift_bound {κ t T R : ℝ} (hκ : 0<κ) (ht : 0≤t)
+    (htT : t≤T) (b : Driver) (hB : ∀ u∈Icc 0 T, |driverValue b u|≤R)
+    {z : ℂ} (hzi : 0<z.im) (hzn : ‖z‖≤1) :
+    ‖inverseFlow κ b t (driverValue b t+z)‖≤R+3+2*a κ*T := by
+  have hw : 0<((driverValue b t:ℂ)+z).im := by simpa using hzi
+  have hp := surviving_flow_positiveSolution ht (inverseFlow_surviving hκ b ht hw)
+  have hre : |(inverseFlow κ b t (driverValue b t+z)).re|≤R+1 := by
+    apply hp.initial_abs_re_le hκ ht
+    · intro u hu
+      exact (hB u ⟨hu.1,hu.2.trans htT⟩).trans (by linarith)
+    · rw [flow_inverseFlow hκ b ht hw]
+      simp only [Complex.add_re,Complex.ofReal_re]
+      exact (abs_add_le _ _).trans
+        (add_le_add (hB t ⟨ht,htT⟩) ((Complex.abs_re_le_norm z).trans hzn))
+  have hi : (inverseFlow κ b t (driverValue b t+z)).im≤1+2*a κ*T+1 :=
+    inverseFlow_im_le hκ ht htT b hw (by simpa using (Complex.im_le_norm z).trans hzn)
+  have hip := inverseFlow_im_pos hκ ht b hw
+  apply (Complex.norm_le_abs_re_add_abs_im _).trans
+  rw [abs_of_pos hip]
+  linarith
+
+ 
+
+theorem angular_uniform_all_windows {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W, ∀ m H : ℕ, ∀ C : ℝ, 0≤C →
+      TendstoUniformlyOn
+        (fun y : ℝ => fun p : ℝ×ℝ => inverseFlow κ b p.1
+          (driverValue b p.1+(y:ℂ)*((p.2:ℂ)+Complex.I)))
+        (fun p => trace κ b (p.1/κ)) (𝓝[>] 0)
+        (Icc (1/((m:ℝ)+1)) H ×ˢ Icc (-C) C) := by
+  filter_upwards [radial_uniform_all_windows hκ hκ8 hW] with b hb
+  intro m H C hC
+  have hc : Continuous (driverValue b) := (map_continuous b).comp continuous_real_toNNReal
+  obtain ⟨R,hR,hRB⟩ := (isCompact_Icc.image hc).isBounded.exists_pos_norm_le
+  have hB : ∀ u∈Icc (0:ℝ) H, |driverValue b u|≤R := by
+    intro u hu
+    simpa only [Real.norm_eq_abs] using hRB _ (mem_image_of_mem _ hu)
+  let B : ℝ := R+3+2*a κ*H
+  let S := Icc (1/((m:ℝ)+1)) (H:ℝ)
+  have ht0 (t : ℝ) (ht : t∈S) : 0≤t :=
+    (by positivity : (0:ℝ)≤1/((m:ℝ)+1)).trans ht.1
+  have hg (t : ℝ) (ht : t∈S) : ‖trace κ b (t/κ)‖≤B := by
+    apply le_of_tendsto ((hb m H).tendsto_at ht).norm
+    filter_upwards [self_mem_nhdsWithin,
+      (show ∀ᶠ y : ℝ in 𝓝[>] 0, y<1 from nhdsWithin_le_nhds (Iio_mem_nhds zero_lt_one))]
+      with y hy hy1
+    exact inverseFlow_shift_bound hκ (ht0 t ht) ht.2 b hB
+      (by simpa using hy) (by simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs,
+        abs_of_pos (show 0<y from hy),Complex.norm_I,mul_one]; exact hy1.le)
+  apply uniform_angular_of_radial (fun t z => inverseFlow κ b t (driverValue b t+z))
+    (fun t => trace κ b (t/κ)) S (M := 2*B) _ _ (hb m H) hC
+  · intro t ht z hz
+    apply ((inverseFlow_hasStrictDerivAt hκ b (ht0 t ht) (by simpa using hz)).hasDerivAt.comp z
+      ((hasDerivAt_id z).const_add (driverValue b t:ℂ))).differentiableAt.differentiableWithinAt
+  · intro t ht z hzi hzn
+    exact (norm_sub_le _ _).trans (by
+      have hh := inverseFlow_shift_bound hκ (ht0 t ht) ht.2 b hB hzi hzn
+      have hh' := hg t ht
+      change ‖inverseFlow κ b t (driverValue b t+z)‖≤B at hh
+      linarith)
+
+end SLEExactGauge.LiteralTrace
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/InitialTrace.lean b/lean/OAI/Probability/SLEGauge/InitialTrace.lean
index b6da55ca53eb0bd59b0e55336cfd1d32b256309e..769246d9bca47f9673ad66235beedcd14d8e23b0 100644
--- a/lean/OAI/Probability/SLEGauge/InitialTrace.lean
+++ b/lean/OAI/Probability/SLEGauge/InitialTrace.lean
@@ -1,409 +1,409 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TraceBoundary
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-noncomputable section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.LiteralTrace
-open TraceChaining
-lemma PositiveSolution.initial_abs_re_le {κ : ℝ} {b : Driver} {z : ℂ} {T R : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) (hT : 0 ≤ T)
-    (hB : ∀ u ∈ Icc 0 T, |driverValue b u| ≤ R) (hfT : |(f T).re| ≤ R) :
-    |z.re| ≤ R := by
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hn (u : ℝ) (hu : u ∈ Icc 0 T) : 0 < ‖f u-driverValue b u‖^2 :=
-    sq_pos_of_pos (norm_pos_iff.mpr (hf.shifted_ne_zero hu))
-  apply abs_le.mpr
-  constructor
-  · have hh := initial_le_of_outward_deriv (R := R) hT
-      (Complex.continuous_re.comp_continuousOn hf.2.1).neg
-      (fun u hu => (hf.re_hasDeriv hu).neg) (by
-        change -(f T).re ≤ R
-        linarith [(abs_le.mp hfT).1])
-      (by
-        intro u hu hg
-        change R < -(f u).re at hg
-        have hBu := (abs_le.mp (hB u hu)).1
-        have hm : 0 < -(a κ*((f u).re-driverValue b u)) := by nlinarith
-        simpa only [neg_div] using div_pos hm (hn u hu))
-    change -(f 0).re ≤ R at hh
-    rw [hf.2.2.1] at hh
-    linarith
-  · have hh := initial_le_of_outward_deriv (R := R) hT
-      (Complex.continuous_re.comp_continuousOn hf.2.1)
-      (fun u hu => hf.re_hasDeriv hu) (abs_le.mp hfT).2
-      (by
-        intro u hu hg
-        change R < (f u).re at hg
-        have hBu := (abs_le.mp (hB u hu)).2
-        exact div_pos (mul_pos ha (by linarith)) (hn u hu))
-    change (f 0).re ≤ R at hh
-    simpa only [hf.2.2.1] using hh
-
-lemma inverseFlow_abs_re_le {κ T R y : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
-    (hy : 0 < y) (b : Driver) (hB : ∀ u ∈ Icc 0 T, |driverValue b u| ≤ R) :
-    |(radialValue κ b T y).re| ≤ R := by
-  have hw : 0 < ((driverValue b T:ℂ)+y*Complex.I).im := by simpa using hy
-  have hf := surviving_flow_positiveSolution hT (inverseFlow_surviving hκ b hT hw)
-  apply hf.initial_abs_re_le hκ hT hB
-  rw [flow_inverseFlow hκ b hT hw]
-  simpa using hB T (right_mem_Icc.mpr hT)
-
-lemma radial_limit_early_bound {κ T R : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
-    (hB : ∀ u ∈ Icc 0 T, |driverValue b u| ≤ R) {z : ℂ}
-    (hz : Tendsto (radialValue κ b T) (𝓝[>] 0) (𝓝 z)) :
-    ‖z‖ ≤ R+Real.sqrt (2*a κ*T) := by
-  have hre : |z.re| ≤ R := by
-    apply le_of_tendsto ((Complex.continuous_re.tendsto z).comp hz |>.abs)
-    filter_upwards [self_mem_nhdsWithin] with y hy
-    exact inverseFlow_abs_re_le hκ hT hy b hB
-  have him0 : 0 ≤ z.im := by
-    apply ge_of_tendsto ((Complex.continuous_im.tendsto z).comp hz)
-    filter_upwards [self_mem_nhdsWithin] with y hy
-    exact (inverseFlow_im_pos hκ hT b (by simpa using hy)).le
-  have hi : z.im^2 ≤ 2*a κ*T := by
-    have hlim : Tendsto (fun y : ℝ => y^2+2*a κ*T) (𝓝[>] 0) (𝓝 (2*a κ*T)) := by
-      simpa using (((tendsto_id'.mpr nhdsWithin_le_nhds :
-        Tendsto (fun y : ℝ => y) (𝓝[>] (0:ℝ)) (𝓝 0)).pow 2).add_const (2*a κ*T))
-    apply le_of_tendsto_of_tendsto (((Complex.continuous_im.tendsto z).comp hz).pow 2) hlim
-    filter_upwards [self_mem_nhdsWithin] with y hy
-    simpa using inverseFlow_im_sq_bound hκ hT b
-      (show 0 < ((driverValue b T:ℂ)+y*Complex.I).im by simpa using hy)
-  have hi' : z.im ≤ Real.sqrt (2*a κ*T) := (Real.le_sqrt him0 (by exact mul_nonneg (mul_nonneg (by norm_num) (div_pos (by norm_num) hκ).le) hT)).mpr hi
-  calc
-    ‖z‖ ≤ |z.re|+|z.im| := Complex.norm_le_abs_re_add_abs_im z
-    _ ≤ R+Real.sqrt (2*a κ*T) := by rw [abs_of_nonneg him0]; linarith
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Topology
-
-lemma inverseFlow_zero (κ : ℝ) (hκ : 0 < κ) (b : Driver) {w : ℂ} (hw : 0 < w.im) :
-    inverseFlow κ b 0 w = w := by
-  have hs := inverseFlow_surviving hκ b (le_refl 0) hw
-  have hf := surviving_flow_positiveSolution (le_refl 0) hs
-  have he := flow_inverseFlow hκ b (le_refl 0) hw
-  rwa [hf.2.2.1] at he
-
- 
-
-lemma trace_zero (κ : ℝ) (hκ : 0 < κ) (b : Driver) : trace κ b 0 = 0 := by
-  unfold trace
-  simp only [mul_zero,driverValue,Real.toNNReal_zero,map_zero,Complex.ofReal_zero,zero_add]
-  apply Filter.Tendsto.limUnder_eq
-  have hc : Tendsto (fun ε : ℝ => (ε:ℂ)*Complex.I) (nhdsWithin 0 (Ioi 0)) (𝓝 (0:ℂ)) := by
-    have hc0 : Continuous (fun ε : ℝ => (ε:ℂ)*Complex.I) :=
-      Complex.continuous_ofReal.mul continuous_const
-    have h : Tendsto (fun ε : ℝ => (ε:ℂ)*Complex.I)
-        (nhdsWithin 0 (Ioi 0)) (𝓝 ((0:ℂ)*Complex.I)) :=
-      (hc0.tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds
-    simpa using h
-  apply hc.congr'
-  filter_upwards [self_mem_nhdsWithin] with ε hε
-  exact (inverseFlow_zero κ hκ b (by simpa using hε)).symm
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-
-lemma trace_limit_zero {κ : ℝ} (hκ : 0 < κ) (b : Driver) :
-    Tendsto (fun y : ℝ => inverseFlow κ b (κ*0)
-      (driverValue b (κ*0)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b 0)) := by
-  rw [trace_zero κ hκ b]
-  simp only [mul_zero,driverValue,Real.toNNReal_zero,map_zero,Complex.ofReal_zero,zero_add]
-  have hc : Tendsto (fun y : ℝ => (y:ℂ)*Complex.I) (𝓝[>] 0) (𝓝 (0:ℂ)) := by
-    simpa only [Complex.ofReal_zero,zero_mul] using
-      ((Complex.continuous_ofReal.tendsto 0).mul_const Complex.I).mono_left nhdsWithin_le_nhds
-  apply hc.congr'
-  filter_upwards [self_mem_nhdsWithin] with y hy
-  exact (inverseFlow_zero κ hκ b (by simpa using hy)).symm
-
-lemma trace_continuousWithinAt_zero {κ : ℝ} (hκ : 0 < κ) (b : Driver)
-    (hlim : ∀ t : ℝ, 0 < t → Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
-        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t))) :
-    ContinuousWithinAt (trace κ b) (Ici 0) 0 := by
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hB : Continuous (driverValue b) := (map_continuous b).comp continuous_real_toNNReal
-  have hB0 : driverValue b 0 = 0 := by simp [driverValue]
-  have hz := trace_zero κ hκ b
-  rw [Metric.continuousWithinAt_iff]
-  intro ε hε
-  obtain ⟨δ,hδ,hδB⟩ := Metric.continuousAt_iff.mp (hB.continuousAt (x := 0))
-    (ε/2) (by linarith)
-  have hi : 0 < ε^2/(8*a κ*κ) := by positivity
-  refine ⟨min (δ/κ) (ε^2/(8*a κ*κ)),lt_min (div_pos hδ hκ) hi,?_⟩
-  intro t ht htd
-  have ht0 : 0 ≤ t := ht
-  have hts : t < min (δ/κ) (ε^2/(8*a κ*κ)) := by
-    simpa only [Real.dist_eq,sub_zero,abs_of_nonneg ht0] using htd
-  rcases ht0.eq_or_lt with he | hp
-  · subst t
-    simp only [dist_self]
-    exact hε
-  have htδ : κ*t < δ := by
-    have hh := (lt_div_iff₀ hκ).mp (hts.trans_le (min_le_left _ _))
-    nlinarith
-  have hdr (u : ℝ) (hu : u ∈ Icc 0 (κ*t)) : |driverValue b u| ≤ ε/2 := by
-    have hh := hδB (show dist u (0:ℝ) < δ from by
-      rw [Real.dist_eq,sub_zero,abs_of_nonneg hu.1]
-      exact hu.2.trans_lt htδ)
-    simpa only [Real.dist_eq,hB0,sub_zero] using hh.le
-  have hbnd := radial_limit_early_bound hκ (mul_nonneg hκ.le ht0) b hdr (hlim t hp)
-  have hroot : Real.sqrt (2*a κ*(κ*t)) < ε/2 := by
-    apply (Real.sqrt_lt (by positivity) (by linarith : 0 ≤ ε/2)).mpr
-    have hh := (lt_div_iff₀ (by positivity : 0 < 8*a κ*κ)).mp
-      (hts.trans_le (min_le_right _ _))
-    nlinarith
-  rw [hz,dist_zero_right]
-  linarith
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open MeasureTheory
-
-lemma surviving_of_le {κ : ℝ} {b : Driver} {z : ℂ} {s t : ℝ}
-    (hs : 0 ≤ s) (hst : s ≤ t) (hz : Surviving κ b z t) : Surviving κ b z s :=
-  ((surviving_flow_positiveSolution (hs.trans hst) hz).restrict hs hst).surviving
-
-lemma radial_limit_mem_closure {κ T s : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s) (hsT : s ≤ T)
-    (b : Driver) {z : ℂ}
-    (hz : Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
-      (𝓝[>] 0) (𝓝 z)) : z ∈ closure {w | Surviving κ b w s} := by
-  apply mem_closure_of_tendsto hz
-  filter_upwards [self_mem_nhdsWithin] with y hy
-  apply surviving_of_le hs hsT
-  exact inverseFlow_surviving hκ b (hs.trans hsT) (by simpa using hy)
-
-lemma radial_limit_not_surviving {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
-    {z : ℂ} (hz : Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
-      (𝓝[>] 0) (𝓝 z)) : ¬ Surviving κ b z T := by
-  intro hs
-  have hg := (flow_continuousAt_initial hT hs).tendsto.comp hz
-  have he : (fun y : ℝ => flow κ b (inverseFlow κ b T (driverValue b T+y*Complex.I)) T) =ᶠ[𝓝[>] 0]
-      (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I) := by
-    filter_upwards [self_mem_nhdsWithin] with y hy
-    exact flow_inverseFlow hκ b hT (by simpa using hy)
-  have hl : Tendsto (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I)
-      (𝓝[>] 0) (𝓝 (driverValue b T:ℂ)) := by
-    simpa only [Complex.ofReal_zero,zero_mul,add_zero] using
-      (((Complex.continuous_ofReal.tendsto 0).mul_const Complex.I).const_add (driverValue b T:ℂ)).mono_left
-        nhdsWithin_le_nhds
-  have hid : flow κ b z T = driverValue b T := tendsto_nhds_unique (hg.congr' he) hl
-  have hpos := (surviving_flow_positiveSolution hT hs).2.2.2.1 T ⟨hT,le_rfl⟩
-  rw [hid,Complex.ofReal_im] at hpos
-  exact (lt_irrefl 0) hpos
-
-lemma radial_limit_mem_frontier {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
-    {z : ℂ} (hz : Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
-      (𝓝[>] 0) (𝓝 z)) : z ∈ frontier {w | Surviving κ b w T} := by
-  rw [(isOpen_surviving κ b hT).frontier_eq]
-  exact ⟨radial_limit_mem_closure hκ hT le_rfl b hz,radial_limit_not_surviving hκ hT b hz⟩
-
- 
-
-theorem trace_countable_boundary {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) (t : ℕ → ℝ) (ht : ∀ n, 0 ≤ t n) :
-    ∀ᵐ b ∂W, ∀ n : ℕ,
-      trace κ b (t n) ∈ frontier {z | Surviving κ b z (κ*t n)} ∧
-        ∀ s : ℝ, 0 ≤ s → s ≤ t n →
-          trace κ b (t n) ∈ closure {z | Surviving κ b z (κ*s)} := by
-  filter_upwards [trace_limit_countable hκ hκ8 hW t ht] with b hb
-  intro n
-  refine ⟨radial_limit_mem_frontier hκ (mul_nonneg hκ.le (ht n)) b (hb n),?_⟩
-  intro s hs hst
-  exact radial_limit_mem_closure hκ (mul_nonneg hκ.le hs)
-    (mul_le_mul_of_nonneg_left hst hκ.le) b (hb n)
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Topology MeasureTheory
-
- 
-
-theorem trace_regular_all_times {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W,
-      (∀ t : ℝ, 0 ≤ t →
-        Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
-          (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t))) ∧
-      ContinuousOn (trace κ b) (Ici 0) ∧
-      (∀ t : ℝ, 0 ≤ t → trace κ b t ∈ frontier {z | Surviving κ b z (κ*t)}) ∧
-      (∀ s : ℝ, 0 ≤ s →
-        (trace κ b) '' Ici s ⊆ closure {z | Surviving κ b z (κ*s)}) := by
-  filter_upwards [trace_limit_all_positive hκ hκ8 hW,
-    trace_continuous_positive hκ hκ8 hW] with b hlim hc
-  have hall (t : ℝ) (ht : 0 ≤ t) :
-      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
-        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t)) := by
-    rcases ht.eq_or_lt with he | hp
-    · subst t
-      exact trace_limit_zero hκ b
-    · exact hlim t hp
-  refine ⟨hall,?_,?_,?_⟩
-  · intro t ht
-    rcases (show 0 ≤ t from ht).eq_or_lt with he | hp
-    · subst t
-      exact trace_continuousWithinAt_zero hκ b hlim
-    · exact (hc.continuousAt (isOpen_Ioi.mem_nhds hp)).continuousWithinAt
-  · intro t ht
-    exact radial_limit_mem_frontier hκ (mul_nonneg hκ.le ht) b (hall t ht)
-  · intro s hs z hz
-    obtain ⟨t,ht,rfl⟩ := hz
-    exact radial_limit_mem_closure hκ (mul_nonneg hκ.le hs)
-      (mul_le_mul_of_nonneg_left ht hκ.le) b (hall t (hs.trans ht))
-
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Topology MeasureTheory
-
- 
-def unboundedHalfPlaneComponent (K : Set ℂ) : Set ℂ :=
-  {z | z ∈ {w : ℂ | 0 < w.im} \ K ∧
-    ¬ Bornology.IsBounded (connectedComponentIn ({w : ℂ | 0 < w.im} \ K) z)}
-
-lemma inverseFlow_image_upper {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
-    inverseFlow κ b T '' {w : ℂ | 0 < w.im} = {z : ℂ | Surviving κ b z T} := by
-  ext z
-  constructor
-  · rintro ⟨w,hw,rfl⟩
-    exact inverseFlow_surviving hκ b hT hw
-  · intro hz
-    refine ⟨flow κ b z T,?_,inverseFlow_flow hT hz⟩
-    exact (surviving_flow_positiveSolution hT hz).2.2.2.1 T ⟨hT,le_rfl⟩
-
-lemma isPreconnected_surviving {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
-    IsPreconnected {z : ℂ | Surviving κ b z T} := by
-  rw [← inverseFlow_image_upper hκ hT b]
-  exact (convex_halfSpace_im_gt 0).isPreconnected.image _
-    (fun w hw => (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.continuousAt.continuousWithinAt)
-
-lemma not_isBounded_surviving {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
-    ¬ Bornology.IsBounded {z : ℂ | Surviving κ b z T} := by
-  intro hb
-  obtain ⟨R,hR,hb⟩ := hb.exists_pos_norm_le
-  let w : ℂ := (R+1)*Complex.I
-  have hw : 0 < w.im := by dsimp [w]; simpa using (show 0 < R+1 by linarith)
-  have hz := inverseFlow_surviving hκ b hT hw
-  have hf := surviving_flow_positiveSolution hT hz
-  have hi := hf.im_antitone hκ (left_mem_Icc.mpr hT) (right_mem_Icc.mpr hT) hT
-  dsimp only at hi
-  rw [hf.2.2.1,flow_inverseFlow hκ b hT hw] at hi
-  have hn := (Complex.im_le_norm (inverseFlow κ b T w)).trans (hb _ hz)
-  have he : w.im = R+1 := by simp [w]
-  rw [he] at hi
-  linarith
-
- 
-
-theorem surviving_subset_trace_component {κ : ℝ} (hκ : 0 < κ) (b : Driver)
-    (hlim : ∀ t : ℝ, 0 ≤ t →
-      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
-        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t)))
-    {t : ℝ} (ht : 0 ≤ t) :
-    {z : ℂ | Surviving κ b z (κ*t)} ⊆
-      unboundedHalfPlaneComponent ((trace κ b) '' Icc 0 t) := by
-  have hT : 0 ≤ κ*t := mul_nonneg hκ.le ht
-  have hsub : {z : ℂ | Surviving κ b z (κ*t)} ⊆
-      {w : ℂ | 0 < w.im} \ ((trace κ b) '' Icc 0 t) := by
-    intro z hz
-    refine ⟨?_,?_⟩
-    · have hh := (surviving_flow_positiveSolution hT hz).2.2.2.1 0
-        (left_mem_Icc.mpr hT)
-      rwa [(surviving_flow_positiveSolution hT hz).2.2.1] at hh
-    · rintro ⟨s,hs,rfl⟩
-      exact radial_limit_not_surviving hκ (mul_nonneg hκ.le hs.1) b (hlim s hs.1)
-        (surviving_of_le (mul_nonneg hκ.le hs.1)
-          (mul_le_mul_of_nonneg_left hs.2 hκ.le) hz)
-  intro z hz
-  refine ⟨hsub hz,?_⟩
-  intro hb
-  exact not_isBounded_surviving hκ hT b (hb.subset
-    ((isPreconnected_surviving hκ hT b).subset_connectedComponentIn hz hsub))
-
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Topology MeasureTheory
-
- 
-
-theorem trace_generation_partial {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W,
-      (∀ t : ℝ, 0 ≤ t →
-        Tendsto (fun ε : ℝ => inverseFlow κ b (κ*t)
-          (driverValue b (κ*t)+ε*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t))) ∧
-      ContinuousOn (trace κ b) (Ici 0) ∧
-      (∀ t : ℝ, 0 ≤ t →
-        {z : ℂ | Surviving κ b z (κ*t)} ⊆
-          unboundedHalfPlaneComponent ((trace κ b) '' Icc 0 t)) ∧
-      (∀ t : ℝ, 0 ≤ t →
-        (trace κ b) '' Ici t ⊆ closure {z : ℂ | Surviving κ b z (κ*t)}) := by
-  filter_upwards [trace_regular_all_times hκ hκ8 hW] with b hb
-  exact ⟨hb.1,hb.2.1,fun t ht => surviving_subset_trace_component hκ b hb.1 ht,
-    hb.2.2.2⟩
-
- 
-
-def MissingReverseTraceDomainObligation : Prop :=
-  ∀ κ : ℝ, 0 < κ → κ < 8 → ∀ W : WienerLaw, IsWiener W →
-    ∀ᵐ b ∂W, ∀ t : ℝ, 0 ≤ t →
-      unboundedHalfPlaneComponent ((trace κ b) '' Icc 0 t) ⊆
-        {z : ℂ | Surviving κ b z (κ*t)}
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TraceBoundary
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+noncomputable section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.LiteralTrace
+open TraceChaining
+lemma PositiveSolution.initial_abs_re_le {κ : ℝ} {b : Driver} {z : ℂ} {T R : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) (hT : 0 ≤ T)
+    (hB : ∀ u ∈ Icc 0 T, |driverValue b u| ≤ R) (hfT : |(f T).re| ≤ R) :
+    |z.re| ≤ R := by
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hn (u : ℝ) (hu : u ∈ Icc 0 T) : 0 < ‖f u-driverValue b u‖^2 :=
+    sq_pos_of_pos (norm_pos_iff.mpr (hf.shifted_ne_zero hu))
+  apply abs_le.mpr
+  constructor
+  · have hh := initial_le_of_outward_deriv (R := R) hT
+      (Complex.continuous_re.comp_continuousOn hf.2.1).neg
+      (fun u hu => (hf.re_hasDeriv hu).neg) (by
+        change -(f T).re ≤ R
+        linarith [(abs_le.mp hfT).1])
+      (by
+        intro u hu hg
+        change R < -(f u).re at hg
+        have hBu := (abs_le.mp (hB u hu)).1
+        have hm : 0 < -(a κ*((f u).re-driverValue b u)) := by nlinarith
+        simpa only [neg_div] using div_pos hm (hn u hu))
+    change -(f 0).re ≤ R at hh
+    rw [hf.2.2.1] at hh
+    linarith
+  · have hh := initial_le_of_outward_deriv (R := R) hT
+      (Complex.continuous_re.comp_continuousOn hf.2.1)
+      (fun u hu => hf.re_hasDeriv hu) (abs_le.mp hfT).2
+      (by
+        intro u hu hg
+        change R < (f u).re at hg
+        have hBu := (abs_le.mp (hB u hu)).2
+        exact div_pos (mul_pos ha (by linarith)) (hn u hu))
+    change (f 0).re ≤ R at hh
+    simpa only [hf.2.2.1] using hh
+
+lemma inverseFlow_abs_re_le {κ T R y : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
+    (hy : 0 < y) (b : Driver) (hB : ∀ u ∈ Icc 0 T, |driverValue b u| ≤ R) :
+    |(radialValue κ b T y).re| ≤ R := by
+  have hw : 0 < ((driverValue b T:ℂ)+y*Complex.I).im := by simpa using hy
+  have hf := surviving_flow_positiveSolution hT (inverseFlow_surviving hκ b hT hw)
+  apply hf.initial_abs_re_le hκ hT hB
+  rw [flow_inverseFlow hκ b hT hw]
+  simpa using hB T (right_mem_Icc.mpr hT)
+
+lemma radial_limit_early_bound {κ T R : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
+    (hB : ∀ u ∈ Icc 0 T, |driverValue b u| ≤ R) {z : ℂ}
+    (hz : Tendsto (radialValue κ b T) (𝓝[>] 0) (𝓝 z)) :
+    ‖z‖ ≤ R+Real.sqrt (2*a κ*T) := by
+  have hre : |z.re| ≤ R := by
+    apply le_of_tendsto ((Complex.continuous_re.tendsto z).comp hz |>.abs)
+    filter_upwards [self_mem_nhdsWithin] with y hy
+    exact inverseFlow_abs_re_le hκ hT hy b hB
+  have him0 : 0 ≤ z.im := by
+    apply ge_of_tendsto ((Complex.continuous_im.tendsto z).comp hz)
+    filter_upwards [self_mem_nhdsWithin] with y hy
+    exact (inverseFlow_im_pos hκ hT b (by simpa using hy)).le
+  have hi : z.im^2 ≤ 2*a κ*T := by
+    have hlim : Tendsto (fun y : ℝ => y^2+2*a κ*T) (𝓝[>] 0) (𝓝 (2*a κ*T)) := by
+      simpa using (((tendsto_id'.mpr nhdsWithin_le_nhds :
+        Tendsto (fun y : ℝ => y) (𝓝[>] (0:ℝ)) (𝓝 0)).pow 2).add_const (2*a κ*T))
+    apply le_of_tendsto_of_tendsto (((Complex.continuous_im.tendsto z).comp hz).pow 2) hlim
+    filter_upwards [self_mem_nhdsWithin] with y hy
+    simpa using inverseFlow_im_sq_bound hκ hT b
+      (show 0 < ((driverValue b T:ℂ)+y*Complex.I).im by simpa using hy)
+  have hi' : z.im ≤ Real.sqrt (2*a κ*T) := (Real.le_sqrt him0 (by exact mul_nonneg (mul_nonneg (by norm_num) (div_pos (by norm_num) hκ).le) hT)).mpr hi
+  calc
+    ‖z‖ ≤ |z.re|+|z.im| := Complex.norm_le_abs_re_add_abs_im z
+    _ ≤ R+Real.sqrt (2*a κ*T) := by rw [abs_of_nonneg him0]; linarith
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Topology
+
+lemma inverseFlow_zero (κ : ℝ) (hκ : 0 < κ) (b : Driver) {w : ℂ} (hw : 0 < w.im) :
+    inverseFlow κ b 0 w = w := by
+  have hs := inverseFlow_surviving hκ b (le_refl 0) hw
+  have hf := surviving_flow_positiveSolution (le_refl 0) hs
+  have he := flow_inverseFlow hκ b (le_refl 0) hw
+  rwa [hf.2.2.1] at he
+
+ 
+
+lemma trace_zero (κ : ℝ) (hκ : 0 < κ) (b : Driver) : trace κ b 0 = 0 := by
+  unfold trace
+  simp only [mul_zero,driverValue,Real.toNNReal_zero,map_zero,Complex.ofReal_zero,zero_add]
+  apply Filter.Tendsto.limUnder_eq
+  have hc : Tendsto (fun ε : ℝ => (ε:ℂ)*Complex.I) (nhdsWithin 0 (Ioi 0)) (𝓝 (0:ℂ)) := by
+    have hc0 : Continuous (fun ε : ℝ => (ε:ℂ)*Complex.I) :=
+      Complex.continuous_ofReal.mul continuous_const
+    have h : Tendsto (fun ε : ℝ => (ε:ℂ)*Complex.I)
+        (nhdsWithin 0 (Ioi 0)) (𝓝 ((0:ℂ)*Complex.I)) :=
+      (hc0.tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds
+    simpa using h
+  apply hc.congr'
+  filter_upwards [self_mem_nhdsWithin] with ε hε
+  exact (inverseFlow_zero κ hκ b (by simpa using hε)).symm
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+
+lemma trace_limit_zero {κ : ℝ} (hκ : 0 < κ) (b : Driver) :
+    Tendsto (fun y : ℝ => inverseFlow κ b (κ*0)
+      (driverValue b (κ*0)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b 0)) := by
+  rw [trace_zero κ hκ b]
+  simp only [mul_zero,driverValue,Real.toNNReal_zero,map_zero,Complex.ofReal_zero,zero_add]
+  have hc : Tendsto (fun y : ℝ => (y:ℂ)*Complex.I) (𝓝[>] 0) (𝓝 (0:ℂ)) := by
+    simpa only [Complex.ofReal_zero,zero_mul] using
+      ((Complex.continuous_ofReal.tendsto 0).mul_const Complex.I).mono_left nhdsWithin_le_nhds
+  apply hc.congr'
+  filter_upwards [self_mem_nhdsWithin] with y hy
+  exact (inverseFlow_zero κ hκ b (by simpa using hy)).symm
+
+lemma trace_continuousWithinAt_zero {κ : ℝ} (hκ : 0 < κ) (b : Driver)
+    (hlim : ∀ t : ℝ, 0 < t → Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
+        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t))) :
+    ContinuousWithinAt (trace κ b) (Ici 0) 0 := by
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hB : Continuous (driverValue b) := (map_continuous b).comp continuous_real_toNNReal
+  have hB0 : driverValue b 0 = 0 := by simp [driverValue]
+  have hz := trace_zero κ hκ b
+  rw [Metric.continuousWithinAt_iff]
+  intro ε hε
+  obtain ⟨δ,hδ,hδB⟩ := Metric.continuousAt_iff.mp (hB.continuousAt (x := 0))
+    (ε/2) (by linarith)
+  have hi : 0 < ε^2/(8*a κ*κ) := by positivity
+  refine ⟨min (δ/κ) (ε^2/(8*a κ*κ)),lt_min (div_pos hδ hκ) hi,?_⟩
+  intro t ht htd
+  have ht0 : 0 ≤ t := ht
+  have hts : t < min (δ/κ) (ε^2/(8*a κ*κ)) := by
+    simpa only [Real.dist_eq,sub_zero,abs_of_nonneg ht0] using htd
+  rcases ht0.eq_or_lt with he | hp
+  · subst t
+    simp only [dist_self]
+    exact hε
+  have htδ : κ*t < δ := by
+    have hh := (lt_div_iff₀ hκ).mp (hts.trans_le (min_le_left _ _))
+    nlinarith
+  have hdr (u : ℝ) (hu : u ∈ Icc 0 (κ*t)) : |driverValue b u| ≤ ε/2 := by
+    have hh := hδB (show dist u (0:ℝ) < δ from by
+      rw [Real.dist_eq,sub_zero,abs_of_nonneg hu.1]
+      exact hu.2.trans_lt htδ)
+    simpa only [Real.dist_eq,hB0,sub_zero] using hh.le
+  have hbnd := radial_limit_early_bound hκ (mul_nonneg hκ.le ht0) b hdr (hlim t hp)
+  have hroot : Real.sqrt (2*a κ*(κ*t)) < ε/2 := by
+    apply (Real.sqrt_lt (by positivity) (by linarith : 0 ≤ ε/2)).mpr
+    have hh := (lt_div_iff₀ (by positivity : 0 < 8*a κ*κ)).mp
+      (hts.trans_le (min_le_right _ _))
+    nlinarith
+  rw [hz,dist_zero_right]
+  linarith
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open MeasureTheory
+
+lemma surviving_of_le {κ : ℝ} {b : Driver} {z : ℂ} {s t : ℝ}
+    (hs : 0 ≤ s) (hst : s ≤ t) (hz : Surviving κ b z t) : Surviving κ b z s :=
+  ((surviving_flow_positiveSolution (hs.trans hst) hz).restrict hs hst).surviving
+
+lemma radial_limit_mem_closure {κ T s : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s) (hsT : s ≤ T)
+    (b : Driver) {z : ℂ}
+    (hz : Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
+      (𝓝[>] 0) (𝓝 z)) : z ∈ closure {w | Surviving κ b w s} := by
+  apply mem_closure_of_tendsto hz
+  filter_upwards [self_mem_nhdsWithin] with y hy
+  apply surviving_of_le hs hsT
+  exact inverseFlow_surviving hκ b (hs.trans hsT) (by simpa using hy)
+
+lemma radial_limit_not_surviving {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
+    {z : ℂ} (hz : Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
+      (𝓝[>] 0) (𝓝 z)) : ¬ Surviving κ b z T := by
+  intro hs
+  have hg := (flow_continuousAt_initial hT hs).tendsto.comp hz
+  have he : (fun y : ℝ => flow κ b (inverseFlow κ b T (driverValue b T+y*Complex.I)) T) =ᶠ[𝓝[>] 0]
+      (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I) := by
+    filter_upwards [self_mem_nhdsWithin] with y hy
+    exact flow_inverseFlow hκ b hT (by simpa using hy)
+  have hl : Tendsto (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I)
+      (𝓝[>] 0) (𝓝 (driverValue b T:ℂ)) := by
+    simpa only [Complex.ofReal_zero,zero_mul,add_zero] using
+      (((Complex.continuous_ofReal.tendsto 0).mul_const Complex.I).const_add (driverValue b T:ℂ)).mono_left
+        nhdsWithin_le_nhds
+  have hid : flow κ b z T = driverValue b T := tendsto_nhds_unique (hg.congr' he) hl
+  have hpos := (surviving_flow_positiveSolution hT hs).2.2.2.1 T ⟨hT,le_rfl⟩
+  rw [hid,Complex.ofReal_im] at hpos
+  exact (lt_irrefl 0) hpos
+
+lemma radial_limit_mem_frontier {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
+    {z : ℂ} (hz : Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
+      (𝓝[>] 0) (𝓝 z)) : z ∈ frontier {w | Surviving κ b w T} := by
+  rw [(isOpen_surviving κ b hT).frontier_eq]
+  exact ⟨radial_limit_mem_closure hκ hT le_rfl b hz,radial_limit_not_surviving hκ hT b hz⟩
+
+ 
+
+theorem trace_countable_boundary {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) (t : ℕ → ℝ) (ht : ∀ n, 0 ≤ t n) :
+    ∀ᵐ b ∂W, ∀ n : ℕ,
+      trace κ b (t n) ∈ frontier {z | Surviving κ b z (κ*t n)} ∧
+        ∀ s : ℝ, 0 ≤ s → s ≤ t n →
+          trace κ b (t n) ∈ closure {z | Surviving κ b z (κ*s)} := by
+  filter_upwards [trace_limit_countable hκ hκ8 hW t ht] with b hb
+  intro n
+  refine ⟨radial_limit_mem_frontier hκ (mul_nonneg hκ.le (ht n)) b (hb n),?_⟩
+  intro s hs hst
+  exact radial_limit_mem_closure hκ (mul_nonneg hκ.le hs)
+    (mul_le_mul_of_nonneg_left hst hκ.le) b (hb n)
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Topology MeasureTheory
+
+ 
+
+theorem trace_regular_all_times {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W,
+      (∀ t : ℝ, 0 ≤ t →
+        Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
+          (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t))) ∧
+      ContinuousOn (trace κ b) (Ici 0) ∧
+      (∀ t : ℝ, 0 ≤ t → trace κ b t ∈ frontier {z | Surviving κ b z (κ*t)}) ∧
+      (∀ s : ℝ, 0 ≤ s →
+        (trace κ b) '' Ici s ⊆ closure {z | Surviving κ b z (κ*s)}) := by
+  filter_upwards [trace_limit_all_positive hκ hκ8 hW,
+    trace_continuous_positive hκ hκ8 hW] with b hlim hc
+  have hall (t : ℝ) (ht : 0 ≤ t) :
+      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
+        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t)) := by
+    rcases ht.eq_or_lt with he | hp
+    · subst t
+      exact trace_limit_zero hκ b
+    · exact hlim t hp
+  refine ⟨hall,?_,?_,?_⟩
+  · intro t ht
+    rcases (show 0 ≤ t from ht).eq_or_lt with he | hp
+    · subst t
+      exact trace_continuousWithinAt_zero hκ b hlim
+    · exact (hc.continuousAt (isOpen_Ioi.mem_nhds hp)).continuousWithinAt
+  · intro t ht
+    exact radial_limit_mem_frontier hκ (mul_nonneg hκ.le ht) b (hall t ht)
+  · intro s hs z hz
+    obtain ⟨t,ht,rfl⟩ := hz
+    exact radial_limit_mem_closure hκ (mul_nonneg hκ.le hs)
+      (mul_le_mul_of_nonneg_left ht hκ.le) b (hall t (hs.trans ht))
+
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Topology MeasureTheory
+
+ 
+def unboundedHalfPlaneComponent (K : Set ℂ) : Set ℂ :=
+  {z | z ∈ {w : ℂ | 0 < w.im} \ K ∧
+    ¬ Bornology.IsBounded (connectedComponentIn ({w : ℂ | 0 < w.im} \ K) z)}
+
+lemma inverseFlow_image_upper {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
+    inverseFlow κ b T '' {w : ℂ | 0 < w.im} = {z : ℂ | Surviving κ b z T} := by
+  ext z
+  constructor
+  · rintro ⟨w,hw,rfl⟩
+    exact inverseFlow_surviving hκ b hT hw
+  · intro hz
+    refine ⟨flow κ b z T,?_,inverseFlow_flow hT hz⟩
+    exact (surviving_flow_positiveSolution hT hz).2.2.2.1 T ⟨hT,le_rfl⟩
+
+lemma isPreconnected_surviving {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
+    IsPreconnected {z : ℂ | Surviving κ b z T} := by
+  rw [← inverseFlow_image_upper hκ hT b]
+  exact (convex_halfSpace_im_gt 0).isPreconnected.image _
+    (fun w hw => (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.continuousAt.continuousWithinAt)
+
+lemma not_isBounded_surviving {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
+    ¬ Bornology.IsBounded {z : ℂ | Surviving κ b z T} := by
+  intro hb
+  obtain ⟨R,hR,hb⟩ := hb.exists_pos_norm_le
+  let w : ℂ := (R+1)*Complex.I
+  have hw : 0 < w.im := by dsimp [w]; simpa using (show 0 < R+1 by linarith)
+  have hz := inverseFlow_surviving hκ b hT hw
+  have hf := surviving_flow_positiveSolution hT hz
+  have hi := hf.im_antitone hκ (left_mem_Icc.mpr hT) (right_mem_Icc.mpr hT) hT
+  dsimp only at hi
+  rw [hf.2.2.1,flow_inverseFlow hκ b hT hw] at hi
+  have hn := (Complex.im_le_norm (inverseFlow κ b T w)).trans (hb _ hz)
+  have he : w.im = R+1 := by simp [w]
+  rw [he] at hi
+  linarith
+
+ 
+
+theorem surviving_subset_trace_component {κ : ℝ} (hκ : 0 < κ) (b : Driver)
+    (hlim : ∀ t : ℝ, 0 ≤ t →
+      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
+        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t)))
+    {t : ℝ} (ht : 0 ≤ t) :
+    {z : ℂ | Surviving κ b z (κ*t)} ⊆
+      unboundedHalfPlaneComponent ((trace κ b) '' Icc 0 t) := by
+  have hT : 0 ≤ κ*t := mul_nonneg hκ.le ht
+  have hsub : {z : ℂ | Surviving κ b z (κ*t)} ⊆
+      {w : ℂ | 0 < w.im} \ ((trace κ b) '' Icc 0 t) := by
+    intro z hz
+    refine ⟨?_,?_⟩
+    · have hh := (surviving_flow_positiveSolution hT hz).2.2.2.1 0
+        (left_mem_Icc.mpr hT)
+      rwa [(surviving_flow_positiveSolution hT hz).2.2.1] at hh
+    · rintro ⟨s,hs,rfl⟩
+      exact radial_limit_not_surviving hκ (mul_nonneg hκ.le hs.1) b (hlim s hs.1)
+        (surviving_of_le (mul_nonneg hκ.le hs.1)
+          (mul_le_mul_of_nonneg_left hs.2 hκ.le) hz)
+  intro z hz
+  refine ⟨hsub hz,?_⟩
+  intro hb
+  exact not_isBounded_surviving hκ hT b (hb.subset
+    ((isPreconnected_surviving hκ hT b).subset_connectedComponentIn hz hsub))
+
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Topology MeasureTheory
+
+ 
+
+theorem trace_generation_partial {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W,
+      (∀ t : ℝ, 0 ≤ t →
+        Tendsto (fun ε : ℝ => inverseFlow κ b (κ*t)
+          (driverValue b (κ*t)+ε*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t))) ∧
+      ContinuousOn (trace κ b) (Ici 0) ∧
+      (∀ t : ℝ, 0 ≤ t →
+        {z : ℂ | Surviving κ b z (κ*t)} ⊆
+          unboundedHalfPlaneComponent ((trace κ b) '' Icc 0 t)) ∧
+      (∀ t : ℝ, 0 ≤ t →
+        (trace κ b) '' Ici t ⊆ closure {z : ℂ | Surviving κ b z (κ*t)}) := by
+  filter_upwards [trace_regular_all_times hκ hκ8 hW] with b hb
+  exact ⟨hb.1,hb.2.1,fun t ht => surviving_subset_trace_component hκ b hb.1 ht,
+    hb.2.2.2⟩
+
+ 
+
+def MissingReverseTraceDomainObligation : Prop :=
+  ∀ κ : ℝ, 0 < κ → κ < 8 → ∀ W : WienerLaw, IsWiener W →
+    ∀ᵐ b ∂W, ∀ t : ℝ, 0 ≤ t →
+      unboundedHalfPlaneComponent ((trace κ b) '' Icc 0 t) ⊆
+        {z : ℂ | Surviving κ b z (κ*t)}
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/InverseDomains.lean b/lean/OAI/Probability/SLEGauge/InverseDomains.lean
index ea5ba48f6c85eefe13831a5eaf2c186e0da4fe25..65817a8b7ad2051eb19eb4a5d544e4585c0f4893 100644
--- a/lean/OAI/Probability/SLEGauge/InverseDomains.lean
+++ b/lean/OAI/Probability/SLEGauge/InverseDomains.lean
@@ -1,421 +1,421 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.EulerFlow
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.LiteralTrace
-lemma PositiveSolution.surviving {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : Surviving κ b z T := by
-  obtain ⟨j,hj,hgap⟩ := hf.exists_cutoff
-  have he := hf.cutoff_eq hj (fun t ht => (hgap t ht).le)
-  obtain ⟨t0,ht0,hmin⟩ := isCompact_Icc.exists_isMinOn ⟨0,le_rfl,hf.1⟩
-    (Complex.continuous_im.comp_continuousOn hf.2.1)
-  refine ⟨j,hj,(hgap t0 ht0).trans_le ?_⟩
-  apply le_csInf (s := (fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)
-    (Set.Nonempty.image _ ⟨0,le_rfl,hf.1⟩)
-  rintro y ⟨t,ht,rfl⟩
-  dsimp only
-  rw [he ht]
-  exact hmin ht
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-
-lemma cutoff_positiveSolution {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} (hT : 0 ≤ T)
-    {j : ℕ} (hj : 1 ≤ j)
-    (hgap : (j:ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)) :
-    PositiveSolution κ b z T (cutoffFlow κ b j z) := by
-  have hc := cutoffFlow_continuous κ b hj z hT
-  have hb := (isCompact_Icc.image_of_continuousOn (Complex.continuous_im.comp_continuousOn hc)).bddBelow
-  have hg t (ht : t ∈ Icc 0 T) : (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im :=
-    hgap.trans_le (csInf_le hb ⟨t,ht,rfl⟩)
-  refine ⟨hT,hc,(cutoffFlow_solution κ b j hj z hT).2.1,?_,?_⟩
-  · intro t ht
-    exact lt_trans (inv_pos.mpr (by exact_mod_cast (Nat.zero_lt_of_lt hj))) (hg t ht)
-  · intro t ht
-    rw [← cutoffField_of_height κ b hj (hg t ht).le]
-    exact (cutoffFlow_solution κ b j hj z hT).2.2 t ht
-
-lemma PositiveSolution.restrict {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : 0 ≤ t) (htT : t ≤ T) :
-    PositiveSolution κ b z t f := by
-  have hi : Icc 0 t ⊆ Icc 0 T := Icc_subset_Icc_right htT
-  exact ⟨ht,hf.2.1.mono hi,hf.2.2.1,fun u hu => hf.2.2.2.1 u (hi hu),
-    fun u hu => (hf.2.2.2.2 u (hi hu)).mono hi⟩
-
-lemma PositiveSolution.flow_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : flow κ b z T = f T := by
-  have hs := hf.surviving
-  have hn := Nat.find_spec hs
-  rw [flow,dite_eq_left hs]
-  exact (cutoff_positiveSolution hf.1 hn.1 hn.2).unique hf ⟨hf.1,le_rfl⟩
-
-lemma PositiveSolution.flow_eqOn {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : EqOn (flow κ b z) f (Icc 0 T) :=
-  fun _ ht => (hf.restrict ht.1 ht.2).flow_eq
-
-lemma surviving_iff_positiveSolution {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} (hT : 0 ≤ T) :
-    Surviving κ b z T ↔ ∃ f, PositiveSolution κ b z T f := by
-  constructor
-  · rintro ⟨j,hj,hgap⟩
-    exact ⟨cutoffFlow κ b j z,cutoff_positiveSolution hT hj hgap⟩
-  · rintro ⟨f,hf⟩
-    exact hf.surviving
-
-lemma surviving_flow_positiveSolution {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hs : Surviving κ b z T) : PositiveSolution κ b z T (flow κ b z) := by
-  obtain ⟨f,hf⟩ := (surviving_iff_positiveSolution hT).mp hs
-  have he := hf.flow_eqOn
-  refine ⟨hT,hf.2.1.congr (fun t ht => he ht),?_,?_,?_⟩
-  · rw [he ⟨le_rfl,hT⟩,hf.2.2.1]
-  · intro t ht
-    rw [he ht]
-    exact hf.2.2.2.1 t ht
-  · intro t ht
-    rw [he ht]
-    exact (hf.2.2.2.2 t ht).congr (fun u hu => he hu) (he ht)
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-
- 
-
-lemma PositiveSolution.extend {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) :
-    ∃ U : ℝ, T < U ∧ ∃ g : ℝ → ℂ, PositiveSolution κ b z U g ∧ EqOn g f (Icc 0 T) := by
-  obtain ⟨j,hj,hgap⟩ := hf.exists_cutoff
-  let g := cutoffFlow κ b j z
-  have he : EqOn g f (Icc 0 T) := hf.cutoff_eq hj (fun t ht => (hgap t ht).le)
-  have hc := cutoffFlow_continuous κ b hj z (show 0 ≤ T+1 by linarith [hf.1])
-  have hy : (j:ℝ)⁻¹ < (g T).im := by rw [he ⟨hf.1,le_rfl⟩]; exact hgap T ⟨hf.1,le_rfl⟩
-  have hci := (Complex.continuous_im.comp_continuousOn hc) T (show T ∈ Icc 0 (T+1) by constructor <;> linarith [hf.1])
-  obtain ⟨δ,hδ,hnear⟩ := Metric.continuousWithinAt_iff.mp hci ((g T).im-(j:ℝ)⁻¹) (sub_pos.mpr hy)
-  let U := T+min (δ/2) (1/2)
-  have hTU : T < U := by dsimp [U]; have := lt_min (by linarith : 0 < δ/2) (by norm_num : (0:ℝ)<1/2); linarith
-  have hU1 : U ≤ T+1 := by dsimp [U]; have := min_le_right (δ/2) (1/2:ℝ); linarith
-  have hU : 0 ≤ U := hf.1.trans hTU.le
-  have hg t (ht : t ∈ Icc 0 U) : (j:ℝ)⁻¹ < (g t).im := by
-    by_cases htT : t ≤ T
-    · rw [he ⟨ht.1,htT⟩]
-      exact hgap _ ⟨ht.1,htT⟩
-    · have hdist : dist t T < δ := by
-        rw [Real.dist_eq,abs_of_nonneg (by linarith)]
-        have hδU : U ≤ T+δ/2 := by dsimp [U]; exact add_le_add le_rfl (min_le_left (δ/2) (1/2:ℝ))
-        linarith [ht.2]
-      have hn := hnear ⟨ht.1,ht.2.trans hU1⟩ hdist
-      rw [Real.dist_eq] at hn
-      have hh := (abs_lt.mp hn).1
-      change -((g T).im-(j:ℝ)⁻¹) < (g t).im-(g T).im at hh
-      linarith
-  refine ⟨U,hTU,g,⟨hU,hc.mono (Icc_subset_Icc_right hU1),?_,?_,?_⟩,he⟩
-  · exact (cutoffFlow_solution κ b j hj z hU).2.1
-  · intro t ht
-    exact (inv_pos.mpr (by exact_mod_cast (Nat.zero_lt_of_lt hj))).trans (hg t ht)
-  · intro t ht
-    rw [← cutoffField_of_height κ b hj (hg t ht).le]
-    exact (cutoffFlow_solution κ b j hj z hU).2.2 t ht
-
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-
-def clockIntegrand (b : Driver) (f : ℝ → ℂ) (t : ℝ) : ℝ :=
-  (f t).im^2/‖f t-driverValue b t‖^4
-
-def rawClock (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℝ :=
-  ∫ u in Icc (0:ℝ) t, clockIntegrand b (flow κ b z) u
-
-lemma PositiveSolution.shifted_ne_zero {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) : f t-driverValue b t ≠ 0 := by
-  intro he
-  have hi := congrArg Complex.im he
-  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-  exact (hf.2.2.2.1 t ht).ne' hi
-
-lemma PositiveSolution.continuous_clockIntegrand {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : ContinuousOn (clockIntegrand b f) (Icc 0 T) := by
-  apply ((Complex.continuous_im.comp_continuousOn hf.2.1).pow 2).div
-    ((hf.2.1.sub (Complex.continuous_ofReal.comp (continuous_driverValue b)).continuousOn).norm.pow 4)
-  intro t ht
-  exact pow_ne_zero _ (norm_ne_zero_iff.mpr (hf.shifted_ne_zero ht))
-
-def timeClip (T t : ℝ) : ℝ := max 0 (min T t)
-lemma continuous_timeClip (T : ℝ) : Continuous (timeClip T) := continuous_const.max (continuous_const.min continuous_id)
-lemma timeClip_mem {T : ℝ} (hT : 0 ≤ T) (t : ℝ) : timeClip T t ∈ Icc 0 T :=
-  ⟨le_max_left _ _,max_le hT (min_le_left _ _)⟩
-lemma timeClip_eq {T t : ℝ} (ht : t ∈ Icc 0 T) : timeClip T t = t :=
-  by rw [timeClip,min_eq_right ht.2,max_eq_right ht.1]
-
- 
-
-def pathClock (b : Driver) (f : ℝ → ℂ) (T t : ℝ) : ℝ :=
-  ∫ u in (0:ℝ)..t, clockIntegrand b f (timeClip T u)
-
-lemma PositiveSolution.continuous_timeClippedIntegrand {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : Continuous (fun t => clockIntegrand b f (timeClip T t)) :=
-  hf.continuous_clockIntegrand.comp_continuous (continuous_timeClip T) (timeClip_mem hf.1)
-
-lemma PositiveSolution.pathClock_deriv {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    HasDerivAt (pathClock b f T) (clockIntegrand b f t) t := by
-  have hc := hf.continuous_timeClippedIntegrand
-  have hd := intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
-    hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-  convert hd using 1 <;> first | rfl | simp only [timeClip_eq ht]
-
-lemma PositiveSolution.pathClock_continuous {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : ContinuousOn (pathClock b f T) (Icc 0 T) :=
-  fun _ ht => (hf.pathClock_deriv ht).continuousAt.continuousWithinAt
-
-lemma PositiveSolution.rawClock_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    rawClock κ b z t = pathClock b f T t := by
-  rw [rawClock,integral_Icc_eq_integral_Ioc,pathClock,intervalIntegral.integral_of_le ht.1]
-  apply setIntegral_congr_fun measurableSet_Ioc
-  intro u hu
-  have huT : u ∈ Icc 0 T := ⟨hu.1.le,hu.2.trans ht.2⟩
-  dsimp only
-  rw [timeClip_eq huT]
-  simp only [clockIntegrand,hf.flow_eqOn huT]
-
-lemma PositiveSolution.pathClock_strictMono {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : StrictMonoOn (pathClock b f T) (Icc 0 T) := by
-  apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _) hf.pathClock_continuous
-    (fun t ht => (hf.pathClock_deriv (interior_subset ht)).hasDerivWithinAt)
-  intro t ht
-  unfold clockIntegrand
-  exact div_pos (sq_pos_of_pos (hf.2.2.2.1 t (interior_subset ht)))
-    (pow_pos (norm_pos_iff.mpr (hf.shifted_ne_zero (interior_subset ht))) _)
-
-lemma PositiveSolution.rawClock_strictMono {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : StrictMonoOn (rawClock κ b z) (Icc 0 T) := by
-  intro u hu v hv huv
-  rw [hf.rawClock_eq hu,hf.rawClock_eq hv]
-  exact hf.pathClock_strictMono hu hv huv
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-
-def rawLevelTime (κ : ℝ) (b : Driver) (z : ℂ) (s : ℝ) : ℝ≥0∞ :=
-  sInf {t | ∃ q : ℚ, 0 < q ∧ t = ENNReal.ofReal (q:ℝ) ∧
-    Surviving κ b z q ∧ s < rawClock κ b z q}
-
-lemma clock_eq_rawClock (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) :
-    clock κ θ b x t = rawClock κ b (diskMapInv θ x) t := rfl
-lemma levelTime_eq_rawLevelTime (κ θ : ℝ) (b : Driver) (s : ℝ) :
-    levelTime κ θ b s = rawLevelTime κ b (diskMapInv θ 0) s := rfl
-lemma rawClock_zero (κ : ℝ) (b : Driver) (z : ℂ) : rawClock κ b z 0 = 0 := by
-  simp [rawClock]
-
-lemma PositiveSolution.rawClock_continuous {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) : ContinuousOn (rawClock κ b z) (Icc 0 T) :=
-  hf.pathClock_continuous.congr (fun _ ht => hf.rawClock_eq ht)
-
- 
-
-lemma PositiveSolution.rawLevelTime_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) :
-    rawLevelTime κ b z (rawClock κ b z T) = ENNReal.ofReal T := by
-  apply le_antisymm
-  · obtain ⟨U,hTU,g,hg,_⟩ := hf.extend
-    apply ENNReal.le_of_forall_pos_le_add
-    intro ε hε _
-    have hεr : (0:ℝ) < ε := by exact_mod_cast hε
-    have hbar : T < min U (T+ε) := lt_min hTU (by linarith)
-    obtain ⟨q,hTq,hq⟩ := exists_rat_btwn hbar
-    have hq0 : (0:ℝ) < q := hf.1.trans_lt hTq
-    have hqU : (q:ℝ) ≤ U := (hq.trans_le (min_le_left _ _)).le
-    have hcq := hg.rawClock_strictMono ⟨hf.1,hTU.le⟩ ⟨hq0.le,hqU⟩ hTq
-    have hle : rawLevelTime κ b z (rawClock κ b z T) ≤ ENNReal.ofReal (q:ℝ) :=
-      sInf_le ⟨q,by exact_mod_cast hq0,rfl,(hg.restrict hq0.le hqU).surviving,hcq⟩
-    refine hle.trans ?_
-    calc
-      ENNReal.ofReal (q:ℝ) ≤ ENNReal.ofReal (T+(ε:ℝ)) :=
-        ENNReal.ofReal_le_ofReal (hq.trans_le (min_le_right _ _)).le
-      _ = ENNReal.ofReal T + ε := by
-        convert ENNReal.ofReal_add hf.1 ε.2 using 1
-        all_goals try rfl
-        exact congrArg (fun x : ℝ≥0∞ => ENNReal.ofReal T + x)
-          (ENNReal.ofReal_coe_nnreal (p := ε)).symm
-  · apply le_sInf
-    rintro _ ⟨q,hq,rfl,_,hclock⟩
-    apply ENNReal.ofReal_le_ofReal
-    by_contra hTq
-    have hq0 : (0:ℝ) ≤ q := by exact_mod_cast hq.le
-    have hqt : (q:ℝ) < T := lt_of_not_ge hTq
-    exact (not_lt_of_ge (hf.rawClock_strictMono ⟨hq0,hqt.le⟩ ⟨hf.1,le_rfl⟩ hqt).le) hclock
-
-lemma rawLevelTime_eq_top_of_no_hit {κ : ℝ} {b : Driver} {z : ℂ} {s : ℝ}
-    (hs : 0 ≤ s)
-    (hn : ¬ ∃ T f, PositiveSolution κ b z T f ∧ rawClock κ b z T = s) :
-    rawLevelTime κ b z s = ∞ := by
-  apply top_unique
-  apply le_sInf
-  rintro _ ⟨q,hq,rfl,hqS,hqs⟩
-  exfalso
-  have hq0 : (0:ℝ) ≤ q := by exact_mod_cast hq.le
-  let hf := surviving_flow_positiveSolution hq0 hqS
-  have hr : s ∈ Icc (rawClock κ b z 0) (rawClock κ b z q) :=
-    ⟨by simpa only [rawClock_zero] using hs,hqs.le⟩
-  obtain ⟨T,hT,hTs⟩ := intermediate_value_Icc hq0 hf.rawClock_continuous hr
-  exact hn ⟨T,flow κ b z,hf.restrict hT.1 hT.2,hTs⟩
-
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric
-open scoped NNReal Topology
-
-lemma cutoffField_lipschitz (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j) (t : ℝ) :
-    LipschitzWith ⟨|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2,by positivity⟩ (cutoffField κ b j t) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  simp only [dist_eq_norm,cutoffField_eq]
-  exact Cutoff.field_sub_le (zpow_pos (by exact_mod_cast (Nat.zero_lt_of_lt hj)) _) _ _ _ _ _
-
- 
-
-lemma PositiveSolution.initial_eq_of_terminal_eq {κ : ℝ} {b : Driver} {z w : ℂ}
-    {T : ℝ} {f g : ℝ → ℂ} (hf : PositiveSolution κ b z T f)
-    (hg : PositiveSolution κ b w T g) (hfg : f T = g T) : z = w := by
-  obtain ⟨j,hj,hjf⟩ := hf.exists_cutoff
-  obtain ⟨k,hk,hkg⟩ := hg.exists_cutoff
-  let n := max j k
-  have hn : 1 ≤ n := hj.trans (le_max_left _ _)
-  have hinj : (n:ℝ)⁻¹ ≤ (j:ℝ)⁻¹ :=
-    inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hj)) (by exact_mod_cast le_max_left j k)
-  have hink : (n:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ :=
-    inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hk)) (by exact_mod_cast le_max_right j k)
-  have hd1 (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt f (cutoffField κ b n t (f t)) (Icc 0 T) t := by
-    rw [cutoffField_of_height κ b hn (hinj.trans (hjf t ht).le)]
-    exact hf.2.2.2.2 t ht
-  have hd2 (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt g (cutoffField κ b n t (g t)) (Icc 0 T) t := by
-    rw [cutoffField_of_height κ b hn (hink.trans (hkg t ht).le)]
-    exact hg.2.2.2.2 t ht
-  have he : EqOn f g (Icc 0 T) := by
-    apply ODE_solution_unique_of_mem_Icc_left
-      (v := cutoffField κ b n) (s := fun _ => univ)
-      (K := ⟨|a κ| *3/((n:ℝ)^(-(2:ℤ)))^2,by positivity⟩)
-      (fun t _ => (cutoffField_lipschitz κ b hn t).lipschitzOnWith)
-      hf.2.1 _ (fun _ _ => mem_univ _) hg.2.1 _ (fun _ _ => mem_univ _) hfg
-    · intro t ht
-      exact (hd1 t ⟨ht.1.le,ht.2⟩).mono_of_mem_nhdsWithin
-        (Filter.mem_of_superset (Icc_mem_nhdsLE ht.1) (Icc_subset_Icc_right ht.2))
-    · intro t ht
-      exact (hd2 t ⟨ht.1.le,ht.2⟩).mono_of_mem_nhdsWithin
-        (Filter.mem_of_superset (Icc_mem_nhdsLE ht.1) (Icc_subset_Icc_right ht.2))
-  exact hf.2.2.1.symm.trans ((he ⟨le_rfl,hf.1⟩).trans hg.2.2.1)
-
-lemma flow_injOn (κ : ℝ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
-    InjOn (fun z => flow κ b z T) {z | Surviving κ b z T} := by
-  intro z hz w hw he
-  exact (surviving_flow_positiveSolution hT hz).initial_eq_of_terminal_eq
-    (surviving_flow_positiveSolution hT hw) he
-
-lemma inverseFlow_spec {κ : ℝ} {b : Driver} {T : ℝ} {w : ℂ}
-    (hw : ∃ z, Surviving κ b z T ∧ flow κ b z T = w) :
-    Surviving κ b (inverseFlow κ b T w) T ∧ flow κ b (inverseFlow κ b T w) T = w := by
-  unfold inverseFlow
-  rw [ite_eq_left hw]
-  exact Classical.epsilon_spec hw
-
-lemma inverseFlow_flow {κ : ℝ} {b : Driver} {T : ℝ} (hT : 0 ≤ T) {z : ℂ}
-    (hz : Surviving κ b z T) : inverseFlow κ b T (flow κ b z T) = z := by
-  have h := inverseFlow_spec (show ∃ w, Surviving κ b w T ∧ flow κ b w T = flow κ b z T from
-    ⟨z,hz,rfl⟩)
-  exact flow_injOn κ b hT h.1 hz h.2
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric MeasureTheory
-open scoped Topology
-
- 
-lemma reverse_height_formula {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
-    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t) :
-    ∀ t ∈ Icc 0 T, y t = y 0 * Real.exp (∫ u in (0:ℝ)..t, r (timeClip T u)) := by
-  let A (t : ℝ) := ∫ u in (0:ℝ)..t, r (timeClip T u)
-  have hc : Continuous (fun u => r (timeClip T u)) :=
-    hr.comp_continuous (continuous_timeClip T) (timeClip_mem hT)
-  have hA (t : ℝ) : HasDerivAt A (r (timeClip T t)) t :=
-    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
-      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-  have hA' : Continuous A := continuous_iff_continuousAt.mpr fun t => (hA t).continuousAt
-  have hd0 (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt (fun u => y u * Real.exp (-A u)) 0 (Icc 0 T) t := by
-    have he := (hd t ht).mul ((hA t).neg.exp.hasDerivWithinAt)
-    rw [timeClip_eq ht] at he
-    convert! he using 1
-    ring
-  have hconst := constant_of_has_deriv_right_zero
-    (hy.mul ((Real.continuous_exp.comp hA'.neg).continuousOn)) (fun t (ht : t ∈ Ico 0 T) =>
-      (hd0 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
-        (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1)))
-  intro t ht
-  have he := hconst t ht
-  dsimp only [Pi.mul_apply,Function.comp_def,Pi.neg_apply] at he
-  have hA0 : A 0 = 0 := by simp [A]
-  rw [hA0,neg_zero,Real.exp_zero,mul_one] at he
-  calc
-    y t = (y t * Real.exp (-A t))*Real.exp (A t) := by
-      rw [mul_assoc,← Real.exp_add,neg_add_cancel,Real.exp_zero,mul_one]
-    _ = y 0 * Real.exp (A t) := by rw [he]
-
-lemma reverse_height_le {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
-    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t)
-    (hy0 : 0 ≤ y 0) (hr0 : ∀ t ∈ Icc 0 T, 0 ≤ r t) (t : ℝ) (ht : t ∈ Icc 0 T) :
-    y 0 ≤ y t := by
-  rw [reverse_height_formula hT hy hr hd t ht]
-  apply le_mul_of_one_le_right hy0
-  apply Real.one_le_exp_iff.mpr
-  apply intervalIntegral.integral_nonneg ht.1
-  intro u _
-  exact hr0 (timeClip T u) (timeClip_mem hT u)
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.Cutoff
-open Set
-lemma field_im (a q : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ) :
-    (field a q b t z).im = -a*z.im/denominator q (z-b t) := by
-  simp [field,inverse,Complex.mul_im,Complex.div_ofReal_im]
-  ring
-
-lemma field_eq_div {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ)
-    (hz : q ≤ z.im) : field a q b t z = (a:ℂ)/(z-b t) := by
-  have hqn : q ≤ ‖z-b t‖ := hz.trans (by simpa using Complex.im_le_norm (z-b t))
-  have hsq : q^2 ≤ ‖z-b t‖^2 := pow_le_pow_left₀ hq.le hqn 2
-  simp only [field,inverse,denominator,max_eq_left hsq,div_eq_mul_inv]
-  rw [Complex.inv_def (z-b t)]
-  simp only [Complex.normSq_eq_norm_sq,Complex.ofReal_inv,starRingEnd_apply]
-end SLEExactGauge.Cutoff
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric MeasureTheory
-open scoped Topology
-
- 
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.EulerFlow
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.LiteralTrace
+lemma PositiveSolution.surviving {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : Surviving κ b z T := by
+  obtain ⟨j,hj,hgap⟩ := hf.exists_cutoff
+  have he := hf.cutoff_eq hj (fun t ht => (hgap t ht).le)
+  obtain ⟨t0,ht0,hmin⟩ := isCompact_Icc.exists_isMinOn ⟨0,le_rfl,hf.1⟩
+    (Complex.continuous_im.comp_continuousOn hf.2.1)
+  refine ⟨j,hj,(hgap t0 ht0).trans_le ?_⟩
+  apply le_csInf (s := (fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)
+    (Set.Nonempty.image _ ⟨0,le_rfl,hf.1⟩)
+  rintro y ⟨t,ht,rfl⟩
+  dsimp only
+  rw [he ht]
+  exact hmin ht
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+
+lemma cutoff_positiveSolution {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} (hT : 0 ≤ T)
+    {j : ℕ} (hj : 1 ≤ j)
+    (hgap : (j:ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)) :
+    PositiveSolution κ b z T (cutoffFlow κ b j z) := by
+  have hc := cutoffFlow_continuous κ b hj z hT
+  have hb := (isCompact_Icc.image_of_continuousOn (Complex.continuous_im.comp_continuousOn hc)).bddBelow
+  have hg t (ht : t ∈ Icc 0 T) : (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im :=
+    hgap.trans_le (csInf_le hb ⟨t,ht,rfl⟩)
+  refine ⟨hT,hc,(cutoffFlow_solution κ b j hj z hT).2.1,?_,?_⟩
+  · intro t ht
+    exact lt_trans (inv_pos.mpr (by exact_mod_cast (Nat.zero_lt_of_lt hj))) (hg t ht)
+  · intro t ht
+    rw [← cutoffField_of_height κ b hj (hg t ht).le]
+    exact (cutoffFlow_solution κ b j hj z hT).2.2 t ht
+
+lemma PositiveSolution.restrict {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : 0 ≤ t) (htT : t ≤ T) :
+    PositiveSolution κ b z t f := by
+  have hi : Icc 0 t ⊆ Icc 0 T := Icc_subset_Icc_right htT
+  exact ⟨ht,hf.2.1.mono hi,hf.2.2.1,fun u hu => hf.2.2.2.1 u (hi hu),
+    fun u hu => (hf.2.2.2.2 u (hi hu)).mono hi⟩
+
+lemma PositiveSolution.flow_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : flow κ b z T = f T := by
+  have hs := hf.surviving
+  have hn := Nat.find_spec hs
+  rw [flow,dite_eq_left hs]
+  exact (cutoff_positiveSolution hf.1 hn.1 hn.2).unique hf ⟨hf.1,le_rfl⟩
+
+lemma PositiveSolution.flow_eqOn {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : EqOn (flow κ b z) f (Icc 0 T) :=
+  fun _ ht => (hf.restrict ht.1 ht.2).flow_eq
+
+lemma surviving_iff_positiveSolution {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} (hT : 0 ≤ T) :
+    Surviving κ b z T ↔ ∃ f, PositiveSolution κ b z T f := by
+  constructor
+  · rintro ⟨j,hj,hgap⟩
+    exact ⟨cutoffFlow κ b j z,cutoff_positiveSolution hT hj hgap⟩
+  · rintro ⟨f,hf⟩
+    exact hf.surviving
+
+lemma surviving_flow_positiveSolution {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hs : Surviving κ b z T) : PositiveSolution κ b z T (flow κ b z) := by
+  obtain ⟨f,hf⟩ := (surviving_iff_positiveSolution hT).mp hs
+  have he := hf.flow_eqOn
+  refine ⟨hT,hf.2.1.congr (fun t ht => he ht),?_,?_,?_⟩
+  · rw [he ⟨le_rfl,hT⟩,hf.2.2.1]
+  · intro t ht
+    rw [he ht]
+    exact hf.2.2.2.1 t ht
+  · intro t ht
+    rw [he ht]
+    exact (hf.2.2.2.2 t ht).congr (fun u hu => he hu) (he ht)
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+
+ 
+
+lemma PositiveSolution.extend {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) :
+    ∃ U : ℝ, T < U ∧ ∃ g : ℝ → ℂ, PositiveSolution κ b z U g ∧ EqOn g f (Icc 0 T) := by
+  obtain ⟨j,hj,hgap⟩ := hf.exists_cutoff
+  let g := cutoffFlow κ b j z
+  have he : EqOn g f (Icc 0 T) := hf.cutoff_eq hj (fun t ht => (hgap t ht).le)
+  have hc := cutoffFlow_continuous κ b hj z (show 0 ≤ T+1 by linarith [hf.1])
+  have hy : (j:ℝ)⁻¹ < (g T).im := by rw [he ⟨hf.1,le_rfl⟩]; exact hgap T ⟨hf.1,le_rfl⟩
+  have hci := (Complex.continuous_im.comp_continuousOn hc) T (show T ∈ Icc 0 (T+1) by constructor <;> linarith [hf.1])
+  obtain ⟨δ,hδ,hnear⟩ := Metric.continuousWithinAt_iff.mp hci ((g T).im-(j:ℝ)⁻¹) (sub_pos.mpr hy)
+  let U := T+min (δ/2) (1/2)
+  have hTU : T < U := by dsimp [U]; have := lt_min (by linarith : 0 < δ/2) (by norm_num : (0:ℝ)<1/2); linarith
+  have hU1 : U ≤ T+1 := by dsimp [U]; have := min_le_right (δ/2) (1/2:ℝ); linarith
+  have hU : 0 ≤ U := hf.1.trans hTU.le
+  have hg t (ht : t ∈ Icc 0 U) : (j:ℝ)⁻¹ < (g t).im := by
+    by_cases htT : t ≤ T
+    · rw [he ⟨ht.1,htT⟩]
+      exact hgap _ ⟨ht.1,htT⟩
+    · have hdist : dist t T < δ := by
+        rw [Real.dist_eq,abs_of_nonneg (by linarith)]
+        have hδU : U ≤ T+δ/2 := by dsimp [U]; exact add_le_add le_rfl (min_le_left (δ/2) (1/2:ℝ))
+        linarith [ht.2]
+      have hn := hnear ⟨ht.1,ht.2.trans hU1⟩ hdist
+      rw [Real.dist_eq] at hn
+      have hh := (abs_lt.mp hn).1
+      change -((g T).im-(j:ℝ)⁻¹) < (g t).im-(g T).im at hh
+      linarith
+  refine ⟨U,hTU,g,⟨hU,hc.mono (Icc_subset_Icc_right hU1),?_,?_,?_⟩,he⟩
+  · exact (cutoffFlow_solution κ b j hj z hU).2.1
+  · intro t ht
+    exact (inv_pos.mpr (by exact_mod_cast (Nat.zero_lt_of_lt hj))).trans (hg t ht)
+  · intro t ht
+    rw [← cutoffField_of_height κ b hj (hg t ht).le]
+    exact (cutoffFlow_solution κ b j hj z hU).2.2 t ht
+
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+
+def clockIntegrand (b : Driver) (f : ℝ → ℂ) (t : ℝ) : ℝ :=
+  (f t).im^2/‖f t-driverValue b t‖^4
+
+def rawClock (κ : ℝ) (b : Driver) (z : ℂ) (t : ℝ) : ℝ :=
+  ∫ u in Icc (0:ℝ) t, clockIntegrand b (flow κ b z) u
+
+lemma PositiveSolution.shifted_ne_zero {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) : f t-driverValue b t ≠ 0 := by
+  intro he
+  have hi := congrArg Complex.im he
+  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+  exact (hf.2.2.2.1 t ht).ne' hi
+
+lemma PositiveSolution.continuous_clockIntegrand {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : ContinuousOn (clockIntegrand b f) (Icc 0 T) := by
+  apply ((Complex.continuous_im.comp_continuousOn hf.2.1).pow 2).div
+    ((hf.2.1.sub (Complex.continuous_ofReal.comp (continuous_driverValue b)).continuousOn).norm.pow 4)
+  intro t ht
+  exact pow_ne_zero _ (norm_ne_zero_iff.mpr (hf.shifted_ne_zero ht))
+
+def timeClip (T t : ℝ) : ℝ := max 0 (min T t)
+lemma continuous_timeClip (T : ℝ) : Continuous (timeClip T) := continuous_const.max (continuous_const.min continuous_id)
+lemma timeClip_mem {T : ℝ} (hT : 0 ≤ T) (t : ℝ) : timeClip T t ∈ Icc 0 T :=
+  ⟨le_max_left _ _,max_le hT (min_le_left _ _)⟩
+lemma timeClip_eq {T t : ℝ} (ht : t ∈ Icc 0 T) : timeClip T t = t :=
+  by rw [timeClip,min_eq_right ht.2,max_eq_right ht.1]
+
+ 
+
+def pathClock (b : Driver) (f : ℝ → ℂ) (T t : ℝ) : ℝ :=
+  ∫ u in (0:ℝ)..t, clockIntegrand b f (timeClip T u)
+
+lemma PositiveSolution.continuous_timeClippedIntegrand {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : Continuous (fun t => clockIntegrand b f (timeClip T t)) :=
+  hf.continuous_clockIntegrand.comp_continuous (continuous_timeClip T) (timeClip_mem hf.1)
+
+lemma PositiveSolution.pathClock_deriv {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    HasDerivAt (pathClock b f T) (clockIntegrand b f t) t := by
+  have hc := hf.continuous_timeClippedIntegrand
+  have hd := intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
+    hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+  convert hd using 1 <;> first | rfl | simp only [timeClip_eq ht]
+
+lemma PositiveSolution.pathClock_continuous {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : ContinuousOn (pathClock b f T) (Icc 0 T) :=
+  fun _ ht => (hf.pathClock_deriv ht).continuousAt.continuousWithinAt
+
+lemma PositiveSolution.rawClock_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    rawClock κ b z t = pathClock b f T t := by
+  rw [rawClock,integral_Icc_eq_integral_Ioc,pathClock,intervalIntegral.integral_of_le ht.1]
+  apply setIntegral_congr_fun measurableSet_Ioc
+  intro u hu
+  have huT : u ∈ Icc 0 T := ⟨hu.1.le,hu.2.trans ht.2⟩
+  dsimp only
+  rw [timeClip_eq huT]
+  simp only [clockIntegrand,hf.flow_eqOn huT]
+
+lemma PositiveSolution.pathClock_strictMono {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : StrictMonoOn (pathClock b f T) (Icc 0 T) := by
+  apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _) hf.pathClock_continuous
+    (fun t ht => (hf.pathClock_deriv (interior_subset ht)).hasDerivWithinAt)
+  intro t ht
+  unfold clockIntegrand
+  exact div_pos (sq_pos_of_pos (hf.2.2.2.1 t (interior_subset ht)))
+    (pow_pos (norm_pos_iff.mpr (hf.shifted_ne_zero (interior_subset ht))) _)
+
+lemma PositiveSolution.rawClock_strictMono {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : StrictMonoOn (rawClock κ b z) (Icc 0 T) := by
+  intro u hu v hv huv
+  rw [hf.rawClock_eq hu,hf.rawClock_eq hv]
+  exact hf.pathClock_strictMono hu hv huv
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+
+def rawLevelTime (κ : ℝ) (b : Driver) (z : ℂ) (s : ℝ) : ℝ≥0∞ :=
+  sInf {t | ∃ q : ℚ, 0 < q ∧ t = ENNReal.ofReal (q:ℝ) ∧
+    Surviving κ b z q ∧ s < rawClock κ b z q}
+
+lemma clock_eq_rawClock (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ) :
+    clock κ θ b x t = rawClock κ b (diskMapInv θ x) t := rfl
+lemma levelTime_eq_rawLevelTime (κ θ : ℝ) (b : Driver) (s : ℝ) :
+    levelTime κ θ b s = rawLevelTime κ b (diskMapInv θ 0) s := rfl
+lemma rawClock_zero (κ : ℝ) (b : Driver) (z : ℂ) : rawClock κ b z 0 = 0 := by
+  simp [rawClock]
+
+lemma PositiveSolution.rawClock_continuous {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) : ContinuousOn (rawClock κ b z) (Icc 0 T) :=
+  hf.pathClock_continuous.congr (fun _ ht => hf.rawClock_eq ht)
+
+ 
+
+lemma PositiveSolution.rawLevelTime_eq {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) :
+    rawLevelTime κ b z (rawClock κ b z T) = ENNReal.ofReal T := by
+  apply le_antisymm
+  · obtain ⟨U,hTU,g,hg,_⟩ := hf.extend
+    apply ENNReal.le_of_forall_pos_le_add
+    intro ε hε _
+    have hεr : (0:ℝ) < ε := by exact_mod_cast hε
+    have hbar : T < min U (T+ε) := lt_min hTU (by linarith)
+    obtain ⟨q,hTq,hq⟩ := exists_rat_btwn hbar
+    have hq0 : (0:ℝ) < q := hf.1.trans_lt hTq
+    have hqU : (q:ℝ) ≤ U := (hq.trans_le (min_le_left _ _)).le
+    have hcq := hg.rawClock_strictMono ⟨hf.1,hTU.le⟩ ⟨hq0.le,hqU⟩ hTq
+    have hle : rawLevelTime κ b z (rawClock κ b z T) ≤ ENNReal.ofReal (q:ℝ) :=
+      sInf_le ⟨q,by exact_mod_cast hq0,rfl,(hg.restrict hq0.le hqU).surviving,hcq⟩
+    refine hle.trans ?_
+    calc
+      ENNReal.ofReal (q:ℝ) ≤ ENNReal.ofReal (T+(ε:ℝ)) :=
+        ENNReal.ofReal_le_ofReal (hq.trans_le (min_le_right _ _)).le
+      _ = ENNReal.ofReal T + ε := by
+        convert ENNReal.ofReal_add hf.1 ε.2 using 1
+        all_goals try rfl
+        exact congrArg (fun x : ℝ≥0∞ => ENNReal.ofReal T + x)
+          (ENNReal.ofReal_coe_nnreal (p := ε)).symm
+  · apply le_sInf
+    rintro _ ⟨q,hq,rfl,_,hclock⟩
+    apply ENNReal.ofReal_le_ofReal
+    by_contra hTq
+    have hq0 : (0:ℝ) ≤ q := by exact_mod_cast hq.le
+    have hqt : (q:ℝ) < T := lt_of_not_ge hTq
+    exact (not_lt_of_ge (hf.rawClock_strictMono ⟨hq0,hqt.le⟩ ⟨hf.1,le_rfl⟩ hqt).le) hclock
+
+lemma rawLevelTime_eq_top_of_no_hit {κ : ℝ} {b : Driver} {z : ℂ} {s : ℝ}
+    (hs : 0 ≤ s)
+    (hn : ¬ ∃ T f, PositiveSolution κ b z T f ∧ rawClock κ b z T = s) :
+    rawLevelTime κ b z s = ∞ := by
+  apply top_unique
+  apply le_sInf
+  rintro _ ⟨q,hq,rfl,hqS,hqs⟩
+  exfalso
+  have hq0 : (0:ℝ) ≤ q := by exact_mod_cast hq.le
+  let hf := surviving_flow_positiveSolution hq0 hqS
+  have hr : s ∈ Icc (rawClock κ b z 0) (rawClock κ b z q) :=
+    ⟨by simpa only [rawClock_zero] using hs,hqs.le⟩
+  obtain ⟨T,hT,hTs⟩ := intermediate_value_Icc hq0 hf.rawClock_continuous hr
+  exact hn ⟨T,flow κ b z,hf.restrict hT.1 hT.2,hTs⟩
+
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric
+open scoped NNReal Topology
+
+lemma cutoffField_lipschitz (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j) (t : ℝ) :
+    LipschitzWith ⟨|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2,by positivity⟩ (cutoffField κ b j t) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  simp only [dist_eq_norm,cutoffField_eq]
+  exact Cutoff.field_sub_le (zpow_pos (by exact_mod_cast (Nat.zero_lt_of_lt hj)) _) _ _ _ _ _
+
+ 
+
+lemma PositiveSolution.initial_eq_of_terminal_eq {κ : ℝ} {b : Driver} {z w : ℂ}
+    {T : ℝ} {f g : ℝ → ℂ} (hf : PositiveSolution κ b z T f)
+    (hg : PositiveSolution κ b w T g) (hfg : f T = g T) : z = w := by
+  obtain ⟨j,hj,hjf⟩ := hf.exists_cutoff
+  obtain ⟨k,hk,hkg⟩ := hg.exists_cutoff
+  let n := max j k
+  have hn : 1 ≤ n := hj.trans (le_max_left _ _)
+  have hinj : (n:ℝ)⁻¹ ≤ (j:ℝ)⁻¹ :=
+    inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hj)) (by exact_mod_cast le_max_left j k)
+  have hink : (n:ℝ)⁻¹ ≤ (k:ℝ)⁻¹ :=
+    inv_anti₀ (by exact_mod_cast (Nat.zero_lt_of_lt hk)) (by exact_mod_cast le_max_right j k)
+  have hd1 (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt f (cutoffField κ b n t (f t)) (Icc 0 T) t := by
+    rw [cutoffField_of_height κ b hn (hinj.trans (hjf t ht).le)]
+    exact hf.2.2.2.2 t ht
+  have hd2 (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt g (cutoffField κ b n t (g t)) (Icc 0 T) t := by
+    rw [cutoffField_of_height κ b hn (hink.trans (hkg t ht).le)]
+    exact hg.2.2.2.2 t ht
+  have he : EqOn f g (Icc 0 T) := by
+    apply ODE_solution_unique_of_mem_Icc_left
+      (v := cutoffField κ b n) (s := fun _ => univ)
+      (K := ⟨|a κ| *3/((n:ℝ)^(-(2:ℤ)))^2,by positivity⟩)
+      (fun t _ => (cutoffField_lipschitz κ b hn t).lipschitzOnWith)
+      hf.2.1 _ (fun _ _ => mem_univ _) hg.2.1 _ (fun _ _ => mem_univ _) hfg
+    · intro t ht
+      exact (hd1 t ⟨ht.1.le,ht.2⟩).mono_of_mem_nhdsWithin
+        (Filter.mem_of_superset (Icc_mem_nhdsLE ht.1) (Icc_subset_Icc_right ht.2))
+    · intro t ht
+      exact (hd2 t ⟨ht.1.le,ht.2⟩).mono_of_mem_nhdsWithin
+        (Filter.mem_of_superset (Icc_mem_nhdsLE ht.1) (Icc_subset_Icc_right ht.2))
+  exact hf.2.2.1.symm.trans ((he ⟨le_rfl,hf.1⟩).trans hg.2.2.1)
+
+lemma flow_injOn (κ : ℝ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
+    InjOn (fun z => flow κ b z T) {z | Surviving κ b z T} := by
+  intro z hz w hw he
+  exact (surviving_flow_positiveSolution hT hz).initial_eq_of_terminal_eq
+    (surviving_flow_positiveSolution hT hw) he
+
+lemma inverseFlow_spec {κ : ℝ} {b : Driver} {T : ℝ} {w : ℂ}
+    (hw : ∃ z, Surviving κ b z T ∧ flow κ b z T = w) :
+    Surviving κ b (inverseFlow κ b T w) T ∧ flow κ b (inverseFlow κ b T w) T = w := by
+  unfold inverseFlow
+  rw [ite_eq_left hw]
+  exact Classical.epsilon_spec hw
+
+lemma inverseFlow_flow {κ : ℝ} {b : Driver} {T : ℝ} (hT : 0 ≤ T) {z : ℂ}
+    (hz : Surviving κ b z T) : inverseFlow κ b T (flow κ b z T) = z := by
+  have h := inverseFlow_spec (show ∃ w, Surviving κ b w T ∧ flow κ b w T = flow κ b z T from
+    ⟨z,hz,rfl⟩)
+  exact flow_injOn κ b hT h.1 hz h.2
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric MeasureTheory
+open scoped Topology
+
+ 
+lemma reverse_height_formula {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
+    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t) :
+    ∀ t ∈ Icc 0 T, y t = y 0 * Real.exp (∫ u in (0:ℝ)..t, r (timeClip T u)) := by
+  let A (t : ℝ) := ∫ u in (0:ℝ)..t, r (timeClip T u)
+  have hc : Continuous (fun u => r (timeClip T u)) :=
+    hr.comp_continuous (continuous_timeClip T) (timeClip_mem hT)
+  have hA (t : ℝ) : HasDerivAt A (r (timeClip T t)) t :=
+    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
+      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+  have hA' : Continuous A := continuous_iff_continuousAt.mpr fun t => (hA t).continuousAt
+  have hd0 (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt (fun u => y u * Real.exp (-A u)) 0 (Icc 0 T) t := by
+    have he := (hd t ht).mul ((hA t).neg.exp.hasDerivWithinAt)
+    rw [timeClip_eq ht] at he
+    convert! he using 1
+    ring
+  have hconst := constant_of_has_deriv_right_zero
+    (hy.mul ((Real.continuous_exp.comp hA'.neg).continuousOn)) (fun t (ht : t ∈ Ico 0 T) =>
+      (hd0 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
+        (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1)))
+  intro t ht
+  have he := hconst t ht
+  dsimp only [Pi.mul_apply,Function.comp_def,Pi.neg_apply] at he
+  have hA0 : A 0 = 0 := by simp [A]
+  rw [hA0,neg_zero,Real.exp_zero,mul_one] at he
+  calc
+    y t = (y t * Real.exp (-A t))*Real.exp (A t) := by
+      rw [mul_assoc,← Real.exp_add,neg_add_cancel,Real.exp_zero,mul_one]
+    _ = y 0 * Real.exp (A t) := by rw [he]
+
+lemma reverse_height_le {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
+    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t)
+    (hy0 : 0 ≤ y 0) (hr0 : ∀ t ∈ Icc 0 T, 0 ≤ r t) (t : ℝ) (ht : t ∈ Icc 0 T) :
+    y 0 ≤ y t := by
+  rw [reverse_height_formula hT hy hr hd t ht]
+  apply le_mul_of_one_le_right hy0
+  apply Real.one_le_exp_iff.mpr
+  apply intervalIntegral.integral_nonneg ht.1
+  intro u _
+  exact hr0 (timeClip T u) (timeClip_mem hT u)
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.Cutoff
+open Set
+lemma field_im (a q : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ) :
+    (field a q b t z).im = -a*z.im/denominator q (z-b t) := by
+  simp [field,inverse,Complex.mul_im,Complex.div_ofReal_im]
+  ring
+
+lemma field_eq_div {q : ℝ} (hq : 0 < q) (a : ℝ) (b : ℝ → ℝ) (t : ℝ) (z : ℂ)
+    (hz : q ≤ z.im) : field a q b t z = (a:ℂ)/(z-b t) := by
+  have hqn : q ≤ ‖z-b t‖ := hz.trans (by simpa using Complex.im_le_norm (z-b t))
+  have hsq : q^2 ≤ ‖z-b t‖^2 := pow_le_pow_left₀ hq.le hqn 2
+  simp only [field,inverse,denominator,max_eq_left hsq,div_eq_mul_inv]
+  rw [Complex.inv_def (z-b t)]
+  simp only [Complex.normSq_eq_norm_sq,Complex.ofReal_inv,starRingEnd_apply]
+end SLEExactGauge.Cutoff
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric MeasureTheory
+open scoped Topology
+
+ 
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/InverseRegularity.lean b/lean/OAI/Probability/SLEGauge/InverseRegularity.lean
index 0f4c72ca7b73eb298fc98c4caa3b14f71494e3bb..104da31863e0f2f58f272c2a9f71d64221c4b229 100644
--- a/lean/OAI/Probability/SLEGauge/InverseRegularity.lean
+++ b/lean/OAI/Probability/SLEGauge/InverseRegularity.lean
@@ -1,465 +1,465 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.InverseDomains
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric MeasureTheory
-open scoped Topology
-theorem exists_inverseFlow (κ : ℝ) (hκ : 0 < κ) (b : Driver) {T : ℝ}
-    (hT : 0 ≤ T) (w : ℂ) (hw : 0 < w.im) :
-    ∃ z, Surviving κ b z T ∧ flow κ b z T = w := by
-  let q : ℝ := w.im/2
-  have hq : 0 < q := by dsimp [q]; positivity
-  let br (t : ℝ) : ℝ := driverValue b (T-t)
-  have hbr : Continuous br := (continuous_driverValue b).comp (continuous_const.sub continuous_id)
-  obtain ⟨g,hg0,hgd⟩ := Cutoff.exists_solution_on_compact hq (-a κ) hbr w hT
-  have hgc : ContinuousOn g (Icc 0 T) := fun t ht => (hgd t ht).continuousWithinAt
-  let r (t : ℝ) : ℝ := a κ / Cutoff.denominator q (g t-br t)
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hr : ContinuousOn r (Icc 0 T) := by
-    apply continuousOn_const.div
-    · exact ((continuous_norm.comp_continuousOn
-        (hgc.sub ((Complex.continuous_ofReal.comp hbr).continuousOn))).pow 2).sup continuousOn_const
-    · intro t _
-      exact (Cutoff.denominator_pos hq _).ne'
-  have hyd (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt (fun u => (g u).im) (r t*(g t).im) (Icc 0 T) t := by
-    have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hgd t ht)
-    convert! hi using 1
-    simp only [Complex.imCLM_apply,Cutoff.field_im,neg_neg,r,div_mul_eq_mul_div]
-  have hheight (t : ℝ) (ht : t ∈ Icc 0 T) : w.im ≤ (g t).im := by
-    have hh := reverse_height_le hT (Complex.continuous_im.comp_continuousOn hgc) hr hyd
-      (by simpa [hg0] using hw.le) (fun u _ => div_nonneg ha.le (Cutoff.denominator_pos hq _).le) t ht
-    simpa [hg0] using hh
-  have hrev (t : ℝ) (ht : t ∈ Icc 0 T) : T-t ∈ Icc 0 T := by
-    constructor <;> linarith [ht.1,ht.2]
-  let f (t : ℝ) : ℂ := g (T-t)
-  have hf : PositiveSolution κ b (g T) T f := by
-    refine ⟨hT,hgc.comp (continuous_const.sub continuous_id).continuousOn hrev,?_,?_,?_⟩
-    · simp [f]
-    · intro t ht
-      exact hw.trans_le (hheight (T-t) (hrev t ht))
-    · intro t ht
-      have hcut := Cutoff.field_eq_div hq (-a κ) br (T-t) (g (T-t))
-        ((by dsimp [q]; linarith : q ≤ w.im).trans (hheight (T-t) (hrev t ht)))
-      have hd := (hgd (T-t) (hrev t ht)).scomp t
-        (((hasDerivAt_const t T).sub (hasDerivAt_id t)).hasDerivWithinAt) hrev
-      rw [hcut] at hd
-      convert! hd using 1
-      simp [f,br]
-      ring
-  refine ⟨g T,hf.surviving,?_⟩
-  rw [hf.flow_eq]
-  simpa [f] using hg0
-
-lemma inverseFlow_surviving {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ} (hT : 0 ≤ T)
-    {w : ℂ} (hw : 0 < w.im) : Surviving κ b (inverseFlow κ b T w) T :=
-  (inverseFlow_spec (exists_inverseFlow κ hκ b hT w hw)).1
-
-lemma flow_inverseFlow {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ} (hT : 0 ≤ T)
-    {w : ℂ} (hw : 0 < w.im) : flow κ b (inverseFlow κ b T w) T = w :=
-  (inverseFlow_spec (exists_inverseFlow κ hκ b hT w hw)).2
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric
-open scoped NNReal Topology
-
-lemma cutoffFlow_dist_le (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
-    (z w : ℂ) {T : ℝ} (hT : 0 ≤ T) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    dist (cutoffFlow κ b j z t) (cutoffFlow κ b j w t) ≤
-      dist z w * Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T) := by
-  have hd (x : ℂ) (u : ℝ) (hu : u ∈ Ico 0 T) :=
-    ((cutoffFlow_solution κ b j hj x hT).2.2 u ⟨hu.1,hu.2.le⟩).mono_of_mem_nhdsWithin
-      (Filter.mem_of_superset (Icc_mem_nhdsGE hu.2) (Icc_subset_Icc_left hu.1))
-  have hh := dist_le_of_trajectories_ODE (fun u => cutoffField_lipschitz κ b hj u)
-    (cutoffFlow_continuous κ b hj z hT) (hd z)
-    (cutoffFlow_continuous κ b hj w hT) (hd w)
-    (show dist (cutoffFlow κ b j z 0) (cutoffFlow κ b j w 0) ≤ dist z w by
-      rw [(cutoffFlow_solution κ b j hj z hT).2.1,(cutoffFlow_solution κ b j hj w hT).2.1]) t ht
-  simp only [sub_zero] at hh
-  change dist _ _ ≤ dist z w * Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*t) at hh
-  have hK : 0 ≤ |a κ| *3/((j:ℝ)^(-(2:ℤ)))^2 := by positivity
-  exact hh.trans (mul_le_mul_of_nonneg_left
-    (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left ht.2 hK)) dist_nonneg)
-
-lemma cutoffFlow_continuous_initial (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
-    {T : ℝ} (hT : 0 ≤ T) : Continuous (fun z => cutoffFlow κ b j z T) := by
-  apply LipschitzWith.continuous (K := ⟨Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T),by positivity⟩)
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  change dist _ _ ≤ Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T) * dist z w
-  simpa only [mul_comm] using cutoffFlow_dist_le κ b hj z w hT ⟨hT,le_rfl⟩
-
- 
-
-lemma surviving_eventually_cutoff_height {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
-    ∃ j : ℕ, 1 ≤ j ∧ ∀ᶠ w in 𝓝 z,
-      Surviving κ b w T ∧ (∀ t ∈ Icc 0 T, flow κ b w t = cutoffFlow κ b j w t) ∧
-      ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ < (flow κ b w t).im := by
-  obtain ⟨j,hj,hgap⟩ := hz
-  let m := sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)
-  let C := Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T)
-  let ε := (m-(j:ℝ)⁻¹)/2
-  have hε : 0 < ε := by dsimp [ε,m]; linarith
-  have hC : 0 < C := Real.exp_pos _
-  have hb := (isCompact_Icc.image_of_continuousOn
-    (Complex.continuous_im.comp_continuousOn (cutoffFlow_continuous κ b hj z hT))).bddBelow
-  refine ⟨j,hj,?_⟩
-  filter_upwards [Metric.ball_mem_nhds z (div_pos hε hC)] with w hw
-  have hd : dist z w*C < ε := by
-    rw [← lt_div_iff₀ hC,dist_comm]
-    exact hw
-  have hg (t : ℝ) (ht : t ∈ Icc 0 T) : m-ε ≤ (cutoffFlow κ b j w t).im := by
-    have h0 : m ≤ (cutoffFlow κ b j z t).im := csInf_le hb ⟨t,ht,rfl⟩
-    have hh := cutoffFlow_dist_le κ b hj z w hT ht
-    have hi : (cutoffFlow κ b j z t).im - (cutoffFlow κ b j w t).im ≤
-        dist (cutoffFlow κ b j z t) (cutoffFlow κ b j w t) := by
-      simpa only [dist_eq_norm,Complex.sub_im] using
-        Complex.im_le_norm (cutoffFlow κ b j z t-cutoffFlow κ b j w t)
-    change dist _ _ ≤ dist z w*C at hh
-    linarith
-  have hgapw : (j:ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j w u).im) '' Icc 0 T) := by
-    apply lt_of_lt_of_le (show (j:ℝ)⁻¹ < m-ε by dsimp [ε]; linarith)
-    apply le_csInf (Set.Nonempty.image _ (show (Icc (0:ℝ) T).Nonempty from ⟨0,le_rfl,hT⟩))
-    rintro y ⟨t,ht,rfl⟩
-    exact hg t ht
-  have hp := cutoff_positiveSolution hT hj hgapw
-  refine ⟨hp.surviving,fun t ht => hp.flow_eqOn ht,?_⟩
-  intro t ht
-  rw [hp.flow_eqOn ht]
-  exact lt_of_lt_of_le (show (j:ℝ)⁻¹ < m-ε by dsimp [ε]; linarith) (hg t ht)
-
-lemma surviving_eventually_cutoff {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
-    ∃ j : ℕ, 1 ≤ j ∧ ∀ᶠ w in 𝓝 z,
-      Surviving κ b w T ∧ ∀ t ∈ Icc 0 T, flow κ b w t = cutoffFlow κ b j w t := by
-  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff_height hT hz
-  exact ⟨j,hj,he.mono fun w hw => ⟨hw.1,hw.2.1⟩⟩
-
-lemma isOpen_surviving (κ : ℝ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
-    IsOpen {z | Surviving κ b z T} := by
-  rw [isOpen_iff_mem_nhds]
-  intro z hz
-  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff hT hz
-  exact he.mono fun w hw => hw.1
-
-lemma flow_continuousAt_initial {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) : ContinuousAt (fun w => flow κ b w T) z := by
-  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff hT hz
-  apply (cutoffFlow_continuous_initial κ b hj hT).continuousAt.congr
-  exact he.mono fun w hw => (hw.2 T ⟨hT,le_rfl⟩).symm
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric MeasureTheory
-open scoped Topology
-
-lemma complex_linear_formula {T : ℝ} {y r : ℝ → ℂ} (hT : 0 ≤ T)
-    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t) :
-    ∀ t ∈ Icc 0 T, y t = y 0 * Complex.exp (∫ u in (0:ℝ)..t, r (timeClip T u)) := by
-  let A (t : ℝ) := ∫ u in (0:ℝ)..t, r (timeClip T u)
-  have hc : Continuous (fun u => r (timeClip T u)) :=
-    hr.comp_continuous (continuous_timeClip T) (timeClip_mem hT)
-  have hA (t : ℝ) : HasDerivAt A (r (timeClip T t)) t :=
-    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
-      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-  have hA' : Continuous A := continuous_iff_continuousAt.mpr fun t => (hA t).continuousAt
-  have hd0 (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt (fun u => y u * Complex.exp (-A u)) 0 (Icc 0 T) t := by
-    have he := (hd t ht).mul ((hA t).neg.cexp.hasDerivWithinAt)
-    rw [timeClip_eq ht] at he
-    convert! he using 1
-    ring
-  have hconst := constant_of_has_deriv_right_zero
-    (hy.mul ((Complex.continuous_exp.comp hA'.neg).continuousOn)) (fun t (ht : t ∈ Ico 0 T) =>
-      (hd0 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
-        (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1)))
-  intro t ht
-  have he := hconst t ht
-  dsimp only [Pi.mul_apply,Function.comp_def,Pi.neg_apply] at he
-  have hA0 : A 0 = 0 := by simp [A]
-  rw [hA0,neg_zero,Complex.exp_zero,mul_one] at he
-  calc
-    y t = (y t * Complex.exp (-A t))*Complex.exp (A t) := by
-      rw [mul_assoc,← Complex.exp_add,neg_add_cancel,Complex.exp_zero,mul_one]
-    _ = y 0 * Complex.exp (A t) := by rw [he]
-
-lemma surviving_centered_ne_zero {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    flow κ b z t-driverValue b t ≠ 0 := by
-  have hi := (surviving_flow_positiveSolution hT hz).2.2.2.1 t ht
-  intro he
-  have := congrArg Complex.im he
-  simp only [Complex.sub_im,Complex.ofReal_im,Complex.zero_im,sub_zero] at this
-  linarith
-
- 
-lemma flow_difference_formula {κ : ℝ} {b : Driver} {z w : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) (hw : Surviving κ b w T) :
-    flow κ b w T-flow κ b z T = (w-z)*Complex.exp
-      (∫ u in (0:ℝ)..T, -(a κ:ℂ)/
-        ((flow κ b z (timeClip T u)-driverValue b (timeClip T u))*
-         (flow κ b w (timeClip T u)-driverValue b (timeClip T u)))) := by
-  let f := flow κ b z
-  let g := flow κ b w
-  let r (u : ℝ) : ℂ := -(a κ:ℂ)/((f u-driverValue b u)*(g u-driverValue b u))
-  have hf := surviving_flow_positiveSolution hT hz
-  have hg := surviving_flow_positiveSolution hT hw
-  have hcB : ContinuousOn (fun u => (driverValue b u:ℂ)) (Icc 0 T) :=
-    (Complex.continuous_ofReal.comp (continuous_driverValue b)).continuousOn
-  have hr : ContinuousOn r (Icc 0 T) :=
-    continuousOn_const.div ((hf.2.1.sub hcB).mul (hg.2.1.sub hcB)) (fun t ht =>
-      mul_ne_zero (surviving_centered_ne_zero hT hz ht) (surviving_centered_ne_zero hT hw ht))
-  have hd (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt (fun u => g u-f u) (r t*(g t-f t)) (Icc 0 T) t := by
-    have h := (hg.2.2.2.2 t ht).sub (hf.2.2.2.2 t ht)
-    have hnz := surviving_centered_ne_zero hT hz ht
-    have hnw := surviving_centered_ne_zero hT hw ht
-    convert! h using 1
-    dsimp [r,f,g]
-    field_simp [hnz,hnw]
-    ring
-  have he := complex_linear_formula hT (hg.2.1.sub hf.2.1) hr hd T ⟨hT,le_rfl⟩
-  have hf0 : f 0 = z := hf.2.2.1
-  have hg0 : g 0 = w := hg.2.2.1
-  simpa only [Pi.sub_apply,hf.2.2.1,hg.2.2.1] using he
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric MeasureTheory
-open scoped Topology
-
-lemma surviving_centered_continuous_clip {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
-    Continuous (fun u => flow κ b z (timeClip T u)-driverValue b (timeClip T u)) :=
-  ((surviving_flow_positiveSolution hT hz).2.1.comp_continuous (continuous_timeClip T)
-    (timeClip_mem hT)).sub
-    (Complex.continuous_ofReal.comp ((continuous_driverValue b).comp (continuous_timeClip T)))
-
-def flowDifferenceExponent (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (w : ℂ) : ℂ :=
-  ∫ u in (0:ℝ)..T, -(a κ:ℂ)/
-    ((flow κ b z (timeClip T u)-driverValue b (timeClip T u))*
-     (flow κ b w (timeClip T u)-driverValue b (timeClip T u)))
-
-lemma flowDifferenceExponent_continuousAt {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
-    ContinuousAt (flowDifferenceExponent κ b z T) z := by
-  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff_height hT hz
-  have hz' := he.self_of_nhds
-  have hj0 : (0:ℝ) < j := by exact_mod_cast (Nat.zero_lt_of_lt hj)
-  have hq : 0 < (j:ℝ)⁻¹ := inv_pos.mpr hj0
-  have hf := surviving_flow_positiveSolution hT hz
-  have hzn (u : ℝ) : (j:ℝ)⁻¹ ≤
-      ‖flow κ b z (timeClip T u)-driverValue b (timeClip T u)‖ := by
-    exact (hz'.2.2 _ (timeClip_mem hT u)).le.trans
-      (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using
-        Complex.im_le_norm (flow κ b z (timeClip T u)-driverValue b (timeClip T u)))
-  unfold flowDifferenceExponent
-  apply intervalIntegral.continuousAt_of_dominated_interval
-    (bound := fun _ => |a κ|/((j:ℝ)⁻¹*(j:ℝ)⁻¹))
-  · filter_upwards [he] with w hw
-    exact (continuous_const.div
-      ((surviving_centered_continuous_clip hT hz).mul (surviving_centered_continuous_clip hT hw.1))
-      (fun u => mul_ne_zero (surviving_centered_ne_zero hT hz (timeClip_mem hT u))
-        (surviving_centered_ne_zero hT hw.1 (timeClip_mem hT u)))).aestronglyMeasurable
-  · filter_upwards [he] with w hw
-    apply Filter.Eventually.of_forall
-    intro u _
-    have hwn : (j:ℝ)⁻¹ ≤ ‖flow κ b w (timeClip T u)-driverValue b (timeClip T u)‖ := by
-      exact (hw.2.2 _ (timeClip_mem hT u)).le.trans
-        (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using
-          Complex.im_le_norm (flow κ b w (timeClip T u)-driverValue b (timeClip T u)))
-    rw [norm_div,norm_neg,Complex.norm_real,Real.norm_eq_abs,norm_mul]
-    exact div_le_div_of_nonneg_left (abs_nonneg _) (mul_pos hq hq)
-      (mul_le_mul (hzn u) hwn hq.le (norm_nonneg _))
-  · exact continuous_const.intervalIntegrable 0 T
-  · apply Filter.Eventually.of_forall
-    intro u _
-    let t := timeClip T u
-    have ht : t ∈ Icc 0 T := timeClip_mem hT u
-    have hzt := (hf.restrict ht.1 ht.2).surviving
-    apply continuousAt_const.div
-      (continuousAt_const.mul ((flow_continuousAt_initial ht.1 hzt).sub continuousAt_const))
-    exact mul_ne_zero (surviving_centered_ne_zero hT hz ht) (surviving_centered_ne_zero hT hz ht)
-
- 
-
-theorem flow_hasDerivAt_initial {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
-    HasDerivAt (fun w => flow κ b w T)
-      (Complex.exp (∫ u in (0:ℝ)..T, -(a κ:ℂ)/
-        (flow κ b z (timeClip T u)-driverValue b (timeClip T u))^2)) z := by
-  have hc := Complex.continuous_exp.continuousAt.comp (flowDifferenceExponent_continuousAt hT hz)
-  have hh := hc.tendsto.mono_left (nhdsWithin_le_nhds (s := {z}ᶜ))
-  apply hasDerivAt_iff_tendsto_slope.mpr
-  have he : ∀ᶠ w in 𝓝 z, Surviving κ b w T := (isOpen_surviving κ b hT).mem_nhds hz
-  have heq : (fun w => Complex.exp (flowDifferenceExponent κ b z T w)) =ᶠ[𝓝[≠] z]
-      slope (fun w => flow κ b w T) z := by
-    filter_upwards [he.filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with w hw hne
-    have hn : w-z ≠ 0 := sub_ne_zero.mpr (by simpa using hne)
-    rw [slope_def_field,flow_difference_formula hT hz hw]
-    change _ = ((w-z)*Complex.exp (flowDifferenceExponent κ b z T w))/(w-z)
-    field_simp
-  simpa only [Function.comp_def,flowDifferenceExponent,pow_two] using hh.congr' heq
-
-lemma flow_deriv_ne_zero {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) : deriv (fun w => flow κ b w T) z ≠ 0 := by
-  rw [(flow_hasDerivAt_initial hT hz).deriv]
-  exact Complex.exp_ne_zero _
-
-lemma flow_differentiableOn (κ : ℝ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
-    DifferentiableOn ℂ (fun z => flow κ b z T) {z | Surviving κ b z T} := by
-  intro z hz
-  exact (flow_hasDerivAt_initial hT hz).differentiableAt.differentiableWithinAt
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric
-open scoped Topology
-
-lemma flow_hasStrictDerivAt_initial {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
-    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
-    HasStrictDerivAt (fun w => flow κ b w T) (deriv (fun w => flow κ b w T) z) z :=
-  ((flow_differentiableOn κ b hT).analyticAt ((isOpen_surviving κ b hT).mem_nhds hz)).hasStrictDerivAt
-
-lemma inverseFlow_hasStrictDerivAt {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ}
-    (hT : 0 ≤ T) {w : ℂ} (hw : 0 < w.im) :
-    HasStrictDerivAt (inverseFlow κ b T)
-      (deriv (fun z => flow κ b z T) (inverseFlow κ b T w))⁻¹ w := by
-  have hz := inverseFlow_surviving hκ b hT hw
-  have hf := flow_hasStrictDerivAt_initial hT hz
-  have he : ∀ᶠ z in 𝓝 (inverseFlow κ b T w), inverseFlow κ b T (flow κ b z T) = z := by
-    filter_upwards [(isOpen_surviving κ b hT).mem_nhds hz] with z hz'
-    exact inverseFlow_flow hT hz'
-  have hh := hf.to_local_left_inverse (flow_deriv_ne_zero hT hz) he
-  rwa [flow_inverseFlow hκ b hT hw] at hh
-
- 
-
-theorem flow_conformal_bijection {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
-    IsOpen {z | Surviving κ b z T} ∧
-    BijOn (fun z => flow κ b z T) {z | Surviving κ b z T} {w | 0 < w.im} ∧
-    DifferentiableOn ℂ (fun z => flow κ b z T) {z | Surviving κ b z T} ∧
-    DifferentiableOn ℂ (inverseFlow κ b T) {w | 0 < w.im} := by
-  refine ⟨isOpen_surviving κ b hT,⟨?_,flow_injOn κ b hT,?_⟩,
-    flow_differentiableOn κ b hT,?_⟩
-  · intro z hz
-    exact (surviving_flow_positiveSolution hT hz).2.2.2.1 T ⟨hT,le_rfl⟩
-  · intro w hw
-    exact exists_inverseFlow κ hκ b hT w hw
-  · intro w hw
-    exact (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.differentiableAt.differentiableWithinAt
-end SLEExactGauge.LiteralTrace
-
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.BrownianCalculus
-
-lemma integral_sq_gaussian (v : ℝ≥0) :
-    (∫ x : ℝ, x ^ 2 ∂gaussianReal 0 v) = v := by
-  have hv := variance_fun_id_gaussianReal (μ := 0) (v := v)
-  rw [variance_eq_integral (X := fun x : ℝ => x) (by fun_prop)] at hv
-  simpa using hv
-
-lemma integral_fourth_gaussian (v : ℝ≥0) :
-    (∫ x : ℝ, x ^ 4 ∂gaussianReal 0 v) = 3 * (v : ℝ)^2 := by
-  let f : ℝ → ℝ := fun t => Real.exp ((v : ℝ) * t^2 / 2)
-  have hf (t : ℝ) : HasDerivAt f ((v:ℝ) * t * f t) t := by
-    convert! ((((hasDerivAt_id t).pow 2).const_mul (v:ℝ)).div_const 2).exp using 1
-    dsimp [f]
-    ring
-  have d0 : deriv f = fun t => (v:ℝ) * t * f t := funext fun t => (hf t).deriv
-  have d1 : deriv (fun t => (v:ℝ) * t * f t) =
-      fun t => ((v:ℝ) + (v:ℝ)^2*t^2) * f t := by
-    ext t
-    convert! (((hasDerivAt_id t).const_mul (v:ℝ)).mul (hf t)).deriv using 1
-    dsimp
-    ring
-  have d2 : deriv (fun t => ((v:ℝ) + (v:ℝ)^2*t^2) * f t) =
-      fun t => (3*(v:ℝ)^2*t + (v:ℝ)^3*t^3) * f t := by
-    ext t
-    convert! (((((hasDerivAt_id t).pow 2).const_mul ((v:ℝ)^2)).const_add (v:ℝ)).mul
-      (hf t)).deriv using 1
-    dsimp
-    ring
-  have d3 : deriv (fun t => (3*(v:ℝ)^2*t + (v:ℝ)^3*t^3) * f t) =
-      fun t => (3*(v:ℝ)^2 + 6*(v:ℝ)^3*t^2 + (v:ℝ)^4*t^4) * f t := by
-    ext t
-    have hp := ((hasDerivAt_id t).const_mul (3*(v:ℝ)^2)).add
-      (((hasDerivAt_id t).pow 3).const_mul ((v:ℝ)^3))
-    convert! (hp.mul (hf t)).deriv using 1
-    dsimp
-    ring
-  have hm := iteratedDeriv_mgf_zero
-    (X := fun x : ℝ => x) (μ := gaussianReal 0 v) (by simp) 4
-  have he : mgf (fun x : ℝ => x) (gaussianReal 0 v) = f := by
-    rw [mgf_fun_id_gaussianReal]
-    simp only [zero_mul, zero_add, f]
-  rw [he] at hm
-  change iteratedDeriv 4 f 0 = (∫ x : ℝ, x^4 ∂gaussianReal 0 v) at hm
-  rw [← hm]
-  norm_num [iteratedDeriv_succ, iteratedDeriv_zero, d0, d1, d2, d3, f]
-
-open scoped ContinuousMapZero
-abbrev Driver := C(ℝ≥0, ℝ)₀
-instance : MeasurableSpace Driver := borel Driver
-instance : BorelSpace Driver := ⟨rfl⟩
-
-def increment (s h : ℝ≥0) (b : Driver) : ℝ := b (s+h) - b s
-
-def history (s : ℝ≥0) (b : Driver) : Iic s → ℝ := fun t => b t
-
-lemma measurable_increment (s h : ℝ≥0) : Measurable (increment s h) := by
-  unfold increment
-  exact (continuous_eval_const (s+h)).measurable.sub
-    (continuous_eval_const s).measurable
-
-lemma measurable_history (s : ℝ≥0) : Measurable (history s) := by
-  exact Measurable.of_eval (fun time => (continuous_eval_const (time : ℝ≥0)).measurable)
-
-variable {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-include hW
-
-lemma increment_law (s h : ℝ≥0) : HasLaw (increment s h) (gaussianReal 0 h) W := by
-  exact (hW.toIsPreBrownianReal.shift s).hasLaw_eval h
-
-lemma increment_mean (s h : ℝ≥0) : (∫ b, increment s h b ∂W) = 0 := by
-  rw [(increment_law hW s h).integral_eq, integral_id_gaussianReal]
-
-lemma increment_sq (s h : ℝ≥0) : (∫ b, (increment s h b)^2 ∂W) = h := by
-  exact ((increment_law hW s h).integral_comp (f := fun x : ℝ => x^2) (by fun_prop)).trans
-    (integral_sq_gaussian h)
-
-lemma increment_fourth (s h : ℝ≥0) :
-    (∫ b, (increment s h b)^4 ∂W) = 3*(h:ℝ)^2 := by
-  exact ((increment_law hW s h).integral_comp (f := fun x : ℝ => x^4) (by fun_prop)).trans
-    (integral_fourth_gaussian h)
-
-end SLEExactGauge.BrownianCalculus
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.InverseDomains
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric MeasureTheory
+open scoped Topology
+theorem exists_inverseFlow (κ : ℝ) (hκ : 0 < κ) (b : Driver) {T : ℝ}
+    (hT : 0 ≤ T) (w : ℂ) (hw : 0 < w.im) :
+    ∃ z, Surviving κ b z T ∧ flow κ b z T = w := by
+  let q : ℝ := w.im/2
+  have hq : 0 < q := by dsimp [q]; positivity
+  let br (t : ℝ) : ℝ := driverValue b (T-t)
+  have hbr : Continuous br := (continuous_driverValue b).comp (continuous_const.sub continuous_id)
+  obtain ⟨g,hg0,hgd⟩ := Cutoff.exists_solution_on_compact hq (-a κ) hbr w hT
+  have hgc : ContinuousOn g (Icc 0 T) := fun t ht => (hgd t ht).continuousWithinAt
+  let r (t : ℝ) : ℝ := a κ / Cutoff.denominator q (g t-br t)
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hr : ContinuousOn r (Icc 0 T) := by
+    apply continuousOn_const.div
+    · exact ((continuous_norm.comp_continuousOn
+        (hgc.sub ((Complex.continuous_ofReal.comp hbr).continuousOn))).pow 2).sup continuousOn_const
+    · intro t _
+      exact (Cutoff.denominator_pos hq _).ne'
+  have hyd (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt (fun u => (g u).im) (r t*(g t).im) (Icc 0 T) t := by
+    have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hgd t ht)
+    convert! hi using 1
+    simp only [Complex.imCLM_apply,Cutoff.field_im,neg_neg,r,div_mul_eq_mul_div]
+  have hheight (t : ℝ) (ht : t ∈ Icc 0 T) : w.im ≤ (g t).im := by
+    have hh := reverse_height_le hT (Complex.continuous_im.comp_continuousOn hgc) hr hyd
+      (by simpa [hg0] using hw.le) (fun u _ => div_nonneg ha.le (Cutoff.denominator_pos hq _).le) t ht
+    simpa [hg0] using hh
+  have hrev (t : ℝ) (ht : t ∈ Icc 0 T) : T-t ∈ Icc 0 T := by
+    constructor <;> linarith [ht.1,ht.2]
+  let f (t : ℝ) : ℂ := g (T-t)
+  have hf : PositiveSolution κ b (g T) T f := by
+    refine ⟨hT,hgc.comp (continuous_const.sub continuous_id).continuousOn hrev,?_,?_,?_⟩
+    · simp [f]
+    · intro t ht
+      exact hw.trans_le (hheight (T-t) (hrev t ht))
+    · intro t ht
+      have hcut := Cutoff.field_eq_div hq (-a κ) br (T-t) (g (T-t))
+        ((by dsimp [q]; linarith : q ≤ w.im).trans (hheight (T-t) (hrev t ht)))
+      have hd := (hgd (T-t) (hrev t ht)).scomp t
+        (((hasDerivAt_const t T).sub (hasDerivAt_id t)).hasDerivWithinAt) hrev
+      rw [hcut] at hd
+      convert! hd using 1
+      simp [f,br]
+      ring
+  refine ⟨g T,hf.surviving,?_⟩
+  rw [hf.flow_eq]
+  simpa [f] using hg0
+
+lemma inverseFlow_surviving {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ} (hT : 0 ≤ T)
+    {w : ℂ} (hw : 0 < w.im) : Surviving κ b (inverseFlow κ b T w) T :=
+  (inverseFlow_spec (exists_inverseFlow κ hκ b hT w hw)).1
+
+lemma flow_inverseFlow {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ} (hT : 0 ≤ T)
+    {w : ℂ} (hw : 0 < w.im) : flow κ b (inverseFlow κ b T w) T = w :=
+  (inverseFlow_spec (exists_inverseFlow κ hκ b hT w hw)).2
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric
+open scoped NNReal Topology
+
+lemma cutoffFlow_dist_le (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
+    (z w : ℂ) {T : ℝ} (hT : 0 ≤ T) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    dist (cutoffFlow κ b j z t) (cutoffFlow κ b j w t) ≤
+      dist z w * Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T) := by
+  have hd (x : ℂ) (u : ℝ) (hu : u ∈ Ico 0 T) :=
+    ((cutoffFlow_solution κ b j hj x hT).2.2 u ⟨hu.1,hu.2.le⟩).mono_of_mem_nhdsWithin
+      (Filter.mem_of_superset (Icc_mem_nhdsGE hu.2) (Icc_subset_Icc_left hu.1))
+  have hh := dist_le_of_trajectories_ODE (fun u => cutoffField_lipschitz κ b hj u)
+    (cutoffFlow_continuous κ b hj z hT) (hd z)
+    (cutoffFlow_continuous κ b hj w hT) (hd w)
+    (show dist (cutoffFlow κ b j z 0) (cutoffFlow κ b j w 0) ≤ dist z w by
+      rw [(cutoffFlow_solution κ b j hj z hT).2.1,(cutoffFlow_solution κ b j hj w hT).2.1]) t ht
+  simp only [sub_zero] at hh
+  change dist _ _ ≤ dist z w * Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*t) at hh
+  have hK : 0 ≤ |a κ| *3/((j:ℝ)^(-(2:ℤ)))^2 := by positivity
+  exact hh.trans (mul_le_mul_of_nonneg_left
+    (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left ht.2 hK)) dist_nonneg)
+
+lemma cutoffFlow_continuous_initial (κ : ℝ) (b : Driver) {j : ℕ} (hj : 1 ≤ j)
+    {T : ℝ} (hT : 0 ≤ T) : Continuous (fun z => cutoffFlow κ b j z T) := by
+  apply LipschitzWith.continuous (K := ⟨Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T),by positivity⟩)
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  change dist _ _ ≤ Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T) * dist z w
+  simpa only [mul_comm] using cutoffFlow_dist_le κ b hj z w hT ⟨hT,le_rfl⟩
+
+ 
+
+lemma surviving_eventually_cutoff_height {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
+    ∃ j : ℕ, 1 ≤ j ∧ ∀ᶠ w in 𝓝 z,
+      Surviving κ b w T ∧ (∀ t ∈ Icc 0 T, flow κ b w t = cutoffFlow κ b j w t) ∧
+      ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ < (flow κ b w t).im := by
+  obtain ⟨j,hj,hgap⟩ := hz
+  let m := sInf ((fun u : ℝ => (cutoffFlow κ b j z u).im) '' Icc 0 T)
+  let C := Real.exp ((|a κ| *3/((j:ℝ)^(-(2:ℤ)))^2)*T)
+  let ε := (m-(j:ℝ)⁻¹)/2
+  have hε : 0 < ε := by dsimp [ε,m]; linarith
+  have hC : 0 < C := Real.exp_pos _
+  have hb := (isCompact_Icc.image_of_continuousOn
+    (Complex.continuous_im.comp_continuousOn (cutoffFlow_continuous κ b hj z hT))).bddBelow
+  refine ⟨j,hj,?_⟩
+  filter_upwards [Metric.ball_mem_nhds z (div_pos hε hC)] with w hw
+  have hd : dist z w*C < ε := by
+    rw [← lt_div_iff₀ hC,dist_comm]
+    exact hw
+  have hg (t : ℝ) (ht : t ∈ Icc 0 T) : m-ε ≤ (cutoffFlow κ b j w t).im := by
+    have h0 : m ≤ (cutoffFlow κ b j z t).im := csInf_le hb ⟨t,ht,rfl⟩
+    have hh := cutoffFlow_dist_le κ b hj z w hT ht
+    have hi : (cutoffFlow κ b j z t).im - (cutoffFlow κ b j w t).im ≤
+        dist (cutoffFlow κ b j z t) (cutoffFlow κ b j w t) := by
+      simpa only [dist_eq_norm,Complex.sub_im] using
+        Complex.im_le_norm (cutoffFlow κ b j z t-cutoffFlow κ b j w t)
+    change dist _ _ ≤ dist z w*C at hh
+    linarith
+  have hgapw : (j:ℝ)⁻¹ < sInf ((fun u : ℝ => (cutoffFlow κ b j w u).im) '' Icc 0 T) := by
+    apply lt_of_lt_of_le (show (j:ℝ)⁻¹ < m-ε by dsimp [ε]; linarith)
+    apply le_csInf (Set.Nonempty.image _ (show (Icc (0:ℝ) T).Nonempty from ⟨0,le_rfl,hT⟩))
+    rintro y ⟨t,ht,rfl⟩
+    exact hg t ht
+  have hp := cutoff_positiveSolution hT hj hgapw
+  refine ⟨hp.surviving,fun t ht => hp.flow_eqOn ht,?_⟩
+  intro t ht
+  rw [hp.flow_eqOn ht]
+  exact lt_of_lt_of_le (show (j:ℝ)⁻¹ < m-ε by dsimp [ε]; linarith) (hg t ht)
+
+lemma surviving_eventually_cutoff {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
+    ∃ j : ℕ, 1 ≤ j ∧ ∀ᶠ w in 𝓝 z,
+      Surviving κ b w T ∧ ∀ t ∈ Icc 0 T, flow κ b w t = cutoffFlow κ b j w t := by
+  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff_height hT hz
+  exact ⟨j,hj,he.mono fun w hw => ⟨hw.1,hw.2.1⟩⟩
+
+lemma isOpen_surviving (κ : ℝ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
+    IsOpen {z | Surviving κ b z T} := by
+  rw [isOpen_iff_mem_nhds]
+  intro z hz
+  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff hT hz
+  exact he.mono fun w hw => hw.1
+
+lemma flow_continuousAt_initial {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) : ContinuousAt (fun w => flow κ b w T) z := by
+  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff hT hz
+  apply (cutoffFlow_continuous_initial κ b hj hT).continuousAt.congr
+  exact he.mono fun w hw => (hw.2 T ⟨hT,le_rfl⟩).symm
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric MeasureTheory
+open scoped Topology
+
+lemma complex_linear_formula {T : ℝ} {y r : ℝ → ℂ} (hT : 0 ≤ T)
+    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t) :
+    ∀ t ∈ Icc 0 T, y t = y 0 * Complex.exp (∫ u in (0:ℝ)..t, r (timeClip T u)) := by
+  let A (t : ℝ) := ∫ u in (0:ℝ)..t, r (timeClip T u)
+  have hc : Continuous (fun u => r (timeClip T u)) :=
+    hr.comp_continuous (continuous_timeClip T) (timeClip_mem hT)
+  have hA (t : ℝ) : HasDerivAt A (r (timeClip T t)) t :=
+    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
+      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+  have hA' : Continuous A := continuous_iff_continuousAt.mpr fun t => (hA t).continuousAt
+  have hd0 (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt (fun u => y u * Complex.exp (-A u)) 0 (Icc 0 T) t := by
+    have he := (hd t ht).mul ((hA t).neg.cexp.hasDerivWithinAt)
+    rw [timeClip_eq ht] at he
+    convert! he using 1
+    ring
+  have hconst := constant_of_has_deriv_right_zero
+    (hy.mul ((Complex.continuous_exp.comp hA'.neg).continuousOn)) (fun t (ht : t ∈ Ico 0 T) =>
+      (hd0 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
+        (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1)))
+  intro t ht
+  have he := hconst t ht
+  dsimp only [Pi.mul_apply,Function.comp_def,Pi.neg_apply] at he
+  have hA0 : A 0 = 0 := by simp [A]
+  rw [hA0,neg_zero,Complex.exp_zero,mul_one] at he
+  calc
+    y t = (y t * Complex.exp (-A t))*Complex.exp (A t) := by
+      rw [mul_assoc,← Complex.exp_add,neg_add_cancel,Complex.exp_zero,mul_one]
+    _ = y 0 * Complex.exp (A t) := by rw [he]
+
+lemma surviving_centered_ne_zero {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    flow κ b z t-driverValue b t ≠ 0 := by
+  have hi := (surviving_flow_positiveSolution hT hz).2.2.2.1 t ht
+  intro he
+  have := congrArg Complex.im he
+  simp only [Complex.sub_im,Complex.ofReal_im,Complex.zero_im,sub_zero] at this
+  linarith
+
+ 
+lemma flow_difference_formula {κ : ℝ} {b : Driver} {z w : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) (hw : Surviving κ b w T) :
+    flow κ b w T-flow κ b z T = (w-z)*Complex.exp
+      (∫ u in (0:ℝ)..T, -(a κ:ℂ)/
+        ((flow κ b z (timeClip T u)-driverValue b (timeClip T u))*
+         (flow κ b w (timeClip T u)-driverValue b (timeClip T u)))) := by
+  let f := flow κ b z
+  let g := flow κ b w
+  let r (u : ℝ) : ℂ := -(a κ:ℂ)/((f u-driverValue b u)*(g u-driverValue b u))
+  have hf := surviving_flow_positiveSolution hT hz
+  have hg := surviving_flow_positiveSolution hT hw
+  have hcB : ContinuousOn (fun u => (driverValue b u:ℂ)) (Icc 0 T) :=
+    (Complex.continuous_ofReal.comp (continuous_driverValue b)).continuousOn
+  have hr : ContinuousOn r (Icc 0 T) :=
+    continuousOn_const.div ((hf.2.1.sub hcB).mul (hg.2.1.sub hcB)) (fun t ht =>
+      mul_ne_zero (surviving_centered_ne_zero hT hz ht) (surviving_centered_ne_zero hT hw ht))
+  have hd (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt (fun u => g u-f u) (r t*(g t-f t)) (Icc 0 T) t := by
+    have h := (hg.2.2.2.2 t ht).sub (hf.2.2.2.2 t ht)
+    have hnz := surviving_centered_ne_zero hT hz ht
+    have hnw := surviving_centered_ne_zero hT hw ht
+    convert! h using 1
+    dsimp [r,f,g]
+    field_simp [hnz,hnw]
+    ring
+  have he := complex_linear_formula hT (hg.2.1.sub hf.2.1) hr hd T ⟨hT,le_rfl⟩
+  have hf0 : f 0 = z := hf.2.2.1
+  have hg0 : g 0 = w := hg.2.2.1
+  simpa only [Pi.sub_apply,hf.2.2.1,hg.2.2.1] using he
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric MeasureTheory
+open scoped Topology
+
+lemma surviving_centered_continuous_clip {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
+    Continuous (fun u => flow κ b z (timeClip T u)-driverValue b (timeClip T u)) :=
+  ((surviving_flow_positiveSolution hT hz).2.1.comp_continuous (continuous_timeClip T)
+    (timeClip_mem hT)).sub
+    (Complex.continuous_ofReal.comp ((continuous_driverValue b).comp (continuous_timeClip T)))
+
+def flowDifferenceExponent (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (w : ℂ) : ℂ :=
+  ∫ u in (0:ℝ)..T, -(a κ:ℂ)/
+    ((flow κ b z (timeClip T u)-driverValue b (timeClip T u))*
+     (flow κ b w (timeClip T u)-driverValue b (timeClip T u)))
+
+lemma flowDifferenceExponent_continuousAt {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
+    ContinuousAt (flowDifferenceExponent κ b z T) z := by
+  obtain ⟨j,hj,he⟩ := surviving_eventually_cutoff_height hT hz
+  have hz' := he.self_of_nhds
+  have hj0 : (0:ℝ) < j := by exact_mod_cast (Nat.zero_lt_of_lt hj)
+  have hq : 0 < (j:ℝ)⁻¹ := inv_pos.mpr hj0
+  have hf := surviving_flow_positiveSolution hT hz
+  have hzn (u : ℝ) : (j:ℝ)⁻¹ ≤
+      ‖flow κ b z (timeClip T u)-driverValue b (timeClip T u)‖ := by
+    exact (hz'.2.2 _ (timeClip_mem hT u)).le.trans
+      (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using
+        Complex.im_le_norm (flow κ b z (timeClip T u)-driverValue b (timeClip T u)))
+  unfold flowDifferenceExponent
+  apply intervalIntegral.continuousAt_of_dominated_interval
+    (bound := fun _ => |a κ|/((j:ℝ)⁻¹*(j:ℝ)⁻¹))
+  · filter_upwards [he] with w hw
+    exact (continuous_const.div
+      ((surviving_centered_continuous_clip hT hz).mul (surviving_centered_continuous_clip hT hw.1))
+      (fun u => mul_ne_zero (surviving_centered_ne_zero hT hz (timeClip_mem hT u))
+        (surviving_centered_ne_zero hT hw.1 (timeClip_mem hT u)))).aestronglyMeasurable
+  · filter_upwards [he] with w hw
+    apply Filter.Eventually.of_forall
+    intro u _
+    have hwn : (j:ℝ)⁻¹ ≤ ‖flow κ b w (timeClip T u)-driverValue b (timeClip T u)‖ := by
+      exact (hw.2.2 _ (timeClip_mem hT u)).le.trans
+        (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using
+          Complex.im_le_norm (flow κ b w (timeClip T u)-driverValue b (timeClip T u)))
+    rw [norm_div,norm_neg,Complex.norm_real,Real.norm_eq_abs,norm_mul]
+    exact div_le_div_of_nonneg_left (abs_nonneg _) (mul_pos hq hq)
+      (mul_le_mul (hzn u) hwn hq.le (norm_nonneg _))
+  · exact continuous_const.intervalIntegrable 0 T
+  · apply Filter.Eventually.of_forall
+    intro u _
+    let t := timeClip T u
+    have ht : t ∈ Icc 0 T := timeClip_mem hT u
+    have hzt := (hf.restrict ht.1 ht.2).surviving
+    apply continuousAt_const.div
+      (continuousAt_const.mul ((flow_continuousAt_initial ht.1 hzt).sub continuousAt_const))
+    exact mul_ne_zero (surviving_centered_ne_zero hT hz ht) (surviving_centered_ne_zero hT hz ht)
+
+ 
+
+theorem flow_hasDerivAt_initial {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
+    HasDerivAt (fun w => flow κ b w T)
+      (Complex.exp (∫ u in (0:ℝ)..T, -(a κ:ℂ)/
+        (flow κ b z (timeClip T u)-driverValue b (timeClip T u))^2)) z := by
+  have hc := Complex.continuous_exp.continuousAt.comp (flowDifferenceExponent_continuousAt hT hz)
+  have hh := hc.tendsto.mono_left (nhdsWithin_le_nhds (s := {z}ᶜ))
+  apply hasDerivAt_iff_tendsto_slope.mpr
+  have he : ∀ᶠ w in 𝓝 z, Surviving κ b w T := (isOpen_surviving κ b hT).mem_nhds hz
+  have heq : (fun w => Complex.exp (flowDifferenceExponent κ b z T w)) =ᶠ[𝓝[≠] z]
+      slope (fun w => flow κ b w T) z := by
+    filter_upwards [he.filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with w hw hne
+    have hn : w-z ≠ 0 := sub_ne_zero.mpr (by simpa using hne)
+    rw [slope_def_field,flow_difference_formula hT hz hw]
+    change _ = ((w-z)*Complex.exp (flowDifferenceExponent κ b z T w))/(w-z)
+    field_simp
+  simpa only [Function.comp_def,flowDifferenceExponent,pow_two] using hh.congr' heq
+
+lemma flow_deriv_ne_zero {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) : deriv (fun w => flow κ b w T) z ≠ 0 := by
+  rw [(flow_hasDerivAt_initial hT hz).deriv]
+  exact Complex.exp_ne_zero _
+
+lemma flow_differentiableOn (κ : ℝ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
+    DifferentiableOn ℂ (fun z => flow κ b z T) {z | Surviving κ b z T} := by
+  intro z hz
+  exact (flow_hasDerivAt_initial hT hz).differentiableAt.differentiableWithinAt
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric
+open scoped Topology
+
+lemma flow_hasStrictDerivAt_initial {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ}
+    (hT : 0 ≤ T) (hz : Surviving κ b z T) :
+    HasStrictDerivAt (fun w => flow κ b w T) (deriv (fun w => flow κ b w T) z) z :=
+  ((flow_differentiableOn κ b hT).analyticAt ((isOpen_surviving κ b hT).mem_nhds hz)).hasStrictDerivAt
+
+lemma inverseFlow_hasStrictDerivAt {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ}
+    (hT : 0 ≤ T) {w : ℂ} (hw : 0 < w.im) :
+    HasStrictDerivAt (inverseFlow κ b T)
+      (deriv (fun z => flow κ b z T) (inverseFlow κ b T w))⁻¹ w := by
+  have hz := inverseFlow_surviving hκ b hT hw
+  have hf := flow_hasStrictDerivAt_initial hT hz
+  have he : ∀ᶠ z in 𝓝 (inverseFlow κ b T w), inverseFlow κ b T (flow κ b z T) = z := by
+    filter_upwards [(isOpen_surviving κ b hT).mem_nhds hz] with z hz'
+    exact inverseFlow_flow hT hz'
+  have hh := hf.to_local_left_inverse (flow_deriv_ne_zero hT hz) he
+  rwa [flow_inverseFlow hκ b hT hw] at hh
+
+ 
+
+theorem flow_conformal_bijection {κ : ℝ} (hκ : 0 < κ) (b : Driver) {T : ℝ} (hT : 0 ≤ T) :
+    IsOpen {z | Surviving κ b z T} ∧
+    BijOn (fun z => flow κ b z T) {z | Surviving κ b z T} {w | 0 < w.im} ∧
+    DifferentiableOn ℂ (fun z => flow κ b z T) {z | Surviving κ b z T} ∧
+    DifferentiableOn ℂ (inverseFlow κ b T) {w | 0 < w.im} := by
+  refine ⟨isOpen_surviving κ b hT,⟨?_,flow_injOn κ b hT,?_⟩,
+    flow_differentiableOn κ b hT,?_⟩
+  · intro z hz
+    exact (surviving_flow_positiveSolution hT hz).2.2.2.1 T ⟨hT,le_rfl⟩
+  · intro w hw
+    exact exists_inverseFlow κ hκ b hT w hw
+  · intro w hw
+    exact (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.differentiableAt.differentiableWithinAt
+end SLEExactGauge.LiteralTrace
+
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.BrownianCalculus
+
+lemma integral_sq_gaussian (v : ℝ≥0) :
+    (∫ x : ℝ, x ^ 2 ∂gaussianReal 0 v) = v := by
+  have hv := variance_fun_id_gaussianReal (μ := 0) (v := v)
+  rw [variance_eq_integral (X := fun x : ℝ => x) (by fun_prop)] at hv
+  simpa using hv
+
+lemma integral_fourth_gaussian (v : ℝ≥0) :
+    (∫ x : ℝ, x ^ 4 ∂gaussianReal 0 v) = 3 * (v : ℝ)^2 := by
+  let f : ℝ → ℝ := fun t => Real.exp ((v : ℝ) * t^2 / 2)
+  have hf (t : ℝ) : HasDerivAt f ((v:ℝ) * t * f t) t := by
+    convert! ((((hasDerivAt_id t).pow 2).const_mul (v:ℝ)).div_const 2).exp using 1
+    dsimp [f]
+    ring
+  have d0 : deriv f = fun t => (v:ℝ) * t * f t := funext fun t => (hf t).deriv
+  have d1 : deriv (fun t => (v:ℝ) * t * f t) =
+      fun t => ((v:ℝ) + (v:ℝ)^2*t^2) * f t := by
+    ext t
+    convert! (((hasDerivAt_id t).const_mul (v:ℝ)).mul (hf t)).deriv using 1
+    dsimp
+    ring
+  have d2 : deriv (fun t => ((v:ℝ) + (v:ℝ)^2*t^2) * f t) =
+      fun t => (3*(v:ℝ)^2*t + (v:ℝ)^3*t^3) * f t := by
+    ext t
+    convert! (((((hasDerivAt_id t).pow 2).const_mul ((v:ℝ)^2)).const_add (v:ℝ)).mul
+      (hf t)).deriv using 1
+    dsimp
+    ring
+  have d3 : deriv (fun t => (3*(v:ℝ)^2*t + (v:ℝ)^3*t^3) * f t) =
+      fun t => (3*(v:ℝ)^2 + 6*(v:ℝ)^3*t^2 + (v:ℝ)^4*t^4) * f t := by
+    ext t
+    have hp := ((hasDerivAt_id t).const_mul (3*(v:ℝ)^2)).add
+      (((hasDerivAt_id t).pow 3).const_mul ((v:ℝ)^3))
+    convert! (hp.mul (hf t)).deriv using 1
+    dsimp
+    ring
+  have hm := iteratedDeriv_mgf_zero
+    (X := fun x : ℝ => x) (μ := gaussianReal 0 v) (by simp) 4
+  have he : mgf (fun x : ℝ => x) (gaussianReal 0 v) = f := by
+    rw [mgf_fun_id_gaussianReal]
+    simp only [zero_mul, zero_add, f]
+  rw [he] at hm
+  change iteratedDeriv 4 f 0 = (∫ x : ℝ, x^4 ∂gaussianReal 0 v) at hm
+  rw [← hm]
+  norm_num [iteratedDeriv_succ, iteratedDeriv_zero, d0, d1, d2, d3, f]
+
+open scoped ContinuousMapZero
+abbrev Driver := C(ℝ≥0, ℝ)₀
+instance : MeasurableSpace Driver := borel Driver
+instance : BorelSpace Driver := ⟨rfl⟩
+
+def increment (s h : ℝ≥0) (b : Driver) : ℝ := b (s+h) - b s
+
+def history (s : ℝ≥0) (b : Driver) : Iic s → ℝ := fun t => b t
+
+lemma measurable_increment (s h : ℝ≥0) : Measurable (increment s h) := by
+  unfold increment
+  exact (continuous_eval_const (s+h)).measurable.sub
+    (continuous_eval_const s).measurable
+
+lemma measurable_history (s : ℝ≥0) : Measurable (history s) := by
+  exact Measurable.of_eval (fun time => (continuous_eval_const (time : ℝ≥0)).measurable)
+
+variable {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+include hW
+
+lemma increment_law (s h : ℝ≥0) : HasLaw (increment s h) (gaussianReal 0 h) W := by
+  exact (hW.toIsPreBrownianReal.shift s).hasLaw_eval h
+
+lemma increment_mean (s h : ℝ≥0) : (∫ b, increment s h b ∂W) = 0 := by
+  rw [(increment_law hW s h).integral_eq, integral_id_gaussianReal]
+
+lemma increment_sq (s h : ℝ≥0) : (∫ b, (increment s h b)^2 ∂W) = h := by
+  exact ((increment_law hW s h).integral_comp (f := fun x : ℝ => x^2) (by fun_prop)).trans
+    (integral_sq_gaussian h)
+
+lemma increment_fourth (s h : ℝ≥0) :
+    (∫ b, (increment s h b)^4 ∂W) = 3*(h:ℝ)^2 := by
+  exact ((increment_law hW s h).integral_comp (f := fun x : ℝ => x^4) (by fun_prop)).trans
+    (integral_fourth_gaussian h)
+
+end SLEExactGauge.BrownianCalculus
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/JacobiKernel.lean b/lean/OAI/Probability/SLEGauge/JacobiKernel.lean
index 4d3b09e25d0662d1a56133ec80eee689e8d07643..4dc52a19e29147df75463475a3af42852e7840e6 100644
--- a/lean/OAI/Probability/SLEGauge/JacobiKernel.lean
+++ b/lean/OAI/Probability/SLEGauge/JacobiKernel.lean
@@ -1,435 +1,435 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.AngleGenerators
-import OAI.Probability.SLEGauge.AngleMeasure
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open Polynomial MeasureTheory Filter Set
-open scoped ENNReal Topology
-namespace SLEExactGauge.Jacobi
-
-def eigenvalue (nu : ℝ) (n : ℕ) : ℝ := (n:ℝ)*(n+nu)/2
-def majorantBase (nu : ℝ) : ℝ := 4*(nu+2)^2
-def mixingTime (nu : ℝ) : ℝ := 4*majorantBase nu
-
-def mode (nu s : ℝ) (n : ℕ) (v w : Interval) : ℝ :=
-  Real.exp (-s*eigenvalue nu n) * (normSq nu 0/normSq nu n) *
-    (poly nu n).eval (v:ℝ) * (poly nu n).eval (w:ℝ)
-
-def kernel (nu s : ℝ) (v w : Interval) : ℝ := ∑' n, mode nu s n v w
-
-lemma eigenvalue_nonneg (nu : ℝ) (hnu : 0≤nu) (n : ℕ) : 0≤eigenvalue nu n := by
-  unfold eigenvalue
-  positivity
-
-lemma eigenvalue_ge (nu : ℝ) (hnu : 1≤nu) (n : ℕ) : (n:ℝ)≤eigenvalue nu n := by
-  cases n with
-  | zero => simp [eigenvalue]
-  | succ n =>
-    have hn : (0:ℝ)≤n := Nat.cast_nonneg n
-    dsimp [eigenvalue]
-    push_cast
-    nlinarith [mul_nonneg hn (sub_nonneg.mpr hnu),sq_nonneg (n:ℝ)]
-
-lemma norm_ratio_le (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
-    normSq nu 0/normSq nu n ≤ ((nu+2)^2)^n := by
-  have hd : 0<((nu+2)^2)^n := by positivity
-  rw [div_le_iff₀ (normSq_pos nu hnu n)]
-  have hn := (div_le_iff₀ hd).mp (normSq_lower nu hnu n)
-  simpa only [mul_comm] using hn
-
-lemma mode_bound (nu : ℝ) (hnu : 1<nu) (s : ℝ) (n : ℕ) (v w : Interval) :
-    ‖mode nu s n v w‖≤Real.exp (-s*eigenvalue nu n)*(majorantBase nu)^n := by
-  have hn := normSq_pos nu hnu n
-  have hz := normSq_zero_pos nu hnu.le
-  have hv : |(v:ℝ)|≤1 := abs_le.mpr v.2
-  have hw : |(w:ℝ)|≤1 := abs_le.mpr w.2
-  rw [mode,norm_mul,norm_mul,norm_mul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),
-    Real.norm_eq_abs,abs_of_pos (div_pos hz hn),Real.norm_eq_abs,Real.norm_eq_abs]
-  calc
-    _ ≤ Real.exp (-s*eigenvalue nu n)*((nu+2)^2)^n*2^n*2^n := by
-      gcongr
-      · exact norm_ratio_le nu hnu n
-      · exact poly_eval_bound nu (by linarith) n v hv
-      · exact poly_eval_bound nu (by linarith) n w hw
-    _ = _ := by
-      unfold majorantBase
-      rw [mul_pow,show (4:ℝ)=2*2 by norm_num,mul_pow]
-      ring
-
-lemma geometric_factor (nu : ℝ) (_hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) :
-    Real.exp (-s)*majorantBase nu≤1/4 := by
-  rw [Real.exp_neg,inv_mul_le_iff₀ (Real.exp_pos s)]
-  have he := Real.add_one_le_exp s
-  dsimp [mixingTime] at hs
-  linarith
-
- 
-lemma mode_geometric_bound (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (n : ℕ) (v w : Interval) : ‖mode nu s n v w‖≤(1/4:ℝ)^n := by
-  have hb : 0 < majorantBase nu := by unfold majorantBase; positivity
-  have hs0 : 0 ≤ s := le_trans (by unfold mixingTime; positivity) hs
-  have h1 : Real.exp (-s*eigenvalue nu n)≤Real.exp (-s*(n:ℝ)) :=
-    Real.exp_le_exp.mpr (mul_le_mul_of_nonpos_left (eigenvalue_ge nu hnu.le n) (neg_nonpos.mpr hs0))
-  calc
-    _ ≤ Real.exp (-s*eigenvalue nu n)*(majorantBase nu)^n := mode_bound nu hnu s n v w
-    _ ≤ Real.exp (-s*(n:ℝ))*(majorantBase nu)^n := mul_le_mul_of_nonneg_right h1 (by positivity)
-    _ = (Real.exp (-s)*majorantBase nu)^n := by rw [mul_comm (-s),Real.exp_nat_mul,mul_pow]
-    _ ≤ _ := pow_le_pow_left₀ (by positivity) (geometric_factor nu hnu s hs) _
-
-lemma mode_summable (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) (v w : Interval) :
-    Summable (fun n => mode nu s n v w) :=
-  Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4) (by norm_num))
-    (fun n => mode_geometric_bound nu hnu s hs n v w)
-
-lemma mode_continuous (nu s : ℝ) (n : ℕ) (v : Interval) : Continuous (mode nu s n v) := by
-  exact continuous_const.mul ((poly nu n).continuous.comp continuous_subtype_val)
-
-lemma kernel_continuous (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) (v : Interval) :
-    Continuous (kernel nu s v) :=
-  continuous_tsum (fun n => mode_continuous nu s n v)
-    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4) (by norm_num))
-    (fun n w => mode_geometric_bound nu hnu s hs n v w)
-
-lemma mode_zero (nu : ℝ) (hnu : 1<nu) (s : ℝ) (v w : Interval) : mode nu s 0 v w=1 := by
-  simp only [mode,eigenvalue,Nat.cast_zero,zero_mul,zero_div,mul_zero,
-    Real.exp_zero,div_self (normSq_zero_pos nu hnu.le).ne',poly_zero,eval_one,one_mul]
-
- 
-lemma kernel_bounds (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) (v w : Interval) :
-    1/2≤kernel nu s v w ∧ kernel nu s v w≤2 := by
-  have htail : ‖∑' n, mode nu s (n+1) v w‖≤(1/3:ℝ) := by
-    have hh : HasSum (fun n : ℕ => (1/4:ℝ)^(n+1)) (1/3:ℝ) := by
-      have hg : HasSum (fun n : ℕ => (1/4:ℝ)^n) ((1-(1/4:ℝ))⁻¹) :=
-        hasSum_geometric_of_lt_one (by norm_num) (by norm_num)
-      have hh := hg.mul_right (1/4:ℝ)
-      norm_num only [show (1-(1/4:ℝ))⁻¹*(1/4)=1/3 by norm_num] at hh
-      simpa only [pow_succ] using hh
-    exact tsum_of_norm_bounded hh (fun n => mode_geometric_bound nu hnu s hs (n+1) v w)
-  rw [Real.norm_eq_abs,abs_le] at htail
-  have he := (mode_summable nu hnu s hs v w).sum_add_tsum_nat_add 1
-  simp only [Finset.sum_range_one,mode_zero nu hnu s v w] at he
-  unfold kernel
-  rw [←he]
-  constructor <;> linarith [htail.1,htail.2]
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Polynomial MeasureTheory Filter Set
-open scoped ENNReal Topology CompactlySupported
-namespace SLEExactGauge.Jacobi
-
-lemma integral_mode_poly (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (n m : ℕ) (v : Interval) :
-    (∫ w : Interval, mode nu s n v w * (poly nu m).eval (w:ℝ) ∂pi nu) =
-      if n=m then Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) else 0 := by
-  have he : (fun w : Interval => mode nu s n v w * (poly nu m).eval (w:ℝ)) =
-      fun w : Interval => (Real.exp (-s*eigenvalue nu n)*(normSq nu 0/normSq nu n)*
-        (poly nu n).eval (v:ℝ))*((poly nu n).eval (w:ℝ)*(poly nu m).eval (w:ℝ)) := by
-    funext w; unfold mode; ring
-  rw [he,integral_const_mul,integral_pi_poly nu hnu]
-  split_ifs with hnm
-  · subst n
-    have hn := (normSq_pos nu hnu m).ne'
-    have hz := (normSq_zero_pos nu hnu.le).ne'
-    field_simp
-  · ring
-
-lemma integral_kernel_poly (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (m : ℕ) (v : Interval) :
-    (∫ w : Interval, kernel nu s v w * (poly nu m).eval (w:ℝ) ∂pi nu) =
-      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) := by
-  let := pi_probability nu hnu.le
-  let F (n : ℕ) (w : Interval) := mode nu s n v w * (poly nu m).eval (w:ℝ)
-  have hc (n : ℕ) : Continuous (F n) := (mode_continuous nu s n v).mul
-    ((poly nu m).continuous.comp continuous_subtype_val)
-  have hb (n : ℕ) (w : Interval) : ‖F n w‖ ≤ (1/4:ℝ)^n*2^m := by
-    dsimp only [F]
-    rw [norm_mul]
-    exact mul_le_mul (mode_geometric_bound nu hnu s hs n v w)
-      (poly_eval_bound nu (by linarith) m w (abs_le.mpr w.2))
-      (norm_nonneg _) (by positivity)
-  have hg : Summable (fun n : ℕ => (1/4:ℝ)^n*2^m) :=
-    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4) (by norm_num)).mul_right _
-  have hi := hasSum_integral_of_dominated_convergence (μ := pi nu) (F := F)
-    (f := fun w => kernel nu s v w * (poly nu m).eval (w:ℝ))
-    (fun n (_ : Interval) => (1/4:ℝ)^n*2^m)
-    (fun n => (hc n).aestronglyMeasurable)
-    (fun n => Eventually.of_forall (hb n)) (Eventually.of_forall fun _ => hg)
-    (integrable_const _) (Eventually.of_forall fun w =>
-      (mode_summable nu hnu s hs v w).hasSum.mul_right ((poly nu m).eval (w:ℝ)))
-  have he (n : ℕ) : (∫ w, F n w ∂pi nu) =
-      if n=m then Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) else 0 :=
-    integral_mode_poly nu hnu s n m v
-  have ht := hi.tsum_eq.symm
-  rw [show (∑' n, ∫ w, F n w ∂pi nu) =
-      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) by simp only [he,tsum_ite_eq]] at ht
-  exact ht
-
-lemma kernel_integrable (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (v : Interval) : Integrable (kernel nu s v) (pi nu) := by
-  let := pi_probability nu hnu.le
-  exact (kernel_continuous nu hnu s hs v).integrable_of_hasCompactSupport
-    (HasCompactSupport.of_compactSpace _)
-
-lemma integral_kernel (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (v : Interval) : (∫ w, kernel nu s v w ∂pi nu)=1 := by
-  simpa only [poly_zero,eval_one,mul_one,eigenvalue,Nat.cast_zero,zero_mul,zero_div,
-    mul_zero,Real.exp_zero] using integral_kernel_poly nu hnu s hs 0 v
-
- 
-def transition (nu s : ℝ) (v : Interval) : Measure Interval :=
-  (pi nu).withDensity (fun w => ENNReal.ofReal (kernel nu s v w))
-
-lemma transition_probability (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (v : Interval) : IsProbabilityMeasure (transition nu s v) := by
-  constructor
-  rw [transition,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,
-    ←ofReal_integral_eq_lintegral_ofReal (kernel_integrable nu hnu s hs v)
-    (Eventually.of_forall fun w => le_trans (by norm_num : (0:ℝ)≤1/2)
-      (kernel_bounds nu hnu s hs v w).1),integral_kernel nu hnu s hs v]
-  norm_num
-
-lemma integral_transition_poly (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (v : Interval) (m : ℕ) :
-    (∫ w : Interval, (poly nu m).eval (w:ℝ) ∂transition nu s v) =
-      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) := by
-  rw [transition,integral_withDensity_eq_integral_toReal_smul
-    (kernel_continuous nu hnu s hs v).measurable.ennreal_ofReal
-    (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
-  simp_rw [ENNReal.toReal_ofReal (le_trans (by norm_num : (0:ℝ)≤1/2)
-    (kernel_bounds nu hnu s hs v _).1),smul_eq_mul]
-  exact integral_kernel_poly nu hnu s hs m v
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open Polynomial MeasureTheory Filter Set
-open scoped ENNReal Topology CompactlySupported
-namespace SLEExactGauge.Jacobi
-
-lemma polynomial_integrable (μ : Measure Interval) [IsFiniteMeasure μ] (p : ℝ[X]) :
-    Integrable (fun x : Interval => p.eval (x:ℝ)) μ :=
-  (p.continuous.comp continuous_subtype_val).integrable_of_hasCompactSupport
-    (HasCompactSupport.of_compactSpace _)
-
- 
-lemma measure_eq_of_polynomial_integral (μ ν : Measure Interval)
-    [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
-    (he : ∀ p : ℝ[X], (∫ x : Interval, p.eval (x:ℝ) ∂μ)=∫ x : Interval, p.eval (x:ℝ) ∂ν) :
-    μ=ν := by
-  apply Measure.ext_of_integral_eq_on_compactlySupported
-  intro f
-  apply eq_of_forall_dist_le
-  intro ε hε
-  obtain ⟨p,hp⟩ := exists_polynomial_near_continuousMap (-1) 1
-    f.toContinuousMap (ε/2) (by positivity)
-  have hb (x : Interval) : ‖p.eval (x:ℝ)-f x‖≤ε/2 :=
-    ((p.toContinuousMapOn Interval-f.toContinuousMap).norm_coe_le_norm x).trans hp.le
-  have hbμ : ‖(∫ x : Interval,p.eval (x:ℝ) ∂μ)-(∫ x,f x∂μ)‖≤ε/2 := by
-    rw [←integral_sub (polynomial_integrable μ p) f.integrable]
-    simpa only [probReal_univ,mul_one] using norm_integral_le_of_norm_le_const
-      (μ:=μ) (Eventually.of_forall hb)
-  have hbν : ‖(∫ x : Interval,p.eval (x:ℝ) ∂ν)-(∫ x,f x∂ν)‖≤ε/2 := by
-    rw [←integral_sub (polynomial_integrable ν p) f.integrable]
-    simpa only [probReal_univ,mul_one] using norm_integral_le_of_norm_le_const
-      (μ:=ν) (Eventually.of_forall hb)
-  calc
-    _ ≤ dist (∫ x,f x∂μ) (∫ x : Interval,p.eval (x:ℝ) ∂μ)+
-        dist (∫ x : Interval,p.eval (x:ℝ) ∂μ) (∫ x,f x∂ν) := dist_triangle _ _ _
-    _ ≤ ε/2+ε/2 := by
-      apply add_le_add
-      · simpa only [dist_eq_norm,norm_sub_rev] using hbμ
-      · simpa only [he p,dist_eq_norm] using hbν
-    _ = ε := by ring
-
-def polynomialIntegral (μ : Measure Interval) [IsFiniteMeasure μ] : ℝ[X] →ₗ[ℝ] ℝ where
-  toFun p := ∫ x : Interval, p.eval (x:ℝ) ∂μ
-  map_add' p q := by
-    simp only [eval_add]
-    exact integral_add (polynomial_integrable μ p) (polynomial_integrable μ q)
-  map_smul' c p := by
-    simp only [eval_smul,smul_eq_mul,RingHom.id_apply]
-    exact integral_const_mul _ _
-
-lemma measure_eq_of_poly_integral (nu : ℝ) (hnu : 0 < nu)
-    (μ ν : Measure Interval) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
-    (he : ∀ n, (∫ x : Interval,(poly nu n).eval (x:ℝ) ∂μ)=
-      ∫ x : Interval,(poly nu n).eval (x:ℝ) ∂ν) : μ=ν := by
-  have heq : polynomialIntegral μ=polynomialIntegral ν := by
-    apply (basis nu hnu).ext
-    intro n
-    simpa only [basis_apply, polynomialIntegral, LinearMap.coe_mk, AddHom.coe_mk] using he n
-  apply measure_eq_of_polynomial_integral
-  intro p
-  exact congrArg (fun L : ℝ[X] →ₗ[ℝ] ℝ => L p) heq
-
- 
-
-lemma transition_identified (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (v : Interval) (μ : Measure Interval) [IsProbabilityMeasure μ]
-    (he : ∀ m, (∫ w : Interval,(poly nu m).eval (w:ℝ) ∂μ)=
-      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ)) : μ=transition nu s v := by
-  let := transition_probability nu hnu s hs v
-  apply measure_eq_of_poly_integral nu (by linarith)
-  intro m
-  rw [he m,integral_transition_poly nu hnu s hs v m]
-
- 
-
-lemma transition_lintegral_bounds (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
-    (v : Interval) (f : Interval→ℝ≥0∞) (hf : Measurable f) :
-    (1/2:ℝ≥0∞)*(∫⁻w,f w∂pi nu) ≤ ∫⁻w,f w∂transition nu s v ∧
-      (∫⁻w,f w∂transition nu s v) ≤ 2*(∫⁻w,f w∂pi nu) := by
-  rw [transition,lintegral_withDensity_eq_lintegral_mul _
-    (kernel_continuous nu hnu s hs v).measurable.ennreal_ofReal hf]
-  constructor
-  · rw [←lintegral_const_mul _ hf]
-    apply lintegral_mono
-    intro w
-    apply mul_le_mul_left
-    have h := ENNReal.ofReal_le_ofReal (kernel_bounds nu hnu s hs v w).1
-    norm_num only [ENNReal.ofReal_div_of_pos (by norm_num : (0:ℝ)<2), ENNReal.ofReal_one, ENNReal.ofReal_ofNat, one_div] at h ⊢
-    exact h
-  · rw [←lintegral_const_mul _ hf]
-    apply lintegral_mono
-    intro w
-    apply mul_le_mul_left
-    have h := ENNReal.ofReal_le_ofReal (kernel_bounds nu hnu s hs v w).2
-    norm_num only [ENNReal.ofReal_div_of_pos (by norm_num : (0:ℝ)<2), ENNReal.ofReal_one, ENNReal.ofReal_ofNat, one_div] at h ⊢
-    exact h
-
-end SLEExactGauge.Jacobi
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Polynomial MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology Interval
-namespace SLEExactGauge.Jacobi
-open CenteredMoments
-
-lemma weight_cos_sin (nu : ℝ) (hnu : 1<nu) (θ : ℝ) (hθ : θ∈Icc 0 Real.pi) :
-    weight nu (Real.cos θ)*Real.sin θ=(Real.sin θ)^nu := by
-  have hs : 0≤Real.sin θ := Real.sin_nonneg_of_mem_Icc hθ
-  have hid : 1-Real.cos θ^2=Real.sin θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
-  rw [weight,hid,←Real.rpow_natCast_mul hs]
-  norm_num only [Nat.cast_ofNat]
-  rw [show (2:ℝ)*((nu-1)/2)=nu-1 by ring]
-  calc
-    (Real.sin θ)^(nu-1)*Real.sin θ = (Real.sin θ)^(nu-1)*(Real.sin θ)^(1:ℝ) := by rw [Real.rpow_one]
-    _ = (Real.sin θ)^(nu-1+1) := (Real.rpow_add' hs (by linarith : nu-1+1≠0)).symm
-    _ = (Real.sin θ)^nu := by congr 1; ring
-
-lemma integral_sin_cos (nu : ℝ) (hnu : 1<nu) (p : ℝ[X]) :
-    (∫θ in 0..Real.pi,(Real.sin θ)^nu*p.eval (Real.cos θ))=
-      ∫v in (-1:ℝ)..1,weight nu v*p.eval v := by
-  have hsub := intervalIntegral.integral_comp_mul_deriv (a:=0) (b:=Real.pi)
-    (f:=Real.cos) (fun θ _ => Real.hasDerivAt_cos θ)
-    Real.continuous_sin.neg.continuousOn ((continuous_weight nu hnu.le).mul p.continuous)
-  simp only [Real.cos_zero,Real.cos_pi,Function.comp_apply,Pi.mul_apply] at hsub
-  rw [intervalIntegral.integral_symm (a := (-1:ℝ)) (b := 1)] at hsub
-  have he : (∫θ in 0..Real.pi,weight nu (Real.cos θ)*p.eval (Real.cos θ)*(-Real.sin θ))=
-      -(∫θ in 0..Real.pi,(Real.sin θ)^nu*p.eval (Real.cos θ)) := by
-    rw [←intervalIntegral.integral_neg]
-    apply intervalIntegral.integral_congr
-    intro θ hθ
-    have hθ' : θ∈Icc 0 Real.pi := by
-      simpa only [uIcc_of_le Real.pi_pos.le] using hθ
-    dsimp only
-    rw [show weight nu (Real.cos θ)*p.eval (Real.cos θ)*(-Real.sin θ)=
-      -(weight nu (Real.cos θ)*Real.sin θ)*p.eval (Real.cos θ) by ring,
-      weight_cos_sin nu hnu θ hθ']
-    ring
-  rw [he] at hsub
-  exact neg_injective hsub
-
-lemma integral_angleArea (f : ℝ→ℝ) :
-    (∫θ : Angle,f θ∂angleArea)=∫θ in 0..Real.pi,f θ := by
-  rw [angleArea,integral_subtype_comap measurableSet_Ioo,←integral_Ioc_eq_integral_Ioo,
-    intervalIntegral.integral_of_le Real.pi_pos.le]
-
-lemma angleNormalization_jacobi (a : ℝ) (ha : 1<4*a) :
-    angleNormalization a=ENNReal.ofReal (normSq (4*a) 0) := by
-  have ha0 : 0≤4*a := by linarith
-  have hm : AEStronglyMeasurable (fun θ : Angle => (Real.sin θ)^(4*a)) angleArea := by
-    fun_prop
-  have hb (θ : Angle) : ‖(Real.sin θ)^(4*a)‖≤1 := by
-    rw [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg
-      (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le _)]
-    exact Real.rpow_le_one (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le (Real.sin_le_one _) (by linarith)
-  have hi : Integrable (fun θ : Angle => (Real.sin θ)^(4*a)) angleArea :=
-    Integrable.of_bound hm 1 (Eventually.of_forall hb)
-  unfold angleNormalization
-  simp only [angleWeight]
-  rw [←ofReal_integral_eq_lintegral_ofReal hi
-    (Eventually.of_forall (fun θ : Angle => Real.rpow_nonneg
-      (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le _)),integral_angleArea (fun θ : ℝ => (Real.sin θ)^(4*a))]
-  have hh := integral_sin_cos (4*a) ha (1:ℝ[X])
-  simp only [eval_one,mul_one] at hh
-  rw [hh]
-  simp only [normSq,inner,poly_zero,eval_one,mul_one]
-
-def angleCos (θ : Angle) : Interval := ⟨Real.cos θ,Real.neg_one_le_cos _,Real.cos_le_one _⟩
-
-lemma angleCos_measurable : Measurable angleCos := by
-  unfold angleCos
-  fun_prop
-
-lemma integral_angleLaw_poly (a : ℝ) (ha : 1<4*a) (p : ℝ[X]) :
-    (∫θ : Angle,p.eval (Real.cos θ)∂angleLaw a)=
-      (∫v in (-1:ℝ)..1,weight (4*a) v*p.eval v)/normSq (4*a) 0 := by
-  rw [angleLaw,integral_smul_measure,integral_withDensity_eq_integral_toReal_smul
-    (angleWeight_measurable a) (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
-  have hn (θ : Angle) : 0≤(Real.sin θ)^(4*a) := Real.rpow_nonneg
-    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le _
-  simp_rw [angleWeight,ENNReal.toReal_ofReal (hn _),smul_eq_mul]
-  rw [angleNormalization_jacobi a ha,ENNReal.toReal_inv,
-    ENNReal.toReal_ofReal (normSq_zero_pos (4*a) ha.le).le,integral_angleArea (fun θ : ℝ => (Real.sin θ)^(4*a)*p.eval (Real.cos θ)),integral_sin_cos (4*a) ha p]
-  ring
-
- 
-lemma map_angleLaw_eq_pi (a : ℝ) (ha : 1<4*a) :
-    (angleLaw a).map angleCos=pi (4*a) := by
-  have := angleLaw_probability a (by linarith)
-  have := pi_probability (4*a) ha.le
-  apply measure_eq_of_polynomial_integral
-  intro p
-  have hf : AEStronglyMeasurable (fun v : Interval => p.eval (v:ℝ)) ((angleLaw a).map angleCos) :=
-    (p.continuous.comp continuous_subtype_val).aestronglyMeasurable
-  rw [integral_map angleCos_measurable.aemeasurable hf]
-  change (∫θ : Angle,p.eval (Real.cos θ)∂angleLaw a)=_
-  rw [integral_angleLaw_poly a ha]
-  have hh := integral_pi_mul (4*a) ha.le p (1:ℝ[X])
-  simpa only [eval_one,mul_one,inner] using hh.symm
-
-end SLEExactGauge.Jacobi
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.AngleGenerators
+import OAI.Probability.SLEGauge.AngleMeasure
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open Polynomial MeasureTheory Filter Set
+open scoped ENNReal Topology
+namespace SLEExactGauge.Jacobi
+
+def eigenvalue (nu : ℝ) (n : ℕ) : ℝ := (n:ℝ)*(n+nu)/2
+def majorantBase (nu : ℝ) : ℝ := 4*(nu+2)^2
+def mixingTime (nu : ℝ) : ℝ := 4*majorantBase nu
+
+def mode (nu s : ℝ) (n : ℕ) (v w : Interval) : ℝ :=
+  Real.exp (-s*eigenvalue nu n) * (normSq nu 0/normSq nu n) *
+    (poly nu n).eval (v:ℝ) * (poly nu n).eval (w:ℝ)
+
+def kernel (nu s : ℝ) (v w : Interval) : ℝ := ∑' n, mode nu s n v w
+
+lemma eigenvalue_nonneg (nu : ℝ) (hnu : 0≤nu) (n : ℕ) : 0≤eigenvalue nu n := by
+  unfold eigenvalue
+  positivity
+
+lemma eigenvalue_ge (nu : ℝ) (hnu : 1≤nu) (n : ℕ) : (n:ℝ)≤eigenvalue nu n := by
+  cases n with
+  | zero => simp [eigenvalue]
+  | succ n =>
+    have hn : (0:ℝ)≤n := Nat.cast_nonneg n
+    dsimp [eigenvalue]
+    push_cast
+    nlinarith [mul_nonneg hn (sub_nonneg.mpr hnu),sq_nonneg (n:ℝ)]
+
+lemma norm_ratio_le (nu : ℝ) (hnu : 1<nu) (n : ℕ) :
+    normSq nu 0/normSq nu n ≤ ((nu+2)^2)^n := by
+  have hd : 0<((nu+2)^2)^n := by positivity
+  rw [div_le_iff₀ (normSq_pos nu hnu n)]
+  have hn := (div_le_iff₀ hd).mp (normSq_lower nu hnu n)
+  simpa only [mul_comm] using hn
+
+lemma mode_bound (nu : ℝ) (hnu : 1<nu) (s : ℝ) (n : ℕ) (v w : Interval) :
+    ‖mode nu s n v w‖≤Real.exp (-s*eigenvalue nu n)*(majorantBase nu)^n := by
+  have hn := normSq_pos nu hnu n
+  have hz := normSq_zero_pos nu hnu.le
+  have hv : |(v:ℝ)|≤1 := abs_le.mpr v.2
+  have hw : |(w:ℝ)|≤1 := abs_le.mpr w.2
+  rw [mode,norm_mul,norm_mul,norm_mul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),
+    Real.norm_eq_abs,abs_of_pos (div_pos hz hn),Real.norm_eq_abs,Real.norm_eq_abs]
+  calc
+    _ ≤ Real.exp (-s*eigenvalue nu n)*((nu+2)^2)^n*2^n*2^n := by
+      gcongr
+      · exact norm_ratio_le nu hnu n
+      · exact poly_eval_bound nu (by linarith) n v hv
+      · exact poly_eval_bound nu (by linarith) n w hw
+    _ = _ := by
+      unfold majorantBase
+      rw [mul_pow,show (4:ℝ)=2*2 by norm_num,mul_pow]
+      ring
+
+lemma geometric_factor (nu : ℝ) (_hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) :
+    Real.exp (-s)*majorantBase nu≤1/4 := by
+  rw [Real.exp_neg,inv_mul_le_iff₀ (Real.exp_pos s)]
+  have he := Real.add_one_le_exp s
+  dsimp [mixingTime] at hs
+  linarith
+
+ 
+lemma mode_geometric_bound (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (n : ℕ) (v w : Interval) : ‖mode nu s n v w‖≤(1/4:ℝ)^n := by
+  have hb : 0 < majorantBase nu := by unfold majorantBase; positivity
+  have hs0 : 0 ≤ s := le_trans (by unfold mixingTime; positivity) hs
+  have h1 : Real.exp (-s*eigenvalue nu n)≤Real.exp (-s*(n:ℝ)) :=
+    Real.exp_le_exp.mpr (mul_le_mul_of_nonpos_left (eigenvalue_ge nu hnu.le n) (neg_nonpos.mpr hs0))
+  calc
+    _ ≤ Real.exp (-s*eigenvalue nu n)*(majorantBase nu)^n := mode_bound nu hnu s n v w
+    _ ≤ Real.exp (-s*(n:ℝ))*(majorantBase nu)^n := mul_le_mul_of_nonneg_right h1 (by positivity)
+    _ = (Real.exp (-s)*majorantBase nu)^n := by rw [mul_comm (-s),Real.exp_nat_mul,mul_pow]
+    _ ≤ _ := pow_le_pow_left₀ (by positivity) (geometric_factor nu hnu s hs) _
+
+lemma mode_summable (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) (v w : Interval) :
+    Summable (fun n => mode nu s n v w) :=
+  Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4) (by norm_num))
+    (fun n => mode_geometric_bound nu hnu s hs n v w)
+
+lemma mode_continuous (nu s : ℝ) (n : ℕ) (v : Interval) : Continuous (mode nu s n v) := by
+  exact continuous_const.mul ((poly nu n).continuous.comp continuous_subtype_val)
+
+lemma kernel_continuous (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) (v : Interval) :
+    Continuous (kernel nu s v) :=
+  continuous_tsum (fun n => mode_continuous nu s n v)
+    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4) (by norm_num))
+    (fun n w => mode_geometric_bound nu hnu s hs n v w)
+
+lemma mode_zero (nu : ℝ) (hnu : 1<nu) (s : ℝ) (v w : Interval) : mode nu s 0 v w=1 := by
+  simp only [mode,eigenvalue,Nat.cast_zero,zero_mul,zero_div,mul_zero,
+    Real.exp_zero,div_self (normSq_zero_pos nu hnu.le).ne',poly_zero,eval_one,one_mul]
+
+ 
+lemma kernel_bounds (nu : ℝ) (hnu : 1<nu) (s : ℝ) (hs : mixingTime nu ≤ s) (v w : Interval) :
+    1/2≤kernel nu s v w ∧ kernel nu s v w≤2 := by
+  have htail : ‖∑' n, mode nu s (n+1) v w‖≤(1/3:ℝ) := by
+    have hh : HasSum (fun n : ℕ => (1/4:ℝ)^(n+1)) (1/3:ℝ) := by
+      have hg : HasSum (fun n : ℕ => (1/4:ℝ)^n) ((1-(1/4:ℝ))⁻¹) :=
+        hasSum_geometric_of_lt_one (by norm_num) (by norm_num)
+      have hh := hg.mul_right (1/4:ℝ)
+      norm_num only [show (1-(1/4:ℝ))⁻¹*(1/4)=1/3 by norm_num] at hh
+      simpa only [pow_succ] using hh
+    exact tsum_of_norm_bounded hh (fun n => mode_geometric_bound nu hnu s hs (n+1) v w)
+  rw [Real.norm_eq_abs,abs_le] at htail
+  have he := (mode_summable nu hnu s hs v w).sum_add_tsum_nat_add 1
+  simp only [Finset.sum_range_one,mode_zero nu hnu s v w] at he
+  unfold kernel
+  rw [←he]
+  constructor <;> linarith [htail.1,htail.2]
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Polynomial MeasureTheory Filter Set
+open scoped ENNReal Topology CompactlySupported
+namespace SLEExactGauge.Jacobi
+
+lemma integral_mode_poly (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (n m : ℕ) (v : Interval) :
+    (∫ w : Interval, mode nu s n v w * (poly nu m).eval (w:ℝ) ∂pi nu) =
+      if n=m then Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) else 0 := by
+  have he : (fun w : Interval => mode nu s n v w * (poly nu m).eval (w:ℝ)) =
+      fun w : Interval => (Real.exp (-s*eigenvalue nu n)*(normSq nu 0/normSq nu n)*
+        (poly nu n).eval (v:ℝ))*((poly nu n).eval (w:ℝ)*(poly nu m).eval (w:ℝ)) := by
+    funext w; unfold mode; ring
+  rw [he,integral_const_mul,integral_pi_poly nu hnu]
+  split_ifs with hnm
+  · subst n
+    have hn := (normSq_pos nu hnu m).ne'
+    have hz := (normSq_zero_pos nu hnu.le).ne'
+    field_simp
+  · ring
+
+lemma integral_kernel_poly (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (m : ℕ) (v : Interval) :
+    (∫ w : Interval, kernel nu s v w * (poly nu m).eval (w:ℝ) ∂pi nu) =
+      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) := by
+  let := pi_probability nu hnu.le
+  let F (n : ℕ) (w : Interval) := mode nu s n v w * (poly nu m).eval (w:ℝ)
+  have hc (n : ℕ) : Continuous (F n) := (mode_continuous nu s n v).mul
+    ((poly nu m).continuous.comp continuous_subtype_val)
+  have hb (n : ℕ) (w : Interval) : ‖F n w‖ ≤ (1/4:ℝ)^n*2^m := by
+    dsimp only [F]
+    rw [norm_mul]
+    exact mul_le_mul (mode_geometric_bound nu hnu s hs n v w)
+      (poly_eval_bound nu (by linarith) m w (abs_le.mpr w.2))
+      (norm_nonneg _) (by positivity)
+  have hg : Summable (fun n : ℕ => (1/4:ℝ)^n*2^m) :=
+    (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4) (by norm_num)).mul_right _
+  have hi := hasSum_integral_of_dominated_convergence (μ := pi nu) (F := F)
+    (f := fun w => kernel nu s v w * (poly nu m).eval (w:ℝ))
+    (fun n (_ : Interval) => (1/4:ℝ)^n*2^m)
+    (fun n => (hc n).aestronglyMeasurable)
+    (fun n => Eventually.of_forall (hb n)) (Eventually.of_forall fun _ => hg)
+    (integrable_const _) (Eventually.of_forall fun w =>
+      (mode_summable nu hnu s hs v w).hasSum.mul_right ((poly nu m).eval (w:ℝ)))
+  have he (n : ℕ) : (∫ w, F n w ∂pi nu) =
+      if n=m then Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) else 0 :=
+    integral_mode_poly nu hnu s n m v
+  have ht := hi.tsum_eq.symm
+  rw [show (∑' n, ∫ w, F n w ∂pi nu) =
+      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) by simp only [he,tsum_ite_eq]] at ht
+  exact ht
+
+lemma kernel_integrable (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (v : Interval) : Integrable (kernel nu s v) (pi nu) := by
+  let := pi_probability nu hnu.le
+  exact (kernel_continuous nu hnu s hs v).integrable_of_hasCompactSupport
+    (HasCompactSupport.of_compactSpace _)
+
+lemma integral_kernel (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (v : Interval) : (∫ w, kernel nu s v w ∂pi nu)=1 := by
+  simpa only [poly_zero,eval_one,mul_one,eigenvalue,Nat.cast_zero,zero_mul,zero_div,
+    mul_zero,Real.exp_zero] using integral_kernel_poly nu hnu s hs 0 v
+
+ 
+def transition (nu s : ℝ) (v : Interval) : Measure Interval :=
+  (pi nu).withDensity (fun w => ENNReal.ofReal (kernel nu s v w))
+
+lemma transition_probability (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (v : Interval) : IsProbabilityMeasure (transition nu s v) := by
+  constructor
+  rw [transition,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ,
+    ←ofReal_integral_eq_lintegral_ofReal (kernel_integrable nu hnu s hs v)
+    (Eventually.of_forall fun w => le_trans (by norm_num : (0:ℝ)≤1/2)
+      (kernel_bounds nu hnu s hs v w).1),integral_kernel nu hnu s hs v]
+  norm_num
+
+lemma integral_transition_poly (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (v : Interval) (m : ℕ) :
+    (∫ w : Interval, (poly nu m).eval (w:ℝ) ∂transition nu s v) =
+      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ) := by
+  rw [transition,integral_withDensity_eq_integral_toReal_smul
+    (kernel_continuous nu hnu s hs v).measurable.ennreal_ofReal
+    (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
+  simp_rw [ENNReal.toReal_ofReal (le_trans (by norm_num : (0:ℝ)≤1/2)
+    (kernel_bounds nu hnu s hs v _).1),smul_eq_mul]
+  exact integral_kernel_poly nu hnu s hs m v
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open Polynomial MeasureTheory Filter Set
+open scoped ENNReal Topology CompactlySupported
+namespace SLEExactGauge.Jacobi
+
+lemma polynomial_integrable (μ : Measure Interval) [IsFiniteMeasure μ] (p : ℝ[X]) :
+    Integrable (fun x : Interval => p.eval (x:ℝ)) μ :=
+  (p.continuous.comp continuous_subtype_val).integrable_of_hasCompactSupport
+    (HasCompactSupport.of_compactSpace _)
+
+ 
+lemma measure_eq_of_polynomial_integral (μ ν : Measure Interval)
+    [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    (he : ∀ p : ℝ[X], (∫ x : Interval, p.eval (x:ℝ) ∂μ)=∫ x : Interval, p.eval (x:ℝ) ∂ν) :
+    μ=ν := by
+  apply Measure.ext_of_integral_eq_on_compactlySupported
+  intro f
+  apply eq_of_forall_dist_le
+  intro ε hε
+  obtain ⟨p,hp⟩ := exists_polynomial_near_continuousMap (-1) 1
+    f.toContinuousMap (ε/2) (by positivity)
+  have hb (x : Interval) : ‖p.eval (x:ℝ)-f x‖≤ε/2 :=
+    ((p.toContinuousMapOn Interval-f.toContinuousMap).norm_coe_le_norm x).trans hp.le
+  have hbμ : ‖(∫ x : Interval,p.eval (x:ℝ) ∂μ)-(∫ x,f x∂μ)‖≤ε/2 := by
+    rw [←integral_sub (polynomial_integrable μ p) f.integrable]
+    simpa only [probReal_univ,mul_one] using norm_integral_le_of_norm_le_const
+      (μ:=μ) (Eventually.of_forall hb)
+  have hbν : ‖(∫ x : Interval,p.eval (x:ℝ) ∂ν)-(∫ x,f x∂ν)‖≤ε/2 := by
+    rw [←integral_sub (polynomial_integrable ν p) f.integrable]
+    simpa only [probReal_univ,mul_one] using norm_integral_le_of_norm_le_const
+      (μ:=ν) (Eventually.of_forall hb)
+  calc
+    _ ≤ dist (∫ x,f x∂μ) (∫ x : Interval,p.eval (x:ℝ) ∂μ)+
+        dist (∫ x : Interval,p.eval (x:ℝ) ∂μ) (∫ x,f x∂ν) := dist_triangle _ _ _
+    _ ≤ ε/2+ε/2 := by
+      apply add_le_add
+      · simpa only [dist_eq_norm,norm_sub_rev] using hbμ
+      · simpa only [he p,dist_eq_norm] using hbν
+    _ = ε := by ring
+
+def polynomialIntegral (μ : Measure Interval) [IsFiniteMeasure μ] : ℝ[X] →ₗ[ℝ] ℝ where
+  toFun p := ∫ x : Interval, p.eval (x:ℝ) ∂μ
+  map_add' p q := by
+    simp only [eval_add]
+    exact integral_add (polynomial_integrable μ p) (polynomial_integrable μ q)
+  map_smul' c p := by
+    simp only [eval_smul,smul_eq_mul,RingHom.id_apply]
+    exact integral_const_mul _ _
+
+lemma measure_eq_of_poly_integral (nu : ℝ) (hnu : 0 < nu)
+    (μ ν : Measure Interval) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    (he : ∀ n, (∫ x : Interval,(poly nu n).eval (x:ℝ) ∂μ)=
+      ∫ x : Interval,(poly nu n).eval (x:ℝ) ∂ν) : μ=ν := by
+  have heq : polynomialIntegral μ=polynomialIntegral ν := by
+    apply (basis nu hnu).ext
+    intro n
+    simpa only [basis_apply, polynomialIntegral, LinearMap.coe_mk, AddHom.coe_mk] using he n
+  apply measure_eq_of_polynomial_integral
+  intro p
+  exact congrArg (fun L : ℝ[X] →ₗ[ℝ] ℝ => L p) heq
+
+ 
+
+lemma transition_identified (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (v : Interval) (μ : Measure Interval) [IsProbabilityMeasure μ]
+    (he : ∀ m, (∫ w : Interval,(poly nu m).eval (w:ℝ) ∂μ)=
+      Real.exp (-s*eigenvalue nu m)*(poly nu m).eval (v:ℝ)) : μ=transition nu s v := by
+  let := transition_probability nu hnu s hs v
+  apply measure_eq_of_poly_integral nu (by linarith)
+  intro m
+  rw [he m,integral_transition_poly nu hnu s hs v m]
+
+ 
+
+lemma transition_lintegral_bounds (nu : ℝ) (hnu : 1 < nu) (s : ℝ) (hs : mixingTime nu ≤ s)
+    (v : Interval) (f : Interval→ℝ≥0∞) (hf : Measurable f) :
+    (1/2:ℝ≥0∞)*(∫⁻w,f w∂pi nu) ≤ ∫⁻w,f w∂transition nu s v ∧
+      (∫⁻w,f w∂transition nu s v) ≤ 2*(∫⁻w,f w∂pi nu) := by
+  rw [transition,lintegral_withDensity_eq_lintegral_mul _
+    (kernel_continuous nu hnu s hs v).measurable.ennreal_ofReal hf]
+  constructor
+  · rw [←lintegral_const_mul _ hf]
+    apply lintegral_mono
+    intro w
+    apply mul_le_mul_left
+    have h := ENNReal.ofReal_le_ofReal (kernel_bounds nu hnu s hs v w).1
+    norm_num only [ENNReal.ofReal_div_of_pos (by norm_num : (0:ℝ)<2), ENNReal.ofReal_one, ENNReal.ofReal_ofNat, one_div] at h ⊢
+    exact h
+  · rw [←lintegral_const_mul _ hf]
+    apply lintegral_mono
+    intro w
+    apply mul_le_mul_left
+    have h := ENNReal.ofReal_le_ofReal (kernel_bounds nu hnu s hs v w).2
+    norm_num only [ENNReal.ofReal_div_of_pos (by norm_num : (0:ℝ)<2), ENNReal.ofReal_one, ENNReal.ofReal_ofNat, one_div] at h ⊢
+    exact h
+
+end SLEExactGauge.Jacobi
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Polynomial MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology Interval
+namespace SLEExactGauge.Jacobi
+open CenteredMoments
+
+lemma weight_cos_sin (nu : ℝ) (hnu : 1<nu) (θ : ℝ) (hθ : θ∈Icc 0 Real.pi) :
+    weight nu (Real.cos θ)*Real.sin θ=(Real.sin θ)^nu := by
+  have hs : 0≤Real.sin θ := Real.sin_nonneg_of_mem_Icc hθ
+  have hid : 1-Real.cos θ^2=Real.sin θ^2 := by nlinarith [Real.sin_sq_add_cos_sq θ]
+  rw [weight,hid,←Real.rpow_natCast_mul hs]
+  norm_num only [Nat.cast_ofNat]
+  rw [show (2:ℝ)*((nu-1)/2)=nu-1 by ring]
+  calc
+    (Real.sin θ)^(nu-1)*Real.sin θ = (Real.sin θ)^(nu-1)*(Real.sin θ)^(1:ℝ) := by rw [Real.rpow_one]
+    _ = (Real.sin θ)^(nu-1+1) := (Real.rpow_add' hs (by linarith : nu-1+1≠0)).symm
+    _ = (Real.sin θ)^nu := by congr 1; ring
+
+lemma integral_sin_cos (nu : ℝ) (hnu : 1<nu) (p : ℝ[X]) :
+    (∫θ in 0..Real.pi,(Real.sin θ)^nu*p.eval (Real.cos θ))=
+      ∫v in (-1:ℝ)..1,weight nu v*p.eval v := by
+  have hsub := intervalIntegral.integral_comp_mul_deriv (a:=0) (b:=Real.pi)
+    (f:=Real.cos) (fun θ _ => Real.hasDerivAt_cos θ)
+    Real.continuous_sin.neg.continuousOn ((continuous_weight nu hnu.le).mul p.continuous)
+  simp only [Real.cos_zero,Real.cos_pi,Function.comp_apply,Pi.mul_apply] at hsub
+  rw [intervalIntegral.integral_symm (a := (-1:ℝ)) (b := 1)] at hsub
+  have he : (∫θ in 0..Real.pi,weight nu (Real.cos θ)*p.eval (Real.cos θ)*(-Real.sin θ))=
+      -(∫θ in 0..Real.pi,(Real.sin θ)^nu*p.eval (Real.cos θ)) := by
+    rw [←intervalIntegral.integral_neg]
+    apply intervalIntegral.integral_congr
+    intro θ hθ
+    have hθ' : θ∈Icc 0 Real.pi := by
+      simpa only [uIcc_of_le Real.pi_pos.le] using hθ
+    dsimp only
+    rw [show weight nu (Real.cos θ)*p.eval (Real.cos θ)*(-Real.sin θ)=
+      -(weight nu (Real.cos θ)*Real.sin θ)*p.eval (Real.cos θ) by ring,
+      weight_cos_sin nu hnu θ hθ']
+    ring
+  rw [he] at hsub
+  exact neg_injective hsub
+
+lemma integral_angleArea (f : ℝ→ℝ) :
+    (∫θ : Angle,f θ∂angleArea)=∫θ in 0..Real.pi,f θ := by
+  rw [angleArea,integral_subtype_comap measurableSet_Ioo,←integral_Ioc_eq_integral_Ioo,
+    intervalIntegral.integral_of_le Real.pi_pos.le]
+
+lemma angleNormalization_jacobi (a : ℝ) (ha : 1<4*a) :
+    angleNormalization a=ENNReal.ofReal (normSq (4*a) 0) := by
+  have ha0 : 0≤4*a := by linarith
+  have hm : AEStronglyMeasurable (fun θ : Angle => (Real.sin θ)^(4*a)) angleArea := by
+    fun_prop
+  have hb (θ : Angle) : ‖(Real.sin θ)^(4*a)‖≤1 := by
+    rw [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg
+      (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le _)]
+    exact Real.rpow_le_one (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le (Real.sin_le_one _) (by linarith)
+  have hi : Integrable (fun θ : Angle => (Real.sin θ)^(4*a)) angleArea :=
+    Integrable.of_bound hm 1 (Eventually.of_forall hb)
+  unfold angleNormalization
+  simp only [angleWeight]
+  rw [←ofReal_integral_eq_lintegral_ofReal hi
+    (Eventually.of_forall (fun θ : Angle => Real.rpow_nonneg
+      (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le _)),integral_angleArea (fun θ : ℝ => (Real.sin θ)^(4*a))]
+  have hh := integral_sin_cos (4*a) ha (1:ℝ[X])
+  simp only [eval_one,mul_one] at hh
+  rw [hh]
+  simp only [normSq,inner,poly_zero,eval_one,mul_one]
+
+def angleCos (θ : Angle) : Interval := ⟨Real.cos θ,Real.neg_one_le_cos _,Real.cos_le_one _⟩
+
+lemma angleCos_measurable : Measurable angleCos := by
+  unfold angleCos
+  fun_prop
+
+lemma integral_angleLaw_poly (a : ℝ) (ha : 1<4*a) (p : ℝ[X]) :
+    (∫θ : Angle,p.eval (Real.cos θ)∂angleLaw a)=
+      (∫v in (-1:ℝ)..1,weight (4*a) v*p.eval v)/normSq (4*a) 0 := by
+  rw [angleLaw,integral_smul_measure,integral_withDensity_eq_integral_toReal_smul
+    (angleWeight_measurable a) (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)]
+  have hn (θ : Angle) : 0≤(Real.sin θ)^(4*a) := Real.rpow_nonneg
+    (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le _
+  simp_rw [angleWeight,ENNReal.toReal_ofReal (hn _),smul_eq_mul]
+  rw [angleNormalization_jacobi a ha,ENNReal.toReal_inv,
+    ENNReal.toReal_ofReal (normSq_zero_pos (4*a) ha.le).le,integral_angleArea (fun θ : ℝ => (Real.sin θ)^(4*a)*p.eval (Real.cos θ)),integral_sin_cos (4*a) ha p]
+  ring
+
+ 
+lemma map_angleLaw_eq_pi (a : ℝ) (ha : 1<4*a) :
+    (angleLaw a).map angleCos=pi (4*a) := by
+  have := angleLaw_probability a (by linarith)
+  have := pi_probability (4*a) ha.le
+  apply measure_eq_of_polynomial_integral
+  intro p
+  have hf : AEStronglyMeasurable (fun v : Interval => p.eval (v:ℝ)) ((angleLaw a).map angleCos) :=
+    (p.continuous.comp continuous_subtype_val).aestronglyMeasurable
+  rw [integral_map angleCos_measurable.aemeasurable hf]
+  change (∫θ : Angle,p.eval (Real.cos θ)∂angleLaw a)=_
+  rw [integral_angleLaw_poly a ha]
+  have hh := integral_pi_mul (4*a) ha.le p (1:ℝ[X])
+  simpa only [eval_one,mul_one,inner] using hh.symm
+
+end SLEExactGauge.Jacobi
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/KernelBounds.lean b/lean/OAI/Probability/SLEGauge/KernelBounds.lean
index 39420697aa3e936cf4736e17d8da5a79553f5072..6a7186254343e06047a64191ffe98a491da3c7f6 100644
--- a/lean/OAI/Probability/SLEGauge/KernelBounds.lean
+++ b/lean/OAI/Probability/SLEGauge/KernelBounds.lean
@@ -1,433 +1,433 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.KernelMeasurability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-namespace ExactSLEKernel
-lemma kernel_cons_bound {m : ℕ} {α b : ℝ} (ha : 0 ≤ α) (hb : 0 < b)
-    (z : ℂ) (x : Fin m → ℂ) :
-    kernel α b (Fin.cons z x) ≤
-      ipow α (1/2)^m * kernel α b x *
-        (ipow α b + ∑ i, ipow α (dist z (x i)) +
-          ENNReal.ofReal b ^ α * ∑ i, ipow (2*α) (dist z (x i))) := by
-  classical
-  have hD : 1 ≤ ipow α (1/2) := by
-    have := ipow_anti ha (show (1:ℝ)/2 ≤ 1 by norm_num)
-    simpa [ipow] using this
-  by_cases hn : ∃ i, dist z (x i) < cappedDistance b x i / 2
-  · obtain ⟨i,hi⟩ := hn
-    have hp : (∏ j : Fin m, ipow α (cappedDistance b (Fin.cons z x) j.succ)) ≤
-        ipow α (1/2)^m * kernel α b x * (ENNReal.ofReal b ^ α * ipow α (dist z (x i))) := by
-      calc
-        _ ≤ ∏ j : Fin m, (ipow α (1/2) * ipow α (cappedDistance b x j)) *
-            (if j=i then ENNReal.ofReal b ^ α * ipow α (dist z (x i)) else 1) := by
-          apply Finset.prod_le_prod
-          intro j hj
-          by_cases hji : j=i
-          · subst j
-            rw [ite_eq_left rfl,cappedDistance_cons_succ,min_eq_right (by have := dist_nonneg (x:=z) (y:=x i); linarith)]
-            calc
-              _ = 1 * ipow α (dist z (x i)) := by simp
-              _ ≤ (ipow α (1/2) * (ipow α (cappedDistance b x i) * ENNReal.ofReal b ^ α)) * ipow α (dist z (x i)) := by
-                exact mul_le_mul' (one_le_mul_of_one_le_of_one_le hD (cap_compensated ha hb x i)) le_rfl
-              _ = _ := by ac_rfl
-          · rw [ite_eq_right hji,mul_one]
-            apply far_cap_factor ha hb.le
-            by_contra h
-            exact hji (half_cap_unique b z x j i (lt_of_not_ge h) hi)
-        _ = _ := by rw [Finset.prod_mul_distrib, Finset.prod_mul_distrib]; simp [Finset.prod_ite_eq',kernel,ipow]
-    rw [kernel,Fin.prod_univ_succ]
-    change ipow α (cappedDistance b (Fin.cons z x) 0) * _ ≤ _
-    rw [near_cap_nearest b z x i hi]
-    calc
-      _ ≤ ipow α (dist z (x i)) *
-          (ipow α (1/2)^m * kernel α b x * (ENNReal.ofReal b ^ α * ipow α (dist z (x i)))) := mul_le_mul' le_rfl hp
-      _ = ipow α (1/2)^m * kernel α b x * (ENNReal.ofReal b ^ α * ipow (2*α) (dist z (x i))) := by rw [ipow_twice]; ac_rfl
-      _ ≤ _ := by
-        apply mul_le_mul' le_rfl
-        exact (mul_le_mul' le_rfl (Finset.single_le_sum (fun k _ => (by positivity : (0:ℝ≥0∞) ≤ ipow (2*α) (dist z (x k)))) (Finset.mem_univ i))).trans le_add_self
-  · push Not at hn
-    have hp : (∏ j : Fin m, ipow α (cappedDistance b (Fin.cons z x) j.succ)) ≤
-        ipow α (1/2)^m * kernel α b x := by
-      calc
-        _ ≤ ∏ j : Fin m, ipow α (1/2) * ipow α (cappedDistance b x j) :=
-          Finset.prod_le_prod (fun j _ => far_cap_factor ha hb.le z x j (hn j))
-        _ = _ := by simp [Finset.prod_mul_distrib,kernel,ipow]
-    rw [kernel,Fin.prod_univ_succ]
-    change ipow α (cappedDistance b (Fin.cons z x) 0) * _ ≤ _
-    calc
-      _ ≤ (ipow α b + ∑ i, ipow α (dist z (x i))) *
-          (ipow α (1/2)^m * kernel α b x) := mul_le_mul' (cap_cons_factor_sum ha b z x) hp
-      _ ≤ _ := by rw [mul_comm]; exact mul_le_mul' le_rfl le_self_add
-
-open Metric
-
-lemma singular_ball_finite {p : ℝ} (_hp : 0 < p) (hp2 : p < 2) (R : ℝ) :
-    (∫⁻ x : ℂ in ball 0 R, edist x 0 ^ (-p)) < ∞ := by
-  have hi : IntegrableOn (fun x : ℂ => ‖x‖ ^ (-p)) (ball 0 R) := by
-    refine integrableOn_ball_of_norm_le_rpow (C:=1) (α:=p) (by simp) (by simpa using hp2) ?_ ?_
-    · filter_upwards [] with x
-      simp [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (norm_nonneg _) _)]
-    · exact (measurable_norm.pow_const _).aestronglyMeasurable
-  have hn : ∀ᵐ x : ℂ ∂volume.restrict (ball 0 R), x ≠ 0 :=
-    ae_restrict_of_ae (by simp [ae_iff])
-  have he : (fun x : ℂ => edist x 0 ^ (-p)) =ᵐ[volume.restrict (ball 0 R)]
-      fun x => ENNReal.ofReal (‖x‖ ^ (-p)) := by
-    filter_upwards [hn] with x hx
-    rw [edist_dist,dist_zero_right,ENNReal.ofReal_rpow_of_pos (norm_pos_iff.mpr hx)]
-  rw [lintegral_congr_ae he]
-  exact hi.lintegral_lt_top
-lemma singular_ball_translate (p : ℝ) (z : ℂ) (R : ℝ) :
-    (∫⁻ x : ℂ in ball z R, edist x z ^ (-p)) = ∫⁻ x : ℂ in ball 0 R, edist x 0 ^ (-p) := by
-  rw [← lintegral_indicator measurableSet_ball,← lintegral_indicator measurableSet_ball]
-  convert lintegral_sub_right_eq_self (μ:=volume) ((ball (0:ℂ) R).indicator (fun x => edist x 0 ^ (-p))) z using 1
-  congr 1
-  funext x
-  simp [Set.indicator,mem_ball,dist_eq_norm,sub_zero,edist_dist]
-lemma singular_uniform {p : ℝ} (hp : 0 < p) (hp2 : p < 2) (K : Set ℂ)
-    (hKb : Bornology.IsBounded K) :
-    ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ z : ℂ, (∫⁻ x in K, edist x z ^ (-p)) ≤ C := by
-  let C := volume K + ∫⁻ x : ℂ in ball 0 1, edist x 0 ^ (-p)
-  refine ⟨C,ENNReal.add_lt_top.mpr ⟨hKb.measure_lt_top,(singular_ball_finite hp hp2 1)⟩,?_⟩
-  intro z
-  calc
-    _ ≤ ∫⁻ x in K, 1 + (ball z 1).indicator (fun x => edist x z ^ (-p)) x := by
-      apply lintegral_mono
-      intro x
-      dsimp only
-      by_cases hx : x ∈ ball z 1
-      · simp [indicator_of_mem hx]
-      · rw [indicator_of_notMem hx,add_zero]
-        have hd : (1:ℝ≥0∞) ≤ edist x z := by
-          rw [edist_dist,ENNReal.one_le_ofReal]
-          simpa [mem_ball,dist_comm] using hx
-        calc
-          _ ≤ edist x z ^ (0:ℝ) := ENNReal.rpow_le_rpow_of_exponent_le hd (by linarith)
-          _ = 1 := by simp
-    _ = volume K + ∫⁻ x in K, (ball z 1).indicator (fun x => edist x z ^ (-p)) x := by
-      rw [lintegral_add_left measurable_const,lintegral_const,Measure.restrict_apply_univ,one_mul]
-    _ ≤ volume K + ∫⁻ x, (ball z 1).indicator (fun x => edist x z ^ (-p)) x :=
-      add_le_add le_rfl (lintegral_mono' Measure.restrict_le_self (fun _ => le_rfl))
-    _ = C := by rw [lintegral_indicator measurableSet_ball,singular_ball_translate]
-lemma conditional_kernel_bound {α b : ℝ} (ha : 0 < α) (ha1 : α < 1) (hb : 0 < b)
-    (m : ℕ) (K : Set ℂ) (hKb : Bornology.IsBounded K) :
-    ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ x : Fin m → ℂ,
-      (∫⁻ z in K, kernel α b (Fin.cons z x)) ≤ C * kernel α b x := by
-  obtain ⟨A,hA,hAb⟩ := singular_uniform ha (by linarith) K hKb
-  obtain ⟨B,hB,hBb⟩ := singular_uniform (show 0 < 2*α by linarith) (by linarith) K hKb
-  let Q := ipow α b * volume K + (m:ℝ≥0∞)*A + ENNReal.ofReal b ^ α * ((m:ℝ≥0∞)*B)
-  let C := ipow α (1/2)^m * Q
-  have hQ : Q < ∞ := by
-    apply ENNReal.add_lt_top.mpr
-    refine ⟨ENNReal.add_lt_top.mpr ⟨?_,?_⟩,?_⟩
-    · exact ENNReal.mul_lt_top (ipow_finite hb) hKb.measure_lt_top
-    · exact ENNReal.mul_lt_top (by simp) hA
-    · exact ENNReal.mul_lt_top (lt_top_iff_ne_top.mpr (ENNReal.rpow_ne_top_of_nonneg ha.le ENNReal.ofReal_ne_top)) (ENNReal.mul_lt_top (by simp) hB)
-  refine ⟨C,ENNReal.mul_lt_top (ENNReal.pow_lt_top (ipow_finite (by norm_num))) hQ,?_⟩
-  intro x
-  have hm (p : ℝ) (i : Fin m) : Measurable (fun z : ℂ => ipow p (dist z (x i))) := by unfold ipow; fun_prop
-  have hsum (p : ℝ) : Measurable (fun z : ℂ => ∑ i, ipow p (dist z (x i))) := Finset.measurable_sum _ (fun _ _ => hm _ _)
-  have hs : (∫⁻ z in K, ipow α b + ∑ i, ipow α (dist z (x i)) +
-        ENNReal.ofReal b ^ α * ∑ i, ipow (2*α) (dist z (x i))) ≤ Q := by
-    rw [lintegral_add_left (f:=fun z : ℂ => ipow α b + ∑ i, ipow α (dist z (x i))) (by fun_prop),lintegral_add_left (f:=fun _ : ℂ => ipow α b) measurable_const,
-        lintegral_const,Measure.restrict_apply_univ,
-        lintegral_finsetSum _ (fun _ _ => hm _ _),lintegral_const_mul _ (hsum _),
-        lintegral_finsetSum _ (fun _ _ => hm _ _)]
-    apply add_le_add
-    · apply add_le_add le_rfl
-      calc
-        _ ≤ ∑ i : Fin m, A := Finset.sum_le_sum (fun i _ => by simpa [ipow,edist_dist] using hAb (x i))
-        _ = _ := by simp
-    · apply mul_le_mul' le_rfl
-      calc
-        _ ≤ ∑ i : Fin m, B := Finset.sum_le_sum (fun i _ => by simpa [ipow,edist_dist] using hBb (x i))
-        _ = _ := by simp
-  calc
-    _ ≤ ∫⁻ z in K, ipow α (1/2)^m * kernel α b x *
-        (ipow α b + ∑ i, ipow α (dist z (x i)) +
-          ENNReal.ofReal b ^ α * ∑ i, ipow (2*α) (dist z (x i))) := lintegral_mono (fun z => kernel_cons_bound ha.le hb z x)
-    _ = _ := lintegral_const_mul _ ((measurable_const.add (hsum α)).add (measurable_const.mul (hsum _)))
-    _ ≤ (ipow α (1/2)^m * kernel α b x) * Q := mul_le_mul' le_rfl hs
-    _ = C * kernel α b x := by dsimp [C]; ac_rfl
-lemma cap_equiv {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ]
-    (e : ι ≃ κ) (b : ℝ) (x : κ → ℂ) (i : ι) :
-    cappedDistance b (x ∘ e) i = cappedDistance b x (e i) := by
-  apply le_antisymm
-  · apply (le_cappedDistance_iff _ _ _ _).mpr
-    refine ⟨cappedDistance_le _ _ _,?_⟩
-    intro j hj
-    have := cappedDistance_le_dist b (x ∘ e) i (e.symm j) (fun h => hj (by rw [←e.apply_symm_apply j,h]))
-    simpa using this
-  · apply (le_cappedDistance_iff _ _ _ _).mpr
-    exact ⟨cappedDistance_le _ _ _,fun j hj => cappedDistance_le_dist b x (e i) (e j) (fun h => hj (e.injective h))⟩
-lemma kernel_equiv {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ]
-    (e : ι ≃ κ) (α b : ℝ) (x : κ → ℂ) : kernel α b (x ∘ e) = kernel α b x := by
-  unfold kernel
-  simp_rw [cap_equiv]
-  exact e.prod_comp (fun i => ENNReal.ofReal (cappedDistance b x i) ^ (-α))
-lemma kernel_swap_cons {m : ℕ} (α b : ℝ) (z y : ℂ) (x : Fin m → ℂ) :
-    kernel α b (Fin.cons z (Fin.cons y x)) = kernel α b (Fin.cons y (Fin.cons z x)) := by
-  classical
-  have he : Fin.cons z (Fin.cons y x) =
-      (Fin.cons y (Fin.cons z x)) ∘ Equiv.swap 0 (1 : Fin (m+2)) := by
-    funext i
-    refine Fin.cases ?_ (fun j => ?_) i
-    · simp
-    · refine Fin.cases ?_ (fun k => ?_) j
-      · simp
-      · simp [Equiv.swap_apply_of_ne_of_ne,Fin.succ_succ_ne_one]
-  rw [he,kernel_equiv]
-lemma lintegral_cons {m : ℕ} (μ : Measure ℂ) [SigmaFinite μ]
-    (f : (Fin (m+1) → ℂ) → ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻ x, f x ∂Measure.pi (fun _ => μ)) =
-      ∫⁻ x : Fin m → ℂ, ∫⁻ z : ℂ, f (Fin.cons z x) ∂μ ∂Measure.pi (fun _ => μ) := by
-  let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (m+1) => ℂ) 0
-  have he := (measurePreserving_piFinSuccAbove (fun _ : Fin (m+1) => μ) 0).symm
-  have hi := he.lintegral_comp hf
-  rw [←hi]
-  change (∫⁻ a : ℂ × (Fin m → ℂ), (f ∘ e.symm) a ∂μ.prod (Measure.pi fun _ => μ)) = _
-  rw [lintegral_prod_symm _ (hf.comp e.symm.measurable).aemeasurable]
-  apply lintegral_congr
-  intro x
-  apply lintegral_congr
-  intro z
-  simp [e,MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv]
-
-theorem capped_distance_kernel_integral (α b : ℝ) (hα : 0 < α) (hα1 : α < 1)
-    (hb : 0 < b) (m : ℕ) (K : Set ℂ) (_hK : MeasurableSet K) (hKb : Bornology.IsBounded K) :
-    (∫⁻ x : Fin m → ℂ, kernel α b x ∂Measure.pi (fun _ => volume.restrict K)) < ∞ := by
-  induction m with
-  | zero => simp [kernel]
-  | succ m ih =>
-    obtain ⟨C,hC,hCb⟩ := conditional_kernel_bound hα hα1 hb m K hKb
-    rw [lintegral_cons _ _ (measurable_kernel hα.le b)]
-    calc
-      _ ≤ ∫⁻ x : Fin m → ℂ, C * kernel α b x ∂Measure.pi (fun _ => volume.restrict K) := lintegral_mono hCb
-      _ = C * ∫⁻ x : Fin m → ℂ, kernel α b x ∂Measure.pi (fun _ => volume.restrict K) := lintegral_const_mul _ (measurable_kernel hα.le b)
-      _ < ∞ := ENNReal.mul_lt_top hC ih
-
-theorem capped_distance_fixed_center (α b : ℝ) (hα : 0 < α) (hα1 : α < 1)
-    (hb : 0 < b) (m : ℕ) (K K₀ : Set ℂ) (_hK : MeasurableSet K)
-    (hKb : Bornology.IsBounded K) (_hK₀b : Bornology.IsBounded K₀) :
-    ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ z ∈ K₀,
-      (∫⁻ x : Fin m → ℂ, kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K)) ≤ C := by
-  have hmain : ∀ m : ℕ, ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ z : ℂ,
-      (∫⁻ x : Fin m → ℂ, kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K)) ≤ C := by
-    intro m
-    induction m with
-    | zero =>
-      refine ⟨ipow α b,ipow_finite hb,fun z => ?_⟩
-      simp [kernel,cappedDistance,ipow]
-    | succ m ih =>
-      obtain ⟨A,hA,hAb⟩ := ih
-      obtain ⟨C,hC,hCb⟩ := conditional_kernel_bound hα hα1 hb (m+1) K hKb
-      refine ⟨C*A,ENNReal.mul_lt_top hC hA,fun z => ?_⟩
-      have hm : Measurable (fun x : Fin (m+1) → ℂ => kernel α b (Fin.cons z x)) :=
-        (measurable_kernel hα.le b).comp (by fun_prop)
-      rw [lintegral_cons _ _ hm]
-      simp_rw [kernel_swap_cons α b z]
-      calc
-        _ ≤ ∫⁻ x : Fin m → ℂ, C * kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K) := lintegral_mono (fun x => hCb (Fin.cons z x))
-        _ = C * ∫⁻ x : Fin m → ℂ, kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K) := lintegral_const_mul _ ((measurable_kernel hα.le b).comp (by fun_prop))
-        _ ≤ _ := mul_le_mul' le_rfl (hAb z)
-  obtain ⟨C,hC,hCb⟩ := hmain m
-  exact ⟨C,hC,fun z _ => hCb z⟩
-end ExactSLEKernel
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain
-open scoped Classical NNReal ENNReal
-noncomputable def fullRadialDensity (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (target : ℝ) : ℝ :=
-  Real.exp (-alpha a * Real.log 2) * radialWeight a b z (target/2)
-lemma fullRadialDensity_bound (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (target : ℝ) (ht : 0<target) (htz : target≤2*z.im) (b : ExactSLEGenerator.Driver) :
-    0≤fullRadialDensity a b.1 z target ∧ fullRadialDensity a b.1 z target≤target^(-alpha a) := by
-  have h := radialWeight_bound a ha z hz (target/2) (by positivity) (by linarith) b
-  refine ⟨mul_nonneg (Real.exp_pos _).le h.1,?_⟩
-  calc
-    _ ≤ Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log (target/2)) :=
-      mul_le_mul_of_nonneg_left h.2 (Real.exp_pos _).le
-    _ = target^(-alpha a) := by
-      rw [←Real.exp_add,Real.log_div ht.ne' (by norm_num : (2:ℝ)≠0),Real.rpow_def_of_pos ht]
-      congr 1; ring
-
-lemma selective_hit_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
-    ∃ C : ℝ, 0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ z : Fin m → ℂ, Function.Injective z → (∀ i, 0<(z i).im) →
-    ∀ cap target : Fin m → ℝ,
-      (∀ i, 0<cap i ∧ cap i≤2*(z i).im) →
-      (∀ i j, i≠j → cap i≤‖z i-z j‖) →
-      ∀ I : Finset (Fin m), (∀ i∈I, 0<target i ∧ target i≤cap i) →
-      W {b : ExactSLEGenerator.Driver | ∀ i∈I, radialTime a b.1 (z i) (target i/2)<∞} ≤
-        ENNReal.ofReal (C*∏ i∈I, (target i/cap i)^(alpha a)) := by
-  choose D hD hDb using fun k => actual_multipoint_hit_bound a ha k
-  let C := 1+∑ k∈Finset.range (m+1), D k
-  have hCle (k : ℕ) (hk : k ≤ m) : D k ≤ C := by
-    have hs : D k≤∑ j∈Finset.range (m+1), D j :=
-      Finset.single_le_sum (fun j _ => (hD j).le) (Finset.mem_range.mpr (by omega))
-    dsimp [C]; linarith
-  have hCp : 0<C := (hD 0).trans_le (hCle 0 (Nat.zero_le m))
-  refine ⟨C,hCp,?_⟩
-  intro W hprob hW z hinj hz cap target hc hsep I ht
-  let e := (Fintype.equivFin I).symm
-  have h : Fintype.card I ≤ m := by simpa using I.card_le_univ
-  have hb := hDb (Fintype.card I) W hW (fun i => z (e i))
-    (hinj.comp (Subtype.val_injective.comp e.injective)) (fun i => hz (e i))
-    (fun i => cap (e i)) (fun i => target (e i)) (fun i => hc (e i))
-    (fun i j hij => hsep (e i) (e j) (fun he => hij (e.injective (Subtype.ext he))))
-    (fun i => ht (e i) (e i).2)
-  have he : {b : ExactSLEGenerator.Driver | ∀ i : Fin (Fintype.card I), radialTime a b.1 (z (e i)) (target (e i)/2)<∞} =
-      {b : ExactSLEGenerator.Driver | ∀ i∈I, radialTime a b.1 (z i) (target i/2)<∞} := by
-    ext b; simp only [mem_ofPred_eq]
-    constructor
-    · intro H i hi; obtain ⟨j,hj⟩ := e.surjective ⟨i,hi⟩; simpa [hj] using H j
-    · intro H i; exact H (e i) (e i).2
-  have hp : (∏ i : Fin (Fintype.card I), (target (e i)/cap (e i))^(alpha a)) =
-      ∏ i∈I, (target i/cap i)^(alpha a) := by
-    rw [e.prod_comp (fun i : I => (target i/cap i)^(alpha a))]
-    exact Finset.prod_coe_sort I (fun i => (target i/cap i)^(alpha a))
-  rw [he,hp] at hb
-  refine hb.trans (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right (hCle _ h) ?_))
-  exact Finset.prod_nonneg fun i hi => Real.rpow_nonneg (le_of_lt (div_pos (ht i hi).1 (hc i).1)) _
-
-theorem actual_density_product_bound (a : ℝ) (ha : 1/4 < a) (m : ℕ) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ z : Fin m → ℂ, Function.Injective z → (∀ i, 0 < (z i).im) →
-    ∀ cap target : Fin m → ℝ,
-      (∀ i, 0<cap i ∧ cap i≤2*(z i).im) →
-      (∀ i j, i≠j → cap i≤‖z i-z j‖) →
-      (∀ i, 0<target i ∧ target i≤2*(z i).im) →
-      (∫⁻ b, ∏ i, ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i)) ∂W) ≤
-        ENNReal.ofReal (C*∏ i, (cap i)^(-alpha a)) := by
-  obtain ⟨C,hC,hCb⟩ := selective_hit_bound a ha m
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW z hinj hz cap target hc hsep ht
-  let I := Finset.univ.filter (fun i => target i≤cap i)
-  let E := {b : ExactSLEGenerator.Driver | ∀ i∈I, radialTime a b.1 (z i) (target i/2)<∞}
-  let A := ∏ i, (target i)^(-alpha a)
-  have hA : 0≤A := Finset.prod_nonneg fun i _ => Real.rpow_nonneg (ht i).1.le _
-  have hhit : W E≤ENNReal.ofReal (C*∏ i∈I, (target i/cap i)^(alpha a)) :=
-    hCb W hW z hinj hz cap target hc hsep I (fun i hi => ⟨(ht i).1,(Finset.mem_filter.mp hi).2⟩)
-  have hpoint (b : ExactSLEGenerator.Driver) :
-      (∏ i, ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i))) ≤ E.indicator (fun _ => ENNReal.ofReal A) b := by
-    by_cases hb : b∈E
-    · rw [Set.indicator_of_mem hb,ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (ht i).1.le _)]
-      exact Finset.prod_le_prod₀ (fun _ _ => bot_le) (fun i _ => ENNReal.ofReal_le_ofReal
-        (fullRadialDensity_bound a ha (z i) (hz i) (target i) (ht i).1 (ht i).2 b).2)
-    · rw [Set.indicator_of_notMem hb]
-      have he : ∃ i∈I, radialTime a b.1 (z i) (target i/2)=∞ := by
-        by_contra hn
-        apply hb
-        intro i hi
-        exact lt_top_iff_ne_top.mpr (fun he => hn ⟨i,hi,he⟩)
-      obtain ⟨i,hi,he⟩ := he
-      have hz0 : ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i))=0 := by
-        simp [fullRadialDensity,radialWeight,he]
-      exact le_of_eq (Finset.prod_eq_zero (Finset.mem_univ i) hz0)
-  have halg : A*(∏ i∈I, (target i/cap i)^(alpha a)) ≤ ∏ i, (cap i)^(-alpha a) := by
-    unfold A I
-    rw [Finset.prod_filter,←Finset.prod_mul_distrib]
-    apply Finset.prod_le_prod₀
-    · intro i _
-      apply mul_nonneg (Real.rpow_nonneg (ht i).1.le _)
-      split_ifs
-      · exact Real.rpow_nonneg (div_nonneg (ht i).1.le (hc i).1.le) _
-      · norm_num
-    · intro i _
-      by_cases hi : target i≤cap i
-      · rw [ite_eq_left hi,Real.div_rpow (ht i).1.le (hc i).1.le,
-          Real.rpow_neg (ht i).1.le,Real.rpow_neg (hc i).1.le]
-        have hp := Real.rpow_pos_of_pos (ht i).1 (alpha a)
-        field_simp
-        rfl
-      · rw [ite_eq_right hi,mul_one]
-        exact Real.rpow_le_rpow_of_nonpos (hc i).1 (le_of_not_ge hi) (neg_nonpos.mpr (alpha_pos ha).le)
-  calc
-    _ ≤ ∫⁻ b, E.indicator (fun _ => ENNReal.ofReal A) b ∂W := lintegral_mono hpoint
-    _ ≤ ENNReal.ofReal A*W E := lintegral_indicator_const_le E _
-    _ ≤ ENNReal.ofReal A*ENNReal.ofReal (C*∏ i∈I, (target i/cap i)^(alpha a)) := mul_le_mul' le_rfl hhit
-    _ = ENNReal.ofReal (C*(A*∏ i∈I, (target i/cap i)^(alpha a))) := by rw [←ENNReal.ofReal_mul hA]; congr 1; ring
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left halg hC.le)
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain ExactSLEKernel
-open scoped Classical NNReal ENNReal
-noncomputable def massDensity (a q : ℝ) (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ≥0∞ :=
-  if 0<z.im then ENNReal.ofReal (fullRadialDensity a b.1 z (2*z.im*q)) else 0
-lemma massDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1) :
-    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => massDensity a q p.1 p.2) := by
-  let U : Set (ExactSLEGenerator.Driver × ℂ) := {p | 0<p.2.im}
-  have hU : MeasurableSet U := measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
-  have hm : Measurable (fun p : U => ENNReal.ofReal
-      (Real.exp (-alpha a*Real.log 2)*radialWeight a p.1.1.1 p.1.2 (p.1.2.im*q))) := by
-    have hp : Measurable (fun p : U => (p.1.1, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im}))) := by fun_prop
-    exact ((measurable_const.mul ((joint_radialWeight_measurable a ha q hq hq1).comp hp)).ennreal_ofReal)
-  have h := hm.dite (g:=fun _ => 0) measurable_const hU
-  convert h using 1
-  funext p
-  simp only [massDensity,fullRadialDensity]
-  have he : 2*p.2.im*q/2=p.2.im*q := by ring
-  simp only [he]
-  rfl
-
-lemma lintegral_power_pi (μ : Measure ℂ) [SigmaFinite μ] (f : ℂ → ℝ≥0∞)
-    (hf : Measurable f) (m : ℕ) :
-    (∫⁻ z, f z ∂μ)^m = ∫⁻ x : Fin m → ℂ, ∏ i, f (x i) ∂Measure.pi (fun _ => μ) := by
-  induction m with
-  | zero => simp
-  | succ m ih =>
-    rw [lintegral_cons _ _ (by fun_prop)]
-    simp_rw [Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ,lintegral_mul_const _ hf]
-    rw [lintegral_const_mul _ (by fun_prop),←ih,pow_succ']
-
-lemma kernel_top_of_not_injective (α c : ℝ) (ha : 0<α) (hc : 0<c)
-    {m : ℕ} (x : Fin m → ℂ) (hx : ¬Function.Injective x) : kernel α c x=∞ := by
-  classical
-  have hh : ∃ i j, x i=x j ∧ i≠j := by
-    by_contra hn
-    apply hx
-    intro i j hij
-    by_contra hne
-    exact hn ⟨i,j,hij,hne⟩
-  obtain ⟨i,j,hij,hne⟩ := hh
-  have hz : cappedDistance c x i=0 := by
-    apply le_antisymm _ (cappedDistance_nonneg hc.le x i)
-    have h := cappedDistance_le_dist c x i j hne.symm
-    simpa [hij] using h
-  have hp : ∀ k, ENNReal.ofReal (cappedDistance c x k)^(-α) ≠ 0 := fun k =>
-    ne_of_gt (ipow_pos ha.le)
-  unfold kernel
-  rw [←Finset.mul_prod_erase Finset.univ _ (Finset.mem_univ i),hz,ENNReal.ofReal_zero,
-    ENNReal.zero_rpow_def]
-  simp only [neg_pos,not_lt.mpr ha.le,ite_false,neg_eq_zero,ha.ne']
-  exact ENNReal.top_mul (Finset.prod_ne_zero_iff.mpr (fun k _ => hp k))
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.KernelMeasurability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+namespace ExactSLEKernel
+lemma kernel_cons_bound {m : ℕ} {α b : ℝ} (ha : 0 ≤ α) (hb : 0 < b)
+    (z : ℂ) (x : Fin m → ℂ) :
+    kernel α b (Fin.cons z x) ≤
+      ipow α (1/2)^m * kernel α b x *
+        (ipow α b + ∑ i, ipow α (dist z (x i)) +
+          ENNReal.ofReal b ^ α * ∑ i, ipow (2*α) (dist z (x i))) := by
+  classical
+  have hD : 1 ≤ ipow α (1/2) := by
+    have := ipow_anti ha (show (1:ℝ)/2 ≤ 1 by norm_num)
+    simpa [ipow] using this
+  by_cases hn : ∃ i, dist z (x i) < cappedDistance b x i / 2
+  · obtain ⟨i,hi⟩ := hn
+    have hp : (∏ j : Fin m, ipow α (cappedDistance b (Fin.cons z x) j.succ)) ≤
+        ipow α (1/2)^m * kernel α b x * (ENNReal.ofReal b ^ α * ipow α (dist z (x i))) := by
+      calc
+        _ ≤ ∏ j : Fin m, (ipow α (1/2) * ipow α (cappedDistance b x j)) *
+            (if j=i then ENNReal.ofReal b ^ α * ipow α (dist z (x i)) else 1) := by
+          apply Finset.prod_le_prod
+          intro j hj
+          by_cases hji : j=i
+          · subst j
+            rw [ite_eq_left rfl,cappedDistance_cons_succ,min_eq_right (by have := dist_nonneg (x:=z) (y:=x i); linarith)]
+            calc
+              _ = 1 * ipow α (dist z (x i)) := by simp
+              _ ≤ (ipow α (1/2) * (ipow α (cappedDistance b x i) * ENNReal.ofReal b ^ α)) * ipow α (dist z (x i)) := by
+                exact mul_le_mul' (one_le_mul_of_one_le_of_one_le hD (cap_compensated ha hb x i)) le_rfl
+              _ = _ := by ac_rfl
+          · rw [ite_eq_right hji,mul_one]
+            apply far_cap_factor ha hb.le
+            by_contra h
+            exact hji (half_cap_unique b z x j i (lt_of_not_ge h) hi)
+        _ = _ := by rw [Finset.prod_mul_distrib, Finset.prod_mul_distrib]; simp [Finset.prod_ite_eq',kernel,ipow]
+    rw [kernel,Fin.prod_univ_succ]
+    change ipow α (cappedDistance b (Fin.cons z x) 0) * _ ≤ _
+    rw [near_cap_nearest b z x i hi]
+    calc
+      _ ≤ ipow α (dist z (x i)) *
+          (ipow α (1/2)^m * kernel α b x * (ENNReal.ofReal b ^ α * ipow α (dist z (x i)))) := mul_le_mul' le_rfl hp
+      _ = ipow α (1/2)^m * kernel α b x * (ENNReal.ofReal b ^ α * ipow (2*α) (dist z (x i))) := by rw [ipow_twice]; ac_rfl
+      _ ≤ _ := by
+        apply mul_le_mul' le_rfl
+        exact (mul_le_mul' le_rfl (Finset.single_le_sum (fun k _ => (by positivity : (0:ℝ≥0∞) ≤ ipow (2*α) (dist z (x k)))) (Finset.mem_univ i))).trans le_add_self
+  · push Not at hn
+    have hp : (∏ j : Fin m, ipow α (cappedDistance b (Fin.cons z x) j.succ)) ≤
+        ipow α (1/2)^m * kernel α b x := by
+      calc
+        _ ≤ ∏ j : Fin m, ipow α (1/2) * ipow α (cappedDistance b x j) :=
+          Finset.prod_le_prod (fun j _ => far_cap_factor ha hb.le z x j (hn j))
+        _ = _ := by simp [Finset.prod_mul_distrib,kernel,ipow]
+    rw [kernel,Fin.prod_univ_succ]
+    change ipow α (cappedDistance b (Fin.cons z x) 0) * _ ≤ _
+    calc
+      _ ≤ (ipow α b + ∑ i, ipow α (dist z (x i))) *
+          (ipow α (1/2)^m * kernel α b x) := mul_le_mul' (cap_cons_factor_sum ha b z x) hp
+      _ ≤ _ := by rw [mul_comm]; exact mul_le_mul' le_rfl le_self_add
+
+open Metric
+
+lemma singular_ball_finite {p : ℝ} (_hp : 0 < p) (hp2 : p < 2) (R : ℝ) :
+    (∫⁻ x : ℂ in ball 0 R, edist x 0 ^ (-p)) < ∞ := by
+  have hi : IntegrableOn (fun x : ℂ => ‖x‖ ^ (-p)) (ball 0 R) := by
+    refine integrableOn_ball_of_norm_le_rpow (C:=1) (α:=p) (by simp) (by simpa using hp2) ?_ ?_
+    · filter_upwards [] with x
+      simp [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (norm_nonneg _) _)]
+    · exact (measurable_norm.pow_const _).aestronglyMeasurable
+  have hn : ∀ᵐ x : ℂ ∂volume.restrict (ball 0 R), x ≠ 0 :=
+    ae_restrict_of_ae (by simp [ae_iff])
+  have he : (fun x : ℂ => edist x 0 ^ (-p)) =ᵐ[volume.restrict (ball 0 R)]
+      fun x => ENNReal.ofReal (‖x‖ ^ (-p)) := by
+    filter_upwards [hn] with x hx
+    rw [edist_dist,dist_zero_right,ENNReal.ofReal_rpow_of_pos (norm_pos_iff.mpr hx)]
+  rw [lintegral_congr_ae he]
+  exact hi.lintegral_lt_top
+lemma singular_ball_translate (p : ℝ) (z : ℂ) (R : ℝ) :
+    (∫⁻ x : ℂ in ball z R, edist x z ^ (-p)) = ∫⁻ x : ℂ in ball 0 R, edist x 0 ^ (-p) := by
+  rw [← lintegral_indicator measurableSet_ball,← lintegral_indicator measurableSet_ball]
+  convert lintegral_sub_right_eq_self (μ:=volume) ((ball (0:ℂ) R).indicator (fun x => edist x 0 ^ (-p))) z using 1
+  congr 1
+  funext x
+  simp [Set.indicator,mem_ball,dist_eq_norm,sub_zero,edist_dist]
+lemma singular_uniform {p : ℝ} (hp : 0 < p) (hp2 : p < 2) (K : Set ℂ)
+    (hKb : Bornology.IsBounded K) :
+    ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ z : ℂ, (∫⁻ x in K, edist x z ^ (-p)) ≤ C := by
+  let C := volume K + ∫⁻ x : ℂ in ball 0 1, edist x 0 ^ (-p)
+  refine ⟨C,ENNReal.add_lt_top.mpr ⟨hKb.measure_lt_top,(singular_ball_finite hp hp2 1)⟩,?_⟩
+  intro z
+  calc
+    _ ≤ ∫⁻ x in K, 1 + (ball z 1).indicator (fun x => edist x z ^ (-p)) x := by
+      apply lintegral_mono
+      intro x
+      dsimp only
+      by_cases hx : x ∈ ball z 1
+      · simp [indicator_of_mem hx]
+      · rw [indicator_of_notMem hx,add_zero]
+        have hd : (1:ℝ≥0∞) ≤ edist x z := by
+          rw [edist_dist,ENNReal.one_le_ofReal]
+          simpa [mem_ball,dist_comm] using hx
+        calc
+          _ ≤ edist x z ^ (0:ℝ) := ENNReal.rpow_le_rpow_of_exponent_le hd (by linarith)
+          _ = 1 := by simp
+    _ = volume K + ∫⁻ x in K, (ball z 1).indicator (fun x => edist x z ^ (-p)) x := by
+      rw [lintegral_add_left measurable_const,lintegral_const,Measure.restrict_apply_univ,one_mul]
+    _ ≤ volume K + ∫⁻ x, (ball z 1).indicator (fun x => edist x z ^ (-p)) x :=
+      add_le_add le_rfl (lintegral_mono' Measure.restrict_le_self (fun _ => le_rfl))
+    _ = C := by rw [lintegral_indicator measurableSet_ball,singular_ball_translate]
+lemma conditional_kernel_bound {α b : ℝ} (ha : 0 < α) (ha1 : α < 1) (hb : 0 < b)
+    (m : ℕ) (K : Set ℂ) (hKb : Bornology.IsBounded K) :
+    ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ x : Fin m → ℂ,
+      (∫⁻ z in K, kernel α b (Fin.cons z x)) ≤ C * kernel α b x := by
+  obtain ⟨A,hA,hAb⟩ := singular_uniform ha (by linarith) K hKb
+  obtain ⟨B,hB,hBb⟩ := singular_uniform (show 0 < 2*α by linarith) (by linarith) K hKb
+  let Q := ipow α b * volume K + (m:ℝ≥0∞)*A + ENNReal.ofReal b ^ α * ((m:ℝ≥0∞)*B)
+  let C := ipow α (1/2)^m * Q
+  have hQ : Q < ∞ := by
+    apply ENNReal.add_lt_top.mpr
+    refine ⟨ENNReal.add_lt_top.mpr ⟨?_,?_⟩,?_⟩
+    · exact ENNReal.mul_lt_top (ipow_finite hb) hKb.measure_lt_top
+    · exact ENNReal.mul_lt_top (by simp) hA
+    · exact ENNReal.mul_lt_top (lt_top_iff_ne_top.mpr (ENNReal.rpow_ne_top_of_nonneg ha.le ENNReal.ofReal_ne_top)) (ENNReal.mul_lt_top (by simp) hB)
+  refine ⟨C,ENNReal.mul_lt_top (ENNReal.pow_lt_top (ipow_finite (by norm_num))) hQ,?_⟩
+  intro x
+  have hm (p : ℝ) (i : Fin m) : Measurable (fun z : ℂ => ipow p (dist z (x i))) := by unfold ipow; fun_prop
+  have hsum (p : ℝ) : Measurable (fun z : ℂ => ∑ i, ipow p (dist z (x i))) := Finset.measurable_sum _ (fun _ _ => hm _ _)
+  have hs : (∫⁻ z in K, ipow α b + ∑ i, ipow α (dist z (x i)) +
+        ENNReal.ofReal b ^ α * ∑ i, ipow (2*α) (dist z (x i))) ≤ Q := by
+    rw [lintegral_add_left (f:=fun z : ℂ => ipow α b + ∑ i, ipow α (dist z (x i))) (by fun_prop),lintegral_add_left (f:=fun _ : ℂ => ipow α b) measurable_const,
+        lintegral_const,Measure.restrict_apply_univ,
+        lintegral_finsetSum _ (fun _ _ => hm _ _),lintegral_const_mul _ (hsum _),
+        lintegral_finsetSum _ (fun _ _ => hm _ _)]
+    apply add_le_add
+    · apply add_le_add le_rfl
+      calc
+        _ ≤ ∑ i : Fin m, A := Finset.sum_le_sum (fun i _ => by simpa [ipow,edist_dist] using hAb (x i))
+        _ = _ := by simp
+    · apply mul_le_mul' le_rfl
+      calc
+        _ ≤ ∑ i : Fin m, B := Finset.sum_le_sum (fun i _ => by simpa [ipow,edist_dist] using hBb (x i))
+        _ = _ := by simp
+  calc
+    _ ≤ ∫⁻ z in K, ipow α (1/2)^m * kernel α b x *
+        (ipow α b + ∑ i, ipow α (dist z (x i)) +
+          ENNReal.ofReal b ^ α * ∑ i, ipow (2*α) (dist z (x i))) := lintegral_mono (fun z => kernel_cons_bound ha.le hb z x)
+    _ = _ := lintegral_const_mul _ ((measurable_const.add (hsum α)).add (measurable_const.mul (hsum _)))
+    _ ≤ (ipow α (1/2)^m * kernel α b x) * Q := mul_le_mul' le_rfl hs
+    _ = C * kernel α b x := by dsimp [C]; ac_rfl
+lemma cap_equiv {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ]
+    (e : ι ≃ κ) (b : ℝ) (x : κ → ℂ) (i : ι) :
+    cappedDistance b (x ∘ e) i = cappedDistance b x (e i) := by
+  apply le_antisymm
+  · apply (le_cappedDistance_iff _ _ _ _).mpr
+    refine ⟨cappedDistance_le _ _ _,?_⟩
+    intro j hj
+    have := cappedDistance_le_dist b (x ∘ e) i (e.symm j) (fun h => hj (by rw [←e.apply_symm_apply j,h]))
+    simpa using this
+  · apply (le_cappedDistance_iff _ _ _ _).mpr
+    exact ⟨cappedDistance_le _ _ _,fun j hj => cappedDistance_le_dist b x (e i) (e j) (fun h => hj (e.injective h))⟩
+lemma kernel_equiv {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ]
+    (e : ι ≃ κ) (α b : ℝ) (x : κ → ℂ) : kernel α b (x ∘ e) = kernel α b x := by
+  unfold kernel
+  simp_rw [cap_equiv]
+  exact e.prod_comp (fun i => ENNReal.ofReal (cappedDistance b x i) ^ (-α))
+lemma kernel_swap_cons {m : ℕ} (α b : ℝ) (z y : ℂ) (x : Fin m → ℂ) :
+    kernel α b (Fin.cons z (Fin.cons y x)) = kernel α b (Fin.cons y (Fin.cons z x)) := by
+  classical
+  have he : Fin.cons z (Fin.cons y x) =
+      (Fin.cons y (Fin.cons z x)) ∘ Equiv.swap 0 (1 : Fin (m+2)) := by
+    funext i
+    refine Fin.cases ?_ (fun j => ?_) i
+    · simp
+    · refine Fin.cases ?_ (fun k => ?_) j
+      · simp
+      · simp [Equiv.swap_apply_of_ne_of_ne,Fin.succ_succ_ne_one]
+  rw [he,kernel_equiv]
+lemma lintegral_cons {m : ℕ} (μ : Measure ℂ) [SigmaFinite μ]
+    (f : (Fin (m+1) → ℂ) → ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻ x, f x ∂Measure.pi (fun _ => μ)) =
+      ∫⁻ x : Fin m → ℂ, ∫⁻ z : ℂ, f (Fin.cons z x) ∂μ ∂Measure.pi (fun _ => μ) := by
+  let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (m+1) => ℂ) 0
+  have he := (measurePreserving_piFinSuccAbove (fun _ : Fin (m+1) => μ) 0).symm
+  have hi := he.lintegral_comp hf
+  rw [←hi]
+  change (∫⁻ a : ℂ × (Fin m → ℂ), (f ∘ e.symm) a ∂μ.prod (Measure.pi fun _ => μ)) = _
+  rw [lintegral_prod_symm _ (hf.comp e.symm.measurable).aemeasurable]
+  apply lintegral_congr
+  intro x
+  apply lintegral_congr
+  intro z
+  simp [e,MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv]
+
+theorem capped_distance_kernel_integral (α b : ℝ) (hα : 0 < α) (hα1 : α < 1)
+    (hb : 0 < b) (m : ℕ) (K : Set ℂ) (_hK : MeasurableSet K) (hKb : Bornology.IsBounded K) :
+    (∫⁻ x : Fin m → ℂ, kernel α b x ∂Measure.pi (fun _ => volume.restrict K)) < ∞ := by
+  induction m with
+  | zero => simp [kernel]
+  | succ m ih =>
+    obtain ⟨C,hC,hCb⟩ := conditional_kernel_bound hα hα1 hb m K hKb
+    rw [lintegral_cons _ _ (measurable_kernel hα.le b)]
+    calc
+      _ ≤ ∫⁻ x : Fin m → ℂ, C * kernel α b x ∂Measure.pi (fun _ => volume.restrict K) := lintegral_mono hCb
+      _ = C * ∫⁻ x : Fin m → ℂ, kernel α b x ∂Measure.pi (fun _ => volume.restrict K) := lintegral_const_mul _ (measurable_kernel hα.le b)
+      _ < ∞ := ENNReal.mul_lt_top hC ih
+
+theorem capped_distance_fixed_center (α b : ℝ) (hα : 0 < α) (hα1 : α < 1)
+    (hb : 0 < b) (m : ℕ) (K K₀ : Set ℂ) (_hK : MeasurableSet K)
+    (hKb : Bornology.IsBounded K) (_hK₀b : Bornology.IsBounded K₀) :
+    ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ z ∈ K₀,
+      (∫⁻ x : Fin m → ℂ, kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K)) ≤ C := by
+  have hmain : ∀ m : ℕ, ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ z : ℂ,
+      (∫⁻ x : Fin m → ℂ, kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K)) ≤ C := by
+    intro m
+    induction m with
+    | zero =>
+      refine ⟨ipow α b,ipow_finite hb,fun z => ?_⟩
+      simp [kernel,cappedDistance,ipow]
+    | succ m ih =>
+      obtain ⟨A,hA,hAb⟩ := ih
+      obtain ⟨C,hC,hCb⟩ := conditional_kernel_bound hα hα1 hb (m+1) K hKb
+      refine ⟨C*A,ENNReal.mul_lt_top hC hA,fun z => ?_⟩
+      have hm : Measurable (fun x : Fin (m+1) → ℂ => kernel α b (Fin.cons z x)) :=
+        (measurable_kernel hα.le b).comp (by fun_prop)
+      rw [lintegral_cons _ _ hm]
+      simp_rw [kernel_swap_cons α b z]
+      calc
+        _ ≤ ∫⁻ x : Fin m → ℂ, C * kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K) := lintegral_mono (fun x => hCb (Fin.cons z x))
+        _ = C * ∫⁻ x : Fin m → ℂ, kernel α b (Fin.cons z x) ∂Measure.pi (fun _ => volume.restrict K) := lintegral_const_mul _ ((measurable_kernel hα.le b).comp (by fun_prop))
+        _ ≤ _ := mul_le_mul' le_rfl (hAb z)
+  obtain ⟨C,hC,hCb⟩ := hmain m
+  exact ⟨C,hC,fun z _ => hCb z⟩
+end ExactSLEKernel
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain
+open scoped Classical NNReal ENNReal
+noncomputable def fullRadialDensity (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (target : ℝ) : ℝ :=
+  Real.exp (-alpha a * Real.log 2) * radialWeight a b z (target/2)
+lemma fullRadialDensity_bound (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (target : ℝ) (ht : 0<target) (htz : target≤2*z.im) (b : ExactSLEGenerator.Driver) :
+    0≤fullRadialDensity a b.1 z target ∧ fullRadialDensity a b.1 z target≤target^(-alpha a) := by
+  have h := radialWeight_bound a ha z hz (target/2) (by positivity) (by linarith) b
+  refine ⟨mul_nonneg (Real.exp_pos _).le h.1,?_⟩
+  calc
+    _ ≤ Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log (target/2)) :=
+      mul_le_mul_of_nonneg_left h.2 (Real.exp_pos _).le
+    _ = target^(-alpha a) := by
+      rw [←Real.exp_add,Real.log_div ht.ne' (by norm_num : (2:ℝ)≠0),Real.rpow_def_of_pos ht]
+      congr 1; ring
+
+lemma selective_hit_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
+    ∃ C : ℝ, 0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ z : Fin m → ℂ, Function.Injective z → (∀ i, 0<(z i).im) →
+    ∀ cap target : Fin m → ℝ,
+      (∀ i, 0<cap i ∧ cap i≤2*(z i).im) →
+      (∀ i j, i≠j → cap i≤‖z i-z j‖) →
+      ∀ I : Finset (Fin m), (∀ i∈I, 0<target i ∧ target i≤cap i) →
+      W {b : ExactSLEGenerator.Driver | ∀ i∈I, radialTime a b.1 (z i) (target i/2)<∞} ≤
+        ENNReal.ofReal (C*∏ i∈I, (target i/cap i)^(alpha a)) := by
+  choose D hD hDb using fun k => actual_multipoint_hit_bound a ha k
+  let C := 1+∑ k∈Finset.range (m+1), D k
+  have hCle (k : ℕ) (hk : k ≤ m) : D k ≤ C := by
+    have hs : D k≤∑ j∈Finset.range (m+1), D j :=
+      Finset.single_le_sum (fun j _ => (hD j).le) (Finset.mem_range.mpr (by omega))
+    dsimp [C]; linarith
+  have hCp : 0<C := (hD 0).trans_le (hCle 0 (Nat.zero_le m))
+  refine ⟨C,hCp,?_⟩
+  intro W hprob hW z hinj hz cap target hc hsep I ht
+  let e := (Fintype.equivFin I).symm
+  have h : Fintype.card I ≤ m := by simpa using I.card_le_univ
+  have hb := hDb (Fintype.card I) W hW (fun i => z (e i))
+    (hinj.comp (Subtype.val_injective.comp e.injective)) (fun i => hz (e i))
+    (fun i => cap (e i)) (fun i => target (e i)) (fun i => hc (e i))
+    (fun i j hij => hsep (e i) (e j) (fun he => hij (e.injective (Subtype.ext he))))
+    (fun i => ht (e i) (e i).2)
+  have he : {b : ExactSLEGenerator.Driver | ∀ i : Fin (Fintype.card I), radialTime a b.1 (z (e i)) (target (e i)/2)<∞} =
+      {b : ExactSLEGenerator.Driver | ∀ i∈I, radialTime a b.1 (z i) (target i/2)<∞} := by
+    ext b; simp only [mem_ofPred_eq]
+    constructor
+    · intro H i hi; obtain ⟨j,hj⟩ := e.surjective ⟨i,hi⟩; simpa [hj] using H j
+    · intro H i; exact H (e i) (e i).2
+  have hp : (∏ i : Fin (Fintype.card I), (target (e i)/cap (e i))^(alpha a)) =
+      ∏ i∈I, (target i/cap i)^(alpha a) := by
+    rw [e.prod_comp (fun i : I => (target i/cap i)^(alpha a))]
+    exact Finset.prod_coe_sort I (fun i => (target i/cap i)^(alpha a))
+  rw [he,hp] at hb
+  refine hb.trans (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right (hCle _ h) ?_))
+  exact Finset.prod_nonneg fun i hi => Real.rpow_nonneg (le_of_lt (div_pos (ht i hi).1 (hc i).1)) _
+
+theorem actual_density_product_bound (a : ℝ) (ha : 1/4 < a) (m : ℕ) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ z : Fin m → ℂ, Function.Injective z → (∀ i, 0 < (z i).im) →
+    ∀ cap target : Fin m → ℝ,
+      (∀ i, 0<cap i ∧ cap i≤2*(z i).im) →
+      (∀ i j, i≠j → cap i≤‖z i-z j‖) →
+      (∀ i, 0<target i ∧ target i≤2*(z i).im) →
+      (∫⁻ b, ∏ i, ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i)) ∂W) ≤
+        ENNReal.ofReal (C*∏ i, (cap i)^(-alpha a)) := by
+  obtain ⟨C,hC,hCb⟩ := selective_hit_bound a ha m
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW z hinj hz cap target hc hsep ht
+  let I := Finset.univ.filter (fun i => target i≤cap i)
+  let E := {b : ExactSLEGenerator.Driver | ∀ i∈I, radialTime a b.1 (z i) (target i/2)<∞}
+  let A := ∏ i, (target i)^(-alpha a)
+  have hA : 0≤A := Finset.prod_nonneg fun i _ => Real.rpow_nonneg (ht i).1.le _
+  have hhit : W E≤ENNReal.ofReal (C*∏ i∈I, (target i/cap i)^(alpha a)) :=
+    hCb W hW z hinj hz cap target hc hsep I (fun i hi => ⟨(ht i).1,(Finset.mem_filter.mp hi).2⟩)
+  have hpoint (b : ExactSLEGenerator.Driver) :
+      (∏ i, ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i))) ≤ E.indicator (fun _ => ENNReal.ofReal A) b := by
+    by_cases hb : b∈E
+    · rw [Set.indicator_of_mem hb,ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (ht i).1.le _)]
+      exact Finset.prod_le_prod₀ (fun _ _ => bot_le) (fun i _ => ENNReal.ofReal_le_ofReal
+        (fullRadialDensity_bound a ha (z i) (hz i) (target i) (ht i).1 (ht i).2 b).2)
+    · rw [Set.indicator_of_notMem hb]
+      have he : ∃ i∈I, radialTime a b.1 (z i) (target i/2)=∞ := by
+        by_contra hn
+        apply hb
+        intro i hi
+        exact lt_top_iff_ne_top.mpr (fun he => hn ⟨i,hi,he⟩)
+      obtain ⟨i,hi,he⟩ := he
+      have hz0 : ENNReal.ofReal (fullRadialDensity a b.1 (z i) (target i))=0 := by
+        simp [fullRadialDensity,radialWeight,he]
+      exact le_of_eq (Finset.prod_eq_zero (Finset.mem_univ i) hz0)
+  have halg : A*(∏ i∈I, (target i/cap i)^(alpha a)) ≤ ∏ i, (cap i)^(-alpha a) := by
+    unfold A I
+    rw [Finset.prod_filter,←Finset.prod_mul_distrib]
+    apply Finset.prod_le_prod₀
+    · intro i _
+      apply mul_nonneg (Real.rpow_nonneg (ht i).1.le _)
+      split_ifs
+      · exact Real.rpow_nonneg (div_nonneg (ht i).1.le (hc i).1.le) _
+      · norm_num
+    · intro i _
+      by_cases hi : target i≤cap i
+      · rw [ite_eq_left hi,Real.div_rpow (ht i).1.le (hc i).1.le,
+          Real.rpow_neg (ht i).1.le,Real.rpow_neg (hc i).1.le]
+        have hp := Real.rpow_pos_of_pos (ht i).1 (alpha a)
+        field_simp
+        rfl
+      · rw [ite_eq_right hi,mul_one]
+        exact Real.rpow_le_rpow_of_nonpos (hc i).1 (le_of_not_ge hi) (neg_nonpos.mpr (alpha_pos ha).le)
+  calc
+    _ ≤ ∫⁻ b, E.indicator (fun _ => ENNReal.ofReal A) b ∂W := lintegral_mono hpoint
+    _ ≤ ENNReal.ofReal A*W E := lintegral_indicator_const_le E _
+    _ ≤ ENNReal.ofReal A*ENNReal.ofReal (C*∏ i∈I, (target i/cap i)^(alpha a)) := mul_le_mul' le_rfl hhit
+    _ = ENNReal.ofReal (C*(A*∏ i∈I, (target i/cap i)^(alpha a))) := by rw [←ENNReal.ofReal_mul hA]; congr 1; ring
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left halg hC.le)
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain ExactSLEKernel
+open scoped Classical NNReal ENNReal
+noncomputable def massDensity (a q : ℝ) (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ≥0∞ :=
+  if 0<z.im then ENNReal.ofReal (fullRadialDensity a b.1 z (2*z.im*q)) else 0
+lemma massDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1) :
+    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => massDensity a q p.1 p.2) := by
+  let U : Set (ExactSLEGenerator.Driver × ℂ) := {p | 0<p.2.im}
+  have hU : MeasurableSet U := measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
+  have hm : Measurable (fun p : U => ENNReal.ofReal
+      (Real.exp (-alpha a*Real.log 2)*radialWeight a p.1.1.1 p.1.2 (p.1.2.im*q))) := by
+    have hp : Measurable (fun p : U => (p.1.1, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im}))) := by fun_prop
+    exact ((measurable_const.mul ((joint_radialWeight_measurable a ha q hq hq1).comp hp)).ennreal_ofReal)
+  have h := hm.dite (g:=fun _ => 0) measurable_const hU
+  convert h using 1
+  funext p
+  simp only [massDensity,fullRadialDensity]
+  have he : 2*p.2.im*q/2=p.2.im*q := by ring
+  simp only [he]
+  rfl
+
+lemma lintegral_power_pi (μ : Measure ℂ) [SigmaFinite μ] (f : ℂ → ℝ≥0∞)
+    (hf : Measurable f) (m : ℕ) :
+    (∫⁻ z, f z ∂μ)^m = ∫⁻ x : Fin m → ℂ, ∏ i, f (x i) ∂Measure.pi (fun _ => μ) := by
+  induction m with
+  | zero => simp
+  | succ m ih =>
+    rw [lintegral_cons _ _ (by fun_prop)]
+    simp_rw [Fin.prod_univ_succ,Fin.cons_zero,Fin.cons_succ,lintegral_mul_const _ hf]
+    rw [lintegral_const_mul _ (by fun_prop),←ih,pow_succ']
+
+lemma kernel_top_of_not_injective (α c : ℝ) (ha : 0<α) (hc : 0<c)
+    {m : ℕ} (x : Fin m → ℂ) (hx : ¬Function.Injective x) : kernel α c x=∞ := by
+  classical
+  have hh : ∃ i j, x i=x j ∧ i≠j := by
+    by_contra hn
+    apply hx
+    intro i j hij
+    by_contra hne
+    exact hn ⟨i,j,hij,hne⟩
+  obtain ⟨i,j,hij,hne⟩ := hh
+  have hz : cappedDistance c x i=0 := by
+    apply le_antisymm _ (cappedDistance_nonneg hc.le x i)
+    have h := cappedDistance_le_dist c x i j hne.symm
+    simpa [hij] using h
+  have hp : ∀ k, ENNReal.ofReal (cappedDistance c x k)^(-α) ≠ 0 := fun k =>
+    ne_of_gt (ipow_pos ha.le)
+  unfold kernel
+  rw [←Finset.mul_prod_erase Finset.univ _ (Finset.mem_univ i),hz,ENNReal.ofReal_zero,
+    ENNReal.zero_rpow_def]
+  simp only [neg_pos,not_lt.mpr ha.le,ite_false,neg_eq_zero,ha.ne']
+  exact ENNReal.top_mul (Finset.prod_ne_zero_iff.mpr (fun k _ => hp k))
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/KernelMeasurability.lean b/lean/OAI/Probability/SLEGauge/KernelMeasurability.lean
index 637aa351e542ecbc3a902baba6d1fb1a0079c437..c824e9428079fd7c6de92c30f906302da2415597 100644
--- a/lean/OAI/Probability/SLEGauge/KernelMeasurability.lean
+++ b/lean/OAI/Probability/SLEGauge/KernelMeasurability.lean
@@ -1,437 +1,437 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ProductIntegrability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEFlow
-variable {Ω : Type*} [MeasurableSpace Ω]
-lemma localState_parameter_measurable (a : ℝ) (ε : Ω → ℝ) (hε : ∀ ω, 0 < ε ω)
-    (z : Ω → ℂ) (b : Ω → ExactSLEGenerator.Driver)
-    (hεm : Measurable ε) (hzm : Measurable z) (hbm : Measurable b) (t : ℝ≥0) :
-    Measurable (fun ω => localState a (ε ω) (hε ω) (z ω) t (b ω)) := by
-  let A : Ω → State → State := fun ω => cutoffDrift a (ε ω)
-  let L : Ω → ℝ≥0 := fun ω => ⟨max (|a|/ε ω) (|a|/(ε ω)^2), by positivity⟩
-  let K : Ω → ℝ≥0 := fun ω => ⟨max (2*|a|/(ε ω)^2) (4*|a|/(ε ω)^3), by positivity⟩
-  have hA : ∀ ω, LipschitzWith (K ω) (A ω) := fun ω => cutoffDrift_lipschitz a (ε ω) (hε ω)
-  have hb : ∀ ω v, ‖A ω v‖ ≤ L ω := fun ω => cutoffDrift_bound a (ε ω) (hε ω)
-  have hmA : Measurable (Function.uncurry A) := by
-    change Measurable (fun w : Ω × State => cutoffDrift a (ε w.1) w.2)
-    unfold cutoffDrift cutoffVector heightClamp
-    simp only [Complex.mk_eq_add_mul_I]
-    fun_prop
-  let u : ℝ≥0 → Ω → State := fun t ω => (b ω).1 t • noise
-  have huc (ω : Ω) : Continuous (fun t => u t ω) := (b ω).1.continuous.smul continuous_const
-  have hum (t : ℝ≥0) : Measurable (u t) := ((driver_eval_measurable t).measurable.comp hbm).smul_const noise
-  obtain ⟨F,hFc,hFm,hFe⟩ := measurable_parameter_flow A L K hA hb hmA u huc hum
-    (fun ω => (z ω,0)) (hzm.prodMk measurable_const)
-  have he (ω : Ω) (t : ℝ≥0) : F t ω = localState a (ε ω) (hε ω) (z ω) t (b ω) - u t ω := by
-    apply integral_flow_unique (hA ω) (huc ω) (huc ω) (hFc ω)
-      ((localState_continuous a (ε ω) (hε ω) (z ω) (b ω)).sub (huc ω)) (hFe ω)
-      (fun s => ?_) (T := t) (fun _ _ => rfl) le_rfl
-    simp only [Pi.sub_apply,sub_add_cancel]
-    rw [localState_eq_zero]
-    dsimp [u,A]
-    abel
-  convert (hFm t).add (hum t) using 1
-  funext ω
-  change localState a (ε ω) (hε ω) (z ω) t (b ω) = F t ω + u t ω
-  rw [he]
-  simp
-
-lemma localState_parameter_joint_measurable (a : ℝ) (ε : Ω → ℝ) (hε : ∀ ω, 0 < ε ω)
-    (z : Ω → ℂ) (b : Ω → ExactSLEGenerator.Driver)
-    (hεm : Measurable ε) (hzm : Measurable z) (hbm : Measurable b) :
-    Measurable (fun p : ℝ≥0 × Ω => localState a (ε p.2) (hε p.2) (z p.2) p.1 (b p.2)) := by
-  exact measurable_uncurry_of_continuous_of_measurable
-    (fun ω => localState_continuous a (ε ω) (hε ω) (z ω) (b ω))
-    (fun t => localState_parameter_measurable a ε hε z b hεm hzm hbm t)
-end ExactSLEGreen
-
-namespace ExactSLEHitting
-variable {Ω : Type*} [MeasurableSpace Ω]
-lemma boundedContact_parameter_measurable (f : ℝ≥0 → Ω → ℝ)
-    (hc : ∀ ω, Continuous (fun t => f t ω)) (hm : ∀ t, Measurable (f t))
-    (B : Ω → ℝ≥0) (hB : Measurable B) :
-    Measurable (fun ω => hittingBtwn f (Iic 0) 0 (B ω) ω) := by
-  apply measurable_of_Iic
-  intro t
-  have he : (fun ω => hittingBtwn f (Iic 0) 0 (B ω) ω) ⁻¹' Iic t =
-      {ω | B ω ≤ t} ∪ {ω | ∃ u ≤ t, f u ω ≤ 0} := by
-    ext ω
-    simp only [mem_preimage,mem_Iic,mem_union,mem_ofPred_eq]
-    by_cases h : B ω ≤ t
-    · exact iff_of_true ((hittingBtwn_le ω).trans h) (Or.inl h)
-    · rw [or_iff_right h]
-      exact boundedContact_le_iff f hc ω (B ω) t (lt_of_not_ge h)
-  rw [he]
-  exact (measurableSet_le hB measurable_const).union
-    (measurable_has_contact (Filtration.const ℝ≥0 (inferInstance : MeasurableSpace Ω) le_rfl) f hc (fun t => hm t) t)
-end ExactSLEHitting
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEHitting
-variable {Ω : Type*} [MeasurableSpace Ω]
-lemma regionStop_parameter_measurable (a : ℝ) (ε : Ω → ℝ) (hε : ∀ ω, 0 < ε ω)
-    (z : Ω → ℂ) (b : Ω → ExactSLEGenerator.Driver) (N ρ δ : Ω → ℝ) (B : Ω → ℝ≥0)
-    (hεm : Measurable ε) (hzm : Measurable z) (hbm : Measurable b)
-    (hN : Measurable N) (hρ : Measurable ρ) (hδ : Measurable δ) (hB : Measurable B) :
-    Measurable (fun ω => regionStop a (ε ω) (hε ω) (z ω) (N ω) (ρ ω) (δ ω) (B ω) (b ω)) := by
-  apply boundedContact_parameter_measurable
-    (fun t ω => regionMonitor (ε ω) (N ω) (ρ ω) (δ ω)
-      (localState a (ε ω) (hε ω) (z ω) t (b ω)))
-    (fun ω => (regionMonitor_continuous _ _ _ _).comp (localState_continuous _ _ _ _ _)) _ B hB
-  intro t
-  have hv := localState_parameter_measurable a ε hε z b hεm hzm hbm t
-  have hy : Measurable (fun ω => Y (localState a (ε ω) (hε ω) (z ω) t (b ω))) :=
-    Y.continuous.measurable.comp hv
-  have hr : Measurable (fun ω => physicalRadius (localState a (ε ω) (hε ω) (z ω) t (b ω))) :=
-    physicalRadius_continuous.measurable.comp hv
-  unfold regionMonitor
-  exact (hy.sub hεm).min ((hN.sub hv.norm).min ((hr.sub hρ).min (hy.sub (hδ.mul hv.fst.norm))))
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-noncomputable def explicitRadialLocalization (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ δ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (hδ : 0 < δ)
-    (hs : δ*‖z‖ ≤ z.im) : RadialLocalization a z ρ δ := by
-  let m := heightFloor z.im ρ δ
-  let C := logCeiling z.im ρ δ
-  have hm : 0 < m := heightFloor_pos _ _ _
-  have hC : 0 ≤ C := logCeiling_nonneg _ _ _
-  have hmz : m ≤ z.im := heightFloor_le hz hρ hr
-  let N : ℝ≥0 := ⟨z.im/δ+C+1,by positivity⟩
-  let B : ℝ≥0 := ⟨z.im^2/(2*a*δ^2)+1,by positivity⟩
-  have hinit : 0 ≤ regionMonitor (m/2) N ρ δ (z,0) := by
-    apply (regionMonitor_nonneg_iff _ _ _ _ _).mpr
-    have hnorm : ‖z‖ ≤ z.im/δ := (le_div_iff₀ hδ).mpr (by simpa only [mul_comm] using hs)
-    refine ⟨?_,?_,?_,hs⟩
-    · change m/2 ≤ z.im
-      linarith
-    · simpa only [Prod.norm_def,Real.norm_eq_abs,abs_zero,max_eq_left (norm_nonneg z)] using
-        (show ‖z‖ ≤ (N : ℝ) by change ‖z‖ ≤ z.im/δ+C+1; linarith)
-    · simpa only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one] using hr
-  refine ⟨m/2,by positivity,N,B,hinit,by change m/2 < m; linarith,?_,?_⟩
-  · change z.im/δ+C < z.im/δ+C+1
-    linarith
-  · change z.im^2 < 2*a*δ^2*(z.im^2/(2*a*δ^2)+1)
-    have hp : 0 < 2*a*δ^2 := by positivity
-    field_simp
-    nlinarith
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-lemma localizationApproximation_tendsto (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a)
-    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (_hr : ρ ≤ z.im)
-    (b : ExactSLEGenerator.Driver)
-    (L : ∀ n, RadialLocalization a z ρ (angleThreshold z n)) :
-    Tendsto (fun n => physicalGreen a (localState a (L n).ε (L n).hε z
-      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b)) atTop
-      (𝓝 (radialWeight a b.1 z ρ)) := by
-  classical
-  let T (n : ℕ) := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
-  let V (n : ℕ) := localState a (L n).ε (L n).hε z (T n) b
-  have hexit (n : ℕ) : physicalRadius (V n) = ρ ∨ Y (V n) = angleThreshold z n*‖(V n).1‖ :=
-    regionStop_true_exit a (L n).ε ha0 (L n).hε z (L n).N ρ (angleThreshold z n)
-      hρ (angleThreshold_pos z hz n) (L n).B (L n).initial (L n).small (L n).large (L n).time b
-  have hactual (n : ℕ) : alive a b.1 (T n) z ∧ actualState a b.1 (T n) z = V n :=
-    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b _ le_rfl
-  by_cases hex : ∃ t : ℝ≥0, alive a b.1 t z ∧ physicalRadius (actualState a b.1 t z) = ρ
-  · obtain ⟨t,ht,htr⟩ := hex
-    obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
-    have he : ∀ᶠ n in atTop, physicalGreen a (localState a (L n).ε (L n).hε z
-      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b) = radialWeight a b.1 z ρ := by
-      filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
-      have ht' : T n ≤ t := regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-        (L n).B (L n).initial b t htr
-      have hη' := hηs (T n) ht'
-      rw [(hactual n).2] at hη'
-      rcases hexit n with hR|hS
-      · exact regionStop_weight_eq a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-          (L n).B (L n).initial b hR
-      · have hsafe := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-          (L n).B (L n).initial b (T n) le_rfl
-        have hnp : 0 < ‖(V n).1‖ := ((L n).hε.trans_le hsafe.1).trans_le (Complex.im_le_norm _)
-        have hd : Y (V n)/‖(V n).1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-        rw [hd] at hη'
-        exact False.elim (not_le.mpr hn hη')
-    exact Filter.Tendsto.congr' (he.mono fun n hn => hn.symm) tendsto_const_nhds
-  · have hw : radialWeight a b.1 z ρ = 0 := by rw [radialWeight,ite_eq_left (radialTime_top_of_no_hit a b.1 z ρ hex)]
-    rw [hw]
-    have hbound (n : ℕ) : physicalGreen a (localState a (L n).ε (L n).hε z
-      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b) ≤
-        Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a) := by
-      rcases hexit n with hR|hS
-      · exact False.elim (hex ⟨T n,(hactual n).1,by simpa only [(hactual n).2] using hR⟩)
-      · have hs := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-          (L n).B (L n).initial b (T n) le_rfl
-        exact physicalGreen_angle_bound a ρ (angleThreshold z n) ha hρ (V n)
-          ((L n).hε.trans_le hs.1) hs.2.2.1 hS
-    have hp : 0 < p a := by unfold p; linarith
-    have hu : Tendsto (fun n => Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using ((angleThreshold_tendsto z).rpow_const_nhds_zero hp).const_mul
-        (Real.exp (-alpha a*Real.log ρ))
-    exact squeeze_zero (fun n => (Real.exp_pos _).le) hbound hu
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-theorem joint_radialWeight_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
-    (hq : 0 < q) (hq1 : q ≤ 1) :
-    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im} =>
-      radialWeight a p.1.1 p.2.1 (p.2.1.im*q)) := by
-  have ha0 : 0<a := by linarith
-  let Ω := ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}
-  let z : Ω → ℂ := fun p => p.2.1
-  let b : Ω → ExactSLEGenerator.Driver := Prod.fst
-  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
-  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
-  have hbM : Measurable b := measurable_fst
-  let ρ : Ω → ℝ := fun ω => (z ω).im*q
-  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
-  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
-  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
-  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
-    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
-  have hρM : Measurable ρ := hzM.im.mul_const q
-  have hδM (n : ℕ) : Measurable (δ n) := by
-    dsimp [δ,angleThreshold]
-    exact (hzM.im.div hzM.norm).div_const _
-  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
-    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
-    unfold heightFloor
-    fun_prop
-  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
-    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    unfold logCeiling heightFloor
-    fun_prop
-  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
-    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    fun_prop
-  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
-    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
-  have hTM (n : ℕ) : Measurable (T n) :=
-    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
-      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
-  have hPM : Measurable (physicalGreen a) := by
-    have hx := X.continuous.measurable
-    have hy := Y.continuous.measurable
-    have hj := J.continuous.measurable
-    unfold physicalGreen logGreen Q
-    fun_prop
-  let F (n : ℕ) (ω : Ω) := physicalGreen a (localState a (L n ω).ε (L n ω).hε (z ω) (T n ω) (b ω))
-  have hFM (n : ℕ) : Measurable (F n) := hPM.comp
-    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b (hεM n) hzM hbM).comp
-      ((hTM n).prodMk measurable_id))
-  apply measurable_of_tendsto_metrizable hFM
-  apply tendsto_pi_nhds.mpr
-  intro ω
-  exact localizationApproximation_tendsto a ha ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω) (b ω)
-    (fun n => L n ω)
-end ExactSLEGreen
-
- 
-
-open MeasureTheory Set
-open scoped ENNReal
-namespace ExactSLEKernel
-noncomputable def cappedDistance {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (b : ℝ) (x : ι → ℂ) (i : ι) : ℝ :=
-  (insert b ((Finset.univ.erase i).image fun j => dist (x i) (x j))).min' (by simp)
-noncomputable def kernel {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α b : ℝ) (x : ι → ℂ) : ℝ≥0∞ :=
-  ∏ i, (ENNReal.ofReal (cappedDistance b x i)) ^ (-α)
-
-section Basic
-variable {ι : Type*} [Fintype ι] [DecidableEq ι]
-lemma le_cappedDistance_iff (b r : ℝ) (x : ι → ℂ) (i : ι) :
-    r ≤ cappedDistance b x i ↔ r ≤ b ∧ ∀ j, j ≠ i → r ≤ dist (x i) (x j) := by
-  classical
-  unfold cappedDistance
-  constructor
-  · intro h
-    refine ⟨h.trans (Finset.min'_le _ _ (by simp)),?_⟩
-    intro j hj
-    exact h.trans (Finset.min'_le _ _ (by simp only [Finset.mem_insert,Finset.mem_image,Finset.mem_erase,Finset.mem_univ,and_true]; exact Or.inr ⟨j,hj,rfl⟩))
-  · rintro ⟨hb,h⟩
-    apply Finset.le_min'
-    intro y hy
-    rcases Finset.mem_insert.mp hy with rfl|hy
-    · exact hb
-    obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hy
-    exact h j (Finset.mem_erase.mp hj).1
-lemma cappedDistance_le (b : ℝ) (x : ι → ℂ) (i : ι) : cappedDistance b x i ≤ b :=
-  ((le_cappedDistance_iff b _ x i).mp le_rfl).1
-lemma cappedDistance_le_dist (b : ℝ) (x : ι → ℂ) (i j : ι) (hji : j ≠ i) :
-    cappedDistance b x i ≤ dist (x i) (x j) :=
-  ((le_cappedDistance_iff b _ x i).mp le_rfl).2 j hji
-lemma cappedDistance_pos {b : ℝ} (hb : 0 < b) (x : ι → ℂ) (hx : Function.Injective x) (i : ι) :
-    0 < cappedDistance b x i := by
-  classical
-  have h := Finset.min'_mem (insert b ((Finset.univ.erase i).image fun j => dist (x i) (x j))) (by simp)
-  change cappedDistance b x i ∈ _ at h
-  rcases Finset.mem_insert.mp h with h|h
-  · rwa [h]
-  · obtain ⟨j,hj,h⟩ := Finset.mem_image.mp h
-    rw [←h]
-    exact dist_pos.mpr (fun he => (Finset.mem_erase.mp hj).1 (hx he).symm)
-lemma cappedDistance_nonneg {b : ℝ} (hb : 0 ≤ b) (x : ι → ℂ) (i : ι) :
-    0 ≤ cappedDistance b x i := (le_cappedDistance_iff _ _ _ _).mpr ⟨hb,fun _ _ => dist_nonneg⟩
-end Basic
-lemma cappedDistance_cons_succ {m : ℕ} (b : ℝ) (z : ℂ) (x : Fin m → ℂ) (i : Fin m) :
-    cappedDistance b (Fin.cons z x) i.succ = min (cappedDistance b x i) (dist z (x i)) := by
-  apply le_antisymm
-  · apply le_min
-    · apply (le_cappedDistance_iff _ _ _ _).mpr
-      refine ⟨cappedDistance_le _ _ _,fun j hj => ?_⟩
-      simpa using cappedDistance_le_dist b (Fin.cons z x) i.succ j.succ (by simpa using hj)
-    · simpa [dist_comm] using cappedDistance_le_dist b (Fin.cons z x) i.succ 0 (Ne.symm (Fin.succ_ne_zero i))
-  · apply (le_cappedDistance_iff _ _ _ _).mpr
-    refine ⟨(min_le_left _ _).trans (cappedDistance_le _ _ _),?_⟩
-    intro j hj
-    revert hj
-    refine Fin.cases (fun hj => ?_) (fun k hj => ?_) j
-    · simp [dist_comm]
-    · have hki : k ≠ i := by intro he; subst k; exact hj rfl
-      exact (min_le_left _ _).trans (by simpa using cappedDistance_le_dist b x i k hki)
-lemma half_cap_unique {m : ℕ} (b : ℝ) (z : ℂ) (x : Fin m → ℂ)
-    (i j : Fin m) (hi : dist z (x i) < cappedDistance b x i / 2)
-    (hj : dist z (x j) < cappedDistance b x j / 2) : i = j := by
-  by_contra hij
-  have h1 := cappedDistance_le_dist b x i j (Ne.symm hij)
-  have h2 := cappedDistance_le_dist b x j i hij
-  have h3 := dist_triangle (x i) z (x j)
-  rw [dist_comm (x i) z] at h3
-  rw [dist_comm (x j) (x i)] at h2
-  linarith
-lemma near_cap_nearest {m : ℕ} (b : ℝ) (z : ℂ) (x : Fin m → ℂ)
-    (i : Fin m) (hi : dist z (x i) < cappedDistance b x i / 2) :
-    cappedDistance b (Fin.cons z x) 0 = dist z (x i) := by
-  apply le_antisymm
-  · simpa using cappedDistance_le_dist b (Fin.cons z x) 0 i.succ (by simp)
-  · apply (le_cappedDistance_iff _ _ _ _).mpr
-    refine ⟨by have := cappedDistance_le b x i; have := dist_nonneg (x:=z) (y:=x i); linarith,?_⟩
-    intro j hj
-    revert hj
-    refine Fin.cases (fun hj => False.elim (hj rfl)) (fun k hj => ?_) j
-    simp only [Fin.cons_zero,Fin.cons_succ]
-    by_cases hki : k=i
-    · simp [hki]
-    · have h1 := cappedDistance_le_dist b x i k hki
-      have h2 := dist_triangle (x i) z (x k)
-      rw [dist_comm (x i) z] at h2
-      linarith
-noncomputable def ipow (α r : ℝ) : ℝ≥0∞ := ENNReal.ofReal r ^ (-α)
-lemma ipow_anti {α r s : ℝ} (ha : 0 ≤ α) (hrs : r ≤ s) : ipow α s ≤ ipow α r := by
-  unfold ipow
-  rw [ENNReal.rpow_neg,ENNReal.rpow_neg]
-  exact ENNReal.inv_le_inv.mpr (ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal hrs) ha)
-lemma ipow_pos {α r : ℝ} (ha : 0 ≤ α) : 0 < ipow α r := by
-  rw [ipow,ENNReal.rpow_neg,ENNReal.inv_pos]
-  exact ENNReal.rpow_ne_top_of_nonneg ha ENNReal.ofReal_ne_top
-lemma ipow_finite {α r : ℝ} (hr : 0 < r) : ipow α r < ∞ := by
-  exact lt_top_iff_ne_top.mpr (ENNReal.rpow_ne_top_of_ne_zero (ne_of_gt (ENNReal.ofReal_pos.mpr hr)) ENNReal.ofReal_ne_top)
-lemma ipow_mul (α : ℝ) {r : ℝ} (hr : 0 ≤ r) (s : ℝ) :
-    ipow α (r*s) = ipow α r * ipow α s := by
-  rw [ipow,ENNReal.ofReal_mul hr,ENNReal.mul_rpow_of_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top]
-  rfl
-lemma ipow_twice (α r : ℝ) : ipow (2*α) r = ipow α r * ipow α r := by
-  unfold ipow
-  rw [show -(2*α)=(-α)*(2:ℕ) by norm_num; ring,ENNReal.rpow_mul,ENNReal.rpow_natCast,pow_two]
-lemma cap_factor_sup {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {α : ℝ} (ha : 0 ≤ α) (b : ℝ) (x : ι → ℂ) (i : ι) :
-    ipow α (cappedDistance b x i) = ipow α b ⊔ ⨆ j, if j=i then 0 else ipow α (dist (x i) (x j)) := by
-  classical
-  apply le_antisymm
-  · have h := Finset.min'_mem (insert b ((Finset.univ.erase i).image fun j => dist (x i) (x j))) (by simp)
-    change cappedDistance b x i ∈ _ at h
-    rcases Finset.mem_insert.mp h with h|h
-    · rw [h]; exact le_sup_left
-    · obtain ⟨j,hj,hj'⟩ := Finset.mem_image.mp h
-      rw [←hj']
-      exact (le_iSup_of_le j (by simp [(Finset.mem_erase.mp hj).1])).trans le_sup_right
-  · apply sup_le
-    · exact ipow_anti ha (cappedDistance_le _ _ _)
-    · apply iSup_le
-      intro j
-      split_ifs with h
-      · exact bot_le
-      · exact ipow_anti ha (cappedDistance_le_dist _ _ _ _ h)
-lemma measurable_kernel {ι : Type*} [Fintype ι] [DecidableEq ι]
-    {α : ℝ} (ha : 0 ≤ α) (b : ℝ) : Measurable (kernel (ι:=ι) α b) := by
-  classical
-  unfold kernel
-  apply Finset.measurable_prod
-  intro i hi
-  change Measurable (fun x : ι → ℂ => ipow α (cappedDistance b x i))
-  simp_rw [cap_factor_sup ha]
-  apply Measurable.sup measurable_const
-  apply Measurable.iSup
-  intro j
-  split_ifs
-  · exact measurable_const
-  · unfold ipow
-    fun_prop
-lemma cap_cons_factor_sum {m : ℕ} {α : ℝ} (ha : 0 ≤ α) (b : ℝ)
-    (z : ℂ) (x : Fin m → ℂ) :
-    ipow α (cappedDistance b (Fin.cons z x) 0) ≤ ipow α b + ∑ i, ipow α (dist z (x i)) := by
-  rw [cap_factor_sup ha]
-  apply sup_le
-  · exact le_self_add
-  · apply iSup_le; intro i
-    refine Fin.cases ?_ (fun j => ?_) i
-    · simp
-    · simp only [Fin.succ_ne_zero,ite_false,Fin.cons_zero,Fin.cons_succ]
-      exact (Finset.single_le_sum (fun k _ => (by positivity : (0:ℝ≥0∞) ≤ ipow α (dist z (x k)))) (Finset.mem_univ j)).trans le_add_self
-lemma far_cap_factor {m : ℕ} {α b : ℝ} (ha : 0 ≤ α) (hb : 0 ≤ b)
-    (z : ℂ) (x : Fin m → ℂ) (i : Fin m)
-    (hi : cappedDistance b x i / 2 ≤ dist z (x i)) :
-    ipow α (cappedDistance b (Fin.cons z x) i.succ) ≤ ipow α (1/2) * ipow α (cappedDistance b x i) := by
-  have hn := cappedDistance_nonneg hb x i
-  have h : cappedDistance b x i / 2 ≤ cappedDistance b (Fin.cons z x) i.succ := by
-    rw [cappedDistance_cons_succ]
-    exact le_min (by linarith) hi
-  have hh := ipow_anti ha h
-  convert hh using 1
-  rw [show cappedDistance b x i / 2 = cappedDistance b x i * (1/2) by ring, ipow_mul α hn,mul_comm]
-lemma cap_compensated {m : ℕ} {α b : ℝ} (ha : 0 ≤ α) (hb : 0 < b)
-    (x : Fin m → ℂ) (i : Fin m) :
-    1 ≤ ipow α (cappedDistance b x i) * ENNReal.ofReal b ^ α := by
-  have h := mul_le_mul' (ipow_anti ha (cappedDistance_le b x i)) (le_refl (ENNReal.ofReal b ^ α))
-  have he : ipow α b * ENNReal.ofReal b ^ α = 1 := by
-    rw [ipow,←ENNReal.rpow_add _ _ (ne_of_gt (ENNReal.ofReal_pos.mpr hb)) ENNReal.ofReal_ne_top]
-    simp
-  rwa [he] at h
-end ExactSLEKernel
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ProductIntegrability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEFlow
+variable {Ω : Type*} [MeasurableSpace Ω]
+lemma localState_parameter_measurable (a : ℝ) (ε : Ω → ℝ) (hε : ∀ ω, 0 < ε ω)
+    (z : Ω → ℂ) (b : Ω → ExactSLEGenerator.Driver)
+    (hεm : Measurable ε) (hzm : Measurable z) (hbm : Measurable b) (t : ℝ≥0) :
+    Measurable (fun ω => localState a (ε ω) (hε ω) (z ω) t (b ω)) := by
+  let A : Ω → State → State := fun ω => cutoffDrift a (ε ω)
+  let L : Ω → ℝ≥0 := fun ω => ⟨max (|a|/ε ω) (|a|/(ε ω)^2), by positivity⟩
+  let K : Ω → ℝ≥0 := fun ω => ⟨max (2*|a|/(ε ω)^2) (4*|a|/(ε ω)^3), by positivity⟩
+  have hA : ∀ ω, LipschitzWith (K ω) (A ω) := fun ω => cutoffDrift_lipschitz a (ε ω) (hε ω)
+  have hb : ∀ ω v, ‖A ω v‖ ≤ L ω := fun ω => cutoffDrift_bound a (ε ω) (hε ω)
+  have hmA : Measurable (Function.uncurry A) := by
+    change Measurable (fun w : Ω × State => cutoffDrift a (ε w.1) w.2)
+    unfold cutoffDrift cutoffVector heightClamp
+    simp only [Complex.mk_eq_add_mul_I]
+    fun_prop
+  let u : ℝ≥0 → Ω → State := fun t ω => (b ω).1 t • noise
+  have huc (ω : Ω) : Continuous (fun t => u t ω) := (b ω).1.continuous.smul continuous_const
+  have hum (t : ℝ≥0) : Measurable (u t) := ((driver_eval_measurable t).measurable.comp hbm).smul_const noise
+  obtain ⟨F,hFc,hFm,hFe⟩ := measurable_parameter_flow A L K hA hb hmA u huc hum
+    (fun ω => (z ω,0)) (hzm.prodMk measurable_const)
+  have he (ω : Ω) (t : ℝ≥0) : F t ω = localState a (ε ω) (hε ω) (z ω) t (b ω) - u t ω := by
+    apply integral_flow_unique (hA ω) (huc ω) (huc ω) (hFc ω)
+      ((localState_continuous a (ε ω) (hε ω) (z ω) (b ω)).sub (huc ω)) (hFe ω)
+      (fun s => ?_) (T := t) (fun _ _ => rfl) le_rfl
+    simp only [Pi.sub_apply,sub_add_cancel]
+    rw [localState_eq_zero]
+    dsimp [u,A]
+    abel
+  convert (hFm t).add (hum t) using 1
+  funext ω
+  change localState a (ε ω) (hε ω) (z ω) t (b ω) = F t ω + u t ω
+  rw [he]
+  simp
+
+lemma localState_parameter_joint_measurable (a : ℝ) (ε : Ω → ℝ) (hε : ∀ ω, 0 < ε ω)
+    (z : Ω → ℂ) (b : Ω → ExactSLEGenerator.Driver)
+    (hεm : Measurable ε) (hzm : Measurable z) (hbm : Measurable b) :
+    Measurable (fun p : ℝ≥0 × Ω => localState a (ε p.2) (hε p.2) (z p.2) p.1 (b p.2)) := by
+  exact measurable_uncurry_of_continuous_of_measurable
+    (fun ω => localState_continuous a (ε ω) (hε ω) (z ω) (b ω))
+    (fun t => localState_parameter_measurable a ε hε z b hεm hzm hbm t)
+end ExactSLEGreen
+
+namespace ExactSLEHitting
+variable {Ω : Type*} [MeasurableSpace Ω]
+lemma boundedContact_parameter_measurable (f : ℝ≥0 → Ω → ℝ)
+    (hc : ∀ ω, Continuous (fun t => f t ω)) (hm : ∀ t, Measurable (f t))
+    (B : Ω → ℝ≥0) (hB : Measurable B) :
+    Measurable (fun ω => hittingBtwn f (Iic 0) 0 (B ω) ω) := by
+  apply measurable_of_Iic
+  intro t
+  have he : (fun ω => hittingBtwn f (Iic 0) 0 (B ω) ω) ⁻¹' Iic t =
+      {ω | B ω ≤ t} ∪ {ω | ∃ u ≤ t, f u ω ≤ 0} := by
+    ext ω
+    simp only [mem_preimage,mem_Iic,mem_union,mem_ofPred_eq]
+    by_cases h : B ω ≤ t
+    · exact iff_of_true ((hittingBtwn_le ω).trans h) (Or.inl h)
+    · rw [or_iff_right h]
+      exact boundedContact_le_iff f hc ω (B ω) t (lt_of_not_ge h)
+  rw [he]
+  exact (measurableSet_le hB measurable_const).union
+    (measurable_has_contact (Filtration.const ℝ≥0 (inferInstance : MeasurableSpace Ω) le_rfl) f hc (fun t => hm t) t)
+end ExactSLEHitting
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEHitting
+variable {Ω : Type*} [MeasurableSpace Ω]
+lemma regionStop_parameter_measurable (a : ℝ) (ε : Ω → ℝ) (hε : ∀ ω, 0 < ε ω)
+    (z : Ω → ℂ) (b : Ω → ExactSLEGenerator.Driver) (N ρ δ : Ω → ℝ) (B : Ω → ℝ≥0)
+    (hεm : Measurable ε) (hzm : Measurable z) (hbm : Measurable b)
+    (hN : Measurable N) (hρ : Measurable ρ) (hδ : Measurable δ) (hB : Measurable B) :
+    Measurable (fun ω => regionStop a (ε ω) (hε ω) (z ω) (N ω) (ρ ω) (δ ω) (B ω) (b ω)) := by
+  apply boundedContact_parameter_measurable
+    (fun t ω => regionMonitor (ε ω) (N ω) (ρ ω) (δ ω)
+      (localState a (ε ω) (hε ω) (z ω) t (b ω)))
+    (fun ω => (regionMonitor_continuous _ _ _ _).comp (localState_continuous _ _ _ _ _)) _ B hB
+  intro t
+  have hv := localState_parameter_measurable a ε hε z b hεm hzm hbm t
+  have hy : Measurable (fun ω => Y (localState a (ε ω) (hε ω) (z ω) t (b ω))) :=
+    Y.continuous.measurable.comp hv
+  have hr : Measurable (fun ω => physicalRadius (localState a (ε ω) (hε ω) (z ω) t (b ω))) :=
+    physicalRadius_continuous.measurable.comp hv
+  unfold regionMonitor
+  exact (hy.sub hεm).min ((hN.sub hv.norm).min ((hr.sub hρ).min (hy.sub (hδ.mul hv.fst.norm))))
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+noncomputable def explicitRadialLocalization (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ δ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (hδ : 0 < δ)
+    (hs : δ*‖z‖ ≤ z.im) : RadialLocalization a z ρ δ := by
+  let m := heightFloor z.im ρ δ
+  let C := logCeiling z.im ρ δ
+  have hm : 0 < m := heightFloor_pos _ _ _
+  have hC : 0 ≤ C := logCeiling_nonneg _ _ _
+  have hmz : m ≤ z.im := heightFloor_le hz hρ hr
+  let N : ℝ≥0 := ⟨z.im/δ+C+1,by positivity⟩
+  let B : ℝ≥0 := ⟨z.im^2/(2*a*δ^2)+1,by positivity⟩
+  have hinit : 0 ≤ regionMonitor (m/2) N ρ δ (z,0) := by
+    apply (regionMonitor_nonneg_iff _ _ _ _ _).mpr
+    have hnorm : ‖z‖ ≤ z.im/δ := (le_div_iff₀ hδ).mpr (by simpa only [mul_comm] using hs)
+    refine ⟨?_,?_,?_,hs⟩
+    · change m/2 ≤ z.im
+      linarith
+    · simpa only [Prod.norm_def,Real.norm_eq_abs,abs_zero,max_eq_left (norm_nonneg z)] using
+        (show ‖z‖ ≤ (N : ℝ) by change ‖z‖ ≤ z.im/δ+C+1; linarith)
+    · simpa only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one] using hr
+  refine ⟨m/2,by positivity,N,B,hinit,by change m/2 < m; linarith,?_,?_⟩
+  · change z.im/δ+C < z.im/δ+C+1
+    linarith
+  · change z.im^2 < 2*a*δ^2*(z.im^2/(2*a*δ^2)+1)
+    have hp : 0 < 2*a*δ^2 := by positivity
+    field_simp
+    nlinarith
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+lemma localizationApproximation_tendsto (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a)
+    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (_hr : ρ ≤ z.im)
+    (b : ExactSLEGenerator.Driver)
+    (L : ∀ n, RadialLocalization a z ρ (angleThreshold z n)) :
+    Tendsto (fun n => physicalGreen a (localState a (L n).ε (L n).hε z
+      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b)) atTop
+      (𝓝 (radialWeight a b.1 z ρ)) := by
+  classical
+  let T (n : ℕ) := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
+  let V (n : ℕ) := localState a (L n).ε (L n).hε z (T n) b
+  have hexit (n : ℕ) : physicalRadius (V n) = ρ ∨ Y (V n) = angleThreshold z n*‖(V n).1‖ :=
+    regionStop_true_exit a (L n).ε ha0 (L n).hε z (L n).N ρ (angleThreshold z n)
+      hρ (angleThreshold_pos z hz n) (L n).B (L n).initial (L n).small (L n).large (L n).time b
+  have hactual (n : ℕ) : alive a b.1 (T n) z ∧ actualState a b.1 (T n) z = V n :=
+    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b _ le_rfl
+  by_cases hex : ∃ t : ℝ≥0, alive a b.1 t z ∧ physicalRadius (actualState a b.1 t z) = ρ
+  · obtain ⟨t,ht,htr⟩ := hex
+    obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
+    have he : ∀ᶠ n in atTop, physicalGreen a (localState a (L n).ε (L n).hε z
+      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b) = radialWeight a b.1 z ρ := by
+      filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
+      have ht' : T n ≤ t := regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+        (L n).B (L n).initial b t htr
+      have hη' := hηs (T n) ht'
+      rw [(hactual n).2] at hη'
+      rcases hexit n with hR|hS
+      · exact regionStop_weight_eq a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+          (L n).B (L n).initial b hR
+      · have hsafe := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+          (L n).B (L n).initial b (T n) le_rfl
+        have hnp : 0 < ‖(V n).1‖ := ((L n).hε.trans_le hsafe.1).trans_le (Complex.im_le_norm _)
+        have hd : Y (V n)/‖(V n).1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+        rw [hd] at hη'
+        exact False.elim (not_le.mpr hn hη')
+    exact Filter.Tendsto.congr' (he.mono fun n hn => hn.symm) tendsto_const_nhds
+  · have hw : radialWeight a b.1 z ρ = 0 := by rw [radialWeight,ite_eq_left (radialTime_top_of_no_hit a b.1 z ρ hex)]
+    rw [hw]
+    have hbound (n : ℕ) : physicalGreen a (localState a (L n).ε (L n).hε z
+      (regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b) b) ≤
+        Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a) := by
+      rcases hexit n with hR|hS
+      · exact False.elim (hex ⟨T n,(hactual n).1,by simpa only [(hactual n).2] using hR⟩)
+      · have hs := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+          (L n).B (L n).initial b (T n) le_rfl
+        exact physicalGreen_angle_bound a ρ (angleThreshold z n) ha hρ (V n)
+          ((L n).hε.trans_le hs.1) hs.2.2.1 hS
+    have hp : 0 < p a := by unfold p; linarith
+    have hu : Tendsto (fun n => Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using ((angleThreshold_tendsto z).rpow_const_nhds_zero hp).const_mul
+        (Real.exp (-alpha a*Real.log ρ))
+    exact squeeze_zero (fun n => (Real.exp_pos _).le) hbound hu
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+theorem joint_radialWeight_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
+    (hq : 0 < q) (hq1 : q ≤ 1) :
+    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im} =>
+      radialWeight a p.1.1 p.2.1 (p.2.1.im*q)) := by
+  have ha0 : 0<a := by linarith
+  let Ω := ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}
+  let z : Ω → ℂ := fun p => p.2.1
+  let b : Ω → ExactSLEGenerator.Driver := Prod.fst
+  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
+  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
+  have hbM : Measurable b := measurable_fst
+  let ρ : Ω → ℝ := fun ω => (z ω).im*q
+  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
+  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
+  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
+  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
+    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
+  have hρM : Measurable ρ := hzM.im.mul_const q
+  have hδM (n : ℕ) : Measurable (δ n) := by
+    dsimp [δ,angleThreshold]
+    exact (hzM.im.div hzM.norm).div_const _
+  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
+    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
+    unfold heightFloor
+    fun_prop
+  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
+    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    unfold logCeiling heightFloor
+    fun_prop
+  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
+    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    fun_prop
+  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
+    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
+  have hTM (n : ℕ) : Measurable (T n) :=
+    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
+      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
+  have hPM : Measurable (physicalGreen a) := by
+    have hx := X.continuous.measurable
+    have hy := Y.continuous.measurable
+    have hj := J.continuous.measurable
+    unfold physicalGreen logGreen Q
+    fun_prop
+  let F (n : ℕ) (ω : Ω) := physicalGreen a (localState a (L n ω).ε (L n ω).hε (z ω) (T n ω) (b ω))
+  have hFM (n : ℕ) : Measurable (F n) := hPM.comp
+    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b (hεM n) hzM hbM).comp
+      ((hTM n).prodMk measurable_id))
+  apply measurable_of_tendsto_metrizable hFM
+  apply tendsto_pi_nhds.mpr
+  intro ω
+  exact localizationApproximation_tendsto a ha ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω) (b ω)
+    (fun n => L n ω)
+end ExactSLEGreen
+
+ 
+
+open MeasureTheory Set
+open scoped ENNReal
+namespace ExactSLEKernel
+noncomputable def cappedDistance {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (b : ℝ) (x : ι → ℂ) (i : ι) : ℝ :=
+  (insert b ((Finset.univ.erase i).image fun j => dist (x i) (x j))).min' (by simp)
+noncomputable def kernel {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α b : ℝ) (x : ι → ℂ) : ℝ≥0∞ :=
+  ∏ i, (ENNReal.ofReal (cappedDistance b x i)) ^ (-α)
+
+section Basic
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+lemma le_cappedDistance_iff (b r : ℝ) (x : ι → ℂ) (i : ι) :
+    r ≤ cappedDistance b x i ↔ r ≤ b ∧ ∀ j, j ≠ i → r ≤ dist (x i) (x j) := by
+  classical
+  unfold cappedDistance
+  constructor
+  · intro h
+    refine ⟨h.trans (Finset.min'_le _ _ (by simp)),?_⟩
+    intro j hj
+    exact h.trans (Finset.min'_le _ _ (by simp only [Finset.mem_insert,Finset.mem_image,Finset.mem_erase,Finset.mem_univ,and_true]; exact Or.inr ⟨j,hj,rfl⟩))
+  · rintro ⟨hb,h⟩
+    apply Finset.le_min'
+    intro y hy
+    rcases Finset.mem_insert.mp hy with rfl|hy
+    · exact hb
+    obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hy
+    exact h j (Finset.mem_erase.mp hj).1
+lemma cappedDistance_le (b : ℝ) (x : ι → ℂ) (i : ι) : cappedDistance b x i ≤ b :=
+  ((le_cappedDistance_iff b _ x i).mp le_rfl).1
+lemma cappedDistance_le_dist (b : ℝ) (x : ι → ℂ) (i j : ι) (hji : j ≠ i) :
+    cappedDistance b x i ≤ dist (x i) (x j) :=
+  ((le_cappedDistance_iff b _ x i).mp le_rfl).2 j hji
+lemma cappedDistance_pos {b : ℝ} (hb : 0 < b) (x : ι → ℂ) (hx : Function.Injective x) (i : ι) :
+    0 < cappedDistance b x i := by
+  classical
+  have h := Finset.min'_mem (insert b ((Finset.univ.erase i).image fun j => dist (x i) (x j))) (by simp)
+  change cappedDistance b x i ∈ _ at h
+  rcases Finset.mem_insert.mp h with h|h
+  · rwa [h]
+  · obtain ⟨j,hj,h⟩ := Finset.mem_image.mp h
+    rw [←h]
+    exact dist_pos.mpr (fun he => (Finset.mem_erase.mp hj).1 (hx he).symm)
+lemma cappedDistance_nonneg {b : ℝ} (hb : 0 ≤ b) (x : ι → ℂ) (i : ι) :
+    0 ≤ cappedDistance b x i := (le_cappedDistance_iff _ _ _ _).mpr ⟨hb,fun _ _ => dist_nonneg⟩
+end Basic
+lemma cappedDistance_cons_succ {m : ℕ} (b : ℝ) (z : ℂ) (x : Fin m → ℂ) (i : Fin m) :
+    cappedDistance b (Fin.cons z x) i.succ = min (cappedDistance b x i) (dist z (x i)) := by
+  apply le_antisymm
+  · apply le_min
+    · apply (le_cappedDistance_iff _ _ _ _).mpr
+      refine ⟨cappedDistance_le _ _ _,fun j hj => ?_⟩
+      simpa using cappedDistance_le_dist b (Fin.cons z x) i.succ j.succ (by simpa using hj)
+    · simpa [dist_comm] using cappedDistance_le_dist b (Fin.cons z x) i.succ 0 (Ne.symm (Fin.succ_ne_zero i))
+  · apply (le_cappedDistance_iff _ _ _ _).mpr
+    refine ⟨(min_le_left _ _).trans (cappedDistance_le _ _ _),?_⟩
+    intro j hj
+    revert hj
+    refine Fin.cases (fun hj => ?_) (fun k hj => ?_) j
+    · simp [dist_comm]
+    · have hki : k ≠ i := by intro he; subst k; exact hj rfl
+      exact (min_le_left _ _).trans (by simpa using cappedDistance_le_dist b x i k hki)
+lemma half_cap_unique {m : ℕ} (b : ℝ) (z : ℂ) (x : Fin m → ℂ)
+    (i j : Fin m) (hi : dist z (x i) < cappedDistance b x i / 2)
+    (hj : dist z (x j) < cappedDistance b x j / 2) : i = j := by
+  by_contra hij
+  have h1 := cappedDistance_le_dist b x i j (Ne.symm hij)
+  have h2 := cappedDistance_le_dist b x j i hij
+  have h3 := dist_triangle (x i) z (x j)
+  rw [dist_comm (x i) z] at h3
+  rw [dist_comm (x j) (x i)] at h2
+  linarith
+lemma near_cap_nearest {m : ℕ} (b : ℝ) (z : ℂ) (x : Fin m → ℂ)
+    (i : Fin m) (hi : dist z (x i) < cappedDistance b x i / 2) :
+    cappedDistance b (Fin.cons z x) 0 = dist z (x i) := by
+  apply le_antisymm
+  · simpa using cappedDistance_le_dist b (Fin.cons z x) 0 i.succ (by simp)
+  · apply (le_cappedDistance_iff _ _ _ _).mpr
+    refine ⟨by have := cappedDistance_le b x i; have := dist_nonneg (x:=z) (y:=x i); linarith,?_⟩
+    intro j hj
+    revert hj
+    refine Fin.cases (fun hj => False.elim (hj rfl)) (fun k hj => ?_) j
+    simp only [Fin.cons_zero,Fin.cons_succ]
+    by_cases hki : k=i
+    · simp [hki]
+    · have h1 := cappedDistance_le_dist b x i k hki
+      have h2 := dist_triangle (x i) z (x k)
+      rw [dist_comm (x i) z] at h2
+      linarith
+noncomputable def ipow (α r : ℝ) : ℝ≥0∞ := ENNReal.ofReal r ^ (-α)
+lemma ipow_anti {α r s : ℝ} (ha : 0 ≤ α) (hrs : r ≤ s) : ipow α s ≤ ipow α r := by
+  unfold ipow
+  rw [ENNReal.rpow_neg,ENNReal.rpow_neg]
+  exact ENNReal.inv_le_inv.mpr (ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal hrs) ha)
+lemma ipow_pos {α r : ℝ} (ha : 0 ≤ α) : 0 < ipow α r := by
+  rw [ipow,ENNReal.rpow_neg,ENNReal.inv_pos]
+  exact ENNReal.rpow_ne_top_of_nonneg ha ENNReal.ofReal_ne_top
+lemma ipow_finite {α r : ℝ} (hr : 0 < r) : ipow α r < ∞ := by
+  exact lt_top_iff_ne_top.mpr (ENNReal.rpow_ne_top_of_ne_zero (ne_of_gt (ENNReal.ofReal_pos.mpr hr)) ENNReal.ofReal_ne_top)
+lemma ipow_mul (α : ℝ) {r : ℝ} (hr : 0 ≤ r) (s : ℝ) :
+    ipow α (r*s) = ipow α r * ipow α s := by
+  rw [ipow,ENNReal.ofReal_mul hr,ENNReal.mul_rpow_of_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top]
+  rfl
+lemma ipow_twice (α r : ℝ) : ipow (2*α) r = ipow α r * ipow α r := by
+  unfold ipow
+  rw [show -(2*α)=(-α)*(2:ℕ) by norm_num; ring,ENNReal.rpow_mul,ENNReal.rpow_natCast,pow_two]
+lemma cap_factor_sup {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {α : ℝ} (ha : 0 ≤ α) (b : ℝ) (x : ι → ℂ) (i : ι) :
+    ipow α (cappedDistance b x i) = ipow α b ⊔ ⨆ j, if j=i then 0 else ipow α (dist (x i) (x j)) := by
+  classical
+  apply le_antisymm
+  · have h := Finset.min'_mem (insert b ((Finset.univ.erase i).image fun j => dist (x i) (x j))) (by simp)
+    change cappedDistance b x i ∈ _ at h
+    rcases Finset.mem_insert.mp h with h|h
+    · rw [h]; exact le_sup_left
+    · obtain ⟨j,hj,hj'⟩ := Finset.mem_image.mp h
+      rw [←hj']
+      exact (le_iSup_of_le j (by simp [(Finset.mem_erase.mp hj).1])).trans le_sup_right
+  · apply sup_le
+    · exact ipow_anti ha (cappedDistance_le _ _ _)
+    · apply iSup_le
+      intro j
+      split_ifs with h
+      · exact bot_le
+      · exact ipow_anti ha (cappedDistance_le_dist _ _ _ _ h)
+lemma measurable_kernel {ι : Type*} [Fintype ι] [DecidableEq ι]
+    {α : ℝ} (ha : 0 ≤ α) (b : ℝ) : Measurable (kernel (ι:=ι) α b) := by
+  classical
+  unfold kernel
+  apply Finset.measurable_prod
+  intro i hi
+  change Measurable (fun x : ι → ℂ => ipow α (cappedDistance b x i))
+  simp_rw [cap_factor_sup ha]
+  apply Measurable.sup measurable_const
+  apply Measurable.iSup
+  intro j
+  split_ifs
+  · exact measurable_const
+  · unfold ipow
+    fun_prop
+lemma cap_cons_factor_sum {m : ℕ} {α : ℝ} (ha : 0 ≤ α) (b : ℝ)
+    (z : ℂ) (x : Fin m → ℂ) :
+    ipow α (cappedDistance b (Fin.cons z x) 0) ≤ ipow α b + ∑ i, ipow α (dist z (x i)) := by
+  rw [cap_factor_sup ha]
+  apply sup_le
+  · exact le_self_add
+  · apply iSup_le; intro i
+    refine Fin.cases ?_ (fun j => ?_) i
+    · simp
+    · simp only [Fin.succ_ne_zero,ite_false,Fin.cons_zero,Fin.cons_succ]
+      exact (Finset.single_le_sum (fun k _ => (by positivity : (0:ℝ≥0∞) ≤ ipow α (dist z (x k)))) (Finset.mem_univ j)).trans le_add_self
+lemma far_cap_factor {m : ℕ} {α b : ℝ} (ha : 0 ≤ α) (hb : 0 ≤ b)
+    (z : ℂ) (x : Fin m → ℂ) (i : Fin m)
+    (hi : cappedDistance b x i / 2 ≤ dist z (x i)) :
+    ipow α (cappedDistance b (Fin.cons z x) i.succ) ≤ ipow α (1/2) * ipow α (cappedDistance b x i) := by
+  have hn := cappedDistance_nonneg hb x i
+  have h : cappedDistance b x i / 2 ≤ cappedDistance b (Fin.cons z x) i.succ := by
+    rw [cappedDistance_cons_succ]
+    exact le_min (by linarith) hi
+  have hh := ipow_anti ha h
+  convert hh using 1
+  rw [show cappedDistance b x i / 2 = cappedDistance b x i * (1/2) by ring, ipow_mul α hn,mul_comm]
+lemma cap_compensated {m : ℕ} {α b : ℝ} (ha : 0 ≤ α) (hb : 0 < b)
+    (x : Fin m → ℂ) (i : Fin m) :
+    1 ≤ ipow α (cappedDistance b x i) * ENNReal.ofReal b ^ α := by
+  have h := mul_le_mul' (ipow_anti ha (cappedDistance_le b x i)) (le_refl (ENNReal.ofReal b ^ α))
+  have he : ipow α b * ENNReal.ofReal b ^ α = 1 := by
+    rw [ipow,←ENNReal.rpow_add _ _ (ne_of_gt (ENNReal.ofReal_pos.mpr hb)) ENNReal.ofReal_ne_top]
+    simp
+  rwa [he] at h
+end ExactSLEKernel
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/LocalFiniteness.lean b/lean/OAI/Probability/SLEGauge/LocalFiniteness.lean
index 7aad4b4942e7d0c54c9d4d2824345389b1d890cb..99ffdbfec2f2228ae3c131e677c16de2950fa4dd 100644
--- a/lean/OAI/Probability/SLEGauge/LocalFiniteness.lean
+++ b/lean/OAI/Probability/SLEGauge/LocalFiniteness.lean
@@ -1,449 +1,449 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialMeasurability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-open Filter Topology
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-lemma actualUpperMass_finiteOnCompacts (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    IsFiniteMeasureOnCompacts (actualUpperMass a q t b) := by
-  constructor
-  intro K hK
-  obtain ⟨c,hc,hcz⟩ := upper_compact_lower K hK
-  have hm : Measurable (fun z : UpperPlane => timeMassDensity a q t b z.1) :=
-    (timeMassDensity_measurable a ha q hq hq1 t).comp
-      (measurable_const.prodMk measurable_subtype_coe)
-  rw [actualUpperMass,withDensity_apply _ hK.measurableSet]
-  apply lt_of_le_of_lt (setLIntegral_mono' hK.measurableSet
-    (fun z hz => timeMassDensity_local_bound a ha q hq hq1 t b c hc z.1 (hcz z hz)))
-  rw [lintegral_const]
-  apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top
-  rw [Measure.restrict_apply_univ]
-  rw [upperArea,upper_embedding.comap_apply]
-  exact (hK.image continuous_subtype_val).measure_lt_top
-
-noncomputable def upperExtension (f : C_c(UpperPlane, ℝ)) : ℂ → ℝ :=
-  Function.extend Subtype.val f 0
-
-lemma upperExtension_apply (f : C_c(UpperPlane, ℝ)) (z : UpperPlane) :
-    upperExtension f z.1=f z := Subtype.val_injective.extend_apply _ _ _
-
-lemma upperExtension_measurable (f : C_c(UpperPlane, ℝ)) : Measurable (upperExtension f) :=
-  upper_embedding.measurable_extend f.continuous.measurable measurable_const
-
-lemma upperExtension_zero (f : C_c(UpperPlane, ℝ)) (z : ℂ)
-    (hz : z∉Subtype.val '' tsupport f) : upperExtension f z=0 := by
-  by_cases h : 0<z.im
-  · rw [show z = (⟨z,h⟩ : UpperPlane).1 from rfl, upperExtension_apply]
-    exact image_eq_zero_of_notMem_tsupport (fun H => hz ⟨⟨z,h⟩,H,rfl⟩)
-  · apply Function.extend_apply'
-    rintro ⟨w,rfl⟩
-    exact h w.2
-
-lemma actualUpperMass_eval (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane, ℝ)) :
-    (∫ z, f z ∂actualUpperMass a q t b) =
-      actualMassEval a q t (Subtype.val '' tsupport f) (upperExtension f) b := by
-  have hm : Measurable (fun z : UpperPlane => timeMassDensity a q t b z.1) :=
-    (timeMassDensity_measurable a ha q hq hq1 t).comp (measurable_const.prodMk measurable_subtype_coe)
-  rw [actualUpperMass,integral_withDensity_eq_integral_toReal_smul hm
-    (Filter.Eventually.of_forall (fun z => timeMassDensity_finite a q t b z.1))]
-  simp only [smul_eq_mul]
-  simp_rw [← upperExtension_apply f]
-  rw [upperArea]
-  trans ∫ z in {z : ℂ | 0<z.im}, (timeMassDensity a q t b z).toReal * upperExtension f z
-  · exact integral_subtype_comap (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet
-      (fun z : ℂ => (timeMassDensity a q t b z).toReal * upperExtension f z)
-  · rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
-    unfold actualMassEval
-    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
-    · apply integral_congr_ae
-      exact Filter.Eventually.of_forall (fun z => mul_comm _ _)
-    · rw [upperExtension_zero f z hz,zero_mul]
-    · rw [upperExtension_zero f z (by rintro ⟨w,hw,rfl⟩; exact hz w.2), mul_zero]
-
-lemma upperExtension_bounded (f : C_c(UpperPlane, ℝ)) :
-    ∃ B : ℝ, 0≤B ∧ ∀ z∈Subtype.val '' tsupport f, ‖upperExtension f z‖≤B := by
-  obtain ⟨B,hB⟩ := f.hasCompactSupport.exists_bound_of_continuousOn f.continuous.continuousOn
-  refine ⟨max B 0, le_max_right _ _, ?_⟩
-  rintro z ⟨w,hw,rfl⟩
-  rw [upperExtension_apply]
-  exact (hB w hw).trans (le_max_left _ _)
-
-noncomputable def positive_mass_combination
-    (μ : ℕ → Measure UpperPlane) (hμ : ∀ j, IsFiniteMeasureOnCompacts (μ j))
-    (s : Finset ℕ) (c : ℕ → ℝ) (hc : ∀ j∈s,0≤c j) : C_c(UpperPlane, ℝ) →ₚ[ℝ] ℝ := by
-  letI := hμ
-  exact PositiveLinearMap.mk₀
-    (∑ j∈s, c j • (CompactlySupportedContinuousMap.integralPositiveLinearMap (μ j)).toLinearMap)
-    (by
-      intro f hf
-      simp only [LinearMap.sum_apply,LinearMap.smul_apply,
-        smul_eq_mul]
-      exact Finset.sum_nonneg (fun j hj => mul_nonneg (hc j hj) (integral_nonneg hf)))
-
-lemma positive_mass_combination_apply
-    (μ : ℕ → Measure UpperPlane) (hμ : ∀ j, IsFiniteMeasureOnCompacts (μ j))
-    (s : Finset ℕ) (c : ℕ → ℝ) (hc : ∀ j∈s,0≤c j) (f : C_c(UpperPlane, ℝ)) :
-    positive_mass_combination μ hμ s c hc f = ∑ j∈s, c j * ∫z, f z∂μ j := by
-  change (∑ j∈s, c j • (CompactlySupportedContinuousMap.integralPositiveLinearMap (μ j)).toLinearMap) f = _
-  simp only [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul]
-  rfl
-theorem actual_radon_mass_extraction (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      ∀ᵐ b ∂W, ∃ μ : ℕ → Measure UpperPlane,
-        (∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular) ∧
-        ∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ i))
-    := by
-  classical
-  obtain ⟨tests,htests⟩ := determining_tests_radon_limit UpperPlane
-  let F : ℕ → C_c(UpperPlane, ℝ) := fun i => tests (Nat.unpair i).2
-  let K : ℕ → Set ℂ := fun i => Subtype.val '' tsupport (F i)
-  have hK (i : ℕ) : IsCompact (K i) := (F i).hasCompactSupport.image continuous_subtype_val
-  choose c hc hcz using fun i => upper_compact_lower (tsupport (F i)) (F i).hasCompactSupport
-  have hcK (i : ℕ) (z : ℂ) (hz : z∈K i) : c i≤2*z.im := by
-    obtain ⟨w,hw,rfl⟩ := hz
-    exact hcz i w hw
-  obtain ⟨X,s,coeff,v,hX,hs,hcoeff,hsum,hconv,hae⟩ := actual_common_mass_extraction a ha W hW
-    K (fun i => (hK i).measurableSet) (fun i => (hK i).isBounded) c hc hcK
-    (fun i => upperExtension (F i)) (fun i => upperExtension_measurable (F i))
-    (fun i => upperExtension_bounded (F i)) (fun i => t (Nat.unpair i).1)
-  refine ⟨s,coeff,hs,hcoeff,hsum,?_⟩
-  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  filter_upwards [hae] with b hb
-  let hμ (i j : ℕ) := actualUpperMass_finiteOnCompacts a ha (q j) (hq j) (hq1 j) (t i) b
-  let L : ℕ → ℕ → (C_c(UpperPlane, ℝ) →ₚ[ℝ] ℝ) := fun i k =>
-    positive_mass_combination (fun j => actualUpperMass a (q j) (t i) b)
-      (hμ i) (s k) (coeff k) (hcoeff k)
-  have hi (i : ℕ) : ∃ μ : Measure UpperPlane, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
-      ∀ f : C_c(UpperPlane, ℝ), Tendsto (fun k => L i k f) atTop (𝓝 (∫z,f z∂μ)) := by
-    apply htests (L i)
-    intro n
-    refine ⟨v (Nat.pair i n) b, ?_⟩
-    have h := hb (Nat.pair i n)
-    simpa only [L,positive_mass_combination_apply,actualUpperMass_eval a ha _ (hq _) (hq1 _),
-      F,K,Nat.unpair_pair] using h
-  choose μ hμlim using hi
-  refine ⟨μ,fun i => ⟨(hμlim i).1,(hμlim i).2.1⟩,fun i f => ?_⟩
-  simpa only [L,positive_mass_combination_apply] using (hμlim i).2.2 f
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-
-lemma radon_le_of_positive_integrals {X : Type*} [TopologicalSpace X] [T2Space X]
-    [LocallyCompactSpace X] [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    (μ ν : Measure X) [μ.Regular] [ν.Regular]
-    (h : ∀ f : C_c(X, ℝ), (∀ x, 0≤f x) → ∫x,f x∂μ ≤ ∫x,f x∂ν) : μ≤ν := by
-  let A := CompactlySupportedContinuousMap.integralPositiveLinearMap μ
-  let B := CompactlySupportedContinuousMap.integralPositiveLinearMap ν
-  let D : C_c(X, ℝ) →ₚ[ℝ] ℝ := PositiveLinearMap.mk₀ (B.toLinearMap-A.toLinearMap)
-    (fun f hf => sub_nonneg.mpr (h f hf))
-  let δ := RealRMK.rieszMeasure D
-  let : TopologicalSpace.PseudoMetrizableSpace X := inferInstance
-  let : SigmaCompactSpace X := inferInstance
-  let : IsFiniteMeasureOnCompacts δ := inferInstance
-  let : IsFiniteMeasureOnCompacts (μ+δ) := ⟨fun K hK => by
-    rw [Measure.add_apply]
-    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩⟩
-  have he : μ+δ=ν := by
-    apply Measure.ext_of_integral_eq_on_compactlySupported
-    intro f
-    rw [integral_add_measure f.integrable f.integrable]
-    have hd : ∫x,f x∂δ=∫x,f x∂ν-∫x,f x∂μ := by
-      exact RealRMK.integral_rieszMeasure D f
-    rw [hd]
-    ring
-  rw [←he]
-  exact le_add_of_nonneg_right bot_le
-
-lemma actualUpperMass_mono_time (a q : ℝ) (b : ExactSLEGenerator.Driver) :
-    Monotone (fun t => actualUpperMass a q t b) := by
-  intro s t hst
-  apply withDensity_mono
-  filter_upwards [] with z
-  unfold timeMassDensity
-  split_ifs with hs ht
-  · exact le_rfl
-  · exact False.elim (ht (hs.trans hst))
-  · exact bot_le
-  · exact le_rfl
-
-lemma ordered_radon_limits
-    (a : ℝ) (ha : 1/4<a) (b : ExactSLEGenerator.Driver) (t : ℕ → ℝ≥0∞)
-    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤coeff k j)
-    (μ : ℕ → Measure UpperPlane)
-    (hμ : ∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular)
-    (hlim : ∀ i (f : C_c(UpperPlane, ℝ)),
-      Tendsto (fun k => ∑ j∈s k, coeff k j *
-          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-        atTop (𝓝 (∫z,f z∂μ i))) :
-    ∀ i j, t i≤t j → μ i≤μ j := by
-  intro i j ht
-  let := (hμ i).1
-  let := (hμ j).1
-  let := (hμ i).2
-  let := (hμ j).2
-  apply radon_le_of_positive_integrals
-  intro f hf
-  apply le_of_tendsto_of_tendsto' (hlim i f) (hlim j f)
-  intro k
-  apply Finset.sum_le_sum
-  intro n hn
-  apply mul_le_mul_of_nonneg_left _ (hc k n hn)
-  have hq1 : Real.exp (-2*a*(n+1))≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hn0 : (0:ℝ)≤n := Nat.cast_nonneg _
-    nlinarith
-  let := actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _) hq1 (t j) b
-  exact integral_mono_measure (actualUpperMass_mono_time a _ b ht)
-    (Filter.Eventually.of_forall hf) f.integrable
-
-lemma actualUpperMass_eval_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (f : C_c(UpperPlane, ℝ)) :
-    Measurable (fun b => ∫z,f z∂actualUpperMass a q t b) := by
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  exact actualMassEval_measurable a ha q hq hq1 t _ _ (upperExtension_measurable f)
-
- 
-
-theorem actual_ordered_radon_extraction (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        AEStronglyMeasurable (fun b => ∫ z, f z ∂μ b i) W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i)))
- := by
-  classical
-  obtain ⟨s,c,hs,hc,hsum,h⟩ := actual_radon_mass_extraction a ha W hW t
-  let P (b : ExactSLEGenerator.Driver) (μ : ℕ → Measure UpperPlane) : Prop :=
-    (∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular) ∧
-    ∀ i (f : C_c(UpperPlane, ℝ)),
-      Tendsto (fun k => ∑ j∈s k, c k j *
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-        atTop (𝓝 (∫z,f z∂μ i))
-  let μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane := fun b =>
-    if H : ∃ ν, P b ν then H.choose else fun _ => 0
-  have hμ (b) (i) : IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular := by
-    dsimp [μ]
-    split_ifs with H
-    · exact H.choose_spec.1 i
-    · exact ⟨inferInstance,inferInstance⟩
-  have hae : ∀ᵐ b ∂W, P b (μ b) := by
-    filter_upwards [h] with b hb
-    dsimp [μ]
-    rw [dite_eq_left hb]
-    exact hb.choose_spec
-  refine ⟨s,c,μ,hs,hc,hsum,hμ,?_,?_⟩
-  · intro i f
-    apply aestronglyMeasurable_of_tendsto_ae atTop
-      (f := fun k b => ∑ j∈s k, c k j *
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-    · intro k
-      suffices H : AEStronglyMeasurable (∑ j∈s k, fun b => c k j *
-          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) W by
-        convert H using 1
-        funext b
-        simp only [Finset.sum_apply]
-      apply Finset.aestronglyMeasurable_sum
-      intro j hj
-      have hq1 : Real.exp (-2*a*(j+1))≤1 := by
-        apply Real.exp_le_one_iff.mpr
-        have ha0 : 0<a := by linarith
-        have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-        nlinarith
-      exact (actualUpperMass_eval_measurable a ha _ (Real.exp_pos _) hq1 (t i) f).aestronglyMeasurable.const_mul _
-    · filter_upwards [hae] with b hb
-      exact hb.2 i f
-  · filter_upwards [hae] with b hb
-    exact ⟨ordered_radon_limits a ha b t s c hc (μ b) hb.1 hb.2,hb.2⟩
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-
-lemma actualMassEval_Lp_bound (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (m : ℕ) (hm : 1 ≤ m)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z, z∈K → c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ, 0≤B ∧ ∀ z∈K, ‖f z‖≤B) :
-    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 → ∀ t : ℝ≥0∞,
-      MemLp (actualMassEval a q t K f) (m : ℝ≥0∞) W ∧ eLpNorm (actualMassEval a q t K f) (m : ℝ≥0∞) W ≤ C := by
-  obtain ⟨C,hC,hCb⟩ := actual_mass_moment_bound a ha m
-  obtain ⟨hAfin,hB⟩ := hCb W hW K hK hKb c hc hcK (fun z => ‖f z‖) hf.norm
-    (fun _ _ => norm_nonneg _) hfB
-  let A := ∫⁻ x : Fin m → ℂ, (∏ i, ENNReal.ofReal ‖f (x i)‖)*
-    ExactSLEKernel.kernel (alpha a) c x ∂Measure.pi (fun _ => volume.restrict K)
-  let D := (ENNReal.ofReal C*A)^(1/(m:ℝ))
-  have hD : D<∞ := ENNReal.rpow_lt_top_of_nonneg (by positivity) (ENNReal.mul_lt_top ENNReal.ofReal_lt_top hAfin).ne
-  refine ⟨D,hD,?_⟩
-  intro q hq hq1 t
-  have hnorm : eLpNorm (actualMassEval a q t K f) (m : ℝ≥0∞) W≤D := by
-    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by exact_mod_cast (Nat.ne_zero_of_lt hm))
-      (by simp) (actualMassEval_measurable a ha q hq hq1 t K f hf).aestronglyMeasurable]
-    rw [ENNReal.toReal_natCast]
-    apply ENNReal.rpow_le_rpow _ (by positivity)
-    calc
-      _ ≤ ∫⁻ b, (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^m ∂W := by
-        apply lintegral_mono
-        intro b
-        change ‖actualMassEval a q t K f b‖ₑ^(m:ℝ) ≤
-          (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^m
-        rw [ENNReal.rpow_natCast]
-        exact pow_le_pow_left' (actualMassEval_enorm_bound a q t K f b) m
-      _ ≤ _ := hB q hq hq1
-  exact ⟨hnorm.trans_lt hD,hnorm⟩
-
-lemma convex_eLpNorm_bound {Ω : Type*} [MeasurableSpace Ω] (W : Measure Ω)
-    (p C : ℝ≥0∞) (hp : 1≤p) (s : Finset ℕ) (c : ℕ → ℝ)
-    (hc : ∀ j∈s, 0≤c j) (hsum : ∑ j∈s,c j=1)
-    (F : ℕ → Ω → ℝ) (_hF : ∀ j∈s, AEStronglyMeasurable (F j) W)
-    (hbound : ∀ j∈s, eLpNorm (F j) p W≤C) :
-    eLpNorm (fun b => ∑ j∈s,c j*F j b) p W≤C := by
-  have he : (fun b => ∑ j∈s,c j*F j b) = ∑ j∈s,c j • F j := by
-    funext b
-    simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul]
-  rw [he]
-  calc
-    _ ≤ ∑ j∈s,eLpNorm (c j • F j) p W :=
-      eLpNorm_sum_le hp
-    _ = ∑ j∈s,ENNReal.ofReal (c j)*eLpNorm (F j) p W := by
-      apply Finset.sum_congr rfl
-      intro j hj
-      rw [eLpNorm_const_smul,←ofReal_norm,Real.norm_eq_abs,abs_of_nonneg (hc j hj)]
-    _ ≤ ∑ j∈s,ENNReal.ofReal (c j)*C := by
-      apply Finset.sum_le_sum
-      intro j hj
-      gcongr
-      exact hbound j hj
-    _ = C := by
-      rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg hc,hsum,ENNReal.ofReal_one,one_mul]
-
-lemma actualUpperMass_Lp_bound (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (m : ℕ) (hm : 1 ≤ m) (f : C_c(UpperPlane, ℝ)) :
-    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 → ∀ t : ℝ≥0∞,
-      MemLp (fun b => ∫z,f z∂actualUpperMass a q t b) (m : ℝ≥0∞) W ∧
-      eLpNorm (fun b => ∫z,f z∂actualUpperMass a q t b) (m : ℝ≥0∞) W ≤ C := by
-  let K := Subtype.val '' tsupport f
-  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
-  obtain ⟨c,hc,hcz⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
-  have hcK (z : ℂ) (hz : z∈K) : c≤2*z.im := by
-    obtain ⟨w,hw,rfl⟩ := hz
-    exact hcz w hw
-  obtain ⟨C,hC,h⟩ := actualMassEval_Lp_bound a ha W hW m hm K hK.measurableSet hK.isBounded
-    c hc hcK (upperExtension f) (upperExtension_measurable f) (upperExtension_bounded f)
-  refine ⟨C,hC,fun q hq hq1 t => ?_⟩
-  simpa only [actualUpperMass_eval a ha q hq hq1] using h q hq hq1 t
-
-lemma actual_radon_limit_memLp (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0∞) (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-    (hc : ∀ k j, j∈s k → 0≤coeff k j) (hsum : ∀ k, ∑ j∈s k,coeff k j=1)
-    (μ : ExactSLEGenerator.Driver → Measure UpperPlane)
-    (f : C_c(UpperPlane, ℝ))
-    (hlim : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,coeff k j *
-      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂μ b)))
-    (m : ℕ) : MemLp (fun b => ∫z,f z∂μ b) (m : ℝ≥0∞) W := by
-  let q (j : ℕ) := Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun k b =>
-    ∑ j∈s k,coeff k j * ∫z,f z∂actualUpperMass a (q j) t b
-  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
-    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
-      (actualUpperMass_eval_measurable a ha (q j) (hq j) (hq1 j) t f).aestronglyMeasurable.const_mul (coeff k j))
-    convert H using 1
-    funext b
-    simp only [F,Finset.sum_apply]
-  have hmeas : AEStronglyMeasurable (fun b => ∫z,f z∂μ b) W :=
-    aestronglyMeasurable_of_tendsto_ae atTop hF hlim
-  by_cases hm : m=0
-  · simpa [hm] using (memLp_zero_iff_aestronglyMeasurable.mpr hmeas)
-  have hmpos : 1 ≤ m := Nat.one_le_iff_ne_zero.mpr hm
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW m hmpos f
-  have hbound (k : ℕ) : eLpNorm (F k) (m : ℝ≥0∞) W≤C :=
-    convex_eLpNorm_bound W _ C (by exact_mod_cast hmpos) (s k) (coeff k) (hc k) (hsum k)
-      (fun j b => ∫z,f z∂actualUpperMass a (q j) t b)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).1.aestronglyMeasurable)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).2)
-  exact (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF hmeas hlim).trans_lt hC
-
- 
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialMeasurability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+open Filter Topology
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+lemma actualUpperMass_finiteOnCompacts (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    IsFiniteMeasureOnCompacts (actualUpperMass a q t b) := by
+  constructor
+  intro K hK
+  obtain ⟨c,hc,hcz⟩ := upper_compact_lower K hK
+  have hm : Measurable (fun z : UpperPlane => timeMassDensity a q t b z.1) :=
+    (timeMassDensity_measurable a ha q hq hq1 t).comp
+      (measurable_const.prodMk measurable_subtype_coe)
+  rw [actualUpperMass,withDensity_apply _ hK.measurableSet]
+  apply lt_of_le_of_lt (setLIntegral_mono' hK.measurableSet
+    (fun z hz => timeMassDensity_local_bound a ha q hq hq1 t b c hc z.1 (hcz z hz)))
+  rw [lintegral_const]
+  apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top
+  rw [Measure.restrict_apply_univ]
+  rw [upperArea,upper_embedding.comap_apply]
+  exact (hK.image continuous_subtype_val).measure_lt_top
+
+noncomputable def upperExtension (f : C_c(UpperPlane, ℝ)) : ℂ → ℝ :=
+  Function.extend Subtype.val f 0
+
+lemma upperExtension_apply (f : C_c(UpperPlane, ℝ)) (z : UpperPlane) :
+    upperExtension f z.1=f z := Subtype.val_injective.extend_apply _ _ _
+
+lemma upperExtension_measurable (f : C_c(UpperPlane, ℝ)) : Measurable (upperExtension f) :=
+  upper_embedding.measurable_extend f.continuous.measurable measurable_const
+
+lemma upperExtension_zero (f : C_c(UpperPlane, ℝ)) (z : ℂ)
+    (hz : z∉Subtype.val '' tsupport f) : upperExtension f z=0 := by
+  by_cases h : 0<z.im
+  · rw [show z = (⟨z,h⟩ : UpperPlane).1 from rfl, upperExtension_apply]
+    exact image_eq_zero_of_notMem_tsupport (fun H => hz ⟨⟨z,h⟩,H,rfl⟩)
+  · apply Function.extend_apply'
+    rintro ⟨w,rfl⟩
+    exact h w.2
+
+lemma actualUpperMass_eval (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane, ℝ)) :
+    (∫ z, f z ∂actualUpperMass a q t b) =
+      actualMassEval a q t (Subtype.val '' tsupport f) (upperExtension f) b := by
+  have hm : Measurable (fun z : UpperPlane => timeMassDensity a q t b z.1) :=
+    (timeMassDensity_measurable a ha q hq hq1 t).comp (measurable_const.prodMk measurable_subtype_coe)
+  rw [actualUpperMass,integral_withDensity_eq_integral_toReal_smul hm
+    (Filter.Eventually.of_forall (fun z => timeMassDensity_finite a q t b z.1))]
+  simp only [smul_eq_mul]
+  simp_rw [← upperExtension_apply f]
+  rw [upperArea]
+  trans ∫ z in {z : ℂ | 0<z.im}, (timeMassDensity a q t b z).toReal * upperExtension f z
+  · exact integral_subtype_comap (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet
+      (fun z : ℂ => (timeMassDensity a q t b z).toReal * upperExtension f z)
+  · rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
+    unfold actualMassEval
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
+    · apply integral_congr_ae
+      exact Filter.Eventually.of_forall (fun z => mul_comm _ _)
+    · rw [upperExtension_zero f z hz,zero_mul]
+    · rw [upperExtension_zero f z (by rintro ⟨w,hw,rfl⟩; exact hz w.2), mul_zero]
+
+lemma upperExtension_bounded (f : C_c(UpperPlane, ℝ)) :
+    ∃ B : ℝ, 0≤B ∧ ∀ z∈Subtype.val '' tsupport f, ‖upperExtension f z‖≤B := by
+  obtain ⟨B,hB⟩ := f.hasCompactSupport.exists_bound_of_continuousOn f.continuous.continuousOn
+  refine ⟨max B 0, le_max_right _ _, ?_⟩
+  rintro z ⟨w,hw,rfl⟩
+  rw [upperExtension_apply]
+  exact (hB w hw).trans (le_max_left _ _)
+
+noncomputable def positive_mass_combination
+    (μ : ℕ → Measure UpperPlane) (hμ : ∀ j, IsFiniteMeasureOnCompacts (μ j))
+    (s : Finset ℕ) (c : ℕ → ℝ) (hc : ∀ j∈s,0≤c j) : C_c(UpperPlane, ℝ) →ₚ[ℝ] ℝ := by
+  letI := hμ
+  exact PositiveLinearMap.mk₀
+    (∑ j∈s, c j • (CompactlySupportedContinuousMap.integralPositiveLinearMap (μ j)).toLinearMap)
+    (by
+      intro f hf
+      simp only [LinearMap.sum_apply,LinearMap.smul_apply,
+        smul_eq_mul]
+      exact Finset.sum_nonneg (fun j hj => mul_nonneg (hc j hj) (integral_nonneg hf)))
+
+lemma positive_mass_combination_apply
+    (μ : ℕ → Measure UpperPlane) (hμ : ∀ j, IsFiniteMeasureOnCompacts (μ j))
+    (s : Finset ℕ) (c : ℕ → ℝ) (hc : ∀ j∈s,0≤c j) (f : C_c(UpperPlane, ℝ)) :
+    positive_mass_combination μ hμ s c hc f = ∑ j∈s, c j * ∫z, f z∂μ j := by
+  change (∑ j∈s, c j • (CompactlySupportedContinuousMap.integralPositiveLinearMap (μ j)).toLinearMap) f = _
+  simp only [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul]
+  rfl
+theorem actual_radon_mass_extraction (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      ∀ᵐ b ∂W, ∃ μ : ℕ → Measure UpperPlane,
+        (∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular) ∧
+        ∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ i))
+    := by
+  classical
+  obtain ⟨tests,htests⟩ := determining_tests_radon_limit UpperPlane
+  let F : ℕ → C_c(UpperPlane, ℝ) := fun i => tests (Nat.unpair i).2
+  let K : ℕ → Set ℂ := fun i => Subtype.val '' tsupport (F i)
+  have hK (i : ℕ) : IsCompact (K i) := (F i).hasCompactSupport.image continuous_subtype_val
+  choose c hc hcz using fun i => upper_compact_lower (tsupport (F i)) (F i).hasCompactSupport
+  have hcK (i : ℕ) (z : ℂ) (hz : z∈K i) : c i≤2*z.im := by
+    obtain ⟨w,hw,rfl⟩ := hz
+    exact hcz i w hw
+  obtain ⟨X,s,coeff,v,hX,hs,hcoeff,hsum,hconv,hae⟩ := actual_common_mass_extraction a ha W hW
+    K (fun i => (hK i).measurableSet) (fun i => (hK i).isBounded) c hc hcK
+    (fun i => upperExtension (F i)) (fun i => upperExtension_measurable (F i))
+    (fun i => upperExtension_bounded (F i)) (fun i => t (Nat.unpair i).1)
+  refine ⟨s,coeff,hs,hcoeff,hsum,?_⟩
+  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  filter_upwards [hae] with b hb
+  let hμ (i j : ℕ) := actualUpperMass_finiteOnCompacts a ha (q j) (hq j) (hq1 j) (t i) b
+  let L : ℕ → ℕ → (C_c(UpperPlane, ℝ) →ₚ[ℝ] ℝ) := fun i k =>
+    positive_mass_combination (fun j => actualUpperMass a (q j) (t i) b)
+      (hμ i) (s k) (coeff k) (hcoeff k)
+  have hi (i : ℕ) : ∃ μ : Measure UpperPlane, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
+      ∀ f : C_c(UpperPlane, ℝ), Tendsto (fun k => L i k f) atTop (𝓝 (∫z,f z∂μ)) := by
+    apply htests (L i)
+    intro n
+    refine ⟨v (Nat.pair i n) b, ?_⟩
+    have h := hb (Nat.pair i n)
+    simpa only [L,positive_mass_combination_apply,actualUpperMass_eval a ha _ (hq _) (hq1 _),
+      F,K,Nat.unpair_pair] using h
+  choose μ hμlim using hi
+  refine ⟨μ,fun i => ⟨(hμlim i).1,(hμlim i).2.1⟩,fun i f => ?_⟩
+  simpa only [L,positive_mass_combination_apply] using (hμlim i).2.2 f
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+
+lemma radon_le_of_positive_integrals {X : Type*} [TopologicalSpace X] [T2Space X]
+    [LocallyCompactSpace X] [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    (μ ν : Measure X) [μ.Regular] [ν.Regular]
+    (h : ∀ f : C_c(X, ℝ), (∀ x, 0≤f x) → ∫x,f x∂μ ≤ ∫x,f x∂ν) : μ≤ν := by
+  let A := CompactlySupportedContinuousMap.integralPositiveLinearMap μ
+  let B := CompactlySupportedContinuousMap.integralPositiveLinearMap ν
+  let D : C_c(X, ℝ) →ₚ[ℝ] ℝ := PositiveLinearMap.mk₀ (B.toLinearMap-A.toLinearMap)
+    (fun f hf => sub_nonneg.mpr (h f hf))
+  let δ := RealRMK.rieszMeasure D
+  let : TopologicalSpace.PseudoMetrizableSpace X := inferInstance
+  let : SigmaCompactSpace X := inferInstance
+  let : IsFiniteMeasureOnCompacts δ := inferInstance
+  let : IsFiniteMeasureOnCompacts (μ+δ) := ⟨fun K hK => by
+    rw [Measure.add_apply]
+    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩⟩
+  have he : μ+δ=ν := by
+    apply Measure.ext_of_integral_eq_on_compactlySupported
+    intro f
+    rw [integral_add_measure f.integrable f.integrable]
+    have hd : ∫x,f x∂δ=∫x,f x∂ν-∫x,f x∂μ := by
+      exact RealRMK.integral_rieszMeasure D f
+    rw [hd]
+    ring
+  rw [←he]
+  exact le_add_of_nonneg_right bot_le
+
+lemma actualUpperMass_mono_time (a q : ℝ) (b : ExactSLEGenerator.Driver) :
+    Monotone (fun t => actualUpperMass a q t b) := by
+  intro s t hst
+  apply withDensity_mono
+  filter_upwards [] with z
+  unfold timeMassDensity
+  split_ifs with hs ht
+  · exact le_rfl
+  · exact False.elim (ht (hs.trans hst))
+  · exact bot_le
+  · exact le_rfl
+
+lemma ordered_radon_limits
+    (a : ℝ) (ha : 1/4<a) (b : ExactSLEGenerator.Driver) (t : ℕ → ℝ≥0∞)
+    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤coeff k j)
+    (μ : ℕ → Measure UpperPlane)
+    (hμ : ∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular)
+    (hlim : ∀ i (f : C_c(UpperPlane, ℝ)),
+      Tendsto (fun k => ∑ j∈s k, coeff k j *
+          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+        atTop (𝓝 (∫z,f z∂μ i))) :
+    ∀ i j, t i≤t j → μ i≤μ j := by
+  intro i j ht
+  let := (hμ i).1
+  let := (hμ j).1
+  let := (hμ i).2
+  let := (hμ j).2
+  apply radon_le_of_positive_integrals
+  intro f hf
+  apply le_of_tendsto_of_tendsto' (hlim i f) (hlim j f)
+  intro k
+  apply Finset.sum_le_sum
+  intro n hn
+  apply mul_le_mul_of_nonneg_left _ (hc k n hn)
+  have hq1 : Real.exp (-2*a*(n+1))≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hn0 : (0:ℝ)≤n := Nat.cast_nonneg _
+    nlinarith
+  let := actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _) hq1 (t j) b
+  exact integral_mono_measure (actualUpperMass_mono_time a _ b ht)
+    (Filter.Eventually.of_forall hf) f.integrable
+
+lemma actualUpperMass_eval_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (f : C_c(UpperPlane, ℝ)) :
+    Measurable (fun b => ∫z,f z∂actualUpperMass a q t b) := by
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  exact actualMassEval_measurable a ha q hq hq1 t _ _ (upperExtension_measurable f)
+
+ 
+
+theorem actual_ordered_radon_extraction (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        AEStronglyMeasurable (fun b => ∫ z, f z ∂μ b i) W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i)))
+ := by
+  classical
+  obtain ⟨s,c,hs,hc,hsum,h⟩ := actual_radon_mass_extraction a ha W hW t
+  let P (b : ExactSLEGenerator.Driver) (μ : ℕ → Measure UpperPlane) : Prop :=
+    (∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular) ∧
+    ∀ i (f : C_c(UpperPlane, ℝ)),
+      Tendsto (fun k => ∑ j∈s k, c k j *
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+        atTop (𝓝 (∫z,f z∂μ i))
+  let μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane := fun b =>
+    if H : ∃ ν, P b ν then H.choose else fun _ => 0
+  have hμ (b) (i) : IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular := by
+    dsimp [μ]
+    split_ifs with H
+    · exact H.choose_spec.1 i
+    · exact ⟨inferInstance,inferInstance⟩
+  have hae : ∀ᵐ b ∂W, P b (μ b) := by
+    filter_upwards [h] with b hb
+    dsimp [μ]
+    rw [dite_eq_left hb]
+    exact hb.choose_spec
+  refine ⟨s,c,μ,hs,hc,hsum,hμ,?_,?_⟩
+  · intro i f
+    apply aestronglyMeasurable_of_tendsto_ae atTop
+      (f := fun k b => ∑ j∈s k, c k j *
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+    · intro k
+      suffices H : AEStronglyMeasurable (∑ j∈s k, fun b => c k j *
+          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) W by
+        convert H using 1
+        funext b
+        simp only [Finset.sum_apply]
+      apply Finset.aestronglyMeasurable_sum
+      intro j hj
+      have hq1 : Real.exp (-2*a*(j+1))≤1 := by
+        apply Real.exp_le_one_iff.mpr
+        have ha0 : 0<a := by linarith
+        have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+        nlinarith
+      exact (actualUpperMass_eval_measurable a ha _ (Real.exp_pos _) hq1 (t i) f).aestronglyMeasurable.const_mul _
+    · filter_upwards [hae] with b hb
+      exact hb.2 i f
+  · filter_upwards [hae] with b hb
+    exact ⟨ordered_radon_limits a ha b t s c hc (μ b) hb.1 hb.2,hb.2⟩
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+
+lemma actualMassEval_Lp_bound (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (m : ℕ) (hm : 1 ≤ m)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z, z∈K → c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ, 0≤B ∧ ∀ z∈K, ‖f z‖≤B) :
+    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 → ∀ t : ℝ≥0∞,
+      MemLp (actualMassEval a q t K f) (m : ℝ≥0∞) W ∧ eLpNorm (actualMassEval a q t K f) (m : ℝ≥0∞) W ≤ C := by
+  obtain ⟨C,hC,hCb⟩ := actual_mass_moment_bound a ha m
+  obtain ⟨hAfin,hB⟩ := hCb W hW K hK hKb c hc hcK (fun z => ‖f z‖) hf.norm
+    (fun _ _ => norm_nonneg _) hfB
+  let A := ∫⁻ x : Fin m → ℂ, (∏ i, ENNReal.ofReal ‖f (x i)‖)*
+    ExactSLEKernel.kernel (alpha a) c x ∂Measure.pi (fun _ => volume.restrict K)
+  let D := (ENNReal.ofReal C*A)^(1/(m:ℝ))
+  have hD : D<∞ := ENNReal.rpow_lt_top_of_nonneg (by positivity) (ENNReal.mul_lt_top ENNReal.ofReal_lt_top hAfin).ne
+  refine ⟨D,hD,?_⟩
+  intro q hq hq1 t
+  have hnorm : eLpNorm (actualMassEval a q t K f) (m : ℝ≥0∞) W≤D := by
+    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by exact_mod_cast (Nat.ne_zero_of_lt hm))
+      (by simp) (actualMassEval_measurable a ha q hq hq1 t K f hf).aestronglyMeasurable]
+    rw [ENNReal.toReal_natCast]
+    apply ENNReal.rpow_le_rpow _ (by positivity)
+    calc
+      _ ≤ ∫⁻ b, (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^m ∂W := by
+        apply lintegral_mono
+        intro b
+        change ‖actualMassEval a q t K f b‖ₑ^(m:ℝ) ≤
+          (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^m
+        rw [ENNReal.rpow_natCast]
+        exact pow_le_pow_left' (actualMassEval_enorm_bound a q t K f b) m
+      _ ≤ _ := hB q hq hq1
+  exact ⟨hnorm.trans_lt hD,hnorm⟩
+
+lemma convex_eLpNorm_bound {Ω : Type*} [MeasurableSpace Ω] (W : Measure Ω)
+    (p C : ℝ≥0∞) (hp : 1≤p) (s : Finset ℕ) (c : ℕ → ℝ)
+    (hc : ∀ j∈s, 0≤c j) (hsum : ∑ j∈s,c j=1)
+    (F : ℕ → Ω → ℝ) (_hF : ∀ j∈s, AEStronglyMeasurable (F j) W)
+    (hbound : ∀ j∈s, eLpNorm (F j) p W≤C) :
+    eLpNorm (fun b => ∑ j∈s,c j*F j b) p W≤C := by
+  have he : (fun b => ∑ j∈s,c j*F j b) = ∑ j∈s,c j • F j := by
+    funext b
+    simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul]
+  rw [he]
+  calc
+    _ ≤ ∑ j∈s,eLpNorm (c j • F j) p W :=
+      eLpNorm_sum_le hp
+    _ = ∑ j∈s,ENNReal.ofReal (c j)*eLpNorm (F j) p W := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      rw [eLpNorm_const_smul,←ofReal_norm,Real.norm_eq_abs,abs_of_nonneg (hc j hj)]
+    _ ≤ ∑ j∈s,ENNReal.ofReal (c j)*C := by
+      apply Finset.sum_le_sum
+      intro j hj
+      gcongr
+      exact hbound j hj
+    _ = C := by
+      rw [←Finset.sum_mul,←ENNReal.ofReal_sum_of_nonneg hc,hsum,ENNReal.ofReal_one,one_mul]
+
+lemma actualUpperMass_Lp_bound (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (m : ℕ) (hm : 1 ≤ m) (f : C_c(UpperPlane, ℝ)) :
+    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 → ∀ t : ℝ≥0∞,
+      MemLp (fun b => ∫z,f z∂actualUpperMass a q t b) (m : ℝ≥0∞) W ∧
+      eLpNorm (fun b => ∫z,f z∂actualUpperMass a q t b) (m : ℝ≥0∞) W ≤ C := by
+  let K := Subtype.val '' tsupport f
+  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
+  obtain ⟨c,hc,hcz⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
+  have hcK (z : ℂ) (hz : z∈K) : c≤2*z.im := by
+    obtain ⟨w,hw,rfl⟩ := hz
+    exact hcz w hw
+  obtain ⟨C,hC,h⟩ := actualMassEval_Lp_bound a ha W hW m hm K hK.measurableSet hK.isBounded
+    c hc hcK (upperExtension f) (upperExtension_measurable f) (upperExtension_bounded f)
+  refine ⟨C,hC,fun q hq hq1 t => ?_⟩
+  simpa only [actualUpperMass_eval a ha q hq hq1] using h q hq hq1 t
+
+lemma actual_radon_limit_memLp (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0∞) (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+    (hc : ∀ k j, j∈s k → 0≤coeff k j) (hsum : ∀ k, ∑ j∈s k,coeff k j=1)
+    (μ : ExactSLEGenerator.Driver → Measure UpperPlane)
+    (f : C_c(UpperPlane, ℝ))
+    (hlim : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,coeff k j *
+      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂μ b)))
+    (m : ℕ) : MemLp (fun b => ∫z,f z∂μ b) (m : ℝ≥0∞) W := by
+  let q (j : ℕ) := Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun k b =>
+    ∑ j∈s k,coeff k j * ∫z,f z∂actualUpperMass a (q j) t b
+  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
+    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
+      (actualUpperMass_eval_measurable a ha (q j) (hq j) (hq1 j) t f).aestronglyMeasurable.const_mul (coeff k j))
+    convert H using 1
+    funext b
+    simp only [F,Finset.sum_apply]
+  have hmeas : AEStronglyMeasurable (fun b => ∫z,f z∂μ b) W :=
+    aestronglyMeasurable_of_tendsto_ae atTop hF hlim
+  by_cases hm : m=0
+  · simpa [hm] using (memLp_zero_iff_aestronglyMeasurable.mpr hmeas)
+  have hmpos : 1 ≤ m := Nat.one_le_iff_ne_zero.mpr hm
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW m hmpos f
+  have hbound (k : ℕ) : eLpNorm (F k) (m : ℝ≥0∞) W≤C :=
+    convex_eLpNorm_bound W _ C (by exact_mod_cast hmpos) (s k) (coeff k) (hc k) (hsum k)
+      (fun j b => ∫z,f z∂actualUpperMass a (q j) t b)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).1.aestronglyMeasurable)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).2)
+  exact (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF hmeas hlim).trans_lt hC
+
+ 
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/LoewnerScaling.lean b/lean/OAI/Probability/SLEGauge/LoewnerScaling.lean
index 139d0619283fc88e384b6c3b133488e70850cceb..06f250bcac9b8b28dfd48b537aca37ddfa02817f 100644
--- a/lean/OAI/Probability/SLEGauge/LoewnerScaling.lean
+++ b/lean/OAI/Probability/SLEGauge/LoewnerScaling.lean
@@ -1,481 +1,481 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.GaugeBounds
-import OAI.Probability.SLEGauge.InitialTrace
-import OAI.Probability.SLEGauge.AngleGenerators
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open ExactSLEMain ExactSLEGreen
-
- 
-def scaled (c : ℝ≥0) (b : ℝ≥0 → ℝ) (t : ℝ≥0) : ℝ :=
-  (c : ℝ)⁻¹ * b (c^2*t)
-
-@[simp] lemma scaled_scaled {c : ℝ≥0} (hc : c≠0) (b : ℝ≥0 → ℝ) :
-    scaled c⁻¹ (scaled c b)=b := by
-  ext t
-  simp only [scaled, NNReal.coe_inv, inv_inv, inv_pow]
-  have htime : c^2*(c^2)⁻¹*t=t := by rw [mul_inv_cancel₀ (pow_ne_zero _ hc),one_mul]
-  simp only [←mul_assoc,htime,mul_inv_cancel₀ (NNReal.coe_ne_zero.mpr hc),one_mul]
-
-lemma scaled_continuous (c : ℝ≥0) {b : ℝ≥0 → ℝ} (hb : Continuous b) :
-    Continuous (scaled c b) := continuous_const.mul (hb.comp (continuous_const.mul continuous_id))
-
- 
-def scaleDriver (c : ℝ≥0) (b : ExactSLEGenerator.Driver) : ExactSLEGenerator.Driver :=
-  ⟨⟨scaled c b.1,scaled_continuous c b.1.continuous⟩,by simp [scaled,b.2]⟩
-
-@[simp] lemma scaleDriver_apply (c : ℝ≥0) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    (scaleDriver c b).1 t=(c:ℝ)⁻¹*b.1 (c^2*t) := rfl
-
-@[simp] lemma scaleDriver_inv {c : ℝ≥0} (hc : c≠0) (b : ExactSLEGenerator.Driver) :
-    scaleDriver c⁻¹ (scaleDriver c b)=b := by
-  apply Subtype.ext
-  apply ContinuousMap.ext
-  exact congrFun (scaled_scaled hc b.1)
-
-lemma continuous_scaleDriver (c : ℝ≥0) : Continuous (scaleDriver c) := by
-  apply Continuous.subtype_mk
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  change Continuous (fun p : ExactSLEGenerator.Driver × ℝ≥0 =>
-    (c:ℝ)⁻¹*p.1.1 (c^2*p.2))
-  exact continuous_const.mul (continuous_eval.comp
-    ((continuous_subtype_val.comp continuous_fst).prodMk (continuous_const.mul continuous_snd)))
-
-lemma odeUntil_scaled {a : ℝ} {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0} {f : ℝ → ℂ}
-    (hf : odeUntil a b (((c:ℝ):ℂ)*z) (c^2*t) f) :
-    odeUntil a (scaled c b) z t (fun u => ((c:ℝ):ℂ)⁻¹*f ((c:ℝ)^2*u)) := by
-  have hcR : 0<(c:ℝ) := hc
-  have hcC : ((c:ℝ):ℂ)≠0 := Complex.ofReal_ne_zero.mpr hcR.ne'
-  refine ⟨?_,fun u hu => ?_⟩
-  · simp only [mul_zero,hf.1,←mul_assoc,inv_mul_cancel₀ hcC,one_mul]
-  have hmaps : MapsTo (fun u : ℝ => (c:ℝ)^2*u) (Icc 0 (t:ℝ))
-      (Icc 0 ((c^2*t:ℝ≥0):ℝ)) := by
-    intro v hv
-    simp only [NNReal.coe_mul,NNReal.coe_pow,mem_Icc]
-    exact ⟨mul_nonneg (sq_nonneg _) hv.1,mul_le_mul_of_nonneg_left hv.2 (sq_nonneg _)⟩
-  have hnn : ((c:ℝ)^2*u).toNNReal=c^2*u.toNNReal := by
-    apply NNReal.coe_injective
-    simp only [NNReal.coe_mul,NNReal.coe_pow,
-      Real.coe_toNNReal _ (mul_nonneg (sq_nonneg _) hu.1),Real.coe_toNNReal _ hu.1]
-  constructor
-  · simpa [←Complex.ofReal_inv,Complex.mul_im] using
-      mul_pos (inv_pos.mpr hcR) (hf.2 _ (hmaps hu)).1
-  · have hd := (hf.2 _ (hmaps hu)).2.scomp u
-      ((hasDerivAt_id u).const_mul ((c:ℝ)^2)).hasDerivWithinAt hmaps
-    have hd' := hd.const_mul ((c:ℝ):ℂ)⁻¹
-    convert! hd' using 1
-    change (a:ℂ) / (((c:ℝ):ℂ)⁻¹*f ((c:ℝ)^2*u) -
-      (((c:ℝ)⁻¹*b (c^2*u.toNNReal):ℝ):ℂ)) = _
-    rw [hnn]
-    simp only [Complex.ofReal_mul,Complex.ofReal_inv,Complex.real_smul,Complex.ofReal_pow]
-    field_simp
-    norm_num
-
-lemma alive_scaled_iff (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) :
-    alive a (scaled c b) t z ↔ alive a b (c^2*t) (((c:ℝ):ℂ)*z) := by
-  constructor
-  · rintro ⟨f,hf⟩
-    have hct : (c⁻¹)^2*(c^2*t)=t := by simp [hc.ne']
-    have hcz : (((c⁻¹:ℝ≥0):ℝ):ℂ)*(((c:ℝ):ℂ)*z)=z := by
-      simp [hc.ne']
-    have hs := odeUntil_scaled (inv_pos.mpr hc)
-      (by rw [hct,hcz]; exact hf : odeUntil a (scaled c b)
-        ((((c⁻¹:ℝ≥0):ℝ):ℂ)*(((c:ℝ):ℂ)*z)) ((c⁻¹)^2*(c^2*t)) f)
-    rw [scaled_scaled hc.ne'] at hs
-    exact ⟨_,hs⟩
-  · rintro ⟨f,hf⟩
-    exact ⟨_,odeUntil_scaled hc hf⟩
-
- 
-
-lemma loewnerMap_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) :
-    loewnerMap a (scaled c b) t z =
-      ((c:ℝ):ℂ)⁻¹*loewnerMap a b (c^2*t) (((c:ℝ):ℂ)*z) := by
-  by_cases h : alive a b (c^2*t) (((c:ℝ):ℂ)*z)
-  · obtain ⟨f,hf⟩ := h
-    rw [loewnerMap_eq (odeUntil_scaled hc hf),loewnerMap_eq hf]
-    simp only [NNReal.coe_mul,NNReal.coe_pow]
-  · have hs := (alive_scaled_iff a hc b t z).not.mpr h
-    simp only [loewnerMap,dite_eq_right h,dite_eq_right hs,mul_zero]
-
-lemma deriv_loewnerMap_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (t : ℝ≥0) (z : ℂ)
-    (hz : alive a (scaled c b) t z) :
-    deriv (loewnerMap a (scaled c b) t) z =
-      deriv (loewnerMap a b (c^2*t)) (((c:ℝ):ℂ)*z) := by
-  have h := (alive_scaled_iff a hc b t z).mp hz
-  have hder := ((hasDerivAt_loewnerMap hb h).comp z
-    ((hasDerivAt_id z).const_mul ((c:ℝ):ℂ))).const_mul ((c:ℝ):ℂ)⁻¹
-  have hf : (fun w => ((c:ℝ):ℂ)⁻¹*
-      (loewnerMap a b (c^2*t) ∘ fun w => ((c:ℝ):ℂ)*w) w) =
-      loewnerMap a (scaled c b) t := by
-    funext w
-    exact (loewnerMap_scaled a hc b t w).symm
-  rw [hf] at hder
-  rw [hder.deriv,(hasDerivAt_loewnerMap hb h).deriv]
-  simp only [mul_one]
-  field_simp [Complex.ofReal_ne_zero.mpr (NNReal.coe_ne_zero.mpr hc.ne')]
-
-lemma actualState_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) (t : ℝ≥0) (z : ℂ)
-    (hz : alive a (scaled c b) t z) :
-    actualState a (scaled c b) t z =
-      (((c:ℝ):ℂ)⁻¹*(actualState a b (c^2*t) (((c:ℝ):ℂ)*z)).1,
-       (actualState a b (c^2*t) (((c:ℝ):ℂ)*z)).2) := by
-  apply Prod.ext
-  · simp only [actualState,loewnerMap_scaled a hc,scaled,
-      Complex.ofReal_mul,Complex.ofReal_inv,mul_sub]
-  · simp only [actualState,deriv_loewnerMap_scaled a hc hb t z hz]
-
-end SLEExactGauge.LoewnerScaling
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal ContinuousMapZero
-namespace SLEExactGauge.FlowIdentification
-
-abbrev TDriver := Target.Driver
-abbrev GDriver := ExactSLEGenerator.Driver
-
- 
-def driverHomeomorph : TDriver ≃ₜ GDriver where
-  toFun b := ⟨(b : C(ℝ≥0,ℝ)),map_zero b⟩
-  invFun b := ⟨b.1,b.2⟩
-  left_inv b := by ext t; rfl
-  right_inv b := by ext t; rfl
-  continuous_toFun := ContinuousMapZero.isEmbedding_toContinuousMap.continuous.subtype_mk _
-  continuous_invFun := ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff.mpr
-    continuous_subtype_val
-
-@[simp] lemma driverHomeomorph_apply (b : TDriver) (t : ℝ≥0) :
-    (driverHomeomorph b).1 t=b t := rfl
-
-@[simp] lemma driverHomeomorph_symm_apply (b : GDriver) (t : ℝ≥0) :
-    driverHomeomorph.symm b t=b.1 t := rfl
-
-lemma measurable_driverHomeomorph : Measurable driverHomeomorph :=
-  driverHomeomorph.continuous.measurable
-
-lemma measurable_driverHomeomorph_symm : Measurable driverHomeomorph.symm :=
-  driverHomeomorph.symm.continuous.measurable
-
-def driverMeasurableEquiv : TDriver ≃ᵐ GDriver :=
-  ⟨driverHomeomorph.toEquiv, measurable_driverHomeomorph,measurable_driverHomeomorph_symm⟩
-
-lemma brownian_map_driver (W : Measure TDriver) (hW : Target.IsWiener W) :
-    IsBrownianReal (fun t (b : GDriver) => b.1 t) (W.map driverHomeomorph) where
-  hasLaw I := by
-    refine ⟨(Measurable.of_eval (fun t : I =>
-      (ExactSLEGenerator.driver_eval_measurable t).measurable)).aemeasurable,?_⟩
-    change (W.map driverHomeomorph).map (fun b : GDriver => fun t : I => b.1 t)=_
-    rw [Measure.map_map (Measurable.of_eval (fun t : I =>
-      (ExactSLEGenerator.driver_eval_measurable t).measurable)) measurable_driverHomeomorph]
-    exact (hW.hasLaw I).map_eq
-  cont := Eventually.of_forall (fun b => b.1.continuous)
-
-lemma positiveSolution_iff (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ≥0) (f : ℝ → ℂ) :
-    LiteralTrace.PositiveSolution κ b z t f ↔
-      ExactSLEMain.odeUntil (Target.a κ) (driverHomeomorph b).1 z t f := by
-  constructor
-  · intro hf
-    exact ⟨hf.2.2.1,fun u hu => ⟨hf.2.2.2.1 u hu,hf.2.2.2.2 u hu⟩⟩
-  · intro hf
-    exact ⟨t.coe_nonneg,ExactSLEMain.odeUntil_continuousOn hf,hf.1,
-      fun u hu => (hf.2 u hu).1,fun u hu => (hf.2 u hu).2⟩
-
-lemma eulerSteps_eq (κ : ℝ) (b : TDriver) (j N : ℕ) (z : ℂ) (k : ℕ) :
-    Target.eulerSteps κ b j N z k=LiteralTrace.eulerSteps κ b j N z k := by
-  induction k with
-  | zero => rfl
-  | succ k ih =>
-    simp only [Target.eulerSteps,LiteralTrace.eulerSteps,ih]
-    rfl
-
-lemma eulerFlow_eq (κ : ℝ) (b : TDriver) (j N : ℕ) (z : ℂ) (t : ℝ) :
-    Target.eulerFlow κ b j N z t=LiteralTrace.eulerFlow κ b j N z t := by
-  simp only [Target.eulerFlow,LiteralTrace.eulerFlow,eulerSteps_eq]
-  rfl
-
-lemma cutoffFlow_eq (κ : ℝ) (b : TDriver) (j : ℕ) (z : ℂ) (t : ℝ) :
-    Target.cutoffFlow κ b j z t=LiteralTrace.cutoffFlow κ b j z t := by
-  simp only [Target.cutoffFlow,LiteralTrace.cutoffFlow,eulerFlow_eq]
-
-lemma surviving_eq (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ) :
-    Target.Surviving κ b z t = LiteralTrace.Surviving κ b z t := by
-  simp only [Target.Surviving,LiteralTrace.Surviving,cutoffFlow_eq]
-
-lemma surviving_iff_alive (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ≥0) :
-    Target.Surviving κ b z t ↔ ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 t z := by
-  rw [surviving_eq,LiteralTrace.surviving_iff_positiveSolution t.coe_nonneg]
-  exact exists_congr (positiveSolution_iff κ b z t)
-
-lemma flow_eq_literal (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ) :
-    Target.flow κ b z t=LiteralTrace.flow κ b z t := by
-  classical
-  unfold Target.flow LiteralTrace.flow
-  split_ifs with ht hl hl
-  · have he : Nat.find ht=Nat.find hl := by
-      apply le_antisymm
-      · apply Nat.find_min' ht
-        simpa only [cutoffFlow_eq] using Nat.find_spec hl
-      · apply Nat.find_min' hl
-        simpa only [cutoffFlow_eq] using Nat.find_spec ht
-    rw [he,cutoffFlow_eq]
-  · exact False.elim (hl ((surviving_eq κ b z t).mp ht))
-  · exact False.elim (ht ((surviving_eq κ b z t).mpr hl))
-  · rfl
-
-lemma flow_eq_loewnerMap (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ≥0) :
-    Target.flow κ b z t=ExactSLEMain.loewnerMap (Target.a κ) (driverHomeomorph b).1 t z := by
-  by_cases ht : Target.Surviving κ b z t
-  · have ht' : LiteralTrace.Surviving κ b z t := (surviving_eq κ b z t).mp ht
-    have hp := LiteralTrace.surviving_flow_positiveSolution t.coe_nonneg ht'
-    rw [flow_eq_literal]
-    exact (ExactSLEMain.loewnerMap_eq ((positiveSolution_iff κ b z t _).mp hp)).symm
-  · have hn : ¬ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 t z :=
-      fun h => ht ((surviving_iff_alive κ b z t).mpr h)
-    simp [Target.flow,ExactSLEMain.loewnerMap,ht,hn]
-
-lemma flow_eq_loewnerMap_real (κ : ℝ) (b : TDriver) (z : ℂ) {t : ℝ} (ht : 0 ≤ t) :
-    Target.flow κ b z t=ExactSLEMain.loewnerMap (Target.a κ) (driverHomeomorph b).1 t.toNNReal z := by
-  simpa only [Real.coe_toNNReal _ ht] using flow_eq_loewnerMap κ b z t.toNNReal
-
-lemma inverseFlow_eq_literal (κ : ℝ) (b : TDriver) (t : ℝ) (w : ℂ) :
-    Target.inverseFlow κ b t w=LiteralTrace.inverseFlow κ b t w := by
-  classical
-  have he : (fun z : ℂ => Target.Surviving κ b z t ∧ Target.flow κ b z t=w)=
-      (fun z : ℂ => LiteralTrace.Surviving κ b z t ∧ LiteralTrace.flow κ b z t=w) := by
-    funext z
-    rw [surviving_eq,flow_eq_literal]
-  unfold Target.inverseFlow LiteralTrace.inverseFlow
-  rw [he]
-
-lemma trace_eq_literal (κ : ℝ) (b : TDriver) :
-    Target.trace κ b=LiteralTrace.trace κ b := by
-  funext t
-  simp only [Target.trace,LiteralTrace.trace,inverseFlow_eq_literal]
-  rfl
-
- 
-
-theorem trace_regular_actual_domains {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : Measure TDriver} (hW : Target.IsWiener W) :
-    ∀ᵐ b ∂W,
-      ContinuousOn (Target.trace κ b) (Ici 0) ∧
-      (∀ t : ℝ, 0 ≤ t → Target.trace κ b t ∈
-        frontier {z | ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 (κ*t).toNNReal z}) ∧
-      (∀ s : ℝ, 0 ≤ s → (Target.trace κ b) '' Ici s ⊆
-        closure {z | ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 (κ*s).toNNReal z}) := by
-  have hdomain (b : TDriver) (t : ℝ) (ht : 0 ≤ t) :
-      {z | LiteralTrace.Surviving κ b z (κ*t)}=
-        {z | ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 (κ*t).toNNReal z} := by
-    ext z
-    simp only [Set.mem_ofPred_eq]
-    simpa only [surviving_eq,Real.coe_toNNReal _ (mul_nonneg hκ.le ht)] using
-      surviving_iff_alive κ b z (κ*t).toNNReal
-  filter_upwards [LiteralTrace.trace_regular_all_times hκ hκ8 hW] with b hb
-  simp only [trace_eq_literal]
-  refine ⟨hb.2.1,?_,?_⟩
-  · intro t ht
-    rw [←hdomain b t ht]
-    exact hb.2.2.1 t ht
-  · intro s hs
-    rw [←hdomain b s hs]
-    exact hb.2.2.2 s hs
-
-end SLEExactGauge.FlowIdentification
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open FlowIdentification
-
-lemma brownian_map_inverse_driver (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W) :
-    Target.IsWiener (W.map driverHomeomorph.symm) where
-  hasLaw I := by
-    refine ⟨(Measurable.of_eval (fun t : I =>
-      (continuous_eval_const (t:ℝ≥0)).measurable)).aemeasurable,?_⟩
-    change (W.map driverHomeomorph.symm).map
-      (fun b : TDriver => fun t : I => b t)=_
-    rw [Measure.map_map (Measurable.of_eval (fun t : I =>
-      (continuous_eval_const (t:ℝ≥0)).measurable)) measurable_driverHomeomorph_symm]
-    exact (hW.hasLaw I).map_eq
-  cont := Eventually.of_forall (fun b => b.continuous)
-
-lemma brownian_evaluation_law_unique {W V : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (hV : IsBrownianReal (fun t (b : GDriver) => b.1 t) V) : W=V := by
-  apply driverMeasurableEquiv.symm.measurableEmbedding.map_injective
-  exact BrownianLaw.evaluation_law_unique (brownian_map_inverse_driver W hW)
-    (brownian_map_inverse_driver V hV)
-
-lemma brownian_scaled {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {c : ℝ≥0} (hc : 0<c) :
-    IsBrownianReal (fun t (b : GDriver) => (scaleDriver c b).1 t) W := by
-  simpa [scaleDriver_apply,Real.sqrt_sq c.coe_nonneg] using hW.smul (pow_ne_zero 2 hc.ne')
-
-lemma brownian_map_scaled {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {c : ℝ≥0} (hc : 0<c) :
-    IsBrownianReal (fun t (b : GDriver) => b.1 t) (W.map (scaleDriver c)) where
-  hasLaw I := by
-    refine ⟨(Measurable.of_eval (fun t : I =>
-      (ExactSLEGenerator.driver_eval_measurable t).measurable)).aemeasurable,?_⟩
-    change (W.map (scaleDriver c)).map (fun b : GDriver => fun t : I => b.1 t)=_
-    rw [Measure.map_map (Measurable.of_eval (fun t : I =>
-      (ExactSLEGenerator.driver_eval_measurable t).measurable)) (continuous_scaleDriver c).measurable]
-    exact ((brownian_scaled hW hc).hasLaw I).map_eq
-  cont := Eventually.of_forall (fun b => b.1.continuous)
-
-lemma map_scaleDriver_eq {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {c : ℝ≥0} (hc : 0<c) : W.map (scaleDriver c)=W :=
-  brownian_evaluation_law_unique (brownian_map_scaled hW hc) hW
-
-lemma measurePreserving_scaleDriver {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {c : ℝ≥0} (hc : 0<c) : MeasurePreserving (scaleDriver c) W W :=
-  ⟨(continuous_scaleDriver c).measurable,map_scaleDriver_eq hW hc⟩
-
-end SLEExactGauge.LoewnerScaling
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace ExactSLEMain
-
-lemma odeUntil_join {a : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
-    {f g : ℝ → ℂ} (hf : odeUntil a b z s f)
-    (hg : odeUntil a (shiftDriver b s) (f s - (b s : ℂ)) t g) :
-    odeUntil a b z (s+t) (fun u => if u ≤ s then f u else g (u-s)+(b s : ℂ)) := by
-  classical
-  let k : ℝ → ℂ := fun u => if u ≤ s then f u else g (u-s)+(b s : ℂ)
-  have hk₁ : EqOn k f (Icc 0 (s:ℝ)) := fun u hu => ite_eq_left hu.2
-  have hk₂ : EqOn k (fun u => g (u-s)+(b s : ℂ)) (Icc (s:ℝ) (s+t)) := by
-    intro u hu
-    dsimp only [k]
-    split_ifs with h
-    · have he : u=(s:ℝ) := le_antisymm h hu.1
-      simp only [he,sub_self,hg.1,sub_add_cancel]
-    · rfl
-  have hmaps : MapsTo (fun u : ℝ => u-s) (Icc (s:ℝ) (s+t)) (Icc 0 (t:ℝ)) := by
-    intro u hu
-    constructor <;> linarith [hu.1,hu.2]
-  have hd₁ (u : ℝ) (hu : u ∈ Icc 0 (s:ℝ)) :
-      HasDerivWithinAt k ((a:ℂ)/(k u-(b u.toNNReal:ℂ))) (Icc 0 (s:ℝ)) u := by
-    rw [hk₁ hu]
-    exact (hf.2 u hu).2.congr_of_mem hk₁ hu
-  have hd₂ (u : ℝ) (hu : u ∈ Icc (s:ℝ) (s+t)) :
-      HasDerivWithinAt k ((a:ℂ)/(k u-(b u.toNNReal:ℂ))) (Icc (s:ℝ) (s+t)) u := by
-    have hnn : s+(u-(s:ℝ)).toNNReal=u.toNNReal := by
-      apply NNReal.coe_injective
-      rw [NNReal.coe_add,Real.coe_toNNReal _ (sub_nonneg.mpr hu.1),
-        Real.coe_toNNReal _ (s.coe_nonneg.trans hu.1)]
-      ring
-    have h := ((hg.2 _ (hmaps hu)).2.scomp u
-      ((hasDerivAt_id u).sub_const (s:ℝ)).hasDerivWithinAt hmaps).add_const (b s:ℂ)
-    simp only [one_smul,shiftDriver,hnn,Complex.ofReal_sub] at h
-    rw [hk₂ hu]
-    convert h.congr_of_mem hk₂ hu using 1; congr 1; ring
-  refine ⟨by change k 0=z; rw [hk₁ (by simp),hf.1],fun u hu => ?_⟩
-  have hu' : u ∈ Icc 0 ((s:ℝ)+t) := hu
-  change 0<(k u).im ∧ HasDerivWithinAt k _ _ u
-  constructor
-  · by_cases hs : u≤(s:ℝ)
-    · rw [hk₁ ⟨hu.1,hs⟩]
-      exact (hf.2 u ⟨hu.1,hs⟩).1
-    · have hu₂ : u ∈ Icc (s:ℝ) (s+t) := ⟨(le_of_not_ge hs),hu.2⟩
-      rw [hk₂ hu₂]
-      simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using (hg.2 _ (hmaps hu₂)).1
-  · rcases lt_trichotomy u (s:ℝ) with hs | hs | hs
-    · apply (hd₁ u ⟨hu.1,hs.le⟩).mono_of_mem_nhdsWithin
-      filter_upwards [self_mem_nhdsWithin,
-        nhdsWithin_le_nhds (Iio_mem_nhds hs)] with v hv hvs
-      exact ⟨hv.1,hvs.le⟩
-    · subst u
-      have h := (hd₁ (s:ℝ) ⟨s.coe_nonneg,le_rfl⟩).union
-        (hd₂ (s:ℝ) ⟨le_rfl,le_add_of_nonneg_right t.coe_nonneg⟩)
-      rw [Icc_union_Icc_eq_Icc s.coe_nonneg (le_add_of_nonneg_right t.coe_nonneg)] at h
-      exact h
-    · apply (hd₂ u ⟨hs.le,hu.2⟩).mono_of_mem_nhdsWithin
-      filter_upwards [self_mem_nhdsWithin,
-        nhdsWithin_le_nhds (Ioi_mem_nhds hs)] with v hv hsv
-      exact ⟨hsv.le,hv.2⟩
-
- 
-theorem alive_restart_iff (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (s t : ℝ≥0) :
-    alive a b (s+t) z ↔ alive a b s z ∧
-      alive a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ)) := by
-  constructor
-  · intro h
-    exact ⟨alive_mono h (by exact le_add_of_nonneg_right bot_le),
-      (loewnerMap_restart a b z s t h).1⟩
-  · rintro ⟨⟨f,hf⟩,g,hg⟩
-    rw [loewnerMap_eq hf] at hg
-    exact ⟨_,odeUntil_join hf hg⟩
-
-lemma continuous_shiftDriver {b : ℝ≥0 → ℝ} (hb : Continuous b) (s : ℝ≥0) :
-    Continuous (shiftDriver b s) :=
-  (hb.comp (continuous_const.add continuous_id)).sub continuous_const
-
- 
-
-lemma deriv_loewnerMap_restart {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {s t : ℝ≥0} (hz : alive a b (s+t) z) :
-    deriv (loewnerMap a b (s+t)) z =
-      deriv (loewnerMap a (shiftDriver b s) t) (loewnerMap a b s z-(b s:ℂ)) *
-        deriv (loewnerMap a b s) z := by
-  have hs := (alive_restart_iff a b z s t).mp hz
-  have hd := ((hasDerivAt_loewnerMap (continuous_shiftDriver hb s) hs.2).comp z
-    ((hasDerivAt_loewnerMap hb hs.1).sub_const (b s:ℂ))).add_const (b s:ℂ)
-  have he : (fun w => loewnerMap a (shiftDriver b s) t (loewnerMap a b s w-(b s:ℂ))+(b s:ℂ))
-      =ᶠ[𝓝 z] loewnerMap a b (s+t) := by
-    filter_upwards [(isOpen_alive hb (s+t)).mem_nhds hz] with w hw
-    rw [(loewnerMap_restart a b w s t hw).2,sub_add_cancel]
-  rw [(hd.congr_of_eventuallyEq he.symm).deriv,
-    (hasDerivAt_loewnerMap hb hs.1).deriv,
-    (hasDerivAt_loewnerMap (continuous_shiftDriver hb s) hs.2).deriv]
-
-end ExactSLEMain
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.GaugeBounds
+import OAI.Probability.SLEGauge.InitialTrace
+import OAI.Probability.SLEGauge.AngleGenerators
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open ExactSLEMain ExactSLEGreen
+
+ 
+def scaled (c : ℝ≥0) (b : ℝ≥0 → ℝ) (t : ℝ≥0) : ℝ :=
+  (c : ℝ)⁻¹ * b (c^2*t)
+
+@[simp] lemma scaled_scaled {c : ℝ≥0} (hc : c≠0) (b : ℝ≥0 → ℝ) :
+    scaled c⁻¹ (scaled c b)=b := by
+  ext t
+  simp only [scaled, NNReal.coe_inv, inv_inv, inv_pow]
+  have htime : c^2*(c^2)⁻¹*t=t := by rw [mul_inv_cancel₀ (pow_ne_zero _ hc),one_mul]
+  simp only [←mul_assoc,htime,mul_inv_cancel₀ (NNReal.coe_ne_zero.mpr hc),one_mul]
+
+lemma scaled_continuous (c : ℝ≥0) {b : ℝ≥0 → ℝ} (hb : Continuous b) :
+    Continuous (scaled c b) := continuous_const.mul (hb.comp (continuous_const.mul continuous_id))
+
+ 
+def scaleDriver (c : ℝ≥0) (b : ExactSLEGenerator.Driver) : ExactSLEGenerator.Driver :=
+  ⟨⟨scaled c b.1,scaled_continuous c b.1.continuous⟩,by simp [scaled,b.2]⟩
+
+@[simp] lemma scaleDriver_apply (c : ℝ≥0) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    (scaleDriver c b).1 t=(c:ℝ)⁻¹*b.1 (c^2*t) := rfl
+
+@[simp] lemma scaleDriver_inv {c : ℝ≥0} (hc : c≠0) (b : ExactSLEGenerator.Driver) :
+    scaleDriver c⁻¹ (scaleDriver c b)=b := by
+  apply Subtype.ext
+  apply ContinuousMap.ext
+  exact congrFun (scaled_scaled hc b.1)
+
+lemma continuous_scaleDriver (c : ℝ≥0) : Continuous (scaleDriver c) := by
+  apply Continuous.subtype_mk
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  change Continuous (fun p : ExactSLEGenerator.Driver × ℝ≥0 =>
+    (c:ℝ)⁻¹*p.1.1 (c^2*p.2))
+  exact continuous_const.mul (continuous_eval.comp
+    ((continuous_subtype_val.comp continuous_fst).prodMk (continuous_const.mul continuous_snd)))
+
+lemma odeUntil_scaled {a : ℝ} {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0} {f : ℝ → ℂ}
+    (hf : odeUntil a b (((c:ℝ):ℂ)*z) (c^2*t) f) :
+    odeUntil a (scaled c b) z t (fun u => ((c:ℝ):ℂ)⁻¹*f ((c:ℝ)^2*u)) := by
+  have hcR : 0<(c:ℝ) := hc
+  have hcC : ((c:ℝ):ℂ)≠0 := Complex.ofReal_ne_zero.mpr hcR.ne'
+  refine ⟨?_,fun u hu => ?_⟩
+  · simp only [mul_zero,hf.1,←mul_assoc,inv_mul_cancel₀ hcC,one_mul]
+  have hmaps : MapsTo (fun u : ℝ => (c:ℝ)^2*u) (Icc 0 (t:ℝ))
+      (Icc 0 ((c^2*t:ℝ≥0):ℝ)) := by
+    intro v hv
+    simp only [NNReal.coe_mul,NNReal.coe_pow,mem_Icc]
+    exact ⟨mul_nonneg (sq_nonneg _) hv.1,mul_le_mul_of_nonneg_left hv.2 (sq_nonneg _)⟩
+  have hnn : ((c:ℝ)^2*u).toNNReal=c^2*u.toNNReal := by
+    apply NNReal.coe_injective
+    simp only [NNReal.coe_mul,NNReal.coe_pow,
+      Real.coe_toNNReal _ (mul_nonneg (sq_nonneg _) hu.1),Real.coe_toNNReal _ hu.1]
+  constructor
+  · simpa [←Complex.ofReal_inv,Complex.mul_im] using
+      mul_pos (inv_pos.mpr hcR) (hf.2 _ (hmaps hu)).1
+  · have hd := (hf.2 _ (hmaps hu)).2.scomp u
+      ((hasDerivAt_id u).const_mul ((c:ℝ)^2)).hasDerivWithinAt hmaps
+    have hd' := hd.const_mul ((c:ℝ):ℂ)⁻¹
+    convert! hd' using 1
+    change (a:ℂ) / (((c:ℝ):ℂ)⁻¹*f ((c:ℝ)^2*u) -
+      (((c:ℝ)⁻¹*b (c^2*u.toNNReal):ℝ):ℂ)) = _
+    rw [hnn]
+    simp only [Complex.ofReal_mul,Complex.ofReal_inv,Complex.real_smul,Complex.ofReal_pow]
+    field_simp
+    norm_num
+
+lemma alive_scaled_iff (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) :
+    alive a (scaled c b) t z ↔ alive a b (c^2*t) (((c:ℝ):ℂ)*z) := by
+  constructor
+  · rintro ⟨f,hf⟩
+    have hct : (c⁻¹)^2*(c^2*t)=t := by simp [hc.ne']
+    have hcz : (((c⁻¹:ℝ≥0):ℝ):ℂ)*(((c:ℝ):ℂ)*z)=z := by
+      simp [hc.ne']
+    have hs := odeUntil_scaled (inv_pos.mpr hc)
+      (by rw [hct,hcz]; exact hf : odeUntil a (scaled c b)
+        ((((c⁻¹:ℝ≥0):ℝ):ℂ)*(((c:ℝ):ℂ)*z)) ((c⁻¹)^2*(c^2*t)) f)
+    rw [scaled_scaled hc.ne'] at hs
+    exact ⟨_,hs⟩
+  · rintro ⟨f,hf⟩
+    exact ⟨_,odeUntil_scaled hc hf⟩
+
+ 
+
+lemma loewnerMap_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) :
+    loewnerMap a (scaled c b) t z =
+      ((c:ℝ):ℂ)⁻¹*loewnerMap a b (c^2*t) (((c:ℝ):ℂ)*z) := by
+  by_cases h : alive a b (c^2*t) (((c:ℝ):ℂ)*z)
+  · obtain ⟨f,hf⟩ := h
+    rw [loewnerMap_eq (odeUntil_scaled hc hf),loewnerMap_eq hf]
+    simp only [NNReal.coe_mul,NNReal.coe_pow]
+  · have hs := (alive_scaled_iff a hc b t z).not.mpr h
+    simp only [loewnerMap,dite_eq_right h,dite_eq_right hs,mul_zero]
+
+lemma deriv_loewnerMap_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (t : ℝ≥0) (z : ℂ)
+    (hz : alive a (scaled c b) t z) :
+    deriv (loewnerMap a (scaled c b) t) z =
+      deriv (loewnerMap a b (c^2*t)) (((c:ℝ):ℂ)*z) := by
+  have h := (alive_scaled_iff a hc b t z).mp hz
+  have hder := ((hasDerivAt_loewnerMap hb h).comp z
+    ((hasDerivAt_id z).const_mul ((c:ℝ):ℂ))).const_mul ((c:ℝ):ℂ)⁻¹
+  have hf : (fun w => ((c:ℝ):ℂ)⁻¹*
+      (loewnerMap a b (c^2*t) ∘ fun w => ((c:ℝ):ℂ)*w) w) =
+      loewnerMap a (scaled c b) t := by
+    funext w
+    exact (loewnerMap_scaled a hc b t w).symm
+  rw [hf] at hder
+  rw [hder.deriv,(hasDerivAt_loewnerMap hb h).deriv]
+  simp only [mul_one]
+  field_simp [Complex.ofReal_ne_zero.mpr (NNReal.coe_ne_zero.mpr hc.ne')]
+
+lemma actualState_scaled (a : ℝ) {c : ℝ≥0} (hc : 0<c)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) (t : ℝ≥0) (z : ℂ)
+    (hz : alive a (scaled c b) t z) :
+    actualState a (scaled c b) t z =
+      (((c:ℝ):ℂ)⁻¹*(actualState a b (c^2*t) (((c:ℝ):ℂ)*z)).1,
+       (actualState a b (c^2*t) (((c:ℝ):ℂ)*z)).2) := by
+  apply Prod.ext
+  · simp only [actualState,loewnerMap_scaled a hc,scaled,
+      Complex.ofReal_mul,Complex.ofReal_inv,mul_sub]
+  · simp only [actualState,deriv_loewnerMap_scaled a hc hb t z hz]
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal ContinuousMapZero
+namespace SLEExactGauge.FlowIdentification
+
+abbrev TDriver := Target.Driver
+abbrev GDriver := ExactSLEGenerator.Driver
+
+ 
+def driverHomeomorph : TDriver ≃ₜ GDriver where
+  toFun b := ⟨(b : C(ℝ≥0,ℝ)),map_zero b⟩
+  invFun b := ⟨b.1,b.2⟩
+  left_inv b := by ext t; rfl
+  right_inv b := by ext t; rfl
+  continuous_toFun := ContinuousMapZero.isEmbedding_toContinuousMap.continuous.subtype_mk _
+  continuous_invFun := ContinuousMapZero.isEmbedding_toContinuousMap.continuous_iff.mpr
+    continuous_subtype_val
+
+@[simp] lemma driverHomeomorph_apply (b : TDriver) (t : ℝ≥0) :
+    (driverHomeomorph b).1 t=b t := rfl
+
+@[simp] lemma driverHomeomorph_symm_apply (b : GDriver) (t : ℝ≥0) :
+    driverHomeomorph.symm b t=b.1 t := rfl
+
+lemma measurable_driverHomeomorph : Measurable driverHomeomorph :=
+  driverHomeomorph.continuous.measurable
+
+lemma measurable_driverHomeomorph_symm : Measurable driverHomeomorph.symm :=
+  driverHomeomorph.symm.continuous.measurable
+
+def driverMeasurableEquiv : TDriver ≃ᵐ GDriver :=
+  ⟨driverHomeomorph.toEquiv, measurable_driverHomeomorph,measurable_driverHomeomorph_symm⟩
+
+lemma brownian_map_driver (W : Measure TDriver) (hW : Target.IsWiener W) :
+    IsBrownianReal (fun t (b : GDriver) => b.1 t) (W.map driverHomeomorph) where
+  hasLaw I := by
+    refine ⟨(Measurable.of_eval (fun t : I =>
+      (ExactSLEGenerator.driver_eval_measurable t).measurable)).aemeasurable,?_⟩
+    change (W.map driverHomeomorph).map (fun b : GDriver => fun t : I => b.1 t)=_
+    rw [Measure.map_map (Measurable.of_eval (fun t : I =>
+      (ExactSLEGenerator.driver_eval_measurable t).measurable)) measurable_driverHomeomorph]
+    exact (hW.hasLaw I).map_eq
+  cont := Eventually.of_forall (fun b => b.1.continuous)
+
+lemma positiveSolution_iff (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ≥0) (f : ℝ → ℂ) :
+    LiteralTrace.PositiveSolution κ b z t f ↔
+      ExactSLEMain.odeUntil (Target.a κ) (driverHomeomorph b).1 z t f := by
+  constructor
+  · intro hf
+    exact ⟨hf.2.2.1,fun u hu => ⟨hf.2.2.2.1 u hu,hf.2.2.2.2 u hu⟩⟩
+  · intro hf
+    exact ⟨t.coe_nonneg,ExactSLEMain.odeUntil_continuousOn hf,hf.1,
+      fun u hu => (hf.2 u hu).1,fun u hu => (hf.2 u hu).2⟩
+
+lemma eulerSteps_eq (κ : ℝ) (b : TDriver) (j N : ℕ) (z : ℂ) (k : ℕ) :
+    Target.eulerSteps κ b j N z k=LiteralTrace.eulerSteps κ b j N z k := by
+  induction k with
+  | zero => rfl
+  | succ k ih =>
+    simp only [Target.eulerSteps,LiteralTrace.eulerSteps,ih]
+    rfl
+
+lemma eulerFlow_eq (κ : ℝ) (b : TDriver) (j N : ℕ) (z : ℂ) (t : ℝ) :
+    Target.eulerFlow κ b j N z t=LiteralTrace.eulerFlow κ b j N z t := by
+  simp only [Target.eulerFlow,LiteralTrace.eulerFlow,eulerSteps_eq]
+  rfl
+
+lemma cutoffFlow_eq (κ : ℝ) (b : TDriver) (j : ℕ) (z : ℂ) (t : ℝ) :
+    Target.cutoffFlow κ b j z t=LiteralTrace.cutoffFlow κ b j z t := by
+  simp only [Target.cutoffFlow,LiteralTrace.cutoffFlow,eulerFlow_eq]
+
+lemma surviving_eq (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ) :
+    Target.Surviving κ b z t = LiteralTrace.Surviving κ b z t := by
+  simp only [Target.Surviving,LiteralTrace.Surviving,cutoffFlow_eq]
+
+lemma surviving_iff_alive (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ≥0) :
+    Target.Surviving κ b z t ↔ ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 t z := by
+  rw [surviving_eq,LiteralTrace.surviving_iff_positiveSolution t.coe_nonneg]
+  exact exists_congr (positiveSolution_iff κ b z t)
+
+lemma flow_eq_literal (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ) :
+    Target.flow κ b z t=LiteralTrace.flow κ b z t := by
+  classical
+  unfold Target.flow LiteralTrace.flow
+  split_ifs with ht hl hl
+  · have he : Nat.find ht=Nat.find hl := by
+      apply le_antisymm
+      · apply Nat.find_min' ht
+        simpa only [cutoffFlow_eq] using Nat.find_spec hl
+      · apply Nat.find_min' hl
+        simpa only [cutoffFlow_eq] using Nat.find_spec ht
+    rw [he,cutoffFlow_eq]
+  · exact False.elim (hl ((surviving_eq κ b z t).mp ht))
+  · exact False.elim (ht ((surviving_eq κ b z t).mpr hl))
+  · rfl
+
+lemma flow_eq_loewnerMap (κ : ℝ) (b : TDriver) (z : ℂ) (t : ℝ≥0) :
+    Target.flow κ b z t=ExactSLEMain.loewnerMap (Target.a κ) (driverHomeomorph b).1 t z := by
+  by_cases ht : Target.Surviving κ b z t
+  · have ht' : LiteralTrace.Surviving κ b z t := (surviving_eq κ b z t).mp ht
+    have hp := LiteralTrace.surviving_flow_positiveSolution t.coe_nonneg ht'
+    rw [flow_eq_literal]
+    exact (ExactSLEMain.loewnerMap_eq ((positiveSolution_iff κ b z t _).mp hp)).symm
+  · have hn : ¬ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 t z :=
+      fun h => ht ((surviving_iff_alive κ b z t).mpr h)
+    simp [Target.flow,ExactSLEMain.loewnerMap,ht,hn]
+
+lemma flow_eq_loewnerMap_real (κ : ℝ) (b : TDriver) (z : ℂ) {t : ℝ} (ht : 0 ≤ t) :
+    Target.flow κ b z t=ExactSLEMain.loewnerMap (Target.a κ) (driverHomeomorph b).1 t.toNNReal z := by
+  simpa only [Real.coe_toNNReal _ ht] using flow_eq_loewnerMap κ b z t.toNNReal
+
+lemma inverseFlow_eq_literal (κ : ℝ) (b : TDriver) (t : ℝ) (w : ℂ) :
+    Target.inverseFlow κ b t w=LiteralTrace.inverseFlow κ b t w := by
+  classical
+  have he : (fun z : ℂ => Target.Surviving κ b z t ∧ Target.flow κ b z t=w)=
+      (fun z : ℂ => LiteralTrace.Surviving κ b z t ∧ LiteralTrace.flow κ b z t=w) := by
+    funext z
+    rw [surviving_eq,flow_eq_literal]
+  unfold Target.inverseFlow LiteralTrace.inverseFlow
+  rw [he]
+
+lemma trace_eq_literal (κ : ℝ) (b : TDriver) :
+    Target.trace κ b=LiteralTrace.trace κ b := by
+  funext t
+  simp only [Target.trace,LiteralTrace.trace,inverseFlow_eq_literal]
+  rfl
+
+ 
+
+theorem trace_regular_actual_domains {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : Measure TDriver} (hW : Target.IsWiener W) :
+    ∀ᵐ b ∂W,
+      ContinuousOn (Target.trace κ b) (Ici 0) ∧
+      (∀ t : ℝ, 0 ≤ t → Target.trace κ b t ∈
+        frontier {z | ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 (κ*t).toNNReal z}) ∧
+      (∀ s : ℝ, 0 ≤ s → (Target.trace κ b) '' Ici s ⊆
+        closure {z | ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 (κ*s).toNNReal z}) := by
+  have hdomain (b : TDriver) (t : ℝ) (ht : 0 ≤ t) :
+      {z | LiteralTrace.Surviving κ b z (κ*t)}=
+        {z | ExactSLEMain.alive (Target.a κ) (driverHomeomorph b).1 (κ*t).toNNReal z} := by
+    ext z
+    simp only [Set.mem_ofPred_eq]
+    simpa only [surviving_eq,Real.coe_toNNReal _ (mul_nonneg hκ.le ht)] using
+      surviving_iff_alive κ b z (κ*t).toNNReal
+  filter_upwards [LiteralTrace.trace_regular_all_times hκ hκ8 hW] with b hb
+  simp only [trace_eq_literal]
+  refine ⟨hb.2.1,?_,?_⟩
+  · intro t ht
+    rw [←hdomain b t ht]
+    exact hb.2.2.1 t ht
+  · intro s hs
+    rw [←hdomain b s hs]
+    exact hb.2.2.2 s hs
+
+end SLEExactGauge.FlowIdentification
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open FlowIdentification
+
+lemma brownian_map_inverse_driver (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W) :
+    Target.IsWiener (W.map driverHomeomorph.symm) where
+  hasLaw I := by
+    refine ⟨(Measurable.of_eval (fun t : I =>
+      (continuous_eval_const (t:ℝ≥0)).measurable)).aemeasurable,?_⟩
+    change (W.map driverHomeomorph.symm).map
+      (fun b : TDriver => fun t : I => b t)=_
+    rw [Measure.map_map (Measurable.of_eval (fun t : I =>
+      (continuous_eval_const (t:ℝ≥0)).measurable)) measurable_driverHomeomorph_symm]
+    exact (hW.hasLaw I).map_eq
+  cont := Eventually.of_forall (fun b => b.continuous)
+
+lemma brownian_evaluation_law_unique {W V : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (hV : IsBrownianReal (fun t (b : GDriver) => b.1 t) V) : W=V := by
+  apply driverMeasurableEquiv.symm.measurableEmbedding.map_injective
+  exact BrownianLaw.evaluation_law_unique (brownian_map_inverse_driver W hW)
+    (brownian_map_inverse_driver V hV)
+
+lemma brownian_scaled {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {c : ℝ≥0} (hc : 0<c) :
+    IsBrownianReal (fun t (b : GDriver) => (scaleDriver c b).1 t) W := by
+  simpa [scaleDriver_apply,Real.sqrt_sq c.coe_nonneg] using hW.smul (pow_ne_zero 2 hc.ne')
+
+lemma brownian_map_scaled {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {c : ℝ≥0} (hc : 0<c) :
+    IsBrownianReal (fun t (b : GDriver) => b.1 t) (W.map (scaleDriver c)) where
+  hasLaw I := by
+    refine ⟨(Measurable.of_eval (fun t : I =>
+      (ExactSLEGenerator.driver_eval_measurable t).measurable)).aemeasurable,?_⟩
+    change (W.map (scaleDriver c)).map (fun b : GDriver => fun t : I => b.1 t)=_
+    rw [Measure.map_map (Measurable.of_eval (fun t : I =>
+      (ExactSLEGenerator.driver_eval_measurable t).measurable)) (continuous_scaleDriver c).measurable]
+    exact ((brownian_scaled hW hc).hasLaw I).map_eq
+  cont := Eventually.of_forall (fun b => b.1.continuous)
+
+lemma map_scaleDriver_eq {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {c : ℝ≥0} (hc : 0<c) : W.map (scaleDriver c)=W :=
+  brownian_evaluation_law_unique (brownian_map_scaled hW hc) hW
+
+lemma measurePreserving_scaleDriver {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {c : ℝ≥0} (hc : 0<c) : MeasurePreserving (scaleDriver c) W W :=
+  ⟨(continuous_scaleDriver c).measurable,map_scaleDriver_eq hW hc⟩
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace ExactSLEMain
+
+lemma odeUntil_join {a : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {s t : ℝ≥0}
+    {f g : ℝ → ℂ} (hf : odeUntil a b z s f)
+    (hg : odeUntil a (shiftDriver b s) (f s - (b s : ℂ)) t g) :
+    odeUntil a b z (s+t) (fun u => if u ≤ s then f u else g (u-s)+(b s : ℂ)) := by
+  classical
+  let k : ℝ → ℂ := fun u => if u ≤ s then f u else g (u-s)+(b s : ℂ)
+  have hk₁ : EqOn k f (Icc 0 (s:ℝ)) := fun u hu => ite_eq_left hu.2
+  have hk₂ : EqOn k (fun u => g (u-s)+(b s : ℂ)) (Icc (s:ℝ) (s+t)) := by
+    intro u hu
+    dsimp only [k]
+    split_ifs with h
+    · have he : u=(s:ℝ) := le_antisymm h hu.1
+      simp only [he,sub_self,hg.1,sub_add_cancel]
+    · rfl
+  have hmaps : MapsTo (fun u : ℝ => u-s) (Icc (s:ℝ) (s+t)) (Icc 0 (t:ℝ)) := by
+    intro u hu
+    constructor <;> linarith [hu.1,hu.2]
+  have hd₁ (u : ℝ) (hu : u ∈ Icc 0 (s:ℝ)) :
+      HasDerivWithinAt k ((a:ℂ)/(k u-(b u.toNNReal:ℂ))) (Icc 0 (s:ℝ)) u := by
+    rw [hk₁ hu]
+    exact (hf.2 u hu).2.congr_of_mem hk₁ hu
+  have hd₂ (u : ℝ) (hu : u ∈ Icc (s:ℝ) (s+t)) :
+      HasDerivWithinAt k ((a:ℂ)/(k u-(b u.toNNReal:ℂ))) (Icc (s:ℝ) (s+t)) u := by
+    have hnn : s+(u-(s:ℝ)).toNNReal=u.toNNReal := by
+      apply NNReal.coe_injective
+      rw [NNReal.coe_add,Real.coe_toNNReal _ (sub_nonneg.mpr hu.1),
+        Real.coe_toNNReal _ (s.coe_nonneg.trans hu.1)]
+      ring
+    have h := ((hg.2 _ (hmaps hu)).2.scomp u
+      ((hasDerivAt_id u).sub_const (s:ℝ)).hasDerivWithinAt hmaps).add_const (b s:ℂ)
+    simp only [one_smul,shiftDriver,hnn,Complex.ofReal_sub] at h
+    rw [hk₂ hu]
+    convert h.congr_of_mem hk₂ hu using 1; congr 1; ring
+  refine ⟨by change k 0=z; rw [hk₁ (by simp),hf.1],fun u hu => ?_⟩
+  have hu' : u ∈ Icc 0 ((s:ℝ)+t) := hu
+  change 0<(k u).im ∧ HasDerivWithinAt k _ _ u
+  constructor
+  · by_cases hs : u≤(s:ℝ)
+    · rw [hk₁ ⟨hu.1,hs⟩]
+      exact (hf.2 u ⟨hu.1,hs⟩).1
+    · have hu₂ : u ∈ Icc (s:ℝ) (s+t) := ⟨(le_of_not_ge hs),hu.2⟩
+      rw [hk₂ hu₂]
+      simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using (hg.2 _ (hmaps hu₂)).1
+  · rcases lt_trichotomy u (s:ℝ) with hs | hs | hs
+    · apply (hd₁ u ⟨hu.1,hs.le⟩).mono_of_mem_nhdsWithin
+      filter_upwards [self_mem_nhdsWithin,
+        nhdsWithin_le_nhds (Iio_mem_nhds hs)] with v hv hvs
+      exact ⟨hv.1,hvs.le⟩
+    · subst u
+      have h := (hd₁ (s:ℝ) ⟨s.coe_nonneg,le_rfl⟩).union
+        (hd₂ (s:ℝ) ⟨le_rfl,le_add_of_nonneg_right t.coe_nonneg⟩)
+      rw [Icc_union_Icc_eq_Icc s.coe_nonneg (le_add_of_nonneg_right t.coe_nonneg)] at h
+      exact h
+    · apply (hd₂ u ⟨hs.le,hu.2⟩).mono_of_mem_nhdsWithin
+      filter_upwards [self_mem_nhdsWithin,
+        nhdsWithin_le_nhds (Ioi_mem_nhds hs)] with v hv hsv
+      exact ⟨hsv.le,hv.2⟩
+
+ 
+theorem alive_restart_iff (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (s t : ℝ≥0) :
+    alive a b (s+t) z ↔ alive a b s z ∧
+      alive a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ)) := by
+  constructor
+  · intro h
+    exact ⟨alive_mono h (by exact le_add_of_nonneg_right bot_le),
+      (loewnerMap_restart a b z s t h).1⟩
+  · rintro ⟨⟨f,hf⟩,g,hg⟩
+    rw [loewnerMap_eq hf] at hg
+    exact ⟨_,odeUntil_join hf hg⟩
+
+lemma continuous_shiftDriver {b : ℝ≥0 → ℝ} (hb : Continuous b) (s : ℝ≥0) :
+    Continuous (shiftDriver b s) :=
+  (hb.comp (continuous_const.add continuous_id)).sub continuous_const
+
+ 
+
+lemma deriv_loewnerMap_restart {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {s t : ℝ≥0} (hz : alive a b (s+t) z) :
+    deriv (loewnerMap a b (s+t)) z =
+      deriv (loewnerMap a (shiftDriver b s) t) (loewnerMap a b s z-(b s:ℂ)) *
+        deriv (loewnerMap a b s) z := by
+  have hs := (alive_restart_iff a b z s t).mp hz
+  have hd := ((hasDerivAt_loewnerMap (continuous_shiftDriver hb s) hs.2).comp z
+    ((hasDerivAt_loewnerMap hb hs.1).sub_const (b s:ℂ))).add_const (b s:ℂ)
+  have he : (fun w => loewnerMap a (shiftDriver b s) t (loewnerMap a b s w-(b s:ℂ))+(b s:ℂ))
+      =ᶠ[𝓝 z] loewnerMap a b (s+t) := by
+    filter_upwards [(isOpen_alive hb (s+t)).mem_nhds hz] with w hw
+    rw [(loewnerMap_restart a b w s t hw).2,sub_add_cancel]
+  rw [(hd.congr_of_eventuallyEq he.symm).deriv,
+    (hasDerivAt_loewnerMap hb hs.1).deriv,
+    (hasDerivAt_loewnerMap (continuous_shiftDriver hb s) hs.2).deriv]
+
+end ExactSLEMain
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/LoewnerSolutions.lean b/lean/OAI/Probability/SLEGauge/LoewnerSolutions.lean
index 5bf971cf8d78412656956bb2e20088b033d7cfaa..2d6ebffa2f3e09c5efce8cae6c45396ba1f8bcc5 100644
--- a/lean/OAI/Probability/SLEGauge/LoewnerSolutions.lean
+++ b/lean/OAI/Probability/SLEGauge/LoewnerSolutions.lean
@@ -1,416 +1,416 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.BrownianMartingales
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-namespace ExactSLEMain
-theorem loewner_local_existence (A : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b)
-    (z : ℂ) (hz : 0 < z.im) : ∃ t : ℝ≥0, 0 < t ∧ alive A b t z := by
-  let ε : ℝ := z.im / 2
-  have hε : 0 < ε := by dsimp [ε]; positivity
-  let L : ℝ := (|A| + 1) / ε
-  have hL : 0 < L := div_pos (by positivity) hε
-  let τ : ℝ := ε / L
-  have hτ : 0 < τ := div_pos hε hL
-  let v : ℝ → ℂ → ℂ := fun u x => (A : ℂ) / (x - (b u.toNNReal : ℂ))
-  have hheight : ∀ x ∈ Metric.closedBall z ε, ε ≤ x.im := by
-    intro x hx
-    have hx' : ‖x - z‖ ≤ ε := by simpa only [Metric.mem_closedBall, dist_eq_norm] using hx
-    have hh := Complex.im_le_norm (z - x)
-    rw [Complex.sub_im, norm_sub_rev] at hh
-    dsimp [ε] at *
-    linarith
-  let εn : ℝ≥0 := ⟨ε, hε.le⟩
-  let Ln : ℝ≥0 := ⟨L, hL.le⟩
-  let Kn : ℝ≥0 := ⟨|A| / ε ^ 2, by positivity⟩
-  let t₀ : Set.Icc (0 : ℝ) τ := ⟨0, ⟨le_rfl, hτ.le⟩⟩
-  have hp : IsPicardLindelof v t₀ z εn 0 Ln Kn := by
-    constructor
-    · intro u hu
-      apply LipschitzOnWith.of_dist_le_mul
-      intro x hx y hy
-      rw [dist_eq_norm, dist_eq_norm]
-      exact loewner_vector_difference A (b u.toNNReal) ε hε x y (hheight x hx) (hheight y hy)
-    · intro x hx
-      apply Continuous.continuousOn
-      apply Continuous.div continuous_const
-        (continuous_const.sub (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))
-      intro u
-      have hi : 0 < (x - (b u.toNNReal : ℂ)).im := by
-        simpa using hε.trans_le (hheight x hx)
-      intro he
-      change x - (b u.toNNReal : ℂ) = 0 at he
-      rw [he] at hi
-      simp at hi
-    · intro u hu x hx
-      change ‖(A : ℂ) / (x - (b u.toNNReal : ℂ))‖ ≤ L
-      have hlo : ε ≤ ‖x - (b u.toNNReal : ℂ)‖ := by
-        calc
-          ε ≤ x.im := hheight x hx
-          _ = (x - (b u.toNNReal : ℂ)).im := by simp
-          _ ≤ _ := Complex.im_le_norm _
-      rw [norm_div, Complex.norm_real, Real.norm_eq_abs]
-      calc
-        |A| / ‖x - (b u.toNNReal : ℂ)‖ ≤ |A| / ε :=
-          div_le_div_of_nonneg_left (abs_nonneg _) hε hlo
-        _ ≤ (|A| + 1) / ε := by gcongr; linarith
-    · change L * max (τ - 0) (0 - 0) ≤ εn - (0 : ℝ≥0)
-      simp only [sub_zero, max_eq_left hτ.le, NNReal.coe_zero, εn]
-      dsimp [τ]
-      rw [mul_div_cancel₀ _ hL.ne']
-      exact le_rfl
-  have hx : z ∈ Metric.closedBall z (0 : ℝ≥0) := by simp
-  obtain ⟨f, hf⟩ := ODE.FunSpace.exists_isFixedPt_next hp hx
-  have hinit : f.compProj t₀ = z := by
-    rw [ODE.FunSpace.compProj_val, ← hf, ODE.FunSpace.next_apply₀]
-  have hderiv : ∀ u ∈ Set.Icc (0 : ℝ) τ,
-      HasDerivWithinAt f.compProj (v u (f.compProj u)) (Set.Icc 0 τ) u := by
-    intro u hu
-    apply ODE.hasDerivWithinAt_picard_Icc t₀.2 hp.continuousOn_uncurry
-      f.continuous_compProj.continuousOn
-      (fun _ _ => f.compProj_mem_closedBall hp.mul_max_le) z hu |>.congr_of_mem _ hu
-    intro u' hu'
-    nth_rw 1 [← hf]
-    rw [ODE.FunSpace.compProj_of_mem hu', ODE.FunSpace.next_apply]
-  refine ⟨⟨τ, hτ.le⟩, hτ, f.compProj, hinit, ?_⟩
-  intro u hu
-  refine ⟨hε.trans_le (hheight _ (f.compProj_mem_closedBall hp.mul_max_le)), hderiv u hu⟩
-
-end ExactSLEMain
-
- 
-
-open MeasureTheory Filter Set Metric Function ODE
-open scoped NNReal Topology
-
-namespace ExactSLEMain
-open Set Metric
-
- 
-def heightClamp (ε : ℝ) (z : ℂ) : ℂ := ⟨z.re, max ε z.im⟩
-
-lemma heightClamp_eq {ε : ℝ} {z : ℂ} (hz : ε ≤ z.im) : heightClamp ε z = z := by
-  apply Complex.ext <;> simp [heightClamp, max_eq_right hz]
-
-lemma heightClamp_lipschitz (ε : ℝ) : LipschitzWith 2 (heightClamp ε) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  rw [dist_eq_norm, dist_eq_norm]
-  have hIm : LipschitzWith 1 Complex.im := LipschitzWith.of_dist_le_mul (fun z w => by
-    simpa only [Real.dist_eq, dist_eq_norm, Real.norm_eq_abs, NNReal.coe_one, _root_.one_mul, Complex.sub_im] using
-      Complex.abs_im_le_norm (z-w))
-  have him := (hIm.const_max ε).dist_le_mul z w
-  have hre := Complex.abs_re_le_norm (z-w)
-  have him' : |max ε z.im - max ε w.im| ≤ ‖z-w‖ := by
-    simpa only [Real.dist_eq, dist_eq_norm, Real.norm_eq_abs, NNReal.coe_one, _root_.one_mul] using him
-  calc
-    ‖heightClamp ε z - heightClamp ε w‖ ≤
-        |(heightClamp ε z - heightClamp ε w).re| + |(heightClamp ε z - heightClamp ε w).im| :=
-      Complex.norm_le_abs_re_add_abs_im _
-    _ ≤ ‖z-w‖ + ‖z-w‖ := add_le_add hre him'
-    _ = (2 : ℝ≥0) * ‖z-w‖ := by norm_num; ring
-
-noncomputable def cutoffVector (A ε : ℝ) (z : ℂ) : ℂ := (A : ℂ) / heightClamp ε z
-
-lemma cutoffVector_bound (A ε : ℝ) (hε : 0 < ε) (z : ℂ) :
-    ‖cutoffVector A ε z‖ ≤ |A|/ε := by
-  rw [cutoffVector, norm_div, Complex.norm_real, Real.norm_eq_abs]
-  exact div_le_div_of_nonneg_left (abs_nonneg _) hε
-    ((show ε ≤ (heightClamp ε z).im from le_max_left _ _).trans (Complex.im_le_norm _))
-
-lemma cutoffVector_lipschitz (A ε : ℝ) (hε : 0 < ε) :
-    LipschitzWith ⟨2*|A|/ε^2, by positivity⟩ (cutoffVector A ε) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  rw [dist_eq_norm, dist_eq_norm]
-  have hv := loewner_vector_difference A 0 ε hε (heightClamp ε z) (heightClamp ε w)
-    (le_max_left _ _) (le_max_left _ _)
-  simp only [Complex.ofReal_zero, sub_zero] at hv
-  calc
-    ‖cutoffVector A ε z - cutoffVector A ε w‖ ≤
-        (|A|/ε^2)*‖heightClamp ε z - heightClamp ε w‖ := hv
-    _ ≤ (|A|/ε^2)*(2*‖z-w‖) := mul_le_mul_of_nonneg_left
-      (by simpa only [dist_eq_norm, NNReal.coe_ofNat] using (heightClamp_lipschitz ε).dist_le_mul z w)
-      (by positivity)
-    _ = _ := by change _ = (2*|A|/ε^2)*‖z-w‖; ring
-
- 
-
-theorem exists_cutoff_loewner (A ε : ℝ) (hε : 0 < ε) (b : ℝ≥0 → ℝ)
-    (hb : Continuous b) :
-    ∃ F : ℝ≥0 → ℂ → ℂ,
-      (∀ z, Continuous (fun t => F t z)) ∧
-      (∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
-        cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ))) := by
-  let L : ℝ≥0 := ⟨|A|/ε, by positivity⟩
-  let K : ℝ≥0 := ⟨2*|A|/ε^2, by positivity⟩
-  have hu : ∀ z : ℂ, Continuous (fun t : ℝ≥0 => -(b t : ℂ)) :=
-    fun _ => (Complex.continuous_ofReal.comp hb).neg
-  obtain ⟨F,hFc,hFm,hFe⟩ := ExactSLEFlow.measurable_additive_flow
-    (cutoffVector A ε) L K (cutoffVector_lipschitz A ε hε)
-    (cutoffVector_bound A ε hε) (fun t (_ : ℂ) => -(b t : ℂ)) hu
-    (fun _ => measurable_const) id measurable_id
-  exact ⟨F,hFc,by simpa only [id_eq, sub_eq_add_neg] using hFe⟩
-
-lemma cutoff_loewner_hasDerivWithinAt {A ε : ℝ} (hε : 0 < ε)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
-    (hFc : ∀ z, Continuous (fun t => F t z))
-    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
-      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
-    (z : ℂ) (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) T) :
-    HasDerivWithinAt (fun u : ℝ => F u.toNNReal z)
-      (cutoffVector A ε (F t.toNNReal z - (b t.toNNReal : ℂ))) (Icc (0 : ℝ) T) t := by
-  simpa only [sub_eq_add_neg, Function.comp_def, Pi.neg_apply] using
-    ExactSLEFlow.integral_flow_hasDerivWithinAt (cutoffVector_lipschitz A ε hε).continuous
-      ((Complex.continuous_ofReal.comp hb).neg) (hFc z) (by
-        simpa only [sub_eq_add_neg, Function.comp_def, Pi.neg_apply] using hFe z) T t ht
-
-lemma cutoff_loewner_dist {A ε : ℝ} (hε : 0 < ε)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
-    (hFc : ∀ z, Continuous (fun t => F t z))
-    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
-      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
-    (z w : ℂ) (T : ℝ≥0) :
-    dist (F T z) (F T w) ≤ dist z w * Real.exp ((2*|A|/ε^2)*(T : ℝ)) := by
-  have hd (z : ℂ) (t : ℝ) (ht : t ∈ Ico (0 : ℝ) T) :=
-    (cutoff_loewner_hasDerivWithinAt hε hb hFc hFe z T t (mem_Icc_of_Ico ht)).mono_of_mem_nhdsWithin
-      (Icc_mem_nhdsGE_of_mem ht)
-  have hv (t : ℝ) : LipschitzWith ⟨2*|A|/ε^2, by positivity⟩
-      (fun z : ℂ => cutoffVector A ε (z - (b t.toNNReal : ℂ))) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro z w
-    change dist (cutoffVector A ε (z - (b t.toNNReal : ℂ)))
-      (cutoffVector A ε (w - (b t.toNNReal : ℂ))) ≤ (2*|A|/ε^2)*dist z w
-    convert (cutoffVector_lipschitz A ε hε).dist_le_mul
-      (z-(b t.toNNReal : ℂ)) (w-(b t.toNNReal : ℂ)) using 1
-    simp only [dist_sub_right]
-    rfl
-  have h0 (z : ℂ) : F 0 z = z := by simpa using hFe z 0
-  have hg := dist_le_of_trajectories_ODE hv
-    ((hFc z).comp continuous_real_toNNReal).continuousOn (hd z)
-    ((hFc w).comp continuous_real_toNNReal).continuousOn (hd w)
-    (show dist (F (0 : ℝ).toNNReal z) (F (0 : ℝ).toNNReal w) ≤ dist z w by simp [h0])
-    (T : ℝ) (show (T : ℝ) ∈ Icc (0 : ℝ) T by simp)
-  change dist (F (T : ℝ).toNNReal z) (F (T : ℝ).toNNReal w) ≤
-    dist z w * Real.exp ((2*|A|/ε^2)*((T : ℝ)-0)) at hg
-  simpa only [Real.toNNReal_coe, sub_zero] using hg
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set Metric
-lemma cutoff_loewner_eq_solution {A ε : ℝ} (hε : 0 < ε)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
-    (hFc : ∀ z, Continuous (fun t => F t z))
-    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
-      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil A b z T f)
-    (hlower : ∀ u ∈ Icc (0 : ℝ) T, ε ≤ (f u).im) :
-    ∀ u ∈ Icc (0 : ℝ) T, F u.toNNReal z = f u := by
-  have hfd := cutoff_loewner_hasDerivWithinAt hε hb hFc hFe z T
-  have hgd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt f (cutoffVector A ε (f u + (-(b u.toNNReal : ℂ)))) (Icc (0 : ℝ) T) u := by
-    have hc : heightClamp ε (f u - (b u.toNNReal : ℂ)) = f u - (b u.toNNReal : ℂ) :=
-      heightClamp_eq (by simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hlower u hu)
-    simpa only [cutoffVector, ← sub_eq_add_neg, hc] using (hf.2 u hu).2
-  apply ExactSLEFlow.additive_flow_unique (cutoffVector_lipschitz A ε hε)
-    ((hFc z).comp continuous_real_toNNReal).continuousOn (odeUntil_continuousOn hf)
-    (fun u hu => by simpa only [sub_eq_add_neg,Function.comp_def] using hfd u hu) hgd
-  simpa [hf.1, Function.comp_def] using hFe z 0
-
- 
-
-theorem loewner_local_cutoff {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hz : alive A b T z) :
-    ∃ (ε δ : ℝ) (F : ℝ≥0 → ℂ → ℂ), 0 < ε ∧ 0 < δ ∧
-      (∀ w, Continuous (fun t => F t w)) ∧
-      (∀ w t, F t w = w + ∫ u in (0 : ℝ)..(t : ℝ),
-        cutoffVector A ε (F u.toNNReal w - (b u.toNNReal : ℂ))) ∧
-      (∀ w ∈ ball z δ, odeUntil A b w T (fun u => F u.toNNReal w)) ∧
-      (∀ w ∈ ball z δ, ∀ u ≤ T, loewnerMap A b u w = F u w) ∧
-      (∀ w ∈ ball z δ, ∀ u ≤ T, ε < (F u w).im) := by
-  obtain ⟨f,hf⟩ := hz
-  obtain ⟨η,hη,hmin⟩ := odeUntil_height_lower hf
-  let ε := η/2
-  have hε : 0 < ε := half_pos hη
-  let K : ℝ := 2*|A|/ε^2
-  let δ := ε / Real.exp (K*(T : ℝ))
-  have hδ : 0 < δ := div_pos hε (Real.exp_pos _)
-  obtain ⟨F,hFc,hFe⟩ := exists_cutoff_loewner A ε hε b hb
-  have hFf := cutoff_loewner_eq_solution hε hb hFc hFe hf
-    (fun u hu => (show ε ≤ η by dsimp [ε]; linarith).trans (hmin u hu))
-  have hIm (w : ℂ) (hw : w ∈ ball z δ) (u : ℝ≥0) (hu : u ≤ T) : ε < (F u w).im := by
-    have hd := cutoff_loewner_dist hε hb hFc hFe w z u
-    have hdt : dist (F u w) (F u z) < ε := by
-      calc
-        dist (F u w) (F u z) ≤ dist w z * Real.exp (K*(u : ℝ)) := hd
-        _ ≤ dist w z * Real.exp (K*(T : ℝ)) := mul_le_mul_of_nonneg_left
-          (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (show (u : ℝ) ≤ T from hu)
-            (by dsimp [K]; positivity))) dist_nonneg
-        _ < δ * Real.exp (K*(T : ℝ)) := mul_lt_mul_of_pos_right hw (Real.exp_pos _)
-        _ = ε := div_mul_cancel₀ _ (Real.exp_ne_zero _)
-    have him := Complex.im_le_norm (F u z - F u w)
-    have hbase : η ≤ (F u z).im := by
-      rw [show F u z = f u by simpa only [Real.toNNReal_coe] using hFf u ⟨u.coe_nonneg,hu⟩]
-      exact hmin u ⟨u.coe_nonneg,hu⟩
-    rw [Complex.sub_im, ← dist_eq_norm, dist_comm] at him
-    dsimp [ε] at *
-    linarith
-  have hode (w : ℂ) (hw : w ∈ ball z δ) :
-      odeUntil A b w T (fun u => F u.toNNReal w) := by
-    refine ⟨by simpa using hFe w 0, fun u hu => ?_⟩
-    have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    have hpos := hIm w hw u.toNNReal huT
-    refine ⟨hε.trans hpos, ?_⟩
-    have hd := cutoff_loewner_hasDerivWithinAt hε hb hFc hFe w T u hu
-    have hc : heightClamp ε (F u.toNNReal w - (b u.toNNReal : ℂ)) =
-        F u.toNNReal w - (b u.toNNReal : ℂ) :=
-      heightClamp_eq (by simpa using hpos.le)
-    simpa only [cutoffVector,hc] using hd
-  refine ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,?_,hIm⟩
-  intro w hw u hu
-  simpa only [Real.toNNReal_coe] using loewnerMap_eq (odeUntil_mono (hode w hw) hu)
-
-lemma isOpen_alive {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b) (T : ℝ≥0) :
-    IsOpen {z | alive A b T z} := by
-  rw [Metric.isOpen_iff]
-  intro z hz
-  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hz
-  exact ⟨δ,hδ,fun w hw => ⟨_,hode w hw⟩⟩
-
-lemma continuousAt_loewnerMap {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) : ContinuousAt (loewnerMap A b T) z := by
-  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hz
-  have hLip : LipschitzWith ⟨Real.exp ((2*|A|/ε^2)*(T : ℝ)), (Real.exp_pos _).le⟩ (F T) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro w v
-    change dist (F T w) (F T v) ≤ Real.exp ((2*|A|/ε^2)*(T : ℝ))*dist w v
-    simpa only [_root_.mul_comm] using cutoff_loewner_dist hε hb hFc hFe w v T
-  apply hLip.continuous.continuousAt.congr
-  filter_upwards [Metric.ball_mem_nhds z hδ] with w hw
-  exact (hmap w hw T le_rfl).symm
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set
-
-lemma scalar_linear_solution (H c : ℝ → ℂ) (hc : Continuous c) (T : ℝ≥0)
-    (hH : ∀ u ∈ Icc (0 : ℝ) T, HasDerivWithinAt H (c u * H u) (Icc (0 : ℝ) T) u) :
-    H T = H 0 * Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), c u) := by
-  let P : ℝ → ℂ := fun t => H t * Complex.exp (-(∫ u in (0 : ℝ)..t, c u))
-  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt P 0 (Icc (0 : ℝ) T) u := by
-    have hi := intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 u)
-      hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-    convert (hH u hu).mul (hi.neg.cexp.hasDerivWithinAt) using 1
-    first | rfl | ring
-  have hnorm := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd
-    (fun u hu => by simp : ∀ u ∈ Icc (0 : ℝ) T, ‖(0 : ℂ)‖ ≤ 0)
-    (convex_Icc (0 : ℝ) T) (show (0 : ℝ) ∈ Icc (0 : ℝ) T by simp)
-    (show (T : ℝ) ∈ Icc (0 : ℝ) T by simp)
-  have hp : P T = P 0 := sub_eq_zero.mp (norm_eq_zero.mp
-    (le_antisymm (by simpa only [zero_mul] using hnorm) (norm_nonneg _)))
-  have he : H T * (Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), c u))⁻¹ = H 0 := by
-    simpa [P, Complex.exp_neg] using hp
-  exact (mul_inv_eq_iff_eq_mul₀ (Complex.exp_ne_zero _)).mp he
-
-lemma scalar_linear_solution_on (H c : ℝ → ℂ) (T : ℝ≥0)
-    (hc : ContinuousOn c (Icc (0 : ℝ) T))
-    (hH : ∀ u ∈ Icc (0 : ℝ) T, HasDerivWithinAt H (c u * H u) (Icc (0 : ℝ) T) u) :
-    H T = H 0 * Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), c u) := by
-  let c' : ℝ → ℂ := fun u => c (projIcc (0 : ℝ) T T.coe_nonneg u)
-  have hc' : Continuous c' := hc.domRestrict.comp continuous_projIcc
-  have he (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : c' u = c u := by
-    dsimp [c']; rw [projIcc_of_mem _ hu]
-  have h := scalar_linear_solution H c' hc' T (fun u hu => by rw [he u hu]; exact hH u hu)
-  rw [h]
-  congr 2
-  apply intervalIntegral.integral_congr
-  intro u hu
-  exact he u (by simpa only [uIcc_of_le T.coe_nonneg] using hu)
-
- 
-theorem loewner_solution_difference {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
-    (hf : odeUntil A b z T f) (hg : odeUntil A b w T g) :
-    f T - g T = (z-w) * Complex.exp (∫ u in (0 : ℝ)..(T : ℝ),
-      -(A : ℂ) / ((f u - (b u.toNNReal : ℂ)) * (g u - (b u.toNNReal : ℂ)))) := by
-  let c : ℝ → ℂ := fun u => -(A : ℂ) /
-    ((f u - (b u.toNNReal : ℂ)) * (g u - (b u.toNNReal : ℂ)))
-  have hfn (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : f u - (b u.toNNReal : ℂ) ≠ 0 := by
-    intro he
-    have hy := (hf.2 u hu).1
-    have hi := congrArg Complex.im he
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-    linarith
-  have hgn (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : g u - (b u.toNNReal : ℂ) ≠ 0 := by
-    intro he
-    have hy := (hg.2 u hu).1
-    have hi := congrArg Complex.im he
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-    linarith
-  have hbc : Continuous (fun u : ℝ => (b u.toNNReal : ℂ)) :=
-    Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)
-  have hc : ContinuousOn c (Icc (0 : ℝ) T) := continuousOn_const.div
-    (((odeUntil_continuousOn hf).sub hbc.continuousOn).mul
-      ((odeUntil_continuousOn hg).sub hbc.continuousOn))
-      (fun u hu => mul_ne_zero (hfn u hu) (hgn u hu))
-  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : HasDerivWithinAt (fun u => f u-g u)
-      (c u*(f u-g u)) (Icc (0 : ℝ) T) u := by
-    convert (hf.2 u hu).2.sub (hg.2 u hu).2 using 1
-    try rfl
-    dsimp [c]
-    field_simp [hfn u hu, hgn u hu]
-    ring
-  simpa only [hf.1,hg.1] using scalar_linear_solution_on (fun u => f u-g u) c T hc hd
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set Metric
-
-lemma heightClamp_norm_lower (ε : ℝ) (z : ℂ) : ε ≤ ‖heightClamp ε z‖ :=
-  (show ε ≤ (heightClamp ε z).im from le_max_left _ _).trans (Complex.im_le_norm _)
-
-lemma heightClamp_ne_zero {ε : ℝ} (hε : 0 < ε) (z : ℂ) : heightClamp ε z ≠ 0 :=
-  norm_pos_iff.mp (hε.trans_le (heightClamp_norm_lower ε z))
-
-lemma invprod_heightClamp_bound (A ε : ℝ) (hε : 0 < ε) (z w : ℂ) :
-    ‖-(A : ℂ)/(heightClamp ε z * heightClamp ε w)‖ ≤ |A|/ε^2 := by
-  rw [norm_div,norm_neg,Complex.norm_real,Real.norm_eq_abs,norm_mul]
-  exact div_le_div_of_nonneg_left (abs_nonneg _) (sq_pos_of_pos hε)
-    (by simpa only [pow_two] using (mul_le_mul (heightClamp_norm_lower ε z)
-      (heightClamp_norm_lower ε w) hε.le (norm_nonneg _)))
-
-lemma cutoff_loewner_continuous_start {A ε : ℝ} (hε : 0 < ε)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
-    (hFc : ∀ z, Continuous (fun t => F t z))
-    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
-      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ))) (T : ℝ≥0) :
-    Continuous (F T) := by
-  have hLip : LipschitzWith ⟨Real.exp ((2*|A|/ε^2)*(T : ℝ)), (Real.exp_pos _).le⟩ (F T) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro w v
-    change dist (F T w) (F T v) ≤ Real.exp ((2*|A|/ε^2)*(T : ℝ))*dist w v
-    simpa only [_root_.mul_comm] using cutoff_loewner_dist hε hb hFc hFe w v T
-  exact hLip.continuous
-
- 
-
-end ExactSLEMain
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.BrownianMartingales
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+namespace ExactSLEMain
+theorem loewner_local_existence (A : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b)
+    (z : ℂ) (hz : 0 < z.im) : ∃ t : ℝ≥0, 0 < t ∧ alive A b t z := by
+  let ε : ℝ := z.im / 2
+  have hε : 0 < ε := by dsimp [ε]; positivity
+  let L : ℝ := (|A| + 1) / ε
+  have hL : 0 < L := div_pos (by positivity) hε
+  let τ : ℝ := ε / L
+  have hτ : 0 < τ := div_pos hε hL
+  let v : ℝ → ℂ → ℂ := fun u x => (A : ℂ) / (x - (b u.toNNReal : ℂ))
+  have hheight : ∀ x ∈ Metric.closedBall z ε, ε ≤ x.im := by
+    intro x hx
+    have hx' : ‖x - z‖ ≤ ε := by simpa only [Metric.mem_closedBall, dist_eq_norm] using hx
+    have hh := Complex.im_le_norm (z - x)
+    rw [Complex.sub_im, norm_sub_rev] at hh
+    dsimp [ε] at *
+    linarith
+  let εn : ℝ≥0 := ⟨ε, hε.le⟩
+  let Ln : ℝ≥0 := ⟨L, hL.le⟩
+  let Kn : ℝ≥0 := ⟨|A| / ε ^ 2, by positivity⟩
+  let t₀ : Set.Icc (0 : ℝ) τ := ⟨0, ⟨le_rfl, hτ.le⟩⟩
+  have hp : IsPicardLindelof v t₀ z εn 0 Ln Kn := by
+    constructor
+    · intro u hu
+      apply LipschitzOnWith.of_dist_le_mul
+      intro x hx y hy
+      rw [dist_eq_norm, dist_eq_norm]
+      exact loewner_vector_difference A (b u.toNNReal) ε hε x y (hheight x hx) (hheight y hy)
+    · intro x hx
+      apply Continuous.continuousOn
+      apply Continuous.div continuous_const
+        (continuous_const.sub (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))
+      intro u
+      have hi : 0 < (x - (b u.toNNReal : ℂ)).im := by
+        simpa using hε.trans_le (hheight x hx)
+      intro he
+      change x - (b u.toNNReal : ℂ) = 0 at he
+      rw [he] at hi
+      simp at hi
+    · intro u hu x hx
+      change ‖(A : ℂ) / (x - (b u.toNNReal : ℂ))‖ ≤ L
+      have hlo : ε ≤ ‖x - (b u.toNNReal : ℂ)‖ := by
+        calc
+          ε ≤ x.im := hheight x hx
+          _ = (x - (b u.toNNReal : ℂ)).im := by simp
+          _ ≤ _ := Complex.im_le_norm _
+      rw [norm_div, Complex.norm_real, Real.norm_eq_abs]
+      calc
+        |A| / ‖x - (b u.toNNReal : ℂ)‖ ≤ |A| / ε :=
+          div_le_div_of_nonneg_left (abs_nonneg _) hε hlo
+        _ ≤ (|A| + 1) / ε := by gcongr; linarith
+    · change L * max (τ - 0) (0 - 0) ≤ εn - (0 : ℝ≥0)
+      simp only [sub_zero, max_eq_left hτ.le, NNReal.coe_zero, εn]
+      dsimp [τ]
+      rw [mul_div_cancel₀ _ hL.ne']
+      exact le_rfl
+  have hx : z ∈ Metric.closedBall z (0 : ℝ≥0) := by simp
+  obtain ⟨f, hf⟩ := ODE.FunSpace.exists_isFixedPt_next hp hx
+  have hinit : f.compProj t₀ = z := by
+    rw [ODE.FunSpace.compProj_val, ← hf, ODE.FunSpace.next_apply₀]
+  have hderiv : ∀ u ∈ Set.Icc (0 : ℝ) τ,
+      HasDerivWithinAt f.compProj (v u (f.compProj u)) (Set.Icc 0 τ) u := by
+    intro u hu
+    apply ODE.hasDerivWithinAt_picard_Icc t₀.2 hp.continuousOn_uncurry
+      f.continuous_compProj.continuousOn
+      (fun _ _ => f.compProj_mem_closedBall hp.mul_max_le) z hu |>.congr_of_mem _ hu
+    intro u' hu'
+    nth_rw 1 [← hf]
+    rw [ODE.FunSpace.compProj_of_mem hu', ODE.FunSpace.next_apply]
+  refine ⟨⟨τ, hτ.le⟩, hτ, f.compProj, hinit, ?_⟩
+  intro u hu
+  refine ⟨hε.trans_le (hheight _ (f.compProj_mem_closedBall hp.mul_max_le)), hderiv u hu⟩
+
+end ExactSLEMain
+
+ 
+
+open MeasureTheory Filter Set Metric Function ODE
+open scoped NNReal Topology
+
+namespace ExactSLEMain
+open Set Metric
+
+ 
+def heightClamp (ε : ℝ) (z : ℂ) : ℂ := ⟨z.re, max ε z.im⟩
+
+lemma heightClamp_eq {ε : ℝ} {z : ℂ} (hz : ε ≤ z.im) : heightClamp ε z = z := by
+  apply Complex.ext <;> simp [heightClamp, max_eq_right hz]
+
+lemma heightClamp_lipschitz (ε : ℝ) : LipschitzWith 2 (heightClamp ε) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  rw [dist_eq_norm, dist_eq_norm]
+  have hIm : LipschitzWith 1 Complex.im := LipschitzWith.of_dist_le_mul (fun z w => by
+    simpa only [Real.dist_eq, dist_eq_norm, Real.norm_eq_abs, NNReal.coe_one, _root_.one_mul, Complex.sub_im] using
+      Complex.abs_im_le_norm (z-w))
+  have him := (hIm.const_max ε).dist_le_mul z w
+  have hre := Complex.abs_re_le_norm (z-w)
+  have him' : |max ε z.im - max ε w.im| ≤ ‖z-w‖ := by
+    simpa only [Real.dist_eq, dist_eq_norm, Real.norm_eq_abs, NNReal.coe_one, _root_.one_mul] using him
+  calc
+    ‖heightClamp ε z - heightClamp ε w‖ ≤
+        |(heightClamp ε z - heightClamp ε w).re| + |(heightClamp ε z - heightClamp ε w).im| :=
+      Complex.norm_le_abs_re_add_abs_im _
+    _ ≤ ‖z-w‖ + ‖z-w‖ := add_le_add hre him'
+    _ = (2 : ℝ≥0) * ‖z-w‖ := by norm_num; ring
+
+noncomputable def cutoffVector (A ε : ℝ) (z : ℂ) : ℂ := (A : ℂ) / heightClamp ε z
+
+lemma cutoffVector_bound (A ε : ℝ) (hε : 0 < ε) (z : ℂ) :
+    ‖cutoffVector A ε z‖ ≤ |A|/ε := by
+  rw [cutoffVector, norm_div, Complex.norm_real, Real.norm_eq_abs]
+  exact div_le_div_of_nonneg_left (abs_nonneg _) hε
+    ((show ε ≤ (heightClamp ε z).im from le_max_left _ _).trans (Complex.im_le_norm _))
+
+lemma cutoffVector_lipschitz (A ε : ℝ) (hε : 0 < ε) :
+    LipschitzWith ⟨2*|A|/ε^2, by positivity⟩ (cutoffVector A ε) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  rw [dist_eq_norm, dist_eq_norm]
+  have hv := loewner_vector_difference A 0 ε hε (heightClamp ε z) (heightClamp ε w)
+    (le_max_left _ _) (le_max_left _ _)
+  simp only [Complex.ofReal_zero, sub_zero] at hv
+  calc
+    ‖cutoffVector A ε z - cutoffVector A ε w‖ ≤
+        (|A|/ε^2)*‖heightClamp ε z - heightClamp ε w‖ := hv
+    _ ≤ (|A|/ε^2)*(2*‖z-w‖) := mul_le_mul_of_nonneg_left
+      (by simpa only [dist_eq_norm, NNReal.coe_ofNat] using (heightClamp_lipschitz ε).dist_le_mul z w)
+      (by positivity)
+    _ = _ := by change _ = (2*|A|/ε^2)*‖z-w‖; ring
+
+ 
+
+theorem exists_cutoff_loewner (A ε : ℝ) (hε : 0 < ε) (b : ℝ≥0 → ℝ)
+    (hb : Continuous b) :
+    ∃ F : ℝ≥0 → ℂ → ℂ,
+      (∀ z, Continuous (fun t => F t z)) ∧
+      (∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
+        cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ))) := by
+  let L : ℝ≥0 := ⟨|A|/ε, by positivity⟩
+  let K : ℝ≥0 := ⟨2*|A|/ε^2, by positivity⟩
+  have hu : ∀ z : ℂ, Continuous (fun t : ℝ≥0 => -(b t : ℂ)) :=
+    fun _ => (Complex.continuous_ofReal.comp hb).neg
+  obtain ⟨F,hFc,hFm,hFe⟩ := ExactSLEFlow.measurable_additive_flow
+    (cutoffVector A ε) L K (cutoffVector_lipschitz A ε hε)
+    (cutoffVector_bound A ε hε) (fun t (_ : ℂ) => -(b t : ℂ)) hu
+    (fun _ => measurable_const) id measurable_id
+  exact ⟨F,hFc,by simpa only [id_eq, sub_eq_add_neg] using hFe⟩
+
+lemma cutoff_loewner_hasDerivWithinAt {A ε : ℝ} (hε : 0 < ε)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
+    (hFc : ∀ z, Continuous (fun t => F t z))
+    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
+      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
+    (z : ℂ) (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) T) :
+    HasDerivWithinAt (fun u : ℝ => F u.toNNReal z)
+      (cutoffVector A ε (F t.toNNReal z - (b t.toNNReal : ℂ))) (Icc (0 : ℝ) T) t := by
+  simpa only [sub_eq_add_neg, Function.comp_def, Pi.neg_apply] using
+    ExactSLEFlow.integral_flow_hasDerivWithinAt (cutoffVector_lipschitz A ε hε).continuous
+      ((Complex.continuous_ofReal.comp hb).neg) (hFc z) (by
+        simpa only [sub_eq_add_neg, Function.comp_def, Pi.neg_apply] using hFe z) T t ht
+
+lemma cutoff_loewner_dist {A ε : ℝ} (hε : 0 < ε)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
+    (hFc : ∀ z, Continuous (fun t => F t z))
+    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
+      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
+    (z w : ℂ) (T : ℝ≥0) :
+    dist (F T z) (F T w) ≤ dist z w * Real.exp ((2*|A|/ε^2)*(T : ℝ)) := by
+  have hd (z : ℂ) (t : ℝ) (ht : t ∈ Ico (0 : ℝ) T) :=
+    (cutoff_loewner_hasDerivWithinAt hε hb hFc hFe z T t (mem_Icc_of_Ico ht)).mono_of_mem_nhdsWithin
+      (Icc_mem_nhdsGE_of_mem ht)
+  have hv (t : ℝ) : LipschitzWith ⟨2*|A|/ε^2, by positivity⟩
+      (fun z : ℂ => cutoffVector A ε (z - (b t.toNNReal : ℂ))) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro z w
+    change dist (cutoffVector A ε (z - (b t.toNNReal : ℂ)))
+      (cutoffVector A ε (w - (b t.toNNReal : ℂ))) ≤ (2*|A|/ε^2)*dist z w
+    convert (cutoffVector_lipschitz A ε hε).dist_le_mul
+      (z-(b t.toNNReal : ℂ)) (w-(b t.toNNReal : ℂ)) using 1
+    simp only [dist_sub_right]
+    rfl
+  have h0 (z : ℂ) : F 0 z = z := by simpa using hFe z 0
+  have hg := dist_le_of_trajectories_ODE hv
+    ((hFc z).comp continuous_real_toNNReal).continuousOn (hd z)
+    ((hFc w).comp continuous_real_toNNReal).continuousOn (hd w)
+    (show dist (F (0 : ℝ).toNNReal z) (F (0 : ℝ).toNNReal w) ≤ dist z w by simp [h0])
+    (T : ℝ) (show (T : ℝ) ∈ Icc (0 : ℝ) T by simp)
+  change dist (F (T : ℝ).toNNReal z) (F (T : ℝ).toNNReal w) ≤
+    dist z w * Real.exp ((2*|A|/ε^2)*((T : ℝ)-0)) at hg
+  simpa only [Real.toNNReal_coe, sub_zero] using hg
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set Metric
+lemma cutoff_loewner_eq_solution {A ε : ℝ} (hε : 0 < ε)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
+    (hFc : ∀ z, Continuous (fun t => F t z))
+    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
+      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ)))
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil A b z T f)
+    (hlower : ∀ u ∈ Icc (0 : ℝ) T, ε ≤ (f u).im) :
+    ∀ u ∈ Icc (0 : ℝ) T, F u.toNNReal z = f u := by
+  have hfd := cutoff_loewner_hasDerivWithinAt hε hb hFc hFe z T
+  have hgd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt f (cutoffVector A ε (f u + (-(b u.toNNReal : ℂ)))) (Icc (0 : ℝ) T) u := by
+    have hc : heightClamp ε (f u - (b u.toNNReal : ℂ)) = f u - (b u.toNNReal : ℂ) :=
+      heightClamp_eq (by simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hlower u hu)
+    simpa only [cutoffVector, ← sub_eq_add_neg, hc] using (hf.2 u hu).2
+  apply ExactSLEFlow.additive_flow_unique (cutoffVector_lipschitz A ε hε)
+    ((hFc z).comp continuous_real_toNNReal).continuousOn (odeUntil_continuousOn hf)
+    (fun u hu => by simpa only [sub_eq_add_neg,Function.comp_def] using hfd u hu) hgd
+  simpa [hf.1, Function.comp_def] using hFe z 0
+
+ 
+
+theorem loewner_local_cutoff {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hz : alive A b T z) :
+    ∃ (ε δ : ℝ) (F : ℝ≥0 → ℂ → ℂ), 0 < ε ∧ 0 < δ ∧
+      (∀ w, Continuous (fun t => F t w)) ∧
+      (∀ w t, F t w = w + ∫ u in (0 : ℝ)..(t : ℝ),
+        cutoffVector A ε (F u.toNNReal w - (b u.toNNReal : ℂ))) ∧
+      (∀ w ∈ ball z δ, odeUntil A b w T (fun u => F u.toNNReal w)) ∧
+      (∀ w ∈ ball z δ, ∀ u ≤ T, loewnerMap A b u w = F u w) ∧
+      (∀ w ∈ ball z δ, ∀ u ≤ T, ε < (F u w).im) := by
+  obtain ⟨f,hf⟩ := hz
+  obtain ⟨η,hη,hmin⟩ := odeUntil_height_lower hf
+  let ε := η/2
+  have hε : 0 < ε := half_pos hη
+  let K : ℝ := 2*|A|/ε^2
+  let δ := ε / Real.exp (K*(T : ℝ))
+  have hδ : 0 < δ := div_pos hε (Real.exp_pos _)
+  obtain ⟨F,hFc,hFe⟩ := exists_cutoff_loewner A ε hε b hb
+  have hFf := cutoff_loewner_eq_solution hε hb hFc hFe hf
+    (fun u hu => (show ε ≤ η by dsimp [ε]; linarith).trans (hmin u hu))
+  have hIm (w : ℂ) (hw : w ∈ ball z δ) (u : ℝ≥0) (hu : u ≤ T) : ε < (F u w).im := by
+    have hd := cutoff_loewner_dist hε hb hFc hFe w z u
+    have hdt : dist (F u w) (F u z) < ε := by
+      calc
+        dist (F u w) (F u z) ≤ dist w z * Real.exp (K*(u : ℝ)) := hd
+        _ ≤ dist w z * Real.exp (K*(T : ℝ)) := mul_le_mul_of_nonneg_left
+          (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (show (u : ℝ) ≤ T from hu)
+            (by dsimp [K]; positivity))) dist_nonneg
+        _ < δ * Real.exp (K*(T : ℝ)) := mul_lt_mul_of_pos_right hw (Real.exp_pos _)
+        _ = ε := div_mul_cancel₀ _ (Real.exp_ne_zero _)
+    have him := Complex.im_le_norm (F u z - F u w)
+    have hbase : η ≤ (F u z).im := by
+      rw [show F u z = f u by simpa only [Real.toNNReal_coe] using hFf u ⟨u.coe_nonneg,hu⟩]
+      exact hmin u ⟨u.coe_nonneg,hu⟩
+    rw [Complex.sub_im, ← dist_eq_norm, dist_comm] at him
+    dsimp [ε] at *
+    linarith
+  have hode (w : ℂ) (hw : w ∈ ball z δ) :
+      odeUntil A b w T (fun u => F u.toNNReal w) := by
+    refine ⟨by simpa using hFe w 0, fun u hu => ?_⟩
+    have huT : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    have hpos := hIm w hw u.toNNReal huT
+    refine ⟨hε.trans hpos, ?_⟩
+    have hd := cutoff_loewner_hasDerivWithinAt hε hb hFc hFe w T u hu
+    have hc : heightClamp ε (F u.toNNReal w - (b u.toNNReal : ℂ)) =
+        F u.toNNReal w - (b u.toNNReal : ℂ) :=
+      heightClamp_eq (by simpa using hpos.le)
+    simpa only [cutoffVector,hc] using hd
+  refine ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,?_,hIm⟩
+  intro w hw u hu
+  simpa only [Real.toNNReal_coe] using loewnerMap_eq (odeUntil_mono (hode w hw) hu)
+
+lemma isOpen_alive {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b) (T : ℝ≥0) :
+    IsOpen {z | alive A b T z} := by
+  rw [Metric.isOpen_iff]
+  intro z hz
+  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hz
+  exact ⟨δ,hδ,fun w hw => ⟨_,hode w hw⟩⟩
+
+lemma continuousAt_loewnerMap {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {T : ℝ≥0} {z : ℂ} (hz : alive A b T z) : ContinuousAt (loewnerMap A b T) z := by
+  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hz
+  have hLip : LipschitzWith ⟨Real.exp ((2*|A|/ε^2)*(T : ℝ)), (Real.exp_pos _).le⟩ (F T) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro w v
+    change dist (F T w) (F T v) ≤ Real.exp ((2*|A|/ε^2)*(T : ℝ))*dist w v
+    simpa only [_root_.mul_comm] using cutoff_loewner_dist hε hb hFc hFe w v T
+  apply hLip.continuous.continuousAt.congr
+  filter_upwards [Metric.ball_mem_nhds z hδ] with w hw
+  exact (hmap w hw T le_rfl).symm
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set
+
+lemma scalar_linear_solution (H c : ℝ → ℂ) (hc : Continuous c) (T : ℝ≥0)
+    (hH : ∀ u ∈ Icc (0 : ℝ) T, HasDerivWithinAt H (c u * H u) (Icc (0 : ℝ) T) u) :
+    H T = H 0 * Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), c u) := by
+  let P : ℝ → ℂ := fun t => H t * Complex.exp (-(∫ u in (0 : ℝ)..t, c u))
+  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt P 0 (Icc (0 : ℝ) T) u := by
+    have hi := intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 u)
+      hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+    convert (hH u hu).mul (hi.neg.cexp.hasDerivWithinAt) using 1
+    first | rfl | ring
+  have hnorm := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd
+    (fun u hu => by simp : ∀ u ∈ Icc (0 : ℝ) T, ‖(0 : ℂ)‖ ≤ 0)
+    (convex_Icc (0 : ℝ) T) (show (0 : ℝ) ∈ Icc (0 : ℝ) T by simp)
+    (show (T : ℝ) ∈ Icc (0 : ℝ) T by simp)
+  have hp : P T = P 0 := sub_eq_zero.mp (norm_eq_zero.mp
+    (le_antisymm (by simpa only [zero_mul] using hnorm) (norm_nonneg _)))
+  have he : H T * (Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), c u))⁻¹ = H 0 := by
+    simpa [P, Complex.exp_neg] using hp
+  exact (mul_inv_eq_iff_eq_mul₀ (Complex.exp_ne_zero _)).mp he
+
+lemma scalar_linear_solution_on (H c : ℝ → ℂ) (T : ℝ≥0)
+    (hc : ContinuousOn c (Icc (0 : ℝ) T))
+    (hH : ∀ u ∈ Icc (0 : ℝ) T, HasDerivWithinAt H (c u * H u) (Icc (0 : ℝ) T) u) :
+    H T = H 0 * Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), c u) := by
+  let c' : ℝ → ℂ := fun u => c (projIcc (0 : ℝ) T T.coe_nonneg u)
+  have hc' : Continuous c' := hc.domRestrict.comp continuous_projIcc
+  have he (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : c' u = c u := by
+    dsimp [c']; rw [projIcc_of_mem _ hu]
+  have h := scalar_linear_solution H c' hc' T (fun u hu => by rw [he u hu]; exact hH u hu)
+  rw [h]
+  congr 2
+  apply intervalIntegral.integral_congr
+  intro u hu
+  exact he u (by simpa only [uIcc_of_le T.coe_nonneg] using hu)
+
+ 
+theorem loewner_solution_difference {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
+    (hf : odeUntil A b z T f) (hg : odeUntil A b w T g) :
+    f T - g T = (z-w) * Complex.exp (∫ u in (0 : ℝ)..(T : ℝ),
+      -(A : ℂ) / ((f u - (b u.toNNReal : ℂ)) * (g u - (b u.toNNReal : ℂ)))) := by
+  let c : ℝ → ℂ := fun u => -(A : ℂ) /
+    ((f u - (b u.toNNReal : ℂ)) * (g u - (b u.toNNReal : ℂ)))
+  have hfn (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : f u - (b u.toNNReal : ℂ) ≠ 0 := by
+    intro he
+    have hy := (hf.2 u hu).1
+    have hi := congrArg Complex.im he
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+    linarith
+  have hgn (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : g u - (b u.toNNReal : ℂ) ≠ 0 := by
+    intro he
+    have hy := (hg.2 u hu).1
+    have hi := congrArg Complex.im he
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+    linarith
+  have hbc : Continuous (fun u : ℝ => (b u.toNNReal : ℂ)) :=
+    Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)
+  have hc : ContinuousOn c (Icc (0 : ℝ) T) := continuousOn_const.div
+    (((odeUntil_continuousOn hf).sub hbc.continuousOn).mul
+      ((odeUntil_continuousOn hg).sub hbc.continuousOn))
+      (fun u hu => mul_ne_zero (hfn u hu) (hgn u hu))
+  have hd (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : HasDerivWithinAt (fun u => f u-g u)
+      (c u*(f u-g u)) (Icc (0 : ℝ) T) u := by
+    convert (hf.2 u hu).2.sub (hg.2 u hu).2 using 1
+    try rfl
+    dsimp [c]
+    field_simp [hfn u hu, hgn u hu]
+    ring
+  simpa only [hf.1,hg.1] using scalar_linear_solution_on (fun u => f u-g u) c T hc hd
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set Metric
+
+lemma heightClamp_norm_lower (ε : ℝ) (z : ℂ) : ε ≤ ‖heightClamp ε z‖ :=
+  (show ε ≤ (heightClamp ε z).im from le_max_left _ _).trans (Complex.im_le_norm _)
+
+lemma heightClamp_ne_zero {ε : ℝ} (hε : 0 < ε) (z : ℂ) : heightClamp ε z ≠ 0 :=
+  norm_pos_iff.mp (hε.trans_le (heightClamp_norm_lower ε z))
+
+lemma invprod_heightClamp_bound (A ε : ℝ) (hε : 0 < ε) (z w : ℂ) :
+    ‖-(A : ℂ)/(heightClamp ε z * heightClamp ε w)‖ ≤ |A|/ε^2 := by
+  rw [norm_div,norm_neg,Complex.norm_real,Real.norm_eq_abs,norm_mul]
+  exact div_le_div_of_nonneg_left (abs_nonneg _) (sq_pos_of_pos hε)
+    (by simpa only [pow_two] using (mul_le_mul (heightClamp_norm_lower ε z)
+      (heightClamp_norm_lower ε w) hε.le (norm_nonneg _)))
+
+lemma cutoff_loewner_continuous_start {A ε : ℝ} (hε : 0 < ε)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {F : ℝ≥0 → ℂ → ℂ}
+    (hFc : ∀ z, Continuous (fun t => F t z))
+    (hFe : ∀ z t, F t z = z + ∫ u in (0 : ℝ)..(t : ℝ),
+      cutoffVector A ε (F u.toNNReal z - (b u.toNNReal : ℂ))) (T : ℝ≥0) :
+    Continuous (F T) := by
+  have hLip : LipschitzWith ⟨Real.exp ((2*|A|/ε^2)*(T : ℝ)), (Real.exp_pos _).le⟩ (F T) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro w v
+    change dist (F T w) (F T v) ≤ Real.exp ((2*|A|/ε^2)*(T : ℝ))*dist w v
+    simpa only [_root_.mul_comm] using cutoff_loewner_dist hε hb hFc hFe w v T
+  exact hLip.continuous
+
+ 
+
+end ExactSLEMain
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/Main.lean b/lean/OAI/Probability/SLEGauge/Main.lean
index 9a22a52cfa09e7b53b3dce34c8c4556e620cca9e..1d8b6201ea1d41f50b735673f14dfac628fea979 100644
--- a/lean/OAI/Probability/SLEGauge/Main.lean
+++ b/lean/OAI/Probability/SLEGauge/Main.lean
@@ -1 +1 @@
-import OAI.Probability.SLEGauge.ExactGaugeEndpoint
+import OAI.Probability.SLEGauge.ExactGaugeEndpoint
diff --git a/lean/OAI/Probability/SLEGauge/MarkedMomentBound.lean b/lean/OAI/Probability/SLEGauge/MarkedMomentBound.lean
index 5daa383e313202543ddacf3a85063003d2e387ed..e02f5c828efa0400820247c399ac21e217fa9f74 100644
--- a/lean/OAI/Probability/SLEGauge/MarkedMomentBound.lean
+++ b/lean/OAI/Probability/SLEGauge/MarkedMomentBound.lean
@@ -1,509 +1,509 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ConformalRestartMap
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEKernel
-
-lemma target_cap_cancellation {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α : ℝ) (hα : 0≤α) (cap target : ι→ℝ)
-    (hc : ∀i,0<cap i) (ht : ∀i,0<target i) :
-    (∏i,target i^(-α))*(∏i∈Finset.univ.filter (fun i => target i≤cap i),
-      (target i/cap i)^α) ≤ ∏i,cap i^(-α) := by
-  rw [Finset.prod_filter,←Finset.prod_mul_distrib]
-  apply Finset.prod_le_prod₀
-  · intro i _
-    apply mul_nonneg (Real.rpow_nonneg (ht i).le _)
-    split_ifs
-    · exact Real.rpow_nonneg (div_nonneg (ht i).le (hc i).le) _
-    · norm_num
-  · intro i _
-    by_cases hi : target i≤cap i
-    · rw [ite_eq_left hi,Real.div_rpow (ht i).le (hc i).le,
-        Real.rpow_neg (ht i).le,Real.rpow_neg (hc i).le]
-      have hp := Real.rpow_pos_of_pos (ht i) α
-      field_simp
-      rfl
-    · rw [ite_eq_right hi,mul_one]
-      exact Real.rpow_le_rpow_of_nonpos (hc i) (le_of_not_ge hi) (neg_nonpos.mpr hα)
-
- 
-
-theorem actual_hit_density_product_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
-    ∃C : ℝ, 0<C ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-      ∀z : Fin (m+1)→ℂ, Function.Injective z → (∀i,0<(z i).im) →
-      ∀cap target : Fin (m+1)→ℝ,
-        (∀i,0<cap i ∧ cap i≤2*(z i).im) →
-        (∀i j,i≠j → cap i≤‖z i-z j‖) →
-        (∀i,0<target i ∧ target i≤2*(z i).im) →
-        (∫⁻b in {b | radialTime a b.1 (z 0) (target 0/2)<∞},
-          ∏i : Fin m, ENNReal.ofReal (fullRadialDensity a b.1 (z i.succ) (target i.succ))∂W) ≤
-          ENNReal.ofReal (C*(target 0)^(alpha a)*(∏i,cap i^(-alpha a))) := by
-  obtain ⟨C,hC,hCb⟩ := selective_hit_bound a ha (m+1)
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW z hinj hz cap target hc hsep ht
-  let I := Finset.univ.filter (fun i => target i≤cap i)
-  let E := {b : ExactSLEGenerator.Driver | ∀i∈I,radialTime a b.1 (z i) (target i/2)<∞}
-  let E₀ := {b : ExactSLEGenerator.Driver | radialTime a b.1 (z 0) (target 0/2)<∞}
-  let A := ∏i : Fin m, (target i.succ)^(-alpha a)
-  have hA : 0≤A := Finset.prod_nonneg (fun i _ => Real.rpow_nonneg (ht _).1.le _)
-  have hhit : W E≤ENNReal.ofReal (C*∏i∈I,(target i/cap i)^(alpha a)) :=
-    hCb W hW z hinj hz cap target hc hsep I
-      (fun i hi => ⟨(ht i).1,(Finset.mem_filter.mp hi).2⟩)
-  have hS := radialTime_stopping a (lt_trans (by norm_num) ha) (z 0) (hz 0)
-      (target 0/2) (by have := (ht 0).1; positivity) (by have := (ht 0).2; linarith)
-  have hE₀ : MeasurableSet E₀ := hS.measurableSpace_le _ (hS.measurable measurableSet_Iio)
-  have hpoint (b : ExactSLEGenerator.Driver) :
-      E₀.indicator (fun b => ∏i : Fin m,
-        ENNReal.ofReal (fullRadialDensity a b.1 (z i.succ) (target i.succ))) b ≤
-      E.indicator (fun _ => ENNReal.ofReal A) b := by
-    by_cases h0 : b∈E₀
-    · rw [Set.indicator_of_mem h0]
-      by_cases hb : b∈E
-      · rw [Set.indicator_of_mem hb,ENNReal.ofReal_prod_of_nonneg
-          (fun i _ => Real.rpow_nonneg (ht i.succ).1.le _)]
-        exact Finset.prod_le_prod₀ (fun _ _ => bot_le) (fun i _ => ENNReal.ofReal_le_ofReal
-          (fullRadialDensity_bound a ha (z i.succ) (hz _) (target i.succ)
-            (ht _).1 (ht _).2 b).2)
-      · rw [Set.indicator_of_notMem hb]
-        have he : ∃i∈I,radialTime a b.1 (z i) (target i/2)=∞ := by
-          by_contra hn
-          apply hb
-          intro i hi
-          exact lt_top_iff_ne_top.mpr (fun he => hn ⟨i,hi,he⟩)
-        obtain ⟨i,hi,he⟩ := he
-        have hi0 : i≠0 := by
-          intro h
-          subst i
-          exact (ne_of_lt (show radialTime a b.1 (z 0) (target 0/2)<∞ from h0)) he
-        obtain ⟨j,rfl⟩ := Fin.exists_succ_eq.mpr hi0
-        have hj : ENNReal.ofReal (fullRadialDensity a b.1 (z j.succ) (target j.succ))=0 := by
-          simp [fullRadialDensity,radialWeight,he]
-        exact le_of_eq (Finset.prod_eq_zero (Finset.mem_univ j) hj)
-    · rw [Set.indicator_of_notMem h0]
-      exact bot_le
-  have halg : A*(∏i∈I,(target i/cap i)^(alpha a)) ≤
-      (target 0)^(alpha a)*∏i,cap i^(-alpha a) := by
-    have he : A=(target 0)^(alpha a)*(∏i,target i^(-alpha a)) := by
-      rw [Fin.prod_univ_succ,←mul_assoc,←Real.rpow_add (ht 0).1]
-      simp only [add_neg_cancel,Real.rpow_zero,one_mul,A]
-    rw [he,mul_assoc]
-    exact mul_le_mul_of_nonneg_left
-      (target_cap_cancellation (alpha a) (alpha_pos ha).le cap target
-        (fun i => (hc i).1) (fun i => (ht i).1))
-      (Real.rpow_nonneg (ht 0).1.le _)
-  calc
-    _ = ∫⁻b,E₀.indicator (fun b => ∏i : Fin m,
-        ENNReal.ofReal (fullRadialDensity a b.1 (z i.succ) (target i.succ))) b∂W :=
-      (lintegral_indicator hE₀ _).symm
-    _ ≤ ∫⁻b,E.indicator (fun _ => ENNReal.ofReal A) b∂W := lintegral_mono hpoint
-    _ ≤ ENNReal.ofReal A*W E := lintegral_indicator_const_le E _
-    _ ≤ ENNReal.ofReal A*ENNReal.ofReal (C*∏i∈I,(target i/cap i)^(alpha a)) :=
-      mul_le_mul' le_rfl hhit
-    _ = ENNReal.ofReal (C*(A*∏i∈I,(target i/cap i)^(alpha a))) := by
-      rw [←ENNReal.ofReal_mul hA]
-      congr 1
-      ring
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (by nlinarith [mul_le_mul_of_nonneg_left halg hC.le])
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEKernel
-
- 
-
-theorem actual_hit_mass_moment_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
-    ∃ C : ℝ, 0<C ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-      ∀K : Set ℂ, MeasurableSet K → Bornology.IsBounded K →
-      ∀c : ℝ, 0<c → (∀x∈K,c≤2*x.im) →
-      ∀z : ℂ, c≤2*z.im →
-      ∀f : ℂ→ℝ, Measurable f → (∀x∈K,0≤f x) →
-      (∃B : ℝ,0≤B ∧ ∀x∈K,f x≤B) →
-      let A := ∫⁻x : Fin m→ℂ, (∏i,ENNReal.ofReal (f (x i)))*
-        kernel (alpha a) c (Fin.cons z x)∂Measure.pi (fun _ => volume.restrict K)
-      A<∞ ∧ ∀ρ : ℝ, 0<ρ → ρ≤z.im → ∀q : ℝ,0<q → q≤1 →
-        (∫⁻b in {b | radialTime a b.1 z ρ<∞},
-          (∫⁻x in K, ENNReal.ofReal (f x)*massDensity a q b x)^m∂W) ≤
-          ENNReal.ofReal (C*(2*ρ)^(alpha a))*A := by
-  obtain ⟨C,hC,hCb⟩ := actual_hit_density_product_bound a ha m
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW K hK hKb c hc hcK z hzc f hf hf0 hfB
-  obtain ⟨B,hB,hBf⟩ := hfB
-  let μ := volume.restrict K
-  let ν := Measure.pi (fun _ : Fin m => μ)
-  have hKae : ∀ᵐ x∂ν,∀i,x i∈K := by
-    apply ae_all_iff.mpr
-    intro i
-    exact (Measure.tendsto_eval_ae_ae (μ:=fun _ : Fin m=>μ) (i:=i)).eventually
-      (ae_restrict_mem hK)
-  have halpha : alpha a<1 := by
-    have ha0 : 0<a := by linarith
-    have hp : 0<1/(4*a) := by positivity
-    dsimp [alpha]; linarith
-  obtain ⟨D,hD,hDb⟩ := capped_distance_fixed_center (alpha a) c (alpha_pos ha)
-    halpha hc m K {z} hK hKb (isCompact_singleton.isBounded)
-  have hkfin : (∫⁻x : Fin m→ℂ,kernel (alpha a) c (Fin.cons z x)∂ν)<∞ :=
-    lt_of_le_of_lt (hDb z (mem_singleton z)) hD
-  have hkm : Measurable (fun x : Fin m→ℂ => kernel (alpha a) c (Fin.cons z x)) :=
-    (measurable_kernel (alpha_pos ha).le c).comp (by fun_prop)
-  have hAfin : (∫⁻x,(∏i,ENNReal.ofReal (f (x i)))*kernel (alpha a) c
-      (Fin.cons z x)∂ν)<∞ := by
-    calc
-      _ ≤ ∫⁻x,(ENNReal.ofReal B)^m*kernel (alpha a) c (Fin.cons z x)∂ν := by
-        apply lintegral_mono_ae
-        filter_upwards [hKae] with x hx
-        apply mul_le_mul' _ le_rfl
-        have hp := Finset.prod_le_prod₀ (fun _ _ => (bot_le : (0:ℝ≥0∞)≤_))
-          (fun i (_ : i∈Finset.univ) => ENNReal.ofReal_le_ofReal (hBf (x i) (hx i)))
-        simpa using hp
-      _ = (ENNReal.ofReal B)^m*∫⁻x,kernel (alpha a) c (Fin.cons z x)∂ν :=
-        lintegral_const_mul _ hkm
-      _ < ∞ := ENNReal.mul_lt_top (ENNReal.pow_lt_top ENNReal.ofReal_lt_top) hkfin
-  refine ⟨hAfin,?_⟩
-  intro ρ hρ hρz q hq hq1
-  have hz : 0<z.im := by linarith
-  let E := {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ<∞}
-  have hd := massDensity_measurable a ha q hq hq1
-  have hdz (b : ExactSLEGenerator.Driver) : Measurable (massDensity a q b) :=
-    hd.comp (measurable_const.prodMk measurable_id)
-  have hdm (x : Fin m→ℂ) : Measurable (fun b => ∏i,massDensity a q b (x i)) := by
-    apply Finset.measurable_prod
-    intro i hi
-    exact hd.comp (measurable_id.prodMk measurable_const)
-  have hprod (x : Fin m→ℂ) (hx : ∀i,x i∈K) :
-      (∫⁻b in E,∏i,massDensity a q b (x i)∂W) ≤
-        ENNReal.ofReal (C*(2*ρ)^(alpha a))*kernel (alpha a) c (Fin.cons z x) := by
-    let Z : Fin (m+1)→ℂ := Fin.cons z x
-    have hZ (i : Fin (m+1)) : 0<(Z i).im := by
-      refine Fin.cases hz (fun j => ?_) i
-      have := hcK (x j) (hx j)
-      simp only [Z, Fin.cons_succ]
-      linarith
-    have hcz (i : Fin (m+1)) : c≤2*(Z i).im := by
-      exact Fin.cases hzc (fun j => hcK (x j) (hx j)) i
-    by_cases hinj : Function.Injective Z
-    · let cap := fun i => cappedDistance c Z i
-      let target : Fin (m+1)→ℝ := Fin.cons (2*ρ) (fun i => 2*(x i).im*q)
-      have hcap (i) : 0<cap i := cappedDistance_pos hc Z hinj i
-      have ht (i) : 0<target i ∧ target i≤2*(Z i).im := by
-        refine Fin.cases ?_ (fun j => ?_) i
-        · simp only [target, Z, Fin.cons_zero]
-          constructor <;> linarith
-        · simp only [target, Z, Fin.cons_succ]
-          have hxi := hZ j.succ
-          simp only [Z,Fin.cons_succ] at hxi
-          constructor
-          · positivity
-          · nlinarith
-      have hh := hCb W hW Z hinj hZ cap target
-        (fun i => ⟨hcap i,(cappedDistance_le c Z i).trans (hcz i)⟩)
-        (fun i j hij => by simpa [dist_eq_norm] using
-          cappedDistance_le_dist c Z i j hij.symm) ht
-      have he (b : ExactSLEGenerator.Driver) (i) : massDensity a q b (x i)=
-          ENNReal.ofReal (fullRadialDensity a b.1 (Z i.succ) (target i.succ)) := by
-        have hxi : 0 < (x i).im := by simpa [Z] using hZ i.succ
-        simp [massDensity, hxi, Z, target]
-      simp_rw [he]
-      have hk : ENNReal.ofReal (C*(target 0)^(alpha a)*∏i,cap i^(-alpha a))=
-          ENNReal.ofReal (C*(2*ρ)^(alpha a))*kernel (alpha a) c (Fin.cons z x) := by
-        rw [ENNReal.ofReal_mul (mul_nonneg hC.le (Real.rpow_nonneg (ht 0).1.le _)),
-          ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (hcap i).le _)]
-        congr 1
-        unfold kernel
-        apply Finset.prod_congr rfl
-        intro i hi
-        exact (ENNReal.ofReal_rpow_of_pos (hcap i)).symm
-      simpa only [Z,target,Fin.cons_zero,mul_div_cancel_left₀ _ (two_ne_zero : (2:ℝ)≠0)]
-        using hh.trans_eq hk
-    · rw [kernel_top_of_not_injective (alpha a) c (alpha_pos ha) hc Z hinj,
-        ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr
-          (mul_pos hC (Real.rpow_pos_of_pos (by positivity) _))))]
-      exact le_top
-  have hjoint : Measurable (fun p : ExactSLEGenerator.Driver × (Fin m→ℂ) =>
-      ∏i,ENNReal.ofReal (f (p.2 i))*massDensity a q p.1 (p.2 i)) := by
-    apply Finset.measurable_prod
-    intro i hi
-    exact ((hf.comp ((measurable_pi_apply i).comp measurable_snd)).ennreal_ofReal).mul
-      (hd.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd)))
-  calc
-    _ = ∫⁻b in E,∫⁻x : Fin m→ℂ,∏i,ENNReal.ofReal (f (x i))*massDensity a q b (x i)∂ν∂W := by
-      apply lintegral_congr
-      intro b
-      exact lintegral_power_pi μ _ (hf.ennreal_ofReal.mul (hdz b)) m
-    _ = ∫⁻x : Fin m→ℂ,∫⁻b in E,∏i,ENNReal.ofReal (f (x i))*massDensity a q b (x i)∂W∂ν :=
-      lintegral_lintegral_swap hjoint.aemeasurable
-    _ ≤ ∫⁻x,ENNReal.ofReal (C*(2*ρ)^(alpha a))*
-        ((∏i,ENNReal.ofReal (f (x i)))*kernel (alpha a) c (Fin.cons z x))∂ν := by
-      apply lintegral_mono_ae
-      filter_upwards [hKae] with x hx
-      simp_rw [Finset.prod_mul_distrib]
-      rw [lintegral_const_mul _ (hdm x)]
-      calc
-        _ ≤ (∏i,ENNReal.ofReal (f (x i)))*
-            (ENNReal.ofReal (C*(2*ρ)^(alpha a))*kernel (alpha a) c (Fin.cons z x)) :=
-          mul_le_mul' le_rfl (hprod x hx)
-        _ = _ := by ring
-    _ = _ := by
-      have hfm : Measurable (fun x : Fin m→ℂ => ∏i,ENNReal.ofReal (f (x i))) := by fun_prop
-      exact lintegral_const_mul _ (hfm.mul hkm)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEKernel
-
-lemma convex_limit_moment_bound {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (m : ℕ) (hm : 1 ≤ m) (A : ℝ≥0∞)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hsum : ∀k,∑j∈s k,c k j=1)
-    (F : ℕ→Ω→ℝ) (hF : ∀j,AEStronglyMeasurable (F j) P)
-    (hbound : ∀j,(∫⁻b,‖F j b‖ₑ^m∂P)≤A)
-    (G : Ω→ℝ)
-    (hlim : ∀ᵐ b∂P,Tendsto (fun k=>∑j∈s k,c k j*F j b) atTop (𝓝 (G b))) :
-    (∫⁻b,‖G b‖ₑ^m∂P)≤A := by
-  have hm0 : (m:ℝ)≠0 := by exact_mod_cast Nat.ne_zero_of_lt hm
-  let D := A^(1/(m:ℝ))
-  have hn (j) : eLpNorm (F j) (m:ℝ≥0∞) P≤D := by
-    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal
-      (by exact_mod_cast Nat.ne_zero_of_lt hm) (by simp) (hF j),
-      ENNReal.toReal_natCast]
-    simpa only [ENNReal.rpow_natCast] using
-      ENNReal.rpow_le_rpow (hbound j) (show (0:ℝ)≤1/(m:ℝ) by positivity)
-  have hFn (k) : AEStronglyMeasurable (fun b=>∑j∈s k,c k j*F j b) P := by
-    have h := Finset.aestronglyMeasurable_sum (μ:=P) (s k)
-      (fun j hj => (hF j).const_mul (c k j))
-    convert h using 1
-    funext b
-    simp only [Finset.sum_apply]
-  have hnb (k) : eLpNorm (fun b=>∑j∈s k,c k j*F j b) (m:ℝ≥0∞) P≤D :=
-    convex_eLpNorm_bound P _ D (by exact_mod_cast hm) (s k) (c k) (hc k) (hsum k)
-      F (fun j _ => hF j) (fun j _ => hn j)
-  have hGm := aestronglyMeasurable_of_tendsto_ae atTop hFn hlim
-  have hg : eLpNorm G (m:ℝ≥0∞) P≤D :=
-    Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hnb) hFn hGm hlim
-  have hid : (eLpNorm G (m:ℝ≥0∞) P)^m=∫⁻b,‖G b‖ₑ^m∂P := by
-    simpa using (eLpNorm_nnreal_pow_eq_lintegral (f:=G) (μ:=P) (p:=(m:ℝ≥0))
-      (by exact_mod_cast Nat.ne_zero_of_lt hm) hGm)
-  rw [←hid]
-  calc
-    _ ≤ D^m := pow_le_pow_left' hg m
-    _ = A := by
-      dsimp [D]
-      rw [←ENNReal.rpow_natCast,←ENNReal.rpow_mul,one_div_mul_cancel hm0,ENNReal.rpow_one]
-
- 
-
-theorem actual_hit_radon_moment_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) (hm : 1 ≤ m) :
-    ∃C : ℝ,0<C ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀f : C_c(UpperPlane,ℝ), ∀c : ℝ,0<c →
-      (∀x∈tsupport f,c≤2*x.1.im) → ∀z : ℂ,c≤2*z.im →
-      let A := ∫⁻x : Fin m→ℂ,(∏i,ENNReal.ofReal ‖upperExtension f (x i)‖)*
-        kernel (alpha a) c (Fin.cons z x)∂Measure.pi
-          (fun _ => volume.restrict (Subtype.val '' tsupport f))
-      A<∞ ∧ ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀(t : ℝ≥0∞) (μ : ExactSLEGenerator.Driver→Measure UpperPlane),
-      (∀ᵐ b∂W,Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫x,f x∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
-          atTop (𝓝 (∫x,f x∂μ b))) →
-      ∀ρ : ℝ,0<ρ → ρ≤z.im →
-        (∫⁻b in {b | radialTime a b.1 z ρ<∞},‖∫x,f x∂μ b‖ₑ^m∂W)≤
-          ENNReal.ofReal (C*(2*ρ)^(alpha a))*A := by
-  obtain ⟨C,hC,hCb⟩ := actual_hit_mass_moment_bound a ha m
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW f c hc hcf z hzc
-  let K := Subtype.val '' tsupport f
-  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
-  have hcK (x : ℂ) (hx : x∈K) : c≤2*x.im := by
-    obtain ⟨y,hy,rfl⟩ := hx
-    exact hcf y hy
-  obtain ⟨hAfin,hB⟩ := hCb W hW K hK.measurableSet hK.isBounded c hc hcK z hzc
-    (fun x=>‖upperExtension f x‖) (upperExtension_measurable f).norm
-    (fun _ _ => norm_nonneg _) (upperExtension_bounded f)
-  refine ⟨hAfin,?_⟩
-  intro s coeff hcoeff hsum t μ hlim ρ hρ hρz
-  let E := {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ<∞}
-  let q (j : ℕ) := Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  apply convex_limit_moment_bound (W.restrict E) m hm _ s coeff hcoeff hsum
-    (fun j b=>∫x,f x∂actualUpperMass a (q j) t b)
-    (fun j=>(actualUpperMass_eval_measurable a ha (q j) (hq j) (hq1 j) t f).aestronglyMeasurable.restrict)
-    _ _ (ae_restrict_of_ae hlim)
-  intro j
-  calc
-    _ ≤ ∫⁻b in E,(∫⁻x in K,ENNReal.ofReal ‖upperExtension f x‖*massDensity a (q j) b x)^m∂W := by
-      apply lintegral_mono
-      intro b
-      dsimp only
-      rw [actualUpperMass_eval a ha (q j) (hq j) (hq1 j)]
-      exact pow_le_pow_left' (actualMassEval_enorm_bound a (q j) t K (upperExtension f) b) m
-    _ ≤ _ := hB ρ hρ hρz (q j) (hq j) (hq1 j)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.KernelSimilarity
-open ExactSLEKernel
-
- 
-def affine (x : ℂ) (y : ℝ) (hy : 0<y) : ℂ ≃ₜ ℂ :=
-  (Homeomorph.mulLeft₀ (y:ℂ) (Complex.ofReal_ne_zero.mpr hy.ne')).trans
-    (Homeomorph.addLeft x)
-
-@[simp] lemma affine_apply (x : ℂ) {y : ℝ} (hy : 0<y) (z : ℂ) :
-    affine x y hy z=x+(y:ℂ)*z := rfl
-
-lemma dist_affine (x : ℂ) {y : ℝ} (hy : 0<y) (u v : ℂ) :
-    dist (affine x y hy u) (affine x y hy v)=y*dist u v := by
-  simp only [affine_apply,dist_eq_norm,add_sub_add_left_eq_sub,←mul_sub,norm_mul,
-    Complex.norm_real,Real.norm_eq_abs,abs_of_pos hy]
-
-lemma cappedDistance_affine {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (x : ℂ) {y : ℝ} (hy : 0<y) (c : ℝ) (z : ι→ℂ) (i : ι) :
-    cappedDistance (y*c) (affine x y hy ∘ z) i=y*cappedDistance c z i := by
-  apply le_antisymm
-  · rw [mul_comm y (cappedDistance c z i)]
-    apply (div_le_iff₀ hy).mp
-    rw [le_cappedDistance_iff]
-    refine ⟨(div_le_iff₀ hy).mpr ?_,fun j hji=>(div_le_iff₀ hy).mpr ?_⟩
-    · simpa [mul_comm] using cappedDistance_le (y*c) (affine x y hy ∘ z) i
-    · simpa only [Function.comp_apply,dist_affine,mul_comm] using
-        cappedDistance_le_dist (y*c) (affine x y hy ∘ z) i j hji
-  · rw [le_cappedDistance_iff]
-    refine ⟨mul_le_mul_of_nonneg_left (cappedDistance_le c z i) hy.le,?_⟩
-    intro j hji
-    simp only [Function.comp_apply,dist_affine]
-    exact mul_le_mul_of_nonneg_left (cappedDistance_le_dist c z i j hji) hy.le
-
-lemma kernel_affine {ι : Type*} [Fintype ι] [DecidableEq ι]
-    (α c : ℝ) (x : ℂ) {y : ℝ} (hy : 0<y) (z : ι→ℂ) :
-    kernel α (y*c) (affine x y hy ∘ z)=
-      (ENNReal.ofReal y)^(-α*(Fintype.card ι:ℝ))*kernel α c z := by
-  unfold kernel
-  simp_rw [cappedDistance_affine x hy c z,ENNReal.ofReal_mul hy.le,
-    ENNReal.mul_rpow_of_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top,
-    Finset.prod_mul_distrib,Finset.prod_const,Finset.card_univ]
-  rw [←ENNReal.rpow_natCast,←ENNReal.rpow_mul]
-
-lemma pi_const_smul {X : Type*} [MeasurableSpace X] (μ : Measure X) [SigmaFinite μ]
-    (r : ℝ≥0) (m : ℕ) :
-    Measure.pi (fun _ : Fin m=>r • μ)=(r:ℝ≥0∞)^m • Measure.pi (fun _ : Fin m=>μ) := by
-  apply Measure.pi_eq
-  intro s hs
-  simp only [Measure.smul_apply,smul_eq_mul,Measure.pi_pi,Finset.prod_mul_distrib,
-    Finset.prod_const,Finset.card_univ,Fintype.card_fin,ENNReal.smul_def]
-
-lemma affine_area (x : ℂ) {y : ℝ} (hy : 0<y) {K : Set ℂ} (hK : MeasurableSet K) :
-    volume.restrict (affine x y hy '' K)=
-      (NNReal.mk (y^2) (sq_nonneg y)) • (volume.restrict K).map (affine x y hy) := by
-  let T := affine x y hy
-  have hder (z : ℂ) : HasDerivAt T (y:ℂ) z := by
-    change HasDerivAt (fun w : ℂ=>x+(y:ℂ)*w) (y:ℂ) z
-    simpa only [mul_one,id_eq] using ((hasDerivAt_id z).const_mul (y:ℂ)).const_add x
-  ext A hA
-  have h := Conformal.lintegral_image_complex hK
-    (fun z _ => (hder z).differentiableAt) T.injective.injOn (A.indicator (fun _=>1))
-  change (∫⁻z in T '' K,A.indicator (fun _=>(1:ℝ≥0∞)) z)=_ at h
-  simp_rw [fun z=> (hder z).deriv,Complex.norm_real,Real.norm_eq_abs,sq_abs] at h
-  have hAcomp : Measurable (fun z=>A.indicator (fun _=>(1:ℝ≥0∞)) (T z)) :=
-    (measurable_const.indicator hA).comp T.measurable
-  rw [lintegral_const_mul _ hAcomp] at h
-  rw [←lintegral_map (measurable_const.indicator hA) T.measurable] at h
-  rw [lintegral_indicator hA,lintegral_indicator hA,setLIntegral_one,setLIntegral_one] at h
-  rw [Measure.smul_apply,ENNReal.smul_def]
-  exact h.trans (by congr 1; simp only [ENNReal.ofReal,Real.toNNReal_of_nonneg (sq_nonneg y)])
-
-lemma affine_pi_integral (x : ℂ) {y : ℝ} (hy : 0<y) {K : Set ℂ} (hK : MeasurableSet K)
-    (m : ℕ) (F : (Fin m→ℂ)→ℝ≥0∞) (hF : Measurable F) :
-    (∫⁻z,F z∂Measure.pi (fun _=>volume.restrict (affine x y hy '' K)))=
-      (ENNReal.ofReal y)^(2*m)*
-        ∫⁻z,F (affine x y hy ∘ z)∂Measure.pi (fun _=>volume.restrict K) := by
-  let T := affine x y hy
-  have : SigmaFinite ((volume.restrict K).map T) := T.toMeasurableEquiv.sigmaFinite_map
-  rw [affine_area x hy hK,pi_const_smul,lintegral_smul_measure,
-    ←Measure.pi_map_pi (fun _ : Fin m=>T.measurable.aemeasurable)]
-  rw [lintegral_map hF (by fun_prop)]
-  change ((NNReal.mk (y^2) (sq_nonneg y)):ℝ≥0∞)^m*_= _
-  rw [show ((NNReal.mk (y^2) (sq_nonneg y)):ℝ≥0∞)=(ENNReal.ofReal y)^2 by
-    rw [←ENNReal.ofReal_pow hy.le]; simp only [ENNReal.ofReal,Real.toNNReal_of_nonneg (sq_nonneg y)],
-    ←pow_mul]
-  rfl
-
- 
-
-theorem fixed_center_integral_affine (α c : ℝ) (hα : 0≤α)
-    (x z₀ : ℂ) {y : ℝ} (hy : 0<y) {K : Set ℂ} (hK : MeasurableSet K) (m : ℕ) :
-    (∫⁻z : Fin m→ℂ,kernel α (y*c) (Fin.cons (affine x y hy z₀) z)
-      ∂Measure.pi (fun _=>volume.restrict (affine x y hy '' K)))=
-      (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1))*
-        ∫⁻z : Fin m→ℂ,kernel α c (Fin.cons z₀ z)
-          ∂Measure.pi (fun _=>volume.restrict K) := by
-  have hm : Measurable (fun z : Fin m→ℂ=>kernel α (y*c) (Fin.cons (affine x y hy z₀) z)) :=
-    (measurable_kernel hα _).comp (by fun_prop)
-  rw [affine_pi_integral x hy hK m _ hm]
-  have he (z : Fin m→ℂ) : Fin.cons (affine x y hy z₀) (affine x y hy ∘ z)=
-      affine x y hy ∘ Fin.cons z₀ z := by
-    funext i
-    refine Fin.cases ?_ (fun j=>?_) i <;> simp
-  simp_rw [he,kernel_affine,Fintype.card_fin]
-  have hf : Measurable (fun z : Fin m→ℂ=>kernel α c (Fin.cons z₀ z)) :=
-    (measurable_kernel hα c).comp (by fun_prop)
-  rw [lintegral_const_mul _ hf,←mul_assoc,
-    ←ENNReal.rpow_natCast,←ENNReal.rpow_add _ _ (ne_of_gt (ENNReal.ofReal_pos.mpr hy)) ENNReal.ofReal_ne_top]
-  congr 2
-  push_cast
-  ring
-
-end SLEExactGauge.KernelSimilarity
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ConformalRestartMap
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEKernel
+
+lemma target_cap_cancellation {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α : ℝ) (hα : 0≤α) (cap target : ι→ℝ)
+    (hc : ∀i,0<cap i) (ht : ∀i,0<target i) :
+    (∏i,target i^(-α))*(∏i∈Finset.univ.filter (fun i => target i≤cap i),
+      (target i/cap i)^α) ≤ ∏i,cap i^(-α) := by
+  rw [Finset.prod_filter,←Finset.prod_mul_distrib]
+  apply Finset.prod_le_prod₀
+  · intro i _
+    apply mul_nonneg (Real.rpow_nonneg (ht i).le _)
+    split_ifs
+    · exact Real.rpow_nonneg (div_nonneg (ht i).le (hc i).le) _
+    · norm_num
+  · intro i _
+    by_cases hi : target i≤cap i
+    · rw [ite_eq_left hi,Real.div_rpow (ht i).le (hc i).le,
+        Real.rpow_neg (ht i).le,Real.rpow_neg (hc i).le]
+      have hp := Real.rpow_pos_of_pos (ht i) α
+      field_simp
+      rfl
+    · rw [ite_eq_right hi,mul_one]
+      exact Real.rpow_le_rpow_of_nonpos (hc i) (le_of_not_ge hi) (neg_nonpos.mpr hα)
+
+ 
+
+theorem actual_hit_density_product_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
+    ∃C : ℝ, 0<C ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+      ∀z : Fin (m+1)→ℂ, Function.Injective z → (∀i,0<(z i).im) →
+      ∀cap target : Fin (m+1)→ℝ,
+        (∀i,0<cap i ∧ cap i≤2*(z i).im) →
+        (∀i j,i≠j → cap i≤‖z i-z j‖) →
+        (∀i,0<target i ∧ target i≤2*(z i).im) →
+        (∫⁻b in {b | radialTime a b.1 (z 0) (target 0/2)<∞},
+          ∏i : Fin m, ENNReal.ofReal (fullRadialDensity a b.1 (z i.succ) (target i.succ))∂W) ≤
+          ENNReal.ofReal (C*(target 0)^(alpha a)*(∏i,cap i^(-alpha a))) := by
+  obtain ⟨C,hC,hCb⟩ := selective_hit_bound a ha (m+1)
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW z hinj hz cap target hc hsep ht
+  let I := Finset.univ.filter (fun i => target i≤cap i)
+  let E := {b : ExactSLEGenerator.Driver | ∀i∈I,radialTime a b.1 (z i) (target i/2)<∞}
+  let E₀ := {b : ExactSLEGenerator.Driver | radialTime a b.1 (z 0) (target 0/2)<∞}
+  let A := ∏i : Fin m, (target i.succ)^(-alpha a)
+  have hA : 0≤A := Finset.prod_nonneg (fun i _ => Real.rpow_nonneg (ht _).1.le _)
+  have hhit : W E≤ENNReal.ofReal (C*∏i∈I,(target i/cap i)^(alpha a)) :=
+    hCb W hW z hinj hz cap target hc hsep I
+      (fun i hi => ⟨(ht i).1,(Finset.mem_filter.mp hi).2⟩)
+  have hS := radialTime_stopping a (lt_trans (by norm_num) ha) (z 0) (hz 0)
+      (target 0/2) (by have := (ht 0).1; positivity) (by have := (ht 0).2; linarith)
+  have hE₀ : MeasurableSet E₀ := hS.measurableSpace_le _ (hS.measurable measurableSet_Iio)
+  have hpoint (b : ExactSLEGenerator.Driver) :
+      E₀.indicator (fun b => ∏i : Fin m,
+        ENNReal.ofReal (fullRadialDensity a b.1 (z i.succ) (target i.succ))) b ≤
+      E.indicator (fun _ => ENNReal.ofReal A) b := by
+    by_cases h0 : b∈E₀
+    · rw [Set.indicator_of_mem h0]
+      by_cases hb : b∈E
+      · rw [Set.indicator_of_mem hb,ENNReal.ofReal_prod_of_nonneg
+          (fun i _ => Real.rpow_nonneg (ht i.succ).1.le _)]
+        exact Finset.prod_le_prod₀ (fun _ _ => bot_le) (fun i _ => ENNReal.ofReal_le_ofReal
+          (fullRadialDensity_bound a ha (z i.succ) (hz _) (target i.succ)
+            (ht _).1 (ht _).2 b).2)
+      · rw [Set.indicator_of_notMem hb]
+        have he : ∃i∈I,radialTime a b.1 (z i) (target i/2)=∞ := by
+          by_contra hn
+          apply hb
+          intro i hi
+          exact lt_top_iff_ne_top.mpr (fun he => hn ⟨i,hi,he⟩)
+        obtain ⟨i,hi,he⟩ := he
+        have hi0 : i≠0 := by
+          intro h
+          subst i
+          exact (ne_of_lt (show radialTime a b.1 (z 0) (target 0/2)<∞ from h0)) he
+        obtain ⟨j,rfl⟩ := Fin.exists_succ_eq.mpr hi0
+        have hj : ENNReal.ofReal (fullRadialDensity a b.1 (z j.succ) (target j.succ))=0 := by
+          simp [fullRadialDensity,radialWeight,he]
+        exact le_of_eq (Finset.prod_eq_zero (Finset.mem_univ j) hj)
+    · rw [Set.indicator_of_notMem h0]
+      exact bot_le
+  have halg : A*(∏i∈I,(target i/cap i)^(alpha a)) ≤
+      (target 0)^(alpha a)*∏i,cap i^(-alpha a) := by
+    have he : A=(target 0)^(alpha a)*(∏i,target i^(-alpha a)) := by
+      rw [Fin.prod_univ_succ,←mul_assoc,←Real.rpow_add (ht 0).1]
+      simp only [add_neg_cancel,Real.rpow_zero,one_mul,A]
+    rw [he,mul_assoc]
+    exact mul_le_mul_of_nonneg_left
+      (target_cap_cancellation (alpha a) (alpha_pos ha).le cap target
+        (fun i => (hc i).1) (fun i => (ht i).1))
+      (Real.rpow_nonneg (ht 0).1.le _)
+  calc
+    _ = ∫⁻b,E₀.indicator (fun b => ∏i : Fin m,
+        ENNReal.ofReal (fullRadialDensity a b.1 (z i.succ) (target i.succ))) b∂W :=
+      (lintegral_indicator hE₀ _).symm
+    _ ≤ ∫⁻b,E.indicator (fun _ => ENNReal.ofReal A) b∂W := lintegral_mono hpoint
+    _ ≤ ENNReal.ofReal A*W E := lintegral_indicator_const_le E _
+    _ ≤ ENNReal.ofReal A*ENNReal.ofReal (C*∏i∈I,(target i/cap i)^(alpha a)) :=
+      mul_le_mul' le_rfl hhit
+    _ = ENNReal.ofReal (C*(A*∏i∈I,(target i/cap i)^(alpha a))) := by
+      rw [←ENNReal.ofReal_mul hA]
+      congr 1
+      ring
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (by nlinarith [mul_le_mul_of_nonneg_left halg hC.le])
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEKernel
+
+ 
+
+theorem actual_hit_mass_moment_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
+    ∃ C : ℝ, 0<C ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+      ∀K : Set ℂ, MeasurableSet K → Bornology.IsBounded K →
+      ∀c : ℝ, 0<c → (∀x∈K,c≤2*x.im) →
+      ∀z : ℂ, c≤2*z.im →
+      ∀f : ℂ→ℝ, Measurable f → (∀x∈K,0≤f x) →
+      (∃B : ℝ,0≤B ∧ ∀x∈K,f x≤B) →
+      let A := ∫⁻x : Fin m→ℂ, (∏i,ENNReal.ofReal (f (x i)))*
+        kernel (alpha a) c (Fin.cons z x)∂Measure.pi (fun _ => volume.restrict K)
+      A<∞ ∧ ∀ρ : ℝ, 0<ρ → ρ≤z.im → ∀q : ℝ,0<q → q≤1 →
+        (∫⁻b in {b | radialTime a b.1 z ρ<∞},
+          (∫⁻x in K, ENNReal.ofReal (f x)*massDensity a q b x)^m∂W) ≤
+          ENNReal.ofReal (C*(2*ρ)^(alpha a))*A := by
+  obtain ⟨C,hC,hCb⟩ := actual_hit_density_product_bound a ha m
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW K hK hKb c hc hcK z hzc f hf hf0 hfB
+  obtain ⟨B,hB,hBf⟩ := hfB
+  let μ := volume.restrict K
+  let ν := Measure.pi (fun _ : Fin m => μ)
+  have hKae : ∀ᵐ x∂ν,∀i,x i∈K := by
+    apply ae_all_iff.mpr
+    intro i
+    exact (Measure.tendsto_eval_ae_ae (μ:=fun _ : Fin m=>μ) (i:=i)).eventually
+      (ae_restrict_mem hK)
+  have halpha : alpha a<1 := by
+    have ha0 : 0<a := by linarith
+    have hp : 0<1/(4*a) := by positivity
+    dsimp [alpha]; linarith
+  obtain ⟨D,hD,hDb⟩ := capped_distance_fixed_center (alpha a) c (alpha_pos ha)
+    halpha hc m K {z} hK hKb (isCompact_singleton.isBounded)
+  have hkfin : (∫⁻x : Fin m→ℂ,kernel (alpha a) c (Fin.cons z x)∂ν)<∞ :=
+    lt_of_le_of_lt (hDb z (mem_singleton z)) hD
+  have hkm : Measurable (fun x : Fin m→ℂ => kernel (alpha a) c (Fin.cons z x)) :=
+    (measurable_kernel (alpha_pos ha).le c).comp (by fun_prop)
+  have hAfin : (∫⁻x,(∏i,ENNReal.ofReal (f (x i)))*kernel (alpha a) c
+      (Fin.cons z x)∂ν)<∞ := by
+    calc
+      _ ≤ ∫⁻x,(ENNReal.ofReal B)^m*kernel (alpha a) c (Fin.cons z x)∂ν := by
+        apply lintegral_mono_ae
+        filter_upwards [hKae] with x hx
+        apply mul_le_mul' _ le_rfl
+        have hp := Finset.prod_le_prod₀ (fun _ _ => (bot_le : (0:ℝ≥0∞)≤_))
+          (fun i (_ : i∈Finset.univ) => ENNReal.ofReal_le_ofReal (hBf (x i) (hx i)))
+        simpa using hp
+      _ = (ENNReal.ofReal B)^m*∫⁻x,kernel (alpha a) c (Fin.cons z x)∂ν :=
+        lintegral_const_mul _ hkm
+      _ < ∞ := ENNReal.mul_lt_top (ENNReal.pow_lt_top ENNReal.ofReal_lt_top) hkfin
+  refine ⟨hAfin,?_⟩
+  intro ρ hρ hρz q hq hq1
+  have hz : 0<z.im := by linarith
+  let E := {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ<∞}
+  have hd := massDensity_measurable a ha q hq hq1
+  have hdz (b : ExactSLEGenerator.Driver) : Measurable (massDensity a q b) :=
+    hd.comp (measurable_const.prodMk measurable_id)
+  have hdm (x : Fin m→ℂ) : Measurable (fun b => ∏i,massDensity a q b (x i)) := by
+    apply Finset.measurable_prod
+    intro i hi
+    exact hd.comp (measurable_id.prodMk measurable_const)
+  have hprod (x : Fin m→ℂ) (hx : ∀i,x i∈K) :
+      (∫⁻b in E,∏i,massDensity a q b (x i)∂W) ≤
+        ENNReal.ofReal (C*(2*ρ)^(alpha a))*kernel (alpha a) c (Fin.cons z x) := by
+    let Z : Fin (m+1)→ℂ := Fin.cons z x
+    have hZ (i : Fin (m+1)) : 0<(Z i).im := by
+      refine Fin.cases hz (fun j => ?_) i
+      have := hcK (x j) (hx j)
+      simp only [Z, Fin.cons_succ]
+      linarith
+    have hcz (i : Fin (m+1)) : c≤2*(Z i).im := by
+      exact Fin.cases hzc (fun j => hcK (x j) (hx j)) i
+    by_cases hinj : Function.Injective Z
+    · let cap := fun i => cappedDistance c Z i
+      let target : Fin (m+1)→ℝ := Fin.cons (2*ρ) (fun i => 2*(x i).im*q)
+      have hcap (i) : 0<cap i := cappedDistance_pos hc Z hinj i
+      have ht (i) : 0<target i ∧ target i≤2*(Z i).im := by
+        refine Fin.cases ?_ (fun j => ?_) i
+        · simp only [target, Z, Fin.cons_zero]
+          constructor <;> linarith
+        · simp only [target, Z, Fin.cons_succ]
+          have hxi := hZ j.succ
+          simp only [Z,Fin.cons_succ] at hxi
+          constructor
+          · positivity
+          · nlinarith
+      have hh := hCb W hW Z hinj hZ cap target
+        (fun i => ⟨hcap i,(cappedDistance_le c Z i).trans (hcz i)⟩)
+        (fun i j hij => by simpa [dist_eq_norm] using
+          cappedDistance_le_dist c Z i j hij.symm) ht
+      have he (b : ExactSLEGenerator.Driver) (i) : massDensity a q b (x i)=
+          ENNReal.ofReal (fullRadialDensity a b.1 (Z i.succ) (target i.succ)) := by
+        have hxi : 0 < (x i).im := by simpa [Z] using hZ i.succ
+        simp [massDensity, hxi, Z, target]
+      simp_rw [he]
+      have hk : ENNReal.ofReal (C*(target 0)^(alpha a)*∏i,cap i^(-alpha a))=
+          ENNReal.ofReal (C*(2*ρ)^(alpha a))*kernel (alpha a) c (Fin.cons z x) := by
+        rw [ENNReal.ofReal_mul (mul_nonneg hC.le (Real.rpow_nonneg (ht 0).1.le _)),
+          ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (hcap i).le _)]
+        congr 1
+        unfold kernel
+        apply Finset.prod_congr rfl
+        intro i hi
+        exact (ENNReal.ofReal_rpow_of_pos (hcap i)).symm
+      simpa only [Z,target,Fin.cons_zero,mul_div_cancel_left₀ _ (two_ne_zero : (2:ℝ)≠0)]
+        using hh.trans_eq hk
+    · rw [kernel_top_of_not_injective (alpha a) c (alpha_pos ha) hc Z hinj,
+        ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr
+          (mul_pos hC (Real.rpow_pos_of_pos (by positivity) _))))]
+      exact le_top
+  have hjoint : Measurable (fun p : ExactSLEGenerator.Driver × (Fin m→ℂ) =>
+      ∏i,ENNReal.ofReal (f (p.2 i))*massDensity a q p.1 (p.2 i)) := by
+    apply Finset.measurable_prod
+    intro i hi
+    exact ((hf.comp ((measurable_pi_apply i).comp measurable_snd)).ennreal_ofReal).mul
+      (hd.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd)))
+  calc
+    _ = ∫⁻b in E,∫⁻x : Fin m→ℂ,∏i,ENNReal.ofReal (f (x i))*massDensity a q b (x i)∂ν∂W := by
+      apply lintegral_congr
+      intro b
+      exact lintegral_power_pi μ _ (hf.ennreal_ofReal.mul (hdz b)) m
+    _ = ∫⁻x : Fin m→ℂ,∫⁻b in E,∏i,ENNReal.ofReal (f (x i))*massDensity a q b (x i)∂W∂ν :=
+      lintegral_lintegral_swap hjoint.aemeasurable
+    _ ≤ ∫⁻x,ENNReal.ofReal (C*(2*ρ)^(alpha a))*
+        ((∏i,ENNReal.ofReal (f (x i)))*kernel (alpha a) c (Fin.cons z x))∂ν := by
+      apply lintegral_mono_ae
+      filter_upwards [hKae] with x hx
+      simp_rw [Finset.prod_mul_distrib]
+      rw [lintegral_const_mul _ (hdm x)]
+      calc
+        _ ≤ (∏i,ENNReal.ofReal (f (x i)))*
+            (ENNReal.ofReal (C*(2*ρ)^(alpha a))*kernel (alpha a) c (Fin.cons z x)) :=
+          mul_le_mul' le_rfl (hprod x hx)
+        _ = _ := by ring
+    _ = _ := by
+      have hfm : Measurable (fun x : Fin m→ℂ => ∏i,ENNReal.ofReal (f (x i))) := by fun_prop
+      exact lintegral_const_mul _ (hfm.mul hkm)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEKernel
+
+lemma convex_limit_moment_bound {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (m : ℕ) (hm : 1 ≤ m) (A : ℝ≥0∞)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hsum : ∀k,∑j∈s k,c k j=1)
+    (F : ℕ→Ω→ℝ) (hF : ∀j,AEStronglyMeasurable (F j) P)
+    (hbound : ∀j,(∫⁻b,‖F j b‖ₑ^m∂P)≤A)
+    (G : Ω→ℝ)
+    (hlim : ∀ᵐ b∂P,Tendsto (fun k=>∑j∈s k,c k j*F j b) atTop (𝓝 (G b))) :
+    (∫⁻b,‖G b‖ₑ^m∂P)≤A := by
+  have hm0 : (m:ℝ)≠0 := by exact_mod_cast Nat.ne_zero_of_lt hm
+  let D := A^(1/(m:ℝ))
+  have hn (j) : eLpNorm (F j) (m:ℝ≥0∞) P≤D := by
+    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal
+      (by exact_mod_cast Nat.ne_zero_of_lt hm) (by simp) (hF j),
+      ENNReal.toReal_natCast]
+    simpa only [ENNReal.rpow_natCast] using
+      ENNReal.rpow_le_rpow (hbound j) (show (0:ℝ)≤1/(m:ℝ) by positivity)
+  have hFn (k) : AEStronglyMeasurable (fun b=>∑j∈s k,c k j*F j b) P := by
+    have h := Finset.aestronglyMeasurable_sum (μ:=P) (s k)
+      (fun j hj => (hF j).const_mul (c k j))
+    convert h using 1
+    funext b
+    simp only [Finset.sum_apply]
+  have hnb (k) : eLpNorm (fun b=>∑j∈s k,c k j*F j b) (m:ℝ≥0∞) P≤D :=
+    convex_eLpNorm_bound P _ D (by exact_mod_cast hm) (s k) (c k) (hc k) (hsum k)
+      F (fun j _ => hF j) (fun j _ => hn j)
+  have hGm := aestronglyMeasurable_of_tendsto_ae atTop hFn hlim
+  have hg : eLpNorm G (m:ℝ≥0∞) P≤D :=
+    Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hnb) hFn hGm hlim
+  have hid : (eLpNorm G (m:ℝ≥0∞) P)^m=∫⁻b,‖G b‖ₑ^m∂P := by
+    simpa using (eLpNorm_nnreal_pow_eq_lintegral (f:=G) (μ:=P) (p:=(m:ℝ≥0))
+      (by exact_mod_cast Nat.ne_zero_of_lt hm) hGm)
+  rw [←hid]
+  calc
+    _ ≤ D^m := pow_le_pow_left' hg m
+    _ = A := by
+      dsimp [D]
+      rw [←ENNReal.rpow_natCast,←ENNReal.rpow_mul,one_div_mul_cancel hm0,ENNReal.rpow_one]
+
+ 
+
+theorem actual_hit_radon_moment_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) (hm : 1 ≤ m) :
+    ∃C : ℝ,0<C ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀f : C_c(UpperPlane,ℝ), ∀c : ℝ,0<c →
+      (∀x∈tsupport f,c≤2*x.1.im) → ∀z : ℂ,c≤2*z.im →
+      let A := ∫⁻x : Fin m→ℂ,(∏i,ENNReal.ofReal ‖upperExtension f (x i)‖)*
+        kernel (alpha a) c (Fin.cons z x)∂Measure.pi
+          (fun _ => volume.restrict (Subtype.val '' tsupport f))
+      A<∞ ∧ ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀(t : ℝ≥0∞) (μ : ExactSLEGenerator.Driver→Measure UpperPlane),
+      (∀ᵐ b∂W,Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫x,f x∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
+          atTop (𝓝 (∫x,f x∂μ b))) →
+      ∀ρ : ℝ,0<ρ → ρ≤z.im →
+        (∫⁻b in {b | radialTime a b.1 z ρ<∞},‖∫x,f x∂μ b‖ₑ^m∂W)≤
+          ENNReal.ofReal (C*(2*ρ)^(alpha a))*A := by
+  obtain ⟨C,hC,hCb⟩ := actual_hit_mass_moment_bound a ha m
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW f c hc hcf z hzc
+  let K := Subtype.val '' tsupport f
+  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
+  have hcK (x : ℂ) (hx : x∈K) : c≤2*x.im := by
+    obtain ⟨y,hy,rfl⟩ := hx
+    exact hcf y hy
+  obtain ⟨hAfin,hB⟩ := hCb W hW K hK.measurableSet hK.isBounded c hc hcK z hzc
+    (fun x=>‖upperExtension f x‖) (upperExtension_measurable f).norm
+    (fun _ _ => norm_nonneg _) (upperExtension_bounded f)
+  refine ⟨hAfin,?_⟩
+  intro s coeff hcoeff hsum t μ hlim ρ hρ hρz
+  let E := {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ<∞}
+  let q (j : ℕ) := Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  apply convex_limit_moment_bound (W.restrict E) m hm _ s coeff hcoeff hsum
+    (fun j b=>∫x,f x∂actualUpperMass a (q j) t b)
+    (fun j=>(actualUpperMass_eval_measurable a ha (q j) (hq j) (hq1 j) t f).aestronglyMeasurable.restrict)
+    _ _ (ae_restrict_of_ae hlim)
+  intro j
+  calc
+    _ ≤ ∫⁻b in E,(∫⁻x in K,ENNReal.ofReal ‖upperExtension f x‖*massDensity a (q j) b x)^m∂W := by
+      apply lintegral_mono
+      intro b
+      dsimp only
+      rw [actualUpperMass_eval a ha (q j) (hq j) (hq1 j)]
+      exact pow_le_pow_left' (actualMassEval_enorm_bound a (q j) t K (upperExtension f) b) m
+    _ ≤ _ := hB ρ hρ hρz (q j) (hq j) (hq1 j)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.KernelSimilarity
+open ExactSLEKernel
+
+ 
+def affine (x : ℂ) (y : ℝ) (hy : 0<y) : ℂ ≃ₜ ℂ :=
+  (Homeomorph.mulLeft₀ (y:ℂ) (Complex.ofReal_ne_zero.mpr hy.ne')).trans
+    (Homeomorph.addLeft x)
+
+@[simp] lemma affine_apply (x : ℂ) {y : ℝ} (hy : 0<y) (z : ℂ) :
+    affine x y hy z=x+(y:ℂ)*z := rfl
+
+lemma dist_affine (x : ℂ) {y : ℝ} (hy : 0<y) (u v : ℂ) :
+    dist (affine x y hy u) (affine x y hy v)=y*dist u v := by
+  simp only [affine_apply,dist_eq_norm,add_sub_add_left_eq_sub,←mul_sub,norm_mul,
+    Complex.norm_real,Real.norm_eq_abs,abs_of_pos hy]
+
+lemma cappedDistance_affine {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (x : ℂ) {y : ℝ} (hy : 0<y) (c : ℝ) (z : ι→ℂ) (i : ι) :
+    cappedDistance (y*c) (affine x y hy ∘ z) i=y*cappedDistance c z i := by
+  apply le_antisymm
+  · rw [mul_comm y (cappedDistance c z i)]
+    apply (div_le_iff₀ hy).mp
+    rw [le_cappedDistance_iff]
+    refine ⟨(div_le_iff₀ hy).mpr ?_,fun j hji=>(div_le_iff₀ hy).mpr ?_⟩
+    · simpa [mul_comm] using cappedDistance_le (y*c) (affine x y hy ∘ z) i
+    · simpa only [Function.comp_apply,dist_affine,mul_comm] using
+        cappedDistance_le_dist (y*c) (affine x y hy ∘ z) i j hji
+  · rw [le_cappedDistance_iff]
+    refine ⟨mul_le_mul_of_nonneg_left (cappedDistance_le c z i) hy.le,?_⟩
+    intro j hji
+    simp only [Function.comp_apply,dist_affine]
+    exact mul_le_mul_of_nonneg_left (cappedDistance_le_dist c z i j hji) hy.le
+
+lemma kernel_affine {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (α c : ℝ) (x : ℂ) {y : ℝ} (hy : 0<y) (z : ι→ℂ) :
+    kernel α (y*c) (affine x y hy ∘ z)=
+      (ENNReal.ofReal y)^(-α*(Fintype.card ι:ℝ))*kernel α c z := by
+  unfold kernel
+  simp_rw [cappedDistance_affine x hy c z,ENNReal.ofReal_mul hy.le,
+    ENNReal.mul_rpow_of_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top,
+    Finset.prod_mul_distrib,Finset.prod_const,Finset.card_univ]
+  rw [←ENNReal.rpow_natCast,←ENNReal.rpow_mul]
+
+lemma pi_const_smul {X : Type*} [MeasurableSpace X] (μ : Measure X) [SigmaFinite μ]
+    (r : ℝ≥0) (m : ℕ) :
+    Measure.pi (fun _ : Fin m=>r • μ)=(r:ℝ≥0∞)^m • Measure.pi (fun _ : Fin m=>μ) := by
+  apply Measure.pi_eq
+  intro s hs
+  simp only [Measure.smul_apply,smul_eq_mul,Measure.pi_pi,Finset.prod_mul_distrib,
+    Finset.prod_const,Finset.card_univ,Fintype.card_fin,ENNReal.smul_def]
+
+lemma affine_area (x : ℂ) {y : ℝ} (hy : 0<y) {K : Set ℂ} (hK : MeasurableSet K) :
+    volume.restrict (affine x y hy '' K)=
+      (NNReal.mk (y^2) (sq_nonneg y)) • (volume.restrict K).map (affine x y hy) := by
+  let T := affine x y hy
+  have hder (z : ℂ) : HasDerivAt T (y:ℂ) z := by
+    change HasDerivAt (fun w : ℂ=>x+(y:ℂ)*w) (y:ℂ) z
+    simpa only [mul_one,id_eq] using ((hasDerivAt_id z).const_mul (y:ℂ)).const_add x
+  ext A hA
+  have h := Conformal.lintegral_image_complex hK
+    (fun z _ => (hder z).differentiableAt) T.injective.injOn (A.indicator (fun _=>1))
+  change (∫⁻z in T '' K,A.indicator (fun _=>(1:ℝ≥0∞)) z)=_ at h
+  simp_rw [fun z=> (hder z).deriv,Complex.norm_real,Real.norm_eq_abs,sq_abs] at h
+  have hAcomp : Measurable (fun z=>A.indicator (fun _=>(1:ℝ≥0∞)) (T z)) :=
+    (measurable_const.indicator hA).comp T.measurable
+  rw [lintegral_const_mul _ hAcomp] at h
+  rw [←lintegral_map (measurable_const.indicator hA) T.measurable] at h
+  rw [lintegral_indicator hA,lintegral_indicator hA,setLIntegral_one,setLIntegral_one] at h
+  rw [Measure.smul_apply,ENNReal.smul_def]
+  exact h.trans (by congr 1; simp only [ENNReal.ofReal,Real.toNNReal_of_nonneg (sq_nonneg y)])
+
+lemma affine_pi_integral (x : ℂ) {y : ℝ} (hy : 0<y) {K : Set ℂ} (hK : MeasurableSet K)
+    (m : ℕ) (F : (Fin m→ℂ)→ℝ≥0∞) (hF : Measurable F) :
+    (∫⁻z,F z∂Measure.pi (fun _=>volume.restrict (affine x y hy '' K)))=
+      (ENNReal.ofReal y)^(2*m)*
+        ∫⁻z,F (affine x y hy ∘ z)∂Measure.pi (fun _=>volume.restrict K) := by
+  let T := affine x y hy
+  have : SigmaFinite ((volume.restrict K).map T) := T.toMeasurableEquiv.sigmaFinite_map
+  rw [affine_area x hy hK,pi_const_smul,lintegral_smul_measure,
+    ←Measure.pi_map_pi (fun _ : Fin m=>T.measurable.aemeasurable)]
+  rw [lintegral_map hF (by fun_prop)]
+  change ((NNReal.mk (y^2) (sq_nonneg y)):ℝ≥0∞)^m*_= _
+  rw [show ((NNReal.mk (y^2) (sq_nonneg y)):ℝ≥0∞)=(ENNReal.ofReal y)^2 by
+    rw [←ENNReal.ofReal_pow hy.le]; simp only [ENNReal.ofReal,Real.toNNReal_of_nonneg (sq_nonneg y)],
+    ←pow_mul]
+  rfl
+
+ 
+
+theorem fixed_center_integral_affine (α c : ℝ) (hα : 0≤α)
+    (x z₀ : ℂ) {y : ℝ} (hy : 0<y) {K : Set ℂ} (hK : MeasurableSet K) (m : ℕ) :
+    (∫⁻z : Fin m→ℂ,kernel α (y*c) (Fin.cons (affine x y hy z₀) z)
+      ∂Measure.pi (fun _=>volume.restrict (affine x y hy '' K)))=
+      (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1))*
+        ∫⁻z : Fin m→ℂ,kernel α c (Fin.cons z₀ z)
+          ∂Measure.pi (fun _=>volume.restrict K) := by
+  have hm : Measurable (fun z : Fin m→ℂ=>kernel α (y*c) (Fin.cons (affine x y hy z₀) z)) :=
+    (measurable_kernel hα _).comp (by fun_prop)
+  rw [affine_pi_integral x hy hK m _ hm]
+  have he (z : Fin m→ℂ) : Fin.cons (affine x y hy z₀) (affine x y hy ∘ z)=
+      affine x y hy ∘ Fin.cons z₀ z := by
+    funext i
+    refine Fin.cases ?_ (fun j=>?_) i <;> simp
+  simp_rw [he,kernel_affine,Fintype.card_fin]
+  have hf : Measurable (fun z : Fin m→ℂ=>kernel α c (Fin.cons z₀ z)) :=
+    (measurable_kernel hα c).comp (by fun_prop)
+  rw [lintegral_const_mul _ hf,←mul_assoc,
+    ←ENNReal.rpow_natCast,←ENNReal.rpow_add _ _ (ne_of_gt (ENNReal.ofReal_pos.mpr hy)) ENNReal.ofReal_ne_top]
+  congr 2
+  push_cast
+  ring
+
+end SLEExactGauge.KernelSimilarity
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/MassMoments.lean b/lean/OAI/Probability/SLEGauge/MassMoments.lean
index 78be2f77a2e67972b94649ef8521c80093e4e604..ea64dd9058ba53d8de1bc7bfc8d22fbad63f8085 100644
--- a/lean/OAI/Probability/SLEGauge/MassMoments.lean
+++ b/lean/OAI/Probability/SLEGauge/MassMoments.lean
@@ -1,402 +1,402 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.KernelBounds
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain ExactSLEKernel
-open scoped Classical NNReal ENNReal
-theorem actual_mass_moment_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
-    ∃ C : ℝ, 0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ (K : Set ℂ), MeasurableSet K → Bornology.IsBounded K →
-    ∀ c : ℝ, 0<c → (∀ z∈K, c≤2*z.im) →
-    ∀ f : ℂ → ℝ, Measurable f → (∀ z∈K, 0≤f z) →
-    (∃ B : ℝ, 0≤B ∧ ∀ z∈K, f z≤B) →
-    let A := ∫⁻ x : Fin m → ℂ, (∏ i, ENNReal.ofReal (f (x i))) * kernel (alpha a) c x
-      ∂Measure.pi (fun _ => volume.restrict K)
-    A<∞ ∧ ∀ q : ℝ, 0<q → q≤1 →
-      (∫⁻ b, (∫⁻ z in K, ENNReal.ofReal (f z)*massDensity a q b z)^m ∂W) ≤
-        ENNReal.ofReal C*A := by
-  obtain ⟨C,hC,hCb⟩ := actual_density_product_bound a ha m
-  refine ⟨C,hC,?_⟩
-  intro W hprob hW K hK hKb c hc hcK f hf hf0 hfB
-  obtain ⟨B,hB,hBf⟩ := hfB
-  let μ := volume.restrict K
-  let ν := Measure.pi (fun _ : Fin m => μ)
-  have hKae : ∀ᵐ x ∂ν, ∀ i, x i∈K := by
-    apply ae_all_iff.mpr
-    intro i
-    exact (Measure.tendsto_eval_ae_ae (μ:=fun _ : Fin m => μ) (i:=i)).eventually (ae_restrict_mem hK)
-  have halpha : alpha a<1 := by
-    have ha0 : 0<a := by linarith
-    have hp : 0<1/(4*a) := by positivity
-    dsimp [alpha]; linarith
-  have hkfin := capped_distance_kernel_integral (alpha a) c (alpha_pos ha) halpha hc m K hK hKb
-  have hkm := measurable_kernel (alpha_pos ha).le c (ι:=Fin m)
-  have hAfin : (∫⁻ x, (∏ i, ENNReal.ofReal (f (x i))) * kernel (alpha a) c x ∂ν)<∞ := by
-    calc
-      _ ≤ ∫⁻ x, (ENNReal.ofReal B)^m * kernel (alpha a) c x ∂ν := by
-        apply lintegral_mono_ae
-        filter_upwards [hKae] with x hx
-        apply mul_le_mul' _ le_rfl
-        have hp := Finset.prod_le_prod₀ (fun _ _ => (bot_le : (0:ℝ≥0∞)≤_))
-          (fun i (_ : i∈Finset.univ) => ENNReal.ofReal_le_ofReal (hBf (x i) (hx i)))
-        simpa using hp
-      _ = (ENNReal.ofReal B)^m * ∫⁻ x, kernel (alpha a) c x ∂ν := lintegral_const_mul _ hkm
-      _ < ∞ := ENNReal.mul_lt_top (ENNReal.pow_lt_top ENNReal.ofReal_lt_top) hkfin
-  refine ⟨hAfin,?_⟩
-  intro q hq hq1
-  have hd := massDensity_measurable a ha q hq hq1
-  have hdz (b : ExactSLEGenerator.Driver) : Measurable (massDensity a q b) := hd.comp (measurable_const.prodMk measurable_id)
-  have hdm (x : Fin m → ℂ) : Measurable (fun b => ∏ i, massDensity a q b (x i)) := by
-    apply Finset.measurable_prod
-    intro i hi
-    exact hd.comp (measurable_id.prodMk measurable_const)
-  have hprod (x : Fin m → ℂ) (hx : ∀ i,x i∈K) :
-      (∫⁻ b, ∏ i, massDensity a q b (x i) ∂W) ≤ ENNReal.ofReal C*kernel (alpha a) c x := by
-    have hz (i) : 0<(x i).im := by have := hcK (x i) (hx i); linarith
-    by_cases hinj : Function.Injective x
-    · let cap := fun i => cappedDistance c x i
-      let target := fun i => 2*(x i).im*q
-      have hcap (i) : 0<cap i := cappedDistance_pos hc x hinj i
-      have ht (i) : 0<target i ∧ target i≤2*(x i).im := by
-        refine ⟨mul_pos (mul_pos (by norm_num) (hz i)) hq,?_⟩
-        dsimp [target]
-        nlinarith [hz i]
-      have hh := hCb W hW x hinj hz cap target
-        (fun i => ⟨hcap i,(cappedDistance_le c x i).trans (hcK (x i) (hx i))⟩)
-        (fun i j hij => by simpa [dist_eq_norm] using cappedDistance_le_dist c x i j hij.symm) ht
-      have he (b : ExactSLEGenerator.Driver) (i) : massDensity a q b (x i)=
-          ENNReal.ofReal (fullRadialDensity a b.1 (x i) (target i)) := by simp [massDensity,hz i,target]
-      simp_rw [he]
-      have hk : ENNReal.ofReal (C*∏ i, cap i^(-alpha a))=ENNReal.ofReal C*kernel (alpha a) c x := by
-        rw [ENNReal.ofReal_mul hC.le,ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (hcap i).le _)]
-        congr 1
-        unfold kernel
-        apply Finset.prod_congr rfl
-        intro i hi
-        exact (ENNReal.ofReal_rpow_of_pos (hcap i)).symm
-      exact hh.trans_eq hk
-    · rw [kernel_top_of_not_injective (alpha a) c (alpha_pos ha) hc x hinj,
-        ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr hC))]
-      exact le_top
-  have hjoint : Measurable (fun p : ExactSLEGenerator.Driver × (Fin m → ℂ) =>
-      ∏ i, ENNReal.ofReal (f (p.2 i))*massDensity a q p.1 (p.2 i)) := by
-    apply Finset.measurable_prod
-    intro i hi
-    exact ((hf.comp ((measurable_pi_apply i).comp measurable_snd)).ennreal_ofReal).mul
-      (hd.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd)))
-  calc
-    _ = ∫⁻ b, ∫⁻ x : Fin m → ℂ, ∏ i, ENNReal.ofReal (f (x i))*massDensity a q b (x i) ∂ν ∂W := by
-      apply lintegral_congr
-      intro b
-      exact lintegral_power_pi μ _ (hf.ennreal_ofReal.mul (hdz b)) m
-    _ = ∫⁻ x : Fin m → ℂ, ∫⁻ b, ∏ i, ENNReal.ofReal (f (x i))*massDensity a q b (x i) ∂W ∂ν :=
-      lintegral_lintegral_swap hjoint.aemeasurable
-    _ ≤ ∫⁻ x, ENNReal.ofReal C*((∏ i, ENNReal.ofReal (f (x i)))*kernel (alpha a) c x) ∂ν := by
-      apply lintegral_mono_ae
-      filter_upwards [hKae] with x hx
-      simp_rw [Finset.prod_mul_distrib]
-      rw [lintegral_const_mul _ (hdm x)]
-      calc
-        _ ≤ (∏ i, ENNReal.ofReal (f (x i)))*(ENNReal.ofReal C*kernel (alpha a) c x) := mul_le_mul' le_rfl (hprod x hx)
-        _ = _ := by ring
-    _ = _ := by
-      have hfm : Measurable (fun x : Fin m → ℂ => ∏ i, ENNReal.ofReal (f (x i))) := by fun_prop
-      exact lintegral_const_mul _ (hfm.mul hkm)
-end ExactSLEGreen
-
-open Filter Topology
-noncomputable section
-namespace ExactSLEMassExtraction
-
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
-
-theorem closed_convex_tail_intersection (X : ℕ → E)
-    (C : ℝ) (hX : ∀ j, ‖X j‖ ≤ C) :
-    ∃ v, ∀ k, v ∈ closure (convexHull ℝ (X '' Set.Ici k)) := by
-  let K := fun k => closure (convexHull ℝ (X '' Set.Ici k))
-  have hne (k) : (K k).Nonempty :=
-    ⟨X k, subset_closure (subset_convexHull ℝ _ ⟨k, Set.mem_Ici.mpr le_rfl, rfl⟩)⟩
-  have hcv (k) : Convex ℝ (K k) := (convex_convexHull ℝ _).closure
-  have hcl (k) : IsClosed (K k) := isClosed_closure
-  have hanti : Antitone K := by
-    intro k l hkl
-    exact closure_mono (convexHull_mono (Set.image_mono (Set.Ici_subset_Ici.mpr hkl)))
-  have hex (k) := exists_norm_eq_iInf_of_complete_convex (hne k) (hcl k).isComplete (hcv k) (0 : E)
-  choose y hy hmin using hex
-  have hle (k) (w) (hw : w ∈ K k) : ‖y k‖ ≤ ‖w‖ := by
-    have hb : BddBelow (Set.range (fun w : K k => ‖(0 : E) - w‖)) :=
-      ⟨0, by rintro _ ⟨i, rfl⟩; exact norm_nonneg _⟩
-    have := ciInf_le hb (⟨w, hw⟩ : K k)
-    rw [← hmin] at this
-    simpa using this
-  have hbd (k) : ‖y k‖ ≤ C :=
-    (hle k (X k) (subset_closure (subset_convexHull ℝ _ ⟨k, Set.mem_Ici.mpr le_rfl, rfl⟩))).trans (hX k)
-  have hm : Monotone (fun k => ‖y k‖ ^ 2) := by
-    intro k l hkl
-    exact pow_le_pow_left₀ (norm_nonneg _) (hle k _ (hanti hkl (hy l))) 2
-  have hb : BddAbove (Set.range (fun k => ‖y k‖ ^ 2)) := by
-    refine ⟨C ^ 2, ?_⟩
-    rintro _ ⟨k, rfl⟩
-    exact pow_le_pow_left₀ (norm_nonneg _) (hbd k) 2
-  have hdiff (k l) (hkl : k ≤ l) : ‖y l - y k‖ ^ 2 ≤ ‖y l‖ ^ 2 - ‖y k‖ ^ 2 := by
-    have hi := (norm_eq_iInf_iff_real_inner_le_zero (hcv k) (hy k)).mp (hmin k)
-      (y l) (hanti hkl (hy l))
-    simp only [zero_sub, inner_neg_left, inner_sub_right, real_inner_self_eq_norm_sq] at hi
-    rw [norm_sub_sq_real, ← real_inner_comm (y l) (y k)]
-    linarith
-  have hcy : CauchySeq y := by
-    have hc := (tendsto_atTop_ciSup hm hb).cauchySeq
-    rw [Metric.cauchySeq_iff] at hc ⊢
-    intro ε hε
-    obtain ⟨N, hN⟩ := hc (ε ^ 2) (sq_pos_of_pos hε)
-    refine ⟨N, fun m hm n hn => ?_⟩
-    have hmn := hN m hm n hn
-    rw [Real.dist_eq, abs_sub_lt_iff] at hmn
-    rw [dist_eq_norm]
-    rcases le_total m n with h | h
-    · have := hdiff m n h
-      rw [norm_sub_rev (y n) (y m)] at this
-      nlinarith [norm_nonneg (y m - y n)]
-    · have := hdiff n m h
-      nlinarith [norm_nonneg (y m - y n)]
-  obtain ⟨v, hv⟩ := cauchySeq_tendsto_of_complete hcy
-  refine ⟨v, fun k => (hcl k).mem_of_tendsto hv ?_⟩
-  exact (eventually_ge_atTop k).mono fun n hn => hanti hn (hy n)
-
-omit [CompleteSpace E] in
-theorem convex_tail_coefficients (X : ℕ → E) (k : ℕ) (v : E)
-    (hv : v ∈ convexHull ℝ (X '' Set.Ici k)) :
-    ∃ (s : Finset ℕ) (c : ℕ → ℝ),
-      (∀ j ∈ s, k ≤ j) ∧ (∀ j ∈ s, 0 ≤ c j) ∧
-      (∑ j ∈ s, c j = 1) ∧ ∑ j ∈ s, c j • X j = v := by
-  classical
-  obtain ⟨ι, inst, w, z, hw, hw1, hz, hzv⟩ := mem_convexHull_iff_exists_fintype.mp hv
-  choose f hf hzf using hz
-  let s := Finset.univ.image f
-  let c := fun j => ∑ i ∈ Finset.univ.filter (fun i => f i = j), w i
-  have hmap : ∀ i ∈ (Finset.univ : Finset ι), f i ∈ s := by simp [s]
-  refine ⟨s, c, ?_, ?_, ?_, ?_⟩
-  · intro j hj
-    obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hj
-    exact hf i
-  · intro j _
-    exact Finset.sum_nonneg fun i _ => hw i
-  · exact (Finset.sum_fiberwise_of_maps_to hmap w).trans hw1
-  · calc
-      ∑ j ∈ s, c j • X j = ∑ j ∈ s, ∑ i ∈ Finset.univ.filter (fun i => f i = j), w i • z i := by
-        apply Finset.sum_congr rfl
-        intro j _
-        rw [show c j = _ from rfl, Finset.sum_smul]
-        apply Finset.sum_congr rfl
-        intro i hi
-        congr 1
-        rw [← hzf i, (Finset.mem_filter.mp hi).2]
-      _ = ∑ i, w i • z i := Finset.sum_fiberwise_of_maps_to hmap _
-      _ = v := hzv
-
-theorem convex_tails_hilbert (X : ℕ → E)
-    (hX : ∃ C : ℝ, ∀ j, ‖X j‖ ≤ C) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (v : E),
-      (∀ k j, j ∈ s k → k ≤ j) ∧
-      (∀ k j, j ∈ s k → 0 ≤ c k j) ∧
-      (∀ k, ∑ j ∈ s k, c k j = 1) ∧
-      Tendsto (fun k => ∑ j ∈ s k, c k j • X j) atTop (𝓝 v) := by
-  rcases hX with ⟨C, hC⟩
-  obtain ⟨v, hv⟩ := closed_convex_tail_intersection X C hC
-  have hex (k) := Metric.mem_closure_iff.mp (hv k) (1 / ((k : ℝ) + 1)) (by positivity)
-  choose u hu hdist using hex
-  have hcoef (k) := convex_tail_coefficients X k (u k) (hu k)
-  choose s c hs hc hsum heq using hcoef
-  refine ⟨s, c, v, hs, hc, hsum, ?_⟩
-  have ht : Tendsto (fun k => dist (u k) v) atTop (𝓝 (0 : ℝ)) := by
-    apply squeeze_zero (fun k => dist_nonneg) _ tendsto_one_div_add_atTop_nhds_zero_nat
-    intro k
-    rw [dist_comm]
-    exact (hdist k).le
-  simpa only [heq] using (tendsto_iff_dist_tendsto_zero.mpr ht)
-
- 
-
-theorem common_convex_tails
-    (X : ℕ → ℕ → E) (hX : ∀ i, ∃ C : ℝ, ∀ j, ‖X j i‖ ≤ C) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (v : ℕ → E),
-      (∀ k j, j ∈ s k → k ≤ j) ∧
-      (∀ k j, j ∈ s k → 0 ≤ c k j) ∧
-      (∀ k, ∑ j ∈ s k, c k j = 1) ∧
-      (∀ i, Tendsto (fun k => ∑ j ∈ s k, c k j • X j i) atTop (𝓝 (v i))) := by
-  classical
-  choose C hC using hX
-  have hC0 (i) : 0 ≤ C i := (norm_nonneg (X 0 i)).trans (hC i 0)
-  let w := fun i => (1 / 2 : ℝ) ^ i / (C i + 1)
-  have hw (i) : 0 < w i := div_pos (pow_pos (by norm_num) _) (by linarith [hC0 i])
-  have hnorm (j i) : ‖w i • X j i‖ ≤ (1 / 2 : ℝ) ^ i := by
-    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (hw i)]
-    change (1 / 2 : ℝ) ^ i / (C i + 1) * ‖X j i‖ ≤ _
-    rw [div_mul_eq_mul_div, div_le_iff₀ (by linarith [hC0 i])]
-    exact mul_le_mul_of_nonneg_left (by linarith [hC i j]) (by positivity)
-  have hsq (j i) : ‖w i • X j i‖ ^ 2 ≤ (1 / 2 : ℝ) ^ i := by
-    have hu : (1 / 2 : ℝ) ^ i ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
-    have hp : 0 ≤ (1 / 2 : ℝ) ^ i := by positivity
-    have hn := hnorm j i
-    nlinarith [norm_nonneg (w i • X j i)]
-  have hsumm (j) : Summable (fun i => ‖w i • X j i‖ ^ 2) :=
-    .of_nonneg_of_le (fun _ => sq_nonneg _) (hsq j)
-      (summable_geometric_of_abs_lt_one (by norm_num))
-  let Y : ℕ → lp (fun _ : ℕ => E) 2 :=
-    fun j => ⟨fun i => w i • X j i, memℓp_gen (by simpa using hsumm j)⟩
-  have hY : ∀ j, ‖Y j‖ ≤ 2 := by
-    intro j
-    apply lp.norm_le_of_tsum_le (by norm_num) (by norm_num)
-    change ∑' i, ‖w i • X j i‖ ^ (2 : ℝ) ≤ (2 : ℝ) ^ (2 : ℝ)
-    simp only [Real.rpow_two]
-    have hh := (hsumm j).tsum_le_tsum (hsq j)
-      (summable_geometric_of_abs_lt_one (by norm_num : |(1 / 2 : ℝ)| < 1))
-    have he : ∑' i : ℕ, (1 / 2 : ℝ) ^ i = 2 := by
-      rw [tsum_geometric_of_abs_lt_one (by norm_num)]
-      norm_num
-    change ∑' i, ‖w i • X j i‖ ^ 2 ≤ 2 ^ 2
-    rw [he] at hh
-    linarith
-  obtain ⟨s, c, v, hs, hc, hsum, ht⟩ := convex_tails_hilbert Y ⟨2, hY⟩
-  refine ⟨s, c, fun i => (w i)⁻¹ • v i, hs, hc, hsum, fun i => ?_⟩
-  have he := ((lp.evalCLM ℝ (fun _ : ℕ => E) 2 i).continuous.tendsto v).comp ht
-  have him := he.const_smul ((w i)⁻¹)
-  convert him using 1
-  · ext k
-    simp only [Function.comp_apply, map_sum, map_smul]
-    change ∑ j ∈ s k, c k j • X j i =
-      (w i)⁻¹ • ∑ j ∈ s k, c k j • (w i • X j i)
-    rw [Finset.smul_sum]
-    apply Finset.sum_congr rfl
-    intro j _
-    rw [← smul_assoc, smul_smul]
-    have hh : (w i)⁻¹ * c k j * w i = c k j := by
-      field_simp [(hw i).ne']
-    rw [smul_eq_mul, hh]
-  · rfl
-
-open MeasureTheory
-open scoped ENNReal
-
-theorem simultaneous_ae_subsequence {Ω F : Type*} [MeasurableSpace Ω]
-    [PseudoEMetricSpace F] (μ : Measure Ω) (f : ℕ → ℕ → Ω → F) (g : ℕ → Ω → F)
-    (hf : ∀ i, TendstoInMeasure μ (fun n => f n i) atTop (g i)) :
-    ∃ ns : ℕ → ℕ, StrictMono ns ∧
-      ∀ᵐ ω ∂μ, ∀ i, Tendsto (fun n => f (ns n) i ω) atTop (𝓝 (g i ω)) := by
-  classical
-  let r := fun i => ExistsSeqTendstoAe.seqTendstoAeSeq (hf i)
-  have hr (i) : StrictMono (r i) := ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono (hf i)
-  let b := fun n => (Finset.range (n+1)).sup (fun i => r i n)
-  have hb (n i) (hi : i ≤ n) : r i n ≤ b n := by
-    exact Finset.le_sup (f := fun i => r i n) (Finset.mem_range.mpr (by omega))
-  have hbm : Monotone b := by
-    intro n m hnm
-    apply Finset.sup_le
-    intro i hi
-    exact ((hr i).monotone hnm).trans (hb m i (by simp only [Finset.mem_range] at hi; omega))
-  let ns := fun n => n + b n
-  have hns : StrictMono ns := by
-    intro n m hnm
-    have hm := hbm hnm.le
-    dsimp [ns]
-    omega
-  refine ⟨ns, hns, ae_all_iff.mpr fun i => ?_⟩
-  let S := fun n => {ω | (2 : ℝ≥0∞)⁻¹ ^ n ≤ edist (f (ns (n+i)) i ω) (g i ω)}
-  have hS (n) : μ (S n) ≤ (2 : ℝ≥0∞)⁻¹ ^ n := by
-    apply ExistsSeqTendstoAe.seqTendstoAeSeq_spec (hf i) n (ns (n+i))
-    exact ((hr i).monotone (by omega : n ≤ n+i)).trans
-      ((hb (n+i) i (by omega)).trans (Nat.le_add_left _ _))
-  have hfin : (∑' n, μ (S n)) ≠ ∞ := by
-    apply ne_top_of_le_ne_top _ (ENNReal.tsum_le_tsum hS)
-    simpa only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv] using
-      (ENNReal.ofNat_ne_top (n := 2))
-  filter_upwards [ae_eventually_notMem hfin] with ω hω
-  apply (tendsto_add_atTop_iff_nat i).mp
-  apply EMetric.tendsto_atTop.mpr
-  intro ε hε
-  obtain ⟨N, hN⟩ := eventually_atTop.mp hω
-  obtain ⟨K, hK⟩ := eventually_atTop.mp
-    ((ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one
-      (by norm_num : (2 : ℝ≥0∞)⁻¹ < 1)).eventually (gt_mem_nhds hε))
-  refine ⟨max N K, fun n hn => ?_⟩
-  exact (not_le.mp (hN n ((le_max_left _ _).trans hn))).trans
-    (hK n ((le_max_right _ _).trans hn))
-
-theorem coe_Lp_sum_smul {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (s : Finset ℕ) (c : ℕ → ℝ) (X : ℕ → Lp ℝ 2 μ) :
-    (fun ω => (∑ j ∈ s, c j • X j) ω) =ᵐ[μ] (fun ω => ∑ j ∈ s, c j * X j ω) := by
-  classical
-  induction s using Finset.induction_on with
-  | empty =>
-    simp only [Finset.sum_empty]
-    filter_upwards [Lp.coeFn_zero ℝ 2 μ] with ω hω
-    exact hω
-  | @insert j s hj ih =>
-    simp only [Finset.sum_insert hj]
-    filter_upwards [Lp.coeFn_add (c j • X j) (∑ i ∈ s, c i • X i),
-      Lp.coeFn_smul (c j) (X j), ih] with ω h1 h2 h3
-    simpa only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, h2, h3] using h1
-
- 
-theorem common_L2_convex_tails
-    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (X : ℕ → ℕ → Lp ℝ 2 μ) (hX : ∀ i, ∃ C : ℝ, ∀ j, ‖X j i‖ ≤ C) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (v : ℕ → Lp ℝ 2 μ),
-      (∀ k j, j ∈ s k → k ≤ j) ∧
-      (∀ k j, j ∈ s k → 0 ≤ c k j) ∧
-      (∀ k, ∑ j ∈ s k, c k j = 1) ∧
-      (∀ i, Tendsto (fun k => ∑ j ∈ s k, c k j • X j i) atTop (𝓝 (v i))) ∧
-      (∀ᵐ ω ∂μ, ∀ i,
-        Tendsto (fun k => ∑ j ∈ s k, c k j * X j i ω) atTop (𝓝 (v i ω))) := by
-  obtain ⟨s, c, v, hs, hc, hsum, ht⟩ := common_convex_tails X hX
-  let f := fun k i ω => (∑ j ∈ s k, c k j • X j i) ω
-  obtain ⟨ns, hns, hae⟩ := simultaneous_ae_subsequence μ f (fun i ω => v i ω)
-    (fun i => tendstoInMeasure_of_tendsto_Lp (ht i))
-  refine ⟨s ∘ ns, c ∘ ns, v, ?_, ?_, ?_, ?_, ?_⟩
-  · intro k j hj
-    exact (hns.id_le k).trans (hs (ns k) j hj)
-  · exact fun k => hc (ns k)
-  · exact fun k => hsum (ns k)
-  · exact fun i => (ht i).comp hns.tendsto_atTop
-  · have he : ∀ᵐ ω ∂μ, ∀ k i, f (ns k) i ω = ∑ j ∈ s (ns k), c (ns k) j * X j i ω :=
-      ae_all_iff.mpr fun k => ae_all_iff.mpr fun i => coe_Lp_sum_smul μ _ _ _
-    filter_upwards [hae, he] with ω hω heω
-    intro i
-    simpa only [Function.comp_apply, ← heω] using hω i
-
-end ExactSLEMassExtraction
-
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.KernelBounds
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain ExactSLEKernel
+open scoped Classical NNReal ENNReal
+theorem actual_mass_moment_bound (a : ℝ) (ha : 1/4<a) (m : ℕ) :
+    ∃ C : ℝ, 0<C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ (K : Set ℂ), MeasurableSet K → Bornology.IsBounded K →
+    ∀ c : ℝ, 0<c → (∀ z∈K, c≤2*z.im) →
+    ∀ f : ℂ → ℝ, Measurable f → (∀ z∈K, 0≤f z) →
+    (∃ B : ℝ, 0≤B ∧ ∀ z∈K, f z≤B) →
+    let A := ∫⁻ x : Fin m → ℂ, (∏ i, ENNReal.ofReal (f (x i))) * kernel (alpha a) c x
+      ∂Measure.pi (fun _ => volume.restrict K)
+    A<∞ ∧ ∀ q : ℝ, 0<q → q≤1 →
+      (∫⁻ b, (∫⁻ z in K, ENNReal.ofReal (f z)*massDensity a q b z)^m ∂W) ≤
+        ENNReal.ofReal C*A := by
+  obtain ⟨C,hC,hCb⟩ := actual_density_product_bound a ha m
+  refine ⟨C,hC,?_⟩
+  intro W hprob hW K hK hKb c hc hcK f hf hf0 hfB
+  obtain ⟨B,hB,hBf⟩ := hfB
+  let μ := volume.restrict K
+  let ν := Measure.pi (fun _ : Fin m => μ)
+  have hKae : ∀ᵐ x ∂ν, ∀ i, x i∈K := by
+    apply ae_all_iff.mpr
+    intro i
+    exact (Measure.tendsto_eval_ae_ae (μ:=fun _ : Fin m => μ) (i:=i)).eventually (ae_restrict_mem hK)
+  have halpha : alpha a<1 := by
+    have ha0 : 0<a := by linarith
+    have hp : 0<1/(4*a) := by positivity
+    dsimp [alpha]; linarith
+  have hkfin := capped_distance_kernel_integral (alpha a) c (alpha_pos ha) halpha hc m K hK hKb
+  have hkm := measurable_kernel (alpha_pos ha).le c (ι:=Fin m)
+  have hAfin : (∫⁻ x, (∏ i, ENNReal.ofReal (f (x i))) * kernel (alpha a) c x ∂ν)<∞ := by
+    calc
+      _ ≤ ∫⁻ x, (ENNReal.ofReal B)^m * kernel (alpha a) c x ∂ν := by
+        apply lintegral_mono_ae
+        filter_upwards [hKae] with x hx
+        apply mul_le_mul' _ le_rfl
+        have hp := Finset.prod_le_prod₀ (fun _ _ => (bot_le : (0:ℝ≥0∞)≤_))
+          (fun i (_ : i∈Finset.univ) => ENNReal.ofReal_le_ofReal (hBf (x i) (hx i)))
+        simpa using hp
+      _ = (ENNReal.ofReal B)^m * ∫⁻ x, kernel (alpha a) c x ∂ν := lintegral_const_mul _ hkm
+      _ < ∞ := ENNReal.mul_lt_top (ENNReal.pow_lt_top ENNReal.ofReal_lt_top) hkfin
+  refine ⟨hAfin,?_⟩
+  intro q hq hq1
+  have hd := massDensity_measurable a ha q hq hq1
+  have hdz (b : ExactSLEGenerator.Driver) : Measurable (massDensity a q b) := hd.comp (measurable_const.prodMk measurable_id)
+  have hdm (x : Fin m → ℂ) : Measurable (fun b => ∏ i, massDensity a q b (x i)) := by
+    apply Finset.measurable_prod
+    intro i hi
+    exact hd.comp (measurable_id.prodMk measurable_const)
+  have hprod (x : Fin m → ℂ) (hx : ∀ i,x i∈K) :
+      (∫⁻ b, ∏ i, massDensity a q b (x i) ∂W) ≤ ENNReal.ofReal C*kernel (alpha a) c x := by
+    have hz (i) : 0<(x i).im := by have := hcK (x i) (hx i); linarith
+    by_cases hinj : Function.Injective x
+    · let cap := fun i => cappedDistance c x i
+      let target := fun i => 2*(x i).im*q
+      have hcap (i) : 0<cap i := cappedDistance_pos hc x hinj i
+      have ht (i) : 0<target i ∧ target i≤2*(x i).im := by
+        refine ⟨mul_pos (mul_pos (by norm_num) (hz i)) hq,?_⟩
+        dsimp [target]
+        nlinarith [hz i]
+      have hh := hCb W hW x hinj hz cap target
+        (fun i => ⟨hcap i,(cappedDistance_le c x i).trans (hcK (x i) (hx i))⟩)
+        (fun i j hij => by simpa [dist_eq_norm] using cappedDistance_le_dist c x i j hij.symm) ht
+      have he (b : ExactSLEGenerator.Driver) (i) : massDensity a q b (x i)=
+          ENNReal.ofReal (fullRadialDensity a b.1 (x i) (target i)) := by simp [massDensity,hz i,target]
+      simp_rw [he]
+      have hk : ENNReal.ofReal (C*∏ i, cap i^(-alpha a))=ENNReal.ofReal C*kernel (alpha a) c x := by
+        rw [ENNReal.ofReal_mul hC.le,ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (hcap i).le _)]
+        congr 1
+        unfold kernel
+        apply Finset.prod_congr rfl
+        intro i hi
+        exact (ENNReal.ofReal_rpow_of_pos (hcap i)).symm
+      exact hh.trans_eq hk
+    · rw [kernel_top_of_not_injective (alpha a) c (alpha_pos ha) hc x hinj,
+        ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr hC))]
+      exact le_top
+  have hjoint : Measurable (fun p : ExactSLEGenerator.Driver × (Fin m → ℂ) =>
+      ∏ i, ENNReal.ofReal (f (p.2 i))*massDensity a q p.1 (p.2 i)) := by
+    apply Finset.measurable_prod
+    intro i hi
+    exact ((hf.comp ((measurable_pi_apply i).comp measurable_snd)).ennreal_ofReal).mul
+      (hd.comp (measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd)))
+  calc
+    _ = ∫⁻ b, ∫⁻ x : Fin m → ℂ, ∏ i, ENNReal.ofReal (f (x i))*massDensity a q b (x i) ∂ν ∂W := by
+      apply lintegral_congr
+      intro b
+      exact lintegral_power_pi μ _ (hf.ennreal_ofReal.mul (hdz b)) m
+    _ = ∫⁻ x : Fin m → ℂ, ∫⁻ b, ∏ i, ENNReal.ofReal (f (x i))*massDensity a q b (x i) ∂W ∂ν :=
+      lintegral_lintegral_swap hjoint.aemeasurable
+    _ ≤ ∫⁻ x, ENNReal.ofReal C*((∏ i, ENNReal.ofReal (f (x i)))*kernel (alpha a) c x) ∂ν := by
+      apply lintegral_mono_ae
+      filter_upwards [hKae] with x hx
+      simp_rw [Finset.prod_mul_distrib]
+      rw [lintegral_const_mul _ (hdm x)]
+      calc
+        _ ≤ (∏ i, ENNReal.ofReal (f (x i)))*(ENNReal.ofReal C*kernel (alpha a) c x) := mul_le_mul' le_rfl (hprod x hx)
+        _ = _ := by ring
+    _ = _ := by
+      have hfm : Measurable (fun x : Fin m → ℂ => ∏ i, ENNReal.ofReal (f (x i))) := by fun_prop
+      exact lintegral_const_mul _ (hfm.mul hkm)
+end ExactSLEGreen
+
+open Filter Topology
+noncomputable section
+namespace ExactSLEMassExtraction
+
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+
+theorem closed_convex_tail_intersection (X : ℕ → E)
+    (C : ℝ) (hX : ∀ j, ‖X j‖ ≤ C) :
+    ∃ v, ∀ k, v ∈ closure (convexHull ℝ (X '' Set.Ici k)) := by
+  let K := fun k => closure (convexHull ℝ (X '' Set.Ici k))
+  have hne (k) : (K k).Nonempty :=
+    ⟨X k, subset_closure (subset_convexHull ℝ _ ⟨k, Set.mem_Ici.mpr le_rfl, rfl⟩)⟩
+  have hcv (k) : Convex ℝ (K k) := (convex_convexHull ℝ _).closure
+  have hcl (k) : IsClosed (K k) := isClosed_closure
+  have hanti : Antitone K := by
+    intro k l hkl
+    exact closure_mono (convexHull_mono (Set.image_mono (Set.Ici_subset_Ici.mpr hkl)))
+  have hex (k) := exists_norm_eq_iInf_of_complete_convex (hne k) (hcl k).isComplete (hcv k) (0 : E)
+  choose y hy hmin using hex
+  have hle (k) (w) (hw : w ∈ K k) : ‖y k‖ ≤ ‖w‖ := by
+    have hb : BddBelow (Set.range (fun w : K k => ‖(0 : E) - w‖)) :=
+      ⟨0, by rintro _ ⟨i, rfl⟩; exact norm_nonneg _⟩
+    have := ciInf_le hb (⟨w, hw⟩ : K k)
+    rw [← hmin] at this
+    simpa using this
+  have hbd (k) : ‖y k‖ ≤ C :=
+    (hle k (X k) (subset_closure (subset_convexHull ℝ _ ⟨k, Set.mem_Ici.mpr le_rfl, rfl⟩))).trans (hX k)
+  have hm : Monotone (fun k => ‖y k‖ ^ 2) := by
+    intro k l hkl
+    exact pow_le_pow_left₀ (norm_nonneg _) (hle k _ (hanti hkl (hy l))) 2
+  have hb : BddAbove (Set.range (fun k => ‖y k‖ ^ 2)) := by
+    refine ⟨C ^ 2, ?_⟩
+    rintro _ ⟨k, rfl⟩
+    exact pow_le_pow_left₀ (norm_nonneg _) (hbd k) 2
+  have hdiff (k l) (hkl : k ≤ l) : ‖y l - y k‖ ^ 2 ≤ ‖y l‖ ^ 2 - ‖y k‖ ^ 2 := by
+    have hi := (norm_eq_iInf_iff_real_inner_le_zero (hcv k) (hy k)).mp (hmin k)
+      (y l) (hanti hkl (hy l))
+    simp only [zero_sub, inner_neg_left, inner_sub_right, real_inner_self_eq_norm_sq] at hi
+    rw [norm_sub_sq_real, ← real_inner_comm (y l) (y k)]
+    linarith
+  have hcy : CauchySeq y := by
+    have hc := (tendsto_atTop_ciSup hm hb).cauchySeq
+    rw [Metric.cauchySeq_iff] at hc ⊢
+    intro ε hε
+    obtain ⟨N, hN⟩ := hc (ε ^ 2) (sq_pos_of_pos hε)
+    refine ⟨N, fun m hm n hn => ?_⟩
+    have hmn := hN m hm n hn
+    rw [Real.dist_eq, abs_sub_lt_iff] at hmn
+    rw [dist_eq_norm]
+    rcases le_total m n with h | h
+    · have := hdiff m n h
+      rw [norm_sub_rev (y n) (y m)] at this
+      nlinarith [norm_nonneg (y m - y n)]
+    · have := hdiff n m h
+      nlinarith [norm_nonneg (y m - y n)]
+  obtain ⟨v, hv⟩ := cauchySeq_tendsto_of_complete hcy
+  refine ⟨v, fun k => (hcl k).mem_of_tendsto hv ?_⟩
+  exact (eventually_ge_atTop k).mono fun n hn => hanti hn (hy n)
+
+omit [CompleteSpace E] in
+theorem convex_tail_coefficients (X : ℕ → E) (k : ℕ) (v : E)
+    (hv : v ∈ convexHull ℝ (X '' Set.Ici k)) :
+    ∃ (s : Finset ℕ) (c : ℕ → ℝ),
+      (∀ j ∈ s, k ≤ j) ∧ (∀ j ∈ s, 0 ≤ c j) ∧
+      (∑ j ∈ s, c j = 1) ∧ ∑ j ∈ s, c j • X j = v := by
+  classical
+  obtain ⟨ι, inst, w, z, hw, hw1, hz, hzv⟩ := mem_convexHull_iff_exists_fintype.mp hv
+  choose f hf hzf using hz
+  let s := Finset.univ.image f
+  let c := fun j => ∑ i ∈ Finset.univ.filter (fun i => f i = j), w i
+  have hmap : ∀ i ∈ (Finset.univ : Finset ι), f i ∈ s := by simp [s]
+  refine ⟨s, c, ?_, ?_, ?_, ?_⟩
+  · intro j hj
+    obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hj
+    exact hf i
+  · intro j _
+    exact Finset.sum_nonneg fun i _ => hw i
+  · exact (Finset.sum_fiberwise_of_maps_to hmap w).trans hw1
+  · calc
+      ∑ j ∈ s, c j • X j = ∑ j ∈ s, ∑ i ∈ Finset.univ.filter (fun i => f i = j), w i • z i := by
+        apply Finset.sum_congr rfl
+        intro j _
+        rw [show c j = _ from rfl, Finset.sum_smul]
+        apply Finset.sum_congr rfl
+        intro i hi
+        congr 1
+        rw [← hzf i, (Finset.mem_filter.mp hi).2]
+      _ = ∑ i, w i • z i := Finset.sum_fiberwise_of_maps_to hmap _
+      _ = v := hzv
+
+theorem convex_tails_hilbert (X : ℕ → E)
+    (hX : ∃ C : ℝ, ∀ j, ‖X j‖ ≤ C) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (v : E),
+      (∀ k j, j ∈ s k → k ≤ j) ∧
+      (∀ k j, j ∈ s k → 0 ≤ c k j) ∧
+      (∀ k, ∑ j ∈ s k, c k j = 1) ∧
+      Tendsto (fun k => ∑ j ∈ s k, c k j • X j) atTop (𝓝 v) := by
+  rcases hX with ⟨C, hC⟩
+  obtain ⟨v, hv⟩ := closed_convex_tail_intersection X C hC
+  have hex (k) := Metric.mem_closure_iff.mp (hv k) (1 / ((k : ℝ) + 1)) (by positivity)
+  choose u hu hdist using hex
+  have hcoef (k) := convex_tail_coefficients X k (u k) (hu k)
+  choose s c hs hc hsum heq using hcoef
+  refine ⟨s, c, v, hs, hc, hsum, ?_⟩
+  have ht : Tendsto (fun k => dist (u k) v) atTop (𝓝 (0 : ℝ)) := by
+    apply squeeze_zero (fun k => dist_nonneg) _ tendsto_one_div_add_atTop_nhds_zero_nat
+    intro k
+    rw [dist_comm]
+    exact (hdist k).le
+  simpa only [heq] using (tendsto_iff_dist_tendsto_zero.mpr ht)
+
+ 
+
+theorem common_convex_tails
+    (X : ℕ → ℕ → E) (hX : ∀ i, ∃ C : ℝ, ∀ j, ‖X j i‖ ≤ C) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (v : ℕ → E),
+      (∀ k j, j ∈ s k → k ≤ j) ∧
+      (∀ k j, j ∈ s k → 0 ≤ c k j) ∧
+      (∀ k, ∑ j ∈ s k, c k j = 1) ∧
+      (∀ i, Tendsto (fun k => ∑ j ∈ s k, c k j • X j i) atTop (𝓝 (v i))) := by
+  classical
+  choose C hC using hX
+  have hC0 (i) : 0 ≤ C i := (norm_nonneg (X 0 i)).trans (hC i 0)
+  let w := fun i => (1 / 2 : ℝ) ^ i / (C i + 1)
+  have hw (i) : 0 < w i := div_pos (pow_pos (by norm_num) _) (by linarith [hC0 i])
+  have hnorm (j i) : ‖w i • X j i‖ ≤ (1 / 2 : ℝ) ^ i := by
+    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (hw i)]
+    change (1 / 2 : ℝ) ^ i / (C i + 1) * ‖X j i‖ ≤ _
+    rw [div_mul_eq_mul_div, div_le_iff₀ (by linarith [hC0 i])]
+    exact mul_le_mul_of_nonneg_left (by linarith [hC i j]) (by positivity)
+  have hsq (j i) : ‖w i • X j i‖ ^ 2 ≤ (1 / 2 : ℝ) ^ i := by
+    have hu : (1 / 2 : ℝ) ^ i ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
+    have hp : 0 ≤ (1 / 2 : ℝ) ^ i := by positivity
+    have hn := hnorm j i
+    nlinarith [norm_nonneg (w i • X j i)]
+  have hsumm (j) : Summable (fun i => ‖w i • X j i‖ ^ 2) :=
+    .of_nonneg_of_le (fun _ => sq_nonneg _) (hsq j)
+      (summable_geometric_of_abs_lt_one (by norm_num))
+  let Y : ℕ → lp (fun _ : ℕ => E) 2 :=
+    fun j => ⟨fun i => w i • X j i, memℓp_gen (by simpa using hsumm j)⟩
+  have hY : ∀ j, ‖Y j‖ ≤ 2 := by
+    intro j
+    apply lp.norm_le_of_tsum_le (by norm_num) (by norm_num)
+    change ∑' i, ‖w i • X j i‖ ^ (2 : ℝ) ≤ (2 : ℝ) ^ (2 : ℝ)
+    simp only [Real.rpow_two]
+    have hh := (hsumm j).tsum_le_tsum (hsq j)
+      (summable_geometric_of_abs_lt_one (by norm_num : |(1 / 2 : ℝ)| < 1))
+    have he : ∑' i : ℕ, (1 / 2 : ℝ) ^ i = 2 := by
+      rw [tsum_geometric_of_abs_lt_one (by norm_num)]
+      norm_num
+    change ∑' i, ‖w i • X j i‖ ^ 2 ≤ 2 ^ 2
+    rw [he] at hh
+    linarith
+  obtain ⟨s, c, v, hs, hc, hsum, ht⟩ := convex_tails_hilbert Y ⟨2, hY⟩
+  refine ⟨s, c, fun i => (w i)⁻¹ • v i, hs, hc, hsum, fun i => ?_⟩
+  have he := ((lp.evalCLM ℝ (fun _ : ℕ => E) 2 i).continuous.tendsto v).comp ht
+  have him := he.const_smul ((w i)⁻¹)
+  convert him using 1
+  · ext k
+    simp only [Function.comp_apply, map_sum, map_smul]
+    change ∑ j ∈ s k, c k j • X j i =
+      (w i)⁻¹ • ∑ j ∈ s k, c k j • (w i • X j i)
+    rw [Finset.smul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    rw [← smul_assoc, smul_smul]
+    have hh : (w i)⁻¹ * c k j * w i = c k j := by
+      field_simp [(hw i).ne']
+    rw [smul_eq_mul, hh]
+  · rfl
+
+open MeasureTheory
+open scoped ENNReal
+
+theorem simultaneous_ae_subsequence {Ω F : Type*} [MeasurableSpace Ω]
+    [PseudoEMetricSpace F] (μ : Measure Ω) (f : ℕ → ℕ → Ω → F) (g : ℕ → Ω → F)
+    (hf : ∀ i, TendstoInMeasure μ (fun n => f n i) atTop (g i)) :
+    ∃ ns : ℕ → ℕ, StrictMono ns ∧
+      ∀ᵐ ω ∂μ, ∀ i, Tendsto (fun n => f (ns n) i ω) atTop (𝓝 (g i ω)) := by
+  classical
+  let r := fun i => ExistsSeqTendstoAe.seqTendstoAeSeq (hf i)
+  have hr (i) : StrictMono (r i) := ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono (hf i)
+  let b := fun n => (Finset.range (n+1)).sup (fun i => r i n)
+  have hb (n i) (hi : i ≤ n) : r i n ≤ b n := by
+    exact Finset.le_sup (f := fun i => r i n) (Finset.mem_range.mpr (by omega))
+  have hbm : Monotone b := by
+    intro n m hnm
+    apply Finset.sup_le
+    intro i hi
+    exact ((hr i).monotone hnm).trans (hb m i (by simp only [Finset.mem_range] at hi; omega))
+  let ns := fun n => n + b n
+  have hns : StrictMono ns := by
+    intro n m hnm
+    have hm := hbm hnm.le
+    dsimp [ns]
+    omega
+  refine ⟨ns, hns, ae_all_iff.mpr fun i => ?_⟩
+  let S := fun n => {ω | (2 : ℝ≥0∞)⁻¹ ^ n ≤ edist (f (ns (n+i)) i ω) (g i ω)}
+  have hS (n) : μ (S n) ≤ (2 : ℝ≥0∞)⁻¹ ^ n := by
+    apply ExistsSeqTendstoAe.seqTendstoAeSeq_spec (hf i) n (ns (n+i))
+    exact ((hr i).monotone (by omega : n ≤ n+i)).trans
+      ((hb (n+i) i (by omega)).trans (Nat.le_add_left _ _))
+  have hfin : (∑' n, μ (S n)) ≠ ∞ := by
+    apply ne_top_of_le_ne_top _ (ENNReal.tsum_le_tsum hS)
+    simpa only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv] using
+      (ENNReal.ofNat_ne_top (n := 2))
+  filter_upwards [ae_eventually_notMem hfin] with ω hω
+  apply (tendsto_add_atTop_iff_nat i).mp
+  apply EMetric.tendsto_atTop.mpr
+  intro ε hε
+  obtain ⟨N, hN⟩ := eventually_atTop.mp hω
+  obtain ⟨K, hK⟩ := eventually_atTop.mp
+    ((ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one
+      (by norm_num : (2 : ℝ≥0∞)⁻¹ < 1)).eventually (gt_mem_nhds hε))
+  refine ⟨max N K, fun n hn => ?_⟩
+  exact (not_le.mp (hN n ((le_max_left _ _).trans hn))).trans
+    (hK n ((le_max_right _ _).trans hn))
+
+theorem coe_Lp_sum_smul {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (s : Finset ℕ) (c : ℕ → ℝ) (X : ℕ → Lp ℝ 2 μ) :
+    (fun ω => (∑ j ∈ s, c j • X j) ω) =ᵐ[μ] (fun ω => ∑ j ∈ s, c j * X j ω) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.sum_empty]
+    filter_upwards [Lp.coeFn_zero ℝ 2 μ] with ω hω
+    exact hω
+  | @insert j s hj ih =>
+    simp only [Finset.sum_insert hj]
+    filter_upwards [Lp.coeFn_add (c j • X j) (∑ i ∈ s, c i • X i),
+      Lp.coeFn_smul (c j) (X j), ih] with ω h1 h2 h3
+    simpa only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, h2, h3] using h1
+
+ 
+theorem common_L2_convex_tails
+    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (X : ℕ → ℕ → Lp ℝ 2 μ) (hX : ∀ i, ∃ C : ℝ, ∀ j, ‖X j i‖ ≤ C) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (v : ℕ → Lp ℝ 2 μ),
+      (∀ k j, j ∈ s k → k ≤ j) ∧
+      (∀ k j, j ∈ s k → 0 ≤ c k j) ∧
+      (∀ k, ∑ j ∈ s k, c k j = 1) ∧
+      (∀ i, Tendsto (fun k => ∑ j ∈ s k, c k j • X j i) atTop (𝓝 (v i))) ∧
+      (∀ᵐ ω ∂μ, ∀ i,
+        Tendsto (fun k => ∑ j ∈ s k, c k j * X j i ω) atTop (𝓝 (v i ω))) := by
+  obtain ⟨s, c, v, hs, hc, hsum, ht⟩ := common_convex_tails X hX
+  let f := fun k i ω => (∑ j ∈ s k, c k j • X j i) ω
+  obtain ⟨ns, hns, hae⟩ := simultaneous_ae_subsequence μ f (fun i ω => v i ω)
+    (fun i => tendstoInMeasure_of_tendsto_Lp (ht i))
+  refine ⟨s ∘ ns, c ∘ ns, v, ?_, ?_, ?_, ?_, ?_⟩
+  · intro k j hj
+    exact (hns.id_le k).trans (hs (ns k) j hj)
+  · exact fun k => hc (ns k)
+  · exact fun k => hsum (ns k)
+  · exact fun i => (ht i).comp hns.tendsto_atTop
+  · have he : ∀ᵐ ω ∂μ, ∀ k i, f (ns k) i ω = ∑ j ∈ s (ns k), c (ns k) j * X j i ω :=
+      ae_all_iff.mpr fun k => ae_all_iff.mpr fun i => coe_Lp_sum_smul μ _ _ _
+    filter_upwards [hae, he] with ω hω heω
+    intro i
+    simpa only [Function.comp_apply, ← heω] using hω i
+
+end ExactSLEMassExtraction
+
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/MassVersionsCore.lean b/lean/OAI/Probability/SLEGauge/MassVersionsCore.lean
index 959d0ed3f412926c4aca977fdae0a382c8ddae1c..4911e06bfa45551a5f3706b492b3afacfb08d01d 100644
--- a/lean/OAI/Probability/SLEGauge/MassVersionsCore.lean
+++ b/lean/OAI/Probability/SLEGauge/MassVersionsCore.lean
@@ -1,430 +1,430 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.GreenIdentities
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped ENNReal NNReal
-
-namespace SLEExactGauge.MassVersions
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
- 
-theorem open_countable_tests (U : Set X) (hU : IsOpen U) :
-    ∃ f : ℕ → C_c(X, ℝ),
-      (∀ n x, f n x ∈ Icc 0 1) ∧ (∀ n, EqOn (f n) 0 Uᶜ) ∧
-      ∀ μ : Measure X, IsFiniteMeasureOnCompacts μ →
-        μ U = ⨆ n, ENNReal.ofReal (∫ x, f n x ∂μ) := by
-  classical
-  let : LocallyCompactSpace U := hU.locallyCompactSpace
-  let K := CompactExhaustion.choice U
-  let A : ℕ → Set X := fun n => Subtype.val '' K n
-  have hAc (n : ℕ) : IsCompact (A n) := (K.isCompact n).image continuous_subtype_val
-  have hAU (n : ℕ) : A n ⊆ U := by
-    rintro x ⟨y, hy, rfl⟩
-    exact y.2
-  choose g hg1 hg0 hgc hg01 using fun n =>
-    exists_continuous_one_zero_of_isCompact (hAc n) hU.isClosed_compl
-      (disjoint_compl_right_iff_subset.mpr (hAU n))
-  let f : ℕ → C_c(X, ℝ) := fun n => ⟨g n, hgc n⟩
-  refine ⟨f, hg01, hg0, fun μ hμ => ?_⟩
-  let := hμ
-  have hmono : Monotone A := fun n m h => image_mono (K.subset h)
-  have hcover : ⋃ n, A n = U := by
-    ext x
-    constructor
-    · intro hx
-      obtain ⟨n, hn⟩ := mem_iUnion.mp hx
-      exact hAU n hn
-    · intro hx
-      obtain ⟨n, hn⟩ := K.exists_mem ⟨x, hx⟩
-      exact mem_iUnion.mpr ⟨n, ⟨⟨x, hx⟩, hn, rfl⟩⟩
-  apply le_antisymm
-  · rw [← hcover, hmono.measure_iUnion]
-    apply iSup_mono
-    intro n
-    apply (f n).integrable.measure_le_integral
-    · exact Eventually.of_forall (fun x => (hg01 n x).1)
-    · intro x hx
-      exact le_of_eq ((hg1 n hx).symm)
-  · apply iSup_le
-    intro n
-    exact integral_le_measure (fun x _ => (hg01 n x).2) (fun x hx => (hg0 n hx).le)
-
- 
-
-theorem measurable_measure_of_compact_tests {Ω : Type*} [MeasurableSpace Ω]
-    (μ : Ω → Measure X) [∀ ω, IsFiniteMeasureOnCompacts (μ ω)]
-    (hμ : ∀ f : C_c(X, ℝ), Measurable (fun ω => ∫ x, f x ∂μ ω)) :
-    Measurable μ := by
-  classical
-  have hopen (U : Set X) (hU : IsOpen U) : Measurable (fun ω => μ ω U) := by
-    obtain ⟨f, -, -, hf⟩ := open_countable_tests U hU
-    have he (ω : Ω) := hf (μ ω) inferInstance
-    simp_rw [he]
-    exact Measurable.iSup (fun n => (hμ (f n)).ennreal_ofReal)
-  let K := CompactExhaustion.choice X
-  let V : ℕ → Set X := fun n => interior (K n)
-  have hV (n : ℕ) : IsOpen (V n) := isOpen_interior
-  have hVmono : Monotone V := fun n m h => interior_mono (K.subset h)
-  have hVcover : ⋃ n, V n = univ := by
-    apply iUnion_eq_univ_iff.mpr
-    intro x
-    obtain ⟨n, hn⟩ := K.exists_mem x
-    exact ⟨n+1, K.subset_interior_succ n hn⟩
-  have hm (n : ℕ) : Measurable (fun ω => (μ ω).restrict (V n)) := by
-    have (ω : Ω) : IsFiniteMeasure ((μ ω).restrict (V n)) := by
-      refine ⟨?_⟩
-      rw [Measure.restrict_apply_univ]
-      exact (measure_mono interior_subset).trans_lt (K.isCompact n).measure_lt_top
-    refine Measurable.measure_of_isPiSystem
-      (μ := fun ω => (μ ω).restrict (V n))
-      (S := {s : Set X | IsOpen s}) ?_
-      (fun s hs t ht _ => hs.inter ht) ?_ ?_
-    · exact BorelSpace.measurable_eq (α := X)
-    · intro U hU
-      simp_rw [Measure.restrict_apply hU.measurableSet]
-      exact hopen (U ∩ V n) (hU.inter (hV n))
-    · simp_rw [Measure.restrict_apply_univ]
-      exact hopen (V n) (hV n)
-  apply Measure.measurable_of_measurable_coe
-  intro U hU
-  have he (ω : Ω) : μ ω U = ⨆ n, ((μ ω).restrict (V n)) U := by
-    simp_rw [Measure.restrict_apply hU]
-    have hmono : Monotone (fun n => U ∩ V n) := fun n m h => inter_subset_inter_right U (hVmono h)
-    rw [← hmono.measure_iUnion, ← inter_iUnion, hVcover, inter_univ]
-  simp_rw [he]
-  exact Measurable.iSup (fun n => (Measure.measurable_coe hU).comp (hm n))
-
- 
-noncomputable def determiningTests : ℕ → C_c(X, ℝ) :=
-  Classical.choose (ExactSLEMassExtraction.determining_tests_radon_limit X)
-
- 
-
-def TestConverges (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) : Prop :=
-  ∀ i, ∃ v : ℝ, Tendsto (fun j => L j (determiningTests i)) atTop (𝓝 v)
-
-lemma exists_radon_of_testConverges (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
-    (h : TestConverges L) :
-    ∃ μ : Measure X, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
-      ∀ f : C_c(X, ℝ), Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂μ)) :=
-  (Classical.choose_spec (ExactSLEMassExtraction.determining_tests_radon_limit X)) L h
-
- 
-
-noncomputable def radonLimit (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) : Measure X := by
-  classical
-  exact if h : TestConverges L then Classical.choose (exists_radon_of_testConverges L h) else 0
-
-lemma radonLimit_finite (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) :
-    IsFiniteMeasureOnCompacts (radonLimit L) := by
-  classical
-  unfold radonLimit
-  split_ifs with h
-  · exact (Classical.choose_spec (exists_radon_of_testConverges L h)).1
-  · infer_instance
-
-lemma radonLimit_regular (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) :
-    (radonLimit L).Regular := by
-  classical
-  unfold radonLimit
-  split_ifs with h
-  · exact (Classical.choose_spec (exists_radon_of_testConverges L h)).2.1
-  · infer_instance
-
-lemma tendsto_radonLimit (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) (h : TestConverges L)
-    (f : C_c(X, ℝ)) :
-    Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂radonLimit L)) := by
-  classical
-  simpa only [radonLimit, dite_eq_left h] using
-    (Classical.choose_spec (exists_radon_of_testConverges L h)).2.2 f
-
-lemma radonLimit_eq (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
-    (μ : Measure X) [μ.Regular]
-    (h : ∀ f : C_c(X, ℝ), Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂μ))) :
-    radonLimit L = μ := by
-  have hr := radonLimit_regular L
-  have hconv : TestConverges L := fun i => ⟨_, h (determiningTests i)⟩
-  exact Measure.ext_of_integral_eq_on_compactlySupported
-    (fun f => tendsto_nhds_unique (tendsto_radonLimit L hconv f) (h f))
-
-lemma measurableSet_testConverges {Ω : Type*} [MeasurableSpace Ω]
-    (L : Ω → ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
-    (hL : ∀ j (f : C_c(X, ℝ)), Measurable (fun ω => L ω j f)) :
-    MeasurableSet {ω | TestConverges (L ω)} := by
-  simp only [TestConverges, ofPred_forall]
-  exact MeasurableSet.iInter (fun i => measurableSet_exists_tendsto
-    (fun j => hL j (determiningTests i)))
-
- 
-
-theorem measurable_radonLimit {Ω : Type*} [MeasurableSpace Ω]
-    (L : Ω → ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
-    (hL : ∀ j (f : C_c(X, ℝ)), Measurable (fun ω => L ω j f)) :
-    Measurable (fun ω => radonLimit (L ω)) := by
-  classical
-  have (ω : Ω) := radonLimit_finite (L ω)
-  apply measurable_measure_of_compact_tests
-  intro f
-  let G := {ω | TestConverges (L ω)}
-  have he : (fun ω => ∫ x, f x ∂radonLimit (L ω)) =
-      G.piecewise (fun ω => limsup (fun j => L ω j f) atTop) (fun _ => 0) := by
-    funext ω
-    by_cases h : ω ∈ G
-    · rw [piecewise_eq_of_mem G _ _ h]
-      exact (tendsto_radonLimit (L ω) h f).limsup_eq.symm
-    · simp [radonLimit, G] at h ⊢
-      simp [h]
-  rw [he]
-  exact (Measurable.limsup (fun j => hL j f)).piecewise
-    (measurableSet_testConverges L hL) measurable_const
-
- 
-
-theorem radonLimit_ae_eq {Ω : Type*} [MeasurableSpace Ω] (W : Measure Ω)
-    {I : Type*} (L : I → Ω → ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
-    (μ : I → Ω → Measure X) [∀ i ω, (μ i ω).Regular]
-    (h : ∀ᵐ ω ∂W, ∀ i (f : C_c(X, ℝ)),
-      Tendsto (fun j => L i ω j f) atTop (𝓝 (∫ x, f x ∂μ i ω))) :
-    ∀ᵐ ω ∂W, ∀ i, radonLimit (L i ω) = μ i ω := by
-  filter_upwards [h] with ω hω
-  exact fun i => radonLimit_eq (L i ω) (μ i ω) (hω i)
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma level_step_le_one (a : ℝ) (ha : 1/4 < a) (j : ℕ) :
-    Real.exp (-2*a*((j:ℝ)+1)) ≤ 1 := by
-  apply Real.exp_le_one_iff.mpr
-  have ha0 : 0 < a := by linarith
-  have hj0 : (0:ℝ) ≤ j := Nat.cast_nonneg _
-  nlinarith
-
- 
-noncomputable def actualMassFunctionals (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : ℕ → (C_c(UpperPlane, ℝ) →ₚ[ℝ] ℝ) :=
-  fun k => positive_mass_combination
-    (fun j => actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b)
-    (fun j => actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _)
-      (level_step_le_one a ha j) t b) (s k) (c k) (hc k)
-
-lemma actualMassFunctionals_apply (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (k : ℕ) (f : C_c(UpperPlane, ℝ)) :
-    actualMassFunctionals a ha s c hc t b k f =
-      ∑ j ∈ s k, c k j * ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b :=
-  positive_mass_combination_apply _ _ _ _ _ _
-
- 
-
-noncomputable def recoveredMass (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : Measure UpperPlane :=
-  radonLimit (actualMassFunctionals a ha s c hc t b)
-
-lemma recoveredMass_measurable (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
-    (t : ℝ≥0∞) : Measurable (recoveredMass a ha s c hc t) := by
-  apply measurable_radonLimit
-  intro k f
-  simp_rw [actualMassFunctionals_apply]
-  apply Finset.measurable_sum
-  intro j hj
-  exact measurable_const.mul (actualUpperMass_eval_measurable a ha _ (Real.exp_pos _)
-    (level_step_le_one a ha j) t f)
-
-lemma recoveredMass_raw_measurable (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
-    (t : ℝ≥0) : Measurable[rawFiltration t] (recoveredMass a ha s c hc t) := by
-  apply @measurable_radonLimit _ _ _ _ _ _ _ _ (rawFiltration t)
-  intro k f
-  simp_rw [actualMassFunctionals_apply]
-  apply Finset.measurable_sum
-  intro j hj
-  exact measurable_const.mul (actualUpperMass_eval_raw_measurable a ha _ (Real.exp_pos _)
-    (level_step_le_one a ha j) t f)
-
- 
-
-theorem actual_radon_measurable_conditional (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i, Measurable (fun b => μ b i)) ∧
-      (∀ i (t₀ : ℝ≥0), t i=t₀ → Measurable[rawFiltration t₀] (fun b => μ b i)) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, coeff k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
-          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
-      (∀ i j (t₀ : ℝ≥0), t i=t₀ → t j=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        ∀ U : Set ExactSLEGenerator.Driver, MeasurableSet[rawFiltration t₀] U →
-        (∫b in U, (∫z,f z∂μ b j)-(∫z,f z∂μ b i) ∂W) =
-          ∫b in U, (∫z : UpperPlane,f z*survivingGreen a t₀ b z.1 ∂upperArea) ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i))) := by
-  obtain ⟨s,c,ν,hs,hc,hsum,hν,hmom,hL1,hmean,-,hcond,h⟩ :=
-    actual_radon_compact_conditional a ha W hW t
-  let μ := fun b i => recoveredMass a ha s c hc (t i) b
-  have he : ∀ᵐ b ∂W, ∀ i, μ b i = ν b i := by
-    filter_upwards [h] with b hb
-    intro i
-    have := (hν b i).2
-    apply radonLimit_eq
-    intro f
-    simpa only [actualMassFunctionals_apply] using hb.2 i f
-  have heI (i : ℕ) (f : C_c(UpperPlane, ℝ)) :
-      (fun b => ∫z, f z∂μ b i) =ᵐ[W] (fun b => ∫z, f z∂ν b i) := by
-    filter_upwards [he] with b hb
-    rw [hb i]
-  refine ⟨s,c,μ,hs,hc,hsum,?_,?_,?_,?_,?_,?_,?_,?_⟩
-  · exact fun b i => ⟨radonLimit_finite _, radonLimit_regular _⟩
-  · exact fun i => recoveredMass_measurable a ha s c hc (t i)
-  · intro i t₀ hi
-    dsimp [μ]
-    rw [hi]
-    exact recoveredMass_raw_measurable a ha s c hc t₀
-  · intro i f m
-    exact ((memLp_congr_ae (heI i f)).mpr (hmom i f m))
-  · intro i f
-    have hnorm (k : ℕ) :
-        eLpNorm (fun b => (∑ j∈s k, c k j *
-          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂μ b i) 1 W =
-        eLpNorm (fun b => (∑ j∈s k, c k j *
-          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂ν b i) 1 W :=
-      eLpNorm_congr_ae (EventuallyEq.sub (Filter.EventuallyEq.refl _ _) (heI i f))
-    simpa only [hnorm] using hL1 i f
-  · intro i hi f
-    exact (integral_congr_ae (heI i f)).trans (hmean i hi f)
-  · intro i j t₀ hi hj f U hU
-    exact (integral_congr_ae (ae_restrict_of_ae ((heI j f).sub (heI i f)))).trans
-      (hcond i j t₀ hi hj f U hU)
-  · filter_upwards [he,h] with b hb hb'
-    simpa only [hb] using hb'
-
-end ExactSLEGreen
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-lemma timeMassDensity_joint_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) :
-    Measurable (fun p : (ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ =>
-      timeMassDensity a q p.1.1 p.1.2 p.2) := by
-  let U : Set ((ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ) := {p | 0<p.2.im}
-  have hU : MeasurableSet U :=
-    measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
-  have hm : Measurable (fun p : U => ENNReal.ofReal (Real.exp (-alpha a*Real.log 2)*
-      (if radialTime a p.1.1.2.1 p.1.2 (p.1.2.im*q)≤p.1.1.1 then
-        radialWeight a p.1.1.2.1 p.1.2 (p.1.2.im*q) else 0))) := by
-    have hp : Measurable (fun p : U =>
-        ((p.1.1.2, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im})),p.1.1.1)) := by fun_prop
-    exact (measurable_const.mul
-      ((joint_radial_approximant_measurable a ha q hq hq1).comp hp)).ennreal_ofReal
-  have h := hm.dite (g:=fun _ => 0) measurable_const hU
-  convert h using 1
-  funext p
-  have he : 2*p.2.im*q/2=p.2.im*q := by ring
-  by_cases hz : 0<p.2.im <;>
-    by_cases ht : radialTime a p.1.2.1 p.2 (p.2.im*q)≤p.1.1 <;>
-    simp [timeMassDensity,massDensity,fullRadialDensity,U,hz,ht,he]
-
- 
-
-lemma timeMassDensity_raw_progressive_joint (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (T : ℝ≥0) :
-    letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration T
-    Measurable (fun p : (Iic T × ExactSLEGenerator.Driver) × ℂ =>
-      timeMassDensity a q p.1.1.1 p.1.2 p.2) := by
-  have h := timeMassDensity_joint_measurable a ha q hq hq1
-  have hb := pastDriver_raw_measurable T
-  let mD : MeasurableSpace ExactSLEGenerator.Driver := inferInstance
-  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration T
-  have hp : @Measurable ((Iic T × ExactSLEGenerator.Driver) × ℂ)
-      ((ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ) inferInstance
-      ((ENNReal.measurableSpace.prod mD).prod (borel ℂ))
-      (fun p => (((p.1.1.1 : ℝ≥0∞), pastDriver T p.1.2),p.2)) :=
-    (((measurable_subtype_coe.comp (measurable_fst.comp measurable_fst)).coe_nnreal_ennreal).prodMk
-      (hb.comp (measurable_snd.comp measurable_fst))).prodMk measurable_snd
-  convert h.comp hp using 1
-  funext p
-  exact timeMassDensity_driver_congr a (by linarith) q hq hq1
-    p.1.2 (pastDriver T p.1.2) p.1.1.1
-    (fun s hs => (pastDriver_eq T p.1.2 s (hs.trans p.1.1.2)).symm) p.2
-
-lemma actualMassEval_progressive (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (K : Set ℂ) (f : ℂ → ℝ) (hf : Measurable f) :
-    IsProgressive rawFiltration (fun t : ℝ≥0 => actualMassEval a q t K f) := by
-  intro T
-  have hD := timeMassDensity_raw_progressive_joint a ha q hq hq1 T
-  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration T
-  have hm : Measurable (fun p : (Iic T × ExactSLEGenerator.Driver) × ℂ =>
-      f p.2*(timeMassDensity a q p.1.1.1 p.1.2 p.2).toReal) :=
-    (hf.comp measurable_snd).mul hD.ennreal_toReal
-  exact hm.stronglyMeasurable.integral_prod_right'.measurable
-
-lemma actualUpperMass_eval_progressive (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (f : C_c(UpperPlane,ℝ)) :
-    IsProgressive rawFiltration
-      (fun t : ℝ≥0 => fun b => ∫z, f z ∂actualUpperMass a q t b) := by
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  exact actualMassEval_progressive a ha q hq hq1 _ _ (upperExtension_measurable f)
-
- 
-
-theorem recoveredMass_progressive (a : ℝ) (ha : 1/4 < a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j) :
-    IsProgressive rawFiltration
-      (fun t : ℝ≥0 => recoveredMass a ha s c hc t) := by
-  intro T
-  apply @measurable_radonLimit _ _ _ _ _ _ _ _
-    (Subtype.instMeasurableSpace.prod (rawFiltration T))
-  intro k f
-  simp_rw [actualMassFunctionals_apply]
-  apply Finset.measurable_sum
-  intro j hj
-  exact measurable_const.mul
-    (actualUpperMass_eval_progressive a ha _ (Real.exp_pos _)
-      (level_step_le_one a ha j) f T)
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.GreenIdentities
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped ENNReal NNReal
+
+namespace SLEExactGauge.MassVersions
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+ 
+theorem open_countable_tests (U : Set X) (hU : IsOpen U) :
+    ∃ f : ℕ → C_c(X, ℝ),
+      (∀ n x, f n x ∈ Icc 0 1) ∧ (∀ n, EqOn (f n) 0 Uᶜ) ∧
+      ∀ μ : Measure X, IsFiniteMeasureOnCompacts μ →
+        μ U = ⨆ n, ENNReal.ofReal (∫ x, f n x ∂μ) := by
+  classical
+  let : LocallyCompactSpace U := hU.locallyCompactSpace
+  let K := CompactExhaustion.choice U
+  let A : ℕ → Set X := fun n => Subtype.val '' K n
+  have hAc (n : ℕ) : IsCompact (A n) := (K.isCompact n).image continuous_subtype_val
+  have hAU (n : ℕ) : A n ⊆ U := by
+    rintro x ⟨y, hy, rfl⟩
+    exact y.2
+  choose g hg1 hg0 hgc hg01 using fun n =>
+    exists_continuous_one_zero_of_isCompact (hAc n) hU.isClosed_compl
+      (disjoint_compl_right_iff_subset.mpr (hAU n))
+  let f : ℕ → C_c(X, ℝ) := fun n => ⟨g n, hgc n⟩
+  refine ⟨f, hg01, hg0, fun μ hμ => ?_⟩
+  let := hμ
+  have hmono : Monotone A := fun n m h => image_mono (K.subset h)
+  have hcover : ⋃ n, A n = U := by
+    ext x
+    constructor
+    · intro hx
+      obtain ⟨n, hn⟩ := mem_iUnion.mp hx
+      exact hAU n hn
+    · intro hx
+      obtain ⟨n, hn⟩ := K.exists_mem ⟨x, hx⟩
+      exact mem_iUnion.mpr ⟨n, ⟨⟨x, hx⟩, hn, rfl⟩⟩
+  apply le_antisymm
+  · rw [← hcover, hmono.measure_iUnion]
+    apply iSup_mono
+    intro n
+    apply (f n).integrable.measure_le_integral
+    · exact Eventually.of_forall (fun x => (hg01 n x).1)
+    · intro x hx
+      exact le_of_eq ((hg1 n hx).symm)
+  · apply iSup_le
+    intro n
+    exact integral_le_measure (fun x _ => (hg01 n x).2) (fun x hx => (hg0 n hx).le)
+
+ 
+
+theorem measurable_measure_of_compact_tests {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Ω → Measure X) [∀ ω, IsFiniteMeasureOnCompacts (μ ω)]
+    (hμ : ∀ f : C_c(X, ℝ), Measurable (fun ω => ∫ x, f x ∂μ ω)) :
+    Measurable μ := by
+  classical
+  have hopen (U : Set X) (hU : IsOpen U) : Measurable (fun ω => μ ω U) := by
+    obtain ⟨f, -, -, hf⟩ := open_countable_tests U hU
+    have he (ω : Ω) := hf (μ ω) inferInstance
+    simp_rw [he]
+    exact Measurable.iSup (fun n => (hμ (f n)).ennreal_ofReal)
+  let K := CompactExhaustion.choice X
+  let V : ℕ → Set X := fun n => interior (K n)
+  have hV (n : ℕ) : IsOpen (V n) := isOpen_interior
+  have hVmono : Monotone V := fun n m h => interior_mono (K.subset h)
+  have hVcover : ⋃ n, V n = univ := by
+    apply iUnion_eq_univ_iff.mpr
+    intro x
+    obtain ⟨n, hn⟩ := K.exists_mem x
+    exact ⟨n+1, K.subset_interior_succ n hn⟩
+  have hm (n : ℕ) : Measurable (fun ω => (μ ω).restrict (V n)) := by
+    have (ω : Ω) : IsFiniteMeasure ((μ ω).restrict (V n)) := by
+      refine ⟨?_⟩
+      rw [Measure.restrict_apply_univ]
+      exact (measure_mono interior_subset).trans_lt (K.isCompact n).measure_lt_top
+    refine Measurable.measure_of_isPiSystem
+      (μ := fun ω => (μ ω).restrict (V n))
+      (S := {s : Set X | IsOpen s}) ?_
+      (fun s hs t ht _ => hs.inter ht) ?_ ?_
+    · exact BorelSpace.measurable_eq (α := X)
+    · intro U hU
+      simp_rw [Measure.restrict_apply hU.measurableSet]
+      exact hopen (U ∩ V n) (hU.inter (hV n))
+    · simp_rw [Measure.restrict_apply_univ]
+      exact hopen (V n) (hV n)
+  apply Measure.measurable_of_measurable_coe
+  intro U hU
+  have he (ω : Ω) : μ ω U = ⨆ n, ((μ ω).restrict (V n)) U := by
+    simp_rw [Measure.restrict_apply hU]
+    have hmono : Monotone (fun n => U ∩ V n) := fun n m h => inter_subset_inter_right U (hVmono h)
+    rw [← hmono.measure_iUnion, ← inter_iUnion, hVcover, inter_univ]
+  simp_rw [he]
+  exact Measurable.iSup (fun n => (Measure.measurable_coe hU).comp (hm n))
+
+ 
+noncomputable def determiningTests : ℕ → C_c(X, ℝ) :=
+  Classical.choose (ExactSLEMassExtraction.determining_tests_radon_limit X)
+
+ 
+
+def TestConverges (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) : Prop :=
+  ∀ i, ∃ v : ℝ, Tendsto (fun j => L j (determiningTests i)) atTop (𝓝 v)
+
+lemma exists_radon_of_testConverges (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
+    (h : TestConverges L) :
+    ∃ μ : Measure X, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
+      ∀ f : C_c(X, ℝ), Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂μ)) :=
+  (Classical.choose_spec (ExactSLEMassExtraction.determining_tests_radon_limit X)) L h
+
+ 
+
+noncomputable def radonLimit (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) : Measure X := by
+  classical
+  exact if h : TestConverges L then Classical.choose (exists_radon_of_testConverges L h) else 0
+
+lemma radonLimit_finite (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) :
+    IsFiniteMeasureOnCompacts (radonLimit L) := by
+  classical
+  unfold radonLimit
+  split_ifs with h
+  · exact (Classical.choose_spec (exists_radon_of_testConverges L h)).1
+  · infer_instance
+
+lemma radonLimit_regular (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) :
+    (radonLimit L).Regular := by
+  classical
+  unfold radonLimit
+  split_ifs with h
+  · exact (Classical.choose_spec (exists_radon_of_testConverges L h)).2.1
+  · infer_instance
+
+lemma tendsto_radonLimit (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ)) (h : TestConverges L)
+    (f : C_c(X, ℝ)) :
+    Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂radonLimit L)) := by
+  classical
+  simpa only [radonLimit, dite_eq_left h] using
+    (Classical.choose_spec (exists_radon_of_testConverges L h)).2.2 f
+
+lemma radonLimit_eq (L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
+    (μ : Measure X) [μ.Regular]
+    (h : ∀ f : C_c(X, ℝ), Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂μ))) :
+    radonLimit L = μ := by
+  have hr := radonLimit_regular L
+  have hconv : TestConverges L := fun i => ⟨_, h (determiningTests i)⟩
+  exact Measure.ext_of_integral_eq_on_compactlySupported
+    (fun f => tendsto_nhds_unique (tendsto_radonLimit L hconv f) (h f))
+
+lemma measurableSet_testConverges {Ω : Type*} [MeasurableSpace Ω]
+    (L : Ω → ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
+    (hL : ∀ j (f : C_c(X, ℝ)), Measurable (fun ω => L ω j f)) :
+    MeasurableSet {ω | TestConverges (L ω)} := by
+  simp only [TestConverges, ofPred_forall]
+  exact MeasurableSet.iInter (fun i => measurableSet_exists_tendsto
+    (fun j => hL j (determiningTests i)))
+
+ 
+
+theorem measurable_radonLimit {Ω : Type*} [MeasurableSpace Ω]
+    (L : Ω → ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
+    (hL : ∀ j (f : C_c(X, ℝ)), Measurable (fun ω => L ω j f)) :
+    Measurable (fun ω => radonLimit (L ω)) := by
+  classical
+  have (ω : Ω) := radonLimit_finite (L ω)
+  apply measurable_measure_of_compact_tests
+  intro f
+  let G := {ω | TestConverges (L ω)}
+  have he : (fun ω => ∫ x, f x ∂radonLimit (L ω)) =
+      G.piecewise (fun ω => limsup (fun j => L ω j f) atTop) (fun _ => 0) := by
+    funext ω
+    by_cases h : ω ∈ G
+    · rw [piecewise_eq_of_mem G _ _ h]
+      exact (tendsto_radonLimit (L ω) h f).limsup_eq.symm
+    · simp [radonLimit, G] at h ⊢
+      simp [h]
+  rw [he]
+  exact (Measurable.limsup (fun j => hL j f)).piecewise
+    (measurableSet_testConverges L hL) measurable_const
+
+ 
+
+theorem radonLimit_ae_eq {Ω : Type*} [MeasurableSpace Ω] (W : Measure Ω)
+    {I : Type*} (L : I → Ω → ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ))
+    (μ : I → Ω → Measure X) [∀ i ω, (μ i ω).Regular]
+    (h : ∀ᵐ ω ∂W, ∀ i (f : C_c(X, ℝ)),
+      Tendsto (fun j => L i ω j f) atTop (𝓝 (∫ x, f x ∂μ i ω))) :
+    ∀ᵐ ω ∂W, ∀ i, radonLimit (L i ω) = μ i ω := by
+  filter_upwards [h] with ω hω
+  exact fun i => radonLimit_eq (L i ω) (μ i ω) (hω i)
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma level_step_le_one (a : ℝ) (ha : 1/4 < a) (j : ℕ) :
+    Real.exp (-2*a*((j:ℝ)+1)) ≤ 1 := by
+  apply Real.exp_le_one_iff.mpr
+  have ha0 : 0 < a := by linarith
+  have hj0 : (0:ℝ) ≤ j := Nat.cast_nonneg _
+  nlinarith
+
+ 
+noncomputable def actualMassFunctionals (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : ℕ → (C_c(UpperPlane, ℝ) →ₚ[ℝ] ℝ) :=
+  fun k => positive_mass_combination
+    (fun j => actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b)
+    (fun j => actualUpperMass_finiteOnCompacts a ha _ (Real.exp_pos _)
+      (level_step_le_one a ha j) t b) (s k) (c k) (hc k)
+
+lemma actualMassFunctionals_apply (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (k : ℕ) (f : C_c(UpperPlane, ℝ)) :
+    actualMassFunctionals a ha s c hc t b k f =
+      ∑ j ∈ s k, c k j * ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*((j:ℝ)+1))) t b :=
+  positive_mass_combination_apply _ _ _ _ _ _
+
+ 
+
+noncomputable def recoveredMass (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : Measure UpperPlane :=
+  radonLimit (actualMassFunctionals a ha s c hc t b)
+
+lemma recoveredMass_measurable (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
+    (t : ℝ≥0∞) : Measurable (recoveredMass a ha s c hc t) := by
+  apply measurable_radonLimit
+  intro k f
+  simp_rw [actualMassFunctionals_apply]
+  apply Finset.measurable_sum
+  intro j hj
+  exact measurable_const.mul (actualUpperMass_eval_measurable a ha _ (Real.exp_pos _)
+    (level_step_le_one a ha j) t f)
+
+lemma recoveredMass_raw_measurable (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j ∈ s k → 0 ≤ c k j)
+    (t : ℝ≥0) : Measurable[rawFiltration t] (recoveredMass a ha s c hc t) := by
+  apply @measurable_radonLimit _ _ _ _ _ _ _ _ (rawFiltration t)
+  intro k f
+  simp_rw [actualMassFunctionals_apply]
+  apply Finset.measurable_sum
+  intro j hj
+  exact measurable_const.mul (actualUpperMass_eval_raw_measurable a ha _ (Real.exp_pos _)
+    (level_step_le_one a ha j) t f)
+
+ 
+
+theorem actual_radon_measurable_conditional (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i, Measurable (fun b => μ b i)) ∧
+      (∀ i (t₀ : ℝ≥0), t i=t₀ → Measurable[rawFiltration t₀] (fun b => μ b i)) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, coeff k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
+          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
+      (∀ i j (t₀ : ℝ≥0), t i=t₀ → t j=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        ∀ U : Set ExactSLEGenerator.Driver, MeasurableSet[rawFiltration t₀] U →
+        (∫b in U, (∫z,f z∂μ b j)-(∫z,f z∂μ b i) ∂W) =
+          ∫b in U, (∫z : UpperPlane,f z*survivingGreen a t₀ b z.1 ∂upperArea) ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i))) := by
+  obtain ⟨s,c,ν,hs,hc,hsum,hν,hmom,hL1,hmean,-,hcond,h⟩ :=
+    actual_radon_compact_conditional a ha W hW t
+  let μ := fun b i => recoveredMass a ha s c hc (t i) b
+  have he : ∀ᵐ b ∂W, ∀ i, μ b i = ν b i := by
+    filter_upwards [h] with b hb
+    intro i
+    have := (hν b i).2
+    apply radonLimit_eq
+    intro f
+    simpa only [actualMassFunctionals_apply] using hb.2 i f
+  have heI (i : ℕ) (f : C_c(UpperPlane, ℝ)) :
+      (fun b => ∫z, f z∂μ b i) =ᵐ[W] (fun b => ∫z, f z∂ν b i) := by
+    filter_upwards [he] with b hb
+    rw [hb i]
+  refine ⟨s,c,μ,hs,hc,hsum,?_,?_,?_,?_,?_,?_,?_,?_⟩
+  · exact fun b i => ⟨radonLimit_finite _, radonLimit_regular _⟩
+  · exact fun i => recoveredMass_measurable a ha s c hc (t i)
+  · intro i t₀ hi
+    dsimp [μ]
+    rw [hi]
+    exact recoveredMass_raw_measurable a ha s c hc t₀
+  · intro i f m
+    exact ((memLp_congr_ae (heI i f)).mpr (hmom i f m))
+  · intro i f
+    have hnorm (k : ℕ) :
+        eLpNorm (fun b => (∑ j∈s k, c k j *
+          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂μ b i) 1 W =
+        eLpNorm (fun b => (∑ j∈s k, c k j *
+          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂ν b i) 1 W :=
+      eLpNorm_congr_ae (EventuallyEq.sub (Filter.EventuallyEq.refl _ _) (heI i f))
+    simpa only [hnorm] using hL1 i f
+  · intro i hi f
+    exact (integral_congr_ae (heI i f)).trans (hmean i hi f)
+  · intro i j t₀ hi hj f U hU
+    exact (integral_congr_ae (ae_restrict_of_ae ((heI j f).sub (heI i f)))).trans
+      (hcond i j t₀ hi hj f U hU)
+  · filter_upwards [he,h] with b hb hb'
+    simpa only [hb] using hb'
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+lemma timeMassDensity_joint_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) :
+    Measurable (fun p : (ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ =>
+      timeMassDensity a q p.1.1 p.1.2 p.2) := by
+  let U : Set ((ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ) := {p | 0<p.2.im}
+  have hU : MeasurableSet U :=
+    measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
+  have hm : Measurable (fun p : U => ENNReal.ofReal (Real.exp (-alpha a*Real.log 2)*
+      (if radialTime a p.1.1.2.1 p.1.2 (p.1.2.im*q)≤p.1.1.1 then
+        radialWeight a p.1.1.2.1 p.1.2 (p.1.2.im*q) else 0))) := by
+    have hp : Measurable (fun p : U =>
+        ((p.1.1.2, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im})),p.1.1.1)) := by fun_prop
+    exact (measurable_const.mul
+      ((joint_radial_approximant_measurable a ha q hq hq1).comp hp)).ennreal_ofReal
+  have h := hm.dite (g:=fun _ => 0) measurable_const hU
+  convert h using 1
+  funext p
+  have he : 2*p.2.im*q/2=p.2.im*q := by ring
+  by_cases hz : 0<p.2.im <;>
+    by_cases ht : radialTime a p.1.2.1 p.2 (p.2.im*q)≤p.1.1 <;>
+    simp [timeMassDensity,massDensity,fullRadialDensity,U,hz,ht,he]
+
+ 
+
+lemma timeMassDensity_raw_progressive_joint (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (T : ℝ≥0) :
+    letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration T
+    Measurable (fun p : (Iic T × ExactSLEGenerator.Driver) × ℂ =>
+      timeMassDensity a q p.1.1.1 p.1.2 p.2) := by
+  have h := timeMassDensity_joint_measurable a ha q hq hq1
+  have hb := pastDriver_raw_measurable T
+  let mD : MeasurableSpace ExactSLEGenerator.Driver := inferInstance
+  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration T
+  have hp : @Measurable ((Iic T × ExactSLEGenerator.Driver) × ℂ)
+      ((ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ) inferInstance
+      ((ENNReal.measurableSpace.prod mD).prod (borel ℂ))
+      (fun p => (((p.1.1.1 : ℝ≥0∞), pastDriver T p.1.2),p.2)) :=
+    (((measurable_subtype_coe.comp (measurable_fst.comp measurable_fst)).coe_nnreal_ennreal).prodMk
+      (hb.comp (measurable_snd.comp measurable_fst))).prodMk measurable_snd
+  convert h.comp hp using 1
+  funext p
+  exact timeMassDensity_driver_congr a (by linarith) q hq hq1
+    p.1.2 (pastDriver T p.1.2) p.1.1.1
+    (fun s hs => (pastDriver_eq T p.1.2 s (hs.trans p.1.1.2)).symm) p.2
+
+lemma actualMassEval_progressive (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (K : Set ℂ) (f : ℂ → ℝ) (hf : Measurable f) :
+    IsProgressive rawFiltration (fun t : ℝ≥0 => actualMassEval a q t K f) := by
+  intro T
+  have hD := timeMassDensity_raw_progressive_joint a ha q hq hq1 T
+  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration T
+  have hm : Measurable (fun p : (Iic T × ExactSLEGenerator.Driver) × ℂ =>
+      f p.2*(timeMassDensity a q p.1.1.1 p.1.2 p.2).toReal) :=
+    (hf.comp measurable_snd).mul hD.ennreal_toReal
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+
+lemma actualUpperMass_eval_progressive (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (f : C_c(UpperPlane,ℝ)) :
+    IsProgressive rawFiltration
+      (fun t : ℝ≥0 => fun b => ∫z, f z ∂actualUpperMass a q t b) := by
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  exact actualMassEval_progressive a ha q hq hq1 _ _ (upperExtension_measurable f)
+
+ 
+
+theorem recoveredMass_progressive (a : ℝ) (ha : 1/4 < a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j) :
+    IsProgressive rawFiltration
+      (fun t : ℝ≥0 => recoveredMass a ha s c hc t) := by
+  intro T
+  apply @measurable_radonLimit _ _ _ _ _ _ _ _
+    (Subtype.instMeasurableSpace.prod (rawFiltration T))
+  intro k f
+  simp_rw [actualMassFunctionals_apply]
+  apply Finset.measurable_sum
+  intro j hj
+  exact measurable_const.mul
+    (actualUpperMass_eval_progressive a ha _ (Real.exp_pos _)
+      (level_step_le_one a ha j) f T)
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ModelCandidates.lean b/lean/OAI/Probability/SLEGauge/ModelCandidates.lean
index 174345bf30f45a84b6e93ea19cd0e54336c0bebb..0e79ef80812adfa40a03354b0d2a47fa064c3d64 100644
--- a/lean/OAI/Probability/SLEGauge/ModelCandidates.lean
+++ b/lean/OAI/Probability/SLEGauge/ModelCandidates.lean
@@ -1,578 +1,578 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.AdaptedFailure
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart LoewnerGeometry CompactSequence MomentBatches
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-def candidateScale : ℝ := max 1 (400*subdiskReach (subdiskRadius M.radiusIndex)*
-  subdiskDistortion (subdiskRadius M.radiusIndex))
-
-def candidateLower : ℝ := (400*diskLower (subdiskRadius M.radiusIndex))^(2-alpha a)
-
-lemma candidateScale_one : 1 ≤ M.candidateScale := le_max_left ..
-lemma candidateScale_pos : 0 < M.candidateScale := zero_lt_one.trans_le M.candidateScale_one
-lemma candidateLower_pos : 0 < M.candidateLower := by
-  apply Real.rpow_pos_of_pos
-  have := diskLower_pos (subdiskRadius M.radiusIndex)
-  positivity
-
-def spatialMass (b : GDriver) : Measure ℂ := spatialRestriction (M.mass ∞ b) univ
-
-lemma spatialMass_apply (b : GDriver) (A : Set ℂ) (hA : MeasurableSet A) :
-    M.spatialMass b A=M.mass ∞ b ((Subtype.val : UpperPlane→ℂ) ⁻¹' A) := by
-  rw [spatialMass,spatialRestriction_apply _ _ _ hA,univ_inter]
-
-lemma spatialMass_measurable (A : Set ℂ) (hA : MeasurableSet A) :
-    Measurable (fun b=>M.spatialMass b A) :=
-  spatialRestriction_measurable _ M.canonical.terminal_measurable _ MeasurableSet.univ _ hA
-
- 
-def candidate (n i : ℕ) (z : ℂ) (b : GDriver) : Prop :=
-  ENNReal.ofReal (M.candidateLower*M.testThreshold n i*(batchRadius a (M.batchLevel n) n i)^(2-alpha a)) ≤
-    M.spatialMass b (closedBall z (M.candidateScale*batchRadius a (M.batchLevel n) n i))
-
-lemma candidate_measurable (n i : ℕ) (z : ℂ) : MeasurableSet {b | M.candidate n i z b} :=
-  measurableSet_le measurable_const (M.spatialMass_measurable _ measurableSet_closedBall)
-
- 
-
-lemma success_implies_candidate (z : ℂ) (hz : 0<z.im) (n i : ℕ)
-    (hy : 400*batchRadius a (M.batchLevel n) n i ≤ z.im) :
-    ∀ᵐ b∂W,ENNReal.ofReal (M.testThreshold n i) ≤
-      M.stoppedTest z (400*batchRadius a (M.batchLevel n) n i) (M.batchLevel n) b →
-      M.candidate n i z b := by
-  let r := batchRadius a (M.batchLevel n) n i
-  let q := subdiskRadius M.radiusIndex
-  have hr : 0<r := batchRadius_pos ..
-  have hσ : 0<400*r := by positivity
-  have hq : 0≤q := (subdiskRadius_pos _).le
-  have hq1 : q<1 := subdiskRadius_lt_one _
-  filter_upwards [M.canonical.path,actual_radial_spatial_bounds κ a ha W
-    M.supports M.coeff M.nonneg M.canonical z hz (400*r) hσ hy q hq hq1] with b hb hbound
-  intro hs
-  have hT : radialTime a b.1 z (400*r*Real.exp (-2*a*M.batchLevel n))≠∞ := by
-    intro he
-    have hztest : M.stoppedTest z (400*r) (M.batchLevel n) b=0 := by
-      simp only [stoppedTest,finiteStoppedTest,he,ite_eq_left]
-    rw [hztest] at hs
-    exact (not_le_of_gt (ENNReal.ofReal_pos.mpr (M.testThreshold_pos n i))) hs
-  have hS : radialTime a b.1 z (400*r)≠∞ := ne_top_of_le_ne_top hT
-    (radialTime_antitone a (by linarith) z hz _ _
-      (testEndpoint_pos (a:=a) _ hσ _) (testEndpoint_le (ha:=ha) _ hσ.le _) hy b)
-  rw [M.stoppedTest_eq z hz (400*r) hσ hy _ b hT] at hs
-  have hloc := (radialDiskRegion_localization a (by linarith) z hz (400*r) hσ hy
-    (half_le_subdiskRadius M.radiusIndex) hq1 b hS).2
-  have hl : radialDiskRegion a (by linarith) z (400*r) q b ⊆
-      (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z (M.candidateScale*r) := by
-    refine hloc.trans (preimage_mono (closedBall_subset_closedBall ?_))
-    have hA : 400*subdiskReach q*subdiskDistortion q ≤ M.candidateScale := le_max_right ..
-    nlinarith [mul_le_mul_of_nonneg_right hA hr.le]
-  have hmass := (mul_le_mul (le_refl
-    (ENNReal.ofReal ((diskLower q*(400*r))^(2-alpha a)))) hs zero_le zero_le).trans ((hbound hS).2 (M.batchLevel n)).1
-  have hmass' := hmass.trans (((hb.monotone le_top) _).trans (measure_mono hl))
-  change ENNReal.ofReal (M.candidateLower*M.testThreshold n i*r^(2-alpha a)) ≤ _
-  rw [M.spatialMass_apply _ _ measurableSet_closedBall]
-  calc
-    _ = ENNReal.ofReal ((diskLower q*(400*r))^(2-alpha a))*ENNReal.ofReal (M.testThreshold n i) := by
-      rw [←ENNReal.ofReal_mul (Real.rpow_nonneg (mul_nonneg (diskLower_pos q).le hσ.le) _)]
-      congr 1
-      rw [show diskLower q*(400*r)=(400*diskLower q)*r by ring,
-        Real.mul_rpow (by have := diskLower_pos q; positivity) hr.le]
-      dsimp [candidateLower,q]
-      ring
-    _ ≤ _ := hmass'
-
-lemma batch_candidates (z : ℂ) (hz : 0<z.im) (n : ℕ)
-    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ))≤z.im) :
-    ∀ᵐ b∂W,(¬∃i<M.testCount n,M.candidate n i z b) → b∈M.batchFailures z n := by
-  have hs : ∀ᵐ b∂W,∀i : ℕ, ENNReal.ofReal (M.testThreshold n i) ≤
-      M.stoppedTest z (400*batchRadius a (M.batchLevel n) n i) (M.batchLevel n) b →
-        M.candidate n i z b := ae_all_iff.mpr (fun i=>M.success_implies_candidate z hz n i
-    ((batchBoundary_start a z.im (by linarith) hz _ _ hr hy i).1.trans
-      (batchBoundary_le a z.im (by linarith) hz _ _ hr hy i)))
-  filter_upwards [hs] with b hb hn i hi hsu
-  exact hn ⟨i,hi,hb i hsu⟩
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-
-lemma diskAt_norm_le_one {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) :
-    ‖diskAt w v‖≤1 := by
-  have hn : ‖v-w‖≤‖v-star w‖ := by
-    rw [←sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)]
-    simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
-      Complex.star_def,Complex.conj_re,Complex.conj_im,sub_neg_eq_add]
-    nlinarith [mul_pos hw hv]
-  simpa only [diskAt,norm_div] using div_le_one_of_le₀ hn (norm_nonneg _)
-
- 
-theorem halfPlane_harnack_ball {f : ℂ→ℂ} {z v : ℂ} {r : ℝ} (hr : 0<r)
-    (hf : DifferentiableOn ℂ f (ball z r))
-    (hpos : ∀w∈ball z r,0<(f w).im) (hv : dist v z≤r/2) :
-    (f v).im≤3*(f z).im := by
-  have hz : z∈ball z r := mem_ball_self hr
-  have hv' : v∈ball z r := lt_of_le_of_lt hv (by linarith)
-  let g : ℂ→ℂ := fun w=>diskAt (f z) (f w)
-  have hd : DifferentiableOn ℂ g (ball z r) := by
-    intro w hw
-    exact ((diskAt_hasDerivAt (hpos z hz) (hpos w hw)).differentiableAt.comp_differentiableWithinAt
-      w (hf w hw))
-  have hzero : g z=0 := by simp [g,diskAt]
-  have hm : MapsTo g (ball z r) (closedBall (g z) 1) := by
-    intro w hw
-    simpa only [mem_closedBall,hzero,dist_zero_right] using diskAt_norm_le_one (hpos z hz) (hpos w hw)
-  have hs := Complex.dist_le_div_mul_dist_of_mapsTo_ball hd hm hv'
-  rw [hzero,dist_zero_right] at hs
-  have hg : ‖g v‖≤1/2 := hs.trans (by
-    calc
-      _ ≤ (1/r)*(r/2) := mul_le_mul_of_nonneg_left hv (by positivity)
-      _ = _ := by field_simp)
-  have hden : 0<‖f v-star (f z)‖ := norm_pos_iff.mpr (diskAt_denominator_ne (hpos z hz) (hpos v hv'))
-  have hnum : ‖f v-f z‖≤(1/2)*‖f v-star (f z)‖ := by
-    exact (div_le_iff₀ hden).mp (show ‖f v-f z‖/‖f v-star (f z)‖≤1/2 by
-      simpa only [g,diskAt,norm_div] using hg)
-  have hu : ‖f v-star (f z)‖≤‖f v-f z‖+2*(f z).im := by
-    calc
-      _ = ‖(f v-f z)+(f z-star (f z))‖ := by congr 1; ring
-      _ ≤ ‖f v-f z‖+‖f z-star (f z)‖ := norm_add_le _ _
-      _ = _ := by
-        rw [show f z-star (f z)=((2*(f z).im:ℝ):ℂ)*Complex.I by
-          apply Complex.ext <;> simp; ring,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-          abs_of_pos (mul_pos (by norm_num) (hpos z hz)),Complex.norm_I,mul_one]
-  have hi := Complex.im_le_norm (f v-f z)
-  rw [Complex.sub_im] at hi
-  linarith
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain ExactSLEGreen LoewnerRestart FlowIdentification
-
- 
-
-theorem alive_of_height_floor (a : ℝ) {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {ε : ℝ} (hε : 0<ε) (hz : ε≤z.im)
-    (hfloor : ∀s≤T,alive a b s z → ε≤(loewnerMap a b s z).im) :
-    alive a b T z := by
-  have hη : 0<ε/2 := half_pos hε
-  obtain ⟨F,hFc,hFe⟩ := exists_cutoff_loewner a (ε/2) hη b hb
-  have hF0 : F 0 z=z := by simpa using hFe z 0
-  let H : ℝ→ℝ := fun t=>(F t.toNNReal z).im
-  have hc : Continuous H := Complex.continuous_im.comp ((hFc z).comp continuous_real_toNNReal)
-  have he0 : H 0=z.im := by simp [H,hF0]
-  have hode (s : ℝ≥0) (hs : ∀u∈Icc (0:ℝ) s,ε/2≤H u) :
-      odeUntil a b z s (fun u=>F u.toNNReal z) := by
-    refine ⟨by simpa using hF0,?_⟩
-    intro u hu
-    have hi := hs u hu
-    refine ⟨hη.trans_le hi,?_⟩
-    have hd := cutoff_loewner_hasDerivWithinAt hη hb hFc hFe z s u hu
-    have hh : heightClamp (ε/2) (F u.toNNReal z-(b u.toNNReal:ℂ))=
-        F u.toNNReal z-(b u.toNNReal:ℂ) := heightClamp_eq (by simpa [H] using hi)
-    simpa only [cutoffVector,hh] using hd
-  have hs : ∀u∈Icc (0:ℝ) T,ε/2≤H u := by
-    by_contra! hn
-    obtain ⟨u,hu,hlow⟩ := hn
-    let S : Set ℝ := Icc 0 T ∩ {t | H t≤ε/2}
-    have hS : IsCompact S := isCompact_Icc.inter_right (isClosed_le hc continuous_const)
-    obtain ⟨s,hs⟩ := hS.exists_isLeast ⟨u,hu,hlow.le⟩
-    have hs0 : 0<s := by
-      have hh := hs.1
-      change s∈Icc 0 (T:ℝ) ∧ H s≤ε/2 at hh
-      rcases hh.1.1.eq_or_lt with h|h
-      · rw [←h,he0] at hh
-        linarith [hh.2]
-      · exact h
-    have hpref : Ico 0 s ⊆ {t | ε/2≤H t} := by
-      intro t ht
-      change ε/2≤H t
-      by_contra! ht'
-      have hb : t∈S := ⟨⟨ht.1,ht.2.le.trans hs.1.1.2⟩,ht'.le⟩
-      exact (not_le.mpr ht.2) (hs.2 hb)
-    have hpref' : Icc 0 s ⊆ {t | ε/2≤H t} := by
-      have hh := closure_minimal hpref (isClosed_le continuous_const hc)
-      rwa [closure_Ico hs0.ne] at hh
-    let s' : ℝ≥0 := ⟨s,hs0.le⟩
-    have ho := hode s' (fun t ht=>hpref' ht)
-    have hh := hfloor s' hs.1.1.2 ⟨_,ho⟩
-    rw [loewnerMap_eq ho] at hh
-    change ε≤H s at hh
-    have hsH : H s≤ε/2 := hs.1.2
-    linarith
-  exact ⟨_,hode T hs⟩
-
- 
-lemma loewnerMap_height_antitone {a : ℝ} (ha : 0≤a) {b : ℝ≥0→ℝ}
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {s t : ℝ≥0} (hs : s≤t) (ht : t≤T) :
-    (loewnerMap a b t z).im≤(loewnerMap a b s z).im := by
-  obtain ⟨f,hf⟩ := hT
-  have hc := Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)
-  have hd (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
-      HasDerivWithinAt (fun u=>(f u).im)
-        (-a*(f u).im/Complex.normSq (f u-(b u.toNNReal:ℂ))) (Icc (0:ℝ) T) u := by
-    have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u (hf.2 u hu).2
-    convert! hh using 1
-    simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
-      zero_mul,zero_div,zero_sub,Complex.sub_im,sub_zero]
-    ring
-  have hm : AntitoneOn (fun u=>(f u).im) (Icc (0:ℝ) T) :=
-    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T:ℝ)) hc
-      (fun u hu=>(hd u (interior_subset hu)).mono interior_subset)
-      (fun u hu=>div_nonpos_of_nonpos_of_nonneg
-        (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ha) (hf.2 u (interior_subset hu)).1.le)
-        (Complex.normSq_nonneg _))
-  rw [loewnerMap_eq (odeUntil_mono hf ht),loewnerMap_eq (odeUntil_mono hf (hs.trans ht))]
-  exact hm ⟨s.coe_nonneg,show (s:ℝ)≤T from hs.trans ht⟩ ⟨t.coe_nonneg,ht⟩ hs
-
- 
-
-theorem center_alive_of_ball_survival {a : ℝ} (ha : 0<a) (b : GDriver)
-    {z v : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) {T : ℝ≥0}
-    (hball : ∀s≤T,alive a b.1 s z → ball z r ⊆ {w | alive a b.1 s w})
-    (hv : dist v z≤r/2) (hT : alive a b.1 T v) : alive a b.1 T z := by
-  let ε : ℝ := min z.im ((loewnerMap a b.1 T v).im/3)
-  have hε : 0<ε := lt_min hz (div_pos (loewnerMap_im_pos hT) (by norm_num))
-  apply alive_of_height_floor a b.1.continuous hε (min_le_left _ _)
-  intro s hs hsz
-  have hd : DifferentiableOn ℂ (loewnerMap a b.1 s) (ball z r) :=
-    fun w hw=>(hasDerivAt_loewnerMap b.1.continuous (hball s hs hsz hw)).differentiableAt.differentiableWithinAt
-  have hh := halfPlane_harnack_ball hr hd
-    (fun w hw=>loewnerMap_im_pos (hball s hs hsz hw)) hv
-  have hm := loewnerMap_height_antitone ha.le hT hs le_rfl
-  exact (min_le_right z.im _).trans ((div_le_iff₀ (by norm_num : (0:ℝ)<3)).mpr (by linarith))
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain ExactSLEGreen LoewnerRestart FlowIdentification
-
- 
-
-theorem frontier_avoids_ball_of_no_crossing {a : ℝ} (ha : 0<a) (b : GDriver)
-    {z : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) (hy : 800*r≤z.im)
-    (hno : radialTime a b.1 z (800*r)=∞) (T : ℝ≥0) :
-    Disjoint (ball z (2*r)) (frontier {v | alive a b.1 T v}) := by
-  have hball (s : ℝ≥0) (hs : alive a b.1 s z) :
-      ball z (4*r) ⊆ {v | alive a b.1 s v} := by
-    have hR := actual_radius_before_target a ha z hz (800*r)
-      (by positivity) hy b s hs (by rw [hno]; exact le_top)
-    exact (ball_subset_closedBall.trans (closedBall_subset_closedBall (by linarith))).trans
-      (radius_ball_subset_alive ha b s hs)
-  by_cases hT : alive a b.1 T z
-  · apply Set.disjoint_left.mpr
-    intro v hv hf
-    have hv' : v∈ball z (4*r) := ball_subset_ball (by linarith) hv
-    rw [(isOpen_alive b.1.continuous T).frontier_eq] at hf
-    exact hf.2 (hball T hT hv')
-  · have hd : Disjoint (ball z (2*r)) {v | alive a b.1 T v} := by
-      apply Set.disjoint_left.mpr
-      intro v hv hvT
-      apply hT
-      apply center_alive_of_ball_survival ha b hz (by positivity : 0<4*r)
-        (fun s _ hs=>hball s hs) _ hvT
-      have hh := mem_ball.mp hv
-      linarith
-    apply Set.disjoint_left.mpr
-    intro v hv hf
-    have hc := frontier_subset_closure hf
-    obtain ⟨w,hw,hws⟩ := mem_closure_iff_nhds.mp hc (ball z (2*r)) (isOpen_ball.mem_nhds hv)
-    exact Set.disjoint_left.mp hd hw hws
-
- 
-
-theorem radial_crossing_of_frontier_visit {a : ℝ} (ha : 0<a) (b : GDriver)
-    {z : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) (hy : 800*r≤z.im)
-    {v : ℂ} {T : ℝ≥0} (hv : v∈ball z (2*r))
-    (hf : v∈frontier {w | alive a b.1 T w}) : radialTime a b.1 z (800*r)≠∞ := by
-  intro hno
-  exact Set.disjoint_left.mp (frontier_avoids_ball_of_no_crossing ha b hz hr hy hno T) hv hf
-
- 
-
-theorem trace_visit_probability_large {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W)
-    {z : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) (hy : 800*r≤z.im) :
-    W {b | ∃t : ℝ, 0≤t ∧ Target.trace κ b t∈ball z (2*r)} ≤
-      Jacobi.visitConstant (Target.a κ)*ENNReal.ofReal ((800*r/z.im)^(alpha (Target.a κ))) := by
-  have hd := trace_regular_actual_domains hκ hκ8 hW
-  have ha := TraceMassIntegration.a_gt_quarter hκ hκ8
-  have hsub : {b : TDriver | ∃t : ℝ,0≤t ∧ Target.trace κ b t∈ball z (2*r)} ≤ᵐ[W]
-      {b | radialTime (Target.a κ) (driverHomeomorph b).1 z (800*r)≠∞} := by
-    filter_upwards [hd] with b hb hvisit
-    obtain ⟨t,ht,hbt⟩ := hvisit
-    exact radial_crossing_of_frontier_visit (by linarith) (driverHomeomorph b) hz hr hy hbt (hb.2.1 t ht)
-  have hR := radialTime_stopping (Target.a κ) (by linarith) z hz (800*r) (by positivity) hy
-  have hmeas : MeasurableSet {b : GDriver | radialTime (Target.a κ) b.1 z (800*r)≠∞} :=
-    (hR.measurable' (measurableSet_singleton ∞)).compl
-  calc
-    _ ≤ W {b | radialTime (Target.a κ) (driverHomeomorph b).1 z (800*r)≠∞} := measure_mono_ae hsub
-    _ = (W.map driverHomeomorph) {b | radialTime (Target.a κ) b.1 z (800*r)≠∞} :=
-      (Measure.map_apply measurable_driverHomeomorph hmeas).symm
-    _ ≤ _ := by
-      let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
-      let : IsProbabilityMeasure (W.map driverHomeomorph) := inferInstance
-      exact Jacobi.radial_hit_probability_radius _ (brownian_map_driver W hW)
-        _ ha z hz (800*r) (by positivity) hy
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology Classical
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart LoewnerGeometry CompactSequence MomentBatches
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-include ha in
-lemma alpha_lt_one : alpha a<1 := by
-  unfold alpha
-  have ha0 : 0<a := by linarith
-  have : 0<1/(4*a) := by positivity
-  linarith
-
-def cleanupEvent (z : ℂ) (n : ℕ) : Set GDriver :=
-  if BatchReady a n ∧ Real.exp (-(n:ℝ)) ≤ z.im then
-    M.batchFailures z n ∩ {b | radialTime a b.1 z (800*M.finalRadius n)≠∞}
-  else if 800*M.finalRadius n≤z.im then
-    {b | radialTime a b.1 z (800*M.finalRadius n)≠∞}
-  else univ
-
-lemma finalRadius_height (z : ℂ) (hz : 0<z.im) (n : ℕ)
-    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
-    800*M.finalRadius n ≤ z.im :=
-  (batch_final_le a z.im (by linarith) hz _ _ _ (M.testCount_pos n hr.1) hr).trans
-    (batchBoundary_le a z.im (by linarith) hz _ _ hr hy _)
-
-lemma cleanupEvent_measurable (z : ℂ) (hz : 0<z.im) (n : ℕ) :
-    MeasurableSet (M.cleanupEvent z n) := by
-  have hh (hy : 800*M.finalRadius n≤z.im) :
-      MeasurableSet {b : GDriver | radialTime a b.1 z (800*M.finalRadius n)≠∞} :=
-    ((radialTime_stopping a (by linarith) z hz _
-      (by have := M.finalRadius_pos n; positivity) hy).measurable' (measurableSet_singleton ∞)).compl
-  unfold cleanupEvent
-  split_ifs with h hy
-  · exact (M.batch_failures_measurable z hz n h.1 h.2).inter (hh (M.finalRadius_height z hz n h.1 h.2))
-  · exact hh hy
-  · exact MeasurableSet.univ
-
-def cellConstant (_ : CenteredMomentModel κ a ha W) : ℝ := max 1 (Jacobi.visitConstant a).toReal * (800:ℝ)^(alpha a)
-
-lemma cellConstant_pos : 0<M.cellConstant :=
-  mul_pos (zero_lt_one.trans_le (le_max_left ..)) (Real.rpow_pos_of_pos (by norm_num) _)
-
-lemma cell_factor (r y : ℝ) (hr : 0<r) (hy : 0<y) :
-    M.cellConstant*r^(alpha a)*y^(-alpha a) =
-      max 1 (Jacobi.visitConstant a).toReal*((800*r/y)^(alpha a)) := by
-  rw [Real.div_rpow (by positivity : 0≤800*r) hy.le,
-    Real.mul_rpow (by norm_num : (0:ℝ)≤800) hr.le,
-    div_eq_mul_inv,←Real.rpow_neg hy.le]
-  unfold cellConstant
-  ring
-
-lemma cleanup_near [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0<z.im) (n : ℕ) :
-    W.real (M.cleanupEvent z n) ≤
-      M.cellConstant*(M.finalRadius n)^(alpha a)*z.im^(-alpha a) := by
-  have hrpos := M.finalRadius_pos n
-  rw [M.cell_factor _ _ (M.finalRadius_pos n) hz]
-  have hhit (hy : 800*M.finalRadius n≤z.im) :
-      W.real {b | radialTime a b.1 z (800*M.finalRadius n)≠∞} ≤
-        max 1 (Jacobi.visitConstant a).toReal*((800*M.finalRadius n/z.im)^(alpha a)) := by
-    have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top (Jacobi.visitConstant_lt_top a).ne ENNReal.ofReal_ne_top)
-      (Jacobi.radial_hit_probability_radius W hW a ha z hz _
-        (by have := M.finalRadius_pos n; positivity) hy)
-    rw [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity)] at hh
-    exact hh.trans (mul_le_mul_of_nonneg_right (le_max_right ..) (by positivity))
-  unfold cleanupEvent
-  split_ifs with h hy
-  · exact (measureReal_mono inter_subset_right).trans (hhit (M.finalRadius_height z hz n h.1 h.2))
-  · exact hhit hy
-  · change (W univ).toReal ≤ _
-    rw [measure_univ, ENNReal.toReal_one]
-    have hr : 1≤800*M.finalRadius n/z.im := (one_le_div hz).mpr (not_le.mp hy).le
-    exact (mul_le_mul (le_max_left 1 _) (Real.one_le_rpow hr (alpha_pos ha).le) zero_le_one (zero_le_one.trans (le_max_left ..))) |>.trans' (by norm_num)
-
-lemma cleanup_far [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0<z.im) (n : ℕ) (hr : BatchReady a n)
-    (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
-    W.real (M.cleanupEvent z n) ≤
-      M.cellConstant*(M.finalRadius n)^(alpha a)*z.im^(-alpha a)*
-        Real.exp (-((2:ℝ)^n/16)) := by
-  have hrpos := M.finalRadius_pos n
-  rw [cleanupEvent,ite_eq_left ⟨hr,hy⟩,M.cell_factor _ _ (M.finalRadius_pos n) hz]
-  have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top
-    (ENNReal.mul_ne_top (Jacobi.visitConstant_lt_top a).ne ENNReal.ofReal_ne_top) ENNReal.ofReal_ne_top)
-    (M.batch_failure_chordal hW z hz n hr hy)
-  rw [ENNReal.toReal_mul,ENNReal.toReal_mul,
-    ENNReal.toReal_ofReal (by positivity),ENNReal.toReal_ofReal (Real.exp_pos _).le] at hh
-  exact hh.trans (mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_right (le_max_right ..) (by positivity)) (Real.exp_pos _).le)
-
- 
-
-lemma cleanup_covers (z : ℂ) (hz : 0<z.im) (n : ℕ) :
-    ∀ᵐ b∂W,∀T : ℝ≥0,∀v∈closedBall z (M.finalRadius n),
-      v∈frontier {w | alive a b.1 T w} →
-      (∃i<M.testCount n,M.candidate n i z b) ∨ b∈M.cleanupEvent z n := by
-  by_cases hh : BatchReady a n ∧ Real.exp (-(n:ℝ)) ≤ z.im
-  · filter_upwards [M.batch_candidates z hz n hh.1 hh.2] with b hb T v hv hf
-    by_cases hc : ∃i<M.testCount n,M.candidate n i z b
-    · exact Or.inl hc
-    · refine Or.inr ?_
-      rw [cleanupEvent,ite_eq_left hh]
-      refine ⟨hb hc,?_⟩
-      exact radial_crossing_of_frontier_visit (by linarith) b hz (M.finalRadius_pos n)
-        (M.finalRadius_height z hz n hh.1 hh.2)
-        ((closedBall_subset_ball (by linarith [M.finalRadius_pos n])) hv) hf
-  · apply Eventually.of_forall
-    intro b T v hv hf
-    right
-    rw [cleanupEvent,ite_eq_right hh]
-    split_ifs with hy
-    · exact radial_crossing_of_frontier_visit (by linarith) b hz (M.finalRadius_pos n) hy
-        ((closedBall_subset_ball (by linarith [M.finalRadius_pos n])) hv) hf
-    · exact mem_univ b
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.CoverCost
-open Deterministic
-
-lemma hausdorff_le_liminf_bound {ι : ℕ→Type*} [∀n,Countable (ι n)]
-    (h : ℝ→ℝ) (E : Set ℂ) (r : ℕ→ℝ≥0∞) (hr : Tendsto r atTop (𝓝 0))
-    (U : ∀n,ι n→Set ℂ) (C : ℕ→ℝ≥0∞)
-    (hdiam : ∀ᶠ n in atTop,∀i,ediam (U n i) ≤ r n)
-    (hcover : ∀ᶠ n in atTop,E ⊆ ⋃i,U n i)
-    (hcost : ∀ᶠ n in atTop,(∑'i,diameterCost h (ediam (U n i))) ≤ C n) :
-    Target.hausdorff h E ≤ liminf C atTop :=
-  (hausdorff_le_liminf_cost h E r hr U hdiam hcover).trans (liminf_le_liminf hcost)
-
- 
-
-lemma hausdorff_le_liminf_of_exists {ι : ℕ→Type*} [∀n,Countable (ι n)]
-    (h : ℝ→ℝ) (E : Set ℂ) (r : ℕ→ℝ≥0∞) (hr : Tendsto r atTop (𝓝 0))
-    (C : ℕ→ℝ≥0∞)
-    (hcovers : ∀ᶠ n in atTop,∃U : ι n→Set ℂ,
-      (∀i,ediam (U i)≤r n) ∧ E⊆⋃i,U i ∧
-      (∑'i,diameterCost h (ediam (U i)))≤C n) :
-    Target.hausdorff h E ≤ liminf C atTop := by
-  classical
-  let U : ∀n,ι n→Set ℂ := fun n=>Classical.epsilon (fun U : ι n→Set ℂ=>
-    (∀i,ediam (U i)≤r n) ∧ E⊆⋃i,U i ∧
-    (∑'i,diameterCost h (ediam (U i)))≤C n)
-  have hU : ∀ᶠ n in atTop,(∀i,ediam (U n i)≤r n) ∧ E⊆⋃i,U n i ∧
-      (∑'i,diameterCost h (ediam (U n i)))≤C n := by
-    filter_upwards [hcovers] with n hn
-    exact Classical.epsilon_spec hn
-  exact hausdorff_le_liminf_bound h E r hr U C (hU.mono fun _ hn=>hn.1)
-    (hU.mono fun _ hn=>hn.2.1) (hU.mono fun _ hn=>hn.2.2)
-
- 
-
-theorem liminf_cost_bound {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (C : ℕ→Ω→ℝ≥0∞) (hC : ∀n,Measurable (C n)) (B : ℝ≥0∞)
-    (hB : B<∞) (hbound : ∀ᶠ n in atTop,(∫⁻ω,C n ω∂P) ≤ B) :
-    Measurable (fun ω=>liminf (fun n=>C n ω) atTop) ∧
-    (∫⁻ω,liminf (fun n=>C n ω) atTop∂P) ≤ B ∧
-    ∀ᵐ ω∂P,liminf (fun n=>C n ω) atTop<∞ := by
-  have hmeas := Measurable.liminf hC
-  have hI : (∫⁻ω,liminf (fun n=>C n ω) atTop∂P) ≤ B :=
-    (lintegral_liminf_le hC).trans (liminf_le_of_frequently_le hbound.frequently)
-  exact ⟨hmeas,hI,ae_lt_top hmeas (hI.trans_lt hB).ne⟩
-
- 
-
-theorem ae_hausdorff_finite_of_covers {Ω : Type*} [MeasurableSpace Ω]
-    {ι : ℕ→Type*} [∀n,Countable (ι n)]
-    (P : Measure Ω) (h : ℝ→ℝ) (E : Ω→Set ℂ)
-    (r : ℕ→ℝ≥0∞) (hr : Tendsto r atTop (𝓝 0))
-    (U : ∀n,Ω→ι n→Set ℂ) (C : ℕ→Ω→ℝ≥0∞)
-    (hC : ∀n,Measurable (C n)) (B : ℝ≥0∞) (hB : B<∞)
-    (hbound : ∀ᶠ n in atTop,(∫⁻ω,C n ω∂P) ≤ B)
-    (hdiam : ∀ᵐ ω∂P,∀ᶠ n in atTop,∀i,ediam (U n ω i) ≤ r n)
-    (hcover : ∀ᵐ ω∂P,∀ᶠ n in atTop,E ω ⊆ ⋃i,U n ω i)
-    (hcost : ∀ᵐ ω∂P,∀ᶠ n in atTop,(∑'i,diameterCost h (ediam (U n ω i))) ≤ C n ω) :
-    (∀ᵐ ω∂P,Target.hausdorff h (E ω) ≤ liminf (fun n=>C n ω) atTop) ∧
-    (∫⁻ω,liminf (fun n=>C n ω) atTop∂P) ≤ B ∧
-    ∀ᵐ ω∂P,Target.hausdorff h (E ω)<∞ := by
-  obtain ⟨hmeas,hI,hfin⟩ := liminf_cost_bound P C hC B hB hbound
-  have hdom : ∀ᵐ ω∂P,Target.hausdorff h (E ω) ≤ liminf (fun n=>C n ω) atTop := by
-    filter_upwards [hdiam,hcover,hcost] with ω hd hc hb
-    exact hausdorff_le_liminf_bound h (E ω) r hr (fun n=>U n ω) (fun n=>C n ω) hd hc hb
-  exact ⟨hdom,hI,by filter_upwards [hdom,hfin] with ω hd hf; exact hd.trans_lt hf⟩
-
-end SLEExactGauge.CoverCost
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.AdaptedFailure
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart LoewnerGeometry CompactSequence MomentBatches
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+def candidateScale : ℝ := max 1 (400*subdiskReach (subdiskRadius M.radiusIndex)*
+  subdiskDistortion (subdiskRadius M.radiusIndex))
+
+def candidateLower : ℝ := (400*diskLower (subdiskRadius M.radiusIndex))^(2-alpha a)
+
+lemma candidateScale_one : 1 ≤ M.candidateScale := le_max_left ..
+lemma candidateScale_pos : 0 < M.candidateScale := zero_lt_one.trans_le M.candidateScale_one
+lemma candidateLower_pos : 0 < M.candidateLower := by
+  apply Real.rpow_pos_of_pos
+  have := diskLower_pos (subdiskRadius M.radiusIndex)
+  positivity
+
+def spatialMass (b : GDriver) : Measure ℂ := spatialRestriction (M.mass ∞ b) univ
+
+lemma spatialMass_apply (b : GDriver) (A : Set ℂ) (hA : MeasurableSet A) :
+    M.spatialMass b A=M.mass ∞ b ((Subtype.val : UpperPlane→ℂ) ⁻¹' A) := by
+  rw [spatialMass,spatialRestriction_apply _ _ _ hA,univ_inter]
+
+lemma spatialMass_measurable (A : Set ℂ) (hA : MeasurableSet A) :
+    Measurable (fun b=>M.spatialMass b A) :=
+  spatialRestriction_measurable _ M.canonical.terminal_measurable _ MeasurableSet.univ _ hA
+
+ 
+def candidate (n i : ℕ) (z : ℂ) (b : GDriver) : Prop :=
+  ENNReal.ofReal (M.candidateLower*M.testThreshold n i*(batchRadius a (M.batchLevel n) n i)^(2-alpha a)) ≤
+    M.spatialMass b (closedBall z (M.candidateScale*batchRadius a (M.batchLevel n) n i))
+
+lemma candidate_measurable (n i : ℕ) (z : ℂ) : MeasurableSet {b | M.candidate n i z b} :=
+  measurableSet_le measurable_const (M.spatialMass_measurable _ measurableSet_closedBall)
+
+ 
+
+lemma success_implies_candidate (z : ℂ) (hz : 0<z.im) (n i : ℕ)
+    (hy : 400*batchRadius a (M.batchLevel n) n i ≤ z.im) :
+    ∀ᵐ b∂W,ENNReal.ofReal (M.testThreshold n i) ≤
+      M.stoppedTest z (400*batchRadius a (M.batchLevel n) n i) (M.batchLevel n) b →
+      M.candidate n i z b := by
+  let r := batchRadius a (M.batchLevel n) n i
+  let q := subdiskRadius M.radiusIndex
+  have hr : 0<r := batchRadius_pos ..
+  have hσ : 0<400*r := by positivity
+  have hq : 0≤q := (subdiskRadius_pos _).le
+  have hq1 : q<1 := subdiskRadius_lt_one _
+  filter_upwards [M.canonical.path,actual_radial_spatial_bounds κ a ha W
+    M.supports M.coeff M.nonneg M.canonical z hz (400*r) hσ hy q hq hq1] with b hb hbound
+  intro hs
+  have hT : radialTime a b.1 z (400*r*Real.exp (-2*a*M.batchLevel n))≠∞ := by
+    intro he
+    have hztest : M.stoppedTest z (400*r) (M.batchLevel n) b=0 := by
+      simp only [stoppedTest,finiteStoppedTest,he,ite_eq_left]
+    rw [hztest] at hs
+    exact (not_le_of_gt (ENNReal.ofReal_pos.mpr (M.testThreshold_pos n i))) hs
+  have hS : radialTime a b.1 z (400*r)≠∞ := ne_top_of_le_ne_top hT
+    (radialTime_antitone a (by linarith) z hz _ _
+      (testEndpoint_pos (a:=a) _ hσ _) (testEndpoint_le (ha:=ha) _ hσ.le _) hy b)
+  rw [M.stoppedTest_eq z hz (400*r) hσ hy _ b hT] at hs
+  have hloc := (radialDiskRegion_localization a (by linarith) z hz (400*r) hσ hy
+    (half_le_subdiskRadius M.radiusIndex) hq1 b hS).2
+  have hl : radialDiskRegion a (by linarith) z (400*r) q b ⊆
+      (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z (M.candidateScale*r) := by
+    refine hloc.trans (preimage_mono (closedBall_subset_closedBall ?_))
+    have hA : 400*subdiskReach q*subdiskDistortion q ≤ M.candidateScale := le_max_right ..
+    nlinarith [mul_le_mul_of_nonneg_right hA hr.le]
+  have hmass := (mul_le_mul (le_refl
+    (ENNReal.ofReal ((diskLower q*(400*r))^(2-alpha a)))) hs zero_le zero_le).trans ((hbound hS).2 (M.batchLevel n)).1
+  have hmass' := hmass.trans (((hb.monotone le_top) _).trans (measure_mono hl))
+  change ENNReal.ofReal (M.candidateLower*M.testThreshold n i*r^(2-alpha a)) ≤ _
+  rw [M.spatialMass_apply _ _ measurableSet_closedBall]
+  calc
+    _ = ENNReal.ofReal ((diskLower q*(400*r))^(2-alpha a))*ENNReal.ofReal (M.testThreshold n i) := by
+      rw [←ENNReal.ofReal_mul (Real.rpow_nonneg (mul_nonneg (diskLower_pos q).le hσ.le) _)]
+      congr 1
+      rw [show diskLower q*(400*r)=(400*diskLower q)*r by ring,
+        Real.mul_rpow (by have := diskLower_pos q; positivity) hr.le]
+      dsimp [candidateLower,q]
+      ring
+    _ ≤ _ := hmass'
+
+lemma batch_candidates (z : ℂ) (hz : 0<z.im) (n : ℕ)
+    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ))≤z.im) :
+    ∀ᵐ b∂W,(¬∃i<M.testCount n,M.candidate n i z b) → b∈M.batchFailures z n := by
+  have hs : ∀ᵐ b∂W,∀i : ℕ, ENNReal.ofReal (M.testThreshold n i) ≤
+      M.stoppedTest z (400*batchRadius a (M.batchLevel n) n i) (M.batchLevel n) b →
+        M.candidate n i z b := ae_all_iff.mpr (fun i=>M.success_implies_candidate z hz n i
+    ((batchBoundary_start a z.im (by linarith) hz _ _ hr hy i).1.trans
+      (batchBoundary_le a z.im (by linarith) hz _ _ hr hy i)))
+  filter_upwards [hs] with b hb hn i hi hsu
+  exact hn ⟨i,hi,hb i hsu⟩
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+
+lemma diskAt_norm_le_one {w v : ℂ} (hw : 0<w.im) (hv : 0<v.im) :
+    ‖diskAt w v‖≤1 := by
+  have hn : ‖v-w‖≤‖v-star w‖ := by
+    rw [←sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)]
+    simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
+      Complex.star_def,Complex.conj_re,Complex.conj_im,sub_neg_eq_add]
+    nlinarith [mul_pos hw hv]
+  simpa only [diskAt,norm_div] using div_le_one_of_le₀ hn (norm_nonneg _)
+
+ 
+theorem halfPlane_harnack_ball {f : ℂ→ℂ} {z v : ℂ} {r : ℝ} (hr : 0<r)
+    (hf : DifferentiableOn ℂ f (ball z r))
+    (hpos : ∀w∈ball z r,0<(f w).im) (hv : dist v z≤r/2) :
+    (f v).im≤3*(f z).im := by
+  have hz : z∈ball z r := mem_ball_self hr
+  have hv' : v∈ball z r := lt_of_le_of_lt hv (by linarith)
+  let g : ℂ→ℂ := fun w=>diskAt (f z) (f w)
+  have hd : DifferentiableOn ℂ g (ball z r) := by
+    intro w hw
+    exact ((diskAt_hasDerivAt (hpos z hz) (hpos w hw)).differentiableAt.comp_differentiableWithinAt
+      w (hf w hw))
+  have hzero : g z=0 := by simp [g,diskAt]
+  have hm : MapsTo g (ball z r) (closedBall (g z) 1) := by
+    intro w hw
+    simpa only [mem_closedBall,hzero,dist_zero_right] using diskAt_norm_le_one (hpos z hz) (hpos w hw)
+  have hs := Complex.dist_le_div_mul_dist_of_mapsTo_ball hd hm hv'
+  rw [hzero,dist_zero_right] at hs
+  have hg : ‖g v‖≤1/2 := hs.trans (by
+    calc
+      _ ≤ (1/r)*(r/2) := mul_le_mul_of_nonneg_left hv (by positivity)
+      _ = _ := by field_simp)
+  have hden : 0<‖f v-star (f z)‖ := norm_pos_iff.mpr (diskAt_denominator_ne (hpos z hz) (hpos v hv'))
+  have hnum : ‖f v-f z‖≤(1/2)*‖f v-star (f z)‖ := by
+    exact (div_le_iff₀ hden).mp (show ‖f v-f z‖/‖f v-star (f z)‖≤1/2 by
+      simpa only [g,diskAt,norm_div] using hg)
+  have hu : ‖f v-star (f z)‖≤‖f v-f z‖+2*(f z).im := by
+    calc
+      _ = ‖(f v-f z)+(f z-star (f z))‖ := by congr 1; ring
+      _ ≤ ‖f v-f z‖+‖f z-star (f z)‖ := norm_add_le _ _
+      _ = _ := by
+        rw [show f z-star (f z)=((2*(f z).im:ℝ):ℂ)*Complex.I by
+          apply Complex.ext <;> simp; ring,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+          abs_of_pos (mul_pos (by norm_num) (hpos z hz)),Complex.norm_I,mul_one]
+  have hi := Complex.im_le_norm (f v-f z)
+  rw [Complex.sub_im] at hi
+  linarith
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain ExactSLEGreen LoewnerRestart FlowIdentification
+
+ 
+
+theorem alive_of_height_floor (a : ℝ) {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {ε : ℝ} (hε : 0<ε) (hz : ε≤z.im)
+    (hfloor : ∀s≤T,alive a b s z → ε≤(loewnerMap a b s z).im) :
+    alive a b T z := by
+  have hη : 0<ε/2 := half_pos hε
+  obtain ⟨F,hFc,hFe⟩ := exists_cutoff_loewner a (ε/2) hη b hb
+  have hF0 : F 0 z=z := by simpa using hFe z 0
+  let H : ℝ→ℝ := fun t=>(F t.toNNReal z).im
+  have hc : Continuous H := Complex.continuous_im.comp ((hFc z).comp continuous_real_toNNReal)
+  have he0 : H 0=z.im := by simp [H,hF0]
+  have hode (s : ℝ≥0) (hs : ∀u∈Icc (0:ℝ) s,ε/2≤H u) :
+      odeUntil a b z s (fun u=>F u.toNNReal z) := by
+    refine ⟨by simpa using hF0,?_⟩
+    intro u hu
+    have hi := hs u hu
+    refine ⟨hη.trans_le hi,?_⟩
+    have hd := cutoff_loewner_hasDerivWithinAt hη hb hFc hFe z s u hu
+    have hh : heightClamp (ε/2) (F u.toNNReal z-(b u.toNNReal:ℂ))=
+        F u.toNNReal z-(b u.toNNReal:ℂ) := heightClamp_eq (by simpa [H] using hi)
+    simpa only [cutoffVector,hh] using hd
+  have hs : ∀u∈Icc (0:ℝ) T,ε/2≤H u := by
+    by_contra! hn
+    obtain ⟨u,hu,hlow⟩ := hn
+    let S : Set ℝ := Icc 0 T ∩ {t | H t≤ε/2}
+    have hS : IsCompact S := isCompact_Icc.inter_right (isClosed_le hc continuous_const)
+    obtain ⟨s,hs⟩ := hS.exists_isLeast ⟨u,hu,hlow.le⟩
+    have hs0 : 0<s := by
+      have hh := hs.1
+      change s∈Icc 0 (T:ℝ) ∧ H s≤ε/2 at hh
+      rcases hh.1.1.eq_or_lt with h|h
+      · rw [←h,he0] at hh
+        linarith [hh.2]
+      · exact h
+    have hpref : Ico 0 s ⊆ {t | ε/2≤H t} := by
+      intro t ht
+      change ε/2≤H t
+      by_contra! ht'
+      have hb : t∈S := ⟨⟨ht.1,ht.2.le.trans hs.1.1.2⟩,ht'.le⟩
+      exact (not_le.mpr ht.2) (hs.2 hb)
+    have hpref' : Icc 0 s ⊆ {t | ε/2≤H t} := by
+      have hh := closure_minimal hpref (isClosed_le continuous_const hc)
+      rwa [closure_Ico hs0.ne] at hh
+    let s' : ℝ≥0 := ⟨s,hs0.le⟩
+    have ho := hode s' (fun t ht=>hpref' ht)
+    have hh := hfloor s' hs.1.1.2 ⟨_,ho⟩
+    rw [loewnerMap_eq ho] at hh
+    change ε≤H s at hh
+    have hsH : H s≤ε/2 := hs.1.2
+    linarith
+  exact ⟨_,hode T hs⟩
+
+ 
+lemma loewnerMap_height_antitone {a : ℝ} (ha : 0≤a) {b : ℝ≥0→ℝ}
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {s t : ℝ≥0} (hs : s≤t) (ht : t≤T) :
+    (loewnerMap a b t z).im≤(loewnerMap a b s z).im := by
+  obtain ⟨f,hf⟩ := hT
+  have hc := Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)
+  have hd (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
+      HasDerivWithinAt (fun u=>(f u).im)
+        (-a*(f u).im/Complex.normSq (f u-(b u.toNNReal:ℂ))) (Icc (0:ℝ) T) u := by
+    have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u (hf.2 u hu).2
+    convert! hh using 1
+    simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
+      zero_mul,zero_div,zero_sub,Complex.sub_im,sub_zero]
+    ring
+  have hm : AntitoneOn (fun u=>(f u).im) (Icc (0:ℝ) T) :=
+    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T:ℝ)) hc
+      (fun u hu=>(hd u (interior_subset hu)).mono interior_subset)
+      (fun u hu=>div_nonpos_of_nonpos_of_nonneg
+        (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ha) (hf.2 u (interior_subset hu)).1.le)
+        (Complex.normSq_nonneg _))
+  rw [loewnerMap_eq (odeUntil_mono hf ht),loewnerMap_eq (odeUntil_mono hf (hs.trans ht))]
+  exact hm ⟨s.coe_nonneg,show (s:ℝ)≤T from hs.trans ht⟩ ⟨t.coe_nonneg,ht⟩ hs
+
+ 
+
+theorem center_alive_of_ball_survival {a : ℝ} (ha : 0<a) (b : GDriver)
+    {z v : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) {T : ℝ≥0}
+    (hball : ∀s≤T,alive a b.1 s z → ball z r ⊆ {w | alive a b.1 s w})
+    (hv : dist v z≤r/2) (hT : alive a b.1 T v) : alive a b.1 T z := by
+  let ε : ℝ := min z.im ((loewnerMap a b.1 T v).im/3)
+  have hε : 0<ε := lt_min hz (div_pos (loewnerMap_im_pos hT) (by norm_num))
+  apply alive_of_height_floor a b.1.continuous hε (min_le_left _ _)
+  intro s hs hsz
+  have hd : DifferentiableOn ℂ (loewnerMap a b.1 s) (ball z r) :=
+    fun w hw=>(hasDerivAt_loewnerMap b.1.continuous (hball s hs hsz hw)).differentiableAt.differentiableWithinAt
+  have hh := halfPlane_harnack_ball hr hd
+    (fun w hw=>loewnerMap_im_pos (hball s hs hsz hw)) hv
+  have hm := loewnerMap_height_antitone ha.le hT hs le_rfl
+  exact (min_le_right z.im _).trans ((div_le_iff₀ (by norm_num : (0:ℝ)<3)).mpr (by linarith))
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain ExactSLEGreen LoewnerRestart FlowIdentification
+
+ 
+
+theorem frontier_avoids_ball_of_no_crossing {a : ℝ} (ha : 0<a) (b : GDriver)
+    {z : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) (hy : 800*r≤z.im)
+    (hno : radialTime a b.1 z (800*r)=∞) (T : ℝ≥0) :
+    Disjoint (ball z (2*r)) (frontier {v | alive a b.1 T v}) := by
+  have hball (s : ℝ≥0) (hs : alive a b.1 s z) :
+      ball z (4*r) ⊆ {v | alive a b.1 s v} := by
+    have hR := actual_radius_before_target a ha z hz (800*r)
+      (by positivity) hy b s hs (by rw [hno]; exact le_top)
+    exact (ball_subset_closedBall.trans (closedBall_subset_closedBall (by linarith))).trans
+      (radius_ball_subset_alive ha b s hs)
+  by_cases hT : alive a b.1 T z
+  · apply Set.disjoint_left.mpr
+    intro v hv hf
+    have hv' : v∈ball z (4*r) := ball_subset_ball (by linarith) hv
+    rw [(isOpen_alive b.1.continuous T).frontier_eq] at hf
+    exact hf.2 (hball T hT hv')
+  · have hd : Disjoint (ball z (2*r)) {v | alive a b.1 T v} := by
+      apply Set.disjoint_left.mpr
+      intro v hv hvT
+      apply hT
+      apply center_alive_of_ball_survival ha b hz (by positivity : 0<4*r)
+        (fun s _ hs=>hball s hs) _ hvT
+      have hh := mem_ball.mp hv
+      linarith
+    apply Set.disjoint_left.mpr
+    intro v hv hf
+    have hc := frontier_subset_closure hf
+    obtain ⟨w,hw,hws⟩ := mem_closure_iff_nhds.mp hc (ball z (2*r)) (isOpen_ball.mem_nhds hv)
+    exact Set.disjoint_left.mp hd hw hws
+
+ 
+
+theorem radial_crossing_of_frontier_visit {a : ℝ} (ha : 0<a) (b : GDriver)
+    {z : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) (hy : 800*r≤z.im)
+    {v : ℂ} {T : ℝ≥0} (hv : v∈ball z (2*r))
+    (hf : v∈frontier {w | alive a b.1 T w}) : radialTime a b.1 z (800*r)≠∞ := by
+  intro hno
+  exact Set.disjoint_left.mp (frontier_avoids_ball_of_no_crossing ha b hz hr hy hno T) hv hf
+
+ 
+
+theorem trace_visit_probability_large {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W)
+    {z : ℂ} (hz : 0<z.im) {r : ℝ} (hr : 0<r) (hy : 800*r≤z.im) :
+    W {b | ∃t : ℝ, 0≤t ∧ Target.trace κ b t∈ball z (2*r)} ≤
+      Jacobi.visitConstant (Target.a κ)*ENNReal.ofReal ((800*r/z.im)^(alpha (Target.a κ))) := by
+  have hd := trace_regular_actual_domains hκ hκ8 hW
+  have ha := TraceMassIntegration.a_gt_quarter hκ hκ8
+  have hsub : {b : TDriver | ∃t : ℝ,0≤t ∧ Target.trace κ b t∈ball z (2*r)} ≤ᵐ[W]
+      {b | radialTime (Target.a κ) (driverHomeomorph b).1 z (800*r)≠∞} := by
+    filter_upwards [hd] with b hb hvisit
+    obtain ⟨t,ht,hbt⟩ := hvisit
+    exact radial_crossing_of_frontier_visit (by linarith) (driverHomeomorph b) hz hr hy hbt (hb.2.1 t ht)
+  have hR := radialTime_stopping (Target.a κ) (by linarith) z hz (800*r) (by positivity) hy
+  have hmeas : MeasurableSet {b : GDriver | radialTime (Target.a κ) b.1 z (800*r)≠∞} :=
+    (hR.measurable' (measurableSet_singleton ∞)).compl
+  calc
+    _ ≤ W {b | radialTime (Target.a κ) (driverHomeomorph b).1 z (800*r)≠∞} := measure_mono_ae hsub
+    _ = (W.map driverHomeomorph) {b | radialTime (Target.a κ) b.1 z (800*r)≠∞} :=
+      (Measure.map_apply measurable_driverHomeomorph hmeas).symm
+    _ ≤ _ := by
+      let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure
+      let : IsProbabilityMeasure (W.map driverHomeomorph) := inferInstance
+      exact Jacobi.radial_hit_probability_radius _ (brownian_map_driver W hW)
+        _ ha z hz (800*r) (by positivity) hy
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology Classical
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart LoewnerGeometry CompactSequence MomentBatches
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+include ha in
+lemma alpha_lt_one : alpha a<1 := by
+  unfold alpha
+  have ha0 : 0<a := by linarith
+  have : 0<1/(4*a) := by positivity
+  linarith
+
+def cleanupEvent (z : ℂ) (n : ℕ) : Set GDriver :=
+  if BatchReady a n ∧ Real.exp (-(n:ℝ)) ≤ z.im then
+    M.batchFailures z n ∩ {b | radialTime a b.1 z (800*M.finalRadius n)≠∞}
+  else if 800*M.finalRadius n≤z.im then
+    {b | radialTime a b.1 z (800*M.finalRadius n)≠∞}
+  else univ
+
+lemma finalRadius_height (z : ℂ) (hz : 0<z.im) (n : ℕ)
+    (hr : BatchReady a n) (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
+    800*M.finalRadius n ≤ z.im :=
+  (batch_final_le a z.im (by linarith) hz _ _ _ (M.testCount_pos n hr.1) hr).trans
+    (batchBoundary_le a z.im (by linarith) hz _ _ hr hy _)
+
+lemma cleanupEvent_measurable (z : ℂ) (hz : 0<z.im) (n : ℕ) :
+    MeasurableSet (M.cleanupEvent z n) := by
+  have hh (hy : 800*M.finalRadius n≤z.im) :
+      MeasurableSet {b : GDriver | radialTime a b.1 z (800*M.finalRadius n)≠∞} :=
+    ((radialTime_stopping a (by linarith) z hz _
+      (by have := M.finalRadius_pos n; positivity) hy).measurable' (measurableSet_singleton ∞)).compl
+  unfold cleanupEvent
+  split_ifs with h hy
+  · exact (M.batch_failures_measurable z hz n h.1 h.2).inter (hh (M.finalRadius_height z hz n h.1 h.2))
+  · exact hh hy
+  · exact MeasurableSet.univ
+
+def cellConstant (_ : CenteredMomentModel κ a ha W) : ℝ := max 1 (Jacobi.visitConstant a).toReal * (800:ℝ)^(alpha a)
+
+lemma cellConstant_pos : 0<M.cellConstant :=
+  mul_pos (zero_lt_one.trans_le (le_max_left ..)) (Real.rpow_pos_of_pos (by norm_num) _)
+
+lemma cell_factor (r y : ℝ) (hr : 0<r) (hy : 0<y) :
+    M.cellConstant*r^(alpha a)*y^(-alpha a) =
+      max 1 (Jacobi.visitConstant a).toReal*((800*r/y)^(alpha a)) := by
+  rw [Real.div_rpow (by positivity : 0≤800*r) hy.le,
+    Real.mul_rpow (by norm_num : (0:ℝ)≤800) hr.le,
+    div_eq_mul_inv,←Real.rpow_neg hy.le]
+  unfold cellConstant
+  ring
+
+lemma cleanup_near [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0<z.im) (n : ℕ) :
+    W.real (M.cleanupEvent z n) ≤
+      M.cellConstant*(M.finalRadius n)^(alpha a)*z.im^(-alpha a) := by
+  have hrpos := M.finalRadius_pos n
+  rw [M.cell_factor _ _ (M.finalRadius_pos n) hz]
+  have hhit (hy : 800*M.finalRadius n≤z.im) :
+      W.real {b | radialTime a b.1 z (800*M.finalRadius n)≠∞} ≤
+        max 1 (Jacobi.visitConstant a).toReal*((800*M.finalRadius n/z.im)^(alpha a)) := by
+    have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top (Jacobi.visitConstant_lt_top a).ne ENNReal.ofReal_ne_top)
+      (Jacobi.radial_hit_probability_radius W hW a ha z hz _
+        (by have := M.finalRadius_pos n; positivity) hy)
+    rw [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity)] at hh
+    exact hh.trans (mul_le_mul_of_nonneg_right (le_max_right ..) (by positivity))
+  unfold cleanupEvent
+  split_ifs with h hy
+  · exact (measureReal_mono inter_subset_right).trans (hhit (M.finalRadius_height z hz n h.1 h.2))
+  · exact hhit hy
+  · change (W univ).toReal ≤ _
+    rw [measure_univ, ENNReal.toReal_one]
+    have hr : 1≤800*M.finalRadius n/z.im := (one_le_div hz).mpr (not_le.mp hy).le
+    exact (mul_le_mul (le_max_left 1 _) (Real.one_le_rpow hr (alpha_pos ha).le) zero_le_one (zero_le_one.trans (le_max_left ..))) |>.trans' (by norm_num)
+
+lemma cleanup_far [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0<z.im) (n : ℕ) (hr : BatchReady a n)
+    (hy : Real.exp (-(n:ℝ)) ≤ z.im) :
+    W.real (M.cleanupEvent z n) ≤
+      M.cellConstant*(M.finalRadius n)^(alpha a)*z.im^(-alpha a)*
+        Real.exp (-((2:ℝ)^n/16)) := by
+  have hrpos := M.finalRadius_pos n
+  rw [cleanupEvent,ite_eq_left ⟨hr,hy⟩,M.cell_factor _ _ (M.finalRadius_pos n) hz]
+  have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top
+    (ENNReal.mul_ne_top (Jacobi.visitConstant_lt_top a).ne ENNReal.ofReal_ne_top) ENNReal.ofReal_ne_top)
+    (M.batch_failure_chordal hW z hz n hr hy)
+  rw [ENNReal.toReal_mul,ENNReal.toReal_mul,
+    ENNReal.toReal_ofReal (by positivity),ENNReal.toReal_ofReal (Real.exp_pos _).le] at hh
+  exact hh.trans (mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_right (le_max_right ..) (by positivity)) (Real.exp_pos _).le)
+
+ 
+
+lemma cleanup_covers (z : ℂ) (hz : 0<z.im) (n : ℕ) :
+    ∀ᵐ b∂W,∀T : ℝ≥0,∀v∈closedBall z (M.finalRadius n),
+      v∈frontier {w | alive a b.1 T w} →
+      (∃i<M.testCount n,M.candidate n i z b) ∨ b∈M.cleanupEvent z n := by
+  by_cases hh : BatchReady a n ∧ Real.exp (-(n:ℝ)) ≤ z.im
+  · filter_upwards [M.batch_candidates z hz n hh.1 hh.2] with b hb T v hv hf
+    by_cases hc : ∃i<M.testCount n,M.candidate n i z b
+    · exact Or.inl hc
+    · refine Or.inr ?_
+      rw [cleanupEvent,ite_eq_left hh]
+      refine ⟨hb hc,?_⟩
+      exact radial_crossing_of_frontier_visit (by linarith) b hz (M.finalRadius_pos n)
+        (M.finalRadius_height z hz n hh.1 hh.2)
+        ((closedBall_subset_ball (by linarith [M.finalRadius_pos n])) hv) hf
+  · apply Eventually.of_forall
+    intro b T v hv hf
+    right
+    rw [cleanupEvent,ite_eq_right hh]
+    split_ifs with hy
+    · exact radial_crossing_of_frontier_visit (by linarith) b hz (M.finalRadius_pos n) hy
+        ((closedBall_subset_ball (by linarith [M.finalRadius_pos n])) hv) hf
+    · exact mem_univ b
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.CoverCost
+open Deterministic
+
+lemma hausdorff_le_liminf_bound {ι : ℕ→Type*} [∀n,Countable (ι n)]
+    (h : ℝ→ℝ) (E : Set ℂ) (r : ℕ→ℝ≥0∞) (hr : Tendsto r atTop (𝓝 0))
+    (U : ∀n,ι n→Set ℂ) (C : ℕ→ℝ≥0∞)
+    (hdiam : ∀ᶠ n in atTop,∀i,ediam (U n i) ≤ r n)
+    (hcover : ∀ᶠ n in atTop,E ⊆ ⋃i,U n i)
+    (hcost : ∀ᶠ n in atTop,(∑'i,diameterCost h (ediam (U n i))) ≤ C n) :
+    Target.hausdorff h E ≤ liminf C atTop :=
+  (hausdorff_le_liminf_cost h E r hr U hdiam hcover).trans (liminf_le_liminf hcost)
+
+ 
+
+lemma hausdorff_le_liminf_of_exists {ι : ℕ→Type*} [∀n,Countable (ι n)]
+    (h : ℝ→ℝ) (E : Set ℂ) (r : ℕ→ℝ≥0∞) (hr : Tendsto r atTop (𝓝 0))
+    (C : ℕ→ℝ≥0∞)
+    (hcovers : ∀ᶠ n in atTop,∃U : ι n→Set ℂ,
+      (∀i,ediam (U i)≤r n) ∧ E⊆⋃i,U i ∧
+      (∑'i,diameterCost h (ediam (U i)))≤C n) :
+    Target.hausdorff h E ≤ liminf C atTop := by
+  classical
+  let U : ∀n,ι n→Set ℂ := fun n=>Classical.epsilon (fun U : ι n→Set ℂ=>
+    (∀i,ediam (U i)≤r n) ∧ E⊆⋃i,U i ∧
+    (∑'i,diameterCost h (ediam (U i)))≤C n)
+  have hU : ∀ᶠ n in atTop,(∀i,ediam (U n i)≤r n) ∧ E⊆⋃i,U n i ∧
+      (∑'i,diameterCost h (ediam (U n i)))≤C n := by
+    filter_upwards [hcovers] with n hn
+    exact Classical.epsilon_spec hn
+  exact hausdorff_le_liminf_bound h E r hr U C (hU.mono fun _ hn=>hn.1)
+    (hU.mono fun _ hn=>hn.2.1) (hU.mono fun _ hn=>hn.2.2)
+
+ 
+
+theorem liminf_cost_bound {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (C : ℕ→Ω→ℝ≥0∞) (hC : ∀n,Measurable (C n)) (B : ℝ≥0∞)
+    (hB : B<∞) (hbound : ∀ᶠ n in atTop,(∫⁻ω,C n ω∂P) ≤ B) :
+    Measurable (fun ω=>liminf (fun n=>C n ω) atTop) ∧
+    (∫⁻ω,liminf (fun n=>C n ω) atTop∂P) ≤ B ∧
+    ∀ᵐ ω∂P,liminf (fun n=>C n ω) atTop<∞ := by
+  have hmeas := Measurable.liminf hC
+  have hI : (∫⁻ω,liminf (fun n=>C n ω) atTop∂P) ≤ B :=
+    (lintegral_liminf_le hC).trans (liminf_le_of_frequently_le hbound.frequently)
+  exact ⟨hmeas,hI,ae_lt_top hmeas (hI.trans_lt hB).ne⟩
+
+ 
+
+theorem ae_hausdorff_finite_of_covers {Ω : Type*} [MeasurableSpace Ω]
+    {ι : ℕ→Type*} [∀n,Countable (ι n)]
+    (P : Measure Ω) (h : ℝ→ℝ) (E : Ω→Set ℂ)
+    (r : ℕ→ℝ≥0∞) (hr : Tendsto r atTop (𝓝 0))
+    (U : ∀n,Ω→ι n→Set ℂ) (C : ℕ→Ω→ℝ≥0∞)
+    (hC : ∀n,Measurable (C n)) (B : ℝ≥0∞) (hB : B<∞)
+    (hbound : ∀ᶠ n in atTop,(∫⁻ω,C n ω∂P) ≤ B)
+    (hdiam : ∀ᵐ ω∂P,∀ᶠ n in atTop,∀i,ediam (U n ω i) ≤ r n)
+    (hcover : ∀ᵐ ω∂P,∀ᶠ n in atTop,E ω ⊆ ⋃i,U n ω i)
+    (hcost : ∀ᵐ ω∂P,∀ᶠ n in atTop,(∑'i,diameterCost h (ediam (U n ω i))) ≤ C n ω) :
+    (∀ᵐ ω∂P,Target.hausdorff h (E ω) ≤ liminf (fun n=>C n ω) atTop) ∧
+    (∫⁻ω,liminf (fun n=>C n ω) atTop∂P) ≤ B ∧
+    ∀ᵐ ω∂P,Target.hausdorff h (E ω)<∞ := by
+  obtain ⟨hmeas,hI,hfin⟩ := liminf_cost_bound P C hC B hB hbound
+  have hdom : ∀ᵐ ω∂P,Target.hausdorff h (E ω) ≤ liminf (fun n=>C n ω) atTop := by
+    filter_upwards [hdiam,hcover,hcost] with ω hd hc hb
+    exact hausdorff_le_liminf_bound h (E ω) r hr (fun n=>U n ω) (fun n=>C n ω) hd hc hb
+  exact ⟨hdom,hI,by filter_upwards [hdom,hfin] with ω hd hf; exact hd.trans_lt hf⟩
+
+end SLEExactGauge.CoverCost
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ModelFiniteTests.lean b/lean/OAI/Probability/SLEGauge/ModelFiniteTests.lean
index 00beac4ba7214a9928fdbd857d666dfb4387d64c..a92f65aec7908904d2b9568a503362111fb297e4 100644
--- a/lean/OAI/Probability/SLEGauge/ModelFiniteTests.lean
+++ b/lean/OAI/Probability/SLEGauge/ModelFiniteTests.lean
@@ -1,559 +1,559 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TailBatches
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator LoewnerRestart CompactSequence
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-lemma subdisk_measurable (l : ℕ) : Measurable (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) := by
-  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (M.mass v b) := M.canonical.finite_compact v b
-  exact subdiskMass_measurable a ha M.mass M.mass_measurable _ l
-
-lemma subdisk_raw_measurable (θ : Angle) (l : ℕ) :
-    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>subdiskMass a M.mass (subdiskRadius M.radiusIndex) l (θ,b)) := by
-  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (M.mass v b) := M.canonical.finite_compact v b
-  exact subdiskMass_raw_measurable _ a ha M.mass θ l
-    (M.canonical.optional_measurable _ (centeredTime_stopping a ha θ l))
-
-lemma subdisk_ge_law (l : ℕ) (H : ℝ≥0∞) :
-    (finiteAngleLaw W a l) {p | H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}=
-      M.law {x | H ≤ x l} := by
-  have he := lintegral_coordinate M.law (finiteAngleLaw W a l)
-    (subdiskMass a M.mass (subdiskRadius M.radiusIndex)) M.subdisk_measurable l (M.finite_laws l)
-    ((Ici H).indicator (fun _=>(1:ℝ≥0∞))) (measurable_const.indicator measurableSet_Ici)
-  have he₁ : (fun x : ℕ→ℝ≥0∞=>(Ici H).indicator (fun _=>(1:ℝ≥0∞)) (x l))=
-      {x : ℕ→ℝ≥0∞ | H ≤ x l}.indicator (fun _=>(1:ℝ≥0∞)) := by ext x; rfl
-  have he₂ : (fun p=>(Ici H).indicator (fun _=>(1:ℝ≥0∞))
-      (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p))=
-      {p | H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}.indicator (fun _=>(1:ℝ≥0∞)) := by ext x; rfl
-  rw [he₁,he₂,lintegral_indicator (measurableSet_le measurable_const (measurable_pi_apply l)),
-    lintegral_indicator (measurableSet_le measurable_const (M.subdisk_measurable l)),lintegral_one,lintegral_one,Measure.restrict_apply_univ,Measure.restrict_apply_univ] at he
-  exact he.symm
-
-lemma subdisk_tail_le (l : ℕ) (H : ℝ) (_hH : 0 ≤ H) :
-    (finiteAngleLaw W a l) {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p} ≤ 
-      M.law {x | H ≤ sequenceMassReal x} := by
-  rw [M.subdisk_ge_law]
-  apply measure_mono_ae
-  filter_upwards [M.increasing_finite] with x hx
-  intro hh
-  exact (ENNReal.ofReal_le_iff_le_toReal hx.2.ne).mp (hh.trans (le_iSup x l))
-
- 
-
-lemma normalized_tail_le [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0<z.im) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.im*Real.exp (-2*a*s))
-    (l : ℕ) (H : ℝ) (hH : 0 ≤ H) :
-    (radialLaw W a z (σ*Real.exp (-2*a*l)))
-      {b | ENNReal.ofReal H ≤ LoewnerRestart.normalizedTest a z σ
-        (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b} ≤ 
-      2*M.law {x | H ≤ sequenceMassReal x} := by
-  let E : Set (Angle×GDriver) := {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}
-  have hE : MeasurableSet E := measurableSet_le measurable_const (M.subdisk_measurable l)
-  have hm := normalizedTest_measurable a ha z hz σ hσ hσz _ (M.subdisk_measurable l)
-  have hF : MeasurableSet {b | ENNReal.ofReal H ≤ LoewnerRestart.normalizedTest a z σ
-      (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b} :=
-    measurableSet_le measurable_const hm
-  have hh := (normalizedTest_conditional_bounds W hW a ha z hz z.im σ hσ hσz le_rfl
-    s hs hgap l univ MeasurableSet.univ (E.indicator (fun _=>(1:ℝ≥0∞)))
-      (measurable_const.indicator hE)).2
-  have he : LoewnerRestart.normalizedTest a z σ (E.indicator (fun _=>(1:ℝ≥0∞)))=
-      {b | ENNReal.ofReal H ≤ LoewnerRestart.normalizedTest a z σ
-        (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b}.indicator (fun _=>(1:ℝ≥0∞)) := by
-    funext b
-    rfl
-  have := radialLaw_probability W hW a ha z hz (σ*Real.exp (-2*a*l))
-    (mul_pos hσ (Real.exp_pos _)) ((mul_le_of_le_one_right hσ.le
-      (Real.exp_le_one_iff.mpr (by have := Nat.cast_nonneg (α:=ℝ) l; nlinarith))).trans hσz)
-  rw [he,setLIntegral_univ,lintegral_indicator hF,lintegral_indicator hE,lintegral_one,lintegral_one,
-    Measure.restrict_apply_univ,Measure.restrict_apply_univ,measure_univ,mul_one] at hh
-  exact hh.trans (by gcongr; exact M.subdisk_tail_le l H hH)
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain
-
-lemma hasDerivWithinAt_primitive_Icc {f : ℝ→ℂ} {T u : ℝ} (hT : 0≤T)
-    (hf : ContinuousOn f (Icc 0 T)) (hu : u∈Icc 0 T) :
-    HasDerivWithinAt (fun v=>∫s in (0:ℝ)..v,f s) (f u) (Icc 0 T) u := by
-  let F : ℝ→ℂ := fun s=>f (projIcc 0 T hT s)
-  have hF : Continuous F := hf.domRestrict.comp continuous_projIcc
-  have he (s : ℝ) (hs : s∈Icc 0 T) : F s=f s := by
-    dsimp [F]; rw [projIcc_of_mem _ hs]
-  have hi (v : ℝ) (hv : v∈Icc 0 T) : (∫s in (0:ℝ)..v,F s)=∫s in (0:ℝ)..v,f s := by
-    apply intervalIntegral.integral_congr
-    intro s hs
-    rw [uIcc_of_le hv.1] at hs
-    exact he s ⟨hs.1,hs.2.trans hv.2⟩
-  have hd := intervalIntegral.integral_hasDerivAt_right
-    (hF.intervalIntegrable 0 u) hF.aestronglyMeasurable.stronglyMeasurableAtFilter hF.continuousAt
-  rw [he u hu] at hd
-  exact hd.hasDerivWithinAt.congr_of_mem (fun v hv=>(hi v hv).symm) hu
-
- 
-def logVariation (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (u : ℝ) : ℂ :=
-  ∫s in (0:ℝ)..u,-(a:ℂ)/(loewnerMap a b s.toNNReal z-(b s.toNNReal:ℂ))^2
-
-def firstVariation (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (u : ℝ) : ℂ :=
-  Complex.exp (logVariation a b z u)
-
-lemma firstVariation_eq_deriv {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {u : ℝ≥0} (hu : alive a b u z) :
-    firstVariation a b z u=deriv (loewnerMap a b u) z :=
-  (hasDerivAt_loewnerMap hb hu).deriv.symm
-
-lemma logVariation_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (logVariation a b z)
-      (-(a:ℂ)/(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^2) (Icc 0 (T:ℝ)) u := by
-  obtain ⟨f,hf⟩ := hT
-  apply hasDerivWithinAt_primitive_Icc T.coe_nonneg _ hu
-  have hc : ContinuousOn (fun s:ℝ=>-(a:ℂ)/(f s-(b s.toNNReal:ℂ))^2) (Icc 0 (T:ℝ)) :=
-    continuousOn_const.div ((odeUntil_center_continuousOn hb hf).pow 2)
-      (fun s hs=>pow_ne_zero _ (odeUntil_center_ne_zero hf hs))
-  exact hc.congr (fun s hs=>by rw [loewnerMap_eq_solution_time hf hs])
-
-lemma firstVariation_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (firstVariation a b z)
-      (-(a:ℂ)/(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^2 * firstVariation a b z u)
-      (Icc 0 (T:ℝ)) u := by
-  convert! (logVariation_hasDerivWithinAt hb hT hu).cexp using 1; first | rfl | (dsimp only [firstVariation]; ring)
-
-lemma firstVariation_continuousOn {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
-    ContinuousOn (firstVariation a b z) (Icc 0 (T:ℝ)) :=
-  fun _ hu=>(firstVariation_hasDerivWithinAt hb hT hu).continuousWithinAt
-
- 
-def secondVariationIntegral (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (t : ℝ) : ℂ :=
-  ∫u in (0:ℝ)..t,2*(a:ℂ)*firstVariation a b z u /
-    (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^3
-
-lemma secondVariationIntegral_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (secondVariationIntegral a b z)
-      (2*(a:ℂ)*firstVariation a b z u /
-        (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^3) (Icc 0 (T:ℝ)) u := by
-  have hc := firstVariation_continuousOn hb hT
-  obtain ⟨f,hf⟩ := hT
-  apply hasDerivWithinAt_primitive_Icc T.coe_nonneg _ hu
-  have hd : ContinuousOn (fun s=>2*(a:ℂ)*firstVariation a b z s/(f s-(b s.toNNReal:ℂ))^3)
-      (Icc 0 (T:ℝ)) := (continuousOn_const.mul hc).div ((odeUntil_center_continuousOn hb hf).pow 3)
-    (fun s hs=>pow_ne_zero _ (odeUntil_center_ne_zero hf hs))
-  exact hd.congr (fun s hs=>by rw [loewnerMap_eq_solution_time hf hs])
-
-lemma secondVariationIntegral_continuousOn {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
-    ContinuousOn (secondVariationIntegral a b z) (Icc 0 (T:ℝ)) :=
-  fun _ hu=>(secondVariationIntegral_hasDerivWithinAt hb hT hu).continuousWithinAt
-
- 
-
-theorem hasDerivAt_deriv_loewnerMap {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
-    HasDerivAt (deriv (loewnerMap a b T))
-      (firstVariation a b z T * secondVariationIntegral a b z T) z := by
-  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hT
-  let Z : ℂ→ℝ→ℂ := fun w u=>heightClamp ε (F u.toNNReal w-(b u.toNNReal:ℂ))
-  let K : ℂ→ℝ→ℂ := fun w u=>-(a:ℂ)/(Z w u)^2
-  let D : ℂ→ℝ→ℂ := fun w u=>Complex.exp (∫v in (0:ℝ)..u,K w v)
-  let P : ℂ→ℝ→ℂ := fun w u=>2*(a:ℂ)*D w u/(Z w u)^3
-  have hZ (w : ℂ) : Continuous (Z w) := (heightClamp_lipschitz ε).continuous.comp
-    (((hFc w).comp continuous_real_toNNReal).sub
-      (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))
-  have hZn (w : ℂ) (u : ℝ) : Z w u≠0 := heightClamp_ne_zero hε _
-  have hK (w : ℂ) : Continuous (K w) := continuous_const.div ((hZ w).pow 2)
-    (fun u=>pow_ne_zero _ (hZn w u))
-  have hD (w : ℂ) : Continuous (D w) := by
-    apply Complex.continuous_exp.comp
-    exact intervalIntegral.continuous_primitive (hK w).intervalIntegrable 0
-  have hP (w : ℂ) : Continuous (P w) := (continuous_const.mul (hD w)).div ((hZ w).pow 3)
-    (fun u=>pow_ne_zero _ (hZn w u))
-  have htime {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) : u.toNNReal≤T := by
-    simpa using Real.toNNReal_le_toNNReal hu.2
-  have hZe (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) :
-      Z w u=loewnerMap a b u.toNNReal w-(b u.toNNReal:ℂ) := by
-    rw [hmap w hw _ (htime hu)]
-    exact heightClamp_eq (by simpa using (hIm w hw _ (htime hu)).le)
-  have hDe (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) :
-      D w u=firstVariation a b w u := by
-    dsimp only [D,firstVariation,logVariation]
-    apply congrArg Complex.exp
-    apply intervalIntegral.integral_congr
-    intro v hv
-    rw [uIcc_of_le hu.1] at hv
-    change -(a:ℂ)/(Z w v)^2= _
-    rw [hZe w hw v ⟨hv.1,hv.2.trans hu.2⟩]
-  have hDd (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) :
-      D w u=deriv (loewnerMap a b u.toNNReal) w := by
-    rw [hDe w hw u hu]
-    convert! firstVariation_eq_deriv hb ⟨_,odeUntil_mono (hode w hw) (htime hu)⟩ using 1
-    rw [Real.coe_toNNReal _ hu.1]
-  have hd (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) (w : ℂ) (hw : w∈ball z δ) :
-      HasDerivAt (fun w=>K w u) (P w u) w := by
-    have h := ((hasDerivAt_loewnerMap hb
-      (show alive a b u.toNNReal w from ⟨_,odeUntil_mono (hode w hw) (htime hu)⟩)).sub_const
-        (b u.toNNReal:ℂ)).pow 2
-    rw [←(hasDerivAt_loewnerMap hb
-      (show alive a b u.toNNReal w from ⟨_,odeUntil_mono (hode w hw) (htime hu)⟩)).deriv,
-      ←hDd w hw u hu] at h
-    change HasDerivAt (fun x=>(loewnerMap a b u.toNNReal x-(b u.toNNReal:ℂ))^2)
-      ((2:ℂ)*(loewnerMap a b u.toNNReal w-(b u.toNNReal:ℂ))^(2-1)*D w u) w at h
-    have hq := (hasDerivAt_const w (-(a:ℂ))).div h
-      (by rw [←hZe w hw u hu]; exact pow_ne_zero _ (hZn w u))
-    have he : (fun x=>K x u)=ᶠ[𝓝 w]
-        (fun x=>-(a:ℂ)/(loewnerMap a b u.toNNReal x-(b u.toNNReal:ℂ))^2) := by
-      filter_upwards [isOpen_ball.mem_nhds hw] with x hx
-      dsimp only [K]; rw [hZe x hx u hu]
-    convert! hq.congr_of_eventuallyEq he using 1
-    dsimp only [P]
-    rw [hZe w hw u hu]
-    simp only [Nat.reduceSub,pow_one]
-    field_simp
-    ring
-  let L : ℝ := Real.exp ((2*|a|/ε^2)*(T:ℝ))
-  have hL : 0<L := Real.exp_pos _
-  have hDb (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) : ‖D w u‖≤L := by
-    rw [hDd w hw u hu]
-    have hLip : LipschitzOnWith ⟨L,hL.le⟩ (loewnerMap a b u.toNNReal) (ball z δ) := by
-      apply LipschitzOnWith.of_dist_le_mul
-      intro x hx y hy
-      change dist (loewnerMap a b u.toNNReal x) (loewnerMap a b u.toNNReal y)≤L*dist x y
-      rw [hmap x hx _ (htime hu),hmap y hy _ (htime hu)]
-      apply (cutoff_loewner_dist hε hb hFc hFe x y u.toNNReal).trans
-      rw [Real.coe_toNNReal _ hu.1]
-      exact (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr
-        (mul_le_mul_of_nonneg_left hu.2 (by positivity))) dist_nonneg).trans_eq (mul_comm _ _)
-    exact norm_deriv_le_of_lipschitzOn (isOpen_ball.mem_nhds hw) hLip
-  have hPb (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) (w : ℂ) (hw : w∈ball z δ) :
-      ‖P w u‖≤2 * |a| * L/ε^3 := by
-    have hn : ε≤‖Z w u‖ := heightClamp_norm_lower ε _
-    have hn0 : 0<‖Z w u‖ := hε.trans_le hn
-    dsimp only [P]
-    rw [norm_div,norm_mul,norm_mul,norm_pow,Complex.norm_real,Real.norm_eq_abs,Complex.norm_ofNat]
-    exact div_le_div₀ (by positivity) (by gcongr; exact hDb w hw u hu)
-      (by positivity) (by gcongr)
-  have hdiff := intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
-    (F := K) (F' := P) (bound := fun _=>2 * |a| * L/ε^3) (a := 0) (b := (T:ℝ))
-    (ball_mem_nhds z hδ)
-    (Eventually.of_forall (fun w=>(hK w).aestronglyMeasurable.restrict))
-    ((hK z).intervalIntegrable 0 T) (hP z).aestronglyMeasurable.restrict
-    (ae_of_all volume (fun u hu=>hPb u (by rw [uIoc_of_le T.coe_nonneg] at hu; exact Ioc_subset_Icc_self hu)))
-    intervalIntegrable_const
-    (ae_of_all volume (fun u hu=>hd u (by rw [uIoc_of_le T.coe_nonneg] at hu; exact Ioc_subset_Icc_self hu)))
-  have he : (fun w=>D w T)=ᶠ[𝓝 z] (deriv (loewnerMap a b T)) := by
-    filter_upwards [ball_mem_nhds z hδ] with w hw
-    simpa only [Real.toNNReal_coe] using hDd w hw T ⟨T.coe_nonneg,le_rfl⟩
-  have hfinal := hdiff.2.cexp.congr_of_eventuallyEq he.symm
-  have hpi : (∫u in (0:ℝ)..(T:ℝ),P z u)=secondVariationIntegral a b z T := by
-    apply intervalIntegral.integral_congr
-    intro u hu
-    rw [uIcc_of_le T.coe_nonneg] at hu
-    dsimp only [P,secondVariationIntegral]
-    rw [hDe z (mem_ball_self hδ) u hu,hZe z (mem_ball_self hδ) u hu]
-  rw [hpi] at hfinal
-  convert! hfinal using 1
-  change firstVariation a b z T * secondVariationIntegral a b z T =
-    D z T * secondVariationIntegral a b z T
-  rw [hDe z (mem_ball_self hδ) T ⟨T.coe_nonneg,le_rfl⟩]
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Topology
-open scoped RealInnerProductSpace InnerProductSpace
-namespace SLEExactGauge.LoewnerGeometry
-
-lemma real_inner_mul_self (v c : ℂ) : ⟪v,c*v⟫_ℝ=c.re*‖v‖^2 := by
-  rw [real_inner_eq_re_inner ℂ]
-  change (inner ℂ v (c • v)).re = _
-  rw [inner_smul_right,inner_self_eq_norm_sq_to_K]
-  change (c*(‖v‖:ℂ)^2).re = _
-  simp only [pow_two,Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im]
-  ring
-
-lemma norm_le_of_affine_dissipative {B c f : ℝ→ℂ} {T R : ℝ}
-    (hT : 0≤T) (hR : 0≤R)
-    (hd : ∀ u∈Icc 0 T, HasDerivWithinAt B (c u*B u+f u) (Icc 0 T) u)
-    (h0 : ‖B 0‖≤R)
-    (hc : ∀ u∈Icc 0 T,(c u).re<0)
-    (hf : ∀ u∈Icc 0 T,‖f u‖≤-R*(c u).re) : ‖B T‖≤R := by
-  have cont : ContinuousOn (fun u=>‖B u‖^2) (Icc 0 T) :=
-    fun u hu=>(hd u hu).norm_sq.continuousWithinAt
-  have deriv (u : ℝ) (hu : u∈Ico 0 T) :
-      HasDerivWithinAt (fun u=>‖B u‖^2)
-        (2*⟪B u,c u*B u+f u⟫_ℝ) (Ici u) u := by
-    apply (hd u ⟨hu.1,hu.2.le⟩).norm_sq.mono_of_mem_nhdsWithin
-    filter_upwards [self_mem_nhdsWithin,
-      (show ∀ᶠ v:ℝ in 𝓝[Ici u] u,v<T from nhdsWithin_le_nhds (Iio_mem_nhds hu.2))]
-      with v hv hvT
-    exact ⟨hu.1.trans hv,hvT.le⟩
-  apply le_of_forall_pos_le_add
-  intro ε hε
-  have hpos : 0<R+ε := by linarith
-  have barrier := image_le_of_deriv_right_lt_deriv_boundary cont deriv
-    (B := fun _=> (R+ε)^2) (B' := fun _=>0)
-    (by nlinarith [norm_nonneg (B 0)])
-    (fun _=>hasDerivAt_const _ _)
-    (by
-      intro u hu he
-      have hh : ‖B u‖=R+ε := (sq_eq_sq₀ (norm_nonneg _) hpos.le).mp he
-      have hcf := hc u ⟨hu.1,hu.2.le⟩
-      have hb := (real_inner_le_norm (B u) (f u)).trans
-        (mul_le_mul_of_nonneg_left (hf u ⟨hu.1,hu.2.le⟩) (norm_nonneg _))
-      rw [inner_add_right,real_inner_mul_self]
-      rw [hh] at hb ⊢
-      have hn : (c u).re*(R+ε)*ε<0 := mul_neg_of_neg_of_pos
-        (mul_neg_of_neg_of_pos hcf hpos) hε
-      nlinarith)
-  have ht := barrier (right_mem_Icc.mpr hT)
-  nlinarith [norm_nonneg (B T)]
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain LoewnerRestart
-
-lemma loewner_time_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ}
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (fun t=>loewnerMap a b t.toNNReal z)
-      ((a:ℂ)/(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) u := by
-  obtain ⟨f,hf⟩ := hT
-  rw [loewnerMap_eq_solution_time hf hu]
-  exact (hf.2 u hu).2.congr_of_mem (fun v hv=>loewnerMap_eq_solution_time hf hv) hu
-
-lemma loewner_height_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ}
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (fun t=>(loewnerMap a b t.toNNReal z).im)
-      (-a*(loewnerMap a b u.toNNReal z).im /
-        Complex.normSq (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) u := by
-  have hd := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u
-    (loewner_time_hasDerivWithinAt hT hu)
-  convert! hd using 1
-  simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
-    Complex.sub_im,sub_zero,zero_mul,zero_div,zero_sub]
-  ring
-
-def coefficientDrift (a : ℝ) (Z : ℂ) : ℂ :=
-  -(a:ℂ)/(Complex.normSq Z:ℂ)+(a:ℂ)/Z^2
-
-def coefficientForcing (a : ℝ) (Z : ℂ) : ℂ :=
-  -coefficientDrift a Z-Complex.I*(2*(a:ℂ)*(Z.im:ℂ)/Z^3)
-
-lemma coefficientDrift_re (a : ℝ) {Z : ℂ} (hZ : Z≠0) :
-    (coefficientDrift a Z).re= -(2*a*Z.im^2/(Complex.normSq Z)^2) := by
-  have hn : Complex.normSq Z≠0 := mt Complex.normSq_eq_zero.mp hZ
-  dsimp only [coefficientDrift]
-  simp only [Complex.add_re,Complex.neg_re,Complex.div_re,Complex.ofReal_re,
-    Complex.ofReal_im,Complex.normSq_ofReal,
-    pow_two,Complex.mul_re,Complex.mul_im]
-  field_simp
-  simp only [Complex.normSq_apply,pow_two,Complex.mul_re,Complex.mul_im]
-  ring
-
-lemma coefficientForcing_eq (a : ℝ) {Z : ℂ} (hZ : Z≠0) :
-    coefficientForcing a Z =
-      (-(4*a*Z.im^2/(Complex.normSq Z)^2):ℝ)*(star Z/Z) := by
-  have hN : (Complex.normSq Z:ℂ)=Z*star Z := (Complex.mul_conj Z).symm
-  have hY : (Z.im:ℂ)=(Z-star Z)/(2*Complex.I) := Complex.im_eq_sub_conj Z
-  have hZn : star Z≠0 := star_ne_zero.mpr hZ
-  simp only [Complex.ofReal_neg,Complex.ofReal_div,Complex.ofReal_mul,Complex.ofReal_pow,
-    Complex.ofReal_ofNat]
-  dsimp only [coefficientForcing,coefficientDrift]
-  rw [hN,hY]
-  field_simp
-  simp only [Complex.I_sq]
-  ring_nf
-
-lemma coefficientForcing_norm {a : ℝ} (ha : 0<a) {Z : ℂ} (hZ : Z≠0) :
-    ‖coefficientForcing a Z‖ = -2*(coefficientDrift a Z).re := by
-  rw [coefficientForcing_eq a hZ,norm_mul,norm_div,norm_star,div_self (norm_ne_zero_iff.mpr hZ),
-    mul_one,Complex.norm_real,Real.norm_eq_abs,abs_neg,abs_of_nonneg (by positivity),
-    coefficientDrift_re a hZ]
-  ring
-
- 
-
-def secondCoefficient (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (u : ℝ) : ℂ :=
-  1-Complex.I*((loewnerMap a b u.toNNReal z).im * secondVariationIntegral a b z u /
-    firstVariation a b z u)
-
-lemma secondCoefficient_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
-    HasDerivWithinAt (secondCoefficient a b z)
-      (coefficientDrift a (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))*secondCoefficient a b z u+
-        coefficientForcing a (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) u := by
-  have htime : u.toNNReal≤T := by simpa using Real.toNNReal_le_toNNReal hu.2
-  have hz : loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ)≠0 := by
-    obtain ⟨f,hf⟩ := hT
-    rw [loewnerMap_eq_solution_time hf hu]
-    exact odeUntil_center_ne_zero hf hu
-  have hD : firstVariation a b z u≠0 := Complex.exp_ne_zero _
-  have hy := (loewner_height_hasDerivWithinAt hT hu).ofReal_comp
-  have hp := secondVariationIntegral_hasDerivWithinAt hb hT hu
-  have hd := firstVariation_hasDerivWithinAt hb hT hu
-  have h := (((hy.mul hp).div hd hD).const_mul Complex.I).const_sub 1
-  convert! h using 1
-  dsimp only [secondCoefficient,coefficientDrift,coefficientForcing,Pi.mul_apply]
-  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.ofReal_neg,
-    Complex.ofReal_mul,Complex.ofReal_div]
-  field_simp
-  ring
-
-lemma secondCoefficient_zero (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) :
-    secondCoefficient a b z 0=1 := by
-  simp [secondCoefficient,secondVariationIntegral]
-
- 
-theorem secondCoefficient_norm_le {a : ℝ} (ha : 0<a) {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) : ‖secondCoefficient a b z T‖≤2 := by
-  apply norm_le_of_affine_dissipative T.coe_nonneg (by norm_num)
-    (fun u hu=>secondCoefficient_hasDerivWithinAt hb hT hu)
-  · rw [secondCoefficient_zero]; norm_num
-  · intro u hu
-    obtain ⟨f,hf⟩ := hT
-    have hy : 0<(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ)).im := by
-      rw [loewnerMap_eq_solution_time hf hu]
-      simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using (hf.2 u hu).1
-    have hn := odeUntil_center_ne_zero hf hu
-    rw [←loewnerMap_eq_solution_time hf hu] at hn
-    rw [coefficientDrift_re a hn]
-    exact neg_neg_of_pos (div_pos (by positivity) (sq_pos_of_ne_zero
-      (mt Complex.normSq_eq_zero.mp hn)))
-  · intro u hu
-    obtain ⟨f,hf⟩ := hT
-    have hn := odeUntil_center_ne_zero hf hu
-    rw [←loewnerMap_eq_solution_time hf hu] at hn
-    exact (coefficientForcing_norm ha hn).le
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerGeometry
-open ExactSLEMain LoewnerRestart FlowIdentification
-
-lemma hasDerivAt_deriv_inverseMap {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    HasDerivAt (deriv (inverseMap a b T))
-      (-secondVariationIntegral a b.1 (inverseMap a b T w) T /
-        (firstVariation a b.1 (inverseMap a b T w) T)^2) w := by
-  have hz := inverseMap_alive ha b T hw
-  have hg := hasDerivAt_deriv_loewnerMap b.1.continuous hz
-  have hf := (inverseMap_hasStrictDerivAt ha b T hw).hasDerivAt
-  rw [←firstVariation_eq_deriv b.1.continuous hz] at hf
-  have hD : firstVariation a b.1 (inverseMap a b T w) T≠0 := Complex.exp_ne_zero _
-  have hd := (hg.comp w hf).inv (by
-    dsimp only [Function.comp_def]; rw [←firstVariation_eq_deriv b.1.continuous hz]; exact hD)
-  have he : deriv (inverseMap a b T)=ᶠ[𝓝 w]
-      (fun w=> (deriv (loewnerMap a b.1 T) (inverseMap a b T w))⁻¹) := by
-    filter_upwards [(isOpen_lt continuous_const Complex.continuous_im).mem_nhds hw] with v hv
-    exact deriv_inverseMap ha b T hv
-  convert! hd.congr_of_eventuallyEq he using 1
-  dsimp only [Function.comp_def]
-  rw [←firstVariation_eq_deriv b.1.continuous hz]
-  field_simp
-
-lemma inverseMap_second_coefficient {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    1+Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/deriv (inverseMap a b T) w)=
-      secondCoefficient a b.1 (inverseMap a b T w) T := by
-  rw [(hasDerivAt_deriv_inverseMap ha b T hw).deriv,deriv_inverseMap ha b T hw,
-    ←firstVariation_eq_deriv b.1.continuous (inverseMap_alive ha b T hw)]
-  have hD : firstVariation a b.1 (inverseMap a b T w) T≠0 := Complex.exp_ne_zero _
-  have hy := congrArg Complex.im (centered_inverseMap ha b T hw)
-  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] at hy
-  dsimp only [secondCoefficient]
-  rw [Real.toNNReal_coe,hy]
-  field_simp
-  ring
-
- 
-
-lemma inverseMap_second_coefficient_norm_le {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    ‖1+Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/deriv (inverseMap a b T) w)‖≤2 := by
-  rw [inverseMap_second_coefficient ha b T hw]
-  exact secondCoefficient_norm_le ha b.1.continuous (inverseMap_alive ha b T hw)
-
- 
-theorem inverseMap_preSchwarzian_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
-    {w : ℂ} (hw : 0<w.im) :
-    ‖deriv (deriv (inverseMap a b T)) w‖ ≤ (3/w.im)*‖deriv (inverseMap a b T) w‖ := by
-  have hn := inverseMap_second_coefficient_norm_le ha b T hw
-  have h := norm_sub_le (1+Complex.I*(w.im:ℂ)*
-    (deriv (deriv (inverseMap a b T)) w/deriv (inverseMap a b T) w)) (1:ℂ)
-  have htriangle : ‖Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/
-      deriv (inverseMap a b T) w)‖≤3 := by
-    calc
-      _ ≤ ‖1+Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/
-        deriv (inverseMap a b T) w)‖+‖(1:ℂ)‖ := by simpa only [add_sub_cancel_left] using h
-      _ ≤ 3 := by rw [norm_one]; linarith
-  rw [norm_mul,norm_mul,Complex.norm_I,one_mul,Complex.norm_real,Real.norm_eq_abs,
-    abs_of_pos hw,norm_div] at htriangle
-  have hd : 0<‖deriv (inverseMap a b T) w‖ := norm_pos_iff.mpr (deriv_inverseMap_ne_zero ha b T hw)
-  rw [div_mul_eq_mul_div]
-  apply (le_div_iff₀ hw).mpr
-  rw [←mul_div_assoc] at htriangle
-  have ha' := (div_le_iff₀ hd).mp htriangle
-  nlinarith
-
-end SLEExactGauge.LoewnerGeometry
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TailBatches
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator LoewnerRestart CompactSequence
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+lemma subdisk_measurable (l : ℕ) : Measurable (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) := by
+  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (M.mass v b) := M.canonical.finite_compact v b
+  exact subdiskMass_measurable a ha M.mass M.mass_measurable _ l
+
+lemma subdisk_raw_measurable (θ : Angle) (l : ℕ) :
+    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>subdiskMass a M.mass (subdiskRadius M.radiusIndex) l (θ,b)) := by
+  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (M.mass v b) := M.canonical.finite_compact v b
+  exact subdiskMass_raw_measurable _ a ha M.mass θ l
+    (M.canonical.optional_measurable _ (centeredTime_stopping a ha θ l))
+
+lemma subdisk_ge_law (l : ℕ) (H : ℝ≥0∞) :
+    (finiteAngleLaw W a l) {p | H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}=
+      M.law {x | H ≤ x l} := by
+  have he := lintegral_coordinate M.law (finiteAngleLaw W a l)
+    (subdiskMass a M.mass (subdiskRadius M.radiusIndex)) M.subdisk_measurable l (M.finite_laws l)
+    ((Ici H).indicator (fun _=>(1:ℝ≥0∞))) (measurable_const.indicator measurableSet_Ici)
+  have he₁ : (fun x : ℕ→ℝ≥0∞=>(Ici H).indicator (fun _=>(1:ℝ≥0∞)) (x l))=
+      {x : ℕ→ℝ≥0∞ | H ≤ x l}.indicator (fun _=>(1:ℝ≥0∞)) := by ext x; rfl
+  have he₂ : (fun p=>(Ici H).indicator (fun _=>(1:ℝ≥0∞))
+      (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p))=
+      {p | H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}.indicator (fun _=>(1:ℝ≥0∞)) := by ext x; rfl
+  rw [he₁,he₂,lintegral_indicator (measurableSet_le measurable_const (measurable_pi_apply l)),
+    lintegral_indicator (measurableSet_le measurable_const (M.subdisk_measurable l)),lintegral_one,lintegral_one,Measure.restrict_apply_univ,Measure.restrict_apply_univ] at he
+  exact he.symm
+
+lemma subdisk_tail_le (l : ℕ) (H : ℝ) (_hH : 0 ≤ H) :
+    (finiteAngleLaw W a l) {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p} ≤ 
+      M.law {x | H ≤ sequenceMassReal x} := by
+  rw [M.subdisk_ge_law]
+  apply measure_mono_ae
+  filter_upwards [M.increasing_finite] with x hx
+  intro hh
+  exact (ENNReal.ofReal_le_iff_le_toReal hx.2.ne).mp (hh.trans (le_iSup x l))
+
+ 
+
+lemma normalized_tail_le [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0<z.im) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.im*Real.exp (-2*a*s))
+    (l : ℕ) (H : ℝ) (hH : 0 ≤ H) :
+    (radialLaw W a z (σ*Real.exp (-2*a*l)))
+      {b | ENNReal.ofReal H ≤ LoewnerRestart.normalizedTest a z σ
+        (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b} ≤ 
+      2*M.law {x | H ≤ sequenceMassReal x} := by
+  let E : Set (Angle×GDriver) := {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}
+  have hE : MeasurableSet E := measurableSet_le measurable_const (M.subdisk_measurable l)
+  have hm := normalizedTest_measurable a ha z hz σ hσ hσz _ (M.subdisk_measurable l)
+  have hF : MeasurableSet {b | ENNReal.ofReal H ≤ LoewnerRestart.normalizedTest a z σ
+      (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b} :=
+    measurableSet_le measurable_const hm
+  have hh := (normalizedTest_conditional_bounds W hW a ha z hz z.im σ hσ hσz le_rfl
+    s hs hgap l univ MeasurableSet.univ (E.indicator (fun _=>(1:ℝ≥0∞)))
+      (measurable_const.indicator hE)).2
+  have he : LoewnerRestart.normalizedTest a z σ (E.indicator (fun _=>(1:ℝ≥0∞)))=
+      {b | ENNReal.ofReal H ≤ LoewnerRestart.normalizedTest a z σ
+        (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b}.indicator (fun _=>(1:ℝ≥0∞)) := by
+    funext b
+    rfl
+  have := radialLaw_probability W hW a ha z hz (σ*Real.exp (-2*a*l))
+    (mul_pos hσ (Real.exp_pos _)) ((mul_le_of_le_one_right hσ.le
+      (Real.exp_le_one_iff.mpr (by have := Nat.cast_nonneg (α:=ℝ) l; nlinarith))).trans hσz)
+  rw [he,setLIntegral_univ,lintegral_indicator hF,lintegral_indicator hE,lintegral_one,lintegral_one,
+    Measure.restrict_apply_univ,Measure.restrict_apply_univ,measure_univ,mul_one] at hh
+  exact hh.trans (by gcongr; exact M.subdisk_tail_le l H hH)
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain
+
+lemma hasDerivWithinAt_primitive_Icc {f : ℝ→ℂ} {T u : ℝ} (hT : 0≤T)
+    (hf : ContinuousOn f (Icc 0 T)) (hu : u∈Icc 0 T) :
+    HasDerivWithinAt (fun v=>∫s in (0:ℝ)..v,f s) (f u) (Icc 0 T) u := by
+  let F : ℝ→ℂ := fun s=>f (projIcc 0 T hT s)
+  have hF : Continuous F := hf.domRestrict.comp continuous_projIcc
+  have he (s : ℝ) (hs : s∈Icc 0 T) : F s=f s := by
+    dsimp [F]; rw [projIcc_of_mem _ hs]
+  have hi (v : ℝ) (hv : v∈Icc 0 T) : (∫s in (0:ℝ)..v,F s)=∫s in (0:ℝ)..v,f s := by
+    apply intervalIntegral.integral_congr
+    intro s hs
+    rw [uIcc_of_le hv.1] at hs
+    exact he s ⟨hs.1,hs.2.trans hv.2⟩
+  have hd := intervalIntegral.integral_hasDerivAt_right
+    (hF.intervalIntegrable 0 u) hF.aestronglyMeasurable.stronglyMeasurableAtFilter hF.continuousAt
+  rw [he u hu] at hd
+  exact hd.hasDerivWithinAt.congr_of_mem (fun v hv=>(hi v hv).symm) hu
+
+ 
+def logVariation (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (u : ℝ) : ℂ :=
+  ∫s in (0:ℝ)..u,-(a:ℂ)/(loewnerMap a b s.toNNReal z-(b s.toNNReal:ℂ))^2
+
+def firstVariation (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (u : ℝ) : ℂ :=
+  Complex.exp (logVariation a b z u)
+
+lemma firstVariation_eq_deriv {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {u : ℝ≥0} (hu : alive a b u z) :
+    firstVariation a b z u=deriv (loewnerMap a b u) z :=
+  (hasDerivAt_loewnerMap hb hu).deriv.symm
+
+lemma logVariation_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (logVariation a b z)
+      (-(a:ℂ)/(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^2) (Icc 0 (T:ℝ)) u := by
+  obtain ⟨f,hf⟩ := hT
+  apply hasDerivWithinAt_primitive_Icc T.coe_nonneg _ hu
+  have hc : ContinuousOn (fun s:ℝ=>-(a:ℂ)/(f s-(b s.toNNReal:ℂ))^2) (Icc 0 (T:ℝ)) :=
+    continuousOn_const.div ((odeUntil_center_continuousOn hb hf).pow 2)
+      (fun s hs=>pow_ne_zero _ (odeUntil_center_ne_zero hf hs))
+  exact hc.congr (fun s hs=>by rw [loewnerMap_eq_solution_time hf hs])
+
+lemma firstVariation_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (firstVariation a b z)
+      (-(a:ℂ)/(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^2 * firstVariation a b z u)
+      (Icc 0 (T:ℝ)) u := by
+  convert! (logVariation_hasDerivWithinAt hb hT hu).cexp using 1; first | rfl | (dsimp only [firstVariation]; ring)
+
+lemma firstVariation_continuousOn {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
+    ContinuousOn (firstVariation a b z) (Icc 0 (T:ℝ)) :=
+  fun _ hu=>(firstVariation_hasDerivWithinAt hb hT hu).continuousWithinAt
+
+ 
+def secondVariationIntegral (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (t : ℝ) : ℂ :=
+  ∫u in (0:ℝ)..t,2*(a:ℂ)*firstVariation a b z u /
+    (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^3
+
+lemma secondVariationIntegral_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (secondVariationIntegral a b z)
+      (2*(a:ℂ)*firstVariation a b z u /
+        (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))^3) (Icc 0 (T:ℝ)) u := by
+  have hc := firstVariation_continuousOn hb hT
+  obtain ⟨f,hf⟩ := hT
+  apply hasDerivWithinAt_primitive_Icc T.coe_nonneg _ hu
+  have hd : ContinuousOn (fun s=>2*(a:ℂ)*firstVariation a b z s/(f s-(b s.toNNReal:ℂ))^3)
+      (Icc 0 (T:ℝ)) := (continuousOn_const.mul hc).div ((odeUntil_center_continuousOn hb hf).pow 3)
+    (fun s hs=>pow_ne_zero _ (odeUntil_center_ne_zero hf hs))
+  exact hd.congr (fun s hs=>by rw [loewnerMap_eq_solution_time hf hs])
+
+lemma secondVariationIntegral_continuousOn {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
+    ContinuousOn (secondVariationIntegral a b z) (Icc 0 (T:ℝ)) :=
+  fun _ hu=>(secondVariationIntegral_hasDerivWithinAt hb hT hu).continuousWithinAt
+
+ 
+
+theorem hasDerivAt_deriv_loewnerMap {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
+    HasDerivAt (deriv (loewnerMap a b T))
+      (firstVariation a b z T * secondVariationIntegral a b z T) z := by
+  obtain ⟨ε,δ,F,hε,hδ,hFc,hFe,hode,hmap,hIm⟩ := loewner_local_cutoff hb hT
+  let Z : ℂ→ℝ→ℂ := fun w u=>heightClamp ε (F u.toNNReal w-(b u.toNNReal:ℂ))
+  let K : ℂ→ℝ→ℂ := fun w u=>-(a:ℂ)/(Z w u)^2
+  let D : ℂ→ℝ→ℂ := fun w u=>Complex.exp (∫v in (0:ℝ)..u,K w v)
+  let P : ℂ→ℝ→ℂ := fun w u=>2*(a:ℂ)*D w u/(Z w u)^3
+  have hZ (w : ℂ) : Continuous (Z w) := (heightClamp_lipschitz ε).continuous.comp
+    (((hFc w).comp continuous_real_toNNReal).sub
+      (Complex.continuous_ofReal.comp (hb.comp continuous_real_toNNReal)))
+  have hZn (w : ℂ) (u : ℝ) : Z w u≠0 := heightClamp_ne_zero hε _
+  have hK (w : ℂ) : Continuous (K w) := continuous_const.div ((hZ w).pow 2)
+    (fun u=>pow_ne_zero _ (hZn w u))
+  have hD (w : ℂ) : Continuous (D w) := by
+    apply Complex.continuous_exp.comp
+    exact intervalIntegral.continuous_primitive (hK w).intervalIntegrable 0
+  have hP (w : ℂ) : Continuous (P w) := (continuous_const.mul (hD w)).div ((hZ w).pow 3)
+    (fun u=>pow_ne_zero _ (hZn w u))
+  have htime {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) : u.toNNReal≤T := by
+    simpa using Real.toNNReal_le_toNNReal hu.2
+  have hZe (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) :
+      Z w u=loewnerMap a b u.toNNReal w-(b u.toNNReal:ℂ) := by
+    rw [hmap w hw _ (htime hu)]
+    exact heightClamp_eq (by simpa using (hIm w hw _ (htime hu)).le)
+  have hDe (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) :
+      D w u=firstVariation a b w u := by
+    dsimp only [D,firstVariation,logVariation]
+    apply congrArg Complex.exp
+    apply intervalIntegral.integral_congr
+    intro v hv
+    rw [uIcc_of_le hu.1] at hv
+    change -(a:ℂ)/(Z w v)^2= _
+    rw [hZe w hw v ⟨hv.1,hv.2.trans hu.2⟩]
+  have hDd (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) :
+      D w u=deriv (loewnerMap a b u.toNNReal) w := by
+    rw [hDe w hw u hu]
+    convert! firstVariation_eq_deriv hb ⟨_,odeUntil_mono (hode w hw) (htime hu)⟩ using 1
+    rw [Real.coe_toNNReal _ hu.1]
+  have hd (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) (w : ℂ) (hw : w∈ball z δ) :
+      HasDerivAt (fun w=>K w u) (P w u) w := by
+    have h := ((hasDerivAt_loewnerMap hb
+      (show alive a b u.toNNReal w from ⟨_,odeUntil_mono (hode w hw) (htime hu)⟩)).sub_const
+        (b u.toNNReal:ℂ)).pow 2
+    rw [←(hasDerivAt_loewnerMap hb
+      (show alive a b u.toNNReal w from ⟨_,odeUntil_mono (hode w hw) (htime hu)⟩)).deriv,
+      ←hDd w hw u hu] at h
+    change HasDerivAt (fun x=>(loewnerMap a b u.toNNReal x-(b u.toNNReal:ℂ))^2)
+      ((2:ℂ)*(loewnerMap a b u.toNNReal w-(b u.toNNReal:ℂ))^(2-1)*D w u) w at h
+    have hq := (hasDerivAt_const w (-(a:ℂ))).div h
+      (by rw [←hZe w hw u hu]; exact pow_ne_zero _ (hZn w u))
+    have he : (fun x=>K x u)=ᶠ[𝓝 w]
+        (fun x=>-(a:ℂ)/(loewnerMap a b u.toNNReal x-(b u.toNNReal:ℂ))^2) := by
+      filter_upwards [isOpen_ball.mem_nhds hw] with x hx
+      dsimp only [K]; rw [hZe x hx u hu]
+    convert! hq.congr_of_eventuallyEq he using 1
+    dsimp only [P]
+    rw [hZe w hw u hu]
+    simp only [Nat.reduceSub,pow_one]
+    field_simp
+    ring
+  let L : ℝ := Real.exp ((2*|a|/ε^2)*(T:ℝ))
+  have hL : 0<L := Real.exp_pos _
+  have hDb (w : ℂ) (hw : w∈ball z δ) (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) : ‖D w u‖≤L := by
+    rw [hDd w hw u hu]
+    have hLip : LipschitzOnWith ⟨L,hL.le⟩ (loewnerMap a b u.toNNReal) (ball z δ) := by
+      apply LipschitzOnWith.of_dist_le_mul
+      intro x hx y hy
+      change dist (loewnerMap a b u.toNNReal x) (loewnerMap a b u.toNNReal y)≤L*dist x y
+      rw [hmap x hx _ (htime hu),hmap y hy _ (htime hu)]
+      apply (cutoff_loewner_dist hε hb hFc hFe x y u.toNNReal).trans
+      rw [Real.coe_toNNReal _ hu.1]
+      exact (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr
+        (mul_le_mul_of_nonneg_left hu.2 (by positivity))) dist_nonneg).trans_eq (mul_comm _ _)
+    exact norm_deriv_le_of_lipschitzOn (isOpen_ball.mem_nhds hw) hLip
+  have hPb (u : ℝ) (hu : u∈Icc 0 (T:ℝ)) (w : ℂ) (hw : w∈ball z δ) :
+      ‖P w u‖≤2 * |a| * L/ε^3 := by
+    have hn : ε≤‖Z w u‖ := heightClamp_norm_lower ε _
+    have hn0 : 0<‖Z w u‖ := hε.trans_le hn
+    dsimp only [P]
+    rw [norm_div,norm_mul,norm_mul,norm_pow,Complex.norm_real,Real.norm_eq_abs,Complex.norm_ofNat]
+    exact div_le_div₀ (by positivity) (by gcongr; exact hDb w hw u hu)
+      (by positivity) (by gcongr)
+  have hdiff := intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (F := K) (F' := P) (bound := fun _=>2 * |a| * L/ε^3) (a := 0) (b := (T:ℝ))
+    (ball_mem_nhds z hδ)
+    (Eventually.of_forall (fun w=>(hK w).aestronglyMeasurable.restrict))
+    ((hK z).intervalIntegrable 0 T) (hP z).aestronglyMeasurable.restrict
+    (ae_of_all volume (fun u hu=>hPb u (by rw [uIoc_of_le T.coe_nonneg] at hu; exact Ioc_subset_Icc_self hu)))
+    intervalIntegrable_const
+    (ae_of_all volume (fun u hu=>hd u (by rw [uIoc_of_le T.coe_nonneg] at hu; exact Ioc_subset_Icc_self hu)))
+  have he : (fun w=>D w T)=ᶠ[𝓝 z] (deriv (loewnerMap a b T)) := by
+    filter_upwards [ball_mem_nhds z hδ] with w hw
+    simpa only [Real.toNNReal_coe] using hDd w hw T ⟨T.coe_nonneg,le_rfl⟩
+  have hfinal := hdiff.2.cexp.congr_of_eventuallyEq he.symm
+  have hpi : (∫u in (0:ℝ)..(T:ℝ),P z u)=secondVariationIntegral a b z T := by
+    apply intervalIntegral.integral_congr
+    intro u hu
+    rw [uIcc_of_le T.coe_nonneg] at hu
+    dsimp only [P,secondVariationIntegral]
+    rw [hDe z (mem_ball_self hδ) u hu,hZe z (mem_ball_self hδ) u hu]
+  rw [hpi] at hfinal
+  convert! hfinal using 1
+  change firstVariation a b z T * secondVariationIntegral a b z T =
+    D z T * secondVariationIntegral a b z T
+  rw [hDe z (mem_ball_self hδ) T ⟨T.coe_nonneg,le_rfl⟩]
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped RealInnerProductSpace InnerProductSpace
+namespace SLEExactGauge.LoewnerGeometry
+
+lemma real_inner_mul_self (v c : ℂ) : ⟪v,c*v⟫_ℝ=c.re*‖v‖^2 := by
+  rw [real_inner_eq_re_inner ℂ]
+  change (inner ℂ v (c • v)).re = _
+  rw [inner_smul_right,inner_self_eq_norm_sq_to_K]
+  change (c*(‖v‖:ℂ)^2).re = _
+  simp only [pow_two,Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im]
+  ring
+
+lemma norm_le_of_affine_dissipative {B c f : ℝ→ℂ} {T R : ℝ}
+    (hT : 0≤T) (hR : 0≤R)
+    (hd : ∀ u∈Icc 0 T, HasDerivWithinAt B (c u*B u+f u) (Icc 0 T) u)
+    (h0 : ‖B 0‖≤R)
+    (hc : ∀ u∈Icc 0 T,(c u).re<0)
+    (hf : ∀ u∈Icc 0 T,‖f u‖≤-R*(c u).re) : ‖B T‖≤R := by
+  have cont : ContinuousOn (fun u=>‖B u‖^2) (Icc 0 T) :=
+    fun u hu=>(hd u hu).norm_sq.continuousWithinAt
+  have deriv (u : ℝ) (hu : u∈Ico 0 T) :
+      HasDerivWithinAt (fun u=>‖B u‖^2)
+        (2*⟪B u,c u*B u+f u⟫_ℝ) (Ici u) u := by
+    apply (hd u ⟨hu.1,hu.2.le⟩).norm_sq.mono_of_mem_nhdsWithin
+    filter_upwards [self_mem_nhdsWithin,
+      (show ∀ᶠ v:ℝ in 𝓝[Ici u] u,v<T from nhdsWithin_le_nhds (Iio_mem_nhds hu.2))]
+      with v hv hvT
+    exact ⟨hu.1.trans hv,hvT.le⟩
+  apply le_of_forall_pos_le_add
+  intro ε hε
+  have hpos : 0<R+ε := by linarith
+  have barrier := image_le_of_deriv_right_lt_deriv_boundary cont deriv
+    (B := fun _=> (R+ε)^2) (B' := fun _=>0)
+    (by nlinarith [norm_nonneg (B 0)])
+    (fun _=>hasDerivAt_const _ _)
+    (by
+      intro u hu he
+      have hh : ‖B u‖=R+ε := (sq_eq_sq₀ (norm_nonneg _) hpos.le).mp he
+      have hcf := hc u ⟨hu.1,hu.2.le⟩
+      have hb := (real_inner_le_norm (B u) (f u)).trans
+        (mul_le_mul_of_nonneg_left (hf u ⟨hu.1,hu.2.le⟩) (norm_nonneg _))
+      rw [inner_add_right,real_inner_mul_self]
+      rw [hh] at hb ⊢
+      have hn : (c u).re*(R+ε)*ε<0 := mul_neg_of_neg_of_pos
+        (mul_neg_of_neg_of_pos hcf hpos) hε
+      nlinarith)
+  have ht := barrier (right_mem_Icc.mpr hT)
+  nlinarith [norm_nonneg (B T)]
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain LoewnerRestart
+
+lemma loewner_time_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ}
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (fun t=>loewnerMap a b t.toNNReal z)
+      ((a:ℂ)/(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) u := by
+  obtain ⟨f,hf⟩ := hT
+  rw [loewnerMap_eq_solution_time hf hu]
+  exact (hf.2 u hu).2.congr_of_mem (fun v hv=>loewnerMap_eq_solution_time hf hv) hu
+
+lemma loewner_height_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ}
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (fun t=>(loewnerMap a b t.toNNReal z).im)
+      (-a*(loewnerMap a b u.toNNReal z).im /
+        Complex.normSq (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) u := by
+  have hd := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u
+    (loewner_time_hasDerivWithinAt hT hu)
+  convert! hd using 1
+  simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
+    Complex.sub_im,sub_zero,zero_mul,zero_div,zero_sub]
+  ring
+
+def coefficientDrift (a : ℝ) (Z : ℂ) : ℂ :=
+  -(a:ℂ)/(Complex.normSq Z:ℂ)+(a:ℂ)/Z^2
+
+def coefficientForcing (a : ℝ) (Z : ℂ) : ℂ :=
+  -coefficientDrift a Z-Complex.I*(2*(a:ℂ)*(Z.im:ℂ)/Z^3)
+
+lemma coefficientDrift_re (a : ℝ) {Z : ℂ} (hZ : Z≠0) :
+    (coefficientDrift a Z).re= -(2*a*Z.im^2/(Complex.normSq Z)^2) := by
+  have hn : Complex.normSq Z≠0 := mt Complex.normSq_eq_zero.mp hZ
+  dsimp only [coefficientDrift]
+  simp only [Complex.add_re,Complex.neg_re,Complex.div_re,Complex.ofReal_re,
+    Complex.ofReal_im,Complex.normSq_ofReal,
+    pow_two,Complex.mul_re,Complex.mul_im]
+  field_simp
+  simp only [Complex.normSq_apply,pow_two,Complex.mul_re,Complex.mul_im]
+  ring
+
+lemma coefficientForcing_eq (a : ℝ) {Z : ℂ} (hZ : Z≠0) :
+    coefficientForcing a Z =
+      (-(4*a*Z.im^2/(Complex.normSq Z)^2):ℝ)*(star Z/Z) := by
+  have hN : (Complex.normSq Z:ℂ)=Z*star Z := (Complex.mul_conj Z).symm
+  have hY : (Z.im:ℂ)=(Z-star Z)/(2*Complex.I) := Complex.im_eq_sub_conj Z
+  have hZn : star Z≠0 := star_ne_zero.mpr hZ
+  simp only [Complex.ofReal_neg,Complex.ofReal_div,Complex.ofReal_mul,Complex.ofReal_pow,
+    Complex.ofReal_ofNat]
+  dsimp only [coefficientForcing,coefficientDrift]
+  rw [hN,hY]
+  field_simp
+  simp only [Complex.I_sq]
+  ring_nf
+
+lemma coefficientForcing_norm {a : ℝ} (ha : 0<a) {Z : ℂ} (hZ : Z≠0) :
+    ‖coefficientForcing a Z‖ = -2*(coefficientDrift a Z).re := by
+  rw [coefficientForcing_eq a hZ,norm_mul,norm_div,norm_star,div_self (norm_ne_zero_iff.mpr hZ),
+    mul_one,Complex.norm_real,Real.norm_eq_abs,abs_neg,abs_of_nonneg (by positivity),
+    coefficientDrift_re a hZ]
+  ring
+
+ 
+
+def secondCoefficient (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) (u : ℝ) : ℂ :=
+  1-Complex.I*((loewnerMap a b u.toNNReal z).im * secondVariationIntegral a b z u /
+    firstVariation a b z u)
+
+lemma secondCoefficient_hasDerivWithinAt {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) {u : ℝ} (hu : u∈Icc 0 (T:ℝ)) :
+    HasDerivWithinAt (secondCoefficient a b z)
+      (coefficientDrift a (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))*secondCoefficient a b z u+
+        coefficientForcing a (loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ))) (Icc 0 (T:ℝ)) u := by
+  have htime : u.toNNReal≤T := by simpa using Real.toNNReal_le_toNNReal hu.2
+  have hz : loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ)≠0 := by
+    obtain ⟨f,hf⟩ := hT
+    rw [loewnerMap_eq_solution_time hf hu]
+    exact odeUntil_center_ne_zero hf hu
+  have hD : firstVariation a b z u≠0 := Complex.exp_ne_zero _
+  have hy := (loewner_height_hasDerivWithinAt hT hu).ofReal_comp
+  have hp := secondVariationIntegral_hasDerivWithinAt hb hT hu
+  have hd := firstVariation_hasDerivWithinAt hb hT hu
+  have h := (((hy.mul hp).div hd hD).const_mul Complex.I).const_sub 1
+  convert! h using 1
+  dsimp only [secondCoefficient,coefficientDrift,coefficientForcing,Pi.mul_apply]
+  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.ofReal_neg,
+    Complex.ofReal_mul,Complex.ofReal_div]
+  field_simp
+  ring
+
+lemma secondCoefficient_zero (a : ℝ) (b : ℝ≥0→ℝ) (z : ℂ) :
+    secondCoefficient a b z 0=1 := by
+  simp [secondCoefficient,secondVariationIntegral]
+
+ 
+theorem secondCoefficient_norm_le {a : ℝ} (ha : 0<a) {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) : ‖secondCoefficient a b z T‖≤2 := by
+  apply norm_le_of_affine_dissipative T.coe_nonneg (by norm_num)
+    (fun u hu=>secondCoefficient_hasDerivWithinAt hb hT hu)
+  · rw [secondCoefficient_zero]; norm_num
+  · intro u hu
+    obtain ⟨f,hf⟩ := hT
+    have hy : 0<(loewnerMap a b u.toNNReal z-(b u.toNNReal:ℂ)).im := by
+      rw [loewnerMap_eq_solution_time hf hu]
+      simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using (hf.2 u hu).1
+    have hn := odeUntil_center_ne_zero hf hu
+    rw [←loewnerMap_eq_solution_time hf hu] at hn
+    rw [coefficientDrift_re a hn]
+    exact neg_neg_of_pos (div_pos (by positivity) (sq_pos_of_ne_zero
+      (mt Complex.normSq_eq_zero.mp hn)))
+  · intro u hu
+    obtain ⟨f,hf⟩ := hT
+    have hn := odeUntil_center_ne_zero hf hu
+    rw [←loewnerMap_eq_solution_time hf hu] at hn
+    exact (coefficientForcing_norm ha hn).le
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerGeometry
+open ExactSLEMain LoewnerRestart FlowIdentification
+
+lemma hasDerivAt_deriv_inverseMap {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    HasDerivAt (deriv (inverseMap a b T))
+      (-secondVariationIntegral a b.1 (inverseMap a b T w) T /
+        (firstVariation a b.1 (inverseMap a b T w) T)^2) w := by
+  have hz := inverseMap_alive ha b T hw
+  have hg := hasDerivAt_deriv_loewnerMap b.1.continuous hz
+  have hf := (inverseMap_hasStrictDerivAt ha b T hw).hasDerivAt
+  rw [←firstVariation_eq_deriv b.1.continuous hz] at hf
+  have hD : firstVariation a b.1 (inverseMap a b T w) T≠0 := Complex.exp_ne_zero _
+  have hd := (hg.comp w hf).inv (by
+    dsimp only [Function.comp_def]; rw [←firstVariation_eq_deriv b.1.continuous hz]; exact hD)
+  have he : deriv (inverseMap a b T)=ᶠ[𝓝 w]
+      (fun w=> (deriv (loewnerMap a b.1 T) (inverseMap a b T w))⁻¹) := by
+    filter_upwards [(isOpen_lt continuous_const Complex.continuous_im).mem_nhds hw] with v hv
+    exact deriv_inverseMap ha b T hv
+  convert! hd.congr_of_eventuallyEq he using 1
+  dsimp only [Function.comp_def]
+  rw [←firstVariation_eq_deriv b.1.continuous hz]
+  field_simp
+
+lemma inverseMap_second_coefficient {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    1+Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/deriv (inverseMap a b T) w)=
+      secondCoefficient a b.1 (inverseMap a b T w) T := by
+  rw [(hasDerivAt_deriv_inverseMap ha b T hw).deriv,deriv_inverseMap ha b T hw,
+    ←firstVariation_eq_deriv b.1.continuous (inverseMap_alive ha b T hw)]
+  have hD : firstVariation a b.1 (inverseMap a b T w) T≠0 := Complex.exp_ne_zero _
+  have hy := congrArg Complex.im (centered_inverseMap ha b T hw)
+  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] at hy
+  dsimp only [secondCoefficient]
+  rw [Real.toNNReal_coe,hy]
+  field_simp
+  ring
+
+ 
+
+lemma inverseMap_second_coefficient_norm_le {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    ‖1+Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/deriv (inverseMap a b T) w)‖≤2 := by
+  rw [inverseMap_second_coefficient ha b T hw]
+  exact secondCoefficient_norm_le ha b.1.continuous (inverseMap_alive ha b T hw)
+
+ 
+theorem inverseMap_preSchwarzian_bound {a : ℝ} (ha : 0<a) (b : GDriver) (T : ℝ≥0)
+    {w : ℂ} (hw : 0<w.im) :
+    ‖deriv (deriv (inverseMap a b T)) w‖ ≤ (3/w.im)*‖deriv (inverseMap a b T) w‖ := by
+  have hn := inverseMap_second_coefficient_norm_le ha b T hw
+  have h := norm_sub_le (1+Complex.I*(w.im:ℂ)*
+    (deriv (deriv (inverseMap a b T)) w/deriv (inverseMap a b T) w)) (1:ℂ)
+  have htriangle : ‖Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/
+      deriv (inverseMap a b T) w)‖≤3 := by
+    calc
+      _ ≤ ‖1+Complex.I*(w.im:ℂ)*(deriv (deriv (inverseMap a b T)) w/
+        deriv (inverseMap a b T) w)‖+‖(1:ℂ)‖ := by simpa only [add_sub_cancel_left] using h
+      _ ≤ 3 := by rw [norm_one]; linarith
+  rw [norm_mul,norm_mul,Complex.norm_I,one_mul,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_pos hw,norm_div] at htriangle
+  have hd : 0<‖deriv (inverseMap a b T) w‖ := norm_pos_iff.mpr (deriv_inverseMap_ne_zero ha b T hw)
+  rw [div_mul_eq_mul_div]
+  apply (le_div_iff₀ hw).mpr
+  rw [←mul_div_assoc] at htriangle
+  have ha' := (div_le_iff₀ hd).mp htriangle
+  nlinarith
+
+end SLEExactGauge.LoewnerGeometry
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/MomentBatches.lean b/lean/OAI/Probability/SLEGauge/MomentBatches.lean
index ed09fe83c798735d36bd8e0a24621cccc98d7b07..c2d0af9550f7481607d977468ed4b648090415c1 100644
--- a/lean/OAI/Probability/SLEGauge/MomentBatches.lean
+++ b/lean/OAI/Probability/SLEGauge/MomentBatches.lean
@@ -1,605 +1,605 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.GaugeBounds
-import OAI.Probability.SLEGauge.RadiusComparison
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-namespace SLEExactGauge.MomentBatches
-
- 
-def moment {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (L : Ω → ℝ)
-    (n : ℕ) : ℝ := ∫ ω, (L ω)^n ∂P
-
-lemma mean_pow_le_moment {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-    [IsProbabilityMeasure P] (L : Ω → ℝ) (hL : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (h1 : Integrable L P) (n : ℕ) (hn : Integrable (fun ω => (L ω)^n) P) :
-    (moment P L 1)^n ≤ moment P L n := by
-  simpa only [moment,pow_one] using
-    (convexOn_pow (𝕜 := ℝ) n).map_integral_le (by fun_prop) isClosed_Ici hL h1 hn
-
- 
-def cutoff (m : ℕ → ℝ) (n : ℕ) : ℝ := 4*m (n+1)/m n
-
- 
-def multiplicity (m : ℕ → ℝ) (n : ℕ) (k : ℤ) : ℕ :=
-  if (2:ℝ)^k ≤ cutoff m n then ⌊(4:ℝ)^n*((2:ℝ)^k)^n/m n⌋₊ else 0
-
-lemma multiplicity_bounds {m : ℕ → ℝ} {n : ℕ} {k : ℤ}
-    (hn : 0 < n) (hm1 : 0 < m 1) (hmn : 0 < m n) (hJ : (m 1)^n ≤ m n)
-    (hk : multiplicity m n k ≠ 0) : m 1/4 ≤ (2:ℝ)^k ∧ (2:ℝ)^k ≤ cutoff m n := by
-  unfold multiplicity at hk
-  split_ifs at hk with hcut
-  · have hw : 1 ≤ (4:ℝ)^n*((2:ℝ)^k)^n/m n :=
-      Nat.floor_pos.mp (Nat.pos_of_ne_zero hk)
-    have hp : (m 1)^n ≤ (4*(2:ℝ)^k)^n := by
-      rw [mul_pow]
-      exact hJ.trans (by simpa only [one_mul] using (le_div_iff₀ hmn).mp hw)
-    have hml := (pow_le_pow_iff_left₀ hm1.le (by positivity) hn.ne').mp hp
-    exact ⟨by linarith,hcut⟩
-  · exact (hk rfl).elim
-
-lemma finite_multiplicity_support {m : ℕ → ℝ} {n : ℕ}
-    (hn : 0 < n) (hm1 : 0 < m 1) (hmn : 0 < m n) (hmnext : 0 < m (n+1))
-    (hJ : (m 1)^n ≤ m n) : {k : ℤ | multiplicity m n k ≠ 0}.Finite := by
-  obtain ⟨l,hl,hl'⟩ := exists_mem_Ico_zpow (show 0 < m 1/4 by positivity) (by norm_num : (1:ℝ) < 2)
-  obtain ⟨u,hu,hu'⟩ := exists_mem_Ico_zpow
-    (show 0 < cutoff m n by unfold cutoff; positivity) (by norm_num : (1:ℝ) < 2)
-  refine (finite_Icc l u).subset ?_
-  intro k hk
-  have hb := multiplicity_bounds hn hm1 hmn hJ hk
-  refine ⟨(zpow_le_zpow_iff_right₀ (by norm_num : (1:ℝ) < 2)).mp (hl.trans hb.1),?_⟩
-  have hlt : k < u+1 := (zpow_lt_zpow_iff_right₀ (by norm_num : (1:ℝ) < 2)).mp
-    (hb.2.trans_lt hu')
-  omega
-
- 
-
-def support (m : ℕ → ℝ) (n : ℕ) : Finset ℤ := by
-  classical
-  exact if h : {k : ℤ | multiplicity m n k ≠ 0}.Finite then h.toFinset else ∅
-
-lemma mem_support {m : ℕ → ℝ} {n : ℕ}
-    (hfin : {k : ℤ | multiplicity m n k ≠ 0}.Finite) (k : ℤ) :
-    k∈support m n ↔ multiplicity m n k ≠ 0 := by
-  simp only [support,dite_eq_left hfin,Finite.mem_toFinset,mem_ofPred_eq]
-
- 
-
-def noncleanup (m : ℕ → ℝ) (n : ℕ) : List ℝ :=
-  ((support m n).sort (· ≤ ·)).flatMap (fun k => List.replicate (multiplicity m n k) ((2:ℝ)^k))
-
-def batch (m : ℕ → ℝ) (n : ℕ) : List ℝ := noncleanup m n ++ [(n:ℝ)]
-
-lemma batch_length_pos (m : ℕ → ℝ) (n : ℕ) : 0 < (batch m n).length := by
-  simp [batch]
-
-lemma batch_entry_lower {m : ℕ → ℝ} {n : ℕ}
-    (hn : 0 < n) (hm1 : 0 < m 1) (hmn : 0 < m n) (hJ : (m 1)^n ≤ m n)
-    {b : ℝ} (hb : b∈batch m n) : min 1 (m 1/4) ≤ b := by
-  simp only [batch,List.mem_append,List.mem_singleton] at hb
-  rcases hb with hb | rfl
-  · obtain ⟨k,hks,hkb⟩ := List.mem_flatMap.mp hb
-    obtain ⟨hk,rfl⟩ := List.mem_replicate.mp hkb
-    exact (min_le_right _ _).trans (multiplicity_bounds hn hm1 hmn hJ hk).1
-  · exact (min_le_left _ _).trans (by exact_mod_cast hn)
-
- 
-lemma batch_cleanup (m : ℕ → ℝ) (n : ℕ) :
-    (batch m n)[(noncleanup m n).length]'(by simp [batch])=(n:ℝ) := by
-  simp [batch,List.getElem_append_right]
-
- 
-
-def badLevels (v : ℕ → ℝ) (M : ℝ) : Set ℕ := {j | 1 ≤ j ∧ v j ≤ 3*M/4}
-
-lemma finite_badLevels {v : ℕ → ℝ} {M : ℝ} (hM : 0 < M)
-    (hv : Tendsto v atTop (𝓝 M)) : (badLevels v M).Finite := by
-  have hev : ∀ᶠ j in atTop, 3*M/4 < v j := hv.eventually_const_lt (by linarith)
-  obtain ⟨N,hN⟩ := eventually_atTop.mp hev
-  refine (finite_Iio N).subset ?_
-  intro j hj
-  change j < N
-  by_contra! hh
-  exact (not_lt_of_ge hj.2) (hN j hh)
-
- 
-
-def level (v : ℕ → ℝ) (M : ℝ) : ℕ := by
-  classical
-  exact 1 + if h : (badLevels v M).Finite then h.toFinset.card else 0
-
-lemma level_pos (v : ℕ → ℝ) (M : ℝ) : 0 < level v M := by
-  unfold level
-  omega
-
-lemma level_moment_gt {v : ℕ → ℝ} {M : ℝ} (hM : 0 < M) (hv : Monotone v)
-    (hlim : Tendsto v atTop (𝓝 M)) : 3*M/4 < v (level v M) := by
-  classical
-  have hfin := finite_badLevels hM hlim
-  by_contra! hh
-  have hsub : Finset.Icc 1 (level v M) ⊆ hfin.toFinset := by
-    intro j hj
-    rw [Finite.mem_toFinset]
-    have hj' := Finset.mem_Icc.mp hj
-    exact ⟨hj'.1,(hv hj'.2).trans hh⟩
-  have hcard := Finset.card_le_card hsub
-  rw [Nat.card_Icc] at hcard
-  simp only [level,dite_eq_left hfin] at hcard
-  omega
-
- 
-abbrev EntryIndex (m : ℕ → ℝ) := Σ n : {n : ℕ // 0 < n}, Fin (batch m n.1).length
-
-instance (m : ℕ → ℝ) : Nonempty (EntryIndex m) :=
-  ⟨⟨⟨1,by norm_num⟩,⟨0,batch_length_pos m 1⟩⟩⟩
-
-def threshold (m : ℕ → ℝ) (i : EntryIndex m) : ℝ := (batch m i.1.1)[i.2.1]
-
- 
-
-def radius (m : ℕ → ℝ) (a : ℝ) (ell : ℕ → ℕ) (i : EntryIndex m) : ℝ :=
-  Real.exp (-((i.1.1:ℝ)^2)-2*a*(i.2.1:ℝ)*((ell i.1.1:ℝ)+(i.1.1:ℝ)))
-
-lemma radius_pos (m : ℕ → ℝ) (a : ℝ) (ell : ℕ → ℕ) (i : EntryIndex m) :
-    0 < radius m a ell i := Real.exp_pos _
-
-lemma radius_le (m : ℕ → ℝ) {a : ℝ} (ha : 0 ≤ a) (ell : ℕ → ℕ) (i : EntryIndex m) :
-    radius m a ell i ≤ Real.exp (-((i.1.1:ℝ)^2)) := by
-  unfold radius
-  apply Real.exp_le_exp.mpr
-  have : 0 ≤ 2*a*(i.2.1:ℝ)*((ell i.1.1:ℝ)+(i.1.1:ℝ)) := by positivity
-  linarith
-
-def cleanupIndex (m : ℕ → ℝ) (n : ℕ) (hn : 0 < n) : EntryIndex m :=
-  ⟨⟨n,hn⟩,⟨(noncleanup m n).length,by simp [batch]⟩⟩
-
-lemma cleanup_threshold (m : ℕ → ℝ) (n : ℕ) (hn : 0 < n) :
-    threshold m (cleanupIndex m n hn)=(n:ℝ) := batch_cleanup m n
-
-lemma cleanup_cost_small (m : ℕ → ℝ) {a d : ℝ} (ha : 0 ≤ a) (hd : 0 < d)
-    (ell : ℕ → ℕ) (ε : ℝ) (hε : 0 < ε) :
-    ∃ i : EntryIndex m, threshold m i * (radius m a ell i)^d < ε := by
-  have hlim : Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-d*(n:ℝ))) atTop (𝓝 0) := by
-    have h := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 d hd).comp
-      (tendsto_natCast_atTop_atTop (R := ℝ))
-    change Tendsto (fun n : ℕ => (n:ℝ)^(1:ℝ)*Real.exp (-d*(n:ℝ))) atTop (𝓝 0) at h
-    simpa only [Real.rpow_one] using h
-  obtain ⟨n,hn,hnε⟩ := ((eventually_ge_atTop 1).and (hlim.eventually_lt_const hε)).exists
-  have hnp : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hn
-  have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
-  have hρ : (radius m a ell (cleanupIndex m n hnp))^d ≤ Real.exp (-d*(n:ℝ)) := by
-    calc
-      (radius m a ell (cleanupIndex m n hnp))^d ≤ (Real.exp (-((n:ℝ)^2)))^d :=
-        Real.rpow_le_rpow (radius_pos m a ell _).le (radius_le m ha ell _) hd.le
-      _ = Real.exp (-((n:ℝ)^2)*d) := (Real.exp_mul _ _).symm
-      _ ≤ Real.exp (-d*(n:ℝ)) := by
-        apply Real.exp_le_exp.mpr
-        nlinarith [mul_nonneg hd.le (show (n:ℝ) ≤ (n:ℝ)^2 by nlinarith |> sub_nonneg.mpr)]
-  refine ⟨cleanupIndex m n hnp,?_⟩
-  rw [cleanup_threshold]
-  exact (mul_le_mul_of_nonneg_left hρ (Nat.cast_nonneg n)).trans_lt hnε
-
- 
-def momentGauge (m : ℕ → ℝ) (a d : ℝ) (ell : ℕ → ℕ) : ℝ → ℝ :=
-  QuadraticGauge.gauge d (threshold m) (radius m a ell)
-
- 
-
-theorem momentGauge_admissible (m : ℕ → ℝ) (hm : ∀ n, 0 < n → 0 < m n)
-    (hJ : ∀ n, 0 < n → (m 1)^n ≤ m n) {a d : ℝ}
-    (ha : 0 ≤ a) (hd : 0 < d) (hd2 : d ≤ 2) (ell : ℕ → ℕ) :
-    Target.IsHausdorffGauge (momentGauge m a d ell) := by
-  have hm1 := hm 1 Nat.zero_lt_one
-  have hc : 0 < min 1 (m 1/4) := lt_min zero_lt_one (by positivity)
-  have hb (i : EntryIndex m) : min 1 (m 1/4) ≤ threshold m i :=
-    batch_entry_lower i.1.2 (hm 1 Nat.zero_lt_one) (hm _ i.1.2) (hJ _ i.1.2)
-      (List.getElem_mem i.2.2)
-  exact QuadraticGauge.isHausdorffGauge (fun i => hc.le.trans (hb i))
-    (radius_pos m a ell) hd.le hd2 hc hb (cleanup_cost_small m ha hd ell)
-
-lemma momentGauge_doubling (m : ℕ → ℝ) (hm : ∀ n, 0 < n → 0 < m n)
-    (hJ : ∀ n, 0 < n → (m 1)^n ≤ m n) (a d : ℝ) (ell : ℕ → ℕ)
-    {C r : ℝ} (hC : 1 ≤ C) : momentGauge m a d ell (C*r) ≤ C^2*momentGauge m a d ell r := by
-  have hm1 := hm 1 Nat.zero_lt_one
-  have hc : 0 ≤ min 1 (m 1/4) := le_min zero_le_one (by positivity)
-  have hb (i : EntryIndex m) : 0 ≤ threshold m i := hc.trans
-    (batch_entry_lower i.1.2 (hm 1 Nat.zero_lt_one) (hm _ i.1.2) (hJ _ i.1.2)
-      (List.getElem_mem i.2.2))
-  exact QuadraticGauge.gauge_scale hb (radius_pos m a ell) hC
-
-end SLEExactGauge.MomentBatches
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open Set Filter Topology
-namespace SLEExactGauge.MomentBatches
-
-lemma dyadic_predecessor (n : ℕ) (k : ℤ) :
-    ((2:ℝ)^(k-1))^n = (1/2:ℝ)^n * ((2:ℝ)^k)^n := by
-  rw [zpow_sub₀ (by norm_num : (2:ℝ)≠0),zpow_one,div_pow,div_pow,one_pow]
-  ring
-
-lemma dyadic_sum_le (n : ℕ) (hn : 0 < n) (S : Finset ℤ) (K : ℤ)
-    (hK : ∀ k∈S, k ≤ K) :
-    (∑ k∈S, ((2:ℝ)^k)^n) ≤ ((2:ℝ)^K)^n/(1-(1/2:ℝ)^n) := by
-  have hq : (1/2:ℝ)^n < 1 := pow_lt_one₀ (by norm_num) (by norm_num) hn.ne'
-  have hden : 0 < 1-(1/2:ℝ)^n := sub_pos.mpr hq
-  induction S using Finset.induction_on_max generalizing K with
-  | empty => simp only [Finset.sum_empty]; positivity
-  | insert a S hmax ih =>
-    have haS : a∉S := by intro ha; exact lt_irrefl a (hmax a ha)
-    rw [Finset.sum_insert haS]
-    have hs := ih (a-1) (fun k hk => by have := hmax k hk; omega)
-    have hstep : ((2:ℝ)^a)^n + ((2:ℝ)^(a-1))^n/(1-(1/2:ℝ)^n) =
-        ((2:ℝ)^a)^n/(1-(1/2:ℝ)^n) := by
-      rw [dyadic_predecessor]
-      field_simp
-      ring
-    calc
-      ((2:ℝ)^a)^n + ∑ k∈S, ((2:ℝ)^k)^n ≤
-        ((2:ℝ)^a)^n + ((2:ℝ)^(a-1))^n/(1-(1/2:ℝ)^n) := add_le_add le_rfl hs
-      _ = ((2:ℝ)^a)^n/(1-(1/2:ℝ)^n) := hstep
-      _ ≤ ((2:ℝ)^K)^n/(1-(1/2:ℝ)^n) := by
-        apply div_le_div_of_nonneg_right _ hden.le
-        apply pow_le_pow_left₀ (by positivity)
-        exact zpow_le_zpow_right₀ (by norm_num) (hK a (Finset.mem_insert_self _ _))
-
- 
-
-lemma dyadic_sum_below (n : ℕ) (hn : 0 < n) (S : Finset ℤ) (x : ℝ) (hx : 0 ≤ x) :
-    (∑ k∈S with (2:ℝ)^k ≤ x, ((2:ℝ)^k)^n) ≤ x^n/(1-(1/2:ℝ)^n) := by
-  classical
-  rcases hx.eq_or_lt with rfl | hxp
-  · simp only [show ∀ k : ℤ, ¬(2:ℝ)^k≤0 from fun k => not_le_of_gt (by positivity),
-      Finset.filter_false,Finset.sum_empty,zero_pow hn.ne',zero_div,le_refl]
-  · obtain ⟨K,hKlo,hKhi⟩ := exists_mem_Ico_zpow hxp (by norm_num : (1:ℝ)<2)
-    have hb : ∀ k∈S.filter (fun k => (2:ℝ)^k ≤ x), k ≤ K := by
-      intro k hk
-      have hkle := (Finset.mem_filter.mp hk).2
-      have hlt : k < K+1 := (zpow_lt_zpow_iff_right₀ (by norm_num : (1:ℝ)<2)).mp
-        (hkle.trans_lt hKhi)
-      omega
-    calc
-      (∑ k∈S with (2:ℝ)^k ≤ x, ((2:ℝ)^k)^n) ≤ ((2:ℝ)^K)^n/(1-(1/2:ℝ)^n) :=
-        dyadic_sum_le n hn _ K hb
-      _ ≤ x^n/(1-(1/2:ℝ)^n) := div_le_div_of_nonneg_right
-        (pow_le_pow_left₀ (by positivity) hKlo n)
-        (by have := pow_lt_one₀ (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)<1) hn.ne'; linarith)
-
-lemma multiplicity_le_weight {m : ℕ → ℝ} {n : ℕ} (hm : 0 < m n) (k : ℤ) :
-    (multiplicity m n k:ℝ) ≤ (4:ℝ)^n*((2:ℝ)^k)^n/m n := by
-  unfold multiplicity
-  split_ifs
-  · exact Nat.floor_le (by positivity)
-  · simp only [Nat.cast_zero]
-    positivity
-
- 
-def counts (m : ℕ → ℝ) (n : ℕ) (x : ℝ) : ℝ :=
-  ∑ k∈support m n, if (2:ℝ)^k ≤ x then (multiplicity m n k:ℝ) else 0
-
-lemma counts_nonneg (m : ℕ → ℝ) (n : ℕ) (x : ℝ) : 0 ≤ counts m n x := by
-  unfold counts
-  exact Finset.sum_nonneg (fun k hk => by split_ifs <;> positivity)
-
-lemma counts_eq_sum_filter (m : ℕ → ℝ) (n : ℕ) (x : ℝ) :
-    counts m n x = ∑ k∈support m n with (2:ℝ)^k ≤ x, (multiplicity m n k:ℝ) := by
-  classical
-  exact (Finset.sum_filter (s := support m n) (fun k => (2:ℝ)^k≤x) (fun k => (multiplicity m n k:ℝ))).symm
-
-lemma counts_upper {m : ℕ → ℝ} {n : ℕ} (hn : 0 < n) (hmn : 0 < m n)
-    {x : ℝ} (hx : 0 ≤ x) :
-    counts m n x ≤ ((4:ℝ)^n/m n) * (x^n/(1-(1/2:ℝ)^n)) := by
-  classical
-  rw [counts_eq_sum_filter]
-  calc
-    (∑ k∈support m n with (2:ℝ)^k ≤ x, (multiplicity m n k:ℝ)) ≤
-        ∑ k∈support m n with (2:ℝ)^k ≤ x, ((4:ℝ)^n/m n)*((2:ℝ)^k)^n := by
-      apply Finset.sum_le_sum
-      intro k hk
-      simpa only [div_mul_eq_mul_div] using multiplicity_le_weight hmn k
-    _ = ((4:ℝ)^n/m n) * (∑ k∈support m n with (2:ℝ)^k ≤ x, ((2:ℝ)^k)^n) := by
-      rw [Finset.mul_sum]
-    _ ≤ ((4:ℝ)^n/m n) * (x^n/(1-(1/2:ℝ)^n)) :=
-      mul_le_mul_of_nonneg_left (dyadic_sum_below n hn _ x hx) (by positivity)
-
-lemma multiplicity_le_counts {m : ℕ → ℝ} {n : ℕ}
-    (hfin : {k : ℤ | multiplicity m n k≠0}.Finite) (k : ℤ) {x : ℝ} (hkx : (2:ℝ)^k ≤ x) :
-    (multiplicity m n k:ℝ) ≤ counts m n x := by
-  classical
-  by_cases hk : multiplicity m n k=0
-  · simpa only [hk,Nat.cast_zero] using counts_nonneg m n x
-  · calc
-      (multiplicity m n k:ℝ) = (if (2:ℝ)^k ≤ x then (multiplicity m n k:ℝ) else 0) := by rw [ite_eq_left hkx]
-      _ ≤ ∑ j∈support m n, if (2:ℝ)^j ≤ x then (multiplicity m n j:ℝ) else 0 :=
-        Finset.single_le_sum (f := fun j => if (2:ℝ)^j≤x then (multiplicity m n j:ℝ) else 0)
-          (fun j hj => by split_ifs <;> positivity) ((mem_support hfin k).mpr hk)
-
- 
-
-lemma counts_lower {m : ℕ → ℝ} {n : ℕ} (hn : 0 < n) (hmn : 0 < m n)
-    (hfin : {k : ℤ | multiplicity m n k≠0}.Finite)
-    {x : ℝ} (hx : 0 ≤ x) (hcut : x ≤ cutoff m n) :
-    (2:ℝ)^n*x^n/m n-1 ≤ counts m n x := by
-  rcases hx.eq_or_lt with rfl | hxp
-  · simp only [zero_pow hn.ne',mul_zero,zero_div,zero_sub]
-    exact (by norm_num : (-1:ℝ)≤0).trans (counts_nonneg m n 0)
-  · obtain ⟨k,hkx,hxk⟩ := exists_mem_Ico_zpow hxp (by norm_num : (1:ℝ)<2)
-    have hx2 : x ≤ 2*(2:ℝ)^k := by
-      rw [zpow_add₀ (by norm_num : (2:ℝ)≠0),zpow_one] at hxk
-      linarith
-    have hpow : (2:ℝ)^n*x^n ≤ (4:ℝ)^n*((2:ℝ)^k)^n := by
-      rw [← mul_pow,← mul_pow]
-      exact pow_le_pow_left₀ (by positivity) (by linarith) n
-    have hw : (2:ℝ)^n*x^n/m n-1 ≤ (multiplicity m n k:ℝ) := by
-      rw [multiplicity,ite_eq_left (hkx.trans hcut)]
-      have hf := Nat.lt_floor_add_one ((4:ℝ)^n*((2:ℝ)^k)^n/m n)
-      have hdiv := div_le_div_of_nonneg_right hpow hmn.le
-      linarith
-    exact hw.trans (multiplicity_le_counts hfin k hkx)
-
- 
-
-lemma finite_entries_before (m : ℕ → ℝ) (N : ℕ) :
-    {i : EntryIndex m | i.1.1 < N}.Finite := by
-  have hf (n : ℕ) : {i : EntryIndex m | i.1.1=n}.Finite := by
-    by_cases hn : 0 < n
-    · have him : {i : EntryIndex m | i.1.1=n} ⊆
-        range (fun k : Fin (batch m n).length => (⟨⟨n,hn⟩,k⟩ : EntryIndex m)) := by
-        rintro ⟨⟨j,hj⟩,k⟩ h
-        change j=n at h
-        subst j
-        exact ⟨k,rfl⟩
-      exact (finite_range _).subset him
-    · have he : {i : EntryIndex m | i.1.1=n} = ∅ := by
-        apply eq_empty_iff_forall_notMem.mpr
-        intro i hi
-        exact hn (hi ▸ i.1.2)
-      rw [he]
-      exact finite_empty
-  have hunion : (⋃ n : Fin N, {i : EntryIndex m | i.1.1=n.1}).Finite := finite_iUnion (fun n => hf n.1)
-  apply hunion.subset
-  intro i hi
-  exact mem_iUnion.mpr ⟨⟨i.1.1,hi⟩,rfl⟩
-
-theorem momentGauge_finite_exclusions (m : ℕ → ℝ)
-    (hm : ∀ n, 0 < n → 0 < m n) (hJ : ∀ n, 0 < n → (m 1)^n ≤ m n)
-    (a d : ℝ) (ell : ℕ → ℕ) (n₀ : ℕ) (F : ℝ → ℝ) (M C : ℝ)
-    (hF : ∀ r, 0 < r → 0 ≤ F r ∧ F r ≤ M)
-    (hbound : ∀ i : EntryIndex m, n₀ ≤ i.1.1 → ∀ r, 0 < r →
-      F r ≤ C*QuadraticGauge.entry d (threshold m i) (radius m a ell i) r) :
-    ∃ C' : ℝ, 0 < C' ∧ ∀ r, 0 < r → F r ≤ C'*momentGauge m a d ell r := by
-  have hm1 := hm 1 Nat.zero_lt_one
-  have hc : 0 < min 1 (m 1/4) := lt_min zero_lt_one (by positivity)
-  have hb (i : EntryIndex m) : 0 < threshold m i := hc.trans_le
-    (batch_entry_lower i.1.2 hm1 (hm _ i.1.2) (hJ _ i.1.2) (List.getElem_mem i.2.2))
-  apply QuadraticGauge.finite_exclusions d (threshold m) (radius m a ell) hb
-    (radius_pos m a ell) (finite_entries_before m n₀).toFinset F M C hF
-  intro i hi
-  apply hbound i
-  exact le_of_not_gt (by simpa only [Finite.mem_toFinset,mem_ofPred_eq] using hi)
-
-end SLEExactGauge.MomentBatches
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-namespace SLEExactGauge.MomentBatches
-
-variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-
-lemma integrable_counts (m : ℕ → ℝ) (n : ℕ) {X : Ω → ℝ} (hX : Measurable X) :
-    Integrable (fun ω => counts m n (X ω)) P := by
-  apply integrable_finsetSum
-  intro k hk
-  exact (integrable_const (multiplicity m n k:ℝ)).indicator (measurableSet_le measurable_const hX)
-
-lemma integral_counts (m : ℕ → ℝ) (n : ℕ) {X : Ω → ℝ} (hX : Measurable X) :
-    (∫ ω, counts m n (X ω) ∂P) =
-      ∑ k∈support m n, (multiplicity m n k:ℝ)*P.real {ω | (2:ℝ)^k ≤ X ω} := by
-  rw [show (fun ω => counts m n (X ω)) =
-    (fun ω => ∑ k∈support m n, {ω | (2:ℝ)^k ≤ X ω}.indicator
-      (fun _ => (multiplicity m n k:ℝ)) ω) from rfl]
-  rw [integral_finsetSum _ (fun k hk => (integrable_const (multiplicity m n k:ℝ)).indicator
-    (measurableSet_le measurable_const hX))]
-  apply Finset.sum_congr rfl
-  intro k hk
-  rw [integral_indicator_const _ (measurableSet_le measurable_const hX),smul_eq_mul,mul_comm]
-
- 
-
-theorem upper_tail_finite (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (n : ℕ) (hn : 0 < n) (hLn : Integrable (fun ω => (L ω)^n) P)
-    (hmn : 0 < moment P L n) :
-    (∑ k∈support (moment P L) n, (multiplicity (moment P L) n k:ℝ)*
-      P.real {ω | 16*(2:ℝ)^k ≤ L ω}) ≤ (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) := by
-  let m := moment P L
-  let C := ((4:ℝ)^n/m n)/(16:ℝ)^n/(1-(1/2:ℝ)^n)
-  have hmaj : (fun ω => counts m n (L ω/16)) ≤ᵐ[P] (fun ω => C*(L ω)^n) := by
-    filter_upwards [hL0] with ω hω
-    calc
-      counts m n (L ω/16) ≤ ((4:ℝ)^n/m n)*((L ω/16)^n/(1-(1/2:ℝ)^n)) :=
-        counts_upper hn hmn (by positivity)
-      _ = C*(L ω)^n := by dsimp only [C]; rw [div_pow]; ring
-  have hI := integral_mono_ae (integrable_counts P m n (hL.div_const 16)) (hLn.const_mul C) hmaj
-  rw [integral_counts P m n (hL.div_const 16),integral_const_mul] at hI
-  have hsets (k : ℤ) : {ω | (2:ℝ)^k ≤ L ω/16} = {ω | 16*(2:ℝ)^k ≤ L ω} := by
-    ext ω
-    simp only [mem_ofPred_eq,le_div_iff₀ (by norm_num : (0:ℝ)<16),mul_comm]
-  simp_rw [hsets] at hI
-  have he : C*(∫ ω,(L ω)^n∂P) = (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) := by
-    change (((4:ℝ)^n/m n)/(16:ℝ)^n/(1-(1/2:ℝ)^n))*m n = _
-    field_simp [show m n ≠ 0 from hmn.ne']
-  exact he ▸ hI
-
-lemma sum_eq_tsum_multiplicity {m : ℕ → ℝ} {n : ℕ}
-    (hfin : {k : ℤ | multiplicity m n k≠0}.Finite) (p : ℤ → ℝ) :
-    (∑ k∈support m n, (multiplicity m n k:ℝ)*p k) =
-      ∑' k : ℤ, (multiplicity m n k:ℝ)*p k := by
-  symm
-  apply tsum_eq_sum
-  intro k hk
-  have hn : multiplicity m n k=0 := not_not.mp (by rwa [mem_support hfin] at hk)
-  simp only [hn,Nat.cast_zero,zero_mul]
-
- 
-
-lemma moment_sequence_properties (L : Ω → ℝ) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
-    (∀ n, 0 < n → 0 < moment P L n) ∧
-    (∀ n, 0 < n → (moment P L 1)^n ≤ moment P L n) := by
-  have hJ (n : ℕ) : (moment P L 1)^n ≤ moment P L n :=
-    mean_pow_le_moment P L hL0 (by simpa only [pow_one] using hint 1) n (hint n)
-  exact ⟨fun n hn => (pow_pos hmean n).trans_le (hJ n),fun n hn => hJ n⟩
-
-theorem upper_tail (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1)
-    (n : ℕ) (hn : 0 < n) :
-    (∑' k : ℤ, (multiplicity (moment P L) n k:ℝ)*P.real {ω | 16*(2:ℝ)^k ≤ L ω}) ≤
-      (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) := by
-  obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
-  rw [← sum_eq_tsum_multiplicity
-    (finite_multiplicity_support hn hmean (hm n hn) (hm (n+1) (by omega)) (hJ n hn))]
-  exact upper_tail_finite P L hL hL0 n hn (hint n) (hm n hn)
-
- 
-
-omit [IsProbabilityMeasure P] in
-lemma truncated_moment_lower [IsProbabilityMeasure P] (L Y : Ω → ℝ) (hY : Measurable Y)
-    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω) (n : ℕ)
-    (hLn : Integrable (fun ω => (L ω)^(n+1)) P)
-    (hYn : Integrable (fun ω => (Y ω)^n) P)
-    (hmn : 0 < moment P L n) (hmnext : 0 < moment P L (n+1))
-    (happrox : 3*moment P L n/4 ≤ moment P Y n) :
-    moment P L n/2 ≤
-      ∫ ω, {ω | Y ω ≤ cutoff (moment P L) n}.indicator (fun ω => (Y ω)^n) ω ∂P := by
-  let T := cutoff (moment P L) n
-  let G := {ω | Y ω ≤ T}
-  have hT : 0 < T := by dsimp [T,cutoff]; positivity
-  have hG : MeasurableSet G := measurableSet_le hY measurable_const
-  have hpoint : (fun ω => (Y ω)^n) ≤ᵐ[P]
-      (fun ω => G.indicator (fun ω => (Y ω)^n) ω + (L ω)^(n+1)/T) := by
-    filter_upwards [hYL] with ω hω
-    have hL0 : 0 ≤ L ω := hω.1.trans hω.2
-    by_cases hg : ω∈G
-    · rw [indicator_of_mem hg]
-      exact le_add_of_nonneg_right (by positivity)
-    · rw [indicator_of_notMem hg,zero_add]
-      apply (le_div_iff₀ hT).mpr
-      calc
-        (Y ω)^n*T ≤ (Y ω)^n*Y ω :=
-          mul_le_mul_of_nonneg_left (le_of_not_ge hg) (pow_nonneg hω.1 _)
-        _ = (Y ω)^(n+1) := (pow_succ _ _).symm
-        _ ≤ (L ω)^(n+1) := pow_le_pow_left₀ hω.1 hω.2 _
-  have hI := integral_mono_ae hYn ((hYn.indicator hG).add (hLn.div_const T)) hpoint
-  simp only [Pi.add_apply] at hI
-  rw [integral_add (hYn.indicator hG) (hLn.div_const T),integral_div] at hI
-  have he : (∫ ω,(L ω)^(n+1)∂P)/T = moment P L n/4 := by
-    change moment P L (n+1)/(4*moment P L (n+1)/moment P L n) = _
-    field_simp [hmn.ne', hmnext.ne']
-  rw [he] at hI
-  change moment P Y n ≤ _ at hI
-  change moment P L n/2 ≤ ∫ ω,G.indicator (fun ω => (Y ω)^n) ω∂P
-  linarith
-
- 
-
-theorem success_strong (L Y : Ω → ℝ) (hY : Measurable Y)
-    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω) (n : ℕ) (hn : 0 < n)
-    (hLn : Integrable (fun ω => (L ω)^(n+1)) P)
-    (hYn : Integrable (fun ω => (Y ω)^n) P)
-    (hmn : 0 < moment P L n) (hmnext : 0 < moment P L (n+1))
-    (hfin : {k : ℤ | multiplicity (moment P L) n k≠0}.Finite)
-    (happrox : 3*moment P L n/4 ≤ moment P Y n) :
-    (2:ℝ)^n/2-1 ≤ ∑ k∈support (moment P L) n, (multiplicity (moment P L) n k:ℝ)*
-      P.real {ω | (2:ℝ)^k ≤ Y ω} := by
-  let m := moment P L
-  let G := {ω | Y ω ≤ cutoff m n}
-  let f := G.indicator (fun ω => (Y ω)^n)
-  let D := (2:ℝ)^n/m n
-  have hG : MeasurableSet G := measurableSet_le hY measurable_const
-  have hD : 0 ≤ D := by dsimp [D,m]; positivity
-  have hpoint : (fun ω => D*f ω-1) ≤ᵐ[P] (fun ω => counts m n (Y ω)) := by
-    filter_upwards [hYL] with ω hω
-    by_cases hg : ω∈G
-    · have he : f ω = (Y ω)^n := indicator_of_mem hg _
-      rw [he]
-      simpa only [D,div_mul_eq_mul_div] using counts_lower hn hmn hfin hω.1 hg
-    · have he : f ω=0 := indicator_of_notMem hg _
-      rw [he,mul_zero,zero_sub]
-      exact (by norm_num : (-1:ℝ)≤0).trans (counts_nonneg m n (Y ω))
-  have hI := integral_mono_ae (((hYn.indicator hG).const_mul D).sub (integrable_const 1))
-    (integrable_counts P m n hY) hpoint
-  simp only [Pi.sub_apply] at hI
-  rw [integral_sub ((hYn.indicator hG).const_mul D) (integrable_const 1),
-    integral_const_mul,integral_const,probReal_univ,one_smul,integral_counts P m n hY] at hI
-  have htr := truncated_moment_lower P L Y hY hYL n hLn hYn hmn hmnext happrox
-  have hmul := mul_le_mul_of_nonneg_left htr hD
-  have he : D*(m n/2) = (2:ℝ)^n/2 := by dsimp [D]; field_simp [show m n ≠ 0 from hmn.ne']
-  rw [he] at hmul
-  exact (sub_le_sub_right hmul 1).trans hI
-
-lemma success_numerics (n : ℕ) (hn : 3 ≤ n) :
-    (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) ≤ (2:ℝ)^n/2-1 ∧
-    (2:ℝ)^n/8 ≤ (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) := by
-  have hlow : (8:ℝ) ≤ 2^n := by
-    have hp := pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hn
-    norm_num at hp ⊢
-    exact hp
-  have hq : (1/2:ℝ)^n ≤ 1/2 := by
-    simpa using pow_le_pow_of_le_one (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)≤1)
-      (show 1 ≤ n by omega)
-  have hden : 1 ≤ 2*(1-(1/2:ℝ)^n) := by linarith
-  have hsub0 : 0 ≤ 1/(2*(1-(1/2:ℝ)^n)) := by positivity
-  have hsub1 : 1/(2*(1-(1/2:ℝ)^n)) ≤ 1 := (div_le_one (by linarith)).mpr hden
-  have hpow : (2:ℝ)^(n-2) = (2:ℝ)^n/4 := by
-    obtain ⟨j,hj⟩ := Nat.exists_eq_add_of_le (show 2 ≤ n by omega)
-    rw [hj,Nat.add_sub_cancel_left,pow_add]
-    norm_num
-  rw [hpow]
-  constructor <;> linarith
-
- 
-
-theorem success (L Y : Ω → ℝ) (hY : Measurable Y)
-    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1)
-    (n : ℕ) (hn : 3 ≤ n) (hYn : Integrable (fun ω => (Y ω)^n) P)
-    (happrox : 3*moment P L n/4 ≤ moment P Y n) :
-    (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) ≤
-      (∑' k : ℤ, (multiplicity (moment P L) n k:ℝ)*P.real {ω | (2:ℝ)^k ≤ Y ω}) ∧
-    (2:ℝ)^n/8 ≤ (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) := by
-  have hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω := hYL.mono (fun ω hω => hω.1.trans hω.2)
-  obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
-  have hnp : 0 < n := by omega
-  have hfin := finite_multiplicity_support hnp hmean (hm n hnp) (hm (n+1) (by omega)) (hJ n hnp)
-  have hb := success_strong P L Y hY hYL n hnp (hint (n+1)) hYn
-    (hm n hnp) (hm (n+1) (by omega)) hfin happrox
-  rw [sum_eq_tsum_multiplicity hfin] at hb
-  exact ⟨(success_numerics n hn).1.trans hb,(success_numerics n hn).2⟩
-
-end SLEExactGauge.MomentBatches
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.GaugeBounds
+import OAI.Probability.SLEGauge.RadiusComparison
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+namespace SLEExactGauge.MomentBatches
+
+ 
+def moment {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (L : Ω → ℝ)
+    (n : ℕ) : ℝ := ∫ ω, (L ω)^n ∂P
+
+lemma mean_pow_le_moment {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+    [IsProbabilityMeasure P] (L : Ω → ℝ) (hL : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (h1 : Integrable L P) (n : ℕ) (hn : Integrable (fun ω => (L ω)^n) P) :
+    (moment P L 1)^n ≤ moment P L n := by
+  simpa only [moment,pow_one] using
+    (convexOn_pow (𝕜 := ℝ) n).map_integral_le (by fun_prop) isClosed_Ici hL h1 hn
+
+ 
+def cutoff (m : ℕ → ℝ) (n : ℕ) : ℝ := 4*m (n+1)/m n
+
+ 
+def multiplicity (m : ℕ → ℝ) (n : ℕ) (k : ℤ) : ℕ :=
+  if (2:ℝ)^k ≤ cutoff m n then ⌊(4:ℝ)^n*((2:ℝ)^k)^n/m n⌋₊ else 0
+
+lemma multiplicity_bounds {m : ℕ → ℝ} {n : ℕ} {k : ℤ}
+    (hn : 0 < n) (hm1 : 0 < m 1) (hmn : 0 < m n) (hJ : (m 1)^n ≤ m n)
+    (hk : multiplicity m n k ≠ 0) : m 1/4 ≤ (2:ℝ)^k ∧ (2:ℝ)^k ≤ cutoff m n := by
+  unfold multiplicity at hk
+  split_ifs at hk with hcut
+  · have hw : 1 ≤ (4:ℝ)^n*((2:ℝ)^k)^n/m n :=
+      Nat.floor_pos.mp (Nat.pos_of_ne_zero hk)
+    have hp : (m 1)^n ≤ (4*(2:ℝ)^k)^n := by
+      rw [mul_pow]
+      exact hJ.trans (by simpa only [one_mul] using (le_div_iff₀ hmn).mp hw)
+    have hml := (pow_le_pow_iff_left₀ hm1.le (by positivity) hn.ne').mp hp
+    exact ⟨by linarith,hcut⟩
+  · exact (hk rfl).elim
+
+lemma finite_multiplicity_support {m : ℕ → ℝ} {n : ℕ}
+    (hn : 0 < n) (hm1 : 0 < m 1) (hmn : 0 < m n) (hmnext : 0 < m (n+1))
+    (hJ : (m 1)^n ≤ m n) : {k : ℤ | multiplicity m n k ≠ 0}.Finite := by
+  obtain ⟨l,hl,hl'⟩ := exists_mem_Ico_zpow (show 0 < m 1/4 by positivity) (by norm_num : (1:ℝ) < 2)
+  obtain ⟨u,hu,hu'⟩ := exists_mem_Ico_zpow
+    (show 0 < cutoff m n by unfold cutoff; positivity) (by norm_num : (1:ℝ) < 2)
+  refine (finite_Icc l u).subset ?_
+  intro k hk
+  have hb := multiplicity_bounds hn hm1 hmn hJ hk
+  refine ⟨(zpow_le_zpow_iff_right₀ (by norm_num : (1:ℝ) < 2)).mp (hl.trans hb.1),?_⟩
+  have hlt : k < u+1 := (zpow_lt_zpow_iff_right₀ (by norm_num : (1:ℝ) < 2)).mp
+    (hb.2.trans_lt hu')
+  omega
+
+ 
+
+def support (m : ℕ → ℝ) (n : ℕ) : Finset ℤ := by
+  classical
+  exact if h : {k : ℤ | multiplicity m n k ≠ 0}.Finite then h.toFinset else ∅
+
+lemma mem_support {m : ℕ → ℝ} {n : ℕ}
+    (hfin : {k : ℤ | multiplicity m n k ≠ 0}.Finite) (k : ℤ) :
+    k∈support m n ↔ multiplicity m n k ≠ 0 := by
+  simp only [support,dite_eq_left hfin,Finite.mem_toFinset,mem_ofPred_eq]
+
+ 
+
+def noncleanup (m : ℕ → ℝ) (n : ℕ) : List ℝ :=
+  ((support m n).sort (· ≤ ·)).flatMap (fun k => List.replicate (multiplicity m n k) ((2:ℝ)^k))
+
+def batch (m : ℕ → ℝ) (n : ℕ) : List ℝ := noncleanup m n ++ [(n:ℝ)]
+
+lemma batch_length_pos (m : ℕ → ℝ) (n : ℕ) : 0 < (batch m n).length := by
+  simp [batch]
+
+lemma batch_entry_lower {m : ℕ → ℝ} {n : ℕ}
+    (hn : 0 < n) (hm1 : 0 < m 1) (hmn : 0 < m n) (hJ : (m 1)^n ≤ m n)
+    {b : ℝ} (hb : b∈batch m n) : min 1 (m 1/4) ≤ b := by
+  simp only [batch,List.mem_append,List.mem_singleton] at hb
+  rcases hb with hb | rfl
+  · obtain ⟨k,hks,hkb⟩ := List.mem_flatMap.mp hb
+    obtain ⟨hk,rfl⟩ := List.mem_replicate.mp hkb
+    exact (min_le_right _ _).trans (multiplicity_bounds hn hm1 hmn hJ hk).1
+  · exact (min_le_left _ _).trans (by exact_mod_cast hn)
+
+ 
+lemma batch_cleanup (m : ℕ → ℝ) (n : ℕ) :
+    (batch m n)[(noncleanup m n).length]'(by simp [batch])=(n:ℝ) := by
+  simp [batch,List.getElem_append_right]
+
+ 
+
+def badLevels (v : ℕ → ℝ) (M : ℝ) : Set ℕ := {j | 1 ≤ j ∧ v j ≤ 3*M/4}
+
+lemma finite_badLevels {v : ℕ → ℝ} {M : ℝ} (hM : 0 < M)
+    (hv : Tendsto v atTop (𝓝 M)) : (badLevels v M).Finite := by
+  have hev : ∀ᶠ j in atTop, 3*M/4 < v j := hv.eventually_const_lt (by linarith)
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hev
+  refine (finite_Iio N).subset ?_
+  intro j hj
+  change j < N
+  by_contra! hh
+  exact (not_lt_of_ge hj.2) (hN j hh)
+
+ 
+
+def level (v : ℕ → ℝ) (M : ℝ) : ℕ := by
+  classical
+  exact 1 + if h : (badLevels v M).Finite then h.toFinset.card else 0
+
+lemma level_pos (v : ℕ → ℝ) (M : ℝ) : 0 < level v M := by
+  unfold level
+  omega
+
+lemma level_moment_gt {v : ℕ → ℝ} {M : ℝ} (hM : 0 < M) (hv : Monotone v)
+    (hlim : Tendsto v atTop (𝓝 M)) : 3*M/4 < v (level v M) := by
+  classical
+  have hfin := finite_badLevels hM hlim
+  by_contra! hh
+  have hsub : Finset.Icc 1 (level v M) ⊆ hfin.toFinset := by
+    intro j hj
+    rw [Finite.mem_toFinset]
+    have hj' := Finset.mem_Icc.mp hj
+    exact ⟨hj'.1,(hv hj'.2).trans hh⟩
+  have hcard := Finset.card_le_card hsub
+  rw [Nat.card_Icc] at hcard
+  simp only [level,dite_eq_left hfin] at hcard
+  omega
+
+ 
+abbrev EntryIndex (m : ℕ → ℝ) := Σ n : {n : ℕ // 0 < n}, Fin (batch m n.1).length
+
+instance (m : ℕ → ℝ) : Nonempty (EntryIndex m) :=
+  ⟨⟨⟨1,by norm_num⟩,⟨0,batch_length_pos m 1⟩⟩⟩
+
+def threshold (m : ℕ → ℝ) (i : EntryIndex m) : ℝ := (batch m i.1.1)[i.2.1]
+
+ 
+
+def radius (m : ℕ → ℝ) (a : ℝ) (ell : ℕ → ℕ) (i : EntryIndex m) : ℝ :=
+  Real.exp (-((i.1.1:ℝ)^2)-2*a*(i.2.1:ℝ)*((ell i.1.1:ℝ)+(i.1.1:ℝ)))
+
+lemma radius_pos (m : ℕ → ℝ) (a : ℝ) (ell : ℕ → ℕ) (i : EntryIndex m) :
+    0 < radius m a ell i := Real.exp_pos _
+
+lemma radius_le (m : ℕ → ℝ) {a : ℝ} (ha : 0 ≤ a) (ell : ℕ → ℕ) (i : EntryIndex m) :
+    radius m a ell i ≤ Real.exp (-((i.1.1:ℝ)^2)) := by
+  unfold radius
+  apply Real.exp_le_exp.mpr
+  have : 0 ≤ 2*a*(i.2.1:ℝ)*((ell i.1.1:ℝ)+(i.1.1:ℝ)) := by positivity
+  linarith
+
+def cleanupIndex (m : ℕ → ℝ) (n : ℕ) (hn : 0 < n) : EntryIndex m :=
+  ⟨⟨n,hn⟩,⟨(noncleanup m n).length,by simp [batch]⟩⟩
+
+lemma cleanup_threshold (m : ℕ → ℝ) (n : ℕ) (hn : 0 < n) :
+    threshold m (cleanupIndex m n hn)=(n:ℝ) := batch_cleanup m n
+
+lemma cleanup_cost_small (m : ℕ → ℝ) {a d : ℝ} (ha : 0 ≤ a) (hd : 0 < d)
+    (ell : ℕ → ℕ) (ε : ℝ) (hε : 0 < ε) :
+    ∃ i : EntryIndex m, threshold m i * (radius m a ell i)^d < ε := by
+  have hlim : Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-d*(n:ℝ))) atTop (𝓝 0) := by
+    have h := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 d hd).comp
+      (tendsto_natCast_atTop_atTop (R := ℝ))
+    change Tendsto (fun n : ℕ => (n:ℝ)^(1:ℝ)*Real.exp (-d*(n:ℝ))) atTop (𝓝 0) at h
+    simpa only [Real.rpow_one] using h
+  obtain ⟨n,hn,hnε⟩ := ((eventually_ge_atTop 1).and (hlim.eventually_lt_const hε)).exists
+  have hnp : 0 < n := lt_of_lt_of_le Nat.zero_lt_one hn
+  have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
+  have hρ : (radius m a ell (cleanupIndex m n hnp))^d ≤ Real.exp (-d*(n:ℝ)) := by
+    calc
+      (radius m a ell (cleanupIndex m n hnp))^d ≤ (Real.exp (-((n:ℝ)^2)))^d :=
+        Real.rpow_le_rpow (radius_pos m a ell _).le (radius_le m ha ell _) hd.le
+      _ = Real.exp (-((n:ℝ)^2)*d) := (Real.exp_mul _ _).symm
+      _ ≤ Real.exp (-d*(n:ℝ)) := by
+        apply Real.exp_le_exp.mpr
+        nlinarith [mul_nonneg hd.le (show (n:ℝ) ≤ (n:ℝ)^2 by nlinarith |> sub_nonneg.mpr)]
+  refine ⟨cleanupIndex m n hnp,?_⟩
+  rw [cleanup_threshold]
+  exact (mul_le_mul_of_nonneg_left hρ (Nat.cast_nonneg n)).trans_lt hnε
+
+ 
+def momentGauge (m : ℕ → ℝ) (a d : ℝ) (ell : ℕ → ℕ) : ℝ → ℝ :=
+  QuadraticGauge.gauge d (threshold m) (radius m a ell)
+
+ 
+
+theorem momentGauge_admissible (m : ℕ → ℝ) (hm : ∀ n, 0 < n → 0 < m n)
+    (hJ : ∀ n, 0 < n → (m 1)^n ≤ m n) {a d : ℝ}
+    (ha : 0 ≤ a) (hd : 0 < d) (hd2 : d ≤ 2) (ell : ℕ → ℕ) :
+    Target.IsHausdorffGauge (momentGauge m a d ell) := by
+  have hm1 := hm 1 Nat.zero_lt_one
+  have hc : 0 < min 1 (m 1/4) := lt_min zero_lt_one (by positivity)
+  have hb (i : EntryIndex m) : min 1 (m 1/4) ≤ threshold m i :=
+    batch_entry_lower i.1.2 (hm 1 Nat.zero_lt_one) (hm _ i.1.2) (hJ _ i.1.2)
+      (List.getElem_mem i.2.2)
+  exact QuadraticGauge.isHausdorffGauge (fun i => hc.le.trans (hb i))
+    (radius_pos m a ell) hd.le hd2 hc hb (cleanup_cost_small m ha hd ell)
+
+lemma momentGauge_doubling (m : ℕ → ℝ) (hm : ∀ n, 0 < n → 0 < m n)
+    (hJ : ∀ n, 0 < n → (m 1)^n ≤ m n) (a d : ℝ) (ell : ℕ → ℕ)
+    {C r : ℝ} (hC : 1 ≤ C) : momentGauge m a d ell (C*r) ≤ C^2*momentGauge m a d ell r := by
+  have hm1 := hm 1 Nat.zero_lt_one
+  have hc : 0 ≤ min 1 (m 1/4) := le_min zero_le_one (by positivity)
+  have hb (i : EntryIndex m) : 0 ≤ threshold m i := hc.trans
+    (batch_entry_lower i.1.2 (hm 1 Nat.zero_lt_one) (hm _ i.1.2) (hJ _ i.1.2)
+      (List.getElem_mem i.2.2))
+  exact QuadraticGauge.gauge_scale hb (radius_pos m a ell) hC
+
+end SLEExactGauge.MomentBatches
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open Set Filter Topology
+namespace SLEExactGauge.MomentBatches
+
+lemma dyadic_predecessor (n : ℕ) (k : ℤ) :
+    ((2:ℝ)^(k-1))^n = (1/2:ℝ)^n * ((2:ℝ)^k)^n := by
+  rw [zpow_sub₀ (by norm_num : (2:ℝ)≠0),zpow_one,div_pow,div_pow,one_pow]
+  ring
+
+lemma dyadic_sum_le (n : ℕ) (hn : 0 < n) (S : Finset ℤ) (K : ℤ)
+    (hK : ∀ k∈S, k ≤ K) :
+    (∑ k∈S, ((2:ℝ)^k)^n) ≤ ((2:ℝ)^K)^n/(1-(1/2:ℝ)^n) := by
+  have hq : (1/2:ℝ)^n < 1 := pow_lt_one₀ (by norm_num) (by norm_num) hn.ne'
+  have hden : 0 < 1-(1/2:ℝ)^n := sub_pos.mpr hq
+  induction S using Finset.induction_on_max generalizing K with
+  | empty => simp only [Finset.sum_empty]; positivity
+  | insert a S hmax ih =>
+    have haS : a∉S := by intro ha; exact lt_irrefl a (hmax a ha)
+    rw [Finset.sum_insert haS]
+    have hs := ih (a-1) (fun k hk => by have := hmax k hk; omega)
+    have hstep : ((2:ℝ)^a)^n + ((2:ℝ)^(a-1))^n/(1-(1/2:ℝ)^n) =
+        ((2:ℝ)^a)^n/(1-(1/2:ℝ)^n) := by
+      rw [dyadic_predecessor]
+      field_simp
+      ring
+    calc
+      ((2:ℝ)^a)^n + ∑ k∈S, ((2:ℝ)^k)^n ≤
+        ((2:ℝ)^a)^n + ((2:ℝ)^(a-1))^n/(1-(1/2:ℝ)^n) := add_le_add le_rfl hs
+      _ = ((2:ℝ)^a)^n/(1-(1/2:ℝ)^n) := hstep
+      _ ≤ ((2:ℝ)^K)^n/(1-(1/2:ℝ)^n) := by
+        apply div_le_div_of_nonneg_right _ hden.le
+        apply pow_le_pow_left₀ (by positivity)
+        exact zpow_le_zpow_right₀ (by norm_num) (hK a (Finset.mem_insert_self _ _))
+
+ 
+
+lemma dyadic_sum_below (n : ℕ) (hn : 0 < n) (S : Finset ℤ) (x : ℝ) (hx : 0 ≤ x) :
+    (∑ k∈S with (2:ℝ)^k ≤ x, ((2:ℝ)^k)^n) ≤ x^n/(1-(1/2:ℝ)^n) := by
+  classical
+  rcases hx.eq_or_lt with rfl | hxp
+  · simp only [show ∀ k : ℤ, ¬(2:ℝ)^k≤0 from fun k => not_le_of_gt (by positivity),
+      Finset.filter_false,Finset.sum_empty,zero_pow hn.ne',zero_div,le_refl]
+  · obtain ⟨K,hKlo,hKhi⟩ := exists_mem_Ico_zpow hxp (by norm_num : (1:ℝ)<2)
+    have hb : ∀ k∈S.filter (fun k => (2:ℝ)^k ≤ x), k ≤ K := by
+      intro k hk
+      have hkle := (Finset.mem_filter.mp hk).2
+      have hlt : k < K+1 := (zpow_lt_zpow_iff_right₀ (by norm_num : (1:ℝ)<2)).mp
+        (hkle.trans_lt hKhi)
+      omega
+    calc
+      (∑ k∈S with (2:ℝ)^k ≤ x, ((2:ℝ)^k)^n) ≤ ((2:ℝ)^K)^n/(1-(1/2:ℝ)^n) :=
+        dyadic_sum_le n hn _ K hb
+      _ ≤ x^n/(1-(1/2:ℝ)^n) := div_le_div_of_nonneg_right
+        (pow_le_pow_left₀ (by positivity) hKlo n)
+        (by have := pow_lt_one₀ (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)<1) hn.ne'; linarith)
+
+lemma multiplicity_le_weight {m : ℕ → ℝ} {n : ℕ} (hm : 0 < m n) (k : ℤ) :
+    (multiplicity m n k:ℝ) ≤ (4:ℝ)^n*((2:ℝ)^k)^n/m n := by
+  unfold multiplicity
+  split_ifs
+  · exact Nat.floor_le (by positivity)
+  · simp only [Nat.cast_zero]
+    positivity
+
+ 
+def counts (m : ℕ → ℝ) (n : ℕ) (x : ℝ) : ℝ :=
+  ∑ k∈support m n, if (2:ℝ)^k ≤ x then (multiplicity m n k:ℝ) else 0
+
+lemma counts_nonneg (m : ℕ → ℝ) (n : ℕ) (x : ℝ) : 0 ≤ counts m n x := by
+  unfold counts
+  exact Finset.sum_nonneg (fun k hk => by split_ifs <;> positivity)
+
+lemma counts_eq_sum_filter (m : ℕ → ℝ) (n : ℕ) (x : ℝ) :
+    counts m n x = ∑ k∈support m n with (2:ℝ)^k ≤ x, (multiplicity m n k:ℝ) := by
+  classical
+  exact (Finset.sum_filter (s := support m n) (fun k => (2:ℝ)^k≤x) (fun k => (multiplicity m n k:ℝ))).symm
+
+lemma counts_upper {m : ℕ → ℝ} {n : ℕ} (hn : 0 < n) (hmn : 0 < m n)
+    {x : ℝ} (hx : 0 ≤ x) :
+    counts m n x ≤ ((4:ℝ)^n/m n) * (x^n/(1-(1/2:ℝ)^n)) := by
+  classical
+  rw [counts_eq_sum_filter]
+  calc
+    (∑ k∈support m n with (2:ℝ)^k ≤ x, (multiplicity m n k:ℝ)) ≤
+        ∑ k∈support m n with (2:ℝ)^k ≤ x, ((4:ℝ)^n/m n)*((2:ℝ)^k)^n := by
+      apply Finset.sum_le_sum
+      intro k hk
+      simpa only [div_mul_eq_mul_div] using multiplicity_le_weight hmn k
+    _ = ((4:ℝ)^n/m n) * (∑ k∈support m n with (2:ℝ)^k ≤ x, ((2:ℝ)^k)^n) := by
+      rw [Finset.mul_sum]
+    _ ≤ ((4:ℝ)^n/m n) * (x^n/(1-(1/2:ℝ)^n)) :=
+      mul_le_mul_of_nonneg_left (dyadic_sum_below n hn _ x hx) (by positivity)
+
+lemma multiplicity_le_counts {m : ℕ → ℝ} {n : ℕ}
+    (hfin : {k : ℤ | multiplicity m n k≠0}.Finite) (k : ℤ) {x : ℝ} (hkx : (2:ℝ)^k ≤ x) :
+    (multiplicity m n k:ℝ) ≤ counts m n x := by
+  classical
+  by_cases hk : multiplicity m n k=0
+  · simpa only [hk,Nat.cast_zero] using counts_nonneg m n x
+  · calc
+      (multiplicity m n k:ℝ) = (if (2:ℝ)^k ≤ x then (multiplicity m n k:ℝ) else 0) := by rw [ite_eq_left hkx]
+      _ ≤ ∑ j∈support m n, if (2:ℝ)^j ≤ x then (multiplicity m n j:ℝ) else 0 :=
+        Finset.single_le_sum (f := fun j => if (2:ℝ)^j≤x then (multiplicity m n j:ℝ) else 0)
+          (fun j hj => by split_ifs <;> positivity) ((mem_support hfin k).mpr hk)
+
+ 
+
+lemma counts_lower {m : ℕ → ℝ} {n : ℕ} (hn : 0 < n) (hmn : 0 < m n)
+    (hfin : {k : ℤ | multiplicity m n k≠0}.Finite)
+    {x : ℝ} (hx : 0 ≤ x) (hcut : x ≤ cutoff m n) :
+    (2:ℝ)^n*x^n/m n-1 ≤ counts m n x := by
+  rcases hx.eq_or_lt with rfl | hxp
+  · simp only [zero_pow hn.ne',mul_zero,zero_div,zero_sub]
+    exact (by norm_num : (-1:ℝ)≤0).trans (counts_nonneg m n 0)
+  · obtain ⟨k,hkx,hxk⟩ := exists_mem_Ico_zpow hxp (by norm_num : (1:ℝ)<2)
+    have hx2 : x ≤ 2*(2:ℝ)^k := by
+      rw [zpow_add₀ (by norm_num : (2:ℝ)≠0),zpow_one] at hxk
+      linarith
+    have hpow : (2:ℝ)^n*x^n ≤ (4:ℝ)^n*((2:ℝ)^k)^n := by
+      rw [← mul_pow,← mul_pow]
+      exact pow_le_pow_left₀ (by positivity) (by linarith) n
+    have hw : (2:ℝ)^n*x^n/m n-1 ≤ (multiplicity m n k:ℝ) := by
+      rw [multiplicity,ite_eq_left (hkx.trans hcut)]
+      have hf := Nat.lt_floor_add_one ((4:ℝ)^n*((2:ℝ)^k)^n/m n)
+      have hdiv := div_le_div_of_nonneg_right hpow hmn.le
+      linarith
+    exact hw.trans (multiplicity_le_counts hfin k hkx)
+
+ 
+
+lemma finite_entries_before (m : ℕ → ℝ) (N : ℕ) :
+    {i : EntryIndex m | i.1.1 < N}.Finite := by
+  have hf (n : ℕ) : {i : EntryIndex m | i.1.1=n}.Finite := by
+    by_cases hn : 0 < n
+    · have him : {i : EntryIndex m | i.1.1=n} ⊆
+        range (fun k : Fin (batch m n).length => (⟨⟨n,hn⟩,k⟩ : EntryIndex m)) := by
+        rintro ⟨⟨j,hj⟩,k⟩ h
+        change j=n at h
+        subst j
+        exact ⟨k,rfl⟩
+      exact (finite_range _).subset him
+    · have he : {i : EntryIndex m | i.1.1=n} = ∅ := by
+        apply eq_empty_iff_forall_notMem.mpr
+        intro i hi
+        exact hn (hi ▸ i.1.2)
+      rw [he]
+      exact finite_empty
+  have hunion : (⋃ n : Fin N, {i : EntryIndex m | i.1.1=n.1}).Finite := finite_iUnion (fun n => hf n.1)
+  apply hunion.subset
+  intro i hi
+  exact mem_iUnion.mpr ⟨⟨i.1.1,hi⟩,rfl⟩
+
+theorem momentGauge_finite_exclusions (m : ℕ → ℝ)
+    (hm : ∀ n, 0 < n → 0 < m n) (hJ : ∀ n, 0 < n → (m 1)^n ≤ m n)
+    (a d : ℝ) (ell : ℕ → ℕ) (n₀ : ℕ) (F : ℝ → ℝ) (M C : ℝ)
+    (hF : ∀ r, 0 < r → 0 ≤ F r ∧ F r ≤ M)
+    (hbound : ∀ i : EntryIndex m, n₀ ≤ i.1.1 → ∀ r, 0 < r →
+      F r ≤ C*QuadraticGauge.entry d (threshold m i) (radius m a ell i) r) :
+    ∃ C' : ℝ, 0 < C' ∧ ∀ r, 0 < r → F r ≤ C'*momentGauge m a d ell r := by
+  have hm1 := hm 1 Nat.zero_lt_one
+  have hc : 0 < min 1 (m 1/4) := lt_min zero_lt_one (by positivity)
+  have hb (i : EntryIndex m) : 0 < threshold m i := hc.trans_le
+    (batch_entry_lower i.1.2 hm1 (hm _ i.1.2) (hJ _ i.1.2) (List.getElem_mem i.2.2))
+  apply QuadraticGauge.finite_exclusions d (threshold m) (radius m a ell) hb
+    (radius_pos m a ell) (finite_entries_before m n₀).toFinset F M C hF
+  intro i hi
+  apply hbound i
+  exact le_of_not_gt (by simpa only [Finite.mem_toFinset,mem_ofPred_eq] using hi)
+
+end SLEExactGauge.MomentBatches
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+namespace SLEExactGauge.MomentBatches
+
+variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+
+lemma integrable_counts (m : ℕ → ℝ) (n : ℕ) {X : Ω → ℝ} (hX : Measurable X) :
+    Integrable (fun ω => counts m n (X ω)) P := by
+  apply integrable_finsetSum
+  intro k hk
+  exact (integrable_const (multiplicity m n k:ℝ)).indicator (measurableSet_le measurable_const hX)
+
+lemma integral_counts (m : ℕ → ℝ) (n : ℕ) {X : Ω → ℝ} (hX : Measurable X) :
+    (∫ ω, counts m n (X ω) ∂P) =
+      ∑ k∈support m n, (multiplicity m n k:ℝ)*P.real {ω | (2:ℝ)^k ≤ X ω} := by
+  rw [show (fun ω => counts m n (X ω)) =
+    (fun ω => ∑ k∈support m n, {ω | (2:ℝ)^k ≤ X ω}.indicator
+      (fun _ => (multiplicity m n k:ℝ)) ω) from rfl]
+  rw [integral_finsetSum _ (fun k hk => (integrable_const (multiplicity m n k:ℝ)).indicator
+    (measurableSet_le measurable_const hX))]
+  apply Finset.sum_congr rfl
+  intro k hk
+  rw [integral_indicator_const _ (measurableSet_le measurable_const hX),smul_eq_mul,mul_comm]
+
+ 
+
+theorem upper_tail_finite (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (n : ℕ) (hn : 0 < n) (hLn : Integrable (fun ω => (L ω)^n) P)
+    (hmn : 0 < moment P L n) :
+    (∑ k∈support (moment P L) n, (multiplicity (moment P L) n k:ℝ)*
+      P.real {ω | 16*(2:ℝ)^k ≤ L ω}) ≤ (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) := by
+  let m := moment P L
+  let C := ((4:ℝ)^n/m n)/(16:ℝ)^n/(1-(1/2:ℝ)^n)
+  have hmaj : (fun ω => counts m n (L ω/16)) ≤ᵐ[P] (fun ω => C*(L ω)^n) := by
+    filter_upwards [hL0] with ω hω
+    calc
+      counts m n (L ω/16) ≤ ((4:ℝ)^n/m n)*((L ω/16)^n/(1-(1/2:ℝ)^n)) :=
+        counts_upper hn hmn (by positivity)
+      _ = C*(L ω)^n := by dsimp only [C]; rw [div_pow]; ring
+  have hI := integral_mono_ae (integrable_counts P m n (hL.div_const 16)) (hLn.const_mul C) hmaj
+  rw [integral_counts P m n (hL.div_const 16),integral_const_mul] at hI
+  have hsets (k : ℤ) : {ω | (2:ℝ)^k ≤ L ω/16} = {ω | 16*(2:ℝ)^k ≤ L ω} := by
+    ext ω
+    simp only [mem_ofPred_eq,le_div_iff₀ (by norm_num : (0:ℝ)<16),mul_comm]
+  simp_rw [hsets] at hI
+  have he : C*(∫ ω,(L ω)^n∂P) = (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) := by
+    change (((4:ℝ)^n/m n)/(16:ℝ)^n/(1-(1/2:ℝ)^n))*m n = _
+    field_simp [show m n ≠ 0 from hmn.ne']
+  exact he ▸ hI
+
+lemma sum_eq_tsum_multiplicity {m : ℕ → ℝ} {n : ℕ}
+    (hfin : {k : ℤ | multiplicity m n k≠0}.Finite) (p : ℤ → ℝ) :
+    (∑ k∈support m n, (multiplicity m n k:ℝ)*p k) =
+      ∑' k : ℤ, (multiplicity m n k:ℝ)*p k := by
+  symm
+  apply tsum_eq_sum
+  intro k hk
+  have hn : multiplicity m n k=0 := not_not.mp (by rwa [mem_support hfin] at hk)
+  simp only [hn,Nat.cast_zero,zero_mul]
+
+ 
+
+lemma moment_sequence_properties (L : Ω → ℝ) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
+    (∀ n, 0 < n → 0 < moment P L n) ∧
+    (∀ n, 0 < n → (moment P L 1)^n ≤ moment P L n) := by
+  have hJ (n : ℕ) : (moment P L 1)^n ≤ moment P L n :=
+    mean_pow_le_moment P L hL0 (by simpa only [pow_one] using hint 1) n (hint n)
+  exact ⟨fun n hn => (pow_pos hmean n).trans_le (hJ n),fun n hn => hJ n⟩
+
+theorem upper_tail (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1)
+    (n : ℕ) (hn : 0 < n) :
+    (∑' k : ℤ, (multiplicity (moment P L) n k:ℝ)*P.real {ω | 16*(2:ℝ)^k ≤ L ω}) ≤
+      (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) := by
+  obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
+  rw [← sum_eq_tsum_multiplicity
+    (finite_multiplicity_support hn hmean (hm n hn) (hm (n+1) (by omega)) (hJ n hn))]
+  exact upper_tail_finite P L hL hL0 n hn (hint n) (hm n hn)
+
+ 
+
+omit [IsProbabilityMeasure P] in
+lemma truncated_moment_lower [IsProbabilityMeasure P] (L Y : Ω → ℝ) (hY : Measurable Y)
+    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω) (n : ℕ)
+    (hLn : Integrable (fun ω => (L ω)^(n+1)) P)
+    (hYn : Integrable (fun ω => (Y ω)^n) P)
+    (hmn : 0 < moment P L n) (hmnext : 0 < moment P L (n+1))
+    (happrox : 3*moment P L n/4 ≤ moment P Y n) :
+    moment P L n/2 ≤
+      ∫ ω, {ω | Y ω ≤ cutoff (moment P L) n}.indicator (fun ω => (Y ω)^n) ω ∂P := by
+  let T := cutoff (moment P L) n
+  let G := {ω | Y ω ≤ T}
+  have hT : 0 < T := by dsimp [T,cutoff]; positivity
+  have hG : MeasurableSet G := measurableSet_le hY measurable_const
+  have hpoint : (fun ω => (Y ω)^n) ≤ᵐ[P]
+      (fun ω => G.indicator (fun ω => (Y ω)^n) ω + (L ω)^(n+1)/T) := by
+    filter_upwards [hYL] with ω hω
+    have hL0 : 0 ≤ L ω := hω.1.trans hω.2
+    by_cases hg : ω∈G
+    · rw [indicator_of_mem hg]
+      exact le_add_of_nonneg_right (by positivity)
+    · rw [indicator_of_notMem hg,zero_add]
+      apply (le_div_iff₀ hT).mpr
+      calc
+        (Y ω)^n*T ≤ (Y ω)^n*Y ω :=
+          mul_le_mul_of_nonneg_left (le_of_not_ge hg) (pow_nonneg hω.1 _)
+        _ = (Y ω)^(n+1) := (pow_succ _ _).symm
+        _ ≤ (L ω)^(n+1) := pow_le_pow_left₀ hω.1 hω.2 _
+  have hI := integral_mono_ae hYn ((hYn.indicator hG).add (hLn.div_const T)) hpoint
+  simp only [Pi.add_apply] at hI
+  rw [integral_add (hYn.indicator hG) (hLn.div_const T),integral_div] at hI
+  have he : (∫ ω,(L ω)^(n+1)∂P)/T = moment P L n/4 := by
+    change moment P L (n+1)/(4*moment P L (n+1)/moment P L n) = _
+    field_simp [hmn.ne', hmnext.ne']
+  rw [he] at hI
+  change moment P Y n ≤ _ at hI
+  change moment P L n/2 ≤ ∫ ω,G.indicator (fun ω => (Y ω)^n) ω∂P
+  linarith
+
+ 
+
+theorem success_strong (L Y : Ω → ℝ) (hY : Measurable Y)
+    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω) (n : ℕ) (hn : 0 < n)
+    (hLn : Integrable (fun ω => (L ω)^(n+1)) P)
+    (hYn : Integrable (fun ω => (Y ω)^n) P)
+    (hmn : 0 < moment P L n) (hmnext : 0 < moment P L (n+1))
+    (hfin : {k : ℤ | multiplicity (moment P L) n k≠0}.Finite)
+    (happrox : 3*moment P L n/4 ≤ moment P Y n) :
+    (2:ℝ)^n/2-1 ≤ ∑ k∈support (moment P L) n, (multiplicity (moment P L) n k:ℝ)*
+      P.real {ω | (2:ℝ)^k ≤ Y ω} := by
+  let m := moment P L
+  let G := {ω | Y ω ≤ cutoff m n}
+  let f := G.indicator (fun ω => (Y ω)^n)
+  let D := (2:ℝ)^n/m n
+  have hG : MeasurableSet G := measurableSet_le hY measurable_const
+  have hD : 0 ≤ D := by dsimp [D,m]; positivity
+  have hpoint : (fun ω => D*f ω-1) ≤ᵐ[P] (fun ω => counts m n (Y ω)) := by
+    filter_upwards [hYL] with ω hω
+    by_cases hg : ω∈G
+    · have he : f ω = (Y ω)^n := indicator_of_mem hg _
+      rw [he]
+      simpa only [D,div_mul_eq_mul_div] using counts_lower hn hmn hfin hω.1 hg
+    · have he : f ω=0 := indicator_of_notMem hg _
+      rw [he,mul_zero,zero_sub]
+      exact (by norm_num : (-1:ℝ)≤0).trans (counts_nonneg m n (Y ω))
+  have hI := integral_mono_ae (((hYn.indicator hG).const_mul D).sub (integrable_const 1))
+    (integrable_counts P m n hY) hpoint
+  simp only [Pi.sub_apply] at hI
+  rw [integral_sub ((hYn.indicator hG).const_mul D) (integrable_const 1),
+    integral_const_mul,integral_const,probReal_univ,one_smul,integral_counts P m n hY] at hI
+  have htr := truncated_moment_lower P L Y hY hYL n hLn hYn hmn hmnext happrox
+  have hmul := mul_le_mul_of_nonneg_left htr hD
+  have he : D*(m n/2) = (2:ℝ)^n/2 := by dsimp [D]; field_simp [show m n ≠ 0 from hmn.ne']
+  rw [he] at hmul
+  exact (sub_le_sub_right hmul 1).trans hI
+
+lemma success_numerics (n : ℕ) (hn : 3 ≤ n) :
+    (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) ≤ (2:ℝ)^n/2-1 ∧
+    (2:ℝ)^n/8 ≤ (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) := by
+  have hlow : (8:ℝ) ≤ 2^n := by
+    have hp := pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hn
+    norm_num at hp ⊢
+    exact hp
+  have hq : (1/2:ℝ)^n ≤ 1/2 := by
+    simpa using pow_le_pow_of_le_one (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)≤1)
+      (show 1 ≤ n by omega)
+  have hden : 1 ≤ 2*(1-(1/2:ℝ)^n) := by linarith
+  have hsub0 : 0 ≤ 1/(2*(1-(1/2:ℝ)^n)) := by positivity
+  have hsub1 : 1/(2*(1-(1/2:ℝ)^n)) ≤ 1 := (div_le_one (by linarith)).mpr hden
+  have hpow : (2:ℝ)^(n-2) = (2:ℝ)^n/4 := by
+    obtain ⟨j,hj⟩ := Nat.exists_eq_add_of_le (show 2 ≤ n by omega)
+    rw [hj,Nat.add_sub_cancel_left,pow_add]
+    norm_num
+  rw [hpow]
+  constructor <;> linarith
+
+ 
+
+theorem success (L Y : Ω → ℝ) (hY : Measurable Y)
+    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1)
+    (n : ℕ) (hn : 3 ≤ n) (hYn : Integrable (fun ω => (Y ω)^n) P)
+    (happrox : 3*moment P L n/4 ≤ moment P Y n) :
+    (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) ≤
+      (∑' k : ℤ, (multiplicity (moment P L) n k:ℝ)*P.real {ω | (2:ℝ)^k ≤ Y ω}) ∧
+    (2:ℝ)^n/8 ≤ (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) := by
+  have hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω := hYL.mono (fun ω hω => hω.1.trans hω.2)
+  obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
+  have hnp : 0 < n := by omega
+  have hfin := finite_multiplicity_support hnp hmean (hm n hnp) (hm (n+1) (by omega)) (hJ n hnp)
+  have hb := success_strong P L Y hY hYL n hnp (hint (n+1)) hYn
+    (hm n hnp) (hm (n+1) (by omega)) hfin happrox
+  rw [sum_eq_tsum_multiplicity hfin] at hb
+  exact ⟨(success_numerics n hn).1.trans hb,(success_numerics n hn).2⟩
+
+end SLEExactGauge.MomentBatches
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/NormalizedDiskMass.lean b/lean/OAI/Probability/SLEGauge/NormalizedDiskMass.lean
index fe7fbd60189cb4317133d95608eb4c6dc7b46a25..6a2f32ab986b404edeeeb6ff4b865af065a53b0f 100644
--- a/lean/OAI/Probability/SLEGauge/NormalizedDiskMass.lean
+++ b/lean/OAI/Probability/SLEGauge/NormalizedDiskMass.lean
@@ -1,537 +1,537 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.HalfPlaneDistortion
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
-open LoewnerGeometry MassVersions
-
-lemma diskAt_fromDisk (θ : ℝ) : diskAt (fromDisk θ 0)=diskMap θ := by
-  funext v
-  rw [fromDisk_zero]
-  have he : star (Complex.exp (Complex.I*θ))=Complex.exp (-Complex.I*θ) := by
-    rw [Complex.star_def,←Complex.exp_conj]
-    simp
-  simp only [diskAt,he,diskMap]
-
-lemma diskAt_scaled (c : ℝ) (hc : 0<c) (w v : ℂ) :
-    diskAt ((c:ℂ)*w) ((c:ℂ)*v)=diskAt w v := by
-  simp only [diskAt,Complex.star_def,map_mul,Complex.conj_ofReal,←mul_sub]
-  exact mul_div_mul_left _ _ (Complex.ofReal_ne_zero.mpr hc.ne')
-
-lemma diskAt_deriv_norm_scaled (c : ℝ) (hc : 0<c) (w v : UpperPlane) :
-    ‖deriv (diskAt ((c:ℂ)*w.1)) ((c:ℂ)*v.1)‖=c⁻¹*‖deriv (diskAt w.1) v.1‖ := by
-  have hw : 0<((c:ℂ)*w.1).im := by simpa [Complex.mul_im] using mul_pos hc w.2
-  have hv : 0<((c:ℂ)*v.1).im := by simpa [Complex.mul_im] using mul_pos hc v.2
-  rw [diskAt_deriv_norm hw hv,diskAt_deriv_norm w.2 v.2]
-  simp only [Complex.star_def,map_mul,Complex.conj_ofReal,←mul_sub,norm_mul,Complex.norm_real,
-    Real.norm_eq_abs,abs_of_pos hc,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,
-    zero_mul,add_zero]
-  field_simp
-
- 
-def pointDiskRegion (w : UpperPlane) (q : ℝ) : Set UpperPlane := {v | ‖diskAt w.1 v.1‖≤q}
-
-def pointDiskWeight (a : ℝ) (w : UpperPlane) (v : UpperPlane) : ℝ≥0∞ :=
-  ENNReal.ofReal (‖deriv (diskAt w.1) v.1‖^(2-alpha a))
-
-def pointDiskMass (a : ℝ) (w : UpperPlane) (q : ℝ) (ν : Measure UpperPlane) : ℝ≥0∞ :=
-  ∫⁻v in pointDiskRegion w q,pointDiskWeight a w v∂ν
-
-lemma pointDiskRegion_measurable (w : UpperPlane) (q : ℝ) : MeasurableSet (pointDiskRegion w q) := by
-  unfold pointDiskRegion diskAt
-  exact measurableSet_le (by fun_prop) measurable_const
-
-lemma pointDiskWeight_measurable (a : ℝ) (w : UpperPlane) : Measurable (pointDiskWeight a w) := by
-  have he : pointDiskWeight a w=fun v : UpperPlane=>ENNReal.ofReal
-      ((2*w.1.im/‖v.1-star w.1‖^2)^(2-alpha a)) := by
-    funext v
-    rw [pointDiskWeight,diskAt_deriv_norm w.2 v.2]
-  rw [he]
-  fun_prop
-
-lemma pointDiskMass_diskMass (a θ q : ℝ) (ν : Measure UpperPlane)
-    (hθ : 0<θ) (hθπ : θ<Real.pi) :
-    pointDiskMass a ⟨fromDisk θ 0,by
-      rw [fromDisk_zero_im]
-      exact Real.sin_pos_of_pos_of_lt_pi hθ hθπ⟩ q ν =
-      diskMass a θ ν (Metric.closedBall 0 q) := by
-  rw [diskMass,Measure.map_apply (diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet,
-    withDensity_apply _ ((diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet)]
-  simp only [pointDiskMass,pointDiskWeight,pointDiskRegion,diskAt_fromDisk,diskWeight,
-    Set.preimage,Metric.mem_closedBall,dist_zero_right]
-
-lemma pointDiskMass_smul (a : ℝ) (w : UpperPlane) (q : ℝ) (ν : Measure UpperPlane) (c : ℝ≥0∞) :
-    pointDiskMass a w q (c•ν)=c*pointDiskMass a w q ν := by
-  rw [pointDiskMass,Measure.restrict_smul,lintegral_smul_measure,smul_eq_mul]
-  rfl
-
-lemma pointDiskMass_map_dilation (a : ℝ) {c : ℝ≥0} (hc : 0<c) (w : UpperPlane)
-    (q : ℝ) (ν : Measure UpperPlane) :
-    pointDiskMass a (upperDilation c hc w) q (ν.map (upperDilation c hc))=
-      ENNReal.ofReal ((c:ℝ)^(alpha a-2))*pointDiskMass a w q ν := by
-  rw [pointDiskMass,←lintegral_indicator (pointDiskRegion_measurable _ q),
-    (upperDilation c hc).measurableEmbedding.lintegral_map]
-  have he (v : UpperPlane) :
-      (pointDiskRegion (upperDilation c hc w) q).indicator
-        (pointDiskWeight a (upperDilation c hc w)) (upperDilation c hc v)=
-        ENNReal.ofReal ((c:ℝ)^(alpha a-2))*(pointDiskRegion w q).indicator
-          (pointDiskWeight a w) v := by
-    have hreg : upperDilation c hc v∈pointDiskRegion (upperDilation c hc w) q ↔
-        v∈pointDiskRegion w q := by
-      change ‖diskAt (((c:ℝ):ℂ)*w.1) (((c:ℝ):ℂ)*v.1)‖≤q ↔ _
-      rw [diskAt_scaled (c:ℝ) hc]
-      rfl
-    by_cases hv : v∈pointDiskRegion w q
-    · rw [indicator_of_mem (hreg.mpr hv),indicator_of_mem hv]
-      rw [pointDiskWeight,upperDilation_apply,upperDilation_apply,
-        diskAt_deriv_norm_scaled _ (show 0<(c:ℝ) from hc) w v,Real.mul_rpow (inv_nonneg.mpr c.coe_nonneg) (norm_nonneg _),
-        Real.inv_rpow c.coe_nonneg,←Real.rpow_neg c.coe_nonneg,neg_sub,
-        ENNReal.ofReal_mul (Real.rpow_nonneg c.coe_nonneg _)]
-      rfl
-    · rw [indicator_of_notMem (hreg.not.mpr hv),indicator_of_notMem hv,mul_zero]
-  simp_rw [he]
-  rw [lintegral_const_mul _ ((pointDiskWeight_measurable a w).indicator (pointDiskRegion_measurable w q)),
-    lintegral_indicator (pointDiskRegion_measurable w q)]
-  rfl
-
- 
-lemma pointDiskMass_recovered_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (s : ℕ→Finset ℕ) (v : ℕ→ℕ→ℝ) (hv : ∀k j,j∈s k→0≤v k j)
-    (t : ℝ≥0∞) (b : GDriver) (w : UpperPlane) (q : ℝ) :
-    pointDiskMass a w q (recoveredMass a ha s v hv t (scaleDriver c b))=
-      pointDiskMass a (upperDilation c hc w) q
-        (recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b) := by
-  have he := pointDiskMass_map_dilation a hc w q (recoveredMass a ha s v hv t (scaleDriver c b))
-  rw [map_recoveredMass_scaled a ha hc,pointDiskMass_smul] at he
-  exact (ENNReal.mul_right_inj
-    (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _)).ne'
-    ENNReal.ofReal_ne_top).mp he.symm
-
- 
-
-lemma normalized_subdiskMass_eq (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (v : ℕ→ℕ→ℝ) (hv : ∀k j,j∈s k→0≤v k j)
-    (w : UpperPlane) (b : GDriver) (q : ℝ) (l : ℕ) :
-    subdiskMass a (recoveredMass a ha s v hv) q l (pointAngle w,normalizeDriver w b)=
-      pointDiskMass a w q (recoveredMass a ha s v hv
-        ((‖w.1‖₊:ℝ≥0∞)^2*levelTime a (pointAngle w).1 (normalizeDriver w b).1 l) b) := by
-  let w₀ : UpperPlane := ⟨fromDisk (pointAngle w).1 0,by
-    rw [fromDisk_zero_im]
-    exact Real.sin_pos_of_pos_of_lt_pi (pointAngle w).2.1 (pointAngle w).2.2⟩
-  have he : upperDilation ‖w.1‖₊ (point_norm_pos w) w₀=w := by
-    apply Subtype.ext
-    exact norm_mul_fromDisk_pointAngle w
-  change diskMass a (pointAngle w).1 _ (Metric.closedBall 0 q)=_
-  rw [←pointDiskMass_diskMass a (pointAngle w).1 q _ (pointAngle w).2.1 (pointAngle w).2.2]
-  change pointDiskMass a w₀ q (recoveredMass a ha s v hv _ (scaleDriver ‖w.1‖₊ b))=_
-  rw [pointDiskMass_recovered_scaled a ha (point_norm_pos w),he]
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
-open LoewnerGeometry MassVersions TraceMassContinuity
-
-lemma inverseUpper_measurableEmbedding (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0) :
-    MeasurableEmbedding (inverseUpper a (lt_trans (by norm_num) ha) b t) := by
-  have hs : MeasurableSet {z : UpperPlane | alive a b.1 t z.1} :=
-    (measurableSet_alive_joint a ha).preimage
-      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
-  exact (MeasurableEmbedding.subtype_coe hs).comp
-    (restartHomeomorph a (lt_trans (by norm_num) ha) b t).measurableEmbedding
-
- 
-
-def spatialDiskRegion (a : ℝ) (ha : 0<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) (q : ℝ) : Set UpperPlane :=
-  inverseUpper a ha b t '' pointDiskRegion w q
-
-lemma spatialDiskRegion_measurable (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) (q : ℝ) :
-    MeasurableSet (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q) :=
-  (inverseUpper_measurableEmbedding a ha b t).measurableSet_image.mpr (pointDiskRegion_measurable w q)
-
-lemma spatialDiskRegion_subset_alive (a : ℝ) (ha : 0<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) (q : ℝ) :
-    spatialDiskRegion a ha b t w q ⊆ {z : UpperPlane | alive a b.1 t z.1} := by
-  rintro _ ⟨v,_,rfl⟩
-  exact inverseMap_alive ha b t v.2
-
-lemma subtype_spatialDiskRegion (a : ℝ) (ha : 0<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) (q : ℝ) :
-    Subtype.val '' spatialDiskRegion a ha b t w q=inverseMap a b t '' centeredRegion w.1 q := by
-  ext z
-  constructor
-  · rintro ⟨u,⟨v,hv,rfl⟩,rfl⟩
-    exact ⟨v.1,⟨v.2,hv⟩,rfl⟩
-  · rintro ⟨v,hv,rfl⟩
-    exact ⟨inverseUpper a ha b t ⟨v,hv.1⟩,⟨⟨v,hv.1⟩,hv.2,rfl⟩,rfl⟩
-
-lemma restartPush_spatialDisk (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) (q : ℝ) (ν : Measure UpperPlane) :
-    restartPush a (lt_trans (by norm_num) ha) b t ν
-      (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q)=
-      ∫⁻v in pointDiskRegion w q,restartWeight a b t v∂ν := by
-  rw [restartPush,(inverseUpper_measurableEmbedding a ha b t).map_apply,
-    spatialDiskRegion,preimage_image_eq _ (inverseUpper_injective _ _ _ _),
-    withDensity_apply _ (pointDiskRegion_measurable w q)]
-
- 
-
-def diskLower (q : ℝ) : ℝ := (2*subdiskDistortion q)⁻¹
-def diskUpper (q : ℝ) : ℝ := subdiskDistortion q*(subdiskReach q+2)^2/2
-
-lemma diskLower_pos (q : ℝ) : 0<diskLower q := by
-  have hd := subdiskDistortion_pos q
-  unfold diskLower
-  positivity
-lemma diskUpper_pos {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 0<diskUpper q := by
-  have hd := subdiskDistortion_pos q
-  have hr := subdiskReach_nonneg hq hq1
-  unfold diskUpper
-  positivity
-
-lemma restartWeight_disk_bounds (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) {q : ℝ} (hq : 0≤q) (hq1 : q<1) (v : UpperPlane)
-    (hv : v∈pointDiskRegion w q) :
-    ENNReal.ofReal ((diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
-        pointDiskWeight a w v ≤ restartWeight a b t v ∧
-    restartWeight a b t v ≤
-      ENNReal.ofReal ((diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
-        pointDiskWeight a w v := by
-  have ha0 : 0<a := by linarith
-  have hd : 0<‖deriv (diskAt w.1) v.1‖ := by
-    rw [diskAt_deriv_norm w.2 v.2]
-    exact div_pos (mul_pos (by norm_num) w.2)
-      (sq_pos_of_pos (norm_pos_iff.mpr (diskAt_denominator_ne w.2 v.2)))
-  have hb := centered_disk_deriv_ratio ha0 b t w.2 hq hq1 ⟨v.2,hv⟩
-  have hlow : diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖)*‖deriv (diskAt w.1) v.1‖≤
-      ‖deriv (inverseMap a b t) v.1‖ := by
-    have hh := (le_div_iff₀ hd).mp hb.1
-    simpa only [diskLower,div_eq_mul_inv,mul_comm,mul_left_comm,mul_assoc] using hh
-  have hupp : ‖deriv (inverseMap a b t) v.1‖≤
-      (diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))*‖deriv (diskAt w.1) v.1‖ :=
-    (div_le_iff₀ hd).mp hb.2
-  have hl0 : 0≤diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖) :=
-    mul_nonneg (diskLower_pos q).le (mul_nonneg w.2.le (norm_nonneg _))
-  have hu0 : 0≤diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖) :=
-    mul_nonneg (diskUpper_pos hq hq1).le (mul_nonneg w.2.le (norm_nonneg _))
-  constructor
-  · have hh := ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (mul_nonneg hl0 hd.le)
-      hlow (dimension_pos a ha).le)
-    rw [Real.mul_rpow hl0 hd.le,ENNReal.ofReal_mul (Real.rpow_nonneg hl0 _)] at hh
-    exact hh
-  · have hh := ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (norm_nonneg _) hupp (dimension_pos a ha).le)
-    rw [Real.mul_rpow hu0 hd.le,ENNReal.ofReal_mul (Real.rpow_nonneg hu0 _)] at hh
-    exact hh
-
- 
-
-theorem restartPush_disk_bounds (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
-    (w : UpperPlane) {q : ℝ} (hq : 0≤q) (hq1 : q<1) (ν : Measure UpperPlane) :
-    ENNReal.ofReal ((diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
-        pointDiskMass a w q ν ≤
-      restartPush a (lt_trans (by norm_num) ha) b t ν
-        (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q) ∧
-    restartPush a (lt_trans (by norm_num) ha) b t ν
-        (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q) ≤
-      ENNReal.ofReal ((diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
-        pointDiskMass a w q ν := by
-  rw [restartPush_spatialDisk a ha,pointDiskMass,←lintegral_const_mul _ (pointDiskWeight_measurable a w),
-    ←lintegral_const_mul _ (pointDiskWeight_measurable a w)]
-  constructor
-  · exact setLIntegral_mono' (pointDiskRegion_measurable w q)
-      (fun v hv=>(restartWeight_disk_bounds a ha b t w hq hq1 v hv).1)
-  · exact setLIntegral_mono' (pointDiskRegion_measurable w q)
-      (fun v hv=>(restartWeight_disk_bounds a ha b t w hq hq1 v hv).2)
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
-open LoewnerGeometry MassVersions TraceMassContinuity
-
- 
-
-def radialDiskRegion (a : ℝ) (ha : 0<a) (z : ℂ) (σ q : ℝ) (b : GDriver) : Set UpperPlane :=
-  spatialDiskRegion a ha b (radialTime a b.1 z σ).toNNReal (radialPoint a z σ b) q
-
-lemma radialPoint_spatial_radius (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (b : GDriver)
-    (hb : radialTime a b.1 z σ≠∞) :
-    (radialPoint a z σ b).1.im *
-      ‖deriv (inverseMap a b (radialTime a b.1 z σ).toNNReal) (radialPoint a z σ b).1‖=σ := by
-  have hs := radialTime_attained a ha z hz σ hσ hσz b hb
-  rw [radialPoint_eq a ha z hz σ hσ hσz b hb]
-  change (loewnerMap a b.1 _ z-(b.1 _:ℂ)).im*‖deriv (inverseMap a b _)
-    (loewnerMap a b.1 _ z-(b.1 _:ℂ))‖=σ
-  rw [inverseMap_spatial_radius ha b _ hs.1,hs.2]
-
-lemma radialDiskRegion_localization (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) {q : ℝ} (hq : 1/2≤q) (hq1 : q<1)
-    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
-    (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z (σ/200) ⊆ radialDiskRegion a ha z σ q b ∧
-    radialDiskRegion a ha z σ q b ⊆
-      (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z (subdiskReach q*subdiskDistortion q*σ) := by
-  have hs := radialTime_attained a ha z hz σ hσ hσz b hb
-  have hh := centered_localization_at_alive ha b _ hs.1 hq hq1
-  rw [hs.2] at hh
-  have he : (Subtype.val : UpperPlane→ℂ) '' radialDiskRegion a ha z σ q b=
-      inverseMap a b (radialTime a b.1 z σ).toNNReal ''
-        centeredRegion (loewnerMap a b.1 (radialTime a b.1 z σ).toNNReal z-
-          (b.1 (radialTime a b.1 z σ).toNNReal:ℂ)) q := by
-    rw [radialDiskRegion,subtype_spatialDiskRegion,radialPoint_eq a ha z hz σ hσ hσz b hb]
-    rfl
-  constructor
-  · intro u hu
-    obtain ⟨v,hv,heq⟩ := (he ▸ hh.1 hu)
-    exact (Subtype.ext heq : v=u) ▸ hv
-  · intro u hu
-    exact hh.2 (he ▸ mem_image_of_mem Subtype.val hu)
-
- 
-
-theorem actual_radial_spatial_bounds (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (D : CanonicalMassData κ a ha W (recoveredMass a ha s c hc))
-    (z : ℂ) (hz : 0<z.im) (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im)
-    (q : ℝ) (hq : 0≤q) (hq1 : q<1) :
-    let μ := recoveredMass a ha s c hc
-    let S := fun b : GDriver=>radialTime a b.1 z σ
-    let R := radialDiskRegion a (lt_trans (by norm_num) ha) z σ q
-    ∀ᵐ b∂W,S b≠∞→μ (S b) b (R b)=0 ∧ ∀l : ℕ,
-      ENNReal.ofReal ((diskLower q*σ)^(2-alpha a))*normalizedTest a z σ (subdiskMass a μ q l) b ≤
-        μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b (R b) ∧
-      μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b (R b) ≤
-        ENNReal.ofReal ((diskUpper q*σ)^(2-alpha a))*normalizedTest a z σ (subdiskMass a μ q l) b := by
-  dsimp only
-  let μ := recoveredMass a ha s c hc
-  let S := fun b : GDriver=>radialTime a b.1 z σ
-  let ha0 : 0<a := lt_trans (by norm_num) ha
-  have hS := radialTime_stopping a ha0 z hz σ hσ hσz
-  filter_upwards [D.restart S hS] with b hb
-  intro hbf
-  have hrest := hb hbf
-  let R := radialDiskRegion a ha0 z σ q b
-  have hR : MeasurableSet R := spatialDiskRegion_measurable a ha b _ _ q
-  have hRA : R⊆{u : UpperPlane | alive a b.1 (S b).toNNReal u.1} :=
-    spatialDiskRegion_subset_alive a ha0 b _ _ q
-  have hzero : μ (S b) b R=0 := by
-    have hh := congrArg (fun ν : Measure UpperPlane=>ν R) hrest.1
-    rwa [Measure.restrict_apply hR,inter_eq_left.mpr hRA,Measure.coe_zero,Pi.zero_apply] at hh
-  refine ⟨hzero,?_⟩
-  intro l
-  let w := radialPoint a z σ b
-  let t := (‖w.1‖₊:ℝ≥0∞)^2*levelTime a (pointAngle w).1 (normalizeDriver w (restartAt S b)).1 l
-  have ht : S b+t=radialTime a b.1 z (σ*Real.exp (-2*a*l)) :=
-    normalized_levelTime_restart a ha0 z hz σ hσ hσz l b hbf
-  have he : μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b R=
-      restartPush a ha0 b (S b).toNNReal (μ t (restartAt S b)) R := by
-    have hh := congrArg (fun ν : Measure UpperPlane=>ν R) (hrest.2 t)
-    rwa [Measure.restrict_apply hR,inter_eq_left.mpr hRA,ht] at hh
-  have hbounds := restartPush_disk_bounds a ha b (S b).toNNReal w hq hq1 (μ t (restartAt S b))
-  rw [radialPoint_spatial_radius a ha0 z hz σ hσ hσz b hbf] at hbounds
-  have hnorm : normalizedTest a z σ (subdiskMass a μ q l) b=
-      pointDiskMass a w q (μ t (restartAt S b)) :=
-    normalized_subdiskMass_eq a ha s c hc w (restartAt S b) q l
-  change ENNReal.ofReal ((diskLower q*σ)^(2-alpha a))*normalizedTest a z σ (subdiskMass a μ q l) b≤
-      μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b R ∧ _
-  rw [hnorm,he]
-  exact hbounds
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.PastMass
-open FlowIdentification ExactSLEGreen TraceMassContinuity TraceRestriction
-
-lemma denseRange_nnrat_nnreal : DenseRange (fun q : ℚ≥0 => (q : ℝ≥0)) := by
-  apply dense_of_exists_between
-  intro a b hab
-  obtain ⟨q,haq,hqb⟩ := exists_rat_btwn (show (a : ℝ)<(b : ℝ) from hab)
-  have hq : 0≤q := Rat.cast_nonneg.mp ((a.2.trans haq.le) : (0 : ℝ)≤q)
-  let r : ℚ≥0 := ⟨q,hq⟩
-  refine ⟨(r : ℝ≥0),⟨r,rfl⟩,?_,?_⟩
-  · change (a : ℝ) < (r : ℝ)
-    simpa only [r,NNRat.cast_mk] using haq
-  · change (r : ℝ) < (b : ℝ)
-    simpa only [r,NNRat.cast_mk] using hqb
-
- 
-
-lemma accumulated_continuous (κ : ℝ) (hκ : 0<κ) (b : GDriver)
-    (μ : ℝ≥0∞→Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q≠∞) :
-    Continuous (fun t : ℝ≥0 => (μ t Q).toReal) := by
-  let := hμ.atomless
-  let f : UpperPlane→ℝ := Q.indicator (fun _=>1)
-  have hf : Integrable f (μ ∞) := (integrable_indicator_iff hQ).mpr (integrableOn_const hfin)
-  have hγ := continuous_internalTrace hκ _ hμ.continuous_trace
-  have hc := continuous_integral_visited (μ ∞) (fun z : UpperPlane=>z.1)
-    measurable_subtype_coe Subtype.val_injective _ hγ f hf
-  have he (t : ℝ≥0) : (∫z in visited (fun z : UpperPlane=>z.1)
-      (internalTrace κ (driverHomeomorph.symm b)) t,f z∂μ ∞) = (μ t Q).toReal := by
-    rw [←hμ.restriction]
-    simp only [f,integral_indicator hQ,integral_const,Measure.real,
-      Measure.restrict_apply_univ,smul_eq_mul,mul_one]
-  simpa only [he] using hc
-
-lemma accumulated_zero (κ : ℝ) (b : GDriver) (μ : ℝ≥0∞→Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ) : μ 0=0 := by
-  simpa only [ENNReal.coe_zero,not_lt_zero,ofPred_false,Measure.restrict_empty] using
-    hμ.strict_firstVisit_identity 0
-
- 
-
-lemma exists_rational_threshold (κ : ℝ) (hκ : 0<κ) (b : GDriver)
-    (μ : ℝ≥0∞→Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q≠∞)
-    (H U : ℝ) (hH : 0≤H) (hHU : H<U) (hUM : U<(μ ∞ Q).toReal) :
-    ∃q : ℚ≥0,H<(μ ((q : ℝ≥0) : ℝ≥0∞) Q).toReal ∧
-      (μ ((q : ℝ≥0) : ℝ≥0∞) Q).toReal<U := by
-  let M : ℝ≥0→ℝ := fun t=>(μ t Q).toReal
-  have hM : Continuous M := accumulated_continuous κ hκ b μ hμ Q hQ hfin
-  have hzero : M 0=0 := by simp [M,accumulated_zero κ b μ hμ]
-  have hlim : Tendsto (fun n : ℕ=>M (n:ℝ≥0)) atTop (𝓝 (μ ∞ Q).toReal) :=
-    (ENNReal.tendsto_toReal hfin).comp (hμ.exhaustion Q hQ)
-  obtain ⟨n,hn⟩ := (hlim.eventually (eventually_gt_nhds hUM)).exists
-  have hmid : (H+U)/2∈Icc (M 0) (M (n:ℝ≥0)) := by rw [hzero]; constructor <;> linarith
-  obtain ⟨t,ht⟩ := intermediate_value_univ (0:ℝ≥0) (n:ℝ≥0) hM hmid
-  exact denseRange_nnrat_nnreal.exists_mem_open (isOpen_Ioo.preimage hM)
-    ⟨t,by change H<M t ∧ M t<U; rw [ht]; constructor <;> linarith⟩
-
- 
-
-lemma mass_le_twice_strict_past (κ : ℝ) (hκ : 0<κ) (b : GDriver)
-    (μ : ℝ≥0∞→Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q≠∞)
-    (H : ℝ) (hH : 0≤H) (hbad : 2*ENNReal.ofReal H<μ ∞ Q) :
-    μ ∞ Q ≤ 2*μ ∞ (Q∩{z | ∃q : ℚ≥0,
-      ENNReal.ofReal H<μ ((q:ℝ≥0):ℝ≥0∞) Q ∧ ((q:ℝ≥0):ℝ≥0∞)<visitTime κ b z}) := by
-  have hbadReal : 2*H<(μ ∞ Q).toReal := by
-    apply (ENNReal.ofReal_lt_iff_lt_toReal (mul_nonneg (by norm_num) hH) hfin).mp
-    simpa only [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2),ENNReal.ofReal_ofNat] using hbad
-  have hMpos : 0<(μ ∞ Q).toReal := by linarith
-  obtain ⟨q,hqH,hqM⟩ := exists_rational_threshold κ hκ b μ hμ Q hQ hfin
-    H ((μ ∞ Q).toReal/2) hH (by linarith) (by linarith)
-  have hqle : μ ((q:ℝ≥0):ℝ≥0∞) Q≤μ ∞ Q := hμ.monotone le_top Q
-  have hqfin : μ ((q:ℝ≥0):ℝ≥0∞) Q≠∞ := ne_top_of_le_ne_top hfin hqle
-  have hqH' : ENNReal.ofReal H<μ ((q:ℝ≥0):ℝ≥0∞) Q :=
-    (ENNReal.ofReal_lt_iff_lt_toReal hH hqfin).mpr hqH
-  have hhalf : μ ∞ Q≤2*(μ ∞ Q-μ ((q:ℝ≥0):ℝ≥0∞) Q) := by
-    apply (ENNReal.toReal_le_toReal hfin (ENNReal.mul_ne_top (by norm_num)
-      (ne_top_of_le_ne_top hfin tsub_le_self))).mp
-    rw [ENNReal.toReal_mul,ENNReal.toReal_sub_of_le hqle hfin]
-    norm_num only [ENNReal.toReal_ofNat]
-    linarith
-  calc
-    μ ∞ Q≤2*(μ ∞ Q-μ ((q:ℝ≥0):ℝ≥0∞) Q) := hhalf
-    _=2*μ ∞ (Q∩{z | ((q:ℝ≥0):ℝ≥0∞)<visitTime κ b z}) := by
-      rw [stopped_strictPast_mass κ b μ hμ Q hQ hfin]
-    _≤_ := by
-      apply mul_le_mul_right
-      apply measure_mono
-      intro z hz
-      exact ⟨hz.1,q,hqH',hz.2⟩
-
- 
-
-def rationalTestEnumeration : ℕ→ℚ≥0 := Classical.choose (exists_surjective_nat ℚ≥0)
-
-lemma rationalTestEnumeration_surjective : Function.Surjective rationalTestEnumeration :=
-  Classical.choose_spec (exists_surjective_nat ℚ≥0)
-
-def rationalPastKernel (W : Measure GDriver) (a : ℝ)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane) (Q : Set UpperPlane) (H : ℝ≥0∞) :
-    UpperPlane→ℝ≥0∞ :=
-  countablePastKernel rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0)) W a μ Q H
-
-namespace TracePalmData
-variable {κ a : ℝ} {W : Measure GDriver} {μ : ℝ≥0∞→GDriver→Measure UpperPlane}
-    (D : TracePalmData κ a W μ)
-include D
-
-lemma rational_Palm (ha : 1/4<a) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ≥0∞) :
-    (∫⁻b,μ ∞ b (Q∩{z | ∃q : ℚ≥0,H<μ ((q:ℝ≥0):ℝ≥0∞) b Q ∧
-      ((q:ℝ≥0):ℝ≥0∞)<visitTime κ b z})∂W)=
-      ∫⁻z in Q, ENNReal.ofReal (initialGreen a z)*rationalPastKernel W a μ Q H z∂upperArea :=
-  D.countable_Palm ha hW rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0))
-    rationalTestEnumeration_surjective Q hQ H
-
-lemma rational_kernel_measurable (ha : 1/4<a) [SFinite W]
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (H : ℝ≥0∞) :
-    Measurable (rationalPastKernel W a μ Q H) :=
-  D.countable_kernel_measurable ha rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0)) Q hQ H
-
- 
-
-lemma expected_bad_mass_le (hκ : 0<κ) (ha : 1/4<a) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ) (hH : 0≤H) :
-    (∫⁻b, {b | 2*ENNReal.ofReal H<μ ∞ b Q}.indicator (fun b=>μ ∞ b Q) b∂W) ≤
-      2*∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*
-        rationalPastKernel W a μ Q (ENNReal.ofReal H) z∂upperArea := by
-  have hm := D.countable_past_aemeasurable rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0))
-    rationalTestEnumeration_surjective Q hQ (ENNReal.ofReal H)
-  rw [←D.rational_Palm ha hW Q hQ (ENNReal.ofReal H),←lintegral_const_mul'' _ hm]
-  apply lintegral_mono_ae
-  filter_upwards [D.path] with b hb
-  by_cases hbad : 2*ENNReal.ofReal H<μ ∞ b Q
-  · simp only [indicator,mem_ofPred_eq,ite_eq_left hbad]
-    exact mass_le_twice_strict_past κ hκ b (fun v=>μ v b) hb Q hQ.measurableSet
-      (D.compact_finite Q hQ ∞ b) H hH hbad
-  · simp only [indicator,mem_ofPred_eq,ite_eq_right hbad]
-    exact bot_le
-
-end TracePalmData
-
-end SLEExactGauge.PastMass
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.HalfPlaneDistortion
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
+open LoewnerGeometry MassVersions
+
+lemma diskAt_fromDisk (θ : ℝ) : diskAt (fromDisk θ 0)=diskMap θ := by
+  funext v
+  rw [fromDisk_zero]
+  have he : star (Complex.exp (Complex.I*θ))=Complex.exp (-Complex.I*θ) := by
+    rw [Complex.star_def,←Complex.exp_conj]
+    simp
+  simp only [diskAt,he,diskMap]
+
+lemma diskAt_scaled (c : ℝ) (hc : 0<c) (w v : ℂ) :
+    diskAt ((c:ℂ)*w) ((c:ℂ)*v)=diskAt w v := by
+  simp only [diskAt,Complex.star_def,map_mul,Complex.conj_ofReal,←mul_sub]
+  exact mul_div_mul_left _ _ (Complex.ofReal_ne_zero.mpr hc.ne')
+
+lemma diskAt_deriv_norm_scaled (c : ℝ) (hc : 0<c) (w v : UpperPlane) :
+    ‖deriv (diskAt ((c:ℂ)*w.1)) ((c:ℂ)*v.1)‖=c⁻¹*‖deriv (diskAt w.1) v.1‖ := by
+  have hw : 0<((c:ℂ)*w.1).im := by simpa [Complex.mul_im] using mul_pos hc w.2
+  have hv : 0<((c:ℂ)*v.1).im := by simpa [Complex.mul_im] using mul_pos hc v.2
+  rw [diskAt_deriv_norm hw hv,diskAt_deriv_norm w.2 v.2]
+  simp only [Complex.star_def,map_mul,Complex.conj_ofReal,←mul_sub,norm_mul,Complex.norm_real,
+    Real.norm_eq_abs,abs_of_pos hc,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,
+    zero_mul,add_zero]
+  field_simp
+
+ 
+def pointDiskRegion (w : UpperPlane) (q : ℝ) : Set UpperPlane := {v | ‖diskAt w.1 v.1‖≤q}
+
+def pointDiskWeight (a : ℝ) (w : UpperPlane) (v : UpperPlane) : ℝ≥0∞ :=
+  ENNReal.ofReal (‖deriv (diskAt w.1) v.1‖^(2-alpha a))
+
+def pointDiskMass (a : ℝ) (w : UpperPlane) (q : ℝ) (ν : Measure UpperPlane) : ℝ≥0∞ :=
+  ∫⁻v in pointDiskRegion w q,pointDiskWeight a w v∂ν
+
+lemma pointDiskRegion_measurable (w : UpperPlane) (q : ℝ) : MeasurableSet (pointDiskRegion w q) := by
+  unfold pointDiskRegion diskAt
+  exact measurableSet_le (by fun_prop) measurable_const
+
+lemma pointDiskWeight_measurable (a : ℝ) (w : UpperPlane) : Measurable (pointDiskWeight a w) := by
+  have he : pointDiskWeight a w=fun v : UpperPlane=>ENNReal.ofReal
+      ((2*w.1.im/‖v.1-star w.1‖^2)^(2-alpha a)) := by
+    funext v
+    rw [pointDiskWeight,diskAt_deriv_norm w.2 v.2]
+  rw [he]
+  fun_prop
+
+lemma pointDiskMass_diskMass (a θ q : ℝ) (ν : Measure UpperPlane)
+    (hθ : 0<θ) (hθπ : θ<Real.pi) :
+    pointDiskMass a ⟨fromDisk θ 0,by
+      rw [fromDisk_zero_im]
+      exact Real.sin_pos_of_pos_of_lt_pi hθ hθπ⟩ q ν =
+      diskMass a θ ν (Metric.closedBall 0 q) := by
+  rw [diskMass,Measure.map_apply (diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet,
+    withDensity_apply _ ((diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet)]
+  simp only [pointDiskMass,pointDiskWeight,pointDiskRegion,diskAt_fromDisk,diskWeight,
+    Set.preimage,Metric.mem_closedBall,dist_zero_right]
+
+lemma pointDiskMass_smul (a : ℝ) (w : UpperPlane) (q : ℝ) (ν : Measure UpperPlane) (c : ℝ≥0∞) :
+    pointDiskMass a w q (c•ν)=c*pointDiskMass a w q ν := by
+  rw [pointDiskMass,Measure.restrict_smul,lintegral_smul_measure,smul_eq_mul]
+  rfl
+
+lemma pointDiskMass_map_dilation (a : ℝ) {c : ℝ≥0} (hc : 0<c) (w : UpperPlane)
+    (q : ℝ) (ν : Measure UpperPlane) :
+    pointDiskMass a (upperDilation c hc w) q (ν.map (upperDilation c hc))=
+      ENNReal.ofReal ((c:ℝ)^(alpha a-2))*pointDiskMass a w q ν := by
+  rw [pointDiskMass,←lintegral_indicator (pointDiskRegion_measurable _ q),
+    (upperDilation c hc).measurableEmbedding.lintegral_map]
+  have he (v : UpperPlane) :
+      (pointDiskRegion (upperDilation c hc w) q).indicator
+        (pointDiskWeight a (upperDilation c hc w)) (upperDilation c hc v)=
+        ENNReal.ofReal ((c:ℝ)^(alpha a-2))*(pointDiskRegion w q).indicator
+          (pointDiskWeight a w) v := by
+    have hreg : upperDilation c hc v∈pointDiskRegion (upperDilation c hc w) q ↔
+        v∈pointDiskRegion w q := by
+      change ‖diskAt (((c:ℝ):ℂ)*w.1) (((c:ℝ):ℂ)*v.1)‖≤q ↔ _
+      rw [diskAt_scaled (c:ℝ) hc]
+      rfl
+    by_cases hv : v∈pointDiskRegion w q
+    · rw [indicator_of_mem (hreg.mpr hv),indicator_of_mem hv]
+      rw [pointDiskWeight,upperDilation_apply,upperDilation_apply,
+        diskAt_deriv_norm_scaled _ (show 0<(c:ℝ) from hc) w v,Real.mul_rpow (inv_nonneg.mpr c.coe_nonneg) (norm_nonneg _),
+        Real.inv_rpow c.coe_nonneg,←Real.rpow_neg c.coe_nonneg,neg_sub,
+        ENNReal.ofReal_mul (Real.rpow_nonneg c.coe_nonneg _)]
+      rfl
+    · rw [indicator_of_notMem (hreg.not.mpr hv),indicator_of_notMem hv,mul_zero]
+  simp_rw [he]
+  rw [lintegral_const_mul _ ((pointDiskWeight_measurable a w).indicator (pointDiskRegion_measurable w q)),
+    lintegral_indicator (pointDiskRegion_measurable w q)]
+  rfl
+
+ 
+lemma pointDiskMass_recovered_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (s : ℕ→Finset ℕ) (v : ℕ→ℕ→ℝ) (hv : ∀k j,j∈s k→0≤v k j)
+    (t : ℝ≥0∞) (b : GDriver) (w : UpperPlane) (q : ℝ) :
+    pointDiskMass a w q (recoveredMass a ha s v hv t (scaleDriver c b))=
+      pointDiskMass a (upperDilation c hc w) q
+        (recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b) := by
+  have he := pointDiskMass_map_dilation a hc w q (recoveredMass a ha s v hv t (scaleDriver c b))
+  rw [map_recoveredMass_scaled a ha hc,pointDiskMass_smul] at he
+  exact (ENNReal.mul_right_inj
+    (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _)).ne'
+    ENNReal.ofReal_ne_top).mp he.symm
+
+ 
+
+lemma normalized_subdiskMass_eq (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (v : ℕ→ℕ→ℝ) (hv : ∀k j,j∈s k→0≤v k j)
+    (w : UpperPlane) (b : GDriver) (q : ℝ) (l : ℕ) :
+    subdiskMass a (recoveredMass a ha s v hv) q l (pointAngle w,normalizeDriver w b)=
+      pointDiskMass a w q (recoveredMass a ha s v hv
+        ((‖w.1‖₊:ℝ≥0∞)^2*levelTime a (pointAngle w).1 (normalizeDriver w b).1 l) b) := by
+  let w₀ : UpperPlane := ⟨fromDisk (pointAngle w).1 0,by
+    rw [fromDisk_zero_im]
+    exact Real.sin_pos_of_pos_of_lt_pi (pointAngle w).2.1 (pointAngle w).2.2⟩
+  have he : upperDilation ‖w.1‖₊ (point_norm_pos w) w₀=w := by
+    apply Subtype.ext
+    exact norm_mul_fromDisk_pointAngle w
+  change diskMass a (pointAngle w).1 _ (Metric.closedBall 0 q)=_
+  rw [←pointDiskMass_diskMass a (pointAngle w).1 q _ (pointAngle w).2.1 (pointAngle w).2.2]
+  change pointDiskMass a w₀ q (recoveredMass a ha s v hv _ (scaleDriver ‖w.1‖₊ b))=_
+  rw [pointDiskMass_recovered_scaled a ha (point_norm_pos w),he]
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
+open LoewnerGeometry MassVersions TraceMassContinuity
+
+lemma inverseUpper_measurableEmbedding (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0) :
+    MeasurableEmbedding (inverseUpper a (lt_trans (by norm_num) ha) b t) := by
+  have hs : MeasurableSet {z : UpperPlane | alive a b.1 t z.1} :=
+    (measurableSet_alive_joint a ha).preimage
+      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
+  exact (MeasurableEmbedding.subtype_coe hs).comp
+    (restartHomeomorph a (lt_trans (by norm_num) ha) b t).measurableEmbedding
+
+ 
+
+def spatialDiskRegion (a : ℝ) (ha : 0<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) (q : ℝ) : Set UpperPlane :=
+  inverseUpper a ha b t '' pointDiskRegion w q
+
+lemma spatialDiskRegion_measurable (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) (q : ℝ) :
+    MeasurableSet (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q) :=
+  (inverseUpper_measurableEmbedding a ha b t).measurableSet_image.mpr (pointDiskRegion_measurable w q)
+
+lemma spatialDiskRegion_subset_alive (a : ℝ) (ha : 0<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) (q : ℝ) :
+    spatialDiskRegion a ha b t w q ⊆ {z : UpperPlane | alive a b.1 t z.1} := by
+  rintro _ ⟨v,_,rfl⟩
+  exact inverseMap_alive ha b t v.2
+
+lemma subtype_spatialDiskRegion (a : ℝ) (ha : 0<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) (q : ℝ) :
+    Subtype.val '' spatialDiskRegion a ha b t w q=inverseMap a b t '' centeredRegion w.1 q := by
+  ext z
+  constructor
+  · rintro ⟨u,⟨v,hv,rfl⟩,rfl⟩
+    exact ⟨v.1,⟨v.2,hv⟩,rfl⟩
+  · rintro ⟨v,hv,rfl⟩
+    exact ⟨inverseUpper a ha b t ⟨v,hv.1⟩,⟨⟨v,hv.1⟩,hv.2,rfl⟩,rfl⟩
+
+lemma restartPush_spatialDisk (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) (q : ℝ) (ν : Measure UpperPlane) :
+    restartPush a (lt_trans (by norm_num) ha) b t ν
+      (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q)=
+      ∫⁻v in pointDiskRegion w q,restartWeight a b t v∂ν := by
+  rw [restartPush,(inverseUpper_measurableEmbedding a ha b t).map_apply,
+    spatialDiskRegion,preimage_image_eq _ (inverseUpper_injective _ _ _ _),
+    withDensity_apply _ (pointDiskRegion_measurable w q)]
+
+ 
+
+def diskLower (q : ℝ) : ℝ := (2*subdiskDistortion q)⁻¹
+def diskUpper (q : ℝ) : ℝ := subdiskDistortion q*(subdiskReach q+2)^2/2
+
+lemma diskLower_pos (q : ℝ) : 0<diskLower q := by
+  have hd := subdiskDistortion_pos q
+  unfold diskLower
+  positivity
+lemma diskUpper_pos {q : ℝ} (hq : 0≤q) (hq1 : q<1) : 0<diskUpper q := by
+  have hd := subdiskDistortion_pos q
+  have hr := subdiskReach_nonneg hq hq1
+  unfold diskUpper
+  positivity
+
+lemma restartWeight_disk_bounds (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) {q : ℝ} (hq : 0≤q) (hq1 : q<1) (v : UpperPlane)
+    (hv : v∈pointDiskRegion w q) :
+    ENNReal.ofReal ((diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
+        pointDiskWeight a w v ≤ restartWeight a b t v ∧
+    restartWeight a b t v ≤
+      ENNReal.ofReal ((diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
+        pointDiskWeight a w v := by
+  have ha0 : 0<a := by linarith
+  have hd : 0<‖deriv (diskAt w.1) v.1‖ := by
+    rw [diskAt_deriv_norm w.2 v.2]
+    exact div_pos (mul_pos (by norm_num) w.2)
+      (sq_pos_of_pos (norm_pos_iff.mpr (diskAt_denominator_ne w.2 v.2)))
+  have hb := centered_disk_deriv_ratio ha0 b t w.2 hq hq1 ⟨v.2,hv⟩
+  have hlow : diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖)*‖deriv (diskAt w.1) v.1‖≤
+      ‖deriv (inverseMap a b t) v.1‖ := by
+    have hh := (le_div_iff₀ hd).mp hb.1
+    simpa only [diskLower,div_eq_mul_inv,mul_comm,mul_left_comm,mul_assoc] using hh
+  have hupp : ‖deriv (inverseMap a b t) v.1‖≤
+      (diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))*‖deriv (diskAt w.1) v.1‖ :=
+    (div_le_iff₀ hd).mp hb.2
+  have hl0 : 0≤diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖) :=
+    mul_nonneg (diskLower_pos q).le (mul_nonneg w.2.le (norm_nonneg _))
+  have hu0 : 0≤diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖) :=
+    mul_nonneg (diskUpper_pos hq hq1).le (mul_nonneg w.2.le (norm_nonneg _))
+  constructor
+  · have hh := ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (mul_nonneg hl0 hd.le)
+      hlow (dimension_pos a ha).le)
+    rw [Real.mul_rpow hl0 hd.le,ENNReal.ofReal_mul (Real.rpow_nonneg hl0 _)] at hh
+    exact hh
+  · have hh := ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (norm_nonneg _) hupp (dimension_pos a ha).le)
+    rw [Real.mul_rpow hu0 hd.le,ENNReal.ofReal_mul (Real.rpow_nonneg hu0 _)] at hh
+    exact hh
+
+ 
+
+theorem restartPush_disk_bounds (a : ℝ) (ha : 1/4<a) (b : GDriver) (t : ℝ≥0)
+    (w : UpperPlane) {q : ℝ} (hq : 0≤q) (hq1 : q<1) (ν : Measure UpperPlane) :
+    ENNReal.ofReal ((diskLower q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
+        pointDiskMass a w q ν ≤
+      restartPush a (lt_trans (by norm_num) ha) b t ν
+        (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q) ∧
+    restartPush a (lt_trans (by norm_num) ha) b t ν
+        (spatialDiskRegion a (lt_trans (by norm_num) ha) b t w q) ≤
+      ENNReal.ofReal ((diskUpper q*(w.1.im*‖deriv (inverseMap a b t) w.1‖))^(2-alpha a)) *
+        pointDiskMass a w q ν := by
+  rw [restartPush_spatialDisk a ha,pointDiskMass,←lintegral_const_mul _ (pointDiskWeight_measurable a w),
+    ←lintegral_const_mul _ (pointDiskWeight_measurable a w)]
+  constructor
+  · exact setLIntegral_mono' (pointDiskRegion_measurable w q)
+      (fun v hv=>(restartWeight_disk_bounds a ha b t w hq hq1 v hv).1)
+  · exact setLIntegral_mono' (pointDiskRegion_measurable w q)
+      (fun v hv=>(restartWeight_disk_bounds a ha b t w hq hq1 v hv).2)
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
+open LoewnerGeometry MassVersions TraceMassContinuity
+
+ 
+
+def radialDiskRegion (a : ℝ) (ha : 0<a) (z : ℂ) (σ q : ℝ) (b : GDriver) : Set UpperPlane :=
+  spatialDiskRegion a ha b (radialTime a b.1 z σ).toNNReal (radialPoint a z σ b) q
+
+lemma radialPoint_spatial_radius (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (b : GDriver)
+    (hb : radialTime a b.1 z σ≠∞) :
+    (radialPoint a z σ b).1.im *
+      ‖deriv (inverseMap a b (radialTime a b.1 z σ).toNNReal) (radialPoint a z σ b).1‖=σ := by
+  have hs := radialTime_attained a ha z hz σ hσ hσz b hb
+  rw [radialPoint_eq a ha z hz σ hσ hσz b hb]
+  change (loewnerMap a b.1 _ z-(b.1 _:ℂ)).im*‖deriv (inverseMap a b _)
+    (loewnerMap a b.1 _ z-(b.1 _:ℂ))‖=σ
+  rw [inverseMap_spatial_radius ha b _ hs.1,hs.2]
+
+lemma radialDiskRegion_localization (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) {q : ℝ} (hq : 1/2≤q) (hq1 : q<1)
+    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
+    (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z (σ/200) ⊆ radialDiskRegion a ha z σ q b ∧
+    radialDiskRegion a ha z σ q b ⊆
+      (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z (subdiskReach q*subdiskDistortion q*σ) := by
+  have hs := radialTime_attained a ha z hz σ hσ hσz b hb
+  have hh := centered_localization_at_alive ha b _ hs.1 hq hq1
+  rw [hs.2] at hh
+  have he : (Subtype.val : UpperPlane→ℂ) '' radialDiskRegion a ha z σ q b=
+      inverseMap a b (radialTime a b.1 z σ).toNNReal ''
+        centeredRegion (loewnerMap a b.1 (radialTime a b.1 z σ).toNNReal z-
+          (b.1 (radialTime a b.1 z σ).toNNReal:ℂ)) q := by
+    rw [radialDiskRegion,subtype_spatialDiskRegion,radialPoint_eq a ha z hz σ hσ hσz b hb]
+    rfl
+  constructor
+  · intro u hu
+    obtain ⟨v,hv,heq⟩ := (he ▸ hh.1 hu)
+    exact (Subtype.ext heq : v=u) ▸ hv
+  · intro u hu
+    exact hh.2 (he ▸ mem_image_of_mem Subtype.val hu)
+
+ 
+
+theorem actual_radial_spatial_bounds (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (D : CanonicalMassData κ a ha W (recoveredMass a ha s c hc))
+    (z : ℂ) (hz : 0<z.im) (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im)
+    (q : ℝ) (hq : 0≤q) (hq1 : q<1) :
+    let μ := recoveredMass a ha s c hc
+    let S := fun b : GDriver=>radialTime a b.1 z σ
+    let R := radialDiskRegion a (lt_trans (by norm_num) ha) z σ q
+    ∀ᵐ b∂W,S b≠∞→μ (S b) b (R b)=0 ∧ ∀l : ℕ,
+      ENNReal.ofReal ((diskLower q*σ)^(2-alpha a))*normalizedTest a z σ (subdiskMass a μ q l) b ≤
+        μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b (R b) ∧
+      μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b (R b) ≤
+        ENNReal.ofReal ((diskUpper q*σ)^(2-alpha a))*normalizedTest a z σ (subdiskMass a μ q l) b := by
+  dsimp only
+  let μ := recoveredMass a ha s c hc
+  let S := fun b : GDriver=>radialTime a b.1 z σ
+  let ha0 : 0<a := lt_trans (by norm_num) ha
+  have hS := radialTime_stopping a ha0 z hz σ hσ hσz
+  filter_upwards [D.restart S hS] with b hb
+  intro hbf
+  have hrest := hb hbf
+  let R := radialDiskRegion a ha0 z σ q b
+  have hR : MeasurableSet R := spatialDiskRegion_measurable a ha b _ _ q
+  have hRA : R⊆{u : UpperPlane | alive a b.1 (S b).toNNReal u.1} :=
+    spatialDiskRegion_subset_alive a ha0 b _ _ q
+  have hzero : μ (S b) b R=0 := by
+    have hh := congrArg (fun ν : Measure UpperPlane=>ν R) hrest.1
+    rwa [Measure.restrict_apply hR,inter_eq_left.mpr hRA,Measure.coe_zero,Pi.zero_apply] at hh
+  refine ⟨hzero,?_⟩
+  intro l
+  let w := radialPoint a z σ b
+  let t := (‖w.1‖₊:ℝ≥0∞)^2*levelTime a (pointAngle w).1 (normalizeDriver w (restartAt S b)).1 l
+  have ht : S b+t=radialTime a b.1 z (σ*Real.exp (-2*a*l)) :=
+    normalized_levelTime_restart a ha0 z hz σ hσ hσz l b hbf
+  have he : μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b R=
+      restartPush a ha0 b (S b).toNNReal (μ t (restartAt S b)) R := by
+    have hh := congrArg (fun ν : Measure UpperPlane=>ν R) (hrest.2 t)
+    rwa [Measure.restrict_apply hR,inter_eq_left.mpr hRA,ht] at hh
+  have hbounds := restartPush_disk_bounds a ha b (S b).toNNReal w hq hq1 (μ t (restartAt S b))
+  rw [radialPoint_spatial_radius a ha0 z hz σ hσ hσz b hbf] at hbounds
+  have hnorm : normalizedTest a z σ (subdiskMass a μ q l) b=
+      pointDiskMass a w q (μ t (restartAt S b)) :=
+    normalized_subdiskMass_eq a ha s c hc w (restartAt S b) q l
+  change ENNReal.ofReal ((diskLower q*σ)^(2-alpha a))*normalizedTest a z σ (subdiskMass a μ q l) b≤
+      μ (radialTime a b.1 z (σ*Real.exp (-2*a*l))) b R ∧ _
+  rw [hnorm,he]
+  exact hbounds
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.PastMass
+open FlowIdentification ExactSLEGreen TraceMassContinuity TraceRestriction
+
+lemma denseRange_nnrat_nnreal : DenseRange (fun q : ℚ≥0 => (q : ℝ≥0)) := by
+  apply dense_of_exists_between
+  intro a b hab
+  obtain ⟨q,haq,hqb⟩ := exists_rat_btwn (show (a : ℝ)<(b : ℝ) from hab)
+  have hq : 0≤q := Rat.cast_nonneg.mp ((a.2.trans haq.le) : (0 : ℝ)≤q)
+  let r : ℚ≥0 := ⟨q,hq⟩
+  refine ⟨(r : ℝ≥0),⟨r,rfl⟩,?_,?_⟩
+  · change (a : ℝ) < (r : ℝ)
+    simpa only [r,NNRat.cast_mk] using haq
+  · change (r : ℝ) < (b : ℝ)
+    simpa only [r,NNRat.cast_mk] using hqb
+
+ 
+
+lemma accumulated_continuous (κ : ℝ) (hκ : 0<κ) (b : GDriver)
+    (μ : ℝ≥0∞→Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q≠∞) :
+    Continuous (fun t : ℝ≥0 => (μ t Q).toReal) := by
+  let := hμ.atomless
+  let f : UpperPlane→ℝ := Q.indicator (fun _=>1)
+  have hf : Integrable f (μ ∞) := (integrable_indicator_iff hQ).mpr (integrableOn_const hfin)
+  have hγ := continuous_internalTrace hκ _ hμ.continuous_trace
+  have hc := continuous_integral_visited (μ ∞) (fun z : UpperPlane=>z.1)
+    measurable_subtype_coe Subtype.val_injective _ hγ f hf
+  have he (t : ℝ≥0) : (∫z in visited (fun z : UpperPlane=>z.1)
+      (internalTrace κ (driverHomeomorph.symm b)) t,f z∂μ ∞) = (μ t Q).toReal := by
+    rw [←hμ.restriction]
+    simp only [f,integral_indicator hQ,integral_const,Measure.real,
+      Measure.restrict_apply_univ,smul_eq_mul,mul_one]
+  simpa only [he] using hc
+
+lemma accumulated_zero (κ : ℝ) (b : GDriver) (μ : ℝ≥0∞→Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ) : μ 0=0 := by
+  simpa only [ENNReal.coe_zero,not_lt_zero,ofPred_false,Measure.restrict_empty] using
+    hμ.strict_firstVisit_identity 0
+
+ 
+
+lemma exists_rational_threshold (κ : ℝ) (hκ : 0<κ) (b : GDriver)
+    (μ : ℝ≥0∞→Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q≠∞)
+    (H U : ℝ) (hH : 0≤H) (hHU : H<U) (hUM : U<(μ ∞ Q).toReal) :
+    ∃q : ℚ≥0,H<(μ ((q : ℝ≥0) : ℝ≥0∞) Q).toReal ∧
+      (μ ((q : ℝ≥0) : ℝ≥0∞) Q).toReal<U := by
+  let M : ℝ≥0→ℝ := fun t=>(μ t Q).toReal
+  have hM : Continuous M := accumulated_continuous κ hκ b μ hμ Q hQ hfin
+  have hzero : M 0=0 := by simp [M,accumulated_zero κ b μ hμ]
+  have hlim : Tendsto (fun n : ℕ=>M (n:ℝ≥0)) atTop (𝓝 (μ ∞ Q).toReal) :=
+    (ENNReal.tendsto_toReal hfin).comp (hμ.exhaustion Q hQ)
+  obtain ⟨n,hn⟩ := (hlim.eventually (eventually_gt_nhds hUM)).exists
+  have hmid : (H+U)/2∈Icc (M 0) (M (n:ℝ≥0)) := by rw [hzero]; constructor <;> linarith
+  obtain ⟨t,ht⟩ := intermediate_value_univ (0:ℝ≥0) (n:ℝ≥0) hM hmid
+  exact denseRange_nnrat_nnreal.exists_mem_open (isOpen_Ioo.preimage hM)
+    ⟨t,by change H<M t ∧ M t<U; rw [ht]; constructor <;> linarith⟩
+
+ 
+
+lemma mass_le_twice_strict_past (κ : ℝ) (hκ : 0<κ) (b : GDriver)
+    (μ : ℝ≥0∞→Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q≠∞)
+    (H : ℝ) (hH : 0≤H) (hbad : 2*ENNReal.ofReal H<μ ∞ Q) :
+    μ ∞ Q ≤ 2*μ ∞ (Q∩{z | ∃q : ℚ≥0,
+      ENNReal.ofReal H<μ ((q:ℝ≥0):ℝ≥0∞) Q ∧ ((q:ℝ≥0):ℝ≥0∞)<visitTime κ b z}) := by
+  have hbadReal : 2*H<(μ ∞ Q).toReal := by
+    apply (ENNReal.ofReal_lt_iff_lt_toReal (mul_nonneg (by norm_num) hH) hfin).mp
+    simpa only [ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2),ENNReal.ofReal_ofNat] using hbad
+  have hMpos : 0<(μ ∞ Q).toReal := by linarith
+  obtain ⟨q,hqH,hqM⟩ := exists_rational_threshold κ hκ b μ hμ Q hQ hfin
+    H ((μ ∞ Q).toReal/2) hH (by linarith) (by linarith)
+  have hqle : μ ((q:ℝ≥0):ℝ≥0∞) Q≤μ ∞ Q := hμ.monotone le_top Q
+  have hqfin : μ ((q:ℝ≥0):ℝ≥0∞) Q≠∞ := ne_top_of_le_ne_top hfin hqle
+  have hqH' : ENNReal.ofReal H<μ ((q:ℝ≥0):ℝ≥0∞) Q :=
+    (ENNReal.ofReal_lt_iff_lt_toReal hH hqfin).mpr hqH
+  have hhalf : μ ∞ Q≤2*(μ ∞ Q-μ ((q:ℝ≥0):ℝ≥0∞) Q) := by
+    apply (ENNReal.toReal_le_toReal hfin (ENNReal.mul_ne_top (by norm_num)
+      (ne_top_of_le_ne_top hfin tsub_le_self))).mp
+    rw [ENNReal.toReal_mul,ENNReal.toReal_sub_of_le hqle hfin]
+    norm_num only [ENNReal.toReal_ofNat]
+    linarith
+  calc
+    μ ∞ Q≤2*(μ ∞ Q-μ ((q:ℝ≥0):ℝ≥0∞) Q) := hhalf
+    _=2*μ ∞ (Q∩{z | ((q:ℝ≥0):ℝ≥0∞)<visitTime κ b z}) := by
+      rw [stopped_strictPast_mass κ b μ hμ Q hQ hfin]
+    _≤_ := by
+      apply mul_le_mul_right
+      apply measure_mono
+      intro z hz
+      exact ⟨hz.1,q,hqH',hz.2⟩
+
+ 
+
+def rationalTestEnumeration : ℕ→ℚ≥0 := Classical.choose (exists_surjective_nat ℚ≥0)
+
+lemma rationalTestEnumeration_surjective : Function.Surjective rationalTestEnumeration :=
+  Classical.choose_spec (exists_surjective_nat ℚ≥0)
+
+def rationalPastKernel (W : Measure GDriver) (a : ℝ)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane) (Q : Set UpperPlane) (H : ℝ≥0∞) :
+    UpperPlane→ℝ≥0∞ :=
+  countablePastKernel rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0)) W a μ Q H
+
+namespace TracePalmData
+variable {κ a : ℝ} {W : Measure GDriver} {μ : ℝ≥0∞→GDriver→Measure UpperPlane}
+    (D : TracePalmData κ a W μ)
+include D
+
+lemma rational_Palm (ha : 1/4<a) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ≥0∞) :
+    (∫⁻b,μ ∞ b (Q∩{z | ∃q : ℚ≥0,H<μ ((q:ℝ≥0):ℝ≥0∞) b Q ∧
+      ((q:ℝ≥0):ℝ≥0∞)<visitTime κ b z})∂W)=
+      ∫⁻z in Q, ENNReal.ofReal (initialGreen a z)*rationalPastKernel W a μ Q H z∂upperArea :=
+  D.countable_Palm ha hW rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0))
+    rationalTestEnumeration_surjective Q hQ H
+
+lemma rational_kernel_measurable (ha : 1/4<a) [SFinite W]
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (H : ℝ≥0∞) :
+    Measurable (rationalPastKernel W a μ Q H) :=
+  D.countable_kernel_measurable ha rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0)) Q hQ H
+
+ 
+
+lemma expected_bad_mass_le (hκ : 0<κ) (ha : 1/4<a) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ) (hH : 0≤H) :
+    (∫⁻b, {b | 2*ENNReal.ofReal H<μ ∞ b Q}.indicator (fun b=>μ ∞ b Q) b∂W) ≤
+      2*∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*
+        rationalPastKernel W a μ Q (ENNReal.ofReal H) z∂upperArea := by
+  have hm := D.countable_past_aemeasurable rationalTestEnumeration (fun q : ℚ≥0=>(q:ℝ≥0))
+    rationalTestEnumeration_surjective Q hQ (ENNReal.ofReal H)
+  rw [←D.rational_Palm ha hW Q hQ (ENNReal.ofReal H),←lintegral_const_mul'' _ hm]
+  apply lintegral_mono_ae
+  filter_upwards [D.path] with b hb
+  by_cases hbad : 2*ENNReal.ofReal H<μ ∞ b Q
+  · simp only [indicator,mem_ofPred_eq,ite_eq_left hbad]
+    exact mass_le_twice_strict_past κ hκ b (fun v=>μ v b) hb Q hQ.measurableSet
+      (D.compact_finite Q hQ ∞ b) H hH hbad
+  · simp only [indicator,mem_ofPred_eq,ite_eq_right hbad]
+    exact bot_le
+
+end TracePalmData
+
+end SLEExactGauge.PastMass
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/NormalizedMarkedMoment.lean b/lean/OAI/Probability/SLEGauge/NormalizedMarkedMoment.lean
index 6e676321ce090d9d9df97c9f8fd668f2dfb9c7d0..9d070ec033122d1814c75de9e46379db394ec02c 100644
--- a/lean/OAI/Probability/SLEGauge/NormalizedMarkedMoment.lean
+++ b/lean/OAI/Probability/SLEGauge/NormalizedMarkedMoment.lean
@@ -1,557 +1,557 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.MarkedMomentBound
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.KernelSimilarity
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen ExactSLEKernel
-
-lemma pi_restrict_mono {K L : Set ℂ} (h : K⊆L) (m : ℕ) :
-    Measure.pi (fun _ : Fin m=>volume.restrict K) ≤
-      Measure.pi (fun _ : Fin m=>volume.restrict L) := by
-  rw [←Measure.restrict_pi_pi,←Measure.restrict_pi_pi]
-  exact Measure.restrict_mono (Set.pi_mono (fun _ _=>h)) le_rfl
-
-lemma normalization_cancel (α C B : ℝ) (hC : 0≤C) (hB : 0≤B)
-    {y r : ℝ} (hy : 0<y) (hr : 0<r) (m : ℕ) :
-    ENNReal.ofReal (C*(2*(y*r))^α)*
-      ((ENNReal.ofReal (B*y^(α-2)))^m*
-        (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1)))=
-      ENNReal.ofReal (C*2^α)*(ENNReal.ofReal B)^m*ENNReal.ofReal (r^α) := by
-  have hy0 : ENNReal.ofReal y≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr hy)
-  have hypow : (ENNReal.ofReal y)^α*
-      (((ENNReal.ofReal y)^(α-2))^m*
-        (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1)))=1 := by
-    rw [←ENNReal.rpow_natCast,←ENNReal.rpow_mul,
-      ←ENNReal.rpow_add _ _ hy0 ENNReal.ofReal_ne_top,
-      ←ENNReal.rpow_add _ _ hy0 ENNReal.ofReal_ne_top,
-      show α+((α-2)*(m:ℝ)+(2*(m:ℝ)-α*((m:ℝ)+1)))=0 by ring,
-      ENNReal.rpow_zero]
-  rw [show 2*(y*r)=y*(2*r) by ring,
-    Real.mul_rpow hy.le (by positivity),Real.mul_rpow (by norm_num : (0:ℝ)≤2) hr.le,
-    ENNReal.ofReal_mul hC, ENNReal.ofReal_mul (Real.rpow_nonneg hy.le _),
-    ENNReal.ofReal_mul (Real.rpow_nonneg (by norm_num : (0:ℝ)≤2) _),
-    ENNReal.ofReal_mul hB,mul_pow,
-    ENNReal.ofReal_mul hC]
-  simp_rw [←ENNReal.ofReal_rpow_of_pos hy]
-  calc
-    _ = (ENNReal.ofReal (C)*ENNReal.ofReal (2^α)*(ENNReal.ofReal B)^m*
-        ENNReal.ofReal (r^α))*
-      ((ENNReal.ofReal y)^α*(((ENNReal.ofReal y)^(α-2))^m*
-        (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1)))) := by ring
-    _ = _ := by rw [hypow,mul_one]
-
-lemma affine_im_real (x : ℝ) {y : ℝ} (hy : 0<y) (z : ℂ) :
-    (affine (x:ℂ) y hy z).im=y*z.im := by
-  simp [Complex.mul_im]
-
- 
-
-theorem actual_normalized_hit_moment_bound (a : ℝ) (ha : 1/4<a)
-    (m : ℕ) (hm : 1 ≤ m) (K : Set ℂ) (hK : MeasurableSet K)
-    (hKb : Bornology.IsBounded K) (c : ℝ) (hc : 0<c) (hc2 : c≤2)
-    (hcK : ∀z∈K,c≤2*z.im) (B : ℝ) (hB : 0≤B) :
-    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀(x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ)),
-      (Subtype.val '' tsupport f ⊆ affine (x:ℂ) y hy '' K) →
-      (∀z : UpperPlane,‖f z‖≤B*y^(alpha a-2)) →
-      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀(t : ℝ≥0∞) (μ : ExactSLEGenerator.Driver→Measure UpperPlane),
-      (∀ᵐ b∂W,Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
-          atTop (𝓝 (∫z,f z∂μ b))) →
-      ∀r : ℝ,0<r → r≤1 →
-        (∫⁻b in {b | radialTime a b.1 (affine (x:ℂ) y hy Complex.I) (y*r)<∞},
-          ‖∫z,f z∂μ b‖ₑ^m∂W)≤D*ENNReal.ofReal (r^(alpha a)) := by
-  obtain ⟨C,hC,hCb⟩ := actual_hit_radon_moment_bound a ha m hm
-  have halpha : alpha a<1 := by
-    have ha0 : 0<a := by linarith
-    have hp : 0<1/(4*a) := by positivity
-    dsimp [alpha]; linarith
-  obtain ⟨D₀,hD₀,hD₀b⟩ := capped_distance_fixed_center (alpha a) c
-    (alpha_pos ha) halpha hc m K {Complex.I} hK hKb isCompact_singleton.isBounded
-  let I := ∫⁻z : Fin m→ℂ,kernel (alpha a) c (Fin.cons Complex.I z)
-    ∂Measure.pi (fun _=>volume.restrict K)
-  have hI : I<∞ := (hD₀b Complex.I (mem_singleton _)).trans_lt hD₀
-  refine ⟨ENNReal.ofReal (C*2^(alpha a))*(ENNReal.ofReal B)^m*I,
-    ENNReal.mul_lt_top (ENNReal.mul_lt_top ENNReal.ofReal_lt_top
-      (ENNReal.pow_lt_top ENNReal.ofReal_lt_top)) hI,?_⟩
-  intro W hprob hW x y hy f hsupport hf s coeff hcoeff hsum t μ hlim r hr hr1
-  let T := affine (x:ℂ) y hy
-  let Kf := Subtype.val '' tsupport f
-  let ν := Measure.pi (fun _ : Fin m=>volume.restrict Kf)
-  have hKf : IsCompact Kf := f.hasCompactSupport.image continuous_subtype_val
-  have hcap (z : UpperPlane) (hz : z∈tsupport f) : y*c≤2*z.1.im := by
-    obtain ⟨w,hw,hew⟩ := hsupport ⟨z,hz,rfl⟩
-    have hi := hcK w hw
-    rw [←hew,affine_im_real]
-    nlinarith
-  have hzc : y*c≤2*(T Complex.I).im := by
-    simp only [T,affine_im_real,Complex.I_im,mul_one]
-    nlinarith
-  obtain ⟨_,hbound⟩ := hCb W hW f (y*c) (mul_pos hy hc) hcap (T Complex.I) hzc
-  have hmker : Measurable (fun z : Fin m→ℂ=>kernel (alpha a) (y*c)
-      (Fin.cons (T Complex.I) z)) :=
-    (measurable_kernel (alpha_pos ha).le _).comp (by fun_prop)
-  have hKae : ∀ᵐ z∂ν,∀i,z i∈Kf := by
-    apply ae_all_iff.mpr
-    intro i
-    exact (Measure.tendsto_eval_ae_ae (μ:=fun _ : Fin m=>volume.restrict Kf) (i:=i)).eventually
-      (ae_restrict_mem hKf.measurableSet)
-  have hA :
-      (∫⁻z : Fin m→ℂ,(∏i,ENNReal.ofReal ‖upperExtension f (z i)‖)*
-        kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν)≤
-      (ENNReal.ofReal (B*y^(alpha a-2)))^m*
-        ((ENNReal.ofReal y)^(2*(m:ℝ)-alpha a*((m:ℝ)+1))*I) := by
-    calc
-      _ ≤ ∫⁻z : Fin m→ℂ,(ENNReal.ofReal (B*y^(alpha a-2)))^m*
-          kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν := by
-        apply lintegral_mono_ae
-        filter_upwards [hKae] with z hz
-        apply mul_le_mul' _ le_rfl
-        have hprod : (∏i : Fin m,ENNReal.ofReal ‖upperExtension f (z i)‖)≤
-            ∏i : Fin m,ENNReal.ofReal (B*y^(alpha a-2)) := by
-          apply Finset.prod_le_prod₀ (fun _ _ => bot_le)
-          intro i hi
-          obtain ⟨w,hw,hew⟩ := hz i
-          exact ENNReal.ofReal_le_ofReal (by rw [←hew,upperExtension_apply]; exact hf w)
-        simpa using hprod
-      _ = (ENNReal.ofReal (B*y^(alpha a-2)))^m*
-          ∫⁻z : Fin m→ℂ,kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν :=
-        lintegral_const_mul _ hmker
-      _ ≤ (ENNReal.ofReal (B*y^(alpha a-2)))^m*
-          ∫⁻z : Fin m→ℂ,kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)
-          ∂Measure.pi (fun _=>volume.restrict (T '' K)) := by
-        apply mul_le_mul' le_rfl
-        exact lintegral_mono' (pi_restrict_mono hsupport m) le_rfl
-      _ = _ := by rw [fixed_center_integral_affine (alpha a) c (alpha_pos ha).le (x:ℂ) Complex.I hy hK]
-  calc
-    _ ≤ ENNReal.ofReal (C*(2*(y*r))^(alpha a))*
-        ∫⁻z : Fin m→ℂ,(∏i,ENNReal.ofReal ‖upperExtension f (z i)‖)*
-          kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν :=
-      hbound s coeff hcoeff hsum t μ hlim (y*r) (mul_pos hy hr) (by
-        simp only [T,affine_im_real,Complex.I_im,mul_one]
-        exact mul_le_of_le_one_right hy.le hr1)
-    _ ≤ ENNReal.ofReal (C*(2*(y*r))^(alpha a))*
-        ((ENNReal.ofReal (B*y^(alpha a-2)))^m*
-          ((ENNReal.ofReal y)^(2*(m:ℝ)-alpha a*((m:ℝ)+1))*I)) :=
-      mul_le_mul' le_rfl hA
-    _ = (ENNReal.ofReal (C*(2*(y*r))^(alpha a))*
-        ((ENNReal.ofReal (B*y^(alpha a-2)))^m*
-          (ENNReal.ofReal y)^(2*(m:ℝ)-alpha a*((m:ℝ)+1))))*I := by ring
-    _ = _ := by rw [normalization_cancel (alpha a) C B hC.le hB hy hr m]; ring
-
-end SLEExactGauge.KernelSimilarity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
-
-lemma radialWeight_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (r : ℝ) (hr : 0<r) (hr1 : r≤1) :
-    Measurable (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z (z.im*r)) :=
-  (joint_radialWeight_measurable a ha r hr hr1).comp
-    (measurable_id.prodMk (measurable_const (a:= (⟨z,hz⟩ : UpperPlane))))
-
- 
-
-lemma radialLaw_lintegral_le_hit (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (r : ℝ) (hr : 0<r) (hr1 : r≤1)
-    (g : ExactSLEGenerator.Driver→ℝ≥0∞) (hg : Measurable g) :
-    (∫⁻b,g b∂radialLaw W a z (z.im*r))≤
-      ENNReal.ofReal ((z.im*r)^(-alpha a)/physicalGreen a (z,0))*
-        ∫⁻b in {b | radialTime a b.1 z (z.im*r)<∞},g b∂W := by
-  have hρ : 0<z.im*r := mul_pos hz hr
-  have hρz : z.im*r≤z.im := mul_le_of_le_one_right hz.le hr1
-  have hG : 0<physicalGreen a (z,0) := Real.exp_pos _
-  have hm := ((radialWeight_measurable a ha z hz r hr hr1).div_const
-    (physicalGreen a (z,0))).ennreal_ofReal
-  have hS := radialTime_stopping a (by linarith) z hz _ hρ hρz
-  have hmT : Measurable (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z (z.im*r)) := hS.measurable'
-  have hE : MeasurableSet {b : ExactSLEGenerator.Driver | radialTime a b.1 z (z.im*r)<∞} :=
-    measurableSet_lt hmT measurable_const
-  rw [radialLaw,lintegral_withDensity_eq_lintegral_mul W hm hg,
-    ←lintegral_indicator hE,←lintegral_const_mul _ (hg.indicator hE)]
-  apply lintegral_mono
-  intro b
-  change _*g b≤_
-  by_cases hb : radialTime a b.1 z (z.im*r)<∞
-  · simp only [Set.indicator,Set.mem_ofPred_eq,hb,ite_true]
-    apply mul_le_mul' _ le_rfl
-    apply ENNReal.ofReal_le_ofReal
-    apply div_le_div_of_nonneg_right _ hG.le
-    have h := (radialWeight_bound a ha z hz _ hρ hρz b).2
-    calc
-      _ ≤ Real.exp (-alpha a*Real.log (z.im*r)) := h
-      _ = _ := by rw [Real.rpow_def_of_pos hρ]; congr 1; ring
-  · simp only [Set.indicator,Set.mem_ofPred_eq,hb,ite_false,mul_zero]
-    have ht : radialTime a b.1 z (z.im*r)=∞ := top_le_iff.mp (not_lt.mp hb)
-    simp [radialWeight,ht]
-
-lemma center_affine (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) :
-    fromDisk θ 0=affine (Real.cos θ:ℂ) (Real.sin θ)
-      (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) Complex.I := by
-  rw [fromDisk_zero,mul_comm Complex.I,Complex.exp_ofReal_mul_I]
-  rfl
-
-lemma tilt_cancel (α p : ℝ) {y r : ℝ} (hy : 0<y) (hr : 0<r) :
-    ENNReal.ofReal ((y*r)^(-α)/(y^(-α)*y^p))*ENNReal.ofReal (r^α)=
-      ENNReal.ofReal (y^(-p)) := by
-  rw [←ENNReal.ofReal_mul (by positivity)]
-  congr 1
-  rw [Real.mul_rpow hy.le hr.le,Real.rpow_neg hy.le p,
-    show y^(-α)*r^(-α)/(y^(-α)*y^p)*r^α=(r^(-α)*r^α)/(y^p) by
-      have h : y^(-α)≠0 := (Real.rpow_pos_of_pos hy _).ne'
-      field_simp,
-    ←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero,one_div]
-
- 
-
-theorem actual_normalized_tilted_moment_bound (a : ℝ) (ha : 1/4<a)
-    (m : ℕ) (hm : 1 ≤ m) (K : Set ℂ) (hK : MeasurableSet K)
-    (hKb : Bornology.IsBounded K) (c : ℝ) (hc : 0<c) (hc2 : c≤2)
-    (hcK : ∀z∈K,c≤2*z.im) (B : ℝ) (hB : 0≤B) :
-    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀(θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (f : C_c(UpperPlane,ℝ)),
-      (Subtype.val '' tsupport f ⊆ affine (Real.cos θ:ℂ) (Real.sin θ)
-        (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) '' K) →
-      (∀z : UpperPlane,‖f z‖≤B*(Real.sin θ)^(alpha a-2)) →
-      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀(t : ℝ≥0∞) (μ : ExactSLEGenerator.Driver→Measure UpperPlane),
-      Measurable (fun b=> ∫z,f z∂μ b) →
-      (∀ᵐ b∂W,Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
-          atTop (𝓝 (∫z,f z∂μ b))) →
-      ∀l : ℕ,(∫⁻b,‖∫z,f z∂μ b‖ₑ^m∂radialLaw W a (fromDisk θ 0)
-          (Real.sin θ*Real.exp (-2*a*l)))≤D*ENNReal.ofReal ((Real.sin θ)^(-p a)) := by
-  obtain ⟨D,hD,hbound⟩ := actual_normalized_hit_moment_bound a ha m hm K hK hKb c hc hc2 hcK B hB
-  refine ⟨D,hD,?_⟩
-  intro W hprob hW θ hθ hθπ f hsupport hf s coeff hcoeff hsum t μ hμ hlim l
-  have hy := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  have hz : 0<(fromDisk θ 0).im := by rw [fromDisk_zero_im]; exact hy
-  let r : ℝ := Real.exp (-2*a*l)
-  have hr : 0<r := Real.exp_pos _
-  have hr1 : r≤1 := Real.exp_le_one_iff.mpr (by
-    have h : 0≤2*a*(l:ℝ) := mul_nonneg (by linarith) (Nat.cast_nonneg l)
-    nlinarith)
-  have hhit := hbound W hW (Real.cos θ) (Real.sin θ) hy f hsupport hf s coeff
-    hcoeff hsum t μ hlim r hr hr1
-  rw [←center_affine θ hθ hθπ] at hhit
-  have ht := radialLaw_lintegral_le_hit W a ha (fromDisk θ 0) hz r hr hr1
-    (fun b=>‖∫z,f z∂μ b‖ₑ^m) (hμ.enorm.pow_const m)
-  rw [fromDisk_zero_im,radial_initial_green a θ hθ hθπ] at ht
-  refine ht.trans ((mul_le_mul' le_rfl hhit).trans_eq ?_)
-  rw [←mul_left_comm D,tilt_cancel (alpha a) (p a) hy hr]
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGreen KernelSimilarity
-
- 
-def upperAffine (x y : ℝ) (hy : 0<y) : UpperPlane ≃ₜ UpperPlane where
-  toFun z := ⟨(x:ℂ)+(y:ℂ)*z.1,by simpa [Complex.mul_im] using mul_pos hy z.2⟩
-  invFun z := ⟨((y:ℂ)⁻¹)*(z.1-(x:ℂ)),by
-    simpa [←Complex.ofReal_inv,Complex.mul_im] using mul_pos (inv_pos.mpr hy) z.2⟩
-  left_inv z := by
-    apply Subtype.ext
-    simp only [add_sub_cancel_left,←mul_assoc,inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr hy.ne'),one_mul]
-  right_inv z := by
-    apply Subtype.ext
-    simp only [←mul_assoc,mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr hy.ne'),one_mul,add_sub_cancel]
-  continuous_toFun := (continuous_const.add (continuous_const.mul continuous_subtype_val)).subtype_mk _
-  continuous_invFun := (continuous_const.mul (continuous_subtype_val.sub continuous_const)).subtype_mk _
-
-@[simp] lemma upperAffine_apply (x y : ℝ) (hy : 0<y) (z : UpperPlane) :
-    (upperAffine x y hy z).1=affine (x:ℂ) y hy z.1 := rfl
-
- 
-def normalizedTest (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ)) : C_c(UpperPlane,ℝ) where
-  toFun z := y^(alpha a-2)*f ((upperAffine x y hy).symm z)
-  continuous_toFun := continuous_const.mul (f.continuous.comp (upperAffine x y hy).symm.continuous)
-  hasCompactSupport' := (f.hasCompactSupport.comp_homeomorph (upperAffine x y hy).symm).mul_left
-
-lemma normalizedTest_support (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ)) :
-    Subtype.val '' tsupport (normalizedTest a x y hy f) ⊆
-      affine (x:ℂ) y hy '' (Subtype.val '' tsupport f) := by
-  rintro z ⟨w,hw,rfl⟩
-  have hsup : tsupport (normalizedTest a x y hy f) ⊆
-      tsupport (f ∘ (upperAffine x y hy).symm) := tsupport_mul_subset_right
-  have hh := tsupport_comp_subset_preimage f (upperAffine x y hy).symm.continuous (hsup hw)
-  refine ⟨((upperAffine x y hy).symm w).1,⟨_,hh,rfl⟩,?_⟩
-  exact congrArg Subtype.val ((upperAffine x y hy).apply_symm_apply w)
-
-lemma normalizedTest_nonneg (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ))
-    (hf : ∀z,0≤f z) (z : UpperPlane) : 0≤normalizedTest a x y hy f z :=
-  mul_nonneg (Real.rpow_nonneg hy.le _) (hf _)
-
-lemma normalizedTest_bound (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ))
-    {B : ℝ} (hf : ∀z,‖f z‖≤B) (z : UpperPlane) :
-    ‖normalizedTest a x y hy f z‖≤B*y^(alpha a-2) := by
-  change ‖y^(alpha a-2)*f _‖≤_
-  rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg hy.le _),mul_comm]
-  exact mul_le_mul_of_nonneg_right (hf _) (Real.rpow_nonneg hy.le _)
-
-lemma disk_negative_endpoint (θ : ℝ) :
-    Complex.exp (-Complex.I*θ)=(Real.cos θ:ℂ)-(Real.sin θ:ℂ)*Complex.I := by
-  rw [show -Complex.I*(θ:ℂ)=((-θ:ℝ):ℂ)*Complex.I by push_cast; ring,
-    Complex.exp_ofReal_mul_I,Real.cos_neg,Real.sin_neg]
-  push_cast
-  ring
-
-lemma disk_denominator_ne (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
-    z.1-Complex.exp (-Complex.I*θ)≠0 := by
-  intro he
-  have hi := congrArg Complex.im he
-  rw [disk_negative_endpoint] at hi
-  have hs:=Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  simp only [Complex.sub_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
-    Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_sub,sub_neg_eq_add,Complex.zero_im] at hi
-  linarith [z.2]
-
-lemma diskMap_affine (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : ℂ) :
-    diskMap θ (affine (Real.cos θ:ℂ) (Real.sin θ)
-      (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) z)=(z-Complex.I)/(z+Complex.I) := by
-  have hy : (Real.sin θ:ℂ)≠0 := Complex.ofReal_ne_zero.mpr
-    (Real.sin_pos_of_pos_of_lt_pi hθ hθπ).ne'
-  rw [diskMap,affine_apply,mul_comm Complex.I,Complex.exp_ofReal_mul_I,disk_negative_endpoint]
-  rw [show (Real.cos θ:ℂ)+(Real.sin θ:ℂ)*z-((Real.cos θ:ℂ)+(Real.sin θ:ℂ)*Complex.I)=
-      (Real.sin θ:ℂ)*(z-Complex.I) by ring,
-    show (Real.cos θ:ℂ)+(Real.sin θ:ℂ)*z-((Real.cos θ:ℂ)-(Real.sin θ:ℂ)*Complex.I)=
-      (Real.sin θ:ℂ)*(z+Complex.I) by ring,mul_div_mul_left _ _ hy]
-
-lemma disk_deriv (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
-    deriv (diskMap θ) z.1=((2*Real.sin θ:ℝ):ℂ)*Complex.I/
-      (z.1-Complex.exp (-Complex.I*θ))^2 := by
-  have hd := (((hasDerivAt_id z.1).sub_const (Complex.exp (Complex.I*θ))).div
-    ((hasDerivAt_id z.1).sub_const (Complex.exp (-Complex.I*θ)))
-      (disk_denominator_ne θ hθ hθπ z)).deriv
-  change deriv (diskMap θ) z.1=_ at hd
-  rw [hd]
-  simp only [id_eq,one_mul,mul_one]
-  rw [show z.1-Complex.exp (-Complex.I*θ)-(z.1-Complex.exp (Complex.I*θ))=
-    Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ) by ring,
-    disk_endpoints_difference]
-
-lemma disk_deriv_bound (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
-    ‖deriv (diskMap θ) z.1‖≤2/(Real.sin θ) := by
-  have hy := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  have hn : Real.sin θ≤‖z.1-Complex.exp (-Complex.I*θ)‖ := by
-    apply le_trans _ (Complex.im_le_norm _)
-    rw [disk_negative_endpoint]
-    simp only [Complex.sub_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
-      Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_sub,sub_neg_eq_add]
-    linarith [z.2]
-  have hn0 : 0<‖z.1-Complex.exp (-Complex.I*θ)‖ := hy.trans_le hn
-  rw [disk_deriv θ hθ hθπ z,norm_div,norm_mul,norm_pow,Complex.norm_real,
-    Real.norm_eq_abs,abs_of_pos (by positivity : 0<2*Real.sin θ),Complex.norm_I,mul_one]
-  apply (div_le_div_iff₀ (sq_pos_of_pos hn0) hy).mpr
-  nlinarith
-
-lemma cayley_inner (z : UpperPlane) (hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤1/4) :
-    dist z.1 Complex.I≤2/3 := by
-  have hn0 : 0<‖z.1+Complex.I‖ := by
-    apply lt_of_lt_of_le _ (Complex.im_le_norm _)
-    simpa only [Complex.add_im,Complex.I_im] using add_pos z.2 (by norm_num : (0:ℝ)<1)
-  rw [norm_div] at hz
-  have h := (div_le_iff₀ hn0).mp hz
-  have hn : ‖z.1+Complex.I‖≤‖z.1-Complex.I‖+2 := by
-    calc
-      _ = ‖(z.1-Complex.I)+(Complex.I+Complex.I)‖ := by congr 1; ring
-      _ ≤ ‖z.1-Complex.I‖+‖Complex.I+Complex.I‖ := norm_add_le _ _
-      _ ≤ ‖z.1-Complex.I‖+2 := by
-        have := norm_add_le Complex.I Complex.I
-        simp only [Complex.norm_I] at this
-        linarith
-  rw [dist_eq_norm]
-  nlinarith
-
- 
-
-lemma disk_preimage_inner (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane)
-    (hz : diskMap θ z.1∈Metric.closedBall (0:ℂ) (1/4)) :
-    dist ((upperAffine (Real.cos θ) (Real.sin θ)
-      (Real.sin_pos_of_pos_of_lt_pi hθ hθπ)).symm z).1 Complex.I≤2/3 := by
-  apply cayley_inner
-  have he := diskMap_affine θ hθ hθπ
-    (((upperAffine (Real.cos θ) (Real.sin θ) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ)).symm z).1)
-  rw [←upperAffine_apply,Homeomorph.apply_symm_apply] at he
-  simpa only [Metric.mem_closedBall,dist_zero_right,he] using hz
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
-
-lemma height_of_ball {z : ℂ} {r : ℝ} (hz : dist z Complex.I≤r) : 1-r≤z.im := by
-  have h : 1-z.im≤dist z Complex.I := by
-    simpa only [Complex.sub_im,Complex.I_im,dist_eq_norm,norm_sub_rev] using
-      Complex.im_le_norm (Complex.I-z)
-  linarith
-
-lemma upper_inner_compact : IsCompact {z : UpperPlane | dist z.1 Complex.I≤2/3} := by
-  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
-  have he : (Subtype.val : UpperPlane→ℂ) '' {z | dist z.1 Complex.I≤2/3}=
-      Metric.closedBall Complex.I (2/3) := by
-    ext z
-    constructor
-    · rintro ⟨w,hw,rfl⟩; exact hw
-    · intro hz
-      have hi := height_of_ball hz
-      exact ⟨⟨z,by linarith⟩,hz,rfl⟩
-  rw [he]
-  exact isCompact_closedBall _ _
-
-lemma cutoff_exists : ∃f : C_c(UpperPlane,ℝ),
-    (∀z,0≤f z ∧ f z≤1) ∧
-    (∀z : UpperPlane, dist z.1 Complex.I≤2/3 → f z=1) ∧
-    (∀z∈tsupport f,dist z.1 Complex.I<3/4) := by
-  obtain ⟨f,hf,hfc,hfs,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen
-    upper_inner_compact (Metric.isOpen_ball.preimage (continuous_subtype_val (p:=fun z : ℂ=>0<z.im)))
-    (show {z : UpperPlane | dist z.1 Complex.I≤2/3} ⊆
-        (Subtype.val : UpperPlane→ℂ) ⁻¹' Metric.ball Complex.I (3/4) from fun z hz=>by
-      change dist z.1 Complex.I<3/4
-      exact lt_of_le_of_lt hz (by norm_num))
-  exact ⟨⟨f,hfc⟩,h01,fun z hz=>hf hz,fun z hz=>hfs hz⟩
-
-def diskCutoff : C_c(UpperPlane,ℝ) := Classical.choose cutoff_exists
-
-lemma diskCutoff_nonneg (z : UpperPlane) : 0≤diskCutoff z :=
-  (Classical.choose_spec cutoff_exists).1 z |>.1
-lemma diskCutoff_le_one (z : UpperPlane) : diskCutoff z≤1 :=
-  (Classical.choose_spec cutoff_exists).1 z |>.2
-lemma diskCutoff_one (z : UpperPlane) (hz : dist z.1 Complex.I≤2/3) : diskCutoff z=1 :=
-  (Classical.choose_spec cutoff_exists).2.1 z hz
-lemma diskCutoff_support (z : UpperPlane) (hz : z∈tsupport diskCutoff) :
-    dist z.1 Complex.I<3/4 := (Classical.choose_spec cutoff_exists).2.2 z hz
-
- 
-
-def diskWeight (a θ : ℝ) (z : UpperPlane) : ℝ := ‖deriv (diskMap θ) z.1‖^(2-alpha a)
-
-def diskMass (a θ : ℝ) (ν : Measure UpperPlane) : Measure ℂ :=
-  (ν.withDensity (fun z=>ENNReal.ofReal (diskWeight a θ z))).map (fun z=>diskMap θ z.1)
-
-lemma diskMap_measurable (θ : ℝ) : Measurable (fun z : UpperPlane=>diskMap θ z.1) := by
-  unfold diskMap
-  fun_prop
-
-lemma diskWeight_measurable (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) :
-    Measurable (diskWeight a θ) := by
-  have he : diskWeight a θ=fun z : UpperPlane=>
-      ‖((2*Real.sin θ:ℝ):ℂ)*Complex.I/(z.1-Complex.exp (-Complex.I*θ))^2‖^(2-alpha a) := by
-    funext z; rw [diskWeight,disk_deriv θ hθ hθπ z]
-  rw [he]
-  fun_prop
-
-lemma dimension_pos (a : ℝ) (ha : 1/4<a) : 0<2-alpha a := by
-  have ha0 : 0<a := by linarith
-  have hp : 0<1/(4*a) := by positivity
-  dsimp [alpha]
-  linarith
-
- 
-def diskTest (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) : C_c(UpperPlane,ℝ) :=
-  (2^(2-alpha a):ℝ) • normalizedTest a (Real.cos θ) (Real.sin θ)
-    (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) diskCutoff
-
-lemma diskTest_nonneg (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
-    0≤diskTest a θ hθ hθπ z :=
-  mul_nonneg (Real.rpow_nonneg (by norm_num) _)
-    (normalizedTest_nonneg a _ _ _ diskCutoff diskCutoff_nonneg z)
-
-lemma diskTest_bound (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
-    ‖diskTest a θ hθ hθπ z‖≤2^(2-alpha a)*(Real.sin θ)^(alpha a-2) := by
-  change ‖(2^(2-alpha a):ℝ)*normalizedTest a _ _ _ diskCutoff z‖≤_
-  rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (by norm_num) _)]
-  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
-  simpa only [one_mul] using normalizedTest_bound a _ _ _ diskCutoff (B:=1)
-    (fun z=>by rw [Real.norm_eq_abs,abs_of_nonneg (diskCutoff_nonneg z)]; exact diskCutoff_le_one z) z
-
-lemma diskTest_support (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) :
-    Subtype.val '' tsupport (diskTest a θ hθ hθπ) ⊆
-      affine (Real.cos θ:ℂ) (Real.sin θ) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) ''
-        (Subtype.val '' tsupport diskCutoff) := by
-  apply (image_mono (tsupport_smul_subset_right (fun _ : UpperPlane => (2^(2-alpha a):ℝ))
-    (normalizedTest a (Real.cos θ) (Real.sin θ) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) diskCutoff))).trans
-  exact normalizedTest_support a _ _ _ diskCutoff
-
-lemma diskWeight_le_diskTest (a θ : ℝ) (ha : 1/4<a) (hθ : 0<θ) (hθπ : θ<Real.pi)
-    (z : UpperPlane) (hz : diskMap θ z.1∈Metric.closedBall (0:ℂ) (1/4)) :
-    diskWeight a θ z≤diskTest a θ hθ hθπ z := by
-  have hy := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  have he := diskCutoff_one _ (disk_preimage_inner θ hθ hθπ z hz)
-  change ‖deriv (diskMap θ) z.1‖^(2-alpha a)≤
-    2^(2-alpha a)*((Real.sin θ)^(alpha a-2)*diskCutoff _)
-  rw [he,mul_one]
-  calc
-    _ ≤ (2/(Real.sin θ))^(2-alpha a) := Real.rpow_le_rpow (norm_nonneg _)
-      (disk_deriv_bound θ hθ hθπ z) (dimension_pos a ha).le
-    _ = _ := by
-      rw [div_eq_mul_inv,Real.mul_rpow (by norm_num : (0:ℝ)≤2) (inv_nonneg.mpr hy.le),
-        Real.inv_rpow hy.le,←Real.rpow_neg hy.le,neg_sub]
-
-lemma diskMass_inner_le_test (a θ : ℝ) (ha : 1/4<a) (hθ : 0<θ) (hθπ : θ<Real.pi)
-    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] :
-    diskMass a θ ν (Metric.closedBall (0:ℂ) (1/4))≤
-      ENNReal.ofReal (∫z,diskTest a θ hθ hθπ z∂ν) := by
-  rw [diskMass,Measure.map_apply (diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet,
-    withDensity_apply _ ((diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet),
-    ofReal_integral_eq_lintegral_ofReal (diskTest a θ hθ hθπ).integrable
-      (Eventually.of_forall (diskTest_nonneg a θ hθ hθπ)),
-    ←lintegral_indicator ((diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet)]
-  apply lintegral_mono
-  intro z
-  by_cases hz : diskMap θ z.1∈Metric.closedBall (0:ℂ) (1/4)
-  · simp only [Set.indicator, Set.mem_preimage, hz, ite_true]
-    exact ENNReal.ofReal_le_ofReal (diskWeight_le_diskTest a θ ha hθ hθπ z hz)
-  · simp only [Set.indicator, Set.mem_preimage, hz, ite_false]; positivity
-
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.MarkedMomentBound
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.KernelSimilarity
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen ExactSLEKernel
+
+lemma pi_restrict_mono {K L : Set ℂ} (h : K⊆L) (m : ℕ) :
+    Measure.pi (fun _ : Fin m=>volume.restrict K) ≤
+      Measure.pi (fun _ : Fin m=>volume.restrict L) := by
+  rw [←Measure.restrict_pi_pi,←Measure.restrict_pi_pi]
+  exact Measure.restrict_mono (Set.pi_mono (fun _ _=>h)) le_rfl
+
+lemma normalization_cancel (α C B : ℝ) (hC : 0≤C) (hB : 0≤B)
+    {y r : ℝ} (hy : 0<y) (hr : 0<r) (m : ℕ) :
+    ENNReal.ofReal (C*(2*(y*r))^α)*
+      ((ENNReal.ofReal (B*y^(α-2)))^m*
+        (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1)))=
+      ENNReal.ofReal (C*2^α)*(ENNReal.ofReal B)^m*ENNReal.ofReal (r^α) := by
+  have hy0 : ENNReal.ofReal y≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr hy)
+  have hypow : (ENNReal.ofReal y)^α*
+      (((ENNReal.ofReal y)^(α-2))^m*
+        (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1)))=1 := by
+    rw [←ENNReal.rpow_natCast,←ENNReal.rpow_mul,
+      ←ENNReal.rpow_add _ _ hy0 ENNReal.ofReal_ne_top,
+      ←ENNReal.rpow_add _ _ hy0 ENNReal.ofReal_ne_top,
+      show α+((α-2)*(m:ℝ)+(2*(m:ℝ)-α*((m:ℝ)+1)))=0 by ring,
+      ENNReal.rpow_zero]
+  rw [show 2*(y*r)=y*(2*r) by ring,
+    Real.mul_rpow hy.le (by positivity),Real.mul_rpow (by norm_num : (0:ℝ)≤2) hr.le,
+    ENNReal.ofReal_mul hC, ENNReal.ofReal_mul (Real.rpow_nonneg hy.le _),
+    ENNReal.ofReal_mul (Real.rpow_nonneg (by norm_num : (0:ℝ)≤2) _),
+    ENNReal.ofReal_mul hB,mul_pow,
+    ENNReal.ofReal_mul hC]
+  simp_rw [←ENNReal.ofReal_rpow_of_pos hy]
+  calc
+    _ = (ENNReal.ofReal (C)*ENNReal.ofReal (2^α)*(ENNReal.ofReal B)^m*
+        ENNReal.ofReal (r^α))*
+      ((ENNReal.ofReal y)^α*(((ENNReal.ofReal y)^(α-2))^m*
+        (ENNReal.ofReal y)^(2*(m:ℝ)-α*((m:ℝ)+1)))) := by ring
+    _ = _ := by rw [hypow,mul_one]
+
+lemma affine_im_real (x : ℝ) {y : ℝ} (hy : 0<y) (z : ℂ) :
+    (affine (x:ℂ) y hy z).im=y*z.im := by
+  simp [Complex.mul_im]
+
+ 
+
+theorem actual_normalized_hit_moment_bound (a : ℝ) (ha : 1/4<a)
+    (m : ℕ) (hm : 1 ≤ m) (K : Set ℂ) (hK : MeasurableSet K)
+    (hKb : Bornology.IsBounded K) (c : ℝ) (hc : 0<c) (hc2 : c≤2)
+    (hcK : ∀z∈K,c≤2*z.im) (B : ℝ) (hB : 0≤B) :
+    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀(x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ)),
+      (Subtype.val '' tsupport f ⊆ affine (x:ℂ) y hy '' K) →
+      (∀z : UpperPlane,‖f z‖≤B*y^(alpha a-2)) →
+      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀(t : ℝ≥0∞) (μ : ExactSLEGenerator.Driver→Measure UpperPlane),
+      (∀ᵐ b∂W,Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
+          atTop (𝓝 (∫z,f z∂μ b))) →
+      ∀r : ℝ,0<r → r≤1 →
+        (∫⁻b in {b | radialTime a b.1 (affine (x:ℂ) y hy Complex.I) (y*r)<∞},
+          ‖∫z,f z∂μ b‖ₑ^m∂W)≤D*ENNReal.ofReal (r^(alpha a)) := by
+  obtain ⟨C,hC,hCb⟩ := actual_hit_radon_moment_bound a ha m hm
+  have halpha : alpha a<1 := by
+    have ha0 : 0<a := by linarith
+    have hp : 0<1/(4*a) := by positivity
+    dsimp [alpha]; linarith
+  obtain ⟨D₀,hD₀,hD₀b⟩ := capped_distance_fixed_center (alpha a) c
+    (alpha_pos ha) halpha hc m K {Complex.I} hK hKb isCompact_singleton.isBounded
+  let I := ∫⁻z : Fin m→ℂ,kernel (alpha a) c (Fin.cons Complex.I z)
+    ∂Measure.pi (fun _=>volume.restrict K)
+  have hI : I<∞ := (hD₀b Complex.I (mem_singleton _)).trans_lt hD₀
+  refine ⟨ENNReal.ofReal (C*2^(alpha a))*(ENNReal.ofReal B)^m*I,
+    ENNReal.mul_lt_top (ENNReal.mul_lt_top ENNReal.ofReal_lt_top
+      (ENNReal.pow_lt_top ENNReal.ofReal_lt_top)) hI,?_⟩
+  intro W hprob hW x y hy f hsupport hf s coeff hcoeff hsum t μ hlim r hr hr1
+  let T := affine (x:ℂ) y hy
+  let Kf := Subtype.val '' tsupport f
+  let ν := Measure.pi (fun _ : Fin m=>volume.restrict Kf)
+  have hKf : IsCompact Kf := f.hasCompactSupport.image continuous_subtype_val
+  have hcap (z : UpperPlane) (hz : z∈tsupport f) : y*c≤2*z.1.im := by
+    obtain ⟨w,hw,hew⟩ := hsupport ⟨z,hz,rfl⟩
+    have hi := hcK w hw
+    rw [←hew,affine_im_real]
+    nlinarith
+  have hzc : y*c≤2*(T Complex.I).im := by
+    simp only [T,affine_im_real,Complex.I_im,mul_one]
+    nlinarith
+  obtain ⟨_,hbound⟩ := hCb W hW f (y*c) (mul_pos hy hc) hcap (T Complex.I) hzc
+  have hmker : Measurable (fun z : Fin m→ℂ=>kernel (alpha a) (y*c)
+      (Fin.cons (T Complex.I) z)) :=
+    (measurable_kernel (alpha_pos ha).le _).comp (by fun_prop)
+  have hKae : ∀ᵐ z∂ν,∀i,z i∈Kf := by
+    apply ae_all_iff.mpr
+    intro i
+    exact (Measure.tendsto_eval_ae_ae (μ:=fun _ : Fin m=>volume.restrict Kf) (i:=i)).eventually
+      (ae_restrict_mem hKf.measurableSet)
+  have hA :
+      (∫⁻z : Fin m→ℂ,(∏i,ENNReal.ofReal ‖upperExtension f (z i)‖)*
+        kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν)≤
+      (ENNReal.ofReal (B*y^(alpha a-2)))^m*
+        ((ENNReal.ofReal y)^(2*(m:ℝ)-alpha a*((m:ℝ)+1))*I) := by
+    calc
+      _ ≤ ∫⁻z : Fin m→ℂ,(ENNReal.ofReal (B*y^(alpha a-2)))^m*
+          kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν := by
+        apply lintegral_mono_ae
+        filter_upwards [hKae] with z hz
+        apply mul_le_mul' _ le_rfl
+        have hprod : (∏i : Fin m,ENNReal.ofReal ‖upperExtension f (z i)‖)≤
+            ∏i : Fin m,ENNReal.ofReal (B*y^(alpha a-2)) := by
+          apply Finset.prod_le_prod₀ (fun _ _ => bot_le)
+          intro i hi
+          obtain ⟨w,hw,hew⟩ := hz i
+          exact ENNReal.ofReal_le_ofReal (by rw [←hew,upperExtension_apply]; exact hf w)
+        simpa using hprod
+      _ = (ENNReal.ofReal (B*y^(alpha a-2)))^m*
+          ∫⁻z : Fin m→ℂ,kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν :=
+        lintegral_const_mul _ hmker
+      _ ≤ (ENNReal.ofReal (B*y^(alpha a-2)))^m*
+          ∫⁻z : Fin m→ℂ,kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)
+          ∂Measure.pi (fun _=>volume.restrict (T '' K)) := by
+        apply mul_le_mul' le_rfl
+        exact lintegral_mono' (pi_restrict_mono hsupport m) le_rfl
+      _ = _ := by rw [fixed_center_integral_affine (alpha a) c (alpha_pos ha).le (x:ℂ) Complex.I hy hK]
+  calc
+    _ ≤ ENNReal.ofReal (C*(2*(y*r))^(alpha a))*
+        ∫⁻z : Fin m→ℂ,(∏i,ENNReal.ofReal ‖upperExtension f (z i)‖)*
+          kernel (alpha a) (y*c) (Fin.cons (T Complex.I) z)∂ν :=
+      hbound s coeff hcoeff hsum t μ hlim (y*r) (mul_pos hy hr) (by
+        simp only [T,affine_im_real,Complex.I_im,mul_one]
+        exact mul_le_of_le_one_right hy.le hr1)
+    _ ≤ ENNReal.ofReal (C*(2*(y*r))^(alpha a))*
+        ((ENNReal.ofReal (B*y^(alpha a-2)))^m*
+          ((ENNReal.ofReal y)^(2*(m:ℝ)-alpha a*((m:ℝ)+1))*I)) :=
+      mul_le_mul' le_rfl hA
+    _ = (ENNReal.ofReal (C*(2*(y*r))^(alpha a))*
+        ((ENNReal.ofReal (B*y^(alpha a-2)))^m*
+          (ENNReal.ofReal y)^(2*(m:ℝ)-alpha a*((m:ℝ)+1))))*I := by ring
+    _ = _ := by rw [normalization_cancel (alpha a) C B hC.le hB hy hr m]; ring
+
+end SLEExactGauge.KernelSimilarity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
+
+lemma radialWeight_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (r : ℝ) (hr : 0<r) (hr1 : r≤1) :
+    Measurable (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z (z.im*r)) :=
+  (joint_radialWeight_measurable a ha r hr hr1).comp
+    (measurable_id.prodMk (measurable_const (a:= (⟨z,hz⟩ : UpperPlane))))
+
+ 
+
+lemma radialLaw_lintegral_le_hit (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (r : ℝ) (hr : 0<r) (hr1 : r≤1)
+    (g : ExactSLEGenerator.Driver→ℝ≥0∞) (hg : Measurable g) :
+    (∫⁻b,g b∂radialLaw W a z (z.im*r))≤
+      ENNReal.ofReal ((z.im*r)^(-alpha a)/physicalGreen a (z,0))*
+        ∫⁻b in {b | radialTime a b.1 z (z.im*r)<∞},g b∂W := by
+  have hρ : 0<z.im*r := mul_pos hz hr
+  have hρz : z.im*r≤z.im := mul_le_of_le_one_right hz.le hr1
+  have hG : 0<physicalGreen a (z,0) := Real.exp_pos _
+  have hm := ((radialWeight_measurable a ha z hz r hr hr1).div_const
+    (physicalGreen a (z,0))).ennreal_ofReal
+  have hS := radialTime_stopping a (by linarith) z hz _ hρ hρz
+  have hmT : Measurable (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z (z.im*r)) := hS.measurable'
+  have hE : MeasurableSet {b : ExactSLEGenerator.Driver | radialTime a b.1 z (z.im*r)<∞} :=
+    measurableSet_lt hmT measurable_const
+  rw [radialLaw,lintegral_withDensity_eq_lintegral_mul W hm hg,
+    ←lintegral_indicator hE,←lintegral_const_mul _ (hg.indicator hE)]
+  apply lintegral_mono
+  intro b
+  change _*g b≤_
+  by_cases hb : radialTime a b.1 z (z.im*r)<∞
+  · simp only [Set.indicator,Set.mem_ofPred_eq,hb,ite_true]
+    apply mul_le_mul' _ le_rfl
+    apply ENNReal.ofReal_le_ofReal
+    apply div_le_div_of_nonneg_right _ hG.le
+    have h := (radialWeight_bound a ha z hz _ hρ hρz b).2
+    calc
+      _ ≤ Real.exp (-alpha a*Real.log (z.im*r)) := h
+      _ = _ := by rw [Real.rpow_def_of_pos hρ]; congr 1; ring
+  · simp only [Set.indicator,Set.mem_ofPred_eq,hb,ite_false,mul_zero]
+    have ht : radialTime a b.1 z (z.im*r)=∞ := top_le_iff.mp (not_lt.mp hb)
+    simp [radialWeight,ht]
+
+lemma center_affine (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) :
+    fromDisk θ 0=affine (Real.cos θ:ℂ) (Real.sin θ)
+      (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) Complex.I := by
+  rw [fromDisk_zero,mul_comm Complex.I,Complex.exp_ofReal_mul_I]
+  rfl
+
+lemma tilt_cancel (α p : ℝ) {y r : ℝ} (hy : 0<y) (hr : 0<r) :
+    ENNReal.ofReal ((y*r)^(-α)/(y^(-α)*y^p))*ENNReal.ofReal (r^α)=
+      ENNReal.ofReal (y^(-p)) := by
+  rw [←ENNReal.ofReal_mul (by positivity)]
+  congr 1
+  rw [Real.mul_rpow hy.le hr.le,Real.rpow_neg hy.le p,
+    show y^(-α)*r^(-α)/(y^(-α)*y^p)*r^α=(r^(-α)*r^α)/(y^p) by
+      have h : y^(-α)≠0 := (Real.rpow_pos_of_pos hy _).ne'
+      field_simp,
+    ←Real.rpow_add hr,neg_add_cancel,Real.rpow_zero,one_div]
+
+ 
+
+theorem actual_normalized_tilted_moment_bound (a : ℝ) (ha : 1/4<a)
+    (m : ℕ) (hm : 1 ≤ m) (K : Set ℂ) (hK : MeasurableSet K)
+    (hKb : Bornology.IsBounded K) (c : ℝ) (hc : 0<c) (hc2 : c≤2)
+    (hcK : ∀z∈K,c≤2*z.im) (B : ℝ) (hB : 0≤B) :
+    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀(θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (f : C_c(UpperPlane,ℝ)),
+      (Subtype.val '' tsupport f ⊆ affine (Real.cos θ:ℂ) (Real.sin θ)
+        (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) '' K) →
+      (∀z : UpperPlane,‖f z‖≤B*(Real.sin θ)^(alpha a-2)) →
+      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀(t : ℝ≥0∞) (μ : ExactSLEGenerator.Driver→Measure UpperPlane),
+      Measurable (fun b=> ∫z,f z∂μ b) →
+      (∀ᵐ b∂W,Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
+          atTop (𝓝 (∫z,f z∂μ b))) →
+      ∀l : ℕ,(∫⁻b,‖∫z,f z∂μ b‖ₑ^m∂radialLaw W a (fromDisk θ 0)
+          (Real.sin θ*Real.exp (-2*a*l)))≤D*ENNReal.ofReal ((Real.sin θ)^(-p a)) := by
+  obtain ⟨D,hD,hbound⟩ := actual_normalized_hit_moment_bound a ha m hm K hK hKb c hc hc2 hcK B hB
+  refine ⟨D,hD,?_⟩
+  intro W hprob hW θ hθ hθπ f hsupport hf s coeff hcoeff hsum t μ hμ hlim l
+  have hy := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  have hz : 0<(fromDisk θ 0).im := by rw [fromDisk_zero_im]; exact hy
+  let r : ℝ := Real.exp (-2*a*l)
+  have hr : 0<r := Real.exp_pos _
+  have hr1 : r≤1 := Real.exp_le_one_iff.mpr (by
+    have h : 0≤2*a*(l:ℝ) := mul_nonneg (by linarith) (Nat.cast_nonneg l)
+    nlinarith)
+  have hhit := hbound W hW (Real.cos θ) (Real.sin θ) hy f hsupport hf s coeff
+    hcoeff hsum t μ hlim r hr hr1
+  rw [←center_affine θ hθ hθπ] at hhit
+  have ht := radialLaw_lintegral_le_hit W a ha (fromDisk θ 0) hz r hr hr1
+    (fun b=>‖∫z,f z∂μ b‖ₑ^m) (hμ.enorm.pow_const m)
+  rw [fromDisk_zero_im,radial_initial_green a θ hθ hθπ] at ht
+  refine ht.trans ((mul_le_mul' le_rfl hhit).trans_eq ?_)
+  rw [←mul_left_comm D,tilt_cancel (alpha a) (p a) hy hr]
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGreen KernelSimilarity
+
+ 
+def upperAffine (x y : ℝ) (hy : 0<y) : UpperPlane ≃ₜ UpperPlane where
+  toFun z := ⟨(x:ℂ)+(y:ℂ)*z.1,by simpa [Complex.mul_im] using mul_pos hy z.2⟩
+  invFun z := ⟨((y:ℂ)⁻¹)*(z.1-(x:ℂ)),by
+    simpa [←Complex.ofReal_inv,Complex.mul_im] using mul_pos (inv_pos.mpr hy) z.2⟩
+  left_inv z := by
+    apply Subtype.ext
+    simp only [add_sub_cancel_left,←mul_assoc,inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr hy.ne'),one_mul]
+  right_inv z := by
+    apply Subtype.ext
+    simp only [←mul_assoc,mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr hy.ne'),one_mul,add_sub_cancel]
+  continuous_toFun := (continuous_const.add (continuous_const.mul continuous_subtype_val)).subtype_mk _
+  continuous_invFun := (continuous_const.mul (continuous_subtype_val.sub continuous_const)).subtype_mk _
+
+@[simp] lemma upperAffine_apply (x y : ℝ) (hy : 0<y) (z : UpperPlane) :
+    (upperAffine x y hy z).1=affine (x:ℂ) y hy z.1 := rfl
+
+ 
+def normalizedTest (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ)) : C_c(UpperPlane,ℝ) where
+  toFun z := y^(alpha a-2)*f ((upperAffine x y hy).symm z)
+  continuous_toFun := continuous_const.mul (f.continuous.comp (upperAffine x y hy).symm.continuous)
+  hasCompactSupport' := (f.hasCompactSupport.comp_homeomorph (upperAffine x y hy).symm).mul_left
+
+lemma normalizedTest_support (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ)) :
+    Subtype.val '' tsupport (normalizedTest a x y hy f) ⊆
+      affine (x:ℂ) y hy '' (Subtype.val '' tsupport f) := by
+  rintro z ⟨w,hw,rfl⟩
+  have hsup : tsupport (normalizedTest a x y hy f) ⊆
+      tsupport (f ∘ (upperAffine x y hy).symm) := tsupport_mul_subset_right
+  have hh := tsupport_comp_subset_preimage f (upperAffine x y hy).symm.continuous (hsup hw)
+  refine ⟨((upperAffine x y hy).symm w).1,⟨_,hh,rfl⟩,?_⟩
+  exact congrArg Subtype.val ((upperAffine x y hy).apply_symm_apply w)
+
+lemma normalizedTest_nonneg (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ))
+    (hf : ∀z,0≤f z) (z : UpperPlane) : 0≤normalizedTest a x y hy f z :=
+  mul_nonneg (Real.rpow_nonneg hy.le _) (hf _)
+
+lemma normalizedTest_bound (a x y : ℝ) (hy : 0<y) (f : C_c(UpperPlane,ℝ))
+    {B : ℝ} (hf : ∀z,‖f z‖≤B) (z : UpperPlane) :
+    ‖normalizedTest a x y hy f z‖≤B*y^(alpha a-2) := by
+  change ‖y^(alpha a-2)*f _‖≤_
+  rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg hy.le _),mul_comm]
+  exact mul_le_mul_of_nonneg_right (hf _) (Real.rpow_nonneg hy.le _)
+
+lemma disk_negative_endpoint (θ : ℝ) :
+    Complex.exp (-Complex.I*θ)=(Real.cos θ:ℂ)-(Real.sin θ:ℂ)*Complex.I := by
+  rw [show -Complex.I*(θ:ℂ)=((-θ:ℝ):ℂ)*Complex.I by push_cast; ring,
+    Complex.exp_ofReal_mul_I,Real.cos_neg,Real.sin_neg]
+  push_cast
+  ring
+
+lemma disk_denominator_ne (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
+    z.1-Complex.exp (-Complex.I*θ)≠0 := by
+  intro he
+  have hi := congrArg Complex.im he
+  rw [disk_negative_endpoint] at hi
+  have hs:=Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  simp only [Complex.sub_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
+    Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_sub,sub_neg_eq_add,Complex.zero_im] at hi
+  linarith [z.2]
+
+lemma diskMap_affine (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : ℂ) :
+    diskMap θ (affine (Real.cos θ:ℂ) (Real.sin θ)
+      (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) z)=(z-Complex.I)/(z+Complex.I) := by
+  have hy : (Real.sin θ:ℂ)≠0 := Complex.ofReal_ne_zero.mpr
+    (Real.sin_pos_of_pos_of_lt_pi hθ hθπ).ne'
+  rw [diskMap,affine_apply,mul_comm Complex.I,Complex.exp_ofReal_mul_I,disk_negative_endpoint]
+  rw [show (Real.cos θ:ℂ)+(Real.sin θ:ℂ)*z-((Real.cos θ:ℂ)+(Real.sin θ:ℂ)*Complex.I)=
+      (Real.sin θ:ℂ)*(z-Complex.I) by ring,
+    show (Real.cos θ:ℂ)+(Real.sin θ:ℂ)*z-((Real.cos θ:ℂ)-(Real.sin θ:ℂ)*Complex.I)=
+      (Real.sin θ:ℂ)*(z+Complex.I) by ring,mul_div_mul_left _ _ hy]
+
+lemma disk_deriv (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
+    deriv (diskMap θ) z.1=((2*Real.sin θ:ℝ):ℂ)*Complex.I/
+      (z.1-Complex.exp (-Complex.I*θ))^2 := by
+  have hd := (((hasDerivAt_id z.1).sub_const (Complex.exp (Complex.I*θ))).div
+    ((hasDerivAt_id z.1).sub_const (Complex.exp (-Complex.I*θ)))
+      (disk_denominator_ne θ hθ hθπ z)).deriv
+  change deriv (diskMap θ) z.1=_ at hd
+  rw [hd]
+  simp only [id_eq,one_mul,mul_one]
+  rw [show z.1-Complex.exp (-Complex.I*θ)-(z.1-Complex.exp (Complex.I*θ))=
+    Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ) by ring,
+    disk_endpoints_difference]
+
+lemma disk_deriv_bound (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
+    ‖deriv (diskMap θ) z.1‖≤2/(Real.sin θ) := by
+  have hy := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  have hn : Real.sin θ≤‖z.1-Complex.exp (-Complex.I*θ)‖ := by
+    apply le_trans _ (Complex.im_le_norm _)
+    rw [disk_negative_endpoint]
+    simp only [Complex.sub_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re,
+      Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_sub,sub_neg_eq_add]
+    linarith [z.2]
+  have hn0 : 0<‖z.1-Complex.exp (-Complex.I*θ)‖ := hy.trans_le hn
+  rw [disk_deriv θ hθ hθπ z,norm_div,norm_mul,norm_pow,Complex.norm_real,
+    Real.norm_eq_abs,abs_of_pos (by positivity : 0<2*Real.sin θ),Complex.norm_I,mul_one]
+  apply (div_le_div_iff₀ (sq_pos_of_pos hn0) hy).mpr
+  nlinarith
+
+lemma cayley_inner (z : UpperPlane) (hz : ‖(z.1-Complex.I)/(z.1+Complex.I)‖≤1/4) :
+    dist z.1 Complex.I≤2/3 := by
+  have hn0 : 0<‖z.1+Complex.I‖ := by
+    apply lt_of_lt_of_le _ (Complex.im_le_norm _)
+    simpa only [Complex.add_im,Complex.I_im] using add_pos z.2 (by norm_num : (0:ℝ)<1)
+  rw [norm_div] at hz
+  have h := (div_le_iff₀ hn0).mp hz
+  have hn : ‖z.1+Complex.I‖≤‖z.1-Complex.I‖+2 := by
+    calc
+      _ = ‖(z.1-Complex.I)+(Complex.I+Complex.I)‖ := by congr 1; ring
+      _ ≤ ‖z.1-Complex.I‖+‖Complex.I+Complex.I‖ := norm_add_le _ _
+      _ ≤ ‖z.1-Complex.I‖+2 := by
+        have := norm_add_le Complex.I Complex.I
+        simp only [Complex.norm_I] at this
+        linarith
+  rw [dist_eq_norm]
+  nlinarith
+
+ 
+
+lemma disk_preimage_inner (θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane)
+    (hz : diskMap θ z.1∈Metric.closedBall (0:ℂ) (1/4)) :
+    dist ((upperAffine (Real.cos θ) (Real.sin θ)
+      (Real.sin_pos_of_pos_of_lt_pi hθ hθπ)).symm z).1 Complex.I≤2/3 := by
+  apply cayley_inner
+  have he := diskMap_affine θ hθ hθπ
+    (((upperAffine (Real.cos θ) (Real.sin θ) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ)).symm z).1)
+  rw [←upperAffine_apply,Homeomorph.apply_symm_apply] at he
+  simpa only [Metric.mem_closedBall,dist_zero_right,he] using hz
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
+
+lemma height_of_ball {z : ℂ} {r : ℝ} (hz : dist z Complex.I≤r) : 1-r≤z.im := by
+  have h : 1-z.im≤dist z Complex.I := by
+    simpa only [Complex.sub_im,Complex.I_im,dist_eq_norm,norm_sub_rev] using
+      Complex.im_le_norm (Complex.I-z)
+  linarith
+
+lemma upper_inner_compact : IsCompact {z : UpperPlane | dist z.1 Complex.I≤2/3} := by
+  apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
+  have he : (Subtype.val : UpperPlane→ℂ) '' {z | dist z.1 Complex.I≤2/3}=
+      Metric.closedBall Complex.I (2/3) := by
+    ext z
+    constructor
+    · rintro ⟨w,hw,rfl⟩; exact hw
+    · intro hz
+      have hi := height_of_ball hz
+      exact ⟨⟨z,by linarith⟩,hz,rfl⟩
+  rw [he]
+  exact isCompact_closedBall _ _
+
+lemma cutoff_exists : ∃f : C_c(UpperPlane,ℝ),
+    (∀z,0≤f z ∧ f z≤1) ∧
+    (∀z : UpperPlane, dist z.1 Complex.I≤2/3 → f z=1) ∧
+    (∀z∈tsupport f,dist z.1 Complex.I<3/4) := by
+  obtain ⟨f,hf,hfc,hfs,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen
+    upper_inner_compact (Metric.isOpen_ball.preimage (continuous_subtype_val (p:=fun z : ℂ=>0<z.im)))
+    (show {z : UpperPlane | dist z.1 Complex.I≤2/3} ⊆
+        (Subtype.val : UpperPlane→ℂ) ⁻¹' Metric.ball Complex.I (3/4) from fun z hz=>by
+      change dist z.1 Complex.I<3/4
+      exact lt_of_le_of_lt hz (by norm_num))
+  exact ⟨⟨f,hfc⟩,h01,fun z hz=>hf hz,fun z hz=>hfs hz⟩
+
+def diskCutoff : C_c(UpperPlane,ℝ) := Classical.choose cutoff_exists
+
+lemma diskCutoff_nonneg (z : UpperPlane) : 0≤diskCutoff z :=
+  (Classical.choose_spec cutoff_exists).1 z |>.1
+lemma diskCutoff_le_one (z : UpperPlane) : diskCutoff z≤1 :=
+  (Classical.choose_spec cutoff_exists).1 z |>.2
+lemma diskCutoff_one (z : UpperPlane) (hz : dist z.1 Complex.I≤2/3) : diskCutoff z=1 :=
+  (Classical.choose_spec cutoff_exists).2.1 z hz
+lemma diskCutoff_support (z : UpperPlane) (hz : z∈tsupport diskCutoff) :
+    dist z.1 Complex.I<3/4 := (Classical.choose_spec cutoff_exists).2.2 z hz
+
+ 
+
+def diskWeight (a θ : ℝ) (z : UpperPlane) : ℝ := ‖deriv (diskMap θ) z.1‖^(2-alpha a)
+
+def diskMass (a θ : ℝ) (ν : Measure UpperPlane) : Measure ℂ :=
+  (ν.withDensity (fun z=>ENNReal.ofReal (diskWeight a θ z))).map (fun z=>diskMap θ z.1)
+
+lemma diskMap_measurable (θ : ℝ) : Measurable (fun z : UpperPlane=>diskMap θ z.1) := by
+  unfold diskMap
+  fun_prop
+
+lemma diskWeight_measurable (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) :
+    Measurable (diskWeight a θ) := by
+  have he : diskWeight a θ=fun z : UpperPlane=>
+      ‖((2*Real.sin θ:ℝ):ℂ)*Complex.I/(z.1-Complex.exp (-Complex.I*θ))^2‖^(2-alpha a) := by
+    funext z; rw [diskWeight,disk_deriv θ hθ hθπ z]
+  rw [he]
+  fun_prop
+
+lemma dimension_pos (a : ℝ) (ha : 1/4<a) : 0<2-alpha a := by
+  have ha0 : 0<a := by linarith
+  have hp : 0<1/(4*a) := by positivity
+  dsimp [alpha]
+  linarith
+
+ 
+def diskTest (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) : C_c(UpperPlane,ℝ) :=
+  (2^(2-alpha a):ℝ) • normalizedTest a (Real.cos θ) (Real.sin θ)
+    (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) diskCutoff
+
+lemma diskTest_nonneg (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
+    0≤diskTest a θ hθ hθπ z :=
+  mul_nonneg (Real.rpow_nonneg (by norm_num) _)
+    (normalizedTest_nonneg a _ _ _ diskCutoff diskCutoff_nonneg z)
+
+lemma diskTest_bound (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) (z : UpperPlane) :
+    ‖diskTest a θ hθ hθπ z‖≤2^(2-alpha a)*(Real.sin θ)^(alpha a-2) := by
+  change ‖(2^(2-alpha a):ℝ)*normalizedTest a _ _ _ diskCutoff z‖≤_
+  rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (by norm_num) _)]
+  apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by norm_num) _)
+  simpa only [one_mul] using normalizedTest_bound a _ _ _ diskCutoff (B:=1)
+    (fun z=>by rw [Real.norm_eq_abs,abs_of_nonneg (diskCutoff_nonneg z)]; exact diskCutoff_le_one z) z
+
+lemma diskTest_support (a θ : ℝ) (hθ : 0<θ) (hθπ : θ<Real.pi) :
+    Subtype.val '' tsupport (diskTest a θ hθ hθπ) ⊆
+      affine (Real.cos θ:ℂ) (Real.sin θ) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) ''
+        (Subtype.val '' tsupport diskCutoff) := by
+  apply (image_mono (tsupport_smul_subset_right (fun _ : UpperPlane => (2^(2-alpha a):ℝ))
+    (normalizedTest a (Real.cos θ) (Real.sin θ) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ) diskCutoff))).trans
+  exact normalizedTest_support a _ _ _ diskCutoff
+
+lemma diskWeight_le_diskTest (a θ : ℝ) (ha : 1/4<a) (hθ : 0<θ) (hθπ : θ<Real.pi)
+    (z : UpperPlane) (hz : diskMap θ z.1∈Metric.closedBall (0:ℂ) (1/4)) :
+    diskWeight a θ z≤diskTest a θ hθ hθπ z := by
+  have hy := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  have he := diskCutoff_one _ (disk_preimage_inner θ hθ hθπ z hz)
+  change ‖deriv (diskMap θ) z.1‖^(2-alpha a)≤
+    2^(2-alpha a)*((Real.sin θ)^(alpha a-2)*diskCutoff _)
+  rw [he,mul_one]
+  calc
+    _ ≤ (2/(Real.sin θ))^(2-alpha a) := Real.rpow_le_rpow (norm_nonneg _)
+      (disk_deriv_bound θ hθ hθπ z) (dimension_pos a ha).le
+    _ = _ := by
+      rw [div_eq_mul_inv,Real.mul_rpow (by norm_num : (0:ℝ)≤2) (inv_nonneg.mpr hy.le),
+        Real.inv_rpow hy.le,←Real.rpow_neg hy.le,neg_sub]
+
+lemma diskMass_inner_le_test (a θ : ℝ) (ha : 1/4<a) (hθ : 0<θ) (hθπ : θ<Real.pi)
+    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] :
+    diskMass a θ ν (Metric.closedBall (0:ℂ) (1/4))≤
+      ENNReal.ofReal (∫z,diskTest a θ hθ hθπ z∂ν) := by
+  rw [diskMass,Measure.map_apply (diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet,
+    withDensity_apply _ ((diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet),
+    ofReal_integral_eq_lintegral_ofReal (diskTest a θ hθ hθπ).integrable
+      (Eventually.of_forall (diskTest_nonneg a θ hθ hθπ)),
+    ←lintegral_indicator ((diskMap_measurable θ) Metric.isClosed_closedBall.measurableSet)]
+  apply lintegral_mono
+  intro z
+  by_cases hz : diskMap θ z.1∈Metric.closedBall (0:ℂ) (1/4)
+  · simp only [Set.indicator, Set.mem_preimage, hz, ite_true]
+    exact ENNReal.ofReal_le_ofReal (diskWeight_le_diskTest a θ ha hθ hθπ z hz)
+  · simp only [Set.indicator, Set.mem_preimage, hz, ite_false]; positivity
+
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/NormalizedRadialTests.lean b/lean/OAI/Probability/SLEGauge/NormalizedRadialTests.lean
index 343baf84663f31f97c264377d6db8c6089ef9260..7bd3df75abba1f7bd6f037ecaa123781d4c7d5bb 100644
--- a/lean/OAI/Probability/SLEGauge/NormalizedRadialTests.lean
+++ b/lean/OAI/Probability/SLEGauge/NormalizedRadialTests.lean
@@ -1,533 +1,533 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialConditionalMixing
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments
-
-lemma radialLaw_normalize_integral (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (w : UpperPlane) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (f : GDriver→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻b,f (normalizeDriver w b)∂radialLaw W a w.1 (w.1.im*q))=
-      ∫⁻b,f b∂radialLaw W a (fromDisk (pointAngle w).1 0)
-        (Real.sin (pointAngle w).1*q) := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi (pointAngle w).2.1 (pointAngle w).2.2
-  have hz : 0<(fromDisk (pointAngle w).1 0).im := by simpa only [fromDisk_zero_im] using hs
-  have he : (‖w.1‖:ℝ)*(Real.sin (pointAngle w).1*q)=w.1.im*q := by
-    rw [sin_pointAngle]
-    field_simp [(point_norm_pos w).ne']
-  have hr : Real.sin (pointAngle w).1*q≤(fromDisk (pointAngle w).1 0).im := by
-    rw [fromDisk_zero_im]
-    exact mul_le_of_le_one_right hs.le hq1
-  have hh := radialLaw_scale_integral W hW a ha (c:=‖w.1‖₊) (point_norm_pos w)
-    _ hz _ (mul_pos hs hq) hr f hf
-  simp only [coe_nnnorm] at hh
-  rw [norm_mul_fromDisk_pointAngle,he] at hh
-  exact hh
-
- 
-def finiteTestKernel (W : Measure GDriver) (a : ℝ) (l : ℕ)
-    (f : Angle×GDriver→ℝ≥0∞) (θ : Angle) : ℝ≥0∞ :=
-  ∫⁻b,f (θ,b)∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))
-
-lemma finiteTestKernel_eq (W : Measure GDriver) (a : ℝ) (ha : 1/4<a) (l : ℕ)
-    (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) (θ : Angle) :
-    finiteTestKernel W a l f θ=
-      ∫⁻b,ENNReal.ofReal (tiltWeight a θ.1 b.1 l)*f (θ,b)∂W := by
-  have hm := (centeredWeight_measurable a ha l).ennreal_ofReal.comp
-    ((measurable_const (a:=θ)).prodMk measurable_id)
-  have he : (fun b : GDriver => ENNReal.ofReal (tiltWeight a θ.1 b.1 l))=
-      radiusDensity a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
-    funext b
-    rw [radiusDensity,tiltWeight_radialWeight a θ.1 ha θ.2.1 θ.2.2 b l]
-  unfold finiteTestKernel radialLaw
-  change (∫⁻b,f (θ,b)∂W.withDensity
-    (radiusDensity a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))))=_
-  rw [←he]
-  exact lintegral_withDensity_eq_lintegral_mul W hm
-    (hf.comp (measurable_const.prodMk measurable_id))
-
-lemma finiteTestKernel_measurable (W : Measure GDriver) [SFinite W]
-    (a : ℝ) (ha : 1/4<a) (l : ℕ) (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
-    Measurable (finiteTestKernel W a l f) := by
-  simp_rw [funext (finiteTestKernel_eq W a ha l f hf)]
-  exact ((centeredWeight_measurable a ha l).ennreal_ofReal.mul hf).lintegral_prod_right'
-
-lemma finiteTestKernel_mixed (W : Measure GDriver) [SFinite W]
-    (a : ℝ) (ha : 1/4<a) (l : ℕ) (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻θ,finiteTestKernel W a l f θ∂angleLaw a)=∫⁻p,f p∂finiteAngleLaw W a l :=
-  (finiteAngleLaw_lintegral W a ha l f hf).symm
-
-lemma pointAngle_radialPoint (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (b : GDriver)
-    (hb : radialTime a b.1 z σ≠∞) :
-    pointAngle (radialPoint a z σ b)=Jacobi.finiteAngle a z σ b := by
-  apply Subtype.ext
-  change physicalTheta ((radialPoint a z σ b).1,0)=physicalTheta (radialState a b.1 z σ)
-  rw [radialPoint_eq a ha z hz σ hσ hσz b hb,radialState,ite_eq_right hb]
-  rfl
-
- 
-def normalizedTest (a : ℝ) (z : ℂ) (σ : ℝ) (f : Angle×GDriver→ℝ≥0∞)
-    (b : GDriver) : ℝ≥0∞ :=
-  f (pointAngle (radialPoint a z σ b),normalizeDriver (radialPoint a z σ b)
-    (restartAt (fun c=>radialTime a c.1 z σ) b))
-
-lemma normalizedTest_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
-    Measurable (normalizedTest a z σ f) := by
-  have hS := radialTime_stopping a (by linarith) z hz σ hσ hσz
-  have hw := (radialPoint_raw_measurable a ha z hz σ hσ hσz).mono hS.measurableSpace_le le_rfl
-  exact hf.comp ((pointAngle_measurable.comp hw).prodMk
-    (continuous_normalizeDriver.measurable.comp (hw.prodMk (measurable_restartAt hS))))
-
- 
-
-theorem normalizedTest_conditional (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im)
-    (l : ℕ) (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace] A)
-    (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻b in A,normalizedTest a z σ f b∂radialLaw W a z (σ*Real.exp (-2*a*l)))=
-      ∫⁻b in A,finiteTestKernel W a l f (Jacobi.finiteAngle a z σ b)∂radialLaw W a z σ := by
-  let F : UpperPlane×GDriver→ℝ≥0∞ := fun p=>f (pointAngle p.1,normalizeDriver p.1 p.2)
-  have hF : Measurable F := hf.comp ((pointAngle_measurable.comp measurable_fst).prodMk
-    continuous_normalizeDriver.measurable)
-  have hq1 : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
-    have ha0 : 0<a := by linarith
-    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
-    nlinarith)
-  have hq : 0<Real.exp (-2*a*l) := Real.exp_pos _
-  have hρ := mul_pos hσ hq
-  have hr := mul_le_of_le_one_right hσ.le hq1
-  change (∫⁻b in A,F (radialPoint a z σ b,restartAt (fun c=>radialTime a c.1 z σ) b)
-    ∂radialLaw W a z (σ*Real.exp (-2*a*l)))=_
-  rw [radialLaw_restart W hW a ha z hz σ _ hρ hr hσz A hA F hF]
-  have hrat : σ*Real.exp (-2*a*l)/σ=Real.exp (-2*a*l) := mul_div_cancel_left₀ _ hσ.ne'
-  simp_rw [hrat]
-  apply lintegral_congr_ae
-  filter_upwards [ae_restrict_of_ae (radial_hit_ae W a ha z hz σ hσ hσz)] with b hb
-  change (∫⁻c,f (pointAngle (radialPoint a z σ b),normalizeDriver (radialPoint a z σ b) c)
-    ∂radialLaw W a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*Real.exp (-2*a*l)))=_
-  rw [radialLaw_normalize_integral W hW a ha _ _ hq hq1
-    (fun c=>f (pointAngle (radialPoint a z σ b),c))
-    (hf.comp (measurable_const.prodMk measurable_id)),
-    pointAngle_radialPoint a (by linarith) z hz σ hσ hσz b hb]
-  rfl
-
- 
-
-theorem normalizedTest_conditional_bounds (W : Measure GDriver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ τ : ℝ) (hτ : 0<τ) (hτσ : τ≤σ) (hσ : σ≤z.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : τ=σ*Real.exp (-2*a*s))
-    (l : ℕ) (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
-      (hτ.trans_le hτσ) hσ).measurableSpace] A)
-    (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
-    (1/2:ℝ≥0∞)*(∫⁻p,f p∂finiteAngleLaw W a l)*radialLaw W a z (τ*Real.exp (-2*a*l)) A ≤
-      ∫⁻b in A,normalizedTest a z τ f b∂radialLaw W a z (τ*Real.exp (-2*a*l)) ∧
-    (∫⁻b in A,normalizedTest a z τ f b∂radialLaw W a z (τ*Real.exp (-2*a*l))) ≤
-      2*(∫⁻p,f p∂finiteAngleLaw W a l)*radialLaw W a z (τ*Real.exp (-2*a*l)) A := by
-  have ha0 : 0<a := by linarith
-  have hT := radialTime_stopping a ha0 z hz τ hτ (hτσ.trans hσ)
-  have hS := radialTime_stopping a ha0 z hz σ (hτ.trans_le hτσ) hσ
-  have hAτ : MeasurableSet[hT.measurableSpace] A :=
-    (hS.measurableSpace_mono hT (radialTime_antitone a ha0 z hz τ σ hτ hτσ hσ)) _ hA
-  have hq1 : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
-    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
-    nlinarith)
-  have hr := mul_le_of_le_one_right hτ.le hq1
-  rw [normalizedTest_conditional W hW a ha z hz τ hτ (hτσ.trans hσ) l A hAτ f hf,
-    radialLaw_consistent_le W hW a ha z hz τ _ (mul_pos hτ (Real.exp_pos _)) hr
-      (hτσ.trans hσ) A hAτ]
-  have hh := finiteAngle_conditional_bounds W hW a ha z hz σ τ hτ hτσ hσ s hs hgap A hA
-    (finiteTestKernel W a l f) (finiteTestKernel_measurable W a ha l f hf)
-  rwa [finiteTestKernel_mixed W a ha l f hf] at hh
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-
-lemma centered_radius_le (a : ℝ) (ha : 0≤a) (θ : Angle) (l : ℕ) :
-    Real.sin θ.1*Real.exp (-2*a*l)≤(fromDisk θ.1 0).im := by
-  rw [fromDisk_zero_im]
-  apply mul_le_of_le_one_right (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le
-  apply Real.exp_le_one_iff.mpr
-  have hl : (0:ℝ)≤l := Nat.cast_nonneg _
-  nlinarith
-
-lemma centered_radius_pos (a : ℝ) (θ : Angle) (l : ℕ) :
-    0<Real.sin θ.1*Real.exp (-2*a*l) :=
-  mul_pos (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) (Real.exp_pos _)
-
-lemma centered_radius_strict (a : ℝ) (ha : 0<a) (θ : Angle) {l k : ℕ} (hlk : l<k) :
-    Real.sin θ.1*Real.exp (-2*a*k)<Real.sin θ.1*Real.exp (-2*a*l) := by
-  apply mul_lt_mul_of_pos_left _ (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
-  apply Real.exp_lt_exp.mpr
-  have hk : (l:ℝ)<k := by exact_mod_cast hlk
-  nlinarith
-
-lemma centeredTime_stopping (a : ℝ) (ha : 1/4<a) (θ : Angle) (l : ℕ) :
-    IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l) := by
-  have he : (fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l)=fun b=>
-      radialTime a b.1 (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
-    funext b
-    exact levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous l
-  rw [he]
-  exact radialTime_stopping a (by linarith) _
-    (by rw [fromDisk_zero_im]; exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
-    _ (centered_radius_pos a θ l) (centered_radius_le a (by linarith) θ l)
-
-lemma centeredTime_monotone (a : ℝ) (ha : 1/4<a) (θ : Angle)
-    (b : ExactSLEGenerator.Driver) : Monotone (levelTime a θ.1 b.1) := by
-  intro l k hlk
-  rcases hlk.eq_or_lt with rfl|hlk
-  · exact le_rfl
-  simp only [levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous]
-  exact radialTime_antitone a (by linarith) _
-    (by rw [fromDisk_zero_im]; exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
-    _ _ (centered_radius_pos a θ k) (centered_radius_strict a (by linarith) θ hlk).le
-    (centered_radius_le a (by linarith) θ l) b
-
-lemma centeredMass_monotone (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (b : ExactSLEGenerator.Driver) (hμ : Monotone (fun v=>μ v b)) (θ : Angle) :
-    Monotone (fun l=>centeredMass a μ l (θ,b)) := by
-  intro l k hlk
-  exact Measure.le_iff'.mp (diskMass_mono
-    (hμ (centeredTime_monotone a ha θ b hlk)) a θ.1) _
-
-lemma centeredMass_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) (l : ℕ)
-    (hμ : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b)) :
-    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>centeredMass a μ l (θ,b)) := by
-  let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
-  have hn := diskMass_joint_measurable a (fun _ : ExactSLEGenerator.Driver=>θ) measurable_const
-    (fun b : ExactSLEGenerator.Driver=>μ (levelTime a θ.1 b.1 l) b) hμ
-  exact (Measure.measurable_coe Metric.isClosed_closedBall.measurableSet).comp hn
-
- 
-lemma centered_laws_lintegral_consistent (W : Measure ExactSLEGenerator.Driver)
-    [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (θ : Angle) {l k : ℕ} (hlk : l≤k)
-    (f : ExactSLEGenerator.Driver→ℝ≥0∞)
-    (hf : Measurable[(centeredTime_stopping a ha θ l).measurableSpace] f) :
-    (∫⁻b,f b∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*k)))=
-      ∫⁻b,f b∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
-  rcases hlk.eq_or_lt with rfl|hlk
-  · rfl
-  obtain ⟨_,_,hS,he⟩ := finite_radial_laws_consistent W hW a ha _
-    (by rw [fromDisk_zero_im]; exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
-    _ _ (centered_radius_pos a θ k) (centered_radius_strict a (by linarith) θ hlk)
-    (centered_radius_le a (by linarith) θ l)
-  have heS : (fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l)=fun b=>
-      radialTime a b.1 (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
-    funext b
-    exact levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous l
-  have hm : (centeredTime_stopping a ha θ l).measurableSpace=hS.measurableSpace := by
-    congr 1
-  have hf' : Measurable[hS.measurableSpace] f := by rwa [hm] at hf
-  have hme : (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*k))).trim
-      hS.measurableSpace_le =
-      (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))).trim
-      hS.measurableSpace_le := by
-    refine @Measure.ext _ hS.measurableSpace _ _ (fun U hU=> ?_)
-    rw [trim_measurableSet_eq _ hU,trim_measurableSet_eq _ hU]
-    exact he U hU
-  calc
-    _ = ∫⁻b,f b∂(radialLaw W a (fromDisk θ.1 0)
-        (Real.sin θ.1*Real.exp (-2*a*k))).trim hS.measurableSpace_le :=
-      (lintegral_trim hS.measurableSpace_le hf').symm
-    _ = _ := by rw [hme,lintegral_trim hS.measurableSpace_le hf']
-
-lemma finiteAngleLaw_lintegral_consistent (W : Measure ExactSLEGenerator.Driver)
-    [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) {l k : ℕ} (hlk : l≤k)
-    (f : Angle×ExactSLEGenerator.Driver→ℝ≥0∞) (hf : Measurable f)
-    (hr : ∀θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace] (fun b=>f (θ,b))) :
-    (∫⁻p,f p∂finiteAngleLaw W a k)=∫⁻p,f p∂finiteAngleLaw W a l := by
-  rw [finiteAngleLaw_lintegral W a ha k f hf,finiteAngleLaw_lintegral W a ha l f hf]
-  exact lintegral_congr (fun θ=>centered_laws_lintegral_consistent W hW a ha θ hlk _ (hr θ))
-
- 
-
-theorem finiteMoment_monotone (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b))
-    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) (m : ℕ) :
-    Monotone (finiteMoment W a μ m) := by
-  intro l k hlk
-  rw [finiteMoment,←finiteAngleLaw_lintegral_consistent W hW a ha hlk _
-    ((centeredMass_measurable a ha μ hμ l).pow_const m)
-    (fun θ=>(centeredMass_raw_measurable a ha μ θ l (hμraw l θ)).pow_const m)]
-  apply lintegral_mono_ae
-  have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
-    Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
-  filter_upwards [(finiteAngleLaw_absolutelyContinuous W a k).ae_le hb] with p hp
-  exact pow_le_pow_left' (centeredMass_monotone a ha μ p.2 hp p.1 hlk) m
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
-
-lemma raw_stopped_function_congr {E : Type*} [MeasurableSpace E] [MeasurableSingletonClass E]
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {F : GDriver→E} (hF : Measurable[hS.measurableSpace] F)
-    {b c : GDriver} {u : ℝ≥0} (hb : S b≤u) (h : ∀v≤u,b.1 v=c.1 v) : F b=F c := by
-  have hh := raw_stopped_event_congr hS (hF (measurableSet_singleton (F b))) hb h
-  have hc := hh.mp (show b∈F ⁻¹' {F b} from rfl)
-  exact hc.symm
-
- 
-
-def finiteStoppedTest (S : GDriver→ℝ≥0∞) (F : GDriver→ℝ≥0∞) (b : GDriver) : ℝ≥0∞ :=
-  if S b=∞ then 0 else F (pastDriver (S b).toNNReal b)
-
-lemma finiteStoppedTest_raw_measurable {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) (F : GDriver→ℝ≥0∞) (hF : Measurable F) :
-    Measurable[hS.measurableSpace] (finiteStoppedTest S F) := by
-  exact Measurable.ite (hS.measurable (measurableSet_singleton ∞)) measurable_const
-    (hF.comp (pastDriver_stopping_measurable S hS))
-
- 
-
-lemma normalized_levelTime_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (l : ℕ) (b : GDriver)
-    (hb : radialTime a b.1 z σ≠∞) :
-    let S := fun c : GDriver=>radialTime a c.1 z σ
-    let w := radialPoint a z σ b
-    S b + (‖w.1‖₊:ℝ≥0∞)^2 * levelTime a (pointAngle w).1
-      (normalizeDriver w (restartAt S b)).1 l = radialTime a b.1 z (σ*Real.exp (-2*a*l)) := by
-  dsimp only
-  let S := fun c : GDriver=>radialTime a c.1 z σ
-  let w := radialPoint a z σ b
-  have hs := (pointAngle w).2.1
-  have hsπ := (pointAngle w).2.2
-  have hq : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
-    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
-    nlinarith)
-  have hc := radialTime_scaled a (c:=‖w.1‖₊) (point_norm_pos w)
-    (restartAt S b).1.continuous (fromDisk (pointAngle w).1 0)
-    (Real.sin (pointAngle w).1*Real.exp (-2*a*l))
-  have hr : ‖w.1‖*(Real.sin (pointAngle w).1*Real.exp (-2*a*l))=w.1.im*Real.exp (-2*a*l) := by
-    rw [sin_pointAngle]
-    field_simp [(point_norm_pos w).ne']
-  simp only [coe_nnnorm] at hc
-  rw [norm_mul_fromDisk_pointAngle,hr] at hc
-  change S b+(‖w.1‖₊:ℝ≥0∞)^2*levelTime a (pointAngle w).1
-    (normalizeDriver w (restartAt S b)).1 l=_
-  rw [levelTime_radialTime a (pointAngle w).1 hs hsπ _
-    (normalizeDriver w (restartAt S b)).1.continuous]
-  change S b+(‖w.1‖₊:ℝ≥0∞)^2*radialTime a (scaled ‖w.1‖₊ (restartAt S b).1)
-    (fromDisk (pointAngle w).1 0) (Real.sin (pointAngle w).1*Real.exp (-2*a*l))=_
-  rw [hc]
-  have hh := radialTime_restart a ha z hz σ (σ*Real.exp (-2*a*l))
-    (mul_pos hσ (Real.exp_pos _)) (mul_le_of_le_one_right hσ.le hq) hσz b hb
-  dsimp only at hh
-  rw [←radialPoint_eq a ha z hz σ hσ hσz b hb,mul_div_cancel_left₀ _ hσ.ne'] at hh
-  simpa only [S,ENNReal.coe_toNNReal hb,restartAt,w] using hh
-
-lemma normalizedTest_history_congr (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (l : ℕ)
-    (f : Angle×GDriver→ℝ≥0∞)
-    (hfraw : ∀θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b : GDriver=>f (θ,b)))
-    {b c : GDriver} {u : ℝ≥0}
-    (hb : radialTime a b.1 z (σ*Real.exp (-2*a*l))≤u)
-    (hbc : ∀v≤u,b.1 v=c.1 v) :
-    normalizedTest a z σ f b=normalizedTest a z σ f c := by
-  let S := fun c : GDriver=>radialTime a c.1 z σ
-  let T := fun c : GDriver=>radialTime a c.1 z (σ*Real.exp (-2*a*l))
-  have ha0 : 0<a := by linarith
-  have hq : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
-    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
-    nlinarith)
-  have hST := radialTime_antitone a ha0 z hz (σ*Real.exp (-2*a*l)) σ
-    (mul_pos hσ (Real.exp_pos _)) (mul_le_of_le_one_right hσ.le hq) hσz b
-  have hS := radialTime_stopping a ha0 z hz σ hσ hσz
-  have hSb : S b≤u := hST.trans hb
-  have hbf : S b≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hSb
-  have hbs : (S b).toNNReal≤u := ENNReal.coe_le_coe.mp (by
-    simpa only [ENNReal.coe_toNNReal hbf] using hSb)
-  have hSc : S c=S b := by
-    have hh := raw_stop_congr hS (ENNReal.coe_toNNReal hbf).symm
-      (fun v hv=>hbc v (hv.trans hbs))
-    simpa only [ENNReal.coe_toNNReal hbf] using hh
-  have hw : radialPoint a z σ b=radialPoint a z σ c :=
-    raw_stopped_function_congr hS (radialPoint_raw_measurable a ha z hz σ hσ hσz) hSb hbc
-  let w := radialPoint a z σ b
-  let b' := normalizeDriver w (restartAt S b)
-  let c' := normalizeDriver w (restartAt S c)
-  let L := levelTime a (pointAngle w).1 b'.1 l
-  have hsum : S b+(‖w.1‖₊:ℝ≥0∞)^2*L=T b :=
-    normalized_levelTime_restart a ha0 z hz σ hσ hσz l b hbf
-  have hLf : L≠∞ := by
-    intro he
-    have hn : (‖w.1‖₊:ℝ≥0∞)^2≠0 := pow_ne_zero 2 (by
-      exact ENNReal.coe_ne_zero.mpr (NNReal.coe_ne_zero.mp (point_norm_pos w).ne'))
-    rw [he,ENNReal.mul_top hn,add_top] at hsum
-    exact (ne_top_of_le_ne_top ENNReal.coe_ne_top hb) hsum.symm
-  have htime : (S b).toNNReal+‖w.1‖₊^2*L.toNNReal≤u := by
-    apply ENNReal.coe_le_coe.mp
-    simp only [ENNReal.coe_add,ENNReal.coe_mul,ENNReal.coe_pow,ENNReal.coe_toNNReal hLf,
-      ENNReal.coe_toNNReal hbf]
-    rw [hsum]
-    exact hb
-  have hhistory : ∀v≤L.toNNReal,b'.1 v=c'.1 v := by
-    intro v hv
-    change ‖w.1‖⁻¹*((restartAt S b).1 (‖w.1‖₊^2*v))=
-      ‖w.1‖⁻¹*((restartAt S c).1 (‖w.1‖₊^2*v))
-    simp only [restartAt,restart_apply,hSc]
-    have ht : (S b).toNNReal+‖w.1‖₊^2*v≤u :=
-      (add_le_add le_rfl (mul_le_mul_of_nonneg_left hv (by positivity))).trans htime
-    rw [hbc _ ht,hbc _ hbs]
-  have hh := raw_stopped_function_congr (centeredTime_stopping a ha (pointAngle w) l)
-    (hfraw (pointAngle w)) (le_of_eq (ENNReal.coe_toNNReal hLf).symm) hhistory
-  change f (pointAngle w,b')=f (pointAngle (radialPoint a z σ c),
-    normalizeDriver (radialPoint a z σ c) (restartAt S c))
-  rw [←hw]
-  exact hh
-
- 
-
-lemma finiteStoppedTest_normalizedTest_eq (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (l : ℕ)
-    (f : Angle×GDriver→ℝ≥0∞)
-    (hfraw : ∀θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b : GDriver=>f (θ,b)))
-    (b : GDriver) (hb : radialTime a b.1 z (σ*Real.exp (-2*a*l))≠∞) :
-    finiteStoppedTest (fun c : GDriver=>radialTime a c.1 z (σ*Real.exp (-2*a*l)))
-      (normalizedTest a z σ f) b=normalizedTest a z σ f b := by
-  rw [finiteStoppedTest,ite_eq_right hb]
-  exact (normalizedTest_history_congr a ha z hz σ hσ hσz l f hfraw
-    (le_of_eq (ENNReal.coe_toNNReal hb).symm) (fun v hv=>(pastDriver_eq _ b v hv).symm)).symm
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology
-open scoped ENNReal NNReal Classical
-namespace SLEExactGauge.CompactSequence
-
- 
-def initialVector {E : Type*} (n : ℕ) (x : ℕ→E) : Fin (n+1)→E := fun i=>x i
-
-lemma continuous_initialVector {E : Type*} [TopologicalSpace E] (n : ℕ) :
-    Continuous (initialVector (E:=E) n) := continuous_pi (fun _=>continuous_apply _)
-
-lemma measurable_initialVector {E : Type*} [MeasurableSpace E] (n : ℕ) :
-    Measurable (initialVector (E:=E) n) := Measurable.of_eval (fun _=>measurable_pi_apply _)
-
- 
-
-theorem exists_sequence_law {E Ω : Type*} [TopologicalSpace E] [CompactSpace E]
-    [T2Space E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
-    [MeasurableSpace Ω]
-    (P : ℕ→Measure Ω) [∀n,IsProbabilityMeasure (P n)]
-    (X : ℕ→Ω→E) (hX : ∀n,Measurable (X n))
-    (hcons : ∀n k,n≤k →
-      (P k).map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω)=
-      (P n).map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω)) :
-    ∃Q : Measure (ℕ→E),IsProbabilityMeasure Q ∧
-      ∀n,Q.map (initialVector n)=(P n).map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω) := by
-  have hXm : Measurable (fun ω : Ω=>fun n=>X n ω) := Measurable.of_eval hX
-  let ν : ℕ→ProbabilityMeasure (ℕ→E) := fun n=>
-    ⟨(P n).map (fun ω : Ω=>fun i=>X i ω),inferInstance⟩
-  obtain ⟨Q,hQ⟩ := exists_clusterPt_of_compactSpace (Filter.map ν atTop)
-  refine ⟨Q,Q.prop,?_⟩
-  intro n
-  have he (k : ℕ) (hkn : n≤k) :
-      (ν k).map (initialVector n)=
-        (ν n).map (initialVector n) := by
-    apply ProbabilityMeasure.toMeasure_injective
-    change ((P k).map (fun ω : Ω=>fun i=>X i ω)).map (initialVector n)=
-      ((P n).map (fun ω : Ω=>fun i=>X i ω)).map (initialVector n)
-    rw [Measure.map_map (measurable_initialVector n) hXm,Measure.map_map (measurable_initialVector n) hXm]
-    exact hcons n k hkn
-  have hc : IsClosed {q : ProbabilityMeasure (ℕ→E) |
-      q.map (initialVector n)=(ν n).map (initialVector n)} :=
-    isClosed_eq (ProbabilityMeasure.continuous_map (continuous_initialVector n)) continuous_const
-  have hq := hc.mem_of_mapClusterPt hQ (Filter.eventually_atTop.mpr ⟨n,he⟩)
-  have hqm := congrArg ProbabilityMeasure.toMeasure hq
-  change Q.toMeasure.map (initialVector n)=
-    ((P n).map (fun ω : Ω=>fun i=>X i ω)).map (initialVector n) at hqm
-  rw [Measure.map_map (measurable_initialVector n) hXm] at hqm
-  exact hqm
-
- 
-lemma lintegral_initialVector {E Ω : Type*} [MeasurableSpace E] [MeasurableSpace Ω]
-    (Q : Measure (ℕ→E)) (P : Measure Ω) (X : ℕ→Ω→E)
-    (hX : ∀n,Measurable (X n)) (n : ℕ)
-    (hQ : Q.map (initialVector n)=P.map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω))
-    (f : (Fin (n+1)→E)→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻x,f (initialVector n x)∂Q)=∫⁻ω,f (fun i=>X i ω)∂P := by
-  rw [←lintegral_map hf (measurable_initialVector n),hQ,
-    lintegral_map hf (Measurable.of_eval (fun i=>hX i))]
-
-lemma lintegral_coordinate {E Ω : Type*} [MeasurableSpace E] [MeasurableSpace Ω]
-    (Q : Measure (ℕ→E)) (P : Measure Ω) (X : ℕ→Ω→E)
-    (hX : ∀n,Measurable (X n)) (n : ℕ)
-    (hQ : Q.map (initialVector n)=P.map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω))
-    (f : E→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻x,f (x n)∂Q)=∫⁻ω,f (X n ω)∂P :=
-  lintegral_initialVector Q P X hX n hQ (fun x=>f (x ⟨n,by omega⟩))
-    (hf.comp (measurable_pi_apply _))
-
-end SLEExactGauge.CompactSequence
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialConditionalMixing
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments
+
+lemma radialLaw_normalize_integral (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (w : UpperPlane) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (f : GDriver→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻b,f (normalizeDriver w b)∂radialLaw W a w.1 (w.1.im*q))=
+      ∫⁻b,f b∂radialLaw W a (fromDisk (pointAngle w).1 0)
+        (Real.sin (pointAngle w).1*q) := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi (pointAngle w).2.1 (pointAngle w).2.2
+  have hz : 0<(fromDisk (pointAngle w).1 0).im := by simpa only [fromDisk_zero_im] using hs
+  have he : (‖w.1‖:ℝ)*(Real.sin (pointAngle w).1*q)=w.1.im*q := by
+    rw [sin_pointAngle]
+    field_simp [(point_norm_pos w).ne']
+  have hr : Real.sin (pointAngle w).1*q≤(fromDisk (pointAngle w).1 0).im := by
+    rw [fromDisk_zero_im]
+    exact mul_le_of_le_one_right hs.le hq1
+  have hh := radialLaw_scale_integral W hW a ha (c:=‖w.1‖₊) (point_norm_pos w)
+    _ hz _ (mul_pos hs hq) hr f hf
+  simp only [coe_nnnorm] at hh
+  rw [norm_mul_fromDisk_pointAngle,he] at hh
+  exact hh
+
+ 
+def finiteTestKernel (W : Measure GDriver) (a : ℝ) (l : ℕ)
+    (f : Angle×GDriver→ℝ≥0∞) (θ : Angle) : ℝ≥0∞ :=
+  ∫⁻b,f (θ,b)∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))
+
+lemma finiteTestKernel_eq (W : Measure GDriver) (a : ℝ) (ha : 1/4<a) (l : ℕ)
+    (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) (θ : Angle) :
+    finiteTestKernel W a l f θ=
+      ∫⁻b,ENNReal.ofReal (tiltWeight a θ.1 b.1 l)*f (θ,b)∂W := by
+  have hm := (centeredWeight_measurable a ha l).ennreal_ofReal.comp
+    ((measurable_const (a:=θ)).prodMk measurable_id)
+  have he : (fun b : GDriver => ENNReal.ofReal (tiltWeight a θ.1 b.1 l))=
+      radiusDensity a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
+    funext b
+    rw [radiusDensity,tiltWeight_radialWeight a θ.1 ha θ.2.1 θ.2.2 b l]
+  unfold finiteTestKernel radialLaw
+  change (∫⁻b,f (θ,b)∂W.withDensity
+    (radiusDensity a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))))=_
+  rw [←he]
+  exact lintegral_withDensity_eq_lintegral_mul W hm
+    (hf.comp (measurable_const.prodMk measurable_id))
+
+lemma finiteTestKernel_measurable (W : Measure GDriver) [SFinite W]
+    (a : ℝ) (ha : 1/4<a) (l : ℕ) (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
+    Measurable (finiteTestKernel W a l f) := by
+  simp_rw [funext (finiteTestKernel_eq W a ha l f hf)]
+  exact ((centeredWeight_measurable a ha l).ennreal_ofReal.mul hf).lintegral_prod_right'
+
+lemma finiteTestKernel_mixed (W : Measure GDriver) [SFinite W]
+    (a : ℝ) (ha : 1/4<a) (l : ℕ) (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻θ,finiteTestKernel W a l f θ∂angleLaw a)=∫⁻p,f p∂finiteAngleLaw W a l :=
+  (finiteAngleLaw_lintegral W a ha l f hf).symm
+
+lemma pointAngle_radialPoint (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (b : GDriver)
+    (hb : radialTime a b.1 z σ≠∞) :
+    pointAngle (radialPoint a z σ b)=Jacobi.finiteAngle a z σ b := by
+  apply Subtype.ext
+  change physicalTheta ((radialPoint a z σ b).1,0)=physicalTheta (radialState a b.1 z σ)
+  rw [radialPoint_eq a ha z hz σ hσ hσz b hb,radialState,ite_eq_right hb]
+  rfl
+
+ 
+def normalizedTest (a : ℝ) (z : ℂ) (σ : ℝ) (f : Angle×GDriver→ℝ≥0∞)
+    (b : GDriver) : ℝ≥0∞ :=
+  f (pointAngle (radialPoint a z σ b),normalizeDriver (radialPoint a z σ b)
+    (restartAt (fun c=>radialTime a c.1 z σ) b))
+
+lemma normalizedTest_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
+    Measurable (normalizedTest a z σ f) := by
+  have hS := radialTime_stopping a (by linarith) z hz σ hσ hσz
+  have hw := (radialPoint_raw_measurable a ha z hz σ hσ hσz).mono hS.measurableSpace_le le_rfl
+  exact hf.comp ((pointAngle_measurable.comp hw).prodMk
+    (continuous_normalizeDriver.measurable.comp (hw.prodMk (measurable_restartAt hS))))
+
+ 
+
+theorem normalizedTest_conditional (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im) (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im)
+    (l : ℕ) (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace] A)
+    (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻b in A,normalizedTest a z σ f b∂radialLaw W a z (σ*Real.exp (-2*a*l)))=
+      ∫⁻b in A,finiteTestKernel W a l f (Jacobi.finiteAngle a z σ b)∂radialLaw W a z σ := by
+  let F : UpperPlane×GDriver→ℝ≥0∞ := fun p=>f (pointAngle p.1,normalizeDriver p.1 p.2)
+  have hF : Measurable F := hf.comp ((pointAngle_measurable.comp measurable_fst).prodMk
+    continuous_normalizeDriver.measurable)
+  have hq1 : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
+    have ha0 : 0<a := by linarith
+    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
+    nlinarith)
+  have hq : 0<Real.exp (-2*a*l) := Real.exp_pos _
+  have hρ := mul_pos hσ hq
+  have hr := mul_le_of_le_one_right hσ.le hq1
+  change (∫⁻b in A,F (radialPoint a z σ b,restartAt (fun c=>radialTime a c.1 z σ) b)
+    ∂radialLaw W a z (σ*Real.exp (-2*a*l)))=_
+  rw [radialLaw_restart W hW a ha z hz σ _ hρ hr hσz A hA F hF]
+  have hrat : σ*Real.exp (-2*a*l)/σ=Real.exp (-2*a*l) := mul_div_cancel_left₀ _ hσ.ne'
+  simp_rw [hrat]
+  apply lintegral_congr_ae
+  filter_upwards [ae_restrict_of_ae (radial_hit_ae W a ha z hz σ hσ hσz)] with b hb
+  change (∫⁻c,f (pointAngle (radialPoint a z σ b),normalizeDriver (radialPoint a z σ b) c)
+    ∂radialLaw W a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*Real.exp (-2*a*l)))=_
+  rw [radialLaw_normalize_integral W hW a ha _ _ hq hq1
+    (fun c=>f (pointAngle (radialPoint a z σ b),c))
+    (hf.comp (measurable_const.prodMk measurable_id)),
+    pointAngle_radialPoint a (by linarith) z hz σ hσ hσz b hb]
+  rfl
+
+ 
+
+theorem normalizedTest_conditional_bounds (W : Measure GDriver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ τ : ℝ) (hτ : 0<τ) (hτσ : τ≤σ) (hσ : σ≤z.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : τ=σ*Real.exp (-2*a*s))
+    (l : ℕ) (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
+      (hτ.trans_le hτσ) hσ).measurableSpace] A)
+    (f : Angle×GDriver→ℝ≥0∞) (hf : Measurable f) :
+    (1/2:ℝ≥0∞)*(∫⁻p,f p∂finiteAngleLaw W a l)*radialLaw W a z (τ*Real.exp (-2*a*l)) A ≤
+      ∫⁻b in A,normalizedTest a z τ f b∂radialLaw W a z (τ*Real.exp (-2*a*l)) ∧
+    (∫⁻b in A,normalizedTest a z τ f b∂radialLaw W a z (τ*Real.exp (-2*a*l))) ≤
+      2*(∫⁻p,f p∂finiteAngleLaw W a l)*radialLaw W a z (τ*Real.exp (-2*a*l)) A := by
+  have ha0 : 0<a := by linarith
+  have hT := radialTime_stopping a ha0 z hz τ hτ (hτσ.trans hσ)
+  have hS := radialTime_stopping a ha0 z hz σ (hτ.trans_le hτσ) hσ
+  have hAτ : MeasurableSet[hT.measurableSpace] A :=
+    (hS.measurableSpace_mono hT (radialTime_antitone a ha0 z hz τ σ hτ hτσ hσ)) _ hA
+  have hq1 : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
+    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
+    nlinarith)
+  have hr := mul_le_of_le_one_right hτ.le hq1
+  rw [normalizedTest_conditional W hW a ha z hz τ hτ (hτσ.trans hσ) l A hAτ f hf,
+    radialLaw_consistent_le W hW a ha z hz τ _ (mul_pos hτ (Real.exp_pos _)) hr
+      (hτσ.trans hσ) A hAτ]
+  have hh := finiteAngle_conditional_bounds W hW a ha z hz σ τ hτ hτσ hσ s hs hgap A hA
+    (finiteTestKernel W a l f) (finiteTestKernel_measurable W a ha l f hf)
+  rwa [finiteTestKernel_mixed W a ha l f hf] at hh
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+
+lemma centered_radius_le (a : ℝ) (ha : 0≤a) (θ : Angle) (l : ℕ) :
+    Real.sin θ.1*Real.exp (-2*a*l)≤(fromDisk θ.1 0).im := by
+  rw [fromDisk_zero_im]
+  apply mul_le_of_le_one_right (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2).le
+  apply Real.exp_le_one_iff.mpr
+  have hl : (0:ℝ)≤l := Nat.cast_nonneg _
+  nlinarith
+
+lemma centered_radius_pos (a : ℝ) (θ : Angle) (l : ℕ) :
+    0<Real.sin θ.1*Real.exp (-2*a*l) :=
+  mul_pos (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2) (Real.exp_pos _)
+
+lemma centered_radius_strict (a : ℝ) (ha : 0<a) (θ : Angle) {l k : ℕ} (hlk : l<k) :
+    Real.sin θ.1*Real.exp (-2*a*k)<Real.sin θ.1*Real.exp (-2*a*l) := by
+  apply mul_lt_mul_of_pos_left _ (Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
+  apply Real.exp_lt_exp.mpr
+  have hk : (l:ℝ)<k := by exact_mod_cast hlk
+  nlinarith
+
+lemma centeredTime_stopping (a : ℝ) (ha : 1/4<a) (θ : Angle) (l : ℕ) :
+    IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l) := by
+  have he : (fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l)=fun b=>
+      radialTime a b.1 (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
+    funext b
+    exact levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous l
+  rw [he]
+  exact radialTime_stopping a (by linarith) _
+    (by rw [fromDisk_zero_im]; exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
+    _ (centered_radius_pos a θ l) (centered_radius_le a (by linarith) θ l)
+
+lemma centeredTime_monotone (a : ℝ) (ha : 1/4<a) (θ : Angle)
+    (b : ExactSLEGenerator.Driver) : Monotone (levelTime a θ.1 b.1) := by
+  intro l k hlk
+  rcases hlk.eq_or_lt with rfl|hlk
+  · exact le_rfl
+  simp only [levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous]
+  exact radialTime_antitone a (by linarith) _
+    (by rw [fromDisk_zero_im]; exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
+    _ _ (centered_radius_pos a θ k) (centered_radius_strict a (by linarith) θ hlk).le
+    (centered_radius_le a (by linarith) θ l) b
+
+lemma centeredMass_monotone (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (b : ExactSLEGenerator.Driver) (hμ : Monotone (fun v=>μ v b)) (θ : Angle) :
+    Monotone (fun l=>centeredMass a μ l (θ,b)) := by
+  intro l k hlk
+  exact Measure.le_iff'.mp (diskMass_mono
+    (hμ (centeredTime_monotone a ha θ b hlk)) a θ.1) _
+
+lemma centeredMass_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) (l : ℕ)
+    (hμ : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b)) :
+    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>centeredMass a μ l (θ,b)) := by
+  let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
+  have hn := diskMass_joint_measurable a (fun _ : ExactSLEGenerator.Driver=>θ) measurable_const
+    (fun b : ExactSLEGenerator.Driver=>μ (levelTime a θ.1 b.1 l) b) hμ
+  exact (Measure.measurable_coe Metric.isClosed_closedBall.measurableSet).comp hn
+
+ 
+lemma centered_laws_lintegral_consistent (W : Measure ExactSLEGenerator.Driver)
+    [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (θ : Angle) {l k : ℕ} (hlk : l≤k)
+    (f : ExactSLEGenerator.Driver→ℝ≥0∞)
+    (hf : Measurable[(centeredTime_stopping a ha θ l).measurableSpace] f) :
+    (∫⁻b,f b∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*k)))=
+      ∫⁻b,f b∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
+  rcases hlk.eq_or_lt with rfl|hlk
+  · rfl
+  obtain ⟨_,_,hS,he⟩ := finite_radial_laws_consistent W hW a ha _
+    (by rw [fromDisk_zero_im]; exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2)
+    _ _ (centered_radius_pos a θ k) (centered_radius_strict a (by linarith) θ hlk)
+    (centered_radius_le a (by linarith) θ l)
+  have heS : (fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l)=fun b=>
+      radialTime a b.1 (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l)) := by
+    funext b
+    exact levelTime_radialTime a θ.1 θ.2.1 θ.2.2 b.1 b.1.continuous l
+  have hm : (centeredTime_stopping a ha θ l).measurableSpace=hS.measurableSpace := by
+    congr 1
+  have hf' : Measurable[hS.measurableSpace] f := by rwa [hm] at hf
+  have hme : (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*k))).trim
+      hS.measurableSpace_le =
+      (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))).trim
+      hS.measurableSpace_le := by
+    refine @Measure.ext _ hS.measurableSpace _ _ (fun U hU=> ?_)
+    rw [trim_measurableSet_eq _ hU,trim_measurableSet_eq _ hU]
+    exact he U hU
+  calc
+    _ = ∫⁻b,f b∂(radialLaw W a (fromDisk θ.1 0)
+        (Real.sin θ.1*Real.exp (-2*a*k))).trim hS.measurableSpace_le :=
+      (lintegral_trim hS.measurableSpace_le hf').symm
+    _ = _ := by rw [hme,lintegral_trim hS.measurableSpace_le hf']
+
+lemma finiteAngleLaw_lintegral_consistent (W : Measure ExactSLEGenerator.Driver)
+    [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) {l k : ℕ} (hlk : l≤k)
+    (f : Angle×ExactSLEGenerator.Driver→ℝ≥0∞) (hf : Measurable f)
+    (hr : ∀θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace] (fun b=>f (θ,b))) :
+    (∫⁻p,f p∂finiteAngleLaw W a k)=∫⁻p,f p∂finiteAngleLaw W a l := by
+  rw [finiteAngleLaw_lintegral W a ha k f hf,finiteAngleLaw_lintegral W a ha l f hf]
+  exact lintegral_congr (fun θ=>centered_laws_lintegral_consistent W hW a ha θ hlk _ (hr θ))
+
+ 
+
+theorem finiteMoment_monotone (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b))
+    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) (m : ℕ) :
+    Monotone (finiteMoment W a μ m) := by
+  intro l k hlk
+  rw [finiteMoment,←finiteAngleLaw_lintegral_consistent W hW a ha hlk _
+    ((centeredMass_measurable a ha μ hμ l).pow_const m)
+    (fun θ=>(centeredMass_raw_measurable a ha μ θ l (hμraw l θ)).pow_const m)]
+  apply lintegral_mono_ae
+  have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
+    Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
+  filter_upwards [(finiteAngleLaw_absolutelyContinuous W a k).ae_le hb] with p hp
+  exact pow_le_pow_left' (centeredMass_monotone a ha μ p.2 hp p.1 hlk) m
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
+
+lemma raw_stopped_function_congr {E : Type*} [MeasurableSpace E] [MeasurableSingletonClass E]
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {F : GDriver→E} (hF : Measurable[hS.measurableSpace] F)
+    {b c : GDriver} {u : ℝ≥0} (hb : S b≤u) (h : ∀v≤u,b.1 v=c.1 v) : F b=F c := by
+  have hh := raw_stopped_event_congr hS (hF (measurableSet_singleton (F b))) hb h
+  have hc := hh.mp (show b∈F ⁻¹' {F b} from rfl)
+  exact hc.symm
+
+ 
+
+def finiteStoppedTest (S : GDriver→ℝ≥0∞) (F : GDriver→ℝ≥0∞) (b : GDriver) : ℝ≥0∞ :=
+  if S b=∞ then 0 else F (pastDriver (S b).toNNReal b)
+
+lemma finiteStoppedTest_raw_measurable {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) (F : GDriver→ℝ≥0∞) (hF : Measurable F) :
+    Measurable[hS.measurableSpace] (finiteStoppedTest S F) := by
+  exact Measurable.ite (hS.measurable (measurableSet_singleton ∞)) measurable_const
+    (hF.comp (pastDriver_stopping_measurable S hS))
+
+ 
+
+lemma normalized_levelTime_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (l : ℕ) (b : GDriver)
+    (hb : radialTime a b.1 z σ≠∞) :
+    let S := fun c : GDriver=>radialTime a c.1 z σ
+    let w := radialPoint a z σ b
+    S b + (‖w.1‖₊:ℝ≥0∞)^2 * levelTime a (pointAngle w).1
+      (normalizeDriver w (restartAt S b)).1 l = radialTime a b.1 z (σ*Real.exp (-2*a*l)) := by
+  dsimp only
+  let S := fun c : GDriver=>radialTime a c.1 z σ
+  let w := radialPoint a z σ b
+  have hs := (pointAngle w).2.1
+  have hsπ := (pointAngle w).2.2
+  have hq : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
+    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
+    nlinarith)
+  have hc := radialTime_scaled a (c:=‖w.1‖₊) (point_norm_pos w)
+    (restartAt S b).1.continuous (fromDisk (pointAngle w).1 0)
+    (Real.sin (pointAngle w).1*Real.exp (-2*a*l))
+  have hr : ‖w.1‖*(Real.sin (pointAngle w).1*Real.exp (-2*a*l))=w.1.im*Real.exp (-2*a*l) := by
+    rw [sin_pointAngle]
+    field_simp [(point_norm_pos w).ne']
+  simp only [coe_nnnorm] at hc
+  rw [norm_mul_fromDisk_pointAngle,hr] at hc
+  change S b+(‖w.1‖₊:ℝ≥0∞)^2*levelTime a (pointAngle w).1
+    (normalizeDriver w (restartAt S b)).1 l=_
+  rw [levelTime_radialTime a (pointAngle w).1 hs hsπ _
+    (normalizeDriver w (restartAt S b)).1.continuous]
+  change S b+(‖w.1‖₊:ℝ≥0∞)^2*radialTime a (scaled ‖w.1‖₊ (restartAt S b).1)
+    (fromDisk (pointAngle w).1 0) (Real.sin (pointAngle w).1*Real.exp (-2*a*l))=_
+  rw [hc]
+  have hh := radialTime_restart a ha z hz σ (σ*Real.exp (-2*a*l))
+    (mul_pos hσ (Real.exp_pos _)) (mul_le_of_le_one_right hσ.le hq) hσz b hb
+  dsimp only at hh
+  rw [←radialPoint_eq a ha z hz σ hσ hσz b hb,mul_div_cancel_left₀ _ hσ.ne'] at hh
+  simpa only [S,ENNReal.coe_toNNReal hb,restartAt,w] using hh
+
+lemma normalizedTest_history_congr (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (l : ℕ)
+    (f : Angle×GDriver→ℝ≥0∞)
+    (hfraw : ∀θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b : GDriver=>f (θ,b)))
+    {b c : GDriver} {u : ℝ≥0}
+    (hb : radialTime a b.1 z (σ*Real.exp (-2*a*l))≤u)
+    (hbc : ∀v≤u,b.1 v=c.1 v) :
+    normalizedTest a z σ f b=normalizedTest a z σ f c := by
+  let S := fun c : GDriver=>radialTime a c.1 z σ
+  let T := fun c : GDriver=>radialTime a c.1 z (σ*Real.exp (-2*a*l))
+  have ha0 : 0<a := by linarith
+  have hq : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
+    have hl : (0:ℝ)≤l := Nat.cast_nonneg l
+    nlinarith)
+  have hST := radialTime_antitone a ha0 z hz (σ*Real.exp (-2*a*l)) σ
+    (mul_pos hσ (Real.exp_pos _)) (mul_le_of_le_one_right hσ.le hq) hσz b
+  have hS := radialTime_stopping a ha0 z hz σ hσ hσz
+  have hSb : S b≤u := hST.trans hb
+  have hbf : S b≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hSb
+  have hbs : (S b).toNNReal≤u := ENNReal.coe_le_coe.mp (by
+    simpa only [ENNReal.coe_toNNReal hbf] using hSb)
+  have hSc : S c=S b := by
+    have hh := raw_stop_congr hS (ENNReal.coe_toNNReal hbf).symm
+      (fun v hv=>hbc v (hv.trans hbs))
+    simpa only [ENNReal.coe_toNNReal hbf] using hh
+  have hw : radialPoint a z σ b=radialPoint a z σ c :=
+    raw_stopped_function_congr hS (radialPoint_raw_measurable a ha z hz σ hσ hσz) hSb hbc
+  let w := radialPoint a z σ b
+  let b' := normalizeDriver w (restartAt S b)
+  let c' := normalizeDriver w (restartAt S c)
+  let L := levelTime a (pointAngle w).1 b'.1 l
+  have hsum : S b+(‖w.1‖₊:ℝ≥0∞)^2*L=T b :=
+    normalized_levelTime_restart a ha0 z hz σ hσ hσz l b hbf
+  have hLf : L≠∞ := by
+    intro he
+    have hn : (‖w.1‖₊:ℝ≥0∞)^2≠0 := pow_ne_zero 2 (by
+      exact ENNReal.coe_ne_zero.mpr (NNReal.coe_ne_zero.mp (point_norm_pos w).ne'))
+    rw [he,ENNReal.mul_top hn,add_top] at hsum
+    exact (ne_top_of_le_ne_top ENNReal.coe_ne_top hb) hsum.symm
+  have htime : (S b).toNNReal+‖w.1‖₊^2*L.toNNReal≤u := by
+    apply ENNReal.coe_le_coe.mp
+    simp only [ENNReal.coe_add,ENNReal.coe_mul,ENNReal.coe_pow,ENNReal.coe_toNNReal hLf,
+      ENNReal.coe_toNNReal hbf]
+    rw [hsum]
+    exact hb
+  have hhistory : ∀v≤L.toNNReal,b'.1 v=c'.1 v := by
+    intro v hv
+    change ‖w.1‖⁻¹*((restartAt S b).1 (‖w.1‖₊^2*v))=
+      ‖w.1‖⁻¹*((restartAt S c).1 (‖w.1‖₊^2*v))
+    simp only [restartAt,restart_apply,hSc]
+    have ht : (S b).toNNReal+‖w.1‖₊^2*v≤u :=
+      (add_le_add le_rfl (mul_le_mul_of_nonneg_left hv (by positivity))).trans htime
+    rw [hbc _ ht,hbc _ hbs]
+  have hh := raw_stopped_function_congr (centeredTime_stopping a ha (pointAngle w) l)
+    (hfraw (pointAngle w)) (le_of_eq (ENNReal.coe_toNNReal hLf).symm) hhistory
+  change f (pointAngle w,b')=f (pointAngle (radialPoint a z σ c),
+    normalizeDriver (radialPoint a z σ c) (restartAt S c))
+  rw [←hw]
+  exact hh
+
+ 
+
+lemma finiteStoppedTest_normalizedTest_eq (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (l : ℕ)
+    (f : Angle×GDriver→ℝ≥0∞)
+    (hfraw : ∀θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b : GDriver=>f (θ,b)))
+    (b : GDriver) (hb : radialTime a b.1 z (σ*Real.exp (-2*a*l))≠∞) :
+    finiteStoppedTest (fun c : GDriver=>radialTime a c.1 z (σ*Real.exp (-2*a*l)))
+      (normalizedTest a z σ f) b=normalizedTest a z σ f b := by
+  rw [finiteStoppedTest,ite_eq_right hb]
+  exact (normalizedTest_history_congr a ha z hz σ hσ hσz l f hfraw
+    (le_of_eq (ENNReal.coe_toNNReal hb).symm) (fun v hv=>(pastDriver_eq _ b v hv).symm)).symm
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology
+open scoped ENNReal NNReal Classical
+namespace SLEExactGauge.CompactSequence
+
+ 
+def initialVector {E : Type*} (n : ℕ) (x : ℕ→E) : Fin (n+1)→E := fun i=>x i
+
+lemma continuous_initialVector {E : Type*} [TopologicalSpace E] (n : ℕ) :
+    Continuous (initialVector (E:=E) n) := continuous_pi (fun _=>continuous_apply _)
+
+lemma measurable_initialVector {E : Type*} [MeasurableSpace E] (n : ℕ) :
+    Measurable (initialVector (E:=E) n) := Measurable.of_eval (fun _=>measurable_pi_apply _)
+
+ 
+
+theorem exists_sequence_law {E Ω : Type*} [TopologicalSpace E] [CompactSpace E]
+    [T2Space E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E]
+    [MeasurableSpace Ω]
+    (P : ℕ→Measure Ω) [∀n,IsProbabilityMeasure (P n)]
+    (X : ℕ→Ω→E) (hX : ∀n,Measurable (X n))
+    (hcons : ∀n k,n≤k →
+      (P k).map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω)=
+      (P n).map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω)) :
+    ∃Q : Measure (ℕ→E),IsProbabilityMeasure Q ∧
+      ∀n,Q.map (initialVector n)=(P n).map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω) := by
+  have hXm : Measurable (fun ω : Ω=>fun n=>X n ω) := Measurable.of_eval hX
+  let ν : ℕ→ProbabilityMeasure (ℕ→E) := fun n=>
+    ⟨(P n).map (fun ω : Ω=>fun i=>X i ω),inferInstance⟩
+  obtain ⟨Q,hQ⟩ := exists_clusterPt_of_compactSpace (Filter.map ν atTop)
+  refine ⟨Q,Q.prop,?_⟩
+  intro n
+  have he (k : ℕ) (hkn : n≤k) :
+      (ν k).map (initialVector n)=
+        (ν n).map (initialVector n) := by
+    apply ProbabilityMeasure.toMeasure_injective
+    change ((P k).map (fun ω : Ω=>fun i=>X i ω)).map (initialVector n)=
+      ((P n).map (fun ω : Ω=>fun i=>X i ω)).map (initialVector n)
+    rw [Measure.map_map (measurable_initialVector n) hXm,Measure.map_map (measurable_initialVector n) hXm]
+    exact hcons n k hkn
+  have hc : IsClosed {q : ProbabilityMeasure (ℕ→E) |
+      q.map (initialVector n)=(ν n).map (initialVector n)} :=
+    isClosed_eq (ProbabilityMeasure.continuous_map (continuous_initialVector n)) continuous_const
+  have hq := hc.mem_of_mapClusterPt hQ (Filter.eventually_atTop.mpr ⟨n,he⟩)
+  have hqm := congrArg ProbabilityMeasure.toMeasure hq
+  change Q.toMeasure.map (initialVector n)=
+    ((P n).map (fun ω : Ω=>fun i=>X i ω)).map (initialVector n) at hqm
+  rw [Measure.map_map (measurable_initialVector n) hXm] at hqm
+  exact hqm
+
+ 
+lemma lintegral_initialVector {E Ω : Type*} [MeasurableSpace E] [MeasurableSpace Ω]
+    (Q : Measure (ℕ→E)) (P : Measure Ω) (X : ℕ→Ω→E)
+    (hX : ∀n,Measurable (X n)) (n : ℕ)
+    (hQ : Q.map (initialVector n)=P.map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω))
+    (f : (Fin (n+1)→E)→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻x,f (initialVector n x)∂Q)=∫⁻ω,f (fun i=>X i ω)∂P := by
+  rw [←lintegral_map hf (measurable_initialVector n),hQ,
+    lintegral_map hf (Measurable.of_eval (fun i=>hX i))]
+
+lemma lintegral_coordinate {E Ω : Type*} [MeasurableSpace E] [MeasurableSpace Ω]
+    (Q : Measure (ℕ→E)) (P : Measure Ω) (X : ℕ→Ω→E)
+    (hX : ∀n,Measurable (X n)) (n : ℕ)
+    (hQ : Q.map (initialVector n)=P.map (fun ω : Ω=>fun i : Fin (n+1)=>X i ω))
+    (f : E→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻x,f (x n)∂Q)=∫⁻ω,f (X n ω)∂P :=
+  lintegral_initialVector Q P X hX n hQ (fun x=>f (x ⟨n,by omega⟩))
+    (hf.comp (measurable_pi_apply _))
+
+end SLEExactGauge.CompactSequence
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/OptionalEstimates.lean b/lean/OAI/Probability/SLEGauge/OptionalEstimates.lean
index c36bfd8603bdedc5bf72f55a580d3a638e4304e6..974682bbafdead4409b11d614da94b2bda6ed63b 100644
--- a/lean/OAI/Probability/SLEGauge/OptionalEstimates.lean
+++ b/lean/OAI/Probability/SLEGauge/OptionalEstimates.lean
@@ -1,452 +1,452 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ProductGenerators
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator Metric
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
- 
-
-theorem discounted_regime_optional {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (Z : ℝ≥0 → Driver → RegimeState m) (hZc : ∀ b, Continuous (fun t => Z t b))
-    (hZm : ∀ t, Measurable[rawFiltration t] (Z t))
-    (A : RegimeState m → RegimeState m) (hAc : Continuous A)
-    (K : ℝ) (hK : 0≤K) (hAb : ∀ z, ‖A z‖≤K)
-    (hZeq : ∀ (u h : ℝ≥0) b, Z (u+h) b = Z u b + increment u h b • (multiNoise m,0) +
-      ∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (Z t.toNNReal b))
-    (S T : Driver → ℝ≥0) (B N : ℝ≥0) (η : ℝ) (hη : 0<η)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
-    (hsafe : ∀ b t, S b≤t → t≤T b → ‖Z t b‖≤N ∧
-      ∀ i∈s, η≤Y ((Z t b).1 i) ∧ physicalDelta a (cap i) (σ i) (off i) ((Z t b).1 i)≤1)
-    (hA : ∀ b t, S b≤t → t≤T b →
-      (A (Z t b)).2 = regimeDiscount s a ε cap σ off (Z t b).1 ∧
-      ∀ i∈s, (A (Z t b)).1 i = physicalDrift a ((Z t b).1 i))
-    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    ∫ b in U, discountedProduct s a cap σ off (Z (T b) b) ∂W ≤
-      ∫ b in U, discountedProduct s a cap σ off (Z (S b) b) ∂W := by
-
-  let f := regimeCutoff s a η N cap σ off
-  have he : ‖(multiNoise m, (0 : ℝ))‖≤1 := by
-    apply max_le _ (by norm_num)
-    apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤1)).mpr
-    intro i
-    simp [multiNoise,noise,Prod.norm_def]
-  have hg (b : Driver) (t : ℝ≥0) (hts : S b≤t) (htt : t≤T b) :
-      generator f A (multiNoise m,0) (Z t b)≤0 := by
-    have hx := hsafe b t hts htt
-    have hnear := regimeCutoff_eq_near s a η hη N cap σ off (Z t b) hx.1
-      (fun i hi => (hx.2 i hi).1)
-    have hn : (fun w => fderiv ℝ f w (multiNoise m,0)) =ᶠ[𝓝 (Z t b)]
-        (fun w => fderiv ℝ (discountedProduct s a cap σ off) w (multiNoise m,0)) :=
-      hnear.fderiv.mono fun w hw => congrArg (fun D : RegimeState m →L[ℝ] ℝ => D (multiNoise m,0)) hw
-    change (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ f w (multiNoise m,0)) (Z t b) (multiNoise m,0)+
-      fderiv ℝ f (Z t b) (A (Z t b))≤0
-    rw [hn.fderiv_eq,hnear.fderiv_eq]
-    exact discountedProduct_nonpos s ha hε hsize cap σ off (Z t b) (A (Z t b))
-      (fun i hi => hη.trans_le (hx.2 i hi).1) (fun i hi => (hx.2 i hi).2) hσ (hA b t hts htt)
-  have h := compact_test_optional_inequality W hW Z hZc hZm A hAc K hK hAb
-    (multiNoise m,0) he hZeq f (regimeCutoff_smooth s a η hη N cap σ off)
-    (regimeCutoff_compact s a η N cap σ off) S T B hS hT hST hTB hg U hU
-  have hfS (b : Driver) : f (Z (S b) b)=discountedProduct s a cap σ off (Z (S b) b) :=
-    (regimeCutoff_eq_near s a η hη N cap σ off _ (hsafe b _ le_rfl (hST b)).1
-      (fun i hi => ((hsafe b _ le_rfl (hST b)).2 i hi).1)).eq_of_nhds
-  have hfT (b : Driver) : f (Z (T b) b)=discountedProduct s a cap σ off (Z (T b) b) :=
-    (regimeCutoff_eq_near s a η hη N cap σ off _ (hsafe b _ (hST b) le_rfl).1
-      (fun i hi => ((hsafe b _ (hST b) le_rfl).2 i hi).1)).eq_of_nhds
-  simpa only [hfS,hfT] using h
-end ExactSLEGreen
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem compact_test_optional_on_event
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
-    (S T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
-    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U)
-    (hneg : ∀ b∈U, ∀ t, S b ≤ t → t ≤ T b → generator f A e (X t b) ≤ 0) :
-    ∫ b in U, f (X (T b) b) ∂W ≤ ∫ b in U, f (X (S b) b) ∂W := by
-  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
-    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
-  have hM : Martingale M rawFiltration W :=
-    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
-  have hgc (b : Driver) : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
-    (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
-  have hmc (b : Driver) : Continuous (fun t => M t b) :=
-    (hf.continuous.comp (hXc b)).sub
-      ((intervalIntegral.continuous_primitive (fun s t => (hgc b).intervalIntegrable s t) 0).comp NNReal.continuous_coe)
-  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
-  have hmb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
-    calc
-      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
-      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
-        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
-          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
-            (fun u _ => hgb (X u.toNNReal b)))
-      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
-  have hFa : StronglyAdapted rawFiltration (fun t b => f (X t b)) := fun t =>
-    (hf.continuous.measurable.comp (hXm t)).stronglyMeasurable
-  have hFm (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
-      Measurable (fun b => f (X (R b) b)) :=
-    (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
-      (fun b => hf.continuous.comp (hXc b))) hR).mono hR.measurableSpace_le le_rfl
-  have hFi (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
-      Integrable (fun b => f (X (R b) b)) (W.restrict U) :=
-    Integrable.of_bound (hFm R hR).aestronglyMeasurable C (Filter.Eventually.of_forall fun b => hfb _)
-  have hMi (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) (hRB : ∀ b, R b≤B) :
-      Integrable (fun b => M (R b) b) (W.restrict U) :=
-    Integrable.of_bound (ExactSLEOptional.stopped_measurable rawFiltration hM hmc R hR).aestronglyMeasurable
-      (C+D*(B+1)) (Filter.Eventually.of_forall fun b => hmb _ ((hRB b).trans (le_add_of_nonneg_right zero_le_one)) b)
-  have hpw (b : Driver) (hb : b∈U) : f (X (T b) b) - f (X (S b) b) ≤ M (T b) b - M (S b) b := by
-    have hi := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
-      ((hgc b).intervalIntegrable 0 (S b)) ((hgc b).intervalIntegrable (S b) (T b))
-    have hn : (∫ u in (S b : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) ≤ 0 := by
-      rw [intervalIntegral.integral_of_le (by exact_mod_cast hST b)]
-      apply setIntegral_nonpos measurableSet_Ioc
-      intro u hu
-      apply hneg b hb u.toNNReal
-      · exact_mod_cast (show (S b : ℝ) ≤ (u.toNNReal : ℝ) by rw [Real.coe_toNNReal _ ((S b).coe_nonneg.trans hu.1.le)]; exact hu.1.le)
-      · exact (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    dsimp only [M]
-    linarith
-  have hcomp := integral_mono_ae ((hFi T hT).sub (hFi S hS))
-    ((hMi T hT hTB).sub (hMi S hS (fun b => (hST b).trans (hTB b))))
-    ((ae_restrict_mem (hS.measurableSpace_le U hU)).mono fun b hb => hpw b hb)
-  simp only [Pi.sub_apply] at hcomp
-  rw [integral_sub (hFi T hT) (hFi S hS),
-    integral_sub (hMi T hT hTB) (hMi S hS (fun b => (hST b).trans (hTB b)))] at hcomp
-  have heq := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hmc B
-    (C+D*(B+1)) hmb S T hS hT hST hTB U hU
-  linarith
-end ExactSLEGenerator
-
-namespace ExactSLEPasting
-open MeasureTheory
-variable {Ω : Type*} {m m₀ : MeasurableSpace Ω} {μ : Measure[m₀] Ω} [IsFiniteMeasure μ]
- 
-
-lemma integral_weighted_le (hm : m ≤ m₀) {f g w : Ω → ℝ}
-    (hf : Integrable f μ) (hg : Integrable g μ)
-    (hw : StronglyMeasurable[m] w) (hw0 : ∀ ω, 0≤w ω)
-    {C : ℝ} (hwb : ∀ ω, ‖w ω‖≤C)
-    (h : ∀ U, MeasurableSet[m] U → ∫ ω in U, f ω ∂μ ≤ ∫ ω in U, g ω ∂μ) :
-    ∫ ω, w ω*f ω ∂μ ≤ ∫ ω, w ω*g ω ∂μ := by
-  have hfm : StronglyMeasurable[m] (μ[f|m]) := stronglyMeasurable_condExp
-  have hgm : StronglyMeasurable[m] (μ[g|m]) := stronglyMeasurable_condExp
-  have hf' : Integrable (μ[f|m]) (μ.trim hm) := Integrable.trim hm integrable_condExp hfm
-  have hg' : Integrable (μ[g|m]) (μ.trim hm) := Integrable.trim hm integrable_condExp hgm
-  have hle : μ[f|m] ≤ᵐ[μ] μ[g|m] := by
-    apply ae_of_ae_trim hm
-    apply ae_le_of_forall_setIntegral_le hf' hg'
-    intro U hU _
-    rw [← setIntegral_trim hm hfm hU,← setIntegral_trim hm hgm hU,
-      setIntegral_condExp hm hf hU,setIntegral_condExp hm hg hU]
-    exact h U hU
-  have hwm : AEStronglyMeasurable w μ := (hw.mono hm).aestronglyMeasurable
-  have hwf := hf.bdd_mul hwm (Filter.Eventually.of_forall hwb)
-  have hwg := hg.bdd_mul hwm (Filter.Eventually.of_forall hwb)
-  have hwf' := (integrable_condExp (f := f) (m := m) (μ := μ)).bdd_mul hwm (Filter.Eventually.of_forall hwb)
-  have hwg' := (integrable_condExp (f := g) (m := m) (μ := μ)).bdd_mul hwm (Filter.Eventually.of_forall hwb)
-  have hEqf : ∫ ω, w ω*f ω ∂μ = ∫ ω, w ω*(μ[f|m]) ω ∂μ := by
-    rw [← integral_condExp hm]
-    exact integral_congr_ae (condExp_mul_of_stronglyMeasurable_left hw hwf hf)
-  have hEqg : ∫ ω, w ω*g ω ∂μ = ∫ ω, w ω*(μ[g|m]) ω ∂μ := by
-    rw [← integral_condExp hm]
-    exact integral_congr_ae (condExp_mul_of_stronglyMeasurable_left hw hwg hg)
-  rw [hEqf,hEqg]
-  exact integral_mono_ae hwf' hwg' (hle.mono fun ω hω => mul_le_mul_of_nonneg_left hω (hw0 ω))
-end ExactSLEPasting
-namespace ExactSLEPasting
-open MeasureTheory Set
-variable {Ω : Type*} {m m₀ : MeasurableSpace Ω} {μ : Measure[m₀] Ω} [IsFiniteMeasure μ]
-lemma integral_weighted_le_on_event (hm : m ≤ m₀) {f g w : Ω → ℝ} {E : Set Ω}
-    (hE : MeasurableSet[m] E) (hf : Integrable f μ) (hg : Integrable g μ)
-    (hw : StronglyMeasurable[m] w) (hw0 : ∀ ω, 0≤w ω)
-    {C : ℝ} (hwb : ∀ ω, ‖w ω‖≤C) (hws : ∀ ω∉E, w ω=0)
-    (h : ∀ U, MeasurableSet[m] U → ∫ ω in E∩U, f ω ∂μ ≤ ∫ ω in E∩U, g ω ∂μ) :
-    ∫ ω, w ω*f ω ∂μ ≤ ∫ ω, w ω*g ω ∂μ := by
-  have hE' : MeasurableSet[m₀] E := hm E hE
-  have hcomp : ∫ ω, w ω*E.indicator f ω ∂μ ≤ ∫ ω, w ω*E.indicator g ω ∂μ := by
-    apply integral_weighted_le hm (hf.indicator hE') (hg.indicator hE') hw hw0 hwb
-    intro U hU
-    simp_rw [integral_indicator hE',Measure.restrict_restrict hE']
-    exact h U hU
-  have he (q : Ω → ℝ) (ω : Ω) : w ω*E.indicator q ω=w ω*q ω := by
-    by_cases hq : ω∈E
-    · simp [hq]
-    · simp [hq,hws ω hq]
-  simpa only [he] using hcomp
-end ExactSLEPasting
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
- 
-
-theorem discounted_regime_weighted_optional {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (Z : ℝ≥0 → Driver → RegimeState m) (hZc : ∀ b, Continuous (fun t => Z t b))
-    (hZm : ∀ t, Measurable[rawFiltration t] (Z t))
-    (A : RegimeState m → RegimeState m) (hAc : Continuous A)
-    (K : ℝ) (hK : 0≤K) (hAb : ∀ z, ‖A z‖≤K)
-    (hZeq : ∀ (u h : ℝ≥0) b, Z (u+h) b = Z u b + increment u h b • (multiNoise m,0) +
-      ∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (Z t.toNNReal b))
-    (S T : Driver → ℝ≥0) (B N : ℝ≥0) (η : ℝ) (hη : 0<η)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
-    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
-    (w : Driver → ℝ) (hw : StronglyMeasurable[hS.measurableSpace] w)
-    (hw0 : ∀ b, 0≤w b) (C : ℝ) (hwb : ∀ b, ‖w b‖≤C)
-    (hws : ∀ b∉E, w b=0)
-    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b → ‖Z t b‖≤N ∧
-      ∀ i∈s, η≤Y ((Z t b).1 i) ∧ physicalDelta a (cap i) (σ i) (off i) ((Z t b).1 i)≤1)
-    (hA : ∀ b∈E, ∀ t, S b≤t → t≤T b →
-      (A (Z t b)).2 = regimeDiscount s a ε cap σ off (Z t b).1 ∧
-      ∀ i∈s, (A (Z t b)).1 i = physicalDrift a ((Z t b).1 i))
-    : ∫ b, w b*discountedProduct s a cap σ off (Z (T b) b) ∂W ≤
-      ∫ b, w b*discountedProduct s a cap σ off (Z (S b) b) ∂W := by
-  let f := regimeCutoff s a η N cap σ off
-  have he : ‖(multiNoise m, (0 : ℝ))‖≤1 := by
-    apply max_le _ (by norm_num)
-    apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤1)).mpr
-    intro i
-    simp [multiNoise,noise,Prod.norm_def]
-  have hg (b : Driver) (hb : b∈E) (t : ℝ≥0) (hts : S b≤t) (htt : t≤T b) :
-      generator f A (multiNoise m,0) (Z t b)≤0 := by
-    have hx := hsafe b hb t hts htt
-    have hnear := regimeCutoff_eq_near s a η hη N cap σ off (Z t b) hx.1
-      (fun i hi => (hx.2 i hi).1)
-    have hn : (fun w => fderiv ℝ f w (multiNoise m,0)) =ᶠ[𝓝 (Z t b)]
-        (fun w => fderiv ℝ (discountedProduct s a cap σ off) w (multiNoise m,0)) :=
-      hnear.fderiv.mono fun w hw => congrArg (fun D : RegimeState m →L[ℝ] ℝ => D (multiNoise m,0)) hw
-    change (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ f w (multiNoise m,0)) (Z t b) (multiNoise m,0)+
-      fderiv ℝ f (Z t b) (A (Z t b))≤0
-    rw [hn.fderiv_eq,hnear.fderiv_eq]
-    exact discountedProduct_nonpos s ha hε hsize cap σ off (Z t b) (A (Z t b))
-      (fun i hi => hη.trans_le (hx.2 i hi).1) (fun i hi => (hx.2 i hi).2) hσ (hA b hb t hts htt)
-  have hfc := regimeCutoff_smooth s a η hη N cap σ off
-  have hfcompact := regimeCutoff_compact s a η N cap σ off
-  have hFa : StronglyAdapted rawFiltration (fun t b => f (Z t b)) := fun t =>
-    (hfc.continuous.measurable.comp (hZm t)).stronglyMeasurable
-  obtain ⟨D,hDb⟩ := hfcompact.exists_bound_of_continuous hfc.continuous
-  have hFi (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
-      Integrable (fun b => f (Z (R b) b)) W := by
-    have hm := (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
-      (fun b => hfc.continuous.comp (hZc b))) hR).mono hR.measurableSpace_le le_rfl
-    exact Integrable.of_bound hm.aestronglyMeasurable D (Filter.Eventually.of_forall fun b => hDb _)
-  have hcomp : ∫ b, w b*f (Z (T b) b) ∂W ≤ ∫ b, w b*f (Z (S b) b) ∂W := by
-    apply ExactSLEPasting.integral_weighted_le_on_event hS.measurableSpace_le hE
-      (hFi T hT) (hFi S hS) hw hw0 hwb hws
-    intro U hU
-    apply compact_test_optional_on_event W hW Z hZc hZm A hAc K hK hAb
-      (multiNoise m,0) he hZeq f hfc hfcompact S T B hS hT hST hTB
-      (E∩U) (hE.inter hU)
-    intro b hb t hts htt
-    exact hg b hb.1 t hts htt
-  have heq (b : Driver) (t : ℝ≥0) (hts : S b≤t) (htt : t≤T b) :
-      w b*f (Z t b)=w b*discountedProduct s a cap σ off (Z t b) := by
-    by_cases hb : b∈E
-    · change w b*regimeCutoff s a η N cap σ off (Z t b)=_
-      rw [(regimeCutoff_eq_near s a η hη N cap σ off _ (hsafe b hb t hts htt).1
-        (fun i hi => ((hsafe b hb t hts htt).2 i hi).1)).eq_of_nhds]
-    · simp [hws b hb]
-  simpa only [heq _ _ (hST _) le_rfl,heq _ _ le_rfl (hST _)] using hcomp
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLECalculus ExactSLEFiniteTilt Metric
-noncomputable def scalarClamp (L x : ℝ) : ℝ := max (-L) (min L x)
-lemma scalarClamp_continuous (L : ℝ) : Continuous (scalarClamp L) :=
-  continuous_const.max (continuous_const.min continuous_id)
-lemma scalarClamp_bounds (L x : ℝ) (hL : 0≤L) : -L ≤ scalarClamp L x ∧ scalarClamp L x ≤ L := by
-  exact ⟨le_max_left _ _,max_le (by linarith) (min_le_left _ _)⟩
-lemma scalarClamp_eq (L x : ℝ) (hx : |x|≤L) : scalarClamp L x=x := by
-  rw [scalarClamp,min_eq_right (abs_le.mp hx).2,max_eq_right (abs_le.mp hx).1]
-lemma scalarClamp_idem (L x : ℝ) (hL : 0≤L) : scalarClamp L (scalarClamp L x)=scalarClamp L x :=
-  scalarClamp_eq L _ (abs_le.mpr (scalarClamp_bounds L x hL))
-noncomputable def regimeClamp (η : ℝ) (N : ℝ≥0) (v : State) : State :=
-  (⟨scalarClamp N (X v),max η (min ((N:ℝ)+η+1) (Y v))⟩,scalarClamp N (J v))
-lemma regimeClamp_continuous (η : ℝ) (N : ℝ≥0) : Continuous (regimeClamp η N) := by
-  have hc : Continuous (fun v : State =>
-      ((scalarClamp N (X v) : ℂ) + (Complex.ofReal (max η (min ((N:ℝ)+η+1) (Y v))))*Complex.I,
-        scalarClamp N (J v))) :=
-    ((Complex.continuous_ofReal.comp ((scalarClamp_continuous N).comp X.continuous)).add
-      ((Complex.continuous_ofReal.comp
-        (continuous_const.max (continuous_const.min Y.continuous))).mul_const Complex.I)).prodMk
-          ((scalarClamp_continuous N).comp J.continuous)
-  convert hc using 1
-  funext v
-  apply Prod.ext
-  · apply Complex.ext <;> simp [regimeClamp]
-  · rfl
-lemma regimeClamp_height (η : ℝ) (N : ℝ≥0) (v : State) : η≤Y (regimeClamp η N v) :=
-  le_max_left _ _
-lemma regimeClamp_idem (η : ℝ) (N : ℝ≥0) (v : State) :
-    regimeClamp η N (regimeClamp η N v)=regimeClamp η N v := by
-  have h : η≤(N:ℝ)+η+1 := by linarith [N.coe_nonneg]
-  apply Prod.ext
-  · apply Complex.ext
-    · exact scalarClamp_idem _ _ N.coe_nonneg
-    · change max η (min ((N:ℝ)+η+1) (max η (min ((N:ℝ)+η+1) (Y v)))) = _
-      rw [min_eq_right (max_le h (min_le_left _ _)),max_eq_right (le_max_left _ _)]
-      rfl
-  · exact scalarClamp_idem _ _ N.coe_nonneg
-lemma regimeClamp_bound (η : ℝ) (N : ℝ≥0) (v : State) :
-    ‖regimeClamp η N v‖≤2*(N:ℝ)+|η|+1 := by
-  have hr : |scalarClamp N (X v)|≤N := abs_le.mpr (scalarClamp_bounds N _ N.coe_nonneg)
-  have hj : |scalarClamp N (J v)|≤N := abs_le.mpr (scalarClamp_bounds N _ N.coe_nonneg)
-  have hy : |max η (min ((N:ℝ)+η+1) (Y v))|≤(N:ℝ)+|η|+1 := by
-    apply abs_le.mpr
-    constructor
-    · calc
-        -((N:ℝ)+|η|+1) ≤ -|η| := by linarith [N.coe_nonneg]
-        _ ≤ η := neg_abs_le η
-        _ ≤ _ := le_max_left _ _
-    · apply max_le
-      · exact (le_abs_self η).trans (by linarith [N.coe_nonneg])
-      · exact (min_le_left _ _).trans (by linarith [le_abs_self η])
-  rw [Prod.norm_def]
-  apply max_le
-  · exact (Complex.norm_le_abs_re_add_abs_im _).trans (by
-      change |scalarClamp N (X v)|+|max η (min ((N:ℝ)+η+1) (Y v))|≤_
-      linarith)
-  · change ‖scalarClamp N (J v)‖≤_
-    rw [Real.norm_eq_abs]
-    linarith [abs_nonneg η,N.coe_nonneg]
-lemma regimeClamp_eq (η : ℝ) (N : ℝ≥0) (v : State) (hη : 0<η)
-    (hn : ‖v‖≤N) (hy : η≤Y v) : regimeClamp η N v=v := by
-  have hx : |X v|≤N := (Complex.abs_re_le_norm v.1).trans ((le_max_left ‖v.1‖ ‖v.2‖).trans hn)
-  have hj : |J v|≤N := (le_max_right ‖v.1‖ ‖v.2‖).trans hn
-  have hu : Y v≤(N:ℝ)+η+1 := by
-    have h : Y v ≤ N := (Complex.im_le_norm v.1).trans ((le_max_left ‖v.1‖ ‖v.2‖).trans hn)
-    linarith
-  apply Prod.ext
-  · apply Complex.ext
-    · exact scalarClamp_eq _ _ hx
-    · change max η (min ((N:ℝ)+η+1) (Y v))=Y v
-      rw [min_eq_right hu,max_eq_right hy]
-  · exact scalarClamp_eq _ _ hj
-
-lemma regimeDiscount_continuousAt {m : ℕ} (s : Finset (Fin m)) (a ε : ℝ)
-    (cap σ off : Fin m → ℝ) (v : Fin m → State) (hy : ∀ i∈s, 0<Y (v i)) :
-    ContinuousAt (regimeDiscount s a ε cap σ off) v := by
-  have hc (i) (hi : i∈s) : ContinuousAt (fun w : Fin m → State => physicalClock a (cap i) (w i)) v :=
-    (physicalClock_smooth a (cap i) (v i) (hy i hi)).continuousAt.comp_of_eq (continuous_apply i).continuousAt rfl
-  have hd (i) (hi : i∈s) : ContinuousAt (fun w : Fin m → State => physicalDelta a (cap i) (σ i) (off i) (w i)) v :=
-    (continuousAt_const.add (continuousAt_const.mul (hc i hi))).rexp
-  have hV (i) (hi : i∈s) : ContinuousAt (fun w : Fin m → State => physicalV (w i)) v :=
-    ((Y.continuous.continuousAt.div ((X.continuous.pow 2).add (Y.continuous.pow 2)).continuousAt (Q_pos (hy i hi)).ne').comp_of_eq
-      (continuous_apply i).continuousAt rfl)
-  unfold regimeDiscount
-  apply ContinuousAt.add
-  · exact continuousAt_const.mul (tendsto_finsetSum s fun i hi => (hd i hi).mul ((hV i hi).pow 2))
-  · apply ContinuousAt.mul continuousAt_const
-    apply tendsto_finsetSum s
-    intro i hi
-    apply tendsto_finsetSum (s.erase i)
-    intro j hj
-    exact ((((hd i hi).mul (hd j (Finset.mem_of_mem_erase hj))).rpow_const
-      (p := -regM a) (Or.inl (mul_pos (Real.exp_pos _) (Real.exp_pos _)).ne')).mul
-        (hV i hi)).mul (hV j (Finset.mem_of_mem_erase hj))
-
-noncomputable def regimeDiscountCut {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) (v : Fin m → State) : ℝ :=
-  regimeDiscount s a ε cap σ off (fun i => regimeClamp η N (v i))
-lemma regimeDiscountCut_continuous {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) : Continuous (regimeDiscountCut s a ε η N cap σ off) := by
-  apply continuous_iff_continuousAt.mpr
-  intro v
-  exact (regimeDiscount_continuousAt s a ε cap σ off _
-    (fun i _ => hη.trans_le (regimeClamp_height η N (v i)) )).comp_of_eq
-      ((continuous_pi fun i => (regimeClamp_continuous η N).comp (continuous_apply i)).continuousAt) rfl
-lemma regimeDiscountCut_bound {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) : ∃ K : ℝ, 0≤K ∧ ∀ v, ‖regimeDiscountCut s a ε η N cap σ off v‖≤K := by
-  have hc := regimeDiscountCut_continuous s a ε η hη N cap σ off
-  obtain ⟨K,hK⟩ := (isCompact_closedBall (0 : Fin m → State) (2*(N:ℝ)+|η|+1)).exists_bound_of_continuousOn hc.continuousOn
-  refine ⟨max K 0,le_max_right _ _,fun v => ?_⟩
-  have hv : (fun i => regimeClamp η N (v i))∈closedBall (0 : Fin m → State) (2*(N:ℝ)+|η|+1) := by
-    rw [mem_closedBall_zero_iff]
-    apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
-    intro i
-    exact regimeClamp_bound η N (v i)
-  have h := hK (fun i => regimeClamp η N (v i)) hv
-  have he : regimeDiscountCut s a ε η N cap σ off (fun i => regimeClamp η N (v i))=
-      regimeDiscountCut s a ε η N cap σ off v := by
-    simp only [regimeDiscountCut,regimeClamp_idem]
-  exact (he ▸ h).trans (le_max_left _ _)
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLECalculus ExactSLEFiniteTilt
-noncomputable def vectorState {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ)
-    (t : ℝ≥0) (b : Driver) : Fin m → State := fun i => localState a η hη (z i) t b
-lemma vectorState_continuous {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ) (b : Driver) :
-    Continuous (fun t => vectorState a η hη z t b) :=
-  continuous_pi fun i => localState_continuous a η hη (z i) b
-lemma vectorState_adapted {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ) (t : ℝ≥0) :
-    Measurable[rawFiltration t] (vectorState a η hη z t) := by
-  let : MeasurableSpace Driver := rawFiltration t
-  exact Measurable.of_eval fun index => localState_adapted a η hη (z index) t
-noncomputable def regimeDrift {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) (v : RegimeState m) : RegimeState m :=
-  (fun i => cutoffDrift a η (v.1 i),regimeDiscountCut s a ε η N cap σ off v.1)
-lemma regimeDrift_continuous {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) : Continuous (regimeDrift s a ε η N cap σ off) :=
-  (continuous_pi fun i => (cutoffDrift_lipschitz a η hη).continuous.comp
-    ((continuous_apply i).comp continuous_fst)).prodMk
-      ((regimeDiscountCut_continuous s a ε η hη N cap σ off).comp continuous_fst)
-lemma regimeDrift_bound {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) : ∃ K : ℝ, 0≤K ∧ ∀ v, ‖regimeDrift s a ε η N cap σ off v‖≤K := by
-  obtain ⟨K,hK,hKb⟩ := regimeDiscountCut_bound s a ε η hη N cap σ off
-  refine ⟨max (max (|a|/η) (|a|/η^2)) K,(le_max_right _ _).trans' hK,fun v => ?_⟩
-  apply max_le
-  · apply (pi_norm_le_iff_of_nonneg ((le_max_left _ _).trans' (by positivity))).mpr
-    intro i
-    exact (cutoffDrift_bound a η hη _).trans (le_max_left _ _)
-  · exact (hKb _).trans (le_max_right _ _)
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ProductGenerators
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator Metric
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
+ 
+
+theorem discounted_regime_optional {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (Z : ℝ≥0 → Driver → RegimeState m) (hZc : ∀ b, Continuous (fun t => Z t b))
+    (hZm : ∀ t, Measurable[rawFiltration t] (Z t))
+    (A : RegimeState m → RegimeState m) (hAc : Continuous A)
+    (K : ℝ) (hK : 0≤K) (hAb : ∀ z, ‖A z‖≤K)
+    (hZeq : ∀ (u h : ℝ≥0) b, Z (u+h) b = Z u b + increment u h b • (multiNoise m,0) +
+      ∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (Z t.toNNReal b))
+    (S T : Driver → ℝ≥0) (B N : ℝ≥0) (η : ℝ) (hη : 0<η)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
+    (hsafe : ∀ b t, S b≤t → t≤T b → ‖Z t b‖≤N ∧
+      ∀ i∈s, η≤Y ((Z t b).1 i) ∧ physicalDelta a (cap i) (σ i) (off i) ((Z t b).1 i)≤1)
+    (hA : ∀ b t, S b≤t → t≤T b →
+      (A (Z t b)).2 = regimeDiscount s a ε cap σ off (Z t b).1 ∧
+      ∀ i∈s, (A (Z t b)).1 i = physicalDrift a ((Z t b).1 i))
+    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    ∫ b in U, discountedProduct s a cap σ off (Z (T b) b) ∂W ≤
+      ∫ b in U, discountedProduct s a cap σ off (Z (S b) b) ∂W := by
+
+  let f := regimeCutoff s a η N cap σ off
+  have he : ‖(multiNoise m, (0 : ℝ))‖≤1 := by
+    apply max_le _ (by norm_num)
+    apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤1)).mpr
+    intro i
+    simp [multiNoise,noise,Prod.norm_def]
+  have hg (b : Driver) (t : ℝ≥0) (hts : S b≤t) (htt : t≤T b) :
+      generator f A (multiNoise m,0) (Z t b)≤0 := by
+    have hx := hsafe b t hts htt
+    have hnear := regimeCutoff_eq_near s a η hη N cap σ off (Z t b) hx.1
+      (fun i hi => (hx.2 i hi).1)
+    have hn : (fun w => fderiv ℝ f w (multiNoise m,0)) =ᶠ[𝓝 (Z t b)]
+        (fun w => fderiv ℝ (discountedProduct s a cap σ off) w (multiNoise m,0)) :=
+      hnear.fderiv.mono fun w hw => congrArg (fun D : RegimeState m →L[ℝ] ℝ => D (multiNoise m,0)) hw
+    change (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ f w (multiNoise m,0)) (Z t b) (multiNoise m,0)+
+      fderiv ℝ f (Z t b) (A (Z t b))≤0
+    rw [hn.fderiv_eq,hnear.fderiv_eq]
+    exact discountedProduct_nonpos s ha hε hsize cap σ off (Z t b) (A (Z t b))
+      (fun i hi => hη.trans_le (hx.2 i hi).1) (fun i hi => (hx.2 i hi).2) hσ (hA b t hts htt)
+  have h := compact_test_optional_inequality W hW Z hZc hZm A hAc K hK hAb
+    (multiNoise m,0) he hZeq f (regimeCutoff_smooth s a η hη N cap σ off)
+    (regimeCutoff_compact s a η N cap σ off) S T B hS hT hST hTB hg U hU
+  have hfS (b : Driver) : f (Z (S b) b)=discountedProduct s a cap σ off (Z (S b) b) :=
+    (regimeCutoff_eq_near s a η hη N cap σ off _ (hsafe b _ le_rfl (hST b)).1
+      (fun i hi => ((hsafe b _ le_rfl (hST b)).2 i hi).1)).eq_of_nhds
+  have hfT (b : Driver) : f (Z (T b) b)=discountedProduct s a cap σ off (Z (T b) b) :=
+    (regimeCutoff_eq_near s a η hη N cap σ off _ (hsafe b _ (hST b) le_rfl).1
+      (fun i hi => ((hsafe b _ (hST b) le_rfl).2 i hi).1)).eq_of_nhds
+  simpa only [hfS,hfT] using h
+end ExactSLEGreen
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem compact_test_optional_on_event
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
+    (S T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
+    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U)
+    (hneg : ∀ b∈U, ∀ t, S b ≤ t → t ≤ T b → generator f A e (X t b) ≤ 0) :
+    ∫ b in U, f (X (T b) b) ∂W ≤ ∫ b in U, f (X (S b) b) ∂W := by
+  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
+    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
+  have hM : Martingale M rawFiltration W :=
+    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
+  have hgc (b : Driver) : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
+    (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
+  have hmc (b : Driver) : Continuous (fun t => M t b) :=
+    (hf.continuous.comp (hXc b)).sub
+      ((intervalIntegral.continuous_primitive (fun s t => (hgc b).intervalIntegrable s t) 0).comp NNReal.continuous_coe)
+  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
+  have hmb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
+    calc
+      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
+      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
+        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
+          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
+            (fun u _ => hgb (X u.toNNReal b)))
+      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
+  have hFa : StronglyAdapted rawFiltration (fun t b => f (X t b)) := fun t =>
+    (hf.continuous.measurable.comp (hXm t)).stronglyMeasurable
+  have hFm (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
+      Measurable (fun b => f (X (R b) b)) :=
+    (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
+      (fun b => hf.continuous.comp (hXc b))) hR).mono hR.measurableSpace_le le_rfl
+  have hFi (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
+      Integrable (fun b => f (X (R b) b)) (W.restrict U) :=
+    Integrable.of_bound (hFm R hR).aestronglyMeasurable C (Filter.Eventually.of_forall fun b => hfb _)
+  have hMi (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) (hRB : ∀ b, R b≤B) :
+      Integrable (fun b => M (R b) b) (W.restrict U) :=
+    Integrable.of_bound (ExactSLEOptional.stopped_measurable rawFiltration hM hmc R hR).aestronglyMeasurable
+      (C+D*(B+1)) (Filter.Eventually.of_forall fun b => hmb _ ((hRB b).trans (le_add_of_nonneg_right zero_le_one)) b)
+  have hpw (b : Driver) (hb : b∈U) : f (X (T b) b) - f (X (S b) b) ≤ M (T b) b - M (S b) b := by
+    have hi := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
+      ((hgc b).intervalIntegrable 0 (S b)) ((hgc b).intervalIntegrable (S b) (T b))
+    have hn : (∫ u in (S b : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) ≤ 0 := by
+      rw [intervalIntegral.integral_of_le (by exact_mod_cast hST b)]
+      apply setIntegral_nonpos measurableSet_Ioc
+      intro u hu
+      apply hneg b hb u.toNNReal
+      · exact_mod_cast (show (S b : ℝ) ≤ (u.toNNReal : ℝ) by rw [Real.coe_toNNReal _ ((S b).coe_nonneg.trans hu.1.le)]; exact hu.1.le)
+      · exact (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    dsimp only [M]
+    linarith
+  have hcomp := integral_mono_ae ((hFi T hT).sub (hFi S hS))
+    ((hMi T hT hTB).sub (hMi S hS (fun b => (hST b).trans (hTB b))))
+    ((ae_restrict_mem (hS.measurableSpace_le U hU)).mono fun b hb => hpw b hb)
+  simp only [Pi.sub_apply] at hcomp
+  rw [integral_sub (hFi T hT) (hFi S hS),
+    integral_sub (hMi T hT hTB) (hMi S hS (fun b => (hST b).trans (hTB b)))] at hcomp
+  have heq := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hmc B
+    (C+D*(B+1)) hmb S T hS hT hST hTB U hU
+  linarith
+end ExactSLEGenerator
+
+namespace ExactSLEPasting
+open MeasureTheory
+variable {Ω : Type*} {m m₀ : MeasurableSpace Ω} {μ : Measure[m₀] Ω} [IsFiniteMeasure μ]
+ 
+
+lemma integral_weighted_le (hm : m ≤ m₀) {f g w : Ω → ℝ}
+    (hf : Integrable f μ) (hg : Integrable g μ)
+    (hw : StronglyMeasurable[m] w) (hw0 : ∀ ω, 0≤w ω)
+    {C : ℝ} (hwb : ∀ ω, ‖w ω‖≤C)
+    (h : ∀ U, MeasurableSet[m] U → ∫ ω in U, f ω ∂μ ≤ ∫ ω in U, g ω ∂μ) :
+    ∫ ω, w ω*f ω ∂μ ≤ ∫ ω, w ω*g ω ∂μ := by
+  have hfm : StronglyMeasurable[m] (μ[f|m]) := stronglyMeasurable_condExp
+  have hgm : StronglyMeasurable[m] (μ[g|m]) := stronglyMeasurable_condExp
+  have hf' : Integrable (μ[f|m]) (μ.trim hm) := Integrable.trim hm integrable_condExp hfm
+  have hg' : Integrable (μ[g|m]) (μ.trim hm) := Integrable.trim hm integrable_condExp hgm
+  have hle : μ[f|m] ≤ᵐ[μ] μ[g|m] := by
+    apply ae_of_ae_trim hm
+    apply ae_le_of_forall_setIntegral_le hf' hg'
+    intro U hU _
+    rw [← setIntegral_trim hm hfm hU,← setIntegral_trim hm hgm hU,
+      setIntegral_condExp hm hf hU,setIntegral_condExp hm hg hU]
+    exact h U hU
+  have hwm : AEStronglyMeasurable w μ := (hw.mono hm).aestronglyMeasurable
+  have hwf := hf.bdd_mul hwm (Filter.Eventually.of_forall hwb)
+  have hwg := hg.bdd_mul hwm (Filter.Eventually.of_forall hwb)
+  have hwf' := (integrable_condExp (f := f) (m := m) (μ := μ)).bdd_mul hwm (Filter.Eventually.of_forall hwb)
+  have hwg' := (integrable_condExp (f := g) (m := m) (μ := μ)).bdd_mul hwm (Filter.Eventually.of_forall hwb)
+  have hEqf : ∫ ω, w ω*f ω ∂μ = ∫ ω, w ω*(μ[f|m]) ω ∂μ := by
+    rw [← integral_condExp hm]
+    exact integral_congr_ae (condExp_mul_of_stronglyMeasurable_left hw hwf hf)
+  have hEqg : ∫ ω, w ω*g ω ∂μ = ∫ ω, w ω*(μ[g|m]) ω ∂μ := by
+    rw [← integral_condExp hm]
+    exact integral_congr_ae (condExp_mul_of_stronglyMeasurable_left hw hwg hg)
+  rw [hEqf,hEqg]
+  exact integral_mono_ae hwf' hwg' (hle.mono fun ω hω => mul_le_mul_of_nonneg_left hω (hw0 ω))
+end ExactSLEPasting
+namespace ExactSLEPasting
+open MeasureTheory Set
+variable {Ω : Type*} {m m₀ : MeasurableSpace Ω} {μ : Measure[m₀] Ω} [IsFiniteMeasure μ]
+lemma integral_weighted_le_on_event (hm : m ≤ m₀) {f g w : Ω → ℝ} {E : Set Ω}
+    (hE : MeasurableSet[m] E) (hf : Integrable f μ) (hg : Integrable g μ)
+    (hw : StronglyMeasurable[m] w) (hw0 : ∀ ω, 0≤w ω)
+    {C : ℝ} (hwb : ∀ ω, ‖w ω‖≤C) (hws : ∀ ω∉E, w ω=0)
+    (h : ∀ U, MeasurableSet[m] U → ∫ ω in E∩U, f ω ∂μ ≤ ∫ ω in E∩U, g ω ∂μ) :
+    ∫ ω, w ω*f ω ∂μ ≤ ∫ ω, w ω*g ω ∂μ := by
+  have hE' : MeasurableSet[m₀] E := hm E hE
+  have hcomp : ∫ ω, w ω*E.indicator f ω ∂μ ≤ ∫ ω, w ω*E.indicator g ω ∂μ := by
+    apply integral_weighted_le hm (hf.indicator hE') (hg.indicator hE') hw hw0 hwb
+    intro U hU
+    simp_rw [integral_indicator hE',Measure.restrict_restrict hE']
+    exact h U hU
+  have he (q : Ω → ℝ) (ω : Ω) : w ω*E.indicator q ω=w ω*q ω := by
+    by_cases hq : ω∈E
+    · simp [hq]
+    · simp [hq,hws ω hq]
+  simpa only [he] using hcomp
+end ExactSLEPasting
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
+ 
+
+theorem discounted_regime_weighted_optional {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (Z : ℝ≥0 → Driver → RegimeState m) (hZc : ∀ b, Continuous (fun t => Z t b))
+    (hZm : ∀ t, Measurable[rawFiltration t] (Z t))
+    (A : RegimeState m → RegimeState m) (hAc : Continuous A)
+    (K : ℝ) (hK : 0≤K) (hAb : ∀ z, ‖A z‖≤K)
+    (hZeq : ∀ (u h : ℝ≥0) b, Z (u+h) b = Z u b + increment u h b • (multiNoise m,0) +
+      ∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (Z t.toNNReal b))
+    (S T : Driver → ℝ≥0) (B N : ℝ≥0) (η : ℝ) (hη : 0<η)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
+    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
+    (w : Driver → ℝ) (hw : StronglyMeasurable[hS.measurableSpace] w)
+    (hw0 : ∀ b, 0≤w b) (C : ℝ) (hwb : ∀ b, ‖w b‖≤C)
+    (hws : ∀ b∉E, w b=0)
+    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b → ‖Z t b‖≤N ∧
+      ∀ i∈s, η≤Y ((Z t b).1 i) ∧ physicalDelta a (cap i) (σ i) (off i) ((Z t b).1 i)≤1)
+    (hA : ∀ b∈E, ∀ t, S b≤t → t≤T b →
+      (A (Z t b)).2 = regimeDiscount s a ε cap σ off (Z t b).1 ∧
+      ∀ i∈s, (A (Z t b)).1 i = physicalDrift a ((Z t b).1 i))
+    : ∫ b, w b*discountedProduct s a cap σ off (Z (T b) b) ∂W ≤
+      ∫ b, w b*discountedProduct s a cap σ off (Z (S b) b) ∂W := by
+  let f := regimeCutoff s a η N cap σ off
+  have he : ‖(multiNoise m, (0 : ℝ))‖≤1 := by
+    apply max_le _ (by norm_num)
+    apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤1)).mpr
+    intro i
+    simp [multiNoise,noise,Prod.norm_def]
+  have hg (b : Driver) (hb : b∈E) (t : ℝ≥0) (hts : S b≤t) (htt : t≤T b) :
+      generator f A (multiNoise m,0) (Z t b)≤0 := by
+    have hx := hsafe b hb t hts htt
+    have hnear := regimeCutoff_eq_near s a η hη N cap σ off (Z t b) hx.1
+      (fun i hi => (hx.2 i hi).1)
+    have hn : (fun w => fderiv ℝ f w (multiNoise m,0)) =ᶠ[𝓝 (Z t b)]
+        (fun w => fderiv ℝ (discountedProduct s a cap σ off) w (multiNoise m,0)) :=
+      hnear.fderiv.mono fun w hw => congrArg (fun D : RegimeState m →L[ℝ] ℝ => D (multiNoise m,0)) hw
+    change (1/2:ℝ)*fderiv ℝ (fun w => fderiv ℝ f w (multiNoise m,0)) (Z t b) (multiNoise m,0)+
+      fderiv ℝ f (Z t b) (A (Z t b))≤0
+    rw [hn.fderiv_eq,hnear.fderiv_eq]
+    exact discountedProduct_nonpos s ha hε hsize cap σ off (Z t b) (A (Z t b))
+      (fun i hi => hη.trans_le (hx.2 i hi).1) (fun i hi => (hx.2 i hi).2) hσ (hA b hb t hts htt)
+  have hfc := regimeCutoff_smooth s a η hη N cap σ off
+  have hfcompact := regimeCutoff_compact s a η N cap σ off
+  have hFa : StronglyAdapted rawFiltration (fun t b => f (Z t b)) := fun t =>
+    (hfc.continuous.measurable.comp (hZm t)).stronglyMeasurable
+  obtain ⟨D,hDb⟩ := hfcompact.exists_bound_of_continuous hfc.continuous
+  have hFi (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
+      Integrable (fun b => f (Z (R b) b)) W := by
+    have hm := (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
+      (fun b => hfc.continuous.comp (hZc b))) hR).mono hR.measurableSpace_le le_rfl
+    exact Integrable.of_bound hm.aestronglyMeasurable D (Filter.Eventually.of_forall fun b => hDb _)
+  have hcomp : ∫ b, w b*f (Z (T b) b) ∂W ≤ ∫ b, w b*f (Z (S b) b) ∂W := by
+    apply ExactSLEPasting.integral_weighted_le_on_event hS.measurableSpace_le hE
+      (hFi T hT) (hFi S hS) hw hw0 hwb hws
+    intro U hU
+    apply compact_test_optional_on_event W hW Z hZc hZm A hAc K hK hAb
+      (multiNoise m,0) he hZeq f hfc hfcompact S T B hS hT hST hTB
+      (E∩U) (hE.inter hU)
+    intro b hb t hts htt
+    exact hg b hb.1 t hts htt
+  have heq (b : Driver) (t : ℝ≥0) (hts : S b≤t) (htt : t≤T b) :
+      w b*f (Z t b)=w b*discountedProduct s a cap σ off (Z t b) := by
+    by_cases hb : b∈E
+    · change w b*regimeCutoff s a η N cap σ off (Z t b)=_
+      rw [(regimeCutoff_eq_near s a η hη N cap σ off _ (hsafe b hb t hts htt).1
+        (fun i hi => ((hsafe b hb t hts htt).2 i hi).1)).eq_of_nhds]
+    · simp [hws b hb]
+  simpa only [heq _ _ (hST _) le_rfl,heq _ _ le_rfl (hST _)] using hcomp
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLECalculus ExactSLEFiniteTilt Metric
+noncomputable def scalarClamp (L x : ℝ) : ℝ := max (-L) (min L x)
+lemma scalarClamp_continuous (L : ℝ) : Continuous (scalarClamp L) :=
+  continuous_const.max (continuous_const.min continuous_id)
+lemma scalarClamp_bounds (L x : ℝ) (hL : 0≤L) : -L ≤ scalarClamp L x ∧ scalarClamp L x ≤ L := by
+  exact ⟨le_max_left _ _,max_le (by linarith) (min_le_left _ _)⟩
+lemma scalarClamp_eq (L x : ℝ) (hx : |x|≤L) : scalarClamp L x=x := by
+  rw [scalarClamp,min_eq_right (abs_le.mp hx).2,max_eq_right (abs_le.mp hx).1]
+lemma scalarClamp_idem (L x : ℝ) (hL : 0≤L) : scalarClamp L (scalarClamp L x)=scalarClamp L x :=
+  scalarClamp_eq L _ (abs_le.mpr (scalarClamp_bounds L x hL))
+noncomputable def regimeClamp (η : ℝ) (N : ℝ≥0) (v : State) : State :=
+  (⟨scalarClamp N (X v),max η (min ((N:ℝ)+η+1) (Y v))⟩,scalarClamp N (J v))
+lemma regimeClamp_continuous (η : ℝ) (N : ℝ≥0) : Continuous (regimeClamp η N) := by
+  have hc : Continuous (fun v : State =>
+      ((scalarClamp N (X v) : ℂ) + (Complex.ofReal (max η (min ((N:ℝ)+η+1) (Y v))))*Complex.I,
+        scalarClamp N (J v))) :=
+    ((Complex.continuous_ofReal.comp ((scalarClamp_continuous N).comp X.continuous)).add
+      ((Complex.continuous_ofReal.comp
+        (continuous_const.max (continuous_const.min Y.continuous))).mul_const Complex.I)).prodMk
+          ((scalarClamp_continuous N).comp J.continuous)
+  convert hc using 1
+  funext v
+  apply Prod.ext
+  · apply Complex.ext <;> simp [regimeClamp]
+  · rfl
+lemma regimeClamp_height (η : ℝ) (N : ℝ≥0) (v : State) : η≤Y (regimeClamp η N v) :=
+  le_max_left _ _
+lemma regimeClamp_idem (η : ℝ) (N : ℝ≥0) (v : State) :
+    regimeClamp η N (regimeClamp η N v)=regimeClamp η N v := by
+  have h : η≤(N:ℝ)+η+1 := by linarith [N.coe_nonneg]
+  apply Prod.ext
+  · apply Complex.ext
+    · exact scalarClamp_idem _ _ N.coe_nonneg
+    · change max η (min ((N:ℝ)+η+1) (max η (min ((N:ℝ)+η+1) (Y v)))) = _
+      rw [min_eq_right (max_le h (min_le_left _ _)),max_eq_right (le_max_left _ _)]
+      rfl
+  · exact scalarClamp_idem _ _ N.coe_nonneg
+lemma regimeClamp_bound (η : ℝ) (N : ℝ≥0) (v : State) :
+    ‖regimeClamp η N v‖≤2*(N:ℝ)+|η|+1 := by
+  have hr : |scalarClamp N (X v)|≤N := abs_le.mpr (scalarClamp_bounds N _ N.coe_nonneg)
+  have hj : |scalarClamp N (J v)|≤N := abs_le.mpr (scalarClamp_bounds N _ N.coe_nonneg)
+  have hy : |max η (min ((N:ℝ)+η+1) (Y v))|≤(N:ℝ)+|η|+1 := by
+    apply abs_le.mpr
+    constructor
+    · calc
+        -((N:ℝ)+|η|+1) ≤ -|η| := by linarith [N.coe_nonneg]
+        _ ≤ η := neg_abs_le η
+        _ ≤ _ := le_max_left _ _
+    · apply max_le
+      · exact (le_abs_self η).trans (by linarith [N.coe_nonneg])
+      · exact (min_le_left _ _).trans (by linarith [le_abs_self η])
+  rw [Prod.norm_def]
+  apply max_le
+  · exact (Complex.norm_le_abs_re_add_abs_im _).trans (by
+      change |scalarClamp N (X v)|+|max η (min ((N:ℝ)+η+1) (Y v))|≤_
+      linarith)
+  · change ‖scalarClamp N (J v)‖≤_
+    rw [Real.norm_eq_abs]
+    linarith [abs_nonneg η,N.coe_nonneg]
+lemma regimeClamp_eq (η : ℝ) (N : ℝ≥0) (v : State) (hη : 0<η)
+    (hn : ‖v‖≤N) (hy : η≤Y v) : regimeClamp η N v=v := by
+  have hx : |X v|≤N := (Complex.abs_re_le_norm v.1).trans ((le_max_left ‖v.1‖ ‖v.2‖).trans hn)
+  have hj : |J v|≤N := (le_max_right ‖v.1‖ ‖v.2‖).trans hn
+  have hu : Y v≤(N:ℝ)+η+1 := by
+    have h : Y v ≤ N := (Complex.im_le_norm v.1).trans ((le_max_left ‖v.1‖ ‖v.2‖).trans hn)
+    linarith
+  apply Prod.ext
+  · apply Complex.ext
+    · exact scalarClamp_eq _ _ hx
+    · change max η (min ((N:ℝ)+η+1) (Y v))=Y v
+      rw [min_eq_right hu,max_eq_right hy]
+  · exact scalarClamp_eq _ _ hj
+
+lemma regimeDiscount_continuousAt {m : ℕ} (s : Finset (Fin m)) (a ε : ℝ)
+    (cap σ off : Fin m → ℝ) (v : Fin m → State) (hy : ∀ i∈s, 0<Y (v i)) :
+    ContinuousAt (regimeDiscount s a ε cap σ off) v := by
+  have hc (i) (hi : i∈s) : ContinuousAt (fun w : Fin m → State => physicalClock a (cap i) (w i)) v :=
+    (physicalClock_smooth a (cap i) (v i) (hy i hi)).continuousAt.comp_of_eq (continuous_apply i).continuousAt rfl
+  have hd (i) (hi : i∈s) : ContinuousAt (fun w : Fin m → State => physicalDelta a (cap i) (σ i) (off i) (w i)) v :=
+    (continuousAt_const.add (continuousAt_const.mul (hc i hi))).rexp
+  have hV (i) (hi : i∈s) : ContinuousAt (fun w : Fin m → State => physicalV (w i)) v :=
+    ((Y.continuous.continuousAt.div ((X.continuous.pow 2).add (Y.continuous.pow 2)).continuousAt (Q_pos (hy i hi)).ne').comp_of_eq
+      (continuous_apply i).continuousAt rfl)
+  unfold regimeDiscount
+  apply ContinuousAt.add
+  · exact continuousAt_const.mul (tendsto_finsetSum s fun i hi => (hd i hi).mul ((hV i hi).pow 2))
+  · apply ContinuousAt.mul continuousAt_const
+    apply tendsto_finsetSum s
+    intro i hi
+    apply tendsto_finsetSum (s.erase i)
+    intro j hj
+    exact ((((hd i hi).mul (hd j (Finset.mem_of_mem_erase hj))).rpow_const
+      (p := -regM a) (Or.inl (mul_pos (Real.exp_pos _) (Real.exp_pos _)).ne')).mul
+        (hV i hi)).mul (hV j (Finset.mem_of_mem_erase hj))
+
+noncomputable def regimeDiscountCut {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) (v : Fin m → State) : ℝ :=
+  regimeDiscount s a ε cap σ off (fun i => regimeClamp η N (v i))
+lemma regimeDiscountCut_continuous {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) : Continuous (regimeDiscountCut s a ε η N cap σ off) := by
+  apply continuous_iff_continuousAt.mpr
+  intro v
+  exact (regimeDiscount_continuousAt s a ε cap σ off _
+    (fun i _ => hη.trans_le (regimeClamp_height η N (v i)) )).comp_of_eq
+      ((continuous_pi fun i => (regimeClamp_continuous η N).comp (continuous_apply i)).continuousAt) rfl
+lemma regimeDiscountCut_bound {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) : ∃ K : ℝ, 0≤K ∧ ∀ v, ‖regimeDiscountCut s a ε η N cap σ off v‖≤K := by
+  have hc := regimeDiscountCut_continuous s a ε η hη N cap σ off
+  obtain ⟨K,hK⟩ := (isCompact_closedBall (0 : Fin m → State) (2*(N:ℝ)+|η|+1)).exists_bound_of_continuousOn hc.continuousOn
+  refine ⟨max K 0,le_max_right _ _,fun v => ?_⟩
+  have hv : (fun i => regimeClamp η N (v i))∈closedBall (0 : Fin m → State) (2*(N:ℝ)+|η|+1) := by
+    rw [mem_closedBall_zero_iff]
+    apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
+    intro i
+    exact regimeClamp_bound η N (v i)
+  have h := hK (fun i => regimeClamp η N (v i)) hv
+  have he : regimeDiscountCut s a ε η N cap σ off (fun i => regimeClamp η N (v i))=
+      regimeDiscountCut s a ε η N cap σ off v := by
+    simp only [regimeDiscountCut,regimeClamp_idem]
+  exact (he ▸ h).trans (le_max_left _ _)
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLECalculus ExactSLEFiniteTilt
+noncomputable def vectorState {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ)
+    (t : ℝ≥0) (b : Driver) : Fin m → State := fun i => localState a η hη (z i) t b
+lemma vectorState_continuous {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ) (b : Driver) :
+    Continuous (fun t => vectorState a η hη z t b) :=
+  continuous_pi fun i => localState_continuous a η hη (z i) b
+lemma vectorState_adapted {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ) (t : ℝ≥0) :
+    Measurable[rawFiltration t] (vectorState a η hη z t) := by
+  let : MeasurableSpace Driver := rawFiltration t
+  exact Measurable.of_eval fun index => localState_adapted a η hη (z index) t
+noncomputable def regimeDrift {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) (v : RegimeState m) : RegimeState m :=
+  (fun i => cutoffDrift a η (v.1 i),regimeDiscountCut s a ε η N cap σ off v.1)
+lemma regimeDrift_continuous {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) : Continuous (regimeDrift s a ε η N cap σ off) :=
+  (continuous_pi fun i => (cutoffDrift_lipschitz a η hη).continuous.comp
+    ((continuous_apply i).comp continuous_fst)).prodMk
+      ((regimeDiscountCut_continuous s a ε η hη N cap σ off).comp continuous_fst)
+lemma regimeDrift_bound {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) : ∃ K : ℝ, 0≤K ∧ ∀ v, ‖regimeDrift s a ε η N cap σ off v‖≤K := by
+  obtain ⟨K,hK,hKb⟩ := regimeDiscountCut_bound s a ε η hη N cap σ off
+  refine ⟨max (max (|a|/η) (|a|/η^2)) K,(le_max_right _ _).trans' hK,fun v => ?_⟩
+  apply max_le
+  · apply (pi_norm_le_iff_of_nonneg ((le_max_left _ _).trans' (by positivity))).mpr
+    intro i
+    exact (cutoffDrift_bound a η hη _).trans (le_max_left _ _)
+  · exact (hKb _).trans (le_max_right _ _)
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/OptionalGreen.lean b/lean/OAI/Probability/SLEGauge/OptionalGreen.lean
index c3df4a61f06a63784104bbef985bcd57495f5bb4..c4a08e1b5be499bbeeb9c144834e4e26611335ad 100644
--- a/lean/OAI/Probability/SLEGauge/OptionalGreen.lean
+++ b/lean/OAI/Probability/SLEGauge/OptionalGreen.lean
@@ -1,588 +1,588 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.FutureSupport
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
- 
-
-def stoppedSurvivingGreen (a : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
-  if t = ∞ then 0 else survivingGreen a t.toNNReal b z
-
- 
-def stoppedRemainingDensity (a q : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
-  if t = ∞ then 0 else remainingMassDensity a q t.toNNReal b z
-
-lemma stoppedSurvivingGreen_nonneg (a : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    0 ≤ stoppedSurvivingGreen a t b z := by
-  unfold stoppedSurvivingGreen
-  split_ifs
-  · exact le_rfl
-  · exact survivingGreen_nonneg _ _ _ _
-
-lemma stoppedRemainingDensity_nonneg (a q : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    0 ≤ stoppedRemainingDensity a q t b z := by
-  unfold stoppedRemainingDensity
-  split_ifs
-  · exact le_rfl
-  · exact remainingMassDensity_nonneg _ _ _ _ _
-
-lemma stoppedRemainingDensity_le_surviving (a q : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0 < z.im) :
-    stoppedRemainingDensity a q t b z ≤ stoppedSurvivingGreen a t b z := by
-  unfold stoppedRemainingDensity stoppedSurvivingGreen
-  split_ifs
-  · exact le_rfl
-  · exact remainingMassDensity_le_surviving a q t.toNNReal b z hz
-
-lemma stopped_remaining_to_surviving (a : ℝ) (ha : 1/4 < a) (t : ℝ≥0∞)
-    (q : ℕ → ℝ) (hq : ∀ n, 0 < q n ∧ q n ≤ 1) (hlim : Tendsto q atTop (𝓝 0))
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0 < z.im) :
-    Tendsto (fun n => stoppedRemainingDensity a (q n) t b z) atTop
-      (𝓝 (stoppedSurvivingGreen a t b z)) := by
-  unfold stoppedRemainingDensity stoppedSurvivingGreen
-  by_cases ht : t = ∞
-  · simp only [ht,ite_true]
-    exact tendsto_const_nhds
-  · simp only [ite_eq_right ht]
-    exact remaining_to_surviving a ha t.toNNReal q hq hlim b z hz
-
-lemma stoppedRemainingDensity_eq (a q : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0 < z.im) :
-    stoppedRemainingDensity a q t b z = Real.exp (-alpha a*Real.log 2) *
-      stoppedRadialRemaining a b.1 z (z.im*q) t := by
-  unfold stoppedRemainingDensity stoppedRadialRemaining remainingMassDensity
-  split_ifs <;> simp_all only [mul_zero]
-
-lemma stoppedRadialTerminal_integrable (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.1.im)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Integrable (fun b => if S b < radialTime a b.1 z.1 ρ
-      then radialWeight a b.1 z.1 ρ else 0) W := by
-  have hmS : Measurable S := hS.measurable'
-  have hmT : Measurable (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z.1 ρ) :=
-    (radialTime_stopping a (by linarith) z.1 z.2 ρ hρ hr).measurable'
-  have hm : MeasurableSet {b : ExactSLEGenerator.Driver | S b < radialTime a b.1 z.1 ρ} :=
-    measurableSet_lt hmS hmT
-  apply ((radialWeight_mean W hW a ha z.1 z.2 ρ hρ hr).1.indicator hm).congr
-  filter_upwards [] with b
-  simp only [indicator_apply,mem_ofPred_eq]
-
-lemma stoppedRemainingDensity_mean_bound (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Integrable (fun b => stoppedRemainingDensity a q (S b) b z.1) W ∧
-      (∫ b, stoppedRemainingDensity a q (S b) b z.1 ∂W) ≤ initialGreen a z := by
-  have hρ : 0 < z.1.im*q := mul_pos z.2 hq
-  have hr : z.1.im*q ≤ z.1.im := mul_le_of_le_one_right z.2.le hq1
-  obtain ⟨hi,he⟩ := radial_conditional_weight_all W hW a ha z.1 z.2
-    (z.1.im*q) hρ hr S hS univ MeasurableSet.univ
-  simp only [setIntegral_univ] at he
-  simp only [stoppedRemainingDensity_eq a q _ _ _ z.2]
-  refine ⟨hi.const_mul _, ?_⟩
-  rw [integral_const_mul, ← he]
-  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-  rw [← (radialWeight_mean W hW a ha z.1 z.2 _ hρ hr).2]
-  apply integral_mono (stoppedRadialTerminal_integrable a ha W hW z _ hρ hr S hS)
-    (radialWeight_mean W hW a ha z.1 z.2 _ hρ hr).1
-  intro b
-  dsimp only
-  split_ifs
-  · exact le_rfl
-  · exact (radialWeight_bound a ha z.1 z.2 _ hρ hr b).1
-
- 
-
-theorem survivingGreen_optional_bound (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) :
-    Integrable (fun b => stoppedSurvivingGreen a (S b) b z.1) W ∧
-      (∫ b, stoppedSurvivingGreen a (S b) b z.1 ∂W) ≤ initialGreen a z := by
-  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
-  have hq (n : ℕ) : 0 < q n ∧ q n ≤ 1 := by
-    constructor
-    · exact div_pos zero_lt_one (by positivity)
-    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
-  have hlim : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun n b =>
-    stoppedRemainingDensity a (q n) (S b) b z.1
-  let G : ExactSLEGenerator.Driver → ℝ := fun b => stoppedSurvivingGreen a (S b) b z.1
-  have hFi (n : ℕ) := (stoppedRemainingDensity_mean_bound a ha W hW z (q n)
-    (hq n).1 (hq n).2 S hS).1
-  have hFm (n : ℕ) : AEStronglyMeasurable (F n) W := (hFi n).aestronglyMeasurable
-  have hconv : ∀ᵐ b ∂W, Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
-    Eventually.of_forall (fun b => stopped_remaining_to_surviving a ha (S b) q hq hlim b z.1 z.2)
-  have hbound (n : ℕ) : eLpNorm (F n) 1 W ≤ ENNReal.ofReal (initialGreen a z) := by
-    rw [eLpNorm_one_eq_lintegral_enorm (hFm n), ← ofReal_integral_norm_eq_lintegral_enorm (hFi n)]
-    have he : (fun b => ‖F n b‖) = F n := by
-      funext b
-      exact Real.norm_of_nonneg (stoppedRemainingDensity_nonneg a (q n) (S b) b z.1)
-    rw [he]
-    exact ENNReal.ofReal_le_ofReal
-      (stoppedRemainingDensity_mean_bound a ha W hW z (q n) (hq n).1 (hq n).2 S hS).2
-  have hGm := aestronglyMeasurable_of_tendsto_ae atTop hFm hconv
-  have hGb := Lp.eLpNorm_le_of_ae_tendsto (Eventually.of_forall hbound) hFm hGm hconv
-  have hGi : Integrable G W :=
-    memLp_one_iff_integrable.mp (hGb.trans_lt ENNReal.ofReal_lt_top)
-  refine ⟨hGi, ?_⟩
-  rw [eLpNorm_one_eq_lintegral_enorm hGm, ← ofReal_integral_norm_eq_lintegral_enorm hGi] at hGb
-  have he : (fun b => ‖G b‖) = G := by
-    funext b
-    exact Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg a (S b) b z.1)
-  rw [he] at hGb
-  exact (ENNReal.ofReal_le_ofReal_iff (initialGreen_pos a z).le).mp hGb
-
-lemma stopped_remaining_integral_tendsto (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S)
-    (q : ℕ → ℝ) (hq : ∀ n, 0 < q n ∧ q n ≤ 1) (hlim : Tendsto q atTop (𝓝 0))
-    (U : Set ExactSLEGenerator.Driver) :
-    Tendsto (fun n => ∫ b in U, stoppedRemainingDensity a (q n) (S b) b z.1 ∂W)
-      atTop (𝓝 (∫ b in U, stoppedSurvivingGreen a (S b) b z.1 ∂W)) := by
-  apply tendsto_integral_of_dominated_convergence
-    (fun b => stoppedSurvivingGreen a (S b) b z.1)
-  · intro n
-    exact (stoppedRemainingDensity_mean_bound a ha W hW z (q n)
-      (hq n).1 (hq n).2 S hS).1.aestronglyMeasurable.restrict
-  · exact (survivingGreen_optional_bound a ha W hW z S hS).1.integrableOn
-  · intro n
-    filter_upwards [] with b
-    rw [Real.norm_of_nonneg (stoppedRemainingDensity_nonneg a (q n) (S b) b z.1)]
-    exact stoppedRemainingDensity_le_surviving a (q n) (S b) b z.1 z.2
-  · exact Eventually.of_forall (fun b =>
-      stopped_remaining_to_surviving a ha (S b) q hq hlim b z.1 z.2)
-
- 
-
-theorem survivingGreen_optional_comparison (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (S T : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (hT : IsStoppingTime rawFiltration T)
-    (hst : S ≤ T) (U : Set ExactSLEGenerator.Driver)
-    (hU : MeasurableSet[hS.measurableSpace] U) :
-    (∫ b in U, stoppedSurvivingGreen a (T b) b z.1 ∂W) ≤
-      ∫ b in U, stoppedSurvivingGreen a (S b) b z.1 ∂W := by
-  have hUT : MeasurableSet[hT.measurableSpace] U := (hS.measurableSpace_mono hT hst) _ hU
-  have hc (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
-      (∫ b in U, stoppedRemainingDensity a q (T b) b z.1 ∂W) ≤
-        ∫ b in U, stoppedRemainingDensity a q (S b) b z.1 ∂W := by
-    have hρ : 0 < z.1.im*q := mul_pos z.2 hq
-    have hr : z.1.im*q ≤ z.1.im := mul_le_of_le_one_right z.2.le hq1
-    have heS := (radial_conditional_weight_all W hW a ha z.1 z.2
-      (z.1.im*q) hρ hr S hS U hU).2
-    have heT := (radial_conditional_weight_all W hW a ha z.1 z.2
-      (z.1.im*q) hρ hr T hT U hUT).2
-    simp only [stoppedRemainingDensity_eq a q _ _ _ z.2,integral_const_mul]
-    rw [← heS, ← heT]
-    apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-    apply integral_mono
-      (stoppedRadialTerminal_integrable a ha W hW z _ hρ hr T hT).integrableOn
-      (stoppedRadialTerminal_integrable a ha W hW z _ hρ hr S hS).integrableOn
-    intro b
-    dsimp only
-    split_ifs with ht hs hs
-    · exact le_rfl
-    · exact False.elim (hs ((hst b).trans_lt ht))
-    · exact (radialWeight_bound a ha z.1 z.2 _ hρ hr b).1
-    · exact le_rfl
-  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
-  have hq (n : ℕ) : 0 < q n ∧ q n ≤ 1 := by
-    constructor
-    · exact div_pos zero_lt_one (by positivity)
-    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
-  have hlim : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-  exact le_of_tendsto_of_tendsto'
-    (stopped_remaining_integral_tendsto a ha W hW z T hT q hq hlim U)
-    (stopped_remaining_integral_tendsto a ha W hW z S hS q hq hlim U)
-    (fun n => hc (q n) (hq n).1 (hq n).2)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma radialRemaining_time_joint_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
-    (hq : 0 < q) (hq1 : q ≤ 1) :
-    Measurable (fun p : ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane) =>
-      radialRemaining a p.2.1.1 p.2.2.1 (p.2.2.1.im*q) p.1) := by
-  have ha0 : 0<a := by linarith
-  let Ω := ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane)
-  let z : Ω → ℂ := fun p => p.2.2.1
-  let b : Ω → ExactSLEGenerator.Driver := fun p => p.2.1
-  let t : Ω → ℝ≥0 := Prod.fst
-  have hz (ω : Ω) : 0<(z ω).im := ω.2.2.2
-  have hzM : Measurable z := measurable_subtype_coe.comp (measurable_snd.comp measurable_snd)
-  have hbM : Measurable b := measurable_fst.comp measurable_snd
-  have htM : Measurable t := measurable_fst
-  let ρ : Ω → ℝ := fun ω => (z ω).im*q
-  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
-  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
-  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
-  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
-    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
-  have hρM : Measurable ρ := hzM.im.mul_const q
-  have hδM (n : ℕ) : Measurable (δ n) := by
-    dsimp [δ,angleThreshold]
-    exact (hzM.im.div hzM.norm).div_const _
-  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
-    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
-    unfold heightFloor
-    fun_prop
-  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
-    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    unfold logCeiling heightFloor
-    fun_prop
-  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
-    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    fun_prop
-  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
-    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
-  have hTM (n : ℕ) : Measurable (T n) :=
-    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
-      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
-  have hPM : Measurable (physicalGreen a) := by
-    have hx := X.continuous.measurable
-    have hy := Y.continuous.measurable
-    have hj := J.continuous.measurable
-    unfold physicalGreen logGreen Q
-    fun_prop
-  let F (n : ℕ) (ω : Ω) := if t ω < T n ω then physicalGreen a (localState a (L n ω).ε (L n ω).hε (z ω) (t ω) (b ω)) else 0
-  have hFM (n : ℕ) : Measurable (F n) := (hPM.comp
-    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b (hεM n) hzM hbM).comp
-      (htM.prodMk measurable_id))).ite (measurableSet_lt htM (hTM n)) measurable_const
-  apply measurable_of_tendsto_metrizable hFM
-  apply tendsto_pi_nhds.mpr
-  intro ω
-  exact localizationRemaining_tendsto a ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω)
-    (fun n => L n ω) (b ω) (t ω)
-
-end ExactSLEGreen
-
-namespace ExactSLEGenerator
-
- 
-
-lemma pastDriver_stopping_measurable
-    (S : Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (fun b => pastDriver (S b).toNNReal b) := by
-  classical
-  have : PolishSpace Driver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
-  obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq (α := ℝ≥0)
-  let e : Driver → (ℕ → ℝ) := fun b n => b.1 (u n)
-  have hec : Continuous e := continuous_pi fun n =>
-    (continuous_eval_const (u n)).comp continuous_subtype_val
-  have hei : Function.Injective e := by
-    intro b c h
-    apply Subtype.ext
-    apply DFunLike.coe_injective
-    exact hu.equalizer b.1.continuous c.1.continuous h
-  apply hec.measurableEmbedding hei |>.measurable_comp_iff.mp
-  have hA : StronglyAdapted rawFiltration (fun t (b : Driver) => b.1 t) :=
-    Filtration.stronglyAdapted_natural driver_eval_measurable
-  have hP := hA.isStronglyProgressive_of_continuous (fun b => b.1.continuous)
-  have hval (n : ℕ) : Measurable[hS.measurableSpace]
-      (fun b => b.1 (ExactSLEGreen.capTime S (u n) b)) := by
-    have h := (measurable_stoppedValue hP (ExactSLEGreen.capTime_stopping S hS (u n))).mono
-      (ExactSLEGreen.stopped_cap_measurable S hS (u n)) le_rfl
-    exact h
-  have htop : MeasurableSet[hS.measurableSpace] {b | S b = ∞} :=
-    hS.measurable (measurableSet_singleton ∞)
-  let : MeasurableSpace Driver := hS.measurableSpace
-  apply measurable_pi_iff.mpr
-  intro n
-  have h : Measurable (fun b => if S b = ∞ then (0 : ℝ)
-      else b.1 (ExactSLEGreen.capTime S (u n) b)) :=
-    Measurable.ite htop measurable_const (hval n)
-  convert h using 1
-  funext b
-  change b.1 (min (u n) (S b).toNNReal) = _
-  by_cases hb : S b = ∞
-  · simp only [hb,ENNReal.toNNReal_top,min_zero,ite_true,b.2]
-  · rw [ite_eq_right hb]
-    congr 1
-    unfold ExactSLEGreen.capTime
-    conv_rhs => rw [← ENNReal.coe_toNNReal hb]
-    simp only [← ENNReal.coe_min, ENNReal.toNNReal_coe, min_comm]
-
-end ExactSLEGenerator
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma remainingMassDensity_time_joint_measurable (a : ℝ) (ha : 1/4 < a)
-    (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
-    Measurable (fun p : ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane) =>
-      remainingMassDensity a q p.1 p.2.1 p.2.2.1) := by
-  have h := (radialRemaining_time_joint_measurable a ha q hq hq1).const_mul
-    (Real.exp (-alpha a*Real.log 2))
-  convert h using 1
-  funext p
-  exact ite_eq_left p.2.2.2
-
-lemma survivingGreen_time_joint_measurable (a : ℝ) (ha : 1/4 < a) :
-    Measurable (fun p : ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane) =>
-      survivingGreen a p.1 p.2.1 p.2.2.1) := by
-  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ)+1)
-  have hq (n : ℕ) : 0<q n ∧ q n≤1 := by
-    constructor
-    · exact div_pos zero_lt_one (by positivity)
-    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
-  apply measurable_of_tendsto_metrizable
-    (fun n => remainingMassDensity_time_joint_measurable a ha (q n) (hq n).1 (hq n).2)
-  apply tendsto_pi_nhds.mpr
-  intro p
-  exact remaining_to_surviving a ha p.1 q hq
-    tendsto_one_div_add_atTop_nhds_zero_nat p.2.1 p.2.2.1 p.2.2.2
-
-lemma survivingGreen_driver_congr (a : ℝ) (b c : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (h : ∀ s≤t, b.1 s=c.1 s) (z : ℂ) :
-    survivingGreen a t b z = survivingGreen a t c z := by
-  simp only [survivingGreen,alive_driver_congr h,actualState_driver_congr h]
-
- 
-
-theorem stoppedSurvivingGreen_raw_joint (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    letI : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      stoppedSurvivingGreen a (S p.1) p.1 p.2.1) := by
-  have hp : @Measurable (ExactSLEGenerator.Driver × UpperPlane)
-      (ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane))
-      (hS.measurableSpace.prod inferInstance) inferInstance
-      (fun p => ((S p.1).toNNReal,(pastDriver (S p.1).toNNReal p.1,p.2))) := by
-    exact ((ENNReal.measurable_toNNReal.comp hS.measurable).comp
-      (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
-      (((pastDriver_stopping_measurable S hS).comp
-        (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
-        (@measurable_snd _ _ hS.measurableSpace inferInstance))
-  have hM := (survivingGreen_time_joint_measurable a ha).comp hp
-  have htop : MeasurableSet[hS.measurableSpace] {b | S b=∞} :=
-    hS.measurable (measurableSet_singleton ∞)
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      if S p.1=∞ then 0 else
-        survivingGreen a (S p.1).toNNReal (pastDriver (S p.1).toNNReal p.1) p.2.1) :=
-    Measurable.ite (measurable_fst htop) measurable_const hM
-  convert hm using 1
-  funext p
-  unfold stoppedSurvivingGreen
-  congr 1
-  exact survivingGreen_driver_congr a p.1 _ _
-    (fun s hs => (pastDriver_eq (S p.1).toNNReal p.1 s hs).symm) p.2.1
-
-lemma stoppedSurvivingGreen_joint_measurable (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      stoppedSurvivingGreen a (S p.1) p.1 p.2.1) :=
-  (stoppedSurvivingGreen_raw_joint a ha S hS).mono
-    (SLEExactGauge.MassVersions.prod_sigma_le hS.measurableSpace_le _) le_rfl
-
- 
-
-theorem stoppedRemainingDensity_raw_joint (a : ℝ) (ha : 1/4 < a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    letI : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      stoppedRemainingDensity a q (S p.1) p.1 p.2.1) := by
-  have hp : @Measurable (ExactSLEGenerator.Driver × UpperPlane)
-      (ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane))
-      (hS.measurableSpace.prod inferInstance) inferInstance
-      (fun p => ((S p.1).toNNReal,(pastDriver (S p.1).toNNReal p.1,p.2))) := by
-    exact ((ENNReal.measurable_toNNReal.comp hS.measurable).comp
-      (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
-      (((pastDriver_stopping_measurable S hS).comp
-        (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
-        (@measurable_snd _ _ hS.measurableSpace inferInstance))
-  have hM := (remainingMassDensity_time_joint_measurable a ha q hq hq1).comp hp
-  have htop : MeasurableSet[hS.measurableSpace] {b | S b=∞} :=
-    hS.measurable (measurableSet_singleton ∞)
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      if S p.1=∞ then 0 else
-        remainingMassDensity a q (S p.1).toNNReal (pastDriver (S p.1).toNNReal p.1) p.2.1) :=
-    Measurable.ite (measurable_fst htop) measurable_const hM
-  convert hm using 1
-  funext p
-  unfold stoppedRemainingDensity
-  congr 1
-  simp only [remainingMassDensity, ite_eq_left p.2.2]
-  congr 1
-  exact radialRemaining_driver_congr a (by linarith) p.2.1 p.2.2 _
-    (mul_pos p.2.2 hq) (mul_le_of_le_one_right p.2.2.le hq1) p.1 _ _
-    (fun s hs => (pastDriver_eq (S p.1).toNNReal p.1 s hs).symm)
-
-lemma stoppedRemainingDensity_joint_measurable (a : ℝ) (ha : 1/4 < a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      stoppedRemainingDensity a q (S p.1) p.1 p.2.1) :=
-  (stoppedRemainingDensity_raw_joint a ha q hq hq1 S hS).mono
-    (SLEExactGauge.MassVersions.prod_sigma_le hS.measurableSpace_le _) le_rfl
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFlow
-
-lemma localState_fst_eq_solution {a ε : ℝ} (hε : 0<ε)
-    (b : ExactSLEGenerator.Driver) {z : ℂ} {T : ℝ≥0} {f : ℝ→ℂ}
-    (hf : odeUntil a b.1 z T f)
-    (hlower : ∀u∈Icc (0:ℝ) T, ε≤(f u).im) :
-    ∀u∈Icc (0:ℝ) T,
-      (localState a ε hε z u.toNNReal b).1+(b.1 u.toNNReal:ℂ)=f u := by
-  let g : ℝ→ℂ := fun u => (localState a ε hε z u.toNNReal b).1+(b.1 u.toNNReal:ℂ)
-  let v : ℝ→ℂ := fun u => -(b.1 u.toNNReal:ℂ)
-  have hgc : Continuous g :=
-    ((localState_continuous a ε hε z b).fst.comp continuous_real_toNNReal).add
-      (Complex.continuous_ofReal.comp (b.1.continuous.comp continuous_real_toNNReal))
-  have hgd (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
-      HasDerivWithinAt g (cutoffVector a ε (g u+v u)) (Icc (0:ℝ) T) u := by
-    have hd := (ContinuousLinearMap.fst ℝ ℂ ℝ).hasFDerivAt.comp_hasDerivWithinAt u
-      (localState_noiseFree_deriv a ε hε z b T u hu)
-    simpa only [g,v,ContinuousLinearMap.coe_fst',Prod.fst_sub,Prod.fst_smul,
-      noise,Prod.smul_mk,Prod.fst,Complex.real_smul,mul_neg,mul_one,sub_neg_eq_add,add_neg_cancel_right,
-      cutoffDrift,Function.comp_def] using hd
-  have hfd (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
-      HasDerivWithinAt f (cutoffVector a ε (f u+v u)) (Icc (0:ℝ) T) u := by
-    have hy : ε≤(f u-(b.1 u.toNNReal:ℂ)).im := by
-      simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using hlower u hu
-    simpa only [v,←sub_eq_add_neg,cutoffVector,heightClamp_eq hy] using (hf.2 u hu).2
-  exact additive_flow_unique (cutoffVector_lipschitz a ε hε) hgc.continuousOn
-    (odeUntil_continuousOn hf) hgd hfd (by simp [localState_zero,hf.1,b.2])
-
-lemma localState_eq_actual_of_height {a ε : ℝ} (hε : 0<ε)
-    (b : ExactSLEGenerator.Driver) {z : ℂ} {T : ℝ≥0} {f : ℝ→ℂ}
-    (hf : odeUntil a b.1 z T f)
-    (hlower : ∀u∈Icc (0:ℝ) T, ε≤(f u).im) :
-    ∀t≤T, localState a ε hε z t b=actualState a b.1 t z := by
-  have he := localState_fst_eq_solution hε b hf hlower
-  have hy (t : ℝ≥0) (ht : t≤T) : ε≤(localState a ε hε z t b).1.im := by
-    have hh := congrArg Complex.im (he t ⟨t.coe_nonneg,ht⟩)
-    simp only [Real.toNNReal_coe,Complex.add_im,Complex.ofReal_im,add_zero] at hh
-    rw [hh]
-    exact hlower t ⟨t.coe_nonneg,ht⟩
-  intro t ht
-  have h := state_eq_loewner_on_safe a ε hε b.1 b.1.continuous b.2 z
-    (fun u => localState a ε hε z u b) (localState_continuous a ε hε z b)
-    (fun u => localState_eq_zero a ε hε z u b) t (fun u hu => hy u (hu.trans ht))
-  exact Prod.ext (h.2.1 t le_rfl).symm h.2.2.symm
-
-lemma localState_eventually_eq_actual (a : ℝ) (b : ExactSLEGenerator.Driver)
-    {z : ℂ} {t : ℝ≥0} (hz : alive a b.1 t z) :
-    ∀ᶠ n : ℕ in atTop,
-      localState a (1/((n:ℝ)+1)) (by positivity) z t b=actualState a b.1 t z := by
-  obtain ⟨f,hf⟩ := hz
-  obtain ⟨η,hη,hηf⟩ := odeUntil_height_lower hf
-  filter_upwards [tendsto_one_div_add_atTop_nhds_zero_nat.eventually (gt_mem_nhds hη)] with n hn
-  exact localState_eq_actual_of_height (by positivity) b hf
-    (fun u hu => hn.le.trans (hηf u hu)) t le_rfl
-
-lemma survivingGreen_pos_iff (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    0<survivingGreen a t b z ↔ alive a b.1 t z := by
-  classical
-  unfold survivingGreen
-  split_ifs
-  · exact iff_of_true (mul_pos (Real.exp_pos _) (Real.exp_pos _)) ‹_›
-  · simp_all
-
-lemma measurableSet_alive_joint (a : ℝ) (ha : 1/4<a) :
-    MeasurableSet {p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane) |
-      alive a p.2.1.1 p.1 p.2.2.1} := by
-  simp_rw [←survivingGreen_pos_iff]
-  exact measurableSet_lt measurable_const (survivingGreen_time_joint_measurable a ha)
-
- 
-def killedState (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) : State :=
-  if alive a b.1 t z then actualState a b.1 t z else 0
-
-lemma killedState_time_joint_measurable (a : ℝ) (ha : 1/4<a) :
-    Measurable (fun p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane) =>
-      killedState a p.1 p.2.1 p.2.2.1) := by
-  classical
-  apply measurable_of_tendsto_metrizable (f:=fun (n : ℕ) p =>
-    if alive a p.2.1.1 p.1 p.2.2.1 then
-      localState a (1/((n:ℝ)+1)) (by positivity) p.2.2.1 p.1 p.2.1 else 0)
-  · intro n
-    exact (localState_parameter_joint_measurable a (fun _ => 1/((n:ℝ)+1))
-      (fun _ => by positivity) (fun p : ExactSLEGenerator.Driver×UpperPlane => p.2.1)
-      Prod.fst measurable_const (measurable_subtype_coe.comp measurable_snd)
-      measurable_fst).ite (measurableSet_alive_joint a ha) measurable_const
-  · apply tendsto_pi_nhds.mpr
-    intro p
-    by_cases hp : alive a p.2.1.1 p.1 p.2.2.1
-    · simp only [ite_eq_left hp,killedState]
-      exact tendsto_const_nhds.congr' ((localState_eventually_eq_actual a p.2.1 hp).mono (fun _ h => h.symm))
-    · simp only [ite_eq_right hp,killedState]
-      exact tendsto_const_nhds
-
-lemma loewnerMap_time_joint_measurable (a : ℝ) (ha : 1/4<a) :
-    Measurable (fun p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane) =>
-      loewnerMap a p.2.1.1 p.1 p.2.2.1) := by
-  classical
-  have he (p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane)) :
-      loewnerMap a p.2.1.1 p.1 p.2.2.1 =
-        if alive a p.2.1.1 p.1 p.2.2.1 then
-          (killedState a p.1 p.2.1 p.2.2.1).1+(p.2.1.1 p.1:ℂ) else 0 := by
-    by_cases h : alive a p.2.1.1 p.1 p.2.2.1
-    · simp only [ite_eq_left h,killedState,actualState,sub_add_cancel]
-    · simp only [ite_eq_right h,loewnerMap,dite_eq_right h]
-  simp_rw [he]
-  exact ((killedState_time_joint_measurable a ha).fst.add
-    ((Complex.continuous_ofReal.comp (continuous_subtype_val.comp (continuous_fst.comp
-      continuous_snd) |>.eval continuous_fst)).measurable)).ite
-        (measurableSet_alive_joint a ha) measurable_const
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.FutureSupport
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+ 
+
+def stoppedSurvivingGreen (a : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
+  if t = ∞ then 0 else survivingGreen a t.toNNReal b z
+
+ 
+def stoppedRemainingDensity (a q : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
+  if t = ∞ then 0 else remainingMassDensity a q t.toNNReal b z
+
+lemma stoppedSurvivingGreen_nonneg (a : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    0 ≤ stoppedSurvivingGreen a t b z := by
+  unfold stoppedSurvivingGreen
+  split_ifs
+  · exact le_rfl
+  · exact survivingGreen_nonneg _ _ _ _
+
+lemma stoppedRemainingDensity_nonneg (a q : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    0 ≤ stoppedRemainingDensity a q t b z := by
+  unfold stoppedRemainingDensity
+  split_ifs
+  · exact le_rfl
+  · exact remainingMassDensity_nonneg _ _ _ _ _
+
+lemma stoppedRemainingDensity_le_surviving (a q : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0 < z.im) :
+    stoppedRemainingDensity a q t b z ≤ stoppedSurvivingGreen a t b z := by
+  unfold stoppedRemainingDensity stoppedSurvivingGreen
+  split_ifs
+  · exact le_rfl
+  · exact remainingMassDensity_le_surviving a q t.toNNReal b z hz
+
+lemma stopped_remaining_to_surviving (a : ℝ) (ha : 1/4 < a) (t : ℝ≥0∞)
+    (q : ℕ → ℝ) (hq : ∀ n, 0 < q n ∧ q n ≤ 1) (hlim : Tendsto q atTop (𝓝 0))
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0 < z.im) :
+    Tendsto (fun n => stoppedRemainingDensity a (q n) t b z) atTop
+      (𝓝 (stoppedSurvivingGreen a t b z)) := by
+  unfold stoppedRemainingDensity stoppedSurvivingGreen
+  by_cases ht : t = ∞
+  · simp only [ht,ite_true]
+    exact tendsto_const_nhds
+  · simp only [ite_eq_right ht]
+    exact remaining_to_surviving a ha t.toNNReal q hq hlim b z hz
+
+lemma stoppedRemainingDensity_eq (a q : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0 < z.im) :
+    stoppedRemainingDensity a q t b z = Real.exp (-alpha a*Real.log 2) *
+      stoppedRadialRemaining a b.1 z (z.im*q) t := by
+  unfold stoppedRemainingDensity stoppedRadialRemaining remainingMassDensity
+  split_ifs <;> simp_all only [mul_zero]
+
+lemma stoppedRadialTerminal_integrable (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.1.im)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Integrable (fun b => if S b < radialTime a b.1 z.1 ρ
+      then radialWeight a b.1 z.1 ρ else 0) W := by
+  have hmS : Measurable S := hS.measurable'
+  have hmT : Measurable (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z.1 ρ) :=
+    (radialTime_stopping a (by linarith) z.1 z.2 ρ hρ hr).measurable'
+  have hm : MeasurableSet {b : ExactSLEGenerator.Driver | S b < radialTime a b.1 z.1 ρ} :=
+    measurableSet_lt hmS hmT
+  apply ((radialWeight_mean W hW a ha z.1 z.2 ρ hρ hr).1.indicator hm).congr
+  filter_upwards [] with b
+  simp only [indicator_apply,mem_ofPred_eq]
+
+lemma stoppedRemainingDensity_mean_bound (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Integrable (fun b => stoppedRemainingDensity a q (S b) b z.1) W ∧
+      (∫ b, stoppedRemainingDensity a q (S b) b z.1 ∂W) ≤ initialGreen a z := by
+  have hρ : 0 < z.1.im*q := mul_pos z.2 hq
+  have hr : z.1.im*q ≤ z.1.im := mul_le_of_le_one_right z.2.le hq1
+  obtain ⟨hi,he⟩ := radial_conditional_weight_all W hW a ha z.1 z.2
+    (z.1.im*q) hρ hr S hS univ MeasurableSet.univ
+  simp only [setIntegral_univ] at he
+  simp only [stoppedRemainingDensity_eq a q _ _ _ z.2]
+  refine ⟨hi.const_mul _, ?_⟩
+  rw [integral_const_mul, ← he]
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+  rw [← (radialWeight_mean W hW a ha z.1 z.2 _ hρ hr).2]
+  apply integral_mono (stoppedRadialTerminal_integrable a ha W hW z _ hρ hr S hS)
+    (radialWeight_mean W hW a ha z.1 z.2 _ hρ hr).1
+  intro b
+  dsimp only
+  split_ifs
+  · exact le_rfl
+  · exact (radialWeight_bound a ha z.1 z.2 _ hρ hr b).1
+
+ 
+
+theorem survivingGreen_optional_bound (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) :
+    Integrable (fun b => stoppedSurvivingGreen a (S b) b z.1) W ∧
+      (∫ b, stoppedSurvivingGreen a (S b) b z.1 ∂W) ≤ initialGreen a z := by
+  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
+  have hq (n : ℕ) : 0 < q n ∧ q n ≤ 1 := by
+    constructor
+    · exact div_pos zero_lt_one (by positivity)
+    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
+  have hlim : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun n b =>
+    stoppedRemainingDensity a (q n) (S b) b z.1
+  let G : ExactSLEGenerator.Driver → ℝ := fun b => stoppedSurvivingGreen a (S b) b z.1
+  have hFi (n : ℕ) := (stoppedRemainingDensity_mean_bound a ha W hW z (q n)
+    (hq n).1 (hq n).2 S hS).1
+  have hFm (n : ℕ) : AEStronglyMeasurable (F n) W := (hFi n).aestronglyMeasurable
+  have hconv : ∀ᵐ b ∂W, Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
+    Eventually.of_forall (fun b => stopped_remaining_to_surviving a ha (S b) q hq hlim b z.1 z.2)
+  have hbound (n : ℕ) : eLpNorm (F n) 1 W ≤ ENNReal.ofReal (initialGreen a z) := by
+    rw [eLpNorm_one_eq_lintegral_enorm (hFm n), ← ofReal_integral_norm_eq_lintegral_enorm (hFi n)]
+    have he : (fun b => ‖F n b‖) = F n := by
+      funext b
+      exact Real.norm_of_nonneg (stoppedRemainingDensity_nonneg a (q n) (S b) b z.1)
+    rw [he]
+    exact ENNReal.ofReal_le_ofReal
+      (stoppedRemainingDensity_mean_bound a ha W hW z (q n) (hq n).1 (hq n).2 S hS).2
+  have hGm := aestronglyMeasurable_of_tendsto_ae atTop hFm hconv
+  have hGb := Lp.eLpNorm_le_of_ae_tendsto (Eventually.of_forall hbound) hFm hGm hconv
+  have hGi : Integrable G W :=
+    memLp_one_iff_integrable.mp (hGb.trans_lt ENNReal.ofReal_lt_top)
+  refine ⟨hGi, ?_⟩
+  rw [eLpNorm_one_eq_lintegral_enorm hGm, ← ofReal_integral_norm_eq_lintegral_enorm hGi] at hGb
+  have he : (fun b => ‖G b‖) = G := by
+    funext b
+    exact Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg a (S b) b z.1)
+  rw [he] at hGb
+  exact (ENNReal.ofReal_le_ofReal_iff (initialGreen_pos a z).le).mp hGb
+
+lemma stopped_remaining_integral_tendsto (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S)
+    (q : ℕ → ℝ) (hq : ∀ n, 0 < q n ∧ q n ≤ 1) (hlim : Tendsto q atTop (𝓝 0))
+    (U : Set ExactSLEGenerator.Driver) :
+    Tendsto (fun n => ∫ b in U, stoppedRemainingDensity a (q n) (S b) b z.1 ∂W)
+      atTop (𝓝 (∫ b in U, stoppedSurvivingGreen a (S b) b z.1 ∂W)) := by
+  apply tendsto_integral_of_dominated_convergence
+    (fun b => stoppedSurvivingGreen a (S b) b z.1)
+  · intro n
+    exact (stoppedRemainingDensity_mean_bound a ha W hW z (q n)
+      (hq n).1 (hq n).2 S hS).1.aestronglyMeasurable.restrict
+  · exact (survivingGreen_optional_bound a ha W hW z S hS).1.integrableOn
+  · intro n
+    filter_upwards [] with b
+    rw [Real.norm_of_nonneg (stoppedRemainingDensity_nonneg a (q n) (S b) b z.1)]
+    exact stoppedRemainingDensity_le_surviving a (q n) (S b) b z.1 z.2
+  · exact Eventually.of_forall (fun b =>
+      stopped_remaining_to_surviving a ha (S b) q hq hlim b z.1 z.2)
+
+ 
+
+theorem survivingGreen_optional_comparison (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (S T : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (hT : IsStoppingTime rawFiltration T)
+    (hst : S ≤ T) (U : Set ExactSLEGenerator.Driver)
+    (hU : MeasurableSet[hS.measurableSpace] U) :
+    (∫ b in U, stoppedSurvivingGreen a (T b) b z.1 ∂W) ≤
+      ∫ b in U, stoppedSurvivingGreen a (S b) b z.1 ∂W := by
+  have hUT : MeasurableSet[hT.measurableSpace] U := (hS.measurableSpace_mono hT hst) _ hU
+  have hc (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
+      (∫ b in U, stoppedRemainingDensity a q (T b) b z.1 ∂W) ≤
+        ∫ b in U, stoppedRemainingDensity a q (S b) b z.1 ∂W := by
+    have hρ : 0 < z.1.im*q := mul_pos z.2 hq
+    have hr : z.1.im*q ≤ z.1.im := mul_le_of_le_one_right z.2.le hq1
+    have heS := (radial_conditional_weight_all W hW a ha z.1 z.2
+      (z.1.im*q) hρ hr S hS U hU).2
+    have heT := (radial_conditional_weight_all W hW a ha z.1 z.2
+      (z.1.im*q) hρ hr T hT U hUT).2
+    simp only [stoppedRemainingDensity_eq a q _ _ _ z.2,integral_const_mul]
+    rw [← heS, ← heT]
+    apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+    apply integral_mono
+      (stoppedRadialTerminal_integrable a ha W hW z _ hρ hr T hT).integrableOn
+      (stoppedRadialTerminal_integrable a ha W hW z _ hρ hr S hS).integrableOn
+    intro b
+    dsimp only
+    split_ifs with ht hs hs
+    · exact le_rfl
+    · exact False.elim (hs ((hst b).trans_lt ht))
+    · exact (radialWeight_bound a ha z.1 z.2 _ hρ hr b).1
+    · exact le_rfl
+  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1)
+  have hq (n : ℕ) : 0 < q n ∧ q n ≤ 1 := by
+    constructor
+    · exact div_pos zero_lt_one (by positivity)
+    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
+  have hlim : Tendsto q atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  exact le_of_tendsto_of_tendsto'
+    (stopped_remaining_integral_tendsto a ha W hW z T hT q hq hlim U)
+    (stopped_remaining_integral_tendsto a ha W hW z S hS q hq hlim U)
+    (fun n => hc (q n) (hq n).1 (hq n).2)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma radialRemaining_time_joint_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
+    (hq : 0 < q) (hq1 : q ≤ 1) :
+    Measurable (fun p : ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane) =>
+      radialRemaining a p.2.1.1 p.2.2.1 (p.2.2.1.im*q) p.1) := by
+  have ha0 : 0<a := by linarith
+  let Ω := ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane)
+  let z : Ω → ℂ := fun p => p.2.2.1
+  let b : Ω → ExactSLEGenerator.Driver := fun p => p.2.1
+  let t : Ω → ℝ≥0 := Prod.fst
+  have hz (ω : Ω) : 0<(z ω).im := ω.2.2.2
+  have hzM : Measurable z := measurable_subtype_coe.comp (measurable_snd.comp measurable_snd)
+  have hbM : Measurable b := measurable_fst.comp measurable_snd
+  have htM : Measurable t := measurable_fst
+  let ρ : Ω → ℝ := fun ω => (z ω).im*q
+  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
+  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
+  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
+  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
+    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
+  have hρM : Measurable ρ := hzM.im.mul_const q
+  have hδM (n : ℕ) : Measurable (δ n) := by
+    dsimp [δ,angleThreshold]
+    exact (hzM.im.div hzM.norm).div_const _
+  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
+    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
+    unfold heightFloor
+    fun_prop
+  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
+    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    unfold logCeiling heightFloor
+    fun_prop
+  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
+    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    fun_prop
+  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
+    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
+  have hTM (n : ℕ) : Measurable (T n) :=
+    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
+      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
+  have hPM : Measurable (physicalGreen a) := by
+    have hx := X.continuous.measurable
+    have hy := Y.continuous.measurable
+    have hj := J.continuous.measurable
+    unfold physicalGreen logGreen Q
+    fun_prop
+  let F (n : ℕ) (ω : Ω) := if t ω < T n ω then physicalGreen a (localState a (L n ω).ε (L n ω).hε (z ω) (t ω) (b ω)) else 0
+  have hFM (n : ℕ) : Measurable (F n) := (hPM.comp
+    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b (hεM n) hzM hbM).comp
+      (htM.prodMk measurable_id))).ite (measurableSet_lt htM (hTM n)) measurable_const
+  apply measurable_of_tendsto_metrizable hFM
+  apply tendsto_pi_nhds.mpr
+  intro ω
+  exact localizationRemaining_tendsto a ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω)
+    (fun n => L n ω) (b ω) (t ω)
+
+end ExactSLEGreen
+
+namespace ExactSLEGenerator
+
+ 
+
+lemma pastDriver_stopping_measurable
+    (S : Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (fun b => pastDriver (S b).toNNReal b) := by
+  classical
+  have : PolishSpace Driver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
+  obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq (α := ℝ≥0)
+  let e : Driver → (ℕ → ℝ) := fun b n => b.1 (u n)
+  have hec : Continuous e := continuous_pi fun n =>
+    (continuous_eval_const (u n)).comp continuous_subtype_val
+  have hei : Function.Injective e := by
+    intro b c h
+    apply Subtype.ext
+    apply DFunLike.coe_injective
+    exact hu.equalizer b.1.continuous c.1.continuous h
+  apply hec.measurableEmbedding hei |>.measurable_comp_iff.mp
+  have hA : StronglyAdapted rawFiltration (fun t (b : Driver) => b.1 t) :=
+    Filtration.stronglyAdapted_natural driver_eval_measurable
+  have hP := hA.isStronglyProgressive_of_continuous (fun b => b.1.continuous)
+  have hval (n : ℕ) : Measurable[hS.measurableSpace]
+      (fun b => b.1 (ExactSLEGreen.capTime S (u n) b)) := by
+    have h := (measurable_stoppedValue hP (ExactSLEGreen.capTime_stopping S hS (u n))).mono
+      (ExactSLEGreen.stopped_cap_measurable S hS (u n)) le_rfl
+    exact h
+  have htop : MeasurableSet[hS.measurableSpace] {b | S b = ∞} :=
+    hS.measurable (measurableSet_singleton ∞)
+  let : MeasurableSpace Driver := hS.measurableSpace
+  apply measurable_pi_iff.mpr
+  intro n
+  have h : Measurable (fun b => if S b = ∞ then (0 : ℝ)
+      else b.1 (ExactSLEGreen.capTime S (u n) b)) :=
+    Measurable.ite htop measurable_const (hval n)
+  convert h using 1
+  funext b
+  change b.1 (min (u n) (S b).toNNReal) = _
+  by_cases hb : S b = ∞
+  · simp only [hb,ENNReal.toNNReal_top,min_zero,ite_true,b.2]
+  · rw [ite_eq_right hb]
+    congr 1
+    unfold ExactSLEGreen.capTime
+    conv_rhs => rw [← ENNReal.coe_toNNReal hb]
+    simp only [← ENNReal.coe_min, ENNReal.toNNReal_coe, min_comm]
+
+end ExactSLEGenerator
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma remainingMassDensity_time_joint_measurable (a : ℝ) (ha : 1/4 < a)
+    (q : ℝ) (hq : 0 < q) (hq1 : q ≤ 1) :
+    Measurable (fun p : ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane) =>
+      remainingMassDensity a q p.1 p.2.1 p.2.2.1) := by
+  have h := (radialRemaining_time_joint_measurable a ha q hq hq1).const_mul
+    (Real.exp (-alpha a*Real.log 2))
+  convert h using 1
+  funext p
+  exact ite_eq_left p.2.2.2
+
+lemma survivingGreen_time_joint_measurable (a : ℝ) (ha : 1/4 < a) :
+    Measurable (fun p : ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane) =>
+      survivingGreen a p.1 p.2.1 p.2.2.1) := by
+  let q : ℕ → ℝ := fun n => 1 / ((n : ℝ)+1)
+  have hq (n : ℕ) : 0<q n ∧ q n≤1 := by
+    constructor
+    · exact div_pos zero_lt_one (by positivity)
+    · apply div_le_one_of_le₀ (by linarith [Nat.cast_nonneg (α := ℝ) n]) (by positivity)
+  apply measurable_of_tendsto_metrizable
+    (fun n => remainingMassDensity_time_joint_measurable a ha (q n) (hq n).1 (hq n).2)
+  apply tendsto_pi_nhds.mpr
+  intro p
+  exact remaining_to_surviving a ha p.1 q hq
+    tendsto_one_div_add_atTop_nhds_zero_nat p.2.1 p.2.2.1 p.2.2.2
+
+lemma survivingGreen_driver_congr (a : ℝ) (b c : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (h : ∀ s≤t, b.1 s=c.1 s) (z : ℂ) :
+    survivingGreen a t b z = survivingGreen a t c z := by
+  simp only [survivingGreen,alive_driver_congr h,actualState_driver_congr h]
+
+ 
+
+theorem stoppedSurvivingGreen_raw_joint (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    letI : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      stoppedSurvivingGreen a (S p.1) p.1 p.2.1) := by
+  have hp : @Measurable (ExactSLEGenerator.Driver × UpperPlane)
+      (ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane))
+      (hS.measurableSpace.prod inferInstance) inferInstance
+      (fun p => ((S p.1).toNNReal,(pastDriver (S p.1).toNNReal p.1,p.2))) := by
+    exact ((ENNReal.measurable_toNNReal.comp hS.measurable).comp
+      (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
+      (((pastDriver_stopping_measurable S hS).comp
+        (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
+        (@measurable_snd _ _ hS.measurableSpace inferInstance))
+  have hM := (survivingGreen_time_joint_measurable a ha).comp hp
+  have htop : MeasurableSet[hS.measurableSpace] {b | S b=∞} :=
+    hS.measurable (measurableSet_singleton ∞)
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      if S p.1=∞ then 0 else
+        survivingGreen a (S p.1).toNNReal (pastDriver (S p.1).toNNReal p.1) p.2.1) :=
+    Measurable.ite (measurable_fst htop) measurable_const hM
+  convert hm using 1
+  funext p
+  unfold stoppedSurvivingGreen
+  congr 1
+  exact survivingGreen_driver_congr a p.1 _ _
+    (fun s hs => (pastDriver_eq (S p.1).toNNReal p.1 s hs).symm) p.2.1
+
+lemma stoppedSurvivingGreen_joint_measurable (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      stoppedSurvivingGreen a (S p.1) p.1 p.2.1) :=
+  (stoppedSurvivingGreen_raw_joint a ha S hS).mono
+    (SLEExactGauge.MassVersions.prod_sigma_le hS.measurableSpace_le _) le_rfl
+
+ 
+
+theorem stoppedRemainingDensity_raw_joint (a : ℝ) (ha : 1/4 < a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    letI : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      stoppedRemainingDensity a q (S p.1) p.1 p.2.1) := by
+  have hp : @Measurable (ExactSLEGenerator.Driver × UpperPlane)
+      (ℝ≥0 × (ExactSLEGenerator.Driver × UpperPlane))
+      (hS.measurableSpace.prod inferInstance) inferInstance
+      (fun p => ((S p.1).toNNReal,(pastDriver (S p.1).toNNReal p.1,p.2))) := by
+    exact ((ENNReal.measurable_toNNReal.comp hS.measurable).comp
+      (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
+      (((pastDriver_stopping_measurable S hS).comp
+        (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
+        (@measurable_snd _ _ hS.measurableSpace inferInstance))
+  have hM := (remainingMassDensity_time_joint_measurable a ha q hq hq1).comp hp
+  have htop : MeasurableSet[hS.measurableSpace] {b | S b=∞} :=
+    hS.measurable (measurableSet_singleton ∞)
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      if S p.1=∞ then 0 else
+        remainingMassDensity a q (S p.1).toNNReal (pastDriver (S p.1).toNNReal p.1) p.2.1) :=
+    Measurable.ite (measurable_fst htop) measurable_const hM
+  convert hm using 1
+  funext p
+  unfold stoppedRemainingDensity
+  congr 1
+  simp only [remainingMassDensity, ite_eq_left p.2.2]
+  congr 1
+  exact radialRemaining_driver_congr a (by linarith) p.2.1 p.2.2 _
+    (mul_pos p.2.2 hq) (mul_le_of_le_one_right p.2.2.le hq1) p.1 _ _
+    (fun s hs => (pastDriver_eq (S p.1).toNNReal p.1 s hs).symm)
+
+lemma stoppedRemainingDensity_joint_measurable (a : ℝ) (ha : 1/4 < a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      stoppedRemainingDensity a q (S p.1) p.1 p.2.1) :=
+  (stoppedRemainingDensity_raw_joint a ha q hq hq1 S hS).mono
+    (SLEExactGauge.MassVersions.prod_sigma_le hS.measurableSpace_le _) le_rfl
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFlow
+
+lemma localState_fst_eq_solution {a ε : ℝ} (hε : 0<ε)
+    (b : ExactSLEGenerator.Driver) {z : ℂ} {T : ℝ≥0} {f : ℝ→ℂ}
+    (hf : odeUntil a b.1 z T f)
+    (hlower : ∀u∈Icc (0:ℝ) T, ε≤(f u).im) :
+    ∀u∈Icc (0:ℝ) T,
+      (localState a ε hε z u.toNNReal b).1+(b.1 u.toNNReal:ℂ)=f u := by
+  let g : ℝ→ℂ := fun u => (localState a ε hε z u.toNNReal b).1+(b.1 u.toNNReal:ℂ)
+  let v : ℝ→ℂ := fun u => -(b.1 u.toNNReal:ℂ)
+  have hgc : Continuous g :=
+    ((localState_continuous a ε hε z b).fst.comp continuous_real_toNNReal).add
+      (Complex.continuous_ofReal.comp (b.1.continuous.comp continuous_real_toNNReal))
+  have hgd (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
+      HasDerivWithinAt g (cutoffVector a ε (g u+v u)) (Icc (0:ℝ) T) u := by
+    have hd := (ContinuousLinearMap.fst ℝ ℂ ℝ).hasFDerivAt.comp_hasDerivWithinAt u
+      (localState_noiseFree_deriv a ε hε z b T u hu)
+    simpa only [g,v,ContinuousLinearMap.coe_fst',Prod.fst_sub,Prod.fst_smul,
+      noise,Prod.smul_mk,Prod.fst,Complex.real_smul,mul_neg,mul_one,sub_neg_eq_add,add_neg_cancel_right,
+      cutoffDrift,Function.comp_def] using hd
+  have hfd (u : ℝ) (hu : u∈Icc (0:ℝ) T) :
+      HasDerivWithinAt f (cutoffVector a ε (f u+v u)) (Icc (0:ℝ) T) u := by
+    have hy : ε≤(f u-(b.1 u.toNNReal:ℂ)).im := by
+      simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using hlower u hu
+    simpa only [v,←sub_eq_add_neg,cutoffVector,heightClamp_eq hy] using (hf.2 u hu).2
+  exact additive_flow_unique (cutoffVector_lipschitz a ε hε) hgc.continuousOn
+    (odeUntil_continuousOn hf) hgd hfd (by simp [localState_zero,hf.1,b.2])
+
+lemma localState_eq_actual_of_height {a ε : ℝ} (hε : 0<ε)
+    (b : ExactSLEGenerator.Driver) {z : ℂ} {T : ℝ≥0} {f : ℝ→ℂ}
+    (hf : odeUntil a b.1 z T f)
+    (hlower : ∀u∈Icc (0:ℝ) T, ε≤(f u).im) :
+    ∀t≤T, localState a ε hε z t b=actualState a b.1 t z := by
+  have he := localState_fst_eq_solution hε b hf hlower
+  have hy (t : ℝ≥0) (ht : t≤T) : ε≤(localState a ε hε z t b).1.im := by
+    have hh := congrArg Complex.im (he t ⟨t.coe_nonneg,ht⟩)
+    simp only [Real.toNNReal_coe,Complex.add_im,Complex.ofReal_im,add_zero] at hh
+    rw [hh]
+    exact hlower t ⟨t.coe_nonneg,ht⟩
+  intro t ht
+  have h := state_eq_loewner_on_safe a ε hε b.1 b.1.continuous b.2 z
+    (fun u => localState a ε hε z u b) (localState_continuous a ε hε z b)
+    (fun u => localState_eq_zero a ε hε z u b) t (fun u hu => hy u (hu.trans ht))
+  exact Prod.ext (h.2.1 t le_rfl).symm h.2.2.symm
+
+lemma localState_eventually_eq_actual (a : ℝ) (b : ExactSLEGenerator.Driver)
+    {z : ℂ} {t : ℝ≥0} (hz : alive a b.1 t z) :
+    ∀ᶠ n : ℕ in atTop,
+      localState a (1/((n:ℝ)+1)) (by positivity) z t b=actualState a b.1 t z := by
+  obtain ⟨f,hf⟩ := hz
+  obtain ⟨η,hη,hηf⟩ := odeUntil_height_lower hf
+  filter_upwards [tendsto_one_div_add_atTop_nhds_zero_nat.eventually (gt_mem_nhds hη)] with n hn
+  exact localState_eq_actual_of_height (by positivity) b hf
+    (fun u hu => hn.le.trans (hηf u hu)) t le_rfl
+
+lemma survivingGreen_pos_iff (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    0<survivingGreen a t b z ↔ alive a b.1 t z := by
+  classical
+  unfold survivingGreen
+  split_ifs
+  · exact iff_of_true (mul_pos (Real.exp_pos _) (Real.exp_pos _)) ‹_›
+  · simp_all
+
+lemma measurableSet_alive_joint (a : ℝ) (ha : 1/4<a) :
+    MeasurableSet {p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane) |
+      alive a p.2.1.1 p.1 p.2.2.1} := by
+  simp_rw [←survivingGreen_pos_iff]
+  exact measurableSet_lt measurable_const (survivingGreen_time_joint_measurable a ha)
+
+ 
+def killedState (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) : State :=
+  if alive a b.1 t z then actualState a b.1 t z else 0
+
+lemma killedState_time_joint_measurable (a : ℝ) (ha : 1/4<a) :
+    Measurable (fun p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane) =>
+      killedState a p.1 p.2.1 p.2.2.1) := by
+  classical
+  apply measurable_of_tendsto_metrizable (f:=fun (n : ℕ) p =>
+    if alive a p.2.1.1 p.1 p.2.2.1 then
+      localState a (1/((n:ℝ)+1)) (by positivity) p.2.2.1 p.1 p.2.1 else 0)
+  · intro n
+    exact (localState_parameter_joint_measurable a (fun _ => 1/((n:ℝ)+1))
+      (fun _ => by positivity) (fun p : ExactSLEGenerator.Driver×UpperPlane => p.2.1)
+      Prod.fst measurable_const (measurable_subtype_coe.comp measurable_snd)
+      measurable_fst).ite (measurableSet_alive_joint a ha) measurable_const
+  · apply tendsto_pi_nhds.mpr
+    intro p
+    by_cases hp : alive a p.2.1.1 p.1 p.2.2.1
+    · simp only [ite_eq_left hp,killedState]
+      exact tendsto_const_nhds.congr' ((localState_eventually_eq_actual a p.2.1 hp).mono (fun _ h => h.symm))
+    · simp only [ite_eq_right hp,killedState]
+      exact tendsto_const_nhds
+
+lemma loewnerMap_time_joint_measurable (a : ℝ) (ha : 1/4<a) :
+    Measurable (fun p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane) =>
+      loewnerMap a p.2.1.1 p.1 p.2.2.1) := by
+  classical
+  have he (p : ℝ≥0×(ExactSLEGenerator.Driver×UpperPlane)) :
+      loewnerMap a p.2.1.1 p.1 p.2.2.1 =
+        if alive a p.2.1.1 p.1 p.2.2.1 then
+          (killedState a p.1 p.2.1 p.2.2.1).1+(p.2.1.1 p.1:ℂ) else 0 := by
+    by_cases h : alive a p.2.1.1 p.1 p.2.2.1
+    · simp only [ite_eq_left h,killedState,actualState,sub_add_cancel]
+    · simp only [ite_eq_right h,loewnerMap,dite_eq_right h]
+  simp_rw [he]
+  exact ((killedState_time_joint_measurable a ha).fst.add
+    ((Complex.continuous_ofReal.comp (continuous_subtype_val.comp (continuous_fst.comp
+      continuous_snd) |>.eval continuous_fst)).measurable)).ite
+        (measurableSet_alive_joint a ha) measurable_const
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/OptionalPastPalm.lean b/lean/OAI/Probability/SLEGauge/OptionalPastPalm.lean
index d6840cebca6d920a33ead372113114d17ba51728..e7eca73291f37a5082c1ad4c786cf457f1e4c058 100644
--- a/lean/OAI/Probability/SLEGauge/OptionalPastPalm.lean
+++ b/lean/OAI/Probability/SLEGauge/OptionalPastPalm.lean
@@ -1,449 +1,449 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.SubdiskMoments
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.PastMass
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
-open TraceMassContinuity MassVersions TraceMassIntegration
-
-lemma optional_green_set_balance (a : ℝ) (S : GDriver→ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (W : Measure GDriver)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hH : ∀H : GDriver × UpperPlane→ℝ≥0∞,
-      Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
-      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=
-        (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
-        ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) :
-    (∫⁻b, μ ∞ b Q∂W) = (∫⁻b, μ (S b) b Q∂W)+
-      ∫⁻b,∫⁻z in Q, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W := by
-  let H : GDriver × UpperPlane→ℝ≥0∞ := fun p=> Q.indicator (fun _=>1) p.2
-  have hm : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H :=
-    (measurable_const.indicator hQ).comp measurable_snd
-  have he (ν : Measure UpperPlane) (b : GDriver) : (∫⁻z,H (b,z)∂ν)=ν Q := by
-    simp only [H,lintegral_indicator hQ,lintegral_one,Measure.restrict_apply_univ]
-  have hg (b : GDriver) :
-      (∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea)=
-        ∫⁻z in Q,ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea := by
-    rw [show (fun z => H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1))=
-      Q.indicator (fun z=>ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)) from by
-        funext z; by_cases hz : z∈Q <;> simp [H,hz], lintegral_indicator hQ]
-  simpa only [he,hg] using hH H hm
-
-lemma expected_green_past_kernel (a : ℝ) (ha : 1/4<a) (W : Measure GDriver) [SFinite W]
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (Q : Set UpperPlane) :
-    (∫⁻b,∫⁻z in Q,ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W)=
-      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a S z∂upperArea := by
-  simp_rw [initial_mul_pastGreenKernel]
-  exact lintegral_lintegral_swap (stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal.aemeasurable
-
- 
-
-lemma strict_past_mass_ae (κ : ℝ) (W : Measure GDriver)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hp : ∀ᵐ b∂W, TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b))
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : ∀b, μ ∞ b Q≠∞)
-    (S : GDriver→ℝ≥0∞) :
-    (fun b=>μ ∞ b (Q∩{z | S b<visitTime κ b z})) =ᵐ[W]
-      (fun b=>μ ∞ b Q-μ (S b) b Q) :=
-  hp.mono fun b hb=>stopped_strictPast_mass κ b (fun v=>μ v b) hb Q hQ (hfin b) (S b)
-
-lemma strict_past_mass_aemeasurable (κ : ℝ) (W : Measure GDriver)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hp : ∀ᵐ b∂W, TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b))
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : ∀b, μ ∞ b Q≠∞)
-    (S : GDriver→ℝ≥0∞) (hm : Measurable (μ ∞))
-    (hsm : Measurable (fun b=>μ (S b) b)) :
-    AEMeasurable (fun b=>μ ∞ b (Q∩{z | S b<visitTime κ b z})) W :=
-  ((Measure.measurable_coe hQ).comp hm |>.sub ((Measure.measurable_coe hQ).comp hsm)).aemeasurable.congr
-    (strict_past_mass_ae κ W μ hp Q hQ hfin S).symm
-
- 
-
-lemma strict_past_Palm (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver) [SFinite W]
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hp : ∀ᵐ b∂W, TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b))
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (hsm : Measurable (fun b=>μ (S b) b))
-    (hH : ∀H : GDriver × UpperPlane→ℝ≥0∞,
-      Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
-      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=
-        (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
-        ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : ∀b, μ ∞ b Q≠∞)
-    (hf : (∫⁻b,μ ∞ b Q∂W)≠∞) :
-    (∫⁻b,μ ∞ b (Q∩{z | S b<visitTime κ b z})∂W)=
-      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a S z∂upperArea := by
-  have hle : (fun b=>μ (S b) b Q) ≤ᵐ[W] (fun b=>μ ∞ b Q) :=
-    hp.mono fun b hb=>hb.monotone le_top Q
-  have hsf : (∫⁻b,μ (S b) b Q∂W)≠∞ := ne_top_of_le_ne_top hf (lintegral_mono_ae hle)
-  have hmS : Measurable (fun b=>μ (S b) b Q) := (Measure.measurable_coe hQ).comp hsm
-  rw [lintegral_congr_ae (strict_past_mass_ae κ W μ hp Q hQ hfin S),
-    lintegral_sub hmS hsf hle,
-    optional_green_set_balance a S hS W μ hH Q hQ, ENNReal.add_sub_cancel_left hsf]
-  exact expected_green_past_kernel a ha W S hS Q
-
-end SLEExactGauge.PastMass
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.PastMass
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
-open TraceMassContinuity MassVersions TraceMassIntegration
-
- 
-
-structure TracePalmData (κ a : ℝ) (W : Measure GDriver)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane) : Prop where
-  terminal_measurable : Measurable (μ ∞)
-  finite_compact : ∀v b, IsFiniteMeasureOnCompacts (μ v b)
-  adapted : Adapted rawFiltration (fun t : ℝ≥0=>μ t)
-  mean : μ ∞ ∘ₘ W=initialGreenMeasure a
-  path : ∀ᵐ b∂W,TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b)
-  optional_measurable : ∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
-    Measurable[hS.measurableSpace] (fun b=>μ (S b) b)
-  optional_kernel : ∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (H : GDriver × UpperPlane→ℝ≥0∞),
-    Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
-      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=
-        (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
-        ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W
-
- 
-
-def countablePastKernel {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0)
-    (W : Measure GDriver) (a : ℝ) (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (Q : Set UpperPlane) (H : ℝ≥0∞) (z : UpperPlane) : ℝ≥0∞ :=
-  ⨆n : ℕ, pastGreenKernel W a
-    (finiteTestTime ((Finset.range n).image (q∘e)) (fun t b=>μ t b Q) H) z
-
-namespace TracePalmData
-variable {κ a : ℝ} {W : Measure GDriver} {μ : ℝ≥0∞→GDriver→Measure UpperPlane}
-    (D : TracePalmData κ a W μ)
-include D
-
-lemma evaluation_adapted (Q : Set UpperPlane) (hQ : MeasurableSet Q) :
-    Adapted rawFiltration (fun t : ℝ≥0=>fun b=>μ t b Q) :=
-  fun t=>(Measure.measurable_coe hQ).comp (D.adapted t)
-
-lemma test_stopping (Q : Set UpperPlane) (hQ : MeasurableSet Q) (F : Finset ℝ≥0) (H : ℝ≥0∞) :
-    IsStoppingTime rawFiltration (finiteTestTime F (fun t b=>μ t b Q) H) :=
-  finiteTestTime_stopping _ _ (D.evaluation_adapted Q hQ) F H
-
-lemma compact_expected_finite (Q : Set UpperPlane) (hQ : IsCompact Q) :
-    (∫⁻b,μ ∞ b Q∂W)≠∞ := by
-  rw [←Measure.bind_apply hQ.measurableSet D.terminal_measurable.aemeasurable,D.mean]
-  exact hQ.measure_lt_top.ne
-
-lemma compact_finite (Q : Set UpperPlane) (hQ : IsCompact Q) (v : ℝ≥0∞) (b : GDriver) :
-    μ v b Q≠∞ := by
-  let := D.finite_compact v b
-  exact hQ.measure_lt_top.ne
-
-lemma stopped_Palm (ha : 1/4<a) [SFinite W] (S : GDriver→ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (Q : Set UpperPlane) (hQ : IsCompact Q) :
-    (∫⁻b,μ ∞ b (Q∩{z | S b<visitTime κ b z})∂W)=
-      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a S z∂upperArea :=
-  strict_past_Palm κ a ha W μ D.path S hS
-    ((D.optional_measurable S hS).mono hS.measurableSpace_le le_rfl)
-    (D.optional_kernel S hS) Q hQ.measurableSet (D.compact_finite Q hQ ∞)
-    (D.compact_expected_finite Q hQ)
-
-lemma finite_Palm (ha : 1/4<a) [SFinite W] (Q : Set UpperPlane) (hQ : IsCompact Q)
-    (F : Finset ℝ≥0) (H : ℝ≥0∞) :
-    (∫⁻b,μ ∞ b (Q∩{z | ∃q∈F,H<μ q b Q ∧ (q:ℝ≥0∞)<visitTime κ b z})∂W)=
-      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*
-        pastGreenKernel W a (finiteTestTime F (fun t b=>μ t b Q) H) z∂upperArea := by
-  simpa only [finiteTestTime_lt_iff] using
-    D.stopped_Palm ha _ (D.test_stopping Q hQ.measurableSet F H) Q hQ
-
-lemma compact_past_aemeasurable (Q : Set UpperPlane) (hQ : IsCompact Q)
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    AEMeasurable (fun b=>μ ∞ b (Q∩{z | S b<visitTime κ b z})) W :=
-  strict_past_mass_aemeasurable κ W μ D.path Q hQ.measurableSet
-    (D.compact_finite Q hQ ∞) S D.terminal_measurable
-    ((D.optional_measurable S hS).mono hS.measurableSpace_le le_rfl)
-
-lemma countable_kernel_measurable (ha : 1/4<a) [SFinite W]
-    {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0) (Q : Set UpperPlane) (hQ : MeasurableSet Q)
-    (H : ℝ≥0∞) : Measurable (countablePastKernel e q W a μ Q H) :=
-  .iSup fun _=>pastGreenKernel_measurable W a ha _ (D.test_stopping Q hQ _ H)
-
- 
-
-lemma countable_Palm (ha : 1/4<a) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0) (he : Function.Surjective e)
-    (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ≥0∞) :
-    (∫⁻b,μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})∂W)=
-      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*countablePastKernel e q W a μ Q H z∂upperArea := by
-  let S (n : ℕ) := finiteTestTime ((Finset.range n).image (q∘e)) (fun t b=>μ t b Q) H
-  have hS (n : ℕ) : IsStoppingTime rawFiltration (S n) := D.test_stopping Q hQ.measurableSet _ H
-  let A (n : ℕ) (b : GDriver) := μ ∞ b (Q∩{z | S n b<visitTime κ b z})
-  have hAm (n : ℕ) : AEMeasurable (A n) W := D.compact_past_aemeasurable Q hQ (S n) (hS n)
-  have hmono : ∀b,Monotone (fun n=>A n b) := by
-    intro b n k hnk
-    apply measure_mono
-    intro z hz
-    exact ⟨hz.1,(finiteTestTime_sequence_antitone e q (fun t b=>μ t b Q) H b hnk).trans_lt hz.2⟩
-  have heq : (fun b=>μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})) =ᵐ[W]
-      (fun b=>⨆n,A n b) := D.path.mono fun b hb=>
-    all_past_mass_eq_iSup e q he κ b (fun v=>μ v b) hb Q hQ.measurableSet H
-  have hKmono : Monotone (fun n z=>ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a (S n) z) := by
-    intro n k hnk z
-    apply mul_le_mul_right
-    exact pastGreenKernel_antitone W hW a ha (S k) (S n) (hS k) (hS n)
-      (fun b=>finiteTestTime_sequence_antitone e q (fun t b=>μ t b Q) H b hnk) z
-  rw [lintegral_congr_ae heq,lintegral_iSup' hAm (Eventually.of_forall hmono)]
-  simp only [A,D.stopped_Palm ha _ (hS _) Q hQ]
-  simp only [countablePastKernel,ENNReal.mul_iSup]
-  exact (lintegral_iSup (fun n=>(initialGreen_continuous a).measurable.ennreal_ofReal.mul
-    (pastGreenKernel_measurable W a ha (S n) (hS n))) hKmono).symm
-
-lemma countable_past_aemeasurable {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0)
-    (he : Function.Surjective e) (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ≥0∞) :
-    AEMeasurable (fun b=>μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})) W := by
-  have heq : (fun b=>μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})) =ᵐ[W]
-      (fun b=>⨆n : ℕ,μ ∞ b (Q∩{z | finiteTestTime ((Finset.range n).image (q∘e))
-        (fun t b=>μ t b Q) H b<visitTime κ b z})) := D.path.mono fun b hb=>
-    all_past_mass_eq_iSup e q he κ b (fun v=>μ v b) hb Q hQ.measurableSet H
-  exact (AEMeasurable.iSup (fun n=>D.compact_past_aemeasurable Q hQ _
-    (D.test_stopping Q hQ.measurableSet _ H))).congr heq.symm
-
-end TracePalmData
-
- 
-
-theorem actual_Palm_data (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
-      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
-      TracePalmData κ (Target.a κ) (W.map driverHomeomorph)
-        (recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc) := by
-  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
-    actual_canonical_trace_mass κ hκ hκ8 W hW
-  refine ⟨s,c,hc,hs,hsum,hmeas ∞,fun v b=>(hreg v b).1,hprog.adapted,hmean,?_,
-    fun S hS=>(hstop S hS).1,fun S hS=>(hstop S hS).2.2.2.2.1⟩
-  apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-  filter_upwards [hpath] with b hb
-  change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
-    (fun v=>recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v (driverHomeomorph b))
-  simpa only [Homeomorph.symm_apply_apply] using hb.1
-
-end SLEExactGauge.PastMass
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceMassContinuity
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions TraceMassIntegration
-open LoewnerRestart PastMass
-
-structure CanonicalMassData (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane) : Prop extends TracePalmData κ a W μ where
-  regular : ∀v b,(μ v b).Regular
-  initial_zero : ∀b, μ 0 b=0
-  progressive : IsProgressive rawFiltration (fun t : ℝ≥0=>μ t)
-  optional_test_moments : ∀(S : GDriver→ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) (m : ℕ), MemLp (fun b=>∫z,f z∂μ (S b) b) (m:ℝ≥0∞) W
-  optional_compact_moments : ∀(S : GDriver→ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S)
-    (K : Set UpperPlane), IsCompact K→∀m : ℕ,MemLp (fun b=>(μ (S b) b K).toReal) (m:ℝ≥0∞) W
-  restart : ∀(S : GDriver→ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S),
-    ∀ᵐ b∂W,S b≠∞→(μ (S b) b).restrict {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}=0 ∧
-      ∀v : ℝ≥0∞,(μ (S b+v) b).restrict {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}=
-        restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ v (restartAt S b))
-  positive_increments : ∀ᵐ b∂W,∀u v : ℝ≥0,u<v→
-    0<(μ v b-μ u b) univ ∧ ∃K : Set UpperPlane,IsCompact K ∧
-      0<(μ v b-μ u b) K ∧ (μ v b-μ u b) K<∞
-
- 
-
-theorem actual_canonical_mass_data (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    ∃(s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
-      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      CanonicalMassData κ (Target.a κ) (a_gt_quarter hκ hκ8) (W.map driverHomeomorph) μ ∧
-      (∀(S : GDriver→ℝ≥0∞),IsStoppingTime rawFiltration S→∀f : C_c(UpperPlane,ℝ),
-        Tendsto (fun k=>eLpNorm (fun b=>(∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-          (Real.exp (-2*(Target.a κ)*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b))
-            1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
-      (∀ᵐ b∂(W.map driverHomeomorph),∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),
-        Tendsto (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-          (Real.exp (-2*(Target.a κ)*(j+1))) v b) atTop (𝓝 (∫z,f z∂μ v b))) := by
-  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
-    actual_canonical_trace_mass κ hκ hκ8 W hW
-  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  have hp : ∀ᵐ b∂(W.map driverHomeomorph),
-      TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b) ∧
-      (∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),Tendsto
-        (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-          (Real.exp (-2*(Target.a κ)*(j+1))) v b) atTop (𝓝 (∫z,f z∂μ v b))) ∧
-      (∀u v : ℝ≥0,u<v→0<(μ v b-μ u b) univ ∧ ∃K : Set UpperPlane,IsCompact K ∧
-        0<(μ v b-μ u b) K ∧ (μ v b-μ u b) K<∞) := by
-    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-    filter_upwards [hpath] with b hb
-    change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
-      (fun v=>μ v (driverHomeomorph b)) ∧ _
-    simp only [show driverMeasurableEquiv b=driverHomeomorph b from rfl,
-      Homeomorph.symm_apply_apply]
-    exact hb
-  refine ⟨s,c,hc,hs,hsum,?_,?_,hp.mono (fun b hb=>hb.2.1)⟩
-  · exact {
-      terminal_measurable := hmeas ∞
-      finite_compact := fun v b=>(hreg v b).1
-      adapted := hprog.adapted
-      mean := hmean
-      path := hp.mono (fun b hb=>hb.1)
-      optional_measurable := fun S hS=>(hstop S hS).1
-      optional_kernel := fun S hS=>(hstop S hS).2.2.2.2.1
-      regular := fun v b=>(hreg v b).2
-      initial_zero := hzero
-      progressive := hprog
-      optional_test_moments := fun S hS=>(hstop S hS).2.1
-      optional_compact_moments := fun S hS=>(hstop S hS).2.2.1
-      restart := fun S hS=>(hstop S hS).2.2.2.2.2
-      positive_increments := hp.mono (fun b hb=>hb.2.2) }
-  · exact fun S hS=>(hstop S hS).2.2.2.1
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-open TraceMassIntegration TraceMassContinuity CompactSequence
-
- 
-
-theorem actual_subdisk_positive_moments (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
-      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      CanonicalMassData κ (Target.a κ) (a_gt_quarter hκ hκ8) (W.map driverHomeomorph) μ ∧
-      (∀v b,IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) ∧
-      (∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
-        Measurable[hS.measurableSpace] (fun b=>μ (S b) b)) ∧
-      (∀ᵐ b∂W,TraceMassFacts κ b (fun v=>μ v (driverHomeomorph b)) ∧
-        (∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),Tendsto
-          (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-             (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
-          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
-        (∀u v : ℝ≥0,u<v →
-          0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) univ)) ∧
-      ∃n₀ : ℕ, let r := subdiskRadius n₀
-      ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
-        (∀n,Q.map (initialVector n)=(finiteAngleLaw (W.map driverHomeomorph) (Target.a κ) n).map
-          (fun p=>fun i : Fin (n+1)=>subdiskMass (Target.a κ) μ r i p)) ∧
-        (∀ᵐ x∂Q,Monotone x ∧ Tendsto x atTop (𝓝 (diskSequenceMass x)) ∧ diskSequenceMass x<∞) ∧
-        (∀ m : ℕ, 1 ≤ m →
-          0<(∫⁻x,(diskSequenceMass x)^m∂Q) ∧
-          (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ ∧
-          (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,subdiskMoment (W.map driverHomeomorph) (Target.a κ) μ r m l ∧
-          Tendsto (subdiskMoment (W.map driverHomeomorph) (Target.a κ) μ r m) atTop
-            (𝓝 (∫⁻x,(diskSequenceMass x)^m∂Q))) := by
-  obtain ⟨s,c,hc,hs,hsum,D,hL,hvAll⟩ :=
-    actual_canonical_mass_data κ hκ hκ8 W hW
-  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  let V := W.map driverHomeomorph
-  have hBW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) V := brownian_map_driver W hW
-  have : IsProbabilityMeasure V := hBW.isGaussianProcess.isProbabilityMeasure
-  have hreg : ∀v b,IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular :=
-    fun v b=>⟨D.finite_compact v b,D.regular v b⟩
-  have (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (μ v b) := D.finite_compact v b
-  have hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) :=
-    recoveredMass_joint_measurable (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  have hraw (l : ℕ) (θ : Angle) :
-      Measurable[(centeredTime_stopping (Target.a κ) (a_gt_quarter hκ hκ8) θ l).measurableSpace]
-        (fun b=>μ (levelTime (Target.a κ) θ.1 b.1 l) b) := D.optional_measurable _ _
-  have hp : ∀ᵐ b∂V,TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b) := D.path
-  have hv : ∀ᵐ b∂V,∀f : C_c(UpperPlane,ℝ),Tendsto
-      (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) ∞ b) atTop (𝓝 (∫z,f z∂μ ∞ b)) :=
-    hvAll.mono (fun b hb=>hb ∞)
-  have hmono : ∀ᵐ b∂V,Monotone (fun v=>μ v b) := hp.mono (fun b hb=>hb.monotone)
-  obtain ⟨n₀,hn₀⟩ := exists_subdisk_positive_mean V hBW (Target.a κ)
-    (a_gt_quarter hκ hκ8) μ hμ (hp.mono (fun b hb=>hb.initial_positive))
-  let r := subdiskRadius n₀
-  have hr : 0≤r := (subdiskRadius_pos n₀).le
-  have hr1 : r<1 := subdiskRadius_lt_one n₀
-  obtain ⟨Q,hQ,hlaw,hQmono⟩ := exists_subdisk_sequence_law V hBW r (Target.a κ)
-    (a_gt_quarter hκ hκ8) μ hμ hraw hmono
-  have he (m : ℕ) : (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,subdiskMoment V (Target.a κ) μ r m l :=
-    subdiskSequence_moment V r (Target.a κ) (a_gt_quarter hκ hκ8) μ hμ Q hlaw hQmono m
-  have hf (m : ℕ) (hm : 1 ≤ m) : (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ := by
-    obtain ⟨D,hD,hDb⟩ := common_limit_subdisk_mixed_moment_bound r hr hr1 (Target.a κ) (a_gt_quarter hκ hκ8) m hm
-    rw [he]
-    apply lt_of_le_of_lt (iSup_le (fun l=> ?_)) hD
-    exact hDb V hBW s c hc hsum μ (fun v b=>(hreg v b).1) hμ
-      (hmono.mono (fun b hb v=>hb le_top)) hv l
-  have hpos1 : 0<∫⁻x,diskSequenceMass x∂Q := by
-    have hh := he 1
-    simp only [pow_one] at hh
-    rw [hh]
-    apply hn₀.trans_le
-    simpa only [subdiskMoment,pow_one] using
-      le_iSup (fun l=>subdiskMoment V (Target.a κ) μ r 1 l) 1
-  have hpos (m : ℕ) (hm : 1 ≤ m) : 0<∫⁻x,(diskSequenceMass x)^m∂Q := by
-    have heq : Function.support (fun x=>diskSequenceMass x^m)=Function.support diskSequenceMass := by
-      ext x; simp only [Function.mem_support, ne_eq, pow_eq_zero_iff (by omega : m≠0)]
-    rw [lintegral_pos_iff_support (diskSequenceMass_measurable.pow_const m), heq]
-    exact (lintegral_pos_iff_support diskSequenceMass_measurable).mp hpos1
-  have hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞ := ae_lt_top (diskSequenceMass_measurable)
-    (show (∫⁻x,diskSequenceMass x∂Q)≠∞ by simpa only [pow_one] using (hf 1 le_rfl).ne)
-  refine ⟨s,c,hc,hs,hsum,D,hreg,hμ,D.optional_measurable,?_,n₀,Q,hQ,hlaw,?_,?_⟩
-  · have hboth := hp.and (hvAll.and D.positive_increments)
-    have ht := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp hboth
-    filter_upwards [ht] with b hb
-    have he : driverMeasurableEquiv b=driverHomeomorph b := rfl
-    simp only [he,Homeomorph.symm_apply_apply] at hb
-    exact ⟨hb.1,hb.2.1,fun u v huv=>(hb.2.2 u v huv).1⟩
-  · filter_upwards [hQmono,hfin] with x hx hxf
-    exact ⟨hx,diskSequenceMass_tendsto hx,hxf⟩
-  · intro m hm
-    refine ⟨hpos m hm,hf m hm,he m,?_⟩
-    rw [he]
-    exact tendsto_atTop_iSup (subdiskMoment_monotone V hBW r (Target.a κ)
-      (a_gt_quarter hκ hκ8) μ hμ hraw hmono m)
-
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.SubdiskMoments
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.PastMass
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
+open TraceMassContinuity MassVersions TraceMassIntegration
+
+lemma optional_green_set_balance (a : ℝ) (S : GDriver→ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (W : Measure GDriver)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hH : ∀H : GDriver × UpperPlane→ℝ≥0∞,
+      Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
+      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=
+        (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
+        ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) :
+    (∫⁻b, μ ∞ b Q∂W) = (∫⁻b, μ (S b) b Q∂W)+
+      ∫⁻b,∫⁻z in Q, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W := by
+  let H : GDriver × UpperPlane→ℝ≥0∞ := fun p=> Q.indicator (fun _=>1) p.2
+  have hm : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H :=
+    (measurable_const.indicator hQ).comp measurable_snd
+  have he (ν : Measure UpperPlane) (b : GDriver) : (∫⁻z,H (b,z)∂ν)=ν Q := by
+    simp only [H,lintegral_indicator hQ,lintegral_one,Measure.restrict_apply_univ]
+  have hg (b : GDriver) :
+      (∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea)=
+        ∫⁻z in Q,ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea := by
+    rw [show (fun z => H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1))=
+      Q.indicator (fun z=>ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)) from by
+        funext z; by_cases hz : z∈Q <;> simp [H,hz], lintegral_indicator hQ]
+  simpa only [he,hg] using hH H hm
+
+lemma expected_green_past_kernel (a : ℝ) (ha : 1/4<a) (W : Measure GDriver) [SFinite W]
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (Q : Set UpperPlane) :
+    (∫⁻b,∫⁻z in Q,ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W)=
+      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a S z∂upperArea := by
+  simp_rw [initial_mul_pastGreenKernel]
+  exact lintegral_lintegral_swap (stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal.aemeasurable
+
+ 
+
+lemma strict_past_mass_ae (κ : ℝ) (W : Measure GDriver)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hp : ∀ᵐ b∂W, TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b))
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : ∀b, μ ∞ b Q≠∞)
+    (S : GDriver→ℝ≥0∞) :
+    (fun b=>μ ∞ b (Q∩{z | S b<visitTime κ b z})) =ᵐ[W]
+      (fun b=>μ ∞ b Q-μ (S b) b Q) :=
+  hp.mono fun b hb=>stopped_strictPast_mass κ b (fun v=>μ v b) hb Q hQ (hfin b) (S b)
+
+lemma strict_past_mass_aemeasurable (κ : ℝ) (W : Measure GDriver)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hp : ∀ᵐ b∂W, TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b))
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : ∀b, μ ∞ b Q≠∞)
+    (S : GDriver→ℝ≥0∞) (hm : Measurable (μ ∞))
+    (hsm : Measurable (fun b=>μ (S b) b)) :
+    AEMeasurable (fun b=>μ ∞ b (Q∩{z | S b<visitTime κ b z})) W :=
+  ((Measure.measurable_coe hQ).comp hm |>.sub ((Measure.measurable_coe hQ).comp hsm)).aemeasurable.congr
+    (strict_past_mass_ae κ W μ hp Q hQ hfin S).symm
+
+ 
+
+lemma strict_past_Palm (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver) [SFinite W]
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hp : ∀ᵐ b∂W, TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b))
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (hsm : Measurable (fun b=>μ (S b) b))
+    (hH : ∀H : GDriver × UpperPlane→ℝ≥0∞,
+      Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
+      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=
+        (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
+        ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : ∀b, μ ∞ b Q≠∞)
+    (hf : (∫⁻b,μ ∞ b Q∂W)≠∞) :
+    (∫⁻b,μ ∞ b (Q∩{z | S b<visitTime κ b z})∂W)=
+      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a S z∂upperArea := by
+  have hle : (fun b=>μ (S b) b Q) ≤ᵐ[W] (fun b=>μ ∞ b Q) :=
+    hp.mono fun b hb=>hb.monotone le_top Q
+  have hsf : (∫⁻b,μ (S b) b Q∂W)≠∞ := ne_top_of_le_ne_top hf (lintegral_mono_ae hle)
+  have hmS : Measurable (fun b=>μ (S b) b Q) := (Measure.measurable_coe hQ).comp hsm
+  rw [lintegral_congr_ae (strict_past_mass_ae κ W μ hp Q hQ hfin S),
+    lintegral_sub hmS hsf hle,
+    optional_green_set_balance a S hS W μ hH Q hQ, ENNReal.add_sub_cancel_left hsf]
+  exact expected_green_past_kernel a ha W S hS Q
+
+end SLEExactGauge.PastMass
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.PastMass
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
+open TraceMassContinuity MassVersions TraceMassIntegration
+
+ 
+
+structure TracePalmData (κ a : ℝ) (W : Measure GDriver)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane) : Prop where
+  terminal_measurable : Measurable (μ ∞)
+  finite_compact : ∀v b, IsFiniteMeasureOnCompacts (μ v b)
+  adapted : Adapted rawFiltration (fun t : ℝ≥0=>μ t)
+  mean : μ ∞ ∘ₘ W=initialGreenMeasure a
+  path : ∀ᵐ b∂W,TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b)
+  optional_measurable : ∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
+    Measurable[hS.measurableSpace] (fun b=>μ (S b) b)
+  optional_kernel : ∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (H : GDriver × UpperPlane→ℝ≥0∞),
+    Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
+      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=
+        (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂W)+
+        ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)∂upperArea∂W
+
+ 
+
+def countablePastKernel {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0)
+    (W : Measure GDriver) (a : ℝ) (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (Q : Set UpperPlane) (H : ℝ≥0∞) (z : UpperPlane) : ℝ≥0∞ :=
+  ⨆n : ℕ, pastGreenKernel W a
+    (finiteTestTime ((Finset.range n).image (q∘e)) (fun t b=>μ t b Q) H) z
+
+namespace TracePalmData
+variable {κ a : ℝ} {W : Measure GDriver} {μ : ℝ≥0∞→GDriver→Measure UpperPlane}
+    (D : TracePalmData κ a W μ)
+include D
+
+lemma evaluation_adapted (Q : Set UpperPlane) (hQ : MeasurableSet Q) :
+    Adapted rawFiltration (fun t : ℝ≥0=>fun b=>μ t b Q) :=
+  fun t=>(Measure.measurable_coe hQ).comp (D.adapted t)
+
+lemma test_stopping (Q : Set UpperPlane) (hQ : MeasurableSet Q) (F : Finset ℝ≥0) (H : ℝ≥0∞) :
+    IsStoppingTime rawFiltration (finiteTestTime F (fun t b=>μ t b Q) H) :=
+  finiteTestTime_stopping _ _ (D.evaluation_adapted Q hQ) F H
+
+lemma compact_expected_finite (Q : Set UpperPlane) (hQ : IsCompact Q) :
+    (∫⁻b,μ ∞ b Q∂W)≠∞ := by
+  rw [←Measure.bind_apply hQ.measurableSet D.terminal_measurable.aemeasurable,D.mean]
+  exact hQ.measure_lt_top.ne
+
+lemma compact_finite (Q : Set UpperPlane) (hQ : IsCompact Q) (v : ℝ≥0∞) (b : GDriver) :
+    μ v b Q≠∞ := by
+  let := D.finite_compact v b
+  exact hQ.measure_lt_top.ne
+
+lemma stopped_Palm (ha : 1/4<a) [SFinite W] (S : GDriver→ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (Q : Set UpperPlane) (hQ : IsCompact Q) :
+    (∫⁻b,μ ∞ b (Q∩{z | S b<visitTime κ b z})∂W)=
+      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a S z∂upperArea :=
+  strict_past_Palm κ a ha W μ D.path S hS
+    ((D.optional_measurable S hS).mono hS.measurableSpace_le le_rfl)
+    (D.optional_kernel S hS) Q hQ.measurableSet (D.compact_finite Q hQ ∞)
+    (D.compact_expected_finite Q hQ)
+
+lemma finite_Palm (ha : 1/4<a) [SFinite W] (Q : Set UpperPlane) (hQ : IsCompact Q)
+    (F : Finset ℝ≥0) (H : ℝ≥0∞) :
+    (∫⁻b,μ ∞ b (Q∩{z | ∃q∈F,H<μ q b Q ∧ (q:ℝ≥0∞)<visitTime κ b z})∂W)=
+      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*
+        pastGreenKernel W a (finiteTestTime F (fun t b=>μ t b Q) H) z∂upperArea := by
+  simpa only [finiteTestTime_lt_iff] using
+    D.stopped_Palm ha _ (D.test_stopping Q hQ.measurableSet F H) Q hQ
+
+lemma compact_past_aemeasurable (Q : Set UpperPlane) (hQ : IsCompact Q)
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    AEMeasurable (fun b=>μ ∞ b (Q∩{z | S b<visitTime κ b z})) W :=
+  strict_past_mass_aemeasurable κ W μ D.path Q hQ.measurableSet
+    (D.compact_finite Q hQ ∞) S D.terminal_measurable
+    ((D.optional_measurable S hS).mono hS.measurableSpace_le le_rfl)
+
+lemma countable_kernel_measurable (ha : 1/4<a) [SFinite W]
+    {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0) (Q : Set UpperPlane) (hQ : MeasurableSet Q)
+    (H : ℝ≥0∞) : Measurable (countablePastKernel e q W a μ Q H) :=
+  .iSup fun _=>pastGreenKernel_measurable W a ha _ (D.test_stopping Q hQ _ H)
+
+ 
+
+lemma countable_Palm (ha : 1/4<a) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0) (he : Function.Surjective e)
+    (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ≥0∞) :
+    (∫⁻b,μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})∂W)=
+      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*countablePastKernel e q W a μ Q H z∂upperArea := by
+  let S (n : ℕ) := finiteTestTime ((Finset.range n).image (q∘e)) (fun t b=>μ t b Q) H
+  have hS (n : ℕ) : IsStoppingTime rawFiltration (S n) := D.test_stopping Q hQ.measurableSet _ H
+  let A (n : ℕ) (b : GDriver) := μ ∞ b (Q∩{z | S n b<visitTime κ b z})
+  have hAm (n : ℕ) : AEMeasurable (A n) W := D.compact_past_aemeasurable Q hQ (S n) (hS n)
+  have hmono : ∀b,Monotone (fun n=>A n b) := by
+    intro b n k hnk
+    apply measure_mono
+    intro z hz
+    exact ⟨hz.1,(finiteTestTime_sequence_antitone e q (fun t b=>μ t b Q) H b hnk).trans_lt hz.2⟩
+  have heq : (fun b=>μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})) =ᵐ[W]
+      (fun b=>⨆n,A n b) := D.path.mono fun b hb=>
+    all_past_mass_eq_iSup e q he κ b (fun v=>μ v b) hb Q hQ.measurableSet H
+  have hKmono : Monotone (fun n z=>ENNReal.ofReal (initialGreen a z)*pastGreenKernel W a (S n) z) := by
+    intro n k hnk z
+    apply mul_le_mul_right
+    exact pastGreenKernel_antitone W hW a ha (S k) (S n) (hS k) (hS n)
+      (fun b=>finiteTestTime_sequence_antitone e q (fun t b=>μ t b Q) H b hnk) z
+  rw [lintegral_congr_ae heq,lintegral_iSup' hAm (Eventually.of_forall hmono)]
+  simp only [A,D.stopped_Palm ha _ (hS _) Q hQ]
+  simp only [countablePastKernel,ENNReal.mul_iSup]
+  exact (lintegral_iSup (fun n=>(initialGreen_continuous a).measurable.ennreal_ofReal.mul
+    (pastGreenKernel_measurable W a ha (S n) (hS n))) hKmono).symm
+
+lemma countable_past_aemeasurable {ι : Type*} (e : ℕ→ι) (q : ι→ℝ≥0)
+    (he : Function.Surjective e) (Q : Set UpperPlane) (hQ : IsCompact Q) (H : ℝ≥0∞) :
+    AEMeasurable (fun b=>μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})) W := by
+  have heq : (fun b=>μ ∞ b (Q∩{z | ∃i,H<μ (q i) b Q ∧ (q i:ℝ≥0∞)<visitTime κ b z})) =ᵐ[W]
+      (fun b=>⨆n : ℕ,μ ∞ b (Q∩{z | finiteTestTime ((Finset.range n).image (q∘e))
+        (fun t b=>μ t b Q) H b<visitTime κ b z})) := D.path.mono fun b hb=>
+    all_past_mass_eq_iSup e q he κ b (fun v=>μ v b) hb Q hQ.measurableSet H
+  exact (AEMeasurable.iSup (fun n=>D.compact_past_aemeasurable Q hQ _
+    (D.test_stopping Q hQ.measurableSet _ H))).congr heq.symm
+
+end TracePalmData
+
+ 
+
+theorem actual_Palm_data (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
+      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
+      TracePalmData κ (Target.a κ) (W.map driverHomeomorph)
+        (recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc) := by
+  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
+    actual_canonical_trace_mass κ hκ hκ8 W hW
+  refine ⟨s,c,hc,hs,hsum,hmeas ∞,fun v b=>(hreg v b).1,hprog.adapted,hmean,?_,
+    fun S hS=>(hstop S hS).1,fun S hS=>(hstop S hS).2.2.2.2.1⟩
+  apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+  filter_upwards [hpath] with b hb
+  change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
+    (fun v=>recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v (driverHomeomorph b))
+  simpa only [Homeomorph.symm_apply_apply] using hb.1
+
+end SLEExactGauge.PastMass
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceMassContinuity
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions TraceMassIntegration
+open LoewnerRestart PastMass
+
+structure CanonicalMassData (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane) : Prop extends TracePalmData κ a W μ where
+  regular : ∀v b,(μ v b).Regular
+  initial_zero : ∀b, μ 0 b=0
+  progressive : IsProgressive rawFiltration (fun t : ℝ≥0=>μ t)
+  optional_test_moments : ∀(S : GDriver→ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) (m : ℕ), MemLp (fun b=>∫z,f z∂μ (S b) b) (m:ℝ≥0∞) W
+  optional_compact_moments : ∀(S : GDriver→ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S)
+    (K : Set UpperPlane), IsCompact K→∀m : ℕ,MemLp (fun b=>(μ (S b) b K).toReal) (m:ℝ≥0∞) W
+  restart : ∀(S : GDriver→ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S),
+    ∀ᵐ b∂W,S b≠∞→(μ (S b) b).restrict {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}=0 ∧
+      ∀v : ℝ≥0∞,(μ (S b+v) b).restrict {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}=
+        restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ v (restartAt S b))
+  positive_increments : ∀ᵐ b∂W,∀u v : ℝ≥0,u<v→
+    0<(μ v b-μ u b) univ ∧ ∃K : Set UpperPlane,IsCompact K ∧
+      0<(μ v b-μ u b) K ∧ (μ v b-μ u b) K<∞
+
+ 
+
+theorem actual_canonical_mass_data (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    ∃(s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
+      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      CanonicalMassData κ (Target.a κ) (a_gt_quarter hκ hκ8) (W.map driverHomeomorph) μ ∧
+      (∀(S : GDriver→ℝ≥0∞),IsStoppingTime rawFiltration S→∀f : C_c(UpperPlane,ℝ),
+        Tendsto (fun k=>eLpNorm (fun b=>(∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+          (Real.exp (-2*(Target.a κ)*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b))
+            1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
+      (∀ᵐ b∂(W.map driverHomeomorph),∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),
+        Tendsto (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+          (Real.exp (-2*(Target.a κ)*(j+1))) v b) atTop (𝓝 (∫z,f z∂μ v b))) := by
+  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
+    actual_canonical_trace_mass κ hκ hκ8 W hW
+  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  have hp : ∀ᵐ b∂(W.map driverHomeomorph),
+      TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b) ∧
+      (∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),Tendsto
+        (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+          (Real.exp (-2*(Target.a κ)*(j+1))) v b) atTop (𝓝 (∫z,f z∂μ v b))) ∧
+      (∀u v : ℝ≥0,u<v→0<(μ v b-μ u b) univ ∧ ∃K : Set UpperPlane,IsCompact K ∧
+        0<(μ v b-μ u b) K ∧ (μ v b-μ u b) K<∞) := by
+    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+    filter_upwards [hpath] with b hb
+    change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
+      (fun v=>μ v (driverHomeomorph b)) ∧ _
+    simp only [show driverMeasurableEquiv b=driverHomeomorph b from rfl,
+      Homeomorph.symm_apply_apply]
+    exact hb
+  refine ⟨s,c,hc,hs,hsum,?_,?_,hp.mono (fun b hb=>hb.2.1)⟩
+  · exact {
+      terminal_measurable := hmeas ∞
+      finite_compact := fun v b=>(hreg v b).1
+      adapted := hprog.adapted
+      mean := hmean
+      path := hp.mono (fun b hb=>hb.1)
+      optional_measurable := fun S hS=>(hstop S hS).1
+      optional_kernel := fun S hS=>(hstop S hS).2.2.2.2.1
+      regular := fun v b=>(hreg v b).2
+      initial_zero := hzero
+      progressive := hprog
+      optional_test_moments := fun S hS=>(hstop S hS).2.1
+      optional_compact_moments := fun S hS=>(hstop S hS).2.2.1
+      restart := fun S hS=>(hstop S hS).2.2.2.2.2
+      positive_increments := hp.mono (fun b hb=>hb.2.2) }
+  · exact fun S hS=>(hstop S hS).2.2.2.1
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+open TraceMassIntegration TraceMassContinuity CompactSequence
+
+ 
+
+theorem actual_subdisk_positive_moments (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
+      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      CanonicalMassData κ (Target.a κ) (a_gt_quarter hκ hκ8) (W.map driverHomeomorph) μ ∧
+      (∀v b,IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) ∧
+      (∀(S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
+        Measurable[hS.measurableSpace] (fun b=>μ (S b) b)) ∧
+      (∀ᵐ b∂W,TraceMassFacts κ b (fun v=>μ v (driverHomeomorph b)) ∧
+        (∀v : ℝ≥0∞,∀f : C_c(UpperPlane,ℝ),Tendsto
+          (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+             (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
+          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
+        (∀u v : ℝ≥0,u<v →
+          0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) univ)) ∧
+      ∃n₀ : ℕ, let r := subdiskRadius n₀
+      ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
+        (∀n,Q.map (initialVector n)=(finiteAngleLaw (W.map driverHomeomorph) (Target.a κ) n).map
+          (fun p=>fun i : Fin (n+1)=>subdiskMass (Target.a κ) μ r i p)) ∧
+        (∀ᵐ x∂Q,Monotone x ∧ Tendsto x atTop (𝓝 (diskSequenceMass x)) ∧ diskSequenceMass x<∞) ∧
+        (∀ m : ℕ, 1 ≤ m →
+          0<(∫⁻x,(diskSequenceMass x)^m∂Q) ∧
+          (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ ∧
+          (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,subdiskMoment (W.map driverHomeomorph) (Target.a κ) μ r m l ∧
+          Tendsto (subdiskMoment (W.map driverHomeomorph) (Target.a κ) μ r m) atTop
+            (𝓝 (∫⁻x,(diskSequenceMass x)^m∂Q))) := by
+  obtain ⟨s,c,hc,hs,hsum,D,hL,hvAll⟩ :=
+    actual_canonical_mass_data κ hκ hκ8 W hW
+  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  let V := W.map driverHomeomorph
+  have hBW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) V := brownian_map_driver W hW
+  have : IsProbabilityMeasure V := hBW.isGaussianProcess.isProbabilityMeasure
+  have hreg : ∀v b,IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular :=
+    fun v b=>⟨D.finite_compact v b,D.regular v b⟩
+  have (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (μ v b) := D.finite_compact v b
+  have hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2) :=
+    recoveredMass_joint_measurable (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  have hraw (l : ℕ) (θ : Angle) :
+      Measurable[(centeredTime_stopping (Target.a κ) (a_gt_quarter hκ hκ8) θ l).measurableSpace]
+        (fun b=>μ (levelTime (Target.a κ) θ.1 b.1 l) b) := D.optional_measurable _ _
+  have hp : ∀ᵐ b∂V,TraceMassFacts κ (driverHomeomorph.symm b) (fun v=>μ v b) := D.path
+  have hv : ∀ᵐ b∂V,∀f : C_c(UpperPlane,ℝ),Tendsto
+      (fun k=>∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) ∞ b) atTop (𝓝 (∫z,f z∂μ ∞ b)) :=
+    hvAll.mono (fun b hb=>hb ∞)
+  have hmono : ∀ᵐ b∂V,Monotone (fun v=>μ v b) := hp.mono (fun b hb=>hb.monotone)
+  obtain ⟨n₀,hn₀⟩ := exists_subdisk_positive_mean V hBW (Target.a κ)
+    (a_gt_quarter hκ hκ8) μ hμ (hp.mono (fun b hb=>hb.initial_positive))
+  let r := subdiskRadius n₀
+  have hr : 0≤r := (subdiskRadius_pos n₀).le
+  have hr1 : r<1 := subdiskRadius_lt_one n₀
+  obtain ⟨Q,hQ,hlaw,hQmono⟩ := exists_subdisk_sequence_law V hBW r (Target.a κ)
+    (a_gt_quarter hκ hκ8) μ hμ hraw hmono
+  have he (m : ℕ) : (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,subdiskMoment V (Target.a κ) μ r m l :=
+    subdiskSequence_moment V r (Target.a κ) (a_gt_quarter hκ hκ8) μ hμ Q hlaw hQmono m
+  have hf (m : ℕ) (hm : 1 ≤ m) : (∫⁻x,(diskSequenceMass x)^m∂Q)<∞ := by
+    obtain ⟨D,hD,hDb⟩ := common_limit_subdisk_mixed_moment_bound r hr hr1 (Target.a κ) (a_gt_quarter hκ hκ8) m hm
+    rw [he]
+    apply lt_of_le_of_lt (iSup_le (fun l=> ?_)) hD
+    exact hDb V hBW s c hc hsum μ (fun v b=>(hreg v b).1) hμ
+      (hmono.mono (fun b hb v=>hb le_top)) hv l
+  have hpos1 : 0<∫⁻x,diskSequenceMass x∂Q := by
+    have hh := he 1
+    simp only [pow_one] at hh
+    rw [hh]
+    apply hn₀.trans_le
+    simpa only [subdiskMoment,pow_one] using
+      le_iSup (fun l=>subdiskMoment V (Target.a κ) μ r 1 l) 1
+  have hpos (m : ℕ) (hm : 1 ≤ m) : 0<∫⁻x,(diskSequenceMass x)^m∂Q := by
+    have heq : Function.support (fun x=>diskSequenceMass x^m)=Function.support diskSequenceMass := by
+      ext x; simp only [Function.mem_support, ne_eq, pow_eq_zero_iff (by omega : m≠0)]
+    rw [lintegral_pos_iff_support (diskSequenceMass_measurable.pow_const m), heq]
+    exact (lintegral_pos_iff_support diskSequenceMass_measurable).mp hpos1
+  have hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞ := ae_lt_top (diskSequenceMass_measurable)
+    (show (∫⁻x,diskSequenceMass x∂Q)≠∞ by simpa only [pow_one] using (hf 1 le_rfl).ne)
+  refine ⟨s,c,hc,hs,hsum,D,hreg,hμ,D.optional_measurable,?_,n₀,Q,hQ,hlaw,?_,?_⟩
+  · have hboth := hp.and (hvAll.and D.positive_increments)
+    have ht := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp hboth
+    filter_upwards [ht] with b hb
+    have he : driverMeasurableEquiv b=driverHomeomorph b := rfl
+    simp only [he,Homeomorph.symm_apply_apply] at hb
+    exact ⟨hb.1,hb.2.1,fun u v huv=>(hb.2.2 u v huv).1⟩
+  · filter_upwards [hQmono,hfin] with x hx hxf
+    exact ⟨hx,diskSequenceMass_tendsto hx,hxf⟩
+  · intro m hm
+    refine ⟨hpos m hm,hf m hm,he m,?_⟩
+    rw [he]
+    exact tendsto_atTop_iSup (subdiskMoment_monotone V hBW r (Target.a κ)
+      (a_gt_quarter hκ hκ8) μ hμ hraw hmono m)
+
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/PastHullSupport.lean b/lean/OAI/Probability/SLEGauge/PastHullSupport.lean
index 79ea63975d7c712bdefb1e663fe1fd6a0698f7cf..d2fa5c13c9496205fbb94ace5c2a0fdb6cf43033 100644
--- a/lean/OAI/Probability/SLEGauge/PastHullSupport.lean
+++ b/lean/OAI/Probability/SLEGauge/PastHullSupport.lean
@@ -1,508 +1,508 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.HeightRadius
-import OAI.Probability.SLEGauge.AngularControlMass
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma fullLoewnerRadius_continuousOn (a : ℝ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    ContinuousOn (fun z => fullLoewnerRadius a b.1 z t) {z | alive a b.1 t z} := by
-  have hd : DifferentiableOn ℂ (loewnerMap a b.1 t) {z | alive a b.1 t z} :=
-    fun _ hz => (hasDerivAt_loewnerMap b.1.continuous hz).differentiableAt.differentiableWithinAt
-  have hc := hd.continuousOn
-  exact (continuousOn_const.mul (Complex.continuous_im.comp_continuousOn hc)).div
-    ((hd.analyticOnNhd (isOpen_alive b.1.continuous t)).deriv.continuousOn.norm)
-    (fun z hz => norm_ne_zero_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hz))
-
-lemma physicalRadius_actual_continuousOn (a : ℝ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    ContinuousOn (fun z => physicalRadius (actualState a b.1 t z)) {z | alive a b.1 t z} := by
-  have hc := (fullLoewnerRadius_continuousOn a b t).div_const 2
-  apply hc.congr
-  intro z hz
-  have he := actual_radius_eq_full b.1.continuous hz
-  linarith
-
- 
-
-lemma surviving_compact_radius_margin (a : ℝ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (K : Set UpperPlane) (hK : IsCompact K)
-    (hKs : ∀ z∈K, alive a b.1 t z.1) :
-    ∃ ε : ℝ, 0<ε ∧ ∀ z∈K, ε≤physicalRadius (actualState a b.1 t z.1)/z.1.im := by
-  by_cases hne : K.Nonempty
-  · have hc : ContinuousOn (fun z : UpperPlane =>
-        physicalRadius (actualState a b.1 t z.1)/z.1.im) K := by
-      apply ContinuousOn.div
-        ((physicalRadius_actual_continuousOn a b t).comp continuous_subtype_val.continuousOn hKs)
-        (Complex.continuous_im.comp continuous_subtype_val).continuousOn
-      exact fun z _ => z.2.ne'
-    obtain ⟨z,hz,hmin⟩ := hK.exists_isMinOn hne hc
-    exact ⟨_,div_pos (physicalRadius_actual_pos a b z.1 t (hKs z hz)) z.2,
-      fun w hw => hmin hw⟩
-  · exact ⟨1,zero_lt_one,fun z hz => False.elim (hne ⟨z,hz⟩)⟩
-
-lemma timeMassDensity_zero_of_radius_lt (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (z : UpperPlane) (ht : alive a b.1 t z.1)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (hqr : z.1.im*q<physicalRadius (actualState a b.1 t z.1)) :
-    timeMassDensity a q t b z.1=0 := by
-  have hn : ¬radialTime a b.1 z.1 (z.1.im*q)≤(t : ℝ≥0∞) := by
-    intro hle
-    have hfin : radialTime a b.1 z.1 (z.1.im*q)≠∞ :=
-      ne_top_of_le_ne_top ENNReal.coe_ne_top hle
-    have htt : (radialTime a b.1 z.1 (z.1.im*q)).toNNReal≤t :=
-      ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hle)
-    have he := (radialTime_attained a ha z.1 z.2 _ (mul_pos z.2 hq)
-      (mul_le_of_le_one_right z.2.le hq1) b hfin).2
-    have hr := physicalRadius_actual_anti a ha b z.1 _ t htt ht
-    rw [he] at hr
-    exact (not_le_of_gt hqr) hr
-  exact ite_eq_right hn
-
-lemma timeMassDensity_eventually_zero_on_surviving_compact (a : ℝ) (ha : 0<a)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (K : Set UpperPlane) (hK : IsCompact K)
-    (hKs : ∀ z∈K, alive a b.1 t z.1)
-    (q : ℕ → ℝ) (hq : ∀ j, 0<q j ∧ q j≤1) (hlim : Tendsto q atTop (𝓝 0)) :
-    ∀ᶠ j in atTop, ∀ z∈K, timeMassDensity a (q j) t b z.1=0 := by
-  obtain ⟨ε,hε,hεK⟩ := surviving_compact_radius_margin a b t K hK hKs
-  filter_upwards [hlim.eventually (gt_mem_nhds hε)] with j hj
-  intro z hz
-  exact timeMassDensity_zero_of_radius_lt a ha b t z (hKs z hz) (q j) (hq j).1 (hq j).2
-    (by simpa only [mul_comm] using (lt_div_iff₀ z.2).mp (hj.trans_le (hεK z hz)))
-
-lemma actualUpperMass_eval_eventually_zero_on_surviving (a : ℝ) (ha : 1/4<a)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (f : C_c(UpperPlane,ℝ))
-    (hf : ∀ z∈tsupport f, alive a b.1 t z.1)
-    (q : ℕ → ℝ) (hq : ∀ j, 0<q j ∧ q j≤1) (hlim : Tendsto q atTop (𝓝 0)) :
-    ∀ᶠ j in atTop, (∫z,f z∂actualUpperMass a (q j) t b)=0 := by
-  filter_upwards [timeMassDensity_eventually_zero_on_surviving_compact a (by linarith)
-    b t (tsupport f) f.hasCompactSupport hf q hq hlim] with j hj
-  have hm : Measurable (fun z : UpperPlane => timeMassDensity a (q j) t b z.1) :=
-    (timeMassDensity_measurable a ha (q j) (hq j).1 (hq j).2 t).comp
-      (measurable_const.prodMk measurable_subtype_coe)
-  rw [actualUpperMass,integral_withDensity_eq_integral_toReal_smul hm
-    (Eventually.of_forall (fun z => timeMassDensity_finite a (q j) t b z.1))]
-  apply integral_eq_zero_of_ae
-  apply Eventually.of_forall
-  intro z
-  by_cases hz : z∈tsupport f
-  · simp only [hj z hz,ENNReal.toReal_zero,zero_smul,Pi.zero_apply]
-  · simp only [image_eq_zero_of_notMem_tsupport hz,smul_zero,Pi.zero_apply]
-
-lemma mass_levels_tendsto_zero (a : ℝ) (ha : 0<a) :
-    Tendsto (fun j : ℕ => Real.exp (-2*a*(j+1))) atTop (𝓝 0) := by
-  apply Real.tendsto_exp_atBot.comp
-  apply Tendsto.const_mul_atTop_of_neg (by linarith : -2*a<0)
-  apply Filter.tendsto_atTop.mpr
-  intro B
-  filter_upwards [(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop).eventually
-    (eventually_ge_atTop B)] with n hn
-  linarith
-
- 
-
-lemma actual_limit_integral_zero_on_surviving (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hs : ∀ k j, j∈s k → k≤j)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (μ : Measure UpperPlane)
-    (f : C_c(UpperPlane,ℝ)) (hf : ∀ z∈tsupport f, alive a b.1 t z.1)
-    (hlim : Tendsto (fun k => ∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a
-      (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂μ))) :
-    (∫z,f z∂μ)=0 := by
-  have hq (j : ℕ) : 0<Real.exp (-2*a*(j+1)) ∧ Real.exp (-2*a*(j+1))≤1 :=
-    ⟨Real.exp_pos _,(level_step_lt_one a ha j).le⟩
-  obtain ⟨N,hN⟩ := eventually_atTop.mp (actualUpperMass_eval_eventually_zero_on_surviving
-    a ha b t f hf _ hq (mass_levels_tendsto_zero a (by linarith)))
-  apply tendsto_nhds_unique hlim
-  apply tendsto_const_nhds.congr'
-  filter_upwards [eventually_ge_atTop N] with k hk
-  symm
-  apply Finset.sum_eq_zero
-  intro j hj
-  rw [hN j (hk.trans (hs k j hj)),mul_zero]
-
-end ExactSLEGreen
-namespace SLEExactGauge.MassVersions
-
- 
-
-lemma open_null_of_compact_tests {X : Type*} [TopologicalSpace X] [T2Space X]
-    [LocallyCompactSpace X] [MeasurableSpace X] [BorelSpace X]
-    (μ : Measure X) [μ.Regular]
-    (U : Set X) (hU : IsOpen U)
-    (htest : ∀ f : C_c(X,ℝ), tsupport f⊆U → (∫x,f x∂μ)=0)
-    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) : μ U=0 := by
-  rw [hU.measure_eq_iSup_isCompact]
-  simp only [ENNReal.iSup_eq_zero]
-  intro K hKU hK
-  obtain ⟨f,hfK,hfc,hfU,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK hU hKU
-  let g : C_c(X,ℝ) := ⟨f,hfc⟩
-  have hz : (fun x => f x)=ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg
-    (fun x => (hf01 x).1) (f.continuous.integrable_of_hasCompactSupport hfc)).mp
-      (htest g hfU)
-  rw [measure_eq_zero_iff_ae_notMem]
-  filter_upwards [hz] with x hx hKx
-  have he := hfK hKx
-  change f x=0 at hx
-  change f x=1 at he
-  linarith
-
-end SLEExactGauge.MassVersions
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_limit_past_hull_support (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hs : ∀ k j, j∈s k → k≤j)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (μ : Measure UpperPlane)
-    [μ.Regular]
-    (hlim : ∀ f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
-        atTop (𝓝 (∫z,f z∂μ)))
-    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) :
-    μ {z : UpperPlane | alive a b.1 t z.1}=0 := by
-  apply open_null_of_compact_tests μ _ ((isOpen_alive b.1.continuous t).preimage continuous_subtype_val)
-  exact fun f hf => actual_limit_integral_zero_on_surviving a ha s c hs b t μ f hf (hlim f)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceLevelLocalization
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
-
- 
-def AngularRegular (κ : ℝ) (b : TDriver) : Prop :=
-  ∀ m H : ℕ, ∀ C : ℝ, 0≤C →
-    TendstoUniformlyOn
-      (fun y : ℝ => fun p : ℝ×ℝ => LiteralTrace.inverseFlow κ b p.1
-        (LiteralTrace.driverValue b p.1+(y:ℂ)*((p.2:ℂ)+Complex.I)))
-      (fun p => LiteralTrace.trace κ b (p.1/κ)) (𝓝[>] 0)
-      (Icc (1/((m:ℝ)+1)) H ×ˢ Icc (-C) C)
-
-lemma angularRegular_ae {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    {W : Measure TDriver} (hW : Target.IsWiener W) : ∀ᵐ b ∂W, AngularRegular κ b :=
-  LiteralTrace.angular_uniform_all_windows hκ hκ8 hW
-
-lemma inverse_actual_state (κ : ℝ) (b : TDriver) (z : UpperPlane)
-    (t : ℝ≥0) (ht : alive (Target.a κ) (driverHomeomorph b).1 t z.1) :
-    let v := actualState (Target.a κ) (driverHomeomorph b).1 t z.1
-    LiteralTrace.inverseFlow κ b t
-      (LiteralTrace.driverValue b t+(Y v:ℂ)*((X v/Y v:ℂ)+Complex.I))=z.1 := by
-  let v := actualState (Target.a κ) (driverHomeomorph b).1 t z.1
-  have hy : 0<Y v := by
-    have he := (actualState_evolution (Target.a κ) (driverHomeomorph b) z t ht).yp t
-      ⟨t.coe_nonneg,le_rfl⟩
-    simpa only [Real.toNNReal_coe] using he
-  have hs : LiteralTrace.Surviving κ b z.1 t :=
-    (surviving_eq κ b z.1 t).mp ((surviving_iff_alive κ b z.1 t).mpr ht)
-  have he : (LiteralTrace.driverValue b t:ℂ)+(Y v:ℂ)*((X v/Y v:ℂ)+Complex.I)=
-      LiteralTrace.flow κ b z.1 t := by
-    rw [←flow_eq_literal,flow_eq_loewnerMap]
-    have hxy : Y v*(X v/Y v)=X v := by field_simp
-    apply Complex.ext
-    · simpa [v,X,Y,actualState,LiteralTrace.driverValue] using
-        congrArg (fun r : ℝ => b t+r) hxy
-    · simp [v,X,Y,actualState]
-  change LiteralTrace.inverseFlow κ b t _=z.1
-  rw [he]
-  exact LiteralTrace.inverseFlow_flow t.coe_nonneg hs
-
- 
-
-theorem level_near_trace (κ : ℝ) (hκ : 0<κ) (b : TDriver)
-    (hb : AngularRegular κ b) (T : ℝ≥0) (η M C ε : ℝ)
-    (hη : 0<η) (_hM : 0<M) (hC : 0≤C) (hε : 0<ε) :
-    ∀ᶠ j : ℕ in atTop, ∀ z : UpperPlane, η≤z.1.im → z.1.im≤M →
-      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
-      τ≤(T:ℝ≥0∞) →
-      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
-      |X v/Y v|≤C →
-      dist z.1 (Target.trace κ b ((τ.toNNReal:ℝ)/κ))<ε := by
-  have ha : 0<Target.a κ := by dsimp [Target.a]; positivity
-  obtain ⟨m,hm⟩ := exists_nat_one_div_lt (show 0<η^2/(4*Target.a κ) by positivity)
-  obtain ⟨H,hH⟩ := exists_nat_gt (T:ℝ)
-  obtain ⟨δ,hδ,hδlim⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp
-    (Metric.tendstoUniformlyOn_iff.mp (hb m H C hC) ε hε)
-  have hg : Tendsto (fun j : ℕ => Real.exp (-Target.a κ*(j+1))) atTop (𝓝 0) := by
-    have he := mass_levels_tendsto_zero (Target.a κ/2) (by positivity)
-    convert he using 1
-    funext j
-    congr 1
-    ring
-  have hg' : Tendsto (fun j : ℕ => M*Real.exp (-Target.a κ*(j+1))) atTop (𝓝 0) := by
-    simpa using hg.const_mul M
-  filter_upwards [(mass_levels_tendsto_zero (Target.a κ) ha).eventually
-      (gt_mem_nhds (show (0:ℝ)<1/2 by norm_num)), hg'.eventually (gt_mem_nhds hδ)] with j hj hj'
-  intro z hzη hzM τ hτ v hv
-  have hτf : τ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hτ
-  have hτT : τ.toNNReal≤T := ENNReal.coe_le_coe.mp (by
-    simpa only [ENNReal.coe_toNNReal hτf] using hτ)
-  have hr : 0<z.1.im*Real.exp (-2*Target.a κ*(j+1)) := mul_pos z.2 (Real.exp_pos _)
-  have hr2 : z.1.im*Real.exp (-2*Target.a κ*(j+1))≤z.1.im/2 := by
-    nlinarith only [mul_le_mul_of_nonneg_left hj.le z.2.le]
-  have hτlower := radialTime_half_lower (Target.a κ) ha (driverHomeomorph b) z _ hr hr2 hτf
-  have hτalive := (radialTime_attained (Target.a κ) ha z.1 z.2 _ hr
-    (by linarith [z.2]) (driverHomeomorph b) hτf).1
-  have hyp : 0<Y v := by
-    simpa only [Real.toNNReal_coe] using
-      (actualState_evolution (Target.a κ) (driverHomeomorph b) z τ.toNNReal hτalive).yp
-        (τ.toNNReal:ℝ) ⟨τ.toNNReal.coe_nonneg,le_rfl⟩
-  have hyv : Y v≤z.1.im*Real.exp (-Target.a κ*(j+1)) := by
-    have hh := level_height_bound (Target.a κ) ha (driverHomeomorph b) z (j+1) (by positivity)
-      (by simpa [τ,levelRadius] using hτf)
-    simpa [v,τ,levelRadius] using hh
-  have hyδ : Y v<δ := hyv.trans_lt ((mul_le_mul_of_nonneg_right hzM (Real.exp_pos _).le).trans_lt hj')
-  have htl : 1/((m:ℝ)+1)≤(τ.toNNReal:ℝ) := by
-    apply hm.le.trans
-    apply le_trans _ hτlower
-    exact div_le_div_of_nonneg_right (pow_le_pow_left₀ hη.le hzη 2) (by positivity)
-  have htu : (τ.toNNReal:ℝ)≤H := (show (τ.toNNReal:ℝ)≤T from hτT).trans hH.le
-  have h := hδlim ⟨hyp,hyδ⟩ (τ.toNNReal,X v/Y v) ⟨⟨htl,htu⟩,abs_le.mp hv⟩
-  simp only [Complex.ofReal_div] at h
-  rw [inverse_actual_state κ b z τ.toNNReal hτalive] at h
-  simpa only [trace_eq_literal,dist_comm] using h
-
-end SLEExactGauge.TraceLevelLocalization
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-lemma sub_add_cancel_of_locally_finite (μ ν : Measure X)
-    [IsFiniteMeasureOnCompacts μ] (h : μ≤ν) : ν-μ+μ=ν := by
-  let K := CompactExhaustion.choice X
-  apply Measure.ext_of_iUnion_eq_univ K.iUnion_eq
-  intro n
-  have : IsFiniteMeasure (μ.restrict (K n)) :=
-    ⟨by simpa only [Measure.restrict_apply_univ] using (K.isCompact n).measure_lt_top (μ := μ)⟩
-  rw [Measure.restrict_add,
-    Measure.restrict_sub_eq_restrict_sub_restrict (K.isCompact n).measurableSet]
-  exact Measure.sub_add_cancel_of_le (Measure.restrict_mono Subset.rfl h)
-
-lemma restriction_of_complementary_support (μ ν : Measure X)
-    [IsFiniteMeasureOnCompacts μ] (h : μ≤ν) (D : Set X) (_hD : MeasurableSet D)
-    (hμ : μ Dᶜ=0) (hrem : (ν-μ) D=0) : μ=ν.restrict D := by
-  have hμD : μ.restrict D=μ := Measure.restrict_eq_self_of_ae_mem (by
-    simpa only [mem_compl_iff,not_not] using measure_eq_zero_iff_ae_notMem.mp hμ)
-  have hremD : (ν-μ).restrict D=0 := Measure.restrict_eq_zero.mpr hrem
-  have he := congrArg (fun ρ : Measure X => ρ.restrict D)
-    (sub_add_cancel_of_locally_finite μ ν h)
-  simpa only [Measure.restrict_add,hremD,hμD,zero_add] using he
-
-end SLEExactGauge.MassVersions
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_raw_mass_hull_accounting (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
-        (∀ i, NullSingletonClass (μ (t i) b)) ∧
-        (∀ i (v : ℝ≥0), t i=v → μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0) ∧
-        (∀ i j, t i≠∞ → t j=∞ →
-          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
-        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-          μ (t i) b=(μ (t j) b).restrict {z : UpperPlane | ¬alive a b.1 v z.1}) := by
-  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
-    actual_raw_atomless_mass_endpoint a ha W hW t
-  let μ := recoveredMass a ha s c hc
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := (hμ v b).1
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : (μ v b).Regular := (hμ v b).2
-  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
-  filter_upwards [hconv] with b hb
-  rcases hb with ⟨horder,hvague,hfuture,hatom⟩
-  have hpast (i : ℕ) (v : ℝ≥0) (hi : t i=v) : μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0 := by
-    apply actual_limit_past_hull_support a ha s c hs b v (μ (t i) b)
-    intro f
-    have he := hvague i f
-    simpa only [hi] using he
-  refine ⟨horder,hvague,hatom,hpast,hfuture,?_⟩
-  intro i j v hi hj
-  have hdead := hfuture i j (hi ▸ ENNReal.coe_ne_top) hj
-  have hreal : (t i).toNNReal=v := by rw [hi,ENNReal.toNNReal_coe]
-  rw [hreal] at hdead
-  have hU : MeasurableSet {z : UpperPlane | alive a b.1 v z.1} := by
-    exact ((isOpen_alive b.1.continuous v).preimage
-      (continuous_subtype_val : Continuous (Subtype.val : UpperPlane → ℂ))).measurableSet
-  apply restriction_of_complementary_support (μ (t i) b) (μ (t j) b)
-    (horder i j (by rw [hj]; exact le_top)) {z : UpperPlane | ¬alive a b.1 v z.1}
-    hU.compl _ hdead
-  simpa only [show {z : UpperPlane | ¬alive a b.1 v z.1} =
-    ({z : UpperPlane | alive a b.1 v z.1})ᶜ from rfl, compl_compl] using hpast i v hi
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceMassIntegration
-open FlowIdentification ExactSLEGenerator ExactSLEMain ExactSLEGreen
-
-lemma a_gt_quarter {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) :
-    1/4<Target.a κ := by
-  unfold Target.a
-  apply (lt_div_iff₀ hκ).mpr
-  linarith
-
- 
-
-lemma past_trace_subset_dead {κ : ℝ} (hκ : 0<κ) (b : TDriver)
-    (hlim : ∀ v : ℝ, 0≤v → Tendsto (fun y : ℝ => LiteralTrace.inverseFlow κ b (κ*v)
-      (LiteralTrace.driverValue b (κ*v)+y*Complex.I)) (𝓝[>] 0)
-        (𝓝 (LiteralTrace.trace κ b v))) {t : ℝ} (ht : 0≤t) :
-    (Target.trace κ b) '' Icc 0 t ⊆
-      {z | ¬alive (Target.a κ) (driverHomeomorph b).1 (κ*t).toNNReal z} := by
-  rintro z ⟨v,hv,rfl⟩ hz
-  have hs : LiteralTrace.Surviving κ b (LiteralTrace.trace κ b v) (κ*t) := by
-    simpa only [surviving_eq,trace_eq_literal,
-      Real.coe_toNNReal _ (mul_nonneg hκ.le ht)] using
-        (surviving_iff_alive κ b (Target.trace κ b v) (κ*t).toNNReal).mpr hz
-  exact LiteralTrace.radial_limit_not_surviving hκ (mul_nonneg hκ.le hv.1) b (hlim v hv.1)
-    (LiteralTrace.surviving_of_le (mul_nonneg hκ.le hv.1)
-      (mul_le_mul_of_nonneg_left hv.2 hκ.le) hs)
-
- 
-
-theorem actual_mass_remaining_avoids_past {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧ (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass (Target.a κ)
-              (Real.exp (-2*(Target.a κ)*(j+1))) (t i) b) - ∫z,f z∂μ (t i) b)
-              1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ)) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂(W.map driverHomeomorph)) =
-          (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂(W.map driverHomeomorph)) +
-            ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen (Target.a κ) v b z.1)
-              ∂upperArea ∂(W.map driverHomeomorph)) ∧
-      ∀ᵐ b ∂W,
-        ContinuousOn (Target.trace κ b) (Ici 0) ∧
-        (∀ i j, t i≤t j → μ (t i) (driverHomeomorph b)≤μ (t j) (driverHomeomorph b)) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1)))
-              (t i) (driverHomeomorph b)) atTop (𝓝 (∫z,f z∂μ (t i) (driverHomeomorph b)))) ∧
-        (∀ i, NullSingletonClass (μ (t i) (driverHomeomorph b))) ∧
-        (∀ i (v : ℝ≥0), t i=v →
-          μ (t i) (driverHomeomorph b)
-            {z : UpperPlane | alive (Target.a κ) (driverHomeomorph b).1 v z.1}=0) ∧
-        (∀ i j, t i≠∞ → t j=∞ →
-          (μ (t j) (driverHomeomorph b)-μ (t i) (driverHomeomorph b))
-            {z : UpperPlane | ¬alive (Target.a κ) (driverHomeomorph b).1 (t i).toNNReal z.1}=0) ∧
-        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-          μ (t i) (driverHomeomorph b) =
-            (μ (t j) (driverHomeomorph b)).restrict
-              {z : UpperPlane | ¬alive (Target.a κ) (driverHomeomorph b).1 v z.1}) ∧
-        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-          (μ (t j) (driverHomeomorph b)-μ (t i) (driverHomeomorph b))
-            {z : UpperPlane | z.1∈(Target.trace κ b) '' Icc 0 ((v:ℝ)/κ)}=0) := by
-  have hBW := brownian_map_driver W hW
-  have : IsProbabilityMeasure (W.map driverHomeomorph) := hBW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨s,c,hc,hs,hcs,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hcond,hconv⟩ :=
-    actual_raw_mass_hull_accounting (Target.a κ) (a_gt_quarter hκ hκ8)
-      (W.map driverHomeomorph) hBW t
-  refine ⟨s,c,hc,hs,hcs,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hcond,?_⟩
-  filter_upwards [ae_of_ae_map measurable_driverHomeomorph.aemeasurable hconv,
-    LiteralTrace.trace_regular_all_times hκ hκ8 hW] with b hb htrace
-  rcases hb with ⟨horder, hvague, hatom, hpast, hfuture, haccount⟩
-  refine ⟨?_,horder,hvague,hatom,hpast,hfuture,haccount,?_⟩
-  · simpa only [trace_eq_literal] using htrace.2.1
-  · intro i j v hi hj
-    have hint : t i ≠ ∞ := hi ▸ ENNReal.coe_ne_top
-    have hf := hfuture i j hint hj
-    have hreal : (t i).toNNReal = v := by rw [hi,ENNReal.toNNReal_coe]
-    rw [hreal] at hf
-    have hp := past_trace_subset_dead hκ b htrace.1 (t := (v:ℝ)/κ) (by positivity)
-    have he : (κ*((v:ℝ)/κ)).toNNReal=v := by
-      rw [mul_div_cancel₀ _ hκ.ne',Real.toNNReal_coe]
-    rw [he] at hp
-    exact measure_mono_null (fun z hz => hp hz) hf
-
-end SLEExactGauge.TraceMassIntegration
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.HeightRadius
+import OAI.Probability.SLEGauge.AngularControlMass
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma fullLoewnerRadius_continuousOn (a : ℝ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    ContinuousOn (fun z => fullLoewnerRadius a b.1 z t) {z | alive a b.1 t z} := by
+  have hd : DifferentiableOn ℂ (loewnerMap a b.1 t) {z | alive a b.1 t z} :=
+    fun _ hz => (hasDerivAt_loewnerMap b.1.continuous hz).differentiableAt.differentiableWithinAt
+  have hc := hd.continuousOn
+  exact (continuousOn_const.mul (Complex.continuous_im.comp_continuousOn hc)).div
+    ((hd.analyticOnNhd (isOpen_alive b.1.continuous t)).deriv.continuousOn.norm)
+    (fun z hz => norm_ne_zero_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hz))
+
+lemma physicalRadius_actual_continuousOn (a : ℝ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    ContinuousOn (fun z => physicalRadius (actualState a b.1 t z)) {z | alive a b.1 t z} := by
+  have hc := (fullLoewnerRadius_continuousOn a b t).div_const 2
+  apply hc.congr
+  intro z hz
+  have he := actual_radius_eq_full b.1.continuous hz
+  linarith
+
+ 
+
+lemma surviving_compact_radius_margin (a : ℝ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (K : Set UpperPlane) (hK : IsCompact K)
+    (hKs : ∀ z∈K, alive a b.1 t z.1) :
+    ∃ ε : ℝ, 0<ε ∧ ∀ z∈K, ε≤physicalRadius (actualState a b.1 t z.1)/z.1.im := by
+  by_cases hne : K.Nonempty
+  · have hc : ContinuousOn (fun z : UpperPlane =>
+        physicalRadius (actualState a b.1 t z.1)/z.1.im) K := by
+      apply ContinuousOn.div
+        ((physicalRadius_actual_continuousOn a b t).comp continuous_subtype_val.continuousOn hKs)
+        (Complex.continuous_im.comp continuous_subtype_val).continuousOn
+      exact fun z _ => z.2.ne'
+    obtain ⟨z,hz,hmin⟩ := hK.exists_isMinOn hne hc
+    exact ⟨_,div_pos (physicalRadius_actual_pos a b z.1 t (hKs z hz)) z.2,
+      fun w hw => hmin hw⟩
+  · exact ⟨1,zero_lt_one,fun z hz => False.elim (hne ⟨z,hz⟩)⟩
+
+lemma timeMassDensity_zero_of_radius_lt (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (z : UpperPlane) (ht : alive a b.1 t z.1)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (hqr : z.1.im*q<physicalRadius (actualState a b.1 t z.1)) :
+    timeMassDensity a q t b z.1=0 := by
+  have hn : ¬radialTime a b.1 z.1 (z.1.im*q)≤(t : ℝ≥0∞) := by
+    intro hle
+    have hfin : radialTime a b.1 z.1 (z.1.im*q)≠∞ :=
+      ne_top_of_le_ne_top ENNReal.coe_ne_top hle
+    have htt : (radialTime a b.1 z.1 (z.1.im*q)).toNNReal≤t :=
+      ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hle)
+    have he := (radialTime_attained a ha z.1 z.2 _ (mul_pos z.2 hq)
+      (mul_le_of_le_one_right z.2.le hq1) b hfin).2
+    have hr := physicalRadius_actual_anti a ha b z.1 _ t htt ht
+    rw [he] at hr
+    exact (not_le_of_gt hqr) hr
+  exact ite_eq_right hn
+
+lemma timeMassDensity_eventually_zero_on_surviving_compact (a : ℝ) (ha : 0<a)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (K : Set UpperPlane) (hK : IsCompact K)
+    (hKs : ∀ z∈K, alive a b.1 t z.1)
+    (q : ℕ → ℝ) (hq : ∀ j, 0<q j ∧ q j≤1) (hlim : Tendsto q atTop (𝓝 0)) :
+    ∀ᶠ j in atTop, ∀ z∈K, timeMassDensity a (q j) t b z.1=0 := by
+  obtain ⟨ε,hε,hεK⟩ := surviving_compact_radius_margin a b t K hK hKs
+  filter_upwards [hlim.eventually (gt_mem_nhds hε)] with j hj
+  intro z hz
+  exact timeMassDensity_zero_of_radius_lt a ha b t z (hKs z hz) (q j) (hq j).1 (hq j).2
+    (by simpa only [mul_comm] using (lt_div_iff₀ z.2).mp (hj.trans_le (hεK z hz)))
+
+lemma actualUpperMass_eval_eventually_zero_on_surviving (a : ℝ) (ha : 1/4<a)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (f : C_c(UpperPlane,ℝ))
+    (hf : ∀ z∈tsupport f, alive a b.1 t z.1)
+    (q : ℕ → ℝ) (hq : ∀ j, 0<q j ∧ q j≤1) (hlim : Tendsto q atTop (𝓝 0)) :
+    ∀ᶠ j in atTop, (∫z,f z∂actualUpperMass a (q j) t b)=0 := by
+  filter_upwards [timeMassDensity_eventually_zero_on_surviving_compact a (by linarith)
+    b t (tsupport f) f.hasCompactSupport hf q hq hlim] with j hj
+  have hm : Measurable (fun z : UpperPlane => timeMassDensity a (q j) t b z.1) :=
+    (timeMassDensity_measurable a ha (q j) (hq j).1 (hq j).2 t).comp
+      (measurable_const.prodMk measurable_subtype_coe)
+  rw [actualUpperMass,integral_withDensity_eq_integral_toReal_smul hm
+    (Eventually.of_forall (fun z => timeMassDensity_finite a (q j) t b z.1))]
+  apply integral_eq_zero_of_ae
+  apply Eventually.of_forall
+  intro z
+  by_cases hz : z∈tsupport f
+  · simp only [hj z hz,ENNReal.toReal_zero,zero_smul,Pi.zero_apply]
+  · simp only [image_eq_zero_of_notMem_tsupport hz,smul_zero,Pi.zero_apply]
+
+lemma mass_levels_tendsto_zero (a : ℝ) (ha : 0<a) :
+    Tendsto (fun j : ℕ => Real.exp (-2*a*(j+1))) atTop (𝓝 0) := by
+  apply Real.tendsto_exp_atBot.comp
+  apply Tendsto.const_mul_atTop_of_neg (by linarith : -2*a<0)
+  apply Filter.tendsto_atTop.mpr
+  intro B
+  filter_upwards [(tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop).eventually
+    (eventually_ge_atTop B)] with n hn
+  linarith
+
+ 
+
+lemma actual_limit_integral_zero_on_surviving (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hs : ∀ k j, j∈s k → k≤j)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (μ : Measure UpperPlane)
+    (f : C_c(UpperPlane,ℝ)) (hf : ∀ z∈tsupport f, alive a b.1 t z.1)
+    (hlim : Tendsto (fun k => ∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a
+      (Real.exp (-2*a*(j+1))) t b) atTop (𝓝 (∫z,f z∂μ))) :
+    (∫z,f z∂μ)=0 := by
+  have hq (j : ℕ) : 0<Real.exp (-2*a*(j+1)) ∧ Real.exp (-2*a*(j+1))≤1 :=
+    ⟨Real.exp_pos _,(level_step_lt_one a ha j).le⟩
+  obtain ⟨N,hN⟩ := eventually_atTop.mp (actualUpperMass_eval_eventually_zero_on_surviving
+    a ha b t f hf _ hq (mass_levels_tendsto_zero a (by linarith)))
+  apply tendsto_nhds_unique hlim
+  apply tendsto_const_nhds.congr'
+  filter_upwards [eventually_ge_atTop N] with k hk
+  symm
+  apply Finset.sum_eq_zero
+  intro j hj
+  rw [hN j (hk.trans (hs k j hj)),mul_zero]
+
+end ExactSLEGreen
+namespace SLEExactGauge.MassVersions
+
+ 
+
+lemma open_null_of_compact_tests {X : Type*} [TopologicalSpace X] [T2Space X]
+    [LocallyCompactSpace X] [MeasurableSpace X] [BorelSpace X]
+    (μ : Measure X) [μ.Regular]
+    (U : Set X) (hU : IsOpen U)
+    (htest : ∀ f : C_c(X,ℝ), tsupport f⊆U → (∫x,f x∂μ)=0)
+    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) : μ U=0 := by
+  rw [hU.measure_eq_iSup_isCompact]
+  simp only [ENNReal.iSup_eq_zero]
+  intro K hKU hK
+  obtain ⟨f,hfK,hfc,hfU,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK hU hKU
+  let g : C_c(X,ℝ) := ⟨f,hfc⟩
+  have hz : (fun x => f x)=ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg
+    (fun x => (hf01 x).1) (f.continuous.integrable_of_hasCompactSupport hfc)).mp
+      (htest g hfU)
+  rw [measure_eq_zero_iff_ae_notMem]
+  filter_upwards [hz] with x hx hKx
+  have he := hfK hKx
+  change f x=0 at hx
+  change f x=1 at he
+  linarith
+
+end SLEExactGauge.MassVersions
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_limit_past_hull_support (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hs : ∀ k j, j∈s k → k≤j)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (μ : Measure UpperPlane)
+    [μ.Regular]
+    (hlim : ∀ f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) t b)
+        atTop (𝓝 (∫z,f z∂μ)))
+    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) :
+    μ {z : UpperPlane | alive a b.1 t z.1}=0 := by
+  apply open_null_of_compact_tests μ _ ((isOpen_alive b.1.continuous t).preimage continuous_subtype_val)
+  exact fun f hf => actual_limit_integral_zero_on_surviving a ha s c hs b t μ f hf (hlim f)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceLevelLocalization
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
+
+ 
+def AngularRegular (κ : ℝ) (b : TDriver) : Prop :=
+  ∀ m H : ℕ, ∀ C : ℝ, 0≤C →
+    TendstoUniformlyOn
+      (fun y : ℝ => fun p : ℝ×ℝ => LiteralTrace.inverseFlow κ b p.1
+        (LiteralTrace.driverValue b p.1+(y:ℂ)*((p.2:ℂ)+Complex.I)))
+      (fun p => LiteralTrace.trace κ b (p.1/κ)) (𝓝[>] 0)
+      (Icc (1/((m:ℝ)+1)) H ×ˢ Icc (-C) C)
+
+lemma angularRegular_ae {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    {W : Measure TDriver} (hW : Target.IsWiener W) : ∀ᵐ b ∂W, AngularRegular κ b :=
+  LiteralTrace.angular_uniform_all_windows hκ hκ8 hW
+
+lemma inverse_actual_state (κ : ℝ) (b : TDriver) (z : UpperPlane)
+    (t : ℝ≥0) (ht : alive (Target.a κ) (driverHomeomorph b).1 t z.1) :
+    let v := actualState (Target.a κ) (driverHomeomorph b).1 t z.1
+    LiteralTrace.inverseFlow κ b t
+      (LiteralTrace.driverValue b t+(Y v:ℂ)*((X v/Y v:ℂ)+Complex.I))=z.1 := by
+  let v := actualState (Target.a κ) (driverHomeomorph b).1 t z.1
+  have hy : 0<Y v := by
+    have he := (actualState_evolution (Target.a κ) (driverHomeomorph b) z t ht).yp t
+      ⟨t.coe_nonneg,le_rfl⟩
+    simpa only [Real.toNNReal_coe] using he
+  have hs : LiteralTrace.Surviving κ b z.1 t :=
+    (surviving_eq κ b z.1 t).mp ((surviving_iff_alive κ b z.1 t).mpr ht)
+  have he : (LiteralTrace.driverValue b t:ℂ)+(Y v:ℂ)*((X v/Y v:ℂ)+Complex.I)=
+      LiteralTrace.flow κ b z.1 t := by
+    rw [←flow_eq_literal,flow_eq_loewnerMap]
+    have hxy : Y v*(X v/Y v)=X v := by field_simp
+    apply Complex.ext
+    · simpa [v,X,Y,actualState,LiteralTrace.driverValue] using
+        congrArg (fun r : ℝ => b t+r) hxy
+    · simp [v,X,Y,actualState]
+  change LiteralTrace.inverseFlow κ b t _=z.1
+  rw [he]
+  exact LiteralTrace.inverseFlow_flow t.coe_nonneg hs
+
+ 
+
+theorem level_near_trace (κ : ℝ) (hκ : 0<κ) (b : TDriver)
+    (hb : AngularRegular κ b) (T : ℝ≥0) (η M C ε : ℝ)
+    (hη : 0<η) (_hM : 0<M) (hC : 0≤C) (hε : 0<ε) :
+    ∀ᶠ j : ℕ in atTop, ∀ z : UpperPlane, η≤z.1.im → z.1.im≤M →
+      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
+      τ≤(T:ℝ≥0∞) →
+      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
+      |X v/Y v|≤C →
+      dist z.1 (Target.trace κ b ((τ.toNNReal:ℝ)/κ))<ε := by
+  have ha : 0<Target.a κ := by dsimp [Target.a]; positivity
+  obtain ⟨m,hm⟩ := exists_nat_one_div_lt (show 0<η^2/(4*Target.a κ) by positivity)
+  obtain ⟨H,hH⟩ := exists_nat_gt (T:ℝ)
+  obtain ⟨δ,hδ,hδlim⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp
+    (Metric.tendstoUniformlyOn_iff.mp (hb m H C hC) ε hε)
+  have hg : Tendsto (fun j : ℕ => Real.exp (-Target.a κ*(j+1))) atTop (𝓝 0) := by
+    have he := mass_levels_tendsto_zero (Target.a κ/2) (by positivity)
+    convert he using 1
+    funext j
+    congr 1
+    ring
+  have hg' : Tendsto (fun j : ℕ => M*Real.exp (-Target.a κ*(j+1))) atTop (𝓝 0) := by
+    simpa using hg.const_mul M
+  filter_upwards [(mass_levels_tendsto_zero (Target.a κ) ha).eventually
+      (gt_mem_nhds (show (0:ℝ)<1/2 by norm_num)), hg'.eventually (gt_mem_nhds hδ)] with j hj hj'
+  intro z hzη hzM τ hτ v hv
+  have hτf : τ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hτ
+  have hτT : τ.toNNReal≤T := ENNReal.coe_le_coe.mp (by
+    simpa only [ENNReal.coe_toNNReal hτf] using hτ)
+  have hr : 0<z.1.im*Real.exp (-2*Target.a κ*(j+1)) := mul_pos z.2 (Real.exp_pos _)
+  have hr2 : z.1.im*Real.exp (-2*Target.a κ*(j+1))≤z.1.im/2 := by
+    nlinarith only [mul_le_mul_of_nonneg_left hj.le z.2.le]
+  have hτlower := radialTime_half_lower (Target.a κ) ha (driverHomeomorph b) z _ hr hr2 hτf
+  have hτalive := (radialTime_attained (Target.a κ) ha z.1 z.2 _ hr
+    (by linarith [z.2]) (driverHomeomorph b) hτf).1
+  have hyp : 0<Y v := by
+    simpa only [Real.toNNReal_coe] using
+      (actualState_evolution (Target.a κ) (driverHomeomorph b) z τ.toNNReal hτalive).yp
+        (τ.toNNReal:ℝ) ⟨τ.toNNReal.coe_nonneg,le_rfl⟩
+  have hyv : Y v≤z.1.im*Real.exp (-Target.a κ*(j+1)) := by
+    have hh := level_height_bound (Target.a κ) ha (driverHomeomorph b) z (j+1) (by positivity)
+      (by simpa [τ,levelRadius] using hτf)
+    simpa [v,τ,levelRadius] using hh
+  have hyδ : Y v<δ := hyv.trans_lt ((mul_le_mul_of_nonneg_right hzM (Real.exp_pos _).le).trans_lt hj')
+  have htl : 1/((m:ℝ)+1)≤(τ.toNNReal:ℝ) := by
+    apply hm.le.trans
+    apply le_trans _ hτlower
+    exact div_le_div_of_nonneg_right (pow_le_pow_left₀ hη.le hzη 2) (by positivity)
+  have htu : (τ.toNNReal:ℝ)≤H := (show (τ.toNNReal:ℝ)≤T from hτT).trans hH.le
+  have h := hδlim ⟨hyp,hyδ⟩ (τ.toNNReal,X v/Y v) ⟨⟨htl,htu⟩,abs_le.mp hv⟩
+  simp only [Complex.ofReal_div] at h
+  rw [inverse_actual_state κ b z τ.toNNReal hτalive] at h
+  simpa only [trace_eq_literal,dist_comm] using h
+
+end SLEExactGauge.TraceLevelLocalization
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+lemma sub_add_cancel_of_locally_finite (μ ν : Measure X)
+    [IsFiniteMeasureOnCompacts μ] (h : μ≤ν) : ν-μ+μ=ν := by
+  let K := CompactExhaustion.choice X
+  apply Measure.ext_of_iUnion_eq_univ K.iUnion_eq
+  intro n
+  have : IsFiniteMeasure (μ.restrict (K n)) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using (K.isCompact n).measure_lt_top (μ := μ)⟩
+  rw [Measure.restrict_add,
+    Measure.restrict_sub_eq_restrict_sub_restrict (K.isCompact n).measurableSet]
+  exact Measure.sub_add_cancel_of_le (Measure.restrict_mono Subset.rfl h)
+
+lemma restriction_of_complementary_support (μ ν : Measure X)
+    [IsFiniteMeasureOnCompacts μ] (h : μ≤ν) (D : Set X) (_hD : MeasurableSet D)
+    (hμ : μ Dᶜ=0) (hrem : (ν-μ) D=0) : μ=ν.restrict D := by
+  have hμD : μ.restrict D=μ := Measure.restrict_eq_self_of_ae_mem (by
+    simpa only [mem_compl_iff,not_not] using measure_eq_zero_iff_ae_notMem.mp hμ)
+  have hremD : (ν-μ).restrict D=0 := Measure.restrict_eq_zero.mpr hrem
+  have he := congrArg (fun ρ : Measure X => ρ.restrict D)
+    (sub_add_cancel_of_locally_finite μ ν h)
+  simpa only [Measure.restrict_add,hremD,hμD,zero_add] using he
+
+end SLEExactGauge.MassVersions
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_raw_mass_hull_accounting (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
+        (∀ i, NullSingletonClass (μ (t i) b)) ∧
+        (∀ i (v : ℝ≥0), t i=v → μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0) ∧
+        (∀ i j, t i≠∞ → t j=∞ →
+          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
+        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+          μ (t i) b=(μ (t j) b).restrict {z : UpperPlane | ¬alive a b.1 v z.1}) := by
+  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
+    actual_raw_atomless_mass_endpoint a ha W hW t
+  let μ := recoveredMass a ha s c hc
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := (hμ v b).1
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : (μ v b).Regular := (hμ v b).2
+  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
+  filter_upwards [hconv] with b hb
+  rcases hb with ⟨horder,hvague,hfuture,hatom⟩
+  have hpast (i : ℕ) (v : ℝ≥0) (hi : t i=v) : μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0 := by
+    apply actual_limit_past_hull_support a ha s c hs b v (μ (t i) b)
+    intro f
+    have he := hvague i f
+    simpa only [hi] using he
+  refine ⟨horder,hvague,hatom,hpast,hfuture,?_⟩
+  intro i j v hi hj
+  have hdead := hfuture i j (hi ▸ ENNReal.coe_ne_top) hj
+  have hreal : (t i).toNNReal=v := by rw [hi,ENNReal.toNNReal_coe]
+  rw [hreal] at hdead
+  have hU : MeasurableSet {z : UpperPlane | alive a b.1 v z.1} := by
+    exact ((isOpen_alive b.1.continuous v).preimage
+      (continuous_subtype_val : Continuous (Subtype.val : UpperPlane → ℂ))).measurableSet
+  apply restriction_of_complementary_support (μ (t i) b) (μ (t j) b)
+    (horder i j (by rw [hj]; exact le_top)) {z : UpperPlane | ¬alive a b.1 v z.1}
+    hU.compl _ hdead
+  simpa only [show {z : UpperPlane | ¬alive a b.1 v z.1} =
+    ({z : UpperPlane | alive a b.1 v z.1})ᶜ from rfl, compl_compl] using hpast i v hi
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceMassIntegration
+open FlowIdentification ExactSLEGenerator ExactSLEMain ExactSLEGreen
+
+lemma a_gt_quarter {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) :
+    1/4<Target.a κ := by
+  unfold Target.a
+  apply (lt_div_iff₀ hκ).mpr
+  linarith
+
+ 
+
+lemma past_trace_subset_dead {κ : ℝ} (hκ : 0<κ) (b : TDriver)
+    (hlim : ∀ v : ℝ, 0≤v → Tendsto (fun y : ℝ => LiteralTrace.inverseFlow κ b (κ*v)
+      (LiteralTrace.driverValue b (κ*v)+y*Complex.I)) (𝓝[>] 0)
+        (𝓝 (LiteralTrace.trace κ b v))) {t : ℝ} (ht : 0≤t) :
+    (Target.trace κ b) '' Icc 0 t ⊆
+      {z | ¬alive (Target.a κ) (driverHomeomorph b).1 (κ*t).toNNReal z} := by
+  rintro z ⟨v,hv,rfl⟩ hz
+  have hs : LiteralTrace.Surviving κ b (LiteralTrace.trace κ b v) (κ*t) := by
+    simpa only [surviving_eq,trace_eq_literal,
+      Real.coe_toNNReal _ (mul_nonneg hκ.le ht)] using
+        (surviving_iff_alive κ b (Target.trace κ b v) (κ*t).toNNReal).mpr hz
+  exact LiteralTrace.radial_limit_not_surviving hκ (mul_nonneg hκ.le hv.1) b (hlim v hv.1)
+    (LiteralTrace.surviving_of_le (mul_nonneg hκ.le hv.1)
+      (mul_le_mul_of_nonneg_left hv.2 hκ.le) hs)
+
+ 
+
+theorem actual_mass_remaining_avoids_past {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧ (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass (Target.a κ)
+              (Real.exp (-2*(Target.a κ)*(j+1))) (t i) b) - ∫z,f z∂μ (t i) b)
+              1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ)) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂(W.map driverHomeomorph)) =
+          (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂(W.map driverHomeomorph)) +
+            ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen (Target.a κ) v b z.1)
+              ∂upperArea ∂(W.map driverHomeomorph)) ∧
+      ∀ᵐ b ∂W,
+        ContinuousOn (Target.trace κ b) (Ici 0) ∧
+        (∀ i j, t i≤t j → μ (t i) (driverHomeomorph b)≤μ (t j) (driverHomeomorph b)) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1)))
+              (t i) (driverHomeomorph b)) atTop (𝓝 (∫z,f z∂μ (t i) (driverHomeomorph b)))) ∧
+        (∀ i, NullSingletonClass (μ (t i) (driverHomeomorph b))) ∧
+        (∀ i (v : ℝ≥0), t i=v →
+          μ (t i) (driverHomeomorph b)
+            {z : UpperPlane | alive (Target.a κ) (driverHomeomorph b).1 v z.1}=0) ∧
+        (∀ i j, t i≠∞ → t j=∞ →
+          (μ (t j) (driverHomeomorph b)-μ (t i) (driverHomeomorph b))
+            {z : UpperPlane | ¬alive (Target.a κ) (driverHomeomorph b).1 (t i).toNNReal z.1}=0) ∧
+        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+          μ (t i) (driverHomeomorph b) =
+            (μ (t j) (driverHomeomorph b)).restrict
+              {z : UpperPlane | ¬alive (Target.a κ) (driverHomeomorph b).1 v z.1}) ∧
+        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+          (μ (t j) (driverHomeomorph b)-μ (t i) (driverHomeomorph b))
+            {z : UpperPlane | z.1∈(Target.trace κ b) '' Icc 0 ((v:ℝ)/κ)}=0) := by
+  have hBW := brownian_map_driver W hW
+  have : IsProbabilityMeasure (W.map driverHomeomorph) := hBW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨s,c,hc,hs,hcs,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hcond,hconv⟩ :=
+    actual_raw_mass_hull_accounting (Target.a κ) (a_gt_quarter hκ hκ8)
+      (W.map driverHomeomorph) hBW t
+  refine ⟨s,c,hc,hs,hcs,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hcond,?_⟩
+  filter_upwards [ae_of_ae_map measurable_driverHomeomorph.aemeasurable hconv,
+    LiteralTrace.trace_regular_all_times hκ hκ8 hW] with b hb htrace
+  rcases hb with ⟨horder, hvague, hatom, hpast, hfuture, haccount⟩
+  refine ⟨?_,horder,hvague,hatom,hpast,hfuture,haccount,?_⟩
+  · simpa only [trace_eq_literal] using htrace.2.1
+  · intro i j v hi hj
+    have hint : t i ≠ ∞ := hi ▸ ENNReal.coe_ne_top
+    have hf := hfuture i j hint hj
+    have hreal : (t i).toNNReal = v := by rw [hi,ENNReal.toNNReal_coe]
+    rw [hreal] at hf
+    have hp := past_trace_subset_dead hκ b htrace.1 (t := (v:ℝ)/κ) (by positivity)
+    have he : (κ*((v:ℝ)/κ)).toNNReal=v := by
+      rw [mul_div_cancel₀ _ hκ.ne',Real.toNNReal_coe]
+    rw [he] at hp
+    exact measure_mono_null (fun z hz => hp hz) hf
+
+end SLEExactGauge.TraceMassIntegration
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/PhaseIntegrals.lean b/lean/OAI/Probability/SLEGauge/PhaseIntegrals.lean
index 12f539168f9db1b77a72eb768286b507c2dd4317..b08f4f02653197f8efb607f87f19c45d33e44265 100644
--- a/lean/OAI/Probability/SLEGauge/PhaseIntegrals.lean
+++ b/lean/OAI/Probability/SLEGauge/PhaseIntegrals.lean
@@ -1,457 +1,457 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.PhaseStopping
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-namespace ExactSLEPasting
-lemma integral_finite_partition {Ω P : Type*} [MeasurableSpace Ω] [Fintype P]
-    {μ : Measure Ω} (phase : Ω → P) (f : Ω → ℝ)
-    (hf : ∀ p, Integrable (fun ω => if phase ω=p then f ω else 0) μ) :
-    (∫ ω, f ω ∂μ)=∑ p, ∫ ω, (if phase ω=p then f ω else 0) ∂μ := by
-  rw [←integral_finsetSum _ (fun p _ => hf p)]
-  congr 1
-  funext ω
-  simp
-lemma integral_le_of_finite_partition {Ω P : Type*} [MeasurableSpace Ω] [Fintype P]
-    {μ : Measure Ω} (phase : Ω → P) (f g : Ω → ℝ)
-    (hf : ∀ p, Integrable (fun ω => if phase ω=p then f ω else 0) μ)
-    (hg : ∀ p, Integrable (fun ω => if phase ω=p then g ω else 0) μ)
-    (hle : ∀ p, (∫ ω, (if phase ω=p then f ω else 0) ∂μ)≤
-      ∫ ω, (if phase ω=p then g ω else 0) ∂μ) : (∫ ω, f ω ∂μ)≤∫ ω, g ω ∂μ := by
-  rw [integral_finite_partition phase f hf,integral_finite_partition phase g hg]
-  exact Finset.sum_le_sum fun p _ => hle p
-end ExactSLEPasting
-
- 
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma clamped_delta_continuous (a η : ℝ) (hη : 0<η) (N : ℝ≥0) (cap σ off : ℝ) :
-    Continuous (fun v : State => physicalDelta a cap σ off (regimeClamp η N v)) := by
-  apply continuous_iff_continuousAt.mpr
-  intro v
-  have hy := hη.trans_le (regimeClamp_height η N v)
-  have hc := (physicalClock_smooth a cap (regimeClamp η N v) hy).continuousAt.comp_of_eq
-    (regimeClamp_continuous η N).continuousAt rfl
-  exact (continuousAt_const.add (continuousAt_const.mul hc)).rexp
-lemma clamped_speed_continuous (η : ℝ) (hη : 0<η) (N : ℝ≥0) :
-    Continuous (fun v : State => physicalV (regimeClamp η N v)) := by
-  apply continuous_iff_continuousAt.mpr
-  intro v
-  have hy := hη.trans_le (regimeClamp_height η N v)
-  exact (Y.continuous.continuousAt.div ((X.continuous.pow 2).add (Y.continuous.pow 2)).continuousAt
-    (Q_pos hy).ne').comp_of_eq (regimeClamp_continuous η N).continuousAt rfl
-lemma regimeState_discount_le {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (N : ℝ≥0) (cap σ off : Fin m → ℝ)
-    (z : Fin m → ℂ) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0)
-    (J₀ J₁ : ℝ) (hJ₀ : 0≤J₀) (hJ₁ : 0≤J₁)
-    (hsingle : ∀ i∈s, (∫ t in (S:ℝ)..(T:ℝ),
-      physicalDelta a (cap i) (σ i) (off i) (regimeClamp η N (vectorState a η hη z t.toNNReal b i))*
-      physicalV (regimeClamp η N (vectorState a η hη z t.toNNReal b i))^2)≤J₀)
-    (hpair : ∀ i∈s, ∀ j∈s.erase i, (∫ t in (S:ℝ)..(T:ℝ),
-      (physicalDelta a (cap i) (σ i) (off i) (regimeClamp η N (vectorState a η hη z t.toNNReal b i))*
-       physicalDelta a (cap j) (σ j) (off j) (regimeClamp η N (vectorState a η hη z t.toNNReal b j)))^(-regM a)*
-       physicalV (regimeClamp η N (vectorState a η hη z t.toNNReal b i))*
-       physicalV (regimeClamp η N (vectorState a η hη z t.toNNReal b j)))≤J₁) :
-    (regimeState s a ε η hη N cap σ off z T b).2-
-      (regimeState s a ε η hη N cap σ off z S b).2≤
-      regC a*(m*J₀)+regYoungC a ε/2*(m*(m*J₁)) := by
-  let V := fun (t : ℝ) (i : Fin m) => regimeClamp η N (vectorState a η hη z t.toNNReal b i)
-  have hd (i : Fin m) : Continuous (fun t : ℝ => physicalDelta a (cap i) (σ i) (off i) (V t i)) :=
-    (clamped_delta_continuous a η hη N (cap i) (σ i) (off i)).comp
-      ((localState_continuous a η hη (z i) b).comp continuous_real_toNNReal)
-  have hv (i : Fin m) : Continuous (fun t : ℝ => physicalV (V t i)) :=
-    (clamped_speed_continuous η hη N).comp ((localState_continuous a η hη (z i) b).comp continuous_real_toNNReal)
-  have hf (i : Fin m) : Continuous (fun t : ℝ => physicalDelta a (cap i) (σ i) (off i) (V t i)*physicalV (V t i)^2) := (hd i).mul ((hv i).pow 2)
-  have hg (i j : Fin m) : Continuous (fun t : ℝ =>
-      (physicalDelta a (cap i) (σ i) (off i) (V t i)*physicalDelta a (cap j) (σ j) (off j) (V t j))^(-regM a)*
-      physicalV (V t i)*physicalV (V t j)) :=
-    (((hd i).mul (hd j)).rpow_const (fun t => Or.inl (mul_pos (Real.exp_pos _) (Real.exp_pos _)).ne')).mul (hv i) |>.mul (hv j)
-  have hk : Continuous (fun t : ℝ => regimeDiscountCut s a ε η N cap σ off (vectorState a η hη z t.toNNReal b)) :=
-    (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp ((vectorState_continuous a η hη z b).comp continuous_real_toNNReal)
-  have he := intervalIntegral.integral_add_adjacent_intervals (μ:=volume) (hk.intervalIntegrable 0 S) (hk.intervalIntegrable S T)
-  have hsum (r : Finset (Fin m)) {J : ℝ} (hJ : 0≤J) (f : Fin m → ℝ) (h : ∀ i∈r,f i≤J) :
-      (∑ i ∈ r, f i) ≤ (m:ℝ)*J := by
-    calc
-      _ ≤ ∑ i ∈ r, J := Finset.sum_le_sum h
-      _ = (r.card:ℝ)*J := by simp
-      _ ≤ m*J := mul_le_mul_of_nonneg_right (by exact_mod_cast (show r.card ≤ m by simpa only [Fintype.card_fin] using r.card_le_univ)) hJ
-  change (∫ t in (0:ℝ)..(T:ℝ), _)-(∫ t in (0:ℝ)..(S:ℝ), _)≤_
-  rw [←he,add_sub_cancel_left]
-  unfold regimeDiscountCut regimeDiscount
-  rw [intervalIntegral.integral_add]
-  · rw [intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul,
-      intervalIntegral.integral_finsetSum (f:=fun i t => physicalDelta a (cap i) (σ i) (off i) (V t i)*physicalV (V t i)^2) (fun i hi => (hf i).intervalIntegrable _ _),
-      intervalIntegral.integral_finsetSum (fun i hi => (continuous_finsetSum _ fun j hj => hg i j).intervalIntegrable _ _)]
-    apply add_le_add
-    · exact mul_le_mul_of_nonneg_left (hsum s hJ₀ _ hsingle) (regC_pos ha).le
-    · apply mul_le_mul_of_nonneg_left _ (by have := regEta_lt_half ha; have := regC2_pos ha; have := regEta_pos ha; unfold regYoungC; apply div_nonneg; apply mul_nonneg; apply mul_nonneg; linarith; positivity; positivity; norm_num)
-      apply hsum s (mul_nonneg (by positivity) hJ₁)
-      intro i hi
-      rw [intervalIntegral.integral_finsetSum (fun j hj => (hg i j).intervalIntegrable _ _)]
-      exact hsum (s.erase i) hJ₁ _ (hpair i hi)
-  · exact ((continuous_finsetSum s fun i hi => hf i).const_mul _).intervalIntegrable _ _
-  · exact ((continuous_finsetSum s fun i hi => continuous_finsetSum _ fun j hj => hg i j).const_mul _).intervalIntegrable _ _
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
-noncomputable def pairDiscountConstant (a : ℝ) : ℝ :=
-  Real.exp (2*(regM a*regBeta a))/(2*a*(1-Real.exp (2*(regM a*regBeta a)-a)))
-lemma pairDiscountConstant_pos {a : ℝ} (ha : 1/4<a) : 0<pairDiscountConstant a := by
-  have ha0 : 0<a := by linarith
-  have he : Real.exp (2*(regM a*regBeta a)-a)<1 := Real.exp_lt_one_iff.mpr (by nlinarith [regBeta_discount ha])
-  unfold pairDiscountConstant
-  positivity
-noncomputable def phaseDiscountConstant (m : ℕ) (a ε : ℝ) : ℝ :=
-  regC a*(m*(1/regBeta a))+regYoungC a ε/2*(m*(m*pairDiscountConstant a))
-lemma phaseDiscountConstant_nonneg (m : ℕ) {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε) :
-    0≤phaseDiscountConstant m a ε := by
-  have hC₂ := regC2_pos ha
-  have hη₀ := regEta_pos ha
-  have hY : 0≤regYoungC a ε := by
-    unfold regYoungC
-    apply mul_nonneg
-    · apply mul_nonneg
-      · linarith [regEta_lt_half ha]
-      · positivity
-    · positivity
-  unfold phaseDiscountConstant
-  have := (pairDiscountConstant_pos ha).le
-  have := (regBeta_pos ha).le
-  have := (regC_pos ha).le
-  positivity
-lemma actual_phase_discount {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
-    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
-    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (M : ℝ≥0) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0)
-    (hST : S≤T)
-    (hboundary : ∀ i, levelTime a b (z i) (cap i) (U i)≤(S:ℝ≥0∞) ∨
-      (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i))
-    (hsafe : ∀ t, S≤t → t≤T → ‖vectorState a η hη z t b‖≤M ∧
-      ∀ i∈phaseActive (actualPhase a z cap U S b),
-      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
-      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
-    (regimeState (phaseActive (actualPhase a z cap U S b)) a ε η hη M cap
-      (phaseSigma a (actualPhase a z cap U S b)) (phaseOffset a U (actualPhase a z cap U S b)) z T b).2-
-    (regimeState (phaseActive (actualPhase a z cap U S b)) a ε η hη M cap
-      (phaseSigma a (actualPhase a z cap U S b)) (phaseOffset a U (actualPhase a z cap U S b)) z S b).2≤
-      phaseDiscountConstant m a ε := by
-  have ha0 : 0<a := by linarith
-  let p := actualPhase a z cap U S b
-  let A := fun i => (levelTime a b (z i) (cap i) 0).toNNReal
-  have htimes (i) (hi : i∈phaseActive p) := active_phase_interval
-    (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith))
-    (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith))
-    (hboundary i) ((phaseActive_mem p i).mp hi)
-  have hA (i) (hi : i∈phaseActive p) : A i≤S :=
-    ENNReal.coe_le_coe.mp (ENNReal.coe_toNNReal_le_self.trans (htimes i hi).1)
-  have hR (i) (hi : i∈phaseActive p) : fullLoewnerRadius a b.1 (z i) (A i)=cap i :=
-    levelTime_fullRadius a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b
-      (ne_top_of_le_ne_top (by simp) (htimes i hi).1)
-  have hclockT (i) (hi : i∈phaseActive p) : physicalClock a (cap i) (actualState a b.1 T (z i))≤U i :=
-    levelTime_clock_le a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b T
-      ((hsafe T hST le_rfl).2 i hi).1 (htimes i hi).2.1
-  have hclockS (i) (hi : i∈phaseActive p) (hp : p i=2) :
-      U i/2≤physicalClock a (cap i) (actualState a b.1 S (z i)) :=
-    levelTime_clock_ge a ha0 (z i) (hz i) (cap i) (U i/2) (hc i) (hr i) (by have := hu i; linarith) b S
-      ((hsafe S le_rfl hST).2 i hi).1 ((htimes i hi).2.2 hp)
-  have he (t : ℝ) (ht : t∈uIcc (S:ℝ) (T:ℝ)) (i) (hi : i∈phaseActive p) :
-      regimeClamp η M (vectorState a η hη z t.toNNReal b i)=actualState a b.1 t.toNNReal (z i) := by
-    have ht' : t∈Icc (S:ℝ) T := by simpa only [uIcc_of_le (show (S:ℝ)≤T from hST)] using ht
-    have hlo : S≤t.toNNReal := by exact_mod_cast (show (S:ℝ)≤(t.toNNReal:ℝ) by rw [Real.coe_toNNReal _ (S.coe_nonneg.trans ht'.1)]; exact ht'.1)
-    have hhi : t.toNNReal≤T := by exact_mod_cast (show (t.toNNReal:ℝ)≤T by rw [Real.coe_toNNReal _ (S.coe_nonneg.trans ht'.1)]; exact ht'.2)
-    have hs := hsafe t.toNNReal hlo hhi
-    rw [regimeClamp_eq η M _ hη ((norm_le_pi_norm _ i).trans hs.1) (hs.2 i hi).2.1]
-    exact (hs.2 i hi).2.2
-  apply regimeState_discount_le (phaseActive p) ha hε hη M cap (phaseSigma a p) (phaseOffset a U p) z b S T
-    (1/regBeta a) (pairDiscountConstant a) (by have := regBeta_pos ha; positivity) (pairDiscountConstant_pos ha).le
-  · intro i hi
-    obtain ⟨f,hf⟩ := ((hsafe T hST le_rfl).2 i hi).1
-    have hh := actual_regularized_single_discount ha b.1.continuous hST (hA i hi) hf (hc i) (hR i hi)
-      (U:=U i) (σ:=phaseSigma a p i) (off:=phaseOffset a U p i)
-    have hp : (phaseSigma a p i=-regBeta a ∧ phaseOffset a U p i=0) ∨
-      (phaseSigma a p i=regBeta a ∧ phaseOffset a U p i=-regBeta a*U i ∧ activeODEClock b.1 T (A i) f T≤U i) := by
-      by_cases hh : p i=2
-      · right
-        refine ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh],?_⟩
-        rw [←actual_clock_eq ha0.ne' (hc i) b.1.continuous hf ((hA i hi).trans hST) le_rfl (hR i hi)]
-        exact hclockT i hi
-      · exact Or.inl ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh]⟩
-    apply (le_of_eq ?_).trans (hh hp)
-    apply intervalIntegral.integral_congr
-    intro t ht
-    dsimp only
-    rw [he t ht i hi]
-  · intro i hi j hj
-    have hj' := Finset.mem_of_mem_erase hj
-    obtain ⟨f,hf⟩ := ((hsafe T hST le_rfl).2 i hi).1
-    obtain ⟨g,hg⟩ := ((hsafe T hST le_rfl).2 j hj').1
-    have hij : i≠j := (Finset.ne_of_mem_erase hj).symm
-    have hp (k : Fin m) (hk : k∈phaseActive p) (f : ℝ → ℂ) (hf : odeUntil a b.1 (z k) T f) :
-      (phaseSigma a p k=-regBeta a ∧ phaseOffset a U p k=0) ∨
-      (phaseSigma a p k=regBeta a ∧ phaseOffset a U p k=-regBeta a*U k ∧ U k/2≤activeODEClock b.1 T (A k) f S) := by
-      by_cases hh : p k=2
-      · right
-        refine ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh],?_⟩
-        rw [←actual_clock_eq ha0.ne' (hc k) b.1.continuous hf ((hA k hk).trans hST) hST (hR k hk)]
-        exact hclockS k hk hh
-      · exact Or.inl ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh]⟩
-    have hh := actual_regularized_pair_discount ha b.1.continuous hST (hA i hi) (hA j hj') hf hg
-      (hinj.ne hij) (hc i) (hc j) (hR i hi) (hR j hj') (hsep i j hij)
-      (by simpa only [norm_sub_rev] using hsep j i hij.symm) (hp i hi f hf) (hp j hj' g hg)
-    apply (le_of_eq ?_).trans hh
-    apply intervalIntegral.integral_congr
-    intro t ht
-    dsimp only
-    rw [he t ht i hi,he t ht j hj']
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory ProbabilityTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma stopped_phaseProduct_measurable {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
-    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable (fun b => phaseProduct a cap U (actualPhase a z cap U (S b) b)
-      (vectorState a η hη z (S b) b)) := by
-  have hp := (actualPhase_measurable a ha z hz cap U hc hr hu S hS).mono hS.measurableSpace_le le_rfl
-  have hv := (vectorState_stopped_measurable a η hη z S hS).mono hS.measurableSpace_le le_rfl
-  have he : (fun b => phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (S b) b))=
-      fun b => ∑ p : PhaseVector m, if actualPhase a z cap U (S b) b=p then
-        phaseProduct a cap U p (vectorState a η hη z (S b) b) else 0 := by funext b; simp
-  rw [he]
-  apply Finset.measurable_sum
-  intro p hp'
-  exact ((phaseProduct_measurable a cap U p).comp hv).ite (hp (measurableSet_singleton p)) measurable_const
-lemma actual_phase_expectation_step {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
-    (hsize : ε*((m:ℝ)-1)≤regK a/2)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
-    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
-    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (M : ℝ≥0)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S T : ExactSLEGenerator.Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b,S b≤T b) (hTB : ∀ b,T b≤B)
-    (hboundary : ∀ b i, ∀ u∈({0,U i/2,U i}:Finset ℝ),
-      levelTime a b (z i) (cap i) u≤(S b:ℝ≥0∞) ∨ (T b:ℝ≥0∞)≤levelTime a b (z i) (cap i) u)
-    (hsafe : ∀ b t, t≤T b → (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
-      ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
-      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
-      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
-      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
-    (∫ b, phaseProduct a cap U (actualPhase a z cap U (T b) b) (vectorState a η hη z (T b) b) ∂W)≤
-      Real.exp (phaseDiscountConstant m a ε)*
-      (∫ b, phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (S b) b) ∂W) := by
-  classical
-  have ha0 : 0<a := by linarith
-  let P := fun (Q : ExactSLEGenerator.Driver → ℝ≥0) (b : ExactSLEGenerator.Driver) => phaseProduct a cap U (actualPhase a z cap U (Q b) b) (vectorState a η hη z (Q b) b)
-  let E := fun p : PhaseVector m => {b | actualPhase a z cap U (S b) b=p}
-  have hE (p) : MeasurableSet[hS.measurableSpace] (E p) :=
-    actualPhase_measurable a ha0 z hz cap U hc hr hu S hS (measurableSet_singleton p)
-  have hE' (p) : MeasurableSet (E p) := hS.measurableSpace_le _ (hE p)
-  have htrans (b) : P T b≤phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (T b) b) := by
-    apply actual_phase_transition ha z hz cap U hc hr hu η hη b (S b) (T b) (hST b) (hboundary b)
-    intro i hi
-    exact ((hsafe b (T b) le_rfl ⟨i,hi⟩).2 i hi).2.2
-  have hint (p : PhaseVector m) (Q : ExactSLEGenerator.Driver → ℝ≥0)
-      (hQ : IsStoppingTime rawFiltration (fun b => (Q b : WithTop ℝ≥0)))
-      (hSQ : ∀ b,S b≤Q b) (hQT : ∀ b,Q b≤T b) :
-      Integrable (fun b => if b∈E p then phaseProduct a cap U p (vectorState a η hη z (Q b) b) else 0) W := by
-    by_cases hp : (phaseActive p).Nonempty
-    · have hi : Integrable (fun b => if b∈E p then regularizedProduct (phaseActive p) a cap (phaseSigma a p)
-          (phaseOffset a U p) (vectorState a η hη z (Q b) b) else 0) W := by
-        have hs : ∀ b∈E p, ‖vectorState a η hη z (Q b) b‖≤M ∧ ∀ i∈phaseActive p, η≤Y (vectorState a η hη z (Q b) b i) := by
-          intro b hb
-          have he : actualPhase a z cap U (S b) b=p := hb
-          have hn : (phaseActive (actualPhase a z cap U (S b) b)).Nonempty := he ▸ hp
-          have hh := actual_phase_interval_safe ha z hz cap U hc hr hu η hη M b (S b) (T b) (hST b) hn
-            (fun i => hboundary b i (U i) (by simp)) (hsafe b)
-            (Q b) (hSQ b) (hQT b)
-          rw [he] at hh
-          exact ⟨hh.1,fun i hi => (hh.2 i hi).1⟩
-        convert regularizedProduct_indicator_integrable (phaseActive p) a η hη M cap (phaseSigma a p) (phaseOffset a U p) z W Q hQ (E p) (hE' p) hs using 1
-        funext b
-        by_cases hb : b∈E p <;> simp only [ite_eq_left,ite_eq_right,hb,not_false_eq_true]
-      convert hi.const_mul (phaseConstant a U p) using 1
-      funext b
-      dsimp only [phaseProduct]
-      split_ifs <;> simp
-    · have he : phaseActive p=∅ := Finset.not_nonempty_iff_eq_empty.mp hp
-      simp only [phaseProduct,he,regularizedProduct,Finset.prod_empty,mul_one]
-      have hic : Integrable (fun _ : ExactSLEGenerator.Driver => phaseConstant a U p) W := integrable_const _
-      convert hic.indicator (hE' p) using 1
-      all_goals first | rfl | (funext b; by_cases hb : b∈E p <;> simp [hb,Set.indicator])
-  have hPi (p) : Integrable (fun b => if b∈E p then P T b else 0) W := by
-    have hm : Measurable (fun b => if b∈E p then P T b else 0) :=
-      (stopped_phaseProduct_measurable a ha0 z hz cap U hc hr hu η hη T hT).ite (hE' p) measurable_const
-    apply (hint p T hT hST (fun _ => le_rfl)).mono' hm.aestronglyMeasurable
-    apply Eventually.of_forall
-    intro b
-    by_cases hb : b∈E p
-    · simp only [ite_eq_left hb,Real.norm_eq_abs,abs_of_pos (show 0<P T b from phaseProduct_pos a cap U _ _)]
-      simpa only [show actualPhase a z cap U (S b) b=p from hb] using htrans b
-    · simp only [ite_eq_right hb,norm_zero,le_refl]
-  have hSi (p) : Integrable (fun b => if b∈E p then P S b else 0) W := by
-    convert hint p S hS (fun _ => le_rfl) hST using 1
-    funext b
-    by_cases hb : b∈E p
-    · simp only [ite_eq_left hb,P,show actualPhase a z cap U (S b) b=p from hb]
-    · simp only [ite_eq_right hb]
-  have hlocal (p) : (∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (T b) b) else 0) ∂W)≤
-      Real.exp (phaseDiscountConstant m a ε)*(∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (S b) b) else 0) ∂W) := by
-    by_cases hp : (phaseActive p).Nonempty
-    · have hsiz : ε*(((phaseActive p).card:ℝ)-1)≤regK a/2 := by
-        have hcard : ((phaseActive p).card:ℝ)≤ m := by exact_mod_cast (show (phaseActive p).card ≤ m by simpa using (phaseActive p).card_le_univ)
-        exact (mul_le_mul_of_nonneg_left (sub_le_sub_right hcard 1) hε.le).trans hsize
-      have hh := vector_regime_event_comparison (phaseActive p) ha hε hη hsiz M cap (phaseSigma a p) (phaseOffset a U p)
-        (fun i hi => phaseSigma_bound ha p i) z W hW S T B hS hT hST hTB (E p) (hE p)
-      have hsafe' : ∀ b∈E p, ∀ t, S b≤t → t≤T b → ‖vectorState a η hη z t b‖≤M ∧
-        ∀ i∈phaseActive p, η≤Y (vectorState a η hη z t b i) ∧
-        physicalDelta a (cap i) (phaseSigma a p i) (phaseOffset a U p i) (vectorState a η hη z t b i)≤1 := by
-        intro b hb
-        have he : actualPhase a z cap U (S b) b=p := hb
-        have hh := actual_phase_interval_safe ha z hz cap U hc hr hu η hη M b (S b) (T b) (hST b)
-          (he ▸ hp) (fun i => hboundary b i (U i) (by simp)) (hsafe b)
-        simpa only [he] using hh
-      have hD : ∀ b∈E p,
-        (regimeState (phaseActive p) a ε η hη M cap (phaseSigma a p) (phaseOffset a U p) z (T b) b).2-
-        (regimeState (phaseActive p) a ε η hη M cap (phaseSigma a p) (phaseOffset a U p) z (S b) b).2≤phaseDiscountConstant m a ε := by
-        intro b hb
-        have he : actualPhase a z cap U (S b) b=p := hb
-        have h := actual_phase_discount ha hε z hz hinj cap U hc hr hsep hu η hη M b (S b) (T b) (hST b)
-          (fun i => hboundary b i (U i) (by simp))
-        apply (he ▸ h)
-        intro t hst htt
-        have h1 := hsafe' b hb t hst htt
-        refine ⟨h1.1,?_⟩
-        intro i hi
-        have hi' : i∈phaseActive (actualPhase a z cap U (S b) b) := he ▸ hi
-        have htarg := (active_phase_interval
-          (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith))
-          (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith))
-          (hboundary b i (U i) (by simp)) ((phaseActive_mem _ i).mp hi')).2.1
-        have hti := (ENNReal.coe_le_coe.mpr htt).trans htarg
-        exact (hsafe b t htt ⟨i,hti⟩).2 i hti
-      have hh' := mul_le_mul_of_nonneg_left (hh hsafe' _ hD) (phaseConstant_pos a U p).le
-      simpa only [←integral_const_mul,phaseProduct,mul_ite,mul_zero,mul_assoc,mul_left_comm] using hh'
-    · have he : phaseActive p=∅ := Finset.not_nonempty_iff_eq_empty.mp hp
-      have hnon : 0≤∫ b, (if b∈E p then phaseConstant a U p else 0) ∂W := integral_nonneg fun b => by have := phaseConstant_pos a U p; split_ifs <;> positivity
-      simp only [phaseProduct,he,regularizedProduct,Finset.prod_empty,mul_one]
-      exact le_mul_of_one_le_left hnon (Real.one_le_exp_iff.mpr (phaseDiscountConstant_nonneg m ha hε))
-  rw [←integral_const_mul]
-  apply ExactSLEPasting.integral_le_of_finite_partition (fun b => actualPhase a z cap U (S b) b)
-  · intro p
-    convert hPi p using 1
-    funext b
-    by_cases hb : b∈E p <;> simp [E] at hb ⊢ <;> simp [hb,P]
-  · intro p
-    convert (hSi p).const_mul (Real.exp (phaseDiscountConstant m a ε)) using 1
-    funext b
-    by_cases hb : b∈E p <;> simp [E] at hb ⊢ <;> simp [hb,P]
-  · intro p
-    have h1 : (∫ b, (if b∈E p then P T b else 0) ∂W) ≤
-        ∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (T b) b) else 0) ∂W :=
-      integral_mono (hPi p) (hint p T hT hST (fun _ => le_rfl)) (fun b => by
-        split_ifs with hb
-        · simpa only [show actualPhase a z cap U (S b) b=p from hb] using htrans b
-        · exact le_rfl)
-    have h2 : (∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (S b) b) else 0) ∂W) =
-        ∫ b, (if b∈E p then P S b else 0) ∂W := by
-      congr 1
-      funext b
-      by_cases hb : b∈E p
-      · simp only [ite_eq_left hb,P,show actualPhase a z cap U (S b) b=p from hb]
-      · simp only [ite_eq_right hb]
-    have h3 := h1.trans (hlocal p)
-    rw [h2,←integral_const_mul] at h3
-    convert h3 using 1 <;> congr 1 <;> funext b <;>
-      (by_cases hb : b∈E p <;> simp [E] at hb ⊢ <;> simp [hb,P])
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma actualState_zero {a : ℝ} (ha : 0<a) (b : ExactSLEGenerator.Driver) {z : ℂ} (hz : 0<z.im) :
-    actualState a b.1 0 z=(z,0) := by
-  have h0 := radialStop_actual a ha z hz (z.im/2) (by positivity) (by linarith) 0 b 0 (show (0:ℝ≥0)≤_ from bot_le)
-  obtain ⟨f,hf⟩ := h0
-  have hm : loewnerMap a b.1 0 z=z := (loewnerMap_eq hf).trans (by simpa using hf.1)
-  have hd : deriv (loewnerMap a b.1 0) z=1 := by
-    simpa using (hasDerivAt_loewnerMap b.1.continuous (show alive a b.1 0 z from ⟨f,hf⟩)).deriv
-  simp [actualState,hm,hd,b.2]
-lemma actual_phaseProduct_initial {m : ℕ} {a : ℝ} (ha : 1/4<a)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
-    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (b : ExactSLEGenerator.Driver) :
-    phaseProduct a cap U (actualPhase a z cap U 0 b) (vectorState a η hη z 0 b)≤1 := by
-  have ha0 : 0<a := by linarith
-  rw [phaseProduct_eq_prod]
-  apply Finset.prod_le_one₀
-  · intro i hi
-    exact (phaseFactor_pos _ _ _ _ _).le
-  · intro i hi
-    have h0 := radialStop_actual a ha0 (z i) (hz i) ((z i).im/2) (by have := hz i; positivity) (by have := hz i; linarith) 0 b 0 (show (0:ℝ≥0)≤_ from bot_le)
-    have hc0 : physicalClock a (cap i) (actualState a b.1 0 (z i))≤0 := by
-      rw [actualState_zero ha0 b (hz i),physicalClock_radius a (cap i) (hc i) _ (hz i)]
-      simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one]
-      apply div_nonpos_of_nonpos_of_nonneg _ (by positivity)
-      exact Real.log_nonpos (by have := hc i; have := hz i; positivity) ((div_le_one (by have := hz i; positivity)).mpr (hr i))
-    have hAB := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith)
-    have hBC := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith)
-    have he : vectorState a η hη z 0 b i=actualState a b.1 0 (z i) := by
-      simp only [vectorState,localState_zero,actualState_zero ha0 b (hz i)]
-    rw [he]
-    generalize hp : actualPhase a z cap U 0 b i=p
-    fin_cases p
-    · norm_num [phaseFactor,Fin.ext_iff]
-    · have hat := ((phase_one_iff _ _ _ 0 hAB hBC).mp hp).1
-      have hge := levelTime_clock_ge a ha0 (z i) (hz i) (cap i) 0 (hc i) (hr i) le_rfl b 0 h0 hat
-      have heq := le_antisymm hc0 hge
-      simpa [phaseFactor,Fin.ext_iff] using physicalRegularized_activation (σ := -regBeta a) (off := 0) ha _ heq rfl
-    · have hbt := ((phase_two_iff _ _ _ 0 hAB hBC).mp hp).1
-      have hge := levelTime_clock_ge a ha0 (z i) (hz i) (cap i) (U i/2) (hc i) (hr i) (by have := hu i; linarith) b 0 h0 hbt
-      have hu0 : U i=0 := by have := hu i; linarith
-      have heq : physicalClock a (cap i) (actualState a b.1 0 (z i))=0 := by linarith
-      simpa [phaseFactor,Fin.ext_iff,hu0] using physicalRegularized_activation (σ := regBeta a) (off := 0) ha _ heq rfl
-    · have hct := (phase_three_iff _ _ _ 0).mp hp
-      have hge := levelTime_clock_ge a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b 0 h0 hct
-      have hu0 : U i=0 := by have := hu i; linarith
-      norm_num [phaseFactor,Fin.ext_iff,hu0]
-lemma actual_phaseProduct_completed {m : ℕ} (a : ℝ) (z : Fin m → ℂ) (cap U : Fin m → ℝ)
-    (η : ℝ) (hη : 0<η) (b : ExactSLEGenerator.Driver) (T : ℝ≥0)
-    (h : ∀ i,levelTime a b (z i) (cap i) (U i)≤(T:ℝ≥0∞)) :
-    phaseProduct a cap U (actualPhase a z cap U T b) (vectorState a η hη z T b)=
-      ∏ i, Real.exp (regP a*U i/2)/3 := by
-  rw [phaseProduct_eq_prod]
-  apply Finset.prod_congr rfl
-  intro i hi
-  have hp : actualPhase a z cap U T b i=3 := (phase_three_iff _ _ _ T).mpr (h i)
-  simp [phaseFactor,hp,Fin.ext_iff]
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.PhaseStopping
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+namespace ExactSLEPasting
+lemma integral_finite_partition {Ω P : Type*} [MeasurableSpace Ω] [Fintype P]
+    {μ : Measure Ω} (phase : Ω → P) (f : Ω → ℝ)
+    (hf : ∀ p, Integrable (fun ω => if phase ω=p then f ω else 0) μ) :
+    (∫ ω, f ω ∂μ)=∑ p, ∫ ω, (if phase ω=p then f ω else 0) ∂μ := by
+  rw [←integral_finsetSum _ (fun p _ => hf p)]
+  congr 1
+  funext ω
+  simp
+lemma integral_le_of_finite_partition {Ω P : Type*} [MeasurableSpace Ω] [Fintype P]
+    {μ : Measure Ω} (phase : Ω → P) (f g : Ω → ℝ)
+    (hf : ∀ p, Integrable (fun ω => if phase ω=p then f ω else 0) μ)
+    (hg : ∀ p, Integrable (fun ω => if phase ω=p then g ω else 0) μ)
+    (hle : ∀ p, (∫ ω, (if phase ω=p then f ω else 0) ∂μ)≤
+      ∫ ω, (if phase ω=p then g ω else 0) ∂μ) : (∫ ω, f ω ∂μ)≤∫ ω, g ω ∂μ := by
+  rw [integral_finite_partition phase f hf,integral_finite_partition phase g hg]
+  exact Finset.sum_le_sum fun p _ => hle p
+end ExactSLEPasting
+
+ 
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma clamped_delta_continuous (a η : ℝ) (hη : 0<η) (N : ℝ≥0) (cap σ off : ℝ) :
+    Continuous (fun v : State => physicalDelta a cap σ off (regimeClamp η N v)) := by
+  apply continuous_iff_continuousAt.mpr
+  intro v
+  have hy := hη.trans_le (regimeClamp_height η N v)
+  have hc := (physicalClock_smooth a cap (regimeClamp η N v) hy).continuousAt.comp_of_eq
+    (regimeClamp_continuous η N).continuousAt rfl
+  exact (continuousAt_const.add (continuousAt_const.mul hc)).rexp
+lemma clamped_speed_continuous (η : ℝ) (hη : 0<η) (N : ℝ≥0) :
+    Continuous (fun v : State => physicalV (regimeClamp η N v)) := by
+  apply continuous_iff_continuousAt.mpr
+  intro v
+  have hy := hη.trans_le (regimeClamp_height η N v)
+  exact (Y.continuous.continuousAt.div ((X.continuous.pow 2).add (Y.continuous.pow 2)).continuousAt
+    (Q_pos hy).ne').comp_of_eq (regimeClamp_continuous η N).continuousAt rfl
+lemma regimeState_discount_le {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (N : ℝ≥0) (cap σ off : Fin m → ℝ)
+    (z : Fin m → ℂ) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0)
+    (J₀ J₁ : ℝ) (hJ₀ : 0≤J₀) (hJ₁ : 0≤J₁)
+    (hsingle : ∀ i∈s, (∫ t in (S:ℝ)..(T:ℝ),
+      physicalDelta a (cap i) (σ i) (off i) (regimeClamp η N (vectorState a η hη z t.toNNReal b i))*
+      physicalV (regimeClamp η N (vectorState a η hη z t.toNNReal b i))^2)≤J₀)
+    (hpair : ∀ i∈s, ∀ j∈s.erase i, (∫ t in (S:ℝ)..(T:ℝ),
+      (physicalDelta a (cap i) (σ i) (off i) (regimeClamp η N (vectorState a η hη z t.toNNReal b i))*
+       physicalDelta a (cap j) (σ j) (off j) (regimeClamp η N (vectorState a η hη z t.toNNReal b j)))^(-regM a)*
+       physicalV (regimeClamp η N (vectorState a η hη z t.toNNReal b i))*
+       physicalV (regimeClamp η N (vectorState a η hη z t.toNNReal b j)))≤J₁) :
+    (regimeState s a ε η hη N cap σ off z T b).2-
+      (regimeState s a ε η hη N cap σ off z S b).2≤
+      regC a*(m*J₀)+regYoungC a ε/2*(m*(m*J₁)) := by
+  let V := fun (t : ℝ) (i : Fin m) => regimeClamp η N (vectorState a η hη z t.toNNReal b i)
+  have hd (i : Fin m) : Continuous (fun t : ℝ => physicalDelta a (cap i) (σ i) (off i) (V t i)) :=
+    (clamped_delta_continuous a η hη N (cap i) (σ i) (off i)).comp
+      ((localState_continuous a η hη (z i) b).comp continuous_real_toNNReal)
+  have hv (i : Fin m) : Continuous (fun t : ℝ => physicalV (V t i)) :=
+    (clamped_speed_continuous η hη N).comp ((localState_continuous a η hη (z i) b).comp continuous_real_toNNReal)
+  have hf (i : Fin m) : Continuous (fun t : ℝ => physicalDelta a (cap i) (σ i) (off i) (V t i)*physicalV (V t i)^2) := (hd i).mul ((hv i).pow 2)
+  have hg (i j : Fin m) : Continuous (fun t : ℝ =>
+      (physicalDelta a (cap i) (σ i) (off i) (V t i)*physicalDelta a (cap j) (σ j) (off j) (V t j))^(-regM a)*
+      physicalV (V t i)*physicalV (V t j)) :=
+    (((hd i).mul (hd j)).rpow_const (fun t => Or.inl (mul_pos (Real.exp_pos _) (Real.exp_pos _)).ne')).mul (hv i) |>.mul (hv j)
+  have hk : Continuous (fun t : ℝ => regimeDiscountCut s a ε η N cap σ off (vectorState a η hη z t.toNNReal b)) :=
+    (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp ((vectorState_continuous a η hη z b).comp continuous_real_toNNReal)
+  have he := intervalIntegral.integral_add_adjacent_intervals (μ:=volume) (hk.intervalIntegrable 0 S) (hk.intervalIntegrable S T)
+  have hsum (r : Finset (Fin m)) {J : ℝ} (hJ : 0≤J) (f : Fin m → ℝ) (h : ∀ i∈r,f i≤J) :
+      (∑ i ∈ r, f i) ≤ (m:ℝ)*J := by
+    calc
+      _ ≤ ∑ i ∈ r, J := Finset.sum_le_sum h
+      _ = (r.card:ℝ)*J := by simp
+      _ ≤ m*J := mul_le_mul_of_nonneg_right (by exact_mod_cast (show r.card ≤ m by simpa only [Fintype.card_fin] using r.card_le_univ)) hJ
+  change (∫ t in (0:ℝ)..(T:ℝ), _)-(∫ t in (0:ℝ)..(S:ℝ), _)≤_
+  rw [←he,add_sub_cancel_left]
+  unfold regimeDiscountCut regimeDiscount
+  rw [intervalIntegral.integral_add]
+  · rw [intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul,
+      intervalIntegral.integral_finsetSum (f:=fun i t => physicalDelta a (cap i) (σ i) (off i) (V t i)*physicalV (V t i)^2) (fun i hi => (hf i).intervalIntegrable _ _),
+      intervalIntegral.integral_finsetSum (fun i hi => (continuous_finsetSum _ fun j hj => hg i j).intervalIntegrable _ _)]
+    apply add_le_add
+    · exact mul_le_mul_of_nonneg_left (hsum s hJ₀ _ hsingle) (regC_pos ha).le
+    · apply mul_le_mul_of_nonneg_left _ (by have := regEta_lt_half ha; have := regC2_pos ha; have := regEta_pos ha; unfold regYoungC; apply div_nonneg; apply mul_nonneg; apply mul_nonneg; linarith; positivity; positivity; norm_num)
+      apply hsum s (mul_nonneg (by positivity) hJ₁)
+      intro i hi
+      rw [intervalIntegral.integral_finsetSum (fun j hj => (hg i j).intervalIntegrable _ _)]
+      exact hsum (s.erase i) hJ₁ _ (hpair i hi)
+  · exact ((continuous_finsetSum s fun i hi => hf i).const_mul _).intervalIntegrable _ _
+  · exact ((continuous_finsetSum s fun i hi => continuous_finsetSum _ fun j hj => hg i j).const_mul _).intervalIntegrable _ _
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+noncomputable def pairDiscountConstant (a : ℝ) : ℝ :=
+  Real.exp (2*(regM a*regBeta a))/(2*a*(1-Real.exp (2*(regM a*regBeta a)-a)))
+lemma pairDiscountConstant_pos {a : ℝ} (ha : 1/4<a) : 0<pairDiscountConstant a := by
+  have ha0 : 0<a := by linarith
+  have he : Real.exp (2*(regM a*regBeta a)-a)<1 := Real.exp_lt_one_iff.mpr (by nlinarith [regBeta_discount ha])
+  unfold pairDiscountConstant
+  positivity
+noncomputable def phaseDiscountConstant (m : ℕ) (a ε : ℝ) : ℝ :=
+  regC a*(m*(1/regBeta a))+regYoungC a ε/2*(m*(m*pairDiscountConstant a))
+lemma phaseDiscountConstant_nonneg (m : ℕ) {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε) :
+    0≤phaseDiscountConstant m a ε := by
+  have hC₂ := regC2_pos ha
+  have hη₀ := regEta_pos ha
+  have hY : 0≤regYoungC a ε := by
+    unfold regYoungC
+    apply mul_nonneg
+    · apply mul_nonneg
+      · linarith [regEta_lt_half ha]
+      · positivity
+    · positivity
+  unfold phaseDiscountConstant
+  have := (pairDiscountConstant_pos ha).le
+  have := (regBeta_pos ha).le
+  have := (regC_pos ha).le
+  positivity
+lemma actual_phase_discount {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
+    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
+    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (M : ℝ≥0) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0)
+    (hST : S≤T)
+    (hboundary : ∀ i, levelTime a b (z i) (cap i) (U i)≤(S:ℝ≥0∞) ∨
+      (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i))
+    (hsafe : ∀ t, S≤t → t≤T → ‖vectorState a η hη z t b‖≤M ∧
+      ∀ i∈phaseActive (actualPhase a z cap U S b),
+      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
+      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
+    (regimeState (phaseActive (actualPhase a z cap U S b)) a ε η hη M cap
+      (phaseSigma a (actualPhase a z cap U S b)) (phaseOffset a U (actualPhase a z cap U S b)) z T b).2-
+    (regimeState (phaseActive (actualPhase a z cap U S b)) a ε η hη M cap
+      (phaseSigma a (actualPhase a z cap U S b)) (phaseOffset a U (actualPhase a z cap U S b)) z S b).2≤
+      phaseDiscountConstant m a ε := by
+  have ha0 : 0<a := by linarith
+  let p := actualPhase a z cap U S b
+  let A := fun i => (levelTime a b (z i) (cap i) 0).toNNReal
+  have htimes (i) (hi : i∈phaseActive p) := active_phase_interval
+    (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith))
+    (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith))
+    (hboundary i) ((phaseActive_mem p i).mp hi)
+  have hA (i) (hi : i∈phaseActive p) : A i≤S :=
+    ENNReal.coe_le_coe.mp (ENNReal.coe_toNNReal_le_self.trans (htimes i hi).1)
+  have hR (i) (hi : i∈phaseActive p) : fullLoewnerRadius a b.1 (z i) (A i)=cap i :=
+    levelTime_fullRadius a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b
+      (ne_top_of_le_ne_top (by simp) (htimes i hi).1)
+  have hclockT (i) (hi : i∈phaseActive p) : physicalClock a (cap i) (actualState a b.1 T (z i))≤U i :=
+    levelTime_clock_le a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b T
+      ((hsafe T hST le_rfl).2 i hi).1 (htimes i hi).2.1
+  have hclockS (i) (hi : i∈phaseActive p) (hp : p i=2) :
+      U i/2≤physicalClock a (cap i) (actualState a b.1 S (z i)) :=
+    levelTime_clock_ge a ha0 (z i) (hz i) (cap i) (U i/2) (hc i) (hr i) (by have := hu i; linarith) b S
+      ((hsafe S le_rfl hST).2 i hi).1 ((htimes i hi).2.2 hp)
+  have he (t : ℝ) (ht : t∈uIcc (S:ℝ) (T:ℝ)) (i) (hi : i∈phaseActive p) :
+      regimeClamp η M (vectorState a η hη z t.toNNReal b i)=actualState a b.1 t.toNNReal (z i) := by
+    have ht' : t∈Icc (S:ℝ) T := by simpa only [uIcc_of_le (show (S:ℝ)≤T from hST)] using ht
+    have hlo : S≤t.toNNReal := by exact_mod_cast (show (S:ℝ)≤(t.toNNReal:ℝ) by rw [Real.coe_toNNReal _ (S.coe_nonneg.trans ht'.1)]; exact ht'.1)
+    have hhi : t.toNNReal≤T := by exact_mod_cast (show (t.toNNReal:ℝ)≤T by rw [Real.coe_toNNReal _ (S.coe_nonneg.trans ht'.1)]; exact ht'.2)
+    have hs := hsafe t.toNNReal hlo hhi
+    rw [regimeClamp_eq η M _ hη ((norm_le_pi_norm _ i).trans hs.1) (hs.2 i hi).2.1]
+    exact (hs.2 i hi).2.2
+  apply regimeState_discount_le (phaseActive p) ha hε hη M cap (phaseSigma a p) (phaseOffset a U p) z b S T
+    (1/regBeta a) (pairDiscountConstant a) (by have := regBeta_pos ha; positivity) (pairDiscountConstant_pos ha).le
+  · intro i hi
+    obtain ⟨f,hf⟩ := ((hsafe T hST le_rfl).2 i hi).1
+    have hh := actual_regularized_single_discount ha b.1.continuous hST (hA i hi) hf (hc i) (hR i hi)
+      (U:=U i) (σ:=phaseSigma a p i) (off:=phaseOffset a U p i)
+    have hp : (phaseSigma a p i=-regBeta a ∧ phaseOffset a U p i=0) ∨
+      (phaseSigma a p i=regBeta a ∧ phaseOffset a U p i=-regBeta a*U i ∧ activeODEClock b.1 T (A i) f T≤U i) := by
+      by_cases hh : p i=2
+      · right
+        refine ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh],?_⟩
+        rw [←actual_clock_eq ha0.ne' (hc i) b.1.continuous hf ((hA i hi).trans hST) le_rfl (hR i hi)]
+        exact hclockT i hi
+      · exact Or.inl ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh]⟩
+    apply (le_of_eq ?_).trans (hh hp)
+    apply intervalIntegral.integral_congr
+    intro t ht
+    dsimp only
+    rw [he t ht i hi]
+  · intro i hi j hj
+    have hj' := Finset.mem_of_mem_erase hj
+    obtain ⟨f,hf⟩ := ((hsafe T hST le_rfl).2 i hi).1
+    obtain ⟨g,hg⟩ := ((hsafe T hST le_rfl).2 j hj').1
+    have hij : i≠j := (Finset.ne_of_mem_erase hj).symm
+    have hp (k : Fin m) (hk : k∈phaseActive p) (f : ℝ → ℂ) (hf : odeUntil a b.1 (z k) T f) :
+      (phaseSigma a p k=-regBeta a ∧ phaseOffset a U p k=0) ∨
+      (phaseSigma a p k=regBeta a ∧ phaseOffset a U p k=-regBeta a*U k ∧ U k/2≤activeODEClock b.1 T (A k) f S) := by
+      by_cases hh : p k=2
+      · right
+        refine ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh],?_⟩
+        rw [←actual_clock_eq ha0.ne' (hc k) b.1.continuous hf ((hA k hk).trans hST) hST (hR k hk)]
+        exact hclockS k hk hh
+      · exact Or.inl ⟨by simp [phaseSigma,hh],by simp [phaseOffset,hh]⟩
+    have hh := actual_regularized_pair_discount ha b.1.continuous hST (hA i hi) (hA j hj') hf hg
+      (hinj.ne hij) (hc i) (hc j) (hR i hi) (hR j hj') (hsep i j hij)
+      (by simpa only [norm_sub_rev] using hsep j i hij.symm) (hp i hi f hf) (hp j hj' g hg)
+    apply (le_of_eq ?_).trans hh
+    apply intervalIntegral.integral_congr
+    intro t ht
+    dsimp only
+    rw [he t ht i hi,he t ht j hj']
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory ProbabilityTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma stopped_phaseProduct_measurable {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
+    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable (fun b => phaseProduct a cap U (actualPhase a z cap U (S b) b)
+      (vectorState a η hη z (S b) b)) := by
+  have hp := (actualPhase_measurable a ha z hz cap U hc hr hu S hS).mono hS.measurableSpace_le le_rfl
+  have hv := (vectorState_stopped_measurable a η hη z S hS).mono hS.measurableSpace_le le_rfl
+  have he : (fun b => phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (S b) b))=
+      fun b => ∑ p : PhaseVector m, if actualPhase a z cap U (S b) b=p then
+        phaseProduct a cap U p (vectorState a η hη z (S b) b) else 0 := by funext b; simp
+  rw [he]
+  apply Finset.measurable_sum
+  intro p hp'
+  exact ((phaseProduct_measurable a cap U p).comp hv).ite (hp (measurableSet_singleton p)) measurable_const
+lemma actual_phase_expectation_step {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
+    (hsize : ε*((m:ℝ)-1)≤regK a/2)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
+    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
+    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (M : ℝ≥0)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S T : ExactSLEGenerator.Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b,S b≤T b) (hTB : ∀ b,T b≤B)
+    (hboundary : ∀ b i, ∀ u∈({0,U i/2,U i}:Finset ℝ),
+      levelTime a b (z i) (cap i) u≤(S b:ℝ≥0∞) ∨ (T b:ℝ≥0∞)≤levelTime a b (z i) (cap i) u)
+    (hsafe : ∀ b t, t≤T b → (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
+      ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
+      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
+      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
+      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
+    (∫ b, phaseProduct a cap U (actualPhase a z cap U (T b) b) (vectorState a η hη z (T b) b) ∂W)≤
+      Real.exp (phaseDiscountConstant m a ε)*
+      (∫ b, phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (S b) b) ∂W) := by
+  classical
+  have ha0 : 0<a := by linarith
+  let P := fun (Q : ExactSLEGenerator.Driver → ℝ≥0) (b : ExactSLEGenerator.Driver) => phaseProduct a cap U (actualPhase a z cap U (Q b) b) (vectorState a η hη z (Q b) b)
+  let E := fun p : PhaseVector m => {b | actualPhase a z cap U (S b) b=p}
+  have hE (p) : MeasurableSet[hS.measurableSpace] (E p) :=
+    actualPhase_measurable a ha0 z hz cap U hc hr hu S hS (measurableSet_singleton p)
+  have hE' (p) : MeasurableSet (E p) := hS.measurableSpace_le _ (hE p)
+  have htrans (b) : P T b≤phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (T b) b) := by
+    apply actual_phase_transition ha z hz cap U hc hr hu η hη b (S b) (T b) (hST b) (hboundary b)
+    intro i hi
+    exact ((hsafe b (T b) le_rfl ⟨i,hi⟩).2 i hi).2.2
+  have hint (p : PhaseVector m) (Q : ExactSLEGenerator.Driver → ℝ≥0)
+      (hQ : IsStoppingTime rawFiltration (fun b => (Q b : WithTop ℝ≥0)))
+      (hSQ : ∀ b,S b≤Q b) (hQT : ∀ b,Q b≤T b) :
+      Integrable (fun b => if b∈E p then phaseProduct a cap U p (vectorState a η hη z (Q b) b) else 0) W := by
+    by_cases hp : (phaseActive p).Nonempty
+    · have hi : Integrable (fun b => if b∈E p then regularizedProduct (phaseActive p) a cap (phaseSigma a p)
+          (phaseOffset a U p) (vectorState a η hη z (Q b) b) else 0) W := by
+        have hs : ∀ b∈E p, ‖vectorState a η hη z (Q b) b‖≤M ∧ ∀ i∈phaseActive p, η≤Y (vectorState a η hη z (Q b) b i) := by
+          intro b hb
+          have he : actualPhase a z cap U (S b) b=p := hb
+          have hn : (phaseActive (actualPhase a z cap U (S b) b)).Nonempty := he ▸ hp
+          have hh := actual_phase_interval_safe ha z hz cap U hc hr hu η hη M b (S b) (T b) (hST b) hn
+            (fun i => hboundary b i (U i) (by simp)) (hsafe b)
+            (Q b) (hSQ b) (hQT b)
+          rw [he] at hh
+          exact ⟨hh.1,fun i hi => (hh.2 i hi).1⟩
+        convert regularizedProduct_indicator_integrable (phaseActive p) a η hη M cap (phaseSigma a p) (phaseOffset a U p) z W Q hQ (E p) (hE' p) hs using 1
+        funext b
+        by_cases hb : b∈E p <;> simp only [ite_eq_left,ite_eq_right,hb,not_false_eq_true]
+      convert hi.const_mul (phaseConstant a U p) using 1
+      funext b
+      dsimp only [phaseProduct]
+      split_ifs <;> simp
+    · have he : phaseActive p=∅ := Finset.not_nonempty_iff_eq_empty.mp hp
+      simp only [phaseProduct,he,regularizedProduct,Finset.prod_empty,mul_one]
+      have hic : Integrable (fun _ : ExactSLEGenerator.Driver => phaseConstant a U p) W := integrable_const _
+      convert hic.indicator (hE' p) using 1
+      all_goals first | rfl | (funext b; by_cases hb : b∈E p <;> simp [hb,Set.indicator])
+  have hPi (p) : Integrable (fun b => if b∈E p then P T b else 0) W := by
+    have hm : Measurable (fun b => if b∈E p then P T b else 0) :=
+      (stopped_phaseProduct_measurable a ha0 z hz cap U hc hr hu η hη T hT).ite (hE' p) measurable_const
+    apply (hint p T hT hST (fun _ => le_rfl)).mono' hm.aestronglyMeasurable
+    apply Eventually.of_forall
+    intro b
+    by_cases hb : b∈E p
+    · simp only [ite_eq_left hb,Real.norm_eq_abs,abs_of_pos (show 0<P T b from phaseProduct_pos a cap U _ _)]
+      simpa only [show actualPhase a z cap U (S b) b=p from hb] using htrans b
+    · simp only [ite_eq_right hb,norm_zero,le_refl]
+  have hSi (p) : Integrable (fun b => if b∈E p then P S b else 0) W := by
+    convert hint p S hS (fun _ => le_rfl) hST using 1
+    funext b
+    by_cases hb : b∈E p
+    · simp only [ite_eq_left hb,P,show actualPhase a z cap U (S b) b=p from hb]
+    · simp only [ite_eq_right hb]
+  have hlocal (p) : (∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (T b) b) else 0) ∂W)≤
+      Real.exp (phaseDiscountConstant m a ε)*(∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (S b) b) else 0) ∂W) := by
+    by_cases hp : (phaseActive p).Nonempty
+    · have hsiz : ε*(((phaseActive p).card:ℝ)-1)≤regK a/2 := by
+        have hcard : ((phaseActive p).card:ℝ)≤ m := by exact_mod_cast (show (phaseActive p).card ≤ m by simpa using (phaseActive p).card_le_univ)
+        exact (mul_le_mul_of_nonneg_left (sub_le_sub_right hcard 1) hε.le).trans hsize
+      have hh := vector_regime_event_comparison (phaseActive p) ha hε hη hsiz M cap (phaseSigma a p) (phaseOffset a U p)
+        (fun i hi => phaseSigma_bound ha p i) z W hW S T B hS hT hST hTB (E p) (hE p)
+      have hsafe' : ∀ b∈E p, ∀ t, S b≤t → t≤T b → ‖vectorState a η hη z t b‖≤M ∧
+        ∀ i∈phaseActive p, η≤Y (vectorState a η hη z t b i) ∧
+        physicalDelta a (cap i) (phaseSigma a p i) (phaseOffset a U p i) (vectorState a η hη z t b i)≤1 := by
+        intro b hb
+        have he : actualPhase a z cap U (S b) b=p := hb
+        have hh := actual_phase_interval_safe ha z hz cap U hc hr hu η hη M b (S b) (T b) (hST b)
+          (he ▸ hp) (fun i => hboundary b i (U i) (by simp)) (hsafe b)
+        simpa only [he] using hh
+      have hD : ∀ b∈E p,
+        (regimeState (phaseActive p) a ε η hη M cap (phaseSigma a p) (phaseOffset a U p) z (T b) b).2-
+        (regimeState (phaseActive p) a ε η hη M cap (phaseSigma a p) (phaseOffset a U p) z (S b) b).2≤phaseDiscountConstant m a ε := by
+        intro b hb
+        have he : actualPhase a z cap U (S b) b=p := hb
+        have h := actual_phase_discount ha hε z hz hinj cap U hc hr hsep hu η hη M b (S b) (T b) (hST b)
+          (fun i => hboundary b i (U i) (by simp))
+        apply (he ▸ h)
+        intro t hst htt
+        have h1 := hsafe' b hb t hst htt
+        refine ⟨h1.1,?_⟩
+        intro i hi
+        have hi' : i∈phaseActive (actualPhase a z cap U (S b) b) := he ▸ hi
+        have htarg := (active_phase_interval
+          (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith))
+          (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith))
+          (hboundary b i (U i) (by simp)) ((phaseActive_mem _ i).mp hi')).2.1
+        have hti := (ENNReal.coe_le_coe.mpr htt).trans htarg
+        exact (hsafe b t htt ⟨i,hti⟩).2 i hti
+      have hh' := mul_le_mul_of_nonneg_left (hh hsafe' _ hD) (phaseConstant_pos a U p).le
+      simpa only [←integral_const_mul,phaseProduct,mul_ite,mul_zero,mul_assoc,mul_left_comm] using hh'
+    · have he : phaseActive p=∅ := Finset.not_nonempty_iff_eq_empty.mp hp
+      have hnon : 0≤∫ b, (if b∈E p then phaseConstant a U p else 0) ∂W := integral_nonneg fun b => by have := phaseConstant_pos a U p; split_ifs <;> positivity
+      simp only [phaseProduct,he,regularizedProduct,Finset.prod_empty,mul_one]
+      exact le_mul_of_one_le_left hnon (Real.one_le_exp_iff.mpr (phaseDiscountConstant_nonneg m ha hε))
+  rw [←integral_const_mul]
+  apply ExactSLEPasting.integral_le_of_finite_partition (fun b => actualPhase a z cap U (S b) b)
+  · intro p
+    convert hPi p using 1
+    funext b
+    by_cases hb : b∈E p <;> simp [E] at hb ⊢ <;> simp [hb,P]
+  · intro p
+    convert (hSi p).const_mul (Real.exp (phaseDiscountConstant m a ε)) using 1
+    funext b
+    by_cases hb : b∈E p <;> simp [E] at hb ⊢ <;> simp [hb,P]
+  · intro p
+    have h1 : (∫ b, (if b∈E p then P T b else 0) ∂W) ≤
+        ∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (T b) b) else 0) ∂W :=
+      integral_mono (hPi p) (hint p T hT hST (fun _ => le_rfl)) (fun b => by
+        split_ifs with hb
+        · simpa only [show actualPhase a z cap U (S b) b=p from hb] using htrans b
+        · exact le_rfl)
+    have h2 : (∫ b, (if b∈E p then phaseProduct a cap U p (vectorState a η hη z (S b) b) else 0) ∂W) =
+        ∫ b, (if b∈E p then P S b else 0) ∂W := by
+      congr 1
+      funext b
+      by_cases hb : b∈E p
+      · simp only [ite_eq_left hb,P,show actualPhase a z cap U (S b) b=p from hb]
+      · simp only [ite_eq_right hb]
+    have h3 := h1.trans (hlocal p)
+    rw [h2,←integral_const_mul] at h3
+    convert h3 using 1 <;> congr 1 <;> funext b <;>
+      (by_cases hb : b∈E p <;> simp [E] at hb ⊢ <;> simp [hb,P])
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma actualState_zero {a : ℝ} (ha : 0<a) (b : ExactSLEGenerator.Driver) {z : ℂ} (hz : 0<z.im) :
+    actualState a b.1 0 z=(z,0) := by
+  have h0 := radialStop_actual a ha z hz (z.im/2) (by positivity) (by linarith) 0 b 0 (show (0:ℝ≥0)≤_ from bot_le)
+  obtain ⟨f,hf⟩ := h0
+  have hm : loewnerMap a b.1 0 z=z := (loewnerMap_eq hf).trans (by simpa using hf.1)
+  have hd : deriv (loewnerMap a b.1 0) z=1 := by
+    simpa using (hasDerivAt_loewnerMap b.1.continuous (show alive a b.1 0 z from ⟨f,hf⟩)).deriv
+  simp [actualState,hm,hd,b.2]
+lemma actual_phaseProduct_initial {m : ℕ} {a : ℝ} (ha : 1/4<a)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
+    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (b : ExactSLEGenerator.Driver) :
+    phaseProduct a cap U (actualPhase a z cap U 0 b) (vectorState a η hη z 0 b)≤1 := by
+  have ha0 : 0<a := by linarith
+  rw [phaseProduct_eq_prod]
+  apply Finset.prod_le_one₀
+  · intro i hi
+    exact (phaseFactor_pos _ _ _ _ _).le
+  · intro i hi
+    have h0 := radialStop_actual a ha0 (z i) (hz i) ((z i).im/2) (by have := hz i; positivity) (by have := hz i; linarith) 0 b 0 (show (0:ℝ≥0)≤_ from bot_le)
+    have hc0 : physicalClock a (cap i) (actualState a b.1 0 (z i))≤0 := by
+      rw [actualState_zero ha0 b (hz i),physicalClock_radius a (cap i) (hc i) _ (hz i)]
+      simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one]
+      apply div_nonpos_of_nonpos_of_nonneg _ (by positivity)
+      exact Real.log_nonpos (by have := hc i; have := hz i; positivity) ((div_le_one (by have := hz i; positivity)).mpr (hr i))
+    have hAB := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith)
+    have hBC := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith)
+    have he : vectorState a η hη z 0 b i=actualState a b.1 0 (z i) := by
+      simp only [vectorState,localState_zero,actualState_zero ha0 b (hz i)]
+    rw [he]
+    generalize hp : actualPhase a z cap U 0 b i=p
+    fin_cases p
+    · norm_num [phaseFactor,Fin.ext_iff]
+    · have hat := ((phase_one_iff _ _ _ 0 hAB hBC).mp hp).1
+      have hge := levelTime_clock_ge a ha0 (z i) (hz i) (cap i) 0 (hc i) (hr i) le_rfl b 0 h0 hat
+      have heq := le_antisymm hc0 hge
+      simpa [phaseFactor,Fin.ext_iff] using physicalRegularized_activation (σ := -regBeta a) (off := 0) ha _ heq rfl
+    · have hbt := ((phase_two_iff _ _ _ 0 hAB hBC).mp hp).1
+      have hge := levelTime_clock_ge a ha0 (z i) (hz i) (cap i) (U i/2) (hc i) (hr i) (by have := hu i; linarith) b 0 h0 hbt
+      have hu0 : U i=0 := by have := hu i; linarith
+      have heq : physicalClock a (cap i) (actualState a b.1 0 (z i))=0 := by linarith
+      simpa [phaseFactor,Fin.ext_iff,hu0] using physicalRegularized_activation (σ := regBeta a) (off := 0) ha _ heq rfl
+    · have hct := (phase_three_iff _ _ _ 0).mp hp
+      have hge := levelTime_clock_ge a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b 0 h0 hct
+      have hu0 : U i=0 := by have := hu i; linarith
+      norm_num [phaseFactor,Fin.ext_iff,hu0]
+lemma actual_phaseProduct_completed {m : ℕ} (a : ℝ) (z : Fin m → ℂ) (cap U : Fin m → ℝ)
+    (η : ℝ) (hη : 0<η) (b : ExactSLEGenerator.Driver) (T : ℝ≥0)
+    (h : ∀ i,levelTime a b (z i) (cap i) (U i)≤(T:ℝ≥0∞)) :
+    phaseProduct a cap U (actualPhase a z cap U T b) (vectorState a η hη z T b)=
+      ∏ i, Real.exp (regP a*U i/2)/3 := by
+  rw [phaseProduct_eq_prod]
+  apply Finset.prod_congr rfl
+  intro i hi
+  have hp : actualPhase a z cap U T b i=3 := (phase_three_iff _ _ _ T).mpr (h i)
+  simp [phaseFactor,hp,Fin.ext_iff]
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/PhaseStopping.lean b/lean/OAI/Probability/SLEGauge/PhaseStopping.lean
index c079a53b0048768ef6743e98c13a0cecb982af85..1ccbcda7d0a40dfe4836d86a29e9d5a11cd1a913 100644
--- a/lean/OAI/Probability/SLEGauge/PhaseStopping.lean
+++ b/lean/OAI/Probability/SLEGauge/PhaseStopping.lean
@@ -1,454 +1,454 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.CutoffExhaustion
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-open scoped Classical
-lemma phaseActive_mem {m : ℕ} (p : PhaseVector m) (i : Fin m) :
-    i∈phaseActive p ↔ p i=1 ∨ p i=2 := by simp [phaseActive]
-lemma phaseConstant_pos {m : ℕ} (a : ℝ) (U : Fin m → ℝ) (p : PhaseVector m) :
-    0<phaseConstant a U p := by
-  apply Finset.prod_pos
-  intro i hi
-  split_ifs <;> positivity
-lemma phaseSigma_bound {m : ℕ} {a : ℝ} (ha : 1/4<a) (p : PhaseVector m) (i : Fin m) :
-    |phaseSigma a p i|≤regBeta a := by
-  unfold phaseSigma
-  split_ifs <;> simp [abs_of_pos (regBeta_pos ha)]
-lemma physicalRegularized_pos (a cap σ off : ℝ) (v : State) :
-    0<physicalRegularized a cap σ off v := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2
-  unfold physicalRegularized regFactor sinePower clockExp physicalDelta
-  positivity
-lemma regularizedProduct_pos {m : ℕ} (s : Finset (Fin m)) (a : ℝ)
-    (cap σ off : Fin m → ℝ) (v : Fin m → State) :
-    0<regularizedProduct s a cap σ off v :=
-  Finset.prod_pos fun i _ => physicalRegularized_pos a (cap i) (σ i) (off i) (v i)
-lemma phaseProduct_pos {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ)
-    (p : PhaseVector m) (v : Fin m → State) : 0<phaseProduct a cap U p v :=
-  mul_pos (phaseConstant_pos a U p) (regularizedProduct_pos _ a cap _ _ v)
-lemma physicalRegularized_activation {a cap σ off : ℝ} (ha : 1/4<a) (v : State)
-    (hu : physicalClock a cap v=0) (ho : off=0) : physicalRegularized a cap σ off v≤1 := by
-  have hθ := physicalTheta_mem v
-  have hb := (regFactor_bounds ha (by norm_num : (0:ℝ)<1) le_rfl
-    (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)).2.2.2
-  simpa [physicalRegularized,clockExp,physicalDelta,hu,ho,div_eq_mul_inv,mul_comm] using (div_le_div_of_nonneg_right hb (by norm_num : (0:ℝ)≤3))
-lemma physicalRegularized_completion {a cap σ off U : ℝ} (ha : 1/4<a) (v : State)
-    (hu : physicalClock a cap v=U) (ho : off+σ*U=0) :
-    Real.exp (regP a*U/2)/3≤physicalRegularized a cap σ off v := by
-  have hθ := physicalTheta_mem v
-  have hb := (regFactor_bounds ha (by norm_num : (0:ℝ)<1) le_rfl
-    (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)).2.1
-  have h := mul_le_mul_of_nonneg_left hb (show 0≤Real.exp (regP a*U/2)/3 by positivity)
-  simpa [physicalRegularized,clockExp,physicalDelta,hu,ho,mul_div_assoc,div_mul_eq_mul_div] using h
-lemma physicalRegularized_midpoint (a cap U : ℝ) (v : State)
-    (hu : physicalClock a cap v=U/2) :
-    physicalRegularized a cap (regBeta a) (-regBeta a*U) v=
-      physicalRegularized a cap (-regBeta a) 0 v := by
-  unfold physicalRegularized physicalDelta
-  rw [hu]
-  congr 3
-  ring
-lemma physicalRegularized_measurable (a cap σ off : ℝ) : Measurable (physicalRegularized a cap σ off) := by
-  unfold physicalRegularized clockExp physicalDelta physicalClock regFactor sinePower physicalTheta
-  have hX := X.continuous.measurable
-  have hY := Y.continuous.measurable
-  have hJ := J.continuous.measurable
-  fun_prop
-lemma phaseProduct_measurable {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ) (p : PhaseVector m) :
-    Measurable (phaseProduct a cap U p) := by
-  unfold phaseProduct regularizedProduct
-  apply measurable_const.mul
-  exact Finset.measurable_prod _ fun i _ => (physicalRegularized_measurable a (cap i) _ _).comp (measurable_pi_apply i)
-end ExactSLEGreen
-
- 
-namespace ExactSLEStops
-open scoped ENNReal NNReal
-lemma phase_mono (A B C : ℝ≥0∞) : Monotone (phase A B C) := by
-  intro S T hST
-  have hst : (S:ℝ≥0∞)≤T := ENNReal.coe_le_coe.mpr hST
-  unfold phase
-  split_ifs <;> norm_num [Fin.le_def] at * <;>
-    have hst' : (S:ℝ≥0∞)≤T := ENNReal.coe_le_coe.mpr hST
-  all_goals exfalso; order
-lemma boundary_eq {τ : ℝ≥0∞} {S T : ℝ≥0}
-    (h : τ≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤τ) (hS : (S:ℝ≥0∞)<τ) (hT : τ≤(T:ℝ≥0∞)) : τ=(T:ℝ≥0∞) :=
-  le_antisymm hT (h.resolve_left (not_le.mpr hS))
-end ExactSLEStops
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops
-open scoped Classical ENNReal NNReal
-noncomputable def phaseFactor (a cap U : ℝ) (p : Fin 4) (v : State) : ℝ :=
-  if p=0 then 1 else if p=3 then Real.exp (regP a*U/2)/3 else
-    physicalRegularized a cap (if p=2 then regBeta a else -regBeta a)
-      (if p=2 then -regBeta a*U else 0) v
-lemma phaseProduct_eq_prod {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ) (p : PhaseVector m)
-    (v : Fin m → State) : phaseProduct a cap U p v=∏ i, phaseFactor a (cap i) (U i) (p i) (v i) := by
-  unfold phaseProduct phaseConstant regularizedProduct phaseActive
-  rw [Finset.prod_filter,←Finset.prod_mul_distrib]
-  apply Finset.prod_congr rfl
-  intro i hi
-  unfold phaseSigma phaseOffset phaseFactor
-  generalize hp : p i=j
-  fin_cases j <;> norm_num [Fin.ext_iff]
-lemma phaseFactor_pos (a cap U : ℝ) (p : Fin 4) (v : State) : 0<phaseFactor a cap U p v := by
-  unfold phaseFactor
-  split_ifs
-  · norm_num
-  · positivity
-  · exact physicalRegularized_pos _ _ _ _ _
-  · exact physicalRegularized_pos _ _ _ _ _
- 
-lemma phaseFactor_transition {a cap U : ℝ} (ha : 1/4<a) (v : State)
-    (A B C : ℝ≥0∞) (S T : ℝ≥0) (hAB : A≤B) (hBC : B≤C) (hST : S≤T)
-    (hA : A≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤A)
-    (hB : B≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤B)
-    (hC : C≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤C)
-    (huA : A=(T:ℝ≥0∞) → physicalClock a cap v=0)
-    (huB : B=(T:ℝ≥0∞) → physicalClock a cap v=U/2)
-    (huC : C=(T:ℝ≥0∞) → physicalClock a cap v=U) :
-    phaseFactor a cap U (phase A B C T) v≤phaseFactor a cap U (phase A B C S) v := by
-  generalize hp : phase A B C S=p
-  generalize hq : phase A B C T=q
-  have hpq : p≤q := by rw [←hp,←hq]; exact phase_mono A B C hST
-  fin_cases p <;> fin_cases q <;> try (norm_num [Fin.le_def] at hpq) <;> try exact le_rfl
-  · have hsa := (phase_zero_iff A B C S hAB hBC).mp hp
-    have hat := ((phase_one_iff A B C T hAB hBC).mp hq).1
-    have hu := huA (boundary_eq hA hsa hat)
-    simpa [phaseFactor,Fin.ext_iff] using physicalRegularized_activation (σ := -regBeta a) (off := 0) ha v hu rfl
-  · have hsa := (phase_zero_iff A B C S hAB hBC).mp hp
-    have hbt := ((phase_two_iff A B C T hAB hBC).mp hq).1
-    have hua := huA (boundary_eq hA hsa (hAB.trans hbt))
-    have hub := huB (boundary_eq hB (hsa.trans_le hAB) hbt)
-    have hu0 : U=0 := by linarith
-    simpa [phaseFactor,Fin.ext_iff,hu0] using
-      physicalRegularized_activation (σ := regBeta a) (off := 0) ha v hua rfl
-  · have hsa := (phase_zero_iff A B C S hAB hBC).mp hp
-    have hct := (phase_three_iff A B C T).mp hq
-    have hua := huA (boundary_eq hA hsa (hAB.trans (hBC.trans hct)))
-    have huc := huC (boundary_eq hC (hsa.trans_le (hAB.trans hBC)) hct)
-    have hu0 : U=0 := by linarith
-    norm_num [phaseFactor,Fin.ext_iff,hu0]
-  · have hsb := ((phase_one_iff A B C S hAB hBC).mp hp).2
-    have hbt := ((phase_two_iff A B C T hAB hBC).mp hq).1
-    have hu := huB (boundary_eq hB hsb hbt)
-    simpa [phaseFactor,Fin.ext_iff] using (physicalRegularized_midpoint a cap U v hu).le
-  · have hsb := ((phase_one_iff A B C S hAB hBC).mp hp).2
-    have hct := (phase_three_iff A B C T).mp hq
-    have hub := huB (boundary_eq hB hsb (hBC.trans hct))
-    have huc := huC (boundary_eq hC (hsb.trans_le hBC) hct)
-    have hu0 : U=0 := by linarith
-    have h := physicalRegularized_completion (σ := -regBeta a) (off := 0) ha v huc (by rw [hu0]; ring)
-    simpa [phaseFactor,Fin.ext_iff] using h
-  · have hsc := ((phase_two_iff A B C S hAB hBC).mp hp).2
-    have hct := (phase_three_iff A B C T).mp hq
-    have hu := huC (boundary_eq hC hsc hct)
-    have h := physicalRegularized_completion (σ := regBeta a) (off := -regBeta a*U) ha v hu (by ring)
-    simpa [phaseFactor,Fin.ext_iff] using h
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
-noncomputable def actualPhase {m : ℕ} (a : ℝ) (z : Fin m → ℂ) (cap U : Fin m → ℝ)
-    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : PhaseVector m := fun i =>
-  phase (levelTime a b (z i) (cap i) 0) (levelTime a b (z i) (cap i) (U i/2))
-    (levelTime a b (z i) (cap i) (U i)) t
-lemma actualPhase_measurable {m : ℕ} (a : ℝ) (ha : 0<a) (z : Fin m → ℂ)
-    (hz : ∀ i, 0<(z i).im) (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i)
-    (hr : ∀ i, cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable[hS.measurableSpace] (fun b => actualPhase a z cap U (S b) b) := by
-  apply @Measurable.of_eval _ _ _ hS.measurableSpace _
-  intro i
-  apply phase_measurable rawFiltration _ _ _ S _ _ _ hS
-  all_goals apply radialTime_stopping a ha (z i) (hz i) _ (levelRadius_pos _ _ _ (hc i))
-  · exact (levelRadius_le ha (hc i) le_rfl).trans (by have := hr i; linarith)
-  · exact (levelRadius_le ha (hc i) (by have := hu i; linarith : 0≤U i/2)).trans (by have := hr i; linarith)
-  · exact (levelRadius_le ha (hc i) (hu i)).trans (by have := hr i; linarith)
-lemma active_phase_interval {A B C : ℝ≥0∞} {S T : ℝ≥0} (hAB : A≤B) (hBC : B≤C)
-    (hC : C≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤C)
-    (hi : phase A B C S=1 ∨ phase A B C S=2) :
-    A≤(S:ℝ≥0∞) ∧ (T:ℝ≥0∞)≤C ∧ (phase A B C S=2 → B≤(S:ℝ≥0∞)) := by
-  rcases hi with hi|hi
-  · have hp := (phase_one_iff A B C S hAB hBC).mp hi
-    refine ⟨hp.1,hC.resolve_left (not_le.mpr (hp.2.trans_le hBC)),?_⟩
-    intro hh
-    have : (1:Fin 4)=2 := hi.symm.trans hh
-    norm_num [Fin.ext_iff] at this
-  · have hp := (phase_two_iff A B C S hAB hBC).mp hi
-    exact ⟨hAB.trans hp.1,hC.resolve_left (not_le.mpr hp.2),fun _ => hp.1⟩
-lemma phase_delta_le {a cap U : ℝ} (ha : 1/4<a) (p : Fin 4) (v : State)
-    (hlo : 0≤physicalClock a cap v) (hhi : physicalClock a cap v≤U) :
-    physicalDelta a cap (if p=2 then regBeta a else -regBeta a)
-      (if p=2 then -regBeta a*U else 0) v≤1 := by
-  unfold physicalDelta
-  apply Real.exp_le_one_iff.mpr
-  split_ifs
-  · nlinarith [regBeta_pos ha]
-  · nlinarith [regBeta_pos ha]
- 
-
-lemma actual_phase_interval_safe {m : ℕ} {a : ℝ} (ha : 1/4<a)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
-    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (M : ℝ≥0) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0)
-    (_hST : S≤T) (hne : (phaseActive (actualPhase a z cap U S b)).Nonempty)
-    (hboundary : ∀ i, levelTime a b (z i) (cap i) (U i)≤(S:ℝ≥0∞) ∨
-      (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i))
-    (hsafe : ∀ t≤T, (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
-      ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
-      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
-      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
-      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
-    ∀ t, S≤t → t≤T → ‖vectorState a η hη z t b‖≤M ∧
-      ∀ i∈phaseActive (actualPhase a z cap U S b),
-      η≤Y (vectorState a η hη z t b i) ∧
-      physicalDelta a (cap i) (phaseSigma a (actualPhase a z cap U S b) i)
-        (phaseOffset a U (actualPhase a z cap U S b) i) (vectorState a η hη z t b i)≤1 := by
-  have ha0 : 0<a := by linarith
-  have htimes (i) (hi : i∈phaseActive (actualPhase a z cap U S b)) :=
-    active_phase_interval
-      (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith))
-      (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith))
-      (hboundary i) ((phaseActive_mem (actualPhase a z cap U S b) i).mp hi)
-  intro t hts htt
-  obtain ⟨j,hj⟩ := hne
-  have hti (i) (hi : i∈phaseActive (actualPhase a z cap U S b)) :
-      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) :=
-    (ENNReal.coe_le_coe.mpr htt).trans (htimes i hi).2.1
-  have hs := hsafe t htt ⟨j,hti j hj⟩
-  refine ⟨hs.1,fun i hi => ⟨(hs.2 i (hti i hi)).2.1,?_⟩⟩
-  rw [(hs.2 i (hti i hi)).2.2]
-  apply phase_delta_le ha
-  · exact levelTime_clock_ge a ha0 (z i) (hz i) (cap i) 0 (hc i) (hr i) le_rfl b t
-      (hs.2 i (hti i hi)).1 ((htimes i hi).1.trans (ENNReal.coe_le_coe.mpr hts))
-  · exact levelTime_clock_le a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b t
-      (hs.2 i (hti i hi)).1 (hti i hi)
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma levelTime_clock_at_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (he : levelTime a b z cap u=(t:ℝ≥0∞)) :
-    physicalClock a cap (actualState a b.1 t z)=u := by
-  have h := levelTime_clock a ha z hz cap u hc hr hu b (by rw [he]; simp)
-  simpa only [he,ENNReal.toNNReal_coe] using h
-lemma actual_phase_transition {m : ℕ} {a : ℝ} (ha : 1/4<a)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
-    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0) (hST : S≤T)
-    (hboundary : ∀ i, ∀ u∈({0,U i/2,U i}:Finset ℝ),
-      levelTime a b (z i) (cap i) u≤(S:ℝ≥0∞) ∨
-      (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) u)
-    (hagree : ∀ i, (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
-      vectorState a η hη z T b i=actualState a b.1 T (z i)) :
-    phaseProduct a cap U (actualPhase a z cap U T b) (vectorState a η hη z T b)≤
-    phaseProduct a cap U (actualPhase a z cap U S b) (vectorState a η hη z T b) := by
-  have ha0 : 0<a := by linarith
-  rw [phaseProduct_eq_prod,phaseProduct_eq_prod]
-  apply Finset.prod_le_prod₀
-  · intro i hi
-    exact (phaseFactor_pos _ _ _ _ _).le
-  · intro i hi
-    have hAB := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith)
-    have hBC := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith)
-    apply phaseFactor_transition ha _ _ _ _ S T hAB hBC hST
-      (hboundary i 0 (by simp)) (hboundary i (U i/2) (by simp)) (hboundary i (U i) (by simp))
-    · intro he
-      rw [hagree i (by rw [←he]; exact hAB.trans hBC)]
-      exact levelTime_clock_at_eq a ha0 (z i) (hz i) (cap i) 0 (hc i) (hr i) le_rfl b T he
-    · intro he
-      rw [hagree i (by rw [←he]; exact hBC)]
-      exact levelTime_clock_at_eq a ha0 (z i) (hz i) (cap i) (U i/2) (hc i) (hr i) (by have := hu i; linarith) b T he
-    · intro he
-      rw [hagree i he.symm.le]
-      exact levelTime_clock_at_eq a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b T he
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ProbabilityTheory MeasureTheory ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus Set Filter
-open scoped Classical NNReal ENNReal
-lemma vectorState_stopped_measurable {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ)
-    (S : Driver → ℝ≥0) (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable[hS.measurableSpace] (fun b => vectorState a η hη z (S b) b) := by
-  have hA : StronglyAdapted rawFiltration (vectorState a η hη z) :=
-    fun t => (vectorState_adapted a η hη z t).stronglyMeasurable
-  exact measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
-    (vectorState_continuous a η hη z)) hS
-lemma regimeState_stopped_measurable {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ)
-    (hη : 0<η) (N : ℝ≥0) (cap σ off : Fin m → ℝ) (z : Fin m → ℂ)
-    (S : Driver → ℝ≥0) (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable[hS.measurableSpace] (fun b => regimeState s a ε η hη N cap σ off z (S b) b) := by
-  have hA : StronglyAdapted rawFiltration (regimeState s a ε η hη N cap σ off z) :=
-    fun t => (regimeState_adapted s a ε η hη N cap σ off z t).stronglyMeasurable
-  exact measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
-    (regimeState_continuous s a ε η hη N cap σ off z)) hS
-lemma regularizedProduct_measurable {m : ℕ} (s : Finset (Fin m)) (a : ℝ)
-    (cap σ off : Fin m → ℝ) : Measurable (regularizedProduct s a cap σ off) := by
-  unfold regularizedProduct
-  exact Finset.measurable_prod _ fun i _ =>
-    (physicalRegularized_measurable a (cap i) (σ i) (off i)).comp (measurable_pi_apply i)
-lemma regularizedProduct_bound {m : ℕ} (s : Finset (Fin m)) (a η : ℝ) (hη : 0<η)
-    (N : ℝ≥0) (cap σ off : Fin m → ℝ) : ∃ C : ℝ, ∀ v : Fin m → State,
-    ‖v‖≤N → (∀ i∈s, η≤Y (v i)) → ‖regularizedProduct s a cap σ off v‖≤C := by
-  obtain ⟨C,hC⟩ := (regimeCutoff_compact s a η N cap σ off).exists_bound_of_continuous
-    (regimeCutoff_smooth s a η hη N cap σ off).continuous
-  refine ⟨C,fun v hv hy => ?_⟩
-  have he := (regimeCutoff_eq_near s a η hη N cap σ off (v,0)
-    (max_le hv (by simp)) hy).eq_of_nhds
-  simpa only [he,discountedProduct,neg_zero,Real.exp_zero,one_mul] using hC (v,0)
-lemma regimeState_discount_bound {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ)
-    (hη : 0<η) (N B : ℝ≥0) (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) :
-    ∃ L : ℝ, 0≤L ∧ ∀ (b : Driver) (t : ℝ≥0), t≤B →
-      |(regimeState s a ε η hη N cap σ off z t b).2|≤L := by
-  obtain ⟨K,hK,hKb⟩ := regimeDiscountCut_bound s a ε η hη N cap σ off
-  refine ⟨K*B,by positivity,fun b t ht => ?_⟩
-  change ‖∫ u in (0:ℝ)..(t:ℝ), regimeDiscountCut s a ε η N cap σ off
-    (vectorState a η hη z u.toNNReal b)‖≤_
-  calc
-    _ ≤ K*|(t:ℝ)-0| := intervalIntegral.norm_integral_le_of_norm_le_const fun u _ => hKb _
-    _ ≤ K*B := by rw [sub_zero,abs_of_nonneg t.coe_nonneg]; exact mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hK
-lemma regularizedProduct_indicator_integrable {m : ℕ} (s : Finset (Fin m)) (a η : ℝ)
-    (hη : 0<η) (N : ℝ≥0) (cap σ off : Fin m → ℝ) (z : Fin m → ℂ)
-    (W : Measure Driver) [IsProbabilityMeasure W] (S : Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (E : Set Driver) (hE : MeasurableSet E)
-    (hsafe : ∀ b∈E, ‖vectorState a η hη z (S b) b‖≤N ∧
-      ∀ i∈s, η≤Y (vectorState a η hη z (S b) b i)) :
-    Integrable (fun b => if b∈E then regularizedProduct s a cap σ off
-      (vectorState a η hη z (S b) b) else 0) W := by
-  obtain ⟨C,hC⟩ := regularizedProduct_bound s a η hη N cap σ off
-  have hm := (regularizedProduct_measurable s a cap σ off).comp
-    ((vectorState_stopped_measurable a η hη z S hS).mono hS.measurableSpace_le le_rfl)
-  apply Integrable.of_bound (hm.ite hE measurable_const).aestronglyMeasurable (max C 0)
-  apply Eventually.of_forall
-  intro b
-  split_ifs with hb
-  · exact (hC _ (hsafe b hb).1 (hsafe b hb).2).trans (le_max_left _ _)
-  · simpa only [norm_zero] using le_max_right C 0
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus ProbabilityTheory MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
- 
-
-theorem vector_regime_event_comparison {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (M : ℝ≥0) (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (z : Fin m → ℂ) (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (S T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
-    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
-    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b →
-      ‖vectorState a η hη z t b‖≤M ∧
-      ∀ i∈s, η≤Y (vectorState a η hη z t b i) ∧
-        physicalDelta a (cap i) (σ i) (off i) (vectorState a η hη z t b i)≤1)
-    (D : ℝ) (hD : ∀ b∈E,
-      (regimeState s a ε η hη M cap σ off z (T b) b).2-
-      (regimeState s a ε η hη M cap σ off z (S b) b).2≤D) :
-    (∫ b, (if b∈E then regularizedProduct s a cap σ off (vectorState a η hη z (T b) b) else 0) ∂W) ≤
-    Real.exp D*(∫ b, (if b∈E then regularizedProduct s a cap σ off (vectorState a η hη z (S b) b) else 0) ∂W) := by
-  let R := fun t b => (regimeState s a ε η hη M cap σ off z t b).2
-  let P := fun t b => regularizedProduct s a cap σ off (vectorState a η hη z t b)
-  obtain ⟨L,hL,hLb⟩ := regimeState_discount_bound s a ε η hη M B cap σ off z
-  have hSB : ∀ b, S b≤B := fun b => (hST b).trans (hTB b)
-  have hRS : Measurable[hS.measurableSpace] (fun b => R (S b) b) :=
-    (regimeState_stopped_measurable s a ε η hη M cap σ off z S hS).snd
-  have hRT : Measurable (fun b => R (T b) b) :=
-    ((regimeState_stopped_measurable s a ε η hη M cap σ off z T hT).mono hT.measurableSpace_le le_rfl).snd
-  have hRs : Measurable (fun b => R (S b) b) := hRS.mono hS.measurableSpace_le le_rfl
-  let w : Driver → ℝ := fun b => if b∈E then Real.exp (R (S b) b) else 0
-  have hw : StronglyMeasurable[hS.measurableSpace] w := (hRS.exp.ite hE measurable_const).stronglyMeasurable
-  have hw0 : ∀ b, 0≤w b := by intro b; dsimp [w]; split_ifs <;> positivity
-  have hwb : ∀ b, ‖w b‖≤Real.exp L := by
-    intro b
-    dsimp [w]
-    split_ifs
-    · simp only [abs_of_pos (Real.exp_pos _)]
-      exact Real.exp_le_exp.mpr ((le_abs_self _).trans (hLb b (S b) (hSB b)))
-    · simpa using (Real.exp_pos L).le
-  have hopt := vector_regime_weighted_optional_bounded s ha hε hη hsize M cap σ off hσ z W hW
-    S T B hS hT hST hTB E hE w hw hw0 (Real.exp L) hwb
-    (by intro b hb; simp [w,hb]) hsafe
-  have hE' : MeasurableSet E := hS.measurableSpace_le E hE
-  have hPi (Q : Driver → ℝ≥0) (hQ : IsStoppingTime rawFiltration (fun b => (Q b : WithTop ℝ≥0)))
-      (hSQ : ∀ b,S b≤Q b) (hQT : ∀ b,Q b≤T b) :
-      Integrable (fun b => if b∈E then P (Q b) b else 0) W := by
-    apply regularizedProduct_indicator_integrable s a η hη M cap σ off z W Q hQ E hE'
-    intro b hb
-    exact ⟨(hsafe b hb _ (hSQ b) (hQT b)).1,fun i hi => (hsafe b hb _ (hSQ b) (hQT b)).2 i hi |>.1⟩
-  have hFi := hPi T hT hST (fun _ => le_rfl)
-  have hGi := hPi S hS (fun _ => le_rfl) hST
-  let F := fun b => if b∈E then Real.exp (R (S b) b-R (T b) b)*P (T b) b else 0
-  have hF_eq : F=fun b => Real.exp (R (S b) b-R (T b) b)*(if b∈E then P (T b) b else 0) := by
-    funext b; dsimp [F]; split_ifs <;> simp
-  have hFint : Integrable F W := by
-    rw [hF_eq]
-    apply hFi.bdd_mul (hRs.sub hRT).exp.aestronglyMeasurable
-    apply Eventually.of_forall
-    intro b
-    simp only [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
-    apply Real.exp_le_exp.mpr (show R (S b) b-R (T b) b≤2*L from ?_)
-    have h1 := (abs_le.mp (hLb b (S b) (hSB b))).2
-    have h2 := (abs_le.mp (hLb b (T b) (hTB b))).1
-    change R (S b) b≤L at h1
-    change -L≤R (T b) b at h2
-    linarith
-  have heT : (fun b => w b*discountedProduct s a cap σ off
-      (regimeState s a ε η hη M cap σ off z (T b) b))=F := by
-    funext b
-    dsimp [w,F,discountedProduct,P,R,regimeState]
-    split_ifs <;> simp [Real.exp_sub,div_eq_mul_inv,Real.exp_neg,mul_assoc]
-  have heS : (fun b => w b*discountedProduct s a cap σ off
-      (regimeState s a ε η hη M cap σ off z (S b) b))=(fun b => if b∈E then P (S b) b else 0) := by
-    funext b
-    dsimp [w,discountedProduct,P,R,regimeState]
-    split_ifs <;> simp [Real.exp_neg,Real.exp_ne_zero]
-  rw [heT,heS] at hopt
-  calc
-    _ ≤ ∫ b, Real.exp D*F b ∂W := integral_mono hFi (hFint.const_mul _) (fun b => by
-      dsimp [F]
-      split_ifs with hb
-      · have h : 1≤Real.exp D*Real.exp (R (S b) b-R (T b) b) := by
-          rw [←Real.exp_add]
-          apply Real.one_le_exp_iff.mpr
-          have hd := hD b hb
-          change R (T b) b-R (S b) b≤D at hd
-          linarith
-        have h0 := (regularizedProduct_pos s a cap σ off (vectorState a η hη z (T b) b)).le
-        simpa only [one_mul,mul_assoc] using mul_le_mul_of_nonneg_right h h0
-      · simp)
-    _ = Real.exp D*(∫ b,F b ∂W) := integral_const_mul _ _
-    _ ≤ _ := mul_le_mul_of_nonneg_left hopt (Real.exp_pos D).le
-end ExactSLEGreen
-open MeasureTheory
-open scoped Classical
-namespace ExactSLEPasting
-end ExactSLEPasting
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.CutoffExhaustion
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+open scoped Classical
+lemma phaseActive_mem {m : ℕ} (p : PhaseVector m) (i : Fin m) :
+    i∈phaseActive p ↔ p i=1 ∨ p i=2 := by simp [phaseActive]
+lemma phaseConstant_pos {m : ℕ} (a : ℝ) (U : Fin m → ℝ) (p : PhaseVector m) :
+    0<phaseConstant a U p := by
+  apply Finset.prod_pos
+  intro i hi
+  split_ifs <;> positivity
+lemma phaseSigma_bound {m : ℕ} {a : ℝ} (ha : 1/4<a) (p : PhaseVector m) (i : Fin m) :
+    |phaseSigma a p i|≤regBeta a := by
+  unfold phaseSigma
+  split_ifs <;> simp [abs_of_pos (regBeta_pos ha)]
+lemma physicalRegularized_pos (a cap σ off : ℝ) (v : State) :
+    0<physicalRegularized a cap σ off v := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2
+  unfold physicalRegularized regFactor sinePower clockExp physicalDelta
+  positivity
+lemma regularizedProduct_pos {m : ℕ} (s : Finset (Fin m)) (a : ℝ)
+    (cap σ off : Fin m → ℝ) (v : Fin m → State) :
+    0<regularizedProduct s a cap σ off v :=
+  Finset.prod_pos fun i _ => physicalRegularized_pos a (cap i) (σ i) (off i) (v i)
+lemma phaseProduct_pos {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ)
+    (p : PhaseVector m) (v : Fin m → State) : 0<phaseProduct a cap U p v :=
+  mul_pos (phaseConstant_pos a U p) (regularizedProduct_pos _ a cap _ _ v)
+lemma physicalRegularized_activation {a cap σ off : ℝ} (ha : 1/4<a) (v : State)
+    (hu : physicalClock a cap v=0) (ho : off=0) : physicalRegularized a cap σ off v≤1 := by
+  have hθ := physicalTheta_mem v
+  have hb := (regFactor_bounds ha (by norm_num : (0:ℝ)<1) le_rfl
+    (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)).2.2.2
+  simpa [physicalRegularized,clockExp,physicalDelta,hu,ho,div_eq_mul_inv,mul_comm] using (div_le_div_of_nonneg_right hb (by norm_num : (0:ℝ)≤3))
+lemma physicalRegularized_completion {a cap σ off U : ℝ} (ha : 1/4<a) (v : State)
+    (hu : physicalClock a cap v=U) (ho : off+σ*U=0) :
+    Real.exp (regP a*U/2)/3≤physicalRegularized a cap σ off v := by
+  have hθ := physicalTheta_mem v
+  have hb := (regFactor_bounds ha (by norm_num : (0:ℝ)<1) le_rfl
+    (Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2)).2.1
+  have h := mul_le_mul_of_nonneg_left hb (show 0≤Real.exp (regP a*U/2)/3 by positivity)
+  simpa [physicalRegularized,clockExp,physicalDelta,hu,ho,mul_div_assoc,div_mul_eq_mul_div] using h
+lemma physicalRegularized_midpoint (a cap U : ℝ) (v : State)
+    (hu : physicalClock a cap v=U/2) :
+    physicalRegularized a cap (regBeta a) (-regBeta a*U) v=
+      physicalRegularized a cap (-regBeta a) 0 v := by
+  unfold physicalRegularized physicalDelta
+  rw [hu]
+  congr 3
+  ring
+lemma physicalRegularized_measurable (a cap σ off : ℝ) : Measurable (physicalRegularized a cap σ off) := by
+  unfold physicalRegularized clockExp physicalDelta physicalClock regFactor sinePower physicalTheta
+  have hX := X.continuous.measurable
+  have hY := Y.continuous.measurable
+  have hJ := J.continuous.measurable
+  fun_prop
+lemma phaseProduct_measurable {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ) (p : PhaseVector m) :
+    Measurable (phaseProduct a cap U p) := by
+  unfold phaseProduct regularizedProduct
+  apply measurable_const.mul
+  exact Finset.measurable_prod _ fun i _ => (physicalRegularized_measurable a (cap i) _ _).comp (measurable_pi_apply i)
+end ExactSLEGreen
+
+ 
+namespace ExactSLEStops
+open scoped ENNReal NNReal
+lemma phase_mono (A B C : ℝ≥0∞) : Monotone (phase A B C) := by
+  intro S T hST
+  have hst : (S:ℝ≥0∞)≤T := ENNReal.coe_le_coe.mpr hST
+  unfold phase
+  split_ifs <;> norm_num [Fin.le_def] at * <;>
+    have hst' : (S:ℝ≥0∞)≤T := ENNReal.coe_le_coe.mpr hST
+  all_goals exfalso; order
+lemma boundary_eq {τ : ℝ≥0∞} {S T : ℝ≥0}
+    (h : τ≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤τ) (hS : (S:ℝ≥0∞)<τ) (hT : τ≤(T:ℝ≥0∞)) : τ=(T:ℝ≥0∞) :=
+  le_antisymm hT (h.resolve_left (not_le.mpr hS))
+end ExactSLEStops
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops
+open scoped Classical ENNReal NNReal
+noncomputable def phaseFactor (a cap U : ℝ) (p : Fin 4) (v : State) : ℝ :=
+  if p=0 then 1 else if p=3 then Real.exp (regP a*U/2)/3 else
+    physicalRegularized a cap (if p=2 then regBeta a else -regBeta a)
+      (if p=2 then -regBeta a*U else 0) v
+lemma phaseProduct_eq_prod {m : ℕ} (a : ℝ) (cap U : Fin m → ℝ) (p : PhaseVector m)
+    (v : Fin m → State) : phaseProduct a cap U p v=∏ i, phaseFactor a (cap i) (U i) (p i) (v i) := by
+  unfold phaseProduct phaseConstant regularizedProduct phaseActive
+  rw [Finset.prod_filter,←Finset.prod_mul_distrib]
+  apply Finset.prod_congr rfl
+  intro i hi
+  unfold phaseSigma phaseOffset phaseFactor
+  generalize hp : p i=j
+  fin_cases j <;> norm_num [Fin.ext_iff]
+lemma phaseFactor_pos (a cap U : ℝ) (p : Fin 4) (v : State) : 0<phaseFactor a cap U p v := by
+  unfold phaseFactor
+  split_ifs
+  · norm_num
+  · positivity
+  · exact physicalRegularized_pos _ _ _ _ _
+  · exact physicalRegularized_pos _ _ _ _ _
+ 
+lemma phaseFactor_transition {a cap U : ℝ} (ha : 1/4<a) (v : State)
+    (A B C : ℝ≥0∞) (S T : ℝ≥0) (hAB : A≤B) (hBC : B≤C) (hST : S≤T)
+    (hA : A≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤A)
+    (hB : B≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤B)
+    (hC : C≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤C)
+    (huA : A=(T:ℝ≥0∞) → physicalClock a cap v=0)
+    (huB : B=(T:ℝ≥0∞) → physicalClock a cap v=U/2)
+    (huC : C=(T:ℝ≥0∞) → physicalClock a cap v=U) :
+    phaseFactor a cap U (phase A B C T) v≤phaseFactor a cap U (phase A B C S) v := by
+  generalize hp : phase A B C S=p
+  generalize hq : phase A B C T=q
+  have hpq : p≤q := by rw [←hp,←hq]; exact phase_mono A B C hST
+  fin_cases p <;> fin_cases q <;> try (norm_num [Fin.le_def] at hpq) <;> try exact le_rfl
+  · have hsa := (phase_zero_iff A B C S hAB hBC).mp hp
+    have hat := ((phase_one_iff A B C T hAB hBC).mp hq).1
+    have hu := huA (boundary_eq hA hsa hat)
+    simpa [phaseFactor,Fin.ext_iff] using physicalRegularized_activation (σ := -regBeta a) (off := 0) ha v hu rfl
+  · have hsa := (phase_zero_iff A B C S hAB hBC).mp hp
+    have hbt := ((phase_two_iff A B C T hAB hBC).mp hq).1
+    have hua := huA (boundary_eq hA hsa (hAB.trans hbt))
+    have hub := huB (boundary_eq hB (hsa.trans_le hAB) hbt)
+    have hu0 : U=0 := by linarith
+    simpa [phaseFactor,Fin.ext_iff,hu0] using
+      physicalRegularized_activation (σ := regBeta a) (off := 0) ha v hua rfl
+  · have hsa := (phase_zero_iff A B C S hAB hBC).mp hp
+    have hct := (phase_three_iff A B C T).mp hq
+    have hua := huA (boundary_eq hA hsa (hAB.trans (hBC.trans hct)))
+    have huc := huC (boundary_eq hC (hsa.trans_le (hAB.trans hBC)) hct)
+    have hu0 : U=0 := by linarith
+    norm_num [phaseFactor,Fin.ext_iff,hu0]
+  · have hsb := ((phase_one_iff A B C S hAB hBC).mp hp).2
+    have hbt := ((phase_two_iff A B C T hAB hBC).mp hq).1
+    have hu := huB (boundary_eq hB hsb hbt)
+    simpa [phaseFactor,Fin.ext_iff] using (physicalRegularized_midpoint a cap U v hu).le
+  · have hsb := ((phase_one_iff A B C S hAB hBC).mp hp).2
+    have hct := (phase_three_iff A B C T).mp hq
+    have hub := huB (boundary_eq hB hsb (hBC.trans hct))
+    have huc := huC (boundary_eq hC (hsb.trans_le hBC) hct)
+    have hu0 : U=0 := by linarith
+    have h := physicalRegularized_completion (σ := -regBeta a) (off := 0) ha v huc (by rw [hu0]; ring)
+    simpa [phaseFactor,Fin.ext_iff] using h
+  · have hsc := ((phase_two_iff A B C S hAB hBC).mp hp).2
+    have hct := (phase_three_iff A B C T).mp hq
+    have hu := huC (boundary_eq hC hsc hct)
+    have h := physicalRegularized_completion (σ := regBeta a) (off := -regBeta a*U) ha v hu (by ring)
+    simpa [phaseFactor,Fin.ext_iff] using h
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+noncomputable def actualPhase {m : ℕ} (a : ℝ) (z : Fin m → ℂ) (cap U : Fin m → ℝ)
+    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : PhaseVector m := fun i =>
+  phase (levelTime a b (z i) (cap i) 0) (levelTime a b (z i) (cap i) (U i/2))
+    (levelTime a b (z i) (cap i) (U i)) t
+lemma actualPhase_measurable {m : ℕ} (a : ℝ) (ha : 0<a) (z : Fin m → ℂ)
+    (hz : ∀ i, 0<(z i).im) (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i)
+    (hr : ∀ i, cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable[hS.measurableSpace] (fun b => actualPhase a z cap U (S b) b) := by
+  apply @Measurable.of_eval _ _ _ hS.measurableSpace _
+  intro i
+  apply phase_measurable rawFiltration _ _ _ S _ _ _ hS
+  all_goals apply radialTime_stopping a ha (z i) (hz i) _ (levelRadius_pos _ _ _ (hc i))
+  · exact (levelRadius_le ha (hc i) le_rfl).trans (by have := hr i; linarith)
+  · exact (levelRadius_le ha (hc i) (by have := hu i; linarith : 0≤U i/2)).trans (by have := hr i; linarith)
+  · exact (levelRadius_le ha (hc i) (hu i)).trans (by have := hr i; linarith)
+lemma active_phase_interval {A B C : ℝ≥0∞} {S T : ℝ≥0} (hAB : A≤B) (hBC : B≤C)
+    (hC : C≤(S:ℝ≥0∞) ∨ (T:ℝ≥0∞)≤C)
+    (hi : phase A B C S=1 ∨ phase A B C S=2) :
+    A≤(S:ℝ≥0∞) ∧ (T:ℝ≥0∞)≤C ∧ (phase A B C S=2 → B≤(S:ℝ≥0∞)) := by
+  rcases hi with hi|hi
+  · have hp := (phase_one_iff A B C S hAB hBC).mp hi
+    refine ⟨hp.1,hC.resolve_left (not_le.mpr (hp.2.trans_le hBC)),?_⟩
+    intro hh
+    have : (1:Fin 4)=2 := hi.symm.trans hh
+    norm_num [Fin.ext_iff] at this
+  · have hp := (phase_two_iff A B C S hAB hBC).mp hi
+    exact ⟨hAB.trans hp.1,hC.resolve_left (not_le.mpr hp.2),fun _ => hp.1⟩
+lemma phase_delta_le {a cap U : ℝ} (ha : 1/4<a) (p : Fin 4) (v : State)
+    (hlo : 0≤physicalClock a cap v) (hhi : physicalClock a cap v≤U) :
+    physicalDelta a cap (if p=2 then regBeta a else -regBeta a)
+      (if p=2 then -regBeta a*U else 0) v≤1 := by
+  unfold physicalDelta
+  apply Real.exp_le_one_iff.mpr
+  split_ifs
+  · nlinarith [regBeta_pos ha]
+  · nlinarith [regBeta_pos ha]
+ 
+
+lemma actual_phase_interval_safe {m : ℕ} {a : ℝ} (ha : 1/4<a)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
+    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (M : ℝ≥0) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0)
+    (_hST : S≤T) (hne : (phaseActive (actualPhase a z cap U S b)).Nonempty)
+    (hboundary : ∀ i, levelTime a b (z i) (cap i) (U i)≤(S:ℝ≥0∞) ∨
+      (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i))
+    (hsafe : ∀ t≤T, (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
+      ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
+      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
+      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
+      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
+    ∀ t, S≤t → t≤T → ‖vectorState a η hη z t b‖≤M ∧
+      ∀ i∈phaseActive (actualPhase a z cap U S b),
+      η≤Y (vectorState a η hη z t b i) ∧
+      physicalDelta a (cap i) (phaseSigma a (actualPhase a z cap U S b) i)
+        (phaseOffset a U (actualPhase a z cap U S b) i) (vectorState a η hη z t b i)≤1 := by
+  have ha0 : 0<a := by linarith
+  have htimes (i) (hi : i∈phaseActive (actualPhase a z cap U S b)) :=
+    active_phase_interval
+      (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith))
+      (levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith))
+      (hboundary i) ((phaseActive_mem (actualPhase a z cap U S b) i).mp hi)
+  intro t hts htt
+  obtain ⟨j,hj⟩ := hne
+  have hti (i) (hi : i∈phaseActive (actualPhase a z cap U S b)) :
+      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) :=
+    (ENNReal.coe_le_coe.mpr htt).trans (htimes i hi).2.1
+  have hs := hsafe t htt ⟨j,hti j hj⟩
+  refine ⟨hs.1,fun i hi => ⟨(hs.2 i (hti i hi)).2.1,?_⟩⟩
+  rw [(hs.2 i (hti i hi)).2.2]
+  apply phase_delta_le ha
+  · exact levelTime_clock_ge a ha0 (z i) (hz i) (cap i) 0 (hc i) (hr i) le_rfl b t
+      (hs.2 i (hti i hi)).1 ((htimes i hi).1.trans (ENNReal.coe_le_coe.mpr hts))
+  · exact levelTime_clock_le a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b t
+      (hs.2 i (hti i hi)).1 (hti i hi)
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma levelTime_clock_at_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (cap u : ℝ) (hc : 0<cap) (hr : cap≤2*z.im) (hu : 0≤u) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (he : levelTime a b z cap u=(t:ℝ≥0∞)) :
+    physicalClock a cap (actualState a b.1 t z)=u := by
+  have h := levelTime_clock a ha z hz cap u hc hr hu b (by rw [he]; simp)
+  simpa only [he,ENNReal.toNNReal_coe] using h
+lemma actual_phase_transition {m : ℕ} {a : ℝ} (ha : 1/4<a)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
+    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (b : ExactSLEGenerator.Driver) (S T : ℝ≥0) (hST : S≤T)
+    (hboundary : ∀ i, ∀ u∈({0,U i/2,U i}:Finset ℝ),
+      levelTime a b (z i) (cap i) u≤(S:ℝ≥0∞) ∨
+      (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) u)
+    (hagree : ∀ i, (T:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
+      vectorState a η hη z T b i=actualState a b.1 T (z i)) :
+    phaseProduct a cap U (actualPhase a z cap U T b) (vectorState a η hη z T b)≤
+    phaseProduct a cap U (actualPhase a z cap U S b) (vectorState a η hη z T b) := by
+  have ha0 : 0<a := by linarith
+  rw [phaseProduct_eq_prod,phaseProduct_eq_prod]
+  apply Finset.prod_le_prod₀
+  · intro i hi
+    exact (phaseFactor_pos _ _ _ _ _).le
+  · intro i hi
+    have hAB := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith)
+    have hBC := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith)
+    apply phaseFactor_transition ha _ _ _ _ S T hAB hBC hST
+      (hboundary i 0 (by simp)) (hboundary i (U i/2) (by simp)) (hboundary i (U i) (by simp))
+    · intro he
+      rw [hagree i (by rw [←he]; exact hAB.trans hBC)]
+      exact levelTime_clock_at_eq a ha0 (z i) (hz i) (cap i) 0 (hc i) (hr i) le_rfl b T he
+    · intro he
+      rw [hagree i (by rw [←he]; exact hBC)]
+      exact levelTime_clock_at_eq a ha0 (z i) (hz i) (cap i) (U i/2) (hc i) (hr i) (by have := hu i; linarith) b T he
+    · intro he
+      rw [hagree i he.symm.le]
+      exact levelTime_clock_at_eq a ha0 (z i) (hz i) (cap i) (U i) (hc i) (hr i) (hu i) b T he
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ProbabilityTheory MeasureTheory ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus Set Filter
+open scoped Classical NNReal ENNReal
+lemma vectorState_stopped_measurable {m : ℕ} (a η : ℝ) (hη : 0<η) (z : Fin m → ℂ)
+    (S : Driver → ℝ≥0) (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable[hS.measurableSpace] (fun b => vectorState a η hη z (S b) b) := by
+  have hA : StronglyAdapted rawFiltration (vectorState a η hη z) :=
+    fun t => (vectorState_adapted a η hη z t).stronglyMeasurable
+  exact measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
+    (vectorState_continuous a η hη z)) hS
+lemma regimeState_stopped_measurable {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ)
+    (hη : 0<η) (N : ℝ≥0) (cap σ off : Fin m → ℝ) (z : Fin m → ℂ)
+    (S : Driver → ℝ≥0) (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable[hS.measurableSpace] (fun b => regimeState s a ε η hη N cap σ off z (S b) b) := by
+  have hA : StronglyAdapted rawFiltration (regimeState s a ε η hη N cap σ off z) :=
+    fun t => (regimeState_adapted s a ε η hη N cap σ off z t).stronglyMeasurable
+  exact measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
+    (regimeState_continuous s a ε η hη N cap σ off z)) hS
+lemma regularizedProduct_measurable {m : ℕ} (s : Finset (Fin m)) (a : ℝ)
+    (cap σ off : Fin m → ℝ) : Measurable (regularizedProduct s a cap σ off) := by
+  unfold regularizedProduct
+  exact Finset.measurable_prod _ fun i _ =>
+    (physicalRegularized_measurable a (cap i) (σ i) (off i)).comp (measurable_pi_apply i)
+lemma regularizedProduct_bound {m : ℕ} (s : Finset (Fin m)) (a η : ℝ) (hη : 0<η)
+    (N : ℝ≥0) (cap σ off : Fin m → ℝ) : ∃ C : ℝ, ∀ v : Fin m → State,
+    ‖v‖≤N → (∀ i∈s, η≤Y (v i)) → ‖regularizedProduct s a cap σ off v‖≤C := by
+  obtain ⟨C,hC⟩ := (regimeCutoff_compact s a η N cap σ off).exists_bound_of_continuous
+    (regimeCutoff_smooth s a η hη N cap σ off).continuous
+  refine ⟨C,fun v hv hy => ?_⟩
+  have he := (regimeCutoff_eq_near s a η hη N cap σ off (v,0)
+    (max_le hv (by simp)) hy).eq_of_nhds
+  simpa only [he,discountedProduct,neg_zero,Real.exp_zero,one_mul] using hC (v,0)
+lemma regimeState_discount_bound {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ)
+    (hη : 0<η) (N B : ℝ≥0) (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) :
+    ∃ L : ℝ, 0≤L ∧ ∀ (b : Driver) (t : ℝ≥0), t≤B →
+      |(regimeState s a ε η hη N cap σ off z t b).2|≤L := by
+  obtain ⟨K,hK,hKb⟩ := regimeDiscountCut_bound s a ε η hη N cap σ off
+  refine ⟨K*B,by positivity,fun b t ht => ?_⟩
+  change ‖∫ u in (0:ℝ)..(t:ℝ), regimeDiscountCut s a ε η N cap σ off
+    (vectorState a η hη z u.toNNReal b)‖≤_
+  calc
+    _ ≤ K*|(t:ℝ)-0| := intervalIntegral.norm_integral_le_of_norm_le_const fun u _ => hKb _
+    _ ≤ K*B := by rw [sub_zero,abs_of_nonneg t.coe_nonneg]; exact mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hK
+lemma regularizedProduct_indicator_integrable {m : ℕ} (s : Finset (Fin m)) (a η : ℝ)
+    (hη : 0<η) (N : ℝ≥0) (cap σ off : Fin m → ℝ) (z : Fin m → ℂ)
+    (W : Measure Driver) [IsProbabilityMeasure W] (S : Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (E : Set Driver) (hE : MeasurableSet E)
+    (hsafe : ∀ b∈E, ‖vectorState a η hη z (S b) b‖≤N ∧
+      ∀ i∈s, η≤Y (vectorState a η hη z (S b) b i)) :
+    Integrable (fun b => if b∈E then regularizedProduct s a cap σ off
+      (vectorState a η hη z (S b) b) else 0) W := by
+  obtain ⟨C,hC⟩ := regularizedProduct_bound s a η hη N cap σ off
+  have hm := (regularizedProduct_measurable s a cap σ off).comp
+    ((vectorState_stopped_measurable a η hη z S hS).mono hS.measurableSpace_le le_rfl)
+  apply Integrable.of_bound (hm.ite hE measurable_const).aestronglyMeasurable (max C 0)
+  apply Eventually.of_forall
+  intro b
+  split_ifs with hb
+  · exact (hC _ (hsafe b hb).1 (hsafe b hb).2).trans (le_max_left _ _)
+  · simpa only [norm_zero] using le_max_right C 0
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus ProbabilityTheory MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+ 
+
+theorem vector_regime_event_comparison {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (M : ℝ≥0) (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (z : Fin m → ℂ) (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (S T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
+    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
+    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b →
+      ‖vectorState a η hη z t b‖≤M ∧
+      ∀ i∈s, η≤Y (vectorState a η hη z t b i) ∧
+        physicalDelta a (cap i) (σ i) (off i) (vectorState a η hη z t b i)≤1)
+    (D : ℝ) (hD : ∀ b∈E,
+      (regimeState s a ε η hη M cap σ off z (T b) b).2-
+      (regimeState s a ε η hη M cap σ off z (S b) b).2≤D) :
+    (∫ b, (if b∈E then regularizedProduct s a cap σ off (vectorState a η hη z (T b) b) else 0) ∂W) ≤
+    Real.exp D*(∫ b, (if b∈E then regularizedProduct s a cap σ off (vectorState a η hη z (S b) b) else 0) ∂W) := by
+  let R := fun t b => (regimeState s a ε η hη M cap σ off z t b).2
+  let P := fun t b => regularizedProduct s a cap σ off (vectorState a η hη z t b)
+  obtain ⟨L,hL,hLb⟩ := regimeState_discount_bound s a ε η hη M B cap σ off z
+  have hSB : ∀ b, S b≤B := fun b => (hST b).trans (hTB b)
+  have hRS : Measurable[hS.measurableSpace] (fun b => R (S b) b) :=
+    (regimeState_stopped_measurable s a ε η hη M cap σ off z S hS).snd
+  have hRT : Measurable (fun b => R (T b) b) :=
+    ((regimeState_stopped_measurable s a ε η hη M cap σ off z T hT).mono hT.measurableSpace_le le_rfl).snd
+  have hRs : Measurable (fun b => R (S b) b) := hRS.mono hS.measurableSpace_le le_rfl
+  let w : Driver → ℝ := fun b => if b∈E then Real.exp (R (S b) b) else 0
+  have hw : StronglyMeasurable[hS.measurableSpace] w := (hRS.exp.ite hE measurable_const).stronglyMeasurable
+  have hw0 : ∀ b, 0≤w b := by intro b; dsimp [w]; split_ifs <;> positivity
+  have hwb : ∀ b, ‖w b‖≤Real.exp L := by
+    intro b
+    dsimp [w]
+    split_ifs
+    · simp only [abs_of_pos (Real.exp_pos _)]
+      exact Real.exp_le_exp.mpr ((le_abs_self _).trans (hLb b (S b) (hSB b)))
+    · simpa using (Real.exp_pos L).le
+  have hopt := vector_regime_weighted_optional_bounded s ha hε hη hsize M cap σ off hσ z W hW
+    S T B hS hT hST hTB E hE w hw hw0 (Real.exp L) hwb
+    (by intro b hb; simp [w,hb]) hsafe
+  have hE' : MeasurableSet E := hS.measurableSpace_le E hE
+  have hPi (Q : Driver → ℝ≥0) (hQ : IsStoppingTime rawFiltration (fun b => (Q b : WithTop ℝ≥0)))
+      (hSQ : ∀ b,S b≤Q b) (hQT : ∀ b,Q b≤T b) :
+      Integrable (fun b => if b∈E then P (Q b) b else 0) W := by
+    apply regularizedProduct_indicator_integrable s a η hη M cap σ off z W Q hQ E hE'
+    intro b hb
+    exact ⟨(hsafe b hb _ (hSQ b) (hQT b)).1,fun i hi => (hsafe b hb _ (hSQ b) (hQT b)).2 i hi |>.1⟩
+  have hFi := hPi T hT hST (fun _ => le_rfl)
+  have hGi := hPi S hS (fun _ => le_rfl) hST
+  let F := fun b => if b∈E then Real.exp (R (S b) b-R (T b) b)*P (T b) b else 0
+  have hF_eq : F=fun b => Real.exp (R (S b) b-R (T b) b)*(if b∈E then P (T b) b else 0) := by
+    funext b; dsimp [F]; split_ifs <;> simp
+  have hFint : Integrable F W := by
+    rw [hF_eq]
+    apply hFi.bdd_mul (hRs.sub hRT).exp.aestronglyMeasurable
+    apply Eventually.of_forall
+    intro b
+    simp only [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
+    apply Real.exp_le_exp.mpr (show R (S b) b-R (T b) b≤2*L from ?_)
+    have h1 := (abs_le.mp (hLb b (S b) (hSB b))).2
+    have h2 := (abs_le.mp (hLb b (T b) (hTB b))).1
+    change R (S b) b≤L at h1
+    change -L≤R (T b) b at h2
+    linarith
+  have heT : (fun b => w b*discountedProduct s a cap σ off
+      (regimeState s a ε η hη M cap σ off z (T b) b))=F := by
+    funext b
+    dsimp [w,F,discountedProduct,P,R,regimeState]
+    split_ifs <;> simp [Real.exp_sub,div_eq_mul_inv,Real.exp_neg,mul_assoc]
+  have heS : (fun b => w b*discountedProduct s a cap σ off
+      (regimeState s a ε η hη M cap σ off z (S b) b))=(fun b => if b∈E then P (S b) b else 0) := by
+    funext b
+    dsimp [w,discountedProduct,P,R,regimeState]
+    split_ifs <;> simp [Real.exp_neg,Real.exp_ne_zero]
+  rw [heT,heS] at hopt
+  calc
+    _ ≤ ∫ b, Real.exp D*F b ∂W := integral_mono hFi (hFint.const_mul _) (fun b => by
+      dsimp [F]
+      split_ifs with hb
+      · have h : 1≤Real.exp D*Real.exp (R (S b) b-R (T b) b) := by
+          rw [←Real.exp_add]
+          apply Real.one_le_exp_iff.mpr
+          have hd := hD b hb
+          change R (T b) b-R (S b) b≤D at hd
+          linarith
+        have h0 := (regularizedProduct_pos s a cap σ off (vectorState a η hη z (T b) b)).le
+        simpa only [one_mul,mul_assoc] using mul_le_mul_of_nonneg_right h h0
+      · simp)
+    _ = Real.exp D*(∫ b,F b ∂W) := integral_const_mul _ _
+    _ ≤ _ := mul_le_mul_of_nonneg_left hopt (Real.exp_pos D).le
+end ExactSLEGreen
+open MeasureTheory
+open scoped Classical
+namespace ExactSLEPasting
+end ExactSLEPasting
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/PlanarGrid.lean b/lean/OAI/Probability/SLEGauge/PlanarGrid.lean
index 734766ed5ea0f285b47eeeb8e9015bf4d6ff6056..3e85b059a4dbc507b57cb962e1fce7013bb7fffd 100644
--- a/lean/OAI/Probability/SLEGauge/PlanarGrid.lean
+++ b/lean/OAI/Probability/SLEGauge/PlanarGrid.lean
@@ -1,471 +1,471 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.GaugeBounds
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped ENNReal NNReal
-namespace SLEExactGauge.PlanarGrid
-
- 
-def cell (r : ℝ) (p : ℤ×ℤ) : Set ℂ :=
-  {z | (p.1:ℝ)*r ≤ z.re ∧ z.re ≤ ((p.1:ℝ)+1)*r ∧
-    (p.2:ℝ)*r ≤ z.im ∧ z.im ≤ ((p.2:ℝ)+1)*r}
-
-lemma cell_measurable (r : ℝ) (p : ℤ×ℤ) : MeasurableSet (cell r p) := by
-  unfold cell
-  measurability
-
- 
-lemma mem_floor_cell (r : ℝ) (hr : 0 < r) (z : ℂ) :
-    z ∈ cell r (⌊z.re/r⌋,⌊z.im/r⌋) := by
-  have lower (x : ℝ) : (⌊x/r⌋:ℝ)*r ≤ x := (le_div_iff₀ hr).mp (Int.floor_le _)
-  have upper (x : ℝ) : x ≤ ((⌊x/r⌋:ℝ)+1)*r :=
-    (div_le_iff₀ hr).mp (Int.lt_floor_add_one _).le
-  exact ⟨lower _,upper _,lower _,upper _⟩
-
- 
-
-def nearby (r s : ℝ) (z : ℂ) : Finset (ℤ×ℤ) :=
-  (Finset.Icc ⌊(z.re-s)/r⌋ ⌊(z.re+s)/r⌋).product
-    (Finset.Icc ⌊(z.im-s)/r⌋ ⌊(z.im+s)/r⌋)
-
-lemma floor_interval_card (r : ℝ) (hr : 0 < r) (x s : ℝ) (hs : 0 ≤ s) :
-    ((Finset.Icc ⌊(x-s)/r⌋ ⌊(x+s)/r⌋).card:ℝ)  ≤  2*(s/r)+2 := by
-  have hlo : (x-s)/r ≤ (x+s)/r := div_le_div_of_nonneg_right (by linarith) hr.le
-  have hf : ⌊(x-s)/r⌋  ≤  ⌊(x+s)/r⌋ := Int.floor_mono hlo
-  have hc := Int.card_Icc_of_le _ _ (Int.le_add_one hf)
-  have hc' : ((Finset.Icc ⌊(x-s)/r⌋ ⌊(x+s)/r⌋).card:ℝ) =
-      (⌊(x+s)/r⌋:ℝ)+1-(⌊(x-s)/r⌋:ℝ) := by exact_mod_cast hc
-  rw [hc']
-  have h1 := Int.floor_le ((x+s)/r)
-  have h2 := Int.lt_floor_add_one ((x-s)/r)
-  have he : (x+s)/r-(x-s)/r=2*(s/r) := by ring
-  linarith
-
-lemma nearby_card (r : ℝ) (hr : 0 < r) (s : ℝ) (hs : 0 ≤ s) (z : ℂ) :
-    ((nearby r s z).card:ℝ)  ≤  16*max 1 ((s/r)^2) := by
-  have h1 := floor_interval_card r hr z.re s hs
-  have h2 := floor_interval_card r hr z.im s hs
-  have hq : 0 ≤ s/r := div_nonneg hs hr.le
-  have hprod : ((nearby r s z).card:ℝ) ≤ (2*(s/r)+2)^2 := by
-    rw [nearby,Finset.product_eq_sprod,Finset.card_product,Nat.cast_mul,pow_two]
-    exact mul_le_mul h1 h2 (Nat.cast_nonneg _) (by positivity)
-  refine hprod.trans ?_
-  by_cases hq1 : s/r ≤ 1
-  · have hsq : (s/r)^2 ≤ 1 := by nlinarith
-    rw [max_eq_left hsq]
-    nlinarith
-  · have hq1' : 1 ≤ s/r := (le_of_not_ge hq1)
-    have hsq : 1 ≤ (s/r)^2 := by nlinarith
-    rw [max_eq_right hsq]
-    nlinarith [sq_nonneg (s/r-1)]
-
-lemma subset_nearby (r : ℝ) (hr : 0 < r) (s : ℝ) (hs : 0 ≤ s)
-    (U : Set ℂ) (hU : ediam U ≤ ENNReal.ofReal s) (z : ℂ) (hz : z ∈ U) :
-    U ⊆ ⋃p ∈ nearby r s z,cell r p := by
-  intro w hw
-  have hd : dist w z ≤ s := by
-    apply (ENNReal.ofReal_le_ofReal_iff hs).mp
-    rw [←edist_dist]
-    exact (edist_le_ediam_of_mem hw hz).trans hU
-  rw [dist_eq_norm] at hd
-  have hre : |w.re-z.re| ≤ s := (Complex.abs_re_le_norm (w-z)).trans hd
-  have him : |w.im-z.im| ≤ s := (Complex.abs_im_le_norm (w-z)).trans hd
-  rcases abs_le.mp hre with ⟨hre1,hre2⟩
-  rcases abs_le.mp him with ⟨him1,him2⟩
-  have hindex : (⌊w.re/r⌋,⌊w.im/r⌋) ∈ nearby r s z := by
-    simp only [nearby,Finset.product_eq_sprod,Finset.mem_product,Finset.mem_Icc]
-    constructor  <;> constructor  <;> apply Int.floor_mono  <;>
-      apply div_le_div_of_nonneg_right _ hr.le  <;> linarith
-  exact mem_iUnion.mpr ⟨_,mem_iUnion.mpr ⟨hindex,mem_floor_cell r hr w⟩⟩
-
- 
-
-theorem outerMass_le (ν : OuterMeasure ℂ) (r : ℝ) (hr : 0 < r)
-    (B : ℝ) (hB : 0 ≤ B) (hcell : ∀p,ν (cell r p) ≤ ENNReal.ofReal B)
-    (s : ℝ) (hs : 0 ≤ s) (U : Set ℂ) (hU : ediam U ≤ ENNReal.ofReal s) :
-    ν U ≤ ENNReal.ofReal (16*B*max 1 ((s/r)^2)) := by
-  classical
-  by_cases hn : U.Nonempty
-  · obtain ⟨z,hz⟩ := hn
-    calc
-      ν U ≤ ν (⋃p ∈ nearby r s z,cell r p) := ν.mono (subset_nearby r hr s hs U hU z hz)
-      _ ≤ ∑p ∈ nearby r s z,ν (cell r p) := measure_biUnion_finset_le _ _
-      _ ≤ ∑_p ∈ nearby r s z,ENNReal.ofReal B := Finset.sum_le_sum (fun p _=>hcell p)
-      _=ENNReal.ofReal (((nearby r s z).card:ℝ)*B) := by
-        simp only [Finset.sum_const,nsmul_eq_mul,ENNReal.ofReal_mul (Nat.cast_nonneg _),
-          ENNReal.ofReal_natCast]
-      _ ≤ _ := ENNReal.ofReal_le_ofReal (by
-        nlinarith [mul_le_mul_of_nonneg_right (nearby_card r hr s hs z) hB])
-  · simp [Set.not_nonempty_iff_eq_empty.mp hn]
-
-end SLEExactGauge.PlanarGrid
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology Classical
-namespace SLEExactGauge.PlanarGrid
-
-lemma cell_closed (r : ℝ) (p : ℤ×ℤ) : IsClosed (cell r p) := by
-  exact (isClosed_le continuous_const Complex.continuous_re).inter
-    ((isClosed_le Complex.continuous_re continuous_const).inter
-      ((isClosed_le continuous_const Complex.continuous_im).inter
-        (isClosed_le Complex.continuous_im continuous_const)))
-
-lemma cell_dist (r : ℝ) (_hr : 0≤r) (p : ℤ×ℤ) {z w : ℂ}
-    (hz : z∈cell r p) (hw : w∈cell r p) : dist w z≤2*r := by
-  have hre : |w.re-z.re|≤r := abs_le.mpr ⟨by linarith [hw.1,hz.2.1],by linarith [hz.1,hw.2.1]⟩
-  have him : |w.im-z.im|≤r := abs_le.mpr ⟨by linarith [hw.2.2.1,hz.2.2.2],by linarith [hz.2.2.1,hw.2.2.2]⟩
-  rw [dist_eq_norm]
-  exact (Complex.norm_le_abs_re_add_abs_im (w-z)).trans (by simpa only [two_mul,Complex.sub_re,Complex.sub_im] using (add_le_add hre him))
-
- 
-lemma index_mem_pair (r : ℝ) (hr : 0<r) (x : ℝ) (p : ℤ)
-    (hp : (p:ℝ)*r≤x) (hq : x≤((p:ℝ)+1)*r) :
-    p∈({⌊x/r⌋,⌊x/r⌋-1} : Finset ℤ) := by
-  have hl : p≤⌊x/r⌋ := Int.le_floor.mpr ((le_div_iff₀ hr).mpr hp)
-  have hu : ⌊x/r⌋≤p+1 := by
-    have hh := (div_le_iff₀ hr).mpr hq
-    have hf := Int.floor_mono hh
-    have he : ((p+1:ℤ):ℝ)=(p:ℝ)+1 := by norm_cast
-    rw [←he,Int.floor_intCast] at hf
-    exact hf
-  simp only [Finset.mem_insert,Finset.mem_singleton]
-  omega
-
-def containingIndices (r : ℝ) (z : ℂ) : Finset (ℤ×ℤ) :=
-  ({⌊z.re/r⌋,⌊z.re/r⌋-1} : Finset ℤ).product {⌊z.im/r⌋,⌊z.im/r⌋-1}
-
-lemma mem_containingIndices (r : ℝ) (hr : 0<r) (z : ℂ) (p : ℤ×ℤ)
-    (hp : z∈cell r p) : p∈containingIndices r z := by
-  exact Finset.mem_product.mpr ⟨index_mem_pair r hr z.re p.1 hp.1 hp.2.1,
-    index_mem_pair r hr z.im p.2 hp.2.2.1 hp.2.2.2⟩
-
-lemma containingIndices_card (r : ℝ) (z : ℂ) : (containingIndices r z).card≤4 := by
-  have h (i : ℤ) : ({i,i-1} : Finset ℤ).card≤2 := Finset.card_insert_le .. |>.trans (by simp)
-  rw [containingIndices,Finset.product_eq_sprod,Finset.card_product]
-  exact (Nat.mul_le_mul (h _) (h _)).trans (by norm_num)
-
- 
-lemma tsum_cell_mass (ν : Measure ℂ) (r : ℝ) (hr : 0<r) :
-    (∑'p : ℤ×ℤ,ν (cell r p)) ≤ 4*ν univ := by
-  have he (p : ℤ×ℤ) : ν (cell r p)=∫⁻z,(cell r p).indicator (fun _=>1) z∂ν :=
-    (lintegral_indicator_one (cell_measurable _ _)).symm
-  simp_rw [he]
-  rw [←lintegral_tsum (fun p=>(measurable_const.indicator (cell_measurable r p)).aemeasurable)]
-  calc
-    _≤∫⁻z,(4:ℝ≥0∞)∂ν := by
-      apply lintegral_mono
-      intro z
-      have ht : (∑'p : ℤ×ℤ,(cell r p).indicator (fun _=>(1:ℝ≥0∞)) z)=
-          ∑p∈containingIndices r z,(cell r p).indicator (fun _=>(1:ℝ≥0∞)) z := by
-        apply tsum_eq_sum
-        intro p hp
-        exact Set.indicator_of_notMem (fun h=>hp (mem_containingIndices r hr z p h)) _
-      change (∑'p : ℤ×ℤ,(cell r p).indicator (fun _=>(1:ℝ≥0∞)) z)≤4
-      rw [ht]
-      calc
-        _≤∑_p∈containingIndices r z,(1:ℝ≥0∞) := by
-          apply Finset.sum_le_sum
-          intro p _
-          exact Set.indicator_le_self' (fun _ _=>zero_le) _
-        _=((containingIndices r z).card:ℝ≥0∞) := by simp
-        _≤4 := by exact_mod_cast containingIndices_card r z
-    _=4*ν univ := lintegral_const _
-
-end SLEExactGauge.PlanarGrid
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Metric
-open scoped ENNReal NNReal
-namespace SLEExactGauge.PlanarGrid
-
- 
-def maximalMass (ν : Measure ℂ) (s : ℝ) : ℝ :=
-  sSup {v | ∃U : Set ℂ, ediam U ≤ ENNReal.ofReal s ∧ v=(ν U).toReal}
-
-lemma maximalMass_nonempty (ν : Measure ℂ) (s : ℝ) :
-    Set.Nonempty {v | ∃U : Set ℂ, ediam U ≤ ENNReal.ofReal s ∧ v=(ν U).toReal} :=
-  ⟨0,∅,by simp,by simp⟩
-
-lemma maximalMass_bdd (ν : Measure ℂ) [IsFiniteMeasure ν] (s : ℝ) :
-    BddAbove {v | ∃U : Set ℂ, ediam U ≤ ENNReal.ofReal s ∧ v=(ν U).toReal} := by
-  refine ⟨(ν univ).toReal,?_⟩
-  rintro v ⟨U,hU,rfl⟩
-  exact ENNReal.toReal_mono (measure_ne_top ν univ) (measure_mono (subset_univ _))
-
-lemma maximalMass_bounds (ν : Measure ℂ) [IsFiniteMeasure ν] (s : ℝ) :
-    0 ≤ maximalMass ν s ∧ maximalMass ν s ≤ (ν univ).toReal := by
-  constructor
-  · exact le_csSup (maximalMass_bdd ν s) ⟨∅,by simp,by simp⟩
-  · apply csSup_le (maximalMass_nonempty ν s)
-    rintro v ⟨U,hU,rfl⟩
-    exact ENNReal.toReal_mono (measure_ne_top ν univ) (measure_mono (subset_univ _))
-
-lemma mass_le_maximalMass (ν : Measure ℂ) [IsFiniteMeasure ν]
-    (s : ℝ) (U : Set ℂ) (hU : ediam U ≤ ENNReal.ofReal s) :
-    (ν U).toReal ≤ maximalMass ν s :=
-  le_csSup (maximalMass_bdd ν s) ⟨U,hU,rfl⟩
-
-lemma maximalMass_grid (ν : Measure ℂ) [IsFiniteMeasure ν] (r : ℝ) (hr : 0<r)
-    (B : ℝ) (hB : 0 ≤ B) (hcell : ∀p,ν (cell r p) ≤ ENNReal.ofReal B)
-    (s : ℝ) (hs : 0 ≤ s) :
-    maximalMass ν s ≤ 16*B*max 1 ((s/r)^2) := by
-  apply csSup_le (maximalMass_nonempty ν s)
-  rintro v ⟨U,hU,rfl⟩
-  exact ENNReal.toReal_le_of_le_ofReal (by positivity)
-    (outerMass_le ν.toOuterMeasure r hr B hB hcell s hs U hU)
-
- 
-
-theorem full_density {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι→ℝ)
-    (hb : ∀i,0<b i) (hρ : ∀i,0<ρ i) (E : Finset ι)
-    (ν : Measure ℂ) [IsFiniteMeasure ν] [NullSingletonClass ν]
-    (K : ℝ) (hK : 0 ≤ K)
-    (hcell : ∀i,i∉E → ∀p,ν (cell (ρ i) p) ≤ ENNReal.ofReal (K*b i*(ρ i)^d)) :
-    ∃C : ℝ, 0<C ∧ ∀U : Set ℂ, ediam U ≤ 1 →
-      ν U ≤ ENNReal.ofReal C * Deterministic.diameterCost (QuadraticGauge.gauge d b ρ) (ediam U) := by
-  obtain ⟨C,hC,hbound⟩ := QuadraticGauge.finite_exclusions d b ρ hb hρ E
-    (maximalMass ν) (ν univ).toReal (16*K) (fun s _=>maximalMass_bounds ν s)
-    (by
-      intro i hi s hs
-      have H := maximalMass_grid ν (ρ i) (hρ i) (K*b i*(ρ i)^d) (mul_nonneg (mul_nonneg hK (hb i).le) (Real.rpow_nonneg (hρ i).le _))
-        (hcell i hi) s hs.le
-      dsimp only [QuadraticGauge.entry]
-      convert H using 1; ring)
-  refine ⟨C,hC,?_⟩
-  intro U hU
-  have hfin : ediam U≠∞ := ne_top_of_le_ne_top (by norm_num) hU
-  by_cases hz : ediam U=0
-  · have he : ν U=0 := (ediam_eq_zero_iff.mp hz).measure_zero ν
-    rw [he]
-    exact bot_le
-  · have hs : 0<(ediam U).toReal := ENNReal.toReal_pos hz hfin
-    have hm := (mass_le_maximalMass ν (ediam U).toReal U (by
-      rw [ENNReal.ofReal_toReal hfin])).trans (hbound _ hs)
-    have he := ENNReal.ofReal_le_ofReal hm
-    rw [ENNReal.ofReal_toReal (measure_ne_top ν U),ENNReal.ofReal_mul hC.le] at he
-    simpa only [Deterministic.diameterCost,ite_eq_right hfin] using he
-
- 
-
-theorem hausdorff_pos_of_grid {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι→ℝ)
-    (hb : ∀i,0<b i) (hρ : ∀i,0<ρ i) (E : Finset ι)
-    (ν : Measure ℂ) [IsFiniteMeasure ν] [NullSingletonClass ν]
-    (K : ℝ) (hK : 0 ≤ K)
-    (hcell : ∀i,i∉E → ∀p,ν (cell (ρ i) p) ≤ ENNReal.ofReal (K*b i*(ρ i)^d))
-    (A : Set ℂ) (hA : 0<ν A) : 0<Target.hausdorff (QuadraticGauge.gauge d b ρ) A := by
-  obtain ⟨C,hC,hbound⟩ := full_density d b ρ hb hρ E ν K hK hcell
-  exact Deterministic.hausdorff_pos_of_mass_bound (δ := 1) (ν := ν.toOuterMeasure)
-    (by norm_num) (ENNReal.ofReal_pos.mpr hC).ne' ENNReal.ofReal_ne_top hbound hA
-
-end SLEExactGauge.PlanarGrid
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped ENNReal NNReal Classical
-namespace SLEExactGauge.PlanarGrid
-
- 
-def badRegion (μ : Measure ℂ) (r B : ℝ) : Set ℂ :=
-  ⋃ (p : ℤ×ℤ) (_ : ENNReal.ofReal B<μ (cell r p)),cell r p
-
-lemma badRegion_measurable (μ : Measure ℂ) (r B : ℝ) : MeasurableSet (badRegion μ r B) :=
-  MeasurableSet.iUnion (fun p=>MeasurableSet.iUnion (fun _=>cell_measurable r p))
-
- 
-def retained {ι X : Type*} (bad : ι→Set X) (E : Finset ι) : Set X :=
-  {z | ∀i,i∉E → z∉bad i}
-
-lemma retained_measurable {ι X : Type*} [Countable ι] [MeasurableSpace X]
-    (bad : ι→Set X) (hbad : ∀i,MeasurableSet (bad i)) (E : Finset ι) :
-    MeasurableSet (retained bad E) := by
-  have he : retained bad E = ⋂i,⋂(_ : i∉E),(bad i)ᶜ := by ext z; simp [retained]
-  rw [he]
-  exact MeasurableSet.iInter (fun i=>MeasurableSet.iInter (fun _=>(hbad i).compl))
-
-lemma exists_positive_retained {ι X : Type*} [Countable ι] [MeasurableSpace X]
-    (σ : Measure X) (bad : ι→Set X) (hfinite : ∀ᵐ z∂σ,{i | z∈bad i}.Finite)
-    (A : Set X) (hA : 0<σ A) :
-    ∃E : Finset ι,0<σ (A∩retained bad E) := by
-  have hc : ∀ᵐ z∂σ,z∈⋃E : Finset ι,retained bad E := by
-    filter_upwards [hfinite] with z hz
-    refine mem_iUnion.mpr ⟨hz.toFinset,?_⟩
-    intro i hi hmem
-    exact hi (hz.mem_toFinset.mpr hmem)
-  by_contra! hh
-  have hzero : σ (A∩⋃E : Finset ι,retained bad E)=0 := by
-    rw [Set.inter_iUnion]
-    exact measure_iUnion_null_iff.mpr (fun E=>le_antisymm (hh E) bot_le)
-  have he : (A : Set X)=ᵐ[σ] (A∩⋃E : Finset ι,retained bad E : Set X) := by
-    filter_upwards [hc] with z hz
-    simp only [mem_inter_iff,hz,and_true]
-  exact hA.ne' ((measure_congr he).trans hzero)
-
-lemma retained_cell_bound {ι : Type*} (μ σ : Measure ℂ) (hσ : σ≤μ)
-    (ρ B : ι→ℝ) (E : Finset ι) (i : ι) (hi : i∉E) (p : ℤ×ℤ) :
-    (σ.restrict (retained (fun i=>badRegion μ (ρ i) (B i)) E)) (cell (ρ i) p) ≤
-      ENNReal.ofReal (B i) := by
-  by_cases hbad : ENNReal.ofReal (B i)<μ (cell (ρ i) p)
-  · rw [Measure.restrict_apply (cell_measurable _ _)]
-    have he : cell (ρ i) p∩retained (fun i=>badRegion μ (ρ i) (B i)) E=∅ := by
-      apply eq_empty_iff_forall_notMem.mpr
-      intro z hz
-      exact hz.2 i hi (mem_iUnion.mpr ⟨p,mem_iUnion.mpr ⟨hbad,hz.1⟩⟩)
-    rw [he,measure_empty]
-    exact bot_le
-  · exact (Measure.restrict_le_self.trans hσ _).trans (le_of_not_gt hbad)
-
- 
-
-theorem hausdorff_pos_of_finitely_bad {ι : Type*} [Nonempty ι] [Countable ι]
-    (d : ℝ) (b ρ : ι→ℝ) (hb : ∀i,0<b i) (hρ : ∀i,0<ρ i)
-    (μ σ : Measure ℂ) (hσ : σ≤μ) [IsFiniteMeasure σ] [NullSingletonClass σ]
-    (K : ℝ) (hK : 0 ≤ K)
-    (hfinite : ∀ᵐ z∂σ,{i | z∈badRegion μ (ρ i) (K*b i*(ρ i)^d)}.Finite)
-    (A : Set ℂ) (hA : 0<σ A) : 0<Target.hausdorff (QuadraticGauge.gauge d b ρ) A := by
-  let bad := fun i=>badRegion μ (ρ i) (K*b i*(ρ i)^d)
-  obtain ⟨E,hE⟩ := exists_positive_retained σ bad hfinite A hA
-  let ν := σ.restrict (retained bad E)
-  have hνA : 0<ν A := by
-    dsimp only [ν]
-    rw [Measure.restrict_apply' (retained_measurable bad (fun i=>badRegion_measurable _ _ _) E)]
-    exact hE
-  exact hausdorff_pos_of_grid d b ρ hb hρ E ν K hK
-    (fun i hi p=>retained_cell_bound μ σ hσ ρ (fun j=>K*b j*(ρ j)^d) E i hi p) A hνA
-
- 
-
-theorem ae_ae_finite_bad {Ω X ι : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    [Countable ι] (P : Measure Ω) (μ : Ω→Measure X) (bad : ι→Ω→Set X)
-    (hm : ∀i,AEMeasurable (fun ω=>μ ω (bad i ω)) P)
-    (hsum : (∑'i,∫⁻ω,μ ω (bad i ω)∂P)≠∞) :
-    ∀ᵐ ω∂P,∀ᵐ z∂μ ω,{i | z∈bad i ω}.Finite := by
-  have h : (∫⁻ω,∑'i,μ ω (bad i ω)∂P)≠∞ := by rwa [lintegral_tsum hm]
-  filter_upwards [ae_lt_top' (AEMeasurable.tsum hm) h] with ω hω
-  exact ae_finite_setOfPred_mem hω.ne
-
-end SLEExactGauge.PlanarGrid
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped ENNReal NNReal Topology Classical
-namespace SLEExactGauge.PlanarGrid
-
- 
-
-def testedBadCell {Ω ι : Type*} [MeasurableSpace Ω]
-    (μ : Ω→Measure ℂ) (ρ B : ι→ℝ) (good : Set ι)
-    (j : ι×(ℤ×ℤ)) (ω : Ω) : Set ℂ :=
-  if j.1∈good ∧ ENNReal.ofReal (B j.1) < μ ω (cell (ρ j.1) j.2)
-    then cell (ρ j.1) j.2 else ∅
-
-lemma testedBadCell_mass {Ω ι : Type*} [MeasurableSpace Ω]
-    (μ : Ω→Measure ℂ) (ρ B : ι→ℝ) (good : Set ι)
-    (j : ι×(ℤ×ℤ)) (ω : Ω) :
-    μ ω (testedBadCell μ ρ B good j ω)=
-      if j.1∈good then
-        {ω | ENNReal.ofReal (B j.1) < μ ω (cell (ρ j.1) j.2)}.indicator
-          (fun ω=>μ ω (cell (ρ j.1) j.2)) ω else 0 := by
-  by_cases hg : j.1∈good <;>
-    by_cases hb : ENNReal.ofReal (B j.1) < μ ω (cell (ρ j.1) j.2) <;>
-    simp [testedBadCell,hg,hb]
-
- 
-
-theorem ae_finitely_bad_of_spatial_tails {Ω ι : Type*}
-    [MeasurableSpace Ω] [Countable ι]
-    (P : Measure Ω) (μ : Ω→Measure ℂ) (ν : Measure ℂ) [IsFiniteMeasure ν]
-    (ρ B : ι→ℝ) (hρ : ∀i,0 < ρ i)
-    (hm : ∀i p,Measurable (fun ω=>μ ω (cell (ρ i) p)))
-    (good : Set ι) (hgood : goodᶜ.Finite)
-    (C : ℝ≥0∞) (hC : C≠∞) (tail : ι→ℝ≥0∞) (htail : (∑'i,tail i)≠∞)
-    (hbound : ∀i∈good,∀p : ℤ×ℤ,
-      (∫⁻ω,{ω | ENNReal.ofReal (B i) < μ ω (cell (ρ i) p)}.indicator
-        (fun ω=>μ ω (cell (ρ i) p)) ω∂P) ≤ C*ν (cell (ρ i) p)*tail i) :
-    ∀ᵐ ω∂P,∀ᵐ z∂μ ω,{i | z∈badRegion (μ ω) (ρ i) (B i)}.Finite := by
-  let bad := testedBadCell μ ρ B good
-  have hbad (j : ι×(ℤ×ℤ)) :
-      Measurable (fun ω=>μ ω (bad j ω)) := by
-    simp only [bad,testedBadCell_mass]
-    by_cases hj : j.1∈good
-    · simp only [ite_eq_left hj]
-      exact (hm j.1 j.2).indicator (measurableSet_lt measurable_const (hm j.1 j.2))
-    · simp only [ite_eq_right hj]
-      exact measurable_const
-  have hi (j : ι×(ℤ×ℤ)) : (∫⁻ω,μ ω (bad j ω)∂P) ≤
-      C*ν (cell (ρ j.1) j.2)*tail j.1 := by
-    simp only [bad,testedBadCell_mass]
-    by_cases hj : j.1∈good
-    · simp only [ite_eq_left hj]
-      exact hbound j.1 hj j.2
-    · simp only [ite_eq_right hj,lintegral_zero]
-      exact bot_le
-  have hsum : (∑'j : ι×(ℤ×ℤ),∫⁻ω,μ ω (bad j ω)∂P)≠∞ := by
-    apply ne_top_of_le_ne_top (show (C*(4*ν univ))*(∑'i,tail i)≠∞ from by finiteness)
-    calc
-      _≤∑'j : ι×(ℤ×ℤ),C*ν (cell (ρ j.1) j.2)*tail j.1 := ENNReal.tsum_le_tsum hi
-      _=∑'i,C*(∑'p : ℤ×ℤ,ν (cell (ρ i) p))*tail i := by
-        rw [ENNReal.tsum_prod']
-        simp_rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
-      _≤∑'i,C*(4*ν univ)*tail i := by
-        apply ENNReal.tsum_le_tsum
-        intro i
-        gcongr
-        exact tsum_cell_mass ν (ρ i) (hρ i)
-      _=(C*(4*ν univ))*(∑'i,tail i) := ENNReal.tsum_mul_left
-  filter_upwards [ae_ae_finite_bad P μ bad (fun j=>(hbad j).aemeasurable) hsum] with ω hω
-  filter_upwards [hω] with z hz
-  apply (hgood.union (hz.image Prod.fst)).subset
-  intro i hi
-  by_cases hg : i∈good
-  · right
-    obtain ⟨p,hp⟩ := mem_iUnion.mp hi
-    obtain ⟨hbad,hcell⟩ := mem_iUnion.mp hp
-    refine ⟨(i,p),?_,rfl⟩
-    change z∈testedBadCell μ ρ B good (i,p) ω
-    simp only [testedBadCell,ite_eq_left (And.intro hg hbad)]
-    exact hcell
-  · exact Or.inl hg
-
-end SLEExactGauge.PlanarGrid
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.GaugeBounds
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped ENNReal NNReal
+namespace SLEExactGauge.PlanarGrid
+
+ 
+def cell (r : ℝ) (p : ℤ×ℤ) : Set ℂ :=
+  {z | (p.1:ℝ)*r ≤ z.re ∧ z.re ≤ ((p.1:ℝ)+1)*r ∧
+    (p.2:ℝ)*r ≤ z.im ∧ z.im ≤ ((p.2:ℝ)+1)*r}
+
+lemma cell_measurable (r : ℝ) (p : ℤ×ℤ) : MeasurableSet (cell r p) := by
+  unfold cell
+  measurability
+
+ 
+lemma mem_floor_cell (r : ℝ) (hr : 0 < r) (z : ℂ) :
+    z ∈ cell r (⌊z.re/r⌋,⌊z.im/r⌋) := by
+  have lower (x : ℝ) : (⌊x/r⌋:ℝ)*r ≤ x := (le_div_iff₀ hr).mp (Int.floor_le _)
+  have upper (x : ℝ) : x ≤ ((⌊x/r⌋:ℝ)+1)*r :=
+    (div_le_iff₀ hr).mp (Int.lt_floor_add_one _).le
+  exact ⟨lower _,upper _,lower _,upper _⟩
+
+ 
+
+def nearby (r s : ℝ) (z : ℂ) : Finset (ℤ×ℤ) :=
+  (Finset.Icc ⌊(z.re-s)/r⌋ ⌊(z.re+s)/r⌋).product
+    (Finset.Icc ⌊(z.im-s)/r⌋ ⌊(z.im+s)/r⌋)
+
+lemma floor_interval_card (r : ℝ) (hr : 0 < r) (x s : ℝ) (hs : 0 ≤ s) :
+    ((Finset.Icc ⌊(x-s)/r⌋ ⌊(x+s)/r⌋).card:ℝ)  ≤  2*(s/r)+2 := by
+  have hlo : (x-s)/r ≤ (x+s)/r := div_le_div_of_nonneg_right (by linarith) hr.le
+  have hf : ⌊(x-s)/r⌋  ≤  ⌊(x+s)/r⌋ := Int.floor_mono hlo
+  have hc := Int.card_Icc_of_le _ _ (Int.le_add_one hf)
+  have hc' : ((Finset.Icc ⌊(x-s)/r⌋ ⌊(x+s)/r⌋).card:ℝ) =
+      (⌊(x+s)/r⌋:ℝ)+1-(⌊(x-s)/r⌋:ℝ) := by exact_mod_cast hc
+  rw [hc']
+  have h1 := Int.floor_le ((x+s)/r)
+  have h2 := Int.lt_floor_add_one ((x-s)/r)
+  have he : (x+s)/r-(x-s)/r=2*(s/r) := by ring
+  linarith
+
+lemma nearby_card (r : ℝ) (hr : 0 < r) (s : ℝ) (hs : 0 ≤ s) (z : ℂ) :
+    ((nearby r s z).card:ℝ)  ≤  16*max 1 ((s/r)^2) := by
+  have h1 := floor_interval_card r hr z.re s hs
+  have h2 := floor_interval_card r hr z.im s hs
+  have hq : 0 ≤ s/r := div_nonneg hs hr.le
+  have hprod : ((nearby r s z).card:ℝ) ≤ (2*(s/r)+2)^2 := by
+    rw [nearby,Finset.product_eq_sprod,Finset.card_product,Nat.cast_mul,pow_two]
+    exact mul_le_mul h1 h2 (Nat.cast_nonneg _) (by positivity)
+  refine hprod.trans ?_
+  by_cases hq1 : s/r ≤ 1
+  · have hsq : (s/r)^2 ≤ 1 := by nlinarith
+    rw [max_eq_left hsq]
+    nlinarith
+  · have hq1' : 1 ≤ s/r := (le_of_not_ge hq1)
+    have hsq : 1 ≤ (s/r)^2 := by nlinarith
+    rw [max_eq_right hsq]
+    nlinarith [sq_nonneg (s/r-1)]
+
+lemma subset_nearby (r : ℝ) (hr : 0 < r) (s : ℝ) (hs : 0 ≤ s)
+    (U : Set ℂ) (hU : ediam U ≤ ENNReal.ofReal s) (z : ℂ) (hz : z ∈ U) :
+    U ⊆ ⋃p ∈ nearby r s z,cell r p := by
+  intro w hw
+  have hd : dist w z ≤ s := by
+    apply (ENNReal.ofReal_le_ofReal_iff hs).mp
+    rw [←edist_dist]
+    exact (edist_le_ediam_of_mem hw hz).trans hU
+  rw [dist_eq_norm] at hd
+  have hre : |w.re-z.re| ≤ s := (Complex.abs_re_le_norm (w-z)).trans hd
+  have him : |w.im-z.im| ≤ s := (Complex.abs_im_le_norm (w-z)).trans hd
+  rcases abs_le.mp hre with ⟨hre1,hre2⟩
+  rcases abs_le.mp him with ⟨him1,him2⟩
+  have hindex : (⌊w.re/r⌋,⌊w.im/r⌋) ∈ nearby r s z := by
+    simp only [nearby,Finset.product_eq_sprod,Finset.mem_product,Finset.mem_Icc]
+    constructor  <;> constructor  <;> apply Int.floor_mono  <;>
+      apply div_le_div_of_nonneg_right _ hr.le  <;> linarith
+  exact mem_iUnion.mpr ⟨_,mem_iUnion.mpr ⟨hindex,mem_floor_cell r hr w⟩⟩
+
+ 
+
+theorem outerMass_le (ν : OuterMeasure ℂ) (r : ℝ) (hr : 0 < r)
+    (B : ℝ) (hB : 0 ≤ B) (hcell : ∀p,ν (cell r p) ≤ ENNReal.ofReal B)
+    (s : ℝ) (hs : 0 ≤ s) (U : Set ℂ) (hU : ediam U ≤ ENNReal.ofReal s) :
+    ν U ≤ ENNReal.ofReal (16*B*max 1 ((s/r)^2)) := by
+  classical
+  by_cases hn : U.Nonempty
+  · obtain ⟨z,hz⟩ := hn
+    calc
+      ν U ≤ ν (⋃p ∈ nearby r s z,cell r p) := ν.mono (subset_nearby r hr s hs U hU z hz)
+      _ ≤ ∑p ∈ nearby r s z,ν (cell r p) := measure_biUnion_finset_le _ _
+      _ ≤ ∑_p ∈ nearby r s z,ENNReal.ofReal B := Finset.sum_le_sum (fun p _=>hcell p)
+      _=ENNReal.ofReal (((nearby r s z).card:ℝ)*B) := by
+        simp only [Finset.sum_const,nsmul_eq_mul,ENNReal.ofReal_mul (Nat.cast_nonneg _),
+          ENNReal.ofReal_natCast]
+      _ ≤ _ := ENNReal.ofReal_le_ofReal (by
+        nlinarith [mul_le_mul_of_nonneg_right (nearby_card r hr s hs z) hB])
+  · simp [Set.not_nonempty_iff_eq_empty.mp hn]
+
+end SLEExactGauge.PlanarGrid
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology Classical
+namespace SLEExactGauge.PlanarGrid
+
+lemma cell_closed (r : ℝ) (p : ℤ×ℤ) : IsClosed (cell r p) := by
+  exact (isClosed_le continuous_const Complex.continuous_re).inter
+    ((isClosed_le Complex.continuous_re continuous_const).inter
+      ((isClosed_le continuous_const Complex.continuous_im).inter
+        (isClosed_le Complex.continuous_im continuous_const)))
+
+lemma cell_dist (r : ℝ) (_hr : 0≤r) (p : ℤ×ℤ) {z w : ℂ}
+    (hz : z∈cell r p) (hw : w∈cell r p) : dist w z≤2*r := by
+  have hre : |w.re-z.re|≤r := abs_le.mpr ⟨by linarith [hw.1,hz.2.1],by linarith [hz.1,hw.2.1]⟩
+  have him : |w.im-z.im|≤r := abs_le.mpr ⟨by linarith [hw.2.2.1,hz.2.2.2],by linarith [hz.2.2.1,hw.2.2.2]⟩
+  rw [dist_eq_norm]
+  exact (Complex.norm_le_abs_re_add_abs_im (w-z)).trans (by simpa only [two_mul,Complex.sub_re,Complex.sub_im] using (add_le_add hre him))
+
+ 
+lemma index_mem_pair (r : ℝ) (hr : 0<r) (x : ℝ) (p : ℤ)
+    (hp : (p:ℝ)*r≤x) (hq : x≤((p:ℝ)+1)*r) :
+    p∈({⌊x/r⌋,⌊x/r⌋-1} : Finset ℤ) := by
+  have hl : p≤⌊x/r⌋ := Int.le_floor.mpr ((le_div_iff₀ hr).mpr hp)
+  have hu : ⌊x/r⌋≤p+1 := by
+    have hh := (div_le_iff₀ hr).mpr hq
+    have hf := Int.floor_mono hh
+    have he : ((p+1:ℤ):ℝ)=(p:ℝ)+1 := by norm_cast
+    rw [←he,Int.floor_intCast] at hf
+    exact hf
+  simp only [Finset.mem_insert,Finset.mem_singleton]
+  omega
+
+def containingIndices (r : ℝ) (z : ℂ) : Finset (ℤ×ℤ) :=
+  ({⌊z.re/r⌋,⌊z.re/r⌋-1} : Finset ℤ).product {⌊z.im/r⌋,⌊z.im/r⌋-1}
+
+lemma mem_containingIndices (r : ℝ) (hr : 0<r) (z : ℂ) (p : ℤ×ℤ)
+    (hp : z∈cell r p) : p∈containingIndices r z := by
+  exact Finset.mem_product.mpr ⟨index_mem_pair r hr z.re p.1 hp.1 hp.2.1,
+    index_mem_pair r hr z.im p.2 hp.2.2.1 hp.2.2.2⟩
+
+lemma containingIndices_card (r : ℝ) (z : ℂ) : (containingIndices r z).card≤4 := by
+  have h (i : ℤ) : ({i,i-1} : Finset ℤ).card≤2 := Finset.card_insert_le .. |>.trans (by simp)
+  rw [containingIndices,Finset.product_eq_sprod,Finset.card_product]
+  exact (Nat.mul_le_mul (h _) (h _)).trans (by norm_num)
+
+ 
+lemma tsum_cell_mass (ν : Measure ℂ) (r : ℝ) (hr : 0<r) :
+    (∑'p : ℤ×ℤ,ν (cell r p)) ≤ 4*ν univ := by
+  have he (p : ℤ×ℤ) : ν (cell r p)=∫⁻z,(cell r p).indicator (fun _=>1) z∂ν :=
+    (lintegral_indicator_one (cell_measurable _ _)).symm
+  simp_rw [he]
+  rw [←lintegral_tsum (fun p=>(measurable_const.indicator (cell_measurable r p)).aemeasurable)]
+  calc
+    _≤∫⁻z,(4:ℝ≥0∞)∂ν := by
+      apply lintegral_mono
+      intro z
+      have ht : (∑'p : ℤ×ℤ,(cell r p).indicator (fun _=>(1:ℝ≥0∞)) z)=
+          ∑p∈containingIndices r z,(cell r p).indicator (fun _=>(1:ℝ≥0∞)) z := by
+        apply tsum_eq_sum
+        intro p hp
+        exact Set.indicator_of_notMem (fun h=>hp (mem_containingIndices r hr z p h)) _
+      change (∑'p : ℤ×ℤ,(cell r p).indicator (fun _=>(1:ℝ≥0∞)) z)≤4
+      rw [ht]
+      calc
+        _≤∑_p∈containingIndices r z,(1:ℝ≥0∞) := by
+          apply Finset.sum_le_sum
+          intro p _
+          exact Set.indicator_le_self' (fun _ _=>zero_le) _
+        _=((containingIndices r z).card:ℝ≥0∞) := by simp
+        _≤4 := by exact_mod_cast containingIndices_card r z
+    _=4*ν univ := lintegral_const _
+
+end SLEExactGauge.PlanarGrid
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Metric
+open scoped ENNReal NNReal
+namespace SLEExactGauge.PlanarGrid
+
+ 
+def maximalMass (ν : Measure ℂ) (s : ℝ) : ℝ :=
+  sSup {v | ∃U : Set ℂ, ediam U ≤ ENNReal.ofReal s ∧ v=(ν U).toReal}
+
+lemma maximalMass_nonempty (ν : Measure ℂ) (s : ℝ) :
+    Set.Nonempty {v | ∃U : Set ℂ, ediam U ≤ ENNReal.ofReal s ∧ v=(ν U).toReal} :=
+  ⟨0,∅,by simp,by simp⟩
+
+lemma maximalMass_bdd (ν : Measure ℂ) [IsFiniteMeasure ν] (s : ℝ) :
+    BddAbove {v | ∃U : Set ℂ, ediam U ≤ ENNReal.ofReal s ∧ v=(ν U).toReal} := by
+  refine ⟨(ν univ).toReal,?_⟩
+  rintro v ⟨U,hU,rfl⟩
+  exact ENNReal.toReal_mono (measure_ne_top ν univ) (measure_mono (subset_univ _))
+
+lemma maximalMass_bounds (ν : Measure ℂ) [IsFiniteMeasure ν] (s : ℝ) :
+    0 ≤ maximalMass ν s ∧ maximalMass ν s ≤ (ν univ).toReal := by
+  constructor
+  · exact le_csSup (maximalMass_bdd ν s) ⟨∅,by simp,by simp⟩
+  · apply csSup_le (maximalMass_nonempty ν s)
+    rintro v ⟨U,hU,rfl⟩
+    exact ENNReal.toReal_mono (measure_ne_top ν univ) (measure_mono (subset_univ _))
+
+lemma mass_le_maximalMass (ν : Measure ℂ) [IsFiniteMeasure ν]
+    (s : ℝ) (U : Set ℂ) (hU : ediam U ≤ ENNReal.ofReal s) :
+    (ν U).toReal ≤ maximalMass ν s :=
+  le_csSup (maximalMass_bdd ν s) ⟨U,hU,rfl⟩
+
+lemma maximalMass_grid (ν : Measure ℂ) [IsFiniteMeasure ν] (r : ℝ) (hr : 0<r)
+    (B : ℝ) (hB : 0 ≤ B) (hcell : ∀p,ν (cell r p) ≤ ENNReal.ofReal B)
+    (s : ℝ) (hs : 0 ≤ s) :
+    maximalMass ν s ≤ 16*B*max 1 ((s/r)^2) := by
+  apply csSup_le (maximalMass_nonempty ν s)
+  rintro v ⟨U,hU,rfl⟩
+  exact ENNReal.toReal_le_of_le_ofReal (by positivity)
+    (outerMass_le ν.toOuterMeasure r hr B hB hcell s hs U hU)
+
+ 
+
+theorem full_density {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι→ℝ)
+    (hb : ∀i,0<b i) (hρ : ∀i,0<ρ i) (E : Finset ι)
+    (ν : Measure ℂ) [IsFiniteMeasure ν] [NullSingletonClass ν]
+    (K : ℝ) (hK : 0 ≤ K)
+    (hcell : ∀i,i∉E → ∀p,ν (cell (ρ i) p) ≤ ENNReal.ofReal (K*b i*(ρ i)^d)) :
+    ∃C : ℝ, 0<C ∧ ∀U : Set ℂ, ediam U ≤ 1 →
+      ν U ≤ ENNReal.ofReal C * Deterministic.diameterCost (QuadraticGauge.gauge d b ρ) (ediam U) := by
+  obtain ⟨C,hC,hbound⟩ := QuadraticGauge.finite_exclusions d b ρ hb hρ E
+    (maximalMass ν) (ν univ).toReal (16*K) (fun s _=>maximalMass_bounds ν s)
+    (by
+      intro i hi s hs
+      have H := maximalMass_grid ν (ρ i) (hρ i) (K*b i*(ρ i)^d) (mul_nonneg (mul_nonneg hK (hb i).le) (Real.rpow_nonneg (hρ i).le _))
+        (hcell i hi) s hs.le
+      dsimp only [QuadraticGauge.entry]
+      convert H using 1; ring)
+  refine ⟨C,hC,?_⟩
+  intro U hU
+  have hfin : ediam U≠∞ := ne_top_of_le_ne_top (by norm_num) hU
+  by_cases hz : ediam U=0
+  · have he : ν U=0 := (ediam_eq_zero_iff.mp hz).measure_zero ν
+    rw [he]
+    exact bot_le
+  · have hs : 0<(ediam U).toReal := ENNReal.toReal_pos hz hfin
+    have hm := (mass_le_maximalMass ν (ediam U).toReal U (by
+      rw [ENNReal.ofReal_toReal hfin])).trans (hbound _ hs)
+    have he := ENNReal.ofReal_le_ofReal hm
+    rw [ENNReal.ofReal_toReal (measure_ne_top ν U),ENNReal.ofReal_mul hC.le] at he
+    simpa only [Deterministic.diameterCost,ite_eq_right hfin] using he
+
+ 
+
+theorem hausdorff_pos_of_grid {ι : Type*} [Nonempty ι] (d : ℝ) (b ρ : ι→ℝ)
+    (hb : ∀i,0<b i) (hρ : ∀i,0<ρ i) (E : Finset ι)
+    (ν : Measure ℂ) [IsFiniteMeasure ν] [NullSingletonClass ν]
+    (K : ℝ) (hK : 0 ≤ K)
+    (hcell : ∀i,i∉E → ∀p,ν (cell (ρ i) p) ≤ ENNReal.ofReal (K*b i*(ρ i)^d))
+    (A : Set ℂ) (hA : 0<ν A) : 0<Target.hausdorff (QuadraticGauge.gauge d b ρ) A := by
+  obtain ⟨C,hC,hbound⟩ := full_density d b ρ hb hρ E ν K hK hcell
+  exact Deterministic.hausdorff_pos_of_mass_bound (δ := 1) (ν := ν.toOuterMeasure)
+    (by norm_num) (ENNReal.ofReal_pos.mpr hC).ne' ENNReal.ofReal_ne_top hbound hA
+
+end SLEExactGauge.PlanarGrid
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped ENNReal NNReal Classical
+namespace SLEExactGauge.PlanarGrid
+
+ 
+def badRegion (μ : Measure ℂ) (r B : ℝ) : Set ℂ :=
+  ⋃ (p : ℤ×ℤ) (_ : ENNReal.ofReal B<μ (cell r p)),cell r p
+
+lemma badRegion_measurable (μ : Measure ℂ) (r B : ℝ) : MeasurableSet (badRegion μ r B) :=
+  MeasurableSet.iUnion (fun p=>MeasurableSet.iUnion (fun _=>cell_measurable r p))
+
+ 
+def retained {ι X : Type*} (bad : ι→Set X) (E : Finset ι) : Set X :=
+  {z | ∀i,i∉E → z∉bad i}
+
+lemma retained_measurable {ι X : Type*} [Countable ι] [MeasurableSpace X]
+    (bad : ι→Set X) (hbad : ∀i,MeasurableSet (bad i)) (E : Finset ι) :
+    MeasurableSet (retained bad E) := by
+  have he : retained bad E = ⋂i,⋂(_ : i∉E),(bad i)ᶜ := by ext z; simp [retained]
+  rw [he]
+  exact MeasurableSet.iInter (fun i=>MeasurableSet.iInter (fun _=>(hbad i).compl))
+
+lemma exists_positive_retained {ι X : Type*} [Countable ι] [MeasurableSpace X]
+    (σ : Measure X) (bad : ι→Set X) (hfinite : ∀ᵐ z∂σ,{i | z∈bad i}.Finite)
+    (A : Set X) (hA : 0<σ A) :
+    ∃E : Finset ι,0<σ (A∩retained bad E) := by
+  have hc : ∀ᵐ z∂σ,z∈⋃E : Finset ι,retained bad E := by
+    filter_upwards [hfinite] with z hz
+    refine mem_iUnion.mpr ⟨hz.toFinset,?_⟩
+    intro i hi hmem
+    exact hi (hz.mem_toFinset.mpr hmem)
+  by_contra! hh
+  have hzero : σ (A∩⋃E : Finset ι,retained bad E)=0 := by
+    rw [Set.inter_iUnion]
+    exact measure_iUnion_null_iff.mpr (fun E=>le_antisymm (hh E) bot_le)
+  have he : (A : Set X)=ᵐ[σ] (A∩⋃E : Finset ι,retained bad E : Set X) := by
+    filter_upwards [hc] with z hz
+    simp only [mem_inter_iff,hz,and_true]
+  exact hA.ne' ((measure_congr he).trans hzero)
+
+lemma retained_cell_bound {ι : Type*} (μ σ : Measure ℂ) (hσ : σ≤μ)
+    (ρ B : ι→ℝ) (E : Finset ι) (i : ι) (hi : i∉E) (p : ℤ×ℤ) :
+    (σ.restrict (retained (fun i=>badRegion μ (ρ i) (B i)) E)) (cell (ρ i) p) ≤
+      ENNReal.ofReal (B i) := by
+  by_cases hbad : ENNReal.ofReal (B i)<μ (cell (ρ i) p)
+  · rw [Measure.restrict_apply (cell_measurable _ _)]
+    have he : cell (ρ i) p∩retained (fun i=>badRegion μ (ρ i) (B i)) E=∅ := by
+      apply eq_empty_iff_forall_notMem.mpr
+      intro z hz
+      exact hz.2 i hi (mem_iUnion.mpr ⟨p,mem_iUnion.mpr ⟨hbad,hz.1⟩⟩)
+    rw [he,measure_empty]
+    exact bot_le
+  · exact (Measure.restrict_le_self.trans hσ _).trans (le_of_not_gt hbad)
+
+ 
+
+theorem hausdorff_pos_of_finitely_bad {ι : Type*} [Nonempty ι] [Countable ι]
+    (d : ℝ) (b ρ : ι→ℝ) (hb : ∀i,0<b i) (hρ : ∀i,0<ρ i)
+    (μ σ : Measure ℂ) (hσ : σ≤μ) [IsFiniteMeasure σ] [NullSingletonClass σ]
+    (K : ℝ) (hK : 0 ≤ K)
+    (hfinite : ∀ᵐ z∂σ,{i | z∈badRegion μ (ρ i) (K*b i*(ρ i)^d)}.Finite)
+    (A : Set ℂ) (hA : 0<σ A) : 0<Target.hausdorff (QuadraticGauge.gauge d b ρ) A := by
+  let bad := fun i=>badRegion μ (ρ i) (K*b i*(ρ i)^d)
+  obtain ⟨E,hE⟩ := exists_positive_retained σ bad hfinite A hA
+  let ν := σ.restrict (retained bad E)
+  have hνA : 0<ν A := by
+    dsimp only [ν]
+    rw [Measure.restrict_apply' (retained_measurable bad (fun i=>badRegion_measurable _ _ _) E)]
+    exact hE
+  exact hausdorff_pos_of_grid d b ρ hb hρ E ν K hK
+    (fun i hi p=>retained_cell_bound μ σ hσ ρ (fun j=>K*b j*(ρ j)^d) E i hi p) A hνA
+
+ 
+
+theorem ae_ae_finite_bad {Ω X ι : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    [Countable ι] (P : Measure Ω) (μ : Ω→Measure X) (bad : ι→Ω→Set X)
+    (hm : ∀i,AEMeasurable (fun ω=>μ ω (bad i ω)) P)
+    (hsum : (∑'i,∫⁻ω,μ ω (bad i ω)∂P)≠∞) :
+    ∀ᵐ ω∂P,∀ᵐ z∂μ ω,{i | z∈bad i ω}.Finite := by
+  have h : (∫⁻ω,∑'i,μ ω (bad i ω)∂P)≠∞ := by rwa [lintegral_tsum hm]
+  filter_upwards [ae_lt_top' (AEMeasurable.tsum hm) h] with ω hω
+  exact ae_finite_setOfPred_mem hω.ne
+
+end SLEExactGauge.PlanarGrid
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped ENNReal NNReal Topology Classical
+namespace SLEExactGauge.PlanarGrid
+
+ 
+
+def testedBadCell {Ω ι : Type*} [MeasurableSpace Ω]
+    (μ : Ω→Measure ℂ) (ρ B : ι→ℝ) (good : Set ι)
+    (j : ι×(ℤ×ℤ)) (ω : Ω) : Set ℂ :=
+  if j.1∈good ∧ ENNReal.ofReal (B j.1) < μ ω (cell (ρ j.1) j.2)
+    then cell (ρ j.1) j.2 else ∅
+
+lemma testedBadCell_mass {Ω ι : Type*} [MeasurableSpace Ω]
+    (μ : Ω→Measure ℂ) (ρ B : ι→ℝ) (good : Set ι)
+    (j : ι×(ℤ×ℤ)) (ω : Ω) :
+    μ ω (testedBadCell μ ρ B good j ω)=
+      if j.1∈good then
+        {ω | ENNReal.ofReal (B j.1) < μ ω (cell (ρ j.1) j.2)}.indicator
+          (fun ω=>μ ω (cell (ρ j.1) j.2)) ω else 0 := by
+  by_cases hg : j.1∈good <;>
+    by_cases hb : ENNReal.ofReal (B j.1) < μ ω (cell (ρ j.1) j.2) <;>
+    simp [testedBadCell,hg,hb]
+
+ 
+
+theorem ae_finitely_bad_of_spatial_tails {Ω ι : Type*}
+    [MeasurableSpace Ω] [Countable ι]
+    (P : Measure Ω) (μ : Ω→Measure ℂ) (ν : Measure ℂ) [IsFiniteMeasure ν]
+    (ρ B : ι→ℝ) (hρ : ∀i,0 < ρ i)
+    (hm : ∀i p,Measurable (fun ω=>μ ω (cell (ρ i) p)))
+    (good : Set ι) (hgood : goodᶜ.Finite)
+    (C : ℝ≥0∞) (hC : C≠∞) (tail : ι→ℝ≥0∞) (htail : (∑'i,tail i)≠∞)
+    (hbound : ∀i∈good,∀p : ℤ×ℤ,
+      (∫⁻ω,{ω | ENNReal.ofReal (B i) < μ ω (cell (ρ i) p)}.indicator
+        (fun ω=>μ ω (cell (ρ i) p)) ω∂P) ≤ C*ν (cell (ρ i) p)*tail i) :
+    ∀ᵐ ω∂P,∀ᵐ z∂μ ω,{i | z∈badRegion (μ ω) (ρ i) (B i)}.Finite := by
+  let bad := testedBadCell μ ρ B good
+  have hbad (j : ι×(ℤ×ℤ)) :
+      Measurable (fun ω=>μ ω (bad j ω)) := by
+    simp only [bad,testedBadCell_mass]
+    by_cases hj : j.1∈good
+    · simp only [ite_eq_left hj]
+      exact (hm j.1 j.2).indicator (measurableSet_lt measurable_const (hm j.1 j.2))
+    · simp only [ite_eq_right hj]
+      exact measurable_const
+  have hi (j : ι×(ℤ×ℤ)) : (∫⁻ω,μ ω (bad j ω)∂P) ≤
+      C*ν (cell (ρ j.1) j.2)*tail j.1 := by
+    simp only [bad,testedBadCell_mass]
+    by_cases hj : j.1∈good
+    · simp only [ite_eq_left hj]
+      exact hbound j.1 hj j.2
+    · simp only [ite_eq_right hj,lintegral_zero]
+      exact bot_le
+  have hsum : (∑'j : ι×(ℤ×ℤ),∫⁻ω,μ ω (bad j ω)∂P)≠∞ := by
+    apply ne_top_of_le_ne_top (show (C*(4*ν univ))*(∑'i,tail i)≠∞ from by finiteness)
+    calc
+      _≤∑'j : ι×(ℤ×ℤ),C*ν (cell (ρ j.1) j.2)*tail j.1 := ENNReal.tsum_le_tsum hi
+      _=∑'i,C*(∑'p : ℤ×ℤ,ν (cell (ρ i) p))*tail i := by
+        rw [ENNReal.tsum_prod']
+        simp_rw [ENNReal.tsum_mul_right,ENNReal.tsum_mul_left]
+      _≤∑'i,C*(4*ν univ)*tail i := by
+        apply ENNReal.tsum_le_tsum
+        intro i
+        gcongr
+        exact tsum_cell_mass ν (ρ i) (hρ i)
+      _=(C*(4*ν univ))*(∑'i,tail i) := ENNReal.tsum_mul_left
+  filter_upwards [ae_ae_finite_bad P μ bad (fun j=>(hbad j).aemeasurable) hsum] with ω hω
+  filter_upwards [hω] with z hz
+  apply (hgood.union (hz.image Prod.fst)).subset
+  intro i hi
+  by_cases hg : i∈good
+  · right
+    obtain ⟨p,hp⟩ := mem_iUnion.mp hi
+    obtain ⟨hbad,hcell⟩ := mem_iUnion.mp hp
+    refine ⟨(i,p),?_,rfl⟩
+    change z∈testedBadCell μ ρ B good (i,p) ω
+    simp only [testedBadCell,ite_eq_left (And.intro hg hbad)]
+    exact hcell
+  · exact Or.inl hg
+
+end SLEExactGauge.PlanarGrid
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/PostStopPotential.lean b/lean/OAI/Probability/SLEGauge/PostStopPotential.lean
index 1b32cb8c5d9f3bbdc447fb8212e86ba80631ddd0..b59b694b9663fc6944cbe29413893111bf4aa9e6 100644
--- a/lean/OAI/Probability/SLEGauge/PostStopPotential.lean
+++ b/lean/OAI/Probability/SLEGauge/PostStopPotential.lean
@@ -1,505 +1,505 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RandomRestrictionKernel
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
-
-lemma lintegral_finite_stop_guard (W : Measure GDriver)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    (M : GDriver→Measure UpperPlane) (A : Set GDriver) (f : UpperPlane→ℝ≥0∞) :
-    (∫⁻b in A,∫⁻z,f z∂(if S b=∞ then 0 else M b)∂W)=
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂M b∂W := by
-  have hfinite : MeasurableSet {b | S b≠∞} :=
-    (hS.measurable' (measurableSet_singleton ∞)).compl
-  rw [inter_comm A {b | S b≠∞},←Measure.restrict_restrict hfinite,←lintegral_indicator hfinite]
-  apply lintegral_congr
-  intro b
-  by_cases hb : S b=∞ <;> simp [hb]
-
- 
-
-theorem recoveredPostStop_potential (a : ℝ) (ha : 1/4<a)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass a ha s c hc v b)))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {t : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[(afterStopFiltration hS) t] A)
-    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    let ν := recoveredPostStop a ha s c hc S
-    (∫⁻b in A,∫⁻z,f z∂ν ∞ b∂W)=
-      (∫⁻b in A,∫⁻z,f z∂ν t b∂W)+
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
-        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
-  intro ν
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let μ := recoveredMass a ha s c hc
-  let T := shiftStop S t
-  have hT := shiftStop_isStoppingTime hS t
-  let R := stoppedSurvivingKernel a ha T hT
-  have hcond (H : GDriver×UpperPlane→ℝ≥0∞)
-      (hH : Measurable[hT.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
-      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=(∫⁻b,∫⁻z,H (b,z)∂μ (T b) b∂W)+
-        ∫⁻b,∫⁻z,H (b,z)∂ R b∂W := by
-    have hi := (recoveredMass_all_stopping a ha W hW s c hc hs hsum hconv T hT).2.2.2.2 H hH
-    have hn : Measurable H := hH.mono (prod_sigma_le hT.measurableSpace_le _) le_rfl
-    have he (b : GDriver) : (∫⁻z,H (b,z)∂ R b)=
-        ∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (T b) b z.1)∂upperArea := by
-      rw [stoppedSurvivingKernel_apply]
-      exact (lintegral_withDensity_eq_lintegral_mul upperArea
-        (stoppedSurvivingGreen_spatial_measurable a ha T hT b).ennreal_ofReal
-        (hn.comp (measurable_const.prodMk measurable_id))).trans
-        (lintegral_congr fun _ => mul_comm _ _)
-    simpa only [he,μ] using hi
-  let D : GDriver→Set UpperPlane := fun b => {z | alive a b.1 (S b).toNNReal z.1}
-  have hD : MeasurableSet[hT.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)]
-      {p : GDriver×UpperPlane | p.2∈D p.1} :=
-    (prod_sigma_le (measurableSpace_le_shiftStop hS t) _) _ (measurableSet_alive_atStop_raw a ha hS)
-  have hB : MeasurableSet[hT.measurableSpace] (A∩{b | S b≠∞}) :=
-    hA.inter ((measurableSpace_le_shiftStop hS t) _ (hS.measurable (measurableSet_singleton ∞)).compl)
-  have h := conditional_random_restrict_on hT.measurableSpace_le W (μ ∞)
-    (fun b => μ (T b) b) R hcond D hD hB hf
-  have he (v : ℝ≥0∞) : (∫⁻b in A,∫⁻z,f z∂ν v b∂W)=
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(μ (S b+v) b).restrict (D b)∂W :=
-    lintegral_finite_stop_guard W hS _ A f
-  rw [he,he]
-  simp only [add_top] 
-  refine h.trans (congrArg (fun x => (∫⁻b in A∩{b | S b≠∞},
-    ∫⁻z,f z∂(μ (T b) b).restrict (D b)∂W)+x) ?_)
-  apply setLIntegral_congr_fun (hT.measurableSpace_le _ hB)
-  intro b hb
-  have hbt : T b≠∞ := ENNReal.add_ne_top.mpr ⟨hb.2,ENNReal.coe_ne_top⟩
-  have heR : R b=survivingKernel a ha ((S b).toNNReal+t) b := by
-    rw [stoppedSurvivingKernel_apply,survivingKernel_apply]
-    congr 1
-    funext z
-    simp only [stoppedSurvivingGreen,T,shiftStop,ite_eq_right (show S b+(t:ℝ≥0∞)≠∞ from hbt),
-      ENNReal.toNNReal_add hb.2 ENNReal.coe_ne_top,ENNReal.toNNReal_coe]
-  change (∫⁻z,f z∂(R b).restrict (D b))=_
-  rw [heR]
-
- 
-theorem recoveredRestart_potential (a : ℝ) (ha : 1/4<a)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass a ha s c hc v b)))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {t : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[(afterStopFiltration hS) t] A)
-    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    let ν := recoveredRestart a ha s c hc S
-    (∫⁻b in A,∫⁻z,f z∂ν ∞ b∂W)=
-      (∫⁻b in A,∫⁻z,f z∂ν t b∂W)+
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
-        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
-  intro ν
-  have he (v : ℝ≥0∞) : (∫⁻b in A,∫⁻z,f z∂ν v b∂W)=
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
-        b (S b).toNNReal (recoveredMass a ha s c hc v (restartAt S b))∂W :=
-    lintegral_finite_stop_guard W hS _ A f
-  rw [he,he]
-  exact recoveredMass_restarted_potential a ha W hW s c hc hs hsum hconv hS hA hf
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology CompactlySupported
-namespace SLEExactGauge.MassVersions
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-lemma measure_eq_of_determiningTests (μ ν : Measure X)
-    [μ.Regular] [ν.Regular]
-    (he : ∀ i, (∫x, determiningTests i x ∂μ) = ∫x, determiningTests i x ∂ν) :
-    μ=ν := by
-  let L : ℕ → (C_c(X,ℝ) →ₚ[ℝ] ℝ) := fun n =>
-    if n%2=0 then CompactlySupportedContinuousMap.integralPositiveLinearMap μ
-      else CompactlySupportedContinuousMap.integralPositiveLinearMap ν
-  have hc : TestConverges L := by
-    intro i
-    refine ⟨∫x, determiningTests i x ∂μ,?_⟩
-    have hh (n : ℕ) : L n (determiningTests i) = ∫x, determiningTests i x ∂μ := by
-      dsimp only [L]
-      split_ifs
-      · rfl
-      · exact (he i).symm
-    simpa only [hh] using (tendsto_const_nhds : Tendsto
-      (fun _ : ℕ => ∫x, determiningTests i x ∂μ) atTop (𝓝 (∫x, determiningTests i x ∂μ)))
-  have hmul : Tendsto (fun n : ℕ => 2*n) atTop atTop := by
-    apply tendsto_atTop.mpr
-    intro N
-    filter_upwards [eventually_ge_atTop N] with n hn
-    omega
-  have hadd : Tendsto (fun n : ℕ => 2*n+1) atTop atTop :=
-    (tendsto_add_atTop_nat 1).comp hmul
-  apply Measure.ext_of_integral_eq_on_compactlySupported
-  intro f
-  have heven := (tendsto_radonLimit L hc f).comp hmul
-  have hodd := (tendsto_radonLimit L hc f).comp hadd
-  have hem (n : ℕ) : L (2*n) f=∫x,f x∂μ := by simp [L]
-  have hov (n : ℕ) : L (2*n+1) f=∫x,f x∂ν := by simp [L]
-  simp only [Function.comp_def,hem] at heven
-  simp only [Function.comp_def,hov] at hodd
-  exact (tendsto_nhds_unique tendsto_const_nhds heven).trans
-    (tendsto_nhds_unique tendsto_const_nhds hodd).symm
-
- 
-theorem ae_all_measure_eq_of_test_eq {Ω I : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (μ ν : I → Ω → Measure X)
-    [∀ t ω, (μ t ω).Regular] [∀ t ω, (ν t ω).Regular]
-    (he : ∀ i, ∀ᵐ ω ∂P, ∀ t, (∫x,determiningTests i x∂μ t ω) =
-      ∫x,determiningTests i x∂ν t ω) :
-    ∀ᵐ ω ∂P, ∀ t, μ t ω=ν t ω := by
-  filter_upwards [ae_all_iff.mpr he] with ω hω t
-  exact measure_eq_of_determiningTests _ _ (fun i => hω i t)
-
- 
-def positiveTest (f : C_c(X,ℝ)) : C_c(X,ℝ) :=
-  ⟨⟨fun x => max (f x) 0, f.continuous.max continuous_const⟩,
-    f.hasCompactSupport.sup (.zero)⟩
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-    [MeasurableSpace X] [BorelSpace X] in
-lemma positiveTest_nonneg [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    (f : C_c(X,ℝ)) (x : X) : 0≤positiveTest f x :=
-  le_max_right _ _
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-    [MeasurableSpace X] [BorelSpace X] in
-lemma positiveTest_sub [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    (f : C_c(X,ℝ)) (x : X) :
-    positiveTest f x-positiveTest (-f) x=f x := by
-  change max (f x) 0-max (-(f x)) 0=f x
-  exact posPart_sub_negPart (f x)
-
-lemma integral_positiveTest_sub (f : C_c(X,ℝ)) (μ : Measure X)
-    [IsFiniteMeasureOnCompacts μ] :
-    (∫x,positiveTest f x∂μ)-(∫x,positiveTest (-f) x∂μ)=∫x,f x∂μ := by
-  rw [← integral_sub (positiveTest f).integrable (positiveTest (-f)).integrable]
-  simp only [positiveTest_sub]
-
-end SLEExactGauge.MassVersions
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.MeshEnergy
-
- 
-def grid (T : ℝ≥0) (n i : ℕ) : ℝ≥0 := T * i / ((n:ℝ≥0)+1)
-
-@[simp] lemma grid_zero (T : ℝ≥0) (n : ℕ) : grid T n 0=0 := by simp [grid]
-
-@[simp] lemma grid_last (T : ℝ≥0) (n : ℕ) : grid T n (n+1)=T := by
-  have h : (n:ℝ≥0)+1≠0 := ne_of_gt (by positivity)
-  simp [grid,Nat.cast_add,Nat.cast_one,h]
-
-lemma grid_mono (T : ℝ≥0) (n : ℕ) : Monotone (grid T n) := by
-  intro i j hij
-  dsimp [grid]
-  gcongr
-
-lemma grid_mem (T : ℝ≥0) (n i : ℕ) (hi : i≤n+1) : grid T n i∈Icc 0 T :=
-  ⟨zero_le, (grid_mono T n hi).trans_eq (grid_last T n)⟩
-
-lemma dist_grid_succ (T : ℝ≥0) (n i : ℕ) :
-    dist (grid T n (i+1)) (grid T n i) = (T:ℝ)/((n:ℝ)+1) := by
-  rw [NNReal.dist_eq]
-  simp only [grid,NNReal.coe_div,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_add,
-    NNReal.coe_one,Nat.cast_add,Nat.cast_one]
-  have he : (T:ℝ)*((i:ℝ)+1)/((n:ℝ)+1)-T*i/((n:ℝ)+1)=T/((n:ℝ)+1) := by ring
-  rw [he,abs_of_nonneg (by positivity)]
-
-lemma tendsto_mesh (T : ℝ≥0) :
-    Tendsto (fun n : ℕ => (T:ℝ)/((n:ℝ)+1)) atTop (𝓝 0) := by
-  simpa [div_eq_mul_inv] using
-    (tendsto_const_nhds.mul (tendsto_one_div_add_atTop_nhds_zero_nat :
-      Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝 0)))
-
- 
-def energy (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) : ℝ :=
-  ∑ i ∈ Finset.range (n+1), (M (grid T n (i+1))-M (grid T n i))^2
-
-lemma energy_nonneg (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) : 0≤energy M T n :=
-  Finset.sum_nonneg fun _ _ => sq_nonneg _
-
- 
-lemma energy_le (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
-    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) (n : ℕ) :
-    energy M T n ≤ (C T-C 0)^2 := by
-  let A : ℕ → ℝ := fun i => C (grid T n (i+1))-C (grid T n i)
-  have ha (i : ℕ) (hi : i∈Finset.range (n+1)) :
-      |M (grid T n (i+1))-M (grid T n i)|≤A i :=
-    hcontrol _ _ (grid_mono T n (Nat.le_succ _))
-      (grid_mem T n (i+1) (by simpa using Finset.mem_range.mp hi)).2
-  have han (i : ℕ) (hi : i∈Finset.range (n+1)) : 0≤A i :=
-    (abs_nonneg _).trans (ha i hi)
-  calc
-    energy M T n ≤ ∑ i ∈ Finset.range (n+1), (A i)^2 := by
-      apply Finset.sum_le_sum
-      intro i hi
-      calc
-        _ = |M (grid T n (i+1))-M (grid T n i)|^2 := (sq_abs _).symm
-        _ ≤ (A i)^2 := (sq_le_sq₀ (abs_nonneg _) (han i hi)).mpr (ha i hi)
-    _ ≤ (∑ i ∈ Finset.range (n+1), A i)^2 := Finset.sum_sq_le_sq_sum_of_nonneg han
-    _ = (C T-C 0)^2 := by
-      rw [show (∑ i ∈ Finset.range (n+1), A i)=C (grid T n (n+1))-C (grid T n 0) from
-        Finset.sum_range_sub (fun i => C (grid T n i)) (n+1)]
-      simp
-
- 
-theorem tendsto_energy_zero (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
-    (hc : ContinuousOn C (Icc 0 T))
-    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) :
-    Tendsto (energy M T) atTop (𝓝 0) := by
-  let V := C T-C 0
-  have hv : 0≤V := (abs_nonneg _).trans (hcontrol 0 T (zero_le) le_rfl)
-  have huc := isCompact_Icc.uniformContinuousOn_of_continuous hc
-  apply Metric.tendsto_atTop.mpr
-  intro ε hε
-  have hη : 0<ε/(V+1) := div_pos hε (by linarith)
-  obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuousOn_iff.mp huc (ε/(V+1)) hη
-  obtain ⟨N,hN⟩ := eventually_atTop.mp ((tendsto_mesh T).eventually (gt_mem_nhds hδ))
-  refine ⟨N,fun n hn => ?_⟩
-  let A : ℕ → ℝ := fun i => C (grid T n (i+1))-C (grid T n i)
-  have ha (i : ℕ) (hi : i∈Finset.range (n+1)) :
-      |M (grid T n (i+1))-M (grid T n i)|≤A i :=
-    hcontrol _ _ (grid_mono T n (Nat.le_succ _))
-      (grid_mem T n (i+1) (by simpa using Finset.mem_range.mp hi)).2
-  have hb (i : ℕ) (hi : i∈Finset.range (n+1)) : A i<ε/(V+1) := by
-    have hi' : i<n+1 := Finset.mem_range.mp hi
-    have h := hd _ (grid_mem T n (i+1) hi') _ (grid_mem T n i hi'.le)
-      (by rw [dist_grid_succ]; exact hN n hn)
-    exact (le_abs_self (A i)).trans_lt h
-  have he : energy M T n≤ε/(V+1)*V := by
-    calc
-      _ ≤ ∑ i ∈ Finset.range (n+1), (ε/(V+1))* A i := by
-        apply Finset.sum_le_sum
-        intro i hi
-        have h0 : 0≤A i := (abs_nonneg _).trans (ha i hi)
-        calc
-          _ = |M (grid T n (i+1))-M (grid T n i)|^2 := (sq_abs _).symm
-          _ ≤ (A i)^2 := (sq_le_sq₀ (abs_nonneg _) h0).mpr (ha i hi)
-          _ ≤ ε/(V+1)*A i := by rw [pow_two]; exact mul_le_mul_of_nonneg_right (hb i hi).le h0
-      _ = _ := by
-        rw [← Finset.mul_sum, show (∑ i ∈ Finset.range (n+1), A i)=
-          C (grid T n (n+1))-C (grid T n 0) from
-            Finset.sum_range_sub (fun i => C (grid T n i)) (n+1)]
-        simp [V]
-  rw [Real.dist_eq,sub_zero,abs_of_nonneg (energy_nonneg _ _ _)]
-  refine he.trans_lt ?_
-  apply (div_mul_eq_mul_div _ _ _).trans_lt
-  exact (div_lt_iff₀ (by linarith : 0<V+1)).mpr (by nlinarith)
-
-end SLEExactGauge.MeshEnergy
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.TaylorVariation
-open MeshEnergy
-
- 
-lemma remainder_le_sq (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
-    {K : ℝ≥0} (hg : LipschitzWith K g) (x y : ℝ) :
-    |f y - f x - g x * (y-x)| ≤ (K:ℝ) * (y-x)^2 := by
-  have h := (convex_uIcc (𝕜 := ℝ) x y).norm_image_sub_le_of_norm_hasDerivWithin_le
-    (f := fun z => f z-g x*z) (f' := fun z => g z-g x)
-    (C := (K:ℝ)*|y-x|)
-    (fun z _ => (by
-      convert! ((hd z).sub ((hasDerivAt_id z).const_mul (g x))).hasDerivWithinAt using 1
-      simp))
-    (fun z hz => (hg.norm_sub_le z x).trans (mul_le_mul_of_nonneg_left
-      (norm_sub_le_of_mem_segment (by simpa [segment_eq_uIcc] using hz)) K.2))
-    left_mem_uIcc right_mem_uIcc
-  have he : f y-g x*y-(f x-g x*x)=f y-f x-g x*(y-x) := by ring
-  calc
-    _ ≤ (K:ℝ) * |y-x| * |y-x| := by simpa only [Real.norm_eq_abs,he] using h
-    _ = _ := by rw [mul_assoc, ← pow_two, sq_abs]
-
-lemma lipschitz_of_deriv_bound (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
-    {L : ℝ≥0} (hg : ∀ x, |g x|≤L) : LipschitzWith L f := by
-  apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hd x).differentiableAt)
-  intro x
-  change |deriv f x|≤(L:ℝ)
-  rw [(hd x).deriv]
-  exact hg x
-
-lemma remainder_le_abs (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
-    {L : ℝ≥0} (hg : ∀ x, |g x|≤L) (x y : ℝ) :
-    |f y-f x-g x*(y-x)|≤(2*(L:ℝ))*|y-x| := by
-  have hf := (lipschitz_of_deriv_bound f g hd hg).norm_sub_le y x
-  calc
-    _ ≤ |f y-f x|+|g x*(y-x)| := abs_sub _ _
-    _ ≤ L*|y-x|+L*|y-x| := by
-      rw [abs_mul]
-      exact add_le_add hf (mul_le_mul_of_nonneg_right (hg x) (abs_nonneg _))
-    _ = _ := by ring
-
- 
-def error (f g : ℝ → ℝ) (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) : ℝ :=
-  f (M T)-f (M 0) - ∑ i ∈ Finset.range (n+1),
-    g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))
-
-lemma error_eq_sum (f g : ℝ → ℝ) (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) :
-    error f g M T n = ∑ i ∈ Finset.range (n+1),
-      (f (M (grid T n (i+1)))-f (M (grid T n i)) -
-        g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))) := by
-  rw [Finset.sum_sub_distrib, Finset.sum_range_sub (fun i => f (M (grid T n i))) (n+1)]
-  simp [error]
-
-lemma abs_error_le_energy (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
-    {K : ℝ≥0} (hg : LipschitzWith K g) (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) :
-    |error f g M T n|≤(K:ℝ)*energy M T n := by
-  rw [error_eq_sum]
-  calc
-    _ ≤ ∑ i ∈ Finset.range (n+1), |f (M (grid T n (i+1)))-f (M (grid T n i)) -
-        g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))| :=
-      Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ i ∈ Finset.range (n+1), (K:ℝ) * (M (grid T n (i+1))-M (grid T n i))^2 :=
-      Finset.sum_le_sum (fun i _ => remainder_le_sq f g hd hg _ _)
-    _ = _ := by rw [← Finset.mul_sum]; rfl
-
-lemma abs_error_le_variation (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
-    {L : ℝ≥0} (hg : ∀ x, |g x|≤L) (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
-    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) (n : ℕ) :
-    |error f g M T n|≤(2*(L:ℝ))*(C T-C 0) := by
-  rw [error_eq_sum]
-  calc
-    _ ≤ ∑ i ∈ Finset.range (n+1), |f (M (grid T n (i+1)))-f (M (grid T n i)) -
-        g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))| :=
-      Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ i ∈ Finset.range (n+1), (2*(L:ℝ)) * (C (grid T n (i+1))-C (grid T n i)) := by
-      apply Finset.sum_le_sum
-      intro i hi
-      exact (remainder_le_abs f g hd hg _ _).trans (mul_le_mul_of_nonneg_left
-        (hcontrol _ _ (grid_mono T n (Nat.le_succ _))
-          (grid_mem T n (i+1) (by simpa using Finset.mem_range.mp hi)).2) (by positivity))
-    _ = _ := by
-      rw [← Finset.mul_sum, Finset.sum_range_sub (fun i => C (grid T n i)) (n+1)]
-      simp
-
-lemma tendsto_error_zero (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
-    {K : ℝ≥0} (hg : LipschitzWith K g) (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
-    (hc : ContinuousOn C (Icc 0 T))
-    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) :
-    Tendsto (error f g M T) atTop (𝓝 0) := by
-  apply squeeze_zero_norm (fun n => ?_)
-    (show Tendsto (fun n => (K:ℝ)*energy M T n) atTop (𝓝 0) by
-      simpa using (tendsto_energy_zero M C T hc hcontrol).const_mul (K:ℝ))
-  exact abs_error_le_energy f g hd hg M T n
-
- 
-
-def detector (x : ℝ) : ℝ := Real.sqrt (1+x^2)-1
-
-def detectorDeriv (x : ℝ) : ℝ := x / Real.sqrt (1+x^2)
-
-lemma one_le_root (x : ℝ) : 1≤Real.sqrt (1+x^2) :=
-  Real.one_le_sqrt.mpr (by nlinarith [sq_nonneg x])
-
-lemma hasDerivAt_root (x : ℝ) : HasDerivAt (fun x : ℝ => Real.sqrt (1+x^2))
-    (x/Real.sqrt (1+x^2)) x := by
-  have h : 1+x^2≠0 := ne_of_gt (by positivity)
-  convert! (((hasDerivAt_id x).pow 2).const_add 1).sqrt h using 1
-  dsimp
-  ring
-
-lemma hasDerivAt_detector (x : ℝ) : HasDerivAt detector (detectorDeriv x) x :=
-  (hasDerivAt_root x).sub_const 1
-
-lemma hasDerivAt_detectorDeriv (x : ℝ) :
-    HasDerivAt detectorDeriv (1/(Real.sqrt (1+x^2))^3) x := by
-  have hp : 0<Real.sqrt (1+x^2) := lt_of_lt_of_le zero_lt_one (one_le_root x)
-  have hs := Real.sq_sqrt (show 0≤1+x^2 by positivity)
-  have hx : HasDerivAt detectorDeriv
-      ((Real.sqrt (1+x^2)-x*(x/Real.sqrt (1+x^2)))/(Real.sqrt (1+x^2))^2) x := by
-    convert! (hasDerivAt_id x).div (hasDerivAt_root x) (ne_of_gt hp) using 1
-    simp
-  convert! hx using 1
-  field_simp
-  nlinarith [hs]
-
-lemma abs_detectorDeriv_le (x : ℝ) : |detectorDeriv x|≤1 := by
-  have hp : 0<Real.sqrt (1+x^2) := lt_of_lt_of_le zero_lt_one (one_le_root x)
-  have hs := Real.sq_sqrt (show 0≤1+x^2 by positivity)
-  rw [detectorDeriv,abs_div,abs_of_pos hp,div_le_iff₀ hp,one_mul]
-  nlinarith [sq_abs x,abs_nonneg x]
-
-lemma lipschitz_detector : LipschitzWith 1 detector :=
-  lipschitz_of_deriv_bound detector detectorDeriv hasDerivAt_detector abs_detectorDeriv_le
-
-lemma lipschitz_detectorDeriv : LipschitzWith 1 detectorDeriv := by
-  apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hasDerivAt_detectorDeriv x).differentiableAt)
-  intro x
-  change |deriv detectorDeriv x|≤(1:ℝ)
-  rw [(hasDerivAt_detectorDeriv x).deriv,abs_of_nonneg (by positivity)]
-  have h := one_le_root x
-  exact (div_le_one (by positivity)).mpr (one_le_pow₀ h)
-
-@[simp] lemma detector_zero : detector 0=0 := by simp [detector]
-
-lemma detector_nonneg (x : ℝ) : 0≤detector x := sub_nonneg.mpr (one_le_root x)
-
-@[simp] lemma detector_eq_zero_iff (x : ℝ) : detector x=0 ↔ x=0 := by
-  constructor
-  · intro h
-    have hs := Real.sq_sqrt (show 0≤1+x^2 by positivity)
-    have he : Real.sqrt (1+x^2)=1 := sub_eq_zero.mp h
-    rw [he] at hs
-    nlinarith [sq_nonneg x]
-  · rintro rfl
-    exact detector_zero
-
-lemma abs_detector_le (x : ℝ) : |detector x|≤|x| := by
-  simpa using lipschitz_detector.norm_sub_le x 0
-
-end SLEExactGauge.TaylorVariation
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RandomRestrictionKernel
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
+
+lemma lintegral_finite_stop_guard (W : Measure GDriver)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    (M : GDriver→Measure UpperPlane) (A : Set GDriver) (f : UpperPlane→ℝ≥0∞) :
+    (∫⁻b in A,∫⁻z,f z∂(if S b=∞ then 0 else M b)∂W)=
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂M b∂W := by
+  have hfinite : MeasurableSet {b | S b≠∞} :=
+    (hS.measurable' (measurableSet_singleton ∞)).compl
+  rw [inter_comm A {b | S b≠∞},←Measure.restrict_restrict hfinite,←lintegral_indicator hfinite]
+  apply lintegral_congr
+  intro b
+  by_cases hb : S b=∞ <;> simp [hb]
+
+ 
+
+theorem recoveredPostStop_potential (a : ℝ) (ha : 1/4<a)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass a ha s c hc v b)))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {t : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[(afterStopFiltration hS) t] A)
+    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    let ν := recoveredPostStop a ha s c hc S
+    (∫⁻b in A,∫⁻z,f z∂ν ∞ b∂W)=
+      (∫⁻b in A,∫⁻z,f z∂ν t b∂W)+
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
+        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
+  intro ν
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let μ := recoveredMass a ha s c hc
+  let T := shiftStop S t
+  have hT := shiftStop_isStoppingTime hS t
+  let R := stoppedSurvivingKernel a ha T hT
+  have hcond (H : GDriver×UpperPlane→ℝ≥0∞)
+      (hH : Measurable[hT.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
+      (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂W)=(∫⁻b,∫⁻z,H (b,z)∂μ (T b) b∂W)+
+        ∫⁻b,∫⁻z,H (b,z)∂ R b∂W := by
+    have hi := (recoveredMass_all_stopping a ha W hW s c hc hs hsum hconv T hT).2.2.2.2 H hH
+    have hn : Measurable H := hH.mono (prod_sigma_le hT.measurableSpace_le _) le_rfl
+    have he (b : GDriver) : (∫⁻z,H (b,z)∂ R b)=
+        ∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (T b) b z.1)∂upperArea := by
+      rw [stoppedSurvivingKernel_apply]
+      exact (lintegral_withDensity_eq_lintegral_mul upperArea
+        (stoppedSurvivingGreen_spatial_measurable a ha T hT b).ennreal_ofReal
+        (hn.comp (measurable_const.prodMk measurable_id))).trans
+        (lintegral_congr fun _ => mul_comm _ _)
+    simpa only [he,μ] using hi
+  let D : GDriver→Set UpperPlane := fun b => {z | alive a b.1 (S b).toNNReal z.1}
+  have hD : MeasurableSet[hT.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)]
+      {p : GDriver×UpperPlane | p.2∈D p.1} :=
+    (prod_sigma_le (measurableSpace_le_shiftStop hS t) _) _ (measurableSet_alive_atStop_raw a ha hS)
+  have hB : MeasurableSet[hT.measurableSpace] (A∩{b | S b≠∞}) :=
+    hA.inter ((measurableSpace_le_shiftStop hS t) _ (hS.measurable (measurableSet_singleton ∞)).compl)
+  have h := conditional_random_restrict_on hT.measurableSpace_le W (μ ∞)
+    (fun b => μ (T b) b) R hcond D hD hB hf
+  have he (v : ℝ≥0∞) : (∫⁻b in A,∫⁻z,f z∂ν v b∂W)=
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(μ (S b+v) b).restrict (D b)∂W :=
+    lintegral_finite_stop_guard W hS _ A f
+  rw [he,he]
+  simp only [add_top] 
+  refine h.trans (congrArg (fun x => (∫⁻b in A∩{b | S b≠∞},
+    ∫⁻z,f z∂(μ (T b) b).restrict (D b)∂W)+x) ?_)
+  apply setLIntegral_congr_fun (hT.measurableSpace_le _ hB)
+  intro b hb
+  have hbt : T b≠∞ := ENNReal.add_ne_top.mpr ⟨hb.2,ENNReal.coe_ne_top⟩
+  have heR : R b=survivingKernel a ha ((S b).toNNReal+t) b := by
+    rw [stoppedSurvivingKernel_apply,survivingKernel_apply]
+    congr 1
+    funext z
+    simp only [stoppedSurvivingGreen,T,shiftStop,ite_eq_right (show S b+(t:ℝ≥0∞)≠∞ from hbt),
+      ENNReal.toNNReal_add hb.2 ENNReal.coe_ne_top,ENNReal.toNNReal_coe]
+  change (∫⁻z,f z∂(R b).restrict (D b))=_
+  rw [heR]
+
+ 
+theorem recoveredRestart_potential (a : ℝ) (ha : 1/4<a)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass a ha s c hc v b)))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {t : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[(afterStopFiltration hS) t] A)
+    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    let ν := recoveredRestart a ha s c hc S
+    (∫⁻b in A,∫⁻z,f z∂ν ∞ b∂W)=
+      (∫⁻b in A,∫⁻z,f z∂ν t b∂W)+
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
+        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
+  intro ν
+  have he (v : ℝ≥0∞) : (∫⁻b in A,∫⁻z,f z∂ν v b∂W)=
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
+        b (S b).toNNReal (recoveredMass a ha s c hc v (restartAt S b))∂W :=
+    lintegral_finite_stop_guard W hS _ A f
+  rw [he,he]
+  exact recoveredMass_restarted_potential a ha W hW s c hc hs hsum hconv hS hA hf
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology CompactlySupported
+namespace SLEExactGauge.MassVersions
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+lemma measure_eq_of_determiningTests (μ ν : Measure X)
+    [μ.Regular] [ν.Regular]
+    (he : ∀ i, (∫x, determiningTests i x ∂μ) = ∫x, determiningTests i x ∂ν) :
+    μ=ν := by
+  let L : ℕ → (C_c(X,ℝ) →ₚ[ℝ] ℝ) := fun n =>
+    if n%2=0 then CompactlySupportedContinuousMap.integralPositiveLinearMap μ
+      else CompactlySupportedContinuousMap.integralPositiveLinearMap ν
+  have hc : TestConverges L := by
+    intro i
+    refine ⟨∫x, determiningTests i x ∂μ,?_⟩
+    have hh (n : ℕ) : L n (determiningTests i) = ∫x, determiningTests i x ∂μ := by
+      dsimp only [L]
+      split_ifs
+      · rfl
+      · exact (he i).symm
+    simpa only [hh] using (tendsto_const_nhds : Tendsto
+      (fun _ : ℕ => ∫x, determiningTests i x ∂μ) atTop (𝓝 (∫x, determiningTests i x ∂μ)))
+  have hmul : Tendsto (fun n : ℕ => 2*n) atTop atTop := by
+    apply tendsto_atTop.mpr
+    intro N
+    filter_upwards [eventually_ge_atTop N] with n hn
+    omega
+  have hadd : Tendsto (fun n : ℕ => 2*n+1) atTop atTop :=
+    (tendsto_add_atTop_nat 1).comp hmul
+  apply Measure.ext_of_integral_eq_on_compactlySupported
+  intro f
+  have heven := (tendsto_radonLimit L hc f).comp hmul
+  have hodd := (tendsto_radonLimit L hc f).comp hadd
+  have hem (n : ℕ) : L (2*n) f=∫x,f x∂μ := by simp [L]
+  have hov (n : ℕ) : L (2*n+1) f=∫x,f x∂ν := by simp [L]
+  simp only [Function.comp_def,hem] at heven
+  simp only [Function.comp_def,hov] at hodd
+  exact (tendsto_nhds_unique tendsto_const_nhds heven).trans
+    (tendsto_nhds_unique tendsto_const_nhds hodd).symm
+
+ 
+theorem ae_all_measure_eq_of_test_eq {Ω I : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (μ ν : I → Ω → Measure X)
+    [∀ t ω, (μ t ω).Regular] [∀ t ω, (ν t ω).Regular]
+    (he : ∀ i, ∀ᵐ ω ∂P, ∀ t, (∫x,determiningTests i x∂μ t ω) =
+      ∫x,determiningTests i x∂ν t ω) :
+    ∀ᵐ ω ∂P, ∀ t, μ t ω=ν t ω := by
+  filter_upwards [ae_all_iff.mpr he] with ω hω t
+  exact measure_eq_of_determiningTests _ _ (fun i => hω i t)
+
+ 
+def positiveTest (f : C_c(X,ℝ)) : C_c(X,ℝ) :=
+  ⟨⟨fun x => max (f x) 0, f.continuous.max continuous_const⟩,
+    f.hasCompactSupport.sup (.zero)⟩
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] [BorelSpace X] in
+lemma positiveTest_nonneg [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    (f : C_c(X,ℝ)) (x : X) : 0≤positiveTest f x :=
+  le_max_right _ _
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] [BorelSpace X] in
+lemma positiveTest_sub [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    (f : C_c(X,ℝ)) (x : X) :
+    positiveTest f x-positiveTest (-f) x=f x := by
+  change max (f x) 0-max (-(f x)) 0=f x
+  exact posPart_sub_negPart (f x)
+
+lemma integral_positiveTest_sub (f : C_c(X,ℝ)) (μ : Measure X)
+    [IsFiniteMeasureOnCompacts μ] :
+    (∫x,positiveTest f x∂μ)-(∫x,positiveTest (-f) x∂μ)=∫x,f x∂μ := by
+  rw [← integral_sub (positiveTest f).integrable (positiveTest (-f)).integrable]
+  simp only [positiveTest_sub]
+
+end SLEExactGauge.MassVersions
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.MeshEnergy
+
+ 
+def grid (T : ℝ≥0) (n i : ℕ) : ℝ≥0 := T * i / ((n:ℝ≥0)+1)
+
+@[simp] lemma grid_zero (T : ℝ≥0) (n : ℕ) : grid T n 0=0 := by simp [grid]
+
+@[simp] lemma grid_last (T : ℝ≥0) (n : ℕ) : grid T n (n+1)=T := by
+  have h : (n:ℝ≥0)+1≠0 := ne_of_gt (by positivity)
+  simp [grid,Nat.cast_add,Nat.cast_one,h]
+
+lemma grid_mono (T : ℝ≥0) (n : ℕ) : Monotone (grid T n) := by
+  intro i j hij
+  dsimp [grid]
+  gcongr
+
+lemma grid_mem (T : ℝ≥0) (n i : ℕ) (hi : i≤n+1) : grid T n i∈Icc 0 T :=
+  ⟨zero_le, (grid_mono T n hi).trans_eq (grid_last T n)⟩
+
+lemma dist_grid_succ (T : ℝ≥0) (n i : ℕ) :
+    dist (grid T n (i+1)) (grid T n i) = (T:ℝ)/((n:ℝ)+1) := by
+  rw [NNReal.dist_eq]
+  simp only [grid,NNReal.coe_div,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_add,
+    NNReal.coe_one,Nat.cast_add,Nat.cast_one]
+  have he : (T:ℝ)*((i:ℝ)+1)/((n:ℝ)+1)-T*i/((n:ℝ)+1)=T/((n:ℝ)+1) := by ring
+  rw [he,abs_of_nonneg (by positivity)]
+
+lemma tendsto_mesh (T : ℝ≥0) :
+    Tendsto (fun n : ℕ => (T:ℝ)/((n:ℝ)+1)) atTop (𝓝 0) := by
+  simpa [div_eq_mul_inv] using
+    (tendsto_const_nhds.mul (tendsto_one_div_add_atTop_nhds_zero_nat :
+      Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝 0)))
+
+ 
+def energy (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) : ℝ :=
+  ∑ i ∈ Finset.range (n+1), (M (grid T n (i+1))-M (grid T n i))^2
+
+lemma energy_nonneg (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) : 0≤energy M T n :=
+  Finset.sum_nonneg fun _ _ => sq_nonneg _
+
+ 
+lemma energy_le (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
+    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) (n : ℕ) :
+    energy M T n ≤ (C T-C 0)^2 := by
+  let A : ℕ → ℝ := fun i => C (grid T n (i+1))-C (grid T n i)
+  have ha (i : ℕ) (hi : i∈Finset.range (n+1)) :
+      |M (grid T n (i+1))-M (grid T n i)|≤A i :=
+    hcontrol _ _ (grid_mono T n (Nat.le_succ _))
+      (grid_mem T n (i+1) (by simpa using Finset.mem_range.mp hi)).2
+  have han (i : ℕ) (hi : i∈Finset.range (n+1)) : 0≤A i :=
+    (abs_nonneg _).trans (ha i hi)
+  calc
+    energy M T n ≤ ∑ i ∈ Finset.range (n+1), (A i)^2 := by
+      apply Finset.sum_le_sum
+      intro i hi
+      calc
+        _ = |M (grid T n (i+1))-M (grid T n i)|^2 := (sq_abs _).symm
+        _ ≤ (A i)^2 := (sq_le_sq₀ (abs_nonneg _) (han i hi)).mpr (ha i hi)
+    _ ≤ (∑ i ∈ Finset.range (n+1), A i)^2 := Finset.sum_sq_le_sq_sum_of_nonneg han
+    _ = (C T-C 0)^2 := by
+      rw [show (∑ i ∈ Finset.range (n+1), A i)=C (grid T n (n+1))-C (grid T n 0) from
+        Finset.sum_range_sub (fun i => C (grid T n i)) (n+1)]
+      simp
+
+ 
+theorem tendsto_energy_zero (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
+    (hc : ContinuousOn C (Icc 0 T))
+    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) :
+    Tendsto (energy M T) atTop (𝓝 0) := by
+  let V := C T-C 0
+  have hv : 0≤V := (abs_nonneg _).trans (hcontrol 0 T (zero_le) le_rfl)
+  have huc := isCompact_Icc.uniformContinuousOn_of_continuous hc
+  apply Metric.tendsto_atTop.mpr
+  intro ε hε
+  have hη : 0<ε/(V+1) := div_pos hε (by linarith)
+  obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuousOn_iff.mp huc (ε/(V+1)) hη
+  obtain ⟨N,hN⟩ := eventually_atTop.mp ((tendsto_mesh T).eventually (gt_mem_nhds hδ))
+  refine ⟨N,fun n hn => ?_⟩
+  let A : ℕ → ℝ := fun i => C (grid T n (i+1))-C (grid T n i)
+  have ha (i : ℕ) (hi : i∈Finset.range (n+1)) :
+      |M (grid T n (i+1))-M (grid T n i)|≤A i :=
+    hcontrol _ _ (grid_mono T n (Nat.le_succ _))
+      (grid_mem T n (i+1) (by simpa using Finset.mem_range.mp hi)).2
+  have hb (i : ℕ) (hi : i∈Finset.range (n+1)) : A i<ε/(V+1) := by
+    have hi' : i<n+1 := Finset.mem_range.mp hi
+    have h := hd _ (grid_mem T n (i+1) hi') _ (grid_mem T n i hi'.le)
+      (by rw [dist_grid_succ]; exact hN n hn)
+    exact (le_abs_self (A i)).trans_lt h
+  have he : energy M T n≤ε/(V+1)*V := by
+    calc
+      _ ≤ ∑ i ∈ Finset.range (n+1), (ε/(V+1))* A i := by
+        apply Finset.sum_le_sum
+        intro i hi
+        have h0 : 0≤A i := (abs_nonneg _).trans (ha i hi)
+        calc
+          _ = |M (grid T n (i+1))-M (grid T n i)|^2 := (sq_abs _).symm
+          _ ≤ (A i)^2 := (sq_le_sq₀ (abs_nonneg _) h0).mpr (ha i hi)
+          _ ≤ ε/(V+1)*A i := by rw [pow_two]; exact mul_le_mul_of_nonneg_right (hb i hi).le h0
+      _ = _ := by
+        rw [← Finset.mul_sum, show (∑ i ∈ Finset.range (n+1), A i)=
+          C (grid T n (n+1))-C (grid T n 0) from
+            Finset.sum_range_sub (fun i => C (grid T n i)) (n+1)]
+        simp [V]
+  rw [Real.dist_eq,sub_zero,abs_of_nonneg (energy_nonneg _ _ _)]
+  refine he.trans_lt ?_
+  apply (div_mul_eq_mul_div _ _ _).trans_lt
+  exact (div_lt_iff₀ (by linarith : 0<V+1)).mpr (by nlinarith)
+
+end SLEExactGauge.MeshEnergy
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.TaylorVariation
+open MeshEnergy
+
+ 
+lemma remainder_le_sq (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
+    {K : ℝ≥0} (hg : LipschitzWith K g) (x y : ℝ) :
+    |f y - f x - g x * (y-x)| ≤ (K:ℝ) * (y-x)^2 := by
+  have h := (convex_uIcc (𝕜 := ℝ) x y).norm_image_sub_le_of_norm_hasDerivWithin_le
+    (f := fun z => f z-g x*z) (f' := fun z => g z-g x)
+    (C := (K:ℝ)*|y-x|)
+    (fun z _ => (by
+      convert! ((hd z).sub ((hasDerivAt_id z).const_mul (g x))).hasDerivWithinAt using 1
+      simp))
+    (fun z hz => (hg.norm_sub_le z x).trans (mul_le_mul_of_nonneg_left
+      (norm_sub_le_of_mem_segment (by simpa [segment_eq_uIcc] using hz)) K.2))
+    left_mem_uIcc right_mem_uIcc
+  have he : f y-g x*y-(f x-g x*x)=f y-f x-g x*(y-x) := by ring
+  calc
+    _ ≤ (K:ℝ) * |y-x| * |y-x| := by simpa only [Real.norm_eq_abs,he] using h
+    _ = _ := by rw [mul_assoc, ← pow_two, sq_abs]
+
+lemma lipschitz_of_deriv_bound (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
+    {L : ℝ≥0} (hg : ∀ x, |g x|≤L) : LipschitzWith L f := by
+  apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hd x).differentiableAt)
+  intro x
+  change |deriv f x|≤(L:ℝ)
+  rw [(hd x).deriv]
+  exact hg x
+
+lemma remainder_le_abs (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
+    {L : ℝ≥0} (hg : ∀ x, |g x|≤L) (x y : ℝ) :
+    |f y-f x-g x*(y-x)|≤(2*(L:ℝ))*|y-x| := by
+  have hf := (lipschitz_of_deriv_bound f g hd hg).norm_sub_le y x
+  calc
+    _ ≤ |f y-f x|+|g x*(y-x)| := abs_sub _ _
+    _ ≤ L*|y-x|+L*|y-x| := by
+      rw [abs_mul]
+      exact add_le_add hf (mul_le_mul_of_nonneg_right (hg x) (abs_nonneg _))
+    _ = _ := by ring
+
+ 
+def error (f g : ℝ → ℝ) (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) : ℝ :=
+  f (M T)-f (M 0) - ∑ i ∈ Finset.range (n+1),
+    g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))
+
+lemma error_eq_sum (f g : ℝ → ℝ) (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) :
+    error f g M T n = ∑ i ∈ Finset.range (n+1),
+      (f (M (grid T n (i+1)))-f (M (grid T n i)) -
+        g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))) := by
+  rw [Finset.sum_sub_distrib, Finset.sum_range_sub (fun i => f (M (grid T n i))) (n+1)]
+  simp [error]
+
+lemma abs_error_le_energy (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
+    {K : ℝ≥0} (hg : LipschitzWith K g) (M : ℝ≥0 → ℝ) (T : ℝ≥0) (n : ℕ) :
+    |error f g M T n|≤(K:ℝ)*energy M T n := by
+  rw [error_eq_sum]
+  calc
+    _ ≤ ∑ i ∈ Finset.range (n+1), |f (M (grid T n (i+1)))-f (M (grid T n i)) -
+        g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ i ∈ Finset.range (n+1), (K:ℝ) * (M (grid T n (i+1))-M (grid T n i))^2 :=
+      Finset.sum_le_sum (fun i _ => remainder_le_sq f g hd hg _ _)
+    _ = _ := by rw [← Finset.mul_sum]; rfl
+
+lemma abs_error_le_variation (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
+    {L : ℝ≥0} (hg : ∀ x, |g x|≤L) (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
+    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) (n : ℕ) :
+    |error f g M T n|≤(2*(L:ℝ))*(C T-C 0) := by
+  rw [error_eq_sum]
+  calc
+    _ ≤ ∑ i ∈ Finset.range (n+1), |f (M (grid T n (i+1)))-f (M (grid T n i)) -
+        g (M (grid T n i)) * (M (grid T n (i+1))-M (grid T n i))| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ i ∈ Finset.range (n+1), (2*(L:ℝ)) * (C (grid T n (i+1))-C (grid T n i)) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      exact (remainder_le_abs f g hd hg _ _).trans (mul_le_mul_of_nonneg_left
+        (hcontrol _ _ (grid_mono T n (Nat.le_succ _))
+          (grid_mem T n (i+1) (by simpa using Finset.mem_range.mp hi)).2) (by positivity))
+    _ = _ := by
+      rw [← Finset.mul_sum, Finset.sum_range_sub (fun i => C (grid T n i)) (n+1)]
+      simp
+
+lemma tendsto_error_zero (f g : ℝ → ℝ) (hd : ∀ x, HasDerivAt f (g x) x)
+    {K : ℝ≥0} (hg : LipschitzWith K g) (M C : ℝ≥0 → ℝ) (T : ℝ≥0)
+    (hc : ContinuousOn C (Icc 0 T))
+    (hcontrol : ∀ s t, s≤t → t≤T → |M t-M s|≤C t-C s) :
+    Tendsto (error f g M T) atTop (𝓝 0) := by
+  apply squeeze_zero_norm (fun n => ?_)
+    (show Tendsto (fun n => (K:ℝ)*energy M T n) atTop (𝓝 0) by
+      simpa using (tendsto_energy_zero M C T hc hcontrol).const_mul (K:ℝ))
+  exact abs_error_le_energy f g hd hg M T n
+
+ 
+
+def detector (x : ℝ) : ℝ := Real.sqrt (1+x^2)-1
+
+def detectorDeriv (x : ℝ) : ℝ := x / Real.sqrt (1+x^2)
+
+lemma one_le_root (x : ℝ) : 1≤Real.sqrt (1+x^2) :=
+  Real.one_le_sqrt.mpr (by nlinarith [sq_nonneg x])
+
+lemma hasDerivAt_root (x : ℝ) : HasDerivAt (fun x : ℝ => Real.sqrt (1+x^2))
+    (x/Real.sqrt (1+x^2)) x := by
+  have h : 1+x^2≠0 := ne_of_gt (by positivity)
+  convert! (((hasDerivAt_id x).pow 2).const_add 1).sqrt h using 1
+  dsimp
+  ring
+
+lemma hasDerivAt_detector (x : ℝ) : HasDerivAt detector (detectorDeriv x) x :=
+  (hasDerivAt_root x).sub_const 1
+
+lemma hasDerivAt_detectorDeriv (x : ℝ) :
+    HasDerivAt detectorDeriv (1/(Real.sqrt (1+x^2))^3) x := by
+  have hp : 0<Real.sqrt (1+x^2) := lt_of_lt_of_le zero_lt_one (one_le_root x)
+  have hs := Real.sq_sqrt (show 0≤1+x^2 by positivity)
+  have hx : HasDerivAt detectorDeriv
+      ((Real.sqrt (1+x^2)-x*(x/Real.sqrt (1+x^2)))/(Real.sqrt (1+x^2))^2) x := by
+    convert! (hasDerivAt_id x).div (hasDerivAt_root x) (ne_of_gt hp) using 1
+    simp
+  convert! hx using 1
+  field_simp
+  nlinarith [hs]
+
+lemma abs_detectorDeriv_le (x : ℝ) : |detectorDeriv x|≤1 := by
+  have hp : 0<Real.sqrt (1+x^2) := lt_of_lt_of_le zero_lt_one (one_le_root x)
+  have hs := Real.sq_sqrt (show 0≤1+x^2 by positivity)
+  rw [detectorDeriv,abs_div,abs_of_pos hp,div_le_iff₀ hp,one_mul]
+  nlinarith [sq_abs x,abs_nonneg x]
+
+lemma lipschitz_detector : LipschitzWith 1 detector :=
+  lipschitz_of_deriv_bound detector detectorDeriv hasDerivAt_detector abs_detectorDeriv_le
+
+lemma lipschitz_detectorDeriv : LipschitzWith 1 detectorDeriv := by
+  apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hasDerivAt_detectorDeriv x).differentiableAt)
+  intro x
+  change |deriv detectorDeriv x|≤(1:ℝ)
+  rw [(hasDerivAt_detectorDeriv x).deriv,abs_of_nonneg (by positivity)]
+  have h := one_le_root x
+  exact (div_le_one (by positivity)).mpr (one_le_pow₀ h)
+
+@[simp] lemma detector_zero : detector 0=0 := by simp [detector]
+
+lemma detector_nonneg (x : ℝ) : 0≤detector x := sub_nonneg.mpr (one_le_root x)
+
+@[simp] lemma detector_eq_zero_iff (x : ℝ) : detector x=0 ↔ x=0 := by
+  constructor
+  · intro h
+    have hs := Real.sq_sqrt (show 0≤1+x^2 by positivity)
+    have he : Real.sqrt (1+x^2)=1 := sub_eq_zero.mp h
+    rw [he] at hs
+    nlinarith [sq_nonneg x]
+  · rintro rfl
+    exact detector_zero
+
+lemma abs_detector_le (x : ℝ) : |detector x|≤|x| := by
+  simpa using lipschitz_detector.norm_sub_le x 0
+
+end SLEExactGauge.TaylorVariation
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ProductGenerators.lean b/lean/OAI/Probability/SLEGauge/ProductGenerators.lean
index fc724f40cb64405016585dc1e3f40727841f17a6..e69ef826cc29ec95962b81ee2255a9d140612a27 100644
--- a/lean/OAI/Probability/SLEGauge/ProductGenerators.lean
+++ b/lean/OAI/Probability/SLEGauge/ProductGenerators.lean
@@ -1,456 +1,456 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialFactors
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-namespace ExactSLECalculus
-variable {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-lemma normalized_product_generator (s : Finset ι) (f : ι → E → ℝ) (x A e : E)
-    (hf : ∀ i ∈ s, ContDiffAt ℝ 2 (f i) x) (hp : ∀ i ∈ s, f i x ≠ 0) :
-    N (fun y => ∏ i ∈ s, f i y) e x / (∏ i ∈ s, f i x) = ∑ i ∈ s, N (f i) e x / f i x ∧
-    L (fun y => ∏ i ∈ s, f i y) A e x / (∏ i ∈ s, f i x) =
-      (∑ i ∈ s, L (f i) A e x / f i x) +
-      ((∑ i ∈ s, N (f i) e x / f i x)^2 - ∑ i ∈ s, (N (f i) e x / f i x)^2)/2 := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simp
-  | @insert i s hi ih =>
-    have hfi := hf i (Finset.mem_insert_self _ _)
-    have hfs : ∀ j∈s, ContDiffAt ℝ 2 (f j) x := fun j hj => hf j (Finset.mem_insert_of_mem hj)
-    have hpi := hp i (Finset.mem_insert_self _ _)
-    have hps : ∀ j∈s, f j x ≠ 0 := fun j hj => hp j (Finset.mem_insert_of_mem hj)
-    have hs := contDiffAt_prod hfs
-    have hnz : (∏ j∈s, f j x) ≠ 0 := Finset.prod_ne_zero_iff.mpr hps
-    obtain ⟨ihN,ihL⟩ := ih hfs hps
-    simp only [Finset.prod_insert hi,Finset.sum_insert hi]
-    rw [N_mul (hfi.differentiableAt (by norm_num)) (hs.differentiableAt (by norm_num)),L_mul hfi hs]
-    constructor
-    · calc
-        _ = N (f i) e x / f i x + N (fun y => ∏j∈s,f j y) e x / (∏j∈s,f j x) := by field_simp; ring
-        _ = _ := by rw [ihN]
-    · have he : (f i x * L (fun y => ∏j∈s,f j y) A e x +
-          (∏j∈s,f j x)*L (f i) A e x + N (f i) e x*N (fun y => ∏j∈s,f j y) e x) /
-          (f i x*(∏j∈s,f j x)) =
-          L (f i) A e x/f i x + L (fun y => ∏j∈s,f j y) A e x/(∏j∈s,f j x) +
-          (N (f i) e x/f i x)*(N (fun y => ∏j∈s,f j y) e x/(∏j∈s,f j x)) := by field_simp; ring
-      rw [he,ihN,ihL]
-      ring
-end ExactSLECalculus
-
-namespace ExactSLECalculus
-variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [NormedAddCommGroup F] [NormedSpace ℝ F]
-lemma N_comp_clm (c : E →L[ℝ] F) {f : F → ℝ} {x e : E}
-    (hf : DifferentiableAt ℝ f (c x)) :
-    N (f ∘ c) e x = N f (c e) (c x) := by
-  rw [N,(hf.hasFDerivAt.comp x c.hasFDerivAt).fderiv]
-  rfl
-lemma L_comp_clm (c : E →L[ℝ] F) {f : F → ℝ} {x A e : E}
-    (hf : ContDiffAt ℝ 2 f (c x)) :
-    L (f ∘ c) A e x = L f (c A) (c e) (c x) := by
-  have hfd := hf.differentiableAt (by norm_num)
-  have heq : N (f ∘ c) e =ᶠ[𝓝 x] (N f (c e) ∘ c) := by
-    filter_upwards [c.continuous.continuousAt.preimage_mem_nhds (hf.eventually (by norm_num))] with y hy
-    change ContDiffAt ℝ 2 f (c y) at hy
-    exact N_comp_clm c (hy.differentiableAt (by norm_num))
-  unfold L
-  rw [N,heq.fderiv_eq,((N_differentiableAt (e := c e) hf).hasFDerivAt.comp x c.hasFDerivAt).fderiv,
-    (hfd.hasFDerivAt.comp x c.hasFDerivAt).fderiv]
-  rfl
-end ExactSLECalculus
-
-namespace ExactSLEFiniteTilt
-lemma pair_noise_form {a δ₁ δ₂ D₁ D₂ V₁ V₂ : ℝ}
-    (hδ₁ : 0<δ₁) (hδ₂ : 0<δ₂) (hD₁ : 0≤D₁) (hD₂ : 0≤D₂)
-    (hV₁ : 0<V₁) (hV₂ : 0<V₂) :
-    ((regP a+regQ a)*3^(regEta a)*δ₁^(-1-regEta a)*D₁^(regEta a)*V₁)*
-      ((regP a+regQ a)*3^(regEta a)*δ₂^(-1-regEta a)*D₂^(regEta a)*V₂) =
-    regC2 a*(δ₁*δ₂)^(-1-regEta a)*(D₁*V₁^2)^(regEta a)*(D₂*V₂^2)^(regEta a)*
-      (V₁*V₂)^(1-2*regEta a) := by
-  have hv₁ : V₁^(2*regEta a)*V₁^(1-2*regEta a)=V₁ := by
-    rw [←Real.rpow_add hV₁,show 2*regEta a+(1-2*regEta a)=1 by ring,Real.rpow_one]
-  have hv₂ : V₂^(2*regEta a)*V₂^(1-2*regEta a)=V₂ := by
-    rw [←Real.rpow_add hV₂,show 2*regEta a+(1-2*regEta a)=1 by ring,Real.rpow_one]
-  have h3 : (3:ℝ)^(2*regEta a)=3^(regEta a)*3^(regEta a) := by
-    rw [show 2*regEta a=regEta a+regEta a by ring,Real.rpow_add (by norm_num : (0:ℝ)<3)]
-  symm
-  rw [regC2,Real.mul_rpow hδ₁.le hδ₂.le,Real.mul_rpow hD₁ (sq_nonneg V₁),
-    Real.mul_rpow hD₂ (sq_nonneg V₂),Real.mul_rpow hV₁.le hV₂.le,
-    ←Real.rpow_natCast_mul hV₁.le 2 (regEta a),←Real.rpow_natCast_mul hV₂.le 2 (regEta a),h3]
-  calc
-    _ = ((regP a+regQ a)*3^(regEta a)*δ₁^(-1-regEta a)*D₁^(regEta a))*
-        ((regP a+regQ a)*3^(regEta a)*δ₂^(-1-regEta a)*D₂^(regEta a))*
-        (V₁^(2*regEta a)*V₁^(1-2*regEta a))*(V₂^(2*regEta a)*V₂^(1-2*regEta a)) := by ring_nf
-    _ = _ := by rw [hv₁,hv₂]; ring
-end ExactSLEFiniteTilt
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt
-lemma actual_noise_pair {a ε cap₁ cap₂ σ₁ σ₂ off₁ off₂ : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (v₁ v₂ : State) (hY₁ : 0<Y v₁) (hY₂ : 0<Y v₂)
-    (hδ₁ : physicalDelta a cap₁ σ₁ off₁ v₁≤1) (hδ₂ : physicalDelta a cap₂ σ₂ off₂ v₂≤1)
-    (hσ₁ : |σ₁|≤regBeta a) (hσ₂ : |σ₂|≤regBeta a) :
-    (N (physicalRegularized a cap₁ σ₁ off₁) noise v₁/physicalRegularized a cap₁ σ₁ off₁ v₁)*
-      (N (physicalRegularized a cap₂ σ₂ off₂) noise v₂/physicalRegularized a cap₂ σ₂ off₂ v₂) ≤
-    ε*(regD a (physicalDelta a cap₁ σ₁ off₁ v₁) (physicalTheta v₁)*(physicalV v₁)^2 +
-      regD a (physicalDelta a cap₂ σ₂ off₂ v₂) (physicalTheta v₂)*(physicalV v₂)^2) +
-    regYoungC a ε*(physicalDelta a cap₁ σ₁ off₁ v₁*physicalDelta a cap₂ σ₂ off₂ v₂)^(-regM a)*
-      physicalV v₁*physicalV v₂ := by
-  have hf₁ := actual_regularized_factor ha v₁ hY₁ hδ₁ hσ₁
-  have hf₂ := actual_regularized_factor ha v₂ hY₂ hδ₂ hσ₂
-  have hdp₁ : 0<physicalDelta a cap₁ σ₁ off₁ v₁ := Real.exp_pos _
-  have hdp₂ : 0<physicalDelta a cap₂ σ₂ off₂ v₂ := Real.exp_pos _
-  have hV₁ : 0<physicalV v₁ := div_pos hY₁ (Q_pos hY₁)
-  have hV₂ : 0<physicalV v₂ := div_pos hY₂ (Q_pos hY₂)
-  have hs₁ := physicalTheta_mem v₁
-  have hs₂ := physicalTheta_mem v₂
-  have hD₁ : 0≤regD a (physicalDelta a cap₁ σ₁ off₁ v₁) (physicalTheta v₁) := by
-    dsimp only [regD]
-    exact mul_nonneg (div_nonneg hdp₁.le ((regFactor_bounds ha hdp₁ hδ₁
-      (Real.sin_pos_of_pos_of_lt_pi hs₁.1 hs₁.2)).1.le)) (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hs₁.1 hs₁.2).le _)
-  have hD₂ : 0≤regD a (physicalDelta a cap₂ σ₂ off₂ v₂) (physicalTheta v₂) := by
-    dsimp only [regD]
-    exact mul_nonneg (div_nonneg hdp₂.le ((regFactor_bounds ha hdp₂ hδ₂
-      (Real.sin_pos_of_pos_of_lt_pi hs₂.1 hs₂.2)).1.le)) (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hs₂.1 hs₂.2).le _)
-  have hb₁ := hf₁.2.2
-  have hb₂ := hf₂.2.2
-  have hn₁ : 0≤|N (physicalRegularized a cap₁ σ₁ off₁) noise v₁| /
-      physicalRegularized a cap₁ σ₁ off₁ v₁ := div_nonneg (abs_nonneg _) hf₁.1.le
-  have hn₂ : 0≤|N (physicalRegularized a cap₂ σ₂ off₂) noise v₂| /
-      physicalRegularized a cap₂ σ₂ off₂ v₂ := div_nonneg (abs_nonneg _) hf₂.1.le
-  have hp := mul_le_mul hb₁ hb₂ hn₂ (hn₁.trans hb₁)
-  have habs := le_abs_self ((N (physicalRegularized a cap₁ σ₁ off₁) noise v₁ /
-      physicalRegularized a cap₁ σ₁ off₁ v₁)*(N (physicalRegularized a cap₂ σ₂ off₂) noise v₂ /
-      physicalRegularized a cap₂ σ₂ off₂ v₂))
-  rw [abs_mul,abs_div,abs_div,abs_of_pos hf₁.1,abs_of_pos hf₂.1] at habs
-  apply (habs.trans hp).trans
-  rw [pair_noise_form hdp₁ hdp₂ hD₁ hD₂ hV₁ hV₂]
-  exact regularized_pair_absorption ha hε hdp₁ hdp₂ hD₁ hD₂ hV₁.le hV₂.le
-end ExactSLEGreen
-
-namespace ExactSLECalculus
-variable {ι : Type*} [DecidableEq ι]
-lemma sum_square_off_diagonal (s : Finset ι) (b : ι → ℝ) :
-    (∑i∈s,b i)^2-(∑i∈s,(b i)^2) = ∑i∈s,∑j∈s.erase i,b i*b j := by
-  have h : ∑i∈s,∑j∈s.erase i,b i*b j = ∑i∈s, ((∑j∈s,b i*b j)-b i*b i) := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    exact Finset.sum_erase_eq_sub hi
-  rw [h,Finset.sum_sub_distrib,←Finset.sum_mul_sum]
-  simp only [pow_two]
-lemma sum_pair_endpoints (s : Finset ι) (X : ι → ℝ) :
-    (∑i∈s,∑j∈s.erase i,(X i+X j)) = 2*((s.card:ℝ)-1)*(∑i∈s,X i) := by
-  have h : (∑i∈s,∑j∈s.erase i,(X i+X j)) =
-      ∑i∈s,((∑j∈s,(X i+X j))-(X i+X i)) := by
-    apply Finset.sum_congr rfl
-    intro i hi
-    exact Finset.sum_erase_eq_sub hi
-  rw [h]
-  simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,Finset.sum_const,nsmul_eq_mul]
-  rw [←Finset.mul_sum]
-  ring
-
-lemma sum_generator_absorption (s : Finset ι) (l b X R : ι → ℝ) (Y : ι → ι → ℝ)
-    (c C ε : ℝ) (hX : ∀i∈s,0≤X i) (hsize : ε*((s.card:ℝ)-1)≤c)
-    (hl : ∀i∈s,l i≤R i-c*X i)
-    (hb : ∀i∈s,∀j∈s.erase i,b i*b j ≤ ε*(X i+X j)+C*Y i j) :
-    (∑i∈s,l i)+((∑i∈s,b i)^2-∑i∈s,(b i)^2)/2 ≤
-      (∑i∈s,R i)+C/2*(∑i∈s,∑j∈s.erase i,Y i j) := by
-  have hl' := Finset.sum_le_sum hl
-  have hb' : (∑i∈s,∑j∈s.erase i,b i*b j) ≤
-      ∑i∈s,∑j∈s.erase i,(ε*(X i+X j)+C*Y i j) := by
-    apply Finset.sum_le_sum
-    intro i hi
-    exact Finset.sum_le_sum (hb i hi)
-  have he : (∑i∈s,∑j∈s.erase i,(ε*(X i+X j)+C*Y i j)) =
-      ε*(∑i∈s,∑j∈s.erase i,(X i+X j))+C*(∑i∈s,∑j∈s.erase i,Y i j) := by
-    simp only [Finset.sum_add_distrib,←Finset.mul_sum]
-  rw [he,sum_pair_endpoints s X] at hb'
-  simp only [Finset.sum_sub_distrib,←Finset.mul_sum] at hl'
-  have hx : 0≤∑i∈s,X i := Finset.sum_nonneg hX
-  have hsize' := mul_le_mul_of_nonneg_right hsize hx
-  rw [sum_square_off_diagonal]
-  nlinarith only [hl',hb',hsize']
-end ExactSLECalculus
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt
-noncomputable def multiDrift {m : ℕ} (a : ℝ) (v : Fin m → State) : Fin m → State :=
-  fun i => physicalDrift a (v i)
-def multiNoise (m : ℕ) : Fin m → State := fun _ => noise
-noncomputable def regularizedProduct {m : ℕ} (s : Finset (Fin m))
-    (a : ℝ) (cap σ off : Fin m → ℝ) (v : Fin m → State) : ℝ :=
-  ∏i∈s, physicalRegularized a (cap i) (σ i) (off i) (v i)
- 
-theorem regularized_product_drift {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (cap σ off : Fin m → ℝ) (v : Fin m → State)
-    (hY : ∀ i∈s, 0<Y (v i))
-    (hδ : ∀ i∈s, physicalDelta a (cap i) (σ i) (off i) (v i)≤1)
-    (hσ : ∀ i∈s, |σ i|≤regBeta a) :
-    0<regularizedProduct s a cap σ off v ∧
-    L (regularizedProduct s a cap σ off) (multiDrift a v) (multiNoise m) v /
-      regularizedProduct s a cap σ off v ≤
-      regC a*(∑i∈s,physicalDelta a (cap i) (σ i) (off i) (v i)*(physicalV (v i))^2) +
-      regYoungC a ε/2*(∑i∈s,∑j∈s.erase i,
-        (physicalDelta a (cap i) (σ i) (off i) (v i)*
-          physicalDelta a (cap j) (σ j) (off j) (v j))^(-regM a)*
-          physicalV (v i)*physicalV (v j)) := by
-
-  classical
-  let F : Fin m → (Fin m → State) → ℝ :=
-    fun i w => physicalRegularized a (cap i) (σ i) (off i) (w i)
-  have hF (i) (hi : i∈s) : ContDiffAt ℝ 2 (F i) v := by
-    exact (physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)).comp v
-      (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff.contDiffAt
-  have hpos (i) (hi : i∈s) : 0<F i v :=
-    (actual_regularized_factor ha (v i) (hY i hi) (hδ i hi) (hσ i hi)).1
-  have hL (i) (hi : i∈s) : L (F i) (multiDrift a v) (multiNoise m) v =
-      L (physicalRegularized a (cap i) (σ i) (off i)) (physicalDrift a (v i)) noise (v i) := by
-    exact L_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)
-      (physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi))
-  have hN (i) (hi : i∈s) : N (F i) (multiNoise m) v =
-      N (physicalRegularized a (cap i) (σ i) (off i)) noise (v i) := by
-    exact N_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)
-      ((physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)).differentiableAt (by norm_num))
-  have hD (i) (hi : i∈s) :
-      0≤regD a (physicalDelta a (cap i) (σ i) (off i) (v i)) (physicalTheta (v i)) := by
-    have hθ := physicalTheta_mem (v i)
-    have hsin := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-    have hd : 0<physicalDelta a (cap i) (σ i) (off i) (v i) := Real.exp_pos _
-    dsimp only [regD]
-    exact mul_nonneg (div_nonneg hd.le (regFactor_bounds ha hd (hδ i hi) hsin).1.le)
-      (Real.rpow_nonneg hsin.le _)
-  constructor
-  · exact Finset.prod_pos hpos
-  · change L (fun w => ∏i∈s,F i w) (multiDrift a v) (multiNoise m) v / (∏i∈s,F i v) ≤ _
-    rw [(normalized_product_generator s F v (multiDrift a v) (multiNoise m) hF
-      (fun i hi => (hpos i hi).ne')).2]
-    have habs := sum_generator_absorption s
-      (fun i => L (F i) (multiDrift a v) (multiNoise m) v / F i v)
-      (fun i => N (F i) (multiNoise m) v / F i v)
-      (fun i => regD a (physicalDelta a (cap i) (σ i) (off i) (v i)) (physicalTheta (v i))*(physicalV (v i))^2)
-      (fun i => regC a*physicalDelta a (cap i) (σ i) (off i) (v i)*(physicalV (v i))^2)
-      (fun i j => (physicalDelta a (cap i) (σ i) (off i) (v i)*
-        physicalDelta a (cap j) (σ j) (off j) (v j))^(-regM a)*physicalV (v i)*physicalV (v j))
-      (regK a/2) (regYoungC a ε) ε
-      (fun i hi => mul_nonneg (hD i hi) (sq_nonneg _)) hsize
-    refine le_trans (habs ?_ ?_) ?_
-    · intro i hi
-      rw [hL i hi]
-      have h := (actual_regularized_factor ha (v i) (hY i hi) (hδ i hi) (hσ i hi)).2.1
-      convert h using 1
-      ring
-    · intro i hi j hj
-      have hjs := Finset.mem_of_mem_erase hj
-      rw [hN i hi,hN j hjs]
-      have h := actual_noise_pair ha hε (v i) (v j) (hY i hi) (hY j hjs)
-        (hδ i hi) (hδ j hjs) (hσ i hi) (hσ j hjs)
-      convert h using 1
-      ring
-    · apply le_of_eq
-      congr 1
-      simp only [mul_assoc,Finset.mul_sum]
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
-abbrev RegimeState (m : ℕ) := (Fin m → State) × ℝ
-noncomputable def regimeDiscount {m : ℕ} (s : Finset (Fin m))
-    (a ε : ℝ) (cap σ off : Fin m → ℝ) (v : Fin m → State) : ℝ :=
-  regC a*(∑i∈s,physicalDelta a (cap i) (σ i) (off i) (v i)*(physicalV (v i))^2) +
-  regYoungC a ε/2*(∑i∈s,∑j∈s.erase i,
-    (physicalDelta a (cap i) (σ i) (off i) (v i)*
-      physicalDelta a (cap j) (σ j) (off j) (v j))^(-regM a)*
-      physicalV (v i)*physicalV (v j))
-noncomputable def discountedProduct {m : ℕ} (s : Finset (Fin m))
-    (a : ℝ) (cap σ off : Fin m → ℝ) (v : RegimeState m) : ℝ :=
-  Real.exp (-v.2)*regularizedProduct s a cap σ off v.1
-end ExactSLEGreen
-
-namespace ExactSLECalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-lemma discounted_generator {f : E → ℝ} {x A e : E} (r k : ℝ)
-    (hf : ContDiffAt ℝ 2 f x) :
-    L (fun y : E×ℝ => Real.exp (-y.2)*f y.1) (A,k) (e,0) (x,r) =
-      Real.exp (-r)*(L f A e x-k*f x) := by
-  let g : E×ℝ → ℝ := fun y => Real.exp (-y.2)
-  have hd (y : E×ℝ) : HasFDerivAt g
-      (Real.exp (-y.2) • (-ContinuousLinearMap.snd ℝ E ℝ)) y := by
-    exact (hasFDerivAt_snd (𝕜 := ℝ) (p := y)).neg.exp
-  have hn (y : E×ℝ) : N g (e,0) y = 0 := by
-    rw [N,(hd y).fderiv]
-    simp
-  have hl : L g (A,k) (e,0) (x,r) = -Real.exp (-r)*k := by
-    simp only [L,show N g (e,0) = (fun _ => 0) from funext hn,N_const,mul_zero,zero_add]
-    rw [(hd (x,r)).fderiv]
-    simp
-  have hgs : ContDiffAt ℝ 2 g (x,r) := contDiffAt_snd.neg.exp
-  have hfs : ContDiffAt ℝ 2 (fun y : E×ℝ => f y.1) (x,r) := hf.comp (x,r) contDiffAt_fst
-  rw [L_mul hgs hfs]
-  change g (x,r)*L (f ∘ ContinuousLinearMap.fst ℝ E ℝ) (A,k) (e,0) (x,r)+
-    f x*L g (A,k) (e,0) (x,r)+N g (e,0) (x,r)*N (fun y : E×ℝ => f y.1) (e,0) (x,r) = _
-  rw [L_comp_clm _ hf,hl,hn]
-  dsimp [g]
-  ring
-end ExactSLECalculus
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator Metric
-lemma physicalRegularized_smooth_infty (a cap σ off : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalRegularized a cap σ off) v := by
-  have hu : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalClock a cap) v :=
-    ((contDiffAt_const.add J.contDiff.contDiffAt).sub
-      (Y.contDiff.contDiffAt.log hv.ne')).div_const (2*a)
-  have ht : ContDiffAt ℝ (↑(⊤ : ℕ∞)) physicalTheta v :=
-    contDiffAt_const.sub ((X.contDiff.contDiffAt.div Y.contDiff.contDiffAt hv.ne').arctan)
-  have hp := physicalTheta_mem v
-  have hs : Real.sin (physicalTheta v) ≠ 0 := (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2).ne'
-  exact (((contDiffAt_const.mul hu).exp.div_const 3).mul
-    ((ht.sin.rpow_const_of_ne hs).add
-      (((contDiffAt_const.add (contDiffAt_const.mul hu)).exp).mul
-        (contDiffAt_const.add (ht.sin.rpow_const_of_ne hs)))))
-lemma regularizedProduct_smooth_infty {m : ℕ} (s : Finset (Fin m))
-    (a : ℝ) (cap σ off : Fin m → ℝ) (v : Fin m → State)
-    (hY : ∀ i∈s, 0<Y (v i)) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (regularizedProduct s a cap σ off) v := by
-  apply contDiffAt_prod
-  intro i hi
-  exact (physicalRegularized_smooth_infty a (cap i) (σ i) (off i) (v i) (hY i hi)).comp v
-    (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff.contDiffAt
-lemma regularizedProduct_generator_agrees {m : ℕ} (s : Finset (Fin m)) {a : ℝ}
-    (ha : 1/4<a) (cap σ off : Fin m → ℝ) (v B : Fin m → State)
-    (hY : ∀ i∈s, 0<Y (v i))
-    (hδ : ∀ i∈s, physicalDelta a (cap i) (σ i) (off i) (v i)≤1)
-    (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (hB : ∀ i∈s, B i=physicalDrift a (v i)) :
-    L (regularizedProduct s a cap σ off) B (multiNoise m) v =
-      L (regularizedProduct s a cap σ off) (multiDrift a v) (multiNoise m) v := by
-  let F : Fin m → (Fin m → State) → ℝ := fun i w => physicalRegularized a (cap i) (σ i) (off i) (w i)
-  have hF (i) (hi : i∈s) : ContDiffAt ℝ 2 (F i) v :=
-    (physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)).comp v
-      (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff.contDiffAt
-  have hpos (i) (hi : i∈s) : F i v≠0 :=
-    (actual_regularized_factor ha (v i) (hY i hi) (hδ i hi) (hσ i hi)).1.ne'
-  have hP : regularizedProduct s a cap σ off v≠0 := Finset.prod_ne_zero_iff.mpr hpos
-  apply (div_left_inj' hP).mp
-  change L (fun w => ∏ i ∈ s, F i w) B (multiNoise m) v / (∏ i ∈ s, F i v) =
-    L (fun w => ∏ i ∈ s, F i w) (multiDrift a v) (multiNoise m) v / (∏ i ∈ s, F i v)
-  rw [(normalized_product_generator s F v B (multiNoise m) hF hpos).2,
-      (normalized_product_generator s F v (multiDrift a v) (multiNoise m) hF hpos).2]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro i hi
-  congr 1
-  have hf := physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)
-  change L (physicalRegularized a (cap i) (σ i) (off i) ∘ (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)) B _ v = _
-  rw [L_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State) hf]
-  change _ = L (physicalRegularized a (cap i) (σ i) (off i) ∘ (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)) (multiDrift a v) _ v
-  rw [L_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State) hf]
-  change L _ (B i) noise (v i) = L _ (physicalDrift a (v i)) noise (v i)
-  rw [hB i hi]
-lemma discountedProduct_nonpos {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (cap σ off : Fin m → ℝ) (v B : RegimeState m)
-    (hY : ∀ i∈s, 0<Y (v.1 i))
-    (hδ : ∀ i∈s, physicalDelta a (cap i) (σ i) (off i) (v.1 i)≤1)
-    (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (hB : B.2=regimeDiscount s a ε cap σ off v.1 ∧
-      ∀ i∈s, B.1 i=physicalDrift a (v.1 i)) :
-    L (discountedProduct s a cap σ off) B (multiNoise m,0) v ≤ 0 := by
-  have hf : ContDiffAt ℝ 2 (regularizedProduct s a cap σ off) v.1 :=
-    (regularizedProduct_smooth_infty s a cap σ off v.1 hY).of_le (by exact WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))
-  change L (fun y : RegimeState m => Real.exp (-y.2)*regularizedProduct s a cap σ off y.1)
-    (B.1,B.2) (multiNoise m,0) (v.1,v.2) ≤ 0
-  rw [discounted_generator _ _ hf,regularizedProduct_generator_agrees s ha cap σ off v.1 B.1 hY hδ hσ hB.2,hB.1]
-  have h := regularized_product_drift s ha hε hsize cap σ off v.1 hY hδ hσ
-  apply mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le
-  exact sub_nonpos.mpr ((div_le_iff₀ h.1).mp h.2)
-
-noncomputable def regularizedHeightTurn (η : ℝ) (v : State) : ℝ :=
-  Real.smoothTransition (4*Y v/η-1)
-lemma regularizedHeightTurn_smooth (η : ℝ) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (regularizedHeightTurn η) :=
-  Real.smoothTransition.contDiff.comp (((contDiff_const.mul Y.contDiff).div_const η).sub contDiff_const)
-lemma regularizedHeightTurn_zero {η : ℝ} (hη : 0<η) (v : State) (hv : Y v≤η/4) :
-    regularizedHeightTurn η v=0 := by
-  apply Real.smoothTransition.zero_of_nonpos
-  rw [sub_nonpos,div_le_iff₀ hη]
-  linarith
-lemma regularizedHeightTurn_one {η : ℝ} (hη : 0<η) (v : State) (hv : η/2≤Y v) :
-    regularizedHeightTurn η v=1 := by
-  apply Real.smoothTransition.one_of_one_le
-  rw [le_sub_iff_add_le,le_div_iff₀ hη]
-  linarith
-lemma height_regularized_smooth (a η cap σ off : ℝ) (hη : 0<η) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (fun v => regularizedHeightTurn η v*physicalRegularized a cap σ off v) := by
-  rw [contDiff_iff_contDiffAt]
-  intro v
-  by_cases hv : 0<Y v
-  · exact (regularizedHeightTurn_smooth η).contDiffAt.mul (physicalRegularized_smooth_infty a cap σ off v hv)
-  · apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
-    have hv' : Y v<η/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
-    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
-    simp only [regularizedHeightTurn_zero hη w hw.le,zero_mul]
-
-def regimeBump (m : ℕ) (N : ℝ≥0) : ContDiffBump (0 : RegimeState m) where
-  rIn := N+1
-  rOut := N+2
-  rIn_pos := by positivity
-  rIn_lt_rOut := by linarith
-noncomputable def regimeCutoff {m : ℕ} (s : Finset (Fin m))
-    (a η : ℝ) (N : ℝ≥0) (cap σ off : Fin m → ℝ) (v : RegimeState m) : ℝ :=
-  regimeBump m N v * (Real.exp (-v.2)*∏i∈s,
-    regularizedHeightTurn η (v.1 i)*physicalRegularized a (cap i) (σ i) (off i) (v.1 i))
-lemma regimeCutoff_smooth {m : ℕ} (s : Finset (Fin m)) (a η : ℝ) (hη : 0<η)
-    (N : ℝ≥0) (cap σ off : Fin m → ℝ) :
-    ContDiff ℝ (↑(⊤ : ℕ∞)) (regimeCutoff s a η N cap σ off) := by
-  apply (regimeBump m N).contDiff.mul
-  apply contDiff_snd.neg.exp.mul
-  apply contDiff_prod
-  intro i hi
-  exact (height_regularized_smooth a η (cap i) (σ i) (off i) hη).comp
-    (((ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff).comp contDiff_fst)
-lemma regimeCutoff_compact {m : ℕ} (s : Finset (Fin m)) (a η : ℝ)
-    (N : ℝ≥0) (cap σ off : Fin m → ℝ) :
-    HasCompactSupport (regimeCutoff s a η N cap σ off) :=
-  (regimeBump m N).hasCompactSupport.mul_right
-lemma regimeCutoff_eq_near {m : ℕ} (s : Finset (Fin m)) (a η : ℝ) (hη : 0<η)
-    (N : ℝ≥0) (cap σ off : Fin m → ℝ) (v : RegimeState m) (hv : ‖v‖≤N)
-    (hY : ∀ i∈s, η≤Y (v.1 i)) :
-    regimeCutoff s a η N cap σ off =ᶠ[𝓝 v] discountedProduct s a cap σ off := by
-  have hb : v∈ball (0 : RegimeState m) (regimeBump m N).rIn := by
-    simp only [mem_ball,dist_zero_right,regimeBump]
-    linarith
-  have hturn : ∀ i∈s, ∀ᶠ w in 𝓝 v, regularizedHeightTurn η (w.1 i)=1 := by
-    intro i hi
-    have hy : η/2<Y (v.1 i) := lt_of_lt_of_le (by linarith) (hY i hi)
-    have hc : Continuous (fun w : RegimeState m => Y (w.1 i)) :=
-      Y.continuous.comp ((continuous_apply i).comp continuous_fst)
-    filter_upwards [hc.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy)] with w hw
-    exact regularizedHeightTurn_one hη (w.1 i) hw.le
-  filter_upwards [(regimeBump m N).eventuallyEq_one_of_mem_ball hb,
-    (Filter.eventually_all_finset s).mpr hturn] with w hw htw
-  simp only [regimeCutoff,hw,Pi.one_apply,one_mul,discountedProduct,regularizedProduct]
-  congr 1
-  apply Finset.prod_congr rfl
-  intro i hi
-  rw [htw i hi,one_mul]
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialFactors
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+namespace ExactSLECalculus
+variable {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+lemma normalized_product_generator (s : Finset ι) (f : ι → E → ℝ) (x A e : E)
+    (hf : ∀ i ∈ s, ContDiffAt ℝ 2 (f i) x) (hp : ∀ i ∈ s, f i x ≠ 0) :
+    N (fun y => ∏ i ∈ s, f i y) e x / (∏ i ∈ s, f i x) = ∑ i ∈ s, N (f i) e x / f i x ∧
+    L (fun y => ∏ i ∈ s, f i y) A e x / (∏ i ∈ s, f i x) =
+      (∑ i ∈ s, L (f i) A e x / f i x) +
+      ((∑ i ∈ s, N (f i) e x / f i x)^2 - ∑ i ∈ s, (N (f i) e x / f i x)^2)/2 := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert i s hi ih =>
+    have hfi := hf i (Finset.mem_insert_self _ _)
+    have hfs : ∀ j∈s, ContDiffAt ℝ 2 (f j) x := fun j hj => hf j (Finset.mem_insert_of_mem hj)
+    have hpi := hp i (Finset.mem_insert_self _ _)
+    have hps : ∀ j∈s, f j x ≠ 0 := fun j hj => hp j (Finset.mem_insert_of_mem hj)
+    have hs := contDiffAt_prod hfs
+    have hnz : (∏ j∈s, f j x) ≠ 0 := Finset.prod_ne_zero_iff.mpr hps
+    obtain ⟨ihN,ihL⟩ := ih hfs hps
+    simp only [Finset.prod_insert hi,Finset.sum_insert hi]
+    rw [N_mul (hfi.differentiableAt (by norm_num)) (hs.differentiableAt (by norm_num)),L_mul hfi hs]
+    constructor
+    · calc
+        _ = N (f i) e x / f i x + N (fun y => ∏j∈s,f j y) e x / (∏j∈s,f j x) := by field_simp; ring
+        _ = _ := by rw [ihN]
+    · have he : (f i x * L (fun y => ∏j∈s,f j y) A e x +
+          (∏j∈s,f j x)*L (f i) A e x + N (f i) e x*N (fun y => ∏j∈s,f j y) e x) /
+          (f i x*(∏j∈s,f j x)) =
+          L (f i) A e x/f i x + L (fun y => ∏j∈s,f j y) A e x/(∏j∈s,f j x) +
+          (N (f i) e x/f i x)*(N (fun y => ∏j∈s,f j y) e x/(∏j∈s,f j x)) := by field_simp; ring
+      rw [he,ihN,ihL]
+      ring
+end ExactSLECalculus
+
+namespace ExactSLECalculus
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+lemma N_comp_clm (c : E →L[ℝ] F) {f : F → ℝ} {x e : E}
+    (hf : DifferentiableAt ℝ f (c x)) :
+    N (f ∘ c) e x = N f (c e) (c x) := by
+  rw [N,(hf.hasFDerivAt.comp x c.hasFDerivAt).fderiv]
+  rfl
+lemma L_comp_clm (c : E →L[ℝ] F) {f : F → ℝ} {x A e : E}
+    (hf : ContDiffAt ℝ 2 f (c x)) :
+    L (f ∘ c) A e x = L f (c A) (c e) (c x) := by
+  have hfd := hf.differentiableAt (by norm_num)
+  have heq : N (f ∘ c) e =ᶠ[𝓝 x] (N f (c e) ∘ c) := by
+    filter_upwards [c.continuous.continuousAt.preimage_mem_nhds (hf.eventually (by norm_num))] with y hy
+    change ContDiffAt ℝ 2 f (c y) at hy
+    exact N_comp_clm c (hy.differentiableAt (by norm_num))
+  unfold L
+  rw [N,heq.fderiv_eq,((N_differentiableAt (e := c e) hf).hasFDerivAt.comp x c.hasFDerivAt).fderiv,
+    (hfd.hasFDerivAt.comp x c.hasFDerivAt).fderiv]
+  rfl
+end ExactSLECalculus
+
+namespace ExactSLEFiniteTilt
+lemma pair_noise_form {a δ₁ δ₂ D₁ D₂ V₁ V₂ : ℝ}
+    (hδ₁ : 0<δ₁) (hδ₂ : 0<δ₂) (hD₁ : 0≤D₁) (hD₂ : 0≤D₂)
+    (hV₁ : 0<V₁) (hV₂ : 0<V₂) :
+    ((regP a+regQ a)*3^(regEta a)*δ₁^(-1-regEta a)*D₁^(regEta a)*V₁)*
+      ((regP a+regQ a)*3^(regEta a)*δ₂^(-1-regEta a)*D₂^(regEta a)*V₂) =
+    regC2 a*(δ₁*δ₂)^(-1-regEta a)*(D₁*V₁^2)^(regEta a)*(D₂*V₂^2)^(regEta a)*
+      (V₁*V₂)^(1-2*regEta a) := by
+  have hv₁ : V₁^(2*regEta a)*V₁^(1-2*regEta a)=V₁ := by
+    rw [←Real.rpow_add hV₁,show 2*regEta a+(1-2*regEta a)=1 by ring,Real.rpow_one]
+  have hv₂ : V₂^(2*regEta a)*V₂^(1-2*regEta a)=V₂ := by
+    rw [←Real.rpow_add hV₂,show 2*regEta a+(1-2*regEta a)=1 by ring,Real.rpow_one]
+  have h3 : (3:ℝ)^(2*regEta a)=3^(regEta a)*3^(regEta a) := by
+    rw [show 2*regEta a=regEta a+regEta a by ring,Real.rpow_add (by norm_num : (0:ℝ)<3)]
+  symm
+  rw [regC2,Real.mul_rpow hδ₁.le hδ₂.le,Real.mul_rpow hD₁ (sq_nonneg V₁),
+    Real.mul_rpow hD₂ (sq_nonneg V₂),Real.mul_rpow hV₁.le hV₂.le,
+    ←Real.rpow_natCast_mul hV₁.le 2 (regEta a),←Real.rpow_natCast_mul hV₂.le 2 (regEta a),h3]
+  calc
+    _ = ((regP a+regQ a)*3^(regEta a)*δ₁^(-1-regEta a)*D₁^(regEta a))*
+        ((regP a+regQ a)*3^(regEta a)*δ₂^(-1-regEta a)*D₂^(regEta a))*
+        (V₁^(2*regEta a)*V₁^(1-2*regEta a))*(V₂^(2*regEta a)*V₂^(1-2*regEta a)) := by ring_nf
+    _ = _ := by rw [hv₁,hv₂]; ring
+end ExactSLEFiniteTilt
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt
+lemma actual_noise_pair {a ε cap₁ cap₂ σ₁ σ₂ off₁ off₂ : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (v₁ v₂ : State) (hY₁ : 0<Y v₁) (hY₂ : 0<Y v₂)
+    (hδ₁ : physicalDelta a cap₁ σ₁ off₁ v₁≤1) (hδ₂ : physicalDelta a cap₂ σ₂ off₂ v₂≤1)
+    (hσ₁ : |σ₁|≤regBeta a) (hσ₂ : |σ₂|≤regBeta a) :
+    (N (physicalRegularized a cap₁ σ₁ off₁) noise v₁/physicalRegularized a cap₁ σ₁ off₁ v₁)*
+      (N (physicalRegularized a cap₂ σ₂ off₂) noise v₂/physicalRegularized a cap₂ σ₂ off₂ v₂) ≤
+    ε*(regD a (physicalDelta a cap₁ σ₁ off₁ v₁) (physicalTheta v₁)*(physicalV v₁)^2 +
+      regD a (physicalDelta a cap₂ σ₂ off₂ v₂) (physicalTheta v₂)*(physicalV v₂)^2) +
+    regYoungC a ε*(physicalDelta a cap₁ σ₁ off₁ v₁*physicalDelta a cap₂ σ₂ off₂ v₂)^(-regM a)*
+      physicalV v₁*physicalV v₂ := by
+  have hf₁ := actual_regularized_factor ha v₁ hY₁ hδ₁ hσ₁
+  have hf₂ := actual_regularized_factor ha v₂ hY₂ hδ₂ hσ₂
+  have hdp₁ : 0<physicalDelta a cap₁ σ₁ off₁ v₁ := Real.exp_pos _
+  have hdp₂ : 0<physicalDelta a cap₂ σ₂ off₂ v₂ := Real.exp_pos _
+  have hV₁ : 0<physicalV v₁ := div_pos hY₁ (Q_pos hY₁)
+  have hV₂ : 0<physicalV v₂ := div_pos hY₂ (Q_pos hY₂)
+  have hs₁ := physicalTheta_mem v₁
+  have hs₂ := physicalTheta_mem v₂
+  have hD₁ : 0≤regD a (physicalDelta a cap₁ σ₁ off₁ v₁) (physicalTheta v₁) := by
+    dsimp only [regD]
+    exact mul_nonneg (div_nonneg hdp₁.le ((regFactor_bounds ha hdp₁ hδ₁
+      (Real.sin_pos_of_pos_of_lt_pi hs₁.1 hs₁.2)).1.le)) (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hs₁.1 hs₁.2).le _)
+  have hD₂ : 0≤regD a (physicalDelta a cap₂ σ₂ off₂ v₂) (physicalTheta v₂) := by
+    dsimp only [regD]
+    exact mul_nonneg (div_nonneg hdp₂.le ((regFactor_bounds ha hdp₂ hδ₂
+      (Real.sin_pos_of_pos_of_lt_pi hs₂.1 hs₂.2)).1.le)) (Real.rpow_nonneg (Real.sin_pos_of_pos_of_lt_pi hs₂.1 hs₂.2).le _)
+  have hb₁ := hf₁.2.2
+  have hb₂ := hf₂.2.2
+  have hn₁ : 0≤|N (physicalRegularized a cap₁ σ₁ off₁) noise v₁| /
+      physicalRegularized a cap₁ σ₁ off₁ v₁ := div_nonneg (abs_nonneg _) hf₁.1.le
+  have hn₂ : 0≤|N (physicalRegularized a cap₂ σ₂ off₂) noise v₂| /
+      physicalRegularized a cap₂ σ₂ off₂ v₂ := div_nonneg (abs_nonneg _) hf₂.1.le
+  have hp := mul_le_mul hb₁ hb₂ hn₂ (hn₁.trans hb₁)
+  have habs := le_abs_self ((N (physicalRegularized a cap₁ σ₁ off₁) noise v₁ /
+      physicalRegularized a cap₁ σ₁ off₁ v₁)*(N (physicalRegularized a cap₂ σ₂ off₂) noise v₂ /
+      physicalRegularized a cap₂ σ₂ off₂ v₂))
+  rw [abs_mul,abs_div,abs_div,abs_of_pos hf₁.1,abs_of_pos hf₂.1] at habs
+  apply (habs.trans hp).trans
+  rw [pair_noise_form hdp₁ hdp₂ hD₁ hD₂ hV₁ hV₂]
+  exact regularized_pair_absorption ha hε hdp₁ hdp₂ hD₁ hD₂ hV₁.le hV₂.le
+end ExactSLEGreen
+
+namespace ExactSLECalculus
+variable {ι : Type*} [DecidableEq ι]
+lemma sum_square_off_diagonal (s : Finset ι) (b : ι → ℝ) :
+    (∑i∈s,b i)^2-(∑i∈s,(b i)^2) = ∑i∈s,∑j∈s.erase i,b i*b j := by
+  have h : ∑i∈s,∑j∈s.erase i,b i*b j = ∑i∈s, ((∑j∈s,b i*b j)-b i*b i) := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    exact Finset.sum_erase_eq_sub hi
+  rw [h,Finset.sum_sub_distrib,←Finset.sum_mul_sum]
+  simp only [pow_two]
+lemma sum_pair_endpoints (s : Finset ι) (X : ι → ℝ) :
+    (∑i∈s,∑j∈s.erase i,(X i+X j)) = 2*((s.card:ℝ)-1)*(∑i∈s,X i) := by
+  have h : (∑i∈s,∑j∈s.erase i,(X i+X j)) =
+      ∑i∈s,((∑j∈s,(X i+X j))-(X i+X i)) := by
+    apply Finset.sum_congr rfl
+    intro i hi
+    exact Finset.sum_erase_eq_sub hi
+  rw [h]
+  simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,Finset.sum_const,nsmul_eq_mul]
+  rw [←Finset.mul_sum]
+  ring
+
+lemma sum_generator_absorption (s : Finset ι) (l b X R : ι → ℝ) (Y : ι → ι → ℝ)
+    (c C ε : ℝ) (hX : ∀i∈s,0≤X i) (hsize : ε*((s.card:ℝ)-1)≤c)
+    (hl : ∀i∈s,l i≤R i-c*X i)
+    (hb : ∀i∈s,∀j∈s.erase i,b i*b j ≤ ε*(X i+X j)+C*Y i j) :
+    (∑i∈s,l i)+((∑i∈s,b i)^2-∑i∈s,(b i)^2)/2 ≤
+      (∑i∈s,R i)+C/2*(∑i∈s,∑j∈s.erase i,Y i j) := by
+  have hl' := Finset.sum_le_sum hl
+  have hb' : (∑i∈s,∑j∈s.erase i,b i*b j) ≤
+      ∑i∈s,∑j∈s.erase i,(ε*(X i+X j)+C*Y i j) := by
+    apply Finset.sum_le_sum
+    intro i hi
+    exact Finset.sum_le_sum (hb i hi)
+  have he : (∑i∈s,∑j∈s.erase i,(ε*(X i+X j)+C*Y i j)) =
+      ε*(∑i∈s,∑j∈s.erase i,(X i+X j))+C*(∑i∈s,∑j∈s.erase i,Y i j) := by
+    simp only [Finset.sum_add_distrib,←Finset.mul_sum]
+  rw [he,sum_pair_endpoints s X] at hb'
+  simp only [Finset.sum_sub_distrib,←Finset.mul_sum] at hl'
+  have hx : 0≤∑i∈s,X i := Finset.sum_nonneg hX
+  have hsize' := mul_le_mul_of_nonneg_right hsize hx
+  rw [sum_square_off_diagonal]
+  nlinarith only [hl',hb',hsize']
+end ExactSLECalculus
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt
+noncomputable def multiDrift {m : ℕ} (a : ℝ) (v : Fin m → State) : Fin m → State :=
+  fun i => physicalDrift a (v i)
+def multiNoise (m : ℕ) : Fin m → State := fun _ => noise
+noncomputable def regularizedProduct {m : ℕ} (s : Finset (Fin m))
+    (a : ℝ) (cap σ off : Fin m → ℝ) (v : Fin m → State) : ℝ :=
+  ∏i∈s, physicalRegularized a (cap i) (σ i) (off i) (v i)
+ 
+theorem regularized_product_drift {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (cap σ off : Fin m → ℝ) (v : Fin m → State)
+    (hY : ∀ i∈s, 0<Y (v i))
+    (hδ : ∀ i∈s, physicalDelta a (cap i) (σ i) (off i) (v i)≤1)
+    (hσ : ∀ i∈s, |σ i|≤regBeta a) :
+    0<regularizedProduct s a cap σ off v ∧
+    L (regularizedProduct s a cap σ off) (multiDrift a v) (multiNoise m) v /
+      regularizedProduct s a cap σ off v ≤
+      regC a*(∑i∈s,physicalDelta a (cap i) (σ i) (off i) (v i)*(physicalV (v i))^2) +
+      regYoungC a ε/2*(∑i∈s,∑j∈s.erase i,
+        (physicalDelta a (cap i) (σ i) (off i) (v i)*
+          physicalDelta a (cap j) (σ j) (off j) (v j))^(-regM a)*
+          physicalV (v i)*physicalV (v j)) := by
+
+  classical
+  let F : Fin m → (Fin m → State) → ℝ :=
+    fun i w => physicalRegularized a (cap i) (σ i) (off i) (w i)
+  have hF (i) (hi : i∈s) : ContDiffAt ℝ 2 (F i) v := by
+    exact (physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)).comp v
+      (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff.contDiffAt
+  have hpos (i) (hi : i∈s) : 0<F i v :=
+    (actual_regularized_factor ha (v i) (hY i hi) (hδ i hi) (hσ i hi)).1
+  have hL (i) (hi : i∈s) : L (F i) (multiDrift a v) (multiNoise m) v =
+      L (physicalRegularized a (cap i) (σ i) (off i)) (physicalDrift a (v i)) noise (v i) := by
+    exact L_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)
+      (physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi))
+  have hN (i) (hi : i∈s) : N (F i) (multiNoise m) v =
+      N (physicalRegularized a (cap i) (σ i) (off i)) noise (v i) := by
+    exact N_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)
+      ((physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)).differentiableAt (by norm_num))
+  have hD (i) (hi : i∈s) :
+      0≤regD a (physicalDelta a (cap i) (σ i) (off i) (v i)) (physicalTheta (v i)) := by
+    have hθ := physicalTheta_mem (v i)
+    have hsin := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+    have hd : 0<physicalDelta a (cap i) (σ i) (off i) (v i) := Real.exp_pos _
+    dsimp only [regD]
+    exact mul_nonneg (div_nonneg hd.le (regFactor_bounds ha hd (hδ i hi) hsin).1.le)
+      (Real.rpow_nonneg hsin.le _)
+  constructor
+  · exact Finset.prod_pos hpos
+  · change L (fun w => ∏i∈s,F i w) (multiDrift a v) (multiNoise m) v / (∏i∈s,F i v) ≤ _
+    rw [(normalized_product_generator s F v (multiDrift a v) (multiNoise m) hF
+      (fun i hi => (hpos i hi).ne')).2]
+    have habs := sum_generator_absorption s
+      (fun i => L (F i) (multiDrift a v) (multiNoise m) v / F i v)
+      (fun i => N (F i) (multiNoise m) v / F i v)
+      (fun i => regD a (physicalDelta a (cap i) (σ i) (off i) (v i)) (physicalTheta (v i))*(physicalV (v i))^2)
+      (fun i => regC a*physicalDelta a (cap i) (σ i) (off i) (v i)*(physicalV (v i))^2)
+      (fun i j => (physicalDelta a (cap i) (σ i) (off i) (v i)*
+        physicalDelta a (cap j) (σ j) (off j) (v j))^(-regM a)*physicalV (v i)*physicalV (v j))
+      (regK a/2) (regYoungC a ε) ε
+      (fun i hi => mul_nonneg (hD i hi) (sq_nonneg _)) hsize
+    refine le_trans (habs ?_ ?_) ?_
+    · intro i hi
+      rw [hL i hi]
+      have h := (actual_regularized_factor ha (v i) (hY i hi) (hδ i hi) (hσ i hi)).2.1
+      convert h using 1
+      ring
+    · intro i hi j hj
+      have hjs := Finset.mem_of_mem_erase hj
+      rw [hN i hi,hN j hjs]
+      have h := actual_noise_pair ha hε (v i) (v j) (hY i hi) (hY j hjs)
+        (hδ i hi) (hδ j hjs) (hσ i hi) (hσ j hjs)
+      convert h using 1
+      ring
+    · apply le_of_eq
+      congr 1
+      simp only [mul_assoc,Finset.mul_sum]
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator
+abbrev RegimeState (m : ℕ) := (Fin m → State) × ℝ
+noncomputable def regimeDiscount {m : ℕ} (s : Finset (Fin m))
+    (a ε : ℝ) (cap σ off : Fin m → ℝ) (v : Fin m → State) : ℝ :=
+  regC a*(∑i∈s,physicalDelta a (cap i) (σ i) (off i) (v i)*(physicalV (v i))^2) +
+  regYoungC a ε/2*(∑i∈s,∑j∈s.erase i,
+    (physicalDelta a (cap i) (σ i) (off i) (v i)*
+      physicalDelta a (cap j) (σ j) (off j) (v j))^(-regM a)*
+      physicalV (v i)*physicalV (v j))
+noncomputable def discountedProduct {m : ℕ} (s : Finset (Fin m))
+    (a : ℝ) (cap σ off : Fin m → ℝ) (v : RegimeState m) : ℝ :=
+  Real.exp (-v.2)*regularizedProduct s a cap σ off v.1
+end ExactSLEGreen
+
+namespace ExactSLECalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+lemma discounted_generator {f : E → ℝ} {x A e : E} (r k : ℝ)
+    (hf : ContDiffAt ℝ 2 f x) :
+    L (fun y : E×ℝ => Real.exp (-y.2)*f y.1) (A,k) (e,0) (x,r) =
+      Real.exp (-r)*(L f A e x-k*f x) := by
+  let g : E×ℝ → ℝ := fun y => Real.exp (-y.2)
+  have hd (y : E×ℝ) : HasFDerivAt g
+      (Real.exp (-y.2) • (-ContinuousLinearMap.snd ℝ E ℝ)) y := by
+    exact (hasFDerivAt_snd (𝕜 := ℝ) (p := y)).neg.exp
+  have hn (y : E×ℝ) : N g (e,0) y = 0 := by
+    rw [N,(hd y).fderiv]
+    simp
+  have hl : L g (A,k) (e,0) (x,r) = -Real.exp (-r)*k := by
+    simp only [L,show N g (e,0) = (fun _ => 0) from funext hn,N_const,mul_zero,zero_add]
+    rw [(hd (x,r)).fderiv]
+    simp
+  have hgs : ContDiffAt ℝ 2 g (x,r) := contDiffAt_snd.neg.exp
+  have hfs : ContDiffAt ℝ 2 (fun y : E×ℝ => f y.1) (x,r) := hf.comp (x,r) contDiffAt_fst
+  rw [L_mul hgs hfs]
+  change g (x,r)*L (f ∘ ContinuousLinearMap.fst ℝ E ℝ) (A,k) (e,0) (x,r)+
+    f x*L g (A,k) (e,0) (x,r)+N g (e,0) (x,r)*N (fun y : E×ℝ => f y.1) (e,0) (x,r) = _
+  rw [L_comp_clm _ hf,hl,hn]
+  dsimp [g]
+  ring
+end ExactSLECalculus
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator Metric
+lemma physicalRegularized_smooth_infty (a cap σ off : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalRegularized a cap σ off) v := by
+  have hu : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (physicalClock a cap) v :=
+    ((contDiffAt_const.add J.contDiff.contDiffAt).sub
+      (Y.contDiff.contDiffAt.log hv.ne')).div_const (2*a)
+  have ht : ContDiffAt ℝ (↑(⊤ : ℕ∞)) physicalTheta v :=
+    contDiffAt_const.sub ((X.contDiff.contDiffAt.div Y.contDiff.contDiffAt hv.ne').arctan)
+  have hp := physicalTheta_mem v
+  have hs : Real.sin (physicalTheta v) ≠ 0 := (Real.sin_pos_of_pos_of_lt_pi hp.1 hp.2).ne'
+  exact (((contDiffAt_const.mul hu).exp.div_const 3).mul
+    ((ht.sin.rpow_const_of_ne hs).add
+      (((contDiffAt_const.add (contDiffAt_const.mul hu)).exp).mul
+        (contDiffAt_const.add (ht.sin.rpow_const_of_ne hs)))))
+lemma regularizedProduct_smooth_infty {m : ℕ} (s : Finset (Fin m))
+    (a : ℝ) (cap σ off : Fin m → ℝ) (v : Fin m → State)
+    (hY : ∀ i∈s, 0<Y (v i)) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (regularizedProduct s a cap σ off) v := by
+  apply contDiffAt_prod
+  intro i hi
+  exact (physicalRegularized_smooth_infty a (cap i) (σ i) (off i) (v i) (hY i hi)).comp v
+    (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff.contDiffAt
+lemma regularizedProduct_generator_agrees {m : ℕ} (s : Finset (Fin m)) {a : ℝ}
+    (ha : 1/4<a) (cap σ off : Fin m → ℝ) (v B : Fin m → State)
+    (hY : ∀ i∈s, 0<Y (v i))
+    (hδ : ∀ i∈s, physicalDelta a (cap i) (σ i) (off i) (v i)≤1)
+    (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (hB : ∀ i∈s, B i=physicalDrift a (v i)) :
+    L (regularizedProduct s a cap σ off) B (multiNoise m) v =
+      L (regularizedProduct s a cap σ off) (multiDrift a v) (multiNoise m) v := by
+  let F : Fin m → (Fin m → State) → ℝ := fun i w => physicalRegularized a (cap i) (σ i) (off i) (w i)
+  have hF (i) (hi : i∈s) : ContDiffAt ℝ 2 (F i) v :=
+    (physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)).comp v
+      (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff.contDiffAt
+  have hpos (i) (hi : i∈s) : F i v≠0 :=
+    (actual_regularized_factor ha (v i) (hY i hi) (hδ i hi) (hσ i hi)).1.ne'
+  have hP : regularizedProduct s a cap σ off v≠0 := Finset.prod_ne_zero_iff.mpr hpos
+  apply (div_left_inj' hP).mp
+  change L (fun w => ∏ i ∈ s, F i w) B (multiNoise m) v / (∏ i ∈ s, F i v) =
+    L (fun w => ∏ i ∈ s, F i w) (multiDrift a v) (multiNoise m) v / (∏ i ∈ s, F i v)
+  rw [(normalized_product_generator s F v B (multiNoise m) hF hpos).2,
+      (normalized_product_generator s F v (multiDrift a v) (multiNoise m) hF hpos).2]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i hi
+  congr 1
+  have hf := physicalRegularized_smooth a (cap i) (σ i) (off i) (v i) (hY i hi)
+  change L (physicalRegularized a (cap i) (σ i) (off i) ∘ (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)) B _ v = _
+  rw [L_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State) hf]
+  change _ = L (physicalRegularized a (cap i) (σ i) (off i) ∘ (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State)) (multiDrift a v) _ v
+  rw [L_comp_clm (ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State) hf]
+  change L _ (B i) noise (v i) = L _ (physicalDrift a (v i)) noise (v i)
+  rw [hB i hi]
+lemma discountedProduct_nonpos {m : ℕ} (s : Finset (Fin m)) {a ε : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (cap σ off : Fin m → ℝ) (v B : RegimeState m)
+    (hY : ∀ i∈s, 0<Y (v.1 i))
+    (hδ : ∀ i∈s, physicalDelta a (cap i) (σ i) (off i) (v.1 i)≤1)
+    (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (hB : B.2=regimeDiscount s a ε cap σ off v.1 ∧
+      ∀ i∈s, B.1 i=physicalDrift a (v.1 i)) :
+    L (discountedProduct s a cap σ off) B (multiNoise m,0) v ≤ 0 := by
+  have hf : ContDiffAt ℝ 2 (regularizedProduct s a cap σ off) v.1 :=
+    (regularizedProduct_smooth_infty s a cap σ off v.1 hY).of_le (by exact WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))
+  change L (fun y : RegimeState m => Real.exp (-y.2)*regularizedProduct s a cap σ off y.1)
+    (B.1,B.2) (multiNoise m,0) (v.1,v.2) ≤ 0
+  rw [discounted_generator _ _ hf,regularizedProduct_generator_agrees s ha cap σ off v.1 B.1 hY hδ hσ hB.2,hB.1]
+  have h := regularized_product_drift s ha hε hsize cap σ off v.1 hY hδ hσ
+  apply mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le
+  exact sub_nonpos.mpr ((div_le_iff₀ h.1).mp h.2)
+
+noncomputable def regularizedHeightTurn (η : ℝ) (v : State) : ℝ :=
+  Real.smoothTransition (4*Y v/η-1)
+lemma regularizedHeightTurn_smooth (η : ℝ) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (regularizedHeightTurn η) :=
+  Real.smoothTransition.contDiff.comp (((contDiff_const.mul Y.contDiff).div_const η).sub contDiff_const)
+lemma regularizedHeightTurn_zero {η : ℝ} (hη : 0<η) (v : State) (hv : Y v≤η/4) :
+    regularizedHeightTurn η v=0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  rw [sub_nonpos,div_le_iff₀ hη]
+  linarith
+lemma regularizedHeightTurn_one {η : ℝ} (hη : 0<η) (v : State) (hv : η/2≤Y v) :
+    regularizedHeightTurn η v=1 := by
+  apply Real.smoothTransition.one_of_one_le
+  rw [le_sub_iff_add_le,le_div_iff₀ hη]
+  linarith
+lemma height_regularized_smooth (a η cap σ off : ℝ) (hη : 0<η) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (fun v => regularizedHeightTurn η v*physicalRegularized a cap σ off v) := by
+  rw [contDiff_iff_contDiffAt]
+  intro v
+  by_cases hv : 0<Y v
+  · exact (regularizedHeightTurn_smooth η).contDiffAt.mul (physicalRegularized_smooth_infty a cap σ off v hv)
+  · apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
+    have hv' : Y v<η/4 := lt_of_le_of_lt (le_of_not_gt hv) (by positivity)
+    filter_upwards [Y.continuous.continuousAt.preimage_mem_nhds (Iio_mem_nhds hv')] with w hw
+    simp only [regularizedHeightTurn_zero hη w hw.le,zero_mul]
+
+def regimeBump (m : ℕ) (N : ℝ≥0) : ContDiffBump (0 : RegimeState m) where
+  rIn := N+1
+  rOut := N+2
+  rIn_pos := by positivity
+  rIn_lt_rOut := by linarith
+noncomputable def regimeCutoff {m : ℕ} (s : Finset (Fin m))
+    (a η : ℝ) (N : ℝ≥0) (cap σ off : Fin m → ℝ) (v : RegimeState m) : ℝ :=
+  regimeBump m N v * (Real.exp (-v.2)*∏i∈s,
+    regularizedHeightTurn η (v.1 i)*physicalRegularized a (cap i) (σ i) (off i) (v.1 i))
+lemma regimeCutoff_smooth {m : ℕ} (s : Finset (Fin m)) (a η : ℝ) (hη : 0<η)
+    (N : ℝ≥0) (cap σ off : Fin m → ℝ) :
+    ContDiff ℝ (↑(⊤ : ℕ∞)) (regimeCutoff s a η N cap σ off) := by
+  apply (regimeBump m N).contDiff.mul
+  apply contDiff_snd.neg.exp.mul
+  apply contDiff_prod
+  intro i hi
+  exact (height_regularized_smooth a η (cap i) (σ i) (off i) hη).comp
+    (((ContinuousLinearMap.proj i : (Fin m → State) →L[ℝ] State).contDiff).comp contDiff_fst)
+lemma regimeCutoff_compact {m : ℕ} (s : Finset (Fin m)) (a η : ℝ)
+    (N : ℝ≥0) (cap σ off : Fin m → ℝ) :
+    HasCompactSupport (regimeCutoff s a η N cap σ off) :=
+  (regimeBump m N).hasCompactSupport.mul_right
+lemma regimeCutoff_eq_near {m : ℕ} (s : Finset (Fin m)) (a η : ℝ) (hη : 0<η)
+    (N : ℝ≥0) (cap σ off : Fin m → ℝ) (v : RegimeState m) (hv : ‖v‖≤N)
+    (hY : ∀ i∈s, η≤Y (v.1 i)) :
+    regimeCutoff s a η N cap σ off =ᶠ[𝓝 v] discountedProduct s a cap σ off := by
+  have hb : v∈ball (0 : RegimeState m) (regimeBump m N).rIn := by
+    simp only [mem_ball,dist_zero_right,regimeBump]
+    linarith
+  have hturn : ∀ i∈s, ∀ᶠ w in 𝓝 v, regularizedHeightTurn η (w.1 i)=1 := by
+    intro i hi
+    have hy : η/2<Y (v.1 i) := lt_of_lt_of_le (by linarith) (hY i hi)
+    have hc : Continuous (fun w : RegimeState m => Y (w.1 i)) :=
+      Y.continuous.comp ((continuous_apply i).comp continuous_fst)
+    filter_upwards [hc.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy)] with w hw
+    exact regularizedHeightTurn_one hη (w.1 i) hw.le
+  filter_upwards [(regimeBump m N).eventuallyEq_one_of_mem_ball hb,
+    (Filter.eventually_all_finset s).mpr hturn] with w hw htw
+  simp only [regimeCutoff,hw,Pi.one_apply,one_mul,discountedProduct,regularizedProduct]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro i hi
+  rw [htw i hi,one_mul]
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ProductIntegrability.lean b/lean/OAI/Probability/SLEGauge/ProductIntegrability.lean
index cf2b38865ad33bb0f0e2121e112d3ae9396cad13..331775212fb7d3f3227ef885a772f6b86883e4df 100644
--- a/lean/OAI/Probability/SLEGauge/ProductIntegrability.lean
+++ b/lean/OAI/Probability/SLEGauge/ProductIntegrability.lean
@@ -1,463 +1,463 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.PhaseIntegrals
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma stopped_phaseProduct_integrable {m : ℕ} {a : ℝ} (ha : 1/4<a)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
-    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (M : ℝ≥0)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hsafe : ∀ b, (∃ i,(S b:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
-      ‖vectorState a η hη z (S b) b‖≤M ∧ ∀ i,
-      (S b:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) → η≤Y (vectorState a η hη z (S b) b i)) :
-    Integrable (fun b => phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (S b) b)) W := by
-  have ha0 : 0<a := by linarith
-  let p := fun b => actualPhase a z cap U (S b) b
-  have hm := (actualPhase_measurable a ha0 z hz cap U hc hr hu S hS).mono hS.measurableSpace_le le_rfl
-  have he : (fun b => phaseProduct a cap U (p b) (vectorState a η hη z (S b) b))=
-      fun b => ∑ q : PhaseVector m, if p b=q then phaseProduct a cap U q (vectorState a η hη z (S b) b) else 0 := by funext b; simp
-  rw [he]
-  apply integrable_finsetSum
-  intro q hq
-  by_cases hn : (phaseActive q).Nonempty
-  · have hi : Integrable (fun b => if p b=q then regularizedProduct (phaseActive q) a cap (phaseSigma a q)
-        (phaseOffset a U q) (vectorState a η hη z (S b) b) else 0) W := by
-      have hs : ∀ b∈{b | p b=q}, ‖vectorState a η hη z (S b) b‖≤M ∧ ∀ i∈phaseActive q, η≤Y (vectorState a η hη z (S b) b i) := by
-        intro b hb
-        have hmem (i) (hi : i∈phaseActive q) : (S b:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) := by
-          have hp : p b i=1 ∨ p b i=2 := by rw [hb]; exact (phaseActive_mem q i).mp hi
-          have hab := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith)
-          have hbc := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith)
-          rcases hp with hp|hp
-          · exact (((phase_one_iff _ _ _ _ hab hbc).mp hp).2.trans_le hbc).le
-          · exact ((phase_two_iff _ _ _ _ hab hbc).mp hp).2.le
-        obtain ⟨i,hi⟩ := hn
-        have h := hsafe b ⟨i,hmem i hi⟩
-        exact ⟨h.1,fun j hj => h.2 j (hmem j hj)⟩
-      convert regularizedProduct_indicator_integrable (phaseActive q) a η hη M cap (phaseSigma a q) (phaseOffset a U q) z W S hS {b | p b=q} (hm (measurableSet_singleton q)) hs using 1
-      all_goals first | rfl | (funext b; by_cases hb : p b=q <;> simp [hb])
-    convert hi.const_mul (phaseConstant a U q) using 1
-    all_goals first | rfl | (funext b; dsimp only [phaseProduct]; split_ifs <;> simp)
-  · have hq0 : phaseActive q=∅ := Finset.not_nonempty_iff_eq_empty.mp hn
-    simp only [phaseProduct,hq0,regularizedProduct,Finset.prod_empty,mul_one]
-    have hi : Integrable (fun _ : ExactSLEGenerator.Driver => phaseConstant a U q) W := integrable_const _
-    convert hi.indicator (hm (measurableSet_singleton q)) using 1
-    all_goals first | rfl | (funext b; by_cases hb : p b=q <;> simp [p] at hb ⊢ <;> simp [hb,Set.indicator])
-end ExactSLEGreen
-namespace ExactSLEStops
-open scoped NNReal
-variable {Ω : Type*} [MeasurableSpace Ω]
-omit [MeasurableSpace Ω] in
-lemma sort_first_zero (l : List (Ω → ℝ≥0)) (h0 : (fun _ => 0)∈l) (ω : Ω) :
-    ((sort l)[0]'(by rw [sort_length]; exact List.length_pos_of_mem h0)) ω=0 := by
-  let v := (sort l).map (fun τ => τ ω)
-  have hz : 0∈v := (sort_perm l ω).mem_iff.mpr (List.mem_map_of_mem h0)
-  obtain ⟨i,hi,he⟩ := List.getElem_of_mem hz
-  have hv := List.pairwise_iff_getElem.mp (sort_sorted l ω)
-  apply le_antisymm _ (show (0:ℝ≥0)≤_ from bot_le)
-  cases i with
-  | zero => simpa only [v,List.getElem_map] using he.le
-  | succ i =>
-    have h := hv 0 (i+1) (by simpa [v,sort_length] using List.length_pos_of_mem h0) hi (Nat.zero_lt_succ i)
-    rw [he] at h
-    simpa only [v,List.getElem_map] using h
-omit [MeasurableSpace Ω] in
-lemma sort_last_bound (l : List (Ω → ℝ≥0)) (B : Ω → ℝ≥0) (hB : B∈l)
-    (hb : ∀ τ∈l, ∀ ω,τ ω≤B ω) (ω : Ω) (N : ℕ) (hN : (sort l).length=N+1) :
-    ((sort l)[N]'(by omega)) ω=B ω := by
-  let v := (sort l).map (fun τ => τ ω)
-  have hz : B ω∈v := (sort_perm l ω).mem_iff.mpr (List.mem_map_of_mem hB)
-  obtain ⟨i,hi,he⟩ := List.getElem_of_mem hz
-  have hv := List.pairwise_iff_getElem.mp (sort_sorted l ω)
-  apply le_antisymm (sort_bounded l B hb _ (List.getElem_mem _) ω)
-  have hiN : i≤N := by
-    have hvlen : v.length=N+1 := by simp [v,hN]
-    omega
-  rcases hiN.eq_or_lt with rfl|hlt
-  · simpa only [v,List.getElem_map] using he.ge
-  · have h := hv i N hi (by simp [hN]) hlt
-    rw [he] at h
-    simpa only [v,List.getElem_map] using h
-end ExactSLEStops
-
- 
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory ProbabilityTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma actual_phase_expectation_bound {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
-    (hsize : ε*((m:ℝ)-1)≤regK a/2)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
-    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
-    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
-    (η : ℝ) (hη : 0<η) (M : ℝ≥0)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (B : ExactSLEGenerator.Driver → ℝ≥0) (L : ℝ≥0)
-    (hB : IsStoppingTime rawFiltration (fun b => (B b : WithTop ℝ≥0)))
-    (hBL : ∀ b,B b≤L)
-    (hsafe : ∀ b t, t≤B b → (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
-      ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
-      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
-      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
-      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
-    (∫ b, phaseProduct a cap U (actualPhase a z cap U (B b) b) (vectorState a η hη z (B b) b) ∂W)≤
-      Real.exp (phaseDiscountConstant m a ε)^(3*m+1) := by
-  classical
-  have ha0 : 0<a := by linarith
-  let C := fun (i : Fin m) (u : ℝ) => clip (fun b => levelTime a b (z i) (cap i) u) B
-  let l : List (ExactSLEGenerator.Driver → ℝ≥0) := [(fun _ => 0),B] ++
-    List.ofFn (fun i => C i 0) ++ List.ofFn (fun i => C i (U i/2)) ++ List.ofFn (fun i => C i (U i))
-  have hlen : (sort l).length=3*m+2 := by simp only [sort_length,l,List.length_append,List.length_cons,List.length_nil,List.length_ofFn]; omega
-  have hmem (τ) : τ∈l ↔ τ=(fun _ => 0) ∨ τ=B ∨
-      (∃ i,τ=C i 0) ∨ (∃ i,τ=C i (U i/2)) ∨ (∃ i,τ=C i (U i)) := by
-    simp only [l,List.mem_append,List.mem_cons,List.not_mem_nil,List.mem_ofFn,or_false]
-    simp only [or_assoc]
-    constructor
-    · rintro (h|h|⟨i,h⟩|⟨i,h⟩|⟨i,h⟩)
-      · exact Or.inl h
-      · exact Or.inr (Or.inl h)
-      · exact Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩))
-      · exact Or.inr (Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩)))
-      · exact Or.inr (Or.inr (Or.inr (Or.inr ⟨i,h.symm⟩)))
-    · rintro (h|h|⟨i,h⟩|⟨i,h⟩|⟨i,h⟩)
-      · exact Or.inl h
-      · exact Or.inr (Or.inl h)
-      · exact Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩))
-      · exact Or.inr (Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩)))
-      · exact Or.inr (Or.inr (Or.inr (Or.inr ⟨i,h.symm⟩)))
-  have hbnd : ∀ τ∈l, ∀ b,τ b≤B b := by
-    intro τ ht b
-    rcases (hmem τ).mp ht with rfl|rfl|⟨i,rfl⟩|⟨i,rfl⟩|⟨i,rfl⟩
-    · exact bot_le
-    · exact le_rfl
-    all_goals exact clip_le _ _ _
-  have hstop : ∀ τ∈l, IsStoppingTime rawFiltration (fun b => (τ b : WithTop ℝ≥0)) := by
-    intro τ ht
-    rcases (hmem τ).mp ht with rfl|rfl|⟨i,rfl⟩|⟨i,rfl⟩|⟨i,rfl⟩
-    · exact isStoppingTime_const _ _
-    · exact hB
-    all_goals apply clip_stopping rawFiltration _ B _ hB
-    all_goals apply radialTime_stopping a ha0 (z i) (hz i) _ (levelRadius_pos _ _ _ (hc i))
-    · exact (levelRadius_le ha0 (hc i) le_rfl).trans (by have := hr i; linarith)
-    · exact (levelRadius_le ha0 (hc i) (by have := hu i; linarith : 0≤U i/2)).trans (by have := hr i; linarith)
-    · exact (levelRadius_le ha0 (hc i) (hu i)).trans (by have := hr i; linarith)
-  let S := fun (k : Fin (3*m+2)) => (sort l)[k.val]'(by rw [hlen]; exact k.isLt)
-  have hS (k) : IsStoppingTime rawFiltration (fun b => (S k b : WithTop ℝ≥0)) :=
-    sort_stopping rawFiltration l hstop _ (List.getElem_mem _)
-  have hSB (k) (b) : S k b≤B b := sort_bounded l B hbnd _ (List.getElem_mem _) b
-  have h0 : S 0=(fun _ => 0) := by
-    funext b
-    convert sort_first_zero l ((hmem _).mpr (Or.inl rfl)) b using 1
-    simp [S]
-    rfl
-  have hlast : S ⟨3*m+1,by omega⟩=B := by
-    funext b
-    exact sort_last_bound l B ((hmem _).mpr (Or.inr (Or.inl rfl))) hbnd b (3*m+1) (by omega)
-  let P := fun (Q : ExactSLEGenerator.Driver → ℝ≥0) (b : ExactSLEGenerator.Driver) =>
-    phaseProduct a cap U (actualPhase a z cap U (Q b) b) (vectorState a η hη z (Q b) b)
-  have hind : ∀ k : Fin (3*m+2), (∫ b,P (S k) b ∂W)≤Real.exp (phaseDiscountConstant m a ε)^k.val := by
-    intro k
-    induction k using Fin.induction with
-    | zero =>
-      rw [h0,Fin.val_zero,pow_zero]
-      have hi := stopped_phaseProduct_integrable ha z hz cap U hc hr hu η hη M W (fun _ => 0)
-        (isStoppingTime_const _ _) (fun b he => by
-          have hh := hsafe b 0 bot_le he
-          exact ⟨hh.1,fun i hi => (hh.2 i hi).2.1⟩)
-      calc
-        _ ≤ ∫ _ : ExactSLEGenerator.Driver, (1:ℝ) ∂W := integral_mono hi (integrable_const _) (fun b => actual_phaseProduct_initial ha z hz cap U hc hr hu η hη b)
-        _ = 1 := by simp
-    | succ k ih =>
-      have hk : k.val+1<(sort l).length := by rw [hlen]; omega
-      have hST (b) : S k.castSucc b≤S k.succ b := by
-        have hh := List.pairwise_iff_getElem.mp (sort_sorted l b) k.val (k.val+1)
-          (by simp only [List.length_map]; omega) (by simpa only [List.length_map] using hk) (by omega)
-        simpa only [List.getElem_map,S,Fin.val_castSucc,Fin.val_succ] using hh
-      have hboundary : ∀ b i, ∀ u∈({0,U i/2,U i}:Finset ℝ),
-          levelTime a b (z i) (cap i) u≤(S k.castSucc b:ℝ≥0∞) ∨ (S k.succ b:ℝ≥0∞)≤levelTime a b (z i) (cap i) u := by
-        intro b i u hu'
-        apply no_actual_change_between l B (fun b => levelTime a b (z i) (cap i) u) _ b k.val hk (hSB k.succ b)
-        change C i u∈l
-        simp only [Finset.mem_insert,Finset.mem_singleton] at hu'
-        rcases hu' with rfl|rfl|rfl
-        · exact (hmem _).mpr (Or.inr (Or.inr (Or.inl ⟨i,rfl⟩)))
-        · exact (hmem _).mpr (Or.inr (Or.inr (Or.inr (Or.inl ⟨i,rfl⟩))))
-        · exact (hmem _).mpr (Or.inr (Or.inr (Or.inr (Or.inr ⟨i,rfl⟩))))
-      have hs := actual_phase_expectation_step ha hε hsize z hz hinj cap U hc hr hsep hu η hη M W hW
-        (S k.castSucc) (S k.succ) L (hS _) (hS _) hST (fun b => (hSB _ b).trans (hBL b)) hboundary
-        (fun b t ht he => hsafe b t (ht.trans (hSB _ b)) he)
-      calc
-        _ ≤ Real.exp (phaseDiscountConstant m a ε)*(∫ b,P (S k.castSucc) b ∂W) := hs
-        _ ≤ Real.exp (phaseDiscountConstant m a ε)*Real.exp (phaseDiscountConstant m a ε)^k.val := mul_le_mul_of_nonneg_left ih (Real.exp_pos _).le
-        _ = _ := by simp only [Fin.val_succ,pow_succ]; ring
-  simpa only [hlast] using hind ⟨3*m+1,by omega⟩
-end ExactSLEGreen
-
- 
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory ProbabilityTheory Set Filter
-open scoped Classical NNReal ENNReal
-lemma actual_all_clocks_bound {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
-    (hsize : ε*((m:ℝ)-1)≤regK a/2)
-    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
-    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
-    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W) :
-    W {b | ∀ i,levelTime a b (z i) (cap i) (U i)<∞}≤
-      ENNReal.ofReal (Real.exp (phaseDiscountConstant m a ε)^(3*m+1) /
-        (∏ i,Real.exp (regP a*U i/2)/3)) := by
-  classical
-  have ha0 : 0<a := by linarith
-  let ρ := fun i => levelRadius a (cap i) (U i)
-  have hρ (i) : 0<ρ i := levelRadius_pos _ _ _ (hc i)
-  have hρz (i) : ρ i≤(z i).im := (levelRadius_le ha0 (hc i) (hu i)).trans (by have := hr i; linarith)
-  let B := fun n => unfinishedCutoff a ha0 z hz ρ hρ hρz n
-  have hB (n) : IsStoppingTime rawFiltration (fun b => (B n b : WithTop ℝ≥0)) := unfinishedCutoff_stopping a ha0 z hz ρ hρ hρz n
-  have hBN (n) (b) : B n b≤(n:ℝ≥0) := by
-    apply ENNReal.coe_le_coe.mp
-    rw [show (B n b:ℝ≥0∞)=unfinishedCutoffTop a ha0 z hz ρ hρ hρz n b from unfinishedCutoff_coe a ha0 z hz ρ hρ hρz n b]
-    exact unfinishedCutoffTop_le a ha0 z hz ρ hρ hρz n b
-  let E := fun n => {b | ∀ i,levelTime a b (z i) (cap i) (U i)≤(B n b:ℝ≥0∞)}
-  let H := ∏ i,Real.exp (regP a*U i/2)/3
-  let A := Real.exp (phaseDiscountConstant m a ε)^(3*m+1)
-  have hH : 0<H := Finset.prod_pos fun i _ => div_pos (Real.exp_pos _) (by norm_num)
-  have hbound (n) : W (E n)≤ENNReal.ofReal (A/H) := by
-    obtain ⟨η,hη,M,hs⟩ := actual_common_cutoff_safe a ha0 z hz ρ hρ hρz n
-    have hsafe : ∀ b t, t≤B n b → (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
-        ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
-        (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
-        alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
-        vectorState a η hη z t b i=actualState a b.1 t (z i) := by
-      intro b t ht he
-      have hh := hs b t ht he
-      refine ⟨hh.1,?_⟩
-      intro i hi
-      exact ⟨(actual_unfinished_cutoff a ha0 z hz ρ hρ hρz).2.1 n b t i ht hi,hh.2 i hi⟩
-    let P := fun b => phaseProduct a cap U (actualPhase a z cap U (B n b) b) (vectorState a η hη z (B n b) b)
-    have hi : Integrable P W := stopped_phaseProduct_integrable ha z hz cap U hc hr hu η hη M W (B n) (hB n)
-      (fun b he => by have h := hsafe b (B n b) le_rfl he; exact ⟨h.1,fun i hi => (h.2 i hi).2.1⟩)
-    have he : (∫ b,P b ∂W)≤A := actual_phase_expectation_bound ha hε hsize z hz hinj cap U hc hr hsep hu η hη M W hW (B n) n (hB n) (hBN n) hsafe
-    have hsub : E n⊆{b | H≤P b} := by
-      intro b hb
-      exact (actual_phaseProduct_completed a z cap U η hη b (B n b) hb).ge
-    have hm := mul_meas_ge_le_integral_of_nonneg (Eventually.of_forall fun b => (phaseProduct_pos a cap U _ _).le) hi H
-    have hh : W.real (E n)≤A/H := by
-      apply (le_div_iff₀ hH).mpr
-      calc
-        W.real (E n)*H = H*W.real (E n) := mul_comm _ _
-        _ ≤ H*W.real {b | H≤P b} := mul_le_mul_of_nonneg_left (measureReal_mono hsub) hH.le
-        _ ≤ A := hm.trans he
-    rw [←ofReal_measureReal (μ := W) (s := E n)]
-    exact ENNReal.ofReal_le_ofReal hh
-  let F := fun N : ℕ => {b | ∀ n≥N,b∈E n}
-  have hF : Monotone F := by
-    intro N M hNM b hb n hn
-    exact hb n (hNM.trans hn)
-  have hsub : {b | ∀ i,levelTime a b (z i) (cap i) (U i)<∞}⊆⋃ N,F N := by
-    intro b hb
-    have he := (actual_unfinished_cutoff a ha0 z hz ρ hρ hρz).2.2 b hb
-    obtain ⟨N,hN⟩ := eventually_atTop.mp he
-    apply mem_iUnion.mpr
-    refine ⟨N,?_⟩
-    intro n hn i
-    exact (hN n hn i).le
-  calc
-    _ ≤ W (⋃ N,F N) := measure_mono hsub
-    _ = ⨆ N,W (F N) := hF.measure_iUnion
-    _ ≤ ENNReal.ofReal (A/H) := iSup_le fun N => (measure_mono (show F N⊆E N from fun b hb => hb N le_rfl)).trans (hbound N)
-end ExactSLEGreen
-
- 
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain ExactSLEFiniteTilt
-open scoped Classical NNReal ENNReal
- 
-
-theorem actual_multipoint_hit_bound (a : ℝ) (ha : 1/4 < a) (m : ℕ) :
-    ∃ C : ℝ, 0 < C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
-    ∀ z : Fin m → ℂ, Function.Injective z → (∀ i, 0 < (z i).im) →
-    ∀ cap target : Fin m → ℝ,
-      (∀ i, 0<cap i ∧ cap i≤2*(z i).im) →
-      (∀ i j, i≠j → cap i≤‖z i-z j‖) →
-      (∀ i, 0<target i ∧ target i≤cap i) →
-      W {b | ∀ i, radialTime a b.1 (z i) (target i/2) < ∞} ≤
-        ENNReal.ofReal (C*∏ i, (target i/cap i)^(alpha a)) := by
-  classical
-  have ha0 : 0<a := by linarith
-  let ε := regK a/(2*((m:ℝ)+1))
-  have hε : 0<ε := div_pos (regK_pos ha) (by positivity)
-  have hsize : ε*((m:ℝ)-1)≤regK a/2 := by
-    calc
-      _ ≤ ε*((m:ℝ)+1) := mul_le_mul_of_nonneg_left (by linarith) hε.le
-      _ = _ := by dsimp [ε]; field_simp
-  let A := Real.exp (phaseDiscountConstant m a ε)^(3*m+1)
-  refine ⟨3^m*A,by dsimp [A]; positivity,?_⟩
-  intro W hprob hW z hinj hz cap target hc hsep ht
-  let U := fun i => -Real.log (target i/cap i)/(2*a)
-  have hrpos (i) : 0<target i/cap i := div_pos (ht i).1 (hc i).1
-  have hrle (i) : target i/cap i≤1 := (div_le_one (hc i).1).mpr (ht i).2
-  have hU (i) : 0≤U i := div_nonneg (neg_nonneg.mpr (Real.log_nonpos (hrpos i).le (hrle i))) (by positivity)
-  have hρ (i) : levelRadius a (cap i) (U i)=target i/2 := by
-    have he : -2*a*U i=Real.log (target i/cap i) := by dsimp [U]; field_simp
-    rw [levelRadius,he,Real.exp_log (hrpos i)]
-    field_simp [(hc i).1.ne']
-  have h := actual_all_clocks_bound ha hε hsize z hz hinj cap U (fun i => (hc i).1) (fun i => (hc i).2) hsep hU W hW
-  have hevent : {b | ∀ i,levelTime a b (z i) (cap i) (U i)<∞}={b | ∀ i,radialTime a b.1 (z i) (target i/2)<∞} := by
-    simp only [levelTime,hρ]
-  rw [hevent] at h
-  have hexp (i) : (Real.exp (regP a*U i/2)/3)⁻¹=3*(target i/cap i)^(alpha a) := by
-    have he : -(regP a*U i/2)=Real.log (target i/cap i)*alpha a := by
-      dsimp [U,regP,alpha]
-      field_simp
-      ring
-    rw [inv_div,div_eq_mul_inv,←Real.exp_neg,he,Real.rpow_def_of_pos (hrpos i)]
-  have hden : (∏ i,Real.exp (regP a*U i/2)/3)⁻¹=3^m*∏ i,(target i/cap i)^(alpha a) := by
-    rw [←Finset.prod_inv_distrib]
-    simp_rw [hexp]
-    rw [Finset.prod_mul_distrib]
-    simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
-  rw [div_eq_mul_inv,hden] at h
-  simpa only [mul_assoc,mul_left_comm,A] using h
-end ExactSLEGreen
-
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-namespace ExactSLEFlow
-variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [MeasurableSpace Ω]
-theorem measurable_parameter_flow_on_interval
-    (A : Ω → E → E) (L K : Ω → ℝ≥0)
-    (hA : ∀ ω, LipschitzWith (K ω) (A ω)) (hb : ∀ ω x, ‖A ω x‖ ≤ L ω)
-    (hmA : Measurable (Function.uncurry A))
-    (u : ℝ → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
-    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) (T : ℝ≥0) :
-    ∃ Y : ℝ → Ω → E,
-      (∀ ω, Continuous (fun t => Y t ω)) ∧
-      (∀ t, Measurable (Y t)) ∧
-      (∀ ω t, t ∈ Icc (0 : ℝ) T →
-        Y t ω = x ω + ∫ v in 0..t, A ω (Y v ω + u v ω)) ∧
-      (∀ ω t, t ∈ Icc (0 : ℝ) T →
-        HasDerivWithinAt (fun v => Y v ω) (A ω (Y t ω + u t ω)) (Icc (0 : ℝ) T) t) := by
-  let t₀ : Icc (0 : ℝ) T := ⟨0, by simp⟩
-  have hf (ω : Ω) : IsPicardLindelof (fun t y => A ω (y + u t ω))
-      t₀ (x ω) (L ω*T) 0 (L ω) (K ω) := bounded_forcing_picard (A ω) (L ω) (K ω) (hA ω) (hb ω) _ (huc ω) _ _
-  have hfix (ω : Ω) := ODE.FunSpace.exists_isFixedPt_next (hf ω)
-    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
-  choose α hα using hfix
-  let Y : ℝ → Ω → E := fun t ω => (α ω).compProj t
-  let β : (ω : Ω) → ODE.FunSpace t₀ (x ω) 0 (L ω) := fun ω => default
-  let N (ω : Ω) := ODE.FunSpace.next (hf ω)
-    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
-  have hm (n : ℕ) (t : ℝ) : Measurable (fun ω => ((N ω)^[n] (β ω)).compProj t) := by
-    induction n generalizing t with
-    | zero => change Measurable x; exact hxm
-    | succ n ih =>
-      have hc (ω : Ω) : Continuous (fun v => A ω (((N ω)^[n] (β ω)).compProj v + u v ω)) :=
-        (hA ω).continuous.comp ((ODE.FunSpace.continuous_compProj _).add (huc ω))
-      have hi := stronglyMeasurable_intervalIntegral_of_continuous
-        (fun v ω => A ω (((N ω)^[n] (β ω)).compProj v + u v ω)) hc
-        (fun v => hmA.comp (measurable_id.prodMk ((ih v).add (hum v))))
-        0 (projIcc (0 : ℝ) T T.coe_nonneg t)
-      simp only [iterate_succ_apply']
-      change Measurable (fun ω => x ω + ∫ v in (0 : ℝ)..
-        (projIcc (0 : ℝ) T T.coe_nonneg t : ℝ),
-          A ω (((N ω)^[n] (β ω)).compProj v + u v ω))
-      exact hxm.add hi.measurable
-  refine ⟨Y, fun ω => (α ω).continuous_compProj, fun t => ?_, ?_, ?_⟩
-  · exact measurable_of_tendsto_metrizable (fun n => hm n t)
-      (tendsto_pi_nhds.mpr (fun ω => picard_iterates_tendsto (hf ω) (α ω) (hα ω) (β ω) t))
-  · intro ω t ht
-    have he := (ODE.FunSpace.isFixedPt_next_iff (hf ω)
-      (mem_closedBall_self (by simp))).mp (hα ω) ⟨t, ht⟩
-    change (α ω).compProj t = x ω + ∫ v in 0..t, A ω ((α ω).compProj v + u v ω)
-    rw [ODE.FunSpace.compProj_of_mem ht]
-    exact he
-  · intro ω t ht
-    apply ODE.hasDerivWithinAt_picard_Icc t₀.2 (hf ω).continuousOn_uncurry
-      (α ω).continuous_compProj.continuousOn
-      (fun _ _ => (α ω).compProj_mem_closedBall (hf ω).mul_max_le) (x ω) ht |>.congr_of_mem _ ht
-    intro v hv
-    change (α ω).compProj v = _
-    nth_rw 1 [← hα ω]
-    rw [ODE.FunSpace.compProj_of_mem hv]
-    rfl
-
-theorem measurable_parameter_flow
-    (A : Ω → E → E) (L K : Ω → ℝ≥0)
-    (hA : ∀ ω, LipschitzWith (K ω) (A ω)) (hb : ∀ ω x, ‖A ω x‖ ≤ L ω)
-    (hmA : Measurable (Function.uncurry A))
-    (u : ℝ≥0 → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
-    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) :
-    ∃ Y : ℝ≥0 → Ω → E,
-      (∀ ω, Continuous (fun t => Y t ω)) ∧
-      (∀ t, Measurable (Y t)) ∧
-      (∀ ω t, Y t ω = x ω + ∫ v in (0 : ℝ)..(t : ℝ),
-        A ω (Y v.toNNReal ω + u v.toNNReal ω)) := by
-  have hex (T : ℝ≥0) := measurable_parameter_flow_on_interval A L K hA hb hmA
-    (fun t ω => u t.toNNReal ω) (fun ω => (huc ω).comp continuous_real_toNNReal)
-    (fun t => hum _) x hxm T
-  choose F hFc hFm hFe hFd using hex
-  have hF0 (T : ℝ≥0) (ω : Ω) : F T 0 ω = x ω := by
-    simpa using hFe T ω 0 (by simp)
-  have hcompat {S T : ℝ≥0} (hST : S ≤ T) (ω : Ω) (v : ℝ) (hv : v ∈ Icc 0 (S : ℝ)) :
-      F S v ω = F T v ω := by
-    have hsub : Icc (0 : ℝ) S ⊆ Icc (0 : ℝ) T := Icc_subset_Icc_right hST
-    apply additive_flow_unique (hA ω) (hFc S ω).continuousOn (hFc T ω).continuousOn
-      (hFd S ω) (fun t ht => (hFd T ω t (hsub ht)).mono hsub)
-      (by rw [hF0, hF0]) hv
-  let Y : ℝ≥0 → Ω → E := fun t ω => F t t ω
-  have hYF {T t : ℝ≥0} (ht : t ≤ T) (ω : Ω) : Y t ω = F T t ω :=
-    hcompat ht ω t (by simp)
-  refine ⟨Y, ?_, fun t => hFm t t, ?_⟩
-  · intro ω
-    apply continuous_iff_continuousAt.mpr
-    intro t
-    have hc : ContinuousAt (fun v : ℝ≥0 => F (t+1) v ω) t :=
-      ((hFc (t+1) ω).comp continuous_subtype_val).continuousAt
-    apply hc.congr
-    filter_upwards [eventually_lt_nhds (show t < t+1 by simp)] with v hv
-    exact (hYF hv.le ω).symm
-  · intro ω t
-    rw [hYF le_rfl ω, hFe t ω t (by simp)]
-    congr 1
-    apply intervalIntegral.integral_congr
-    intro v hv
-    have hv' : v ∈ Icc (0 : ℝ) t := by simpa [uIcc_of_le t.coe_nonneg] using hv
-    have hvt : v.toNNReal ≤ t := (Real.toNNReal_le_toNNReal hv'.2).trans_eq (by simp)
-    dsimp only
-    rw [hYF hvt ω, Real.coe_toNNReal _ hv'.1]
-
-end ExactSLEFlow
-
-namespace ExactSLEGreen
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.PhaseIntegrals
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma stopped_phaseProduct_integrable {m : ℕ} {a : ℝ} (ha : 1/4<a)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (cap U : Fin m → ℝ)
+    (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (M : ℝ≥0)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hsafe : ∀ b, (∃ i,(S b:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
+      ‖vectorState a η hη z (S b) b‖≤M ∧ ∀ i,
+      (S b:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) → η≤Y (vectorState a η hη z (S b) b i)) :
+    Integrable (fun b => phaseProduct a cap U (actualPhase a z cap U (S b) b) (vectorState a η hη z (S b) b)) W := by
+  have ha0 : 0<a := by linarith
+  let p := fun b => actualPhase a z cap U (S b) b
+  have hm := (actualPhase_measurable a ha0 z hz cap U hc hr hu S hS).mono hS.measurableSpace_le le_rfl
+  have he : (fun b => phaseProduct a cap U (p b) (vectorState a η hη z (S b) b))=
+      fun b => ∑ q : PhaseVector m, if p b=q then phaseProduct a cap U q (vectorState a η hη z (S b) b) else 0 := by funext b; simp
+  rw [he]
+  apply integrable_finsetSum
+  intro q hq
+  by_cases hn : (phaseActive q).Nonempty
+  · have hi : Integrable (fun b => if p b=q then regularizedProduct (phaseActive q) a cap (phaseSigma a q)
+        (phaseOffset a U q) (vectorState a η hη z (S b) b) else 0) W := by
+      have hs : ∀ b∈{b | p b=q}, ‖vectorState a η hη z (S b) b‖≤M ∧ ∀ i∈phaseActive q, η≤Y (vectorState a η hη z (S b) b i) := by
+        intro b hb
+        have hmem (i) (hi : i∈phaseActive q) : (S b:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) := by
+          have hp : p b i=1 ∨ p b i=2 := by rw [hb]; exact (phaseActive_mem q i).mp hi
+          have hab := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b 0 (U i/2) le_rfl (by have := hu i; linarith)
+          have hbc := levelTime_mono a ha0 (z i) (hz i) (cap i) (hc i) (hr i) b (U i/2) (U i) (by have := hu i; linarith) (by have := hu i; linarith)
+          rcases hp with hp|hp
+          · exact (((phase_one_iff _ _ _ _ hab hbc).mp hp).2.trans_le hbc).le
+          · exact ((phase_two_iff _ _ _ _ hab hbc).mp hp).2.le
+        obtain ⟨i,hi⟩ := hn
+        have h := hsafe b ⟨i,hmem i hi⟩
+        exact ⟨h.1,fun j hj => h.2 j (hmem j hj)⟩
+      convert regularizedProduct_indicator_integrable (phaseActive q) a η hη M cap (phaseSigma a q) (phaseOffset a U q) z W S hS {b | p b=q} (hm (measurableSet_singleton q)) hs using 1
+      all_goals first | rfl | (funext b; by_cases hb : p b=q <;> simp [hb])
+    convert hi.const_mul (phaseConstant a U q) using 1
+    all_goals first | rfl | (funext b; dsimp only [phaseProduct]; split_ifs <;> simp)
+  · have hq0 : phaseActive q=∅ := Finset.not_nonempty_iff_eq_empty.mp hn
+    simp only [phaseProduct,hq0,regularizedProduct,Finset.prod_empty,mul_one]
+    have hi : Integrable (fun _ : ExactSLEGenerator.Driver => phaseConstant a U q) W := integrable_const _
+    convert hi.indicator (hm (measurableSet_singleton q)) using 1
+    all_goals first | rfl | (funext b; by_cases hb : p b=q <;> simp [p] at hb ⊢ <;> simp [hb,Set.indicator])
+end ExactSLEGreen
+namespace ExactSLEStops
+open scoped NNReal
+variable {Ω : Type*} [MeasurableSpace Ω]
+omit [MeasurableSpace Ω] in
+lemma sort_first_zero (l : List (Ω → ℝ≥0)) (h0 : (fun _ => 0)∈l) (ω : Ω) :
+    ((sort l)[0]'(by rw [sort_length]; exact List.length_pos_of_mem h0)) ω=0 := by
+  let v := (sort l).map (fun τ => τ ω)
+  have hz : 0∈v := (sort_perm l ω).mem_iff.mpr (List.mem_map_of_mem h0)
+  obtain ⟨i,hi,he⟩ := List.getElem_of_mem hz
+  have hv := List.pairwise_iff_getElem.mp (sort_sorted l ω)
+  apply le_antisymm _ (show (0:ℝ≥0)≤_ from bot_le)
+  cases i with
+  | zero => simpa only [v,List.getElem_map] using he.le
+  | succ i =>
+    have h := hv 0 (i+1) (by simpa [v,sort_length] using List.length_pos_of_mem h0) hi (Nat.zero_lt_succ i)
+    rw [he] at h
+    simpa only [v,List.getElem_map] using h
+omit [MeasurableSpace Ω] in
+lemma sort_last_bound (l : List (Ω → ℝ≥0)) (B : Ω → ℝ≥0) (hB : B∈l)
+    (hb : ∀ τ∈l, ∀ ω,τ ω≤B ω) (ω : Ω) (N : ℕ) (hN : (sort l).length=N+1) :
+    ((sort l)[N]'(by omega)) ω=B ω := by
+  let v := (sort l).map (fun τ => τ ω)
+  have hz : B ω∈v := (sort_perm l ω).mem_iff.mpr (List.mem_map_of_mem hB)
+  obtain ⟨i,hi,he⟩ := List.getElem_of_mem hz
+  have hv := List.pairwise_iff_getElem.mp (sort_sorted l ω)
+  apply le_antisymm (sort_bounded l B hb _ (List.getElem_mem _) ω)
+  have hiN : i≤N := by
+    have hvlen : v.length=N+1 := by simp [v,hN]
+    omega
+  rcases hiN.eq_or_lt with rfl|hlt
+  · simpa only [v,List.getElem_map] using he.ge
+  · have h := hv i N hi (by simp [hN]) hlt
+    rw [he] at h
+    simpa only [v,List.getElem_map] using h
+end ExactSLEStops
+
+ 
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory ProbabilityTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma actual_phase_expectation_bound {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
+    (hsize : ε*((m:ℝ)-1)≤regK a/2)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
+    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
+    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
+    (η : ℝ) (hη : 0<η) (M : ℝ≥0)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (B : ExactSLEGenerator.Driver → ℝ≥0) (L : ℝ≥0)
+    (hB : IsStoppingTime rawFiltration (fun b => (B b : WithTop ℝ≥0)))
+    (hBL : ∀ b,B b≤L)
+    (hsafe : ∀ b t, t≤B b → (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
+      ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
+      (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
+      alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
+      vectorState a η hη z t b i=actualState a b.1 t (z i)) :
+    (∫ b, phaseProduct a cap U (actualPhase a z cap U (B b) b) (vectorState a η hη z (B b) b) ∂W)≤
+      Real.exp (phaseDiscountConstant m a ε)^(3*m+1) := by
+  classical
+  have ha0 : 0<a := by linarith
+  let C := fun (i : Fin m) (u : ℝ) => clip (fun b => levelTime a b (z i) (cap i) u) B
+  let l : List (ExactSLEGenerator.Driver → ℝ≥0) := [(fun _ => 0),B] ++
+    List.ofFn (fun i => C i 0) ++ List.ofFn (fun i => C i (U i/2)) ++ List.ofFn (fun i => C i (U i))
+  have hlen : (sort l).length=3*m+2 := by simp only [sort_length,l,List.length_append,List.length_cons,List.length_nil,List.length_ofFn]; omega
+  have hmem (τ) : τ∈l ↔ τ=(fun _ => 0) ∨ τ=B ∨
+      (∃ i,τ=C i 0) ∨ (∃ i,τ=C i (U i/2)) ∨ (∃ i,τ=C i (U i)) := by
+    simp only [l,List.mem_append,List.mem_cons,List.not_mem_nil,List.mem_ofFn,or_false]
+    simp only [or_assoc]
+    constructor
+    · rintro (h|h|⟨i,h⟩|⟨i,h⟩|⟨i,h⟩)
+      · exact Or.inl h
+      · exact Or.inr (Or.inl h)
+      · exact Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩))
+      · exact Or.inr (Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩)))
+      · exact Or.inr (Or.inr (Or.inr (Or.inr ⟨i,h.symm⟩)))
+    · rintro (h|h|⟨i,h⟩|⟨i,h⟩|⟨i,h⟩)
+      · exact Or.inl h
+      · exact Or.inr (Or.inl h)
+      · exact Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩))
+      · exact Or.inr (Or.inr (Or.inr (Or.inl ⟨i,h.symm⟩)))
+      · exact Or.inr (Or.inr (Or.inr (Or.inr ⟨i,h.symm⟩)))
+  have hbnd : ∀ τ∈l, ∀ b,τ b≤B b := by
+    intro τ ht b
+    rcases (hmem τ).mp ht with rfl|rfl|⟨i,rfl⟩|⟨i,rfl⟩|⟨i,rfl⟩
+    · exact bot_le
+    · exact le_rfl
+    all_goals exact clip_le _ _ _
+  have hstop : ∀ τ∈l, IsStoppingTime rawFiltration (fun b => (τ b : WithTop ℝ≥0)) := by
+    intro τ ht
+    rcases (hmem τ).mp ht with rfl|rfl|⟨i,rfl⟩|⟨i,rfl⟩|⟨i,rfl⟩
+    · exact isStoppingTime_const _ _
+    · exact hB
+    all_goals apply clip_stopping rawFiltration _ B _ hB
+    all_goals apply radialTime_stopping a ha0 (z i) (hz i) _ (levelRadius_pos _ _ _ (hc i))
+    · exact (levelRadius_le ha0 (hc i) le_rfl).trans (by have := hr i; linarith)
+    · exact (levelRadius_le ha0 (hc i) (by have := hu i; linarith : 0≤U i/2)).trans (by have := hr i; linarith)
+    · exact (levelRadius_le ha0 (hc i) (hu i)).trans (by have := hr i; linarith)
+  let S := fun (k : Fin (3*m+2)) => (sort l)[k.val]'(by rw [hlen]; exact k.isLt)
+  have hS (k) : IsStoppingTime rawFiltration (fun b => (S k b : WithTop ℝ≥0)) :=
+    sort_stopping rawFiltration l hstop _ (List.getElem_mem _)
+  have hSB (k) (b) : S k b≤B b := sort_bounded l B hbnd _ (List.getElem_mem _) b
+  have h0 : S 0=(fun _ => 0) := by
+    funext b
+    convert sort_first_zero l ((hmem _).mpr (Or.inl rfl)) b using 1
+    simp [S]
+    rfl
+  have hlast : S ⟨3*m+1,by omega⟩=B := by
+    funext b
+    exact sort_last_bound l B ((hmem _).mpr (Or.inr (Or.inl rfl))) hbnd b (3*m+1) (by omega)
+  let P := fun (Q : ExactSLEGenerator.Driver → ℝ≥0) (b : ExactSLEGenerator.Driver) =>
+    phaseProduct a cap U (actualPhase a z cap U (Q b) b) (vectorState a η hη z (Q b) b)
+  have hind : ∀ k : Fin (3*m+2), (∫ b,P (S k) b ∂W)≤Real.exp (phaseDiscountConstant m a ε)^k.val := by
+    intro k
+    induction k using Fin.induction with
+    | zero =>
+      rw [h0,Fin.val_zero,pow_zero]
+      have hi := stopped_phaseProduct_integrable ha z hz cap U hc hr hu η hη M W (fun _ => 0)
+        (isStoppingTime_const _ _) (fun b he => by
+          have hh := hsafe b 0 bot_le he
+          exact ⟨hh.1,fun i hi => (hh.2 i hi).2.1⟩)
+      calc
+        _ ≤ ∫ _ : ExactSLEGenerator.Driver, (1:ℝ) ∂W := integral_mono hi (integrable_const _) (fun b => actual_phaseProduct_initial ha z hz cap U hc hr hu η hη b)
+        _ = 1 := by simp
+    | succ k ih =>
+      have hk : k.val+1<(sort l).length := by rw [hlen]; omega
+      have hST (b) : S k.castSucc b≤S k.succ b := by
+        have hh := List.pairwise_iff_getElem.mp (sort_sorted l b) k.val (k.val+1)
+          (by simp only [List.length_map]; omega) (by simpa only [List.length_map] using hk) (by omega)
+        simpa only [List.getElem_map,S,Fin.val_castSucc,Fin.val_succ] using hh
+      have hboundary : ∀ b i, ∀ u∈({0,U i/2,U i}:Finset ℝ),
+          levelTime a b (z i) (cap i) u≤(S k.castSucc b:ℝ≥0∞) ∨ (S k.succ b:ℝ≥0∞)≤levelTime a b (z i) (cap i) u := by
+        intro b i u hu'
+        apply no_actual_change_between l B (fun b => levelTime a b (z i) (cap i) u) _ b k.val hk (hSB k.succ b)
+        change C i u∈l
+        simp only [Finset.mem_insert,Finset.mem_singleton] at hu'
+        rcases hu' with rfl|rfl|rfl
+        · exact (hmem _).mpr (Or.inr (Or.inr (Or.inl ⟨i,rfl⟩)))
+        · exact (hmem _).mpr (Or.inr (Or.inr (Or.inr (Or.inl ⟨i,rfl⟩))))
+        · exact (hmem _).mpr (Or.inr (Or.inr (Or.inr (Or.inr ⟨i,rfl⟩))))
+      have hs := actual_phase_expectation_step ha hε hsize z hz hinj cap U hc hr hsep hu η hη M W hW
+        (S k.castSucc) (S k.succ) L (hS _) (hS _) hST (fun b => (hSB _ b).trans (hBL b)) hboundary
+        (fun b t ht he => hsafe b t (ht.trans (hSB _ b)) he)
+      calc
+        _ ≤ Real.exp (phaseDiscountConstant m a ε)*(∫ b,P (S k.castSucc) b ∂W) := hs
+        _ ≤ Real.exp (phaseDiscountConstant m a ε)*Real.exp (phaseDiscountConstant m a ε)^k.val := mul_le_mul_of_nonneg_left ih (Real.exp_pos _).le
+        _ = _ := by simp only [Fin.val_succ,pow_succ]; ring
+  simpa only [hlast] using hind ⟨3*m+1,by omega⟩
+end ExactSLEGreen
+
+ 
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt ExactSLEStops MeasureTheory ProbabilityTheory Set Filter
+open scoped Classical NNReal ENNReal
+lemma actual_all_clocks_bound {m : ℕ} {a ε : ℝ} (ha : 1/4<a) (hε : 0<ε)
+    (hsize : ε*((m:ℝ)-1)≤regK a/2)
+    (z : Fin m → ℂ) (hz : ∀ i,0<(z i).im) (hinj : Function.Injective z)
+    (cap U : Fin m → ℝ) (hc : ∀ i,0<cap i) (hr : ∀ i,cap i≤2*(z i).im)
+    (hsep : ∀ i j,i≠j → cap i≤‖z i-z j‖) (hu : ∀ i,0≤U i)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W) :
+    W {b | ∀ i,levelTime a b (z i) (cap i) (U i)<∞}≤
+      ENNReal.ofReal (Real.exp (phaseDiscountConstant m a ε)^(3*m+1) /
+        (∏ i,Real.exp (regP a*U i/2)/3)) := by
+  classical
+  have ha0 : 0<a := by linarith
+  let ρ := fun i => levelRadius a (cap i) (U i)
+  have hρ (i) : 0<ρ i := levelRadius_pos _ _ _ (hc i)
+  have hρz (i) : ρ i≤(z i).im := (levelRadius_le ha0 (hc i) (hu i)).trans (by have := hr i; linarith)
+  let B := fun n => unfinishedCutoff a ha0 z hz ρ hρ hρz n
+  have hB (n) : IsStoppingTime rawFiltration (fun b => (B n b : WithTop ℝ≥0)) := unfinishedCutoff_stopping a ha0 z hz ρ hρ hρz n
+  have hBN (n) (b) : B n b≤(n:ℝ≥0) := by
+    apply ENNReal.coe_le_coe.mp
+    rw [show (B n b:ℝ≥0∞)=unfinishedCutoffTop a ha0 z hz ρ hρ hρz n b from unfinishedCutoff_coe a ha0 z hz ρ hρ hρz n b]
+    exact unfinishedCutoffTop_le a ha0 z hz ρ hρ hρz n b
+  let E := fun n => {b | ∀ i,levelTime a b (z i) (cap i) (U i)≤(B n b:ℝ≥0∞)}
+  let H := ∏ i,Real.exp (regP a*U i/2)/3
+  let A := Real.exp (phaseDiscountConstant m a ε)^(3*m+1)
+  have hH : 0<H := Finset.prod_pos fun i _ => div_pos (Real.exp_pos _) (by norm_num)
+  have hbound (n) : W (E n)≤ENNReal.ofReal (A/H) := by
+    obtain ⟨η,hη,M,hs⟩ := actual_common_cutoff_safe a ha0 z hz ρ hρ hρz n
+    have hsafe : ∀ b t, t≤B n b → (∃ i, (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i)) →
+        ‖vectorState a η hη z t b‖≤M ∧ ∀ i,
+        (t:ℝ≥0∞)≤levelTime a b (z i) (cap i) (U i) →
+        alive a b.1 t (z i) ∧ η≤Y (vectorState a η hη z t b i) ∧
+        vectorState a η hη z t b i=actualState a b.1 t (z i) := by
+      intro b t ht he
+      have hh := hs b t ht he
+      refine ⟨hh.1,?_⟩
+      intro i hi
+      exact ⟨(actual_unfinished_cutoff a ha0 z hz ρ hρ hρz).2.1 n b t i ht hi,hh.2 i hi⟩
+    let P := fun b => phaseProduct a cap U (actualPhase a z cap U (B n b) b) (vectorState a η hη z (B n b) b)
+    have hi : Integrable P W := stopped_phaseProduct_integrable ha z hz cap U hc hr hu η hη M W (B n) (hB n)
+      (fun b he => by have h := hsafe b (B n b) le_rfl he; exact ⟨h.1,fun i hi => (h.2 i hi).2.1⟩)
+    have he : (∫ b,P b ∂W)≤A := actual_phase_expectation_bound ha hε hsize z hz hinj cap U hc hr hsep hu η hη M W hW (B n) n (hB n) (hBN n) hsafe
+    have hsub : E n⊆{b | H≤P b} := by
+      intro b hb
+      exact (actual_phaseProduct_completed a z cap U η hη b (B n b) hb).ge
+    have hm := mul_meas_ge_le_integral_of_nonneg (Eventually.of_forall fun b => (phaseProduct_pos a cap U _ _).le) hi H
+    have hh : W.real (E n)≤A/H := by
+      apply (le_div_iff₀ hH).mpr
+      calc
+        W.real (E n)*H = H*W.real (E n) := mul_comm _ _
+        _ ≤ H*W.real {b | H≤P b} := mul_le_mul_of_nonneg_left (measureReal_mono hsub) hH.le
+        _ ≤ A := hm.trans he
+    rw [←ofReal_measureReal (μ := W) (s := E n)]
+    exact ENNReal.ofReal_le_ofReal hh
+  let F := fun N : ℕ => {b | ∀ n≥N,b∈E n}
+  have hF : Monotone F := by
+    intro N M hNM b hb n hn
+    exact hb n (hNM.trans hn)
+  have hsub : {b | ∀ i,levelTime a b (z i) (cap i) (U i)<∞}⊆⋃ N,F N := by
+    intro b hb
+    have he := (actual_unfinished_cutoff a ha0 z hz ρ hρ hρz).2.2 b hb
+    obtain ⟨N,hN⟩ := eventually_atTop.mp he
+    apply mem_iUnion.mpr
+    refine ⟨N,?_⟩
+    intro n hn i
+    exact (hN n hn i).le
+  calc
+    _ ≤ W (⋃ N,F N) := measure_mono hsub
+    _ = ⨆ N,W (F N) := hF.measure_iUnion
+    _ ≤ ENNReal.ofReal (A/H) := iSup_le fun N => (measure_mono (show F N⊆E N from fun b hb => hb N le_rfl)).trans (hbound N)
+end ExactSLEGreen
+
+ 
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Set ExactSLEGenerator ExactSLEMain ExactSLEFiniteTilt
+open scoped Classical NNReal ENNReal
+ 
+
+theorem actual_multipoint_hit_bound (a : ℝ) (ha : 1/4 < a) (m : ℕ) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+    IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W →
+    ∀ z : Fin m → ℂ, Function.Injective z → (∀ i, 0 < (z i).im) →
+    ∀ cap target : Fin m → ℝ,
+      (∀ i, 0<cap i ∧ cap i≤2*(z i).im) →
+      (∀ i j, i≠j → cap i≤‖z i-z j‖) →
+      (∀ i, 0<target i ∧ target i≤cap i) →
+      W {b | ∀ i, radialTime a b.1 (z i) (target i/2) < ∞} ≤
+        ENNReal.ofReal (C*∏ i, (target i/cap i)^(alpha a)) := by
+  classical
+  have ha0 : 0<a := by linarith
+  let ε := regK a/(2*((m:ℝ)+1))
+  have hε : 0<ε := div_pos (regK_pos ha) (by positivity)
+  have hsize : ε*((m:ℝ)-1)≤regK a/2 := by
+    calc
+      _ ≤ ε*((m:ℝ)+1) := mul_le_mul_of_nonneg_left (by linarith) hε.le
+      _ = _ := by dsimp [ε]; field_simp
+  let A := Real.exp (phaseDiscountConstant m a ε)^(3*m+1)
+  refine ⟨3^m*A,by dsimp [A]; positivity,?_⟩
+  intro W hprob hW z hinj hz cap target hc hsep ht
+  let U := fun i => -Real.log (target i/cap i)/(2*a)
+  have hrpos (i) : 0<target i/cap i := div_pos (ht i).1 (hc i).1
+  have hrle (i) : target i/cap i≤1 := (div_le_one (hc i).1).mpr (ht i).2
+  have hU (i) : 0≤U i := div_nonneg (neg_nonneg.mpr (Real.log_nonpos (hrpos i).le (hrle i))) (by positivity)
+  have hρ (i) : levelRadius a (cap i) (U i)=target i/2 := by
+    have he : -2*a*U i=Real.log (target i/cap i) := by dsimp [U]; field_simp
+    rw [levelRadius,he,Real.exp_log (hrpos i)]
+    field_simp [(hc i).1.ne']
+  have h := actual_all_clocks_bound ha hε hsize z hz hinj cap U (fun i => (hc i).1) (fun i => (hc i).2) hsep hU W hW
+  have hevent : {b | ∀ i,levelTime a b (z i) (cap i) (U i)<∞}={b | ∀ i,radialTime a b.1 (z i) (target i/2)<∞} := by
+    simp only [levelTime,hρ]
+  rw [hevent] at h
+  have hexp (i) : (Real.exp (regP a*U i/2)/3)⁻¹=3*(target i/cap i)^(alpha a) := by
+    have he : -(regP a*U i/2)=Real.log (target i/cap i)*alpha a := by
+      dsimp [U,regP,alpha]
+      field_simp
+      ring
+    rw [inv_div,div_eq_mul_inv,←Real.exp_neg,he,Real.rpow_def_of_pos (hrpos i)]
+  have hden : (∏ i,Real.exp (regP a*U i/2)/3)⁻¹=3^m*∏ i,(target i/cap i)^(alpha a) := by
+    rw [←Finset.prod_inv_distrib]
+    simp_rw [hexp]
+    rw [Finset.prod_mul_distrib]
+    simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin]
+  rw [div_eq_mul_inv,hden] at h
+  simpa only [mul_assoc,mul_left_comm,A] using h
+end ExactSLEGreen
+
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+namespace ExactSLEFlow
+variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [MeasurableSpace Ω]
+theorem measurable_parameter_flow_on_interval
+    (A : Ω → E → E) (L K : Ω → ℝ≥0)
+    (hA : ∀ ω, LipschitzWith (K ω) (A ω)) (hb : ∀ ω x, ‖A ω x‖ ≤ L ω)
+    (hmA : Measurable (Function.uncurry A))
+    (u : ℝ → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
+    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) (T : ℝ≥0) :
+    ∃ Y : ℝ → Ω → E,
+      (∀ ω, Continuous (fun t => Y t ω)) ∧
+      (∀ t, Measurable (Y t)) ∧
+      (∀ ω t, t ∈ Icc (0 : ℝ) T →
+        Y t ω = x ω + ∫ v in 0..t, A ω (Y v ω + u v ω)) ∧
+      (∀ ω t, t ∈ Icc (0 : ℝ) T →
+        HasDerivWithinAt (fun v => Y v ω) (A ω (Y t ω + u t ω)) (Icc (0 : ℝ) T) t) := by
+  let t₀ : Icc (0 : ℝ) T := ⟨0, by simp⟩
+  have hf (ω : Ω) : IsPicardLindelof (fun t y => A ω (y + u t ω))
+      t₀ (x ω) (L ω*T) 0 (L ω) (K ω) := bounded_forcing_picard (A ω) (L ω) (K ω) (hA ω) (hb ω) _ (huc ω) _ _
+  have hfix (ω : Ω) := ODE.FunSpace.exists_isFixedPt_next (hf ω)
+    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
+  choose α hα using hfix
+  let Y : ℝ → Ω → E := fun t ω => (α ω).compProj t
+  let β : (ω : Ω) → ODE.FunSpace t₀ (x ω) 0 (L ω) := fun ω => default
+  let N (ω : Ω) := ODE.FunSpace.next (hf ω)
+    (mem_closedBall_self (by simp) : x ω ∈ closedBall (x ω) (0 : ℝ))
+  have hm (n : ℕ) (t : ℝ) : Measurable (fun ω => ((N ω)^[n] (β ω)).compProj t) := by
+    induction n generalizing t with
+    | zero => change Measurable x; exact hxm
+    | succ n ih =>
+      have hc (ω : Ω) : Continuous (fun v => A ω (((N ω)^[n] (β ω)).compProj v + u v ω)) :=
+        (hA ω).continuous.comp ((ODE.FunSpace.continuous_compProj _).add (huc ω))
+      have hi := stronglyMeasurable_intervalIntegral_of_continuous
+        (fun v ω => A ω (((N ω)^[n] (β ω)).compProj v + u v ω)) hc
+        (fun v => hmA.comp (measurable_id.prodMk ((ih v).add (hum v))))
+        0 (projIcc (0 : ℝ) T T.coe_nonneg t)
+      simp only [iterate_succ_apply']
+      change Measurable (fun ω => x ω + ∫ v in (0 : ℝ)..
+        (projIcc (0 : ℝ) T T.coe_nonneg t : ℝ),
+          A ω (((N ω)^[n] (β ω)).compProj v + u v ω))
+      exact hxm.add hi.measurable
+  refine ⟨Y, fun ω => (α ω).continuous_compProj, fun t => ?_, ?_, ?_⟩
+  · exact measurable_of_tendsto_metrizable (fun n => hm n t)
+      (tendsto_pi_nhds.mpr (fun ω => picard_iterates_tendsto (hf ω) (α ω) (hα ω) (β ω) t))
+  · intro ω t ht
+    have he := (ODE.FunSpace.isFixedPt_next_iff (hf ω)
+      (mem_closedBall_self (by simp))).mp (hα ω) ⟨t, ht⟩
+    change (α ω).compProj t = x ω + ∫ v in 0..t, A ω ((α ω).compProj v + u v ω)
+    rw [ODE.FunSpace.compProj_of_mem ht]
+    exact he
+  · intro ω t ht
+    apply ODE.hasDerivWithinAt_picard_Icc t₀.2 (hf ω).continuousOn_uncurry
+      (α ω).continuous_compProj.continuousOn
+      (fun _ _ => (α ω).compProj_mem_closedBall (hf ω).mul_max_le) (x ω) ht |>.congr_of_mem _ ht
+    intro v hv
+    change (α ω).compProj v = _
+    nth_rw 1 [← hα ω]
+    rw [ODE.FunSpace.compProj_of_mem hv]
+    rfl
+
+theorem measurable_parameter_flow
+    (A : Ω → E → E) (L K : Ω → ℝ≥0)
+    (hA : ∀ ω, LipschitzWith (K ω) (A ω)) (hb : ∀ ω x, ‖A ω x‖ ≤ L ω)
+    (hmA : Measurable (Function.uncurry A))
+    (u : ℝ≥0 → Ω → E) (huc : ∀ ω, Continuous (fun t => u t ω))
+    (hum : ∀ t, Measurable (u t)) (x : Ω → E) (hxm : Measurable x) :
+    ∃ Y : ℝ≥0 → Ω → E,
+      (∀ ω, Continuous (fun t => Y t ω)) ∧
+      (∀ t, Measurable (Y t)) ∧
+      (∀ ω t, Y t ω = x ω + ∫ v in (0 : ℝ)..(t : ℝ),
+        A ω (Y v.toNNReal ω + u v.toNNReal ω)) := by
+  have hex (T : ℝ≥0) := measurable_parameter_flow_on_interval A L K hA hb hmA
+    (fun t ω => u t.toNNReal ω) (fun ω => (huc ω).comp continuous_real_toNNReal)
+    (fun t => hum _) x hxm T
+  choose F hFc hFm hFe hFd using hex
+  have hF0 (T : ℝ≥0) (ω : Ω) : F T 0 ω = x ω := by
+    simpa using hFe T ω 0 (by simp)
+  have hcompat {S T : ℝ≥0} (hST : S ≤ T) (ω : Ω) (v : ℝ) (hv : v ∈ Icc 0 (S : ℝ)) :
+      F S v ω = F T v ω := by
+    have hsub : Icc (0 : ℝ) S ⊆ Icc (0 : ℝ) T := Icc_subset_Icc_right hST
+    apply additive_flow_unique (hA ω) (hFc S ω).continuousOn (hFc T ω).continuousOn
+      (hFd S ω) (fun t ht => (hFd T ω t (hsub ht)).mono hsub)
+      (by rw [hF0, hF0]) hv
+  let Y : ℝ≥0 → Ω → E := fun t ω => F t t ω
+  have hYF {T t : ℝ≥0} (ht : t ≤ T) (ω : Ω) : Y t ω = F T t ω :=
+    hcompat ht ω t (by simp)
+  refine ⟨Y, ?_, fun t => hFm t t, ?_⟩
+  · intro ω
+    apply continuous_iff_continuousAt.mpr
+    intro t
+    have hc : ContinuousAt (fun v : ℝ≥0 => F (t+1) v ω) t :=
+      ((hFc (t+1) ω).comp continuous_subtype_val).continuousAt
+    apply hc.congr
+    filter_upwards [eventually_lt_nhds (show t < t+1 by simp)] with v hv
+    exact (hYF hv.le ω).symm
+  · intro ω t
+    rw [hYF le_rfl ω, hFe t ω t (by simp)]
+    congr 1
+    apply intervalIntegral.integral_congr
+    intro v hv
+    have hv' : v ∈ Icc (0 : ℝ) t := by simpa [uIcc_of_le t.coe_nonneg] using hv
+    have hvt : v.toNNReal ≤ t := (Real.toNNReal_le_toNNReal hv'.2).trans_eq (by simp)
+    dsimp only
+    rw [hYF hvt ω, Real.coe_toNNReal _ hv'.1]
+
+end ExactSLEFlow
+
+namespace ExactSLEGreen
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialBounds.lean b/lean/OAI/Probability/SLEGauge/RadialBounds.lean
index 239be6507527cded8bfa6e18010955afad49dba7..34331a47dfcf7a2941324ee3d313716fcffb96e5 100644
--- a/lean/OAI/Probability/SLEGauge/RadialBounds.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialBounds.lean
@@ -1,450 +1,450 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.CutoffDrift
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-namespace ExactSLEBarrier
-variable {a : ℝ} {T : ℝ≥0} {y j q : ℝ → ℝ}
-lemma Evolution.logRadius_le (h : Evolution a T y j q) (ha : 0 ≤ a)
-    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
-    Real.log (y t)-j t ≤ Real.log (y 0)-j 0 := by
-  have hc : ContinuousOn (fun t => Real.log (y t)-j t) (Icc 0 (T : ℝ)) :=
-    (h.yc.log (fun u hu => (h.yp u hu).ne')).sub h.jc
-  have hm : AntitoneOn (fun t => Real.log (y t)-j t) (Icc 0 (T : ℝ)) :=
-    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) hc
-      (fun u hu => (h.logRadius_deriv (interior_subset hu)).mono interior_subset)
-      (fun u hu => div_nonpos_of_nonpos_of_nonneg (by nlinarith [sq_nonneg (y u)]) (sq_nonneg _))
-  exact hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
-
- 
-
-theorem Evolution.height_barrier (h : Evolution a T y j q) (ha : 0 ≤ a)
-    (δ ρ : ℝ) (hδ : 0 < δ) (_hρ : 0 < ρ) (hj0 : j 0 = 0)
-    (hs : ∀ t ∈ Icc 0 (T : ℝ), δ^2 * q t ≤ (y t)^2)
-    (hr : ∀ t ∈ Icc 0 (T : ℝ), Real.log ρ ≤ Real.log (y t)-j t)
-    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
-    Real.exp (Real.log (y 0)+(Real.log ρ-Real.log (y 0))/(2*δ^2)) ≤ y t := by
-  let F : ℝ → ℝ := fun u => Real.log (y u) - (Real.log (y u)-j u)/(2*δ^2)
-  have hFc : ContinuousOn F (Icc 0 (T : ℝ)) :=
-    (h.yc.log (fun u hu => (h.yp u hu).ne')).sub
-      (((h.yc.log (fun u hu => (h.yp u hu).ne')).sub h.jc).div_const _)
-  have hFd (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) :
-      HasDerivWithinAt F (a*((y u)^2-δ^2*q u)/(δ^2*(q u)^2)) (Icc 0 (T : ℝ)) u := by
-    convert (h.logHeight_deriv hu).sub ((h.logRadius_deriv hu).div_const (2*δ^2)) using 1
-    try rfl
-    field_simp [hδ.ne',(h.qp u hu).ne']
-    ring
-  have hm : MonotoneOn F (Icc 0 (T : ℝ)) :=
-    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 (T : ℝ)) hFc
-      (fun u hu => (hFd u (interior_subset hu)).mono interior_subset)
-      (fun u hu => div_nonneg (mul_nonneg ha (sub_nonneg.mpr (hs u (interior_subset hu)))) (by positivity))
-  have hi := hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
-  dsimp only [F] at hi
-  rw [hj0,sub_zero] at hi
-  have hdr := div_le_div_of_nonneg_right (hr t ht) (by positivity : 0 ≤ 2*δ^2)
-  have he : Real.log (y 0)+(Real.log ρ-Real.log (y 0))/(2*δ^2) ≤ Real.log (y t) := by
-    rw [sub_div]
-    linarith
-  simpa only [Real.exp_log (h.yp t ht)] using Real.exp_le_exp.mpr he
-
- 
-
-theorem Evolution.time_barrier (h : Evolution a T y j q) (ha : 0 ≤ a)
-    (δ : ℝ) (hs : ∀ t ∈ Icc 0 (T : ℝ), δ^2 * q t ≤ (y t)^2)
-    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : 2*a*δ^2*t ≤ (y 0)^2 := by
-  let F : ℝ → ℝ := fun u => (y u)^2 + 2*a*δ^2*u
-  have hFc : ContinuousOn F (Icc 0 (T : ℝ)) :=
-    (h.yc.pow 2).add (continuousOn_const.mul continuousOn_id)
-  have hFd (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) :
-      HasDerivWithinAt F (2*a*(δ^2-(y u)^2/q u)) (Icc 0 (T : ℝ)) u := by
-    convert ((h.yd u hu).pow 2).add ((hasDerivWithinAt_id u _).const_mul (2*a*δ^2)) using 1 <;> try rfl
-    simp only [Nat.reduceSub,pow_one,Nat.cast_ofNat]
-    ring
-  have hm : AntitoneOn F (Icc 0 (T : ℝ)) :=
-    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) hFc
-      (fun u hu => (hFd u (interior_subset hu)).mono interior_subset)
-      (fun u hu => mul_nonpos_of_nonneg_of_nonpos (by positivity)
-        (sub_nonpos.mpr ((le_div_iff₀ (h.qp u (interior_subset hu))).mpr (hs u (interior_subset hu)))))
-  have hi := hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
-  dsimp only [F] at hi
-  nlinarith [sq_nonneg (y t)]
-
-end ExactSLEBarrier
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma localState_projection_deriv (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (b : ExactSLEGenerator.Driver) (L : State →L[ℝ] ℝ) (hL : L noise = 0)
-    (T : ℝ≥0) (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-    HasDerivWithinAt (fun u => L (localState a ε hε z u.toNNReal b))
-      (L (cutoffDrift a ε (localState a ε hε z u.toNNReal b))) (Icc (0 : ℝ) T) u := by
-  let V := fun t => localState a ε hε z t b
-  have hVc : Continuous V := localState_continuous a ε hε z b
-  have hAc : Continuous (fun u : ℝ => cutoffDrift a ε (V u.toNNReal)) :=
-    (cutoffDrift_lipschitz a ε hε).continuous.comp (hVc.comp continuous_real_toNNReal)
-  have hgc : Continuous (fun u : ℝ => L (cutoffDrift a ε (V u.toNNReal))) := L.continuous.comp hAc
-  have he (t : ℝ≥0) : L (V t) = L (z,0) +
-      ∫ u in (0 : ℝ)..(t : ℝ), L (cutoffDrift a ε (V u.toNNReal)) := by
-    have h := congrArg L (localState_eq_zero a ε hε z t b)
-    simp only [map_add,map_smul,hL,smul_zero,add_zero] at h
-    rw [← L.intervalIntegral_comp_comm (hAc.intervalIntegrable 0 t)] at h
-    exact h
-  have hd := (intervalIntegral.integral_hasDerivAt_right (hgc.intervalIntegrable 0 u)
-    hgc.aestronglyMeasurable.stronglyMeasurableAtFilter hgc.continuousAt).const_add (L (z,0))
-  apply hd.hasDerivWithinAt.congr_of_mem _ hu
-  intro v hv
-  simpa only [Real.coe_toNNReal _ hv.1] using he v.toNNReal
-
-lemma Y_physicalDrift (a : ℝ) (v : State) :
-    Y (physicalDrift a v) = -a*Y v/Q v := by
-  simp only [Y_apply,physicalDrift,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
-    zero_mul,zero_div,zero_sub,← Q_eq]
-  ring
-lemma J_physicalDrift (a : ℝ) (v : State) :
-    J (physicalDrift a v) = -a*(Q v-2*(Y v)^2)/(Q v)^2 := by
-  simp only [J_apply,physicalDrift,Complex.div_re,Complex.neg_re,Complex.ofReal_re,
-    Complex.neg_im,Complex.ofReal_im,neg_zero,zero_mul,zero_div,add_zero,map_pow,
-    ← Q_eq]
-  simp only [pow_two,Complex.mul_re]
-  unfold Q
-  simp only [X_apply,Y_apply]
-  ring
-
-lemma localState_evolution (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (b : ExactSLEGenerator.Driver) (T : ℝ≥0)
-    (hy : ∀ t ≤ T, ε ≤ Y (localState a ε hε z t b)) :
-    ExactSLEBarrier.Evolution a T
-      (fun u => Y (localState a ε hε z u.toNNReal b))
-      (fun u => J (localState a ε hε z u.toNNReal b))
-      (fun u => Q (localState a ε hε z u.toNNReal b)) := by
-  have hVc := (localState_continuous a ε hε z b).comp continuous_real_toNNReal
-  have ht (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : u.toNNReal ≤ T :=
-    (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-  have hp (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : 0 < Y (localState a ε hε z u.toNNReal b) :=
-    hε.trans_le (hy _ (ht u hu))
-  refine ⟨(Y.continuous.comp hVc).continuousOn,(J.continuous.comp hVc).continuousOn,
-    hp,fun u hu => Q_pos (hp u hu),?_,?_⟩
-  · intro u hu
-    have hd := localState_projection_deriv a ε hε z b Y (by simp [noise]) T u hu
-    rw [cutoffDrift_eq_physical a ε _ (hy _ (ht u hu))] at hd
-    exact (show HasDerivWithinAt _ (Y (physicalDrift a _)) _ _ from hd).congr_deriv (Y_physicalDrift a _)
-  · intro u hu
-    have hd := localState_projection_deriv a ε hε z b J (by simp [noise]) T u hu
-    rw [cutoffDrift_eq_physical a ε _ (hy _ (ht u hu))] at hd
-    exact (show HasDerivWithinAt _ (J (physicalDrift a _)) _ _ from hd).congr_deriv (J_physicalDrift a _)
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEHitting
-
-noncomputable def regionMonitor (ε N ρ δ : ℝ) (v : State) : ℝ :=
-  min (Y v-ε) (min (N-‖v‖) (min (physicalRadius v-ρ) (Y v-δ*‖v.1‖)))
-lemma regionMonitor_continuous (ε N ρ δ : ℝ) : Continuous (regionMonitor ε N ρ δ) :=
-  (Y.continuous.sub continuous_const).min ((continuous_const.sub continuous_norm).min
-    ((physicalRadius_continuous.sub continuous_const).min
-      (Y.continuous.sub (continuous_const.mul (continuous_fst.norm)))))
-lemma regionMonitor_nonneg_iff (ε N ρ δ : ℝ) (v : State) :
-    0 ≤ regionMonitor ε N ρ δ v ↔
-      ε ≤ Y v ∧ ‖v‖ ≤ N ∧ ρ ≤ physicalRadius v ∧ δ*‖v.1‖ ≤ Y v := by
-  simp only [regionMonitor,le_min_iff,sub_nonneg]
-lemma regionMonitor_nonpos_iff (ε N ρ δ : ℝ) (v : State) :
-    regionMonitor ε N ρ δ v ≤ 0 ↔
-      Y v ≤ ε ∨ N ≤ ‖v‖ ∨ physicalRadius v ≤ ρ ∨ Y v ≤ δ*‖v.1‖ := by
-  simp only [regionMonitor,min_le_iff,sub_nonpos]
-
-noncomputable def regionStop (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) : ExactSLEGenerator.Driver → ℝ≥0 :=
-  hittingBtwn (fun t b => regionMonitor ε N ρ δ (localState a ε hε z t b)) (Iic 0) 0 B
-
-lemma regionStop_stopping (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) :
-    IsStoppingTime rawFiltration (fun b => (regionStop a ε hε z N ρ δ B b : WithTop ℝ≥0)) := by
-  exact boundedContact_stopping rawFiltration _
-    (fun b => (regionMonitor_continuous ε N ρ δ).comp (localState_continuous a ε hε z b))
-    (fun t => (regionMonitor_continuous ε N ρ δ).measurable.comp (localState_adapted a ε hε z t)) B
-
-lemma regionStop_le (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    regionStop a ε hε z N ρ δ B b ≤ B := hittingBtwn_le b
-
-lemma regionStop_safe (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ regionStop a ε hε z N ρ δ B b) :
-    ε ≤ Y (localState a ε hε z t b) ∧ ‖localState a ε hε z t b‖ ≤ N ∧
-      ρ ≤ physicalRadius (localState a ε hε z t b) ∧
-      δ*‖(localState a ε hε z t b).1‖ ≤ Y (localState a ε hε z t b) := by
-  apply (regionMonitor_nonneg_iff ε N ρ δ _).mp
-  exact nonneg_before_boundedContact _
-    (fun b => (regionMonitor_continuous ε N ρ δ).comp (localState_continuous a ε hε z b))
-    (fun b => by simpa only [Function.comp_apply,localState_zero] using h0) B b t ht
-
-lemma regionStop_contact (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver) (ht : regionStop a ε hε z N ρ δ B b < B) :
-    let v := localState a ε hε z (regionStop a ε hε z N ρ δ B b) b
-    Y v = ε ∨ ‖v‖ = N ∨ physicalRadius v = ρ ∨ Y v = δ*‖v.1‖ := by
-  dsimp only
-  have hsafe := regionStop_safe a ε hε z N ρ δ B h0 b _ le_rfl
-  have hz := boundedContact_eq_zero _
-    (fun b => (regionMonitor_continuous ε N ρ δ).comp (localState_continuous a ε hε z b))
-    (fun b => by simpa only [Function.comp_apply,localState_zero] using h0) B b ht
-  rcases (regionMonitor_nonpos_iff ε N ρ δ _).mp (le_of_eq hz) with h|h|h|h
-  · exact Or.inl (le_antisymm h hsafe.1)
-  · exact Or.inr (Or.inl (le_antisymm hsafe.2.1 h))
-  · exact Or.inr (Or.inr (Or.inl (le_antisymm h hsafe.2.2.1)))
-  · exact Or.inr (Or.inr (Or.inr (le_antisymm h hsafe.2.2.2)))
-
-lemma localState_stopped_measurable (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable (fun b => localState a ε hε z (S b) b) := by
-  have hA : StronglyAdapted rawFiltration (localState a ε hε z) :=
-    fun t => (localState_adapted a ε hε z t).stronglyMeasurable
-  exact (measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
-    (localState_continuous a ε hε z)) hS).mono hS.measurableSpace_le le_rfl
-
- 
-
-lemma regionStop_green_mean (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a ε : ℝ) (ha : a ≠ 0) (hε : 0 < ε) (z : ℂ)
-    (N B : ℝ≥0) (ρ δ : ℝ) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0)) :
-    Integrable (fun b => physicalGreen a (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b)) W ∧
-    (∫ b, physicalGreen a (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) ∂W) = physicalGreen a (z,0) := by
-  let S := regionStop a ε hε z N ρ δ B
-  have hS := regionStop_stopping a ε hε z N ρ δ B
-  have he (b : ExactSLEGenerator.Driver) :
-      greenCutoff a ε N (localState a ε hε z (S b) b) =
-      physicalGreen a (localState a ε hε z (S b) b) := by
-    have hs := regionStop_safe a ε hε z N ρ δ B h0 b (S b) le_rfl
-    exact (greenCutoff_eq_near a ε hε N _ hs.2.1 hs.1).eq_of_nhds
-  constructor
-  · have hm : Measurable (fun b => greenCutoff a ε N (localState a ε hε z (S b) b)) :=
-      (greenCutoff_smooth a ε hε N).continuous.measurable.comp (localState_stopped_measurable a ε hε z S hS)
-    obtain ⟨C,hC⟩ := (greenCutoff_compact a ε N).exists_bound_of_continuous
-      (greenCutoff_smooth a ε hε N).continuous
-    have hi : Integrable (fun b => greenCutoff a ε N (localState a ε hε z (S b) b)) W := Integrable.of_bound hm.aestronglyMeasurable C (Filter.Eventually.of_forall fun b => hC _)
-    simpa only [he] using hi
-  · have ho := localGreen_optional W hW a ε ha hε z N B (fun _ => 0) S
-      (isStoppingTime_const rawFiltration 0) hS (fun _ => bot_le)
-      (regionStop_le a ε hε z N ρ δ B)
-      (fun b t ht => ⟨(regionStop_safe a ε hε z N ρ δ B h0 b t ht).1,
-        (regionStop_safe a ε hε z N ρ δ B h0 b t ht).2.1⟩)
-      univ MeasurableSet.univ
-    simpa only [localState_zero,Measure.restrict_univ,integral_const,probReal_univ,one_smul] using ho.symm
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-noncomputable def heightFloor (y0 ρ δ : ℝ) : ℝ :=
-  Real.exp (Real.log y0+(Real.log ρ-Real.log y0)/(2*δ^2))
-noncomputable def logCeiling (y0 ρ δ : ℝ) : ℝ :=
-  |Real.log (heightFloor y0 ρ δ)| + 2*|Real.log y0| + |Real.log ρ|
-
-lemma heightFloor_pos (y0 ρ δ : ℝ) : 0 < heightFloor y0 ρ δ := Real.exp_pos _
-lemma logCeiling_nonneg (y0 ρ δ : ℝ) : 0 ≤ logCeiling y0 ρ δ := by
-  unfold logCeiling
-  positivity
-
-lemma heightFloor_le {y0 ρ δ : ℝ} (hy0 : 0 < y0) (hρ : 0 < ρ) (hr : ρ ≤ y0) :
-    heightFloor y0 ρ δ ≤ y0 := by
-  calc
-    heightFloor y0 ρ δ ≤ Real.exp (Real.log y0) := ?_
-    _ = y0 := Real.exp_log hy0
-  unfold heightFloor
-  apply Real.exp_le_exp.mpr
-  have he := Real.log_le_log hρ hr
-  have hd : (Real.log ρ-Real.log y0)/(2*δ^2) ≤ 0 :=
-    div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr he) (by positivity)
-  linarith
-
- 
-
-lemma regionStop_barriers (a ε : ℝ) (ha : 0 ≤ a) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (hρ : 0 < ρ) (hδ : 0 < δ) (B : ℝ≥0)
-    (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (ht : t ≤ regionStop a ε hε z N ρ δ B b) :
-    heightFloor z.im ρ δ ≤ Y (localState a ε hε z t b) ∧
-      Y (localState a ε hε z t b) ≤ z.im ∧
-      |J (localState a ε hε z t b)| ≤ logCeiling z.im ρ δ ∧
-      ‖localState a ε hε z t b‖ ≤ z.im/δ + logCeiling z.im ρ δ ∧
-      2*a*δ^2*(t : ℝ) ≤ z.im^2 := by
-  let T := regionStop a ε hε z N ρ δ B b
-  let V := fun u => localState a ε hε z u b
-  have hsafe (u : ℝ≥0) (hu : u ≤ T) := regionStop_safe a ε hε z N ρ δ B h0 b u hu
-  have he := localState_evolution a ε hε z b T (fun u hu => (hsafe u hu).1)
-  have hg : ∀ u ∈ Icc (0 : ℝ) T, δ^2 * Q (V u.toNNReal) ≤ (Y (V u.toNNReal))^2 := by
-    intro u hu
-    have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    have hs := (hsafe _ hut).2.2.2
-    have hh := pow_le_pow_left₀ (mul_nonneg hδ.le (norm_nonneg _)) hs 2
-    simpa only [Q_eq,Complex.normSq_eq_norm_sq,mul_pow] using hh
-  have hr : ∀ u ∈ Icc (0 : ℝ) T, Real.log ρ ≤ Real.log (Y (V u.toNNReal))-J (V u.toNNReal) := by
-    intro u hu
-    have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    have hs := hsafe _ hut
-    have hh := Real.log_le_log hρ hs.2.2.1
-    have hyp : 0 < Y (V u.toNNReal) := hε.trans_le hs.1
-    change Real.log ρ ≤ Real.log (physicalRadius (V u.toNNReal)) at hh
-    rw [physicalRadius,Real.log_mul hyp.ne' (Real.exp_ne_zero _),Real.log_exp] at hh
-    simpa only [sub_eq_add_neg] using hh
-  have hj0 : J (V (0 : ℝ).toNNReal) = 0 := by simp [V,localState_zero]
-  have hym : heightFloor z.im ρ δ ≤ Y (V t) := by
-    have hh := he.height_barrier ha δ ρ hδ hρ hj0 hg hr t ⟨t.coe_nonneg,ht⟩
-    simpa only [Real.toNNReal_coe,Real.toNNReal_zero,localState_zero,Y_apply,heightFloor] using hh
-  have hyM : Y (V t) ≤ z.im := by
-    have hh := he.height_le ha t ⟨t.coe_nonneg,ht⟩
-    simpa only [Real.toNNReal_coe,Real.toNNReal_zero,localState_zero,Y_apply] using hh
-  have hy0 : 0 < z.im := by
-    have hs := (regionMonitor_nonneg_iff ε N ρ δ (z,0)).mp h0
-    exact hε.trans_le hs.1
-  have htpos : 0 < Y (V t) := (heightFloor_pos _ _ _).trans_le hym
-  have hlr := hr t ⟨t.coe_nonneg,ht⟩
-  have hLR : Real.log (Y (V t))-J (V t) ≤ Real.log z.im := by
-    have hh := he.logRadius_le ha t ⟨t.coe_nonneg,ht⟩
-    simpa only [Real.toNNReal_coe,Real.toNNReal_zero,localState_zero,Y_apply,J_apply,sub_zero] using hh
-  simp only [Real.toNNReal_coe] at hlr
-  have hlY := Real.log_le_log htpos hyM
-  have hly := Real.log_le_log (heightFloor_pos z.im ρ δ) hym
-  have hj : |J (V t)| ≤ logCeiling z.im ρ δ := by
-    apply abs_le.mpr
-    unfold logCeiling
-    constructor <;> linarith [le_abs_self (Real.log z.im),neg_le_abs (Real.log z.im),
-      le_abs_self (Real.log ρ),neg_le_abs (Real.log ρ),
-      le_abs_self (Real.log (heightFloor z.im ρ δ)),neg_le_abs (Real.log (heightFloor z.im ρ δ)),
-      abs_nonneg (Real.log z.im),abs_nonneg (Real.log ρ),abs_nonneg (Real.log (heightFloor z.im ρ δ))]
-  have hv : ‖V t‖ ≤ z.im/δ+logCeiling z.im ρ δ := by
-    have hzn : ‖(V t).1‖ ≤ z.im/δ := (le_div_iff₀ hδ).mpr (by
-      rw [mul_comm]
-      exact (hsafe t ht).2.2.2.trans hyM)
-    change max ‖(V t).1‖ ‖(V t).2‖ ≤ _
-    apply max_le
-    · linarith [logCeiling_nonneg z.im ρ δ]
-    · change |J (V t)| ≤ _
-      linarith [div_pos hy0 hδ]
-  have htime : 2*a*δ^2*(t : ℝ) ≤ z.im^2 := by
-    have hh := he.time_barrier ha δ hg t ⟨t.coe_nonneg,ht⟩
-    simpa only [Real.toNNReal_zero,localState_zero,Y_apply] using hh
-  exact ⟨hym,hyM,hj,hv,htime⟩
-
- 
-
-lemma regionStop_true_exit (a ε : ℝ) (ha : 0 < a) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (hρ : 0 < ρ) (hδ : 0 < δ) (B : ℝ≥0)
-    (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (hεm : ε < heightFloor z.im ρ δ)
-    (hN : z.im/δ+logCeiling z.im ρ δ < N)
-    (hB : z.im^2 < 2*a*δ^2*(B : ℝ)) (b : ExactSLEGenerator.Driver) :
-    let v := localState a ε hε z (regionStop a ε hε z N ρ δ B b) b
-    physicalRadius v = ρ ∨ Y v = δ*‖v.1‖ := by
-  let T := regionStop a ε hε z N ρ δ B b
-  have hs := regionStop_barriers a ε ha.le hε z N ρ δ hρ hδ B h0 b T le_rfl
-  have hTB : T < B := by
-    have hc : 0 < 2*a*δ^2 := by positivity
-    exact_mod_cast (show (T : ℝ) < B by nlinarith)
-  have hc := regionStop_contact a ε hε z N ρ δ B h0 b hTB
-  rcases hc with h|h|h|h
-  · exfalso
-    have hh := hs.1
-    rw [h] at hh
-    exact (not_le.mpr hεm) hh
-  · exfalso
-    have hh := hs.2.2.2.1
-    rw [h] at hh
-    exact (not_le.mpr hN) hh
-  · exact Or.inl h
-  · exact Or.inr h
-
-lemma regionStop_loewner (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver) :
-    let T := regionStop a ε hε z N ρ δ B b
-    alive a b.1 T z ∧
-      (∀ t ≤ T, loewnerMap a b.1 t z - (b.1 t : ℂ) = (localState a ε hε z t b).1) ∧
-      Real.log ‖deriv (loewnerMap a b.1 T) z‖ = (localState a ε hε z T b).2 := by
-  exact state_eq_loewner_on_safe a ε hε b.1 b.1.continuous b.2 z _
-    (localState_continuous a ε hε z b) (fun t => localState_eq_zero a ε hε z t b) _
-    (fun t ht => (regionStop_safe a ε hε z N ρ δ B h0 b t ht).1)
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
- 
-
-noncomputable def actualState (a : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) : State :=
-  (loewnerMap a b t z - (b t : ℂ), Real.log ‖deriv (loewnerMap a b t) z‖)
-noncomputable def radialTime (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) : ℝ≥0∞ :=
-  sInf {u : ℝ≥0∞ | ∃ t : ℝ≥0, u = t ∧ alive a b t z ∧ physicalRadius (actualState a b t z) = ρ}
-noncomputable def radialWeight (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) : ℝ :=
-  if radialTime a b z ρ = ∞ then 0 else
-    physicalGreen a (actualState a b (radialTime a b z ρ).toNNReal z)
-
-lemma regionStop_actualState (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (ht : t ≤ regionStop a ε hε z N ρ δ B b) :
-    alive a b.1 t z ∧ actualState a b.1 t z = localState a ε hε z t b := by
-  have h := state_eq_loewner_on_safe a ε hε b.1 b.1.continuous b.2 z _
-    (localState_continuous a ε hε z b) (fun u => localState_eq_zero a ε hε z u b) t
-    (fun u hu => (regionStop_safe a ε hε z N ρ δ B h0 b u (hu.trans ht)).1)
-  exact ⟨h.1, Prod.ext (h.2.1 t le_rfl) h.2.2⟩
-
-lemma regionStop_le_hit (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (ht : physicalRadius (actualState a b.1 t z) = ρ) :
-    regionStop a ε hε z N ρ δ B b ≤ t := by
-  by_contra hn
-  have hlt := lt_of_not_ge hn
-  have he := (regionStop_actualState a ε hε z N ρ δ B h0 b t hlt.le).2
-  have hc : regionMonitor ε N ρ δ (localState a ε hε z t b) ≤ 0 := by
-    apply (regionMonitor_nonpos_iff _ _ _ _ _).mpr
-    exact Or.inr (Or.inr (Or.inl (by simpa only [← he,ht] using (le_rfl : ρ ≤ ρ))))
-  exact hn (hittingBtwn_le_of_mem bot_le
-    (hlt.le.trans (regionStop_le a ε hε z N ρ δ B b)) hc)
-
-lemma regionStop_eq_radialTime (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver)
-    (hr : physicalRadius (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) = ρ) :
-    radialTime a b.1 z ρ = (regionStop a ε hε z N ρ δ B b : ℝ≥0∞) := by
-  let T := regionStop a ε hε z N ρ δ B b
-  have he := regionStop_actualState a ε hε z N ρ δ B h0 b T le_rfl
-  apply le_antisymm
-  · exact sInf_le ⟨T,rfl,he.1,by simpa only [he.2] using hr⟩
-  · apply le_sInf
-    rintro u ⟨t,rfl,_,ht⟩
-    exact ENNReal.coe_le_coe.mpr (regionStop_le_hit a ε hε z N ρ δ B h0 b t ht)
-
-lemma radialTime_top_of_no_hit (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ)
-    (h : ¬ ∃ t : ℝ≥0, alive a b t z ∧ physicalRadius (actualState a b t z) = ρ) :
-    radialTime a b z ρ = ∞ := by
-  have he : {u : ℝ≥0∞ | ∃ t : ℝ≥0, u = t ∧ alive a b t z ∧ physicalRadius (actualState a b t z) = ρ} = ∅ := by
-    apply eq_empty_iff_forall_notMem.mpr
-    rintro u ⟨t,_,ht⟩
-    exact h ⟨t,ht⟩
-  simp only [radialTime,he,sInf_empty]
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.CutoffDrift
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+namespace ExactSLEBarrier
+variable {a : ℝ} {T : ℝ≥0} {y j q : ℝ → ℝ}
+lemma Evolution.logRadius_le (h : Evolution a T y j q) (ha : 0 ≤ a)
+    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
+    Real.log (y t)-j t ≤ Real.log (y 0)-j 0 := by
+  have hc : ContinuousOn (fun t => Real.log (y t)-j t) (Icc 0 (T : ℝ)) :=
+    (h.yc.log (fun u hu => (h.yp u hu).ne')).sub h.jc
+  have hm : AntitoneOn (fun t => Real.log (y t)-j t) (Icc 0 (T : ℝ)) :=
+    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) hc
+      (fun u hu => (h.logRadius_deriv (interior_subset hu)).mono interior_subset)
+      (fun u hu => div_nonpos_of_nonpos_of_nonneg (by nlinarith [sq_nonneg (y u)]) (sq_nonneg _))
+  exact hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
+
+ 
+
+theorem Evolution.height_barrier (h : Evolution a T y j q) (ha : 0 ≤ a)
+    (δ ρ : ℝ) (hδ : 0 < δ) (_hρ : 0 < ρ) (hj0 : j 0 = 0)
+    (hs : ∀ t ∈ Icc 0 (T : ℝ), δ^2 * q t ≤ (y t)^2)
+    (hr : ∀ t ∈ Icc 0 (T : ℝ), Real.log ρ ≤ Real.log (y t)-j t)
+    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) :
+    Real.exp (Real.log (y 0)+(Real.log ρ-Real.log (y 0))/(2*δ^2)) ≤ y t := by
+  let F : ℝ → ℝ := fun u => Real.log (y u) - (Real.log (y u)-j u)/(2*δ^2)
+  have hFc : ContinuousOn F (Icc 0 (T : ℝ)) :=
+    (h.yc.log (fun u hu => (h.yp u hu).ne')).sub
+      (((h.yc.log (fun u hu => (h.yp u hu).ne')).sub h.jc).div_const _)
+  have hFd (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) :
+      HasDerivWithinAt F (a*((y u)^2-δ^2*q u)/(δ^2*(q u)^2)) (Icc 0 (T : ℝ)) u := by
+    convert (h.logHeight_deriv hu).sub ((h.logRadius_deriv hu).div_const (2*δ^2)) using 1
+    try rfl
+    field_simp [hδ.ne',(h.qp u hu).ne']
+    ring
+  have hm : MonotoneOn F (Icc 0 (T : ℝ)) :=
+    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 (T : ℝ)) hFc
+      (fun u hu => (hFd u (interior_subset hu)).mono interior_subset)
+      (fun u hu => div_nonneg (mul_nonneg ha (sub_nonneg.mpr (hs u (interior_subset hu)))) (by positivity))
+  have hi := hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
+  dsimp only [F] at hi
+  rw [hj0,sub_zero] at hi
+  have hdr := div_le_div_of_nonneg_right (hr t ht) (by positivity : 0 ≤ 2*δ^2)
+  have he : Real.log (y 0)+(Real.log ρ-Real.log (y 0))/(2*δ^2) ≤ Real.log (y t) := by
+    rw [sub_div]
+    linarith
+  simpa only [Real.exp_log (h.yp t ht)] using Real.exp_le_exp.mpr he
+
+ 
+
+theorem Evolution.time_barrier (h : Evolution a T y j q) (ha : 0 ≤ a)
+    (δ : ℝ) (hs : ∀ t ∈ Icc 0 (T : ℝ), δ^2 * q t ≤ (y t)^2)
+    (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : 2*a*δ^2*t ≤ (y 0)^2 := by
+  let F : ℝ → ℝ := fun u => (y u)^2 + 2*a*δ^2*u
+  have hFc : ContinuousOn F (Icc 0 (T : ℝ)) :=
+    (h.yc.pow 2).add (continuousOn_const.mul continuousOn_id)
+  have hFd (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) :
+      HasDerivWithinAt F (2*a*(δ^2-(y u)^2/q u)) (Icc 0 (T : ℝ)) u := by
+    convert ((h.yd u hu).pow 2).add ((hasDerivWithinAt_id u _).const_mul (2*a*δ^2)) using 1 <;> try rfl
+    simp only [Nat.reduceSub,pow_one,Nat.cast_ofNat]
+    ring
+  have hm : AntitoneOn F (Icc 0 (T : ℝ)) :=
+    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 (T : ℝ)) hFc
+      (fun u hu => (hFd u (interior_subset hu)).mono interior_subset)
+      (fun u hu => mul_nonpos_of_nonneg_of_nonpos (by positivity)
+        (sub_nonpos.mpr ((le_div_iff₀ (h.qp u (interior_subset hu))).mpr (hs u (interior_subset hu)))))
+  have hi := hm ⟨le_rfl,T.coe_nonneg⟩ ht ht.1
+  dsimp only [F] at hi
+  nlinarith [sq_nonneg (y t)]
+
+end ExactSLEBarrier
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma localState_projection_deriv (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (b : ExactSLEGenerator.Driver) (L : State →L[ℝ] ℝ) (hL : L noise = 0)
+    (T : ℝ≥0) (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+    HasDerivWithinAt (fun u => L (localState a ε hε z u.toNNReal b))
+      (L (cutoffDrift a ε (localState a ε hε z u.toNNReal b))) (Icc (0 : ℝ) T) u := by
+  let V := fun t => localState a ε hε z t b
+  have hVc : Continuous V := localState_continuous a ε hε z b
+  have hAc : Continuous (fun u : ℝ => cutoffDrift a ε (V u.toNNReal)) :=
+    (cutoffDrift_lipschitz a ε hε).continuous.comp (hVc.comp continuous_real_toNNReal)
+  have hgc : Continuous (fun u : ℝ => L (cutoffDrift a ε (V u.toNNReal))) := L.continuous.comp hAc
+  have he (t : ℝ≥0) : L (V t) = L (z,0) +
+      ∫ u in (0 : ℝ)..(t : ℝ), L (cutoffDrift a ε (V u.toNNReal)) := by
+    have h := congrArg L (localState_eq_zero a ε hε z t b)
+    simp only [map_add,map_smul,hL,smul_zero,add_zero] at h
+    rw [← L.intervalIntegral_comp_comm (hAc.intervalIntegrable 0 t)] at h
+    exact h
+  have hd := (intervalIntegral.integral_hasDerivAt_right (hgc.intervalIntegrable 0 u)
+    hgc.aestronglyMeasurable.stronglyMeasurableAtFilter hgc.continuousAt).const_add (L (z,0))
+  apply hd.hasDerivWithinAt.congr_of_mem _ hu
+  intro v hv
+  simpa only [Real.coe_toNNReal _ hv.1] using he v.toNNReal
+
+lemma Y_physicalDrift (a : ℝ) (v : State) :
+    Y (physicalDrift a v) = -a*Y v/Q v := by
+  simp only [Y_apply,physicalDrift,Complex.div_im,Complex.ofReal_re,Complex.ofReal_im,
+    zero_mul,zero_div,zero_sub,← Q_eq]
+  ring
+lemma J_physicalDrift (a : ℝ) (v : State) :
+    J (physicalDrift a v) = -a*(Q v-2*(Y v)^2)/(Q v)^2 := by
+  simp only [J_apply,physicalDrift,Complex.div_re,Complex.neg_re,Complex.ofReal_re,
+    Complex.neg_im,Complex.ofReal_im,neg_zero,zero_mul,zero_div,add_zero,map_pow,
+    ← Q_eq]
+  simp only [pow_two,Complex.mul_re]
+  unfold Q
+  simp only [X_apply,Y_apply]
+  ring
+
+lemma localState_evolution (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (b : ExactSLEGenerator.Driver) (T : ℝ≥0)
+    (hy : ∀ t ≤ T, ε ≤ Y (localState a ε hε z t b)) :
+    ExactSLEBarrier.Evolution a T
+      (fun u => Y (localState a ε hε z u.toNNReal b))
+      (fun u => J (localState a ε hε z u.toNNReal b))
+      (fun u => Q (localState a ε hε z u.toNNReal b)) := by
+  have hVc := (localState_continuous a ε hε z b).comp continuous_real_toNNReal
+  have ht (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : u.toNNReal ≤ T :=
+    (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+  have hp (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : 0 < Y (localState a ε hε z u.toNNReal b) :=
+    hε.trans_le (hy _ (ht u hu))
+  refine ⟨(Y.continuous.comp hVc).continuousOn,(J.continuous.comp hVc).continuousOn,
+    hp,fun u hu => Q_pos (hp u hu),?_,?_⟩
+  · intro u hu
+    have hd := localState_projection_deriv a ε hε z b Y (by simp [noise]) T u hu
+    rw [cutoffDrift_eq_physical a ε _ (hy _ (ht u hu))] at hd
+    exact (show HasDerivWithinAt _ (Y (physicalDrift a _)) _ _ from hd).congr_deriv (Y_physicalDrift a _)
+  · intro u hu
+    have hd := localState_projection_deriv a ε hε z b J (by simp [noise]) T u hu
+    rw [cutoffDrift_eq_physical a ε _ (hy _ (ht u hu))] at hd
+    exact (show HasDerivWithinAt _ (J (physicalDrift a _)) _ _ from hd).congr_deriv (J_physicalDrift a _)
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEHitting
+
+noncomputable def regionMonitor (ε N ρ δ : ℝ) (v : State) : ℝ :=
+  min (Y v-ε) (min (N-‖v‖) (min (physicalRadius v-ρ) (Y v-δ*‖v.1‖)))
+lemma regionMonitor_continuous (ε N ρ δ : ℝ) : Continuous (regionMonitor ε N ρ δ) :=
+  (Y.continuous.sub continuous_const).min ((continuous_const.sub continuous_norm).min
+    ((physicalRadius_continuous.sub continuous_const).min
+      (Y.continuous.sub (continuous_const.mul (continuous_fst.norm)))))
+lemma regionMonitor_nonneg_iff (ε N ρ δ : ℝ) (v : State) :
+    0 ≤ regionMonitor ε N ρ δ v ↔
+      ε ≤ Y v ∧ ‖v‖ ≤ N ∧ ρ ≤ physicalRadius v ∧ δ*‖v.1‖ ≤ Y v := by
+  simp only [regionMonitor,le_min_iff,sub_nonneg]
+lemma regionMonitor_nonpos_iff (ε N ρ δ : ℝ) (v : State) :
+    regionMonitor ε N ρ δ v ≤ 0 ↔
+      Y v ≤ ε ∨ N ≤ ‖v‖ ∨ physicalRadius v ≤ ρ ∨ Y v ≤ δ*‖v.1‖ := by
+  simp only [regionMonitor,min_le_iff,sub_nonpos]
+
+noncomputable def regionStop (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) : ExactSLEGenerator.Driver → ℝ≥0 :=
+  hittingBtwn (fun t b => regionMonitor ε N ρ δ (localState a ε hε z t b)) (Iic 0) 0 B
+
+lemma regionStop_stopping (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) :
+    IsStoppingTime rawFiltration (fun b => (regionStop a ε hε z N ρ δ B b : WithTop ℝ≥0)) := by
+  exact boundedContact_stopping rawFiltration _
+    (fun b => (regionMonitor_continuous ε N ρ δ).comp (localState_continuous a ε hε z b))
+    (fun t => (regionMonitor_continuous ε N ρ δ).measurable.comp (localState_adapted a ε hε z t)) B
+
+lemma regionStop_le (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    regionStop a ε hε z N ρ δ B b ≤ B := hittingBtwn_le b
+
+lemma regionStop_safe (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ regionStop a ε hε z N ρ δ B b) :
+    ε ≤ Y (localState a ε hε z t b) ∧ ‖localState a ε hε z t b‖ ≤ N ∧
+      ρ ≤ physicalRadius (localState a ε hε z t b) ∧
+      δ*‖(localState a ε hε z t b).1‖ ≤ Y (localState a ε hε z t b) := by
+  apply (regionMonitor_nonneg_iff ε N ρ δ _).mp
+  exact nonneg_before_boundedContact _
+    (fun b => (regionMonitor_continuous ε N ρ δ).comp (localState_continuous a ε hε z b))
+    (fun b => by simpa only [Function.comp_apply,localState_zero] using h0) B b t ht
+
+lemma regionStop_contact (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver) (ht : regionStop a ε hε z N ρ δ B b < B) :
+    let v := localState a ε hε z (regionStop a ε hε z N ρ δ B b) b
+    Y v = ε ∨ ‖v‖ = N ∨ physicalRadius v = ρ ∨ Y v = δ*‖v.1‖ := by
+  dsimp only
+  have hsafe := regionStop_safe a ε hε z N ρ δ B h0 b _ le_rfl
+  have hz := boundedContact_eq_zero _
+    (fun b => (regionMonitor_continuous ε N ρ δ).comp (localState_continuous a ε hε z b))
+    (fun b => by simpa only [Function.comp_apply,localState_zero] using h0) B b ht
+  rcases (regionMonitor_nonpos_iff ε N ρ δ _).mp (le_of_eq hz) with h|h|h|h
+  · exact Or.inl (le_antisymm h hsafe.1)
+  · exact Or.inr (Or.inl (le_antisymm hsafe.2.1 h))
+  · exact Or.inr (Or.inr (Or.inl (le_antisymm h hsafe.2.2.1)))
+  · exact Or.inr (Or.inr (Or.inr (le_antisymm h hsafe.2.2.2)))
+
+lemma localState_stopped_measurable (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable (fun b => localState a ε hε z (S b) b) := by
+  have hA : StronglyAdapted rawFiltration (localState a ε hε z) :=
+    fun t => (localState_adapted a ε hε z t).stronglyMeasurable
+  exact (measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
+    (localState_continuous a ε hε z)) hS).mono hS.measurableSpace_le le_rfl
+
+ 
+
+lemma regionStop_green_mean (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a ε : ℝ) (ha : a ≠ 0) (hε : 0 < ε) (z : ℂ)
+    (N B : ℝ≥0) (ρ δ : ℝ) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0)) :
+    Integrable (fun b => physicalGreen a (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b)) W ∧
+    (∫ b, physicalGreen a (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) ∂W) = physicalGreen a (z,0) := by
+  let S := regionStop a ε hε z N ρ δ B
+  have hS := regionStop_stopping a ε hε z N ρ δ B
+  have he (b : ExactSLEGenerator.Driver) :
+      greenCutoff a ε N (localState a ε hε z (S b) b) =
+      physicalGreen a (localState a ε hε z (S b) b) := by
+    have hs := regionStop_safe a ε hε z N ρ δ B h0 b (S b) le_rfl
+    exact (greenCutoff_eq_near a ε hε N _ hs.2.1 hs.1).eq_of_nhds
+  constructor
+  · have hm : Measurable (fun b => greenCutoff a ε N (localState a ε hε z (S b) b)) :=
+      (greenCutoff_smooth a ε hε N).continuous.measurable.comp (localState_stopped_measurable a ε hε z S hS)
+    obtain ⟨C,hC⟩ := (greenCutoff_compact a ε N).exists_bound_of_continuous
+      (greenCutoff_smooth a ε hε N).continuous
+    have hi : Integrable (fun b => greenCutoff a ε N (localState a ε hε z (S b) b)) W := Integrable.of_bound hm.aestronglyMeasurable C (Filter.Eventually.of_forall fun b => hC _)
+    simpa only [he] using hi
+  · have ho := localGreen_optional W hW a ε ha hε z N B (fun _ => 0) S
+      (isStoppingTime_const rawFiltration 0) hS (fun _ => bot_le)
+      (regionStop_le a ε hε z N ρ δ B)
+      (fun b t ht => ⟨(regionStop_safe a ε hε z N ρ δ B h0 b t ht).1,
+        (regionStop_safe a ε hε z N ρ δ B h0 b t ht).2.1⟩)
+      univ MeasurableSet.univ
+    simpa only [localState_zero,Measure.restrict_univ,integral_const,probReal_univ,one_smul] using ho.symm
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+noncomputable def heightFloor (y0 ρ δ : ℝ) : ℝ :=
+  Real.exp (Real.log y0+(Real.log ρ-Real.log y0)/(2*δ^2))
+noncomputable def logCeiling (y0 ρ δ : ℝ) : ℝ :=
+  |Real.log (heightFloor y0 ρ δ)| + 2*|Real.log y0| + |Real.log ρ|
+
+lemma heightFloor_pos (y0 ρ δ : ℝ) : 0 < heightFloor y0 ρ δ := Real.exp_pos _
+lemma logCeiling_nonneg (y0 ρ δ : ℝ) : 0 ≤ logCeiling y0 ρ δ := by
+  unfold logCeiling
+  positivity
+
+lemma heightFloor_le {y0 ρ δ : ℝ} (hy0 : 0 < y0) (hρ : 0 < ρ) (hr : ρ ≤ y0) :
+    heightFloor y0 ρ δ ≤ y0 := by
+  calc
+    heightFloor y0 ρ δ ≤ Real.exp (Real.log y0) := ?_
+    _ = y0 := Real.exp_log hy0
+  unfold heightFloor
+  apply Real.exp_le_exp.mpr
+  have he := Real.log_le_log hρ hr
+  have hd : (Real.log ρ-Real.log y0)/(2*δ^2) ≤ 0 :=
+    div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr he) (by positivity)
+  linarith
+
+ 
+
+lemma regionStop_barriers (a ε : ℝ) (ha : 0 ≤ a) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (hρ : 0 < ρ) (hδ : 0 < δ) (B : ℝ≥0)
+    (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (ht : t ≤ regionStop a ε hε z N ρ δ B b) :
+    heightFloor z.im ρ δ ≤ Y (localState a ε hε z t b) ∧
+      Y (localState a ε hε z t b) ≤ z.im ∧
+      |J (localState a ε hε z t b)| ≤ logCeiling z.im ρ δ ∧
+      ‖localState a ε hε z t b‖ ≤ z.im/δ + logCeiling z.im ρ δ ∧
+      2*a*δ^2*(t : ℝ) ≤ z.im^2 := by
+  let T := regionStop a ε hε z N ρ δ B b
+  let V := fun u => localState a ε hε z u b
+  have hsafe (u : ℝ≥0) (hu : u ≤ T) := regionStop_safe a ε hε z N ρ δ B h0 b u hu
+  have he := localState_evolution a ε hε z b T (fun u hu => (hsafe u hu).1)
+  have hg : ∀ u ∈ Icc (0 : ℝ) T, δ^2 * Q (V u.toNNReal) ≤ (Y (V u.toNNReal))^2 := by
+    intro u hu
+    have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    have hs := (hsafe _ hut).2.2.2
+    have hh := pow_le_pow_left₀ (mul_nonneg hδ.le (norm_nonneg _)) hs 2
+    simpa only [Q_eq,Complex.normSq_eq_norm_sq,mul_pow] using hh
+  have hr : ∀ u ∈ Icc (0 : ℝ) T, Real.log ρ ≤ Real.log (Y (V u.toNNReal))-J (V u.toNNReal) := by
+    intro u hu
+    have hut : u.toNNReal ≤ T := (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    have hs := hsafe _ hut
+    have hh := Real.log_le_log hρ hs.2.2.1
+    have hyp : 0 < Y (V u.toNNReal) := hε.trans_le hs.1
+    change Real.log ρ ≤ Real.log (physicalRadius (V u.toNNReal)) at hh
+    rw [physicalRadius,Real.log_mul hyp.ne' (Real.exp_ne_zero _),Real.log_exp] at hh
+    simpa only [sub_eq_add_neg] using hh
+  have hj0 : J (V (0 : ℝ).toNNReal) = 0 := by simp [V,localState_zero]
+  have hym : heightFloor z.im ρ δ ≤ Y (V t) := by
+    have hh := he.height_barrier ha δ ρ hδ hρ hj0 hg hr t ⟨t.coe_nonneg,ht⟩
+    simpa only [Real.toNNReal_coe,Real.toNNReal_zero,localState_zero,Y_apply,heightFloor] using hh
+  have hyM : Y (V t) ≤ z.im := by
+    have hh := he.height_le ha t ⟨t.coe_nonneg,ht⟩
+    simpa only [Real.toNNReal_coe,Real.toNNReal_zero,localState_zero,Y_apply] using hh
+  have hy0 : 0 < z.im := by
+    have hs := (regionMonitor_nonneg_iff ε N ρ δ (z,0)).mp h0
+    exact hε.trans_le hs.1
+  have htpos : 0 < Y (V t) := (heightFloor_pos _ _ _).trans_le hym
+  have hlr := hr t ⟨t.coe_nonneg,ht⟩
+  have hLR : Real.log (Y (V t))-J (V t) ≤ Real.log z.im := by
+    have hh := he.logRadius_le ha t ⟨t.coe_nonneg,ht⟩
+    simpa only [Real.toNNReal_coe,Real.toNNReal_zero,localState_zero,Y_apply,J_apply,sub_zero] using hh
+  simp only [Real.toNNReal_coe] at hlr
+  have hlY := Real.log_le_log htpos hyM
+  have hly := Real.log_le_log (heightFloor_pos z.im ρ δ) hym
+  have hj : |J (V t)| ≤ logCeiling z.im ρ δ := by
+    apply abs_le.mpr
+    unfold logCeiling
+    constructor <;> linarith [le_abs_self (Real.log z.im),neg_le_abs (Real.log z.im),
+      le_abs_self (Real.log ρ),neg_le_abs (Real.log ρ),
+      le_abs_self (Real.log (heightFloor z.im ρ δ)),neg_le_abs (Real.log (heightFloor z.im ρ δ)),
+      abs_nonneg (Real.log z.im),abs_nonneg (Real.log ρ),abs_nonneg (Real.log (heightFloor z.im ρ δ))]
+  have hv : ‖V t‖ ≤ z.im/δ+logCeiling z.im ρ δ := by
+    have hzn : ‖(V t).1‖ ≤ z.im/δ := (le_div_iff₀ hδ).mpr (by
+      rw [mul_comm]
+      exact (hsafe t ht).2.2.2.trans hyM)
+    change max ‖(V t).1‖ ‖(V t).2‖ ≤ _
+    apply max_le
+    · linarith [logCeiling_nonneg z.im ρ δ]
+    · change |J (V t)| ≤ _
+      linarith [div_pos hy0 hδ]
+  have htime : 2*a*δ^2*(t : ℝ) ≤ z.im^2 := by
+    have hh := he.time_barrier ha δ hg t ⟨t.coe_nonneg,ht⟩
+    simpa only [Real.toNNReal_zero,localState_zero,Y_apply] using hh
+  exact ⟨hym,hyM,hj,hv,htime⟩
+
+ 
+
+lemma regionStop_true_exit (a ε : ℝ) (ha : 0 < a) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (hρ : 0 < ρ) (hδ : 0 < δ) (B : ℝ≥0)
+    (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (hεm : ε < heightFloor z.im ρ δ)
+    (hN : z.im/δ+logCeiling z.im ρ δ < N)
+    (hB : z.im^2 < 2*a*δ^2*(B : ℝ)) (b : ExactSLEGenerator.Driver) :
+    let v := localState a ε hε z (regionStop a ε hε z N ρ δ B b) b
+    physicalRadius v = ρ ∨ Y v = δ*‖v.1‖ := by
+  let T := regionStop a ε hε z N ρ δ B b
+  have hs := regionStop_barriers a ε ha.le hε z N ρ δ hρ hδ B h0 b T le_rfl
+  have hTB : T < B := by
+    have hc : 0 < 2*a*δ^2 := by positivity
+    exact_mod_cast (show (T : ℝ) < B by nlinarith)
+  have hc := regionStop_contact a ε hε z N ρ δ B h0 b hTB
+  rcases hc with h|h|h|h
+  · exfalso
+    have hh := hs.1
+    rw [h] at hh
+    exact (not_le.mpr hεm) hh
+  · exfalso
+    have hh := hs.2.2.2.1
+    rw [h] at hh
+    exact (not_le.mpr hN) hh
+  · exact Or.inl h
+  · exact Or.inr h
+
+lemma regionStop_loewner (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver) :
+    let T := regionStop a ε hε z N ρ δ B b
+    alive a b.1 T z ∧
+      (∀ t ≤ T, loewnerMap a b.1 t z - (b.1 t : ℂ) = (localState a ε hε z t b).1) ∧
+      Real.log ‖deriv (loewnerMap a b.1 T) z‖ = (localState a ε hε z T b).2 := by
+  exact state_eq_loewner_on_safe a ε hε b.1 b.1.continuous b.2 z _
+    (localState_continuous a ε hε z b) (fun t => localState_eq_zero a ε hε z t b) _
+    (fun t ht => (regionStop_safe a ε hε z N ρ δ B h0 b t ht).1)
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+ 
+
+noncomputable def actualState (a : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (z : ℂ) : State :=
+  (loewnerMap a b t z - (b t : ℂ), Real.log ‖deriv (loewnerMap a b t) z‖)
+noncomputable def radialTime (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) : ℝ≥0∞ :=
+  sInf {u : ℝ≥0∞ | ∃ t : ℝ≥0, u = t ∧ alive a b t z ∧ physicalRadius (actualState a b t z) = ρ}
+noncomputable def radialWeight (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) : ℝ :=
+  if radialTime a b z ρ = ∞ then 0 else
+    physicalGreen a (actualState a b (radialTime a b z ρ).toNNReal z)
+
+lemma regionStop_actualState (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (ht : t ≤ regionStop a ε hε z N ρ δ B b) :
+    alive a b.1 t z ∧ actualState a b.1 t z = localState a ε hε z t b := by
+  have h := state_eq_loewner_on_safe a ε hε b.1 b.1.continuous b.2 z _
+    (localState_continuous a ε hε z b) (fun u => localState_eq_zero a ε hε z u b) t
+    (fun u hu => (regionStop_safe a ε hε z N ρ δ B h0 b u (hu.trans ht)).1)
+  exact ⟨h.1, Prod.ext (h.2.1 t le_rfl) h.2.2⟩
+
+lemma regionStop_le_hit (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (ht : physicalRadius (actualState a b.1 t z) = ρ) :
+    regionStop a ε hε z N ρ δ B b ≤ t := by
+  by_contra hn
+  have hlt := lt_of_not_ge hn
+  have he := (regionStop_actualState a ε hε z N ρ δ B h0 b t hlt.le).2
+  have hc : regionMonitor ε N ρ δ (localState a ε hε z t b) ≤ 0 := by
+    apply (regionMonitor_nonpos_iff _ _ _ _ _).mpr
+    exact Or.inr (Or.inr (Or.inl (by simpa only [← he,ht] using (le_rfl : ρ ≤ ρ))))
+  exact hn (hittingBtwn_le_of_mem bot_le
+    (hlt.le.trans (regionStop_le a ε hε z N ρ δ B b)) hc)
+
+lemma regionStop_eq_radialTime (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver)
+    (hr : physicalRadius (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) = ρ) :
+    radialTime a b.1 z ρ = (regionStop a ε hε z N ρ δ B b : ℝ≥0∞) := by
+  let T := regionStop a ε hε z N ρ δ B b
+  have he := regionStop_actualState a ε hε z N ρ δ B h0 b T le_rfl
+  apply le_antisymm
+  · exact sInf_le ⟨T,rfl,he.1,by simpa only [he.2] using hr⟩
+  · apply le_sInf
+    rintro u ⟨t,rfl,_,ht⟩
+    exact ENNReal.coe_le_coe.mpr (regionStop_le_hit a ε hε z N ρ δ B h0 b t ht)
+
+lemma radialTime_top_of_no_hit (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ)
+    (h : ¬ ∃ t : ℝ≥0, alive a b t z ∧ physicalRadius (actualState a b t z) = ρ) :
+    radialTime a b z ρ = ∞ := by
+  have he : {u : ℝ≥0∞ | ∃ t : ℝ≥0, u = t ∧ alive a b t z ∧ physicalRadius (actualState a b t z) = ρ} = ∅ := by
+    apply eq_empty_iff_forall_notMem.mpr
+    rintro u ⟨t,_,ht⟩
+    exact h ⟨t,ht⟩
+  simp only [radialTime,he,sInf_empty]
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialConditionalMixing.lean b/lean/OAI/Probability/SLEGauge/RadialConditionalMixing.lean
index 471f9d1f317a500429c417d3888d5a86bb8b038c..e7980a03773b3a2f566d6236894bc43698b6c5e2 100644
--- a/lean/OAI/Probability/SLEGauge/RadialConditionalMixing.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialConditionalMixing.lean
@@ -1,501 +1,501 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RecoveredInteriorRestart
-import OAI.Probability.SLEGauge.RadialRestartRaw
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments
-
-lemma radialLaw_consistent_le (W : Measure GDriver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
-      (hρ.trans_le hρσ) hσ).measurableSpace] A) :
-    radialLaw W a z ρ A=radialLaw W a z σ A := by
-  rcases hρσ.eq_or_lt with rfl|hρσ
-  · rfl
-  · obtain ⟨_,_,hS,he⟩ := finite_radial_laws_consistent W hW a ha z hz ρ σ hρ hρσ hσ
-    exact he A hA
-
- 
-
-theorem finiteAngle_conditional_integral_bounds (W : Measure GDriver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=σ*Real.exp (-2*a*s))
-    (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
-      (hρ.trans_le hρσ) hσ).measurableSpace] A)
-    (f : Angle→ℝ≥0∞) (hf : Measurable f) :
-    (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z σ A ≤
-      ∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ ∧
-    (∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ) ≤
-      2*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z σ A := by
-  have hs0 : 0 ≤ s := le_trans (by unfold Jacobi.mixingTime Jacobi.majorantBase; positivity) hs
-  have hσ0 := hρ.trans_le hρσ
-  have he (w : UpperPlane) : w.1.im*(ρ/σ)=levelRadius a (2*w.1.im) s := by
-    rw [hgap]
-    unfold levelRadius
-    field_simp
-  have hb (w : UpperPlane) := Jacobi.finiteAngleLaw_integral_bounds W hW a ha w.1 w.2 s hs f hf
-  have hb' (w : UpperPlane) :
-      (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a) ≤
-        ∫⁻c,f (Jacobi.finiteAngle a w.1 (w.1.im*(ρ/σ)) c)∂radialLaw W a w.1 (w.1.im*(ρ/σ)) ∧
-      (∫⁻c,f (Jacobi.finiteAngle a w.1 (w.1.im*(ρ/σ)) c)∂radialLaw W a w.1 (w.1.im*(ρ/σ))) ≤
-        2*(∫⁻θ,f θ∂angleLaw a) := by
-    rw [he w]
-    simpa only [Jacobi.finiteAngleLaw,lintegral_map hf
-      (Jacobi.finiteAngle_measurable a (by linarith) w.1 w.2 _
-        (levelRadius_pos a _ s (mul_pos (by norm_num) w.2))
-        (Jacobi.levelRadius_initial_le a (by linarith) w.1 w.2 s hs0))] using hb w
-  rw [finiteAngle_conditional W hW a ha z hz σ ρ hρ hρσ hσ A hA f hf]
-  constructor
-  · simpa only [lintegral_const,Measure.restrict_apply_univ] using
-      lintegral_mono (μ := (radialLaw W a z σ).restrict A)
-        (fun b => (hb' (radialPoint a z σ b)).1)
-  · simpa only [lintegral_const,Measure.restrict_apply_univ] using
-      lintegral_mono (μ := (radialLaw W a z σ).restrict A)
-        (fun b => (hb' (radialPoint a z σ b)).2)
-
- 
-
-theorem finiteAngle_conditional_bounds (W : Measure GDriver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=σ*Real.exp (-2*a*s))
-    (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
-      (hρ.trans_le hρσ) hσ).measurableSpace] A)
-    (f : Angle→ℝ≥0∞) (hf : Measurable f) :
-    (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z ρ A ≤
-      ∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ ∧
-    (∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ) ≤
-      2*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z ρ A := by
-  rw [radialLaw_consistent_le W hW a ha z hz σ ρ hρ hρσ hσ A hA]
-  exact finiteAngle_conditional_integral_bounds W hW a ha z hz σ ρ hρ hρσ hσ s hs hgap A hA f hf
-
- 
-
-lemma radial_change_law_event (W : Measure GDriver)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (A : Set GDriver) (hA : MeasurableSet A) :
-    W (A∩{b | radialTime a b.1 z ρ≠∞}) =
-      ENNReal.ofReal (ρ^(alpha a)*physicalGreen a (z,0))*
-        ∫⁻b in A,ENNReal.ofReal ((Real.sin (Jacobi.finiteAngle a z ρ b : ℝ))^(-p a))
-          ∂radialLaw W a z ρ := by
-  have he := Jacobi.radial_change_law W a ha z hz ρ hρ hr (A.indicator (fun _=>1))
-    (measurable_const.indicator hA)
-  have hi : (fun b => ENNReal.ofReal
-      ((Real.sin (Jacobi.finiteAngle a z ρ b : ℝ))^(-p a))*A.indicator (fun _=>1) b) =
-      A.indicator (fun b => ENNReal.ofReal
-        ((Real.sin (Jacobi.finiteAngle a z ρ b : ℝ))^(-p a))) := by
-    funext b
-    by_cases hb : b∈A <;> simp [hb]
-  simp only [hi,lintegral_indicator hA,Measure.restrict_restrict hA,lintegral_one,
-    Measure.restrict_apply_univ] at he
-  simpa only [inter_comm] using he
-
- 
-
-theorem radial_change_law_past_bound (W : Measure GDriver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=σ*Real.exp (-2*a*s))
-    (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
-      (hρ.trans_le hρσ) hσ).measurableSpace] A) :
-    W (A∩{b | radialTime a b.1 z ρ≠∞}) ≤
-      Jacobi.visitConstant a*ENNReal.ofReal ((ρ/z.im)^(alpha a))*radialLaw W a z ρ A := by
-  have hA' : MeasurableSet A := (radialTime_stopping a (by linarith) z hz σ
-    (hρ.trans_le hρσ) hσ).measurableSpace_le _ hA
-  rw [radial_change_law_event W a ha z hz ρ hρ (hρσ.trans hσ) A hA']
-  have hb := (finiteAngle_conditional_bounds W hW a ha z hz σ ρ hρ hρσ hσ s hs hgap A hA
-    (fun θ=>ENNReal.ofReal ((Real.sin θ)^(-p a))) (by fun_prop)).2
-  have hG : ENNReal.ofReal (ρ^(alpha a)*physicalGreen a (z,0)) ≤
-      ENNReal.ofReal ((ρ/z.im)^(alpha a)) := by
-    apply ENNReal.ofReal_le_ofReal
-    rw [Real.div_rpow hρ.le hz.le,div_eq_mul_inv,←Real.rpow_neg hz.le]
-    exact mul_le_mul_of_nonneg_left (Jacobi.physicalGreen_initial_bound a ha z hz)
-      (Real.rpow_nonneg hρ.le _)
-  calc
-    _ ≤ ENNReal.ofReal ((ρ/z.im)^(alpha a))*
-        (2*(∫⁻θ : Angle,ENNReal.ofReal ((Real.sin θ)^(-p a))∂angleLaw a)*radialLaw W a z ρ A) :=
-      mul_le_mul hG hb zero_le zero_le
-    _ ≤ _ := by
-      rw [←mul_assoc,mul_comm (ENNReal.ofReal _) (2*_)]
-      exact mul_le_mul_of_nonneg_right
-        (mul_le_mul_of_nonneg_right (le_add_right le_rfl) zero_le) zero_le
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-
-lemma actualUpperMass_eval_joint_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (f : C_c(UpperPlane,ℝ)) :
-    Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>∫z,f z∂actualUpperMass a q p.1 p.2) := by
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  have hm : Measurable (fun p : (ℝ≥0∞×ExactSLEGenerator.Driver) × ℂ =>
-      upperExtension f p.2*(timeMassDensity a q p.1.1 p.1.2 p.2).toReal) :=
-    ((upperExtension_measurable f).comp measurable_snd).mul
-      (timeMassDensity_joint_measurable a ha q hq hq1).ennreal_toReal
-  exact hm.stronglyMeasurable.integral_prod_right'.measurable
-
-lemma recoveredMass_joint_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j) :
-    Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>recoveredMass a ha s c hc p.1 p.2) := by
-  apply measurable_radonLimit
-  intro k f
-  simp_rw [actualMassFunctionals_apply]
-  apply Finset.measurable_sum
-  intro j hj
-  exact measurable_const.mul (actualUpperMass_eval_joint_measurable a ha _
-    (Real.exp_pos _) (level_step_le_one a ha j) f)
-
-lemma centeredTime_measurable (a : ℝ) (ha : 1/4<a) (l : ℕ) :
-    Measurable (fun p : Angle×ExactSLEGenerator.Driver=>levelTime a p.1.1 p.2.1 l) := by
-  have hz : Measurable (fun θ : Angle => (⟨fromDisk θ.1 0,by
-      rw [fromDisk_zero_im]
-      exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2⟩ : UpperPlane)) := by
-    apply Measurable.subtype_mk
-    simp only [fromDisk_zero]
-    fun_prop
-  have hr : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
-    have hl : (0:ℝ)≤l := Nat.cast_nonneg _
-    have ha0 : 0<a := by linarith
-    nlinarith)
-  convert (joint_radialTime_measurable a ha _ (Real.exp_pos _) hr).comp
-    (measurable_snd.prodMk (hz.comp measurable_fst)) using 1
-  funext p
-  simp only [levelTime_radialTime a p.1.1 p.1.2.1 p.1.2.2 p.2.1 p.2.1.continuous,
-    Function.comp_apply,fromDisk_zero_im]
-
-lemma centeredWeight_measurable (a : ℝ) (ha : 1/4<a) (l : ℕ) :
-    Measurable (fun p : Angle×ExactSLEGenerator.Driver=>tiltWeight a p.1.1 p.2.1 l) := by
-  have hz : Measurable (fun θ : Angle => (⟨fromDisk θ.1 0,by
-      rw [fromDisk_zero_im]
-      exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2⟩ : UpperPlane)) := by
-    apply Measurable.subtype_mk
-    simp only [fromDisk_zero]
-    fun_prop
-  have hr : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
-    have hl : (0:ℝ)≤l := Nat.cast_nonneg _
-    have ha0 : 0<a := by linarith
-    nlinarith)
-  have hG : Measurable (fun θ : Angle=>physicalGreen a (fromDisk θ.1 0,0)) := by
-    have he : (fun θ : Angle=>physicalGreen a (fromDisk θ.1 0,0))=
-        fun θ : Angle => Real.sin θ.1^(-alpha a)*Real.sin θ.1^(p a) := by
-      funext θ
-      exact radial_initial_green a θ.1 θ.2.1 θ.2.2
-    rw [he]
-    fun_prop
-  convert ((joint_radialWeight_measurable a ha _ (Real.exp_pos _) hr).comp
-    (measurable_snd.prodMk (hz.comp measurable_fst))).div (hG.comp measurable_fst) using 1
-  funext p
-  simp only [tiltWeight_radialWeight a p.1.1 ha p.1.2.1 p.1.2.2 p.2 l,Function.comp_apply,Pi.div_apply,fromDisk_zero_im]
-
-lemma diskMap_joint_measurable :
-    Measurable (fun p : Angle×UpperPlane => diskMap p.1.1 p.2.1) := by
-  unfold diskMap
-  fun_prop
-
-lemma diskWeight_joint_measurable (a : ℝ) :
-    Measurable (fun p : Angle×UpperPlane => diskWeight a p.1.1 p.2) := by
-  have he : (fun p : Angle×UpperPlane=>diskWeight a p.1.1 p.2)=
-      fun p : Angle×UpperPlane =>
-        ‖((2*Real.sin p.1.1:ℝ):ℂ)*Complex.I/(p.2.1-Complex.exp (-Complex.I*p.1.1))^2‖^(2-alpha a) := by
-    funext p
-    rw [diskWeight,disk_deriv p.1.1 p.1.2.1 p.1.2.2 p.2]
-  rw [he]
-  fun_prop
-
- 
-
-lemma diskMass_joint_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (a : ℝ) (θ : Ω→Angle) (hθ : Measurable θ) (ν : Ω→Measure UpperPlane)
-    (hν : Measurable ν) [∀ω,IsFiniteMeasureOnCompacts (ν ω)] :
-    Measurable (fun ω=>diskMass a (θ ω).1 (ν ω)) := by
-  let k : Kernel Ω UpperPlane := ⟨ν,hν⟩
-  have : IsSFiniteKernel k := isSFiniteKernel_of_locally_finite k
-  apply Measure.measurable_of_measurable_coe
-  intro A hA
-  have hm : Measurable (fun p : Ω×UpperPlane =>
-      A.indicator (fun _=> (1:ℝ≥0∞)) (diskMap (θ p.1).1 p.2.1)*
-        ENNReal.ofReal (diskWeight a (θ p.1).1 p.2)) :=
-    ((measurable_const.indicator hA).comp (diskMap_joint_measurable.comp
-      ((hθ.comp measurable_fst).prodMk measurable_snd))).mul
-      ((diskWeight_joint_measurable a).comp
-        ((hθ.comp measurable_fst).prodMk measurable_snd)).ennreal_ofReal
-  have he (ω : Ω) : diskMass a (θ ω).1 (ν ω) A =
-      ∫⁻z,A.indicator (fun _=>(1:ℝ≥0∞)) (diskMap (θ ω).1 z.1)*
-        ENNReal.ofReal (diskWeight a (θ ω).1 z)∂k ω := by
-    rw [diskMass,Measure.map_apply (diskMap_measurable _) hA,
-      withDensity_apply _ ((diskMap_measurable _) hA),
-      ←lintegral_indicator ((diskMap_measurable _) hA)]
-    apply lintegral_congr
-    intro z
-    by_cases hz : diskMap (θ ω).1 z.1∈A <;>
-      simp [Set.indicator,Set.mem_preimage,hz]
-  simp_rw [he]
-  exact hm.lintegral_kernel_prod_right
-
- 
-
-def centeredMass (a : ℝ) (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (l : ℕ) (p : Angle×ExactSLEGenerator.Driver) : ℝ≥0∞ :=
-  diskMass a p.1.1 (μ (levelTime a p.1.1 p.2.1 l) p.2) (Metric.closedBall (0:ℂ) (1/4))
-
-lemma centeredMass_measurable (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (l : ℕ) :
-    Measurable (centeredMass a μ l) := by
-  have hn := diskMass_joint_measurable a Prod.fst measurable_fst
-    (fun p : Angle×ExactSLEGenerator.Driver=>μ (levelTime a p.1.1 p.2.1 l) p.2)
-    (hμ.comp ((centeredTime_measurable a ha l).prodMk measurable_snd))
-  exact (Measure.measurable_coe (Metric.isClosed_closedBall).measurableSet).comp hn
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-
- 
-
-def finiteAngleLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (l : ℕ) :
-    Measure (Angle×ExactSLEGenerator.Driver) :=
-  ((angleLaw a).prod W).withDensity (fun p=>ENNReal.ofReal (tiltWeight a p.1.1 p.2.1 l))
-
-lemma finiteAngleLaw_absolutelyContinuous (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (l : ℕ) : finiteAngleLaw W a l≪(angleLaw a).prod W :=
-  withDensity_absolutelyContinuous _ _
-
-lemma finiteAngleLaw_lintegral (W : Measure ExactSLEGenerator.Driver) [SFinite W]
-    (a : ℝ) (ha : 1/4<a) (l : ℕ)
-    (f : Angle×ExactSLEGenerator.Driver→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻p,f p∂finiteAngleLaw W a l)=∫⁻θ : Angle,∫⁻b,f (θ,b)
-      ∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))∂angleLaw a := by
-  have hD := (centeredWeight_measurable a ha l).ennreal_ofReal
-  rw [finiteAngleLaw,lintegral_withDensity_eq_lintegral_mul _ hD hf,
-    lintegral_prod _ ((hD.mul hf).aemeasurable)]
-  apply lintegral_congr
-  intro θ
-  have hd := hD.comp ((measurable_const (a:=θ)).prodMk measurable_id)
-  have ht := hf.comp ((measurable_const (a:=θ)).prodMk measurable_id)
-  have he : (fun b : ExactSLEGenerator.Driver=>ENNReal.ofReal (tiltWeight a θ.1 b.1 l))=
-      fun b : ExactSLEGenerator.Driver=>ENNReal.ofReal (radialWeight a b.1 (fromDisk θ.1 0)
-        (Real.sin θ.1*Real.exp (-2*a*l))/physicalGreen a (fromDisk θ.1 0,0)) := by
-    funext b
-    rw [tiltWeight_radialWeight a θ.1 ha θ.2.1 θ.2.2 b l]
-  change (∫⁻b,ENNReal.ofReal (tiltWeight a θ.1 b.1 l)*f (θ,b)∂W)=_
-  rw [radialLaw,←he]
-  exact (lintegral_withDensity_eq_lintegral_mul W hd ht).symm
-
-lemma finiteAngleLaw_probability (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (l : ℕ) : IsProbabilityMeasure (finiteAngleLaw W a l) := by
-  have ha0 : 0<a := by linarith
-  have : IsProbabilityMeasure (angleLaw a) := angleLaw_probability a ha0.le
-  have hp (θ : Angle) : IsProbabilityMeasure
-      (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))) := by
-    have hs := Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
-    apply radialLaw_probability W hW a ha _ (by simpa only [fromDisk_zero_im] using hs)
-    · exact mul_pos hs (Real.exp_pos _)
-    · rw [fromDisk_zero_im]
-      apply mul_le_of_le_one_right hs.le
-      apply Real.exp_le_one_iff.mpr
-      have hl : (0:ℝ)≤l := Nat.cast_nonneg _
-      nlinarith
-  have (θ : Angle) : IsProbabilityMeasure
-      (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))) := hp θ
-  rw [isProbabilityMeasure_iff,←lintegral_one,
-    finiteAngleLaw_lintegral W a ha l _ measurable_const]
-  simp only [lintegral_one, measure_univ]
-
- 
-
-def finiteMoment (W : Measure ExactSLEGenerator.Driver) (a : ℝ)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane) (m l : ℕ) : ℝ≥0∞ :=
-  ∫⁻p,(centeredMass a μ l p)^m∂finiteAngleLaw W a l
-
- 
-
-theorem common_limit_mixed_moment_bound (a : ℝ) (ha : 1/4<a)
-    (m : ℕ) (hm : 1 ≤ m) :
-    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
-      (∀v b,IsFiniteMeasureOnCompacts (μ v b)) →
-      Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2) →
-      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
-      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
-          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
-      ∀l, finiteMoment W a μ m l≤D := by
-  obtain ⟨C,hC,hCb⟩ := common_limit_disk_tilted_moment a ha m hm
-  let J := ∫⁻θ : Angle,ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a
-  refine ⟨C*J,ENNReal.mul_lt_top hC (angle_inverse_power_finite a),?_⟩
-  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim l
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := hμfin v b
-  rw [finiteMoment,finiteAngleLaw_lintegral W a ha l _
-    ((centeredMass_measurable a ha μ hμ l).pow_const m)]
-  calc
-    _ ≤ ∫⁻θ : Angle,C*ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a := by
-      apply lintegral_mono
-      intro θ
-      let T := fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l
-      exact hCb W hW s coeff hcoeff hsum μ (hμfin ∞)
-        (hμ.comp (measurable_const.prodMk measurable_id)) hmono hlim
-        θ.1 θ.2.1 θ.2.2 l T
-    _ = C*J := lintegral_const_mul _ (by fun_prop)
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
-
- 
-def pointAngle (w : UpperPlane) : Angle := Jacobi.angleState (w.1,0)
-
-lemma pointAngle_measurable : Measurable pointAngle :=
-  Jacobi.angleState_measurable.comp (measurable_subtype_coe.prodMk measurable_const)
-
-lemma point_norm_pos (w : UpperPlane) : 0<‖w.1‖ := w.2.trans_le (Complex.im_le_norm _)
-
-lemma fromDisk_pointAngle (w : UpperPlane) :
-    fromDisk (pointAngle w).1 0 = ((‖w.1‖:ℝ):ℂ)⁻¹*w.1 := by
-  have hn := point_norm_pos w
-  have hs := Jacobi.sin_physicalTheta (w.1,0) w.2
-  have hc := physicalTheta_cot (w.1,0)
-  have hsin := Real.sin_pos_of_pos_of_lt_pi (pointAngle w).2.1 (pointAngle w).2.2
-  change Real.sin (pointAngle w).1 = w.1.im/‖w.1‖ at hs
-  change Real.cot (pointAngle w).1 = w.1.re/w.1.im at hc
-  rw [Real.cot_eq_cos_div_sin,hs,div_eq_div_iff (div_ne_zero w.2.ne' hn.ne') w.2.ne'] at hc
-  have hcos : Real.cos (pointAngle w).1=w.1.re/‖w.1‖ := by
-    apply (eq_div_iff hn.ne').mpr
-    field_simp at hc
-    exact (mul_left_cancel₀ w.2.ne' (by nlinarith :
-      w.1.im * (Real.cos (pointAngle w).1 * ‖w.1‖) = w.1.im * w.1.re))
-  rw [fromDisk_zero,mul_comm,Complex.exp_mul_I,←Complex.ofReal_inv]
-  simp only [←Complex.ofReal_cos, ←Complex.ofReal_sin]
-  apply Complex.ext <;> simp only [Complex.add_re,Complex.add_im,Complex.mul_re,Complex.mul_im,
-    Complex.ofReal_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,mul_zero,zero_mul,
-    add_zero,zero_add,mul_one,sub_zero]
-  · rw [hcos]; ring
-  · rw [hs]; ring
-
-lemma norm_mul_fromDisk_pointAngle (w : UpperPlane) :
-    ((‖w.1‖:ℝ):ℂ)*fromDisk (pointAngle w).1 0=w.1 := by
-  rw [fromDisk_pointAngle,←mul_assoc,mul_inv_cancel₀
-    (Complex.ofReal_ne_zero.mpr (point_norm_pos w).ne'),one_mul]
-
-lemma sin_pointAngle (w : UpperPlane) : Real.sin (pointAngle w).1=w.1.im/‖w.1‖ :=
-  Jacobi.sin_physicalTheta (w.1,0) w.2
-
- 
-def normalizeDriver (w : UpperPlane) (b : GDriver) : GDriver :=
-  scaleDriver ‖w.1‖₊ b
-
-lemma continuous_normalizeDriver : Continuous (fun p : UpperPlane×GDriver => normalizeDriver p.1 p.2) := by
-  apply Continuous.subtype_mk
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  change Continuous (fun p : (UpperPlane×GDriver)×ℝ≥0 =>
-    ‖p.1.1.1‖⁻¹*p.1.2.1 (‖p.1.1.1‖₊^2*p.2))
-  have hn : Continuous (fun p : (UpperPlane×GDriver)×ℝ≥0 => ‖p.1.1.1‖) := by fun_prop
-  have ht : Continuous (fun p : (UpperPlane×GDriver)×ℝ≥0 =>
-      ‖p.1.1.1‖₊^2*p.2) := by
-    exact ((hn.pow 2).mul (NNReal.continuous_coe.comp continuous_snd)).subtype_mk _
-  exact (hn.inv₀ (fun p=>(point_norm_pos p.1.1).ne')).mul
-    (continuous_eval.comp ((continuous_subtype_val.comp continuous_fst.snd).prodMk
-      ht))
-
-lemma radiusDensity_scaled (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
-    (b : GDriver) (z : ℂ) (hz : 0<z.im) (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
-    radiusDensity a z ρ (scaleDriver c b)=
-      radiusDensity a (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) b := by
-  have he : (z,(0:ℝ))=scaleState c ((((c:ℝ):ℂ)*z),0) := by
-    simp only [scaleState,←mul_assoc,inv_mul_cancel₀
-      (Complex.ofReal_ne_zero.mpr (NNReal.coe_pos.mpr hc).ne'),one_mul]
-  have hy : 0<Y ((((c:ℝ):ℂ)*z),0) := by
-    simpa [Complex.mul_im] using mul_pos (show 0<(c:ℝ) from hc) hz
-  unfold radiusDensity
-  rw [radialWeight_scaled a ha hc b z hz ρ hρ hr,he,scaleState_green a hc _ hy,
-    mul_div_mul_left _ _ (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) (alpha a)).ne']
-
- 
-
-lemma radialLaw_scale_integral (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (z : ℂ) (hz : 0<z.im) (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im)
-    (f : GDriver→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻b,f (scaleDriver c b)∂radialLaw W a (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ))=
-      ∫⁻b,f b∂radialLaw W a z ρ := by
-  have hz' : 0<(((c:ℝ):ℂ)*z).im := by
-    simpa [Complex.mul_im] using mul_pos (show 0<(c:ℝ) from hc) hz
-  have hρ' : 0<(c:ℝ)*ρ := mul_pos hc hρ
-  have hr' : (c:ℝ)*ρ≤(((c:ℝ):ℂ)*z).im := by
-    simpa [Complex.mul_im] using mul_le_mul_of_nonneg_left hr c.coe_nonneg
-  change (∫⁻b,f (scaleDriver c b)∂W.withDensity (radiusDensity a (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ)))=
-    ∫⁻b,f b∂W.withDensity (radiusDensity a z ρ)
-  have hleft := lintegral_withDensity_eq_lintegral_mul W
-    (radiusDensity_measurable a ha _ hz' _ hρ' hr') (hf.comp (continuous_scaleDriver c).measurable)
-  simp only [Function.comp_def] at hleft
-  rw [hleft, lintegral_withDensity_eq_lintegral_mul W (radiusDensity_measurable a ha z hz ρ hρ hr) hf]
-  simp only [Pi.mul_apply]
-  simp_rw [←radiusDensity_scaled a (by linarith) hc _ z hz ρ hρ hr]
-  exact (measurePreserving_scaleDriver hW hc).lintegral_comp
-    ((radiusDensity_measurable a ha z hz ρ hρ hr).mul hf)
-
-end SLEExactGauge.LoewnerRestart
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RecoveredInteriorRestart
+import OAI.Probability.SLEGauge.RadialRestartRaw
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments
+
+lemma radialLaw_consistent_le (W : Measure GDriver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
+      (hρ.trans_le hρσ) hσ).measurableSpace] A) :
+    radialLaw W a z ρ A=radialLaw W a z σ A := by
+  rcases hρσ.eq_or_lt with rfl|hρσ
+  · rfl
+  · obtain ⟨_,_,hS,he⟩ := finite_radial_laws_consistent W hW a ha z hz ρ σ hρ hρσ hσ
+    exact he A hA
+
+ 
+
+theorem finiteAngle_conditional_integral_bounds (W : Measure GDriver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=σ*Real.exp (-2*a*s))
+    (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
+      (hρ.trans_le hρσ) hσ).measurableSpace] A)
+    (f : Angle→ℝ≥0∞) (hf : Measurable f) :
+    (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z σ A ≤
+      ∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ ∧
+    (∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ) ≤
+      2*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z σ A := by
+  have hs0 : 0 ≤ s := le_trans (by unfold Jacobi.mixingTime Jacobi.majorantBase; positivity) hs
+  have hσ0 := hρ.trans_le hρσ
+  have he (w : UpperPlane) : w.1.im*(ρ/σ)=levelRadius a (2*w.1.im) s := by
+    rw [hgap]
+    unfold levelRadius
+    field_simp
+  have hb (w : UpperPlane) := Jacobi.finiteAngleLaw_integral_bounds W hW a ha w.1 w.2 s hs f hf
+  have hb' (w : UpperPlane) :
+      (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a) ≤
+        ∫⁻c,f (Jacobi.finiteAngle a w.1 (w.1.im*(ρ/σ)) c)∂radialLaw W a w.1 (w.1.im*(ρ/σ)) ∧
+      (∫⁻c,f (Jacobi.finiteAngle a w.1 (w.1.im*(ρ/σ)) c)∂radialLaw W a w.1 (w.1.im*(ρ/σ))) ≤
+        2*(∫⁻θ,f θ∂angleLaw a) := by
+    rw [he w]
+    simpa only [Jacobi.finiteAngleLaw,lintegral_map hf
+      (Jacobi.finiteAngle_measurable a (by linarith) w.1 w.2 _
+        (levelRadius_pos a _ s (mul_pos (by norm_num) w.2))
+        (Jacobi.levelRadius_initial_le a (by linarith) w.1 w.2 s hs0))] using hb w
+  rw [finiteAngle_conditional W hW a ha z hz σ ρ hρ hρσ hσ A hA f hf]
+  constructor
+  · simpa only [lintegral_const,Measure.restrict_apply_univ] using
+      lintegral_mono (μ := (radialLaw W a z σ).restrict A)
+        (fun b => (hb' (radialPoint a z σ b)).1)
+  · simpa only [lintegral_const,Measure.restrict_apply_univ] using
+      lintegral_mono (μ := (radialLaw W a z σ).restrict A)
+        (fun b => (hb' (radialPoint a z σ b)).2)
+
+ 
+
+theorem finiteAngle_conditional_bounds (W : Measure GDriver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=σ*Real.exp (-2*a*s))
+    (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
+      (hρ.trans_le hρσ) hσ).measurableSpace] A)
+    (f : Angle→ℝ≥0∞) (hf : Measurable f) :
+    (1/2:ℝ≥0∞)*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z ρ A ≤
+      ∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ ∧
+    (∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ) ≤
+      2*(∫⁻θ,f θ∂angleLaw a)*radialLaw W a z ρ A := by
+  rw [radialLaw_consistent_le W hW a ha z hz σ ρ hρ hρσ hσ A hA]
+  exact finiteAngle_conditional_integral_bounds W hW a ha z hz σ ρ hρ hρσ hσ s hs hgap A hA f hf
+
+ 
+
+lemma radial_change_law_event (W : Measure GDriver)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (A : Set GDriver) (hA : MeasurableSet A) :
+    W (A∩{b | radialTime a b.1 z ρ≠∞}) =
+      ENNReal.ofReal (ρ^(alpha a)*physicalGreen a (z,0))*
+        ∫⁻b in A,ENNReal.ofReal ((Real.sin (Jacobi.finiteAngle a z ρ b : ℝ))^(-p a))
+          ∂radialLaw W a z ρ := by
+  have he := Jacobi.radial_change_law W a ha z hz ρ hρ hr (A.indicator (fun _=>1))
+    (measurable_const.indicator hA)
+  have hi : (fun b => ENNReal.ofReal
+      ((Real.sin (Jacobi.finiteAngle a z ρ b : ℝ))^(-p a))*A.indicator (fun _=>1) b) =
+      A.indicator (fun b => ENNReal.ofReal
+        ((Real.sin (Jacobi.finiteAngle a z ρ b : ℝ))^(-p a))) := by
+    funext b
+    by_cases hb : b∈A <;> simp [hb]
+  simp only [hi,lintegral_indicator hA,Measure.restrict_restrict hA,lintegral_one,
+    Measure.restrict_apply_univ] at he
+  simpa only [inter_comm] using he
+
+ 
+
+theorem radial_change_law_past_bound (W : Measure GDriver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : ρ=σ*Real.exp (-2*a*s))
+    (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
+      (hρ.trans_le hρσ) hσ).measurableSpace] A) :
+    W (A∩{b | radialTime a b.1 z ρ≠∞}) ≤
+      Jacobi.visitConstant a*ENNReal.ofReal ((ρ/z.im)^(alpha a))*radialLaw W a z ρ A := by
+  have hA' : MeasurableSet A := (radialTime_stopping a (by linarith) z hz σ
+    (hρ.trans_le hρσ) hσ).measurableSpace_le _ hA
+  rw [radial_change_law_event W a ha z hz ρ hρ (hρσ.trans hσ) A hA']
+  have hb := (finiteAngle_conditional_bounds W hW a ha z hz σ ρ hρ hρσ hσ s hs hgap A hA
+    (fun θ=>ENNReal.ofReal ((Real.sin θ)^(-p a))) (by fun_prop)).2
+  have hG : ENNReal.ofReal (ρ^(alpha a)*physicalGreen a (z,0)) ≤
+      ENNReal.ofReal ((ρ/z.im)^(alpha a)) := by
+    apply ENNReal.ofReal_le_ofReal
+    rw [Real.div_rpow hρ.le hz.le,div_eq_mul_inv,←Real.rpow_neg hz.le]
+    exact mul_le_mul_of_nonneg_left (Jacobi.physicalGreen_initial_bound a ha z hz)
+      (Real.rpow_nonneg hρ.le _)
+  calc
+    _ ≤ ENNReal.ofReal ((ρ/z.im)^(alpha a))*
+        (2*(∫⁻θ : Angle,ENNReal.ofReal ((Real.sin θ)^(-p a))∂angleLaw a)*radialLaw W a z ρ A) :=
+      mul_le_mul hG hb zero_le zero_le
+    _ ≤ _ := by
+      rw [←mul_assoc,mul_comm (ENNReal.ofReal _) (2*_)]
+      exact mul_le_mul_of_nonneg_right
+        (mul_le_mul_of_nonneg_right (le_add_right le_rfl) zero_le) zero_le
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+
+lemma actualUpperMass_eval_joint_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (f : C_c(UpperPlane,ℝ)) :
+    Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>∫z,f z∂actualUpperMass a q p.1 p.2) := by
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  have hm : Measurable (fun p : (ℝ≥0∞×ExactSLEGenerator.Driver) × ℂ =>
+      upperExtension f p.2*(timeMassDensity a q p.1.1 p.1.2 p.2).toReal) :=
+    ((upperExtension_measurable f).comp measurable_snd).mul
+      (timeMassDensity_joint_measurable a ha q hq hq1).ennreal_toReal
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+
+lemma recoveredMass_joint_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j) :
+    Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>recoveredMass a ha s c hc p.1 p.2) := by
+  apply measurable_radonLimit
+  intro k f
+  simp_rw [actualMassFunctionals_apply]
+  apply Finset.measurable_sum
+  intro j hj
+  exact measurable_const.mul (actualUpperMass_eval_joint_measurable a ha _
+    (Real.exp_pos _) (level_step_le_one a ha j) f)
+
+lemma centeredTime_measurable (a : ℝ) (ha : 1/4<a) (l : ℕ) :
+    Measurable (fun p : Angle×ExactSLEGenerator.Driver=>levelTime a p.1.1 p.2.1 l) := by
+  have hz : Measurable (fun θ : Angle => (⟨fromDisk θ.1 0,by
+      rw [fromDisk_zero_im]
+      exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2⟩ : UpperPlane)) := by
+    apply Measurable.subtype_mk
+    simp only [fromDisk_zero]
+    fun_prop
+  have hr : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
+    have hl : (0:ℝ)≤l := Nat.cast_nonneg _
+    have ha0 : 0<a := by linarith
+    nlinarith)
+  convert (joint_radialTime_measurable a ha _ (Real.exp_pos _) hr).comp
+    (measurable_snd.prodMk (hz.comp measurable_fst)) using 1
+  funext p
+  simp only [levelTime_radialTime a p.1.1 p.1.2.1 p.1.2.2 p.2.1 p.2.1.continuous,
+    Function.comp_apply,fromDisk_zero_im]
+
+lemma centeredWeight_measurable (a : ℝ) (ha : 1/4<a) (l : ℕ) :
+    Measurable (fun p : Angle×ExactSLEGenerator.Driver=>tiltWeight a p.1.1 p.2.1 l) := by
+  have hz : Measurable (fun θ : Angle => (⟨fromDisk θ.1 0,by
+      rw [fromDisk_zero_im]
+      exact Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2⟩ : UpperPlane)) := by
+    apply Measurable.subtype_mk
+    simp only [fromDisk_zero]
+    fun_prop
+  have hr : Real.exp (-2*a*l)≤1 := Real.exp_le_one_iff.mpr (by
+    have hl : (0:ℝ)≤l := Nat.cast_nonneg _
+    have ha0 : 0<a := by linarith
+    nlinarith)
+  have hG : Measurable (fun θ : Angle=>physicalGreen a (fromDisk θ.1 0,0)) := by
+    have he : (fun θ : Angle=>physicalGreen a (fromDisk θ.1 0,0))=
+        fun θ : Angle => Real.sin θ.1^(-alpha a)*Real.sin θ.1^(p a) := by
+      funext θ
+      exact radial_initial_green a θ.1 θ.2.1 θ.2.2
+    rw [he]
+    fun_prop
+  convert ((joint_radialWeight_measurable a ha _ (Real.exp_pos _) hr).comp
+    (measurable_snd.prodMk (hz.comp measurable_fst))).div (hG.comp measurable_fst) using 1
+  funext p
+  simp only [tiltWeight_radialWeight a p.1.1 ha p.1.2.1 p.1.2.2 p.2 l,Function.comp_apply,Pi.div_apply,fromDisk_zero_im]
+
+lemma diskMap_joint_measurable :
+    Measurable (fun p : Angle×UpperPlane => diskMap p.1.1 p.2.1) := by
+  unfold diskMap
+  fun_prop
+
+lemma diskWeight_joint_measurable (a : ℝ) :
+    Measurable (fun p : Angle×UpperPlane => diskWeight a p.1.1 p.2) := by
+  have he : (fun p : Angle×UpperPlane=>diskWeight a p.1.1 p.2)=
+      fun p : Angle×UpperPlane =>
+        ‖((2*Real.sin p.1.1:ℝ):ℂ)*Complex.I/(p.2.1-Complex.exp (-Complex.I*p.1.1))^2‖^(2-alpha a) := by
+    funext p
+    rw [diskWeight,disk_deriv p.1.1 p.1.2.1 p.1.2.2 p.2]
+  rw [he]
+  fun_prop
+
+ 
+
+lemma diskMass_joint_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (a : ℝ) (θ : Ω→Angle) (hθ : Measurable θ) (ν : Ω→Measure UpperPlane)
+    (hν : Measurable ν) [∀ω,IsFiniteMeasureOnCompacts (ν ω)] :
+    Measurable (fun ω=>diskMass a (θ ω).1 (ν ω)) := by
+  let k : Kernel Ω UpperPlane := ⟨ν,hν⟩
+  have : IsSFiniteKernel k := isSFiniteKernel_of_locally_finite k
+  apply Measure.measurable_of_measurable_coe
+  intro A hA
+  have hm : Measurable (fun p : Ω×UpperPlane =>
+      A.indicator (fun _=> (1:ℝ≥0∞)) (diskMap (θ p.1).1 p.2.1)*
+        ENNReal.ofReal (diskWeight a (θ p.1).1 p.2)) :=
+    ((measurable_const.indicator hA).comp (diskMap_joint_measurable.comp
+      ((hθ.comp measurable_fst).prodMk measurable_snd))).mul
+      ((diskWeight_joint_measurable a).comp
+        ((hθ.comp measurable_fst).prodMk measurable_snd)).ennreal_ofReal
+  have he (ω : Ω) : diskMass a (θ ω).1 (ν ω) A =
+      ∫⁻z,A.indicator (fun _=>(1:ℝ≥0∞)) (diskMap (θ ω).1 z.1)*
+        ENNReal.ofReal (diskWeight a (θ ω).1 z)∂k ω := by
+    rw [diskMass,Measure.map_apply (diskMap_measurable _) hA,
+      withDensity_apply _ ((diskMap_measurable _) hA),
+      ←lintegral_indicator ((diskMap_measurable _) hA)]
+    apply lintegral_congr
+    intro z
+    by_cases hz : diskMap (θ ω).1 z.1∈A <;>
+      simp [Set.indicator,Set.mem_preimage,hz]
+  simp_rw [he]
+  exact hm.lintegral_kernel_prod_right
+
+ 
+
+def centeredMass (a : ℝ) (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (l : ℕ) (p : Angle×ExactSLEGenerator.Driver) : ℝ≥0∞ :=
+  diskMass a p.1.1 (μ (levelTime a p.1.1 p.2.1 l) p.2) (Metric.closedBall (0:ℂ) (1/4))
+
+lemma centeredMass_measurable (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (l : ℕ) :
+    Measurable (centeredMass a μ l) := by
+  have hn := diskMass_joint_measurable a Prod.fst measurable_fst
+    (fun p : Angle×ExactSLEGenerator.Driver=>μ (levelTime a p.1.1 p.2.1 l) p.2)
+    (hμ.comp ((centeredTime_measurable a ha l).prodMk measurable_snd))
+  exact (Measure.measurable_coe (Metric.isClosed_closedBall).measurableSet).comp hn
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+
+ 
+
+def finiteAngleLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (l : ℕ) :
+    Measure (Angle×ExactSLEGenerator.Driver) :=
+  ((angleLaw a).prod W).withDensity (fun p=>ENNReal.ofReal (tiltWeight a p.1.1 p.2.1 l))
+
+lemma finiteAngleLaw_absolutelyContinuous (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (l : ℕ) : finiteAngleLaw W a l≪(angleLaw a).prod W :=
+  withDensity_absolutelyContinuous _ _
+
+lemma finiteAngleLaw_lintegral (W : Measure ExactSLEGenerator.Driver) [SFinite W]
+    (a : ℝ) (ha : 1/4<a) (l : ℕ)
+    (f : Angle×ExactSLEGenerator.Driver→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻p,f p∂finiteAngleLaw W a l)=∫⁻θ : Angle,∫⁻b,f (θ,b)
+      ∂radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))∂angleLaw a := by
+  have hD := (centeredWeight_measurable a ha l).ennreal_ofReal
+  rw [finiteAngleLaw,lintegral_withDensity_eq_lintegral_mul _ hD hf,
+    lintegral_prod _ ((hD.mul hf).aemeasurable)]
+  apply lintegral_congr
+  intro θ
+  have hd := hD.comp ((measurable_const (a:=θ)).prodMk measurable_id)
+  have ht := hf.comp ((measurable_const (a:=θ)).prodMk measurable_id)
+  have he : (fun b : ExactSLEGenerator.Driver=>ENNReal.ofReal (tiltWeight a θ.1 b.1 l))=
+      fun b : ExactSLEGenerator.Driver=>ENNReal.ofReal (radialWeight a b.1 (fromDisk θ.1 0)
+        (Real.sin θ.1*Real.exp (-2*a*l))/physicalGreen a (fromDisk θ.1 0,0)) := by
+    funext b
+    rw [tiltWeight_radialWeight a θ.1 ha θ.2.1 θ.2.2 b l]
+  change (∫⁻b,ENNReal.ofReal (tiltWeight a θ.1 b.1 l)*f (θ,b)∂W)=_
+  rw [radialLaw,←he]
+  exact (lintegral_withDensity_eq_lintegral_mul W hd ht).symm
+
+lemma finiteAngleLaw_probability (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (l : ℕ) : IsProbabilityMeasure (finiteAngleLaw W a l) := by
+  have ha0 : 0<a := by linarith
+  have : IsProbabilityMeasure (angleLaw a) := angleLaw_probability a ha0.le
+  have hp (θ : Angle) : IsProbabilityMeasure
+      (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))) := by
+    have hs := Real.sin_pos_of_pos_of_lt_pi θ.2.1 θ.2.2
+    apply radialLaw_probability W hW a ha _ (by simpa only [fromDisk_zero_im] using hs)
+    · exact mul_pos hs (Real.exp_pos _)
+    · rw [fromDisk_zero_im]
+      apply mul_le_of_le_one_right hs.le
+      apply Real.exp_le_one_iff.mpr
+      have hl : (0:ℝ)≤l := Nat.cast_nonneg _
+      nlinarith
+  have (θ : Angle) : IsProbabilityMeasure
+      (radialLaw W a (fromDisk θ.1 0) (Real.sin θ.1*Real.exp (-2*a*l))) := hp θ
+  rw [isProbabilityMeasure_iff,←lintegral_one,
+    finiteAngleLaw_lintegral W a ha l _ measurable_const]
+  simp only [lintegral_one, measure_univ]
+
+ 
+
+def finiteMoment (W : Measure ExactSLEGenerator.Driver) (a : ℝ)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane) (m l : ℕ) : ℝ≥0∞ :=
+  ∫⁻p,(centeredMass a μ l p)^m∂finiteAngleLaw W a l
+
+ 
+
+theorem common_limit_mixed_moment_bound (a : ℝ) (ha : 1/4<a)
+    (m : ℕ) (hm : 1 ≤ m) :
+    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
+      (∀v b,IsFiniteMeasureOnCompacts (μ v b)) →
+      Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2) →
+      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
+      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
+          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
+      ∀l, finiteMoment W a μ m l≤D := by
+  obtain ⟨C,hC,hCb⟩ := common_limit_disk_tilted_moment a ha m hm
+  let J := ∫⁻θ : Angle,ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a
+  refine ⟨C*J,ENNReal.mul_lt_top hC (angle_inverse_power_finite a),?_⟩
+  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim l
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := hμfin v b
+  rw [finiteMoment,finiteAngleLaw_lintegral W a ha l _
+    ((centeredMass_measurable a ha μ hμ l).pow_const m)]
+  calc
+    _ ≤ ∫⁻θ : Angle,C*ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a := by
+      apply lintegral_mono
+      intro θ
+      let T := fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l
+      exact hCb W hW s coeff hcoeff hsum μ (hμfin ∞)
+        (hμ.comp (measurable_const.prodMk measurable_id)) hmono hlim
+        θ.1 θ.2.1 θ.2.2 l T
+    _ = C*J := lintegral_const_mul _ (by fun_prop)
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator CenteredMoments LoewnerScaling
+
+ 
+def pointAngle (w : UpperPlane) : Angle := Jacobi.angleState (w.1,0)
+
+lemma pointAngle_measurable : Measurable pointAngle :=
+  Jacobi.angleState_measurable.comp (measurable_subtype_coe.prodMk measurable_const)
+
+lemma point_norm_pos (w : UpperPlane) : 0<‖w.1‖ := w.2.trans_le (Complex.im_le_norm _)
+
+lemma fromDisk_pointAngle (w : UpperPlane) :
+    fromDisk (pointAngle w).1 0 = ((‖w.1‖:ℝ):ℂ)⁻¹*w.1 := by
+  have hn := point_norm_pos w
+  have hs := Jacobi.sin_physicalTheta (w.1,0) w.2
+  have hc := physicalTheta_cot (w.1,0)
+  have hsin := Real.sin_pos_of_pos_of_lt_pi (pointAngle w).2.1 (pointAngle w).2.2
+  change Real.sin (pointAngle w).1 = w.1.im/‖w.1‖ at hs
+  change Real.cot (pointAngle w).1 = w.1.re/w.1.im at hc
+  rw [Real.cot_eq_cos_div_sin,hs,div_eq_div_iff (div_ne_zero w.2.ne' hn.ne') w.2.ne'] at hc
+  have hcos : Real.cos (pointAngle w).1=w.1.re/‖w.1‖ := by
+    apply (eq_div_iff hn.ne').mpr
+    field_simp at hc
+    exact (mul_left_cancel₀ w.2.ne' (by nlinarith :
+      w.1.im * (Real.cos (pointAngle w).1 * ‖w.1‖) = w.1.im * w.1.re))
+  rw [fromDisk_zero,mul_comm,Complex.exp_mul_I,←Complex.ofReal_inv]
+  simp only [←Complex.ofReal_cos, ←Complex.ofReal_sin]
+  apply Complex.ext <;> simp only [Complex.add_re,Complex.add_im,Complex.mul_re,Complex.mul_im,
+    Complex.ofReal_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,mul_zero,zero_mul,
+    add_zero,zero_add,mul_one,sub_zero]
+  · rw [hcos]; ring
+  · rw [hs]; ring
+
+lemma norm_mul_fromDisk_pointAngle (w : UpperPlane) :
+    ((‖w.1‖:ℝ):ℂ)*fromDisk (pointAngle w).1 0=w.1 := by
+  rw [fromDisk_pointAngle,←mul_assoc,mul_inv_cancel₀
+    (Complex.ofReal_ne_zero.mpr (point_norm_pos w).ne'),one_mul]
+
+lemma sin_pointAngle (w : UpperPlane) : Real.sin (pointAngle w).1=w.1.im/‖w.1‖ :=
+  Jacobi.sin_physicalTheta (w.1,0) w.2
+
+ 
+def normalizeDriver (w : UpperPlane) (b : GDriver) : GDriver :=
+  scaleDriver ‖w.1‖₊ b
+
+lemma continuous_normalizeDriver : Continuous (fun p : UpperPlane×GDriver => normalizeDriver p.1 p.2) := by
+  apply Continuous.subtype_mk
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  change Continuous (fun p : (UpperPlane×GDriver)×ℝ≥0 =>
+    ‖p.1.1.1‖⁻¹*p.1.2.1 (‖p.1.1.1‖₊^2*p.2))
+  have hn : Continuous (fun p : (UpperPlane×GDriver)×ℝ≥0 => ‖p.1.1.1‖) := by fun_prop
+  have ht : Continuous (fun p : (UpperPlane×GDriver)×ℝ≥0 =>
+      ‖p.1.1.1‖₊^2*p.2) := by
+    exact ((hn.pow 2).mul (NNReal.continuous_coe.comp continuous_snd)).subtype_mk _
+  exact (hn.inv₀ (fun p=>(point_norm_pos p.1.1).ne')).mul
+    (continuous_eval.comp ((continuous_subtype_val.comp continuous_fst.snd).prodMk
+      ht))
+
+lemma radiusDensity_scaled (a : ℝ) (ha : 0<a) {c : ℝ≥0} (hc : 0<c)
+    (b : GDriver) (z : ℂ) (hz : 0<z.im) (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
+    radiusDensity a z ρ (scaleDriver c b)=
+      radiusDensity a (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ) b := by
+  have he : (z,(0:ℝ))=scaleState c ((((c:ℝ):ℂ)*z),0) := by
+    simp only [scaleState,←mul_assoc,inv_mul_cancel₀
+      (Complex.ofReal_ne_zero.mpr (NNReal.coe_pos.mpr hc).ne'),one_mul]
+  have hy : 0<Y ((((c:ℝ):ℂ)*z),0) := by
+    simpa [Complex.mul_im] using mul_pos (show 0<(c:ℝ) from hc) hz
+  unfold radiusDensity
+  rw [radialWeight_scaled a ha hc b z hz ρ hρ hr,he,scaleState_green a hc _ hy,
+    mul_div_mul_left _ _ (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) (alpha a)).ne']
+
+ 
+
+lemma radialLaw_scale_integral (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (z : ℂ) (hz : 0<z.im) (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im)
+    (f : GDriver→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻b,f (scaleDriver c b)∂radialLaw W a (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ))=
+      ∫⁻b,f b∂radialLaw W a z ρ := by
+  have hz' : 0<(((c:ℝ):ℂ)*z).im := by
+    simpa [Complex.mul_im] using mul_pos (show 0<(c:ℝ) from hc) hz
+  have hρ' : 0<(c:ℝ)*ρ := mul_pos hc hρ
+  have hr' : (c:ℝ)*ρ≤(((c:ℝ):ℂ)*z).im := by
+    simpa [Complex.mul_im] using mul_le_mul_of_nonneg_left hr c.coe_nonneg
+  change (∫⁻b,f (scaleDriver c b)∂W.withDensity (radiusDensity a (((c:ℝ):ℂ)*z) ((c:ℝ)*ρ)))=
+    ∫⁻b,f b∂W.withDensity (radiusDensity a z ρ)
+  have hleft := lintegral_withDensity_eq_lintegral_mul W
+    (radiusDensity_measurable a ha _ hz' _ hρ' hr') (hf.comp (continuous_scaleDriver c).measurable)
+  simp only [Function.comp_def] at hleft
+  rw [hleft, lintegral_withDensity_eq_lintegral_mul W (radiusDensity_measurable a ha z hz ρ hρ hr) hf]
+  simp only [Pi.mul_apply]
+  simp_rw [←radiusDensity_scaled a (by linarith) hc _ z hz ρ hρ hr]
+  exact (measurePreserving_scaleDriver hW hc).lintegral_comp
+    ((radiusDensity_measurable a ha z hz ρ hρ hr).mul hf)
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialFactors.lean b/lean/OAI/Probability/SLEGauge/RadialFactors.lean
index fec75206131920787c1b1799a5d123be98088265..4e5e50e450b95f65298c2a090b8ac32bef3c8d5b 100644
--- a/lean/OAI/Probability/SLEGauge/RadialFactors.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialFactors.lean
@@ -1,437 +1,437 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.AngleGenerators
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-namespace ExactSLEFiniteTilt
-open Filter
-open scoped Topology
-
-lemma regFactor_hasDerivAt (a δ θ : ℝ) (hθ : 0 < Real.sin θ) :
-    HasDerivAt (regFactor a δ)
-      (Real.cos θ*regP a*Real.sin θ^(regP a-1) +
-        δ*(Real.cos θ*regQ a*Real.sin θ^(regQ a-1))) θ := by
-  convert (sinePower_hasDerivAt (regP a) θ hθ).add
-      (((sinePower_hasDerivAt (regQ a) θ hθ).const_add 1).const_mul δ) using 1
-  rfl
-
-lemma sineDerivative_hasDerivAt (r θ : ℝ) (hθ : 0 < Real.sin θ) :
-    HasDerivAt (fun x => Real.cos x*r*Real.sin x^(r-1))
-      (deriv (deriv (sinePower r)) θ) θ := by
-  rw [sinePower_secondDeriv r θ hθ]
-  convert ((Real.hasDerivAt_cos θ).mul_const r).mul
-    ((Real.hasDerivAt_sin θ).rpow_const (p := r-1) (Or.inl hθ.ne')) using 1
-  congr 2
-  ring_nf
-
-lemma regFactor_secondDeriv (a δ θ : ℝ) (hθ : 0 < Real.sin θ) :
-    deriv (deriv (regFactor a δ)) θ = deriv (deriv (sinePower (regP a))) θ +
-      δ*deriv (deriv (sinePower (regQ a))) θ := by
-  have he : deriv (regFactor a δ) =ᶠ[𝓝 θ]
-      (fun x => Real.cos x*regP a*Real.sin x^(regP a-1) +
-        δ*(Real.cos x*regQ a*Real.sin x^(regQ a-1))) := by
-    have hp : ∀ᶠ x in 𝓝 θ, 0 < Real.sin x :=
-      (isOpen_lt continuous_const Real.continuous_sin).mem_nhds hθ
-    filter_upwards [hp] with x hx using (regFactor_hasDerivAt a δ x hx).deriv
-  rw [he.deriv_eq]
-  exact ((sineDerivative_hasDerivAt (regP a) θ hθ).add
-    ((sineDerivative_hasDerivAt (regQ a) θ hθ).const_mul δ)).deriv
-
-lemma regFactor_generator (a δ θ : ℝ) (hθ : 0 < Real.sin θ) :
-    angleGenerator (1-2*a) (regFactor a δ) θ =
-      angleGenerator (1-2*a) (sinePower (regP a)) θ +
-      δ*angleGenerator (1-2*a) (sinePower (regQ a)) θ := by
-  dsimp only [angleGenerator]
-  rw [regFactor_secondDeriv a δ θ hθ,(regFactor_hasDerivAt a δ θ hθ).deriv,
-    (sinePower_hasDerivAt (regP a) θ hθ).deriv,
-    (sinePower_hasDerivAt (regQ a) θ hθ).deriv]
-  ring
-
-lemma sinePower_mul_cot_sq (r θ : ℝ) (hθ : 0 < Real.sin θ) :
-    sinePower r θ*Real.cot θ^2 = Real.sin θ^(r-2)-sinePower r θ := by
-  dsimp only [sinePower]
-  rw [Real.cot_eq_cos_div_sin,div_pow,Real.rpow_sub hθ,Real.rpow_two]
-  field_simp
-  nlinarith [Real.sin_sq_add_cos_sq θ]
-
-lemma regFactor_exact_drift (a δ σ θ : ℝ) (hθ : 0 < Real.sin θ) :
-    regNormalizedDrift a δ σ θ = δ/regFactor a δ θ *
-      (regP a/2+(regP a-regQ a)*(1+regQ a)/2*Real.sin θ^(regQ a) -
-        regK a*Real.sin θ^(regQ a-2) + σ*(1+Real.sin θ^(regQ a))) := by
-  have hG := angleGenerator_sinePower (1-2*a) (regQ a) θ hθ
-  have hP := angleGenerator_sinePower (1-2*a) (regP a) θ hθ
-  have hpq : regP a-1+2*(1-2*a) = 0 := by dsimp [regP]; ring
-  have hqp : regQ a-1+2*(1-2*a) = regQ a-regP a := by dsimp [regP]; ring
-  simp only [hpq,hqp] at hP hG
-  rw [show sinePower (regQ a) θ *
-    (-regQ a/2+regQ a*(regQ a-regP a)/2*Real.cot θ^2) =
-      -regQ a/2*sinePower (regQ a) θ + regQ a*(regQ a-regP a)/2*
-        (sinePower (regQ a) θ*Real.cot θ^2) by ring,
-    sinePower_mul_cot_sq _ _ hθ] at hG
-  dsimp only [regNormalizedDrift]
-  rw [regFactor_generator a δ θ hθ,hP,hG]
-  dsimp only [regFactor,regK,sinePower]
-  ring
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLEFiniteTilt
-
-lemma regFactor_bounds {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hθ : 0 < Real.sin θ) :
-    0 < regFactor a δ θ ∧ δ ≤ regFactor a δ θ ∧
-    Real.sin θ^(regP a) ≤ regFactor a δ θ ∧ regFactor a δ θ ≤ 3 := by
-  have hs1 := Real.sin_le_one θ
-  have hp := Real.rpow_pos_of_pos hθ (regP a)
-  have hq := Real.rpow_pos_of_pos hθ (regQ a)
-  have hp1 := Real.rpow_le_one hθ.le hs1 (regP_pos ha).le
-  have hq1 := Real.rpow_le_one hθ.le hs1 (regQ_pos ha).le
-  dsimp only [regFactor,sinePower]
-  have hmul := mul_nonneg hδ.le hq.le
-  have hm1 := mul_le_mul_of_nonneg_left hq1 hδ.le
-  constructor
-  · positivity
-  constructor
-  · nlinarith
-  constructor
-  · nlinarith
-  · nlinarith
-
-lemma regDrift_bound {a δ σ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hσ : |σ| ≤ regBeta a) (hθ : 0 < Real.sin θ) :
-    regNormalizedDrift a δ σ θ ≤ regC a*δ-regK a/2*regD a δ θ := by
-  obtain ⟨hfp,hδf,hspf,hf3⟩ := regFactor_bounds ha hδ hδ1 hθ
-  have hp := regP_pos ha
-  have hq := regQ_pos ha
-  have hq1 := regQ_lt_one ha
-  have hpq := sub_pos.mpr (regQ_lt_p ha)
-  have hk := regK_pos ha
-  have hA := regA_pos ha
-  have hs0 := Real.rpow_pos_of_pos hθ (regQ a)
-  have hs1 := Real.rpow_le_one hθ.le (Real.sin_le_one θ) hq.le
-  have hσ1 : σ ≤ 1 := (le_abs_self σ).trans (hσ.trans (regBeta_le_one a))
-  have hσterm := mul_le_mul_of_nonneg_right hσ1 (show 0 ≤ 1+Real.sin θ^(regQ a) by positivity)
-  have hqterm := mul_le_mul_of_nonneg_left hs1
-    (show 0 ≤ (regP a-regQ a)*(1+regQ a)/2 by positivity)
-  have hbracket : regP a/2+(regP a-regQ a)*(1+regQ a)/2*Real.sin θ^(regQ a) -
-      regK a*Real.sin θ^(regQ a-2) + σ*(1+Real.sin θ^(regQ a)) ≤
-      regA a-regK a*Real.sin θ^(regQ a-2) := by
-    dsimp only [regA]; linarith
-  have hx : 0 ≤ δ/regFactor a δ θ := (div_pos hδ hfp).le
-  have hraw := mul_le_mul_of_nonneg_left hbracket hx
-  rw [←regFactor_exact_drift a δ σ θ hθ] at hraw
-  have hD : 0 ≤ regD a δ θ := mul_nonneg hx (Real.rpow_nonneg hθ.le _)
-  by_cases hsmall : Real.sin θ < regH a
-  · have hpow := Real.rpow_le_rpow_of_nonpos hθ hsmall.le (show regQ a-2 ≤ 0 by linarith)
-    rw [regH_power ha] at hpow
-    have hthresh : regA a ≤ regK a/2*Real.sin θ^(regQ a-2) := by
-      have hh := (div_le_iff₀ hk).mp hpow
-      nlinarith
-    have hh := mul_le_mul_of_nonneg_left hthresh hx
-    have hCδ : 0 ≤ regC a*δ := mul_nonneg (regC_pos ha).le hδ.le
-    dsimp only [regD]
-    nlinarith
-  · have hpow := Real.rpow_le_rpow (regH_pos ha).le (le_of_not_gt hsmall) hp.le
-    have hpowf := hpow.trans hspf
-    have hdiv := div_le_div_of_nonneg_left hA.le (Real.rpow_pos_of_pos (regH_pos ha) (regP a)) hpowf
-    have hh := mul_le_mul_of_nonneg_left hdiv hδ.le
-    have hC : regC a = regA a/(regH a^(regP a)) := by
-      dsimp [regC]; rw [Real.rpow_neg (regH_pos ha).le]; ring
-    rw [←hC] at hh
-    have hKD := mul_nonneg hk.le hD
-    dsimp only [regD] at hD hKD ⊢
-    simp only [div_eq_mul_inv] at hraw hh hKD ⊢
-    nlinarith only [hraw,hh,hKD]
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLEFiniteTilt
-
-lemma regFactor_derivative_simple {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hθ : 0 < Real.sin θ) :
-    |deriv (regFactor a δ) θ| ≤ (regP a+regQ a)*Real.sin θ^(regQ a-1) := by
-  have hp := regP_pos ha
-  have hq := regQ_pos ha
-  have hpq := regQ_lt_p ha
-  have hs1 := Real.sin_le_one θ
-  have hsp := Real.rpow_pos_of_pos hθ (regP a-1)
-  have hsq := Real.rpow_pos_of_pos hθ (regQ a-1)
-  have hpow := Real.rpow_le_rpow_of_exponent_ge hθ hs1 (show regQ a-1 ≤ regP a-1 by linarith)
-  have hpp := mul_le_mul_of_nonneg_left hpow hp.le
-  have hδp := mul_le_mul_of_nonneg_right hδ1 (mul_nonneg hq.le hsq.le)
-  have hcos := mul_le_mul_of_nonneg_right (Real.abs_cos_le_one θ)
-    (show 0 ≤ regP a*Real.sin θ^(regP a-1)+δ*regQ a*Real.sin θ^(regQ a-1) by positivity)
-  rw [(regFactor_hasDerivAt a δ θ hθ).deriv]
-  rw [show Real.cos θ*regP a*Real.sin θ^(regP a-1)+
-    δ*(Real.cos θ*regQ a*Real.sin θ^(regQ a-1)) =
-      Real.cos θ*(regP a*Real.sin θ^(regP a-1)+δ*regQ a*Real.sin θ^(regQ a-1)) by ring]
-  rw [abs_mul,abs_of_nonneg (show 0 ≤ regP a*Real.sin θ^(regP a-1)+
-    δ*regQ a*Real.sin θ^(regQ a-1) by positivity)]
-  nlinarith
-
-lemma regFactor_interpolation {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hθ : 0 < Real.sin θ) :
-    Real.sin θ^(regQ a-1)/regFactor a δ θ ≤
-      3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) := by
-  obtain ⟨hfp,hδf,hspf,hf3⟩ := regFactor_bounds ha hδ hδ1 hθ
-  have hη := regEta_pos ha
-  have hs := Real.rpow_pos_of_pos hθ (regQ a-1)
-  have hiden : 3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) =
-      (3/regFactor a δ θ)^(regEta a)*Real.sin θ^(regQ a-1)/δ := by
-    dsimp only [regD]
-    rw [Real.mul_rpow (div_nonneg hδ.le hfp.le) (Real.rpow_nonneg hθ.le _),
-      Real.div_rpow hδ.le hfp.le,←Real.rpow_mul hθ.le,
-      mul_comm (regQ a-2) (regEta a),regEta_identity ha,
-      Real.rpow_sub hδ,Real.rpow_neg_one,
-      Real.div_rpow (by norm_num : (0 : ℝ) ≤ 3) hfp.le]
-    field_simp
-  have hpow : 1 ≤ (3/regFactor a δ θ)^(regEta a) :=
-    Real.one_le_rpow ((one_le_div hfp).mpr hf3) hη.le
-  have hdiv := div_le_div_of_nonneg_left hs.le hδ hδf
-  rw [hiden]
-  calc
-    _ ≤ Real.sin θ^(regQ a-1)/δ := hdiv
-    _ ≤ _ := div_le_div_of_nonneg_right
-      (by nlinarith [mul_le_mul_of_nonneg_right hpow hs.le]) hδ.le
-
-lemma regDerivative_bound {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
-    (hθ : 0 < Real.sin θ) :
-    |deriv (regFactor a δ) θ|/regFactor a δ θ ≤
-      (regP a+regQ a)*3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) := by
-  have hf := (regFactor_bounds ha hδ hδ1 hθ).1
-  have hsimple := div_le_div_of_nonneg_right (regFactor_derivative_simple ha hδ hδ1 hθ) hf.le
-  have hinter := mul_le_mul_of_nonneg_left (regFactor_interpolation ha hδ hδ1 hθ)
-    (add_nonneg (regP_pos ha).le (regQ_pos ha).le)
-  calc
-    _ ≤ (regP a+regQ a)*Real.sin θ^(regQ a-1)/regFactor a δ θ := hsimple
-    _ = (regP a+regQ a)*(Real.sin θ^(regQ a-1)/regFactor a δ θ) := by ring
-    _ ≤ _ := hinter
-    _ = _ := by ring
-
- 
-theorem regularized_factor_estimates {a δ σ θ : ℝ} (ha : 1/4 < a)
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hσ : |σ| ≤ regBeta a)
-    (hθ : θ ∈ Set.Ioo 0 Real.pi) :
-    0 < regBeta a ∧ 2*regM a*regBeta a < a ∧
-    0 < regK a ∧ 0 < regC a ∧
-    regNormalizedDrift a δ σ θ ≤ regC a*δ-regK a/2*regD a δ θ ∧
-    |deriv (regFactor a δ) θ|/regFactor a δ θ ≤
-      (regP a+regQ a)*3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  exact ⟨regBeta_pos ha,regBeta_discount ha,regK_pos ha,regC_pos ha,
-    regDrift_bound ha hδ hδ1 hσ hs,regDerivative_bound ha hδ hδ1 hs⟩
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLEGreen
-open ExactSLECalculus ExactSLEFiniteTilt
-noncomputable def physicalDelta (a cap σ off : ℝ) (v : State) : ℝ :=
-  Real.exp (off+σ*physicalClock a cap v)
-noncomputable def physicalRegularized (a cap σ off : ℝ) (v : State) : ℝ :=
-  clockExp a cap (regP a/2) v/3*regFactor a (physicalDelta a cap σ off v) (physicalTheta v)
-lemma clockExp_shift (a cap c σ off : ℝ) (v : State) :
-    clockExp a cap c v*physicalDelta a cap σ off v = Real.exp off*clockExp a cap (c+σ) v := by
-  dsimp only [clockExp,physicalDelta]
-  rw [← Real.exp_add,← Real.exp_add]
-  congr 1
-  ring
-lemma physicalRegularized_decomposition (a cap σ off : ℝ) :
-    physicalRegularized a cap σ off = fun v => (1/3:ℝ)*
-      (clockExp a cap (regP a/2) v*angleLift (sinePower (regP a)) v +
-        Real.exp off*(clockExp a cap (regP a/2+σ) v +
-          clockExp a cap (regP a/2+σ) v*angleLift (sinePower (regQ a)) v)) := by
-  funext v
-  have h := clockExp_shift a cap (regP a/2) σ off v
-  dsimp only [physicalRegularized,regFactor,angleLift]
-  nlinarith only [h,h |> congrArg (fun x => x*sinePower (regQ a) (physicalTheta v))]
-lemma physicalRegularized_smooth (a cap σ off : ℝ) (v : State) (hv : 0<Y v) :
-    ContDiffAt ℝ 2 (physicalRegularized a cap σ off) v := by
-  have hθ := physicalTheta_mem v
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  rw [physicalRegularized_decomposition]
-  apply contDiffAt_const.mul
-  exact (((clockExp_smooth a cap (regP a/2) v hv).mul
-    (angleLift_smooth _ v hv (sinePower_smooth _ _ hs))).add
-    (contDiffAt_const.mul ((clockExp_smooth a cap (regP a/2+σ) v hv).add
-      ((clockExp_smooth a cap (regP a/2+σ) v hv).mul
-        (angleLift_smooth _ v hv (sinePower_smooth _ _ hs))))))
-lemma physicalRegularized_L (a cap σ off : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
-    L (physicalRegularized a cap σ off) (physicalDrift a v) noise v =
-      clockExp a cap (regP a/2) v/3*
-      (angleGenerator (1-2*a) (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v) +
-        regP a/2*regFactor a (physicalDelta a cap σ off v) (physicalTheta v) +
-        σ*physicalDelta a cap σ off v*(1+sinePower (regQ a) (physicalTheta v)))*(physicalV v)^2 := by
-  have hθ := physicalTheta_mem v
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  have hp := (clockExp_smooth a cap (regP a/2) v hv).mul
-    (angleLift_smooth _ v hv (sinePower_smooth (regP a) _ hs))
-  have hq := (clockExp_smooth a cap (regP a/2+σ) v hv).mul
-    (angleLift_smooth _ v hv (sinePower_smooth (regQ a) _ hs))
-  have hc := clockExp_smooth a cap (regP a/2+σ) v hv
-  rw [physicalRegularized_decomposition,L_const_mul (hp.add (contDiffAt_const.mul (hc.add hq))),
-    L_add hp (contDiffAt_const.mul (hc.add hq)),L_const_mul (hc.add hq),L_add hc hq,
-    clockExp_L a cap (regP a/2+σ) ha v hv,
-    exponential_angle_generator _ a cap (regP a/2) ha v hv (sinePower_smooth _ _ hs),
-    exponential_angle_generator _ a cap (regP a/2+σ) ha v hv (sinePower_smooth _ _ hs),
-    regFactor_generator _ _ _ hs]
-  dsimp only [regFactor]
-  have h := clockExp_shift a cap (regP a/2) σ off v
-  linear_combination -(physicalV v)^2/3 *
-    ((regP a/2+σ)*(1+sinePower (regQ a) (physicalTheta v)) +
-      angleGenerator (1-2*a) (sinePower (regQ a)) (physicalTheta v)) * h
-lemma physicalRegularized_N (a cap σ off : ℝ) (v : State) (hv : 0<Y v) :
-    N (physicalRegularized a cap σ off) noise v =
-      clockExp a cap (regP a/2) v/3*
-        deriv (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v)*physicalV v := by
-  have hθ := physicalTheta_mem v
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  have he := (clockExp_smooth a cap (regP a/2) v hv).differentiableAt (by norm_num)
-  have hc := (clockExp_smooth a cap (regP a/2+σ) v hv).differentiableAt (by norm_num)
-  have hp := (angleLift_smooth _ v hv (sinePower_smooth (regP a) _ hs)).differentiableAt (by norm_num)
-  have hq := (angleLift_smooth _ v hv (sinePower_smooth (regQ a) _ hs)).differentiableAt (by norm_num)
-  have hconst : DifferentiableAt ℝ (fun _ : State => Real.exp off) v := differentiableAt_const _
-  erw [physicalRegularized_decomposition,N_const_mul ((he.mul hp).add (hconst.mul (hc.add (hc.mul hq)))) (1/3:ℝ),
-    N_add (he.mul hp) (hconst.mul (hc.add (hc.mul hq))),N_const_mul (hc.add (hc.mul hq)) (Real.exp off),
-    N_add hc (hc.mul hq),N_mul he hp,N_mul hc hq,
-    clockExp_N a cap (regP a/2) v hv,clockExp_N a cap (regP a/2+σ) v hv,
-    angleLift_N _ v hv (sinePower_hasDerivAt (regP a) _ hs).differentiableAt,
-    angleLift_N _ v hv (sinePower_hasDerivAt (regQ a) _ hs).differentiableAt,
-    (regFactor_hasDerivAt a _ _ hs).deriv,
-    (sinePower_hasDerivAt (regP a) _ hs).deriv,(sinePower_hasDerivAt (regQ a) _ hs).deriv]
-  have h := clockExp_shift a cap (regP a/2) σ off v
-  linear_combination -(Real.cos (physicalTheta v)*regQ a*
-    Real.sin (physicalTheta v)^(regQ a-1)*physicalV v/3)*h
-
- 
-theorem actual_regularized_factor {a cap σ off : ℝ} (ha : 1/4<a)
-    (v : State) (hv : 0<Y v) (hδ : physicalDelta a cap σ off v≤1) (hσ : |σ|≤regBeta a) :
-    0<physicalRegularized a cap σ off v ∧
-    L (physicalRegularized a cap σ off) (physicalDrift a v) noise v / physicalRegularized a cap σ off v ≤
-      (regC a*physicalDelta a cap σ off v-regK a/2*regD a (physicalDelta a cap σ off v) (physicalTheta v))*(physicalV v)^2 ∧
-    |N (physicalRegularized a cap σ off) noise v|/physicalRegularized a cap σ off v ≤
-      (regP a+regQ a)*3^(regEta a)*(physicalDelta a cap σ off v)^(-1-regEta a)*
-        (regD a (physicalDelta a cap σ off v) (physicalTheta v))^(regEta a)*physicalV v := by
-  have hθ := physicalTheta_mem v
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
-  have hδp : 0<physicalDelta a cap σ off v := Real.exp_pos _
-  have hf := (regFactor_bounds ha hδp hδ hs).1
-  have he : 0<clockExp a cap (regP a/2) v := Real.exp_pos _
-  have hV : 0<physicalV v := div_pos hv (Q_pos hv)
-  have hpos : 0<physicalRegularized a cap σ off v := by
-    dsimp only [physicalRegularized]; positivity
-  refine ⟨hpos,?_,?_⟩
-  · have hL : L (physicalRegularized a cap σ off) (physicalDrift a v) noise v /
-        physicalRegularized a cap σ off v =
-          regNormalizedDrift a (physicalDelta a cap σ off v) σ (physicalTheta v)*(physicalV v)^2 := by
-      rw [physicalRegularized_L a cap σ off (by linarith) v hv]
-      dsimp only [physicalRegularized,regNormalizedDrift]
-      field_simp
-    rw [hL]
-    exact mul_le_mul_of_nonneg_right (regDrift_bound ha hδp hδ hσ hs) (sq_nonneg _)
-  · rw [physicalRegularized_N a cap σ off v hv,abs_mul,abs_mul,
-      abs_of_pos (div_pos he (by norm_num)),abs_of_pos hV]
-    have hN : clockExp a cap (regP a/2) v/3 *
-        |deriv (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v)| * physicalV v /
-        physicalRegularized a cap σ off v =
-        |deriv (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v)| /
-          regFactor a (physicalDelta a cap σ off v) (physicalTheta v)*physicalV v := by
-      dsimp only [physicalRegularized]; field_simp
-    rw [hN]
-    exact mul_le_mul_of_nonneg_right (regDerivative_bound ha hδp hδ hs) hV.le
-
-end ExactSLEGreen
-
-namespace ExactSLEFiniteTilt
-
-lemma weighted_pair_young {η ε C X Y Z : ℝ} (hη : 0<η) (hη2 : η<1/2)
-    (hε : 0<ε) (hC : 0<C) (hX : 0≤X) (hY : 0≤Y) (hZ : 0≤Z) :
-    C*X^η*Y^η*Z^(1-2*η) ≤ ε*(X+Y) +
-      (1-2*η)*C^(1/(1-2*η))*(η/ε)^(2*η/(1-2*η))*Z := by
-  let r : ℝ := 1-2*η
-  have hr : 0<r := by dsimp [r]; linarith
-  let A : ℝ := (C*(η/ε)^(2*η))^(1/r)
-  have hA : 0<A := by dsimp [A]; positivity
-  have hg := Real.geom_mean_le_arith_mean3_weighted hη.le hη.le hr.le
-    (show 0≤ε/η*X by positivity) (show 0≤ε/η*Y by positivity)
-    (show 0≤A*Z by positivity) (show η+η+r=1 by dsimp [r]; ring)
-  have hAr : A^r = C*(η/ε)^(2*η) := by
-    dsimp [A]
-    rw [← Real.rpow_mul (by positivity : 0≤C*(η/ε)^(2*η)),div_mul_cancel₀ _ hr.ne',Real.rpow_one]
-  have hcancel : (ε/η)^η*(ε/η)^η*(η/ε)^(2*η)=1 := by
-    rw [← Real.rpow_add (by positivity),show η+η=2*η by ring,
-      ← Real.mul_rpow (by positivity) (by positivity)]
-    have he : ε/η*(η/ε)=1 := by field_simp
-    rw [he,Real.one_rpow]
-  have hleft : (ε/η*X)^η*(ε/η*Y)^η*(A*Z)^r = C*X^η*Y^η*Z^r := by
-    rw [Real.mul_rpow (by positivity) hX,Real.mul_rpow (by positivity) hY,
-      Real.mul_rpow hA.le hZ,hAr]
-    calc
-      _ = C*X^η*Y^η*Z^r*((ε/η)^η*(ε/η)^η*(η/ε)^(2*η)) := by ring
-      _ = _ := by rw [hcancel,mul_one]
-  rw [hleft] at hg
-  have hAe : A = C^(1/r)*(η/ε)^(2*η/r) := by
-    dsimp [A]
-    rw [Real.mul_rpow hC.le (by positivity),← Real.rpow_mul (by positivity)]
-    congr 2
-    ring
-  have hs : η*(ε/η*X)+η*(ε/η*Y)=ε*(X+Y) := by field_simp
-  rw [hs,hAe] at hg
-  simpa only [r,mul_assoc] using hg
-
-noncomputable def regC2 (a : ℝ) : ℝ := (regP a+regQ a)^2 * 3^(2*regEta a)
-noncomputable def regYoungC (a ε : ℝ) : ℝ :=
-  (1-2*regEta a) * regC2 a ^ (1/(1-2*regEta a)) *
-    (regEta a/ε) ^ (2*regEta a/(1-2*regEta a))
-
-lemma regC2_pos {a : ℝ} (ha : 1/4<a) : 0<regC2 a := by
-  have hp := regP_pos ha
-  have hq := regQ_pos ha
-  dsimp [regC2]
-  positivity
-
- 
-theorem regularized_pair_absorption {a ε δᵢ δⱼ Dᵢ Dⱼ Vᵢ Vⱼ : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hδᵢ : 0<δᵢ) (hδⱼ : 0<δⱼ)
-    (hDᵢ : 0≤Dᵢ) (hDⱼ : 0≤Dⱼ) (hVᵢ : 0≤Vᵢ) (hVⱼ : 0≤Vⱼ) :
-    regC2 a * (δᵢ*δⱼ)^(-1-regEta a) *
-      (Dᵢ*Vᵢ^2)^(regEta a)*(Dⱼ*Vⱼ^2)^(regEta a)*(Vᵢ*Vⱼ)^(1-2*regEta a) ≤
-    ε*(Dᵢ*Vᵢ^2+Dⱼ*Vⱼ^2) + regYoungC a ε*(δᵢ*δⱼ)^(-regM a)*Vᵢ*Vⱼ := by
-  have hC := regC2_pos ha
-  have hd : 0<δᵢ*δⱼ := mul_pos hδᵢ hδⱼ
-  have hy := weighted_pair_young (regEta_pos ha) (regEta_lt_half ha) hε
-    (show 0<regC2 a*(δᵢ*δⱼ)^(-1-regEta a) by positivity)
-    (show 0≤Dᵢ*Vᵢ^2 by positivity) (show 0≤Dⱼ*Vⱼ^2 by positivity)
-    (show 0≤Vᵢ*Vⱼ by positivity)
-  have he : (regC2 a*(δᵢ*δⱼ)^(-1-regEta a))^(1/(1-2*regEta a)) =
-      regC2 a^(1/(1-2*regEta a))*(δᵢ*δⱼ)^(-regM a) := by
-    rw [Real.mul_rpow hC.le (by positivity),← Real.rpow_mul hd.le]
-    congr 2
-    dsimp [regM]
-    ring
-  rw [he] at hy
-  dsimp only [regYoungC]
-  convert hy using 1
-  ring
-
-end ExactSLEFiniteTilt
-
-namespace ExactSLECalculus
-variable {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-end ExactSLECalculus
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.AngleGenerators
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+namespace ExactSLEFiniteTilt
+open Filter
+open scoped Topology
+
+lemma regFactor_hasDerivAt (a δ θ : ℝ) (hθ : 0 < Real.sin θ) :
+    HasDerivAt (regFactor a δ)
+      (Real.cos θ*regP a*Real.sin θ^(regP a-1) +
+        δ*(Real.cos θ*regQ a*Real.sin θ^(regQ a-1))) θ := by
+  convert (sinePower_hasDerivAt (regP a) θ hθ).add
+      (((sinePower_hasDerivAt (regQ a) θ hθ).const_add 1).const_mul δ) using 1
+  rfl
+
+lemma sineDerivative_hasDerivAt (r θ : ℝ) (hθ : 0 < Real.sin θ) :
+    HasDerivAt (fun x => Real.cos x*r*Real.sin x^(r-1))
+      (deriv (deriv (sinePower r)) θ) θ := by
+  rw [sinePower_secondDeriv r θ hθ]
+  convert ((Real.hasDerivAt_cos θ).mul_const r).mul
+    ((Real.hasDerivAt_sin θ).rpow_const (p := r-1) (Or.inl hθ.ne')) using 1
+  congr 2
+  ring_nf
+
+lemma regFactor_secondDeriv (a δ θ : ℝ) (hθ : 0 < Real.sin θ) :
+    deriv (deriv (regFactor a δ)) θ = deriv (deriv (sinePower (regP a))) θ +
+      δ*deriv (deriv (sinePower (regQ a))) θ := by
+  have he : deriv (regFactor a δ) =ᶠ[𝓝 θ]
+      (fun x => Real.cos x*regP a*Real.sin x^(regP a-1) +
+        δ*(Real.cos x*regQ a*Real.sin x^(regQ a-1))) := by
+    have hp : ∀ᶠ x in 𝓝 θ, 0 < Real.sin x :=
+      (isOpen_lt continuous_const Real.continuous_sin).mem_nhds hθ
+    filter_upwards [hp] with x hx using (regFactor_hasDerivAt a δ x hx).deriv
+  rw [he.deriv_eq]
+  exact ((sineDerivative_hasDerivAt (regP a) θ hθ).add
+    ((sineDerivative_hasDerivAt (regQ a) θ hθ).const_mul δ)).deriv
+
+lemma regFactor_generator (a δ θ : ℝ) (hθ : 0 < Real.sin θ) :
+    angleGenerator (1-2*a) (regFactor a δ) θ =
+      angleGenerator (1-2*a) (sinePower (regP a)) θ +
+      δ*angleGenerator (1-2*a) (sinePower (regQ a)) θ := by
+  dsimp only [angleGenerator]
+  rw [regFactor_secondDeriv a δ θ hθ,(regFactor_hasDerivAt a δ θ hθ).deriv,
+    (sinePower_hasDerivAt (regP a) θ hθ).deriv,
+    (sinePower_hasDerivAt (regQ a) θ hθ).deriv]
+  ring
+
+lemma sinePower_mul_cot_sq (r θ : ℝ) (hθ : 0 < Real.sin θ) :
+    sinePower r θ*Real.cot θ^2 = Real.sin θ^(r-2)-sinePower r θ := by
+  dsimp only [sinePower]
+  rw [Real.cot_eq_cos_div_sin,div_pow,Real.rpow_sub hθ,Real.rpow_two]
+  field_simp
+  nlinarith [Real.sin_sq_add_cos_sq θ]
+
+lemma regFactor_exact_drift (a δ σ θ : ℝ) (hθ : 0 < Real.sin θ) :
+    regNormalizedDrift a δ σ θ = δ/regFactor a δ θ *
+      (regP a/2+(regP a-regQ a)*(1+regQ a)/2*Real.sin θ^(regQ a) -
+        regK a*Real.sin θ^(regQ a-2) + σ*(1+Real.sin θ^(regQ a))) := by
+  have hG := angleGenerator_sinePower (1-2*a) (regQ a) θ hθ
+  have hP := angleGenerator_sinePower (1-2*a) (regP a) θ hθ
+  have hpq : regP a-1+2*(1-2*a) = 0 := by dsimp [regP]; ring
+  have hqp : regQ a-1+2*(1-2*a) = regQ a-regP a := by dsimp [regP]; ring
+  simp only [hpq,hqp] at hP hG
+  rw [show sinePower (regQ a) θ *
+    (-regQ a/2+regQ a*(regQ a-regP a)/2*Real.cot θ^2) =
+      -regQ a/2*sinePower (regQ a) θ + regQ a*(regQ a-regP a)/2*
+        (sinePower (regQ a) θ*Real.cot θ^2) by ring,
+    sinePower_mul_cot_sq _ _ hθ] at hG
+  dsimp only [regNormalizedDrift]
+  rw [regFactor_generator a δ θ hθ,hP,hG]
+  dsimp only [regFactor,regK,sinePower]
+  ring
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLEFiniteTilt
+
+lemma regFactor_bounds {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hθ : 0 < Real.sin θ) :
+    0 < regFactor a δ θ ∧ δ ≤ regFactor a δ θ ∧
+    Real.sin θ^(regP a) ≤ regFactor a δ θ ∧ regFactor a δ θ ≤ 3 := by
+  have hs1 := Real.sin_le_one θ
+  have hp := Real.rpow_pos_of_pos hθ (regP a)
+  have hq := Real.rpow_pos_of_pos hθ (regQ a)
+  have hp1 := Real.rpow_le_one hθ.le hs1 (regP_pos ha).le
+  have hq1 := Real.rpow_le_one hθ.le hs1 (regQ_pos ha).le
+  dsimp only [regFactor,sinePower]
+  have hmul := mul_nonneg hδ.le hq.le
+  have hm1 := mul_le_mul_of_nonneg_left hq1 hδ.le
+  constructor
+  · positivity
+  constructor
+  · nlinarith
+  constructor
+  · nlinarith
+  · nlinarith
+
+lemma regDrift_bound {a δ σ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hσ : |σ| ≤ regBeta a) (hθ : 0 < Real.sin θ) :
+    regNormalizedDrift a δ σ θ ≤ regC a*δ-regK a/2*regD a δ θ := by
+  obtain ⟨hfp,hδf,hspf,hf3⟩ := regFactor_bounds ha hδ hδ1 hθ
+  have hp := regP_pos ha
+  have hq := regQ_pos ha
+  have hq1 := regQ_lt_one ha
+  have hpq := sub_pos.mpr (regQ_lt_p ha)
+  have hk := regK_pos ha
+  have hA := regA_pos ha
+  have hs0 := Real.rpow_pos_of_pos hθ (regQ a)
+  have hs1 := Real.rpow_le_one hθ.le (Real.sin_le_one θ) hq.le
+  have hσ1 : σ ≤ 1 := (le_abs_self σ).trans (hσ.trans (regBeta_le_one a))
+  have hσterm := mul_le_mul_of_nonneg_right hσ1 (show 0 ≤ 1+Real.sin θ^(regQ a) by positivity)
+  have hqterm := mul_le_mul_of_nonneg_left hs1
+    (show 0 ≤ (regP a-regQ a)*(1+regQ a)/2 by positivity)
+  have hbracket : regP a/2+(regP a-regQ a)*(1+regQ a)/2*Real.sin θ^(regQ a) -
+      regK a*Real.sin θ^(regQ a-2) + σ*(1+Real.sin θ^(regQ a)) ≤
+      regA a-regK a*Real.sin θ^(regQ a-2) := by
+    dsimp only [regA]; linarith
+  have hx : 0 ≤ δ/regFactor a δ θ := (div_pos hδ hfp).le
+  have hraw := mul_le_mul_of_nonneg_left hbracket hx
+  rw [←regFactor_exact_drift a δ σ θ hθ] at hraw
+  have hD : 0 ≤ regD a δ θ := mul_nonneg hx (Real.rpow_nonneg hθ.le _)
+  by_cases hsmall : Real.sin θ < regH a
+  · have hpow := Real.rpow_le_rpow_of_nonpos hθ hsmall.le (show regQ a-2 ≤ 0 by linarith)
+    rw [regH_power ha] at hpow
+    have hthresh : regA a ≤ regK a/2*Real.sin θ^(regQ a-2) := by
+      have hh := (div_le_iff₀ hk).mp hpow
+      nlinarith
+    have hh := mul_le_mul_of_nonneg_left hthresh hx
+    have hCδ : 0 ≤ regC a*δ := mul_nonneg (regC_pos ha).le hδ.le
+    dsimp only [regD]
+    nlinarith
+  · have hpow := Real.rpow_le_rpow (regH_pos ha).le (le_of_not_gt hsmall) hp.le
+    have hpowf := hpow.trans hspf
+    have hdiv := div_le_div_of_nonneg_left hA.le (Real.rpow_pos_of_pos (regH_pos ha) (regP a)) hpowf
+    have hh := mul_le_mul_of_nonneg_left hdiv hδ.le
+    have hC : regC a = regA a/(regH a^(regP a)) := by
+      dsimp [regC]; rw [Real.rpow_neg (regH_pos ha).le]; ring
+    rw [←hC] at hh
+    have hKD := mul_nonneg hk.le hD
+    dsimp only [regD] at hD hKD ⊢
+    simp only [div_eq_mul_inv] at hraw hh hKD ⊢
+    nlinarith only [hraw,hh,hKD]
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLEFiniteTilt
+
+lemma regFactor_derivative_simple {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hθ : 0 < Real.sin θ) :
+    |deriv (regFactor a δ) θ| ≤ (regP a+regQ a)*Real.sin θ^(regQ a-1) := by
+  have hp := regP_pos ha
+  have hq := regQ_pos ha
+  have hpq := regQ_lt_p ha
+  have hs1 := Real.sin_le_one θ
+  have hsp := Real.rpow_pos_of_pos hθ (regP a-1)
+  have hsq := Real.rpow_pos_of_pos hθ (regQ a-1)
+  have hpow := Real.rpow_le_rpow_of_exponent_ge hθ hs1 (show regQ a-1 ≤ regP a-1 by linarith)
+  have hpp := mul_le_mul_of_nonneg_left hpow hp.le
+  have hδp := mul_le_mul_of_nonneg_right hδ1 (mul_nonneg hq.le hsq.le)
+  have hcos := mul_le_mul_of_nonneg_right (Real.abs_cos_le_one θ)
+    (show 0 ≤ regP a*Real.sin θ^(regP a-1)+δ*regQ a*Real.sin θ^(regQ a-1) by positivity)
+  rw [(regFactor_hasDerivAt a δ θ hθ).deriv]
+  rw [show Real.cos θ*regP a*Real.sin θ^(regP a-1)+
+    δ*(Real.cos θ*regQ a*Real.sin θ^(regQ a-1)) =
+      Real.cos θ*(regP a*Real.sin θ^(regP a-1)+δ*regQ a*Real.sin θ^(regQ a-1)) by ring]
+  rw [abs_mul,abs_of_nonneg (show 0 ≤ regP a*Real.sin θ^(regP a-1)+
+    δ*regQ a*Real.sin θ^(regQ a-1) by positivity)]
+  nlinarith
+
+lemma regFactor_interpolation {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hθ : 0 < Real.sin θ) :
+    Real.sin θ^(regQ a-1)/regFactor a δ θ ≤
+      3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) := by
+  obtain ⟨hfp,hδf,hspf,hf3⟩ := regFactor_bounds ha hδ hδ1 hθ
+  have hη := regEta_pos ha
+  have hs := Real.rpow_pos_of_pos hθ (regQ a-1)
+  have hiden : 3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) =
+      (3/regFactor a δ θ)^(regEta a)*Real.sin θ^(regQ a-1)/δ := by
+    dsimp only [regD]
+    rw [Real.mul_rpow (div_nonneg hδ.le hfp.le) (Real.rpow_nonneg hθ.le _),
+      Real.div_rpow hδ.le hfp.le,←Real.rpow_mul hθ.le,
+      mul_comm (regQ a-2) (regEta a),regEta_identity ha,
+      Real.rpow_sub hδ,Real.rpow_neg_one,
+      Real.div_rpow (by norm_num : (0 : ℝ) ≤ 3) hfp.le]
+    field_simp
+  have hpow : 1 ≤ (3/regFactor a δ θ)^(regEta a) :=
+    Real.one_le_rpow ((one_le_div hfp).mpr hf3) hη.le
+  have hdiv := div_le_div_of_nonneg_left hs.le hδ hδf
+  rw [hiden]
+  calc
+    _ ≤ Real.sin θ^(regQ a-1)/δ := hdiv
+    _ ≤ _ := div_le_div_of_nonneg_right
+      (by nlinarith [mul_le_mul_of_nonneg_right hpow hs.le]) hδ.le
+
+lemma regDerivative_bound {a δ θ : ℝ} (ha : 1/4 < a) (hδ : 0 < δ) (hδ1 : δ ≤ 1)
+    (hθ : 0 < Real.sin θ) :
+    |deriv (regFactor a δ) θ|/regFactor a δ θ ≤
+      (regP a+regQ a)*3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) := by
+  have hf := (regFactor_bounds ha hδ hδ1 hθ).1
+  have hsimple := div_le_div_of_nonneg_right (regFactor_derivative_simple ha hδ hδ1 hθ) hf.le
+  have hinter := mul_le_mul_of_nonneg_left (regFactor_interpolation ha hδ hδ1 hθ)
+    (add_nonneg (regP_pos ha).le (regQ_pos ha).le)
+  calc
+    _ ≤ (regP a+regQ a)*Real.sin θ^(regQ a-1)/regFactor a δ θ := hsimple
+    _ = (regP a+regQ a)*(Real.sin θ^(regQ a-1)/regFactor a δ θ) := by ring
+    _ ≤ _ := hinter
+    _ = _ := by ring
+
+ 
+theorem regularized_factor_estimates {a δ σ θ : ℝ} (ha : 1/4 < a)
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hσ : |σ| ≤ regBeta a)
+    (hθ : θ ∈ Set.Ioo 0 Real.pi) :
+    0 < regBeta a ∧ 2*regM a*regBeta a < a ∧
+    0 < regK a ∧ 0 < regC a ∧
+    regNormalizedDrift a δ σ θ ≤ regC a*δ-regK a/2*regD a δ θ ∧
+    |deriv (regFactor a δ) θ|/regFactor a δ θ ≤
+      (regP a+regQ a)*3^(regEta a)*δ^(-1-regEta a)*(regD a δ θ)^(regEta a) := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  exact ⟨regBeta_pos ha,regBeta_discount ha,regK_pos ha,regC_pos ha,
+    regDrift_bound ha hδ hδ1 hσ hs,regDerivative_bound ha hδ hδ1 hs⟩
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLEGreen
+open ExactSLECalculus ExactSLEFiniteTilt
+noncomputable def physicalDelta (a cap σ off : ℝ) (v : State) : ℝ :=
+  Real.exp (off+σ*physicalClock a cap v)
+noncomputable def physicalRegularized (a cap σ off : ℝ) (v : State) : ℝ :=
+  clockExp a cap (regP a/2) v/3*regFactor a (physicalDelta a cap σ off v) (physicalTheta v)
+lemma clockExp_shift (a cap c σ off : ℝ) (v : State) :
+    clockExp a cap c v*physicalDelta a cap σ off v = Real.exp off*clockExp a cap (c+σ) v := by
+  dsimp only [clockExp,physicalDelta]
+  rw [← Real.exp_add,← Real.exp_add]
+  congr 1
+  ring
+lemma physicalRegularized_decomposition (a cap σ off : ℝ) :
+    physicalRegularized a cap σ off = fun v => (1/3:ℝ)*
+      (clockExp a cap (regP a/2) v*angleLift (sinePower (regP a)) v +
+        Real.exp off*(clockExp a cap (regP a/2+σ) v +
+          clockExp a cap (regP a/2+σ) v*angleLift (sinePower (regQ a)) v)) := by
+  funext v
+  have h := clockExp_shift a cap (regP a/2) σ off v
+  dsimp only [physicalRegularized,regFactor,angleLift]
+  nlinarith only [h,h |> congrArg (fun x => x*sinePower (regQ a) (physicalTheta v))]
+lemma physicalRegularized_smooth (a cap σ off : ℝ) (v : State) (hv : 0<Y v) :
+    ContDiffAt ℝ 2 (physicalRegularized a cap σ off) v := by
+  have hθ := physicalTheta_mem v
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  rw [physicalRegularized_decomposition]
+  apply contDiffAt_const.mul
+  exact (((clockExp_smooth a cap (regP a/2) v hv).mul
+    (angleLift_smooth _ v hv (sinePower_smooth _ _ hs))).add
+    (contDiffAt_const.mul ((clockExp_smooth a cap (regP a/2+σ) v hv).add
+      ((clockExp_smooth a cap (regP a/2+σ) v hv).mul
+        (angleLift_smooth _ v hv (sinePower_smooth _ _ hs))))))
+lemma physicalRegularized_L (a cap σ off : ℝ) (ha : a≠0) (v : State) (hv : 0<Y v) :
+    L (physicalRegularized a cap σ off) (physicalDrift a v) noise v =
+      clockExp a cap (regP a/2) v/3*
+      (angleGenerator (1-2*a) (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v) +
+        regP a/2*regFactor a (physicalDelta a cap σ off v) (physicalTheta v) +
+        σ*physicalDelta a cap σ off v*(1+sinePower (regQ a) (physicalTheta v)))*(physicalV v)^2 := by
+  have hθ := physicalTheta_mem v
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  have hp := (clockExp_smooth a cap (regP a/2) v hv).mul
+    (angleLift_smooth _ v hv (sinePower_smooth (regP a) _ hs))
+  have hq := (clockExp_smooth a cap (regP a/2+σ) v hv).mul
+    (angleLift_smooth _ v hv (sinePower_smooth (regQ a) _ hs))
+  have hc := clockExp_smooth a cap (regP a/2+σ) v hv
+  rw [physicalRegularized_decomposition,L_const_mul (hp.add (contDiffAt_const.mul (hc.add hq))),
+    L_add hp (contDiffAt_const.mul (hc.add hq)),L_const_mul (hc.add hq),L_add hc hq,
+    clockExp_L a cap (regP a/2+σ) ha v hv,
+    exponential_angle_generator _ a cap (regP a/2) ha v hv (sinePower_smooth _ _ hs),
+    exponential_angle_generator _ a cap (regP a/2+σ) ha v hv (sinePower_smooth _ _ hs),
+    regFactor_generator _ _ _ hs]
+  dsimp only [regFactor]
+  have h := clockExp_shift a cap (regP a/2) σ off v
+  linear_combination -(physicalV v)^2/3 *
+    ((regP a/2+σ)*(1+sinePower (regQ a) (physicalTheta v)) +
+      angleGenerator (1-2*a) (sinePower (regQ a)) (physicalTheta v)) * h
+lemma physicalRegularized_N (a cap σ off : ℝ) (v : State) (hv : 0<Y v) :
+    N (physicalRegularized a cap σ off) noise v =
+      clockExp a cap (regP a/2) v/3*
+        deriv (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v)*physicalV v := by
+  have hθ := physicalTheta_mem v
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  have he := (clockExp_smooth a cap (regP a/2) v hv).differentiableAt (by norm_num)
+  have hc := (clockExp_smooth a cap (regP a/2+σ) v hv).differentiableAt (by norm_num)
+  have hp := (angleLift_smooth _ v hv (sinePower_smooth (regP a) _ hs)).differentiableAt (by norm_num)
+  have hq := (angleLift_smooth _ v hv (sinePower_smooth (regQ a) _ hs)).differentiableAt (by norm_num)
+  have hconst : DifferentiableAt ℝ (fun _ : State => Real.exp off) v := differentiableAt_const _
+  erw [physicalRegularized_decomposition,N_const_mul ((he.mul hp).add (hconst.mul (hc.add (hc.mul hq)))) (1/3:ℝ),
+    N_add (he.mul hp) (hconst.mul (hc.add (hc.mul hq))),N_const_mul (hc.add (hc.mul hq)) (Real.exp off),
+    N_add hc (hc.mul hq),N_mul he hp,N_mul hc hq,
+    clockExp_N a cap (regP a/2) v hv,clockExp_N a cap (regP a/2+σ) v hv,
+    angleLift_N _ v hv (sinePower_hasDerivAt (regP a) _ hs).differentiableAt,
+    angleLift_N _ v hv (sinePower_hasDerivAt (regQ a) _ hs).differentiableAt,
+    (regFactor_hasDerivAt a _ _ hs).deriv,
+    (sinePower_hasDerivAt (regP a) _ hs).deriv,(sinePower_hasDerivAt (regQ a) _ hs).deriv]
+  have h := clockExp_shift a cap (regP a/2) σ off v
+  linear_combination -(Real.cos (physicalTheta v)*regQ a*
+    Real.sin (physicalTheta v)^(regQ a-1)*physicalV v/3)*h
+
+ 
+theorem actual_regularized_factor {a cap σ off : ℝ} (ha : 1/4<a)
+    (v : State) (hv : 0<Y v) (hδ : physicalDelta a cap σ off v≤1) (hσ : |σ|≤regBeta a) :
+    0<physicalRegularized a cap σ off v ∧
+    L (physicalRegularized a cap σ off) (physicalDrift a v) noise v / physicalRegularized a cap σ off v ≤
+      (regC a*physicalDelta a cap σ off v-regK a/2*regD a (physicalDelta a cap σ off v) (physicalTheta v))*(physicalV v)^2 ∧
+    |N (physicalRegularized a cap σ off) noise v|/physicalRegularized a cap σ off v ≤
+      (regP a+regQ a)*3^(regEta a)*(physicalDelta a cap σ off v)^(-1-regEta a)*
+        (regD a (physicalDelta a cap σ off v) (physicalTheta v))^(regEta a)*physicalV v := by
+  have hθ := physicalTheta_mem v
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ.1 hθ.2
+  have hδp : 0<physicalDelta a cap σ off v := Real.exp_pos _
+  have hf := (regFactor_bounds ha hδp hδ hs).1
+  have he : 0<clockExp a cap (regP a/2) v := Real.exp_pos _
+  have hV : 0<physicalV v := div_pos hv (Q_pos hv)
+  have hpos : 0<physicalRegularized a cap σ off v := by
+    dsimp only [physicalRegularized]; positivity
+  refine ⟨hpos,?_,?_⟩
+  · have hL : L (physicalRegularized a cap σ off) (physicalDrift a v) noise v /
+        physicalRegularized a cap σ off v =
+          regNormalizedDrift a (physicalDelta a cap σ off v) σ (physicalTheta v)*(physicalV v)^2 := by
+      rw [physicalRegularized_L a cap σ off (by linarith) v hv]
+      dsimp only [physicalRegularized,regNormalizedDrift]
+      field_simp
+    rw [hL]
+    exact mul_le_mul_of_nonneg_right (regDrift_bound ha hδp hδ hσ hs) (sq_nonneg _)
+  · rw [physicalRegularized_N a cap σ off v hv,abs_mul,abs_mul,
+      abs_of_pos (div_pos he (by norm_num)),abs_of_pos hV]
+    have hN : clockExp a cap (regP a/2) v/3 *
+        |deriv (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v)| * physicalV v /
+        physicalRegularized a cap σ off v =
+        |deriv (regFactor a (physicalDelta a cap σ off v)) (physicalTheta v)| /
+          regFactor a (physicalDelta a cap σ off v) (physicalTheta v)*physicalV v := by
+      dsimp only [physicalRegularized]; field_simp
+    rw [hN]
+    exact mul_le_mul_of_nonneg_right (regDerivative_bound ha hδp hδ hs) hV.le
+
+end ExactSLEGreen
+
+namespace ExactSLEFiniteTilt
+
+lemma weighted_pair_young {η ε C X Y Z : ℝ} (hη : 0<η) (hη2 : η<1/2)
+    (hε : 0<ε) (hC : 0<C) (hX : 0≤X) (hY : 0≤Y) (hZ : 0≤Z) :
+    C*X^η*Y^η*Z^(1-2*η) ≤ ε*(X+Y) +
+      (1-2*η)*C^(1/(1-2*η))*(η/ε)^(2*η/(1-2*η))*Z := by
+  let r : ℝ := 1-2*η
+  have hr : 0<r := by dsimp [r]; linarith
+  let A : ℝ := (C*(η/ε)^(2*η))^(1/r)
+  have hA : 0<A := by dsimp [A]; positivity
+  have hg := Real.geom_mean_le_arith_mean3_weighted hη.le hη.le hr.le
+    (show 0≤ε/η*X by positivity) (show 0≤ε/η*Y by positivity)
+    (show 0≤A*Z by positivity) (show η+η+r=1 by dsimp [r]; ring)
+  have hAr : A^r = C*(η/ε)^(2*η) := by
+    dsimp [A]
+    rw [← Real.rpow_mul (by positivity : 0≤C*(η/ε)^(2*η)),div_mul_cancel₀ _ hr.ne',Real.rpow_one]
+  have hcancel : (ε/η)^η*(ε/η)^η*(η/ε)^(2*η)=1 := by
+    rw [← Real.rpow_add (by positivity),show η+η=2*η by ring,
+      ← Real.mul_rpow (by positivity) (by positivity)]
+    have he : ε/η*(η/ε)=1 := by field_simp
+    rw [he,Real.one_rpow]
+  have hleft : (ε/η*X)^η*(ε/η*Y)^η*(A*Z)^r = C*X^η*Y^η*Z^r := by
+    rw [Real.mul_rpow (by positivity) hX,Real.mul_rpow (by positivity) hY,
+      Real.mul_rpow hA.le hZ,hAr]
+    calc
+      _ = C*X^η*Y^η*Z^r*((ε/η)^η*(ε/η)^η*(η/ε)^(2*η)) := by ring
+      _ = _ := by rw [hcancel,mul_one]
+  rw [hleft] at hg
+  have hAe : A = C^(1/r)*(η/ε)^(2*η/r) := by
+    dsimp [A]
+    rw [Real.mul_rpow hC.le (by positivity),← Real.rpow_mul (by positivity)]
+    congr 2
+    ring
+  have hs : η*(ε/η*X)+η*(ε/η*Y)=ε*(X+Y) := by field_simp
+  rw [hs,hAe] at hg
+  simpa only [r,mul_assoc] using hg
+
+noncomputable def regC2 (a : ℝ) : ℝ := (regP a+regQ a)^2 * 3^(2*regEta a)
+noncomputable def regYoungC (a ε : ℝ) : ℝ :=
+  (1-2*regEta a) * regC2 a ^ (1/(1-2*regEta a)) *
+    (regEta a/ε) ^ (2*regEta a/(1-2*regEta a))
+
+lemma regC2_pos {a : ℝ} (ha : 1/4<a) : 0<regC2 a := by
+  have hp := regP_pos ha
+  have hq := regQ_pos ha
+  dsimp [regC2]
+  positivity
+
+ 
+theorem regularized_pair_absorption {a ε δᵢ δⱼ Dᵢ Dⱼ Vᵢ Vⱼ : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hδᵢ : 0<δᵢ) (hδⱼ : 0<δⱼ)
+    (hDᵢ : 0≤Dᵢ) (hDⱼ : 0≤Dⱼ) (hVᵢ : 0≤Vᵢ) (hVⱼ : 0≤Vⱼ) :
+    regC2 a * (δᵢ*δⱼ)^(-1-regEta a) *
+      (Dᵢ*Vᵢ^2)^(regEta a)*(Dⱼ*Vⱼ^2)^(regEta a)*(Vᵢ*Vⱼ)^(1-2*regEta a) ≤
+    ε*(Dᵢ*Vᵢ^2+Dⱼ*Vⱼ^2) + regYoungC a ε*(δᵢ*δⱼ)^(-regM a)*Vᵢ*Vⱼ := by
+  have hC := regC2_pos ha
+  have hd : 0<δᵢ*δⱼ := mul_pos hδᵢ hδⱼ
+  have hy := weighted_pair_young (regEta_pos ha) (regEta_lt_half ha) hε
+    (show 0<regC2 a*(δᵢ*δⱼ)^(-1-regEta a) by positivity)
+    (show 0≤Dᵢ*Vᵢ^2 by positivity) (show 0≤Dⱼ*Vⱼ^2 by positivity)
+    (show 0≤Vᵢ*Vⱼ by positivity)
+  have he : (regC2 a*(δᵢ*δⱼ)^(-1-regEta a))^(1/(1-2*regEta a)) =
+      regC2 a^(1/(1-2*regEta a))*(δᵢ*δⱼ)^(-regM a) := by
+    rw [Real.mul_rpow hC.le (by positivity),← Real.rpow_mul hd.le]
+    congr 2
+    dsimp [regM]
+    ring
+  rw [he] at hy
+  dsimp only [regYoungC]
+  convert hy using 1
+  ring
+
+end ExactSLEFiniteTilt
+
+namespace ExactSLECalculus
+variable {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+end ExactSLECalculus
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialMeasurability.lean b/lean/OAI/Probability/SLEGauge/RadialMeasurability.lean
index 8e5c63aa003889a9fa92d4f6b96e2debf3f822a3..51d34291f8ae89ba1093f1d66ec8ae3c6b174615 100644
--- a/lean/OAI/Probability/SLEGauge/RadialMeasurability.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialMeasurability.lean
@@ -1,467 +1,467 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.MassMoments
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-open Filter Topology
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-theorem joint_radialTime_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
-    (hq : 0 < q) (hq1 : q ≤ 1) :
-    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im} =>
-      radialTime a p.1.1 p.2.1 (p.2.1.im*q)) := by
-  have ha0 : 0<a := by linarith
-  let Ω := ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}
-  let z : Ω → ℂ := fun p => p.2.1
-  let b : Ω → ExactSLEGenerator.Driver := Prod.fst
-  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
-  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
-  have hbM : Measurable b := measurable_fst
-  let ρ : Ω → ℝ := fun ω => (z ω).im*q
-  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
-  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
-  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
-  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
-    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
-  have hρM : Measurable ρ := hzM.im.mul_const q
-  have hδM (n : ℕ) : Measurable (δ n) := by
-    dsimp [δ,angleThreshold]
-    exact (hzM.im.div hzM.norm).div_const _
-  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
-    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
-    unfold heightFloor
-    fun_prop
-  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
-    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    unfold logCeiling heightFloor
-    fun_prop
-  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
-    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    fun_prop
-  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
-    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
-  have hTM (n : ℕ) : Measurable (T n) :=
-    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
-      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
-  let V (n : ℕ) (ω : Ω) := localState a (L n ω).ε (L n ω).hε (z ω) (T n ω) (b ω)
-  have hVM (n : ℕ) : Measurable (V n) :=
-    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε)
-      z b (hεM n) hzM hbM).comp ((hTM n).prodMk measurable_id))
-  apply measurable_of_Iic
-  intro t
-  by_cases ht : t=∞
-  · subst t
-    simp
-  have he : (fun ω : Ω => radialTime a (b ω).1 (z ω) (ρ ω)) ⁻¹' Iic t =
-      ⋃ n : ℕ, {ω | T n ω ≤ t.toNNReal} ∩ {ω | physicalRadius (V n ω) = ρ ω} := by
-    ext ω
-    simp only [mem_preimage,mem_Iic,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
-    constructor
-    · intro hrt
-      have hfin : radialTime a (b ω).1 (z ω) (ρ ω) ≠ ∞ := ne_top_of_le_ne_top ht hrt
-      let r := (radialTime a (b ω).1 (z ω) (ρ ω)).toNNReal
-      have hhit := radialTime_attained a ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω) (b ω) hfin
-      obtain ⟨η,hη,hηs⟩ := actual_sine_lower a (b ω).1 (b ω).1.continuous (z ω) r hhit.1
-      obtain ⟨n,hn⟩ := ((angleThreshold_tendsto (z ω)).eventually (gt_mem_nhds hη)).exists
-      have hTr : T n ω≤r := regionStop_le_hit a (L n ω).ε (L n ω).hε (z ω)
-        (L n ω).N (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) r hhit.2
-      have hrt' : r≤t.toNNReal := ENNReal.toNNReal_mono ht hrt
-      refine ⟨n,hTr.trans hrt',?_⟩
-      rcases regionStop_true_exit a (L n ω).ε ha0 (L n ω).hε (z ω) (L n ω).N
-        (ρ ω) (δ n ω) (hρ ω) (angleThreshold_pos (z ω) (hz ω) n) (L n ω).B
-        (L n ω).initial (L n ω).small (L n ω).large (L n ω).time (b ω) with hR|hS
-      · exact hR
-      · have hact := regionStop_actualState a (L n ω).ε (L n ω).hε (z ω) (L n ω).N
-          (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) (T n ω) le_rfl
-        have hs := regionStop_safe a (L n ω).ε (L n ω).hε (z ω) (L n ω).N
-          (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) (T n ω) le_rfl
-        have hp : 0 < ‖(V n ω).1‖ := ((L n ω).hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
-        have hd : Y (V n ω)/‖(V n ω).1‖ = δ n ω := (div_eq_iff hp.ne').mpr hS
-        have hη' := hηs (T n ω) hTr
-        rw [hact.2,hd] at hη'
-        exact False.elim (not_le.mpr hn hη')
-    · rintro ⟨n,hnt,hR⟩
-      have heq := regionStop_eq_radialTime a (L n ω).ε (L n ω).hε (z ω) (L n ω).N
-        (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) hR
-      rw [heq]
-      exact (ENNReal.coe_le_coe.mpr hnt).trans_eq (ENNReal.coe_toNNReal ht)
-  rw [he]
-  exact MeasurableSet.iUnion (fun n => (measurableSet_le (hTM n) measurable_const).inter
-    (measurableSet_eq_fun (physicalRadius_continuous.measurable.comp (hVM n)) hρM))
-
-theorem joint_radial_approximant_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
-    (hq : 0 < q) (hq1 : q ≤ 1) :
-    Measurable (fun p : (ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}) × ℝ≥0∞ =>
-      if radialTime a p.1.1.1 p.1.2.1 (p.1.2.1.im*q) ≤ p.2 then
-        radialWeight a p.1.1.1 p.1.2.1 (p.1.2.1.im*q) else 0) := by
-  exact ((joint_radialWeight_measurable a ha q hq hq1).comp measurable_fst).ite
-    (measurableSet_le ((joint_radialTime_measurable a ha q hq hq1).comp measurable_fst) measurable_snd)
-    measurable_const
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped Classical NNReal ENNReal
-noncomputable def timeMassDensity (a q : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ≥0∞ :=
-  if radialTime a b.1 z (z.im*q) ≤ t then massDensity a q b z else 0
-noncomputable def actualMassEval (a q : ℝ) (t : ℝ≥0∞)
-    (K : Set ℂ) (f : ℂ → ℝ) (b : ExactSLEGenerator.Driver) : ℝ :=
-  ∫ z in K, f z * (timeMassDensity a q t b z).toReal
-lemma timeMassDensity_le (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    timeMassDensity a q t b z ≤ massDensity a q b z := by
-  unfold timeMassDensity
-  split_ifs <;> simp
-
-lemma timeMassDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0∞) :
-    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t p.1 p.2) := by
-  let U : Set (ExactSLEGenerator.Driver × ℂ) := {p | 0<p.2.im}
-  have hU : MeasurableSet U := measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
-  have hm : Measurable (fun p : U => ENNReal.ofReal (Real.exp (-alpha a*Real.log 2)*
-      (if radialTime a p.1.1.1 p.1.2 (p.1.2.im*q)≤t then
-        radialWeight a p.1.1.1 p.1.2 (p.1.2.im*q) else 0))) := by
-    have hp : Measurable (fun p : U => ((p.1.1, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im})),t)) := by fun_prop
-    exact (measurable_const.mul ((joint_radial_approximant_measurable a ha q hq hq1).comp hp)).ennreal_ofReal
-  have h := hm.dite (g:=fun _ => 0) measurable_const hU
-  convert h using 1
-  funext p
-  have he : 2*p.2.im*q/2=p.2.im*q := by ring
-  by_cases hz : 0<p.2.im <;> by_cases ht : radialTime a p.1.1 p.2 (p.2.im*q)≤t <;>
-    simp [timeMassDensity,massDensity,fullRadialDensity,U,hz,ht,he]
-
-lemma actualMassEval_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0∞) (K : Set ℂ) (f : ℂ → ℝ) (hf : Measurable f) :
-    Measurable (actualMassEval a q t K f) := by
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      f p.2 * (timeMassDensity a q t p.1 p.2).toReal) :=
-    (hf.comp measurable_snd).mul (timeMassDensity_measurable a ha q hq hq1 t).ennreal_toReal
-  exact hm.stronglyMeasurable.integral_prod_right.measurable
-
-lemma actualMassEval_enorm_bound (a q : ℝ) (t : ℝ≥0∞) (K : Set ℂ) (f : ℂ → ℝ)
-    (b : ExactSLEGenerator.Driver) :
-    ‖actualMassEval a q t K f b‖ₑ ≤
-      ∫⁻ z in K, ENNReal.ofReal ‖f z‖ * massDensity a q b z := by
-  refine (enorm_integral_le_lintegral_enorm _).trans ?_
-  apply lintegral_mono
-  intro z
-  dsimp only
-  rw [enorm_mul, Real.enorm_eq_ofReal_abs, ←Real.norm_eq_abs,
-    Real.enorm_eq_ofReal ENNReal.toReal_nonneg]
-  exact mul_le_mul' le_rfl (ENNReal.ofReal_toReal_le.trans (timeMassDensity_le a q t b z))
-
-lemma actualMassEval_L2_bound (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z, z∈K → c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ, 0≤B ∧ ∀ z∈K, ‖f z‖≤B) :
-    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 → ∀ t : ℝ≥0∞,
-      MemLp (actualMassEval a q t K f) 2 W ∧ eLpNorm (actualMassEval a q t K f) 2 W ≤ C := by
-  obtain ⟨C,hC,hCb⟩ := actual_mass_moment_bound a ha 2
-  obtain ⟨hAfin,hB⟩ := hCb W hW K hK hKb c hc hcK (fun z => ‖f z‖) hf.norm
-    (fun _ _ => norm_nonneg _) hfB
-  let A := ∫⁻ x : Fin 2 → ℂ, (∏ i, ENNReal.ofReal ‖f (x i)‖)*
-    ExactSLEKernel.kernel (alpha a) c x ∂Measure.pi (fun _ => volume.restrict K)
-  let D := (ENNReal.ofReal C*A)^(1/(2:ℝ))
-  have hD : D<∞ := ENNReal.rpow_lt_top_of_nonneg (by norm_num) (ENNReal.mul_lt_top ENNReal.ofReal_lt_top hAfin).ne
-  refine ⟨D,hD,?_⟩
-  intro q hq hq1 t
-  have hnorm : eLpNorm (actualMassEval a q t K f) 2 W≤D := by
-    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num)
-      (actualMassEval_measurable a ha q hq hq1 t K f hf).aestronglyMeasurable]
-    norm_num only [ENNReal.toReal_ofNat]
-    apply ENNReal.rpow_le_rpow _ (by norm_num)
-    calc
-      _ ≤ ∫⁻ b, (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^2 ∂W := by
-        apply lintegral_mono
-        intro b
-        change ‖actualMassEval a q t K f b‖ₑ^(2:ℝ) ≤
-          (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^(2:ℕ)
-        rw [ENNReal.rpow_two]
-        exact pow_le_pow_left' (actualMassEval_enorm_bound a q t K f b) 2
-      _ ≤ _ := hB q hq hq1
-  exact ⟨hnorm.trans_lt hD,hnorm⟩
-
-theorem actualMassEval_integrable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0∞) (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z, z∈K → c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ, 0≤B ∧ ∀ z∈K, ‖f z‖≤B)
-    (b : ExactSLEGenerator.Driver) :
-    Integrable (fun z => f z*(timeMassDensity a q t b z).toReal) (volume.restrict K) := by
-  obtain ⟨B,hB,hBf⟩ := hfB
-  let D := (c*q)^(-alpha a)
-  have hD : 0≤D := Real.rpow_nonneg (mul_pos hc hq).le _
-  have hd (z : ℂ) (hzK : z∈K) : (timeMassDensity a q t b z).toReal≤D := by
-    have hz : 0<z.im := by have := hcK z hzK; linarith
-    have hit := fullRadialDensity_bound a ha z hz (2*z.im*q) (by positivity) (by nlinarith) b
-    have htar : (2*z.im*q)^(-alpha a)≤D :=
-      Real.rpow_le_rpow_of_nonpos (mul_pos hc hq) (mul_le_mul_of_nonneg_right (hcK z hzK) hq.le)
-        (neg_nonpos.mpr (alpha_pos ha).le)
-    have hm : massDensity a q b z≤ENNReal.ofReal D := by
-      simp only [massDensity,ite_eq_left hz]
-      exact ENNReal.ofReal_le_ofReal (hit.2.trans htar)
-    have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top ((timeMassDensity_le a q t b z).trans hm)
-    simpa only [ENNReal.toReal_ofReal hD] using hh
-  have : IsFiniteMeasure (volume.restrict K) := ⟨by simpa using hKb.measure_lt_top⟩
-  have hm : Measurable (fun z => timeMassDensity a q t b z) :=
-    (timeMassDensity_measurable a ha q hq hq1 t).comp (measurable_const.prodMk measurable_id)
-  apply Integrable.of_bound ((hf.mul hm.ennreal_toReal).aestronglyMeasurable) (B*D)
-  filter_upwards [ae_restrict_mem hK] with z hz
-  change ‖f z*(timeMassDensity a q t b z).toReal‖ ≤ B*D
-  rw [norm_mul,Real.norm_of_nonneg ENNReal.toReal_nonneg]
-  exact mul_le_mul (hBf z hz) (hd z hz) ENNReal.toReal_nonneg hB
-theorem actual_common_mass_extraction (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (K : ℕ → Set ℂ) (hK : ∀ i, MeasurableSet (K i))
-    (hKb : ∀ i, Bornology.IsBounded (K i))
-    (c : ℕ → ℝ) (hc : ∀ i, 0<c i) (hcK : ∀ i z, z∈K i → c i≤2*z.im)
-    (f : ℕ → ℂ → ℝ) (hf : ∀ i, Measurable (f i))
-    (hfB : ∀ i, ∃ B : ℝ, 0≤B ∧ ∀ z∈K i, ‖f i z‖≤B)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (X : ℕ → ℕ → Lp ℝ 2 W) (s : ℕ → Finset ℕ)
-      (coeff : ℕ → ℕ → ℝ) (v : ℕ → Lp ℝ 2 W),
-      (∀ j i, (fun b => X j i b) =ᵐ[W]
-        actualMassEval a (Real.exp (-2*a*(j+1))) (t i) (K i) (f i)) ∧
-      (∀ k j, j∈s k → k≤j) ∧
-      (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ i, Tendsto (fun k => ∑ j∈s k, coeff k j • X j i) atTop (𝓝 (v i))) ∧
-      (∀ᵐ b ∂W, ∀ i, Tendsto (fun k => ∑ j∈s k, coeff k j *
-        actualMassEval a (Real.exp (-2*a*(j+1))) (t i) (K i) (f i) b)
-        atTop (𝓝 (v i b)))
-    := by
-  choose C hC hBound using fun i => actualMassEval_L2_bound a ha W hW (K i) (hK i) (hKb i)
-    (c i) (hc i) (hcK i) (f i) (hf i) (hfB i)
-  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  let hLp := fun j i => (hBound i (q j) (hq j) (hq1 j) (t i)).1
-  let X := fun j i => (hLp j i).toLp (actualMassEval a (q j) (t i) (K i) (f i))
-  have hX (i : ℕ) : ∃ B : ℝ, ∀ j, ‖X j i‖≤B := by
-    refine ⟨(C i).toReal,?_⟩
-    intro j
-    rw [Lp.norm_toLp]
-    exact ENNReal.toReal_mono (hC i).ne (hBound i (q j) (hq j) (hq1 j) (t i)).2
-  obtain ⟨s,coeff,v,hs,hcoeff,hsum,hconv,hae⟩ := common_L2_convex_tails W X hX
-  have he j i : (fun b => X j i b) =ᵐ[W] actualMassEval a (q j) (t i) (K i) (f i) :=
-    (hLp j i).coeFn_toLp
-  refine ⟨X,s,coeff,v,he,hs,hcoeff,hsum,hconv,?_⟩
-  have heall : ∀ᵐ b ∂W, ∀ j i, X j i b=actualMassEval a (q j) (t i) (K i) (f i) b :=
-    ae_all_iff.mpr fun j => ae_all_iff.mpr fun i => he j i
-  filter_upwards [hae,heall] with b hb heq
-  intro i
-  simpa only [heq] using hb i
-
-end ExactSLEGreen
-
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEMassExtraction
-variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-omit [T2Space X] [MeasurableSpace X] [BorelSpace X] in
-theorem supported_dense_tests (K : Set X) (hK : IsCompact K) :
-    ∃ g : ℕ → C_c(X, ℝ), (∀ i, tsupport (g i) ⊆ K) ∧
-      ∀ f : C_c(X, ℝ), tsupport f ⊆ K → ∀ ε > 0,
-        ∃ i, ∀ x, |f x - g i x| < ε := by
-  classical
-  let S := {f : C(X, ℝ) // tsupport f ⊆ K}
-  have : SecondCountableTopology C(X, ℝ) := inferInstance
-  have : SecondCountableTopology S := TopologicalSpace.Subtype.secondCountableTopology _
-  have : Nonempty S := ⟨⟨0, by simp⟩⟩
-  let d := TopologicalSpace.denseSeq S
-  let g : ℕ → C_c(X, ℝ) := fun i =>
-    ⟨(d i).1, hK.of_isClosed_subset (isClosed_tsupport _) (d i).2⟩
-  refine ⟨g, fun i => (d i).2, ?_⟩
-  intro f hf ε hε
-  let s : S := ⟨f.toContinuousMap, hf⟩
-  have hs : s ∈ closure (range d) := TopologicalSpace.denseRange_denseSeq S s
-  obtain ⟨v, hv, ht⟩ := mem_closure_iff_seq_limit.mp hs
-  have ht' : Tendsto (fun n => (v n).1) atTop (𝓝 s.1) :=
-    continuous_subtype_val.continuousAt.tendsto.comp ht
-  have hu := (ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn.mp ht') K hK
-  obtain ⟨n, hn⟩ := (Metric.tendstoUniformlyOn_iff.mp hu ε hε).exists
-  obtain ⟨i, hi⟩ := hv n
-  refine ⟨i, fun x => ?_⟩
-  by_cases hx : x ∈ K
-  · have H := hn x hx
-    rw [← hi] at H
-    exact H
-  · have hf0 : f x = 0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hf h))
-    have hg0 : g i x = 0 := image_eq_zero_of_notMem_tsupport (fun h => hx ((d i).2 h))
-    simpa [hf0, hg0] using hε
-
-omit [MeasurableSpace X] [BorelSpace X] in
-theorem cutoff_test (K : Set X) (hK : IsCompact K) :
-    ∃ g : C_c(X, ℝ), (∀ x, 0 ≤ g x) ∧ ∀ x ∈ K, g x = 1 := by
-  obtain ⟨g, hg, _, hc, hn⟩ := exists_continuous_one_zero_of_isCompact hK
-    isClosed_empty (disjoint_empty K)
-  exact ⟨⟨g, hc⟩, fun x => (hn x).1, fun x hx => hg hx⟩
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-    [MeasurableSpace X] [BorelSpace X] in
-theorem positive_map_error (L : C_c(X, ℝ) →ₚ[ℝ] ℝ)
-    (f h g : C_c(X, ℝ)) (K : Set X) (hf : tsupport f ⊆ K)
-    (hh : tsupport h ⊆ K) (hg : ∀ x, 0 ≤ g x) (hg1 : ∀ x ∈ K, g x = 1)
-    {ε : ℝ} (hε : 0 ≤ ε) (he : ∀ x, |f x - h x| ≤ ε) :
-    |L f - L h| ≤ ε * L g := by
-  have hp : f - h ≤ ε • g := by
-    intro x
-    by_cases hx : x ∈ K
-    · simpa [hg1 x hx] using (abs_le.mp (he x)).2
-    · have hf0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hf H))
-      have hh0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hh H))
-      simpa [hf0, hh0] using mul_nonneg hε (hg x)
-  have hm : -(ε • g) ≤ f - h := by
-    intro x
-    by_cases hx : x ∈ K
-    · simpa [hg1 x hx] using (abs_le.mp (he x)).1
-    · have hf0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hf H))
-      have hh0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hh H))
-      simpa [hf0, hh0] using neg_nonpos.mpr (mul_nonneg hε (hg x))
-  exact abs_le.mpr ⟨by simpa using L.monotone' hm, by simpa using L.monotone' hp⟩
-
-theorem determining_tests_radon_limit
-    (X : Type*) [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] :
-    ∃ tests : ℕ → C_c(X, ℝ), ∀ L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ),
-      (∀ i, ∃ v : ℝ, Tendsto (fun j => L j (tests i)) atTop (𝓝 v)) →
-      ∃ μ : Measure X, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
-        ∀ f : C_c(X, ℝ), Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂μ)) := by
-  classical
-  let K := CompactExhaustion.choice X
-  choose d hd hdense using fun n => supported_dense_tests (K n) (K.isCompact n)
-  choose g hg hg1 using fun n => cutoff_test (K n) (K.isCompact n)
-  let e := Denumerable.eqv ((ℕ × ℕ) ⊕ ℕ)
-  let tests : ℕ → C_c(X, ℝ) := fun i => Sum.elim (fun p => d p.1 p.2) g (e.symm i)
-  refine ⟨tests, ?_⟩
-  intro L hL
-  have hdL (n i : ℕ) : ∃ v : ℝ, Tendsto (fun j => L j (d n i)) atTop (𝓝 v) := by
-    simpa [tests] using hL (e (.inl (n,i)))
-  have hgL (n : ℕ) : ∃ v : ℝ, Tendsto (fun j => L j (g n)) atTop (𝓝 v) := by
-    simpa [tests] using hL (e (.inr n))
-  have hc (f : C_c(X, ℝ)) : CauchySeq (fun j => L j f) := by
-    obtain ⟨n, hn⟩ := K.exists_superset_of_isCompact f.hasCompactSupport
-    obtain ⟨v, hv⟩ := hgL n
-    obtain ⟨C, hC⟩ := hv.bddAbove_range
-    let D := max C 1
-    have hD : 0 < D := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
-    have hDj (j : ℕ) : L j (g n) ≤ D := (hC (mem_range_self j)).trans (le_max_left _ _)
-    apply Metric.cauchySeq_iff.mpr
-    intro ε hε
-    let δ := ε / (4*D)
-    have hδ : 0 < δ := div_pos hε (mul_pos (by norm_num) hD)
-    obtain ⟨i, hi⟩ := hdense n f hn δ hδ
-    have herr (j : ℕ) : |L j f-L j (d n i)| ≤ ε/4 := by
-      have H := positive_map_error (L j) f (d n i) (g n) (K n) hn (hd n i)
-        (hg n) (hg1 n) hδ.le (fun x => (hi x).le)
-      have hh : δ*D=ε/4 := by dsimp [δ]; field_simp
-      exact H.trans ((mul_le_mul_of_nonneg_left (hDj j) hδ.le).trans_eq hh)
-    obtain ⟨w, hw⟩ := hdL n i
-    obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.mp hw.cauchySeq (ε/2) (by positivity)
-    refine ⟨N, fun j hj k hk => ?_⟩
-    have hjk := hN j hj k hk
-    rw [Real.dist_eq] at hjk ⊢
-    have H := abs_sub_le (L j f) (L j (d n i)) (L k f)
-    have H' := abs_sub_le (L j (d n i)) (L k (d n i)) (L k f)
-    have he := herr k
-    rw [abs_sub_comm] at he
-    linarith [herr j]
-  choose V hV using fun f : C_c(X, ℝ) => cauchySeq_tendsto_of_complete (hc f)
-  let Λ : C_c(X, ℝ) →ₚ[ℝ] ℝ := PositiveLinearMap.mk₀
-    { toFun := V
-      map_add' := fun f h => by
-        apply tendsto_nhds_unique (hV (f+h))
-        simpa only [map_add] using (hV f).add (hV h)
-      map_smul' := fun r f => by
-        apply tendsto_nhds_unique (hV (r • f))
-        simpa only [map_smul, smul_eq_mul, RingHom.id_apply] using (hV f).const_mul r }
-    (fun f hf => ge_of_tendsto' (hV f) (fun j => (L j).map_nonneg hf))
-  refine ⟨RealRMK.rieszMeasure Λ, inferInstance, inferInstance, fun f => ?_⟩
-  rw [RealRMK.integral_rieszMeasure]
-  exact hV f
-
-end ExactSLEMassExtraction
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-abbrev UpperPlane := {z : ℂ // 0 < z.im}
-noncomputable def upperArea : Measure UpperPlane := volume.comap Subtype.val
-noncomputable def actualUpperMass (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    Measure UpperPlane := upperArea.withDensity (fun z => timeMassDensity a q t b z.1)
-lemma upperPlane_open : IsOpen {z : ℂ | 0 < z.im} := isOpen_lt continuous_const Complex.continuous_im
-
-instance upperPlane_locallyCompact : LocallyCompactSpace UpperPlane :=
-  upperPlane_open.isOpenEmbedding_subtypeVal.locallyCompactSpace
-
-lemma upper_embedding : MeasurableEmbedding (Subtype.val : UpperPlane → ℂ) :=
-  MeasurableEmbedding.subtype_coe upperPlane_open.measurableSet
-
-lemma upper_compact_lower (K : Set UpperPlane) (hK : IsCompact K) :
-    ∃ c : ℝ, 0<c ∧ ∀ z∈K, c≤2*z.1.im := by
-  by_cases hn : K.Nonempty
-  · obtain ⟨z,hz,hmin⟩ := hK.exists_isMinOn hn
-      ((continuous_const.mul (Complex.continuous_im.comp continuous_subtype_val)).continuousOn)
-    exact ⟨2*z.1.im, mul_pos (by norm_num) z.2, fun w hw => hmin hw⟩
-  · exact ⟨1, by norm_num, fun z hz => False.elim (hn ⟨z,hz⟩)⟩
-
-lemma timeMassDensity_finite (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    timeMassDensity a q t b z < ⊤ := by
-  apply lt_of_le_of_lt (timeMassDensity_le a q t b z)
-  unfold massDensity
-  split_ifs <;> simp
-
-lemma timeMassDensity_local_bound (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (c : ℝ) (hc : 0<c)
-    (z : ℂ) (hz : c≤2*z.im) :
-    timeMassDensity a q t b z≤ENNReal.ofReal ((c*q)^(-alpha a)) := by
-  have hz0 : 0<z.im := by linarith
-  have hit := fullRadialDensity_bound a ha z hz0 (2*z.im*q) (by positivity) (by nlinarith) b
-  have htar : (2*z.im*q)^(-alpha a)≤(c*q)^(-alpha a) :=
-    Real.rpow_le_rpow_of_nonpos (mul_pos hc hq) (mul_le_mul_of_nonneg_right hz hq.le)
-       (neg_nonpos.mpr (alpha_pos ha).le)
-  apply (timeMassDensity_le a q t b z).trans
-  simp only [massDensity,ite_eq_left hz0]
-  exact ENNReal.ofReal_le_ofReal (hit.2.trans htar)
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.MassMoments
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+open Filter Topology
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+theorem joint_radialTime_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
+    (hq : 0 < q) (hq1 : q ≤ 1) :
+    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im} =>
+      radialTime a p.1.1 p.2.1 (p.2.1.im*q)) := by
+  have ha0 : 0<a := by linarith
+  let Ω := ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}
+  let z : Ω → ℂ := fun p => p.2.1
+  let b : Ω → ExactSLEGenerator.Driver := Prod.fst
+  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
+  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
+  have hbM : Measurable b := measurable_fst
+  let ρ : Ω → ℝ := fun ω => (z ω).im*q
+  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
+  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
+  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
+  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
+    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
+  have hρM : Measurable ρ := hzM.im.mul_const q
+  have hδM (n : ℕ) : Measurable (δ n) := by
+    dsimp [δ,angleThreshold]
+    exact (hzM.im.div hzM.norm).div_const _
+  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
+    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
+    unfold heightFloor
+    fun_prop
+  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
+    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    unfold logCeiling heightFloor
+    fun_prop
+  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
+    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    fun_prop
+  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
+    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
+  have hTM (n : ℕ) : Measurable (T n) :=
+    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
+      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
+  let V (n : ℕ) (ω : Ω) := localState a (L n ω).ε (L n ω).hε (z ω) (T n ω) (b ω)
+  have hVM (n : ℕ) : Measurable (V n) :=
+    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε)
+      z b (hεM n) hzM hbM).comp ((hTM n).prodMk measurable_id))
+  apply measurable_of_Iic
+  intro t
+  by_cases ht : t=∞
+  · subst t
+    simp
+  have he : (fun ω : Ω => radialTime a (b ω).1 (z ω) (ρ ω)) ⁻¹' Iic t =
+      ⋃ n : ℕ, {ω | T n ω ≤ t.toNNReal} ∩ {ω | physicalRadius (V n ω) = ρ ω} := by
+    ext ω
+    simp only [mem_preimage,mem_Iic,mem_iUnion,mem_inter_iff,mem_ofPred_eq]
+    constructor
+    · intro hrt
+      have hfin : radialTime a (b ω).1 (z ω) (ρ ω) ≠ ∞ := ne_top_of_le_ne_top ht hrt
+      let r := (radialTime a (b ω).1 (z ω) (ρ ω)).toNNReal
+      have hhit := radialTime_attained a ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω) (b ω) hfin
+      obtain ⟨η,hη,hηs⟩ := actual_sine_lower a (b ω).1 (b ω).1.continuous (z ω) r hhit.1
+      obtain ⟨n,hn⟩ := ((angleThreshold_tendsto (z ω)).eventually (gt_mem_nhds hη)).exists
+      have hTr : T n ω≤r := regionStop_le_hit a (L n ω).ε (L n ω).hε (z ω)
+        (L n ω).N (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) r hhit.2
+      have hrt' : r≤t.toNNReal := ENNReal.toNNReal_mono ht hrt
+      refine ⟨n,hTr.trans hrt',?_⟩
+      rcases regionStop_true_exit a (L n ω).ε ha0 (L n ω).hε (z ω) (L n ω).N
+        (ρ ω) (δ n ω) (hρ ω) (angleThreshold_pos (z ω) (hz ω) n) (L n ω).B
+        (L n ω).initial (L n ω).small (L n ω).large (L n ω).time (b ω) with hR|hS
+      · exact hR
+      · have hact := regionStop_actualState a (L n ω).ε (L n ω).hε (z ω) (L n ω).N
+          (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) (T n ω) le_rfl
+        have hs := regionStop_safe a (L n ω).ε (L n ω).hε (z ω) (L n ω).N
+          (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) (T n ω) le_rfl
+        have hp : 0 < ‖(V n ω).1‖ := ((L n ω).hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
+        have hd : Y (V n ω)/‖(V n ω).1‖ = δ n ω := (div_eq_iff hp.ne').mpr hS
+        have hη' := hηs (T n ω) hTr
+        rw [hact.2,hd] at hη'
+        exact False.elim (not_le.mpr hn hη')
+    · rintro ⟨n,hnt,hR⟩
+      have heq := regionStop_eq_radialTime a (L n ω).ε (L n ω).hε (z ω) (L n ω).N
+        (ρ ω) (δ n ω) (L n ω).B (L n ω).initial (b ω) hR
+      rw [heq]
+      exact (ENNReal.coe_le_coe.mpr hnt).trans_eq (ENNReal.coe_toNNReal ht)
+  rw [he]
+  exact MeasurableSet.iUnion (fun n => (measurableSet_le (hTM n) measurable_const).inter
+    (measurableSet_eq_fun (physicalRadius_continuous.measurable.comp (hVM n)) hρM))
+
+theorem joint_radial_approximant_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
+    (hq : 0 < q) (hq1 : q ≤ 1) :
+    Measurable (fun p : (ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}) × ℝ≥0∞ =>
+      if radialTime a p.1.1.1 p.1.2.1 (p.1.2.1.im*q) ≤ p.2 then
+        radialWeight a p.1.1.1 p.1.2.1 (p.1.2.1.im*q) else 0) := by
+  exact ((joint_radialWeight_measurable a ha q hq hq1).comp measurable_fst).ite
+    (measurableSet_le ((joint_radialTime_measurable a ha q hq hq1).comp measurable_fst) measurable_snd)
+    measurable_const
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped Classical NNReal ENNReal
+noncomputable def timeMassDensity (a q : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ≥0∞ :=
+  if radialTime a b.1 z (z.im*q) ≤ t then massDensity a q b z else 0
+noncomputable def actualMassEval (a q : ℝ) (t : ℝ≥0∞)
+    (K : Set ℂ) (f : ℂ → ℝ) (b : ExactSLEGenerator.Driver) : ℝ :=
+  ∫ z in K, f z * (timeMassDensity a q t b z).toReal
+lemma timeMassDensity_le (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    timeMassDensity a q t b z ≤ massDensity a q b z := by
+  unfold timeMassDensity
+  split_ifs <;> simp
+
+lemma timeMassDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0∞) :
+    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t p.1 p.2) := by
+  let U : Set (ExactSLEGenerator.Driver × ℂ) := {p | 0<p.2.im}
+  have hU : MeasurableSet U := measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
+  have hm : Measurable (fun p : U => ENNReal.ofReal (Real.exp (-alpha a*Real.log 2)*
+      (if radialTime a p.1.1.1 p.1.2 (p.1.2.im*q)≤t then
+        radialWeight a p.1.1.1 p.1.2 (p.1.2.im*q) else 0))) := by
+    have hp : Measurable (fun p : U => ((p.1.1, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im})),t)) := by fun_prop
+    exact (measurable_const.mul ((joint_radial_approximant_measurable a ha q hq hq1).comp hp)).ennreal_ofReal
+  have h := hm.dite (g:=fun _ => 0) measurable_const hU
+  convert h using 1
+  funext p
+  have he : 2*p.2.im*q/2=p.2.im*q := by ring
+  by_cases hz : 0<p.2.im <;> by_cases ht : radialTime a p.1.1 p.2 (p.2.im*q)≤t <;>
+    simp [timeMassDensity,massDensity,fullRadialDensity,U,hz,ht,he]
+
+lemma actualMassEval_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0∞) (K : Set ℂ) (f : ℂ → ℝ) (hf : Measurable f) :
+    Measurable (actualMassEval a q t K f) := by
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      f p.2 * (timeMassDensity a q t p.1 p.2).toReal) :=
+    (hf.comp measurable_snd).mul (timeMassDensity_measurable a ha q hq hq1 t).ennreal_toReal
+  exact hm.stronglyMeasurable.integral_prod_right.measurable
+
+lemma actualMassEval_enorm_bound (a q : ℝ) (t : ℝ≥0∞) (K : Set ℂ) (f : ℂ → ℝ)
+    (b : ExactSLEGenerator.Driver) :
+    ‖actualMassEval a q t K f b‖ₑ ≤
+      ∫⁻ z in K, ENNReal.ofReal ‖f z‖ * massDensity a q b z := by
+  refine (enorm_integral_le_lintegral_enorm _).trans ?_
+  apply lintegral_mono
+  intro z
+  dsimp only
+  rw [enorm_mul, Real.enorm_eq_ofReal_abs, ←Real.norm_eq_abs,
+    Real.enorm_eq_ofReal ENNReal.toReal_nonneg]
+  exact mul_le_mul' le_rfl (ENNReal.ofReal_toReal_le.trans (timeMassDensity_le a q t b z))
+
+lemma actualMassEval_L2_bound (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z, z∈K → c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ, 0≤B ∧ ∀ z∈K, ‖f z‖≤B) :
+    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 → ∀ t : ℝ≥0∞,
+      MemLp (actualMassEval a q t K f) 2 W ∧ eLpNorm (actualMassEval a q t K f) 2 W ≤ C := by
+  obtain ⟨C,hC,hCb⟩ := actual_mass_moment_bound a ha 2
+  obtain ⟨hAfin,hB⟩ := hCb W hW K hK hKb c hc hcK (fun z => ‖f z‖) hf.norm
+    (fun _ _ => norm_nonneg _) hfB
+  let A := ∫⁻ x : Fin 2 → ℂ, (∏ i, ENNReal.ofReal ‖f (x i)‖)*
+    ExactSLEKernel.kernel (alpha a) c x ∂Measure.pi (fun _ => volume.restrict K)
+  let D := (ENNReal.ofReal C*A)^(1/(2:ℝ))
+  have hD : D<∞ := ENNReal.rpow_lt_top_of_nonneg (by norm_num) (ENNReal.mul_lt_top ENNReal.ofReal_lt_top hAfin).ne
+  refine ⟨D,hD,?_⟩
+  intro q hq hq1 t
+  have hnorm : eLpNorm (actualMassEval a q t K f) 2 W≤D := by
+    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num)
+      (actualMassEval_measurable a ha q hq hq1 t K f hf).aestronglyMeasurable]
+    norm_num only [ENNReal.toReal_ofNat]
+    apply ENNReal.rpow_le_rpow _ (by norm_num)
+    calc
+      _ ≤ ∫⁻ b, (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^2 ∂W := by
+        apply lintegral_mono
+        intro b
+        change ‖actualMassEval a q t K f b‖ₑ^(2:ℝ) ≤
+          (∫⁻ z in K, ENNReal.ofReal ‖f z‖*massDensity a q b z)^(2:ℕ)
+        rw [ENNReal.rpow_two]
+        exact pow_le_pow_left' (actualMassEval_enorm_bound a q t K f b) 2
+      _ ≤ _ := hB q hq hq1
+  exact ⟨hnorm.trans_lt hD,hnorm⟩
+
+theorem actualMassEval_integrable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0∞) (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z, z∈K → c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ, 0≤B ∧ ∀ z∈K, ‖f z‖≤B)
+    (b : ExactSLEGenerator.Driver) :
+    Integrable (fun z => f z*(timeMassDensity a q t b z).toReal) (volume.restrict K) := by
+  obtain ⟨B,hB,hBf⟩ := hfB
+  let D := (c*q)^(-alpha a)
+  have hD : 0≤D := Real.rpow_nonneg (mul_pos hc hq).le _
+  have hd (z : ℂ) (hzK : z∈K) : (timeMassDensity a q t b z).toReal≤D := by
+    have hz : 0<z.im := by have := hcK z hzK; linarith
+    have hit := fullRadialDensity_bound a ha z hz (2*z.im*q) (by positivity) (by nlinarith) b
+    have htar : (2*z.im*q)^(-alpha a)≤D :=
+      Real.rpow_le_rpow_of_nonpos (mul_pos hc hq) (mul_le_mul_of_nonneg_right (hcK z hzK) hq.le)
+        (neg_nonpos.mpr (alpha_pos ha).le)
+    have hm : massDensity a q b z≤ENNReal.ofReal D := by
+      simp only [massDensity,ite_eq_left hz]
+      exact ENNReal.ofReal_le_ofReal (hit.2.trans htar)
+    have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top ((timeMassDensity_le a q t b z).trans hm)
+    simpa only [ENNReal.toReal_ofReal hD] using hh
+  have : IsFiniteMeasure (volume.restrict K) := ⟨by simpa using hKb.measure_lt_top⟩
+  have hm : Measurable (fun z => timeMassDensity a q t b z) :=
+    (timeMassDensity_measurable a ha q hq hq1 t).comp (measurable_const.prodMk measurable_id)
+  apply Integrable.of_bound ((hf.mul hm.ennreal_toReal).aestronglyMeasurable) (B*D)
+  filter_upwards [ae_restrict_mem hK] with z hz
+  change ‖f z*(timeMassDensity a q t b z).toReal‖ ≤ B*D
+  rw [norm_mul,Real.norm_of_nonneg ENNReal.toReal_nonneg]
+  exact mul_le_mul (hBf z hz) (hd z hz) ENNReal.toReal_nonneg hB
+theorem actual_common_mass_extraction (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (K : ℕ → Set ℂ) (hK : ∀ i, MeasurableSet (K i))
+    (hKb : ∀ i, Bornology.IsBounded (K i))
+    (c : ℕ → ℝ) (hc : ∀ i, 0<c i) (hcK : ∀ i z, z∈K i → c i≤2*z.im)
+    (f : ℕ → ℂ → ℝ) (hf : ∀ i, Measurable (f i))
+    (hfB : ∀ i, ∃ B : ℝ, 0≤B ∧ ∀ z∈K i, ‖f i z‖≤B)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (X : ℕ → ℕ → Lp ℝ 2 W) (s : ℕ → Finset ℕ)
+      (coeff : ℕ → ℕ → ℝ) (v : ℕ → Lp ℝ 2 W),
+      (∀ j i, (fun b => X j i b) =ᵐ[W]
+        actualMassEval a (Real.exp (-2*a*(j+1))) (t i) (K i) (f i)) ∧
+      (∀ k j, j∈s k → k≤j) ∧
+      (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ i, Tendsto (fun k => ∑ j∈s k, coeff k j • X j i) atTop (𝓝 (v i))) ∧
+      (∀ᵐ b ∂W, ∀ i, Tendsto (fun k => ∑ j∈s k, coeff k j *
+        actualMassEval a (Real.exp (-2*a*(j+1))) (t i) (K i) (f i) b)
+        atTop (𝓝 (v i b)))
+    := by
+  choose C hC hBound using fun i => actualMassEval_L2_bound a ha W hW (K i) (hK i) (hKb i)
+    (c i) (hc i) (hcK i) (f i) (hf i) (hfB i)
+  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  let hLp := fun j i => (hBound i (q j) (hq j) (hq1 j) (t i)).1
+  let X := fun j i => (hLp j i).toLp (actualMassEval a (q j) (t i) (K i) (f i))
+  have hX (i : ℕ) : ∃ B : ℝ, ∀ j, ‖X j i‖≤B := by
+    refine ⟨(C i).toReal,?_⟩
+    intro j
+    rw [Lp.norm_toLp]
+    exact ENNReal.toReal_mono (hC i).ne (hBound i (q j) (hq j) (hq1 j) (t i)).2
+  obtain ⟨s,coeff,v,hs,hcoeff,hsum,hconv,hae⟩ := common_L2_convex_tails W X hX
+  have he j i : (fun b => X j i b) =ᵐ[W] actualMassEval a (q j) (t i) (K i) (f i) :=
+    (hLp j i).coeFn_toLp
+  refine ⟨X,s,coeff,v,he,hs,hcoeff,hsum,hconv,?_⟩
+  have heall : ∀ᵐ b ∂W, ∀ j i, X j i b=actualMassEval a (q j) (t i) (K i) (f i) b :=
+    ae_all_iff.mpr fun j => ae_all_iff.mpr fun i => he j i
+  filter_upwards [hae,heall] with b hb heq
+  intro i
+  simpa only [heq] using hb i
+
+end ExactSLEGreen
+
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEMassExtraction
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+omit [T2Space X] [MeasurableSpace X] [BorelSpace X] in
+theorem supported_dense_tests (K : Set X) (hK : IsCompact K) :
+    ∃ g : ℕ → C_c(X, ℝ), (∀ i, tsupport (g i) ⊆ K) ∧
+      ∀ f : C_c(X, ℝ), tsupport f ⊆ K → ∀ ε > 0,
+        ∃ i, ∀ x, |f x - g i x| < ε := by
+  classical
+  let S := {f : C(X, ℝ) // tsupport f ⊆ K}
+  have : SecondCountableTopology C(X, ℝ) := inferInstance
+  have : SecondCountableTopology S := TopologicalSpace.Subtype.secondCountableTopology _
+  have : Nonempty S := ⟨⟨0, by simp⟩⟩
+  let d := TopologicalSpace.denseSeq S
+  let g : ℕ → C_c(X, ℝ) := fun i =>
+    ⟨(d i).1, hK.of_isClosed_subset (isClosed_tsupport _) (d i).2⟩
+  refine ⟨g, fun i => (d i).2, ?_⟩
+  intro f hf ε hε
+  let s : S := ⟨f.toContinuousMap, hf⟩
+  have hs : s ∈ closure (range d) := TopologicalSpace.denseRange_denseSeq S s
+  obtain ⟨v, hv, ht⟩ := mem_closure_iff_seq_limit.mp hs
+  have ht' : Tendsto (fun n => (v n).1) atTop (𝓝 s.1) :=
+    continuous_subtype_val.continuousAt.tendsto.comp ht
+  have hu := (ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn.mp ht') K hK
+  obtain ⟨n, hn⟩ := (Metric.tendstoUniformlyOn_iff.mp hu ε hε).exists
+  obtain ⟨i, hi⟩ := hv n
+  refine ⟨i, fun x => ?_⟩
+  by_cases hx : x ∈ K
+  · have H := hn x hx
+    rw [← hi] at H
+    exact H
+  · have hf0 : f x = 0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hf h))
+    have hg0 : g i x = 0 := image_eq_zero_of_notMem_tsupport (fun h => hx ((d i).2 h))
+    simpa [hf0, hg0] using hε
+
+omit [MeasurableSpace X] [BorelSpace X] in
+theorem cutoff_test (K : Set X) (hK : IsCompact K) :
+    ∃ g : C_c(X, ℝ), (∀ x, 0 ≤ g x) ∧ ∀ x ∈ K, g x = 1 := by
+  obtain ⟨g, hg, _, hc, hn⟩ := exists_continuous_one_zero_of_isCompact hK
+    isClosed_empty (disjoint_empty K)
+  exact ⟨⟨g, hc⟩, fun x => (hn x).1, fun x hx => hg hx⟩
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] [BorelSpace X] in
+theorem positive_map_error (L : C_c(X, ℝ) →ₚ[ℝ] ℝ)
+    (f h g : C_c(X, ℝ)) (K : Set X) (hf : tsupport f ⊆ K)
+    (hh : tsupport h ⊆ K) (hg : ∀ x, 0 ≤ g x) (hg1 : ∀ x ∈ K, g x = 1)
+    {ε : ℝ} (hε : 0 ≤ ε) (he : ∀ x, |f x - h x| ≤ ε) :
+    |L f - L h| ≤ ε * L g := by
+  have hp : f - h ≤ ε • g := by
+    intro x
+    by_cases hx : x ∈ K
+    · simpa [hg1 x hx] using (abs_le.mp (he x)).2
+    · have hf0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hf H))
+      have hh0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hh H))
+      simpa [hf0, hh0] using mul_nonneg hε (hg x)
+  have hm : -(ε • g) ≤ f - h := by
+    intro x
+    by_cases hx : x ∈ K
+    · simpa [hg1 x hx] using (abs_le.mp (he x)).1
+    · have hf0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hf H))
+      have hh0 := image_eq_zero_of_notMem_tsupport (fun H => hx (hh H))
+      simpa [hf0, hh0] using neg_nonpos.mpr (mul_nonneg hε (hg x))
+  exact abs_le.mpr ⟨by simpa using L.monotone' hm, by simpa using L.monotone' hp⟩
+
+theorem determining_tests_radon_limit
+    (X : Type*) [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] :
+    ∃ tests : ℕ → C_c(X, ℝ), ∀ L : ℕ → (C_c(X, ℝ) →ₚ[ℝ] ℝ),
+      (∀ i, ∃ v : ℝ, Tendsto (fun j => L j (tests i)) atTop (𝓝 v)) →
+      ∃ μ : Measure X, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
+        ∀ f : C_c(X, ℝ), Tendsto (fun j => L j f) atTop (𝓝 (∫ x, f x ∂μ)) := by
+  classical
+  let K := CompactExhaustion.choice X
+  choose d hd hdense using fun n => supported_dense_tests (K n) (K.isCompact n)
+  choose g hg hg1 using fun n => cutoff_test (K n) (K.isCompact n)
+  let e := Denumerable.eqv ((ℕ × ℕ) ⊕ ℕ)
+  let tests : ℕ → C_c(X, ℝ) := fun i => Sum.elim (fun p => d p.1 p.2) g (e.symm i)
+  refine ⟨tests, ?_⟩
+  intro L hL
+  have hdL (n i : ℕ) : ∃ v : ℝ, Tendsto (fun j => L j (d n i)) atTop (𝓝 v) := by
+    simpa [tests] using hL (e (.inl (n,i)))
+  have hgL (n : ℕ) : ∃ v : ℝ, Tendsto (fun j => L j (g n)) atTop (𝓝 v) := by
+    simpa [tests] using hL (e (.inr n))
+  have hc (f : C_c(X, ℝ)) : CauchySeq (fun j => L j f) := by
+    obtain ⟨n, hn⟩ := K.exists_superset_of_isCompact f.hasCompactSupport
+    obtain ⟨v, hv⟩ := hgL n
+    obtain ⟨C, hC⟩ := hv.bddAbove_range
+    let D := max C 1
+    have hD : 0 < D := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+    have hDj (j : ℕ) : L j (g n) ≤ D := (hC (mem_range_self j)).trans (le_max_left _ _)
+    apply Metric.cauchySeq_iff.mpr
+    intro ε hε
+    let δ := ε / (4*D)
+    have hδ : 0 < δ := div_pos hε (mul_pos (by norm_num) hD)
+    obtain ⟨i, hi⟩ := hdense n f hn δ hδ
+    have herr (j : ℕ) : |L j f-L j (d n i)| ≤ ε/4 := by
+      have H := positive_map_error (L j) f (d n i) (g n) (K n) hn (hd n i)
+        (hg n) (hg1 n) hδ.le (fun x => (hi x).le)
+      have hh : δ*D=ε/4 := by dsimp [δ]; field_simp
+      exact H.trans ((mul_le_mul_of_nonneg_left (hDj j) hδ.le).trans_eq hh)
+    obtain ⟨w, hw⟩ := hdL n i
+    obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.mp hw.cauchySeq (ε/2) (by positivity)
+    refine ⟨N, fun j hj k hk => ?_⟩
+    have hjk := hN j hj k hk
+    rw [Real.dist_eq] at hjk ⊢
+    have H := abs_sub_le (L j f) (L j (d n i)) (L k f)
+    have H' := abs_sub_le (L j (d n i)) (L k (d n i)) (L k f)
+    have he := herr k
+    rw [abs_sub_comm] at he
+    linarith [herr j]
+  choose V hV using fun f : C_c(X, ℝ) => cauchySeq_tendsto_of_complete (hc f)
+  let Λ : C_c(X, ℝ) →ₚ[ℝ] ℝ := PositiveLinearMap.mk₀
+    { toFun := V
+      map_add' := fun f h => by
+        apply tendsto_nhds_unique (hV (f+h))
+        simpa only [map_add] using (hV f).add (hV h)
+      map_smul' := fun r f => by
+        apply tendsto_nhds_unique (hV (r • f))
+        simpa only [map_smul, smul_eq_mul, RingHom.id_apply] using (hV f).const_mul r }
+    (fun f hf => ge_of_tendsto' (hV f) (fun j => (L j).map_nonneg hf))
+  refine ⟨RealRMK.rieszMeasure Λ, inferInstance, inferInstance, fun f => ?_⟩
+  rw [RealRMK.integral_rieszMeasure]
+  exact hV f
+
+end ExactSLEMassExtraction
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+abbrev UpperPlane := {z : ℂ // 0 < z.im}
+noncomputable def upperArea : Measure UpperPlane := volume.comap Subtype.val
+noncomputable def actualUpperMass (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    Measure UpperPlane := upperArea.withDensity (fun z => timeMassDensity a q t b z.1)
+lemma upperPlane_open : IsOpen {z : ℂ | 0 < z.im} := isOpen_lt continuous_const Complex.continuous_im
+
+instance upperPlane_locallyCompact : LocallyCompactSpace UpperPlane :=
+  upperPlane_open.isOpenEmbedding_subtypeVal.locallyCompactSpace
+
+lemma upper_embedding : MeasurableEmbedding (Subtype.val : UpperPlane → ℂ) :=
+  MeasurableEmbedding.subtype_coe upperPlane_open.measurableSet
+
+lemma upper_compact_lower (K : Set UpperPlane) (hK : IsCompact K) :
+    ∃ c : ℝ, 0<c ∧ ∀ z∈K, c≤2*z.1.im := by
+  by_cases hn : K.Nonempty
+  · obtain ⟨z,hz,hmin⟩ := hK.exists_isMinOn hn
+      ((continuous_const.mul (Complex.continuous_im.comp continuous_subtype_val)).continuousOn)
+    exact ⟨2*z.1.im, mul_pos (by norm_num) z.2, fun w hw => hmin hw⟩
+  · exact ⟨1, by norm_num, fun z hz => False.elim (hn ⟨z,hz⟩)⟩
+
+lemma timeMassDensity_finite (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    timeMassDensity a q t b z < ⊤ := by
+  apply lt_of_le_of_lt (timeMassDensity_le a q t b z)
+  unfold massDensity
+  split_ifs <;> simp
+
+lemma timeMassDensity_local_bound (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (c : ℝ) (hc : 0<c)
+    (z : ℂ) (hz : c≤2*z.im) :
+    timeMassDensity a q t b z≤ENNReal.ofReal ((c*q)^(-alpha a)) := by
+  have hz0 : 0<z.im := by linarith
+  have hit := fullRadialDensity_bound a ha z hz0 (2*z.im*q) (by positivity) (by nlinarith) b
+  have htar : (2*z.im*q)^(-alpha a)≤(c*q)^(-alpha a) :=
+    Real.rpow_le_rpow_of_nonpos (mul_pos hc hq) (mul_le_mul_of_nonneg_right hz hq.le)
+       (neg_nonpos.mpr (alpha_pos ha).le)
+  apply (timeMassDensity_le a q t b z).trans
+  simp only [massDensity,ite_eq_left hz0]
+  exact ENNReal.ofReal_le_ofReal (hit.2.trans htar)
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialMoments.lean b/lean/OAI/Probability/SLEGauge/RadialMoments.lean
index 6f4d6a2ce0a081f1f0664a7a31c5610af015c28f..3f8f7e6b8fac31544c2e155f841c0e3d120acbb4 100644
--- a/lean/OAI/Probability/SLEGauge/RadialMoments.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialMoments.lean
@@ -1,471 +1,471 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ReverseFlow
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-noncomputable section
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory
-lemma radial_length_moment {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T)
-    {W : WienerLaw} (hW : IsWiener W) :
-    (∫⁻ b : Driver, (radialLength κ b T y)^(lambda (a κ)) ∂W) ≤
-      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(lambda (a κ)+zeta (a κ))) := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hT0 : 0 ≤ T := hδ.le.trans hT
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  have hz := (parameters ha).1
-  have hl := lambda_gt_one (show 0 < a κ by linarith)
-  have hp0 : 0 < lambda (a κ) := by linarith
-  let F : Driver → ℝ → ℝ≥0∞ := fun b u => ENNReal.ofReal ‖radialCoeff κ b T u‖
-  have hm : Measurable (Function.uncurry F) :=
-    (measurable_radial_norm hκ hT0).ennreal_ofReal.comp measurable_swap
-  have hmass : (volume.restrict (Ioc (0:ℝ) y)) univ = ENNReal.ofReal y := by simp
-  calc
-    _ ≤ ∫⁻ b : Driver, (ENNReal.ofReal y)^(lambda (a κ)-1)*
-        (∫⁻ u in Ioc (0:ℝ) y, (F b u)^(lambda (a κ))) ∂W := by
-      apply lintegral_mono
-      intro b
-      simpa [radialLength,F,hmass,Function.uncurry] using lintegral_pow_le_mass
-        (volume.restrict (Ioc (0:ℝ) y)) ((hm.comp (measurable_const.prodMk measurable_id)).aemeasurable) hl
-    _ = (ENNReal.ofReal y)^(lambda (a κ)-1)*
-        (∫⁻ u in Ioc (0:ℝ) y, ∫⁻ b : Driver, (F b u)^(lambda (a κ)) ∂W) := by
-      rw [lintegral_const_mul' _ _ (by finiteness),
-        lintegral_lintegral_swap (hm.pow_const _).aemeasurable]
-    _ ≤ (ENNReal.ofReal y)^(lambda (a κ)-1)*
-        (∫⁻ _u in Ioc (0:ℝ) y, ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))) := by
-      apply mul_le_mul' le_rfl
-      apply setLIntegral_mono' measurableSet_Ioc
-      intro u hu
-      have he : (∫⁻ b : Driver, (F b u)^(lambda (a κ)) ∂W) =
-          ∫⁻ b : Driver, ENNReal.ofReal
-            (‖deriv (inverseFlow κ b T) (driverValue b T+u*Complex.I)‖^(lambda (a κ))) ∂W := by
-        congr 1
-        funext b
-        simp only [F,radialCoeff,ite_eq_left hu.1,norm_mul,Complex.norm_I,mul_one]
-        exact ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hp0.le
-      rw [he]
-      apply (inverse_deriv_moment_away hκ hκ8 hu.1 hδ hT hW).trans
-      apply ENNReal.ofReal_le_ofReal
-      gcongr <;> linarith [hu.1,hu.2]
-    _ = _ := by
-      rw [lintegral_const,hmass,ENNReal.ofReal_rpow_of_nonneg hy.le (by linarith : 0 ≤ lambda (a κ)-1),
-        ← ENNReal.ofReal_mul (by positivity),← ENNReal.ofReal_mul (by positivity)]
-      congr 1
-      calc
-        y^(lambda (a κ)-1)*((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ))*y) =
-          (3/δ)^(zeta (a κ)/2)*(y^(lambda (a κ)-1)*y^(zeta (a κ))*y^((1:ℝ))) := by rw [Real.rpow_one]; ring
-        _ = (3/δ)^(zeta (a κ)/2)*y^(lambda (a κ)+zeta (a κ)) := by
-          rw [← Real.rpow_add hy,← Real.rpow_add hy]
-          congr 2
-          ring
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory
-
-lemma measurable_radialLength {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (y : ℝ) :
-    Measurable (fun b : Driver => radialLength κ b T y) :=
-  ((measurable_radial_norm hκ hT).ennreal_ofReal.comp measurable_swap).lintegral_prod_right'
-
-lemma radial_length_beta_tail {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    W {b : Driver | ENNReal.ofReal (y^(traceBeta κ)) ≤ radialLength κ b T y} ≤
-      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(2+traceEta κ)) := by
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  have hp := trace_exponents hκ hκ8
-  have hl := (parameters ha).2.1
-  have hK : 0 < y^(traceBeta κ) := Real.rpow_pos_of_pos hy _
-  have hKP : 0 < (y^(traceBeta κ))^(lambda (a κ)) := Real.rpow_pos_of_pos hK _
-  have hm := (measurable_radialLength hκ (hδ.le.trans hT) y).pow_const (lambda (a κ))
-  calc
-    _ ≤ W {b : Driver | ENNReal.ofReal ((y^(traceBeta κ))^(lambda (a κ))) ≤
-        (radialLength κ b T y)^(lambda (a κ))} := by
-      apply measure_mono
-      intro b hb
-      rw [← ENNReal.ofReal_rpow_of_nonneg hK.le hl.le]
-      exact ENNReal.rpow_le_rpow hb hl.le
-    _ ≤ (∫⁻ b : Driver, (radialLength κ b T y)^(lambda (a κ)) ∂W)/
-        ENNReal.ofReal ((y^(traceBeta κ))^(lambda (a κ))) :=
-      meas_ge_le_lintegral_div hm.aemeasurable (ne_of_gt (ENNReal.ofReal_pos.mpr hKP)) ENNReal.ofReal_ne_top
-    _ ≤ ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(lambda (a κ)+zeta (a κ)))/
-        ENNReal.ofReal ((y^(traceBeta κ))^(lambda (a κ))) :=
-      ENNReal.div_le_div_right (radial_length_moment hκ hκ8 hy hδ hT hW) _
-    _ = _ := by
-      rw [← ENNReal.ofReal_div_of_pos hKP,← Real.rpow_mul hy.le,mul_div_assoc,← Real.rpow_sub hy]
-      congr 3
-      linarith [hp.2.2.2]
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory
-lemma refined_ratio (N : ℕ) (η : ℝ) :
-    (2:ℝ)^(2*N+1)*(((1/2:ℝ)^N)^(2+η)) = (1/2:ℝ)^((N:ℝ)*η-1) := by
-  rw [← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 1/2)]
-  have he : (2:ℝ)^(2*N+1) = (1/2:ℝ)^(-((2*N+1:ℕ):ℝ)) := by
-    rw [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 1/2),Real.rpow_natCast,← inv_pow]
-    norm_num
-  rw [he,← Real.rpow_add (by norm_num : (0:ℝ) < 1/2)]
-  congr 1
-  push_cast
-  ring
-lemma refined_sum_term (M N n : ℕ) (C η : ℝ) :
-    (((M*(2^(2*N+1))^n:ℕ):ℝ))*(C*(((1/2:ℝ)^N)^n)^(2+η)) =
-      ((M:ℝ)*C)*(((1/2:ℝ)^((N:ℝ)*η-1))^n) := by
-  rw [← refined_ratio N η,mul_pow]
-  push_cast
-  rw [← Real.rpow_natCast_mul (by positivity : 0 ≤ (1/2:ℝ)^N),
-    mul_comm (n:ℝ) (2+η),Real.rpow_mul_natCast (by positivity : 0 ≤ (1/2:ℝ)^N)]
-  ring
-
-def refinedHeight (N n : ℕ) : ℝ := ((1/2:ℝ)^N)^n
-
-def refinedGridTime (δ : ℝ) (N n k : ℕ) : ℝ := δ+(k:ℝ)/((2:ℝ)^(2*N+1))^n
-
-lemma refinedGridTime_ge (δ : ℝ) (N n k : ℕ) : δ ≤ refinedGridTime δ N n k := by
-  unfold refinedGridTime
-  exact le_add_of_nonneg_right (by positivity)
-
-def radialGridBad (κ δ : ℝ) (M N n : ℕ) : Set Driver :=
-  ⋃ k : Fin (M*(2^(2*N+1))^n), {b : Driver |
-    ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) ≤
-      radialLength κ b (refinedGridTime δ N n k) (refinedHeight N n)}
-
-lemma radialGridBad_measure {κ δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hδ : 0 < δ) {W : WienerLaw} (hW : IsWiener W) (M N n : ℕ) :
-    W (radialGridBad κ δ M N n) ≤
-      ENNReal.ofReal (((M:ℝ)*(3/δ)^(zeta (a κ)/2))*
-        (((1/2:ℝ)^((N:ℝ)*traceEta κ-1))^n)) := by
-  have hy : 0 < refinedHeight N n := by unfold refinedHeight; positivity
-  calc
-    _ ≤ ∑ k : Fin (M*(2^(2*N+1))^n), W {b : Driver |
-        ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) ≤
-          radialLength κ b (refinedGridTime δ N n k) (refinedHeight N n)} :=
-      measure_iUnion_fintype_le W _
-    _ ≤ ∑ _k : Fin (M*(2^(2*N+1))^n), ENNReal.ofReal
-        ((3/δ)^(zeta (a κ)/2)*(refinedHeight N n)^(2+traceEta κ)) := by
-      apply Finset.sum_le_sum
-      intro k _
-      exact radial_length_beta_tail hκ hκ8 hy hδ (refinedGridTime_ge δ N n k) hW
-    _ = ENNReal.ofReal (((M*(2^(2*N+1))^n:ℕ):ℝ)*
-        ((3/δ)^(zeta (a κ)/2)*(refinedHeight N n)^(2+traceEta κ))) := by
-      rw [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,
-        ← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (Nat.cast_nonneg _)]
-    _ = _ := by rw [refinedHeight,refined_sum_term]
-
-lemma radialGridBad_summable {κ δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hδ : 0 < δ) {W : WienerLaw} (hW : IsWiener W) (M N : ℕ)
-    (hN : 1 < (N:ℝ)*traceEta κ) :
-    (∑' n : ℕ, W (radialGridBad κ δ M N n)) ≠ (∞:ℝ≥0∞) := by
-  have hr : |(1/2:ℝ)^((N:ℝ)*traceEta κ-1)| < 1 := by
-    rw [abs_of_pos (Real.rpow_pos_of_pos (by norm_num) _)]
-    exact Real.rpow_lt_one (by norm_num) (by norm_num) (by linarith)
-  apply ne_top_of_le_ne_top
-    (((summable_geometric_of_abs_lt_one hr).mul_left
-      ((M:ℝ)*(3/δ)^(zeta (a κ)/2))).tsum_ofReal_ne_top)
-  exact ENNReal.tsum_le_tsum (radialGridBad_measure hκ hκ8 hδ hW M N)
-
- 
-
-theorem radial_length_refined_eventually {κ δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hδ : 0 < δ) {W : WienerLaw} (hW : IsWiener W) (M N : ℕ)
-    (hN : 1 < (N:ℝ)*traceEta κ) :
-    ∀ᵐ b ∂W, ∀ᶠ n : ℕ in atTop, ∀ k : Fin (M*(2^(2*N+1))^n),
-      radialLength κ b (refinedGridTime δ N n k) (refinedHeight N n) <
-        ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) := by
-  filter_upwards [ae_eventually_notMem (radialGridBad_summable hκ hκ8 hδ hW M N hN)] with b hb
-  filter_upwards [hb] with n hn
-  intro k
-  apply lt_of_not_ge
-  intro h
-  exact hn (mem_iUnion.mpr ⟨k,h⟩)
-
- 
-
-theorem radial_length_all_refined_grids {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∃ N : ℕ, 0 < N ∧ 1 < (N:ℝ)*traceEta κ ∧
-      ∀ᵐ b ∂W, ∀ m M : ℕ, ∀ᶠ n : ℕ in atTop, ∀ k : Fin (M*(2^(2*N+1))^n),
-        radialLength κ b (refinedGridTime (1/((m:ℝ)+1)) N n k) (refinedHeight N n) <
-          ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) := by
-  have hp := trace_exponents hκ hκ8
-  obtain ⟨N,hN⟩ := exists_nat_gt (1/traceEta κ)
-  have hN' : 1 < (N:ℝ)*traceEta κ := (div_lt_iff₀ hp.2.2.1).mp hN
-  have hN0 : 0 < N := by
-    by_contra hh
-    have : N = 0 := Nat.eq_zero_of_not_pos hh
-    subst N
-    norm_num at hN'
-  refine ⟨N,hN0,hN',?_⟩
-  apply ae_all_iff.mpr
-  intro m
-  apply ae_all_iff.mpr
-  intro M
-  exact radial_length_refined_eventually hκ hκ8 (by positivity) hW M N hN'
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open MeasureTheory
-
- 
-
-lemma radial_increment_le {κ T y z : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
-    (hz : 0 < z) (hzy : z ≤ y) (b : Driver) :
-    ENNReal.ofReal ‖inverseFlow κ b T (driverValue b T+y*Complex.I)-
-      inverseFlow κ b T (driverValue b T+z*Complex.I)‖ ≤ radialLength κ b T y := by
-  have hsub : Icc z y ⊆ Ioi (0:ℝ) := fun u hu => hz.trans_le hu.1
-  have hc := (continuousOn_radialCoeff hκ hT b).mono hsub
-  have hi : IntervalIntegrable (radialCoeff κ b T) volume z y :=
-    hc.intervalIntegrable_of_Icc hzy
-  have hd (u : ℝ) (hu : u ∈ uIcc z y) :
-      HasDerivAt (fun v : ℝ => inverseFlow κ b T (driverValue b T+v*Complex.I))
-        (radialCoeff κ b T u) u := by
-    rw [uIcc_of_le hzy] at hu
-    exact hasDerivAt_radial hκ hT (hsub hu) b
-  rw [← intervalIntegral.integral_eq_sub_of_hasDerivAt hd hi,
-    intervalIntegral.integral_of_le hzy,ofReal_norm]
-  apply (enorm_integral_le_lintegral_enorm _).trans
-  change (∫⁻ u in Ioc z y, ‖radialCoeff κ b T u‖ₑ) ≤ radialLength κ b T y
-  simp only [radialLength,ofReal_norm]
-  exact lintegral_mono_set (Ioc_subset_Ioc hz.le le_rfl)
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open Set Filter Metric
-open scoped Topology
-
-lemma halfPlane_sub_conj_ne_zero {w v : ℂ} (hw : 0 < w.im) (hv : 0 < v.im) :
-    w-star v ≠ 0 := by
-  intro he
-  have hi := congrArg Complex.im he
-  simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,Complex.zero_im] at hi
-  linarith
-
-lemma halfPlane_disk_norm_lt_one {w v : ℂ} (hw : 0 < w.im) (hv : 0 < v.im) :
-    ‖(w-v)/(w-star v)‖ < 1 := by
-  have hs : ‖w-v‖^2 < ‖w-star v‖^2 := by
-    rw [← Complex.normSq_eq_norm_sq,← Complex.normSq_eq_norm_sq]
-    simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
-      Complex.star_def,Complex.conj_re,Complex.conj_im]
-    nlinarith [mul_pos hw hv]
-  rw [norm_div,div_lt_one (norm_pos_iff.mpr (halfPlane_sub_conj_ne_zero hw hv))]
-  nlinarith [norm_nonneg (w-v),norm_nonneg (w-star v)]
-
-lemma halfPlane_self_sub_conj_norm {w : ℂ} (hw : 0 < w.im) :
-    ‖w-star w‖ = 2*w.im := by
-  have he : w-star w = (2*w.im:ℂ)*Complex.I := by
-    apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im]; ring
-  rw [he,norm_mul,Complex.norm_I,mul_one]
-  norm_cast
-  rw [Real.norm_eq_abs,abs_of_nonneg (by positivity)]
-
- 
-
-lemma ball_radius_le_conformal_radius {f : ℂ → ℂ} {z : ℂ} {r : ℝ} (hr : 0 < r)
-    (hf : DifferentiableOn ℂ f (ball z r))
-    (hH : ∀ w ∈ ball z r, 0 < (f w).im) (hd : deriv f z ≠ 0) :
-    r ≤ 2*(f z).im / ‖deriv f z‖ := by
-  have hz : z ∈ ball z r := by simpa using hr
-  have hy := hH z hz
-  let F (w : ℂ) : ℂ := (f w-f z)/(f w-star (f z))
-  have hn (w : ℂ) (hw : w ∈ ball z r) : f w-star (f z) ≠ 0 :=
-    halfPlane_sub_conj_ne_zero (hH w hw) hy
-  have hF : DifferentiableOn ℂ F (ball z r) :=
-    (hf.sub_const (f z)).div (hf.sub_const (star (f z))) hn
-  have hFz : F z = 0 := by simp [F]
-  have hm : MapsTo F (ball z r) (closedBall (F z) 1) := by
-    intro w hw
-    rw [hFz,mem_closedBall,dist_zero_right]
-    exact (halfPlane_disk_norm_lt_one (hH w hw) hy).le
-  have hh := Complex.norm_deriv_le_div_of_mapsTo_ball hF hm hr
-  have hfd := (hf.differentiableAt (ball_mem_nhds z hr)).hasDerivAt
-  have hFd : HasDerivAt F (deriv f z/(f z-star (f z))) z := by
-    have h := (hfd.sub_const (f z)).div (hfd.sub_const (star (f z))) (hn z hz)
-    convert! h using 1; first | rfl | (simp only [sub_self,zero_mul,sub_zero]; field_simp)
-  rw [hFd.deriv,norm_div,halfPlane_self_sub_conj_norm hy] at hh
-  have hi := (div_le_div_iff₀ (show 0 < 2*(f z).im by positivity) hr).mp hh
-  apply (le_div_iff₀ (norm_pos_iff.mpr hd)).mpr
-  nlinarith
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-
-lemma PositiveSolution.im_hasDeriv {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    HasDerivWithinAt (fun u => (f u).im)
-      (-a κ*(f t).im/‖f t-driverValue b t‖^2) (Icc 0 T) t := by
-  have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hf.2.2.2.2 t ht)
-  convert! hi using 1; first | rfl |
-    (simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_im,Complex.ofReal_re,
-     Complex.sub_im,sub_zero,Complex.normSq_eq_norm_sq,zero_mul,zero_div,zero_sub]; ring)
-
-lemma PositiveSolution.im_antitone {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) :
-    AntitoneOn (fun u => (f u).im) (Icc 0 T) := by
-  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
-    (Complex.continuous_im.comp_continuousOn hf.2.1)
-    (fun t ht => (hf.im_hasDeriv (interior_subset ht)).mono interior_subset)
-  intro t ht
-  apply div_nonpos_of_nonpos_of_nonneg _ (sq_nonneg _)
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hy := hf.2.2.2.1 t (interior_subset ht)
-  nlinarith
-
-lemma PositiveSolution.im_sq_bound {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) {s t : ℝ}
-    (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T) (hst : s ≤ t) :
-    (f s).im^2 ≤ (f t).im^2+2*a κ*(t-s) := by
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hm : MonotoneOn (fun u => (f u).im^2+2*a κ*u) (Icc 0 T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
-      ((Complex.continuous_im.comp_continuousOn hf.2.1).pow 2 |>.add
-        ((continuous_const.mul continuous_id).continuousOn))
-      (fun u hu => (((hf.im_hasDeriv (interior_subset hu)).pow 2).add
-        ((hasDerivAt_id u).const_mul (2*a κ)).hasDerivWithinAt).mono interior_subset)
-    intro u hu
-    have ht := interior_subset hu
-    have hn := Complex.im_le_norm (f u-driverValue b u)
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] at hn
-    have hp := hf.2.2.2.1 u ht
-    have hs : (f u).im^2 ≤ ‖f u-driverValue b u‖^2 := pow_le_pow_left₀ hp.le hn 2
-    have hn0 : 0 < ‖f u-driverValue b u‖^2 := sq_pos_of_pos (hp.trans_le hn)
-    norm_num only [Nat.cast_ofNat,pow_one,mul_one]
-    convert! (div_nonneg (mul_nonneg (show 0 ≤ 2*a κ by positivity) (sub_nonneg.mpr hs)) hn0.le) using 1; field_simp [ne_of_gt (hp.trans_le hn)]; ring
-  have hh := hm hs ht hst
-  dsimp only at hh
-  linarith
-
-lemma inverseFlow_im_sq_bound {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
-    (b : Driver) {w : ℂ} (hw : 0 < w.im) :
-    (inverseFlow κ b T w).im^2 ≤ w.im^2+2*a κ*T := by
-  have hf := surviving_flow_positiveSolution hT (inverseFlow_surviving hκ b hT hw)
-  have hh := hf.im_sq_bound hκ (left_mem_Icc.mpr hT) (right_mem_Icc.mpr hT) hT
-  simpa only [hf.2.2.1,flow_inverseFlow hκ b hT hw,sub_zero] using hh
-
-lemma inverseFlow_im_pos {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
-    (b : Driver) {w : ℂ} (hw : 0 < w.im) : 0 < (inverseFlow κ b T w).im := by
-  have hf := surviving_flow_positiveSolution hT (inverseFlow_surviving hκ b hT hw)
-  simpa only [hf.2.2.1] using hf.2.2.2.1 0 (left_mem_Icc.mpr hT)
-
-lemma inverseFlow_deriv_absolute {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
-    (b : Driver) {w : ℂ} (hw : 0 < w.im) :
-    ‖deriv (inverseFlow κ b T) w‖ ≤ 2*(inverseFlow κ b T w).im/w.im := by
-  have hs : ball w w.im ⊆ {u : ℂ | 0 < u.im} := by
-    intro u hu
-    rw [mem_ball,dist_eq_norm] at hu
-    have hh := Complex.im_le_norm (w-u)
-    rw [norm_sub_rev,Complex.sub_im] at hh
-    change 0 < u.im
-    linarith
-  have hd : deriv (inverseFlow κ b T) w ≠ 0 := by
-    rw [(inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.deriv]
-    exact inv_ne_zero (flow_deriv_ne_zero hT (inverseFlow_surviving hκ b hT hw))
-  have hh := ball_radius_le_conformal_radius hw
-    ((flow_conformal_bijection hκ b hT).2.2.2.mono hs)
-    (fun u hu => inverseFlow_im_pos hκ hT b (hs hu)) hd
-  have hh' := (le_div_iff₀ (norm_pos_iff.mpr hd)).mp hh
-  apply (le_div_iff₀ hw).mpr
-  nlinarith
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-
-lemma PositiveSolution.displacement_le {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) {s t : ℝ}
-    (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T) (hst : s ≤ t) :
-    ‖f t-f s‖ ≤ (a κ/(f t).im)*(t-s) := by
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hy := hf.2.2.2.1 t ht
-  have hin : Icc s t ⊆ Icc 0 T := Icc_subset_Icc hs.1 ht.2
-  have hh := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
-    (fun u hu => (hf.2.2.2.2 u (hin hu)).mono hin)
-    (show ∀ u ∈ Icc s t, ‖(a κ:ℂ)/(f u-driverValue b u)‖ ≤ a κ/(f t).im from by
-      intro u hu
-      rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos ha]
-      apply div_le_div_of_nonneg_left ha.le hy
-      apply (hf.im_antitone hκ (hin hu) ht hu.2).trans
-      simpa using Complex.im_le_norm (f u-driverValue b u))
-    (convex_Icc s t) (left_mem_Icc.mpr hst) (right_mem_Icc.mpr hst)
-  simpa only [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hst)] using hh
-
-lemma inverseFlow_im_le {κ t T R : ℝ} (hκ : 0 < κ) (ht : 0 ≤ t) (htT : t ≤ T)
-    (b : Driver) {w : ℂ} (hw : 0 < w.im) (hR : w.im ≤ R) :
-    (inverseFlow κ b t w).im ≤ R+2*a κ*T+1 := by
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hT : 0 ≤ T := ht.trans htT
-  have hR0 : 0 ≤ R := hw.le.trans hR
-  have hA : 0 ≤ 2*a κ*T := by positivity
-  have him := inverseFlow_im_sq_bound hκ ht b hw
-  have hnon := inverseFlow_im_pos hκ ht b hw
-  have hup : w.im^2 ≤ R^2 := pow_le_pow_left₀ hw.le hR 2
-  have hat : 2*a κ*t ≤ 2*a κ*T := mul_le_mul_of_nonneg_left htT (by positivity)
-  nlinarith [sq_nonneg (2*a κ*T),mul_nonneg hR0 hA]
-
-lemma inverseFlow_lipschitz_strip {κ t T y R : ℝ} (hκ : 0 < κ) (ht : 0 ≤ t)
-    (htT : t ≤ T) (hy : 0 < y) (b : Driver) {v w : ℂ}
-    (hv : y ≤ v.im ∧ v.im ≤ R) (hw : y ≤ w.im ∧ w.im ≤ R) :
-    ‖inverseFlow κ b t v-inverseFlow κ b t w‖ ≤
-      (2*(R+2*a κ*T+1)/y)*‖v-w‖ := by
-  have hs : Convex ℝ {u : ℂ | y ≤ u.im ∧ u.im ≤ R} :=
-    (convex_Icc y R).linear_preimage Complex.imCLM.toLinearMap
-  apply Convex.norm_image_sub_le_of_norm_deriv_le (𝕜 := ℂ)
-    (fun u hu => (inverseFlow_hasStrictDerivAt hκ b ht (hy.trans_le hu.1)).hasDerivAt.differentiableAt) _ hs hw hv
-  intro u hu
-  apply (inverseFlow_deriv_absolute hκ ht b (hy.trans_le hu.1)).trans
-  calc
-    _ ≤ 2*(R+2*a κ*T+1)/u.im := by
-      apply div_le_div_of_nonneg_right _ (hy.le.trans hu.1)
-      exact mul_le_mul_of_nonneg_left (inverseFlow_im_le hκ ht htT b (hy.trans_le hu.1) hu.2) (by norm_num)
-    _ ≤ 2*(R+2*a κ*T+1)/y := by
-      apply div_le_div_of_nonneg_left _ hy hu.1
-      have ha : 0 < a κ := div_pos (by norm_num) hκ
-      have hT : 0 ≤ T := ht.trans htT
-      have hR0 : 0 < R := hy.trans_le (hu.1.trans hu.2)
-      positivity
-
- 
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ReverseFlow
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+noncomputable section
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory
+lemma radial_length_moment {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T)
+    {W : WienerLaw} (hW : IsWiener W) :
+    (∫⁻ b : Driver, (radialLength κ b T y)^(lambda (a κ)) ∂W) ≤
+      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(lambda (a κ)+zeta (a κ))) := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hT0 : 0 ≤ T := hδ.le.trans hT
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  have hz := (parameters ha).1
+  have hl := lambda_gt_one (show 0 < a κ by linarith)
+  have hp0 : 0 < lambda (a κ) := by linarith
+  let F : Driver → ℝ → ℝ≥0∞ := fun b u => ENNReal.ofReal ‖radialCoeff κ b T u‖
+  have hm : Measurable (Function.uncurry F) :=
+    (measurable_radial_norm hκ hT0).ennreal_ofReal.comp measurable_swap
+  have hmass : (volume.restrict (Ioc (0:ℝ) y)) univ = ENNReal.ofReal y := by simp
+  calc
+    _ ≤ ∫⁻ b : Driver, (ENNReal.ofReal y)^(lambda (a κ)-1)*
+        (∫⁻ u in Ioc (0:ℝ) y, (F b u)^(lambda (a κ))) ∂W := by
+      apply lintegral_mono
+      intro b
+      simpa [radialLength,F,hmass,Function.uncurry] using lintegral_pow_le_mass
+        (volume.restrict (Ioc (0:ℝ) y)) ((hm.comp (measurable_const.prodMk measurable_id)).aemeasurable) hl
+    _ = (ENNReal.ofReal y)^(lambda (a κ)-1)*
+        (∫⁻ u in Ioc (0:ℝ) y, ∫⁻ b : Driver, (F b u)^(lambda (a κ)) ∂W) := by
+      rw [lintegral_const_mul' _ _ (by finiteness),
+        lintegral_lintegral_swap (hm.pow_const _).aemeasurable]
+    _ ≤ (ENNReal.ofReal y)^(lambda (a κ)-1)*
+        (∫⁻ _u in Ioc (0:ℝ) y, ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))) := by
+      apply mul_le_mul' le_rfl
+      apply setLIntegral_mono' measurableSet_Ioc
+      intro u hu
+      have he : (∫⁻ b : Driver, (F b u)^(lambda (a κ)) ∂W) =
+          ∫⁻ b : Driver, ENNReal.ofReal
+            (‖deriv (inverseFlow κ b T) (driverValue b T+u*Complex.I)‖^(lambda (a κ))) ∂W := by
+        congr 1
+        funext b
+        simp only [F,radialCoeff,ite_eq_left hu.1,norm_mul,Complex.norm_I,mul_one]
+        exact ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hp0.le
+      rw [he]
+      apply (inverse_deriv_moment_away hκ hκ8 hu.1 hδ hT hW).trans
+      apply ENNReal.ofReal_le_ofReal
+      gcongr <;> linarith [hu.1,hu.2]
+    _ = _ := by
+      rw [lintegral_const,hmass,ENNReal.ofReal_rpow_of_nonneg hy.le (by linarith : 0 ≤ lambda (a κ)-1),
+        ← ENNReal.ofReal_mul (by positivity),← ENNReal.ofReal_mul (by positivity)]
+      congr 1
+      calc
+        y^(lambda (a κ)-1)*((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ))*y) =
+          (3/δ)^(zeta (a κ)/2)*(y^(lambda (a κ)-1)*y^(zeta (a κ))*y^((1:ℝ))) := by rw [Real.rpow_one]; ring
+        _ = (3/δ)^(zeta (a κ)/2)*y^(lambda (a κ)+zeta (a κ)) := by
+          rw [← Real.rpow_add hy,← Real.rpow_add hy]
+          congr 2
+          ring
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory
+
+lemma measurable_radialLength {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (y : ℝ) :
+    Measurable (fun b : Driver => radialLength κ b T y) :=
+  ((measurable_radial_norm hκ hT).ennreal_ofReal.comp measurable_swap).lintegral_prod_right'
+
+lemma radial_length_beta_tail {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    W {b : Driver | ENNReal.ofReal (y^(traceBeta κ)) ≤ radialLength κ b T y} ≤
+      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(2+traceEta κ)) := by
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  have hp := trace_exponents hκ hκ8
+  have hl := (parameters ha).2.1
+  have hK : 0 < y^(traceBeta κ) := Real.rpow_pos_of_pos hy _
+  have hKP : 0 < (y^(traceBeta κ))^(lambda (a κ)) := Real.rpow_pos_of_pos hK _
+  have hm := (measurable_radialLength hκ (hδ.le.trans hT) y).pow_const (lambda (a κ))
+  calc
+    _ ≤ W {b : Driver | ENNReal.ofReal ((y^(traceBeta κ))^(lambda (a κ))) ≤
+        (radialLength κ b T y)^(lambda (a κ))} := by
+      apply measure_mono
+      intro b hb
+      rw [← ENNReal.ofReal_rpow_of_nonneg hK.le hl.le]
+      exact ENNReal.rpow_le_rpow hb hl.le
+    _ ≤ (∫⁻ b : Driver, (radialLength κ b T y)^(lambda (a κ)) ∂W)/
+        ENNReal.ofReal ((y^(traceBeta κ))^(lambda (a κ))) :=
+      meas_ge_le_lintegral_div hm.aemeasurable (ne_of_gt (ENNReal.ofReal_pos.mpr hKP)) ENNReal.ofReal_ne_top
+    _ ≤ ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(lambda (a κ)+zeta (a κ)))/
+        ENNReal.ofReal ((y^(traceBeta κ))^(lambda (a κ))) :=
+      ENNReal.div_le_div_right (radial_length_moment hκ hκ8 hy hδ hT hW) _
+    _ = _ := by
+      rw [← ENNReal.ofReal_div_of_pos hKP,← Real.rpow_mul hy.le,mul_div_assoc,← Real.rpow_sub hy]
+      congr 3
+      linarith [hp.2.2.2]
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory
+lemma refined_ratio (N : ℕ) (η : ℝ) :
+    (2:ℝ)^(2*N+1)*(((1/2:ℝ)^N)^(2+η)) = (1/2:ℝ)^((N:ℝ)*η-1) := by
+  rw [← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 1/2)]
+  have he : (2:ℝ)^(2*N+1) = (1/2:ℝ)^(-((2*N+1:ℕ):ℝ)) := by
+    rw [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 1/2),Real.rpow_natCast,← inv_pow]
+    norm_num
+  rw [he,← Real.rpow_add (by norm_num : (0:ℝ) < 1/2)]
+  congr 1
+  push_cast
+  ring
+lemma refined_sum_term (M N n : ℕ) (C η : ℝ) :
+    (((M*(2^(2*N+1))^n:ℕ):ℝ))*(C*(((1/2:ℝ)^N)^n)^(2+η)) =
+      ((M:ℝ)*C)*(((1/2:ℝ)^((N:ℝ)*η-1))^n) := by
+  rw [← refined_ratio N η,mul_pow]
+  push_cast
+  rw [← Real.rpow_natCast_mul (by positivity : 0 ≤ (1/2:ℝ)^N),
+    mul_comm (n:ℝ) (2+η),Real.rpow_mul_natCast (by positivity : 0 ≤ (1/2:ℝ)^N)]
+  ring
+
+def refinedHeight (N n : ℕ) : ℝ := ((1/2:ℝ)^N)^n
+
+def refinedGridTime (δ : ℝ) (N n k : ℕ) : ℝ := δ+(k:ℝ)/((2:ℝ)^(2*N+1))^n
+
+lemma refinedGridTime_ge (δ : ℝ) (N n k : ℕ) : δ ≤ refinedGridTime δ N n k := by
+  unfold refinedGridTime
+  exact le_add_of_nonneg_right (by positivity)
+
+def radialGridBad (κ δ : ℝ) (M N n : ℕ) : Set Driver :=
+  ⋃ k : Fin (M*(2^(2*N+1))^n), {b : Driver |
+    ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) ≤
+      radialLength κ b (refinedGridTime δ N n k) (refinedHeight N n)}
+
+lemma radialGridBad_measure {κ δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hδ : 0 < δ) {W : WienerLaw} (hW : IsWiener W) (M N n : ℕ) :
+    W (radialGridBad κ δ M N n) ≤
+      ENNReal.ofReal (((M:ℝ)*(3/δ)^(zeta (a κ)/2))*
+        (((1/2:ℝ)^((N:ℝ)*traceEta κ-1))^n)) := by
+  have hy : 0 < refinedHeight N n := by unfold refinedHeight; positivity
+  calc
+    _ ≤ ∑ k : Fin (M*(2^(2*N+1))^n), W {b : Driver |
+        ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) ≤
+          radialLength κ b (refinedGridTime δ N n k) (refinedHeight N n)} :=
+      measure_iUnion_fintype_le W _
+    _ ≤ ∑ _k : Fin (M*(2^(2*N+1))^n), ENNReal.ofReal
+        ((3/δ)^(zeta (a κ)/2)*(refinedHeight N n)^(2+traceEta κ)) := by
+      apply Finset.sum_le_sum
+      intro k _
+      exact radial_length_beta_tail hκ hκ8 hy hδ (refinedGridTime_ge δ N n k) hW
+    _ = ENNReal.ofReal (((M*(2^(2*N+1))^n:ℕ):ℝ)*
+        ((3/δ)^(zeta (a κ)/2)*(refinedHeight N n)^(2+traceEta κ))) := by
+      rw [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,
+        ← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (Nat.cast_nonneg _)]
+    _ = _ := by rw [refinedHeight,refined_sum_term]
+
+lemma radialGridBad_summable {κ δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hδ : 0 < δ) {W : WienerLaw} (hW : IsWiener W) (M N : ℕ)
+    (hN : 1 < (N:ℝ)*traceEta κ) :
+    (∑' n : ℕ, W (radialGridBad κ δ M N n)) ≠ (∞:ℝ≥0∞) := by
+  have hr : |(1/2:ℝ)^((N:ℝ)*traceEta κ-1)| < 1 := by
+    rw [abs_of_pos (Real.rpow_pos_of_pos (by norm_num) _)]
+    exact Real.rpow_lt_one (by norm_num) (by norm_num) (by linarith)
+  apply ne_top_of_le_ne_top
+    (((summable_geometric_of_abs_lt_one hr).mul_left
+      ((M:ℝ)*(3/δ)^(zeta (a κ)/2))).tsum_ofReal_ne_top)
+  exact ENNReal.tsum_le_tsum (radialGridBad_measure hκ hκ8 hδ hW M N)
+
+ 
+
+theorem radial_length_refined_eventually {κ δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hδ : 0 < δ) {W : WienerLaw} (hW : IsWiener W) (M N : ℕ)
+    (hN : 1 < (N:ℝ)*traceEta κ) :
+    ∀ᵐ b ∂W, ∀ᶠ n : ℕ in atTop, ∀ k : Fin (M*(2^(2*N+1))^n),
+      radialLength κ b (refinedGridTime δ N n k) (refinedHeight N n) <
+        ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) := by
+  filter_upwards [ae_eventually_notMem (radialGridBad_summable hκ hκ8 hδ hW M N hN)] with b hb
+  filter_upwards [hb] with n hn
+  intro k
+  apply lt_of_not_ge
+  intro h
+  exact hn (mem_iUnion.mpr ⟨k,h⟩)
+
+ 
+
+theorem radial_length_all_refined_grids {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∃ N : ℕ, 0 < N ∧ 1 < (N:ℝ)*traceEta κ ∧
+      ∀ᵐ b ∂W, ∀ m M : ℕ, ∀ᶠ n : ℕ in atTop, ∀ k : Fin (M*(2^(2*N+1))^n),
+        radialLength κ b (refinedGridTime (1/((m:ℝ)+1)) N n k) (refinedHeight N n) <
+          ENNReal.ofReal ((refinedHeight N n)^(traceBeta κ)) := by
+  have hp := trace_exponents hκ hκ8
+  obtain ⟨N,hN⟩ := exists_nat_gt (1/traceEta κ)
+  have hN' : 1 < (N:ℝ)*traceEta κ := (div_lt_iff₀ hp.2.2.1).mp hN
+  have hN0 : 0 < N := by
+    by_contra hh
+    have : N = 0 := Nat.eq_zero_of_not_pos hh
+    subst N
+    norm_num at hN'
+  refine ⟨N,hN0,hN',?_⟩
+  apply ae_all_iff.mpr
+  intro m
+  apply ae_all_iff.mpr
+  intro M
+  exact radial_length_refined_eventually hκ hκ8 (by positivity) hW M N hN'
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open MeasureTheory
+
+ 
+
+lemma radial_increment_le {κ T y z : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
+    (hz : 0 < z) (hzy : z ≤ y) (b : Driver) :
+    ENNReal.ofReal ‖inverseFlow κ b T (driverValue b T+y*Complex.I)-
+      inverseFlow κ b T (driverValue b T+z*Complex.I)‖ ≤ radialLength κ b T y := by
+  have hsub : Icc z y ⊆ Ioi (0:ℝ) := fun u hu => hz.trans_le hu.1
+  have hc := (continuousOn_radialCoeff hκ hT b).mono hsub
+  have hi : IntervalIntegrable (radialCoeff κ b T) volume z y :=
+    hc.intervalIntegrable_of_Icc hzy
+  have hd (u : ℝ) (hu : u ∈ uIcc z y) :
+      HasDerivAt (fun v : ℝ => inverseFlow κ b T (driverValue b T+v*Complex.I))
+        (radialCoeff κ b T u) u := by
+    rw [uIcc_of_le hzy] at hu
+    exact hasDerivAt_radial hκ hT (hsub hu) b
+  rw [← intervalIntegral.integral_eq_sub_of_hasDerivAt hd hi,
+    intervalIntegral.integral_of_le hzy,ofReal_norm]
+  apply (enorm_integral_le_lintegral_enorm _).trans
+  change (∫⁻ u in Ioc z y, ‖radialCoeff κ b T u‖ₑ) ≤ radialLength κ b T y
+  simp only [radialLength,ofReal_norm]
+  exact lintegral_mono_set (Ioc_subset_Ioc hz.le le_rfl)
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open Set Filter Metric
+open scoped Topology
+
+lemma halfPlane_sub_conj_ne_zero {w v : ℂ} (hw : 0 < w.im) (hv : 0 < v.im) :
+    w-star v ≠ 0 := by
+  intro he
+  have hi := congrArg Complex.im he
+  simp only [Complex.sub_im,Complex.star_def,Complex.conj_im,Complex.zero_im] at hi
+  linarith
+
+lemma halfPlane_disk_norm_lt_one {w v : ℂ} (hw : 0 < w.im) (hv : 0 < v.im) :
+    ‖(w-v)/(w-star v)‖ < 1 := by
+  have hs : ‖w-v‖^2 < ‖w-star v‖^2 := by
+    rw [← Complex.normSq_eq_norm_sq,← Complex.normSq_eq_norm_sq]
+    simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
+      Complex.star_def,Complex.conj_re,Complex.conj_im]
+    nlinarith [mul_pos hw hv]
+  rw [norm_div,div_lt_one (norm_pos_iff.mpr (halfPlane_sub_conj_ne_zero hw hv))]
+  nlinarith [norm_nonneg (w-v),norm_nonneg (w-star v)]
+
+lemma halfPlane_self_sub_conj_norm {w : ℂ} (hw : 0 < w.im) :
+    ‖w-star w‖ = 2*w.im := by
+  have he : w-star w = (2*w.im:ℂ)*Complex.I := by
+    apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im]; ring
+  rw [he,norm_mul,Complex.norm_I,mul_one]
+  norm_cast
+  rw [Real.norm_eq_abs,abs_of_nonneg (by positivity)]
+
+ 
+
+lemma ball_radius_le_conformal_radius {f : ℂ → ℂ} {z : ℂ} {r : ℝ} (hr : 0 < r)
+    (hf : DifferentiableOn ℂ f (ball z r))
+    (hH : ∀ w ∈ ball z r, 0 < (f w).im) (hd : deriv f z ≠ 0) :
+    r ≤ 2*(f z).im / ‖deriv f z‖ := by
+  have hz : z ∈ ball z r := by simpa using hr
+  have hy := hH z hz
+  let F (w : ℂ) : ℂ := (f w-f z)/(f w-star (f z))
+  have hn (w : ℂ) (hw : w ∈ ball z r) : f w-star (f z) ≠ 0 :=
+    halfPlane_sub_conj_ne_zero (hH w hw) hy
+  have hF : DifferentiableOn ℂ F (ball z r) :=
+    (hf.sub_const (f z)).div (hf.sub_const (star (f z))) hn
+  have hFz : F z = 0 := by simp [F]
+  have hm : MapsTo F (ball z r) (closedBall (F z) 1) := by
+    intro w hw
+    rw [hFz,mem_closedBall,dist_zero_right]
+    exact (halfPlane_disk_norm_lt_one (hH w hw) hy).le
+  have hh := Complex.norm_deriv_le_div_of_mapsTo_ball hF hm hr
+  have hfd := (hf.differentiableAt (ball_mem_nhds z hr)).hasDerivAt
+  have hFd : HasDerivAt F (deriv f z/(f z-star (f z))) z := by
+    have h := (hfd.sub_const (f z)).div (hfd.sub_const (star (f z))) (hn z hz)
+    convert! h using 1; first | rfl | (simp only [sub_self,zero_mul,sub_zero]; field_simp)
+  rw [hFd.deriv,norm_div,halfPlane_self_sub_conj_norm hy] at hh
+  have hi := (div_le_div_iff₀ (show 0 < 2*(f z).im by positivity) hr).mp hh
+  apply (le_div_iff₀ (norm_pos_iff.mpr hd)).mpr
+  nlinarith
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+
+lemma PositiveSolution.im_hasDeriv {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    HasDerivWithinAt (fun u => (f u).im)
+      (-a κ*(f t).im/‖f t-driverValue b t‖^2) (Icc 0 T) t := by
+  have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hf.2.2.2.2 t ht)
+  convert! hi using 1; first | rfl |
+    (simp only [Complex.imCLM_apply,Complex.div_im,Complex.ofReal_im,Complex.ofReal_re,
+     Complex.sub_im,sub_zero,Complex.normSq_eq_norm_sq,zero_mul,zero_div,zero_sub]; ring)
+
+lemma PositiveSolution.im_antitone {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) :
+    AntitoneOn (fun u => (f u).im) (Icc 0 T) := by
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
+    (Complex.continuous_im.comp_continuousOn hf.2.1)
+    (fun t ht => (hf.im_hasDeriv (interior_subset ht)).mono interior_subset)
+  intro t ht
+  apply div_nonpos_of_nonpos_of_nonneg _ (sq_nonneg _)
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hy := hf.2.2.2.1 t (interior_subset ht)
+  nlinarith
+
+lemma PositiveSolution.im_sq_bound {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) {s t : ℝ}
+    (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T) (hst : s ≤ t) :
+    (f s).im^2 ≤ (f t).im^2+2*a κ*(t-s) := by
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hm : MonotoneOn (fun u => (f u).im^2+2*a κ*u) (Icc 0 T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _)
+      ((Complex.continuous_im.comp_continuousOn hf.2.1).pow 2 |>.add
+        ((continuous_const.mul continuous_id).continuousOn))
+      (fun u hu => (((hf.im_hasDeriv (interior_subset hu)).pow 2).add
+        ((hasDerivAt_id u).const_mul (2*a κ)).hasDerivWithinAt).mono interior_subset)
+    intro u hu
+    have ht := interior_subset hu
+    have hn := Complex.im_le_norm (f u-driverValue b u)
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] at hn
+    have hp := hf.2.2.2.1 u ht
+    have hs : (f u).im^2 ≤ ‖f u-driverValue b u‖^2 := pow_le_pow_left₀ hp.le hn 2
+    have hn0 : 0 < ‖f u-driverValue b u‖^2 := sq_pos_of_pos (hp.trans_le hn)
+    norm_num only [Nat.cast_ofNat,pow_one,mul_one]
+    convert! (div_nonneg (mul_nonneg (show 0 ≤ 2*a κ by positivity) (sub_nonneg.mpr hs)) hn0.le) using 1; field_simp [ne_of_gt (hp.trans_le hn)]; ring
+  have hh := hm hs ht hst
+  dsimp only at hh
+  linarith
+
+lemma inverseFlow_im_sq_bound {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
+    (b : Driver) {w : ℂ} (hw : 0 < w.im) :
+    (inverseFlow κ b T w).im^2 ≤ w.im^2+2*a κ*T := by
+  have hf := surviving_flow_positiveSolution hT (inverseFlow_surviving hκ b hT hw)
+  have hh := hf.im_sq_bound hκ (left_mem_Icc.mpr hT) (right_mem_Icc.mpr hT) hT
+  simpa only [hf.2.2.1,flow_inverseFlow hκ b hT hw,sub_zero] using hh
+
+lemma inverseFlow_im_pos {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
+    (b : Driver) {w : ℂ} (hw : 0 < w.im) : 0 < (inverseFlow κ b T w).im := by
+  have hf := surviving_flow_positiveSolution hT (inverseFlow_surviving hκ b hT hw)
+  simpa only [hf.2.2.1] using hf.2.2.2.1 0 (left_mem_Icc.mpr hT)
+
+lemma inverseFlow_deriv_absolute {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T)
+    (b : Driver) {w : ℂ} (hw : 0 < w.im) :
+    ‖deriv (inverseFlow κ b T) w‖ ≤ 2*(inverseFlow κ b T w).im/w.im := by
+  have hs : ball w w.im ⊆ {u : ℂ | 0 < u.im} := by
+    intro u hu
+    rw [mem_ball,dist_eq_norm] at hu
+    have hh := Complex.im_le_norm (w-u)
+    rw [norm_sub_rev,Complex.sub_im] at hh
+    change 0 < u.im
+    linarith
+  have hd : deriv (inverseFlow κ b T) w ≠ 0 := by
+    rw [(inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.deriv]
+    exact inv_ne_zero (flow_deriv_ne_zero hT (inverseFlow_surviving hκ b hT hw))
+  have hh := ball_radius_le_conformal_radius hw
+    ((flow_conformal_bijection hκ b hT).2.2.2.mono hs)
+    (fun u hu => inverseFlow_im_pos hκ hT b (hs hu)) hd
+  have hh' := (le_div_iff₀ (norm_pos_iff.mpr hd)).mp hh
+  apply (le_div_iff₀ hw).mpr
+  nlinarith
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+
+lemma PositiveSolution.displacement_le {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) (hκ : 0 < κ) {s t : ℝ}
+    (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T) (hst : s ≤ t) :
+    ‖f t-f s‖ ≤ (a κ/(f t).im)*(t-s) := by
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hy := hf.2.2.2.1 t ht
+  have hin : Icc s t ⊆ Icc 0 T := Icc_subset_Icc hs.1 ht.2
+  have hh := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
+    (fun u hu => (hf.2.2.2.2 u (hin hu)).mono hin)
+    (show ∀ u ∈ Icc s t, ‖(a κ:ℂ)/(f u-driverValue b u)‖ ≤ a κ/(f t).im from by
+      intro u hu
+      rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos ha]
+      apply div_le_div_of_nonneg_left ha.le hy
+      apply (hf.im_antitone hκ (hin hu) ht hu.2).trans
+      simpa using Complex.im_le_norm (f u-driverValue b u))
+    (convex_Icc s t) (left_mem_Icc.mpr hst) (right_mem_Icc.mpr hst)
+  simpa only [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hst)] using hh
+
+lemma inverseFlow_im_le {κ t T R : ℝ} (hκ : 0 < κ) (ht : 0 ≤ t) (htT : t ≤ T)
+    (b : Driver) {w : ℂ} (hw : 0 < w.im) (hR : w.im ≤ R) :
+    (inverseFlow κ b t w).im ≤ R+2*a κ*T+1 := by
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hT : 0 ≤ T := ht.trans htT
+  have hR0 : 0 ≤ R := hw.le.trans hR
+  have hA : 0 ≤ 2*a κ*T := by positivity
+  have him := inverseFlow_im_sq_bound hκ ht b hw
+  have hnon := inverseFlow_im_pos hκ ht b hw
+  have hup : w.im^2 ≤ R^2 := pow_le_pow_left₀ hw.le hR 2
+  have hat : 2*a κ*t ≤ 2*a κ*T := mul_le_mul_of_nonneg_left htT (by positivity)
+  nlinarith [sq_nonneg (2*a κ*T),mul_nonneg hR0 hA]
+
+lemma inverseFlow_lipschitz_strip {κ t T y R : ℝ} (hκ : 0 < κ) (ht : 0 ≤ t)
+    (htT : t ≤ T) (hy : 0 < y) (b : Driver) {v w : ℂ}
+    (hv : y ≤ v.im ∧ v.im ≤ R) (hw : y ≤ w.im ∧ w.im ≤ R) :
+    ‖inverseFlow κ b t v-inverseFlow κ b t w‖ ≤
+      (2*(R+2*a κ*T+1)/y)*‖v-w‖ := by
+  have hs : Convex ℝ {u : ℂ | y ≤ u.im ∧ u.im ≤ R} :=
+    (convex_Icc y R).linear_preimage Complex.imCLM.toLinearMap
+  apply Convex.norm_image_sub_le_of_norm_deriv_le (𝕜 := ℂ)
+    (fun u hu => (inverseFlow_hasStrictDerivAt hκ b ht (hy.trans_le hu.1)).hasDerivAt.differentiableAt) _ hs hw hv
+  intro u hu
+  apply (inverseFlow_deriv_absolute hκ ht b (hy.trans_le hu.1)).trans
+  calc
+    _ ≤ 2*(R+2*a κ*T+1)/u.im := by
+      apply div_le_div_of_nonneg_right _ (hy.le.trans hu.1)
+      exact mul_le_mul_of_nonneg_left (inverseFlow_im_le hκ ht htT b (hy.trans_le hu.1) hu.2) (by norm_num)
+    _ ≤ 2*(R+2*a κ*T+1)/y := by
+      apply div_le_div_of_nonneg_left _ hy hu.1
+      have ha : 0 < a κ := div_pos (by norm_num) hκ
+      have hT : 0 ≤ T := ht.trans htT
+      have hR0 : 0 < R := hy.trans_le (hu.1.trans hu.2)
+      positivity
+
+ 
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialRestart.lean b/lean/OAI/Probability/SLEGauge/RadialRestart.lean
index 7d4d361c0269644ad93e2338d7fb81022c0a9ff8..343203d1fb6adc8ae93b81271e3aaecc801f2ef8 100644
--- a/lean/OAI/Probability/SLEGauge/RadialRestart.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialRestart.lean
@@ -1,459 +1,459 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.AngleLyapunov
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-
-lemma physicalRadius_actual_eq_inv_deriv (a : ℝ) (b : GDriver) (s : ℝ≥0)
-    (z : ℂ) (hs : alive a b.1 s z) :
-    physicalRadius (actualState a b.1 s z) =
-      ‖deriv (loewnerMap a b.1 s) z‖⁻¹ * (actualState a b.1 s z).1.im := by
-  unfold physicalRadius actualState
-  simp only [Y_apply, J_apply, Real.exp_neg]
-  rw [Real.exp_log (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hs))]
-  ring
-
-lemma radialTime_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
-    let s := (radialTime a b.1 z σ).toNNReal
-    let w := (actualState a b.1 s z).1
-    (s : ℝ≥0∞) + radialTime a (restart s b).1 w (w.im*(ρ/σ)) =
-      radialTime a b.1 z ρ := by
-  let s := (radialTime a b.1 z σ).toNNReal
-  let w := (actualState a b.1 s z).1
-  let c := ‖deriv (loewnerMap a b.1 s) z‖
-  have hσ0 := hρ.trans_le hρσ
-  have hs := radialTime_attained a ha z hz σ hσ0 hσ b hb
-  have hc : 0<c := norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hs.1)
-  have hrel : c⁻¹*w.im=σ :=
-    (physicalRadius_actual_eq_inv_deriv a b s z hs.1).symm.trans hs.2
-  have hr : c⁻¹*(w.im*(ρ/σ))=ρ := by rw [←mul_assoc,hrel]; field_simp
-  have hle (t : ℝ≥0) (ht : alive a b.1 t z)
-      (hR : physicalRadius (actualState a b.1 t z)=ρ) : s≤t := by
-    apply ENNReal.coe_le_coe.mp
-    rw [show (s:ℝ≥0∞)=radialTime a b.1 z σ from ENNReal.coe_toNNReal hb]
-    exact (radialTime_antitone a ha z hz ρ σ hρ hρσ hσ b).trans
-      (radialTime_le_of_hit a z ρ b t ht hR)
-  have he : (fun u : ℝ≥0∞ => (s:ℝ≥0∞)+u) ''
-      {u : ℝ≥0∞ | ∃t : ℝ≥0, u=t ∧ alive a (restart s b).1 t w ∧
-        physicalRadius (actualState a (restart s b).1 t w)=w.im*(ρ/σ)} =
-      {u : ℝ≥0∞ | ∃t : ℝ≥0, u=t ∧ alive a b.1 t z ∧
-        physicalRadius (actualState a b.1 t z)=ρ} := by
-    ext u
-    constructor
-    · rintro ⟨u,⟨t,rfl,ht,hR⟩,rfl⟩
-      have ht' : alive a b.1 (s+t) z := (alive_restart_iff a b.1 z s t).mpr ⟨hs.1,ht⟩
-      refine ⟨s+t,by simp,ht',?_⟩
-      rw [physicalRadius_restart b.1.continuous ht']
-      change c⁻¹*physicalRadius (actualState a (restart s b).1 t w)=ρ
-      rw [hR,hr]
-    · rintro ⟨t,rfl,ht,hR⟩
-      have hst := hle t ht hR
-      let v := t-s
-      have heq : s+v=t := add_tsub_cancel_of_le hst
-      have ht' : alive a (restart s b).1 v w :=
-        ((alive_restart_iff a b.1 z s v).mp (heq ▸ ht)).2
-      refine ⟨v,⟨v,rfl,ht',?_⟩,by simp [←ENNReal.coe_add,heq]⟩
-      apply (mul_left_cancel₀ (inv_ne_zero hc.ne'))
-      rw [hr]
-      calc
-        c⁻¹*physicalRadius (actualState a (restart s b).1 v w) =
-            physicalRadius (actualState a b.1 (s+v) z) :=
-          (physicalRadius_restart b.1.continuous (heq ▸ ht)).symm
-        _ = ρ := by rw [heq,hR]
-  change (s:ℝ≥0∞)+sInf _=sInf _
-  rw [ENNReal.add_sInf,←sInf_image,he]
-
-lemma radialTime_restart_top_iff (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
-    let s := (radialTime a b.1 z σ).toNNReal
-    let w := (actualState a b.1 s z).1
-    radialTime a (restart s b).1 w (w.im*(ρ/σ))=∞ ↔ radialTime a b.1 z ρ=∞ := by
-  rw [←radialTime_restart a ha z hz σ ρ hρ hρσ hσ b hb]
-  simp
-
- 
-
-lemma radialWeight_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
-    let s := (radialTime a b.1 z σ).toNNReal
-    let w := (actualState a b.1 s z).1
-    radialWeight a b.1 z ρ = radialWeight a b.1 z σ *
-      (radialWeight a (restart s b).1 w (w.im*(ρ/σ)) / physicalGreen a (w,0)) := by
-  let s := (radialTime a b.1 z σ).toNNReal
-  let w := (actualState a b.1 s z).1
-  let τ := radialTime a (restart s b).1 w (w.im*(ρ/σ))
-  change radialWeight a b.1 z ρ = radialWeight a b.1 z σ *
-    (radialWeight a (restart s b).1 w (w.im*(ρ/σ)) / physicalGreen a (w,0))
-  have htop := radialTime_restart_top_iff a ha z hz σ ρ hρ hρσ hσ b hb
-  have hs := radialTime_attained a ha z hz σ (hρ.trans_le hρσ) hσ b hb
-  by_cases hf : radialTime a b.1 z ρ=∞
-  · rw [show radialWeight a b.1 z ρ = 0 from ite_eq_left hf,
-      show radialWeight a (restart s b).1 w (w.im*(ρ/σ)) = 0 from ite_eq_left (htop.mpr hf)]
-    simp
-  · have hτ : τ≠∞ := fun h => hf (htop.mp h)
-    have ht : s+τ.toNNReal=(radialTime a b.1 z ρ).toNNReal := by
-      have hh := congrArg ENNReal.toNNReal (radialTime_restart a ha z hz σ ρ hρ hρσ hσ b hb)
-      change ((s:ℝ≥0∞)+τ).toNNReal=(radialTime a b.1 z ρ).toNNReal at hh
-      simpa only [ENNReal.toNNReal_add ENNReal.coe_ne_top hτ,ENNReal.toNNReal_coe] using hh
-    have hfull := (radialTime_attained a ha z hz ρ hρ (hρσ.trans hσ) b hf).1
-    rw [show radialWeight a b.1 z ρ = _ from ite_eq_right hf,
-      show radialWeight a b.1 z σ = _ from ite_eq_right hb,
-      show radialWeight a (restart s b).1 w (w.im*(ρ/σ)) = _ from ite_eq_right hτ]
-    rw [←ht,actualState_restart b.1.continuous (ht ▸ hfull),physicalGreen_add_log]
-    have hi : physicalGreen a (actualState a b.1 s z) =
-        Real.exp (alpha a*(actualState a b.1 s z).2)*physicalGreen a (w,0) := by
-      convert physicalGreen_add_log a (w,0) (actualState a b.1 s z).2 using 1
-      simp [w]
-    change Real.exp (alpha a*(actualState a b.1 s z).2)*
-      physicalGreen a (actualState a (restart s b).1 τ.toNNReal w) = _
-    rw [hi]
-    have hG : physicalGreen a (w,0) ≠ 0 := (Real.exp_pos _).ne'
-    field_simp [hG]
-    simp only [τ,mul_div_assoc]
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma actualState_continuousOn {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
-    ContinuousOn (fun u : ℝ => actualState a b u.toNNReal z) (Icc 0 (T : ℝ)) := by
-  obtain ⟨f,hf⟩ := hT
-  let Z : ℝ → ℂ := fun u => f u - (b u.toNNReal : ℂ)
-  let A : ℝ → ℝ := fun u => (-(a : ℂ) / (Z u)^2).re
-  have hZ : ContinuousOn Z (Icc 0 (T : ℝ)) := odeUntil_center_continuousOn hb hf
-  have hA : ContinuousOn A (Icc 0 (T : ℝ)) :=
-    Complex.continuous_re.comp_continuousOn (continuousOn_const.div (hZ.pow 2)
-      (fun u hu => pow_ne_zero _ (odeUntil_center_ne_zero hf hu)))
-  have hAi : IntervalIntegrable A volume 0 T := hA.intervalIntegrable_of_Icc T.coe_nonneg
-  have hprim : ContinuousOn (fun u : ℝ => ∫ v in (0 : ℝ)..u, A v)
-      (Icc 0 (T : ℝ)) := by
-    simpa only [uIcc_of_le T.coe_nonneg] using
-      intervalIntegral.continuousOn_primitive_interval' hAi
-        (show (0 : ℝ) ∈ uIcc 0 (T : ℝ) from left_mem_uIcc)
-  have hC := hZ.prodMk hprim
-  apply hC.congr
-  intro u hu
-  have huT : u.toNNReal ≤ T := by simpa using Real.toNNReal_le_toNNReal hu.2
-  apply Prod.ext
-  · simp only [actualState,Z,loewnerMap_eq_solution_time hf hu]
-  · change Real.log ‖deriv (loewnerMap a b u.toNNReal) z‖ = _
-    rw [log_norm_deriv_loewnerMap hb ⟨f,odeUntil_mono hf huT⟩,
-      Real.coe_toNNReal _ hu.1]
-    apply intervalIntegral.integral_congr
-    intro v hv
-    have hv' : v ∈ Icc 0 (T : ℝ) :=
-      (Icc_subset_Icc_right hu.2) (by simpa only [uIcc_of_le hu.1] using hv)
-    simp only [Z,loewnerMap_eq_solution_time hf hv']
-
-lemma actualState_bounded_until {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
-    ∃ η C : ℝ, 0 < η ∧ ∀ t : ℝ≥0, t ≤ T →
-      η ≤ Y (actualState a b t z) ∧ ‖actualState a b t z‖ ≤ C := by
-  obtain ⟨f,hf⟩ := hT
-  obtain ⟨η,hη,hmin⟩ := odeUntil_height_lower hf
-  obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn
-    (actualState_continuousOn hb ⟨f,hf⟩)
-  refine ⟨η,C,hη,?_⟩
-  intro t ht
-  have hmem : (t : ℝ) ∈ Icc 0 (T : ℝ) := ⟨t.coe_nonneg,ht⟩
-  constructor
-  · have he : loewnerMap a b t z = f t := by
-      simpa only [Real.toNNReal_coe] using loewnerMap_eq_solution_time hf hmem
-    simpa only [actualState,Y_apply,he,Complex.sub_im,Complex.ofReal_im,sub_zero]
-      using hmin (t : ℝ) hmem
-  · simpa only [Real.toNNReal_coe] using hC (t : ℝ) hmem
-
-noncomputable def exhaustionHeight (z : UpperPlane) (n : ℕ) : ℝ :=
-  z.1.im / ((n : ℝ) + 2)
-
-lemma exhaustionHeight_pos (z : UpperPlane) (n : ℕ) : 0 < exhaustionHeight z n :=
-  div_pos z.2 (by positivity)
-
-lemma exhaustionHeight_le (z : UpperPlane) (n : ℕ) : exhaustionHeight z n ≤ z.1.im := by
-  exact div_le_self z.2.le (by linarith [Nat.cast_nonneg (α := ℝ) n])
-
-lemma exhaustionHeight_tendsto (z : UpperPlane) :
-    Tendsto (exhaustionHeight z) atTop (𝓝 0) := by
-  have h : Tendsto (fun n : ℕ => 1 / ((n : ℝ) + 2)) atTop (𝓝 0) :=
-    by simpa only [Function.comp_def, Nat.cast_add,Nat.cast_ofNat] using
-      (tendsto_one_div_atTop_nhds_zero_nat (𝕜 := ℝ)).comp (tendsto_add_atTop_nat 2)
-  change Tendsto (fun n : ℕ => z.1.im / ((n : ℝ) + 2)) atTop (𝓝 0)
-  simpa only [mul_one_div,mul_zero] using h.const_mul z.1.im
-
-noncomputable def exhaustionNorm (z : UpperPlane) (n : ℕ) : ℝ≥0 :=
-  ⟨‖z.1‖ + n + 1, by positivity⟩
-
-noncomputable def exhaustionStop (a : ℝ) (z : UpperPlane) (n : ℕ) : ExactSLEGenerator.Driver → ℝ≥0 :=
-  regionStop a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1
-    (exhaustionNorm z n) 0 0 n
-
-lemma exhaustionRegion_initial (z : UpperPlane) (n : ℕ) :
-    0 ≤ regionMonitor (exhaustionHeight z n) (exhaustionNorm z n) 0 0 (z.1,0) := by
-  apply (regionMonitor_nonneg_iff _ _ _ _ _).mpr
-  refine ⟨exhaustionHeight_le z n, ?_, ?_, ?_⟩
-  · simp only [Prod.norm_def,norm_zero,max_eq_left (norm_nonneg z.1)]
-    change ‖z.1‖ ≤ ‖z.1‖ + (n : ℝ) + 1
-    linarith [Nat.cast_nonneg (α := ℝ) n]
-  · simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one]
-    exact z.2.le
-  · simp only [zero_mul,Y_apply]
-    exact z.2.le
-
-lemma exhaustionStop_stopping (a : ℝ) (z : UpperPlane) (n : ℕ) :
-    IsStoppingTime rawFiltration (fun b => (exhaustionStop a z n b : WithTop ℝ≥0)) :=
-  regionStop_stopping a _ _ z.1 _ 0 0 n
-
-lemma exhaustionStop_le (a : ℝ) (z : UpperPlane) (n : ℕ) (b : ExactSLEGenerator.Driver) :
-    exhaustionStop a z n b ≤ n := regionStop_le a _ _ z.1 _ 0 0 n b
-
-lemma exhaustionStop_agrees (a : ℝ) (z : UpperPlane) (n : ℕ)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ exhaustionStop a z n b) :
-    alive a b.1 t z.1 ∧ actualState a b.1 t z.1 =
-      localState a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1 t b :=
-  regionStop_actualState a _ _ z.1 _ 0 0 n (exhaustionRegion_initial z n) b t ht
-
- 
-
-lemma exhaustionStop_eventually (a : ℝ) (z : UpperPlane) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (ht : alive a b.1 t z.1) :
-    ∀ᶠ n in atTop, t ≤ exhaustionStop a z n b := by
-  obtain ⟨η,C,hη,hbound⟩ := actualState_bounded_until b.1.continuous ht
-  have hheight : ∀ᶠ n in atTop, exhaustionHeight z n < η :=
-    (exhaustionHeight_tendsto z).eventually (gt_mem_nhds hη)
-  have hlarge : ∀ᶠ n : ℕ in atTop, max (t : ℝ) C < n :=
-    (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_gt_atTop _)
-  filter_upwards [hheight,hlarge] with n hn hlarge
-  by_contra hnot
-  have hs : exhaustionStop a z n b < t := lt_of_not_ge hnot
-  have hsn : exhaustionStop a z n b < n :=
-    hs.trans (by exact_mod_cast (lt_of_le_of_lt (le_max_left _ _) hlarge))
-  have he := (exhaustionStop_agrees a z n b _ le_rfl).2
-  have hb := hbound (exhaustionStop a z n b) hs.le
-  have hcontact := regionStop_contact a _ (exhaustionHeight_pos z n) z.1
-    (exhaustionNorm z n) 0 0 n (exhaustionRegion_initial z n) b hsn
-  simp only [exhaustionStop] at he hb
-  rw [← he] at hcontact
-  rcases hcontact with h | h | h | h
-  · linarith only [hb.1, h, hn]
-  · have hCN : C < (exhaustionNorm z n : ℝ) := by
-      change C < ‖z.1‖ + (n : ℝ) + 1
-      have hCn := lt_of_le_of_lt (le_max_right (t : ℝ) C) hlarge
-      linarith [norm_nonneg z.1]
-    linarith only [hb.2,h,hCN]
-  · have hp := physicalRadius_actual_pos a b z.1 _ (exhaustionStop_agrees a z n b _ le_rfl).1
-    simp only [exhaustionStop] at hp
-    rw [h] at hp
-    exact lt_irrefl _ hp
-  · simp only [zero_mul] at h
-    linarith only [hb.1,h,hη]
-
-lemma exhaustionStop_measurable (a : ℝ) (z : UpperPlane) (n : ℕ) :
-    Measurable (exhaustionStop a z n) := by
-  apply measurable_of_Iic
-  intro t
-  simpa only [preimage, mem_Iic, WithTop.coe_le_coe] using
-    rawFiltration.le t _ ((exhaustionStop_stopping a z n).measurableSet_le t)
-
- 
-
-noncomputable def killedObservable (a : ℝ) (z : UpperPlane) (F : State → ℝ)
-    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : ℝ :=
-  if alive a b.1 t z.1 then F (actualState a b.1 t z.1) else 0
-
-noncomputable def localizedObservable (a : ℝ) (z : UpperPlane) (F : State → ℝ)
-    (n : ℕ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : ℝ :=
-  if t ≤ exhaustionStop a z n b then
-    F (localState a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1 t b) else 0
-
-lemma localizedObservable_joint_measurable (a : ℝ) (z : UpperPlane)
-    (F : State → ℝ) (hF : Measurable F) (n : ℕ) :
-    Measurable (fun p : ℝ≥0 × ExactSLEGenerator.Driver => localizedObservable a z F n p.1 p.2) := by
-  exact Measurable.ite
-    (measurableSet_le measurable_fst ((exhaustionStop_measurable a z n).comp measurable_snd))
-    (hF.comp (localState_parameter_joint_measurable a (fun _ => exhaustionHeight z n)
-      (fun _ => exhaustionHeight_pos z n) (fun _ => z.1) id
-      measurable_const measurable_const measurable_id)) measurable_const
-
-lemma localizedObservable_tendsto (a : ℝ) (z : UpperPlane) (F : State → ℝ)
-    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    Tendsto (fun n => localizedObservable a z F n t b) atTop (𝓝 (killedObservable a z F t b)) := by
-  by_cases ht : alive a b.1 t z.1
-  · apply tendsto_const_nhds.congr'
-    filter_upwards [exhaustionStop_eventually a z b t ht] with n hn
-    simp only [localizedObservable,ite_eq_left hn,killedObservable,ite_eq_left ht,
-      (exhaustionStop_agrees a z n b t hn).2]
-  · have hn (n : ℕ) : ¬t ≤ exhaustionStop a z n b :=
-      fun hn => ht (exhaustionStop_agrees a z n b t hn).1
-    simpa only [localizedObservable,ite_eq_right (hn _),killedObservable,ite_eq_right ht] using
-      (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))
-
-lemma killedObservable_joint_measurable (a : ℝ) (z : UpperPlane)
-    (F : State → ℝ) (hF : Measurable F) :
-    Measurable (fun p : ℝ≥0 × ExactSLEGenerator.Driver => killedObservable a z F p.1 p.2) :=
-  measurable_of_tendsto_metrizable (localizedObservable_joint_measurable a z F hF)
-    (tendsto_pi_nhds.mpr (fun p => localizedObservable_tendsto a z F p.1 p.2))
-
-lemma killedObservable_nonneg (a : ℝ) (z : UpperPlane)
-    (F : State → ℝ) (hF : ∀ v, 0 < Y v → 0 ≤ F v)
-    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : 0 ≤ killedObservable a z F t b := by
-  unfold killedObservable
-  split_ifs with ht
-  · obtain ⟨f,hf⟩ := ht
-    apply hF
-    have he : loewnerMap a b.1 t z.1 = f t := by
-      simpa only [Real.toNNReal_coe] using
-        loewnerMap_eq_solution_time hf (right_mem_Icc.mpr t.coe_nonneg)
-    simpa only [actualState,Y_apply,he,Complex.sub_im,Complex.ofReal_im,
-      sub_zero] using (hf.2 (t : ℝ) (right_mem_Icc.mpr t.coe_nonneg)).1
-  · exact le_rfl
-
-lemma greenLifetime_measurable (a : ℝ) :
-    Measurable (fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v) := by
-  unfold physicalGreen logGreen lifetimeTest Q X Y J
-  fun_prop
-
-lemma lifetimeCost_measurable (a : ℝ) : Measurable (lifetimeCost a) := by
-  unfold lifetimeCost physicalGreen logGreen Q X Y J
-  fun_prop
-
- 
-
-theorem global_lifetime_test_budget (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) (t : ℝ≥0) :
-    (∫⁻ b, ENNReal.ofReal (killedObservable a z
-      (fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v) t b) ∂W) ≤
-      ENNReal.ofReal (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) := by
-  let F : State → ℝ := fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v
-  have hpos (v : State) (hv : 0 < Y v) : 0 ≤ F v :=
-    mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg _ _ hv.le)
-  have hbound (n : ℕ) : (∫⁻ b, ENNReal.ofReal (localizedObservable a z F n t b) ∂W) ≤
-      ENNReal.ofReal (F (z.1,0)) := by
-    let ε := exhaustionHeight z n
-    let V := localState a ε (exhaustionHeight_pos z n) z.1
-    let S : ExactSLEGenerator.Driver → ℝ≥0 := fun b => min (exhaustionStop a z n b) t
-    have hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)) := by
-      simpa only [S, WithTop.coe_min] using (exhaustionStop_stopping a z n).min_const t
-    have hsafe (b : ExactSLEGenerator.Driver) (u : ℝ≥0) (hu : u ≤ S b) :
-        ε ≤ Y (V u b) ∧ ‖V u b‖ ≤ exhaustionNorm z n := by
-      have h := regionStop_safe a ε (exhaustionHeight_pos z n) z.1
-        (exhaustionNorm z n) 0 0 n (exhaustionRegion_initial z n) b u
-        (hu.trans (min_le_left _ _))
-      exact ⟨h.1,h.2.1⟩
-    have hb := local_lifetime_budget W hW a ε ha (exhaustionHeight_pos z n)
-      z.1 (exhaustionNorm z n) n S hS
-      (fun b => (min_le_left _ _).trans (exhaustionStop_le a z n b)) hsafe
-    calc
-      _ ≤ ∫⁻ b, ENNReal.ofReal (F (V (S b) b)) ∂W := by
-        apply lintegral_mono
-        intro b
-        apply ENNReal.ofReal_le_ofReal
-        by_cases ht : t ≤ exhaustionStop a z n b
-        · simp only [localizedObservable,ite_eq_left ht,S,min_eq_right ht,F,V,ε,le_refl]
-        · rw [localizedObservable,ite_eq_right ht]
-          exact hpos _ ((exhaustionHeight_pos z n).trans_le (hsafe b (S b) le_rfl).1)
-      _ = ENNReal.ofReal (∫ b, F (V (S b) b) ∂W) :=
-        (ofReal_integral_eq_lintegral_ofReal hb.1 (Eventually.of_forall (fun b =>
-          hpos _ ((exhaustionHeight_pos z n).trans_le (hsafe b (S b) le_rfl).1)))).symm
-      _ ≤ _ := ENNReal.ofReal_le_ofReal hb.2.1
-  have hlim (b : ExactSLEGenerator.Driver) :
-      liminf (fun n => ENNReal.ofReal (localizedObservable a z F n t b)) atTop =
-        ENNReal.ofReal (killedObservable a z F t b) :=
-    (ENNReal.tendsto_ofReal (localizedObservable_tendsto a z F t b)).liminf_eq
-  change (∫⁻ b, ENNReal.ofReal (killedObservable a z F t b) ∂W) ≤ _
-  simp_rw [← hlim]
-  exact (lintegral_liminf_le (fun n =>
-    ((localizedObservable_joint_measurable a z F (greenLifetime_measurable a) n).comp
-      (measurable_const.prodMk measurable_id)).ennreal_ofReal)).trans
-    (liminf_le_of_frequently_le (Eventually.of_forall hbound).frequently)
-
-lemma localizedObservable_occupation (a : ℝ) (z : UpperPlane) (F : State → ℝ)
-    (n : ℕ) (b : ExactSLEGenerator.Driver) :
-    (∫⁻ u in Ioi (0 : ℝ), ENNReal.ofReal (localizedObservable a z F n u.toNNReal b)) =
-      ∫⁻ u in Ioc (0 : ℝ) (exhaustionStop a z n b : ℝ),
-        ENNReal.ofReal (F (localState a (exhaustionHeight z n)
-          (exhaustionHeight_pos z n) z.1 u.toNNReal b)) := by
-  have he (u : ℝ) : ENNReal.ofReal (localizedObservable a z F n u.toNNReal b) =
-      (Iic (exhaustionStop a z n b : ℝ)).indicator
-        (fun u : ℝ => ENNReal.ofReal (F (localState a (exhaustionHeight z n)
-          (exhaustionHeight_pos z n) z.1 u.toNNReal b))) u := by
-    by_cases hu : u ≤ (exhaustionStop a z n b : ℝ)
-    · simp only [localizedObservable,Real.toNNReal_le_iff_le_coe,indicator,mem_Iic,hu,ite_true]
-    · simp only [localizedObservable,Real.toNNReal_le_iff_le_coe,indicator,mem_Iic,hu,ite_false,
-        ENNReal.ofReal_zero]
-  simp_rw [he]
-  rw [setLIntegral_indicator measurableSet_Iic, inter_comm, Ioi_inter_Iic]
-
- 
-
-theorem global_lifetime_occupation_budget (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (z : UpperPlane) :
-    (∫⁻ b, ∫⁻ u in Ioi (0 : ℝ),
-      ENNReal.ofReal (killedObservable a z (lifetimeCost a) u.toNNReal b) ∂volume ∂W) ≤
-      ENNReal.ofReal (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) := by
-  let G : ℕ → ExactSLEGenerator.Driver × ℝ → ℝ≥0∞ := fun n p =>
-    ENNReal.ofReal (localizedObservable a z (lifetimeCost a) n p.2.toNNReal p.1)
-  let H : ExactSLEGenerator.Driver × ℝ → ℝ≥0∞ := fun p =>
-    ENNReal.ofReal (killedObservable a z (lifetimeCost a) p.2.toNNReal p.1)
-  have hG (n : ℕ) : Measurable (G n) :=
-    ((localizedObservable_joint_measurable a z (lifetimeCost a) (lifetimeCost_measurable a) n).comp
-      ((measurable_real_toNNReal.comp measurable_snd).prodMk measurable_fst)).ennreal_ofReal
-  have hH : Measurable H :=
-    ((killedObservable_joint_measurable a z (lifetimeCost a) (lifetimeCost_measurable a)).comp
-      ((measurable_real_toNNReal.comp measurable_snd).prodMk measurable_fst)).ennreal_ofReal
-  have hbound (n : ℕ) :
-      (∫⁻ p, G n p ∂W.prod (volume.restrict (Ioi (0 : ℝ)))) ≤
-        ENNReal.ofReal (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) := by
-    rw [lintegral_prod _ (hG n).aemeasurable]
-    change (∫⁻ b, ∫⁻ u in Ioi (0 : ℝ),
-      ENNReal.ofReal (localizedObservable a z (lifetimeCost a) n u.toNNReal b) ∂volume ∂W) ≤ _
-    simp_rw [localizedObservable_occupation]
-    apply (local_lifetime_budget W hW a (exhaustionHeight z n) ha (exhaustionHeight_pos z n)
-      z.1 (exhaustionNorm z n) n (exhaustionStop a z n) (exhaustionStop_stopping a z n)
-      (exhaustionStop_le a z n) ?_).2.2
-    intro b u hu
-    have h := regionStop_safe a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1
-      (exhaustionNorm z n) 0 0 n (exhaustionRegion_initial z n) b u hu
-    exact ⟨h.1,h.2.1⟩
-  have hlim (p : ExactSLEGenerator.Driver × ℝ) :
-      liminf (fun n => G n p) atTop = H p :=
-    (ENNReal.tendsto_ofReal (localizedObservable_tendsto a z (lifetimeCost a) p.2.toNNReal p.1)).liminf_eq
-  rw [← lintegral_prod H hH.aemeasurable]
-  simp_rw [← hlim]
-  exact (lintegral_liminf_le hG).trans
-    (liminf_le_of_frequently_le (Eventually.of_forall hbound).frequently)
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.AngleLyapunov
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+
+lemma physicalRadius_actual_eq_inv_deriv (a : ℝ) (b : GDriver) (s : ℝ≥0)
+    (z : ℂ) (hs : alive a b.1 s z) :
+    physicalRadius (actualState a b.1 s z) =
+      ‖deriv (loewnerMap a b.1 s) z‖⁻¹ * (actualState a b.1 s z).1.im := by
+  unfold physicalRadius actualState
+  simp only [Y_apply, J_apply, Real.exp_neg]
+  rw [Real.exp_log (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hs))]
+  ring
+
+lemma radialTime_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
+    let s := (radialTime a b.1 z σ).toNNReal
+    let w := (actualState a b.1 s z).1
+    (s : ℝ≥0∞) + radialTime a (restart s b).1 w (w.im*(ρ/σ)) =
+      radialTime a b.1 z ρ := by
+  let s := (radialTime a b.1 z σ).toNNReal
+  let w := (actualState a b.1 s z).1
+  let c := ‖deriv (loewnerMap a b.1 s) z‖
+  have hσ0 := hρ.trans_le hρσ
+  have hs := radialTime_attained a ha z hz σ hσ0 hσ b hb
+  have hc : 0<c := norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hs.1)
+  have hrel : c⁻¹*w.im=σ :=
+    (physicalRadius_actual_eq_inv_deriv a b s z hs.1).symm.trans hs.2
+  have hr : c⁻¹*(w.im*(ρ/σ))=ρ := by rw [←mul_assoc,hrel]; field_simp
+  have hle (t : ℝ≥0) (ht : alive a b.1 t z)
+      (hR : physicalRadius (actualState a b.1 t z)=ρ) : s≤t := by
+    apply ENNReal.coe_le_coe.mp
+    rw [show (s:ℝ≥0∞)=radialTime a b.1 z σ from ENNReal.coe_toNNReal hb]
+    exact (radialTime_antitone a ha z hz ρ σ hρ hρσ hσ b).trans
+      (radialTime_le_of_hit a z ρ b t ht hR)
+  have he : (fun u : ℝ≥0∞ => (s:ℝ≥0∞)+u) ''
+      {u : ℝ≥0∞ | ∃t : ℝ≥0, u=t ∧ alive a (restart s b).1 t w ∧
+        physicalRadius (actualState a (restart s b).1 t w)=w.im*(ρ/σ)} =
+      {u : ℝ≥0∞ | ∃t : ℝ≥0, u=t ∧ alive a b.1 t z ∧
+        physicalRadius (actualState a b.1 t z)=ρ} := by
+    ext u
+    constructor
+    · rintro ⟨u,⟨t,rfl,ht,hR⟩,rfl⟩
+      have ht' : alive a b.1 (s+t) z := (alive_restart_iff a b.1 z s t).mpr ⟨hs.1,ht⟩
+      refine ⟨s+t,by simp,ht',?_⟩
+      rw [physicalRadius_restart b.1.continuous ht']
+      change c⁻¹*physicalRadius (actualState a (restart s b).1 t w)=ρ
+      rw [hR,hr]
+    · rintro ⟨t,rfl,ht,hR⟩
+      have hst := hle t ht hR
+      let v := t-s
+      have heq : s+v=t := add_tsub_cancel_of_le hst
+      have ht' : alive a (restart s b).1 v w :=
+        ((alive_restart_iff a b.1 z s v).mp (heq ▸ ht)).2
+      refine ⟨v,⟨v,rfl,ht',?_⟩,by simp [←ENNReal.coe_add,heq]⟩
+      apply (mul_left_cancel₀ (inv_ne_zero hc.ne'))
+      rw [hr]
+      calc
+        c⁻¹*physicalRadius (actualState a (restart s b).1 v w) =
+            physicalRadius (actualState a b.1 (s+v) z) :=
+          (physicalRadius_restart b.1.continuous (heq ▸ ht)).symm
+        _ = ρ := by rw [heq,hR]
+  change (s:ℝ≥0∞)+sInf _=sInf _
+  rw [ENNReal.add_sInf,←sInf_image,he]
+
+lemma radialTime_restart_top_iff (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
+    let s := (radialTime a b.1 z σ).toNNReal
+    let w := (actualState a b.1 s z).1
+    radialTime a (restart s b).1 w (w.im*(ρ/σ))=∞ ↔ radialTime a b.1 z ρ=∞ := by
+  rw [←radialTime_restart a ha z hz σ ρ hρ hρσ hσ b hb]
+  simp
+
+ 
+
+lemma radialWeight_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (b : GDriver) (hb : radialTime a b.1 z σ≠∞) :
+    let s := (radialTime a b.1 z σ).toNNReal
+    let w := (actualState a b.1 s z).1
+    radialWeight a b.1 z ρ = radialWeight a b.1 z σ *
+      (radialWeight a (restart s b).1 w (w.im*(ρ/σ)) / physicalGreen a (w,0)) := by
+  let s := (radialTime a b.1 z σ).toNNReal
+  let w := (actualState a b.1 s z).1
+  let τ := radialTime a (restart s b).1 w (w.im*(ρ/σ))
+  change radialWeight a b.1 z ρ = radialWeight a b.1 z σ *
+    (radialWeight a (restart s b).1 w (w.im*(ρ/σ)) / physicalGreen a (w,0))
+  have htop := radialTime_restart_top_iff a ha z hz σ ρ hρ hρσ hσ b hb
+  have hs := radialTime_attained a ha z hz σ (hρ.trans_le hρσ) hσ b hb
+  by_cases hf : radialTime a b.1 z ρ=∞
+  · rw [show radialWeight a b.1 z ρ = 0 from ite_eq_left hf,
+      show radialWeight a (restart s b).1 w (w.im*(ρ/σ)) = 0 from ite_eq_left (htop.mpr hf)]
+    simp
+  · have hτ : τ≠∞ := fun h => hf (htop.mp h)
+    have ht : s+τ.toNNReal=(radialTime a b.1 z ρ).toNNReal := by
+      have hh := congrArg ENNReal.toNNReal (radialTime_restart a ha z hz σ ρ hρ hρσ hσ b hb)
+      change ((s:ℝ≥0∞)+τ).toNNReal=(radialTime a b.1 z ρ).toNNReal at hh
+      simpa only [ENNReal.toNNReal_add ENNReal.coe_ne_top hτ,ENNReal.toNNReal_coe] using hh
+    have hfull := (radialTime_attained a ha z hz ρ hρ (hρσ.trans hσ) b hf).1
+    rw [show radialWeight a b.1 z ρ = _ from ite_eq_right hf,
+      show radialWeight a b.1 z σ = _ from ite_eq_right hb,
+      show radialWeight a (restart s b).1 w (w.im*(ρ/σ)) = _ from ite_eq_right hτ]
+    rw [←ht,actualState_restart b.1.continuous (ht ▸ hfull),physicalGreen_add_log]
+    have hi : physicalGreen a (actualState a b.1 s z) =
+        Real.exp (alpha a*(actualState a b.1 s z).2)*physicalGreen a (w,0) := by
+      convert physicalGreen_add_log a (w,0) (actualState a b.1 s z).2 using 1
+      simp [w]
+    change Real.exp (alpha a*(actualState a b.1 s z).2)*
+      physicalGreen a (actualState a (restart s b).1 τ.toNNReal w) = _
+    rw [hi]
+    have hG : physicalGreen a (w,0) ≠ 0 := (Real.exp_pos _).ne'
+    field_simp [hG]
+    simp only [τ,mul_div_assoc]
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma actualState_continuousOn {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
+    ContinuousOn (fun u : ℝ => actualState a b u.toNNReal z) (Icc 0 (T : ℝ)) := by
+  obtain ⟨f,hf⟩ := hT
+  let Z : ℝ → ℂ := fun u => f u - (b u.toNNReal : ℂ)
+  let A : ℝ → ℝ := fun u => (-(a : ℂ) / (Z u)^2).re
+  have hZ : ContinuousOn Z (Icc 0 (T : ℝ)) := odeUntil_center_continuousOn hb hf
+  have hA : ContinuousOn A (Icc 0 (T : ℝ)) :=
+    Complex.continuous_re.comp_continuousOn (continuousOn_const.div (hZ.pow 2)
+      (fun u hu => pow_ne_zero _ (odeUntil_center_ne_zero hf hu)))
+  have hAi : IntervalIntegrable A volume 0 T := hA.intervalIntegrable_of_Icc T.coe_nonneg
+  have hprim : ContinuousOn (fun u : ℝ => ∫ v in (0 : ℝ)..u, A v)
+      (Icc 0 (T : ℝ)) := by
+    simpa only [uIcc_of_le T.coe_nonneg] using
+      intervalIntegral.continuousOn_primitive_interval' hAi
+        (show (0 : ℝ) ∈ uIcc 0 (T : ℝ) from left_mem_uIcc)
+  have hC := hZ.prodMk hprim
+  apply hC.congr
+  intro u hu
+  have huT : u.toNNReal ≤ T := by simpa using Real.toNNReal_le_toNNReal hu.2
+  apply Prod.ext
+  · simp only [actualState,Z,loewnerMap_eq_solution_time hf hu]
+  · change Real.log ‖deriv (loewnerMap a b u.toNNReal) z‖ = _
+    rw [log_norm_deriv_loewnerMap hb ⟨f,odeUntil_mono hf huT⟩,
+      Real.coe_toNNReal _ hu.1]
+    apply intervalIntegral.integral_congr
+    intro v hv
+    have hv' : v ∈ Icc 0 (T : ℝ) :=
+      (Icc_subset_Icc_right hu.2) (by simpa only [uIcc_of_le hu.1] using hv)
+    simp only [Z,loewnerMap_eq_solution_time hf hv']
+
+lemma actualState_bounded_until {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} (hT : alive a b T z) :
+    ∃ η C : ℝ, 0 < η ∧ ∀ t : ℝ≥0, t ≤ T →
+      η ≤ Y (actualState a b t z) ∧ ‖actualState a b t z‖ ≤ C := by
+  obtain ⟨f,hf⟩ := hT
+  obtain ⟨η,hη,hmin⟩ := odeUntil_height_lower hf
+  obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn
+    (actualState_continuousOn hb ⟨f,hf⟩)
+  refine ⟨η,C,hη,?_⟩
+  intro t ht
+  have hmem : (t : ℝ) ∈ Icc 0 (T : ℝ) := ⟨t.coe_nonneg,ht⟩
+  constructor
+  · have he : loewnerMap a b t z = f t := by
+      simpa only [Real.toNNReal_coe] using loewnerMap_eq_solution_time hf hmem
+    simpa only [actualState,Y_apply,he,Complex.sub_im,Complex.ofReal_im,sub_zero]
+      using hmin (t : ℝ) hmem
+  · simpa only [Real.toNNReal_coe] using hC (t : ℝ) hmem
+
+noncomputable def exhaustionHeight (z : UpperPlane) (n : ℕ) : ℝ :=
+  z.1.im / ((n : ℝ) + 2)
+
+lemma exhaustionHeight_pos (z : UpperPlane) (n : ℕ) : 0 < exhaustionHeight z n :=
+  div_pos z.2 (by positivity)
+
+lemma exhaustionHeight_le (z : UpperPlane) (n : ℕ) : exhaustionHeight z n ≤ z.1.im := by
+  exact div_le_self z.2.le (by linarith [Nat.cast_nonneg (α := ℝ) n])
+
+lemma exhaustionHeight_tendsto (z : UpperPlane) :
+    Tendsto (exhaustionHeight z) atTop (𝓝 0) := by
+  have h : Tendsto (fun n : ℕ => 1 / ((n : ℝ) + 2)) atTop (𝓝 0) :=
+    by simpa only [Function.comp_def, Nat.cast_add,Nat.cast_ofNat] using
+      (tendsto_one_div_atTop_nhds_zero_nat (𝕜 := ℝ)).comp (tendsto_add_atTop_nat 2)
+  change Tendsto (fun n : ℕ => z.1.im / ((n : ℝ) + 2)) atTop (𝓝 0)
+  simpa only [mul_one_div,mul_zero] using h.const_mul z.1.im
+
+noncomputable def exhaustionNorm (z : UpperPlane) (n : ℕ) : ℝ≥0 :=
+  ⟨‖z.1‖ + n + 1, by positivity⟩
+
+noncomputable def exhaustionStop (a : ℝ) (z : UpperPlane) (n : ℕ) : ExactSLEGenerator.Driver → ℝ≥0 :=
+  regionStop a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1
+    (exhaustionNorm z n) 0 0 n
+
+lemma exhaustionRegion_initial (z : UpperPlane) (n : ℕ) :
+    0 ≤ regionMonitor (exhaustionHeight z n) (exhaustionNorm z n) 0 0 (z.1,0) := by
+  apply (regionMonitor_nonneg_iff _ _ _ _ _).mpr
+  refine ⟨exhaustionHeight_le z n, ?_, ?_, ?_⟩
+  · simp only [Prod.norm_def,norm_zero,max_eq_left (norm_nonneg z.1)]
+    change ‖z.1‖ ≤ ‖z.1‖ + (n : ℝ) + 1
+    linarith [Nat.cast_nonneg (α := ℝ) n]
+  · simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one]
+    exact z.2.le
+  · simp only [zero_mul,Y_apply]
+    exact z.2.le
+
+lemma exhaustionStop_stopping (a : ℝ) (z : UpperPlane) (n : ℕ) :
+    IsStoppingTime rawFiltration (fun b => (exhaustionStop a z n b : WithTop ℝ≥0)) :=
+  regionStop_stopping a _ _ z.1 _ 0 0 n
+
+lemma exhaustionStop_le (a : ℝ) (z : UpperPlane) (n : ℕ) (b : ExactSLEGenerator.Driver) :
+    exhaustionStop a z n b ≤ n := regionStop_le a _ _ z.1 _ 0 0 n b
+
+lemma exhaustionStop_agrees (a : ℝ) (z : UpperPlane) (n : ℕ)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (ht : t ≤ exhaustionStop a z n b) :
+    alive a b.1 t z.1 ∧ actualState a b.1 t z.1 =
+      localState a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1 t b :=
+  regionStop_actualState a _ _ z.1 _ 0 0 n (exhaustionRegion_initial z n) b t ht
+
+ 
+
+lemma exhaustionStop_eventually (a : ℝ) (z : UpperPlane) (b : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (ht : alive a b.1 t z.1) :
+    ∀ᶠ n in atTop, t ≤ exhaustionStop a z n b := by
+  obtain ⟨η,C,hη,hbound⟩ := actualState_bounded_until b.1.continuous ht
+  have hheight : ∀ᶠ n in atTop, exhaustionHeight z n < η :=
+    (exhaustionHeight_tendsto z).eventually (gt_mem_nhds hη)
+  have hlarge : ∀ᶠ n : ℕ in atTop, max (t : ℝ) C < n :=
+    (tendsto_natCast_atTop_atTop (R := ℝ)).eventually (eventually_gt_atTop _)
+  filter_upwards [hheight,hlarge] with n hn hlarge
+  by_contra hnot
+  have hs : exhaustionStop a z n b < t := lt_of_not_ge hnot
+  have hsn : exhaustionStop a z n b < n :=
+    hs.trans (by exact_mod_cast (lt_of_le_of_lt (le_max_left _ _) hlarge))
+  have he := (exhaustionStop_agrees a z n b _ le_rfl).2
+  have hb := hbound (exhaustionStop a z n b) hs.le
+  have hcontact := regionStop_contact a _ (exhaustionHeight_pos z n) z.1
+    (exhaustionNorm z n) 0 0 n (exhaustionRegion_initial z n) b hsn
+  simp only [exhaustionStop] at he hb
+  rw [← he] at hcontact
+  rcases hcontact with h | h | h | h
+  · linarith only [hb.1, h, hn]
+  · have hCN : C < (exhaustionNorm z n : ℝ) := by
+      change C < ‖z.1‖ + (n : ℝ) + 1
+      have hCn := lt_of_le_of_lt (le_max_right (t : ℝ) C) hlarge
+      linarith [norm_nonneg z.1]
+    linarith only [hb.2,h,hCN]
+  · have hp := physicalRadius_actual_pos a b z.1 _ (exhaustionStop_agrees a z n b _ le_rfl).1
+    simp only [exhaustionStop] at hp
+    rw [h] at hp
+    exact lt_irrefl _ hp
+  · simp only [zero_mul] at h
+    linarith only [hb.1,h,hη]
+
+lemma exhaustionStop_measurable (a : ℝ) (z : UpperPlane) (n : ℕ) :
+    Measurable (exhaustionStop a z n) := by
+  apply measurable_of_Iic
+  intro t
+  simpa only [preimage, mem_Iic, WithTop.coe_le_coe] using
+    rawFiltration.le t _ ((exhaustionStop_stopping a z n).measurableSet_le t)
+
+ 
+
+noncomputable def killedObservable (a : ℝ) (z : UpperPlane) (F : State → ℝ)
+    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : ℝ :=
+  if alive a b.1 t z.1 then F (actualState a b.1 t z.1) else 0
+
+noncomputable def localizedObservable (a : ℝ) (z : UpperPlane) (F : State → ℝ)
+    (n : ℕ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : ℝ :=
+  if t ≤ exhaustionStop a z n b then
+    F (localState a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1 t b) else 0
+
+lemma localizedObservable_joint_measurable (a : ℝ) (z : UpperPlane)
+    (F : State → ℝ) (hF : Measurable F) (n : ℕ) :
+    Measurable (fun p : ℝ≥0 × ExactSLEGenerator.Driver => localizedObservable a z F n p.1 p.2) := by
+  exact Measurable.ite
+    (measurableSet_le measurable_fst ((exhaustionStop_measurable a z n).comp measurable_snd))
+    (hF.comp (localState_parameter_joint_measurable a (fun _ => exhaustionHeight z n)
+      (fun _ => exhaustionHeight_pos z n) (fun _ => z.1) id
+      measurable_const measurable_const measurable_id)) measurable_const
+
+lemma localizedObservable_tendsto (a : ℝ) (z : UpperPlane) (F : State → ℝ)
+    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    Tendsto (fun n => localizedObservable a z F n t b) atTop (𝓝 (killedObservable a z F t b)) := by
+  by_cases ht : alive a b.1 t z.1
+  · apply tendsto_const_nhds.congr'
+    filter_upwards [exhaustionStop_eventually a z b t ht] with n hn
+    simp only [localizedObservable,ite_eq_left hn,killedObservable,ite_eq_left ht,
+      (exhaustionStop_agrees a z n b t hn).2]
+  · have hn (n : ℕ) : ¬t ≤ exhaustionStop a z n b :=
+      fun hn => ht (exhaustionStop_agrees a z n b t hn).1
+    simpa only [localizedObservable,ite_eq_right (hn _),killedObservable,ite_eq_right ht] using
+      (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))
+
+lemma killedObservable_joint_measurable (a : ℝ) (z : UpperPlane)
+    (F : State → ℝ) (hF : Measurable F) :
+    Measurable (fun p : ℝ≥0 × ExactSLEGenerator.Driver => killedObservable a z F p.1 p.2) :=
+  measurable_of_tendsto_metrizable (localizedObservable_joint_measurable a z F hF)
+    (tendsto_pi_nhds.mpr (fun p => localizedObservable_tendsto a z F p.1 p.2))
+
+lemma killedObservable_nonneg (a : ℝ) (z : UpperPlane)
+    (F : State → ℝ) (hF : ∀ v, 0 < Y v → 0 ≤ F v)
+    (t : ℝ≥0) (b : ExactSLEGenerator.Driver) : 0 ≤ killedObservable a z F t b := by
+  unfold killedObservable
+  split_ifs with ht
+  · obtain ⟨f,hf⟩ := ht
+    apply hF
+    have he : loewnerMap a b.1 t z.1 = f t := by
+      simpa only [Real.toNNReal_coe] using
+        loewnerMap_eq_solution_time hf (right_mem_Icc.mpr t.coe_nonneg)
+    simpa only [actualState,Y_apply,he,Complex.sub_im,Complex.ofReal_im,
+      sub_zero] using (hf.2 (t : ℝ) (right_mem_Icc.mpr t.coe_nonneg)).1
+  · exact le_rfl
+
+lemma greenLifetime_measurable (a : ℝ) :
+    Measurable (fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v) := by
+  unfold physicalGreen logGreen lifetimeTest Q X Y J
+  fun_prop
+
+lemma lifetimeCost_measurable (a : ℝ) : Measurable (lifetimeCost a) := by
+  unfold lifetimeCost physicalGreen logGreen Q X Y J
+  fun_prop
+
+ 
+
+theorem global_lifetime_test_budget (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) (t : ℝ≥0) :
+    (∫⁻ b, ENNReal.ofReal (killedObservable a z
+      (fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v) t b) ∂W) ≤
+      ENNReal.ofReal (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) := by
+  let F : State → ℝ := fun v => physicalGreen a v * lifetimeTest (lifetimeExponent a) v
+  have hpos (v : State) (hv : 0 < Y v) : 0 ≤ F v :=
+    mul_nonneg (Real.exp_pos _).le (lifetimeTest_nonneg _ _ hv.le)
+  have hbound (n : ℕ) : (∫⁻ b, ENNReal.ofReal (localizedObservable a z F n t b) ∂W) ≤
+      ENNReal.ofReal (F (z.1,0)) := by
+    let ε := exhaustionHeight z n
+    let V := localState a ε (exhaustionHeight_pos z n) z.1
+    let S : ExactSLEGenerator.Driver → ℝ≥0 := fun b => min (exhaustionStop a z n b) t
+    have hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)) := by
+      simpa only [S, WithTop.coe_min] using (exhaustionStop_stopping a z n).min_const t
+    have hsafe (b : ExactSLEGenerator.Driver) (u : ℝ≥0) (hu : u ≤ S b) :
+        ε ≤ Y (V u b) ∧ ‖V u b‖ ≤ exhaustionNorm z n := by
+      have h := regionStop_safe a ε (exhaustionHeight_pos z n) z.1
+        (exhaustionNorm z n) 0 0 n (exhaustionRegion_initial z n) b u
+        (hu.trans (min_le_left _ _))
+      exact ⟨h.1,h.2.1⟩
+    have hb := local_lifetime_budget W hW a ε ha (exhaustionHeight_pos z n)
+      z.1 (exhaustionNorm z n) n S hS
+      (fun b => (min_le_left _ _).trans (exhaustionStop_le a z n b)) hsafe
+    calc
+      _ ≤ ∫⁻ b, ENNReal.ofReal (F (V (S b) b)) ∂W := by
+        apply lintegral_mono
+        intro b
+        apply ENNReal.ofReal_le_ofReal
+        by_cases ht : t ≤ exhaustionStop a z n b
+        · simp only [localizedObservable,ite_eq_left ht,S,min_eq_right ht,F,V,ε,le_refl]
+        · rw [localizedObservable,ite_eq_right ht]
+          exact hpos _ ((exhaustionHeight_pos z n).trans_le (hsafe b (S b) le_rfl).1)
+      _ = ENNReal.ofReal (∫ b, F (V (S b) b) ∂W) :=
+        (ofReal_integral_eq_lintegral_ofReal hb.1 (Eventually.of_forall (fun b =>
+          hpos _ ((exhaustionHeight_pos z n).trans_le (hsafe b (S b) le_rfl).1)))).symm
+      _ ≤ _ := ENNReal.ofReal_le_ofReal hb.2.1
+  have hlim (b : ExactSLEGenerator.Driver) :
+      liminf (fun n => ENNReal.ofReal (localizedObservable a z F n t b)) atTop =
+        ENNReal.ofReal (killedObservable a z F t b) :=
+    (ENNReal.tendsto_ofReal (localizedObservable_tendsto a z F t b)).liminf_eq
+  change (∫⁻ b, ENNReal.ofReal (killedObservable a z F t b) ∂W) ≤ _
+  simp_rw [← hlim]
+  exact (lintegral_liminf_le (fun n =>
+    ((localizedObservable_joint_measurable a z F (greenLifetime_measurable a) n).comp
+      (measurable_const.prodMk measurable_id)).ennreal_ofReal)).trans
+    (liminf_le_of_frequently_le (Eventually.of_forall hbound).frequently)
+
+lemma localizedObservable_occupation (a : ℝ) (z : UpperPlane) (F : State → ℝ)
+    (n : ℕ) (b : ExactSLEGenerator.Driver) :
+    (∫⁻ u in Ioi (0 : ℝ), ENNReal.ofReal (localizedObservable a z F n u.toNNReal b)) =
+      ∫⁻ u in Ioc (0 : ℝ) (exhaustionStop a z n b : ℝ),
+        ENNReal.ofReal (F (localState a (exhaustionHeight z n)
+          (exhaustionHeight_pos z n) z.1 u.toNNReal b)) := by
+  have he (u : ℝ) : ENNReal.ofReal (localizedObservable a z F n u.toNNReal b) =
+      (Iic (exhaustionStop a z n b : ℝ)).indicator
+        (fun u : ℝ => ENNReal.ofReal (F (localState a (exhaustionHeight z n)
+          (exhaustionHeight_pos z n) z.1 u.toNNReal b))) u := by
+    by_cases hu : u ≤ (exhaustionStop a z n b : ℝ)
+    · simp only [localizedObservable,Real.toNNReal_le_iff_le_coe,indicator,mem_Iic,hu,ite_true]
+    · simp only [localizedObservable,Real.toNNReal_le_iff_le_coe,indicator,mem_Iic,hu,ite_false,
+        ENNReal.ofReal_zero]
+  simp_rw [he]
+  rw [setLIntegral_indicator measurableSet_Iic, inter_comm, Ioi_inter_Iic]
+
+ 
+
+theorem global_lifetime_occupation_budget (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (z : UpperPlane) :
+    (∫⁻ b, ∫⁻ u in Ioi (0 : ℝ),
+      ENNReal.ofReal (killedObservable a z (lifetimeCost a) u.toNNReal b) ∂volume ∂W) ≤
+      ENNReal.ofReal (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) := by
+  let G : ℕ → ExactSLEGenerator.Driver × ℝ → ℝ≥0∞ := fun n p =>
+    ENNReal.ofReal (localizedObservable a z (lifetimeCost a) n p.2.toNNReal p.1)
+  let H : ExactSLEGenerator.Driver × ℝ → ℝ≥0∞ := fun p =>
+    ENNReal.ofReal (killedObservable a z (lifetimeCost a) p.2.toNNReal p.1)
+  have hG (n : ℕ) : Measurable (G n) :=
+    ((localizedObservable_joint_measurable a z (lifetimeCost a) (lifetimeCost_measurable a) n).comp
+      ((measurable_real_toNNReal.comp measurable_snd).prodMk measurable_fst)).ennreal_ofReal
+  have hH : Measurable H :=
+    ((killedObservable_joint_measurable a z (lifetimeCost a) (lifetimeCost_measurable a)).comp
+      ((measurable_real_toNNReal.comp measurable_snd).prodMk measurable_fst)).ennreal_ofReal
+  have hbound (n : ℕ) :
+      (∫⁻ p, G n p ∂W.prod (volume.restrict (Ioi (0 : ℝ)))) ≤
+        ENNReal.ofReal (physicalGreen a (z.1,0) * lifetimeTest (lifetimeExponent a) (z.1,0)) := by
+    rw [lintegral_prod _ (hG n).aemeasurable]
+    change (∫⁻ b, ∫⁻ u in Ioi (0 : ℝ),
+      ENNReal.ofReal (localizedObservable a z (lifetimeCost a) n u.toNNReal b) ∂volume ∂W) ≤ _
+    simp_rw [localizedObservable_occupation]
+    apply (local_lifetime_budget W hW a (exhaustionHeight z n) ha (exhaustionHeight_pos z n)
+      z.1 (exhaustionNorm z n) n (exhaustionStop a z n) (exhaustionStop_stopping a z n)
+      (exhaustionStop_le a z n) ?_).2.2
+    intro b u hu
+    have h := regionStop_safe a (exhaustionHeight z n) (exhaustionHeight_pos z n) z.1
+      (exhaustionNorm z n) 0 0 n (exhaustionRegion_initial z n) b u hu
+    exact ⟨h.1,h.2.1⟩
+  have hlim (p : ExactSLEGenerator.Driver × ℝ) :
+      liminf (fun n => G n p) atTop = H p :=
+    (ENNReal.tendsto_ofReal (localizedObservable_tendsto a z (lifetimeCost a) p.2.toNNReal p.1)).liminf_eq
+  rw [← lintegral_prod H hH.aemeasurable]
+  simp_rw [← hlim]
+  exact (lintegral_liminf_le hG).trans
+    (liminf_le_of_frequently_le (Eventually.of_forall hbound).frequently)
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialRestartRaw.lean b/lean/OAI/Probability/SLEGauge/RadialRestartRaw.lean
index 4c6cd3016989142eacb1ec4d84362327e41e9bb3..66cd0ddd9dcd6d7bb6c10afd011e5dd2ba2da459 100644
--- a/lean/OAI/Probability/SLEGauge/RadialRestartRaw.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialRestartRaw.lean
@@ -1,495 +1,495 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RawRestartTests
-import OAI.Probability.SLEGauge.FiniteAngleEigen
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Classical NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-
-lemma radialState_raw_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) :
-    Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
-      (fun b : GDriver => radialState a b.1 z σ) := by
-  let S := fun b : GDriver => radialTime a b.1 z σ
-  have hS := radialTime_stopping a (by linarith) z hz σ hσ hσz
-  have hD := stoppedData_raw_measurable hS
-  have hm : Measurable[hS.measurableSpace] (fun b : GDriver =>
-      killedState a (stoppedData S b).1 (stoppedData S b).2 z) :=
-    (killedState_time_joint_measurable a ha).comp
-      (hD.fst.prodMk (hD.snd.prodMk (measurable_const (a := (⟨z,hz⟩ : UpperPlane)))))
-  have he (b : GDriver) : radialState a b.1 z σ =
-      if S b=∞ then 0 else killedState a (stoppedData S b).1 (stoppedData S b).2 z := by
-    by_cases hb : S b=∞
-    · exact (ite_eq_left hb).trans (ite_eq_left hb).symm
-    · have hs := (radialTime_attained a (by linarith) z hz σ hσ hσz b hb).1
-      have hc := pastDriver_eq (S b).toNNReal b
-      have hp := (alive_driver_congr hc).mpr hs
-      rw [radialState,ite_eq_right hb,ite_eq_right hb]
-      change actualState a b.1 (S b).toNNReal z =
-        if alive a (pastDriver (S b).toNNReal b).1 (S b).toNNReal z then
-          actualState a (pastDriver (S b).toNNReal b).1 (S b).toNNReal z else 0
-      rw [ite_eq_left hp,actualState_driver_congr hc]
-  simp_rw [he]
-  exact Measurable.ite (hS.measurable (measurableSet_singleton ∞)) measurable_const hm
-
- 
-
-def radialPoint (a : ℝ) (z : ℂ) (σ : ℝ) (b : GDriver) : UpperPlane :=
-  if h : 0<(radialState a b.1 z σ).1.im then ⟨(radialState a b.1 z σ).1,h⟩
-  else ⟨Complex.I,by simp⟩
-
-lemma radialPoint_raw_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) :
-    Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
-      (radialPoint a z σ) := by
-  have hm := radialState_raw_measurable a ha z hz σ hσ hσz
-  suffices Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
-      (fun b : GDriver => (radialPoint a z σ b : ℂ)) from this.subtype_mk
-  simp only [radialPoint,apply_dite,dite_eq_ite]
-  exact hm.fst.ite (measurableSet_lt measurable_const hm.fst.im) measurable_const
-
-lemma radialPoint_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (b : GDriver)
-    (hb : radialTime a b.1 z σ≠∞) :
-    (radialPoint a z σ b).1=(actualState a b.1 (radialTime a b.1 z σ).toNNReal z).1 := by
-  have hs := (radialTime_attained a ha z hz σ hσ hσz b hb).1
-  have hp : 0<(radialState a b.1 z σ).1.im := by
-    simpa only [radialState,ite_eq_right hb,Y_apply] using LoewnerScaling.Y_actual_pos hs
-  rw [radialPoint,dite_eq_left hp]
-  simp only [radialState,ite_eq_right hb]
-
- 
-def radiusDensity (a : ℝ) (z : ℂ) (σ : ℝ) (b : GDriver) : ℝ≥0∞ :=
-  ENNReal.ofReal (radialWeight a b.1 z σ / physicalGreen a (z,0))
-
-lemma radiusDensity_raw_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) :
-    Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
-      (radiusDensity a z σ) := by
-  have hS := radialTime_stopping a (by linarith) z hz σ hσ hσz
-  have hm : Measurable[hS.measurableSpace] (fun b : GDriver => radialWeight a b.1 z σ) := by
-    have he (b : GDriver) : radialWeight a b.1 z σ =
-        if radialTime a b.1 z σ=∞ then 0 else physicalGreen a (radialState a b.1 z σ) := by
-      unfold radialWeight radialState
-      split_ifs <;> rfl
-    simp_rw [he]
-    apply Measurable.ite (hS.measurable (measurableSet_singleton ∞)) measurable_const
-    exact (show Measurable (physicalGreen a) by unfold physicalGreen logGreen Q; fun_prop).comp
-      (radialState_raw_measurable a ha z hz σ hσ hσz)
-  exact (hm.div_const _).ennreal_ofReal
-
-lemma radiusDensity_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) : Measurable (radiusDensity a z σ) :=
-  (radiusDensity_raw_measurable a ha z hz σ hσ hσz).mono
-    (radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace_le le_rfl
-
-lemma joint_radiusDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) :
-    Measurable (fun p : GDriver×UpperPlane => radiusDensity a p.2.1 (p.2.1.im*q) p.1) := by
-  apply Measurable.ennreal_ofReal
-  apply (joint_radialWeight_measurable a ha q hq hq1).div
-  unfold physicalGreen logGreen Q
-  fun_prop
-
-lemma radiusDensity_eq_zero (a : ℝ) (z : ℂ) (σ : ℝ) (b : GDriver)
-    (hb : radialTime a b.1 z σ=∞) : radiusDensity a z σ b=0 := by
-  simp only [radiusDensity,radialWeight,ite_eq_left hb,zero_div,ENNReal.ofReal_zero]
-
-lemma radiusDensity_restart (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im) (b : GDriver) :
-    radiusDensity a z ρ b = radiusDensity a z σ b *
-      radiusDensity a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
-        (restartAt (fun c => radialTime a c.1 z σ) b) := by
-  have ha0 : 0<a := by linarith
-  by_cases hb : radialTime a b.1 z σ=∞
-  · have hf : radialTime a b.1 z ρ=∞ := top_le_iff.mp
-      (hb ▸ radialTime_antitone a ha0 z hz ρ σ hρ hρσ hσ b)
-    rw [radiusDensity_eq_zero a z σ b hb,radiusDensity_eq_zero a z ρ b hf,zero_mul]
-  · rw [radialPoint_eq a ha0 z hz σ (hρ.trans_le hρσ) hσ b hb]
-    unfold radiusDensity restartAt
-    rw [radialWeight_restart a ha0 z hz σ ρ hρ hρσ hσ b hb]
-    have hp : 0≤radialWeight a b.1 z σ / physicalGreen a (z,0) :=
-      div_nonneg (radialWeight_bound a ha z hz σ (hρ.trans_le hρσ) hσ b).1 (Real.exp_pos _).le
-    rw [←ENNReal.ofReal_mul hp]
-    congr 1
-    ring
-
-lemma finiteAngle_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im) (b : GDriver)
-    (hf : radialTime a b.1 z ρ≠∞) :
-    Jacobi.finiteAngle a z ρ b =
-      Jacobi.finiteAngle a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
-        (restartAt (fun c => radialTime a c.1 z σ) b) := by
-  have hb := ne_top_of_le_ne_top hf (radialTime_antitone a ha z hz ρ σ hρ hρσ hσ b)
-  let s := (radialTime a b.1 z σ).toNNReal
-  let w := (actualState a b.1 s z).1
-  let τ := radialTime a (restart s b).1 w (w.im*(ρ/σ))
-  have hτ : τ≠∞ := (radialTime_restart_top_iff a ha z hz σ ρ hρ hρσ hσ b hb).not.mpr hf
-  have ht : s+τ.toNNReal=(radialTime a b.1 z ρ).toNNReal := by
-    have hh := congrArg ENNReal.toNNReal (radialTime_restart a ha z hz σ ρ hρ hρσ hσ b hb)
-    change ((s:ℝ≥0∞)+τ).toNNReal=(radialTime a b.1 z ρ).toNNReal at hh
-    simpa only [ENNReal.toNNReal_add ENNReal.coe_ne_top hτ,ENNReal.toNNReal_coe] using hh
-  have hs := (radialTime_attained a ha z hz ρ hρ (hρσ.trans hσ) b hf).1
-  rw [radialPoint_eq a ha z hz σ (hρ.trans_le hρσ) hσ b hb]
-  apply Subtype.ext
-  change physicalTheta (radialState a b.1 z ρ) =
-    physicalTheta (radialState a (restart s b).1 w (w.im*(ρ/σ)))
-  rw [radialState,ite_eq_right hf,radialState,ite_eq_right hτ,←ht,
-    actualState_restart b.1.continuous (ht ▸ hs)]
-  rfl
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-
-lemma setLIntegral_eq_inter_of_zero {Ω : Type*} [MeasurableSpace Ω]
-    (μ : Measure Ω) (A B : Set Ω) (hB : MeasurableSet B) (f : Ω→ℝ≥0∞)
-    (hf : ∀ b, b∉B → f b=0) : (∫⁻b in A,f b∂μ)=∫⁻b in A∩B,f b∂μ := by
-  have he : f=B.indicator f := by
-    funext b
-    by_cases hb : b∈B
-    · simp only [indicator_of_mem hb]
-    · simp only [indicator_of_notMem hb,hf b hb]
-  calc
-    _ = ∫⁻b in A,B.indicator f b∂μ := by rw [←he]
-    _ = ∫⁻b in A∩B,f b∂μ := by
-      rw [lintegral_indicator hB,Measure.restrict_restrict hB,inter_comm]
-
-lemma radial_hit_ae (W : Measure GDriver) (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
-    ∀ᵐ b∂radialLaw W a z ρ,radialTime a b.1 z ρ≠∞ := by
-  rw [radialLaw]
-  apply (ae_withDensity_iff (radiusDensity_measurable a ha z hz ρ hρ hr)).mpr
-  exact Eventually.of_forall (fun b hb hf => hb (radiusDensity_eq_zero a z ρ b hf))
-
- 
-
-theorem radialLaw_restart (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ (hρ.trans_le hρσ) hσ).measurableSpace] A)
-    (F : UpperPlane×GDriver→ℝ≥0∞) (hF : Measurable F) :
-    (∫⁻b in A,F (radialPoint a z σ b,restartAt (fun c => radialTime a c.1 z σ) b)
-      ∂radialLaw W a z ρ) =
-    ∫⁻b in A,∫⁻c,F (radialPoint a z σ b,c)
-      ∂radialLaw W a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
-      ∂radialLaw W a z σ := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hσ0 := hρ.trans_le hρσ
-  have hS := radialTime_stopping a (by linarith) z hz σ hσ0 hσ
-  let S : GDriver→ℝ≥0∞ := fun b => radialTime a b.1 z σ
-  let Z := radialPoint a z σ
-  let D : GDriver→ℝ≥0∞×UpperPlane := fun b => (radiusDensity a z σ b,Z b)
-  have hD : Measurable[hS.measurableSpace] D :=
-    (radiusDensity_raw_measurable a ha z hz σ hσ0 hσ).prodMk
-      (radialPoint_raw_measurable a ha z hz σ hσ0 hσ)
-  have hZ : Measurable Z := (radialPoint_raw_measurable a ha z hz σ hσ0 hσ).mono
-    hS.measurableSpace_le le_rfl
-  have hY := measurable_restartAt hS
-  have hA' : MeasurableSet A := hS.measurableSpace_le _ hA
-  have hhit : MeasurableSet {b : GDriver | S b≠∞} :=
-    (hS.measurable' (measurableSet_singleton ∞)).compl
-  have hq : 0<ρ/σ := div_pos hρ hσ0
-  have hq1 : ρ/σ≤1 := (div_le_one hσ0).mpr hρσ
-  let K : UpperPlane×GDriver→ℝ≥0∞ := fun p =>
-    radiusDensity a p.1.1 (p.1.1.im*(ρ/σ)) p.2 * F p
-  have hK : Measurable K :=
-    ((joint_radiusDensity_measurable a ha (ρ/σ) hq hq1).comp measurable_swap).mul hF
-  let H : (ℝ≥0∞×UpperPlane)×GDriver→ℝ≥0∞ := fun p => p.1.1*K (p.1.2,p.2)
-  have hH : Measurable H := measurable_fst.fst.mul
-    (hK.comp (measurable_fst.snd.prodMk measurable_snd))
-  have he (b : GDriver) : radiusDensity a z ρ b * F (Z b,restartAt S b)=H (D b,restartAt S b) := by
-    rw [radiusDensity_restart a ha z hz σ ρ hρ hρσ hσ]
-    exact mul_assoc _ _ _
-  have hi (b : GDriver) : (∫⁻c,H (D b,c)∂W)=radiusDensity a z σ b *
-      ∫⁻c,F (Z b,c)∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ)) := by
-    have hm : Measurable (fun c : GDriver => F (Z b,c)) :=
-      hF.comp (measurable_const.prodMk measurable_id)
-    have he' : (∫⁻c,F (Z b,c)∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ)))=
-        ∫⁻c,K (Z b,c)∂W :=
-      lintegral_withDensity_eq_lintegral_mul W
-        (radiusDensity_measurable a ha (Z b).1 (Z b).2 _ (mul_pos (Z b).2 hq)
-          (mul_le_of_le_one_right (Z b).2.le hq1)) hm
-    rw [he']
-    exact lintegral_const_mul _ (hK.comp (measurable_const.prodMk measurable_id))
-  have hkm : Measurable (fun b => ∫⁻c,F (Z b,c)
-      ∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ))) := by
-    have he' (b : GDriver) : (∫⁻c,F (Z b,c)
-        ∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ)))=∫⁻c,K (Z b,c)∂W := by
-      exact lintegral_withDensity_eq_lintegral_mul W
-        (radiusDensity_measurable a ha (Z b).1 (Z b).2 _ (mul_pos (Z b).2 hq)
-          (mul_le_of_le_one_right (Z b).2.le hq1)) (hF.comp (measurable_const.prodMk measurable_id))
-    simp_rw [he']
-    exact hK.lintegral_prod_right'.comp hZ
-  change (∫⁻b in A,F (Z b,restartAt S b)∂W.withDensity (radiusDensity a z ρ)) =
-    ∫⁻b in A,∫⁻c,F (Z b,c)∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ))
-      ∂W.withDensity (radiusDensity a z σ)
-  have hfm : Measurable (fun b : GDriver => F (Z b,restartAt S b)) := hF.comp (hZ.prodMk hY)
-  rw [setLIntegral_withDensity_eq_setLIntegral_mul W (radiusDensity_measurable a ha z hz ρ hρ (hρσ.trans hσ))
-      hfm hA',
-    setLIntegral_withDensity_eq_setLIntegral_mul W (radiusDensity_measurable a ha z hz σ hσ0 hσ) hkm hA']
-  simp only [Pi.mul_apply,he,←hi]
-  calc
-    _ = ∫⁻b in A∩{b | S b≠∞},H (D b,restartAt S b)∂W :=
-      setLIntegral_eq_inter_of_zero W A _ hhit _ (fun b hb => by
-        have hb' : S b=∞ := not_not.mp hb
-        simp only [H,D,radiusDensity_eq_zero a z σ b hb',zero_mul])
-    _ = ∫⁻b in A∩{b | S b≠∞},∫⁻c,H (D b,c)∂W ∂W :=
-      lintegral_data_restartAt_on hW hS hD hA hH
-    _ = ∫⁻b in A,∫⁻c,H (D b,c)∂W ∂W :=
-      (setLIntegral_eq_inter_of_zero W A _ hhit _ (fun b hb => by
-        have hb' : S b=∞ := not_not.mp hb
-        simp only [H,D,radiusDensity_eq_zero a z σ b hb',zero_mul,lintegral_zero])).symm
-
- 
-theorem finiteAngle_conditional (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
-    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
-    (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ (hρ.trans_le hρσ) hσ).measurableSpace] A)
-    (f : CenteredMoments.Angle→ℝ≥0∞) (hf : Measurable f) :
-    (∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ) =
-    ∫⁻b in A,∫⁻c,f (Jacobi.finiteAngle a (radialPoint a z σ b).1
-      ((radialPoint a z σ b).1.im*(ρ/σ)) c)
-      ∂radialLaw W a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
-      ∂radialLaw W a z σ := by
-  have hq : 0<ρ/σ := div_pos hρ (hρ.trans_le hρσ)
-  have hq1 : ρ/σ≤1 := (div_le_one (hρ.trans_le hρσ)).mpr hρσ
-  let F : UpperPlane×GDriver→ℝ≥0∞ := fun p => f (Jacobi.finiteAngle a p.1.1 (p.1.1.im*(ρ/σ)) p.2)
-  have hF : Measurable F := hf.comp (Jacobi.angleState_measurable.comp
-    ((joint_radialState_measurable a (by linarith) (ρ/σ) hq hq1).comp measurable_swap))
-  rw [←radialLaw_restart W hW a ha z hz σ ρ hρ hρσ hσ A hA F hF]
-  apply lintegral_congr_ae
-  filter_upwards [ae_restrict_of_ae (radial_hit_ae W a ha z hz ρ hρ (hρσ.trans hσ))] with b hb
-  exact congrArg f (finiteAngle_restart a (by linarith) z hz σ ρ hρ hρσ hσ b hb)
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.Jacobi
-open CenteredMoments ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
-
-lemma sin_physicalTheta (v : State) (hy : 0 < Y v) :
-    Real.sin (physicalTheta v) = Y v / ‖v.1‖ := by
-  have hn : 0 < ‖v.1‖ := hy.trans_le (Complex.im_le_norm _)
-  have hs := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2
-  have hc := physicalTheta_cot v
-  rw [Real.cot_eq_cos_div_sin, div_eq_div_iff hs.ne' hy.ne'] at hc
-  have he : (X v)^2 + (Y v)^2 = ‖v.1‖^2 := by
-    simpa only [X_apply, Y_apply, Complex.normSq_apply, pow_two] using Complex.normSq_eq_norm_sq v.1
-  have hid := Real.sin_sq_add_cos_sq (physicalTheta v)
-  have heq : (Real.sin (physicalTheta v) * ‖v.1‖)^2 = (Y v)^2 := by
-    nlinarith [sq_nonneg (Real.sin (physicalTheta v) * X v - Real.cos (physicalTheta v) * Y v),
-      congrArg (fun x : ℝ => x^2) hc,
-      congrArg (fun x : ℝ => x * (Y v)^2) hid,
-      congrArg (fun x : ℝ => x * (Real.sin (physicalTheta v))^2) he]
-  apply (eq_div_iff hn.ne').mpr
-  nlinarith [mul_pos hs hn]
-
- 
-lemma radialWeight_inverse_sine (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
-    (hb : radialTime a b.1 z ρ ≠ ∞) :
-    radialWeight a b.1 z ρ * (Real.sin (finiteAngle a z ρ b : ℝ)) ^ (-p a) =
-      ρ ^ (-alpha a) := by
-  have ht := radialTime_attained a ha z hz ρ hρ hr b hb
-  have hy := LoewnerScaling.Y_actual_pos ht.1
-  have hs := Real.sin_pos_of_pos_of_lt_pi (finiteAngle a z ρ b).2.1
-    (finiteAngle a z ρ b).2.2
-  have hθ : Real.sin (finiteAngle a z ρ b : ℝ) =
-      Y (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z) /
-        ‖(actualState a b.1 (radialTime a b.1 z ρ).toNNReal z).1‖ := by
-    simpa only [finiteAngle, angleState, radialState, ite_eq_right hb] using sin_physicalTheta _ hy
-  rw [radialWeight, ite_eq_right hb, physicalGreen_eq_radius_sine a _ hy, ht.2, ← hθ,
-    mul_assoc, ← Real.rpow_add hs]
-  simp
-
-lemma inverse_radial_density (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver) :
-    (ρ^(alpha a) * physicalGreen a (z,0)) *
-        ((radialWeight a b.1 z ρ / physicalGreen a (z,0)) *
-          (Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) =
-      if radialTime a b.1 z ρ ≠ ∞ then 1 else 0 := by
-  by_cases hb : radialTime a b.1 z ρ ≠ ∞
-  · rw [ite_eq_left hb]
-    have hp : physicalGreen a (z,0) ≠ 0 := (Real.exp_pos _).ne'
-    calc
-      _ = ρ^(alpha a) * (radialWeight a b.1 z ρ *
-          (Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) := by field_simp
-      _ = 1 := by
-        rw [radialWeight_inverse_sine a ha z hz ρ hρ hr b hb, ← Real.rpow_add hρ]
-        simp
-  · simp only [ite_eq_right hb, radialWeight, ite_eq_left (not_not.mp hb), zero_div, zero_mul, mul_zero]
-
- 
-
-theorem radial_change_law (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (g : ExactSLEGenerator.Driver → ℝ≥0∞) (hg : Measurable g) :
-    (∫⁻b in {b | radialTime a b.1 z ρ ≠ ∞}, g b ∂W) =
-      ENNReal.ofReal (ρ^(alpha a) * physicalGreen a (z,0)) *
-        ∫⁻b, ENNReal.ofReal ((Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) * g b
-          ∂radialLaw W a z ρ := by
-  have ha0 : 0 < a := by linarith
-  have hhit : MeasurableSet {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ ≠ ∞} :=
-    ((radialTime_stopping a ha0 z hz ρ hρ hr).measurable' (measurableSet_singleton ∞)).compl
-  have hs : Measurable (fun b : ExactSLEGenerator.Driver =>
-      ENNReal.ofReal ((Real.sin (finiteAngle a z ρ b : ℝ))^(-p a))) := by
-    exact (show Measurable (fun θ : Angle => ENNReal.ofReal ((Real.sin θ)^(-p a))) by fun_prop).comp
-      (finiteAngle_measurable a ha0 z hz ρ hρ hr)
-  have hd : Measurable (fun b : ExactSLEGenerator.Driver =>
-      ENNReal.ofReal (radialWeight a b.1 z ρ / physicalGreen a (z,0))) :=
-    ((radialWeight_measurable a ha z hz ρ hρ hr).div_const _).ennreal_ofReal
-  have hf : Measurable (fun b =>
-      ENNReal.ofReal ((Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) * g b) := hs.mul hg
-  rw [radialLaw, lintegral_withDensity_eq_lintegral_mul W hd hf, ← lintegral_const_mul _ (hd.mul hf)]
-  rw [← lintegral_indicator hhit]
-  apply lintegral_congr
-  intro b
-  have hpos : 0 ≤ (radialWeight a b.1 z ρ / physicalGreen a (z,0)) :=
-    div_nonneg (radialWeight_bound a ha z hz ρ hρ hr b).1 (Real.exp_pos _).le
-  have hc : 0 ≤ ρ^(alpha a) * physicalGreen a (z,0) :=
-    mul_nonneg (Real.rpow_nonneg hρ.le _) (Real.exp_pos _).le
-  have he := congrArg ENNReal.ofReal (inverse_radial_density a ha0 z hz ρ hρ hr b)
-  rw [ENNReal.ofReal_mul hc, ENNReal.ofReal_mul hpos] at he
-  simp only [Pi.mul_apply]
-  rw [← mul_assoc _ _ (g b), ← mul_assoc _ _ (g b), he]
-  by_cases hb : radialTime a b.1 z ρ ≠ ∞ <;> simp [hb]
-
-lemma radial_hit_probability_eq (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (s : ℝ) (hs : 0 ≤ s) :
-    W {b | radialTime a b.1 z (levelRadius a (2*z.im) s) ≠ ∞} =
-      ENNReal.ofReal ((levelRadius a (2*z.im) s)^(alpha a) * physicalGreen a (z,0)) *
-        ∫⁻θ : Angle, ENNReal.ofReal ((Real.sin θ)^(-p a)) ∂finiteAngleLaw W a z s := by
-  have hρ := levelRadius_pos a (2*z.im) s (by positivity : 0 < 2*z.im)
-  have hr := levelRadius_initial_le a (by linarith) z hz s hs
-  have he := radial_change_law W a ha z hz _ hρ hr (fun _ => 1) measurable_const
-  simp only [lintegral_one, Measure.restrict_apply_univ, mul_one] at he
-  rw [finiteAngleLaw, lintegral_map (by fun_prop)
-    (finiteAngle_measurable a (by linarith) z hz _ hρ hr)]
-  exact he
-
-lemma physicalGreen_initial_bound (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im) :
-    physicalGreen a (z,0) ≤ z.im^(-alpha a) := by
-  have hr : physicalRadius (z,0) = z.im := by simp [physicalRadius, Y_apply, J_apply]
-  rw [Real.rpow_def_of_pos hz]
-  simpa only [mul_comm] using physicalGreen_bound a z.im ha hz (z,0) hz hr.ge
-
- 
-
-theorem radial_hit_probability_long (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (s : ℝ) (hs : mixingTime (4*a) ≤ s) :
-    W {b | radialTime a b.1 z (levelRadius a (2*z.im) s) ≠ ∞} ≤
-      ENNReal.ofReal ((levelRadius a (2*z.im) s / z.im)^(alpha a)) *
-        (2 * ∫⁻θ : Angle, ENNReal.ofReal ((Real.sin θ)^(-p a)) ∂angleLaw a) := by
-  have hs0 : 0 ≤ s := le_trans (by unfold mixingTime majorantBase; positivity) hs
-  have hρ := levelRadius_pos a (2*z.im) s (by positivity : 0 < 2*z.im)
-  rw [radial_hit_probability_eq W a ha z hz s hs0]
-  apply mul_le_mul
-  · apply ENNReal.ofReal_le_ofReal
-    rw [Real.div_rpow hρ.le hz.le, div_eq_mul_inv, ← Real.rpow_neg hz.le]
-    exact mul_le_mul_of_nonneg_left (physicalGreen_initial_bound a ha z hz)
-      (Real.rpow_nonneg hρ.le _)
-  · exact (finiteAngleLaw_integral_bounds W hW a ha z hz s hs _ (by fun_prop)).2
-  all_goals positivity
-
- 
-def visitConstant (a : ℝ) : ℝ≥0∞ :=
-  2 * (∫⁻θ : Angle, ENNReal.ofReal ((Real.sin θ)^(-p a)) ∂angleLaw a) +
-    ENNReal.ofReal (Real.exp (2*a*mixingTime (4*a)*alpha a))
-
-lemma visitConstant_lt_top (a : ℝ) : visitConstant a < ∞ :=
-  ENNReal.add_lt_top.mpr ⟨ENNReal.mul_lt_top (by norm_num) (angle_inverse_power_finite a),
-    ENNReal.ofReal_lt_top⟩
-
-lemma visitConstant_pos (a : ℝ) : 0 < visitConstant a :=
-  lt_of_lt_of_le (ENNReal.ofReal_pos.mpr (Real.exp_pos _)) (le_add_left le_rfl)
-
-lemma levelRadius_ratio (a : ℝ) (z : ℂ) (hz : 0 < z.im) (s : ℝ) :
-    levelRadius a (2*z.im) s / z.im = Real.exp (-2*a*s) := by
-  unfold levelRadius
-  field_simp
-
- 
-
-theorem radial_hit_probability (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im) (s : ℝ) (_hs : 0 ≤ s) :
-    W {b | radialTime a b.1 z (levelRadius a (2*z.im) s) ≠ ∞} ≤
-      visitConstant a * ENNReal.ofReal ((levelRadius a (2*z.im) s / z.im)^(alpha a)) := by
-  by_cases hm : mixingTime (4*a) ≤ s
-  · exact (radial_hit_probability_long W hW a ha z hz s hm).trans (by
-      rw [mul_comm]
-      exact mul_le_mul_of_nonneg_right (le_add_right le_rfl) zero_le)
-  · have hh : ENNReal.ofReal (Real.exp (2*a*mixingTime (4*a)*alpha a)) *
-        ENNReal.ofReal ((levelRadius a (2*z.im) s / z.im)^(alpha a)) ≥ 1 := by
-      rw [levelRadius_ratio a z hz s, ← ENNReal.ofReal_mul (Real.exp_pos _).le,
-        ← Real.exp_mul, ← Real.exp_add, ← ENNReal.ofReal_one]
-      apply ENNReal.ofReal_le_ofReal
-      rw [Real.one_le_exp_iff]
-      have hp := alpha_pos ha
-      nlinarith [mul_nonneg (show 0 ≤ 2*a by linarith)
-        (sub_nonneg.mpr (not_le.mp hm).le)]
-    exact (prob_le_one (μ := W)).trans (hh.trans
-      (mul_le_mul_of_nonneg_right (le_add_left le_rfl) zero_le))
-
- 
-theorem radial_hit_probability_radius (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
-    W {b | radialTime a b.1 z ρ ≠ ∞} ≤ visitConstant a * ENNReal.ofReal ((ρ/z.im)^(alpha a)) := by
-  let s : ℝ := -(Real.log (ρ/z.im))/(2*a)
-  have ha0 : 0 < a := by linarith
-  have hq : 0 < ρ/z.im := div_pos hρ hz
-  have hs : 0 ≤ s := div_nonneg (neg_nonneg.mpr (Real.log_nonpos hq.le ((div_le_one hz).mpr hr)))
-    (by positivity)
-  have he : levelRadius a (2*z.im) s = ρ := by
-    have hexp : -2*a*s = Real.log (ρ/z.im) := by dsimp [s]; field_simp
-    unfold levelRadius
-    rw [hexp, Real.exp_log hq]
-    field_simp
-  simpa only [he] using radial_hit_probability W hW a ha z hz s hs
-
-end SLEExactGauge.Jacobi
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RawRestartTests
+import OAI.Probability.SLEGauge.FiniteAngleEigen
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Classical NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+
+lemma radialState_raw_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) :
+    Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
+      (fun b : GDriver => radialState a b.1 z σ) := by
+  let S := fun b : GDriver => radialTime a b.1 z σ
+  have hS := radialTime_stopping a (by linarith) z hz σ hσ hσz
+  have hD := stoppedData_raw_measurable hS
+  have hm : Measurable[hS.measurableSpace] (fun b : GDriver =>
+      killedState a (stoppedData S b).1 (stoppedData S b).2 z) :=
+    (killedState_time_joint_measurable a ha).comp
+      (hD.fst.prodMk (hD.snd.prodMk (measurable_const (a := (⟨z,hz⟩ : UpperPlane)))))
+  have he (b : GDriver) : radialState a b.1 z σ =
+      if S b=∞ then 0 else killedState a (stoppedData S b).1 (stoppedData S b).2 z := by
+    by_cases hb : S b=∞
+    · exact (ite_eq_left hb).trans (ite_eq_left hb).symm
+    · have hs := (radialTime_attained a (by linarith) z hz σ hσ hσz b hb).1
+      have hc := pastDriver_eq (S b).toNNReal b
+      have hp := (alive_driver_congr hc).mpr hs
+      rw [radialState,ite_eq_right hb,ite_eq_right hb]
+      change actualState a b.1 (S b).toNNReal z =
+        if alive a (pastDriver (S b).toNNReal b).1 (S b).toNNReal z then
+          actualState a (pastDriver (S b).toNNReal b).1 (S b).toNNReal z else 0
+      rw [ite_eq_left hp,actualState_driver_congr hc]
+  simp_rw [he]
+  exact Measurable.ite (hS.measurable (measurableSet_singleton ∞)) measurable_const hm
+
+ 
+
+def radialPoint (a : ℝ) (z : ℂ) (σ : ℝ) (b : GDriver) : UpperPlane :=
+  if h : 0<(radialState a b.1 z σ).1.im then ⟨(radialState a b.1 z σ).1,h⟩
+  else ⟨Complex.I,by simp⟩
+
+lemma radialPoint_raw_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) :
+    Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
+      (radialPoint a z σ) := by
+  have hm := radialState_raw_measurable a ha z hz σ hσ hσz
+  suffices Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
+      (fun b : GDriver => (radialPoint a z σ b : ℂ)) from this.subtype_mk
+  simp only [radialPoint,apply_dite,dite_eq_ite]
+  exact hm.fst.ite (measurableSet_lt measurable_const hm.fst.im) measurable_const
+
+lemma radialPoint_eq (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) (b : GDriver)
+    (hb : radialTime a b.1 z σ≠∞) :
+    (radialPoint a z σ b).1=(actualState a b.1 (radialTime a b.1 z σ).toNNReal z).1 := by
+  have hs := (radialTime_attained a ha z hz σ hσ hσz b hb).1
+  have hp : 0<(radialState a b.1 z σ).1.im := by
+    simpa only [radialState,ite_eq_right hb,Y_apply] using LoewnerScaling.Y_actual_pos hs
+  rw [radialPoint,dite_eq_left hp]
+  simp only [radialState,ite_eq_right hb]
+
+ 
+def radiusDensity (a : ℝ) (z : ℂ) (σ : ℝ) (b : GDriver) : ℝ≥0∞ :=
+  ENNReal.ofReal (radialWeight a b.1 z σ / physicalGreen a (z,0))
+
+lemma radiusDensity_raw_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) :
+    Measurable[(radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace]
+      (radiusDensity a z σ) := by
+  have hS := radialTime_stopping a (by linarith) z hz σ hσ hσz
+  have hm : Measurable[hS.measurableSpace] (fun b : GDriver => radialWeight a b.1 z σ) := by
+    have he (b : GDriver) : radialWeight a b.1 z σ =
+        if radialTime a b.1 z σ=∞ then 0 else physicalGreen a (radialState a b.1 z σ) := by
+      unfold radialWeight radialState
+      split_ifs <;> rfl
+    simp_rw [he]
+    apply Measurable.ite (hS.measurable (measurableSet_singleton ∞)) measurable_const
+    exact (show Measurable (physicalGreen a) by unfold physicalGreen logGreen Q; fun_prop).comp
+      (radialState_raw_measurable a ha z hz σ hσ hσz)
+  exact (hm.div_const _).ennreal_ofReal
+
+lemma radiusDensity_measurable (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ : ℝ) (hσ : 0<σ) (hσz : σ≤z.im) : Measurable (radiusDensity a z σ) :=
+  (radiusDensity_raw_measurable a ha z hz σ hσ hσz).mono
+    (radialTime_stopping a (by linarith) z hz σ hσ hσz).measurableSpace_le le_rfl
+
+lemma joint_radiusDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) :
+    Measurable (fun p : GDriver×UpperPlane => radiusDensity a p.2.1 (p.2.1.im*q) p.1) := by
+  apply Measurable.ennreal_ofReal
+  apply (joint_radialWeight_measurable a ha q hq hq1).div
+  unfold physicalGreen logGreen Q
+  fun_prop
+
+lemma radiusDensity_eq_zero (a : ℝ) (z : ℂ) (σ : ℝ) (b : GDriver)
+    (hb : radialTime a b.1 z σ=∞) : radiusDensity a z σ b=0 := by
+  simp only [radiusDensity,radialWeight,ite_eq_left hb,zero_div,ENNReal.ofReal_zero]
+
+lemma radiusDensity_restart (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im) (b : GDriver) :
+    radiusDensity a z ρ b = radiusDensity a z σ b *
+      radiusDensity a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
+        (restartAt (fun c => radialTime a c.1 z σ) b) := by
+  have ha0 : 0<a := by linarith
+  by_cases hb : radialTime a b.1 z σ=∞
+  · have hf : radialTime a b.1 z ρ=∞ := top_le_iff.mp
+      (hb ▸ radialTime_antitone a ha0 z hz ρ σ hρ hρσ hσ b)
+    rw [radiusDensity_eq_zero a z σ b hb,radiusDensity_eq_zero a z ρ b hf,zero_mul]
+  · rw [radialPoint_eq a ha0 z hz σ (hρ.trans_le hρσ) hσ b hb]
+    unfold radiusDensity restartAt
+    rw [radialWeight_restart a ha0 z hz σ ρ hρ hρσ hσ b hb]
+    have hp : 0≤radialWeight a b.1 z σ / physicalGreen a (z,0) :=
+      div_nonneg (radialWeight_bound a ha z hz σ (hρ.trans_le hρσ) hσ b).1 (Real.exp_pos _).le
+    rw [←ENNReal.ofReal_mul hp]
+    congr 1
+    ring
+
+lemma finiteAngle_restart (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im) (b : GDriver)
+    (hf : radialTime a b.1 z ρ≠∞) :
+    Jacobi.finiteAngle a z ρ b =
+      Jacobi.finiteAngle a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
+        (restartAt (fun c => radialTime a c.1 z σ) b) := by
+  have hb := ne_top_of_le_ne_top hf (radialTime_antitone a ha z hz ρ σ hρ hρσ hσ b)
+  let s := (radialTime a b.1 z σ).toNNReal
+  let w := (actualState a b.1 s z).1
+  let τ := radialTime a (restart s b).1 w (w.im*(ρ/σ))
+  have hτ : τ≠∞ := (radialTime_restart_top_iff a ha z hz σ ρ hρ hρσ hσ b hb).not.mpr hf
+  have ht : s+τ.toNNReal=(radialTime a b.1 z ρ).toNNReal := by
+    have hh := congrArg ENNReal.toNNReal (radialTime_restart a ha z hz σ ρ hρ hρσ hσ b hb)
+    change ((s:ℝ≥0∞)+τ).toNNReal=(radialTime a b.1 z ρ).toNNReal at hh
+    simpa only [ENNReal.toNNReal_add ENNReal.coe_ne_top hτ,ENNReal.toNNReal_coe] using hh
+  have hs := (radialTime_attained a ha z hz ρ hρ (hρσ.trans hσ) b hf).1
+  rw [radialPoint_eq a ha z hz σ (hρ.trans_le hρσ) hσ b hb]
+  apply Subtype.ext
+  change physicalTheta (radialState a b.1 z ρ) =
+    physicalTheta (radialState a (restart s b).1 w (w.im*(ρ/σ)))
+  rw [radialState,ite_eq_right hf,radialState,ite_eq_right hτ,←ht,
+    actualState_restart b.1.continuous (ht ▸ hs)]
+  rfl
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+
+lemma setLIntegral_eq_inter_of_zero {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Measure Ω) (A B : Set Ω) (hB : MeasurableSet B) (f : Ω→ℝ≥0∞)
+    (hf : ∀ b, b∉B → f b=0) : (∫⁻b in A,f b∂μ)=∫⁻b in A∩B,f b∂μ := by
+  have he : f=B.indicator f := by
+    funext b
+    by_cases hb : b∈B
+    · simp only [indicator_of_mem hb]
+    · simp only [indicator_of_notMem hb,hf b hb]
+  calc
+    _ = ∫⁻b in A,B.indicator f b∂μ := by rw [←he]
+    _ = ∫⁻b in A∩B,f b∂μ := by
+      rw [lintegral_indicator hB,Measure.restrict_restrict hB,inter_comm]
+
+lemma radial_hit_ae (W : Measure GDriver) (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) :
+    ∀ᵐ b∂radialLaw W a z ρ,radialTime a b.1 z ρ≠∞ := by
+  rw [radialLaw]
+  apply (ae_withDensity_iff (radiusDensity_measurable a ha z hz ρ hρ hr)).mpr
+  exact Eventually.of_forall (fun b hb hf => hb (radiusDensity_eq_zero a z ρ b hf))
+
+ 
+
+theorem radialLaw_restart (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ (hρ.trans_le hρσ) hσ).measurableSpace] A)
+    (F : UpperPlane×GDriver→ℝ≥0∞) (hF : Measurable F) :
+    (∫⁻b in A,F (radialPoint a z σ b,restartAt (fun c => radialTime a c.1 z σ) b)
+      ∂radialLaw W a z ρ) =
+    ∫⁻b in A,∫⁻c,F (radialPoint a z σ b,c)
+      ∂radialLaw W a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
+      ∂radialLaw W a z σ := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hσ0 := hρ.trans_le hρσ
+  have hS := radialTime_stopping a (by linarith) z hz σ hσ0 hσ
+  let S : GDriver→ℝ≥0∞ := fun b => radialTime a b.1 z σ
+  let Z := radialPoint a z σ
+  let D : GDriver→ℝ≥0∞×UpperPlane := fun b => (radiusDensity a z σ b,Z b)
+  have hD : Measurable[hS.measurableSpace] D :=
+    (radiusDensity_raw_measurable a ha z hz σ hσ0 hσ).prodMk
+      (radialPoint_raw_measurable a ha z hz σ hσ0 hσ)
+  have hZ : Measurable Z := (radialPoint_raw_measurable a ha z hz σ hσ0 hσ).mono
+    hS.measurableSpace_le le_rfl
+  have hY := measurable_restartAt hS
+  have hA' : MeasurableSet A := hS.measurableSpace_le _ hA
+  have hhit : MeasurableSet {b : GDriver | S b≠∞} :=
+    (hS.measurable' (measurableSet_singleton ∞)).compl
+  have hq : 0<ρ/σ := div_pos hρ hσ0
+  have hq1 : ρ/σ≤1 := (div_le_one hσ0).mpr hρσ
+  let K : UpperPlane×GDriver→ℝ≥0∞ := fun p =>
+    radiusDensity a p.1.1 (p.1.1.im*(ρ/σ)) p.2 * F p
+  have hK : Measurable K :=
+    ((joint_radiusDensity_measurable a ha (ρ/σ) hq hq1).comp measurable_swap).mul hF
+  let H : (ℝ≥0∞×UpperPlane)×GDriver→ℝ≥0∞ := fun p => p.1.1*K (p.1.2,p.2)
+  have hH : Measurable H := measurable_fst.fst.mul
+    (hK.comp (measurable_fst.snd.prodMk measurable_snd))
+  have he (b : GDriver) : radiusDensity a z ρ b * F (Z b,restartAt S b)=H (D b,restartAt S b) := by
+    rw [radiusDensity_restart a ha z hz σ ρ hρ hρσ hσ]
+    exact mul_assoc _ _ _
+  have hi (b : GDriver) : (∫⁻c,H (D b,c)∂W)=radiusDensity a z σ b *
+      ∫⁻c,F (Z b,c)∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ)) := by
+    have hm : Measurable (fun c : GDriver => F (Z b,c)) :=
+      hF.comp (measurable_const.prodMk measurable_id)
+    have he' : (∫⁻c,F (Z b,c)∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ)))=
+        ∫⁻c,K (Z b,c)∂W :=
+      lintegral_withDensity_eq_lintegral_mul W
+        (radiusDensity_measurable a ha (Z b).1 (Z b).2 _ (mul_pos (Z b).2 hq)
+          (mul_le_of_le_one_right (Z b).2.le hq1)) hm
+    rw [he']
+    exact lintegral_const_mul _ (hK.comp (measurable_const.prodMk measurable_id))
+  have hkm : Measurable (fun b => ∫⁻c,F (Z b,c)
+      ∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ))) := by
+    have he' (b : GDriver) : (∫⁻c,F (Z b,c)
+        ∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ)))=∫⁻c,K (Z b,c)∂W := by
+      exact lintegral_withDensity_eq_lintegral_mul W
+        (radiusDensity_measurable a ha (Z b).1 (Z b).2 _ (mul_pos (Z b).2 hq)
+          (mul_le_of_le_one_right (Z b).2.le hq1)) (hF.comp (measurable_const.prodMk measurable_id))
+    simp_rw [he']
+    exact hK.lintegral_prod_right'.comp hZ
+  change (∫⁻b in A,F (Z b,restartAt S b)∂W.withDensity (radiusDensity a z ρ)) =
+    ∫⁻b in A,∫⁻c,F (Z b,c)∂radialLaw W a (Z b).1 ((Z b).1.im*(ρ/σ))
+      ∂W.withDensity (radiusDensity a z σ)
+  have hfm : Measurable (fun b : GDriver => F (Z b,restartAt S b)) := hF.comp (hZ.prodMk hY)
+  rw [setLIntegral_withDensity_eq_setLIntegral_mul W (radiusDensity_measurable a ha z hz ρ hρ (hρσ.trans hσ))
+      hfm hA',
+    setLIntegral_withDensity_eq_setLIntegral_mul W (radiusDensity_measurable a ha z hz σ hσ0 hσ) hkm hA']
+  simp only [Pi.mul_apply,he,←hi]
+  calc
+    _ = ∫⁻b in A∩{b | S b≠∞},H (D b,restartAt S b)∂W :=
+      setLIntegral_eq_inter_of_zero W A _ hhit _ (fun b hb => by
+        have hb' : S b=∞ := not_not.mp hb
+        simp only [H,D,radiusDensity_eq_zero a z σ b hb',zero_mul])
+    _ = ∫⁻b in A∩{b | S b≠∞},∫⁻c,H (D b,c)∂W ∂W :=
+      lintegral_data_restartAt_on hW hS hD hA hH
+    _ = ∫⁻b in A,∫⁻c,H (D b,c)∂W ∂W :=
+      (setLIntegral_eq_inter_of_zero W A _ hhit _ (fun b hb => by
+        have hb' : S b=∞ := not_not.mp hb
+        simp only [H,D,radiusDensity_eq_zero a z σ b hb',zero_mul,lintegral_zero])).symm
+
+ 
+theorem finiteAngle_conditional (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (z : ℂ) (hz : 0<z.im)
+    (σ ρ : ℝ) (hρ : 0<ρ) (hρσ : ρ≤σ) (hσ : σ≤z.im)
+    (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ (hρ.trans_le hρσ) hσ).measurableSpace] A)
+    (f : CenteredMoments.Angle→ℝ≥0∞) (hf : Measurable f) :
+    (∫⁻b in A,f (Jacobi.finiteAngle a z ρ b)∂radialLaw W a z ρ) =
+    ∫⁻b in A,∫⁻c,f (Jacobi.finiteAngle a (radialPoint a z σ b).1
+      ((radialPoint a z σ b).1.im*(ρ/σ)) c)
+      ∂radialLaw W a (radialPoint a z σ b).1 ((radialPoint a z σ b).1.im*(ρ/σ))
+      ∂radialLaw W a z σ := by
+  have hq : 0<ρ/σ := div_pos hρ (hρ.trans_le hρσ)
+  have hq1 : ρ/σ≤1 := (div_le_one (hρ.trans_le hρσ)).mpr hρσ
+  let F : UpperPlane×GDriver→ℝ≥0∞ := fun p => f (Jacobi.finiteAngle a p.1.1 (p.1.1.im*(ρ/σ)) p.2)
+  have hF : Measurable F := hf.comp (Jacobi.angleState_measurable.comp
+    ((joint_radialState_measurable a (by linarith) (ρ/σ) hq hq1).comp measurable_swap))
+  rw [←radialLaw_restart W hW a ha z hz σ ρ hρ hρσ hσ A hA F hF]
+  apply lintegral_congr_ae
+  filter_upwards [ae_restrict_of_ae (radial_hit_ae W a ha z hz ρ hρ (hρσ.trans hσ))] with b hb
+  exact congrArg f (finiteAngle_restart a (by linarith) z hz σ ρ hρ hρσ hσ b hb)
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.Jacobi
+open CenteredMoments ExactSLEGreen ExactSLECalculus ExactSLEFiniteTilt ExactSLEGenerator ExactSLEMain
+
+lemma sin_physicalTheta (v : State) (hy : 0 < Y v) :
+    Real.sin (physicalTheta v) = Y v / ‖v.1‖ := by
+  have hn : 0 < ‖v.1‖ := hy.trans_le (Complex.im_le_norm _)
+  have hs := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2
+  have hc := physicalTheta_cot v
+  rw [Real.cot_eq_cos_div_sin, div_eq_div_iff hs.ne' hy.ne'] at hc
+  have he : (X v)^2 + (Y v)^2 = ‖v.1‖^2 := by
+    simpa only [X_apply, Y_apply, Complex.normSq_apply, pow_two] using Complex.normSq_eq_norm_sq v.1
+  have hid := Real.sin_sq_add_cos_sq (physicalTheta v)
+  have heq : (Real.sin (physicalTheta v) * ‖v.1‖)^2 = (Y v)^2 := by
+    nlinarith [sq_nonneg (Real.sin (physicalTheta v) * X v - Real.cos (physicalTheta v) * Y v),
+      congrArg (fun x : ℝ => x^2) hc,
+      congrArg (fun x : ℝ => x * (Y v)^2) hid,
+      congrArg (fun x : ℝ => x * (Real.sin (physicalTheta v))^2) he]
+  apply (eq_div_iff hn.ne').mpr
+  nlinarith [mul_pos hs hn]
+
+ 
+lemma radialWeight_inverse_sine (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
+    (hb : radialTime a b.1 z ρ ≠ ∞) :
+    radialWeight a b.1 z ρ * (Real.sin (finiteAngle a z ρ b : ℝ)) ^ (-p a) =
+      ρ ^ (-alpha a) := by
+  have ht := radialTime_attained a ha z hz ρ hρ hr b hb
+  have hy := LoewnerScaling.Y_actual_pos ht.1
+  have hs := Real.sin_pos_of_pos_of_lt_pi (finiteAngle a z ρ b).2.1
+    (finiteAngle a z ρ b).2.2
+  have hθ : Real.sin (finiteAngle a z ρ b : ℝ) =
+      Y (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z) /
+        ‖(actualState a b.1 (radialTime a b.1 z ρ).toNNReal z).1‖ := by
+    simpa only [finiteAngle, angleState, radialState, ite_eq_right hb] using sin_physicalTheta _ hy
+  rw [radialWeight, ite_eq_right hb, physicalGreen_eq_radius_sine a _ hy, ht.2, ← hθ,
+    mul_assoc, ← Real.rpow_add hs]
+  simp
+
+lemma inverse_radial_density (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver) :
+    (ρ^(alpha a) * physicalGreen a (z,0)) *
+        ((radialWeight a b.1 z ρ / physicalGreen a (z,0)) *
+          (Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) =
+      if radialTime a b.1 z ρ ≠ ∞ then 1 else 0 := by
+  by_cases hb : radialTime a b.1 z ρ ≠ ∞
+  · rw [ite_eq_left hb]
+    have hp : physicalGreen a (z,0) ≠ 0 := (Real.exp_pos _).ne'
+    calc
+      _ = ρ^(alpha a) * (radialWeight a b.1 z ρ *
+          (Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) := by field_simp
+      _ = 1 := by
+        rw [radialWeight_inverse_sine a ha z hz ρ hρ hr b hb, ← Real.rpow_add hρ]
+        simp
+  · simp only [ite_eq_right hb, radialWeight, ite_eq_left (not_not.mp hb), zero_div, zero_mul, mul_zero]
+
+ 
+
+theorem radial_change_law (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (g : ExactSLEGenerator.Driver → ℝ≥0∞) (hg : Measurable g) :
+    (∫⁻b in {b | radialTime a b.1 z ρ ≠ ∞}, g b ∂W) =
+      ENNReal.ofReal (ρ^(alpha a) * physicalGreen a (z,0)) *
+        ∫⁻b, ENNReal.ofReal ((Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) * g b
+          ∂radialLaw W a z ρ := by
+  have ha0 : 0 < a := by linarith
+  have hhit : MeasurableSet {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ ≠ ∞} :=
+    ((radialTime_stopping a ha0 z hz ρ hρ hr).measurable' (measurableSet_singleton ∞)).compl
+  have hs : Measurable (fun b : ExactSLEGenerator.Driver =>
+      ENNReal.ofReal ((Real.sin (finiteAngle a z ρ b : ℝ))^(-p a))) := by
+    exact (show Measurable (fun θ : Angle => ENNReal.ofReal ((Real.sin θ)^(-p a))) by fun_prop).comp
+      (finiteAngle_measurable a ha0 z hz ρ hρ hr)
+  have hd : Measurable (fun b : ExactSLEGenerator.Driver =>
+      ENNReal.ofReal (radialWeight a b.1 z ρ / physicalGreen a (z,0))) :=
+    ((radialWeight_measurable a ha z hz ρ hρ hr).div_const _).ennreal_ofReal
+  have hf : Measurable (fun b =>
+      ENNReal.ofReal ((Real.sin (finiteAngle a z ρ b : ℝ))^(-p a)) * g b) := hs.mul hg
+  rw [radialLaw, lintegral_withDensity_eq_lintegral_mul W hd hf, ← lintegral_const_mul _ (hd.mul hf)]
+  rw [← lintegral_indicator hhit]
+  apply lintegral_congr
+  intro b
+  have hpos : 0 ≤ (radialWeight a b.1 z ρ / physicalGreen a (z,0)) :=
+    div_nonneg (radialWeight_bound a ha z hz ρ hρ hr b).1 (Real.exp_pos _).le
+  have hc : 0 ≤ ρ^(alpha a) * physicalGreen a (z,0) :=
+    mul_nonneg (Real.rpow_nonneg hρ.le _) (Real.exp_pos _).le
+  have he := congrArg ENNReal.ofReal (inverse_radial_density a ha0 z hz ρ hρ hr b)
+  rw [ENNReal.ofReal_mul hc, ENNReal.ofReal_mul hpos] at he
+  simp only [Pi.mul_apply]
+  rw [← mul_assoc _ _ (g b), ← mul_assoc _ _ (g b), he]
+  by_cases hb : radialTime a b.1 z ρ ≠ ∞ <;> simp [hb]
+
+lemma radial_hit_probability_eq (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (s : ℝ) (hs : 0 ≤ s) :
+    W {b | radialTime a b.1 z (levelRadius a (2*z.im) s) ≠ ∞} =
+      ENNReal.ofReal ((levelRadius a (2*z.im) s)^(alpha a) * physicalGreen a (z,0)) *
+        ∫⁻θ : Angle, ENNReal.ofReal ((Real.sin θ)^(-p a)) ∂finiteAngleLaw W a z s := by
+  have hρ := levelRadius_pos a (2*z.im) s (by positivity : 0 < 2*z.im)
+  have hr := levelRadius_initial_le a (by linarith) z hz s hs
+  have he := radial_change_law W a ha z hz _ hρ hr (fun _ => 1) measurable_const
+  simp only [lintegral_one, Measure.restrict_apply_univ, mul_one] at he
+  rw [finiteAngleLaw, lintegral_map (by fun_prop)
+    (finiteAngle_measurable a (by linarith) z hz _ hρ hr)]
+  exact he
+
+lemma physicalGreen_initial_bound (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im) :
+    physicalGreen a (z,0) ≤ z.im^(-alpha a) := by
+  have hr : physicalRadius (z,0) = z.im := by simp [physicalRadius, Y_apply, J_apply]
+  rw [Real.rpow_def_of_pos hz]
+  simpa only [mul_comm] using physicalGreen_bound a z.im ha hz (z,0) hz hr.ge
+
+ 
+
+theorem radial_hit_probability_long (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (s : ℝ) (hs : mixingTime (4*a) ≤ s) :
+    W {b | radialTime a b.1 z (levelRadius a (2*z.im) s) ≠ ∞} ≤
+      ENNReal.ofReal ((levelRadius a (2*z.im) s / z.im)^(alpha a)) *
+        (2 * ∫⁻θ : Angle, ENNReal.ofReal ((Real.sin θ)^(-p a)) ∂angleLaw a) := by
+  have hs0 : 0 ≤ s := le_trans (by unfold mixingTime majorantBase; positivity) hs
+  have hρ := levelRadius_pos a (2*z.im) s (by positivity : 0 < 2*z.im)
+  rw [radial_hit_probability_eq W a ha z hz s hs0]
+  apply mul_le_mul
+  · apply ENNReal.ofReal_le_ofReal
+    rw [Real.div_rpow hρ.le hz.le, div_eq_mul_inv, ← Real.rpow_neg hz.le]
+    exact mul_le_mul_of_nonneg_left (physicalGreen_initial_bound a ha z hz)
+      (Real.rpow_nonneg hρ.le _)
+  · exact (finiteAngleLaw_integral_bounds W hW a ha z hz s hs _ (by fun_prop)).2
+  all_goals positivity
+
+ 
+def visitConstant (a : ℝ) : ℝ≥0∞ :=
+  2 * (∫⁻θ : Angle, ENNReal.ofReal ((Real.sin θ)^(-p a)) ∂angleLaw a) +
+    ENNReal.ofReal (Real.exp (2*a*mixingTime (4*a)*alpha a))
+
+lemma visitConstant_lt_top (a : ℝ) : visitConstant a < ∞ :=
+  ENNReal.add_lt_top.mpr ⟨ENNReal.mul_lt_top (by norm_num) (angle_inverse_power_finite a),
+    ENNReal.ofReal_lt_top⟩
+
+lemma visitConstant_pos (a : ℝ) : 0 < visitConstant a :=
+  lt_of_lt_of_le (ENNReal.ofReal_pos.mpr (Real.exp_pos _)) (le_add_left le_rfl)
+
+lemma levelRadius_ratio (a : ℝ) (z : ℂ) (hz : 0 < z.im) (s : ℝ) :
+    levelRadius a (2*z.im) s / z.im = Real.exp (-2*a*s) := by
+  unfold levelRadius
+  field_simp
+
+ 
+
+theorem radial_hit_probability (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im) (s : ℝ) (_hs : 0 ≤ s) :
+    W {b | radialTime a b.1 z (levelRadius a (2*z.im) s) ≠ ∞} ≤
+      visitConstant a * ENNReal.ofReal ((levelRadius a (2*z.im) s / z.im)^(alpha a)) := by
+  by_cases hm : mixingTime (4*a) ≤ s
+  · exact (radial_hit_probability_long W hW a ha z hz s hm).trans (by
+      rw [mul_comm]
+      exact mul_le_mul_of_nonneg_right (le_add_right le_rfl) zero_le)
+  · have hh : ENNReal.ofReal (Real.exp (2*a*mixingTime (4*a)*alpha a)) *
+        ENNReal.ofReal ((levelRadius a (2*z.im) s / z.im)^(alpha a)) ≥ 1 := by
+      rw [levelRadius_ratio a z hz s, ← ENNReal.ofReal_mul (Real.exp_pos _).le,
+        ← Real.exp_mul, ← Real.exp_add, ← ENNReal.ofReal_one]
+      apply ENNReal.ofReal_le_ofReal
+      rw [Real.one_le_exp_iff]
+      have hp := alpha_pos ha
+      nlinarith [mul_nonneg (show 0 ≤ 2*a by linarith)
+        (sub_nonneg.mpr (not_le.mp hm).le)]
+    exact (prob_le_one (μ := W)).trans (hh.trans
+      (mul_le_mul_of_nonneg_right (le_add_left le_rfl) zero_le))
+
+ 
+theorem radial_hit_probability_radius (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
+    W {b | radialTime a b.1 z ρ ≠ ∞} ≤ visitConstant a * ENNReal.ofReal ((ρ/z.im)^(alpha a)) := by
+  let s : ℝ := -(Real.log (ρ/z.im))/(2*a)
+  have ha0 : 0 < a := by linarith
+  have hq : 0 < ρ/z.im := div_pos hρ hz
+  have hs : 0 ≤ s := div_nonneg (neg_nonneg.mpr (Real.log_nonpos hq.le ((div_le_one hz).mpr hr)))
+    (by positivity)
+  have he : levelRadius a (2*z.im) s = ρ := by
+    have hexp : -2*a*s = Real.log (ρ/z.im) := by dsimp [s]; field_simp
+    unfold levelRadius
+    rw [hexp, Real.exp_log hq]
+    field_simp
+  simpa only [he] using radial_hit_probability W hW a ha z hz s hs
+
+end SLEExactGauge.Jacobi
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadialStopping.lean b/lean/OAI/Probability/SLEGauge/RadialStopping.lean
index 69e06591eed09555c212927b57d4e97f99ca107e..89a08e2a2f8a9774fd3fbf53a0ff555c7f69f842 100644
--- a/lean/OAI/Probability/SLEGauge/RadialStopping.lean
+++ b/lean/OAI/Probability/SLEGauge/RadialStopping.lean
@@ -1,433 +1,433 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TiltedMeasures
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open scoped Classical
-lemma capTime_eq (S : ExactSLEGenerator.Driver → ℝ≥0∞) (n : ℝ≥0) (b : ExactSLEGenerator.Driver)
-    (h : S b ≤ n) : capTime S n b = (S b).toNNReal := by simp only [capTime,min_eq_left h]
-lemma stopped_before_cap (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (n : ℝ≥0) (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    MeasurableSet[(capTime_stopping S hS n).measurableSpace] (U ∩ {b | S b < n}) := by
-  have hc : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (n : WithTop ℝ≥0)) := isStoppingTime_const _ _
-  have he : U ∩ {b | S b < n} = (U ∩ {b | S b ≤ n}) \ {b | S b = n} := by
-    ext b
-    simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,Set.mem_sdiff]
-    exact ⟨fun h => ⟨⟨h.1,h.2.le⟩,h.2.ne⟩,fun h => ⟨h.1.1,lt_of_le_of_ne h.1.2 h.2⟩⟩
-  have hm := (hS.measurableSet_inter_le hc U hU).diff (hS.measurableSet_eq_stopping_time_min hc)
-  have heq : (fun b => (capTime S n b : WithTop ℝ≥0)) = fun b => min (S b) (n : ℝ≥0∞) := funext (capTime_coe S n)
-  rw [he]
-  exact ((hS.min hc).measurableSpace_mono (capTime_stopping S hS n) (fun b => by
-    change min (S b) (n : ℝ≥0∞) ≤ (capTime S n b : ℝ≥0∞)
-    rw [capTime_coe])) _ hm
-
-lemma stopped_cap_measurable (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (n : ℝ≥0) : (capTime_stopping S hS n).measurableSpace ≤ hS.measurableSpace :=
-  (capTime_stopping S hS n).measurableSpace_mono hS (fun b => by
-    change (capTime S n b : ℝ≥0∞) ≤ S b
-    rw [capTime_coe]
-    exact min_le_left _ _)
-lemma stopping_before_measurable (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (n : ℝ≥0) : MeasurableSet {b | S b < n} :=
-  hS.measurableSpace_le _ (hS.measurableSet_lt' n)
-
-lemma eventually_before_nat {v : ℝ≥0∞} (hv : v ≠ ∞) : ∀ᶠ n : ℕ in atTop, v < ((n : ℝ≥0) : ℝ≥0∞) := by
-  obtain ⟨N,hN⟩ := exists_nat_gt v.toNNReal
-  filter_upwards [eventually_ge_atTop N] with n hn
-  rw [← ENNReal.coe_toNNReal hv,ENNReal.coe_lt_coe]
-  exact hN.trans_le (Nat.cast_le.mpr hn)
-
- 
-
-theorem radial_conditional_weight_all (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    Integrable (fun b => stoppedRadialRemaining a b.1 z ρ (S b)) W ∧
-    (∫ b in U, (if S b < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) ∂W) =
-      ∫ b in U, stoppedRadialRemaining a b.1 z ρ (S b) ∂W := by
-  let C : ℕ → Set ExactSLEGenerator.Driver := fun n => {b | S b < (n : ℝ≥0)}
-  let F := fun n => (C n).indicator (fun b => radialRemaining a b.1 z ρ (capTime S n b))
-  let H := fun n => (C n).indicator (fun b => if (capTime S n b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0)
-  let G := fun b : ExactSLEGenerator.Driver => stoppedRadialRemaining a b.1 z ρ (S b)
-  let K := fun b : ExactSLEGenerator.Driver => if S b < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0
-  let c := Real.exp (-alpha a*Real.log ρ)
-  have hFi (n : ℕ) : Integrable (F n) W :=
-    (radial_conditional_weight W hW a ha z hz ρ hρ hr (capTime S n) (capTime_stopping S hS n)
-      Set.univ MeasurableSet.univ).1.indicator (stopping_before_measurable S hS n)
-  have hHi (n : ℕ) : Integrable (H n) W :=
-    (radial_terminal_integrable W hW a ha z hz ρ hρ hr (capTime S n) (capTime_stopping S hS n)).indicator
-      (stopping_before_measurable S hS n)
-  have hFc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => F n b) atTop (𝓝 (G b)) := by
-    by_cases hb : S b = ∞
-    · have he : (fun n => F n b) = fun _ : ℕ => 0 := by
-        funext n
-        apply Set.indicator_of_notMem
-        simp [C,hb]
-      have hg : G b = 0 := by simp [G,stoppedRadialRemaining,hb]
-      rw [he,hg]
-      exact tendsto_const_nhds
-    · apply Filter.Tendsto.congr' _ tendsto_const_nhds
-      filter_upwards [eventually_before_nat hb] with n hn
-      dsimp only [G,F,stoppedRadialRemaining]
-      rw [ite_eq_right hb,Set.indicator_of_mem (show b ∈ C n from hn),capTime_eq S n b hn.le]
-  have hHc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => H n b) atTop (𝓝 (K b)) := by
-    by_cases hb : S b = ∞
-    · have he : (fun n => H n b) = fun _ : ℕ => 0 := by
-        funext n
-        apply Set.indicator_of_notMem
-        simp [C,hb]
-      have hk : K b = 0 := by simp [K,hb]
-      rw [he,hk]
-      exact tendsto_const_nhds
-    · apply Filter.Tendsto.congr' _ tendsto_const_nhds
-      filter_upwards [eventually_before_nat hb] with n hn
-      dsimp only [K,H]
-      rw [Set.indicator_of_mem (show b ∈ C n from hn),capTime_coe,min_eq_left hn.le]
-  have hFb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖F n b‖ ≤ c := by
-    by_cases hb : b ∈ C n
-    · dsimp only [F]
-      rw [Set.indicator_of_mem hb]
-      exact radialRemaining_bound a ha z hz ρ hρ hr b (capTime S n b)
-    · simpa only [F,Set.indicator_of_notMem hb,norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
-  have hHb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖H n b‖ ≤ c := by
-    by_cases hb : b ∈ C n
-    · dsimp only [H]
-      rw [Set.indicator_of_mem hb]
-      split_ifs
-      · have h := radialWeight_bound a ha z hz ρ hρ hr b
-        simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
-      · simpa only [norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
-    · simpa only [H,Set.indicator_of_notMem hb,norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
-  have hGi : Integrable G W := Integrable.of_bound
-    (aestronglyMeasurable_of_tendsto_ae atTop (fun n => (hFi n).aestronglyMeasurable) (Eventually.of_forall hFc)) c
-    (Eventually.of_forall fun b => le_of_tendsto (hFc b).norm (Eventually.of_forall fun n => hFb n b))
-  refine ⟨hGi,?_⟩
-  have he (n : ℕ) : (∫ b in U, F n b ∂W) = ∫ b in U, H n b ∂W := by
-    have hC := stopping_before_measurable S hS n
-    dsimp only [F,H]
-    rw [integral_indicator hC,integral_indicator hC,Measure.restrict_restrict hC,Set.inter_comm (C n) U]
-    exact (radial_conditional_weight W hW a ha z hz ρ hρ hr (capTime S n) (capTime_stopping S hS n)
-      (U ∩ C n) (stopped_before_cap S hS n U hU)).2.symm
-  have hFl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => c)
-    (fun n => (hFi n).aestronglyMeasurable.restrict) (integrable_const c)
-    (fun n => Eventually.of_forall (hFb n)) (Eventually.of_forall hFc)
-  have hHl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => c)
-    (fun n => (hHi n).aestronglyMeasurable.restrict) (integrable_const c)
-    (fun n => Eventually.of_forall (hHb n)) (Eventually.of_forall hHc)
-  have heq : (fun n => ∫ b in U, F n b ∂W) = fun n => ∫ b in U, H n b ∂W := funext he
-  rw [heq] at hFl
-  exact tendsto_nhds_unique hHl hFl
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open scoped Classical
-
-noncomputable def radialLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (z : ℂ)
-    (ρ : ℝ) : Measure ExactSLEGenerator.Driver :=
-  W.withDensity (fun b => ENNReal.ofReal (radialWeight a b.1 z ρ / physicalGreen a (z,0)))
-
-lemma radialTime_stopping (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
-    IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z ρ) := by
-  intro t
-  let T := radialStop a ha z hz ρ hρ hr
-  let L := fun n => radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let F := fun n (b : ExactSLEGenerator.Driver) =>
-    physicalRadius (localState a (L n).ε (L n).hε z (min (T n b) t) b)
-  have hFm (n : ℕ) : Measurable[rawFiltration t] (F n) := by
-    have hT := radialStop_stopping a ha z hz ρ hρ hr n
-    have hmin : IsStoppingTime rawFiltration (fun b => (min (T n b) t : WithTop ℝ≥0)) := by
-      simpa only [WithTop.coe_min] using hT.min (isStoppingTime_const rawFiltration t)
-    have hle : hmin.measurableSpace ≤ rawFiltration t :=
-      hmin.measurableSpace_le_of_le_const (fun b => WithTop.coe_le_coe.mpr (min_le_right _ _))
-    have hA : StronglyAdapted rawFiltration (localState a (L n).ε (L n).hε z) :=
-      fun u => (localState_adapted a (L n).ε (L n).hε z u).stronglyMeasurable
-    exact physicalRadius_continuous.measurable.comp
-      ((measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
-        (localState_continuous a (L n).ε (L n).hε z)) hmin).mono hle le_rfl)
-  have he : {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ ≤ (t : ℝ≥0∞)} =
-      ⋃ n : ℕ, {b | T n b ≤ t} ∩ {b | F n b = ρ} := by
-    ext b
-    simp only [Set.mem_ofPred_eq,Set.mem_iUnion,Set.mem_inter_iff]
-    constructor
-    · intro hb
-      have hfin : radialTime a b.1 z ρ ≠ ∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hb
-      obtain ⟨n,hn⟩ := (radialStop_eventually_eq a ha z hz ρ hρ hr b hfin).exists
-      have hnt : T n b ≤ t := by
-        apply ENNReal.coe_le_coe.mp
-        exact hn.trans_le hb
-      refine ⟨n,hnt,?_⟩
-      dsimp only [F]
-      rw [min_eq_left hnt]
-      have hr' := (radialTime_attained a ha z hz ρ hρ hr b hfin).2
-      rw [← hn,ENNReal.toNNReal_coe] at hr'
-      have hact := (regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ
-        (angleThreshold z n) (L n).B (L n).initial b (T n b) le_rfl).2
-      rw [hact] at hr'
-      exact hr'
-    · rintro ⟨n,hnt,hR⟩
-      dsimp only [F] at hR
-      rw [min_eq_left hnt] at hR
-      have htime := regionStop_eq_radialTime a (L n).ε (L n).hε z (L n).N ρ
-        (angleThreshold z n) (L n).B (L n).initial b hR
-      rw [htime]
-      exact ENNReal.coe_le_coe.mpr hnt
-  change MeasurableSet[rawFiltration t] {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ ≤ (t : ℝ≥0∞)}
-  rw [he]
-  apply MeasurableSet.iUnion
-  intro n
-  have hT := radialStop_stopping a ha z hz ρ hρ hr n
-  have htm : MeasurableSet[rawFiltration t] {b | T n b ≤ t} := by
-    simpa only [WithTop.coe_le_coe] using hT t
-  exact htm.inter (measurableSet_eq_fun (hFm n) measurable_const)
-
-lemma radialTime_le_of_hit (a : ℝ) (z : ℂ) (ρ : ℝ) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (ht : alive a b.1 t z) (hR : physicalRadius (actualState a b.1 t z) = ρ) :
-    radialTime a b.1 z ρ ≤ (t : ℝ≥0∞) :=
-  sInf_le ⟨t,rfl,ht,hR⟩
-
-lemma radialTime_strict_of_finite (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im)
-    (b : ExactSLEGenerator.Driver) (hfin : radialTime a b.1 z ρ ≠ ∞) :
-    radialTime a b.1 z σ < radialTime a b.1 z ρ := by
-  have hr : ρ ≤ z.im := hρσ.le.trans hσ
-  obtain ⟨n,hn⟩ := (radialStop_eventually_eq a ha z hz ρ hρ hr b hfin).exists
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T := radialStop a ha z hz ρ hρ hr n b
-  let f := fun t : ℝ≥0 => physicalRadius (localState a L.ε L.hε z t b)
-  have hfc : Continuous f := physicalRadius_continuous.comp (localState_continuous a L.ε L.hε z b)
-  have hR : f T = ρ := by
-    have h := (radialTime_attained a ha z hz ρ hρ hr b hfin).2
-    rw [← hn,ENNReal.toNNReal_coe] at h
-    rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl).2] at h
-    exact h
-  have h0 : f 0 = z.im := by simp [f,localState_zero,physicalRadius,Y,J]
-  obtain ⟨u,hu,huR⟩ := intermediate_value_Icc' (show (0 : ℝ≥0) ≤ T from bot_le) hfc.continuousOn
-    (show σ ∈ Icc (f T) (f 0) by rw [hR,h0]; exact ⟨hρσ.le,hσ⟩)
-  have huT : u < T := hu.2.lt_of_ne (by intro he; rw [he,hR] at huR; exact hρσ.ne huR)
-  have hact := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b u hu.2
-  have hle : radialTime a b.1 z σ ≤ (u : ℝ≥0∞) :=
-    radialTime_le_of_hit a z σ b u hact.1 (by rw [hact.2]; exact huR)
-  exact hle.trans_lt ((ENNReal.coe_lt_coe.mpr huT).trans_eq hn)
-
-lemma radialTime_strict_of_first_finite (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im)
-    (b : ExactSLEGenerator.Driver) (hfin : radialTime a b.1 z σ ≠ ∞) :
-    radialTime a b.1 z σ < radialTime a b.1 z ρ := by
-  by_cases h : radialTime a b.1 z ρ = ∞
-  · rw [h]
-    exact lt_top_iff_ne_top.mpr hfin
-  · exact radialTime_strict_of_finite a ha z hz ρ σ hρ hρσ hσ b h
-
-lemma radialWeight_consistency
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im)
-    (hS : IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z σ))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    (∫ b in U, radialWeight a b.1 z ρ ∂W) = ∫ b in U, radialWeight a b.1 z σ ∂W := by
-  have ha0 : 0 < a := by linarith
-  have hsmall (b : ExactSLEGenerator.Driver) :
-      (if radialTime a b.1 z σ < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) = radialWeight a b.1 z ρ := by
-    by_cases hf : radialTime a b.1 z ρ = ∞
-    · simp [radialWeight,hf]
-    · rw [ite_eq_left (radialTime_strict_of_finite a ha0 z hz ρ σ hρ hρσ hσ b hf)]
-  have hlarge (b : ExactSLEGenerator.Driver) :
-      stoppedRadialRemaining a b.1 z ρ (radialTime a b.1 z σ) = radialWeight a b.1 z σ := by
-    by_cases hf : radialTime a b.1 z σ = ∞
-    · simp [stoppedRadialRemaining,radialWeight,hf]
-    · have hat := (radialTime_attained a ha0 z hz σ (hρ.trans hρσ) hσ b hf).1
-      have hlt := radialTime_strict_of_first_finite a ha0 z hz ρ σ hρ hρσ hσ b hf
-      rw [stoppedRadialRemaining,ite_eq_right hf,radialRemaining,radialWeight,ite_eq_right hf]
-      rw [ite_eq_left ⟨hat,by simpa only [ENNReal.coe_toNNReal hf] using hlt⟩]
-  have hc := (radial_conditional_weight_all W hW a ha z hz ρ hρ (hρσ.le.trans hσ)
-    (fun b => radialTime a b.1 z σ) hS U hU).2
-  simpa only [hsmall,hlarge] using hc
-
-lemma radialLaw_apply
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet U) :
-    radialLaw W a z ρ U = ENNReal.ofReal ((∫ b in U, radialWeight a b.1 z ρ ∂W) / physicalGreen a (z,0)) := by
-  rw [radialLaw,MeasureTheory.withDensity_apply _ hU]
-  have hi := ((radialWeight_mean W hW a ha z hz ρ hρ hr).1.restrict (s := U)).div_const (physicalGreen a (z,0))
-  have hn : 0 ≤ᵐ[W.restrict U] (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ / physicalGreen a (z,0)) :=
-    Eventually.of_forall (fun b => div_nonneg (radialWeight_bound a ha z hz ρ hρ hr b).1 (Real.exp_pos _).le)
-  rw [← ofReal_integral_eq_lintegral_ofReal hi hn,integral_div]
-
-lemma radialLaw_probability
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
-    IsProbabilityMeasure (radialLaw W a z ρ) := by
-  rw [isProbabilityMeasure_iff,radialLaw_apply W hW a ha z hz ρ hρ hr Set.univ MeasurableSet.univ,
-    Measure.restrict_univ,(radialWeight_mean W hW a ha z hz ρ hρ hr).2,
-    div_self (show physicalGreen a (z,0) ≠ 0 from (Real.exp_pos _).ne'),ENNReal.ofReal_one]
-
-theorem finite_radial_laws_consistent
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im) :
-    IsProbabilityMeasure (radialLaw W a z ρ) ∧ IsProbabilityMeasure (radialLaw W a z σ) ∧
-    ∃ hS : IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z σ),
-      ∀ U, MeasurableSet[hS.measurableSpace] U → radialLaw W a z ρ U = radialLaw W a z σ U := by
-  have hσ0 : 0 < σ := hρ.trans hρσ
-  have hr : ρ ≤ z.im := hρσ.le.trans hσ
-  have ha0 : 0 < a := by linarith
-  have hS := radialTime_stopping a ha0 z hz σ hσ0 hσ
-  refine ⟨radialLaw_probability W hW a ha z hz ρ hρ hr,
-    radialLaw_probability W hW a ha z hz σ hσ0 hσ,hS,?_⟩
-  intro U hU
-  have hU' : MeasurableSet U := hS.measurableSpace_le _ hU
-  rw [radialLaw_apply W hW a ha z hz ρ hρ hr U hU',
-    radialLaw_apply W hW a ha z hz σ hσ0 hσ U hU',
-    radialWeight_consistency W hW a ha z hz ρ σ hρ hρσ hσ hS U hU]
-
-end ExactSLEGreen
-
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-theorem compact_test_optional_inequality
-    (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
-    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
-    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
-    (e : E) (he : ‖e‖ ≤ 1)
-    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
-      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
-    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
-    (S T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
-    (hneg : ∀ b t, S b ≤ t → t ≤ T b → generator f A e (X t b) ≤ 0)
-    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    ∫ b in U, f (X (T b) b) ∂W ≤ ∫ b in U, f (X (S b) b) ∂W := by
-  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
-    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
-  have hM : Martingale M rawFiltration W :=
-    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
-  have hgc (b : Driver) : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
-    (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
-  have hmc (b : Driver) : Continuous (fun t => M t b) :=
-    (hf.continuous.comp (hXc b)).sub
-      ((intervalIntegral.continuous_primitive (fun s t => (hgc b).intervalIntegrable s t) 0).comp NNReal.continuous_coe)
-  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
-  have hmb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
-    calc
-      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
-      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
-        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
-          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
-            (fun u _ => hgb (X u.toNNReal b)))
-      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
-  have hFa : StronglyAdapted rawFiltration (fun t b => f (X t b)) := fun t =>
-    (hf.continuous.measurable.comp (hXm t)).stronglyMeasurable
-  have hFm (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
-      Measurable (fun b => f (X (R b) b)) :=
-    (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
-      (fun b => hf.continuous.comp (hXc b))) hR).mono hR.measurableSpace_le le_rfl
-  have hFi (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
-      Integrable (fun b => f (X (R b) b)) (W.restrict U) :=
-    Integrable.of_bound (hFm R hR).aestronglyMeasurable C (Filter.Eventually.of_forall fun b => hfb _)
-  have hMi (R : Driver → ℝ≥0)
-      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) (hRB : ∀ b, R b≤B) :
-      Integrable (fun b => M (R b) b) (W.restrict U) :=
-    Integrable.of_bound (ExactSLEOptional.stopped_measurable rawFiltration hM hmc R hR).aestronglyMeasurable
-      (C+D*(B+1)) (Filter.Eventually.of_forall fun b => hmb _ ((hRB b).trans (le_add_of_nonneg_right zero_le_one)) b)
-  have hpw (b : Driver) : f (X (T b) b) - f (X (S b) b) ≤ M (T b) b - M (S b) b := by
-    have hi := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
-      ((hgc b).intervalIntegrable 0 (S b)) ((hgc b).intervalIntegrable (S b) (T b))
-    have hn : (∫ u in (S b : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) ≤ 0 := by
-      rw [intervalIntegral.integral_of_le (by exact_mod_cast hST b)]
-      apply setIntegral_nonpos measurableSet_Ioc
-      intro u hu
-      apply hneg b u.toNNReal
-      · exact_mod_cast (show (S b : ℝ) ≤ (u.toNNReal : ℝ) by rw [Real.coe_toNNReal _ ((S b).coe_nonneg.trans hu.1.le)]; exact hu.1.le)
-      · exact (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
-    dsimp only [M]
-    linarith
-  have hcomp := integral_mono ((hFi T hT).sub (hFi S hS))
-    ((hMi T hT hTB).sub (hMi S hS (fun b => (hST b).trans (hTB b)))) hpw
-  simp only [Pi.sub_apply] at hcomp
-  rw [integral_sub (hFi T hT) (hFi S hS),
-    integral_sub (hMi T hT hTB) (hMi S hS (fun b => (hST b).trans (hTB b)))] at hcomp
-  have heq := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hmc B
-    (C+D*(B+1)) hmb S T hS hT hST hTB U hU
-  linarith
-end ExactSLEGenerator
-
-namespace ExactSLEGreen
-noncomputable def physicalTheta (v : State) : ℝ := Real.pi/2-Real.arctan (X v/Y v)
-noncomputable def thetaDeriv (v : State) : State →L[ℝ] ℝ :=
-  -(Y v/Q v) • X + (X v/Q v) • Y
-noncomputable def physicalV (v : State) : ℝ := Y v/Q v
-noncomputable def velocityDeriv (v : State) : State →L[ℝ] ℝ :=
-  (1/Q v) • Y - (Y v/(Q v)^2) • ((2*X v) • X+(2*Y v) • Y)
-
-lemma physicalTheta_mem (v : State) : physicalTheta v ∈ Ioo 0 Real.pi := by
-  have h := Real.arctan_mem_Ioo (X v/Y v)
-  dsimp only [physicalTheta]
-  constructor <;> linarith [h.1,h.2]
-
-lemma physicalTheta_cot (v : State) : Real.cot (physicalTheta v) = X v/Y v := by
-  rw [physicalTheta,Real.cot_eq_cos_div_sin,Real.cos_pi_div_two_sub,Real.sin_pi_div_two_sub,
-    ← Real.tan_eq_sin_div_cos,Real.tan_arctan]
-
-lemma hasFDerivAt_physicalTheta (v : State) (hv : 0<Y v) :
-    HasFDerivAt physicalTheta (thetaDeriv v) v := by
-  have hy := hv.ne'
-  have hq := (Q_pos hv).ne'
-  have ht : 1+(X v/Y v)^2≠0 := ne_of_gt (by positivity)
-  have hYinv := (hasDerivAt_inv hy).comp_hasFDerivAt v Y.hasFDerivAt
-  have hd := ((X.hasFDerivAt.mul hYinv).arctan).const_sub (Real.pi/2)
-  convert hd using 1 <;> try rfl
-  apply ContinuousLinearMap.ext
-  intro w
-  simp only [thetaDeriv,add_apply,smul_apply,
-    neg_apply,smul_eq_mul,Pi.mul_apply,Function.comp_apply]
-  field_simp
-  dsimp only [Q]
-  ring
-
-lemma thetaDeriv_noise (v : State) : thetaDeriv v noise = physicalV v := by
-  simp [thetaDeriv,noise,physicalV,X,Y]
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TiltedMeasures
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open scoped Classical
+lemma capTime_eq (S : ExactSLEGenerator.Driver → ℝ≥0∞) (n : ℝ≥0) (b : ExactSLEGenerator.Driver)
+    (h : S b ≤ n) : capTime S n b = (S b).toNNReal := by simp only [capTime,min_eq_left h]
+lemma stopped_before_cap (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (n : ℝ≥0) (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    MeasurableSet[(capTime_stopping S hS n).measurableSpace] (U ∩ {b | S b < n}) := by
+  have hc : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (n : WithTop ℝ≥0)) := isStoppingTime_const _ _
+  have he : U ∩ {b | S b < n} = (U ∩ {b | S b ≤ n}) \ {b | S b = n} := by
+    ext b
+    simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,Set.mem_sdiff]
+    exact ⟨fun h => ⟨⟨h.1,h.2.le⟩,h.2.ne⟩,fun h => ⟨h.1.1,lt_of_le_of_ne h.1.2 h.2⟩⟩
+  have hm := (hS.measurableSet_inter_le hc U hU).diff (hS.measurableSet_eq_stopping_time_min hc)
+  have heq : (fun b => (capTime S n b : WithTop ℝ≥0)) = fun b => min (S b) (n : ℝ≥0∞) := funext (capTime_coe S n)
+  rw [he]
+  exact ((hS.min hc).measurableSpace_mono (capTime_stopping S hS n) (fun b => by
+    change min (S b) (n : ℝ≥0∞) ≤ (capTime S n b : ℝ≥0∞)
+    rw [capTime_coe])) _ hm
+
+lemma stopped_cap_measurable (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (n : ℝ≥0) : (capTime_stopping S hS n).measurableSpace ≤ hS.measurableSpace :=
+  (capTime_stopping S hS n).measurableSpace_mono hS (fun b => by
+    change (capTime S n b : ℝ≥0∞) ≤ S b
+    rw [capTime_coe]
+    exact min_le_left _ _)
+lemma stopping_before_measurable (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (n : ℝ≥0) : MeasurableSet {b | S b < n} :=
+  hS.measurableSpace_le _ (hS.measurableSet_lt' n)
+
+lemma eventually_before_nat {v : ℝ≥0∞} (hv : v ≠ ∞) : ∀ᶠ n : ℕ in atTop, v < ((n : ℝ≥0) : ℝ≥0∞) := by
+  obtain ⟨N,hN⟩ := exists_nat_gt v.toNNReal
+  filter_upwards [eventually_ge_atTop N] with n hn
+  rw [← ENNReal.coe_toNNReal hv,ENNReal.coe_lt_coe]
+  exact hN.trans_le (Nat.cast_le.mpr hn)
+
+ 
+
+theorem radial_conditional_weight_all (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    Integrable (fun b => stoppedRadialRemaining a b.1 z ρ (S b)) W ∧
+    (∫ b in U, (if S b < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) ∂W) =
+      ∫ b in U, stoppedRadialRemaining a b.1 z ρ (S b) ∂W := by
+  let C : ℕ → Set ExactSLEGenerator.Driver := fun n => {b | S b < (n : ℝ≥0)}
+  let F := fun n => (C n).indicator (fun b => radialRemaining a b.1 z ρ (capTime S n b))
+  let H := fun n => (C n).indicator (fun b => if (capTime S n b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0)
+  let G := fun b : ExactSLEGenerator.Driver => stoppedRadialRemaining a b.1 z ρ (S b)
+  let K := fun b : ExactSLEGenerator.Driver => if S b < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0
+  let c := Real.exp (-alpha a*Real.log ρ)
+  have hFi (n : ℕ) : Integrable (F n) W :=
+    (radial_conditional_weight W hW a ha z hz ρ hρ hr (capTime S n) (capTime_stopping S hS n)
+      Set.univ MeasurableSet.univ).1.indicator (stopping_before_measurable S hS n)
+  have hHi (n : ℕ) : Integrable (H n) W :=
+    (radial_terminal_integrable W hW a ha z hz ρ hρ hr (capTime S n) (capTime_stopping S hS n)).indicator
+      (stopping_before_measurable S hS n)
+  have hFc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => F n b) atTop (𝓝 (G b)) := by
+    by_cases hb : S b = ∞
+    · have he : (fun n => F n b) = fun _ : ℕ => 0 := by
+        funext n
+        apply Set.indicator_of_notMem
+        simp [C,hb]
+      have hg : G b = 0 := by simp [G,stoppedRadialRemaining,hb]
+      rw [he,hg]
+      exact tendsto_const_nhds
+    · apply Filter.Tendsto.congr' _ tendsto_const_nhds
+      filter_upwards [eventually_before_nat hb] with n hn
+      dsimp only [G,F,stoppedRadialRemaining]
+      rw [ite_eq_right hb,Set.indicator_of_mem (show b ∈ C n from hn),capTime_eq S n b hn.le]
+  have hHc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => H n b) atTop (𝓝 (K b)) := by
+    by_cases hb : S b = ∞
+    · have he : (fun n => H n b) = fun _ : ℕ => 0 := by
+        funext n
+        apply Set.indicator_of_notMem
+        simp [C,hb]
+      have hk : K b = 0 := by simp [K,hb]
+      rw [he,hk]
+      exact tendsto_const_nhds
+    · apply Filter.Tendsto.congr' _ tendsto_const_nhds
+      filter_upwards [eventually_before_nat hb] with n hn
+      dsimp only [K,H]
+      rw [Set.indicator_of_mem (show b ∈ C n from hn),capTime_coe,min_eq_left hn.le]
+  have hFb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖F n b‖ ≤ c := by
+    by_cases hb : b ∈ C n
+    · dsimp only [F]
+      rw [Set.indicator_of_mem hb]
+      exact radialRemaining_bound a ha z hz ρ hρ hr b (capTime S n b)
+    · simpa only [F,Set.indicator_of_notMem hb,norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
+  have hHb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖H n b‖ ≤ c := by
+    by_cases hb : b ∈ C n
+    · dsimp only [H]
+      rw [Set.indicator_of_mem hb]
+      split_ifs
+      · have h := radialWeight_bound a ha z hz ρ hρ hr b
+        simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
+      · simpa only [norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
+    · simpa only [H,Set.indicator_of_notMem hb,norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
+  have hGi : Integrable G W := Integrable.of_bound
+    (aestronglyMeasurable_of_tendsto_ae atTop (fun n => (hFi n).aestronglyMeasurable) (Eventually.of_forall hFc)) c
+    (Eventually.of_forall fun b => le_of_tendsto (hFc b).norm (Eventually.of_forall fun n => hFb n b))
+  refine ⟨hGi,?_⟩
+  have he (n : ℕ) : (∫ b in U, F n b ∂W) = ∫ b in U, H n b ∂W := by
+    have hC := stopping_before_measurable S hS n
+    dsimp only [F,H]
+    rw [integral_indicator hC,integral_indicator hC,Measure.restrict_restrict hC,Set.inter_comm (C n) U]
+    exact (radial_conditional_weight W hW a ha z hz ρ hρ hr (capTime S n) (capTime_stopping S hS n)
+      (U ∩ C n) (stopped_before_cap S hS n U hU)).2.symm
+  have hFl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => c)
+    (fun n => (hFi n).aestronglyMeasurable.restrict) (integrable_const c)
+    (fun n => Eventually.of_forall (hFb n)) (Eventually.of_forall hFc)
+  have hHl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => c)
+    (fun n => (hHi n).aestronglyMeasurable.restrict) (integrable_const c)
+    (fun n => Eventually.of_forall (hHb n)) (Eventually.of_forall hHc)
+  have heq : (fun n => ∫ b in U, F n b ∂W) = fun n => ∫ b in U, H n b ∂W := funext he
+  rw [heq] at hFl
+  exact tendsto_nhds_unique hHl hFl
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open scoped Classical
+
+noncomputable def radialLaw (W : Measure ExactSLEGenerator.Driver) (a : ℝ) (z : ℂ)
+    (ρ : ℝ) : Measure ExactSLEGenerator.Driver :=
+  W.withDensity (fun b => ENNReal.ofReal (radialWeight a b.1 z ρ / physicalGreen a (z,0)))
+
+lemma radialTime_stopping (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
+    IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z ρ) := by
+  intro t
+  let T := radialStop a ha z hz ρ hρ hr
+  let L := fun n => radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let F := fun n (b : ExactSLEGenerator.Driver) =>
+    physicalRadius (localState a (L n).ε (L n).hε z (min (T n b) t) b)
+  have hFm (n : ℕ) : Measurable[rawFiltration t] (F n) := by
+    have hT := radialStop_stopping a ha z hz ρ hρ hr n
+    have hmin : IsStoppingTime rawFiltration (fun b => (min (T n b) t : WithTop ℝ≥0)) := by
+      simpa only [WithTop.coe_min] using hT.min (isStoppingTime_const rawFiltration t)
+    have hle : hmin.measurableSpace ≤ rawFiltration t :=
+      hmin.measurableSpace_le_of_le_const (fun b => WithTop.coe_le_coe.mpr (min_le_right _ _))
+    have hA : StronglyAdapted rawFiltration (localState a (L n).ε (L n).hε z) :=
+      fun u => (localState_adapted a (L n).ε (L n).hε z u).stronglyMeasurable
+    exact physicalRadius_continuous.measurable.comp
+      ((measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
+        (localState_continuous a (L n).ε (L n).hε z)) hmin).mono hle le_rfl)
+  have he : {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ ≤ (t : ℝ≥0∞)} =
+      ⋃ n : ℕ, {b | T n b ≤ t} ∩ {b | F n b = ρ} := by
+    ext b
+    simp only [Set.mem_ofPred_eq,Set.mem_iUnion,Set.mem_inter_iff]
+    constructor
+    · intro hb
+      have hfin : radialTime a b.1 z ρ ≠ ∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hb
+      obtain ⟨n,hn⟩ := (radialStop_eventually_eq a ha z hz ρ hρ hr b hfin).exists
+      have hnt : T n b ≤ t := by
+        apply ENNReal.coe_le_coe.mp
+        exact hn.trans_le hb
+      refine ⟨n,hnt,?_⟩
+      dsimp only [F]
+      rw [min_eq_left hnt]
+      have hr' := (radialTime_attained a ha z hz ρ hρ hr b hfin).2
+      rw [← hn,ENNReal.toNNReal_coe] at hr'
+      have hact := (regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ
+        (angleThreshold z n) (L n).B (L n).initial b (T n b) le_rfl).2
+      rw [hact] at hr'
+      exact hr'
+    · rintro ⟨n,hnt,hR⟩
+      dsimp only [F] at hR
+      rw [min_eq_left hnt] at hR
+      have htime := regionStop_eq_radialTime a (L n).ε (L n).hε z (L n).N ρ
+        (angleThreshold z n) (L n).B (L n).initial b hR
+      rw [htime]
+      exact ENNReal.coe_le_coe.mpr hnt
+  change MeasurableSet[rawFiltration t] {b : ExactSLEGenerator.Driver | radialTime a b.1 z ρ ≤ (t : ℝ≥0∞)}
+  rw [he]
+  apply MeasurableSet.iUnion
+  intro n
+  have hT := radialStop_stopping a ha z hz ρ hρ hr n
+  have htm : MeasurableSet[rawFiltration t] {b | T n b ≤ t} := by
+    simpa only [WithTop.coe_le_coe] using hT t
+  exact htm.inter (measurableSet_eq_fun (hFm n) measurable_const)
+
+lemma radialTime_le_of_hit (a : ℝ) (z : ℂ) (ρ : ℝ) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (ht : alive a b.1 t z) (hR : physicalRadius (actualState a b.1 t z) = ρ) :
+    radialTime a b.1 z ρ ≤ (t : ℝ≥0∞) :=
+  sInf_le ⟨t,rfl,ht,hR⟩
+
+lemma radialTime_strict_of_finite (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im)
+    (b : ExactSLEGenerator.Driver) (hfin : radialTime a b.1 z ρ ≠ ∞) :
+    radialTime a b.1 z σ < radialTime a b.1 z ρ := by
+  have hr : ρ ≤ z.im := hρσ.le.trans hσ
+  obtain ⟨n,hn⟩ := (radialStop_eventually_eq a ha z hz ρ hρ hr b hfin).exists
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T := radialStop a ha z hz ρ hρ hr n b
+  let f := fun t : ℝ≥0 => physicalRadius (localState a L.ε L.hε z t b)
+  have hfc : Continuous f := physicalRadius_continuous.comp (localState_continuous a L.ε L.hε z b)
+  have hR : f T = ρ := by
+    have h := (radialTime_attained a ha z hz ρ hρ hr b hfin).2
+    rw [← hn,ENNReal.toNNReal_coe] at h
+    rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl).2] at h
+    exact h
+  have h0 : f 0 = z.im := by simp [f,localState_zero,physicalRadius,Y,J]
+  obtain ⟨u,hu,huR⟩ := intermediate_value_Icc' (show (0 : ℝ≥0) ≤ T from bot_le) hfc.continuousOn
+    (show σ ∈ Icc (f T) (f 0) by rw [hR,h0]; exact ⟨hρσ.le,hσ⟩)
+  have huT : u < T := hu.2.lt_of_ne (by intro he; rw [he,hR] at huR; exact hρσ.ne huR)
+  have hact := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b u hu.2
+  have hle : radialTime a b.1 z σ ≤ (u : ℝ≥0∞) :=
+    radialTime_le_of_hit a z σ b u hact.1 (by rw [hact.2]; exact huR)
+  exact hle.trans_lt ((ENNReal.coe_lt_coe.mpr huT).trans_eq hn)
+
+lemma radialTime_strict_of_first_finite (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im)
+    (b : ExactSLEGenerator.Driver) (hfin : radialTime a b.1 z σ ≠ ∞) :
+    radialTime a b.1 z σ < radialTime a b.1 z ρ := by
+  by_cases h : radialTime a b.1 z ρ = ∞
+  · rw [h]
+    exact lt_top_iff_ne_top.mpr hfin
+  · exact radialTime_strict_of_finite a ha z hz ρ σ hρ hρσ hσ b h
+
+lemma radialWeight_consistency
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im)
+    (hS : IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z σ))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    (∫ b in U, radialWeight a b.1 z ρ ∂W) = ∫ b in U, radialWeight a b.1 z σ ∂W := by
+  have ha0 : 0 < a := by linarith
+  have hsmall (b : ExactSLEGenerator.Driver) :
+      (if radialTime a b.1 z σ < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) = radialWeight a b.1 z ρ := by
+    by_cases hf : radialTime a b.1 z ρ = ∞
+    · simp [radialWeight,hf]
+    · rw [ite_eq_left (radialTime_strict_of_finite a ha0 z hz ρ σ hρ hρσ hσ b hf)]
+  have hlarge (b : ExactSLEGenerator.Driver) :
+      stoppedRadialRemaining a b.1 z ρ (radialTime a b.1 z σ) = radialWeight a b.1 z σ := by
+    by_cases hf : radialTime a b.1 z σ = ∞
+    · simp [stoppedRadialRemaining,radialWeight,hf]
+    · have hat := (radialTime_attained a ha0 z hz σ (hρ.trans hρσ) hσ b hf).1
+      have hlt := radialTime_strict_of_first_finite a ha0 z hz ρ σ hρ hρσ hσ b hf
+      rw [stoppedRadialRemaining,ite_eq_right hf,radialRemaining,radialWeight,ite_eq_right hf]
+      rw [ite_eq_left ⟨hat,by simpa only [ENNReal.coe_toNNReal hf] using hlt⟩]
+  have hc := (radial_conditional_weight_all W hW a ha z hz ρ hρ (hρσ.le.trans hσ)
+    (fun b => radialTime a b.1 z σ) hS U hU).2
+  simpa only [hsmall,hlarge] using hc
+
+lemma radialLaw_apply
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet U) :
+    radialLaw W a z ρ U = ENNReal.ofReal ((∫ b in U, radialWeight a b.1 z ρ ∂W) / physicalGreen a (z,0)) := by
+  rw [radialLaw,MeasureTheory.withDensity_apply _ hU]
+  have hi := ((radialWeight_mean W hW a ha z hz ρ hρ hr).1.restrict (s := U)).div_const (physicalGreen a (z,0))
+  have hn : 0 ≤ᵐ[W.restrict U] (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ / physicalGreen a (z,0)) :=
+    Eventually.of_forall (fun b => div_nonneg (radialWeight_bound a ha z hz ρ hρ hr b).1 (Real.exp_pos _).le)
+  rw [← ofReal_integral_eq_lintegral_ofReal hi hn,integral_div]
+
+lemma radialLaw_probability
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
+    IsProbabilityMeasure (radialLaw W a z ρ) := by
+  rw [isProbabilityMeasure_iff,radialLaw_apply W hW a ha z hz ρ hρ hr Set.univ MeasurableSet.univ,
+    Measure.restrict_univ,(radialWeight_mean W hW a ha z hz ρ hρ hr).2,
+    div_self (show physicalGreen a (z,0) ≠ 0 from (Real.exp_pos _).ne'),ENNReal.ofReal_one]
+
+theorem finite_radial_laws_consistent
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ σ : ℝ) (hρ : 0 < ρ) (hρσ : ρ < σ) (hσ : σ ≤ z.im) :
+    IsProbabilityMeasure (radialLaw W a z ρ) ∧ IsProbabilityMeasure (radialLaw W a z σ) ∧
+    ∃ hS : IsStoppingTime rawFiltration (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z σ),
+      ∀ U, MeasurableSet[hS.measurableSpace] U → radialLaw W a z ρ U = radialLaw W a z σ U := by
+  have hσ0 : 0 < σ := hρ.trans hρσ
+  have hr : ρ ≤ z.im := hρσ.le.trans hσ
+  have ha0 : 0 < a := by linarith
+  have hS := radialTime_stopping a ha0 z hz σ hσ0 hσ
+  refine ⟨radialLaw_probability W hW a ha z hz ρ hρ hr,
+    radialLaw_probability W hW a ha z hz σ hσ0 hσ,hS,?_⟩
+  intro U hU
+  have hU' : MeasurableSet U := hS.measurableSpace_le _ hU
+  rw [radialLaw_apply W hW a ha z hz ρ hρ hr U hU',
+    radialLaw_apply W hW a ha z hz σ hσ0 hσ U hU',
+    radialWeight_consistency W hW a ha z hz ρ σ hρ hρσ hσ hS U hU]
+
+end ExactSLEGreen
+
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+theorem compact_test_optional_inequality
+    (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (X : ℝ≥0 → Driver → E) (hXc : ∀ b, Continuous (fun t => X t b))
+    (hXm : ∀ t, Measurable[rawFiltration t] (X t))
+    (A : E → E) (hAc : Continuous A) (K : ℝ) (hK : 0 ≤ K) (hAb : ∀ x, ‖A x‖ ≤ K)
+    (e : E) (he : ‖e‖ ≤ 1)
+    (hXeq : ∀ (s h : ℝ≥0) b, X (s+h) b = X s b + increment s h b • e +
+      ∫ t in (s : ℝ)..((s+h : ℝ≥0) : ℝ), A (X t.toNNReal b))
+    (f : E → ℝ) (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hfc : HasCompactSupport f)
+    (S T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b ≤ T b) (hTB : ∀ b, T b ≤ B)
+    (hneg : ∀ b t, S b ≤ t → t ≤ T b → generator f A e (X t b) ≤ 0)
+    (U : Set Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    ∫ b in U, f (X (T b) b) ∂W ≤ ∫ b in U, f (X (S b) b) ∂W := by
+  let M : ℝ≥0 → Driver → ℝ := fun t b => f (X t b) -
+    ∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)
+  have hM : Martingale M rawFiltration W :=
+    compact_smooth_brownian_martingale W hW X hXc hXm A hAc K hK hAb e he hXeq f hf hfc
+  have hgc (b : Driver) : Continuous (fun u : ℝ => generator f A e (X u.toNNReal b)) :=
+    (generator_continuous f hf A hAc e).comp ((hXc b).comp continuous_real_toNNReal)
+  have hmc (b : Driver) : Continuous (fun t => M t b) :=
+    (hf.continuous.comp (hXc b)).sub
+      ((intervalIntegral.continuous_primitive (fun s t => (hgc b).intervalIntegrable s t) 0).comp NNReal.continuous_coe)
+  obtain ⟨C,D,hC,hD,hfb,hgb⟩ := compact_generator_bounds f hf hfc A K hK hAb e
+  have hmb (t : ℝ≥0) (ht : t ≤ B+1) (b : Driver) : ‖M t b‖ ≤ C+D*(B+1) := by
+    calc
+      _ ≤ ‖f (X t b)‖ + ‖∫ u in (0 : ℝ)..(t : ℝ), generator f A e (X u.toNNReal b)‖ := norm_sub_le _ _
+      _ ≤ C+D*(t : ℝ) := add_le_add (hfb _) (by
+        simpa only [sub_zero,abs_of_nonneg t.coe_nonneg] using
+          intervalIntegral.norm_integral_le_of_norm_le_const (a := (0 : ℝ)) (b := (t : ℝ))
+            (fun u _ => hgb (X u.toNNReal b)))
+      _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hD)
+  have hFa : StronglyAdapted rawFiltration (fun t b => f (X t b)) := fun t =>
+    (hf.continuous.measurable.comp (hXm t)).stronglyMeasurable
+  have hFm (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
+      Measurable (fun b => f (X (R b) b)) :=
+    (measurable_stoppedValue (hFa.isStronglyProgressive_of_continuous
+      (fun b => hf.continuous.comp (hXc b))) hR).mono hR.measurableSpace_le le_rfl
+  have hFi (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) :
+      Integrable (fun b => f (X (R b) b)) (W.restrict U) :=
+    Integrable.of_bound (hFm R hR).aestronglyMeasurable C (Filter.Eventually.of_forall fun b => hfb _)
+  have hMi (R : Driver → ℝ≥0)
+      (hR : IsStoppingTime rawFiltration (fun b => (R b : WithTop ℝ≥0))) (hRB : ∀ b, R b≤B) :
+      Integrable (fun b => M (R b) b) (W.restrict U) :=
+    Integrable.of_bound (ExactSLEOptional.stopped_measurable rawFiltration hM hmc R hR).aestronglyMeasurable
+      (C+D*(B+1)) (Filter.Eventually.of_forall fun b => hmb _ ((hRB b).trans (le_add_of_nonneg_right zero_le_one)) b)
+  have hpw (b : Driver) : f (X (T b) b) - f (X (S b) b) ≤ M (T b) b - M (S b) b := by
+    have hi := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
+      ((hgc b).intervalIntegrable 0 (S b)) ((hgc b).intervalIntegrable (S b) (T b))
+    have hn : (∫ u in (S b : ℝ)..(T b : ℝ), generator f A e (X u.toNNReal b)) ≤ 0 := by
+      rw [intervalIntegral.integral_of_le (by exact_mod_cast hST b)]
+      apply setIntegral_nonpos measurableSet_Ioc
+      intro u hu
+      apply hneg b u.toNNReal
+      · exact_mod_cast (show (S b : ℝ) ≤ (u.toNNReal : ℝ) by rw [Real.coe_toNNReal _ ((S b).coe_nonneg.trans hu.1.le)]; exact hu.1.le)
+      · exact (Real.toNNReal_le_toNNReal hu.2).trans_eq (by simp)
+    dsimp only [M]
+    linarith
+  have hcomp := integral_mono ((hFi T hT).sub (hFi S hS))
+    ((hMi T hT hTB).sub (hMi S hS (fun b => (hST b).trans (hTB b)))) hpw
+  simp only [Pi.sub_apply] at hcomp
+  rw [integral_sub (hFi T hT) (hFi S hS),
+    integral_sub (hMi T hT hTB) (hMi S hS (fun b => (hST b).trans (hTB b)))] at hcomp
+  have heq := ExactSLEOptional.setIntegral_optional_sampling rawFiltration hM hmc B
+    (C+D*(B+1)) hmb S T hS hT hST hTB U hU
+  linarith
+end ExactSLEGenerator
+
+namespace ExactSLEGreen
+noncomputable def physicalTheta (v : State) : ℝ := Real.pi/2-Real.arctan (X v/Y v)
+noncomputable def thetaDeriv (v : State) : State →L[ℝ] ℝ :=
+  -(Y v/Q v) • X + (X v/Q v) • Y
+noncomputable def physicalV (v : State) : ℝ := Y v/Q v
+noncomputable def velocityDeriv (v : State) : State →L[ℝ] ℝ :=
+  (1/Q v) • Y - (Y v/(Q v)^2) • ((2*X v) • X+(2*Y v) • Y)
+
+lemma physicalTheta_mem (v : State) : physicalTheta v ∈ Ioo 0 Real.pi := by
+  have h := Real.arctan_mem_Ioo (X v/Y v)
+  dsimp only [physicalTheta]
+  constructor <;> linarith [h.1,h.2]
+
+lemma physicalTheta_cot (v : State) : Real.cot (physicalTheta v) = X v/Y v := by
+  rw [physicalTheta,Real.cot_eq_cos_div_sin,Real.cos_pi_div_two_sub,Real.sin_pi_div_two_sub,
+    ← Real.tan_eq_sin_div_cos,Real.tan_arctan]
+
+lemma hasFDerivAt_physicalTheta (v : State) (hv : 0<Y v) :
+    HasFDerivAt physicalTheta (thetaDeriv v) v := by
+  have hy := hv.ne'
+  have hq := (Q_pos hv).ne'
+  have ht : 1+(X v/Y v)^2≠0 := ne_of_gt (by positivity)
+  have hYinv := (hasDerivAt_inv hy).comp_hasFDerivAt v Y.hasFDerivAt
+  have hd := ((X.hasFDerivAt.mul hYinv).arctan).const_sub (Real.pi/2)
+  convert hd using 1 <;> try rfl
+  apply ContinuousLinearMap.ext
+  intro w
+  simp only [thetaDeriv,add_apply,smul_apply,
+    neg_apply,smul_eq_mul,Pi.mul_apply,Function.comp_apply]
+  field_simp
+  dsimp only [Q]
+  ring
+
+lemma thetaDeriv_noise (v : State) : thetaDeriv v noise = physicalV v := by
+  simp [thetaDeriv,noise,physicalV,X,Y]
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadiusComparison.lean b/lean/OAI/Probability/SLEGauge/RadiusComparison.lean
index f41ce7d9d525d9e77154b0c7208939083fbed8fe..d873f67a1304b936cb90cca9c213016267107e05 100644
--- a/lean/OAI/Probability/SLEGauge/RadiusComparison.lean
+++ b/lean/OAI/Probability/SLEGauge/RadiusComparison.lean
@@ -1,451 +1,451 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TwoPointGrowth
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-lemma actual_radius_eq_full {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {t : ℝ≥0} (ht : alive a b t z) :
-    2*physicalRadius (actualState a b t z)=fullLoewnerRadius a b z t := by
-  have hd := norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb ht)
-  simp only [physicalRadius,actualState,Y_apply,J_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,
-    Real.exp_neg,Real.exp_log hd,fullLoewnerRadius,div_eq_mul_inv]
-  ring
-lemma physicalClock_radius (a cap : ℝ) (hcap : 0<cap) (v : State) (hv : 0<Y v) :
-    physicalClock a cap v=Real.log (cap/(2*physicalRadius v))/(2*a) := by
-  rw [physicalClock,physicalRadius,Real.log_div hcap.ne' (by norm_num : (2:ℝ)≠0),
-    Real.log_div hcap.ne' (show 2*(Y v*Real.exp (-J v))≠0 by positivity),
-    Real.log_mul (by norm_num : (2:ℝ)≠0) (by positivity),
-    Real.log_mul hv.ne' (Real.exp_ne_zero _),Real.log_exp]
-  ring
-lemma actual_clock_eq {a cap : ℝ} (ha : a≠0) (hcap : 0<cap)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z : ℂ} {T σ t : ℝ≥0} {f : ℝ → ℂ}
-    (hf : odeUntil a b z T f) (hσ : σ≤T) (ht : t≤T)
-    (hR : fullLoewnerRadius a b z σ=cap) :
-    physicalClock a cap (actualState a b t z)=activeODEClock b T σ f t := by
-  have hy : 0<Y (actualState a b t z) := by
-    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq (odeUntil_mono hf ht)]
-      using (hf.2 t ⟨t.coe_nonneg,by exact_mod_cast ht⟩).1
-  rw [physicalClock_radius a cap hcap _ hy,actual_radius_eq_full hb ⟨f,odeUntil_mono hf ht⟩,
-    fullRadius_active_clock hb hf hσ ht,hR]
-  have he : cap/(cap*Real.exp (-2*a*activeODEClock b T σ f t))=
-      Real.exp (2*a*activeODEClock b T σ f t) := by
-    rw [show -2*a*activeODEClock b T σ f t = -(2*a*activeODEClock b T σ f t) by ring,
-      Real.exp_neg,div_mul_eq_div_div,div_self hcap.ne',one_div,inv_inv]
-  rw [he,Real.log_exp]
-  field_simp
-lemma actual_speed_eq {a : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {T t : ℝ≥0} {f : ℝ → ℂ}
-    (hf : odeUntil a b z T f) (ht : t≤T) :
-    physicalV (actualState a b t z)=odeSpeed b T f t := by
-  rw [odeSpeed_eq f ⟨t.coe_nonneg,by exact_mod_cast ht⟩]
-  simp only [physicalV,actualState,Y_apply,Q_eq,loewnerMap_eq (odeUntil_mono hf ht),
-    Complex.sub_im,Complex.ofReal_im,sub_zero,Real.toNNReal_coe]
-end ExactSLEGreen
-namespace ExactSLEMain
-lemma exp_clock_integral {a σ off : ℝ} (hσ : σ≠0)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z : ℂ} {T A S : ℝ≥0} {f : ℝ → ℂ}
-    (hf : odeUntil a b z T f) :
-    (∫ t in (S:ℝ)..(T:ℝ), Real.exp (off+σ*activeODEClock b T A f t)*odeSpeed b T f t^2)=
-      (Real.exp (off+σ*activeODEClock b T A f T)-
-        Real.exp (off+σ*activeODEClock b T A f S))/σ := by
-  let u := activeODEClock b T A f
-  let v := odeSpeed b T f
-  have hc : Continuous u := continuous_iff_continuousAt.mpr fun t => (activeODEClock_deriv hb hf A t).continuousAt
-  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
-    (f:=fun t => Real.exp (off+σ*u t)/σ)
-    (f':=fun t => Real.exp (off+σ*u t)*v t^2)
-    (a:=(S:ℝ)) (b:=(T:ℝ))
-    (fun t _ => by
-      exact (((((activeODEClock_deriv hb hf A t).const_mul σ).const_add off).exp).div_const σ).congr_deriv (by
-        dsimp only [u,v]; field_simp [hσ]))
-    (((Real.continuous_exp.comp (continuous_const.add (continuous_const.mul hc))).mul
-      ((odeSpeed_continuous hb hf).pow 2)).intervalIntegrable _ _)
-  simpa only [sub_div] using he
-end ExactSLEMain
-namespace ExactSLEMain
-lemma activeODEClock_monotone {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T A : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f) :
-    Monotone (activeODEClock b T A f) := by
-  exact monotone_of_hasDerivAt_nonneg (fun t => activeODEClock_deriv hb hf A t) (fun t => sq_nonneg _)
-end ExactSLEMain
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
- 
-
-theorem actual_regularized_pair_discount {a : ℝ} (ha : 1/4<a)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z w : ℂ} {f g : ℝ → ℂ}
-    {T S A B : ℝ≥0} (hST : S≤T) (hA : A≤S) (hB : B≤S)
-    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
-    {cap₁ cap₂ U₁ U₂ σ₁ σ₂ off₁ off₂ : ℝ}
-    (hcap₁ : 0<cap₁) (hcap₂ : 0<cap₂)
-    (hR₁ : fullLoewnerRadius a b z A=cap₁) (hR₂ : fullLoewnerRadius a b w B=cap₂)
-    (hsep₁ : cap₁≤‖z-w‖) (hsep₂ : cap₂≤‖z-w‖)
-    (hp₁ : (σ₁=-regBeta a ∧ off₁=0) ∨
-      (σ₁=regBeta a ∧ off₁=-regBeta a*U₁ ∧ U₁/2≤activeODEClock b T A f S))
-    (hp₂ : (σ₂=-regBeta a ∧ off₂=0) ∨
-      (σ₂=regBeta a ∧ off₂=-regBeta a*U₂ ∧ U₂/2≤activeODEClock b T B g S)) :
-    (∫ t in (S:ℝ)..(T:ℝ),
-      (physicalDelta a cap₁ σ₁ off₁ (actualState a b t.toNNReal z)*
-       physicalDelta a cap₂ σ₂ off₂ (actualState a b t.toNNReal w))^(-regM a)*
-       physicalV (actualState a b t.toNNReal z)*physicalV (actualState a b t.toNNReal w)) ≤
-      Real.exp (2*(regM a*regBeta a))/(2*a*(1-Real.exp (2*(regM a*regBeta a)-a))) := by
-  have ha0 : 0<a := by linarith
-  have hM := regM_pos ha
-  have hβ := regBeta_pos ha
-  let u := activeODEClock b T A f
-  let v := activeODEClock b T B g
-  let V := odeSpeed b T f
-  let W := odeSpeed b T g
-  have huc : Continuous u := continuous_iff_continuousAt.mpr fun t => (activeODEClock_deriv hb hf A t).continuousAt
-  have hvc : Continuous v := continuous_iff_continuousAt.mpr fun t => (activeODEClock_deriv hb hg B t).continuousAt
-  let F := fun t => Real.exp ((off₁+σ₁*u t+off₂+σ₂*v t)*(-regM a))*V t*W t
-  let G := fun t => Real.exp ((regM a*regBeta a)*(u t+v t))*V t*W t
-  have hFc : Continuous F := (((Real.continuous_exp.comp
-    (((continuous_const.add (continuous_const.mul huc)).add continuous_const).add
-      (continuous_const.mul hvc) |>.mul continuous_const)).mul (odeSpeed_continuous hb hf)).mul
-        (odeSpeed_continuous hb hg))
-  have hGc : Continuous G := ((Real.continuous_exp.comp
-    (continuous_const.mul (huc.add hvc))).mul (odeSpeed_continuous hb hf)).mul (odeSpeed_continuous hb hg)
-  have hST' : (S:ℝ)≤T := by exact_mod_cast hST
-  have he : (∫ t in (S:ℝ)..(T:ℝ),
-      (physicalDelta a cap₁ σ₁ off₁ (actualState a b t.toNNReal z)*
-       physicalDelta a cap₂ σ₂ off₂ (actualState a b t.toNNReal w))^(-regM a)*
-       physicalV (actualState a b t.toNNReal z)*physicalV (actualState a b t.toNNReal w))=
-      ∫ t in (S:ℝ)..(T:ℝ), F t := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    have ht' : t∈Icc (S:ℝ) T := by simpa only [uIcc_of_le hST'] using ht
-    have ht0 := S.coe_nonneg.trans ht'.1
-    have htT : t.toNNReal≤T := by exact_mod_cast (show (t.toNNReal:ℝ)≤T by simpa only [Real.coe_toNNReal _ ht0] using ht'.2)
-    simp only [physicalDelta]
-    rw [actual_clock_eq ha0.ne' hcap₁ hb hf (hA.trans hST) htT hR₁,
-      actual_clock_eq ha0.ne' hcap₂ hb hg (hB.trans hST) htT hR₂,
-      actual_speed_eq hf htT,actual_speed_eq hg htT,Real.coe_toNNReal _ ht0,←Real.exp_add,
-      Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
-    dsimp only [F,u,v,V,W]
-    congr 3
-    ring
-  rw [he]
-  calc
-    _ ≤ ∫ t in (S:ℝ)..(T:ℝ), G t := by
-      apply intervalIntegral.integral_mono_on hST' (hFc.intervalIntegrable _ _) (hGc.intervalIntegrable _ _)
-      intro t ht
-      have ht0 : 0≤t := S.coe_nonneg.trans ht.1
-      have hV : 0≤V t := by
-        change 0≤odeSpeed b T f t
-        rw [odeSpeed_eq f ⟨ht0,ht.2⟩]
-        exact div_nonneg ((hf.2 t ⟨ht0,ht.2⟩).1.le) (Complex.normSq_nonneg _)
-      have hW : 0≤W t := by
-        change 0≤odeSpeed b T g t
-        rw [odeSpeed_eq g ⟨ht0,ht.2⟩]
-        exact div_nonneg ((hg.2 t ⟨ht0,ht.2⟩).1.le) (Complex.normSq_nonneg _)
-      have h1 : -(off₁+σ₁*u t)≤regBeta a*u t := by
-        rcases hp₁ with ⟨rfl,rfl⟩|⟨rfl,rfl,hmid⟩
-        · simp
-        · have hu : U₁/2≤u t := hmid.trans (activeODEClock_monotone hb hf ht.1)
-          nlinarith [mul_le_mul_of_nonneg_left hu hβ.le]
-      have h2 : -(off₂+σ₂*v t)≤regBeta a*v t := by
-        rcases hp₂ with ⟨rfl,rfl⟩|⟨rfl,rfl,hmid⟩
-        · simp
-        · have hv : U₂/2≤v t := hmid.trans (activeODEClock_monotone hb hg ht.1)
-          nlinarith [mul_le_mul_of_nonneg_left hv hβ.le]
-      apply mul_le_mul_of_nonneg_right _ hW
-      apply mul_le_mul_of_nonneg_right _ hV
-      apply Real.exp_le_exp.mpr
-      have hh := mul_le_mul_of_nonneg_left (add_le_add h1 h2) hM.le
-      nlinarith only [hh]
-    _ ≤ _ := actual_pair_interaction ha0 (mul_pos hM hβ).le
-      (by nlinarith [regBeta_discount ha]) hb hST hA hB hf hg hzw
-      (hR₁.le.trans hsep₁) (hR₂.le.trans hsep₂)
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-lemma actual_regularized_single_discount {a : ℝ} (ha : 1/4<a)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z : ℂ} {f : ℝ → ℂ}
-    {T S A : ℝ≥0} (hST : S≤T) (hA : A≤S) (hf : odeUntil a b z T f)
-    {cap U σ off : ℝ} (hcap : 0<cap) (hR : fullLoewnerRadius a b z A=cap)
-    (hp : (σ=-regBeta a ∧ off=0) ∨
-      (σ=regBeta a ∧ off=-regBeta a*U ∧ activeODEClock b T A f T≤U)) :
-    (∫ t in (S:ℝ)..(T:ℝ), physicalDelta a cap σ off (actualState a b t.toNNReal z)*
-      physicalV (actualState a b t.toNNReal z)^2) ≤ 1/regBeta a := by
-  have ha0 : 0<a := by linarith
-  have hβ := regBeta_pos ha
-  have hST' : (S:ℝ)≤T := by exact_mod_cast hST
-  let u := activeODEClock b T A f
-  have he : (∫ t in (S:ℝ)..(T:ℝ), physicalDelta a cap σ off (actualState a b t.toNNReal z)*
-      physicalV (actualState a b t.toNNReal z)^2)=
-      ∫ t in (S:ℝ)..(T:ℝ), Real.exp (off+σ*u t)*odeSpeed b T f t^2 := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    have ht' : t∈Icc (S:ℝ) T := by simpa only [uIcc_of_le hST'] using ht
-    have ht0 := S.coe_nonneg.trans ht'.1
-    have htT : t.toNNReal≤T := by exact_mod_cast (show (t.toNNReal:ℝ)≤T by simpa only [Real.coe_toNNReal _ ht0] using ht'.2)
-    simp only [physicalDelta]
-    rw [actual_clock_eq ha0.ne' hcap hb hf (hA.trans hST) htT hR,
-      actual_speed_eq hf htT,Real.coe_toNNReal _ ht0]
-  rw [he]
-  rcases hp with ⟨rfl,rfl⟩|⟨rfl,rfl,hU⟩
-  · rw [exp_clock_integral (neg_ne_zero.mpr hβ.ne') hb hf]
-    have hu : 0≤u S := activeODEClock_nonneg (by exact_mod_cast hA)
-    have hex : Real.exp (0+-regBeta a*u S)≤1 := Real.exp_le_one_iff.mpr (by nlinarith)
-    have hnon := (Real.exp_pos (0+-regBeta a*u T)).le
-    rw [div_neg,←neg_div,neg_sub]
-    exact (div_le_div_iff_of_pos_right hβ).mpr (by linarith)
-  · rw [exp_clock_integral hβ.ne' hb hf]
-    have hex : Real.exp (-regBeta a*U+regBeta a*u T)≤1 := Real.exp_le_one_iff.mpr (by
-      have h := mul_le_mul_of_nonneg_left hU hβ.le
-      dsimp only [u] at *
-      linarith)
-    have hnon := (Real.exp_pos (-regBeta a*U+regBeta a*u S)).le
-    exact (div_le_div_iff_of_pos_right hβ).mpr (by linarith)
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open scoped Classical
- 
-
-noncomputable def unfinishedCutoffTop {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
-    (n : ℕ) (b : ExactSLEGenerator.Driver) : ℝ≥0∞ :=
-  min (n:ℝ≥0∞) (Finset.univ.inf fun i =>
-    if (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i)
-    then ∞ else (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞))
-noncomputable def unfinishedCutoff {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
-    (n : ℕ) (b : ExactSLEGenerator.Driver) : ℝ≥0 := (unfinishedCutoffTop a ha z hz ρ hρ hr n b).toNNReal
-end ExactSLEGreen
-
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-namespace ExactSLEStops
-variable {Ω : Type*} [MeasurableSpace Ω]
- 
-def insert (τ : Ω → ℝ≥0) : List (Ω → ℝ≥0) → List (Ω → ℝ≥0)
-  | [] => [τ]
-  | σ::l => (fun ω => min (τ ω) (σ ω)) :: insert (fun ω => max (τ ω) (σ ω)) l
-def sort : List (Ω → ℝ≥0) → List (Ω → ℝ≥0)
-  | [] => []
-  | τ::l => insert τ (sort l)
-omit [MeasurableSpace Ω] in
-lemma insert_length (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) :
-    (insert τ l).length=l.length+1 := by
-  induction l generalizing τ with
-  | nil => rfl
-  | cons σ l ih => simp only [insert,List.length_cons,ih]
-omit [MeasurableSpace Ω] in
-lemma sort_length (l : List (Ω → ℝ≥0)) : (sort l).length=l.length := by
-  induction l with
-  | nil => rfl
-  | cons τ l ih => simp only [sort,insert_length,List.length_cons,ih]
-lemma insert_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0))
-    (hτ : IsStoppingTime F (fun ω => (τ ω : WithTop ℝ≥0)))
-    (hl : ∀ σ ∈ l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0))) :
-    ∀ σ ∈ insert τ l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0)) := by
-  induction l generalizing τ with
-  | nil => simpa only [insert,List.mem_singleton] using fun σ (h : σ=τ) => h ▸ hτ
-  | cons π l ih =>
-    intro σ hσ
-    have hπ := hl π (by simp)
-    rcases List.mem_cons.mp hσ with h|h
-    · subst σ
-      simpa only [WithTop.coe_min] using hτ.min hπ
-    · apply ih (fun ω => max (τ ω) (π ω)) ?_ (fun σ h => hl σ (by simp [h])) σ h
-      simpa only [WithTop.coe_max] using hτ.max hπ
-lemma sort_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (l : List (Ω → ℝ≥0))
-    (hl : ∀ σ ∈ l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0))) :
-    ∀ σ ∈ sort l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0)) := by
-  induction l with
-  | nil => simp [sort]
-  | cons τ l ih =>
-    exact insert_stopping F τ (sort l) (hl τ (by simp)) (ih (fun σ h => hl σ (by simp [h])))
-omit [MeasurableSpace Ω] in
-lemma insert_perm (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (ω : Ω) :
-    ((insert τ l).map (fun σ => σ ω)).Perm ((τ::l).map (fun σ => σ ω)) := by
-  induction l generalizing τ with
-  | nil => exact List.Perm.refl _
-  | cons σ l ih =>
-    simp only [insert,List.map_cons]
-    have h := (ih (fun ω => max (τ ω) (σ ω))).cons (min (τ ω) (σ ω))
-    rcases le_total (τ ω) (σ ω) with ht|ht
-    · simpa only [List.map_cons,min_eq_left ht,max_eq_right ht] using h
-    · simpa only [List.map_cons,min_eq_right ht,max_eq_left ht] using h.trans (List.Perm.swap _ _ _)
-omit [MeasurableSpace Ω] in
-lemma sort_perm (l : List (Ω → ℝ≥0)) (ω : Ω) :
-    ((sort l).map (fun σ => σ ω)).Perm (l.map (fun σ => σ ω)) := by
-  induction l with
-  | nil => exact List.Perm.refl _
-  | cons τ l ih => exact (insert_perm τ (sort l) ω).trans (ih.cons (τ ω))
-omit [MeasurableSpace Ω] in
-lemma insert_sorted (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (ω : Ω)
-    (hl : ((l.map (fun σ => σ ω))).Pairwise (· ≤ ·)) :
-    ((insert τ l).map (fun σ => σ ω)).Pairwise (· ≤ ·) := by
-  induction l generalizing τ with
-  | nil => simp [insert]
-  | cons σ l ih =>
-    simp only [List.map_cons,List.pairwise_cons] at hl
-    simp only [insert,List.map_cons,List.pairwise_cons]
-    refine ⟨?_,ih _ hl.2⟩
-    intro x hx
-    have hx' := (insert_perm (fun ω => max (τ ω) (σ ω)) l ω).mem_iff.mp hx
-    simp only [List.map_cons,List.mem_cons] at hx'
-    rcases hx' with rfl|hx'
-    · exact (min_le_left _ _).trans (le_max_left _ _)
-    · exact (min_le_right _ _).trans (hl.1 x hx')
-omit [MeasurableSpace Ω] in
-lemma sort_sorted (l : List (Ω → ℝ≥0)) (ω : Ω) :
-    ((sort l).map (fun σ => σ ω)).Pairwise (· ≤ ·) := by
-  induction l with
-  | nil => simp [sort]
-  | cons τ l ih => exact insert_sorted τ (sort l) ω ih
- 
-omit [MeasurableSpace Ω] in
-lemma no_change_between (l : List (Ω → ℝ≥0)) (ω : Ω) (k : ℕ)
-    (hk : k+1<(sort l).length) (τ : Ω → ℝ≥0) (hτ : τ∈l) :
-    τ ω ≤ (sort l)[k] ω ∨ (sort l)[k+1] ω ≤ τ ω := by
-  let v := (sort l).map (fun σ => σ ω)
-  have hv : v.Pairwise (· ≤ ·) := sort_sorted l ω
-  have ht : τ ω∈v := (sort_perm l ω).mem_iff.mpr (List.mem_map_of_mem hτ)
-  obtain ⟨i,hi,he⟩ := List.getElem_of_mem ht
-  have hm := List.pairwise_iff_getElem.mp hv
-  by_cases hik : i≤k
-  · left
-    rcases hik.eq_or_lt with rfl|hi'
-    · simpa [v] using he.symm.le
-    · have h := hm i k hi (by simpa [v] using (Nat.lt_of_succ_lt hk)) hi'
-      rw [he] at h
-      simpa only [v,List.getElem_map] using h
-  · right
-    have hki : k+1 ≤ i := Nat.succ_le_of_lt (lt_of_not_ge hik)
-    rcases hki.eq_or_lt with heq|hlt
-    · subst i
-      simpa [v] using he.le
-    · have h := hm (k+1) i (by simpa [v] using hk) hi hlt
-      rw [he] at h
-      simpa only [v,List.getElem_map] using h
-end ExactSLEStops
-namespace ExactSLEStops
-open Set Filter
-open scoped Classical
-variable {Ω : Type*} [MeasurableSpace Ω]
-lemma deferred_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (τ : Ω → ℝ≥0∞) (hτ : IsStoppingTime F τ) (A : Set Ω)
-    (hA : MeasurableSet[hτ.measurableSpace] A) :
-    IsStoppingTime F (fun ω => if ω∈A then ∞ else τ ω) := by
-  intro t
-  change MeasurableSet[F t] {ω | (if ω∈A then ∞ else τ ω)≤(t:ℝ≥0∞)}
-  have he : {ω | (if ω∈A then ∞ else τ ω)≤(t:ℝ≥0∞)}=Aᶜ∩{ω | τ ω≤(t:ℝ≥0∞)} := by
-    ext ω
-    by_cases h : ω∈A <;> simp [h]
-  rw [he]
-  exact (hτ.measurableSet Aᶜ).mp hA.compl |>.2 t
-lemma finset_inf_stopping {ι : Type*} (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
-    (s : Finset ι) (τ : ι → Ω → ℝ≥0∞)
-    (hτ : ∀ i∈s, IsStoppingTime F (τ i)) :
-    IsStoppingTime F (fun ω => s.inf fun i => τ i ω) := by
-  induction s using Finset.induction_on with
-  | empty =>
-    intro t
-    change MeasurableSet[F t] {ω | ((∅ : Finset ι).inf fun i => τ i ω)≤(t:ℝ≥0∞)}
-    simp
-  | @insert i s hi ih =>
-    have h := (hτ i (Finset.mem_insert_self _ _)).min (ih fun j hj => hτ j (Finset.mem_insert_of_mem hj))
-    intro t
-    change MeasurableSet[F t] {ω | ((Insert.insert i s).inf fun i => τ i ω)≤(t:ℝ≥0∞)}
-    have hh := h t
-    change MeasurableSet[F t] {ω | min (τ i ω) (s.inf fun i => τ i ω)≤(t:ℝ≥0∞)} at hh
-    simpa only [Finset.inf_insert] using hh
-omit [MeasurableSpace Ω] in
-lemma sort_bounded (l : List (Ω → ℝ≥0)) (B : Ω → ℝ≥0)
-    (hl : ∀ τ∈l, ∀ ω, τ ω≤B ω) : ∀ τ∈sort l, ∀ ω, τ ω≤B ω := by
-  intro τ hτ ω
-  have h : τ ω∈l.map (fun σ => σ ω) :=
-    (sort_perm l ω).mem_iff.mp (List.mem_map_of_mem hτ)
-  obtain ⟨σ,hσ,he⟩ := List.mem_map.mp h
-  exact he ▸ hl σ hσ ω
-end ExactSLEStops
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEStops
-open scoped Classical
-lemma unfinishedCutoffTop_le {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
-    (n : ℕ) (b : ExactSLEGenerator.Driver) : unfinishedCutoffTop a ha z hz ρ hρ hr n b≤n := min_le_left _ _
-lemma unfinishedCutoff_coe {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
-    (n : ℕ) (b : ExactSLEGenerator.Driver) :
-    (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)=unfinishedCutoffTop a ha z hz ρ hρ hr n b := by
-  exact ENNReal.coe_toNNReal (ne_top_of_le_ne_top (by simp) (unfinishedCutoffTop_le a ha z hz ρ hρ hr n b))
-lemma unfinishedCutoff_stopping {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im) (n : ℕ) :
-    IsStoppingTime rawFiltration (fun b => (unfinishedCutoff a ha z hz ρ hρ hr n b : WithTop ℝ≥0)) := by
-  have htop : IsStoppingTime rawFiltration (unfinishedCutoffTop a ha z hz ρ hρ hr n) := by
-    apply (isStoppingTime_const rawFiltration (n:ℝ≥0)).min
-    apply finset_inf_stopping
-    intro i hi
-    let A := {b | (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i)}
-    have htau := radialStop_stopping a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n
-    have hA : MeasurableSet[htau.measurableSpace] A := htau.measurableSet_eq_stopping_time
-      (radialTime_stopping a ha (z i) (hz i) (ρ i) (hρ i) (hr i))
-    intro t
-    change MeasurableSet[rawFiltration t] {b | (if (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i) then ∞ else (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞))≤(t:ℝ≥0∞)}
-    have hd := ((htau.measurableSet Aᶜ).mp hA.compl).2 t
-    convert hd using 1
-    ext b
-    by_cases he : b∈A
-    · simp only [A,Set.mem_ofPred_eq] at he
-      simp [he,A]
-      try exact WithTop.coe_le_coe.symm
-    · simp only [A,Set.mem_ofPred_eq] at he
-      simp [he,A]
-      try exact WithTop.coe_le_coe.symm
-
-  intro t
-  have h := htop t
-  change MeasurableSet[rawFiltration t] {b | unfinishedCutoffTop a ha z hz ρ hρ hr n b≤(t:ℝ≥0∞)} at h
-  change MeasurableSet[rawFiltration t] {b | (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)≤(t:ℝ≥0∞)}
-  simpa only [unfinishedCutoff_coe] using h
-lemma unfinishedCutoff_before_individual {m : ℕ} (a : ℝ) (ha : 0<a)
-    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
-    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
-    (n : ℕ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (i : Fin m)
-    (ht : t≤unfinishedCutoff a ha z hz ρ hρ hr n b)
-    (hti : (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i)) :
-    t≤radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b := by
-  by_cases he : (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i)
-  · exact ENNReal.coe_le_coe.mp (by rw [he]; exact hti)
-  · have hb : unfinishedCutoffTop a ha z hz ρ hρ hr n b≤
-        (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞) := by
-      apply (min_le_right _ _).trans
-      have h := Finset.inf_le (s:=Finset.univ) (f:=fun j =>
-        if (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞)=radialTime a b.1 (z j) (ρ j)
-        then ∞ else (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞)) (Finset.mem_univ i)
-      simpa only [ite_eq_right he] using h
-    exact ENNReal.coe_le_coe.mp ((ENNReal.coe_le_coe.mpr ht).trans (by rw [unfinishedCutoff_coe]; exact hb))
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TwoPointGrowth
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+lemma actual_radius_eq_full {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {t : ℝ≥0} (ht : alive a b t z) :
+    2*physicalRadius (actualState a b t z)=fullLoewnerRadius a b z t := by
+  have hd := norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb ht)
+  simp only [physicalRadius,actualState,Y_apply,J_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,
+    Real.exp_neg,Real.exp_log hd,fullLoewnerRadius,div_eq_mul_inv]
+  ring
+lemma physicalClock_radius (a cap : ℝ) (hcap : 0<cap) (v : State) (hv : 0<Y v) :
+    physicalClock a cap v=Real.log (cap/(2*physicalRadius v))/(2*a) := by
+  rw [physicalClock,physicalRadius,Real.log_div hcap.ne' (by norm_num : (2:ℝ)≠0),
+    Real.log_div hcap.ne' (show 2*(Y v*Real.exp (-J v))≠0 by positivity),
+    Real.log_mul (by norm_num : (2:ℝ)≠0) (by positivity),
+    Real.log_mul hv.ne' (Real.exp_ne_zero _),Real.log_exp]
+  ring
+lemma actual_clock_eq {a cap : ℝ} (ha : a≠0) (hcap : 0<cap)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z : ℂ} {T σ t : ℝ≥0} {f : ℝ → ℂ}
+    (hf : odeUntil a b z T f) (hσ : σ≤T) (ht : t≤T)
+    (hR : fullLoewnerRadius a b z σ=cap) :
+    physicalClock a cap (actualState a b t z)=activeODEClock b T σ f t := by
+  have hy : 0<Y (actualState a b t z) := by
+    simpa only [actualState,Y_apply,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq (odeUntil_mono hf ht)]
+      using (hf.2 t ⟨t.coe_nonneg,by exact_mod_cast ht⟩).1
+  rw [physicalClock_radius a cap hcap _ hy,actual_radius_eq_full hb ⟨f,odeUntil_mono hf ht⟩,
+    fullRadius_active_clock hb hf hσ ht,hR]
+  have he : cap/(cap*Real.exp (-2*a*activeODEClock b T σ f t))=
+      Real.exp (2*a*activeODEClock b T σ f t) := by
+    rw [show -2*a*activeODEClock b T σ f t = -(2*a*activeODEClock b T σ f t) by ring,
+      Real.exp_neg,div_mul_eq_div_div,div_self hcap.ne',one_div,inv_inv]
+  rw [he,Real.log_exp]
+  field_simp
+lemma actual_speed_eq {a : ℝ} {b : ℝ≥0 → ℝ} {z : ℂ} {T t : ℝ≥0} {f : ℝ → ℂ}
+    (hf : odeUntil a b z T f) (ht : t≤T) :
+    physicalV (actualState a b t z)=odeSpeed b T f t := by
+  rw [odeSpeed_eq f ⟨t.coe_nonneg,by exact_mod_cast ht⟩]
+  simp only [physicalV,actualState,Y_apply,Q_eq,loewnerMap_eq (odeUntil_mono hf ht),
+    Complex.sub_im,Complex.ofReal_im,sub_zero,Real.toNNReal_coe]
+end ExactSLEGreen
+namespace ExactSLEMain
+lemma exp_clock_integral {a σ off : ℝ} (hσ : σ≠0)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z : ℂ} {T A S : ℝ≥0} {f : ℝ → ℂ}
+    (hf : odeUntil a b z T f) :
+    (∫ t in (S:ℝ)..(T:ℝ), Real.exp (off+σ*activeODEClock b T A f t)*odeSpeed b T f t^2)=
+      (Real.exp (off+σ*activeODEClock b T A f T)-
+        Real.exp (off+σ*activeODEClock b T A f S))/σ := by
+  let u := activeODEClock b T A f
+  let v := odeSpeed b T f
+  have hc : Continuous u := continuous_iff_continuousAt.mpr fun t => (activeODEClock_deriv hb hf A t).continuousAt
+  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (f:=fun t => Real.exp (off+σ*u t)/σ)
+    (f':=fun t => Real.exp (off+σ*u t)*v t^2)
+    (a:=(S:ℝ)) (b:=(T:ℝ))
+    (fun t _ => by
+      exact (((((activeODEClock_deriv hb hf A t).const_mul σ).const_add off).exp).div_const σ).congr_deriv (by
+        dsimp only [u,v]; field_simp [hσ]))
+    (((Real.continuous_exp.comp (continuous_const.add (continuous_const.mul hc))).mul
+      ((odeSpeed_continuous hb hf).pow 2)).intervalIntegrable _ _)
+  simpa only [sub_div] using he
+end ExactSLEMain
+namespace ExactSLEMain
+lemma activeODEClock_monotone {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T A : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f) :
+    Monotone (activeODEClock b T A f) := by
+  exact monotone_of_hasDerivAt_nonneg (fun t => activeODEClock_deriv hb hf A t) (fun t => sq_nonneg _)
+end ExactSLEMain
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+ 
+
+theorem actual_regularized_pair_discount {a : ℝ} (ha : 1/4<a)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z w : ℂ} {f g : ℝ → ℂ}
+    {T S A B : ℝ≥0} (hST : S≤T) (hA : A≤S) (hB : B≤S)
+    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
+    {cap₁ cap₂ U₁ U₂ σ₁ σ₂ off₁ off₂ : ℝ}
+    (hcap₁ : 0<cap₁) (hcap₂ : 0<cap₂)
+    (hR₁ : fullLoewnerRadius a b z A=cap₁) (hR₂ : fullLoewnerRadius a b w B=cap₂)
+    (hsep₁ : cap₁≤‖z-w‖) (hsep₂ : cap₂≤‖z-w‖)
+    (hp₁ : (σ₁=-regBeta a ∧ off₁=0) ∨
+      (σ₁=regBeta a ∧ off₁=-regBeta a*U₁ ∧ U₁/2≤activeODEClock b T A f S))
+    (hp₂ : (σ₂=-regBeta a ∧ off₂=0) ∨
+      (σ₂=regBeta a ∧ off₂=-regBeta a*U₂ ∧ U₂/2≤activeODEClock b T B g S)) :
+    (∫ t in (S:ℝ)..(T:ℝ),
+      (physicalDelta a cap₁ σ₁ off₁ (actualState a b t.toNNReal z)*
+       physicalDelta a cap₂ σ₂ off₂ (actualState a b t.toNNReal w))^(-regM a)*
+       physicalV (actualState a b t.toNNReal z)*physicalV (actualState a b t.toNNReal w)) ≤
+      Real.exp (2*(regM a*regBeta a))/(2*a*(1-Real.exp (2*(regM a*regBeta a)-a))) := by
+  have ha0 : 0<a := by linarith
+  have hM := regM_pos ha
+  have hβ := regBeta_pos ha
+  let u := activeODEClock b T A f
+  let v := activeODEClock b T B g
+  let V := odeSpeed b T f
+  let W := odeSpeed b T g
+  have huc : Continuous u := continuous_iff_continuousAt.mpr fun t => (activeODEClock_deriv hb hf A t).continuousAt
+  have hvc : Continuous v := continuous_iff_continuousAt.mpr fun t => (activeODEClock_deriv hb hg B t).continuousAt
+  let F := fun t => Real.exp ((off₁+σ₁*u t+off₂+σ₂*v t)*(-regM a))*V t*W t
+  let G := fun t => Real.exp ((regM a*regBeta a)*(u t+v t))*V t*W t
+  have hFc : Continuous F := (((Real.continuous_exp.comp
+    (((continuous_const.add (continuous_const.mul huc)).add continuous_const).add
+      (continuous_const.mul hvc) |>.mul continuous_const)).mul (odeSpeed_continuous hb hf)).mul
+        (odeSpeed_continuous hb hg))
+  have hGc : Continuous G := ((Real.continuous_exp.comp
+    (continuous_const.mul (huc.add hvc))).mul (odeSpeed_continuous hb hf)).mul (odeSpeed_continuous hb hg)
+  have hST' : (S:ℝ)≤T := by exact_mod_cast hST
+  have he : (∫ t in (S:ℝ)..(T:ℝ),
+      (physicalDelta a cap₁ σ₁ off₁ (actualState a b t.toNNReal z)*
+       physicalDelta a cap₂ σ₂ off₂ (actualState a b t.toNNReal w))^(-regM a)*
+       physicalV (actualState a b t.toNNReal z)*physicalV (actualState a b t.toNNReal w))=
+      ∫ t in (S:ℝ)..(T:ℝ), F t := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    have ht' : t∈Icc (S:ℝ) T := by simpa only [uIcc_of_le hST'] using ht
+    have ht0 := S.coe_nonneg.trans ht'.1
+    have htT : t.toNNReal≤T := by exact_mod_cast (show (t.toNNReal:ℝ)≤T by simpa only [Real.coe_toNNReal _ ht0] using ht'.2)
+    simp only [physicalDelta]
+    rw [actual_clock_eq ha0.ne' hcap₁ hb hf (hA.trans hST) htT hR₁,
+      actual_clock_eq ha0.ne' hcap₂ hb hg (hB.trans hST) htT hR₂,
+      actual_speed_eq hf htT,actual_speed_eq hg htT,Real.coe_toNNReal _ ht0,←Real.exp_add,
+      Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp]
+    dsimp only [F,u,v,V,W]
+    congr 3
+    ring
+  rw [he]
+  calc
+    _ ≤ ∫ t in (S:ℝ)..(T:ℝ), G t := by
+      apply intervalIntegral.integral_mono_on hST' (hFc.intervalIntegrable _ _) (hGc.intervalIntegrable _ _)
+      intro t ht
+      have ht0 : 0≤t := S.coe_nonneg.trans ht.1
+      have hV : 0≤V t := by
+        change 0≤odeSpeed b T f t
+        rw [odeSpeed_eq f ⟨ht0,ht.2⟩]
+        exact div_nonneg ((hf.2 t ⟨ht0,ht.2⟩).1.le) (Complex.normSq_nonneg _)
+      have hW : 0≤W t := by
+        change 0≤odeSpeed b T g t
+        rw [odeSpeed_eq g ⟨ht0,ht.2⟩]
+        exact div_nonneg ((hg.2 t ⟨ht0,ht.2⟩).1.le) (Complex.normSq_nonneg _)
+      have h1 : -(off₁+σ₁*u t)≤regBeta a*u t := by
+        rcases hp₁ with ⟨rfl,rfl⟩|⟨rfl,rfl,hmid⟩
+        · simp
+        · have hu : U₁/2≤u t := hmid.trans (activeODEClock_monotone hb hf ht.1)
+          nlinarith [mul_le_mul_of_nonneg_left hu hβ.le]
+      have h2 : -(off₂+σ₂*v t)≤regBeta a*v t := by
+        rcases hp₂ with ⟨rfl,rfl⟩|⟨rfl,rfl,hmid⟩
+        · simp
+        · have hv : U₂/2≤v t := hmid.trans (activeODEClock_monotone hb hg ht.1)
+          nlinarith [mul_le_mul_of_nonneg_left hv hβ.le]
+      apply mul_le_mul_of_nonneg_right _ hW
+      apply mul_le_mul_of_nonneg_right _ hV
+      apply Real.exp_le_exp.mpr
+      have hh := mul_le_mul_of_nonneg_left (add_le_add h1 h2) hM.le
+      nlinarith only [hh]
+    _ ≤ _ := actual_pair_interaction ha0 (mul_pos hM hβ).le
+      (by nlinarith [regBeta_discount ha]) hb hST hA hB hf hg hzw
+      (hR₁.le.trans hsep₁) (hR₂.le.trans hsep₂)
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+lemma actual_regularized_single_discount {a : ℝ} (ha : 1/4<a)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z : ℂ} {f : ℝ → ℂ}
+    {T S A : ℝ≥0} (hST : S≤T) (hA : A≤S) (hf : odeUntil a b z T f)
+    {cap U σ off : ℝ} (hcap : 0<cap) (hR : fullLoewnerRadius a b z A=cap)
+    (hp : (σ=-regBeta a ∧ off=0) ∨
+      (σ=regBeta a ∧ off=-regBeta a*U ∧ activeODEClock b T A f T≤U)) :
+    (∫ t in (S:ℝ)..(T:ℝ), physicalDelta a cap σ off (actualState a b t.toNNReal z)*
+      physicalV (actualState a b t.toNNReal z)^2) ≤ 1/regBeta a := by
+  have ha0 : 0<a := by linarith
+  have hβ := regBeta_pos ha
+  have hST' : (S:ℝ)≤T := by exact_mod_cast hST
+  let u := activeODEClock b T A f
+  have he : (∫ t in (S:ℝ)..(T:ℝ), physicalDelta a cap σ off (actualState a b t.toNNReal z)*
+      physicalV (actualState a b t.toNNReal z)^2)=
+      ∫ t in (S:ℝ)..(T:ℝ), Real.exp (off+σ*u t)*odeSpeed b T f t^2 := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    have ht' : t∈Icc (S:ℝ) T := by simpa only [uIcc_of_le hST'] using ht
+    have ht0 := S.coe_nonneg.trans ht'.1
+    have htT : t.toNNReal≤T := by exact_mod_cast (show (t.toNNReal:ℝ)≤T by simpa only [Real.coe_toNNReal _ ht0] using ht'.2)
+    simp only [physicalDelta]
+    rw [actual_clock_eq ha0.ne' hcap hb hf (hA.trans hST) htT hR,
+      actual_speed_eq hf htT,Real.coe_toNNReal _ ht0]
+  rw [he]
+  rcases hp with ⟨rfl,rfl⟩|⟨rfl,rfl,hU⟩
+  · rw [exp_clock_integral (neg_ne_zero.mpr hβ.ne') hb hf]
+    have hu : 0≤u S := activeODEClock_nonneg (by exact_mod_cast hA)
+    have hex : Real.exp (0+-regBeta a*u S)≤1 := Real.exp_le_one_iff.mpr (by nlinarith)
+    have hnon := (Real.exp_pos (0+-regBeta a*u T)).le
+    rw [div_neg,←neg_div,neg_sub]
+    exact (div_le_div_iff_of_pos_right hβ).mpr (by linarith)
+  · rw [exp_clock_integral hβ.ne' hb hf]
+    have hex : Real.exp (-regBeta a*U+regBeta a*u T)≤1 := Real.exp_le_one_iff.mpr (by
+      have h := mul_le_mul_of_nonneg_left hU hβ.le
+      dsimp only [u] at *
+      linarith)
+    have hnon := (Real.exp_pos (-regBeta a*U+regBeta a*u S)).le
+    exact (div_le_div_iff_of_pos_right hβ).mpr (by linarith)
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open scoped Classical
+ 
+
+noncomputable def unfinishedCutoffTop {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
+    (n : ℕ) (b : ExactSLEGenerator.Driver) : ℝ≥0∞ :=
+  min (n:ℝ≥0∞) (Finset.univ.inf fun i =>
+    if (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i)
+    then ∞ else (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞))
+noncomputable def unfinishedCutoff {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
+    (n : ℕ) (b : ExactSLEGenerator.Driver) : ℝ≥0 := (unfinishedCutoffTop a ha z hz ρ hρ hr n b).toNNReal
+end ExactSLEGreen
+
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+namespace ExactSLEStops
+variable {Ω : Type*} [MeasurableSpace Ω]
+ 
+def insert (τ : Ω → ℝ≥0) : List (Ω → ℝ≥0) → List (Ω → ℝ≥0)
+  | [] => [τ]
+  | σ::l => (fun ω => min (τ ω) (σ ω)) :: insert (fun ω => max (τ ω) (σ ω)) l
+def sort : List (Ω → ℝ≥0) → List (Ω → ℝ≥0)
+  | [] => []
+  | τ::l => insert τ (sort l)
+omit [MeasurableSpace Ω] in
+lemma insert_length (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) :
+    (insert τ l).length=l.length+1 := by
+  induction l generalizing τ with
+  | nil => rfl
+  | cons σ l ih => simp only [insert,List.length_cons,ih]
+omit [MeasurableSpace Ω] in
+lemma sort_length (l : List (Ω → ℝ≥0)) : (sort l).length=l.length := by
+  induction l with
+  | nil => rfl
+  | cons τ l ih => simp only [sort,insert_length,List.length_cons,ih]
+lemma insert_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0))
+    (hτ : IsStoppingTime F (fun ω => (τ ω : WithTop ℝ≥0)))
+    (hl : ∀ σ ∈ l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0))) :
+    ∀ σ ∈ insert τ l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0)) := by
+  induction l generalizing τ with
+  | nil => simpa only [insert,List.mem_singleton] using fun σ (h : σ=τ) => h ▸ hτ
+  | cons π l ih =>
+    intro σ hσ
+    have hπ := hl π (by simp)
+    rcases List.mem_cons.mp hσ with h|h
+    · subst σ
+      simpa only [WithTop.coe_min] using hτ.min hπ
+    · apply ih (fun ω => max (τ ω) (π ω)) ?_ (fun σ h => hl σ (by simp [h])) σ h
+      simpa only [WithTop.coe_max] using hτ.max hπ
+lemma sort_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (l : List (Ω → ℝ≥0))
+    (hl : ∀ σ ∈ l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0))) :
+    ∀ σ ∈ sort l, IsStoppingTime F (fun ω => (σ ω : WithTop ℝ≥0)) := by
+  induction l with
+  | nil => simp [sort]
+  | cons τ l ih =>
+    exact insert_stopping F τ (sort l) (hl τ (by simp)) (ih (fun σ h => hl σ (by simp [h])))
+omit [MeasurableSpace Ω] in
+lemma insert_perm (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (ω : Ω) :
+    ((insert τ l).map (fun σ => σ ω)).Perm ((τ::l).map (fun σ => σ ω)) := by
+  induction l generalizing τ with
+  | nil => exact List.Perm.refl _
+  | cons σ l ih =>
+    simp only [insert,List.map_cons]
+    have h := (ih (fun ω => max (τ ω) (σ ω))).cons (min (τ ω) (σ ω))
+    rcases le_total (τ ω) (σ ω) with ht|ht
+    · simpa only [List.map_cons,min_eq_left ht,max_eq_right ht] using h
+    · simpa only [List.map_cons,min_eq_right ht,max_eq_left ht] using h.trans (List.Perm.swap _ _ _)
+omit [MeasurableSpace Ω] in
+lemma sort_perm (l : List (Ω → ℝ≥0)) (ω : Ω) :
+    ((sort l).map (fun σ => σ ω)).Perm (l.map (fun σ => σ ω)) := by
+  induction l with
+  | nil => exact List.Perm.refl _
+  | cons τ l ih => exact (insert_perm τ (sort l) ω).trans (ih.cons (τ ω))
+omit [MeasurableSpace Ω] in
+lemma insert_sorted (τ : Ω → ℝ≥0) (l : List (Ω → ℝ≥0)) (ω : Ω)
+    (hl : ((l.map (fun σ => σ ω))).Pairwise (· ≤ ·)) :
+    ((insert τ l).map (fun σ => σ ω)).Pairwise (· ≤ ·) := by
+  induction l generalizing τ with
+  | nil => simp [insert]
+  | cons σ l ih =>
+    simp only [List.map_cons,List.pairwise_cons] at hl
+    simp only [insert,List.map_cons,List.pairwise_cons]
+    refine ⟨?_,ih _ hl.2⟩
+    intro x hx
+    have hx' := (insert_perm (fun ω => max (τ ω) (σ ω)) l ω).mem_iff.mp hx
+    simp only [List.map_cons,List.mem_cons] at hx'
+    rcases hx' with rfl|hx'
+    · exact (min_le_left _ _).trans (le_max_left _ _)
+    · exact (min_le_right _ _).trans (hl.1 x hx')
+omit [MeasurableSpace Ω] in
+lemma sort_sorted (l : List (Ω → ℝ≥0)) (ω : Ω) :
+    ((sort l).map (fun σ => σ ω)).Pairwise (· ≤ ·) := by
+  induction l with
+  | nil => simp [sort]
+  | cons τ l ih => exact insert_sorted τ (sort l) ω ih
+ 
+omit [MeasurableSpace Ω] in
+lemma no_change_between (l : List (Ω → ℝ≥0)) (ω : Ω) (k : ℕ)
+    (hk : k+1<(sort l).length) (τ : Ω → ℝ≥0) (hτ : τ∈l) :
+    τ ω ≤ (sort l)[k] ω ∨ (sort l)[k+1] ω ≤ τ ω := by
+  let v := (sort l).map (fun σ => σ ω)
+  have hv : v.Pairwise (· ≤ ·) := sort_sorted l ω
+  have ht : τ ω∈v := (sort_perm l ω).mem_iff.mpr (List.mem_map_of_mem hτ)
+  obtain ⟨i,hi,he⟩ := List.getElem_of_mem ht
+  have hm := List.pairwise_iff_getElem.mp hv
+  by_cases hik : i≤k
+  · left
+    rcases hik.eq_or_lt with rfl|hi'
+    · simpa [v] using he.symm.le
+    · have h := hm i k hi (by simpa [v] using (Nat.lt_of_succ_lt hk)) hi'
+      rw [he] at h
+      simpa only [v,List.getElem_map] using h
+  · right
+    have hki : k+1 ≤ i := Nat.succ_le_of_lt (lt_of_not_ge hik)
+    rcases hki.eq_or_lt with heq|hlt
+    · subst i
+      simpa [v] using he.le
+    · have h := hm (k+1) i (by simpa [v] using hk) hi hlt
+      rw [he] at h
+      simpa only [v,List.getElem_map] using h
+end ExactSLEStops
+namespace ExactSLEStops
+open Set Filter
+open scoped Classical
+variable {Ω : Type*} [MeasurableSpace Ω]
+lemma deferred_stopping (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (τ : Ω → ℝ≥0∞) (hτ : IsStoppingTime F τ) (A : Set Ω)
+    (hA : MeasurableSet[hτ.measurableSpace] A) :
+    IsStoppingTime F (fun ω => if ω∈A then ∞ else τ ω) := by
+  intro t
+  change MeasurableSet[F t] {ω | (if ω∈A then ∞ else τ ω)≤(t:ℝ≥0∞)}
+  have he : {ω | (if ω∈A then ∞ else τ ω)≤(t:ℝ≥0∞)}=Aᶜ∩{ω | τ ω≤(t:ℝ≥0∞)} := by
+    ext ω
+    by_cases h : ω∈A <;> simp [h]
+  rw [he]
+  exact (hτ.measurableSet Aᶜ).mp hA.compl |>.2 t
+lemma finset_inf_stopping {ι : Type*} (F : Filtration ℝ≥0 ‹MeasurableSpace Ω›)
+    (s : Finset ι) (τ : ι → Ω → ℝ≥0∞)
+    (hτ : ∀ i∈s, IsStoppingTime F (τ i)) :
+    IsStoppingTime F (fun ω => s.inf fun i => τ i ω) := by
+  induction s using Finset.induction_on with
+  | empty =>
+    intro t
+    change MeasurableSet[F t] {ω | ((∅ : Finset ι).inf fun i => τ i ω)≤(t:ℝ≥0∞)}
+    simp
+  | @insert i s hi ih =>
+    have h := (hτ i (Finset.mem_insert_self _ _)).min (ih fun j hj => hτ j (Finset.mem_insert_of_mem hj))
+    intro t
+    change MeasurableSet[F t] {ω | ((Insert.insert i s).inf fun i => τ i ω)≤(t:ℝ≥0∞)}
+    have hh := h t
+    change MeasurableSet[F t] {ω | min (τ i ω) (s.inf fun i => τ i ω)≤(t:ℝ≥0∞)} at hh
+    simpa only [Finset.inf_insert] using hh
+omit [MeasurableSpace Ω] in
+lemma sort_bounded (l : List (Ω → ℝ≥0)) (B : Ω → ℝ≥0)
+    (hl : ∀ τ∈l, ∀ ω, τ ω≤B ω) : ∀ τ∈sort l, ∀ ω, τ ω≤B ω := by
+  intro τ hτ ω
+  have h : τ ω∈l.map (fun σ => σ ω) :=
+    (sort_perm l ω).mem_iff.mp (List.mem_map_of_mem hτ)
+  obtain ⟨σ,hσ,he⟩ := List.mem_map.mp h
+  exact he ▸ hl σ hσ ω
+end ExactSLEStops
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEStops
+open scoped Classical
+lemma unfinishedCutoffTop_le {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
+    (n : ℕ) (b : ExactSLEGenerator.Driver) : unfinishedCutoffTop a ha z hz ρ hρ hr n b≤n := min_le_left _ _
+lemma unfinishedCutoff_coe {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
+    (n : ℕ) (b : ExactSLEGenerator.Driver) :
+    (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)=unfinishedCutoffTop a ha z hz ρ hρ hr n b := by
+  exact ENNReal.coe_toNNReal (ne_top_of_le_ne_top (by simp) (unfinishedCutoffTop_le a ha z hz ρ hρ hr n b))
+lemma unfinishedCutoff_stopping {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im) (n : ℕ) :
+    IsStoppingTime rawFiltration (fun b => (unfinishedCutoff a ha z hz ρ hρ hr n b : WithTop ℝ≥0)) := by
+  have htop : IsStoppingTime rawFiltration (unfinishedCutoffTop a ha z hz ρ hρ hr n) := by
+    apply (isStoppingTime_const rawFiltration (n:ℝ≥0)).min
+    apply finset_inf_stopping
+    intro i hi
+    let A := {b | (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i)}
+    have htau := radialStop_stopping a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n
+    have hA : MeasurableSet[htau.measurableSpace] A := htau.measurableSet_eq_stopping_time
+      (radialTime_stopping a ha (z i) (hz i) (ρ i) (hρ i) (hr i))
+    intro t
+    change MeasurableSet[rawFiltration t] {b | (if (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i) then ∞ else (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞))≤(t:ℝ≥0∞)}
+    have hd := ((htau.measurableSet Aᶜ).mp hA.compl).2 t
+    convert hd using 1
+    ext b
+    by_cases he : b∈A
+    · simp only [A,Set.mem_ofPred_eq] at he
+      simp [he,A]
+      try exact WithTop.coe_le_coe.symm
+    · simp only [A,Set.mem_ofPred_eq] at he
+      simp [he,A]
+      try exact WithTop.coe_le_coe.symm
+
+  intro t
+  have h := htop t
+  change MeasurableSet[rawFiltration t] {b | unfinishedCutoffTop a ha z hz ρ hρ hr n b≤(t:ℝ≥0∞)} at h
+  change MeasurableSet[rawFiltration t] {b | (unfinishedCutoff a ha z hz ρ hρ hr n b : ℝ≥0∞)≤(t:ℝ≥0∞)}
+  simpa only [unfinishedCutoff_coe] using h
+lemma unfinishedCutoff_before_individual {m : ℕ} (a : ℝ) (ha : 0<a)
+    (z : Fin m → ℂ) (hz : ∀ i, 0<(z i).im)
+    (ρ : Fin m → ℝ) (hρ : ∀ i, 0<ρ i) (hr : ∀ i, ρ i≤(z i).im)
+    (n : ℕ) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) (i : Fin m)
+    (ht : t≤unfinishedCutoff a ha z hz ρ hρ hr n b)
+    (hti : (t:ℝ≥0∞)≤radialTime a b.1 (z i) (ρ i)) :
+    t≤radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b := by
+  by_cases he : (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞)=radialTime a b.1 (z i) (ρ i)
+  · exact ENNReal.coe_le_coe.mp (by rw [he]; exact hti)
+  · have hb : unfinishedCutoffTop a ha z hz ρ hρ hr n b≤
+        (radialStop a ha (z i) (hz i) (ρ i) (hρ i) (hr i) n b : ℝ≥0∞) := by
+      apply (min_le_right _ _).trans
+      have h := Finset.inf_le (s:=Finset.univ) (f:=fun j =>
+        if (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞)=radialTime a b.1 (z j) (ρ j)
+        then ∞ else (radialStop a ha (z j) (hz j) (ρ j) (hρ j) (hr j) n b : ℝ≥0∞)) (Finset.mem_univ i)
+      simpa only [ite_eq_right he] using h
+    exact ENNReal.coe_le_coe.mp ((ENNReal.coe_le_coe.mpr ht).trans (by rw [unfinishedCutoff_coe]; exact hb))
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadonKernels.lean b/lean/OAI/Probability/SLEGauge/RadonKernels.lean
index b42431a75021ea5d277012d657582c26a01d9eca..1f446e2cf978cf07c7b6d88ab3402425f5cd7ce9 100644
--- a/lean/OAI/Probability/SLEGauge/RadonKernels.lean
+++ b/lean/OAI/Probability/SLEGauge/RadonKernels.lean
@@ -1,587 +1,587 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.MassVersionsCore
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Set Filter CompactlySupported
-open scoped ENNReal NNReal Topology
-
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
-
- 
-
-theorem isSFiniteKernel_of_pointwise_finite (κ : Kernel Ω X)
-    [∀ ω, IsFiniteMeasure (κ ω)] : IsSFiniteKernel κ := by
-  let r : Ω → ℝ≥0∞ := fun ω => κ ω univ + 1
-  have hr : Measurable r := (κ.measurable_coe MeasurableSet.univ).add_const 1
-  have hr0 (ω : Ω) : r ω ≠ 0 := by simp [r]
-  have hrtop (ω : Ω) : r ω ≠ ∞ := by simp [r, measure_ne_top]
-  let η : Kernel Ω X := ⟨fun ω => (r ω)⁻¹ • κ ω, by
-    apply Measure.measurable_of_measurable_coe
-    intro A hA
-    simp only [Measure.smul_apply, smul_eq_mul]
-    exact hr.inv.mul (κ.measurable_coe hA)⟩
-  have hη : IsFiniteKernel η := by
-    refine ⟨⟨1, ENNReal.one_lt_top, fun ω => ?_⟩⟩
-    change (r ω)⁻¹ * κ ω univ ≤ 1
-    calc
-      _ ≤ (r ω)⁻¹ * r ω := mul_le_mul_of_nonneg_left (le_add_right le_rfl) zero_le
-      _ = 1 := ENNReal.inv_mul_cancel (hr0 ω) (hrtop ω)
-  have he : κ = η.withDensity (fun ω _ => r ω) := by
-    apply Kernel.ext
-    intro ω
-    rw [Kernel.withDensity_apply η (f := fun ω (_ : X) => r ω)
-      (hr.comp measurable_fst), withDensity_const]
-    change κ ω = r ω • ((r ω)⁻¹ • κ ω)
-    rw [smul_smul, ENNReal.mul_inv_cancel (hr0 ω) (hrtop ω), one_smul]
-  rw [he]
-  exact Kernel.IsSFiniteKernel.withDensity η (fun ω _ => hrtop ω)
-
-variable [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [BorelSpace X]
-
- 
-
-theorem isSFiniteKernel_of_locally_finite (κ : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] : IsSFiniteKernel κ := by
-  let K := CompactExhaustion.choice X
-  let V := disjointed (fun n => K n)
-  have hV (n : ℕ) : MeasurableSet (V n) :=
-    MeasurableSet.disjointed (fun n => (K.isCompact n).measurableSet) n
-  let η : ℕ → Kernel Ω X := fun n => κ.restrict (hV n)
-  have hη (n : ℕ) : IsSFiniteKernel (η n) := by
-    have (ω : Ω) : IsFiniteMeasure (η n ω) := by
-      refine ⟨?_⟩
-      rw [show η n ω = (κ ω).restrict (V n) from rfl, Measure.restrict_apply_univ]
-      exact (measure_mono (disjointed_subset _ _)).trans_lt (K.isCompact n).measure_lt_top
-    exact isSFiniteKernel_of_pointwise_finite (η n)
-  have he : Kernel.sum η = κ := by
-    apply Kernel.ext
-    intro ω
-    change Measure.sum (fun n => (κ ω).restrict (V n)) = κ ω
-    rw [← Measure.restrict_iUnion (disjoint_disjointed _) (fun n => hV n)]
-    simp only [iUnion_disjointed, K.iUnion_eq, Measure.restrict_univ]
-  rw [← he]
-  exact Kernel.isSFiniteKernel_sum
-
- 
-
-theorem average_finiteOnCompacts (κ : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] (W : Measure Ω)
-    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W) :
-    IsFiniteMeasureOnCompacts (κ ∘ₘ W) := by
-  constructor
-  intro K hK
-  obtain ⟨g,hg1,-,hgc,hg01⟩ := exists_continuous_one_zero_of_isCompact
-    hK isClosed_empty (disjoint_empty K)
-  let f : C_c(X,ℝ) := ⟨g,hgc⟩
-  rw [Measure.bind_apply hK.measurableSet κ.aemeasurable]
-  apply lt_of_le_of_lt _ (hκ f).lintegral_lt_top
-  apply lintegral_mono
-  intro ω
-  exact f.integrable.measure_le_integral (Eventually.of_forall (fun x => (hg01 x).1))
-    (fun x hx => le_of_eq ((hg1 hx).symm))
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
-lemma average_integrable_compact_test (κ : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] (W : Measure Ω)
-    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
-    (f : C_c(X,ℝ)) : Integrable f (κ ∘ₘ W) := by
-  let g : C_c(X,ℝ) := ⟨⟨fun x => ‖f x‖, f.continuous.norm⟩, f.hasCompactSupport.norm⟩
-  apply (Measure.integrable_comp_iff f.continuous.measurable.aestronglyMeasurable).mpr
-  exact ⟨Eventually.of_forall (fun _ => f.integrable), hκ g⟩
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
-lemma average_compact_eval (κ : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] (W : Measure Ω)
-    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
-    (f : C_c(X,ℝ)) :
-    (∫x, f x ∂(κ ∘ₘ W)) = ∫ω, (∫x, f x ∂κ ω) ∂W := by
-  have hf := average_integrable_compact_test κ W hκ f
-  rw [Measure.comp_eq_comp_const_apply] at hf ⊢
-  exact Kernel.integral_comp hf
-
-variable [TopologicalSpace.PseudoMetrizableSpace X]
-
- 
-
-theorem average_eq_of_compact_tests (κ η : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] [∀ ω, IsFiniteMeasureOnCompacts (η ω)]
-    (W : Measure Ω)
-    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
-    (hη : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂η ω) W)
-    (he : ∀ f : C_c(X,ℝ), (∫ω, (∫x, f x ∂κ ω) ∂W) = ∫ω, (∫x, f x ∂η ω) ∂W) :
-    κ ∘ₘ W = η ∘ₘ W := by
-  have := average_finiteOnCompacts κ W hκ
-  have := average_finiteOnCompacts η W hη
-  apply Measure.ext_of_integral_eq_on_compactlySupported
-  intro f
-  simpa only [average_compact_eval _ _ hκ, average_compact_eval _ _ hη] using he f
-
-omit mΩ [MeasurableSpace X] [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-  [SecondCountableTopology X] [BorelSpace X] [TopologicalSpace.PseudoMetrizableSpace X] in
-lemma prod_sigma_le {m m₀ : MeasurableSpace Ω} (hm : m ≤ m₀)
-    (n : MeasurableSpace X) : m.prod n ≤ m₀.prod n :=
-  sup_le_sup (MeasurableSpace.comap_mono hm) le_rfl
-
- 
-
-theorem conditional_product_eq_of_compact_tests (κ η : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] [∀ ω, IsFiniteMeasureOnCompacts (η ω)]
-    (W : Measure Ω) [SFinite W]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
-    (hη : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂η ω) W)
-    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
-      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W) :
-    (W ⊗ₘ κ).trim (prod_sigma_le hm _) = (W ⊗ₘ η).trim (prod_sigma_le hm _) := by
-  let : MeasurableSpace Ω := mΩ
-  have := isSFiniteKernel_of_locally_finite κ
-  have := isSFiniteKernel_of_locally_finite η
-  have havg (U : Set Ω) (hU : MeasurableSet[m] U) :
-      κ ∘ₘ (W.restrict U) = η ∘ₘ (W.restrict U) :=
-    average_eq_of_compact_tests κ η (W.restrict U)
-      (fun f => (hκ f).restrict) (fun f => (hη f).restrict) (he U hU)
-  have hrect (U : Set Ω) (hU : MeasurableSet[m] U) (A : Set X) (hA : MeasurableSet A) :
-      (W ⊗ₘ κ).trim (prod_sigma_le hm _) (U ×ˢ A) =
-        (W ⊗ₘ η).trim (prod_sigma_le hm _) (U ×ˢ A) := by
-    rw [trim_measurableSet_eq _ (hU.prod hA), trim_measurableSet_eq _ (hU.prod hA),
-      Measure.compProd_apply_prod (hm _ hU) hA, Measure.compProd_apply_prod (hm _ hU) hA,
-      ← Measure.bind_apply hA κ.aemeasurable, ← Measure.bind_apply hA η.aemeasurable, havg U hU]
-  let K := CompactExhaustion.choice X
-  have := average_finiteOnCompacts κ W hκ
-  have hfin (n : ℕ) : (W ⊗ₘ κ).trim (prod_sigma_le hm _) (univ ×ˢ K n) ≠ ∞ := by
-    rw [trim_measurableSet_eq _ (MeasurableSet.univ.prod (K.isCompact n).measurableSet),
-      Measure.compProd_apply_prod MeasurableSet.univ (K.isCompact n).measurableSet,
-      setLIntegral_univ, ← Measure.bind_apply (K.isCompact n).measurableSet κ.aemeasurable]
-    exact (K.isCompact n).measure_lt_top.ne
-  let : MeasurableSpace Ω := m
-  apply Measure.ext_of_generateFrom_of_iUnion
-    (image2 (· ×ˢ ·) {U : Set Ω | MeasurableSet U} {A : Set X | MeasurableSet A})
-    (fun n => univ ×ˢ K n) generateFrom_prod.symm isPiSystem_prod
-  · ext p
-    simp only [mem_iUnion, mem_prod, mem_univ, true_and, iff_true]
-    exact K.exists_mem p.2
-  · intro n
-    exact ⟨univ, MeasurableSet.univ, K n, (K.isCompact n).measurableSet, rfl⟩
-  · exact hfin
-  · rintro _ ⟨U, hU, A, hA, rfl⟩
-    exact hrect U hU A hA
-
- 
-
-theorem conditional_lintegral_eq_of_compact_tests (κ η : Kernel Ω X)
-    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] [∀ ω, IsFiniteMeasureOnCompacts (η ω)]
-    (W : Measure Ω) [SFinite W]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
-    (hη : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂η ω) W)
-    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
-      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W)
-    (H : Ω × X → ℝ≥0∞) (hH : Measurable[m.prod ‹MeasurableSpace X›] H) :
-    (∫⁻ω, ∫⁻x, H (ω,x) ∂κ ω ∂W) = ∫⁻ω, ∫⁻x, H (ω,x) ∂η ω ∂W := by
-  let : MeasurableSpace Ω := mΩ
-  have := isSFiniteKernel_of_locally_finite κ
-  have := isSFiniteKernel_of_locally_finite η
-  have hH₀ : Measurable H := hH.mono (prod_sigma_le hm _) le_rfl
-  rw [← Measure.lintegral_compProd hH₀, ← Measure.lintegral_compProd hH₀,
-    ← lintegral_trim (prod_sigma_le hm _) hH,
-    ← lintegral_trim (μ := W ⊗ₘ η) (prod_sigma_le hm _) hH,
-    conditional_product_eq_of_compact_tests κ η W m hm hκ hη he]
-
-end SLEExactGauge.MassVersions
-
-namespace SLEExactGauge.MassVersions
-
-variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [BorelSpace X]
-    [TopologicalSpace.PseudoMetrizableSpace X]
-
-omit [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-  [BorelSpace X] [TopologicalSpace.PseudoMetrizableSpace X] in
-lemma average_mono (κ : Kernel Ω X) {W V : Measure Ω} (h : V ≤ W) : κ ∘ₘ V ≤ κ ∘ₘ W := by
-  apply Measure.le_iff.mpr
-  intro A hA
-  rw [Measure.bind_apply hA κ.aemeasurable, Measure.bind_apply hA κ.aemeasurable]
-  exact lintegral_mono' h le_rfl
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X]
-  [TopologicalSpace.PseudoMetrizableSpace X] in
-lemma average_restrict_finite (κ : Kernel Ω X) (W : Measure Ω)
-    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] (U : Set Ω) :
-    IsFiniteMeasureOnCompacts (κ ∘ₘ W.restrict U) := by
-  constructor
-  intro K hK
-  exact ((average_mono κ Measure.restrict_le_self) K).trans_lt hK.measure_lt_top
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-  [TopologicalSpace.PseudoMetrizableSpace X] in
-lemma average_compact_eval_of_average_finite (κ : Kernel Ω X) (W : Measure Ω)
-    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] (f : C_c(X, ℝ)) :
-    (∫x, f x ∂(κ ∘ₘ W)) = ∫ω, (∫x, f x ∂κ ω) ∂W := by
-  have hf : Integrable f (κ ∘ₘ W) := f.integrable
-  rw [Measure.comp_eq_comp_const_apply] at hf ⊢
-  exact Kernel.integral_comp hf
-
- 
-
-theorem conditional_product_eq_of_average_finite (κ η : Kernel Ω X)
-    [IsSFiniteKernel κ] [IsSFiniteKernel η] (W : Measure Ω) [SFinite W]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] [IsFiniteMeasureOnCompacts (η ∘ₘ W)]
-    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
-      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W) :
-    (W ⊗ₘ κ).trim (prod_sigma_le hm _) = (W ⊗ₘ η).trim (prod_sigma_le hm _) := by
-  let : MeasurableSpace Ω := mΩ
-  have havg (U : Set Ω) (hU : MeasurableSet[m] U) :
-      κ ∘ₘ (W.restrict U) = η ∘ₘ (W.restrict U) := by
-    have := average_restrict_finite κ W U
-    have := average_restrict_finite η W U
-    apply Measure.ext_of_integral_eq_on_compactlySupported
-    intro f
-    simpa only [average_compact_eval_of_average_finite] using he U hU f
-  have hrect (U : Set Ω) (hU : MeasurableSet[m] U) (A : Set X) (hA : MeasurableSet A) :
-      (W ⊗ₘ κ).trim (prod_sigma_le hm _) (U ×ˢ A) =
-        (W ⊗ₘ η).trim (prod_sigma_le hm _) (U ×ˢ A) := by
-    rw [trim_measurableSet_eq _ (hU.prod hA), trim_measurableSet_eq _ (hU.prod hA),
-      Measure.compProd_apply_prod (hm _ hU) hA, Measure.compProd_apply_prod (hm _ hU) hA,
-      ← Measure.bind_apply hA κ.aemeasurable, ← Measure.bind_apply hA η.aemeasurable, havg U hU]
-  let K := CompactExhaustion.choice X
-  have hfin (n : ℕ) : (W ⊗ₘ κ).trim (prod_sigma_le hm _) (univ ×ˢ K n) ≠ ∞ := by
-    rw [trim_measurableSet_eq _ (MeasurableSet.univ.prod (K.isCompact n).measurableSet),
-      Measure.compProd_apply_prod MeasurableSet.univ (K.isCompact n).measurableSet,
-      setLIntegral_univ, ← Measure.bind_apply (K.isCompact n).measurableSet κ.aemeasurable]
-    exact (K.isCompact n).measure_lt_top.ne
-  let : MeasurableSpace Ω := m
-  apply Measure.ext_of_generateFrom_of_iUnion
-    (image2 (· ×ˢ ·) {U : Set Ω | MeasurableSet U} {A : Set X | MeasurableSet A})
-    (fun n => univ ×ˢ K n) generateFrom_prod.symm isPiSystem_prod
-  · ext p
-    simp only [mem_iUnion, mem_prod, mem_univ, true_and, iff_true]
-    exact K.exists_mem p.2
-  · intro n
-    exact ⟨univ, MeasurableSet.univ, K n, (K.isCompact n).measurableSet, rfl⟩
-  · exact hfin
-  · rintro _ ⟨U, hU, A, hA, rfl⟩
-    exact hrect U hU A hA
-
-theorem conditional_lintegral_eq_of_average_finite (κ η : Kernel Ω X)
-    [IsSFiniteKernel κ] [IsSFiniteKernel η] (W : Measure Ω) [SFinite W]
-    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
-    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] [IsFiniteMeasureOnCompacts (η ∘ₘ W)]
-    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
-      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W)
-    (H : Ω × X → ℝ≥0∞) (hH : Measurable[m.prod ‹MeasurableSpace X›] H) :
-    (∫⁻ω, ∫⁻x, H (ω,x) ∂κ ω ∂W) = ∫⁻ω, ∫⁻x, H (ω,x) ∂η ω ∂W := by
-  let : MeasurableSpace Ω := mΩ
-  have hH₀ : Measurable H := hH.mono (prod_sigma_le hm _) le_rfl
-  rw [← Measure.lintegral_compProd hH₀, ← Measure.lintegral_compProd hH₀,
-    ← lintegral_trim (prod_sigma_le hm _) hH,
-    ← lintegral_trim (μ := W ⊗ₘ η) (prod_sigma_le hm _) hH,
-    conditional_product_eq_of_average_finite κ η W m hm he]
-
-end SLEExactGauge.MassVersions
-
-namespace SLEExactGauge.MassVersions
-open MeasureTheory ProbabilityTheory Set CompactlySupported
-variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [BorelSpace X]
-
-omit [BorelSpace X] in
-lemma finiteOnCompacts_of_integrable_tests (ν : Measure X)
-    (hν : ∀ f : C_c(X,ℝ), Integrable f ν) : IsFiniteMeasureOnCompacts ν := by
-  constructor
-  intro K hK
-  obtain ⟨g,hg1,-,hgc,hg01⟩ := exists_continuous_one_zero_of_isCompact
-    hK isClosed_empty (disjoint_empty K)
-  let f : C_c(X,ℝ) := ⟨g,hgc⟩
-  exact ((hν f).measure_le_integral (Eventually.of_forall (fun x => (hg01 x).1))
-    (fun x hx => le_of_eq ((hg1 hx).symm))).trans_lt ENNReal.ofReal_lt_top
-
-omit [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] [BorelSpace X] in
-lemma average_add (κ η : Kernel Ω X) (W : Measure Ω) :
-    (κ + η) ∘ₘ W = κ ∘ₘ W + η ∘ₘ W := by
-  rw [Measure.comp_eq_comp_const_apply, Kernel.comp_add_left,
-    add_apply, ← Measure.comp_eq_comp_const_apply, ← Measure.comp_eq_comp_const_apply]
-
-end SLEExactGauge.MassVersions
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-instance upperArea_finiteOnCompacts : IsFiniteMeasureOnCompacts upperArea := by
-  constructor
-  intro K hK
-  rw [upperArea, upper_embedding.comap_apply]
-  exact (hK.image continuous_subtype_val).measure_lt_top
-
-lemma survivingGreen_joint_measurable (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => survivingGreen a t p.1 p.2.1) :=
-  (actual_surviving_green a ha t).1.mono (prod_sigma_le (rawFiltration.le t) _) le_rfl
-
-lemma survivingGreen_spatial_measurable (a : ℝ) (ha : 1/4<a) (t : ℝ≥0)
-    (b : ExactSLEGenerator.Driver) : Measurable (fun z : UpperPlane => survivingGreen a t b z.1) :=
-  (survivingGreen_joint_measurable a ha t).comp (measurable_const.prodMk measurable_id)
-
- 
-noncomputable def survivingKernel (a : ℝ) (_ha : 1/4<a) (t : ℝ≥0) :
-    Kernel ExactSLEGenerator.Driver UpperPlane :=
-  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
-    (fun b z => ENNReal.ofReal (survivingGreen a t b z.1))
-
-instance survivingKernel_sfinite (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
-    IsSFiniteKernel (survivingKernel a ha t) :=
-  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
-
-lemma survivingKernel_apply (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    survivingKernel a ha t b = upperArea.withDensity (fun z => ENNReal.ofReal (survivingGreen a t b z.1)) :=
-  Kernel.withDensity_apply _ (survivingGreen_joint_measurable a ha t).ennreal_ofReal b
-
-lemma survivingKernel_compact_eval (a : ℝ) (ha : 1/4<a) (t : ℝ≥0)
-    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
-    (∫z, f z ∂survivingKernel a ha t b) = ∫z, f z * survivingGreen a t b z.1 ∂upperArea := by
-  rw [survivingKernel_apply, integral_withDensity_eq_integral_toReal_smul]
-  · simp only [ENNReal.toReal_ofReal (survivingGreen_nonneg _ _ _ _), smul_eq_mul, mul_comm]
-  · exact (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal
-  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
-
-lemma surviving_compact_spatial_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
-    Integrable (fun p : UpperPlane × ExactSLEGenerator.Driver => f p.1*survivingGreen a t p.2 p.1.1)
-      ((upperArea.restrict (tsupport f)).prod W) := by
-  obtain ⟨c,hc,hcK⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
-  obtain ⟨B,hB,hBf⟩ := upperExtension_bounded f
-  exact (actual_remaining_spatial_L1 a ha W hW t (tsupport f) f.hasCompactSupport.measurableSet
-    f.hasCompactSupport.measure_lt_top (c/2) (by positivity)
-    (fun z hz => by have := hcK z hz; linarith) f f.continuous.measurable B hB
-    (fun z hz => by simpa only [upperExtension_apply] using hBf z.1 ⟨z,hz,rfl⟩)
-    (fun n : ℕ => 1/((n:ℝ)+1))
-    (fun n => ⟨by positivity, (div_le_one (by positivity)).mpr (by linarith [Nat.cast_nonneg (α:=ℝ) n])⟩)
-    tendsto_one_div_add_atTop_nhds_zero_nat).1
-
-lemma survivingKernel_compact_mean_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
-    Integrable (fun b => ∫z, f z ∂survivingKernel a ha t b) W := by
-  simp_rw [survivingKernel_compact_eval]
-  convert (surviving_compact_spatial_integrable a ha W hW t f).integral_prod_right using 1
-  funext b
-  exact (setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by
-    rw [image_eq_zero_of_notMem_tsupport hz, zero_mul])).symm
-
-lemma survivingKernel_compact_ae_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
-    ∀ᵐ b ∂W, Integrable f (survivingKernel a ha t b) := by
-  filter_upwards [(surviving_compact_spatial_integrable a ha W hW t f).prod_left_ae] with b hb
-  rw [survivingKernel_apply, integrable_withDensity_iff_integrable_smul'
-    (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal
-    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
-  simp only [ENNReal.toReal_ofReal (survivingGreen_nonneg _ _ _ _), smul_eq_mul]
-  have hs : Function.support (fun z : UpperPlane => f z * survivingGreen a t b z.1) ⊆ tsupport f :=
-    (Function.support_mul_subset_left _ _).trans (subset_tsupport f)
-  simpa only [mul_comm] using (integrableOn_iff_integrable_of_support_subset hs).mp hb
-
-lemma survivingKernel_average_compact_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
-    Integrable f (survivingKernel a ha t ∘ₘ W) := by
-  let g : C_c(UpperPlane,ℝ) := ⟨⟨fun z => ‖f z‖,f.continuous.norm⟩,f.hasCompactSupport.norm⟩
-  apply (Measure.integrable_comp_iff f.continuous.measurable.aestronglyMeasurable).mpr
-  exact ⟨survivingKernel_compact_ae_integrable a ha W hW t f,
-    survivingKernel_compact_mean_integrable a ha W hW t g⟩
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma survivingKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) : IsFiniteMeasureOnCompacts (survivingKernel a ha t ∘ₘ W) :=
-  finiteOnCompacts_of_integrable_tests _ (survivingKernel_average_compact_integrable a ha W hW t)
-
- 
-
-theorem compact_conditional_to_random_tests (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0) (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
-    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂μ b) W)
-    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂ν b) W)
-    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
-      MeasurableSet[rawFiltration t] U →
-      (∫b in U, (∫z, f z ∂ν b) - (∫z, f z ∂μ b) ∂W) =
-        ∫b in U, (∫z, f z*survivingGreen a t b z.1 ∂upperArea) ∂W)
-    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
-    (hH : Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H) :
-    (∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ b ∂W) +
-      ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a t b z.1) ∂upperArea ∂W := by
-  have := isSFiniteKernel_of_locally_finite μ
-  have := isSFiniteKernel_of_locally_finite ν
-  have := average_finiteOnCompacts μ W hμ
-  have := average_finiteOnCompacts ν W hν
-  have := survivingKernel_average_finiteOnCompacts a ha W hW t
-  have : IsFiniteMeasureOnCompacts ((μ + survivingKernel a ha t) ∘ₘ W) := by
-    rw [SLEExactGauge.MassVersions.average_add]
-    constructor
-    intro K hK
-    rw [Measure.add_apply]
-    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩
-  have hidentity := conditional_lintegral_eq_of_average_finite ν (μ + survivingKernel a ha t) W
-    (rawFiltration t) (rawFiltration.le t) ?_ H hH
-  · rw [hidentity]
-    have hH₀ : Measurable H := hH.mono (prod_sigma_le (rawFiltration.le t) _) le_rfl
-    have heval (b : ExactSLEGenerator.Driver) :
-        (∫⁻z, H (b,z) ∂(μ+survivingKernel a ha t) b) =
-          (∫⁻z, H (b,z) ∂μ b) + ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a t b z.1) ∂upperArea := by
-      have hHb : Measurable (fun z : UpperPlane => H (b,z)) :=
-        hH₀.comp ((measurable_const (a := b)).prodMk measurable_id)
-      rw [add_apply, lintegral_add_measure, survivingKernel_apply]
-      congr 1
-      exact (lintegral_withDensity_eq_lintegral_mul upperArea
-        (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal hHb).trans
-          (lintegral_congr (fun _ => mul_comm _ _))
-    simp_rw [heval]
-    rw [lintegral_add_left]
-    exact hH₀.lintegral_kernel_prod_right'
-  · intro U hU f
-    have hadd : (fun b => ∫z, f z ∂(μ+survivingKernel a ha t) b) =ᵐ[W]
-        (fun b => (∫z, f z ∂μ b)+(∫z, f z ∂survivingKernel a ha t b)) := by
-      filter_upwards [survivingKernel_compact_ae_integrable a ha W hW t f] with b hb
-      exact integral_add_measure f.integrable hb
-    rw [integral_congr_ae (ae_restrict_of_ae hadd),
-      integral_add (hμ f).restrict (survivingKernel_compact_mean_integrable a ha W hW t f).restrict]
-    have hf := he f U hU
-    rw [integral_sub (hν f).restrict (hμ f).restrict] at hf
-    simp_rw [survivingKernel_compact_eval]
-    linarith
-
-end ExactSLEGreen
-end
- 
-
- 
-section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_raw_mass_extraction (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        (∫b, (∫z,f z∂μ (t i) b) ∂W) = ∫z : UpperPlane,
-          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        ∀ U : Set ExactSLEGenerator.Driver, MeasurableSet[rawFiltration v] U →
-        (∫b in U, (∫z,f z∂μ (t j) b)-(∫z,f z∂μ (t i) b) ∂W) =
-          ∫b in U, (∫z : UpperPlane,f z*survivingGreen a v b z.1 ∂upperArea) ∂W) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫z,f z∂μ (t i) b))) := by
-  obtain ⟨s,c,ν,hs,hc,hsum,hν,hmom,hL1,hmean,-,hcond,h⟩ :=
-    actual_radon_compact_conditional a ha W hW t
-  let μ := recoveredMass a ha s c hc
-  have hμ (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-      IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular :=
-    ⟨radonLimit_finite _, radonLimit_regular _⟩
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := (hμ v b).1
-  have hmeas (v : ℝ≥0∞) : Measurable (μ v) := recoveredMass_measurable a ha s c hc v
-  have he : ∀ᵐ b ∂W, ∀ i, μ (t i) b = ν b i := by
-    filter_upwards [h] with b hb
-    intro i
-    have := (hν b i).2
-    apply radonLimit_eq
-    intro f
-    simpa only [actualMassFunctionals_apply] using hb.2 i f
-  have heI (i : ℕ) (f : C_c(UpperPlane,ℝ)) :
-      (fun b => ∫z,f z∂μ (t i) b) =ᵐ[W] (fun b => ∫z,f z∂ν b i) := by
-    filter_upwards [he] with b hb
-    rw [hb i]
-  have hmoment (i : ℕ) (f : C_c(UpperPlane,ℝ)) (m : ℕ) :
-      MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W :=
-    (memLp_congr_ae (heI i f)).mpr (hmom i f m)
-  have hcompact (i j : ℕ) (v : ℝ≥0) (hi : t i=v) (hj : t j=∞)
-      (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
-      (hU : MeasurableSet[rawFiltration v] U) :
-      (∫b in U, (∫z,f z∂μ (t j) b)-(∫z,f z∂μ (t i) b) ∂W) =
-        ∫b in U, (∫z : UpperPlane,f z*survivingGreen a v b z.1 ∂upperArea) ∂W :=
-    (integral_congr_ae (ae_restrict_of_ae ((heI j f).sub (heI i f)))).trans
-      (hcond i j v hi hj f U hU)
-  refine ⟨s,c,hc,hs,hsum,hμ,hmeas,recoveredMass_progressive a ha s c hc,hmoment,?_,?_,hcompact,?_,?_⟩
-  · intro i f
-    have hnorm (k : ℕ) :
-        eLpNorm (fun b => (∑ j∈s k, c k j *
-          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂μ (t i) b) 1 W =
-        eLpNorm (fun b => (∑ j∈s k, c k j *
-          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂ν b i) 1 W :=
-      eLpNorm_congr_ae (EventuallyEq.sub (Filter.EventuallyEq.refl _ _) (heI i f))
-    dsimp only [μ] at hnorm
-    simpa only [hnorm] using hL1 i f
-  · intro i hi f
-    exact (integral_congr_ae (heI i f)).trans (hmean i hi f)
-  · intro i j v hi hj H hH
-    let K : ℝ≥0∞ → Kernel ExactSLEGenerator.Driver UpperPlane := fun r => ⟨μ r,hmeas r⟩
-    exact compact_conditional_to_random_tests a ha W hW v (K (t i)) (K (t j))
-      (fun f => by
-        change Integrable (fun b => ∫z,f z∂μ (t i) b) W
-        exact memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using hmoment i f 1))
-      (fun f => by
-        change Integrable (fun b => ∫z,f z∂μ (t j) b) W
-        exact memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using hmoment j f 1))
-      (hcompact i j v hi hj) H hH
-  · filter_upwards [he,h] with b hb hb'
-    dsimp only [μ] at hb
-    simpa only [hb] using hb'
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.MassVersionsCore
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Set Filter CompactlySupported
+open scoped ENNReal NNReal Topology
+
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
+
+ 
+
+theorem isSFiniteKernel_of_pointwise_finite (κ : Kernel Ω X)
+    [∀ ω, IsFiniteMeasure (κ ω)] : IsSFiniteKernel κ := by
+  let r : Ω → ℝ≥0∞ := fun ω => κ ω univ + 1
+  have hr : Measurable r := (κ.measurable_coe MeasurableSet.univ).add_const 1
+  have hr0 (ω : Ω) : r ω ≠ 0 := by simp [r]
+  have hrtop (ω : Ω) : r ω ≠ ∞ := by simp [r, measure_ne_top]
+  let η : Kernel Ω X := ⟨fun ω => (r ω)⁻¹ • κ ω, by
+    apply Measure.measurable_of_measurable_coe
+    intro A hA
+    simp only [Measure.smul_apply, smul_eq_mul]
+    exact hr.inv.mul (κ.measurable_coe hA)⟩
+  have hη : IsFiniteKernel η := by
+    refine ⟨⟨1, ENNReal.one_lt_top, fun ω => ?_⟩⟩
+    change (r ω)⁻¹ * κ ω univ ≤ 1
+    calc
+      _ ≤ (r ω)⁻¹ * r ω := mul_le_mul_of_nonneg_left (le_add_right le_rfl) zero_le
+      _ = 1 := ENNReal.inv_mul_cancel (hr0 ω) (hrtop ω)
+  have he : κ = η.withDensity (fun ω _ => r ω) := by
+    apply Kernel.ext
+    intro ω
+    rw [Kernel.withDensity_apply η (f := fun ω (_ : X) => r ω)
+      (hr.comp measurable_fst), withDensity_const]
+    change κ ω = r ω • ((r ω)⁻¹ • κ ω)
+    rw [smul_smul, ENNReal.mul_inv_cancel (hr0 ω) (hrtop ω), one_smul]
+  rw [he]
+  exact Kernel.IsSFiniteKernel.withDensity η (fun ω _ => hrtop ω)
+
+variable [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [BorelSpace X]
+
+ 
+
+theorem isSFiniteKernel_of_locally_finite (κ : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] : IsSFiniteKernel κ := by
+  let K := CompactExhaustion.choice X
+  let V := disjointed (fun n => K n)
+  have hV (n : ℕ) : MeasurableSet (V n) :=
+    MeasurableSet.disjointed (fun n => (K.isCompact n).measurableSet) n
+  let η : ℕ → Kernel Ω X := fun n => κ.restrict (hV n)
+  have hη (n : ℕ) : IsSFiniteKernel (η n) := by
+    have (ω : Ω) : IsFiniteMeasure (η n ω) := by
+      refine ⟨?_⟩
+      rw [show η n ω = (κ ω).restrict (V n) from rfl, Measure.restrict_apply_univ]
+      exact (measure_mono (disjointed_subset _ _)).trans_lt (K.isCompact n).measure_lt_top
+    exact isSFiniteKernel_of_pointwise_finite (η n)
+  have he : Kernel.sum η = κ := by
+    apply Kernel.ext
+    intro ω
+    change Measure.sum (fun n => (κ ω).restrict (V n)) = κ ω
+    rw [← Measure.restrict_iUnion (disjoint_disjointed _) (fun n => hV n)]
+    simp only [iUnion_disjointed, K.iUnion_eq, Measure.restrict_univ]
+  rw [← he]
+  exact Kernel.isSFiniteKernel_sum
+
+ 
+
+theorem average_finiteOnCompacts (κ : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] (W : Measure Ω)
+    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W) :
+    IsFiniteMeasureOnCompacts (κ ∘ₘ W) := by
+  constructor
+  intro K hK
+  obtain ⟨g,hg1,-,hgc,hg01⟩ := exists_continuous_one_zero_of_isCompact
+    hK isClosed_empty (disjoint_empty K)
+  let f : C_c(X,ℝ) := ⟨g,hgc⟩
+  rw [Measure.bind_apply hK.measurableSet κ.aemeasurable]
+  apply lt_of_le_of_lt _ (hκ f).lintegral_lt_top
+  apply lintegral_mono
+  intro ω
+  exact f.integrable.measure_le_integral (Eventually.of_forall (fun x => (hg01 x).1))
+    (fun x hx => le_of_eq ((hg1 hx).symm))
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
+lemma average_integrable_compact_test (κ : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] (W : Measure Ω)
+    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
+    (f : C_c(X,ℝ)) : Integrable f (κ ∘ₘ W) := by
+  let g : C_c(X,ℝ) := ⟨⟨fun x => ‖f x‖, f.continuous.norm⟩, f.hasCompactSupport.norm⟩
+  apply (Measure.integrable_comp_iff f.continuous.measurable.aestronglyMeasurable).mpr
+  exact ⟨Eventually.of_forall (fun _ => f.integrable), hκ g⟩
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
+lemma average_compact_eval (κ : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] (W : Measure Ω)
+    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
+    (f : C_c(X,ℝ)) :
+    (∫x, f x ∂(κ ∘ₘ W)) = ∫ω, (∫x, f x ∂κ ω) ∂W := by
+  have hf := average_integrable_compact_test κ W hκ f
+  rw [Measure.comp_eq_comp_const_apply] at hf ⊢
+  exact Kernel.integral_comp hf
+
+variable [TopologicalSpace.PseudoMetrizableSpace X]
+
+ 
+
+theorem average_eq_of_compact_tests (κ η : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] [∀ ω, IsFiniteMeasureOnCompacts (η ω)]
+    (W : Measure Ω)
+    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
+    (hη : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂η ω) W)
+    (he : ∀ f : C_c(X,ℝ), (∫ω, (∫x, f x ∂κ ω) ∂W) = ∫ω, (∫x, f x ∂η ω) ∂W) :
+    κ ∘ₘ W = η ∘ₘ W := by
+  have := average_finiteOnCompacts κ W hκ
+  have := average_finiteOnCompacts η W hη
+  apply Measure.ext_of_integral_eq_on_compactlySupported
+  intro f
+  simpa only [average_compact_eval _ _ hκ, average_compact_eval _ _ hη] using he f
+
+omit mΩ [MeasurableSpace X] [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+  [SecondCountableTopology X] [BorelSpace X] [TopologicalSpace.PseudoMetrizableSpace X] in
+lemma prod_sigma_le {m m₀ : MeasurableSpace Ω} (hm : m ≤ m₀)
+    (n : MeasurableSpace X) : m.prod n ≤ m₀.prod n :=
+  sup_le_sup (MeasurableSpace.comap_mono hm) le_rfl
+
+ 
+
+theorem conditional_product_eq_of_compact_tests (κ η : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] [∀ ω, IsFiniteMeasureOnCompacts (η ω)]
+    (W : Measure Ω) [SFinite W]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
+    (hη : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂η ω) W)
+    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
+      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W) :
+    (W ⊗ₘ κ).trim (prod_sigma_le hm _) = (W ⊗ₘ η).trim (prod_sigma_le hm _) := by
+  let : MeasurableSpace Ω := mΩ
+  have := isSFiniteKernel_of_locally_finite κ
+  have := isSFiniteKernel_of_locally_finite η
+  have havg (U : Set Ω) (hU : MeasurableSet[m] U) :
+      κ ∘ₘ (W.restrict U) = η ∘ₘ (W.restrict U) :=
+    average_eq_of_compact_tests κ η (W.restrict U)
+      (fun f => (hκ f).restrict) (fun f => (hη f).restrict) (he U hU)
+  have hrect (U : Set Ω) (hU : MeasurableSet[m] U) (A : Set X) (hA : MeasurableSet A) :
+      (W ⊗ₘ κ).trim (prod_sigma_le hm _) (U ×ˢ A) =
+        (W ⊗ₘ η).trim (prod_sigma_le hm _) (U ×ˢ A) := by
+    rw [trim_measurableSet_eq _ (hU.prod hA), trim_measurableSet_eq _ (hU.prod hA),
+      Measure.compProd_apply_prod (hm _ hU) hA, Measure.compProd_apply_prod (hm _ hU) hA,
+      ← Measure.bind_apply hA κ.aemeasurable, ← Measure.bind_apply hA η.aemeasurable, havg U hU]
+  let K := CompactExhaustion.choice X
+  have := average_finiteOnCompacts κ W hκ
+  have hfin (n : ℕ) : (W ⊗ₘ κ).trim (prod_sigma_le hm _) (univ ×ˢ K n) ≠ ∞ := by
+    rw [trim_measurableSet_eq _ (MeasurableSet.univ.prod (K.isCompact n).measurableSet),
+      Measure.compProd_apply_prod MeasurableSet.univ (K.isCompact n).measurableSet,
+      setLIntegral_univ, ← Measure.bind_apply (K.isCompact n).measurableSet κ.aemeasurable]
+    exact (K.isCompact n).measure_lt_top.ne
+  let : MeasurableSpace Ω := m
+  apply Measure.ext_of_generateFrom_of_iUnion
+    (image2 (· ×ˢ ·) {U : Set Ω | MeasurableSet U} {A : Set X | MeasurableSet A})
+    (fun n => univ ×ˢ K n) generateFrom_prod.symm isPiSystem_prod
+  · ext p
+    simp only [mem_iUnion, mem_prod, mem_univ, true_and, iff_true]
+    exact K.exists_mem p.2
+  · intro n
+    exact ⟨univ, MeasurableSet.univ, K n, (K.isCompact n).measurableSet, rfl⟩
+  · exact hfin
+  · rintro _ ⟨U, hU, A, hA, rfl⟩
+    exact hrect U hU A hA
+
+ 
+
+theorem conditional_lintegral_eq_of_compact_tests (κ η : Kernel Ω X)
+    [∀ ω, IsFiniteMeasureOnCompacts (κ ω)] [∀ ω, IsFiniteMeasureOnCompacts (η ω)]
+    (W : Measure Ω) [SFinite W]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    (hκ : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂κ ω) W)
+    (hη : ∀ f : C_c(X,ℝ), Integrable (fun ω => ∫x, f x ∂η ω) W)
+    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
+      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W)
+    (H : Ω × X → ℝ≥0∞) (hH : Measurable[m.prod ‹MeasurableSpace X›] H) :
+    (∫⁻ω, ∫⁻x, H (ω,x) ∂κ ω ∂W) = ∫⁻ω, ∫⁻x, H (ω,x) ∂η ω ∂W := by
+  let : MeasurableSpace Ω := mΩ
+  have := isSFiniteKernel_of_locally_finite κ
+  have := isSFiniteKernel_of_locally_finite η
+  have hH₀ : Measurable H := hH.mono (prod_sigma_le hm _) le_rfl
+  rw [← Measure.lintegral_compProd hH₀, ← Measure.lintegral_compProd hH₀,
+    ← lintegral_trim (prod_sigma_le hm _) hH,
+    ← lintegral_trim (μ := W ⊗ₘ η) (prod_sigma_le hm _) hH,
+    conditional_product_eq_of_compact_tests κ η W m hm hκ hη he]
+
+end SLEExactGauge.MassVersions
+
+namespace SLEExactGauge.MassVersions
+
+variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [BorelSpace X]
+    [TopologicalSpace.PseudoMetrizableSpace X]
+
+omit [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+  [BorelSpace X] [TopologicalSpace.PseudoMetrizableSpace X] in
+lemma average_mono (κ : Kernel Ω X) {W V : Measure Ω} (h : V ≤ W) : κ ∘ₘ V ≤ κ ∘ₘ W := by
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [Measure.bind_apply hA κ.aemeasurable, Measure.bind_apply hA κ.aemeasurable]
+  exact lintegral_mono' h le_rfl
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X]
+  [TopologicalSpace.PseudoMetrizableSpace X] in
+lemma average_restrict_finite (κ : Kernel Ω X) (W : Measure Ω)
+    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] (U : Set Ω) :
+    IsFiniteMeasureOnCompacts (κ ∘ₘ W.restrict U) := by
+  constructor
+  intro K hK
+  exact ((average_mono κ Measure.restrict_le_self) K).trans_lt hK.measure_lt_top
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+  [TopologicalSpace.PseudoMetrizableSpace X] in
+lemma average_compact_eval_of_average_finite (κ : Kernel Ω X) (W : Measure Ω)
+    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] (f : C_c(X, ℝ)) :
+    (∫x, f x ∂(κ ∘ₘ W)) = ∫ω, (∫x, f x ∂κ ω) ∂W := by
+  have hf : Integrable f (κ ∘ₘ W) := f.integrable
+  rw [Measure.comp_eq_comp_const_apply] at hf ⊢
+  exact Kernel.integral_comp hf
+
+ 
+
+theorem conditional_product_eq_of_average_finite (κ η : Kernel Ω X)
+    [IsSFiniteKernel κ] [IsSFiniteKernel η] (W : Measure Ω) [SFinite W]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] [IsFiniteMeasureOnCompacts (η ∘ₘ W)]
+    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
+      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W) :
+    (W ⊗ₘ κ).trim (prod_sigma_le hm _) = (W ⊗ₘ η).trim (prod_sigma_le hm _) := by
+  let : MeasurableSpace Ω := mΩ
+  have havg (U : Set Ω) (hU : MeasurableSet[m] U) :
+      κ ∘ₘ (W.restrict U) = η ∘ₘ (W.restrict U) := by
+    have := average_restrict_finite κ W U
+    have := average_restrict_finite η W U
+    apply Measure.ext_of_integral_eq_on_compactlySupported
+    intro f
+    simpa only [average_compact_eval_of_average_finite] using he U hU f
+  have hrect (U : Set Ω) (hU : MeasurableSet[m] U) (A : Set X) (hA : MeasurableSet A) :
+      (W ⊗ₘ κ).trim (prod_sigma_le hm _) (U ×ˢ A) =
+        (W ⊗ₘ η).trim (prod_sigma_le hm _) (U ×ˢ A) := by
+    rw [trim_measurableSet_eq _ (hU.prod hA), trim_measurableSet_eq _ (hU.prod hA),
+      Measure.compProd_apply_prod (hm _ hU) hA, Measure.compProd_apply_prod (hm _ hU) hA,
+      ← Measure.bind_apply hA κ.aemeasurable, ← Measure.bind_apply hA η.aemeasurable, havg U hU]
+  let K := CompactExhaustion.choice X
+  have hfin (n : ℕ) : (W ⊗ₘ κ).trim (prod_sigma_le hm _) (univ ×ˢ K n) ≠ ∞ := by
+    rw [trim_measurableSet_eq _ (MeasurableSet.univ.prod (K.isCompact n).measurableSet),
+      Measure.compProd_apply_prod MeasurableSet.univ (K.isCompact n).measurableSet,
+      setLIntegral_univ, ← Measure.bind_apply (K.isCompact n).measurableSet κ.aemeasurable]
+    exact (K.isCompact n).measure_lt_top.ne
+  let : MeasurableSpace Ω := m
+  apply Measure.ext_of_generateFrom_of_iUnion
+    (image2 (· ×ˢ ·) {U : Set Ω | MeasurableSet U} {A : Set X | MeasurableSet A})
+    (fun n => univ ×ˢ K n) generateFrom_prod.symm isPiSystem_prod
+  · ext p
+    simp only [mem_iUnion, mem_prod, mem_univ, true_and, iff_true]
+    exact K.exists_mem p.2
+  · intro n
+    exact ⟨univ, MeasurableSet.univ, K n, (K.isCompact n).measurableSet, rfl⟩
+  · exact hfin
+  · rintro _ ⟨U, hU, A, hA, rfl⟩
+    exact hrect U hU A hA
+
+theorem conditional_lintegral_eq_of_average_finite (κ η : Kernel Ω X)
+    [IsSFiniteKernel κ] [IsSFiniteKernel η] (W : Measure Ω) [SFinite W]
+    (m : MeasurableSpace Ω) (hm : m ≤ mΩ)
+    [IsFiniteMeasureOnCompacts (κ ∘ₘ W)] [IsFiniteMeasureOnCompacts (η ∘ₘ W)]
+    (he : ∀ U : Set Ω, MeasurableSet[m] U → ∀ f : C_c(X,ℝ),
+      (∫ω in U, (∫x, f x ∂κ ω) ∂W) = ∫ω in U, (∫x, f x ∂η ω) ∂W)
+    (H : Ω × X → ℝ≥0∞) (hH : Measurable[m.prod ‹MeasurableSpace X›] H) :
+    (∫⁻ω, ∫⁻x, H (ω,x) ∂κ ω ∂W) = ∫⁻ω, ∫⁻x, H (ω,x) ∂η ω ∂W := by
+  let : MeasurableSpace Ω := mΩ
+  have hH₀ : Measurable H := hH.mono (prod_sigma_le hm _) le_rfl
+  rw [← Measure.lintegral_compProd hH₀, ← Measure.lintegral_compProd hH₀,
+    ← lintegral_trim (prod_sigma_le hm _) hH,
+    ← lintegral_trim (μ := W ⊗ₘ η) (prod_sigma_le hm _) hH,
+    conditional_product_eq_of_average_finite κ η W m hm he]
+
+end SLEExactGauge.MassVersions
+
+namespace SLEExactGauge.MassVersions
+open MeasureTheory ProbabilityTheory Set CompactlySupported
+variable {Ω X : Type*} [mΩ : MeasurableSpace Ω] [MeasurableSpace X]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [BorelSpace X]
+
+omit [BorelSpace X] in
+lemma finiteOnCompacts_of_integrable_tests (ν : Measure X)
+    (hν : ∀ f : C_c(X,ℝ), Integrable f ν) : IsFiniteMeasureOnCompacts ν := by
+  constructor
+  intro K hK
+  obtain ⟨g,hg1,-,hgc,hg01⟩ := exists_continuous_one_zero_of_isCompact
+    hK isClosed_empty (disjoint_empty K)
+  let f : C_c(X,ℝ) := ⟨g,hgc⟩
+  exact ((hν f).measure_le_integral (Eventually.of_forall (fun x => (hg01 x).1))
+    (fun x hx => le_of_eq ((hg1 hx).symm))).trans_lt ENNReal.ofReal_lt_top
+
+omit [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] [BorelSpace X] in
+lemma average_add (κ η : Kernel Ω X) (W : Measure Ω) :
+    (κ + η) ∘ₘ W = κ ∘ₘ W + η ∘ₘ W := by
+  rw [Measure.comp_eq_comp_const_apply, Kernel.comp_add_left,
+    add_apply, ← Measure.comp_eq_comp_const_apply, ← Measure.comp_eq_comp_const_apply]
+
+end SLEExactGauge.MassVersions
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+instance upperArea_finiteOnCompacts : IsFiniteMeasureOnCompacts upperArea := by
+  constructor
+  intro K hK
+  rw [upperArea, upper_embedding.comap_apply]
+  exact (hK.image continuous_subtype_val).measure_lt_top
+
+lemma survivingGreen_joint_measurable (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => survivingGreen a t p.1 p.2.1) :=
+  (actual_surviving_green a ha t).1.mono (prod_sigma_le (rawFiltration.le t) _) le_rfl
+
+lemma survivingGreen_spatial_measurable (a : ℝ) (ha : 1/4<a) (t : ℝ≥0)
+    (b : ExactSLEGenerator.Driver) : Measurable (fun z : UpperPlane => survivingGreen a t b z.1) :=
+  (survivingGreen_joint_measurable a ha t).comp (measurable_const.prodMk measurable_id)
+
+ 
+noncomputable def survivingKernel (a : ℝ) (_ha : 1/4<a) (t : ℝ≥0) :
+    Kernel ExactSLEGenerator.Driver UpperPlane :=
+  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
+    (fun b z => ENNReal.ofReal (survivingGreen a t b z.1))
+
+instance survivingKernel_sfinite (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) :
+    IsSFiniteKernel (survivingKernel a ha t) :=
+  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
+
+lemma survivingKernel_apply (a : ℝ) (ha : 1/4<a) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    survivingKernel a ha t b = upperArea.withDensity (fun z => ENNReal.ofReal (survivingGreen a t b z.1)) :=
+  Kernel.withDensity_apply _ (survivingGreen_joint_measurable a ha t).ennreal_ofReal b
+
+lemma survivingKernel_compact_eval (a : ℝ) (ha : 1/4<a) (t : ℝ≥0)
+    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
+    (∫z, f z ∂survivingKernel a ha t b) = ∫z, f z * survivingGreen a t b z.1 ∂upperArea := by
+  rw [survivingKernel_apply, integral_withDensity_eq_integral_toReal_smul]
+  · simp only [ENNReal.toReal_ofReal (survivingGreen_nonneg _ _ _ _), smul_eq_mul, mul_comm]
+  · exact (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal
+  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
+
+lemma surviving_compact_spatial_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
+    Integrable (fun p : UpperPlane × ExactSLEGenerator.Driver => f p.1*survivingGreen a t p.2 p.1.1)
+      ((upperArea.restrict (tsupport f)).prod W) := by
+  obtain ⟨c,hc,hcK⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
+  obtain ⟨B,hB,hBf⟩ := upperExtension_bounded f
+  exact (actual_remaining_spatial_L1 a ha W hW t (tsupport f) f.hasCompactSupport.measurableSet
+    f.hasCompactSupport.measure_lt_top (c/2) (by positivity)
+    (fun z hz => by have := hcK z hz; linarith) f f.continuous.measurable B hB
+    (fun z hz => by simpa only [upperExtension_apply] using hBf z.1 ⟨z,hz,rfl⟩)
+    (fun n : ℕ => 1/((n:ℝ)+1))
+    (fun n => ⟨by positivity, (div_le_one (by positivity)).mpr (by linarith [Nat.cast_nonneg (α:=ℝ) n])⟩)
+    tendsto_one_div_add_atTop_nhds_zero_nat).1
+
+lemma survivingKernel_compact_mean_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
+    Integrable (fun b => ∫z, f z ∂survivingKernel a ha t b) W := by
+  simp_rw [survivingKernel_compact_eval]
+  convert (surviving_compact_spatial_integrable a ha W hW t f).integral_prod_right using 1
+  funext b
+  exact (setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by
+    rw [image_eq_zero_of_notMem_tsupport hz, zero_mul])).symm
+
+lemma survivingKernel_compact_ae_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
+    ∀ᵐ b ∂W, Integrable f (survivingKernel a ha t b) := by
+  filter_upwards [(surviving_compact_spatial_integrable a ha W hW t f).prod_left_ae] with b hb
+  rw [survivingKernel_apply, integrable_withDensity_iff_integrable_smul'
+    (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal
+    (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  simp only [ENNReal.toReal_ofReal (survivingGreen_nonneg _ _ _ _), smul_eq_mul]
+  have hs : Function.support (fun z : UpperPlane => f z * survivingGreen a t b z.1) ⊆ tsupport f :=
+    (Function.support_mul_subset_left _ _).trans (subset_tsupport f)
+  simpa only [mul_comm] using (integrableOn_iff_integrable_of_support_subset hs).mp hb
+
+lemma survivingKernel_average_compact_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (f : C_c(UpperPlane,ℝ)) :
+    Integrable f (survivingKernel a ha t ∘ₘ W) := by
+  let g : C_c(UpperPlane,ℝ) := ⟨⟨fun z => ‖f z‖,f.continuous.norm⟩,f.hasCompactSupport.norm⟩
+  apply (Measure.integrable_comp_iff f.continuous.measurable.aestronglyMeasurable).mpr
+  exact ⟨survivingKernel_compact_ae_integrable a ha W hW t f,
+    survivingKernel_compact_mean_integrable a ha W hW t g⟩
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma survivingKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) : IsFiniteMeasureOnCompacts (survivingKernel a ha t ∘ₘ W) :=
+  finiteOnCompacts_of_integrable_tests _ (survivingKernel_average_compact_integrable a ha W hW t)
+
+ 
+
+theorem compact_conditional_to_random_tests (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0) (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
+    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂μ b) W)
+    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z, f z ∂ν b) W)
+    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
+      MeasurableSet[rawFiltration t] U →
+      (∫b in U, (∫z, f z ∂ν b) - (∫z, f z ∂μ b) ∂W) =
+        ∫b in U, (∫z, f z*survivingGreen a t b z.1 ∂upperArea) ∂W)
+    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
+    (hH : Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H) :
+    (∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ b ∂W) +
+      ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a t b z.1) ∂upperArea ∂W := by
+  have := isSFiniteKernel_of_locally_finite μ
+  have := isSFiniteKernel_of_locally_finite ν
+  have := average_finiteOnCompacts μ W hμ
+  have := average_finiteOnCompacts ν W hν
+  have := survivingKernel_average_finiteOnCompacts a ha W hW t
+  have : IsFiniteMeasureOnCompacts ((μ + survivingKernel a ha t) ∘ₘ W) := by
+    rw [SLEExactGauge.MassVersions.average_add]
+    constructor
+    intro K hK
+    rw [Measure.add_apply]
+    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩
+  have hidentity := conditional_lintegral_eq_of_average_finite ν (μ + survivingKernel a ha t) W
+    (rawFiltration t) (rawFiltration.le t) ?_ H hH
+  · rw [hidentity]
+    have hH₀ : Measurable H := hH.mono (prod_sigma_le (rawFiltration.le t) _) le_rfl
+    have heval (b : ExactSLEGenerator.Driver) :
+        (∫⁻z, H (b,z) ∂(μ+survivingKernel a ha t) b) =
+          (∫⁻z, H (b,z) ∂μ b) + ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a t b z.1) ∂upperArea := by
+      have hHb : Measurable (fun z : UpperPlane => H (b,z)) :=
+        hH₀.comp ((measurable_const (a := b)).prodMk measurable_id)
+      rw [add_apply, lintegral_add_measure, survivingKernel_apply]
+      congr 1
+      exact (lintegral_withDensity_eq_lintegral_mul upperArea
+        (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal hHb).trans
+          (lintegral_congr (fun _ => mul_comm _ _))
+    simp_rw [heval]
+    rw [lintegral_add_left]
+    exact hH₀.lintegral_kernel_prod_right'
+  · intro U hU f
+    have hadd : (fun b => ∫z, f z ∂(μ+survivingKernel a ha t) b) =ᵐ[W]
+        (fun b => (∫z, f z ∂μ b)+(∫z, f z ∂survivingKernel a ha t b)) := by
+      filter_upwards [survivingKernel_compact_ae_integrable a ha W hW t f] with b hb
+      exact integral_add_measure f.integrable hb
+    rw [integral_congr_ae (ae_restrict_of_ae hadd),
+      integral_add (hμ f).restrict (survivingKernel_compact_mean_integrable a ha W hW t f).restrict]
+    have hf := he f U hU
+    rw [integral_sub (hν f).restrict (hμ f).restrict] at hf
+    simp_rw [survivingKernel_compact_eval]
+    linarith
+
+end ExactSLEGreen
+end
+ 
+
+ 
+section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_raw_mass_extraction (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        (∫b, (∫z,f z∂μ (t i) b) ∂W) = ∫z : UpperPlane,
+          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        ∀ U : Set ExactSLEGenerator.Driver, MeasurableSet[rawFiltration v] U →
+        (∫b in U, (∫z,f z∂μ (t j) b)-(∫z,f z∂μ (t i) b) ∂W) =
+          ∫b in U, (∫z : UpperPlane,f z*survivingGreen a v b z.1 ∂upperArea) ∂W) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫z,f z∂μ (t i) b))) := by
+  obtain ⟨s,c,ν,hs,hc,hsum,hν,hmom,hL1,hmean,-,hcond,h⟩ :=
+    actual_radon_compact_conditional a ha W hW t
+  let μ := recoveredMass a ha s c hc
+  have hμ (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+      IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular :=
+    ⟨radonLimit_finite _, radonLimit_regular _⟩
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := (hμ v b).1
+  have hmeas (v : ℝ≥0∞) : Measurable (μ v) := recoveredMass_measurable a ha s c hc v
+  have he : ∀ᵐ b ∂W, ∀ i, μ (t i) b = ν b i := by
+    filter_upwards [h] with b hb
+    intro i
+    have := (hν b i).2
+    apply radonLimit_eq
+    intro f
+    simpa only [actualMassFunctionals_apply] using hb.2 i f
+  have heI (i : ℕ) (f : C_c(UpperPlane,ℝ)) :
+      (fun b => ∫z,f z∂μ (t i) b) =ᵐ[W] (fun b => ∫z,f z∂ν b i) := by
+    filter_upwards [he] with b hb
+    rw [hb i]
+  have hmoment (i : ℕ) (f : C_c(UpperPlane,ℝ)) (m : ℕ) :
+      MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W :=
+    (memLp_congr_ae (heI i f)).mpr (hmom i f m)
+  have hcompact (i j : ℕ) (v : ℝ≥0) (hi : t i=v) (hj : t j=∞)
+      (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
+      (hU : MeasurableSet[rawFiltration v] U) :
+      (∫b in U, (∫z,f z∂μ (t j) b)-(∫z,f z∂μ (t i) b) ∂W) =
+        ∫b in U, (∫z : UpperPlane,f z*survivingGreen a v b z.1 ∂upperArea) ∂W :=
+    (integral_congr_ae (ae_restrict_of_ae ((heI j f).sub (heI i f)))).trans
+      (hcond i j v hi hj f U hU)
+  refine ⟨s,c,hc,hs,hsum,hμ,hmeas,recoveredMass_progressive a ha s c hc,hmoment,?_,?_,hcompact,?_,?_⟩
+  · intro i f
+    have hnorm (k : ℕ) :
+        eLpNorm (fun b => (∑ j∈s k, c k j *
+          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂μ (t i) b) 1 W =
+        eLpNorm (fun b => (∑ j∈s k, c k j *
+          ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b) - ∫z,f z∂ν b i) 1 W :=
+      eLpNorm_congr_ae (EventuallyEq.sub (Filter.EventuallyEq.refl _ _) (heI i f))
+    dsimp only [μ] at hnorm
+    simpa only [hnorm] using hL1 i f
+  · intro i hi f
+    exact (integral_congr_ae (heI i f)).trans (hmean i hi f)
+  · intro i j v hi hj H hH
+    let K : ℝ≥0∞ → Kernel ExactSLEGenerator.Driver UpperPlane := fun r => ⟨μ r,hmeas r⟩
+    exact compact_conditional_to_random_tests a ha W hW v (K (t i)) (K (t j))
+      (fun f => by
+        change Integrable (fun b => ∫z,f z∂μ (t i) b) W
+        exact memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using hmoment i f 1))
+      (fun f => by
+        change Integrable (fun b => ∫z,f z∂μ (t j) b) W
+        exact memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using hmoment j f 1))
+      (hcompact i j v hi hj) H hH
+  · filter_upwards [he,h] with b hb hb'
+    dsimp only [μ] at hb
+    simpa only [hb] using hb'
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RadonMoments.lean b/lean/OAI/Probability/SLEGauge/RadonMoments.lean
index cb0d11f4ea97094bf274e43e2665c9767c70173c..78664b6dce527b2f7034db126475f65b05183982 100644
--- a/lean/OAI/Probability/SLEGauge/RadonMoments.lean
+++ b/lean/OAI/Probability/SLEGauge/RadonMoments.lean
@@ -1,472 +1,472 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.LocalFiniteness
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-open Filter Topology
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-theorem actual_radon_all_moments (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        AEStronglyMeasurable (fun b => ∫ z, f z ∂μ b i) W) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i)))
- := by
-  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmeas,h⟩ := actual_ordered_radon_extraction a ha W hW t
-  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmeas,?_,h⟩
-  intro i f m
-  apply actual_radon_limit_memLp a ha W hW (t i) s c hc hsum (fun b => μ b i) f _ m
-  filter_upwards [h] with b hb
-  exact hb.2 i f
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-
-lemma unifIntegrable_of_L2_bound {Ω : Type*} [MeasurableSpace Ω]
-    (W : Measure Ω) (F : ℕ → Ω → ℝ)
-    (hF : ∀ n, AEStronglyMeasurable (F n) W)
-    (C : ℝ≥0∞) (hC : C<∞) (hbound : ∀ n, eLpNorm (F n) 2 W≤C) :
-    UnifIntegrable F 1 W := by
-  apply unifIntegrable_iff.mpr
-  intro ε hε
-  have ht : Tendsto (fun δ : ℝ => C * (ENNReal.ofReal δ)^((1:ℝ)/2)) (𝓝 0) (𝓝 0) := by
-    have h := (ENNReal.continuous_rpow_const (y := (1:ℝ)/2)).continuousAt.tendsto.comp
-      (ENNReal.continuous_ofReal.continuousAt.tendsto (x := (0:ℝ)))
-    have hzero : (ENNReal.ofReal (0:ℝ))^((1:ℝ)/2)=(0:ℝ≥0∞) := by simp
-    rw [hzero] at h
-    simpa using ENNReal.Tendsto.const_mul h (Or.inr hC.ne)
-  obtain ⟨r,hr,hrb⟩ := Metric.eventually_nhds_iff.mp
-    ((tendsto_order.mp ht).2 ε hε)
-  have hδ : 0<r/2 := by positivity
-  have hsml : C*(ENNReal.ofReal (r/2))^((1:ℝ)/2)<ε :=
-    hrb (y := r/2) (by rw [Real.dist_eq, sub_zero, abs_of_pos hδ]; linarith)
-  refine ⟨ENNReal.ofReal (r/2),ENNReal.ofReal_pos.mpr hδ,?_⟩
-  intro n s hWs
-  calc
-    _ ≤ eLpNorm (F n) 2 (W.restrict s) * (W s)^((1:ℝ)/2) := by
-      convert eLpNorm_le_eLpNorm_mul_rpow_measure_univ
-        (μ := W.restrict s) (by norm_num : (1:ℝ≥0∞)≤2) (hF n).restrict using 1
-      norm_num
-    _ ≤ C*(ENNReal.ofReal (r/2))^((1:ℝ)/2) := by
-      gcongr
-      exact (eLpNorm_mono_measure (F n) Measure.restrict_le_self).trans (hbound n)
-    _ ≤ ε := hsml.le
-
-lemma actual_radon_limit_L1 (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℝ≥0∞) (q : ℕ → ℝ) (hq : ∀ j,0<q j) (hq1 : ∀ j,q j≤1)
-    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-    (hc : ∀ k j,j∈s k→0≤coeff k j) (hsum : ∀ k,∑j∈s k,coeff k j=1)
-    (μ : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
-    (hlim : ∀ᵐ b∂W, Tendsto (fun k => ∑j∈s k,coeff k j*
-      ∫z,f z∂actualUpperMass a (q j) t b) atTop (𝓝 (∫z,f z∂μ b))) :
-    Tendsto (fun k => eLpNorm (fun b => (∑j∈s k,coeff k j*
-      ∫z,f z∂actualUpperMass a (q j) t b)-(∫z,f z∂μ b)) 1 W) atTop (𝓝 0) := by
-  let F := fun k b => ∑j∈s k,coeff k j*∫z,f z∂actualUpperMass a (q j) t b
-  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
-    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
-      (actualUpperMass_eval_measurable a ha (q j) (hq j) (hq1 j) t f).aestronglyMeasurable.const_mul (coeff k j))
-    convert H using 1
-    funext b
-    simp only [F,Finset.sum_apply]
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW 2 (by norm_num) f
-  have hbound (k : ℕ) : eLpNorm (F k) 2 W≤C :=
-    convex_eLpNorm_bound W _ C (by norm_num) (s k) (coeff k) (hc k) (hsum k)
-      (fun j b => ∫z,f z∂actualUpperMass a (q j) t b)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).1.aestronglyMeasurable)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).2)
-  have hg2 : MemLp (fun b => ∫z,f z∂μ b) 2 W :=
-    (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF
-      (aestronglyMeasurable_of_tendsto_ae atTop hF hlim) hlim).trans_lt hC
-  have hg := hg2.mono_exponent (by norm_num : (1:ℝ≥0∞)≤2)
-  exact tendsto_Lp_finite_of_tendsto_ae le_rfl (by simp) hF (by simpa using hg)
-    (unifIntegrable_of_L2_bound W F hF C hC hbound) hlim
-
- 
-
-theorem actual_radon_L1_convergence (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, coeff k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i)))
- := by
-  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmeas,hmom,h⟩ := actual_radon_all_moments a ha W hW t
-  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,?_,h⟩
-  intro i f
-  have hq1 (j : ℕ) : Real.exp (-2*a*(j+1))≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  apply actual_radon_limit_L1 a ha W hW (t i)
-    (fun j => Real.exp (-2*a*(j+1)))
-    (fun j => Real.exp_pos _) hq1 s c hc hsum (fun b => μ b i) f
-  filter_upwards [h] with b hb
-  exact hb.2 i f
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
-open scoped NNReal ENNReal
-
-lemma actualMassEval_joint_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B) :
-    Integrable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      f p.2 * (timeMassDensity a q t p.1 p.2).toReal) (W.prod (volume.restrict K)) := by
-  obtain ⟨B,hB,hBf⟩ := hfB
-  have : IsFiniteMeasure (volume.restrict K) := ⟨by simpa using hKb.measure_lt_top⟩
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      f p.2 * (timeMassDensity a q t p.1 p.2).toReal) :=
-    (hf.comp measurable_snd).mul (timeMassDensity_measurable a ha q hq hq1 t).ennreal_toReal
-  apply Integrable.of_bound hm.aestronglyMeasurable (B*((c*q)^(-alpha a)))
-  have hmem : ∀ᵐ p ∂W.prod (volume.restrict K), p.2∈K :=
-    (Measure.ae_prod_iff_ae_ae (hK.preimage measurable_snd)).mpr
-      (Eventually.of_forall fun _ => ae_restrict_mem hK)
-  filter_upwards [hmem] with p hp
-  have hb := timeMassDensity_local_bound a ha q hq hq1 t p.1 c hc p.2 (hcK _ hp)
-  have hD : 0≤(c*q)^(-alpha a) := Real.rpow_nonneg (by positivity) _
-  have hd : (timeMassDensity a q t p.1 p.2).toReal≤(c*q)^(-alpha a) := by
-    exact ENNReal.toReal_le_of_le_ofReal hD hb
-  rw [norm_mul,Real.norm_of_nonneg ENNReal.toReal_nonneg]
-  exact mul_le_mul (hBf _ hp) hd ENNReal.toReal_nonneg hB
-
-lemma terminal_density_mean (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (z : ℂ) (hz : 0<z.im) :
-    (∫b, (timeMassDensity a q ∞ b z).toReal ∂W) =
-      Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0) := by
-  have hnon (b : ExactSLEGenerator.Driver) :=
-    (fullRadialDensity_bound a ha z hz (2*z.im*q) (by positivity) (by nlinarith) b).1
-  simp only [timeMassDensity,le_top,ite_true,massDensity,hz,ENNReal.toReal_ofReal (hnon _)]
-  have he : 2*z.im*q/2=z.im*q := by ring
-  simp only [fullRadialDensity,he]
-  rw [integral_const_mul]
-  congr 1
-  exact (radialWeight_mean W hW a ha z hz (z.im*q) (by positivity) (by nlinarith)).2
-
-lemma actualMassEval_terminal_mean (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B) :
-    (∫b,actualMassEval a q ∞ K f b∂W) =
-      ∫z in K, f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)) := by
-  unfold actualMassEval
-  rw [integral_integral_swap (actualMassEval_joint_integrable a ha W q hq hq1 ∞ K hK hKb c hc hcK f hf hfB)]
-  apply setIntegral_congr_fun hK
-  intro z hz
-  dsimp only
-  rw [integral_const_mul,terminal_density_mean a ha W hW q hq hq1 z (by have := hcK z hz; linarith)]
-
-lemma actualUpperMass_terminal_mean (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (f : C_c(UpperPlane,ℝ)) :
-    (∫b, (∫z,f z∂actualUpperMass a q ∞ b) ∂W) = ∫z : UpperPlane,
-      f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea := by
-  let K := Subtype.val '' tsupport f
-  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
-  obtain ⟨c,hc,hcK⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  rw [actualMassEval_terminal_mean a ha W hW q hq hq1 K hK.measurableSet hK.isBounded c hc
-    (by rintro z ⟨w,hw,rfl⟩; exact hcK w hw) (upperExtension f)
-    (upperExtension_measurable f) (upperExtension_bounded f)]
-  simp_rw [← upperExtension_apply f]
-  trans ∫z in {z : ℂ | 0<z.im},
-    upperExtension f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0))
-  · rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
-    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
-    · rw [upperExtension_zero f z (by rintro ⟨w,hw,rfl⟩; exact hz w.2), zero_mul]
-    · rw [upperExtension_zero f z hz, zero_mul]
-  · exact (integral_subtype_comap (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet
-      (fun z : ℂ => upperExtension f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)))).symm
-
-lemma actual_radon_limit_mean (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-    (hsum : ∀ k,∑j∈s k,coeff k j=1)
-    (μ : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
-    (hg : MemLp (fun b => ∫z,f z∂μ b) 1 W)
-    (hlim : Tendsto (fun k => eLpNorm (fun b => (∑j∈s k,coeff k j*
-      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)-(∫z,f z∂μ b)) 1 W) atTop (𝓝 0)) :
-    (∫b, (∫z,f z∂μ b) ∂W) = ∫z : UpperPlane,
-      f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea := by
-  let q (j : ℕ) := Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  let F := fun k b => ∑j∈s k,coeff k j*∫z,f z∂actualUpperMass a (q j) ∞ b
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW 1 (by norm_num) f
-  have hFi (k : ℕ) : Integrable (F k) W :=
-    integrable_finsetSum (s k) (fun j hj =>
-      ((hCb (q j) (hq j) (hq1 j) ∞).1.integrable (by norm_num)).const_mul (coeff k j))
-  change Tendsto (fun k => eLpNorm (fun b => F k b - ∫z,f z∂μ b) 1 W) atTop (𝓝 0) at hlim
-  have ht := tendsto_integral_of_L1 (fun b => ∫z,f z∂μ b) hg.aestronglyMeasurable
-    (Eventually.of_forall hFi) (by
-      exact hlim.congr fun index => eLpNorm_one_eq_lintegral_enorm
-        ((hFi index).aestronglyMeasurable.sub hg.aestronglyMeasurable))
-  have he (k : ℕ) : (∫b,F k b∂W) = ∫z : UpperPlane,
-      f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea := by
-    dsimp only [F]
-    rw [integral_finsetSum _ (fun j hj =>
-      ((hCb (q j) (hq j) (hq1 j) ∞).1.integrable (by norm_num)).const_mul (coeff k j))]
-    simp_rw [integral_const_mul]
-    simp_rw [actualUpperMass_terminal_mean a ha W hW _ (hq _) (hq1 _) f]
-    rw [← Finset.sum_mul,hsum k,one_mul]
-  exact tendsto_nhds_unique ht (by simpa only [he] using tendsto_const_nhds)
-
- 
-
-theorem actual_radon_terminal_mean (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) :
-    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
-      (∀ k, ∑ j∈s k, coeff k j=1) ∧
-      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane, ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, coeff k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
-        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
-          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
-        (∀ i (f : C_c(UpperPlane, ℝ)),
-          Tendsto (fun k => ∑ j∈s k, coeff k j *
-            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫ z, f z ∂μ b i)))
- := by
-  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,h⟩ := actual_radon_L1_convergence a ha W hW t
-  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,?_,h⟩
-  intro i hi f
-  apply actual_radon_limit_mean a ha W hW s c hsum (fun b => μ b i) f
-    (by simpa using hmom i f 1)
-  simpa only [hi] using hL1 i f
-
-end ExactSLEGreen
-
-namespace ExactSLEGenerator
-open MeasureTheory ProbabilityTheory Filter Topology
-open scoped NNReal ENNReal
-noncomputable def pastDriver (t : ℝ≥0) (b : Driver) : Driver :=
-  ⟨⟨fun s => b.1 (min s t), b.1.continuous.comp (continuous_id.min continuous_const)⟩, by simpa using b.2⟩
-lemma pastDriver_eq (t : ℝ≥0) (b : Driver) (s : ℝ≥0) (hs : s≤t) :
-    (pastDriver t b).1 s = b.1 s := by simp [pastDriver,min_eq_left hs]
-lemma pastDriver_raw_measurable (t : ℝ≥0) : Measurable[rawFiltration t] (pastDriver t) := by
-  have : PolishSpace Driver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
-  obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq (α:=ℝ≥0)
-  let e : Driver → (ℕ → ℝ) := fun b n => b.1 (u n)
-  have hec : Continuous e := continuous_pi fun n => (continuous_eval_const (u n)).comp continuous_subtype_val
-  have hei : Function.Injective e := by
-    intro b c h
-    apply Subtype.ext
-    apply DFunLike.coe_injective
-    exact hu.equalizer b.1.continuous c.1.continuous h
-  have hem := hec.measurableEmbedding hei
-  apply hem.measurable_comp_iff.mp
-  let : MeasurableSpace Driver := rawFiltration t
-  apply measurable_pi_iff.mpr
-  intro n
-  exact ((Filtration.stronglyAdapted_natural driver_eval_measurable) (min (u n) t)).measurable.mono
-    (rawFiltration.mono (min_le_right _ _)) le_rfl
-end ExactSLEGenerator
-namespace ExactSLEMain
-open MeasureTheory ExactSLEGenerator
-open scoped NNReal ENNReal
-lemma odeUntil_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
-    (h : ∀ s≤t, b s=c s) {f : ℝ → ℂ} (hf : odeUntil a b z t f) : odeUntil a c z t f := by
-  refine ⟨hf.1,fun u hu => ?_⟩
-  have he := h u.toNNReal ((Real.toNNReal_le_iff_le_coe).mpr hu.2)
-  simpa only [he] using hf.2 u hu
-lemma alive_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
-    (h : ∀ s≤t, b s=c s) : alive a b t z ↔ alive a c t z := by
-  exact ⟨fun ⟨f,hf⟩ => ⟨f,odeUntil_driver_congr h hf⟩,
-    fun ⟨f,hf⟩ => ⟨f,odeUntil_driver_congr (fun s hs => (h s hs).symm) hf⟩⟩
-lemma loewnerMap_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {t : ℝ≥0}
-    (h : ∀ s≤t, b s=c s) : loewnerMap a b t = loewnerMap a c t := by
-  funext z
-  by_cases hb : alive a b t z
-  · obtain ⟨f,hf⟩ := hb
-    exact (loewnerMap_eq hf).trans (loewnerMap_eq (odeUntil_driver_congr h hf)).symm
-  · have hc := mt (alive_driver_congr h).mpr hb
-    simp [loewnerMap,hb,hc]
-end ExactSLEMain
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal
-lemma actualState_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {t : ℝ≥0} (h : ∀ s≤t,b s=c s) (z : ℂ) :
-    actualState a b t z = actualState a c t z := by
-  simp only [actualState,loewnerMap_driver_congr h,h t le_rfl]
-lemma radialTime_driver_congr_before (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b c : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (h : ∀ s≤t,b.1 s=c.1 s) (hb : radialTime a b.1 z ρ≤t) :
-    radialTime a b.1 z ρ = radialTime a c.1 z ρ := by
-  have hbfin : radialTime a b.1 z ρ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hb
-  have hbr : (radialTime a b.1 z ρ).toNNReal≤t := by
-    exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hbfin] using hb)
-  obtain ⟨hba,hbh⟩ := radialTime_attained a ha z hz ρ hρ hr b hbfin
-  have he := actualState_driver_congr (a:=a) (fun s hs => h s (hs.trans hbr)) z
-  have hc : radialTime a c.1 z ρ≤radialTime a b.1 z ρ := by
-    have hh := radialTime_le_of_hit a z ρ c (radialTime a b.1 z ρ).toNNReal
-      ((alive_driver_congr (fun s hs => h s (hs.trans hbr))).mp hba) (by rw [← he]; exact hbh)
-    simpa only [ENNReal.coe_toNNReal hbfin] using hh
-  have hcfin : radialTime a c.1 z ρ≠∞ := ne_top_of_le_ne_top hbfin hc
-  have hcr : (radialTime a c.1 z ρ).toNNReal≤t := by
-    exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hcfin] using hc.trans hb)
-  obtain ⟨hca,hch⟩ := radialTime_attained a ha z hz ρ hρ hr c hcfin
-  have he' := actualState_driver_congr (a:=a) (fun s hs => (h s (hs.trans hcr)).symm) z
-  have hh := radialTime_le_of_hit a z ρ b (radialTime a c.1 z ρ).toNNReal
-    ((alive_driver_congr (fun s hs => (h s (hs.trans hcr)).symm)).mp hca) (by rw [← he']; exact hch)
-  exact le_antisymm (by simpa only [ENNReal.coe_toNNReal hcfin] using hh) hc
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal
-lemma timeMassDensity_driver_congr (a : ℝ) (ha : 0<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (b c : ExactSLEGenerator.Driver) (t : ℝ≥0) (h : ∀ s≤t,b.1 s=c.1 s) (z : ℂ) :
-    timeMassDensity a q t b z = timeMassDensity a q t c z := by
-  by_cases hz : 0<z.im
-  · have hρ : 0<z.im*q := mul_pos hz hq
-    have hr : z.im*q≤z.im := by nlinarith
-    have hiff : radialTime a b.1 z (z.im*q)≤t ↔ radialTime a c.1 z (z.im*q)≤t := by
-      constructor
-      · intro hb
-        rwa [← radialTime_driver_congr_before a ha z hz _ hρ hr b c t h hb]
-      · intro hc
-        rwa [← radialTime_driver_congr_before a ha z hz _ hρ hr c b t (fun s hs => (h s hs).symm) hc]
-    by_cases hb : radialTime a b.1 z (z.im*q)≤t
-    · have he := radialTime_driver_congr_before a ha z hz _ hρ hr b c t h hb
-      have hfin : radialTime a b.1 z (z.im*q)≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hb
-      have ht : (radialTime a b.1 z (z.im*q)).toNNReal≤t := by
-        exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hb)
-      have hs := actualState_driver_congr (a:=a) (fun s hs => h s (hs.trans ht)) z
-      have hw : radialWeight a b.1 z (z.im*q)=radialWeight a c.1 z (z.im*q) := by
-        unfold radialWeight
-        rw [ite_eq_right hfin,ite_eq_right (ne_of_eq_of_ne he.symm hfin),← he,hs]
-      have hn : 2*z.im*q/2=z.im*q := by ring
-      simp only [timeMassDensity,ite_eq_left hb,ite_eq_left (hiff.mp hb),massDensity,ite_eq_left hz,fullRadialDensity,hn,hw]
-    · simp only [timeMassDensity,ite_eq_right hb,ite_eq_right (mt hiff.mpr hb)]
-  · simp only [timeMassDensity,massDensity,ite_eq_right hz,ite_self]
-lemma timeMassDensity_past_eq (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
-    (hq1 : q≤1) (t : ℝ≥0) :
-    (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t p.1 p.2) =
-    (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t (pastDriver t p.1) p.2) := by
-  funext p
-  exact timeMassDensity_driver_congr a (by linarith) q hq hq1 p.1 (pastDriver t p.1) t
-    (fun s hs => (pastDriver_eq t p.1 s hs).symm) p.2
-lemma rawPastPair_measurable (t : ℝ≥0) :
-    @Measurable (ExactSLEGenerator.Driver × ℂ) (ExactSLEGenerator.Driver × ℂ)
-      ((rawFiltration t).prod (borel ℂ)) inferInstance (fun p => (pastDriver t p.1,p.2)) := by
-  exact ((pastDriver_raw_measurable t).comp
-    (@measurable_fst _ _ (rawFiltration t) (borel ℂ))).prodMk
-    (@measurable_snd _ _ (rawFiltration t) (borel ℂ))
- 
-theorem timeMassDensity_raw_joint (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
-    (hq1 : q≤1) (t : ℝ≥0) :
-    letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t p.1 p.2) := by
-  change @Measurable (ExactSLEGenerator.Driver × ℂ) ℝ≥0∞
-    ((rawFiltration t).prod (borel ℂ)) inferInstance _
-  rw [timeMassDensity_past_eq a ha q hq hq1 t]
-  exact @Measurable.comp (ExactSLEGenerator.Driver × ℂ) (ExactSLEGenerator.Driver × ℂ) ℝ≥0∞
-    ((rawFiltration t).prod (borel ℂ)) inferInstance inferInstance
-    (fun p => timeMassDensity a q (t : ℝ≥0∞) p.1 p.2)
-    (fun p => (pastDriver t p.1,p.2))
-    (timeMassDensity_measurable a ha q hq hq1 (t : ℝ≥0∞)) (rawPastPair_measurable t)
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.LocalFiniteness
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+open Filter Topology
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+theorem actual_radon_all_moments (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        AEStronglyMeasurable (fun b => ∫ z, f z ∂μ b i) W) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i)))
+ := by
+  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmeas,h⟩ := actual_ordered_radon_extraction a ha W hW t
+  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmeas,?_,h⟩
+  intro i f m
+  apply actual_radon_limit_memLp a ha W hW (t i) s c hc hsum (fun b => μ b i) f _ m
+  filter_upwards [h] with b hb
+  exact hb.2 i f
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+
+lemma unifIntegrable_of_L2_bound {Ω : Type*} [MeasurableSpace Ω]
+    (W : Measure Ω) (F : ℕ → Ω → ℝ)
+    (hF : ∀ n, AEStronglyMeasurable (F n) W)
+    (C : ℝ≥0∞) (hC : C<∞) (hbound : ∀ n, eLpNorm (F n) 2 W≤C) :
+    UnifIntegrable F 1 W := by
+  apply unifIntegrable_iff.mpr
+  intro ε hε
+  have ht : Tendsto (fun δ : ℝ => C * (ENNReal.ofReal δ)^((1:ℝ)/2)) (𝓝 0) (𝓝 0) := by
+    have h := (ENNReal.continuous_rpow_const (y := (1:ℝ)/2)).continuousAt.tendsto.comp
+      (ENNReal.continuous_ofReal.continuousAt.tendsto (x := (0:ℝ)))
+    have hzero : (ENNReal.ofReal (0:ℝ))^((1:ℝ)/2)=(0:ℝ≥0∞) := by simp
+    rw [hzero] at h
+    simpa using ENNReal.Tendsto.const_mul h (Or.inr hC.ne)
+  obtain ⟨r,hr,hrb⟩ := Metric.eventually_nhds_iff.mp
+    ((tendsto_order.mp ht).2 ε hε)
+  have hδ : 0<r/2 := by positivity
+  have hsml : C*(ENNReal.ofReal (r/2))^((1:ℝ)/2)<ε :=
+    hrb (y := r/2) (by rw [Real.dist_eq, sub_zero, abs_of_pos hδ]; linarith)
+  refine ⟨ENNReal.ofReal (r/2),ENNReal.ofReal_pos.mpr hδ,?_⟩
+  intro n s hWs
+  calc
+    _ ≤ eLpNorm (F n) 2 (W.restrict s) * (W s)^((1:ℝ)/2) := by
+      convert eLpNorm_le_eLpNorm_mul_rpow_measure_univ
+        (μ := W.restrict s) (by norm_num : (1:ℝ≥0∞)≤2) (hF n).restrict using 1
+      norm_num
+    _ ≤ C*(ENNReal.ofReal (r/2))^((1:ℝ)/2) := by
+      gcongr
+      exact (eLpNorm_mono_measure (F n) Measure.restrict_le_self).trans (hbound n)
+    _ ≤ ε := hsml.le
+
+lemma actual_radon_limit_L1 (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℝ≥0∞) (q : ℕ → ℝ) (hq : ∀ j,0<q j) (hq1 : ∀ j,q j≤1)
+    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+    (hc : ∀ k j,j∈s k→0≤coeff k j) (hsum : ∀ k,∑j∈s k,coeff k j=1)
+    (μ : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
+    (hlim : ∀ᵐ b∂W, Tendsto (fun k => ∑j∈s k,coeff k j*
+      ∫z,f z∂actualUpperMass a (q j) t b) atTop (𝓝 (∫z,f z∂μ b))) :
+    Tendsto (fun k => eLpNorm (fun b => (∑j∈s k,coeff k j*
+      ∫z,f z∂actualUpperMass a (q j) t b)-(∫z,f z∂μ b)) 1 W) atTop (𝓝 0) := by
+  let F := fun k b => ∑j∈s k,coeff k j*∫z,f z∂actualUpperMass a (q j) t b
+  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
+    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
+      (actualUpperMass_eval_measurable a ha (q j) (hq j) (hq1 j) t f).aestronglyMeasurable.const_mul (coeff k j))
+    convert H using 1
+    funext b
+    simp only [F,Finset.sum_apply]
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW 2 (by norm_num) f
+  have hbound (k : ℕ) : eLpNorm (F k) 2 W≤C :=
+    convex_eLpNorm_bound W _ C (by norm_num) (s k) (coeff k) (hc k) (hsum k)
+      (fun j b => ∫z,f z∂actualUpperMass a (q j) t b)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).1.aestronglyMeasurable)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) t).2)
+  have hg2 : MemLp (fun b => ∫z,f z∂μ b) 2 W :=
+    (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF
+      (aestronglyMeasurable_of_tendsto_ae atTop hF hlim) hlim).trans_lt hC
+  have hg := hg2.mono_exponent (by norm_num : (1:ℝ≥0∞)≤2)
+  exact tendsto_Lp_finite_of_tendsto_ae le_rfl (by simp) hF (by simpa using hg)
+    (unifIntegrable_of_L2_bound W F hF C hC hbound) hlim
+
+ 
+
+theorem actual_radon_L1_convergence (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, coeff k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i)))
+ := by
+  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmeas,hmom,h⟩ := actual_radon_all_moments a ha W hW t
+  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,?_,h⟩
+  intro i f
+  have hq1 (j : ℕ) : Real.exp (-2*a*(j+1))≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  apply actual_radon_limit_L1 a ha W hW (t i)
+    (fun j => Real.exp (-2*a*(j+1)))
+    (fun j => Real.exp_pos _) hq1 s c hc hsum (fun b => μ b i) f
+  filter_upwards [h] with b hb
+  exact hb.2 i f
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction
+open scoped NNReal ENNReal
+
+lemma actualMassEval_joint_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B) :
+    Integrable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      f p.2 * (timeMassDensity a q t p.1 p.2).toReal) (W.prod (volume.restrict K)) := by
+  obtain ⟨B,hB,hBf⟩ := hfB
+  have : IsFiniteMeasure (volume.restrict K) := ⟨by simpa using hKb.measure_lt_top⟩
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      f p.2 * (timeMassDensity a q t p.1 p.2).toReal) :=
+    (hf.comp measurable_snd).mul (timeMassDensity_measurable a ha q hq hq1 t).ennreal_toReal
+  apply Integrable.of_bound hm.aestronglyMeasurable (B*((c*q)^(-alpha a)))
+  have hmem : ∀ᵐ p ∂W.prod (volume.restrict K), p.2∈K :=
+    (Measure.ae_prod_iff_ae_ae (hK.preimage measurable_snd)).mpr
+      (Eventually.of_forall fun _ => ae_restrict_mem hK)
+  filter_upwards [hmem] with p hp
+  have hb := timeMassDensity_local_bound a ha q hq hq1 t p.1 c hc p.2 (hcK _ hp)
+  have hD : 0≤(c*q)^(-alpha a) := Real.rpow_nonneg (by positivity) _
+  have hd : (timeMassDensity a q t p.1 p.2).toReal≤(c*q)^(-alpha a) := by
+    exact ENNReal.toReal_le_of_le_ofReal hD hb
+  rw [norm_mul,Real.norm_of_nonneg ENNReal.toReal_nonneg]
+  exact mul_le_mul (hBf _ hp) hd ENNReal.toReal_nonneg hB
+
+lemma terminal_density_mean (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (z : ℂ) (hz : 0<z.im) :
+    (∫b, (timeMassDensity a q ∞ b z).toReal ∂W) =
+      Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0) := by
+  have hnon (b : ExactSLEGenerator.Driver) :=
+    (fullRadialDensity_bound a ha z hz (2*z.im*q) (by positivity) (by nlinarith) b).1
+  simp only [timeMassDensity,le_top,ite_true,massDensity,hz,ENNReal.toReal_ofReal (hnon _)]
+  have he : 2*z.im*q/2=z.im*q := by ring
+  simp only [fullRadialDensity,he]
+  rw [integral_const_mul]
+  congr 1
+  exact (radialWeight_mean W hW a ha z hz (z.im*q) (by positivity) (by nlinarith)).2
+
+lemma actualMassEval_terminal_mean (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B) :
+    (∫b,actualMassEval a q ∞ K f b∂W) =
+      ∫z in K, f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)) := by
+  unfold actualMassEval
+  rw [integral_integral_swap (actualMassEval_joint_integrable a ha W q hq hq1 ∞ K hK hKb c hc hcK f hf hfB)]
+  apply setIntegral_congr_fun hK
+  intro z hz
+  dsimp only
+  rw [integral_const_mul,terminal_density_mean a ha W hW q hq hq1 z (by have := hcK z hz; linarith)]
+
+lemma actualUpperMass_terminal_mean (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (f : C_c(UpperPlane,ℝ)) :
+    (∫b, (∫z,f z∂actualUpperMass a q ∞ b) ∂W) = ∫z : UpperPlane,
+      f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea := by
+  let K := Subtype.val '' tsupport f
+  have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val
+  obtain ⟨c,hc,hcK⟩ := upper_compact_lower (tsupport f) f.hasCompactSupport
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  rw [actualMassEval_terminal_mean a ha W hW q hq hq1 K hK.measurableSet hK.isBounded c hc
+    (by rintro z ⟨w,hw,rfl⟩; exact hcK w hw) (upperExtension f)
+    (upperExtension_measurable f) (upperExtension_bounded f)]
+  simp_rw [← upperExtension_apply f]
+  trans ∫z in {z : ℂ | 0<z.im},
+    upperExtension f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0))
+  · rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => ?_)]
+    · rw [upperExtension_zero f z (by rintro ⟨w,hw,rfl⟩; exact hz w.2), zero_mul]
+    · rw [upperExtension_zero f z hz, zero_mul]
+  · exact (integral_subtype_comap (μ := (volume : Measure ℂ)) upperPlane_open.measurableSet
+      (fun z : ℂ => upperExtension f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0)))).symm
+
+lemma actual_radon_limit_mean (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+    (hsum : ∀ k,∑j∈s k,coeff k j=1)
+    (μ : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
+    (hg : MemLp (fun b => ∫z,f z∂μ b) 1 W)
+    (hlim : Tendsto (fun k => eLpNorm (fun b => (∑j∈s k,coeff k j*
+      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)-(∫z,f z∂μ b)) 1 W) atTop (𝓝 0)) :
+    (∫b, (∫z,f z∂μ b) ∂W) = ∫z : UpperPlane,
+      f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea := by
+  let q (j : ℕ) := Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  let F := fun k b => ∑j∈s k,coeff k j*∫z,f z∂actualUpperMass a (q j) ∞ b
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW 1 (by norm_num) f
+  have hFi (k : ℕ) : Integrable (F k) W :=
+    integrable_finsetSum (s k) (fun j hj =>
+      ((hCb (q j) (hq j) (hq1 j) ∞).1.integrable (by norm_num)).const_mul (coeff k j))
+  change Tendsto (fun k => eLpNorm (fun b => F k b - ∫z,f z∂μ b) 1 W) atTop (𝓝 0) at hlim
+  have ht := tendsto_integral_of_L1 (fun b => ∫z,f z∂μ b) hg.aestronglyMeasurable
+    (Eventually.of_forall hFi) (by
+      exact hlim.congr fun index => eLpNorm_one_eq_lintegral_enorm
+        ((hFi index).aestronglyMeasurable.sub hg.aestronglyMeasurable))
+  have he (k : ℕ) : (∫b,F k b∂W) = ∫z : UpperPlane,
+      f z*(Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea := by
+    dsimp only [F]
+    rw [integral_finsetSum _ (fun j hj =>
+      ((hCb (q j) (hq j) (hq1 j) ∞).1.integrable (by norm_num)).const_mul (coeff k j))]
+    simp_rw [integral_const_mul]
+    simp_rw [actualUpperMass_terminal_mean a ha W hW _ (hq _) (hq1 _) f]
+    rw [← Finset.sum_mul,hsum k,one_mul]
+  exact tendsto_nhds_unique ht (by simpa only [he] using tendsto_const_nhds)
+
+ 
+
+theorem actual_radon_terminal_mean (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) :
+    ∃ (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+      (μ : ExactSLEGenerator.Driver → ℕ → Measure UpperPlane),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤coeff k j) ∧
+      (∀ k, ∑ j∈s k, coeff k j=1) ∧
+      (∀ b i, IsFiniteMeasureOnCompacts (μ b i) ∧ (μ b i).Regular) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ b i) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane, ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, coeff k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ b i) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → ∀ f : C_c(UpperPlane,ℝ),
+        (∫b, (∫z,f z∂μ b i) ∂W) = ∫z : UpperPlane,
+          f z * (Real.exp (-alpha a*Real.log 2)*physicalGreen a (z.1,0)) ∂upperArea) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i ≤ t j → μ b i ≤ μ b j) ∧
+        (∀ i (f : C_c(UpperPlane, ℝ)),
+          Tendsto (fun k => ∑ j∈s k, coeff k j *
+            ∫ z, f z ∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫ z, f z ∂μ b i)))
+ := by
+  obtain ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,h⟩ := actual_radon_L1_convergence a ha W hW t
+  refine ⟨s,c,μ,hs,hc,hsum,hμ,hmom,hL1,?_,h⟩
+  intro i hi f
+  apply actual_radon_limit_mean a ha W hW s c hsum (fun b => μ b i) f
+    (by simpa using hmom i f 1)
+  simpa only [hi] using hL1 i f
+
+end ExactSLEGreen
+
+namespace ExactSLEGenerator
+open MeasureTheory ProbabilityTheory Filter Topology
+open scoped NNReal ENNReal
+noncomputable def pastDriver (t : ℝ≥0) (b : Driver) : Driver :=
+  ⟨⟨fun s => b.1 (min s t), b.1.continuous.comp (continuous_id.min continuous_const)⟩, by simpa using b.2⟩
+lemma pastDriver_eq (t : ℝ≥0) (b : Driver) (s : ℝ≥0) (hs : s≤t) :
+    (pastDriver t b).1 s = b.1 s := by simp [pastDriver,min_eq_left hs]
+lemma pastDriver_raw_measurable (t : ℝ≥0) : Measurable[rawFiltration t] (pastDriver t) := by
+  have : PolishSpace Driver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
+  obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq (α:=ℝ≥0)
+  let e : Driver → (ℕ → ℝ) := fun b n => b.1 (u n)
+  have hec : Continuous e := continuous_pi fun n => (continuous_eval_const (u n)).comp continuous_subtype_val
+  have hei : Function.Injective e := by
+    intro b c h
+    apply Subtype.ext
+    apply DFunLike.coe_injective
+    exact hu.equalizer b.1.continuous c.1.continuous h
+  have hem := hec.measurableEmbedding hei
+  apply hem.measurable_comp_iff.mp
+  let : MeasurableSpace Driver := rawFiltration t
+  apply measurable_pi_iff.mpr
+  intro n
+  exact ((Filtration.stronglyAdapted_natural driver_eval_measurable) (min (u n) t)).measurable.mono
+    (rawFiltration.mono (min_le_right _ _)) le_rfl
+end ExactSLEGenerator
+namespace ExactSLEMain
+open MeasureTheory ExactSLEGenerator
+open scoped NNReal ENNReal
+lemma odeUntil_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
+    (h : ∀ s≤t, b s=c s) {f : ℝ → ℂ} (hf : odeUntil a b z t f) : odeUntil a c z t f := by
+  refine ⟨hf.1,fun u hu => ?_⟩
+  have he := h u.toNNReal ((Real.toNNReal_le_iff_le_coe).mpr hu.2)
+  simpa only [he] using hf.2 u hu
+lemma alive_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {z : ℂ} {t : ℝ≥0}
+    (h : ∀ s≤t, b s=c s) : alive a b t z ↔ alive a c t z := by
+  exact ⟨fun ⟨f,hf⟩ => ⟨f,odeUntil_driver_congr h hf⟩,
+    fun ⟨f,hf⟩ => ⟨f,odeUntil_driver_congr (fun s hs => (h s hs).symm) hf⟩⟩
+lemma loewnerMap_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {t : ℝ≥0}
+    (h : ∀ s≤t, b s=c s) : loewnerMap a b t = loewnerMap a c t := by
+  funext z
+  by_cases hb : alive a b t z
+  · obtain ⟨f,hf⟩ := hb
+    exact (loewnerMap_eq hf).trans (loewnerMap_eq (odeUntil_driver_congr h hf)).symm
+  · have hc := mt (alive_driver_congr h).mpr hb
+    simp [loewnerMap,hb,hc]
+end ExactSLEMain
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal
+lemma actualState_driver_congr {a : ℝ} {b c : ℝ≥0 → ℝ} {t : ℝ≥0} (h : ∀ s≤t,b s=c s) (z : ℂ) :
+    actualState a b t z = actualState a c t z := by
+  simp only [actualState,loewnerMap_driver_congr h,h t le_rfl]
+lemma radialTime_driver_congr_before (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b c : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (h : ∀ s≤t,b.1 s=c.1 s) (hb : radialTime a b.1 z ρ≤t) :
+    radialTime a b.1 z ρ = radialTime a c.1 z ρ := by
+  have hbfin : radialTime a b.1 z ρ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hb
+  have hbr : (radialTime a b.1 z ρ).toNNReal≤t := by
+    exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hbfin] using hb)
+  obtain ⟨hba,hbh⟩ := radialTime_attained a ha z hz ρ hρ hr b hbfin
+  have he := actualState_driver_congr (a:=a) (fun s hs => h s (hs.trans hbr)) z
+  have hc : radialTime a c.1 z ρ≤radialTime a b.1 z ρ := by
+    have hh := radialTime_le_of_hit a z ρ c (radialTime a b.1 z ρ).toNNReal
+      ((alive_driver_congr (fun s hs => h s (hs.trans hbr))).mp hba) (by rw [← he]; exact hbh)
+    simpa only [ENNReal.coe_toNNReal hbfin] using hh
+  have hcfin : radialTime a c.1 z ρ≠∞ := ne_top_of_le_ne_top hbfin hc
+  have hcr : (radialTime a c.1 z ρ).toNNReal≤t := by
+    exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hcfin] using hc.trans hb)
+  obtain ⟨hca,hch⟩ := radialTime_attained a ha z hz ρ hρ hr c hcfin
+  have he' := actualState_driver_congr (a:=a) (fun s hs => (h s (hs.trans hcr)).symm) z
+  have hh := radialTime_le_of_hit a z ρ b (radialTime a c.1 z ρ).toNNReal
+    ((alive_driver_congr (fun s hs => (h s (hs.trans hcr)).symm)).mp hca) (by rw [← he']; exact hch)
+  exact le_antisymm (by simpa only [ENNReal.coe_toNNReal hcfin] using hh) hc
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal
+lemma timeMassDensity_driver_congr (a : ℝ) (ha : 0<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (b c : ExactSLEGenerator.Driver) (t : ℝ≥0) (h : ∀ s≤t,b.1 s=c.1 s) (z : ℂ) :
+    timeMassDensity a q t b z = timeMassDensity a q t c z := by
+  by_cases hz : 0<z.im
+  · have hρ : 0<z.im*q := mul_pos hz hq
+    have hr : z.im*q≤z.im := by nlinarith
+    have hiff : radialTime a b.1 z (z.im*q)≤t ↔ radialTime a c.1 z (z.im*q)≤t := by
+      constructor
+      · intro hb
+        rwa [← radialTime_driver_congr_before a ha z hz _ hρ hr b c t h hb]
+      · intro hc
+        rwa [← radialTime_driver_congr_before a ha z hz _ hρ hr c b t (fun s hs => (h s hs).symm) hc]
+    by_cases hb : radialTime a b.1 z (z.im*q)≤t
+    · have he := radialTime_driver_congr_before a ha z hz _ hρ hr b c t h hb
+      have hfin : radialTime a b.1 z (z.im*q)≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hb
+      have ht : (radialTime a b.1 z (z.im*q)).toNNReal≤t := by
+        exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hb)
+      have hs := actualState_driver_congr (a:=a) (fun s hs => h s (hs.trans ht)) z
+      have hw : radialWeight a b.1 z (z.im*q)=radialWeight a c.1 z (z.im*q) := by
+        unfold radialWeight
+        rw [ite_eq_right hfin,ite_eq_right (ne_of_eq_of_ne he.symm hfin),← he,hs]
+      have hn : 2*z.im*q/2=z.im*q := by ring
+      simp only [timeMassDensity,ite_eq_left hb,ite_eq_left (hiff.mp hb),massDensity,ite_eq_left hz,fullRadialDensity,hn,hw]
+    · simp only [timeMassDensity,ite_eq_right hb,ite_eq_right (mt hiff.mpr hb)]
+  · simp only [timeMassDensity,massDensity,ite_eq_right hz,ite_self]
+lemma timeMassDensity_past_eq (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
+    (hq1 : q≤1) (t : ℝ≥0) :
+    (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t p.1 p.2) =
+    (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t (pastDriver t p.1) p.2) := by
+  funext p
+  exact timeMassDensity_driver_congr a (by linarith) q hq hq1 p.1 (pastDriver t p.1) t
+    (fun s hs => (pastDriver_eq t p.1 s hs).symm) p.2
+lemma rawPastPair_measurable (t : ℝ≥0) :
+    @Measurable (ExactSLEGenerator.Driver × ℂ) (ExactSLEGenerator.Driver × ℂ)
+      ((rawFiltration t).prod (borel ℂ)) inferInstance (fun p => (pastDriver t p.1,p.2)) := by
+  exact ((pastDriver_raw_measurable t).comp
+    (@measurable_fst _ _ (rawFiltration t) (borel ℂ))).prodMk
+    (@measurable_snd _ _ (rawFiltration t) (borel ℂ))
+ 
+theorem timeMassDensity_raw_joint (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
+    (hq1 : q≤1) (t : ℝ≥0) :
+    letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => timeMassDensity a q t p.1 p.2) := by
+  change @Measurable (ExactSLEGenerator.Driver × ℂ) ℝ≥0∞
+    ((rawFiltration t).prod (borel ℂ)) inferInstance _
+  rw [timeMassDensity_past_eq a ha q hq hq1 t]
+  exact @Measurable.comp (ExactSLEGenerator.Driver × ℂ) (ExactSLEGenerator.Driver × ℂ) ℝ≥0∞
+    ((rawFiltration t).prod (borel ℂ)) inferInstance inferInstance
+    (fun p => timeMassDensity a q (t : ℝ≥0∞) p.1 p.2)
+    (fun p => (pastDriver t p.1,p.2))
+    (timeMassDensity_measurable a ha q hq hq1 (t : ℝ≥0∞)) (rawPastPair_measurable t)
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RandomRestrictionKernel.lean b/lean/OAI/Probability/SLEGauge/RandomRestrictionKernel.lean
index 9dd5d76b2beca9adb41808b855a82c673aa2dd64..e4e597a01ee2b40468d15679f33ad819c115a3ab 100644
--- a/lean/OAI/Probability/SLEGauge/RandomRestrictionKernel.lean
+++ b/lean/OAI/Probability/SLEGauge/RandomRestrictionKernel.lean
@@ -1,595 +1,595 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RawRestartTests
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-
-lemma measurable_random_restrict {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (K : Kernel Ω X) [IsSFiniteKernel K] {A : Set (Ω×X)} (hA : MeasurableSet A) :
-    Measurable (fun ω => (K ω).restrict {x | (ω,x)∈A}) := by
-  let F : Ω→X→ℝ≥0∞ := fun ω x => if (ω,x)∈A then 1 else 0
-  have hF : Measurable (Function.uncurry F) := by
-    change Measurable (fun p : Ω×X => if p∈A then (1:ℝ≥0∞) else 0)
-    exact Measurable.ite hA measurable_const measurable_const
-  have he (ω : Ω) : K.withDensity F ω=(K ω).restrict {x | (ω,x)∈A} := by
-    rw [Kernel.withDensity_apply K hF]
-    change (K ω).withDensity ({x | (ω,x)∈A}.indicator (fun _ => 1))=_
-    have hB : MeasurableSet {x | (ω,x)∈A} := hA.preimage (measurable_const.prodMk measurable_id)
-    rw [withDensity_indicator hB]
-    exact MeasureTheory.withDensity_one
-  have he' : (K.withDensity F : Ω→Measure X)=(fun ω => (K ω).restrict {x | (ω,x)∈A}) := funext he
-  rw [←he']
-  exact (K.withDensity F).measurable
-
-end SLEExactGauge.MassVersions
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.TraceRestriction
-variable {X Y : Type*} [MeasurableSpace X] [MetricSpace Y]
-    [MeasurableSpace Y] [BorelSpace Y]
-
-lemma tendsto_integral_visited_atTop (ν : Measure X)
-    (ι : X→Y) (hι : Measurable ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
-    (hfull : ∀ᵐ x ∂ν, ι x∈range γ) (f : X→ℝ) (hf : Integrable f ν) :
-    Tendsto (fun t : ℝ≥0 => ∫x in visited ι γ t,f x∂ν) atTop (𝓝 (∫x,f x∂ν)) := by
-  simp_rw [←integral_indicator (measurableSet_visited ι hι γ hγ _)]
-  apply tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖)
-  · exact Eventually.of_forall (fun t => hf.aestronglyMeasurable.indicator
-      (measurableSet_visited ι hι γ hγ t))
-  · exact Eventually.of_forall (fun _ => Eventually.of_forall fun _ => norm_indicator_le_norm_self _ _)
-  · exact hf.norm
-  · filter_upwards [hfull] with x hx
-    obtain ⟨u,hu⟩ := hx
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_ge_atTop u] with t ht
-    exact (indicator_of_mem (show x∈visited ι γ t from ⟨u,⟨bot_le,ht⟩,hu⟩) f).symm
-
-lemma continuous_integral_restrict_visited (ν : Measure X) [NullSingletonClass ν]
-    (ι : X→Y) (hι : Measurable ι) (hinj : Function.Injective ι)
-    (γ : ℝ≥0→Y) (hγ : Continuous γ) {A : Set X} (hA : MeasurableSet A)
-    (f : X→ℝ) (hf : Integrable f ν) :
-    Continuous (fun t : ℝ≥0 => ∫x,f x∂(ν.restrict (visited ι γ t)).restrict A) := by
-  simp_rw [←integral_indicator hA]
-  exact continuous_integral_visited ν ι hι hinj γ hγ (A.indicator f) (hf.indicator hA)
-
-lemma tendsto_integral_restrict_visited_atTop (ν : Measure X)
-    (ι : X→Y) (hι : Measurable ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
-    (hfull : ∀ᵐ x ∂ν, ι x∈range γ) {A : Set X} (hA : MeasurableSet A)
-    (f : X→ℝ) (hf : Integrable f ν) :
-    Tendsto (fun t : ℝ≥0 => ∫x,f x∂(ν.restrict (visited ι γ t)).restrict A)
-      atTop (𝓝 (∫x in A,f x∂ν)) := by
-  simp_rw [←integral_indicator hA]
-  exact tendsto_integral_visited_atTop ν ι hι γ hγ hfull (A.indicator f) (hf.indicator hA)
-
-end SLEExactGauge.TraceRestriction
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
-open TraceRestriction TraceMassContinuity TraceMassIntegration
-
-lemma measurableSet_alive_atStop_raw (a : ℝ) (ha : 1/4<a)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    MeasurableSet[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)]
-      {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1} := by
-  have hD := stoppedData_raw_measurable hS
-  have hin : @Measurable (GDriver×UpperPlane) (ℝ≥0×(GDriver×UpperPlane))
-      (hS.measurableSpace.prod inferInstance) inferInstance (fun p =>
-      ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) :=
-    (hD.fst.comp measurable_fst).prodMk ((hD.snd.comp measurable_fst).prodMk measurable_snd)
-  have he : {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1}=
-      (fun p : GDriver×UpperPlane => ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) ⁻¹'
-        {p : ℝ≥0×(GDriver×UpperPlane) | alive a p.2.1.1 p.1 p.2.2.1} := by
-    ext p
-    exact (alive_driver_congr (pastDriver_eq (S p.1).toNNReal p.1)).symm
-  rw [he]
-  exact (measurableSet_alive_joint a ha).preimage hin
-
-lemma measurableSet_alive_spatial (a : ℝ) (b : GDriver) (t : ℝ≥0) :
-    MeasurableSet {z : UpperPlane | alive a b.1 t z.1} :=
-  ((isOpen_alive b.1.continuous t).preimage continuous_subtype_val).measurableSet
-
- 
-def recoveredPostStop (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (S : GDriver→ℝ≥0∞) (v : ℝ≥0∞) (b : GDriver) : Measure UpperPlane :=
-  if S b=∞ then 0 else (recoveredMass a ha s c hc (S b+v) b).restrict
-    {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}
-
-lemma recoveredPostStop_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (t : ℝ≥0) :
-    Measurable[(afterStopFiltration hS) t] (recoveredPostStop a ha s c hc S t) := by
-  let : MeasurableSpace GDriver := (afterStopFiltration hS) t
-  let K : Kernel GDriver UpperPlane := ⟨fun b => recoveredMass a ha s c hc (S b+t) b,
-    recoveredMass_stopping_raw_measurable a ha s c hc _ (shiftStop_isStoppingTime hS t)⟩
-  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
-  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
-  have hA : MeasurableSet {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1} :=
-    (prod_sigma_le (measurableSpace_le_shiftStop hS t) _ ) _ (measurableSet_alive_atStop_raw a ha hS)
-  have htop : MeasurableSet {b : GDriver | S b=∞} :=
-    (measurableSpace_le_shiftStop hS t) _ (hS.measurable (measurableSet_singleton ∞))
-  exact Measurable.ite htop measurable_const (measurable_random_restrict K hA)
-
-lemma recoveredPostStop_terminal_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    Measurable (recoveredPostStop a ha s c hc S ∞) := by
-  let K : Kernel GDriver UpperPlane := ⟨recoveredMass a ha s c hc ∞,
-    recoveredMass_measurable a ha s c hc ∞⟩
-  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
-  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
-  have hA : MeasurableSet {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1} :=
-    (prod_sigma_le hS.measurableSpace_le _) _ (measurableSet_alive_atStop_raw a ha hS)
-  have htop : MeasurableSet {b : GDriver | S b=∞} := hS.measurable' (measurableSet_singleton ∞)
-  have hh : Measurable (fun b : GDriver => if S b=∞ then (0:Measure UpperPlane) else
-      (K b).restrict {z | alive a b.1 (S b).toNNReal z.1}) :=
-    Measurable.ite htop measurable_const (measurable_random_restrict K hA)
-  change Measurable (fun b => recoveredPostStop a ha s c hc S ∞ b)
-  simpa only [recoveredPostStop,add_top,K,Kernel.coe_mk] using hh
-
-lemma recoveredPostStop_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (v : ℝ≥0∞) :
-    Measurable (recoveredPostStop a ha s c hc S v) := by
-  induction v using ENNReal.recTopCoe with
-  | top => exact recoveredPostStop_terminal_measurable a ha s c hc hS
-  | coe t =>
-    exact (recoveredPostStop_raw_measurable a ha s c hc hS t).mono
-      ((afterStopFiltration hS).le t) le_rfl
-
-lemma recoveredPostStop_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (S : GDriver→ℝ≥0∞) (v : ℝ≥0∞) (b : GDriver) :
-    IsFiniteMeasureOnCompacts (recoveredPostStop a ha s c hc S v b) := by
-  unfold recoveredPostStop
-  split_ifs
-  · infer_instance
-  · let : IsFiniteMeasureOnCompacts (recoveredMass a ha s c hc (S b+v) b) := radonLimit_finite _
-    infer_instance
-
-lemma recoveredPostStop_compact_stronglyAdapted (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (f : C_c(UpperPlane,ℝ)) :
-    StronglyAdapted (afterStopFiltration hS)
-      (fun t : ℝ≥0 => fun b => ∫z,f z∂recoveredPostStop a ha s c hc S t b) := by
-  intro t
-  let : MeasurableSpace GDriver := (afterStopFiltration hS) t
-  let K : Kernel GDriver UpperPlane := ⟨recoveredPostStop a ha s c hc S t,
-    recoveredPostStop_raw_measurable a ha s c hc hS t⟩
-  exact f.continuous.stronglyMeasurable.integral_kernel (κ:=K)
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-variable {Ω X : Type*} [MeasurableSpace Ω]
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-
-theorem ae_finiteOnCompacts_of_average (K : Kernel Ω X) (W : Measure Ω)
-    [IsFiniteMeasureOnCompacts (K ∘ₘ W)] :
-    ∀ᵐ ω ∂W, IsFiniteMeasureOnCompacts (K ω) := by
-  let C := CompactExhaustion.choice X
-  have hn (n : ℕ) : ∀ᵐ ω ∂W, K ω (C n)<∞ := by
-    apply ae_lt_top (K.measurable_coe (C.isCompact n).measurableSet)
-    rw [←Measure.bind_apply (C.isCompact n).measurableSet K.aemeasurable]
-    exact (C.isCompact n).measure_lt_top.ne
-  filter_upwards [ae_all_iff.mpr hn] with ω hω
-  constructor
-  intro A hA
-  obtain ⟨n,hn⟩ := C.exists_superset_of_isCompact hA
-  exact (measure_mono hn).trans_lt (hω n)
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
-lemma compact_eval_integrable_of_average [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] (K : Kernel Ω X) (W : Measure Ω)
-    [IsFiniteMeasureOnCompacts (K ∘ₘ W)] (f : C_c(X,ℝ)) :
-    Integrable (fun ω => ∫x,f x∂K ω) W := by
-  have hi : Integrable f (K ∘ₘ W) := f.integrable
-  rw [Measure.comp_eq_comp_const_apply] at hi
-  exact hi.integral_comp
-
-lemma ae_compact_integrable_of_average (K : Kernel Ω X) (W : Measure Ω)
-    [IsFiniteMeasureOnCompacts (K ∘ₘ W)] :
-    ∀ᵐ ω ∂W, ∀f : C_c(X,ℝ), Integrable f (K ω) := by
-  filter_upwards [ae_finiteOnCompacts_of_average K W] with ω hω f
-  let := hω
-  exact f.integrable
-
-end SLEExactGauge.MassVersions
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
-open TraceRestriction TraceMassContinuity TraceMassIntegration
-
-lemma recoveredPostStop_terminal_le (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (S : GDriver→ℝ≥0∞) (b : GDriver) :
-    recoveredPostStop a ha s c hc S ∞ b≤recoveredMass a ha s c hc ∞ b := by
-  unfold recoveredPostStop
-  split_ifs
-  · exact bot_le
-  · simp only [add_top]
-    exact Measure.restrict_le_self
-
-lemma recoveredPostStop_terminal_integrable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (W : Measure GDriver)
-    (hmean : recoveredMass a ha s c hc ∞ ∘ₘ W=initialGreenMeasure a)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (f : C_c(UpperPlane,ℝ)) :
-    Integrable (fun b => ∫z,f z∂recoveredPostStop a ha s c hc S ∞ b) W := by
-  let K : Kernel GDriver UpperPlane := ⟨recoveredPostStop a ha s c hc S ∞,
-    recoveredPostStop_terminal_measurable a ha s c hc hS⟩
-  have hle : K ∘ₘ W ≤ initialGreenMeasure a := by
-    rw [←hmean]
-    apply Measure.le_iff.mpr
-    intro A hA
-    rw [Measure.bind_apply hA K.aemeasurable,
-      Measure.bind_apply hA (recoveredMass_measurable a ha s c hc ∞).aemeasurable]
-    exact lintegral_mono (fun b => (recoveredPostStop_terminal_le a ha s c hc S b) A)
-  let : IsFiniteMeasureOnCompacts (K ∘ₘ W) :=
-    ⟨fun A hA => (hle A).trans_lt hA.measure_lt_top⟩
-  exact compact_eval_integrable_of_average K W f
-
-lemma recoveredPostStop_zero_of_convergence (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (hs : ∀k j,j∈s k→k≤j) (S : GDriver→ℝ≥0∞) (b : GDriver)
-    (hconv : ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂recoveredMass a ha s c hc v b))) :
-    recoveredPostStop a ha s c hc S 0 b=0 := by
-  unfold recoveredPostStop
-  split_ifs with hb
-  · rfl
-  · rw [add_zero,←ENNReal.coe_toNNReal hb]
-    let : IsFiniteMeasureOnCompacts (recoveredMass a ha s c hc (S b).toNNReal b) :=
-      radonLimit_finite _
-    apply Measure.restrict_eq_zero.mpr
-    exact actual_limit_past_hull_support a ha s c hs b (S b).toNNReal _
-      (hconv (S b).toNNReal)
-
- 
-
-theorem recoveredPostStop_path_properties (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure GDriver) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
-      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
-    (S : GDriver→ℝ≥0∞) :
-    let ν := recoveredPostStop (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S
-    ∀ᵐ b ∂W, ν 0 b=0 ∧ Monotone (fun v => ν v b) ∧
-      (∀f : C_c(UpperPlane,ℝ), Continuous (fun t : ℝ≥0 => ∫z,f z∂ν t b) ∧
-        Tendsto (fun t : ℝ≥0 => ∫z,f z∂ν t b) atTop (𝓝 (∫z,f z∂ν ∞ b))) := by
-  intro ν
-  let a := Target.a κ
-  have ha : 1/4<a := a_gt_quarter hκ hκ8
-  let μ := recoveredMass a ha s c hc
-  filter_upwards [hpath,hconv] with b hp hcv
-  refine ⟨recoveredPostStop_zero_of_convergence a ha s c hc hs S b hcv,?_,?_⟩
-  · intro v w hvw
-    change (if S b=∞ then 0 else (μ (S b+v) b).restrict _ )≤
-      if S b=∞ then 0 else (μ (S b+w) b).restrict _
-    split_ifs
-    · exact le_rfl
-    · exact Measure.restrict_mono le_rfl (hp.monotone (add_le_add le_rfl hvw))
-  · by_cases hb : S b=∞
-    · have he (v : ℝ≥0∞) : ν v b=0 := by simp only [ν,recoveredPostStop,hb,ite_true]
-      simp only [he,integral_zero_measure]
-      exact fun _ => ⟨continuous_const,tendsto_const_nhds⟩
-    · let A : Set UpperPlane := {z | alive a b.1 (S b).toNNReal z.1}
-      have hA : MeasurableSet A := measurableSet_alive_spatial a b _
-      have he (t : ℝ≥0) : ν t b=(μ (((S b).toNNReal+t : ℝ≥0) : ℝ≥0∞) b).restrict A := by
-        simp only [ν,recoveredPostStop,ite_eq_right hb,μ,ENNReal.coe_add,ENNReal.coe_toNNReal hb,A,a]
-      have het : ν ∞ b=(μ ∞ b).restrict A := by
-        simp only [ν,recoveredPostStop,ite_eq_right hb,add_top,μ,A,a]
-      change TraceMassFacts κ (driverHomeomorph.symm b) (fun v => μ v b) at hp
-      let := hp.atomless
-      let : IsFiniteMeasureOnCompacts (μ ∞ b) := radonLimit_finite _
-      have hγ := continuous_internalTrace hκ _ hp.continuous_trace
-      have hfull : ∀ᵐ z ∂μ ∞ b, z.1∈range (internalTrace κ (driverHomeomorph.symm b)) := by
-        filter_upwards [hp.firstVisit_finite] with z hz
-        exact (firstVisit_lt_top_iff _ _ _).mp hz
-      have hshift : Tendsto (fun t : ℝ≥0 => (S b).toNNReal+t) atTop atTop := by
-        apply tendsto_atTop.mpr
-        intro u
-        filter_upwards [eventually_ge_atTop u] with t ht
-        exact ht.trans (le_add_self)
-      intro f
-      simp_rw [he,het,hp.restriction]
-      exact ⟨(continuous_integral_restrict_visited (μ ∞ b) _ measurable_subtype_coe
-        Subtype.val_injective _ hγ hA f f.integrable).comp (continuous_const.add continuous_id),
-        (tendsto_integral_restrict_visited_atTop (μ ∞ b) _ measurable_subtype_coe _
-          hγ hfull hA f f.integrable).comp hshift⟩
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
-
-lemma restartPush_integral (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (μ : Measure UpperPlane) {f : UpperPlane→ℝ} (hf : Measurable f) :
-    (∫z,f z∂restartPush a (lt_trans (by norm_num) ha) b s μ)=
-      ∫w,(restartWeight a b s w).toReal*f (inverseUpper a (lt_trans (by norm_num) ha) b s w)∂μ := by
-  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
-  rw [restartPush,integral_map hJ.aemeasurable hf.aestronglyMeasurable,
-    integral_withDensity_eq_integral_toReal_smul (restartWeight_measurable a ha b s)
-      (Eventually.of_forall (fun w => lt_top_iff_ne_top.mpr (restartWeight_ne_top a b s w)))]
-  rfl
-
-lemma integrable_restartPush_iff (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (μ : Measure UpperPlane) {f : UpperPlane→ℝ} (hf : Measurable f) :
-    Integrable f (restartPush a (lt_trans (by norm_num) ha) b s μ) ↔
-      Integrable (fun w => (restartWeight a b s w).toReal*
-        f (inverseUpper a (lt_trans (by norm_num) ha) b s w)) μ := by
-  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
-  rw [restartPush,integrable_map_measure hf.aestronglyMeasurable hJ.aemeasurable,
-    integrable_withDensity_iff_integrable_smul' (restartWeight_measurable a ha b s)
-      (Eventually.of_forall (fun w => lt_top_iff_ne_top.mpr (restartWeight_ne_top a b s w)))]
-  rfl
-
-lemma restartPush_mono (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0)
-    {μ ν : Measure UpperPlane} (h : μ≤ν) :
-    restartPush a ha b s μ≤restartPush a ha b s ν := by
-  apply Measure.map_mono _ (continuous_inverseUpper a ha b s).measurable
-  apply Measure.le_iff.mpr
-  intro A hA
-  rw [withDensity_apply _ hA,withDensity_apply _ hA]
-  exact lintegral_mono' (Measure.restrict_mono le_rfl h) le_rfl
-
-lemma ae_restartAt {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {p : GDriver→Prop} (hp : ∀ᵐ b ∂W, p b) :
-    ∀ᵐ b ∂W, S b≠∞ → p (restartAt S b) := by
-  apply (ae_restrict_iff' (hS.measurable' (measurableSet_singleton ∞)).compl).mp
-  apply ae_of_ae_map (measurable_restartAt hS).aemeasurable
-  have he := map_restartAt_restrict hW hS (A:=univ) MeasurableSet.univ
-  rw [univ_inter] at he
-  change ∀ᵐ b ∂(W.restrict {b | S b≠∞}).map (restartAt S), p b
-  rw [he]
-  exact Measure.ae_smul_measure hp _
-
- 
-
-theorem restartCandidateKernel_terminal_regular (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (hμ : μ ∘ₘ W=initialGreenMeasure a)
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    (∀ᵐ b ∂W, IsFiniteMeasureOnCompacts (restartCandidateKernel a ha μ S hS b) ∧
-      (restartCandidateKernel a ha μ S hS b).Regular) ∧
-    ∀f : C_c(UpperPlane,ℝ), Integrable
-      (fun b => ∫z,f z∂restartCandidateKernel a ha μ S hS b) W := by
-  let K := restartCandidateKernel a ha μ S hS
-  have hav : K ∘ₘ W ≤ initialGreenMeasure a :=
-    restartCandidateKernel_average_le_initial a ha μ W hW hμ S hS
-  let : IsFiniteMeasureOnCompacts (K ∘ₘ W) := ⟨fun A hA =>
-    (hav A).trans_lt hA.measure_lt_top⟩
-  refine ⟨?_,compact_eval_integrable_of_average K W⟩
-  filter_upwards [ae_finiteOnCompacts_of_average K W] with b hb
-  let := hb
-  exact ⟨hb,inferInstance⟩
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions TraceRestriction
-open TraceMassContinuity TraceMassIntegration
-
-lemma continuous_restartPush_visited (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (ν : Measure UpperPlane) [NullSingletonClass ν] (γ : ℝ≥0→ℂ) (hγ : Continuous γ)
-    {f : UpperPlane→ℝ} (hf : Measurable f)
-    (hfi : Integrable f (restartPush a (lt_trans (by norm_num) ha) b s ν)) :
-    Continuous (fun t : ℝ≥0 => ∫z,f z∂restartPush a (lt_trans (by norm_num) ha)
-      b s (ν.restrict (visited (fun z : UpperPlane => z.1) γ t))) := by
-  simp_rw [restartPush_integral a ha b s _ hf]
-  exact continuous_integral_visited ν _ measurable_subtype_coe Subtype.val_injective γ hγ _
-    ((integrable_restartPush_iff a ha b s ν hf).mp hfi)
-
-lemma tendsto_restartPush_visited_atTop (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (ν : Measure UpperPlane) (γ : ℝ≥0→ℂ) (hγ : Continuous γ)
-    (hfull : ∀ᵐ z ∂ν, z.1∈range γ) {f : UpperPlane→ℝ} (hf : Measurable f)
-    (hfi : Integrable f (restartPush a (lt_trans (by norm_num) ha) b s ν)) :
-    Tendsto (fun t : ℝ≥0 => ∫z,f z∂restartPush a (lt_trans (by norm_num) ha)
-      b s (ν.restrict (visited (fun z : UpperPlane => z.1) γ t))) atTop
-      (𝓝 (∫z,f z∂restartPush a (lt_trans (by norm_num) ha) b s ν)) := by
-  simp_rw [restartPush_integral a ha b s _ hf]
-  exact tendsto_integral_visited_atTop ν _ measurable_subtype_coe γ hγ hfull _
-    ((integrable_restartPush_iff a ha b s ν hf).mp hfi)
-
- 
-
-theorem recoveredRestart_path_properties (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
-      initialGreenMeasure (Target.a κ))
-    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
-      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    let ν := recoveredRestart (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S
-    (∀f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂ν ∞ b) W) ∧
-    (∀ᵐ b ∂W, (∀v, IsFiniteMeasureOnCompacts (ν v b) ∧ (ν v b).Regular) ∧
-      ν 0 b=0 ∧ Monotone (fun v => ν v b) ∧
-      (∀f : C_c(UpperPlane,ℝ), Continuous (fun t : ℝ≥0 => ∫z,f z∂ν t b) ∧
-        Tendsto (fun t : ℝ≥0 => ∫z,f z∂ν t b) atTop (𝓝 (∫z,f z∂ν ∞ b)))) := by
-  intro ν
-  let a := Target.a κ
-  have ha : 1/4<a := a_gt_quarter hκ hκ8
-  let μ := recoveredMass a ha s c hc
-  let K : Kernel GDriver UpperPlane := ⟨μ ∞,recoveredMass_measurable a ha s c hc ∞⟩
-  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
-  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
-  have hreg := restartCandidateKernel_terminal_regular a ha K W hW hmean S hS
-  have hreg' : ∀ᵐ b ∂W, IsFiniteMeasureOnCompacts (ν ∞ b) ∧ (ν ∞ b).Regular := hreg.1
-  refine ⟨hreg.2,?_⟩
-  filter_upwards [hreg',ae_restartAt hW hS hpath] with b hr hp
-  have hν (v : ℝ≥0∞) : ν v b=if S b=∞ then 0 else restartPush a
-      (lt_trans (by norm_num) ha) b (S b).toNNReal (μ v (restartAt S b)) := rfl
-  by_cases hb : S b=∞
-  · have he (v : ℝ≥0∞) : ν v b=0 := by simp only [hν,hb,ite_true]
-    simp only [he,integral_zero_measure]
-    exact ⟨fun _ => ⟨inferInstance,inferInstance⟩,True.intro,monotone_const,
-      fun _ => ⟨continuous_const,tendsto_const_nhds⟩⟩
-  · have hp := hp hb
-    change TraceMassFacts κ (driverHomeomorph.symm (restartAt S b)) (fun v => μ v (restartAt S b)) at hp
-    have he (v : ℝ≥0∞) : ν v b=restartPush a (lt_trans (by norm_num) ha)
-        b (S b).toNNReal (μ v (restartAt S b)) := by simp only [hν,ite_eq_right hb]
-    have hmono : Monotone (fun v => ν v b) := by
-      intro v w hvw
-      change ν v b≤ν w b
-      rw [he,he]
-      exact restartPush_mono _ _ _ _ (hp.monotone hvw)
-    have hdom (v : ℝ≥0∞) : ν v b≤ν ∞ b := hmono le_top
-    have hlocal (v : ℝ≥0∞) : IsFiniteMeasureOnCompacts (ν v b) := by
-      let := hr.1
-      exact ⟨fun A hA => ((hdom v) A).trans_lt hA.measure_lt_top⟩
-    refine ⟨fun v => ⟨hlocal v,?_⟩,?_,hmono,?_⟩
-    · let := hlocal v
-      infer_instance
-    · rw [he]
-      simp only [μ,recoveredMass_zero a ha s c hc]
-      simp only [restartPush,withDensity_zero_left,Measure.map_zero]
-    · intro f
-      let := hr.1
-      let := hp.atomless
-      have hfi : Integrable f (restartPush a (lt_trans (by norm_num) ha) b
-          (S b).toNNReal (μ ∞ (restartAt S b))) := by rw [←he]; exact f.integrable
-      have hγ := continuous_internalTrace hκ _ hp.continuous_trace
-      have hfull : ∀ᵐ z ∂μ ∞ (restartAt S b), z.1∈range (internalTrace κ (driverHomeomorph.symm (restartAt S b))) := by
-        filter_upwards [hp.firstVisit_finite] with z hz
-        exact (firstVisit_lt_top_iff _ _ _).mp hz
-      simp_rw [he,hp.restriction]
-      exact ⟨continuous_restartPush_visited a ha b _ _ _ hγ f.continuous.measurable hfi,
-        tendsto_restartPush_visited_atTop a ha b _ _ _ hγ hfull f.continuous.measurable hfi⟩
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped Classical ENNReal
-namespace SLEExactGauge.MassVersions
-
-lemma random_restrict_test_lintegral {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
-    (W : Measure Ω) (K : Ω→Measure X) (D : Ω→Set X)
-    (hD : ∀ω,MeasurableSet (D ω)) {B : Set Ω} (hB : MeasurableSet B) (f : X→ℝ≥0∞) :
-    (∫⁻ω,∫⁻x,(if ω∈B then (D ω).indicator f x else 0)∂K ω∂W)=
-      ∫⁻ω in B,∫⁻x,f x∂(K ω).restrict (D ω)∂W := by
-  rw [←lintegral_indicator hB]
-  apply lintegral_congr
-  intro ω
-  by_cases hω : ω∈B
-  · simp only [ite_eq_left hω,indicator_of_mem hω]
-    exact lintegral_indicator (hD ω) f
-  · simp only [ite_eq_right hω,indicator_of_notMem hω,lintegral_zero]
-
-lemma conditional_random_restrict_on {Ω X : Type*} [mΩ : MeasurableSpace Ω]
-    [MeasurableSpace X] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
-    (W : @Measure Ω mΩ) (K L R : Ω→Measure X)
-    (hcond : ∀H : Ω×X→ℝ≥0∞, Measurable[m.prod (inferInstance : MeasurableSpace X)] H →
-      (∫⁻ω,∫⁻x,H (ω,x)∂K ω∂W)=(∫⁻ω,∫⁻x,H (ω,x)∂L ω∂W)+
-        ∫⁻ω,∫⁻x,H (ω,x)∂ R ω∂W)
-    (D : Ω→Set X) (hD : MeasurableSet[m.prod (inferInstance : MeasurableSpace X)]
-      {p : Ω×X | p.2∈D p.1}) {B : Set Ω} (hB : MeasurableSet[m] B)
-    {f : X→ℝ≥0∞} (hf : Measurable f) :
-    (∫⁻ω in B,∫⁻x,f x∂(K ω).restrict (D ω)∂W)=
-      (∫⁻ω in B,∫⁻x,f x∂(L ω).restrict (D ω)∂W)+
-      ∫⁻ω in B,∫⁻x,f x∂(R ω).restrict (D ω)∂W := by
-  let F : Ω×X→ℝ≥0∞ := fun p => if p.2∈D p.1 then f p.2 else 0
-  have hF : Measurable[m.prod (inferInstance : MeasurableSpace X)] F :=
-    Measurable.ite hD (hf.comp measurable_snd) measurable_const
-  let H : Ω×X→ℝ≥0∞ := fun p => if p.1∈B then F p else 0
-  have hH : Measurable[m.prod (inferInstance : MeasurableSpace X)] H :=
-    Measurable.ite (hB.preimage measurable_fst) hF measurable_const
-  have hsec (ω : Ω) : MeasurableSet (D ω) := by
-    exact hD.preimage (measurable_const.prodMk measurable_id)
-  have he (M : Ω→Measure X) : (∫⁻ω,∫⁻x,H (ω,x)∂M ω∂W)=
-      ∫⁻ω in B,∫⁻x,f x∂(M ω).restrict (D ω)∂W := by
-    change (∫⁻ω,∫⁻x,(if ω∈B then (D ω).indicator f x else 0)∂M ω∂W)=_
-    exact @random_restrict_test_lintegral Ω X mΩ _ W M D hsec B (hm B hB) f
-  simpa only [he] using hcond H hH
-
-end SLEExactGauge.MassVersions
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RawRestartTests
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+
+lemma measurable_random_restrict {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (K : Kernel Ω X) [IsSFiniteKernel K] {A : Set (Ω×X)} (hA : MeasurableSet A) :
+    Measurable (fun ω => (K ω).restrict {x | (ω,x)∈A}) := by
+  let F : Ω→X→ℝ≥0∞ := fun ω x => if (ω,x)∈A then 1 else 0
+  have hF : Measurable (Function.uncurry F) := by
+    change Measurable (fun p : Ω×X => if p∈A then (1:ℝ≥0∞) else 0)
+    exact Measurable.ite hA measurable_const measurable_const
+  have he (ω : Ω) : K.withDensity F ω=(K ω).restrict {x | (ω,x)∈A} := by
+    rw [Kernel.withDensity_apply K hF]
+    change (K ω).withDensity ({x | (ω,x)∈A}.indicator (fun _ => 1))=_
+    have hB : MeasurableSet {x | (ω,x)∈A} := hA.preimage (measurable_const.prodMk measurable_id)
+    rw [withDensity_indicator hB]
+    exact MeasureTheory.withDensity_one
+  have he' : (K.withDensity F : Ω→Measure X)=(fun ω => (K ω).restrict {x | (ω,x)∈A}) := funext he
+  rw [←he']
+  exact (K.withDensity F).measurable
+
+end SLEExactGauge.MassVersions
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.TraceRestriction
+variable {X Y : Type*} [MeasurableSpace X] [MetricSpace Y]
+    [MeasurableSpace Y] [BorelSpace Y]
+
+lemma tendsto_integral_visited_atTop (ν : Measure X)
+    (ι : X→Y) (hι : Measurable ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
+    (hfull : ∀ᵐ x ∂ν, ι x∈range γ) (f : X→ℝ) (hf : Integrable f ν) :
+    Tendsto (fun t : ℝ≥0 => ∫x in visited ι γ t,f x∂ν) atTop (𝓝 (∫x,f x∂ν)) := by
+  simp_rw [←integral_indicator (measurableSet_visited ι hι γ hγ _)]
+  apply tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖)
+  · exact Eventually.of_forall (fun t => hf.aestronglyMeasurable.indicator
+      (measurableSet_visited ι hι γ hγ t))
+  · exact Eventually.of_forall (fun _ => Eventually.of_forall fun _ => norm_indicator_le_norm_self _ _)
+  · exact hf.norm
+  · filter_upwards [hfull] with x hx
+    obtain ⟨u,hu⟩ := hx
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_ge_atTop u] with t ht
+    exact (indicator_of_mem (show x∈visited ι γ t from ⟨u,⟨bot_le,ht⟩,hu⟩) f).symm
+
+lemma continuous_integral_restrict_visited (ν : Measure X) [NullSingletonClass ν]
+    (ι : X→Y) (hι : Measurable ι) (hinj : Function.Injective ι)
+    (γ : ℝ≥0→Y) (hγ : Continuous γ) {A : Set X} (hA : MeasurableSet A)
+    (f : X→ℝ) (hf : Integrable f ν) :
+    Continuous (fun t : ℝ≥0 => ∫x,f x∂(ν.restrict (visited ι γ t)).restrict A) := by
+  simp_rw [←integral_indicator hA]
+  exact continuous_integral_visited ν ι hι hinj γ hγ (A.indicator f) (hf.indicator hA)
+
+lemma tendsto_integral_restrict_visited_atTop (ν : Measure X)
+    (ι : X→Y) (hι : Measurable ι) (γ : ℝ≥0→Y) (hγ : Continuous γ)
+    (hfull : ∀ᵐ x ∂ν, ι x∈range γ) {A : Set X} (hA : MeasurableSet A)
+    (f : X→ℝ) (hf : Integrable f ν) :
+    Tendsto (fun t : ℝ≥0 => ∫x,f x∂(ν.restrict (visited ι γ t)).restrict A)
+      atTop (𝓝 (∫x in A,f x∂ν)) := by
+  simp_rw [←integral_indicator hA]
+  exact tendsto_integral_visited_atTop ν ι hι γ hγ hfull (A.indicator f) (hf.indicator hA)
+
+end SLEExactGauge.TraceRestriction
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
+open TraceRestriction TraceMassContinuity TraceMassIntegration
+
+lemma measurableSet_alive_atStop_raw (a : ℝ) (ha : 1/4<a)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    MeasurableSet[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)]
+      {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1} := by
+  have hD := stoppedData_raw_measurable hS
+  have hin : @Measurable (GDriver×UpperPlane) (ℝ≥0×(GDriver×UpperPlane))
+      (hS.measurableSpace.prod inferInstance) inferInstance (fun p =>
+      ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) :=
+    (hD.fst.comp measurable_fst).prodMk ((hD.snd.comp measurable_fst).prodMk measurable_snd)
+  have he : {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1}=
+      (fun p : GDriver×UpperPlane => ((stoppedData S p.1).1,((stoppedData S p.1).2,p.2))) ⁻¹'
+        {p : ℝ≥0×(GDriver×UpperPlane) | alive a p.2.1.1 p.1 p.2.2.1} := by
+    ext p
+    exact (alive_driver_congr (pastDriver_eq (S p.1).toNNReal p.1)).symm
+  rw [he]
+  exact (measurableSet_alive_joint a ha).preimage hin
+
+lemma measurableSet_alive_spatial (a : ℝ) (b : GDriver) (t : ℝ≥0) :
+    MeasurableSet {z : UpperPlane | alive a b.1 t z.1} :=
+  ((isOpen_alive b.1.continuous t).preimage continuous_subtype_val).measurableSet
+
+ 
+def recoveredPostStop (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (S : GDriver→ℝ≥0∞) (v : ℝ≥0∞) (b : GDriver) : Measure UpperPlane :=
+  if S b=∞ then 0 else (recoveredMass a ha s c hc (S b+v) b).restrict
+    {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}
+
+lemma recoveredPostStop_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (t : ℝ≥0) :
+    Measurable[(afterStopFiltration hS) t] (recoveredPostStop a ha s c hc S t) := by
+  let : MeasurableSpace GDriver := (afterStopFiltration hS) t
+  let K : Kernel GDriver UpperPlane := ⟨fun b => recoveredMass a ha s c hc (S b+t) b,
+    recoveredMass_stopping_raw_measurable a ha s c hc _ (shiftStop_isStoppingTime hS t)⟩
+  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
+  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
+  have hA : MeasurableSet {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1} :=
+    (prod_sigma_le (measurableSpace_le_shiftStop hS t) _ ) _ (measurableSet_alive_atStop_raw a ha hS)
+  have htop : MeasurableSet {b : GDriver | S b=∞} :=
+    (measurableSpace_le_shiftStop hS t) _ (hS.measurable (measurableSet_singleton ∞))
+  exact Measurable.ite htop measurable_const (measurable_random_restrict K hA)
+
+lemma recoveredPostStop_terminal_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    Measurable (recoveredPostStop a ha s c hc S ∞) := by
+  let K : Kernel GDriver UpperPlane := ⟨recoveredMass a ha s c hc ∞,
+    recoveredMass_measurable a ha s c hc ∞⟩
+  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
+  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
+  have hA : MeasurableSet {p : GDriver×UpperPlane | alive a p.1.1 (S p.1).toNNReal p.2.1} :=
+    (prod_sigma_le hS.measurableSpace_le _) _ (measurableSet_alive_atStop_raw a ha hS)
+  have htop : MeasurableSet {b : GDriver | S b=∞} := hS.measurable' (measurableSet_singleton ∞)
+  have hh : Measurable (fun b : GDriver => if S b=∞ then (0:Measure UpperPlane) else
+      (K b).restrict {z | alive a b.1 (S b).toNNReal z.1}) :=
+    Measurable.ite htop measurable_const (measurable_random_restrict K hA)
+  change Measurable (fun b => recoveredPostStop a ha s c hc S ∞ b)
+  simpa only [recoveredPostStop,add_top,K,Kernel.coe_mk] using hh
+
+lemma recoveredPostStop_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (v : ℝ≥0∞) :
+    Measurable (recoveredPostStop a ha s c hc S v) := by
+  induction v using ENNReal.recTopCoe with
+  | top => exact recoveredPostStop_terminal_measurable a ha s c hc hS
+  | coe t =>
+    exact (recoveredPostStop_raw_measurable a ha s c hc hS t).mono
+      ((afterStopFiltration hS).le t) le_rfl
+
+lemma recoveredPostStop_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (S : GDriver→ℝ≥0∞) (v : ℝ≥0∞) (b : GDriver) :
+    IsFiniteMeasureOnCompacts (recoveredPostStop a ha s c hc S v b) := by
+  unfold recoveredPostStop
+  split_ifs
+  · infer_instance
+  · let : IsFiniteMeasureOnCompacts (recoveredMass a ha s c hc (S b+v) b) := radonLimit_finite _
+    infer_instance
+
+lemma recoveredPostStop_compact_stronglyAdapted (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (f : C_c(UpperPlane,ℝ)) :
+    StronglyAdapted (afterStopFiltration hS)
+      (fun t : ℝ≥0 => fun b => ∫z,f z∂recoveredPostStop a ha s c hc S t b) := by
+  intro t
+  let : MeasurableSpace GDriver := (afterStopFiltration hS) t
+  let K : Kernel GDriver UpperPlane := ⟨recoveredPostStop a ha s c hc S t,
+    recoveredPostStop_raw_measurable a ha s c hc hS t⟩
+  exact f.continuous.stronglyMeasurable.integral_kernel (κ:=K)
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+variable {Ω X : Type*} [MeasurableSpace Ω]
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+
+theorem ae_finiteOnCompacts_of_average (K : Kernel Ω X) (W : Measure Ω)
+    [IsFiniteMeasureOnCompacts (K ∘ₘ W)] :
+    ∀ᵐ ω ∂W, IsFiniteMeasureOnCompacts (K ω) := by
+  let C := CompactExhaustion.choice X
+  have hn (n : ℕ) : ∀ᵐ ω ∂W, K ω (C n)<∞ := by
+    apply ae_lt_top (K.measurable_coe (C.isCompact n).measurableSet)
+    rw [←Measure.bind_apply (C.isCompact n).measurableSet K.aemeasurable]
+    exact (C.isCompact n).measure_lt_top.ne
+  filter_upwards [ae_all_iff.mpr hn] with ω hω
+  constructor
+  intro A hA
+  obtain ⟨n,hn⟩ := C.exists_superset_of_isCompact hA
+  exact (measure_mono hn).trans_lt (hω n)
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
+lemma compact_eval_integrable_of_average [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] (K : Kernel Ω X) (W : Measure Ω)
+    [IsFiniteMeasureOnCompacts (K ∘ₘ W)] (f : C_c(X,ℝ)) :
+    Integrable (fun ω => ∫x,f x∂K ω) W := by
+  have hi : Integrable f (K ∘ₘ W) := f.integrable
+  rw [Measure.comp_eq_comp_const_apply] at hi
+  exact hi.integral_comp
+
+lemma ae_compact_integrable_of_average (K : Kernel Ω X) (W : Measure Ω)
+    [IsFiniteMeasureOnCompacts (K ∘ₘ W)] :
+    ∀ᵐ ω ∂W, ∀f : C_c(X,ℝ), Integrable f (K ω) := by
+  filter_upwards [ae_finiteOnCompacts_of_average K W] with ω hω f
+  let := hω
+  exact f.integrable
+
+end SLEExactGauge.MassVersions
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
+open TraceRestriction TraceMassContinuity TraceMassIntegration
+
+lemma recoveredPostStop_terminal_le (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (S : GDriver→ℝ≥0∞) (b : GDriver) :
+    recoveredPostStop a ha s c hc S ∞ b≤recoveredMass a ha s c hc ∞ b := by
+  unfold recoveredPostStop
+  split_ifs
+  · exact bot_le
+  · simp only [add_top]
+    exact Measure.restrict_le_self
+
+lemma recoveredPostStop_terminal_integrable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (W : Measure GDriver)
+    (hmean : recoveredMass a ha s c hc ∞ ∘ₘ W=initialGreenMeasure a)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (f : C_c(UpperPlane,ℝ)) :
+    Integrable (fun b => ∫z,f z∂recoveredPostStop a ha s c hc S ∞ b) W := by
+  let K : Kernel GDriver UpperPlane := ⟨recoveredPostStop a ha s c hc S ∞,
+    recoveredPostStop_terminal_measurable a ha s c hc hS⟩
+  have hle : K ∘ₘ W ≤ initialGreenMeasure a := by
+    rw [←hmean]
+    apply Measure.le_iff.mpr
+    intro A hA
+    rw [Measure.bind_apply hA K.aemeasurable,
+      Measure.bind_apply hA (recoveredMass_measurable a ha s c hc ∞).aemeasurable]
+    exact lintegral_mono (fun b => (recoveredPostStop_terminal_le a ha s c hc S b) A)
+  let : IsFiniteMeasureOnCompacts (K ∘ₘ W) :=
+    ⟨fun A hA => (hle A).trans_lt hA.measure_lt_top⟩
+  exact compact_eval_integrable_of_average K W f
+
+lemma recoveredPostStop_zero_of_convergence (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (hs : ∀k j,j∈s k→k≤j) (S : GDriver→ℝ≥0∞) (b : GDriver)
+    (hconv : ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂recoveredMass a ha s c hc v b))) :
+    recoveredPostStop a ha s c hc S 0 b=0 := by
+  unfold recoveredPostStop
+  split_ifs with hb
+  · rfl
+  · rw [add_zero,←ENNReal.coe_toNNReal hb]
+    let : IsFiniteMeasureOnCompacts (recoveredMass a ha s c hc (S b).toNNReal b) :=
+      radonLimit_finite _
+    apply Measure.restrict_eq_zero.mpr
+    exact actual_limit_past_hull_support a ha s c hs b (S b).toNNReal _
+      (hconv (S b).toNNReal)
+
+ 
+
+theorem recoveredPostStop_path_properties (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure GDriver) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
+      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
+    (S : GDriver→ℝ≥0∞) :
+    let ν := recoveredPostStop (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S
+    ∀ᵐ b ∂W, ν 0 b=0 ∧ Monotone (fun v => ν v b) ∧
+      (∀f : C_c(UpperPlane,ℝ), Continuous (fun t : ℝ≥0 => ∫z,f z∂ν t b) ∧
+        Tendsto (fun t : ℝ≥0 => ∫z,f z∂ν t b) atTop (𝓝 (∫z,f z∂ν ∞ b))) := by
+  intro ν
+  let a := Target.a κ
+  have ha : 1/4<a := a_gt_quarter hκ hκ8
+  let μ := recoveredMass a ha s c hc
+  filter_upwards [hpath,hconv] with b hp hcv
+  refine ⟨recoveredPostStop_zero_of_convergence a ha s c hc hs S b hcv,?_,?_⟩
+  · intro v w hvw
+    change (if S b=∞ then 0 else (μ (S b+v) b).restrict _ )≤
+      if S b=∞ then 0 else (μ (S b+w) b).restrict _
+    split_ifs
+    · exact le_rfl
+    · exact Measure.restrict_mono le_rfl (hp.monotone (add_le_add le_rfl hvw))
+  · by_cases hb : S b=∞
+    · have he (v : ℝ≥0∞) : ν v b=0 := by simp only [ν,recoveredPostStop,hb,ite_true]
+      simp only [he,integral_zero_measure]
+      exact fun _ => ⟨continuous_const,tendsto_const_nhds⟩
+    · let A : Set UpperPlane := {z | alive a b.1 (S b).toNNReal z.1}
+      have hA : MeasurableSet A := measurableSet_alive_spatial a b _
+      have he (t : ℝ≥0) : ν t b=(μ (((S b).toNNReal+t : ℝ≥0) : ℝ≥0∞) b).restrict A := by
+        simp only [ν,recoveredPostStop,ite_eq_right hb,μ,ENNReal.coe_add,ENNReal.coe_toNNReal hb,A,a]
+      have het : ν ∞ b=(μ ∞ b).restrict A := by
+        simp only [ν,recoveredPostStop,ite_eq_right hb,add_top,μ,A,a]
+      change TraceMassFacts κ (driverHomeomorph.symm b) (fun v => μ v b) at hp
+      let := hp.atomless
+      let : IsFiniteMeasureOnCompacts (μ ∞ b) := radonLimit_finite _
+      have hγ := continuous_internalTrace hκ _ hp.continuous_trace
+      have hfull : ∀ᵐ z ∂μ ∞ b, z.1∈range (internalTrace κ (driverHomeomorph.symm b)) := by
+        filter_upwards [hp.firstVisit_finite] with z hz
+        exact (firstVisit_lt_top_iff _ _ _).mp hz
+      have hshift : Tendsto (fun t : ℝ≥0 => (S b).toNNReal+t) atTop atTop := by
+        apply tendsto_atTop.mpr
+        intro u
+        filter_upwards [eventually_ge_atTop u] with t ht
+        exact ht.trans (le_add_self)
+      intro f
+      simp_rw [he,het,hp.restriction]
+      exact ⟨(continuous_integral_restrict_visited (μ ∞ b) _ measurable_subtype_coe
+        Subtype.val_injective _ hγ hA f f.integrable).comp (continuous_const.add continuous_id),
+        (tendsto_integral_restrict_visited_atTop (μ ∞ b) _ measurable_subtype_coe _
+          hγ hfull hA f f.integrable).comp hshift⟩
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
+
+lemma restartPush_integral (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (μ : Measure UpperPlane) {f : UpperPlane→ℝ} (hf : Measurable f) :
+    (∫z,f z∂restartPush a (lt_trans (by norm_num) ha) b s μ)=
+      ∫w,(restartWeight a b s w).toReal*f (inverseUpper a (lt_trans (by norm_num) ha) b s w)∂μ := by
+  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
+  rw [restartPush,integral_map hJ.aemeasurable hf.aestronglyMeasurable,
+    integral_withDensity_eq_integral_toReal_smul (restartWeight_measurable a ha b s)
+      (Eventually.of_forall (fun w => lt_top_iff_ne_top.mpr (restartWeight_ne_top a b s w)))]
+  rfl
+
+lemma integrable_restartPush_iff (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (μ : Measure UpperPlane) {f : UpperPlane→ℝ} (hf : Measurable f) :
+    Integrable f (restartPush a (lt_trans (by norm_num) ha) b s μ) ↔
+      Integrable (fun w => (restartWeight a b s w).toReal*
+        f (inverseUpper a (lt_trans (by norm_num) ha) b s w)) μ := by
+  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
+  rw [restartPush,integrable_map_measure hf.aestronglyMeasurable hJ.aemeasurable,
+    integrable_withDensity_iff_integrable_smul' (restartWeight_measurable a ha b s)
+      (Eventually.of_forall (fun w => lt_top_iff_ne_top.mpr (restartWeight_ne_top a b s w)))]
+  rfl
+
+lemma restartPush_mono (a : ℝ) (ha : 0<a) (b : GDriver) (s : ℝ≥0)
+    {μ ν : Measure UpperPlane} (h : μ≤ν) :
+    restartPush a ha b s μ≤restartPush a ha b s ν := by
+  apply Measure.map_mono _ (continuous_inverseUpper a ha b s).measurable
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [withDensity_apply _ hA,withDensity_apply _ hA]
+  exact lintegral_mono' (Measure.restrict_mono le_rfl h) le_rfl
+
+lemma ae_restartAt {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {p : GDriver→Prop} (hp : ∀ᵐ b ∂W, p b) :
+    ∀ᵐ b ∂W, S b≠∞ → p (restartAt S b) := by
+  apply (ae_restrict_iff' (hS.measurable' (measurableSet_singleton ∞)).compl).mp
+  apply ae_of_ae_map (measurable_restartAt hS).aemeasurable
+  have he := map_restartAt_restrict hW hS (A:=univ) MeasurableSet.univ
+  rw [univ_inter] at he
+  change ∀ᵐ b ∂(W.restrict {b | S b≠∞}).map (restartAt S), p b
+  rw [he]
+  exact Measure.ae_smul_measure hp _
+
+ 
+
+theorem restartCandidateKernel_terminal_regular (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (hμ : μ ∘ₘ W=initialGreenMeasure a)
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    (∀ᵐ b ∂W, IsFiniteMeasureOnCompacts (restartCandidateKernel a ha μ S hS b) ∧
+      (restartCandidateKernel a ha μ S hS b).Regular) ∧
+    ∀f : C_c(UpperPlane,ℝ), Integrable
+      (fun b => ∫z,f z∂restartCandidateKernel a ha μ S hS b) W := by
+  let K := restartCandidateKernel a ha μ S hS
+  have hav : K ∘ₘ W ≤ initialGreenMeasure a :=
+    restartCandidateKernel_average_le_initial a ha μ W hW hμ S hS
+  let : IsFiniteMeasureOnCompacts (K ∘ₘ W) := ⟨fun A hA =>
+    (hav A).trans_lt hA.measure_lt_top⟩
+  refine ⟨?_,compact_eval_integrable_of_average K W⟩
+  filter_upwards [ae_finiteOnCompacts_of_average K W] with b hb
+  let := hb
+  exact ⟨hb,inferInstance⟩
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions TraceRestriction
+open TraceMassContinuity TraceMassIntegration
+
+lemma continuous_restartPush_visited (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (ν : Measure UpperPlane) [NullSingletonClass ν] (γ : ℝ≥0→ℂ) (hγ : Continuous γ)
+    {f : UpperPlane→ℝ} (hf : Measurable f)
+    (hfi : Integrable f (restartPush a (lt_trans (by norm_num) ha) b s ν)) :
+    Continuous (fun t : ℝ≥0 => ∫z,f z∂restartPush a (lt_trans (by norm_num) ha)
+      b s (ν.restrict (visited (fun z : UpperPlane => z.1) γ t))) := by
+  simp_rw [restartPush_integral a ha b s _ hf]
+  exact continuous_integral_visited ν _ measurable_subtype_coe Subtype.val_injective γ hγ _
+    ((integrable_restartPush_iff a ha b s ν hf).mp hfi)
+
+lemma tendsto_restartPush_visited_atTop (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (ν : Measure UpperPlane) (γ : ℝ≥0→ℂ) (hγ : Continuous γ)
+    (hfull : ∀ᵐ z ∂ν, z.1∈range γ) {f : UpperPlane→ℝ} (hf : Measurable f)
+    (hfi : Integrable f (restartPush a (lt_trans (by norm_num) ha) b s ν)) :
+    Tendsto (fun t : ℝ≥0 => ∫z,f z∂restartPush a (lt_trans (by norm_num) ha)
+      b s (ν.restrict (visited (fun z : UpperPlane => z.1) γ t))) atTop
+      (𝓝 (∫z,f z∂restartPush a (lt_trans (by norm_num) ha) b s ν)) := by
+  simp_rw [restartPush_integral a ha b s _ hf]
+  exact tendsto_integral_visited_atTop ν _ measurable_subtype_coe γ hγ hfull _
+    ((integrable_restartPush_iff a ha b s ν hf).mp hfi)
+
+ 
+
+theorem recoveredRestart_path_properties (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
+      initialGreenMeasure (Target.a κ))
+    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
+      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    let ν := recoveredRestart (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S
+    (∀f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂ν ∞ b) W) ∧
+    (∀ᵐ b ∂W, (∀v, IsFiniteMeasureOnCompacts (ν v b) ∧ (ν v b).Regular) ∧
+      ν 0 b=0 ∧ Monotone (fun v => ν v b) ∧
+      (∀f : C_c(UpperPlane,ℝ), Continuous (fun t : ℝ≥0 => ∫z,f z∂ν t b) ∧
+        Tendsto (fun t : ℝ≥0 => ∫z,f z∂ν t b) atTop (𝓝 (∫z,f z∂ν ∞ b)))) := by
+  intro ν
+  let a := Target.a κ
+  have ha : 1/4<a := a_gt_quarter hκ hκ8
+  let μ := recoveredMass a ha s c hc
+  let K : Kernel GDriver UpperPlane := ⟨μ ∞,recoveredMass_measurable a ha s c hc ∞⟩
+  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
+  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
+  have hreg := restartCandidateKernel_terminal_regular a ha K W hW hmean S hS
+  have hreg' : ∀ᵐ b ∂W, IsFiniteMeasureOnCompacts (ν ∞ b) ∧ (ν ∞ b).Regular := hreg.1
+  refine ⟨hreg.2,?_⟩
+  filter_upwards [hreg',ae_restartAt hW hS hpath] with b hr hp
+  have hν (v : ℝ≥0∞) : ν v b=if S b=∞ then 0 else restartPush a
+      (lt_trans (by norm_num) ha) b (S b).toNNReal (μ v (restartAt S b)) := rfl
+  by_cases hb : S b=∞
+  · have he (v : ℝ≥0∞) : ν v b=0 := by simp only [hν,hb,ite_true]
+    simp only [he,integral_zero_measure]
+    exact ⟨fun _ => ⟨inferInstance,inferInstance⟩,True.intro,monotone_const,
+      fun _ => ⟨continuous_const,tendsto_const_nhds⟩⟩
+  · have hp := hp hb
+    change TraceMassFacts κ (driverHomeomorph.symm (restartAt S b)) (fun v => μ v (restartAt S b)) at hp
+    have he (v : ℝ≥0∞) : ν v b=restartPush a (lt_trans (by norm_num) ha)
+        b (S b).toNNReal (μ v (restartAt S b)) := by simp only [hν,ite_eq_right hb]
+    have hmono : Monotone (fun v => ν v b) := by
+      intro v w hvw
+      change ν v b≤ν w b
+      rw [he,he]
+      exact restartPush_mono _ _ _ _ (hp.monotone hvw)
+    have hdom (v : ℝ≥0∞) : ν v b≤ν ∞ b := hmono le_top
+    have hlocal (v : ℝ≥0∞) : IsFiniteMeasureOnCompacts (ν v b) := by
+      let := hr.1
+      exact ⟨fun A hA => ((hdom v) A).trans_lt hA.measure_lt_top⟩
+    refine ⟨fun v => ⟨hlocal v,?_⟩,?_,hmono,?_⟩
+    · let := hlocal v
+      infer_instance
+    · rw [he]
+      simp only [μ,recoveredMass_zero a ha s c hc]
+      simp only [restartPush,withDensity_zero_left,Measure.map_zero]
+    · intro f
+      let := hr.1
+      let := hp.atomless
+      have hfi : Integrable f (restartPush a (lt_trans (by norm_num) ha) b
+          (S b).toNNReal (μ ∞ (restartAt S b))) := by rw [←he]; exact f.integrable
+      have hγ := continuous_internalTrace hκ _ hp.continuous_trace
+      have hfull : ∀ᵐ z ∂μ ∞ (restartAt S b), z.1∈range (internalTrace κ (driverHomeomorph.symm (restartAt S b))) := by
+        filter_upwards [hp.firstVisit_finite] with z hz
+        exact (firstVisit_lt_top_iff _ _ _).mp hz
+      simp_rw [he,hp.restriction]
+      exact ⟨continuous_restartPush_visited a ha b _ _ _ hγ f.continuous.measurable hfi,
+        tendsto_restartPush_visited_atTop a ha b _ _ _ hγ hfull f.continuous.measurable hfi⟩
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped Classical ENNReal
+namespace SLEExactGauge.MassVersions
+
+lemma random_restrict_test_lintegral {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
+    (W : Measure Ω) (K : Ω→Measure X) (D : Ω→Set X)
+    (hD : ∀ω,MeasurableSet (D ω)) {B : Set Ω} (hB : MeasurableSet B) (f : X→ℝ≥0∞) :
+    (∫⁻ω,∫⁻x,(if ω∈B then (D ω).indicator f x else 0)∂K ω∂W)=
+      ∫⁻ω in B,∫⁻x,f x∂(K ω).restrict (D ω)∂W := by
+  rw [←lintegral_indicator hB]
+  apply lintegral_congr
+  intro ω
+  by_cases hω : ω∈B
+  · simp only [ite_eq_left hω,indicator_of_mem hω]
+    exact lintegral_indicator (hD ω) f
+  · simp only [ite_eq_right hω,indicator_of_notMem hω,lintegral_zero]
+
+lemma conditional_random_restrict_on {Ω X : Type*} [mΩ : MeasurableSpace Ω]
+    [MeasurableSpace X] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
+    (W : @Measure Ω mΩ) (K L R : Ω→Measure X)
+    (hcond : ∀H : Ω×X→ℝ≥0∞, Measurable[m.prod (inferInstance : MeasurableSpace X)] H →
+      (∫⁻ω,∫⁻x,H (ω,x)∂K ω∂W)=(∫⁻ω,∫⁻x,H (ω,x)∂L ω∂W)+
+        ∫⁻ω,∫⁻x,H (ω,x)∂ R ω∂W)
+    (D : Ω→Set X) (hD : MeasurableSet[m.prod (inferInstance : MeasurableSpace X)]
+      {p : Ω×X | p.2∈D p.1}) {B : Set Ω} (hB : MeasurableSet[m] B)
+    {f : X→ℝ≥0∞} (hf : Measurable f) :
+    (∫⁻ω in B,∫⁻x,f x∂(K ω).restrict (D ω)∂W)=
+      (∫⁻ω in B,∫⁻x,f x∂(L ω).restrict (D ω)∂W)+
+      ∫⁻ω in B,∫⁻x,f x∂(R ω).restrict (D ω)∂W := by
+  let F : Ω×X→ℝ≥0∞ := fun p => if p.2∈D p.1 then f p.2 else 0
+  have hF : Measurable[m.prod (inferInstance : MeasurableSpace X)] F :=
+    Measurable.ite hD (hf.comp measurable_snd) measurable_const
+  let H : Ω×X→ℝ≥0∞ := fun p => if p.1∈B then F p else 0
+  have hH : Measurable[m.prod (inferInstance : MeasurableSpace X)] H :=
+    Measurable.ite (hB.preimage measurable_fst) hF measurable_const
+  have hsec (ω : Ω) : MeasurableSet (D ω) := by
+    exact hD.preimage (measurable_const.prodMk measurable_id)
+  have he (M : Ω→Measure X) : (∫⁻ω,∫⁻x,H (ω,x)∂M ω∂W)=
+      ∫⁻ω in B,∫⁻x,f x∂(M ω).restrict (D ω)∂W := by
+    change (∫⁻ω,∫⁻x,(if ω∈B then (D ω).indicator f x else 0)∂M ω∂W)=_
+    exact @random_restrict_test_lintegral Ω X mΩ _ W M D hsec B (hm B hB) f
+  simpa only [he] using hcond H hH
+
+end SLEExactGauge.MassVersions
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RawBrownianRestart.lean b/lean/OAI/Probability/SLEGauge/RawBrownianRestart.lean
index 5f8f6b0671dc788a5301ffb708cd08782a9a03e9..f47f3d39538dc9e60f2fb7135bea367fabd8da74 100644
--- a/lean/OAI/Probability/SLEGauge/RawBrownianRestart.lean
+++ b/lean/OAI/Probability/SLEGauge/RawBrownianRestart.lean
@@ -1,483 +1,483 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.BrownianRestart
-import OAI.Probability.SLEGauge.SurvivalLocalization
-import OAI.Probability.SLEGauge.LoewnerScaling
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology Function
-open scoped NNReal ENNReal ContinuousMapZero BoundedContinuousFunction
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEGenerator ExactSLEMain
-
- 
-def restart (t : ℝ≥0) (b : GDriver) : GDriver :=
-  driverHomeomorph (BrownianRestart.shift t (driverHomeomorph.symm b))
-
-@[simp] lemma restart_apply (t u : ℝ≥0) (b : GDriver) :
-    (restart t b).1 u=b.1 (t+u)-b.1 t := rfl
-
-lemma continuous_restart_joint : Continuous (fun p : ℝ≥0×GDriver => restart p.1 p.2) :=
-  driverHomeomorph.continuous.comp (BrownianRestart.continuous_shift_joint.comp
-    (continuous_fst.prodMk (driverHomeomorph.symm.continuous.comp continuous_snd)))
-
- 
-def restartAt (S : GDriver→ℝ≥0∞) (b : GDriver) : GDriver := restart (S b).toNNReal b
-
-lemma comap_rawFiltration (t : ℝ≥0) :
-    MeasurableSpace.comap driverHomeomorph (rawFiltration t)=BrownianRestart.filtration t := by
-  simp only [rawFiltration,BrownianRestart.filtration,Filtration.natural_eq_comap,
-    MeasurableSpace.comap_comp]
-  rfl
-
-lemma measurable_driver_raw (t : ℝ≥0) :
-    @Measurable TDriver GDriver (BrownianRestart.filtration t) (rawFiltration t) driverHomeomorph :=
-  measurable_iff_comap_le.mpr (comap_rawFiltration t).le
-
-lemma measurable_driver_raw_all :
-    @Measurable TDriver GDriver (⨆t,BrownianRestart.filtration t) (⨆t,rawFiltration t) driverHomeomorph := by
-  apply measurable_iff_comap_le.mpr
-  simp only [MeasurableSpace.comap_iSup,comap_rawFiltration,le_refl]
-
-lemma stoppingTime_transport {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    IsStoppingTime BrownianRestart.filtration (S ∘ driverHomeomorph) := by
-  intro t
-  exact measurable_driver_raw t (hS t)
-
-lemma measurable_driver_stopped {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    @Measurable TDriver GDriver (stoppingTime_transport hS).measurableSpace hS.measurableSpace
-      driverHomeomorph := by
-  intro A hA
-  refine ⟨measurable_driver_raw_all hA.1,fun t => ?_⟩
-  exact measurable_driver_raw t (hA.2 t)
-
-lemma measurable_restartAt {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    Measurable (restartAt S) :=
-  continuous_restart_joint.measurable.comp
-    ((ENNReal.measurable_toNNReal.comp hS.measurable').prodMk measurable_id)
-
-lemma restartAt_transport (S : GDriver→ℝ≥0∞) (b : TDriver) :
-    driverHomeomorph (BrownianRestart.shiftAt (S ∘ driverHomeomorph) b)=
-      restartAt S (driverHomeomorph b) := by
-  simp only [restartAt,restart,BrownianRestart.shiftAt,Function.comp_apply,
-    Homeomorph.symm_apply_apply]
-
-lemma setIntegral_restartAt {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) (f : GDriver→ᵇℝ) :
-    (∫b in A ∩ {b | S b≠∞}, f (restartAt S b) ∂W)=
-      ∫_b in A ∩ {b | S b≠∞}, (∫b,f b∂W) ∂W := by
-  have h := BrownianRestart.setIntegral_shiftAt_finite
-    (LoewnerScaling.brownian_map_inverse_driver W hW) (stoppingTime_transport hS)
-    (measurable_driver_stopped hS hA) (f.compContinuous ⟨driverHomeomorph,driverHomeomorph.continuous⟩)
-  have hm : MeasurableEmbedding driverHomeomorph.symm :=
-    driverMeasurableEquiv.symm.measurableEmbedding
-  rw [hm.setIntegral_map,hm.setIntegral_map,hm.integral_map] at h
-  change
-    (∫b in driverHomeomorph.symm ⁻¹' (driverHomeomorph ⁻¹' A ∩
-      {b | S (driverHomeomorph b)≠∞}),
-      f (driverHomeomorph (BrownianRestart.shiftAt
-        (fun b => S (driverHomeomorph b)) (driverHomeomorph.symm b))) ∂W)=
-    ∫_b in driverHomeomorph.symm ⁻¹' (driverHomeomorph ⁻¹' A ∩
-      {b | S (driverHomeomorph b)≠∞}),
-      (∫b,f (driverHomeomorph (driverHomeomorph.symm b))∂W) ∂W at h
-  have hA' : driverHomeomorph.symm ⁻¹' (driverHomeomorph ⁻¹' A)=A := by
-    ext b
-    change driverHomeomorph (driverHomeomorph.symm b) ∈ A ↔ b∈A
-    rw [driverHomeomorph.apply_symm_apply]
-  simpa [restartAt,restart,BrownianRestart.shiftAt,hA'] using h
-
- 
-
-theorem map_restartAt_restrict {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) :
-    (W.restrict (A ∩ {b | S b≠∞})).map (restartAt S)=W (A ∩ {b | S b≠∞}) • W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have : IsFiniteMeasure (W (A ∩ {b | S b≠∞}) • W) := W.smul_finite (measure_ne_top W _)
-  have : PolishSpace GDriver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro f
-  rw [integral_map (measurable_restartAt hS).aemeasurable f.continuous.aestronglyMeasurable,
-    setIntegral_restartAt hW hS hA f,integral_smul_measure]
-  simp only [integral_const,measureReal_def,Measure.restrict_apply_univ]
-
- 
-
-theorem map_data_restartAt {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D) :
-    (W.restrict {b | S b≠∞}).map (fun b => (D b,restartAt S b))=
-      ((W.restrict {b | S b≠∞}).map D).prod W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
-  have hY := measurable_restartAt hS
-  apply Measure.ext_prod
-  intro A B hA hB
-  rw [Measure.map_apply (hD'.prodMk hY) (hA.prod hB),Measure.prod_prod,
-    Measure.map_apply hD' hA,Measure.restrict_apply (hD' hA)]
-  have he := congrArg (fun μ : Measure GDriver => μ B) (map_restartAt_restrict hW hS (hD hA))
-  rw [Measure.map_apply hY hB,Measure.restrict_apply (hY hB),Measure.smul_apply,smul_eq_mul] at he
-  rw [Measure.restrict_apply ((hD'.prodMk hY) (hA.prod hB))]
-  convert he using 1
-  congr 1
-  ext b
-  simp only [mem_inter_iff,mem_preimage,mem_prod,mem_ofPred_eq]
-  tauto
-
- 
-theorem lintegral_data_restartAt {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D)
-    {F : X×GDriver→ℝ≥0∞} (hF : Measurable F) :
-    (∫⁻b in {b | S b≠∞},F (D b,restartAt S b) ∂W)=
-      ∫⁻b in {b | S b≠∞},∫⁻c,F (D b,c) ∂W ∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
-  rw [←lintegral_map hF (hD'.prodMk (measurable_restartAt hS)),map_data_restartAt hW hS hD,
-    lintegral_prod _ hF.aemeasurable,lintegral_map hF.lintegral_prod_right' hD']
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEMain
-
-lemma actualState_restart {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {s t : ℝ≥0} {z : ℂ} (hz : alive a b (s+t) z) :
-    actualState a b (s+t) z =
-      ((actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))).1,
-        (actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))).2+
-          (actualState a b s z).2) := by
-  have hs := (alive_restart_iff a b z s t).mp hz
-  apply Prod.ext
-  · simp only [actualState,shiftDriver,Complex.ofReal_sub]
-    rw [(loewnerMap_restart a b z s t hz).2]
-    ring
-  · simp only [actualState,deriv_loewnerMap_restart hb hz,norm_mul,
-      Real.log_mul (norm_ne_zero_iff.mpr (deriv_loewnerMap_ne_zero
-        (continuous_shiftDriver hb s) hs.2))
-        (norm_ne_zero_iff.mpr (deriv_loewnerMap_ne_zero hb hs.1))]
-
-lemma physicalRadius_add_log (v : State) (l : ℝ) :
-    physicalRadius (v.1,v.2+l)=Real.exp (-l)*physicalRadius v := by
-  simp only [physicalRadius,Y_apply,J_apply,neg_add,Real.exp_add]
-  ring
-
-lemma logGreen_add_log (a : ℝ) (v : State) (l : ℝ) :
-    logGreen a (v.1,v.2+l)=alpha a*l+logGreen a v := by
-  simp only [logGreen,J_apply,Y_apply,Q,X_apply]
-  ring
-
-lemma physicalGreen_add_log (a : ℝ) (v : State) (l : ℝ) :
-    physicalGreen a (v.1,v.2+l)=Real.exp (alpha a*l)*physicalGreen a v := by
-  rw [physicalGreen,logGreen_add_log,Real.exp_add]
-  rfl
-
- 
-lemma physicalRadius_restart {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {s t : ℝ≥0} {z : ℂ} (hz : alive a b (s+t) z) :
-    physicalRadius (actualState a b (s+t) z)=
-      ‖deriv (loewnerMap a b s) z‖⁻¹ *
-        physicalRadius (actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))) := by
-  rw [actualState_restart hb hz,physicalRadius_add_log]
-  congr 1
-  rw [actualState,Real.exp_neg,Real.exp_log]
-  exact norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb
-    ((alive_restart_iff a b z s t).mp hz).1)
-
- 
-lemma physicalGreen_restart {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
-    {s t : ℝ≥0} {z : ℂ} (hz : alive a b (s+t) z) :
-    physicalGreen a (actualState a b (s+t) z)=
-      ‖deriv (loewnerMap a b s) z‖^(alpha a) *
-        physicalGreen a (actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))) := by
-  rw [actualState_restart hb hz,physicalGreen_add_log]
-  congr 1
-  rw [Real.rpow_def_of_pos (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb
-    ((alive_restart_iff a b z s t).mp hz).1))]
-  congr 1
-  exact mul_comm _ _
-
-end ExactSLEGreen
-
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen
-
-lemma restart_eq_shiftDriver (s : ℝ≥0) (b : GDriver) :
-    ((restart s b).1 : ℝ≥0→ℝ)=shiftDriver b.1 s := rfl
-
- 
-
-theorem survivingGreen_restart (a : ℝ) (b : GDriver) (s t : ℝ≥0) {z : ℂ}
-    (hz : alive a b.1 s z) :
-    survivingGreen a (s+t) b z = ‖deriv (loewnerMap a b.1 s) z‖^(alpha a) *
-      survivingGreen a t (restart s b) (loewnerMap a b.1 s z-(b.1 s:ℂ)) := by
-  classical
-  have hc : alive a b.1 (s+t) z ↔
-      alive a (restart s b).1 t (loewnerMap a b.1 s z-(b.1 s:ℂ)) := by
-    rw [alive_restart_iff]
-    exact and_iff_right hz
-  by_cases h : alive a b.1 (s+t) z
-  · unfold survivingGreen
-    rw [ite_eq_left h, ite_eq_left (hc.mp h),physicalGreen_restart b.1.continuous h,
-      restart_eq_shiftDriver]
-    ring
-  · simp only [survivingGreen,ite_eq_right h,ite_eq_right (hc.not.mp h),mul_zero]
-
- 
-theorem survivingGreen_restart_ratio (a : ℝ) (b : GDriver) (s t : ℝ≥0) {z : ℂ}
-    (hz : alive a b.1 s z) :
-    survivingGreen a (s+t) b z / survivingGreen a s b z =
-      survivingGreen a t (restart s b) (loewnerMap a b.1 s z-(b.1 s:ℂ)) /
-        survivingGreen a 0 (restart s b) (loewnerMap a b.1 s z-(b.1 s:ℂ)) := by
-  have he := survivingGreen_restart a b s 0 hz
-  simp only [add_zero] at he
-  rw [survivingGreen_restart a b s t hz,he]
-  exact mul_div_mul_left _ _ (Real.rpow_pos_of_pos
-    (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hz)) _).ne'
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-namespace ExactSLECalculus
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-
-lemma N_rpow {f : E → ℝ} {v e : E} (r : ℝ)
-    (hf : DifferentiableAt ℝ f v) (hp : 0 < f v) :
-    N (fun w => f w ^ r) e v = r * f v ^ (r-1) * N f e v := by
-  have hh := Real.contDiffAt_rpow_const (n := 1) (p := r) (Or.inl hp.ne')
-  simpa only [Function.comp_def, Real.deriv_rpow_const] using
-    N_comp (e := e) hf hh.differentiableAt_one
-
-lemma L_rpow {f : E → ℝ} {v A e : E} (r : ℝ)
-    (hf : ContDiffAt ℝ 2 f v) (hp : 0 < f v) :
-    L (fun w => f w ^ r) A e v = r * f v ^ (r-1) * L f A e v +
-      (1/2:ℝ) * (r * (r-1) * f v ^ (r-2)) * N f e v ^ 2 := by
-  have hh := Real.contDiffAt_rpow_const (n := 2) (p := r) (Or.inl hp.ne')
-  have he : deriv (deriv (fun x : ℝ => x ^ r)) (f v) =
-      r * (r-1) * f v ^ (r-2) := by
-    rw [Real.deriv_rpow_const', deriv_const_mul_field, Real.deriv_rpow_const]
-    rw [show r - 1 - 1 = r - 2 by ring]
-    ring
-  simpa only [Function.comp_def, Real.deriv_rpow_const, he] using
-    L_comp (A := A) (e := e) hf hh
-
-end ExactSLECalculus
-
-namespace ExactSLEGreen
-open ExactSLECalculus
-
- 
-
-noncomputable def centeredGenerator (a : ℝ) (f : State → ℝ) (v : State) : ℝ :=
-  L f (physicalDrift a v) noise v + noiseLog a v * N f noise v
-
-lemma green_product_generator (a : ℝ) (ha : a ≠ 0) (f : State → ℝ)
-    (v : State) (hv : 0 < Y v) (hf : ContDiffAt ℝ 2 f v) :
-    L (fun w => physicalGreen a w * f w) (physicalDrift a v) noise v =
-      physicalGreen a v * centeredGenerator a f v := by
-  have hg : ContDiffAt ℝ 2 (physicalGreen a) v :=
-    (contDiffAt_physicalGreen a v hv).of_le (by norm_cast)
-  rw [L_mul hg hf]
-  have hz : L (physicalGreen a) (physicalDrift a v) noise v = 0 :=
-    green_generator_zero a ha v hv
-  rw [hz, show N (physicalGreen a) noise v = physicalGreen a v * noiseLog a v from
-    green_fderiv_noise a v hv]
-  dsimp [centeredGenerator]
-  ring
-
-lemma N_Q (v : State) : N Q noise v = -2 * X v := by
-  rw [N, (hasFDerivAt_Q v).fderiv]
-  simp [noise]
-
-lemma L_Q (a : ℝ) (v : State) :
-    L Q (physicalDrift a v) noise v =
-      1 + 2 * a * (X v ^ 2 - Y v ^ 2) / Q v := by
-  have hn : N Q noise = fun w => -2 * X w := funext N_Q
-  rw [L, hn, N, (X.hasFDerivAt.const_mul (-2 : ℝ)).fderiv,
-    (hasFDerivAt_Q v).fderiv]
-  simp only [_root_.smul_apply, _root_.add_apply, smul_eq_mul,
-    regX_drift, regY_drift]
-  simp only [X_apply, noise, Complex.neg_re, Complex.one_re]
-  ring
-
-lemma N_Y (v : State) : N Y noise v = 0 := by
-  rw [N, Y.hasFDerivAt.fderiv]
-  simp [noise]
-
-lemma L_Y (a : ℝ) (v : State) :
-    L Y (physicalDrift a v) noise v = -a * Y v / Q v := by
-  have hn : N Y noise = fun _ => (0 : ℝ) := funext N_Y
-  rw [L, hn, N, fderiv_const_apply, _root_.zero_apply,
-    Y.hasFDerivAt.fderiv, regY_drift]
-  ring
-
-lemma centeredGenerator_Qpow (a q : ℝ) (v : State) (hv : 0 < Y v) :
-    centeredGenerator a (fun w => Q w ^ (q/2)) v =
-      q * Q v ^ (q/2-2) * ((1/2-a) * Q v + (q/2-2*a) * X v ^ 2) := by
-  have hc : ContDiffAt ℝ 2 Q v := ((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt
-  have hq := Q_pos hv
-  have he : Q v ^ (q/2-1) = Q v ^ (q/2-2) * Q v := by
-    rw [← Real.rpow_add_one hq.ne']
-    congr 1
-    ring
-  rw [centeredGenerator, L_rpow _ hc hq, N_rpow _ (hasFDerivAt_Q v).differentiableAt hq,
-    L_Q, N_Q, he]
-  dsimp only [noiseLog, p]
-  field_simp
-  ring_nf
-  simp only [Q]
-  ring
-
-lemma centeredGenerator_Ypow (a q : ℝ) (v : State) (hv : 0 < Y v) :
-    centeredGenerator a (fun w => Y w ^ q) v = -a * q * Y v ^ q / Q v := by
-  have he : Y v ^ (q-1) * Y v = Y v ^ q := by
-    rw [← Real.rpow_add_one hv.ne']
-    congr 1
-    ring
-  rw [centeredGenerator, L_rpow _ Y.contDiff.contDiffAt hv,
-    N_rpow _ Y.differentiableAt hv, N_Y, L_Y]
-  calc
-    _ = -a * q * (Y v ^ (q-1) * Y v) / Q v := by ring
-    _ = _ := by rw [he]
-
- 
-noncomputable def lifetimeTest (q : ℝ) (v : State) : ℝ :=
-  Q v ^ (q/2) + 2 * Y v ^ q
-
-lemma lifetimeTest_smooth (q : ℝ) (v : State) (hv : 0 < Y v) :
-    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (lifetimeTest q) v := by
-  exact ((((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt.rpow_const_of_ne
-    (Q_pos hv).ne').add (contDiffAt_const.mul (Y.contDiff.contDiffAt.rpow_const_of_ne hv.ne')))
-
-lemma lifetimeTest_nonneg (q : ℝ) (v : State) (hv : 0 ≤ Y v) :
-    0 ≤ lifetimeTest q v := by
-  unfold lifetimeTest Q
-  positivity
-
-lemma lifetimeTest_coercive (q : ℝ) (v : State) (hv : 0 ≤ Y v) :
-    Q v ^ (q/2) ≤ lifetimeTest q v := by
-  dsimp [lifetimeTest]
-  exact le_add_of_nonneg_right (mul_nonneg (by norm_num) (Real.rpow_nonneg hv _))
-
-lemma centeredGenerator_lifetimeTest (a q : ℝ) (v : State) (hv : 0 < Y v) :
-    centeredGenerator a (lifetimeTest q) v =
-      q * Q v ^ (q/2-2) * ((1/2-a) * Q v + (q/2-2*a) * X v ^ 2) -
-      2 * a * q * Y v ^ q / Q v := by
-  have hQ : ContDiffAt ℝ 2 (fun w => Q w ^ (q/2)) v :=
-    (((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt.rpow_const_of_ne (Q_pos hv).ne')
-  have hY : ContDiffAt ℝ 2 (fun w => Y w ^ q) v :=
-    Y.contDiff.contDiffAt.rpow_const_of_ne hv.ne'
-  have split : centeredGenerator a (lifetimeTest q) v =
-      centeredGenerator a (fun w => Q w ^ (q/2)) v +
-      2 * centeredGenerator a (fun w => Y w ^ q) v := by
-    unfold centeredGenerator lifetimeTest
-    rw [L_add hQ (contDiffAt_const.mul hY), L_const_mul hY,
-      N_add (hQ.differentiableAt (by norm_num)) ((contDiffAt_const.mul hY).differentiableAt (by norm_num)),
-      N_const_mul (hY.differentiableAt (by norm_num))]
-    ring
-  rw [split, centeredGenerator_Qpow _ _ _ hv, centeredGenerator_Ypow _ _ _ hv]
-  ring
-
-lemma height_power_comparison (q : ℝ) (hq : q ≤ 2) (v : State) (hv : 0 < Y v) :
-    Y v ^ 2 * Q v ^ (q/2-1) ≤ Y v ^ q := by
-  have hsq : 0 < Y v ^ 2 := sq_pos_of_pos hv
-  have hle : Y v ^ 2 ≤ Q v := by dsimp only [Q]; nlinarith [sq_nonneg (X v)]
-  have h := mul_le_mul_of_nonneg_left
-    (Real.rpow_le_rpow_of_nonpos hsq hle (by linarith : q/2-1 ≤ 0)) hsq.le
-  have he : Y v ^ 2 * (Y v ^ 2) ^ (q/2-1) = Y v ^ q := by
-    rw [← Real.rpow_two (Y v), ← Real.rpow_mul hv.le, ← Real.rpow_add hv]
-    congr 1
-    ring
-  exact h.trans_eq he
-
- 
-
-lemma centeredGenerator_lifetimeTest_le (a q : ℝ) (ha : 0 < a)
-    (hq : 0 < q) (hq2 : q ≤ 2) (v : State) (hv : 0 < Y v) :
-    centeredGenerator a (lifetimeTest q) v ≤
-      -(q * (6*a-1-q)/2) * Q v ^ (q/2-1) := by
-  have hQ := Q_pos hv
-  have hp := height_power_comparison q hq2 v hv
-  have he : Q v ^ (q/2-1) = Q v ^ (q/2-2) * Q v := by
-    rw [← Real.rpow_add_one hQ.ne']
-    congr 1
-    ring
-  have hdiv : Y v ^ 2 * Q v ^ (q/2-2) ≤ Y v ^ q / Q v := by
-    apply (le_div_iff₀ hQ).mpr
-    nlinarith only [hp.trans_eq (by rfl), show Y v ^ 2 * Q v ^ (q/2-1) =
-      (Y v ^ 2 * Q v ^ (q/2-2)) * Q v by rw [he]; ring]
-  rw [centeredGenerator_lifetimeTest _ _ _ hv, he]
-  have hs : X v ^ 2 ≤ Q v := by dsimp only [Q]; nlinarith [sq_nonneg (Y v)]
-  have hn := mul_le_mul_of_nonneg_left hdiv (show 0 ≤ 2*a*q by positivity)
-  have hpos : 0 ≤ q * Q v ^ (q/2-2) := by positivity
-  have hx := mul_le_mul_of_nonneg_left hs (show 0 ≤ q/2 * (q * Q v ^ (q/2-2)) by positivity)
-  dsimp only [Q] at hn hx ⊢
-  simp only [div_eq_mul_inv] at hn hx ⊢
-  nlinarith only [hn, hx]
-
-noncomputable def lifetimeExponent (a : ℝ) : ℝ := min 1 (3*a-1/2)
-noncomputable def lifetimeDrift (a : ℝ) : ℝ :=
-  lifetimeExponent a * (6*a-1-lifetimeExponent a)/2
-
-lemma lifetimeExponent_pos {a : ℝ} (ha : 1/4 < a) : 0 < lifetimeExponent a := by
-  dsimp [lifetimeExponent]
-  exact lt_min (by norm_num) (by linarith)
-
-lemma lifetimeExponent_le_one (a : ℝ) : lifetimeExponent a ≤ 1 := min_le_left _ _
-
-lemma lifetimeDrift_pos {a : ℝ} (ha : 1/4 < a) : 0 < lifetimeDrift a := by
-  have hq := lifetimeExponent_pos ha
-  have hq' : lifetimeExponent a ≤ 3*a-1/2 := min_le_right _ _
-  exact div_pos (mul_pos hq (by linarith)) (by norm_num)
-
- 
-lemma green_lifetime_generator_le (a : ℝ) (ha : 1/4 < a)
-    (v : State) (hv : 0 < Y v) :
-    L (fun w => physicalGreen a w * lifetimeTest (lifetimeExponent a) w)
-      (physicalDrift a v) noise v ≤
-    -lifetimeDrift a * physicalGreen a v * Q v ^ (lifetimeExponent a/2-1) := by
-  rw [green_product_generator a (by linarith) _ _ hv
-    ((lifetimeTest_smooth _ _ hv).of_le (by norm_cast))]
-  have h := mul_le_mul_of_nonneg_left
-    (centeredGenerator_lifetimeTest_le a (lifetimeExponent a) (by linarith)
-      (lifetimeExponent_pos ha) ((lifetimeExponent_le_one a).trans (by norm_num)) v hv)
-    (Real.exp_pos (logGreen a v)).le
-  dsimp only [lifetimeDrift, physicalGreen] at *
-  convert h using 1
-  first | rfl | ring
-
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.BrownianRestart
+import OAI.Probability.SLEGauge.SurvivalLocalization
+import OAI.Probability.SLEGauge.LoewnerScaling
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology Function
+open scoped NNReal ENNReal ContinuousMapZero BoundedContinuousFunction
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEGenerator ExactSLEMain
+
+ 
+def restart (t : ℝ≥0) (b : GDriver) : GDriver :=
+  driverHomeomorph (BrownianRestart.shift t (driverHomeomorph.symm b))
+
+@[simp] lemma restart_apply (t u : ℝ≥0) (b : GDriver) :
+    (restart t b).1 u=b.1 (t+u)-b.1 t := rfl
+
+lemma continuous_restart_joint : Continuous (fun p : ℝ≥0×GDriver => restart p.1 p.2) :=
+  driverHomeomorph.continuous.comp (BrownianRestart.continuous_shift_joint.comp
+    (continuous_fst.prodMk (driverHomeomorph.symm.continuous.comp continuous_snd)))
+
+ 
+def restartAt (S : GDriver→ℝ≥0∞) (b : GDriver) : GDriver := restart (S b).toNNReal b
+
+lemma comap_rawFiltration (t : ℝ≥0) :
+    MeasurableSpace.comap driverHomeomorph (rawFiltration t)=BrownianRestart.filtration t := by
+  simp only [rawFiltration,BrownianRestart.filtration,Filtration.natural_eq_comap,
+    MeasurableSpace.comap_comp]
+  rfl
+
+lemma measurable_driver_raw (t : ℝ≥0) :
+    @Measurable TDriver GDriver (BrownianRestart.filtration t) (rawFiltration t) driverHomeomorph :=
+  measurable_iff_comap_le.mpr (comap_rawFiltration t).le
+
+lemma measurable_driver_raw_all :
+    @Measurable TDriver GDriver (⨆t,BrownianRestart.filtration t) (⨆t,rawFiltration t) driverHomeomorph := by
+  apply measurable_iff_comap_le.mpr
+  simp only [MeasurableSpace.comap_iSup,comap_rawFiltration,le_refl]
+
+lemma stoppingTime_transport {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    IsStoppingTime BrownianRestart.filtration (S ∘ driverHomeomorph) := by
+  intro t
+  exact measurable_driver_raw t (hS t)
+
+lemma measurable_driver_stopped {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    @Measurable TDriver GDriver (stoppingTime_transport hS).measurableSpace hS.measurableSpace
+      driverHomeomorph := by
+  intro A hA
+  refine ⟨measurable_driver_raw_all hA.1,fun t => ?_⟩
+  exact measurable_driver_raw t (hA.2 t)
+
+lemma measurable_restartAt {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    Measurable (restartAt S) :=
+  continuous_restart_joint.measurable.comp
+    ((ENNReal.measurable_toNNReal.comp hS.measurable').prodMk measurable_id)
+
+lemma restartAt_transport (S : GDriver→ℝ≥0∞) (b : TDriver) :
+    driverHomeomorph (BrownianRestart.shiftAt (S ∘ driverHomeomorph) b)=
+      restartAt S (driverHomeomorph b) := by
+  simp only [restartAt,restart,BrownianRestart.shiftAt,Function.comp_apply,
+    Homeomorph.symm_apply_apply]
+
+lemma setIntegral_restartAt {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) (f : GDriver→ᵇℝ) :
+    (∫b in A ∩ {b | S b≠∞}, f (restartAt S b) ∂W)=
+      ∫_b in A ∩ {b | S b≠∞}, (∫b,f b∂W) ∂W := by
+  have h := BrownianRestart.setIntegral_shiftAt_finite
+    (LoewnerScaling.brownian_map_inverse_driver W hW) (stoppingTime_transport hS)
+    (measurable_driver_stopped hS hA) (f.compContinuous ⟨driverHomeomorph,driverHomeomorph.continuous⟩)
+  have hm : MeasurableEmbedding driverHomeomorph.symm :=
+    driverMeasurableEquiv.symm.measurableEmbedding
+  rw [hm.setIntegral_map,hm.setIntegral_map,hm.integral_map] at h
+  change
+    (∫b in driverHomeomorph.symm ⁻¹' (driverHomeomorph ⁻¹' A ∩
+      {b | S (driverHomeomorph b)≠∞}),
+      f (driverHomeomorph (BrownianRestart.shiftAt
+        (fun b => S (driverHomeomorph b)) (driverHomeomorph.symm b))) ∂W)=
+    ∫_b in driverHomeomorph.symm ⁻¹' (driverHomeomorph ⁻¹' A ∩
+      {b | S (driverHomeomorph b)≠∞}),
+      (∫b,f (driverHomeomorph (driverHomeomorph.symm b))∂W) ∂W at h
+  have hA' : driverHomeomorph.symm ⁻¹' (driverHomeomorph ⁻¹' A)=A := by
+    ext b
+    change driverHomeomorph (driverHomeomorph.symm b) ∈ A ↔ b∈A
+    rw [driverHomeomorph.apply_symm_apply]
+  simpa [restartAt,restart,BrownianRestart.shiftAt,hA'] using h
+
+ 
+
+theorem map_restartAt_restrict {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) :
+    (W.restrict (A ∩ {b | S b≠∞})).map (restartAt S)=W (A ∩ {b | S b≠∞}) • W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have : IsFiniteMeasure (W (A ∩ {b | S b≠∞}) • W) := W.smul_finite (measure_ne_top W _)
+  have : PolishSpace GDriver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro f
+  rw [integral_map (measurable_restartAt hS).aemeasurable f.continuous.aestronglyMeasurable,
+    setIntegral_restartAt hW hS hA f,integral_smul_measure]
+  simp only [integral_const,measureReal_def,Measure.restrict_apply_univ]
+
+ 
+
+theorem map_data_restartAt {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D) :
+    (W.restrict {b | S b≠∞}).map (fun b => (D b,restartAt S b))=
+      ((W.restrict {b | S b≠∞}).map D).prod W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
+  have hY := measurable_restartAt hS
+  apply Measure.ext_prod
+  intro A B hA hB
+  rw [Measure.map_apply (hD'.prodMk hY) (hA.prod hB),Measure.prod_prod,
+    Measure.map_apply hD' hA,Measure.restrict_apply (hD' hA)]
+  have he := congrArg (fun μ : Measure GDriver => μ B) (map_restartAt_restrict hW hS (hD hA))
+  rw [Measure.map_apply hY hB,Measure.restrict_apply (hY hB),Measure.smul_apply,smul_eq_mul] at he
+  rw [Measure.restrict_apply ((hD'.prodMk hY) (hA.prod hB))]
+  convert he using 1
+  congr 1
+  ext b
+  simp only [mem_inter_iff,mem_preimage,mem_prod,mem_ofPred_eq]
+  tauto
+
+ 
+theorem lintegral_data_restartAt {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D)
+    {F : X×GDriver→ℝ≥0∞} (hF : Measurable F) :
+    (∫⁻b in {b | S b≠∞},F (D b,restartAt S b) ∂W)=
+      ∫⁻b in {b | S b≠∞},∫⁻c,F (D b,c) ∂W ∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
+  rw [←lintegral_map hF (hD'.prodMk (measurable_restartAt hS)),map_data_restartAt hW hS hD,
+    lintegral_prod _ hF.aemeasurable,lintegral_map hF.lintegral_prod_right' hD']
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEMain
+
+lemma actualState_restart {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {s t : ℝ≥0} {z : ℂ} (hz : alive a b (s+t) z) :
+    actualState a b (s+t) z =
+      ((actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))).1,
+        (actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))).2+
+          (actualState a b s z).2) := by
+  have hs := (alive_restart_iff a b z s t).mp hz
+  apply Prod.ext
+  · simp only [actualState,shiftDriver,Complex.ofReal_sub]
+    rw [(loewnerMap_restart a b z s t hz).2]
+    ring
+  · simp only [actualState,deriv_loewnerMap_restart hb hz,norm_mul,
+      Real.log_mul (norm_ne_zero_iff.mpr (deriv_loewnerMap_ne_zero
+        (continuous_shiftDriver hb s) hs.2))
+        (norm_ne_zero_iff.mpr (deriv_loewnerMap_ne_zero hb hs.1))]
+
+lemma physicalRadius_add_log (v : State) (l : ℝ) :
+    physicalRadius (v.1,v.2+l)=Real.exp (-l)*physicalRadius v := by
+  simp only [physicalRadius,Y_apply,J_apply,neg_add,Real.exp_add]
+  ring
+
+lemma logGreen_add_log (a : ℝ) (v : State) (l : ℝ) :
+    logGreen a (v.1,v.2+l)=alpha a*l+logGreen a v := by
+  simp only [logGreen,J_apply,Y_apply,Q,X_apply]
+  ring
+
+lemma physicalGreen_add_log (a : ℝ) (v : State) (l : ℝ) :
+    physicalGreen a (v.1,v.2+l)=Real.exp (alpha a*l)*physicalGreen a v := by
+  rw [physicalGreen,logGreen_add_log,Real.exp_add]
+  rfl
+
+ 
+lemma physicalRadius_restart {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {s t : ℝ≥0} {z : ℂ} (hz : alive a b (s+t) z) :
+    physicalRadius (actualState a b (s+t) z)=
+      ‖deriv (loewnerMap a b s) z‖⁻¹ *
+        physicalRadius (actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))) := by
+  rw [actualState_restart hb hz,physicalRadius_add_log]
+  congr 1
+  rw [actualState,Real.exp_neg,Real.exp_log]
+  exact norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb
+    ((alive_restart_iff a b z s t).mp hz).1)
+
+ 
+lemma physicalGreen_restart {a : ℝ} {b : ℝ≥0→ℝ} (hb : Continuous b)
+    {s t : ℝ≥0} {z : ℂ} (hz : alive a b (s+t) z) :
+    physicalGreen a (actualState a b (s+t) z)=
+      ‖deriv (loewnerMap a b s) z‖^(alpha a) *
+        physicalGreen a (actualState a (shiftDriver b s) t (loewnerMap a b s z-(b s:ℂ))) := by
+  rw [actualState_restart hb hz,physicalGreen_add_log]
+  congr 1
+  rw [Real.rpow_def_of_pos (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb
+    ((alive_restart_iff a b z s t).mp hz).1))]
+  congr 1
+  exact mul_comm _ _
+
+end ExactSLEGreen
+
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen
+
+lemma restart_eq_shiftDriver (s : ℝ≥0) (b : GDriver) :
+    ((restart s b).1 : ℝ≥0→ℝ)=shiftDriver b.1 s := rfl
+
+ 
+
+theorem survivingGreen_restart (a : ℝ) (b : GDriver) (s t : ℝ≥0) {z : ℂ}
+    (hz : alive a b.1 s z) :
+    survivingGreen a (s+t) b z = ‖deriv (loewnerMap a b.1 s) z‖^(alpha a) *
+      survivingGreen a t (restart s b) (loewnerMap a b.1 s z-(b.1 s:ℂ)) := by
+  classical
+  have hc : alive a b.1 (s+t) z ↔
+      alive a (restart s b).1 t (loewnerMap a b.1 s z-(b.1 s:ℂ)) := by
+    rw [alive_restart_iff]
+    exact and_iff_right hz
+  by_cases h : alive a b.1 (s+t) z
+  · unfold survivingGreen
+    rw [ite_eq_left h, ite_eq_left (hc.mp h),physicalGreen_restart b.1.continuous h,
+      restart_eq_shiftDriver]
+    ring
+  · simp only [survivingGreen,ite_eq_right h,ite_eq_right (hc.not.mp h),mul_zero]
+
+ 
+theorem survivingGreen_restart_ratio (a : ℝ) (b : GDriver) (s t : ℝ≥0) {z : ℂ}
+    (hz : alive a b.1 s z) :
+    survivingGreen a (s+t) b z / survivingGreen a s b z =
+      survivingGreen a t (restart s b) (loewnerMap a b.1 s z-(b.1 s:ℂ)) /
+        survivingGreen a 0 (restart s b) (loewnerMap a b.1 s z-(b.1 s:ℂ)) := by
+  have he := survivingGreen_restart a b s 0 hz
+  simp only [add_zero] at he
+  rw [survivingGreen_restart a b s t hz,he]
+  exact mul_div_mul_left _ _ (Real.rpow_pos_of_pos
+    (norm_pos_iff.mpr (deriv_loewnerMap_ne_zero b.1.continuous hz)) _).ne'
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+namespace ExactSLECalculus
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma N_rpow {f : E → ℝ} {v e : E} (r : ℝ)
+    (hf : DifferentiableAt ℝ f v) (hp : 0 < f v) :
+    N (fun w => f w ^ r) e v = r * f v ^ (r-1) * N f e v := by
+  have hh := Real.contDiffAt_rpow_const (n := 1) (p := r) (Or.inl hp.ne')
+  simpa only [Function.comp_def, Real.deriv_rpow_const] using
+    N_comp (e := e) hf hh.differentiableAt_one
+
+lemma L_rpow {f : E → ℝ} {v A e : E} (r : ℝ)
+    (hf : ContDiffAt ℝ 2 f v) (hp : 0 < f v) :
+    L (fun w => f w ^ r) A e v = r * f v ^ (r-1) * L f A e v +
+      (1/2:ℝ) * (r * (r-1) * f v ^ (r-2)) * N f e v ^ 2 := by
+  have hh := Real.contDiffAt_rpow_const (n := 2) (p := r) (Or.inl hp.ne')
+  have he : deriv (deriv (fun x : ℝ => x ^ r)) (f v) =
+      r * (r-1) * f v ^ (r-2) := by
+    rw [Real.deriv_rpow_const', deriv_const_mul_field, Real.deriv_rpow_const]
+    rw [show r - 1 - 1 = r - 2 by ring]
+    ring
+  simpa only [Function.comp_def, Real.deriv_rpow_const, he] using
+    L_comp (A := A) (e := e) hf hh
+
+end ExactSLECalculus
+
+namespace ExactSLEGreen
+open ExactSLECalculus
+
+ 
+
+noncomputable def centeredGenerator (a : ℝ) (f : State → ℝ) (v : State) : ℝ :=
+  L f (physicalDrift a v) noise v + noiseLog a v * N f noise v
+
+lemma green_product_generator (a : ℝ) (ha : a ≠ 0) (f : State → ℝ)
+    (v : State) (hv : 0 < Y v) (hf : ContDiffAt ℝ 2 f v) :
+    L (fun w => physicalGreen a w * f w) (physicalDrift a v) noise v =
+      physicalGreen a v * centeredGenerator a f v := by
+  have hg : ContDiffAt ℝ 2 (physicalGreen a) v :=
+    (contDiffAt_physicalGreen a v hv).of_le (by norm_cast)
+  rw [L_mul hg hf]
+  have hz : L (physicalGreen a) (physicalDrift a v) noise v = 0 :=
+    green_generator_zero a ha v hv
+  rw [hz, show N (physicalGreen a) noise v = physicalGreen a v * noiseLog a v from
+    green_fderiv_noise a v hv]
+  dsimp [centeredGenerator]
+  ring
+
+lemma N_Q (v : State) : N Q noise v = -2 * X v := by
+  rw [N, (hasFDerivAt_Q v).fderiv]
+  simp [noise]
+
+lemma L_Q (a : ℝ) (v : State) :
+    L Q (physicalDrift a v) noise v =
+      1 + 2 * a * (X v ^ 2 - Y v ^ 2) / Q v := by
+  have hn : N Q noise = fun w => -2 * X w := funext N_Q
+  rw [L, hn, N, (X.hasFDerivAt.const_mul (-2 : ℝ)).fderiv,
+    (hasFDerivAt_Q v).fderiv]
+  simp only [_root_.smul_apply, _root_.add_apply, smul_eq_mul,
+    regX_drift, regY_drift]
+  simp only [X_apply, noise, Complex.neg_re, Complex.one_re]
+  ring
+
+lemma N_Y (v : State) : N Y noise v = 0 := by
+  rw [N, Y.hasFDerivAt.fderiv]
+  simp [noise]
+
+lemma L_Y (a : ℝ) (v : State) :
+    L Y (physicalDrift a v) noise v = -a * Y v / Q v := by
+  have hn : N Y noise = fun _ => (0 : ℝ) := funext N_Y
+  rw [L, hn, N, fderiv_const_apply, _root_.zero_apply,
+    Y.hasFDerivAt.fderiv, regY_drift]
+  ring
+
+lemma centeredGenerator_Qpow (a q : ℝ) (v : State) (hv : 0 < Y v) :
+    centeredGenerator a (fun w => Q w ^ (q/2)) v =
+      q * Q v ^ (q/2-2) * ((1/2-a) * Q v + (q/2-2*a) * X v ^ 2) := by
+  have hc : ContDiffAt ℝ 2 Q v := ((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt
+  have hq := Q_pos hv
+  have he : Q v ^ (q/2-1) = Q v ^ (q/2-2) * Q v := by
+    rw [← Real.rpow_add_one hq.ne']
+    congr 1
+    ring
+  rw [centeredGenerator, L_rpow _ hc hq, N_rpow _ (hasFDerivAt_Q v).differentiableAt hq,
+    L_Q, N_Q, he]
+  dsimp only [noiseLog, p]
+  field_simp
+  ring_nf
+  simp only [Q]
+  ring
+
+lemma centeredGenerator_Ypow (a q : ℝ) (v : State) (hv : 0 < Y v) :
+    centeredGenerator a (fun w => Y w ^ q) v = -a * q * Y v ^ q / Q v := by
+  have he : Y v ^ (q-1) * Y v = Y v ^ q := by
+    rw [← Real.rpow_add_one hv.ne']
+    congr 1
+    ring
+  rw [centeredGenerator, L_rpow _ Y.contDiff.contDiffAt hv,
+    N_rpow _ Y.differentiableAt hv, N_Y, L_Y]
+  calc
+    _ = -a * q * (Y v ^ (q-1) * Y v) / Q v := by ring
+    _ = _ := by rw [he]
+
+ 
+noncomputable def lifetimeTest (q : ℝ) (v : State) : ℝ :=
+  Q v ^ (q/2) + 2 * Y v ^ q
+
+lemma lifetimeTest_smooth (q : ℝ) (v : State) (hv : 0 < Y v) :
+    ContDiffAt ℝ (↑(⊤ : ℕ∞)) (lifetimeTest q) v := by
+  exact ((((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt.rpow_const_of_ne
+    (Q_pos hv).ne').add (contDiffAt_const.mul (Y.contDiff.contDiffAt.rpow_const_of_ne hv.ne')))
+
+lemma lifetimeTest_nonneg (q : ℝ) (v : State) (hv : 0 ≤ Y v) :
+    0 ≤ lifetimeTest q v := by
+  unfold lifetimeTest Q
+  positivity
+
+lemma lifetimeTest_coercive (q : ℝ) (v : State) (hv : 0 ≤ Y v) :
+    Q v ^ (q/2) ≤ lifetimeTest q v := by
+  dsimp [lifetimeTest]
+  exact le_add_of_nonneg_right (mul_nonneg (by norm_num) (Real.rpow_nonneg hv _))
+
+lemma centeredGenerator_lifetimeTest (a q : ℝ) (v : State) (hv : 0 < Y v) :
+    centeredGenerator a (lifetimeTest q) v =
+      q * Q v ^ (q/2-2) * ((1/2-a) * Q v + (q/2-2*a) * X v ^ 2) -
+      2 * a * q * Y v ^ q / Q v := by
+  have hQ : ContDiffAt ℝ 2 (fun w => Q w ^ (q/2)) v :=
+    (((X.contDiff.pow 2).add (Y.contDiff.pow 2)).contDiffAt.rpow_const_of_ne (Q_pos hv).ne')
+  have hY : ContDiffAt ℝ 2 (fun w => Y w ^ q) v :=
+    Y.contDiff.contDiffAt.rpow_const_of_ne hv.ne'
+  have split : centeredGenerator a (lifetimeTest q) v =
+      centeredGenerator a (fun w => Q w ^ (q/2)) v +
+      2 * centeredGenerator a (fun w => Y w ^ q) v := by
+    unfold centeredGenerator lifetimeTest
+    rw [L_add hQ (contDiffAt_const.mul hY), L_const_mul hY,
+      N_add (hQ.differentiableAt (by norm_num)) ((contDiffAt_const.mul hY).differentiableAt (by norm_num)),
+      N_const_mul (hY.differentiableAt (by norm_num))]
+    ring
+  rw [split, centeredGenerator_Qpow _ _ _ hv, centeredGenerator_Ypow _ _ _ hv]
+  ring
+
+lemma height_power_comparison (q : ℝ) (hq : q ≤ 2) (v : State) (hv : 0 < Y v) :
+    Y v ^ 2 * Q v ^ (q/2-1) ≤ Y v ^ q := by
+  have hsq : 0 < Y v ^ 2 := sq_pos_of_pos hv
+  have hle : Y v ^ 2 ≤ Q v := by dsimp only [Q]; nlinarith [sq_nonneg (X v)]
+  have h := mul_le_mul_of_nonneg_left
+    (Real.rpow_le_rpow_of_nonpos hsq hle (by linarith : q/2-1 ≤ 0)) hsq.le
+  have he : Y v ^ 2 * (Y v ^ 2) ^ (q/2-1) = Y v ^ q := by
+    rw [← Real.rpow_two (Y v), ← Real.rpow_mul hv.le, ← Real.rpow_add hv]
+    congr 1
+    ring
+  exact h.trans_eq he
+
+ 
+
+lemma centeredGenerator_lifetimeTest_le (a q : ℝ) (ha : 0 < a)
+    (hq : 0 < q) (hq2 : q ≤ 2) (v : State) (hv : 0 < Y v) :
+    centeredGenerator a (lifetimeTest q) v ≤
+      -(q * (6*a-1-q)/2) * Q v ^ (q/2-1) := by
+  have hQ := Q_pos hv
+  have hp := height_power_comparison q hq2 v hv
+  have he : Q v ^ (q/2-1) = Q v ^ (q/2-2) * Q v := by
+    rw [← Real.rpow_add_one hQ.ne']
+    congr 1
+    ring
+  have hdiv : Y v ^ 2 * Q v ^ (q/2-2) ≤ Y v ^ q / Q v := by
+    apply (le_div_iff₀ hQ).mpr
+    nlinarith only [hp.trans_eq (by rfl), show Y v ^ 2 * Q v ^ (q/2-1) =
+      (Y v ^ 2 * Q v ^ (q/2-2)) * Q v by rw [he]; ring]
+  rw [centeredGenerator_lifetimeTest _ _ _ hv, he]
+  have hs : X v ^ 2 ≤ Q v := by dsimp only [Q]; nlinarith [sq_nonneg (Y v)]
+  have hn := mul_le_mul_of_nonneg_left hdiv (show 0 ≤ 2*a*q by positivity)
+  have hpos : 0 ≤ q * Q v ^ (q/2-2) := by positivity
+  have hx := mul_le_mul_of_nonneg_left hs (show 0 ≤ q/2 * (q * Q v ^ (q/2-2)) by positivity)
+  dsimp only [Q] at hn hx ⊢
+  simp only [div_eq_mul_inv] at hn hx ⊢
+  nlinarith only [hn, hx]
+
+noncomputable def lifetimeExponent (a : ℝ) : ℝ := min 1 (3*a-1/2)
+noncomputable def lifetimeDrift (a : ℝ) : ℝ :=
+  lifetimeExponent a * (6*a-1-lifetimeExponent a)/2
+
+lemma lifetimeExponent_pos {a : ℝ} (ha : 1/4 < a) : 0 < lifetimeExponent a := by
+  dsimp [lifetimeExponent]
+  exact lt_min (by norm_num) (by linarith)
+
+lemma lifetimeExponent_le_one (a : ℝ) : lifetimeExponent a ≤ 1 := min_le_left _ _
+
+lemma lifetimeDrift_pos {a : ℝ} (ha : 1/4 < a) : 0 < lifetimeDrift a := by
+  have hq := lifetimeExponent_pos ha
+  have hq' : lifetimeExponent a ≤ 3*a-1/2 := min_le_right _ _
+  exact div_pos (mul_pos hq (by linarith)) (by norm_num)
+
+ 
+lemma green_lifetime_generator_le (a : ℝ) (ha : 1/4 < a)
+    (v : State) (hv : 0 < Y v) :
+    L (fun w => physicalGreen a w * lifetimeTest (lifetimeExponent a) w)
+      (physicalDrift a v) noise v ≤
+    -lifetimeDrift a * physicalGreen a v * Q v ^ (lifetimeExponent a/2-1) := by
+  rw [green_product_generator a (by linarith) _ _ hv
+    ((lifetimeTest_smooth _ _ hv).of_le (by norm_cast))]
+  have h := mul_le_mul_of_nonneg_left
+    (centeredGenerator_lifetimeTest_le a (lifetimeExponent a) (by linarith)
+      (lifetimeExponent_pos ha) ((lifetimeExponent_le_one a).trans (by norm_num)) v hv)
+    (Real.exp_pos (logGreen a v)).le
+  dsimp only [lifetimeDrift, physicalGreen] at *
+  convert h using 1
+  first | rfl | ring
+
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RawRestartTests.lean b/lean/OAI/Probability/SLEGauge/RawRestartTests.lean
index a5abf468947bec95619227629417ebca35bfaba5..9929f3d2a6d41cd5f57e0563c05a8a637c2b739f 100644
--- a/lean/OAI/Probability/SLEGauge/RawRestartTests.lean
+++ b/lean/OAI/Probability/SLEGauge/RawRestartTests.lean
@@ -1,604 +1,604 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.CommonStoppedMass
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEGenerator ExactSLEMain
-
-theorem map_data_restartAt_on {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) :
-    (W.restrict (A∩{b | S b≠∞})).map (fun b => (D b,restartAt S b))=
-      ((W.restrict (A∩{b | S b≠∞})).map D).prod W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
-  have hY := measurable_restartAt hS
-  apply Measure.ext_prod
-  intro B C hB hC
-  rw [Measure.map_apply (hD'.prodMk hY) (hB.prod hC),Measure.prod_prod,
-    Measure.map_apply hD' hB,Measure.restrict_apply (hD' hB)]
-  have he := congrArg (fun μ : Measure GDriver => μ C)
-    (map_restartAt_restrict hW hS ((hD hB).inter hA))
-  rw [Measure.map_apply hY hC,Measure.restrict_apply (hY hC),
-    Measure.smul_apply,smul_eq_mul] at he
-  rw [Measure.restrict_apply ((hD'.prodMk hY) (hB.prod hC))]
-  convert he using 1 <;> congr 1
-  ext b
-  simp only [mem_inter_iff,mem_preimage,mem_prod,mem_ofPred_eq]
-  tauto
-  simp only [inter_assoc]
-
- 
-
-theorem lintegral_data_restartAt_on {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A)
-    {F : X×GDriver→ℝ≥0∞} (hF : Measurable F) :
-    (∫⁻b in A∩{b | S b≠∞},F (D b,restartAt S b) ∂W)=
-      ∫⁻b in A∩{b | S b≠∞},∫⁻c,F (D b,c) ∂W ∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
-  rw [←lintegral_map hF (hD'.prodMk (measurable_restartAt hS)),
-    map_data_restartAt_on hW hS hD hA,lintegral_prod _ hF.aemeasurable,
-    lintegral_map hF.lintegral_prod_right' hD']
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGenerator
-
-lemma kernel_with_data_measurable {X : Type*} [MeasurableSpace X]
-    (K : Kernel GDriver X) [IsSFiniteKernel K]
-    {F : (ℝ≥0×GDriver)×X→ℝ≥0∞} (hF : Measurable F) :
-    Measurable (fun p : (ℝ≥0×GDriver)×GDriver => ∫⁻z,F (p.1,z)∂K p.2) := by
-  let L : Kernel ((ℝ≥0×GDriver)×GDriver) X := K.comap Prod.snd measurable_snd
-  exact (hF.comp (measurable_fst.fst.prodMk measurable_snd)).lintegral_kernel_prod_right' (κ:=L)
-
- 
-
-lemma lintegral_kernel_restartAt_on_shift {X : Type*} [MeasurableSpace X]
-    (K : Kernel GDriver X) [IsSFiniteKernel K] {W : Measure GDriver}
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {t : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
-    {F : (ℝ≥0×GDriver)×X→ℝ≥0∞} (hF : Measurable F) :
-    (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂K (restartAt S b)∂W)=
-      ∫⁻b in {b | S b≠∞},∫⁻c,(concatenatedEvent S A).indicator
-        (fun p : (ℝ≥0×GDriver)×GDriver => ∫⁻z,F (p.1,z)∂K p.2)
-        (stoppedData S b,c)∂W∂W := by
-  have hAe : MeasurableSet A := (shiftStop_isStoppingTime hS t).measurableSpace_le A hA
-  have hfinite : MeasurableSet {b | S b≠∞} :=
-    (hS.measurable' (measurableSet_singleton ∞)).compl
-  have htest := (kernel_with_data_measurable K hF).indicator
-    (concatenatedEvent_measurable hS hAe)
-  have hi := lintegral_data_restartAt_on hW hS (stoppedData_raw_measurable hS)
-    MeasurableSet.univ htest
-  simp only [univ_inter] at hi
-  rw [←hi,←Measure.restrict_restrict hAe,←lintegral_indicator hAe]
-  apply setLIntegral_congr_fun hfinite
-  intro b hb
-  have he : (stoppedData S b,restartAt S b)∈concatenatedEvent S A ↔ b∈A :=
-    concatenatedEvent_stoppedData hb
-  by_cases h : b∈A
-  · simp only [Set.indicator_of_mem h,Set.indicator_of_mem (he.mpr h)]
-  · simp only [Set.indicator_of_notMem h,Set.indicator_of_notMem (mt he.mp h)]
-
- 
-
-theorem restartAt_conditional_kernel {X : Type*} [MeasurableSpace X]
-    (K L G : Kernel GDriver X) [IsSFiniteKernel K] [IsSFiniteKernel L] [IsSFiniteKernel G]
-    {W : Measure GDriver} (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {t : ℝ≥0}
-    (hcond : ∀H : GDriver×X→ℝ≥0∞,
-      @Measurable (GDriver×X) ℝ≥0∞ ((rawFiltration t).prod inferInstance) inferInstance H →
-      (∫⁻c,∫⁻z,H (c,z)∂K c∂W)=(∫⁻c,∫⁻z,H (c,z)∂L c∂W)+∫⁻c,∫⁻z,H (c,z)∂G c∂W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
-    {F : (ℝ≥0×GDriver)×X→ℝ≥0∞} (hF : Measurable F) :
-    (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂K (restartAt S b)∂W)=
-      (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂L (restartAt S b)∂W)+
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂G (restartAt S b)∂W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let E := concatenatedEvent S A
-  let T (M : Kernel GDriver X) (p : ℝ≥0×GDriver) (c : GDriver) :=
-    E.indicator (fun q : (ℝ≥0×GDriver)×GDriver => ∫⁻z,F (q.1,z)∂M q.2) (p,c)
-  have hsection (p : ℝ≥0×GDriver) : MeasurableSet[rawFiltration t] {c | (p,c)∈E} :=
-    (concatenatedEvent_raw_joint hS hA).preimage (measurable_const.prodMk measurable_id)
-  have hFsection (p : ℝ≥0×GDriver) : Measurable (fun z : X => F (p,z)) :=
-    hF.comp (measurable_const.prodMk measurable_id)
-  have hi (p : ℝ≥0×GDriver) : (∫⁻c,T K p c∂W)=(∫⁻c,T L p c∂W)+∫⁻c,T G p c∂W := by
-    let H : GDriver×X→ℝ≥0∞ := fun q =>
-      ({c | (p,c)∈E}).indicator (fun _ => F (p,q.2)) q.1
-    have hH : @Measurable (GDriver×X) ℝ≥0∞ ((rawFiltration t).prod inferInstance)
-        inferInstance H := by
-      exact ((hFsection p).comp measurable_snd).indicator
-        ((hsection p).preimage measurable_fst)
-    have he (M : Kernel GDriver X) (c : GDriver) : (∫⁻z,H (c,z)∂M c)=T M p c := by
-      by_cases hc : (p,c)∈E
-      · simp only [H,T,Set.indicator_apply,Set.mem_ofPred_eq,hc,ite_true]
-      · simp only [H,T,Set.indicator_apply,Set.mem_ofPred_eq,hc,ite_false,lintegral_zero]
-    simpa only [he] using hcond H hH
-  rw [lintegral_kernel_restartAt_on_shift K hW hS hA hF,
-    lintegral_kernel_restartAt_on_shift L hW hS hA hF,
-    lintegral_kernel_restartAt_on_shift G hW hS hA hF]
-  change (∫⁻b in {b | S b≠∞},∫⁻c,T K (stoppedData S b) c∂W∂W)=
-    (∫⁻b in {b | S b≠∞},∫⁻c,T L (stoppedData S b) c∂W∂W)+
-    ∫⁻b in {b | S b≠∞},∫⁻c,T G (stoppedData S b) c∂W∂W
-  simp_rw [hi]
-  apply lintegral_add_left
-  have hM := ((kernel_with_data_measurable L hF).indicator
-    (concatenatedEvent_measurable hS ((shiftStop_isStoppingTime hS t).measurableSpace_le A hA)))
-  exact hM.lintegral_prod_right'.comp
-    ((stoppedData_raw_measurable hS).mono hS.measurableSpace_le le_rfl)
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-
-lemma survivingGreen_zero (a : ℝ) (ha : 0<a) (b : GDriver) (z : UpperPlane) :
-    survivingGreen a 0 b z.1=initialGreen a z := by
-  have hi : alive a b.1 0 z.1 := radialStop_actual a ha z.1 z.2 (z.1.im/2)
-    (by have := z.2; positivity) (by have := z.2; linarith) 0 b 0 bot_le
-  simp only [survivingGreen,ite_eq_left hi,actualState_zero ha b z.2,initialGreen]
-
-lemma survivingKernel_zero (a : ℝ) (ha : 1/4<a) (b : GDriver) :
-    survivingKernel a ha 0 b=initialGreenMeasure a := by
-  rw [survivingKernel_apply]
-  simp only [survivingGreen_zero a (lt_trans (by norm_num) ha),initialGreenMeasure]
-
-lemma survivingKernel_restrict_alive (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
-    (survivingKernel a ha s b).restrict {z : UpperPlane | alive a b.1 s z.1}=
-      survivingKernel a ha s b := by
-  have hD : MeasurableSet {z : UpperPlane | alive a b.1 s z.1} :=
-    (measurableSet_alive_joint a ha).preimage
-      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
-  rw [survivingKernel_apply,restrict_withDensity hD,←withDensity_indicator hD]
-  congr 1
-  funext z
-  by_cases hz : alive a b.1 s z.1
-  · simp only [Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_left hz]
-  · simp only [Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_right hz,survivingGreen,ENNReal.ofReal_zero]
-
-lemma restartPush_initialGreen (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
-    restartPush a (lt_trans (by norm_num) ha) b s (initialGreenMeasure a)=
-      survivingKernel a ha s b := by
-  have h := restartPush_survivingKernel a ha b s 0
-  rwa [survivingKernel_zero,add_zero,survivingKernel_restrict_alive] at h
-
- 
-
-lemma restartPush_test_joint_measurable (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    Measurable (fun p : (ℝ≥0×GDriver)×GDriver =>
-      ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) p.1.2 p.1.1 (μ p.2)) := by
-  simp_rw [restartPush_lintegral a ha _ _ _ hf]
-  let K : Kernel ((ℝ≥0×GDriver)×GDriver) UpperPlane := μ.comap Prod.snd measurable_snd
-  have hin : Measurable (fun p : ((ℝ≥0×GDriver)×GDriver)×UpperPlane =>
-      (p.1.1.1,(p.1.1.2,p.2))) :=
-    (measurable_fst.fst.fst).prodMk ((measurable_fst.fst.snd).prodMk measurable_snd)
-  exact ((restartWeight_joint_measurable a ha).comp hin |>.mul
-    (hf.comp ((inverseUpper_joint_measurable a ha).comp hin))).lintegral_kernel_prod_right' (κ:=K)
-
-lemma restartPush_mean (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (μ : Kernel GDriver UpperPlane) (W : Measure GDriver)
-    (hμ : μ ∘ₘ W=initialGreenMeasure a) {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    (∫⁻c,∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b s (μ c)∂W)=
-      ∫⁻z,f z∂survivingKernel a ha s b := by
-  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
-  have hg : Measurable (fun z => restartWeight a b s z*
-      f (inverseUpper a (lt_trans (by norm_num) ha) b s z)) :=
-    (restartWeight_measurable a ha b s).mul (hf.comp hJ)
-  simp_rw [restartPush_lintegral a ha b s _ hf]
-  rw [←Measure.lintegral_bind (m:=W) μ.aemeasurable hg.aemeasurable]
-  change (∫⁻z,restartWeight a b s z*f (inverseUpper a (lt_trans (by norm_num) ha) b s z)
-    ∂μ ∘ₘ W)=_
-  rw [hμ,←restartPush_lintegral a ha b s _ hf,restartPush_initialGreen]
-
- 
-
-theorem restartPush_stopped_terminal_mean (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (hμ : μ ∘ₘ W=initialGreenMeasure a)
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A)
-    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
-      b (S b).toNNReal (μ (restartAt S b))∂W)=
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂stoppedSurvivingKernel a ha S hS b∂W := by
-  let F : ((ℝ≥0×GDriver)×GDriver)→ℝ≥0∞ := fun p =>
-    ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) p.1.2 p.1.1 (μ p.2)
-  have h := lintegral_data_restartAt_on hW hS (stoppedData_raw_measurable hS) hA
-    (restartPush_test_joint_measurable a ha μ hf)
-  have he (b c : GDriver) : F (stoppedData S b,c)=
-      ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ c) := by
-    simp only [F,restartPush,inverseUpper_stoppedData,
-      restartWeight_stoppedData a (lt_trans (by norm_num) ha)]
-  change (∫⁻b in A∩{b | S b≠∞},F (stoppedData S b,restartAt S b)∂W)=
-    ∫⁻b in A∩{b | S b≠∞},∫⁻c,F (stoppedData S b,c)∂W∂W at h
-  simp_rw [he] at h
-  rw [h]
-  apply setLIntegral_congr_fun ((hS.measurableSpace_le A hA).inter
-    (hS.measurable' (measurableSet_singleton ∞)).compl)
-  intro b hb
-  dsimp only
-  rw [restartPush_mean a ha _ _ μ W hμ hf,stoppedSurvivingKernel_apply]
-  have hbfin : S b≠∞ := hb.2
-  simp only [stoppedSurvivingGreen,ite_eq_right hbfin,survivingKernel_apply]
-
-lemma restartPush_measure_joint_measurable (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] :
-    Measurable (fun p : (ℝ≥0×GDriver)×GDriver =>
-      restartPush a (lt_trans (by norm_num) ha) p.1.2 p.1.1 (μ p.2)) := by
-  apply Measure.measurable_of_measurable_coe
-  intro A hA
-  have h := restartPush_test_joint_measurable a ha μ
-    (measurable_const.indicator hA : Measurable (A.indicator (fun _ => (1:ℝ≥0∞))))
-  simpa only [lintegral_indicator hA,setLIntegral_one] using h
-
- 
-
-def restartCandidateKernel (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ]
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Kernel GDriver UpperPlane where
-  toFun b := if S b=∞ then 0 else
-    restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ (restartAt S b))
-  measurable' := by
-    have hin : Measurable (fun b => (((S b).toNNReal,b),restartAt S b)) :=
-      ((ENNReal.measurable_toNNReal.comp hS.measurable').prodMk measurable_id).prodMk
-        (measurable_restartAt hS)
-    have h := (restartPush_measure_joint_measurable a ha μ).comp hin
-    change Measurable (fun b => restartPush a (lt_trans (by norm_num) ha)
-      b (S b).toNNReal (μ (restartAt S b))) at h
-    exact Measurable.ite (hS.measurable' (measurableSet_singleton ∞)) measurable_const h
-
-lemma restartCandidateKernel_apply (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ]
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) (b : GDriver) :
-    restartCandidateKernel a ha μ S hS b=if S b=∞ then 0 else
-      restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ (restartAt S b)) := rfl
-
- 
-
-theorem restartCandidateKernel_average_on (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (hμ : μ ∘ₘ W=initialGreenMeasure a)
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) :
-    restartCandidateKernel a ha μ S hS ∘ₘ (W.restrict A)=
-      stoppedSurvivingKernel a ha S hS ∘ₘ (W.restrict A) := by
-  have hfinite : MeasurableSet {b | S b≠∞} :=
-    (hS.measurable' (measurableSet_singleton ∞)).compl
-  ext B hB
-  rw [Measure.bind_apply hB (Kernel.aemeasurable _),Measure.bind_apply hB (Kernel.aemeasurable _)]
-  have he := restartPush_stopped_terminal_mean a ha μ W hW hμ S hS hA
-    (measurable_const.indicator hB : Measurable (B.indicator (fun _ => (1:ℝ≥0∞))))
-  simp only [lintegral_indicator hB,setLIntegral_one] at he
-  calc
-    _ = ∫⁻b in A∩{b | S b≠∞},
-        restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ (restartAt S b)) B ∂W := by
-      rw [inter_comm A {b | S b≠∞},←Measure.restrict_restrict hfinite,←lintegral_indicator hfinite]
-      apply lintegral_congr
-      intro b
-      by_cases hb : S b=∞ <;> simp [restartCandidateKernel_apply,hb]
-    _ = _ := he
-    _ = ∫⁻b in A,stoppedSurvivingKernel a ha S hS b B∂W := by
-      rw [inter_comm A {b | S b≠∞},←Measure.restrict_restrict hfinite,←lintegral_indicator hfinite]
-      apply lintegral_congr
-      intro b
-      by_cases hb : S b=∞
-      · simp [hb,stoppedSurvivingKernel_apply,stoppedSurvivingGreen]
-      · simp [hb]
-
- 
-
-theorem restartCandidateKernel_average_le_initial (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (hμ : μ ∘ₘ W=initialGreenMeasure a)
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    restartCandidateKernel a ha μ S hS ∘ₘ W ≤ initialGreenMeasure a := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have h := restartCandidateKernel_average_on a ha μ W hW hμ S hS MeasurableSet.univ
-  rw [Measure.restrict_univ] at h
-  rw [h]
-  exact stoppedSurvivingKernel_average_le_initial a ha W hW S hS
-
- 
-
-theorem restartCandidateKernel_compact_lintegral_lt_top (a : ℝ) (ha : 1/4<a)
-    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
-    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (hμ : μ ∘ₘ W=initialGreenMeasure a)
-    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    {K : Set UpperPlane} (hK : IsCompact K) :
-    (∫⁻b,restartCandidateKernel a ha μ S hS b K∂W)<∞ := by
-  rw [←Measure.bind_apply hK.measurableSet (Kernel.aemeasurable _)]
-  exact ((restartCandidateKernel_average_le_initial a ha μ W hW hμ S hS) K).trans_lt
-    hK.measure_lt_top
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-
-lemma restartPullback_joint_measurable (a : ℝ) (ha : 1/4<a)
-    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane =>
-      restartWeight a p.1.2 p.1.1 p.2*
-        f (inverseUpper a (lt_trans (by norm_num) ha) p.1.2 p.1.1 p.2)) := by
-  have hin : Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane => (p.1.1,(p.1.2,p.2))) :=
-    measurable_fst.fst.prodMk (measurable_fst.snd.prodMk measurable_snd)
-  have hweight : Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane =>
-      restartWeight a p.1.2 p.1.1 p.2) := by
-    simpa only [Function.comp_def] using (restartWeight_joint_measurable a ha).comp hin
-  have hinverse : Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane =>
-      inverseUpper a (lt_trans (by norm_num) ha) p.1.2 p.1.1 p.2) := by
-    simpa only [Function.comp_def] using (inverseUpper_joint_measurable a ha).comp hin
-  exact hweight.mul (hf.comp hinverse)
-
- 
-
-theorem restartPush_conditional_kernel (a : ℝ) (ha : 1/4<a)
-    (K L : Kernel GDriver UpperPlane) [IsSFiniteKernel K] [IsSFiniteKernel L]
-    {W : Measure GDriver} (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    {t : ℝ≥0}
-    (hcond : ∀H : GDriver×UpperPlane→ℝ≥0∞,
-      Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-      (∫⁻c,∫⁻z,H (c,z)∂K c∂W)=(∫⁻c,∫⁻z,H (c,z)∂L c∂W)+
-        ∫⁻c,∫⁻z,H (c,z)∂survivingKernel a ha t c∂W)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {A : Set GDriver} (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
-    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
-        b (S b).toNNReal (K (restartAt S b))∂W)=
-      (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
-        b (S b).toNNReal (L (restartAt S b))∂W)+
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
-        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
-  let F : (ℝ≥0×GDriver)×UpperPlane→ℝ≥0∞ := fun p =>
-    restartWeight a p.1.2 p.1.1 p.2*
-      f (inverseUpper a (lt_trans (by norm_num) ha) p.1.2 p.1.1 p.2)
-  have h := restartAt_conditional_kernel K L (survivingKernel a ha t) hW hcond hS hA
-    (restartPullback_joint_measurable a ha hf)
-  have he (M : Kernel GDriver UpperPlane) (b : GDriver) :
-      (∫⁻z,F (stoppedData S b,z)∂M (restartAt S b))=
-      ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (M (restartAt S b)) := by
-    rw [restartPush_lintegral a ha _ _ _ hf]
-    simp only [F,inverseUpper_stoppedData,restartWeight_stoppedData a (lt_trans (by norm_num) ha)]
-  change (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂K (restartAt S b)∂W)=
-    (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂L (restartAt S b)∂W)+
-    ∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂survivingKernel a ha t (restartAt S b)∂W at h
-  simp_rw [he] at h
-  simpa only [restartAt,restartPush_survivingKernel] using h
-
- 
-
-theorem recoveredMass_restarted_potential (a : ℝ) (ha : 1/4<a)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass a ha s c hc v b)))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
-    {t : ℝ≥0} {A : Set GDriver}
-    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
-    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
-    let μ := recoveredMass a ha s c hc
-    (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
-        b (S b).toNNReal (μ ∞ (restartAt S b))∂W)=
-      (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
-        b (S b).toNNReal (μ t (restartAt S b))∂W)+
-      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
-        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
-  intro μ
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let K : ℝ≥0∞→Kernel GDriver UpperPlane := fun v =>
-    ⟨μ v,recoveredMass_measurable a ha s c hc v⟩
-  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (K v b) := radonLimit_finite _
-  have (v : ℝ≥0∞) : IsSFiniteKernel (K v) := isSFiniteKernel_of_locally_finite (K v)
-  have hcond (H : GDriver×UpperPlane→ℝ≥0∞)
-      (hH : Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H) :
-      (∫⁻b,∫⁻z,H (b,z)∂K ∞ b∂W)=(∫⁻b,∫⁻z,H (b,z)∂K t b∂W)+
-        ∫⁻b,∫⁻z,H (b,z)∂survivingKernel a ha t b∂W := by
-    have hi := (recoveredMass_all_stopping a ha W hW s c hc hs hsum hconv
-      _ (isStoppingTime_const rawFiltration t)).2.2.2.2 H
-    rw [IsStoppingTime.measurableSpace_const] at hi
-    have hn : Measurable H := hH.mono (prod_sigma_le (rawFiltration.le t) _) le_rfl
-    have he (b : GDriver) : (∫⁻z,H (b,z)∂survivingKernel a ha t b)=
-        ∫⁻z,H (b,z)*ENNReal.ofReal (survivingGreen a t b z.1)∂upperArea := by
-      rw [survivingKernel_apply]
-      exact (lintegral_withDensity_eq_lintegral_mul upperArea
-        (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal
-        (hn.comp (measurable_const.prodMk measurable_id))).trans
-        (lintegral_congr fun _ => mul_comm _ _)
-    simp_rw [he]
-    simpa only [K,Kernel.coe_mk,μ,stoppedSurvivingGreen, ENNReal.some_eq_coe', ENNReal.coe_ne_top, ↓reduceIte,
-      ENNReal.toNNReal_coe] using hi hH
-  exact restartPush_conditional_kernel a ha (K ∞) (K t) hW hcond hS hA hf
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
-
- 
-def afterStopFiltration {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    Filtration ℝ≥0 (inferInstance : MeasurableSpace GDriver) where
-  seq t := (shiftStop_isStoppingTime hS t).measurableSpace
-  mono' := fun t u htu => (shiftStop_isStoppingTime hS t).measurableSpace_mono
-    (shiftStop_isStoppingTime hS u) (fun b => by
-      change S b+(t:ℝ≥0∞) ≤ S b+(u:ℝ≥0∞)
-      exact add_le_add le_rfl (ENNReal.coe_le_coe.mpr htu))
-  le' t := (shiftStop_isStoppingTime hS t).measurableSpace_le
-
-lemma raw_function_congr {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
-    {t : ℝ≥0} {f : GDriver→X} (hf : Measurable[rawFiltration t] f)
-    {b c : GDriver} (h : ∀ u≤t,b.1 u=c.1 u) : f b=f c := by
-  have he := (raw_event_congr (hf (measurableSet_singleton (f b))) h).mp
-    (show b∈f ⁻¹' {f b} from rfl)
-  exact he.symm
-
-lemma raw_measure_congr {X : Type*} [MeasurableSpace X] {t : ℝ≥0}
-    {K : GDriver→Measure X} (hK : Measurable[rawFiltration t] K)
-    {b c : GDriver} (h : ∀ u≤t,b.1 u=c.1 u) : K b=K c := by
-  ext A hA
-  exact raw_function_congr ((Measure.measurable_coe hA).comp hK) h
-
- 
-
-def freshPast (S : GDriver→ℝ≥0∞) (t : ℝ≥0) (b : GDriver) : GDriver :=
-  pastDriver t (restart (S b).toNNReal
-    (pastDriver (shiftStop S t b).toNNReal b))
-
-lemma freshPast_raw_measurable {S : GDriver→ℝ≥0∞}
-    (hS : IsStoppingTime rawFiltration S) (t : ℝ≥0) :
-    Measurable[(afterStopFiltration hS) t] (freshPast S t) := by
-  have htime : Measurable[(afterStopFiltration hS) t]
-      (fun b => (S b).toNNReal) :=
-    (ENNReal.measurable_toNNReal.comp hS.measurable).mono (measurableSpace_le_shiftStop hS t) le_rfl
-  have hfrozen := pastDriver_stopping_measurable (shiftStop S t) (shiftStop_isStoppingTime hS t)
-  exact ((pastDriver_raw_measurable t).mono (rawFiltration.le t) le_rfl).comp
-    (continuous_restart_joint.measurable.comp (htime.prodMk hfrozen))
-
-lemma freshPast_eq (S : GDriver→ℝ≥0∞) (t : ℝ≥0) {b : GDriver} (hb : S b≠∞) :
-    freshPast S t b=pastDriver t (restartAt S b) := by
-  have he : (shiftStop S t b).toNNReal=(S b).toNNReal+t := by
-    rw [shiftStop,ENNReal.toNNReal_add hb ENNReal.coe_ne_top,ENNReal.toNNReal_coe]
-  apply Subtype.ext
-  ext u
-  change (restart (S b).toNNReal (pastDriver (shiftStop S t b).toNNReal b)).1 (min u t)=
-    (restart (S b).toNNReal b).1 (min u t)
-  simp only [restart_apply,he]
-  rw [pastDriver_eq _ b _ (add_le_add (le_refl (S b).toNNReal) (min_le_right u t)),
-    pastDriver_eq _ b _ (le_self_add)]
-
- 
-
-lemma restartCandidateKernel_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (K : Kernel GDriver UpperPlane) [IsSFiniteKernel K] {t : ℝ≥0}
-    (hK : Measurable[rawFiltration t] K)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    Measurable[(afterStopFiltration hS) t] (restartCandidateKernel a ha K S hS) := by
-  classical
-  have hD := (stoppedData_raw_measurable hS).mono (measurableSpace_le_shiftStop hS t) le_rfl
-  have hM := (restartPush_measure_joint_measurable a ha K).comp
-    (hD.prodMk (freshPast_raw_measurable hS t))
-  have htop : MeasurableSet[(afterStopFiltration hS) t] {b | S b=∞} :=
-    (measurableSpace_le_shiftStop hS t) _ (hS.measurable (measurableSet_singleton ∞))
-  have hh : Measurable[(afterStopFiltration hS) t] (fun b =>
-      if S b=∞ then (0:Measure UpperPlane) else restartPush a (lt_trans (by norm_num) ha)
-        (stoppedData S b).2 (stoppedData S b).1 (K (freshPast S t b))) :=
-    Measurable.ite htop measurable_const hM
-  convert hh using 1
-  funext b
-  change (if S b=∞ then 0 else restartPush a _ b (S b).toNNReal (K (restartAt S b)))=
-    if S b=∞ then 0 else restartPush a _ (stoppedData S b).2 (stoppedData S b).1
-      (K (freshPast S t b))
-  by_cases hb : S b=∞
-  · simp only [hb,ite_true]
-  · simp only [ite_eq_right hb,freshPast_eq S t hb]
-    rw [raw_measure_congr hK (pastDriver_eq t (restartAt S b))]
-    simp only [restartPush,inverseUpper_stoppedData,
-      restartWeight_stoppedData a (lt_trans (by norm_num) ha)]
-
- 
-def recoveredRestart (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (S : GDriver→ℝ≥0∞) (v : ℝ≥0∞) (b : GDriver) : Measure UpperPlane :=
-  if S b=∞ then 0 else restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal
-    (recoveredMass a ha s c hc v (restartAt S b))
-
-lemma recoveredRestart_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (v : ℝ≥0∞) :
-    Measurable (recoveredRestart a ha s c hc S v) := by
-  let K : Kernel GDriver UpperPlane :=
-    ⟨recoveredMass a ha s c hc v,recoveredMass_measurable a ha s c hc v⟩
-  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
-  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
-  exact (restartCandidateKernel a ha K S hS).measurable
-
-lemma recoveredRestart_compact_stronglyAdapted (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (f : C_c(UpperPlane,ℝ)) :
-    StronglyAdapted (afterStopFiltration hS)
-      (fun t : ℝ≥0 => fun b => ∫z,f z∂recoveredRestart a ha s c hc S t b) := by
-  intro t
-  let K : Kernel GDriver UpperPlane :=
-    ⟨recoveredMass a ha s c hc t,recoveredMass_measurable a ha s c hc t⟩
-  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
-  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
-  have hh := restartCandidateKernel_raw_measurable a ha K
-    (recoveredMass_raw_measurable a ha s c hc t) hS
-  let : MeasurableSpace GDriver := (afterStopFiltration hS) t
-  let L : @Kernel GDriver UpperPlane ((afterStopFiltration hS) t) inferInstance :=
-    ⟨recoveredRestart a ha s c hc S t,hh⟩
-  exact f.continuous.stronglyMeasurable.integral_kernel (κ:=L)
-
-end SLEExactGauge.LoewnerRestart
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.CommonStoppedMass
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEGenerator ExactSLEMain
+
+theorem map_data_restartAt_on {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) :
+    (W.restrict (A∩{b | S b≠∞})).map (fun b => (D b,restartAt S b))=
+      ((W.restrict (A∩{b | S b≠∞})).map D).prod W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
+  have hY := measurable_restartAt hS
+  apply Measure.ext_prod
+  intro B C hB hC
+  rw [Measure.map_apply (hD'.prodMk hY) (hB.prod hC),Measure.prod_prod,
+    Measure.map_apply hD' hB,Measure.restrict_apply (hD' hB)]
+  have he := congrArg (fun μ : Measure GDriver => μ C)
+    (map_restartAt_restrict hW hS ((hD hB).inter hA))
+  rw [Measure.map_apply hY hC,Measure.restrict_apply (hY hC),
+    Measure.smul_apply,smul_eq_mul] at he
+  rw [Measure.restrict_apply ((hD'.prodMk hY) (hB.prod hC))]
+  convert he using 1 <;> congr 1
+  ext b
+  simp only [mem_inter_iff,mem_preimage,mem_prod,mem_ofPred_eq]
+  tauto
+  simp only [inter_assoc]
+
+ 
+
+theorem lintegral_data_restartAt_on {X : Type*} [MeasurableSpace X] {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {D : GDriver→X} (hD : Measurable[hS.measurableSpace] D)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A)
+    {F : X×GDriver→ℝ≥0∞} (hF : Measurable F) :
+    (∫⁻b in A∩{b | S b≠∞},F (D b,restartAt S b) ∂W)=
+      ∫⁻b in A∩{b | S b≠∞},∫⁻c,F (D b,c) ∂W ∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hD' : Measurable D := hD.mono hS.measurableSpace_le le_rfl
+  rw [←lintegral_map hF (hD'.prodMk (measurable_restartAt hS)),
+    map_data_restartAt_on hW hS hD hA,lintegral_prod _ hF.aemeasurable,
+    lintegral_map hF.lintegral_prod_right' hD']
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGenerator
+
+lemma kernel_with_data_measurable {X : Type*} [MeasurableSpace X]
+    (K : Kernel GDriver X) [IsSFiniteKernel K]
+    {F : (ℝ≥0×GDriver)×X→ℝ≥0∞} (hF : Measurable F) :
+    Measurable (fun p : (ℝ≥0×GDriver)×GDriver => ∫⁻z,F (p.1,z)∂K p.2) := by
+  let L : Kernel ((ℝ≥0×GDriver)×GDriver) X := K.comap Prod.snd measurable_snd
+  exact (hF.comp (measurable_fst.fst.prodMk measurable_snd)).lintegral_kernel_prod_right' (κ:=L)
+
+ 
+
+lemma lintegral_kernel_restartAt_on_shift {X : Type*} [MeasurableSpace X]
+    (K : Kernel GDriver X) [IsSFiniteKernel K] {W : Measure GDriver}
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {t : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
+    {F : (ℝ≥0×GDriver)×X→ℝ≥0∞} (hF : Measurable F) :
+    (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂K (restartAt S b)∂W)=
+      ∫⁻b in {b | S b≠∞},∫⁻c,(concatenatedEvent S A).indicator
+        (fun p : (ℝ≥0×GDriver)×GDriver => ∫⁻z,F (p.1,z)∂K p.2)
+        (stoppedData S b,c)∂W∂W := by
+  have hAe : MeasurableSet A := (shiftStop_isStoppingTime hS t).measurableSpace_le A hA
+  have hfinite : MeasurableSet {b | S b≠∞} :=
+    (hS.measurable' (measurableSet_singleton ∞)).compl
+  have htest := (kernel_with_data_measurable K hF).indicator
+    (concatenatedEvent_measurable hS hAe)
+  have hi := lintegral_data_restartAt_on hW hS (stoppedData_raw_measurable hS)
+    MeasurableSet.univ htest
+  simp only [univ_inter] at hi
+  rw [←hi,←Measure.restrict_restrict hAe,←lintegral_indicator hAe]
+  apply setLIntegral_congr_fun hfinite
+  intro b hb
+  have he : (stoppedData S b,restartAt S b)∈concatenatedEvent S A ↔ b∈A :=
+    concatenatedEvent_stoppedData hb
+  by_cases h : b∈A
+  · simp only [Set.indicator_of_mem h,Set.indicator_of_mem (he.mpr h)]
+  · simp only [Set.indicator_of_notMem h,Set.indicator_of_notMem (mt he.mp h)]
+
+ 
+
+theorem restartAt_conditional_kernel {X : Type*} [MeasurableSpace X]
+    (K L G : Kernel GDriver X) [IsSFiniteKernel K] [IsSFiniteKernel L] [IsSFiniteKernel G]
+    {W : Measure GDriver} (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {t : ℝ≥0}
+    (hcond : ∀H : GDriver×X→ℝ≥0∞,
+      @Measurable (GDriver×X) ℝ≥0∞ ((rawFiltration t).prod inferInstance) inferInstance H →
+      (∫⁻c,∫⁻z,H (c,z)∂K c∂W)=(∫⁻c,∫⁻z,H (c,z)∂L c∂W)+∫⁻c,∫⁻z,H (c,z)∂G c∂W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
+    {F : (ℝ≥0×GDriver)×X→ℝ≥0∞} (hF : Measurable F) :
+    (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂K (restartAt S b)∂W)=
+      (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂L (restartAt S b)∂W)+
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂G (restartAt S b)∂W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let E := concatenatedEvent S A
+  let T (M : Kernel GDriver X) (p : ℝ≥0×GDriver) (c : GDriver) :=
+    E.indicator (fun q : (ℝ≥0×GDriver)×GDriver => ∫⁻z,F (q.1,z)∂M q.2) (p,c)
+  have hsection (p : ℝ≥0×GDriver) : MeasurableSet[rawFiltration t] {c | (p,c)∈E} :=
+    (concatenatedEvent_raw_joint hS hA).preimage (measurable_const.prodMk measurable_id)
+  have hFsection (p : ℝ≥0×GDriver) : Measurable (fun z : X => F (p,z)) :=
+    hF.comp (measurable_const.prodMk measurable_id)
+  have hi (p : ℝ≥0×GDriver) : (∫⁻c,T K p c∂W)=(∫⁻c,T L p c∂W)+∫⁻c,T G p c∂W := by
+    let H : GDriver×X→ℝ≥0∞ := fun q =>
+      ({c | (p,c)∈E}).indicator (fun _ => F (p,q.2)) q.1
+    have hH : @Measurable (GDriver×X) ℝ≥0∞ ((rawFiltration t).prod inferInstance)
+        inferInstance H := by
+      exact ((hFsection p).comp measurable_snd).indicator
+        ((hsection p).preimage measurable_fst)
+    have he (M : Kernel GDriver X) (c : GDriver) : (∫⁻z,H (c,z)∂M c)=T M p c := by
+      by_cases hc : (p,c)∈E
+      · simp only [H,T,Set.indicator_apply,Set.mem_ofPred_eq,hc,ite_true]
+      · simp only [H,T,Set.indicator_apply,Set.mem_ofPred_eq,hc,ite_false,lintegral_zero]
+    simpa only [he] using hcond H hH
+  rw [lintegral_kernel_restartAt_on_shift K hW hS hA hF,
+    lintegral_kernel_restartAt_on_shift L hW hS hA hF,
+    lintegral_kernel_restartAt_on_shift G hW hS hA hF]
+  change (∫⁻b in {b | S b≠∞},∫⁻c,T K (stoppedData S b) c∂W∂W)=
+    (∫⁻b in {b | S b≠∞},∫⁻c,T L (stoppedData S b) c∂W∂W)+
+    ∫⁻b in {b | S b≠∞},∫⁻c,T G (stoppedData S b) c∂W∂W
+  simp_rw [hi]
+  apply lintegral_add_left
+  have hM := ((kernel_with_data_measurable L hF).indicator
+    (concatenatedEvent_measurable hS ((shiftStop_isStoppingTime hS t).measurableSpace_le A hA)))
+  exact hM.lintegral_prod_right'.comp
+    ((stoppedData_raw_measurable hS).mono hS.measurableSpace_le le_rfl)
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+
+lemma survivingGreen_zero (a : ℝ) (ha : 0<a) (b : GDriver) (z : UpperPlane) :
+    survivingGreen a 0 b z.1=initialGreen a z := by
+  have hi : alive a b.1 0 z.1 := radialStop_actual a ha z.1 z.2 (z.1.im/2)
+    (by have := z.2; positivity) (by have := z.2; linarith) 0 b 0 bot_le
+  simp only [survivingGreen,ite_eq_left hi,actualState_zero ha b z.2,initialGreen]
+
+lemma survivingKernel_zero (a : ℝ) (ha : 1/4<a) (b : GDriver) :
+    survivingKernel a ha 0 b=initialGreenMeasure a := by
+  rw [survivingKernel_apply]
+  simp only [survivingGreen_zero a (lt_trans (by norm_num) ha),initialGreenMeasure]
+
+lemma survivingKernel_restrict_alive (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
+    (survivingKernel a ha s b).restrict {z : UpperPlane | alive a b.1 s z.1}=
+      survivingKernel a ha s b := by
+  have hD : MeasurableSet {z : UpperPlane | alive a b.1 s z.1} :=
+    (measurableSet_alive_joint a ha).preimage
+      (measurable_const.prodMk (measurable_const.prodMk measurable_id))
+  rw [survivingKernel_apply,restrict_withDensity hD,←withDensity_indicator hD]
+  congr 1
+  funext z
+  by_cases hz : alive a b.1 s z.1
+  · simp only [Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_left hz]
+  · simp only [Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_right hz,survivingGreen,ENNReal.ofReal_zero]
+
+lemma restartPush_initialGreen (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0) :
+    restartPush a (lt_trans (by norm_num) ha) b s (initialGreenMeasure a)=
+      survivingKernel a ha s b := by
+  have h := restartPush_survivingKernel a ha b s 0
+  rwa [survivingKernel_zero,add_zero,survivingKernel_restrict_alive] at h
+
+ 
+
+lemma restartPush_test_joint_measurable (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    Measurable (fun p : (ℝ≥0×GDriver)×GDriver =>
+      ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) p.1.2 p.1.1 (μ p.2)) := by
+  simp_rw [restartPush_lintegral a ha _ _ _ hf]
+  let K : Kernel ((ℝ≥0×GDriver)×GDriver) UpperPlane := μ.comap Prod.snd measurable_snd
+  have hin : Measurable (fun p : ((ℝ≥0×GDriver)×GDriver)×UpperPlane =>
+      (p.1.1.1,(p.1.1.2,p.2))) :=
+    (measurable_fst.fst.fst).prodMk ((measurable_fst.fst.snd).prodMk measurable_snd)
+  exact ((restartWeight_joint_measurable a ha).comp hin |>.mul
+    (hf.comp ((inverseUpper_joint_measurable a ha).comp hin))).lintegral_kernel_prod_right' (κ:=K)
+
+lemma restartPush_mean (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (μ : Kernel GDriver UpperPlane) (W : Measure GDriver)
+    (hμ : μ ∘ₘ W=initialGreenMeasure a) {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    (∫⁻c,∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b s (μ c)∂W)=
+      ∫⁻z,f z∂survivingKernel a ha s b := by
+  have hJ := (continuous_inverseUpper a (lt_trans (by norm_num) ha) b s).measurable
+  have hg : Measurable (fun z => restartWeight a b s z*
+      f (inverseUpper a (lt_trans (by norm_num) ha) b s z)) :=
+    (restartWeight_measurable a ha b s).mul (hf.comp hJ)
+  simp_rw [restartPush_lintegral a ha b s _ hf]
+  rw [←Measure.lintegral_bind (m:=W) μ.aemeasurable hg.aemeasurable]
+  change (∫⁻z,restartWeight a b s z*f (inverseUpper a (lt_trans (by norm_num) ha) b s z)
+    ∂μ ∘ₘ W)=_
+  rw [hμ,←restartPush_lintegral a ha b s _ hf,restartPush_initialGreen]
+
+ 
+
+theorem restartPush_stopped_terminal_mean (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (hμ : μ ∘ₘ W=initialGreenMeasure a)
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A)
+    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
+      b (S b).toNNReal (μ (restartAt S b))∂W)=
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂stoppedSurvivingKernel a ha S hS b∂W := by
+  let F : ((ℝ≥0×GDriver)×GDriver)→ℝ≥0∞ := fun p =>
+    ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) p.1.2 p.1.1 (μ p.2)
+  have h := lintegral_data_restartAt_on hW hS (stoppedData_raw_measurable hS) hA
+    (restartPush_test_joint_measurable a ha μ hf)
+  have he (b c : GDriver) : F (stoppedData S b,c)=
+      ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ c) := by
+    simp only [F,restartPush,inverseUpper_stoppedData,
+      restartWeight_stoppedData a (lt_trans (by norm_num) ha)]
+  change (∫⁻b in A∩{b | S b≠∞},F (stoppedData S b,restartAt S b)∂W)=
+    ∫⁻b in A∩{b | S b≠∞},∫⁻c,F (stoppedData S b,c)∂W∂W at h
+  simp_rw [he] at h
+  rw [h]
+  apply setLIntegral_congr_fun ((hS.measurableSpace_le A hA).inter
+    (hS.measurable' (measurableSet_singleton ∞)).compl)
+  intro b hb
+  dsimp only
+  rw [restartPush_mean a ha _ _ μ W hμ hf,stoppedSurvivingKernel_apply]
+  have hbfin : S b≠∞ := hb.2
+  simp only [stoppedSurvivingGreen,ite_eq_right hbfin,survivingKernel_apply]
+
+lemma restartPush_measure_joint_measurable (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] :
+    Measurable (fun p : (ℝ≥0×GDriver)×GDriver =>
+      restartPush a (lt_trans (by norm_num) ha) p.1.2 p.1.1 (μ p.2)) := by
+  apply Measure.measurable_of_measurable_coe
+  intro A hA
+  have h := restartPush_test_joint_measurable a ha μ
+    (measurable_const.indicator hA : Measurable (A.indicator (fun _ => (1:ℝ≥0∞))))
+  simpa only [lintegral_indicator hA,setLIntegral_one] using h
+
+ 
+
+def restartCandidateKernel (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ]
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Kernel GDriver UpperPlane where
+  toFun b := if S b=∞ then 0 else
+    restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ (restartAt S b))
+  measurable' := by
+    have hin : Measurable (fun b => (((S b).toNNReal,b),restartAt S b)) :=
+      ((ENNReal.measurable_toNNReal.comp hS.measurable').prodMk measurable_id).prodMk
+        (measurable_restartAt hS)
+    have h := (restartPush_measure_joint_measurable a ha μ).comp hin
+    change Measurable (fun b => restartPush a (lt_trans (by norm_num) ha)
+      b (S b).toNNReal (μ (restartAt S b))) at h
+    exact Measurable.ite (hS.measurable' (measurableSet_singleton ∞)) measurable_const h
+
+lemma restartCandidateKernel_apply (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ]
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) (b : GDriver) :
+    restartCandidateKernel a ha μ S hS b=if S b=∞ then 0 else
+      restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ (restartAt S b)) := rfl
+
+ 
+
+theorem restartCandidateKernel_average_on (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (hμ : μ ∘ₘ W=initialGreenMeasure a)
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[hS.measurableSpace] A) :
+    restartCandidateKernel a ha μ S hS ∘ₘ (W.restrict A)=
+      stoppedSurvivingKernel a ha S hS ∘ₘ (W.restrict A) := by
+  have hfinite : MeasurableSet {b | S b≠∞} :=
+    (hS.measurable' (measurableSet_singleton ∞)).compl
+  ext B hB
+  rw [Measure.bind_apply hB (Kernel.aemeasurable _),Measure.bind_apply hB (Kernel.aemeasurable _)]
+  have he := restartPush_stopped_terminal_mean a ha μ W hW hμ S hS hA
+    (measurable_const.indicator hB : Measurable (B.indicator (fun _ => (1:ℝ≥0∞))))
+  simp only [lintegral_indicator hB,setLIntegral_one] at he
+  calc
+    _ = ∫⁻b in A∩{b | S b≠∞},
+        restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (μ (restartAt S b)) B ∂W := by
+      rw [inter_comm A {b | S b≠∞},←Measure.restrict_restrict hfinite,←lintegral_indicator hfinite]
+      apply lintegral_congr
+      intro b
+      by_cases hb : S b=∞ <;> simp [restartCandidateKernel_apply,hb]
+    _ = _ := he
+    _ = ∫⁻b in A,stoppedSurvivingKernel a ha S hS b B∂W := by
+      rw [inter_comm A {b | S b≠∞},←Measure.restrict_restrict hfinite,←lintegral_indicator hfinite]
+      apply lintegral_congr
+      intro b
+      by_cases hb : S b=∞
+      · simp [hb,stoppedSurvivingKernel_apply,stoppedSurvivingGreen]
+      · simp [hb]
+
+ 
+
+theorem restartCandidateKernel_average_le_initial (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (hμ : μ ∘ₘ W=initialGreenMeasure a)
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    restartCandidateKernel a ha μ S hS ∘ₘ W ≤ initialGreenMeasure a := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have h := restartCandidateKernel_average_on a ha μ W hW hμ S hS MeasurableSet.univ
+  rw [Measure.restrict_univ] at h
+  rw [h]
+  exact stoppedSurvivingKernel_average_le_initial a ha W hW S hS
+
+ 
+
+theorem restartCandidateKernel_compact_lintegral_lt_top (a : ℝ) (ha : 1/4<a)
+    (μ : Kernel GDriver UpperPlane) [IsSFiniteKernel μ] (W : Measure GDriver)
+    (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (hμ : μ ∘ₘ W=initialGreenMeasure a)
+    (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    {K : Set UpperPlane} (hK : IsCompact K) :
+    (∫⁻b,restartCandidateKernel a ha μ S hS b K∂W)<∞ := by
+  rw [←Measure.bind_apply hK.measurableSet (Kernel.aemeasurable _)]
+  exact ((restartCandidateKernel_average_le_initial a ha μ W hW hμ S hS) K).trans_lt
+    hK.measure_lt_top
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+
+lemma restartPullback_joint_measurable (a : ℝ) (ha : 1/4<a)
+    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane =>
+      restartWeight a p.1.2 p.1.1 p.2*
+        f (inverseUpper a (lt_trans (by norm_num) ha) p.1.2 p.1.1 p.2)) := by
+  have hin : Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane => (p.1.1,(p.1.2,p.2))) :=
+    measurable_fst.fst.prodMk (measurable_fst.snd.prodMk measurable_snd)
+  have hweight : Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane =>
+      restartWeight a p.1.2 p.1.1 p.2) := by
+    simpa only [Function.comp_def] using (restartWeight_joint_measurable a ha).comp hin
+  have hinverse : Measurable (fun p : (ℝ≥0×GDriver)×UpperPlane =>
+      inverseUpper a (lt_trans (by norm_num) ha) p.1.2 p.1.1 p.2) := by
+    simpa only [Function.comp_def] using (inverseUpper_joint_measurable a ha).comp hin
+  exact hweight.mul (hf.comp hinverse)
+
+ 
+
+theorem restartPush_conditional_kernel (a : ℝ) (ha : 1/4<a)
+    (K L : Kernel GDriver UpperPlane) [IsSFiniteKernel K] [IsSFiniteKernel L]
+    {W : Measure GDriver} (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    {t : ℝ≥0}
+    (hcond : ∀H : GDriver×UpperPlane→ℝ≥0∞,
+      Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+      (∫⁻c,∫⁻z,H (c,z)∂K c∂W)=(∫⁻c,∫⁻z,H (c,z)∂L c∂W)+
+        ∫⁻c,∫⁻z,H (c,z)∂survivingKernel a ha t c∂W)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {A : Set GDriver} (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
+    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
+        b (S b).toNNReal (K (restartAt S b))∂W)=
+      (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
+        b (S b).toNNReal (L (restartAt S b))∂W)+
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
+        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
+  let F : (ℝ≥0×GDriver)×UpperPlane→ℝ≥0∞ := fun p =>
+    restartWeight a p.1.2 p.1.1 p.2*
+      f (inverseUpper a (lt_trans (by norm_num) ha) p.1.2 p.1.1 p.2)
+  have h := restartAt_conditional_kernel K L (survivingKernel a ha t) hW hcond hS hA
+    (restartPullback_joint_measurable a ha hf)
+  have he (M : Kernel GDriver UpperPlane) (b : GDriver) :
+      (∫⁻z,F (stoppedData S b,z)∂M (restartAt S b))=
+      ∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal (M (restartAt S b)) := by
+    rw [restartPush_lintegral a ha _ _ _ hf]
+    simp only [F,inverseUpper_stoppedData,restartWeight_stoppedData a (lt_trans (by norm_num) ha)]
+  change (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂K (restartAt S b)∂W)=
+    (∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂L (restartAt S b)∂W)+
+    ∫⁻b in A∩{b | S b≠∞},∫⁻z,F (stoppedData S b,z)∂survivingKernel a ha t (restartAt S b)∂W at h
+  simp_rw [he] at h
+  simpa only [restartAt,restartPush_survivingKernel] using h
+
+ 
+
+theorem recoveredMass_restarted_potential (a : ℝ) (ha : 1/4<a)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass a ha s c hc v b)))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S)
+    {t : ℝ≥0} {A : Set GDriver}
+    (hA : MeasurableSet[(shiftStop_isStoppingTime hS t).measurableSpace] A)
+    {f : UpperPlane→ℝ≥0∞} (hf : Measurable f) :
+    let μ := recoveredMass a ha s c hc
+    (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
+        b (S b).toNNReal (μ ∞ (restartAt S b))∂W)=
+      (∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂restartPush a (lt_trans (by norm_num) ha)
+        b (S b).toNNReal (μ t (restartAt S b))∂W)+
+      ∫⁻b in A∩{b | S b≠∞},∫⁻z,f z∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
+        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W := by
+  intro μ
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let K : ℝ≥0∞→Kernel GDriver UpperPlane := fun v =>
+    ⟨μ v,recoveredMass_measurable a ha s c hc v⟩
+  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (K v b) := radonLimit_finite _
+  have (v : ℝ≥0∞) : IsSFiniteKernel (K v) := isSFiniteKernel_of_locally_finite (K v)
+  have hcond (H : GDriver×UpperPlane→ℝ≥0∞)
+      (hH : Measurable[(rawFiltration t).prod (inferInstance : MeasurableSpace UpperPlane)] H) :
+      (∫⁻b,∫⁻z,H (b,z)∂K ∞ b∂W)=(∫⁻b,∫⁻z,H (b,z)∂K t b∂W)+
+        ∫⁻b,∫⁻z,H (b,z)∂survivingKernel a ha t b∂W := by
+    have hi := (recoveredMass_all_stopping a ha W hW s c hc hs hsum hconv
+      _ (isStoppingTime_const rawFiltration t)).2.2.2.2 H
+    rw [IsStoppingTime.measurableSpace_const] at hi
+    have hn : Measurable H := hH.mono (prod_sigma_le (rawFiltration.le t) _) le_rfl
+    have he (b : GDriver) : (∫⁻z,H (b,z)∂survivingKernel a ha t b)=
+        ∫⁻z,H (b,z)*ENNReal.ofReal (survivingGreen a t b z.1)∂upperArea := by
+      rw [survivingKernel_apply]
+      exact (lintegral_withDensity_eq_lintegral_mul upperArea
+        (survivingGreen_spatial_measurable a ha t b).ennreal_ofReal
+        (hn.comp (measurable_const.prodMk measurable_id))).trans
+        (lintegral_congr fun _ => mul_comm _ _)
+    simp_rw [he]
+    simpa only [K,Kernel.coe_mk,μ,stoppedSurvivingGreen, ENNReal.some_eq_coe', ENNReal.coe_ne_top, ↓reduceIte,
+      ENNReal.toNNReal_coe] using hi hH
+  exact restartPush_conditional_kernel a ha (K ∞) (K t) hW hcond hS hA hf
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions
+
+ 
+def afterStopFiltration {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    Filtration ℝ≥0 (inferInstance : MeasurableSpace GDriver) where
+  seq t := (shiftStop_isStoppingTime hS t).measurableSpace
+  mono' := fun t u htu => (shiftStop_isStoppingTime hS t).measurableSpace_mono
+    (shiftStop_isStoppingTime hS u) (fun b => by
+      change S b+(t:ℝ≥0∞) ≤ S b+(u:ℝ≥0∞)
+      exact add_le_add le_rfl (ENNReal.coe_le_coe.mpr htu))
+  le' t := (shiftStop_isStoppingTime hS t).measurableSpace_le
+
+lemma raw_function_congr {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
+    {t : ℝ≥0} {f : GDriver→X} (hf : Measurable[rawFiltration t] f)
+    {b c : GDriver} (h : ∀ u≤t,b.1 u=c.1 u) : f b=f c := by
+  have he := (raw_event_congr (hf (measurableSet_singleton (f b))) h).mp
+    (show b∈f ⁻¹' {f b} from rfl)
+  exact he.symm
+
+lemma raw_measure_congr {X : Type*} [MeasurableSpace X] {t : ℝ≥0}
+    {K : GDriver→Measure X} (hK : Measurable[rawFiltration t] K)
+    {b c : GDriver} (h : ∀ u≤t,b.1 u=c.1 u) : K b=K c := by
+  ext A hA
+  exact raw_function_congr ((Measure.measurable_coe hA).comp hK) h
+
+ 
+
+def freshPast (S : GDriver→ℝ≥0∞) (t : ℝ≥0) (b : GDriver) : GDriver :=
+  pastDriver t (restart (S b).toNNReal
+    (pastDriver (shiftStop S t b).toNNReal b))
+
+lemma freshPast_raw_measurable {S : GDriver→ℝ≥0∞}
+    (hS : IsStoppingTime rawFiltration S) (t : ℝ≥0) :
+    Measurable[(afterStopFiltration hS) t] (freshPast S t) := by
+  have htime : Measurable[(afterStopFiltration hS) t]
+      (fun b => (S b).toNNReal) :=
+    (ENNReal.measurable_toNNReal.comp hS.measurable).mono (measurableSpace_le_shiftStop hS t) le_rfl
+  have hfrozen := pastDriver_stopping_measurable (shiftStop S t) (shiftStop_isStoppingTime hS t)
+  exact ((pastDriver_raw_measurable t).mono (rawFiltration.le t) le_rfl).comp
+    (continuous_restart_joint.measurable.comp (htime.prodMk hfrozen))
+
+lemma freshPast_eq (S : GDriver→ℝ≥0∞) (t : ℝ≥0) {b : GDriver} (hb : S b≠∞) :
+    freshPast S t b=pastDriver t (restartAt S b) := by
+  have he : (shiftStop S t b).toNNReal=(S b).toNNReal+t := by
+    rw [shiftStop,ENNReal.toNNReal_add hb ENNReal.coe_ne_top,ENNReal.toNNReal_coe]
+  apply Subtype.ext
+  ext u
+  change (restart (S b).toNNReal (pastDriver (shiftStop S t b).toNNReal b)).1 (min u t)=
+    (restart (S b).toNNReal b).1 (min u t)
+  simp only [restart_apply,he]
+  rw [pastDriver_eq _ b _ (add_le_add (le_refl (S b).toNNReal) (min_le_right u t)),
+    pastDriver_eq _ b _ (le_self_add)]
+
+ 
+
+lemma restartCandidateKernel_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (K : Kernel GDriver UpperPlane) [IsSFiniteKernel K] {t : ℝ≥0}
+    (hK : Measurable[rawFiltration t] K)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    Measurable[(afterStopFiltration hS) t] (restartCandidateKernel a ha K S hS) := by
+  classical
+  have hD := (stoppedData_raw_measurable hS).mono (measurableSpace_le_shiftStop hS t) le_rfl
+  have hM := (restartPush_measure_joint_measurable a ha K).comp
+    (hD.prodMk (freshPast_raw_measurable hS t))
+  have htop : MeasurableSet[(afterStopFiltration hS) t] {b | S b=∞} :=
+    (measurableSpace_le_shiftStop hS t) _ (hS.measurable (measurableSet_singleton ∞))
+  have hh : Measurable[(afterStopFiltration hS) t] (fun b =>
+      if S b=∞ then (0:Measure UpperPlane) else restartPush a (lt_trans (by norm_num) ha)
+        (stoppedData S b).2 (stoppedData S b).1 (K (freshPast S t b))) :=
+    Measurable.ite htop measurable_const hM
+  convert hh using 1
+  funext b
+  change (if S b=∞ then 0 else restartPush a _ b (S b).toNNReal (K (restartAt S b)))=
+    if S b=∞ then 0 else restartPush a _ (stoppedData S b).2 (stoppedData S b).1
+      (K (freshPast S t b))
+  by_cases hb : S b=∞
+  · simp only [hb,ite_true]
+  · simp only [ite_eq_right hb,freshPast_eq S t hb]
+    rw [raw_measure_congr hK (pastDriver_eq t (restartAt S b))]
+    simp only [restartPush,inverseUpper_stoppedData,
+      restartWeight_stoppedData a (lt_trans (by norm_num) ha)]
+
+ 
+def recoveredRestart (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (S : GDriver→ℝ≥0∞) (v : ℝ≥0∞) (b : GDriver) : Measure UpperPlane :=
+  if S b=∞ then 0 else restartPush a (lt_trans (by norm_num) ha) b (S b).toNNReal
+    (recoveredMass a ha s c hc v (restartAt S b))
+
+lemma recoveredRestart_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (v : ℝ≥0∞) :
+    Measurable (recoveredRestart a ha s c hc S v) := by
+  let K : Kernel GDriver UpperPlane :=
+    ⟨recoveredMass a ha s c hc v,recoveredMass_measurable a ha s c hc v⟩
+  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
+  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
+  exact (restartCandidateKernel a ha K S hS).measurable
+
+lemma recoveredRestart_compact_stronglyAdapted (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) (f : C_c(UpperPlane,ℝ)) :
+    StronglyAdapted (afterStopFiltration hS)
+      (fun t : ℝ≥0 => fun b => ∫z,f z∂recoveredRestart a ha s c hc S t b) := by
+  intro t
+  let K : Kernel GDriver UpperPlane :=
+    ⟨recoveredMass a ha s c hc t,recoveredMass_measurable a ha s c hc t⟩
+  let (b : GDriver) : IsFiniteMeasureOnCompacts (K b) := radonLimit_finite _
+  have : IsSFiniteKernel K := isSFiniteKernel_of_locally_finite K
+  have hh := restartCandidateKernel_raw_measurable a ha K
+    (recoveredMass_raw_measurable a ha s c hc t) hS
+  let : MeasurableSpace GDriver := (afterStopFiltration hS) t
+  let L : @Kernel GDriver UpperPlane ((afterStopFiltration hS) t) inferInstance :=
+    ⟨recoveredRestart a ha s c hc S t,hh⟩
+  exact f.continuous.stronglyMeasurable.integral_kernel (κ:=L)
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RecoveredInteriorRestart.lean b/lean/OAI/Probability/SLEGauge/RecoveredInteriorRestart.lean
index ea92e9fc76aa7b061ce23a430d3eb5cebfb432d1..c8bf532d6eaad630f0710c9af038e7974050a973 100644
--- a/lean/OAI/Probability/SLEGauge/RecoveredInteriorRestart.lean
+++ b/lean/OAI/Probability/SLEGauge/RecoveredInteriorRestart.lean
@@ -1,575 +1,575 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.NormalizedMarkedMoment
-import OAI.Probability.SLEGauge.FiniteVariationMartingale
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
-open TraceRestriction TraceMassContinuity TraceMassIntegration CompensatorUniqueness
-
- 
-
-theorem recoveredPostStop_eq_recoveredRestart (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
-      initialGreenMeasure (Target.a κ))
-    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
-      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    ∀ᵐ b ∂W, ∀v : ℝ≥0∞,
-      recoveredPostStop (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S v b=
-        recoveredRestart (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S v b := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let a := Target.a κ
-  have ha : 1/4<a := a_gt_quarter hκ hκ8
-  let μ := recoveredPostStop a ha s c hc S
-  let ν := recoveredRestart a ha s c hc S
-  let ℱ := afterStopFiltration hS
-  have hμp := recoveredPostStop_path_properties κ hκ hκ8 W s c hc hs hpath hconv S
-  have hνp := recoveredRestart_path_properties κ hκ hκ8 W hW s c hc hmean hpath hS
-  have hμr (v : ℝ≥0∞) (b : GDriver) : (μ v b).Regular := by
-    let := recoveredPostStop_finiteOnCompacts a ha s c hc S v b
-    infer_instance
-  have hνr : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, (ν v b).Regular :=
-    hνp.2.mono (fun b hb v => (hb.1 v).2)
-  have hμo : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => μ v b) := hμp.mono (fun b hb => hb.2.1)
-  have hνo : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => ν v b) := hνp.2.mono (fun b hb => hb.2.2.1)
-  have hμi (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂μ ∞ b) W :=
-    recoveredPostStop_terminal_integrable a ha s c hc W hmean hS f
-  have hνi (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂ν ∞ b) W := hνp.1 f
-  have hμm (f : C_c(UpperPlane,ℝ)) : StronglyAdapted ℱ (fun t : ℝ≥0 => fun b => ∫z,f z∂μ t b) :=
-    recoveredPostStop_compact_stronglyAdapted a ha s c hc hS f
-  have hνm (f : C_c(UpperPlane,ℝ)) : StronglyAdapted ℱ (fun t : ℝ≥0 => fun b => ∫z,f z∂ν t b) :=
-    recoveredRestart_compact_stronglyAdapted a ha s c hc hS f
-  have hμl (v : ℝ≥0∞) (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Integrable f (μ v b) := by
-    apply Eventually.of_forall
-    intro b
-    let := hμr v b
-    exact f.integrable
-  have hνl (v : ℝ≥0∞) (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Integrable f (ν v b) := by
-    filter_upwards [hνr] with b hb
-    let := hb v
-    exact f.integrable
-  have hpot (f : C_c(UpperPlane,ℝ)) (hf : ∀z,0≤f z) (t : ℝ≥0) :
-      W[(fun b => (∫z,f z∂μ ∞ b)-(∫z,f z∂μ t b)) | ℱ t] =ᵐ[W]
-        W[(fun b => (∫z,f z∂ν ∞ b)-(∫z,f z∂ν t b)) | ℱ t] := by
-    have hμti := integrable_test_of_ae_measure_le W (μ t) (μ ∞) hf
-      (((hμm f t).mono (ℱ.le t)).aestronglyMeasurable) (hμl ∞ f) (hμi f)
-      (hμo.mono (fun b hb => hb le_top))
-    have hνti := integrable_test_of_ae_measure_le W (ν t) (ν ∞) hf
-      (((hνm f t).mono (ℱ.le t)).aestronglyMeasurable) (hνl ∞ f) (hνi f)
-      (hνo.mono (fun b hb => hb le_top))
-    let V : Set GDriver→ℝ≥0∞ := fun A => ∫⁻b in A∩{b | S b≠∞},
-      ∫⁻z,ENNReal.ofReal (f z)∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
-        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W
-    exact condExp_sub_eq_of_lintegral_potential (ℱ.le t) W (μ t) (ν t) (μ ∞) (ν ∞) hf
-      hμti hνti (hμi f) (hνi f) (hμl t f) (hνl t f) (hμl ∞ f) (hνl ∞ f) V
-      (fun A hA => recoveredPostStop_potential a ha W hW s c hc hs hsum hconv hS hA
-        f.continuous.measurable.ennreal_ofReal)
-      (fun A hA => recoveredRestart_potential a ha W hW s c hc hs hsum hconv hS hA
-        f.continuous.measurable.ennreal_ofReal)
-  have he : ∀ᵐ b ∂W, ∀t : ℝ≥0, μ t b=ν t b := by
-    apply ae_all_measure_eq_of_ae_regular (ℱ:=ℱ) (fun t : ℝ≥0 => μ t) (fun t : ℝ≥0 => ν t)
-      (μ ∞) (ν ∞)
-      (Eventually.of_forall fun b t => hμr t b)
-      (hνr.mono (fun b hb t => hb t))
-      (Eventually.of_forall (hμr ∞)) (hνr.mono (fun b hb => hb ∞))
-      hμm hνm (hμp.mono fun b hb => hb.1) (hνp.2.mono fun b hb => hb.2.1)
-      (fun f => hμp.mono fun b hb => (hb.2.2 f).1)
-      (fun f => hνp.2.mono fun b hb => (hb.2.2.2 f).1)
-      (hμo.mono fun b hb u v huv => hb (ENNReal.coe_le_coe.mpr huv))
-      (hνo.mono fun b hb u v huv => hb (ENNReal.coe_le_coe.mpr huv))
-      (hμo.mono fun b hb t => hb le_top) (hνo.mono fun b hb t => hb le_top)
-      hμi hνi hpot
-  have het : μ ∞ =ᵐ[W] ν ∞ := ae_terminal_measure_eq_of_ae_regular
-    (fun t : ℝ≥0 => μ t) (fun t : ℝ≥0 => ν t) (μ ∞) (ν ∞)
-    (Eventually.of_forall (hμr ∞)) (hνr.mono (fun b hb => hb ∞)) he
-    (fun f => hμp.mono fun b hb => (hb.2.2 f).2)
-    (fun f => hνp.2.mono fun b hb => (hb.2.2.2 f).2)
-  filter_upwards [he,het] with b hb ht v
-  induction v using ENNReal.recTopCoe with
-  | top => exact ht
-  | coe t => exact hb t
-
- 
-
-theorem recoveredMass_interior_restart (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
-      initialGreenMeasure (Target.a κ))
-    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
-      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-    ∀ᵐ b ∂W, S b≠∞ →
-      (μ (S b) b).restrict {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=0 ∧
-      ∀v : ℝ≥0∞, (μ (S b+v) b).restrict
-          {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=
-        restartPush (Target.a κ) (lt_trans (by norm_num) (a_gt_quarter hκ hκ8)) b
-          (S b).toNNReal (μ v (restartAt S b)) := by
-  intro μ
-  filter_upwards [recoveredPostStop_eq_recoveredRestart κ hκ hκ8 W hW s c hc hs hsum
-    hmean hpath hconv hS,hconv] with b hb hcV hSb
-  refine ⟨?_,?_⟩
-  · have hz := recoveredPostStop_zero_of_convergence (Target.a κ) (a_gt_quarter hκ hκ8)
-      s c hc hs S b hcV
-    simpa only [recoveredPostStop,ite_eq_right hSb,add_zero,μ] using hz
-  · intro v
-    simpa only [recoveredPostStop,recoveredRestart,ite_eq_right hSb,μ] using hb v
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.LoewnerRestart
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
-open TraceMassContinuity TraceMassIntegration
-
-lemma restartPush_positive (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
-    (μ : Measure UpperPlane) (hμ : 0<μ univ) :
-    0<restartPush a (lt_trans (by norm_num) ha) b s μ univ := by
-  apply pos_iff_ne_zero.mpr
-  intro hz
-  rw [restartPush,Measure.map_apply (continuous_inverseUpper a
-    (lt_trans (by norm_num) ha) b s).measurable MeasurableSet.univ,preimage_univ] at hz
-  have h := (withDensity_apply_eq_zero (restartWeight_measurable a ha b s)).mp hz
-  have hs : {w : UpperPlane | restartWeight a b s w≠0}=univ := by
-    ext w
-    simp only [mem_ofPred_eq,mem_univ,iff_true]
-    exact (restartWeight_pos a (lt_trans (by norm_num) ha) b s w).ne'
-  rw [hs,univ_inter] at h
-  exact hμ.ne' h
-
- 
-
-theorem recoveredMass_positive_after_stop (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
-      initialGreenMeasure (Target.a κ))
-    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
-      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
-    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
-    let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-    ∀ᵐ b ∂W, S b≠∞ → ∀v : ℝ≥0∞, 0<v → 0<(μ (S b+v) b-μ (S b) b) univ := by
-  intro μ
-  filter_upwards [recoveredMass_interior_restart κ hκ hκ8 W hW s c hc hs hsum hmean
-    hpath hconv hS,ae_restartAt hW hS hpath] with b hr hp hSb v hv
-  have hr := hr hSb
-  have hp := (hp hSb).initial_positive v hv
-  let H := {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}
-  have hH : MeasurableSet H := measurableSet_alive_spatial _ _ _
-  have hx : (μ (S b+v) b-μ (S b) b).restrict H=
-      restartPush (Target.a κ) (lt_trans (by norm_num) (a_gt_quarter hκ hκ8)) b
-        (S b).toNNReal (μ v (restartAt S b)) := by
-    rw [Measure.restrict_sub_eq_restrict_sub_restrict hH,hr.1,Measure.sub_zero,hr.2 v]
-  have hpos := restartPush_positive (Target.a κ) (a_gt_quarter hκ hκ8) b
-    (S b).toNNReal (μ v (restartAt S b)) hp
-  rw [←hx] at hpos
-  exact hpos.trans_le (Measure.le_iff'.mp Measure.restrict_le_self univ)
-
-lemma measure_sub_antitone_right {X : Type*} [MeasurableSpace X]
-    (μ : Measure X) {ν ξ : Measure X} (h : ν≤ξ) : μ-ξ≤μ-ν := by
-  simp only [Measure.sub_def]
-  apply sInf_le_sInf
-  intro τ hτ
-  exact (show μ≤τ+ν from hτ).trans (add_le_add le_rfl h)
-
- 
-
-theorem recoveredMass_positive_increments (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
-      initialGreenMeasure (Target.a κ))
-    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
-      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
-      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))) :
-    let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-    ∀ᵐ b ∂W, ∀u v : ℝ≥0, u<v →
-      0<(μ v b-μ u b) univ ∧ ∃K : Set UpperPlane, IsCompact K ∧
-        0<(μ v b-μ u b) K ∧ (μ v b-μ u b) K<∞ := by
-  intro μ
-  have hall : ∀ᵐ b ∂W, ∀q : ℚ, ∀v : ℝ≥0∞, 0<v →
-      0<(μ ((Real.toNNReal (q:ℝ):ℝ≥0∞)+v) b-μ (Real.toNNReal (q:ℝ)) b) univ := by
-    apply ae_all_iff.mpr
-    intro q
-    have hh := recoveredMass_positive_after_stop κ hκ hκ8 W hW s c hc hs hsum
-      hmean hpath hconv (S:=fun _ => (Real.toNNReal (q:ℝ):ℝ≥0∞)) (isStoppingTime_const rawFiltration _)
-    filter_upwards [hh] with b hb
-    exact hb ENNReal.coe_ne_top
-  filter_upwards [hall,hpath] with b hb hp u v huv
-  obtain ⟨q,huq,hqv⟩ := exists_rat_btwn (show (u:ℝ)<(v:ℝ) from huv)
-  let p : ℝ≥0 := Real.toNNReal (q:ℝ)
-  have hq0 : 0≤(q:ℝ) := u.coe_nonneg.trans huq.le
-  have huc : u≤p := by
-    rw [←NNReal.coe_le_coe,Real.coe_toNNReal _ hq0]
-    exact huq.le
-  have hpc : p<v := by
-    rw [←NNReal.coe_lt_coe,Real.coe_toNNReal _ hq0]
-    exact hqv
-  have he : (p:ℝ≥0∞)+((v-p:ℝ≥0):ℝ≥0∞)=(v:ℝ≥0∞) := by
-    rw [←ENNReal.coe_add,add_tsub_cancel_of_le hpc.le]
-  have hpos := hb q (v-p) (ENNReal.coe_pos.mpr (tsub_pos_iff_lt.mpr hpc))
-  change 0<(μ ((p:ℝ≥0∞)+((v-p:ℝ≥0):ℝ≥0∞)) b-μ p b) univ at hpos
-  rw [he] at hpos
-  have hpos' : 0<(μ v b-μ u b) univ := hpos.trans_le
-    ((measure_sub_antitone_right (μ v b) (hp.monotone (ENNReal.coe_le_coe.mpr huc))) univ)
-  let : IsFiniteMeasureOnCompacts (μ v b) := radonLimit_finite _
-  let : IsFiniteMeasureOnCompacts (μ v b-μ u b) := ⟨fun K hK =>
-    ((Measure.sub_le : μ v b-μ u b≤μ v b) K).trans_lt hK.measure_lt_top⟩
-  obtain ⟨K,-,hK,hpK⟩ := isOpen_univ.exists_lt_isCompact hpos'
-  exact ⟨hpos',K,hK,hpK,hK.measure_lt_top⟩
-
-end SLEExactGauge.LoewnerRestart
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma recoveredMass_compact_stronglyAdapted (a : ℝ) (ha : 1/4<a)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (f : C_c(UpperPlane,ℝ)) :
-    StronglyAdapted rawFiltration (fun t : ℝ≥0 => fun b =>
-      ∫z,f z∂recoveredMass a ha s c hc t b) := by
-  intro t
-  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-  let K : Kernel ExactSLEGenerator.Driver UpperPlane :=
-    ⟨recoveredMass a ha s c hc t,recoveredMass_raw_measurable a ha s c hc t⟩
-  exact f.continuous.stronglyMeasurable.integral_kernel (κ:=K)
-
- 
-
-theorem recoveredMass_stopped_condExp (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
-    (hsum : ∀k,∑j∈s k,c k j=1)
-    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂recoveredMass a ha s c hc v b)))
-    (S : ExactSLEGenerator.Driver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) :
-    W[(fun b => (∫z,f z∂recoveredMass a ha s c hc ∞ b)-
-      (∫z,f z∂recoveredMass a ha s c hc (S b) b)) | hS.measurableSpace] =ᵐ[W]
-      fun b => ∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea := by
-  let μ := recoveredMass a ha s c hc
-  have hcv (v : ExactSLEGenerator.Driver→ℝ≥0∞) : ∀ᵐ b ∂W, Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
-        (Real.exp (-2*a*(j+1))) (v b) b)
-      atTop (𝓝 (∫z,f z∂μ (v b) b)) := hconv.mono (fun b hb => hb (v b) f)
-  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (∞:ℝ≥0∞)) :=
-    fun t => by simp
-  have hi (v : ExactSLEGenerator.Driver→ℝ≥0∞) (hv : IsStoppingTime rawFiltration v) :
-      Integrable (fun b => ∫z,f z∂μ (v b) b) W :=
-    memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using
-      actual_stopped_radon_limit_memLp a ha W hW v hv s c hc hsum _ f (hcv v) 1)
-  have hg := stoppedSurvivingKernel_compact_mean_integrable a ha W hW S hS f
-  simp_rw [stoppedSurvivingKernel_compact_eval] at hg
-  have hgm : StronglyMeasurable[hS.measurableSpace]
-      (fun b => ∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea) := by
-    let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-    let K : Kernel ExactSLEGenerator.Driver UpperPlane :=
-      ⟨stoppedSurvivingKernel a ha S hS,stoppedSurvivingKernel_raw_measurable a ha S hS⟩
-    have h := f.continuous.stronglyMeasurable.integral_kernel (κ:=K)
-    change StronglyMeasurable (fun b => ∫z,f z∂stoppedSurvivingKernel a ha S hS b) at h
-    simpa only [stoppedSurvivingKernel_compact_eval] using h
-  symm
-  apply ae_eq_condExp_of_forall_setIntegral_eq hS.measurableSpace_le
-    ((hi _ htop).sub (hi S hS)) (fun _ _ _ => hg.integrableOn) _ hgm.aestronglyMeasurable
-  intro U hU _
-  exact (actual_stopped_limit_compact_conditional a ha W hW S hS s c hs hc hsum
-    (fun b => μ (S b) b) (μ ∞) f (hcv S) (hcv (fun _ => ∞)) U hU).symm
-
- 
-
-theorem recoveredMass_extraction_unique (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (s₁ s₂ : ℕ→Finset ℕ) (c₁ c₂ : ℕ→ℕ→ℝ)
-    (hc₁ : ∀k j,j∈s₁ k→0≤c₁ k j) (hc₂ : ∀k j,j∈s₂ k→0≤c₂ k j)
-    (hs₁ : ∀k j,j∈s₁ k→k≤j) (hs₂ : ∀k j,j∈s₂ k→k≤j)
-    (hsum₁ : ∀k,∑j∈s₁ k,c₁ k j=1) (hsum₂ : ∀k,∑j∈s₂ k,c₂ k j=1)
-    (hconv₁ : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s₁ k,c₁ k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂recoveredMass a ha s₁ c₁ hc₁ v b)))
-    (hconv₂ : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s₂ k,c₂ k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂recoveredMass a ha s₂ c₂ hc₂ v b)))
-    (hcont₁ : ∀f : C_c(UpperPlane,ℝ), ∀ᵐ b ∂W, Continuous (fun v : ℝ≥0 =>
-      ∫z,f z∂recoveredMass a ha s₁ c₁ hc₁ v b))
-    (hcont₂ : ∀f : C_c(UpperPlane,ℝ), ∀ᵐ b ∂W, Continuous (fun v : ℝ≥0 =>
-      ∫z,f z∂recoveredMass a ha s₂ c₂ hc₂ v b))
-    (hmono₁ : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => recoveredMass a ha s₁ c₁ hc₁ v b))
-    (hmono₂ : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => recoveredMass a ha s₂ c₂ hc₂ v b))
-    (hexh₁ : ∀ᵐ b ∂W, ∀A : Set UpperPlane, MeasurableSet A → Tendsto
-      (fun n : ℕ => recoveredMass a ha s₁ c₁ hc₁ (n:ℝ≥0) b A)
-      atTop (𝓝 (recoveredMass a ha s₁ c₁ hc₁ ∞ b A)))
-    (hexh₂ : ∀ᵐ b ∂W, ∀A : Set UpperPlane, MeasurableSet A → Tendsto
-      (fun n : ℕ => recoveredMass a ha s₂ c₂ hc₂ (n:ℝ≥0) b A)
-      atTop (𝓝 (recoveredMass a ha s₂ c₂ hc₂ ∞ b A))) :
-    ∀ᵐ b ∂W, ∀v : ℝ≥0∞,
-      recoveredMass a ha s₁ c₁ hc₁ v b=recoveredMass a ha s₂ c₂ hc₂ v b := by
-  let μ := recoveredMass a ha s₁ c₁ hc₁
-  let ν := recoveredMass a ha s₂ c₂ hc₂
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : (μ v b).Regular := radonLimit_regular _
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : (ν v b).Regular := radonLimit_regular _
-  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (∞:ℝ≥0∞)) :=
-    fun t => by simp
-  have hi₁ (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂μ ∞ b) W := by
-    apply memLp_one_iff_integrable.mp
-    simpa only [Nat.cast_one] using (actual_stopped_radon_limit_memLp a ha W hW
-      _ htop s₁ c₁ hc₁ hsum₁ (μ ∞) f (hconv₁.mono (fun b hb => hb ∞ f)) 1)
-  have hi₂ (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂ν ∞ b) W := by
-    apply memLp_one_iff_integrable.mp
-    simpa only [Nat.cast_one] using (actual_stopped_radon_limit_memLp a ha W hW
-      _ htop s₂ c₂ hc₂ hsum₂ (ν ∞) f (hconv₂.mono (fun b hb => hb ∞ f)) 1)
-  have hzero₁ : μ 0=ᵐ[W] 0 := Eventually.of_forall (recoveredMass_zero a ha s₁ c₁ hc₁)
-  have hzero₂ : ν 0=ᵐ[W] 0 := Eventually.of_forall (recoveredMass_zero a ha s₂ c₂ hc₂)
-  have he : ∀ᵐ b ∂W, ∀v : ℝ≥0, μ v b=ν v b := by
-    apply SLEExactGauge.CompensatorUniqueness.ae_all_measure_eq
-      (ℱ:=rawFiltration) (fun v : ℝ≥0 => μ v) (fun v : ℝ≥0 => ν v) (μ ∞) (ν ∞)
-      (recoveredMass_compact_stronglyAdapted a ha s₁ c₁ hc₁)
-      (recoveredMass_compact_stronglyAdapted a ha s₂ c₂ hc₂)
-      hzero₁ hzero₂ hcont₁ hcont₂
-      (hmono₁.mono (fun b hb s t hst => hb (ENNReal.coe_le_coe.mpr hst)))
-      (hmono₂.mono (fun b hb s t hst => hb (ENNReal.coe_le_coe.mpr hst)))
-      (hmono₁.mono (fun b hb v => hb le_top))
-      (hmono₂.mono (fun b hb v => hb le_top)) hi₁ hi₂
-    intro f _ t
-    have h₁ := recoveredMass_stopped_condExp a ha W hW s₁ c₁ hc₁ hs₁ hsum₁ hconv₁
-      _ (isStoppingTime_const rawFiltration t) f
-    have h₂ := recoveredMass_stopped_condExp a ha W hW s₂ c₂ hc₂ hs₂ hsum₂ hconv₂
-      _ (isStoppingTime_const rawFiltration t) f
-    have hh := h₁.trans h₂.symm
-    rw [IsStoppingTime.measurableSpace_const] at hh
-    simpa only [μ, ν, ENNReal.some_eq_coe'] using hh
-  filter_upwards [he,hexh₁,hexh₂] with b hb h₁ h₂ v
-  induction v using ENNReal.recTopCoe with
-  | coe v => exact hb v
-  | top =>
-    apply Measure.ext
-    intro A hA
-    have h : Tendsto (fun n : ℕ => μ (n:ℝ≥0) b A) atTop (𝓝 (μ ∞ b A)) := h₁ A hA
-    simp only [hb] at h
-    exact tendsto_nhds_unique h (h₂ A hA)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceMassContinuity
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions TraceMassIntegration
-open LoewnerRestart
-
- 
-
-theorem actual_canonical_trace_mass (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
-      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      (∀v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀b, μ 0 b=0) ∧ (∀v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      μ ∞ ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ) ∧
-      (∀ (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
-        Measurable[hS.measurableSpace] (fun b => μ (S b) b) ∧
-        (∀f : C_c(UpperPlane,ℝ), ∀m : ℕ,
-          MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
-        (∀K : Set UpperPlane, IsCompact K → ∀m : ℕ,
-          MemLp (fun b => (μ (S b) b K).toReal) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
-        (∀f : C_c(UpperPlane,ℝ), Tendsto (fun k => eLpNorm
-          (fun b => (∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-            (Real.exp (-2*(Target.a κ)*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b))
-              1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
-        (∀H : GDriver × UpperPlane→ℝ≥0∞,
-          Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
-          (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂(W.map driverHomeomorph))=
-            (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂(W.map driverHomeomorph))+
-            ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen (Target.a κ) (S b) b z.1)
-              ∂upperArea∂(W.map driverHomeomorph)) ∧
-        (∀ᵐ b ∂(W.map driverHomeomorph), S b≠∞ →
-          (μ (S b) b).restrict {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=0 ∧
-          ∀v : ℝ≥0∞, (μ (S b+v) b).restrict
-              {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=
-            restartPush (Target.a κ) (lt_trans (by norm_num) (a_gt_quarter hκ hκ8)) b
-              (S b).toNNReal (μ v (restartAt S b)))) ∧
-      ∀ᵐ b ∂W, TraceMassFacts κ b (fun v => μ v (driverHomeomorph b)) ∧
-        (∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-          (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-            (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
-          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
-        (∀u v : ℝ≥0, u<v →
-          0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) univ ∧
-          ∃K : Set UpperPlane, IsCompact K ∧
-            0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) K ∧
-            (μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) K<∞) := by
-  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
-    actual_common_trace_mass κ hκ hκ8 W hW
-  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  have hBW := brownian_map_driver W hW
-  have hp : ∀ᵐ b ∂(W.map driverHomeomorph),
-      TraceMassFacts κ (driverHomeomorph.symm b) (fun v => μ v b) := by
-    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-    filter_upwards [hpath] with b hb
-    change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
-      (fun v => μ v (driverHomeomorph b))
-    simpa only [Homeomorph.symm_apply_apply] using hb.1
-  have hv : ∀ᵐ b ∂(W.map driverHomeomorph), ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
-      atTop (𝓝 (∫z,f z∂μ v b)) := by
-    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
-    exact hpath.mono (fun b hb => hb.2)
-  refine ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,?_,?_⟩
-  · intro S hS
-    rcases hstop S hS with ⟨hsm,hmom,hK,hL,hH⟩
-    exact ⟨hsm,hmom,hK,hL,hH,recoveredMass_interior_restart κ hκ hκ8
-      (W.map driverHomeomorph) hBW s c hc hs hsum hmean hp hv hS⟩
-  · have hpos := recoveredMass_positive_increments κ hκ hκ8
-      (W.map driverHomeomorph) hBW s c hc hs hsum hmean hp hv
-    have hpos' := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp hpos
-    filter_upwards [hpath,hpos'] with b hb hbp
-    exact ⟨hb.1,hb.2,hbp⟩
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
-
-lemma diskMass_mono {ν η : Measure UpperPlane} (h : ν≤η) (a θ : ℝ) :
-    diskMass a θ ν≤diskMass a θ η := by
-  apply Measure.map_mono _ (diskMap_measurable θ)
-  apply Measure.le_iff.mpr
-  intro U hU
-  rw [withDensity_apply _ hU,withDensity_apply _ hU]
-  exact lintegral_mono' (Measure.restrict_mono le_rfl h) le_rfl
-
-lemma diskMass_inner_le_enorm (a θ : ℝ) (ha : 1/4<a) (hθ : 0<θ) (hθπ : θ<Real.pi)
-    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] :
-    diskMass a θ ν (Metric.closedBall (0:ℂ) (1/4))≤‖∫z,diskTest a θ hθ hθπ z∂ν‖ₑ :=
-  (diskMass_inner_le_test a θ ha hθ hθπ ν).trans (Real.ofReal_le_enorm _)
-
- 
-
-theorem common_limit_disk_tilted_moment (a : ℝ) (ha : 1/4<a)
-    (m : ℕ) (hm : 1 ≤ m) :
-    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
-      (∀b,IsFiniteMeasureOnCompacts (μ ∞ b)) → Measurable (μ ∞) →
-      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
-      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
-          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
-      ∀(θ : ℝ) (_hθ : 0<θ) (_hθπ : θ<Real.pi) (l : ℕ)
-        (T : ExactSLEGenerator.Driver→ℝ≥0∞),
-      (∫⁻b,(diskMass a θ (μ (T b) b) (Metric.closedBall (0:ℂ) (1/4)))^m
-        ∂radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)))≤
-        D*ENNReal.ofReal ((Real.sin θ)^(-p a)) := by
-  let K : Set ℂ := Subtype.val '' tsupport diskCutoff
-  have hK : IsCompact K := diskCutoff.hasCompactSupport.image continuous_subtype_val
-  have hcK : ∀z∈K,(1/2:ℝ)≤2*z.im := by
-    rintro z ⟨w,hw,rfl⟩
-    have hh := height_of_ball (diskCutoff_support w hw).le
-    linarith
-  obtain ⟨D,hD,hDb⟩ := actual_normalized_tilted_moment_bound a ha m hm K
-    hK.measurableSet hK.isBounded (1/2) (by norm_num) (by norm_num) hcK
-    (2^(2-alpha a)) (Real.rpow_nonneg (by norm_num) _)
-  refine ⟨D,hD,?_⟩
-  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim θ hθ hθπ l T
-  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ ∞ b) := hμfin b
-  let f := diskTest a θ hθ hθπ
-  have hfm : Measurable (fun b=>∫z,f z∂μ ∞ b) := by
-    let Kμ : Kernel ExactSLEGenerator.Driver UpperPlane := ⟨μ ∞,hμ⟩
-    exact (f.continuous.stronglyMeasurable.integral_kernel (κ:=Kμ)).measurable
-  have hb := hDb W hW θ hθ hθπ f (diskTest_support a θ hθ hθπ)
-    (diskTest_bound a θ hθ hθπ) s coeff hcoeff hsum ∞ (μ ∞) hfm
-    (hlim.mono (fun b hb=>hb f)) l
-  refine le_trans (lintegral_mono_ae ?_) hb
-  have habs : radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) ≪ W :=
-    withDensity_absolutelyContinuous W _
-  filter_upwards [habs.ae_le hmono] with b hb
-  exact pow_le_pow_left' ((Measure.le_iff'.mp (diskMass_mono (hb (T b)) a θ)
-    _).trans (diskMass_inner_le_enorm a θ ha hθ hθπ (μ ∞ b))) m
-
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.NormalizedMarkedMoment
+import OAI.Probability.SLEGauge.FiniteVariationMartingale
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
+open TraceRestriction TraceMassContinuity TraceMassIntegration CompensatorUniqueness
+
+ 
+
+theorem recoveredPostStop_eq_recoveredRestart (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
+      initialGreenMeasure (Target.a κ))
+    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
+      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    ∀ᵐ b ∂W, ∀v : ℝ≥0∞,
+      recoveredPostStop (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S v b=
+        recoveredRestart (Target.a κ) (a_gt_quarter hκ hκ8) s c hc S v b := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let a := Target.a κ
+  have ha : 1/4<a := a_gt_quarter hκ hκ8
+  let μ := recoveredPostStop a ha s c hc S
+  let ν := recoveredRestart a ha s c hc S
+  let ℱ := afterStopFiltration hS
+  have hμp := recoveredPostStop_path_properties κ hκ hκ8 W s c hc hs hpath hconv S
+  have hνp := recoveredRestart_path_properties κ hκ hκ8 W hW s c hc hmean hpath hS
+  have hμr (v : ℝ≥0∞) (b : GDriver) : (μ v b).Regular := by
+    let := recoveredPostStop_finiteOnCompacts a ha s c hc S v b
+    infer_instance
+  have hνr : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, (ν v b).Regular :=
+    hνp.2.mono (fun b hb v => (hb.1 v).2)
+  have hμo : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => μ v b) := hμp.mono (fun b hb => hb.2.1)
+  have hνo : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => ν v b) := hνp.2.mono (fun b hb => hb.2.2.1)
+  have hμi (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂μ ∞ b) W :=
+    recoveredPostStop_terminal_integrable a ha s c hc W hmean hS f
+  have hνi (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂ν ∞ b) W := hνp.1 f
+  have hμm (f : C_c(UpperPlane,ℝ)) : StronglyAdapted ℱ (fun t : ℝ≥0 => fun b => ∫z,f z∂μ t b) :=
+    recoveredPostStop_compact_stronglyAdapted a ha s c hc hS f
+  have hνm (f : C_c(UpperPlane,ℝ)) : StronglyAdapted ℱ (fun t : ℝ≥0 => fun b => ∫z,f z∂ν t b) :=
+    recoveredRestart_compact_stronglyAdapted a ha s c hc hS f
+  have hμl (v : ℝ≥0∞) (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Integrable f (μ v b) := by
+    apply Eventually.of_forall
+    intro b
+    let := hμr v b
+    exact f.integrable
+  have hνl (v : ℝ≥0∞) (f : C_c(UpperPlane,ℝ)) : ∀ᵐ b ∂W, Integrable f (ν v b) := by
+    filter_upwards [hνr] with b hb
+    let := hb v
+    exact f.integrable
+  have hpot (f : C_c(UpperPlane,ℝ)) (hf : ∀z,0≤f z) (t : ℝ≥0) :
+      W[(fun b => (∫z,f z∂μ ∞ b)-(∫z,f z∂μ t b)) | ℱ t] =ᵐ[W]
+        W[(fun b => (∫z,f z∂ν ∞ b)-(∫z,f z∂ν t b)) | ℱ t] := by
+    have hμti := integrable_test_of_ae_measure_le W (μ t) (μ ∞) hf
+      (((hμm f t).mono (ℱ.le t)).aestronglyMeasurable) (hμl ∞ f) (hμi f)
+      (hμo.mono (fun b hb => hb le_top))
+    have hνti := integrable_test_of_ae_measure_le W (ν t) (ν ∞) hf
+      (((hνm f t).mono (ℱ.le t)).aestronglyMeasurable) (hνl ∞ f) (hνi f)
+      (hνo.mono (fun b hb => hb le_top))
+    let V : Set GDriver→ℝ≥0∞ := fun A => ∫⁻b in A∩{b | S b≠∞},
+      ∫⁻z,ENNReal.ofReal (f z)∂(survivingKernel a ha ((S b).toNNReal+t) b).restrict
+        {z : UpperPlane | alive a b.1 (S b).toNNReal z.1}∂W
+    exact condExp_sub_eq_of_lintegral_potential (ℱ.le t) W (μ t) (ν t) (μ ∞) (ν ∞) hf
+      hμti hνti (hμi f) (hνi f) (hμl t f) (hνl t f) (hμl ∞ f) (hνl ∞ f) V
+      (fun A hA => recoveredPostStop_potential a ha W hW s c hc hs hsum hconv hS hA
+        f.continuous.measurable.ennreal_ofReal)
+      (fun A hA => recoveredRestart_potential a ha W hW s c hc hs hsum hconv hS hA
+        f.continuous.measurable.ennreal_ofReal)
+  have he : ∀ᵐ b ∂W, ∀t : ℝ≥0, μ t b=ν t b := by
+    apply ae_all_measure_eq_of_ae_regular (ℱ:=ℱ) (fun t : ℝ≥0 => μ t) (fun t : ℝ≥0 => ν t)
+      (μ ∞) (ν ∞)
+      (Eventually.of_forall fun b t => hμr t b)
+      (hνr.mono (fun b hb t => hb t))
+      (Eventually.of_forall (hμr ∞)) (hνr.mono (fun b hb => hb ∞))
+      hμm hνm (hμp.mono fun b hb => hb.1) (hνp.2.mono fun b hb => hb.2.1)
+      (fun f => hμp.mono fun b hb => (hb.2.2 f).1)
+      (fun f => hνp.2.mono fun b hb => (hb.2.2.2 f).1)
+      (hμo.mono fun b hb u v huv => hb (ENNReal.coe_le_coe.mpr huv))
+      (hνo.mono fun b hb u v huv => hb (ENNReal.coe_le_coe.mpr huv))
+      (hμo.mono fun b hb t => hb le_top) (hνo.mono fun b hb t => hb le_top)
+      hμi hνi hpot
+  have het : μ ∞ =ᵐ[W] ν ∞ := ae_terminal_measure_eq_of_ae_regular
+    (fun t : ℝ≥0 => μ t) (fun t : ℝ≥0 => ν t) (μ ∞) (ν ∞)
+    (Eventually.of_forall (hμr ∞)) (hνr.mono (fun b hb => hb ∞)) he
+    (fun f => hμp.mono fun b hb => (hb.2.2 f).2)
+    (fun f => hνp.2.mono fun b hb => (hb.2.2.2 f).2)
+  filter_upwards [he,het] with b hb ht v
+  induction v using ENNReal.recTopCoe with
+  | top => exact ht
+  | coe t => exact hb t
+
+ 
+
+theorem recoveredMass_interior_restart (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
+      initialGreenMeasure (Target.a κ))
+    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
+      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+    ∀ᵐ b ∂W, S b≠∞ →
+      (μ (S b) b).restrict {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=0 ∧
+      ∀v : ℝ≥0∞, (μ (S b+v) b).restrict
+          {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=
+        restartPush (Target.a κ) (lt_trans (by norm_num) (a_gt_quarter hκ hκ8)) b
+          (S b).toNNReal (μ v (restartAt S b)) := by
+  intro μ
+  filter_upwards [recoveredPostStop_eq_recoveredRestart κ hκ hκ8 W hW s c hc hs hsum
+    hmean hpath hconv hS,hconv] with b hb hcV hSb
+  refine ⟨?_,?_⟩
+  · have hz := recoveredPostStop_zero_of_convergence (Target.a κ) (a_gt_quarter hκ hκ8)
+      s c hc hs S b hcV
+    simpa only [recoveredPostStop,ite_eq_right hSb,add_zero,μ] using hz
+  · intro v
+    simpa only [recoveredPostStop,recoveredRestart,ite_eq_right hSb,μ] using hb v
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.LoewnerRestart
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator MassVersions
+open TraceMassContinuity TraceMassIntegration
+
+lemma restartPush_positive (a : ℝ) (ha : 1/4<a) (b : GDriver) (s : ℝ≥0)
+    (μ : Measure UpperPlane) (hμ : 0<μ univ) :
+    0<restartPush a (lt_trans (by norm_num) ha) b s μ univ := by
+  apply pos_iff_ne_zero.mpr
+  intro hz
+  rw [restartPush,Measure.map_apply (continuous_inverseUpper a
+    (lt_trans (by norm_num) ha) b s).measurable MeasurableSet.univ,preimage_univ] at hz
+  have h := (withDensity_apply_eq_zero (restartWeight_measurable a ha b s)).mp hz
+  have hs : {w : UpperPlane | restartWeight a b s w≠0}=univ := by
+    ext w
+    simp only [mem_ofPred_eq,mem_univ,iff_true]
+    exact (restartWeight_pos a (lt_trans (by norm_num) ha) b s w).ne'
+  rw [hs,univ_inter] at h
+  exact hμ.ne' h
+
+ 
+
+theorem recoveredMass_positive_after_stop (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
+      initialGreenMeasure (Target.a κ))
+    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
+      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b)))
+    {S : GDriver→ℝ≥0∞} (hS : IsStoppingTime rawFiltration S) :
+    let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+    ∀ᵐ b ∂W, S b≠∞ → ∀v : ℝ≥0∞, 0<v → 0<(μ (S b+v) b-μ (S b) b) univ := by
+  intro μ
+  filter_upwards [recoveredMass_interior_restart κ hκ hκ8 W hW s c hc hs hsum hmean
+    hpath hconv hS,ae_restartAt hW hS hpath] with b hr hp hSb v hv
+  have hr := hr hSb
+  have hp := (hp hSb).initial_positive v hv
+  let H := {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}
+  have hH : MeasurableSet H := measurableSet_alive_spatial _ _ _
+  have hx : (μ (S b+v) b-μ (S b) b).restrict H=
+      restartPush (Target.a κ) (lt_trans (by norm_num) (a_gt_quarter hκ hκ8)) b
+        (S b).toNNReal (μ v (restartAt S b)) := by
+    rw [Measure.restrict_sub_eq_restrict_sub_restrict hH,hr.1,Measure.sub_zero,hr.2 v]
+  have hpos := restartPush_positive (Target.a κ) (a_gt_quarter hκ hκ8) b
+    (S b).toNNReal (μ v (restartAt S b)) hp
+  rw [←hx] at hpos
+  exact hpos.trans_le (Measure.le_iff'.mp Measure.restrict_le_self univ)
+
+lemma measure_sub_antitone_right {X : Type*} [MeasurableSpace X]
+    (μ : Measure X) {ν ξ : Measure X} (h : ν≤ξ) : μ-ξ≤μ-ν := by
+  simp only [Measure.sub_def]
+  apply sInf_le_sInf
+  intro τ hτ
+  exact (show μ≤τ+ν from hτ).trans (add_le_add le_rfl h)
+
+ 
+
+theorem recoveredMass_positive_increments (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure GDriver) (hW : IsBrownianReal (fun t (b : GDriver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hmean : recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc ∞ ∘ₘ W=
+      initialGreenMeasure (Target.a κ))
+    (hpath : ∀ᵐ b ∂W, TraceMassFacts κ (driverHomeomorph.symm b)
+      (fun v => recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀g : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,g z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
+      atTop (𝓝 (∫z,g z∂recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc v b))) :
+    let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+    ∀ᵐ b ∂W, ∀u v : ℝ≥0, u<v →
+      0<(μ v b-μ u b) univ ∧ ∃K : Set UpperPlane, IsCompact K ∧
+        0<(μ v b-μ u b) K ∧ (μ v b-μ u b) K<∞ := by
+  intro μ
+  have hall : ∀ᵐ b ∂W, ∀q : ℚ, ∀v : ℝ≥0∞, 0<v →
+      0<(μ ((Real.toNNReal (q:ℝ):ℝ≥0∞)+v) b-μ (Real.toNNReal (q:ℝ)) b) univ := by
+    apply ae_all_iff.mpr
+    intro q
+    have hh := recoveredMass_positive_after_stop κ hκ hκ8 W hW s c hc hs hsum
+      hmean hpath hconv (S:=fun _ => (Real.toNNReal (q:ℝ):ℝ≥0∞)) (isStoppingTime_const rawFiltration _)
+    filter_upwards [hh] with b hb
+    exact hb ENNReal.coe_ne_top
+  filter_upwards [hall,hpath] with b hb hp u v huv
+  obtain ⟨q,huq,hqv⟩ := exists_rat_btwn (show (u:ℝ)<(v:ℝ) from huv)
+  let p : ℝ≥0 := Real.toNNReal (q:ℝ)
+  have hq0 : 0≤(q:ℝ) := u.coe_nonneg.trans huq.le
+  have huc : u≤p := by
+    rw [←NNReal.coe_le_coe,Real.coe_toNNReal _ hq0]
+    exact huq.le
+  have hpc : p<v := by
+    rw [←NNReal.coe_lt_coe,Real.coe_toNNReal _ hq0]
+    exact hqv
+  have he : (p:ℝ≥0∞)+((v-p:ℝ≥0):ℝ≥0∞)=(v:ℝ≥0∞) := by
+    rw [←ENNReal.coe_add,add_tsub_cancel_of_le hpc.le]
+  have hpos := hb q (v-p) (ENNReal.coe_pos.mpr (tsub_pos_iff_lt.mpr hpc))
+  change 0<(μ ((p:ℝ≥0∞)+((v-p:ℝ≥0):ℝ≥0∞)) b-μ p b) univ at hpos
+  rw [he] at hpos
+  have hpos' : 0<(μ v b-μ u b) univ := hpos.trans_le
+    ((measure_sub_antitone_right (μ v b) (hp.monotone (ENNReal.coe_le_coe.mpr huc))) univ)
+  let : IsFiniteMeasureOnCompacts (μ v b) := radonLimit_finite _
+  let : IsFiniteMeasureOnCompacts (μ v b-μ u b) := ⟨fun K hK =>
+    ((Measure.sub_le : μ v b-μ u b≤μ v b) K).trans_lt hK.measure_lt_top⟩
+  obtain ⟨K,-,hK,hpK⟩ := isOpen_univ.exists_lt_isCompact hpos'
+  exact ⟨hpos',K,hK,hpK,hK.measure_lt_top⟩
+
+end SLEExactGauge.LoewnerRestart
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma recoveredMass_compact_stronglyAdapted (a : ℝ) (ha : 1/4<a)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (f : C_c(UpperPlane,ℝ)) :
+    StronglyAdapted rawFiltration (fun t : ℝ≥0 => fun b =>
+      ∫z,f z∂recoveredMass a ha s c hc t b) := by
+  intro t
+  let : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+  let K : Kernel ExactSLEGenerator.Driver UpperPlane :=
+    ⟨recoveredMass a ha s c hc t,recoveredMass_raw_measurable a ha s c hc t⟩
+  exact f.continuous.stronglyMeasurable.integral_kernel (κ:=K)
+
+ 
+
+theorem recoveredMass_stopped_condExp (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀k j,j∈s k→0≤c k j) (hs : ∀k j,j∈s k→k≤j)
+    (hsum : ∀k,∑j∈s k,c k j=1)
+    (hconv : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂recoveredMass a ha s c hc v b)))
+    (S : ExactSLEGenerator.Driver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) :
+    W[(fun b => (∫z,f z∂recoveredMass a ha s c hc ∞ b)-
+      (∫z,f z∂recoveredMass a ha s c hc (S b) b)) | hS.measurableSpace] =ᵐ[W]
+      fun b => ∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea := by
+  let μ := recoveredMass a ha s c hc
+  have hcv (v : ExactSLEGenerator.Driver→ℝ≥0∞) : ∀ᵐ b ∂W, Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass a
+        (Real.exp (-2*a*(j+1))) (v b) b)
+      atTop (𝓝 (∫z,f z∂μ (v b) b)) := hconv.mono (fun b hb => hb (v b) f)
+  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (∞:ℝ≥0∞)) :=
+    fun t => by simp
+  have hi (v : ExactSLEGenerator.Driver→ℝ≥0∞) (hv : IsStoppingTime rawFiltration v) :
+      Integrable (fun b => ∫z,f z∂μ (v b) b) W :=
+    memLp_one_iff_integrable.mp (by simpa only [Nat.cast_one] using
+      actual_stopped_radon_limit_memLp a ha W hW v hv s c hc hsum _ f (hcv v) 1)
+  have hg := stoppedSurvivingKernel_compact_mean_integrable a ha W hW S hS f
+  simp_rw [stoppedSurvivingKernel_compact_eval] at hg
+  have hgm : StronglyMeasurable[hS.measurableSpace]
+      (fun b => ∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea) := by
+    let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+    let K : Kernel ExactSLEGenerator.Driver UpperPlane :=
+      ⟨stoppedSurvivingKernel a ha S hS,stoppedSurvivingKernel_raw_measurable a ha S hS⟩
+    have h := f.continuous.stronglyMeasurable.integral_kernel (κ:=K)
+    change StronglyMeasurable (fun b => ∫z,f z∂stoppedSurvivingKernel a ha S hS b) at h
+    simpa only [stoppedSurvivingKernel_compact_eval] using h
+  symm
+  apply ae_eq_condExp_of_forall_setIntegral_eq hS.measurableSpace_le
+    ((hi _ htop).sub (hi S hS)) (fun _ _ _ => hg.integrableOn) _ hgm.aestronglyMeasurable
+  intro U hU _
+  exact (actual_stopped_limit_compact_conditional a ha W hW S hS s c hs hc hsum
+    (fun b => μ (S b) b) (μ ∞) f (hcv S) (hcv (fun _ => ∞)) U hU).symm
+
+ 
+
+theorem recoveredMass_extraction_unique (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (s₁ s₂ : ℕ→Finset ℕ) (c₁ c₂ : ℕ→ℕ→ℝ)
+    (hc₁ : ∀k j,j∈s₁ k→0≤c₁ k j) (hc₂ : ∀k j,j∈s₂ k→0≤c₂ k j)
+    (hs₁ : ∀k j,j∈s₁ k→k≤j) (hs₂ : ∀k j,j∈s₂ k→k≤j)
+    (hsum₁ : ∀k,∑j∈s₁ k,c₁ k j=1) (hsum₂ : ∀k,∑j∈s₂ k,c₂ k j=1)
+    (hconv₁ : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s₁ k,c₁ k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂recoveredMass a ha s₁ c₁ hc₁ v b)))
+    (hconv₂ : ∀ᵐ b ∂W, ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s₂ k,c₂ k j*∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂recoveredMass a ha s₂ c₂ hc₂ v b)))
+    (hcont₁ : ∀f : C_c(UpperPlane,ℝ), ∀ᵐ b ∂W, Continuous (fun v : ℝ≥0 =>
+      ∫z,f z∂recoveredMass a ha s₁ c₁ hc₁ v b))
+    (hcont₂ : ∀f : C_c(UpperPlane,ℝ), ∀ᵐ b ∂W, Continuous (fun v : ℝ≥0 =>
+      ∫z,f z∂recoveredMass a ha s₂ c₂ hc₂ v b))
+    (hmono₁ : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => recoveredMass a ha s₁ c₁ hc₁ v b))
+    (hmono₂ : ∀ᵐ b ∂W, Monotone (fun v : ℝ≥0∞ => recoveredMass a ha s₂ c₂ hc₂ v b))
+    (hexh₁ : ∀ᵐ b ∂W, ∀A : Set UpperPlane, MeasurableSet A → Tendsto
+      (fun n : ℕ => recoveredMass a ha s₁ c₁ hc₁ (n:ℝ≥0) b A)
+      atTop (𝓝 (recoveredMass a ha s₁ c₁ hc₁ ∞ b A)))
+    (hexh₂ : ∀ᵐ b ∂W, ∀A : Set UpperPlane, MeasurableSet A → Tendsto
+      (fun n : ℕ => recoveredMass a ha s₂ c₂ hc₂ (n:ℝ≥0) b A)
+      atTop (𝓝 (recoveredMass a ha s₂ c₂ hc₂ ∞ b A))) :
+    ∀ᵐ b ∂W, ∀v : ℝ≥0∞,
+      recoveredMass a ha s₁ c₁ hc₁ v b=recoveredMass a ha s₂ c₂ hc₂ v b := by
+  let μ := recoveredMass a ha s₁ c₁ hc₁
+  let ν := recoveredMass a ha s₂ c₂ hc₂
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : (μ v b).Regular := radonLimit_regular _
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : (ν v b).Regular := radonLimit_regular _
+  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (∞:ℝ≥0∞)) :=
+    fun t => by simp
+  have hi₁ (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂μ ∞ b) W := by
+    apply memLp_one_iff_integrable.mp
+    simpa only [Nat.cast_one] using (actual_stopped_radon_limit_memLp a ha W hW
+      _ htop s₁ c₁ hc₁ hsum₁ (μ ∞) f (hconv₁.mono (fun b hb => hb ∞ f)) 1)
+  have hi₂ (f : C_c(UpperPlane,ℝ)) : Integrable (fun b => ∫z,f z∂ν ∞ b) W := by
+    apply memLp_one_iff_integrable.mp
+    simpa only [Nat.cast_one] using (actual_stopped_radon_limit_memLp a ha W hW
+      _ htop s₂ c₂ hc₂ hsum₂ (ν ∞) f (hconv₂.mono (fun b hb => hb ∞ f)) 1)
+  have hzero₁ : μ 0=ᵐ[W] 0 := Eventually.of_forall (recoveredMass_zero a ha s₁ c₁ hc₁)
+  have hzero₂ : ν 0=ᵐ[W] 0 := Eventually.of_forall (recoveredMass_zero a ha s₂ c₂ hc₂)
+  have he : ∀ᵐ b ∂W, ∀v : ℝ≥0, μ v b=ν v b := by
+    apply SLEExactGauge.CompensatorUniqueness.ae_all_measure_eq
+      (ℱ:=rawFiltration) (fun v : ℝ≥0 => μ v) (fun v : ℝ≥0 => ν v) (μ ∞) (ν ∞)
+      (recoveredMass_compact_stronglyAdapted a ha s₁ c₁ hc₁)
+      (recoveredMass_compact_stronglyAdapted a ha s₂ c₂ hc₂)
+      hzero₁ hzero₂ hcont₁ hcont₂
+      (hmono₁.mono (fun b hb s t hst => hb (ENNReal.coe_le_coe.mpr hst)))
+      (hmono₂.mono (fun b hb s t hst => hb (ENNReal.coe_le_coe.mpr hst)))
+      (hmono₁.mono (fun b hb v => hb le_top))
+      (hmono₂.mono (fun b hb v => hb le_top)) hi₁ hi₂
+    intro f _ t
+    have h₁ := recoveredMass_stopped_condExp a ha W hW s₁ c₁ hc₁ hs₁ hsum₁ hconv₁
+      _ (isStoppingTime_const rawFiltration t) f
+    have h₂ := recoveredMass_stopped_condExp a ha W hW s₂ c₂ hc₂ hs₂ hsum₂ hconv₂
+      _ (isStoppingTime_const rawFiltration t) f
+    have hh := h₁.trans h₂.symm
+    rw [IsStoppingTime.measurableSpace_const] at hh
+    simpa only [μ, ν, ENNReal.some_eq_coe'] using hh
+  filter_upwards [he,hexh₁,hexh₂] with b hb h₁ h₂ v
+  induction v using ENNReal.recTopCoe with
+  | coe v => exact hb v
+  | top =>
+    apply Measure.ext
+    intro A hA
+    have h : Tendsto (fun n : ℕ => μ (n:ℝ≥0) b A) atTop (𝓝 (μ ∞ b A)) := h₁ A hA
+    simp only [hb] at h
+    exact tendsto_nhds_unique h (h₂ A hA)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceMassContinuity
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions TraceMassIntegration
+open LoewnerRestart
+
+ 
+
+theorem actual_canonical_trace_mass (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    ∃ (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j),
+      (∀k j,j∈s k→k≤j) ∧ (∀k,∑j∈s k,c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      (∀v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀b, μ 0 b=0) ∧ (∀v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      μ ∞ ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ) ∧
+      (∀ (S : GDriver→ℝ≥0∞) (hS : IsStoppingTime rawFiltration S),
+        Measurable[hS.measurableSpace] (fun b => μ (S b) b) ∧
+        (∀f : C_c(UpperPlane,ℝ), ∀m : ℕ,
+          MemLp (fun b => ∫z,f z∂μ (S b) b) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
+        (∀K : Set UpperPlane, IsCompact K → ∀m : ℕ,
+          MemLp (fun b => (μ (S b) b K).toReal) (m:ℝ≥0∞) (W.map driverHomeomorph)) ∧
+        (∀f : C_c(UpperPlane,ℝ), Tendsto (fun k => eLpNorm
+          (fun b => (∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+            (Real.exp (-2*(Target.a κ)*(j+1))) (S b) b)-(∫z,f z∂μ (S b) b))
+              1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
+        (∀H : GDriver × UpperPlane→ℝ≥0∞,
+          Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H →
+          (∫⁻b,∫⁻z,H (b,z)∂μ ∞ b∂(W.map driverHomeomorph))=
+            (∫⁻b,∫⁻z,H (b,z)∂μ (S b) b∂(W.map driverHomeomorph))+
+            ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen (Target.a κ) (S b) b z.1)
+              ∂upperArea∂(W.map driverHomeomorph)) ∧
+        (∀ᵐ b ∂(W.map driverHomeomorph), S b≠∞ →
+          (μ (S b) b).restrict {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=0 ∧
+          ∀v : ℝ≥0∞, (μ (S b+v) b).restrict
+              {z : UpperPlane | alive (Target.a κ) b.1 (S b).toNNReal z.1}=
+            restartPush (Target.a κ) (lt_trans (by norm_num) (a_gt_quarter hκ hκ8)) b
+              (S b).toNNReal (μ v (restartAt S b)))) ∧
+      ∀ᵐ b ∂W, TraceMassFacts κ b (fun v => μ v (driverHomeomorph b)) ∧
+        (∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+          (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+            (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
+          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
+        (∀u v : ℝ≥0, u<v →
+          0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) univ ∧
+          ∃K : Set UpperPlane, IsCompact K ∧
+            0<(μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) K ∧
+            (μ v (driverHomeomorph b)-μ u (driverHomeomorph b)) K<∞) := by
+  obtain ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,hstop,hpath⟩ :=
+    actual_common_trace_mass κ hκ hκ8 W hW
+  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  have hBW := brownian_map_driver W hW
+  have hp : ∀ᵐ b ∂(W.map driverHomeomorph),
+      TraceMassFacts κ (driverHomeomorph.symm b) (fun v => μ v b) := by
+    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+    filter_upwards [hpath] with b hb
+    change TraceMassFacts κ (driverHomeomorph.symm (driverHomeomorph b))
+      (fun v => μ v (driverHomeomorph b))
+    simpa only [Homeomorph.symm_apply_apply] using hb.1
+  have hv : ∀ᵐ b ∂(W.map driverHomeomorph), ∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) v b)
+      atTop (𝓝 (∫z,f z∂μ v b)) := by
+    apply driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mpr
+    exact hpath.mono (fun b hb => hb.2)
+  refine ⟨s,c,hc,hs,hsum,hreg,hzero,hmeas,hprog,hmean,?_,?_⟩
+  · intro S hS
+    rcases hstop S hS with ⟨hsm,hmom,hK,hL,hH⟩
+    exact ⟨hsm,hmom,hK,hL,hH,recoveredMass_interior_restart κ hκ hκ8
+      (W.map driverHomeomorph) hBW s c hc hs hsum hmean hp hv hS⟩
+  · have hpos := recoveredMass_positive_increments κ hκ hκ8
+      (W.map driverHomeomorph) hBW s c hc hs hsum hmean hp hv
+    have hpos' := driverMeasurableEquiv.measurableEmbedding.ae_map_iff.mp hpos
+    filter_upwards [hpath,hpos'] with b hb hbp
+    exact ⟨hb.1,hb.2,hbp⟩
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
+
+lemma diskMass_mono {ν η : Measure UpperPlane} (h : ν≤η) (a θ : ℝ) :
+    diskMass a θ ν≤diskMass a θ η := by
+  apply Measure.map_mono _ (diskMap_measurable θ)
+  apply Measure.le_iff.mpr
+  intro U hU
+  rw [withDensity_apply _ hU,withDensity_apply _ hU]
+  exact lintegral_mono' (Measure.restrict_mono le_rfl h) le_rfl
+
+lemma diskMass_inner_le_enorm (a θ : ℝ) (ha : 1/4<a) (hθ : 0<θ) (hθπ : θ<Real.pi)
+    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] :
+    diskMass a θ ν (Metric.closedBall (0:ℂ) (1/4))≤‖∫z,diskTest a θ hθ hθπ z∂ν‖ₑ :=
+  (diskMass_inner_le_test a θ ha hθ hθπ ν).trans (Real.ofReal_le_enorm _)
+
+ 
+
+theorem common_limit_disk_tilted_moment (a : ℝ) (ha : 1/4<a)
+    (m : ℕ) (hm : 1 ≤ m) :
+    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
+      (∀b,IsFiniteMeasureOnCompacts (μ ∞ b)) → Measurable (μ ∞) →
+      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
+      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
+          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
+      ∀(θ : ℝ) (_hθ : 0<θ) (_hθπ : θ<Real.pi) (l : ℕ)
+        (T : ExactSLEGenerator.Driver→ℝ≥0∞),
+      (∫⁻b,(diskMass a θ (μ (T b) b) (Metric.closedBall (0:ℂ) (1/4)))^m
+        ∂radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)))≤
+        D*ENNReal.ofReal ((Real.sin θ)^(-p a)) := by
+  let K : Set ℂ := Subtype.val '' tsupport diskCutoff
+  have hK : IsCompact K := diskCutoff.hasCompactSupport.image continuous_subtype_val
+  have hcK : ∀z∈K,(1/2:ℝ)≤2*z.im := by
+    rintro z ⟨w,hw,rfl⟩
+    have hh := height_of_ball (diskCutoff_support w hw).le
+    linarith
+  obtain ⟨D,hD,hDb⟩ := actual_normalized_tilted_moment_bound a ha m hm K
+    hK.measurableSet hK.isBounded (1/2) (by norm_num) (by norm_num) hcK
+    (2^(2-alpha a)) (Real.rpow_nonneg (by norm_num) _)
+  refine ⟨D,hD,?_⟩
+  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim θ hθ hθπ l T
+  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ ∞ b) := hμfin b
+  let f := diskTest a θ hθ hθπ
+  have hfm : Measurable (fun b=>∫z,f z∂μ ∞ b) := by
+    let Kμ : Kernel ExactSLEGenerator.Driver UpperPlane := ⟨μ ∞,hμ⟩
+    exact (f.continuous.stronglyMeasurable.integral_kernel (κ:=Kμ)).measurable
+  have hb := hDb W hW θ hθ hθπ f (diskTest_support a θ hθ hθπ)
+    (diskTest_bound a θ hθ hθπ) s coeff hcoeff hsum ∞ (μ ∞) hfm
+    (hlim.mono (fun b hb=>hb f)) l
+  refine le_trans (lintegral_mono_ae ?_) hb
+  have habs : radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) ≪ W :=
+    withDensity_absolutelyContinuous W _
+  filter_upwards [habs.ae_le hmono] with b hb
+  exact pow_le_pow_left' ((Measure.le_iff'.mp (diskMass_mono (hb (T b)) a θ)
+    _).trans (diskMass_inner_le_enorm a θ ha hθ hθπ (μ ∞ b))) m
+
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ReverseFlow.lean b/lean/OAI/Probability/SLEGauge/ReverseFlow.lean
index 3c49bba349fe889411e9d6ee30a65d997292a89a..2849a4782d55580a1919de082c0ff404eab3031a 100644
--- a/lean/OAI/Probability/SLEGauge/ReverseFlow.lean
+++ b/lean/OAI/Probability/SLEGauge/ReverseFlow.lean
@@ -1,462 +1,462 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ReverseMoments
-import OAI.Probability.SLEGauge.BrownianShifts
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-noncomputable section
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative
-lemma forward_integral_eq_reverse {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
-    (b : Driver) :
-    (∫ u in (0:ℝ)..T, -(a κ:ℂ)/
-      (flow κ b (reversedCurve κ y T b T+driverValue b T) (timeClip T u)-driverValue b (timeClip T u))^2) =
-        -(∫ s in (0:ℝ)..T, reverseCoeff κ y T b s) := by
-  have hf := reverse_reparam_solution hκ hy hT b
-  have hrev (u : ℝ) (hu : u ∈ Icc 0 T) : T-u ∈ Icc 0 T := by constructor <;> linarith [hu.1,hu.2]
-  have heq (u : ℝ) (hu : u ∈ Icc 0 T) :
-      -(a κ:ℂ)/(flow κ b (reversedCurve κ y T b T+driverValue b T) (timeClip T u)-driverValue b (timeClip T u))^2 =
-        -reverseCoeff κ y T b (T-u) := by
-    rw [timeClip_eq hu,hf.flow_eqOn hu]
-    have hβ := BrownianRestart.backward_before (hrev u hu) b
-    simp only [sub_sub_cancel] at hβ
-    simp only [reverseCoeff,hβ,driverValue,Complex.ofReal_sub]
-    have hden : reversedCurve κ y T b (T-u)+(b T.toNNReal:ℂ)-b u.toNNReal =
-        reversedCurve κ y T b (T-u)-((b u.toNNReal:ℂ)-b T.toNNReal) := by ring
-    rw [hden]
-    simp only [div_eq_mul_inv,inv_pow,neg_mul]
-  calc
-    _ = (∫ u in (0:ℝ)..T, -reverseCoeff κ y T b (T-u)) := by
-      apply intervalIntegral.integral_congr
-      intro u hu
-      exact heq u (by simpa only [uIcc_of_le hT] using hu)
-    _ = -(∫ u in (0:ℝ)..T, reverseCoeff κ y T b (T-u)) := intervalIntegral.integral_neg
-    _ = -(∫ s in (0:ℝ)..T, reverseCoeff κ y T b s) := by
-      rw [intervalIntegral.integral_comp_sub_left]
-      simp only [sub_self,sub_zero]
-
- 
-
-theorem inverseFlow_deriv_norm_eq_reverse {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
-    (b : Driver) : ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖ =
-      Real.exp ((reversedState κ y T b T).2.2) := by
-  have hw : 0 < ((driverValue b T:ℂ)+y*Complex.I).im := by simpa using hy
-  have hd := (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.deriv
-  rw [hd,inverseFlow_eq_reversedCurve hκ hy hT b,
-    (flow_hasDerivAt_initial hT (reverse_reparam_solution hκ hy hT b).surviving).deriv,
-    forward_integral_eq_reverse hκ hy hT b,norm_inv,Complex.norm_exp,Complex.neg_re,
-    ← Real.exp_neg,neg_neg,reversed_log_eq_integral hκ hy hT b]
-end SLEExactGauge.LiteralTrace
-
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory ProbabilityTheory
-
-lemma measurable_reverse_log (a y T : ℝ) (hy : 0 < y) :
-    Measurable (fun b : Driver =>
-      (ReverseDerivative.reverseFlow a (y/4) (T+y^2) y b T).2.2) :=
-  (ReverseDerivative.measurable_driven (by positivity : 0 < y/4)
-    a (ReverseDerivative.initial (T+y^2) y) T).fst.snd.snd
-
-lemma measurable_inverse_deriv_norm {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T) :
-    Measurable (fun b : Driver => ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖) := by
-  simp_rw [inverseFlow_deriv_norm_eq_reverse hκ hy hT]
-  exact ((measurable_reverse_log (a κ) y T hy).comp (BrownianRestart.measurable_backward T.toNNReal)).exp
-
- 
-
-theorem inverse_deriv_moment {κ y T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    (∫⁻ b : Driver, ENNReal.ofReal
-      (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W) ≤
-      ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta (a κ)/2)) := by
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  let F := fun b : Driver => ENNReal.ofReal (Real.exp
-    (lambda (a κ)*(ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y b T).2.2))
-  have mF : Measurable F :=
-    (measurable_const.mul (measurable_reverse_log (a κ) y T hy)).exp.ennreal_ofReal
-  have hid (b : Driver) :
-      ENNReal.ofReal (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) =
-        F (BrownianRestart.backward T.toNNReal b) := by
-    rw [inverseFlow_deriv_norm_eq_reverse hκ hy hT,
-      Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,mul_comm]
-    rfl
-  simp_rw [hid]
-  rw [← lintegral_map mF (BrownianRestart.measurable_backward T.toNNReal),
-    BrownianRestart.backward_preserves_law hW]
-  exact ReverseDerivative.reverse_logderiv_moment ha hy hT hW
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory ProbabilityTheory
-
-lemma lambda_gt_one {a : ℝ} (ha : 0 < a) : 1 < lambda a := by
-  unfold lambda
-  have hh : 0 < 3/(16*a) := by positivity
-  linarith
-
-lemma inverse_deriv_first_moment {κ y T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    (∫⁻ b : Driver, ENNReal.ofReal ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖ ∂W) ≤ 2 := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  have hl := lambda_gt_one (show 0 < a κ by linarith)
-  have hp (x : ℝ) (hx : 0 ≤ x) : x ≤ 1+x^(lambda (a κ)) := by
-    by_cases h : x ≤ 1
-    · exact h.trans (le_add_of_nonneg_right (Real.rpow_nonneg hx _))
-    · have he := Real.rpow_le_rpow_of_exponent_le (le_of_not_ge h) hl.le
-      simpa only [Real.rpow_one] using he.trans (le_add_of_nonneg_left (by norm_num : (0:ℝ) ≤ 1))
-  calc
-    _ ≤ ∫⁻ b : Driver, (1+ENNReal.ofReal (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ)))) ∂W := by
-      apply lintegral_mono
-      intro b
-      dsimp only
-      rw [← ENNReal.ofReal_one,← ENNReal.ofReal_add (by norm_num) (Real.rpow_nonneg (norm_nonneg _) _)]
-      exact ENNReal.ofReal_le_ofReal (hp _ (norm_nonneg _))
-    _ = 1+(∫⁻ b : Driver, ENNReal.ofReal (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W) := by
-      rw [lintegral_add_left measurable_const]
-      simp
-    _ ≤ 1+ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta (a κ)/2)) :=
-      add_le_add le_rfl (inverse_deriv_moment hκ hκ8 hy hT hW)
-    _ ≤ 2 := by
-      have hr : 3*y^2/(T+3*y^2) ≤ 1 := (div_le_one (by positivity)).mpr (by linarith)
-      have hh := Real.rpow_le_rpow (by positivity : 0 ≤ 3*y^2/(T+3*y^2)) hr
-        (by linarith [(parameters ha).1] : 0 ≤ zeta (a κ)/2)
-      simp only [Real.one_rpow] at hh
-      have hh' : ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta (a κ)/2)) ≤ 1 := by
-        simpa using ENNReal.ofReal_le_ofReal hh
-      calc _ ≤ 1+1 := add_le_add le_rfl hh'
-           _ = 2 := by norm_num
-
- 
-def radialCoeff (κ : ℝ) (b : Driver) (T y : ℝ) : ℂ :=
-  if 0 < y then deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)*Complex.I else 0
-
-lemma continuousOn_radialCoeff {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
-    ContinuousOn (radialCoeff κ b T) (Ioi 0) := by
-  have hopen : IsOpen {w : ℂ | 0 < w.im} := isOpen_lt continuous_const Complex.continuous_im
-  have hc := (((flow_conformal_bijection hκ b hT).2.2.2).analyticOnNhd hopen).deriv.continuousOn
-  have hline : Continuous (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I) := by fun_prop
-  have hh := (hc.comp hline.continuousOn (show MapsTo (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I) (Ioi 0) {w : ℂ | 0 < w.im} from by intro y hy; simpa using hy)).mul (continuousOn_const (c := Complex.I))
-  apply hh.congr
-  intro y hy
-  rw [radialCoeff,ite_eq_left (show 0 < y from hy)]
-  rfl
-
-lemma measurable_positive_radial_norm {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) :
-    Measurable (fun p : (Ioi (0:ℝ)) × Driver =>
-      ‖deriv (inverseFlow κ p.2 T) (driverValue p.2 T+(p.1:ℝ)*Complex.I)‖) := by
-  exact measurable_uncurry_of_continuous_of_measurable
-    (u := fun (y : Ioi (0:ℝ)) (b : Driver) => ‖deriv (inverseFlow κ b T) (driverValue b T+(y:ℝ)*Complex.I)‖) (by
-    intro b
-    have hh := (continuousOn_radialCoeff hκ hT b).norm.domRestrict
-    have he : ((Ioi (0:ℝ)).domRestrict (fun y => ‖radialCoeff κ b T y‖)) =
-        (fun y : Ioi (0:ℝ) => ‖deriv (inverseFlow κ b T) (driverValue b T+(y:ℝ)*Complex.I)‖) := by
-      funext y
-      change ‖radialCoeff κ b T y‖ = _
-      rw [radialCoeff,ite_eq_left (show 0 < (y:ℝ) from y.property),norm_mul,Complex.norm_I,mul_one]
-    rwa [he] at hh) (by
-    intro y
-    exact measurable_inverse_deriv_norm hκ y.property hT)
-
-lemma measurable_radial_norm {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) :
-    Measurable (fun p : ℝ × Driver => ‖radialCoeff κ p.2 T p.1‖) := by
-  have hm := measurable_positive_radial_norm hκ hT
-  apply measurable_of_restrict_of_restrict_compl (s := {p : ℝ × Driver | 0 < p.1})
-    (measurableSet_lt measurable_const measurable_fst)
-  · have mm : Measurable (fun p : {p : ℝ × Driver | 0 < p.1} =>
-        ((⟨p.val.1,p.property⟩ : Ioi (0:ℝ)),p.val.2)) :=
-      (measurable_subtype_coe.fst.subtype_mk).prodMk measurable_subtype_coe.snd
-    have he : ({p : ℝ × Driver | 0 < p.1}.domRestrict
-        (fun p : ℝ × Driver => ‖radialCoeff κ p.2 T p.1‖)) =
-        (fun p : {p : ℝ × Driver | 0 < p.1} =>
-          ‖deriv (inverseFlow κ p.val.2 T) (driverValue p.val.2 T+p.val.1*Complex.I)‖) := by
-      funext p
-      change ‖radialCoeff κ p.val.2 T p.val.1‖ = _
-      rw [radialCoeff,ite_eq_left (show 0 < p.val.1 from p.property),norm_mul,Complex.norm_I,mul_one]
-    rw [he]
-    have hh := hm.comp mm
-    change Measurable (fun p : {p : ℝ × Driver | 0 < p.1} =>
-      ‖deriv (inverseFlow κ p.val.2 T) (driverValue p.val.2 T+p.val.1*Complex.I)‖) at hh
-    exact hh
-  · have he : ({p : ℝ × Driver | 0 < p.1}ᶜ.domRestrict
-        (fun p : ℝ × Driver => ‖radialCoeff κ p.2 T p.1‖)) =
-        (fun _ : ↥({p : ℝ × Driver | 0 < p.1}ᶜ) => (0:ℝ)) := by
-      funext p
-      change ‖radialCoeff κ p.val.2 T p.val.1‖ = 0
-      rw [radialCoeff,ite_eq_right (show ¬0 < p.val.1 from p.property),norm_zero]
-    rw [he]
-    exact measurable_const
-
-lemma radial_length_ae_finite {κ T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W, (∫⁻ y in Icc (0:ℝ) 1, ENNReal.ofReal ‖radialCoeff κ b T y‖) < (∞:ℝ≥0∞) := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let F := fun b : Driver => ∫⁻ y in Icc (0:ℝ) 1, ENNReal.ofReal ‖radialCoeff κ b T y‖
-  have hm := (measurable_radial_norm hκ hT).ennreal_ofReal
-  have mF : Measurable F := (hm.comp measurable_swap).lintegral_prod_right'
-  apply ae_lt_top mF
-  have hs : (∫⁻ b, F b ∂W) ≤ 2 := by
-    rw [show (∫⁻ b, F b ∂W) = ∫⁻ y in Icc (0:ℝ) 1, ∫⁻ b, ENNReal.ofReal ‖radialCoeff κ b T y‖ ∂W from
-      lintegral_lintegral_swap (hm.comp measurable_swap).aemeasurable]
-    calc
-      _ ≤ ∫⁻ y in Icc (0:ℝ) 1, (2:ℝ≥0∞) := by
-        apply lintegral_mono
-        intro y
-        by_cases hy : 0 < y
-        · simpa only [radialCoeff,ite_eq_left hy,norm_mul,Complex.norm_I,mul_one] using
-            inverse_deriv_first_moment hκ hκ8 hy hT hW
-        · simp only [radialCoeff,ite_eq_right hy,norm_zero,ENNReal.ofReal_zero,lintegral_zero]
-          exact bot_le
-      _ = 2 := by simp
-  exact ne_top_of_le_ne_top (by norm_num) hs
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open MeasureTheory
-
-lemma measurable_radialCoeff {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
-    Measurable (radialCoeff κ b T) := by
-  apply measurable_of_restrict_of_restrict_compl (s := Ioi (0:ℝ)) measurableSet_Ioi
-  · exact (continuousOn_radialCoeff hκ hT b).domRestrict.measurable
-  · have he : ((Ioi (0:ℝ))ᶜ.domRestrict (radialCoeff κ b T)) =
-        (fun _ : ↥((Ioi (0:ℝ))ᶜ) => (0:ℂ)) := by
-      funext y
-      change radialCoeff κ b T y.val = 0
-      rw [radialCoeff,ite_eq_right (show ¬0 < y.val from y.property)]
-    rw [he]
-    exact measurable_const
-
-lemma hasDerivAt_radial {κ T y : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (hy : 0 < y) (b : Driver) :
-    HasDerivAt (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
-      (radialCoeff κ b T y) y := by
-  have hw : 0 < ((driverValue b T:ℂ)+y*Complex.I).im := by simpa using hy
-  have hf := (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.differentiableAt.hasDerivAt
-  have hl : HasDerivAt (fun z : ℂ => (driverValue b T:ℂ)+z*Complex.I) Complex.I (y:ℂ) := by
-    convert! (hasDerivAt_const (y:ℂ) (driverValue b T:ℂ)).add ((hasDerivAt_id (y:ℂ)).mul_const Complex.I) using 1; simp
-  convert! (hf.comp (y:ℂ) hl).comp_ofReal using 1; simp only [radialCoeff,ite_eq_left hy]
-
-lemma radial_limit_of_integrable {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
-    (hi : IntegrableOn (radialCoeff κ b T) (Icc (0:ℝ) 1)) :
-    ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
-      (𝓝[>] 0) (𝓝 z) := by
-  let f := fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I)
-  let v := fun y : ℝ => radialCoeff κ b T y
-  have hi' : IntegrableOn v (uIcc (0:ℝ) 1) := by simpa only [uIcc_of_le zero_le_one] using hi
-  have hcont := intervalIntegral.continuousOn_primitive_interval_left hi'
-  have hevent : ∀ᶠ y : ℝ in 𝓝[>] 0, y ∈ Ioc (0:ℝ) 1 := by
-    filter_upwards [self_mem_nhdsWithin,
-      (show ∀ᶠ y : ℝ in 𝓝[>] 0, y < 1 from nhdsWithin_le_nhds (Iio_mem_nhds zero_lt_one))] with y hy hy1
-    exact ⟨hy,hy1.le⟩
-  have hmap : Tendsto (fun y : ℝ => y) (𝓝[>] 0) (𝓝[uIcc (0:ℝ) 1] 0) := by
-    apply tendsto_nhdsWithin_iff.mpr
-    refine ⟨tendsto_id'.mpr nhdsWithin_le_nhds,?_⟩
-    filter_upwards [hevent] with y hy
-    simpa only [uIcc_of_le zero_le_one] using (show y ∈ Icc (0:ℝ) 1 from ⟨hy.1.le,hy.2⟩)
-  have hlim := ((hcont 0 left_mem_uIcc).tendsto.comp hmap).const_sub (f 1)
-  refine ⟨f 1-(∫ u in (0:ℝ)..1, v u),hlim.congr' ?_⟩
-  filter_upwards [hevent] with y hy
-  have hd (u : ℝ) (hu : u ∈ uIcc y 1) : HasDerivAt f (v u) u := by
-    rw [uIcc_of_le hy.2] at hu
-    exact hasDerivAt_radial hκ hT (hy.1.trans_le hu.1) b
-  have hint : IntervalIntegrable v volume y 1 := by
-    apply IntegrableOn.intervalIntegrable
-    simpa only [uIcc_of_le hy.2] using
-      (hi.mono_set (show Icc y 1 ⊆ Icc (0:ℝ) 1 from Icc_subset_Icc hy.1.le le_rfl))
-  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt hd hint
-  simp only [Function.comp_apply]
-  rw [he]
-  exact sub_sub_cancel _ _
-
- 
-
-theorem radial_limit_ae {κ T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W, ∃ z : ℂ,
-      Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I)) (𝓝[>] 0) (𝓝 z) := by
-  filter_upwards [radial_length_ae_finite hκ hκ8 hT hW] with b hb
-  apply radial_limit_of_integrable hκ hT b
-  exact ⟨(measurable_radialCoeff hκ hT b).aestronglyMeasurable,
-    (hasFiniteIntegral_iff_norm _).mpr hb⟩
-
- 
-
-theorem trace_limit_countable {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) (t : ℕ → ℝ) (ht : ∀ n, 0 ≤ t n) :
-    ∀ᵐ b ∂W, ∀ n : ℕ,
-      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t n) (driverValue b (κ*t n)+y*Complex.I))
-        (𝓝[>] 0) (𝓝 (trace κ b (t n))) := by
-  have hh := ae_all_iff.mpr (fun n => radial_limit_ae hκ hκ8 (mul_nonneg hκ.le (ht n)) hW)
-  filter_upwards [hh] with b hb
-  intro n
-  obtain ⟨z,hz⟩ := hb n
-  have he : trace κ b (t n) = z := by
-    exact hz.limUnder_eq
-  rwa [he]
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory
-
- 
-
-def traceBeta (κ : ℝ) : ℝ :=
-  (lambda (a κ)+zeta (a κ)-2)/(2*lambda (a κ))
-
-def traceEta (κ : ℝ) : ℝ := (lambda (a κ)+zeta (a κ)-2)/2
-
-lemma trace_exponents {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
-    0 < traceBeta κ ∧ traceBeta κ < 1 ∧ 0 < traceEta κ ∧
-      zeta (a κ)-(traceBeta κ-1)*lambda (a κ) = 2+traceEta κ := by
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  have hp := parameters ha
-  have hsum : 0 < lambda (a κ)+zeta (a κ)-2 := by linarith [hp.2.2.1]
-  have hden : 0 < 2*lambda (a κ) := by linarith [hp.2.1]
-  have hlt : zeta (a κ) < lambda (a κ)+2 := by
-    unfold lambda zeta
-    have : 0 < a κ := by linarith
-    have h1 : 0 ≤ 1/(16*a κ) := by positivity
-    have h3 : 0 ≤ 3/(16*a κ) := by positivity
-    linarith
-  refine ⟨div_pos hsum hden, (div_lt_one hden).mpr (by linarith),div_pos hsum (by norm_num),?_⟩
-  unfold traceBeta traceEta
-  field_simp [ne_of_gt hp.2.1]
-  ring
-
-lemma inverse_deriv_moment_away {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    (∫⁻ b : Driver, ENNReal.ofReal
-      (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W) ≤
-      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ))) := by
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  have hz := (parameters ha).1
-  have hT0 : 0 ≤ T := hδ.le.trans hT
-  apply (inverse_deriv_moment hκ hκ8 hy (hδ.le.trans hT) hW).trans
-  apply ENNReal.ofReal_le_ofReal
-  calc
-    _ ≤ ((3/δ)*y^2)^(zeta (a κ)/2) := by
-      apply Real.rpow_le_rpow (by positivity) _ (by positivity)
-      calc
-        _ ≤ 3*y^2/δ := div_le_div_of_nonneg_left (by positivity) hδ (by nlinarith)
-        _ = (3/δ)*y^2 := by ring
-    _ = (3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)) := by
-      rw [Real.mul_rpow (by positivity) (sq_nonneg y),
-        ← Real.rpow_natCast_mul hy.le 2 (zeta (a κ)/2)]
-      congr 1
-      congr 1
-      norm_num
-      ring
-
-lemma inverse_deriv_tail_away {κ y T δ K : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) (hK : 0 < K)
-    {W : WienerLaw} (hW : IsWiener W) :
-    W {b : Driver | K ≤ ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖} ≤
-      ENNReal.ofReal (((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/K^(lambda (a κ))) := by
-  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
-  have hl := (parameters ha).2.1
-  have hm := ((measurable_inverse_deriv_norm hκ hy (hδ.le.trans hT)).pow_const
-    (lambda (a κ))).ennreal_ofReal
-  have hKP : 0 < K^(lambda (a κ)) := Real.rpow_pos_of_pos hK _
-  calc
-    _ ≤ W {b : Driver | ENNReal.ofReal (K^(lambda (a κ))) ≤ ENNReal.ofReal
-        (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ)))} := by
-      apply measure_mono
-      intro b hb
-      exact ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow hK.le hb hl.le)
-    _ ≤ (∫⁻ b : Driver, ENNReal.ofReal
-        (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W)/
-        ENNReal.ofReal (K^(lambda (a κ))) :=
-      meas_ge_le_lintegral_div hm.aemeasurable (ne_of_gt (ENNReal.ofReal_pos.mpr hKP)) ENNReal.ofReal_ne_top
-    _ ≤ ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/
-        ENNReal.ofReal (K^(lambda (a κ))) := by
-      exact ENNReal.div_le_div_right (inverse_deriv_moment_away hκ hκ8 hy hδ hT hW) _
-    _ = ENNReal.ofReal (((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/K^(lambda (a κ))) :=
-      (ENNReal.ofReal_div_of_pos hKP).symm
-
-lemma inverse_deriv_beta_tail {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) {W : WienerLaw} (hW : IsWiener W) :
-    W {b : Driver | y^(traceBeta κ-1) ≤
-      ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖} ≤
-      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(2+traceEta κ)) := by
-  have hp := trace_exponents hκ hκ8
-  have he : ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/(y^(traceBeta κ-1))^(lambda (a κ)) =
-      (3/δ)^(zeta (a κ)/2)*y^(2+traceEta κ) := by
-    rw [← Real.rpow_mul hy.le, mul_div_assoc, ← Real.rpow_sub hy,
-      hp.2.2.2]
-  simpa only [he] using (inverse_deriv_tail_away hκ hκ8 hy hδ hT
-    (Real.rpow_pos_of_pos hy (traceBeta κ-1)) hW)
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open ReverseDerivative MeasureTheory
-
-lemma lintegral_pow_le_mass {α : Type*} [MeasurableSpace α] (μ : Measure α)
-    {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {p : ℝ} (hp : 1 < p) :
-    (∫⁻ x, f x ∂μ)^p ≤ (μ univ)^(p-1)*(∫⁻ x, (f x)^p ∂μ) := by
-  have hp0 : 0 < p := by linarith
-  let q := p/(p-1)
-  have hpq : p.HolderConjugate q := by
-    rw [Real.holderConjugate_iff]
-    refine ⟨hp,?_⟩
-    dsimp [q]
-    field_simp
-    ring
-  have hh := ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf
-    (aemeasurable_const (b := (1:ℝ≥0∞)))
-  simp only [Pi.mul_apply,mul_one,ENNReal.one_rpow,lintegral_const,one_mul] at hh
-  have hh' := ENNReal.rpow_le_rpow hh hp0.le
-  rw [ENNReal.mul_rpow_of_nonneg _ _ hp0.le,← ENNReal.rpow_mul,
-    one_div_mul_cancel (ne_of_gt hp0),ENNReal.rpow_one,← ENNReal.rpow_mul] at hh'
-  have he : (1/q)*p = p-1 := by dsimp [q]; field_simp
-  rw [he,mul_comm] at hh'
-  exact hh'
-
- 
-def radialLength (κ : ℝ) (b : Driver) (T y : ℝ) : ℝ≥0∞ :=
-  ∫⁻ u in Ioc (0:ℝ) y, ENNReal.ofReal ‖radialCoeff κ b T u‖
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ReverseMoments
+import OAI.Probability.SLEGauge.BrownianShifts
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+noncomputable section
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative
+lemma forward_integral_eq_reverse {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
+    (b : Driver) :
+    (∫ u in (0:ℝ)..T, -(a κ:ℂ)/
+      (flow κ b (reversedCurve κ y T b T+driverValue b T) (timeClip T u)-driverValue b (timeClip T u))^2) =
+        -(∫ s in (0:ℝ)..T, reverseCoeff κ y T b s) := by
+  have hf := reverse_reparam_solution hκ hy hT b
+  have hrev (u : ℝ) (hu : u ∈ Icc 0 T) : T-u ∈ Icc 0 T := by constructor <;> linarith [hu.1,hu.2]
+  have heq (u : ℝ) (hu : u ∈ Icc 0 T) :
+      -(a κ:ℂ)/(flow κ b (reversedCurve κ y T b T+driverValue b T) (timeClip T u)-driverValue b (timeClip T u))^2 =
+        -reverseCoeff κ y T b (T-u) := by
+    rw [timeClip_eq hu,hf.flow_eqOn hu]
+    have hβ := BrownianRestart.backward_before (hrev u hu) b
+    simp only [sub_sub_cancel] at hβ
+    simp only [reverseCoeff,hβ,driverValue,Complex.ofReal_sub]
+    have hden : reversedCurve κ y T b (T-u)+(b T.toNNReal:ℂ)-b u.toNNReal =
+        reversedCurve κ y T b (T-u)-((b u.toNNReal:ℂ)-b T.toNNReal) := by ring
+    rw [hden]
+    simp only [div_eq_mul_inv,inv_pow,neg_mul]
+  calc
+    _ = (∫ u in (0:ℝ)..T, -reverseCoeff κ y T b (T-u)) := by
+      apply intervalIntegral.integral_congr
+      intro u hu
+      exact heq u (by simpa only [uIcc_of_le hT] using hu)
+    _ = -(∫ u in (0:ℝ)..T, reverseCoeff κ y T b (T-u)) := intervalIntegral.integral_neg
+    _ = -(∫ s in (0:ℝ)..T, reverseCoeff κ y T b s) := by
+      rw [intervalIntegral.integral_comp_sub_left]
+      simp only [sub_self,sub_zero]
+
+ 
+
+theorem inverseFlow_deriv_norm_eq_reverse {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T)
+    (b : Driver) : ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖ =
+      Real.exp ((reversedState κ y T b T).2.2) := by
+  have hw : 0 < ((driverValue b T:ℂ)+y*Complex.I).im := by simpa using hy
+  have hd := (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.deriv
+  rw [hd,inverseFlow_eq_reversedCurve hκ hy hT b,
+    (flow_hasDerivAt_initial hT (reverse_reparam_solution hκ hy hT b).surviving).deriv,
+    forward_integral_eq_reverse hκ hy hT b,norm_inv,Complex.norm_exp,Complex.neg_re,
+    ← Real.exp_neg,neg_neg,reversed_log_eq_integral hκ hy hT b]
+end SLEExactGauge.LiteralTrace
+
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory ProbabilityTheory
+
+lemma measurable_reverse_log (a y T : ℝ) (hy : 0 < y) :
+    Measurable (fun b : Driver =>
+      (ReverseDerivative.reverseFlow a (y/4) (T+y^2) y b T).2.2) :=
+  (ReverseDerivative.measurable_driven (by positivity : 0 < y/4)
+    a (ReverseDerivative.initial (T+y^2) y) T).fst.snd.snd
+
+lemma measurable_inverse_deriv_norm {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y) (hT : 0 ≤ T) :
+    Measurable (fun b : Driver => ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖) := by
+  simp_rw [inverseFlow_deriv_norm_eq_reverse hκ hy hT]
+  exact ((measurable_reverse_log (a κ) y T hy).comp (BrownianRestart.measurable_backward T.toNNReal)).exp
+
+ 
+
+theorem inverse_deriv_moment {κ y T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    (∫⁻ b : Driver, ENNReal.ofReal
+      (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W) ≤
+      ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta (a κ)/2)) := by
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  let F := fun b : Driver => ENNReal.ofReal (Real.exp
+    (lambda (a κ)*(ReverseDerivative.reverseFlow (a κ) (y/4) (T+y^2) y b T).2.2))
+  have mF : Measurable F :=
+    (measurable_const.mul (measurable_reverse_log (a κ) y T hy)).exp.ennreal_ofReal
+  have hid (b : Driver) :
+      ENNReal.ofReal (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) =
+        F (BrownianRestart.backward T.toNNReal b) := by
+    rw [inverseFlow_deriv_norm_eq_reverse hκ hy hT,
+      Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,mul_comm]
+    rfl
+  simp_rw [hid]
+  rw [← lintegral_map mF (BrownianRestart.measurable_backward T.toNNReal),
+    BrownianRestart.backward_preserves_law hW]
+  exact ReverseDerivative.reverse_logderiv_moment ha hy hT hW
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory ProbabilityTheory
+
+lemma lambda_gt_one {a : ℝ} (ha : 0 < a) : 1 < lambda a := by
+  unfold lambda
+  have hh : 0 < 3/(16*a) := by positivity
+  linarith
+
+lemma inverse_deriv_first_moment {κ y T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    (∫⁻ b : Driver, ENNReal.ofReal ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖ ∂W) ≤ 2 := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  have hl := lambda_gt_one (show 0 < a κ by linarith)
+  have hp (x : ℝ) (hx : 0 ≤ x) : x ≤ 1+x^(lambda (a κ)) := by
+    by_cases h : x ≤ 1
+    · exact h.trans (le_add_of_nonneg_right (Real.rpow_nonneg hx _))
+    · have he := Real.rpow_le_rpow_of_exponent_le (le_of_not_ge h) hl.le
+      simpa only [Real.rpow_one] using he.trans (le_add_of_nonneg_left (by norm_num : (0:ℝ) ≤ 1))
+  calc
+    _ ≤ ∫⁻ b : Driver, (1+ENNReal.ofReal (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ)))) ∂W := by
+      apply lintegral_mono
+      intro b
+      dsimp only
+      rw [← ENNReal.ofReal_one,← ENNReal.ofReal_add (by norm_num) (Real.rpow_nonneg (norm_nonneg _) _)]
+      exact ENNReal.ofReal_le_ofReal (hp _ (norm_nonneg _))
+    _ = 1+(∫⁻ b : Driver, ENNReal.ofReal (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W) := by
+      rw [lintegral_add_left measurable_const]
+      simp
+    _ ≤ 1+ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta (a κ)/2)) :=
+      add_le_add le_rfl (inverse_deriv_moment hκ hκ8 hy hT hW)
+    _ ≤ 2 := by
+      have hr : 3*y^2/(T+3*y^2) ≤ 1 := (div_le_one (by positivity)).mpr (by linarith)
+      have hh := Real.rpow_le_rpow (by positivity : 0 ≤ 3*y^2/(T+3*y^2)) hr
+        (by linarith [(parameters ha).1] : 0 ≤ zeta (a κ)/2)
+      simp only [Real.one_rpow] at hh
+      have hh' : ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta (a κ)/2)) ≤ 1 := by
+        simpa using ENNReal.ofReal_le_ofReal hh
+      calc _ ≤ 1+1 := add_le_add le_rfl hh'
+           _ = 2 := by norm_num
+
+ 
+def radialCoeff (κ : ℝ) (b : Driver) (T y : ℝ) : ℂ :=
+  if 0 < y then deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)*Complex.I else 0
+
+lemma continuousOn_radialCoeff {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
+    ContinuousOn (radialCoeff κ b T) (Ioi 0) := by
+  have hopen : IsOpen {w : ℂ | 0 < w.im} := isOpen_lt continuous_const Complex.continuous_im
+  have hc := (((flow_conformal_bijection hκ b hT).2.2.2).analyticOnNhd hopen).deriv.continuousOn
+  have hline : Continuous (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I) := by fun_prop
+  have hh := (hc.comp hline.continuousOn (show MapsTo (fun y : ℝ => (driverValue b T:ℂ)+y*Complex.I) (Ioi 0) {w : ℂ | 0 < w.im} from by intro y hy; simpa using hy)).mul (continuousOn_const (c := Complex.I))
+  apply hh.congr
+  intro y hy
+  rw [radialCoeff,ite_eq_left (show 0 < y from hy)]
+  rfl
+
+lemma measurable_positive_radial_norm {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) :
+    Measurable (fun p : (Ioi (0:ℝ)) × Driver =>
+      ‖deriv (inverseFlow κ p.2 T) (driverValue p.2 T+(p.1:ℝ)*Complex.I)‖) := by
+  exact measurable_uncurry_of_continuous_of_measurable
+    (u := fun (y : Ioi (0:ℝ)) (b : Driver) => ‖deriv (inverseFlow κ b T) (driverValue b T+(y:ℝ)*Complex.I)‖) (by
+    intro b
+    have hh := (continuousOn_radialCoeff hκ hT b).norm.domRestrict
+    have he : ((Ioi (0:ℝ)).domRestrict (fun y => ‖radialCoeff κ b T y‖)) =
+        (fun y : Ioi (0:ℝ) => ‖deriv (inverseFlow κ b T) (driverValue b T+(y:ℝ)*Complex.I)‖) := by
+      funext y
+      change ‖radialCoeff κ b T y‖ = _
+      rw [radialCoeff,ite_eq_left (show 0 < (y:ℝ) from y.property),norm_mul,Complex.norm_I,mul_one]
+    rwa [he] at hh) (by
+    intro y
+    exact measurable_inverse_deriv_norm hκ y.property hT)
+
+lemma measurable_radial_norm {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) :
+    Measurable (fun p : ℝ × Driver => ‖radialCoeff κ p.2 T p.1‖) := by
+  have hm := measurable_positive_radial_norm hκ hT
+  apply measurable_of_restrict_of_restrict_compl (s := {p : ℝ × Driver | 0 < p.1})
+    (measurableSet_lt measurable_const measurable_fst)
+  · have mm : Measurable (fun p : {p : ℝ × Driver | 0 < p.1} =>
+        ((⟨p.val.1,p.property⟩ : Ioi (0:ℝ)),p.val.2)) :=
+      (measurable_subtype_coe.fst.subtype_mk).prodMk measurable_subtype_coe.snd
+    have he : ({p : ℝ × Driver | 0 < p.1}.domRestrict
+        (fun p : ℝ × Driver => ‖radialCoeff κ p.2 T p.1‖)) =
+        (fun p : {p : ℝ × Driver | 0 < p.1} =>
+          ‖deriv (inverseFlow κ p.val.2 T) (driverValue p.val.2 T+p.val.1*Complex.I)‖) := by
+      funext p
+      change ‖radialCoeff κ p.val.2 T p.val.1‖ = _
+      rw [radialCoeff,ite_eq_left (show 0 < p.val.1 from p.property),norm_mul,Complex.norm_I,mul_one]
+    rw [he]
+    have hh := hm.comp mm
+    change Measurable (fun p : {p : ℝ × Driver | 0 < p.1} =>
+      ‖deriv (inverseFlow κ p.val.2 T) (driverValue p.val.2 T+p.val.1*Complex.I)‖) at hh
+    exact hh
+  · have he : ({p : ℝ × Driver | 0 < p.1}ᶜ.domRestrict
+        (fun p : ℝ × Driver => ‖radialCoeff κ p.2 T p.1‖)) =
+        (fun _ : ↥({p : ℝ × Driver | 0 < p.1}ᶜ) => (0:ℝ)) := by
+      funext p
+      change ‖radialCoeff κ p.val.2 T p.val.1‖ = 0
+      rw [radialCoeff,ite_eq_right (show ¬0 < p.val.1 from p.property),norm_zero]
+    rw [he]
+    exact measurable_const
+
+lemma radial_length_ae_finite {κ T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W, (∫⁻ y in Icc (0:ℝ) 1, ENNReal.ofReal ‖radialCoeff κ b T y‖) < (∞:ℝ≥0∞) := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let F := fun b : Driver => ∫⁻ y in Icc (0:ℝ) 1, ENNReal.ofReal ‖radialCoeff κ b T y‖
+  have hm := (measurable_radial_norm hκ hT).ennreal_ofReal
+  have mF : Measurable F := (hm.comp measurable_swap).lintegral_prod_right'
+  apply ae_lt_top mF
+  have hs : (∫⁻ b, F b ∂W) ≤ 2 := by
+    rw [show (∫⁻ b, F b ∂W) = ∫⁻ y in Icc (0:ℝ) 1, ∫⁻ b, ENNReal.ofReal ‖radialCoeff κ b T y‖ ∂W from
+      lintegral_lintegral_swap (hm.comp measurable_swap).aemeasurable]
+    calc
+      _ ≤ ∫⁻ y in Icc (0:ℝ) 1, (2:ℝ≥0∞) := by
+        apply lintegral_mono
+        intro y
+        by_cases hy : 0 < y
+        · simpa only [radialCoeff,ite_eq_left hy,norm_mul,Complex.norm_I,mul_one] using
+            inverse_deriv_first_moment hκ hκ8 hy hT hW
+        · simp only [radialCoeff,ite_eq_right hy,norm_zero,ENNReal.ofReal_zero,lintegral_zero]
+          exact bot_le
+      _ = 2 := by simp
+  exact ne_top_of_le_ne_top (by norm_num) hs
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open MeasureTheory
+
+lemma measurable_radialCoeff {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver) :
+    Measurable (radialCoeff κ b T) := by
+  apply measurable_of_restrict_of_restrict_compl (s := Ioi (0:ℝ)) measurableSet_Ioi
+  · exact (continuousOn_radialCoeff hκ hT b).domRestrict.measurable
+  · have he : ((Ioi (0:ℝ))ᶜ.domRestrict (radialCoeff κ b T)) =
+        (fun _ : ↥((Ioi (0:ℝ))ᶜ) => (0:ℂ)) := by
+      funext y
+      change radialCoeff κ b T y.val = 0
+      rw [radialCoeff,ite_eq_right (show ¬0 < y.val from y.property)]
+    rw [he]
+    exact measurable_const
+
+lemma hasDerivAt_radial {κ T y : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (hy : 0 < y) (b : Driver) :
+    HasDerivAt (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
+      (radialCoeff κ b T y) y := by
+  have hw : 0 < ((driverValue b T:ℂ)+y*Complex.I).im := by simpa using hy
+  have hf := (inverseFlow_hasStrictDerivAt hκ b hT hw).hasDerivAt.differentiableAt.hasDerivAt
+  have hl : HasDerivAt (fun z : ℂ => (driverValue b T:ℂ)+z*Complex.I) Complex.I (y:ℂ) := by
+    convert! (hasDerivAt_const (y:ℂ) (driverValue b T:ℂ)).add ((hasDerivAt_id (y:ℂ)).mul_const Complex.I) using 1; simp
+  convert! (hf.comp (y:ℂ) hl).comp_ofReal using 1; simp only [radialCoeff,ite_eq_left hy]
+
+lemma radial_limit_of_integrable {κ T : ℝ} (hκ : 0 < κ) (hT : 0 ≤ T) (b : Driver)
+    (hi : IntegrableOn (radialCoeff κ b T) (Icc (0:ℝ) 1)) :
+    ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I))
+      (𝓝[>] 0) (𝓝 z) := by
+  let f := fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I)
+  let v := fun y : ℝ => radialCoeff κ b T y
+  have hi' : IntegrableOn v (uIcc (0:ℝ) 1) := by simpa only [uIcc_of_le zero_le_one] using hi
+  have hcont := intervalIntegral.continuousOn_primitive_interval_left hi'
+  have hevent : ∀ᶠ y : ℝ in 𝓝[>] 0, y ∈ Ioc (0:ℝ) 1 := by
+    filter_upwards [self_mem_nhdsWithin,
+      (show ∀ᶠ y : ℝ in 𝓝[>] 0, y < 1 from nhdsWithin_le_nhds (Iio_mem_nhds zero_lt_one))] with y hy hy1
+    exact ⟨hy,hy1.le⟩
+  have hmap : Tendsto (fun y : ℝ => y) (𝓝[>] 0) (𝓝[uIcc (0:ℝ) 1] 0) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    refine ⟨tendsto_id'.mpr nhdsWithin_le_nhds,?_⟩
+    filter_upwards [hevent] with y hy
+    simpa only [uIcc_of_le zero_le_one] using (show y ∈ Icc (0:ℝ) 1 from ⟨hy.1.le,hy.2⟩)
+  have hlim := ((hcont 0 left_mem_uIcc).tendsto.comp hmap).const_sub (f 1)
+  refine ⟨f 1-(∫ u in (0:ℝ)..1, v u),hlim.congr' ?_⟩
+  filter_upwards [hevent] with y hy
+  have hd (u : ℝ) (hu : u ∈ uIcc y 1) : HasDerivAt f (v u) u := by
+    rw [uIcc_of_le hy.2] at hu
+    exact hasDerivAt_radial hκ hT (hy.1.trans_le hu.1) b
+  have hint : IntervalIntegrable v volume y 1 := by
+    apply IntegrableOn.intervalIntegrable
+    simpa only [uIcc_of_le hy.2] using
+      (hi.mono_set (show Icc y 1 ⊆ Icc (0:ℝ) 1 from Icc_subset_Icc hy.1.le le_rfl))
+  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt hd hint
+  simp only [Function.comp_apply]
+  rw [he]
+  exact sub_sub_cancel _ _
+
+ 
+
+theorem radial_limit_ae {κ T : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hT : 0 ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W, ∃ z : ℂ,
+      Tendsto (fun y : ℝ => inverseFlow κ b T (driverValue b T+y*Complex.I)) (𝓝[>] 0) (𝓝 z) := by
+  filter_upwards [radial_length_ae_finite hκ hκ8 hT hW] with b hb
+  apply radial_limit_of_integrable hκ hT b
+  exact ⟨(measurable_radialCoeff hκ hT b).aestronglyMeasurable,
+    (hasFiniteIntegral_iff_norm _).mpr hb⟩
+
+ 
+
+theorem trace_limit_countable {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) (t : ℕ → ℝ) (ht : ∀ n, 0 ≤ t n) :
+    ∀ᵐ b ∂W, ∀ n : ℕ,
+      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t n) (driverValue b (κ*t n)+y*Complex.I))
+        (𝓝[>] 0) (𝓝 (trace κ b (t n))) := by
+  have hh := ae_all_iff.mpr (fun n => radial_limit_ae hκ hκ8 (mul_nonneg hκ.le (ht n)) hW)
+  filter_upwards [hh] with b hb
+  intro n
+  obtain ⟨z,hz⟩ := hb n
+  have he : trace κ b (t n) = z := by
+    exact hz.limUnder_eq
+  rwa [he]
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory
+
+ 
+
+def traceBeta (κ : ℝ) : ℝ :=
+  (lambda (a κ)+zeta (a κ)-2)/(2*lambda (a κ))
+
+def traceEta (κ : ℝ) : ℝ := (lambda (a κ)+zeta (a κ)-2)/2
+
+lemma trace_exponents {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) :
+    0 < traceBeta κ ∧ traceBeta κ < 1 ∧ 0 < traceEta κ ∧
+      zeta (a κ)-(traceBeta κ-1)*lambda (a κ) = 2+traceEta κ := by
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  have hp := parameters ha
+  have hsum : 0 < lambda (a κ)+zeta (a κ)-2 := by linarith [hp.2.2.1]
+  have hden : 0 < 2*lambda (a κ) := by linarith [hp.2.1]
+  have hlt : zeta (a κ) < lambda (a κ)+2 := by
+    unfold lambda zeta
+    have : 0 < a κ := by linarith
+    have h1 : 0 ≤ 1/(16*a κ) := by positivity
+    have h3 : 0 ≤ 3/(16*a κ) := by positivity
+    linarith
+  refine ⟨div_pos hsum hden, (div_lt_one hden).mpr (by linarith),div_pos hsum (by norm_num),?_⟩
+  unfold traceBeta traceEta
+  field_simp [ne_of_gt hp.2.1]
+  ring
+
+lemma inverse_deriv_moment_away {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    (∫⁻ b : Driver, ENNReal.ofReal
+      (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W) ≤
+      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ))) := by
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  have hz := (parameters ha).1
+  have hT0 : 0 ≤ T := hδ.le.trans hT
+  apply (inverse_deriv_moment hκ hκ8 hy (hδ.le.trans hT) hW).trans
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ ≤ ((3/δ)*y^2)^(zeta (a κ)/2) := by
+      apply Real.rpow_le_rpow (by positivity) _ (by positivity)
+      calc
+        _ ≤ 3*y^2/δ := div_le_div_of_nonneg_left (by positivity) hδ (by nlinarith)
+        _ = (3/δ)*y^2 := by ring
+    _ = (3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)) := by
+      rw [Real.mul_rpow (by positivity) (sq_nonneg y),
+        ← Real.rpow_natCast_mul hy.le 2 (zeta (a κ)/2)]
+      congr 1
+      congr 1
+      norm_num
+      ring
+
+lemma inverse_deriv_tail_away {κ y T δ K : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) (hK : 0 < K)
+    {W : WienerLaw} (hW : IsWiener W) :
+    W {b : Driver | K ≤ ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖} ≤
+      ENNReal.ofReal (((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/K^(lambda (a κ))) := by
+  have ha : 1/4 < a κ := by unfold a; apply (lt_div_iff₀ hκ).mpr; linarith
+  have hl := (parameters ha).2.1
+  have hm := ((measurable_inverse_deriv_norm hκ hy (hδ.le.trans hT)).pow_const
+    (lambda (a κ))).ennreal_ofReal
+  have hKP : 0 < K^(lambda (a κ)) := Real.rpow_pos_of_pos hK _
+  calc
+    _ ≤ W {b : Driver | ENNReal.ofReal (K^(lambda (a κ))) ≤ ENNReal.ofReal
+        (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ)))} := by
+      apply measure_mono
+      intro b hb
+      exact ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow hK.le hb hl.le)
+    _ ≤ (∫⁻ b : Driver, ENNReal.ofReal
+        (‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖^(lambda (a κ))) ∂W)/
+        ENNReal.ofReal (K^(lambda (a κ))) :=
+      meas_ge_le_lintegral_div hm.aemeasurable (ne_of_gt (ENNReal.ofReal_pos.mpr hKP)) ENNReal.ofReal_ne_top
+    _ ≤ ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/
+        ENNReal.ofReal (K^(lambda (a κ))) := by
+      exact ENNReal.div_le_div_right (inverse_deriv_moment_away hκ hκ8 hy hδ hT hW) _
+    _ = ENNReal.ofReal (((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/K^(lambda (a κ))) :=
+      (ENNReal.ofReal_div_of_pos hKP).symm
+
+lemma inverse_deriv_beta_tail {κ y T δ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    (hy : 0 < y) (hδ : 0 < δ) (hT : δ ≤ T) {W : WienerLaw} (hW : IsWiener W) :
+    W {b : Driver | y^(traceBeta κ-1) ≤
+      ‖deriv (inverseFlow κ b T) (driverValue b T+y*Complex.I)‖} ≤
+      ENNReal.ofReal ((3/δ)^(zeta (a κ)/2)*y^(2+traceEta κ)) := by
+  have hp := trace_exponents hκ hκ8
+  have he : ((3/δ)^(zeta (a κ)/2)*y^(zeta (a κ)))/(y^(traceBeta κ-1))^(lambda (a κ)) =
+      (3/δ)^(zeta (a κ)/2)*y^(2+traceEta κ) := by
+    rw [← Real.rpow_mul hy.le, mul_div_assoc, ← Real.rpow_sub hy,
+      hp.2.2.2]
+  simpa only [he] using (inverse_deriv_tail_away hκ hκ8 hy hδ hT
+    (Real.rpow_pos_of_pos hy (traceBeta κ-1)) hW)
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open ReverseDerivative MeasureTheory
+
+lemma lintegral_pow_le_mass {α : Type*} [MeasurableSpace α] (μ : Measure α)
+    {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {p : ℝ} (hp : 1 < p) :
+    (∫⁻ x, f x ∂μ)^p ≤ (μ univ)^(p-1)*(∫⁻ x, (f x)^p ∂μ) := by
+  have hp0 : 0 < p := by linarith
+  let q := p/(p-1)
+  have hpq : p.HolderConjugate q := by
+    rw [Real.holderConjugate_iff]
+    refine ⟨hp,?_⟩
+    dsimp [q]
+    field_simp
+    ring
+  have hh := ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf
+    (aemeasurable_const (b := (1:ℝ≥0∞)))
+  simp only [Pi.mul_apply,mul_one,ENNReal.one_rpow,lintegral_const,one_mul] at hh
+  have hh' := ENNReal.rpow_le_rpow hh hp0.le
+  rw [ENNReal.mul_rpow_of_nonneg _ _ hp0.le,← ENNReal.rpow_mul,
+    one_div_mul_cancel (ne_of_gt hp0),ENNReal.rpow_one,← ENNReal.rpow_mul] at hh'
+  have he : (1/q)*p = p-1 := by dsimp [q]; field_simp
+  rw [he,mul_comm] at hh'
+  exact hh'
+
+ 
+def radialLength (κ : ℝ) (b : Driver) (T y : ℝ) : ℝ≥0∞ :=
+  ∫⁻ u in Ioc (0:ℝ) y, ENNReal.ofReal ‖radialCoeff κ b T u‖
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ReverseGenerators.lean b/lean/OAI/Probability/SLEGauge/ReverseGenerators.lean
index 35c07f9d6cc4a36e3c8a7ddc48e20fb28de704bd..bc7052d8fc21d0eff3326e8f8c7d7069c00050c0 100644
--- a/lean/OAI/Probability/SLEGauge/ReverseGenerators.lean
+++ b/lean/OAI/Probability/SLEGauge/ReverseGenerators.lean
@@ -1,421 +1,421 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StoppingMeasurability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-namespace SLEExactGauge.ReverseDerivative
-lemma line_deriv_test (a b : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im)
-    (v : State) :
-    HasDerivAt (fun t : ℝ => test a (q+t • v) b)
-      (test a q b * (lambda a*v.2.2+Hx a q.1 (q.2.1.re-b) q.2.1.im*v.2.1.re+
-        Hy a q.1 (q.2.1.re-b) q.2.1.im*v.2.1.im-
-        zeta a/(4*D q.1 (q.2.1.re-b) q.2.1.im)*v.1)) 0 := by
-  let x := q.2.1.re-b
-  let y := q.2.1.im
-  let u := q.1
-  have dx : HasDerivAt (fun t : ℝ => x+t*v.2.1.re) v.2.1.re 0 := by
-    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.2.1.re).const_add x
-  have dy : HasDerivAt (fun t : ℝ => y+t*v.2.1.im) v.2.1.im 0 := by
-    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.2.1.im).const_add y
-  have du : HasDerivAt (fun t : ℝ => u+t*v.1) v.1 0 := by
-    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.1).const_add u
-  have dl : HasDerivAt (fun t : ℝ => q.2.2+t*v.2.2) v.2.2 0 := by
-    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.2.2).const_add q.2.2
-  have dr := (dx.pow 2).add (dy.pow 2)
-  have dd := dr.add (du.div_const 2)
-  have hR := (R_pos x hy).ne'
-  have hD := (D_pos hu hy x).ne'
-  have hlogR := dr.log (by simpa [R,y] using hR)
-  have hlogD := dd.log (by simpa [D,R,y,u] using hD)
-  have hlogY := dy.log (by simpa [y] using hy.ne')
-  have hd := ((((dl.const_mul (lambda a)).add (hlogR.const_mul (r a/2))).add
-    (hlogY.const_mul (zeta a-r a))).sub (hlogD.const_mul (zeta a/2))).exp
-  convert! hd using 1
-  · ext t
-    simp [test,F,H,R,D,x,y,u,Complex.add_re,Complex.add_im]
-    congr 4 <;> ring_nf
-  · simp only [zero_mul,add_zero] at *
-    dsimp [test,F,H,Hx,Hy,R,D,x,y,u] at *
-    simp only [zero_mul,add_zero,pow_one]
-    congr 1
-    field_simp [hR,hD,hy.ne']
-    ring
-
-lemma fderiv_test_vector (a b : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) :
-    fderiv ℝ (fun q => test a q b) q (vector a q b) =
-      test a q b * ((-a*(q.2.1.re-b)/R (q.2.1.re-b) q.2.1.im)*
-        Hx a q.1 (q.2.1.re-b) q.2.1.im+
-        (a*q.2.1.im/R (q.2.1.re-b) q.2.1.im)*Hy a q.1 (q.2.1.re-b) q.2.1.im+
-        a*((q.2.1.re-b)^2-q.2.1.im^2)/(R (q.2.1.re-b) q.2.1.im)^2*lambda a+
-        zeta a/(4*D q.1 (q.2.1.re-b) q.2.1.im)) := by
-  have hv := (((hasDerivAt_id (0:ℝ)).smul_const (vector a q b)).const_add q)
-  simp only [id_eq,one_smul] at hv
-  have hq : q + (0:ℝ) • vector a q b = q := by simp
-  have houter := (differentiableAt_test a b hu hy).hasFDerivAt
-  rw [← hq] at houter
-  have hd := houter.comp_hasDerivAt (0:ℝ)
-    (show HasDerivAt (fun t : ℝ => q+t • vector a q b) (vector a q b) 0 from hv)
-  simp only [zero_smul,add_zero] at hd
-  rw [hd.unique (line_deriv_test a b hu hy (vector a q b))]
-  congr 1
-  simp [vector,Complex.div_re,Complex.div_im,Complex.inv_re,Complex.inv_im,
-    Complex.mul_re,pow_two,Complex.normSq_apply,R]
-  simp only [div_eq_mul_inv, mul_inv_rev]
-  ring
-
-lemma second_driver_test (a : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) (b : ℝ) :
-    deriv (deriv (test a q)) b = test a q b *
-      ((Hx a q.1 (q.2.1.re-b) q.2.1.im)^2+Hxx a q.1 (q.2.1.re-b) q.2.1.im) := by
-  have hfirst : deriv (test a q) = fun b => -test a q b * Hx a q.1 (q.2.1.re-b) q.2.1.im := by
-    funext b
-    have hd := ((hasDerivAt_Hx (a := a) (x := q.2.1.re-b) (l := q.2.2) hu hy).exp).comp b
-      ((hasDerivAt_id b).const_sub q.2.1.re)
-    convert! hd.deriv using 1
-    simp only [test,F]
-    ring
-  rw [hfirst]
-  have hf := ((hasDerivAt_Hx (a := a) (x := q.2.1.re-b) (l := q.2.2) hu hy).exp).comp b
-    ((hasDerivAt_id b).const_sub q.2.1.re)
-  have hh := (hasDerivAt_Hxx (a := a) (x := q.2.1.re-b) hu hy).comp b
-    ((hasDerivAt_id b).const_sub q.2.1.re)
-  convert! (hf.neg.mul hh).deriv using 1
-  simp only [test,F,Function.comp_apply,Pi.neg_apply]
-  ring
-
-lemma test_generator_nonpos {a : ℝ} (ha : 1/4 < a) {q : State}
-    (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) (b : ℝ) :
-    fderiv ℝ (fun q => test a q b) q (vector a q b)+deriv (deriv (test a q)) b/2 ≤ 0 := by
-  rw [fderiv_test_vector a b hu hy,second_driver_test a hu hy]
-  have hi := mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos (H a q.1 (q.2.1.re-b) q.2.1.im q.2.2)).le
-    (generator_nonpos (x := q.2.1.re-b) ha hu hy)
-  change test a q b * _ ≤ 0 at hi
-  nlinarith only [hi]
-
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open scoped ContDiff
-
-def domain : Set (State × ℝ) := {q | 0 < q.1.1 ∧ 0 < q.1.2.1.im}
-
-lemma isOpen_domain : IsOpen domain := by
-  apply IsOpen.inter
-  · exact isOpen_lt continuous_const (continuous_fst.comp continuous_fst)
-  · exact isOpen_lt continuous_const
-      (Complex.continuous_im.comp (continuous_fst.comp (continuous_snd.comp continuous_fst)))
-
-lemma contDiffOn_test (a : ℝ) :
-    ContDiffOn ℝ ∞ (Function.uncurry (test a)) domain := by
-  intro q hq
-  apply ContDiffAt.contDiffWithinAt
-  have hre : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.1.re-q.2) q :=
-    (Complex.reCLM.contDiff.contDiffAt.comp q
-      (contDiffAt_fst.comp q (contDiffAt_snd.comp q contDiffAt_fst))).sub contDiffAt_snd
-  have him : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.1.im) q :=
-    Complex.imCLM.contDiff.contDiffAt.comp q
-      (contDiffAt_fst.comp q (contDiffAt_snd.comp q contDiffAt_fst))
-  have hr := (hre.pow 2).add (him.pow 2)
-  have hd := hr.add ((contDiffAt_fst.comp q contDiffAt_fst).div_const 2)
-  have hl : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.2) q :=
-    contDiffAt_snd.comp q (contDiffAt_snd.comp q contDiffAt_fst)
-  have hR := (R_pos (q.1.2.1.re-q.2) hq.2).ne'
-  have hD := (D_pos hq.1.le hq.2 (q.1.2.1.re-q.2)).ne'
-  have he := ((((contDiffAt_const (c := lambda a)).mul hl).add
-    ((contDiffAt_const (c := r a/2)).mul (hr.log hR))).add
-    ((contDiffAt_const (c := zeta a-r a)).mul (him.log hq.2.ne'))).sub
-    ((contDiffAt_const (c := zeta a/2)).mul (hd.log hD))
-  exact he.exp
-
-lemma contDiffOn_vector (a : ℝ) :
-    ContDiffOn ℝ ∞ (Function.uncurry (vector a)) domain := by
-  intro q hq
-  apply ContDiffAt.contDiffWithinAt
-  have hz : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.1-(q.2:ℂ)) q :=
-    (contDiffAt_fst.comp q (contDiffAt_snd.comp q contDiffAt_fst)).sub
-      (Complex.ofRealCLM.contDiff.contDiffAt.comp q contDiffAt_snd)
-  have hn : q.1.2.1-(q.2:ℂ) ≠ 0 := by
-    intro he
-    have hi := congrArg Complex.im he
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-    exact hq.2.ne' hi
-  have hi := hz.inv hn
-  have hg : ContDiffAt ℝ ∞ (fun q : State × ℝ => -(a:ℂ)/(q.1.2.1-q.2)) q := by
-    convert! (contDiffAt_const (c := -(a:ℂ))).mul hi using 1
-  have hl : ContDiffAt ℝ ∞ (fun q : State × ℝ => a*((q.1.2.1-q.2)⁻¹^2).re) q :=
-    contDiffAt_const.mul (Complex.reCLM.contDiff.contDiffAt.comp q (hi.pow 2))
-  exact contDiffAt_const.prodMk (hg.prodMk hl)
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open Cutoff
-
-def globalVector (a q : ℝ) (x : State) (b : ℝ) : State :=
-  (-1, (-(a:ℂ)*inverse q (x.2.1-b), a*(inverse q (x.2.1-b)^2).re))
-
-def bound (a q : ℝ) : ℝ := 1+|a| *q⁻¹+|a| *q⁻¹^2
-
-def lipBound (a q : ℝ) : ℝ := |a| *(3/q^2)+|a| *(6/q^3)
-
-lemma continuous_globalVector {q : ℝ} (hq : 0 < q) (a : ℝ) :
-    Continuous (Function.uncurry (globalVector a q)) := by
-  have hi : Continuous (fun x : State × ℝ => inverse q (x.1.2.1-x.2)) :=
-    (lipschitz_inverse hq).continuous.comp
-      ((continuous_fst.comp (continuous_snd.comp continuous_fst)).sub
-        (Complex.continuous_ofReal.comp continuous_snd))
-  exact continuous_const.prodMk ((continuous_const.mul hi).prodMk
-    (continuous_const.mul (Complex.continuous_re.comp (hi.pow 2))))
-
-lemma norm_globalVector_le {q : ℝ} (hq : 0 < q) (a : ℝ) (x : State) (b : ℝ) :
-    ‖globalVector a q x b‖ ≤ bound a q := by
-  have hi := norm_inverse_le hq (x.2.1-b)
-  have h1 : ‖-(a:ℂ)*inverse q (x.2.1-b)‖ ≤ |a| *q⁻¹ := by
-    rw [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs]
-    exact mul_le_mul_of_nonneg_left hi (abs_nonneg _)
-  have h2 : |a*(inverse q (x.2.1-b)^2).re| ≤ |a| *q⁻¹^2 := by
-    rw [abs_mul]
-    apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
-    exact (Complex.abs_re_le_norm _).trans (by
-      rw [norm_pow]
-      exact pow_le_pow_left₀ (norm_nonneg _) hi 2)
-  change max ‖(-1:ℝ)‖ (max ‖-(a:ℂ)*inverse q (x.2.1-b)‖
-    ‖a*(inverse q (x.2.1-b)^2).re‖) ≤ _
-  simp only [Real.norm_eq_abs,abs_neg,abs_one]
-  apply max_le
-  · dsimp [bound]; nlinarith [show 0 ≤ |a| *q⁻¹ by positivity,show 0 ≤ |a| *q⁻¹^2 by positivity]
-  · apply max_le
-    · exact h1.trans (by dsimp [bound]; nlinarith [show 0 ≤ |a| *q⁻¹^2 by positivity])
-    · exact h2.trans (by dsimp [bound]; nlinarith [show 0 ≤ |a| *q⁻¹ by positivity])
-
-lemma inverse_sq_sub_le {q : ℝ} (hq : 0 < q) (z w : ℂ) :
-    ‖inverse q z^2-inverse q w^2‖ ≤ (6/q^3)*‖z-w‖ := by
-  have hh : inverse q z^2-inverse q w^2 = (inverse q z-inverse q w)*(inverse q z+inverse q w) := by ring
-  rw [hh,norm_mul]
-  have hi : ‖inverse q z+inverse q w‖ ≤ 2*q⁻¹ :=
-    (norm_add_le _ _).trans (by linarith [norm_inverse_le hq z,norm_inverse_le hq w])
-  have h := mul_le_mul (inverse_sub_bound hq z w) hi (norm_nonneg _) (by positivity)
-  exact h.trans_eq (by ring)
-
-lemma globalVector_sub_le {q : ℝ} (hq : 0 < q) (a b : ℝ) (x y : State) :
-    ‖globalVector a q x b-globalVector a q y b‖ ≤ lipBound a q*‖x-y‖ := by
-  have hxy : ‖x.2.1-y.2.1‖ ≤ ‖x-y‖ :=
-    (norm_fst_le (x.2-y.2)).trans (norm_snd_le (x-y))
-  have hz : x.2.1-(b:ℂ)-(y.2.1-b) = x.2.1-y.2.1 := by abel
-  have h1 : ‖-(a:ℂ)*inverse q (x.2.1-b)- (-(a:ℂ)*inverse q (y.2.1-b))‖ ≤
-      |a| *(3/q^2)*‖x-y‖ := by
-    rw [← mul_sub,norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs]
-    have h := inverse_sub_bound hq (x.2.1-b) (y.2.1-b)
-    rw [hz] at h
-    exact (mul_le_mul_of_nonneg_left h (abs_nonneg _)).trans (by
-      calc |a| * ((3/q^2)*‖x.2.1-y.2.1‖) = |a| *(3/q^2)*‖x.2.1-y.2.1‖ := by ring
-           _ ≤ _ := mul_le_mul_of_nonneg_left hxy (by positivity))
-  have h2 : |a*(inverse q (x.2.1-b)^2).re-a*(inverse q (y.2.1-b)^2).re| ≤
-      |a| *(6/q^3)*‖x-y‖ := by
-    rw [← mul_sub,abs_mul,← Complex.sub_re]
-    have h := inverse_sq_sub_le hq (x.2.1-b) (y.2.1-b)
-    rw [hz] at h
-    exact (mul_le_mul_of_nonneg_left ((Complex.abs_re_le_norm _).trans h) (abs_nonneg _)).trans
-      (by calc
-        |a| * ((6/q^3)*‖x.2.1-y.2.1‖) = |a| *(6/q^3)*‖x.2.1-y.2.1‖ := by ring
-        _ ≤ _ := mul_le_mul_of_nonneg_left hxy (by positivity))
-  change max ‖(-1:ℝ)-(-1)‖ (max _ ‖a*(inverse q (x.2.1-b)^2).re-a*(inverse q (y.2.1-b)^2).re‖) ≤ _
-  simp only [sub_self,norm_zero,Real.norm_eq_abs]
-  apply max_le
-  · dsimp [lipBound]; positivity
-  · apply max_le
-    · exact h1.trans (by dsimp [lipBound]; nlinarith [norm_nonneg (x-y),show 0 ≤ |a| *(6/q^3) by positivity])
-    · exact h2.trans (by dsimp [lipBound]; nlinarith [norm_nonneg (x-y),show 0 ≤ |a| *(3/q^2) by positivity])
-
-lemma inverse_eq_of_height {q : ℝ} (hq : 0 < q) (z : ℂ) (b : ℝ) (hz : q ≤ z.im) :
-    inverse q (z-b) = (z-b)⁻¹ := by
-  have hqn : q ≤ ‖z-b‖ := hz.trans (by simpa using Complex.im_le_norm (z-b))
-  have hsq : q^2 ≤ ‖z-b‖^2 := pow_le_pow_left₀ hq.le hqn 2
-  rw [inverse, denominator, max_eq_left hsq,Complex.inv_def]
-  simp only [Complex.normSq_eq_norm_sq,div_eq_mul_inv,Complex.ofReal_inv,starRingEnd_apply]
-
-lemma globalVector_eq {q : ℝ} (hq : 0 < q) (a : ℝ) (x : State) (b : ℝ)
-    (hy : q ≤ x.2.1.im) : globalVector a q x b = vector a x b := by
-  simp only [globalVector,vector,inverse_eq_of_height hq _ _ hy,div_eq_mul_inv]
-
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-def timeClip (T t : ℝ) : ℝ := max 0 (min T t)
-lemma continuous_timeClip (T : ℝ) : Continuous (timeClip T) := continuous_const.max (continuous_const.min continuous_id)
-lemma timeClip_mem {T : ℝ} (hT : 0 ≤ T) (t : ℝ) : timeClip T t ∈ Icc 0 T :=
-  ⟨le_max_left _ _,max_le hT (min_le_left _ _)⟩
-lemma timeClip_eq {T t : ℝ} (ht : t ∈ Icc 0 T) : timeClip T t = t :=
-  by rw [timeClip,min_eq_right ht.2,max_eq_right ht.1]
- 
-lemma reverse_height_formula {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
-    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t) :
-    ∀ t ∈ Icc 0 T, y t = y 0 * Real.exp (∫ u in (0:ℝ)..t, r (timeClip T u)) := by
-  let A (t : ℝ) := ∫ u in (0:ℝ)..t, r (timeClip T u)
-  have hc : Continuous (fun u => r (timeClip T u)) :=
-    hr.comp_continuous (continuous_timeClip T) (timeClip_mem hT)
-  have hA (t : ℝ) : HasDerivAt A (r (timeClip T t)) t :=
-    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
-      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-  have hA' : Continuous A := continuous_iff_continuousAt.mpr fun t => (hA t).continuousAt
-  have hd0 (t : ℝ) (ht : t ∈ Icc 0 T) :
-      HasDerivWithinAt (fun u => y u * Real.exp (-A u)) 0 (Icc 0 T) t := by
-    have he := (hd t ht).mul ((hA t).neg.exp.hasDerivWithinAt)
-    rw [timeClip_eq ht] at he
-    convert! he using 1
-    first | rfl | ring
-  have hconst := constant_of_has_deriv_right_zero
-    (hy.mul ((Real.continuous_exp.comp hA'.neg).continuousOn)) (fun t (ht : t ∈ Ico 0 T) =>
-      (hd0 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
-        (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1)))
-  intro t ht
-  have he := hconst t ht
-  dsimp only [Pi.mul_apply,Function.comp_def,Pi.neg_apply] at he
-  have hA0 : A 0 = 0 := by simp [A]
-  rw [hA0,neg_zero,Real.exp_zero,mul_one] at he
-  calc
-    y t = (y t * Real.exp (-A t))*Real.exp (A t) := by
-      rw [mul_assoc,← Real.exp_add,neg_add_cancel,Real.exp_zero,mul_one]
-    _ = y 0 * Real.exp (A t) := by rw [he]
-
-lemma reverse_height_le {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
-    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t)
-    (hy0 : 0 ≤ y 0) (hr0 : ∀ t ∈ Icc 0 T, 0 ≤ r t) (t : ℝ) (ht : t ∈ Icc 0 T) :
-    y 0 ≤ y t := by
-  rw [reverse_height_formula hT hy hr hd t ht]
-  apply le_mul_of_one_le_right hy0
-  apply Real.one_le_exp_iff.mpr
-  apply intervalIntegral.integral_nonneg ht.1
-  intro u _
-  exact hr0 (timeClip T u) (timeClip_mem hT u)
-
-lemma globalVector_height (a q : ℝ) (x : State) (b : ℝ) :
-    (globalVector a q x b).2.1.im = (a/Cutoff.denominator q (x.2.1-b))*x.2.1.im := by
-  simp [globalVector,Cutoff.inverse,Complex.mul_im,Complex.div_ofReal_im]
-  ring
-
-def initial (U y : ℝ) : State := (U, (⟨0,y⟩,0))
-def reverseFlow (a q U y : ℝ) (b : ℝ≥0 → ℝ) : ℝ → State :=
-  ODEPath.flow (globalVector a q) (initial U y) b
-
-lemma reverseFlow_solution {q : ℝ} (hq : 0 < q) (a U y : ℝ) (b : ℝ≥0 → ℝ)
-    (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T) :
-    reverseFlow a q U y b 0 = initial U y ∧
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (reverseFlow a q U y b)
-      (globalVector a q (reverseFlow a q U y b t) (b t.toNNReal)) (Icc 0 T) t := by
-  exact ODEPath.flow_solution _ (continuous_globalVector hq a) _ b hb
-    (by dsimp [bound]; positivity) (by dsimp [lipBound]; positivity) hT
-    (norm_globalVector_le hq a) (globalVector_sub_le hq a)
-
-lemma reverseFlow_height {q a y : ℝ} (hq : 0 < q) (ha : 0 ≤ a) (hy : 0 ≤ y)
-    (U : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T)
-    (t : ℝ) (ht : t ∈ Icc 0 T) : y ≤ (reverseFlow a q U y b t).2.1.im := by
-  let g := reverseFlow a q U y b
-  have hg := reverseFlow_solution hq a U y b hb hT
-  have hc : ContinuousOn g (Icc 0 T) := fun s hs => (hg.2 s hs).continuousWithinAt
-  let proj : State →L[ℝ] ℝ := Complex.imCLM.comp
-    ((ContinuousLinearMap.fst ℝ ℂ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℂ × ℝ)))
-  let r := fun s => a/Cutoff.denominator q ((g s).2.1-b s.toNNReal)
-  have hd : ∀ s ∈ Icc 0 T, HasDerivWithinAt (fun t => proj (g t)) (r s * proj (g s)) (Icc 0 T) s := by
-    intro s hs
-    have hh := proj.hasFDerivAt.comp_hasDerivWithinAt s (hg.2 s hs)
-    convert! hh using 1
-    exact (globalVector_height a q (g s) (b s.toNNReal)).symm
-  have hr : ContinuousOn r (Icc 0 T) := by
-    apply continuousOn_const.div
-    · unfold Cutoff.denominator
-      exact continuous_max.comp_continuousOn
-        ((((hc.snd.fst.sub ((Complex.continuous_ofReal.comp
-        (hb.comp continuous_real_toNNReal)).continuousOn)).norm).pow 2).prodMk continuousOn_const)
-    · intro s hs; exact (Cutoff.denominator_pos hq _).ne'
-  have he := reverse_height_le hT (proj.continuous.comp_continuousOn hc) hr hd
-    (by
-      change 0 ≤ (reverseFlow a q U y b 0).2.1.im
-      rw [hg.1]
-      exact hy)
-    (fun s hs => div_nonneg ha (Cutoff.denominator_pos hq _).le) t ht
-  change (reverseFlow a q U y b 0).2.1.im ≤ (reverseFlow a q U y b t).2.1.im at he
-  rw [hg.1] at he
-  exact he
-
-lemma reverseFlow_time {q : ℝ} (hq : 0 < q) (a U y : ℝ) (b : ℝ≥0 → ℝ)
-    (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T) (t : ℝ) (ht : t ∈ Icc 0 T) :
-    (reverseFlow a q U y b t).1 = U-t := by
-  let g := reverseFlow a q U y b
-  have hg := reverseFlow_solution hq a U y b hb hT
-  have hc : ContinuousOn g (Icc 0 T) := fun s hs => (hg.2 s hs).continuousWithinAt
-  have hd : ∀ s ∈ Icc 0 T, HasDerivWithinAt (fun s => (g s).1+s) 0 (Icc 0 T) s := by
-    intro s hs
-    have hdg := (ContinuousLinearMap.fst ℝ ℝ (ℂ × ℝ)).hasFDerivAt.comp_hasDerivWithinAt s (hg.2 s hs)
-    convert! hdg.add (hasDerivWithinAt_id s (Icc 0 T)) using 1
-    simp [globalVector]
-  have he := constant_of_has_deriv_right_zero (hc.fst.add continuousOn_id)
-    (fun s (hs : s ∈ Ico 0 T) => (hd s ⟨hs.1,hs.2.le⟩).mono_of_mem_nhdsWithin
-      (Filter.mem_of_superset (Icc_mem_nhdsGE hs.2) (Icc_subset_Icc_left hs.1))) t ht
-  change (reverseFlow a q U y b t).1+t = (reverseFlow a q U y b 0).1+0 at he
-  rw [hg.1] at he
-  change (reverseFlow a q U y b t).1+t = U+0 at he
-  linarith
-
-lemma reverseFlow_actual {q a y : ℝ} (hq : 0 < q) (ha : 0 ≤ a) (hy : q ≤ y)
-    (U : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T) :
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (reverseFlow a q U y b)
-      (vector a (reverseFlow a q U y b t) (b t.toNNReal)) (Icc 0 T) t := by
-  intro t ht
-  have hg := (reverseFlow_solution hq a U y b hb hT).2 t ht
-  rw [globalVector_eq hq a _ _
-    (hy.trans (reverseFlow_height hq ha (hq.le.trans hy) U b hb hT t ht))] at hg
-  exact hg
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open _root_.OAI.SLEExactGauge.Localization BrownianCalculus ODEPath RationalStop
-open MeasureTheory ProbabilityTheory
-
-end SLEExactGauge.ReverseDerivative
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StoppingMeasurability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+namespace SLEExactGauge.ReverseDerivative
+lemma line_deriv_test (a b : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im)
+    (v : State) :
+    HasDerivAt (fun t : ℝ => test a (q+t • v) b)
+      (test a q b * (lambda a*v.2.2+Hx a q.1 (q.2.1.re-b) q.2.1.im*v.2.1.re+
+        Hy a q.1 (q.2.1.re-b) q.2.1.im*v.2.1.im-
+        zeta a/(4*D q.1 (q.2.1.re-b) q.2.1.im)*v.1)) 0 := by
+  let x := q.2.1.re-b
+  let y := q.2.1.im
+  let u := q.1
+  have dx : HasDerivAt (fun t : ℝ => x+t*v.2.1.re) v.2.1.re 0 := by
+    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.2.1.re).const_add x
+  have dy : HasDerivAt (fun t : ℝ => y+t*v.2.1.im) v.2.1.im 0 := by
+    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.2.1.im).const_add y
+  have du : HasDerivAt (fun t : ℝ => u+t*v.1) v.1 0 := by
+    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.1).const_add u
+  have dl : HasDerivAt (fun t : ℝ => q.2.2+t*v.2.2) v.2.2 0 := by
+    simpa using ((hasDerivAt_id (0:ℝ)).mul_const v.2.2).const_add q.2.2
+  have dr := (dx.pow 2).add (dy.pow 2)
+  have dd := dr.add (du.div_const 2)
+  have hR := (R_pos x hy).ne'
+  have hD := (D_pos hu hy x).ne'
+  have hlogR := dr.log (by simpa [R,y] using hR)
+  have hlogD := dd.log (by simpa [D,R,y,u] using hD)
+  have hlogY := dy.log (by simpa [y] using hy.ne')
+  have hd := ((((dl.const_mul (lambda a)).add (hlogR.const_mul (r a/2))).add
+    (hlogY.const_mul (zeta a-r a))).sub (hlogD.const_mul (zeta a/2))).exp
+  convert! hd using 1
+  · ext t
+    simp [test,F,H,R,D,x,y,u,Complex.add_re,Complex.add_im]
+    congr 4 <;> ring_nf
+  · simp only [zero_mul,add_zero] at *
+    dsimp [test,F,H,Hx,Hy,R,D,x,y,u] at *
+    simp only [zero_mul,add_zero,pow_one]
+    congr 1
+    field_simp [hR,hD,hy.ne']
+    ring
+
+lemma fderiv_test_vector (a b : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) :
+    fderiv ℝ (fun q => test a q b) q (vector a q b) =
+      test a q b * ((-a*(q.2.1.re-b)/R (q.2.1.re-b) q.2.1.im)*
+        Hx a q.1 (q.2.1.re-b) q.2.1.im+
+        (a*q.2.1.im/R (q.2.1.re-b) q.2.1.im)*Hy a q.1 (q.2.1.re-b) q.2.1.im+
+        a*((q.2.1.re-b)^2-q.2.1.im^2)/(R (q.2.1.re-b) q.2.1.im)^2*lambda a+
+        zeta a/(4*D q.1 (q.2.1.re-b) q.2.1.im)) := by
+  have hv := (((hasDerivAt_id (0:ℝ)).smul_const (vector a q b)).const_add q)
+  simp only [id_eq,one_smul] at hv
+  have hq : q + (0:ℝ) • vector a q b = q := by simp
+  have houter := (differentiableAt_test a b hu hy).hasFDerivAt
+  rw [← hq] at houter
+  have hd := houter.comp_hasDerivAt (0:ℝ)
+    (show HasDerivAt (fun t : ℝ => q+t • vector a q b) (vector a q b) 0 from hv)
+  simp only [zero_smul,add_zero] at hd
+  rw [hd.unique (line_deriv_test a b hu hy (vector a q b))]
+  congr 1
+  simp [vector,Complex.div_re,Complex.div_im,Complex.inv_re,Complex.inv_im,
+    Complex.mul_re,pow_two,Complex.normSq_apply,R]
+  simp only [div_eq_mul_inv, mul_inv_rev]
+  ring
+
+lemma second_driver_test (a : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) (b : ℝ) :
+    deriv (deriv (test a q)) b = test a q b *
+      ((Hx a q.1 (q.2.1.re-b) q.2.1.im)^2+Hxx a q.1 (q.2.1.re-b) q.2.1.im) := by
+  have hfirst : deriv (test a q) = fun b => -test a q b * Hx a q.1 (q.2.1.re-b) q.2.1.im := by
+    funext b
+    have hd := ((hasDerivAt_Hx (a := a) (x := q.2.1.re-b) (l := q.2.2) hu hy).exp).comp b
+      ((hasDerivAt_id b).const_sub q.2.1.re)
+    convert! hd.deriv using 1
+    simp only [test,F]
+    ring
+  rw [hfirst]
+  have hf := ((hasDerivAt_Hx (a := a) (x := q.2.1.re-b) (l := q.2.2) hu hy).exp).comp b
+    ((hasDerivAt_id b).const_sub q.2.1.re)
+  have hh := (hasDerivAt_Hxx (a := a) (x := q.2.1.re-b) hu hy).comp b
+    ((hasDerivAt_id b).const_sub q.2.1.re)
+  convert! (hf.neg.mul hh).deriv using 1
+  simp only [test,F,Function.comp_apply,Pi.neg_apply]
+  ring
+
+lemma test_generator_nonpos {a : ℝ} (ha : 1/4 < a) {q : State}
+    (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) (b : ℝ) :
+    fderiv ℝ (fun q => test a q b) q (vector a q b)+deriv (deriv (test a q)) b/2 ≤ 0 := by
+  rw [fderiv_test_vector a b hu hy,second_driver_test a hu hy]
+  have hi := mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos (H a q.1 (q.2.1.re-b) q.2.1.im q.2.2)).le
+    (generator_nonpos (x := q.2.1.re-b) ha hu hy)
+  change test a q b * _ ≤ 0 at hi
+  nlinarith only [hi]
+
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open scoped ContDiff
+
+def domain : Set (State × ℝ) := {q | 0 < q.1.1 ∧ 0 < q.1.2.1.im}
+
+lemma isOpen_domain : IsOpen domain := by
+  apply IsOpen.inter
+  · exact isOpen_lt continuous_const (continuous_fst.comp continuous_fst)
+  · exact isOpen_lt continuous_const
+      (Complex.continuous_im.comp (continuous_fst.comp (continuous_snd.comp continuous_fst)))
+
+lemma contDiffOn_test (a : ℝ) :
+    ContDiffOn ℝ ∞ (Function.uncurry (test a)) domain := by
+  intro q hq
+  apply ContDiffAt.contDiffWithinAt
+  have hre : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.1.re-q.2) q :=
+    (Complex.reCLM.contDiff.contDiffAt.comp q
+      (contDiffAt_fst.comp q (contDiffAt_snd.comp q contDiffAt_fst))).sub contDiffAt_snd
+  have him : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.1.im) q :=
+    Complex.imCLM.contDiff.contDiffAt.comp q
+      (contDiffAt_fst.comp q (contDiffAt_snd.comp q contDiffAt_fst))
+  have hr := (hre.pow 2).add (him.pow 2)
+  have hd := hr.add ((contDiffAt_fst.comp q contDiffAt_fst).div_const 2)
+  have hl : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.2) q :=
+    contDiffAt_snd.comp q (contDiffAt_snd.comp q contDiffAt_fst)
+  have hR := (R_pos (q.1.2.1.re-q.2) hq.2).ne'
+  have hD := (D_pos hq.1.le hq.2 (q.1.2.1.re-q.2)).ne'
+  have he := ((((contDiffAt_const (c := lambda a)).mul hl).add
+    ((contDiffAt_const (c := r a/2)).mul (hr.log hR))).add
+    ((contDiffAt_const (c := zeta a-r a)).mul (him.log hq.2.ne'))).sub
+    ((contDiffAt_const (c := zeta a/2)).mul (hd.log hD))
+  exact he.exp
+
+lemma contDiffOn_vector (a : ℝ) :
+    ContDiffOn ℝ ∞ (Function.uncurry (vector a)) domain := by
+  intro q hq
+  apply ContDiffAt.contDiffWithinAt
+  have hz : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2.1-(q.2:ℂ)) q :=
+    (contDiffAt_fst.comp q (contDiffAt_snd.comp q contDiffAt_fst)).sub
+      (Complex.ofRealCLM.contDiff.contDiffAt.comp q contDiffAt_snd)
+  have hn : q.1.2.1-(q.2:ℂ) ≠ 0 := by
+    intro he
+    have hi := congrArg Complex.im he
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+    exact hq.2.ne' hi
+  have hi := hz.inv hn
+  have hg : ContDiffAt ℝ ∞ (fun q : State × ℝ => -(a:ℂ)/(q.1.2.1-q.2)) q := by
+    convert! (contDiffAt_const (c := -(a:ℂ))).mul hi using 1
+  have hl : ContDiffAt ℝ ∞ (fun q : State × ℝ => a*((q.1.2.1-q.2)⁻¹^2).re) q :=
+    contDiffAt_const.mul (Complex.reCLM.contDiff.contDiffAt.comp q (hi.pow 2))
+  exact contDiffAt_const.prodMk (hg.prodMk hl)
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open Cutoff
+
+def globalVector (a q : ℝ) (x : State) (b : ℝ) : State :=
+  (-1, (-(a:ℂ)*inverse q (x.2.1-b), a*(inverse q (x.2.1-b)^2).re))
+
+def bound (a q : ℝ) : ℝ := 1+|a| *q⁻¹+|a| *q⁻¹^2
+
+def lipBound (a q : ℝ) : ℝ := |a| *(3/q^2)+|a| *(6/q^3)
+
+lemma continuous_globalVector {q : ℝ} (hq : 0 < q) (a : ℝ) :
+    Continuous (Function.uncurry (globalVector a q)) := by
+  have hi : Continuous (fun x : State × ℝ => inverse q (x.1.2.1-x.2)) :=
+    (lipschitz_inverse hq).continuous.comp
+      ((continuous_fst.comp (continuous_snd.comp continuous_fst)).sub
+        (Complex.continuous_ofReal.comp continuous_snd))
+  exact continuous_const.prodMk ((continuous_const.mul hi).prodMk
+    (continuous_const.mul (Complex.continuous_re.comp (hi.pow 2))))
+
+lemma norm_globalVector_le {q : ℝ} (hq : 0 < q) (a : ℝ) (x : State) (b : ℝ) :
+    ‖globalVector a q x b‖ ≤ bound a q := by
+  have hi := norm_inverse_le hq (x.2.1-b)
+  have h1 : ‖-(a:ℂ)*inverse q (x.2.1-b)‖ ≤ |a| *q⁻¹ := by
+    rw [norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs]
+    exact mul_le_mul_of_nonneg_left hi (abs_nonneg _)
+  have h2 : |a*(inverse q (x.2.1-b)^2).re| ≤ |a| *q⁻¹^2 := by
+    rw [abs_mul]
+    apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
+    exact (Complex.abs_re_le_norm _).trans (by
+      rw [norm_pow]
+      exact pow_le_pow_left₀ (norm_nonneg _) hi 2)
+  change max ‖(-1:ℝ)‖ (max ‖-(a:ℂ)*inverse q (x.2.1-b)‖
+    ‖a*(inverse q (x.2.1-b)^2).re‖) ≤ _
+  simp only [Real.norm_eq_abs,abs_neg,abs_one]
+  apply max_le
+  · dsimp [bound]; nlinarith [show 0 ≤ |a| *q⁻¹ by positivity,show 0 ≤ |a| *q⁻¹^2 by positivity]
+  · apply max_le
+    · exact h1.trans (by dsimp [bound]; nlinarith [show 0 ≤ |a| *q⁻¹^2 by positivity])
+    · exact h2.trans (by dsimp [bound]; nlinarith [show 0 ≤ |a| *q⁻¹ by positivity])
+
+lemma inverse_sq_sub_le {q : ℝ} (hq : 0 < q) (z w : ℂ) :
+    ‖inverse q z^2-inverse q w^2‖ ≤ (6/q^3)*‖z-w‖ := by
+  have hh : inverse q z^2-inverse q w^2 = (inverse q z-inverse q w)*(inverse q z+inverse q w) := by ring
+  rw [hh,norm_mul]
+  have hi : ‖inverse q z+inverse q w‖ ≤ 2*q⁻¹ :=
+    (norm_add_le _ _).trans (by linarith [norm_inverse_le hq z,norm_inverse_le hq w])
+  have h := mul_le_mul (inverse_sub_bound hq z w) hi (norm_nonneg _) (by positivity)
+  exact h.trans_eq (by ring)
+
+lemma globalVector_sub_le {q : ℝ} (hq : 0 < q) (a b : ℝ) (x y : State) :
+    ‖globalVector a q x b-globalVector a q y b‖ ≤ lipBound a q*‖x-y‖ := by
+  have hxy : ‖x.2.1-y.2.1‖ ≤ ‖x-y‖ :=
+    (norm_fst_le (x.2-y.2)).trans (norm_snd_le (x-y))
+  have hz : x.2.1-(b:ℂ)-(y.2.1-b) = x.2.1-y.2.1 := by abel
+  have h1 : ‖-(a:ℂ)*inverse q (x.2.1-b)- (-(a:ℂ)*inverse q (y.2.1-b))‖ ≤
+      |a| *(3/q^2)*‖x-y‖ := by
+    rw [← mul_sub,norm_mul,norm_neg,Complex.norm_real,Real.norm_eq_abs]
+    have h := inverse_sub_bound hq (x.2.1-b) (y.2.1-b)
+    rw [hz] at h
+    exact (mul_le_mul_of_nonneg_left h (abs_nonneg _)).trans (by
+      calc |a| * ((3/q^2)*‖x.2.1-y.2.1‖) = |a| *(3/q^2)*‖x.2.1-y.2.1‖ := by ring
+           _ ≤ _ := mul_le_mul_of_nonneg_left hxy (by positivity))
+  have h2 : |a*(inverse q (x.2.1-b)^2).re-a*(inverse q (y.2.1-b)^2).re| ≤
+      |a| *(6/q^3)*‖x-y‖ := by
+    rw [← mul_sub,abs_mul,← Complex.sub_re]
+    have h := inverse_sq_sub_le hq (x.2.1-b) (y.2.1-b)
+    rw [hz] at h
+    exact (mul_le_mul_of_nonneg_left ((Complex.abs_re_le_norm _).trans h) (abs_nonneg _)).trans
+      (by calc
+        |a| * ((6/q^3)*‖x.2.1-y.2.1‖) = |a| *(6/q^3)*‖x.2.1-y.2.1‖ := by ring
+        _ ≤ _ := mul_le_mul_of_nonneg_left hxy (by positivity))
+  change max ‖(-1:ℝ)-(-1)‖ (max _ ‖a*(inverse q (x.2.1-b)^2).re-a*(inverse q (y.2.1-b)^2).re‖) ≤ _
+  simp only [sub_self,norm_zero,Real.norm_eq_abs]
+  apply max_le
+  · dsimp [lipBound]; positivity
+  · apply max_le
+    · exact h1.trans (by dsimp [lipBound]; nlinarith [norm_nonneg (x-y),show 0 ≤ |a| *(6/q^3) by positivity])
+    · exact h2.trans (by dsimp [lipBound]; nlinarith [norm_nonneg (x-y),show 0 ≤ |a| *(3/q^2) by positivity])
+
+lemma inverse_eq_of_height {q : ℝ} (hq : 0 < q) (z : ℂ) (b : ℝ) (hz : q ≤ z.im) :
+    inverse q (z-b) = (z-b)⁻¹ := by
+  have hqn : q ≤ ‖z-b‖ := hz.trans (by simpa using Complex.im_le_norm (z-b))
+  have hsq : q^2 ≤ ‖z-b‖^2 := pow_le_pow_left₀ hq.le hqn 2
+  rw [inverse, denominator, max_eq_left hsq,Complex.inv_def]
+  simp only [Complex.normSq_eq_norm_sq,div_eq_mul_inv,Complex.ofReal_inv,starRingEnd_apply]
+
+lemma globalVector_eq {q : ℝ} (hq : 0 < q) (a : ℝ) (x : State) (b : ℝ)
+    (hy : q ≤ x.2.1.im) : globalVector a q x b = vector a x b := by
+  simp only [globalVector,vector,inverse_eq_of_height hq _ _ hy,div_eq_mul_inv]
+
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+def timeClip (T t : ℝ) : ℝ := max 0 (min T t)
+lemma continuous_timeClip (T : ℝ) : Continuous (timeClip T) := continuous_const.max (continuous_const.min continuous_id)
+lemma timeClip_mem {T : ℝ} (hT : 0 ≤ T) (t : ℝ) : timeClip T t ∈ Icc 0 T :=
+  ⟨le_max_left _ _,max_le hT (min_le_left _ _)⟩
+lemma timeClip_eq {T t : ℝ} (ht : t ∈ Icc 0 T) : timeClip T t = t :=
+  by rw [timeClip,min_eq_right ht.2,max_eq_right ht.1]
+ 
+lemma reverse_height_formula {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
+    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t) :
+    ∀ t ∈ Icc 0 T, y t = y 0 * Real.exp (∫ u in (0:ℝ)..t, r (timeClip T u)) := by
+  let A (t : ℝ) := ∫ u in (0:ℝ)..t, r (timeClip T u)
+  have hc : Continuous (fun u => r (timeClip T u)) :=
+    hr.comp_continuous (continuous_timeClip T) (timeClip_mem hT)
+  have hA (t : ℝ) : HasDerivAt A (r (timeClip T t)) t :=
+    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
+      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+  have hA' : Continuous A := continuous_iff_continuousAt.mpr fun t => (hA t).continuousAt
+  have hd0 (t : ℝ) (ht : t ∈ Icc 0 T) :
+      HasDerivWithinAt (fun u => y u * Real.exp (-A u)) 0 (Icc 0 T) t := by
+    have he := (hd t ht).mul ((hA t).neg.exp.hasDerivWithinAt)
+    rw [timeClip_eq ht] at he
+    convert! he using 1
+    first | rfl | ring
+  have hconst := constant_of_has_deriv_right_zero
+    (hy.mul ((Real.continuous_exp.comp hA'.neg).continuousOn)) (fun t (ht : t ∈ Ico 0 T) =>
+      (hd0 t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin
+        (Filter.mem_of_superset (Icc_mem_nhdsGE ht.2) (Icc_subset_Icc_left ht.1)))
+  intro t ht
+  have he := hconst t ht
+  dsimp only [Pi.mul_apply,Function.comp_def,Pi.neg_apply] at he
+  have hA0 : A 0 = 0 := by simp [A]
+  rw [hA0,neg_zero,Real.exp_zero,mul_one] at he
+  calc
+    y t = (y t * Real.exp (-A t))*Real.exp (A t) := by
+      rw [mul_assoc,← Real.exp_add,neg_add_cancel,Real.exp_zero,mul_one]
+    _ = y 0 * Real.exp (A t) := by rw [he]
+
+lemma reverse_height_le {T : ℝ} {y r : ℝ → ℝ} (hT : 0 ≤ T)
+    (hy : ContinuousOn y (Icc 0 T)) (hr : ContinuousOn r (Icc 0 T))
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y (r t*y t) (Icc 0 T) t)
+    (hy0 : 0 ≤ y 0) (hr0 : ∀ t ∈ Icc 0 T, 0 ≤ r t) (t : ℝ) (ht : t ∈ Icc 0 T) :
+    y 0 ≤ y t := by
+  rw [reverse_height_formula hT hy hr hd t ht]
+  apply le_mul_of_one_le_right hy0
+  apply Real.one_le_exp_iff.mpr
+  apply intervalIntegral.integral_nonneg ht.1
+  intro u _
+  exact hr0 (timeClip T u) (timeClip_mem hT u)
+
+lemma globalVector_height (a q : ℝ) (x : State) (b : ℝ) :
+    (globalVector a q x b).2.1.im = (a/Cutoff.denominator q (x.2.1-b))*x.2.1.im := by
+  simp [globalVector,Cutoff.inverse,Complex.mul_im,Complex.div_ofReal_im]
+  ring
+
+def initial (U y : ℝ) : State := (U, (⟨0,y⟩,0))
+def reverseFlow (a q U y : ℝ) (b : ℝ≥0 → ℝ) : ℝ → State :=
+  ODEPath.flow (globalVector a q) (initial U y) b
+
+lemma reverseFlow_solution {q : ℝ} (hq : 0 < q) (a U y : ℝ) (b : ℝ≥0 → ℝ)
+    (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T) :
+    reverseFlow a q U y b 0 = initial U y ∧
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (reverseFlow a q U y b)
+      (globalVector a q (reverseFlow a q U y b t) (b t.toNNReal)) (Icc 0 T) t := by
+  exact ODEPath.flow_solution _ (continuous_globalVector hq a) _ b hb
+    (by dsimp [bound]; positivity) (by dsimp [lipBound]; positivity) hT
+    (norm_globalVector_le hq a) (globalVector_sub_le hq a)
+
+lemma reverseFlow_height {q a y : ℝ} (hq : 0 < q) (ha : 0 ≤ a) (hy : 0 ≤ y)
+    (U : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T)
+    (t : ℝ) (ht : t ∈ Icc 0 T) : y ≤ (reverseFlow a q U y b t).2.1.im := by
+  let g := reverseFlow a q U y b
+  have hg := reverseFlow_solution hq a U y b hb hT
+  have hc : ContinuousOn g (Icc 0 T) := fun s hs => (hg.2 s hs).continuousWithinAt
+  let proj : State →L[ℝ] ℝ := Complex.imCLM.comp
+    ((ContinuousLinearMap.fst ℝ ℂ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℂ × ℝ)))
+  let r := fun s => a/Cutoff.denominator q ((g s).2.1-b s.toNNReal)
+  have hd : ∀ s ∈ Icc 0 T, HasDerivWithinAt (fun t => proj (g t)) (r s * proj (g s)) (Icc 0 T) s := by
+    intro s hs
+    have hh := proj.hasFDerivAt.comp_hasDerivWithinAt s (hg.2 s hs)
+    convert! hh using 1
+    exact (globalVector_height a q (g s) (b s.toNNReal)).symm
+  have hr : ContinuousOn r (Icc 0 T) := by
+    apply continuousOn_const.div
+    · unfold Cutoff.denominator
+      exact continuous_max.comp_continuousOn
+        ((((hc.snd.fst.sub ((Complex.continuous_ofReal.comp
+        (hb.comp continuous_real_toNNReal)).continuousOn)).norm).pow 2).prodMk continuousOn_const)
+    · intro s hs; exact (Cutoff.denominator_pos hq _).ne'
+  have he := reverse_height_le hT (proj.continuous.comp_continuousOn hc) hr hd
+    (by
+      change 0 ≤ (reverseFlow a q U y b 0).2.1.im
+      rw [hg.1]
+      exact hy)
+    (fun s hs => div_nonneg ha (Cutoff.denominator_pos hq _).le) t ht
+  change (reverseFlow a q U y b 0).2.1.im ≤ (reverseFlow a q U y b t).2.1.im at he
+  rw [hg.1] at he
+  exact he
+
+lemma reverseFlow_time {q : ℝ} (hq : 0 < q) (a U y : ℝ) (b : ℝ≥0 → ℝ)
+    (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T) (t : ℝ) (ht : t ∈ Icc 0 T) :
+    (reverseFlow a q U y b t).1 = U-t := by
+  let g := reverseFlow a q U y b
+  have hg := reverseFlow_solution hq a U y b hb hT
+  have hc : ContinuousOn g (Icc 0 T) := fun s hs => (hg.2 s hs).continuousWithinAt
+  have hd : ∀ s ∈ Icc 0 T, HasDerivWithinAt (fun s => (g s).1+s) 0 (Icc 0 T) s := by
+    intro s hs
+    have hdg := (ContinuousLinearMap.fst ℝ ℝ (ℂ × ℝ)).hasFDerivAt.comp_hasDerivWithinAt s (hg.2 s hs)
+    convert! hdg.add (hasDerivWithinAt_id s (Icc 0 T)) using 1
+    simp [globalVector]
+  have he := constant_of_has_deriv_right_zero (hc.fst.add continuousOn_id)
+    (fun s (hs : s ∈ Ico 0 T) => (hd s ⟨hs.1,hs.2.le⟩).mono_of_mem_nhdsWithin
+      (Filter.mem_of_superset (Icc_mem_nhdsGE hs.2) (Icc_subset_Icc_left hs.1))) t ht
+  change (reverseFlow a q U y b t).1+t = (reverseFlow a q U y b 0).1+0 at he
+  rw [hg.1] at he
+  change (reverseFlow a q U y b t).1+t = U+0 at he
+  linarith
+
+lemma reverseFlow_actual {q a y : ℝ} (hq : 0 < q) (ha : 0 ≤ a) (hy : q ≤ y)
+    (U : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b) {T : ℝ} (hT : 0 ≤ T) :
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (reverseFlow a q U y b)
+      (vector a (reverseFlow a q U y b t) (b t.toNNReal)) (Icc 0 T) t := by
+  intro t ht
+  have hg := (reverseFlow_solution hq a U y b hb hT).2 t ht
+  rw [globalVector_eq hq a _ _
+    (hy.trans (reverseFlow_height hq ha (hq.le.trans hy) U b hb hT t ht))] at hg
+  exact hg
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open _root_.OAI.SLEExactGauge.Localization BrownianCalculus ODEPath RationalStop
+open MeasureTheory ProbabilityTheory
+
+end SLEExactGauge.ReverseDerivative
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/ReverseMoments.lean b/lean/OAI/Probability/SLEGauge/ReverseMoments.lean
index 55da58fd62228da62efef307bf89a447da191cc0..d477c220659c4046acd0bdab8751d7d31df615ef 100644
--- a/lean/OAI/Probability/SLEGauge/ReverseMoments.lean
+++ b/lean/OAI/Probability/SLEGauge/ReverseMoments.lean
@@ -1,484 +1,484 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ReverseGenerators
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-namespace SLEExactGauge.ReverseDerivative
-open _root_.OAI.SLEExactGauge.Localization BrownianCalculus ODEPath RationalStop
-open MeasureTheory ProbabilityTheory
-theorem expected_stopped_reverse_aux {a : ℝ} (ha : 1/4 < a)
-    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    {H : ℝ} (hH : 0 ≤ H) (φ : State × ℝ → ℝ) (hφ : Continuous φ)
-    (hK : IsCompact {x | φ x ≤ 2}) (hKO : {x | φ x ≤ 2} ⊆ domain)
-    (v : State → ℝ → State) (hv : Continuous (Function.uncurry v))
-    {M L : ℝ} (hM : 0 ≤ M) (hL : 0 ≤ L)
-    (hbound : ∀ x b, ‖v x b‖ ≤ M)
-    (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hveq : ∀ x, φ x ≤ 2 → v x.1 x.2 = vector a x.1 x.2)
-    (z : State) (hz : φ (z,0) < 1) :
-    let g := fun b : Driver => ODEPath.flow v z b
-    let τ := fun b : Driver => Exit (fun u => φ (g b u,b u.toNNReal)) 1 H
-    ∀ t ≥ 0, (∫ b : Driver, test a (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ test a z 0 := by
-  let g := fun b : Driver => ODEPath.flow v z b
-  let f := fun (b : ℝ≥0 → ℝ) u => φ (ODEPath.flow v z b u,b u.toNNReal)
-  let τ := fun b : Driver => Exit (f b) 1 H
-  have hg (b : Driver) := ODEPath.flow_solution v hv z b b.continuous
-    hM hL (by linarith : 0 ≤ H+1) hbound hlip
-  have hgc (b : Driver) : ContinuousOn (g b) (Icc 0 (H+1)) :=
-    ODEPath.continuousOn_flow v hv z b b.continuous hM hL (by linarith) hbound hlip
-  have hfc (b : Driver) : ContinuousOn (f b) (Icc 0 (H+1)) :=
-    hφ.comp_continuousOn ((hgc b).prodMk ((b.continuous.comp continuous_real_toNNReal).continuousOn))
-  have hfcH (b : Driver) : ContinuousOn (f b) (Icc 0 H) :=
-    (hfc b).mono (Icc_subset_Icc_right (by linarith))
-  have hf0 (b : Driver) : f b 0 < 1 := by
-    simpa only [f,(hg b).1,Real.toNNReal_zero,map_zero] using hz
-  have hτ (b : Driver) : 0 ≤ τ b ∧ τ b < H+1 := by
-    have hb := exit_bounds hH (hfcH b) (R := 1)
-    exact ⟨hb.1,lt_of_le_of_lt hb.2 (by linarith)⟩
-  have hsafe (b : Driver) : ∀ u ∈ Icc 0 (τ b), f b u ≤ 1 :=
-    bound_until_exit hH (hfcH b) (hf0 b)
-  change ∀ t ≥ 0, (∫ b : Driver, test a (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ test a z 0
-  intro t ht
-  let K : Set (State × ℝ) := {q | φ q ≤ 2}
-  have hDo : IsOpen {q | φ q < 2} := isOpen_lt hφ continuous_const
-  have hK1 : IsCompact {q | φ q ≤ 1} :=
-    hK.of_isClosed_subset (isClosed_le hφ continuous_const) (fun q hq => by dsimp at *; linarith)
-  obtain ⟨δ,hδ,hδD⟩ := hK1.exists_thickening_subset_open hDo (fun q hq => by dsimp at *; linarith)
-  have hD (b : Driver) : ∃ δ > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < δ → (x,b u.toNNReal) ∈ K := by
-    refine ⟨δ,hδ,?_⟩
-    intro u hu x hx
-    have hi : φ (x,b u.toNNReal) < 2 := hδD (by
-      apply Metric.mem_thickening_iff.mpr
-      refine ⟨(g b u,b u.toNNReal),hsafe b u hu,?_⟩
-      rw [dist_eq_norm]
-      change max ‖x-g b u‖ ‖b u.toNNReal-b u.toNNReal‖ < δ
-      simpa using hx)
-    exact hi.le
-  have mf (u : ℝ) : Measurable (fun b : ℝ≥0 → ℝ => f b u) :=
-    hφ.measurable.comp ((ODEPath.measurable_flow v hv.measurable z u).prodMk (measurable_pi_apply _))
-  have hfpast : ∀ s ≥ 0, ∀ b c : ℝ≥0 → ℝ,
-      (∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) → f b s = f c s := by
-    intro s hs b c hbc
-    have hb := hbc s.toNNReal (by simpa only [Real.coe_toNNReal',max_eq_left hs] using le_refl s)
-    simp only [f,ODEPath.flow_past v z hs hbc,hb]
-  let S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (State × ℝ)) := fun r =>
-    {q | (r:ℝ) < H ∧ Safe (f q.1) 1 r ∧ q.2 ∈ K}
-  have mS : ∀ r, MeasurableSet (S r) := by
-    intro r
-    exact (MeasurableSet.const _).inter
-      (((measurableSet_safe f mf 1 _).preimage measurable_fst).inter (hK.measurableSet.preimage measurable_snd))
-  have hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r) := by
-    intro r b c hbc x
-    have he : Safe (f b) 1 r ↔ Safe (f c) 1 r := safe_past hfpast r.coe_nonneg
-      (fun u hu => hbc u hu)
-    exact and_congr_right fun _ => and_congr_left fun _ => he
-  have hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K) := by
-    intro b r x
-    change (_ ∧ _ ∧ _) ↔ (_ ∧ _)
-    rw [lt_exit_iff hH (hfcH b) r.coe_nonneg]
-    exact and_assoc.symm
-  apply expected_stopped_local_grid hW ((Function.uncurry (test a))) K domain hK isOpen_domain hKO (contDiffOn_test a)
-    v hv.measurable g z hL hM hbound hlip (fun b => (hg b).1) (fun b => (hg b).2)
-    (fun b => hv.comp_continuousOn ((hgc b).prodMk
-      ((b.continuous.comp continuous_real_toNNReal).continuousOn))) τ hτ hD
-    (fun q hq => by rw [hveq q hq]; exact test_generator_nonpos ha (hKO hq).1.le (hKO hq).2 q.2)
-    S mS hSpast hSeq (fun r q hq => hq.2.2) (by change φ (z,0) ≤ 2; linarith) ht
-
-theorem expected_stopped_reverseFlow {a q : ℝ} (ha : 1/4 < a) (hq : 0 < q)
-    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    {H : ℝ} (hH : 0 ≤ H) (φ : State × ℝ → ℝ) (hφ : Continuous φ)
-    (hK : IsCompact {x | φ x ≤ 2}) (hKO : {x | φ x ≤ 2} ⊆ domain)
-    (hyK : ∀ x, φ x ≤ 2 → q ≤ x.1.2.1.im)
-    (z : State) (hz : φ (z,0) < 1) :
-    let g := fun b : Driver => ODEPath.flow (globalVector a q) z b
-    let τ := fun b : Driver => Exit (fun u => φ (g b u,b u.toNNReal)) 1 H
-    ∀ t ≥ 0, (∫ b : Driver, test a (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ test a z 0 := by
-  exact expected_stopped_reverse_aux ha hW hH φ hφ hK hKO (globalVector a q)
-    (continuous_globalVector hq a) (by dsimp [bound]; positivity)
-    (by dsimp [lipBound]; positivity) (norm_globalVector_le hq a) (globalVector_sub_le hq a)
-    (fun x hx => globalVector_eq hq a _ _ (hyK x hx)) z hz
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open MeasureTheory ProbabilityTheory BrownianCalculus RationalStop
-
-def exhaustion (q : ℝ) (n : ℕ) (x : State × ℝ) : ℝ :=
-  max (‖x‖/((n:ℝ)+1)) (max (3-((n:ℝ)+1)*x.1.1) (3-x.1.2.1.im/q))
-
-lemma continuous_exhaustion (q : ℝ) (n : ℕ) : Continuous (exhaustion q n) := by
-  unfold exhaustion
-  fun_prop
-
-lemma exhaustion_compact (q : ℝ) (n : ℕ) : IsCompact {x | exhaustion q n x ≤ 2} := by
-  apply (isCompact_closedBall (0 : State × ℝ) (2*((n:ℝ)+1))).of_isClosed_subset
-    (isClosed_le (continuous_exhaustion q n) continuous_const)
-  intro x hx
-  have h : ‖x‖/((n:ℝ)+1) ≤ 2 := (le_max_left _ _).trans hx
-  rw [mem_closedBall,dist_zero_right]
-  exact (div_le_iff₀ (by positivity : 0 < (n:ℝ)+1)).mp h
-
-lemma exhaustion_height {q : ℝ} (hq : 0 < q) (n : ℕ) {x : State × ℝ}
-    (hx : exhaustion q n x ≤ 2) : q ≤ x.1.2.1.im := by
-  have h : 3-x.1.2.1.im/q ≤ 2 :=
-    (le_max_right _ _).trans ((le_max_right _ _).trans hx)
-  have hi : 1 ≤ x.1.2.1.im/q := by linarith
-  simpa using (le_div_iff₀ hq).mp hi
-
-lemma exhaustion_domain {q : ℝ} (hq : 0 < q) (n : ℕ) :
-    {x | exhaustion q n x ≤ 2} ⊆ domain := by
-  intro x hx
-  refine ⟨?_,hq.trans_le (exhaustion_height hq n hx)⟩
-  have h : 3-((n:ℝ)+1)*x.1.1 ≤ 2 :=
-    (le_max_left _ _).trans ((le_max_right _ _).trans hx)
-  have hn : 0 < (n:ℝ)+1 := by positivity
-  nlinarith
-
-lemma exhaustion_lt {q : ℝ} (hq : 0 < q) (n : ℕ) {x : State × ℝ}
-    (hn : ‖x‖ < (n:ℝ)+1) (hu : 2 < ((n:ℝ)+1)*x.1.1) (hy : 2*q < x.1.2.1.im) :
-    exhaustion q n x < 1 := by
-  apply max_lt
-  · exact (div_lt_one (by positivity)).mpr hn
-  · apply max_lt
-    · linarith
-    · have hi := (lt_div_iff₀ hq).mpr hy
-      linarith
-
-lemma eventually_exhaustion_lt {q δ : ℝ} (hq : 0 < q) (hδ : 0 < δ)
-    {T : ℝ} (_hT : 0 ≤ T) (f : ℝ → State × ℝ) (hf : ContinuousOn f (Icc 0 T))
-    (hu : ∀ t ∈ Icc 0 T, δ ≤ (f t).1.1)
-    (hy : ∀ t ∈ Icc 0 T, 2*q < (f t).1.2.1.im) :
-    ∀ᶠ n : ℕ in atTop, ∀ t ∈ Icc 0 T, exhaustion q n (f t) < 1 := by
-  obtain ⟨C,hC⟩ := (isCompact_Icc : IsCompact (Icc (0:ℝ) T)).exists_bound_of_continuousOn hf
-  obtain ⟨N,hN⟩ := exists_nat_gt (max C (2/δ))
-  filter_upwards [eventually_ge_atTop N] with n hn
-  intro t ht
-  have hnN : (N:ℝ) ≤ n := by exact_mod_cast hn
-  have hnC : C < (n:ℝ)+1 := by linarith [le_max_left C (2/δ)]
-  have hnδ : 2/δ < (n:ℝ)+1 := by linarith [le_max_right C (2/δ)]
-  apply exhaustion_lt hq n ((hC t ht).trans_lt hnC) _ (hy t ht)
-  have hi : 2 < ((n:ℝ)+1)*δ := (div_lt_iff₀ hδ).mp hnδ
-  exact hi.trans_le (mul_le_mul_of_nonneg_left (hu t ht) (by positivity))
-
-lemma measurable_test (a : ℝ) : Measurable (Function.uncurry (test a)) := by
-  change Measurable (fun x : State × ℝ => Real.exp
-    (lambda a*x.1.2.2+r a/2*Real.log ((x.1.2.1.re-x.2)^2+x.1.2.1.im^2)+
-     (zeta a-r a)*Real.log x.1.2.1.im-zeta a/2*Real.log
-       (((x.1.2.1.re-x.2)^2+x.1.2.1.im^2)+x.1.1/2)))
-  fun_prop
-
-lemma test_pos (a : ℝ) (x : State) (b : ℝ) : 0 < test a x b := Real.exp_pos _
-
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open MeasureTheory ProbabilityTheory BrownianCalculus RationalStop
-
-def driven (a q : ℝ) (z : State) (b : Driver) (t : ℝ) : State × ℝ :=
-  (ODEPath.flow (globalVector a q) z b t,b t.toNNReal)
-
-lemma measurable_driven {q : ℝ} (hq : 0 < q) (a : ℝ) (z : State) (t : ℝ) :
-    Measurable (fun b : Driver => driven a q z b t) := by
-  have mb : Measurable (fun b : Driver => (b : ℝ≥0 → ℝ)) :=
-    Measurable.of_eval fun time => (show Continuous (fun driver : Driver => driver time) from continuous_eval_const time).measurable
-  exact ((ODEPath.measurable_flow _ (continuous_globalVector hq a).measurable z t).comp mb).prodMk
-    (show Continuous (fun b : Driver => b t.toNNReal) from continuous_eval_const _).measurable
-
-lemma continuousOn_driven {q T : ℝ} (hq : 0 < q) (hT : 0 ≤ T) (a : ℝ) (z : State) (b : Driver) :
-    ContinuousOn (driven a q z b) (Icc 0 T) := by
-  exact (ODEPath.continuousOn_flow _ (continuous_globalVector hq a) z b b.continuous
-    (by dsimp [bound]; positivity) (by dsimp [lipBound]; positivity) hT
-    (norm_globalVector_le hq a) (globalVector_sub_le hq a)).prodMk
-    ((b.continuous.comp continuous_real_toNNReal).continuousOn)
-
-lemma exit_eq_horizon {f : ℝ → ℝ} {T : ℝ} (hT : 0 ≤ T)
-    (hf : ContinuousOn f (Icc 0 T)) (hs : ∀ t ∈ Icc 0 T, f t < 1) : Exit f 1 T = T := by
-  have hb := exit_bounds hT hf (R := 1)
-  apply le_antisymm hb.2
-  by_contra hn
-  have he : Exit f 1 T < T := lt_of_not_ge hn
-  have hs' : Safe f 1 (Exit f 1 T) := (safe_iff hb.1
-    (hf.mono (Icc_subset_Icc_right hb.2))).mpr (fun t ht => hs t ⟨ht.1,ht.2.trans hb.2⟩)
-  exact lt_irrefl _ ((lt_exit_iff hT hf hb.1).mpr ⟨he,hs'⟩)
-
-lemma stopped_test_integrable {q : ℝ} (hq : 0 < q) (a : ℝ)
-    {W : Measure Driver} [IsFiniteMeasure W] {T : ℝ} (hT : 0 ≤ T)
-    (φ : State × ℝ → ℝ) (hφ : Continuous φ)
-    (hK : IsCompact {x | φ x ≤ 2}) (hKO : {x | φ x ≤ 2} ⊆ domain)
-    (z : State) (hz : φ (z,0) < 1) :
-    Integrable (fun b : Driver => test a
-      (driven a q z b (Exit (fun u => φ (driven a q z b u)) 1 T)).1
-      (driven a q z b (Exit (fun u => φ (driven a q z b u)) 1 T)).2) W := by
-  let f := fun (b : Driver) u => φ (driven a q z b u)
-  let τ := fun b => Exit (f b) 1 T
-  have hfc (b : Driver) : ContinuousOn (f b) (Icc 0 T) :=
-    hφ.comp_continuousOn (continuousOn_driven hq hT a z b)
-  have mτ : Measurable τ := measurable_exit hT
-    (fun u => hφ.measurable.comp (measurable_driven hq a z u)) hfc
-  have hτ (b : Driver) : τ b ∈ Icc 0 T := exit_bounds hT (hfc b)
-  have md : Measurable (fun b => driven a q z b (τ b)) := measurable_stopped_eval hT
-    (measurable_driven hq a z) (continuousOn_driven hq hT a z) mτ hτ
-  have hf0 (b : Driver) : f b 0 < 1 := by
-    have hg := ODEPath.flow_solution (globalVector a q) (continuous_globalVector hq a) z b b.continuous
-      (by dsimp [bound]; positivity) (by dsimp [lipBound]; positivity) hT
-      (norm_globalVector_le hq a) (globalVector_sub_le hq a)
-    simpa only [f,driven,hg.1,Real.toNNReal_zero,map_zero] using hz
-  have hsafe (b : Driver) : driven a q z b (τ b) ∈ {x | φ x ≤ 2} := by
-    have hb := bound_until_exit hT (hfc b) (hf0 b) (τ b) ⟨(hτ b).1,le_rfl⟩
-    change f b (τ b) ≤ 2
-    linarith
-  obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn ((contDiffOn_test a).continuousOn.mono hKO)
-  exact Integrable.of_bound ((measurable_test a).comp md).aestronglyMeasurable C
-    (ae_of_all _ fun b => hC _ (hsafe b))
-
-lemma lintegral_of_eventually_stopped {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
-    (f : ℕ → Ω → ℝ) (hf : ∀ n, Integrable (f n) μ) (hpos : ∀ n x, 0 ≤ f n x)
-    (F : Ω → ℝ) (hlim : ∀ x, ∀ᶠ n in atTop, f n x = F x)
-    (C : ℝ) (he : ∀ n, (∫ x, f n x ∂μ) ≤ C) :
-    (∫⁻ x, ENNReal.ofReal (F x) ∂μ) ≤ ENNReal.ofReal C := by
-  have hl (x : Ω) : liminf (fun n => ENNReal.ofReal (f n x)) atTop = ENNReal.ofReal (F x) :=
-    (tendsto_const_nhds.congr' ((hlim x).mono fun n hn => congrArg ENNReal.ofReal hn.symm)).liminf_eq
-  calc (∫⁻ x, ENNReal.ofReal (F x) ∂μ)
-      = ∫⁻ x, liminf (fun n => ENNReal.ofReal (f n x)) atTop ∂μ := by simp only [hl]
-    _ ≤ liminf (fun n => ∫⁻ x, ENNReal.ofReal (f n x) ∂μ) atTop :=
-      lintegral_liminf_le' fun n => (hf n).aestronglyMeasurable.aemeasurable.ennreal_ofReal
-    _ ≤ liminf (fun _ : ℕ => ENNReal.ofReal C) atTop := by
-      refine Filter.liminf_le_liminf ?_
-      filter_upwards [] with n
-      rw [← ofReal_integral_eq_lintegral_ofReal (hf n) (ae_of_all _ (hpos n))]
-      exact ENNReal.ofReal_le_ofReal (he n)
-    _ = ENNReal.ofReal C := tendsto_const_nhds.liminf_eq
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open MeasureTheory ProbabilityTheory BrownianCalculus RationalStop
-
- 
-
-theorem reverse_supersolution_bound {a y δ T : ℝ} (ha : 1/4 < a)
-    (hy : 0 < y) (hδ : 0 < δ) (hT : 0 ≤ T)
-    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W) :
-    (∫⁻ b : Driver, ENNReal.ofReal
-      (test a (reverseFlow a (y/4) (T+δ) y b T) (b T.toNNReal)) ∂W) ≤
-        ENNReal.ofReal (test a (initial (T+δ) y) 0) := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  let q := y/4
-  let z := initial (T+δ) y
-  have hq : 0 < q := by dsimp [q]; positivity
-  have ha0 : 0 ≤ a := by linarith
-  have hi : ∀ᶠ n : ℕ in atTop, exhaustion q n (z,0) < 1 := by
-    have hh := eventually_exhaustion_lt hq hδ (T := 0) le_rfl (fun _ => (z,0))
-      continuousOn_const (fun t ht => by change δ ≤ T+δ; linarith)
-      (fun t ht => by change 2*(y/4) < y; linarith)
-    filter_upwards [hh] with n hn
-    exact hn 0 ⟨le_rfl,le_rfl⟩
-  obtain ⟨N,hN⟩ := eventually_atTop.mp hi
-  let φ := fun n => exhaustion q (n+N)
-  have hφ0 (n : ℕ) : φ n (z,0) < 1 := hN (n+N) (by omega)
-  let τ := fun n (b : Driver) => Exit (fun u => φ n (driven a q z b u)) 1 T
-  let f := fun n (b : Driver) => test a (driven a q z b (τ n b)).1 (driven a q z b (τ n b)).2
-  have hτ (n : ℕ) (b : Driver) : τ n b ∈ Icc 0 T := exit_bounds hT
-    ((continuous_exhaustion q (n+N)).comp_continuousOn (continuousOn_driven hq hT a z b))
-  have hf (n : ℕ) : Integrable (f n) W := stopped_test_integrable hq a hT (φ n)
-    (continuous_exhaustion q (n+N)) (exhaustion_compact q (n+N)) (exhaustion_domain hq (n+N)) z (hφ0 n)
-  have he (n : ℕ) : (∫ b, f n b ∂W) ≤ test a z 0 := by
-    have hh := expected_stopped_reverseFlow ha hq hW hT (φ n) (continuous_exhaustion q (n+N))
-      (exhaustion_compact q (n+N)) (exhaustion_domain hq (n+N))
-      (fun x hx => exhaustion_height hq (n+N) hx) z (hφ0 n) T hT
-    have heq (b : Driver) : min T (τ n b) = τ n b := min_eq_right (hτ n b).2
-    change (∫ b : Driver, test a (ODEPath.flow (globalVector a q) z b (min T (τ n b)))
-      (b (min T (τ n b)).toNNReal) ∂W) ≤ test a z 0 at hh
-    simpa only [heq,f,driven] using hh
-  have hlim (b : Driver) : ∀ᶠ n in atTop,
-      f n b = test a (reverseFlow a q (T+δ) y b T) (b T.toNNReal) := by
-    have hh := eventually_exhaustion_lt hq hδ hT (driven a q z b)
-      (continuousOn_driven hq hT a z b)
-      (fun t ht => by
-        change δ ≤ (reverseFlow a q (T+δ) y b t).1
-        rw [reverseFlow_time hq a (T+δ) y b b.continuous hT t ht]
-        linarith [ht.2])
-      (fun t ht => by
-        have hb := reverseFlow_height hq ha0 hy.le (T+δ) b b.continuous hT t ht
-        change 2*q < (reverseFlow a q (T+δ) y b t).2.1.im
-        dsimp [q]
-        linarith)
-    obtain ⟨K,hK⟩ := eventually_atTop.mp hh
-    filter_upwards [eventually_ge_atTop K] with n hn
-    have hexit : τ n b = T := exit_eq_horizon hT
-      ((continuous_exhaustion q (n+N)).comp_continuousOn (continuousOn_driven hq hT a z b))
-      (hK (n+N) (by omega))
-    simp only [f,hexit,driven]
-    rfl
-  exact lintegral_of_eventually_stopped W f hf (fun n b => (test_pos a _ _).le)
-    (fun b => test a (reverseFlow a q (T+δ) y b T) (b T.toNNReal)) hlim (test a z 0) he
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-
-lemma r_sub_zeta_pos {a : ℝ} (ha : 1/4 < a) : 0 < r a-zeta a := by
-  have ha0 : 0 < a := by linarith
-  have he : r a-zeta a = a+1/2+1/(16*a) := by unfold r zeta; ring
-  rw [he]
-  positivity
-
-lemma exp_logderiv_le_test {a : ℝ} (ha : 1/4 < a) (v : State) (b : ℝ)
-    (hu : 0 ≤ v.1) (hy : 0 < v.2.1.im) (huy : v.1 ≤ v.2.1.im^2) :
-    Real.exp (lambda a*v.2.2) ≤ Real.exp (zeta a/2*Real.log (3/2))*test a v b := by
-  let x := v.2.1.re-b
-  let y := v.2.1.im
-  have hR := R_pos x hy
-  have hD := D_pos hu hy x
-  have hlogR : 2*Real.log y ≤ Real.log (R x y) := by
-    have hh := Real.log_le_log (sq_pos_of_pos hy) (show y^2 ≤ R x y from by unfold R; nlinarith [sq_nonneg x])
-    simpa only [Real.log_pow,Nat.cast_ofNat] using hh
-  have hlogD : Real.log (D v.1 x y) ≤ Real.log (R x y)+Real.log (3/2) := by
-    have hle : D v.1 x y ≤ R x y*(3/2) := by
-      dsimp [D,R,x,y]
-      nlinarith [sq_nonneg (v.2.1.re-b)]
-    have hh := Real.log_le_log hD hle
-    rwa [Real.log_mul hR.ne' (by norm_num : (3/2:ℝ) ≠ 0)] at hh
-  have hz := (parameters ha).1
-  have hrz := r_sub_zeta_pos ha
-  have h1 := mul_nonneg hrz.le (sub_nonneg.mpr hlogR)
-  have h2 := mul_nonneg hz.le (sub_nonneg.mpr hlogD)
-  unfold test F
-  rw [← Real.exp_add]
-  apply Real.exp_le_exp.mpr
-  change lambda a*v.2.2 ≤ zeta a/2*Real.log (3/2)+H a v.1 x y v.2.2
-  unfold H
-  nlinarith
-
-lemma initial_test_scale (a : ℝ) {y T : ℝ} (hy : 0 < y) (hT : 0 ≤ T) :
-    Real.exp (zeta a/2*Real.log (3/2))*test a (initial (T+y^2) y) 0 =
-      (3*y^2/(T+3*y^2))^(zeta a/2) := by
-  have hy2 : 0 < y^2 := sq_pos_of_pos hy
-  have hD : 0 < y^2+(T+y^2)/2 := by positivity
-  have hden : 0 < T+3*y^2 := by positivity
-  have hrat : 0 < 3*y^2/(T+3*y^2) := by positivity
-  have he : 3*y^2/(T+3*y^2) = ((3/2:ℝ)*y^2)/(y^2+(T+y^2)/2) := by
-    field_simp
-    ring
-  rw [Real.rpow_def_of_pos hrat,he,
-    Real.log_div (mul_ne_zero (by norm_num) hy2.ne') hD.ne',
-    Real.log_mul (by norm_num : (3/2:ℝ) ≠ 0) hy2.ne',Real.log_pow]
-  simp only [test,initial,F,H,D,R,sub_zero,zero_pow (by omega : (2:ℕ) ≠ 0),zero_add,mul_zero,zero_add]
-  rw [← Real.exp_add,Real.log_pow]
-  congr 1
-  ring
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-open MeasureTheory ProbabilityTheory BrownianCalculus
-
- 
-
-theorem reverse_logderiv_moment {a y T : ℝ} (ha : 1/4 < a)
-    (hy : 0 < y) (hT : 0 ≤ T)
-    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W) :
-    (∫⁻ b : Driver, ENNReal.ofReal (Real.exp
-      (lambda a*(reverseFlow a (y/4) (T+y^2) y b T).2.2)) ∂W) ≤
-        ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta a/2)) := by
-  let C := Real.exp (zeta a/2*Real.log (3/2))
-  let g := fun (b : Driver) => reverseFlow a (y/4) (T+y^2) y b T
-  have hq : 0 < y/4 := by positivity
-  have htime (b : Driver) : (g b).1 = y^2 := by
-    dsimp [g]
-    rw [reverseFlow_time hq a (T+y^2) y b b.continuous hT T ⟨hT,le_rfl⟩]
-    ring
-  have hheight (b : Driver) : y ≤ (g b).2.1.im :=
-    reverseFlow_height hq (by linarith) hy.le (T+y^2) b b.continuous hT T ⟨hT,le_rfl⟩
-  have hpoint (b : Driver) : Real.exp (lambda a*(g b).2.2) ≤ C*test a (g b) (b T.toNNReal) := by
-    apply exp_logderiv_le_test ha _ _
-    · rw [htime]; positivity
-    · exact hy.trans_le (hheight b)
-    · rw [htime]
-      nlinarith [hheight b]
-  calc (∫⁻ b : Driver, ENNReal.ofReal (Real.exp (lambda a*(g b).2.2)) ∂W)
-      ≤ ∫⁻ b : Driver, ENNReal.ofReal (C*test a (g b) (b T.toNNReal)) ∂W :=
-        lintegral_mono fun b => ENNReal.ofReal_le_ofReal (hpoint b)
-    _ = ENNReal.ofReal C*(∫⁻ b : Driver, ENNReal.ofReal (test a (g b) (b T.toNNReal)) ∂W) := by
-      simp only [ENNReal.ofReal_mul (show 0 ≤ C from (Real.exp_pos _).le)]
-      exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-    _ ≤ ENNReal.ofReal C*ENNReal.ofReal (test a (initial (T+y^2) y) 0) :=
-      mul_le_mul' le_rfl (reverse_supersolution_bound ha hy (sq_pos_of_pos hy) hT hW)
-    _ = ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta a/2)) := by
-      rw [← ENNReal.ofReal_mul (show 0 ≤ C from (Real.exp_pos _).le)]
-      exact congrArg ENNReal.ofReal (initial_test_scale a hy hT)
-end SLEExactGauge.ReverseDerivative
-
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-
-noncomputable section
-namespace SLEExactGauge.BrownianRestart
-
-abbrev Driver := C(ℝ≥0, ℝ)₀
-instance : MeasurableSpace Driver := borel Driver
-instance : BorelSpace Driver := ⟨rfl⟩
-instance : PolishSpace Driver := ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace
-
-def code (b : Driver) : ℚ → ℝ := fun q => b (Real.toNNReal (q:ℝ))
-
-lemma measurable_code : Measurable code :=
-  Measurable.of_eval fun _ => (continuous_eval_const _).measurable
-
-lemma dense_rational_time : DenseRange (fun q : ℚ => Real.toNNReal (q:ℝ)) := by
-  have hd : Surjective Real.toNNReal := fun x => ⟨(x:ℝ),Real.toNNReal_coe⟩
-  exact hd.denseRange.comp Rat.denseRange_cast continuous_real_toNNReal
-
-lemma code_injective : Injective code := by
-  intro b c h
-  apply ContinuousMapZero.ext
-  have he := dense_rational_time.equalizer (map_continuous b) (map_continuous c) h
-  exact congrFun he
-
-lemma code_embedding : MeasurableEmbedding code := measurable_code.measurableEmbedding code_injective
-
-end SLEExactGauge.BrownianRestart
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ReverseGenerators
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+namespace SLEExactGauge.ReverseDerivative
+open _root_.OAI.SLEExactGauge.Localization BrownianCalculus ODEPath RationalStop
+open MeasureTheory ProbabilityTheory
+theorem expected_stopped_reverse_aux {a : ℝ} (ha : 1/4 < a)
+    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    {H : ℝ} (hH : 0 ≤ H) (φ : State × ℝ → ℝ) (hφ : Continuous φ)
+    (hK : IsCompact {x | φ x ≤ 2}) (hKO : {x | φ x ≤ 2} ⊆ domain)
+    (v : State → ℝ → State) (hv : Continuous (Function.uncurry v))
+    {M L : ℝ} (hM : 0 ≤ M) (hL : 0 ≤ L)
+    (hbound : ∀ x b, ‖v x b‖ ≤ M)
+    (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hveq : ∀ x, φ x ≤ 2 → v x.1 x.2 = vector a x.1 x.2)
+    (z : State) (hz : φ (z,0) < 1) :
+    let g := fun b : Driver => ODEPath.flow v z b
+    let τ := fun b : Driver => Exit (fun u => φ (g b u,b u.toNNReal)) 1 H
+    ∀ t ≥ 0, (∫ b : Driver, test a (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ test a z 0 := by
+  let g := fun b : Driver => ODEPath.flow v z b
+  let f := fun (b : ℝ≥0 → ℝ) u => φ (ODEPath.flow v z b u,b u.toNNReal)
+  let τ := fun b : Driver => Exit (f b) 1 H
+  have hg (b : Driver) := ODEPath.flow_solution v hv z b b.continuous
+    hM hL (by linarith : 0 ≤ H+1) hbound hlip
+  have hgc (b : Driver) : ContinuousOn (g b) (Icc 0 (H+1)) :=
+    ODEPath.continuousOn_flow v hv z b b.continuous hM hL (by linarith) hbound hlip
+  have hfc (b : Driver) : ContinuousOn (f b) (Icc 0 (H+1)) :=
+    hφ.comp_continuousOn ((hgc b).prodMk ((b.continuous.comp continuous_real_toNNReal).continuousOn))
+  have hfcH (b : Driver) : ContinuousOn (f b) (Icc 0 H) :=
+    (hfc b).mono (Icc_subset_Icc_right (by linarith))
+  have hf0 (b : Driver) : f b 0 < 1 := by
+    simpa only [f,(hg b).1,Real.toNNReal_zero,map_zero] using hz
+  have hτ (b : Driver) : 0 ≤ τ b ∧ τ b < H+1 := by
+    have hb := exit_bounds hH (hfcH b) (R := 1)
+    exact ⟨hb.1,lt_of_le_of_lt hb.2 (by linarith)⟩
+  have hsafe (b : Driver) : ∀ u ∈ Icc 0 (τ b), f b u ≤ 1 :=
+    bound_until_exit hH (hfcH b) (hf0 b)
+  change ∀ t ≥ 0, (∫ b : Driver, test a (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ test a z 0
+  intro t ht
+  let K : Set (State × ℝ) := {q | φ q ≤ 2}
+  have hDo : IsOpen {q | φ q < 2} := isOpen_lt hφ continuous_const
+  have hK1 : IsCompact {q | φ q ≤ 1} :=
+    hK.of_isClosed_subset (isClosed_le hφ continuous_const) (fun q hq => by dsimp at *; linarith)
+  obtain ⟨δ,hδ,hδD⟩ := hK1.exists_thickening_subset_open hDo (fun q hq => by dsimp at *; linarith)
+  have hD (b : Driver) : ∃ δ > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < δ → (x,b u.toNNReal) ∈ K := by
+    refine ⟨δ,hδ,?_⟩
+    intro u hu x hx
+    have hi : φ (x,b u.toNNReal) < 2 := hδD (by
+      apply Metric.mem_thickening_iff.mpr
+      refine ⟨(g b u,b u.toNNReal),hsafe b u hu,?_⟩
+      rw [dist_eq_norm]
+      change max ‖x-g b u‖ ‖b u.toNNReal-b u.toNNReal‖ < δ
+      simpa using hx)
+    exact hi.le
+  have mf (u : ℝ) : Measurable (fun b : ℝ≥0 → ℝ => f b u) :=
+    hφ.measurable.comp ((ODEPath.measurable_flow v hv.measurable z u).prodMk (measurable_pi_apply _))
+  have hfpast : ∀ s ≥ 0, ∀ b c : ℝ≥0 → ℝ,
+      (∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) → f b s = f c s := by
+    intro s hs b c hbc
+    have hb := hbc s.toNNReal (by simpa only [Real.coe_toNNReal',max_eq_left hs] using le_refl s)
+    simp only [f,ODEPath.flow_past v z hs hbc,hb]
+  let S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (State × ℝ)) := fun r =>
+    {q | (r:ℝ) < H ∧ Safe (f q.1) 1 r ∧ q.2 ∈ K}
+  have mS : ∀ r, MeasurableSet (S r) := by
+    intro r
+    exact (MeasurableSet.const _).inter
+      (((measurableSet_safe f mf 1 _).preimage measurable_fst).inter (hK.measurableSet.preimage measurable_snd))
+  have hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r) := by
+    intro r b c hbc x
+    have he : Safe (f b) 1 r ↔ Safe (f c) 1 r := safe_past hfpast r.coe_nonneg
+      (fun u hu => hbc u hu)
+    exact and_congr_right fun _ => and_congr_left fun _ => he
+  have hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K) := by
+    intro b r x
+    change (_ ∧ _ ∧ _) ↔ (_ ∧ _)
+    rw [lt_exit_iff hH (hfcH b) r.coe_nonneg]
+    exact and_assoc.symm
+  apply expected_stopped_local_grid hW ((Function.uncurry (test a))) K domain hK isOpen_domain hKO (contDiffOn_test a)
+    v hv.measurable g z hL hM hbound hlip (fun b => (hg b).1) (fun b => (hg b).2)
+    (fun b => hv.comp_continuousOn ((hgc b).prodMk
+      ((b.continuous.comp continuous_real_toNNReal).continuousOn))) τ hτ hD
+    (fun q hq => by rw [hveq q hq]; exact test_generator_nonpos ha (hKO hq).1.le (hKO hq).2 q.2)
+    S mS hSpast hSeq (fun r q hq => hq.2.2) (by change φ (z,0) ≤ 2; linarith) ht
+
+theorem expected_stopped_reverseFlow {a q : ℝ} (ha : 1/4 < a) (hq : 0 < q)
+    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    {H : ℝ} (hH : 0 ≤ H) (φ : State × ℝ → ℝ) (hφ : Continuous φ)
+    (hK : IsCompact {x | φ x ≤ 2}) (hKO : {x | φ x ≤ 2} ⊆ domain)
+    (hyK : ∀ x, φ x ≤ 2 → q ≤ x.1.2.1.im)
+    (z : State) (hz : φ (z,0) < 1) :
+    let g := fun b : Driver => ODEPath.flow (globalVector a q) z b
+    let τ := fun b : Driver => Exit (fun u => φ (g b u,b u.toNNReal)) 1 H
+    ∀ t ≥ 0, (∫ b : Driver, test a (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ test a z 0 := by
+  exact expected_stopped_reverse_aux ha hW hH φ hφ hK hKO (globalVector a q)
+    (continuous_globalVector hq a) (by dsimp [bound]; positivity)
+    (by dsimp [lipBound]; positivity) (norm_globalVector_le hq a) (globalVector_sub_le hq a)
+    (fun x hx => globalVector_eq hq a _ _ (hyK x hx)) z hz
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open MeasureTheory ProbabilityTheory BrownianCalculus RationalStop
+
+def exhaustion (q : ℝ) (n : ℕ) (x : State × ℝ) : ℝ :=
+  max (‖x‖/((n:ℝ)+1)) (max (3-((n:ℝ)+1)*x.1.1) (3-x.1.2.1.im/q))
+
+lemma continuous_exhaustion (q : ℝ) (n : ℕ) : Continuous (exhaustion q n) := by
+  unfold exhaustion
+  fun_prop
+
+lemma exhaustion_compact (q : ℝ) (n : ℕ) : IsCompact {x | exhaustion q n x ≤ 2} := by
+  apply (isCompact_closedBall (0 : State × ℝ) (2*((n:ℝ)+1))).of_isClosed_subset
+    (isClosed_le (continuous_exhaustion q n) continuous_const)
+  intro x hx
+  have h : ‖x‖/((n:ℝ)+1) ≤ 2 := (le_max_left _ _).trans hx
+  rw [mem_closedBall,dist_zero_right]
+  exact (div_le_iff₀ (by positivity : 0 < (n:ℝ)+1)).mp h
+
+lemma exhaustion_height {q : ℝ} (hq : 0 < q) (n : ℕ) {x : State × ℝ}
+    (hx : exhaustion q n x ≤ 2) : q ≤ x.1.2.1.im := by
+  have h : 3-x.1.2.1.im/q ≤ 2 :=
+    (le_max_right _ _).trans ((le_max_right _ _).trans hx)
+  have hi : 1 ≤ x.1.2.1.im/q := by linarith
+  simpa using (le_div_iff₀ hq).mp hi
+
+lemma exhaustion_domain {q : ℝ} (hq : 0 < q) (n : ℕ) :
+    {x | exhaustion q n x ≤ 2} ⊆ domain := by
+  intro x hx
+  refine ⟨?_,hq.trans_le (exhaustion_height hq n hx)⟩
+  have h : 3-((n:ℝ)+1)*x.1.1 ≤ 2 :=
+    (le_max_left _ _).trans ((le_max_right _ _).trans hx)
+  have hn : 0 < (n:ℝ)+1 := by positivity
+  nlinarith
+
+lemma exhaustion_lt {q : ℝ} (hq : 0 < q) (n : ℕ) {x : State × ℝ}
+    (hn : ‖x‖ < (n:ℝ)+1) (hu : 2 < ((n:ℝ)+1)*x.1.1) (hy : 2*q < x.1.2.1.im) :
+    exhaustion q n x < 1 := by
+  apply max_lt
+  · exact (div_lt_one (by positivity)).mpr hn
+  · apply max_lt
+    · linarith
+    · have hi := (lt_div_iff₀ hq).mpr hy
+      linarith
+
+lemma eventually_exhaustion_lt {q δ : ℝ} (hq : 0 < q) (hδ : 0 < δ)
+    {T : ℝ} (_hT : 0 ≤ T) (f : ℝ → State × ℝ) (hf : ContinuousOn f (Icc 0 T))
+    (hu : ∀ t ∈ Icc 0 T, δ ≤ (f t).1.1)
+    (hy : ∀ t ∈ Icc 0 T, 2*q < (f t).1.2.1.im) :
+    ∀ᶠ n : ℕ in atTop, ∀ t ∈ Icc 0 T, exhaustion q n (f t) < 1 := by
+  obtain ⟨C,hC⟩ := (isCompact_Icc : IsCompact (Icc (0:ℝ) T)).exists_bound_of_continuousOn hf
+  obtain ⟨N,hN⟩ := exists_nat_gt (max C (2/δ))
+  filter_upwards [eventually_ge_atTop N] with n hn
+  intro t ht
+  have hnN : (N:ℝ) ≤ n := by exact_mod_cast hn
+  have hnC : C < (n:ℝ)+1 := by linarith [le_max_left C (2/δ)]
+  have hnδ : 2/δ < (n:ℝ)+1 := by linarith [le_max_right C (2/δ)]
+  apply exhaustion_lt hq n ((hC t ht).trans_lt hnC) _ (hy t ht)
+  have hi : 2 < ((n:ℝ)+1)*δ := (div_lt_iff₀ hδ).mp hnδ
+  exact hi.trans_le (mul_le_mul_of_nonneg_left (hu t ht) (by positivity))
+
+lemma measurable_test (a : ℝ) : Measurable (Function.uncurry (test a)) := by
+  change Measurable (fun x : State × ℝ => Real.exp
+    (lambda a*x.1.2.2+r a/2*Real.log ((x.1.2.1.re-x.2)^2+x.1.2.1.im^2)+
+     (zeta a-r a)*Real.log x.1.2.1.im-zeta a/2*Real.log
+       (((x.1.2.1.re-x.2)^2+x.1.2.1.im^2)+x.1.1/2)))
+  fun_prop
+
+lemma test_pos (a : ℝ) (x : State) (b : ℝ) : 0 < test a x b := Real.exp_pos _
+
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open MeasureTheory ProbabilityTheory BrownianCalculus RationalStop
+
+def driven (a q : ℝ) (z : State) (b : Driver) (t : ℝ) : State × ℝ :=
+  (ODEPath.flow (globalVector a q) z b t,b t.toNNReal)
+
+lemma measurable_driven {q : ℝ} (hq : 0 < q) (a : ℝ) (z : State) (t : ℝ) :
+    Measurable (fun b : Driver => driven a q z b t) := by
+  have mb : Measurable (fun b : Driver => (b : ℝ≥0 → ℝ)) :=
+    Measurable.of_eval fun time => (show Continuous (fun driver : Driver => driver time) from continuous_eval_const time).measurable
+  exact ((ODEPath.measurable_flow _ (continuous_globalVector hq a).measurable z t).comp mb).prodMk
+    (show Continuous (fun b : Driver => b t.toNNReal) from continuous_eval_const _).measurable
+
+lemma continuousOn_driven {q T : ℝ} (hq : 0 < q) (hT : 0 ≤ T) (a : ℝ) (z : State) (b : Driver) :
+    ContinuousOn (driven a q z b) (Icc 0 T) := by
+  exact (ODEPath.continuousOn_flow _ (continuous_globalVector hq a) z b b.continuous
+    (by dsimp [bound]; positivity) (by dsimp [lipBound]; positivity) hT
+    (norm_globalVector_le hq a) (globalVector_sub_le hq a)).prodMk
+    ((b.continuous.comp continuous_real_toNNReal).continuousOn)
+
+lemma exit_eq_horizon {f : ℝ → ℝ} {T : ℝ} (hT : 0 ≤ T)
+    (hf : ContinuousOn f (Icc 0 T)) (hs : ∀ t ∈ Icc 0 T, f t < 1) : Exit f 1 T = T := by
+  have hb := exit_bounds hT hf (R := 1)
+  apply le_antisymm hb.2
+  by_contra hn
+  have he : Exit f 1 T < T := lt_of_not_ge hn
+  have hs' : Safe f 1 (Exit f 1 T) := (safe_iff hb.1
+    (hf.mono (Icc_subset_Icc_right hb.2))).mpr (fun t ht => hs t ⟨ht.1,ht.2.trans hb.2⟩)
+  exact lt_irrefl _ ((lt_exit_iff hT hf hb.1).mpr ⟨he,hs'⟩)
+
+lemma stopped_test_integrable {q : ℝ} (hq : 0 < q) (a : ℝ)
+    {W : Measure Driver} [IsFiniteMeasure W] {T : ℝ} (hT : 0 ≤ T)
+    (φ : State × ℝ → ℝ) (hφ : Continuous φ)
+    (hK : IsCompact {x | φ x ≤ 2}) (hKO : {x | φ x ≤ 2} ⊆ domain)
+    (z : State) (hz : φ (z,0) < 1) :
+    Integrable (fun b : Driver => test a
+      (driven a q z b (Exit (fun u => φ (driven a q z b u)) 1 T)).1
+      (driven a q z b (Exit (fun u => φ (driven a q z b u)) 1 T)).2) W := by
+  let f := fun (b : Driver) u => φ (driven a q z b u)
+  let τ := fun b => Exit (f b) 1 T
+  have hfc (b : Driver) : ContinuousOn (f b) (Icc 0 T) :=
+    hφ.comp_continuousOn (continuousOn_driven hq hT a z b)
+  have mτ : Measurable τ := measurable_exit hT
+    (fun u => hφ.measurable.comp (measurable_driven hq a z u)) hfc
+  have hτ (b : Driver) : τ b ∈ Icc 0 T := exit_bounds hT (hfc b)
+  have md : Measurable (fun b => driven a q z b (τ b)) := measurable_stopped_eval hT
+    (measurable_driven hq a z) (continuousOn_driven hq hT a z) mτ hτ
+  have hf0 (b : Driver) : f b 0 < 1 := by
+    have hg := ODEPath.flow_solution (globalVector a q) (continuous_globalVector hq a) z b b.continuous
+      (by dsimp [bound]; positivity) (by dsimp [lipBound]; positivity) hT
+      (norm_globalVector_le hq a) (globalVector_sub_le hq a)
+    simpa only [f,driven,hg.1,Real.toNNReal_zero,map_zero] using hz
+  have hsafe (b : Driver) : driven a q z b (τ b) ∈ {x | φ x ≤ 2} := by
+    have hb := bound_until_exit hT (hfc b) (hf0 b) (τ b) ⟨(hτ b).1,le_rfl⟩
+    change f b (τ b) ≤ 2
+    linarith
+  obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn ((contDiffOn_test a).continuousOn.mono hKO)
+  exact Integrable.of_bound ((measurable_test a).comp md).aestronglyMeasurable C
+    (ae_of_all _ fun b => hC _ (hsafe b))
+
+lemma lintegral_of_eventually_stopped {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
+    (f : ℕ → Ω → ℝ) (hf : ∀ n, Integrable (f n) μ) (hpos : ∀ n x, 0 ≤ f n x)
+    (F : Ω → ℝ) (hlim : ∀ x, ∀ᶠ n in atTop, f n x = F x)
+    (C : ℝ) (he : ∀ n, (∫ x, f n x ∂μ) ≤ C) :
+    (∫⁻ x, ENNReal.ofReal (F x) ∂μ) ≤ ENNReal.ofReal C := by
+  have hl (x : Ω) : liminf (fun n => ENNReal.ofReal (f n x)) atTop = ENNReal.ofReal (F x) :=
+    (tendsto_const_nhds.congr' ((hlim x).mono fun n hn => congrArg ENNReal.ofReal hn.symm)).liminf_eq
+  calc (∫⁻ x, ENNReal.ofReal (F x) ∂μ)
+      = ∫⁻ x, liminf (fun n => ENNReal.ofReal (f n x)) atTop ∂μ := by simp only [hl]
+    _ ≤ liminf (fun n => ∫⁻ x, ENNReal.ofReal (f n x) ∂μ) atTop :=
+      lintegral_liminf_le' fun n => (hf n).aestronglyMeasurable.aemeasurable.ennreal_ofReal
+    _ ≤ liminf (fun _ : ℕ => ENNReal.ofReal C) atTop := by
+      refine Filter.liminf_le_liminf ?_
+      filter_upwards [] with n
+      rw [← ofReal_integral_eq_lintegral_ofReal (hf n) (ae_of_all _ (hpos n))]
+      exact ENNReal.ofReal_le_ofReal (he n)
+    _ = ENNReal.ofReal C := tendsto_const_nhds.liminf_eq
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open MeasureTheory ProbabilityTheory BrownianCalculus RationalStop
+
+ 
+
+theorem reverse_supersolution_bound {a y δ T : ℝ} (ha : 1/4 < a)
+    (hy : 0 < y) (hδ : 0 < δ) (hT : 0 ≤ T)
+    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W) :
+    (∫⁻ b : Driver, ENNReal.ofReal
+      (test a (reverseFlow a (y/4) (T+δ) y b T) (b T.toNNReal)) ∂W) ≤
+        ENNReal.ofReal (test a (initial (T+δ) y) 0) := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  let q := y/4
+  let z := initial (T+δ) y
+  have hq : 0 < q := by dsimp [q]; positivity
+  have ha0 : 0 ≤ a := by linarith
+  have hi : ∀ᶠ n : ℕ in atTop, exhaustion q n (z,0) < 1 := by
+    have hh := eventually_exhaustion_lt hq hδ (T := 0) le_rfl (fun _ => (z,0))
+      continuousOn_const (fun t ht => by change δ ≤ T+δ; linarith)
+      (fun t ht => by change 2*(y/4) < y; linarith)
+    filter_upwards [hh] with n hn
+    exact hn 0 ⟨le_rfl,le_rfl⟩
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hi
+  let φ := fun n => exhaustion q (n+N)
+  have hφ0 (n : ℕ) : φ n (z,0) < 1 := hN (n+N) (by omega)
+  let τ := fun n (b : Driver) => Exit (fun u => φ n (driven a q z b u)) 1 T
+  let f := fun n (b : Driver) => test a (driven a q z b (τ n b)).1 (driven a q z b (τ n b)).2
+  have hτ (n : ℕ) (b : Driver) : τ n b ∈ Icc 0 T := exit_bounds hT
+    ((continuous_exhaustion q (n+N)).comp_continuousOn (continuousOn_driven hq hT a z b))
+  have hf (n : ℕ) : Integrable (f n) W := stopped_test_integrable hq a hT (φ n)
+    (continuous_exhaustion q (n+N)) (exhaustion_compact q (n+N)) (exhaustion_domain hq (n+N)) z (hφ0 n)
+  have he (n : ℕ) : (∫ b, f n b ∂W) ≤ test a z 0 := by
+    have hh := expected_stopped_reverseFlow ha hq hW hT (φ n) (continuous_exhaustion q (n+N))
+      (exhaustion_compact q (n+N)) (exhaustion_domain hq (n+N))
+      (fun x hx => exhaustion_height hq (n+N) hx) z (hφ0 n) T hT
+    have heq (b : Driver) : min T (τ n b) = τ n b := min_eq_right (hτ n b).2
+    change (∫ b : Driver, test a (ODEPath.flow (globalVector a q) z b (min T (τ n b)))
+      (b (min T (τ n b)).toNNReal) ∂W) ≤ test a z 0 at hh
+    simpa only [heq,f,driven] using hh
+  have hlim (b : Driver) : ∀ᶠ n in atTop,
+      f n b = test a (reverseFlow a q (T+δ) y b T) (b T.toNNReal) := by
+    have hh := eventually_exhaustion_lt hq hδ hT (driven a q z b)
+      (continuousOn_driven hq hT a z b)
+      (fun t ht => by
+        change δ ≤ (reverseFlow a q (T+δ) y b t).1
+        rw [reverseFlow_time hq a (T+δ) y b b.continuous hT t ht]
+        linarith [ht.2])
+      (fun t ht => by
+        have hb := reverseFlow_height hq ha0 hy.le (T+δ) b b.continuous hT t ht
+        change 2*q < (reverseFlow a q (T+δ) y b t).2.1.im
+        dsimp [q]
+        linarith)
+    obtain ⟨K,hK⟩ := eventually_atTop.mp hh
+    filter_upwards [eventually_ge_atTop K] with n hn
+    have hexit : τ n b = T := exit_eq_horizon hT
+      ((continuous_exhaustion q (n+N)).comp_continuousOn (continuousOn_driven hq hT a z b))
+      (hK (n+N) (by omega))
+    simp only [f,hexit,driven]
+    rfl
+  exact lintegral_of_eventually_stopped W f hf (fun n b => (test_pos a _ _).le)
+    (fun b => test a (reverseFlow a q (T+δ) y b T) (b T.toNNReal)) hlim (test a z 0) he
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+
+lemma r_sub_zeta_pos {a : ℝ} (ha : 1/4 < a) : 0 < r a-zeta a := by
+  have ha0 : 0 < a := by linarith
+  have he : r a-zeta a = a+1/2+1/(16*a) := by unfold r zeta; ring
+  rw [he]
+  positivity
+
+lemma exp_logderiv_le_test {a : ℝ} (ha : 1/4 < a) (v : State) (b : ℝ)
+    (hu : 0 ≤ v.1) (hy : 0 < v.2.1.im) (huy : v.1 ≤ v.2.1.im^2) :
+    Real.exp (lambda a*v.2.2) ≤ Real.exp (zeta a/2*Real.log (3/2))*test a v b := by
+  let x := v.2.1.re-b
+  let y := v.2.1.im
+  have hR := R_pos x hy
+  have hD := D_pos hu hy x
+  have hlogR : 2*Real.log y ≤ Real.log (R x y) := by
+    have hh := Real.log_le_log (sq_pos_of_pos hy) (show y^2 ≤ R x y from by unfold R; nlinarith [sq_nonneg x])
+    simpa only [Real.log_pow,Nat.cast_ofNat] using hh
+  have hlogD : Real.log (D v.1 x y) ≤ Real.log (R x y)+Real.log (3/2) := by
+    have hle : D v.1 x y ≤ R x y*(3/2) := by
+      dsimp [D,R,x,y]
+      nlinarith [sq_nonneg (v.2.1.re-b)]
+    have hh := Real.log_le_log hD hle
+    rwa [Real.log_mul hR.ne' (by norm_num : (3/2:ℝ) ≠ 0)] at hh
+  have hz := (parameters ha).1
+  have hrz := r_sub_zeta_pos ha
+  have h1 := mul_nonneg hrz.le (sub_nonneg.mpr hlogR)
+  have h2 := mul_nonneg hz.le (sub_nonneg.mpr hlogD)
+  unfold test F
+  rw [← Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  change lambda a*v.2.2 ≤ zeta a/2*Real.log (3/2)+H a v.1 x y v.2.2
+  unfold H
+  nlinarith
+
+lemma initial_test_scale (a : ℝ) {y T : ℝ} (hy : 0 < y) (hT : 0 ≤ T) :
+    Real.exp (zeta a/2*Real.log (3/2))*test a (initial (T+y^2) y) 0 =
+      (3*y^2/(T+3*y^2))^(zeta a/2) := by
+  have hy2 : 0 < y^2 := sq_pos_of_pos hy
+  have hD : 0 < y^2+(T+y^2)/2 := by positivity
+  have hden : 0 < T+3*y^2 := by positivity
+  have hrat : 0 < 3*y^2/(T+3*y^2) := by positivity
+  have he : 3*y^2/(T+3*y^2) = ((3/2:ℝ)*y^2)/(y^2+(T+y^2)/2) := by
+    field_simp
+    ring
+  rw [Real.rpow_def_of_pos hrat,he,
+    Real.log_div (mul_ne_zero (by norm_num) hy2.ne') hD.ne',
+    Real.log_mul (by norm_num : (3/2:ℝ) ≠ 0) hy2.ne',Real.log_pow]
+  simp only [test,initial,F,H,D,R,sub_zero,zero_pow (by omega : (2:ℕ) ≠ 0),zero_add,mul_zero,zero_add]
+  rw [← Real.exp_add,Real.log_pow]
+  congr 1
+  ring
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+open MeasureTheory ProbabilityTheory BrownianCalculus
+
+ 
+
+theorem reverse_logderiv_moment {a y T : ℝ} (ha : 1/4 < a)
+    (hy : 0 < y) (hT : 0 ≤ T)
+    {W : Measure Driver} (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W) :
+    (∫⁻ b : Driver, ENNReal.ofReal (Real.exp
+      (lambda a*(reverseFlow a (y/4) (T+y^2) y b T).2.2)) ∂W) ≤
+        ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta a/2)) := by
+  let C := Real.exp (zeta a/2*Real.log (3/2))
+  let g := fun (b : Driver) => reverseFlow a (y/4) (T+y^2) y b T
+  have hq : 0 < y/4 := by positivity
+  have htime (b : Driver) : (g b).1 = y^2 := by
+    dsimp [g]
+    rw [reverseFlow_time hq a (T+y^2) y b b.continuous hT T ⟨hT,le_rfl⟩]
+    ring
+  have hheight (b : Driver) : y ≤ (g b).2.1.im :=
+    reverseFlow_height hq (by linarith) hy.le (T+y^2) b b.continuous hT T ⟨hT,le_rfl⟩
+  have hpoint (b : Driver) : Real.exp (lambda a*(g b).2.2) ≤ C*test a (g b) (b T.toNNReal) := by
+    apply exp_logderiv_le_test ha _ _
+    · rw [htime]; positivity
+    · exact hy.trans_le (hheight b)
+    · rw [htime]
+      nlinarith [hheight b]
+  calc (∫⁻ b : Driver, ENNReal.ofReal (Real.exp (lambda a*(g b).2.2)) ∂W)
+      ≤ ∫⁻ b : Driver, ENNReal.ofReal (C*test a (g b) (b T.toNNReal)) ∂W :=
+        lintegral_mono fun b => ENNReal.ofReal_le_ofReal (hpoint b)
+    _ = ENNReal.ofReal C*(∫⁻ b : Driver, ENNReal.ofReal (test a (g b) (b T.toNNReal)) ∂W) := by
+      simp only [ENNReal.ofReal_mul (show 0 ≤ C from (Real.exp_pos _).le)]
+      exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+    _ ≤ ENNReal.ofReal C*ENNReal.ofReal (test a (initial (T+y^2) y) 0) :=
+      mul_le_mul' le_rfl (reverse_supersolution_bound ha hy (sq_pos_of_pos hy) hT hW)
+    _ = ENNReal.ofReal ((3*y^2/(T+3*y^2))^(zeta a/2)) := by
+      rw [← ENNReal.ofReal_mul (show 0 ≤ C from (Real.exp_pos _).le)]
+      exact congrArg ENNReal.ofReal (initial_test_scale a hy hT)
+end SLEExactGauge.ReverseDerivative
+
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+
+noncomputable section
+namespace SLEExactGauge.BrownianRestart
+
+abbrev Driver := C(ℝ≥0, ℝ)₀
+instance : MeasurableSpace Driver := borel Driver
+instance : BorelSpace Driver := ⟨rfl⟩
+instance : PolishSpace Driver := ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace
+
+def code (b : Driver) : ℚ → ℝ := fun q => b (Real.toNNReal (q:ℝ))
+
+lemma measurable_code : Measurable code :=
+  Measurable.of_eval fun _ => (continuous_eval_const _).measurable
+
+lemma dense_rational_time : DenseRange (fun q : ℚ => Real.toNNReal (q:ℝ)) := by
+  have hd : Surjective Real.toNNReal := fun x => ⟨(x:ℝ),Real.toNNReal_coe⟩
+  exact hd.denseRange.comp Rat.denseRange_cast continuous_real_toNNReal
+
+lemma code_injective : Injective code := by
+  intro b c h
+  apply ContinuousMapZero.ext
+  have he := dense_rational_time.equalizer (map_continuous b) (map_continuous c) h
+  exact congrFun he
+
+lemma code_embedding : MeasurableEmbedding code := measurable_code.measurableEmbedding code_injective
+
+end SLEExactGauge.BrownianRestart
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/RiemannIntegrals.lean b/lean/OAI/Probability/SLEGauge/RiemannIntegrals.lean
index 088a23abc1482ebfa45839fbc1eb4c4c6d15af0b..41955ca0f1592edaebe1e8b9e2fa53c94c54b85a 100644
--- a/lean/OAI/Probability/SLEGauge/RiemannIntegrals.lean
+++ b/lean/OAI/Probability/SLEGauge/RiemannIntegrals.lean
@@ -1,401 +1,401 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StochasticCalculus
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-namespace ExactSLEGenerator
-lemma riemann_sum_tendsto (f : ℝ → ℝ) (hf : Continuous f) (T : ℝ) (hT : 0 ≤ T) :
-    Tendsto (fun n : ℕ => ∑ i ∈ Finset.range (n+1),
-      (T / (n+1 : ℕ)) * f ((i : ℝ)*(T / (n+1 : ℕ)))) atTop
-      (𝓝 (∫ s in 0..T, f s)) := by
-  by_cases hT0 : T = 0
-  · subst T
-    simp
-  have hTp : 0 < T := lt_of_le_of_ne hT (Ne.symm hT0)
-  have hu := (isCompact_Icc : IsCompact (Set.Icc 0 T)).uniformContinuousOn_of_continuous hf.continuousOn
-  rw [Metric.uniformContinuousOn_iff] at hu
-  have hmesh : Tendsto (fun n : ℕ => T / (n+1 : ℕ)) atTop (𝓝 0) := by
-    have hh : Tendsto (fun n : ℕ => (n+1 : ℕ)) atTop atTop :=
-      tendsto_add_atTop_nat 1
-    simpa using tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_atTop.comp hh)
-  apply Metric.tendsto_nhds.mpr
-  intro ε hε
-  obtain ⟨δ, hδ, huδ⟩ := hu (ε/(2*T)) (by positivity)
-  filter_upwards [hmesh.eventually_lt_const hδ] with n hn
-  rw [Real.dist_eq]
-  have he := riemann_sum_error_le f hf T hT (n+1) (Nat.succ_pos _) (ε/(2*T))
-    (by positivity) (by
-      intro u hu v hv huv
-      exact le_of_lt (huδ u hu v hv (by simpa only [Real.dist_eq] using huv.trans_lt hn)))
-  calc
-    _ ≤ ε / (2*T) * T := he
-    _ = ε / 2 := by field_simp
-    _ < ε := by linarith
-
-lemma riemann_sum_bound (f : ℝ → ℝ) (T K : ℝ) (hT : 0 ≤ T) (N : ℕ) (hN : 0 < N)
-    (hbound : ∀ t ∈ Set.Icc 0 T, |f t| ≤ K) :
-    |∑ i ∈ Finset.range N, (T/N)*f ((i : ℝ)*(T/N))| ≤ K*T := by
-  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
-  have hh : 0 ≤ T / N := div_nonneg hT hNr.le
-  have hNh : (N : ℝ) * (T / N) = T := by field_simp
-  calc
-    _ ≤ ∑ i ∈ Finset.range N, |(T/N)*f ((i : ℝ)*(T/N))| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ _i ∈ Finset.range N, (T/N)*K := by
-      apply Finset.sum_le_sum
-      intro i hi
-      rw [abs_mul, abs_of_nonneg hh]
-      apply mul_le_mul_of_nonneg_left _ hh
-      apply hbound
-      refine ⟨by positivity, ?_⟩
-      calc
-        _ ≤ (N : ℝ)*(T/N) := mul_le_mul_of_nonneg_right
-          (by exact_mod_cast (Finset.mem_range.mp hi).le) hh
-        _ = T := hNh
-    _ = K*((N : ℝ)*(T/N)) := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
-    _ = K*T := by rw [hNh]
-
-lemma expected_riemann_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    (W : Measure Ω) [IsFiniteMeasure W] (f : ℝ → Ω → ℝ)
-    (hf : ∀ b, Continuous (fun t => f t b))
-    (hm : ∀ t, AEStronglyMeasurable (f t) W)
-    (T K : ℝ) (hT : 0 ≤ T)
-    (hbound : ∀ b t, t ∈ Set.Icc 0 T → |f t b| ≤ K) :
-    Tendsto (fun n : ℕ => ∫ b, ∑ i ∈ Finset.range (n+1),
-      (T/(n+1 : ℕ))*f ((i : ℝ)*(T/(n+1 : ℕ))) b ∂W) atTop
-      (𝓝 (∫ b, (∫ t in 0..T, f t b) ∂W)) := by
-  apply tendsto_integral_of_dominated_convergence (fun _ => K*T)
-  · intro n
-    have hs := Finset.aestronglyMeasurable_sum
-      (Finset.range (n+1)) (fun i hi =>
-        (hm ((i : ℝ)*(T/(n+1 : ℕ)))).const_mul (T/(n+1 : ℕ)))
-    convert hs using 1
-    funext b
-    simp only [Finset.sum_apply]
-  · exact integrable_const _
-  · intro n
-    refine ae_of_all W ?_
-    intro b
-    simpa only [Real.norm_eq_abs] using
-      riemann_sum_bound (fun t => f t b) T K hT (n+1) (Nat.succ_pos _) (hbound b)
-  · exact ae_of_all W (fun b => riemann_sum_tendsto (fun t => f t b) (hf b) T hT)
-
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
-lemma stieltjes_riemann_error_le (D : ℝ → E →L[ℝ] ℝ) (a : ℝ → E)
-    (ha : Continuous a) (T M ε : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M)
-    (N : ℕ) (hN : 0 < N)
-    (hbound : ∀ t ∈ Set.Icc 0 T, ‖D t‖ ≤ M)
-    (hmod : ∀ u ∈ Set.Icc 0 T, ∀ v ∈ Set.Icc 0 T,
-      |u-v| ≤ T/N → ‖a u-a v‖ ≤ ε) :
-    |(∑ i ∈ Finset.range N, D ((i : ℝ)*(T/N))
-        (∫ s in (i : ℝ)*(T/N)..((i+1 : ℕ) : ℝ)*(T/N), a s)) -
-      ∑ i ∈ Finset.range N, (T/N) * D ((i : ℝ)*(T/N)) (a ((i : ℝ)*(T/N)))| ≤ M*ε*T := by
-  let h := T / (N : ℝ)
-  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
-  have hh : 0 ≤ h := div_nonneg hT hNr.le
-  have hNh : (N : ℝ) * h = T := by dsimp [h]; field_simp
-  have hdiff (i : ℕ) : ((i+1 : ℕ) : ℝ)*h-(i : ℝ)*h = h := by push_cast; ring
-  have he (i : ℕ) (hi : i < N) :
-      |D ((i : ℝ)*h) (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s) -
-        h * D ((i : ℝ)*h) (a ((i : ℝ)*h))| ≤ M*ε*h := by
-    have hai : (i : ℝ)*h ∈ Set.Icc 0 T := by
-      refine ⟨by positivity, ?_⟩
-      rw [← hNh]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi.le) hh
-    have hbi : ((i+1 : ℕ) : ℝ)*h ∈ Set.Icc 0 T := by
-      refine ⟨by positivity, ?_⟩
-      rw [← hNh]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi) hh
-    have hab : (i : ℝ)*h ≤ ((i+1 : ℕ) : ℝ)*h := by linarith [hdiff i]
-    have hpt : ∀ s ∈ Set.uIoc ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h),
-        ‖D ((i : ℝ)*h) (a s-a ((i : ℝ)*h))‖ ≤ M*ε := by
-      intro s hs
-      rw [Set.uIoc_of_le hab] at hs
-      have has : ‖a s - a ((i : ℝ)*h)‖ ≤ ε := by
-        apply hmod s ⟨hai.1.trans hs.1.le, hs.2.trans hbi.2⟩ _ hai
-        rw [abs_of_nonneg (sub_nonneg.mpr hs.1.le)]
-        linarith [hs.2, hdiff i]
-      calc
-        _ ≤ ‖D ((i : ℝ)*h)‖ * ‖a s-a ((i : ℝ)*h)‖ := (D _).le_opNorm _
-        _ ≤ M * ‖a s-a ((i : ℝ)*h)‖ := mul_le_mul_of_nonneg_right (hbound _ hai) (norm_nonneg _)
-        _ ≤ M*ε := mul_le_mul_of_nonneg_left has hM
-    have hi₁ := ((D ((i : ℝ)*h)).continuous.comp ha).intervalIntegrable
-      (μ := volume) ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h)
-    simp only [Function.comp_def] at hi₁
-    have hi₂ : IntervalIntegrable (fun _ : ℝ => D ((i : ℝ)*h) (a ((i : ℝ)*h))) volume
-        ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h) := intervalIntegrable_const
-    have hval : (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h,
-        D ((i : ℝ)*h) (a s-a ((i : ℝ)*h))) =
-        D ((i : ℝ)*h) (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s) -
-          h * D ((i : ℝ)*h) (a ((i : ℝ)*h)) := by
-      simp only [map_sub]
-      rw [intervalIntegral.integral_sub hi₁ hi₂,
-        (D _).intervalIntegral_comp_comm (ha.intervalIntegrable _ _),
-        intervalIntegral.integral_const, hdiff, smul_eq_mul]
-    have hb := intervalIntegral.norm_integral_le_of_norm_le_const hpt
-    rw [hval, Real.norm_eq_abs, hdiff, abs_of_nonneg hh] at hb
-    exact hb
-  change |(∑ i ∈ Finset.range N, D ((i : ℝ)*h)
-    (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)) -
-      ∑ i ∈ Finset.range N, h*D ((i : ℝ)*h) (a ((i : ℝ)*h))| ≤ _
-  rw [← Finset.sum_sub_distrib]
-  calc
-    _ ≤ ∑ i ∈ Finset.range N, |D ((i : ℝ)*h)
-        (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s) -
-          h*D ((i : ℝ)*h) (a ((i : ℝ)*h))| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ _i ∈ Finset.range N, M*ε*h := Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
-    _ = M*ε*((N : ℝ)*h) := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
-    _ = M*ε*T := by rw [hNh]
-
-lemma stieltjes_sum_tendsto (D : ℝ → E →L[ℝ] ℝ) (a : ℝ → E)
-    (hD : Continuous D) (ha : Continuous a) (T M : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M)
-    (hbound : ∀ t ∈ Set.Icc 0 T, ‖D t‖ ≤ M) :
-    Tendsto (fun n : ℕ => ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*(T/(n+1 : ℕ)))
-      (∫ s in (i : ℝ)*(T/(n+1 : ℕ))..((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ)), a s)) atTop
-      (𝓝 (∫ t in 0..T, D t (a t))) := by
-  let S := fun n : ℕ => ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*(T/(n+1 : ℕ)))
-    (∫ s in (i : ℝ)*(T/(n+1 : ℕ))..((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ)), a s)
-  let R := fun n : ℕ => ∑ i ∈ Finset.range (n+1), (T/(n+1 : ℕ)) *
-    D ((i : ℝ)*(T/(n+1 : ℕ))) (a ((i : ℝ)*(T/(n+1 : ℕ))))
-  have hR : Tendsto R atTop (𝓝 (∫ t in 0..T, D t (a t))) :=
-    riemann_sum_tendsto (fun t => D t (a t)) (hD.clm_apply ha) T hT
-  have hdiff : Tendsto (fun n => S n-R n) atTop (𝓝 0) := by
-    have hu := (isCompact_Icc : IsCompact (Set.Icc 0 T)).uniformContinuousOn_of_continuous ha.continuousOn
-    rw [Metric.uniformContinuousOn_iff] at hu
-    have hmesh : Tendsto (fun n : ℕ => T/(n+1 : ℕ)) atTop (𝓝 0) :=
-      (tendsto_const_div_atTop_nhds_zero_nat T).comp (tendsto_add_atTop_nat 1)
-    apply Metric.tendsto_nhds.mpr
-    intro ε hε
-    have hh : 0 < M*T+1 := by positivity
-    have hep : 0 < ε / (2*(M*T+1)) := by positivity
-    obtain ⟨δ, hδ, huδ⟩ := hu (ε/(2*(M*T+1))) hep
-    filter_upwards [hmesh.eventually_lt_const hδ] with n hn
-    have herror := stieltjes_riemann_error_le D a ha T M (ε/(2*(M*T+1))) hT hM
-      (n+1) (Nat.succ_pos _) hbound (by
-        intro u hu v hv huv
-        have hub := huδ u hu v hv (by simpa only [Real.dist_eq] using huv.trans_lt hn)
-        exact le_of_lt (by simpa only [dist_eq_norm] using hub))
-    rw [Real.dist_eq, sub_zero]
-    change |S n-R n| ≤ _ at herror
-    calc
-      _ ≤ M*(ε/(2*(M*T+1)))*T := herror
-      _ ≤ (ε/(2*(M*T+1)))*(M*T+1) := by nlinarith [hep.le]
-      _ = ε/2 := by field_simp
-      _ < ε := by linarith
-  have hh := hdiff.add hR
-  simpa only [sub_add_cancel, zero_add] using hh
-
-omit [CompleteSpace E] in
-lemma stieltjes_sum_bound (D : ℝ → E →L[ℝ] ℝ) (a : ℝ → E)
-    (T M K : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M) (_hK : 0 ≤ K)
-    (N : ℕ) (hN : 0 < N)
-    (hD : ∀ t ∈ Set.Icc 0 T, ‖D t‖ ≤ M)
-    (ha : ∀ t ∈ Set.Icc 0 T, ‖a t‖ ≤ K) :
-    |∑ i ∈ Finset.range N, D ((i : ℝ)*(T/N))
-      (∫ s in (i : ℝ)*(T/N)..((i+1 : ℕ) : ℝ)*(T/N), a s)| ≤ M*K*T := by
-  let h := T / (N : ℝ)
-  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
-  have hh : 0 ≤ h := div_nonneg hT hNr.le
-  have hNh : (N : ℝ) * h = T := by dsimp [h]; field_simp
-  have hdiff (i : ℕ) : ((i+1 : ℕ) : ℝ)*h-(i : ℝ)*h = h := by push_cast; ring
-  have he (i : ℕ) (hi : i < N) :
-      |D ((i : ℝ)*h) (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)| ≤ M*K*h := by
-    have hai : (i : ℝ)*h ∈ Set.Icc 0 T := by
-      refine ⟨by positivity, ?_⟩
-      rw [← hNh]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi.le) hh
-    have hbi : ((i+1 : ℕ) : ℝ)*h ∈ Set.Icc 0 T := by
-      refine ⟨by positivity, ?_⟩
-      rw [← hNh]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi) hh
-    have hab : (i : ℝ)*h ≤ ((i+1 : ℕ) : ℝ)*h := by linarith [hdiff i]
-    have hv : ‖∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s‖ ≤ K*h := by
-      have hb := intervalIntegral.norm_integral_le_of_norm_le_const (fun s hs =>
-        ha s (show s ∈ Set.Icc 0 T from by
-          rw [Set.uIoc_of_le hab] at hs
-          exact ⟨hai.1.trans hs.1.le, hs.2.trans hbi.2⟩))
-      simpa only [hdiff, abs_of_nonneg hh] using hb
-    calc
-      _ ≤ ‖D ((i : ℝ)*h)‖ * ‖∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s‖ := by
-        simpa only [Real.norm_eq_abs] using (D ((i : ℝ)*h)).le_opNorm _
-      _ ≤ M * (K*h) := mul_le_mul (hD _ hai) hv (norm_nonneg _) hM
-      _ = _ := by ring
-  change |∑ i ∈ Finset.range N, D ((i : ℝ)*h)
-    (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)| ≤ _
-  calc
-    _ ≤ ∑ i ∈ Finset.range N, |D ((i : ℝ)*h)
-        (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ _i ∈ Finset.range N, M*K*h := Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
-    _ = M*K*((N : ℝ)*h) := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
-    _ = M*K*T := by rw [hNh]
-
-lemma expected_stieltjes_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    (W : Measure Ω) [IsFiniteMeasure W] (D : ℝ → Ω → E →L[ℝ] ℝ) (a : ℝ → Ω → E)
-    (hD : ∀ b, Continuous (fun t => D t b)) (ha : ∀ b, Continuous (fun t => a t b))
-    (hm : ∀ u v, AEStronglyMeasurable (fun b => D u b (∫ s in u..v, a s b)) W)
-    (T M K : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M) (hK : 0 ≤ K)
-    (hDb : ∀ b t, t ∈ Set.Icc 0 T → ‖D t b‖ ≤ M)
-    (hab : ∀ b t, t ∈ Set.Icc 0 T → ‖a t b‖ ≤ K) :
-    Tendsto (fun n : ℕ => ∫ b, ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*(T/(n+1 : ℕ))) b
-      (∫ s in (i : ℝ)*(T/(n+1 : ℕ))..((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ)), a s b) ∂W) atTop
-      (𝓝 (∫ b, (∫ t in 0..T, D t b (a t b)) ∂W)) := by
-  apply tendsto_integral_of_dominated_convergence (fun _ => M*K*T)
-  · intro n
-    have hs := Finset.aestronglyMeasurable_sum (Finset.range (n+1)) (fun i hi =>
-      hm ((i : ℝ)*(T/(n+1 : ℕ))) (((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ))))
-    convert hs using 1
-    funext b
-    simp only [Finset.sum_apply]
-  · exact integrable_const _
-  · intro n
-    refine ae_of_all W ?_
-    intro b
-    simpa only [Real.norm_eq_abs] using stieltjes_sum_bound (fun t => D t b)
-      (fun t => a t b) T M K hT hM hK (n+1) (Nat.succ_pos _) (hDb b) (hab b)
-  · exact ae_of_all W (fun b => stieltjes_sum_tendsto (fun t => D t b)
-      (fun t => a t b) (hD b) (ha b) T M hT hM (hDb b))
-
-end ExactSLEGenerator
-
-open MeasureTheory ProbabilityTheory Filter
-open scoped NNReal Topology
-namespace ExactSLEGenerator
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
- 
-
-theorem dynkin_of_mixed_taylor (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (start : ℝ≥0) (F Z Y : ℝ → Driver → ℝ) (D : ℝ → Driver → E →L[ℝ] ℝ) (A : ℝ → Driver → E)
-    (hF : ∀ t : ℝ≥0, Integrable (F t) W)
-    (hZm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Z t))
-    (hYm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Y t))
-    (hZi : ∀ t : ℝ≥0, Integrable (Z t) W)
-    (hYc : ∀ b, Continuous (fun t => Y t b))
-    (hYm' : ∀ t, AEStronglyMeasurable (Y t) W)
-    (hDc : ∀ b, Continuous (fun t => D t b))
-    (hAc : ∀ b, Continuous (fun t => A t b))
-    (hVm : ∀ u v, AEStronglyMeasurable (fun b => D u b (∫ t in u..v, A t b)) W)
-    (C M K : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M) (hK : 0 ≤ K)
-    (hYb : ∀ b t, |Y t b| ≤ M) (hDb : ∀ b t, ‖D t b‖ ≤ M)
-    (hAb : ∀ b t, ‖A t b‖ ≤ K)
-    (hTaylor : ∀ (s h : ℝ≥0) b,
-      |F (s+h) b - F s b - Z s b * increment (start+s) h b -
-        Y s b * increment (start+s) h b ^ 2 / 2 - D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b)| ≤
-      C * (K * h * (|increment (start+s) h b| + K * h) + |increment (start+s) h b| ^ 3))
-    (T : ℝ≥0) :
-    (∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) =
-      (1/2 : ℝ) * (∫ b, (∫ t in 0..(T : ℝ), Y t b) ∂W) +
-      (∫ b, (∫ t in 0..(T : ℝ), D t b (A t b)) ∂W) := by
-  have hYi (t : ℝ) : Integrable (Y t) W :=
-    Integrable.of_bound (hYm' t) M (ae_of_all W (fun b => by simpa [Real.norm_eq_abs] using hYb b t))
-  have hVi (u v : ℝ) : Integrable (fun b => D u b (∫ t in u..v, A t b)) W := by
-    apply Integrable.of_bound (hVm u v) (M * (K * |v-u|))
-    refine ae_of_all W (fun b => ?_)
-    calc
-      _ ≤ ‖D u b‖ * ‖∫ t in u..v, A t b‖ := (D u b).le_opNorm _
-      _ ≤ M * (K * |v-u|) := mul_le_mul (hDb b u)
-        (intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht => hAb b t))
-        (norm_nonneg _) hM
-  let S : ℕ → ℝ := fun n => ∑ i ∈ Finset.range (n+1),
-    ((T : ℝ)/(n+1 : ℕ)/2 * (∫ b, Y ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b ∂W) +
-      (∫ b, D ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b
-        (∫ t in ((i : ℝ)*((T : ℝ)/(n+1 : ℕ)))..(((i+1 : ℕ) : ℝ)*((T : ℝ)/(n+1 : ℕ))), A t b) ∂W))
-  have hS (n : ℕ) : S n =
-      (1/2 : ℝ) * (∫ b, ∑ i ∈ Finset.range (n+1),
-        ((T : ℝ)/(n+1 : ℕ))*Y ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b ∂W) +
-      (∫ b, ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b
-        (∫ t in ((i : ℝ)*((T : ℝ)/(n+1 : ℕ)))..(((i+1 : ℕ) : ℝ)*((T : ℝ)/(n+1 : ℕ))), A t b) ∂W) := by
-    rw [integral_finsetSum _ (fun i hi => (hYi _).const_mul _),
-      integral_finsetSum _ (fun i hi => hVi _ _)]
-    simp only [integral_const_mul, Finset.mul_sum]
-    dsimp [S]
-    rw [Finset.sum_add_distrib]
-    congr 1
-    apply Finset.sum_congr rfl
-    intro i hi
-    ring
-  have hlim : Tendsto S atTop (𝓝 ((1/2 : ℝ) * (∫ b, (∫ t in 0..(T : ℝ), Y t b) ∂W) +
-      (∫ b, (∫ t in 0..(T : ℝ), D t b (A t b)) ∂W))) := by
-    have hy := expected_riemann_sum_tendsto W Y hYc hYm' T M T.coe_nonneg
-      (fun b t ht => hYb b t)
-    have hv := expected_stieltjes_sum_tendsto W D A hDc hAc hVm T M K T.coe_nonneg hM hK
-      (fun b t ht => hDb b t) (fun b t ht => hAb b t)
-    convert (hy.const_mul (1/2 : ℝ)).add hv using 1
-    funext n
-    exact hS n
-  have herr (n : ℕ) : |(∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) - S n| ≤
-      C * (T : ℝ) * ((K * absMoment 1 + absMoment 3) *
-        Real.sqrt ((T : ℝ)/(n+1 : ℕ)) + K^2*((T : ℝ)/(n+1 : ℕ))) := by
-    let h : ℝ≥0 := T / (n+1 : ℕ)
-    have hnh : ((n+1 : ℕ) : ℝ≥0) * h = T := by dsimp [h]; field_simp
-    have he := expected_taylor_sum W hW start (n+1) h
-      (fun i => F (((i : ℝ≥0)*h : ℝ≥0) : ℝ)) (fun i => Z (((i : ℝ≥0)*h : ℝ≥0) : ℝ)) (fun i => Y (((i : ℝ≥0)*h : ℝ≥0) : ℝ))
-      (fun i b => D (((i : ℝ≥0)*h : ℝ≥0) : ℝ) b (∫ t in (((i : ℝ≥0)*h) : ℝ)..((((i+1 : ℕ) : ℝ≥0)*h) : ℝ), A t b))
-      (fun i => hF _) (fun i => hVi _ _) (fun i => hZm _) (fun i => hYm _)
-      (fun i => hZi _) (fun i => hYi _) C K hC hK (by
-        intro i hi b
-        have ht := hTaylor ((i : ℝ≥0)*h) h b
-        convert ht using 1; push_cast; ring_nf)
-    have hnr : ((n+1 : ℕ) : ℝ) * (h : ℝ) = T := by exact_mod_cast hnh
-    simp only [hnh, Nat.cast_zero, zero_mul, NNReal.coe_zero] at he
-    simp only [NNReal.coe_mul, NNReal.coe_natCast] at he
-    have hSn : (∑ i ∈ Finset.range (n+1),
-      ((h : ℝ)/2 * (∫ b, Y ((i : ℝ)*(h : ℝ)) b ∂W) +
-      (∫ b, D ((i : ℝ)*(h : ℝ)) b (∫ t in ((i : ℝ)*(h : ℝ))..(((i+1 : ℕ) : ℝ)*(h : ℝ)), A t b) ∂W))) = S n := by
-      simp only [S, h, NNReal.coe_div, NNReal.coe_natCast]
-    rw [hSn] at he
-    have hcoef : ((n+1 : ℕ) : ℝ) * C * (h : ℝ) = C*(T : ℝ) := by
-      calc
-        _ = C * (((n+1 : ℕ) : ℝ) * (h : ℝ)) := by ring
-        _ = C*(T : ℝ) := by rw [hnr]
-    rw [hcoef] at he
-    simpa only [h, NNReal.coe_div, NNReal.coe_natCast] using he
-  have he0 : Tendsto (fun n => |(∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) - S n|) atTop (𝓝 0) :=
-    squeeze_zero (fun n => abs_nonneg _) herr (taylor_error_tendsto_zero T C K)
-  have hed : Tendsto S atTop (𝓝 ((∫ b, F T b ∂W) - (∫ b, F 0 b ∂W))) := by
-    rw [tendsto_iff_dist_tendsto_zero]
-    have heq : (fun n => dist (S n) ((∫ b, F T b ∂W) - (∫ b, F 0 b ∂W))) =
-        (fun n => |(∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) - S n|) := by
-      funext n
-      rw [Real.dist_eq, abs_sub_comm]
-    rw [heq]
-    exact he0
-  exact tendsto_nhds_unique hed hlim
-
-end ExactSLEGenerator
-
-namespace ExactSLEGenerator
-variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace Ω]
-
-omit [CompleteSpace E] in
-lemma stronglyMeasurable_intervalIntegral_of_continuous [MeasurableSpace E] [BorelSpace E]
-    [SecondCountableTopology E] (A : ℝ → Ω → E)
-    (hc : ∀ b, Continuous (fun t => A t b)) (hm : ∀ t, Measurable (A t)) (u v : ℝ) :
-    StronglyMeasurable (fun b => ∫ t in u..v, A t b) := by
-  have hj : StronglyMeasurable (Function.uncurry A) :=
-    (measurable_uncurry_of_continuous_of_measurable hc hm).stronglyMeasurable
-  by_cases huv : u ≤ v
-  · simp_rw [intervalIntegral.integral_of_le huv]
-    exact hj.integral_prod_left
-  · simp_rw [intervalIntegral.integral_of_ge (le_of_not_ge huv)]
-    exact hj.integral_prod_left.neg
-
-variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
- 
-
-end ExactSLEGenerator
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StochasticCalculus
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+namespace ExactSLEGenerator
+lemma riemann_sum_tendsto (f : ℝ → ℝ) (hf : Continuous f) (T : ℝ) (hT : 0 ≤ T) :
+    Tendsto (fun n : ℕ => ∑ i ∈ Finset.range (n+1),
+      (T / (n+1 : ℕ)) * f ((i : ℝ)*(T / (n+1 : ℕ)))) atTop
+      (𝓝 (∫ s in 0..T, f s)) := by
+  by_cases hT0 : T = 0
+  · subst T
+    simp
+  have hTp : 0 < T := lt_of_le_of_ne hT (Ne.symm hT0)
+  have hu := (isCompact_Icc : IsCompact (Set.Icc 0 T)).uniformContinuousOn_of_continuous hf.continuousOn
+  rw [Metric.uniformContinuousOn_iff] at hu
+  have hmesh : Tendsto (fun n : ℕ => T / (n+1 : ℕ)) atTop (𝓝 0) := by
+    have hh : Tendsto (fun n : ℕ => (n+1 : ℕ)) atTop atTop :=
+      tendsto_add_atTop_nat 1
+    simpa using tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_atTop.comp hh)
+  apply Metric.tendsto_nhds.mpr
+  intro ε hε
+  obtain ⟨δ, hδ, huδ⟩ := hu (ε/(2*T)) (by positivity)
+  filter_upwards [hmesh.eventually_lt_const hδ] with n hn
+  rw [Real.dist_eq]
+  have he := riemann_sum_error_le f hf T hT (n+1) (Nat.succ_pos _) (ε/(2*T))
+    (by positivity) (by
+      intro u hu v hv huv
+      exact le_of_lt (huδ u hu v hv (by simpa only [Real.dist_eq] using huv.trans_lt hn)))
+  calc
+    _ ≤ ε / (2*T) * T := he
+    _ = ε / 2 := by field_simp
+    _ < ε := by linarith
+
+lemma riemann_sum_bound (f : ℝ → ℝ) (T K : ℝ) (hT : 0 ≤ T) (N : ℕ) (hN : 0 < N)
+    (hbound : ∀ t ∈ Set.Icc 0 T, |f t| ≤ K) :
+    |∑ i ∈ Finset.range N, (T/N)*f ((i : ℝ)*(T/N))| ≤ K*T := by
+  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
+  have hh : 0 ≤ T / N := div_nonneg hT hNr.le
+  have hNh : (N : ℝ) * (T / N) = T := by field_simp
+  calc
+    _ ≤ ∑ i ∈ Finset.range N, |(T/N)*f ((i : ℝ)*(T/N))| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _i ∈ Finset.range N, (T/N)*K := by
+      apply Finset.sum_le_sum
+      intro i hi
+      rw [abs_mul, abs_of_nonneg hh]
+      apply mul_le_mul_of_nonneg_left _ hh
+      apply hbound
+      refine ⟨by positivity, ?_⟩
+      calc
+        _ ≤ (N : ℝ)*(T/N) := mul_le_mul_of_nonneg_right
+          (by exact_mod_cast (Finset.mem_range.mp hi).le) hh
+        _ = T := hNh
+    _ = K*((N : ℝ)*(T/N)) := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
+    _ = K*T := by rw [hNh]
+
+lemma expected_riemann_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    (W : Measure Ω) [IsFiniteMeasure W] (f : ℝ → Ω → ℝ)
+    (hf : ∀ b, Continuous (fun t => f t b))
+    (hm : ∀ t, AEStronglyMeasurable (f t) W)
+    (T K : ℝ) (hT : 0 ≤ T)
+    (hbound : ∀ b t, t ∈ Set.Icc 0 T → |f t b| ≤ K) :
+    Tendsto (fun n : ℕ => ∫ b, ∑ i ∈ Finset.range (n+1),
+      (T/(n+1 : ℕ))*f ((i : ℝ)*(T/(n+1 : ℕ))) b ∂W) atTop
+      (𝓝 (∫ b, (∫ t in 0..T, f t b) ∂W)) := by
+  apply tendsto_integral_of_dominated_convergence (fun _ => K*T)
+  · intro n
+    have hs := Finset.aestronglyMeasurable_sum
+      (Finset.range (n+1)) (fun i hi =>
+        (hm ((i : ℝ)*(T/(n+1 : ℕ)))).const_mul (T/(n+1 : ℕ)))
+    convert hs using 1
+    funext b
+    simp only [Finset.sum_apply]
+  · exact integrable_const _
+  · intro n
+    refine ae_of_all W ?_
+    intro b
+    simpa only [Real.norm_eq_abs] using
+      riemann_sum_bound (fun t => f t b) T K hT (n+1) (Nat.succ_pos _) (hbound b)
+  · exact ae_of_all W (fun b => riemann_sum_tendsto (fun t => f t b) (hf b) T hT)
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+lemma stieltjes_riemann_error_le (D : ℝ → E →L[ℝ] ℝ) (a : ℝ → E)
+    (ha : Continuous a) (T M ε : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M)
+    (N : ℕ) (hN : 0 < N)
+    (hbound : ∀ t ∈ Set.Icc 0 T, ‖D t‖ ≤ M)
+    (hmod : ∀ u ∈ Set.Icc 0 T, ∀ v ∈ Set.Icc 0 T,
+      |u-v| ≤ T/N → ‖a u-a v‖ ≤ ε) :
+    |(∑ i ∈ Finset.range N, D ((i : ℝ)*(T/N))
+        (∫ s in (i : ℝ)*(T/N)..((i+1 : ℕ) : ℝ)*(T/N), a s)) -
+      ∑ i ∈ Finset.range N, (T/N) * D ((i : ℝ)*(T/N)) (a ((i : ℝ)*(T/N)))| ≤ M*ε*T := by
+  let h := T / (N : ℝ)
+  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
+  have hh : 0 ≤ h := div_nonneg hT hNr.le
+  have hNh : (N : ℝ) * h = T := by dsimp [h]; field_simp
+  have hdiff (i : ℕ) : ((i+1 : ℕ) : ℝ)*h-(i : ℝ)*h = h := by push_cast; ring
+  have he (i : ℕ) (hi : i < N) :
+      |D ((i : ℝ)*h) (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s) -
+        h * D ((i : ℝ)*h) (a ((i : ℝ)*h))| ≤ M*ε*h := by
+    have hai : (i : ℝ)*h ∈ Set.Icc 0 T := by
+      refine ⟨by positivity, ?_⟩
+      rw [← hNh]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi.le) hh
+    have hbi : ((i+1 : ℕ) : ℝ)*h ∈ Set.Icc 0 T := by
+      refine ⟨by positivity, ?_⟩
+      rw [← hNh]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi) hh
+    have hab : (i : ℝ)*h ≤ ((i+1 : ℕ) : ℝ)*h := by linarith [hdiff i]
+    have hpt : ∀ s ∈ Set.uIoc ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h),
+        ‖D ((i : ℝ)*h) (a s-a ((i : ℝ)*h))‖ ≤ M*ε := by
+      intro s hs
+      rw [Set.uIoc_of_le hab] at hs
+      have has : ‖a s - a ((i : ℝ)*h)‖ ≤ ε := by
+        apply hmod s ⟨hai.1.trans hs.1.le, hs.2.trans hbi.2⟩ _ hai
+        rw [abs_of_nonneg (sub_nonneg.mpr hs.1.le)]
+        linarith [hs.2, hdiff i]
+      calc
+        _ ≤ ‖D ((i : ℝ)*h)‖ * ‖a s-a ((i : ℝ)*h)‖ := (D _).le_opNorm _
+        _ ≤ M * ‖a s-a ((i : ℝ)*h)‖ := mul_le_mul_of_nonneg_right (hbound _ hai) (norm_nonneg _)
+        _ ≤ M*ε := mul_le_mul_of_nonneg_left has hM
+    have hi₁ := ((D ((i : ℝ)*h)).continuous.comp ha).intervalIntegrable
+      (μ := volume) ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h)
+    simp only [Function.comp_def] at hi₁
+    have hi₂ : IntervalIntegrable (fun _ : ℝ => D ((i : ℝ)*h) (a ((i : ℝ)*h))) volume
+        ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h) := intervalIntegrable_const
+    have hval : (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h,
+        D ((i : ℝ)*h) (a s-a ((i : ℝ)*h))) =
+        D ((i : ℝ)*h) (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s) -
+          h * D ((i : ℝ)*h) (a ((i : ℝ)*h)) := by
+      simp only [map_sub]
+      rw [intervalIntegral.integral_sub hi₁ hi₂,
+        (D _).intervalIntegral_comp_comm (ha.intervalIntegrable _ _),
+        intervalIntegral.integral_const, hdiff, smul_eq_mul]
+    have hb := intervalIntegral.norm_integral_le_of_norm_le_const hpt
+    rw [hval, Real.norm_eq_abs, hdiff, abs_of_nonneg hh] at hb
+    exact hb
+  change |(∑ i ∈ Finset.range N, D ((i : ℝ)*h)
+    (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)) -
+      ∑ i ∈ Finset.range N, h*D ((i : ℝ)*h) (a ((i : ℝ)*h))| ≤ _
+  rw [← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑ i ∈ Finset.range N, |D ((i : ℝ)*h)
+        (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s) -
+          h*D ((i : ℝ)*h) (a ((i : ℝ)*h))| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _i ∈ Finset.range N, M*ε*h := Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
+    _ = M*ε*((N : ℝ)*h) := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
+    _ = M*ε*T := by rw [hNh]
+
+lemma stieltjes_sum_tendsto (D : ℝ → E →L[ℝ] ℝ) (a : ℝ → E)
+    (hD : Continuous D) (ha : Continuous a) (T M : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M)
+    (hbound : ∀ t ∈ Set.Icc 0 T, ‖D t‖ ≤ M) :
+    Tendsto (fun n : ℕ => ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*(T/(n+1 : ℕ)))
+      (∫ s in (i : ℝ)*(T/(n+1 : ℕ))..((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ)), a s)) atTop
+      (𝓝 (∫ t in 0..T, D t (a t))) := by
+  let S := fun n : ℕ => ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*(T/(n+1 : ℕ)))
+    (∫ s in (i : ℝ)*(T/(n+1 : ℕ))..((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ)), a s)
+  let R := fun n : ℕ => ∑ i ∈ Finset.range (n+1), (T/(n+1 : ℕ)) *
+    D ((i : ℝ)*(T/(n+1 : ℕ))) (a ((i : ℝ)*(T/(n+1 : ℕ))))
+  have hR : Tendsto R atTop (𝓝 (∫ t in 0..T, D t (a t))) :=
+    riemann_sum_tendsto (fun t => D t (a t)) (hD.clm_apply ha) T hT
+  have hdiff : Tendsto (fun n => S n-R n) atTop (𝓝 0) := by
+    have hu := (isCompact_Icc : IsCompact (Set.Icc 0 T)).uniformContinuousOn_of_continuous ha.continuousOn
+    rw [Metric.uniformContinuousOn_iff] at hu
+    have hmesh : Tendsto (fun n : ℕ => T/(n+1 : ℕ)) atTop (𝓝 0) :=
+      (tendsto_const_div_atTop_nhds_zero_nat T).comp (tendsto_add_atTop_nat 1)
+    apply Metric.tendsto_nhds.mpr
+    intro ε hε
+    have hh : 0 < M*T+1 := by positivity
+    have hep : 0 < ε / (2*(M*T+1)) := by positivity
+    obtain ⟨δ, hδ, huδ⟩ := hu (ε/(2*(M*T+1))) hep
+    filter_upwards [hmesh.eventually_lt_const hδ] with n hn
+    have herror := stieltjes_riemann_error_le D a ha T M (ε/(2*(M*T+1))) hT hM
+      (n+1) (Nat.succ_pos _) hbound (by
+        intro u hu v hv huv
+        have hub := huδ u hu v hv (by simpa only [Real.dist_eq] using huv.trans_lt hn)
+        exact le_of_lt (by simpa only [dist_eq_norm] using hub))
+    rw [Real.dist_eq, sub_zero]
+    change |S n-R n| ≤ _ at herror
+    calc
+      _ ≤ M*(ε/(2*(M*T+1)))*T := herror
+      _ ≤ (ε/(2*(M*T+1)))*(M*T+1) := by nlinarith [hep.le]
+      _ = ε/2 := by field_simp
+      _ < ε := by linarith
+  have hh := hdiff.add hR
+  simpa only [sub_add_cancel, zero_add] using hh
+
+omit [CompleteSpace E] in
+lemma stieltjes_sum_bound (D : ℝ → E →L[ℝ] ℝ) (a : ℝ → E)
+    (T M K : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M) (_hK : 0 ≤ K)
+    (N : ℕ) (hN : 0 < N)
+    (hD : ∀ t ∈ Set.Icc 0 T, ‖D t‖ ≤ M)
+    (ha : ∀ t ∈ Set.Icc 0 T, ‖a t‖ ≤ K) :
+    |∑ i ∈ Finset.range N, D ((i : ℝ)*(T/N))
+      (∫ s in (i : ℝ)*(T/N)..((i+1 : ℕ) : ℝ)*(T/N), a s)| ≤ M*K*T := by
+  let h := T / (N : ℝ)
+  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
+  have hh : 0 ≤ h := div_nonneg hT hNr.le
+  have hNh : (N : ℝ) * h = T := by dsimp [h]; field_simp
+  have hdiff (i : ℕ) : ((i+1 : ℕ) : ℝ)*h-(i : ℝ)*h = h := by push_cast; ring
+  have he (i : ℕ) (hi : i < N) :
+      |D ((i : ℝ)*h) (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)| ≤ M*K*h := by
+    have hai : (i : ℝ)*h ∈ Set.Icc 0 T := by
+      refine ⟨by positivity, ?_⟩
+      rw [← hNh]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi.le) hh
+    have hbi : ((i+1 : ℕ) : ℝ)*h ∈ Set.Icc 0 T := by
+      refine ⟨by positivity, ?_⟩
+      rw [← hNh]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi) hh
+    have hab : (i : ℝ)*h ≤ ((i+1 : ℕ) : ℝ)*h := by linarith [hdiff i]
+    have hv : ‖∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s‖ ≤ K*h := by
+      have hb := intervalIntegral.norm_integral_le_of_norm_le_const (fun s hs =>
+        ha s (show s ∈ Set.Icc 0 T from by
+          rw [Set.uIoc_of_le hab] at hs
+          exact ⟨hai.1.trans hs.1.le, hs.2.trans hbi.2⟩))
+      simpa only [hdiff, abs_of_nonneg hh] using hb
+    calc
+      _ ≤ ‖D ((i : ℝ)*h)‖ * ‖∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s‖ := by
+        simpa only [Real.norm_eq_abs] using (D ((i : ℝ)*h)).le_opNorm _
+      _ ≤ M * (K*h) := mul_le_mul (hD _ hai) hv (norm_nonneg _) hM
+      _ = _ := by ring
+  change |∑ i ∈ Finset.range N, D ((i : ℝ)*h)
+    (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)| ≤ _
+  calc
+    _ ≤ ∑ i ∈ Finset.range N, |D ((i : ℝ)*h)
+        (∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, a s)| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _i ∈ Finset.range N, M*K*h := Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
+    _ = M*K*((N : ℝ)*h) := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
+    _ = M*K*T := by rw [hNh]
+
+lemma expected_stieltjes_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    (W : Measure Ω) [IsFiniteMeasure W] (D : ℝ → Ω → E →L[ℝ] ℝ) (a : ℝ → Ω → E)
+    (hD : ∀ b, Continuous (fun t => D t b)) (ha : ∀ b, Continuous (fun t => a t b))
+    (hm : ∀ u v, AEStronglyMeasurable (fun b => D u b (∫ s in u..v, a s b)) W)
+    (T M K : ℝ) (hT : 0 ≤ T) (hM : 0 ≤ M) (hK : 0 ≤ K)
+    (hDb : ∀ b t, t ∈ Set.Icc 0 T → ‖D t b‖ ≤ M)
+    (hab : ∀ b t, t ∈ Set.Icc 0 T → ‖a t b‖ ≤ K) :
+    Tendsto (fun n : ℕ => ∫ b, ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*(T/(n+1 : ℕ))) b
+      (∫ s in (i : ℝ)*(T/(n+1 : ℕ))..((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ)), a s b) ∂W) atTop
+      (𝓝 (∫ b, (∫ t in 0..T, D t b (a t b)) ∂W)) := by
+  apply tendsto_integral_of_dominated_convergence (fun _ => M*K*T)
+  · intro n
+    have hs := Finset.aestronglyMeasurable_sum (Finset.range (n+1)) (fun i hi =>
+      hm ((i : ℝ)*(T/(n+1 : ℕ))) (((i+1 : ℕ) : ℝ)*(T/(n+1 : ℕ))))
+    convert hs using 1
+    funext b
+    simp only [Finset.sum_apply]
+  · exact integrable_const _
+  · intro n
+    refine ae_of_all W ?_
+    intro b
+    simpa only [Real.norm_eq_abs] using stieltjes_sum_bound (fun t => D t b)
+      (fun t => a t b) T M K hT hM hK (n+1) (Nat.succ_pos _) (hDb b) (hab b)
+  · exact ae_of_all W (fun b => stieltjes_sum_tendsto (fun t => D t b)
+      (fun t => a t b) (hD b) (ha b) T M hT hM (hDb b))
+
+end ExactSLEGenerator
+
+open MeasureTheory ProbabilityTheory Filter
+open scoped NNReal Topology
+namespace ExactSLEGenerator
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ 
+
+theorem dynkin_of_mixed_taylor (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (start : ℝ≥0) (F Z Y : ℝ → Driver → ℝ) (D : ℝ → Driver → E →L[ℝ] ℝ) (A : ℝ → Driver → E)
+    (hF : ∀ t : ℝ≥0, Integrable (F t) W)
+    (hZm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Z t))
+    (hYm : ∀ t : ℝ≥0, Measurable[rawFiltration (start+t)] (Y t))
+    (hZi : ∀ t : ℝ≥0, Integrable (Z t) W)
+    (hYc : ∀ b, Continuous (fun t => Y t b))
+    (hYm' : ∀ t, AEStronglyMeasurable (Y t) W)
+    (hDc : ∀ b, Continuous (fun t => D t b))
+    (hAc : ∀ b, Continuous (fun t => A t b))
+    (hVm : ∀ u v, AEStronglyMeasurable (fun b => D u b (∫ t in u..v, A t b)) W)
+    (C M K : ℝ) (hC : 0 ≤ C) (hM : 0 ≤ M) (hK : 0 ≤ K)
+    (hYb : ∀ b t, |Y t b| ≤ M) (hDb : ∀ b t, ‖D t b‖ ≤ M)
+    (hAb : ∀ b t, ‖A t b‖ ≤ K)
+    (hTaylor : ∀ (s h : ℝ≥0) b,
+      |F (s+h) b - F s b - Z s b * increment (start+s) h b -
+        Y s b * increment (start+s) h b ^ 2 / 2 - D s b (∫ t in (s : ℝ)..(s+h : ℝ≥0), A t b)| ≤
+      C * (K * h * (|increment (start+s) h b| + K * h) + |increment (start+s) h b| ^ 3))
+    (T : ℝ≥0) :
+    (∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) =
+      (1/2 : ℝ) * (∫ b, (∫ t in 0..(T : ℝ), Y t b) ∂W) +
+      (∫ b, (∫ t in 0..(T : ℝ), D t b (A t b)) ∂W) := by
+  have hYi (t : ℝ) : Integrable (Y t) W :=
+    Integrable.of_bound (hYm' t) M (ae_of_all W (fun b => by simpa [Real.norm_eq_abs] using hYb b t))
+  have hVi (u v : ℝ) : Integrable (fun b => D u b (∫ t in u..v, A t b)) W := by
+    apply Integrable.of_bound (hVm u v) (M * (K * |v-u|))
+    refine ae_of_all W (fun b => ?_)
+    calc
+      _ ≤ ‖D u b‖ * ‖∫ t in u..v, A t b‖ := (D u b).le_opNorm _
+      _ ≤ M * (K * |v-u|) := mul_le_mul (hDb b u)
+        (intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht => hAb b t))
+        (norm_nonneg _) hM
+  let S : ℕ → ℝ := fun n => ∑ i ∈ Finset.range (n+1),
+    ((T : ℝ)/(n+1 : ℕ)/2 * (∫ b, Y ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b ∂W) +
+      (∫ b, D ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b
+        (∫ t in ((i : ℝ)*((T : ℝ)/(n+1 : ℕ)))..(((i+1 : ℕ) : ℝ)*((T : ℝ)/(n+1 : ℕ))), A t b) ∂W))
+  have hS (n : ℕ) : S n =
+      (1/2 : ℝ) * (∫ b, ∑ i ∈ Finset.range (n+1),
+        ((T : ℝ)/(n+1 : ℕ))*Y ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b ∂W) +
+      (∫ b, ∑ i ∈ Finset.range (n+1), D ((i : ℝ)*((T : ℝ)/(n+1 : ℕ))) b
+        (∫ t in ((i : ℝ)*((T : ℝ)/(n+1 : ℕ)))..(((i+1 : ℕ) : ℝ)*((T : ℝ)/(n+1 : ℕ))), A t b) ∂W) := by
+    rw [integral_finsetSum _ (fun i hi => (hYi _).const_mul _),
+      integral_finsetSum _ (fun i hi => hVi _ _)]
+    simp only [integral_const_mul, Finset.mul_sum]
+    dsimp [S]
+    rw [Finset.sum_add_distrib]
+    congr 1
+    apply Finset.sum_congr rfl
+    intro i hi
+    ring
+  have hlim : Tendsto S atTop (𝓝 ((1/2 : ℝ) * (∫ b, (∫ t in 0..(T : ℝ), Y t b) ∂W) +
+      (∫ b, (∫ t in 0..(T : ℝ), D t b (A t b)) ∂W))) := by
+    have hy := expected_riemann_sum_tendsto W Y hYc hYm' T M T.coe_nonneg
+      (fun b t ht => hYb b t)
+    have hv := expected_stieltjes_sum_tendsto W D A hDc hAc hVm T M K T.coe_nonneg hM hK
+      (fun b t ht => hDb b t) (fun b t ht => hAb b t)
+    convert (hy.const_mul (1/2 : ℝ)).add hv using 1
+    funext n
+    exact hS n
+  have herr (n : ℕ) : |(∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) - S n| ≤
+      C * (T : ℝ) * ((K * absMoment 1 + absMoment 3) *
+        Real.sqrt ((T : ℝ)/(n+1 : ℕ)) + K^2*((T : ℝ)/(n+1 : ℕ))) := by
+    let h : ℝ≥0 := T / (n+1 : ℕ)
+    have hnh : ((n+1 : ℕ) : ℝ≥0) * h = T := by dsimp [h]; field_simp
+    have he := expected_taylor_sum W hW start (n+1) h
+      (fun i => F (((i : ℝ≥0)*h : ℝ≥0) : ℝ)) (fun i => Z (((i : ℝ≥0)*h : ℝ≥0) : ℝ)) (fun i => Y (((i : ℝ≥0)*h : ℝ≥0) : ℝ))
+      (fun i b => D (((i : ℝ≥0)*h : ℝ≥0) : ℝ) b (∫ t in (((i : ℝ≥0)*h) : ℝ)..((((i+1 : ℕ) : ℝ≥0)*h) : ℝ), A t b))
+      (fun i => hF _) (fun i => hVi _ _) (fun i => hZm _) (fun i => hYm _)
+      (fun i => hZi _) (fun i => hYi _) C K hC hK (by
+        intro i hi b
+        have ht := hTaylor ((i : ℝ≥0)*h) h b
+        convert ht using 1; push_cast; ring_nf)
+    have hnr : ((n+1 : ℕ) : ℝ) * (h : ℝ) = T := by exact_mod_cast hnh
+    simp only [hnh, Nat.cast_zero, zero_mul, NNReal.coe_zero] at he
+    simp only [NNReal.coe_mul, NNReal.coe_natCast] at he
+    have hSn : (∑ i ∈ Finset.range (n+1),
+      ((h : ℝ)/2 * (∫ b, Y ((i : ℝ)*(h : ℝ)) b ∂W) +
+      (∫ b, D ((i : ℝ)*(h : ℝ)) b (∫ t in ((i : ℝ)*(h : ℝ))..(((i+1 : ℕ) : ℝ)*(h : ℝ)), A t b) ∂W))) = S n := by
+      simp only [S, h, NNReal.coe_div, NNReal.coe_natCast]
+    rw [hSn] at he
+    have hcoef : ((n+1 : ℕ) : ℝ) * C * (h : ℝ) = C*(T : ℝ) := by
+      calc
+        _ = C * (((n+1 : ℕ) : ℝ) * (h : ℝ)) := by ring
+        _ = C*(T : ℝ) := by rw [hnr]
+    rw [hcoef] at he
+    simpa only [h, NNReal.coe_div, NNReal.coe_natCast] using he
+  have he0 : Tendsto (fun n => |(∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) - S n|) atTop (𝓝 0) :=
+    squeeze_zero (fun n => abs_nonneg _) herr (taylor_error_tendsto_zero T C K)
+  have hed : Tendsto S atTop (𝓝 ((∫ b, F T b ∂W) - (∫ b, F 0 b ∂W))) := by
+    rw [tendsto_iff_dist_tendsto_zero]
+    have heq : (fun n => dist (S n) ((∫ b, F T b ∂W) - (∫ b, F 0 b ∂W))) =
+        (fun n => |(∫ b, F T b ∂W) - (∫ b, F 0 b ∂W) - S n|) := by
+      funext n
+      rw [Real.dist_eq, abs_sub_comm]
+    rw [heq]
+    exact he0
+  exact tendsto_nhds_unique hed hlim
+
+end ExactSLEGenerator
+
+namespace ExactSLEGenerator
+variable {E Ω : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace Ω]
+
+omit [CompleteSpace E] in
+lemma stronglyMeasurable_intervalIntegral_of_continuous [MeasurableSpace E] [BorelSpace E]
+    [SecondCountableTopology E] (A : ℝ → Ω → E)
+    (hc : ∀ b, Continuous (fun t => A t b)) (hm : ∀ t, Measurable (A t)) (u v : ℝ) :
+    StronglyMeasurable (fun b => ∫ t in u..v, A t b) := by
+  have hj : StronglyMeasurable (Function.uncurry A) :=
+    (measurable_uncurry_of_continuous_of_measurable hc hm).stronglyMeasurable
+  by_cases huv : u ≤ v
+  · simp_rw [intervalIntegral.integral_of_le huv]
+    exact hj.integral_prod_left
+  · simp_rw [intervalIntegral.integral_of_ge (le_of_not_ge huv)]
+    exact hj.integral_prod_left.neg
+
+variable [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+ 
+
+end ExactSLEGenerator
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/SpatialPastTail.lean b/lean/OAI/Probability/SLEGauge/SpatialPastTail.lean
index bab47676f308d46adb11a6bbf7f1993135ab78da..f195fca28a592f260b369c964fb1aa990e58bdff 100644
--- a/lean/OAI/Probability/SLEGauge/SpatialPastTail.lean
+++ b/lean/OAI/Probability/SLEGauge/SpatialPastTail.lean
@@ -1,545 +1,545 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.PlanarGrid
-import OAI.Probability.SLEGauge.NormalizedDiskMass
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart PastMass CompactSequence
-
-lemma half_le_subdiskRadius (n : ℕ) : (1:ℝ)/2 ≤ subdiskRadius n := by
-  unfold subdiskRadius
-  have h : (2:ℝ) ≤ n+2 := by have := Nat.cast_nonneg (α:=ℝ) n; linarith
-  have hh := one_div_le_one_div_of_le (by norm_num : (0:ℝ)<2) h
-  linarith
-
-lemma radial_exp_tendsto (a : ℝ) (ha : 0<a) :
-    Tendsto (fun l : ℕ=>Real.exp (-2*a*l)) atTop (𝓝 0) := by
-  have H := tendsto_pow_atTop_nhds_zero_of_lt_one (Real.exp_pos (-2*a)).le
-    (Real.exp_lt_one_iff.mpr (by linarith : -2*a<0))
-  simpa only [←Real.exp_nat_mul, mul_comm (↑_ : ℝ) (-2*a)] using H
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
- 
-
-lemma finite_past_tail [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : UpperPlane) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.1.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.1.im*Real.exp (-2*a*s))
-    (Q : Set UpperPlane) (hQ : Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
-    (F : Finset ℝ≥0) (H : ℝ) (hH : 0 ≤ H) (l : ℕ) :
-    (radialLaw W a z.1 (σ*Real.exp (-2*a*l)))
-      {b | finiteTestTime F (fun t b=>M.mass t b Q)
-        (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) b <
-          radialTime a b.1 z.1 (σ*Real.exp (-2*a*l))} ≤
-      2*M.law {x | H ≤ sequenceMassReal x} := by
-  let q := subdiskRadius M.radiusIndex
-  have hq : 0 ≤ q := (subdiskRadius_pos _).le
-  have hq1 : q<1 := subdiskRadius_lt_one _
-  have ha0 : 0<a := by linarith
-  have hρ : 0<σ*Real.exp (-2*a*l) := mul_pos hσ (Real.exp_pos _)
-  have hρσ : σ*Real.exp (-2*a*l) ≤ σ := mul_le_of_le_one_right hσ.le
-    (Real.exp_le_one_iff.mpr (by have := Nat.cast_nonneg (α:=ℝ) l; nlinarith))
-  have hac : radialLaw W a z.1 (σ*Real.exp (-2*a*l)) ≪ W :=
-    withDensity_absolutelyContinuous _ _
-  have hbound := actual_radial_spatial_bounds κ a ha W M.supports M.coeff M.nonneg
-    M.canonical z.1 z.2 σ hσ hσz q hq hq1
-  refine le_trans ?_ (M.normalized_tail_le hW z.1 z.2 σ hσ hσz s hs hgap l H hH)
-  apply measure_mono_ae
-  filter_upwards [hac.ae_le M.canonical.path,hac.ae_le hbound,
-    radial_hit_ae W a ha z.1 z.2 _ hρ (hρσ.trans hσz)] with b hb hbb hhit
-  intro hbf
-  have hS : radialTime a b.1 z.1 σ≠∞ := ne_top_of_le_ne_top hhit
-    (radialTime_antitone a ha0 z.1 z.2 _ σ hρ hρσ hσz b)
-  have hloc := (radialDiskRegion_localization a ha0 z.1 z.2 σ hσ hσz
-    (half_le_subdiskRadius M.radiusIndex) hq1 b hS).1
-  obtain ⟨t,ht,htmass,htT⟩ := (finiteTestTime_lt_iff _ _ _ _ _).mp hbf
-  have hmass := (htmass.trans_le (hb.monotone htT.le Q)).trans_le
-    ((measure_mono (hQ.trans hloc)).trans ((hbb hS).2 l).2)
-  have hpos : 0<(diskUpper q*σ)^(2-alpha a) := Real.rpow_pos_of_pos
-    (mul_pos (diskUpper_pos hq hq1) hσ) _
-  rw [ENNReal.ofReal_mul hpos.le] at hmass
-  exact ((ENNReal.mul_lt_mul_iff_right (ENNReal.ofReal_pos.mpr hpos).ne'
-    ENNReal.ofReal_ne_top).mp hmass).le
-
- 
-lemma past_kernel_tail [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : UpperPlane) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.1.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.1.im*Real.exp (-2*a*s))
-    (Q : Set UpperPlane) (hQm : MeasurableSet Q)
-    (hQ : Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
-    (F : Finset ℝ≥0) (H : ℝ) (hH : 0 ≤ H) :
-    pastGreenKernel W a (finiteTestTime F (fun t b=>M.mass t b Q)
-      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H))) z ≤
-      2*M.law {x | H ≤ sequenceMassReal x} := by
-  let ratio (l : ℕ) := (σ/z.1.im)*Real.exp (-2*a*l)
-  have hr (l : ℕ) : 0<ratio l ∧ ratio l ≤ 1 := by
-    refine ⟨mul_pos (div_pos hσ z.2) (Real.exp_pos _),?_⟩
-    exact (mul_le_of_le_one_right (div_pos hσ z.2).le
-      (Real.exp_le_one_iff.mpr (by have := Nat.cast_nonneg (α:=ℝ) l; nlinarith))).trans
-      ((div_le_one z.2).mpr hσz)
-  have hr0 : Tendsto ratio atTop (𝓝 0) := by
-    simpa only [mul_zero] using (radial_exp_tendsto a (by linarith)).const_mul (σ/z.1.im)
-  have e (l : ℕ) : z.1.im*ratio l=σ*Real.exp (-2*a*l) := by
-    dsimp only [ratio]
-    field_simp [z.2.ne']
-  have ht := radialLaw_strict_past_tendsto W hW a ha _
-    (M.canonical.toTracePalmData.test_stopping Q hQm F
-      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H))) z ratio hr hr0
-  simp only [e] at ht
-  exact le_of_tendsto ht (Eventually.of_forall (fun l=>
-    M.finite_past_tail hW z σ hσ hσz s hs hgap Q hQ F H hH l))
-
-lemma rational_past_kernel_tail [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : UpperPlane) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.1.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.1.im*Real.exp (-2*a*s))
-    (Q : Set UpperPlane) (hQm : MeasurableSet Q)
-    (hQ : Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
-    (H : ℝ) (hH : 0 ≤ H) :
-    rationalPastKernel W a M.mass Q
-      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) z ≤
-      2*M.law {x | H ≤ sequenceMassReal x} := by
-  apply iSup_le
-  intro n
-  exact M.past_kernel_tail hW z σ hσ hσz s hs hgap Q hQm hQ _ H hH
-
- 
-
-lemma expected_spatial_bad_mass [IsProbabilityMeasure W]
-    (hκ : 0<κ) (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (σ : ℝ) (hσ : 0<σ) (Q : Set UpperPlane) (hQ : IsCompact Q)
-    (hsize : ∀z∈Q,Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
-    (hgap : ∀z∈Q,σ ≤ z.1.im ∧ ∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧
-      σ=z.1.im*Real.exp (-2*a*s))
-    (H : ℝ) (hH : 0 ≤ H) :
-    (∫⁻b,{b | 2*ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)<
-      M.mass ∞ b Q}.indicator (fun b=>M.mass ∞ b Q) b∂W) ≤
-      4*initialGreenMeasure a Q*M.law {x | H ≤ sequenceMassReal x} := by
-  have hh := M.canonical.toTracePalmData.expected_bad_mass_le hκ ha hW Q hQ
-    (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)
-    (mul_nonneg (Real.rpow_nonneg (mul_nonneg
-      (diskUpper_pos (subdiskRadius_pos _).le (subdiskRadius_lt_one _)).le hσ.le) _) hH)
-  refine hh.trans ?_
-  have hb : (∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*rationalPastKernel W a M.mass Q
-      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) z∂upperArea) ≤
-      (initialGreenMeasure a Q)*(2*M.law {x | H ≤ sequenceMassReal x}) := by
-    have hb' : (∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*rationalPastKernel W a M.mass Q
-      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) z∂upperArea) ≤
-      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*(2*M.law {x | H ≤ sequenceMassReal x})∂upperArea := by
-      apply setLIntegral_mono' hQ.measurableSet
-      intro z hz
-      gcongr
-      obtain ⟨hsz,s,hs,he⟩ := hgap z hz
-      exact M.rational_past_kernel_tail hW z σ hσ hsz s hs he Q hQ.measurableSet
-        (hsize z hz) H hH
-    refine hb'.trans_eq ?_
-    rw [lintegral_mul_const _ (initialGreen_continuous a).measurable.ennreal_ofReal]
-    rw [initialGreenMeasure,withDensity_apply _ hQ.measurableSet]
-  calc
-    _ ≤ 2*(initialGreenMeasure a Q*(2*M.law {x | H ≤ sequenceMassReal x})) := by gcongr; exact hb
-    _ = _ := by ring
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart PastMass CompactSequence MomentBatches
-
-lemma clock_gap_of_small (a : ℝ) (ha : 0 < a) (T : ℝ) (hT : 0 ≤ T)
-    (y σ : ℝ) (hy : 0 < y) (hσ : 0 < σ) (hsmall : σ ≤ y*Real.exp (-2*a*T)) :
-    σ ≤ y ∧ ∃s : ℝ,T ≤ s ∧ σ=y*Real.exp (-2*a*s) := by
-  have hq := div_pos hσ hy
-  have he : σ/y ≤ Real.exp (-2*a*T) := (div_le_iff₀ hy).mpr (by simpa only [mul_comm] using hsmall)
-  have hlog := Real.log_le_log hq he
-  rw [Real.log_exp] at hlog
-  have hσy : σ ≤ y := hsmall.trans (mul_le_of_le_one_right hy.le
-    (Real.exp_le_one_iff.mpr (by nlinarith)))
-  refine ⟨hσy,-Real.log (σ/y)/(2*a),?_,?_⟩
-  · apply (le_div_iff₀ (by positivity : 0 < 2*a)).mpr
-    nlinarith
-  · have ee : -2*a*(-Real.log (σ/y)/(2*a))=Real.log (σ/y) := by field_simp
-    rw [ee,Real.exp_log hq,mul_div_cancel₀ _ hy.ne']
-
-lemma batchRadius_eventually_small (m : ℕ → ℝ) (a : ℝ) (ha : 0 ≤ a) (ell : ℕ → ℕ)
-    (ε : ℝ) (hε : 0 < ε) : ∃N : ℕ,∀i : EntryIndex m,N ≤ i.1.1 → radius m a ell i ≤ ε := by
-  have ht : Tendsto (fun n : ℕ=>Real.exp (-(n:ℝ))) atTop (𝓝 0) := by
-    simpa using radial_exp_tendsto (1/2) (by norm_num : (0:ℝ) < 1/2)
-  obtain ⟨N,hN⟩ := eventually_atTop.mp (ht.eventually_le_const hε)
-  refine ⟨N,fun i hi=>(radius_le m ha ell i).trans ((Real.exp_le_exp.mpr ?_).trans (hN i.1.1 hi))⟩
-  have hn : (1:ℝ) ≤ i.1.1 := by exact_mod_cast i.1.2
-  nlinarith
-
-lemma compact_batch_mixing (m : ℕ → ℝ) (a : ℝ) (ha : 0 < a) (ell : ℕ → ℕ)
-    (E : Set UpperPlane) (hE : IsCompact E) :
-    ∃N : ℕ,∀i : EntryIndex m,N ≤ i.1.1 → ∀z∈E,
-      400*radius m a ell i ≤ z.1.im ∧ ∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧
-        400*radius m a ell i=z.1.im*Real.exp (-2*a*s) := by
-  obtain ⟨η,hη,hηE⟩ := hE.exists_forall_le'
-    (Complex.continuous_im.comp continuous_subtype_val).continuousOn (fun z _=>z.2)
-  let T := Jacobi.mixingTime (4*a)
-  have hT : 0 ≤ T := by unfold T Jacobi.mixingTime Jacobi.majorantBase; positivity
-  obtain ⟨N,hN⟩ := batchRadius_eventually_small m a ha.le ell
-    (η*Real.exp (-2*a*T)/400) (by positivity)
-  refine ⟨N,fun i hi z hz=>clock_gap_of_small a ha T hT z.1.im
-    (400*radius m a ell i) z.2 (by have := radius_pos m a ell i; positivity) ?_⟩
-  calc
-    400*radius m a ell i ≤ η*Real.exp (-2*a*T) := by linarith [hN i hi]
-    _ ≤ z.1.im*Real.exp (-2*a*T) := mul_le_mul_of_nonneg_right (hηE z hz) (Real.exp_pos _).le
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4 < a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-abbrev Entries := EntryIndex (moment M.law sequenceMassReal)
-
-def entryThreshold (i : M.Entries) : ℝ := threshold (moment M.law sequenceMassReal) i
-
-def entryRadius (i : M.Entries) : ℝ := radius (moment M.law sequenceMassReal) a
-  (momentLevel M.law sequenceMassReal sequenceLevelReal) i
-
-lemma entryRadius_pos (i : M.Entries) : 0 < M.entryRadius i := radius_pos ..
-
-lemma entryThreshold_pos (i : M.Entries) : 0 < M.entryThreshold i := by
-  let := M.probability
-  have hm1 := M.moments_positive 1 le_rfl
-  have hh := batch_entry_lower i.1.2 hm1 (M.moments_positive i.1.1 i.1.2)
-    (mean_pow_le_moment M.law sequenceMassReal
-      (Eventually.of_forall sequenceMassReal_nonneg) (by simpa only [pow_one] using M.integrable_moments 1) i.1.1 (M.integrable_moments _))
-    (List.getElem_mem i.2.2)
-  exact (lt_min zero_lt_one (by positivity : 0 < moment M.law sequenceMassReal 1/4)).trans_le hh
-
-lemma gauge_eq_quadratic (d : ℝ) :
-    M.gauge d=QuadraticGauge.gauge d M.entryThreshold M.entryRadius := rfl
-
-lemma entry_tails_finite : (∑'i : M.Entries,
-    M.law {x | 16*M.entryThreshold i ≤ sequenceMassReal x})≠∞ := by
-  let := M.probability
-  have h := M.tails_summable.tsum_ofReal_ne_top
-  simpa only [entryThreshold,Entries,Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using h
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped ENNReal NNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEGreen
-
- 
-def spatialRestriction (μ : Measure UpperPlane) (E : Set UpperPlane) : Measure ℂ :=
-  (μ.restrict E).map Subtype.val
-
-lemma spatialRestriction_apply (μ : Measure UpperPlane) (E : Set UpperPlane)
-    (A : Set ℂ) (hA : MeasurableSet A) :
-    spatialRestriction μ E A=μ (E∩(Subtype.val : UpperPlane→ℂ) ⁻¹' A) := by
-  rw [spatialRestriction,Measure.map_apply measurable_subtype_coe hA,
-    Measure.restrict_apply (measurable_subtype_coe hA),inter_comm]
-
-lemma spatialRestriction_mono {μ ν : Measure UpperPlane} (h : μ≤ν) (E : Set UpperPlane) :
-    spatialRestriction μ E ≤ spatialRestriction ν E := by
-  exact Measure.map_mono (Measure.restrict_mono le_rfl h) measurable_subtype_coe
-
-lemma spatialRestriction_univ (μ : Measure UpperPlane) (E : Set UpperPlane) :
-    spatialRestriction μ E univ=μ E := by
-  rw [spatialRestriction_apply μ E univ MeasurableSet.univ]
-  simp
-
-lemma spatialRestriction_finite (μ : Measure UpperPlane) (E : Set UpperPlane) (hE : μ E<∞) :
-    IsFiniteMeasure (spatialRestriction μ E) :=
-  ⟨by simpa only [spatialRestriction_univ] using hE⟩
-
-lemma spatialRestriction_atomless (μ : Measure UpperPlane) [NullSingletonClass μ]
-    (E : Set UpperPlane) : NullSingletonClass (spatialRestriction μ E) := by
-  constructor
-  intro z
-  rw [spatialRestriction_apply μ E {z} (measurableSet_singleton z)]
-  exact measure_mono_null inter_subset_right
-    ((Set.subsingleton_singleton.preimage Subtype.val_injective).measure_zero μ)
-
-lemma spatialRestriction_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (μ : Ω→Measure UpperPlane) (hμ : Measurable μ) (E : Set UpperPlane)
-    (hE : MeasurableSet E) (A : Set ℂ) (hA : MeasurableSet A) :
-    Measurable (fun ω=>spatialRestriction (μ ω) E A) := by
-  simp only [spatialRestriction_apply _ _ _ hA]
-  exact (Measure.measurable_coe (hE.inter (measurable_subtype_coe hA))).comp hμ
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart PastMass CompactSequence MomentBatches
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-def gridConstant : ℝ := 32*(400*diskUpper (subdiskRadius M.radiusIndex))^(2-alpha a)
-
-lemma gridConstant_pos : 0 < M.gridConstant := by
-  have hU := diskUpper_pos (subdiskRadius_pos M.radiusIndex).le (subdiskRadius_lt_one M.radiusIndex)
-  exact mul_pos (by norm_num) (Real.rpow_pos_of_pos (by positivity) _)
-
-lemma grid_threshold (i : M.Entries) :
-    2*ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*(400*M.entryRadius i))^(2-alpha a))*(16*M.entryThreshold i))=
-      ENNReal.ofReal (M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a)) := by
-  have hU := diskUpper_pos (subdiskRadius_pos M.radiusIndex).le (subdiskRadius_lt_one M.radiusIndex)
-  have he : diskUpper (subdiskRadius M.radiusIndex)*(400*M.entryRadius i)=
-      (400*diskUpper (subdiskRadius M.radiusIndex))*M.entryRadius i := by ring
-  rw [he, Real.mul_rpow (by positivity : 0 ≤ 400*diskUpper (subdiskRadius M.radiusIndex))
-    (M.entryRadius_pos i).le,←ENNReal.ofReal_ofNat (n:=2),←ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2)]
-  congr 1
-  unfold gridConstant
-  ring
-
- 
-
-theorem compact_finitely_bad [IsProbabilityMeasure W]
-    (hκ : 0<κ) (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (E : Set UpperPlane) (hE : IsCompact E) :
-    ∀ᵐ b∂W,∀ᵐ z∂spatialRestriction (M.mass ∞ b) E,
-      {i : M.Entries | z∈PlanarGrid.badRegion (spatialRestriction (M.mass ∞ b) E)
-        (M.entryRadius i) (M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a))}.Finite := by
-  let ν := spatialRestriction (initialGreenMeasure a) E
-  let : IsFiniteMeasure ν := spatialRestriction_finite _ E hE.measure_lt_top
-  obtain ⟨N,hN⟩ := compact_batch_mixing (moment M.law sequenceMassReal) a (by linarith)
-    (momentLevel M.law sequenceMassReal sequenceLevelReal) E hE
-  let good : Set M.Entries := {i | N ≤ i.1.1}
-  have hg : goodᶜ.Finite := by
-    simpa only [good,compl_ofPred,not_le] using finite_entries_before (moment M.law sequenceMassReal) N
-  apply PlanarGrid.ae_finitely_bad_of_spatial_tails W
-    (fun b=>spatialRestriction (M.mass ∞ b) E) ν M.entryRadius
-    (fun i=>M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a)) M.entryRadius_pos
-    (fun i p=>spatialRestriction_measurable (M.mass ∞) M.canonical.terminal_measurable E
-      hE.measurableSet _ (PlanarGrid.cell_measurable _ _)) good hg 4 (by norm_num)
-    (fun i=>M.law {x | 16*M.entryThreshold i ≤ sequenceMassReal x}) M.entry_tails_finite
-  intro i hi p
-  let Q := E∩(Subtype.val : UpperPlane→ℂ) ⁻¹' PlanarGrid.cell (M.entryRadius i) p
-  have hQ : IsCompact Q := hE.inter_right
-    ((PlanarGrid.cell_closed _ _).preimage continuous_subtype_val)
-  have hs : ∀z∈Q,Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 ((400*M.entryRadius i)/200) := by
-    intro z hz w hw
-    apply mem_closedBall.mpr
-    calc
-      dist w.1 z.1 ≤ 2*M.entryRadius i := PlanarGrid.cell_dist _ (M.entryRadius_pos i).le p hz.2 hw.2
-      _ = (400*M.entryRadius i)/200 := by ring
-  have hh := M.expected_spatial_bad_mass hκ hW (400*M.entryRadius i)
-    (by have := M.entryRadius_pos i; positivity) Q hQ hs
-    (fun z hz=>hN i hi z hz.1) (16*M.entryThreshold i) (by have := M.entryThreshold_pos i; positivity)
-  rw [M.grid_threshold] at hh
-  simpa only [ν,spatialRestriction_apply _ _ _ (PlanarGrid.cell_measurable _ _)] using hh
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart PastMass CompactSequence MomentBatches TraceMassContinuity
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
- 
-
-theorem hausdorff_positive [IsProbabilityMeasure W]
-    (hκ : 0<κ) (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W) :
-    ∀ᵐ b∂W,∀s t : ℝ,0<s → s<t →
-      0<Target.hausdorff (M.gauge (2-alpha a))
-        ((Target.trace κ (driverHomeomorph.symm b)) '' Icc s t) := by
-  let E := CompactExhaustion.choice UpperPlane
-  have hbad : ∀ᵐ b∂W,∀n : ℕ,∀ᵐ z∂spatialRestriction (M.mass ∞ b) (E n),
-      {i : M.Entries | z∈PlanarGrid.badRegion (spatialRestriction (M.mass ∞ b) (E n))
-        (M.entryRadius i) (M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a))}.Finite :=
-    ae_all_iff.mpr (fun n=>M.compact_finitely_bad hκ hW (E n) (E.isCompact n))
-  filter_upwards [hbad,M.canonical.path,M.canonical.positive_increments] with b hb hf hp
-  intro s t hs hst
-  let u := (κ*s).toNNReal
-  let v := (κ*t).toNNReal
-  have hu : (u:ℝ)=κ*s := Real.coe_toNNReal _ (mul_pos hκ hs).le
-  have hv : (v:ℝ)=κ*t := Real.coe_toNNReal _ (mul_pos hκ (hs.trans hst)).le
-  have huv : u<v := by
-    change (u:ℝ)<(v:ℝ)
-    rw [hu,hv]
-    exact mul_lt_mul_of_pos_left hst hκ
-  obtain ⟨_,K,hK,hKpos,_⟩ := hp u v huv
-  obtain ⟨n,hKn⟩ := E.exists_superset_of_isCompact hK
-  let τ := M.mass v b-M.mass u b
-  have hτ : τ ≤ M.mass ∞ b := Measure.sub_le.trans (hf.monotone le_top)
-  let : NullSingletonClass (M.mass ∞ b) := hf.atomless
-  let : NullSingletonClass τ := ⟨fun z=>measure_mono_null (le_refl {z})
-    (le_antisymm ((hτ {z}).trans_eq (measure_singleton z)) bot_le)⟩
-  let : IsFiniteMeasureOnCompacts (M.mass ∞ b) := M.canonical.finite_compact ∞ b
-  let σ := spatialRestriction τ (E n)
-  let μ := spatialRestriction (M.mass ∞ b) (E n)
-  have hσ : σ≤μ := spatialRestriction_mono hτ (E n)
-  let : IsFiniteMeasure σ := spatialRestriction_finite τ (E n)
-    ((hτ (E n)).trans_lt (E.isCompact n).measure_lt_top)
-  let : NullSingletonClass σ := spatialRestriction_atomless τ (E n)
-  let A : Set ℂ := (Target.trace κ (driverHomeomorph.symm b)) '' Icc s t
-  have hA : IsCompact A := isCompact_Icc.image_of_continuousOn
-    (hf.continuous_trace.mono (fun x hx=>hs.le.trans hx.1))
-  have hzero : σ Aᶜ=0 := by
-    rw [spatialRestriction_apply τ (E n) _ hA.measurableSet.compl]
-    apply measure_mono_null inter_subset_right
-    exact hf.capacity_increment_support hκ s t hs.le hst.le
-  have hpos : 0<σ A := by
-    have hposU : 0<σ univ := by
-      rw [spatialRestriction_univ]
-      exact hKpos.trans_le (measure_mono hKn)
-    have he := measure_add_measure_compl (μ:=σ) hA.measurableSet
-    rw [hzero,add_zero] at he
-    exact he ▸ hposU
-  rw [M.gauge_eq_quadratic]
-  exact PlanarGrid.hausdorff_pos_of_finitely_bad (2-alpha a) M.entryThreshold M.entryRadius
-    M.entryThreshold_pos M.entryRadius_pos μ σ hσ M.gridConstant M.gridConstant_pos.le
-    ((Measure.absolutelyContinuous_of_le hσ).ae_le (hb n)) A hpos
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
-open LoewnerRestart PastMass CompactSequence MomentBatches
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
-def stoppedTest (z : ℂ) (τ : ℝ) (l : ℕ) : GDriver→ℝ≥0∞ :=
-  finiteStoppedTest (fun b : GDriver=>radialTime a b.1 z (τ*Real.exp (-2*a*l)))
-    (LoewnerRestart.normalizedTest a z τ (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l))
-
-lemma testEndpoint_pos (τ : ℝ) (hτ : 0<τ) (l : ℕ) : 0<τ*Real.exp (-2*a*l) := by positivity
-
-include ha in
-lemma testEndpoint_le (τ : ℝ) (hτ : 0 ≤ τ) (l : ℕ) : τ*Real.exp (-2*a*l) ≤ τ :=
-  mul_le_of_le_one_right hτ (Real.exp_le_one_iff.mpr (by
-    have hl : (0:ℝ) ≤ l := Nat.cast_nonneg l
-    nlinarith))
-
-lemma stoppedTest_measurable (z : ℂ) (hz : 0<z.im) (τ : ℝ) (hτ : 0<τ)
-    (hτz : τ ≤ z.im) (l : ℕ) :
-    Measurable[(radialTime_stopping a (by linarith) z hz (τ*Real.exp (-2*a*l))
-      ((testEndpoint_pos (a := a)) τ hτ l) (((testEndpoint_le (ha := ha)) τ hτ.le l).trans hτz)).measurableSpace]
-      (M.stoppedTest z τ l) :=
-  finiteStoppedTest_raw_measurable _ _
-    (normalizedTest_measurable a ha z hz τ hτ hτz _ (M.subdisk_measurable l))
-
-lemma stoppedTest_eq (z : ℂ) (hz : 0<z.im) (τ : ℝ) (hτ : 0<τ)
-    (hτz : τ ≤ z.im) (l : ℕ) (b : GDriver)
-    (hb : radialTime a b.1 z (τ*Real.exp (-2*a*l))≠∞) :
-    M.stoppedTest z τ l b=LoewnerRestart.normalizedTest a z τ
-      (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b :=
-  finiteStoppedTest_normalizedTest_eq a ha z hz τ hτ hτz l _
-    (fun θ=>M.subdisk_raw_measurable θ l) b hb
-
-lemma subdisk_ge_real_law (l : ℕ) (H : ℝ) (_hH : 0 ≤ H) :
-    (finiteAngleLaw W a l) {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}=
-      M.law {x | H ≤ sequenceLevelReal l x} := by
-  rw [M.subdisk_ge_law]
-  apply measure_congr
-  filter_upwards [M.increasing_finite] with x hx
-  exact propext (ENNReal.ofReal_le_iff_le_toReal (ne_top_of_le_ne_top hx.2.ne (le_iSup x l)))
-
- 
-
-theorem stopped_success_lower [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
-    (z : ℂ) (hz : 0<z.im) (σ τ : ℝ) (hτ : 0<τ) (hτσ : τ ≤ σ) (hσz : σ ≤ z.im)
-    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : τ=σ*Real.exp (-2*a*s))
-    (l : ℕ) (H : ℝ) (hH : 0 ≤ H) (A : Set GDriver)
-    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
-      (hτ.trans_le hτσ) hσz).measurableSpace] A) :
-    (1/2:ℝ≥0∞)*M.law {x | H ≤ sequenceLevelReal l x}*
-        radialLaw W a z (τ*Real.exp (-2*a*l)) A  ≤ 
-      radialLaw W a z (τ*Real.exp (-2*a*l))
-        (A∩{b | ENNReal.ofReal H ≤ M.stoppedTest z τ l b}) := by
-  let E : Set (Angle×GDriver) :=
-    {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}
-  have hE : MeasurableSet E := measurableSet_le measurable_const (M.subdisk_measurable l)
-  have hh := (normalizedTest_conditional_bounds W hW a ha z hz σ τ hτ hτσ hσz s hs hgap
-    l A hA (E.indicator (fun _=>(1:ℝ≥0∞))) (measurable_const.indicator hE)).1
-  rw [lintegral_indicator hE,lintegral_one,Measure.restrict_apply_univ,
-    M.subdisk_ge_real_law l H hH] at hh
-  refine hh.trans_eq ?_
-  have hm := M.stoppedTest_measurable z hz τ hτ (hτσ.trans hσz) l
-  have hT := radialTime_stopping a (by linarith) z hz (τ*Real.exp (-2*a*l))
-    ((testEndpoint_pos (a := a)) τ hτ l) (((testEndpoint_le (ha := ha)) τ hτ.le l).trans (hτσ.trans hσz))
-  have hS : MeasurableSet {b | ENNReal.ofReal H ≤ M.stoppedTest z τ l b} :=
-    hT.measurableSpace_le _ (measurableSet_le measurable_const hm)
-  have he : (∫⁻b in A,LoewnerRestart.normalizedTest a z τ (E.indicator (fun _=>(1:ℝ≥0∞))) b
-      ∂radialLaw W a z (τ*Real.exp (-2*a*l)))=
-      ∫⁻b in A,{b | ENNReal.ofReal H ≤ M.stoppedTest z τ l b}.indicator
-        (fun _=>(1:ℝ≥0∞)) b∂radialLaw W a z (τ*Real.exp (-2*a*l)) := by
-    apply lintegral_congr_ae
-    filter_upwards [ae_restrict_of_ae (radial_hit_ae W a ha z hz (τ*Real.exp (-2*a*l))
-      ((testEndpoint_pos (a := a)) τ hτ l) (((testEndpoint_le (ha := ha)) τ hτ.le l).trans (hτσ.trans hσz)))] with b hb
-    simp only [Set.indicator,Set.mem_ofPred_eq,M.stoppedTest_eq z hz τ hτ (hτσ.trans hσz) l b hb]
-    simp [LoewnerRestart.normalizedTest,E,Set.indicator]; rfl
-  rw [he,lintegral_indicator hS,lintegral_one,Measure.restrict_apply_univ,Measure.restrict_apply hS,inter_comm]
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.PlanarGrid
+import OAI.Probability.SLEGauge.NormalizedDiskMass
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart PastMass CompactSequence
+
+lemma half_le_subdiskRadius (n : ℕ) : (1:ℝ)/2 ≤ subdiskRadius n := by
+  unfold subdiskRadius
+  have h : (2:ℝ) ≤ n+2 := by have := Nat.cast_nonneg (α:=ℝ) n; linarith
+  have hh := one_div_le_one_div_of_le (by norm_num : (0:ℝ)<2) h
+  linarith
+
+lemma radial_exp_tendsto (a : ℝ) (ha : 0<a) :
+    Tendsto (fun l : ℕ=>Real.exp (-2*a*l)) atTop (𝓝 0) := by
+  have H := tendsto_pow_atTop_nhds_zero_of_lt_one (Real.exp_pos (-2*a)).le
+    (Real.exp_lt_one_iff.mpr (by linarith : -2*a<0))
+  simpa only [←Real.exp_nat_mul, mul_comm (↑_ : ℝ) (-2*a)] using H
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+ 
+
+lemma finite_past_tail [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : UpperPlane) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.1.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.1.im*Real.exp (-2*a*s))
+    (Q : Set UpperPlane) (hQ : Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
+    (F : Finset ℝ≥0) (H : ℝ) (hH : 0 ≤ H) (l : ℕ) :
+    (radialLaw W a z.1 (σ*Real.exp (-2*a*l)))
+      {b | finiteTestTime F (fun t b=>M.mass t b Q)
+        (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) b <
+          radialTime a b.1 z.1 (σ*Real.exp (-2*a*l))} ≤
+      2*M.law {x | H ≤ sequenceMassReal x} := by
+  let q := subdiskRadius M.radiusIndex
+  have hq : 0 ≤ q := (subdiskRadius_pos _).le
+  have hq1 : q<1 := subdiskRadius_lt_one _
+  have ha0 : 0<a := by linarith
+  have hρ : 0<σ*Real.exp (-2*a*l) := mul_pos hσ (Real.exp_pos _)
+  have hρσ : σ*Real.exp (-2*a*l) ≤ σ := mul_le_of_le_one_right hσ.le
+    (Real.exp_le_one_iff.mpr (by have := Nat.cast_nonneg (α:=ℝ) l; nlinarith))
+  have hac : radialLaw W a z.1 (σ*Real.exp (-2*a*l)) ≪ W :=
+    withDensity_absolutelyContinuous _ _
+  have hbound := actual_radial_spatial_bounds κ a ha W M.supports M.coeff M.nonneg
+    M.canonical z.1 z.2 σ hσ hσz q hq hq1
+  refine le_trans ?_ (M.normalized_tail_le hW z.1 z.2 σ hσ hσz s hs hgap l H hH)
+  apply measure_mono_ae
+  filter_upwards [hac.ae_le M.canonical.path,hac.ae_le hbound,
+    radial_hit_ae W a ha z.1 z.2 _ hρ (hρσ.trans hσz)] with b hb hbb hhit
+  intro hbf
+  have hS : radialTime a b.1 z.1 σ≠∞ := ne_top_of_le_ne_top hhit
+    (radialTime_antitone a ha0 z.1 z.2 _ σ hρ hρσ hσz b)
+  have hloc := (radialDiskRegion_localization a ha0 z.1 z.2 σ hσ hσz
+    (half_le_subdiskRadius M.radiusIndex) hq1 b hS).1
+  obtain ⟨t,ht,htmass,htT⟩ := (finiteTestTime_lt_iff _ _ _ _ _).mp hbf
+  have hmass := (htmass.trans_le (hb.monotone htT.le Q)).trans_le
+    ((measure_mono (hQ.trans hloc)).trans ((hbb hS).2 l).2)
+  have hpos : 0<(diskUpper q*σ)^(2-alpha a) := Real.rpow_pos_of_pos
+    (mul_pos (diskUpper_pos hq hq1) hσ) _
+  rw [ENNReal.ofReal_mul hpos.le] at hmass
+  exact ((ENNReal.mul_lt_mul_iff_right (ENNReal.ofReal_pos.mpr hpos).ne'
+    ENNReal.ofReal_ne_top).mp hmass).le
+
+ 
+lemma past_kernel_tail [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : UpperPlane) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.1.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.1.im*Real.exp (-2*a*s))
+    (Q : Set UpperPlane) (hQm : MeasurableSet Q)
+    (hQ : Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
+    (F : Finset ℝ≥0) (H : ℝ) (hH : 0 ≤ H) :
+    pastGreenKernel W a (finiteTestTime F (fun t b=>M.mass t b Q)
+      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H))) z ≤
+      2*M.law {x | H ≤ sequenceMassReal x} := by
+  let ratio (l : ℕ) := (σ/z.1.im)*Real.exp (-2*a*l)
+  have hr (l : ℕ) : 0<ratio l ∧ ratio l ≤ 1 := by
+    refine ⟨mul_pos (div_pos hσ z.2) (Real.exp_pos _),?_⟩
+    exact (mul_le_of_le_one_right (div_pos hσ z.2).le
+      (Real.exp_le_one_iff.mpr (by have := Nat.cast_nonneg (α:=ℝ) l; nlinarith))).trans
+      ((div_le_one z.2).mpr hσz)
+  have hr0 : Tendsto ratio atTop (𝓝 0) := by
+    simpa only [mul_zero] using (radial_exp_tendsto a (by linarith)).const_mul (σ/z.1.im)
+  have e (l : ℕ) : z.1.im*ratio l=σ*Real.exp (-2*a*l) := by
+    dsimp only [ratio]
+    field_simp [z.2.ne']
+  have ht := radialLaw_strict_past_tendsto W hW a ha _
+    (M.canonical.toTracePalmData.test_stopping Q hQm F
+      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H))) z ratio hr hr0
+  simp only [e] at ht
+  exact le_of_tendsto ht (Eventually.of_forall (fun l=>
+    M.finite_past_tail hW z σ hσ hσz s hs hgap Q hQ F H hH l))
+
+lemma rational_past_kernel_tail [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : UpperPlane) (σ : ℝ) (hσ : 0<σ) (hσz : σ ≤ z.1.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : σ=z.1.im*Real.exp (-2*a*s))
+    (Q : Set UpperPlane) (hQm : MeasurableSet Q)
+    (hQ : Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
+    (H : ℝ) (hH : 0 ≤ H) :
+    rationalPastKernel W a M.mass Q
+      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) z ≤
+      2*M.law {x | H ≤ sequenceMassReal x} := by
+  apply iSup_le
+  intro n
+  exact M.past_kernel_tail hW z σ hσ hσz s hs hgap Q hQm hQ _ H hH
+
+ 
+
+lemma expected_spatial_bad_mass [IsProbabilityMeasure W]
+    (hκ : 0<κ) (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (σ : ℝ) (hσ : 0<σ) (Q : Set UpperPlane) (hQ : IsCompact Q)
+    (hsize : ∀z∈Q,Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 (σ/200))
+    (hgap : ∀z∈Q,σ ≤ z.1.im ∧ ∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧
+      σ=z.1.im*Real.exp (-2*a*s))
+    (H : ℝ) (hH : 0 ≤ H) :
+    (∫⁻b,{b | 2*ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)<
+      M.mass ∞ b Q}.indicator (fun b=>M.mass ∞ b Q) b∂W) ≤
+      4*initialGreenMeasure a Q*M.law {x | H ≤ sequenceMassReal x} := by
+  have hh := M.canonical.toTracePalmData.expected_bad_mass_le hκ ha hW Q hQ
+    (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)
+    (mul_nonneg (Real.rpow_nonneg (mul_nonneg
+      (diskUpper_pos (subdiskRadius_pos _).le (subdiskRadius_lt_one _)).le hσ.le) _) hH)
+  refine hh.trans ?_
+  have hb : (∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*rationalPastKernel W a M.mass Q
+      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) z∂upperArea) ≤
+      (initialGreenMeasure a Q)*(2*M.law {x | H ≤ sequenceMassReal x}) := by
+    have hb' : (∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*rationalPastKernel W a M.mass Q
+      (ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*σ)^(2-alpha a))*H)) z∂upperArea) ≤
+      ∫⁻z in Q,ENNReal.ofReal (initialGreen a z)*(2*M.law {x | H ≤ sequenceMassReal x})∂upperArea := by
+      apply setLIntegral_mono' hQ.measurableSet
+      intro z hz
+      gcongr
+      obtain ⟨hsz,s,hs,he⟩ := hgap z hz
+      exact M.rational_past_kernel_tail hW z σ hσ hsz s hs he Q hQ.measurableSet
+        (hsize z hz) H hH
+    refine hb'.trans_eq ?_
+    rw [lintegral_mul_const _ (initialGreen_continuous a).measurable.ennreal_ofReal]
+    rw [initialGreenMeasure,withDensity_apply _ hQ.measurableSet]
+  calc
+    _ ≤ 2*(initialGreenMeasure a Q*(2*M.law {x | H ≤ sequenceMassReal x})) := by gcongr; exact hb
+    _ = _ := by ring
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart PastMass CompactSequence MomentBatches
+
+lemma clock_gap_of_small (a : ℝ) (ha : 0 < a) (T : ℝ) (hT : 0 ≤ T)
+    (y σ : ℝ) (hy : 0 < y) (hσ : 0 < σ) (hsmall : σ ≤ y*Real.exp (-2*a*T)) :
+    σ ≤ y ∧ ∃s : ℝ,T ≤ s ∧ σ=y*Real.exp (-2*a*s) := by
+  have hq := div_pos hσ hy
+  have he : σ/y ≤ Real.exp (-2*a*T) := (div_le_iff₀ hy).mpr (by simpa only [mul_comm] using hsmall)
+  have hlog := Real.log_le_log hq he
+  rw [Real.log_exp] at hlog
+  have hσy : σ ≤ y := hsmall.trans (mul_le_of_le_one_right hy.le
+    (Real.exp_le_one_iff.mpr (by nlinarith)))
+  refine ⟨hσy,-Real.log (σ/y)/(2*a),?_,?_⟩
+  · apply (le_div_iff₀ (by positivity : 0 < 2*a)).mpr
+    nlinarith
+  · have ee : -2*a*(-Real.log (σ/y)/(2*a))=Real.log (σ/y) := by field_simp
+    rw [ee,Real.exp_log hq,mul_div_cancel₀ _ hy.ne']
+
+lemma batchRadius_eventually_small (m : ℕ → ℝ) (a : ℝ) (ha : 0 ≤ a) (ell : ℕ → ℕ)
+    (ε : ℝ) (hε : 0 < ε) : ∃N : ℕ,∀i : EntryIndex m,N ≤ i.1.1 → radius m a ell i ≤ ε := by
+  have ht : Tendsto (fun n : ℕ=>Real.exp (-(n:ℝ))) atTop (𝓝 0) := by
+    simpa using radial_exp_tendsto (1/2) (by norm_num : (0:ℝ) < 1/2)
+  obtain ⟨N,hN⟩ := eventually_atTop.mp (ht.eventually_le_const hε)
+  refine ⟨N,fun i hi=>(radius_le m ha ell i).trans ((Real.exp_le_exp.mpr ?_).trans (hN i.1.1 hi))⟩
+  have hn : (1:ℝ) ≤ i.1.1 := by exact_mod_cast i.1.2
+  nlinarith
+
+lemma compact_batch_mixing (m : ℕ → ℝ) (a : ℝ) (ha : 0 < a) (ell : ℕ → ℕ)
+    (E : Set UpperPlane) (hE : IsCompact E) :
+    ∃N : ℕ,∀i : EntryIndex m,N ≤ i.1.1 → ∀z∈E,
+      400*radius m a ell i ≤ z.1.im ∧ ∃s : ℝ,Jacobi.mixingTime (4*a) ≤ s ∧
+        400*radius m a ell i=z.1.im*Real.exp (-2*a*s) := by
+  obtain ⟨η,hη,hηE⟩ := hE.exists_forall_le'
+    (Complex.continuous_im.comp continuous_subtype_val).continuousOn (fun z _=>z.2)
+  let T := Jacobi.mixingTime (4*a)
+  have hT : 0 ≤ T := by unfold T Jacobi.mixingTime Jacobi.majorantBase; positivity
+  obtain ⟨N,hN⟩ := batchRadius_eventually_small m a ha.le ell
+    (η*Real.exp (-2*a*T)/400) (by positivity)
+  refine ⟨N,fun i hi z hz=>clock_gap_of_small a ha T hT z.1.im
+    (400*radius m a ell i) z.2 (by have := radius_pos m a ell i; positivity) ?_⟩
+  calc
+    400*radius m a ell i ≤ η*Real.exp (-2*a*T) := by linarith [hN i hi]
+    _ ≤ z.1.im*Real.exp (-2*a*T) := mul_le_mul_of_nonneg_right (hηE z hz) (Real.exp_pos _).le
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4 < a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+abbrev Entries := EntryIndex (moment M.law sequenceMassReal)
+
+def entryThreshold (i : M.Entries) : ℝ := threshold (moment M.law sequenceMassReal) i
+
+def entryRadius (i : M.Entries) : ℝ := radius (moment M.law sequenceMassReal) a
+  (momentLevel M.law sequenceMassReal sequenceLevelReal) i
+
+lemma entryRadius_pos (i : M.Entries) : 0 < M.entryRadius i := radius_pos ..
+
+lemma entryThreshold_pos (i : M.Entries) : 0 < M.entryThreshold i := by
+  let := M.probability
+  have hm1 := M.moments_positive 1 le_rfl
+  have hh := batch_entry_lower i.1.2 hm1 (M.moments_positive i.1.1 i.1.2)
+    (mean_pow_le_moment M.law sequenceMassReal
+      (Eventually.of_forall sequenceMassReal_nonneg) (by simpa only [pow_one] using M.integrable_moments 1) i.1.1 (M.integrable_moments _))
+    (List.getElem_mem i.2.2)
+  exact (lt_min zero_lt_one (by positivity : 0 < moment M.law sequenceMassReal 1/4)).trans_le hh
+
+lemma gauge_eq_quadratic (d : ℝ) :
+    M.gauge d=QuadraticGauge.gauge d M.entryThreshold M.entryRadius := rfl
+
+lemma entry_tails_finite : (∑'i : M.Entries,
+    M.law {x | 16*M.entryThreshold i ≤ sequenceMassReal x})≠∞ := by
+  let := M.probability
+  have h := M.tails_summable.tsum_ofReal_ne_top
+  simpa only [entryThreshold,Entries,Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using h
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped ENNReal NNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEGreen
+
+ 
+def spatialRestriction (μ : Measure UpperPlane) (E : Set UpperPlane) : Measure ℂ :=
+  (μ.restrict E).map Subtype.val
+
+lemma spatialRestriction_apply (μ : Measure UpperPlane) (E : Set UpperPlane)
+    (A : Set ℂ) (hA : MeasurableSet A) :
+    spatialRestriction μ E A=μ (E∩(Subtype.val : UpperPlane→ℂ) ⁻¹' A) := by
+  rw [spatialRestriction,Measure.map_apply measurable_subtype_coe hA,
+    Measure.restrict_apply (measurable_subtype_coe hA),inter_comm]
+
+lemma spatialRestriction_mono {μ ν : Measure UpperPlane} (h : μ≤ν) (E : Set UpperPlane) :
+    spatialRestriction μ E ≤ spatialRestriction ν E := by
+  exact Measure.map_mono (Measure.restrict_mono le_rfl h) measurable_subtype_coe
+
+lemma spatialRestriction_univ (μ : Measure UpperPlane) (E : Set UpperPlane) :
+    spatialRestriction μ E univ=μ E := by
+  rw [spatialRestriction_apply μ E univ MeasurableSet.univ]
+  simp
+
+lemma spatialRestriction_finite (μ : Measure UpperPlane) (E : Set UpperPlane) (hE : μ E<∞) :
+    IsFiniteMeasure (spatialRestriction μ E) :=
+  ⟨by simpa only [spatialRestriction_univ] using hE⟩
+
+lemma spatialRestriction_atomless (μ : Measure UpperPlane) [NullSingletonClass μ]
+    (E : Set UpperPlane) : NullSingletonClass (spatialRestriction μ E) := by
+  constructor
+  intro z
+  rw [spatialRestriction_apply μ E {z} (measurableSet_singleton z)]
+  exact measure_mono_null inter_subset_right
+    ((Set.subsingleton_singleton.preimage Subtype.val_injective).measure_zero μ)
+
+lemma spatialRestriction_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Ω→Measure UpperPlane) (hμ : Measurable μ) (E : Set UpperPlane)
+    (hE : MeasurableSet E) (A : Set ℂ) (hA : MeasurableSet A) :
+    Measurable (fun ω=>spatialRestriction (μ ω) E A) := by
+  simp only [spatialRestriction_apply _ _ _ hA]
+  exact (Measure.measurable_coe (hE.inter (measurable_subtype_coe hA))).comp hμ
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart PastMass CompactSequence MomentBatches
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+def gridConstant : ℝ := 32*(400*diskUpper (subdiskRadius M.radiusIndex))^(2-alpha a)
+
+lemma gridConstant_pos : 0 < M.gridConstant := by
+  have hU := diskUpper_pos (subdiskRadius_pos M.radiusIndex).le (subdiskRadius_lt_one M.radiusIndex)
+  exact mul_pos (by norm_num) (Real.rpow_pos_of_pos (by positivity) _)
+
+lemma grid_threshold (i : M.Entries) :
+    2*ENNReal.ofReal (((diskUpper (subdiskRadius M.radiusIndex)*(400*M.entryRadius i))^(2-alpha a))*(16*M.entryThreshold i))=
+      ENNReal.ofReal (M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a)) := by
+  have hU := diskUpper_pos (subdiskRadius_pos M.radiusIndex).le (subdiskRadius_lt_one M.radiusIndex)
+  have he : diskUpper (subdiskRadius M.radiusIndex)*(400*M.entryRadius i)=
+      (400*diskUpper (subdiskRadius M.radiusIndex))*M.entryRadius i := by ring
+  rw [he, Real.mul_rpow (by positivity : 0 ≤ 400*diskUpper (subdiskRadius M.radiusIndex))
+    (M.entryRadius_pos i).le,←ENNReal.ofReal_ofNat (n:=2),←ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2)]
+  congr 1
+  unfold gridConstant
+  ring
+
+ 
+
+theorem compact_finitely_bad [IsProbabilityMeasure W]
+    (hκ : 0<κ) (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (E : Set UpperPlane) (hE : IsCompact E) :
+    ∀ᵐ b∂W,∀ᵐ z∂spatialRestriction (M.mass ∞ b) E,
+      {i : M.Entries | z∈PlanarGrid.badRegion (spatialRestriction (M.mass ∞ b) E)
+        (M.entryRadius i) (M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a))}.Finite := by
+  let ν := spatialRestriction (initialGreenMeasure a) E
+  let : IsFiniteMeasure ν := spatialRestriction_finite _ E hE.measure_lt_top
+  obtain ⟨N,hN⟩ := compact_batch_mixing (moment M.law sequenceMassReal) a (by linarith)
+    (momentLevel M.law sequenceMassReal sequenceLevelReal) E hE
+  let good : Set M.Entries := {i | N ≤ i.1.1}
+  have hg : goodᶜ.Finite := by
+    simpa only [good,compl_ofPred,not_le] using finite_entries_before (moment M.law sequenceMassReal) N
+  apply PlanarGrid.ae_finitely_bad_of_spatial_tails W
+    (fun b=>spatialRestriction (M.mass ∞ b) E) ν M.entryRadius
+    (fun i=>M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a)) M.entryRadius_pos
+    (fun i p=>spatialRestriction_measurable (M.mass ∞) M.canonical.terminal_measurable E
+      hE.measurableSet _ (PlanarGrid.cell_measurable _ _)) good hg 4 (by norm_num)
+    (fun i=>M.law {x | 16*M.entryThreshold i ≤ sequenceMassReal x}) M.entry_tails_finite
+  intro i hi p
+  let Q := E∩(Subtype.val : UpperPlane→ℂ) ⁻¹' PlanarGrid.cell (M.entryRadius i) p
+  have hQ : IsCompact Q := hE.inter_right
+    ((PlanarGrid.cell_closed _ _).preimage continuous_subtype_val)
+  have hs : ∀z∈Q,Q ⊆ (Subtype.val : UpperPlane→ℂ) ⁻¹' closedBall z.1 ((400*M.entryRadius i)/200) := by
+    intro z hz w hw
+    apply mem_closedBall.mpr
+    calc
+      dist w.1 z.1 ≤ 2*M.entryRadius i := PlanarGrid.cell_dist _ (M.entryRadius_pos i).le p hz.2 hw.2
+      _ = (400*M.entryRadius i)/200 := by ring
+  have hh := M.expected_spatial_bad_mass hκ hW (400*M.entryRadius i)
+    (by have := M.entryRadius_pos i; positivity) Q hQ hs
+    (fun z hz=>hN i hi z hz.1) (16*M.entryThreshold i) (by have := M.entryThreshold_pos i; positivity)
+  rw [M.grid_threshold] at hh
+  simpa only [ν,spatialRestriction_apply _ _ _ (PlanarGrid.cell_measurable _ _)] using hh
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart PastMass CompactSequence MomentBatches TraceMassContinuity
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+ 
+
+theorem hausdorff_positive [IsProbabilityMeasure W]
+    (hκ : 0<κ) (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W) :
+    ∀ᵐ b∂W,∀s t : ℝ,0<s → s<t →
+      0<Target.hausdorff (M.gauge (2-alpha a))
+        ((Target.trace κ (driverHomeomorph.symm b)) '' Icc s t) := by
+  let E := CompactExhaustion.choice UpperPlane
+  have hbad : ∀ᵐ b∂W,∀n : ℕ,∀ᵐ z∂spatialRestriction (M.mass ∞ b) (E n),
+      {i : M.Entries | z∈PlanarGrid.badRegion (spatialRestriction (M.mass ∞ b) (E n))
+        (M.entryRadius i) (M.gridConstant*M.entryThreshold i*(M.entryRadius i)^(2-alpha a))}.Finite :=
+    ae_all_iff.mpr (fun n=>M.compact_finitely_bad hκ hW (E n) (E.isCompact n))
+  filter_upwards [hbad,M.canonical.path,M.canonical.positive_increments] with b hb hf hp
+  intro s t hs hst
+  let u := (κ*s).toNNReal
+  let v := (κ*t).toNNReal
+  have hu : (u:ℝ)=κ*s := Real.coe_toNNReal _ (mul_pos hκ hs).le
+  have hv : (v:ℝ)=κ*t := Real.coe_toNNReal _ (mul_pos hκ (hs.trans hst)).le
+  have huv : u<v := by
+    change (u:ℝ)<(v:ℝ)
+    rw [hu,hv]
+    exact mul_lt_mul_of_pos_left hst hκ
+  obtain ⟨_,K,hK,hKpos,_⟩ := hp u v huv
+  obtain ⟨n,hKn⟩ := E.exists_superset_of_isCompact hK
+  let τ := M.mass v b-M.mass u b
+  have hτ : τ ≤ M.mass ∞ b := Measure.sub_le.trans (hf.monotone le_top)
+  let : NullSingletonClass (M.mass ∞ b) := hf.atomless
+  let : NullSingletonClass τ := ⟨fun z=>measure_mono_null (le_refl {z})
+    (le_antisymm ((hτ {z}).trans_eq (measure_singleton z)) bot_le)⟩
+  let : IsFiniteMeasureOnCompacts (M.mass ∞ b) := M.canonical.finite_compact ∞ b
+  let σ := spatialRestriction τ (E n)
+  let μ := spatialRestriction (M.mass ∞ b) (E n)
+  have hσ : σ≤μ := spatialRestriction_mono hτ (E n)
+  let : IsFiniteMeasure σ := spatialRestriction_finite τ (E n)
+    ((hτ (E n)).trans_lt (E.isCompact n).measure_lt_top)
+  let : NullSingletonClass σ := spatialRestriction_atomless τ (E n)
+  let A : Set ℂ := (Target.trace κ (driverHomeomorph.symm b)) '' Icc s t
+  have hA : IsCompact A := isCompact_Icc.image_of_continuousOn
+    (hf.continuous_trace.mono (fun x hx=>hs.le.trans hx.1))
+  have hzero : σ Aᶜ=0 := by
+    rw [spatialRestriction_apply τ (E n) _ hA.measurableSet.compl]
+    apply measure_mono_null inter_subset_right
+    exact hf.capacity_increment_support hκ s t hs.le hst.le
+  have hpos : 0<σ A := by
+    have hposU : 0<σ univ := by
+      rw [spatialRestriction_univ]
+      exact hKpos.trans_le (measure_mono hKn)
+    have he := measure_add_measure_compl (μ:=σ) hA.measurableSet
+    rw [hzero,add_zero] at he
+    exact he ▸ hposU
+  rw [M.gauge_eq_quadratic]
+  exact PlanarGrid.hausdorff_pos_of_finitely_bad (2-alpha a) M.entryThreshold M.entryRadius
+    M.entryThreshold_pos M.entryRadius_pos μ σ hσ M.gridConstant M.gridConstant_pos.le
+    ((Measure.absolutelyContinuous_of_le hσ).ae_le (hb n)) A hpos
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGreen ExactSLEGenerator
+open LoewnerRestart PastMass CompactSequence MomentBatches
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+def stoppedTest (z : ℂ) (τ : ℝ) (l : ℕ) : GDriver→ℝ≥0∞ :=
+  finiteStoppedTest (fun b : GDriver=>radialTime a b.1 z (τ*Real.exp (-2*a*l)))
+    (LoewnerRestart.normalizedTest a z τ (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l))
+
+lemma testEndpoint_pos (τ : ℝ) (hτ : 0<τ) (l : ℕ) : 0<τ*Real.exp (-2*a*l) := by positivity
+
+include ha in
+lemma testEndpoint_le (τ : ℝ) (hτ : 0 ≤ τ) (l : ℕ) : τ*Real.exp (-2*a*l) ≤ τ :=
+  mul_le_of_le_one_right hτ (Real.exp_le_one_iff.mpr (by
+    have hl : (0:ℝ) ≤ l := Nat.cast_nonneg l
+    nlinarith))
+
+lemma stoppedTest_measurable (z : ℂ) (hz : 0<z.im) (τ : ℝ) (hτ : 0<τ)
+    (hτz : τ ≤ z.im) (l : ℕ) :
+    Measurable[(radialTime_stopping a (by linarith) z hz (τ*Real.exp (-2*a*l))
+      ((testEndpoint_pos (a := a)) τ hτ l) (((testEndpoint_le (ha := ha)) τ hτ.le l).trans hτz)).measurableSpace]
+      (M.stoppedTest z τ l) :=
+  finiteStoppedTest_raw_measurable _ _
+    (normalizedTest_measurable a ha z hz τ hτ hτz _ (M.subdisk_measurable l))
+
+lemma stoppedTest_eq (z : ℂ) (hz : 0<z.im) (τ : ℝ) (hτ : 0<τ)
+    (hτz : τ ≤ z.im) (l : ℕ) (b : GDriver)
+    (hb : radialTime a b.1 z (τ*Real.exp (-2*a*l))≠∞) :
+    M.stoppedTest z τ l b=LoewnerRestart.normalizedTest a z τ
+      (subdiskMass a M.mass (subdiskRadius M.radiusIndex) l) b :=
+  finiteStoppedTest_normalizedTest_eq a ha z hz τ hτ hτz l _
+    (fun θ=>M.subdisk_raw_measurable θ l) b hb
+
+lemma subdisk_ge_real_law (l : ℕ) (H : ℝ) (_hH : 0 ≤ H) :
+    (finiteAngleLaw W a l) {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}=
+      M.law {x | H ≤ sequenceLevelReal l x} := by
+  rw [M.subdisk_ge_law]
+  apply measure_congr
+  filter_upwards [M.increasing_finite] with x hx
+  exact propext (ENNReal.ofReal_le_iff_le_toReal (ne_top_of_le_ne_top hx.2.ne (le_iSup x l)))
+
+ 
+
+theorem stopped_success_lower [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : GDriver)=>b.1 t) W)
+    (z : ℂ) (hz : 0<z.im) (σ τ : ℝ) (hτ : 0<τ) (hτσ : τ ≤ σ) (hσz : σ ≤ z.im)
+    (s : ℝ) (hs : Jacobi.mixingTime (4*a) ≤ s) (hgap : τ=σ*Real.exp (-2*a*s))
+    (l : ℕ) (H : ℝ) (hH : 0 ≤ H) (A : Set GDriver)
+    (hA : MeasurableSet[(radialTime_stopping a (by linarith) z hz σ
+      (hτ.trans_le hτσ) hσz).measurableSpace] A) :
+    (1/2:ℝ≥0∞)*M.law {x | H ≤ sequenceLevelReal l x}*
+        radialLaw W a z (τ*Real.exp (-2*a*l)) A  ≤ 
+      radialLaw W a z (τ*Real.exp (-2*a*l))
+        (A∩{b | ENNReal.ofReal H ≤ M.stoppedTest z τ l b}) := by
+  let E : Set (Angle×GDriver) :=
+    {p | ENNReal.ofReal H ≤ subdiskMass a M.mass (subdiskRadius M.radiusIndex) l p}
+  have hE : MeasurableSet E := measurableSet_le measurable_const (M.subdisk_measurable l)
+  have hh := (normalizedTest_conditional_bounds W hW a ha z hz σ τ hτ hτσ hσz s hs hgap
+    l A hA (E.indicator (fun _=>(1:ℝ≥0∞))) (measurable_const.indicator hE)).1
+  rw [lintegral_indicator hE,lintegral_one,Measure.restrict_apply_univ,
+    M.subdisk_ge_real_law l H hH] at hh
+  refine hh.trans_eq ?_
+  have hm := M.stoppedTest_measurable z hz τ hτ (hτσ.trans hσz) l
+  have hT := radialTime_stopping a (by linarith) z hz (τ*Real.exp (-2*a*l))
+    ((testEndpoint_pos (a := a)) τ hτ l) (((testEndpoint_le (ha := ha)) τ hτ.le l).trans (hτσ.trans hσz))
+  have hS : MeasurableSet {b | ENNReal.ofReal H ≤ M.stoppedTest z τ l b} :=
+    hT.measurableSpace_le _ (measurableSet_le measurable_const hm)
+  have he : (∫⁻b in A,LoewnerRestart.normalizedTest a z τ (E.indicator (fun _=>(1:ℝ≥0∞))) b
+      ∂radialLaw W a z (τ*Real.exp (-2*a*l)))=
+      ∫⁻b in A,{b | ENNReal.ofReal H ≤ M.stoppedTest z τ l b}.indicator
+        (fun _=>(1:ℝ≥0∞)) b∂radialLaw W a z (τ*Real.exp (-2*a*l)) := by
+    apply lintegral_congr_ae
+    filter_upwards [ae_restrict_of_ae (radial_hit_ae W a ha z hz (τ*Real.exp (-2*a*l))
+      ((testEndpoint_pos (a := a)) τ hτ l) (((testEndpoint_le (ha := ha)) τ hτ.le l).trans (hτσ.trans hσz)))] with b hb
+    simp only [Set.indicator,Set.mem_ofPred_eq,M.stoppedTest_eq z hz τ hτ (hτσ.trans hσz) l b hb]
+    simp [LoewnerRestart.normalizedTest,E,Set.indicator]; rfl
+  rw [he,lintegral_indicator hS,lintegral_one,Measure.restrict_apply_univ,Measure.restrict_apply hS,inter_comm]
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StateBounds.lean b/lean/OAI/Probability/SLEGauge/StateBounds.lean
index ee52ac2fa208affcc42fd9d5b55e0f9040fb45ba..50aacdd62746ddf06c72b1cde63020db07e91e73 100644
--- a/lean/OAI/Probability/SLEGauge/StateBounds.lean
+++ b/lean/OAI/Probability/SLEGauge/StateBounds.lean
@@ -1,399 +1,399 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.OptionalEstimates
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLECalculus ExactSLEFiniteTilt
-noncomputable def regimeState {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (t : ℝ≥0) (b : Driver) : RegimeState m :=
-  (vectorState a η hη z t b,
-    ∫ u in (0:ℝ)..(t:ℝ), regimeDiscountCut s a ε η N cap σ off (vectorState a η hη z u.toNNReal b))
-lemma regimeState_continuous {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (b : Driver) :
-    Continuous (fun t => regimeState s a ε η hη N cap σ off z t b) := by
-  have hc : Continuous (fun u : ℝ => regimeDiscountCut s a ε η N cap σ off
-      (vectorState a η hη z u.toNNReal b)) :=
-    (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp
-      ((vectorState_continuous a η hη z b).comp continuous_real_toNNReal)
-  exact (vectorState_continuous a η hη z b).prodMk
-    ((intervalIntegral.continuous_primitive hc.intervalIntegrable 0).comp continuous_subtype_val)
-lemma regimeState_adapted {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (t : ℝ≥0) :
-    Measurable[rawFiltration t] (regimeState s a ε η hη N cap σ off z t) := by
-  apply (vectorState_adapted a η hη z t).prodMk
-  apply adapted_intervalIntegral rawFiltration
-    (fun u b => regimeDiscountCut s a ε η N cap σ off (vectorState a η hη z u b))
-  · intro b
-    exact (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp
-      (vectorState_continuous a η hη z b)
-  · intro u
-    exact (regimeDiscountCut_continuous s a ε η hη N cap σ off).measurable.comp
-      (vectorState_adapted a η hη z u)
-lemma regimeState_eq {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
-    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (u h : ℝ≥0) (b : Driver) :
-    regimeState s a ε η hη N cap σ off z (u+h) b =
-      regimeState s a ε η hη N cap σ off z u b+increment u h b • (multiNoise m,0)+
-        ∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), regimeDrift s a ε η N cap σ off
-          (regimeState s a ε η hη N cap σ off z t.toNNReal b) := by
-  let A := regimeDrift s a ε η N cap σ off
-  let V := regimeState s a ε η hη N cap σ off z
-  have hc : Continuous (fun t : ℝ => A (V t.toNNReal b)) :=
-    (regimeDrift_continuous s a ε η hη N cap σ off).comp
-      ((regimeState_continuous s a ε η hη N cap σ off z b).comp continuous_real_toNNReal)
-  have hi := hc.intervalIntegrable (μ := volume) (u:ℝ) ((u+h:ℝ≥0):ℝ)
-  apply Prod.ext
-  · funext i
-    let L : RegimeState m →L[ℝ] State := (ContinuousLinearMap.proj i).comp (ContinuousLinearMap.fst ℝ _ _)
-    have he := L.intervalIntegral_comp_comm hi
-    change localState a η hη (z i) (u+h) b = localState a η hη (z i) u b+
-      increment u h b • noise+L (∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (V t.toNNReal b))
-    rw [← he]
-    exact localState_eq a η hη (z i) u h b
-  · let L := ContinuousLinearMap.snd ℝ (Fin m → State) ℝ
-    have he := L.intervalIntegral_comp_comm hi
-    change (V (u+h) b).2 = (V u b).2+increment u h b • (0:ℝ)+L
-      (∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (V t.toNNReal b))
-    rw [smul_zero,add_zero,← he]
-    have hk : Continuous (fun t : ℝ => regimeDiscountCut s a ε η N cap σ off
-        (vectorState a η hη z t.toNNReal b)) :=
-      (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp
-        ((vectorState_continuous a η hη z b).comp continuous_real_toNNReal)
-    exact (intervalIntegral.integral_add_adjacent_intervals (μ := volume) (hk.intervalIntegrable 0 (u:ℝ))
-      (hk.intervalIntegrable (u:ℝ) ((u+h:ℝ≥0):ℝ))).symm
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus
-lemma regimeDiscount_congr {m : ℕ} (s : Finset (Fin m)) (a ε : ℝ)
-    (cap σ off : Fin m → ℝ) (v w : Fin m → State) (h : ∀ i∈s, v i=w i) :
-    regimeDiscount s a ε cap σ off v=regimeDiscount s a ε cap σ off w := by
-  unfold regimeDiscount
-  congr 1
-  · congr 1
-    apply Finset.sum_congr rfl
-    intro i hi
-    rw [h i hi]
-  · congr 1
-    apply Finset.sum_congr rfl
-    intro i hi
-    apply Finset.sum_congr rfl
-    intro j hj
-    rw [h i hi,h j (Finset.mem_of_mem_erase hj)]
- 
-
-theorem vector_regime_weighted_optional {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (N : ℝ≥0) (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (z : Fin m → ℂ) (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (S T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
-    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
-    (w : Driver → ℝ) (hw : StronglyMeasurable[hS.measurableSpace] w)
-    (hw0 : ∀ b, 0≤w b) (C : ℝ) (hwb : ∀ b, ‖w b‖≤C)
-    (hws : ∀ b∉E, w b=0)
-    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b →
-      ‖regimeState s a ε η hη N cap σ off z t b‖≤N ∧
-      ∀ i∈s, η≤Y (vectorState a η hη z t b i) ∧
-        physicalDelta a (cap i) (σ i) (off i) (vectorState a η hη z t b i)≤1) :
-    ∫ b, w b*discountedProduct s a cap σ off
-      (regimeState s a ε η hη N cap σ off z (T b) b) ∂W ≤
-    ∫ b, w b*discountedProduct s a cap σ off
-      (regimeState s a ε η hη N cap σ off z (S b) b) ∂W := by
-  obtain ⟨K,hK,hKb⟩ := regimeDrift_bound s a ε η hη N cap σ off
-  apply discounted_regime_weighted_optional s ha hε hsize cap σ off hσ W hW
-    (regimeState s a ε η hη N cap σ off z)
-    (regimeState_continuous s a ε η hη N cap σ off z)
-    (regimeState_adapted s a ε η hη N cap σ off z)
-    (regimeDrift s a ε η N cap σ off)
-    (regimeDrift_continuous s a ε η hη N cap σ off) K hK hKb
-    (regimeState_eq s a ε η hη N cap σ off z)
-    S T B N η hη hS hT hST hTB E hE w hw hw0 C hwb hws hsafe
-  intro b hb t hts htt
-  have hn := (hsafe b hb t hts htt).1
-  have hy := (hsafe b hb t hts htt).2
-  constructor
-  · change regimeDiscountCut s a ε η N cap σ off _ = _
-    apply regimeDiscount_congr
-    intro i hi
-    apply regimeClamp_eq η N _ hη _ (hy i hi).1
-    have hv : ‖vectorState a η hη z t b‖≤N :=
-      (le_max_left _ _).trans hn
-    exact (norm_le_pi_norm _ i).trans hv
-  · intro i hi
-    exact cutoffDrift_eq_physical a η _ (hy i hi).1
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus
- 
-
-theorem vector_regime_weighted_optional_bounded {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
-    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
-    (M : ℝ≥0) (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
-    (z : Fin m → ℂ) (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (S T : Driver → ℝ≥0) (B : ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
-    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
-    (w : Driver → ℝ) (hw : StronglyMeasurable[hS.measurableSpace] w)
-    (hw0 : ∀ b, 0≤w b) (C : ℝ) (hwb : ∀ b, ‖w b‖≤C)
-    (hws : ∀ b∉E, w b=0)
-    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b →
-      ‖vectorState a η hη z t b‖≤M ∧
-      ∀ i∈s, η≤Y (vectorState a η hη z t b i) ∧
-        physicalDelta a (cap i) (σ i) (off i) (vectorState a η hη z t b i)≤1) :
-    ∫ b, w b*discountedProduct s a cap σ off
-      (regimeState s a ε η hη M cap σ off z (T b) b) ∂W ≤
-    ∫ b, w b*discountedProduct s a cap σ off
-      (regimeState s a ε η hη M cap σ off z (S b) b) ∂W := by
-  obtain ⟨K,hK,hKb⟩ := regimeDrift_bound s a ε η hη M cap σ off
-  obtain ⟨L,hL,hLb⟩ := regimeDiscountCut_bound s a ε η hη M cap σ off
-  let N : ℝ≥0 := ⟨max M (L*B),le_trans M.coe_nonneg (le_max_left _ _)⟩
-  have hb (b : Driver) (t : ℝ≥0) (ht : t≤B) :
-      ‖(regimeState s a ε η hη M cap σ off z t b).2‖≤L*B := by
-    change ‖∫ u in (0:ℝ)..(t:ℝ), regimeDiscountCut s a ε η M cap σ off
-      (vectorState a η hη z u.toNNReal b)‖≤_
-    calc
-      _ ≤ L*|(t:ℝ)-0| := intervalIntegral.norm_integral_le_of_norm_le_const fun u _ => hLb _
-      _ ≤ L*B := by rw [sub_zero,abs_of_nonneg t.coe_nonneg]; exact mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hL
-  apply discounted_regime_weighted_optional s ha hε hsize cap σ off hσ W hW
-    (regimeState s a ε η hη M cap σ off z)
-    (regimeState_continuous s a ε η hη M cap σ off z)
-    (regimeState_adapted s a ε η hη M cap σ off z)
-    (regimeDrift s a ε η M cap σ off)
-    (regimeDrift_continuous s a ε η hη M cap σ off) K hK hKb
-    (regimeState_eq s a ε η hη M cap σ off z)
-    S T B N η hη hS hT hST hTB E hE w hw hw0 C hwb hws
-  · intro b he t hst htt
-    refine ⟨max_le ?_ ?_,(hsafe b he t hst htt).2⟩
-    · exact (hsafe b he t hst htt).1.trans (le_max_left _ _)
-    · exact (hb b t (htt.trans (hTB b))).trans (le_max_right _ _)
-  · intro b he t hst htt
-    have hs := hsafe b he t hst htt
-    constructor
-    · change regimeDiscountCut s a ε η M cap σ off _ = _
-      apply regimeDiscount_congr
-      intro i hi
-      exact regimeClamp_eq η M _ hη ((norm_le_pi_norm _ i).trans hs.1) (hs.2 i hi).1
-    · intro i hi
-      exact cutoffDrift_eq_physical a η _ (hs.2 i hi).1
-end ExactSLEGreen
-
-namespace ExactSLEMain
-open Set
-
-lemma upper_pair_norm_sq (z w : ℂ) :
-    ‖z-star w‖^2 = ‖z-w‖^2 + 4*z.im*w.im := by
-  simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im,
-    Complex.star_def, Complex.conj_re, Complex.conj_im]
-  ring
-
-lemma upper_pair_norm_lt {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    ‖z-w‖ < ‖z-star w‖ := by
-  have h := upper_pair_norm_sq z w
-  have hp : 0 < 4*z.im*w.im := by positivity
-  nlinarith [norm_nonneg (z-w),norm_nonneg (z-star w)]
-
-lemma upper_harnack {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    z.im * (‖z-star w‖ - ‖z-w‖) ≤ w.im * (‖z-star w‖ + ‖z-w‖) := by
-  have hn : ‖z-w‖^2 = (z.re-w.re)^2+(z.im-w.im)^2 := by
-    rw [Complex.sq_norm]
-    simp [Complex.normSq_apply, pow_two]
-  have hd := upper_pair_norm_sq z w
-  have he : ((z.im+w.im)*‖z-w‖)^2 - ((z.im-w.im)*‖z-star w‖)^2 =
-      4*z.im*w.im*(z.re-w.re)^2 := by
-    rw [mul_pow, mul_pow, hd, hn]
-    ring
-  have hs : ((z.im-w.im)*‖z-star w‖)^2 ≤ ((z.im+w.im)*‖z-w‖)^2 := by
-    have hp : 0 ≤ 4*z.im*w.im*(z.re-w.re)^2 := by positivity
-    linarith
-  have hh : (z.im-w.im)*‖z-star w‖ ≤ (z.im+w.im)*‖z-w‖ :=
-    le_of_sq_le_sq hs (by positivity)
-  nlinarith
-
-lemma upper_velocity_harnack {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    (z.im/Complex.normSq z) * (‖z-star w‖ - ‖z-w‖) ≤
-    (w.im/Complex.normSq w) * (‖z-star w‖ + ‖z-w‖) := by
-  have hzn : z ≠ 0 := by intro h; simp [h] at hz
-  have hwn : w ≠ 0 := by intro h; simp [h] at hw
-  have hzN := Complex.normSq_pos.mpr hzn
-  have hwN := Complex.normSq_pos.mpr hwn
-  have hi : (-z⁻¹).im = z.im/Complex.normSq z := by simp only [Complex.neg_im, Complex.inv_im]; ring
-  have hj : (-w⁻¹).im = w.im/Complex.normSq w := by simp only [Complex.neg_im, Complex.inv_im]; ring
-  have hn : ‖-z⁻¹ - -w⁻¹‖ = ‖z-w‖ / (‖z‖*‖w‖) := by
-    have he : -z⁻¹ - -w⁻¹ = (z-w)/(z*w) := by field_simp; ring
-    rw [he,norm_div,norm_mul]
-  have hd : ‖-z⁻¹ - star (-w⁻¹)‖ = ‖z-star w‖ / (‖z‖*‖w‖) := by
-    have hwc : star w ≠ 0 := star_ne_zero.mpr hwn
-    have he : -z⁻¹ - star (-w⁻¹) = (z-star w)/(z*star w) := by
-      simp only [star_neg,star_inv₀]
-      field_simp
-      ring
-    rw [he,norm_div,norm_mul,norm_star]
-  have h := upper_harnack (z := -z⁻¹) (w := -w⁻¹)
-    (by rw [hi]; positivity) (by rw [hj]; positivity)
-  rw [hi,hj,hn,hd,← sub_div,← add_div,← mul_div_assoc,← mul_div_assoc] at h
-  exact (div_le_div_iff_of_pos_right (mul_pos (norm_pos_iff.mpr hzn) (norm_pos_iff.mpr hwn))).mp h
-
-lemma upper_initial_growth {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    ‖z-w‖ ≤ 2*z.im * (‖z-w‖/‖z-star w‖) / (1-‖z-w‖/‖z-star w‖)^2 := by
-  have hnd := upper_pair_norm_lt hz hw
-  have hd : 0 < ‖z-star w‖ := (norm_nonneg _).trans_lt hnd
-  have hdiff : 0 < ‖z-star w‖ - ‖z-w‖ := sub_pos.mpr hnd
-  have htri : ‖z-star w‖ ≤ ‖z-w‖+2*z.im := by
-    have hh := norm_add_le (star z-w) (z-star z)
-    have he : star z-w+(z-star z) = z-w := by ring
-    have hh := norm_add_le (z-w) (w-star w)
-    have hh' := norm_add_le (star (z-w)) (z-star z)
-    have hnorm : ‖z-star z‖ = 2*z.im := by
-      have he : z-star z = (2*z.im : ℝ)*Complex.I := by
-        apply Complex.ext <;> simp
-        ring
-      rw [he,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one,abs_of_pos (by positivity)]
-    have he : star (z-w)+(z-star z) = z-star w := by simp only [star_sub]; ring
-    rw [he,norm_star,hnorm] at hh'
-    linarith
-  have hs : (‖z-star w‖-‖z-w‖)^2 ≤ 2*z.im*‖z-star w‖ := by
-    calc
-      (‖z-star w‖-‖z-w‖)^2 ≤ (2*z.im)*(‖z-star w‖-‖z-w‖) := by
-        nlinarith [mul_nonneg (show 0 ≤ 2*z.im-(‖z-star w‖-‖z-w‖) by linarith) hdiff.le]
-      _ ≤ _ := mul_le_mul_of_nonneg_left (sub_le_self _ (norm_nonneg _)) (by positivity)
-  have he : 2*z.im * (‖z-w‖/‖z-star w‖) / (1-‖z-w‖/‖z-star w‖)^2 =
-      ‖z-w‖ * (2*z.im*‖z-star w‖)/(‖z-star w‖-‖z-w‖)^2 := by
-    field_simp
-  rw [he]
-  exact (le_div_iff₀ (sq_pos_of_pos hdiff)).mpr (mul_le_mul_of_nonneg_left hs (norm_nonneg _))
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set
-
-lemma complex_linear_norm_deriv {f : ℝ → ℂ} {c : ℂ} {D : Set ℝ} {t : ℝ}
-    (hf : HasDerivWithinAt f (c*f t) D t) (hn : f t ≠ 0) :
-    HasDerivWithinAt (fun t => ‖f t‖) (c.re*‖f t‖) D t := by
-  have hN := norm_pos_iff.mpr hn
-  have h := hf.norm_sq.sqrt (pow_ne_zero 2 hN.ne')
-  simp only [Real.sqrt_sq (norm_nonneg _)] at h
-  convert h using 1
-  try rfl
-  rw [Complex.inner, mul_assoc, Complex.mul_conj, Complex.mul_re]
-  simp only [Complex.ofReal_re,Complex.ofReal_im,mul_zero,sub_zero,Complex.normSq_eq_norm_sq]
-  field_simp
-
-lemma loewner_pair_drift (A : ℝ) {z w : ℂ} (_hz : z ≠ 0) (_hw : w ≠ 0) :
-    (-(A : ℂ)/(z*w)).re - (-(A : ℂ)/(z*star w)).re =
-      2*A*(z.im/Complex.normSq z)*(w.im/Complex.normSq w) := by
-  simp only [div_eq_mul_inv, mul_inv_rev, Complex.mul_re, Complex.mul_im,
-    Complex.neg_re, Complex.neg_im, Complex.ofReal_re, Complex.ofReal_im,
-    Complex.inv_re, Complex.inv_im, Complex.star_def, Complex.conj_re,Complex.conj_im,
-    Complex.normSq_conj]
-  ring
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set
-
-lemma growth_factor_deriv {D : Set ℝ} {t A v w : ℝ} {R r : ℝ → ℝ}
-    (hR : HasDerivWithinAt R (-2*A*v^2*R t) D t)
-    (hr : HasDerivWithinAt r (2*A*v*w*r t) D t) (hn : r t ≠ 1) :
-    HasDerivWithinAt (fun t => R t*r t/(1-r t)^2)
-      (2*A*R t*r t*v/(1-r t)^3 * (w*(1+r t)-v*(1-r t))) D t := by
-  have h := (hR.mul hr).div ((hr.const_sub 1).pow 2) (pow_ne_zero 2 (sub_ne_zero.mpr hn.symm))
-  convert h using 1
-  try rfl
-  simp only [Pi.pow_apply,Pi.mul_apply]
-  field_simp
-  ring
-
-lemma odeUntil_pair_ne {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
-    (hf : odeUntil A b z T f) (hg : odeUntil A b w T g) (hzw : z ≠ w)
-    {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) : f u-g u ≠ 0 := by
-  have hsub : Icc (0 : ℝ) u.toNNReal ⊆ Icc (0 : ℝ) T := by
-    rw [Real.coe_toNNReal _ hu.1]
-    exact Icc_subset_Icc_right hu.2
-  have hf' : odeUntil A b z u.toNNReal f := ⟨hf.1,fun s hs =>
-    ⟨(hf.2 s (hsub hs)).1,((hf.2 s (hsub hs)).2).mono hsub⟩⟩
-  have hg' : odeUntil A b w u.toNNReal g := ⟨hg.1,fun s hs =>
-    ⟨(hg.2 s (hsub hs)).1,((hg.2 s (hsub hs)).2).mono hsub⟩⟩
-  have he := loewner_solution_difference hb hf' hg'
-  rw [Real.coe_toNNReal _ hu.1] at he
-  rw [he]
-  exact mul_ne_zero (sub_ne_zero.mpr hzw) (Complex.exp_ne_zero _)
-
-lemma odeUntil_pair_ratio_deriv {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
-    (hf : odeUntil A b z T f) (hg : odeUntil A b w T g) (hzw : z ≠ w)
-    {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) :
-    HasDerivWithinAt (fun u => ‖f u-g u‖/‖f u-star (g u)‖)
-      (2*A*((f u).im/Complex.normSq (f u-(b u.toNNReal : ℂ)))*
-        ((g u).im/Complex.normSq (g u-(b u.toNNReal : ℂ)))*
-        (‖f u-g u‖/‖f u-star (g u)‖)) (Icc (0 : ℝ) T) u := by
-  let Z := f u-(b u.toNNReal : ℂ)
-  let W := g u-(b u.toNNReal : ℂ)
-  have hZ : Z ≠ 0 := odeUntil_center_ne_zero hf hu
-  have hW : W ≠ 0 := odeUntil_center_ne_zero hg hu
-  have hD : f u-star (g u) ≠ 0 := norm_pos_iff.mp
-    ((norm_nonneg _).trans_lt (upper_pair_norm_lt (hf.2 u hu).1 (hg.2 u hu).1))
-  have hd : HasDerivWithinAt (fun u => f u-g u)
-      ((-(A : ℂ)/(Z*W))*(f u-g u)) (Icc (0 : ℝ) T) u := by
-    convert (hf.2 u hu).2.sub (hg.2 u hu).2 using 1
-    try rfl
-    dsimp [Z,W]
-    field_simp [odeUntil_center_ne_zero hf hu,odeUntil_center_ne_zero hg hu]
-    ring
-  have hc := Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt u (hg.2 u hu).2
-  change HasDerivWithinAt (fun s => star (g s)) (star ((A : ℂ)/W)) _ _ at hc
-  have he : HasDerivWithinAt (fun u => f u-star (g u))
-      ((-(A : ℂ)/(Z*star W))*(f u-star (g u))) (Icc (0 : ℝ) T) u := by
-    convert (hf.2 u hu).2.sub hc using 1
-    try rfl
-    simp only [star_div₀, Complex.star_def, Complex.conj_ofReal]
-    change -(A : ℂ)/(Z*star W)*(f u-star (g u)) = (A : ℂ)/Z-(A : ℂ)/star W
-    have he : f u-star (g u) = Z-star W := by simp [Z,W]
-    rw [he]
-    rw [div_sub_div _ _ hZ (star_ne_zero.mpr hW)]
-    ring
-  have hn := complex_linear_norm_deriv hd (odeUntil_pair_ne hb hf hg hzw hu)
-  have hm := complex_linear_norm_deriv he hD
-  have hh := hn.div hm (norm_ne_zero_iff.mpr hD)
-  convert hh using 1
-  try rfl
-  have hi := loewner_pair_drift A hZ hW
-  have hiz : Z.im = (f u).im := by simp [Z]
-  have hiw : W.im = (g u).im := by simp [W]
-  rw [hiz,hiw] at hi
-  change (2*A*((f u).im/Complex.normSq Z)*((g u).im/Complex.normSq W))*
-    (‖f u-g u‖/‖f u-star (g u)‖) = _
-  rw [← hi]
-  field_simp [norm_ne_zero_iff.mpr hD]
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open Set
-
-end ExactSLEMain
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.OptionalEstimates
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLECalculus ExactSLEFiniteTilt
+noncomputable def regimeState {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (t : ℝ≥0) (b : Driver) : RegimeState m :=
+  (vectorState a η hη z t b,
+    ∫ u in (0:ℝ)..(t:ℝ), regimeDiscountCut s a ε η N cap σ off (vectorState a η hη z u.toNNReal b))
+lemma regimeState_continuous {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (b : Driver) :
+    Continuous (fun t => regimeState s a ε η hη N cap σ off z t b) := by
+  have hc : Continuous (fun u : ℝ => regimeDiscountCut s a ε η N cap σ off
+      (vectorState a η hη z u.toNNReal b)) :=
+    (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp
+      ((vectorState_continuous a η hη z b).comp continuous_real_toNNReal)
+  exact (vectorState_continuous a η hη z b).prodMk
+    ((intervalIntegral.continuous_primitive hc.intervalIntegrable 0).comp continuous_subtype_val)
+lemma regimeState_adapted {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (t : ℝ≥0) :
+    Measurable[rawFiltration t] (regimeState s a ε η hη N cap σ off z t) := by
+  apply (vectorState_adapted a η hη z t).prodMk
+  apply adapted_intervalIntegral rawFiltration
+    (fun u b => regimeDiscountCut s a ε η N cap σ off (vectorState a η hη z u b))
+  · intro b
+    exact (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp
+      (vectorState_continuous a η hη z b)
+  · intro u
+    exact (regimeDiscountCut_continuous s a ε η hη N cap σ off).measurable.comp
+      (vectorState_adapted a η hη z u)
+lemma regimeState_eq {m : ℕ} (s : Finset (Fin m)) (a ε η : ℝ) (hη : 0<η) (N : ℝ≥0)
+    (cap σ off : Fin m → ℝ) (z : Fin m → ℂ) (u h : ℝ≥0) (b : Driver) :
+    regimeState s a ε η hη N cap σ off z (u+h) b =
+      regimeState s a ε η hη N cap σ off z u b+increment u h b • (multiNoise m,0)+
+        ∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), regimeDrift s a ε η N cap σ off
+          (regimeState s a ε η hη N cap σ off z t.toNNReal b) := by
+  let A := regimeDrift s a ε η N cap σ off
+  let V := regimeState s a ε η hη N cap σ off z
+  have hc : Continuous (fun t : ℝ => A (V t.toNNReal b)) :=
+    (regimeDrift_continuous s a ε η hη N cap σ off).comp
+      ((regimeState_continuous s a ε η hη N cap σ off z b).comp continuous_real_toNNReal)
+  have hi := hc.intervalIntegrable (μ := volume) (u:ℝ) ((u+h:ℝ≥0):ℝ)
+  apply Prod.ext
+  · funext i
+    let L : RegimeState m →L[ℝ] State := (ContinuousLinearMap.proj i).comp (ContinuousLinearMap.fst ℝ _ _)
+    have he := L.intervalIntegral_comp_comm hi
+    change localState a η hη (z i) (u+h) b = localState a η hη (z i) u b+
+      increment u h b • noise+L (∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (V t.toNNReal b))
+    rw [← he]
+    exact localState_eq a η hη (z i) u h b
+  · let L := ContinuousLinearMap.snd ℝ (Fin m → State) ℝ
+    have he := L.intervalIntegral_comp_comm hi
+    change (V (u+h) b).2 = (V u b).2+increment u h b • (0:ℝ)+L
+      (∫ t in (u:ℝ)..((u+h:ℝ≥0):ℝ), A (V t.toNNReal b))
+    rw [smul_zero,add_zero,← he]
+    have hk : Continuous (fun t : ℝ => regimeDiscountCut s a ε η N cap σ off
+        (vectorState a η hη z t.toNNReal b)) :=
+      (regimeDiscountCut_continuous s a ε η hη N cap σ off).comp
+        ((vectorState_continuous a η hη z b).comp continuous_real_toNNReal)
+    exact (intervalIntegral.integral_add_adjacent_intervals (μ := volume) (hk.intervalIntegrable 0 (u:ℝ))
+      (hk.intervalIntegrable (u:ℝ) ((u+h:ℝ≥0):ℝ))).symm
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus
+lemma regimeDiscount_congr {m : ℕ} (s : Finset (Fin m)) (a ε : ℝ)
+    (cap σ off : Fin m → ℝ) (v w : Fin m → State) (h : ∀ i∈s, v i=w i) :
+    regimeDiscount s a ε cap σ off v=regimeDiscount s a ε cap σ off w := by
+  unfold regimeDiscount
+  congr 1
+  · congr 1
+    apply Finset.sum_congr rfl
+    intro i hi
+    rw [h i hi]
+  · congr 1
+    apply Finset.sum_congr rfl
+    intro i hi
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [h i hi,h j (Finset.mem_of_mem_erase hj)]
+ 
+
+theorem vector_regime_weighted_optional {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (N : ℝ≥0) (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (z : Fin m → ℂ) (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (S T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
+    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
+    (w : Driver → ℝ) (hw : StronglyMeasurable[hS.measurableSpace] w)
+    (hw0 : ∀ b, 0≤w b) (C : ℝ) (hwb : ∀ b, ‖w b‖≤C)
+    (hws : ∀ b∉E, w b=0)
+    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b →
+      ‖regimeState s a ε η hη N cap σ off z t b‖≤N ∧
+      ∀ i∈s, η≤Y (vectorState a η hη z t b i) ∧
+        physicalDelta a (cap i) (σ i) (off i) (vectorState a η hη z t b i)≤1) :
+    ∫ b, w b*discountedProduct s a cap σ off
+      (regimeState s a ε η hη N cap σ off z (T b) b) ∂W ≤
+    ∫ b, w b*discountedProduct s a cap σ off
+      (regimeState s a ε η hη N cap σ off z (S b) b) ∂W := by
+  obtain ⟨K,hK,hKb⟩ := regimeDrift_bound s a ε η hη N cap σ off
+  apply discounted_regime_weighted_optional s ha hε hsize cap σ off hσ W hW
+    (regimeState s a ε η hη N cap σ off z)
+    (regimeState_continuous s a ε η hη N cap σ off z)
+    (regimeState_adapted s a ε η hη N cap σ off z)
+    (regimeDrift s a ε η N cap σ off)
+    (regimeDrift_continuous s a ε η hη N cap σ off) K hK hKb
+    (regimeState_eq s a ε η hη N cap σ off z)
+    S T B N η hη hS hT hST hTB E hE w hw hw0 C hwb hws hsafe
+  intro b hb t hts htt
+  have hn := (hsafe b hb t hts htt).1
+  have hy := (hsafe b hb t hts htt).2
+  constructor
+  · change regimeDiscountCut s a ε η N cap σ off _ = _
+    apply regimeDiscount_congr
+    intro i hi
+    apply regimeClamp_eq η N _ hη _ (hy i hi).1
+    have hv : ‖vectorState a η hη z t b‖≤N :=
+      (le_max_left _ _).trans hn
+    exact (norm_le_pi_norm _ i).trans hv
+  · intro i hi
+    exact cutoffDrift_eq_physical a η _ (hy i hi).1
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEFiniteTilt ExactSLECalculus
+ 
+
+theorem vector_regime_weighted_optional_bounded {m : ℕ} (s : Finset (Fin m)) {a ε η : ℝ}
+    (ha : 1/4<a) (hε : 0<ε) (hη : 0<η) (hsize : ε*((s.card:ℝ)-1)≤regK a/2)
+    (M : ℝ≥0) (cap σ off : Fin m → ℝ) (hσ : ∀ i∈s, |σ i|≤regBeta a)
+    (z : Fin m → ℂ) (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (S T : Driver → ℝ≥0) (B : ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (hST : ∀ b, S b≤T b) (hTB : ∀ b, T b≤B)
+    (E : Set Driver) (hE : MeasurableSet[hS.measurableSpace] E)
+    (w : Driver → ℝ) (hw : StronglyMeasurable[hS.measurableSpace] w)
+    (hw0 : ∀ b, 0≤w b) (C : ℝ) (hwb : ∀ b, ‖w b‖≤C)
+    (hws : ∀ b∉E, w b=0)
+    (hsafe : ∀ b∈E, ∀ t, S b≤t → t≤T b →
+      ‖vectorState a η hη z t b‖≤M ∧
+      ∀ i∈s, η≤Y (vectorState a η hη z t b i) ∧
+        physicalDelta a (cap i) (σ i) (off i) (vectorState a η hη z t b i)≤1) :
+    ∫ b, w b*discountedProduct s a cap σ off
+      (regimeState s a ε η hη M cap σ off z (T b) b) ∂W ≤
+    ∫ b, w b*discountedProduct s a cap σ off
+      (regimeState s a ε η hη M cap σ off z (S b) b) ∂W := by
+  obtain ⟨K,hK,hKb⟩ := regimeDrift_bound s a ε η hη M cap σ off
+  obtain ⟨L,hL,hLb⟩ := regimeDiscountCut_bound s a ε η hη M cap σ off
+  let N : ℝ≥0 := ⟨max M (L*B),le_trans M.coe_nonneg (le_max_left _ _)⟩
+  have hb (b : Driver) (t : ℝ≥0) (ht : t≤B) :
+      ‖(regimeState s a ε η hη M cap σ off z t b).2‖≤L*B := by
+    change ‖∫ u in (0:ℝ)..(t:ℝ), regimeDiscountCut s a ε η M cap σ off
+      (vectorState a η hη z u.toNNReal b)‖≤_
+    calc
+      _ ≤ L*|(t:ℝ)-0| := intervalIntegral.norm_integral_le_of_norm_le_const fun u _ => hLb _
+      _ ≤ L*B := by rw [sub_zero,abs_of_nonneg t.coe_nonneg]; exact mul_le_mul_of_nonneg_left (by exact_mod_cast ht) hL
+  apply discounted_regime_weighted_optional s ha hε hsize cap σ off hσ W hW
+    (regimeState s a ε η hη M cap σ off z)
+    (regimeState_continuous s a ε η hη M cap σ off z)
+    (regimeState_adapted s a ε η hη M cap σ off z)
+    (regimeDrift s a ε η M cap σ off)
+    (regimeDrift_continuous s a ε η hη M cap σ off) K hK hKb
+    (regimeState_eq s a ε η hη M cap σ off z)
+    S T B N η hη hS hT hST hTB E hE w hw hw0 C hwb hws
+  · intro b he t hst htt
+    refine ⟨max_le ?_ ?_,(hsafe b he t hst htt).2⟩
+    · exact (hsafe b he t hst htt).1.trans (le_max_left _ _)
+    · exact (hb b t (htt.trans (hTB b))).trans (le_max_right _ _)
+  · intro b he t hst htt
+    have hs := hsafe b he t hst htt
+    constructor
+    · change regimeDiscountCut s a ε η M cap σ off _ = _
+      apply regimeDiscount_congr
+      intro i hi
+      exact regimeClamp_eq η M _ hη ((norm_le_pi_norm _ i).trans hs.1) (hs.2 i hi).1
+    · intro i hi
+      exact cutoffDrift_eq_physical a η _ (hs.2 i hi).1
+end ExactSLEGreen
+
+namespace ExactSLEMain
+open Set
+
+lemma upper_pair_norm_sq (z w : ℂ) :
+    ‖z-star w‖^2 = ‖z-w‖^2 + 4*z.im*w.im := by
+  simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im,
+    Complex.star_def, Complex.conj_re, Complex.conj_im]
+  ring
+
+lemma upper_pair_norm_lt {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    ‖z-w‖ < ‖z-star w‖ := by
+  have h := upper_pair_norm_sq z w
+  have hp : 0 < 4*z.im*w.im := by positivity
+  nlinarith [norm_nonneg (z-w),norm_nonneg (z-star w)]
+
+lemma upper_harnack {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    z.im * (‖z-star w‖ - ‖z-w‖) ≤ w.im * (‖z-star w‖ + ‖z-w‖) := by
+  have hn : ‖z-w‖^2 = (z.re-w.re)^2+(z.im-w.im)^2 := by
+    rw [Complex.sq_norm]
+    simp [Complex.normSq_apply, pow_two]
+  have hd := upper_pair_norm_sq z w
+  have he : ((z.im+w.im)*‖z-w‖)^2 - ((z.im-w.im)*‖z-star w‖)^2 =
+      4*z.im*w.im*(z.re-w.re)^2 := by
+    rw [mul_pow, mul_pow, hd, hn]
+    ring
+  have hs : ((z.im-w.im)*‖z-star w‖)^2 ≤ ((z.im+w.im)*‖z-w‖)^2 := by
+    have hp : 0 ≤ 4*z.im*w.im*(z.re-w.re)^2 := by positivity
+    linarith
+  have hh : (z.im-w.im)*‖z-star w‖ ≤ (z.im+w.im)*‖z-w‖ :=
+    le_of_sq_le_sq hs (by positivity)
+  nlinarith
+
+lemma upper_velocity_harnack {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    (z.im/Complex.normSq z) * (‖z-star w‖ - ‖z-w‖) ≤
+    (w.im/Complex.normSq w) * (‖z-star w‖ + ‖z-w‖) := by
+  have hzn : z ≠ 0 := by intro h; simp [h] at hz
+  have hwn : w ≠ 0 := by intro h; simp [h] at hw
+  have hzN := Complex.normSq_pos.mpr hzn
+  have hwN := Complex.normSq_pos.mpr hwn
+  have hi : (-z⁻¹).im = z.im/Complex.normSq z := by simp only [Complex.neg_im, Complex.inv_im]; ring
+  have hj : (-w⁻¹).im = w.im/Complex.normSq w := by simp only [Complex.neg_im, Complex.inv_im]; ring
+  have hn : ‖-z⁻¹ - -w⁻¹‖ = ‖z-w‖ / (‖z‖*‖w‖) := by
+    have he : -z⁻¹ - -w⁻¹ = (z-w)/(z*w) := by field_simp; ring
+    rw [he,norm_div,norm_mul]
+  have hd : ‖-z⁻¹ - star (-w⁻¹)‖ = ‖z-star w‖ / (‖z‖*‖w‖) := by
+    have hwc : star w ≠ 0 := star_ne_zero.mpr hwn
+    have he : -z⁻¹ - star (-w⁻¹) = (z-star w)/(z*star w) := by
+      simp only [star_neg,star_inv₀]
+      field_simp
+      ring
+    rw [he,norm_div,norm_mul,norm_star]
+  have h := upper_harnack (z := -z⁻¹) (w := -w⁻¹)
+    (by rw [hi]; positivity) (by rw [hj]; positivity)
+  rw [hi,hj,hn,hd,← sub_div,← add_div,← mul_div_assoc,← mul_div_assoc] at h
+  exact (div_le_div_iff_of_pos_right (mul_pos (norm_pos_iff.mpr hzn) (norm_pos_iff.mpr hwn))).mp h
+
+lemma upper_initial_growth {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    ‖z-w‖ ≤ 2*z.im * (‖z-w‖/‖z-star w‖) / (1-‖z-w‖/‖z-star w‖)^2 := by
+  have hnd := upper_pair_norm_lt hz hw
+  have hd : 0 < ‖z-star w‖ := (norm_nonneg _).trans_lt hnd
+  have hdiff : 0 < ‖z-star w‖ - ‖z-w‖ := sub_pos.mpr hnd
+  have htri : ‖z-star w‖ ≤ ‖z-w‖+2*z.im := by
+    have hh := norm_add_le (star z-w) (z-star z)
+    have he : star z-w+(z-star z) = z-w := by ring
+    have hh := norm_add_le (z-w) (w-star w)
+    have hh' := norm_add_le (star (z-w)) (z-star z)
+    have hnorm : ‖z-star z‖ = 2*z.im := by
+      have he : z-star z = (2*z.im : ℝ)*Complex.I := by
+        apply Complex.ext <;> simp
+        ring
+      rw [he,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one,abs_of_pos (by positivity)]
+    have he : star (z-w)+(z-star z) = z-star w := by simp only [star_sub]; ring
+    rw [he,norm_star,hnorm] at hh'
+    linarith
+  have hs : (‖z-star w‖-‖z-w‖)^2 ≤ 2*z.im*‖z-star w‖ := by
+    calc
+      (‖z-star w‖-‖z-w‖)^2 ≤ (2*z.im)*(‖z-star w‖-‖z-w‖) := by
+        nlinarith [mul_nonneg (show 0 ≤ 2*z.im-(‖z-star w‖-‖z-w‖) by linarith) hdiff.le]
+      _ ≤ _ := mul_le_mul_of_nonneg_left (sub_le_self _ (norm_nonneg _)) (by positivity)
+  have he : 2*z.im * (‖z-w‖/‖z-star w‖) / (1-‖z-w‖/‖z-star w‖)^2 =
+      ‖z-w‖ * (2*z.im*‖z-star w‖)/(‖z-star w‖-‖z-w‖)^2 := by
+    field_simp
+  rw [he]
+  exact (le_div_iff₀ (sq_pos_of_pos hdiff)).mpr (mul_le_mul_of_nonneg_left hs (norm_nonneg _))
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set
+
+lemma complex_linear_norm_deriv {f : ℝ → ℂ} {c : ℂ} {D : Set ℝ} {t : ℝ}
+    (hf : HasDerivWithinAt f (c*f t) D t) (hn : f t ≠ 0) :
+    HasDerivWithinAt (fun t => ‖f t‖) (c.re*‖f t‖) D t := by
+  have hN := norm_pos_iff.mpr hn
+  have h := hf.norm_sq.sqrt (pow_ne_zero 2 hN.ne')
+  simp only [Real.sqrt_sq (norm_nonneg _)] at h
+  convert h using 1
+  try rfl
+  rw [Complex.inner, mul_assoc, Complex.mul_conj, Complex.mul_re]
+  simp only [Complex.ofReal_re,Complex.ofReal_im,mul_zero,sub_zero,Complex.normSq_eq_norm_sq]
+  field_simp
+
+lemma loewner_pair_drift (A : ℝ) {z w : ℂ} (_hz : z ≠ 0) (_hw : w ≠ 0) :
+    (-(A : ℂ)/(z*w)).re - (-(A : ℂ)/(z*star w)).re =
+      2*A*(z.im/Complex.normSq z)*(w.im/Complex.normSq w) := by
+  simp only [div_eq_mul_inv, mul_inv_rev, Complex.mul_re, Complex.mul_im,
+    Complex.neg_re, Complex.neg_im, Complex.ofReal_re, Complex.ofReal_im,
+    Complex.inv_re, Complex.inv_im, Complex.star_def, Complex.conj_re,Complex.conj_im,
+    Complex.normSq_conj]
+  ring
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set
+
+lemma growth_factor_deriv {D : Set ℝ} {t A v w : ℝ} {R r : ℝ → ℝ}
+    (hR : HasDerivWithinAt R (-2*A*v^2*R t) D t)
+    (hr : HasDerivWithinAt r (2*A*v*w*r t) D t) (hn : r t ≠ 1) :
+    HasDerivWithinAt (fun t => R t*r t/(1-r t)^2)
+      (2*A*R t*r t*v/(1-r t)^3 * (w*(1+r t)-v*(1-r t))) D t := by
+  have h := (hR.mul hr).div ((hr.const_sub 1).pow 2) (pow_ne_zero 2 (sub_ne_zero.mpr hn.symm))
+  convert h using 1
+  try rfl
+  simp only [Pi.pow_apply,Pi.mul_apply]
+  field_simp
+  ring
+
+lemma odeUntil_pair_ne {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
+    (hf : odeUntil A b z T f) (hg : odeUntil A b w T g) (hzw : z ≠ w)
+    {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) : f u-g u ≠ 0 := by
+  have hsub : Icc (0 : ℝ) u.toNNReal ⊆ Icc (0 : ℝ) T := by
+    rw [Real.coe_toNNReal _ hu.1]
+    exact Icc_subset_Icc_right hu.2
+  have hf' : odeUntil A b z u.toNNReal f := ⟨hf.1,fun s hs =>
+    ⟨(hf.2 s (hsub hs)).1,((hf.2 s (hsub hs)).2).mono hsub⟩⟩
+  have hg' : odeUntil A b w u.toNNReal g := ⟨hg.1,fun s hs =>
+    ⟨(hg.2 s (hsub hs)).1,((hg.2 s (hsub hs)).2).mono hsub⟩⟩
+  have he := loewner_solution_difference hb hf' hg'
+  rw [Real.coe_toNNReal _ hu.1] at he
+  rw [he]
+  exact mul_ne_zero (sub_ne_zero.mpr hzw) (Complex.exp_ne_zero _)
+
+lemma odeUntil_pair_ratio_deriv {A : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
+    (hf : odeUntil A b z T f) (hg : odeUntil A b w T g) (hzw : z ≠ w)
+    {u : ℝ} (hu : u ∈ Icc (0 : ℝ) T) :
+    HasDerivWithinAt (fun u => ‖f u-g u‖/‖f u-star (g u)‖)
+      (2*A*((f u).im/Complex.normSq (f u-(b u.toNNReal : ℂ)))*
+        ((g u).im/Complex.normSq (g u-(b u.toNNReal : ℂ)))*
+        (‖f u-g u‖/‖f u-star (g u)‖)) (Icc (0 : ℝ) T) u := by
+  let Z := f u-(b u.toNNReal : ℂ)
+  let W := g u-(b u.toNNReal : ℂ)
+  have hZ : Z ≠ 0 := odeUntil_center_ne_zero hf hu
+  have hW : W ≠ 0 := odeUntil_center_ne_zero hg hu
+  have hD : f u-star (g u) ≠ 0 := norm_pos_iff.mp
+    ((norm_nonneg _).trans_lt (upper_pair_norm_lt (hf.2 u hu).1 (hg.2 u hu).1))
+  have hd : HasDerivWithinAt (fun u => f u-g u)
+      ((-(A : ℂ)/(Z*W))*(f u-g u)) (Icc (0 : ℝ) T) u := by
+    convert (hf.2 u hu).2.sub (hg.2 u hu).2 using 1
+    try rfl
+    dsimp [Z,W]
+    field_simp [odeUntil_center_ne_zero hf hu,odeUntil_center_ne_zero hg hu]
+    ring
+  have hc := Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt u (hg.2 u hu).2
+  change HasDerivWithinAt (fun s => star (g s)) (star ((A : ℂ)/W)) _ _ at hc
+  have he : HasDerivWithinAt (fun u => f u-star (g u))
+      ((-(A : ℂ)/(Z*star W))*(f u-star (g u))) (Icc (0 : ℝ) T) u := by
+    convert (hf.2 u hu).2.sub hc using 1
+    try rfl
+    simp only [star_div₀, Complex.star_def, Complex.conj_ofReal]
+    change -(A : ℂ)/(Z*star W)*(f u-star (g u)) = (A : ℂ)/Z-(A : ℂ)/star W
+    have he : f u-star (g u) = Z-star W := by simp [Z,W]
+    rw [he]
+    rw [div_sub_div _ _ hZ (star_ne_zero.mpr hW)]
+    ring
+  have hn := complex_linear_norm_deriv hd (odeUntil_pair_ne hb hf hg hzw hu)
+  have hm := complex_linear_norm_deriv he hD
+  have hh := hn.div hm (norm_ne_zero_iff.mpr hD)
+  convert hh using 1
+  try rfl
+  have hi := loewner_pair_drift A hZ hW
+  have hiz : Z.im = (f u).im := by simp [Z]
+  have hiw : W.im = (g u).im := by simp [W]
+  rw [hiz,hiw] at hi
+  change (2*A*((f u).im/Complex.normSq Z)*((g u).im/Complex.normSq W))*
+    (‖f u-g u‖/‖f u-star (g u)‖) = _
+  rw [← hi]
+  field_simp [norm_ne_zero_iff.mpr hD]
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open Set
+
+end ExactSLEMain
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StochasticCalculus.lean b/lean/OAI/Probability/SLEGauge/StochasticCalculus.lean
index 535c9c1352424b91931fbfe549cce46bbccb04a4..6ece8a1913575c84b5898fa5236be3760d621034 100644
--- a/lean/OAI/Probability/SLEGauge/StochasticCalculus.lean
+++ b/lean/OAI/Probability/SLEGauge/StochasticCalculus.lean
@@ -1,430 +1,430 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StoppingMeasurability
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
- 
- 
-
- 
-
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-namespace ExactSLEGenerator
-instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _
-instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
-abbrev Driver := {b : C(ℝ≥0, ℝ) // b 0 = 0}
-instance : MeasurableSpace Driver := borel Driver
-instance : BorelSpace Driver := ⟨rfl⟩
-lemma driver_eval_measurable (t : ℝ≥0) : StronglyMeasurable (fun b : Driver => b.1 t) :=
-  ((continuous_eval_const t).comp continuous_subtype_val).stronglyMeasurable
-noncomputable def rawFiltration : Filtration ℝ≥0 (borel Driver) :=
-  Filtration.natural (fun t (b : Driver) => b.1 t) driver_eval_measurable
-
-def increment (s h : ℝ≥0) (b : Driver) : ℝ := b.1 (s+h) - b.1 s
-lemma increment_measurable (s h : ℝ≥0) : Measurable (increment s h) :=
-  (driver_eval_measurable (s+h)).measurable.sub (driver_eval_measurable s).measurable
-lemma increment_law (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
-    HasLaw (increment s h) (gaussianReal 0 h) W := by
-  exact (hW.toIsPreBrownianReal.shift s).hasLaw_eval h
-lemma increment_integrable (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
-    Integrable (increment s h) W :=
-  (increment_law W hW s h).hasGaussianLaw.integrable
-lemma increment_mean (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
-    ∫ b, increment s h b ∂W = 0 := by
-  rw [(increment_law W hW s h).integral_eq, integral_id_gaussianReal]
-lemma increment_abs_moment_integrable (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) (n : ℕ) (hn : n ≠ 0) :
-    Integrable (fun b => |increment s h b| ^ n) W := by
-  simpa only [Real.norm_eq_abs] using
-    ((increment_law W hW s h).hasGaussianLaw.memLp (p := (n : ℝ≥0∞)) (by simp)).integrable_norm_pow hn
-lemma increment_sq_integrable (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
-    Integrable (fun b => increment s h b ^ 2) W := by
-  simpa only [sq_abs] using increment_abs_moment_integrable W hW s h 2 (by norm_num)
-lemma increment_second_moment (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
-    ∫ b, increment s h b ^ 2 ∂W = (h : ℝ) := by
-  have hg : ∫ x, x ^ 2 ∂gaussianReal 0 h = (h : ℝ) := by
-    have hg := variance_fun_id_gaussianReal (μ := 0) (v := h)
-    rw [variance_eq_integral (X := fun x : ℝ => x) measurable_id.aemeasurable,
-      integral_id_gaussianReal] at hg
-    simpa using hg
-  calc
-    _ = ∫ x, x ^ 2 ∂gaussianReal 0 h :=
-      (increment_law W hW s h).integral_comp (measurable_id.pow_const 2).aestronglyMeasurable
-    _ = _ := hg
-
-noncomputable def absMoment (n : ℕ) : ℝ := ∫ x, |x| ^ n ∂gaussianReal 0 1
-lemma absMoment_nonneg (n : ℕ) : 0 ≤ absMoment n :=
-  integral_nonneg fun value => pow_nonneg (abs_nonneg value) _
-lemma gaussian_abs_moment (h : ℝ≥0) (n : ℕ) :
-    ∫ x, |x| ^ n ∂gaussianReal 0 h = (Real.sqrt h) ^ n * absMoment n := by
-  have he : gaussianReal 0 h = (gaussianReal 0 1).map (fun x => Real.sqrt h * x) := by
-    rw [gaussianReal_map_const_mul]
-    congr 1
-    · simp
-    · ext
-      simp only [NNReal.coe_mk, mul_one]
-      exact (Real.sq_sqrt h.coe_nonneg).symm
-  rw [he]
-  have hm : AEMeasurable (fun x : ℝ => Real.sqrt h * x) (gaussianReal 0 1) :=
-    (measurable_const.mul measurable_id).aemeasurable
-  have hf : AEStronglyMeasurable (fun x : ℝ => |x| ^ n)
-      ((gaussianReal 0 1).map (fun x => Real.sqrt h * x)) :=
-    ((measurable_id.abs).pow_const n).aestronglyMeasurable
-  rw [integral_map hm hf]
-  simp_rw [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _), mul_pow]
-  rw [integral_const_mul]
-  rfl
-lemma increment_abs_moment (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) (n : ℕ) :
-    ∫ b, |increment s h b| ^ n ∂W = (Real.sqrt h) ^ n * absMoment n := by
-  calc
-    _ = ∫ x, |x| ^ n ∂gaussianReal 0 h :=
-      (increment_law W hW s h).integral_comp ((measurable_id.abs).pow_const n).aestronglyMeasurable
-    _ = _ := gaussian_abs_moment h n
-
-lemma increment_indep (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0)
-    {Z : Driver → ℝ} (hZ : Measurable[rawFiltration s] Z) :
-    IndepFun Z (increment s h) W := by
-  let fut : Driver → (ℝ≥0 → ℝ) := fun b t => b.1 (s+t) - b.1 s
-  have hi : Indep (rawFiltration s) (MeasurableSpace.comap fut inferInstance) W := by
-    rw [rawFiltration, Filtration.natural_eq_comap]
-    exact (IndepFun_iff_Indep _ _ _).mp (hW.toIsPreBrownianReal.indepFun_shift s).symm
-  apply (IndepFun_iff_Indep _ _ _).mpr
-  apply indep_of_indep_of_le hi hZ.comap_le
-  have hm : @Measurable Driver ℝ (MeasurableSpace.comap fut inferInstance) _
-      (increment s h) := (measurable_pi_apply h).comp (comap_measurable fut)
-  exact hm.comap_le
-
-lemma increment_weighted_mean (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0)
-    {Z : Driver → ℝ} (hZ : Measurable[rawFiltration s] Z) :
-    ∫ b, Z b * increment s h b ∂W = 0 := by
-  rw [(increment_indep W hW s h hZ).integral_fun_mul_eq_mul_integral
-    ((hZ.mono (rawFiltration.le s) le_rfl).aestronglyMeasurable)
-    (increment_measurable s h).aestronglyMeasurable, increment_mean W hW, mul_zero]
-
-lemma increment_weighted_second_moment (W : Measure Driver)
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0)
-    {Z : Driver → ℝ} (hZ : Measurable[rawFiltration s] Z) :
-    ∫ b, Z b * increment s h b ^ 2 ∂W = (h : ℝ) * ∫ b, Z b ∂W := by
-  have hi : IndepFun Z (fun b => increment s h b ^ 2) W :=
-    (increment_indep W hW s h hZ).comp measurable_id (measurable_id.pow_const 2)
-  rw [hi.integral_fun_mul_eq_mul_integral
-    ((hZ.mono (rawFiltration.le s) le_rfl).aestronglyMeasurable)
-    ((increment_measurable s h).pow_const 2).aestronglyMeasurable,
-    increment_second_moment W hW, mul_comm]
-
- 
-
-theorem expected_taylor_step (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (s h : ℝ≥0) (F₀ F₁ Z Y V : Driver → ℝ)
-    (hF₀ : Integrable F₀ W) (hF₁ : Integrable F₁ W) (hV : Integrable V W)
-    (hZm : Measurable[rawFiltration s] Z) (hYm : Measurable[rawFiltration s] Y)
-    (hZi : Integrable Z W) (hYi : Integrable Y W)
-    (C K : ℝ) (_hC : 0 ≤ C) (_hK : 0 ≤ K)
-    (hTaylor : ∀ b, |F₁ b - F₀ b - Z b * increment s h b -
-        Y b * increment s h b ^ 2 / 2 - V b| ≤
-      C * (K * h * (|increment s h b| + K * h) + |increment s h b| ^ 3)) :
-    |(∫ b, F₁ b ∂W) - (∫ b, F₀ b ∂W) - (h : ℝ) / 2 * (∫ b, Y b ∂W) -
-      (∫ b, V b ∂W)| ≤
-      C * (K * h * (Real.sqrt h * absMoment 1 + K * h) + Real.sqrt h ^ 3 * absMoment 3) := by
-  have hprod : Integrable (fun b => Z b * increment s h b) W :=
-    (increment_indep W hW s h hZm).integrable_mul hZi (increment_integrable W hW s h)
-  have hprod₂ : Integrable (fun b => Y b * increment s h b ^ 2 / 2) W := by
-    apply Integrable.div_const
-    exact ((increment_indep W hW s h hYm).comp measurable_id (measurable_id.pow_const 2)).integrable_mul
-      hYi (increment_sq_integrable W hW s h)
-  let R := fun b => F₁ b - F₀ b - Z b * increment s h b - Y b * increment s h b ^ 2 / 2 - V b
-  have hRi : Integrable R W := (((hF₁.sub hF₀).sub hprod).sub hprod₂).sub hV
-  have hright : Integrable (fun b => C *
-      (K * h * (|increment s h b| + K * h) + |increment s h b| ^ 3)) W := by
-    apply Integrable.const_mul
-    apply Integrable.add
-    · exact ((increment_integrable W hW s h).abs.add (integrable_const _)).const_mul _
-    · exact increment_abs_moment_integrable W hW s h 3 (by norm_num)
-  have he : (∫ b, R b ∂W) = (∫ b, F₁ b ∂W) - (∫ b, F₀ b ∂W) -
-      (h : ℝ) / 2 * (∫ b, Y b ∂W) - (∫ b, V b ∂W) := by
-    dsimp [R]
-    have he₃ := integral_sub (((hF₁.sub hF₀).sub hprod).sub hprod₂) hV
-    have he₂ := integral_sub ((hF₁.sub hF₀).sub hprod) hprod₂
-    have he₁ := integral_sub (hF₁.sub hF₀) hprod
-    simp only [Pi.sub_apply] at he₃ he₂ he₁
-    rw [he₃, he₂, he₁, integral_sub hF₁ hF₀,
-      increment_weighted_mean W hW s h hZm, integral_div,
-      increment_weighted_second_moment W hW s h hYm]
-    ring
-  rw [← he]
-  calc
-    _ ≤ ∫ b, |R b| ∂W := by simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm R
-    _ ≤ ∫ b, C * (K * h * (|increment s h b| + K * h) + |increment s h b| ^ 3) ∂W :=
-      integral_mono hRi.abs hright hTaylor
-    _ = _ := by
-      have ha := integral_add
-        (((increment_integrable W hW s h).abs.add (integrable_const (K * (h : ℝ)))).const_mul (K * (h : ℝ)))
-        (increment_abs_moment_integrable W hW s h 3 (by norm_num))
-      have hb := integral_add (increment_integrable W hW s h).abs (integrable_const (K * (h : ℝ)))
-      simp only [Pi.add_apply] at ha hb
-      rw [integral_const_mul, ha, integral_const_mul, hb,
-        integral_const, probReal_univ, smul_eq_mul, one_mul,
-        increment_abs_moment W hW s h 3]
-      have habs := increment_abs_moment W hW s h 1
-      simp only [pow_one] at habs
-      rw [habs]
-
- 
-
-theorem expected_taylor_sum (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
-    (start : ℝ≥0) (N : ℕ) (h : ℝ≥0) (F Z Y V : ℕ → Driver → ℝ)
-    (hF : ∀ i, Integrable (F i) W) (hV : ∀ i, Integrable (V i) W)
-    (hZm : ∀ i : ℕ, Measurable[rawFiltration (start + i * h)] (Z i))
-    (hYm : ∀ i : ℕ, Measurable[rawFiltration (start + i * h)] (Y i))
-    (hZi : ∀ i, Integrable (Z i) W) (hYi : ∀ i, Integrable (Y i) W)
-    (C K : ℝ) (hC : 0 ≤ C) (hK : 0 ≤ K)
-    (hTaylor : ∀ i < N, ∀ b, |F (i+1) b - F i b - Z i b * increment (start+i*h) h b -
-        Y i b * increment (start+i*h) h b ^ 2 / 2 - V i b| ≤
-      C * (K * h * (|increment (start+i*h) h b| + K * h) + |increment (start+i*h) h b| ^ 3)) :
-    |(∫ b, F N b ∂W) - (∫ b, F 0 b ∂W) -
-      ∑ i ∈ Finset.range N, ((h : ℝ) / 2 * (∫ b, Y i b ∂W) + (∫ b, V i b ∂W))| ≤
-      (N : ℝ) * C * h * ((K * absMoment 1 + absMoment 3) * Real.sqrt h + K ^ 2 * h) := by
-  let e : ℕ → ℝ := fun i => (∫ b, F (i+1) b ∂W) - (∫ b, F i b ∂W) -
-      (h : ℝ) / 2 * (∫ b, Y i b ∂W) - (∫ b, V i b ∂W)
-  have he (i : ℕ) (hi : i < N) : |e i| ≤
-      C * h * ((K * absMoment 1 + absMoment 3) * Real.sqrt h + K ^ 2 * h) := by
-    have hs := expected_taylor_step W hW (start+i*h) h (F i) (F (i+1)) (Z i) (Y i) (V i)
-      (hF i) (hF (i+1)) (hV i) (hZm i) (hYm i) (hZi i) (hYi i) C K hC hK (hTaylor i hi)
-    change |e i| ≤ _ at hs
-    convert hs using 1
-    rw [show Real.sqrt h ^ 3 = (h : ℝ) * Real.sqrt h by rw [pow_succ, Real.sq_sqrt h.coe_nonneg]]
-    ring
-  have htel : ∑ i ∈ Finset.range N, e i = (∫ b, F N b ∂W) - (∫ b, F 0 b ∂W) -
-      ∑ i ∈ Finset.range N, ((h : ℝ) / 2 * (∫ b, Y i b ∂W) + (∫ b, V i b ∂W)) := by
-    have heq : e = fun i => ((∫ b, F (i+1) b ∂W) - (∫ b, F i b ∂W)) -
-        ((h : ℝ) / 2 * (∫ b, Y i b ∂W) + (∫ b, V i b ∂W)) := by
-      funext i
-      dsimp [e]
-      ring
-    rw [heq, Finset.sum_sub_distrib, Finset.sum_range_sub (fun i => ∫ b, F i b ∂W) N]
-  rw [← htel]
-  calc
-    _ ≤ ∑ i ∈ Finset.range N, |e i| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ _i ∈ Finset.range N,
-        C * h * ((K * absMoment 1 + absMoment 3) * Real.sqrt h + K ^ 2 * h) :=
-      Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
-    _ = _ := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
-
- 
-lemma taylor_error_tendsto_zero (T C K : ℝ) :
-    Tendsto (fun n : ℕ => C * T * ((K * absMoment 1 + absMoment 3) *
-      Real.sqrt (T / (n+1 : ℕ)) + K ^ 2 * (T / (n+1 : ℕ)))) atTop (𝓝 0) := by
-  have h : Tendsto (fun n : ℕ => T / (n+1 : ℕ)) atTop (𝓝 0) :=
-    (tendsto_const_div_atTop_nhds_zero_nat T).comp (tendsto_add_atTop_nat 1)
-  have hs : Tendsto (fun n : ℕ => Real.sqrt (T / (n+1 : ℕ))) atTop (𝓝 0) := by
-    simpa only [Function.comp_def, Real.sqrt_zero] using (Real.continuous_sqrt.tendsto 0).comp h
-  simpa using ((hs.const_mul (K * absMoment 1 + absMoment 3)).add (h.const_mul (K^2))).const_mul (C*T)
-end ExactSLEGenerator
-
- 
-
-open scoped Topology
-open Set
-namespace ExactSLEGenerator
-
-lemma scalar_taylor_cubic (f : ℝ → ℝ) (hf : ContDiff ℝ 3 f)
-    (C : ℝ) (hC : ∀ x, |iteratedDeriv 3 f x| ≤ C) (w : ℝ) :
-    |f w - f 0 - deriv f 0 * w - iteratedDeriv 2 f 0 * w ^ 2 / 2| ≤
-      C * |w| ^ 3 := by
-  by_cases hw : w = 0
-  · subst w
-    simp
-  have hn : (0 : ℝ) ≠ w := Ne.symm hw
-  obtain ⟨u, hu, he⟩ := taylor_mean_remainder_lagrange_iteratedDeriv (n := 2) hn hf.contDiffOn
-  have hid (n : ℕ) (hnn : n ≤ 3) : iteratedDerivWithin n f (Set.uIcc 0 w) 0 =
-      iteratedDeriv n f 0 := by
-    rw [iteratedDerivWithin_eq_iteratedFDerivWithin, iteratedDeriv_eq_iteratedFDeriv]
-    congr 1
-    exact iteratedFDerivWithin_eq_iteratedFDeriv (uniqueDiffOn_uIcc hn)
-      ((hf.of_le (by exact_mod_cast hnn)).contDiffAt) (left_mem_uIcc)
-  have hp : taylorWithinEval f 2 (Set.uIcc 0 w) 0 w =
-      f 0 + deriv f 0 * w + iteratedDeriv 2 f 0 * w^2/2 := by
-    rw [taylorWithinEval_succ f 1 (Set.uIcc 0 w) 0 w,
-      taylorWithinEval_succ f 0 (Set.uIcc 0 w) 0 w, taylor_within_zero_eval]
-    rw [hid 1 (by norm_num), hid 2 (by norm_num)]
-    simp only [iteratedDeriv_one, sub_zero, Nat.factorial, Nat.cast_mul, Nat.cast_one,
-      zero_add, pow_one, smul_eq_mul]
-    ring
-  rw [hp] at he
-  have hh : |f w - f 0 - deriv f 0 * w - iteratedDeriv 2 f 0 * w ^ 2 / 2| =
-      |iteratedDeriv 3 f u| * |w| ^ 3 / 6 := by
-    have he' : f w - f 0 - deriv f 0 * w - iteratedDeriv 2 f 0 * w ^ 2 / 2 =
-        iteratedDeriv 3 f u * w ^ 3 / 6 := by
-      norm_num at he
-      linarith only [he]
-    rw [he', abs_div, abs_mul, abs_pow]
-    norm_num
-  rw [hh]
-  have hC0 : 0 ≤ C := (abs_nonneg _).trans (hC u)
-  calc
-    _ ≤ C * |w| ^ 3 / 6 := by gcongr; exact hC u
-    _ ≤ _ := by nlinarith [mul_nonneg hC0 (pow_nonneg (abs_nonneg w) 3)]
-
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-
-lemma first_order_remainder (f : E → ℝ) (D : E → E →L[ℝ] ℝ)
-    (hD : ∀ x, HasFDerivAt f (D x) x) (C : ℝ)
-    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖) (x v : E) :
-    |f (x+v) - f x - D x v| ≤ C * ‖v‖^2 := by
-  let F := fun u => f u - D x u
-  have hF : ∀ u ∈ Metric.closedBall x ‖v‖,
-      HasFDerivWithinAt F (D u - D x) (Metric.closedBall x ‖v‖) u := by
-    intro u hu
-    exact ((hD u).sub (D x).hasFDerivAt).hasFDerivWithinAt
-  have hbound : ∀ u ∈ Metric.closedBall x ‖v‖, ‖D u - D x‖ ≤ C * ‖v‖ := by
-    intro u hu
-    have hC : 0 ≤ C ∨ v = 0 := by
-      by_cases hv : v = 0
-      · exact Or.inr hv
-      · left
-        have hp : 0 < ‖v‖ := norm_pos_iff.mpr hv
-        have ht := (norm_nonneg (D (x+v)-D x)).trans (hLip x (x+v))
-        simp only [add_sub_cancel_left] at ht
-        exact nonneg_of_mul_nonneg_left ht hp
-    rcases hC with hC | rfl
-    · exact (hLip x u).trans (mul_le_mul_of_nonneg_left (by simpa [dist_eq_norm] using hu) hC)
-    · have hu' : u = x := by simpa using hu
-      subst u
-      simp
-  have hb := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hF hbound
-    (convex_closedBall x ‖v‖) (Metric.mem_closedBall_self (norm_nonneg v))
-    (show x+v ∈ Metric.closedBall x ‖v‖ by simp [Metric.mem_closedBall, dist_eq_norm])
-  simpa [F, map_add, Real.norm_eq_abs, pow_two, sub_eq_add_neg, add_assoc, add_left_comm, add_comm,
-    mul_assoc] using hb
-
- 
-
-lemma mixed_taylor_remainder (f : E → ℝ) (D : E → E →L[ℝ] ℝ)
-    (hD : ∀ x, HasFDerivAt f (D x) x) (e : E) (he : ‖e‖ ≤ 1)
-    (C : ℝ) (hC : 0 ≤ C)
-    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
-    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
-    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
-    (x v : E) (w : ℝ) :
-    |f (x + w • e + v) - f x - D x e * w -
-      iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0 * w^2/2 - D x v| ≤
-    C * (‖v‖ * (|w| + ‖v‖) + |w|^3) := by
-  have hd : deriv (fun r : ℝ => f (x + r • e)) 0 = D x e := by
-    have hi := ((hasDerivAt_id (0 : ℝ)).smul_const e).const_add x
-    have ho := (hD (x + (0 : ℝ) • e)).comp_hasDerivAt 0 hi
-    simpa only [Function.comp_def, id_eq, zero_smul, one_smul, add_zero] using ho.deriv
-  have hb := scalar_taylor_cubic (fun r : ℝ => f (x + r • e)) (hSmooth x) C
-    (hThird x) w
-  rw [hd] at hb
-  simp only [zero_smul, add_zero] at hb
-  have hv := first_order_remainder f D hD C hLip (x + w • e) v
-  have hc : |(D (x+w•e)-D x) v| ≤ C * |w| * ‖v‖ := by
-    calc
-      _ ≤ ‖D (x+w•e)-D x‖ * ‖v‖ := by
-        simpa only [Real.norm_eq_abs] using (D (x+w•e)-D x).le_opNorm v
-      _ ≤ (C * ‖(x+w•e)-x‖) * ‖v‖ := by gcongr; exact hLip x (x+w•e)
-      _ = (C * (|w| * ‖e‖)) * ‖v‖ := by simp [norm_smul, Real.norm_eq_abs]
-      _ ≤ _ := by
-        have hw := mul_le_mul_of_nonneg_left he (abs_nonneg w)
-        simpa only [mul_one] using mul_le_mul_of_nonneg_right
-          (mul_le_mul_of_nonneg_left hw hC) (norm_nonneg v)
-  have hsum := (abs_add_le (f (x+w•e+v)-f (x+w•e)-D (x+w•e) v)
-    (f (x+w•e)-f x-D x e*w-
-      iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2)).trans (add_le_add hv hb)
-  have hsum' := (abs_add_le ((f (x+w•e+v)-f (x+w•e)-D (x+w•e) v)+
-    (f (x+w•e)-f x-D x e*w-
-      iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2)) ((D (x+w•e)-D x) v)).trans
-        (add_le_add hsum hc)
-  have heq : ((f (x+w•e+v)-f (x+w•e)-D (x+w•e) v)+
-    (f (x+w•e)-f x-D x e*w-
-      iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2)) + ((D (x+w•e)-D x) v) =
-      f (x+w•e+v)-f x-D x e*w-
-        iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2-D x v := by
-    simp only [sub_apply]
-    ring
-  rw [heq] at hsum'
-  calc
-    _ ≤ (C * ‖v‖^2 + C * |w|^3) + C * |w| * ‖v‖ := hsum'
-    _ = _ := by ring
-
-end ExactSLEGenerator
-
- 
-open MeasureTheory Filter
-open scoped Topology
-namespace ExactSLEGenerator
-
-lemma riemann_sum_error_le (f : ℝ → ℝ) (hf : Continuous f) (T : ℝ) (hT : 0 ≤ T)
-    (N : ℕ) (hN : 0 < N) (ε : ℝ) (_hε : 0 ≤ ε)
-    (hmod : ∀ u ∈ Set.Icc 0 T, ∀ v ∈ Set.Icc 0 T,
-      |u-v| ≤ T / N → |f u-f v| ≤ ε) :
-    |(∑ i ∈ Finset.range N, (T/N) * f (i*(T/N))) - ∫ s in 0..T, f s| ≤ ε*T := by
-  let h := T / (N : ℝ)
-  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
-  have hh : 0 ≤ h := div_nonneg hT hNr.le
-  have hNh : (N : ℝ) * h = T := by dsimp [h]; field_simp
-  let G : ℕ → ℝ := fun i => ∫ s in 0..(i : ℝ)*h, f s
-  have hG (i : ℕ) : ∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, f s = G (i+1)-G i := by
-    have ha := intervalIntegral.integral_add_adjacent_intervals
-      (hf.intervalIntegrable (μ := volume) 0 ((i : ℝ)*h))
-      (hf.intervalIntegrable (μ := volume) ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h))
-    dsimp [G]
-    linarith only [ha]
-  have hdiff (i : ℕ) : ((i+1 : ℕ) : ℝ)*h-(i : ℝ)*h = h := by push_cast; ring
-  have he (i : ℕ) (hi : i < N) : |h*f ((i : ℝ)*h) - (G (i+1)-G i)| ≤ ε*h := by
-    have hai : (i : ℝ) * h ∈ Set.Icc 0 T := by
-      refine ⟨mul_nonneg (Nat.cast_nonneg _) hh, ?_⟩
-      rw [← hNh]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi.le) hh
-    have hbi : ((i+1 : ℕ) : ℝ) * h ∈ Set.Icc 0 T := by
-      refine ⟨by positivity, ?_⟩
-      rw [← hNh]
-      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi) hh
-    have hab : (i : ℝ)*h ≤ ((i+1 : ℕ) : ℝ)*h := by linarith [hdiff i]
-    have hbound : ∀ s ∈ Set.uIoc ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h),
-        ‖f ((i : ℝ)*h)-f s‖ ≤ ε := by
-      intro s hs
-      rw [Set.uIoc_of_le hab] at hs
-      rw [Real.norm_eq_abs]
-      apply hmod _ hai _ ⟨hai.1.trans hs.1.le, hs.2.trans hbi.2⟩
-      rw [abs_of_nonpos (by linarith [hs.1])]
-      linarith [hs.2, hdiff i]
-    have hb := intervalIntegral.norm_integral_le_of_norm_le_const hbound
-    have hc : IntervalIntegrable (fun _ : ℝ => f ((i : ℝ)*h)) volume
-        ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h) := intervalIntegrable_const
-    rw [intervalIntegral.integral_sub hc (hf.intervalIntegrable _ _),
-      intervalIntegral.integral_const, smul_eq_mul, hdiff, hG, Real.norm_eq_abs,
-      abs_of_nonneg hh] at hb
-    exact hb
-  have hs : ∑ i ∈ Finset.range N, (G (i+1)-G i) = ∫ s in 0..T, f s := by
-    rw [Finset.sum_range_sub]
-    simp [G, hNh]
-  change |(∑ i ∈ Finset.range N, h*f ((i : ℝ)*h)) - _| ≤ _
-  rw [← hs, ← Finset.sum_sub_distrib]
-  calc
-    _ ≤ ∑ i ∈ Finset.range N, |h*f ((i : ℝ)*h)-(G (i+1)-G i)| :=
-      Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ _i ∈ Finset.range N, ε*h := Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
-    _ = ε*T := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; rw [← hNh]; ring
-
-end ExactSLEGenerator
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StoppingMeasurability
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+ 
+ 
+
+ 
+
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+namespace ExactSLEGenerator
+instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _
+instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩
+abbrev Driver := {b : C(ℝ≥0, ℝ) // b 0 = 0}
+instance : MeasurableSpace Driver := borel Driver
+instance : BorelSpace Driver := ⟨rfl⟩
+lemma driver_eval_measurable (t : ℝ≥0) : StronglyMeasurable (fun b : Driver => b.1 t) :=
+  ((continuous_eval_const t).comp continuous_subtype_val).stronglyMeasurable
+noncomputable def rawFiltration : Filtration ℝ≥0 (borel Driver) :=
+  Filtration.natural (fun t (b : Driver) => b.1 t) driver_eval_measurable
+
+def increment (s h : ℝ≥0) (b : Driver) : ℝ := b.1 (s+h) - b.1 s
+lemma increment_measurable (s h : ℝ≥0) : Measurable (increment s h) :=
+  (driver_eval_measurable (s+h)).measurable.sub (driver_eval_measurable s).measurable
+lemma increment_law (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
+    HasLaw (increment s h) (gaussianReal 0 h) W := by
+  exact (hW.toIsPreBrownianReal.shift s).hasLaw_eval h
+lemma increment_integrable (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
+    Integrable (increment s h) W :=
+  (increment_law W hW s h).hasGaussianLaw.integrable
+lemma increment_mean (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
+    ∫ b, increment s h b ∂W = 0 := by
+  rw [(increment_law W hW s h).integral_eq, integral_id_gaussianReal]
+lemma increment_abs_moment_integrable (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) (n : ℕ) (hn : n ≠ 0) :
+    Integrable (fun b => |increment s h b| ^ n) W := by
+  simpa only [Real.norm_eq_abs] using
+    ((increment_law W hW s h).hasGaussianLaw.memLp (p := (n : ℝ≥0∞)) (by simp)).integrable_norm_pow hn
+lemma increment_sq_integrable (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
+    Integrable (fun b => increment s h b ^ 2) W := by
+  simpa only [sq_abs] using increment_abs_moment_integrable W hW s h 2 (by norm_num)
+lemma increment_second_moment (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) :
+    ∫ b, increment s h b ^ 2 ∂W = (h : ℝ) := by
+  have hg : ∫ x, x ^ 2 ∂gaussianReal 0 h = (h : ℝ) := by
+    have hg := variance_fun_id_gaussianReal (μ := 0) (v := h)
+    rw [variance_eq_integral (X := fun x : ℝ => x) measurable_id.aemeasurable,
+      integral_id_gaussianReal] at hg
+    simpa using hg
+  calc
+    _ = ∫ x, x ^ 2 ∂gaussianReal 0 h :=
+      (increment_law W hW s h).integral_comp (measurable_id.pow_const 2).aestronglyMeasurable
+    _ = _ := hg
+
+noncomputable def absMoment (n : ℕ) : ℝ := ∫ x, |x| ^ n ∂gaussianReal 0 1
+lemma absMoment_nonneg (n : ℕ) : 0 ≤ absMoment n :=
+  integral_nonneg fun value => pow_nonneg (abs_nonneg value) _
+lemma gaussian_abs_moment (h : ℝ≥0) (n : ℕ) :
+    ∫ x, |x| ^ n ∂gaussianReal 0 h = (Real.sqrt h) ^ n * absMoment n := by
+  have he : gaussianReal 0 h = (gaussianReal 0 1).map (fun x => Real.sqrt h * x) := by
+    rw [gaussianReal_map_const_mul]
+    congr 1
+    · simp
+    · ext
+      simp only [NNReal.coe_mk, mul_one]
+      exact (Real.sq_sqrt h.coe_nonneg).symm
+  rw [he]
+  have hm : AEMeasurable (fun x : ℝ => Real.sqrt h * x) (gaussianReal 0 1) :=
+    (measurable_const.mul measurable_id).aemeasurable
+  have hf : AEStronglyMeasurable (fun x : ℝ => |x| ^ n)
+      ((gaussianReal 0 1).map (fun x => Real.sqrt h * x)) :=
+    ((measurable_id.abs).pow_const n).aestronglyMeasurable
+  rw [integral_map hm hf]
+  simp_rw [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _), mul_pow]
+  rw [integral_const_mul]
+  rfl
+lemma increment_abs_moment (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0) (n : ℕ) :
+    ∫ b, |increment s h b| ^ n ∂W = (Real.sqrt h) ^ n * absMoment n := by
+  calc
+    _ = ∫ x, |x| ^ n ∂gaussianReal 0 h :=
+      (increment_law W hW s h).integral_comp ((measurable_id.abs).pow_const n).aestronglyMeasurable
+    _ = _ := gaussian_abs_moment h n
+
+lemma increment_indep (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0)
+    {Z : Driver → ℝ} (hZ : Measurable[rawFiltration s] Z) :
+    IndepFun Z (increment s h) W := by
+  let fut : Driver → (ℝ≥0 → ℝ) := fun b t => b.1 (s+t) - b.1 s
+  have hi : Indep (rawFiltration s) (MeasurableSpace.comap fut inferInstance) W := by
+    rw [rawFiltration, Filtration.natural_eq_comap]
+    exact (IndepFun_iff_Indep _ _ _).mp (hW.toIsPreBrownianReal.indepFun_shift s).symm
+  apply (IndepFun_iff_Indep _ _ _).mpr
+  apply indep_of_indep_of_le hi hZ.comap_le
+  have hm : @Measurable Driver ℝ (MeasurableSpace.comap fut inferInstance) _
+      (increment s h) := (measurable_pi_apply h).comp (comap_measurable fut)
+  exact hm.comap_le
+
+lemma increment_weighted_mean (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0)
+    {Z : Driver → ℝ} (hZ : Measurable[rawFiltration s] Z) :
+    ∫ b, Z b * increment s h b ∂W = 0 := by
+  rw [(increment_indep W hW s h hZ).integral_fun_mul_eq_mul_integral
+    ((hZ.mono (rawFiltration.le s) le_rfl).aestronglyMeasurable)
+    (increment_measurable s h).aestronglyMeasurable, increment_mean W hW, mul_zero]
+
+lemma increment_weighted_second_moment (W : Measure Driver)
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W) (s h : ℝ≥0)
+    {Z : Driver → ℝ} (hZ : Measurable[rawFiltration s] Z) :
+    ∫ b, Z b * increment s h b ^ 2 ∂W = (h : ℝ) * ∫ b, Z b ∂W := by
+  have hi : IndepFun Z (fun b => increment s h b ^ 2) W :=
+    (increment_indep W hW s h hZ).comp measurable_id (measurable_id.pow_const 2)
+  rw [hi.integral_fun_mul_eq_mul_integral
+    ((hZ.mono (rawFiltration.le s) le_rfl).aestronglyMeasurable)
+    ((increment_measurable s h).pow_const 2).aestronglyMeasurable,
+    increment_second_moment W hW, mul_comm]
+
+ 
+
+theorem expected_taylor_step (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (s h : ℝ≥0) (F₀ F₁ Z Y V : Driver → ℝ)
+    (hF₀ : Integrable F₀ W) (hF₁ : Integrable F₁ W) (hV : Integrable V W)
+    (hZm : Measurable[rawFiltration s] Z) (hYm : Measurable[rawFiltration s] Y)
+    (hZi : Integrable Z W) (hYi : Integrable Y W)
+    (C K : ℝ) (_hC : 0 ≤ C) (_hK : 0 ≤ K)
+    (hTaylor : ∀ b, |F₁ b - F₀ b - Z b * increment s h b -
+        Y b * increment s h b ^ 2 / 2 - V b| ≤
+      C * (K * h * (|increment s h b| + K * h) + |increment s h b| ^ 3)) :
+    |(∫ b, F₁ b ∂W) - (∫ b, F₀ b ∂W) - (h : ℝ) / 2 * (∫ b, Y b ∂W) -
+      (∫ b, V b ∂W)| ≤
+      C * (K * h * (Real.sqrt h * absMoment 1 + K * h) + Real.sqrt h ^ 3 * absMoment 3) := by
+  have hprod : Integrable (fun b => Z b * increment s h b) W :=
+    (increment_indep W hW s h hZm).integrable_mul hZi (increment_integrable W hW s h)
+  have hprod₂ : Integrable (fun b => Y b * increment s h b ^ 2 / 2) W := by
+    apply Integrable.div_const
+    exact ((increment_indep W hW s h hYm).comp measurable_id (measurable_id.pow_const 2)).integrable_mul
+      hYi (increment_sq_integrable W hW s h)
+  let R := fun b => F₁ b - F₀ b - Z b * increment s h b - Y b * increment s h b ^ 2 / 2 - V b
+  have hRi : Integrable R W := (((hF₁.sub hF₀).sub hprod).sub hprod₂).sub hV
+  have hright : Integrable (fun b => C *
+      (K * h * (|increment s h b| + K * h) + |increment s h b| ^ 3)) W := by
+    apply Integrable.const_mul
+    apply Integrable.add
+    · exact ((increment_integrable W hW s h).abs.add (integrable_const _)).const_mul _
+    · exact increment_abs_moment_integrable W hW s h 3 (by norm_num)
+  have he : (∫ b, R b ∂W) = (∫ b, F₁ b ∂W) - (∫ b, F₀ b ∂W) -
+      (h : ℝ) / 2 * (∫ b, Y b ∂W) - (∫ b, V b ∂W) := by
+    dsimp [R]
+    have he₃ := integral_sub (((hF₁.sub hF₀).sub hprod).sub hprod₂) hV
+    have he₂ := integral_sub ((hF₁.sub hF₀).sub hprod) hprod₂
+    have he₁ := integral_sub (hF₁.sub hF₀) hprod
+    simp only [Pi.sub_apply] at he₃ he₂ he₁
+    rw [he₃, he₂, he₁, integral_sub hF₁ hF₀,
+      increment_weighted_mean W hW s h hZm, integral_div,
+      increment_weighted_second_moment W hW s h hYm]
+    ring
+  rw [← he]
+  calc
+    _ ≤ ∫ b, |R b| ∂W := by simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm R
+    _ ≤ ∫ b, C * (K * h * (|increment s h b| + K * h) + |increment s h b| ^ 3) ∂W :=
+      integral_mono hRi.abs hright hTaylor
+    _ = _ := by
+      have ha := integral_add
+        (((increment_integrable W hW s h).abs.add (integrable_const (K * (h : ℝ)))).const_mul (K * (h : ℝ)))
+        (increment_abs_moment_integrable W hW s h 3 (by norm_num))
+      have hb := integral_add (increment_integrable W hW s h).abs (integrable_const (K * (h : ℝ)))
+      simp only [Pi.add_apply] at ha hb
+      rw [integral_const_mul, ha, integral_const_mul, hb,
+        integral_const, probReal_univ, smul_eq_mul, one_mul,
+        increment_abs_moment W hW s h 3]
+      have habs := increment_abs_moment W hW s h 1
+      simp only [pow_one] at habs
+      rw [habs]
+
+ 
+
+theorem expected_taylor_sum (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : Driver) => b.1 t) W)
+    (start : ℝ≥0) (N : ℕ) (h : ℝ≥0) (F Z Y V : ℕ → Driver → ℝ)
+    (hF : ∀ i, Integrable (F i) W) (hV : ∀ i, Integrable (V i) W)
+    (hZm : ∀ i : ℕ, Measurable[rawFiltration (start + i * h)] (Z i))
+    (hYm : ∀ i : ℕ, Measurable[rawFiltration (start + i * h)] (Y i))
+    (hZi : ∀ i, Integrable (Z i) W) (hYi : ∀ i, Integrable (Y i) W)
+    (C K : ℝ) (hC : 0 ≤ C) (hK : 0 ≤ K)
+    (hTaylor : ∀ i < N, ∀ b, |F (i+1) b - F i b - Z i b * increment (start+i*h) h b -
+        Y i b * increment (start+i*h) h b ^ 2 / 2 - V i b| ≤
+      C * (K * h * (|increment (start+i*h) h b| + K * h) + |increment (start+i*h) h b| ^ 3)) :
+    |(∫ b, F N b ∂W) - (∫ b, F 0 b ∂W) -
+      ∑ i ∈ Finset.range N, ((h : ℝ) / 2 * (∫ b, Y i b ∂W) + (∫ b, V i b ∂W))| ≤
+      (N : ℝ) * C * h * ((K * absMoment 1 + absMoment 3) * Real.sqrt h + K ^ 2 * h) := by
+  let e : ℕ → ℝ := fun i => (∫ b, F (i+1) b ∂W) - (∫ b, F i b ∂W) -
+      (h : ℝ) / 2 * (∫ b, Y i b ∂W) - (∫ b, V i b ∂W)
+  have he (i : ℕ) (hi : i < N) : |e i| ≤
+      C * h * ((K * absMoment 1 + absMoment 3) * Real.sqrt h + K ^ 2 * h) := by
+    have hs := expected_taylor_step W hW (start+i*h) h (F i) (F (i+1)) (Z i) (Y i) (V i)
+      (hF i) (hF (i+1)) (hV i) (hZm i) (hYm i) (hZi i) (hYi i) C K hC hK (hTaylor i hi)
+    change |e i| ≤ _ at hs
+    convert hs using 1
+    rw [show Real.sqrt h ^ 3 = (h : ℝ) * Real.sqrt h by rw [pow_succ, Real.sq_sqrt h.coe_nonneg]]
+    ring
+  have htel : ∑ i ∈ Finset.range N, e i = (∫ b, F N b ∂W) - (∫ b, F 0 b ∂W) -
+      ∑ i ∈ Finset.range N, ((h : ℝ) / 2 * (∫ b, Y i b ∂W) + (∫ b, V i b ∂W)) := by
+    have heq : e = fun i => ((∫ b, F (i+1) b ∂W) - (∫ b, F i b ∂W)) -
+        ((h : ℝ) / 2 * (∫ b, Y i b ∂W) + (∫ b, V i b ∂W)) := by
+      funext i
+      dsimp [e]
+      ring
+    rw [heq, Finset.sum_sub_distrib, Finset.sum_range_sub (fun i => ∫ b, F i b ∂W) N]
+  rw [← htel]
+  calc
+    _ ≤ ∑ i ∈ Finset.range N, |e i| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _i ∈ Finset.range N,
+        C * h * ((K * absMoment 1 + absMoment 3) * Real.sqrt h + K ^ 2 * h) :=
+      Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
+    _ = _ := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring
+
+ 
+lemma taylor_error_tendsto_zero (T C K : ℝ) :
+    Tendsto (fun n : ℕ => C * T * ((K * absMoment 1 + absMoment 3) *
+      Real.sqrt (T / (n+1 : ℕ)) + K ^ 2 * (T / (n+1 : ℕ)))) atTop (𝓝 0) := by
+  have h : Tendsto (fun n : ℕ => T / (n+1 : ℕ)) atTop (𝓝 0) :=
+    (tendsto_const_div_atTop_nhds_zero_nat T).comp (tendsto_add_atTop_nat 1)
+  have hs : Tendsto (fun n : ℕ => Real.sqrt (T / (n+1 : ℕ))) atTop (𝓝 0) := by
+    simpa only [Function.comp_def, Real.sqrt_zero] using (Real.continuous_sqrt.tendsto 0).comp h
+  simpa using ((hs.const_mul (K * absMoment 1 + absMoment 3)).add (h.const_mul (K^2))).const_mul (C*T)
+end ExactSLEGenerator
+
+ 
+
+open scoped Topology
+open Set
+namespace ExactSLEGenerator
+
+lemma scalar_taylor_cubic (f : ℝ → ℝ) (hf : ContDiff ℝ 3 f)
+    (C : ℝ) (hC : ∀ x, |iteratedDeriv 3 f x| ≤ C) (w : ℝ) :
+    |f w - f 0 - deriv f 0 * w - iteratedDeriv 2 f 0 * w ^ 2 / 2| ≤
+      C * |w| ^ 3 := by
+  by_cases hw : w = 0
+  · subst w
+    simp
+  have hn : (0 : ℝ) ≠ w := Ne.symm hw
+  obtain ⟨u, hu, he⟩ := taylor_mean_remainder_lagrange_iteratedDeriv (n := 2) hn hf.contDiffOn
+  have hid (n : ℕ) (hnn : n ≤ 3) : iteratedDerivWithin n f (Set.uIcc 0 w) 0 =
+      iteratedDeriv n f 0 := by
+    rw [iteratedDerivWithin_eq_iteratedFDerivWithin, iteratedDeriv_eq_iteratedFDeriv]
+    congr 1
+    exact iteratedFDerivWithin_eq_iteratedFDeriv (uniqueDiffOn_uIcc hn)
+      ((hf.of_le (by exact_mod_cast hnn)).contDiffAt) (left_mem_uIcc)
+  have hp : taylorWithinEval f 2 (Set.uIcc 0 w) 0 w =
+      f 0 + deriv f 0 * w + iteratedDeriv 2 f 0 * w^2/2 := by
+    rw [taylorWithinEval_succ f 1 (Set.uIcc 0 w) 0 w,
+      taylorWithinEval_succ f 0 (Set.uIcc 0 w) 0 w, taylor_within_zero_eval]
+    rw [hid 1 (by norm_num), hid 2 (by norm_num)]
+    simp only [iteratedDeriv_one, sub_zero, Nat.factorial, Nat.cast_mul, Nat.cast_one,
+      zero_add, pow_one, smul_eq_mul]
+    ring
+  rw [hp] at he
+  have hh : |f w - f 0 - deriv f 0 * w - iteratedDeriv 2 f 0 * w ^ 2 / 2| =
+      |iteratedDeriv 3 f u| * |w| ^ 3 / 6 := by
+    have he' : f w - f 0 - deriv f 0 * w - iteratedDeriv 2 f 0 * w ^ 2 / 2 =
+        iteratedDeriv 3 f u * w ^ 3 / 6 := by
+      norm_num at he
+      linarith only [he]
+    rw [he', abs_div, abs_mul, abs_pow]
+    norm_num
+  rw [hh]
+  have hC0 : 0 ≤ C := (abs_nonneg _).trans (hC u)
+  calc
+    _ ≤ C * |w| ^ 3 / 6 := by gcongr; exact hC u
+    _ ≤ _ := by nlinarith [mul_nonneg hC0 (pow_nonneg (abs_nonneg w) 3)]
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma first_order_remainder (f : E → ℝ) (D : E → E →L[ℝ] ℝ)
+    (hD : ∀ x, HasFDerivAt f (D x) x) (C : ℝ)
+    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖) (x v : E) :
+    |f (x+v) - f x - D x v| ≤ C * ‖v‖^2 := by
+  let F := fun u => f u - D x u
+  have hF : ∀ u ∈ Metric.closedBall x ‖v‖,
+      HasFDerivWithinAt F (D u - D x) (Metric.closedBall x ‖v‖) u := by
+    intro u hu
+    exact ((hD u).sub (D x).hasFDerivAt).hasFDerivWithinAt
+  have hbound : ∀ u ∈ Metric.closedBall x ‖v‖, ‖D u - D x‖ ≤ C * ‖v‖ := by
+    intro u hu
+    have hC : 0 ≤ C ∨ v = 0 := by
+      by_cases hv : v = 0
+      · exact Or.inr hv
+      · left
+        have hp : 0 < ‖v‖ := norm_pos_iff.mpr hv
+        have ht := (norm_nonneg (D (x+v)-D x)).trans (hLip x (x+v))
+        simp only [add_sub_cancel_left] at ht
+        exact nonneg_of_mul_nonneg_left ht hp
+    rcases hC with hC | rfl
+    · exact (hLip x u).trans (mul_le_mul_of_nonneg_left (by simpa [dist_eq_norm] using hu) hC)
+    · have hu' : u = x := by simpa using hu
+      subst u
+      simp
+  have hb := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hF hbound
+    (convex_closedBall x ‖v‖) (Metric.mem_closedBall_self (norm_nonneg v))
+    (show x+v ∈ Metric.closedBall x ‖v‖ by simp [Metric.mem_closedBall, dist_eq_norm])
+  simpa [F, map_add, Real.norm_eq_abs, pow_two, sub_eq_add_neg, add_assoc, add_left_comm, add_comm,
+    mul_assoc] using hb
+
+ 
+
+lemma mixed_taylor_remainder (f : E → ℝ) (D : E → E →L[ℝ] ℝ)
+    (hD : ∀ x, HasFDerivAt f (D x) x) (e : E) (he : ‖e‖ ≤ 1)
+    (C : ℝ) (hC : 0 ≤ C)
+    (hLip : ∀ x y, ‖D y - D x‖ ≤ C * ‖y-x‖)
+    (hSmooth : ∀ x, ContDiff ℝ 3 (fun w : ℝ => f (x + w • e)))
+    (hThird : ∀ x w, |iteratedDeriv 3 (fun r : ℝ => f (x + r • e)) w| ≤ C)
+    (x v : E) (w : ℝ) :
+    |f (x + w • e + v) - f x - D x e * w -
+      iteratedDeriv 2 (fun r : ℝ => f (x + r • e)) 0 * w^2/2 - D x v| ≤
+    C * (‖v‖ * (|w| + ‖v‖) + |w|^3) := by
+  have hd : deriv (fun r : ℝ => f (x + r • e)) 0 = D x e := by
+    have hi := ((hasDerivAt_id (0 : ℝ)).smul_const e).const_add x
+    have ho := (hD (x + (0 : ℝ) • e)).comp_hasDerivAt 0 hi
+    simpa only [Function.comp_def, id_eq, zero_smul, one_smul, add_zero] using ho.deriv
+  have hb := scalar_taylor_cubic (fun r : ℝ => f (x + r • e)) (hSmooth x) C
+    (hThird x) w
+  rw [hd] at hb
+  simp only [zero_smul, add_zero] at hb
+  have hv := first_order_remainder f D hD C hLip (x + w • e) v
+  have hc : |(D (x+w•e)-D x) v| ≤ C * |w| * ‖v‖ := by
+    calc
+      _ ≤ ‖D (x+w•e)-D x‖ * ‖v‖ := by
+        simpa only [Real.norm_eq_abs] using (D (x+w•e)-D x).le_opNorm v
+      _ ≤ (C * ‖(x+w•e)-x‖) * ‖v‖ := by gcongr; exact hLip x (x+w•e)
+      _ = (C * (|w| * ‖e‖)) * ‖v‖ := by simp [norm_smul, Real.norm_eq_abs]
+      _ ≤ _ := by
+        have hw := mul_le_mul_of_nonneg_left he (abs_nonneg w)
+        simpa only [mul_one] using mul_le_mul_of_nonneg_right
+          (mul_le_mul_of_nonneg_left hw hC) (norm_nonneg v)
+  have hsum := (abs_add_le (f (x+w•e+v)-f (x+w•e)-D (x+w•e) v)
+    (f (x+w•e)-f x-D x e*w-
+      iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2)).trans (add_le_add hv hb)
+  have hsum' := (abs_add_le ((f (x+w•e+v)-f (x+w•e)-D (x+w•e) v)+
+    (f (x+w•e)-f x-D x e*w-
+      iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2)) ((D (x+w•e)-D x) v)).trans
+        (add_le_add hsum hc)
+  have heq : ((f (x+w•e+v)-f (x+w•e)-D (x+w•e) v)+
+    (f (x+w•e)-f x-D x e*w-
+      iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2)) + ((D (x+w•e)-D x) v) =
+      f (x+w•e+v)-f x-D x e*w-
+        iteratedDeriv 2 (fun r : ℝ => f (x+r•e)) 0*w^2/2-D x v := by
+    simp only [sub_apply]
+    ring
+  rw [heq] at hsum'
+  calc
+    _ ≤ (C * ‖v‖^2 + C * |w|^3) + C * |w| * ‖v‖ := hsum'
+    _ = _ := by ring
+
+end ExactSLEGenerator
+
+ 
+open MeasureTheory Filter
+open scoped Topology
+namespace ExactSLEGenerator
+
+lemma riemann_sum_error_le (f : ℝ → ℝ) (hf : Continuous f) (T : ℝ) (hT : 0 ≤ T)
+    (N : ℕ) (hN : 0 < N) (ε : ℝ) (_hε : 0 ≤ ε)
+    (hmod : ∀ u ∈ Set.Icc 0 T, ∀ v ∈ Set.Icc 0 T,
+      |u-v| ≤ T / N → |f u-f v| ≤ ε) :
+    |(∑ i ∈ Finset.range N, (T/N) * f (i*(T/N))) - ∫ s in 0..T, f s| ≤ ε*T := by
+  let h := T / (N : ℝ)
+  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
+  have hh : 0 ≤ h := div_nonneg hT hNr.le
+  have hNh : (N : ℝ) * h = T := by dsimp [h]; field_simp
+  let G : ℕ → ℝ := fun i => ∫ s in 0..(i : ℝ)*h, f s
+  have hG (i : ℕ) : ∫ s in (i : ℝ)*h..((i+1 : ℕ) : ℝ)*h, f s = G (i+1)-G i := by
+    have ha := intervalIntegral.integral_add_adjacent_intervals
+      (hf.intervalIntegrable (μ := volume) 0 ((i : ℝ)*h))
+      (hf.intervalIntegrable (μ := volume) ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h))
+    dsimp [G]
+    linarith only [ha]
+  have hdiff (i : ℕ) : ((i+1 : ℕ) : ℝ)*h-(i : ℝ)*h = h := by push_cast; ring
+  have he (i : ℕ) (hi : i < N) : |h*f ((i : ℝ)*h) - (G (i+1)-G i)| ≤ ε*h := by
+    have hai : (i : ℝ) * h ∈ Set.Icc 0 T := by
+      refine ⟨mul_nonneg (Nat.cast_nonneg _) hh, ?_⟩
+      rw [← hNh]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi.le) hh
+    have hbi : ((i+1 : ℕ) : ℝ) * h ∈ Set.Icc 0 T := by
+      refine ⟨by positivity, ?_⟩
+      rw [← hNh]
+      exact mul_le_mul_of_nonneg_right (by exact_mod_cast hi) hh
+    have hab : (i : ℝ)*h ≤ ((i+1 : ℕ) : ℝ)*h := by linarith [hdiff i]
+    have hbound : ∀ s ∈ Set.uIoc ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h),
+        ‖f ((i : ℝ)*h)-f s‖ ≤ ε := by
+      intro s hs
+      rw [Set.uIoc_of_le hab] at hs
+      rw [Real.norm_eq_abs]
+      apply hmod _ hai _ ⟨hai.1.trans hs.1.le, hs.2.trans hbi.2⟩
+      rw [abs_of_nonpos (by linarith [hs.1])]
+      linarith [hs.2, hdiff i]
+    have hb := intervalIntegral.norm_integral_le_of_norm_le_const hbound
+    have hc : IntervalIntegrable (fun _ : ℝ => f ((i : ℝ)*h)) volume
+        ((i : ℝ)*h) (((i+1 : ℕ) : ℝ)*h) := intervalIntegrable_const
+    rw [intervalIntegral.integral_sub hc (hf.intervalIntegrable _ _),
+      intervalIntegral.integral_const, smul_eq_mul, hdiff, hG, Real.norm_eq_abs,
+      abs_of_nonneg hh] at hb
+    exact hb
+  have hs : ∑ i ∈ Finset.range N, (G (i+1)-G i) = ∫ s in 0..T, f s := by
+    rw [Finset.sum_range_sub]
+    simp [G, hNh]
+  change |(∑ i ∈ Finset.range N, h*f ((i : ℝ)*h)) - _| ≤ _
+  rw [← hs, ← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑ i ∈ Finset.range N, |h*f ((i : ℝ)*h)-(G (i+1)-G i)| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _i ∈ Finset.range N, ε*h := Finset.sum_le_sum fun i hi => he i (Finset.mem_range.mp hi)
+    _ = ε*T := by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; rw [← hNh]; ring
+
+end ExactSLEGenerator
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StoppedEstimates.lean b/lean/OAI/Probability/SLEGauge/StoppedEstimates.lean
index 78828bcd5a44e1fcc8b8ec32c5547c327c791bd8..28bcbbf5cbbf0f2316ad2fb33556ec74042bc4df 100644
--- a/lean/OAI/Probability/SLEGauge/StoppedEstimates.lean
+++ b/lean/OAI/Probability/SLEGauge/StoppedEstimates.lean
@@ -1,469 +1,469 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.EulerMoments
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
-theorem expected_stopped_local {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
-    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
-    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hg0 : ∀ b, g b 0 = z)
-    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
-    (τ : (ℝ≥0 → ℝ) → ℝ) (mτ : Measurable τ)
-    (hτpast : ∀ (s : ℝ≥0) b c, (∀ u ≤ s, b u = c u) → ((s:ℝ) < τ b ↔ (s:ℝ) < τ c))
-    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
-    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
-    (hgen : ∀ q ∈ K,
-      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
-        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 ≤ 0)
-    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
-    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) ≤ f (z,0) := by
-  obtain ⟨G,hG,hGc,hGeq⟩ := Localization.compact_extension hK hO hKO f hf
-  obtain ⟨A,hA⟩ := exists_smoothTest_of_compact G hG hGc
-  have heq (q : E × ℝ) (hq : q ∈ K) : A.F q.1 q.2 = f q := by
-    rw [hA]
-    exact (hGeq q hq).self_of_nhds
-  have hDgen (q : E × ℝ) (hq : q ∈ K) :
-      A.D q.1 q.2 (v q.1 q.2)+deriv (deriv (A.F q.1)) q.2/2 ≤ 0 := by
-    have hx : (fun x => A.F x q.2) =ᶠ[𝓝 q.1] fun x => f (x,q.2) := by
-      rw [hA]
-      exact (hGeq q hq).comp_tendsto (continuousAt_id.prodMk continuousAt_const)
-    have hb : A.F q.1 =ᶠ[𝓝 q.2] fun b => f (q.1,b) := by
-      rw [hA]
-      exact (hGeq q hq).comp_tendsto (continuousAt_const.prodMk continuousAt_id)
-    rw [← (A.hD q.1 q.2).fderiv, hx.fderiv_eq, hb.deriv.deriv_eq]
-    exact hgen q hq
-  have hineq := expected_stopped_smooth hW A v mv g z hL hM hv hlip hg0 hder hcf
-    τ mτ hτpast hτ K hK.measurableSet hD hDgen ht
-  rw [heq (z,0) hz] at hineq
-  convert hineq using 1
-  apply integral_congr_ae
-  filter_upwards [] with b
-  symm
-  apply heq (g b (min t (τ b)), b (min t (τ b)).toNNReal)
-  obtain ⟨ε,hε,hDb⟩ := hD b
-  exact hDb (min t (τ b)) ⟨le_min ht (hτ b).1,min_le_right _ _⟩ _ (by simpa using hε)
-
-end SLEExactGauge.BrownianCalculus
-
- 
-
-noncomputable section
-namespace SLEExactGauge
-
- 
-structure Parameter where
-  κ : ℝ
-  κ_pos : 0 < κ
-  κ_lt_eight : κ < 8
-
-namespace Parameter
-
-def a (P : Parameter) : ℝ := 2 / P.κ
-def d (P : Parameter) : ℝ := 1 + 1 / (4 * P.a)
-def α (P : Parameter) : ℝ := 2 - P.d
-def p (P : Parameter) : ℝ := 4 * P.a - 1
- 
-def q (P : Parameter) : ℝ := (1 / 2 : ℝ) * min 1 P.p
-def η (P : Parameter) : ℝ := (1 - P.q) / (2 - P.q)
-def M (P : Parameter) : ℝ := (1 + P.η) / (1 - 2 * P.η)
-def β (P : Parameter) : ℝ := min 1 (P.a / (4 * P.M))
-def k (P : Parameter) : ℝ := P.q * (P.p - P.q) / 2
-def A (P : Parameter) : ℝ := P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 + 2
-def h₀ (P : Parameter) : ℝ := (P.k / (2 * P.A)) ^ (1 / (2 - P.q))
-def C₁ (P : Parameter) : ℝ := P.A * P.h₀ ^ (-P.p)
-def c₁ (P : Parameter) : ℝ := P.k / 2
-
-def δ (P : Parameter) (U u : ℝ) : ℝ := Real.exp (-P.β * min u (U - u))
- 
-def f (P : Parameter) (δ s : ℝ) : ℝ := s ^ P.p + δ * (1 + s ^ P.q)
-def D (P : Parameter) (δ s : ℝ) : ℝ := δ / P.f δ s * s ^ (P.q - 2)
- 
-def drift (P : Parameter) (δ s σ : ℝ) : ℝ :=
-  δ / P.f δ s * (P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 * s ^ P.q -
-    P.k * s ^ (P.q - 2) + σ * (1 + s ^ P.q))
-
-lemma a_gt_quarter (P : Parameter) : (1 / 4 : ℝ) < P.a := by
-  unfold a
-  apply (lt_div_iff₀ P.κ_pos).2
-  linarith [P.κ_lt_eight]
-
-lemma p_pos (P : Parameter) : 0 < P.p := by
-  unfold p
-  linarith [P.a_gt_quarter]
-
-lemma q_pos (P : Parameter) : 0 < P.q := by
-  unfold q
-  exact mul_pos (by norm_num) (lt_min (by norm_num) P.p_pos)
-
-lemma q_le_half (P : Parameter) : P.q ≤ 1 / 2 := by
-  unfold q
-  linarith [min_le_left (1 : ℝ) P.p]
-
-lemma q_lt_one (P : Parameter) : P.q < 1 := by
-  linarith [P.q_le_half]
-
-lemma q_lt_p (P : Parameter) : P.q < P.p := by
-  unfold q
-  linarith [min_le_right (1 : ℝ) P.p, P.p_pos]
-
-lemma k_pos (P : Parameter) : 0 < P.k := by
-  unfold k
-  exact div_pos (mul_pos P.q_pos (sub_pos.mpr P.q_lt_p)) (by norm_num)
-
-lemma A_pos (P : Parameter) : 0 < P.A := by
-  unfold A
-  have := mul_pos (sub_pos.mpr P.q_lt_p) (by linarith [P.q_pos] : 0 < 1 + P.q)
-  linarith [P.p_pos]
-
-lemma β_le_one (P : Parameter) : P.β ≤ 1 := min_le_left _ _
-
-lemma h₀_pos (P : Parameter) : 0 < P.h₀ := by
-  unfold h₀
-  exact Real.rpow_pos_of_pos (div_pos P.k_pos (by linarith [P.A_pos])) _
-
-lemma h₀_lt_one (P : Parameter) : P.h₀ < 1 := by
-  have hbase : P.k / (2 * P.A) < 1 := by
-    apply (div_lt_one (by linarith [P.A_pos])).2
-    unfold k A
-    have hprod := mul_pos (sub_pos.mpr P.q_lt_p)
-      (by linarith [P.q_pos] : 0 < 2 + P.q)
-    nlinarith [P.p_pos]
-  exact Real.rpow_lt_one (div_nonneg P.k_pos.le (by linarith [P.A_pos])) hbase
-    (one_div_pos.mpr (by linarith [P.q_lt_one]))
-
-lemma h₀_power (P : Parameter) : P.h₀ ^ (P.q - 2) = 2 * P.A / P.k := by
-  have hb : 0 ≤ P.k / (2 * P.A) :=
-    div_nonneg P.k_pos.le (by linarith [P.A_pos])
-  have hq : 2 - P.q ≠ 0 := by linarith [P.q_lt_one]
-  have hexp : (1 / (2 - P.q)) * (P.q - 2) = (-1 : ℝ) := by
-    field_simp
-    ring
-  unfold h₀
-  rw [← Real.rpow_mul hb, hexp, Real.rpow_neg_one]
-  simp only [inv_div]
-
-lemma f_pos (P : Parameter) {δ s : ℝ} (hδ : 0 < δ) (hs : 0 < s) :
-    0 < P.f δ s := by
-  unfold f
-  positivity
-
- 
-def RegularizedDriftObligation : Prop :=
-  ∀ (P : Parameter) (δ s σ : ℝ), 0 < δ → δ ≤ 1 → 0 < s → s ≤ 1 →
-    |σ| ≤ P.β → P.drift δ s σ ≤ P.C₁ * δ - P.c₁ * P.D δ s
-
- 
-theorem regularized_drift_bound : RegularizedDriftObligation := by
-  intro P δ s σ hδ _ hs hs_one hσ
-  have hf : 0 < P.f δ s := P.f_pos hδ hs
-  have hδf : 0 ≤ δ / P.f δ s := (div_pos hδ hf).le
-  have hsq : s ^ P.q ≤ 1 := Real.rpow_le_one hs.le hs_one P.q_pos.le
-  have hsq0 : 0 ≤ s ^ P.q := Real.rpow_nonneg hs.le _
-  have hσ_one : σ ≤ 1 := (le_abs_self σ).trans (hσ.trans P.β_le_one)
-  have hσterm : σ * (1 + s ^ P.q) ≤ 2 := by
-    calc
-      _ ≤ 1 * (1 + s ^ P.q) :=
-        mul_le_mul_of_nonneg_right hσ_one (by positivity)
-      _ ≤ 2 := by linarith
-  have hcoef : 0 ≤ (P.p - P.q) * (1 + P.q) / 2 := by
-    have := sub_pos.mpr P.q_lt_p
-    have := P.q_pos
-    positivity
-  have hbracket : P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 * s ^ P.q +
-      σ * (1 + s ^ P.q) ≤ P.A := by
-    have hmul := mul_le_mul_of_nonneg_left hsq hcoef
-    unfold A
-    linarith
-  have hrew : P.drift δ s σ + P.c₁ * P.D δ s =
-      δ / P.f δ s * (P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 * s ^ P.q +
-        σ * (1 + s ^ P.q) - P.k / 2 * s ^ (P.q - 2)) := by
-    unfold drift c₁ D
-    ring
-  have hbound : P.drift δ s σ + P.c₁ * P.D δ s ≤
-      δ / P.f δ s * (P.A - P.k / 2 * s ^ (P.q - 2)) := by
-    rw [hrew]
-    exact mul_le_mul_of_nonneg_left (sub_le_sub_right hbracket _) hδf
-  have hgoal : P.drift δ s σ + P.c₁ * P.D δ s ≤ P.C₁ * δ := by
-    rcases le_total s P.h₀ with hsmall | hlarge
-    · have hexp : P.q - 2 ≤ 0 := by linarith [P.q_lt_one]
-      have hpow := Real.rpow_le_rpow_of_nonpos hs hsmall hexp
-      rw [P.h₀_power] at hpow
-      have hcancel : 2 * P.A ≤ s ^ (P.q - 2) * P.k :=
-        (div_le_iff₀ P.k_pos).mp hpow
-      have hneg : P.A - P.k / 2 * s ^ (P.q - 2) ≤ 0 := by nlinarith
-      have hzero := mul_nonpos_of_nonneg_of_nonpos hδf hneg
-      have hc : 0 ≤ P.C₁ * δ := by
-        unfold C₁
-        have := P.A_pos
-        have := P.h₀_pos
-        positivity
-      exact hbound.trans (hzero.trans hc)
-    · have hsp : P.h₀ ^ P.p ≤ s ^ P.p :=
-        Real.rpow_le_rpow P.h₀_pos.le hlarge P.p_pos.le
-      have hflow : P.h₀ ^ P.p ≤ P.f δ s := by
-        unfold f
-        have hterm : 0 ≤ δ * (1 + s ^ P.q) := by positivity
-        linarith
-      have hdiv : δ / P.f δ s ≤ δ / P.h₀ ^ P.p :=
-        div_le_div_of_nonneg_left hδ.le (Real.rpow_pos_of_pos P.h₀_pos _) hflow
-      have hdrop : P.A - P.k / 2 * s ^ (P.q - 2) ≤ P.A := by
-        have := P.k_pos
-        have := Real.rpow_pos_of_pos hs (P.q - 2)
-        nlinarith
-      calc
-        _ ≤ δ / P.f δ s * (P.A - P.k / 2 * s ^ (P.q - 2)) := hbound
-        _ ≤ δ / P.f δ s * P.A := mul_le_mul_of_nonneg_left hdrop hδf
-        _ ≤ δ / P.h₀ ^ P.p * P.A := mul_le_mul_of_nonneg_right hdiv P.A_pos.le
-        _ = P.C₁ * δ := by
-          unfold C₁
-          rw [Real.rpow_neg P.h₀_pos.le]
-          ring
-  linarith
-
- 
-lemma η_pos (P : Parameter) : 0 < P.η := by
-  unfold η
-  exact div_pos (by linarith [P.q_lt_one]) (by linarith [P.q_lt_one])
-
-lemma η_lt_half (P : Parameter) : P.η < 1 / 2 := by
-  unfold η
-  apply (div_lt_iff₀ (by linarith [P.q_lt_one] : 0 < 2 - P.q)).2
-  linarith [P.q_pos]
-
-lemma η_exponent (P : Parameter) : (P.q - 2) * P.η = P.q - 1 := by
-  unfold η
-  have hq : 2 - P.q ≠ 0 := by linarith [P.q_lt_one]
-  field_simp
-  ring
-
-lemma M_pos (P : Parameter) : 0 < P.M := by
-  unfold M
-  exact div_pos (by linarith [P.η_pos]) (by linarith [P.η_lt_half])
-
-lemma M_formula (P : Parameter) : P.M = 3 / P.q - 2 := by
-  have hq : P.q ≠ 0 := P.q_pos.ne'
-  have hq2 : 2 - P.q ≠ 0 := by linarith [P.q_lt_one]
-  have hη : 1 - 2 * P.η ≠ 0 := by linarith [P.η_lt_half]
-  unfold M
-  apply (div_eq_iff hη).2
-  unfold η
-  field_simp
-  ring
-
-lemma β_pos (P : Parameter) : 0 < P.β := by
-  unfold β
-  exact lt_min (by norm_num) (div_pos (by linarith [P.a_gt_quarter]) (by
-    have := P.M_pos
-    positivity))
-
-lemma interaction_exponent_lt (P : Parameter) : 2 * (P.M * P.β) < P.a := by
-  have h := min_le_right (1 : ℝ) (P.a / (4 * P.M))
-  change P.β ≤ P.a / (4 * P.M) at h
-  have hM : 0 < 4 * P.M := by have := P.M_pos; positivity
-  have hb := (le_div_iff₀ hM).mp h
-  nlinarith [P.a_gt_quarter]
-
-lemma f_ge_δ (P : Parameter) {δ s : ℝ} (hδ : 0 ≤ δ) (hs : 0 ≤ s) :
-    δ ≤ P.f δ s := by
-  unfold f
-  have := Real.rpow_nonneg hs P.p
-  have := Real.rpow_nonneg hs P.q
-  nlinarith
-
-lemma f_le_three (P : Parameter) {δ s : ℝ} (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1)
-    (hs : 0 ≤ s) (hs1 : s ≤ 1) : P.f δ s ≤ 3 := by
-  have hp := Real.rpow_le_one hs hs1 P.p_pos.le
-  have hq := Real.rpow_le_one hs hs1 P.q_pos.le
-  have hnn := Real.rpow_nonneg hs P.q
-  unfold f
-  nlinarith
-
- 
-def angularDerivative (P : Parameter) (δ s c : ℝ) : ℝ :=
-  (P.p * s ^ (P.p - 1) + δ * P.q * s ^ (P.q - 1)) * c
-
-lemma angularDerivative_bound (P : Parameter) {δ s c : ℝ}
-    (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hs : 0 < s) (hs1 : s ≤ 1) (hc : |c| ≤ 1) :
-    |P.angularDerivative δ s c| ≤ (P.p + P.q) * s ^ (P.q - 1) := by
-  have hpq : s ^ (P.p - 1) ≤ s ^ (P.q - 1) :=
-    Real.rpow_le_rpow_of_exponent_ge hs hs1 (by linarith [P.q_lt_p])
-  have hp0 := Real.rpow_nonneg hs.le (P.p - 1)
-  have hq0 := Real.rpow_nonneg hs.le (P.q - 1)
-  have hcoef : 0 ≤ P.p * s ^ (P.p - 1) + δ * P.q * s ^ (P.q - 1) := by
-    have := P.p_pos
-    have := P.q_pos
-    positivity
-  unfold angularDerivative
-  rw [abs_mul, abs_of_nonneg hcoef]
-  calc
-    _ ≤ (P.p * s ^ (P.p - 1) + δ * P.q * s ^ (P.q - 1)) * 1 :=
-      mul_le_mul_of_nonneg_left hc hcoef
-    _ ≤ (P.p + P.q) * s ^ (P.q - 1) := by
-      have hfirst := mul_le_mul_of_nonneg_left hpq P.p_pos.le
-      have hsecond := mul_le_mul_of_nonneg_right hδ1 (mul_nonneg P.q_pos.le hq0)
-      nlinarith
-
- 
-theorem normalized_angularDerivative_bound (P : Parameter) {δ s c : ℝ}
-    (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hs : 0 < s) (hs1 : s ≤ 1) (hc : |c| ≤ 1) :
-    |P.angularDerivative δ s c| / P.f δ s ≤
-      (P.p + P.q) * (3 : ℝ) ^ P.η * δ ^ (-1 - P.η) * (P.D δ s) ^ P.η := by
-  have hf := P.f_pos hδ hs
-  have hfδ := P.f_ge_δ hδ.le hs.le
-  have hf3 := P.f_le_three hδ.le hδ1 hs.le hs1
-  have hθ := P.angularDerivative_bound hδ.le hδ1 hs hs1 hc
-  have hbase : |P.angularDerivative δ s c| / P.f δ s ≤
-      (P.p + P.q) / δ * s ^ (P.q - 1) := by
-    calc
-      _ ≤ ((P.p + P.q) * s ^ (P.q - 1)) / P.f δ s :=
-        div_le_div_of_nonneg_right hθ hf.le
-      _ ≤ ((P.p + P.q) * s ^ (P.q - 1)) / δ :=
-        div_le_div_of_nonneg_left (by have := P.p_pos; have := P.q_pos; positivity)
-          hδ hfδ
-      _ = (P.p + P.q) / δ * s ^ (P.q - 1) := by ring
-  have hD : (P.D δ s) ^ P.η =
-      δ ^ P.η / (P.f δ s) ^ P.η * s ^ (P.q - 1) := by
-    unfold D
-    rw [Real.mul_rpow (div_pos hδ hf).le (Real.rpow_nonneg hs.le _),
-      Real.div_rpow hδ.le hf.le, ← Real.rpow_mul hs.le, P.η_exponent]
-  have hδpow : δ ^ (-1 - P.η) * δ ^ P.η = δ⁻¹ := by
-    rw [← Real.rpow_add hδ]
-    convert Real.rpow_neg_one δ using 1
-    congr 1
-    ring
-  have hfη : (P.f δ s) ^ P.η ≤ (3 : ℝ) ^ P.η :=
-    Real.rpow_le_rpow hf.le hf3 P.η_pos.le
-  have hratio : 1 ≤ (3 : ℝ) ^ P.η / (P.f δ s) ^ P.η :=
-    (one_le_div (Real.rpow_pos_of_pos hf _)).2 hfη
-  rw [hD]
-  have hrew : (P.p + P.q) * (3 : ℝ) ^ P.η * δ ^ (-1 - P.η) *
-      (δ ^ P.η / (P.f δ s) ^ P.η * s ^ (P.q - 1)) =
-      ((P.p + P.q) / δ * s ^ (P.q - 1)) *
-        ((3 : ℝ) ^ P.η / (P.f δ s) ^ P.η) := by
-    calc
-      _ = (P.p + P.q) * s ^ (P.q - 1) * ((3 : ℝ) ^ P.η /
-        (P.f δ s) ^ P.η) * (δ ^ (-1 - P.η) * δ ^ P.η) := by ring
-      _ = _ := by rw [hδpow]; ring
-  rw [hrew]
-  exact hbase.trans (le_mul_of_one_le_right
-    (by have := P.p_pos; have := P.q_pos; positivity) hratio)
-
- 
-def angleGenerator (P : Parameter) (g : ℝ → ℝ) (θ : ℝ) : ℝ :=
-  deriv (deriv g) θ / 2 + (1 - 2 * P.a) * (Real.cos θ / Real.sin θ) * deriv g θ
-
-lemma hasDerivAt_sinePower (r θ : ℝ) (hs : 0 < Real.sin θ) :
-    HasDerivAt (fun t : ℝ => Real.sin t ^ r)
-      (r * Real.cos θ * Real.sin θ ^ (r - 1)) θ := by
-  convert! (Real.hasDerivAt_sin θ).rpow_const (Or.inl hs.ne') using 1
-  ring
-
-lemma hasDerivAt_sinePower_first (r θ : ℝ) (hs : 0 < Real.sin θ) :
-    HasDerivAt (fun t : ℝ => r * Real.cos t * Real.sin t ^ (r - 1))
-      (r * (r - 1) * Real.cos θ ^ 2 * Real.sin θ ^ (r - 2) - r * Real.sin θ ^ r) θ := by
-  have hp : Real.sin θ ^ (r - 1) * Real.sin θ = Real.sin θ ^ r := by
-    convert! (Real.rpow_add_one hs.ne' (r - 1)).symm using 1
-    congr 1
-    ring
-  convert! ((Real.hasDerivAt_cos θ).const_mul r).mul
-    ((Real.hasDerivAt_sin θ).rpow_const (p := r - 1) (Or.inl hs.ne')) using 1
-  rw [show r - 1 - 1 = r - 2 by ring]
-  rw [← hp]
-  ring
-
-lemma deriv_sinePower (r θ : ℝ) (hs : 0 < Real.sin θ) :
-    deriv (fun t : ℝ => Real.sin t ^ r) θ = r * Real.cos θ * Real.sin θ ^ (r - 1) :=
-  (hasDerivAt_sinePower r θ hs).deriv
-
-lemma secondDeriv_sinePower (r θ : ℝ) (hs : 0 < Real.sin θ) :
-    deriv (deriv (fun t : ℝ => Real.sin t ^ r)) θ =
-      r * (r - 1) * Real.cos θ ^ 2 * Real.sin θ ^ (r - 2) - r * Real.sin θ ^ r := by
-  have heq : deriv (fun t : ℝ => Real.sin t ^ r) =ᶠ[nhds θ]
-      (fun t : ℝ => r * Real.cos t * Real.sin t ^ (r - 1)) := by
-    filter_upwards [Real.continuous_sin.continuousAt.eventually (Ioi_mem_nhds hs)] with t ht
-    exact deriv_sinePower r t ht
-  exact ((hasDerivAt_sinePower_first r θ hs).congr_of_eventuallyEq heq).deriv
-
- 
-theorem sine_generator (P : Parameter) (r θ : ℝ) (hs : 0 < Real.sin θ) :
-    P.angleGenerator (fun t : ℝ => Real.sin t ^ r) θ / (Real.sin θ ^ r) =
-      -r / 2 + r * (r - P.p) / 2 * (Real.cos θ / Real.sin θ) ^ 2 := by
-  have hpow1 : Real.sin θ ^ (r - 1) = Real.sin θ ^ r / Real.sin θ := by
-    rw [Real.rpow_sub hs, Real.rpow_one]
-  have hpow2 : Real.sin θ ^ (r - 2) = Real.sin θ ^ r / Real.sin θ ^ 2 := by
-    rw [Real.rpow_sub hs, Real.rpow_two]
-  have hsp : Real.sin θ ^ r ≠ 0 := (Real.rpow_pos_of_pos hs _).ne'
-  unfold angleGenerator
-  rw [secondDeriv_sinePower r θ hs, deriv_sinePower r θ hs, hpow1, hpow2]
-  unfold p
-  field_simp
-  ring
-
-lemma hasDerivAt_factor (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
-    HasDerivAt (fun t : ℝ => P.f δ (Real.sin t))
-      (P.angularDerivative δ (Real.sin θ) (Real.cos θ)) θ := by
-  convert! (hasDerivAt_sinePower P.p θ hs).add
-    (((hasDerivAt_sinePower P.q θ hs).const_add 1).const_mul δ) using 1
-  unfold angularDerivative
-  ring
-
- 
-def C₂ (P : Parameter) : ℝ := ((P.p + P.q) * (3 : ℝ) ^ P.η) ^ 2
-
- 
-def youngConstant (P : Parameter) (C ε : ℝ) : ℝ :=
-  (1 - 2 * P.η) * C ^ (1 / (1 - 2 * P.η)) *
-    (P.η / ε) ^ (2 * P.η / (1 - 2 * P.η))
-
-lemma C₂_pos (P : Parameter) : 0 < P.C₂ := by
-  unfold C₂
-  have := P.p_pos
-  have := P.q_pos
-  positivity
-
- 
-
-end Parameter
-end SLEExactGauge
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.EulerMoments
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+theorem expected_stopped_local {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
+    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
+    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hg0 : ∀ b, g b 0 = z)
+    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
+    (τ : (ℝ≥0 → ℝ) → ℝ) (mτ : Measurable τ)
+    (hτpast : ∀ (s : ℝ≥0) b c, (∀ u ≤ s, b u = c u) → ((s:ℝ) < τ b ↔ (s:ℝ) < τ c))
+    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
+    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
+    (hgen : ∀ q ∈ K,
+      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
+        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 ≤ 0)
+    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
+    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) ≤ f (z,0) := by
+  obtain ⟨G,hG,hGc,hGeq⟩ := Localization.compact_extension hK hO hKO f hf
+  obtain ⟨A,hA⟩ := exists_smoothTest_of_compact G hG hGc
+  have heq (q : E × ℝ) (hq : q ∈ K) : A.F q.1 q.2 = f q := by
+    rw [hA]
+    exact (hGeq q hq).self_of_nhds
+  have hDgen (q : E × ℝ) (hq : q ∈ K) :
+      A.D q.1 q.2 (v q.1 q.2)+deriv (deriv (A.F q.1)) q.2/2 ≤ 0 := by
+    have hx : (fun x => A.F x q.2) =ᶠ[𝓝 q.1] fun x => f (x,q.2) := by
+      rw [hA]
+      exact (hGeq q hq).comp_tendsto (continuousAt_id.prodMk continuousAt_const)
+    have hb : A.F q.1 =ᶠ[𝓝 q.2] fun b => f (q.1,b) := by
+      rw [hA]
+      exact (hGeq q hq).comp_tendsto (continuousAt_const.prodMk continuousAt_id)
+    rw [← (A.hD q.1 q.2).fderiv, hx.fderiv_eq, hb.deriv.deriv_eq]
+    exact hgen q hq
+  have hineq := expected_stopped_smooth hW A v mv g z hL hM hv hlip hg0 hder hcf
+    τ mτ hτpast hτ K hK.measurableSet hD hDgen ht
+  rw [heq (z,0) hz] at hineq
+  convert hineq using 1
+  apply integral_congr_ae
+  filter_upwards [] with b
+  symm
+  apply heq (g b (min t (τ b)), b (min t (τ b)).toNNReal)
+  obtain ⟨ε,hε,hDb⟩ := hD b
+  exact hDb (min t (τ b)) ⟨le_min ht (hτ b).1,min_le_right _ _⟩ _ (by simpa using hε)
+
+end SLEExactGauge.BrownianCalculus
+
+ 
+
+noncomputable section
+namespace SLEExactGauge
+
+ 
+structure Parameter where
+  κ : ℝ
+  κ_pos : 0 < κ
+  κ_lt_eight : κ < 8
+
+namespace Parameter
+
+def a (P : Parameter) : ℝ := 2 / P.κ
+def d (P : Parameter) : ℝ := 1 + 1 / (4 * P.a)
+def α (P : Parameter) : ℝ := 2 - P.d
+def p (P : Parameter) : ℝ := 4 * P.a - 1
+ 
+def q (P : Parameter) : ℝ := (1 / 2 : ℝ) * min 1 P.p
+def η (P : Parameter) : ℝ := (1 - P.q) / (2 - P.q)
+def M (P : Parameter) : ℝ := (1 + P.η) / (1 - 2 * P.η)
+def β (P : Parameter) : ℝ := min 1 (P.a / (4 * P.M))
+def k (P : Parameter) : ℝ := P.q * (P.p - P.q) / 2
+def A (P : Parameter) : ℝ := P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 + 2
+def h₀ (P : Parameter) : ℝ := (P.k / (2 * P.A)) ^ (1 / (2 - P.q))
+def C₁ (P : Parameter) : ℝ := P.A * P.h₀ ^ (-P.p)
+def c₁ (P : Parameter) : ℝ := P.k / 2
+
+def δ (P : Parameter) (U u : ℝ) : ℝ := Real.exp (-P.β * min u (U - u))
+ 
+def f (P : Parameter) (δ s : ℝ) : ℝ := s ^ P.p + δ * (1 + s ^ P.q)
+def D (P : Parameter) (δ s : ℝ) : ℝ := δ / P.f δ s * s ^ (P.q - 2)
+ 
+def drift (P : Parameter) (δ s σ : ℝ) : ℝ :=
+  δ / P.f δ s * (P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 * s ^ P.q -
+    P.k * s ^ (P.q - 2) + σ * (1 + s ^ P.q))
+
+lemma a_gt_quarter (P : Parameter) : (1 / 4 : ℝ) < P.a := by
+  unfold a
+  apply (lt_div_iff₀ P.κ_pos).2
+  linarith [P.κ_lt_eight]
+
+lemma p_pos (P : Parameter) : 0 < P.p := by
+  unfold p
+  linarith [P.a_gt_quarter]
+
+lemma q_pos (P : Parameter) : 0 < P.q := by
+  unfold q
+  exact mul_pos (by norm_num) (lt_min (by norm_num) P.p_pos)
+
+lemma q_le_half (P : Parameter) : P.q ≤ 1 / 2 := by
+  unfold q
+  linarith [min_le_left (1 : ℝ) P.p]
+
+lemma q_lt_one (P : Parameter) : P.q < 1 := by
+  linarith [P.q_le_half]
+
+lemma q_lt_p (P : Parameter) : P.q < P.p := by
+  unfold q
+  linarith [min_le_right (1 : ℝ) P.p, P.p_pos]
+
+lemma k_pos (P : Parameter) : 0 < P.k := by
+  unfold k
+  exact div_pos (mul_pos P.q_pos (sub_pos.mpr P.q_lt_p)) (by norm_num)
+
+lemma A_pos (P : Parameter) : 0 < P.A := by
+  unfold A
+  have := mul_pos (sub_pos.mpr P.q_lt_p) (by linarith [P.q_pos] : 0 < 1 + P.q)
+  linarith [P.p_pos]
+
+lemma β_le_one (P : Parameter) : P.β ≤ 1 := min_le_left _ _
+
+lemma h₀_pos (P : Parameter) : 0 < P.h₀ := by
+  unfold h₀
+  exact Real.rpow_pos_of_pos (div_pos P.k_pos (by linarith [P.A_pos])) _
+
+lemma h₀_lt_one (P : Parameter) : P.h₀ < 1 := by
+  have hbase : P.k / (2 * P.A) < 1 := by
+    apply (div_lt_one (by linarith [P.A_pos])).2
+    unfold k A
+    have hprod := mul_pos (sub_pos.mpr P.q_lt_p)
+      (by linarith [P.q_pos] : 0 < 2 + P.q)
+    nlinarith [P.p_pos]
+  exact Real.rpow_lt_one (div_nonneg P.k_pos.le (by linarith [P.A_pos])) hbase
+    (one_div_pos.mpr (by linarith [P.q_lt_one]))
+
+lemma h₀_power (P : Parameter) : P.h₀ ^ (P.q - 2) = 2 * P.A / P.k := by
+  have hb : 0 ≤ P.k / (2 * P.A) :=
+    div_nonneg P.k_pos.le (by linarith [P.A_pos])
+  have hq : 2 - P.q ≠ 0 := by linarith [P.q_lt_one]
+  have hexp : (1 / (2 - P.q)) * (P.q - 2) = (-1 : ℝ) := by
+    field_simp
+    ring
+  unfold h₀
+  rw [← Real.rpow_mul hb, hexp, Real.rpow_neg_one]
+  simp only [inv_div]
+
+lemma f_pos (P : Parameter) {δ s : ℝ} (hδ : 0 < δ) (hs : 0 < s) :
+    0 < P.f δ s := by
+  unfold f
+  positivity
+
+ 
+def RegularizedDriftObligation : Prop :=
+  ∀ (P : Parameter) (δ s σ : ℝ), 0 < δ → δ ≤ 1 → 0 < s → s ≤ 1 →
+    |σ| ≤ P.β → P.drift δ s σ ≤ P.C₁ * δ - P.c₁ * P.D δ s
+
+ 
+theorem regularized_drift_bound : RegularizedDriftObligation := by
+  intro P δ s σ hδ _ hs hs_one hσ
+  have hf : 0 < P.f δ s := P.f_pos hδ hs
+  have hδf : 0 ≤ δ / P.f δ s := (div_pos hδ hf).le
+  have hsq : s ^ P.q ≤ 1 := Real.rpow_le_one hs.le hs_one P.q_pos.le
+  have hsq0 : 0 ≤ s ^ P.q := Real.rpow_nonneg hs.le _
+  have hσ_one : σ ≤ 1 := (le_abs_self σ).trans (hσ.trans P.β_le_one)
+  have hσterm : σ * (1 + s ^ P.q) ≤ 2 := by
+    calc
+      _ ≤ 1 * (1 + s ^ P.q) :=
+        mul_le_mul_of_nonneg_right hσ_one (by positivity)
+      _ ≤ 2 := by linarith
+  have hcoef : 0 ≤ (P.p - P.q) * (1 + P.q) / 2 := by
+    have := sub_pos.mpr P.q_lt_p
+    have := P.q_pos
+    positivity
+  have hbracket : P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 * s ^ P.q +
+      σ * (1 + s ^ P.q) ≤ P.A := by
+    have hmul := mul_le_mul_of_nonneg_left hsq hcoef
+    unfold A
+    linarith
+  have hrew : P.drift δ s σ + P.c₁ * P.D δ s =
+      δ / P.f δ s * (P.p / 2 + (P.p - P.q) * (1 + P.q) / 2 * s ^ P.q +
+        σ * (1 + s ^ P.q) - P.k / 2 * s ^ (P.q - 2)) := by
+    unfold drift c₁ D
+    ring
+  have hbound : P.drift δ s σ + P.c₁ * P.D δ s ≤
+      δ / P.f δ s * (P.A - P.k / 2 * s ^ (P.q - 2)) := by
+    rw [hrew]
+    exact mul_le_mul_of_nonneg_left (sub_le_sub_right hbracket _) hδf
+  have hgoal : P.drift δ s σ + P.c₁ * P.D δ s ≤ P.C₁ * δ := by
+    rcases le_total s P.h₀ with hsmall | hlarge
+    · have hexp : P.q - 2 ≤ 0 := by linarith [P.q_lt_one]
+      have hpow := Real.rpow_le_rpow_of_nonpos hs hsmall hexp
+      rw [P.h₀_power] at hpow
+      have hcancel : 2 * P.A ≤ s ^ (P.q - 2) * P.k :=
+        (div_le_iff₀ P.k_pos).mp hpow
+      have hneg : P.A - P.k / 2 * s ^ (P.q - 2) ≤ 0 := by nlinarith
+      have hzero := mul_nonpos_of_nonneg_of_nonpos hδf hneg
+      have hc : 0 ≤ P.C₁ * δ := by
+        unfold C₁
+        have := P.A_pos
+        have := P.h₀_pos
+        positivity
+      exact hbound.trans (hzero.trans hc)
+    · have hsp : P.h₀ ^ P.p ≤ s ^ P.p :=
+        Real.rpow_le_rpow P.h₀_pos.le hlarge P.p_pos.le
+      have hflow : P.h₀ ^ P.p ≤ P.f δ s := by
+        unfold f
+        have hterm : 0 ≤ δ * (1 + s ^ P.q) := by positivity
+        linarith
+      have hdiv : δ / P.f δ s ≤ δ / P.h₀ ^ P.p :=
+        div_le_div_of_nonneg_left hδ.le (Real.rpow_pos_of_pos P.h₀_pos _) hflow
+      have hdrop : P.A - P.k / 2 * s ^ (P.q - 2) ≤ P.A := by
+        have := P.k_pos
+        have := Real.rpow_pos_of_pos hs (P.q - 2)
+        nlinarith
+      calc
+        _ ≤ δ / P.f δ s * (P.A - P.k / 2 * s ^ (P.q - 2)) := hbound
+        _ ≤ δ / P.f δ s * P.A := mul_le_mul_of_nonneg_left hdrop hδf
+        _ ≤ δ / P.h₀ ^ P.p * P.A := mul_le_mul_of_nonneg_right hdiv P.A_pos.le
+        _ = P.C₁ * δ := by
+          unfold C₁
+          rw [Real.rpow_neg P.h₀_pos.le]
+          ring
+  linarith
+
+ 
+lemma η_pos (P : Parameter) : 0 < P.η := by
+  unfold η
+  exact div_pos (by linarith [P.q_lt_one]) (by linarith [P.q_lt_one])
+
+lemma η_lt_half (P : Parameter) : P.η < 1 / 2 := by
+  unfold η
+  apply (div_lt_iff₀ (by linarith [P.q_lt_one] : 0 < 2 - P.q)).2
+  linarith [P.q_pos]
+
+lemma η_exponent (P : Parameter) : (P.q - 2) * P.η = P.q - 1 := by
+  unfold η
+  have hq : 2 - P.q ≠ 0 := by linarith [P.q_lt_one]
+  field_simp
+  ring
+
+lemma M_pos (P : Parameter) : 0 < P.M := by
+  unfold M
+  exact div_pos (by linarith [P.η_pos]) (by linarith [P.η_lt_half])
+
+lemma M_formula (P : Parameter) : P.M = 3 / P.q - 2 := by
+  have hq : P.q ≠ 0 := P.q_pos.ne'
+  have hq2 : 2 - P.q ≠ 0 := by linarith [P.q_lt_one]
+  have hη : 1 - 2 * P.η ≠ 0 := by linarith [P.η_lt_half]
+  unfold M
+  apply (div_eq_iff hη).2
+  unfold η
+  field_simp
+  ring
+
+lemma β_pos (P : Parameter) : 0 < P.β := by
+  unfold β
+  exact lt_min (by norm_num) (div_pos (by linarith [P.a_gt_quarter]) (by
+    have := P.M_pos
+    positivity))
+
+lemma interaction_exponent_lt (P : Parameter) : 2 * (P.M * P.β) < P.a := by
+  have h := min_le_right (1 : ℝ) (P.a / (4 * P.M))
+  change P.β ≤ P.a / (4 * P.M) at h
+  have hM : 0 < 4 * P.M := by have := P.M_pos; positivity
+  have hb := (le_div_iff₀ hM).mp h
+  nlinarith [P.a_gt_quarter]
+
+lemma f_ge_δ (P : Parameter) {δ s : ℝ} (hδ : 0 ≤ δ) (hs : 0 ≤ s) :
+    δ ≤ P.f δ s := by
+  unfold f
+  have := Real.rpow_nonneg hs P.p
+  have := Real.rpow_nonneg hs P.q
+  nlinarith
+
+lemma f_le_three (P : Parameter) {δ s : ℝ} (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ 1) : P.f δ s ≤ 3 := by
+  have hp := Real.rpow_le_one hs hs1 P.p_pos.le
+  have hq := Real.rpow_le_one hs hs1 P.q_pos.le
+  have hnn := Real.rpow_nonneg hs P.q
+  unfold f
+  nlinarith
+
+ 
+def angularDerivative (P : Parameter) (δ s c : ℝ) : ℝ :=
+  (P.p * s ^ (P.p - 1) + δ * P.q * s ^ (P.q - 1)) * c
+
+lemma angularDerivative_bound (P : Parameter) {δ s c : ℝ}
+    (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hs : 0 < s) (hs1 : s ≤ 1) (hc : |c| ≤ 1) :
+    |P.angularDerivative δ s c| ≤ (P.p + P.q) * s ^ (P.q - 1) := by
+  have hpq : s ^ (P.p - 1) ≤ s ^ (P.q - 1) :=
+    Real.rpow_le_rpow_of_exponent_ge hs hs1 (by linarith [P.q_lt_p])
+  have hp0 := Real.rpow_nonneg hs.le (P.p - 1)
+  have hq0 := Real.rpow_nonneg hs.le (P.q - 1)
+  have hcoef : 0 ≤ P.p * s ^ (P.p - 1) + δ * P.q * s ^ (P.q - 1) := by
+    have := P.p_pos
+    have := P.q_pos
+    positivity
+  unfold angularDerivative
+  rw [abs_mul, abs_of_nonneg hcoef]
+  calc
+    _ ≤ (P.p * s ^ (P.p - 1) + δ * P.q * s ^ (P.q - 1)) * 1 :=
+      mul_le_mul_of_nonneg_left hc hcoef
+    _ ≤ (P.p + P.q) * s ^ (P.q - 1) := by
+      have hfirst := mul_le_mul_of_nonneg_left hpq P.p_pos.le
+      have hsecond := mul_le_mul_of_nonneg_right hδ1 (mul_nonneg P.q_pos.le hq0)
+      nlinarith
+
+ 
+theorem normalized_angularDerivative_bound (P : Parameter) {δ s c : ℝ}
+    (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hs : 0 < s) (hs1 : s ≤ 1) (hc : |c| ≤ 1) :
+    |P.angularDerivative δ s c| / P.f δ s ≤
+      (P.p + P.q) * (3 : ℝ) ^ P.η * δ ^ (-1 - P.η) * (P.D δ s) ^ P.η := by
+  have hf := P.f_pos hδ hs
+  have hfδ := P.f_ge_δ hδ.le hs.le
+  have hf3 := P.f_le_three hδ.le hδ1 hs.le hs1
+  have hθ := P.angularDerivative_bound hδ.le hδ1 hs hs1 hc
+  have hbase : |P.angularDerivative δ s c| / P.f δ s ≤
+      (P.p + P.q) / δ * s ^ (P.q - 1) := by
+    calc
+      _ ≤ ((P.p + P.q) * s ^ (P.q - 1)) / P.f δ s :=
+        div_le_div_of_nonneg_right hθ hf.le
+      _ ≤ ((P.p + P.q) * s ^ (P.q - 1)) / δ :=
+        div_le_div_of_nonneg_left (by have := P.p_pos; have := P.q_pos; positivity)
+          hδ hfδ
+      _ = (P.p + P.q) / δ * s ^ (P.q - 1) := by ring
+  have hD : (P.D δ s) ^ P.η =
+      δ ^ P.η / (P.f δ s) ^ P.η * s ^ (P.q - 1) := by
+    unfold D
+    rw [Real.mul_rpow (div_pos hδ hf).le (Real.rpow_nonneg hs.le _),
+      Real.div_rpow hδ.le hf.le, ← Real.rpow_mul hs.le, P.η_exponent]
+  have hδpow : δ ^ (-1 - P.η) * δ ^ P.η = δ⁻¹ := by
+    rw [← Real.rpow_add hδ]
+    convert Real.rpow_neg_one δ using 1
+    congr 1
+    ring
+  have hfη : (P.f δ s) ^ P.η ≤ (3 : ℝ) ^ P.η :=
+    Real.rpow_le_rpow hf.le hf3 P.η_pos.le
+  have hratio : 1 ≤ (3 : ℝ) ^ P.η / (P.f δ s) ^ P.η :=
+    (one_le_div (Real.rpow_pos_of_pos hf _)).2 hfη
+  rw [hD]
+  have hrew : (P.p + P.q) * (3 : ℝ) ^ P.η * δ ^ (-1 - P.η) *
+      (δ ^ P.η / (P.f δ s) ^ P.η * s ^ (P.q - 1)) =
+      ((P.p + P.q) / δ * s ^ (P.q - 1)) *
+        ((3 : ℝ) ^ P.η / (P.f δ s) ^ P.η) := by
+    calc
+      _ = (P.p + P.q) * s ^ (P.q - 1) * ((3 : ℝ) ^ P.η /
+        (P.f δ s) ^ P.η) * (δ ^ (-1 - P.η) * δ ^ P.η) := by ring
+      _ = _ := by rw [hδpow]; ring
+  rw [hrew]
+  exact hbase.trans (le_mul_of_one_le_right
+    (by have := P.p_pos; have := P.q_pos; positivity) hratio)
+
+ 
+def angleGenerator (P : Parameter) (g : ℝ → ℝ) (θ : ℝ) : ℝ :=
+  deriv (deriv g) θ / 2 + (1 - 2 * P.a) * (Real.cos θ / Real.sin θ) * deriv g θ
+
+lemma hasDerivAt_sinePower (r θ : ℝ) (hs : 0 < Real.sin θ) :
+    HasDerivAt (fun t : ℝ => Real.sin t ^ r)
+      (r * Real.cos θ * Real.sin θ ^ (r - 1)) θ := by
+  convert! (Real.hasDerivAt_sin θ).rpow_const (Or.inl hs.ne') using 1
+  ring
+
+lemma hasDerivAt_sinePower_first (r θ : ℝ) (hs : 0 < Real.sin θ) :
+    HasDerivAt (fun t : ℝ => r * Real.cos t * Real.sin t ^ (r - 1))
+      (r * (r - 1) * Real.cos θ ^ 2 * Real.sin θ ^ (r - 2) - r * Real.sin θ ^ r) θ := by
+  have hp : Real.sin θ ^ (r - 1) * Real.sin θ = Real.sin θ ^ r := by
+    convert! (Real.rpow_add_one hs.ne' (r - 1)).symm using 1
+    congr 1
+    ring
+  convert! ((Real.hasDerivAt_cos θ).const_mul r).mul
+    ((Real.hasDerivAt_sin θ).rpow_const (p := r - 1) (Or.inl hs.ne')) using 1
+  rw [show r - 1 - 1 = r - 2 by ring]
+  rw [← hp]
+  ring
+
+lemma deriv_sinePower (r θ : ℝ) (hs : 0 < Real.sin θ) :
+    deriv (fun t : ℝ => Real.sin t ^ r) θ = r * Real.cos θ * Real.sin θ ^ (r - 1) :=
+  (hasDerivAt_sinePower r θ hs).deriv
+
+lemma secondDeriv_sinePower (r θ : ℝ) (hs : 0 < Real.sin θ) :
+    deriv (deriv (fun t : ℝ => Real.sin t ^ r)) θ =
+      r * (r - 1) * Real.cos θ ^ 2 * Real.sin θ ^ (r - 2) - r * Real.sin θ ^ r := by
+  have heq : deriv (fun t : ℝ => Real.sin t ^ r) =ᶠ[nhds θ]
+      (fun t : ℝ => r * Real.cos t * Real.sin t ^ (r - 1)) := by
+    filter_upwards [Real.continuous_sin.continuousAt.eventually (Ioi_mem_nhds hs)] with t ht
+    exact deriv_sinePower r t ht
+  exact ((hasDerivAt_sinePower_first r θ hs).congr_of_eventuallyEq heq).deriv
+
+ 
+theorem sine_generator (P : Parameter) (r θ : ℝ) (hs : 0 < Real.sin θ) :
+    P.angleGenerator (fun t : ℝ => Real.sin t ^ r) θ / (Real.sin θ ^ r) =
+      -r / 2 + r * (r - P.p) / 2 * (Real.cos θ / Real.sin θ) ^ 2 := by
+  have hpow1 : Real.sin θ ^ (r - 1) = Real.sin θ ^ r / Real.sin θ := by
+    rw [Real.rpow_sub hs, Real.rpow_one]
+  have hpow2 : Real.sin θ ^ (r - 2) = Real.sin θ ^ r / Real.sin θ ^ 2 := by
+    rw [Real.rpow_sub hs, Real.rpow_two]
+  have hsp : Real.sin θ ^ r ≠ 0 := (Real.rpow_pos_of_pos hs _).ne'
+  unfold angleGenerator
+  rw [secondDeriv_sinePower r θ hs, deriv_sinePower r θ hs, hpow1, hpow2]
+  unfold p
+  field_simp
+  ring
+
+lemma hasDerivAt_factor (P : Parameter) {δ θ : ℝ} (hs : 0 < Real.sin θ) :
+    HasDerivAt (fun t : ℝ => P.f δ (Real.sin t))
+      (P.angularDerivative δ (Real.sin θ) (Real.cos θ)) θ := by
+  convert! (hasDerivAt_sinePower P.p θ hs).add
+    (((hasDerivAt_sinePower P.q θ hs).const_add 1).const_mul δ) using 1
+  unfold angularDerivative
+  ring
+
+ 
+def C₂ (P : Parameter) : ℝ := ((P.p + P.q) * (3 : ℝ) ^ P.η) ^ 2
+
+ 
+def youngConstant (P : Parameter) (C ε : ℝ) : ℝ :=
+  (1 - 2 * P.η) * C ^ (1 / (1 - 2 * P.η)) *
+    (P.η / ε) ^ (2 * P.η / (1 - 2 * P.η))
+
+lemma C₂_pos (P : Parameter) : 0 < P.C₂ := by
+  unfold C₂
+  have := P.p_pos
+  have := P.q_pos
+  positivity
+
+ 
+
+end Parameter
+end SLEExactGauge
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StoppedFiniteMass.lean b/lean/OAI/Probability/SLEGauge/StoppedFiniteMass.lean
index 26186e0852320960176d36bacbb307b31cac8c79..85d6f1032cad599b957bc5c54e9a5f462ac33694 100644
--- a/lean/OAI/Probability/SLEGauge/StoppedFiniteMass.lean
+++ b/lean/OAI/Probability/SLEGauge/StoppedFiniteMass.lean
@@ -1,483 +1,483 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TraceMassEndpoint
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma timeMassDensity_stopping_joint_measurable (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      timeMassDensity a q (S p.1) p.1 p.2.1) := by
-  exact (timeMassDensity_joint_measurable a ha q hq hq1).comp
-    (((hS.measurable'.comp measurable_fst).prodMk measurable_fst).prodMk
-      (measurable_subtype_coe.comp measurable_snd))
-
- 
-
-def stoppedMassKernel (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
-    Kernel ExactSLEGenerator.Driver UpperPlane :=
-  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
-    (fun b z => timeMassDensity a q (S b) b z.1)
-
-instance stoppedMassKernel_sfinite (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
-    IsSFiniteKernel (stoppedMassKernel a q S) :=
-  Kernel.IsSFiniteKernel.withDensity _ (fun b z => (timeMassDensity_finite a q (S b) b z.1).ne)
-
-lemma stoppedMassKernel_apply (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    stoppedMassKernel a q S b = actualUpperMass a q (S b) b :=
-  Kernel.withDensity_apply _ (timeMassDensity_stopping_joint_measurable a ha q hq hq1 S hS) b
-
-lemma actualUpperMass_measurable_stopping (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable (fun b => actualUpperMass a q (S b) b) := by
-  have he : (fun b => actualUpperMass a q (S b) b) =
-      (fun b => stoppedMassKernel a q S b) :=
-    funext (fun b => (stoppedMassKernel_apply a ha q hq hq1 S hS b).symm)
-  rw [he]
-  exact (stoppedMassKernel a q S).measurable
-
-lemma actualUpperMass_add_unhit (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    actualUpperMass a q ∞ b = actualUpperMass a q (S b) b + unhitMassKernel a q S b := by
-  rw [unhitMassKernel_apply a ha q hq hq1 S hS]
-  unfold actualUpperMass
-  have he : (fun z : UpperPlane => timeMassDensity a q ∞ b z.1) =
-      (fun z => timeMassDensity a q (S b) b z.1) +
-      (fun z => ENNReal.ofReal (unhitMassDensity a q (S b) b z)) :=
-    funext (fun z => timeMassDensity_add_unhit a q (S b) b z)
-  rw [he,withDensity_add_left]
-  exact (timeMassDensity_stopping_joint_measurable a ha q hq hq1 S hS).comp
-    (measurable_const.prodMk measurable_id)
-
- 
-
-theorem actualUpperMass_stopped_random_test (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
-    (hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
-    (∫⁻b, ∫⁻z, H (b,z) ∂actualUpperMass a q ∞ b ∂W) =
-      (∫⁻b, ∫⁻z, H (b,z) ∂actualUpperMass a q (S b) b ∂W) +
-      ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1) ∂upperArea ∂W := by
-  have hH₀ : Measurable H := hH.mono (prod_sigma_le hS.measurableSpace_le _) le_rfl
-  have hmeas : Measurable (fun b => ∫⁻z, H (b,z) ∂actualUpperMass a q (S b) b) := by
-    simpa only [stoppedMassKernel_apply a ha q hq hq1 S hS] using
-      hH₀.lintegral_kernel_prod_right' (κ := stoppedMassKernel a q S)
-  simp_rw [actualUpperMass_add_unhit a ha q hq hq1 S hS, lintegral_add_measure]
-  rw [lintegral_add_left hmeas, unhitMassKernel_random_test a ha W hW q hq hq1 S hS H hH]
-  congr 1
-  apply lintegral_congr
-  intro b
-  rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS,
-    lintegral_withDensity_eq_lintegral_mul]
-  · exact lintegral_congr (fun z => mul_comm _ _)
-  · exact ((stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).comp
-      (measurable_const.prodMk measurable_id)).ennreal_ofReal
-  · exact hH₀.comp (measurable_const.prodMk measurable_id)
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma remainingMassDensity_antitone_level (a : ℝ) (ha : 0<a) (t : ℝ≥0)
-    (b : ExactSLEGenerator.Driver) (z : UpperPlane)
-    (q r : ℝ) (hq : 0<q) (hqr : q≤r) (hr : r≤1) :
-    remainingMassDensity a r t b z.1 ≤ remainingMassDensity a q t b z.1 := by
-  have hT := radialTime_antitone a ha z.1 z.2 _ _ (mul_pos z.2 hq)
-    (mul_le_mul_of_nonneg_left hqr z.2.le) (mul_le_of_le_one_right z.2.le hr) b
-  simp only [remainingMassDensity,ite_eq_left z.2,radialRemaining]
-  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-  split_ifs with hr hq hq
-  · exact le_rfl
-  · exact False.elim (hq ⟨hr.1,hr.2.trans_le hT⟩)
-  · unfold physicalGreen
-    positivity
-  · exact le_rfl
-
-lemma stoppedRemainingDensity_antitone_level (a : ℝ) (ha : 0<a) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : UpperPlane)
-    (q r : ℝ) (hq : 0<q) (hqr : q≤r) (hr : r≤1) :
-    stoppedRemainingDensity a r t b z.1 ≤ stoppedRemainingDensity a q t b z.1 := by
-  unfold stoppedRemainingDensity
-  split_ifs
-  · exact le_rfl
-  · exact remainingMassDensity_antitone_level a ha t.toNNReal b z q r hq hqr hr
-
-lemma remainingMassDensity_eventually_eq_surviving (a : ℝ) (ha : 0<a) (t : ℝ≥0)
-    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
-    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-    ∀ᶠ n in atTop, remainingMassDensity a (q n) t b z.1 = survivingGreen a t b z.1 := by
-  by_cases ht : alive a b.1 t z.1
-  · have hR := physicalRadius_actual_pos a b z.1 t ht
-    have hqR : ∀ᶠ n in atTop, z.1.im*q n<physicalRadius (actualState a b.1 t z.1) := by
-      have hconv : Tendsto (fun n => z.1.im*q n) atTop (𝓝 0) := by simpa using hlim.const_mul z.1.im
-      exact hconv.eventually (gt_mem_nhds hR)
-    filter_upwards [hqR] with n hn
-    have hnT : (t : ℝ≥0∞)<radialTime a b.1 z.1 (z.1.im*q n) := by
-      by_contra h
-      have hTt := le_of_not_gt h
-      have hfin : radialTime a b.1 z.1 (z.1.im*q n)≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hTt
-      have hle : (radialTime a b.1 z.1 (z.1.im*q n)).toNNReal≤t := by
-        exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hTt)
-      have hh := (radialTime_attained a ha z.1 z.2 _ (mul_pos z.2 (hq n).1)
-        (mul_le_of_le_one_right z.2.le (hq n).2) b hfin).2
-      have he := physicalRadius_actual_anti a ha b z.1 _ t hle ht
-      rw [hh] at he
-      exact not_le.mpr hn he
-    simp only [remainingMassDensity,ite_eq_left z.2,radialRemaining,ht,hnT,and_self,ite_true,
-      survivingGreen]
-  · filter_upwards [] with n
-    simp only [remainingMassDensity,ite_eq_left z.2,radialRemaining,ht,false_and,ite_false,
-      survivingGreen,mul_zero]
-
-lemma stoppedRemainingDensity_eventually_eq_surviving (a : ℝ) (ha : 0<a) (t : ℝ≥0∞)
-    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
-    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-    ∀ᶠ n in atTop, stoppedRemainingDensity a (q n) t b z.1 = stoppedSurvivingGreen a t b z.1 := by
-  unfold stoppedRemainingDensity stoppedSurvivingGreen
-  by_cases ht : t=∞
-  · simp only [ht,ite_true,eventually_const]
-  · simp only [ite_eq_right ht]
-    exact remainingMassDensity_eventually_eq_surviving a ha t.toNNReal q hq hlim b z
-
- 
-
-theorem stoppedRemainingDensity_random_test_tendsto (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver)
-    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hanti : Antitone q)
-    (hlim : Tendsto q atTop (𝓝 0))
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞) (hH : Measurable H) :
-    Tendsto (fun n => ∫⁻b, ∫⁻z, H (b,z)*
-        ENNReal.ofReal (stoppedRemainingDensity a (q n) (S b) b z.1) ∂upperArea ∂W)
-      atTop (𝓝 (∫⁻b, ∫⁻z, H (b,z)*
-        ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂upperArea ∂W)) := by
-  have hm (n : ℕ) : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      H p*ENNReal.ofReal (stoppedRemainingDensity a (q n) (S p.1) p.1 p.2.1)) :=
-    hH.mul (stoppedRemainingDensity_joint_measurable a ha (q n) (hq n).1 (hq n).2 S hS).ennreal_ofReal
-  have hmono (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-      Monotone (fun n => H (b,z)*ENNReal.ofReal (stoppedRemainingDensity a (q n) (S b) b z.1)) := by
-    intro i j hij
-    exact mul_le_mul_right (ENNReal.ofReal_le_ofReal
-      (stoppedRemainingDensity_antitone_level a (by linarith) (S b) b z
-        (q j) (q i) (hq j).1 (hanti hij) (hq i).2)) _
-  apply lintegral_tendsto_of_tendsto_of_monotone
-  · intro n
-    exact (hm n).lintegral_prod_right'.aemeasurable
-  · exact Eventually.of_forall (fun b i j hij => lintegral_mono (fun z => hmono b z hij))
-  · filter_upwards [] with b
-    apply lintegral_tendsto_of_tendsto_of_monotone
-    · intro n
-      exact ((hm n).comp (measurable_const.prodMk measurable_id)).aemeasurable
-    · exact Eventually.of_forall (hmono b)
-    · filter_upwards [] with z
-      apply tendsto_const_nhds.congr'
-      filter_upwards [stoppedRemainingDensity_eventually_eq_surviving a (by linarith)
-        (S b) q hq hlim b z] with n hn
-      rw [hn]
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_stopped_remaining_spatial_L1 (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (K : Set UpperPlane) (hK : MeasurableSet K)
-    (hKfin : upperArea K<∞) (c : ℝ) (hc : 0<c) (hcK : ∀ z∈K,c≤z.1.im)
-    (f : UpperPlane → ℝ) (hf : Measurable f) (B : ℝ) (hB : 0≤B) (hfB : ∀z∈K,‖f z‖≤B)
-    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
-    Integrable (fun p : UpperPlane × ExactSLEGenerator.Driver => f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1)
-      ((upperArea.restrict K).prod W) ∧
-    Tendsto (fun n => ∫p : UpperPlane × ExactSLEGenerator.Driver,
-      ‖f p.1*stoppedRemainingDensity a (q n) (S p.2) p.2 p.1.1-f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1‖
-      ∂((upperArea.restrict K).prod W)) atTop (𝓝 0)
- := by
-  let ν := upperArea.restrict K
-  have : IsFiniteMeasure ν := ⟨by simpa [ν] using hKfin⟩
-  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1
-  let H : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*stoppedRemainingDensity a (q n) (S p.2) p.2 p.1.1
-  have hHm (n : ℕ) : Measurable (H n) :=
-    (hf.comp measurable_fst).mul ((stoppedRemainingDensity_joint_measurable a ha (q n) (hq n).1 (hq n).2 S hS).comp measurable_swap)
-  have hconv (p : UpperPlane × ExactSLEGenerator.Driver) : Tendsto (fun n => H n p) atTop (𝓝 (F p)) :=
-    (stopped_remaining_to_surviving a ha (S p.2) q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)
-  have hFm : Measurable F := measurable_of_tendsto_metrizable hHm (tendsto_pi_nhds.mpr hconv)
-  have hi (z : UpperPlane) : Integrable (fun b => F (z,b)) W :=
-    (survivingGreen_optional_bound a ha W hW z S hS).1.const_mul (f z)
-  have he (z : UpperPlane) : (∫b,‖F (z,b)‖∂W) = ‖f z‖*(∫b,stoppedSurvivingGreen a (S b) b z.1∂W) := by
-    simp only [F,norm_mul,Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg _ _ _ _),integral_const_mul]
-  let C := Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log c)
-  have hC : 0≤C := by dsimp [C]; positivity
-  have hb (z : UpperPlane) (hz : z∈K) : (∫b,‖F (z,b)‖∂W)≤B*C := by
-    rw [he]
-    have hG := (survivingGreen_optional_bound a ha W hW z S hS).2
-    have hgreen : physicalGreen a (z.1,0)≤Real.exp (-alpha a*Real.log c) := by
-      apply physicalGreen_bound a c ha hc _ z.2
-      simpa [physicalRadius,Y] using hcK z hz
-    have hpos : 0≤(∫b,stoppedSurvivingGreen a (S b) b z.1∂W) := integral_nonneg (fun b => stoppedSurvivingGreen_nonneg _ _ _ _)
-    exact mul_le_mul (hfB z hz) (hG.trans (mul_le_mul_of_nonneg_left hgreen (Real.exp_pos _).le))
-      hpos hB
-  have hFi : Integrable F (ν.prod W) := by
-    apply (integrable_prod_iff hFm.aestronglyMeasurable).mpr
-    refine ⟨Eventually.of_forall hi,?_⟩
-    apply Integrable.of_bound (hFm.aestronglyMeasurable.norm.integral_prod_right') (B*C)
-    filter_upwards [ae_restrict_mem hK] with z hz
-    rw [Real.norm_of_nonneg (integral_nonneg (fun b => norm_nonneg (F (z,b))))]
-    exact hb z hz
-  refine ⟨hFi,?_⟩
-  have hD : Integrable (fun p => 2*‖F p‖) (ν.prod W) := hFi.norm.const_mul 2
-  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖H n p-F p‖≤2*‖F p‖ := by
-    have hle : ‖H n p‖≤‖F p‖ := by
-      dsimp only [H,F]
-      rw [norm_mul,norm_mul,Real.norm_of_nonneg (stoppedRemainingDensity_nonneg _ _ _ _ _),
-        Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg _ _ _ _)]
-      exact mul_le_mul_of_nonneg_left (stoppedRemainingDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
-    calc ‖H n p-F p‖≤‖H n p‖+‖F p‖ := norm_sub_le _ _
-      _≤2*‖F p‖ := by linarith
-  have hlim' := tendsto_integral_of_dominated_convergence (fun p => 2*‖F p‖)
-    (fun n => ((hHm n).sub hFm).norm.aestronglyMeasurable) hD
-    (fun n => Eventually.of_forall (fun p => by simpa using hbound n p))
-    (Eventually.of_forall (fun p => by simpa using ((hconv p).sub_const (F p)).norm))
-  simpa using hlim'
-
-lemma actualUpper_stopped_remaining_integral_tendsto (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
-    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
-    Tendsto (fun n => ∫b in U, (∫z : UpperPlane,f z*stoppedRemainingDensity a (q n) (S b) b z.1∂upperArea) ∂W)
-      atTop (𝓝 (∫b in U, (∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea) ∂W)) := by
-  let K := tsupport f
-  have hK : IsCompact K := f.hasCompactSupport
-  have hfin : upperArea K<∞ := by
-    rw [upperArea,upper_embedding.comap_apply]
-    exact (hK.image continuous_subtype_val).measure_lt_top
-  have : IsFiniteMeasure (upperArea.restrict K) := ⟨by simpa using hfin⟩
-  obtain ⟨c,hc,hcK⟩ := upper_compact_lower K hK
-  obtain ⟨B,hB,hBf⟩ := upperExtension_bounded f
-  have H := actual_stopped_remaining_spatial_L1 a ha W hW S hS K hK.measurableSet hfin (c/2) (by positivity)
-    (fun z hz => by have := hcK z hz; linarith) f f.continuous.measurable B hB
-    (fun z hz => by simpa only [upperExtension_apply] using hBf z.1 ⟨z,hz,rfl⟩) q hq hlim
-  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1
-  let R : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*stoppedRemainingDensity a (q n) (S p.2) p.2 p.1.1
-  let μ := (upperArea.restrict K).prod (W.restrict U)
-  have hFi : Integrable F μ := H.1.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self)
-  have hRm (n : ℕ) : Measurable (R n) :=
-    (f.continuous.measurable.comp measurable_fst).mul
-      ((stoppedRemainingDensity_joint_measurable a ha (q n) (hq n).1 (hq n).2 S hS).comp measurable_swap)
-  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖R n p‖≤‖F p‖ := by
-    dsimp only [R,F]
-    rw [norm_mul,norm_mul,Real.norm_of_nonneg (stoppedRemainingDensity_nonneg _ _ _ _ _),
-      Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg _ _ _ _)]
-    exact mul_le_mul_of_nonneg_left (stoppedRemainingDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
-  have hRi (n : ℕ) : Integrable (R n) μ := hFi.norm.mono' (hRm n).aestronglyMeasurable
-    (Eventually.of_forall (hbound n))
-  have ht := tendsto_integral_of_dominated_convergence (fun p => ‖F p‖)
-    (fun n => (hRm n).aestronglyMeasurable) hFi.norm (fun n => Eventually.of_forall (hbound n))
-    (Eventually.of_forall (fun p => (stopped_remaining_to_surviving a ha (S p.2) q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)))
-  have he (g : UpperPlane → ℝ) : (∫z in K,f z*g z∂upperArea) = ∫z,f z*g z∂upperArea :=
-    setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by rw [image_eq_zero_of_notMem_tsupport hz,zero_mul])
-  have hRint (n : ℕ) : (∫p,R n p∂μ) = ∫b in U,(∫z : UpperPlane,f z*stoppedRemainingDensity a (q n) (S b) b z.1∂upperArea)∂W := by
-    rw [integral_prod_symm (R n) (hRi n)]
-    apply integral_congr_ae
-    filter_upwards [] with b
-    exact he _
-  have hFint : (∫p,F p∂μ) = ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea)∂W := by
-    rw [integral_prod_symm F hFi]
-    apply integral_congr_ae
-    filter_upwards [] with b
-    exact he _
-  change Tendsto (fun n => ∫p,R n p∂μ) atTop (𝓝 (∫p,F p∂μ)) at ht
-  simpa only [hRint,hFint] using ht
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma actualUpperMass_stopped_norm_bound (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
-    ‖∫z, f z ∂actualUpperMass a q t b‖ ≤
-      ∫z, ‖f z‖ ∂actualUpperMass a q ∞ b := by
-  have := actualUpperMass_finiteOnCompacts a ha q hq hq1 ∞ b
-  exact (norm_integral_le_integral_norm _).trans (integral_mono_measure
-    (actualUpperMass_mono_time a q b le_top) (Eventually.of_forall (fun _ => norm_nonneg _))
-    f.integrable.norm)
-
- 
-
-theorem actualUpperMass_stopped_Lp_bound (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (m : ℕ) (hm : 1 ≤ m) (f : C_c(UpperPlane,ℝ)) :
-    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 →
-      ∀ S : ExactSLEGenerator.Driver → ℝ≥0∞, IsStoppingTime rawFiltration S →
-      MemLp (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) (m : ℝ≥0∞) W ∧
-      eLpNorm (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) (m : ℝ≥0∞) W ≤ C := by
-  let g : C_c(UpperPlane,ℝ) := ⟨⟨fun z => ‖f z‖,f.continuous.norm⟩,f.hasCompactSupport.norm⟩
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW m hm g
-  refine ⟨C,hC,fun q hq hq1 S hS => ?_⟩
-  have hmeas : StronglyMeasurable (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) := by
-    simpa [stoppedMassKernel_apply a ha q hq hq1 S hS] using
-      (f.continuous.measurable.stronglyMeasurable.integral_kernel
-        (κ := stoppedMassKernel a q S))
-  have hb : ∀ᵐ b ∂W, ‖∫z, f z ∂actualUpperMass a q (S b) b‖ ≤
-      ∫z, g z ∂actualUpperMass a q ∞ b :=
-    Eventually.of_forall (fun b => actualUpperMass_stopped_norm_bound a ha q hq hq1 (S b) b f)
-  refine ⟨(hCb q hq hq1 ∞).1.mono' hmeas.aestronglyMeasurable hb,?_⟩
-  exact (eLpNorm_mono_ae hmeas.aestronglyMeasurable (hb.mono (fun _ h => h.trans (le_abs_self _)))).trans
-    (hCb q hq hq1 ∞).2
-
-lemma actualUpperMass_stopped_eval_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) :
-    Integrable (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) W := by
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_stopped_Lp_bound a ha W hW 1 le_rfl f
-  exact (hCb q hq hq1 S hS).1.integrable (by norm_num)
-
-lemma stoppedRemainingKernel_compact_eval (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
-    (∫z, f z ∂stoppedRemainingKernel a q S b) =
-      ∫z, f z*stoppedRemainingDensity a q (S b) b z.1 ∂upperArea := by
-  rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS,
-    integral_withDensity_eq_integral_toReal_smul]
-  · simp only [ENNReal.toReal_ofReal (stoppedRemainingDensity_nonneg _ _ _ _ _), smul_eq_mul, mul_comm]
-  · exact ((stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).comp
-      (measurable_const.prodMk measurable_id)).ennreal_ofReal
-  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
-
- 
-
-theorem actualUpperMass_stopped_finite_kernel (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    (∫b in U, (∫z, f z ∂actualUpperMass a q ∞ b)-(∫z, f z ∂actualUpperMass a q (S b) b) ∂W) =
-      ∫b in U, (∫z : UpperPlane, f z*stoppedRemainingDensity a q (S b) b z.1 ∂upperArea) ∂W := by
-  have := unhitMassKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
-  have := stoppedRemainingKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
-  have := average_restrict_finite (unhitMassKernel a q S) W U
-  have := average_restrict_finite (stoppedRemainingKernel a q S) W U
-  have he (b : ExactSLEGenerator.Driver) :
-      (∫z, f z ∂actualUpperMass a q ∞ b)-(∫z, f z ∂actualUpperMass a q (S b) b) =
-        ∫z, f z ∂unhitMassKernel a q S b := by
-    have := actualUpperMass_finiteOnCompacts a ha q hq hq1 ∞ b
-    have hi : Integrable f (actualUpperMass a q (S b) b + unhitMassKernel a q S b) := by
-      rw [← actualUpperMass_add_unhit a ha q hq hq1 S hS]
-      exact f.integrable
-    rw [actualUpperMass_add_unhit a ha q hq hq1 S hS, integral_add_measure
-      (integrable_add_measure.mp hi).1 (integrable_add_measure.mp hi).2]
-    ring
-  simp_rw [he]
-  rw [← average_compact_eval_of_average_finite,
-    unhitMassKernel_conditional_average a ha W hW q hq hq1 S hS U hU,
-    average_compact_eval_of_average_finite]
-  exact integral_congr_ae (Eventually.of_forall (fun b =>
-    stoppedRemainingKernel_compact_eval a ha q hq hq1 S hS b f))
-
- 
-
-theorem actualUpperMass_stopped_convex_potential_limit (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-    (hs : ∀ k j, j∈s k → k≤j) (hc : ∀ k j, j∈s k → 0≤c k j)
-    (hsum : ∀ k, ∑j∈s k,c k j=1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    Tendsto (fun k => ∫b in U,
-      ∑j∈s k,c k j*((∫z, f z ∂actualUpperMass a (q j) ∞ b)-
-        (∫z, f z ∂actualUpperMass a (q j) (S b) b)) ∂W)
-      atTop (𝓝 (∫b in U, (∫z : UpperPlane, f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W)) := by
-  have hpot := actualUpper_stopped_remaining_integral_tendsto a ha W hW S hS f U q hq hlim
-  have hcombo := scalar_convex_tail_limit s c hs hc hsum _ _ hpot
-  have he (k : ℕ) : (∫b in U,
-      ∑j∈s k,c k j*((∫z, f z ∂actualUpperMass a (q j) ∞ b)-
-        (∫z, f z ∂actualUpperMass a (q j) (S b) b)) ∂W) =
-      ∑j∈s k,c k j*(∫b in U,
-        (∫z : UpperPlane, f z*stoppedRemainingDensity a (q j) (S b) b z.1 ∂upperArea) ∂W) := by
-    rw [integral_finsetSum]
-    · apply Finset.sum_congr rfl
-      intro j hj
-      rw [integral_const_mul, actualUpperMass_stopped_finite_kernel a ha W hW
-        (q j) (hq j).1 (hq j).2 S hS f U hU]
-    · intro j hj
-      exact (((actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j).1 (hq j).2
-        (fun _ => ∞) (fun t => by simp) f).sub
-        (actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j).1 (hq j).2 S hS f)).const_mul
-        (c k j)).restrict
-  simpa only [he] using hcombo
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TraceMassEndpoint
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma timeMassDensity_stopping_joint_measurable (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      timeMassDensity a q (S p.1) p.1 p.2.1) := by
+  exact (timeMassDensity_joint_measurable a ha q hq hq1).comp
+    (((hS.measurable'.comp measurable_fst).prodMk measurable_fst).prodMk
+      (measurable_subtype_coe.comp measurable_snd))
+
+ 
+
+def stoppedMassKernel (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
+    Kernel ExactSLEGenerator.Driver UpperPlane :=
+  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
+    (fun b z => timeMassDensity a q (S b) b z.1)
+
+instance stoppedMassKernel_sfinite (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
+    IsSFiniteKernel (stoppedMassKernel a q S) :=
+  Kernel.IsSFiniteKernel.withDensity _ (fun b z => (timeMassDensity_finite a q (S b) b z.1).ne)
+
+lemma stoppedMassKernel_apply (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    stoppedMassKernel a q S b = actualUpperMass a q (S b) b :=
+  Kernel.withDensity_apply _ (timeMassDensity_stopping_joint_measurable a ha q hq hq1 S hS) b
+
+lemma actualUpperMass_measurable_stopping (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable (fun b => actualUpperMass a q (S b) b) := by
+  have he : (fun b => actualUpperMass a q (S b) b) =
+      (fun b => stoppedMassKernel a q S b) :=
+    funext (fun b => (stoppedMassKernel_apply a ha q hq hq1 S hS b).symm)
+  rw [he]
+  exact (stoppedMassKernel a q S).measurable
+
+lemma actualUpperMass_add_unhit (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    actualUpperMass a q ∞ b = actualUpperMass a q (S b) b + unhitMassKernel a q S b := by
+  rw [unhitMassKernel_apply a ha q hq hq1 S hS]
+  unfold actualUpperMass
+  have he : (fun z : UpperPlane => timeMassDensity a q ∞ b z.1) =
+      (fun z => timeMassDensity a q (S b) b z.1) +
+      (fun z => ENNReal.ofReal (unhitMassDensity a q (S b) b z)) :=
+    funext (fun z => timeMassDensity_add_unhit a q (S b) b z)
+  rw [he,withDensity_add_left]
+  exact (timeMassDensity_stopping_joint_measurable a ha q hq hq1 S hS).comp
+    (measurable_const.prodMk measurable_id)
+
+ 
+
+theorem actualUpperMass_stopped_random_test (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
+    (hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
+    (∫⁻b, ∫⁻z, H (b,z) ∂actualUpperMass a q ∞ b ∂W) =
+      (∫⁻b, ∫⁻z, H (b,z) ∂actualUpperMass a q (S b) b ∂W) +
+      ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1) ∂upperArea ∂W := by
+  have hH₀ : Measurable H := hH.mono (prod_sigma_le hS.measurableSpace_le _) le_rfl
+  have hmeas : Measurable (fun b => ∫⁻z, H (b,z) ∂actualUpperMass a q (S b) b) := by
+    simpa only [stoppedMassKernel_apply a ha q hq hq1 S hS] using
+      hH₀.lintegral_kernel_prod_right' (κ := stoppedMassKernel a q S)
+  simp_rw [actualUpperMass_add_unhit a ha q hq hq1 S hS, lintegral_add_measure]
+  rw [lintegral_add_left hmeas, unhitMassKernel_random_test a ha W hW q hq hq1 S hS H hH]
+  congr 1
+  apply lintegral_congr
+  intro b
+  rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS,
+    lintegral_withDensity_eq_lintegral_mul]
+  · exact lintegral_congr (fun z => mul_comm _ _)
+  · exact ((stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).comp
+      (measurable_const.prodMk measurable_id)).ennreal_ofReal
+  · exact hH₀.comp (measurable_const.prodMk measurable_id)
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma remainingMassDensity_antitone_level (a : ℝ) (ha : 0<a) (t : ℝ≥0)
+    (b : ExactSLEGenerator.Driver) (z : UpperPlane)
+    (q r : ℝ) (hq : 0<q) (hqr : q≤r) (hr : r≤1) :
+    remainingMassDensity a r t b z.1 ≤ remainingMassDensity a q t b z.1 := by
+  have hT := radialTime_antitone a ha z.1 z.2 _ _ (mul_pos z.2 hq)
+    (mul_le_mul_of_nonneg_left hqr z.2.le) (mul_le_of_le_one_right z.2.le hr) b
+  simp only [remainingMassDensity,ite_eq_left z.2,radialRemaining]
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+  split_ifs with hr hq hq
+  · exact le_rfl
+  · exact False.elim (hq ⟨hr.1,hr.2.trans_le hT⟩)
+  · unfold physicalGreen
+    positivity
+  · exact le_rfl
+
+lemma stoppedRemainingDensity_antitone_level (a : ℝ) (ha : 0<a) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : UpperPlane)
+    (q r : ℝ) (hq : 0<q) (hqr : q≤r) (hr : r≤1) :
+    stoppedRemainingDensity a r t b z.1 ≤ stoppedRemainingDensity a q t b z.1 := by
+  unfold stoppedRemainingDensity
+  split_ifs
+  · exact le_rfl
+  · exact remainingMassDensity_antitone_level a ha t.toNNReal b z q r hq hqr hr
+
+lemma remainingMassDensity_eventually_eq_surviving (a : ℝ) (ha : 0<a) (t : ℝ≥0)
+    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
+    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+    ∀ᶠ n in atTop, remainingMassDensity a (q n) t b z.1 = survivingGreen a t b z.1 := by
+  by_cases ht : alive a b.1 t z.1
+  · have hR := physicalRadius_actual_pos a b z.1 t ht
+    have hqR : ∀ᶠ n in atTop, z.1.im*q n<physicalRadius (actualState a b.1 t z.1) := by
+      have hconv : Tendsto (fun n => z.1.im*q n) atTop (𝓝 0) := by simpa using hlim.const_mul z.1.im
+      exact hconv.eventually (gt_mem_nhds hR)
+    filter_upwards [hqR] with n hn
+    have hnT : (t : ℝ≥0∞)<radialTime a b.1 z.1 (z.1.im*q n) := by
+      by_contra h
+      have hTt := le_of_not_gt h
+      have hfin : radialTime a b.1 z.1 (z.1.im*q n)≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hTt
+      have hle : (radialTime a b.1 z.1 (z.1.im*q n)).toNNReal≤t := by
+        exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hTt)
+      have hh := (radialTime_attained a ha z.1 z.2 _ (mul_pos z.2 (hq n).1)
+        (mul_le_of_le_one_right z.2.le (hq n).2) b hfin).2
+      have he := physicalRadius_actual_anti a ha b z.1 _ t hle ht
+      rw [hh] at he
+      exact not_le.mpr hn he
+    simp only [remainingMassDensity,ite_eq_left z.2,radialRemaining,ht,hnT,and_self,ite_true,
+      survivingGreen]
+  · filter_upwards [] with n
+    simp only [remainingMassDensity,ite_eq_left z.2,radialRemaining,ht,false_and,ite_false,
+      survivingGreen,mul_zero]
+
+lemma stoppedRemainingDensity_eventually_eq_surviving (a : ℝ) (ha : 0<a) (t : ℝ≥0∞)
+    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
+    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+    ∀ᶠ n in atTop, stoppedRemainingDensity a (q n) t b z.1 = stoppedSurvivingGreen a t b z.1 := by
+  unfold stoppedRemainingDensity stoppedSurvivingGreen
+  by_cases ht : t=∞
+  · simp only [ht,ite_true,eventually_const]
+  · simp only [ite_eq_right ht]
+    exact remainingMassDensity_eventually_eq_surviving a ha t.toNNReal q hq hlim b z
+
+ 
+
+theorem stoppedRemainingDensity_random_test_tendsto (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver)
+    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hanti : Antitone q)
+    (hlim : Tendsto q atTop (𝓝 0))
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞) (hH : Measurable H) :
+    Tendsto (fun n => ∫⁻b, ∫⁻z, H (b,z)*
+        ENNReal.ofReal (stoppedRemainingDensity a (q n) (S b) b z.1) ∂upperArea ∂W)
+      atTop (𝓝 (∫⁻b, ∫⁻z, H (b,z)*
+        ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂upperArea ∂W)) := by
+  have hm (n : ℕ) : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      H p*ENNReal.ofReal (stoppedRemainingDensity a (q n) (S p.1) p.1 p.2.1)) :=
+    hH.mul (stoppedRemainingDensity_joint_measurable a ha (q n) (hq n).1 (hq n).2 S hS).ennreal_ofReal
+  have hmono (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+      Monotone (fun n => H (b,z)*ENNReal.ofReal (stoppedRemainingDensity a (q n) (S b) b z.1)) := by
+    intro i j hij
+    exact mul_le_mul_right (ENNReal.ofReal_le_ofReal
+      (stoppedRemainingDensity_antitone_level a (by linarith) (S b) b z
+        (q j) (q i) (hq j).1 (hanti hij) (hq i).2)) _
+  apply lintegral_tendsto_of_tendsto_of_monotone
+  · intro n
+    exact (hm n).lintegral_prod_right'.aemeasurable
+  · exact Eventually.of_forall (fun b i j hij => lintegral_mono (fun z => hmono b z hij))
+  · filter_upwards [] with b
+    apply lintegral_tendsto_of_tendsto_of_monotone
+    · intro n
+      exact ((hm n).comp (measurable_const.prodMk measurable_id)).aemeasurable
+    · exact Eventually.of_forall (hmono b)
+    · filter_upwards [] with z
+      apply tendsto_const_nhds.congr'
+      filter_upwards [stoppedRemainingDensity_eventually_eq_surviving a (by linarith)
+        (S b) q hq hlim b z] with n hn
+      rw [hn]
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_stopped_remaining_spatial_L1 (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (K : Set UpperPlane) (hK : MeasurableSet K)
+    (hKfin : upperArea K<∞) (c : ℝ) (hc : 0<c) (hcK : ∀ z∈K,c≤z.1.im)
+    (f : UpperPlane → ℝ) (hf : Measurable f) (B : ℝ) (hB : 0≤B) (hfB : ∀z∈K,‖f z‖≤B)
+    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
+    Integrable (fun p : UpperPlane × ExactSLEGenerator.Driver => f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1)
+      ((upperArea.restrict K).prod W) ∧
+    Tendsto (fun n => ∫p : UpperPlane × ExactSLEGenerator.Driver,
+      ‖f p.1*stoppedRemainingDensity a (q n) (S p.2) p.2 p.1.1-f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1‖
+      ∂((upperArea.restrict K).prod W)) atTop (𝓝 0)
+ := by
+  let ν := upperArea.restrict K
+  have : IsFiniteMeasure ν := ⟨by simpa [ν] using hKfin⟩
+  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1
+  let H : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*stoppedRemainingDensity a (q n) (S p.2) p.2 p.1.1
+  have hHm (n : ℕ) : Measurable (H n) :=
+    (hf.comp measurable_fst).mul ((stoppedRemainingDensity_joint_measurable a ha (q n) (hq n).1 (hq n).2 S hS).comp measurable_swap)
+  have hconv (p : UpperPlane × ExactSLEGenerator.Driver) : Tendsto (fun n => H n p) atTop (𝓝 (F p)) :=
+    (stopped_remaining_to_surviving a ha (S p.2) q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)
+  have hFm : Measurable F := measurable_of_tendsto_metrizable hHm (tendsto_pi_nhds.mpr hconv)
+  have hi (z : UpperPlane) : Integrable (fun b => F (z,b)) W :=
+    (survivingGreen_optional_bound a ha W hW z S hS).1.const_mul (f z)
+  have he (z : UpperPlane) : (∫b,‖F (z,b)‖∂W) = ‖f z‖*(∫b,stoppedSurvivingGreen a (S b) b z.1∂W) := by
+    simp only [F,norm_mul,Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg _ _ _ _),integral_const_mul]
+  let C := Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log c)
+  have hC : 0≤C := by dsimp [C]; positivity
+  have hb (z : UpperPlane) (hz : z∈K) : (∫b,‖F (z,b)‖∂W)≤B*C := by
+    rw [he]
+    have hG := (survivingGreen_optional_bound a ha W hW z S hS).2
+    have hgreen : physicalGreen a (z.1,0)≤Real.exp (-alpha a*Real.log c) := by
+      apply physicalGreen_bound a c ha hc _ z.2
+      simpa [physicalRadius,Y] using hcK z hz
+    have hpos : 0≤(∫b,stoppedSurvivingGreen a (S b) b z.1∂W) := integral_nonneg (fun b => stoppedSurvivingGreen_nonneg _ _ _ _)
+    exact mul_le_mul (hfB z hz) (hG.trans (mul_le_mul_of_nonneg_left hgreen (Real.exp_pos _).le))
+      hpos hB
+  have hFi : Integrable F (ν.prod W) := by
+    apply (integrable_prod_iff hFm.aestronglyMeasurable).mpr
+    refine ⟨Eventually.of_forall hi,?_⟩
+    apply Integrable.of_bound (hFm.aestronglyMeasurable.norm.integral_prod_right') (B*C)
+    filter_upwards [ae_restrict_mem hK] with z hz
+    rw [Real.norm_of_nonneg (integral_nonneg (fun b => norm_nonneg (F (z,b))))]
+    exact hb z hz
+  refine ⟨hFi,?_⟩
+  have hD : Integrable (fun p => 2*‖F p‖) (ν.prod W) := hFi.norm.const_mul 2
+  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖H n p-F p‖≤2*‖F p‖ := by
+    have hle : ‖H n p‖≤‖F p‖ := by
+      dsimp only [H,F]
+      rw [norm_mul,norm_mul,Real.norm_of_nonneg (stoppedRemainingDensity_nonneg _ _ _ _ _),
+        Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg _ _ _ _)]
+      exact mul_le_mul_of_nonneg_left (stoppedRemainingDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
+    calc ‖H n p-F p‖≤‖H n p‖+‖F p‖ := norm_sub_le _ _
+      _≤2*‖F p‖ := by linarith
+  have hlim' := tendsto_integral_of_dominated_convergence (fun p => 2*‖F p‖)
+    (fun n => ((hHm n).sub hFm).norm.aestronglyMeasurable) hD
+    (fun n => Eventually.of_forall (fun p => by simpa using hbound n p))
+    (Eventually.of_forall (fun p => by simpa using ((hconv p).sub_const (F p)).norm))
+  simpa using hlim'
+
+lemma actualUpper_stopped_remaining_integral_tendsto (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) (U : Set ExactSLEGenerator.Driver)
+    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0)) :
+    Tendsto (fun n => ∫b in U, (∫z : UpperPlane,f z*stoppedRemainingDensity a (q n) (S b) b z.1∂upperArea) ∂W)
+      atTop (𝓝 (∫b in U, (∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea) ∂W)) := by
+  let K := tsupport f
+  have hK : IsCompact K := f.hasCompactSupport
+  have hfin : upperArea K<∞ := by
+    rw [upperArea,upper_embedding.comap_apply]
+    exact (hK.image continuous_subtype_val).measure_lt_top
+  have : IsFiniteMeasure (upperArea.restrict K) := ⟨by simpa using hfin⟩
+  obtain ⟨c,hc,hcK⟩ := upper_compact_lower K hK
+  obtain ⟨B,hB,hBf⟩ := upperExtension_bounded f
+  have H := actual_stopped_remaining_spatial_L1 a ha W hW S hS K hK.measurableSet hfin (c/2) (by positivity)
+    (fun z hz => by have := hcK z hz; linarith) f f.continuous.measurable B hB
+    (fun z hz => by simpa only [upperExtension_apply] using hBf z.1 ⟨z,hz,rfl⟩) q hq hlim
+  let F : UpperPlane × ExactSLEGenerator.Driver → ℝ := fun p => f p.1*stoppedSurvivingGreen a (S p.2) p.2 p.1.1
+  let R : ℕ → UpperPlane × ExactSLEGenerator.Driver → ℝ := fun n p => f p.1*stoppedRemainingDensity a (q n) (S p.2) p.2 p.1.1
+  let μ := (upperArea.restrict K).prod (W.restrict U)
+  have hFi : Integrable F μ := H.1.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self)
+  have hRm (n : ℕ) : Measurable (R n) :=
+    (f.continuous.measurable.comp measurable_fst).mul
+      ((stoppedRemainingDensity_joint_measurable a ha (q n) (hq n).1 (hq n).2 S hS).comp measurable_swap)
+  have hbound (n : ℕ) (p : UpperPlane × ExactSLEGenerator.Driver) : ‖R n p‖≤‖F p‖ := by
+    dsimp only [R,F]
+    rw [norm_mul,norm_mul,Real.norm_of_nonneg (stoppedRemainingDensity_nonneg _ _ _ _ _),
+      Real.norm_of_nonneg (stoppedSurvivingGreen_nonneg _ _ _ _)]
+    exact mul_le_mul_of_nonneg_left (stoppedRemainingDensity_le_surviving _ _ _ _ _ p.1.2) (norm_nonneg _)
+  have hRi (n : ℕ) : Integrable (R n) μ := hFi.norm.mono' (hRm n).aestronglyMeasurable
+    (Eventually.of_forall (hbound n))
+  have ht := tendsto_integral_of_dominated_convergence (fun p => ‖F p‖)
+    (fun n => (hRm n).aestronglyMeasurable) hFi.norm (fun n => Eventually.of_forall (hbound n))
+    (Eventually.of_forall (fun p => (stopped_remaining_to_surviving a ha (S p.2) q hq hlim p.2 p.1.1 p.1.2).const_mul (f p.1)))
+  have he (g : UpperPlane → ℝ) : (∫z in K,f z*g z∂upperArea) = ∫z,f z*g z∂upperArea :=
+    setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz => by rw [image_eq_zero_of_notMem_tsupport hz,zero_mul])
+  have hRint (n : ℕ) : (∫p,R n p∂μ) = ∫b in U,(∫z : UpperPlane,f z*stoppedRemainingDensity a (q n) (S b) b z.1∂upperArea)∂W := by
+    rw [integral_prod_symm (R n) (hRi n)]
+    apply integral_congr_ae
+    filter_upwards [] with b
+    exact he _
+  have hFint : (∫p,F p∂μ) = ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1∂upperArea)∂W := by
+    rw [integral_prod_symm F hFi]
+    apply integral_congr_ae
+    filter_upwards [] with b
+    exact he _
+  change Tendsto (fun n => ∫p,R n p∂μ) atTop (𝓝 (∫p,F p∂μ)) at ht
+  simpa only [hRint,hFint] using ht
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma actualUpperMass_stopped_norm_bound (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
+    ‖∫z, f z ∂actualUpperMass a q t b‖ ≤
+      ∫z, ‖f z‖ ∂actualUpperMass a q ∞ b := by
+  have := actualUpperMass_finiteOnCompacts a ha q hq hq1 ∞ b
+  exact (norm_integral_le_integral_norm _).trans (integral_mono_measure
+    (actualUpperMass_mono_time a q b le_top) (Eventually.of_forall (fun _ => norm_nonneg _))
+    f.integrable.norm)
+
+ 
+
+theorem actualUpperMass_stopped_Lp_bound (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (m : ℕ) (hm : 1 ≤ m) (f : C_c(UpperPlane,ℝ)) :
+    ∃ C : ℝ≥0∞, C<∞ ∧ ∀ q : ℝ, 0<q → q≤1 →
+      ∀ S : ExactSLEGenerator.Driver → ℝ≥0∞, IsStoppingTime rawFiltration S →
+      MemLp (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) (m : ℝ≥0∞) W ∧
+      eLpNorm (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) (m : ℝ≥0∞) W ≤ C := by
+  let g : C_c(UpperPlane,ℝ) := ⟨⟨fun z => ‖f z‖,f.continuous.norm⟩,f.hasCompactSupport.norm⟩
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_Lp_bound a ha W hW m hm g
+  refine ⟨C,hC,fun q hq hq1 S hS => ?_⟩
+  have hmeas : StronglyMeasurable (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) := by
+    simpa [stoppedMassKernel_apply a ha q hq hq1 S hS] using
+      (f.continuous.measurable.stronglyMeasurable.integral_kernel
+        (κ := stoppedMassKernel a q S))
+  have hb : ∀ᵐ b ∂W, ‖∫z, f z ∂actualUpperMass a q (S b) b‖ ≤
+      ∫z, g z ∂actualUpperMass a q ∞ b :=
+    Eventually.of_forall (fun b => actualUpperMass_stopped_norm_bound a ha q hq hq1 (S b) b f)
+  refine ⟨(hCb q hq hq1 ∞).1.mono' hmeas.aestronglyMeasurable hb,?_⟩
+  exact (eLpNorm_mono_ae hmeas.aestronglyMeasurable (hb.mono (fun _ h => h.trans (le_abs_self _)))).trans
+    (hCb q hq hq1 ∞).2
+
+lemma actualUpperMass_stopped_eval_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) :
+    Integrable (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) W := by
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_stopped_Lp_bound a ha W hW 1 le_rfl f
+  exact (hCb q hq hq1 S hS).1.integrable (by norm_num)
+
+lemma stoppedRemainingKernel_compact_eval (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
+    (∫z, f z ∂stoppedRemainingKernel a q S b) =
+      ∫z, f z*stoppedRemainingDensity a q (S b) b z.1 ∂upperArea := by
+  rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS,
+    integral_withDensity_eq_integral_toReal_smul]
+  · simp only [ENNReal.toReal_ofReal (stoppedRemainingDensity_nonneg _ _ _ _ _), smul_eq_mul, mul_comm]
+  · exact ((stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).comp
+      (measurable_const.prodMk measurable_id)).ennreal_ofReal
+  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
+
+ 
+
+theorem actualUpperMass_stopped_finite_kernel (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    (∫b in U, (∫z, f z ∂actualUpperMass a q ∞ b)-(∫z, f z ∂actualUpperMass a q (S b) b) ∂W) =
+      ∫b in U, (∫z : UpperPlane, f z*stoppedRemainingDensity a q (S b) b z.1 ∂upperArea) ∂W := by
+  have := unhitMassKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
+  have := stoppedRemainingKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
+  have := average_restrict_finite (unhitMassKernel a q S) W U
+  have := average_restrict_finite (stoppedRemainingKernel a q S) W U
+  have he (b : ExactSLEGenerator.Driver) :
+      (∫z, f z ∂actualUpperMass a q ∞ b)-(∫z, f z ∂actualUpperMass a q (S b) b) =
+        ∫z, f z ∂unhitMassKernel a q S b := by
+    have := actualUpperMass_finiteOnCompacts a ha q hq hq1 ∞ b
+    have hi : Integrable f (actualUpperMass a q (S b) b + unhitMassKernel a q S b) := by
+      rw [← actualUpperMass_add_unhit a ha q hq hq1 S hS]
+      exact f.integrable
+    rw [actualUpperMass_add_unhit a ha q hq hq1 S hS, integral_add_measure
+      (integrable_add_measure.mp hi).1 (integrable_add_measure.mp hi).2]
+    ring
+  simp_rw [he]
+  rw [← average_compact_eval_of_average_finite,
+    unhitMassKernel_conditional_average a ha W hW q hq hq1 S hS U hU,
+    average_compact_eval_of_average_finite]
+  exact integral_congr_ae (Eventually.of_forall (fun b =>
+    stoppedRemainingKernel_compact_eval a ha q hq hq1 S hS b f))
+
+ 
+
+theorem actualUpperMass_stopped_convex_potential_limit (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℕ → ℝ) (hq : ∀ n, 0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+    (hs : ∀ k j, j∈s k → k≤j) (hc : ∀ k j, j∈s k → 0≤c k j)
+    (hsum : ∀ k, ∑j∈s k,c k j=1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    Tendsto (fun k => ∫b in U,
+      ∑j∈s k,c k j*((∫z, f z ∂actualUpperMass a (q j) ∞ b)-
+        (∫z, f z ∂actualUpperMass a (q j) (S b) b)) ∂W)
+      atTop (𝓝 (∫b in U, (∫z : UpperPlane, f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W)) := by
+  have hpot := actualUpper_stopped_remaining_integral_tendsto a ha W hW S hS f U q hq hlim
+  have hcombo := scalar_convex_tail_limit s c hs hc hsum _ _ hpot
+  have he (k : ℕ) : (∫b in U,
+      ∑j∈s k,c k j*((∫z, f z ∂actualUpperMass a (q j) ∞ b)-
+        (∫z, f z ∂actualUpperMass a (q j) (S b) b)) ∂W) =
+      ∑j∈s k,c k j*(∫b in U,
+        (∫z : UpperPlane, f z*stoppedRemainingDensity a (q j) (S b) b z.1 ∂upperArea) ∂W) := by
+    rw [integral_finsetSum]
+    · apply Finset.sum_congr rfl
+      intro j hj
+      rw [integral_const_mul, actualUpperMass_stopped_finite_kernel a ha W hW
+        (q j) (hq j).1 (hq j).2 S hS f U hU]
+    · intro j hj
+      exact (((actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j).1 (hq j).2
+        (fun _ => ∞) (fun t => by simp) f).sub
+        (actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j).1 (hq j).2 S hS f)).const_mul
+        (c k j)).restrict
+  simpa only [he] using hcombo
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StoppedLimits.lean b/lean/OAI/Probability/SLEGauge/StoppedLimits.lean
index 78204e3d71011b78cbb9b012cad2c3586f46a2bf..d2249af8521fa8b951295a514433ce7ebaa55093 100644
--- a/lean/OAI/Probability/SLEGauge/StoppedLimits.lean
+++ b/lean/OAI/Probability/SLEGauge/StoppedLimits.lean
@@ -1,484 +1,484 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.EulerConvergence
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
-
- 
-
-theorem expected_stopped_local_eq {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
-    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
-    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hg0 : ∀ b, g b 0 = z)
-    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
-    (τ : (ℝ≥0 → ℝ) → ℝ) (mτ : Measurable τ)
-    (hτpast : ∀ (s : ℝ≥0) b c, (∀ u ≤ s, b u = c u) → ((s:ℝ) < τ b ↔ (s:ℝ) < τ c))
-    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
-    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
-    (hgen : ∀ q ∈ K,
-      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
-        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 = 0)
-    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
-    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) = f (z,0) := by
-  have hle := expected_stopped_local hW f K O hK hO hKO hf v mv g z hL hM hv hlip
-    hg0 hder hcf τ mτ hτpast hτ hD (fun q hq => (hgen q hq).le) hz ht
-  have hneg : ∀ q ∈ K,
-      fderiv ℝ (fun x => -f (x,q.2)) q.1 (v q.1 q.2) +
-        deriv (deriv (fun b => -f (q.1,b))) q.2 / 2 ≤ 0 := by
-    intro q hq
-    have hfd : DifferentiableAt ℝ (fun x => f (x,q.2)) q.1 :=
-      ((hf q (hKO hq)).contDiffAt (hO.mem_nhds (hKO hq))).differentiableAt (by simp) |>.comp q.1
-        (by fun_prop)
-    rw [fderiv_fun_neg]
-    simp only [neg_apply,deriv.fun_neg',deriv.fun_neg]
-    linarith [hgen q hq]
-  have hge := expected_stopped_local hW (fun q => -f q) K O hK hO hKO hf.neg v mv g z hL hM hv hlip
-    hg0 hder hcf τ mτ hτpast hτ hD hneg hz ht
-  rw [integral_neg] at hge
-  linarith
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.OnePoint
-open Parameter AngleCalculus
-open scoped ContDiff
-abbrev State := ℝ × ℂ
-
-def domain : Set (State × ℝ) := {q | 0 < q.1.2.im}
-def field (P : Parameter) (x : State) (b : ℝ) : State :=
-  ((V (x.2-b))^2, (P.a:ℂ)/(x.2-b))
- 
-def green (P : Parameter) (q : State × ℝ) : ℝ :=
-  Real.exp (P.p*q.1.1/2) * Real.sin (q.1.2-q.2).arg ^ P.p
-
-lemma isOpen_domain : IsOpen domain := by
-  exact isOpen_lt continuous_const (by fun_prop)
-
-lemma green_nonneg (P : Parameter) {q : State × ℝ} (hq : q ∈ domain) : 0 ≤ green P q := by
-  exact mul_nonneg (Real.exp_pos _).le (Real.rpow_nonneg (sin_angle_pos (by simpa [domain] using hq)).le _)
-
-lemma green_le (P : Parameter) {s : ℝ} {q : State × ℝ} (hu : q.1.1 ≤ s)
-    (hq : q ∈ domain) : green P q ≤ Real.exp (P.p*s/2) := by
-  have hs : 0 < Real.sin (q.1.2-(q.2:ℂ)).arg := sin_angle_pos (by simpa [domain] using hq)
-  have hp : Real.sin (q.1.2-(q.2:ℂ)).arg ^ P.p ≤ 1 :=
-    Real.rpow_le_one hs.le (Real.sin_le_one _) P.p_pos.le
-  calc
-    green P q ≤ Real.exp (P.p*q.1.1/2) := by
-      exact mul_le_of_le_one_right (Real.exp_pos _).le hp
-    _ ≤ Real.exp (P.p*s/2) := Real.exp_le_exp.mpr (by nlinarith [P.p_pos])
-
-lemma contDiffOn_green (P : Parameter) : ContDiffOn ℝ ∞ (green P) domain := by
-  intro q hq
-  apply ContDiffAt.contDiffWithinAt
-  have hh : 0 < (q.1.2-(q.2:ℂ)).im := by simpa [domain] using hq
-  have ha := (contDiffAt_arg_real (show q.1.2-(q.2:ℂ) ∈ Complex.slitPlane from Or.inr hh.ne') ∞).comp q
-    (show ContDiffAt ℝ ∞ (fun r : State × ℝ => r.1.2-(r.2:ℂ)) q from
-      (contDiffAt_snd.comp q contDiffAt_fst).sub
-        (Complex.ofRealCLM.contDiff.contDiffAt.comp q contDiffAt_snd))
-  have hs := ha.sin.rpow_const_of_ne (p := P.p) (sin_angle_pos hh).ne'
-  exact (show ContDiffAt ℝ ∞ (fun r : State × ℝ => Real.exp (P.p*r.1.1/2)) q by fun_prop).mul hs
-
-lemma green_generator (P : Parameter) {q : State × ℝ} (hq : q ∈ domain) :
-    fderiv ℝ (fun x => green P (x,q.2)) q.1 (field P q.1 q.2) +
-      deriv (deriv (fun b => green P (q.1,b))) q.2 / 2 = 0 := by
-  rcases q with ⟨⟨u,z⟩,b⟩
-  have hy : 0 < z.im := hq
-  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
-  have hphy := physical_generator_angle P.a (fun θ => Real.sin θ^P.p)
-    (fun θ hθ => (hasDerivAt_sinePower P.p θ hθ).differentiableAt.hasDerivAt)
-    (fun θ hθ => by
-      have he : deriv (fun t => Real.sin t^P.p) =ᶠ[𝓝 θ] fun t => P.p*Real.cos t*Real.sin t^(P.p-1) := by
-        filter_upwards [Real.continuous_sin.continuousAt.eventually (Ioi_mem_nhds hθ)] with t ht
-        exact deriv_sinePower P.p t ht
-      exact ((hasDerivAt_sinePower_first P.p θ hθ).congr_of_eventuallyEq he).differentiableAt.hasDerivAt)
-    hy b
-  have hzero := P.shifted_sine_generator P.p (z-b).arg hs
-  simp only [sub_self,zero_mul,mul_zero,zero_div] at hzero
-  have heq : (fun x : State => green P (x,b)) = fun x => 3*P.physicalFactor (fun _ => 0) x b := by
-    funext x
-    simp only [green,Parameter.physicalFactor,Parameter.f,zero_mul,add_zero]
-    ring
-  have hfd := P.contDiffAt_physical_factor (fun _ => 0) u b z hy ∞ contDiffAt_const
-  rw [heq,fderiv_const_mul (hfd.differentiableAt (by simp))]
-  simp only [smul_apply,smul_eq_mul]
-  rw [P.physical_factor_state_derivative (fun _ => 0) u 0 b _ z _ hy (by simpa using hasDerivAt_const u (0:ℝ))]
-  simp only [Parameter.f,zero_mul,add_zero]
-  have hsecond : deriv (deriv (fun c => green P ((u,z),c))) b =
-      Real.exp (P.p*u/2)*deriv (deriv (fun c : ℝ => Real.sin (z-c).arg^P.p)) b := by
-    change deriv (deriv (fun c : ℝ => Real.exp (P.p*u/2)*Real.sin (z-c).arg^P.p)) b = _
-    rw [deriv_const_mul_field',deriv_const_mul_field]
-  rw [hsecond]
-  unfold physicalGenerator at hphy
-  change _ = (V (z-b))^2*P.angleGenerator (fun θ => Real.sin θ^P.p) (z-b).arg at hphy
-  dsimp [field]
-  linear_combination Real.exp (P.p*u/2)*hphy + Real.exp (P.p*u/2)*(V (z-b))^2*hzero
-end SLEExactGauge.OnePoint
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-omit [CompleteSpace E] in
-theorem expected_stopped_smooth_grid {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hg0 : ∀ b, g b 0 = z)
-    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
-    (τ : Driver → ℝ)
-    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
-    (D : Set (E × ℝ)) (_mD : MeasurableSet D)
-    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ D)
-    (hgen : ∀ x ∈ D, A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
-    (S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mS : ∀ r, MeasurableSet (S r))
-    (hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r))
-    (hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ D))
-    (hSD : ∀ r q, q ∈ S r → q.2 ∈ D)
-    (ht : 0 ≤ t) :
-    (∫ b : Driver, A.F (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ A.F z 0 := by
-  let h : ℕ → ℝ≥0 := fun n => (1/2)^n
-  have hh : ∀ n, 0 < h n := fun n => pow_pos (by norm_num) n
-  have hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0) := by
-    simp only [h,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat]
-    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-  let U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)) := fun n k => S ((k:ℝ≥0)*h n)
-  have mU : ∀ n k, MeasurableSet (U n k) := fun n k => mS _
-  have hu : ∀ n (k : ℕ) b c, (∀ u ≤ (k:ℝ≥0)*h n, b u = c u) → ∀ x,
-      ((b,x) ∈ U n k ↔ (c,x) ∈ U n k) := fun _ _ _ _ => hSpast _ _ _
-  apply expected_frozen_limit hW A v mv U mU h (fun n => ⌊t/(h n:ℝ)⌋₊) (z,0) hu M hv
-    (fun _ _ b x hx => hgen x (hSD _ (b,x) hx))
-    (fun b => (g b (min t (τ b)), b (min t (τ b)).toNNReal)) t hh0
-    (EulerGeneral.mesh_times_converge (fun n => hh n) hh0 ht)
-  filter_upwards [] with b
-  obtain ⟨ε,hε,hDb⟩ := hD b
-  have hp := frozen_nodes_path_limit v (g b) z b b.continuous D U hL hM (hg0 b)
-    (fun _ _ => hlip _) (hder b) (hcf b) (fun u _ => hv _ _) (hτ b).1 (hτ b).2 ht
-    hε hDb hh hh0 (fun n k x => by simpa only [U,NNReal.coe_mul,NNReal.coe_natCast] using hSeq b ((k:ℝ≥0)*h n) x)
-  simpa only [map_zero] using hp
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
-theorem expected_stopped_local_grid {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
-    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
-    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hg0 : ∀ b, g b 0 = z)
-    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
-    (τ : Driver → ℝ)
-    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
-    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
-    (hgen : ∀ q ∈ K,
-      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
-        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 ≤ 0)
-    (S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mS : ∀ r, MeasurableSet (S r))
-    (hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r))
-    (hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K))
-    (hSD : ∀ r q, q ∈ S r → q.2 ∈ K)
-    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
-    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) ≤ f (z,0) := by
-  obtain ⟨G,hG,hGc,hGeq⟩ := Localization.compact_extension hK hO hKO f hf
-  obtain ⟨A,hA⟩ := exists_smoothTest_of_compact G hG hGc
-  have heq (q : E × ℝ) (hq : q ∈ K) : A.F q.1 q.2 = f q := by
-    rw [hA]
-    exact (hGeq q hq).self_of_nhds
-  have hDgen (q : E × ℝ) (hq : q ∈ K) :
-      A.D q.1 q.2 (v q.1 q.2)+deriv (deriv (A.F q.1)) q.2/2 ≤ 0 := by
-    have hx : (fun x => A.F x q.2) =ᶠ[𝓝 q.1] fun x => f (x,q.2) := by
-      rw [hA]
-      exact (hGeq q hq).comp_tendsto (continuousAt_id.prodMk continuousAt_const)
-    have hb : A.F q.1 =ᶠ[𝓝 q.2] fun b => f (q.1,b) := by
-      rw [hA]
-      exact (hGeq q hq).comp_tendsto (continuousAt_const.prodMk continuousAt_id)
-    rw [← (A.hD q.1 q.2).fderiv, hx.fderiv_eq, hb.deriv.deriv_eq]
-    exact hgen q hq
-  have hineq := expected_stopped_smooth_grid hW A v mv g z hL hM hv hlip hg0 hder hcf
-    τ hτ K hK.measurableSet hD hDgen S mS hSpast hSeq hSD ht
-  rw [heq (z,0) hz] at hineq
-  convert hineq using 1
-  apply integral_congr_ae
-  filter_upwards [] with b
-  symm
-  apply heq (g b (min t (τ b)), b (min t (τ b)).toNNReal)
-  obtain ⟨ε,hε,hDb⟩ := hD b
-  exact hDb (min t (τ b)) ⟨le_min ht (hτ b).1,min_le_right _ _⟩ _ (by simpa using hε)
-
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.BrownianCalculus
-open Classical
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
- 
-
-theorem expected_stopped_local_grid_eq {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
-    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
-    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
-    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
-    (hg0 : ∀ b, g b 0 = z)
-    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
-    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
-    (τ : Driver → ℝ)
-    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
-    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
-    (hgen : ∀ q ∈ K,
-      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
-        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 = 0)
-    (S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mS : ∀ r, MeasurableSet (S r))
-    (hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r))
-    (hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K))
-    (hSD : ∀ r q, q ∈ S r → q.2 ∈ K)
-    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
-    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) = f (z,0) := by
-  have hle := expected_stopped_local_grid hW f K O hK hO hKO hf v mv g z hL hM hv hlip
-    hg0 hder hcf τ hτ hD (fun q hq => (hgen q hq).le) S mS hSpast hSeq hSD hz ht
-  have hneg : ∀ q ∈ K,
-      fderiv ℝ (fun x => -f (x,q.2)) q.1 (v q.1 q.2) +
-        deriv (deriv (fun b => -f (q.1,b))) q.2 / 2 ≤ 0 := by
-    intro q hq
-    have hfd : DifferentiableAt ℝ (fun x => f (x,q.2)) q.1 :=
-      ((hf q (hKO hq)).contDiffAt (hO.mem_nhds (hKO hq))).differentiableAt (by simp) |>.comp q.1
-        (by fun_prop)
-    rw [fderiv_fun_neg]
-    simp only [neg_apply,deriv.fun_neg',deriv.fun_neg]
-    linarith [hgen q hq]
-  have hge := expected_stopped_local_grid hW (fun q => -f q) K O hK hO hKO hf.neg v mv g z hL hM hv hlip
-    hg0 hder hcf τ hτ hD hneg S mS hSpast hSeq hSD hz ht
-  rw [integral_neg] at hge
-  linarith
-end SLEExactGauge.BrownianCalculus
-
-namespace SLEExactGauge.Localization
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-
- 
-
-theorem bounded_lipschitz_ode (v : E → ℝ → E)
-    (cv : Continuous (Function.uncurry v)) (b : ℝ → ℝ) (cb : Continuous b)
-    (z : E) {M L T : ℝ} (hM : 0 ≤ M) (hL : 0 ≤ L) (hT : 0 ≤ T)
-    (hbound : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) :
-    ∃ g : ℝ → E, g 0 = z ∧ ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt g (v (g t) (b t)) (Icc 0 T) t := by
-  let M' : ℝ≥0 := ⟨M,hM⟩
-  let L' : ℝ≥0 := ⟨L,hL⟩
-  let R : ℝ≥0 := ⟨M*T,mul_nonneg hM hT⟩
-  have hp : IsPicardLindelof (fun t x => v x (b t)) (tmin := 0) (tmax := T)
-      ⟨0,⟨le_rfl,hT⟩⟩ z R 0 M' L' := by
-    constructor
-    · intro t _
-      apply LipschitzWith.lipschitzOnWith
-      apply LipschitzWith.of_dist_le_mul
-      intro x y
-      change dist (v x (b t)) (v y (b t)) ≤ L * dist x y
-      simpa only [dist_eq_norm] using hlip (b t) x y
-    · intro x _
-      exact (cv.comp (continuous_const.prodMk cb)).continuousOn
-    · intro t _ x _
-      exact hbound x (b t)
-    · change M*max (T-0) (0-0) ≤ M*T-0
-      simp only [sub_zero,sub_self,max_eq_left hT,le_refl]
-  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
-
-end SLEExactGauge.Localization
-
-open MeasureTheory
-namespace SLEExactGauge.ODEPath
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
-  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
-
-def mesh (n : ℕ) : ℝ := (1/2)^n
-lemma mesh_pos (n : ℕ) : 0 < mesh n := pow_pos (by norm_num) n
-lemma mesh_zero : Tendsto mesh atTop (𝓝 0) :=
-  tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
-
-def drift (v : E → ℝ → E) (b : ℝ≥0 → ℝ) (t : ℝ) (x : E) : E := v x (b t.toNNReal)
-def flow (v : E → ℝ → E) (z : E) (b : ℝ≥0 → ℝ) (t : ℝ) : E :=
-  limUnder atTop (fun n => EulerGeneral.interpolate (drift v b) (mesh n) z t)
-
-omit [CompleteSpace E] in
-lemma measurable_nodes (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (h : ℝ) (z : E) (k : ℕ) :
-    Measurable (fun b : ℝ≥0 → ℝ => EulerGeneral.nodes (drift v b) h z k) := by
-  induction k with
-  | zero => exact measurable_const
-  | succ k ih =>
-    exact ih.add (measurable_const.smul (mv.comp (ih.prodMk (measurable_pi_apply _))))
-
-omit [CompleteSpace E] in
-lemma measurable_interpolate (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
-    (h : ℝ) (z : E) (t : ℝ) :
-    Measurable (fun b : ℝ≥0 → ℝ => EulerGeneral.interpolate (drift v b) h z t) := by
-  unfold EulerGeneral.interpolate
-  exact (measurable_const.smul (measurable_nodes v mv h z _)).add
-    (measurable_const.smul (measurable_nodes v mv h z _))
-
-lemma measurable_flow (v : E → ℝ → E) (mv : Measurable (Function.uncurry v)) (z : E) (t : ℝ) :
-    Measurable (fun b : ℝ≥0 → ℝ => flow v z b t) :=
-  (StronglyMeasurable.limUnder (fun n => (measurable_interpolate v mv (mesh n) z t).stronglyMeasurable)).measurable
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma nodes_past (v : E → ℝ → E) {h : ℝ} (hh : 0 < h) (z : E)
-    {b c : ℝ≥0 → ℝ} {s : ℝ} (hs : 0 ≤ s)
-    (he : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u)
-    (k : ℕ) (hk : (k:ℝ)*h ≤ s+h) :
-    EulerGeneral.nodes (drift v b) h z k = EulerGeneral.nodes (drift v c) h z k := by
-  induction k with
-  | zero => rfl
-  | succ k ih =>
-    have hk' : (k:ℝ)*h ≤ s := by push_cast at hk; nlinarith
-    have hb : b (((k:ℝ)*h).toNNReal) = c (((k:ℝ)*h).toNNReal) :=
-      he _ (by simpa only [Real.coe_toNNReal',max_le_iff] using And.intro hk' hs)
-    simp only [EulerGeneral.nodes,drift,ih (by linarith),hb]
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma interpolate_past (v : E → ℝ → E) {h : ℝ} (hh : 0 < h) (z : E)
-    {b c : ℝ≥0 → ℝ} {s : ℝ} (hs : 0 ≤ s)
-    (he : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) :
-    EulerGeneral.interpolate (drift v b) h z s = EulerGeneral.interpolate (drift v c) h z s := by
-  have hk : (⌊s/h⌋₊:ℝ)*h ≤ s := (le_div_iff₀ hh).mp (Nat.floor_le (div_nonneg hs hh.le))
-  unfold EulerGeneral.interpolate
-  dsimp only
-  rw [nodes_past v hh z hs he _ (by linarith),nodes_past v hh z hs he _ (by push_cast; linarith)]
-
-omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma flow_past (v : E → ℝ → E) (z : E) {b c : ℝ≥0 → ℝ} {s : ℝ} (hs : 0 ≤ s)
-    (he : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) : flow v z b s = flow v z c s := by
-  unfold flow
-  congr 1
-  funext n
-  exact interpolate_past v (mesh_pos n) z hs he
-
- 
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-theorem flow_solution (v : E → ℝ → E) (cv : Continuous (Function.uncurry v))
-    (z : E) (b : ℝ≥0 → ℝ) (cb : Continuous b) {M L T : ℝ}
-    (hM : 0 ≤ M) (hL : 0 ≤ L) (hT : 0 ≤ T)
-    (hbound : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) :
-    flow v z b 0 = z ∧ ∀ t ∈ Icc 0 T, HasDerivWithinAt (flow v z b)
-      (v (flow v z b t) (b t.toNNReal)) (Icc 0 T) t := by
-  obtain ⟨g,hg0,hg⟩ := Localization.bounded_lipschitz_ode v cv
-    (fun t => b t.toNNReal) (cb.comp continuous_real_toNNReal) z hM hL (by linarith : 0 ≤ T+1) hbound hlip
-  have hgc : ContinuousOn g (Icc 0 (T+1)) := fun t ht => (hg t ht).continuousWithinAt
-  have hcf : ContinuousOn (fun t => v (g t) (b t.toNNReal)) (Icc 0 (T+1)) :=
-    cv.comp_continuousOn (hgc.prodMk ((cb.comp continuous_real_toNNReal).continuousOn))
-  have hconv : TendstoUniformlyOn (fun n => EulerGeneral.interpolate (drift v b) (mesh n) z)
-      g atTop (Icc 0 T) :=
-    EulerGeneral.converge_uniformOn hL hM hg0 (fun _ _ => hlip _) hg hcf
-      (fun _ _ => hbound _ _) hT (by linarith) mesh_pos mesh_zero
-  have heq : ∀ t ∈ Icc 0 T, flow v z b t = g t :=
-    fun t ht => (hconv.tendsto_at ht).limUnder_eq
-  refine ⟨(heq 0 ⟨le_rfl,hT⟩).trans hg0,?_⟩
-  intro t ht
-  rw [heq t ht]
-  exact ((hg t ⟨ht.1,by linarith [ht.2]⟩).mono
-    (show Icc 0 T ⊆ Icc 0 (T+1) from fun u hu => ⟨hu.1,by linarith [hu.2]⟩)).congr heq (heq t ht)
-
-omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
-lemma continuousOn_flow (v : E → ℝ → E) (cv : Continuous (Function.uncurry v))
-    (z : E) (b : ℝ≥0 → ℝ) (cb : Continuous b) {M L T : ℝ}
-    (hM : 0 ≤ M) (hL : 0 ≤ L) (hT : 0 ≤ T)
-    (hbound : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) :
-    ContinuousOn (flow v z b) (Icc 0 T) :=
-  fun t ht => ((flow_solution v cv z b cb hM hL hT hbound hlip).2 t ht).continuousWithinAt
-
-end SLEExactGauge.ODEPath
-
-namespace SLEExactGauge.RationalStop
-
-def sample (s r : ℝ) : ℝ := max 0 (min s r)
-lemma sample_mem {s : ℝ} (hs : 0 ≤ s) (r : ℝ) : sample s r ∈ Icc 0 s :=
-  ⟨le_max_left _ _,max_le hs (min_le_left _ _)⟩
-lemma sample_self {s u : ℝ} (hu : u ∈ Icc 0 s) : sample s u = u := by
-  simp [sample,min_eq_right hu.2,max_eq_right hu.1]
-lemma continuous_sample (s : ℝ) : Continuous (sample s) := by unfold sample; fun_prop
-
- 
-
-def Safe (f : ℝ → ℝ) (R s : ℝ) : Prop :=
-  ∃ n : ℕ, ∀ r : ℚ, f (sample s r) ≤ R-1/((n:ℝ)+1)
-
-lemma safe_iff {f : ℝ → ℝ} {R s : ℝ} (hs : 0 ≤ s)
-    (hf : ContinuousOn f (Icc 0 s)) : Safe f R s ↔ ∀ u ∈ Icc 0 s, f u < R := by
-  constructor
-  · rintro ⟨n,hn⟩ u hu
-    have hc : Continuous (fun x => f (sample s x)) :=
-      hf.comp_continuous (continuous_sample s) (sample_mem hs)
-    have h : f (sample s u) ≤ R-1/((n:ℝ)+1) := Rat.denseRange_cast.induction_on u
-      (isClosed_le hc continuous_const) hn
-    rw [sample_self hu] at h
-    have hp : 0 < 1/((n:ℝ)+1) := by positivity
-    linarith
-  · intro h
-    obtain ⟨u,hu,hm⟩ := isCompact_Icc.exists_isMaxOn ⟨0,⟨le_rfl,hs⟩⟩ hf
-    obtain ⟨n,hn⟩ := exists_nat_one_div_lt (sub_pos.mpr (h u hu))
-    refine ⟨n,fun r => ?_⟩
-    have hr : f (sample s r) ≤ f u := hm (sample_mem hs (r:ℝ))
-    linarith
-
-end SLEExactGauge.RationalStop
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.EulerConvergence
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+ 
+
+theorem expected_stopped_local_eq {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
+    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
+    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hg0 : ∀ b, g b 0 = z)
+    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
+    (τ : (ℝ≥0 → ℝ) → ℝ) (mτ : Measurable τ)
+    (hτpast : ∀ (s : ℝ≥0) b c, (∀ u ≤ s, b u = c u) → ((s:ℝ) < τ b ↔ (s:ℝ) < τ c))
+    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
+    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
+    (hgen : ∀ q ∈ K,
+      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
+        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 = 0)
+    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
+    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) = f (z,0) := by
+  have hle := expected_stopped_local hW f K O hK hO hKO hf v mv g z hL hM hv hlip
+    hg0 hder hcf τ mτ hτpast hτ hD (fun q hq => (hgen q hq).le) hz ht
+  have hneg : ∀ q ∈ K,
+      fderiv ℝ (fun x => -f (x,q.2)) q.1 (v q.1 q.2) +
+        deriv (deriv (fun b => -f (q.1,b))) q.2 / 2 ≤ 0 := by
+    intro q hq
+    have hfd : DifferentiableAt ℝ (fun x => f (x,q.2)) q.1 :=
+      ((hf q (hKO hq)).contDiffAt (hO.mem_nhds (hKO hq))).differentiableAt (by simp) |>.comp q.1
+        (by fun_prop)
+    rw [fderiv_fun_neg]
+    simp only [neg_apply,deriv.fun_neg',deriv.fun_neg]
+    linarith [hgen q hq]
+  have hge := expected_stopped_local hW (fun q => -f q) K O hK hO hKO hf.neg v mv g z hL hM hv hlip
+    hg0 hder hcf τ mτ hτpast hτ hD hneg hz ht
+  rw [integral_neg] at hge
+  linarith
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.OnePoint
+open Parameter AngleCalculus
+open scoped ContDiff
+abbrev State := ℝ × ℂ
+
+def domain : Set (State × ℝ) := {q | 0 < q.1.2.im}
+def field (P : Parameter) (x : State) (b : ℝ) : State :=
+  ((V (x.2-b))^2, (P.a:ℂ)/(x.2-b))
+ 
+def green (P : Parameter) (q : State × ℝ) : ℝ :=
+  Real.exp (P.p*q.1.1/2) * Real.sin (q.1.2-q.2).arg ^ P.p
+
+lemma isOpen_domain : IsOpen domain := by
+  exact isOpen_lt continuous_const (by fun_prop)
+
+lemma green_nonneg (P : Parameter) {q : State × ℝ} (hq : q ∈ domain) : 0 ≤ green P q := by
+  exact mul_nonneg (Real.exp_pos _).le (Real.rpow_nonneg (sin_angle_pos (by simpa [domain] using hq)).le _)
+
+lemma green_le (P : Parameter) {s : ℝ} {q : State × ℝ} (hu : q.1.1 ≤ s)
+    (hq : q ∈ domain) : green P q ≤ Real.exp (P.p*s/2) := by
+  have hs : 0 < Real.sin (q.1.2-(q.2:ℂ)).arg := sin_angle_pos (by simpa [domain] using hq)
+  have hp : Real.sin (q.1.2-(q.2:ℂ)).arg ^ P.p ≤ 1 :=
+    Real.rpow_le_one hs.le (Real.sin_le_one _) P.p_pos.le
+  calc
+    green P q ≤ Real.exp (P.p*q.1.1/2) := by
+      exact mul_le_of_le_one_right (Real.exp_pos _).le hp
+    _ ≤ Real.exp (P.p*s/2) := Real.exp_le_exp.mpr (by nlinarith [P.p_pos])
+
+lemma contDiffOn_green (P : Parameter) : ContDiffOn ℝ ∞ (green P) domain := by
+  intro q hq
+  apply ContDiffAt.contDiffWithinAt
+  have hh : 0 < (q.1.2-(q.2:ℂ)).im := by simpa [domain] using hq
+  have ha := (contDiffAt_arg_real (show q.1.2-(q.2:ℂ) ∈ Complex.slitPlane from Or.inr hh.ne') ∞).comp q
+    (show ContDiffAt ℝ ∞ (fun r : State × ℝ => r.1.2-(r.2:ℂ)) q from
+      (contDiffAt_snd.comp q contDiffAt_fst).sub
+        (Complex.ofRealCLM.contDiff.contDiffAt.comp q contDiffAt_snd))
+  have hs := ha.sin.rpow_const_of_ne (p := P.p) (sin_angle_pos hh).ne'
+  exact (show ContDiffAt ℝ ∞ (fun r : State × ℝ => Real.exp (P.p*r.1.1/2)) q by fun_prop).mul hs
+
+lemma green_generator (P : Parameter) {q : State × ℝ} (hq : q ∈ domain) :
+    fderiv ℝ (fun x => green P (x,q.2)) q.1 (field P q.1 q.2) +
+      deriv (deriv (fun b => green P (q.1,b))) q.2 / 2 = 0 := by
+  rcases q with ⟨⟨u,z⟩,b⟩
+  have hy : 0 < z.im := hq
+  have hs : 0 < Real.sin (z-(b:ℂ)).arg := sin_angle_pos (by simpa using hy)
+  have hphy := physical_generator_angle P.a (fun θ => Real.sin θ^P.p)
+    (fun θ hθ => (hasDerivAt_sinePower P.p θ hθ).differentiableAt.hasDerivAt)
+    (fun θ hθ => by
+      have he : deriv (fun t => Real.sin t^P.p) =ᶠ[𝓝 θ] fun t => P.p*Real.cos t*Real.sin t^(P.p-1) := by
+        filter_upwards [Real.continuous_sin.continuousAt.eventually (Ioi_mem_nhds hθ)] with t ht
+        exact deriv_sinePower P.p t ht
+      exact ((hasDerivAt_sinePower_first P.p θ hθ).congr_of_eventuallyEq he).differentiableAt.hasDerivAt)
+    hy b
+  have hzero := P.shifted_sine_generator P.p (z-b).arg hs
+  simp only [sub_self,zero_mul,mul_zero,zero_div] at hzero
+  have heq : (fun x : State => green P (x,b)) = fun x => 3*P.physicalFactor (fun _ => 0) x b := by
+    funext x
+    simp only [green,Parameter.physicalFactor,Parameter.f,zero_mul,add_zero]
+    ring
+  have hfd := P.contDiffAt_physical_factor (fun _ => 0) u b z hy ∞ contDiffAt_const
+  rw [heq,fderiv_const_mul (hfd.differentiableAt (by simp))]
+  simp only [smul_apply,smul_eq_mul]
+  rw [P.physical_factor_state_derivative (fun _ => 0) u 0 b _ z _ hy (by simpa using hasDerivAt_const u (0:ℝ))]
+  simp only [Parameter.f,zero_mul,add_zero]
+  have hsecond : deriv (deriv (fun c => green P ((u,z),c))) b =
+      Real.exp (P.p*u/2)*deriv (deriv (fun c : ℝ => Real.sin (z-c).arg^P.p)) b := by
+    change deriv (deriv (fun c : ℝ => Real.exp (P.p*u/2)*Real.sin (z-c).arg^P.p)) b = _
+    rw [deriv_const_mul_field',deriv_const_mul_field]
+  rw [hsecond]
+  unfold physicalGenerator at hphy
+  change _ = (V (z-b))^2*P.angleGenerator (fun θ => Real.sin θ^P.p) (z-b).arg at hphy
+  dsimp [field]
+  linear_combination Real.exp (P.p*u/2)*hphy + Real.exp (P.p*u/2)*(V (z-b))^2*hzero
+end SLEExactGauge.OnePoint
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+omit [CompleteSpace E] in
+theorem expected_stopped_smooth_grid {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (A : SmoothTest E) (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hg0 : ∀ b, g b 0 = z)
+    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
+    (τ : Driver → ℝ)
+    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
+    (D : Set (E × ℝ)) (_mD : MeasurableSet D)
+    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ D)
+    (hgen : ∀ x ∈ D, A.D x.1 x.2 (v x.1 x.2)+deriv (deriv (A.F x.1)) x.2/2 ≤ 0)
+    (S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mS : ∀ r, MeasurableSet (S r))
+    (hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r))
+    (hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ D))
+    (hSD : ∀ r q, q ∈ S r → q.2 ∈ D)
+    (ht : 0 ≤ t) :
+    (∫ b : Driver, A.F (g b (min t (τ b))) (b (min t (τ b)).toNNReal) ∂W) ≤ A.F z 0 := by
+  let h : ℕ → ℝ≥0 := fun n => (1/2)^n
+  have hh : ∀ n, 0 < h n := fun n => pow_pos (by norm_num) n
+  have hh0 : Tendsto (fun n => (h n:ℝ)) atTop (𝓝 0) := by
+    simp only [h,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat]
+    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+  let U : ℕ → ℕ → Set ((ℝ≥0 → ℝ) × (E × ℝ)) := fun n k => S ((k:ℝ≥0)*h n)
+  have mU : ∀ n k, MeasurableSet (U n k) := fun n k => mS _
+  have hu : ∀ n (k : ℕ) b c, (∀ u ≤ (k:ℝ≥0)*h n, b u = c u) → ∀ x,
+      ((b,x) ∈ U n k ↔ (c,x) ∈ U n k) := fun _ _ _ _ => hSpast _ _ _
+  apply expected_frozen_limit hW A v mv U mU h (fun n => ⌊t/(h n:ℝ)⌋₊) (z,0) hu M hv
+    (fun _ _ b x hx => hgen x (hSD _ (b,x) hx))
+    (fun b => (g b (min t (τ b)), b (min t (τ b)).toNNReal)) t hh0
+    (EulerGeneral.mesh_times_converge (fun n => hh n) hh0 ht)
+  filter_upwards [] with b
+  obtain ⟨ε,hε,hDb⟩ := hD b
+  have hp := frozen_nodes_path_limit v (g b) z b b.continuous D U hL hM (hg0 b)
+    (fun _ _ => hlip _) (hder b) (hcf b) (fun u _ => hv _ _) (hτ b).1 (hτ b).2 ht
+    hε hDb hh hh0 (fun n k x => by simpa only [U,NNReal.coe_mul,NNReal.coe_natCast] using hSeq b ((k:ℝ≥0)*h n) x)
+  simpa only [map_zero] using hp
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+theorem expected_stopped_local_grid {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
+    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
+    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hg0 : ∀ b, g b 0 = z)
+    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
+    (τ : Driver → ℝ)
+    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
+    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
+    (hgen : ∀ q ∈ K,
+      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
+        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 ≤ 0)
+    (S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mS : ∀ r, MeasurableSet (S r))
+    (hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r))
+    (hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K))
+    (hSD : ∀ r q, q ∈ S r → q.2 ∈ K)
+    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
+    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) ≤ f (z,0) := by
+  obtain ⟨G,hG,hGc,hGeq⟩ := Localization.compact_extension hK hO hKO f hf
+  obtain ⟨A,hA⟩ := exists_smoothTest_of_compact G hG hGc
+  have heq (q : E × ℝ) (hq : q ∈ K) : A.F q.1 q.2 = f q := by
+    rw [hA]
+    exact (hGeq q hq).self_of_nhds
+  have hDgen (q : E × ℝ) (hq : q ∈ K) :
+      A.D q.1 q.2 (v q.1 q.2)+deriv (deriv (A.F q.1)) q.2/2 ≤ 0 := by
+    have hx : (fun x => A.F x q.2) =ᶠ[𝓝 q.1] fun x => f (x,q.2) := by
+      rw [hA]
+      exact (hGeq q hq).comp_tendsto (continuousAt_id.prodMk continuousAt_const)
+    have hb : A.F q.1 =ᶠ[𝓝 q.2] fun b => f (q.1,b) := by
+      rw [hA]
+      exact (hGeq q hq).comp_tendsto (continuousAt_const.prodMk continuousAt_id)
+    rw [← (A.hD q.1 q.2).fderiv, hx.fderiv_eq, hb.deriv.deriv_eq]
+    exact hgen q hq
+  have hineq := expected_stopped_smooth_grid hW A v mv g z hL hM hv hlip hg0 hder hcf
+    τ hτ K hK.measurableSet hD hDgen S mS hSpast hSeq hSD ht
+  rw [heq (z,0) hz] at hineq
+  convert hineq using 1
+  apply integral_congr_ae
+  filter_upwards [] with b
+  symm
+  apply heq (g b (min t (τ b)), b (min t (τ b)).toNNReal)
+  obtain ⟨ε,hε,hDb⟩ := hD b
+  exact hDb (min t (τ b)) ⟨le_min ht (hτ b).1,min_le_right _ _⟩ _ (by simpa using hε)
+
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.BrownianCalculus
+open Classical
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+ 
+
+theorem expected_stopped_local_grid_eq {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (f : E × ℝ → ℝ) (K O : Set (E × ℝ))
+    (hK : IsCompact K) (hO : IsOpen O) (hKO : K ⊆ O) (hf : ContDiffOn ℝ ∞ f O)
+    (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (g : Driver → ℝ → E) (z : E) {L M T t : ℝ} (hL : 0 ≤ L) (hM : 0 ≤ M)
+    (hv : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖)
+    (hg0 : ∀ b, g b 0 = z)
+    (hder : ∀ b u, u ∈ Icc 0 T → HasDerivWithinAt (g b) (v (g b u) (b u.toNNReal)) (Icc 0 T) u)
+    (hcf : ∀ b, ContinuousOn (fun u => v (g b u) (b u.toNNReal)) (Icc 0 T))
+    (τ : Driver → ℝ)
+    (hτ : ∀ b : Driver, 0 ≤ τ b ∧ τ b < T)
+    (hD : ∀ b : Driver, ∃ ε > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < ε → (x,b u.toNNReal) ∈ K)
+    (hgen : ∀ q ∈ K,
+      fderiv ℝ (fun x => f (x,q.2)) q.1 (v q.1 q.2) +
+        deriv (deriv (fun b => f (q.1,b))) q.2 / 2 = 0)
+    (S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (E × ℝ))) (mS : ∀ r, MeasurableSet (S r))
+    (hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r))
+    (hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K))
+    (hSD : ∀ r q, q ∈ S r → q.2 ∈ K)
+    (hz : (z,0) ∈ K) (ht : 0 ≤ t) :
+    (∫ b : Driver, f (g b (min t (τ b)), b (min t (τ b)).toNNReal) ∂W) = f (z,0) := by
+  have hle := expected_stopped_local_grid hW f K O hK hO hKO hf v mv g z hL hM hv hlip
+    hg0 hder hcf τ hτ hD (fun q hq => (hgen q hq).le) S mS hSpast hSeq hSD hz ht
+  have hneg : ∀ q ∈ K,
+      fderiv ℝ (fun x => -f (x,q.2)) q.1 (v q.1 q.2) +
+        deriv (deriv (fun b => -f (q.1,b))) q.2 / 2 ≤ 0 := by
+    intro q hq
+    have hfd : DifferentiableAt ℝ (fun x => f (x,q.2)) q.1 :=
+      ((hf q (hKO hq)).contDiffAt (hO.mem_nhds (hKO hq))).differentiableAt (by simp) |>.comp q.1
+        (by fun_prop)
+    rw [fderiv_fun_neg]
+    simp only [neg_apply,deriv.fun_neg',deriv.fun_neg]
+    linarith [hgen q hq]
+  have hge := expected_stopped_local_grid hW (fun q => -f q) K O hK hO hKO hf.neg v mv g z hL hM hv hlip
+    hg0 hder hcf τ hτ hD hneg S mS hSpast hSeq hSD hz ht
+  rw [integral_neg] at hge
+  linarith
+end SLEExactGauge.BrownianCalculus
+
+namespace SLEExactGauge.Localization
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ 
+
+theorem bounded_lipschitz_ode (v : E → ℝ → E)
+    (cv : Continuous (Function.uncurry v)) (b : ℝ → ℝ) (cb : Continuous b)
+    (z : E) {M L T : ℝ} (hM : 0 ≤ M) (hL : 0 ≤ L) (hT : 0 ≤ T)
+    (hbound : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) :
+    ∃ g : ℝ → E, g 0 = z ∧ ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt g (v (g t) (b t)) (Icc 0 T) t := by
+  let M' : ℝ≥0 := ⟨M,hM⟩
+  let L' : ℝ≥0 := ⟨L,hL⟩
+  let R : ℝ≥0 := ⟨M*T,mul_nonneg hM hT⟩
+  have hp : IsPicardLindelof (fun t x => v x (b t)) (tmin := 0) (tmax := T)
+      ⟨0,⟨le_rfl,hT⟩⟩ z R 0 M' L' := by
+    constructor
+    · intro t _
+      apply LipschitzWith.lipschitzOnWith
+      apply LipschitzWith.of_dist_le_mul
+      intro x y
+      change dist (v x (b t)) (v y (b t)) ≤ L * dist x y
+      simpa only [dist_eq_norm] using hlip (b t) x y
+    · intro x _
+      exact (cv.comp (continuous_const.prodMk cb)).continuousOn
+    · intro t _ x _
+      exact hbound x (b t)
+    · change M*max (T-0) (0-0) ≤ M*T-0
+      simp only [sub_zero,sub_self,max_eq_left hT,le_refl]
+  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
+
+end SLEExactGauge.Localization
+
+open MeasureTheory
+namespace SLEExactGauge.ODEPath
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
+
+def mesh (n : ℕ) : ℝ := (1/2)^n
+lemma mesh_pos (n : ℕ) : 0 < mesh n := pow_pos (by norm_num) n
+lemma mesh_zero : Tendsto mesh atTop (𝓝 0) :=
+  tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+
+def drift (v : E → ℝ → E) (b : ℝ≥0 → ℝ) (t : ℝ) (x : E) : E := v x (b t.toNNReal)
+def flow (v : E → ℝ → E) (z : E) (b : ℝ≥0 → ℝ) (t : ℝ) : E :=
+  limUnder atTop (fun n => EulerGeneral.interpolate (drift v b) (mesh n) z t)
+
+omit [CompleteSpace E] in
+lemma measurable_nodes (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (h : ℝ) (z : E) (k : ℕ) :
+    Measurable (fun b : ℝ≥0 → ℝ => EulerGeneral.nodes (drift v b) h z k) := by
+  induction k with
+  | zero => exact measurable_const
+  | succ k ih =>
+    exact ih.add (measurable_const.smul (mv.comp (ih.prodMk (measurable_pi_apply _))))
+
+omit [CompleteSpace E] in
+lemma measurable_interpolate (v : E → ℝ → E) (mv : Measurable (Function.uncurry v))
+    (h : ℝ) (z : E) (t : ℝ) :
+    Measurable (fun b : ℝ≥0 → ℝ => EulerGeneral.interpolate (drift v b) h z t) := by
+  unfold EulerGeneral.interpolate
+  exact (measurable_const.smul (measurable_nodes v mv h z _)).add
+    (measurable_const.smul (measurable_nodes v mv h z _))
+
+lemma measurable_flow (v : E → ℝ → E) (mv : Measurable (Function.uncurry v)) (z : E) (t : ℝ) :
+    Measurable (fun b : ℝ≥0 → ℝ => flow v z b t) :=
+  (StronglyMeasurable.limUnder (fun n => (measurable_interpolate v mv (mesh n) z t).stronglyMeasurable)).measurable
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma nodes_past (v : E → ℝ → E) {h : ℝ} (hh : 0 < h) (z : E)
+    {b c : ℝ≥0 → ℝ} {s : ℝ} (hs : 0 ≤ s)
+    (he : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u)
+    (k : ℕ) (hk : (k:ℝ)*h ≤ s+h) :
+    EulerGeneral.nodes (drift v b) h z k = EulerGeneral.nodes (drift v c) h z k := by
+  induction k with
+  | zero => rfl
+  | succ k ih =>
+    have hk' : (k:ℝ)*h ≤ s := by push_cast at hk; nlinarith
+    have hb : b (((k:ℝ)*h).toNNReal) = c (((k:ℝ)*h).toNNReal) :=
+      he _ (by simpa only [Real.coe_toNNReal',max_le_iff] using And.intro hk' hs)
+    simp only [EulerGeneral.nodes,drift,ih (by linarith),hb]
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma interpolate_past (v : E → ℝ → E) {h : ℝ} (hh : 0 < h) (z : E)
+    {b c : ℝ≥0 → ℝ} {s : ℝ} (hs : 0 ≤ s)
+    (he : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) :
+    EulerGeneral.interpolate (drift v b) h z s = EulerGeneral.interpolate (drift v c) h z s := by
+  have hk : (⌊s/h⌋₊:ℝ)*h ≤ s := (le_div_iff₀ hh).mp (Nat.floor_le (div_nonneg hs hh.le))
+  unfold EulerGeneral.interpolate
+  dsimp only
+  rw [nodes_past v hh z hs he _ (by linarith),nodes_past v hh z hs he _ (by push_cast; linarith)]
+
+omit [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma flow_past (v : E → ℝ → E) (z : E) {b c : ℝ≥0 → ℝ} {s : ℝ} (hs : 0 ≤ s)
+    (he : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) : flow v z b s = flow v z c s := by
+  unfold flow
+  congr 1
+  funext n
+  exact interpolate_past v (mesh_pos n) z hs he
+
+ 
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+theorem flow_solution (v : E → ℝ → E) (cv : Continuous (Function.uncurry v))
+    (z : E) (b : ℝ≥0 → ℝ) (cb : Continuous b) {M L T : ℝ}
+    (hM : 0 ≤ M) (hL : 0 ≤ L) (hT : 0 ≤ T)
+    (hbound : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) :
+    flow v z b 0 = z ∧ ∀ t ∈ Icc 0 T, HasDerivWithinAt (flow v z b)
+      (v (flow v z b t) (b t.toNNReal)) (Icc 0 T) t := by
+  obtain ⟨g,hg0,hg⟩ := Localization.bounded_lipschitz_ode v cv
+    (fun t => b t.toNNReal) (cb.comp continuous_real_toNNReal) z hM hL (by linarith : 0 ≤ T+1) hbound hlip
+  have hgc : ContinuousOn g (Icc 0 (T+1)) := fun t ht => (hg t ht).continuousWithinAt
+  have hcf : ContinuousOn (fun t => v (g t) (b t.toNNReal)) (Icc 0 (T+1)) :=
+    cv.comp_continuousOn (hgc.prodMk ((cb.comp continuous_real_toNNReal).continuousOn))
+  have hconv : TendstoUniformlyOn (fun n => EulerGeneral.interpolate (drift v b) (mesh n) z)
+      g atTop (Icc 0 T) :=
+    EulerGeneral.converge_uniformOn hL hM hg0 (fun _ _ => hlip _) hg hcf
+      (fun _ _ => hbound _ _) hT (by linarith) mesh_pos mesh_zero
+  have heq : ∀ t ∈ Icc 0 T, flow v z b t = g t :=
+    fun t ht => (hconv.tendsto_at ht).limUnder_eq
+  refine ⟨(heq 0 ⟨le_rfl,hT⟩).trans hg0,?_⟩
+  intro t ht
+  rw [heq t ht]
+  exact ((hg t ⟨ht.1,by linarith [ht.2]⟩).mono
+    (show Icc 0 T ⊆ Icc 0 (T+1) from fun u hu => ⟨hu.1,by linarith [hu.2]⟩)).congr heq (heq t ht)
+
+omit [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] in
+lemma continuousOn_flow (v : E → ℝ → E) (cv : Continuous (Function.uncurry v))
+    (z : E) (b : ℝ≥0 → ℝ) (cb : Continuous b) {M L T : ℝ}
+    (hM : 0 ≤ M) (hL : 0 ≤ L) (hT : 0 ≤ T)
+    (hbound : ∀ x b, ‖v x b‖ ≤ M) (hlip : ∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) :
+    ContinuousOn (flow v z b) (Icc 0 T) :=
+  fun t ht => ((flow_solution v cv z b cb hM hL hT hbound hlip).2 t ht).continuousWithinAt
+
+end SLEExactGauge.ODEPath
+
+namespace SLEExactGauge.RationalStop
+
+def sample (s r : ℝ) : ℝ := max 0 (min s r)
+lemma sample_mem {s : ℝ} (hs : 0 ≤ s) (r : ℝ) : sample s r ∈ Icc 0 s :=
+  ⟨le_max_left _ _,max_le hs (min_le_left _ _)⟩
+lemma sample_self {s u : ℝ} (hu : u ∈ Icc 0 s) : sample s u = u := by
+  simp [sample,min_eq_right hu.2,max_eq_right hu.1]
+lemma continuous_sample (s : ℝ) : Continuous (sample s) := by unfold sample; fun_prop
+
+ 
+
+def Safe (f : ℝ → ℝ) (R s : ℝ) : Prop :=
+  ∃ n : ℕ, ∀ r : ℚ, f (sample s r) ≤ R-1/((n:ℝ)+1)
+
+lemma safe_iff {f : ℝ → ℝ} {R s : ℝ} (hs : 0 ≤ s)
+    (hf : ContinuousOn f (Icc 0 s)) : Safe f R s ↔ ∀ u ∈ Icc 0 s, f u < R := by
+  constructor
+  · rintro ⟨n,hn⟩ u hu
+    have hc : Continuous (fun x => f (sample s x)) :=
+      hf.comp_continuous (continuous_sample s) (sample_mem hs)
+    have h : f (sample s u) ≤ R-1/((n:ℝ)+1) := Rat.denseRange_cast.induction_on u
+      (isClosed_le hc continuous_const) hn
+    rw [sample_self hu] at h
+    have hp : 0 < 1/((n:ℝ)+1) := by positivity
+    linarith
+  · intro h
+    obtain ⟨u,hu,hm⟩ := isCompact_Icc.exists_isMaxOn ⟨0,⟨le_rfl,hs⟩⟩ hf
+    obtain ⟨n,hn⟩ := exists_nat_one_div_lt (sub_pos.mpr (h u hu))
+    refine ⟨n,fun r => ?_⟩
+    have hr : f (sample s r) ≤ f u := hm (sample_mem hs (r:ℝ))
+    linarith
+
+end SLEExactGauge.RationalStop
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StoppedMassMeasurability.lean b/lean/OAI/Probability/SLEGauge/StoppedMassMeasurability.lean
index 2535e63e6e3de2e5e69150558698c90046a837d8..4312701551b1f92d333ef579456ea79dd1140906 100644
--- a/lean/OAI/Probability/SLEGauge/StoppedMassMeasurability.lean
+++ b/lean/OAI/Probability/SLEGauge/StoppedMassMeasurability.lean
@@ -1,547 +1,547 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StoppedFiniteMass
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-
-namespace ExactSLEGenerator
-
- 
-def stoppedPastDriver (S : Driver → ℝ≥0∞) (b : Driver) : Driver :=
-  if S b = ∞ then b else pastDriver (S b).toNNReal b
-
-lemma stoppedPastDriver_apply (S : Driver → ℝ≥0∞) (b : Driver) (t : ℝ≥0) :
-    (stoppedPastDriver S b).1 t = b.1 (ExactSLEGreen.capTime S t b) := by
-  by_cases hb : S b = ∞
-  · simp [stoppedPastDriver,hb,ExactSLEGreen.capTime]
-  · rw [stoppedPastDriver,ite_eq_right hb]
-    change b.1 (min t (S b).toNNReal) = b.1 (ExactSLEGreen.capTime S t b)
-    congr 1
-    unfold ExactSLEGreen.capTime
-    conv_rhs => rw [← ENNReal.coe_toNNReal hb]
-    simp only [← ENNReal.coe_min,ENNReal.toNNReal_coe,min_comm]
-
-lemma stoppedPastDriver_measurable (S : Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (stoppedPastDriver S) := by
-  have : PolishSpace Driver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
-  obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq (α := ℝ≥0)
-  let e : Driver → (ℕ → ℝ) := fun b n => b.1 (u n)
-  have hec : Continuous e := continuous_pi fun n =>
-    (continuous_eval_const (u n)).comp continuous_subtype_val
-  have hei : Function.Injective e := by
-    intro b c h
-    apply Subtype.ext
-    apply DFunLike.coe_injective
-    exact hu.equalizer b.1.continuous c.1.continuous h
-  apply hec.measurableEmbedding hei |>.measurable_comp_iff.mp
-  have hA : StronglyAdapted rawFiltration (fun t (b : Driver) => b.1 t) :=
-    Filtration.stronglyAdapted_natural driver_eval_measurable
-  have hP := hA.isStronglyProgressive_of_continuous (fun b => b.1.continuous)
-  have hval (n : ℕ) : Measurable[hS.measurableSpace]
-      (fun b => b.1 (ExactSLEGreen.capTime S (u n) b)) := by
-    have h := (measurable_stoppedValue hP (ExactSLEGreen.capTime_stopping S hS (u n))).mono
-      (ExactSLEGreen.stopped_cap_measurable S hS (u n)) le_rfl
-    exact h
-  let : MeasurableSpace Driver := hS.measurableSpace
-  apply measurable_pi_iff.mpr
-  intro n
-  simpa only [Function.comp_apply,e,stoppedPastDriver_apply] using hval n
-
-end ExactSLEGenerator
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma timeMassDensity_stoppedPastDriver (a : ℝ) (ha : 0<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    timeMassDensity a q (S b) b z =
-      timeMassDensity a q (S b) (stoppedPastDriver S b) z := by
-  by_cases hb : S b=∞
-  · simp [stoppedPastDriver,hb]
-  · conv_lhs => rw [← ENNReal.coe_toNNReal hb]
-    conv_rhs => rw [← ENNReal.coe_toNNReal hb]
-    apply timeMassDensity_driver_congr a ha q hq hq1
-    intro t ht
-    simpa only [stoppedPastDriver,ite_eq_right hb] using
-      (pastDriver_eq (S b).toNNReal b t ht).symm
-
- 
-theorem timeMassDensity_stopping_raw_joint (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    letI : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-    Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      timeMassDensity a q (S p.1) p.1 p.2) := by
-  have hp : @Measurable (ExactSLEGenerator.Driver × ℂ)
-      ((ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ)
-      (hS.measurableSpace.prod inferInstance) inferInstance
-      (fun p => ((S p.1,stoppedPastDriver S p.1),p.2)) := by
-    exact ((hS.measurable.comp (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
-      ((stoppedPastDriver_measurable S hS).comp
-        (@measurable_fst _ _ hS.measurableSpace inferInstance))).prodMk
-        (@measurable_snd _ _ hS.measurableSpace inferInstance)
-  have hm := (timeMassDensity_joint_measurable a ha q hq hq1).comp hp
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  have hm' : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      timeMassDensity a q (S p.1) (stoppedPastDriver S p.1) p.2) := hm
-  convert hm' using 1
-  funext p
-  exact timeMassDensity_stoppedPastDriver a (by linarith) q hq hq1 S p.1 p.2
-
-lemma actualUpperMass_stopping_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (fun b => actualUpperMass a q (S b) b) := by
-  have hM := timeMassDensity_stopping_raw_joint a ha q hq hq1 S hS
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      timeMassDensity a q (S p.1) p.1 p.2.1) :=
-    hM.comp (measurable_fst.prodMk (measurable_subtype_coe.comp measurable_snd))
-  apply Measure.measurable_of_measurable_coe
-  intro A hA
-  simp only [actualUpperMass,withDensity_apply _ hA]
-  exact hm.lintegral_prod_right'
-
-lemma actualUpperMass_stopping_eval_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) :
-    Measurable[hS.measurableSpace] (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) := by
-  simp_rw [actualUpperMass_eval a ha q hq hq1]
-  have hD := timeMassDensity_stopping_raw_joint a ha q hq hq1 S hS
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      upperExtension f p.2*(timeMassDensity a q (S p.1) p.1 p.2).toReal) :=
-    ((upperExtension_measurable f).comp measurable_snd).mul hD.ennreal_toReal
-  exact hm.stronglyMeasurable.integral_prod_right'.measurable
-
-lemma actualUpperMass_stopping_eval_measurable (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) :
-    Measurable (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) :=
-  (actualUpperMass_stopping_eval_raw_measurable a ha q hq hq1 S hS f).mono
-    hS.measurableSpace_le le_rfl
-
- 
-
-theorem recoveredMass_stopping_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (fun b => recoveredMass a ha s c hc (S b) b) := by
-  apply @measurable_radonLimit _ _ _ _ _ _ _ _ hS.measurableSpace
-  intro k f
-  simp_rw [actualMassFunctionals_apply]
-  apply Finset.measurable_sum
-  intro j hj
-  exact measurable_const.mul (actualUpperMass_stopping_eval_raw_measurable a ha _ (Real.exp_pos _)
-    (level_step_le_one a ha j) S hS f)
-
-lemma recoveredMass_stopping_measurable (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable (fun b => recoveredMass a ha s c hc (S b) b) :=
-  (recoveredMass_stopping_raw_measurable a ha s c hc S hS).mono hS.measurableSpace_le le_rfl
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
-
-lemma actual_stopped_radon_limit_memLp (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-    (hc : ∀ k j, j∈s k → 0≤coeff k j) (hsum : ∀ k, ∑ j∈s k,coeff k j=1)
-    (μ : ExactSLEGenerator.Driver → Measure UpperPlane)
-    (f : C_c(UpperPlane, ℝ))
-    (hlim : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,coeff k j *
-      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S b) b) atTop (𝓝 (∫z,f z∂μ b)))
-    (m : ℕ) : MemLp (fun b => ∫z,f z∂μ b) (m : ℝ≥0∞) W := by
-  let q (j : ℕ) := Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := by
-    apply Real.exp_le_one_iff.mpr
-    have ha0 : 0<a := by linarith
-    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
-    nlinarith
-  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun k b =>
-    ∑ j∈s k,coeff k j * ∫z,f z∂actualUpperMass a (q j) (S b) b
-  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
-    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
-      (actualUpperMass_stopping_eval_measurable a ha (q j) (hq j) (hq1 j) S hS f).aestronglyMeasurable.const_mul (coeff k j))
-    convert H using 1
-    funext b
-    simp only [F,Finset.sum_apply]
-  have hmeas : AEStronglyMeasurable (fun b => ∫z,f z∂μ b) W :=
-    aestronglyMeasurable_of_tendsto_ae atTop hF hlim
-  by_cases hm : m=0
-  · simpa [hm] using (memLp_zero_iff_aestronglyMeasurable.mpr hmeas)
-  have hmpos : 1 ≤ m := Nat.one_le_iff_ne_zero.mpr hm
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_stopped_Lp_bound a ha W hW m hmpos f
-  have hbound (k : ℕ) : eLpNorm (F k) (m : ℝ≥0∞) W≤C :=
-    convex_eLpNorm_bound W _ C (by exact_mod_cast hmpos) (s k) (coeff k) (hc k) (hsum k)
-      (fun j b => ∫z,f z∂actualUpperMass a (q j) (S b) b)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).1.aestronglyMeasurable)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).2)
-  exact (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF hmeas hlim).trans_lt hC
-
-lemma actual_stopped_radon_limit_L1 (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) (q : ℕ → ℝ) (hq : ∀ j,0<q j) (hq1 : ∀ j,q j≤1)
-    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
-    (hc : ∀ k j,j∈s k→0≤coeff k j) (hsum : ∀ k,∑j∈s k,coeff k j=1)
-    (μ : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
-    (hlim : ∀ᵐ b∂W, Tendsto (fun k => ∑j∈s k,coeff k j*
-      ∫z,f z∂actualUpperMass a (q j) (S b) b) atTop (𝓝 (∫z,f z∂μ b))) :
-    Tendsto (fun k => eLpNorm (fun b => (∑j∈s k,coeff k j*
-      ∫z,f z∂actualUpperMass a (q j) (S b) b)-(∫z,f z∂μ b)) 1 W) atTop (𝓝 0) := by
-  let F := fun k b => ∑j∈s k,coeff k j*∫z,f z∂actualUpperMass a (q j) (S b) b
-  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
-    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
-      (actualUpperMass_stopping_eval_measurable a ha (q j) (hq j) (hq1 j) S hS f).aestronglyMeasurable.const_mul (coeff k j))
-    convert H using 1
-    funext b
-    simp only [F,Finset.sum_apply]
-  obtain ⟨C,hC,hCb⟩ := actualUpperMass_stopped_Lp_bound a ha W hW 2 (by norm_num) f
-  have hbound (k : ℕ) : eLpNorm (F k) 2 W≤C :=
-    convex_eLpNorm_bound W _ C (by norm_num) (s k) (coeff k) (hc k) (hsum k)
-      (fun j b => ∫z,f z∂actualUpperMass a (q j) (S b) b)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).1.aestronglyMeasurable)
-      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).2)
-  have hg2 : MemLp (fun b => ∫z,f z∂μ b) 2 W :=
-    (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF
-      (aestronglyMeasurable_of_tendsto_ae atTop hF hlim) hlim).trans_lt hC
-  have hg := hg2.mono_exponent (by norm_num : (1:ℝ≥0∞)≤2)
-  exact tendsto_Lp_finite_of_tendsto_ae le_rfl (by simp) hF (by simpa using hg)
-    (unifIntegrable_of_L2_bound W F hF C hC hbound) hlim
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
-
-lemma actual_stopped_radon_mass_extraction (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ℕ → ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : ∀ i, IsStoppingTime rawFiltration (S i)) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤c k j) ∧
-      (∀ k, ∑ j∈s k,c k j=1) ∧
-      ∀ᵐ b ∂W, ∃ μ : ℕ → Measure UpperPlane,
-        (∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular) ∧
-        ∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k,c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
-            atTop (𝓝 (∫z,f z∂μ i)) := by
-  obtain ⟨tests,htests⟩ := determining_tests_radon_limit UpperPlane
-  let F : ℕ → C_c(UpperPlane,ℝ) := fun i => tests (Nat.unpair i).2
-  choose C hC hBound using fun i => actualUpperMass_stopped_Lp_bound a ha W hW 2 (by norm_num) (F i)
-  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := level_step_le_one a ha j
-  let A : ℕ → ℕ → ExactSLEGenerator.Driver → ℝ := fun j i b =>
-    ∫z,F i z∂actualUpperMass a (q j) (S (Nat.unpair i).1 b) b
-  have hLp (j i : ℕ) : MemLp (A j i) 2 W := by
-    simpa only [Nat.cast_ofNat] using
-      (hBound i (q j) (hq j) (hq1 j) (S (Nat.unpair i).1) (hS _)).1
-  let X := fun j i => (hLp j i).toLp (A j i)
-  have hX (i : ℕ) : ∃ B : ℝ, ∀ j, ‖X j i‖≤B := by
-    refine ⟨(C i).toReal,fun j => ?_⟩
-    rw [Lp.norm_toLp]
-    exact ENNReal.toReal_mono (hC i).ne
-      (hBound i (q j) (hq j) (hq1 j) (S (Nat.unpair i).1) (hS _)).2
-  obtain ⟨s,c,v,hs,hc,hsum,-,hae⟩ := common_L2_convex_tails W X hX
-  have he : ∀ᵐ b ∂W, ∀ j i, X j i b=A j i b :=
-    ae_all_iff.mpr (fun j => ae_all_iff.mpr (fun i => (hLp j i).coeFn_toLp))
-  refine ⟨s,c,hs,hc,hsum,?_⟩
-  filter_upwards [hae,he] with b hb heq
-  let L (i k : ℕ) : C_c(UpperPlane,ℝ) →ₚ[ℝ] ℝ :=
-    actualMassFunctionals a ha s c hc (S i b) b k
-  have hi (i : ℕ) : ∃ μ : Measure UpperPlane, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
-      ∀ f : C_c(UpperPlane,ℝ), Tendsto (fun k => L i k f) atTop (𝓝 (∫z,f z∂μ)) := by
-    apply htests (L i)
-    intro n
-    refine ⟨v (Nat.pair i n) b,?_⟩
-    have h := hb (Nat.pair i n)
-    simpa only [heq,A,F,L,actualMassFunctionals_apply,Nat.unpair_pair] using h
-  choose μ hμ using hi
-  refine ⟨μ,fun i => ⟨(hμ i).1,(hμ i).2.1⟩,fun i f => ?_⟩
-  simpa only [L,actualMassFunctionals_apply] using (hμ i).2.2 f
-
- 
-
-theorem actual_raw_stopped_extraction (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ℕ → ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : ∀ i, IsStoppingTime rawFiltration (S i)) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j,j∈s k → 0≤c k j),
-      (∀ k j,j∈s k → k≤j) ∧ (∀ k,∑ j∈s k,c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ t b, IsFiniteMeasureOnCompacts (μ t b) ∧ (μ t b).Regular) ∧
-      IsProgressive rawFiltration (fun t : ℝ≥0 => μ t) ∧
-      (∀ i, Measurable[(hS i).measurableSpace] (fun b => μ (S i b) b)) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (S i b) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k,c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
-            - ∫z,f z∂μ (S i b) b) 1 W) atTop (𝓝 0)) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j,S i b≤S j b → μ (S i b) b≤μ (S j b) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k,c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
-            atTop (𝓝 (∫z,f z∂μ (S i b) b))) := by
-  obtain ⟨s,c,hs,hc,hsum,h⟩ := actual_stopped_radon_mass_extraction a ha W hW S hS
-  let μ := recoveredMass a ha s c hc
-  have hμ (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-      IsFiniteMeasureOnCompacts (μ t b) ∧ (μ t b).Regular :=
-    ⟨radonLimit_finite _,radonLimit_regular _⟩
-  have hlim : ∀ᵐ b ∂W, ∀ i (f : C_c(UpperPlane,ℝ)),
-      Tendsto (fun k => ∑ j∈s k,c k j *
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
-        atTop (𝓝 (∫z,f z∂μ (S i b) b)) := by
-    filter_upwards [h] with b hb
-    obtain ⟨ν,hν,hconv⟩ := hb
-    have he (i : ℕ) : μ (S i b) b=ν i := by
-      have := (hν i).2
-      apply radonLimit_eq
-      intro f
-      simpa only [actualMassFunctionals_apply] using hconv i f
-    intro i f
-    rw [he i]
-    exact hconv i f
-  refine ⟨s,c,hc,hs,hsum,hμ,recoveredMass_progressive a ha s c hc,?_,?_,?_,?_⟩
-  · exact fun i => recoveredMass_stopping_raw_measurable a ha s c hc (S i) (hS i)
-  · intro i f m
-    exact actual_stopped_radon_limit_memLp a ha W hW (S i) (hS i) s c hc hsum
-      (fun b => μ (S i b) b) f (hlim.mono (fun b hb => hb i f)) m
-  · intro i f
-    exact actual_stopped_radon_limit_L1 a ha W hW (S i) (hS i)
-      (fun j => Real.exp (-2*a*(j+1))) (fun j => Real.exp_pos _)
-      (level_step_le_one a ha) s c hc hsum (fun b => μ (S i b) b) f
-      (hlim.mono (fun b hb => hb i f))
-  · filter_upwards [hlim] with b hb
-    exact ⟨ordered_radon_limits a ha b (fun i => S i b) s c hc (fun i => μ (S i b) b)
-      (fun i => hμ (S i b) b) hb,hb⟩
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
-
-lemma actual_stopped_limit_compact_conditional (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-    (hs : ∀ k j,j∈s k → k≤j) (hc : ∀ k j,j∈s k → 0≤c k j)
-    (hsum : ∀ k,∑ j∈s k,c k j=1)
-    (μ ν : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
-    (hμ : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,c k j *
-      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S b) b)
-      atTop (𝓝 (∫z,f z∂μ b)))
-    (hν : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,c k j *
-      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
-      atTop (𝓝 (∫z,f z∂ν b)))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
-      ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W := by
-  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
-  have hq (j : ℕ) : 0<q j := Real.exp_pos _
-  have hq1 (j : ℕ) : q j≤1 := level_step_le_one a ha j
-  have hq0 : Tendsto q atTop (𝓝 0) := mass_levels_tendsto_zero a (by linarith)
-  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => ∞) :=
-    fun t => by simp
-  have hiμ : Integrable (fun b => ∫z,f z∂μ b) W := by
-    have := actual_stopped_radon_limit_memLp a ha W hW S hS s c hc hsum μ f hμ 1
-    exact this.integrable (by norm_num)
-  have hiν : Integrable (fun b => ∫z,f z∂ν b) W := by
-    have := actual_stopped_radon_limit_memLp a ha W hW (fun _ => ∞) htop s c hc hsum ν f hν 1
-    exact this.integrable (by norm_num)
-  let F := fun k b => ∑ j∈s k,c k j * ∫z,f z∂actualUpperMass a (q j) (S b) b
-  let E := fun k b => ∑ j∈s k,c k j * ∫z,f z∂actualUpperMass a (q j) ∞ b
-  have hiF (k : ℕ) : Integrable (F k) W :=
-    integrable_finsetSum (s k) (fun j hj =>
-      (actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j) (hq1 j) S hS f).const_mul (c k j))
-  have hiE (k : ℕ) : Integrable (E k) W :=
-    integrable_finsetSum (s k) (fun j hj =>
-      (actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j) (hq1 j) (fun _ => ∞) htop f).const_mul (c k j))
-  have hF := setIntegral_tendsto_of_L1 W F (fun b => ∫z,f z∂μ b) hiF hiμ
-    (actual_stopped_radon_limit_L1 a ha W hW S hS q hq hq1 s c hc hsum μ f hμ) U
-  have hE := setIntegral_tendsto_of_L1 W E (fun b => ∫z,f z∂ν b) hiE hiν
-    (actual_stopped_radon_limit_L1 a ha W hW (fun _ => ∞) htop q hq hq1 s c hc hsum ν f hν) U
-  have hpot := actualUpperMass_stopped_convex_potential_limit a ha W hW q
-    (fun j => ⟨hq j,hq1 j⟩) hq0 s c hs hc hsum S hS f U hU
-  have he (k : ℕ) : (∫b in U, E k b ∂W)-(∫b in U, F k b ∂W) =
-      ∫b in U,∑ j∈s k,c k j*((∫z,f z∂actualUpperMass a (q j) ∞ b)-
-        (∫z,f z∂actualUpperMass a (q j) (S b) b)) ∂W := by
-    rw [←integral_sub (hiE k).restrict (hiF k).restrict]
-    congr 1
-    funext b
-    simp only [E,F,mul_sub,Finset.sum_sub_distrib]
-  rw [integral_sub hiν.restrict hiμ.restrict]
-  exact tendsto_nhds_unique (hE.sub hF) (by simpa only [he] using hpot)
-
- 
-
-theorem stopped_compact_conditional_to_random_tests (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
-    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂μ b) W)
-    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂ν b) W)
-    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
-      MeasurableSet[hS.measurableSpace] U →
-      (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
-        ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W)
-    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
-    (hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
-    (∫⁻b,∫⁻z,H (b,z) ∂ν b ∂W) = (∫⁻b,∫⁻z,H (b,z) ∂μ b ∂W) +
-      ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂upperArea ∂W := by
-  have := isSFiniteKernel_of_locally_finite μ
-  have := isSFiniteKernel_of_locally_finite ν
-  have := average_finiteOnCompacts μ W hμ
-  have := average_finiteOnCompacts ν W hν
-  have := stoppedSurvivingKernel_average_finiteOnCompacts a ha W hW S hS
-  have : IsFiniteMeasureOnCompacts ((μ+stoppedSurvivingKernel a ha S hS) ∘ₘ W) := by
-    rw [SLEExactGauge.MassVersions.average_add]
-    constructor
-    intro K hK
-    rw [Measure.add_apply]
-    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩
-  have hidentity := conditional_lintegral_eq_of_average_finite ν
-    (μ+stoppedSurvivingKernel a ha S hS) W hS.measurableSpace hS.measurableSpace_le ?_ H hH
-  · rw [hidentity]
-    have hH₀ : Measurable H := hH.mono (prod_sigma_le hS.measurableSpace_le _) le_rfl
-    have heval (b : ExactSLEGenerator.Driver) :
-        (∫⁻z,H (b,z) ∂(μ+stoppedSurvivingKernel a ha S hS) b) =
-          (∫⁻z,H (b,z) ∂μ b)+
-            ∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂upperArea := by
-      have hHb : Measurable (fun z : UpperPlane => H (b,z)) :=
-        hH₀.comp ((measurable_const (a:=b)).prodMk measurable_id)
-      rw [add_apply,lintegral_add_measure,stoppedSurvivingKernel_apply]
-      congr 1
-      exact (lintegral_withDensity_eq_lintegral_mul upperArea
-        (stoppedSurvivingGreen_spatial_measurable a ha S hS b).ennreal_ofReal hHb).trans
-          (lintegral_congr (fun _ => mul_comm _ _))
-    simp_rw [heval]
-    rw [lintegral_add_left]
-    exact hH₀.lintegral_kernel_prod_right'
-  · intro U hU f
-    have hadd : (fun b => ∫z,f z∂(μ+stoppedSurvivingKernel a ha S hS) b) =ᵐ[W]
-        (fun b => (∫z,f z∂μ b)+(∫z,f z∂stoppedSurvivingKernel a ha S hS b)) := by
-      filter_upwards [stoppedSurvivingKernel_compact_ae_integrable a ha W hW S hS f] with b hb
-      exact integral_add_measure f.integrable hb
-    rw [integral_congr_ae (ae_restrict_of_ae hadd),
-      integral_add (hμ f).restrict (stoppedSurvivingKernel_compact_mean_integrable a ha W hW S hS f).restrict]
-    have hf := he f U hU
-    rw [integral_sub (hν f).restrict (hμ f).restrict] at hf
-    simp_rw [stoppedSurvivingKernel_compact_eval]
-    linarith
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem stopped_conditional_future_support (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
-    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂μ b) W)
-    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂ν b) W)
-    (hle : ∀ᵐ b ∂W, μ b≤ν b)
-    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
-      MeasurableSet[hS.measurableSpace] U →
-      (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
-        ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W) :
-    ∀ᵐ b ∂W, (ν b-μ b) {z : UpperPlane | stoppedSurvivingGreen a (S b) b z.1=0}=0 := by
-  let D : Set (ExactSLEGenerator.Driver × UpperPlane) :=
-    {p | stoppedSurvivingGreen a (S p.1) p.1 p.2.1=0}
-  have hD : MeasurableSet[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] D :=
-    (stoppedSurvivingGreen_raw_joint a ha S hS) (measurableSet_singleton (0 : ℝ))
-  have hD₀ : MeasurableSet D := prod_sigma_le hS.measurableSpace_le _ _ hD
-  have := average_finiteOnCompacts ν W hν
-  apply random_set_sub_null_of_compact_means μ ν W D hD₀ hle
-  intro K hK
-  let E := D∩Prod.snd ⁻¹' K
-  have hE : MeasurableSet[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] E :=
-    hD.inter (measurable_snd hK.measurableSet)
-  have hE₀ : MeasurableSet E := prod_sigma_le hS.measurableSpace_le _ _ hE
-  let H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞ := E.indicator (fun _ => 1)
-  have hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H :=
-    measurable_const.indicator hE
-  have hid := stopped_compact_conditional_to_random_tests a ha W hW S hS μ ν hμ hν he H hH
-  have hev (ρ : Measure UpperPlane) (b : ExactSLEGenerator.Driver) :
-      (∫⁻z,H (b,z) ∂ρ)=ρ ({z | (b,z)∈D}∩K) := by
-    have hh := lintegral_indicator_const_comp (μ:=ρ)
-      ((measurable_const (a:=b)).prodMk measurable_id) hE₀ (1 : ℝ≥0∞)
-    have hs : (fun z : UpperPlane => (b,id z)) ⁻¹' E={z | (b,z)∈D}∩K := by
-      ext z; rfl
-    rw [hs,one_mul] at hh
-    exact hh
-  have hz (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-      H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)=0 := by
-    by_cases h : (b,z)∈E
-    · rw [show stoppedSurvivingGreen a (S b) b z.1=0 from h.1,ENNReal.ofReal_zero,mul_zero]
-    · simp only [H,indicator_of_notMem h,zero_mul]
-  simpa only [hev,hz,lintegral_zero,add_zero] using hid
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StoppedFiniteMass
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+
+namespace ExactSLEGenerator
+
+ 
+def stoppedPastDriver (S : Driver → ℝ≥0∞) (b : Driver) : Driver :=
+  if S b = ∞ then b else pastDriver (S b).toNNReal b
+
+lemma stoppedPastDriver_apply (S : Driver → ℝ≥0∞) (b : Driver) (t : ℝ≥0) :
+    (stoppedPastDriver S b).1 t = b.1 (ExactSLEGreen.capTime S t b) := by
+  by_cases hb : S b = ∞
+  · simp [stoppedPastDriver,hb,ExactSLEGreen.capTime]
+  · rw [stoppedPastDriver,ite_eq_right hb]
+    change b.1 (min t (S b).toNNReal) = b.1 (ExactSLEGreen.capTime S t b)
+    congr 1
+    unfold ExactSLEGreen.capTime
+    conv_rhs => rw [← ENNReal.coe_toNNReal hb]
+    simp only [← ENNReal.coe_min,ENNReal.toNNReal_coe,min_comm]
+
+lemma stoppedPastDriver_measurable (S : Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (stoppedPastDriver S) := by
+  have : PolishSpace Driver := (isClosed_eq (continuous_eval_const 0) continuous_const).polishSpace
+  obtain ⟨u,hu⟩ := TopologicalSpace.exists_dense_seq (α := ℝ≥0)
+  let e : Driver → (ℕ → ℝ) := fun b n => b.1 (u n)
+  have hec : Continuous e := continuous_pi fun n =>
+    (continuous_eval_const (u n)).comp continuous_subtype_val
+  have hei : Function.Injective e := by
+    intro b c h
+    apply Subtype.ext
+    apply DFunLike.coe_injective
+    exact hu.equalizer b.1.continuous c.1.continuous h
+  apply hec.measurableEmbedding hei |>.measurable_comp_iff.mp
+  have hA : StronglyAdapted rawFiltration (fun t (b : Driver) => b.1 t) :=
+    Filtration.stronglyAdapted_natural driver_eval_measurable
+  have hP := hA.isStronglyProgressive_of_continuous (fun b => b.1.continuous)
+  have hval (n : ℕ) : Measurable[hS.measurableSpace]
+      (fun b => b.1 (ExactSLEGreen.capTime S (u n) b)) := by
+    have h := (measurable_stoppedValue hP (ExactSLEGreen.capTime_stopping S hS (u n))).mono
+      (ExactSLEGreen.stopped_cap_measurable S hS (u n)) le_rfl
+    exact h
+  let : MeasurableSpace Driver := hS.measurableSpace
+  apply measurable_pi_iff.mpr
+  intro n
+  simpa only [Function.comp_apply,e,stoppedPastDriver_apply] using hval n
+
+end ExactSLEGenerator
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma timeMassDensity_stoppedPastDriver (a : ℝ) (ha : 0<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    timeMassDensity a q (S b) b z =
+      timeMassDensity a q (S b) (stoppedPastDriver S b) z := by
+  by_cases hb : S b=∞
+  · simp [stoppedPastDriver,hb]
+  · conv_lhs => rw [← ENNReal.coe_toNNReal hb]
+    conv_rhs => rw [← ENNReal.coe_toNNReal hb]
+    apply timeMassDensity_driver_congr a ha q hq hq1
+    intro t ht
+    simpa only [stoppedPastDriver,ite_eq_right hb] using
+      (pastDriver_eq (S b).toNNReal b t ht).symm
+
+ 
+theorem timeMassDensity_stopping_raw_joint (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    letI : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+    Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      timeMassDensity a q (S p.1) p.1 p.2) := by
+  have hp : @Measurable (ExactSLEGenerator.Driver × ℂ)
+      ((ℝ≥0∞ × ExactSLEGenerator.Driver) × ℂ)
+      (hS.measurableSpace.prod inferInstance) inferInstance
+      (fun p => ((S p.1,stoppedPastDriver S p.1),p.2)) := by
+    exact ((hS.measurable.comp (@measurable_fst _ _ hS.measurableSpace inferInstance)).prodMk
+      ((stoppedPastDriver_measurable S hS).comp
+        (@measurable_fst _ _ hS.measurableSpace inferInstance))).prodMk
+        (@measurable_snd _ _ hS.measurableSpace inferInstance)
+  have hm := (timeMassDensity_joint_measurable a ha q hq hq1).comp hp
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  have hm' : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      timeMassDensity a q (S p.1) (stoppedPastDriver S p.1) p.2) := hm
+  convert hm' using 1
+  funext p
+  exact timeMassDensity_stoppedPastDriver a (by linarith) q hq hq1 S p.1 p.2
+
+lemma actualUpperMass_stopping_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (fun b => actualUpperMass a q (S b) b) := by
+  have hM := timeMassDensity_stopping_raw_joint a ha q hq hq1 S hS
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      timeMassDensity a q (S p.1) p.1 p.2.1) :=
+    hM.comp (measurable_fst.prodMk (measurable_subtype_coe.comp measurable_snd))
+  apply Measure.measurable_of_measurable_coe
+  intro A hA
+  simp only [actualUpperMass,withDensity_apply _ hA]
+  exact hm.lintegral_prod_right'
+
+lemma actualUpperMass_stopping_eval_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) :
+    Measurable[hS.measurableSpace] (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) := by
+  simp_rw [actualUpperMass_eval a ha q hq hq1]
+  have hD := timeMassDensity_stopping_raw_joint a ha q hq hq1 S hS
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      upperExtension f p.2*(timeMassDensity a q (S p.1) p.1 p.2).toReal) :=
+    ((upperExtension_measurable f).comp measurable_snd).mul hD.ennreal_toReal
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+
+lemma actualUpperMass_stopping_eval_measurable (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) :
+    Measurable (fun b => ∫z, f z ∂actualUpperMass a q (S b) b) :=
+  (actualUpperMass_stopping_eval_raw_measurable a ha q hq hq1 S hS f).mono
+    hS.measurableSpace_le le_rfl
+
+ 
+
+theorem recoveredMass_stopping_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (fun b => recoveredMass a ha s c hc (S b) b) := by
+  apply @measurable_radonLimit _ _ _ _ _ _ _ _ hS.measurableSpace
+  intro k f
+  simp_rw [actualMassFunctionals_apply]
+  apply Finset.measurable_sum
+  intro j hj
+  exact measurable_const.mul (actualUpperMass_stopping_eval_raw_measurable a ha _ (Real.exp_pos _)
+    (level_step_le_one a ha j) S hS f)
+
+lemma recoveredMass_stopping_measurable (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable (fun b => recoveredMass a ha s c hc (S b) b) :=
+  (recoveredMass_stopping_raw_measurable a ha s c hc S hS).mono hS.measurableSpace_le le_rfl
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
+
+lemma actual_stopped_radon_limit_memLp (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+    (hc : ∀ k j, j∈s k → 0≤coeff k j) (hsum : ∀ k, ∑ j∈s k,coeff k j=1)
+    (μ : ExactSLEGenerator.Driver → Measure UpperPlane)
+    (f : C_c(UpperPlane, ℝ))
+    (hlim : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,coeff k j *
+      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S b) b) atTop (𝓝 (∫z,f z∂μ b)))
+    (m : ℕ) : MemLp (fun b => ∫z,f z∂μ b) (m : ℝ≥0∞) W := by
+  let q (j : ℕ) := Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := by
+    apply Real.exp_le_one_iff.mpr
+    have ha0 : 0<a := by linarith
+    have hj0 : (0:ℝ)≤j := Nat.cast_nonneg _
+    nlinarith
+  let F : ℕ → ExactSLEGenerator.Driver → ℝ := fun k b =>
+    ∑ j∈s k,coeff k j * ∫z,f z∂actualUpperMass a (q j) (S b) b
+  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
+    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
+      (actualUpperMass_stopping_eval_measurable a ha (q j) (hq j) (hq1 j) S hS f).aestronglyMeasurable.const_mul (coeff k j))
+    convert H using 1
+    funext b
+    simp only [F,Finset.sum_apply]
+  have hmeas : AEStronglyMeasurable (fun b => ∫z,f z∂μ b) W :=
+    aestronglyMeasurable_of_tendsto_ae atTop hF hlim
+  by_cases hm : m=0
+  · simpa [hm] using (memLp_zero_iff_aestronglyMeasurable.mpr hmeas)
+  have hmpos : 1 ≤ m := Nat.one_le_iff_ne_zero.mpr hm
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_stopped_Lp_bound a ha W hW m hmpos f
+  have hbound (k : ℕ) : eLpNorm (F k) (m : ℝ≥0∞) W≤C :=
+    convex_eLpNorm_bound W _ C (by exact_mod_cast hmpos) (s k) (coeff k) (hc k) (hsum k)
+      (fun j b => ∫z,f z∂actualUpperMass a (q j) (S b) b)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).1.aestronglyMeasurable)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).2)
+  exact (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF hmeas hlim).trans_lt hC
+
+lemma actual_stopped_radon_limit_L1 (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) (q : ℕ → ℝ) (hq : ∀ j,0<q j) (hq1 : ∀ j,q j≤1)
+    (s : ℕ → Finset ℕ) (coeff : ℕ → ℕ → ℝ)
+    (hc : ∀ k j,j∈s k→0≤coeff k j) (hsum : ∀ k,∑j∈s k,coeff k j=1)
+    (μ : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
+    (hlim : ∀ᵐ b∂W, Tendsto (fun k => ∑j∈s k,coeff k j*
+      ∫z,f z∂actualUpperMass a (q j) (S b) b) atTop (𝓝 (∫z,f z∂μ b))) :
+    Tendsto (fun k => eLpNorm (fun b => (∑j∈s k,coeff k j*
+      ∫z,f z∂actualUpperMass a (q j) (S b) b)-(∫z,f z∂μ b)) 1 W) atTop (𝓝 0) := by
+  let F := fun k b => ∑j∈s k,coeff k j*∫z,f z∂actualUpperMass a (q j) (S b) b
+  have hF (k : ℕ) : AEStronglyMeasurable (F k) W := by
+    have H := Finset.aestronglyMeasurable_sum (μ := W) (s k) (fun j hj =>
+      (actualUpperMass_stopping_eval_measurable a ha (q j) (hq j) (hq1 j) S hS f).aestronglyMeasurable.const_mul (coeff k j))
+    convert H using 1
+    funext b
+    simp only [F,Finset.sum_apply]
+  obtain ⟨C,hC,hCb⟩ := actualUpperMass_stopped_Lp_bound a ha W hW 2 (by norm_num) f
+  have hbound (k : ℕ) : eLpNorm (F k) 2 W≤C :=
+    convex_eLpNorm_bound W _ C (by norm_num) (s k) (coeff k) (hc k) (hsum k)
+      (fun j b => ∫z,f z∂actualUpperMass a (q j) (S b) b)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).1.aestronglyMeasurable)
+      (fun j hj => (hCb (q j) (hq j) (hq1 j) S hS).2)
+  have hg2 : MemLp (fun b => ∫z,f z∂μ b) 2 W :=
+    (Lp.eLpNorm_le_of_ae_tendsto (Filter.Eventually.of_forall hbound) hF
+      (aestronglyMeasurable_of_tendsto_ae atTop hF hlim) hlim).trans_lt hC
+  have hg := hg2.mono_exponent (by norm_num : (1:ℝ≥0∞)≤2)
+  exact tendsto_Lp_finite_of_tendsto_ae le_rfl (by simp) hF (by simpa using hg)
+    (unifIntegrable_of_L2_bound W F hF C hC hbound) hlim
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
+
+lemma actual_stopped_radon_mass_extraction (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ℕ → ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : ∀ i, IsStoppingTime rawFiltration (S i)) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k j, j∈s k → 0≤c k j) ∧
+      (∀ k, ∑ j∈s k,c k j=1) ∧
+      ∀ᵐ b ∂W, ∃ μ : ℕ → Measure UpperPlane,
+        (∀ i, IsFiniteMeasureOnCompacts (μ i) ∧ (μ i).Regular) ∧
+        ∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k,c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
+            atTop (𝓝 (∫z,f z∂μ i)) := by
+  obtain ⟨tests,htests⟩ := determining_tests_radon_limit UpperPlane
+  let F : ℕ → C_c(UpperPlane,ℝ) := fun i => tests (Nat.unpair i).2
+  choose C hC hBound using fun i => actualUpperMass_stopped_Lp_bound a ha W hW 2 (by norm_num) (F i)
+  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := level_step_le_one a ha j
+  let A : ℕ → ℕ → ExactSLEGenerator.Driver → ℝ := fun j i b =>
+    ∫z,F i z∂actualUpperMass a (q j) (S (Nat.unpair i).1 b) b
+  have hLp (j i : ℕ) : MemLp (A j i) 2 W := by
+    simpa only [Nat.cast_ofNat] using
+      (hBound i (q j) (hq j) (hq1 j) (S (Nat.unpair i).1) (hS _)).1
+  let X := fun j i => (hLp j i).toLp (A j i)
+  have hX (i : ℕ) : ∃ B : ℝ, ∀ j, ‖X j i‖≤B := by
+    refine ⟨(C i).toReal,fun j => ?_⟩
+    rw [Lp.norm_toLp]
+    exact ENNReal.toReal_mono (hC i).ne
+      (hBound i (q j) (hq j) (hq1 j) (S (Nat.unpair i).1) (hS _)).2
+  obtain ⟨s,c,v,hs,hc,hsum,-,hae⟩ := common_L2_convex_tails W X hX
+  have he : ∀ᵐ b ∂W, ∀ j i, X j i b=A j i b :=
+    ae_all_iff.mpr (fun j => ae_all_iff.mpr (fun i => (hLp j i).coeFn_toLp))
+  refine ⟨s,c,hs,hc,hsum,?_⟩
+  filter_upwards [hae,he] with b hb heq
+  let L (i k : ℕ) : C_c(UpperPlane,ℝ) →ₚ[ℝ] ℝ :=
+    actualMassFunctionals a ha s c hc (S i b) b k
+  have hi (i : ℕ) : ∃ μ : Measure UpperPlane, IsFiniteMeasureOnCompacts μ ∧ μ.Regular ∧
+      ∀ f : C_c(UpperPlane,ℝ), Tendsto (fun k => L i k f) atTop (𝓝 (∫z,f z∂μ)) := by
+    apply htests (L i)
+    intro n
+    refine ⟨v (Nat.pair i n) b,?_⟩
+    have h := hb (Nat.pair i n)
+    simpa only [heq,A,F,L,actualMassFunctionals_apply,Nat.unpair_pair] using h
+  choose μ hμ using hi
+  refine ⟨μ,fun i => ⟨(hμ i).1,(hμ i).2.1⟩,fun i f => ?_⟩
+  simpa only [L,actualMassFunctionals_apply] using (hμ i).2.2 f
+
+ 
+
+theorem actual_raw_stopped_extraction (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ℕ → ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : ∀ i, IsStoppingTime rawFiltration (S i)) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j,j∈s k → 0≤c k j),
+      (∀ k j,j∈s k → k≤j) ∧ (∀ k,∑ j∈s k,c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ t b, IsFiniteMeasureOnCompacts (μ t b) ∧ (μ t b).Regular) ∧
+      IsProgressive rawFiltration (fun t : ℝ≥0 => μ t) ∧
+      (∀ i, Measurable[(hS i).measurableSpace] (fun b => μ (S i b) b)) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (S i b) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k,c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
+            - ∫z,f z∂μ (S i b) b) 1 W) atTop (𝓝 0)) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j,S i b≤S j b → μ (S i b) b≤μ (S j b) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k,c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
+            atTop (𝓝 (∫z,f z∂μ (S i b) b))) := by
+  obtain ⟨s,c,hs,hc,hsum,h⟩ := actual_stopped_radon_mass_extraction a ha W hW S hS
+  let μ := recoveredMass a ha s c hc
+  have hμ (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+      IsFiniteMeasureOnCompacts (μ t b) ∧ (μ t b).Regular :=
+    ⟨radonLimit_finite _,radonLimit_regular _⟩
+  have hlim : ∀ᵐ b ∂W, ∀ i (f : C_c(UpperPlane,ℝ)),
+      Tendsto (fun k => ∑ j∈s k,c k j *
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S i b) b)
+        atTop (𝓝 (∫z,f z∂μ (S i b) b)) := by
+    filter_upwards [h] with b hb
+    obtain ⟨ν,hν,hconv⟩ := hb
+    have he (i : ℕ) : μ (S i b) b=ν i := by
+      have := (hν i).2
+      apply radonLimit_eq
+      intro f
+      simpa only [actualMassFunctionals_apply] using hconv i f
+    intro i f
+    rw [he i]
+    exact hconv i f
+  refine ⟨s,c,hc,hs,hsum,hμ,recoveredMass_progressive a ha s c hc,?_,?_,?_,?_⟩
+  · exact fun i => recoveredMass_stopping_raw_measurable a ha s c hc (S i) (hS i)
+  · intro i f m
+    exact actual_stopped_radon_limit_memLp a ha W hW (S i) (hS i) s c hc hsum
+      (fun b => μ (S i b) b) f (hlim.mono (fun b hb => hb i f)) m
+  · intro i f
+    exact actual_stopped_radon_limit_L1 a ha W hW (S i) (hS i)
+      (fun j => Real.exp (-2*a*(j+1))) (fun j => Real.exp_pos _)
+      (level_step_le_one a ha) s c hc hsum (fun b => μ (S i b) b) f
+      (hlim.mono (fun b hb => hb i f))
+  · filter_upwards [hlim] with b hb
+    exact ⟨ordered_radon_limits a ha b (fun i => S i b) s c hc (fun i => μ (S i b) b)
+      (fun i => hμ (S i b) b) hb,hb⟩
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain ExactSLEMassExtraction SLEExactGauge.MassVersions
+
+lemma actual_stopped_limit_compact_conditional (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+    (hs : ∀ k j,j∈s k → k≤j) (hc : ∀ k j,j∈s k → 0≤c k j)
+    (hsum : ∀ k,∑ j∈s k,c k j=1)
+    (μ ν : ExactSLEGenerator.Driver → Measure UpperPlane) (f : C_c(UpperPlane,ℝ))
+    (hμ : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,c k j *
+      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (S b) b)
+      atTop (𝓝 (∫z,f z∂μ b)))
+    (hν : ∀ᵐ b ∂W, Tendsto (fun k => ∑ j∈s k,c k j *
+      ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
+      atTop (𝓝 (∫z,f z∂ν b)))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
+      ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W := by
+  let q := fun j : ℕ => Real.exp (-2*a*(j+1))
+  have hq (j : ℕ) : 0<q j := Real.exp_pos _
+  have hq1 (j : ℕ) : q j≤1 := level_step_le_one a ha j
+  have hq0 : Tendsto q atTop (𝓝 0) := mass_levels_tendsto_zero a (by linarith)
+  have htop : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => ∞) :=
+    fun t => by simp
+  have hiμ : Integrable (fun b => ∫z,f z∂μ b) W := by
+    have := actual_stopped_radon_limit_memLp a ha W hW S hS s c hc hsum μ f hμ 1
+    exact this.integrable (by norm_num)
+  have hiν : Integrable (fun b => ∫z,f z∂ν b) W := by
+    have := actual_stopped_radon_limit_memLp a ha W hW (fun _ => ∞) htop s c hc hsum ν f hν 1
+    exact this.integrable (by norm_num)
+  let F := fun k b => ∑ j∈s k,c k j * ∫z,f z∂actualUpperMass a (q j) (S b) b
+  let E := fun k b => ∑ j∈s k,c k j * ∫z,f z∂actualUpperMass a (q j) ∞ b
+  have hiF (k : ℕ) : Integrable (F k) W :=
+    integrable_finsetSum (s k) (fun j hj =>
+      (actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j) (hq1 j) S hS f).const_mul (c k j))
+  have hiE (k : ℕ) : Integrable (E k) W :=
+    integrable_finsetSum (s k) (fun j hj =>
+      (actualUpperMass_stopped_eval_integrable a ha W hW (q j) (hq j) (hq1 j) (fun _ => ∞) htop f).const_mul (c k j))
+  have hF := setIntegral_tendsto_of_L1 W F (fun b => ∫z,f z∂μ b) hiF hiμ
+    (actual_stopped_radon_limit_L1 a ha W hW S hS q hq hq1 s c hc hsum μ f hμ) U
+  have hE := setIntegral_tendsto_of_L1 W E (fun b => ∫z,f z∂ν b) hiE hiν
+    (actual_stopped_radon_limit_L1 a ha W hW (fun _ => ∞) htop q hq hq1 s c hc hsum ν f hν) U
+  have hpot := actualUpperMass_stopped_convex_potential_limit a ha W hW q
+    (fun j => ⟨hq j,hq1 j⟩) hq0 s c hs hc hsum S hS f U hU
+  have he (k : ℕ) : (∫b in U, E k b ∂W)-(∫b in U, F k b ∂W) =
+      ∫b in U,∑ j∈s k,c k j*((∫z,f z∂actualUpperMass a (q j) ∞ b)-
+        (∫z,f z∂actualUpperMass a (q j) (S b) b)) ∂W := by
+    rw [←integral_sub (hiE k).restrict (hiF k).restrict]
+    congr 1
+    funext b
+    simp only [E,F,mul_sub,Finset.sum_sub_distrib]
+  rw [integral_sub hiν.restrict hiμ.restrict]
+  exact tendsto_nhds_unique (hE.sub hF) (by simpa only [he] using hpot)
+
+ 
+
+theorem stopped_compact_conditional_to_random_tests (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
+    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂μ b) W)
+    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂ν b) W)
+    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
+      MeasurableSet[hS.measurableSpace] U →
+      (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
+        ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W)
+    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
+    (hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
+    (∫⁻b,∫⁻z,H (b,z) ∂ν b ∂W) = (∫⁻b,∫⁻z,H (b,z) ∂μ b ∂W) +
+      ∫⁻b,∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂upperArea ∂W := by
+  have := isSFiniteKernel_of_locally_finite μ
+  have := isSFiniteKernel_of_locally_finite ν
+  have := average_finiteOnCompacts μ W hμ
+  have := average_finiteOnCompacts ν W hν
+  have := stoppedSurvivingKernel_average_finiteOnCompacts a ha W hW S hS
+  have : IsFiniteMeasureOnCompacts ((μ+stoppedSurvivingKernel a ha S hS) ∘ₘ W) := by
+    rw [SLEExactGauge.MassVersions.average_add]
+    constructor
+    intro K hK
+    rw [Measure.add_apply]
+    exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩
+  have hidentity := conditional_lintegral_eq_of_average_finite ν
+    (μ+stoppedSurvivingKernel a ha S hS) W hS.measurableSpace hS.measurableSpace_le ?_ H hH
+  · rw [hidentity]
+    have hH₀ : Measurable H := hH.mono (prod_sigma_le hS.measurableSpace_le _) le_rfl
+    have heval (b : ExactSLEGenerator.Driver) :
+        (∫⁻z,H (b,z) ∂(μ+stoppedSurvivingKernel a ha S hS) b) =
+          (∫⁻z,H (b,z) ∂μ b)+
+            ∫⁻z,H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂upperArea := by
+      have hHb : Measurable (fun z : UpperPlane => H (b,z)) :=
+        hH₀.comp ((measurable_const (a:=b)).prodMk measurable_id)
+      rw [add_apply,lintegral_add_measure,stoppedSurvivingKernel_apply]
+      congr 1
+      exact (lintegral_withDensity_eq_lintegral_mul upperArea
+        (stoppedSurvivingGreen_spatial_measurable a ha S hS b).ennreal_ofReal hHb).trans
+          (lintegral_congr (fun _ => mul_comm _ _))
+    simp_rw [heval]
+    rw [lintegral_add_left]
+    exact hH₀.lintegral_kernel_prod_right'
+  · intro U hU f
+    have hadd : (fun b => ∫z,f z∂(μ+stoppedSurvivingKernel a ha S hS) b) =ᵐ[W]
+        (fun b => (∫z,f z∂μ b)+(∫z,f z∂stoppedSurvivingKernel a ha S hS b)) := by
+      filter_upwards [stoppedSurvivingKernel_compact_ae_integrable a ha W hW S hS f] with b hb
+      exact integral_add_measure f.integrable hb
+    rw [integral_congr_ae (ae_restrict_of_ae hadd),
+      integral_add (hμ f).restrict (stoppedSurvivingKernel_compact_mean_integrable a ha W hW S hS f).restrict]
+    have hf := he f U hU
+    rw [integral_sub (hν f).restrict (hμ f).restrict] at hf
+    simp_rw [stoppedSurvivingKernel_compact_eval]
+    linarith
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem stopped_conditional_future_support (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (μ ν : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (μ b)] [∀ b, IsFiniteMeasureOnCompacts (ν b)]
+    (hμ : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂μ b) W)
+    (hν : ∀ f : C_c(UpperPlane,ℝ), Integrable (fun b => ∫z,f z∂ν b) W)
+    (hle : ∀ᵐ b ∂W, μ b≤ν b)
+    (he : ∀ f : C_c(UpperPlane,ℝ), ∀ U : Set ExactSLEGenerator.Driver,
+      MeasurableSet[hS.measurableSpace] U →
+      (∫b in U,(∫z,f z∂ν b)-(∫z,f z∂μ b) ∂W) =
+        ∫b in U,(∫z : UpperPlane,f z*stoppedSurvivingGreen a (S b) b z.1 ∂upperArea) ∂W) :
+    ∀ᵐ b ∂W, (ν b-μ b) {z : UpperPlane | stoppedSurvivingGreen a (S b) b z.1=0}=0 := by
+  let D : Set (ExactSLEGenerator.Driver × UpperPlane) :=
+    {p | stoppedSurvivingGreen a (S p.1) p.1 p.2.1=0}
+  have hD : MeasurableSet[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] D :=
+    (stoppedSurvivingGreen_raw_joint a ha S hS) (measurableSet_singleton (0 : ℝ))
+  have hD₀ : MeasurableSet D := prod_sigma_le hS.measurableSpace_le _ _ hD
+  have := average_finiteOnCompacts ν W hν
+  apply random_set_sub_null_of_compact_means μ ν W D hD₀ hle
+  intro K hK
+  let E := D∩Prod.snd ⁻¹' K
+  have hE : MeasurableSet[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] E :=
+    hD.inter (measurable_snd hK.measurableSet)
+  have hE₀ : MeasurableSet E := prod_sigma_le hS.measurableSpace_le _ _ hE
+  let H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞ := E.indicator (fun _ => 1)
+  have hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H :=
+    measurable_const.indicator hE
+  have hid := stopped_compact_conditional_to_random_tests a ha W hW S hS μ ν hμ hν he H hH
+  have hev (ρ : Measure UpperPlane) (b : ExactSLEGenerator.Driver) :
+      (∫⁻z,H (b,z) ∂ρ)=ρ ({z | (b,z)∈D}∩K) := by
+    have hh := lintegral_indicator_const_comp (μ:=ρ)
+      ((measurable_const (a:=b)).prodMk measurable_id) hE₀ (1 : ℝ≥0∞)
+    have hs : (fun z : UpperPlane => (b,id z)) ⁻¹' E={z | (b,z)∈D}∩K := by
+      ext z; rfl
+    rw [hs,one_mul] at hh
+    exact hh
+  have hz (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+      H (b,z)*ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)=0 := by
+    by_cases h : (b,z)∈E
+    · rw [show stoppedSurvivingGreen a (S b) b z.1=0 from h.1,ENNReal.ofReal_zero,mul_zero]
+    · simp only [H,indicator_of_notMem h,zero_mul]
+  simpa only [hev,hz,lintegral_zero,add_zero] using hid
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StoppedWeights.lean b/lean/OAI/Probability/SLEGauge/StoppedWeights.lean
index 4390ee2e1db1f86b9453bd025d5f26f346c99903..6ee1c28ef7c9b33216a616b94d06978ff796ed7c 100644
--- a/lean/OAI/Probability/SLEGauge/StoppedWeights.lean
+++ b/lean/OAI/Probability/SLEGauge/StoppedWeights.lean
@@ -1,449 +1,449 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.ReverseGenerators
-import OAI.Probability.SLEGauge.RadialBounds
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-lemma regionStop_weight_eq (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
-    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
-    (b : ExactSLEGenerator.Driver)
-    (hr : physicalRadius (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) = ρ) :
-    physicalGreen a (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) =
-      radialWeight a b.1 z ρ := by
-  have he := regionStop_eq_radialTime a ε hε z N ρ δ B h0 b hr
-  rw [radialWeight,he,ite_eq_right ENNReal.coe_ne_top,ENNReal.toNNReal_coe]
-  rw [(regionStop_actualState a ε hε z N ρ δ B h0 b _ le_rfl).2]
-
-lemma actual_sine_lower (a : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b) (z : ℂ)
-    (T : ℝ≥0) (hT : alive a b T z) :
-    ∃ η : ℝ, 0 < η ∧ ∀ t : ℝ≥0, t ≤ T →
-      η ≤ Y (actualState a b t z) / ‖(actualState a b t z).1‖ := by
-  obtain ⟨f,hf⟩ := hT
-  let s : ℝ → ℝ := fun u => (f u).im / ‖f u-(b u.toNNReal : ℂ)‖
-  have hnp (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : 0 < ‖f u-(b u.toNNReal : ℂ)‖ :=
-    norm_pos_iff.mpr (odeUntil_center_ne_zero hf hu)
-  have hsc : ContinuousOn s (Icc (0 : ℝ) T) :=
-    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)).div
-      (odeUntil_center_continuousOn hb hf).norm (fun u hu => (hnp u hu).ne')
-  obtain ⟨u,hu,hmin⟩ := isCompact_Icc.exists_isMinOn
-    (nonempty_Icc.mpr T.coe_nonneg) hsc
-  refine ⟨s u,div_pos (hf.2 u hu).1 (hnp u hu),?_⟩
-  intro t ht
-  have hv : (t : ℝ) ∈ Icc (0 : ℝ) T := ⟨t.coe_nonneg,ht⟩
-  have hl := hmin hv
-  change s u ≤ s (t : ℝ) at hl
-  have he : loewnerMap a b t z = f t := by
-    simpa only [Real.toNNReal_coe] using loewnerMap_eq_solution_time hf hv
-  simpa only [s,actualState,Y_apply,he,
-    Real.toNNReal_coe,Complex.sub_im,Complex.ofReal_im,sub_zero] using hl
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
- 
-
-structure RadialLocalization (a : ℝ) (z : ℂ) (ρ δ : ℝ) where
-  ε : ℝ
-  hε : 0 < ε
-  N : ℝ≥0
-  B : ℝ≥0
-  initial : 0 ≤ regionMonitor ε N ρ δ (z,0)
-  small : ε < heightFloor z.im ρ δ
-  large : z.im/δ+logCeiling z.im ρ δ < N
-  time : z.im^2 < 2*a*δ^2*(B : ℝ)
-
-lemma exists_radialLocalization (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ δ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (hδ : 0 < δ)
-    (hs : δ*‖z‖ ≤ z.im) : Nonempty (RadialLocalization a z ρ δ) := by
-  let m := heightFloor z.im ρ δ
-  let C := logCeiling z.im ρ δ
-  have hm : 0 < m := heightFloor_pos _ _ _
-  have hC : 0 ≤ C := logCeiling_nonneg _ _ _
-  have hmz : m ≤ z.im := heightFloor_le hz hρ hr
-  let N : ℝ≥0 := ⟨z.im/δ+C+1,by positivity⟩
-  let B : ℝ≥0 := ⟨z.im^2/(2*a*δ^2)+1,by positivity⟩
-  have hinit : 0 ≤ regionMonitor (m/2) N ρ δ (z,0) := by
-    apply (regionMonitor_nonneg_iff _ _ _ _ _).mpr
-    have hnorm : ‖z‖ ≤ z.im/δ := (le_div_iff₀ hδ).mpr (by simpa only [mul_comm] using hs)
-    refine ⟨?_,?_,?_,hs⟩
-    · change m/2 ≤ z.im
-      linarith
-    · simpa only [Prod.norm_def,Real.norm_eq_abs,abs_zero,max_eq_left (norm_nonneg z)] using
-        (show ‖z‖ ≤ (N : ℝ) by change ‖z‖ ≤ z.im/δ+C+1; linarith)
-    · simpa only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one] using hr
-  refine ⟨⟨m/2,by positivity,N,B,hinit,by change m/2 < m; linarith,?_,?_⟩⟩
-  · change z.im/δ+C < z.im/δ+C+1
-    linarith
-  · change z.im^2 < 2*a*δ^2*(z.im^2/(2*a*δ^2)+1)
-    have hp : 0 < 2*a*δ^2 := by positivity
-    field_simp
-    nlinarith
-
-noncomputable def radialLocalization (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ δ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (hδ : 0 < δ)
-    (hs : δ*‖z‖ ≤ z.im) : RadialLocalization a z ρ δ :=
-  Classical.choice (exists_radialLocalization a ha z hz ρ δ hρ hr hδ hs)
-
-noncomputable def angleThreshold (z : ℂ) (n : ℕ) : ℝ :=
-  (z.im/‖z‖) / ((n : ℝ)+1)
-lemma angleThreshold_pos (z : ℂ) (hz : 0 < z.im) (n : ℕ) : 0 < angleThreshold z n := by
-  have hn : 0 < ‖z‖ := hz.trans_le (Complex.im_le_norm z)
-  unfold angleThreshold
-  positivity
-lemma angleThreshold_initial (z : ℂ) (hz : 0 < z.im) (n : ℕ) :
-    angleThreshold z n * ‖z‖ ≤ z.im := by
-  have hn : 0 < ‖z‖ := hz.trans_le (Complex.im_le_norm z)
-  have hle : angleThreshold z n ≤ z.im/‖z‖ := by
-    exact div_le_self (div_nonneg hz.le hn.le) (by nlinarith [Nat.cast_nonneg (α := ℝ) n])
-  exact (mul_le_mul_of_nonneg_right hle hn.le).trans_eq (div_mul_cancel₀ _ hn.ne')
-lemma angleThreshold_tendsto (z : ℂ) : Tendsto (angleThreshold z) atTop (𝓝 0) := by
-  change Tendsto (fun n : ℕ => (z.im/‖z‖)/((n : ℝ)+1)) atTop (𝓝 0)
-  simpa only [mul_one_div,mul_zero] using
-    tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (z.im/‖z‖)
-
-noncomputable def radialApproximation (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) (b : ExactSLEGenerator.Driver) : ℝ :=
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  physicalGreen a (localState a L.ε L.hε z
-    (regionStop a L.ε L.hε z L.N ρ (angleThreshold z n) L.B b) b)
-
-lemma radialApproximation_mean (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) :
-    Integrable (radialApproximation a ha z hz ρ hρ hr n) W ∧
-      (∫ b, radialApproximation a ha z hz ρ hρ hr n b ∂W) = physicalGreen a (z,0) := by
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  exact regionStop_green_mean W hW a L.ε ha.ne' L.hε z L.N L.B ρ (angleThreshold z n) L.initial
-
-lemma radialApproximation_bound (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a) (z : ℂ)
-    (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
-    (b : ExactSLEGenerator.Driver) :
-    0 ≤ radialApproximation a ha0 z hz ρ hρ hr n b ∧
-      radialApproximation a ha0 z hz ρ hρ hr n b ≤ Real.exp (-alpha a*Real.log ρ) := by
-  let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b _ le_rfl
-  exact ⟨(Real.exp_pos _).le,physicalGreen_bound a ρ ha hρ _ (L.hε.trans_le hs.1) hs.2.2.1⟩
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma physicalGreen_angle_bound (a ρ δ : ℝ) (ha : 1/4 < a) (hρ : 0 < ρ)
-    (v : State) (hy : 0 < Y v) (hr : ρ ≤ physicalRadius v)
-    (hs : Y v = δ*‖v.1‖) :
-    physicalGreen a v ≤ Real.exp (-alpha a*Real.log ρ) * δ^(p a) := by
-  have hn : 0 < ‖v.1‖ := hy.trans_le (Complex.im_le_norm _)
-  have hd : Y v / ‖v.1‖ = δ := (div_eq_iff hn.ne').mpr hs
-  rw [physicalGreen_eq_radius_sine a v hy,hd]
-  have hR : 0 < physicalRadius v := hρ.trans_le hr
-  apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (by
-    rw [← hd]; exact div_nonneg hy.le hn.le) _)
-  rw [Real.rpow_def_of_pos hR]
-  apply Real.exp_le_exp.mpr
-  have hlog := Real.log_le_log hρ hr
-  nlinarith [mul_le_mul_of_nonneg_left hlog (alpha_pos ha).le]
-
-lemma radialApproximation_tendsto (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a)
-    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (b : ExactSLEGenerator.Driver) :
-    Tendsto (fun n => radialApproximation a ha0 z hz ρ hρ hr n b) atTop
-      (𝓝 (radialWeight a b.1 z ρ)) := by
-  classical
-  let L (n : ℕ) := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T (n : ℕ) := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
-  let V (n : ℕ) := localState a (L n).ε (L n).hε z (T n) b
-  have hexit (n : ℕ) : physicalRadius (V n) = ρ ∨ Y (V n) = angleThreshold z n*‖(V n).1‖ :=
-    regionStop_true_exit a (L n).ε ha0 (L n).hε z (L n).N ρ (angleThreshold z n)
-      hρ (angleThreshold_pos z hz n) (L n).B (L n).initial (L n).small (L n).large (L n).time b
-  have hactual (n : ℕ) : alive a b.1 (T n) z ∧ actualState a b.1 (T n) z = V n :=
-    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-      (L n).B (L n).initial b _ le_rfl
-  by_cases hex : ∃ t : ℝ≥0, alive a b.1 t z ∧ physicalRadius (actualState a b.1 t z) = ρ
-  · obtain ⟨t,ht,htr⟩ := hex
-    obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
-    have he : ∀ᶠ n in atTop, radialApproximation a ha0 z hz ρ hρ hr n b = radialWeight a b.1 z ρ := by
-      filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
-      have ht' : T n ≤ t := regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-        (L n).B (L n).initial b t htr
-      have hη' := hηs (T n) ht'
-      rw [(hactual n).2] at hη'
-      rcases hexit n with hR|hS
-      · exact regionStop_weight_eq a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-          (L n).B (L n).initial b hR
-      · have hsafe := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-          (L n).B (L n).initial b (T n) le_rfl
-        have hnp : 0 < ‖(V n).1‖ := ((L n).hε.trans_le hsafe.1).trans_le (Complex.im_le_norm _)
-        have hd : Y (V n)/‖(V n).1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-        rw [hd] at hη'
-        exact False.elim (not_le.mpr hn hη')
-    exact Filter.Tendsto.congr' (he.mono fun n hn => hn.symm) tendsto_const_nhds
-  · have hw : radialWeight a b.1 z ρ = 0 := by rw [radialWeight,ite_eq_left (radialTime_top_of_no_hit a b.1 z ρ hex)]
-    rw [hw]
-    have hbound (n : ℕ) : radialApproximation a ha0 z hz ρ hρ hr n b ≤
-        Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a) := by
-      rcases hexit n with hR|hS
-      · exact False.elim (hex ⟨T n,(hactual n).1,by simpa only [(hactual n).2] using hR⟩)
-      · have hs := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
-          (L n).B (L n).initial b (T n) le_rfl
-        exact physicalGreen_angle_bound a ρ (angleThreshold z n) ha hρ (V n)
-          ((L n).hε.trans_le hs.1) hs.2.2.1 hS
-    have hp : 0 < p a := by unfold p; linarith
-    have hu : Tendsto (fun n => Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using ((angleThreshold_tendsto z).rpow_const_nhds_zero hp).const_mul
-        (Real.exp (-alpha a*Real.log ρ))
-    exact squeeze_zero (fun n => (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1) hbound hu
-
- 
-
-theorem radialWeight_mean (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
-    Integrable (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ) W ∧
-      (∫ b : ExactSLEGenerator.Driver, radialWeight a b.1 z ρ ∂W) = physicalGreen a (z,0) := by
-  have ha0 : 0 < a := by linarith
-  let F := radialApproximation a ha0 z hz ρ hρ hr
-  let G := fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ
-  let C := Real.exp (-alpha a*Real.log ρ)
-  have hFi (n : ℕ) : Integrable (F n) W := (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).1
-  have hconv (b : ExactSLEGenerator.Driver) : Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
-    radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
-  have hFm (n : ℕ) := (hFi n).aestronglyMeasurable
-  have hGm : AEStronglyMeasurable G W := aestronglyMeasurable_of_tendsto_ae atTop hFm
-    (Eventually.of_forall hconv)
-  have hFb (n : ℕ) : ∀ᵐ b ∂W, ‖F n b‖ ≤ C := by
-    filter_upwards [] with b
-    have h := radialApproximation_bound a ha ha0 z hz ρ hρ hr n b
-    change ‖radialApproximation a ha0 z hz ρ hρ hr n b‖ ≤ Real.exp (-alpha a*Real.log ρ)
-    simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
-  have hGb : ∀ᵐ b ∂W, ‖G b‖ ≤ C := by
-    filter_upwards [] with b
-    exact le_of_tendsto (hconv b).norm (Eventually.of_forall fun n => by
-      have h := radialApproximation_bound a ha ha0 z hz ρ hρ hr n b
-      change ‖radialApproximation a ha0 z hz ρ hρ hr n b‖ ≤ Real.exp (-alpha a*Real.log ρ)
-      simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2)
-  have hGi : Integrable G W := Integrable.of_bound hGm C hGb
-  refine ⟨hGi,?_⟩
-  have hi := tendsto_integral_of_dominated_convergence (fun _ => C) hFm
-    (integrable_const C) hFb (Eventually.of_forall hconv)
-  have he : (fun n => ∫ b, F n b ∂W) = fun _ : ℕ => physicalGreen a (z,0) :=
-    funext fun n => (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).2
-  rw [he] at hi
-  exact tendsto_nhds_unique hi tendsto_const_nhds
-
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma radialTime_attained (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
-    (hfin : radialTime a b.1 z ρ ≠ ∞) :
-    alive a b.1 (radialTime a b.1 z ρ).toNNReal z ∧
-      physicalRadius (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z) = ρ := by
-  have hex : ∃ t : ℝ≥0, alive a b.1 t z ∧ physicalRadius (actualState a b.1 t z) = ρ := by
-    by_contra hn
-    exact hfin (radialTime_top_of_no_hit a b.1 z ρ hn)
-  obtain ⟨t,ht,htr⟩ := hex
-  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
-  obtain ⟨n,hn⟩ := ((angleThreshold_tendsto z).eventually (gt_mem_nhds hη)).exists
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T := regionStop a L.ε L.hε z L.N ρ (angleThreshold z n) L.B b
-  let V := localState a L.ε L.hε z T b
-  have he := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
-  have ht' : T ≤ t := regionStop_le_hit a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t htr
-  have hR : physicalRadius V = ρ := by
-    rcases regionStop_true_exit a L.ε ha L.hε z L.N ρ (angleThreshold z n)
-      hρ (angleThreshold_pos z hz n) L.B L.initial L.small L.large L.time b with hR|hS
-    · exact hR
-    · have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
-      have hnp : 0 < ‖V.1‖ := (L.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
-      have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-      have hη' := hηs T ht'
-      rw [he.2,hd] at hη'
-      exact False.elim (not_le.mpr hn hη')
-  have htime := regionStop_eq_radialTime a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b hR
-  rw [htime,ENNReal.toNNReal_coe]
-  change alive a b.1 T z ∧ physicalRadius (actualState a b.1 T z) = ρ
-  exact ⟨he.1,by rw [he.2]; exact hR⟩
-
-end ExactSLEGreen
-
-open Filter
-open scoped Topology
-namespace ExactSLEMain
-noncomputable def diskMap (θ : ℝ) (z : ℂ) : ℂ :=
-  (z - Complex.exp (Complex.I * θ)) / (z - Complex.exp (-Complex.I * θ))
-noncomputable def fromDisk (θ : ℝ) (x : ℂ) : ℂ :=
-  (x * Complex.exp (-Complex.I * θ) - Complex.exp (Complex.I * θ)) / (x - 1)
-
-lemma fromDisk_zero (θ : ℝ) : fromDisk θ 0 = Complex.exp (Complex.I*θ) := by simp [fromDisk]
-lemma fromDisk_zero_im (θ : ℝ) : (fromDisk θ 0).im = Real.sin θ := by
-  rw [fromDisk_zero,mul_comm,Complex.exp_ofReal_mul_I_im]
-lemma fromDisk_zero_norm (θ : ℝ) : ‖fromDisk θ 0‖ = 1 := by
-  rw [fromDisk_zero,Complex.norm_exp]
-  simp
-lemma disk_endpoints_difference (θ : ℝ) :
-    Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ) = (2*Real.sin θ : ℝ)*Complex.I := by
-  rw [mul_comm Complex.I,show -Complex.I*(θ : ℂ) = ((-θ : ℝ) : ℂ)*Complex.I by push_cast; ring,
-    Complex.exp_ofReal_mul_I,Complex.exp_ofReal_mul_I,Real.cos_neg,Real.sin_neg]
-  push_cast
-  ring
-lemma disk_endpoints_ne (θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
-    Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ) ≠ 0 := by
-  rw [disk_endpoints_difference]
-  exact mul_ne_zero (Complex.ofReal_ne_zero.mpr (mul_pos (by norm_num) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ)).ne') Complex.I_ne_zero
-lemma disk_deriv_at_zero (θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
-    deriv (diskMap θ) (fromDisk θ 0) =
-      (Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ))⁻¹ := by
-  rw [fromDisk_zero]
-  have hn := disk_endpoints_ne θ hθ hθπ
-  have hd := ((hasDerivAt_id (Complex.exp (Complex.I*θ))).sub_const (Complex.exp (Complex.I*θ))).div
-    ((hasDerivAt_id (Complex.exp (Complex.I*θ))).sub_const (Complex.exp (-Complex.I*θ))) hn
-  change deriv (fun z => (z-Complex.exp (Complex.I*θ))/(z-Complex.exp (-Complex.I*θ))) _ = _
-  have hd' := hd.deriv
-  change deriv (fun z => (z-Complex.exp (Complex.I*θ))/(z-Complex.exp (-Complex.I*θ))) _ = _ at hd'
-  rw [hd']
-  simp only [id_eq,sub_self,zero_mul,sub_zero,one_mul]
-  field_simp [hn]
-lemma disk_deriv_norm (θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
-    ‖deriv (diskMap θ) (fromDisk θ 0)‖ = (2*Real.sin θ)⁻¹ := by
-  rw [disk_deriv_at_zero θ hθ hθπ,disk_endpoints_difference,norm_inv,norm_mul,Complex.norm_real,
-    Complex.norm_I,mul_one,Real.norm_eq_abs,abs_of_pos (mul_pos (by norm_num) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ))]
-end ExactSLEMain
-
-namespace ExactSLEMain
-noncomputable def centeredZ (A : ℝ) (θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℂ :=
-  loewnerMap A b t (fromDisk θ x) - (b t : ℂ)
-noncomputable def sineAngle (A θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℝ :=
-  (centeredZ A θ b t x).im / ‖centeredZ A θ b t x‖
-noncomputable def diskRadius (A θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℝ :=
-  ‖deriv (diskMap θ) (fromDisk θ x)‖ * (2 * (centeredZ A θ b t x).im) /
-    ‖deriv (loewnerMap A b t) (fromDisk θ x)‖
-noncomputable def green (a θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℝ := by
-  classical
-  exact if alive a b t (fromDisk θ x) then
-    diskRadius a θ b t x ^ (-(2 - (1 + 1 / (4 * a)))) * sineAngle a θ b t x ^ (4 * a - 1)
-    else 0
-
-noncomputable def levelTime (a θ : ℝ) (b : ℝ≥0 → ℝ) (l : ℕ) : ℝ≥0∞ :=
-  sInf {u : ℝ≥0∞ | ∃ t : ℝ≥0, u = t ∧ alive a b t (fromDisk θ 0) ∧
-    diskRadius a θ b t 0 = Real.exp (-2 * a * l)}
-noncomputable def tiltWeight (a θ : ℝ) (b : ℝ≥0 → ℝ) (l : ℕ) : ℝ :=
-  if levelTime a θ b l = ∞ then 0 else
-    Real.exp (2 * a * (2 - (1 + 1 / (4 * a))) * l) *
-      sineAngle a θ b (levelTime a θ b l).toNNReal 0 ^ (4 * a - 1) / Real.sin θ ^ (4 * a - 1)
-
-end ExactSLEMain
-
-namespace ExactSLEMain
-open ExactSLEGreen
-
-lemma diskRadius_actualState (a θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi)
-    (b : ℝ≥0 → ℝ) (hb : Continuous b) (t : ℝ≥0)
-    (ht : alive a b t (fromDisk θ 0)) :
-    diskRadius a θ b t 0 = physicalRadius (actualState a b t (fromDisk θ 0)) / Real.sin θ := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  have hd : 0 < ‖deriv (loewnerMap a b t) (fromDisk θ 0)‖ :=
-    norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb ht)
-  unfold diskRadius physicalRadius actualState centeredZ
-  rw [disk_deriv_norm θ hθ hθπ]
-  simp only [Y_apply,J_apply,Real.exp_neg,Real.exp_log hd]
-  field_simp
-
-lemma levelTime_radialTime (a θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi)
-    (b : ℝ≥0 → ℝ) (hb : Continuous b) (l : ℕ) :
-    levelTime a θ b l = radialTime a b (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  unfold levelTime radialTime
-  congr 1
-  ext u
-  constructor
-  · rintro ⟨t,hu,ht,hR⟩
-    refine ⟨t,hu,ht,?_⟩
-    rw [diskRadius_actualState a θ hθ hθπ b hb t ht] at hR
-    simpa only [mul_comm] using (div_eq_iff hs.ne').mp hR
-  · rintro ⟨t,hu,ht,hR⟩
-    refine ⟨t,hu,ht,?_⟩
-    rw [diskRadius_actualState a θ hθ hθπ b hb t ht,hR,mul_div_cancel_left₀ _ hs.ne']
-
-lemma radial_initial_green (a θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
-    physicalGreen a (fromDisk θ 0,0) =
-      (Real.sin θ)^(-alpha a)*(Real.sin θ)^(p a) := by
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  rw [physicalGreen_eq_radius_sine a _ (by simpa only [Y_apply,fromDisk_zero_im] using hs)]
-  simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one,fromDisk_zero_im,
-    fromDisk_zero_norm,div_one]
-
-lemma tiltWeight_radialWeight (a θ : ℝ) (ha : 1/4 < a) (hθ : 0 < θ) (hθπ : θ < Real.pi)
-    (b : Driver) (l : ℕ) :
-    tiltWeight a θ b.1 l =
-      radialWeight a b.1 (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) /
-        physicalGreen a (fromDisk θ 0,0) := by
-  have ha0 : 0 < a := by linarith
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  let ρ := Real.sin θ*Real.exp (-2*a*l)
-  have hρ : 0 < ρ := mul_pos hs (Real.exp_pos _)
-  have hr : ρ ≤ (fromDisk θ 0).im := by
-    rw [fromDisk_zero_im]
-    have he : Real.exp (-2*a*l) ≤ 1 := Real.exp_le_one_iff.mpr (by
-      have h : (0 : ℝ) ≤ 2*a*l := by positivity
-      nlinarith)
-    exact (mul_le_mul_of_nonneg_left he hs.le).trans_eq (mul_one _)
-  have hz : 0 < (fromDisk θ 0).im := by simpa only [fromDisk_zero_im] using hs
-  have htime := levelTime_radialTime a θ hθ hθπ b.1 b.1.continuous l
-  by_cases hfin : radialTime a b.1 (fromDisk θ 0) ρ = ∞
-  · rw [tiltWeight,htime,ite_eq_left hfin,radialWeight,ite_eq_left hfin,zero_div]
-  · have ht := radialTime_attained a ha0 (fromDisk θ 0) hz ρ hρ hr b hfin
-    let T := (radialTime a b.1 (fromDisk θ 0) ρ).toNNReal
-    have hy : 0 < Y (actualState a b.1 T (fromDisk θ 0)) := by
-      obtain ⟨f,hf⟩ := ht.1
-      rw [Y_apply,actualState,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq hf]
-      exact (hf.2 T ⟨T.coe_nonneg,le_rfl⟩).1
-    have heα : 2-(1+1/(4*a)) = alpha a := by unfold alpha; ring
-    rw [tiltWeight,htime,ite_eq_right hfin,radialWeight,ite_eq_right hfin,
-      physicalGreen_eq_radius_sine a _ hy,ht.2,radial_initial_green a θ hθ hθπ,heα]
-    change Real.exp (2*a*alpha a*l)*
-        (Y (actualState a b.1 T (fromDisk θ 0))/‖(actualState a b.1 T (fromDisk θ 0)).1‖)^(p a) /
-        (Real.sin θ)^(p a) =
-      ρ^(-alpha a)*(Y (actualState a b.1 T (fromDisk θ 0))/‖(actualState a b.1 T (fromDisk θ 0)).1‖)^(p a) /
-        ((Real.sin θ)^(-alpha a)*(Real.sin θ)^(p a))
-    have he : ρ^(-alpha a) = (Real.sin θ)^(-alpha a)*Real.exp (2*a*alpha a*l) := by
-      rw [Real.rpow_def_of_pos hρ,Real.rpow_def_of_pos hs]
-      dsimp only [ρ]
-      rw [Real.log_mul hs.ne' (Real.exp_ne_zero _),Real.log_exp,← Real.exp_add]
-      congr 1
-      ring
-    rw [he]
-    have hn : (Real.sin θ)^(-alpha a) ≠ 0 := (Real.rpow_pos_of_pos hs _).ne'
-    field_simp
-
- 
-
-end ExactSLEMain
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.ReverseGenerators
+import OAI.Probability.SLEGauge.RadialBounds
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+lemma regionStop_weight_eq (a ε : ℝ) (hε : 0 < ε) (z : ℂ)
+    (N ρ δ : ℝ) (B : ℝ≥0) (h0 : 0 ≤ regionMonitor ε N ρ δ (z,0))
+    (b : ExactSLEGenerator.Driver)
+    (hr : physicalRadius (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) = ρ) :
+    physicalGreen a (localState a ε hε z (regionStop a ε hε z N ρ δ B b) b) =
+      radialWeight a b.1 z ρ := by
+  have he := regionStop_eq_radialTime a ε hε z N ρ δ B h0 b hr
+  rw [radialWeight,he,ite_eq_right ENNReal.coe_ne_top,ENNReal.toNNReal_coe]
+  rw [(regionStop_actualState a ε hε z N ρ δ B h0 b _ le_rfl).2]
+
+lemma actual_sine_lower (a : ℝ) (b : ℝ≥0 → ℝ) (hb : Continuous b) (z : ℂ)
+    (T : ℝ≥0) (hT : alive a b T z) :
+    ∃ η : ℝ, 0 < η ∧ ∀ t : ℝ≥0, t ≤ T →
+      η ≤ Y (actualState a b t z) / ‖(actualState a b t z).1‖ := by
+  obtain ⟨f,hf⟩ := hT
+  let s : ℝ → ℝ := fun u => (f u).im / ‖f u-(b u.toNNReal : ℂ)‖
+  have hnp (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : 0 < ‖f u-(b u.toNNReal : ℂ)‖ :=
+    norm_pos_iff.mpr (odeUntil_center_ne_zero hf hu)
+  have hsc : ContinuousOn s (Icc (0 : ℝ) T) :=
+    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)).div
+      (odeUntil_center_continuousOn hb hf).norm (fun u hu => (hnp u hu).ne')
+  obtain ⟨u,hu,hmin⟩ := isCompact_Icc.exists_isMinOn
+    (nonempty_Icc.mpr T.coe_nonneg) hsc
+  refine ⟨s u,div_pos (hf.2 u hu).1 (hnp u hu),?_⟩
+  intro t ht
+  have hv : (t : ℝ) ∈ Icc (0 : ℝ) T := ⟨t.coe_nonneg,ht⟩
+  have hl := hmin hv
+  change s u ≤ s (t : ℝ) at hl
+  have he : loewnerMap a b t z = f t := by
+    simpa only [Real.toNNReal_coe] using loewnerMap_eq_solution_time hf hv
+  simpa only [s,actualState,Y_apply,he,
+    Real.toNNReal_coe,Complex.sub_im,Complex.ofReal_im,sub_zero] using hl
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+ 
+
+structure RadialLocalization (a : ℝ) (z : ℂ) (ρ δ : ℝ) where
+  ε : ℝ
+  hε : 0 < ε
+  N : ℝ≥0
+  B : ℝ≥0
+  initial : 0 ≤ regionMonitor ε N ρ δ (z,0)
+  small : ε < heightFloor z.im ρ δ
+  large : z.im/δ+logCeiling z.im ρ δ < N
+  time : z.im^2 < 2*a*δ^2*(B : ℝ)
+
+lemma exists_radialLocalization (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ δ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (hδ : 0 < δ)
+    (hs : δ*‖z‖ ≤ z.im) : Nonempty (RadialLocalization a z ρ δ) := by
+  let m := heightFloor z.im ρ δ
+  let C := logCeiling z.im ρ δ
+  have hm : 0 < m := heightFloor_pos _ _ _
+  have hC : 0 ≤ C := logCeiling_nonneg _ _ _
+  have hmz : m ≤ z.im := heightFloor_le hz hρ hr
+  let N : ℝ≥0 := ⟨z.im/δ+C+1,by positivity⟩
+  let B : ℝ≥0 := ⟨z.im^2/(2*a*δ^2)+1,by positivity⟩
+  have hinit : 0 ≤ regionMonitor (m/2) N ρ δ (z,0) := by
+    apply (regionMonitor_nonneg_iff _ _ _ _ _).mpr
+    have hnorm : ‖z‖ ≤ z.im/δ := (le_div_iff₀ hδ).mpr (by simpa only [mul_comm] using hs)
+    refine ⟨?_,?_,?_,hs⟩
+    · change m/2 ≤ z.im
+      linarith
+    · simpa only [Prod.norm_def,Real.norm_eq_abs,abs_zero,max_eq_left (norm_nonneg z)] using
+        (show ‖z‖ ≤ (N : ℝ) by change ‖z‖ ≤ z.im/δ+C+1; linarith)
+    · simpa only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one] using hr
+  refine ⟨⟨m/2,by positivity,N,B,hinit,by change m/2 < m; linarith,?_,?_⟩⟩
+  · change z.im/δ+C < z.im/δ+C+1
+    linarith
+  · change z.im^2 < 2*a*δ^2*(z.im^2/(2*a*δ^2)+1)
+    have hp : 0 < 2*a*δ^2 := by positivity
+    field_simp
+    nlinarith
+
+noncomputable def radialLocalization (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ δ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (hδ : 0 < δ)
+    (hs : δ*‖z‖ ≤ z.im) : RadialLocalization a z ρ δ :=
+  Classical.choice (exists_radialLocalization a ha z hz ρ δ hρ hr hδ hs)
+
+noncomputable def angleThreshold (z : ℂ) (n : ℕ) : ℝ :=
+  (z.im/‖z‖) / ((n : ℝ)+1)
+lemma angleThreshold_pos (z : ℂ) (hz : 0 < z.im) (n : ℕ) : 0 < angleThreshold z n := by
+  have hn : 0 < ‖z‖ := hz.trans_le (Complex.im_le_norm z)
+  unfold angleThreshold
+  positivity
+lemma angleThreshold_initial (z : ℂ) (hz : 0 < z.im) (n : ℕ) :
+    angleThreshold z n * ‖z‖ ≤ z.im := by
+  have hn : 0 < ‖z‖ := hz.trans_le (Complex.im_le_norm z)
+  have hle : angleThreshold z n ≤ z.im/‖z‖ := by
+    exact div_le_self (div_nonneg hz.le hn.le) (by nlinarith [Nat.cast_nonneg (α := ℝ) n])
+  exact (mul_le_mul_of_nonneg_right hle hn.le).trans_eq (div_mul_cancel₀ _ hn.ne')
+lemma angleThreshold_tendsto (z : ℂ) : Tendsto (angleThreshold z) atTop (𝓝 0) := by
+  change Tendsto (fun n : ℕ => (z.im/‖z‖)/((n : ℝ)+1)) atTop (𝓝 0)
+  simpa only [mul_one_div,mul_zero] using
+    tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (z.im/‖z‖)
+
+noncomputable def radialApproximation (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) (b : ExactSLEGenerator.Driver) : ℝ :=
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  physicalGreen a (localState a L.ε L.hε z
+    (regionStop a L.ε L.hε z L.N ρ (angleThreshold z n) L.B b) b)
+
+lemma radialApproximation_mean (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) :
+    Integrable (radialApproximation a ha z hz ρ hρ hr n) W ∧
+      (∫ b, radialApproximation a ha z hz ρ hρ hr n b ∂W) = physicalGreen a (z,0) := by
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  exact regionStop_green_mean W hW a L.ε ha.ne' L.hε z L.N L.B ρ (angleThreshold z n) L.initial
+
+lemma radialApproximation_bound (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a) (z : ℂ)
+    (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
+    (b : ExactSLEGenerator.Driver) :
+    0 ≤ radialApproximation a ha0 z hz ρ hρ hr n b ∧
+      radialApproximation a ha0 z hz ρ hρ hr n b ≤ Real.exp (-alpha a*Real.log ρ) := by
+  let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b _ le_rfl
+  exact ⟨(Real.exp_pos _).le,physicalGreen_bound a ρ ha hρ _ (L.hε.trans_le hs.1) hs.2.2.1⟩
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma physicalGreen_angle_bound (a ρ δ : ℝ) (ha : 1/4 < a) (hρ : 0 < ρ)
+    (v : State) (hy : 0 < Y v) (hr : ρ ≤ physicalRadius v)
+    (hs : Y v = δ*‖v.1‖) :
+    physicalGreen a v ≤ Real.exp (-alpha a*Real.log ρ) * δ^(p a) := by
+  have hn : 0 < ‖v.1‖ := hy.trans_le (Complex.im_le_norm _)
+  have hd : Y v / ‖v.1‖ = δ := (div_eq_iff hn.ne').mpr hs
+  rw [physicalGreen_eq_radius_sine a v hy,hd]
+  have hR : 0 < physicalRadius v := hρ.trans_le hr
+  apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (by
+    rw [← hd]; exact div_nonneg hy.le hn.le) _)
+  rw [Real.rpow_def_of_pos hR]
+  apply Real.exp_le_exp.mpr
+  have hlog := Real.log_le_log hρ hr
+  nlinarith [mul_le_mul_of_nonneg_left hlog (alpha_pos ha).le]
+
+lemma radialApproximation_tendsto (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a)
+    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (b : ExactSLEGenerator.Driver) :
+    Tendsto (fun n => radialApproximation a ha0 z hz ρ hρ hr n b) atTop
+      (𝓝 (radialWeight a b.1 z ρ)) := by
+  classical
+  let L (n : ℕ) := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T (n : ℕ) := regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
+  let V (n : ℕ) := localState a (L n).ε (L n).hε z (T n) b
+  have hexit (n : ℕ) : physicalRadius (V n) = ρ ∨ Y (V n) = angleThreshold z n*‖(V n).1‖ :=
+    regionStop_true_exit a (L n).ε ha0 (L n).hε z (L n).N ρ (angleThreshold z n)
+      hρ (angleThreshold_pos z hz n) (L n).B (L n).initial (L n).small (L n).large (L n).time b
+  have hactual (n : ℕ) : alive a b.1 (T n) z ∧ actualState a b.1 (T n) z = V n :=
+    regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+      (L n).B (L n).initial b _ le_rfl
+  by_cases hex : ∃ t : ℝ≥0, alive a b.1 t z ∧ physicalRadius (actualState a b.1 t z) = ρ
+  · obtain ⟨t,ht,htr⟩ := hex
+    obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
+    have he : ∀ᶠ n in atTop, radialApproximation a ha0 z hz ρ hρ hr n b = radialWeight a b.1 z ρ := by
+      filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
+      have ht' : T n ≤ t := regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+        (L n).B (L n).initial b t htr
+      have hη' := hηs (T n) ht'
+      rw [(hactual n).2] at hη'
+      rcases hexit n with hR|hS
+      · exact regionStop_weight_eq a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+          (L n).B (L n).initial b hR
+      · have hsafe := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+          (L n).B (L n).initial b (T n) le_rfl
+        have hnp : 0 < ‖(V n).1‖ := ((L n).hε.trans_le hsafe.1).trans_le (Complex.im_le_norm _)
+        have hd : Y (V n)/‖(V n).1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+        rw [hd] at hη'
+        exact False.elim (not_le.mpr hn hη')
+    exact Filter.Tendsto.congr' (he.mono fun n hn => hn.symm) tendsto_const_nhds
+  · have hw : radialWeight a b.1 z ρ = 0 := by rw [radialWeight,ite_eq_left (radialTime_top_of_no_hit a b.1 z ρ hex)]
+    rw [hw]
+    have hbound (n : ℕ) : radialApproximation a ha0 z hz ρ hρ hr n b ≤
+        Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a) := by
+      rcases hexit n with hR|hS
+      · exact False.elim (hex ⟨T n,(hactual n).1,by simpa only [(hactual n).2] using hR⟩)
+      · have hs := regionStop_safe a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n)
+          (L n).B (L n).initial b (T n) le_rfl
+        exact physicalGreen_angle_bound a ρ (angleThreshold z n) ha hρ (V n)
+          ((L n).hε.trans_le hs.1) hs.2.2.1 hS
+    have hp : 0 < p a := by unfold p; linarith
+    have hu : Tendsto (fun n => Real.exp (-alpha a*Real.log ρ) * (angleThreshold z n)^(p a)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using ((angleThreshold_tendsto z).rpow_const_nhds_zero hp).const_mul
+        (Real.exp (-alpha a*Real.log ρ))
+    exact squeeze_zero (fun n => (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1) hbound hu
+
+ 
+
+theorem radialWeight_mean (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) :
+    Integrable (fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ) W ∧
+      (∫ b : ExactSLEGenerator.Driver, radialWeight a b.1 z ρ ∂W) = physicalGreen a (z,0) := by
+  have ha0 : 0 < a := by linarith
+  let F := radialApproximation a ha0 z hz ρ hρ hr
+  let G := fun b : ExactSLEGenerator.Driver => radialWeight a b.1 z ρ
+  let C := Real.exp (-alpha a*Real.log ρ)
+  have hFi (n : ℕ) : Integrable (F n) W := (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).1
+  have hconv (b : ExactSLEGenerator.Driver) : Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
+    radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
+  have hFm (n : ℕ) := (hFi n).aestronglyMeasurable
+  have hGm : AEStronglyMeasurable G W := aestronglyMeasurable_of_tendsto_ae atTop hFm
+    (Eventually.of_forall hconv)
+  have hFb (n : ℕ) : ∀ᵐ b ∂W, ‖F n b‖ ≤ C := by
+    filter_upwards [] with b
+    have h := radialApproximation_bound a ha ha0 z hz ρ hρ hr n b
+    change ‖radialApproximation a ha0 z hz ρ hρ hr n b‖ ≤ Real.exp (-alpha a*Real.log ρ)
+    simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
+  have hGb : ∀ᵐ b ∂W, ‖G b‖ ≤ C := by
+    filter_upwards [] with b
+    exact le_of_tendsto (hconv b).norm (Eventually.of_forall fun n => by
+      have h := radialApproximation_bound a ha ha0 z hz ρ hρ hr n b
+      change ‖radialApproximation a ha0 z hz ρ hρ hr n b‖ ≤ Real.exp (-alpha a*Real.log ρ)
+      simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2)
+  have hGi : Integrable G W := Integrable.of_bound hGm C hGb
+  refine ⟨hGi,?_⟩
+  have hi := tendsto_integral_of_dominated_convergence (fun _ => C) hFm
+    (integrable_const C) hFb (Eventually.of_forall hconv)
+  have he : (fun n => ∫ b, F n b ∂W) = fun _ : ℕ => physicalGreen a (z,0) :=
+    funext fun n => (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).2
+  rw [he] at hi
+  exact tendsto_nhds_unique hi tendsto_const_nhds
+
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma radialTime_attained (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
+    (hfin : radialTime a b.1 z ρ ≠ ∞) :
+    alive a b.1 (radialTime a b.1 z ρ).toNNReal z ∧
+      physicalRadius (actualState a b.1 (radialTime a b.1 z ρ).toNNReal z) = ρ := by
+  have hex : ∃ t : ℝ≥0, alive a b.1 t z ∧ physicalRadius (actualState a b.1 t z) = ρ := by
+    by_contra hn
+    exact hfin (radialTime_top_of_no_hit a b.1 z ρ hn)
+  obtain ⟨t,ht,htr⟩ := hex
+  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
+  obtain ⟨n,hn⟩ := ((angleThreshold_tendsto z).eventually (gt_mem_nhds hη)).exists
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T := regionStop a L.ε L.hε z L.N ρ (angleThreshold z n) L.B b
+  let V := localState a L.ε L.hε z T b
+  have he := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
+  have ht' : T ≤ t := regionStop_le_hit a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t htr
+  have hR : physicalRadius V = ρ := by
+    rcases regionStop_true_exit a L.ε ha L.hε z L.N ρ (angleThreshold z n)
+      hρ (angleThreshold_pos z hz n) L.B L.initial L.small L.large L.time b with hR|hS
+    · exact hR
+    · have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
+      have hnp : 0 < ‖V.1‖ := (L.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
+      have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+      have hη' := hηs T ht'
+      rw [he.2,hd] at hη'
+      exact False.elim (not_le.mpr hn hη')
+  have htime := regionStop_eq_radialTime a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b hR
+  rw [htime,ENNReal.toNNReal_coe]
+  change alive a b.1 T z ∧ physicalRadius (actualState a b.1 T z) = ρ
+  exact ⟨he.1,by rw [he.2]; exact hR⟩
+
+end ExactSLEGreen
+
+open Filter
+open scoped Topology
+namespace ExactSLEMain
+noncomputable def diskMap (θ : ℝ) (z : ℂ) : ℂ :=
+  (z - Complex.exp (Complex.I * θ)) / (z - Complex.exp (-Complex.I * θ))
+noncomputable def fromDisk (θ : ℝ) (x : ℂ) : ℂ :=
+  (x * Complex.exp (-Complex.I * θ) - Complex.exp (Complex.I * θ)) / (x - 1)
+
+lemma fromDisk_zero (θ : ℝ) : fromDisk θ 0 = Complex.exp (Complex.I*θ) := by simp [fromDisk]
+lemma fromDisk_zero_im (θ : ℝ) : (fromDisk θ 0).im = Real.sin θ := by
+  rw [fromDisk_zero,mul_comm,Complex.exp_ofReal_mul_I_im]
+lemma fromDisk_zero_norm (θ : ℝ) : ‖fromDisk θ 0‖ = 1 := by
+  rw [fromDisk_zero,Complex.norm_exp]
+  simp
+lemma disk_endpoints_difference (θ : ℝ) :
+    Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ) = (2*Real.sin θ : ℝ)*Complex.I := by
+  rw [mul_comm Complex.I,show -Complex.I*(θ : ℂ) = ((-θ : ℝ) : ℂ)*Complex.I by push_cast; ring,
+    Complex.exp_ofReal_mul_I,Complex.exp_ofReal_mul_I,Real.cos_neg,Real.sin_neg]
+  push_cast
+  ring
+lemma disk_endpoints_ne (θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
+    Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ) ≠ 0 := by
+  rw [disk_endpoints_difference]
+  exact mul_ne_zero (Complex.ofReal_ne_zero.mpr (mul_pos (by norm_num) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ)).ne') Complex.I_ne_zero
+lemma disk_deriv_at_zero (θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
+    deriv (diskMap θ) (fromDisk θ 0) =
+      (Complex.exp (Complex.I*θ)-Complex.exp (-Complex.I*θ))⁻¹ := by
+  rw [fromDisk_zero]
+  have hn := disk_endpoints_ne θ hθ hθπ
+  have hd := ((hasDerivAt_id (Complex.exp (Complex.I*θ))).sub_const (Complex.exp (Complex.I*θ))).div
+    ((hasDerivAt_id (Complex.exp (Complex.I*θ))).sub_const (Complex.exp (-Complex.I*θ))) hn
+  change deriv (fun z => (z-Complex.exp (Complex.I*θ))/(z-Complex.exp (-Complex.I*θ))) _ = _
+  have hd' := hd.deriv
+  change deriv (fun z => (z-Complex.exp (Complex.I*θ))/(z-Complex.exp (-Complex.I*θ))) _ = _ at hd'
+  rw [hd']
+  simp only [id_eq,sub_self,zero_mul,sub_zero,one_mul]
+  field_simp [hn]
+lemma disk_deriv_norm (θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
+    ‖deriv (diskMap θ) (fromDisk θ 0)‖ = (2*Real.sin θ)⁻¹ := by
+  rw [disk_deriv_at_zero θ hθ hθπ,disk_endpoints_difference,norm_inv,norm_mul,Complex.norm_real,
+    Complex.norm_I,mul_one,Real.norm_eq_abs,abs_of_pos (mul_pos (by norm_num) (Real.sin_pos_of_pos_of_lt_pi hθ hθπ))]
+end ExactSLEMain
+
+namespace ExactSLEMain
+noncomputable def centeredZ (A : ℝ) (θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℂ :=
+  loewnerMap A b t (fromDisk θ x) - (b t : ℂ)
+noncomputable def sineAngle (A θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℝ :=
+  (centeredZ A θ b t x).im / ‖centeredZ A θ b t x‖
+noncomputable def diskRadius (A θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℝ :=
+  ‖deriv (diskMap θ) (fromDisk θ x)‖ * (2 * (centeredZ A θ b t x).im) /
+    ‖deriv (loewnerMap A b t) (fromDisk θ x)‖
+noncomputable def green (a θ : ℝ) (b : ℝ≥0 → ℝ) (t : ℝ≥0) (x : ℂ) : ℝ := by
+  classical
+  exact if alive a b t (fromDisk θ x) then
+    diskRadius a θ b t x ^ (-(2 - (1 + 1 / (4 * a)))) * sineAngle a θ b t x ^ (4 * a - 1)
+    else 0
+
+noncomputable def levelTime (a θ : ℝ) (b : ℝ≥0 → ℝ) (l : ℕ) : ℝ≥0∞ :=
+  sInf {u : ℝ≥0∞ | ∃ t : ℝ≥0, u = t ∧ alive a b t (fromDisk θ 0) ∧
+    diskRadius a θ b t 0 = Real.exp (-2 * a * l)}
+noncomputable def tiltWeight (a θ : ℝ) (b : ℝ≥0 → ℝ) (l : ℕ) : ℝ :=
+  if levelTime a θ b l = ∞ then 0 else
+    Real.exp (2 * a * (2 - (1 + 1 / (4 * a))) * l) *
+      sineAngle a θ b (levelTime a θ b l).toNNReal 0 ^ (4 * a - 1) / Real.sin θ ^ (4 * a - 1)
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+open ExactSLEGreen
+
+lemma diskRadius_actualState (a θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi)
+    (b : ℝ≥0 → ℝ) (hb : Continuous b) (t : ℝ≥0)
+    (ht : alive a b t (fromDisk θ 0)) :
+    diskRadius a θ b t 0 = physicalRadius (actualState a b t (fromDisk θ 0)) / Real.sin θ := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  have hd : 0 < ‖deriv (loewnerMap a b t) (fromDisk θ 0)‖ :=
+    norm_pos_iff.mpr (deriv_loewnerMap_ne_zero hb ht)
+  unfold diskRadius physicalRadius actualState centeredZ
+  rw [disk_deriv_norm θ hθ hθπ]
+  simp only [Y_apply,J_apply,Real.exp_neg,Real.exp_log hd]
+  field_simp
+
+lemma levelTime_radialTime (a θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi)
+    (b : ℝ≥0 → ℝ) (hb : Continuous b) (l : ℕ) :
+    levelTime a θ b l = radialTime a b (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  unfold levelTime radialTime
+  congr 1
+  ext u
+  constructor
+  · rintro ⟨t,hu,ht,hR⟩
+    refine ⟨t,hu,ht,?_⟩
+    rw [diskRadius_actualState a θ hθ hθπ b hb t ht] at hR
+    simpa only [mul_comm] using (div_eq_iff hs.ne').mp hR
+  · rintro ⟨t,hu,ht,hR⟩
+    refine ⟨t,hu,ht,?_⟩
+    rw [diskRadius_actualState a θ hθ hθπ b hb t ht,hR,mul_div_cancel_left₀ _ hs.ne']
+
+lemma radial_initial_green (a θ : ℝ) (hθ : 0 < θ) (hθπ : θ < Real.pi) :
+    physicalGreen a (fromDisk θ 0,0) =
+      (Real.sin θ)^(-alpha a)*(Real.sin θ)^(p a) := by
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  rw [physicalGreen_eq_radius_sine a _ (by simpa only [Y_apply,fromDisk_zero_im] using hs)]
+  simp only [physicalRadius,Y_apply,J_apply,neg_zero,Real.exp_zero,mul_one,fromDisk_zero_im,
+    fromDisk_zero_norm,div_one]
+
+lemma tiltWeight_radialWeight (a θ : ℝ) (ha : 1/4 < a) (hθ : 0 < θ) (hθπ : θ < Real.pi)
+    (b : Driver) (l : ℕ) :
+    tiltWeight a θ b.1 l =
+      radialWeight a b.1 (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) /
+        physicalGreen a (fromDisk θ 0,0) := by
+  have ha0 : 0 < a := by linarith
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  let ρ := Real.sin θ*Real.exp (-2*a*l)
+  have hρ : 0 < ρ := mul_pos hs (Real.exp_pos _)
+  have hr : ρ ≤ (fromDisk θ 0).im := by
+    rw [fromDisk_zero_im]
+    have he : Real.exp (-2*a*l) ≤ 1 := Real.exp_le_one_iff.mpr (by
+      have h : (0 : ℝ) ≤ 2*a*l := by positivity
+      nlinarith)
+    exact (mul_le_mul_of_nonneg_left he hs.le).trans_eq (mul_one _)
+  have hz : 0 < (fromDisk θ 0).im := by simpa only [fromDisk_zero_im] using hs
+  have htime := levelTime_radialTime a θ hθ hθπ b.1 b.1.continuous l
+  by_cases hfin : radialTime a b.1 (fromDisk θ 0) ρ = ∞
+  · rw [tiltWeight,htime,ite_eq_left hfin,radialWeight,ite_eq_left hfin,zero_div]
+  · have ht := radialTime_attained a ha0 (fromDisk θ 0) hz ρ hρ hr b hfin
+    let T := (radialTime a b.1 (fromDisk θ 0) ρ).toNNReal
+    have hy : 0 < Y (actualState a b.1 T (fromDisk θ 0)) := by
+      obtain ⟨f,hf⟩ := ht.1
+      rw [Y_apply,actualState,Complex.sub_im,Complex.ofReal_im,sub_zero,loewnerMap_eq hf]
+      exact (hf.2 T ⟨T.coe_nonneg,le_rfl⟩).1
+    have heα : 2-(1+1/(4*a)) = alpha a := by unfold alpha; ring
+    rw [tiltWeight,htime,ite_eq_right hfin,radialWeight,ite_eq_right hfin,
+      physicalGreen_eq_radius_sine a _ hy,ht.2,radial_initial_green a θ hθ hθπ,heα]
+    change Real.exp (2*a*alpha a*l)*
+        (Y (actualState a b.1 T (fromDisk θ 0))/‖(actualState a b.1 T (fromDisk θ 0)).1‖)^(p a) /
+        (Real.sin θ)^(p a) =
+      ρ^(-alpha a)*(Y (actualState a b.1 T (fromDisk θ 0))/‖(actualState a b.1 T (fromDisk θ 0)).1‖)^(p a) /
+        ((Real.sin θ)^(-alpha a)*(Real.sin θ)^(p a))
+    have he : ρ^(-alpha a) = (Real.sin θ)^(-alpha a)*Real.exp (2*a*alpha a*l) := by
+      rw [Real.rpow_def_of_pos hρ,Real.rpow_def_of_pos hs]
+      dsimp only [ρ]
+      rw [Real.log_mul hs.ne' (Real.exp_ne_zero _),Real.log_exp,← Real.exp_add]
+      congr 1
+      ring
+    rw [he]
+    have hn : (Real.sin θ)^(-alpha a) ≠ 0 := (Real.rpow_pos_of_pos hs _).ne'
+    field_simp
+
+ 
+
+end ExactSLEMain
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/StoppingMeasurability.lean b/lean/OAI/Probability/SLEGauge/StoppingMeasurability.lean
index 0f05f0343359bc1bf934decfa1d985ea465c265c..e3d6c789ff7ab0d9d1989a6f7c7db7add6fb635d 100644
--- a/lean/OAI/Probability/SLEGauge/StoppingMeasurability.lean
+++ b/lean/OAI/Probability/SLEGauge/StoppingMeasurability.lean
@@ -1,467 +1,467 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StoppedLimits
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-namespace SLEExactGauge.RationalStop
-lemma measurableSet_safe {Ω : Type*} [MeasurableSpace Ω] (f : Ω → ℝ → ℝ)
-    (hf : ∀ u, Measurable (fun b => f b u)) (R s : ℝ) :
-    MeasurableSet {b | Safe (f b) R s} := by
-  simp only [Safe,ofPred_exists,ofPred_forall]
-  exact MeasurableSet.iUnion fun n => MeasurableSet.iInter fun r =>
-    measurableSet_le (hf _) measurable_const
-
-lemma safe_past {f : (ℝ≥0 → ℝ) → ℝ → ℝ}
-    (hp : ∀ s ≥ 0, ∀ b c, (∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) → f b s = f c s)
-    {R s : ℝ} (hs : 0 ≤ s) {b c : ℝ≥0 → ℝ}
-    (hbc : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) : Safe (f b) R s ↔ Safe (f c) R s := by
-  have he : ∀ r : ℚ, f b (sample s r) = f c (sample s r) := by
-    intro r
-    exact hp _ (sample_mem hs r).1 b c (fun u hu => hbc u (hu.trans (sample_mem hs r).2))
-  unfold Safe
-  simp only [he]
-
- 
-
-def Exit (f : ℝ → ℝ) (R H : ℝ) : ℝ :=
-  sInf ({u | u ∈ Icc 0 H ∧ R ≤ f u} ∪ {H})
-
-lemma exit_mem {f : ℝ → ℝ} {R H : ℝ} (hH : 0 ≤ H)
-    (hf : ContinuousOn f (Icc 0 H)) :
-    Exit f R H ∈ {u | u ∈ Icc 0 H ∧ R ≤ f u} ∪ {H} := by
-  have hc : IsClosed {u | u ∈ Icc 0 H ∧ R ≤ f u} :=
-    (hf.upperSemicontinuousOn.isCompact_inter_preimage_Ici isCompact_Icc R).isClosed
-  exact (hc.union isClosed_singleton).csInf_mem ⟨H,Or.inr rfl⟩
-    ⟨0,by intro u hu; rcases hu with hu|hu; exact hu.1.1; simpa only [show u = H from hu] using hH⟩
-
-lemma exit_bounds {f : ℝ → ℝ} {R H : ℝ} (hH : 0 ≤ H)
-    (hf : ContinuousOn f (Icc 0 H)) : Exit f R H ∈ Icc 0 H := by
-  rcases exit_mem hH hf with ⟨hu,_⟩ | hu
-  · exact hu
-  · change Exit f R H = H at hu
-    rw [hu]
-    exact ⟨hH,le_rfl⟩
-
-lemma exit_le {f : ℝ → ℝ} {R H u : ℝ} (hH : 0 ≤ H)
-    (hu : u ∈ Icc 0 H) (huR : R ≤ f u) : Exit f R H ≤ u := by
-  exact csInf_le ⟨0,by intro v hv; rcases hv with hv|hv; exact hv.1.1; simpa only [show v = H from hv] using hH⟩
-    (Or.inl ⟨hu,huR⟩)
-
-lemma lt_exit_iff {f : ℝ → ℝ} {R H s : ℝ} (hH : 0 ≤ H)
-    (hf : ContinuousOn f (Icc 0 H)) (hs : 0 ≤ s) :
-    s < Exit f R H ↔ s < H ∧ Safe f R s := by
-  constructor
-  · intro h
-    have hsH := h.trans_le (exit_bounds hH hf).2
-    refine ⟨hsH,(safe_iff hs (hf.mono (Icc_subset_Icc_right hsH.le))).mpr ?_⟩
-    intro u hu
-    by_contra! hn
-    have he := exit_le hH ⟨hu.1,hu.2.trans hsH.le⟩ hn
-    linarith [hu.2]
-  · rintro ⟨hsH,hS⟩
-    have h := (safe_iff hs (hf.mono (Icc_subset_Icc_right hsH.le))).mp hS
-    by_contra! hn
-    rcases exit_mem hH hf with he|he
-    · exact (h _ ⟨he.1.1,hn⟩).not_ge he.2
-    · have he' : Exit f R H = H := he
-      linarith
-
- 
-
-lemma bound_until_exit {f : ℝ → ℝ} {R H : ℝ} (hH : 0 ≤ H)
-    (hf : ContinuousOn f (Icc 0 H)) (h0 : f 0 < R) :
-    ∀ u ∈ Icc 0 (Exit f R H), f u ≤ R := by
-  intro u hu
-  have huH := hu.2.trans (exit_bounds hH hf).2
-  by_cases h : u = 0
-  · simpa only [h] using h0.le
-  have hu0 : 0 < u := lt_of_le_of_ne hu.1 (Ne.symm h)
-  have hd : ∀ t ∈ Ico 0 u, f t ≤ R := by
-    intro t ht
-    have htE := ht.2.trans_le hu.2
-    exact ((safe_iff ht.1 (hf.mono (Icc_subset_Icc_right (ht.2.le.trans huH)))).mp
-      ((lt_exit_iff hH hf ht.1).mp htE).2 t ⟨ht.1,le_rfl⟩).le
-  have hc : IsClosed {t ∈ Icc 0 H | f t ≤ R} :=
-    (hf.lowerSemicontinuousOn.isCompact_inter_preimage_Iic isCompact_Icc R).isClosed
-  have hsub : Ico 0 u ⊆ {t ∈ Icc 0 H | f t ≤ R} :=
-    fun t ht => ⟨⟨ht.1,ht.2.le.trans huH⟩,hd t ht⟩
-  have he := closure_minimal hsub hc
-  rw [closure_Ico hu0.ne] at he
-  exact (he ⟨hu0.le,le_rfl⟩).2
-
-end SLEExactGauge.RationalStop
-
-namespace SLEExactGauge.RationalStop
-open Set MeasureTheory
-
-lemma measurable_exit {Ω : Type*} [MeasurableSpace Ω] {f : Ω → ℝ → ℝ}
-    {H R : ℝ} (hH : 0 ≤ H) (hf : ∀ u, Measurable (fun b => f b u))
-    (hc : ∀ b, ContinuousOn (f b) (Icc 0 H)) :
-    Measurable (fun b => Exit (f b) R H) := by
-  apply measurable_of_Ioi
-  intro s
-  by_cases hs : 0 ≤ s
-  · have he : (fun b => Exit (f b) R H) ⁻¹' Ioi s = {b | s < H ∧ Safe (f b) R s} := by
-      ext b
-      exact lt_exit_iff hH (hc b) hs
-    rw [he]
-    exact (MeasurableSet.const _).inter (measurableSet_safe f hf R s)
-  · have he : (fun b => Exit (f b) R H) ⁻¹' Ioi s = univ := by
-      ext b
-      simp only [mem_preimage,mem_Ioi,mem_univ,iff_true]
-      exact (lt_of_not_ge hs).trans_le (exit_bounds hH (hc b)).1
-    rw [he]
-    exact MeasurableSet.univ
-
- 
-
-lemma measurable_stopped_eval {Ω E : Type*} [MeasurableSpace Ω]
-    [MeasurableSpace E] [TopologicalSpace E] [TopologicalSpace.PseudoMetrizableSpace E] [BorelSpace E]
-    {f : Ω → ℝ → E} {τ : Ω → ℝ} {H : ℝ} (hH : 0 ≤ H)
-    (hf : ∀ u, Measurable (fun b => f b u)) (hc : ∀ b, ContinuousOn (f b) (Icc 0 H))
-    (hτ : Measurable τ) (hτH : ∀ b, τ b ∈ Icc 0 H) :
-    Measurable (fun b => f b (τ b)) := by
-  have hm := measurable_uncurry_of_continuous_of_measurable
-    (u := fun t b => f b (sample H t))
-    (fun b => (hc b).comp_continuous (continuous_sample H) (sample_mem hH))
-    (fun t => hf (sample H t))
-  have he : (fun b => f b (τ b)) = (fun q : ℝ × Ω => f q.2 (sample H q.1)) ∘ (fun b => (τ b,b)) := by
-    funext b
-    exact congrArg (f b) (sample_self (hτH b)).symm
-  rw [he]
-  exact hm.comp (hτ.prodMk measurable_id)
-end SLEExactGauge.RationalStop
-
-namespace SLEExactGauge.AngleCalculus
-open scoped ContDiff
-lemma contDiffAt_V {z : ℂ} (hz : z ≠ 0) : ContDiffAt ℝ ∞ V z := by
-  have hd : ContDiff ℝ ∞ (fun z : ℂ => Complex.normSq z) := by
-    exact (Complex.reCLM.contDiff.mul Complex.reCLM.contDiff).add
-      (Complex.imCLM.contDiff.mul Complex.imCLM.contDiff)
-  exact Complex.imCLM.contDiff.contDiffAt.div hd.contDiffAt (Complex.normSq_pos.mpr hz).ne'
-end SLEExactGauge.AngleCalculus
-namespace SLEExactGauge.OnePoint
-open _root_.OAI.SLEExactGauge.Localization AngleCalculus BrownianCalculus ODEPath RationalStop
-open scoped ContDiff
-
-lemma contDiffOn_field (P : Parameter) : ContDiffOn ℝ ∞ (Function.uncurry (field P)) domain := by
-  intro q hq
-  apply ContDiffAt.contDiffWithinAt
-  have hz : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2-(q.2:ℂ)) q :=
-    (contDiffAt_snd.comp q contDiffAt_fst).sub (Complex.ofRealCLM.contDiff.contDiffAt.comp q contDiffAt_snd)
-  have hn : q.1.2-(q.2:ℂ) ≠ 0 := by
-    intro he
-    have hi := congrArg Complex.im he
-    have hh : 0 < q.1.2.im := hq
-    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-    linarith
-  exact (((contDiffAt_V hn).comp q hz).pow 2).prodMk
-    (by
-      simpa only [div_eq_mul_inv] using (contDiffAt_const (c := (P.a:ℂ))).mul
-        (hz.inv hn))
-
-lemma exists_cutoff_field (P : Parameter) {K : Set (State × ℝ)}
-    (hK : IsCompact K) (hKO : K ⊆ domain) :
-    ∃ v : State → ℝ → State, ∃ M L : ℝ, 0 ≤ M ∧ 0 ≤ L ∧
-      ContDiff ℝ ∞ (Function.uncurry v) ∧
-      (∀ x b, ‖v x b‖ ≤ M) ∧
-      (∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) ∧
-      (∀ q ∈ K, v q.1 q.2 = field P q.1 q.2) := by
-  obtain ⟨V,hV,hVK,hVe⟩ := compact_extension hK isOpen_domain hKO
-    (Function.uncurry (field P)) (contDiffOn_field P)
-  obtain ⟨M,hM⟩ := hVK.exists_bound_of_continuous hV.continuous
-  obtain ⟨L,hL⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hVK hV (by simp)
-  refine ⟨fun x b => V (x,b),max M 0,L,le_max_right _ _,L.coe_nonneg,hV,?_,?_,?_⟩
-  · intro x b
-    exact (hM (x,b)).trans (le_max_left _ _)
-  · intro b x y
-    have H := hL.dist_le_mul (x,b) (y,b)
-    simpa [dist_eq_norm,Prod.norm_def] using H
-  · intro q hq
-    exact (hVe q hq).self_of_nhds
-
- 
-
-theorem exists_stopped_green (P : Parameter) {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    {H : ℝ} (hH : 0 ≤ H) (φ : State × ℝ → ℝ) (hφ : Continuous φ)
-    (hK : IsCompact {q | φ q ≤ 2}) (hKO : {q | φ q ≤ 2} ⊆ domain)
-    (z : State) (hz : φ (z,0) < 1) :
-    ∃ v : State → ℝ → State, ∃ M L : ℝ, 0 ≤ M ∧ 0 ≤ L ∧
-      ContDiff ℝ ∞ (Function.uncurry v) ∧
-      (∀ x b, ‖v x b‖ ≤ M) ∧
-      (∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) ∧
-      (∀ q, φ q ≤ 2 → v q.1 q.2 = field P q.1 q.2) ∧
-      let g := fun b : Driver => ODEPath.flow v z b
-      let τ := fun b : Driver => Exit (fun u => φ (g b u,b u.toNNReal)) 1 H
-      (∀ b u, u ∈ Icc 0 (τ b) → φ (g b u,b u.toNNReal) ≤ 1) ∧
-      ∀ t ≥ 0, (∫ b : Driver, green P
-        (g b (min t (τ b)),b (min t (τ b)).toNNReal) ∂W) = green P (z,0) := by
-  obtain ⟨v,M,L,hM,hL,hv,hbound,hlip,hveq⟩ := exists_cutoff_field P hK hKO
-  let g := fun b : Driver => ODEPath.flow v z b
-  let f := fun (b : ℝ≥0 → ℝ) u => φ (ODEPath.flow v z b u,b u.toNNReal)
-  let τ := fun b : Driver => Exit (f b) 1 H
-  have hg (b : Driver) := ODEPath.flow_solution v hv.continuous z b b.continuous
-    hM hL (by linarith : 0 ≤ H+1) hbound hlip
-  have hgc (b : Driver) : ContinuousOn (g b) (Icc 0 (H+1)) :=
-    ODEPath.continuousOn_flow v hv.continuous z b b.continuous hM hL (by linarith) hbound hlip
-  have hfc (b : Driver) : ContinuousOn (f b) (Icc 0 (H+1)) :=
-    hφ.comp_continuousOn ((hgc b).prodMk ((b.continuous.comp continuous_real_toNNReal).continuousOn))
-  have hfcH (b : Driver) : ContinuousOn (f b) (Icc 0 H) :=
-    (hfc b).mono (Icc_subset_Icc_right (by linarith))
-  have hf0 (b : Driver) : f b 0 < 1 := by
-    simpa only [f,(hg b).1,Real.toNNReal_zero,map_zero] using hz
-  have hτ (b : Driver) : 0 ≤ τ b ∧ τ b < H+1 := by
-    have hb := exit_bounds hH (hfcH b) (R := 1)
-    exact ⟨hb.1,lt_of_le_of_lt hb.2 (by linarith)⟩
-  have hsafe (b : Driver) : ∀ u ∈ Icc 0 (τ b), f b u ≤ 1 :=
-    bound_until_exit hH (hfcH b) (hf0 b)
-  refine ⟨v,M,L,hM,hL,hv,hbound,hlip,hveq,hsafe,?_⟩
-  intro t ht
-  let K : Set (State × ℝ) := {q | φ q ≤ 2}
-  have hDo : IsOpen {q | φ q < 2} := isOpen_lt hφ continuous_const
-  have hK1 : IsCompact {q | φ q ≤ 1} :=
-    hK.of_isClosed_subset (isClosed_le hφ continuous_const) (fun q hq => by dsimp at *; linarith)
-  obtain ⟨δ,hδ,hδD⟩ := hK1.exists_thickening_subset_open hDo (fun q hq => by dsimp at *; linarith)
-  have hD (b : Driver) : ∃ δ > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
-      ‖x-g b u‖ < δ → (x,b u.toNNReal) ∈ K := by
-    refine ⟨δ,hδ,?_⟩
-    intro u hu x hx
-    have hi : φ (x,b u.toNNReal) < 2 := hδD (by
-      apply Metric.mem_thickening_iff.mpr
-      refine ⟨(g b u,b u.toNNReal),hsafe b u hu,?_⟩
-      rw [dist_eq_norm]
-      change max ‖x-g b u‖ ‖b u.toNNReal-b u.toNNReal‖ < δ
-      simpa using hx)
-    exact hi.le
-  have mf (u : ℝ) : Measurable (fun b : ℝ≥0 → ℝ => f b u) :=
-    hφ.measurable.comp ((ODEPath.measurable_flow v hv.continuous.measurable z u).prodMk (measurable_pi_apply _))
-  have hfpast : ∀ s ≥ 0, ∀ b c : ℝ≥0 → ℝ,
-      (∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) → f b s = f c s := by
-    intro s hs b c hbc
-    have hb := hbc s.toNNReal (by simpa only [Real.coe_toNNReal',max_eq_left hs] using le_refl s)
-    simp only [f,ODEPath.flow_past v z hs hbc,hb]
-  let S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (State × ℝ)) := fun r =>
-    {q | (r:ℝ) < H ∧ Safe (f q.1) 1 r ∧ q.2 ∈ K}
-  have mS : ∀ r, MeasurableSet (S r) := by
-    intro r
-    exact (MeasurableSet.const _).inter
-      (((measurableSet_safe f mf 1 _).preimage measurable_fst).inter (hK.measurableSet.preimage measurable_snd))
-  have hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r) := by
-    intro r b c hbc x
-    have he : Safe (f b) 1 r ↔ Safe (f c) 1 r := safe_past hfpast r.coe_nonneg
-      (fun u hu => hbc u hu)
-    exact and_congr_right fun _ => and_congr_left fun _ => he
-  have hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K) := by
-    intro b r x
-    change (_ ∧ _ ∧ _) ↔ (_ ∧ _)
-    rw [lt_exit_iff hH (hfcH b) r.coe_nonneg]
-    exact and_assoc.symm
-  apply expected_stopped_local_grid_eq hW (green P) K domain hK isOpen_domain hKO (contDiffOn_green P)
-    v hv.continuous.measurable g z hL hM hbound hlip (fun b => (hg b).1) (fun b => (hg b).2)
-    (fun b => hv.continuous.comp_continuousOn ((hgc b).prodMk
-      ((b.continuous.comp continuous_real_toNNReal).continuousOn))) τ hτ hD
-    (fun q hq => by rw [hveq q hq]; exact green_generator P (hKO hq))
-    S mS hSpast hSeq (fun r q hq => hq.2.2) (by change φ (z,0) ≤ 2; linarith) ht
-end SLEExactGauge.OnePoint
-
- 
-
-noncomputable section
-open Set Filter Real
-open scoped Topology
-namespace SLEExactGauge.ReverseDerivative
-
-def r (a : ℝ) : ℝ := 2*a+1/2
-def zeta (a : ℝ) : ℝ := a-1/(16*a)
-def lambda (a : ℝ) : ℝ := a+1+3/(16*a)
-def R (x y : ℝ) : ℝ := x^2+y^2
-def D (u x y : ℝ) : ℝ := R x y+u/2
-
-def H (a u x y l : ℝ) : ℝ :=
-  lambda a*l+r a/2*Real.log (R x y)+(zeta a-r a)*Real.log y-
-    zeta a/2*Real.log (D u x y)
-def F (a u x y l : ℝ) : ℝ := Real.exp (H a u x y l)
-def Hx (a u x y : ℝ) : ℝ := r a*x/R x y-zeta a*x/D u x y
-def Hxx (a u x y : ℝ) : ℝ :=
-  r a*(y^2-x^2)/(R x y)^2-zeta a/D u x y+2*zeta a*x^2/(D u x y)^2
-def Hy (a u x y : ℝ) : ℝ :=
-  r a*y/R x y+(zeta a-r a)/y-zeta a*y/D u x y
-
-def residual (a u x y : ℝ) : ℝ :=
-  R x y*((a-1/2+1/(16*a))*x^2+(2*a+1/2)*y^2)+
-    u/2*((2*a+3/2)*x^2+(2*a+1/2)*y^2)
-
-lemma R_pos (x : ℝ) {y : ℝ} (hy : 0 < y) : 0 < R x y := by
-  unfold R
-  positivity
-
-lemma D_pos {u y : ℝ} (hu : 0 ≤ u) (hy : 0 < y) (x : ℝ) : 0 < D u x y := by
-  exact add_pos_of_pos_of_nonneg (R_pos x hy) (div_nonneg hu (by norm_num))
-
-lemma parameters {a : ℝ} (ha : 1/4 < a) :
-    0 < zeta a ∧ 0 < lambda a ∧ 2 < zeta a+lambda a ∧ 1 < r a ∧ r a < 4*a := by
-  have ha0 : 0 < a := by linarith
-  have h16 : 0 < 16*a := by positivity
-  have hz : 0 < zeta a := by
-    unfold zeta
-    have hsq : 1 < 16*a^2 := by nlinarith
-    apply sub_pos.mpr
-    apply (div_lt_iff₀ h16).2
-    nlinarith
-  have hl : 0 < lambda a := by unfold lambda; positivity
-  have he : zeta a+lambda a-2 = (4*a-1)^2/(8*a) := by
-    unfold zeta lambda
-    field_simp
-    ring
-  have hsum : 2 < zeta a+lambda a := by
-    have : 0 < (4*a-1)^2/(8*a) := div_pos (sq_pos_of_pos (by linarith)) (by positivity)
-    linarith
-  exact ⟨hz,hl,hsum,by unfold r; linarith,by unfold r; linarith⟩
-
-lemma residual_nonneg {a u y : ℝ} (ha : 1/4 < a) (hu : 0 ≤ u)
-    (hy : 0 < y) (x : ℝ) : 0 ≤ residual a u x y := by
-  have ha0 : 0 < a := by linarith
-  have hc : 0 ≤ a-1/2+1/(16*a) := by
-    have he : a-1/2+1/(16*a) = (4*a-1)^2/(16*a) := by field_simp; ring
-    rw [he]
-    positivity
-  unfold residual
-  have hR := (R_pos x hy).le
-  positivity
-
-lemma hasDerivAt_Hx {a u x y l : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
-    HasDerivAt (fun w => H a u w y l) (Hx a u x y) x := by
-  have hR : HasDerivAt (fun w : ℝ => R w y) (2*x) x := by
-    convert! ((hasDerivAt_id x).pow 2).add_const (y^2) using 1
-    simp
-  have hD : HasDerivAt (fun w : ℝ => D u w y) (2*x) x := hR.add_const (u/2)
-  have hd := (((hR.log (R_pos x hy).ne').const_mul (r a/2)).const_add (lambda a*l)
-    |>.add_const ((zeta a-r a)*Real.log y)).sub
-      ((hD.log (D_pos hu hy x).ne').const_mul (zeta a/2))
-  convert! hd using 1
-  · dsimp [Hx]; ring
-
-lemma hasDerivAt_Hxx {a u x y : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
-    HasDerivAt (fun w => Hx a u w y) (Hxx a u x y) x := by
-  have hR : HasDerivAt (fun w : ℝ => R w y) (2*x) x := by
-    convert! ((hasDerivAt_id x).pow 2).add_const (y^2) using 1
-    simp
-  have hD : HasDerivAt (fun w : ℝ => D u w y) (2*x) x := hR.add_const (u/2)
-  have hd := (((hasDerivAt_id x).const_mul (r a)).div hR (R_pos x hy).ne').sub
-    (((hasDerivAt_id x).const_mul (zeta a)).div hD (D_pos hu hy x).ne')
-  convert! hd using 1
-  dsimp [Hxx]
-  have hnR := (R_pos x hy).ne'
-  have hnD := (D_pos hu hy x).ne'
-  field_simp
-  simp only [R]
-  ring
-
-lemma deriv_Fx {a u y l : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
-    deriv (fun x => F a u x y l) = fun x => F a u x y l * Hx a u x y := by
-  funext x
-  exact (hasDerivAt_Hx hu hy).exp.deriv
-
-lemma deriv_Fxx {a u x y l : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
-    deriv (deriv (fun w => F a u w y l)) x =
-      F a u x y l * ((Hx a u x y)^2+Hxx a u x y) := by
-  rw [deriv_Fx hu hy]
-  have hd := ((hasDerivAt_Hx (a := a) (x := x) (l := l) hu hy).exp).mul
-    (hasDerivAt_Hxx (a := a) (x := x) hu hy)
-  convert! hd.deriv using 1
-  dsimp [F]
-  ring
-
-lemma generator_identity {a u x y : ℝ} (ha : 0 < a) (hu : 0 ≤ u) (hy : 0 < y) :
-    ((Hx a u x y)^2+Hxx a u x y)/2 +
-      (-a*x/R x y)*Hx a u x y+(a*y/R x y)*Hy a u x y+
-      (a*(x^2-y^2)/(R x y)^2)*lambda a+zeta a/(4*D u x y) =
-      -zeta a*residual a u x y/(2*R x y*(D u x y)^2) := by
-  have ha' := ha.ne'
-  have hR := (R_pos x hy).ne'
-  have hD := (D_pos hu hy x).ne'
-  have hy' := hy.ne'
-  simp only [Hx,Hxx,Hy,lambda,zeta,r,residual]
-  field_simp
-  simp only [D,R]
-  ring
-
-lemma generator_nonpos {a u x y : ℝ} (ha : 1/4 < a) (hu : 0 ≤ u) (hy : 0 < y) :
-    ((Hx a u x y)^2+Hxx a u x y)/2 +
-      (-a*x/R x y)*Hx a u x y+(a*y/R x y)*Hy a u x y+
-      (a*(x^2-y^2)/(R x y)^2)*lambda a+zeta a/(4*D u x y) ≤ 0 := by
-  rw [generator_identity (by linarith) hu hy]
-  apply div_nonpos_of_nonpos_of_nonneg
-  · exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (parameters ha).1.le)
-      (residual_nonneg ha hu hy x)
-  · exact mul_nonneg (mul_nonneg (by norm_num) (R_pos x hy).le) (sq_nonneg _)
-
-end SLEExactGauge.ReverseDerivative
-
-namespace SLEExactGauge.ReverseDerivative
-
-abbrev State := ℝ × ℂ × ℝ
-
-def test (a : ℝ) (q : State) (b : ℝ) : ℝ :=
-  F a q.1 (q.2.1.re-b) q.2.1.im q.2.2
-
-def vector (a : ℝ) (q : State) (b : ℝ) : State :=
-  (-1, (-(a:ℂ)/(q.2.1-b),a*((q.2.1-b)⁻¹^2).re))
-
-lemma differentiableAt_test (a b : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) :
-    DifferentiableAt ℝ (fun q => test a q b) q := by
-  have hR := (R_pos (q.2.1.re-b) hy).ne'
-  have hD := (D_pos hu hy (q.2.1.re-b)).ne'
-  have hy' := hy.ne'
-  have hre : DifferentiableAt ℝ (fun q : State => q.2.1.re) q :=
-    Complex.reCLM.differentiableAt.comp _ (differentiableAt_fst.comp _ differentiableAt_snd)
-  have him : DifferentiableAt ℝ (fun q : State => q.2.1.im) q :=
-    Complex.imCLM.differentiableAt.comp _ (differentiableAt_fst.comp _ differentiableAt_snd)
-  dsimp [test, F, H, D, R] at *
-  fun_prop
-
-end SLEExactGauge.ReverseDerivative
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StoppedLimits
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+namespace SLEExactGauge.RationalStop
+lemma measurableSet_safe {Ω : Type*} [MeasurableSpace Ω] (f : Ω → ℝ → ℝ)
+    (hf : ∀ u, Measurable (fun b => f b u)) (R s : ℝ) :
+    MeasurableSet {b | Safe (f b) R s} := by
+  simp only [Safe,ofPred_exists,ofPred_forall]
+  exact MeasurableSet.iUnion fun n => MeasurableSet.iInter fun r =>
+    measurableSet_le (hf _) measurable_const
+
+lemma safe_past {f : (ℝ≥0 → ℝ) → ℝ → ℝ}
+    (hp : ∀ s ≥ 0, ∀ b c, (∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) → f b s = f c s)
+    {R s : ℝ} (hs : 0 ≤ s) {b c : ℝ≥0 → ℝ}
+    (hbc : ∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) : Safe (f b) R s ↔ Safe (f c) R s := by
+  have he : ∀ r : ℚ, f b (sample s r) = f c (sample s r) := by
+    intro r
+    exact hp _ (sample_mem hs r).1 b c (fun u hu => hbc u (hu.trans (sample_mem hs r).2))
+  unfold Safe
+  simp only [he]
+
+ 
+
+def Exit (f : ℝ → ℝ) (R H : ℝ) : ℝ :=
+  sInf ({u | u ∈ Icc 0 H ∧ R ≤ f u} ∪ {H})
+
+lemma exit_mem {f : ℝ → ℝ} {R H : ℝ} (hH : 0 ≤ H)
+    (hf : ContinuousOn f (Icc 0 H)) :
+    Exit f R H ∈ {u | u ∈ Icc 0 H ∧ R ≤ f u} ∪ {H} := by
+  have hc : IsClosed {u | u ∈ Icc 0 H ∧ R ≤ f u} :=
+    (hf.upperSemicontinuousOn.isCompact_inter_preimage_Ici isCompact_Icc R).isClosed
+  exact (hc.union isClosed_singleton).csInf_mem ⟨H,Or.inr rfl⟩
+    ⟨0,by intro u hu; rcases hu with hu|hu; exact hu.1.1; simpa only [show u = H from hu] using hH⟩
+
+lemma exit_bounds {f : ℝ → ℝ} {R H : ℝ} (hH : 0 ≤ H)
+    (hf : ContinuousOn f (Icc 0 H)) : Exit f R H ∈ Icc 0 H := by
+  rcases exit_mem hH hf with ⟨hu,_⟩ | hu
+  · exact hu
+  · change Exit f R H = H at hu
+    rw [hu]
+    exact ⟨hH,le_rfl⟩
+
+lemma exit_le {f : ℝ → ℝ} {R H u : ℝ} (hH : 0 ≤ H)
+    (hu : u ∈ Icc 0 H) (huR : R ≤ f u) : Exit f R H ≤ u := by
+  exact csInf_le ⟨0,by intro v hv; rcases hv with hv|hv; exact hv.1.1; simpa only [show v = H from hv] using hH⟩
+    (Or.inl ⟨hu,huR⟩)
+
+lemma lt_exit_iff {f : ℝ → ℝ} {R H s : ℝ} (hH : 0 ≤ H)
+    (hf : ContinuousOn f (Icc 0 H)) (hs : 0 ≤ s) :
+    s < Exit f R H ↔ s < H ∧ Safe f R s := by
+  constructor
+  · intro h
+    have hsH := h.trans_le (exit_bounds hH hf).2
+    refine ⟨hsH,(safe_iff hs (hf.mono (Icc_subset_Icc_right hsH.le))).mpr ?_⟩
+    intro u hu
+    by_contra! hn
+    have he := exit_le hH ⟨hu.1,hu.2.trans hsH.le⟩ hn
+    linarith [hu.2]
+  · rintro ⟨hsH,hS⟩
+    have h := (safe_iff hs (hf.mono (Icc_subset_Icc_right hsH.le))).mp hS
+    by_contra! hn
+    rcases exit_mem hH hf with he|he
+    · exact (h _ ⟨he.1.1,hn⟩).not_ge he.2
+    · have he' : Exit f R H = H := he
+      linarith
+
+ 
+
+lemma bound_until_exit {f : ℝ → ℝ} {R H : ℝ} (hH : 0 ≤ H)
+    (hf : ContinuousOn f (Icc 0 H)) (h0 : f 0 < R) :
+    ∀ u ∈ Icc 0 (Exit f R H), f u ≤ R := by
+  intro u hu
+  have huH := hu.2.trans (exit_bounds hH hf).2
+  by_cases h : u = 0
+  · simpa only [h] using h0.le
+  have hu0 : 0 < u := lt_of_le_of_ne hu.1 (Ne.symm h)
+  have hd : ∀ t ∈ Ico 0 u, f t ≤ R := by
+    intro t ht
+    have htE := ht.2.trans_le hu.2
+    exact ((safe_iff ht.1 (hf.mono (Icc_subset_Icc_right (ht.2.le.trans huH)))).mp
+      ((lt_exit_iff hH hf ht.1).mp htE).2 t ⟨ht.1,le_rfl⟩).le
+  have hc : IsClosed {t ∈ Icc 0 H | f t ≤ R} :=
+    (hf.lowerSemicontinuousOn.isCompact_inter_preimage_Iic isCompact_Icc R).isClosed
+  have hsub : Ico 0 u ⊆ {t ∈ Icc 0 H | f t ≤ R} :=
+    fun t ht => ⟨⟨ht.1,ht.2.le.trans huH⟩,hd t ht⟩
+  have he := closure_minimal hsub hc
+  rw [closure_Ico hu0.ne] at he
+  exact (he ⟨hu0.le,le_rfl⟩).2
+
+end SLEExactGauge.RationalStop
+
+namespace SLEExactGauge.RationalStop
+open Set MeasureTheory
+
+lemma measurable_exit {Ω : Type*} [MeasurableSpace Ω] {f : Ω → ℝ → ℝ}
+    {H R : ℝ} (hH : 0 ≤ H) (hf : ∀ u, Measurable (fun b => f b u))
+    (hc : ∀ b, ContinuousOn (f b) (Icc 0 H)) :
+    Measurable (fun b => Exit (f b) R H) := by
+  apply measurable_of_Ioi
+  intro s
+  by_cases hs : 0 ≤ s
+  · have he : (fun b => Exit (f b) R H) ⁻¹' Ioi s = {b | s < H ∧ Safe (f b) R s} := by
+      ext b
+      exact lt_exit_iff hH (hc b) hs
+    rw [he]
+    exact (MeasurableSet.const _).inter (measurableSet_safe f hf R s)
+  · have he : (fun b => Exit (f b) R H) ⁻¹' Ioi s = univ := by
+      ext b
+      simp only [mem_preimage,mem_Ioi,mem_univ,iff_true]
+      exact (lt_of_not_ge hs).trans_le (exit_bounds hH (hc b)).1
+    rw [he]
+    exact MeasurableSet.univ
+
+ 
+
+lemma measurable_stopped_eval {Ω E : Type*} [MeasurableSpace Ω]
+    [MeasurableSpace E] [TopologicalSpace E] [TopologicalSpace.PseudoMetrizableSpace E] [BorelSpace E]
+    {f : Ω → ℝ → E} {τ : Ω → ℝ} {H : ℝ} (hH : 0 ≤ H)
+    (hf : ∀ u, Measurable (fun b => f b u)) (hc : ∀ b, ContinuousOn (f b) (Icc 0 H))
+    (hτ : Measurable τ) (hτH : ∀ b, τ b ∈ Icc 0 H) :
+    Measurable (fun b => f b (τ b)) := by
+  have hm := measurable_uncurry_of_continuous_of_measurable
+    (u := fun t b => f b (sample H t))
+    (fun b => (hc b).comp_continuous (continuous_sample H) (sample_mem hH))
+    (fun t => hf (sample H t))
+  have he : (fun b => f b (τ b)) = (fun q : ℝ × Ω => f q.2 (sample H q.1)) ∘ (fun b => (τ b,b)) := by
+    funext b
+    exact congrArg (f b) (sample_self (hτH b)).symm
+  rw [he]
+  exact hm.comp (hτ.prodMk measurable_id)
+end SLEExactGauge.RationalStop
+
+namespace SLEExactGauge.AngleCalculus
+open scoped ContDiff
+lemma contDiffAt_V {z : ℂ} (hz : z ≠ 0) : ContDiffAt ℝ ∞ V z := by
+  have hd : ContDiff ℝ ∞ (fun z : ℂ => Complex.normSq z) := by
+    exact (Complex.reCLM.contDiff.mul Complex.reCLM.contDiff).add
+      (Complex.imCLM.contDiff.mul Complex.imCLM.contDiff)
+  exact Complex.imCLM.contDiff.contDiffAt.div hd.contDiffAt (Complex.normSq_pos.mpr hz).ne'
+end SLEExactGauge.AngleCalculus
+namespace SLEExactGauge.OnePoint
+open _root_.OAI.SLEExactGauge.Localization AngleCalculus BrownianCalculus ODEPath RationalStop
+open scoped ContDiff
+
+lemma contDiffOn_field (P : Parameter) : ContDiffOn ℝ ∞ (Function.uncurry (field P)) domain := by
+  intro q hq
+  apply ContDiffAt.contDiffWithinAt
+  have hz : ContDiffAt ℝ ∞ (fun q : State × ℝ => q.1.2-(q.2:ℂ)) q :=
+    (contDiffAt_snd.comp q contDiffAt_fst).sub (Complex.ofRealCLM.contDiff.contDiffAt.comp q contDiffAt_snd)
+  have hn : q.1.2-(q.2:ℂ) ≠ 0 := by
+    intro he
+    have hi := congrArg Complex.im he
+    have hh : 0 < q.1.2.im := hq
+    simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+    linarith
+  exact (((contDiffAt_V hn).comp q hz).pow 2).prodMk
+    (by
+      simpa only [div_eq_mul_inv] using (contDiffAt_const (c := (P.a:ℂ))).mul
+        (hz.inv hn))
+
+lemma exists_cutoff_field (P : Parameter) {K : Set (State × ℝ)}
+    (hK : IsCompact K) (hKO : K ⊆ domain) :
+    ∃ v : State → ℝ → State, ∃ M L : ℝ, 0 ≤ M ∧ 0 ≤ L ∧
+      ContDiff ℝ ∞ (Function.uncurry v) ∧
+      (∀ x b, ‖v x b‖ ≤ M) ∧
+      (∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) ∧
+      (∀ q ∈ K, v q.1 q.2 = field P q.1 q.2) := by
+  obtain ⟨V,hV,hVK,hVe⟩ := compact_extension hK isOpen_domain hKO
+    (Function.uncurry (field P)) (contDiffOn_field P)
+  obtain ⟨M,hM⟩ := hVK.exists_bound_of_continuous hV.continuous
+  obtain ⟨L,hL⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hVK hV (by simp)
+  refine ⟨fun x b => V (x,b),max M 0,L,le_max_right _ _,L.coe_nonneg,hV,?_,?_,?_⟩
+  · intro x b
+    exact (hM (x,b)).trans (le_max_left _ _)
+  · intro b x y
+    have H := hL.dist_le_mul (x,b) (y,b)
+    simpa [dist_eq_norm,Prod.norm_def] using H
+  · intro q hq
+    exact (hVe q hq).self_of_nhds
+
+ 
+
+theorem exists_stopped_green (P : Parameter) {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    {H : ℝ} (hH : 0 ≤ H) (φ : State × ℝ → ℝ) (hφ : Continuous φ)
+    (hK : IsCompact {q | φ q ≤ 2}) (hKO : {q | φ q ≤ 2} ⊆ domain)
+    (z : State) (hz : φ (z,0) < 1) :
+    ∃ v : State → ℝ → State, ∃ M L : ℝ, 0 ≤ M ∧ 0 ≤ L ∧
+      ContDiff ℝ ∞ (Function.uncurry v) ∧
+      (∀ x b, ‖v x b‖ ≤ M) ∧
+      (∀ b x y, ‖v x b-v y b‖ ≤ L*‖x-y‖) ∧
+      (∀ q, φ q ≤ 2 → v q.1 q.2 = field P q.1 q.2) ∧
+      let g := fun b : Driver => ODEPath.flow v z b
+      let τ := fun b : Driver => Exit (fun u => φ (g b u,b u.toNNReal)) 1 H
+      (∀ b u, u ∈ Icc 0 (τ b) → φ (g b u,b u.toNNReal) ≤ 1) ∧
+      ∀ t ≥ 0, (∫ b : Driver, green P
+        (g b (min t (τ b)),b (min t (τ b)).toNNReal) ∂W) = green P (z,0) := by
+  obtain ⟨v,M,L,hM,hL,hv,hbound,hlip,hveq⟩ := exists_cutoff_field P hK hKO
+  let g := fun b : Driver => ODEPath.flow v z b
+  let f := fun (b : ℝ≥0 → ℝ) u => φ (ODEPath.flow v z b u,b u.toNNReal)
+  let τ := fun b : Driver => Exit (f b) 1 H
+  have hg (b : Driver) := ODEPath.flow_solution v hv.continuous z b b.continuous
+    hM hL (by linarith : 0 ≤ H+1) hbound hlip
+  have hgc (b : Driver) : ContinuousOn (g b) (Icc 0 (H+1)) :=
+    ODEPath.continuousOn_flow v hv.continuous z b b.continuous hM hL (by linarith) hbound hlip
+  have hfc (b : Driver) : ContinuousOn (f b) (Icc 0 (H+1)) :=
+    hφ.comp_continuousOn ((hgc b).prodMk ((b.continuous.comp continuous_real_toNNReal).continuousOn))
+  have hfcH (b : Driver) : ContinuousOn (f b) (Icc 0 H) :=
+    (hfc b).mono (Icc_subset_Icc_right (by linarith))
+  have hf0 (b : Driver) : f b 0 < 1 := by
+    simpa only [f,(hg b).1,Real.toNNReal_zero,map_zero] using hz
+  have hτ (b : Driver) : 0 ≤ τ b ∧ τ b < H+1 := by
+    have hb := exit_bounds hH (hfcH b) (R := 1)
+    exact ⟨hb.1,lt_of_le_of_lt hb.2 (by linarith)⟩
+  have hsafe (b : Driver) : ∀ u ∈ Icc 0 (τ b), f b u ≤ 1 :=
+    bound_until_exit hH (hfcH b) (hf0 b)
+  refine ⟨v,M,L,hM,hL,hv,hbound,hlip,hveq,hsafe,?_⟩
+  intro t ht
+  let K : Set (State × ℝ) := {q | φ q ≤ 2}
+  have hDo : IsOpen {q | φ q < 2} := isOpen_lt hφ continuous_const
+  have hK1 : IsCompact {q | φ q ≤ 1} :=
+    hK.of_isClosed_subset (isClosed_le hφ continuous_const) (fun q hq => by dsimp at *; linarith)
+  obtain ⟨δ,hδ,hδD⟩ := hK1.exists_thickening_subset_open hDo (fun q hq => by dsimp at *; linarith)
+  have hD (b : Driver) : ∃ δ > 0, ∀ u ∈ Icc 0 (τ b), ∀ x,
+      ‖x-g b u‖ < δ → (x,b u.toNNReal) ∈ K := by
+    refine ⟨δ,hδ,?_⟩
+    intro u hu x hx
+    have hi : φ (x,b u.toNNReal) < 2 := hδD (by
+      apply Metric.mem_thickening_iff.mpr
+      refine ⟨(g b u,b u.toNNReal),hsafe b u hu,?_⟩
+      rw [dist_eq_norm]
+      change max ‖x-g b u‖ ‖b u.toNNReal-b u.toNNReal‖ < δ
+      simpa using hx)
+    exact hi.le
+  have mf (u : ℝ) : Measurable (fun b : ℝ≥0 → ℝ => f b u) :=
+    hφ.measurable.comp ((ODEPath.measurable_flow v hv.continuous.measurable z u).prodMk (measurable_pi_apply _))
+  have hfpast : ∀ s ≥ 0, ∀ b c : ℝ≥0 → ℝ,
+      (∀ u : ℝ≥0, (u:ℝ) ≤ s → b u = c u) → f b s = f c s := by
+    intro s hs b c hbc
+    have hb := hbc s.toNNReal (by simpa only [Real.coe_toNNReal',max_eq_left hs] using le_refl s)
+    simp only [f,ODEPath.flow_past v z hs hbc,hb]
+  let S : ℝ≥0 → Set ((ℝ≥0 → ℝ) × (State × ℝ)) := fun r =>
+    {q | (r:ℝ) < H ∧ Safe (f q.1) 1 r ∧ q.2 ∈ K}
+  have mS : ∀ r, MeasurableSet (S r) := by
+    intro r
+    exact (MeasurableSet.const _).inter
+      (((measurableSet_safe f mf 1 _).preimage measurable_fst).inter (hK.measurableSet.preimage measurable_snd))
+  have hSpast : ∀ r b c, (∀ u ≤ r, b u = c u) → ∀ x, ((b,x) ∈ S r ↔ (c,x) ∈ S r) := by
+    intro r b c hbc x
+    have he : Safe (f b) 1 r ↔ Safe (f c) 1 r := safe_past hfpast r.coe_nonneg
+      (fun u hu => hbc u hu)
+    exact and_congr_right fun _ => and_congr_left fun _ => he
+  have hSeq : ∀ (b : Driver) r x, (((b:ℝ≥0 → ℝ),x) ∈ S r ↔ (r:ℝ) < τ b ∧ x ∈ K) := by
+    intro b r x
+    change (_ ∧ _ ∧ _) ↔ (_ ∧ _)
+    rw [lt_exit_iff hH (hfcH b) r.coe_nonneg]
+    exact and_assoc.symm
+  apply expected_stopped_local_grid_eq hW (green P) K domain hK isOpen_domain hKO (contDiffOn_green P)
+    v hv.continuous.measurable g z hL hM hbound hlip (fun b => (hg b).1) (fun b => (hg b).2)
+    (fun b => hv.continuous.comp_continuousOn ((hgc b).prodMk
+      ((b.continuous.comp continuous_real_toNNReal).continuousOn))) τ hτ hD
+    (fun q hq => by rw [hveq q hq]; exact green_generator P (hKO hq))
+    S mS hSpast hSeq (fun r q hq => hq.2.2) (by change φ (z,0) ≤ 2; linarith) ht
+end SLEExactGauge.OnePoint
+
+ 
+
+noncomputable section
+open Set Filter Real
+open scoped Topology
+namespace SLEExactGauge.ReverseDerivative
+
+def r (a : ℝ) : ℝ := 2*a+1/2
+def zeta (a : ℝ) : ℝ := a-1/(16*a)
+def lambda (a : ℝ) : ℝ := a+1+3/(16*a)
+def R (x y : ℝ) : ℝ := x^2+y^2
+def D (u x y : ℝ) : ℝ := R x y+u/2
+
+def H (a u x y l : ℝ) : ℝ :=
+  lambda a*l+r a/2*Real.log (R x y)+(zeta a-r a)*Real.log y-
+    zeta a/2*Real.log (D u x y)
+def F (a u x y l : ℝ) : ℝ := Real.exp (H a u x y l)
+def Hx (a u x y : ℝ) : ℝ := r a*x/R x y-zeta a*x/D u x y
+def Hxx (a u x y : ℝ) : ℝ :=
+  r a*(y^2-x^2)/(R x y)^2-zeta a/D u x y+2*zeta a*x^2/(D u x y)^2
+def Hy (a u x y : ℝ) : ℝ :=
+  r a*y/R x y+(zeta a-r a)/y-zeta a*y/D u x y
+
+def residual (a u x y : ℝ) : ℝ :=
+  R x y*((a-1/2+1/(16*a))*x^2+(2*a+1/2)*y^2)+
+    u/2*((2*a+3/2)*x^2+(2*a+1/2)*y^2)
+
+lemma R_pos (x : ℝ) {y : ℝ} (hy : 0 < y) : 0 < R x y := by
+  unfold R
+  positivity
+
+lemma D_pos {u y : ℝ} (hu : 0 ≤ u) (hy : 0 < y) (x : ℝ) : 0 < D u x y := by
+  exact add_pos_of_pos_of_nonneg (R_pos x hy) (div_nonneg hu (by norm_num))
+
+lemma parameters {a : ℝ} (ha : 1/4 < a) :
+    0 < zeta a ∧ 0 < lambda a ∧ 2 < zeta a+lambda a ∧ 1 < r a ∧ r a < 4*a := by
+  have ha0 : 0 < a := by linarith
+  have h16 : 0 < 16*a := by positivity
+  have hz : 0 < zeta a := by
+    unfold zeta
+    have hsq : 1 < 16*a^2 := by nlinarith
+    apply sub_pos.mpr
+    apply (div_lt_iff₀ h16).2
+    nlinarith
+  have hl : 0 < lambda a := by unfold lambda; positivity
+  have he : zeta a+lambda a-2 = (4*a-1)^2/(8*a) := by
+    unfold zeta lambda
+    field_simp
+    ring
+  have hsum : 2 < zeta a+lambda a := by
+    have : 0 < (4*a-1)^2/(8*a) := div_pos (sq_pos_of_pos (by linarith)) (by positivity)
+    linarith
+  exact ⟨hz,hl,hsum,by unfold r; linarith,by unfold r; linarith⟩
+
+lemma residual_nonneg {a u y : ℝ} (ha : 1/4 < a) (hu : 0 ≤ u)
+    (hy : 0 < y) (x : ℝ) : 0 ≤ residual a u x y := by
+  have ha0 : 0 < a := by linarith
+  have hc : 0 ≤ a-1/2+1/(16*a) := by
+    have he : a-1/2+1/(16*a) = (4*a-1)^2/(16*a) := by field_simp; ring
+    rw [he]
+    positivity
+  unfold residual
+  have hR := (R_pos x hy).le
+  positivity
+
+lemma hasDerivAt_Hx {a u x y l : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
+    HasDerivAt (fun w => H a u w y l) (Hx a u x y) x := by
+  have hR : HasDerivAt (fun w : ℝ => R w y) (2*x) x := by
+    convert! ((hasDerivAt_id x).pow 2).add_const (y^2) using 1
+    simp
+  have hD : HasDerivAt (fun w : ℝ => D u w y) (2*x) x := hR.add_const (u/2)
+  have hd := (((hR.log (R_pos x hy).ne').const_mul (r a/2)).const_add (lambda a*l)
+    |>.add_const ((zeta a-r a)*Real.log y)).sub
+      ((hD.log (D_pos hu hy x).ne').const_mul (zeta a/2))
+  convert! hd using 1
+  · dsimp [Hx]; ring
+
+lemma hasDerivAt_Hxx {a u x y : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
+    HasDerivAt (fun w => Hx a u w y) (Hxx a u x y) x := by
+  have hR : HasDerivAt (fun w : ℝ => R w y) (2*x) x := by
+    convert! ((hasDerivAt_id x).pow 2).add_const (y^2) using 1
+    simp
+  have hD : HasDerivAt (fun w : ℝ => D u w y) (2*x) x := hR.add_const (u/2)
+  have hd := (((hasDerivAt_id x).const_mul (r a)).div hR (R_pos x hy).ne').sub
+    (((hasDerivAt_id x).const_mul (zeta a)).div hD (D_pos hu hy x).ne')
+  convert! hd using 1
+  dsimp [Hxx]
+  have hnR := (R_pos x hy).ne'
+  have hnD := (D_pos hu hy x).ne'
+  field_simp
+  simp only [R]
+  ring
+
+lemma deriv_Fx {a u y l : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
+    deriv (fun x => F a u x y l) = fun x => F a u x y l * Hx a u x y := by
+  funext x
+  exact (hasDerivAt_Hx hu hy).exp.deriv
+
+lemma deriv_Fxx {a u x y l : ℝ} (hu : 0 ≤ u) (hy : 0 < y) :
+    deriv (deriv (fun w => F a u w y l)) x =
+      F a u x y l * ((Hx a u x y)^2+Hxx a u x y) := by
+  rw [deriv_Fx hu hy]
+  have hd := ((hasDerivAt_Hx (a := a) (x := x) (l := l) hu hy).exp).mul
+    (hasDerivAt_Hxx (a := a) (x := x) hu hy)
+  convert! hd.deriv using 1
+  dsimp [F]
+  ring
+
+lemma generator_identity {a u x y : ℝ} (ha : 0 < a) (hu : 0 ≤ u) (hy : 0 < y) :
+    ((Hx a u x y)^2+Hxx a u x y)/2 +
+      (-a*x/R x y)*Hx a u x y+(a*y/R x y)*Hy a u x y+
+      (a*(x^2-y^2)/(R x y)^2)*lambda a+zeta a/(4*D u x y) =
+      -zeta a*residual a u x y/(2*R x y*(D u x y)^2) := by
+  have ha' := ha.ne'
+  have hR := (R_pos x hy).ne'
+  have hD := (D_pos hu hy x).ne'
+  have hy' := hy.ne'
+  simp only [Hx,Hxx,Hy,lambda,zeta,r,residual]
+  field_simp
+  simp only [D,R]
+  ring
+
+lemma generator_nonpos {a u x y : ℝ} (ha : 1/4 < a) (hu : 0 ≤ u) (hy : 0 < y) :
+    ((Hx a u x y)^2+Hxx a u x y)/2 +
+      (-a*x/R x y)*Hx a u x y+(a*y/R x y)*Hy a u x y+
+      (a*(x^2-y^2)/(R x y)^2)*lambda a+zeta a/(4*D u x y) ≤ 0 := by
+  rw [generator_identity (by linarith) hu hy]
+  apply div_nonpos_of_nonpos_of_nonneg
+  · exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (parameters ha).1.le)
+      (residual_nonneg ha hu hy x)
+  · exact mul_nonneg (mul_nonneg (by norm_num) (R_pos x hy).le) (sq_nonneg _)
+
+end SLEExactGauge.ReverseDerivative
+
+namespace SLEExactGauge.ReverseDerivative
+
+abbrev State := ℝ × ℂ × ℝ
+
+def test (a : ℝ) (q : State) (b : ℝ) : ℝ :=
+  F a q.1 (q.2.1.re-b) q.2.1.im q.2.2
+
+def vector (a : ℝ) (q : State) (b : ℝ) : State :=
+  (-1, (-(a:ℂ)/(q.2.1-b),a*((q.2.1-b)⁻¹^2).re))
+
+lemma differentiableAt_test (a b : ℝ) {q : State} (hu : 0 ≤ q.1) (hy : 0 < q.2.1.im) :
+    DifferentiableAt ℝ (fun q => test a q b) q := by
+  have hR := (R_pos (q.2.1.re-b) hy).ne'
+  have hD := (D_pos hu hy (q.2.1.re-b)).ne'
+  have hy' := hy.ne'
+  have hre : DifferentiableAt ℝ (fun q : State => q.2.1.re) q :=
+    Complex.reCLM.differentiableAt.comp _ (differentiableAt_fst.comp _ differentiableAt_snd)
+  have him : DifferentiableAt ℝ (fun q : State => q.2.1.im) q :=
+    Complex.imCLM.differentiableAt.comp _ (differentiableAt_fst.comp _ differentiableAt_snd)
+  dsimp [test, F, H, D, R] at *
+  fun_prop
+
+end SLEExactGauge.ReverseDerivative
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/SubdiskMoments.lean b/lean/OAI/Probability/SLEGauge/SubdiskMoments.lean
index 3d1a5368a54375491f11abda14323ac778e3cad7..dedec85350a0d8c2d411691f1eea1f6884419044 100644
--- a/lean/OAI/Probability/SLEGauge/SubdiskMoments.lean
+++ b/lean/OAI/Probability/SLEGauge/SubdiskMoments.lean
@@ -1,538 +1,538 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.CenteredSequenceLaw
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
-
- 
-
-theorem common_limit_subdisk_tilted_moment (r : ℝ) (hr : 0≤r) (hr1 : r<1)
-    (a : ℝ) (ha : 1/4<a)
-    (m : ℕ) (hm : 1 ≤ m) :
-    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
-      (∀b,IsFiniteMeasureOnCompacts (μ ∞ b)) → Measurable (μ ∞) →
-      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
-      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
-          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
-      ∀(θ : ℝ) (_hθ : 0<θ) (_hθπ : θ<Real.pi) (l : ℕ)
-        (T : ExactSLEGenerator.Driver→ℝ≥0∞),
-      (∫⁻b,(diskMass a θ (μ (T b) b) (Metric.closedBall (0:ℂ) r))^m
-        ∂radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)))≤
-        D*ENNReal.ofReal ((Real.sin θ)^(-p a)) := by
-  let K : Set ℂ := Subtype.val '' tsupport (subdiskCutoff r hr hr1)
-  have hK : IsCompact K := (subdiskCutoff r hr hr1).hasCompactSupport.image continuous_subtype_val
-  have hcK : ∀z∈K,(1-r^2)/4≤2*z.im := by
-    rintro z ⟨w,hw,rfl⟩
-    have hh := subdiskCutoff_support r hr hr1 w hw
-    linarith
-  obtain ⟨D,hD,hDb⟩ := actual_normalized_tilted_moment_bound a ha m hm K
-    hK.measurableSet hK.isBounded ((1-r^2)/4) (by nlinarith) (by nlinarith [sq_nonneg r]) hcK
-    (2^(2-alpha a)) (Real.rpow_nonneg (by norm_num) _)
-  refine ⟨D,hD,?_⟩
-  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim θ hθ hθπ l T
-  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ ∞ b) := hμfin b
-  let f := subdiskTest a ⟨θ,hθ,hθπ⟩ r hr hr1
-  have hfm : Measurable (fun b=>∫z,f z∂μ ∞ b) := by
-    let Kμ : Kernel ExactSLEGenerator.Driver UpperPlane := ⟨μ ∞,hμ⟩
-    exact (f.continuous.stronglyMeasurable.integral_kernel (κ:=Kμ)).measurable
-  have hb := hDb W hW θ hθ hθπ f (subdiskTest_support a ⟨θ,hθ,hθπ⟩ r hr hr1)
-    (subdiskTest_bound a ⟨θ,hθ,hθπ⟩ r hr hr1) s coeff hcoeff hsum ∞ (μ ∞) hfm
-    (hlim.mono (fun b hb=>hb f)) l
-  refine le_trans (lintegral_mono_ae ?_) hb
-  have habs : radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) ≪ W :=
-    withDensity_absolutelyContinuous W _
-  filter_upwards [habs.ae_le hmono] with b hb
-  exact pow_le_pow_left' ((Measure.le_iff'.mp (diskMass_mono (hb (T b)) a θ)
-    _).trans (subdiskMass_le_enorm a ha ⟨θ,hθ,hθπ⟩ r hr hr1 (μ ∞ b))) m
-
- 
-
-def subdiskMoment (W : Measure ExactSLEGenerator.Driver) (a : ℝ)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane) (r : ℝ) (m l : ℕ) : ℝ≥0∞ :=
-  ∫⁻p,(subdiskMass a μ r l p)^m∂finiteAngleLaw W a l
-
- 
-
-theorem common_limit_subdisk_mixed_moment_bound (r : ℝ) (hr : 0≤r) (hr1 : r<1) (a : ℝ) (ha : 1/4<a)
-    (m : ℕ) (hm : 1 ≤ m) :
-    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
-      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
-      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
-      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
-      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
-      (∀v b,IsFiniteMeasureOnCompacts (μ v b)) →
-      Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2) →
-      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
-      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
-        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
-          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
-      ∀l, subdiskMoment W a μ r m l≤D := by
-  obtain ⟨C,hC,hCb⟩ := common_limit_subdisk_tilted_moment r hr hr1 a ha m hm
-  let J := ∫⁻θ : Angle,ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a
-  refine ⟨C*J,ENNReal.mul_lt_top hC (angle_inverse_power_finite a),?_⟩
-  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim l
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := hμfin v b
-  rw [subdiskMoment,finiteAngleLaw_lintegral W a ha l _
-    ((subdiskMass_measurable a ha μ hμ r l).pow_const m)]
-  calc
-    _ ≤ ∫⁻θ : Angle,C*ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a := by
-      apply lintegral_mono
-      intro θ
-      let T := fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l
-      exact hCb W hW s coeff hcoeff hsum μ (hμfin ∞)
-        (hμ.comp (measurable_const.prodMk measurable_id)) hmono hlim
-        θ.1 θ.2.1 θ.2.2 l T
-    _ = C*J := lintegral_const_mul _ (by fun_prop)
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
-
-lemma subdiskMass_monotone (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (b : ExactSLEGenerator.Driver) (hμ : Monotone (fun v=>μ v b)) (θ : Angle) :
-    Monotone (fun l=>subdiskMass a μ r l (θ,b)) := by
-  intro l k hlk
-  exact Measure.le_iff'.mp (diskMass_mono
-    (hμ (centeredTime_monotone a ha θ b hlk)) a θ.1) _
-
-lemma subdiskMass_raw_measurable (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) (l : ℕ)
-    (hμ : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b)) :
-    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>subdiskMass a μ r l (θ,b)) := by
-  let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
-  have hn := diskMass_joint_measurable a (fun _ : ExactSLEGenerator.Driver=>θ) measurable_const
-    (fun b : ExactSLEGenerator.Driver=>μ (levelTime a θ.1 b.1 l) b) hμ
-  exact (Measure.measurable_coe Metric.isClosed_closedBall.measurableSet).comp hn
-
-lemma subdiskMass_raw_measurable_of_le (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) {i l : ℕ} (hil : i≤l)
-    (hμ : Measurable[(centeredTime_stopping a ha θ i).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 i) b)) :
-    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>subdiskMass a μ r i (θ,b)) :=
-  (subdiskMass_raw_measurable r a ha μ θ i hμ).mono
-    ((centeredTime_stopping a ha θ i).measurableSpace_mono (centeredTime_stopping a ha θ l)
-      (fun b=>centeredTime_monotone a ha θ b hil)) le_rfl
-
-lemma subdisk_vector_consistent (W : Measure ExactSLEGenerator.Driver)
-    [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b))
-    {l k : ℕ} (hlk : l≤k) :
-    (finiteAngleLaw W a k).map (fun p=>fun i : Fin (l+1)=>subdiskMass a μ r i p)=
-      (finiteAngleLaw W a l).map (fun p=>fun i : Fin (l+1)=>subdiskMass a μ r i p) := by
-  have hm : Measurable (fun p=>fun i : Fin (l+1)=>subdiskMass a μ r i p) :=
-    Measurable.of_eval (fun i=>subdiskMass_measurable a ha μ hμ r i)
-  apply Measure.ext
-  intro A hA
-  rw [Measure.map_apply hm hA,Measure.map_apply hm hA]
-  simp only [←lintegral_indicator_one (hm hA)]
-  apply finiteAngleLaw_lintegral_consistent W hW a ha hlk
-  · exact measurable_const.indicator (hm hA)
-  · intro θ
-    let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
-    have hv : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-        (fun b=>fun i : Fin (l+1)=>subdiskMass a μ r i (θ,b)) :=
-      Measurable.of_eval (fun i=>subdiskMass_raw_measurable_of_le r a ha μ θ
-        (by omega) (hμraw i θ))
-    exact measurable_const.indicator (hv hA)
-
- 
-
-theorem exists_subdisk_sequence_law (W : Measure ExactSLEGenerator.Driver)
-    [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b))
-    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) :
-    ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
-      (∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
-        (fun p=>fun i : Fin (n+1)=>subdiskMass a μ r i p)) ∧
-      (∀ᵐ x∂Q,Monotone x) := by
-  have (n : ℕ) : IsProbabilityMeasure (finiteAngleLaw W a n) :=
-    finiteAngleLaw_probability W hW a ha n
-  obtain ⟨Q,hQ,hQlaw⟩ := exists_sequence_law (finiteAngleLaw W a)
-    (subdiskMass a μ r) (subdiskMass_measurable a ha μ hμ r)
-    (fun _ _ hlk=>subdisk_vector_consistent W hW r a ha μ hμ hμraw hlk)
-  refine ⟨Q,hQ,hQlaw,?_⟩
-  have hX (l : ℕ) : ∀ᵐ p∂finiteAngleLaw W a l,Monotone (fun n=>subdiskMass a μ r n p) := by
-    have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
-      Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
-    filter_upwards [(finiteAngleLaw_absolutelyContinuous W a l).ae_le hb] with p hp
-    exact subdiskMass_monotone r a ha μ p.2 hp p.1
-  have hs (n : ℕ) : ∀ᵐ x∂Q,x n≤x (n+1) := by
-    let i : Fin (n+1+1) := ⟨n,by omega⟩
-    let j : Fin (n+1+1) := ⟨n+1,by omega⟩
-    have hm : MeasurableSet {x : Fin (n+1+1)→ℝ≥0∞ | x i≤x j} :=
-      measurableSet_le (measurable_pi_apply _) (measurable_pi_apply _)
-    change ∀ᵐ x∂Q,(initialVector (n+1) x) i≤(initialVector (n+1) x) j
-    rw [←ae_map_iff (measurable_initialVector (n+1)).aemeasurable hm,hQlaw]
-    apply (ae_map_iff (Measurable.of_eval
-      (fun i : Fin (n+1+1)=>subdiskMass_measurable a ha μ hμ r i)).aemeasurable hm).mpr
-    filter_upwards [hX (n+1)] with p hp
-    exact hp (by simp only [i,j]; omega)
-  filter_upwards [ae_all_iff.mpr hs] with x hx
-  exact monotone_nat_of_le_succ hx
-
- 
-
-theorem subdiskSequence_moment (W : Measure ExactSLEGenerator.Driver)
-    (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (Q : Measure (ℕ→ℝ≥0∞))
-    (hQlaw : ∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
-      (fun p=>fun i : Fin (n+1)=>subdiskMass a μ r i p))
-    (hQmono : ∀ᵐ x∂Q,Monotone x) (m : ℕ) :
-    (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,subdiskMoment W a μ r m l := by
-  simp only [diskSequenceMass,ENNReal.iSup_pow]
-  rw [lintegral_iSup' (fun _=>(measurable_pi_apply _).pow_const m |>.aemeasurable)
-    (hQmono.mono (fun _ hx _ _ hij=>pow_le_pow_left' (hx hij) m))]
-  congr with l
-  exact lintegral_coordinate Q (finiteAngleLaw W a l) (subdiskMass a μ r)
-    (subdiskMass_measurable a ha μ hμ r) l (hQlaw l) (fun x=>x^m)
-    (measurable_id.pow_const m)
-
- 
-
-theorem subdiskMoment_monotone (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
-    (r a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
-      (fun b=>μ (levelTime a θ.1 b.1 l) b))
-    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) (m : ℕ) :
-    Monotone (subdiskMoment W a μ r m) := by
-  intro l k hlk
-  rw [subdiskMoment,←finiteAngleLaw_lintegral_consistent W hW a ha hlk _
-    ((subdiskMass_measurable a ha μ hμ r l).pow_const m)
-    (fun θ=>(subdiskMass_raw_measurable r a ha μ θ l (hμraw l θ)).pow_const m)]
-  apply lintegral_mono_ae
-  have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
-    Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
-  filter_upwards [(finiteAngleLaw_absolutelyContinuous W a k).ae_le hb] with p hp
-  exact pow_le_pow_left' (subdiskMass_monotone r a ha μ p.2 hp p.1 hlk) m
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-namespace SLEExactGauge.PastMass
-open ExactSLEGenerator ExactSLEMain ExactSLEGreen
-
- 
-
-def pastGreenKernel (W : Measure ExactSLEGenerator.Driver) (a : ℝ)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (z : UpperPlane) : ℝ≥0∞ :=
-  (∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W) /
-    ENNReal.ofReal (initialGreen a z)
-
-lemma pastGreenKernel_measurable (W : Measure ExactSLEGenerator.Driver) [SFinite W]
-    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) : Measurable (pastGreenKernel W a S) :=
-  ((stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal.lintegral_prod_left').div
-    (initialGreen_continuous a).measurable.ennreal_ofReal
-
-lemma initial_mul_pastGreenKernel (W : Measure ExactSLEGenerator.Driver)
-    (a : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) (z : UpperPlane) :
-    ENNReal.ofReal (initialGreen a z) * pastGreenKernel W a S z =
-      ∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W :=
-  ENNReal.mul_div_cancel (ENNReal.ofReal_pos.mpr (initialGreen_pos a z)).ne'
-    ENNReal.ofReal_ne_top
-
-lemma pastGreenKernel_eq_ofReal (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane) :
-    pastGreenKernel W a S z = ENNReal.ofReal
-      ((∫b, stoppedSurvivingGreen a (S b) b z.1 ∂W) / initialGreen a z) := by
-  rw [ENNReal.ofReal_div_of_pos (initialGreen_pos a z),
-    ofReal_integral_eq_lintegral_ofReal (survivingGreen_optional_bound a ha W hW z S hS).1
-      (Eventually.of_forall (fun b => stoppedSurvivingGreen_nonneg a (S b) b z.1))]
-  rfl
-
-lemma pastGreenKernel_le_one (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane) :
-    pastGreenKernel W a S z ≤ 1 := by
-  rw [pastGreenKernel_eq_ofReal W hW a ha S hS]
-  exact (ENNReal.ofReal_le_ofReal ((div_le_one (initialGreen_pos a z)).mpr
-    (survivingGreen_optional_bound a ha W hW z S hS).2)).trans_eq (by simp)
-
-lemma pastGreenKernel_antitone (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (S T : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (hT : IsStoppingTime rawFiltration T)
-    (hst : S ≤ T) (z : UpperPlane) :
-    pastGreenKernel W a T z ≤ pastGreenKernel W a S z := by
-  rw [pastGreenKernel_eq_ofReal W hW a ha T hT, pastGreenKernel_eq_ofReal W hW a ha S hS]
-  apply ENNReal.ofReal_le_ofReal
-  apply div_le_div_of_nonneg_right _ (initialGreen_pos a z).le
-  simpa only [setIntegral_univ] using
-    survivingGreen_optional_comparison a ha W hW z S T hS hT hst univ MeasurableSet.univ
-
- 
-
-lemma radialLaw_strict_past (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) :
-    (radialLaw W a z.1 (z.1.im*q)) {b | S b < radialTime a b.1 z.1 (z.1.im*q)} =
-      ENNReal.ofReal ((∫b, stoppedRemainingDensity a q (S b) b z.1 ∂W) / initialGreen a z) := by
-  have hρ : 0<z.1.im*q := mul_pos z.2 hq
-  have hr : z.1.im*q≤z.1.im := mul_le_of_le_one_right z.2.le hq1
-  have hT := radialTime_stopping a (by linarith) z.1 z.2 _ hρ hr
-  have hmS : Measurable S := hS.measurable'
-  have hmT : Measurable (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z.1 (z.1.im*q)) :=
-    hT.measurable'
-  have hE : MeasurableSet {b : ExactSLEGenerator.Driver | S b < radialTime a b.1 z.1 (z.1.im*q)} :=
-    measurableSet_lt hmS hmT
-  rw [radialLaw_apply W hW a ha z.1 z.2 _ hρ hr _ hE]
-  have hp := (radial_conditional_weight_all W hW a ha z.1 z.2 _ hρ hr S hS
-    univ MeasurableSet.univ).2
-  simp only [setIntegral_univ] at hp
-  rw [← integral_indicator hE]
-  simp only [indicator_apply, mem_ofPred_eq]
-  rw [hp]
-  congr 1
-  simp only [stoppedRemainingDensity_eq a q _ _ _ z.2, integral_const_mul, initialGreen]
-  exact (mul_div_mul_left _ _ (Real.exp_ne_zero _)).symm
-
- 
-
-theorem radialLaw_strict_past_tendsto (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane)
-    (q : ℕ → ℝ) (hq : ∀n, 0<q n ∧ q n≤1) (hq0 : Tendsto q atTop (𝓝 0)) :
-    Tendsto (fun n => (radialLaw W a z.1 (z.1.im*q n))
-      {b | S b < radialTime a b.1 z.1 (z.1.im*q n)}) atTop
-      (𝓝 (pastGreenKernel W a S z)) := by
-  simp_rw [radialLaw_strict_past W hW a ha S hS z _ (hq _).1 (hq _).2]
-  rw [pastGreenKernel_eq_ofReal W hW a ha S hS]
-  have he := stopped_remaining_integral_tendsto a ha W hW z S hS q hq hq0 univ
-  simp only [setIntegral_univ] at he
-  exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (he.div_const _)
-
-end SLEExactGauge.PastMass
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set
-open scoped Classical NNReal ENNReal
-namespace SLEExactGauge.PastMass
-
-variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
-
- 
-def finiteTestTime (F : Finset ℝ≥0) (M : ℝ≥0 → Ω → ℝ≥0∞)
-    (H : ℝ≥0∞) (ω : Ω) : ℝ≥0∞ :=
-  F.inf (fun q => if H < M q ω then (q : ℝ≥0∞) else ∞)
-
-omit mΩ in
-lemma finiteTestTime_lt_iff [MeasurableSpace Ω] (F : Finset ℝ≥0) (M : ℝ≥0 → Ω → ℝ≥0∞)
-    (H : ℝ≥0∞) (ω : Ω) (v : ℝ≥0∞) :
-    finiteTestTime F M H ω < v ↔ ∃ q ∈ F, H < M q ω ∧ (q : ℝ≥0∞) < v := by
-  rw [finiteTestTime, Finset.inf_lt_iff]
-  congr! 2 with q
-  by_cases h : H < M q ω <;> simp [h]
-
-omit mΩ in
-lemma finiteTestTime_le_iff [MeasurableSpace Ω] (F : Finset ℝ≥0) (M : ℝ≥0 → Ω → ℝ≥0∞)
-    (H : ℝ≥0∞) (ω : Ω) (t : ℝ≥0) :
-    finiteTestTime F M H ω ≤ (t : ℝ≥0∞) ↔ ∃ q ∈ F, q ≤ t ∧ H < M q ω := by
-  rw [finiteTestTime, Finset.inf_le_iff ENNReal.coe_lt_top]
-  congr! 2 with q
-  by_cases h : H < M q ω <;> simp [h]
-
-lemma finiteTestTime_stopping (ℱ : Filtration ℝ≥0 mΩ)
-    (M : ℝ≥0 → Ω → ℝ≥0∞) (hM : Adapted ℱ M)
-    (F : Finset ℝ≥0) (H : ℝ≥0∞) :
-    IsStoppingTime ℱ (finiteTestTime F M H) := by
-  intro t
-  change MeasurableSet[ℱ t] {ω | finiteTestTime F M H ω ≤ (t : ℝ≥0∞)}
-  have he : {ω | finiteTestTime F M H ω ≤ (t : ℝ≥0∞)} =
-      ⋃ q ∈ F, ⋃ (_ : q ≤ t), {ω | H < M q ω} := by
-    ext ω
-    simp only [mem_ofPred_eq, finiteTestTime_le_iff, mem_iUnion, exists_prop]
-  rw [he]
-  apply MeasurableSet.biUnion F.countable_toSet
-  intro q hq
-  apply MeasurableSet.iUnion
-  intro hqt
-  exact ℱ.mono hqt _ (measurableSet_lt measurable_const (hM q))
-
-omit mΩ in
-lemma finiteTestTime_antitone [MeasurableSpace Ω] (M : ℝ≥0 → Ω → ℝ≥0∞) (H : ℝ≥0∞)
-    (ω : Ω) : Antitone (fun F : Finset ℝ≥0 => finiteTestTime F M H ω) := by
-  intro F G hFG
-  exact Finset.inf_mono hFG
-
- 
-
-lemma finiteTestTime_iUnion {ι : Type*} (e : ℕ → ι) (q : ι → ℝ≥0)
-    (he : Function.Surjective e) (M : ℝ≥0 → Ω → ℝ≥0∞) (H : ℝ≥0∞)
-    (ω : Ω) (v : ℝ≥0∞) :
-    (∃ i, H < M (q i) ω ∧ (q i : ℝ≥0∞) < v) ↔
-      ∃ n : ℕ, finiteTestTime ((Finset.range n).image (q ∘ e)) M H ω < v := by
-  simp only [finiteTestTime_lt_iff, Finset.mem_image, Finset.mem_range,
-    Function.comp_apply]
-  constructor
-  · rintro ⟨i,hi⟩
-    obtain ⟨k,rfl⟩ := he i
-    exact ⟨k+1,q (e k),⟨k,Nat.lt_succ_self k,rfl⟩,hi⟩
-  · rintro ⟨n,t,⟨k,hk,rfl⟩,ht⟩
-    exact ⟨e k,ht⟩
-
-lemma finiteTestTime_sequence_antitone {ι : Type*} (e : ℕ → ι) (q : ι → ℝ≥0)
-    (M : ℝ≥0 → Ω → ℝ≥0∞) (H : ℝ≥0∞) (ω : Ω) :
-    Antitone (fun n => finiteTestTime ((Finset.range n).image (q ∘ e)) M H ω) := by
-  intro n k hnk
-  exact finiteTestTime_antitone M H ω (Finset.image_subset_image (Finset.range_mono hnk))
-
-end SLEExactGauge.PastMass
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-namespace SLEExactGauge.PastMass
-open FlowIdentification ExactSLEGreen TraceMassContinuity TraceRestriction
-
- 
-def visitTime (κ : ℝ) (b : GDriver) : UpperPlane → ℝ≥0∞ :=
-  firstVisit (fun z : UpperPlane => z.1) (internalTrace κ (driverHomeomorph.symm b))
-
-lemma stopped_restriction_add (κ : ℝ) (b : GDriver)
-    (μ : ℝ≥0∞ → Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (s : ℝ≥0∞) :
-    μ s Q + μ ∞ (Q ∩ {z | s < visitTime κ b z}) = μ ∞ Q := by
-  induction s using ENNReal.recTopCoe with
-  | top => simp
-  | coe s =>
-    rw [hμ.firstVisit_identity s, Measure.restrict_apply hQ]
-    have he : Q ∩ {z | (s : ℝ≥0∞) < visitTime κ b z} =
-        Q \ {z | visitTime κ b z ≤ (s : ℝ≥0∞)} := by
-      ext z
-      simp only [mem_inter_iff,Set.mem_sdiff,mem_ofPred_eq,not_le]
-    rw [he]
-    exact measure_inter_add_sdiff Q (measurableSet_le hμ.firstVisit_measurable measurable_const)
-
-lemma stopped_strictPast_mass (κ : ℝ) (b : GDriver)
-    (μ : ℝ≥0∞ → Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q ≠ ∞) (s : ℝ≥0∞) :
-    μ ∞ (Q ∩ {z | s < visitTime κ b z}) = μ ∞ Q - μ s Q := by
-  have hsq : μ s Q ≠ ∞ := ne_top_of_le_ne_top hfin (hμ.monotone le_top Q)
-  rw [← stopped_restriction_add κ b μ hμ Q hQ s, ENNReal.add_sub_cancel_left hsq]
-
-lemma finite_past_mass (κ : ℝ) (b : GDriver)
-    (μ : ℝ≥0∞ → Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q ≠ ∞)
-    (F : Finset ℝ≥0) (H : ℝ≥0∞) :
-    μ ∞ (Q ∩ {z | ∃q ∈ F, H < μ q Q ∧ (q : ℝ≥0∞) < visitTime κ b z}) =
-      μ ∞ Q - μ (finiteTestTime F (fun q (_ : Unit) => μ q Q) H ()) Q := by
-  simpa only [finiteTestTime_lt_iff] using
-    stopped_strictPast_mass κ b μ hμ Q hQ hfin
-      (finiteTestTime F (fun q (_ : Unit) => μ q Q) H ())
-
- 
-
-lemma all_past_mass_eq_iSup {ι : Type*} (e : ℕ → ι) (q : ι → ℝ≥0)
-    (he : Function.Surjective e) (κ : ℝ) (b : GDriver)
-    (μ : ℝ≥0∞ → Measure UpperPlane)
-    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
-    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (H : ℝ≥0∞) :
-    μ ∞ (Q ∩ {z | ∃i, H < μ (q i) Q ∧ (q i : ℝ≥0∞) < visitTime κ b z}) =
-      ⨆n : ℕ, μ ∞ (Q ∩ {z |
-        finiteTestTime ((Finset.range n).image (q ∘ e))
-          (fun q (_ : Unit) => μ q Q) H () < visitTime κ b z}) := by
-  let A (n : ℕ) : Set UpperPlane := Q ∩ {z |
-    finiteTestTime ((Finset.range n).image (q ∘ e))
-      (fun q (_ : Unit) => μ q Q) H () < visitTime κ b z}
-  have hA (n : ℕ) : MeasurableSet (A n) := hQ.inter
-    (measurableSet_lt measurable_const hμ.firstVisit_measurable)
-  have hAm : Monotone A := by
-    intro n k hnk z hz
-    exact ⟨hz.1,(finiteTestTime_sequence_antitone e q (fun q (_ : Unit) => μ q Q)
-      H () hnk).trans_lt hz.2⟩
-  have heq : Q ∩ {z | ∃i, H < μ (q i) Q ∧ (q i : ℝ≥0∞) < visitTime κ b z} =
-      ⋃n : ℕ, A n := by
-    ext z
-    simp only [mem_inter_iff,mem_ofPred_eq,mem_iUnion,A,
-      finiteTestTime_iUnion e q he (fun q (_ : Unit) => μ q Q) H ()]
-    exact exists_and_left.symm
-  rw [heq, hAm.measure_iUnion]
-
-end SLEExactGauge.PastMass
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.CenteredSequenceLaw
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen KernelSimilarity
+
+ 
+
+theorem common_limit_subdisk_tilted_moment (r : ℝ) (hr : 0≤r) (hr1 : r<1)
+    (a : ℝ) (ha : 1/4<a)
+    (m : ℕ) (hm : 1 ≤ m) :
+    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
+      (∀b,IsFiniteMeasureOnCompacts (μ ∞ b)) → Measurable (μ ∞) →
+      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
+      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
+          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
+      ∀(θ : ℝ) (_hθ : 0<θ) (_hθπ : θ<Real.pi) (l : ℕ)
+        (T : ExactSLEGenerator.Driver→ℝ≥0∞),
+      (∫⁻b,(diskMass a θ (μ (T b) b) (Metric.closedBall (0:ℂ) r))^m
+        ∂radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)))≤
+        D*ENNReal.ofReal ((Real.sin θ)^(-p a)) := by
+  let K : Set ℂ := Subtype.val '' tsupport (subdiskCutoff r hr hr1)
+  have hK : IsCompact K := (subdiskCutoff r hr hr1).hasCompactSupport.image continuous_subtype_val
+  have hcK : ∀z∈K,(1-r^2)/4≤2*z.im := by
+    rintro z ⟨w,hw,rfl⟩
+    have hh := subdiskCutoff_support r hr hr1 w hw
+    linarith
+  obtain ⟨D,hD,hDb⟩ := actual_normalized_tilted_moment_bound a ha m hm K
+    hK.measurableSet hK.isBounded ((1-r^2)/4) (by nlinarith) (by nlinarith [sq_nonneg r]) hcK
+    (2^(2-alpha a)) (Real.rpow_nonneg (by norm_num) _)
+  refine ⟨D,hD,?_⟩
+  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim θ hθ hθπ l T
+  let (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ ∞ b) := hμfin b
+  let f := subdiskTest a ⟨θ,hθ,hθπ⟩ r hr hr1
+  have hfm : Measurable (fun b=>∫z,f z∂μ ∞ b) := by
+    let Kμ : Kernel ExactSLEGenerator.Driver UpperPlane := ⟨μ ∞,hμ⟩
+    exact (f.continuous.stronglyMeasurable.integral_kernel (κ:=Kμ)).measurable
+  have hb := hDb W hW θ hθ hθπ f (subdiskTest_support a ⟨θ,hθ,hθπ⟩ r hr hr1)
+    (subdiskTest_bound a ⟨θ,hθ,hθπ⟩ r hr hr1) s coeff hcoeff hsum ∞ (μ ∞) hfm
+    (hlim.mono (fun b hb=>hb f)) l
+  refine le_trans (lintegral_mono_ae ?_) hb
+  have habs : radialLaw W a (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) ≪ W :=
+    withDensity_absolutelyContinuous W _
+  filter_upwards [habs.ae_le hmono] with b hb
+  exact pow_le_pow_left' ((Measure.le_iff'.mp (diskMass_mono (hb (T b)) a θ)
+    _).trans (subdiskMass_le_enorm a ha ⟨θ,hθ,hθπ⟩ r hr hr1 (μ ∞ b))) m
+
+ 
+
+def subdiskMoment (W : Measure ExactSLEGenerator.Driver) (a : ℝ)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane) (r : ℝ) (m l : ℕ) : ℝ≥0∞ :=
+  ∫⁻p,(subdiskMass a μ r l p)^m∂finiteAngleLaw W a l
+
+ 
+
+theorem common_limit_subdisk_mixed_moment_bound (r : ℝ) (hr : 0≤r) (hr1 : r<1) (a : ℝ) (ha : 1/4<a)
+    (m : ℕ) (hm : 1 ≤ m) :
+    ∃D : ℝ≥0∞,D<∞ ∧ ∀(W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W],
+      IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W →
+      ∀(s : ℕ→Finset ℕ) (coeff : ℕ→ℕ→ℝ),
+      (∀k j,j∈s k→0≤coeff k j) → (∀k,∑j∈s k,coeff k j=1) →
+      ∀μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane,
+      (∀v b,IsFiniteMeasureOnCompacts (μ v b)) →
+      Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2) →
+      (∀ᵐ b∂W,∀v : ℝ≥0∞,μ v b≤μ ∞ b) →
+      (∀ᵐ b∂W,∀f : C_c(UpperPlane,ℝ),Tendsto (fun k=>∑j∈s k,coeff k j*
+        ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) ∞ b)
+          atTop (𝓝 (∫z,f z∂μ ∞ b))) →
+      ∀l, subdiskMoment W a μ r m l≤D := by
+  obtain ⟨C,hC,hCb⟩ := common_limit_subdisk_tilted_moment r hr hr1 a ha m hm
+  let J := ∫⁻θ : Angle,ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a
+  refine ⟨C*J,ENNReal.mul_lt_top hC (angle_inverse_power_finite a),?_⟩
+  intro W hprob hW s coeff hcoeff hsum μ hμfin hμ hmono hlim l
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := hμfin v b
+  rw [subdiskMoment,finiteAngleLaw_lintegral W a ha l _
+    ((subdiskMass_measurable a ha μ hμ r l).pow_const m)]
+  calc
+    _ ≤ ∫⁻θ : Angle,C*ENNReal.ofReal (Real.sin θ.1^(-p a))∂angleLaw a := by
+      apply lintegral_mono
+      intro θ
+      let T := fun b : ExactSLEGenerator.Driver=>levelTime a θ.1 b.1 l
+      exact hCb W hW s coeff hcoeff hsum μ (hμfin ∞)
+        (hμ.comp (measurable_const.prodMk measurable_id)) hmono hlim
+        θ.1 θ.2.1 θ.2.2 l T
+    _ = C*J := lintegral_const_mul _ (by fun_prop)
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
+
+lemma subdiskMass_monotone (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (b : ExactSLEGenerator.Driver) (hμ : Monotone (fun v=>μ v b)) (θ : Angle) :
+    Monotone (fun l=>subdiskMass a μ r l (θ,b)) := by
+  intro l k hlk
+  exact Measure.le_iff'.mp (diskMass_mono
+    (hμ (centeredTime_monotone a ha θ b hlk)) a θ.1) _
+
+lemma subdiskMass_raw_measurable (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) (l : ℕ)
+    (hμ : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b)) :
+    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>subdiskMass a μ r l (θ,b)) := by
+  let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
+  have hn := diskMass_joint_measurable a (fun _ : ExactSLEGenerator.Driver=>θ) measurable_const
+    (fun b : ExactSLEGenerator.Driver=>μ (levelTime a θ.1 b.1 l) b) hμ
+  exact (Measure.measurable_coe Metric.isClosed_closedBall.measurableSet).comp hn
+
+lemma subdiskMass_raw_measurable_of_le (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)] (θ : Angle) {i l : ℕ} (hil : i≤l)
+    (hμ : Measurable[(centeredTime_stopping a ha θ i).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 i) b)) :
+    Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>subdiskMass a μ r i (θ,b)) :=
+  (subdiskMass_raw_measurable r a ha μ θ i hμ).mono
+    ((centeredTime_stopping a ha θ i).measurableSpace_mono (centeredTime_stopping a ha θ l)
+      (fun b=>centeredTime_monotone a ha θ b hil)) le_rfl
+
+lemma subdisk_vector_consistent (W : Measure ExactSLEGenerator.Driver)
+    [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b))
+    {l k : ℕ} (hlk : l≤k) :
+    (finiteAngleLaw W a k).map (fun p=>fun i : Fin (l+1)=>subdiskMass a μ r i p)=
+      (finiteAngleLaw W a l).map (fun p=>fun i : Fin (l+1)=>subdiskMass a μ r i p) := by
+  have hm : Measurable (fun p=>fun i : Fin (l+1)=>subdiskMass a μ r i p) :=
+    Measurable.of_eval (fun i=>subdiskMass_measurable a ha μ hμ r i)
+  apply Measure.ext
+  intro A hA
+  rw [Measure.map_apply hm hA,Measure.map_apply hm hA]
+  simp only [←lintegral_indicator_one (hm hA)]
+  apply finiteAngleLaw_lintegral_consistent W hW a ha hlk
+  · exact measurable_const.indicator (hm hA)
+  · intro θ
+    let : MeasurableSpace ExactSLEGenerator.Driver := (centeredTime_stopping a ha θ l).measurableSpace
+    have hv : Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+        (fun b=>fun i : Fin (l+1)=>subdiskMass a μ r i (θ,b)) :=
+      Measurable.of_eval (fun i=>subdiskMass_raw_measurable_of_le r a ha μ θ
+        (by omega) (hμraw i θ))
+    exact measurable_const.indicator (hv hA)
+
+ 
+
+theorem exists_subdisk_sequence_law (W : Measure ExactSLEGenerator.Driver)
+    [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b))
+    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) :
+    ∃Q : Measure (ℕ→ℝ≥0∞),IsProbabilityMeasure Q ∧
+      (∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
+        (fun p=>fun i : Fin (n+1)=>subdiskMass a μ r i p)) ∧
+      (∀ᵐ x∂Q,Monotone x) := by
+  have (n : ℕ) : IsProbabilityMeasure (finiteAngleLaw W a n) :=
+    finiteAngleLaw_probability W hW a ha n
+  obtain ⟨Q,hQ,hQlaw⟩ := exists_sequence_law (finiteAngleLaw W a)
+    (subdiskMass a μ r) (subdiskMass_measurable a ha μ hμ r)
+    (fun _ _ hlk=>subdisk_vector_consistent W hW r a ha μ hμ hμraw hlk)
+  refine ⟨Q,hQ,hQlaw,?_⟩
+  have hX (l : ℕ) : ∀ᵐ p∂finiteAngleLaw W a l,Monotone (fun n=>subdiskMass a μ r n p) := by
+    have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
+      Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
+    filter_upwards [(finiteAngleLaw_absolutelyContinuous W a l).ae_le hb] with p hp
+    exact subdiskMass_monotone r a ha μ p.2 hp p.1
+  have hs (n : ℕ) : ∀ᵐ x∂Q,x n≤x (n+1) := by
+    let i : Fin (n+1+1) := ⟨n,by omega⟩
+    let j : Fin (n+1+1) := ⟨n+1,by omega⟩
+    have hm : MeasurableSet {x : Fin (n+1+1)→ℝ≥0∞ | x i≤x j} :=
+      measurableSet_le (measurable_pi_apply _) (measurable_pi_apply _)
+    change ∀ᵐ x∂Q,(initialVector (n+1) x) i≤(initialVector (n+1) x) j
+    rw [←ae_map_iff (measurable_initialVector (n+1)).aemeasurable hm,hQlaw]
+    apply (ae_map_iff (Measurable.of_eval
+      (fun i : Fin (n+1+1)=>subdiskMass_measurable a ha μ hμ r i)).aemeasurable hm).mpr
+    filter_upwards [hX (n+1)] with p hp
+    exact hp (by simp only [i,j]; omega)
+  filter_upwards [ae_all_iff.mpr hs] with x hx
+  exact monotone_nat_of_le_succ hx
+
+ 
+
+theorem subdiskSequence_moment (W : Measure ExactSLEGenerator.Driver)
+    (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (Q : Measure (ℕ→ℝ≥0∞))
+    (hQlaw : ∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
+      (fun p=>fun i : Fin (n+1)=>subdiskMass a μ r i p))
+    (hQmono : ∀ᵐ x∂Q,Monotone x) (m : ℕ) :
+    (∫⁻x,(diskSequenceMass x)^m∂Q)=⨆l,subdiskMoment W a μ r m l := by
+  simp only [diskSequenceMass,ENNReal.iSup_pow]
+  rw [lintegral_iSup' (fun _=>(measurable_pi_apply _).pow_const m |>.aemeasurable)
+    (hQmono.mono (fun _ hx _ _ hij=>pow_le_pow_left' (hx hij) m))]
+  congr with l
+  exact lintegral_coordinate Q (finiteAngleLaw W a l) (subdiskMass a μ r)
+    (subdiskMass_measurable a ha μ hμ r) l (hQlaw l) (fun x=>x^m)
+    (measurable_id.pow_const m)
+
+ 
+
+theorem subdiskMoment_monotone (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver)=>b.1 t) W)
+    (r a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→ExactSLEGenerator.Driver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×ExactSLEGenerator.Driver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (hμraw : ∀l θ,Measurable[(centeredTime_stopping a ha θ l).measurableSpace]
+      (fun b=>μ (levelTime a θ.1 b.1 l) b))
+    (hmono : ∀ᵐ b∂W,Monotone (fun v=>μ v b)) (m : ℕ) :
+    Monotone (subdiskMoment W a μ r m) := by
+  intro l k hlk
+  rw [subdiskMoment,←finiteAngleLaw_lintegral_consistent W hW a ha hlk _
+    ((subdiskMass_measurable a ha μ hμ r l).pow_const m)
+    (fun θ=>(subdiskMass_raw_measurable r a ha μ θ l (hμraw l θ)).pow_const m)]
+  apply lintegral_mono_ae
+  have hb : ∀ᵐ p ∂(angleLaw a).prod W,Monotone (fun v=>μ v p.2) :=
+    Measure.quasiMeasurePreserving_snd.tendsto_ae.eventually hmono
+  filter_upwards [(finiteAngleLaw_absolutelyContinuous W a k).ae_le hb] with p hp
+  exact pow_le_pow_left' (subdiskMass_monotone r a ha μ p.2 hp p.1 hlk) m
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+namespace SLEExactGauge.PastMass
+open ExactSLEGenerator ExactSLEMain ExactSLEGreen
+
+ 
+
+def pastGreenKernel (W : Measure ExactSLEGenerator.Driver) (a : ℝ)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (z : UpperPlane) : ℝ≥0∞ :=
+  (∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W) /
+    ENNReal.ofReal (initialGreen a z)
+
+lemma pastGreenKernel_measurable (W : Measure ExactSLEGenerator.Driver) [SFinite W]
+    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) : Measurable (pastGreenKernel W a S) :=
+  ((stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal.lintegral_prod_left').div
+    (initialGreen_continuous a).measurable.ennreal_ofReal
+
+lemma initial_mul_pastGreenKernel (W : Measure ExactSLEGenerator.Driver)
+    (a : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) (z : UpperPlane) :
+    ENNReal.ofReal (initialGreen a z) * pastGreenKernel W a S z =
+      ∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W :=
+  ENNReal.mul_div_cancel (ENNReal.ofReal_pos.mpr (initialGreen_pos a z)).ne'
+    ENNReal.ofReal_ne_top
+
+lemma pastGreenKernel_eq_ofReal (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane) :
+    pastGreenKernel W a S z = ENNReal.ofReal
+      ((∫b, stoppedSurvivingGreen a (S b) b z.1 ∂W) / initialGreen a z) := by
+  rw [ENNReal.ofReal_div_of_pos (initialGreen_pos a z),
+    ofReal_integral_eq_lintegral_ofReal (survivingGreen_optional_bound a ha W hW z S hS).1
+      (Eventually.of_forall (fun b => stoppedSurvivingGreen_nonneg a (S b) b z.1))]
+  rfl
+
+lemma pastGreenKernel_le_one (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane) :
+    pastGreenKernel W a S z ≤ 1 := by
+  rw [pastGreenKernel_eq_ofReal W hW a ha S hS]
+  exact (ENNReal.ofReal_le_ofReal ((div_le_one (initialGreen_pos a z)).mpr
+    (survivingGreen_optional_bound a ha W hW z S hS).2)).trans_eq (by simp)
+
+lemma pastGreenKernel_antitone (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (S T : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (hT : IsStoppingTime rawFiltration T)
+    (hst : S ≤ T) (z : UpperPlane) :
+    pastGreenKernel W a T z ≤ pastGreenKernel W a S z := by
+  rw [pastGreenKernel_eq_ofReal W hW a ha T hT, pastGreenKernel_eq_ofReal W hW a ha S hS]
+  apply ENNReal.ofReal_le_ofReal
+  apply div_le_div_of_nonneg_right _ (initialGreen_pos a z).le
+  simpa only [setIntegral_univ] using
+    survivingGreen_optional_comparison a ha W hW z S T hS hT hst univ MeasurableSet.univ
+
+ 
+
+lemma radialLaw_strict_past (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) :
+    (radialLaw W a z.1 (z.1.im*q)) {b | S b < radialTime a b.1 z.1 (z.1.im*q)} =
+      ENNReal.ofReal ((∫b, stoppedRemainingDensity a q (S b) b z.1 ∂W) / initialGreen a z) := by
+  have hρ : 0<z.1.im*q := mul_pos z.2 hq
+  have hr : z.1.im*q≤z.1.im := mul_le_of_le_one_right z.2.le hq1
+  have hT := radialTime_stopping a (by linarith) z.1 z.2 _ hρ hr
+  have hmS : Measurable S := hS.measurable'
+  have hmT : Measurable (fun b : ExactSLEGenerator.Driver => radialTime a b.1 z.1 (z.1.im*q)) :=
+    hT.measurable'
+  have hE : MeasurableSet {b : ExactSLEGenerator.Driver | S b < radialTime a b.1 z.1 (z.1.im*q)} :=
+    measurableSet_lt hmS hmT
+  rw [radialLaw_apply W hW a ha z.1 z.2 _ hρ hr _ hE]
+  have hp := (radial_conditional_weight_all W hW a ha z.1 z.2 _ hρ hr S hS
+    univ MeasurableSet.univ).2
+  simp only [setIntegral_univ] at hp
+  rw [← integral_indicator hE]
+  simp only [indicator_apply, mem_ofPred_eq]
+  rw [hp]
+  congr 1
+  simp only [stoppedRemainingDensity_eq a q _ _ _ z.2, integral_const_mul, initialGreen]
+  exact (mul_div_mul_left _ _ (Real.exp_ne_zero _)).symm
+
+ 
+
+theorem radialLaw_strict_past_tendsto (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4<a) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) (z : UpperPlane)
+    (q : ℕ → ℝ) (hq : ∀n, 0<q n ∧ q n≤1) (hq0 : Tendsto q atTop (𝓝 0)) :
+    Tendsto (fun n => (radialLaw W a z.1 (z.1.im*q n))
+      {b | S b < radialTime a b.1 z.1 (z.1.im*q n)}) atTop
+      (𝓝 (pastGreenKernel W a S z)) := by
+  simp_rw [radialLaw_strict_past W hW a ha S hS z _ (hq _).1 (hq _).2]
+  rw [pastGreenKernel_eq_ofReal W hW a ha S hS]
+  have he := stopped_remaining_integral_tendsto a ha W hW z S hS q hq hq0 univ
+  simp only [setIntegral_univ] at he
+  exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (he.div_const _)
+
+end SLEExactGauge.PastMass
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Classical NNReal ENNReal
+namespace SLEExactGauge.PastMass
+
+variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
+
+ 
+def finiteTestTime (F : Finset ℝ≥0) (M : ℝ≥0 → Ω → ℝ≥0∞)
+    (H : ℝ≥0∞) (ω : Ω) : ℝ≥0∞ :=
+  F.inf (fun q => if H < M q ω then (q : ℝ≥0∞) else ∞)
+
+omit mΩ in
+lemma finiteTestTime_lt_iff [MeasurableSpace Ω] (F : Finset ℝ≥0) (M : ℝ≥0 → Ω → ℝ≥0∞)
+    (H : ℝ≥0∞) (ω : Ω) (v : ℝ≥0∞) :
+    finiteTestTime F M H ω < v ↔ ∃ q ∈ F, H < M q ω ∧ (q : ℝ≥0∞) < v := by
+  rw [finiteTestTime, Finset.inf_lt_iff]
+  congr! 2 with q
+  by_cases h : H < M q ω <;> simp [h]
+
+omit mΩ in
+lemma finiteTestTime_le_iff [MeasurableSpace Ω] (F : Finset ℝ≥0) (M : ℝ≥0 → Ω → ℝ≥0∞)
+    (H : ℝ≥0∞) (ω : Ω) (t : ℝ≥0) :
+    finiteTestTime F M H ω ≤ (t : ℝ≥0∞) ↔ ∃ q ∈ F, q ≤ t ∧ H < M q ω := by
+  rw [finiteTestTime, Finset.inf_le_iff ENNReal.coe_lt_top]
+  congr! 2 with q
+  by_cases h : H < M q ω <;> simp [h]
+
+lemma finiteTestTime_stopping (ℱ : Filtration ℝ≥0 mΩ)
+    (M : ℝ≥0 → Ω → ℝ≥0∞) (hM : Adapted ℱ M)
+    (F : Finset ℝ≥0) (H : ℝ≥0∞) :
+    IsStoppingTime ℱ (finiteTestTime F M H) := by
+  intro t
+  change MeasurableSet[ℱ t] {ω | finiteTestTime F M H ω ≤ (t : ℝ≥0∞)}
+  have he : {ω | finiteTestTime F M H ω ≤ (t : ℝ≥0∞)} =
+      ⋃ q ∈ F, ⋃ (_ : q ≤ t), {ω | H < M q ω} := by
+    ext ω
+    simp only [mem_ofPred_eq, finiteTestTime_le_iff, mem_iUnion, exists_prop]
+  rw [he]
+  apply MeasurableSet.biUnion F.countable_toSet
+  intro q hq
+  apply MeasurableSet.iUnion
+  intro hqt
+  exact ℱ.mono hqt _ (measurableSet_lt measurable_const (hM q))
+
+omit mΩ in
+lemma finiteTestTime_antitone [MeasurableSpace Ω] (M : ℝ≥0 → Ω → ℝ≥0∞) (H : ℝ≥0∞)
+    (ω : Ω) : Antitone (fun F : Finset ℝ≥0 => finiteTestTime F M H ω) := by
+  intro F G hFG
+  exact Finset.inf_mono hFG
+
+ 
+
+lemma finiteTestTime_iUnion {ι : Type*} (e : ℕ → ι) (q : ι → ℝ≥0)
+    (he : Function.Surjective e) (M : ℝ≥0 → Ω → ℝ≥0∞) (H : ℝ≥0∞)
+    (ω : Ω) (v : ℝ≥0∞) :
+    (∃ i, H < M (q i) ω ∧ (q i : ℝ≥0∞) < v) ↔
+      ∃ n : ℕ, finiteTestTime ((Finset.range n).image (q ∘ e)) M H ω < v := by
+  simp only [finiteTestTime_lt_iff, Finset.mem_image, Finset.mem_range,
+    Function.comp_apply]
+  constructor
+  · rintro ⟨i,hi⟩
+    obtain ⟨k,rfl⟩ := he i
+    exact ⟨k+1,q (e k),⟨k,Nat.lt_succ_self k,rfl⟩,hi⟩
+  · rintro ⟨n,t,⟨k,hk,rfl⟩,ht⟩
+    exact ⟨e k,ht⟩
+
+lemma finiteTestTime_sequence_antitone {ι : Type*} (e : ℕ → ι) (q : ι → ℝ≥0)
+    (M : ℝ≥0 → Ω → ℝ≥0∞) (H : ℝ≥0∞) (ω : Ω) :
+    Antitone (fun n => finiteTestTime ((Finset.range n).image (q ∘ e)) M H ω) := by
+  intro n k hnk
+  exact finiteTestTime_antitone M H ω (Finset.image_subset_image (Finset.range_mono hnk))
+
+end SLEExactGauge.PastMass
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+namespace SLEExactGauge.PastMass
+open FlowIdentification ExactSLEGreen TraceMassContinuity TraceRestriction
+
+ 
+def visitTime (κ : ℝ) (b : GDriver) : UpperPlane → ℝ≥0∞ :=
+  firstVisit (fun z : UpperPlane => z.1) (internalTrace κ (driverHomeomorph.symm b))
+
+lemma stopped_restriction_add (κ : ℝ) (b : GDriver)
+    (μ : ℝ≥0∞ → Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (s : ℝ≥0∞) :
+    μ s Q + μ ∞ (Q ∩ {z | s < visitTime κ b z}) = μ ∞ Q := by
+  induction s using ENNReal.recTopCoe with
+  | top => simp
+  | coe s =>
+    rw [hμ.firstVisit_identity s, Measure.restrict_apply hQ]
+    have he : Q ∩ {z | (s : ℝ≥0∞) < visitTime κ b z} =
+        Q \ {z | visitTime κ b z ≤ (s : ℝ≥0∞)} := by
+      ext z
+      simp only [mem_inter_iff,Set.mem_sdiff,mem_ofPred_eq,not_le]
+    rw [he]
+    exact measure_inter_add_sdiff Q (measurableSet_le hμ.firstVisit_measurable measurable_const)
+
+lemma stopped_strictPast_mass (κ : ℝ) (b : GDriver)
+    (μ : ℝ≥0∞ → Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q ≠ ∞) (s : ℝ≥0∞) :
+    μ ∞ (Q ∩ {z | s < visitTime κ b z}) = μ ∞ Q - μ s Q := by
+  have hsq : μ s Q ≠ ∞ := ne_top_of_le_ne_top hfin (hμ.monotone le_top Q)
+  rw [← stopped_restriction_add κ b μ hμ Q hQ s, ENNReal.add_sub_cancel_left hsq]
+
+lemma finite_past_mass (κ : ℝ) (b : GDriver)
+    (μ : ℝ≥0∞ → Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (hfin : μ ∞ Q ≠ ∞)
+    (F : Finset ℝ≥0) (H : ℝ≥0∞) :
+    μ ∞ (Q ∩ {z | ∃q ∈ F, H < μ q Q ∧ (q : ℝ≥0∞) < visitTime κ b z}) =
+      μ ∞ Q - μ (finiteTestTime F (fun q (_ : Unit) => μ q Q) H ()) Q := by
+  simpa only [finiteTestTime_lt_iff] using
+    stopped_strictPast_mass κ b μ hμ Q hQ hfin
+      (finiteTestTime F (fun q (_ : Unit) => μ q Q) H ())
+
+ 
+
+lemma all_past_mass_eq_iSup {ι : Type*} (e : ℕ → ι) (q : ι → ℝ≥0)
+    (he : Function.Surjective e) (κ : ℝ) (b : GDriver)
+    (μ : ℝ≥0∞ → Measure UpperPlane)
+    (hμ : TraceMassFacts κ (driverHomeomorph.symm b) μ)
+    (Q : Set UpperPlane) (hQ : MeasurableSet Q) (H : ℝ≥0∞) :
+    μ ∞ (Q ∩ {z | ∃i, H < μ (q i) Q ∧ (q i : ℝ≥0∞) < visitTime κ b z}) =
+      ⨆n : ℕ, μ ∞ (Q ∩ {z |
+        finiteTestTime ((Finset.range n).image (q ∘ e))
+          (fun q (_ : Unit) => μ q Q) H () < visitTime κ b z}) := by
+  let A (n : ℕ) : Set UpperPlane := Q ∩ {z |
+    finiteTestTime ((Finset.range n).image (q ∘ e))
+      (fun q (_ : Unit) => μ q Q) H () < visitTime κ b z}
+  have hA (n : ℕ) : MeasurableSet (A n) := hQ.inter
+    (measurableSet_lt measurable_const hμ.firstVisit_measurable)
+  have hAm : Monotone A := by
+    intro n k hnk z hz
+    exact ⟨hz.1,(finiteTestTime_sequence_antitone e q (fun q (_ : Unit) => μ q Q)
+      H () hnk).trans_lt hz.2⟩
+  have heq : Q ∩ {z | ∃i, H < μ (q i) Q ∧ (q i : ℝ≥0∞) < visitTime κ b z} =
+      ⋃n : ℕ, A n := by
+    ext z
+    simp only [mem_inter_iff,mem_ofPred_eq,mem_iUnion,A,
+      finiteTestTime_iUnion e q he (fun q (_ : Unit) => μ q Q) H ()]
+    exact exists_and_left.symm
+  rw [heq, hAm.measure_iUnion]
+
+end SLEExactGauge.PastMass
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/SurvivalLocalization.lean b/lean/OAI/Probability/SLEGauge/SurvivalLocalization.lean
index 29fe919ffc416ace6e8f24feb911cf748198bfe3..d86331b35ea3ad609782da30499eee21e21c0a26 100644
--- a/lean/OAI/Probability/SLEGauge/SurvivalLocalization.lean
+++ b/lean/OAI/Probability/SLEGauge/SurvivalLocalization.lean
@@ -1,448 +1,448 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadonMoments
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-open MeasureTheory Set
-open MeasureTheory ProbabilityTheory
-open scoped NNReal ENNReal
-open MeasureTheory
-open scoped Classical
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open scoped ENNReal
-open Filter Topology
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Metric Function
-open scoped NNReal ENNReal Topology
-open MeasureTheory Filter Topology Set CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-lemma localizationStop_after_survival (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (_hr : ρ ≤ z.im)
-    (L : ∀ n, RadialLocalization a z ρ (angleThreshold z n)) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (ht : alive a b.1 t z) (htr : (t : ℝ≥0∞) < radialTime a b.1 z ρ) :
-    ∀ᶠ n in atTop, t < regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b := by
-  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
-  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
-  let Ln := L n
-  let T := regionStop a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n) Ln.B b
-  let V := localState a Ln.ε Ln.hε z T b
-  by_contra hnt
-  have hTt : T ≤ t := le_of_not_gt hnt
-  have he := regionStop_actualState a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n) Ln.B Ln.initial b T le_rfl
-  rcases regionStop_true_exit a Ln.ε ha Ln.hε z Ln.N ρ (angleThreshold z n) hρ
-    (angleThreshold_pos z hz n) Ln.B Ln.initial Ln.small Ln.large Ln.time b with hR|hS
-  · have htime := regionStop_eq_radialTime a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n)
-      Ln.B Ln.initial b hR
-    have hle : radialTime a b.1 z ρ ≤ (t : ℝ≥0∞) := by
-      rw [htime]
-      exact ENNReal.coe_le_coe.mpr hTt
-    exact not_le.mpr htr hle
-  · have hs := regionStop_safe a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n) Ln.B Ln.initial b T le_rfl
-    have hnp : 0 < ‖V.1‖ := (Ln.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
-    have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-    have hη' := hηs T hTt
-    rw [he.2,hd] at hη'
-    exact not_le.mpr hn hη'
-
-lemma localizationRemaining_tendsto (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im)
-    (L : ∀ n, RadialLocalization a z ρ (angleThreshold z n))
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    Tendsto (fun n => if t < regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
-      then physicalGreen a (localState a (L n).ε (L n).hε z t b) else 0)
-      atTop (𝓝 (radialRemaining a b.1 z ρ t)) := by
-  by_cases h : alive a b.1 t z ∧ (t : ℝ≥0∞)<radialTime a b.1 z ρ
-  · apply tendsto_const_nhds.congr'
-    filter_upwards [localizationStop_after_survival a ha z hz ρ hρ hr L b t h.1 h.2] with n hn
-    rw [ite_eq_left hn,radialRemaining,ite_eq_left h]
-    exact congrArg (physicalGreen a) (regionStop_actualState a (L n).ε (L n).hε z
-      (L n).N ρ (angleThreshold z n) (L n).B (L n).initial b t hn.le).2
-  · have hv (n : ℕ) : ¬ t < regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b := by
-      intro hn
-      apply h
-      refine ⟨(regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ
-        (angleThreshold z n) (L n).B (L n).initial b t hn.le).1, ?_⟩
-      apply lt_of_lt_of_le (ENNReal.coe_lt_coe.mpr hn)
-      apply le_sInf
-      rintro u ⟨v,rfl,_,hv⟩
-      exact ENNReal.coe_le_coe.mpr (regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ
-        (angleThreshold z n) (L n).B (L n).initial b v hv)
-    simpa only [ite_eq_right (hv _),radialRemaining,ite_eq_right h] using
-      (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped NNReal ENNReal Topology
-
-theorem joint_radialRemaining_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
-    (hq : 0 < q) (hq1 : q ≤ 1) (t : ℝ≥0) :
-    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im} =>
-      radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t) := by
-  have ha0 : 0<a := by linarith
-  let Ω := ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}
-  let z : Ω → ℂ := fun p => p.2.1
-  let b : Ω → ExactSLEGenerator.Driver := Prod.fst
-  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
-  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
-  have hbM : Measurable b := measurable_fst
-  let ρ : Ω → ℝ := fun ω => (z ω).im*q
-  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
-  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
-  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
-  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
-    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
-  have hρM : Measurable ρ := hzM.im.mul_const q
-  have hδM (n : ℕ) : Measurable (δ n) := by
-    dsimp [δ,angleThreshold]
-    exact (hzM.im.div hzM.norm).div_const _
-  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
-    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
-    unfold heightFloor
-    fun_prop
-  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
-    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    unfold logCeiling heightFloor
-    fun_prop
-  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
-    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
-    apply Measurable.subtype_mk
-    fun_prop
-  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
-    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
-  have hTM (n : ℕ) : Measurable (T n) :=
-    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
-      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
-  have hPM : Measurable (physicalGreen a) := by
-    have hx := X.continuous.measurable
-    have hy := Y.continuous.measurable
-    have hj := J.continuous.measurable
-    unfold physicalGreen logGreen Q
-    fun_prop
-  let F (n : ℕ) (ω : Ω) := if t < T n ω then physicalGreen a (localState a (L n ω).ε (L n ω).hε (z ω) t (b ω)) else 0
-  have hFM (n : ℕ) : Measurable (F n) := (hPM.comp
-    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b (hεM n) hzM hbM).comp
-      (measurable_const.prodMk measurable_id))).ite (measurableSet_lt measurable_const (hTM n)) measurable_const
-  apply measurable_of_tendsto_metrizable hFM
-  apply tendsto_pi_nhds.mpr
-  intro ω
-  exact localizationRemaining_tendsto a ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω)
-    (fun n => L n ω) (b ω) t
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-lemma radialRemaining_driver_congr (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
-    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b c : ExactSLEGenerator.Driver) (t : ℝ≥0)
-    (h : ∀ s≤t,b.1 s=c.1 s) : radialRemaining a b.1 z ρ t = radialRemaining a c.1 z ρ t := by
-  have hiff : radialTime a b.1 z ρ≤t ↔ radialTime a c.1 z ρ≤t := by
-    constructor
-    · intro hb
-      rwa [← radialTime_driver_congr_before a ha z hz ρ hρ hr b c t h hb]
-    · intro hc
-      rwa [← radialTime_driver_congr_before a ha z hz ρ hρ hr c b t (fun s hs => (h s hs).symm) hc]
-  have hh : (t : ℝ≥0∞)<radialTime a b.1 z ρ ↔ (t : ℝ≥0∞)<radialTime a c.1 z ρ := by
-    simpa only [not_le] using hiff.not
-  simp only [radialRemaining,alive_driver_congr h,hh,actualState_driver_congr h]
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory ExactSLEGenerator ExactSLEMain
-open scoped NNReal ENNReal
- 
-
-theorem radialRemaining_raw_joint (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
-    (hq1 : q≤1) (t : ℝ≥0) :
-    letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
-    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0<z.im} =>
-      radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t)
-    := by
-  have he : (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0<z.im} =>
-      radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t) =
-    (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0<z.im} =>
-      radialRemaining a (pastDriver t p.1).1 p.2.1 (p.2.1.im*q) t) := by
-    funext p
-    exact radialRemaining_driver_congr a (by linarith) p.2.1 p.2.2 _ (mul_pos p.2.2 hq)
-      (mul_le_of_le_one_right p.2.2.le hq1) p.1 (pastDriver t p.1) t
-      (fun s hs => (pastDriver_eq t p.1 s hs).symm)
-  change @Measurable (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im}) ℝ
-    ((rawFiltration t).prod inferInstance) inferInstance _
-  rw [he]
-  have hp : @Measurable (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im})
-      (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im})
-      ((rawFiltration t).prod inferInstance) inferInstance (fun p => (pastDriver t p.1,p.2)) :=
-    ((pastDriver_raw_measurable t).comp
-      (@measurable_fst _ _ (rawFiltration t) inferInstance)).prodMk
-      (@measurable_snd _ _ (rawFiltration t) inferInstance)
-  exact @Measurable.comp (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im})
-    (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im}) ℝ
-    ((rawFiltration t).prod inferInstance) inferInstance inferInstance
-    (fun p => radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t)
-    (fun p => (pastDriver t p.1,p.2))
-    (joint_radialRemaining_measurable a ha q hq hq1 t) hp
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory ExactSLEGenerator ExactSLEMain
-open scoped NNReal ENNReal
-noncomputable def remainingMassDensity (a q : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
-  if 0<z.im then Real.exp (-alpha a*Real.log 2)*radialRemaining a b.1 z (z.im*q) t else 0
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-lemma remainingMassDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
-    (hq1 : q≤1) (t : ℝ≥0) :
-    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => remainingMassDensity a q t p.1 p.2) := by
-  let U : Set (ExactSLEGenerator.Driver × ℂ) := {p | 0<p.2.im}
-  have hU : MeasurableSet U := measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
-  have hm : Measurable (fun p : U =>
-      Real.exp (-alpha a*Real.log 2)*radialRemaining a p.1.1.1 p.1.2 (p.1.2.im*q) t) := by
-    have hp : Measurable (fun p : U => (p.1.1, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im}))) := by fun_prop
-    exact measurable_const.mul ((joint_radialRemaining_measurable a ha q hq hq1 t).comp hp)
-  exact hm.dite (g:=fun _ => 0) measurable_const hU
-lemma remainingMassDensity_local_bound (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
-    (hq1 : q≤1) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (c : ℝ) (hc : 0<c)
-    (z : ℂ) (hz : c≤2*z.im) : ‖remainingMassDensity a q t b z‖≤(c*q)^(-alpha a) := by
-  have hz0 : 0<z.im := by linarith
-  have hb := radialRemaining_bound a ha z hz0 (z.im*q) (mul_pos hz0 hq)
-    (mul_le_of_le_one_right hz0.le hq1) b t
-  have ht : 0<2*z.im*q := by positivity
-  calc
-    _ = Real.exp (-alpha a*Real.log 2)*‖radialRemaining a b.1 z (z.im*q) t‖ := by
-      simp only [remainingMassDensity,ite_eq_left hz0,norm_mul,Real.norm_of_nonneg (Real.exp_pos _).le]
-    _ ≤ Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log (z.im*q)) :=
-      mul_le_mul_of_nonneg_left hb (Real.exp_pos _).le
-    _ = (2*z.im*q)^(-alpha a) := by
-      rw [←Real.exp_add,Real.rpow_def_of_pos ht]
-      rw [show 2*z.im*q=2*(z.im*q) by ring,Real.log_mul (by norm_num) (mul_pos hz0 hq).ne']
-      congr 1; ring
-    _ ≤ _ := Real.rpow_le_rpow_of_nonpos (mul_pos hc hq)
-      (mul_le_mul_of_nonneg_right hz hq.le) (neg_nonpos.mpr (alpha_pos ha).le)
-lemma remainingMassEval_joint_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsFiniteMeasure W]
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B) :
-    Integrable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      f p.2 * remainingMassDensity a q t p.1 p.2) (W.prod (volume.restrict K)) := by
-  obtain ⟨B,hB,hBf⟩ := hfB
-  have : IsFiniteMeasure (volume.restrict K) := ⟨by simpa using hKb.measure_lt_top⟩
-  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      f p.2 * remainingMassDensity a q t p.1 p.2) :=
-    (hf.comp measurable_snd).mul (remainingMassDensity_measurable a ha q hq hq1 t)
-  apply Integrable.of_bound hm.aestronglyMeasurable (B*((c*q)^(-alpha a)))
-  have hmem : ∀ᵐ p ∂W.prod (volume.restrict K), p.2∈K :=
-    (Measure.ae_prod_iff_ae_ae (hK.preimage measurable_snd)).mpr
-      (Eventually.of_forall fun _ => ae_restrict_mem hK)
-  filter_upwards [hmem] with p hp
-  rw [norm_mul]
-  exact mul_le_mul (hBf _ hp) (remainingMassDensity_local_bound a ha q hq hq1 t p.1 c hc p.2 (hcK _ hp))
-    (norm_nonneg _) hB
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open MeasureTheory ProbabilityTheory Filter Set
-open scoped Classical NNReal ENNReal Topology
-lemma finite_density_conditional (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0) (z : ℂ) (hz : 0<z.im)
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[rawFiltration t] U) :
-    (∫b in U, (timeMassDensity a q ∞ b z).toReal-(timeMassDensity a q t b z).toReal ∂W) =
-      ∫b in U, remainingMassDensity a q t b z ∂W := by
-  have he (b : ExactSLEGenerator.Driver) :
-      (timeMassDensity a q ∞ b z).toReal-(timeMassDensity a q t b z).toReal =
-        Real.exp (-alpha a*Real.log 2)*(if (t : ℝ≥0∞)<radialTime a b.1 z (z.im*q)
-          then radialWeight a b.1 z (z.im*q) else 0) := by
-    have hn := (fullRadialDensity_bound a ha z hz (2*z.im*q) (by positivity) (by nlinarith) b).1
-    have hρ : 2*z.im*q/2=z.im*q := by ring
-    by_cases hh : radialTime a b.1 z (z.im*q)≤t
-    · simp only [timeMassDensity,le_top,ite_true,ite_eq_left hh,massDensity,ite_eq_left hz,
-        ENNReal.toReal_ofReal hn,sub_self,ite_eq_right (not_lt.mpr hh),mul_zero]
-    · simp only [timeMassDensity,le_top,ite_true,ite_eq_right hh,massDensity,ite_eq_left hz,
-        ENNReal.toReal_zero,sub_zero,ite_eq_left (lt_of_not_ge hh),fullRadialDensity,hρ]
-      exact ENNReal.toReal_ofReal (by simpa only [fullRadialDensity,hρ] using hn)
-  simp_rw [he,remainingMassDensity,ite_eq_left hz]
-  rw [integral_const_mul,integral_const_mul]
-  congr 1
-  have hS : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (t : WithTop ℝ≥0)) :=
-    isStoppingTime_const _ _
-  exact (radial_conditional_weight W hW a ha z hz (z.im*q) (mul_pos hz hq)
-    (mul_le_of_le_one_right hz.le hq1) (fun _ => t) hS U
-    ((hS.le_measurableSpace_of_const_le (fun _ => le_rfl)) _ hU)).2
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory ExactSLEGenerator ExactSLEMain
-open scoped NNReal ENNReal
- 
-
-theorem actual_finite_conditional_mass (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0)
-    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
-    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
-    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B)
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[rawFiltration t] U) :
-    (∫b in U, actualMassEval a q ∞ K f b-actualMassEval a q t K f b ∂W) =
-      ∫b in U, (∫z in K, f z*remainingMassDensity a q t b z) ∂W
-    := by
-  have hIm (v : ℝ≥0∞) := actualMassEval_joint_integrable a ha W q hq hq1 v K hK hKb c hc hcK f hf hfB
-  have hI (v : ℝ≥0∞) : Integrable (fun p : ExactSLEGenerator.Driver × ℂ =>
-      f p.2*(timeMassDensity a q v p.1 p.2).toReal) ((W.restrict U).prod (volume.restrict K)) :=
-    (hIm v).mono_measure (Measure.prod_mono Measure.restrict_le_self le_rfl)
-  have hsub : Integrable (fun p : ExactSLEGenerator.Driver × ℂ => f p.2*
-      ((timeMassDensity a q ∞ p.1 p.2).toReal-(timeMassDensity a q t p.1 p.2).toReal))
-      ((W.restrict U).prod (volume.restrict K)) := by
-    convert (hI ∞).sub (hI t) using 1
-    first | rfl | (funext point; dsimp only [Pi.sub_apply]; ring)
-  have he (b : ExactSLEGenerator.Driver) : actualMassEval a q ∞ K f b-actualMassEval a q t K f b =
-      ∫z in K, f z*((timeMassDensity a q ∞ b z).toReal-(timeMassDensity a q t b z).toReal) := by
-    simp only [actualMassEval,mul_sub]
-    exact (integral_sub
-      (actualMassEval_integrable a ha q hq hq1 ∞ K hK hKb c hc hcK f hf hfB b)
-      (actualMassEval_integrable a ha q hq hq1 t K hK hKb c hc hcK f hf hfB b)).symm
-  simp_rw [he]
-  rw [integral_integral_swap hsub]
-  conv_rhs => rw [integral_integral_swap
-    (remainingMassEval_joint_integrable a ha (W.restrict U) q hq hq1 t K hK hKb c hc hcK f hf hfB)]
-  apply setIntegral_congr_fun hK
-  intro z hzK
-  dsimp only
-  rw [integral_const_mul,integral_const_mul]
-  congr 1
-  exact finite_density_conditional a ha W hW q hq hq1 t z (by have := hcK z hzK; linarith) U hU
-end ExactSLEGreen
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open scoped Classical NNReal ENNReal
- 
-noncomputable def survivingGreen (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
-  if alive a b.1 t z then Real.exp (-alpha a*Real.log 2)*physicalGreen a (actualState a b.1 t z) else 0
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
-open scoped NNReal ENNReal Topology
-lemma physicalRadius_actual_pos (a : ℝ) (b : ExactSLEGenerator.Driver) (z : ℂ)
-    (t : ℝ≥0) (ht : alive a b.1 t z) : 0<physicalRadius (actualState a b.1 t z) := by
-  obtain ⟨f,hf⟩ := ht
-  have h := fullRadius_pos b.1.continuous hf le_rfl
-  rw [←actual_radius_eq_full b.1.continuous ⟨f,hf⟩] at h
-  linarith
-lemma physicalRadius_actual_anti (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver) (z : ℂ)
-    (s t : ℝ≥0) (hst : s≤t) (ht : alive a b.1 t z) :
-    physicalRadius (actualState a b.1 t z)≤physicalRadius (actualState a b.1 s z) := by
-  obtain ⟨f,hf⟩ := ht
-  have he := fullRadius_active_clock b.1.continuous hf hst le_rfl
-  have hp := fullRadius_pos b.1.continuous hf hst
-  have hn := activeODEClock_nonneg (b:=b.1) (T:=t) (σ:=s) (f:=f) (show (s:ℝ)≤t from hst)
-  have hle : fullLoewnerRadius a b.1 z t≤fullLoewnerRadius a b.1 z s := by
-    rw [he]
-    exact mul_le_of_le_one_right hp.le (Real.exp_le_one_iff.mpr (by nlinarith))
-  rw [←actual_radius_eq_full b.1.continuous ⟨f,hf⟩,
-    ←actual_radius_eq_full b.1.continuous ⟨f,odeUntil_mono hf hst⟩] at hle
-  linarith
-lemma survivingGreen_nonneg (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    0 ≤ survivingGreen a t b z := by
-  unfold survivingGreen physicalGreen
-  split_ifs <;> positivity
-lemma remainingMassDensity_nonneg (a q : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) :
-    0≤remainingMassDensity a q t b z := by
-  unfold remainingMassDensity radialRemaining physicalGreen
-  split_ifs <;> positivity
-lemma remaining_to_surviving (a : ℝ) (ha : 1/4<a) (t : ℝ≥0)
-    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
-    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im) :
-    Tendsto (fun n => remainingMassDensity a (q n) t b z) atTop (𝓝 (survivingGreen a t b z)) := by
-  by_cases ht : alive a b.1 t z
-  · have hR := physicalRadius_actual_pos a b z t ht
-    have hqR : ∀ᶠ n in atTop, z.im*q n<physicalRadius (actualState a b.1 t z) := by
-      have hconv : Tendsto (fun n => z.im*q n) atTop (𝓝 0) := by simpa using hlim.const_mul z.im
-      exact hconv.eventually (gt_mem_nhds hR)
-    apply tendsto_const_nhds.congr'
-    filter_upwards [hqR] with n hn
-    have hnT : (t : ℝ≥0∞)<radialTime a b.1 z (z.im*q n) := by
-      by_contra h
-      have hTt := le_of_not_gt h
-      have hfin : radialTime a b.1 z (z.im*q n)≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hTt
-      have hle : (radialTime a b.1 z (z.im*q n)).toNNReal≤t := by
-        exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hTt)
-      have hh := (radialTime_attained a (by linarith) z hz _ (mul_pos hz (hq n).1)
-        (mul_le_of_le_one_right hz.le (hq n).2) b hfin).2
-      have he := physicalRadius_actual_anti a (by linarith) b z _ t hle ht
-      rw [hh] at he
-      exact not_le.mpr hn he
-    simp only [remainingMassDensity,ite_eq_left hz,radialRemaining,ht,hnT,and_self,ite_true,survivingGreen]
-  · simpa only [remainingMassDensity,ite_eq_left hz,radialRemaining,ht,false_and,ite_false,survivingGreen,
-      mul_zero] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
-open scoped Classical NNReal ENNReal Topology
-lemma remainingMassDensity_mean_bound (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0) (z : ℂ) (hz : 0<z.im) :
-    Integrable (fun b => remainingMassDensity a q t b z) W ∧
-      (∫b,remainingMassDensity a q t b z∂W)≤Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0) := by
-  have hS : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (t : WithTop ℝ≥0)) :=
-    isStoppingTime_const _ _
-  have H := radial_conditional_weight W hW a ha z hz (z.im*q) (mul_pos hz hq)
-    (mul_le_of_le_one_right hz.le hq1) (fun _ => t) hS univ MeasurableSet.univ
-  have hmean := radialWeight_mean W hW a ha z hz (z.im*q) (mul_pos hz hq)
-    (mul_le_of_le_one_right hz.le hq1)
-  constructor
-  · simpa only [remainingMassDensity,ite_eq_left hz] using H.1.const_mul (Real.exp (-alpha a*Real.log 2))
-  · simp only [remainingMassDensity,ite_eq_left hz]
-    have He := H.2
-    simp only [setIntegral_univ] at He
-    rw [integral_const_mul,←He]
-    apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
-    rw [←hmean.2]
-    apply integral_mono_of_nonneg
-    · filter_upwards [] with b
-      split_ifs
-      · exact (radialWeight_bound a ha z hz _ (mul_pos hz hq)
-          (mul_le_of_le_one_right hz.le hq1) b).1
-      · exact le_rfl
-    · exact hmean.1
-    · filter_upwards [] with b
-      split_ifs
-      · exact le_rfl
-      · exact (radialWeight_bound a ha z hz _ (mul_pos hz hq)
-          (mul_le_of_le_one_right hz.le hq1) b).1
-end ExactSLEGreen
-namespace ExactSLEGreen
-open MeasureTheory ProbabilityTheory Filter ExactSLEGenerator ExactSLEMain
-open scoped Classical NNReal ENNReal Topology
- 
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadonMoments
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+open MeasureTheory Set
+open MeasureTheory ProbabilityTheory
+open scoped NNReal ENNReal
+open MeasureTheory
+open scoped Classical
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open scoped ENNReal
+open Filter Topology
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Metric Function
+open scoped NNReal ENNReal Topology
+open MeasureTheory Filter Topology Set CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+lemma localizationStop_after_survival (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (_hr : ρ ≤ z.im)
+    (L : ∀ n, RadialLocalization a z ρ (angleThreshold z n)) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (ht : alive a b.1 t z) (htr : (t : ℝ≥0∞) < radialTime a b.1 z ρ) :
+    ∀ᶠ n in atTop, t < regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b := by
+  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
+  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
+  let Ln := L n
+  let T := regionStop a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n) Ln.B b
+  let V := localState a Ln.ε Ln.hε z T b
+  by_contra hnt
+  have hTt : T ≤ t := le_of_not_gt hnt
+  have he := regionStop_actualState a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n) Ln.B Ln.initial b T le_rfl
+  rcases regionStop_true_exit a Ln.ε ha Ln.hε z Ln.N ρ (angleThreshold z n) hρ
+    (angleThreshold_pos z hz n) Ln.B Ln.initial Ln.small Ln.large Ln.time b with hR|hS
+  · have htime := regionStop_eq_radialTime a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n)
+      Ln.B Ln.initial b hR
+    have hle : radialTime a b.1 z ρ ≤ (t : ℝ≥0∞) := by
+      rw [htime]
+      exact ENNReal.coe_le_coe.mpr hTt
+    exact not_le.mpr htr hle
+  · have hs := regionStop_safe a Ln.ε Ln.hε z Ln.N ρ (angleThreshold z n) Ln.B Ln.initial b T le_rfl
+    have hnp : 0 < ‖V.1‖ := (Ln.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
+    have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+    have hη' := hηs T hTt
+    rw [he.2,hd] at hη'
+    exact not_le.mpr hn hη'
+
+lemma localizationRemaining_tendsto (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im)
+    (L : ∀ n, RadialLocalization a z ρ (angleThreshold z n))
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    Tendsto (fun n => if t < regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b
+      then physicalGreen a (localState a (L n).ε (L n).hε z t b) else 0)
+      atTop (𝓝 (radialRemaining a b.1 z ρ t)) := by
+  by_cases h : alive a b.1 t z ∧ (t : ℝ≥0∞)<radialTime a b.1 z ρ
+  · apply tendsto_const_nhds.congr'
+    filter_upwards [localizationStop_after_survival a ha z hz ρ hρ hr L b t h.1 h.2] with n hn
+    rw [ite_eq_left hn,radialRemaining,ite_eq_left h]
+    exact congrArg (physicalGreen a) (regionStop_actualState a (L n).ε (L n).hε z
+      (L n).N ρ (angleThreshold z n) (L n).B (L n).initial b t hn.le).2
+  · have hv (n : ℕ) : ¬ t < regionStop a (L n).ε (L n).hε z (L n).N ρ (angleThreshold z n) (L n).B b := by
+      intro hn
+      apply h
+      refine ⟨(regionStop_actualState a (L n).ε (L n).hε z (L n).N ρ
+        (angleThreshold z n) (L n).B (L n).initial b t hn.le).1, ?_⟩
+      apply lt_of_lt_of_le (ENNReal.coe_lt_coe.mpr hn)
+      apply le_sInf
+      rintro u ⟨v,rfl,_,hv⟩
+      exact ENNReal.coe_le_coe.mpr (regionStop_le_hit a (L n).ε (L n).hε z (L n).N ρ
+        (angleThreshold z n) (L n).B (L n).initial b v hv)
+    simpa only [ite_eq_right (hv _),radialRemaining,ite_eq_right h] using
+      (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped NNReal ENNReal Topology
+
+theorem joint_radialRemaining_measurable (a : ℝ) (ha : 1/4 < a) (q : ℝ)
+    (hq : 0 < q) (hq1 : q ≤ 1) (t : ℝ≥0) :
+    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im} =>
+      radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t) := by
+  have ha0 : 0<a := by linarith
+  let Ω := ExactSLEGenerator.Driver × {z : ℂ // 0 < z.im}
+  let z : Ω → ℂ := fun p => p.2.1
+  let b : Ω → ExactSLEGenerator.Driver := Prod.fst
+  have hz (ω : Ω) : 0<(z ω).im := ω.2.2
+  have hzM : Measurable z := measurable_subtype_coe.comp measurable_snd
+  have hbM : Measurable b := measurable_fst
+  let ρ : Ω → ℝ := fun ω => (z ω).im*q
+  have hρ (ω : Ω) : 0<ρ ω := mul_pos (hz ω) hq
+  have hr (ω : Ω) : ρ ω≤(z ω).im := mul_le_of_le_one_right (hz ω).le hq1
+  let δ : ℕ → Ω → ℝ := fun n ω => angleThreshold (z ω) n
+  let L (n : ℕ) (ω : Ω) := explicitRadialLocalization a ha0 (z ω) (hz ω) (ρ ω) (δ n ω)
+    (hρ ω) (hr ω) (angleThreshold_pos (z ω) (hz ω) n) (angleThreshold_initial (z ω) (hz ω) n)
+  have hρM : Measurable ρ := hzM.im.mul_const q
+  have hδM (n : ℕ) : Measurable (δ n) := by
+    dsimp [δ,angleThreshold]
+    exact (hzM.im.div hzM.norm).div_const _
+  have hεM (n : ℕ) : Measurable (fun ω => (L n ω).ε) := by
+    change Measurable (fun ω => heightFloor (z ω).im (ρ ω) (δ n ω)/2)
+    unfold heightFloor
+    fun_prop
+  have hNM (n : ℕ) : Measurable (fun ω => (L n ω).N) := by
+    change Measurable (fun ω => (⟨(z ω).im/δ n ω+logCeiling (z ω).im (ρ ω) (δ n ω)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    unfold logCeiling heightFloor
+    fun_prop
+  have hBM (n : ℕ) : Measurable (fun ω => (L n ω).B) := by
+    change Measurable (fun ω => (⟨(z ω).im^2/(2*a*(δ n ω)^2)+1, _⟩ : ℝ≥0))
+    apply Measurable.subtype_mk
+    fun_prop
+  let T (n : ℕ) (ω : Ω) := regionStop a (L n ω).ε (L n ω).hε (z ω)
+    (L n ω).N (ρ ω) (δ n ω) (L n ω).B (b ω)
+  have hTM (n : ℕ) : Measurable (T n) :=
+    regionStop_parameter_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b _ ρ (δ n) _ (hεM n) hzM hbM
+      (measurable_subtype_coe.comp (hNM n)) hρM (hδM n) (hBM n)
+  have hPM : Measurable (physicalGreen a) := by
+    have hx := X.continuous.measurable
+    have hy := Y.continuous.measurable
+    have hj := J.continuous.measurable
+    unfold physicalGreen logGreen Q
+    fun_prop
+  let F (n : ℕ) (ω : Ω) := if t < T n ω then physicalGreen a (localState a (L n ω).ε (L n ω).hε (z ω) t (b ω)) else 0
+  have hFM (n : ℕ) : Measurable (F n) := (hPM.comp
+    ((localState_parameter_joint_measurable a (fun ω => (L n ω).ε) (fun ω => (L n ω).hε) z b (hεM n) hzM hbM).comp
+      (measurable_const.prodMk measurable_id))).ite (measurableSet_lt measurable_const (hTM n)) measurable_const
+  apply measurable_of_tendsto_metrizable hFM
+  apply tendsto_pi_nhds.mpr
+  intro ω
+  exact localizationRemaining_tendsto a ha0 (z ω) (hz ω) (ρ ω) (hρ ω) (hr ω)
+    (fun n => L n ω) (b ω) t
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+lemma radialRemaining_driver_congr (a : ℝ) (ha : 0<a) (z : ℂ) (hz : 0<z.im)
+    (ρ : ℝ) (hρ : 0<ρ) (hr : ρ≤z.im) (b c : ExactSLEGenerator.Driver) (t : ℝ≥0)
+    (h : ∀ s≤t,b.1 s=c.1 s) : radialRemaining a b.1 z ρ t = radialRemaining a c.1 z ρ t := by
+  have hiff : radialTime a b.1 z ρ≤t ↔ radialTime a c.1 z ρ≤t := by
+    constructor
+    · intro hb
+      rwa [← radialTime_driver_congr_before a ha z hz ρ hρ hr b c t h hb]
+    · intro hc
+      rwa [← radialTime_driver_congr_before a ha z hz ρ hρ hr c b t (fun s hs => (h s hs).symm) hc]
+  have hh : (t : ℝ≥0∞)<radialTime a b.1 z ρ ↔ (t : ℝ≥0∞)<radialTime a c.1 z ρ := by
+    simpa only [not_le] using hiff.not
+  simp only [radialRemaining,alive_driver_congr h,hh,actualState_driver_congr h]
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory ExactSLEGenerator ExactSLEMain
+open scoped NNReal ENNReal
+ 
+
+theorem radialRemaining_raw_joint (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
+    (hq1 : q≤1) (t : ℝ≥0) :
+    letI : MeasurableSpace ExactSLEGenerator.Driver := rawFiltration t
+    Measurable (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0<z.im} =>
+      radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t)
+    := by
+  have he : (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0<z.im} =>
+      radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t) =
+    (fun p : ExactSLEGenerator.Driver × {z : ℂ // 0<z.im} =>
+      radialRemaining a (pastDriver t p.1).1 p.2.1 (p.2.1.im*q) t) := by
+    funext p
+    exact radialRemaining_driver_congr a (by linarith) p.2.1 p.2.2 _ (mul_pos p.2.2 hq)
+      (mul_le_of_le_one_right p.2.2.le hq1) p.1 (pastDriver t p.1) t
+      (fun s hs => (pastDriver_eq t p.1 s hs).symm)
+  change @Measurable (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im}) ℝ
+    ((rawFiltration t).prod inferInstance) inferInstance _
+  rw [he]
+  have hp : @Measurable (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im})
+      (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im})
+      ((rawFiltration t).prod inferInstance) inferInstance (fun p => (pastDriver t p.1,p.2)) :=
+    ((pastDriver_raw_measurable t).comp
+      (@measurable_fst _ _ (rawFiltration t) inferInstance)).prodMk
+      (@measurable_snd _ _ (rawFiltration t) inferInstance)
+  exact @Measurable.comp (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im})
+    (ExactSLEGenerator.Driver × {z : ℂ // 0<z.im}) ℝ
+    ((rawFiltration t).prod inferInstance) inferInstance inferInstance
+    (fun p => radialRemaining a p.1.1 p.2.1 (p.2.1.im*q) t)
+    (fun p => (pastDriver t p.1,p.2))
+    (joint_radialRemaining_measurable a ha q hq hq1 t) hp
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory ExactSLEGenerator ExactSLEMain
+open scoped NNReal ENNReal
+noncomputable def remainingMassDensity (a q : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
+  if 0<z.im then Real.exp (-alpha a*Real.log 2)*radialRemaining a b.1 z (z.im*q) t else 0
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+lemma remainingMassDensity_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
+    (hq1 : q≤1) (t : ℝ≥0) :
+    Measurable (fun p : ExactSLEGenerator.Driver × ℂ => remainingMassDensity a q t p.1 p.2) := by
+  let U : Set (ExactSLEGenerator.Driver × ℂ) := {p | 0<p.2.im}
+  have hU : MeasurableSet U := measurableSet_lt measurable_const (Complex.measurable_im.comp measurable_snd)
+  have hm : Measurable (fun p : U =>
+      Real.exp (-alpha a*Real.log 2)*radialRemaining a p.1.1.1 p.1.2 (p.1.2.im*q) t) := by
+    have hp : Measurable (fun p : U => (p.1.1, (⟨p.1.2,p.2⟩ : {z : ℂ // 0<z.im}))) := by fun_prop
+    exact measurable_const.mul ((joint_radialRemaining_measurable a ha q hq hq1 t).comp hp)
+  exact hm.dite (g:=fun _ => 0) measurable_const hU
+lemma remainingMassDensity_local_bound (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q)
+    (hq1 : q≤1) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (c : ℝ) (hc : 0<c)
+    (z : ℂ) (hz : c≤2*z.im) : ‖remainingMassDensity a q t b z‖≤(c*q)^(-alpha a) := by
+  have hz0 : 0<z.im := by linarith
+  have hb := radialRemaining_bound a ha z hz0 (z.im*q) (mul_pos hz0 hq)
+    (mul_le_of_le_one_right hz0.le hq1) b t
+  have ht : 0<2*z.im*q := by positivity
+  calc
+    _ = Real.exp (-alpha a*Real.log 2)*‖radialRemaining a b.1 z (z.im*q) t‖ := by
+      simp only [remainingMassDensity,ite_eq_left hz0,norm_mul,Real.norm_of_nonneg (Real.exp_pos _).le]
+    _ ≤ Real.exp (-alpha a*Real.log 2)*Real.exp (-alpha a*Real.log (z.im*q)) :=
+      mul_le_mul_of_nonneg_left hb (Real.exp_pos _).le
+    _ = (2*z.im*q)^(-alpha a) := by
+      rw [←Real.exp_add,Real.rpow_def_of_pos ht]
+      rw [show 2*z.im*q=2*(z.im*q) by ring,Real.log_mul (by norm_num) (mul_pos hz0 hq).ne']
+      congr 1; ring
+    _ ≤ _ := Real.rpow_le_rpow_of_nonpos (mul_pos hc hq)
+      (mul_le_mul_of_nonneg_right hz hq.le) (neg_nonpos.mpr (alpha_pos ha).le)
+lemma remainingMassEval_joint_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsFiniteMeasure W]
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B) :
+    Integrable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      f p.2 * remainingMassDensity a q t p.1 p.2) (W.prod (volume.restrict K)) := by
+  obtain ⟨B,hB,hBf⟩ := hfB
+  have : IsFiniteMeasure (volume.restrict K) := ⟨by simpa using hKb.measure_lt_top⟩
+  have hm : Measurable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      f p.2 * remainingMassDensity a q t p.1 p.2) :=
+    (hf.comp measurable_snd).mul (remainingMassDensity_measurable a ha q hq hq1 t)
+  apply Integrable.of_bound hm.aestronglyMeasurable (B*((c*q)^(-alpha a)))
+  have hmem : ∀ᵐ p ∂W.prod (volume.restrict K), p.2∈K :=
+    (Measure.ae_prod_iff_ae_ae (hK.preimage measurable_snd)).mpr
+      (Eventually.of_forall fun _ => ae_restrict_mem hK)
+  filter_upwards [hmem] with p hp
+  rw [norm_mul]
+  exact mul_le_mul (hBf _ hp) (remainingMassDensity_local_bound a ha q hq hq1 t p.1 c hc p.2 (hcK _ hp))
+    (norm_nonneg _) hB
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open MeasureTheory ProbabilityTheory Filter Set
+open scoped Classical NNReal ENNReal Topology
+lemma finite_density_conditional (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0) (z : ℂ) (hz : 0<z.im)
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[rawFiltration t] U) :
+    (∫b in U, (timeMassDensity a q ∞ b z).toReal-(timeMassDensity a q t b z).toReal ∂W) =
+      ∫b in U, remainingMassDensity a q t b z ∂W := by
+  have he (b : ExactSLEGenerator.Driver) :
+      (timeMassDensity a q ∞ b z).toReal-(timeMassDensity a q t b z).toReal =
+        Real.exp (-alpha a*Real.log 2)*(if (t : ℝ≥0∞)<radialTime a b.1 z (z.im*q)
+          then radialWeight a b.1 z (z.im*q) else 0) := by
+    have hn := (fullRadialDensity_bound a ha z hz (2*z.im*q) (by positivity) (by nlinarith) b).1
+    have hρ : 2*z.im*q/2=z.im*q := by ring
+    by_cases hh : radialTime a b.1 z (z.im*q)≤t
+    · simp only [timeMassDensity,le_top,ite_true,ite_eq_left hh,massDensity,ite_eq_left hz,
+        ENNReal.toReal_ofReal hn,sub_self,ite_eq_right (not_lt.mpr hh),mul_zero]
+    · simp only [timeMassDensity,le_top,ite_true,ite_eq_right hh,massDensity,ite_eq_left hz,
+        ENNReal.toReal_zero,sub_zero,ite_eq_left (lt_of_not_ge hh),fullRadialDensity,hρ]
+      exact ENNReal.toReal_ofReal (by simpa only [fullRadialDensity,hρ] using hn)
+  simp_rw [he,remainingMassDensity,ite_eq_left hz]
+  rw [integral_const_mul,integral_const_mul]
+  congr 1
+  have hS : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (t : WithTop ℝ≥0)) :=
+    isStoppingTime_const _ _
+  exact (radial_conditional_weight W hW a ha z hz (z.im*q) (mul_pos hz hq)
+    (mul_le_of_le_one_right hz.le hq1) (fun _ => t) hS U
+    ((hS.le_measurableSpace_of_const_le (fun _ => le_rfl)) _ hU)).2
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory ExactSLEGenerator ExactSLEMain
+open scoped NNReal ENNReal
+ 
+
+theorem actual_finite_conditional_mass (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0)
+    (K : Set ℂ) (hK : MeasurableSet K) (hKb : Bornology.IsBounded K)
+    (c : ℝ) (hc : 0<c) (hcK : ∀ z,z∈K→c≤2*z.im)
+    (f : ℂ → ℝ) (hf : Measurable f) (hfB : ∃ B : ℝ,0≤B ∧ ∀ z∈K,‖f z‖≤B)
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[rawFiltration t] U) :
+    (∫b in U, actualMassEval a q ∞ K f b-actualMassEval a q t K f b ∂W) =
+      ∫b in U, (∫z in K, f z*remainingMassDensity a q t b z) ∂W
+    := by
+  have hIm (v : ℝ≥0∞) := actualMassEval_joint_integrable a ha W q hq hq1 v K hK hKb c hc hcK f hf hfB
+  have hI (v : ℝ≥0∞) : Integrable (fun p : ExactSLEGenerator.Driver × ℂ =>
+      f p.2*(timeMassDensity a q v p.1 p.2).toReal) ((W.restrict U).prod (volume.restrict K)) :=
+    (hIm v).mono_measure (Measure.prod_mono Measure.restrict_le_self le_rfl)
+  have hsub : Integrable (fun p : ExactSLEGenerator.Driver × ℂ => f p.2*
+      ((timeMassDensity a q ∞ p.1 p.2).toReal-(timeMassDensity a q t p.1 p.2).toReal))
+      ((W.restrict U).prod (volume.restrict K)) := by
+    convert (hI ∞).sub (hI t) using 1
+    first | rfl | (funext point; dsimp only [Pi.sub_apply]; ring)
+  have he (b : ExactSLEGenerator.Driver) : actualMassEval a q ∞ K f b-actualMassEval a q t K f b =
+      ∫z in K, f z*((timeMassDensity a q ∞ b z).toReal-(timeMassDensity a q t b z).toReal) := by
+    simp only [actualMassEval,mul_sub]
+    exact (integral_sub
+      (actualMassEval_integrable a ha q hq hq1 ∞ K hK hKb c hc hcK f hf hfB b)
+      (actualMassEval_integrable a ha q hq hq1 t K hK hKb c hc hcK f hf hfB b)).symm
+  simp_rw [he]
+  rw [integral_integral_swap hsub]
+  conv_rhs => rw [integral_integral_swap
+    (remainingMassEval_joint_integrable a ha (W.restrict U) q hq hq1 t K hK hKb c hc hcK f hf hfB)]
+  apply setIntegral_congr_fun hK
+  intro z hzK
+  dsimp only
+  rw [integral_const_mul,integral_const_mul]
+  congr 1
+  exact finite_density_conditional a ha W hW q hq hq1 t z (by have := hcK z hzK; linarith) U hU
+end ExactSLEGreen
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open scoped Classical NNReal ENNReal
+ 
+noncomputable def survivingGreen (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) : ℝ :=
+  if alive a b.1 t z then Real.exp (-alpha a*Real.log 2)*physicalGreen a (actualState a b.1 t z) else 0
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
+open scoped NNReal ENNReal Topology
+lemma physicalRadius_actual_pos (a : ℝ) (b : ExactSLEGenerator.Driver) (z : ℂ)
+    (t : ℝ≥0) (ht : alive a b.1 t z) : 0<physicalRadius (actualState a b.1 t z) := by
+  obtain ⟨f,hf⟩ := ht
+  have h := fullRadius_pos b.1.continuous hf le_rfl
+  rw [←actual_radius_eq_full b.1.continuous ⟨f,hf⟩] at h
+  linarith
+lemma physicalRadius_actual_anti (a : ℝ) (ha : 0<a) (b : ExactSLEGenerator.Driver) (z : ℂ)
+    (s t : ℝ≥0) (hst : s≤t) (ht : alive a b.1 t z) :
+    physicalRadius (actualState a b.1 t z)≤physicalRadius (actualState a b.1 s z) := by
+  obtain ⟨f,hf⟩ := ht
+  have he := fullRadius_active_clock b.1.continuous hf hst le_rfl
+  have hp := fullRadius_pos b.1.continuous hf hst
+  have hn := activeODEClock_nonneg (b:=b.1) (T:=t) (σ:=s) (f:=f) (show (s:ℝ)≤t from hst)
+  have hle : fullLoewnerRadius a b.1 z t≤fullLoewnerRadius a b.1 z s := by
+    rw [he]
+    exact mul_le_of_le_one_right hp.le (Real.exp_le_one_iff.mpr (by nlinarith))
+  rw [←actual_radius_eq_full b.1.continuous ⟨f,hf⟩,
+    ←actual_radius_eq_full b.1.continuous ⟨f,odeUntil_mono hf hst⟩] at hle
+  linarith
+lemma survivingGreen_nonneg (a : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    0 ≤ survivingGreen a t b z := by
+  unfold survivingGreen physicalGreen
+  split_ifs <;> positivity
+lemma remainingMassDensity_nonneg (a q : ℝ) (t : ℝ≥0) (b : ExactSLEGenerator.Driver) (z : ℂ) :
+    0≤remainingMassDensity a q t b z := by
+  unfold remainingMassDensity radialRemaining physicalGreen
+  split_ifs <;> positivity
+lemma remaining_to_surviving (a : ℝ) (ha : 1/4<a) (t : ℝ≥0)
+    (q : ℕ → ℝ) (hq : ∀n,0<q n ∧ q n≤1) (hlim : Tendsto q atTop (𝓝 0))
+    (b : ExactSLEGenerator.Driver) (z : ℂ) (hz : 0<z.im) :
+    Tendsto (fun n => remainingMassDensity a (q n) t b z) atTop (𝓝 (survivingGreen a t b z)) := by
+  by_cases ht : alive a b.1 t z
+  · have hR := physicalRadius_actual_pos a b z t ht
+    have hqR : ∀ᶠ n in atTop, z.im*q n<physicalRadius (actualState a b.1 t z) := by
+      have hconv : Tendsto (fun n => z.im*q n) atTop (𝓝 0) := by simpa using hlim.const_mul z.im
+      exact hconv.eventually (gt_mem_nhds hR)
+    apply tendsto_const_nhds.congr'
+    filter_upwards [hqR] with n hn
+    have hnT : (t : ℝ≥0∞)<radialTime a b.1 z (z.im*q n) := by
+      by_contra h
+      have hTt := le_of_not_gt h
+      have hfin : radialTime a b.1 z (z.im*q n)≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hTt
+      have hle : (radialTime a b.1 z (z.im*q n)).toNNReal≤t := by
+        exact ENNReal.coe_le_coe.mp (by simpa only [ENNReal.coe_toNNReal hfin] using hTt)
+      have hh := (radialTime_attained a (by linarith) z hz _ (mul_pos hz (hq n).1)
+        (mul_le_of_le_one_right hz.le (hq n).2) b hfin).2
+      have he := physicalRadius_actual_anti a (by linarith) b z _ t hle ht
+      rw [hh] at he
+      exact not_le.mpr hn he
+    simp only [remainingMassDensity,ite_eq_left hz,radialRemaining,ht,hnT,and_self,ite_true,survivingGreen]
+  · simpa only [remainingMassDensity,ite_eq_left hz,radialRemaining,ht,false_and,ite_false,survivingGreen,
+      mul_zero] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter Set ExactSLEGenerator ExactSLEMain
+open scoped Classical NNReal ENNReal Topology
+lemma remainingMassDensity_mean_bound (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0) (z : ℂ) (hz : 0<z.im) :
+    Integrable (fun b => remainingMassDensity a q t b z) W ∧
+      (∫b,remainingMassDensity a q t b z∂W)≤Real.exp (-alpha a*Real.log 2)*physicalGreen a (z,0) := by
+  have hS : IsStoppingTime rawFiltration (fun _ : ExactSLEGenerator.Driver => (t : WithTop ℝ≥0)) :=
+    isStoppingTime_const _ _
+  have H := radial_conditional_weight W hW a ha z hz (z.im*q) (mul_pos hz hq)
+    (mul_le_of_le_one_right hz.le hq1) (fun _ => t) hS univ MeasurableSet.univ
+  have hmean := radialWeight_mean W hW a ha z hz (z.im*q) (mul_pos hz hq)
+    (mul_le_of_le_one_right hz.le hq1)
+  constructor
+  · simpa only [remainingMassDensity,ite_eq_left hz] using H.1.const_mul (Real.exp (-alpha a*Real.log 2))
+  · simp only [remainingMassDensity,ite_eq_left hz]
+    have He := H.2
+    simp only [setIntegral_univ] at He
+    rw [integral_const_mul,←He]
+    apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+    rw [←hmean.2]
+    apply integral_mono_of_nonneg
+    · filter_upwards [] with b
+      split_ifs
+      · exact (radialWeight_bound a ha z hz _ (mul_pos hz hq)
+          (mul_le_of_le_one_right hz.le hq1) b).1
+      · exact le_rfl
+    · exact hmean.1
+    · filter_upwards [] with b
+      split_ifs
+      · exact le_rfl
+      · exact (radialWeight_bound a ha z hz _ (mul_pos hz hq)
+          (mul_le_of_le_one_right hz.le hq1) b).1
+end ExactSLEGreen
+namespace ExactSLEGreen
+open MeasureTheory ProbabilityTheory Filter ExactSLEGenerator ExactSLEMain
+open scoped Classical NNReal ENNReal Topology
+ 
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TailBatches.lean b/lean/OAI/Probability/SLEGauge/TailBatches.lean
index 84a22138b0d47c33803e45101593684ba729490d..439ef500af98b280a2d589f62c22a1af84a86cc5 100644
--- a/lean/OAI/Probability/SLEGauge/TailBatches.lean
+++ b/lean/OAI/Probability/SLEGauge/TailBatches.lean
@@ -1,562 +1,562 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.OptionalPastPalm
-import OAI.Probability.SLEGauge.MomentBatches
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped ENNReal
-namespace SLEExactGauge.MomentBatches
-
-lemma sum_repeated_thresholds (l : List ℤ) (N : ℤ → ℕ) (f : ℝ → ℝ) :
-    ((l.flatMap (fun k => List.replicate (N k) ((2:ℝ)^k))).map f).sum =
-      (l.map (fun k => (N k:ℝ)*f ((2:ℝ)^k))).sum := by
-  induction l with
-  | nil => simp
-  | cons k l ih => simpa [List.sum_replicate,nsmul_eq_mul] using congrArg ((N k:ℝ)*f ((2:ℝ)^k)+·) ih
-
-lemma noncleanup_sum (m : ℕ → ℝ) (n : ℕ) (f : ℝ → ℝ) :
-    ((noncleanup m n).map f).sum =
-      ∑ k∈support m n, (multiplicity m n k:ℝ)*f ((2:ℝ)^k) := by
-  rw [noncleanup,sum_repeated_thresholds]
-  simpa only [Finset.sort_toFinset] using
-    (List.sum_toFinset (fun k => (multiplicity m n k:ℝ)*f ((2:ℝ)^k))
-      ((support m n).sort_nodup (· ≤ ·))).symm
-
-lemma batch_sum (m : ℕ → ℝ) (n : ℕ) (f : ℝ → ℝ) :
-    ((batch m n).map f).sum =
-      (∑ k∈support m n, (multiplicity m n k:ℝ)*f ((2:ℝ)^k)) + f n := by
-  simp only [batch,List.map_append,List.map_cons,List.map_nil,List.sum_append,
-    List.sum_cons,List.sum_nil,add_zero,noncleanup_sum]
-
-lemma fin_sum_batch (m : ℕ → ℝ) (n : ℕ) (f : ℝ → ℝ) :
-    (∑ i : Fin (batch m n).length, f ((batch m n)[i.1])) =
-      ((batch m n).map f).sum := by
-  rw [← List.sum_ofFn]
-  change (List.ofFn (f ∘ (fun i : Fin (batch m n).length => (batch m n)[i.1]))).sum = _
-  rw [← List.map_ofFn,List.ofFn_getElem]
-
- 
-
-lemma cleanup_counts_le (S : Finset ℕ) {x : ℝ} (hx : 0 ≤ x) :
-    (∑ n∈S, if 16*((n:ℝ)+1) ≤ x then (1:ℝ) else 0) ≤ x/16 := by
-  classical
-  let A : Finset ℕ := S.filter (fun n : ℕ => 16*((n:ℝ)+1) ≤ x)
-  have hsub : A ⊆ Finset.range ⌊x/16⌋₊ := by
-    intro n hn
-    rw [Finset.mem_range]
-    have hh := (Finset.mem_filter.mp hn).2
-    have hf : n+1 ≤ ⌊x/16⌋₊ := Nat.le_floor (by push_cast; linarith)
-    omega
-  calc
-    (∑ n∈S, if 16*((n:ℝ)+1) ≤ x then (1:ℝ) else 0) = (A.card:ℝ) := by
-      rw [← Finset.sum_filter]
-      simp [A]
-    _ ≤ (⌊x/16⌋₊:ℝ) := by exact_mod_cast (Finset.card_le_card hsub).trans_eq (Finset.card_range _)
-    _ ≤ x/16 := Nat.floor_le (by positivity)
-
-variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-
-lemma cleanup_sum_le (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hInt : Integrable L P) (S : Finset ℕ) :
-    (∑ n∈S, P.real {ω | 16*((n:ℝ)+1) ≤ L ω}) ≤ moment P L 1/16 := by
-  let f (n : ℕ) := {ω | 16*((n:ℝ)+1) ≤ L ω}.indicator (fun _ : Ω => (1:ℝ))
-  have hint (n : ℕ) : Integrable (f n) P :=
-    (integrable_const 1).indicator (measurableSet_le measurable_const hL)
-  have he (n : ℕ) : (∫ ω, f n ω ∂P)=P.real {ω | 16*((n:ℝ)+1) ≤ L ω} := by
-    rw [integral_indicator_const _ (measurableSet_le measurable_const hL),smul_eq_mul,mul_one]
-  have hpoint : (fun ω => ∑ n∈S, f n ω) ≤ᵐ[P] (fun ω => L ω/16) :=
-    hL0.mono fun ω hω => cleanup_counts_le S hω
-  have hI := integral_mono_ae (integrable_finsetSum _ (fun n hn => hint n)) (hInt.div_const 16) hpoint
-  rw [integral_finsetSum _ (fun n hn => hint n),integral_div] at hI
-  simpa only [he,moment,pow_one] using hI
-
- 
-
-theorem cleanup_tail (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hInt : Integrable L P) :
-    Summable (fun n : ℕ => P.real {ω | 16*((n:ℝ)+1) ≤ L ω}) ∧
-    (∑' n : ℕ, P.real {ω | 16*((n:ℝ)+1) ≤ L ω}) ≤ moment P L 1/16 :=
-  ⟨summable_of_sum_le (fun _ => measureReal_nonneg) (cleanup_sum_le P L hL hL0 hInt),
-   Real.tsum_le_of_sum_le (fun _ => measureReal_nonneg) (cleanup_sum_le P L hL hL0 hInt)⟩
-
-lemma upper_tail_bound_geometric (n : ℕ) (hn : 0 < n) :
-    (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) ≤ 2*(1/4:ℝ)^n := by
-  have hq : (1/2:ℝ)^n ≤ 1/2 := by
-    simpa using pow_le_pow_of_le_one (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)≤1)
-      (show 1 ≤ n from hn)
-  have hden : (1/2:ℝ) ≤ 1-(1/2:ℝ)^n := by linarith
-  rw [← div_pow]
-  norm_num only [show (4:ℝ)/16=1/4 by norm_num]
-  calc
-    (1/4:ℝ)^n/(1-(1/2:ℝ)^n) ≤ (1/4:ℝ)^n/(1/2) :=
-      div_le_div_of_nonneg_left (by positivity) (by norm_num) hden
-    _ = 2*(1/4:ℝ)^n := by ring
-
-lemma summable_noncleanup_tails (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
-    Summable (fun n : ℕ =>
-      (((noncleanup (moment P L) (n+1)).map (fun b => P.real {ω | 16*b ≤ L ω})).sum)) := by
-  have hgeom := (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4)
-    (by norm_num : (1/4:ℝ)<1)).mul_left 2
-  have hgeom' : Summable (fun n : ℕ => 2*(1/4:ℝ)^(n+1)) :=
-    hgeom.comp_injective Nat.succ_injective
-  apply Summable.of_nonneg_of_le _ _ hgeom'
-  · intro n
-    exact List.sum_nonneg (by intro x hx; obtain ⟨b,hb,rfl⟩ := List.mem_map.mp hx; exact measureReal_nonneg)
-  · intro n
-    rw [noncleanup_sum]
-    obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
-    exact (upper_tail_finite P L hL hL0 (n+1) (by omega) (hint (n+1)) (hm (n+1) (by omega))).trans
-      (upper_tail_bound_geometric (n+1) (by omega))
-
- 
-
-theorem summable_batch_tails (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
-    Summable (fun n : ℕ =>
-      (((batch (moment P L) (n+1)).map (fun b => P.real {ω | 16*b ≤ L ω})).sum)) := by
-  have hInt : Integrable L P := by simpa only [pow_one] using hint 1
-  have hs := (summable_noncleanup_tails P L hL hL0 hint hmean).add
-    (cleanup_tail P L hL hL0 hInt).1
-  simpa only [batch,List.map_append,List.sum_append,List.map_cons,List.map_nil,List.sum_cons,
-    List.sum_nil,add_zero,Nat.cast_add,Nat.cast_one] using hs
-
- 
-theorem summable_entry_tails (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
-    Summable (fun i : EntryIndex (moment P L) => P.real {ω | 16*threshold (moment P L) i ≤ L ω}) := by
-  have hs := summable_batch_tails P L hL hL0 hint hmean
-  have hinj : Function.Injective (fun n : {n : ℕ // 0 < n} => n.val-1) := by
-    intro n j h
-    have hn := n.property
-    have hj := j.property
-    change n.val-1=j.val-1 at h
-    apply Subtype.ext
-    omega
-  have hs' := hs.comp_injective hinj
-  have hseq : Summable (fun n : {n : ℕ // 0 < n} =>
-      (((batch (moment P L) n.val).map (fun b => P.real {ω | 16*b ≤ L ω})).sum)) := by
-    apply hs'.congr
-    intro n
-    dsimp only [Function.comp_apply]
-    rw [Nat.sub_add_cancel n.property]
-  apply (summable_sigma_of_nonneg (fun i => measureReal_nonneg)).mpr
-  refine ⟨fun n => (hasSum_fintype _).summable,?_⟩
-  apply hseq.congr
-  intro n
-  rw [tsum_fintype]
-  exact (fin_sum_batch (moment P L) n.val (fun b => P.real {ω | 16*b ≤ L ω})).symm
-
- 
-
-theorem tail_summability (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
-    (∑' i : EntryIndex (moment P L), P {ω | 16*threshold (moment P L) i ≤ L ω}) < ∞ := by
-  have h := (summable_entry_tails P L hL hL0 hint hmean).tsum_ofReal_lt_top
-  simpa only [Measure.real,ENNReal.ofReal_toReal (measure_ne_top P _)] using h
-
-end SLEExactGauge.MomentBatches
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped ENNReal
-namespace SLEExactGauge.MomentBatches
-
-variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
-
-lemma increasing_level_bounds (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
-    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
-    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
-    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω))) :
-    ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω ∧ Y j ω ≤ L ω := by
-  filter_upwards [hY0,hmono,hlim] with ω h0 hm hl
-  exact fun j => ⟨h0 j,hm.ge_of_tendsto hl j⟩
-
-lemma integrable_level_pow (L Y : Ω → ℝ) (hY : Measurable Y)
-    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω) (n : ℕ)
-    (hint : Integrable (fun ω => (L ω)^n) P) :
-    Integrable (fun ω => (Y ω)^n) P := by
-  refine hint.mono_nonneg (hY.pow_const n).aestronglyMeasurable ?_ ?_
-  · exact hYL.mono fun ω hω => pow_nonneg hω.1 n
-  · exact hYL.mono fun ω hω => pow_le_pow_left₀ hω.1 hω.2 n
-
- 
-
-theorem increasing_moments (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
-    (hY : ∀ j, Measurable (Y j))
-    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
-    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
-    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω)))
-    (n : ℕ) (hint : Integrable (fun ω => (L ω)^n) P) :
-    (∀ j, Integrable (fun ω => (Y j ω)^n) P) ∧
-    Monotone (fun j => moment P (Y j) n) ∧
-    Tendsto (fun j => moment P (Y j) n) atTop (𝓝 (moment P L n)) := by
-  have hb := increasing_level_bounds P L Y hY0 hmono hlim
-  have hI (j : ℕ) : Integrable (fun ω => (Y j ω)^n) P :=
-    integrable_level_pow P L (Y j) (hY j) (hb.mono fun ω hω => hω j) n hint
-  refine ⟨hI,?_,?_⟩
-  · intro j k hjk
-    apply integral_mono_ae (hI j) (hI k)
-    filter_upwards [hY0,hmono] with ω h0 hm
-    exact pow_le_pow_left₀ (h0 j) (hm hjk) n
-  · apply tendsto_integral_of_dominated_convergence (fun ω => (L ω)^n)
-      (fun j => (hY j |>.pow_const n).aestronglyMeasurable) hint
-    · intro j
-      filter_upwards [hb] with ω hω
-      rw [Real.norm_of_nonneg (pow_nonneg (hω j).1 n)]
-      exact pow_le_pow_left₀ (hω j).1 (hω j).2 n
-    · exact hlim.mono fun ω hω => hω.pow n
-
- 
-
-lemma level_eq_tsum_indicators {v : ℕ → ℝ} {M : ℝ} (hM : 0 < M)
-    (hlim : Tendsto v atTop (𝓝 M)) :
-    level v M = 1 + ∑' j : ℕ, if 1 ≤ j ∧ v j ≤ 3*M/4 then 1 else 0 := by
-  classical
-  have hfin := finite_badLevels hM hlim
-  rw [tsum_eq_sum (s := hfin.toFinset) (fun j hj => ?_)]
-  · simp only [level,dite_eq_left hfin]
-    congr 1
-    calc
-      hfin.toFinset.card = ∑ _ ∈ hfin.toFinset, (1:ℕ) := by simp
-      _ = _ := by
-        apply Finset.sum_congr rfl
-        intro j hj
-        have hj' : 1 ≤ j ∧ v j ≤ 3*M/4 := by
-          simpa only [Finite.mem_toFinset,badLevels,mem_ofPred_eq] using hj
-        simp only [ite_eq_left hj']
-  · have hnj : ¬ (1 ≤ j ∧ v j ≤ 3*M/4) := by
-      simpa only [Finite.mem_toFinset,badLevels,mem_ofPred_eq] using hj
-    exact ite_eq_right hnj
-
- 
-
-def momentLevel (L : Ω → ℝ) (Y : ℕ → Ω → ℝ) (n : ℕ) : ℕ :=
-  level (fun j => moment P (Y j) n) (moment P L n)
-
-def fullMomentGauge (L : Ω → ℝ) (Y : ℕ → Ω → ℝ) (a d : ℝ) : ℝ → ℝ :=
-  momentGauge (moment P L) a d (momentLevel P L Y)
-
-variable [IsProbabilityMeasure P]
-
- 
-
-theorem selected_level_moment (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
-    (hY : ∀ j, Measurable (Y j))
-    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
-    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
-    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω)))
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P)
-    (hmean : 0 < moment P L 1) (n : ℕ) (hn : 0 < n) :
-    0 < momentLevel P L Y n ∧
-    (badLevels (fun j => moment P (Y j) n) (moment P L n)).Finite ∧
-    3*moment P L n/4 < moment P (Y (momentLevel P L Y n)) n := by
-  have hb := increasing_level_bounds P L Y hY0 hmono hlim
-  have hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω := hb.mono fun ω hω => (hω 0).1.trans (hω 0).2
-  have hm := (moment_sequence_properties P L hL0 hint hmean).1 n hn
-  obtain ⟨hI,hmn,hlimn⟩ := increasing_moments P L Y hY hY0 hmono hlim n (hint n)
-  exact ⟨level_pos _ _,finite_badLevels hm hlimn,level_moment_gt hm hmn hlimn⟩
-
- 
-
-theorem success_selected_level (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
-    (hY : ∀ j, Measurable (Y j))
-    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
-    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
-    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω)))
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P)
-    (hmean : 0 < moment P L 1) (n : ℕ) (hn : 3 ≤ n) :
-    (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) ≤
-      (∑' k : ℤ, (multiplicity (moment P L) n k:ℝ)*
-        P.real {ω | (2:ℝ)^k ≤ Y (momentLevel P L Y n) ω}) ∧
-    (2:ℝ)^n/8 ≤ (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) := by
-  have hb := (increasing_level_bounds P L Y hY0 hmono hlim).mono
-    (fun ω hω => hω (momentLevel P L Y n))
-  exact success P L (Y (momentLevel P L Y n)) (hY _) hb hint hmean n hn
-    (integrable_level_pow P L _ (hY _) hb n (hint n))
-    (selected_level_moment P L Y hY hY0 hmono hlim hint hmean n (by omega)).2.2.le
-
- 
-
-theorem fullMomentGauge_properties (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
-    (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
-    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P)
-    (hmean : 0 < moment P L 1) {a d : ℝ}
-    (ha : 0 ≤ a) (hd : 0 < d) (hd2 : d ≤ 2) :
-    Target.IsHausdorffGauge (fullMomentGauge P L Y a d) ∧
-    ∀ C r : ℝ, 1 ≤ C → 0 ≤ r →
-      fullMomentGauge P L Y a d (C*r) ≤ C^2*fullMomentGauge P L Y a d r := by
-  obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
-  exact ⟨momentGauge_admissible _ hm hJ ha hd hd2 _,
-    fun C r hC _ => momentGauge_doubling _ hm hJ a d _ hC⟩
-
-end SLEExactGauge.MomentBatches
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
-open MomentBatches
-
-def sequenceMassReal (x : ℕ→ℝ≥0∞) : ℝ := (diskSequenceMass x).toReal
-def sequenceLevelReal (l : ℕ) (x : ℕ→ℝ≥0∞) : ℝ := (x l).toReal
-
-lemma sequenceMassReal_measurable : Measurable sequenceMassReal :=
-  diskSequenceMass_measurable.ennreal_toReal
-
-lemma sequenceLevelReal_measurable (l : ℕ) : Measurable (sequenceLevelReal l) :=
-  (measurable_pi_apply l).ennreal_toReal
-
-lemma sequenceMassReal_nonneg (x : ℕ→ℝ≥0∞) : 0 ≤ sequenceMassReal x := ENNReal.toReal_nonneg
-lemma sequenceLevelReal_nonneg (l : ℕ) (x : ℕ→ℝ≥0∞) : 0 ≤ sequenceLevelReal l x :=
-  ENNReal.toReal_nonneg
-
-lemma sequenceReal_monotone {x : ℕ→ℝ≥0∞} (hmono : Monotone x)
-    (hfin : diskSequenceMass x<∞) : Monotone (fun l=>sequenceLevelReal l x) := by
-  intro l k hlk
-  exact ENNReal.toReal_mono ((le_iSup x k).trans_lt hfin).ne (hmono hlk)
-
-lemma sequenceReal_tendsto {x : ℕ→ℝ≥0∞} (hmono : Monotone x)
-    (hfin : diskSequenceMass x<∞) :
-    Tendsto (fun l=>sequenceLevelReal l x) atTop (𝓝 (sequenceMassReal x)) :=
-  (ENNReal.tendsto_toReal hfin.ne).comp (diskSequenceMass_tendsto hmono)
-
-lemma sequenceMassReal_integrable_pow (Q : Measure (ℕ→ℝ≥0∞))
-    (m : ℕ) (hf : (∫⁻x,diskSequenceMass x^m∂Q)<∞) :
-    Integrable (fun x=>sequenceMassReal x^m) Q := by
-  simpa only [ENNReal.toReal_pow, sequenceMassReal] using
-    integrable_toReal_of_lintegral_ne_top
-      (diskSequenceMass_measurable.pow_const m).aemeasurable hf.ne
-
-lemma sequenceMassReal_moment (Q : Measure (ℕ→ℝ≥0∞))
-    (hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞) (m : ℕ) :
-    moment Q sequenceMassReal m=(∫⁻x,diskSequenceMass x^m∂Q).toReal := by
-  unfold moment sequenceMassReal
-  simp only [←ENNReal.toReal_pow]
-  exact integral_toReal (diskSequenceMass_measurable.pow_const m).aemeasurable
-    (hfin.mono (fun _ hx=>ENNReal.pow_lt_top hx))
-
-lemma sequenceLevelReal_moment (W : Measure GDriver) (a : ℝ) (ha : 1/4<a)
-    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
-    (hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2))
-    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
-    (r : ℝ) (Q : Measure (ℕ→ℝ≥0∞))
-    (hQlaw : ∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
-      (fun p=>fun i : Fin (n+1)=>subdiskMass a μ r i p))
-    (hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞) (m l : ℕ) :
-    moment Q (sequenceLevelReal l) m=(subdiskMoment W a μ r m l).toReal := by
-  have he := lintegral_coordinate Q (finiteAngleLaw W a l) (subdiskMass a μ r)
-    (subdiskMass_measurable a ha μ hμ r) l (hQlaw l) (fun x=>x^m)
-    (measurable_id.pow_const m)
-  unfold moment sequenceLevelReal
-  simp only [←ENNReal.toReal_pow]
-  rw [integral_toReal ((measurable_pi_apply l).pow_const m).aemeasurable
-    (hfin.mono (fun x hx=>ENNReal.pow_lt_top ((le_iSup x l).trans_lt hx))),he]
-  rfl
-
- 
-
-theorem sequenceReal_moment_data (Q : Measure (ℕ→ℝ≥0∞)) [IsProbabilityMeasure Q]
-    (hmono : ∀ᵐ x∂Q,Monotone x) (hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞)
-    (hmom : ∀m : ℕ, 1 ≤ m → 0<(∫⁻x,diskSequenceMass x^m∂Q) ∧
-      (∫⁻x,diskSequenceMass x^m∂Q)<∞) :
-    (∀ᵐ x∂Q,Monotone (fun l=>sequenceLevelReal l x) ∧
-      Tendsto (fun l=>sequenceLevelReal l x) atTop (𝓝 (sequenceMassReal x))) ∧
-    (∀m : ℕ,Integrable (fun x=>sequenceMassReal x^m) Q) ∧
-    (∀m : ℕ, 1 ≤ m → 0 < moment Q sequenceMassReal m) ∧
-    (∀m : ℕ,Monotone (fun l=>moment Q (sequenceLevelReal l) m) ∧
-      Tendsto (fun l=>moment Q (sequenceLevelReal l) m) atTop
-        (𝓝 (moment Q sequenceMassReal m))) := by
-  have hp : ∀ᵐ x∂Q,Monotone (fun l=>sequenceLevelReal l x) ∧
-      Tendsto (fun l=>sequenceLevelReal l x) atTop (𝓝 (sequenceMassReal x)) := by
-    filter_upwards [hmono,hfin] with x hx hxf
-    exact ⟨sequenceReal_monotone hx hxf,sequenceReal_tendsto hx hxf⟩
-  have hI (m : ℕ) : Integrable (fun x=>sequenceMassReal x^m) Q := by
-    rcases m with _|m
-    · simpa only [pow_zero] using integrable_const (1:ℝ)
-    · exact sequenceMassReal_integrable_pow Q (m+1) (hmom _ (by omega)).2
-  refine ⟨hp,hI,?_,?_⟩
-  · intro m hm
-    rw [sequenceMassReal_moment Q hfin m]
-    exact ENNReal.toReal_pos (hmom m hm).1.ne' (hmom m hm).2.ne
-  · intro m
-    exact (increasing_moments Q sequenceMassReal sequenceLevelReal sequenceLevelReal_measurable
-      (Eventually.of_forall (fun x l=>sequenceLevelReal_nonneg l x))
-      (hp.mono (fun _ hx=>hx.1)) (hp.mono (fun _ hx=>hx.2)) m (hI m)).2
-
-end SLEExactGauge.CenteredMoments
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.CenteredMoments
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
-open TraceMassIntegration TraceMassContinuity MomentBatches
-
- 
-
-structure CenteredMomentModel (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver) where
-  supports : ℕ→Finset ℕ
-  coeff : ℕ→ℕ→ℝ
-  nonneg : ∀k j,j∈supports k → 0≤coeff k j
-  tails : ∀k j,j∈supports k → k≤j
-  normalized : ∀k,∑j∈supports k,coeff k j=1
-  radiusIndex : ℕ
-  law : Measure (ℕ→ℝ≥0∞)
-  probability : IsProbabilityMeasure law
-  finite_laws : ∀n,law.map (initialVector n)=(finiteAngleLaw W a n).map
-    (fun p=>fun i : Fin (n+1)=>subdiskMass a (recoveredMass a ha supports coeff nonneg)
-      (subdiskRadius radiusIndex) i p)
-  increasing_finite : ∀ᵐ x∂law,Monotone x ∧ diskSequenceMass x<∞
-  positive_moments : ∀m : ℕ, 1 ≤ m → 0<(∫⁻x,diskSequenceMass x^m∂law) ∧
-    (∫⁻x,diskSequenceMass x^m∂law)<∞
-  canonical : CanonicalMassData κ a ha W (recoveredMass a ha supports coeff nonneg)
-
- 
-
-theorem actual_centered_moment_model (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W) :
-    Nonempty (CenteredMomentModel κ (Target.a κ) (a_gt_quarter hκ hκ8)
-      (W.map driverHomeomorph)) := by
-  obtain ⟨s,c,hc,hs,hsum,D,hreg,hμ,hraw,hpath,n₀,Q,hQ,hlaw,hseq,hmom⟩ :=
-    actual_subdisk_positive_moments κ hκ hκ8 W hW
-  exact ⟨{
-    supports := s
-    coeff := c
-    nonneg := hc
-    tails := hs
-    normalized := hsum
-    radiusIndex := n₀
-    law := Q
-    probability := hQ
-    finite_laws := hlaw
-    increasing_finite := hseq.mono (fun x hx=>⟨hx.1,hx.2.2⟩)
-    positive_moments := fun m hm=>⟨(hmom m hm).1,(hmom m hm).2.1⟩
-    canonical := D }⟩
-
-namespace CenteredMomentModel
-variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
-
- 
-
-def gauge (d : ℝ) : ℝ→ℝ := fullMomentGauge M.law sequenceMassReal sequenceLevelReal a d
-
-def mass : ℝ≥0∞→GDriver→Measure UpperPlane :=
-  recoveredMass a ha M.supports M.coeff M.nonneg
-
-lemma mass_measurable : Measurable (fun p : ℝ≥0∞×GDriver=>M.mass p.1 p.2) :=
-  recoveredMass_joint_measurable a ha M.supports M.coeff M.nonneg
-
-lemma integrable_moments (m : ℕ) : Integrable (fun x=>sequenceMassReal x^m) M.law := by
-  let := M.probability
-  exact (sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
-    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.1 m
-
-lemma moments_positive (m : ℕ) (hm : 1 ≤ m) : 0 < moment M.law sequenceMassReal m := by
-  let := M.probability
-  exact (sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
-    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.2.1 m hm
-
-lemma moment_finite_level (m l : ℕ) :
-    moment M.law (sequenceLevelReal l) m=
-      (subdiskMoment W a M.mass (subdiskRadius M.radiusIndex) m l).toReal := by
-  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (M.mass v b) := M.canonical.finite_compact v b
-  exact sequenceLevelReal_moment W a ha M.mass M.mass_measurable (subdiskRadius M.radiusIndex)
-    M.law M.finite_laws (M.increasing_finite.mono (fun _ hx=>hx.2)) m l
-
-lemma finite_moments_increasing (m : ℕ) : Monotone
-    (fun l=>moment M.law (sequenceLevelReal l) m) := by
-  let := M.probability
-  exact ((sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
-    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.2.2 m).1
-
-lemma finite_moments_tendsto (m : ℕ) :
-    Tendsto (fun l=>moment M.law (sequenceLevelReal l) m) atTop
-      (𝓝 (moment M.law sequenceMassReal m)) := by
-  let := M.probability
-  exact ((sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
-    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.2.2 m).2
-
-lemma gauge_admissible {d : ℝ} (hd : 0<d) (hd2 : d≤2) : Target.IsHausdorffGauge (M.gauge d) := by
-  let := M.probability
-  exact (fullMomentGauge_properties M.law sequenceMassReal sequenceLevelReal
-    (Eventually.of_forall sequenceMassReal_nonneg) M.integrable_moments
-    (M.moments_positive 1 le_rfl) (by linarith : 0≤a) hd hd2).1
-
-lemma gauge_doubling {d : ℝ} (hd : 0<d) (hd2 : d≤2) (C r : ℝ) (hC : 1≤C) (hr : 0≤r) :
-    M.gauge d (C*r) ≤ C^2*M.gauge d r := by
-  let := M.probability
-  exact (fullMomentGauge_properties M.law sequenceMassReal sequenceLevelReal
-    (Eventually.of_forall sequenceMassReal_nonneg) M.integrable_moments
-    (M.moments_positive 1 le_rfl) (by linarith : 0≤a) hd hd2).2 C r hC hr
-
-lemma selected_moments (n : ℕ) (hn : 0<n) :
-    let l := momentLevel M.law sequenceMassReal sequenceLevelReal n
-    0<l ∧ 3*moment M.law sequenceMassReal n/4 < moment M.law (sequenceLevelReal l) n := by
-  let := M.probability
-  obtain ⟨hp,hI,hm,hlim⟩ := sequenceReal_moment_data M.law
-    (M.increasing_finite.mono (fun _ hx=>hx.1))
-    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments
-  have hh := selected_level_moment M.law sequenceMassReal sequenceLevelReal sequenceLevelReal_measurable
-    (Eventually.of_forall (fun x l=>sequenceLevelReal_nonneg l x))
-    (hp.mono (fun _ hx=>hx.1)) (hp.mono (fun _ hx=>hx.2)) hI (hm 1 le_rfl) n hn
-  exact ⟨hh.1,hh.2.2⟩
-
- 
-
-lemma success (n : ℕ) (hn : 3≤n) :
-    (2:ℝ)^n/8 ≤ ∑'k : ℤ,(multiplicity (moment M.law sequenceMassReal) n k:ℝ)*
-      M.law.real {x | (2:ℝ)^k ≤ sequenceLevelReal
-        (momentLevel M.law sequenceMassReal sequenceLevelReal n) x} := by
-  let := M.probability
-  obtain ⟨hp,hI,hm,hlim⟩ := sequenceReal_moment_data M.law
-    (M.increasing_finite.mono (fun _ hx=>hx.1))
-    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments
-  have hh := success_selected_level M.law sequenceMassReal sequenceLevelReal sequenceLevelReal_measurable
-    (Eventually.of_forall (fun x l=>sequenceLevelReal_nonneg l x))
-    (hp.mono (fun _ hx=>hx.1)) (hp.mono (fun _ hx=>hx.2)) hI (hm 1 le_rfl) n hn
-  exact hh.2.trans hh.1
-
- 
-lemma tails_summable : Summable (fun i : EntryIndex (moment M.law sequenceMassReal)=>
-    M.law.real {x | 16*threshold (moment M.law sequenceMassReal) i ≤ sequenceMassReal x}) := by
-  let := M.probability
-  exact summable_entry_tails M.law sequenceMassReal sequenceMassReal_measurable
-    (Eventually.of_forall sequenceMassReal_nonneg) M.integrable_moments (M.moments_positive 1 le_rfl)
-
-end CenteredMomentModel
-end SLEExactGauge.CenteredMoments
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.OptionalPastPalm
+import OAI.Probability.SLEGauge.MomentBatches
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped ENNReal
+namespace SLEExactGauge.MomentBatches
+
+lemma sum_repeated_thresholds (l : List ℤ) (N : ℤ → ℕ) (f : ℝ → ℝ) :
+    ((l.flatMap (fun k => List.replicate (N k) ((2:ℝ)^k))).map f).sum =
+      (l.map (fun k => (N k:ℝ)*f ((2:ℝ)^k))).sum := by
+  induction l with
+  | nil => simp
+  | cons k l ih => simpa [List.sum_replicate,nsmul_eq_mul] using congrArg ((N k:ℝ)*f ((2:ℝ)^k)+·) ih
+
+lemma noncleanup_sum (m : ℕ → ℝ) (n : ℕ) (f : ℝ → ℝ) :
+    ((noncleanup m n).map f).sum =
+      ∑ k∈support m n, (multiplicity m n k:ℝ)*f ((2:ℝ)^k) := by
+  rw [noncleanup,sum_repeated_thresholds]
+  simpa only [Finset.sort_toFinset] using
+    (List.sum_toFinset (fun k => (multiplicity m n k:ℝ)*f ((2:ℝ)^k))
+      ((support m n).sort_nodup (· ≤ ·))).symm
+
+lemma batch_sum (m : ℕ → ℝ) (n : ℕ) (f : ℝ → ℝ) :
+    ((batch m n).map f).sum =
+      (∑ k∈support m n, (multiplicity m n k:ℝ)*f ((2:ℝ)^k)) + f n := by
+  simp only [batch,List.map_append,List.map_cons,List.map_nil,List.sum_append,
+    List.sum_cons,List.sum_nil,add_zero,noncleanup_sum]
+
+lemma fin_sum_batch (m : ℕ → ℝ) (n : ℕ) (f : ℝ → ℝ) :
+    (∑ i : Fin (batch m n).length, f ((batch m n)[i.1])) =
+      ((batch m n).map f).sum := by
+  rw [← List.sum_ofFn]
+  change (List.ofFn (f ∘ (fun i : Fin (batch m n).length => (batch m n)[i.1]))).sum = _
+  rw [← List.map_ofFn,List.ofFn_getElem]
+
+ 
+
+lemma cleanup_counts_le (S : Finset ℕ) {x : ℝ} (hx : 0 ≤ x) :
+    (∑ n∈S, if 16*((n:ℝ)+1) ≤ x then (1:ℝ) else 0) ≤ x/16 := by
+  classical
+  let A : Finset ℕ := S.filter (fun n : ℕ => 16*((n:ℝ)+1) ≤ x)
+  have hsub : A ⊆ Finset.range ⌊x/16⌋₊ := by
+    intro n hn
+    rw [Finset.mem_range]
+    have hh := (Finset.mem_filter.mp hn).2
+    have hf : n+1 ≤ ⌊x/16⌋₊ := Nat.le_floor (by push_cast; linarith)
+    omega
+  calc
+    (∑ n∈S, if 16*((n:ℝ)+1) ≤ x then (1:ℝ) else 0) = (A.card:ℝ) := by
+      rw [← Finset.sum_filter]
+      simp [A]
+    _ ≤ (⌊x/16⌋₊:ℝ) := by exact_mod_cast (Finset.card_le_card hsub).trans_eq (Finset.card_range _)
+    _ ≤ x/16 := Nat.floor_le (by positivity)
+
+variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+
+lemma cleanup_sum_le (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hInt : Integrable L P) (S : Finset ℕ) :
+    (∑ n∈S, P.real {ω | 16*((n:ℝ)+1) ≤ L ω}) ≤ moment P L 1/16 := by
+  let f (n : ℕ) := {ω | 16*((n:ℝ)+1) ≤ L ω}.indicator (fun _ : Ω => (1:ℝ))
+  have hint (n : ℕ) : Integrable (f n) P :=
+    (integrable_const 1).indicator (measurableSet_le measurable_const hL)
+  have he (n : ℕ) : (∫ ω, f n ω ∂P)=P.real {ω | 16*((n:ℝ)+1) ≤ L ω} := by
+    rw [integral_indicator_const _ (measurableSet_le measurable_const hL),smul_eq_mul,mul_one]
+  have hpoint : (fun ω => ∑ n∈S, f n ω) ≤ᵐ[P] (fun ω => L ω/16) :=
+    hL0.mono fun ω hω => cleanup_counts_le S hω
+  have hI := integral_mono_ae (integrable_finsetSum _ (fun n hn => hint n)) (hInt.div_const 16) hpoint
+  rw [integral_finsetSum _ (fun n hn => hint n),integral_div] at hI
+  simpa only [he,moment,pow_one] using hI
+
+ 
+
+theorem cleanup_tail (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hInt : Integrable L P) :
+    Summable (fun n : ℕ => P.real {ω | 16*((n:ℝ)+1) ≤ L ω}) ∧
+    (∑' n : ℕ, P.real {ω | 16*((n:ℝ)+1) ≤ L ω}) ≤ moment P L 1/16 :=
+  ⟨summable_of_sum_le (fun _ => measureReal_nonneg) (cleanup_sum_le P L hL hL0 hInt),
+   Real.tsum_le_of_sum_le (fun _ => measureReal_nonneg) (cleanup_sum_le P L hL hL0 hInt)⟩
+
+lemma upper_tail_bound_geometric (n : ℕ) (hn : 0 < n) :
+    (4:ℝ)^n/(16:ℝ)^n/(1-(1/2:ℝ)^n) ≤ 2*(1/4:ℝ)^n := by
+  have hq : (1/2:ℝ)^n ≤ 1/2 := by
+    simpa using pow_le_pow_of_le_one (by norm_num : (0:ℝ)≤1/2) (by norm_num : (1/2:ℝ)≤1)
+      (show 1 ≤ n from hn)
+  have hden : (1/2:ℝ) ≤ 1-(1/2:ℝ)^n := by linarith
+  rw [← div_pow]
+  norm_num only [show (4:ℝ)/16=1/4 by norm_num]
+  calc
+    (1/4:ℝ)^n/(1-(1/2:ℝ)^n) ≤ (1/4:ℝ)^n/(1/2) :=
+      div_le_div_of_nonneg_left (by positivity) (by norm_num) hden
+    _ = 2*(1/4:ℝ)^n := by ring
+
+lemma summable_noncleanup_tails (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
+    Summable (fun n : ℕ =>
+      (((noncleanup (moment P L) (n+1)).map (fun b => P.real {ω | 16*b ≤ L ω})).sum)) := by
+  have hgeom := (summable_geometric_of_lt_one (by norm_num : (0:ℝ)≤1/4)
+    (by norm_num : (1/4:ℝ)<1)).mul_left 2
+  have hgeom' : Summable (fun n : ℕ => 2*(1/4:ℝ)^(n+1)) :=
+    hgeom.comp_injective Nat.succ_injective
+  apply Summable.of_nonneg_of_le _ _ hgeom'
+  · intro n
+    exact List.sum_nonneg (by intro x hx; obtain ⟨b,hb,rfl⟩ := List.mem_map.mp hx; exact measureReal_nonneg)
+  · intro n
+    rw [noncleanup_sum]
+    obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
+    exact (upper_tail_finite P L hL hL0 (n+1) (by omega) (hint (n+1)) (hm (n+1) (by omega))).trans
+      (upper_tail_bound_geometric (n+1) (by omega))
+
+ 
+
+theorem summable_batch_tails (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
+    Summable (fun n : ℕ =>
+      (((batch (moment P L) (n+1)).map (fun b => P.real {ω | 16*b ≤ L ω})).sum)) := by
+  have hInt : Integrable L P := by simpa only [pow_one] using hint 1
+  have hs := (summable_noncleanup_tails P L hL hL0 hint hmean).add
+    (cleanup_tail P L hL hL0 hInt).1
+  simpa only [batch,List.map_append,List.sum_append,List.map_cons,List.map_nil,List.sum_cons,
+    List.sum_nil,add_zero,Nat.cast_add,Nat.cast_one] using hs
+
+ 
+theorem summable_entry_tails (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
+    Summable (fun i : EntryIndex (moment P L) => P.real {ω | 16*threshold (moment P L) i ≤ L ω}) := by
+  have hs := summable_batch_tails P L hL hL0 hint hmean
+  have hinj : Function.Injective (fun n : {n : ℕ // 0 < n} => n.val-1) := by
+    intro n j h
+    have hn := n.property
+    have hj := j.property
+    change n.val-1=j.val-1 at h
+    apply Subtype.ext
+    omega
+  have hs' := hs.comp_injective hinj
+  have hseq : Summable (fun n : {n : ℕ // 0 < n} =>
+      (((batch (moment P L) n.val).map (fun b => P.real {ω | 16*b ≤ L ω})).sum)) := by
+    apply hs'.congr
+    intro n
+    dsimp only [Function.comp_apply]
+    rw [Nat.sub_add_cancel n.property]
+  apply (summable_sigma_of_nonneg (fun i => measureReal_nonneg)).mpr
+  refine ⟨fun n => (hasSum_fintype _).summable,?_⟩
+  apply hseq.congr
+  intro n
+  rw [tsum_fintype]
+  exact (fin_sum_batch (moment P L) n.val (fun b => P.real {ω | 16*b ≤ L ω})).symm
+
+ 
+
+theorem tail_summability (L : Ω → ℝ) (hL : Measurable L) (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P) (hmean : 0 < moment P L 1) :
+    (∑' i : EntryIndex (moment P L), P {ω | 16*threshold (moment P L) i ≤ L ω}) < ∞ := by
+  have h := (summable_entry_tails P L hL hL0 hint hmean).tsum_ofReal_lt_top
+  simpa only [Measure.real,ENNReal.ofReal_toReal (measure_ne_top P _)] using h
+
+end SLEExactGauge.MomentBatches
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped ENNReal
+namespace SLEExactGauge.MomentBatches
+
+variable {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
+
+lemma increasing_level_bounds (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
+    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
+    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
+    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω))) :
+    ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω ∧ Y j ω ≤ L ω := by
+  filter_upwards [hY0,hmono,hlim] with ω h0 hm hl
+  exact fun j => ⟨h0 j,hm.ge_of_tendsto hl j⟩
+
+lemma integrable_level_pow (L Y : Ω → ℝ) (hY : Measurable Y)
+    (hYL : ∀ᵐ ω ∂P, 0 ≤ Y ω ∧ Y ω ≤ L ω) (n : ℕ)
+    (hint : Integrable (fun ω => (L ω)^n) P) :
+    Integrable (fun ω => (Y ω)^n) P := by
+  refine hint.mono_nonneg (hY.pow_const n).aestronglyMeasurable ?_ ?_
+  · exact hYL.mono fun ω hω => pow_nonneg hω.1 n
+  · exact hYL.mono fun ω hω => pow_le_pow_left₀ hω.1 hω.2 n
+
+ 
+
+theorem increasing_moments (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
+    (hY : ∀ j, Measurable (Y j))
+    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
+    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
+    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω)))
+    (n : ℕ) (hint : Integrable (fun ω => (L ω)^n) P) :
+    (∀ j, Integrable (fun ω => (Y j ω)^n) P) ∧
+    Monotone (fun j => moment P (Y j) n) ∧
+    Tendsto (fun j => moment P (Y j) n) atTop (𝓝 (moment P L n)) := by
+  have hb := increasing_level_bounds P L Y hY0 hmono hlim
+  have hI (j : ℕ) : Integrable (fun ω => (Y j ω)^n) P :=
+    integrable_level_pow P L (Y j) (hY j) (hb.mono fun ω hω => hω j) n hint
+  refine ⟨hI,?_,?_⟩
+  · intro j k hjk
+    apply integral_mono_ae (hI j) (hI k)
+    filter_upwards [hY0,hmono] with ω h0 hm
+    exact pow_le_pow_left₀ (h0 j) (hm hjk) n
+  · apply tendsto_integral_of_dominated_convergence (fun ω => (L ω)^n)
+      (fun j => (hY j |>.pow_const n).aestronglyMeasurable) hint
+    · intro j
+      filter_upwards [hb] with ω hω
+      rw [Real.norm_of_nonneg (pow_nonneg (hω j).1 n)]
+      exact pow_le_pow_left₀ (hω j).1 (hω j).2 n
+    · exact hlim.mono fun ω hω => hω.pow n
+
+ 
+
+lemma level_eq_tsum_indicators {v : ℕ → ℝ} {M : ℝ} (hM : 0 < M)
+    (hlim : Tendsto v atTop (𝓝 M)) :
+    level v M = 1 + ∑' j : ℕ, if 1 ≤ j ∧ v j ≤ 3*M/4 then 1 else 0 := by
+  classical
+  have hfin := finite_badLevels hM hlim
+  rw [tsum_eq_sum (s := hfin.toFinset) (fun j hj => ?_)]
+  · simp only [level,dite_eq_left hfin]
+    congr 1
+    calc
+      hfin.toFinset.card = ∑ _ ∈ hfin.toFinset, (1:ℕ) := by simp
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro j hj
+        have hj' : 1 ≤ j ∧ v j ≤ 3*M/4 := by
+          simpa only [Finite.mem_toFinset,badLevels,mem_ofPred_eq] using hj
+        simp only [ite_eq_left hj']
+  · have hnj : ¬ (1 ≤ j ∧ v j ≤ 3*M/4) := by
+      simpa only [Finite.mem_toFinset,badLevels,mem_ofPred_eq] using hj
+    exact ite_eq_right hnj
+
+ 
+
+def momentLevel (L : Ω → ℝ) (Y : ℕ → Ω → ℝ) (n : ℕ) : ℕ :=
+  level (fun j => moment P (Y j) n) (moment P L n)
+
+def fullMomentGauge (L : Ω → ℝ) (Y : ℕ → Ω → ℝ) (a d : ℝ) : ℝ → ℝ :=
+  momentGauge (moment P L) a d (momentLevel P L Y)
+
+variable [IsProbabilityMeasure P]
+
+ 
+
+theorem selected_level_moment (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
+    (hY : ∀ j, Measurable (Y j))
+    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
+    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
+    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω)))
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P)
+    (hmean : 0 < moment P L 1) (n : ℕ) (hn : 0 < n) :
+    0 < momentLevel P L Y n ∧
+    (badLevels (fun j => moment P (Y j) n) (moment P L n)).Finite ∧
+    3*moment P L n/4 < moment P (Y (momentLevel P L Y n)) n := by
+  have hb := increasing_level_bounds P L Y hY0 hmono hlim
+  have hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω := hb.mono fun ω hω => (hω 0).1.trans (hω 0).2
+  have hm := (moment_sequence_properties P L hL0 hint hmean).1 n hn
+  obtain ⟨hI,hmn,hlimn⟩ := increasing_moments P L Y hY hY0 hmono hlim n (hint n)
+  exact ⟨level_pos _ _,finite_badLevels hm hlimn,level_moment_gt hm hmn hlimn⟩
+
+ 
+
+theorem success_selected_level (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
+    (hY : ∀ j, Measurable (Y j))
+    (hY0 : ∀ᵐ ω ∂P, ∀ j, 0 ≤ Y j ω)
+    (hmono : ∀ᵐ ω ∂P, Monotone (fun j => Y j ω))
+    (hlim : ∀ᵐ ω ∂P, Tendsto (fun j => Y j ω) atTop (𝓝 (L ω)))
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P)
+    (hmean : 0 < moment P L 1) (n : ℕ) (hn : 3 ≤ n) :
+    (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) ≤
+      (∑' k : ℤ, (multiplicity (moment P L) n k:ℝ)*
+        P.real {ω | (2:ℝ)^k ≤ Y (momentLevel P L Y n) ω}) ∧
+    (2:ℝ)^n/8 ≤ (2:ℝ)^(n-2)-1/(2*(1-(1/2:ℝ)^n)) := by
+  have hb := (increasing_level_bounds P L Y hY0 hmono hlim).mono
+    (fun ω hω => hω (momentLevel P L Y n))
+  exact success P L (Y (momentLevel P L Y n)) (hY _) hb hint hmean n hn
+    (integrable_level_pow P L _ (hY _) hb n (hint n))
+    (selected_level_moment P L Y hY hY0 hmono hlim hint hmean n (by omega)).2.2.le
+
+ 
+
+theorem fullMomentGauge_properties (L : Ω → ℝ) (Y : ℕ → Ω → ℝ)
+    (hL0 : ∀ᵐ ω ∂P, 0 ≤ L ω)
+    (hint : ∀ n, Integrable (fun ω => (L ω)^n) P)
+    (hmean : 0 < moment P L 1) {a d : ℝ}
+    (ha : 0 ≤ a) (hd : 0 < d) (hd2 : d ≤ 2) :
+    Target.IsHausdorffGauge (fullMomentGauge P L Y a d) ∧
+    ∀ C r : ℝ, 1 ≤ C → 0 ≤ r →
+      fullMomentGauge P L Y a d (C*r) ≤ C^2*fullMomentGauge P L Y a d r := by
+  obtain ⟨hm,hJ⟩ := moment_sequence_properties P L hL0 hint hmean
+  exact ⟨momentGauge_admissible _ hm hJ ha hd hd2 _,
+    fun C r hC _ => momentGauge_doubling _ hm hJ a d _ hC⟩
+
+end SLEExactGauge.MomentBatches
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
+open MomentBatches
+
+def sequenceMassReal (x : ℕ→ℝ≥0∞) : ℝ := (diskSequenceMass x).toReal
+def sequenceLevelReal (l : ℕ) (x : ℕ→ℝ≥0∞) : ℝ := (x l).toReal
+
+lemma sequenceMassReal_measurable : Measurable sequenceMassReal :=
+  diskSequenceMass_measurable.ennreal_toReal
+
+lemma sequenceLevelReal_measurable (l : ℕ) : Measurable (sequenceLevelReal l) :=
+  (measurable_pi_apply l).ennreal_toReal
+
+lemma sequenceMassReal_nonneg (x : ℕ→ℝ≥0∞) : 0 ≤ sequenceMassReal x := ENNReal.toReal_nonneg
+lemma sequenceLevelReal_nonneg (l : ℕ) (x : ℕ→ℝ≥0∞) : 0 ≤ sequenceLevelReal l x :=
+  ENNReal.toReal_nonneg
+
+lemma sequenceReal_monotone {x : ℕ→ℝ≥0∞} (hmono : Monotone x)
+    (hfin : diskSequenceMass x<∞) : Monotone (fun l=>sequenceLevelReal l x) := by
+  intro l k hlk
+  exact ENNReal.toReal_mono ((le_iSup x k).trans_lt hfin).ne (hmono hlk)
+
+lemma sequenceReal_tendsto {x : ℕ→ℝ≥0∞} (hmono : Monotone x)
+    (hfin : diskSequenceMass x<∞) :
+    Tendsto (fun l=>sequenceLevelReal l x) atTop (𝓝 (sequenceMassReal x)) :=
+  (ENNReal.tendsto_toReal hfin.ne).comp (diskSequenceMass_tendsto hmono)
+
+lemma sequenceMassReal_integrable_pow (Q : Measure (ℕ→ℝ≥0∞))
+    (m : ℕ) (hf : (∫⁻x,diskSequenceMass x^m∂Q)<∞) :
+    Integrable (fun x=>sequenceMassReal x^m) Q := by
+  simpa only [ENNReal.toReal_pow, sequenceMassReal] using
+    integrable_toReal_of_lintegral_ne_top
+      (diskSequenceMass_measurable.pow_const m).aemeasurable hf.ne
+
+lemma sequenceMassReal_moment (Q : Measure (ℕ→ℝ≥0∞))
+    (hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞) (m : ℕ) :
+    moment Q sequenceMassReal m=(∫⁻x,diskSequenceMass x^m∂Q).toReal := by
+  unfold moment sequenceMassReal
+  simp only [←ENNReal.toReal_pow]
+  exact integral_toReal (diskSequenceMass_measurable.pow_const m).aemeasurable
+    (hfin.mono (fun _ hx=>ENNReal.pow_lt_top hx))
+
+lemma sequenceLevelReal_moment (W : Measure GDriver) (a : ℝ) (ha : 1/4<a)
+    (μ : ℝ≥0∞→GDriver→Measure UpperPlane)
+    (hμ : Measurable (fun p : ℝ≥0∞×GDriver=>μ p.1 p.2))
+    [∀v b,IsFiniteMeasureOnCompacts (μ v b)]
+    (r : ℝ) (Q : Measure (ℕ→ℝ≥0∞))
+    (hQlaw : ∀n,Q.map (initialVector n)=(finiteAngleLaw W a n).map
+      (fun p=>fun i : Fin (n+1)=>subdiskMass a μ r i p))
+    (hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞) (m l : ℕ) :
+    moment Q (sequenceLevelReal l) m=(subdiskMoment W a μ r m l).toReal := by
+  have he := lintegral_coordinate Q (finiteAngleLaw W a l) (subdiskMass a μ r)
+    (subdiskMass_measurable a ha μ hμ r) l (hQlaw l) (fun x=>x^m)
+    (measurable_id.pow_const m)
+  unfold moment sequenceLevelReal
+  simp only [←ENNReal.toReal_pow]
+  rw [integral_toReal ((measurable_pi_apply l).pow_const m).aemeasurable
+    (hfin.mono (fun x hx=>ENNReal.pow_lt_top ((le_iSup x l).trans_lt hx))),he]
+  rfl
+
+ 
+
+theorem sequenceReal_moment_data (Q : Measure (ℕ→ℝ≥0∞)) [IsProbabilityMeasure Q]
+    (hmono : ∀ᵐ x∂Q,Monotone x) (hfin : ∀ᵐ x∂Q,diskSequenceMass x<∞)
+    (hmom : ∀m : ℕ, 1 ≤ m → 0<(∫⁻x,diskSequenceMass x^m∂Q) ∧
+      (∫⁻x,diskSequenceMass x^m∂Q)<∞) :
+    (∀ᵐ x∂Q,Monotone (fun l=>sequenceLevelReal l x) ∧
+      Tendsto (fun l=>sequenceLevelReal l x) atTop (𝓝 (sequenceMassReal x))) ∧
+    (∀m : ℕ,Integrable (fun x=>sequenceMassReal x^m) Q) ∧
+    (∀m : ℕ, 1 ≤ m → 0 < moment Q sequenceMassReal m) ∧
+    (∀m : ℕ,Monotone (fun l=>moment Q (sequenceLevelReal l) m) ∧
+      Tendsto (fun l=>moment Q (sequenceLevelReal l) m) atTop
+        (𝓝 (moment Q sequenceMassReal m))) := by
+  have hp : ∀ᵐ x∂Q,Monotone (fun l=>sequenceLevelReal l x) ∧
+      Tendsto (fun l=>sequenceLevelReal l x) atTop (𝓝 (sequenceMassReal x)) := by
+    filter_upwards [hmono,hfin] with x hx hxf
+    exact ⟨sequenceReal_monotone hx hxf,sequenceReal_tendsto hx hxf⟩
+  have hI (m : ℕ) : Integrable (fun x=>sequenceMassReal x^m) Q := by
+    rcases m with _|m
+    · simpa only [pow_zero] using integrable_const (1:ℝ)
+    · exact sequenceMassReal_integrable_pow Q (m+1) (hmom _ (by omega)).2
+  refine ⟨hp,hI,?_,?_⟩
+  · intro m hm
+    rw [sequenceMassReal_moment Q hfin m]
+    exact ENNReal.toReal_pos (hmom m hm).1.ne' (hmom m hm).2.ne
+  · intro m
+    exact (increasing_moments Q sequenceMassReal sequenceLevelReal sequenceLevelReal_measurable
+      (Eventually.of_forall (fun x l=>sequenceLevelReal_nonneg l x))
+      (hp.mono (fun _ hx=>hx.1)) (hp.mono (fun _ hx=>hx.2)) m (hI m)).2
+
+end SLEExactGauge.CenteredMoments
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.CenteredMoments
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen MassVersions CompactSequence
+open TraceMassIntegration TraceMassContinuity MomentBatches
+
+ 
+
+structure CenteredMomentModel (κ a : ℝ) (ha : 1/4<a) (W : Measure GDriver) where
+  supports : ℕ→Finset ℕ
+  coeff : ℕ→ℕ→ℝ
+  nonneg : ∀k j,j∈supports k → 0≤coeff k j
+  tails : ∀k j,j∈supports k → k≤j
+  normalized : ∀k,∑j∈supports k,coeff k j=1
+  radiusIndex : ℕ
+  law : Measure (ℕ→ℝ≥0∞)
+  probability : IsProbabilityMeasure law
+  finite_laws : ∀n,law.map (initialVector n)=(finiteAngleLaw W a n).map
+    (fun p=>fun i : Fin (n+1)=>subdiskMass a (recoveredMass a ha supports coeff nonneg)
+      (subdiskRadius radiusIndex) i p)
+  increasing_finite : ∀ᵐ x∂law,Monotone x ∧ diskSequenceMass x<∞
+  positive_moments : ∀m : ℕ, 1 ≤ m → 0<(∫⁻x,diskSequenceMass x^m∂law) ∧
+    (∫⁻x,diskSequenceMass x^m∂law)<∞
+  canonical : CanonicalMassData κ a ha W (recoveredMass a ha supports coeff nonneg)
+
+ 
+
+theorem actual_centered_moment_model (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W) :
+    Nonempty (CenteredMomentModel κ (Target.a κ) (a_gt_quarter hκ hκ8)
+      (W.map driverHomeomorph)) := by
+  obtain ⟨s,c,hc,hs,hsum,D,hreg,hμ,hraw,hpath,n₀,Q,hQ,hlaw,hseq,hmom⟩ :=
+    actual_subdisk_positive_moments κ hκ hκ8 W hW
+  exact ⟨{
+    supports := s
+    coeff := c
+    nonneg := hc
+    tails := hs
+    normalized := hsum
+    radiusIndex := n₀
+    law := Q
+    probability := hQ
+    finite_laws := hlaw
+    increasing_finite := hseq.mono (fun x hx=>⟨hx.1,hx.2.2⟩)
+    positive_moments := fun m hm=>⟨(hmom m hm).1,(hmom m hm).2.1⟩
+    canonical := D }⟩
+
+namespace CenteredMomentModel
+variable {κ a : ℝ} {ha : 1/4<a} {W : Measure GDriver} (M : CenteredMomentModel κ a ha W)
+
+ 
+
+def gauge (d : ℝ) : ℝ→ℝ := fullMomentGauge M.law sequenceMassReal sequenceLevelReal a d
+
+def mass : ℝ≥0∞→GDriver→Measure UpperPlane :=
+  recoveredMass a ha M.supports M.coeff M.nonneg
+
+lemma mass_measurable : Measurable (fun p : ℝ≥0∞×GDriver=>M.mass p.1 p.2) :=
+  recoveredMass_joint_measurable a ha M.supports M.coeff M.nonneg
+
+lemma integrable_moments (m : ℕ) : Integrable (fun x=>sequenceMassReal x^m) M.law := by
+  let := M.probability
+  exact (sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
+    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.1 m
+
+lemma moments_positive (m : ℕ) (hm : 1 ≤ m) : 0 < moment M.law sequenceMassReal m := by
+  let := M.probability
+  exact (sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
+    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.2.1 m hm
+
+lemma moment_finite_level (m l : ℕ) :
+    moment M.law (sequenceLevelReal l) m=
+      (subdiskMoment W a M.mass (subdiskRadius M.radiusIndex) m l).toReal := by
+  let (v : ℝ≥0∞) (b : GDriver) : IsFiniteMeasureOnCompacts (M.mass v b) := M.canonical.finite_compact v b
+  exact sequenceLevelReal_moment W a ha M.mass M.mass_measurable (subdiskRadius M.radiusIndex)
+    M.law M.finite_laws (M.increasing_finite.mono (fun _ hx=>hx.2)) m l
+
+lemma finite_moments_increasing (m : ℕ) : Monotone
+    (fun l=>moment M.law (sequenceLevelReal l) m) := by
+  let := M.probability
+  exact ((sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
+    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.2.2 m).1
+
+lemma finite_moments_tendsto (m : ℕ) :
+    Tendsto (fun l=>moment M.law (sequenceLevelReal l) m) atTop
+      (𝓝 (moment M.law sequenceMassReal m)) := by
+  let := M.probability
+  exact ((sequenceReal_moment_data M.law (M.increasing_finite.mono (fun _ hx=>hx.1))
+    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments).2.2.2 m).2
+
+lemma gauge_admissible {d : ℝ} (hd : 0<d) (hd2 : d≤2) : Target.IsHausdorffGauge (M.gauge d) := by
+  let := M.probability
+  exact (fullMomentGauge_properties M.law sequenceMassReal sequenceLevelReal
+    (Eventually.of_forall sequenceMassReal_nonneg) M.integrable_moments
+    (M.moments_positive 1 le_rfl) (by linarith : 0≤a) hd hd2).1
+
+lemma gauge_doubling {d : ℝ} (hd : 0<d) (hd2 : d≤2) (C r : ℝ) (hC : 1≤C) (hr : 0≤r) :
+    M.gauge d (C*r) ≤ C^2*M.gauge d r := by
+  let := M.probability
+  exact (fullMomentGauge_properties M.law sequenceMassReal sequenceLevelReal
+    (Eventually.of_forall sequenceMassReal_nonneg) M.integrable_moments
+    (M.moments_positive 1 le_rfl) (by linarith : 0≤a) hd hd2).2 C r hC hr
+
+lemma selected_moments (n : ℕ) (hn : 0<n) :
+    let l := momentLevel M.law sequenceMassReal sequenceLevelReal n
+    0<l ∧ 3*moment M.law sequenceMassReal n/4 < moment M.law (sequenceLevelReal l) n := by
+  let := M.probability
+  obtain ⟨hp,hI,hm,hlim⟩ := sequenceReal_moment_data M.law
+    (M.increasing_finite.mono (fun _ hx=>hx.1))
+    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments
+  have hh := selected_level_moment M.law sequenceMassReal sequenceLevelReal sequenceLevelReal_measurable
+    (Eventually.of_forall (fun x l=>sequenceLevelReal_nonneg l x))
+    (hp.mono (fun _ hx=>hx.1)) (hp.mono (fun _ hx=>hx.2)) hI (hm 1 le_rfl) n hn
+  exact ⟨hh.1,hh.2.2⟩
+
+ 
+
+lemma success (n : ℕ) (hn : 3≤n) :
+    (2:ℝ)^n/8 ≤ ∑'k : ℤ,(multiplicity (moment M.law sequenceMassReal) n k:ℝ)*
+      M.law.real {x | (2:ℝ)^k ≤ sequenceLevelReal
+        (momentLevel M.law sequenceMassReal sequenceLevelReal n) x} := by
+  let := M.probability
+  obtain ⟨hp,hI,hm,hlim⟩ := sequenceReal_moment_data M.law
+    (M.increasing_finite.mono (fun _ hx=>hx.1))
+    (M.increasing_finite.mono (fun _ hx=>hx.2)) M.positive_moments
+  have hh := success_selected_level M.law sequenceMassReal sequenceLevelReal sequenceLevelReal_measurable
+    (Eventually.of_forall (fun x l=>sequenceLevelReal_nonneg l x))
+    (hp.mono (fun _ hx=>hx.1)) (hp.mono (fun _ hx=>hx.2)) hI (hm 1 le_rfl) n hn
+  exact hh.2.trans hh.1
+
+ 
+lemma tails_summable : Summable (fun i : EntryIndex (moment M.law sequenceMassReal)=>
+    M.law.real {x | 16*threshold (moment M.law sequenceMassReal) i ≤ sequenceMassReal x}) := by
+  let := M.probability
+  exact summable_entry_tails M.law sequenceMassReal sequenceMassReal_measurable
+    (Eventually.of_forall sequenceMassReal_nonneg) M.integrable_moments (M.moments_positive 1 le_rfl)
+
+end CenteredMomentModel
+end SLEExactGauge.CenteredMoments
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TerminalExhaustion.lean b/lean/OAI/Probability/SLEGauge/TerminalExhaustion.lean
index 0b69a8b9d82f6bf17dc626b448df686f7fe23d5b..0c383d1be6bda311230f06378e21f580edbd39f1 100644
--- a/lean/OAI/Probability/SLEGauge/TerminalExhaustion.lean
+++ b/lean/OAI/Probability/SLEGauge/TerminalExhaustion.lean
@@ -1,486 +1,486 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TracePastSupport
-import OAI.Probability.SLEGauge.BrownianGerm
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem terminal_avoids_eternal_survival (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (μ : ℕ → Kernel ExactSLEGenerator.Driver UpperPlane)
-    (ν : Kernel ExactSLEGenerator.Driver UpperPlane)
-    [∀ b, IsFiniteMeasureOnCompacts (ν b)]
-    (hpast : ∀ᵐ b ∂W, ∀ n : ℕ, μ n b {z : UpperPlane | alive a b.1 n z.1}=0)
-    (hkernel : ∀ n : ℕ, ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-      Measurable[(rawFiltration (n : ℝ≥0)).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-      (∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ n b ∂W) +
-        ∫⁻b, ∫⁻z, H (b,z)*ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W) :
-    ∀ᵐ b ∂W, ν b {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1}=0 := by
-  have := isSFiniteKernel_of_locally_finite ν
-  let U : ℕ → Set (ExactSLEGenerator.Driver × UpperPlane) :=
-    fun n => {p | alive a p.1.1 n p.2.1}
-  have hU (n : ℕ) :
-      MeasurableSet[(rawFiltration (n : ℝ≥0)).prod (inferInstance : MeasurableSpace UpperPlane)]
-        (U n) := by
-    simpa only [compl_ofPred,not_not] using (deadSet_raw_measurable a ha n).compl
-  have hU₀ (n : ℕ) : MeasurableSet (U n) := prod_sigma_le (rawFiltration.le n) _ _ (hU n)
-  let D := ⋂ n, U n
-  have hD : MeasurableSet D := MeasurableSet.iInter hU₀
-  have hbound (K : Set UpperPlane) (hK : IsCompact K) (n : ℕ) :
-      (∫⁻b, ν b ({z | (b,z)∈D}∩K) ∂W) ≤ ∫⁻b, ∫⁻z in K,
-        ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W := by
-    let E : Set (ExactSLEGenerator.Driver × UpperPlane) := U n ∩ Prod.snd ⁻¹' K
-    have hE : MeasurableSet[(rawFiltration (n : ℝ≥0)).prod
-        (inferInstance : MeasurableSpace UpperPlane)] E :=
-      (hU n).inter (measurable_snd hK.measurableSet)
-    have hE₀ : MeasurableSet E := prod_sigma_le (rawFiltration.le n) _ _ hE
-    let H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞ := E.indicator (fun _ => 1)
-    have hH : Measurable[(rawFiltration (n : ℝ≥0)).prod
-        (inferInstance : MeasurableSpace UpperPlane)] H := measurable_const.indicator hE
-    have hev (ρ : Measure UpperPlane) (b : ExactSLEGenerator.Driver) :
-        (∫⁻z, H (b,z) ∂ρ) = ρ ({z | (b,z)∈U n}∩K) := by
-      have hh := lintegral_indicator_const_comp (μ := ρ)
-        ((measurable_const (a := b)).prodMk measurable_id) hE₀ (1 : ℝ≥0∞)
-      have hs : (fun z : UpperPlane => (b,id z)) ⁻¹' E = {z | (b,z)∈U n}∩K := by
-        ext z
-        rfl
-      rw [hs,one_mul] at hh
-      exact hh
-    have hzero : (∫⁻b, ∫⁻z, H (b,z) ∂μ n b ∂W)=0 := by
-      have h : (fun b => ∫⁻z, H (b,z) ∂μ n b) =ᵐ[W] (fun _ => 0) := by
-        filter_upwards [hpast] with b hb
-        rw [hev]
-        exact measure_mono_null inter_subset_left (hb n)
-      rw [lintegral_congr_ae h,lintegral_zero]
-    calc
-      _ ≤ ∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W := by
-        apply lintegral_mono
-        intro b
-        dsimp only
-        rw [hev]
-        exact measure_mono (inter_subset_inter_left K (fun z hz => mem_iInter.mp hz n))
-      _ = ∫⁻b, ∫⁻z, H (b,z)*ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W := by
-        rw [hkernel n H hH,hzero,zero_add]
-      _ ≤ _ := by
-        apply lintegral_mono
-        intro b
-        dsimp only
-        rw [← lintegral_indicator hK.measurableSet]
-        apply lintegral_mono
-        intro z
-        by_cases hz : z∈K
-        · rw [indicator_of_mem hz]
-          by_cases he : (b,z)∈E
-          · simp only [H,indicator_of_mem he,one_mul,le_refl]
-          · simp only [H,indicator_of_notMem he,zero_mul,zero_le]
-        · have he : (b,z)∉E := fun he => hz he.2
-          simp only [H,indicator_of_notMem he,indicator_of_notMem hz,zero_mul,le_refl]
-  let K := CompactExhaustion.choice UpperPlane
-  have hnull (k : ℕ) : ∀ᵐ b ∂W, ν b ({z | (b,z)∈D}∩K k)=0 := by
-    apply (lintegral_eq_zero_iff (Kernel.measurable_kernel_prodMk_left
-      (hD.inter (measurable_snd (K.isCompact k).measurableSet)))).mp
-    apply le_antisymm _ (bot_le)
-    exact ge_of_tendsto' (survivingGreen_compact_mean_tendsto_zero a ha W hW (K k)
-      (K.isCompact k)) (hbound (K k) (K.isCompact k))
-  filter_upwards [ae_all_iff.mpr hnull] with b hb
-  have he : (⋃ k, {z : UpperPlane | (b,z)∈D}∩K k) =
-      {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1} := by
-    rw [← inter_iUnion,K.iUnion_eq,inter_univ]
-    ext z
-    simp only [D,U,mem_ofPred_eq,mem_iInter]
-  rw [← he]
-  exact measure_iUnion_null hb
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X]
-    [MeasurableSpace X] [BorelSpace X]
-
-omit [TopologicalSpace X] [T2Space X] [BorelSpace X] in
-lemma restriction_exhaustion (ν : Measure X) (D : ℕ → Set X)
-    (hmono : Monotone D) (hnull : ν (⋃ n, D n)ᶜ=0) :
-    ∀ A : Set X, MeasurableSet A →
-      Tendsto (fun n => ν.restrict (D n) A) atTop (𝓝 (ν A)) := by
-  have hfull : ν.restrict (⋃ n, D n)=ν :=
-    Measure.restrict_eq_self_of_ae_mem (by
-      simpa only [mem_compl_iff,not_not] using measure_eq_zero_iff_ae_notMem.mp hnull)
-  intro A hA
-  have hm : Monotone (fun n => A∩D n) :=
-    fun i j hij => inter_subset_inter_right A (hmono hij)
-  have ht := tendsto_measure_iUnion_atTop (μ := ν) hm
-  have hset : (⋃ n, A∩D n)=A∩⋃ n, D n := by rw [inter_iUnion]
-  simp only [Function.comp_def,hset,← Measure.restrict_apply hA,hfull] at ht
-  exact ht
-
-lemma locally_finite_sub_restrict_apply (ν : Measure X)
-    [IsFiniteMeasureOnCompacts ν] (D : Set X) (hD : MeasurableSet D)
-    (K : Set X) (hK : IsCompact K) :
-    (ν-ν.restrict D) K=ν (K∩Dᶜ) := by
-  let : IsFiniteMeasure (ν.restrict K) :=
-    ⟨by simpa only [Measure.restrict_apply_univ] using hK.measure_lt_top (μ := ν)⟩
-  have he : (ν-ν.restrict D).restrict K = ν.restrict K - (ν.restrict K).restrict D := by
-    rw [Measure.restrict_sub_eq_restrict_sub_restrict hK.measurableSet,
-      Measure.restrict_comm hK.measurableSet]
-  calc
-    _ = (ν.restrict K - (ν.restrict K).restrict D) univ := by
-      rw [← he,Measure.restrict_apply_univ]
-    _ = ν K - ν (K∩D) := by
-      rw [Measure.sub_apply MeasurableSet.univ Measure.restrict_le_self]
-      simp only [Measure.restrict_apply_univ,Measure.restrict_apply hD,inter_comm]
-    _ = ν (K∩Dᶜ) := by
-      have hfin : ν (K∩D) ≠ ∞ := (measure_mono inter_subset_left |>.trans_lt hK.measure_lt_top).ne
-      rw [← measure_inter_add_sdiff K hD,ENNReal.add_sub_cancel_left hfin]
-      rfl
-
- 
-
-lemma remaining_restriction_tendsto_zero (ν : Measure X)
-    [IsFiniteMeasureOnCompacts ν] (D : ℕ → Set X)
-    (hD : ∀ n, MeasurableSet (D n)) (hmono : Monotone D)
-    (hnull : ν (⋃ n, D n)ᶜ=0) :
-    ∀ K : Set X, IsCompact K →
-      Tendsto (fun n => (ν-ν.restrict (D n)) K) atTop (𝓝 0) := by
-  intro K hK
-  let U := fun n => K∩(D n)ᶜ
-  have hU (n : ℕ) : MeasurableSet (U n) := hK.measurableSet.inter (hD n).compl
-  have hm : Antitone U := fun i j hij => inter_subset_inter_right K (compl_subset_compl.mpr (hmono hij))
-  have hf : ∃ n, ν (U n) ≠ ∞ := ⟨0,
-    (measure_mono inter_subset_left |>.trans_lt hK.measure_lt_top).ne⟩
-  have he : (⋂ n, U n)=K∩(⋃ n, D n)ᶜ := by
-    ext x
-    simp only [U,mem_iInter,mem_inter_iff,mem_compl_iff,mem_iUnion,not_exists,forall_and]
-    simp only [forall_const]
-  have hzero : ν (⋂ n, U n)=0 := by
-    rw [he]
-    exact measure_mono_null inter_subset_right hnull
-  have ht := tendsto_measure_iInter_atTop (fun n => (hU n).nullMeasurableSet) hm hf
-  simpa only [Function.comp_def,hzero,locally_finite_sub_restrict_apply ν _ (hD _) K hK]
-    using ht
-
-end SLEExactGauge.MassVersions
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-
-theorem actual_raw_mass_exhaustion (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) (idx : ℕ → ℕ)
-    (hidx : ∀ n : ℕ, t (idx n) = (n : ℝ≥0)) (j : ℕ) (hj : t j=∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
-        (∀ i, NullSingletonClass (μ (t i) b)) ∧
-        (∀ i (v : ℝ≥0), t i=v → μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0) ∧
-        (∀ i j, t i≠∞ → t j=∞ →
-          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
-        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-          μ (t i) b=(μ (t j) b).restrict {z : UpperPlane | ¬alive a b.1 v z.1}) ∧
-        μ ∞ b {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1}=0 ∧
-        (∀ A : Set UpperPlane, MeasurableSet A →
-          Tendsto (fun n : ℕ => μ (n : ℝ≥0) b A) atTop (𝓝 (μ ∞ b A))) ∧
-        (∀ K : Set UpperPlane, IsCompact K →
-          Tendsto (fun n : ℕ => (μ ∞ b-μ (n : ℝ≥0) b) K) atTop (𝓝 0)) := by
-  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
-    actual_raw_mass_hull_accounting a ha W hW t
-  let μ := recoveredMass a ha s c hc
-  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := (hμ v b).1
-  let ν : Kernel ExactSLEGenerator.Driver UpperPlane := ⟨μ ∞,hmeas ∞⟩
-  let θ : ℕ → Kernel ExactSLEGenerator.Driver UpperPlane :=
-    fun n => ⟨μ (n : ℝ≥0),hmeas (n : ℝ≥0)⟩
-  have (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (ν b) := hμ ∞ b |>.1
-  have hpast : ∀ᵐ b ∂W, ∀ n : ℕ, θ n b {z : UpperPlane | alive a b.1 n z.1}=0 := by
-    filter_upwards [hconv] with b hb
-    intro n
-    have h := hb.2.2.2.1 (idx n) n (hidx n)
-    simpa only [hidx n,θ,ν,Kernel.coe_mk,μ] using h
-  have he : ∀ n : ℕ, ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-      Measurable[(rawFiltration (n : ℝ≥0)).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-      (∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂θ n b ∂W) +
-        ∫⁻b, ∫⁻z, H (b,z)*ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W := by
-    intro n H hH₀
-    simpa only [hidx n,hj,θ,ν,Kernel.coe_mk,μ] using hH (idx n) j n (hidx n) hj H hH₀
-  have havoid := terminal_avoids_eternal_survival a ha W hW θ ν hpast he
-  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
-  filter_upwards [hconv,havoid] with b hb hbavoid
-  rcases hb with ⟨horder,hvague,hatom,hpast,hfuture,hrestrict⟩
-  let D : ℕ → Set UpperPlane := fun n => {z | ¬alive a b.1 n z.1}
-  have hD (n : ℕ) : MeasurableSet (D n) :=
-    ((isOpen_alive b.1.continuous n).preimage
-      (continuous_subtype_val : Continuous (Subtype.val : UpperPlane → ℂ))).measurableSet.compl
-  have hm : Monotone D := by
-    intro n m hnm z hz hn
-    exact hz (alive_mono hn (Nat.cast_le.mpr hnm))
-  have hnull : μ ∞ b (⋃ n, D n)ᶜ=0 := by
-    have hset : (⋃ n, D n)ᶜ={z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1} := by
-      ext z
-      simp only [D,mem_compl_iff,mem_iUnion,mem_ofPred_eq,not_exists,not_not]
-    rw [hset]
-    exact hbavoid
-  have hr (n : ℕ) : μ (n : ℝ≥0) b=(μ ∞ b).restrict (D n) := by
-    simpa only [hidx n,hj] using hrestrict (idx n) j n (hidx n) hj
-  refine ⟨horder,hvague,hatom,hpast,hfuture,hrestrict,hbavoid,?_,?_⟩
-  · change ∀ A : Set UpperPlane, MeasurableSet A →
-      Tendsto (fun n : ℕ => μ (n : ℝ≥0) b A) atTop (𝓝 (μ ∞ b A))
-    simp only [hr]
-    exact restriction_exhaustion (μ ∞ b) D hm hnull
-  · change ∀ K : Set UpperPlane, IsCompact K →
-      Tendsto (fun n : ℕ => (μ ∞ b-μ (n : ℝ≥0) b) K) atTop (𝓝 0)
-    simp only [hr]
-    exact remaining_restriction_tendsto_zero (μ ∞ b) D hD hm hnull
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-open SLEExactGauge.LoewnerScaling SLEExactGauge.GermPositivity
-
-lemma upperArea_univ_pos : 0<upperArea univ := by
-  rw [upperArea,upper_embedding.comap_apply,image_univ,Subtype.range_val_subtype]
-  exact upperPlane_open.measure_pos volume ⟨Complex.I,by simp⟩
-
-lemma initialGreenMeasure_univ_pos (a : ℝ) : 0 < initialGreenMeasure a univ := by
-  rw [initialGreenMeasure,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
-  rw [lintegral_pos_iff_support (initialGreen_continuous a).measurable.ennreal_ofReal]
-  have he : Function.support (fun z : UpperPlane => ENNReal.ofReal (initialGreen a z))=univ := by
-    ext z
-    simp only [Function.mem_support,mem_univ,iff_true]
-    exact (ENNReal.ofReal_pos.mpr (initialGreen_pos a z)).ne'
-  rw [he]
-  exact upperArea_univ_pos
-
- 
-
-lemma positive_probability_of_exhaustion {Ω X : Type*}
-    [MeasurableSpace Ω] [MeasurableSpace X]
-    (P : Measure Ω) (μ : ℝ≥0∞ → Ω → Measure X)
-    (hm : Measurable (μ ∞)) (hz : ∀ ω, μ 0 ω=0)
-    (hmean : 0<(μ ∞ ∘ₘ P) univ)
-    (hex : ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => μ (n : ℝ≥0) ω univ)
-      atTop (𝓝 (μ ∞ ω univ))) :
-    ∃ n : ℕ, 0<n ∧ 0<P {ω | 0<μ (n : ℝ≥0) ω univ} := by
-  by_contra! h
-  have hzero (n : ℕ) : ∀ᵐ ω ∂P, μ (n : ℝ≥0) ω univ=0 := by
-    by_cases hn : n=0
-    · subst n
-      exact Eventually.of_forall (fun ω => by simp only [Nat.cast_zero,ENNReal.coe_zero,hz]; rfl)
-    · rw [ae_iff]
-      simpa only [pos_iff_ne_zero] using le_antisymm (h n (Nat.pos_of_ne_zero hn)) bot_le
-  have hzTerminal : ∀ᵐ ω ∂P, μ ∞ ω univ=0 := by
-    filter_upwards [ae_all_iff.mpr hzero,hex] with ω hω he
-    have hh : Tendsto (fun n : ℕ => μ (n : ℝ≥0) ω univ) atTop (𝓝 0) := by
-      simpa only [hω] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ≥0∞)) atTop (𝓝 0))
-    exact tendsto_nhds_unique he hh
-  have hzmean : (μ ∞ ∘ₘ P) univ=0 := by
-    rw [Measure.bind_apply MeasurableSet.univ hm.aemeasurable]
-    rw [lintegral_congr_ae hzTerminal,lintegral_zero]
-  exact hmean.ne' hzmean
-
- 
-
-lemma recoveredMass_ae_smallTime_positive (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
-    (W : Measure ExactSLEGenerator.Driver)
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (hmean : recoveredMass a ha s c hc ∞ ∘ₘ W=initialGreenMeasure a)
-    (hex : ∀ᵐ b ∂W, Tendsto
-      (fun n : ℕ => recoveredMass a ha s c hc (n : ℝ≥0) b univ)
-      atTop (𝓝 (recoveredMass a ha s c hc ∞ b univ)))
-    (hanti : ∀ᵐ b ∂W, Antitone (fun n => recoveredMass a ha s c hc (smallTime n) b univ)) :
-    ∀ᵐ b ∂W, ∀ n : ℕ, 0<recoveredMass a ha s c hc (smallTime n) b univ := by
-  let μ := recoveredMass a ha s c hc
-  obtain ⟨u,hu,hp⟩ := positive_probability_of_exhaustion W μ
-    (recoveredMass_measurable a ha s c hc ∞) (recoveredMass_zero a ha s c hc)
-    (by rw [hmean]; exact initialGreenMeasure_univ_pos a) hex
-  let E (n : ℕ) : Set ExactSLEGenerator.Driver := {b | 0<μ (smallTime n) b univ}
-  have hE (n : ℕ) : MeasurableSet[rawFiltration (smallTime n)] (E n) :=
-    measurableSet_lt measurable_const ((Measure.measurable_coe MeasurableSet.univ).comp
-      (recoveredMass_raw_measurable a ha s c hc (smallTime n)))
-  have heAnti : ∀ᵐ b ∂W, Antitone (fun n => b∈E n) := by
-    filter_upwards [hanti] with b hb n m hnm hmem
-    exact hmem.trans_le (hb hnm)
-  have hlower (n : ℕ) : W {b | 0<μ (u : ℝ≥0) b univ}≤W (E n) := by
-    exact (recoveredMass_positivity_probability a ha s c hc W hW
-      (u : ℝ≥0) (smallTime n) (Nat.cast_pos.mpr hu) (smallTime_pos n)).le
-  exact ae_all_mem (fun t (b : ExactSLEGenerator.Driver) => b.1 t)
-    driver_eval_measurable (fun b => b.1.continuous) (fun b => b.2)
-    W hW.toIsPreBrownianReal smallTime tendsto_smallTime E hE heAnti hp hlower
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-open SLEExactGauge.GermPositivity
-
-theorem actual_raw_mass_positive (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (t : ℕ → ℝ≥0∞) (idx : ℕ → ℕ)
-    (hidx : ∀ n : ℕ, t (idx n) = (n : ℝ≥0))
-    (is : ℕ → ℕ) (his : ∀ n : ℕ, t (is n)=(smallTime n : ℝ≥0∞))
-    (j : ℕ) (hj : t j=∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass a ha s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
-      ∀ᵐ b ∂W,
-        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
-        (∀ i (f : C_c(UpperPlane,ℝ)),
-          Tendsto (fun k => ∑ j∈s k, c k j *
-            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
-            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
-        (∀ i, NullSingletonClass (μ (t i) b)) ∧
-        (∀ i (v : ℝ≥0), t i=v → μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0) ∧
-        (∀ i j, t i≠∞ → t j=∞ →
-          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
-        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-          μ (t i) b=(μ (t j) b).restrict {z : UpperPlane | ¬alive a b.1 v z.1}) ∧
-        μ ∞ b {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1}=0 ∧
-        (∀ A : Set UpperPlane, MeasurableSet A →
-          Tendsto (fun n : ℕ => μ (n : ℝ≥0) b A) atTop (𝓝 (μ ∞ b A))) ∧
-        (∀ K : Set UpperPlane, IsCompact K →
-          Tendsto (fun n : ℕ => (μ ∞ b-μ (n : ℝ≥0) b) K) atTop (𝓝 0)) ∧
-        (∀ i, 0<t i → 0<μ (t i) b univ) ∧
-        (∀ i, 0<t i → ∃ K : Set UpperPlane, IsCompact K ∧
-          0<μ (t i) b K ∧ μ (t i) b K<∞) := by
-  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
-    actual_raw_mass_exhaustion a ha W hW t idx hidx j hj
-  let μ := recoveredMass a ha s c hc
-  have hanti : ∀ᵐ b ∂W, Antitone (fun n => μ (smallTime n) b univ) := by
-    filter_upwards [hconv] with b hb n m hnm
-    have ht : t (is m)≤t (is n) := by
-      rw [his,his]
-      exact_mod_cast smallTime_antitone hnm
-    have hh := hb.1 (is m) (is n) ht
-    rw [his,his] at hh
-    exact Measure.le_iff'.mp hh univ
-  have hex : ∀ᵐ b ∂W, Tendsto (fun n : ℕ => μ (n : ℝ≥0) b univ)
-      atTop (𝓝 (μ ∞ b univ)) := by
-    filter_upwards [hconv] with b hb
-    exact hb.2.2.2.2.2.2.2.1 univ MeasurableSet.univ
-  have hmean' : μ ∞ ∘ₘ W=initialGreenMeasure a := by
-    simpa only [hj] using hmean j hj
-  have hpos := recoveredMass_ae_smallTime_positive a ha s c hc W hW hmean' hex hanti
-  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
-  filter_upwards [hconv,hpos] with b hb hpb
-  have hpositive (i : ℕ) (hi : 0<t i) : 0<μ (t i) b univ := by
-    have hlim : Tendsto (fun n => (smallTime n : ℝ≥0∞)) atTop (𝓝 0) :=
-      ENNReal.continuous_coe.tendsto 0 |>.comp tendsto_smallTime
-    obtain ⟨n,hn⟩ := (hlim.eventually (gt_mem_nhds hi)).exists
-    have hh := hb.1 (is n) i (by simpa only [his] using hn.le)
-    rw [his] at hh
-    exact (hpb n).trans_le (Measure.le_iff'.mp hh univ)
-  rcases hb with ⟨ho,hv,hat,hpa,hfu,hr,he,hx,hk⟩
-  refine ⟨ho,hv,hat,hpa,hfu,hr,he,hx,hk,hpositive,?_⟩
-  intro i hi
-  let : (μ (t i) b).Regular := (hμ _ _).2
-  let : IsFiniteMeasureOnCompacts (μ (t i) b) := (hμ _ _).1
-  obtain ⟨K,-,hK,hpK⟩ := isOpen_univ.exists_lt_isCompact (hpositive i hi)
-  exact ⟨K,hK,hpK,hK.measure_lt_top⟩
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TracePastSupport
+import OAI.Probability.SLEGauge.BrownianGerm
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem terminal_avoids_eternal_survival (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (μ : ℕ → Kernel ExactSLEGenerator.Driver UpperPlane)
+    (ν : Kernel ExactSLEGenerator.Driver UpperPlane)
+    [∀ b, IsFiniteMeasureOnCompacts (ν b)]
+    (hpast : ∀ᵐ b ∂W, ∀ n : ℕ, μ n b {z : UpperPlane | alive a b.1 n z.1}=0)
+    (hkernel : ∀ n : ℕ, ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+      Measurable[(rawFiltration (n : ℝ≥0)).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+      (∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ n b ∂W) +
+        ∫⁻b, ∫⁻z, H (b,z)*ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W) :
+    ∀ᵐ b ∂W, ν b {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1}=0 := by
+  have := isSFiniteKernel_of_locally_finite ν
+  let U : ℕ → Set (ExactSLEGenerator.Driver × UpperPlane) :=
+    fun n => {p | alive a p.1.1 n p.2.1}
+  have hU (n : ℕ) :
+      MeasurableSet[(rawFiltration (n : ℝ≥0)).prod (inferInstance : MeasurableSpace UpperPlane)]
+        (U n) := by
+    simpa only [compl_ofPred,not_not] using (deadSet_raw_measurable a ha n).compl
+  have hU₀ (n : ℕ) : MeasurableSet (U n) := prod_sigma_le (rawFiltration.le n) _ _ (hU n)
+  let D := ⋂ n, U n
+  have hD : MeasurableSet D := MeasurableSet.iInter hU₀
+  have hbound (K : Set UpperPlane) (hK : IsCompact K) (n : ℕ) :
+      (∫⁻b, ν b ({z | (b,z)∈D}∩K) ∂W) ≤ ∫⁻b, ∫⁻z in K,
+        ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W := by
+    let E : Set (ExactSLEGenerator.Driver × UpperPlane) := U n ∩ Prod.snd ⁻¹' K
+    have hE : MeasurableSet[(rawFiltration (n : ℝ≥0)).prod
+        (inferInstance : MeasurableSpace UpperPlane)] E :=
+      (hU n).inter (measurable_snd hK.measurableSet)
+    have hE₀ : MeasurableSet E := prod_sigma_le (rawFiltration.le n) _ _ hE
+    let H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞ := E.indicator (fun _ => 1)
+    have hH : Measurable[(rawFiltration (n : ℝ≥0)).prod
+        (inferInstance : MeasurableSpace UpperPlane)] H := measurable_const.indicator hE
+    have hev (ρ : Measure UpperPlane) (b : ExactSLEGenerator.Driver) :
+        (∫⁻z, H (b,z) ∂ρ) = ρ ({z | (b,z)∈U n}∩K) := by
+      have hh := lintegral_indicator_const_comp (μ := ρ)
+        ((measurable_const (a := b)).prodMk measurable_id) hE₀ (1 : ℝ≥0∞)
+      have hs : (fun z : UpperPlane => (b,id z)) ⁻¹' E = {z | (b,z)∈U n}∩K := by
+        ext z
+        rfl
+      rw [hs,one_mul] at hh
+      exact hh
+    have hzero : (∫⁻b, ∫⁻z, H (b,z) ∂μ n b ∂W)=0 := by
+      have h : (fun b => ∫⁻z, H (b,z) ∂μ n b) =ᵐ[W] (fun _ => 0) := by
+        filter_upwards [hpast] with b hb
+        rw [hev]
+        exact measure_mono_null inter_subset_left (hb n)
+      rw [lintegral_congr_ae h,lintegral_zero]
+    calc
+      _ ≤ ∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W := by
+        apply lintegral_mono
+        intro b
+        dsimp only
+        rw [hev]
+        exact measure_mono (inter_subset_inter_left K (fun z hz => mem_iInter.mp hz n))
+      _ = ∫⁻b, ∫⁻z, H (b,z)*ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W := by
+        rw [hkernel n H hH,hzero,zero_add]
+      _ ≤ _ := by
+        apply lintegral_mono
+        intro b
+        dsimp only
+        rw [← lintegral_indicator hK.measurableSet]
+        apply lintegral_mono
+        intro z
+        by_cases hz : z∈K
+        · rw [indicator_of_mem hz]
+          by_cases he : (b,z)∈E
+          · simp only [H,indicator_of_mem he,one_mul,le_refl]
+          · simp only [H,indicator_of_notMem he,zero_mul,zero_le]
+        · have he : (b,z)∉E := fun he => hz he.2
+          simp only [H,indicator_of_notMem he,indicator_of_notMem hz,zero_mul,le_refl]
+  let K := CompactExhaustion.choice UpperPlane
+  have hnull (k : ℕ) : ∀ᵐ b ∂W, ν b ({z | (b,z)∈D}∩K k)=0 := by
+    apply (lintegral_eq_zero_iff (Kernel.measurable_kernel_prodMk_left
+      (hD.inter (measurable_snd (K.isCompact k).measurableSet)))).mp
+    apply le_antisymm _ (bot_le)
+    exact ge_of_tendsto' (survivingGreen_compact_mean_tendsto_zero a ha W hW (K k)
+      (K.isCompact k)) (hbound (K k) (K.isCompact k))
+  filter_upwards [ae_all_iff.mpr hnull] with b hb
+  have he : (⋃ k, {z : UpperPlane | (b,z)∈D}∩K k) =
+      {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1} := by
+    rw [← inter_iUnion,K.iUnion_eq,inter_univ]
+    ext z
+    simp only [D,U,mem_ofPred_eq,mem_iInter]
+  rw [← he]
+  exact measure_iUnion_null hb
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X]
+    [MeasurableSpace X] [BorelSpace X]
+
+omit [TopologicalSpace X] [T2Space X] [BorelSpace X] in
+lemma restriction_exhaustion (ν : Measure X) (D : ℕ → Set X)
+    (hmono : Monotone D) (hnull : ν (⋃ n, D n)ᶜ=0) :
+    ∀ A : Set X, MeasurableSet A →
+      Tendsto (fun n => ν.restrict (D n) A) atTop (𝓝 (ν A)) := by
+  have hfull : ν.restrict (⋃ n, D n)=ν :=
+    Measure.restrict_eq_self_of_ae_mem (by
+      simpa only [mem_compl_iff,not_not] using measure_eq_zero_iff_ae_notMem.mp hnull)
+  intro A hA
+  have hm : Monotone (fun n => A∩D n) :=
+    fun i j hij => inter_subset_inter_right A (hmono hij)
+  have ht := tendsto_measure_iUnion_atTop (μ := ν) hm
+  have hset : (⋃ n, A∩D n)=A∩⋃ n, D n := by rw [inter_iUnion]
+  simp only [Function.comp_def,hset,← Measure.restrict_apply hA,hfull] at ht
+  exact ht
+
+lemma locally_finite_sub_restrict_apply (ν : Measure X)
+    [IsFiniteMeasureOnCompacts ν] (D : Set X) (hD : MeasurableSet D)
+    (K : Set X) (hK : IsCompact K) :
+    (ν-ν.restrict D) K=ν (K∩Dᶜ) := by
+  let : IsFiniteMeasure (ν.restrict K) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using hK.measure_lt_top (μ := ν)⟩
+  have he : (ν-ν.restrict D).restrict K = ν.restrict K - (ν.restrict K).restrict D := by
+    rw [Measure.restrict_sub_eq_restrict_sub_restrict hK.measurableSet,
+      Measure.restrict_comm hK.measurableSet]
+  calc
+    _ = (ν.restrict K - (ν.restrict K).restrict D) univ := by
+      rw [← he,Measure.restrict_apply_univ]
+    _ = ν K - ν (K∩D) := by
+      rw [Measure.sub_apply MeasurableSet.univ Measure.restrict_le_self]
+      simp only [Measure.restrict_apply_univ,Measure.restrict_apply hD,inter_comm]
+    _ = ν (K∩Dᶜ) := by
+      have hfin : ν (K∩D) ≠ ∞ := (measure_mono inter_subset_left |>.trans_lt hK.measure_lt_top).ne
+      rw [← measure_inter_add_sdiff K hD,ENNReal.add_sub_cancel_left hfin]
+      rfl
+
+ 
+
+lemma remaining_restriction_tendsto_zero (ν : Measure X)
+    [IsFiniteMeasureOnCompacts ν] (D : ℕ → Set X)
+    (hD : ∀ n, MeasurableSet (D n)) (hmono : Monotone D)
+    (hnull : ν (⋃ n, D n)ᶜ=0) :
+    ∀ K : Set X, IsCompact K →
+      Tendsto (fun n => (ν-ν.restrict (D n)) K) atTop (𝓝 0) := by
+  intro K hK
+  let U := fun n => K∩(D n)ᶜ
+  have hU (n : ℕ) : MeasurableSet (U n) := hK.measurableSet.inter (hD n).compl
+  have hm : Antitone U := fun i j hij => inter_subset_inter_right K (compl_subset_compl.mpr (hmono hij))
+  have hf : ∃ n, ν (U n) ≠ ∞ := ⟨0,
+    (measure_mono inter_subset_left |>.trans_lt hK.measure_lt_top).ne⟩
+  have he : (⋂ n, U n)=K∩(⋃ n, D n)ᶜ := by
+    ext x
+    simp only [U,mem_iInter,mem_inter_iff,mem_compl_iff,mem_iUnion,not_exists,forall_and]
+    simp only [forall_const]
+  have hzero : ν (⋂ n, U n)=0 := by
+    rw [he]
+    exact measure_mono_null inter_subset_right hnull
+  have ht := tendsto_measure_iInter_atTop (fun n => (hU n).nullMeasurableSet) hm hf
+  simpa only [Function.comp_def,hzero,locally_finite_sub_restrict_apply ν _ (hD _) K hK]
+    using ht
+
+end SLEExactGauge.MassVersions
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+
+theorem actual_raw_mass_exhaustion (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) (idx : ℕ → ℕ)
+    (hidx : ∀ n : ℕ, t (idx n) = (n : ℝ≥0)) (j : ℕ) (hj : t j=∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
+        (∀ i, NullSingletonClass (μ (t i) b)) ∧
+        (∀ i (v : ℝ≥0), t i=v → μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0) ∧
+        (∀ i j, t i≠∞ → t j=∞ →
+          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
+        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+          μ (t i) b=(μ (t j) b).restrict {z : UpperPlane | ¬alive a b.1 v z.1}) ∧
+        μ ∞ b {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1}=0 ∧
+        (∀ A : Set UpperPlane, MeasurableSet A →
+          Tendsto (fun n : ℕ => μ (n : ℝ≥0) b A) atTop (𝓝 (μ ∞ b A))) ∧
+        (∀ K : Set UpperPlane, IsCompact K →
+          Tendsto (fun n : ℕ => (μ ∞ b-μ (n : ℝ≥0) b) K) atTop (𝓝 0)) := by
+  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
+    actual_raw_mass_hull_accounting a ha W hW t
+  let μ := recoveredMass a ha s c hc
+  let (v : ℝ≥0∞) (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (μ v b) := (hμ v b).1
+  let ν : Kernel ExactSLEGenerator.Driver UpperPlane := ⟨μ ∞,hmeas ∞⟩
+  let θ : ℕ → Kernel ExactSLEGenerator.Driver UpperPlane :=
+    fun n => ⟨μ (n : ℝ≥0),hmeas (n : ℝ≥0)⟩
+  have (b : ExactSLEGenerator.Driver) : IsFiniteMeasureOnCompacts (ν b) := hμ ∞ b |>.1
+  have hpast : ∀ᵐ b ∂W, ∀ n : ℕ, θ n b {z : UpperPlane | alive a b.1 n z.1}=0 := by
+    filter_upwards [hconv] with b hb
+    intro n
+    have h := hb.2.2.2.1 (idx n) n (hidx n)
+    simpa only [hidx n,θ,ν,Kernel.coe_mk,μ] using h
+  have he : ∀ n : ℕ, ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+      Measurable[(rawFiltration (n : ℝ≥0)).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+      (∫⁻b, ∫⁻z, H (b,z) ∂ν b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂θ n b ∂W) +
+        ∫⁻b, ∫⁻z, H (b,z)*ENNReal.ofReal (survivingGreen a n b z.1) ∂upperArea ∂W := by
+    intro n H hH₀
+    simpa only [hidx n,hj,θ,ν,Kernel.coe_mk,μ] using hH (idx n) j n (hidx n) hj H hH₀
+  have havoid := terminal_avoids_eternal_survival a ha W hW θ ν hpast he
+  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
+  filter_upwards [hconv,havoid] with b hb hbavoid
+  rcases hb with ⟨horder,hvague,hatom,hpast,hfuture,hrestrict⟩
+  let D : ℕ → Set UpperPlane := fun n => {z | ¬alive a b.1 n z.1}
+  have hD (n : ℕ) : MeasurableSet (D n) :=
+    ((isOpen_alive b.1.continuous n).preimage
+      (continuous_subtype_val : Continuous (Subtype.val : UpperPlane → ℂ))).measurableSet.compl
+  have hm : Monotone D := by
+    intro n m hnm z hz hn
+    exact hz (alive_mono hn (Nat.cast_le.mpr hnm))
+  have hnull : μ ∞ b (⋃ n, D n)ᶜ=0 := by
+    have hset : (⋃ n, D n)ᶜ={z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1} := by
+      ext z
+      simp only [D,mem_compl_iff,mem_iUnion,mem_ofPred_eq,not_exists,not_not]
+    rw [hset]
+    exact hbavoid
+  have hr (n : ℕ) : μ (n : ℝ≥0) b=(μ ∞ b).restrict (D n) := by
+    simpa only [hidx n,hj] using hrestrict (idx n) j n (hidx n) hj
+  refine ⟨horder,hvague,hatom,hpast,hfuture,hrestrict,hbavoid,?_,?_⟩
+  · change ∀ A : Set UpperPlane, MeasurableSet A →
+      Tendsto (fun n : ℕ => μ (n : ℝ≥0) b A) atTop (𝓝 (μ ∞ b A))
+    simp only [hr]
+    exact restriction_exhaustion (μ ∞ b) D hm hnull
+  · change ∀ K : Set UpperPlane, IsCompact K →
+      Tendsto (fun n : ℕ => (μ ∞ b-μ (n : ℝ≥0) b) K) atTop (𝓝 0)
+    simp only [hr]
+    exact remaining_restriction_tendsto_zero (μ ∞ b) D hD hm hnull
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+open SLEExactGauge.LoewnerScaling SLEExactGauge.GermPositivity
+
+lemma upperArea_univ_pos : 0<upperArea univ := by
+  rw [upperArea,upper_embedding.comap_apply,image_univ,Subtype.range_val_subtype]
+  exact upperPlane_open.measure_pos volume ⟨Complex.I,by simp⟩
+
+lemma initialGreenMeasure_univ_pos (a : ℝ) : 0 < initialGreenMeasure a univ := by
+  rw [initialGreenMeasure,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ]
+  rw [lintegral_pos_iff_support (initialGreen_continuous a).measurable.ennreal_ofReal]
+  have he : Function.support (fun z : UpperPlane => ENNReal.ofReal (initialGreen a z))=univ := by
+    ext z
+    simp only [Function.mem_support,mem_univ,iff_true]
+    exact (ENNReal.ofReal_pos.mpr (initialGreen_pos a z)).ne'
+  rw [he]
+  exact upperArea_univ_pos
+
+ 
+
+lemma positive_probability_of_exhaustion {Ω X : Type*}
+    [MeasurableSpace Ω] [MeasurableSpace X]
+    (P : Measure Ω) (μ : ℝ≥0∞ → Ω → Measure X)
+    (hm : Measurable (μ ∞)) (hz : ∀ ω, μ 0 ω=0)
+    (hmean : 0<(μ ∞ ∘ₘ P) univ)
+    (hex : ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => μ (n : ℝ≥0) ω univ)
+      atTop (𝓝 (μ ∞ ω univ))) :
+    ∃ n : ℕ, 0<n ∧ 0<P {ω | 0<μ (n : ℝ≥0) ω univ} := by
+  by_contra! h
+  have hzero (n : ℕ) : ∀ᵐ ω ∂P, μ (n : ℝ≥0) ω univ=0 := by
+    by_cases hn : n=0
+    · subst n
+      exact Eventually.of_forall (fun ω => by simp only [Nat.cast_zero,ENNReal.coe_zero,hz]; rfl)
+    · rw [ae_iff]
+      simpa only [pos_iff_ne_zero] using le_antisymm (h n (Nat.pos_of_ne_zero hn)) bot_le
+  have hzTerminal : ∀ᵐ ω ∂P, μ ∞ ω univ=0 := by
+    filter_upwards [ae_all_iff.mpr hzero,hex] with ω hω he
+    have hh : Tendsto (fun n : ℕ => μ (n : ℝ≥0) ω univ) atTop (𝓝 0) := by
+      simpa only [hω] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ≥0∞)) atTop (𝓝 0))
+    exact tendsto_nhds_unique he hh
+  have hzmean : (μ ∞ ∘ₘ P) univ=0 := by
+    rw [Measure.bind_apply MeasurableSet.univ hm.aemeasurable]
+    rw [lintegral_congr_ae hzTerminal,lintegral_zero]
+  exact hmean.ne' hzmean
+
+ 
+
+lemma recoveredMass_ae_smallTime_positive (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ) (hc : ∀ k j, j∈s k → 0≤c k j)
+    (W : Measure ExactSLEGenerator.Driver)
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (hmean : recoveredMass a ha s c hc ∞ ∘ₘ W=initialGreenMeasure a)
+    (hex : ∀ᵐ b ∂W, Tendsto
+      (fun n : ℕ => recoveredMass a ha s c hc (n : ℝ≥0) b univ)
+      atTop (𝓝 (recoveredMass a ha s c hc ∞ b univ)))
+    (hanti : ∀ᵐ b ∂W, Antitone (fun n => recoveredMass a ha s c hc (smallTime n) b univ)) :
+    ∀ᵐ b ∂W, ∀ n : ℕ, 0<recoveredMass a ha s c hc (smallTime n) b univ := by
+  let μ := recoveredMass a ha s c hc
+  obtain ⟨u,hu,hp⟩ := positive_probability_of_exhaustion W μ
+    (recoveredMass_measurable a ha s c hc ∞) (recoveredMass_zero a ha s c hc)
+    (by rw [hmean]; exact initialGreenMeasure_univ_pos a) hex
+  let E (n : ℕ) : Set ExactSLEGenerator.Driver := {b | 0<μ (smallTime n) b univ}
+  have hE (n : ℕ) : MeasurableSet[rawFiltration (smallTime n)] (E n) :=
+    measurableSet_lt measurable_const ((Measure.measurable_coe MeasurableSet.univ).comp
+      (recoveredMass_raw_measurable a ha s c hc (smallTime n)))
+  have heAnti : ∀ᵐ b ∂W, Antitone (fun n => b∈E n) := by
+    filter_upwards [hanti] with b hb n m hnm hmem
+    exact hmem.trans_le (hb hnm)
+  have hlower (n : ℕ) : W {b | 0<μ (u : ℝ≥0) b univ}≤W (E n) := by
+    exact (recoveredMass_positivity_probability a ha s c hc W hW
+      (u : ℝ≥0) (smallTime n) (Nat.cast_pos.mpr hu) (smallTime_pos n)).le
+  exact ae_all_mem (fun t (b : ExactSLEGenerator.Driver) => b.1 t)
+    driver_eval_measurable (fun b => b.1.continuous) (fun b => b.2)
+    W hW.toIsPreBrownianReal smallTime tendsto_smallTime E hE heAnti hp hlower
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+open SLEExactGauge.GermPositivity
+
+theorem actual_raw_mass_positive (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (t : ℕ → ℝ≥0∞) (idx : ℕ → ℕ)
+    (hidx : ∀ n : ℕ, t (idx n) = (n : ℝ≥0))
+    (is : ℕ → ℕ) (his : ∀ n : ℕ, t (is n)=(smallTime n : ℝ≥0∞))
+    (j : ℕ) (hj : t j=∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass a ha s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) W) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) W) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 W) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ W=initialGreenMeasure a) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂W) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂W) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen a v b z.1) ∂upperArea ∂W) ∧
+      ∀ᵐ b ∂W,
+        (∀ i j, t i≤t j → μ (t i) b≤μ (t j) b) ∧
+        (∀ i (f : C_c(UpperPlane,ℝ)),
+          Tendsto (fun k => ∑ j∈s k, c k j *
+            ∫z,f z∂actualUpperMass a (Real.exp (-2*a*(j+1))) (t i) b)
+            atTop (𝓝 (∫z,f z∂μ (t i) b))) ∧
+        (∀ i, NullSingletonClass (μ (t i) b)) ∧
+        (∀ i (v : ℝ≥0), t i=v → μ (t i) b {z : UpperPlane | alive a b.1 v z.1}=0) ∧
+        (∀ i j, t i≠∞ → t j=∞ →
+          (μ (t j) b-μ (t i) b) {z : UpperPlane | ¬alive a b.1 (t i).toNNReal z.1}=0) ∧
+        (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+          μ (t i) b=(μ (t j) b).restrict {z : UpperPlane | ¬alive a b.1 v z.1}) ∧
+        μ ∞ b {z : UpperPlane | ∀ n : ℕ, alive a b.1 n z.1}=0 ∧
+        (∀ A : Set UpperPlane, MeasurableSet A →
+          Tendsto (fun n : ℕ => μ (n : ℝ≥0) b A) atTop (𝓝 (μ ∞ b A))) ∧
+        (∀ K : Set UpperPlane, IsCompact K →
+          Tendsto (fun n : ℕ => (μ ∞ b-μ (n : ℝ≥0) b) K) atTop (𝓝 0)) ∧
+        (∀ i, 0<t i → 0<μ (t i) b univ) ∧
+        (∀ i, 0<t i → ∃ K : Set UpperPlane, IsCompact K ∧
+          0<μ (t i) b K ∧ μ (t i) b K<∞) := by
+  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
+    actual_raw_mass_exhaustion a ha W hW t idx hidx j hj
+  let μ := recoveredMass a ha s c hc
+  have hanti : ∀ᵐ b ∂W, Antitone (fun n => μ (smallTime n) b univ) := by
+    filter_upwards [hconv] with b hb n m hnm
+    have ht : t (is m)≤t (is n) := by
+      rw [his,his]
+      exact_mod_cast smallTime_antitone hnm
+    have hh := hb.1 (is m) (is n) ht
+    rw [his,his] at hh
+    exact Measure.le_iff'.mp hh univ
+  have hex : ∀ᵐ b ∂W, Tendsto (fun n : ℕ => μ (n : ℝ≥0) b univ)
+      atTop (𝓝 (μ ∞ b univ)) := by
+    filter_upwards [hconv] with b hb
+    exact hb.2.2.2.2.2.2.2.1 univ MeasurableSet.univ
+  have hmean' : μ ∞ ∘ₘ W=initialGreenMeasure a := by
+    simpa only [hj] using hmean j hj
+  have hpos := recoveredMass_ae_smallTime_positive a ha s c hc W hW hmean' hex hanti
+  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
+  filter_upwards [hconv,hpos] with b hb hpb
+  have hpositive (i : ℕ) (hi : 0<t i) : 0<μ (t i) b univ := by
+    have hlim : Tendsto (fun n => (smallTime n : ℝ≥0∞)) atTop (𝓝 0) :=
+      ENNReal.continuous_coe.tendsto 0 |>.comp tendsto_smallTime
+    obtain ⟨n,hn⟩ := (hlim.eventually (gt_mem_nhds hi)).exists
+    have hh := hb.1 (is n) i (by simpa only [his] using hn.le)
+    rw [his] at hh
+    exact (hpb n).trans_le (Measure.le_iff'.mp hh univ)
+  rcases hb with ⟨ho,hv,hat,hpa,hfu,hr,he,hx,hk⟩
+  refine ⟨ho,hv,hat,hpa,hfu,hr,he,hx,hk,hpositive,?_⟩
+  intro i hi
+  let : (μ (t i) b).Regular := (hμ _ _).2
+  let : IsFiniteMeasureOnCompacts (μ (t i) b) := (hμ _ _).1
+  obtain ⟨K,-,hK,hpK⟩ := isOpen_univ.exists_lt_isCompact (hpositive i hi)
+  exact ⟨K,hK,hpK,hK.measure_lt_top⟩
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TiltedMeasures.lean b/lean/OAI/Probability/SLEGauge/TiltedMeasures.lean
index af0bb25fb4a9338b765877a49568cffd82b9a2aa..a88e48f445cf5f006ed0629b5b7f6a4b4b653d10 100644
--- a/lean/OAI/Probability/SLEGauge/TiltedMeasures.lean
+++ b/lean/OAI/Probability/SLEGauge/TiltedMeasures.lean
@@ -1,453 +1,453 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StoppedWeights
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-namespace ExactSLEMain
-open ExactSLEGreen
-theorem finite_tilt_mean_one (W : Measure Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b.1 t) W)
-    (a θ : ℝ) (ha : 1 / 4 < a) (hθ : 0 < θ) (hθπ : θ < Real.pi) (l : ℕ) :
-    Integrable (fun b : Driver => tiltWeight a θ b.1 l) W ∧
-    (∫ b : Driver, tiltWeight a θ b.1 l ∂W) = 1 := by
-  have ha0 : 0 < a := by linarith
-  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
-  have hz : 0 < (fromDisk θ 0).im := by simpa only [fromDisk_zero_im] using hs
-  have hρ : 0 < Real.sin θ*Real.exp (-2*a*l) := mul_pos hs (Real.exp_pos _)
-  have hr : Real.sin θ*Real.exp (-2*a*l) ≤ (fromDisk θ 0).im := by
-    rw [fromDisk_zero_im]
-    have he : Real.exp (-2*a*l) ≤ 1 := Real.exp_le_one_iff.mpr (by
-      have h : (0 : ℝ) ≤ 2*a*l := by positivity
-      nlinarith)
-    exact (mul_le_mul_of_nonneg_left he hs.le).trans_eq (mul_one _)
-  have hm := radialWeight_mean W hW a ha (fromDisk θ 0) hz _ hρ hr
-  have he : (fun b : Driver => tiltWeight a θ b.1 l) =
-      fun b : Driver => radialWeight a b.1 (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) /
-        physicalGreen a (fromDisk θ 0,0) := funext fun b => tiltWeight_radialWeight a θ ha hθ hθπ b l
-  rw [he]
-  refine ⟨hm.1.div_const _,?_⟩
-  rw [integral_div,hm.2]
-  exact div_self (show physicalGreen a (fromDisk θ 0,0) ≠ 0 from (Real.exp_pos _).ne')
-
-end ExactSLEMain
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open scoped Classical
-
-noncomputable def radialRemaining (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) (t : ℝ≥0) : ℝ :=
-  if alive a b t z ∧ (t : ℝ≥0∞) < radialTime a b z ρ then
-    physicalGreen a (actualState a b t z) else 0
-
-noncomputable def radialStop (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) : ExactSLEGenerator.Driver → ℝ≥0 :=
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  regionStop a L.ε L.hε z L.N ρ (angleThreshold z n) L.B
-
-lemma radialStop_actual (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (ht : t ≤ radialStop a ha z hz ρ hρ hr n b) :
-    alive a b.1 t z := by
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  exact (regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht).1
-
-lemma radialStop_le_radialTime (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) (b : ExactSLEGenerator.Driver) :
-    (radialStop a ha z hz ρ hρ hr n b : ℝ≥0∞) ≤ radialTime a b.1 z ρ := by
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  apply le_sInf
-  rintro u ⟨t,rfl,_,ht⟩
-  exact ENNReal.coe_le_coe.mpr (regionStop_le_hit a L.ε L.hε z L.N ρ (angleThreshold z n)
-    L.B L.initial b t ht)
-
-lemma radialStop_after_survival (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
-    (t : ℝ≥0) (ht : alive a b.1 t z) (htr : (t : ℝ≥0∞) < radialTime a b.1 z ρ) :
-    ∀ᶠ n in atTop, t < radialStop a ha z hz ρ hρ hr n b := by
-  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
-  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T := radialStop a ha z hz ρ hρ hr n b
-  let V := localState a L.ε L.hε z T b
-  by_contra hnt
-  have hTt : T ≤ t := le_of_not_gt hnt
-  have he := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
-  rcases regionStop_true_exit a L.ε ha L.hε z L.N ρ (angleThreshold z n) hρ
-    (angleThreshold_pos z hz n) L.B L.initial L.small L.large L.time b with hR|hS
-  · have htime := regionStop_eq_radialTime a L.ε L.hε z L.N ρ (angleThreshold z n)
-      L.B L.initial b hR
-    have hle : radialTime a b.1 z ρ ≤ (t : ℝ≥0∞) := by
-      rw [htime]
-      exact ENNReal.coe_le_coe.mpr hTt
-    exact not_le.mpr htr hle
-  · have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
-    have hnp : 0 < ‖V.1‖ := (L.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
-    have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-    have hη' := hηs T hTt
-    rw [he.2,hd] at hη'
-    exact not_le.mpr hn hη'
-
-lemma radialStop_eventually_eq (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
-    (hfin : radialTime a b.1 z ρ ≠ ∞) :
-    ∀ᶠ n in atTop, (radialStop a ha z hz ρ hρ hr n b : ℝ≥0∞) = radialTime a b.1 z ρ := by
-  let t := (radialTime a b.1 z ρ).toNNReal
-  have ht := radialTime_attained a ha z hz ρ hρ hr b hfin
-  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht.1
-  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T := radialStop a ha z hz ρ hρ hr n b
-  let V := localState a L.ε L.hε z T b
-  have hTt : T ≤ t := regionStop_le_hit a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht.2
-  have he := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
-  rcases regionStop_true_exit a L.ε ha L.hε z L.N ρ (angleThreshold z n) hρ
-    (angleThreshold_pos z hz n) L.B L.initial L.small L.large L.time b with hR|hS
-  · exact (regionStop_eq_radialTime a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b hR).symm
-  · have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
-    have hnp : 0 < ‖V.1‖ := (L.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
-    have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
-    have hη' := hηs T hTt
-    rw [he.2,hd] at hη'
-    exact False.elim (not_le.mpr hn hη')
-
-lemma radialStop_before_tendsto (a : ℝ) (_ha : 1/4 < a) (ha0 : 0 < a)
-    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    Tendsto (fun n => if t < radialStop a ha0 z hz ρ hρ hr n b then
-        physicalGreen a (actualState a b.1 t z) else 0) atTop
-      (𝓝 (radialRemaining a b.1 z ρ t)) := by
-  by_cases hg : alive a b.1 t z ∧ (t : ℝ≥0∞) < radialTime a b.1 z ρ
-  · rw [radialRemaining,ite_eq_left hg]
-    apply Filter.Tendsto.congr' _ tendsto_const_nhds
-    filter_upwards [radialStop_after_survival a ha0 z hz ρ hρ hr b t hg.1 hg.2] with n hn
-    simp only [ite_eq_left hn]
-  · rw [radialRemaining,ite_eq_right hg]
-    have hc : ∀ n, ¬ t < radialStop a ha0 z hz ρ hρ hr n b := by
-      intro n hn
-      apply hg
-      exact ⟨radialStop_actual a ha0 z hz ρ hρ hr n b t hn.le,
-        (ENNReal.coe_lt_coe.mpr hn).trans_le (radialStop_le_radialTime a ha0 z hz ρ hρ hr n b)⟩
-    simp only [ite_eq_right (hc _)]
-    exact tendsto_const_nhds
-
-lemma radialStop_terminal_tendsto (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a)
-    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    Tendsto (fun n => if t < radialStop a ha0 z hz ρ hρ hr n b then
-        radialApproximation a ha0 z hz ρ hρ hr n b else 0) atTop
-      (𝓝 (if (t : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0)) := by
-  by_cases hfin : radialTime a b.1 z ρ = ∞
-  · have hg : radialWeight a b.1 z ρ = 0 := by rw [radialWeight,ite_eq_left hfin]
-    simp only [hg,ite_self]
-    have hc := radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
-    rw [hg] at hc
-    apply squeeze_zero _ _ hc
-    · intro n
-      split_ifs
-      · exact (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1
-      · exact le_rfl
-    · intro n
-      split_ifs
-      · exact le_rfl
-      · exact (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1
-  · have he : ∀ᶠ n in atTop, (t < radialStop a ha0 z hz ρ hρ hr n b) ↔
-        (t : ℝ≥0∞) < radialTime a b.1 z ρ := by
-      filter_upwards [radialStop_eventually_eq a ha0 z hz ρ hρ hr b hfin] with n hn
-      rw [← hn,ENNReal.coe_lt_coe]
-    by_cases hlt : (t : ℝ≥0∞) < radialTime a b.1 z ρ
-    · rw [ite_eq_left hlt]
-      apply Filter.Tendsto.congr' _ (radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b)
-      filter_upwards [he] with n hn
-      rw [ite_eq_left (hn.mpr hlt)]
-    · rw [ite_eq_right hlt]
-      apply Filter.Tendsto.congr' _ tendsto_const_nhds
-      filter_upwards [he] with n hn
-      rw [ite_eq_right (fun h => hlt (hn.mp h))]
-
-lemma radialStop_stopping (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) :
-    IsStoppingTime rawFiltration (fun b => (radialStop a ha z hz ρ hρ hr n b : WithTop ℝ≥0)) := by
-  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  exact regionStop_stopping a L.ε L.hε z L.N ρ (angleThreshold z n) L.B
-
-lemma stopped_event_lt {S T : ExactSLEGenerator.Driver → ℝ≥0}
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    MeasurableSet[(hS.min hT).measurableSpace] (U ∩ {b | S b < T b}) := by
-  have he : U ∩ {b | S b < T b} =
-      (U ∩ {b | (S b : WithTop ℝ≥0) ≤ T b}) \ {b | (S b : WithTop ℝ≥0) = T b} := by
-    ext b
-    simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,Set.mem_sdiff,WithTop.coe_le_coe,WithTop.coe_eq_coe]
-    exact ⟨fun h => ⟨⟨h.1,h.2.le⟩,h.2.ne⟩,fun h => ⟨h.1.1,lt_of_le_of_ne h.1.2 h.2⟩⟩
-  rw [he]
-  exact (hS.measurableSet_inter_le hT U hU).diff (hS.measurableSet_eq_stopping_time_min hT)
-
-lemma radialStop_optional (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (_ha : 1/4 < a) (ha0 : 0 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    Integrable (fun b => if S b < radialStop a ha0 z hz ρ hρ hr n b then
-      physicalGreen a (actualState a b.1 (S b) z) else 0) W ∧
-    (∫ b in U, (if S b < radialStop a ha0 z hz ρ hρ hr n b then
-        physicalGreen a (actualState a b.1 (S b) z) else 0) ∂W) =
-      ∫ b in U, (if S b < radialStop a ha0 z hz ρ hρ hr n b then
-        radialApproximation a ha0 z hz ρ hρ hr n b else 0) ∂W := by
-  let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T := radialStop a ha0 z hz ρ hρ hr n
-  have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
-  let C : Set ExactSLEGenerator.Driver := {b | S b < T b}
-  have hC : MeasurableSet C := by
-    exact (iSup_le fun t => rawFiltration.le t) _ (by
-      simpa only [WithTop.coe_lt_coe] using measurableSet_lt hS.measurable_iSup hT.measurable_iSup)
-  have hmin : IsStoppingTime rawFiltration (fun b => (min (S b) (T b) : WithTop ℝ≥0)) := by
-    simpa only [WithTop.coe_min] using hS.min hT
-  have hUC : MeasurableSet[hmin.measurableSpace] (U ∩ C) := by
-    simpa only [← WithTop.coe_min] using stopped_event_lt hS hT U hU
-  have hUC' : MeasurableSet (U ∩ C) := hmin.measurableSpace_le _ hUC
-  let f := fun b => physicalGreen a (localState a L.ε L.hε z (min (S b) (T b)) b)
-  have hsafe (b : ExactSLEGenerator.Driver) :
-      L.ε ≤ Y (localState a L.ε L.hε z (min (S b) (T b)) b) ∧
-      ‖localState a L.ε L.hε z (min (S b) (T b)) b‖ ≤ L.N := by
-    have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (min (S b) (T b)) (min_le_right (S b) (T b))
-    exact ⟨hs.1,hs.2.1⟩
-  have hf_eq : f = fun b => greenCutoff a L.ε L.N (localState a L.ε L.hε z (min (S b) (T b)) b) := by
-    funext b
-    exact ((greenCutoff_eq_near a L.ε L.hε L.N _ (hsafe b).2 (hsafe b).1).eq_of_nhds).symm
-  have hfi : Integrable f W := by
-    rw [hf_eq]
-    have hm := (greenCutoff_smooth a L.ε L.hε L.N).continuous.measurable.comp
-      (localState_stopped_measurable a L.ε L.hε z (fun b => min (S b) (T b)) hmin)
-    obtain ⟨c,hc⟩ := (greenCutoff_compact a L.ε L.N).exists_bound_of_continuous
-      (greenCutoff_smooth a L.ε L.hε L.N).continuous
-    exact Integrable.of_bound hm.aestronglyMeasurable c (Eventually.of_forall fun b => hc _)
-  have he (b : ExactSLEGenerator.Driver) (hb : b ∈ C) :
-      physicalGreen a (actualState a b.1 (S b) z) = f b := by
-    have hbt : S b < T b := hb
-    dsimp only [f]
-    rw [min_eq_left hbt.le]
-    rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (S b) hbt.le).2]
-  have hfun : (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) =
-      C.indicator f := by
-    funext b
-    by_cases hb : b ∈ C
-    · rw [Set.indicator_of_mem hb,ite_eq_left (show S b < T b from hb),he b hb]
-    · rw [Set.indicator_of_notMem hb,ite_eq_right (show ¬S b < T b from hb)]
-  constructor
-  · change Integrable (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) W
-    rw [hfun]
-    exact hfi.indicator hC
-  · change (∫ b in U, (if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) ∂W) = _
-    rw [hfun,integral_indicator hC,Measure.restrict_restrict hC,Set.inter_comm C U]
-    have ho := localGreen_optional W hW a L.ε ha0.ne' L.hε z L.N L.B
-      (fun b => min (S b) (T b)) T hmin hT (fun b => min_le_right _ _) (fun b => regionStop_le a L.ε L.hε z L.N ρ (angleThreshold z n) L.B b)
-      (fun b t ht => let hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht
-        ⟨hs.1,hs.2.1⟩) (U ∩ C) hUC
-    change (∫ b in U ∩ C, f b ∂W) = _
-    rw [ho]
-    have hfun' : (fun b => if S b < T b then radialApproximation a ha0 z hz ρ hρ hr n b else 0) =
-        C.indicator (radialApproximation a ha0 z hz ρ hρ hr n) := rfl
-    change _ = ∫ b in U, (if S b < T b then radialApproximation a ha0 z hz ρ hρ hr n b else 0) ∂W
-    rw [hfun',integral_indicator hC,Measure.restrict_restrict hC,Set.inter_comm C U]
-    rfl
-
-lemma radialStop_before_bound (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a) (z : ℂ)
-    (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
-    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    ‖if t < radialStop a ha0 z hz ρ hρ hr n b then
-      physicalGreen a (actualState a b.1 t z) else 0‖ ≤ Real.exp (-alpha a*Real.log ρ) := by
-  split_ifs with ht
-  · let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
-      (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-    rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht.le).2]
-    have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht.le
-    rw [Real.norm_eq_abs,abs_of_pos (show 0 < physicalGreen a (localState a L.ε L.hε z t b) from Real.exp_pos _)]
-    exact physicalGreen_bound a ρ ha hρ _ (L.hε.trans_le hs.1) hs.2.2.1
-  · simpa only [norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
-
-lemma radialStop_before_measurable (a : ℝ) (ha0 : 0 < a) (z : ℂ)
-    (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable[hS.measurableSpace] (fun b => if S b < radialStop a ha0 z hz ρ hρ hr n b then
-      physicalGreen a (actualState a b.1 (S b) z) else 0) := by
-  let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
-    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
-  let T := radialStop a ha0 z hz ρ hρ hr n
-  have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
-  have hmin : IsStoppingTime rawFiltration (fun b => (min (S b) (T b) : WithTop ℝ≥0)) := by
-    simpa only [WithTop.coe_min] using hS.min hT
-  have hle : hmin.measurableSpace ≤ hS.measurableSpace :=
-    hmin.measurableSpace_mono hS (fun b => WithTop.coe_le_coe.mpr (min_le_left _ _))
-  have hC : MeasurableSet[hS.measurableSpace] {b | S b < T b} := by
-    apply hle
-    simpa only [Set.univ_inter,← WithTop.coe_min] using stopped_event_lt hS hT Set.univ MeasurableSet.univ
-  have hA : StronglyAdapted rawFiltration (localState a L.ε L.hε z) :=
-    fun t => (localState_adapted a L.ε L.hε z t).stronglyMeasurable
-  have hv : Measurable[hS.measurableSpace] (fun b => localState a L.ε L.hε z (min (S b) (T b)) b) :=
-    (measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
-      (localState_continuous a L.ε L.hε z)) hmin).mono hle le_rfl
-  have hfun : (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) =
-      {b | S b < T b}.indicator
-        (fun b => greenCutoff a L.ε L.N (localState a L.ε L.hε z (min (S b) (T b)) b)) := by
-    funext b
-    by_cases hb : S b < T b
-    · rw [ite_eq_left hb,Set.indicator_of_mem (show b ∈ {b | S b < T b} from hb),min_eq_left hb.le]
-      have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (S b) hb.le
-      rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (S b) hb.le).2]
-      exact ((greenCutoff_eq_near a L.ε L.hε L.N _ hs.2.1 hs.1).eq_of_nhds).symm
-    · rw [ite_eq_right hb,Set.indicator_of_notMem (show b ∉ {b | S b < T b} from hb)]
-  change Measurable[hS.measurableSpace] (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0)
-  rw [hfun]
-  exact ((greenCutoff_smooth a L.ε L.hε L.N).continuous.measurable.comp hv).indicator hC
-
-lemma radialRemaining_measurable (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Measurable[hS.measurableSpace] (fun b => radialRemaining a b.1 z ρ (S b)) := by
-  have ha0 : 0 < a := by linarith
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  apply measurable_of_tendsto_metrizable
-    (fun n => radialStop_before_measurable a ha0 z hz ρ hρ hr n S hS)
-  exact tendsto_pi_nhds.mpr (fun b => radialStop_before_tendsto a ha ha0 z hz ρ hρ hr b (S b))
-
- 
-
-theorem radial_conditional_weight (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    Integrable (fun b => radialRemaining a b.1 z ρ (S b)) W ∧
-    (∫ b in U, (if (S b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) ∂W) =
-      ∫ b in U, radialRemaining a b.1 z ρ (S b) ∂W := by
-  have ha0 : 0 < a := by linarith
-  let F := fun n (b : ExactSLEGenerator.Driver) => if S b < radialStop a ha0 z hz ρ hρ hr n b
-    then physicalGreen a (actualState a b.1 (S b) z) else 0
-  let H := fun n (b : ExactSLEGenerator.Driver) => if S b < radialStop a ha0 z hz ρ hρ hr n b
-    then radialApproximation a ha0 z hz ρ hρ hr n b else 0
-  let G := fun b : ExactSLEGenerator.Driver => radialRemaining a b.1 z ρ (S b)
-  let K := fun b : ExactSLEGenerator.Driver => if (S b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0
-  let C := Real.exp (-alpha a*Real.log ρ)
-  have hFi (n : ℕ) : Integrable (F n) W :=
-    (radialStop_optional W hW a ha ha0 z hz ρ hρ hr n S hS U hU).1
-  have hHi (n : ℕ) : Integrable (H n) W := by
-    have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
-    have hC : MeasurableSet {b | S b < radialStop a ha0 z hz ρ hρ hr n b} := by
-      exact (iSup_le fun t => rawFiltration.le t) _ (by
-        simpa only [WithTop.coe_lt_coe] using measurableSet_lt hS.measurable_iSup hT.measurable_iSup)
-    exact (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).1.indicator hC
-  have hFc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
-    radialStop_before_tendsto a ha ha0 z hz ρ hρ hr b (S b)
-  have hHc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => H n b) atTop (𝓝 (K b)) :=
-    radialStop_terminal_tendsto a ha ha0 z hz ρ hρ hr b (S b)
-  have hFb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖F n b‖ ≤ C :=
-    radialStop_before_bound a ha ha0 z hz ρ hρ hr n b (S b)
-  have hHb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖H n b‖ ≤ C := by
-    dsimp only [H]
-    split_ifs
-    · have h := radialApproximation_bound a ha ha0 z hz ρ hρ hr n b
-      simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
-    · exact (norm_zero.trans_le (Real.exp_pos _).le)
-  have hGi : Integrable G W := Integrable.of_bound
-    ((radialRemaining_measurable a ha z hz ρ hρ hr S hS).mono hS.measurableSpace_le le_rfl).aestronglyMeasurable C
-    (Eventually.of_forall fun b => le_of_tendsto (hFc b).norm (Eventually.of_forall fun n => hFb n b))
-  refine ⟨hGi,?_⟩
-  have hFl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => C)
-    (fun n => (hFi n).aestronglyMeasurable.restrict) (integrable_const C)
-    (fun n => Eventually.of_forall (hFb n)) (Eventually.of_forall hFc)
-  have hHl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => C)
-    (fun n => (hHi n).aestronglyMeasurable.restrict) (integrable_const C)
-    (fun n => Eventually.of_forall (hHb n)) (Eventually.of_forall hHc)
-  have he : (fun n => ∫ b in U, F n b ∂W) = fun n => ∫ b in U, H n b ∂W :=
-    funext fun n => (radialStop_optional W hW a ha ha0 z hz ρ hρ hr n S hS U hU).2
-  rw [he] at hFl
-  exact tendsto_nhds_unique hHl hFl
-
-end ExactSLEGreen
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-open scoped Classical
-noncomputable def stoppedRadialRemaining (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) (t : ℝ≥0∞) : ℝ :=
-  if t = ∞ then 0 else radialRemaining a b z ρ t.toNNReal
-
-lemma radialRemaining_bound (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
-    ‖radialRemaining a b.1 z ρ t‖ ≤ Real.exp (-alpha a*Real.log ρ) := by
-  have ha0 : 0 < a := by linarith
-  exact le_of_tendsto (radialStop_before_tendsto a ha ha0 z hz ρ hρ hr b t).norm
-    (Eventually.of_forall fun n => radialStop_before_bound a ha ha0 z hz ρ hρ hr n b t)
-
-lemma radialWeight_bound (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver) :
-    0 ≤ radialWeight a b.1 z ρ ∧ radialWeight a b.1 z ρ ≤ Real.exp (-alpha a*Real.log ρ) := by
-  have ha0 : 0 < a := by linarith
-  have hc := radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
-  exact ⟨ge_of_tendsto hc (Eventually.of_forall fun n => (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1),
-    le_of_tendsto hc (Eventually.of_forall fun n => (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).2)⟩
-
-lemma radial_terminal_integrable (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
-    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
-    (S : ExactSLEGenerator.Driver → ℝ≥0)
-    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
-    Integrable (fun b => if (S b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) W := by
-  have ha0 : 0 < a := by linarith
-  let H := fun n (b : ExactSLEGenerator.Driver) => if S b < radialStop a ha0 z hz ρ hρ hr n b
-    then radialApproximation a ha0 z hz ρ hρ hr n b else 0
-  have hHi (n : ℕ) : Integrable (H n) W := by
-    have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
-    have hC : MeasurableSet {b | S b < radialStop a ha0 z hz ρ hρ hr n b} := by
-      exact (iSup_le fun t => rawFiltration.le t) _ (by
-        simpa only [WithTop.coe_lt_coe] using measurableSet_lt hS.measurable_iSup hT.measurable_iSup)
-    exact (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).1.indicator hC
-  apply Integrable.of_bound (aestronglyMeasurable_of_tendsto_ae atTop (fun n => (hHi n).aestronglyMeasurable)
-    (Eventually.of_forall fun b => radialStop_terminal_tendsto a ha ha0 z hz ρ hρ hr b (S b)))
-    (Real.exp (-alpha a*Real.log ρ))
-  filter_upwards [] with b
-  split_ifs
-  · have h := radialWeight_bound a ha z hz ρ hρ hr b
-    simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
-  · simpa only [norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
-
-noncomputable def capTime (S : ExactSLEGenerator.Driver → ℝ≥0∞) (n : ℝ≥0) (b : ExactSLEGenerator.Driver) : ℝ≥0 :=
-  (min (S b) (n : ℝ≥0∞)).toNNReal
-lemma capTime_coe (S : ExactSLEGenerator.Driver → ℝ≥0∞) (n : ℝ≥0) (b : ExactSLEGenerator.Driver) :
-    (capTime S n b : ℝ≥0∞) = min (S b) (n : ℝ≥0∞) :=
-  ENNReal.coe_toNNReal (ne_of_lt (lt_of_le_of_lt (min_le_right _ _) ENNReal.coe_lt_top))
-lemma capTime_stopping (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (n : ℝ≥0) : IsStoppingTime rawFiltration (fun b => (capTime S n b : WithTop ℝ≥0)) := by
-  have heq : (fun b => (capTime S n b : WithTop ℝ≥0)) = fun b => min (S b) (n : ℝ≥0∞) := funext (capTime_coe S n)
-  rw [heq]
-  exact hS.min_const n
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StoppedWeights
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+namespace ExactSLEMain
+open ExactSLEGreen
+theorem finite_tilt_mean_one (W : Measure Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b.1 t) W)
+    (a θ : ℝ) (ha : 1 / 4 < a) (hθ : 0 < θ) (hθπ : θ < Real.pi) (l : ℕ) :
+    Integrable (fun b : Driver => tiltWeight a θ b.1 l) W ∧
+    (∫ b : Driver, tiltWeight a θ b.1 l ∂W) = 1 := by
+  have ha0 : 0 < a := by linarith
+  have hs := Real.sin_pos_of_pos_of_lt_pi hθ hθπ
+  have hz : 0 < (fromDisk θ 0).im := by simpa only [fromDisk_zero_im] using hs
+  have hρ : 0 < Real.sin θ*Real.exp (-2*a*l) := mul_pos hs (Real.exp_pos _)
+  have hr : Real.sin θ*Real.exp (-2*a*l) ≤ (fromDisk θ 0).im := by
+    rw [fromDisk_zero_im]
+    have he : Real.exp (-2*a*l) ≤ 1 := Real.exp_le_one_iff.mpr (by
+      have h : (0 : ℝ) ≤ 2*a*l := by positivity
+      nlinarith)
+    exact (mul_le_mul_of_nonneg_left he hs.le).trans_eq (mul_one _)
+  have hm := radialWeight_mean W hW a ha (fromDisk θ 0) hz _ hρ hr
+  have he : (fun b : Driver => tiltWeight a θ b.1 l) =
+      fun b : Driver => radialWeight a b.1 (fromDisk θ 0) (Real.sin θ*Real.exp (-2*a*l)) /
+        physicalGreen a (fromDisk θ 0,0) := funext fun b => tiltWeight_radialWeight a θ ha hθ hθπ b l
+  rw [he]
+  refine ⟨hm.1.div_const _,?_⟩
+  rw [integral_div,hm.2]
+  exact div_self (show physicalGreen a (fromDisk θ 0,0) ≠ 0 from (Real.exp_pos _).ne')
+
+end ExactSLEMain
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open scoped Classical
+
+noncomputable def radialRemaining (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) (t : ℝ≥0) : ℝ :=
+  if alive a b t z ∧ (t : ℝ≥0∞) < radialTime a b z ρ then
+    physicalGreen a (actualState a b t z) else 0
+
+noncomputable def radialStop (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) : ExactSLEGenerator.Driver → ℝ≥0 :=
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  regionStop a L.ε L.hε z L.N ρ (angleThreshold z n) L.B
+
+lemma radialStop_actual (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (ht : t ≤ radialStop a ha z hz ρ hρ hr n b) :
+    alive a b.1 t z := by
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  exact (regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht).1
+
+lemma radialStop_le_radialTime (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) (b : ExactSLEGenerator.Driver) :
+    (radialStop a ha z hz ρ hρ hr n b : ℝ≥0∞) ≤ radialTime a b.1 z ρ := by
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  apply le_sInf
+  rintro u ⟨t,rfl,_,ht⟩
+  exact ENNReal.coe_le_coe.mpr (regionStop_le_hit a L.ε L.hε z L.N ρ (angleThreshold z n)
+    L.B L.initial b t ht)
+
+lemma radialStop_after_survival (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
+    (t : ℝ≥0) (ht : alive a b.1 t z) (htr : (t : ℝ≥0∞) < radialTime a b.1 z ρ) :
+    ∀ᶠ n in atTop, t < radialStop a ha z hz ρ hρ hr n b := by
+  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht
+  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T := radialStop a ha z hz ρ hρ hr n b
+  let V := localState a L.ε L.hε z T b
+  by_contra hnt
+  have hTt : T ≤ t := le_of_not_gt hnt
+  have he := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
+  rcases regionStop_true_exit a L.ε ha L.hε z L.N ρ (angleThreshold z n) hρ
+    (angleThreshold_pos z hz n) L.B L.initial L.small L.large L.time b with hR|hS
+  · have htime := regionStop_eq_radialTime a L.ε L.hε z L.N ρ (angleThreshold z n)
+      L.B L.initial b hR
+    have hle : radialTime a b.1 z ρ ≤ (t : ℝ≥0∞) := by
+      rw [htime]
+      exact ENNReal.coe_le_coe.mpr hTt
+    exact not_le.mpr htr hle
+  · have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
+    have hnp : 0 < ‖V.1‖ := (L.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
+    have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+    have hη' := hηs T hTt
+    rw [he.2,hd] at hη'
+    exact not_le.mpr hn hη'
+
+lemma radialStop_eventually_eq (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver)
+    (hfin : radialTime a b.1 z ρ ≠ ∞) :
+    ∀ᶠ n in atTop, (radialStop a ha z hz ρ hρ hr n b : ℝ≥0∞) = radialTime a b.1 z ρ := by
+  let t := (radialTime a b.1 z ρ).toNNReal
+  have ht := radialTime_attained a ha z hz ρ hρ hr b hfin
+  obtain ⟨η,hη,hηs⟩ := actual_sine_lower a b.1 b.1.continuous z t ht.1
+  filter_upwards [(angleThreshold_tendsto z).eventually (gt_mem_nhds hη)] with n hn
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T := radialStop a ha z hz ρ hρ hr n b
+  let V := localState a L.ε L.hε z T b
+  have hTt : T ≤ t := regionStop_le_hit a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht.2
+  have he := regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
+  rcases regionStop_true_exit a L.ε ha L.hε z L.N ρ (angleThreshold z n) hρ
+    (angleThreshold_pos z hz n) L.B L.initial L.small L.large L.time b with hR|hS
+  · exact (regionStop_eq_radialTime a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b hR).symm
+  · have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b T le_rfl
+    have hnp : 0 < ‖V.1‖ := (L.hε.trans_le hs.1).trans_le (Complex.im_le_norm _)
+    have hd : Y V/‖V.1‖ = angleThreshold z n := (div_eq_iff hnp.ne').mpr hS
+    have hη' := hηs T hTt
+    rw [he.2,hd] at hη'
+    exact False.elim (not_le.mpr hn hη')
+
+lemma radialStop_before_tendsto (a : ℝ) (_ha : 1/4 < a) (ha0 : 0 < a)
+    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    Tendsto (fun n => if t < radialStop a ha0 z hz ρ hρ hr n b then
+        physicalGreen a (actualState a b.1 t z) else 0) atTop
+      (𝓝 (radialRemaining a b.1 z ρ t)) := by
+  by_cases hg : alive a b.1 t z ∧ (t : ℝ≥0∞) < radialTime a b.1 z ρ
+  · rw [radialRemaining,ite_eq_left hg]
+    apply Filter.Tendsto.congr' _ tendsto_const_nhds
+    filter_upwards [radialStop_after_survival a ha0 z hz ρ hρ hr b t hg.1 hg.2] with n hn
+    simp only [ite_eq_left hn]
+  · rw [radialRemaining,ite_eq_right hg]
+    have hc : ∀ n, ¬ t < radialStop a ha0 z hz ρ hρ hr n b := by
+      intro n hn
+      apply hg
+      exact ⟨radialStop_actual a ha0 z hz ρ hρ hr n b t hn.le,
+        (ENNReal.coe_lt_coe.mpr hn).trans_le (radialStop_le_radialTime a ha0 z hz ρ hρ hr n b)⟩
+    simp only [ite_eq_right (hc _)]
+    exact tendsto_const_nhds
+
+lemma radialStop_terminal_tendsto (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a)
+    (z : ℂ) (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    Tendsto (fun n => if t < radialStop a ha0 z hz ρ hρ hr n b then
+        radialApproximation a ha0 z hz ρ hρ hr n b else 0) atTop
+      (𝓝 (if (t : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0)) := by
+  by_cases hfin : radialTime a b.1 z ρ = ∞
+  · have hg : radialWeight a b.1 z ρ = 0 := by rw [radialWeight,ite_eq_left hfin]
+    simp only [hg,ite_self]
+    have hc := radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
+    rw [hg] at hc
+    apply squeeze_zero _ _ hc
+    · intro n
+      split_ifs
+      · exact (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1
+      · exact le_rfl
+    · intro n
+      split_ifs
+      · exact le_rfl
+      · exact (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1
+  · have he : ∀ᶠ n in atTop, (t < radialStop a ha0 z hz ρ hρ hr n b) ↔
+        (t : ℝ≥0∞) < radialTime a b.1 z ρ := by
+      filter_upwards [radialStop_eventually_eq a ha0 z hz ρ hρ hr b hfin] with n hn
+      rw [← hn,ENNReal.coe_lt_coe]
+    by_cases hlt : (t : ℝ≥0∞) < radialTime a b.1 z ρ
+    · rw [ite_eq_left hlt]
+      apply Filter.Tendsto.congr' _ (radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b)
+      filter_upwards [he] with n hn
+      rw [ite_eq_left (hn.mpr hlt)]
+    · rw [ite_eq_right hlt]
+      apply Filter.Tendsto.congr' _ tendsto_const_nhds
+      filter_upwards [he] with n hn
+      rw [ite_eq_right (fun h => hlt (hn.mp h))]
+
+lemma radialStop_stopping (a : ℝ) (ha : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ) :
+    IsStoppingTime rawFiltration (fun b => (radialStop a ha z hz ρ hρ hr n b : WithTop ℝ≥0)) := by
+  let L := radialLocalization a ha z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  exact regionStop_stopping a L.ε L.hε z L.N ρ (angleThreshold z n) L.B
+
+lemma stopped_event_lt {S T : ExactSLEGenerator.Driver → ℝ≥0}
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (hT : IsStoppingTime rawFiltration (fun b => (T b : WithTop ℝ≥0)))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    MeasurableSet[(hS.min hT).measurableSpace] (U ∩ {b | S b < T b}) := by
+  have he : U ∩ {b | S b < T b} =
+      (U ∩ {b | (S b : WithTop ℝ≥0) ≤ T b}) \ {b | (S b : WithTop ℝ≥0) = T b} := by
+    ext b
+    simp only [Set.mem_inter_iff,Set.mem_ofPred_eq,Set.mem_sdiff,WithTop.coe_le_coe,WithTop.coe_eq_coe]
+    exact ⟨fun h => ⟨⟨h.1,h.2.le⟩,h.2.ne⟩,fun h => ⟨h.1.1,lt_of_le_of_ne h.1.2 h.2⟩⟩
+  rw [he]
+  exact (hS.measurableSet_inter_le hT U hU).diff (hS.measurableSet_eq_stopping_time_min hT)
+
+lemma radialStop_optional (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (_ha : 1/4 < a) (ha0 : 0 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    Integrable (fun b => if S b < radialStop a ha0 z hz ρ hρ hr n b then
+      physicalGreen a (actualState a b.1 (S b) z) else 0) W ∧
+    (∫ b in U, (if S b < radialStop a ha0 z hz ρ hρ hr n b then
+        physicalGreen a (actualState a b.1 (S b) z) else 0) ∂W) =
+      ∫ b in U, (if S b < radialStop a ha0 z hz ρ hρ hr n b then
+        radialApproximation a ha0 z hz ρ hρ hr n b else 0) ∂W := by
+  let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T := radialStop a ha0 z hz ρ hρ hr n
+  have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
+  let C : Set ExactSLEGenerator.Driver := {b | S b < T b}
+  have hC : MeasurableSet C := by
+    exact (iSup_le fun t => rawFiltration.le t) _ (by
+      simpa only [WithTop.coe_lt_coe] using measurableSet_lt hS.measurable_iSup hT.measurable_iSup)
+  have hmin : IsStoppingTime rawFiltration (fun b => (min (S b) (T b) : WithTop ℝ≥0)) := by
+    simpa only [WithTop.coe_min] using hS.min hT
+  have hUC : MeasurableSet[hmin.measurableSpace] (U ∩ C) := by
+    simpa only [← WithTop.coe_min] using stopped_event_lt hS hT U hU
+  have hUC' : MeasurableSet (U ∩ C) := hmin.measurableSpace_le _ hUC
+  let f := fun b => physicalGreen a (localState a L.ε L.hε z (min (S b) (T b)) b)
+  have hsafe (b : ExactSLEGenerator.Driver) :
+      L.ε ≤ Y (localState a L.ε L.hε z (min (S b) (T b)) b) ∧
+      ‖localState a L.ε L.hε z (min (S b) (T b)) b‖ ≤ L.N := by
+    have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (min (S b) (T b)) (min_le_right (S b) (T b))
+    exact ⟨hs.1,hs.2.1⟩
+  have hf_eq : f = fun b => greenCutoff a L.ε L.N (localState a L.ε L.hε z (min (S b) (T b)) b) := by
+    funext b
+    exact ((greenCutoff_eq_near a L.ε L.hε L.N _ (hsafe b).2 (hsafe b).1).eq_of_nhds).symm
+  have hfi : Integrable f W := by
+    rw [hf_eq]
+    have hm := (greenCutoff_smooth a L.ε L.hε L.N).continuous.measurable.comp
+      (localState_stopped_measurable a L.ε L.hε z (fun b => min (S b) (T b)) hmin)
+    obtain ⟨c,hc⟩ := (greenCutoff_compact a L.ε L.N).exists_bound_of_continuous
+      (greenCutoff_smooth a L.ε L.hε L.N).continuous
+    exact Integrable.of_bound hm.aestronglyMeasurable c (Eventually.of_forall fun b => hc _)
+  have he (b : ExactSLEGenerator.Driver) (hb : b ∈ C) :
+      physicalGreen a (actualState a b.1 (S b) z) = f b := by
+    have hbt : S b < T b := hb
+    dsimp only [f]
+    rw [min_eq_left hbt.le]
+    rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (S b) hbt.le).2]
+  have hfun : (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) =
+      C.indicator f := by
+    funext b
+    by_cases hb : b ∈ C
+    · rw [Set.indicator_of_mem hb,ite_eq_left (show S b < T b from hb),he b hb]
+    · rw [Set.indicator_of_notMem hb,ite_eq_right (show ¬S b < T b from hb)]
+  constructor
+  · change Integrable (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) W
+    rw [hfun]
+    exact hfi.indicator hC
+  · change (∫ b in U, (if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) ∂W) = _
+    rw [hfun,integral_indicator hC,Measure.restrict_restrict hC,Set.inter_comm C U]
+    have ho := localGreen_optional W hW a L.ε ha0.ne' L.hε z L.N L.B
+      (fun b => min (S b) (T b)) T hmin hT (fun b => min_le_right _ _) (fun b => regionStop_le a L.ε L.hε z L.N ρ (angleThreshold z n) L.B b)
+      (fun b t ht => let hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht
+        ⟨hs.1,hs.2.1⟩) (U ∩ C) hUC
+    change (∫ b in U ∩ C, f b ∂W) = _
+    rw [ho]
+    have hfun' : (fun b => if S b < T b then radialApproximation a ha0 z hz ρ hρ hr n b else 0) =
+        C.indicator (radialApproximation a ha0 z hz ρ hρ hr n) := rfl
+    change _ = ∫ b in U, (if S b < T b then radialApproximation a ha0 z hz ρ hρ hr n b else 0) ∂W
+    rw [hfun',integral_indicator hC,Measure.restrict_restrict hC,Set.inter_comm C U]
+    rfl
+
+lemma radialStop_before_bound (a : ℝ) (ha : 1/4 < a) (ha0 : 0 < a) (z : ℂ)
+    (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
+    (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    ‖if t < radialStop a ha0 z hz ρ hρ hr n b then
+      physicalGreen a (actualState a b.1 t z) else 0‖ ≤ Real.exp (-alpha a*Real.log ρ) := by
+  split_ifs with ht
+  · let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
+      (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+    rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht.le).2]
+    have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b t ht.le
+    rw [Real.norm_eq_abs,abs_of_pos (show 0 < physicalGreen a (localState a L.ε L.hε z t b) from Real.exp_pos _)]
+    exact physicalGreen_bound a ρ ha hρ _ (L.hε.trans_le hs.1) hs.2.2.1
+  · simpa only [norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
+
+lemma radialStop_before_measurable (a : ℝ) (ha0 : 0 < a) (z : ℂ)
+    (hz : 0 < z.im) (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (n : ℕ)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable[hS.measurableSpace] (fun b => if S b < radialStop a ha0 z hz ρ hρ hr n b then
+      physicalGreen a (actualState a b.1 (S b) z) else 0) := by
+  let L := radialLocalization a ha0 z hz ρ (angleThreshold z n) hρ hr
+    (angleThreshold_pos z hz n) (angleThreshold_initial z hz n)
+  let T := radialStop a ha0 z hz ρ hρ hr n
+  have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
+  have hmin : IsStoppingTime rawFiltration (fun b => (min (S b) (T b) : WithTop ℝ≥0)) := by
+    simpa only [WithTop.coe_min] using hS.min hT
+  have hle : hmin.measurableSpace ≤ hS.measurableSpace :=
+    hmin.measurableSpace_mono hS (fun b => WithTop.coe_le_coe.mpr (min_le_left _ _))
+  have hC : MeasurableSet[hS.measurableSpace] {b | S b < T b} := by
+    apply hle
+    simpa only [Set.univ_inter,← WithTop.coe_min] using stopped_event_lt hS hT Set.univ MeasurableSet.univ
+  have hA : StronglyAdapted rawFiltration (localState a L.ε L.hε z) :=
+    fun t => (localState_adapted a L.ε L.hε z t).stronglyMeasurable
+  have hv : Measurable[hS.measurableSpace] (fun b => localState a L.ε L.hε z (min (S b) (T b)) b) :=
+    (measurable_stoppedValue (hA.isStronglyProgressive_of_continuous
+      (localState_continuous a L.ε L.hε z)) hmin).mono hle le_rfl
+  have hfun : (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0) =
+      {b | S b < T b}.indicator
+        (fun b => greenCutoff a L.ε L.N (localState a L.ε L.hε z (min (S b) (T b)) b)) := by
+    funext b
+    by_cases hb : S b < T b
+    · rw [ite_eq_left hb,Set.indicator_of_mem (show b ∈ {b | S b < T b} from hb),min_eq_left hb.le]
+      have hs := regionStop_safe a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (S b) hb.le
+      rw [(regionStop_actualState a L.ε L.hε z L.N ρ (angleThreshold z n) L.B L.initial b (S b) hb.le).2]
+      exact ((greenCutoff_eq_near a L.ε L.hε L.N _ hs.2.1 hs.1).eq_of_nhds).symm
+    · rw [ite_eq_right hb,Set.indicator_of_notMem (show b ∉ {b | S b < T b} from hb)]
+  change Measurable[hS.measurableSpace] (fun b => if S b < T b then physicalGreen a (actualState a b.1 (S b) z) else 0)
+  rw [hfun]
+  exact ((greenCutoff_smooth a L.ε L.hε L.N).continuous.measurable.comp hv).indicator hC
+
+lemma radialRemaining_measurable (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Measurable[hS.measurableSpace] (fun b => radialRemaining a b.1 z ρ (S b)) := by
+  have ha0 : 0 < a := by linarith
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  apply measurable_of_tendsto_metrizable
+    (fun n => radialStop_before_measurable a ha0 z hz ρ hρ hr n S hS)
+  exact tendsto_pi_nhds.mpr (fun b => radialStop_before_tendsto a ha ha0 z hz ρ hρ hr b (S b))
+
+ 
+
+theorem radial_conditional_weight (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0)))
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    Integrable (fun b => radialRemaining a b.1 z ρ (S b)) W ∧
+    (∫ b in U, (if (S b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) ∂W) =
+      ∫ b in U, radialRemaining a b.1 z ρ (S b) ∂W := by
+  have ha0 : 0 < a := by linarith
+  let F := fun n (b : ExactSLEGenerator.Driver) => if S b < radialStop a ha0 z hz ρ hρ hr n b
+    then physicalGreen a (actualState a b.1 (S b) z) else 0
+  let H := fun n (b : ExactSLEGenerator.Driver) => if S b < radialStop a ha0 z hz ρ hρ hr n b
+    then radialApproximation a ha0 z hz ρ hρ hr n b else 0
+  let G := fun b : ExactSLEGenerator.Driver => radialRemaining a b.1 z ρ (S b)
+  let K := fun b : ExactSLEGenerator.Driver => if (S b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0
+  let C := Real.exp (-alpha a*Real.log ρ)
+  have hFi (n : ℕ) : Integrable (F n) W :=
+    (radialStop_optional W hW a ha ha0 z hz ρ hρ hr n S hS U hU).1
+  have hHi (n : ℕ) : Integrable (H n) W := by
+    have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
+    have hC : MeasurableSet {b | S b < radialStop a ha0 z hz ρ hρ hr n b} := by
+      exact (iSup_le fun t => rawFiltration.le t) _ (by
+        simpa only [WithTop.coe_lt_coe] using measurableSet_lt hS.measurable_iSup hT.measurable_iSup)
+    exact (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).1.indicator hC
+  have hFc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => F n b) atTop (𝓝 (G b)) :=
+    radialStop_before_tendsto a ha ha0 z hz ρ hρ hr b (S b)
+  have hHc (b : ExactSLEGenerator.Driver) : Tendsto (fun n => H n b) atTop (𝓝 (K b)) :=
+    radialStop_terminal_tendsto a ha ha0 z hz ρ hρ hr b (S b)
+  have hFb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖F n b‖ ≤ C :=
+    radialStop_before_bound a ha ha0 z hz ρ hρ hr n b (S b)
+  have hHb (n : ℕ) (b : ExactSLEGenerator.Driver) : ‖H n b‖ ≤ C := by
+    dsimp only [H]
+    split_ifs
+    · have h := radialApproximation_bound a ha ha0 z hz ρ hρ hr n b
+      simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
+    · exact (norm_zero.trans_le (Real.exp_pos _).le)
+  have hGi : Integrable G W := Integrable.of_bound
+    ((radialRemaining_measurable a ha z hz ρ hρ hr S hS).mono hS.measurableSpace_le le_rfl).aestronglyMeasurable C
+    (Eventually.of_forall fun b => le_of_tendsto (hFc b).norm (Eventually.of_forall fun n => hFb n b))
+  refine ⟨hGi,?_⟩
+  have hFl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => C)
+    (fun n => (hFi n).aestronglyMeasurable.restrict) (integrable_const C)
+    (fun n => Eventually.of_forall (hFb n)) (Eventually.of_forall hFc)
+  have hHl := tendsto_integral_of_dominated_convergence (μ := W.restrict U) (fun _ => C)
+    (fun n => (hHi n).aestronglyMeasurable.restrict) (integrable_const C)
+    (fun n => Eventually.of_forall (hHb n)) (Eventually.of_forall hHc)
+  have he : (fun n => ∫ b in U, F n b ∂W) = fun n => ∫ b in U, H n b ∂W :=
+    funext fun n => (radialStop_optional W hW a ha ha0 z hz ρ hρ hr n S hS U hU).2
+  rw [he] at hFl
+  exact tendsto_nhds_unique hHl hFl
+
+end ExactSLEGreen
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+open scoped Classical
+noncomputable def stoppedRadialRemaining (a : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (ρ : ℝ) (t : ℝ≥0∞) : ℝ :=
+  if t = ∞ then 0 else radialRemaining a b z ρ t.toNNReal
+
+lemma radialRemaining_bound (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver) (t : ℝ≥0) :
+    ‖radialRemaining a b.1 z ρ t‖ ≤ Real.exp (-alpha a*Real.log ρ) := by
+  have ha0 : 0 < a := by linarith
+  exact le_of_tendsto (radialStop_before_tendsto a ha ha0 z hz ρ hρ hr b t).norm
+    (Eventually.of_forall fun n => radialStop_before_bound a ha ha0 z hz ρ hρ hr n b t)
+
+lemma radialWeight_bound (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im) (b : ExactSLEGenerator.Driver) :
+    0 ≤ radialWeight a b.1 z ρ ∧ radialWeight a b.1 z ρ ≤ Real.exp (-alpha a*Real.log ρ) := by
+  have ha0 : 0 < a := by linarith
+  have hc := radialApproximation_tendsto a ha ha0 z hz ρ hρ hr b
+  exact ⟨ge_of_tendsto hc (Eventually.of_forall fun n => (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).1),
+    le_of_tendsto hc (Eventually.of_forall fun n => (radialApproximation_bound a ha ha0 z hz ρ hρ hr n b).2)⟩
+
+lemma radial_terminal_integrable (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (a : ℝ) (ha : 1/4 < a) (z : ℂ) (hz : 0 < z.im)
+    (ρ : ℝ) (hρ : 0 < ρ) (hr : ρ ≤ z.im)
+    (S : ExactSLEGenerator.Driver → ℝ≥0)
+    (hS : IsStoppingTime rawFiltration (fun b => (S b : WithTop ℝ≥0))) :
+    Integrable (fun b => if (S b : ℝ≥0∞) < radialTime a b.1 z ρ then radialWeight a b.1 z ρ else 0) W := by
+  have ha0 : 0 < a := by linarith
+  let H := fun n (b : ExactSLEGenerator.Driver) => if S b < radialStop a ha0 z hz ρ hρ hr n b
+    then radialApproximation a ha0 z hz ρ hρ hr n b else 0
+  have hHi (n : ℕ) : Integrable (H n) W := by
+    have hT := radialStop_stopping a ha0 z hz ρ hρ hr n
+    have hC : MeasurableSet {b | S b < radialStop a ha0 z hz ρ hρ hr n b} := by
+      exact (iSup_le fun t => rawFiltration.le t) _ (by
+        simpa only [WithTop.coe_lt_coe] using measurableSet_lt hS.measurable_iSup hT.measurable_iSup)
+    exact (radialApproximation_mean W hW a ha0 z hz ρ hρ hr n).1.indicator hC
+  apply Integrable.of_bound (aestronglyMeasurable_of_tendsto_ae atTop (fun n => (hHi n).aestronglyMeasurable)
+    (Eventually.of_forall fun b => radialStop_terminal_tendsto a ha ha0 z hz ρ hρ hr b (S b)))
+    (Real.exp (-alpha a*Real.log ρ))
+  filter_upwards [] with b
+  split_ifs
+  · have h := radialWeight_bound a ha z hz ρ hρ hr b
+    simpa only [Real.norm_eq_abs,abs_of_nonneg h.1] using h.2
+  · simpa only [norm_zero] using (Real.exp_pos (-alpha a*Real.log ρ)).le
+
+noncomputable def capTime (S : ExactSLEGenerator.Driver → ℝ≥0∞) (n : ℝ≥0) (b : ExactSLEGenerator.Driver) : ℝ≥0 :=
+  (min (S b) (n : ℝ≥0∞)).toNNReal
+lemma capTime_coe (S : ExactSLEGenerator.Driver → ℝ≥0∞) (n : ℝ≥0) (b : ExactSLEGenerator.Driver) :
+    (capTime S n b : ℝ≥0∞) = min (S b) (n : ℝ≥0∞) :=
+  ENNReal.coe_toNNReal (ne_of_lt (lt_of_le_of_lt (min_le_right _ _) ENNReal.coe_lt_top))
+lemma capTime_stopping (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (n : ℝ≥0) : IsStoppingTime rawFiltration (fun b => (capTime S n b : WithTop ℝ≥0)) := by
+  have heq : (fun b => (capTime S n b : WithTop ℝ≥0)) = fun b => min (S b) (n : ℝ≥0∞) := funext (capTime_coe S n)
+  rw [heq]
+  exact hS.min_const n
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TraceBoundary.lean b/lean/OAI/Probability/SLEGauge/TraceBoundary.lean
index f49c36457d334d9219acf84a2c55a080c40fafc4..6ca2a3b21f982cede9bedd7c0f0eaa20e404da84 100644
--- a/lean/OAI/Probability/SLEGauge/TraceBoundary.lean
+++ b/lean/OAI/Probability/SLEGauge/TraceBoundary.lean
@@ -1,485 +1,485 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.TraceContinuity
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-noncomputable section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.LiteralTrace
-open BrownianCalculus
-lemma traceRatio_pos (N : ℕ) : 0 < traceRatio N := by unfold traceRatio; positivity
-
-lemma traceRatio_lt_one {N : ℕ} (hN : 0 < N) : traceRatio N < 1 := by
-  exact pow_lt_one₀ (by norm_num) (by norm_num) (by omega)
-
-lemma traceMeshBase_eq (N : ℕ) : traceMeshBase N = (2^(2*N+1))^4 := by
-  unfold traceMeshBase modulusBase
-  rw [← pow_mul]
-  congr 1
-  omega
-
-lemma traceRate_eq (N : ℕ) : modulusRate (4*N+2) = traceRatio N/2 := by
-  unfold modulusRate traceRatio
-  rw [pow_add]
-  norm_num
-  ring
-
-lemma traceMesh_inv (N : ℕ) : 1/(traceMeshBase N:ℝ) = (traceRatio N)^2/16 := by
-  unfold traceMeshBase modulusBase traceRatio
-  push_cast
-  rw [← pow_mul,show 2*(4*N+2) = (4*N)*2+4 by omega,pow_add,div_pow,one_pow]
-  norm_num
-  ring
-
-lemma traceHeight_eq (N n : ℕ) : refinedHeight N (4*n) = (traceRatio N)^n := by
-  unfold refinedHeight traceRatio
-  rw [← pow_mul,← pow_mul]
-  congr 1
-  ring
-
-lemma traceGridTime_eq (δ : ℝ) (N n k : ℕ) :
-    refinedGridTime δ N (4*n) k = δ+TraceChaining.grid (traceMeshBase N) n k := by
-  unfold refinedGridTime TraceChaining.grid
-  rw [traceMeshBase_eq]
-  push_cast
-  simp only [← pow_mul]
-  congr 3; ring
-
-lemma traceGridCard_eq (M N n : ℕ) :
-    M*(2^(2*N+1))^(4*n) = M*(traceMeshBase N)^n := by
-  rw [traceMeshBase_eq]
-  simp only [← pow_mul]
-  congr 2; ring
-
-lemma traceRate_div (N n : ℕ) :
-    (modulusRate (4*N+2))^n/(traceRatio N)^(n+1) =
-      1/traceRatio N*(1/2:ℝ)^n := by
-  rw [traceRate_eq,div_pow,pow_succ,div_pow,one_pow]
-  field_simp [ne_of_gt (traceRatio_pos N)]
-
-lemma traceMesh_div (N n : ℕ) :
-    (1/(traceMeshBase N:ℝ)^n)/((traceRatio N)^(n+1))^2 =
-      1/(traceRatio N)^2*(1/16:ℝ)^n := by
-  rw [← one_div_pow,traceMesh_inv,div_pow,
-    show (traceRatio N)^(n+1) = (traceRatio N)^n*traceRatio N by rfl,mul_pow,div_pow,one_pow]
-  rw [pow_right_comm (traceRatio N) 2 n]
-  field_simp [ne_of_gt (traceRatio_pos N)]
-
-lemma traceMesh_eventually (N : ℕ) : ∀ᶠ n : ℕ in atTop,
-    1/(traceMeshBase N:ℝ)^n ≤ ((traceRatio N)^(n+1))^2 := by
-  have hh : Tendsto (fun n : ℕ => 1/(traceRatio N)^2*(1/16:ℝ)^n) atTop (𝓝 0) := by
-    simpa only [mul_zero] using (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : 0 ≤ (1/16:ℝ))
-      (by norm_num : (1/16:ℝ) < 1)).const_mul (1/(traceRatio N)^2)
-  filter_upwards [hh.eventually (Iio_mem_nhds zero_lt_one)] with n hn
-  rw [← traceMesh_div] at hn
-  have hr := traceRatio_pos N
-  exact ((div_lt_one (by positivity : 0 < ((traceRatio N)^(n+1))^2)).mp hn).le
-
-lemma traceError_summable {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {N : ℕ} (hN : 0 < N) (H : ℕ) :
-    Summable (traceError κ N H) := by
-  have hp := trace_exponents hκ hκ8
-  have hr : |(traceRatio N)^(traceBeta κ)| < 1 := by
-    rw [abs_of_pos (Real.rpow_pos_of_pos (traceRatio_pos N) _)]
-    exact Real.rpow_lt_one (traceRatio_pos N).le (traceRatio_lt_one hN) hp.1
-  have hs := (summable_geometric_of_abs_lt_one (by norm_num : |(1/2:ℝ)| < 1)).mul_left
-    (brownianModulusConst (4*N+2)/traceRatio N)
-  have ht := (summable_geometric_of_abs_lt_one (by norm_num : |(1/16:ℝ)| < 1)).mul_left
-    (a κ/(traceRatio N)^2)
-  have hu := (((hs.add ht).mul_left (4*(2+a κ+2*a κ*H))).add (summable_geometric_of_abs_lt_one hr))
-  change Summable (fun n : ℕ => 4*(2+a κ+2*a κ*H)*
-    (brownianModulusConst (4*N+2)/traceRatio N*(1/2:ℝ)^n+
-      a κ/(traceRatio N)^2*(1/16:ℝ)^n)+((traceRatio N)^(traceBeta κ))^n)
-  exact hu
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open BrownianCalculus TraceChaining
-
- 
-
-theorem radial_uniform_bands {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∃ N : ℕ, 0 < N ∧ ∀ᵐ b ∂W, ∀ m H : ℕ, ∀ᶠ n : ℕ in atTop,
-      ∀ t : ℝ, 1/((m:ℝ)+1) ≤ t → t ≤ H →
-      ∀ y : ℝ, (traceRatio N)^(n+1) ≤ y → y ≤ (traceRatio N)^n →
-      ‖radialValue κ b t y-radialValue κ b t ((traceRatio N)^n)‖ ≤
-        traceError κ N H n := by
-  obtain ⟨N,hN,hNeta,hgrid⟩ := radial_length_all_refined_grids hκ hκ8 hW
-  refine ⟨N,hN,?_⟩
-  filter_upwards [hgrid,brownian_uniform_modulus hW] with b hb hB
-  intro m H
-  let δ : ℝ := 1/((m:ℝ)+1)
-  have hδ : 0 < δ := by dsimp [δ]; positivity
-  have hr := traceRatio_pos N
-  have hr1 := traceRatio_lt_one hN
-  have hq : 0 < traceMeshBase N := by unfold traceMeshBase modulusBase; positivity
-  have hsub : Tendsto (fun n : ℕ => 4*n) atTop atTop := by
-    apply tendsto_atTop_mono (f := fun n : ℕ => n) (fun n => by omega) tendsto_id
-  have hg := hsub.eventually (hb m H)
-  obtain ⟨n₀,hn₀⟩ := hB (4*N+2) H (by omega)
-  filter_upwards [hg,traceMesh_eventually N,eventually_ge_atTop n₀] with n hn hmesh hn0
-  intro t hδt htH y hvy hyz
-  let k := index (traceMeshBase N) n (t-δ)
-  let s := δ+approx (traceMeshBase N) n (t-δ)
-  have hab := approx_bounds hq n (sub_nonneg.mpr hδt)
-  have hs : 0 ≤ s := by dsimp [s]; linarith [hab.1]
-  have hst : s ≤ t := by dsimp [s]; linarith [hab.2.1]
-  have hclose : t-s ≤ 1/(traceMeshBase N:ℝ)^n := by
-    dsimp [s]; linarith [hab.2.2]
-  have hk : k < H*(traceMeshBase N)^n := by
-    have hp : 0 < (traceMeshBase N:ℝ)^n := pow_pos (by exact_mod_cast hq) _
-    have hf := Nat.floor_le (mul_nonneg hp.le (sub_nonneg.mpr hδt))
-    have hh : (k:ℝ) < (H:ℝ)*(traceMeshBase N:ℝ)^n := by
-      dsimp [k,index]
-      nlinarith
-    exact_mod_cast hh
-  have hgk := hn ⟨k,by rwa [traceGridCard_eq]⟩
-  rw [traceGridTime_eq,traceHeight_eq] at hgk
-  change radialLength κ b s ((traceRatio N)^n) < _ at hgk
-  have hR : radialLength κ b s ((traceRatio N)^n) ≤
-      ENNReal.ofReal (((traceRatio N)^(traceBeta κ))^n) := by
-    have he : ((traceRatio N)^n)^(traceBeta κ) =
-        ((traceRatio N)^(traceBeta κ))^n := by
-      rw [← Real.rpow_natCast_mul hr.le,mul_comm (n:ℝ) (traceBeta κ),
-        Real.rpow_mul_natCast hr.le]
-    rw [he] at hgk
-    exact hgk.le
-  have hbnd := hn₀ n hn0 s t hs hst htH hclose
-  change |driverValue b t-driverValue b s| ≤
-    brownianModulusConst (4*N+2)*(modulusRate (4*N+2))^n at hbnd
-  have hp := radial_interpolation hκ hs hst htH (pow_pos hr _) hvy hyz
-    (pow_le_one₀ hr.le hr1.le) (hclose.trans hmesh) b hbnd hR (by positivity)
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hc : 0 ≤ 2*(2+a κ+2*a κ*H)/(traceRatio N)^(n+1) := by positivity
-  have hstep : a κ*(t-s)/(traceRatio N)^(n+1) ≤
-      a κ*(1/(traceMeshBase N:ℝ)^n)/(traceRatio N)^(n+1) := by
-    gcongr
-  have hp' : ‖radialValue κ b t y-radialValue κ b t ((traceRatio N)^n)‖ ≤
-      2*(2*(2+a κ+2*a κ*H)/(traceRatio N)^(n+1)*
-        (brownianModulusConst (4*N+2)*(modulusRate (4*N+2))^n+
-          a κ*(1/(traceMeshBase N:ℝ)^n)/(traceRatio N)^(n+1)))+
-        ((traceRatio N)^(traceBeta κ))^n := by
-    apply hp.trans
-    gcongr
-  apply hp'.trans_eq
-  dsimp [traceError]
-  rw [show 2*(2*(2+a κ+2*a κ*H)/(traceRatio N)^(n+1)*
-      (brownianModulusConst (4*N+2)*(modulusRate (4*N+2))^n+
-        a κ*(1/(traceMeshBase N:ℝ)^n)/(traceRatio N)^(n+1))) =
-      4*(2+a κ+2*a κ*H)*
-        (brownianModulusConst (4*N+2)*
-          ((modulusRate (4*N+2))^n/(traceRatio N)^(n+1))+
-          a κ*((1/(traceMeshBase N:ℝ)^n)/((traceRatio N)^(n+1))^2)) by ring]
-  rw [traceRate_div,traceMesh_div]
-  ring
-end SLEExactGauge.LiteralTrace
-
- 
-
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.TraceChaining
-
-lemma radial_limit_of_bands {E : Type*} [MetricSpace E] [CompleteSpace E]
-    (f : ℝ → E) {r : ℝ} (hr : 0 < r) (hr1 : r < 1)
-    {e : ℕ → ℝ} (he : Summable e)
-    (hb : ∀ᶠ n : ℕ in atTop, ∀ y : ℝ, r^(n+1) ≤ y → y ≤ r^n →
-      dist (f y) (f (r^n)) ≤ e n) :
-    ∃ z : E, Tendsto f (𝓝[>] 0) (𝓝 z) := by
-  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp hb
-  have hs (n : ℕ) : dist (f (r^(n+n₀))) (f (r^((n+1)+n₀))) ≤ e (n+n₀) := by
-    rw [dist_comm]
-    exact hn₀ (n+n₀) (by omega) _ (by simp [Nat.add_assoc,Nat.add_comm])
-      (by simpa only [show n+1+n₀ = (n+n₀)+1 by omega] using
-        pow_le_pow_of_le_one hr.le hr1.le (Nat.le_succ (n+n₀)))
-  obtain ⟨z,hz⟩ := cauchySeq_tendsto_of_complete
-    (cauchySeq_of_dist_le_of_summable (fun n => e (n+n₀)) hs
-      ((summable_nat_add_iff n₀).mpr he))
-  have hz' : Tendsto (fun n : ℕ => f (r^n)) atTop (𝓝 z) :=
-    (tendsto_add_atTop_iff_nat n₀).mp hz
-  refine ⟨z,Metric.tendsto_nhds.mpr ?_⟩
-  intro ε hε
-  have hε2 : 0 < ε/2 := by linarith
-  have hsmall := (he.tendsto_atTop_zero.eventually (Iio_mem_nhds hε2))
-  have hnear := (Metric.tendsto_atTop.mp hz') (ε/2) hε2
-  obtain ⟨n₁,hn₁⟩ := eventually_atTop.mp hsmall
-  obtain ⟨n₂,hn₂⟩ := hnear
-  let n₃ := max n₀ (max n₁ n₂)
-  have hv : 0 < r^(n₃+1) := pow_pos hr _
-  filter_upwards [self_mem_nhdsWithin,
-    (show ∀ᶠ y : ℝ in 𝓝[>] 0, y < r^(n₃+1) from
-      nhdsWithin_le_nhds (Iio_mem_nhds hv))] with y hy hyv
-  have hy1 : y ≤ 1 := hyv.le.trans (pow_le_one₀ hr.le hr1.le)
-  obtain ⟨n,hnlow,hnup⟩ := exists_nat_pow_near_of_lt_one hy hy1 hr hr1
-  have hn3 : n₃ ≤ n := by
-    by_contra hh
-    have hnlt : n+1 ≤ n₃+1 := by omega
-    have hlo := pow_le_pow_of_le_one hr.le hr1.le hnlt
-    linarith
-  have hband := hn₀ n (le_trans (le_max_left _ _) hn3) y hnlow.le hnup
-  have hen := hn₁ n (le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hn3)
-  have hzn := hn₂ n (le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hn3)
-  exact (dist_triangle _ _ _).trans_lt (by linarith)
-end SLEExactGauge.TraceChaining
-namespace SLEExactGauge.TraceChaining
-
-lemma radial_uniform_limit_of_bands {X E : Type*} [MetricSpace E] [CompleteSpace E]
-    (f : X → ℝ → E) (g : X → E) (S : Set X)
-    {r : ℝ} (hr : 0 < r) (hr1 : r < 1) {e : ℕ → ℝ} (he : Summable e)
-    (hb : ∀ᶠ n : ℕ in atTop, ∀ t ∈ S, ∀ y : ℝ, r^(n+1) ≤ y → y ≤ r^n →
-      dist (f t y) (f t (r^n)) ≤ e n)
-    (hg : ∀ t ∈ S, Tendsto (f t) (𝓝[>] 0) (𝓝 (g t))) :
-    TendstoUniformlyOn (fun n t => f t (r^n)) g atTop S := by
-  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp hb
-  have hgeom : Tendsto (fun n : ℕ => r^n) atTop (𝓝[>] 0) := by
-    apply tendsto_nhdsWithin_iff.mpr
-    exact ⟨tendsto_pow_atTop_nhds_zero_of_lt_one hr.le hr1,
-      Eventually.of_forall (fun n => pow_pos hr n)⟩
-  have hd (n : ℕ) (hn : n₀ ≤ n) (t : X) (ht : t ∈ S) :
-      dist (f t (r^n)) (g t) ≤ ∑' j : ℕ, e (j+n) := by
-    have hstep (j : ℕ) : dist (f t (r^(j+n))) (f t (r^((j+1)+n))) ≤ e (j+n) := by
-      rw [dist_comm]
-      apply hn₀ (j+n) (by omega) t ht
-      · simp [Nat.add_assoc,Nat.add_comm]
-      · simpa only [show j+1+n = (j+n)+1 by omega] using
-          pow_le_pow_of_le_one hr.le hr1.le (Nat.le_succ (j+n))
-    have hh := dist_le_tsum_of_dist_le_of_tendsto₀ (fun j => e (j+n)) hstep
-      ((summable_nat_add_iff n).mpr he)
-      (((hg t ht).comp hgeom).comp (tendsto_add_atTop_nat n))
-    simpa only [Nat.zero_add] using hh
-  rw [Metric.tendstoUniformlyOn_iff]
-  intro ε hε
-  have hh := (tendsto_sum_nat_add e).eventually (Iio_mem_nhds hε)
-  filter_upwards [hh,eventually_ge_atTop n₀] with n hn hn0
-  intro t ht
-  rw [dist_comm]
-  exact (hd n hn0 t ht).trans_lt hn
-end SLEExactGauge.TraceChaining
-namespace SLEExactGauge.LiteralTrace
-open TraceChaining
-
- 
-
-theorem trace_limit_all_positive {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W, ∀ t : ℝ, 0 < t →
-      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
-        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t)) := by
-  obtain ⟨N,hN,hb⟩ := radial_uniform_bands hκ hκ8 hW
-  filter_upwards [hb] with b hb
-  intro t ht
-  have hT : 0 < κ*t := mul_pos hκ ht
-  obtain ⟨m,hm⟩ := exists_nat_gt (1/(κ*t))
-  have hδ : 1/((m:ℝ)+1) ≤ κ*t := by
-    have hmp : 0 < (m:ℝ)+1 := by positivity
-    apply (div_le_iff₀ hmp).mpr
-    have hh := (div_lt_iff₀ hT).mp hm
-    nlinarith
-  obtain ⟨H,hH⟩ := exists_nat_gt (κ*t)
-  obtain ⟨z,hz⟩ := radial_limit_of_bands (radialValue κ b (κ*t))
-    (traceRatio_pos N) (traceRatio_lt_one hN) (traceError_summable hκ hκ8 hN H)
-    (by
-      filter_upwards [hb m H] with n hn
-      intro y hy hy'
-      simpa only [dist_eq_norm] using hn (κ*t) hδ hH.le y hy hy')
-  have he : trace κ b t = z := hz.limUnder_eq
-  rwa [he]
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-
-lemma radial_time_modulus_symmetric {κ T y s t : ℝ} (hκ : 0 < κ)
-    (hy : 0 < y) (hy1 : y ≤ 1) (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T)
-    (hclose : |t-s| ≤ y^2) (b : Driver) :
-    ‖radialValue κ b t y-radialValue κ b s y‖ ≤
-      2*(2+a κ+2*a κ*T)/y*(|driverValue b t-driverValue b s|+a κ*|t-s|/y) := by
-  by_cases hst : s ≤ t
-  · rw [abs_of_nonneg (sub_nonneg.mpr hst)] at hclose ⊢
-    exact inverseFlow_time_modulus hκ hs.1 hst ht.2 hy hy1 hclose b
-  · have hts : t ≤ s := (lt_of_not_ge hst).le
-    rw [abs_sub_comm t s,abs_of_nonneg (sub_nonneg.mpr hts)] at hclose ⊢
-    rw [norm_sub_rev,abs_sub_comm]
-    exact inverseFlow_time_modulus hκ ht.1 hts hs.2 hy hy1 hclose b
-
- 
-
-lemma radialValue_continuousOn {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y)
-    (hy1 : y ≤ 1) (b : Driver) :
-    ContinuousOn (fun t => radialValue κ b t y) (Icc 0 T) := by
-  intro t ht
-  apply tendsto_iff_norm_sub_tendsto_zero.mpr
-  have hB : Continuous (driverValue b) :=
-    (map_continuous b).comp continuous_real_toNNReal
-  have h1 : Tendsto (fun s : ℝ => |driverValue b s-driverValue b t|)
-      (𝓝[Icc 0 T] t) (𝓝 0) := by
-    have hh : Tendsto (driverValue b) (𝓝[Icc 0 T] t) (𝓝 (driverValue b t)) :=
-      (hB.tendsto t).mono_left nhdsWithin_le_nhds
-    simpa using (hh.sub_const (driverValue b t)).abs
-  have h2 : Tendsto (fun s : ℝ => |s-t|) (𝓝[Icc 0 T] t) (𝓝 0) := by
-    have hh : Tendsto (fun s : ℝ => s) (𝓝[Icc 0 T] t) (𝓝 t) :=
-      tendsto_id'.mpr nhdsWithin_le_nhds
-    simpa using (hh.sub_const t).abs
-  have hb := (h1.add ((h2.const_mul (a κ)).div_const y)).const_mul
-    (2*(2+a κ+2*a κ*T)/y)
-  have hb' : Tendsto (fun s : ℝ => 2*(2+a κ+2*a κ*T)/y*
-      (|driverValue b s-driverValue b t|+a κ*|s-t|/y))
-      (𝓝[Icc 0 T] t) (𝓝 0) := by simpa using hb
-  apply squeeze_zero' (Eventually.of_forall (fun _ => norm_nonneg _)) _ hb'
-  filter_upwards [self_mem_nhdsWithin,h2.eventually (Iio_mem_nhds (sq_pos_of_pos hy))] with s hs hclose
-  exact radial_time_modulus_symmetric hκ hy hy1 ht hs hclose.le b
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open TraceChaining
-
- 
-
-theorem trace_continuous_positive {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
-    {W : WienerLaw} (hW : IsWiener W) :
-    ∀ᵐ b ∂W, ContinuousOn (trace κ b) (Ioi 0) := by
-  obtain ⟨N,hN,hb⟩ := radial_uniform_bands hκ hκ8 hW
-  filter_upwards [hb,trace_limit_all_positive hκ hκ8 hW] with b hb hlim
-  let g : ℝ → ℂ := fun T => limUnder (𝓝[>] (0:ℝ)) (radialValue κ b T)
-  have hg (T : ℝ) (hT : 0 < T) : Tendsto (radialValue κ b T) (𝓝[>] 0) (𝓝 (g T)) := by
-    have hh := hlim (T/κ) (div_pos hT hκ)
-    have he : κ*(T/κ) = T := by field_simp
-    rw [he] at hh
-    have he' : g T = trace κ b (T/κ) := hh.limUnder_eq
-    rwa [he']
-  have hc (m H : ℕ) : ContinuousOn g (Icc (1/((m:ℝ)+1)) H) := by
-    have hu := radial_uniform_limit_of_bands (fun T => radialValue κ b T) g
-      (Icc (1/((m:ℝ)+1)) H) (traceRatio_pos N) (traceRatio_lt_one hN)
-      (traceError_summable hκ hκ8 hN H)
-      (by
-        filter_upwards [hb m H] with n hn
-        intro T hT y hy hy'
-        simpa only [dist_eq_norm] using hn T hT.1 hT.2 y hy hy')
-      (fun T hT => hg T ((by positivity : 0 < 1/((m:ℝ)+1)).trans_le hT.1))
-    apply hu.continuousOn
-    apply Eventually.frequently
-    apply Eventually.of_forall
-    intro n
-    exact (radialValue_continuousOn hκ (pow_pos (traceRatio_pos N) n)
-      (pow_le_one₀ (traceRatio_pos N).le (traceRatio_lt_one hN).le) b).mono
-      (Icc_subset_Icc (by positivity) le_rfl)
-  have hc' : ContinuousOn g (Ioi 0) := by
-    apply continuousOn_of_forall_continuousAt
-    intro T hT
-    obtain ⟨m,hm⟩ := exists_nat_gt (1/T)
-    obtain ⟨H,hH⟩ := exists_nat_gt T
-    have hδ : 1/((m:ℝ)+1) < T := by
-      apply (div_lt_iff₀ (by positivity : 0 < (m:ℝ)+1)).mpr
-      have hh := (div_lt_iff₀ hT).mp hm
-      nlinarith
-    exact (hc m H).continuousAt (Icc_mem_nhds hδ hH)
-  have he : (fun t => g (κ*t)) = trace κ b := by
-    funext t
-    rfl
-  rw [← he]
-  exact hc'.comp (continuous_const.mul continuous_id).continuousOn
-    (fun t ht => mul_pos hκ ht)
-end SLEExactGauge.LiteralTrace
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.TraceChaining
-
- 
-
-lemma initial_le_of_outward_deriv {f f' : ℝ → ℝ} {T R : ℝ} (hT : 0 ≤ T)
-    (hf : ContinuousOn f (Icc 0 T))
-    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt f (f' t) (Icc 0 T) t)
-    (hR : f T ≤ R) (hout : ∀ t ∈ Icc 0 T, R < f t → 0 < f' t) :
-    f 0 ≤ R := by
-  have hm : MapsTo (fun t : ℝ => T-t) (Icc 0 T) (Icc 0 T) := by
-    intro t ht
-    constructor <;> linarith [ht.1,ht.2]
-  have hc : ContinuousOn (fun t => f (T-t)) (Icc 0 T) :=
-    hf.comp (continuous_const.sub continuous_id).continuousOn hm
-  have hh (t : ℝ) (ht : t ∈ Ico 0 T) :
-      HasDerivWithinAt (fun t => f (T-t)) (-f' (T-t)) (Ici t) t := by
-    have h := (hd (T-t) (hm ⟨ht.1,ht.2.le⟩)).comp t
-      ((hasDerivAt_id t).const_sub T).hasDerivWithinAt hm
-    have he : Icc (0:ℝ) T ∈ 𝓝[Ici t] t := by
-      filter_upwards [self_mem_nhdsWithin,
-        (show ∀ᶠ u : ℝ in 𝓝[Ici t] t, u < T from
-          nhdsWithin_le_nhds (Iio_mem_nhds ht.2))] with u hu huT
-      exact ⟨ht.1.trans hu,huT.le⟩
-    convert! h.mono_of_mem_nhdsWithin he using 1; simp only [mul_neg,mul_one]
-  apply le_of_forall_pos_le_add
-  intro ε hε
-  have hfence := image_le_of_deriv_right_lt_deriv_boundary hc hh
-    (B := fun _ => R+ε) (B' := fun _ => 0)
-    (by simpa using hR.trans (by linarith : R ≤ R+ε))
-    (fun _ => hasDerivAt_const _ _)
-    (by
-      intro t ht he
-      have hp := hout (T-t) (hm ⟨ht.1,ht.2.le⟩) (by linarith)
-      linarith)
-  simpa using hfence (right_mem_Icc.mpr hT)
-end SLEExactGauge.TraceChaining
-namespace SLEExactGauge.LiteralTrace
-open TraceChaining
-
-lemma PositiveSolution.re_hasDeriv {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
-    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    HasDerivWithinAt (fun u => (f u).re)
-      (a κ*((f t).re-driverValue b t)/‖f t-driverValue b t‖^2) (Icc 0 T) t := by
-  have hi := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t (hf.2.2.2.2 t ht)
-  convert! hi using 1; first | rfl |
-    (simp only [Complex.reCLM_apply,Complex.div_re,Complex.ofReal_im,Complex.ofReal_re,
-     Complex.sub_re,Complex.normSq_eq_norm_sq,zero_mul]; ring)
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.TraceContinuity
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+noncomputable section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.LiteralTrace
+open BrownianCalculus
+lemma traceRatio_pos (N : ℕ) : 0 < traceRatio N := by unfold traceRatio; positivity
+
+lemma traceRatio_lt_one {N : ℕ} (hN : 0 < N) : traceRatio N < 1 := by
+  exact pow_lt_one₀ (by norm_num) (by norm_num) (by omega)
+
+lemma traceMeshBase_eq (N : ℕ) : traceMeshBase N = (2^(2*N+1))^4 := by
+  unfold traceMeshBase modulusBase
+  rw [← pow_mul]
+  congr 1
+  omega
+
+lemma traceRate_eq (N : ℕ) : modulusRate (4*N+2) = traceRatio N/2 := by
+  unfold modulusRate traceRatio
+  rw [pow_add]
+  norm_num
+  ring
+
+lemma traceMesh_inv (N : ℕ) : 1/(traceMeshBase N:ℝ) = (traceRatio N)^2/16 := by
+  unfold traceMeshBase modulusBase traceRatio
+  push_cast
+  rw [← pow_mul,show 2*(4*N+2) = (4*N)*2+4 by omega,pow_add,div_pow,one_pow]
+  norm_num
+  ring
+
+lemma traceHeight_eq (N n : ℕ) : refinedHeight N (4*n) = (traceRatio N)^n := by
+  unfold refinedHeight traceRatio
+  rw [← pow_mul,← pow_mul]
+  congr 1
+  ring
+
+lemma traceGridTime_eq (δ : ℝ) (N n k : ℕ) :
+    refinedGridTime δ N (4*n) k = δ+TraceChaining.grid (traceMeshBase N) n k := by
+  unfold refinedGridTime TraceChaining.grid
+  rw [traceMeshBase_eq]
+  push_cast
+  simp only [← pow_mul]
+  congr 3; ring
+
+lemma traceGridCard_eq (M N n : ℕ) :
+    M*(2^(2*N+1))^(4*n) = M*(traceMeshBase N)^n := by
+  rw [traceMeshBase_eq]
+  simp only [← pow_mul]
+  congr 2; ring
+
+lemma traceRate_div (N n : ℕ) :
+    (modulusRate (4*N+2))^n/(traceRatio N)^(n+1) =
+      1/traceRatio N*(1/2:ℝ)^n := by
+  rw [traceRate_eq,div_pow,pow_succ,div_pow,one_pow]
+  field_simp [ne_of_gt (traceRatio_pos N)]
+
+lemma traceMesh_div (N n : ℕ) :
+    (1/(traceMeshBase N:ℝ)^n)/((traceRatio N)^(n+1))^2 =
+      1/(traceRatio N)^2*(1/16:ℝ)^n := by
+  rw [← one_div_pow,traceMesh_inv,div_pow,
+    show (traceRatio N)^(n+1) = (traceRatio N)^n*traceRatio N by rfl,mul_pow,div_pow,one_pow]
+  rw [pow_right_comm (traceRatio N) 2 n]
+  field_simp [ne_of_gt (traceRatio_pos N)]
+
+lemma traceMesh_eventually (N : ℕ) : ∀ᶠ n : ℕ in atTop,
+    1/(traceMeshBase N:ℝ)^n ≤ ((traceRatio N)^(n+1))^2 := by
+  have hh : Tendsto (fun n : ℕ => 1/(traceRatio N)^2*(1/16:ℝ)^n) atTop (𝓝 0) := by
+    simpa only [mul_zero] using (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : 0 ≤ (1/16:ℝ))
+      (by norm_num : (1/16:ℝ) < 1)).const_mul (1/(traceRatio N)^2)
+  filter_upwards [hh.eventually (Iio_mem_nhds zero_lt_one)] with n hn
+  rw [← traceMesh_div] at hn
+  have hr := traceRatio_pos N
+  exact ((div_lt_one (by positivity : 0 < ((traceRatio N)^(n+1))^2)).mp hn).le
+
+lemma traceError_summable {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) {N : ℕ} (hN : 0 < N) (H : ℕ) :
+    Summable (traceError κ N H) := by
+  have hp := trace_exponents hκ hκ8
+  have hr : |(traceRatio N)^(traceBeta κ)| < 1 := by
+    rw [abs_of_pos (Real.rpow_pos_of_pos (traceRatio_pos N) _)]
+    exact Real.rpow_lt_one (traceRatio_pos N).le (traceRatio_lt_one hN) hp.1
+  have hs := (summable_geometric_of_abs_lt_one (by norm_num : |(1/2:ℝ)| < 1)).mul_left
+    (brownianModulusConst (4*N+2)/traceRatio N)
+  have ht := (summable_geometric_of_abs_lt_one (by norm_num : |(1/16:ℝ)| < 1)).mul_left
+    (a κ/(traceRatio N)^2)
+  have hu := (((hs.add ht).mul_left (4*(2+a κ+2*a κ*H))).add (summable_geometric_of_abs_lt_one hr))
+  change Summable (fun n : ℕ => 4*(2+a κ+2*a κ*H)*
+    (brownianModulusConst (4*N+2)/traceRatio N*(1/2:ℝ)^n+
+      a κ/(traceRatio N)^2*(1/16:ℝ)^n)+((traceRatio N)^(traceBeta κ))^n)
+  exact hu
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open BrownianCalculus TraceChaining
+
+ 
+
+theorem radial_uniform_bands {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∃ N : ℕ, 0 < N ∧ ∀ᵐ b ∂W, ∀ m H : ℕ, ∀ᶠ n : ℕ in atTop,
+      ∀ t : ℝ, 1/((m:ℝ)+1) ≤ t → t ≤ H →
+      ∀ y : ℝ, (traceRatio N)^(n+1) ≤ y → y ≤ (traceRatio N)^n →
+      ‖radialValue κ b t y-radialValue κ b t ((traceRatio N)^n)‖ ≤
+        traceError κ N H n := by
+  obtain ⟨N,hN,hNeta,hgrid⟩ := radial_length_all_refined_grids hκ hκ8 hW
+  refine ⟨N,hN,?_⟩
+  filter_upwards [hgrid,brownian_uniform_modulus hW] with b hb hB
+  intro m H
+  let δ : ℝ := 1/((m:ℝ)+1)
+  have hδ : 0 < δ := by dsimp [δ]; positivity
+  have hr := traceRatio_pos N
+  have hr1 := traceRatio_lt_one hN
+  have hq : 0 < traceMeshBase N := by unfold traceMeshBase modulusBase; positivity
+  have hsub : Tendsto (fun n : ℕ => 4*n) atTop atTop := by
+    apply tendsto_atTop_mono (f := fun n : ℕ => n) (fun n => by omega) tendsto_id
+  have hg := hsub.eventually (hb m H)
+  obtain ⟨n₀,hn₀⟩ := hB (4*N+2) H (by omega)
+  filter_upwards [hg,traceMesh_eventually N,eventually_ge_atTop n₀] with n hn hmesh hn0
+  intro t hδt htH y hvy hyz
+  let k := index (traceMeshBase N) n (t-δ)
+  let s := δ+approx (traceMeshBase N) n (t-δ)
+  have hab := approx_bounds hq n (sub_nonneg.mpr hδt)
+  have hs : 0 ≤ s := by dsimp [s]; linarith [hab.1]
+  have hst : s ≤ t := by dsimp [s]; linarith [hab.2.1]
+  have hclose : t-s ≤ 1/(traceMeshBase N:ℝ)^n := by
+    dsimp [s]; linarith [hab.2.2]
+  have hk : k < H*(traceMeshBase N)^n := by
+    have hp : 0 < (traceMeshBase N:ℝ)^n := pow_pos (by exact_mod_cast hq) _
+    have hf := Nat.floor_le (mul_nonneg hp.le (sub_nonneg.mpr hδt))
+    have hh : (k:ℝ) < (H:ℝ)*(traceMeshBase N:ℝ)^n := by
+      dsimp [k,index]
+      nlinarith
+    exact_mod_cast hh
+  have hgk := hn ⟨k,by rwa [traceGridCard_eq]⟩
+  rw [traceGridTime_eq,traceHeight_eq] at hgk
+  change radialLength κ b s ((traceRatio N)^n) < _ at hgk
+  have hR : radialLength κ b s ((traceRatio N)^n) ≤
+      ENNReal.ofReal (((traceRatio N)^(traceBeta κ))^n) := by
+    have he : ((traceRatio N)^n)^(traceBeta κ) =
+        ((traceRatio N)^(traceBeta κ))^n := by
+      rw [← Real.rpow_natCast_mul hr.le,mul_comm (n:ℝ) (traceBeta κ),
+        Real.rpow_mul_natCast hr.le]
+    rw [he] at hgk
+    exact hgk.le
+  have hbnd := hn₀ n hn0 s t hs hst htH hclose
+  change |driverValue b t-driverValue b s| ≤
+    brownianModulusConst (4*N+2)*(modulusRate (4*N+2))^n at hbnd
+  have hp := radial_interpolation hκ hs hst htH (pow_pos hr _) hvy hyz
+    (pow_le_one₀ hr.le hr1.le) (hclose.trans hmesh) b hbnd hR (by positivity)
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hc : 0 ≤ 2*(2+a κ+2*a κ*H)/(traceRatio N)^(n+1) := by positivity
+  have hstep : a κ*(t-s)/(traceRatio N)^(n+1) ≤
+      a κ*(1/(traceMeshBase N:ℝ)^n)/(traceRatio N)^(n+1) := by
+    gcongr
+  have hp' : ‖radialValue κ b t y-radialValue κ b t ((traceRatio N)^n)‖ ≤
+      2*(2*(2+a κ+2*a κ*H)/(traceRatio N)^(n+1)*
+        (brownianModulusConst (4*N+2)*(modulusRate (4*N+2))^n+
+          a κ*(1/(traceMeshBase N:ℝ)^n)/(traceRatio N)^(n+1)))+
+        ((traceRatio N)^(traceBeta κ))^n := by
+    apply hp.trans
+    gcongr
+  apply hp'.trans_eq
+  dsimp [traceError]
+  rw [show 2*(2*(2+a κ+2*a κ*H)/(traceRatio N)^(n+1)*
+      (brownianModulusConst (4*N+2)*(modulusRate (4*N+2))^n+
+        a κ*(1/(traceMeshBase N:ℝ)^n)/(traceRatio N)^(n+1))) =
+      4*(2+a κ+2*a κ*H)*
+        (brownianModulusConst (4*N+2)*
+          ((modulusRate (4*N+2))^n/(traceRatio N)^(n+1))+
+          a κ*((1/(traceMeshBase N:ℝ)^n)/((traceRatio N)^(n+1))^2)) by ring]
+  rw [traceRate_div,traceMesh_div]
+  ring
+end SLEExactGauge.LiteralTrace
+
+ 
+
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.TraceChaining
+
+lemma radial_limit_of_bands {E : Type*} [MetricSpace E] [CompleteSpace E]
+    (f : ℝ → E) {r : ℝ} (hr : 0 < r) (hr1 : r < 1)
+    {e : ℕ → ℝ} (he : Summable e)
+    (hb : ∀ᶠ n : ℕ in atTop, ∀ y : ℝ, r^(n+1) ≤ y → y ≤ r^n →
+      dist (f y) (f (r^n)) ≤ e n) :
+    ∃ z : E, Tendsto f (𝓝[>] 0) (𝓝 z) := by
+  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp hb
+  have hs (n : ℕ) : dist (f (r^(n+n₀))) (f (r^((n+1)+n₀))) ≤ e (n+n₀) := by
+    rw [dist_comm]
+    exact hn₀ (n+n₀) (by omega) _ (by simp [Nat.add_assoc,Nat.add_comm])
+      (by simpa only [show n+1+n₀ = (n+n₀)+1 by omega] using
+        pow_le_pow_of_le_one hr.le hr1.le (Nat.le_succ (n+n₀)))
+  obtain ⟨z,hz⟩ := cauchySeq_tendsto_of_complete
+    (cauchySeq_of_dist_le_of_summable (fun n => e (n+n₀)) hs
+      ((summable_nat_add_iff n₀).mpr he))
+  have hz' : Tendsto (fun n : ℕ => f (r^n)) atTop (𝓝 z) :=
+    (tendsto_add_atTop_iff_nat n₀).mp hz
+  refine ⟨z,Metric.tendsto_nhds.mpr ?_⟩
+  intro ε hε
+  have hε2 : 0 < ε/2 := by linarith
+  have hsmall := (he.tendsto_atTop_zero.eventually (Iio_mem_nhds hε2))
+  have hnear := (Metric.tendsto_atTop.mp hz') (ε/2) hε2
+  obtain ⟨n₁,hn₁⟩ := eventually_atTop.mp hsmall
+  obtain ⟨n₂,hn₂⟩ := hnear
+  let n₃ := max n₀ (max n₁ n₂)
+  have hv : 0 < r^(n₃+1) := pow_pos hr _
+  filter_upwards [self_mem_nhdsWithin,
+    (show ∀ᶠ y : ℝ in 𝓝[>] 0, y < r^(n₃+1) from
+      nhdsWithin_le_nhds (Iio_mem_nhds hv))] with y hy hyv
+  have hy1 : y ≤ 1 := hyv.le.trans (pow_le_one₀ hr.le hr1.le)
+  obtain ⟨n,hnlow,hnup⟩ := exists_nat_pow_near_of_lt_one hy hy1 hr hr1
+  have hn3 : n₃ ≤ n := by
+    by_contra hh
+    have hnlt : n+1 ≤ n₃+1 := by omega
+    have hlo := pow_le_pow_of_le_one hr.le hr1.le hnlt
+    linarith
+  have hband := hn₀ n (le_trans (le_max_left _ _) hn3) y hnlow.le hnup
+  have hen := hn₁ n (le_trans (le_trans (le_max_left _ _) (le_max_right _ _)) hn3)
+  have hzn := hn₂ n (le_trans (le_trans (le_max_right _ _) (le_max_right _ _)) hn3)
+  exact (dist_triangle _ _ _).trans_lt (by linarith)
+end SLEExactGauge.TraceChaining
+namespace SLEExactGauge.TraceChaining
+
+lemma radial_uniform_limit_of_bands {X E : Type*} [MetricSpace E] [CompleteSpace E]
+    (f : X → ℝ → E) (g : X → E) (S : Set X)
+    {r : ℝ} (hr : 0 < r) (hr1 : r < 1) {e : ℕ → ℝ} (he : Summable e)
+    (hb : ∀ᶠ n : ℕ in atTop, ∀ t ∈ S, ∀ y : ℝ, r^(n+1) ≤ y → y ≤ r^n →
+      dist (f t y) (f t (r^n)) ≤ e n)
+    (hg : ∀ t ∈ S, Tendsto (f t) (𝓝[>] 0) (𝓝 (g t))) :
+    TendstoUniformlyOn (fun n t => f t (r^n)) g atTop S := by
+  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp hb
+  have hgeom : Tendsto (fun n : ℕ => r^n) atTop (𝓝[>] 0) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    exact ⟨tendsto_pow_atTop_nhds_zero_of_lt_one hr.le hr1,
+      Eventually.of_forall (fun n => pow_pos hr n)⟩
+  have hd (n : ℕ) (hn : n₀ ≤ n) (t : X) (ht : t ∈ S) :
+      dist (f t (r^n)) (g t) ≤ ∑' j : ℕ, e (j+n) := by
+    have hstep (j : ℕ) : dist (f t (r^(j+n))) (f t (r^((j+1)+n))) ≤ e (j+n) := by
+      rw [dist_comm]
+      apply hn₀ (j+n) (by omega) t ht
+      · simp [Nat.add_assoc,Nat.add_comm]
+      · simpa only [show j+1+n = (j+n)+1 by omega] using
+          pow_le_pow_of_le_one hr.le hr1.le (Nat.le_succ (j+n))
+    have hh := dist_le_tsum_of_dist_le_of_tendsto₀ (fun j => e (j+n)) hstep
+      ((summable_nat_add_iff n).mpr he)
+      (((hg t ht).comp hgeom).comp (tendsto_add_atTop_nat n))
+    simpa only [Nat.zero_add] using hh
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  have hh := (tendsto_sum_nat_add e).eventually (Iio_mem_nhds hε)
+  filter_upwards [hh,eventually_ge_atTop n₀] with n hn hn0
+  intro t ht
+  rw [dist_comm]
+  exact (hd n hn0 t ht).trans_lt hn
+end SLEExactGauge.TraceChaining
+namespace SLEExactGauge.LiteralTrace
+open TraceChaining
+
+ 
+
+theorem trace_limit_all_positive {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W, ∀ t : ℝ, 0 < t →
+      Tendsto (fun y : ℝ => inverseFlow κ b (κ*t)
+        (driverValue b (κ*t)+y*Complex.I)) (𝓝[>] 0) (𝓝 (trace κ b t)) := by
+  obtain ⟨N,hN,hb⟩ := radial_uniform_bands hκ hκ8 hW
+  filter_upwards [hb] with b hb
+  intro t ht
+  have hT : 0 < κ*t := mul_pos hκ ht
+  obtain ⟨m,hm⟩ := exists_nat_gt (1/(κ*t))
+  have hδ : 1/((m:ℝ)+1) ≤ κ*t := by
+    have hmp : 0 < (m:ℝ)+1 := by positivity
+    apply (div_le_iff₀ hmp).mpr
+    have hh := (div_lt_iff₀ hT).mp hm
+    nlinarith
+  obtain ⟨H,hH⟩ := exists_nat_gt (κ*t)
+  obtain ⟨z,hz⟩ := radial_limit_of_bands (radialValue κ b (κ*t))
+    (traceRatio_pos N) (traceRatio_lt_one hN) (traceError_summable hκ hκ8 hN H)
+    (by
+      filter_upwards [hb m H] with n hn
+      intro y hy hy'
+      simpa only [dist_eq_norm] using hn (κ*t) hδ hH.le y hy hy')
+  have he : trace κ b t = z := hz.limUnder_eq
+  rwa [he]
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+
+lemma radial_time_modulus_symmetric {κ T y s t : ℝ} (hκ : 0 < κ)
+    (hy : 0 < y) (hy1 : y ≤ 1) (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T)
+    (hclose : |t-s| ≤ y^2) (b : Driver) :
+    ‖radialValue κ b t y-radialValue κ b s y‖ ≤
+      2*(2+a κ+2*a κ*T)/y*(|driverValue b t-driverValue b s|+a κ*|t-s|/y) := by
+  by_cases hst : s ≤ t
+  · rw [abs_of_nonneg (sub_nonneg.mpr hst)] at hclose ⊢
+    exact inverseFlow_time_modulus hκ hs.1 hst ht.2 hy hy1 hclose b
+  · have hts : t ≤ s := (lt_of_not_ge hst).le
+    rw [abs_sub_comm t s,abs_of_nonneg (sub_nonneg.mpr hts)] at hclose ⊢
+    rw [norm_sub_rev,abs_sub_comm]
+    exact inverseFlow_time_modulus hκ ht.1 hts hs.2 hy hy1 hclose b
+
+ 
+
+lemma radialValue_continuousOn {κ y T : ℝ} (hκ : 0 < κ) (hy : 0 < y)
+    (hy1 : y ≤ 1) (b : Driver) :
+    ContinuousOn (fun t => radialValue κ b t y) (Icc 0 T) := by
+  intro t ht
+  apply tendsto_iff_norm_sub_tendsto_zero.mpr
+  have hB : Continuous (driverValue b) :=
+    (map_continuous b).comp continuous_real_toNNReal
+  have h1 : Tendsto (fun s : ℝ => |driverValue b s-driverValue b t|)
+      (𝓝[Icc 0 T] t) (𝓝 0) := by
+    have hh : Tendsto (driverValue b) (𝓝[Icc 0 T] t) (𝓝 (driverValue b t)) :=
+      (hB.tendsto t).mono_left nhdsWithin_le_nhds
+    simpa using (hh.sub_const (driverValue b t)).abs
+  have h2 : Tendsto (fun s : ℝ => |s-t|) (𝓝[Icc 0 T] t) (𝓝 0) := by
+    have hh : Tendsto (fun s : ℝ => s) (𝓝[Icc 0 T] t) (𝓝 t) :=
+      tendsto_id'.mpr nhdsWithin_le_nhds
+    simpa using (hh.sub_const t).abs
+  have hb := (h1.add ((h2.const_mul (a κ)).div_const y)).const_mul
+    (2*(2+a κ+2*a κ*T)/y)
+  have hb' : Tendsto (fun s : ℝ => 2*(2+a κ+2*a κ*T)/y*
+      (|driverValue b s-driverValue b t|+a κ*|s-t|/y))
+      (𝓝[Icc 0 T] t) (𝓝 0) := by simpa using hb
+  apply squeeze_zero' (Eventually.of_forall (fun _ => norm_nonneg _)) _ hb'
+  filter_upwards [self_mem_nhdsWithin,h2.eventually (Iio_mem_nhds (sq_pos_of_pos hy))] with s hs hclose
+  exact radial_time_modulus_symmetric hκ hy hy1 ht hs hclose.le b
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open TraceChaining
+
+ 
+
+theorem trace_continuous_positive {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8)
+    {W : WienerLaw} (hW : IsWiener W) :
+    ∀ᵐ b ∂W, ContinuousOn (trace κ b) (Ioi 0) := by
+  obtain ⟨N,hN,hb⟩ := radial_uniform_bands hκ hκ8 hW
+  filter_upwards [hb,trace_limit_all_positive hκ hκ8 hW] with b hb hlim
+  let g : ℝ → ℂ := fun T => limUnder (𝓝[>] (0:ℝ)) (radialValue κ b T)
+  have hg (T : ℝ) (hT : 0 < T) : Tendsto (radialValue κ b T) (𝓝[>] 0) (𝓝 (g T)) := by
+    have hh := hlim (T/κ) (div_pos hT hκ)
+    have he : κ*(T/κ) = T := by field_simp
+    rw [he] at hh
+    have he' : g T = trace κ b (T/κ) := hh.limUnder_eq
+    rwa [he']
+  have hc (m H : ℕ) : ContinuousOn g (Icc (1/((m:ℝ)+1)) H) := by
+    have hu := radial_uniform_limit_of_bands (fun T => radialValue κ b T) g
+      (Icc (1/((m:ℝ)+1)) H) (traceRatio_pos N) (traceRatio_lt_one hN)
+      (traceError_summable hκ hκ8 hN H)
+      (by
+        filter_upwards [hb m H] with n hn
+        intro T hT y hy hy'
+        simpa only [dist_eq_norm] using hn T hT.1 hT.2 y hy hy')
+      (fun T hT => hg T ((by positivity : 0 < 1/((m:ℝ)+1)).trans_le hT.1))
+    apply hu.continuousOn
+    apply Eventually.frequently
+    apply Eventually.of_forall
+    intro n
+    exact (radialValue_continuousOn hκ (pow_pos (traceRatio_pos N) n)
+      (pow_le_one₀ (traceRatio_pos N).le (traceRatio_lt_one hN).le) b).mono
+      (Icc_subset_Icc (by positivity) le_rfl)
+  have hc' : ContinuousOn g (Ioi 0) := by
+    apply continuousOn_of_forall_continuousAt
+    intro T hT
+    obtain ⟨m,hm⟩ := exists_nat_gt (1/T)
+    obtain ⟨H,hH⟩ := exists_nat_gt T
+    have hδ : 1/((m:ℝ)+1) < T := by
+      apply (div_lt_iff₀ (by positivity : 0 < (m:ℝ)+1)).mpr
+      have hh := (div_lt_iff₀ hT).mp hm
+      nlinarith
+    exact (hc m H).continuousAt (Icc_mem_nhds hδ hH)
+  have he : (fun t => g (κ*t)) = trace κ b := by
+    funext t
+    rfl
+  rw [← he]
+  exact hc'.comp (continuous_const.mul continuous_id).continuousOn
+    (fun t ht => mul_pos hκ ht)
+end SLEExactGauge.LiteralTrace
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.TraceChaining
+
+ 
+
+lemma initial_le_of_outward_deriv {f f' : ℝ → ℝ} {T R : ℝ} (hT : 0 ≤ T)
+    (hf : ContinuousOn f (Icc 0 T))
+    (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt f (f' t) (Icc 0 T) t)
+    (hR : f T ≤ R) (hout : ∀ t ∈ Icc 0 T, R < f t → 0 < f' t) :
+    f 0 ≤ R := by
+  have hm : MapsTo (fun t : ℝ => T-t) (Icc 0 T) (Icc 0 T) := by
+    intro t ht
+    constructor <;> linarith [ht.1,ht.2]
+  have hc : ContinuousOn (fun t => f (T-t)) (Icc 0 T) :=
+    hf.comp (continuous_const.sub continuous_id).continuousOn hm
+  have hh (t : ℝ) (ht : t ∈ Ico 0 T) :
+      HasDerivWithinAt (fun t => f (T-t)) (-f' (T-t)) (Ici t) t := by
+    have h := (hd (T-t) (hm ⟨ht.1,ht.2.le⟩)).comp t
+      ((hasDerivAt_id t).const_sub T).hasDerivWithinAt hm
+    have he : Icc (0:ℝ) T ∈ 𝓝[Ici t] t := by
+      filter_upwards [self_mem_nhdsWithin,
+        (show ∀ᶠ u : ℝ in 𝓝[Ici t] t, u < T from
+          nhdsWithin_le_nhds (Iio_mem_nhds ht.2))] with u hu huT
+      exact ⟨ht.1.trans hu,huT.le⟩
+    convert! h.mono_of_mem_nhdsWithin he using 1; simp only [mul_neg,mul_one]
+  apply le_of_forall_pos_le_add
+  intro ε hε
+  have hfence := image_le_of_deriv_right_lt_deriv_boundary hc hh
+    (B := fun _ => R+ε) (B' := fun _ => 0)
+    (by simpa using hR.trans (by linarith : R ≤ R+ε))
+    (fun _ => hasDerivAt_const _ _)
+    (by
+      intro t ht he
+      have hp := hout (T-t) (hm ⟨ht.1,ht.2.le⟩) (by linarith)
+      linarith)
+  simpa using hfence (right_mem_Icc.mpr hT)
+end SLEExactGauge.TraceChaining
+namespace SLEExactGauge.LiteralTrace
+open TraceChaining
+
+lemma PositiveSolution.re_hasDeriv {κ : ℝ} {b : Driver} {z : ℂ} {T : ℝ} {f : ℝ → ℂ}
+    (hf : PositiveSolution κ b z T f) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    HasDerivWithinAt (fun u => (f u).re)
+      (a κ*((f t).re-driverValue b t)/‖f t-driverValue b t‖^2) (Icc 0 T) t := by
+  have hi := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t (hf.2.2.2.2 t ht)
+  convert! hi using 1; first | rfl |
+    (simp only [Complex.reCLM_apply,Complex.div_re,Complex.ofReal_im,Complex.ofReal_re,
+     Complex.sub_re,Complex.normSq_eq_norm_sq,zero_mul]; ring)
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TraceContinuity.lean b/lean/OAI/Probability/SLEGauge/TraceContinuity.lean
index f24cf65065753952e83f1bcfed1500f5610f6fc4..d613c17b3adf39defed424f0cfbc397d440652cf 100644
--- a/lean/OAI/Probability/SLEGauge/TraceContinuity.lean
+++ b/lean/OAI/Probability/SLEGauge/TraceContinuity.lean
@@ -1,486 +1,486 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.RadialMoments
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped NNReal ENNReal Topology
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-open scoped ContDiff Manifold
-noncomputable section
-open Filter Topology
-noncomputable section
-open Complex Real Set Filter
-open scoped Topology
-noncomputable section
-open scoped Topology
-open MeasureTheory
-noncomputable section
-open Set Filter Real
-open scoped Topology
-open Set MeasureTheory ProbabilityTheory Topology Function Filter
-open scoped NNReal ENNReal ContinuousMapZero
-noncomputable section
-namespace SLEExactGauge.LiteralTrace
-theorem inverseFlow_time_modulus {κ s t T y : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
-    (hst : s ≤ t) (htT : t ≤ T) (hy : 0 < y) (hy1 : y ≤ 1)
-    (hΔ : t-s ≤ y^2) (b : Driver) :
-    ‖inverseFlow κ b t (driverValue b t+y*Complex.I)-
-        inverseFlow κ b s (driverValue b s+y*Complex.I)‖ ≤
-      (2*(2+a κ+2*a κ*T)/y)*(|driverValue b t-driverValue b s|+a κ*(t-s)/y) := by
-  let z := inverseFlow κ b t (driverValue b t+y*Complex.I)
-  have ht : 0 ≤ t := hs.trans hst
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hw : 0 < ((driverValue b t:ℂ)+y*Complex.I).im := by simpa using hy
-  have hf := surviving_flow_positiveSolution ht (inverseFlow_surviving hκ b ht hw)
-  change PositiveSolution κ b z t (flow κ b z) at hf
-  let v := flow κ b z s
-  have hfT : flow κ b z t = (driverValue b t:ℂ)+y*Complex.I := flow_inverseFlow hκ b ht hw
-  have his : s ∈ Icc 0 t := ⟨hs,hst⟩
-  have hit : t ∈ Icc 0 t := ⟨ht,le_rfl⟩
-  have hvis : y ≤ v.im := by simpa only [hfT,v,Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,
-    Complex.mul_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero] using hf.im_antitone hκ his hit hst
-  have hdist : ‖v-((driverValue b t:ℂ)+y*Complex.I)‖ ≤ a κ*(t-s)/y := by
-    have hh := hf.displacement_le hκ his hit hst
-    rw [hfT] at hh
-    simpa only [v,norm_sub_rev,Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im,
-      Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero,div_mul_eq_mul_div] using hh
-  have hsmall : a κ*(t-s)/y ≤ a κ*y := by
-    apply (div_le_iff₀ hy).mpr
-    nlinarith [mul_le_mul_of_nonneg_left hΔ ha.le]
-  have hiv : v.im ≤ 1+a κ := by
-    have hh := Complex.im_le_norm (v-((driverValue b t:ℂ)+y*Complex.I))
-    simp only [Complex.sub_im,Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im,
-      Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero] at hh
-    nlinarith [mul_le_mul_of_nonneg_left hy1 ha.le]
-  have hvmap : inverseFlow κ b s v = z := inverseFlow_flow hs (hf.restrict hs hst).surviving
-  have hh := inverseFlow_lipschitz_strip hκ hs (hst.trans htT) hy b
-    (v := v) (w := (driverValue b s:ℂ)+y*Complex.I) ⟨hvis,hiv⟩
-    (by simp only [Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im,Complex.I_im,
-          Complex.I_re,mul_one,mul_zero,zero_add,add_zero]; exact ⟨le_rfl,by linarith⟩)
-  have hinput : ‖v-((driverValue b s:ℂ)+y*Complex.I)‖ ≤
-      |driverValue b t-driverValue b s|+a κ*(t-s)/y := by
-    calc
-      _ ≤ ‖v-((driverValue b t:ℂ)+y*Complex.I)‖+
-          ‖((driverValue b t:ℂ)+y*Complex.I)-((driverValue b s:ℂ)+y*Complex.I)‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
-      _ ≤ _ := by
-        rw [add_sub_add_right_eq_sub,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs]
-        linarith
-  rw [hvmap] at hh
-  have hT0 : 0 ≤ T := ht.trans htT
-  calc
-    _ ≤ (2*(1+a κ+2*a κ*T+1)/y)*‖v-((driverValue b s:ℂ)+y*Complex.I)‖ := hh
-    _ ≤ (2*(1+a κ+2*a κ*T+1)/y)*(|driverValue b t-driverValue b s|+a κ*(t-s)/y) :=
-      mul_le_mul_of_nonneg_left hinput (by positivity)
-    _ = _ := by ring
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.BrownianCalculus
-
-def gaussianMoment (n : ℕ) : ℝ := ∫ x : ℝ, |x|^n ∂gaussianReal 0 1
-
-lemma gaussianMoment_nonneg (n : ℕ) : 0 ≤ gaussianMoment n :=
-  integral_nonneg (fun _ => by positivity)
-
-lemma gaussian_abs_pow_integrable (v : ℝ≥0) (n : ℕ) :
-    Integrable (fun x : ℝ => |x|^n) (gaussianReal 0 v) := by
-  simpa only [id_eq,Real.norm_eq_abs] using
-    (memLp_id_gaussianReal' (μ := 0) (v := v) (n:ℝ≥0∞) (by simp)).integrable_norm_pow'
-
-lemma gaussian_abs_pow_scaled (σ : ℝ≥0) (n : ℕ) :
-    (∫ x : ℝ, |x|^n ∂gaussianReal 0 (σ^2)) = (σ:ℝ)^n*gaussianMoment n := by
-  have he : (gaussianReal 0 1).map (fun x : ℝ => (σ:ℝ)*x) =
-      gaussianReal 0 (σ^2) := by
-    convert gaussianReal_map_const_mul (μ := 0) (v := 1) (σ:ℝ) using 1; simp
-    congr 1
-  rw [← he,integral_map (by fun_prop) (by fun_prop)]
-  simp_rw [abs_mul,abs_of_nonneg σ.coe_nonneg,mul_pow]
-  exact integral_const_mul _ _
-
-lemma increment_abs_pow_scaled {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s σ : ℝ≥0) (n : ℕ) :
-    (∫ b, |increment s (σ^2) b|^n ∂W) = (σ:ℝ)^n*gaussianMoment n := by
-  exact ((increment_law hW s (σ^2)).integral_comp (by fun_prop)).trans
-    (gaussian_abs_pow_scaled σ n)
-
-lemma increment_abs_pow_integrable {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s h : ℝ≥0) (n : ℕ) : Integrable (fun b => |increment s h b|^n) W := by
-  let := hW.isGaussianProcess.isProbabilityMeasure
-  have hm := (increment_law hW s h).hasGaussianLaw.memLp (p := (n : ℝ≥0∞)) (by simp)
-  simpa only [Real.norm_eq_abs] using hm.integrable_norm_pow'
-
-lemma increment_scaled_tail {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (s σ : ℝ≥0) (hσ : 0 < σ) {r : ℝ} (hr : 0 < r) (n : ℕ) :
-    W {b : Driver | (σ:ℝ)*r ≤ |increment s (σ^2) b|} ≤
-      ENNReal.ofReal (gaussianMoment n/r^n) := by
-  have hsr : 0 < ((σ:ℝ)*r)^n := pow_pos (mul_pos (by exact_mod_cast hσ) hr) _
-  calc
-    _ ≤ W {b : Driver | ENNReal.ofReal (((σ:ℝ)*r)^n) ≤
-        ENNReal.ofReal (|increment s (σ^2) b|^n)} := by
-      apply measure_mono
-      intro b hb
-      exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ (by positivity) hb _)
-    _ ≤ (∫⁻ b : Driver, ENNReal.ofReal (|increment s (σ^2) b|^n) ∂W)/
-        ENNReal.ofReal (((σ:ℝ)*r)^n) :=
-      meas_ge_le_lintegral_div (((measurable_increment s (σ^2)).abs.pow_const n).ennreal_ofReal.aemeasurable) (ne_of_gt (ENNReal.ofReal_pos.mpr hsr)) ENNReal.ofReal_ne_top
-    _ = ENNReal.ofReal (((σ:ℝ)^n*gaussianMoment n)/(((σ:ℝ)*r)^n)) := by
-      rw [← ofReal_integral_eq_lintegral_ofReal (increment_abs_pow_integrable hW s (σ^2) n)
-        (ae_of_all _ (fun _ => by positivity)),increment_abs_pow_scaled hW,
-        ← ENNReal.ofReal_div_of_pos hsr]
-    _ = _ := by
-      congr 1
-      rw [mul_pow]
-      field_simp
-end SLEExactGauge.BrownianCalculus
-namespace SLEExactGauge.BrownianCalculus
-
-def modulusBase (K : ℕ) : ℕ := 2^(2*K)
-def modulusStd (K n : ℕ) : ℝ≥0 := ((1/2:ℝ≥0)^K)^n
-def modulusRate (K : ℕ) : ℝ := 2*(1/2:ℝ)^K
-
-def modulusGridBad (K M n : ℕ) : Set Driver :=
-  ⋃ k : Fin (M*(modulusBase K)^n), {b : Driver |
-    (modulusRate K)^n ≤ |increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b|}
-
-lemma modulus_grid_measure {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (K M n : ℕ) : W (modulusGridBad K M n) ≤
-      ENNReal.ofReal (((M:ℝ)*gaussianMoment (2*K+1))*(1/2:ℝ)^n) := by
-  have hs : 0 < modulusStd K n := by unfold modulusStd; positivity
-  have he : (modulusRate K)^n = (modulusStd K n:ℝ)*(2:ℝ)^n := by
-    simp only [modulusRate,modulusStd,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat,mul_pow]
-    ring
-  calc
-    _ ≤ ∑ k : Fin (M*(modulusBase K)^n), W {b : Driver |
-      (modulusRate K)^n ≤ |increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b|} :=
-      measure_iUnion_fintype_le _ _
-    _ ≤ ∑ _k : Fin (M*(modulusBase K)^n), ENNReal.ofReal (gaussianMoment (2*K+1)/((2:ℝ)^n)^(2*K+1)) := by
-      apply Finset.sum_le_sum
-      intro k _
-      rw [he]
-      exact increment_scaled_tail hW _ _ hs (by positivity) _
-    _ = ENNReal.ofReal (((M*(modulusBase K)^n:ℕ):ℝ)*
-        (gaussianMoment (2*K+1)/((2:ℝ)^n)^(2*K+1))) := by
-      rw [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,
-        ← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (Nat.cast_nonneg _)]
-    _ = _ := by
-      congr 1
-      simp only [modulusBase,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat]
-      rw [← pow_mul,← pow_mul,show n*(2*K+1) = (2*K)*n+n by ring,pow_add]
-      field_simp
-      rw [mul_assoc,← mul_pow]
-      norm_num
-
-lemma modulus_grid_eventually {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
-    (K M : ℕ) :
-    ∀ᵐ b ∂W, ∀ᶠ n : ℕ in atTop, ∀ k : Fin (M*(modulusBase K)^n),
-      |increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b| < (modulusRate K)^n := by
-  have hs : (∑' n : ℕ, W (modulusGridBad K M n)) ≠ (∞:ℝ≥0∞) := by
-    apply ne_top_of_le_ne_top
-      (((summable_geometric_of_abs_lt_one (by norm_num : |(1/2:ℝ)| < 1)).mul_left
-        ((M:ℝ)*gaussianMoment (2*K+1))).tsum_ofReal_ne_top)
-    exact ENNReal.tsum_le_tsum (modulus_grid_measure hW K M)
-  filter_upwards [ae_eventually_notMem hs] with b hb
-  filter_upwards [hb] with n hn
-  intro k
-  exact lt_of_not_ge (fun h => hn (mem_iUnion.mpr ⟨k,h⟩))
-
-lemma modulus_std_square (K n : ℕ) :
-    ((modulusStd K n:ℝ≥0):ℝ)^2 = 1/(modulusBase K:ℝ)^n := by
-  simp only [modulusStd,modulusBase,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,
-    NNReal.coe_ofNat,Nat.cast_pow,Nat.cast_ofNat,div_pow,one_pow]
-  rw [← pow_mul,← pow_mul,← pow_mul]
-  congr 2
-  ring
-
-lemma modulus_rate_pos (K : ℕ) : 0 < modulusRate K := by
-  unfold modulusRate; positivity
-
-lemma modulus_rate_lt_one {K : ℕ} (hK : 2 ≤ K) : modulusRate K < 1 := by
-  have hh : (1/2:ℝ)^K ≤ (1/2:ℝ)^2 :=
-    pow_le_pow_of_le_one (by norm_num) (by norm_num) hK
-  dsimp [modulusRate]
-  linarith
-
-end SLEExactGauge.BrownianCalculus
-
- 
-
-noncomputable section
-open MeasureTheory Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.TraceChaining
-
-def grid (q n k : ℕ) : ℝ := (k:ℝ)/(q:ℝ)^n
-
-def index (q n : ℕ) (t : ℝ) : ℕ := ⌊(q:ℝ)^n*t⌋₊
-
-def approx (q n : ℕ) (t : ℝ) : ℝ := grid q n (index q n t)
-
-lemma approx_bounds {q : ℕ} (hq : 0 < q) (n : ℕ) {t : ℝ} (ht : 0 ≤ t) :
-    0 ≤ approx q n t ∧ approx q n t ≤ t ∧ t < approx q n t+1/(q:ℝ)^n := by
-  have hp : 0 < (q:ℝ)^n := pow_pos (by exact_mod_cast hq) _
-  refine ⟨by unfold approx grid; positivity,?_,?_⟩
-  · exact (div_le_iff₀ hp).mpr (by
-      simpa only [index,mul_comm] using Nat.floor_le (mul_nonneg hp.le ht))
-  · have hh := Nat.lt_floor_add_one ((q:ℝ)^n*t)
-    dsimp [approx,grid,index]
-    rw [← add_div]
-    exact (lt_div_iff₀ hp).mpr (by simpa only [mul_comm] using hh)
-
-lemma approx_tendsto {q : ℕ} (hq : 1 < q) {t : ℝ} (ht : 0 ≤ t) :
-    Tendsto (fun n => approx q n t) atTop (𝓝 t) := by
-  have hqp : 0 < q := by omega
-  have hqR : 1 < (q:ℝ) := by exact_mod_cast hq
-  have hz : Tendsto (fun n : ℕ => 1/(q:ℝ)^n) atTop (𝓝 0) := by
-    simpa only [one_div,inv_pow] using tendsto_pow_atTop_nhds_zero_of_lt_one
-      (by positivity : 0 ≤ (q:ℝ)⁻¹) ((inv_lt_one₀ (by positivity)).mpr hqR)
-  apply tendsto_of_tendsto_of_tendsto_of_le_of_le
-    (g := fun n => t-1/(q:ℝ)^n) (h := fun _ => t)
-    (by simpa using tendsto_const_nhds.sub hz) tendsto_const_nhds
-    (fun n => by linarith [(approx_bounds hqp n ht).2.2])
-    (fun n => (approx_bounds hqp n ht).2.1)
-
-lemma index_le {q M n : ℕ} (_hq : 0 < q) {t : ℝ} (ht : t ≤ M) :
-    index q n t ≤ M*q^n := by
-  apply Nat.floor_le_of_le
-  simp only [Nat.cast_mul,Nat.cast_pow]
-  nlinarith [pow_nonneg (Nat.cast_nonneg q : (0:ℝ) ≤ q) n]
-
-lemma index_refine {q n : ℕ} (hq : 0 < q) (t : ℝ) :
-    index q (n+1) t / q = index q n t := by
-  unfold index
-  rw [pow_succ,mul_right_comm _ _ t,Nat.mul_cast_floor_div_cancel (by omega)]
-
-lemma grid_refine {q : ℕ} (hq : 0 < q) (n k : ℕ) :
-    grid q (n+1) (q*k) = grid q n k := by
-  unfold grid
-  push_cast
-  rw [pow_succ]
-  field_simp
-
-lemma grid_distance_le {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
-    (q n L : ℕ) {c : ℝ} (_hc : 0 ≤ c)
-    (hb : ∀ k < L, dist (f (grid q n k)) (f (grid q n (k+1))) ≤ c)
-    {i j : ℕ} (hij : i ≤ j) (hj : j ≤ L) :
-    dist (f (grid q n i)) (f (grid q n j)) ≤ (j-i:ℕ)*c := by
-  calc
-    _ ≤ ∑ k ∈ Finset.Ico i j, c := dist_le_Ico_sum_of_dist_le (f := fun k => f (grid q n k)) hij
-      (fun {k} _hk hk => hb k (hk.trans_le hj))
-    _ = _ := by simp
-
-lemma approx_step {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
-    {q : ℕ} (hq : 0 < q) (M n : ℕ) {t : ℝ} (ht : t ≤ M)
-    {c : ℝ} (hc : 0 ≤ c)
-    (hb : ∀ k < M*q^(n+1), dist (f (grid q (n+1) k))
-      (f (grid q (n+1) (k+1))) ≤ c) :
-    dist (f (approx q n t)) (f (approx q (n+1) t)) ≤ (q:ℝ)*c := by
-  have hi := index_refine (n := n) hq t
-  have hlow : q*index q n t ≤ index q (n+1) t := by
-    rw [← hi,mul_comm]
-    exact Nat.div_mul_le_self _ _
-  have hupp : index q (n+1) t - q*index q n t ≤ q := by
-    have he := Nat.mod_add_div (index q (n+1) t) q
-    have hm := Nat.mod_lt (index q (n+1) t) hq
-    rw [hi] at he
-    omega
-  have h := grid_distance_le f q (n+1) (M*q^(n+1)) hc hb hlow (index_le hq ht)
-  rw [grid_refine hq] at h
-  exact h.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast hupp) hc)
-
-lemma approx_distance {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
-    (hf : Continuous f) {q : ℕ} (hq : 1 < q) (M n₀ : ℕ)
-    {ρ : ℝ} (hρ0 : 0 ≤ ρ) (hρ1 : ρ < 1)
-    (hb : ∀ n, n₀ ≤ n → ∀ k < M*q^n,
-      dist (f (grid q n k)) (f (grid q n (k+1))) ≤ ρ^n)
-    {t : ℝ} (ht0 : 0 ≤ t) (htM : t ≤ M) {n : ℕ} (hn : n₀ ≤ n) :
-    dist (f (approx q n t)) (f t) ≤ (q:ℝ)*ρ^(n+1)/(1-ρ) := by
-  have hl := (hf.tendsto t).comp ((approx_tendsto hq ht0).comp (tendsto_add_atTop_nat n))
-  have hs (j : ℕ) : dist (f (approx q (j+n) t)) (f (approx q (j+1+n) t)) ≤
-      ((q:ℝ)*ρ^(n+1))*ρ^j := by
-    have hh := approx_step f (by omega : 0 < q) M (j+n) htM
-      (pow_nonneg hρ0 (j+n+1)) (hb (j+n+1) (by omega))
-    rw [show j+1+n = j+n+1 by omega]
-    simpa only [show j+n+1 = (n+1)+j by omega,pow_add,mul_assoc] using hh
-  have hh := dist_le_of_le_geometric_of_tendsto₀ ρ ((q:ℝ)*ρ^(n+1)) hρ1 hs hl
-  simpa only [Nat.zero_add] using hh
-
- 
-
-lemma grid_modulus {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
-    (hf : Continuous f) {q : ℕ} (hq : 1 < q) (M n₀ : ℕ)
-    {ρ : ℝ} (hρ0 : 0 ≤ ρ) (hρ1 : ρ < 1)
-    (hb : ∀ n, n₀ ≤ n → ∀ k < M*q^n,
-      dist (f (grid q n k)) (f (grid q n (k+1))) ≤ ρ^n)
-    {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) (ht : t ≤ M) {n : ℕ} (hn : n₀ ≤ n)
-    (hclose : t-s ≤ 1/(q:ℝ)^n) :
-    dist (f s) (f t) ≤ (2+2*(q:ℝ)*ρ/(1-ρ))*ρ^n := by
-  have hqp : 0 < q := by omega
-  have hp : 0 < (q:ℝ)^n := pow_pos (by exact_mod_cast hqp) _
-  have hi : index q n s ≤ index q n t :=
-    Nat.floor_mono (mul_le_mul_of_nonneg_left hst hp.le)
-  have hj : index q n t - index q n s ≤ 2 := by
-    have h1 := Nat.floor_le (mul_nonneg hp.le (hs.trans hst))
-    have h2 := Nat.lt_floor_add_one ((q:ℝ)^n*s)
-    have h3 := (le_div_iff₀ hp).mp hclose
-    have h4 : (index q n t:ℝ) < (index q n s:ℝ)+2 := by
-      dsimp [index]
-      nlinarith
-    have h5 : index q n t < index q n s+2 := by exact_mod_cast h4
-    omega
-  have hg := grid_distance_le f q n (M*q^n) (pow_nonneg hρ0 n)
-    (hb n hn) hi (index_le hqp ht)
-  have hg' : dist (f (approx q n s)) (f (approx q n t)) ≤ 2*ρ^n :=
-    hg.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (pow_nonneg hρ0 n))
-  have hs' := approx_distance f hf hq M n₀ hρ0 hρ1 hb hs (hst.trans ht) hn
-  have ht' := approx_distance f hf hq M n₀ hρ0 hρ1 hb (hs.trans hst) ht hn
-  calc
-    _ ≤ dist (f s) (f (approx q n s)) + dist (f (approx q n s)) (f (approx q n t)) +
-        dist (f (approx q n t)) (f t) := dist_triangle4 _ _ _ _
-    _ ≤ (q:ℝ)*ρ^(n+1)/(1-ρ)+2*ρ^n+(q:ℝ)*ρ^(n+1)/(1-ρ) := by
-      rw [dist_comm (f s)]
-      gcongr
-    _ = _ := by rw [pow_succ]; ring
-end SLEExactGauge.TraceChaining
-namespace SLEExactGauge.BrownianCalculus
-open SLEExactGauge.TraceChaining
-
-lemma increment_grid_eq (K n k : ℕ) (b : Driver) :
-    increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b =
-      b ((grid (modulusBase K) n (k+1)).toNNReal)-
-        b ((grid (modulusBase K) n k).toNNReal) := by
-  have he (j : ℕ) : (grid (modulusBase K) n j).toNNReal =
-      (j:ℝ≥0)*(modulusStd K n)^2 := by
-    apply NNReal.coe_injective
-    rw [Real.coe_toNNReal _ (by unfold grid; positivity)]
-    simp only [NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_pow,modulus_std_square,grid]
-    ring
-  rw [increment,he,he]
-  congr 2
-  push_cast
-  ring
-
- 
-
-theorem brownian_uniform_modulus {W : Measure Driver}
-    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W) :
-    ∀ᵐ b ∂W, ∀ K M : ℕ, 2 ≤ K → ∃ n₀ : ℕ, ∀ n ≥ n₀,
-      ∀ s t : ℝ, 0 ≤ s → s ≤ t → t ≤ M → t-s ≤ 1/(modulusBase K:ℝ)^n →
-      |b t.toNNReal-b s.toNNReal| ≤
-        (2+2*(modulusBase K:ℝ)*modulusRate K/(1-modulusRate K))*(modulusRate K)^n := by
-  have hh := ae_all_iff.mpr (fun K => ae_all_iff.mpr (fun M => modulus_grid_eventually hW K M))
-  filter_upwards [hh] with b hb
-  intro K M hK
-  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp (hb K M)
-  refine ⟨n₀,?_⟩
-  intro n hn s t hs hst ht hclose
-  have hg (m : ℕ) (hm : n₀ ≤ m) (k : ℕ) (hk : k < M*(modulusBase K)^m) :
-      dist (b ((grid (modulusBase K) m k).toNNReal))
-        (b ((grid (modulusBase K) m (k+1)).toNNReal)) ≤ (modulusRate K)^m := by
-    have hi := (hn₀ m hm ⟨k,hk⟩).le
-    rw [increment_grid_eq] at hi
-    simpa only [Real.dist_eq,abs_sub_comm] using hi
-  have hq : 1 < modulusBase K := by
-    dsimp [modulusBase]
-    exact one_lt_pow₀ (by norm_num) (by omega)
-  have hu := grid_modulus (fun t : ℝ => b t.toNNReal)
-    ((map_continuous b).comp continuous_real_toNNReal) hq M n₀
-    (modulus_rate_pos K).le (modulus_rate_lt_one hK) hg hs hst ht hn hclose
-  simpa only [Real.dist_eq,abs_sub_comm] using hu
-end SLEExactGauge.BrownianCalculus
-namespace SLEExactGauge.LiteralTrace
-
-abbrev radialValue (κ : ℝ) (b : Driver) (t y : ℝ) : ℂ :=
-  inverseFlow κ b t (driverValue b t+y*Complex.I)
-
-lemma radial_increment_real {κ s y z R : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
-    (hz : 0 < z) (hzy : z ≤ y) (b : Driver)
-    (hR : radialLength κ b s y ≤ ENNReal.ofReal R) (hR0 : 0 ≤ R) :
-    ‖radialValue κ b s y-radialValue κ b s z‖ ≤ R := by
-  exact (ENNReal.ofReal_le_ofReal_iff hR0).mp ((radial_increment_le hκ hs hz hzy b).trans hR)
-
-lemma time_bound_below {κ s t T y v D : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
-    (hst : s ≤ t) (htT : t ≤ T) (hv : 0 < v) (hvy : v ≤ y) (hy1 : y ≤ 1)
-    (hΔ : t-s ≤ v^2) (b : Driver) (hD : |driverValue b t-driverValue b s| ≤ D) :
-    ‖radialValue κ b t y-radialValue κ b s y‖ ≤
-      (2*(2+a κ+2*a κ*T)/v)*(D+a κ*(t-s)/v) := by
-  have ha : 0 < a κ := div_pos (by norm_num) hκ
-  have hy := hv.trans_le hvy
-  have hT : 0 ≤ T := hs.trans (hst.trans htT)
-  have ht : 0 ≤ t-s := sub_nonneg.mpr hst
-  have hc : 0 ≤ 2*(2+a κ+2*a κ*T) := by positivity
-  apply (inverseFlow_time_modulus hκ hs hst htT hy hy1
-    (hΔ.trans (pow_le_pow_left₀ hv.le hvy 2)) b).trans
-  apply mul_le_mul
-  · exact div_le_div_of_nonneg_left hc hv hvy
-  · exact add_le_add hD (div_le_div_of_nonneg_left (mul_nonneg ha.le ht) hv hvy)
-  · positivity
-  · positivity
-
- 
-
-lemma radial_interpolation {κ s t T v y z D R : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
-    (hst : s ≤ t) (htT : t ≤ T) (hv : 0 < v) (hvy : v ≤ y) (hyz : y ≤ z) (hz1 : z ≤ 1)
-    (hΔ : t-s ≤ v^2) (b : Driver) (hD : |driverValue b t-driverValue b s| ≤ D)
-    (hR : radialLength κ b s z ≤ ENNReal.ofReal R) (hR0 : 0 ≤ R) :
-    ‖radialValue κ b t y-radialValue κ b t z‖ ≤
-      2*((2*(2+a κ+2*a κ*T)/v)*(D+a κ*(t-s)/v))+R := by
-  have h1 := time_bound_below hκ hs hst htT hv hvy (hyz.trans hz1) hΔ b hD
-  have h2 := time_bound_below hκ hs hst htT hv (hvy.trans hyz) hz1 hΔ b hD
-  have h3 := radial_increment_real hκ hs (hv.trans_le hvy) hyz b hR hR0
-  have hh := dist_triangle4 (radialValue κ b t y) (radialValue κ b s y)
-    (radialValue κ b s z) (radialValue κ b t z)
-  simp only [dist_eq_norm] at hh
-  rw [norm_sub_rev] at h2 h3
-  linarith
-end SLEExactGauge.LiteralTrace
-namespace SLEExactGauge.LiteralTrace
-open BrownianCalculus
-
-def traceRatio (N : ℕ) : ℝ := (1/2:ℝ)^(4*N)
-def traceMeshBase (N : ℕ) : ℕ := modulusBase (4*N+2)
-def brownianModulusConst (K : ℕ) : ℝ :=
-  2+2*(modulusBase K:ℝ)*modulusRate K/(1-modulusRate K)
-
-def traceError (κ : ℝ) (N H n : ℕ) : ℝ :=
-  4*(2+a κ+2*a κ*H)*
-    (brownianModulusConst (4*N+2)/traceRatio N*(1/2:ℝ)^n+
-      a κ/(traceRatio N)^2*(1/16:ℝ)^n)+((traceRatio N)^(traceBeta κ))^n
-
-end SLEExactGauge.LiteralTrace
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.RadialMoments
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped NNReal ENNReal Topology
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+open scoped ContDiff Manifold
+noncomputable section
+open Filter Topology
+noncomputable section
+open Complex Real Set Filter
+open scoped Topology
+noncomputable section
+open scoped Topology
+open MeasureTheory
+noncomputable section
+open Set Filter Real
+open scoped Topology
+open Set MeasureTheory ProbabilityTheory Topology Function Filter
+open scoped NNReal ENNReal ContinuousMapZero
+noncomputable section
+namespace SLEExactGauge.LiteralTrace
+theorem inverseFlow_time_modulus {κ s t T y : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
+    (hst : s ≤ t) (htT : t ≤ T) (hy : 0 < y) (hy1 : y ≤ 1)
+    (hΔ : t-s ≤ y^2) (b : Driver) :
+    ‖inverseFlow κ b t (driverValue b t+y*Complex.I)-
+        inverseFlow κ b s (driverValue b s+y*Complex.I)‖ ≤
+      (2*(2+a κ+2*a κ*T)/y)*(|driverValue b t-driverValue b s|+a κ*(t-s)/y) := by
+  let z := inverseFlow κ b t (driverValue b t+y*Complex.I)
+  have ht : 0 ≤ t := hs.trans hst
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hw : 0 < ((driverValue b t:ℂ)+y*Complex.I).im := by simpa using hy
+  have hf := surviving_flow_positiveSolution ht (inverseFlow_surviving hκ b ht hw)
+  change PositiveSolution κ b z t (flow κ b z) at hf
+  let v := flow κ b z s
+  have hfT : flow κ b z t = (driverValue b t:ℂ)+y*Complex.I := flow_inverseFlow hκ b ht hw
+  have his : s ∈ Icc 0 t := ⟨hs,hst⟩
+  have hit : t ∈ Icc 0 t := ⟨ht,le_rfl⟩
+  have hvis : y ≤ v.im := by simpa only [hfT,v,Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,
+    Complex.mul_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero] using hf.im_antitone hκ his hit hst
+  have hdist : ‖v-((driverValue b t:ℂ)+y*Complex.I)‖ ≤ a κ*(t-s)/y := by
+    have hh := hf.displacement_le hκ his hit hst
+    rw [hfT] at hh
+    simpa only [v,norm_sub_rev,Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im,
+      Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero,div_mul_eq_mul_div] using hh
+  have hsmall : a κ*(t-s)/y ≤ a κ*y := by
+    apply (div_le_iff₀ hy).mpr
+    nlinarith [mul_le_mul_of_nonneg_left hΔ ha.le]
+  have hiv : v.im ≤ 1+a κ := by
+    have hh := Complex.im_le_norm (v-((driverValue b t:ℂ)+y*Complex.I))
+    simp only [Complex.sub_im,Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im,
+      Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero] at hh
+    nlinarith [mul_le_mul_of_nonneg_left hy1 ha.le]
+  have hvmap : inverseFlow κ b s v = z := inverseFlow_flow hs (hf.restrict hs hst).surviving
+  have hh := inverseFlow_lipschitz_strip hκ hs (hst.trans htT) hy b
+    (v := v) (w := (driverValue b s:ℂ)+y*Complex.I) ⟨hvis,hiv⟩
+    (by simp only [Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im,Complex.I_im,
+          Complex.I_re,mul_one,mul_zero,zero_add,add_zero]; exact ⟨le_rfl,by linarith⟩)
+  have hinput : ‖v-((driverValue b s:ℂ)+y*Complex.I)‖ ≤
+      |driverValue b t-driverValue b s|+a κ*(t-s)/y := by
+    calc
+      _ ≤ ‖v-((driverValue b t:ℂ)+y*Complex.I)‖+
+          ‖((driverValue b t:ℂ)+y*Complex.I)-((driverValue b s:ℂ)+y*Complex.I)‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
+      _ ≤ _ := by
+        rw [add_sub_add_right_eq_sub,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs]
+        linarith
+  rw [hvmap] at hh
+  have hT0 : 0 ≤ T := ht.trans htT
+  calc
+    _ ≤ (2*(1+a κ+2*a κ*T+1)/y)*‖v-((driverValue b s:ℂ)+y*Complex.I)‖ := hh
+    _ ≤ (2*(1+a κ+2*a κ*T+1)/y)*(|driverValue b t-driverValue b s|+a κ*(t-s)/y) :=
+      mul_le_mul_of_nonneg_left hinput (by positivity)
+    _ = _ := by ring
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.BrownianCalculus
+
+def gaussianMoment (n : ℕ) : ℝ := ∫ x : ℝ, |x|^n ∂gaussianReal 0 1
+
+lemma gaussianMoment_nonneg (n : ℕ) : 0 ≤ gaussianMoment n :=
+  integral_nonneg (fun _ => by positivity)
+
+lemma gaussian_abs_pow_integrable (v : ℝ≥0) (n : ℕ) :
+    Integrable (fun x : ℝ => |x|^n) (gaussianReal 0 v) := by
+  simpa only [id_eq,Real.norm_eq_abs] using
+    (memLp_id_gaussianReal' (μ := 0) (v := v) (n:ℝ≥0∞) (by simp)).integrable_norm_pow'
+
+lemma gaussian_abs_pow_scaled (σ : ℝ≥0) (n : ℕ) :
+    (∫ x : ℝ, |x|^n ∂gaussianReal 0 (σ^2)) = (σ:ℝ)^n*gaussianMoment n := by
+  have he : (gaussianReal 0 1).map (fun x : ℝ => (σ:ℝ)*x) =
+      gaussianReal 0 (σ^2) := by
+    convert gaussianReal_map_const_mul (μ := 0) (v := 1) (σ:ℝ) using 1; simp
+    congr 1
+  rw [← he,integral_map (by fun_prop) (by fun_prop)]
+  simp_rw [abs_mul,abs_of_nonneg σ.coe_nonneg,mul_pow]
+  exact integral_const_mul _ _
+
+lemma increment_abs_pow_scaled {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s σ : ℝ≥0) (n : ℕ) :
+    (∫ b, |increment s (σ^2) b|^n ∂W) = (σ:ℝ)^n*gaussianMoment n := by
+  exact ((increment_law hW s (σ^2)).integral_comp (by fun_prop)).trans
+    (gaussian_abs_pow_scaled σ n)
+
+lemma increment_abs_pow_integrable {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s h : ℝ≥0) (n : ℕ) : Integrable (fun b => |increment s h b|^n) W := by
+  let := hW.isGaussianProcess.isProbabilityMeasure
+  have hm := (increment_law hW s h).hasGaussianLaw.memLp (p := (n : ℝ≥0∞)) (by simp)
+  simpa only [Real.norm_eq_abs] using hm.integrable_norm_pow'
+
+lemma increment_scaled_tail {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (s σ : ℝ≥0) (hσ : 0 < σ) {r : ℝ} (hr : 0 < r) (n : ℕ) :
+    W {b : Driver | (σ:ℝ)*r ≤ |increment s (σ^2) b|} ≤
+      ENNReal.ofReal (gaussianMoment n/r^n) := by
+  have hsr : 0 < ((σ:ℝ)*r)^n := pow_pos (mul_pos (by exact_mod_cast hσ) hr) _
+  calc
+    _ ≤ W {b : Driver | ENNReal.ofReal (((σ:ℝ)*r)^n) ≤
+        ENNReal.ofReal (|increment s (σ^2) b|^n)} := by
+      apply measure_mono
+      intro b hb
+      exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ (by positivity) hb _)
+    _ ≤ (∫⁻ b : Driver, ENNReal.ofReal (|increment s (σ^2) b|^n) ∂W)/
+        ENNReal.ofReal (((σ:ℝ)*r)^n) :=
+      meas_ge_le_lintegral_div (((measurable_increment s (σ^2)).abs.pow_const n).ennreal_ofReal.aemeasurable) (ne_of_gt (ENNReal.ofReal_pos.mpr hsr)) ENNReal.ofReal_ne_top
+    _ = ENNReal.ofReal (((σ:ℝ)^n*gaussianMoment n)/(((σ:ℝ)*r)^n)) := by
+      rw [← ofReal_integral_eq_lintegral_ofReal (increment_abs_pow_integrable hW s (σ^2) n)
+        (ae_of_all _ (fun _ => by positivity)),increment_abs_pow_scaled hW,
+        ← ENNReal.ofReal_div_of_pos hsr]
+    _ = _ := by
+      congr 1
+      rw [mul_pow]
+      field_simp
+end SLEExactGauge.BrownianCalculus
+namespace SLEExactGauge.BrownianCalculus
+
+def modulusBase (K : ℕ) : ℕ := 2^(2*K)
+def modulusStd (K n : ℕ) : ℝ≥0 := ((1/2:ℝ≥0)^K)^n
+def modulusRate (K : ℕ) : ℝ := 2*(1/2:ℝ)^K
+
+def modulusGridBad (K M n : ℕ) : Set Driver :=
+  ⋃ k : Fin (M*(modulusBase K)^n), {b : Driver |
+    (modulusRate K)^n ≤ |increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b|}
+
+lemma modulus_grid_measure {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (K M n : ℕ) : W (modulusGridBad K M n) ≤
+      ENNReal.ofReal (((M:ℝ)*gaussianMoment (2*K+1))*(1/2:ℝ)^n) := by
+  have hs : 0 < modulusStd K n := by unfold modulusStd; positivity
+  have he : (modulusRate K)^n = (modulusStd K n:ℝ)*(2:ℝ)^n := by
+    simp only [modulusRate,modulusStd,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat,mul_pow]
+    ring
+  calc
+    _ ≤ ∑ k : Fin (M*(modulusBase K)^n), W {b : Driver |
+      (modulusRate K)^n ≤ |increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b|} :=
+      measure_iUnion_fintype_le _ _
+    _ ≤ ∑ _k : Fin (M*(modulusBase K)^n), ENNReal.ofReal (gaussianMoment (2*K+1)/((2:ℝ)^n)^(2*K+1)) := by
+      apply Finset.sum_le_sum
+      intro k _
+      rw [he]
+      exact increment_scaled_tail hW _ _ hs (by positivity) _
+    _ = ENNReal.ofReal (((M*(modulusBase K)^n:ℕ):ℝ)*
+        (gaussianMoment (2*K+1)/((2:ℝ)^n)^(2*K+1))) := by
+      rw [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,
+        ← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (Nat.cast_nonneg _)]
+    _ = _ := by
+      congr 1
+      simp only [modulusBase,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat]
+      rw [← pow_mul,← pow_mul,show n*(2*K+1) = (2*K)*n+n by ring,pow_add]
+      field_simp
+      rw [mul_assoc,← mul_pow]
+      norm_num
+
+lemma modulus_grid_eventually {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W)
+    (K M : ℕ) :
+    ∀ᵐ b ∂W, ∀ᶠ n : ℕ in atTop, ∀ k : Fin (M*(modulusBase K)^n),
+      |increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b| < (modulusRate K)^n := by
+  have hs : (∑' n : ℕ, W (modulusGridBad K M n)) ≠ (∞:ℝ≥0∞) := by
+    apply ne_top_of_le_ne_top
+      (((summable_geometric_of_abs_lt_one (by norm_num : |(1/2:ℝ)| < 1)).mul_left
+        ((M:ℝ)*gaussianMoment (2*K+1))).tsum_ofReal_ne_top)
+    exact ENNReal.tsum_le_tsum (modulus_grid_measure hW K M)
+  filter_upwards [ae_eventually_notMem hs] with b hb
+  filter_upwards [hb] with n hn
+  intro k
+  exact lt_of_not_ge (fun h => hn (mem_iUnion.mpr ⟨k,h⟩))
+
+lemma modulus_std_square (K n : ℕ) :
+    ((modulusStd K n:ℝ≥0):ℝ)^2 = 1/(modulusBase K:ℝ)^n := by
+  simp only [modulusStd,modulusBase,NNReal.coe_pow,NNReal.coe_div,NNReal.coe_one,
+    NNReal.coe_ofNat,Nat.cast_pow,Nat.cast_ofNat,div_pow,one_pow]
+  rw [← pow_mul,← pow_mul,← pow_mul]
+  congr 2
+  ring
+
+lemma modulus_rate_pos (K : ℕ) : 0 < modulusRate K := by
+  unfold modulusRate; positivity
+
+lemma modulus_rate_lt_one {K : ℕ} (hK : 2 ≤ K) : modulusRate K < 1 := by
+  have hh : (1/2:ℝ)^K ≤ (1/2:ℝ)^2 :=
+    pow_le_pow_of_le_one (by norm_num) (by norm_num) hK
+  dsimp [modulusRate]
+  linarith
+
+end SLEExactGauge.BrownianCalculus
+
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.TraceChaining
+
+def grid (q n k : ℕ) : ℝ := (k:ℝ)/(q:ℝ)^n
+
+def index (q n : ℕ) (t : ℝ) : ℕ := ⌊(q:ℝ)^n*t⌋₊
+
+def approx (q n : ℕ) (t : ℝ) : ℝ := grid q n (index q n t)
+
+lemma approx_bounds {q : ℕ} (hq : 0 < q) (n : ℕ) {t : ℝ} (ht : 0 ≤ t) :
+    0 ≤ approx q n t ∧ approx q n t ≤ t ∧ t < approx q n t+1/(q:ℝ)^n := by
+  have hp : 0 < (q:ℝ)^n := pow_pos (by exact_mod_cast hq) _
+  refine ⟨by unfold approx grid; positivity,?_,?_⟩
+  · exact (div_le_iff₀ hp).mpr (by
+      simpa only [index,mul_comm] using Nat.floor_le (mul_nonneg hp.le ht))
+  · have hh := Nat.lt_floor_add_one ((q:ℝ)^n*t)
+    dsimp [approx,grid,index]
+    rw [← add_div]
+    exact (lt_div_iff₀ hp).mpr (by simpa only [mul_comm] using hh)
+
+lemma approx_tendsto {q : ℕ} (hq : 1 < q) {t : ℝ} (ht : 0 ≤ t) :
+    Tendsto (fun n => approx q n t) atTop (𝓝 t) := by
+  have hqp : 0 < q := by omega
+  have hqR : 1 < (q:ℝ) := by exact_mod_cast hq
+  have hz : Tendsto (fun n : ℕ => 1/(q:ℝ)^n) atTop (𝓝 0) := by
+    simpa only [one_div,inv_pow] using tendsto_pow_atTop_nhds_zero_of_lt_one
+      (by positivity : 0 ≤ (q:ℝ)⁻¹) ((inv_lt_one₀ (by positivity)).mpr hqR)
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le
+    (g := fun n => t-1/(q:ℝ)^n) (h := fun _ => t)
+    (by simpa using tendsto_const_nhds.sub hz) tendsto_const_nhds
+    (fun n => by linarith [(approx_bounds hqp n ht).2.2])
+    (fun n => (approx_bounds hqp n ht).2.1)
+
+lemma index_le {q M n : ℕ} (_hq : 0 < q) {t : ℝ} (ht : t ≤ M) :
+    index q n t ≤ M*q^n := by
+  apply Nat.floor_le_of_le
+  simp only [Nat.cast_mul,Nat.cast_pow]
+  nlinarith [pow_nonneg (Nat.cast_nonneg q : (0:ℝ) ≤ q) n]
+
+lemma index_refine {q n : ℕ} (hq : 0 < q) (t : ℝ) :
+    index q (n+1) t / q = index q n t := by
+  unfold index
+  rw [pow_succ,mul_right_comm _ _ t,Nat.mul_cast_floor_div_cancel (by omega)]
+
+lemma grid_refine {q : ℕ} (hq : 0 < q) (n k : ℕ) :
+    grid q (n+1) (q*k) = grid q n k := by
+  unfold grid
+  push_cast
+  rw [pow_succ]
+  field_simp
+
+lemma grid_distance_le {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
+    (q n L : ℕ) {c : ℝ} (_hc : 0 ≤ c)
+    (hb : ∀ k < L, dist (f (grid q n k)) (f (grid q n (k+1))) ≤ c)
+    {i j : ℕ} (hij : i ≤ j) (hj : j ≤ L) :
+    dist (f (grid q n i)) (f (grid q n j)) ≤ (j-i:ℕ)*c := by
+  calc
+    _ ≤ ∑ k ∈ Finset.Ico i j, c := dist_le_Ico_sum_of_dist_le (f := fun k => f (grid q n k)) hij
+      (fun {k} _hk hk => hb k (hk.trans_le hj))
+    _ = _ := by simp
+
+lemma approx_step {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
+    {q : ℕ} (hq : 0 < q) (M n : ℕ) {t : ℝ} (ht : t ≤ M)
+    {c : ℝ} (hc : 0 ≤ c)
+    (hb : ∀ k < M*q^(n+1), dist (f (grid q (n+1) k))
+      (f (grid q (n+1) (k+1))) ≤ c) :
+    dist (f (approx q n t)) (f (approx q (n+1) t)) ≤ (q:ℝ)*c := by
+  have hi := index_refine (n := n) hq t
+  have hlow : q*index q n t ≤ index q (n+1) t := by
+    rw [← hi,mul_comm]
+    exact Nat.div_mul_le_self _ _
+  have hupp : index q (n+1) t - q*index q n t ≤ q := by
+    have he := Nat.mod_add_div (index q (n+1) t) q
+    have hm := Nat.mod_lt (index q (n+1) t) hq
+    rw [hi] at he
+    omega
+  have h := grid_distance_le f q (n+1) (M*q^(n+1)) hc hb hlow (index_le hq ht)
+  rw [grid_refine hq] at h
+  exact h.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast hupp) hc)
+
+lemma approx_distance {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
+    (hf : Continuous f) {q : ℕ} (hq : 1 < q) (M n₀ : ℕ)
+    {ρ : ℝ} (hρ0 : 0 ≤ ρ) (hρ1 : ρ < 1)
+    (hb : ∀ n, n₀ ≤ n → ∀ k < M*q^n,
+      dist (f (grid q n k)) (f (grid q n (k+1))) ≤ ρ^n)
+    {t : ℝ} (ht0 : 0 ≤ t) (htM : t ≤ M) {n : ℕ} (hn : n₀ ≤ n) :
+    dist (f (approx q n t)) (f t) ≤ (q:ℝ)*ρ^(n+1)/(1-ρ) := by
+  have hl := (hf.tendsto t).comp ((approx_tendsto hq ht0).comp (tendsto_add_atTop_nat n))
+  have hs (j : ℕ) : dist (f (approx q (j+n) t)) (f (approx q (j+1+n) t)) ≤
+      ((q:ℝ)*ρ^(n+1))*ρ^j := by
+    have hh := approx_step f (by omega : 0 < q) M (j+n) htM
+      (pow_nonneg hρ0 (j+n+1)) (hb (j+n+1) (by omega))
+    rw [show j+1+n = j+n+1 by omega]
+    simpa only [show j+n+1 = (n+1)+j by omega,pow_add,mul_assoc] using hh
+  have hh := dist_le_of_le_geometric_of_tendsto₀ ρ ((q:ℝ)*ρ^(n+1)) hρ1 hs hl
+  simpa only [Nat.zero_add] using hh
+
+ 
+
+lemma grid_modulus {E : Type*} [PseudoMetricSpace E] (f : ℝ → E)
+    (hf : Continuous f) {q : ℕ} (hq : 1 < q) (M n₀ : ℕ)
+    {ρ : ℝ} (hρ0 : 0 ≤ ρ) (hρ1 : ρ < 1)
+    (hb : ∀ n, n₀ ≤ n → ∀ k < M*q^n,
+      dist (f (grid q n k)) (f (grid q n (k+1))) ≤ ρ^n)
+    {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) (ht : t ≤ M) {n : ℕ} (hn : n₀ ≤ n)
+    (hclose : t-s ≤ 1/(q:ℝ)^n) :
+    dist (f s) (f t) ≤ (2+2*(q:ℝ)*ρ/(1-ρ))*ρ^n := by
+  have hqp : 0 < q := by omega
+  have hp : 0 < (q:ℝ)^n := pow_pos (by exact_mod_cast hqp) _
+  have hi : index q n s ≤ index q n t :=
+    Nat.floor_mono (mul_le_mul_of_nonneg_left hst hp.le)
+  have hj : index q n t - index q n s ≤ 2 := by
+    have h1 := Nat.floor_le (mul_nonneg hp.le (hs.trans hst))
+    have h2 := Nat.lt_floor_add_one ((q:ℝ)^n*s)
+    have h3 := (le_div_iff₀ hp).mp hclose
+    have h4 : (index q n t:ℝ) < (index q n s:ℝ)+2 := by
+      dsimp [index]
+      nlinarith
+    have h5 : index q n t < index q n s+2 := by exact_mod_cast h4
+    omega
+  have hg := grid_distance_le f q n (M*q^n) (pow_nonneg hρ0 n)
+    (hb n hn) hi (index_le hqp ht)
+  have hg' : dist (f (approx q n s)) (f (approx q n t)) ≤ 2*ρ^n :=
+    hg.trans (mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (pow_nonneg hρ0 n))
+  have hs' := approx_distance f hf hq M n₀ hρ0 hρ1 hb hs (hst.trans ht) hn
+  have ht' := approx_distance f hf hq M n₀ hρ0 hρ1 hb (hs.trans hst) ht hn
+  calc
+    _ ≤ dist (f s) (f (approx q n s)) + dist (f (approx q n s)) (f (approx q n t)) +
+        dist (f (approx q n t)) (f t) := dist_triangle4 _ _ _ _
+    _ ≤ (q:ℝ)*ρ^(n+1)/(1-ρ)+2*ρ^n+(q:ℝ)*ρ^(n+1)/(1-ρ) := by
+      rw [dist_comm (f s)]
+      gcongr
+    _ = _ := by rw [pow_succ]; ring
+end SLEExactGauge.TraceChaining
+namespace SLEExactGauge.BrownianCalculus
+open SLEExactGauge.TraceChaining
+
+lemma increment_grid_eq (K n k : ℕ) (b : Driver) :
+    increment ((k:ℝ≥0)*(modulusStd K n)^2) ((modulusStd K n)^2) b =
+      b ((grid (modulusBase K) n (k+1)).toNNReal)-
+        b ((grid (modulusBase K) n k).toNNReal) := by
+  have he (j : ℕ) : (grid (modulusBase K) n j).toNNReal =
+      (j:ℝ≥0)*(modulusStd K n)^2 := by
+    apply NNReal.coe_injective
+    rw [Real.coe_toNNReal _ (by unfold grid; positivity)]
+    simp only [NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_pow,modulus_std_square,grid]
+    ring
+  rw [increment,he,he]
+  congr 2
+  push_cast
+  ring
+
+ 
+
+theorem brownian_uniform_modulus {W : Measure Driver}
+    (hW : IsBrownianReal (fun (t : ℝ≥0) (b : Driver) => b t) W) :
+    ∀ᵐ b ∂W, ∀ K M : ℕ, 2 ≤ K → ∃ n₀ : ℕ, ∀ n ≥ n₀,
+      ∀ s t : ℝ, 0 ≤ s → s ≤ t → t ≤ M → t-s ≤ 1/(modulusBase K:ℝ)^n →
+      |b t.toNNReal-b s.toNNReal| ≤
+        (2+2*(modulusBase K:ℝ)*modulusRate K/(1-modulusRate K))*(modulusRate K)^n := by
+  have hh := ae_all_iff.mpr (fun K => ae_all_iff.mpr (fun M => modulus_grid_eventually hW K M))
+  filter_upwards [hh] with b hb
+  intro K M hK
+  obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp (hb K M)
+  refine ⟨n₀,?_⟩
+  intro n hn s t hs hst ht hclose
+  have hg (m : ℕ) (hm : n₀ ≤ m) (k : ℕ) (hk : k < M*(modulusBase K)^m) :
+      dist (b ((grid (modulusBase K) m k).toNNReal))
+        (b ((grid (modulusBase K) m (k+1)).toNNReal)) ≤ (modulusRate K)^m := by
+    have hi := (hn₀ m hm ⟨k,hk⟩).le
+    rw [increment_grid_eq] at hi
+    simpa only [Real.dist_eq,abs_sub_comm] using hi
+  have hq : 1 < modulusBase K := by
+    dsimp [modulusBase]
+    exact one_lt_pow₀ (by norm_num) (by omega)
+  have hu := grid_modulus (fun t : ℝ => b t.toNNReal)
+    ((map_continuous b).comp continuous_real_toNNReal) hq M n₀
+    (modulus_rate_pos K).le (modulus_rate_lt_one hK) hg hs hst ht hn hclose
+  simpa only [Real.dist_eq,abs_sub_comm] using hu
+end SLEExactGauge.BrownianCalculus
+namespace SLEExactGauge.LiteralTrace
+
+abbrev radialValue (κ : ℝ) (b : Driver) (t y : ℝ) : ℂ :=
+  inverseFlow κ b t (driverValue b t+y*Complex.I)
+
+lemma radial_increment_real {κ s y z R : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
+    (hz : 0 < z) (hzy : z ≤ y) (b : Driver)
+    (hR : radialLength κ b s y ≤ ENNReal.ofReal R) (hR0 : 0 ≤ R) :
+    ‖radialValue κ b s y-radialValue κ b s z‖ ≤ R := by
+  exact (ENNReal.ofReal_le_ofReal_iff hR0).mp ((radial_increment_le hκ hs hz hzy b).trans hR)
+
+lemma time_bound_below {κ s t T y v D : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
+    (hst : s ≤ t) (htT : t ≤ T) (hv : 0 < v) (hvy : v ≤ y) (hy1 : y ≤ 1)
+    (hΔ : t-s ≤ v^2) (b : Driver) (hD : |driverValue b t-driverValue b s| ≤ D) :
+    ‖radialValue κ b t y-radialValue κ b s y‖ ≤
+      (2*(2+a κ+2*a κ*T)/v)*(D+a κ*(t-s)/v) := by
+  have ha : 0 < a κ := div_pos (by norm_num) hκ
+  have hy := hv.trans_le hvy
+  have hT : 0 ≤ T := hs.trans (hst.trans htT)
+  have ht : 0 ≤ t-s := sub_nonneg.mpr hst
+  have hc : 0 ≤ 2*(2+a κ+2*a κ*T) := by positivity
+  apply (inverseFlow_time_modulus hκ hs hst htT hy hy1
+    (hΔ.trans (pow_le_pow_left₀ hv.le hvy 2)) b).trans
+  apply mul_le_mul
+  · exact div_le_div_of_nonneg_left hc hv hvy
+  · exact add_le_add hD (div_le_div_of_nonneg_left (mul_nonneg ha.le ht) hv hvy)
+  · positivity
+  · positivity
+
+ 
+
+lemma radial_interpolation {κ s t T v y z D R : ℝ} (hκ : 0 < κ) (hs : 0 ≤ s)
+    (hst : s ≤ t) (htT : t ≤ T) (hv : 0 < v) (hvy : v ≤ y) (hyz : y ≤ z) (hz1 : z ≤ 1)
+    (hΔ : t-s ≤ v^2) (b : Driver) (hD : |driverValue b t-driverValue b s| ≤ D)
+    (hR : radialLength κ b s z ≤ ENNReal.ofReal R) (hR0 : 0 ≤ R) :
+    ‖radialValue κ b t y-radialValue κ b t z‖ ≤
+      2*((2*(2+a κ+2*a κ*T)/v)*(D+a κ*(t-s)/v))+R := by
+  have h1 := time_bound_below hκ hs hst htT hv hvy (hyz.trans hz1) hΔ b hD
+  have h2 := time_bound_below hκ hs hst htT hv (hvy.trans hyz) hz1 hΔ b hD
+  have h3 := radial_increment_real hκ hs (hv.trans_le hvy) hyz b hR hR0
+  have hh := dist_triangle4 (radialValue κ b t y) (radialValue κ b s y)
+    (radialValue κ b s z) (radialValue κ b t z)
+  simp only [dist_eq_norm] at hh
+  rw [norm_sub_rev] at h2 h3
+  linarith
+end SLEExactGauge.LiteralTrace
+namespace SLEExactGauge.LiteralTrace
+open BrownianCalculus
+
+def traceRatio (N : ℕ) : ℝ := (1/2:ℝ)^(4*N)
+def traceMeshBase (N : ℕ) : ℕ := modulusBase (4*N+2)
+def brownianModulusConst (K : ℕ) : ℝ :=
+  2+2*(modulusBase K:ℝ)*modulusRate K/(1-modulusRate K)
+
+def traceError (κ : ℝ) (N H n : ℕ) : ℝ :=
+  4*(2+a κ+2*a κ*H)*
+    (brownianModulusConst (4*N+2)/traceRatio N*(1/2:ℝ)^n+
+      a κ/(traceRatio N)^2*(1/16:ℝ)^n)+((traceRatio N)^(traceBeta κ))^n
+
+end SLEExactGauge.LiteralTrace
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TraceMassEndpoint.lean b/lean/OAI/Probability/SLEGauge/TraceMassEndpoint.lean
index 2d0ce9e7d80c1bb2846df6afa618f53e89ef2282..3f5a586aed64d10105a837b414a3fe503c38d914 100644
--- a/lean/OAI/Probability/SLEGauge/TraceMassEndpoint.lean
+++ b/lean/OAI/Probability/SLEGauge/TraceMassEndpoint.lean
@@ -1,521 +1,521 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.OptionalGreen
-import OAI.Probability.SLEGauge.TerminalExhaustion
-import OAI.Probability.SLEGauge.TraceRestriction
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal symmDiff
-namespace SLEExactGauge.TraceMassContinuity
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
-open TraceMassIntegration TraceRestriction TraceLevelLocalization MassVersions
-open GermPositivity
-
- 
-
-theorem actual_raw_mass_trace (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (W : Measure TDriver) (hW : Target.IsWiener W)
-    (t : ℕ→ℝ≥0∞) (hD : Dense {u : ℝ≥0 | ∃i,t i=u})
-    (idx : ℕ→ℕ) (hidx : ∀n : ℕ,t (idx n)=(n:ℝ≥0))
-    (is : ℕ→ℕ) (his : ∀n : ℕ,t (is n)=(smallTime n:ℝ≥0∞))
-    (j : ℕ) (hj : t j=∞) :
-    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
-      (hc : ∀ k j, j∈s k → 0≤c k j),
-      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
-      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
-      (∀ b, μ 0 b=0) ∧
-      (∀ v, Measurable (μ v)) ∧
-      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
-        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
-      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
-        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
-      (∀ i (f : C_c(UpperPlane,ℝ)),
-        Tendsto (fun k => eLpNorm (fun b =>
-          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1))) (t i) b)
-            - ∫z,f z∂μ (t i) b) 1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
-      (∀ i, t i=∞ → μ (t i) ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ)) ∧
-      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
-        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
-        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
-        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂(W.map driverHomeomorph)) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂(W.map driverHomeomorph)) +
-          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen (Target.a κ) v b z.1) ∂upperArea ∂(W.map driverHomeomorph)) ∧
-      ∀ᵐ b ∂W,
-        ContinuousOn (Target.trace κ b) (Ici 0) ∧
-        NullSingletonClass (μ ∞ (driverHomeomorph b)) ∧
-        (∀v : ℝ≥0, μ v (driverHomeomorph b)=
-          (μ ∞ (driverHomeomorph b)).restrict
-            (visited (fun z : UpperPlane => z.1) (internalTrace κ b) v)) ∧
-        (∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
-          (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-            (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
-          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
-        (∀A : Set UpperPlane, MeasurableSet A → Tendsto
-          (fun n : ℕ => μ (n:ℝ≥0) (driverHomeomorph b) A)
-          atTop (𝓝 (μ ∞ (driverHomeomorph b) A))) ∧
-        (∀K : Set UpperPlane, IsCompact K → Tendsto
-          (fun n : ℕ => (μ ∞ (driverHomeomorph b)-μ (n:ℝ≥0) (driverHomeomorph b)) K)
-          atTop (𝓝 0)) ∧
-        (∀v : ℝ≥0∞, 0<v → 0<μ v (driverHomeomorph b) univ) ∧
-        (∀v : ℝ≥0∞, 0<v → ∃K : Set UpperPlane, IsCompact K ∧
-          0<μ v (driverHomeomorph b) K ∧ μ v (driverHomeomorph b) K<∞) := by
-  have hBW := brownian_map_driver W hW
-  have : IsProbabilityMeasure (W.map driverHomeomorph) := hBW.isGaussianProcess.isProbabilityMeasure
-  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
-    actual_raw_mass_positive (Target.a κ) (a_gt_quarter hκ hκ8)
-      (W.map driverHomeomorph) hBW t idx hidx is his j hj
-  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
-  let K := CompactExhaustion.choice UpperPlane
-  let q := (4*Target.a κ-1)/2
-  have hq : 0<q := by dsimp [q]; linarith [a_gt_quarter hκ hκ8]
-  have hqp : q<4*Target.a κ-1 := by dsimp [q]; linarith [a_gt_quarter hκ hκ8]
-  have hang := convexAngularMass_liminf_finite (Target.a κ) q (a_gt_quarter hκ hκ8)
-    hq hqp (W.map driverHomeomorph) hBW K K.isCompact s c hc hsum
-  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
-  filter_upwards [ae_of_ae_map measurable_driverHomeomorph.aemeasurable hconv,
-    ae_of_ae_map measurable_driverHomeomorph.aemeasurable hang,
-    angularRegular_ae hκ hκ8 hW, LiteralTrace.trace_regular_all_times hκ hκ8 hW]
-    with b hb hba hbr hbt
-  rcases hb with ⟨horder,hvague,hatom,hpast,hfuture,haccount,he,hx,hk,hpos,hposK⟩
-  have hcont : ContinuousOn (Target.trace κ b) (Ici 0) := by
-    simpa only [trace_eq_literal] using hbt.2.1
-  have hγ := continuous_internalTrace hκ b hcont
-  have h0D : (0:ℝ≥0)∈{u : ℝ≥0 | ∃i,t i=u} := ⟨idx 0,by simpa using hidx 0⟩
-  let (u : ℝ≥0∞) : IsFiniteMeasureOnCompacts (μ u (driverHomeomorph b)) := (hμ _ _).1
-  let (u : ℝ≥0∞) : (μ u (driverHomeomorph b)).Regular := (hμ _ _).2
-  let : NullSingletonClass (μ ∞ (driverHomeomorph b)) := by simpa only [hj] using hatom j
-  have hrestrict (i : ℕ) (u : ℝ≥0) (hi : t i=u) :
-      μ u (driverHomeomorph b)=(μ ∞ (driverHomeomorph b)).restrict
-        (visited (fun z : UpperPlane => z.1) (internalTrace κ b) u) := by
-    have hle : μ u (driverHomeomorph b)≤μ ∞ (driverHomeomorph b) := by
-      simpa only [hi,hj] using horder i j (by rw [hj]; exact le_top)
-    apply restriction_of_complementary_support _ _ hle _
-      (measurableSet_visited _ measurable_subtype_coe _ hγ u)
-    · rw [visited_internalTrace hκ]
-      exact actual_limit_past_trace_support κ hκ hκ8 q hq s c hc hs b hbr hcont K hba u
-        (μ u (driverHomeomorph b)) (by
-          intro f
-          simpa only [hi] using hvague i f)
-    · rw [visited_internalTrace hκ]
-      have hf := hfuture i j (hi ▸ ENNReal.coe_ne_top) hj
-      simp only [hi,hj,ENNReal.toNNReal_coe] at hf
-      have hp := past_trace_subset_dead hκ b hbt.1 (t := (u:ℝ)/κ) (by positivity)
-      have hu : (κ*((u:ℝ)/κ)).toNNReal=u := by
-        rw [mul_div_cancel₀ _ hκ.ne',Real.toNNReal_coe]
-      rw [hu] at hp
-      exact measure_mono_null (fun z hz => hp hz) hf
-  have hdc (u : ℝ≥0) (hu : u∈{u : ℝ≥0 | ∃i,t i=u}) (f : C_c(UpperPlane,ℝ)) :
-      Tendsto (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-          (Real.exp (-2*(Target.a κ)*(j+1))) u (driverHomeomorph b))
-        atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) u,
-          f z∂μ ∞ (driverHomeomorph b))) := by
-    obtain ⟨i,hi⟩ := hu
-    have hv := hvague i f
-    change Tendsto _ atTop (𝓝 (∫z,f z∂μ (t i) (driverHomeomorph b))) at hv
-    rw [hi,hrestrict i u hi] at hv
-    exact hv
-  have hall := recoveredMass_all_times κ hκ (a_gt_quarter hκ hκ8) s c hc b hcont
-    (μ ∞ (driverHomeomorph b)) _ hD h0D hdc
-  have hvall := actualMass_tendsto_all_times κ hκ (a_gt_quarter hκ hκ8) s c hc b hcont
-    (μ ∞ (driverHomeomorph b)) _ hD h0D hdc
-  change ∀v : ℝ≥0, μ v (driverHomeomorph b)=
-    (μ ∞ (driverHomeomorph b)).restrict
-      (visited (fun z : UpperPlane => z.1) (internalTrace κ b) v) at hall
-  have hmono : Monotone (fun u : ℝ≥0∞ => μ u (driverHomeomorph b)) := by
-    intro u v huv
-    change μ u (driverHomeomorph b)≤μ v (driverHomeomorph b)
-    induction v using ENNReal.recTopCoe with
-    | top =>
-      induction u using ENNReal.recTopCoe with
-      | top => exact le_rfl
-      | coe u => rw [hall u]; exact Measure.restrict_le_self
-    | coe v =>
-      have hu : u≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top huv
-      rw [←ENNReal.coe_toNNReal hu,hall,hall]
-      exact Measure.restrict_mono (visited_mono _ _ (by
-        apply ENNReal.coe_le_coe.mp
-        simpa only [ENNReal.coe_toNNReal hu] using huv)) le_rfl
-  have hp (u : ℝ≥0∞) (hu : 0<u) : 0<μ u (driverHomeomorph b) univ := by
-    have hlim : Tendsto (fun n => (smallTime n:ℝ≥0∞)) atTop (𝓝 0) :=
-      ENNReal.continuous_coe.tendsto 0 |>.comp tendsto_smallTime
-    obtain ⟨n,hn⟩ := (hlim.eventually (gt_mem_nhds hu)).exists
-    have hnpos : 0<μ (smallTime n) (driverHomeomorph b) univ := by
-      simpa only [his] using hpos (is n) (by rw [his]; exact_mod_cast smallTime_pos n)
-    exact hnpos.trans_le (Measure.le_iff'.mp (hmono hn.le) univ)
-  refine ⟨hcont,inferInstance,hall,?_,hx,hk,hp,?_⟩
-  · intro u
-    induction u using ENNReal.recTopCoe with
-    | top => intro f; simpa only [hj] using hvague j f
-    | coe u =>
-      intro f
-      change Tendsto _ atTop (𝓝 (∫z,f z∂μ u (driverHomeomorph b)))
-      rw [hall u]
-      exact hvall u f
-  · intro u hu
-    obtain ⟨A,-,hA,hpA⟩ := isOpen_univ.exists_lt_isCompact (hp u hu)
-    exact ⟨A,hA,hpA,hA.measure_lt_top⟩
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
-lemma stoppedSurvivingGreen_spatial_measurable (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    Measurable (fun z : UpperPlane => stoppedSurvivingGreen a (S b) b z.1) :=
-  (stoppedSurvivingGreen_joint_measurable a ha S hS).comp
-    (measurable_const.prodMk measurable_id)
-
- 
-def stoppedSurvivingKernel (a : ℝ) (_ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S) :
-    Kernel ExactSLEGenerator.Driver UpperPlane :=
-  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
-    (fun b z => ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1))
-
-instance stoppedSurvivingKernel_sfinite (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    IsSFiniteKernel (stoppedSurvivingKernel a ha S hS) :=
-  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
-
-lemma stoppedSurvivingKernel_apply (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    stoppedSurvivingKernel a ha S hS b = upperArea.withDensity
-      (fun z => ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)) :=
-  Kernel.withDensity_apply _ (stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal b
-
-lemma stoppedSurvivingKernel_raw_measurable (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (stoppedSurvivingKernel a ha S hS) := by
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  apply Measure.measurable_of_measurable_coe
-  intro A hA
-  simp_rw [stoppedSurvivingKernel_apply, withDensity_apply _ hA]
-  exact (stoppedSurvivingGreen_raw_joint a ha S hS).ennreal_ofReal.lintegral_prod_right'
-
-lemma stoppedSurvivingGreen_lintegral_le_initial (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (z : UpperPlane) :
-    (∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W) ≤
-      ENNReal.ofReal (initialGreen a z) := by
-  obtain ⟨hi,hb⟩ := survivingGreen_optional_bound a ha W hW z S hS
-  rw [← ofReal_integral_eq_lintegral_ofReal hi
-    (Eventually.of_forall (fun b => stoppedSurvivingGreen_nonneg a (S b) b z.1))]
-  exact ENNReal.ofReal_le_ofReal hb
-
- 
-
-theorem stoppedSurvivingKernel_average_le_initial (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    stoppedSurvivingKernel a ha S hS ∘ₘ W ≤ initialGreenMeasure a := by
-  apply Measure.le_iff.mpr
-  intro A hA
-  rw [Measure.bind_apply hA (Kernel.aemeasurable _)]
-  simp_rw [stoppedSurvivingKernel_apply, withDensity_apply _ hA]
-  rw [lintegral_lintegral_swap
-    (stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal.aemeasurable]
-  change (∫⁻z in A, ∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W ∂upperArea) ≤
-    (initialGreenMeasure a) A
-  rw [initialGreenMeasure,withDensity_apply _ hA]
-  exact lintegral_mono (stoppedSurvivingGreen_lintegral_le_initial a ha W hW S hS)
-
-lemma stoppedSurvivingKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    IsFiniteMeasureOnCompacts (stoppedSurvivingKernel a ha S hS ∘ₘ W) := by
-  constructor
-  intro K hK
-  exact ((stoppedSurvivingKernel_average_le_initial a ha W hW S hS) K).trans_lt
-    hK.measure_lt_top
-
-lemma stoppedSurvivingKernel_compact_eval (a : ℝ) (ha : 1/4 < a)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
-    (∫z, f z ∂stoppedSurvivingKernel a ha S hS b) =
-      ∫z, f z * stoppedSurvivingGreen a (S b) b z.1 ∂upperArea := by
-  rw [stoppedSurvivingKernel_apply, integral_withDensity_eq_integral_toReal_smul]
-  · simp only [ENNReal.toReal_ofReal (stoppedSurvivingGreen_nonneg _ _ _ _), smul_eq_mul, mul_comm]
-  · exact (stoppedSurvivingGreen_spatial_measurable a ha S hS b).ennreal_ofReal
-  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
-
-lemma stoppedSurvivingKernel_compact_ae_integrable (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) :
-    ∀ᵐ b ∂W, Integrable f (stoppedSurvivingKernel a ha S hS b) := by
-  have := stoppedSurvivingKernel_average_finiteOnCompacts a ha W hW S hS
-  have hf : Integrable f (stoppedSurvivingKernel a ha S hS ∘ₘ W) := f.integrable
-  exact (Measure.integrable_comp_iff f.continuous.measurable.aestronglyMeasurable).mp hf |>.1
-
-lemma stoppedSurvivingKernel_compact_mean_integrable (a : ℝ) (ha : 1/4 < a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (f : C_c(UpperPlane,ℝ)) :
-    Integrable (fun b => ∫z, f z ∂stoppedSurvivingKernel a ha S hS b) W := by
-  have := stoppedSurvivingKernel_average_finiteOnCompacts a ha W hW S hS
-  have hf : Integrable f (stoppedSurvivingKernel a ha S hS ∘ₘ W) := f.integrable
-  rw [Measure.comp_eq_comp_const_apply] at hf
-  exact hf.integral_comp
-
-end ExactSLEGreen
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped Classical NNReal ENNReal
-
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
-
- 
-def unhitMassDensity (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver)
-    (z : UpperPlane) : ℝ :=
-  Real.exp (-alpha a*Real.log 2) *
-    (if t < radialTime a b.1 z.1 (z.1.im*q)
-      then radialWeight a b.1 z.1 (z.1.im*q) else 0)
-
-lemma unhitMassDensity_nonneg (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-    0 ≤ unhitMassDensity a q t b z := by
-  have hn := (radialWeight_bound a ha z.1 z.2 _ (mul_pos z.2 hq)
-    (mul_le_of_le_one_right z.2.le hq1) b).1
-  unfold unhitMassDensity
-  split_ifs <;> positivity
-
-lemma timeMassDensity_add_unhit (a q : ℝ) (t : ℝ≥0∞)
-    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
-    timeMassDensity a q ∞ b z.1 = timeMassDensity a q t b z.1 +
-      ENNReal.ofReal (unhitMassDensity a q t b z) := by
-  have he : 2*z.1.im*q/2=z.1.im*q := by ring
-  by_cases hh : radialTime a b.1 z.1 (z.1.im*q)≤t
-  · simp only [timeMassDensity, le_top, ite_true, ite_eq_left hh, unhitMassDensity,
-      ite_eq_right (not_lt.mpr hh), mul_zero, ENNReal.ofReal_zero, add_zero]
-  · simp only [timeMassDensity, le_top, ite_true, ite_eq_right hh, unhitMassDensity,
-      ite_eq_left (lt_of_not_ge hh), massDensity, ite_eq_left z.2, fullRadialDensity, he, zero_add]
-
-lemma unhitMassDensity_joint_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
-    (hq : 0<q) (hq1 : q≤1) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
-    (hS : IsStoppingTime rawFiltration S) :
-    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
-      unhitMassDensity a q (S p.1) p.1 p.2) := by
-  have hmS : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => S p.1) :=
-    hS.measurable'.comp measurable_fst
-  have hmT := joint_radialTime_measurable a ha q hq hq1
-  have hmW := joint_radialWeight_measurable a ha q hq hq1
-  exact (hmW.ite (measurableSet_lt hmS hmT) measurable_const).const_mul _
-
-lemma unhitMassDensity_integrable (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (z : UpperPlane) : Integrable (fun b => unhitMassDensity a q (S b) b z) W :=
-  (stoppedRadialTerminal_integrable a ha W hW z _ (mul_pos z.2 hq)
-    (mul_le_of_le_one_right z.2.le hq1) S hS).const_mul _
-
-lemma unhitMassDensity_conditional (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (z : UpperPlane) (U : Set ExactSLEGenerator.Driver)
-    (hU : MeasurableSet[hS.measurableSpace] U) :
-    (∫b in U, unhitMassDensity a q (S b) b z ∂W) =
-      ∫b in U, stoppedRemainingDensity a q (S b) b z.1 ∂W := by
-  simp only [unhitMassDensity,stoppedRemainingDensity_eq a q _ _ _ z.2,
-    integral_const_mul]
-  congr 1
-  exact (radial_conditional_weight_all W hW a ha z.1 z.2 _
-    (mul_pos z.2 hq) (mul_le_of_le_one_right z.2.le hq1) S hS U hU).2
-
-lemma unhitMassDensity_lintegral_conditional (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (z : UpperPlane) (U : Set ExactSLEGenerator.Driver)
-    (hU : MeasurableSet[hS.measurableSpace] U) :
-    (∫⁻b in U, ENNReal.ofReal (unhitMassDensity a q (S b) b z) ∂W) =
-      ∫⁻b in U, ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1) ∂W := by
-  rw [← ofReal_integral_eq_lintegral_ofReal
-      (unhitMassDensity_integrable a ha W hW q hq hq1 S hS z).restrict
-      (Eventually.of_forall (fun b => unhitMassDensity_nonneg a ha q hq hq1 (S b) b z)),
-    ← ofReal_integral_eq_lintegral_ofReal
-      (stoppedRemainingDensity_mean_bound a ha W hW z q hq hq1 S hS).1.restrict
-      (Eventually.of_forall (fun b => stoppedRemainingDensity_nonneg a q (S b) b z.1)),
-    unhitMassDensity_conditional a ha W hW q hq hq1 S hS z U hU]
-
-def unhitMassKernel (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
-    Kernel ExactSLEGenerator.Driver UpperPlane :=
-  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
-    (fun b z => ENNReal.ofReal (unhitMassDensity a q (S b) b z))
-
-def stoppedRemainingKernel (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
-    Kernel ExactSLEGenerator.Driver UpperPlane :=
-  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
-    (fun b z => ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1))
-
-instance unhitMassKernel_sfinite (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
-    IsSFiniteKernel (unhitMassKernel a q S) :=
-  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
-
-instance stoppedRemainingKernel_sfinite (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
-    IsSFiniteKernel (stoppedRemainingKernel a q S) :=
-  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
-
-lemma unhitMassKernel_apply (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    unhitMassKernel a q S b = upperArea.withDensity
-      (fun z => ENNReal.ofReal (unhitMassDensity a q (S b) b z)) :=
-  Kernel.withDensity_apply _ (unhitMassDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal b
-
-lemma stoppedRemainingKernel_apply (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    stoppedRemainingKernel a q S b = upperArea.withDensity
-      (fun z => ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1)) :=
-  Kernel.withDensity_apply _ (stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal b
-
-lemma stoppedRemainingKernel_raw_measurable (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    Measurable[hS.measurableSpace] (stoppedRemainingKernel a q S) := by
-  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
-  apply Measure.measurable_of_measurable_coe
-  intro A hA
-  simp_rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS, withDensity_apply _ hA]
-  exact (stoppedRemainingDensity_raw_joint a ha q hq hq1 S hS).ennreal_ofReal.lintegral_prod_right'
-
- 
-
-theorem unhitMassKernel_conditional_average (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
-    unhitMassKernel a q S ∘ₘ W.restrict U =
-      stoppedRemainingKernel a q S ∘ₘ W.restrict U := by
-  apply Measure.ext
-  intro A hA
-  rw [Measure.bind_apply hA (Kernel.aemeasurable _),
-    Measure.bind_apply hA (Kernel.aemeasurable _)]
-  simp_rw [unhitMassKernel_apply a ha q hq hq1 S hS,
-    stoppedRemainingKernel_apply a ha q hq hq1 S hS, withDensity_apply _ hA]
-  calc
-    _ = ∫⁻z in A, ∫⁻b in U, ENNReal.ofReal (unhitMassDensity a q (S b) b z) ∂W ∂upperArea :=
-      lintegral_lintegral_swap
-        (unhitMassDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal.aemeasurable
-    _ = ∫⁻z in A, ∫⁻b in U, ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1) ∂W ∂upperArea :=
-      lintegral_congr (fun z =>
-        unhitMassDensity_lintegral_conditional a ha W hW q hq hq1 S hS z U hU)
-    _ = _ := (lintegral_lintegral_swap
-        (stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal.aemeasurable).symm
-
-lemma stoppedRemainingKernel_le_surviving (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (b : ExactSLEGenerator.Driver) :
-    stoppedRemainingKernel a q S b ≤ stoppedSurvivingKernel a ha S hS b := by
-  rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS,
-    stoppedSurvivingKernel_apply]
-  apply withDensity_mono
-  exact Eventually.of_forall (fun z => ENNReal.ofReal_le_ofReal
-    (stoppedRemainingDensity_le_surviving a q (S b) b z.1 z.2))
-
-lemma stoppedRemainingKernel_average_le_surviving (a : ℝ) (ha : 1/4<a)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (W : Measure ExactSLEGenerator.Driver) :
-    stoppedRemainingKernel a q S ∘ₘ W ≤ stoppedSurvivingKernel a ha S hS ∘ₘ W := by
-  apply Measure.le_iff.mpr
-  intro A hA
-  rw [Measure.bind_apply hA (Kernel.aemeasurable _),
-    Measure.bind_apply hA (Kernel.aemeasurable _)]
-  exact lintegral_mono (fun b => (stoppedRemainingKernel_le_surviving a ha q hq hq1 S hS b) A)
-
-lemma stoppedRemainingKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    IsFiniteMeasureOnCompacts (stoppedRemainingKernel a q S ∘ₘ W) := by
-  constructor
-  intro K hK
-  exact ((stoppedRemainingKernel_average_le_surviving a ha q hq hq1 S hS W).trans
-    (stoppedSurvivingKernel_average_le_initial a ha W hW S hS) K).trans_lt hK.measure_lt_top
-
-lemma unhitMassKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
-    IsFiniteMeasureOnCompacts (unhitMassKernel a q S ∘ₘ W) := by
-  have he := unhitMassKernel_conditional_average a ha W hW q hq hq1 S hS univ MeasurableSet.univ
-  rw [Measure.restrict_univ] at he
-  rw [he]
-  exact stoppedRemainingKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
-
- 
-
-theorem unhitMassKernel_random_test (a : ℝ) (ha : 1/4<a)
-    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
-    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
-    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
-    (hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
-    (∫⁻b, ∫⁻z, H (b,z) ∂unhitMassKernel a q S b ∂W) =
-      ∫⁻b, ∫⁻z, H (b,z) ∂stoppedRemainingKernel a q S b ∂W := by
-  have := unhitMassKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
-  have := stoppedRemainingKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
-  apply conditional_lintegral_eq_of_average_finite _ _ W _ hS.measurableSpace_le
-  · intro U hU f
-    have := average_restrict_finite (unhitMassKernel a q S) W U
-    have := average_restrict_finite (stoppedRemainingKernel a q S) W U
-    rw [← average_compact_eval_of_average_finite,
-      ← average_compact_eval_of_average_finite,
-      unhitMassKernel_conditional_average a ha W hW q hq hq1 S hS U hU]
-  · exact hH
-
-end ExactSLEGreen
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.OptionalGreen
+import OAI.Probability.SLEGauge.TerminalExhaustion
+import OAI.Probability.SLEGauge.TraceRestriction
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal symmDiff
+namespace SLEExactGauge.TraceMassContinuity
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
+open TraceMassIntegration TraceRestriction TraceLevelLocalization MassVersions
+open GermPositivity
+
+ 
+
+theorem actual_raw_mass_trace (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (W : Measure TDriver) (hW : Target.IsWiener W)
+    (t : ℕ→ℝ≥0∞) (hD : Dense {u : ℝ≥0 | ∃i,t i=u})
+    (idx : ℕ→ℕ) (hidx : ∀n : ℕ,t (idx n)=(n:ℝ≥0))
+    (is : ℕ→ℕ) (his : ∀n : ℕ,t (is n)=(smallTime n:ℝ≥0∞))
+    (j : ℕ) (hj : t j=∞) :
+    ∃ (s : ℕ → Finset ℕ) (c : ℕ → ℕ → ℝ)
+      (hc : ∀ k j, j∈s k → 0≤c k j),
+      (∀ k j, j∈s k → k≤j) ∧ (∀ k, ∑ j∈s k, c k j=1) ∧
+      let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+      (∀ v b, IsFiniteMeasureOnCompacts (μ v b) ∧ (μ v b).Regular) ∧
+      (∀ b, μ 0 b=0) ∧
+      (∀ v, Measurable (μ v)) ∧
+      IsProgressive rawFiltration (fun v : ℝ≥0 => μ v) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)) (m : ℕ),
+        MemLp (fun b => ∫z,f z∂μ (t i) b) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
+      (∀ i (K : Set UpperPlane), IsCompact K → ∀ m : ℕ,
+        MemLp (fun b => (μ (t i) b K).toReal) (m : ℝ≥0∞) (W.map driverHomeomorph)) ∧
+      (∀ i (f : C_c(UpperPlane,ℝ)),
+        Tendsto (fun k => eLpNorm (fun b =>
+          (∑ j∈s k, c k j * ∫z,f z∂actualUpperMass (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1))) (t i) b)
+            - ∫z,f z∂μ (t i) b) 1 (W.map driverHomeomorph)) atTop (𝓝 0)) ∧
+      (∀ i, t i=∞ → μ (t i) ∘ₘ (W.map driverHomeomorph)=initialGreenMeasure (Target.a κ)) ∧
+      (∀ i j (v : ℝ≥0), t i=v → t j=∞ →
+        ∀ H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞,
+        Measurable[(rawFiltration v).prod (inferInstance : MeasurableSpace UpperPlane)] H →
+        (∫⁻b, ∫⁻z, H (b,z) ∂μ (t j) b ∂(W.map driverHomeomorph)) = (∫⁻b, ∫⁻z, H (b,z) ∂μ (t i) b ∂(W.map driverHomeomorph)) +
+          ∫⁻b, ∫⁻z, H (b,z) * ENNReal.ofReal (survivingGreen (Target.a κ) v b z.1) ∂upperArea ∂(W.map driverHomeomorph)) ∧
+      ∀ᵐ b ∂W,
+        ContinuousOn (Target.trace κ b) (Ici 0) ∧
+        NullSingletonClass (μ ∞ (driverHomeomorph b)) ∧
+        (∀v : ℝ≥0, μ v (driverHomeomorph b)=
+          (μ ∞ (driverHomeomorph b)).restrict
+            (visited (fun z : UpperPlane => z.1) (internalTrace κ b) v)) ∧
+        (∀v : ℝ≥0∞, ∀f : C_c(UpperPlane,ℝ), Tendsto
+          (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+            (Real.exp (-2*(Target.a κ)*(j+1))) v (driverHomeomorph b))
+          atTop (𝓝 (∫z,f z∂μ v (driverHomeomorph b)))) ∧
+        (∀A : Set UpperPlane, MeasurableSet A → Tendsto
+          (fun n : ℕ => μ (n:ℝ≥0) (driverHomeomorph b) A)
+          atTop (𝓝 (μ ∞ (driverHomeomorph b) A))) ∧
+        (∀K : Set UpperPlane, IsCompact K → Tendsto
+          (fun n : ℕ => (μ ∞ (driverHomeomorph b)-μ (n:ℝ≥0) (driverHomeomorph b)) K)
+          atTop (𝓝 0)) ∧
+        (∀v : ℝ≥0∞, 0<v → 0<μ v (driverHomeomorph b) univ) ∧
+        (∀v : ℝ≥0∞, 0<v → ∃K : Set UpperPlane, IsCompact K ∧
+          0<μ v (driverHomeomorph b) K ∧ μ v (driverHomeomorph b) K<∞) := by
+  have hBW := brownian_map_driver W hW
+  have : IsProbabilityMeasure (W.map driverHomeomorph) := hBW.isGaussianProcess.isProbabilityMeasure
+  obtain ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,hconv⟩ :=
+    actual_raw_mass_positive (Target.a κ) (a_gt_quarter hκ hκ8)
+      (W.map driverHomeomorph) hBW t idx hidx is his j hj
+  let μ := recoveredMass (Target.a κ) (a_gt_quarter hκ hκ8) s c hc
+  let K := CompactExhaustion.choice UpperPlane
+  let q := (4*Target.a κ-1)/2
+  have hq : 0<q := by dsimp [q]; linarith [a_gt_quarter hκ hκ8]
+  have hqp : q<4*Target.a κ-1 := by dsimp [q]; linarith [a_gt_quarter hκ hκ8]
+  have hang := convexAngularMass_liminf_finite (Target.a κ) q (a_gt_quarter hκ hκ8)
+    hq hqp (W.map driverHomeomorph) hBW K K.isCompact s c hc hsum
+  refine ⟨s,c,hc,hs,hsum,hμ,hzero,hmeas,hprog,hmom,hcompact,hL1,hmean,hH,?_⟩
+  filter_upwards [ae_of_ae_map measurable_driverHomeomorph.aemeasurable hconv,
+    ae_of_ae_map measurable_driverHomeomorph.aemeasurable hang,
+    angularRegular_ae hκ hκ8 hW, LiteralTrace.trace_regular_all_times hκ hκ8 hW]
+    with b hb hba hbr hbt
+  rcases hb with ⟨horder,hvague,hatom,hpast,hfuture,haccount,he,hx,hk,hpos,hposK⟩
+  have hcont : ContinuousOn (Target.trace κ b) (Ici 0) := by
+    simpa only [trace_eq_literal] using hbt.2.1
+  have hγ := continuous_internalTrace hκ b hcont
+  have h0D : (0:ℝ≥0)∈{u : ℝ≥0 | ∃i,t i=u} := ⟨idx 0,by simpa using hidx 0⟩
+  let (u : ℝ≥0∞) : IsFiniteMeasureOnCompacts (μ u (driverHomeomorph b)) := (hμ _ _).1
+  let (u : ℝ≥0∞) : (μ u (driverHomeomorph b)).Regular := (hμ _ _).2
+  let : NullSingletonClass (μ ∞ (driverHomeomorph b)) := by simpa only [hj] using hatom j
+  have hrestrict (i : ℕ) (u : ℝ≥0) (hi : t i=u) :
+      μ u (driverHomeomorph b)=(μ ∞ (driverHomeomorph b)).restrict
+        (visited (fun z : UpperPlane => z.1) (internalTrace κ b) u) := by
+    have hle : μ u (driverHomeomorph b)≤μ ∞ (driverHomeomorph b) := by
+      simpa only [hi,hj] using horder i j (by rw [hj]; exact le_top)
+    apply restriction_of_complementary_support _ _ hle _
+      (measurableSet_visited _ measurable_subtype_coe _ hγ u)
+    · rw [visited_internalTrace hκ]
+      exact actual_limit_past_trace_support κ hκ hκ8 q hq s c hc hs b hbr hcont K hba u
+        (μ u (driverHomeomorph b)) (by
+          intro f
+          simpa only [hi] using hvague i f)
+    · rw [visited_internalTrace hκ]
+      have hf := hfuture i j (hi ▸ ENNReal.coe_ne_top) hj
+      simp only [hi,hj,ENNReal.toNNReal_coe] at hf
+      have hp := past_trace_subset_dead hκ b hbt.1 (t := (u:ℝ)/κ) (by positivity)
+      have hu : (κ*((u:ℝ)/κ)).toNNReal=u := by
+        rw [mul_div_cancel₀ _ hκ.ne',Real.toNNReal_coe]
+      rw [hu] at hp
+      exact measure_mono_null (fun z hz => hp hz) hf
+  have hdc (u : ℝ≥0) (hu : u∈{u : ℝ≥0 | ∃i,t i=u}) (f : C_c(UpperPlane,ℝ)) :
+      Tendsto (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+          (Real.exp (-2*(Target.a κ)*(j+1))) u (driverHomeomorph b))
+        atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) u,
+          f z∂μ ∞ (driverHomeomorph b))) := by
+    obtain ⟨i,hi⟩ := hu
+    have hv := hvague i f
+    change Tendsto _ atTop (𝓝 (∫z,f z∂μ (t i) (driverHomeomorph b))) at hv
+    rw [hi,hrestrict i u hi] at hv
+    exact hv
+  have hall := recoveredMass_all_times κ hκ (a_gt_quarter hκ hκ8) s c hc b hcont
+    (μ ∞ (driverHomeomorph b)) _ hD h0D hdc
+  have hvall := actualMass_tendsto_all_times κ hκ (a_gt_quarter hκ hκ8) s c hc b hcont
+    (μ ∞ (driverHomeomorph b)) _ hD h0D hdc
+  change ∀v : ℝ≥0, μ v (driverHomeomorph b)=
+    (μ ∞ (driverHomeomorph b)).restrict
+      (visited (fun z : UpperPlane => z.1) (internalTrace κ b) v) at hall
+  have hmono : Monotone (fun u : ℝ≥0∞ => μ u (driverHomeomorph b)) := by
+    intro u v huv
+    change μ u (driverHomeomorph b)≤μ v (driverHomeomorph b)
+    induction v using ENNReal.recTopCoe with
+    | top =>
+      induction u using ENNReal.recTopCoe with
+      | top => exact le_rfl
+      | coe u => rw [hall u]; exact Measure.restrict_le_self
+    | coe v =>
+      have hu : u≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top huv
+      rw [←ENNReal.coe_toNNReal hu,hall,hall]
+      exact Measure.restrict_mono (visited_mono _ _ (by
+        apply ENNReal.coe_le_coe.mp
+        simpa only [ENNReal.coe_toNNReal hu] using huv)) le_rfl
+  have hp (u : ℝ≥0∞) (hu : 0<u) : 0<μ u (driverHomeomorph b) univ := by
+    have hlim : Tendsto (fun n => (smallTime n:ℝ≥0∞)) atTop (𝓝 0) :=
+      ENNReal.continuous_coe.tendsto 0 |>.comp tendsto_smallTime
+    obtain ⟨n,hn⟩ := (hlim.eventually (gt_mem_nhds hu)).exists
+    have hnpos : 0<μ (smallTime n) (driverHomeomorph b) univ := by
+      simpa only [his] using hpos (is n) (by rw [his]; exact_mod_cast smallTime_pos n)
+    exact hnpos.trans_le (Measure.le_iff'.mp (hmono hn.le) univ)
+  refine ⟨hcont,inferInstance,hall,?_,hx,hk,hp,?_⟩
+  · intro u
+    induction u using ENNReal.recTopCoe with
+    | top => intro f; simpa only [hj] using hvague j f
+    | coe u =>
+      intro f
+      change Tendsto _ atTop (𝓝 (∫z,f z∂μ u (driverHomeomorph b)))
+      rw [hall u]
+      exact hvall u f
+  · intro u hu
+    obtain ⟨A,-,hA,hpA⟩ := isOpen_univ.exists_lt_isCompact (hp u hu)
+    exact ⟨A,hA,hpA,hA.measure_lt_top⟩
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+lemma stoppedSurvivingGreen_spatial_measurable (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    Measurable (fun z : UpperPlane => stoppedSurvivingGreen a (S b) b z.1) :=
+  (stoppedSurvivingGreen_joint_measurable a ha S hS).comp
+    (measurable_const.prodMk measurable_id)
+
+ 
+def stoppedSurvivingKernel (a : ℝ) (_ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (_hS : IsStoppingTime rawFiltration S) :
+    Kernel ExactSLEGenerator.Driver UpperPlane :=
+  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
+    (fun b z => ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1))
+
+instance stoppedSurvivingKernel_sfinite (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    IsSFiniteKernel (stoppedSurvivingKernel a ha S hS) :=
+  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
+
+lemma stoppedSurvivingKernel_apply (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    stoppedSurvivingKernel a ha S hS b = upperArea.withDensity
+      (fun z => ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1)) :=
+  Kernel.withDensity_apply _ (stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal b
+
+lemma stoppedSurvivingKernel_raw_measurable (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (stoppedSurvivingKernel a ha S hS) := by
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  apply Measure.measurable_of_measurable_coe
+  intro A hA
+  simp_rw [stoppedSurvivingKernel_apply, withDensity_apply _ hA]
+  exact (stoppedSurvivingGreen_raw_joint a ha S hS).ennreal_ofReal.lintegral_prod_right'
+
+lemma stoppedSurvivingGreen_lintegral_le_initial (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (z : UpperPlane) :
+    (∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W) ≤
+      ENNReal.ofReal (initialGreen a z) := by
+  obtain ⟨hi,hb⟩ := survivingGreen_optional_bound a ha W hW z S hS
+  rw [← ofReal_integral_eq_lintegral_ofReal hi
+    (Eventually.of_forall (fun b => stoppedSurvivingGreen_nonneg a (S b) b z.1))]
+  exact ENNReal.ofReal_le_ofReal hb
+
+ 
+
+theorem stoppedSurvivingKernel_average_le_initial (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    stoppedSurvivingKernel a ha S hS ∘ₘ W ≤ initialGreenMeasure a := by
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [Measure.bind_apply hA (Kernel.aemeasurable _)]
+  simp_rw [stoppedSurvivingKernel_apply, withDensity_apply _ hA]
+  rw [lintegral_lintegral_swap
+    (stoppedSurvivingGreen_joint_measurable a ha S hS).ennreal_ofReal.aemeasurable]
+  change (∫⁻z in A, ∫⁻b, ENNReal.ofReal (stoppedSurvivingGreen a (S b) b z.1) ∂W ∂upperArea) ≤
+    (initialGreenMeasure a) A
+  rw [initialGreenMeasure,withDensity_apply _ hA]
+  exact lintegral_mono (stoppedSurvivingGreen_lintegral_le_initial a ha W hW S hS)
+
+lemma stoppedSurvivingKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    IsFiniteMeasureOnCompacts (stoppedSurvivingKernel a ha S hS ∘ₘ W) := by
+  constructor
+  intro K hK
+  exact ((stoppedSurvivingKernel_average_le_initial a ha W hW S hS) K).trans_lt
+    hK.measure_lt_top
+
+lemma stoppedSurvivingKernel_compact_eval (a : ℝ) (ha : 1/4 < a)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) (f : C_c(UpperPlane,ℝ)) :
+    (∫z, f z ∂stoppedSurvivingKernel a ha S hS b) =
+      ∫z, f z * stoppedSurvivingGreen a (S b) b z.1 ∂upperArea := by
+  rw [stoppedSurvivingKernel_apply, integral_withDensity_eq_integral_toReal_smul]
+  · simp only [ENNReal.toReal_ofReal (stoppedSurvivingGreen_nonneg _ _ _ _), smul_eq_mul, mul_comm]
+  · exact (stoppedSurvivingGreen_spatial_measurable a ha S hS b).ennreal_ofReal
+  · exact Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)
+
+lemma stoppedSurvivingKernel_compact_ae_integrable (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) :
+    ∀ᵐ b ∂W, Integrable f (stoppedSurvivingKernel a ha S hS b) := by
+  have := stoppedSurvivingKernel_average_finiteOnCompacts a ha W hW S hS
+  have hf : Integrable f (stoppedSurvivingKernel a ha S hS ∘ₘ W) := f.integrable
+  exact (Measure.integrable_comp_iff f.continuous.measurable.aestronglyMeasurable).mp hf |>.1
+
+lemma stoppedSurvivingKernel_compact_mean_integrable (a : ℝ) (ha : 1/4 < a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (f : C_c(UpperPlane,ℝ)) :
+    Integrable (fun b => ∫z, f z ∂stoppedSurvivingKernel a ha S hS b) W := by
+  have := stoppedSurvivingKernel_average_finiteOnCompacts a ha W hW S hS
+  have hf : Integrable f (stoppedSurvivingKernel a ha S hS ∘ₘ W) := f.integrable
+  rw [Measure.comp_eq_comp_const_apply] at hf
+  exact hf.integral_comp
+
+end ExactSLEGreen
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped Classical NNReal ENNReal
+
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain SLEExactGauge.MassVersions
+
+ 
+def unhitMassDensity (a q : ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver)
+    (z : UpperPlane) : ℝ :=
+  Real.exp (-alpha a*Real.log 2) *
+    (if t < radialTime a b.1 z.1 (z.1.im*q)
+      then radialWeight a b.1 z.1 (z.1.im*q) else 0)
+
+lemma unhitMassDensity_nonneg (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+    0 ≤ unhitMassDensity a q t b z := by
+  have hn := (radialWeight_bound a ha z.1 z.2 _ (mul_pos z.2 hq)
+    (mul_le_of_le_one_right z.2.le hq1) b).1
+  unfold unhitMassDensity
+  split_ifs <;> positivity
+
+lemma timeMassDensity_add_unhit (a q : ℝ) (t : ℝ≥0∞)
+    (b : ExactSLEGenerator.Driver) (z : UpperPlane) :
+    timeMassDensity a q ∞ b z.1 = timeMassDensity a q t b z.1 +
+      ENNReal.ofReal (unhitMassDensity a q t b z) := by
+  have he : 2*z.1.im*q/2=z.1.im*q := by ring
+  by_cases hh : radialTime a b.1 z.1 (z.1.im*q)≤t
+  · simp only [timeMassDensity, le_top, ite_true, ite_eq_left hh, unhitMassDensity,
+      ite_eq_right (not_lt.mpr hh), mul_zero, ENNReal.ofReal_zero, add_zero]
+  · simp only [timeMassDensity, le_top, ite_true, ite_eq_right hh, unhitMassDensity,
+      ite_eq_left (lt_of_not_ge hh), massDensity, ite_eq_left z.2, fullRadialDensity, he, zero_add]
+
+lemma unhitMassDensity_joint_measurable (a : ℝ) (ha : 1/4<a) (q : ℝ)
+    (hq : 0<q) (hq1 : q≤1) (S : ExactSLEGenerator.Driver → ℝ≥0∞)
+    (hS : IsStoppingTime rawFiltration S) :
+    Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane =>
+      unhitMassDensity a q (S p.1) p.1 p.2) := by
+  have hmS : Measurable (fun p : ExactSLEGenerator.Driver × UpperPlane => S p.1) :=
+    hS.measurable'.comp measurable_fst
+  have hmT := joint_radialTime_measurable a ha q hq hq1
+  have hmW := joint_radialWeight_measurable a ha q hq hq1
+  exact (hmW.ite (measurableSet_lt hmS hmT) measurable_const).const_mul _
+
+lemma unhitMassDensity_integrable (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (z : UpperPlane) : Integrable (fun b => unhitMassDensity a q (S b) b z) W :=
+  (stoppedRadialTerminal_integrable a ha W hW z _ (mul_pos z.2 hq)
+    (mul_le_of_le_one_right z.2.le hq1) S hS).const_mul _
+
+lemma unhitMassDensity_conditional (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (z : UpperPlane) (U : Set ExactSLEGenerator.Driver)
+    (hU : MeasurableSet[hS.measurableSpace] U) :
+    (∫b in U, unhitMassDensity a q (S b) b z ∂W) =
+      ∫b in U, stoppedRemainingDensity a q (S b) b z.1 ∂W := by
+  simp only [unhitMassDensity,stoppedRemainingDensity_eq a q _ _ _ z.2,
+    integral_const_mul]
+  congr 1
+  exact (radial_conditional_weight_all W hW a ha z.1 z.2 _
+    (mul_pos z.2 hq) (mul_le_of_le_one_right z.2.le hq1) S hS U hU).2
+
+lemma unhitMassDensity_lintegral_conditional (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (z : UpperPlane) (U : Set ExactSLEGenerator.Driver)
+    (hU : MeasurableSet[hS.measurableSpace] U) :
+    (∫⁻b in U, ENNReal.ofReal (unhitMassDensity a q (S b) b z) ∂W) =
+      ∫⁻b in U, ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1) ∂W := by
+  rw [← ofReal_integral_eq_lintegral_ofReal
+      (unhitMassDensity_integrable a ha W hW q hq hq1 S hS z).restrict
+      (Eventually.of_forall (fun b => unhitMassDensity_nonneg a ha q hq hq1 (S b) b z)),
+    ← ofReal_integral_eq_lintegral_ofReal
+      (stoppedRemainingDensity_mean_bound a ha W hW z q hq hq1 S hS).1.restrict
+      (Eventually.of_forall (fun b => stoppedRemainingDensity_nonneg a q (S b) b z.1)),
+    unhitMassDensity_conditional a ha W hW q hq hq1 S hS z U hU]
+
+def unhitMassKernel (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
+    Kernel ExactSLEGenerator.Driver UpperPlane :=
+  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
+    (fun b z => ENNReal.ofReal (unhitMassDensity a q (S b) b z))
+
+def stoppedRemainingKernel (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
+    Kernel ExactSLEGenerator.Driver UpperPlane :=
+  (Kernel.const ExactSLEGenerator.Driver upperArea).withDensity
+    (fun b z => ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1))
+
+instance unhitMassKernel_sfinite (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
+    IsSFiniteKernel (unhitMassKernel a q S) :=
+  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
+
+instance stoppedRemainingKernel_sfinite (a q : ℝ) (S : ExactSLEGenerator.Driver → ℝ≥0∞) :
+    IsSFiniteKernel (stoppedRemainingKernel a q S) :=
+  Kernel.IsSFiniteKernel.withDensity _ (fun _ _ => ENNReal.ofReal_ne_top)
+
+lemma unhitMassKernel_apply (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    unhitMassKernel a q S b = upperArea.withDensity
+      (fun z => ENNReal.ofReal (unhitMassDensity a q (S b) b z)) :=
+  Kernel.withDensity_apply _ (unhitMassDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal b
+
+lemma stoppedRemainingKernel_apply (a : ℝ) (ha : 1/4<a) (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    stoppedRemainingKernel a q S b = upperArea.withDensity
+      (fun z => ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1)) :=
+  Kernel.withDensity_apply _ (stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal b
+
+lemma stoppedRemainingKernel_raw_measurable (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    Measurable[hS.measurableSpace] (stoppedRemainingKernel a q S) := by
+  let : MeasurableSpace ExactSLEGenerator.Driver := hS.measurableSpace
+  apply Measure.measurable_of_measurable_coe
+  intro A hA
+  simp_rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS, withDensity_apply _ hA]
+  exact (stoppedRemainingDensity_raw_joint a ha q hq hq1 S hS).ennreal_ofReal.lintegral_prod_right'
+
+ 
+
+theorem unhitMassKernel_conditional_average (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (U : Set ExactSLEGenerator.Driver) (hU : MeasurableSet[hS.measurableSpace] U) :
+    unhitMassKernel a q S ∘ₘ W.restrict U =
+      stoppedRemainingKernel a q S ∘ₘ W.restrict U := by
+  apply Measure.ext
+  intro A hA
+  rw [Measure.bind_apply hA (Kernel.aemeasurable _),
+    Measure.bind_apply hA (Kernel.aemeasurable _)]
+  simp_rw [unhitMassKernel_apply a ha q hq hq1 S hS,
+    stoppedRemainingKernel_apply a ha q hq hq1 S hS, withDensity_apply _ hA]
+  calc
+    _ = ∫⁻z in A, ∫⁻b in U, ENNReal.ofReal (unhitMassDensity a q (S b) b z) ∂W ∂upperArea :=
+      lintegral_lintegral_swap
+        (unhitMassDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal.aemeasurable
+    _ = ∫⁻z in A, ∫⁻b in U, ENNReal.ofReal (stoppedRemainingDensity a q (S b) b z.1) ∂W ∂upperArea :=
+      lintegral_congr (fun z =>
+        unhitMassDensity_lintegral_conditional a ha W hW q hq hq1 S hS z U hU)
+    _ = _ := (lintegral_lintegral_swap
+        (stoppedRemainingDensity_joint_measurable a ha q hq hq1 S hS).ennreal_ofReal.aemeasurable).symm
+
+lemma stoppedRemainingKernel_le_surviving (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (b : ExactSLEGenerator.Driver) :
+    stoppedRemainingKernel a q S b ≤ stoppedSurvivingKernel a ha S hS b := by
+  rw [stoppedRemainingKernel_apply a ha q hq hq1 S hS,
+    stoppedSurvivingKernel_apply]
+  apply withDensity_mono
+  exact Eventually.of_forall (fun z => ENNReal.ofReal_le_ofReal
+    (stoppedRemainingDensity_le_surviving a q (S b) b z.1 z.2))
+
+lemma stoppedRemainingKernel_average_le_surviving (a : ℝ) (ha : 1/4<a)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (W : Measure ExactSLEGenerator.Driver) :
+    stoppedRemainingKernel a q S ∘ₘ W ≤ stoppedSurvivingKernel a ha S hS ∘ₘ W := by
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [Measure.bind_apply hA (Kernel.aemeasurable _),
+    Measure.bind_apply hA (Kernel.aemeasurable _)]
+  exact lintegral_mono (fun b => (stoppedRemainingKernel_le_surviving a ha q hq hq1 S hS b) A)
+
+lemma stoppedRemainingKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    IsFiniteMeasureOnCompacts (stoppedRemainingKernel a q S ∘ₘ W) := by
+  constructor
+  intro K hK
+  exact ((stoppedRemainingKernel_average_le_surviving a ha q hq hq1 S hS W).trans
+    (stoppedSurvivingKernel_average_le_initial a ha W hW S hS) K).trans_lt hK.measure_lt_top
+
+lemma unhitMassKernel_average_finiteOnCompacts (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S) :
+    IsFiniteMeasureOnCompacts (unhitMassKernel a q S ∘ₘ W) := by
+  have he := unhitMassKernel_conditional_average a ha W hW q hq hq1 S hS univ MeasurableSet.univ
+  rw [Measure.restrict_univ] at he
+  rw [he]
+  exact stoppedRemainingKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
+
+ 
+
+theorem unhitMassKernel_random_test (a : ℝ) (ha : 1/4<a)
+    (W : Measure ExactSLEGenerator.Driver) [IsProbabilityMeasure W]
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1)
+    (S : ExactSLEGenerator.Driver → ℝ≥0∞) (hS : IsStoppingTime rawFiltration S)
+    (H : ExactSLEGenerator.Driver × UpperPlane → ℝ≥0∞)
+    (hH : Measurable[hS.measurableSpace.prod (inferInstance : MeasurableSpace UpperPlane)] H) :
+    (∫⁻b, ∫⁻z, H (b,z) ∂unhitMassKernel a q S b ∂W) =
+      ∫⁻b, ∫⁻z, H (b,z) ∂stoppedRemainingKernel a q S b ∂W := by
+  have := unhitMassKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
+  have := stoppedRemainingKernel_average_finiteOnCompacts a ha W hW q hq hq1 S hS
+  apply conditional_lintegral_eq_of_average_finite _ _ W _ hS.measurableSpace_le
+  · intro U hU f
+    have := average_restrict_finite (unhitMassKernel a q S) W U
+    have := average_restrict_finite (stoppedRemainingKernel a q S) W U
+    rw [← average_compact_eval_of_average_finite,
+      ← average_compact_eval_of_average_finite,
+      unhitMassKernel_conditional_average a ha W hW q hq hq1 S hS U hU]
+  · exact hH
+
+end ExactSLEGreen
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TracePastSupport.lean b/lean/OAI/Probability/SLEGauge/TracePastSupport.lean
index c5c84df750f0e93279e1d5e8774f5449ff6b0170..1a2c86725491c17bf0b8e22f54a8d4dc6738a5ab 100644
--- a/lean/OAI/Probability/SLEGauge/TracePastSupport.lean
+++ b/lean/OAI/Probability/SLEGauge/TracePastSupport.lean
@@ -1,599 +1,599 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.PastHullSupport
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.TraceLevelLocalization
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen ExactSLEFiniteTilt
-
- 
-
-lemma compact_away_angle_large (κ : ℝ) (hκ : 0<κ) (b : TDriver)
-    (hb : AngularRegular κ b) (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
-    (T : ℝ≥0) (K : Set UpperPlane) (hK : IsCompact K)
-    (haway : ∀ z∈K, z.1∉(Target.trace κ b) '' Icc 0 ((T:ℝ)/κ))
-    (C : ℝ) (hC : 0≤C) :
-    ∀ᶠ j : ℕ in atTop, ∀ z∈K,
-      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
-      τ≤(T:ℝ≥0∞) →
-      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
-      C < |X v/Y v| := by
-  by_cases hne : K.Nonempty
-  · have hc : ContinuousOn (fun z : UpperPlane => z.1.im) K :=
-      (Complex.continuous_im.comp continuous_subtype_val).continuousOn
-    obtain ⟨zlo,hzlo,hmin⟩ := hK.exists_isMinOn hne hc
-    obtain ⟨zhi,hzhi,hmax⟩ := hK.exists_isMaxOn hne hc
-    have htr : IsCompact ((Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)) :=
-      isCompact_Icc.image_of_continuousOn (hcont.mono (fun _ hx => hx.1))
-    have hdis : Disjoint ((fun z : UpperPlane => z.1) '' K)
-        ((Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)) := by
-      rw [Set.disjoint_left]
-      rintro _ ⟨z,hz,rfl⟩ htrz
-      exact haway z hz htrz
-    obtain ⟨ε,hε,hsep⟩ := Metric.exists_pos_forall_lt_edist
-      (hK.image continuous_subtype_val) htr.isClosed hdis
-    filter_upwards [level_near_trace κ hκ b hb T zlo.1.im zhi.1.im C ε
-      zlo.2 zhi.2 hC (by exact_mod_cast hε)] with j hj
-    intro z hz τ hτ v
-    by_contra! hangle
-    have hd := hj z (hmin hz) (hmax hz) hτ hangle
-    have hτf : τ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hτ
-    have htT : τ.toNNReal≤T := ENNReal.coe_le_coe.mp (by
-      simpa only [ENNReal.coe_toNNReal hτf] using hτ)
-    have httr : Target.trace κ b ((τ.toNNReal:ℝ)/κ)∈
-        (Target.trace κ b) '' Icc 0 ((T:ℝ)/κ) :=
-      ⟨_,⟨by positivity,div_le_div_of_nonneg_right (show (τ.toNNReal:ℝ)≤T from htT) hκ.le⟩,rfl⟩
-    have hh := hsep z.1 ⟨z,hz,rfl⟩ _ httr
-    rw [edist_nndist,ENNReal.coe_lt_coe] at hh
-    exact (not_lt_of_ge (le_of_lt (show (ε:ℝ)<dist z.1
-      (Target.trace κ b ((τ.toNNReal:ℝ)/κ)) from hh))) hd
-  · exact Eventually.of_forall (fun _ z hz => False.elim (hne ⟨z,hz⟩))
-
-lemma angleLift_ge_cot_rpow (q C : ℝ) (hq : 0≤q) (hC : 0≤C) (v : State)
-    (hv : C≤|X v/Y v|) : C^q≤angleLift (sinePower (-q)) v := by
-  have hp := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2
-  have habs : |X v/Y v|≤(Real.sin (physicalTheta v))⁻¹ := by
-    rw [←physicalTheta_cot,Real.cot_eq_cos_div_sin,abs_div,abs_of_pos hp]
-    simpa only [one_div] using div_le_div_of_nonneg_right
-      (Real.abs_cos_le_one (physicalTheta v)) hp.le
-  calc
-    C^q ≤ ((Real.sin (physicalTheta v))⁻¹)^q := Real.rpow_le_rpow hC (hv.trans habs) hq
-    _ = angleLift (sinePower (-q)) v := by
-      simp only [angleLift,sinePower,Real.rpow_neg_eq_inv_rpow]
-
- 
-lemma timeMassDensity_le_angular (κ q C : ℝ) (hq : 0≤q) (hC : 0<C)
-    (b : TDriver) (T : ℝ≥0) (j : ℕ) (z : UpperPlane)
-    (hangle :
-      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
-      τ≤(T:ℝ≥0∞) →
-      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
-      C≤|X v/Y v|) :
-    timeMassDensity (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1))) T
-      (driverHomeomorph b) z.1 ≤
-      ENNReal.ofReal (C^(-q))*angularDensity (Target.a κ) q j (driverHomeomorph b) z := by
-  let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-    (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
-  by_cases hτ : τ≤(T:ℝ≥0∞)
-  · have hτf : τ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hτ
-    let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
-    have hF := angleLift_ge_cot_rpow q C hq hC.le v (hangle hτ)
-    have hpow : 0<C^(-q) := Real.rpow_pos_of_pos hC _
-    have hprod : C^(-q)*C^q=1 := by
-      rw [Real.rpow_neg hC.le,inv_mul_cancel₀ (Real.rpow_pos_of_pos hC q).ne']
-    have hmul : 1≤C^(-q)*angleLift (sinePower (-q)) v := by
-      rw [←hprod]
-      exact mul_le_mul_of_nonneg_left hF hpow.le
-    have hnorm : 2*z.1.im*Real.exp (-2*Target.a κ*(j+1))/2=
-        z.1.im*Real.exp (-2*Target.a κ*(j+1)) := by ring
-    have hrad : levelRadius (Target.a κ) (2*z.1.im) (j+1)=
-        z.1.im*Real.exp (-2*Target.a κ*(j+1)) := by simp [levelRadius]
-    simp only [timeMassDensity,show radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-      (z.1.im*Real.exp (-2*Target.a κ*(j+1)))≤(T:ℝ≥0∞) from hτ,ite_true,
-      massDensity,z.2,fullRadialDensity,hnorm,radialWeight,
-      show radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-        (z.1.im*Real.exp (-2*Target.a κ*(j+1)))≠∞ from hτf,ite_false,
-      angularDensity,hrad,radialTestWeight]
-    rw [←ENNReal.ofReal_mul hpow.le]
-    apply ENNReal.ofReal_le_ofReal
-    have hg : 0≤Real.exp (-alpha (Target.a κ)*Real.log 2)*physicalGreen (Target.a κ) v :=
-      mul_nonneg (Real.exp_pos _).le (Real.exp_pos _).le
-    convert mul_le_mul_of_nonneg_left hmul hg using 1 <;> first | rfl | (dsimp only [v,τ]; ring)
-  · simp only [timeMassDensity,show ¬radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-      (z.1.im*Real.exp (-2*Target.a κ*(j+1)))≤(T:ℝ≥0∞) from hτ,ite_false,zero_le]
-
-lemma actualUpperMass_ofReal_integral (a r : ℝ) (ha : 1/4<a)
-    (hr : 0<r) (hr1 : r≤1) (T : ℝ≥0∞) (b : ExactSLEGenerator.Driver)
-    (f : C_c(UpperPlane,ℝ)) (hf : ∀z,0≤f z) :
-    ENNReal.ofReal (∫z,f z∂actualUpperMass a r T b)=
-      ∫⁻z, timeMassDensity a r T b z.1*ENNReal.ofReal (f z)∂upperArea := by
-  let := actualUpperMass_finiteOnCompacts a ha r hr hr1 T b
-  have hm : Measurable (fun z : UpperPlane => timeMassDensity a r T b z.1) :=
-    (timeMassDensity_measurable a ha r hr hr1 T).comp
-      (measurable_const.prodMk measurable_subtype_coe)
-  rw [ofReal_integral_eq_lintegral_ofReal f.integrable (Eventually.of_forall hf),
-    actualUpperMass,lintegral_withDensity_eq_lintegral_mul upperArea hm
-      (g := fun z => ENNReal.ofReal (f z)) f.continuous.measurable.ennreal_ofReal]
-  rfl
-
-lemma angularCompactMass_const_mul (a q : ℝ) (ha : 1/4<a) (j : ℕ)
-    (b : ExactSLEGenerator.Driver) (K : Set UpperPlane) (hK : MeasurableSet K)
-    (C : ℝ≥0∞) :
-    C*angularCompactMass a q j K b=
-      ∫⁻z, K.indicator (fun z => C*angularDensity a q j b z) z∂upperArea := by
-  have hm : Measurable (fun z : UpperPlane => angularDensity a q j b z) :=
-    (angularDensity_measurable a q ha j).comp (measurable_const.prodMk measurable_id)
-  rw [angularCompactMass,lintegral_indicator hK,lintegral_const_mul C hm]
-
- 
-
-lemma actualUpperMass_integral_le_angular (κ q C B : ℝ) (ha : 1/4<Target.a κ)
-    (hq : 0≤q) (hC : 0<C) (hB : 0≤B) (b : TDriver) (T : ℝ≥0)
-    (j : ℕ) (K : Set UpperPlane) (hK : MeasurableSet K)
-    (f : C_c(UpperPlane,ℝ)) (hf : ∀ z,0≤f z) (hfK : tsupport f⊆K)
-    (hfB : ∀ z∈tsupport f,f z≤B)
-    (hangle : ∀ z∈tsupport f,
-      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
-        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
-      τ≤(T:ℝ≥0∞) →
-      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
-      C≤|X v/Y v|) :
-    ENNReal.ofReal (∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b))≤
-      ENNReal.ofReal (B*C^(-q))*angularCompactMass (Target.a κ) q j K (driverHomeomorph b) := by
-  rw [actualUpperMass_ofReal_integral (Target.a κ) _ ha (Real.exp_pos _)
-      (level_step_lt_one (Target.a κ) ha j).le T (driverHomeomorph b) f hf,
-    angularCompactMass_const_mul (Target.a κ) q ha j (driverHomeomorph b) K hK]
-  apply lintegral_mono
-  intro z
-  by_cases hz : z∈tsupport f
-  · rw [Set.indicator_of_mem (hfK hz)]
-    change timeMassDensity (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1))) T
-      (driverHomeomorph b) z.1*ENNReal.ofReal (f z)≤_
-    calc
-      _ ≤ (ENNReal.ofReal (C^(-q))*angularDensity (Target.a κ) q j (driverHomeomorph b) z)*
-          ENNReal.ofReal B :=
-        mul_le_mul' (timeMassDensity_le_angular κ q C hq hC b T j z (hangle z hz))
-          (ENNReal.ofReal_le_ofReal (hfB z hz))
-      _ = _ := by rw [ENNReal.ofReal_mul hB]; ac_rfl
-  · simp only [image_eq_zero_of_notMem_tsupport hz,ENNReal.ofReal_zero,
-      mul_zero,zero_le]
-
- 
-
-theorem actual_limit_integral_zero_away_trace (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (q : ℝ) (hq : 0<q) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀ k j,j∈s k→0≤c k j) (hs : ∀ k j,j∈s k→k≤j)
-    (b : TDriver) (hb : AngularRegular κ b)
-    (hcont : ContinuousOn (Target.trace κ b) (Ici 0)) (T : ℝ≥0)
-    (K : Set UpperPlane) (hK : MeasurableSet K)
-    (f : C_c(UpperPlane,ℝ)) (hf : ∀z,0≤f z) (hfK : tsupport f⊆K)
-    (haway : ∀ z∈tsupport f,z.1∉(Target.trace κ b) '' Icc 0 ((T:ℝ)/κ))
-    (μ : Measure UpperPlane)
-    (hlim : Tendsto (fun k => ∑ j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b)) atTop (𝓝 (∫z,f z∂μ)))
-    (hangular : liminf (fun k => convexAngularMass (Target.a κ) q (s k) (c k) K
-        (driverHomeomorph b)) atTop<∞) : (∫z,f z∂μ)=0 := by
-  obtain ⟨B,hB⟩ := f.hasCompactSupport.bddAbove_image f.continuous.continuousOn
-  let D : ℝ := max 0 B
-  have hD : 0≤D := le_max_left _ _
-  have hDf : ∀ z∈tsupport f,f z≤D := fun z hz => (hB ⟨z,hz,rfl⟩).trans (le_max_right _ _)
-  obtain ⟨L,hL,_⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hangular
-  have hfreq := Filter.frequently_lt_of_liminf_lt (h := hL)
-  have hbound (C : ℝ) (hC : 0<C) : ENNReal.ofReal (∫z,f z∂μ)≤
-      ENNReal.ofReal (D*C^(-q))*(L:ℝ≥0∞) := by
-    obtain ⟨N,hN⟩ := eventually_atTop.mp (compact_away_angle_large κ hκ b hb hcont T
-      (tsupport f) f.hasCompactSupport haway C hC.le)
-    have hev : ∀ᶠ k : ℕ in atTop,
-        ENNReal.ofReal (∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-          (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b))≤
-        ENNReal.ofReal (D*C^(-q))*convexAngularMass (Target.a κ) q (s k) (c k) K
-          (driverHomeomorph b) := by
-      filter_upwards [eventually_ge_atTop N] with k hk
-      rw [ENNReal.ofReal_sum_of_nonneg (fun j hj => mul_nonneg (hc k j hj) (integral_nonneg hf)),
-        convexAngularMass,Finset.mul_sum]
-      apply Finset.sum_le_sum
-      intro j hj
-      rw [ENNReal.ofReal_mul (hc k j hj)]
-      have hbj := actualUpperMass_integral_le_angular κ q C D
-        (TraceMassIntegration.a_gt_quarter hκ hκ8) hq.le hC hD b T j K hK f hf hfK hDf
-        (fun z hz hτ => (hN j (hk.trans (hs k j hj)) z hz hτ).le)
-      exact (mul_le_mul' le_rfl hbj).trans_eq (by ac_rfl)
-    apply le_of_tendsto_of_frequently (ENNReal.tendsto_ofReal hlim)
-    exact (hfreq.and_eventually hev).mono (fun k hk => hk.2.trans (mul_le_mul' le_rfl hk.1.le))
-  have hdecay : Tendsto (fun C : ℝ => ENNReal.ofReal ((D*C^(-q))*(L:ℝ))) atTop (𝓝 0) := by
-    simpa only [mul_zero,zero_mul,ENNReal.ofReal_zero] using
-      ENNReal.tendsto_ofReal (((tendsto_rpow_neg_atTop hq).const_mul D).mul_const (L:ℝ))
-  have hz : ENNReal.ofReal (∫z,f z∂μ)≤0 := by
-    apply ge_of_tendsto hdecay
-    filter_upwards [eventually_gt_atTop (0:ℝ)] with C hC
-    rw [ENNReal.ofReal_mul (mul_nonneg hD (Real.rpow_pos_of_pos hC _).le),ENNReal.ofReal_coe_nnreal]
-    exact hbound C hC
-  exact le_antisymm (ENNReal.ofReal_eq_zero.mp (le_antisymm hz bot_le)) (integral_nonneg hf)
-
-end SLEExactGauge.TraceLevelLocalization
-
-namespace SLEExactGauge.MassVersions
-
- 
-
-lemma open_null_of_nonneg_compact_tests {X : Type*} [TopologicalSpace X] [T2Space X]
-    [LocallyCompactSpace X] [MeasurableSpace X] [BorelSpace X]
-    (μ : Measure X) [μ.Regular]
-    (U : Set X) (hU : IsOpen U)
-    (htest : ∀ f : C_c(X,ℝ), (∀x,0≤f x) → tsupport f⊆U → (∫x,f x∂μ)=0)
-    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) : μ U=0 := by
-  rw [hU.measure_eq_iSup_isCompact]
-  simp only [ENNReal.iSup_eq_zero]
-  intro K hKU hK
-  obtain ⟨f,hfK,hfc,hfU,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK hU hKU
-  let g : C_c(X,ℝ) := ⟨f,hfc⟩
-  have hz : (fun x => f x)=ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg
-    (fun x => (hf01 x).1) (f.continuous.integrable_of_hasCompactSupport hfc)).mp
-      (htest g (fun x => (hf01 x).1) hfU)
-  rw [measure_eq_zero_iff_ae_notMem]
-  filter_upwards [hz] with x hx hKx
-  have he := hfK hKx
-  change f x=0 at hx
-  change f x=1 at he
-  linarith
-
-end SLEExactGauge.MassVersions
-
-namespace SLEExactGauge.TraceLevelLocalization
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
-
- 
-
-theorem actual_limit_past_trace_support (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
-    (q : ℝ) (hq : 0<q) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
-    (hc : ∀ k j,j∈s k→0≤c k j) (hs : ∀ k j,j∈s k→k≤j)
-    (b : TDriver) (hb : AngularRegular κ b)
-    (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
-    (K : CompactExhaustion UpperPlane)
-    (hangular : ∀ n,liminf (fun k => convexAngularMass (Target.a κ) q (s k) (c k) (K n)
-        (driverHomeomorph b)) atTop<∞)
-    (T : ℝ≥0) (μ : Measure UpperPlane) [μ.Regular]
-    (hlim : ∀ f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b)) atTop (𝓝 (∫z,f z∂μ)))
-    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) :
-    μ {z : UpperPlane | z.1∉(Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)}=0 := by
-  have htr : IsCompact ((Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)) :=
-    isCompact_Icc.image_of_continuousOn (hcont.mono (fun _ hx => hx.1))
-  apply MassVersions.open_null_of_nonneg_compact_tests μ _
-    (htr.isClosed.isOpen_compl.preimage continuous_subtype_val)
-  intro f hf hfU
-  obtain ⟨n,hn⟩ := K.exists_superset_of_isCompact f.hasCompactSupport
-  exact actual_limit_integral_zero_away_trace κ hκ hκ8 q hq s c hc hs b hb hcont T
-    (K n) (K.isCompact n).measurableSet f hf hn hfU μ (hlim f) (hangular n)
-
-end SLEExactGauge.TraceLevelLocalization
-end
-end
- 
-
- 
-section
- 
-noncomputable section
-open MeasureTheory Set Filter Topology
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open ExactSLEGreen
-
- 
-def upperDilation (c : ℝ≥0) (hc : 0<c) : UpperPlane ≃ₜ UpperPlane where
-  toFun z := ⟨((c:ℝ):ℂ)*z.1,by
-    simpa [Complex.mul_im] using mul_pos (show 0<(c:ℝ) from hc) z.2⟩
-  invFun z := ⟨((c:ℝ):ℂ)⁻¹*z.1,by
-    simpa [←Complex.ofReal_inv,Complex.mul_im] using
-      mul_pos (inv_pos.mpr (show 0<(c:ℝ) from hc)) z.2⟩
-  left_inv z := by
-    apply Subtype.ext
-    simp only [←mul_assoc,inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr
-      (NNReal.coe_ne_zero.mpr hc.ne')),one_mul]
-  right_inv z := by
-    apply Subtype.ext
-    simp only [←mul_assoc,mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr
-      (NNReal.coe_ne_zero.mpr hc.ne')),one_mul]
-  continuous_toFun := (continuous_const.mul continuous_subtype_val).subtype_mk _
-  continuous_invFun := (continuous_const.mul continuous_subtype_val).subtype_mk _
-
-@[simp] lemma upperDilation_apply (c : ℝ≥0) (hc : 0<c) (z : UpperPlane) :
-    (upperDilation c hc z).1=((c:ℝ):ℂ)*z.1 := rfl
-
-@[simp] lemma upperDilation_symm_apply (c : ℝ≥0) (hc : 0<c) (z : UpperPlane) :
-    ((upperDilation c hc).symm z).1=((c:ℝ):ℂ)⁻¹*z.1 := rfl
-
-lemma upperDilation_symm (c : ℝ≥0) (hc : 0<c) :
-    (upperDilation c hc).symm=upperDilation c⁻¹ (inv_pos.mpr hc) := by
-  ext z
-  simp
-
-lemma map_upperArea (c : ℝ≥0) (hc : 0<c) :
-    upperArea.map (upperDilation c hc)=(c:ℝ≥0∞)⁻¹^2 • upperArea := by
-  let f : ℂ → ℂ := fun z => ((c:ℝ):ℂ)*z
-  have hf : Measurable f := (continuous_const.mul continuous_id).measurable
-  have hpre : f ⁻¹' {z : ℂ | 0<z.im}={z : ℂ | 0<z.im} := by
-    ext z
-    simp only [mem_preimage,mem_ofPred_eq,f,Complex.mul_im,Complex.ofReal_re,
-      Complex.ofReal_im,zero_mul,add_zero]
-    exact mul_pos_iff_of_pos_left (show 0<(c:ℝ) from hc)
-  have hmap : (volume : Measure ℂ).map f=(c:ℝ≥0∞)⁻¹^2 • volume := by
-    have hm := Measure.map_addHaar_smul (volume : Measure ℂ)
-      (show (c:ℝ)≠0 from NNReal.coe_ne_zero.mpr hc.ne')
-    simpa only [Complex.real_smul,Complex.finrank_real_complex,abs_of_nonneg
-      (inv_nonneg.mpr (sq_nonneg (c:ℝ))),
-      ENNReal.ofReal_inv_of_pos (sq_pos_of_pos (show 0<(c:ℝ) from hc)),
-      ENNReal.ofReal_pow c.coe_nonneg,ENNReal.ofReal_coe_nnreal,ENNReal.inv_pow,f] using hm
-  apply upper_embedding.map_injective
-  calc
-    (upperArea.map (upperDilation c hc)).map Subtype.val =
-        (upperArea.map (Subtype.val : UpperPlane → ℂ)).map f := by
-      rw [Measure.map_map measurable_subtype_coe (upperDilation c hc).measurable,
-        Measure.map_map hf measurable_subtype_coe]
-      rfl
-    _ = ((volume : Measure ℂ).restrict {z | 0<z.im}).map f := by
-      rw [upperArea,upper_embedding.map_comap,Subtype.range_val_subtype]
-    _ = ((volume : Measure ℂ).map f).restrict {z | 0<z.im} := by
-      rw [Measure.restrict_map hf upperPlane_open.measurableSet,hpre]
-    _ = ((c:ℝ≥0∞)⁻¹^2 • upperArea).map Subtype.val := by
-      rw [hmap,Measure.map_smul _ measurable_subtype_coe.aemeasurable,
-        upperArea,upper_embedding.map_comap,Measure.restrict_smul,
-        Subtype.range_val_subtype]
-
- 
-lemma map_withDensity_comp {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
-    (e : X ≃ᵐ Y) (μ : Measure X) (f : Y → ℝ≥0∞) (hf : Measurable f) :
-    (μ.withDensity (fun x => f (e x))).map e=(μ.map e).withDensity f := by
-  ext A hA
-  rw [Measure.map_apply e.measurable hA,withDensity_apply _ (e.measurable hA),
-    withDensity_apply _ hA,Measure.restrict_map e.measurable hA,
-    lintegral_map hf e.measurable]
-
-end SLEExactGauge.LoewnerScaling
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open ExactSLEMain ExactSLEGreen
-
-lemma scaling_factor (a : ℝ) {c : ℝ≥0} (hc : 0<c) :
-    ENNReal.ofReal ((c:ℝ)^(alpha a))*(c:ℝ≥0∞)⁻¹^2 =
-      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) := by
-  rw [Real.rpow_sub (show 0<(c:ℝ) from hc),Real.rpow_two,
-    ENNReal.ofReal_div_of_pos (sq_pos_of_pos (show 0<(c:ℝ) from hc)),
-    ENNReal.ofReal_pow c.coe_nonneg,ENNReal.ofReal_coe_nnreal,div_eq_mul_inv,ENNReal.inv_pow]
-
-lemma map_actualUpperMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    (actualUpperMass a q t (scaleDriver c b)).map (upperDilation c hc) =
-      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) • actualUpperMass a q ((c:ℝ≥0∞)^2*t) b := by
-  let e := upperDilation c hc
-  let f : UpperPlane → ℝ≥0∞ := fun z => timeMassDensity a q ((c:ℝ≥0∞)^2*t) b z.1
-  have hf : Measurable f := (timeMassDensity_measurable a ha q hq hq1 _).comp
-    (measurable_const.prodMk measurable_subtype_coe)
-  have he : (fun z : UpperPlane => timeMassDensity a q t (scaleDriver c b) z.1) =
-      fun z => ENNReal.ofReal ((c:ℝ)^(alpha a))*f (e z) := by
-    funext z
-    exact timeMassDensity_scaled_upper a (by linarith) hc q hq hq1 t b z.1 z.2
-  simp only [actualUpperMass,he]
-  rw [show (fun z => ENNReal.ofReal ((c:ℝ)^(alpha a))*f (e z)) =
-    ENNReal.ofReal ((c:ℝ)^(alpha a)) • (fun z => f (e z)) from rfl,
-    withDensity_smul' _ _ ENNReal.ofReal_ne_top,
-    Measure.map_smul _ (upperDilation c hc).measurable.aemeasurable]
-  have hmf : (upperArea.withDensity (fun z => f (e z))).map e =
-      (upperArea.map e).withDensity f := map_withDensity_comp e.toMeasurableEquiv upperArea f hf
-  rw [hmf,map_upperArea c hc,withDensity_smul_measure,smul_smul,scaling_factor a hc]
-
-lemma map_actualConvexMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (s : Finset ℕ) (v : ℕ → ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    (actualConvexMass a s v t (scaleDriver c b)).map (upperDilation c hc) =
-      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) • actualConvexMass a s v ((c:ℝ≥0∞)^2*t) b := by
-  simp only [actualConvexMass,Measure.map_finset_sum (upperDilation c hc).measurable.aemeasurable,
-    Measure.map_smul _ (upperDilation c hc).measurable.aemeasurable,
-    map_actualUpperMass_scaled a ha hc _ (Real.exp_pos _) (level_step_le_one a ha _),
-    Finset.smul_sum]
-  apply Finset.sum_congr rfl
-  intro j hj
-  exact smul_comm _ _ _
-
-end SLEExactGauge.LoewnerScaling
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Set Filter Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.MassVersions
-
-variable {X Y : Type*}
-    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    [TopologicalSpace Y] [T2Space Y] [LocallyCompactSpace Y]
-    [SecondCountableTopology Y] [MeasurableSpace Y] [BorelSpace Y]
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
-    [MeasurableSpace X] [BorelSpace X] [T2Space Y] [LocallyCompactSpace Y]
-    [SecondCountableTopology Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma compact_comp_symm_comp [T2Space X] [LocallyCompactSpace X]
-    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
-    [T2Space Y] [LocallyCompactSpace Y] [SecondCountableTopology Y]
-    [MeasurableSpace Y] [BorelSpace Y] (e : X ≃ₜ Y) (f : C_c(X,ℝ)) :
-    (f.comp e.symm.toCocompactMap).comp e.toCocompactMap=f := by
-  ext x
-  exact congrArg f (e.symm_apply_apply x)
-
-lemma testConverges_equiv (e : X ≃ₜ Y) {r : ℝ} (hr : 0<r)
-    (L : ℕ → (C_c(X,ℝ) →ₚ[ℝ] ℝ)) (R : ℕ → (C_c(Y,ℝ) →ₚ[ℝ] ℝ))
-    (h : ∀ k (f : C_c(Y,ℝ)), L k (f.comp e.toCocompactMap)=r*R k f) :
-    TestConverges L ↔ TestConverges R := by
-  constructor
-  · intro hL i
-    have ht := (tendsto_radonLimit L hL
-      ((determiningTests i).comp e.toCocompactMap)).const_mul r⁻¹
-    simp only [h,←mul_assoc,inv_mul_cancel₀ hr.ne',one_mul] at ht
-    exact ⟨_,ht⟩
-  · intro hR i
-    have hb (k : ℕ) : L k (determiningTests i)=
-        r*R k ((determiningTests i).comp e.symm.toCocompactMap) := by
-      simpa only [compact_comp_symm_comp] using h k
-        ((determiningTests i).comp e.symm.toCocompactMap)
-    refine ⟨r * ∫ y, ((determiningTests i).comp e.symm.toCocompactMap) y ∂radonLimit R,?_⟩
-    simpa only [hb] using (tendsto_radonLimit R hR
-      ((determiningTests i).comp e.symm.toCocompactMap)).const_mul r
-
- 
-
-theorem radonLimit_equiv (e : X ≃ₜ Y) {r : ℝ} (hr : 0<r)
-    (L : ℕ → (C_c(X,ℝ) →ₚ[ℝ] ℝ)) (R : ℕ → (C_c(Y,ℝ) →ₚ[ℝ] ℝ))
-    (h : ∀ k (f : C_c(Y,ℝ)), L k (f.comp e.toCocompactMap)=r*R k f) :
-    (radonLimit L).map e=ENNReal.ofReal r • radonLimit R := by
-  classical
-  by_cases hR : TestConverges R
-  · have hL := (testConverges_equiv e hr L R h).mpr hR
-    let := radonLimit_regular L
-    let := radonLimit_regular R
-    let : ((radonLimit L).map e).Regular := Measure.Regular.map e
-    let : (ENNReal.ofReal r • radonLimit R).Regular :=
-      Measure.Regular.smul ENNReal.ofReal_ne_top
-    apply Measure.ext_of_integral_eq_on_compactlySupported
-    intro f
-    rw [integral_smul_measure,ENNReal.toReal_ofReal hr.le,smul_eq_mul]
-    rw [show (∫ y,f y ∂(radonLimit L).map e) =
-      ∫ x,(f.comp e.toCocompactMap) x ∂radonLimit L from
-      integral_map_equiv e.toMeasurableEquiv f]
-    apply tendsto_nhds_unique (tendsto_radonLimit L hL (f.comp e.toCocompactMap))
-    simpa only [h] using (tendsto_radonLimit R hR f).const_mul r
-  · have hL := (testConverges_equiv e hr L R h).not.mpr hR
-    simp only [radonLimit,dite_eq_right hL,dite_eq_right hR,Measure.map_zero,smul_zero]
-
-end SLEExactGauge.MassVersions
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
-open scoped NNReal ENNReal
-namespace SLEExactGauge.LoewnerScaling
-open ExactSLEMain ExactSLEGreen SLEExactGauge.MassVersions
-
-lemma actualMassFunctionals_integral (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (k : ℕ) (f : C_c(UpperPlane,ℝ)) :
-    actualMassFunctionals a ha s v hv t b k f = ∫ z,f z ∂actualConvexMass a (s k) (v k) t b := by
-  rw [actualMassFunctionals_apply,actualConvexMass_integral a ha _ _ (hv k)]
-
-lemma actualMassFunctionals_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (k : ℕ) (f : C_c(UpperPlane,ℝ)) :
-    actualMassFunctionals a ha s v hv t (scaleDriver c b) k
-      (f.comp (upperDilation c hc).toCocompactMap) =
-    (c:ℝ)^(alpha a-2)* actualMassFunctionals a ha s v hv ((c:ℝ≥0∞)^2*t) b k f := by
-  simp only [actualMassFunctionals_integral]
-  calc
-    _ = ∫ z,f z ∂(actualConvexMass a (s k) (v k) t (scaleDriver c b)).map
-        (upperDilation c hc) := (integral_map_equiv (upperDilation c hc).toMeasurableEquiv f).symm
-    _ = _ := by
-      rw [map_actualConvexMass_scaled a ha hc,integral_smul_measure,
-        ENNReal.toReal_ofReal (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _).le,smul_eq_mul]
-
- 
-theorem map_recoveredMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    (recoveredMass a ha s v hv t (scaleDriver c b)).map (upperDilation c hc) =
-      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) • recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b := by
-  apply radonLimit_equiv (upperDilation c hc)
-    (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _) _ _
-  exact actualMassFunctionals_scaled a ha hc s v hv t b
-
- 
-
-theorem recoveredMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (A : Set UpperPlane) :
-    recoveredMass a ha s v hv t (scaleDriver c b) A =
-      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) *
-        recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b ((upperDilation c hc) '' A) := by
-  have h := congrArg (fun μ : Measure UpperPlane => μ ((upperDilation c hc) '' A))
-    (map_recoveredMass_scaled a ha hc s v hv t b)
-  simpa only [(upperDilation c hc).measurableEmbedding.map_apply,
-    Set.preimage_image_eq _ (upperDilation c hc).injective,Measure.smul_apply,smul_eq_mul] using h
-
-lemma recoveredMass_scaled_univ_pos_iff (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
-    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
-    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
-    0 < recoveredMass a ha s v hv t (scaleDriver c b) univ ↔
-      0 < recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b univ := by
-  rw [recoveredMass_scaled a ha hc s v hv t b univ,
-    image_univ,range_eq_univ.mpr (upperDilation c hc).surjective]
-  have hr : 0<ENNReal.ofReal ((c:ℝ)^(alpha a-2)) :=
-    ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _)
-  rw [ENNReal.mul_pos_iff]
-  exact and_iff_right hr
-
- 
-
-theorem recoveredMass_positivity_probability (a : ℝ) (ha : 1/4<a)
-    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
-    (W : Measure ExactSLEGenerator.Driver)
-    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
-    (u t : ℝ≥0) (hu : 0<u) (ht : 0<t) :
-    W {b | 0<recoveredMass a ha s v hv u b univ}=
-      W {b | 0<recoveredMass a ha s v hv t b univ} := by
-  let c := NNReal.sqrt (t/u)
-  have hc : 0<c := NNReal.sqrt_pos.mpr (div_pos ht hu)
-  have hcu : c^2*u=t := by
-    dsimp only [c]
-    rw [NNReal.sq_sqrt,div_mul_cancel₀ _ hu.ne']
-  have hcu' : (c:ℝ≥0∞)^2*(u:ℝ≥0∞)=(t:ℝ≥0∞) := by exact_mod_cast hcu
-  let E := {b | 0<recoveredMass a ha s v hv u b univ}
-  have hE : MeasurableSet E := measurableSet_lt measurable_const
-    ((Measure.measurable_coe MeasurableSet.univ).comp (recoveredMass_measurable a ha s v hv u))
-  calc
-    _ = W ((scaleDriver c) ⁻¹' E) := by
-      exact ((measurePreserving_scaleDriver hW hc).measure_preimage hE.nullMeasurableSet).symm
-    _ = _ := by
-      apply congrArg (fun A : Set ExactSLEGenerator.Driver => W A)
-      apply Set.ext
-      intro b
-      change (0<recoveredMass a ha s v hv u (scaleDriver c b) univ) ↔ _
-      rw [recoveredMass_scaled_univ_pos_iff a ha hc s v hv u b,hcu']
-      rfl
-
-end SLEExactGauge.LoewnerScaling
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.PastHullSupport
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.TraceLevelLocalization
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen ExactSLEFiniteTilt
+
+ 
+
+lemma compact_away_angle_large (κ : ℝ) (hκ : 0<κ) (b : TDriver)
+    (hb : AngularRegular κ b) (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
+    (T : ℝ≥0) (K : Set UpperPlane) (hK : IsCompact K)
+    (haway : ∀ z∈K, z.1∉(Target.trace κ b) '' Icc 0 ((T:ℝ)/κ))
+    (C : ℝ) (hC : 0≤C) :
+    ∀ᶠ j : ℕ in atTop, ∀ z∈K,
+      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
+      τ≤(T:ℝ≥0∞) →
+      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
+      C < |X v/Y v| := by
+  by_cases hne : K.Nonempty
+  · have hc : ContinuousOn (fun z : UpperPlane => z.1.im) K :=
+      (Complex.continuous_im.comp continuous_subtype_val).continuousOn
+    obtain ⟨zlo,hzlo,hmin⟩ := hK.exists_isMinOn hne hc
+    obtain ⟨zhi,hzhi,hmax⟩ := hK.exists_isMaxOn hne hc
+    have htr : IsCompact ((Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)) :=
+      isCompact_Icc.image_of_continuousOn (hcont.mono (fun _ hx => hx.1))
+    have hdis : Disjoint ((fun z : UpperPlane => z.1) '' K)
+        ((Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)) := by
+      rw [Set.disjoint_left]
+      rintro _ ⟨z,hz,rfl⟩ htrz
+      exact haway z hz htrz
+    obtain ⟨ε,hε,hsep⟩ := Metric.exists_pos_forall_lt_edist
+      (hK.image continuous_subtype_val) htr.isClosed hdis
+    filter_upwards [level_near_trace κ hκ b hb T zlo.1.im zhi.1.im C ε
+      zlo.2 zhi.2 hC (by exact_mod_cast hε)] with j hj
+    intro z hz τ hτ v
+    by_contra! hangle
+    have hd := hj z (hmin hz) (hmax hz) hτ hangle
+    have hτf : τ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hτ
+    have htT : τ.toNNReal≤T := ENNReal.coe_le_coe.mp (by
+      simpa only [ENNReal.coe_toNNReal hτf] using hτ)
+    have httr : Target.trace κ b ((τ.toNNReal:ℝ)/κ)∈
+        (Target.trace κ b) '' Icc 0 ((T:ℝ)/κ) :=
+      ⟨_,⟨by positivity,div_le_div_of_nonneg_right (show (τ.toNNReal:ℝ)≤T from htT) hκ.le⟩,rfl⟩
+    have hh := hsep z.1 ⟨z,hz,rfl⟩ _ httr
+    rw [edist_nndist,ENNReal.coe_lt_coe] at hh
+    exact (not_lt_of_ge (le_of_lt (show (ε:ℝ)<dist z.1
+      (Target.trace κ b ((τ.toNNReal:ℝ)/κ)) from hh))) hd
+  · exact Eventually.of_forall (fun _ z hz => False.elim (hne ⟨z,hz⟩))
+
+lemma angleLift_ge_cot_rpow (q C : ℝ) (hq : 0≤q) (hC : 0≤C) (v : State)
+    (hv : C≤|X v/Y v|) : C^q≤angleLift (sinePower (-q)) v := by
+  have hp := Real.sin_pos_of_pos_of_lt_pi (physicalTheta_mem v).1 (physicalTheta_mem v).2
+  have habs : |X v/Y v|≤(Real.sin (physicalTheta v))⁻¹ := by
+    rw [←physicalTheta_cot,Real.cot_eq_cos_div_sin,abs_div,abs_of_pos hp]
+    simpa only [one_div] using div_le_div_of_nonneg_right
+      (Real.abs_cos_le_one (physicalTheta v)) hp.le
+  calc
+    C^q ≤ ((Real.sin (physicalTheta v))⁻¹)^q := Real.rpow_le_rpow hC (hv.trans habs) hq
+    _ = angleLift (sinePower (-q)) v := by
+      simp only [angleLift,sinePower,Real.rpow_neg_eq_inv_rpow]
+
+ 
+lemma timeMassDensity_le_angular (κ q C : ℝ) (hq : 0≤q) (hC : 0<C)
+    (b : TDriver) (T : ℝ≥0) (j : ℕ) (z : UpperPlane)
+    (hangle :
+      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
+      τ≤(T:ℝ≥0∞) →
+      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
+      C≤|X v/Y v|) :
+    timeMassDensity (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1))) T
+      (driverHomeomorph b) z.1 ≤
+      ENNReal.ofReal (C^(-q))*angularDensity (Target.a κ) q j (driverHomeomorph b) z := by
+  let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+    (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
+  by_cases hτ : τ≤(T:ℝ≥0∞)
+  · have hτf : τ≠∞ := ne_top_of_le_ne_top ENNReal.coe_ne_top hτ
+    let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
+    have hF := angleLift_ge_cot_rpow q C hq hC.le v (hangle hτ)
+    have hpow : 0<C^(-q) := Real.rpow_pos_of_pos hC _
+    have hprod : C^(-q)*C^q=1 := by
+      rw [Real.rpow_neg hC.le,inv_mul_cancel₀ (Real.rpow_pos_of_pos hC q).ne']
+    have hmul : 1≤C^(-q)*angleLift (sinePower (-q)) v := by
+      rw [←hprod]
+      exact mul_le_mul_of_nonneg_left hF hpow.le
+    have hnorm : 2*z.1.im*Real.exp (-2*Target.a κ*(j+1))/2=
+        z.1.im*Real.exp (-2*Target.a κ*(j+1)) := by ring
+    have hrad : levelRadius (Target.a κ) (2*z.1.im) (j+1)=
+        z.1.im*Real.exp (-2*Target.a κ*(j+1)) := by simp [levelRadius]
+    simp only [timeMassDensity,show radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+      (z.1.im*Real.exp (-2*Target.a κ*(j+1)))≤(T:ℝ≥0∞) from hτ,ite_true,
+      massDensity,z.2,fullRadialDensity,hnorm,radialWeight,
+      show radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+        (z.1.im*Real.exp (-2*Target.a κ*(j+1)))≠∞ from hτf,ite_false,
+      angularDensity,hrad,radialTestWeight]
+    rw [←ENNReal.ofReal_mul hpow.le]
+    apply ENNReal.ofReal_le_ofReal
+    have hg : 0≤Real.exp (-alpha (Target.a κ)*Real.log 2)*physicalGreen (Target.a κ) v :=
+      mul_nonneg (Real.exp_pos _).le (Real.exp_pos _).le
+    convert mul_le_mul_of_nonneg_left hmul hg using 1 <;> first | rfl | (dsimp only [v,τ]; ring)
+  · simp only [timeMassDensity,show ¬radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+      (z.1.im*Real.exp (-2*Target.a κ*(j+1)))≤(T:ℝ≥0∞) from hτ,ite_false,zero_le]
+
+lemma actualUpperMass_ofReal_integral (a r : ℝ) (ha : 1/4<a)
+    (hr : 0<r) (hr1 : r≤1) (T : ℝ≥0∞) (b : ExactSLEGenerator.Driver)
+    (f : C_c(UpperPlane,ℝ)) (hf : ∀z,0≤f z) :
+    ENNReal.ofReal (∫z,f z∂actualUpperMass a r T b)=
+      ∫⁻z, timeMassDensity a r T b z.1*ENNReal.ofReal (f z)∂upperArea := by
+  let := actualUpperMass_finiteOnCompacts a ha r hr hr1 T b
+  have hm : Measurable (fun z : UpperPlane => timeMassDensity a r T b z.1) :=
+    (timeMassDensity_measurable a ha r hr hr1 T).comp
+      (measurable_const.prodMk measurable_subtype_coe)
+  rw [ofReal_integral_eq_lintegral_ofReal f.integrable (Eventually.of_forall hf),
+    actualUpperMass,lintegral_withDensity_eq_lintegral_mul upperArea hm
+      (g := fun z => ENNReal.ofReal (f z)) f.continuous.measurable.ennreal_ofReal]
+  rfl
+
+lemma angularCompactMass_const_mul (a q : ℝ) (ha : 1/4<a) (j : ℕ)
+    (b : ExactSLEGenerator.Driver) (K : Set UpperPlane) (hK : MeasurableSet K)
+    (C : ℝ≥0∞) :
+    C*angularCompactMass a q j K b=
+      ∫⁻z, K.indicator (fun z => C*angularDensity a q j b z) z∂upperArea := by
+  have hm : Measurable (fun z : UpperPlane => angularDensity a q j b z) :=
+    (angularDensity_measurable a q ha j).comp (measurable_const.prodMk measurable_id)
+  rw [angularCompactMass,lintegral_indicator hK,lintegral_const_mul C hm]
+
+ 
+
+lemma actualUpperMass_integral_le_angular (κ q C B : ℝ) (ha : 1/4<Target.a κ)
+    (hq : 0≤q) (hC : 0<C) (hB : 0≤B) (b : TDriver) (T : ℝ≥0)
+    (j : ℕ) (K : Set UpperPlane) (hK : MeasurableSet K)
+    (f : C_c(UpperPlane,ℝ)) (hf : ∀ z,0≤f z) (hfK : tsupport f⊆K)
+    (hfB : ∀ z∈tsupport f,f z≤B)
+    (hangle : ∀ z∈tsupport f,
+      let τ := radialTime (Target.a κ) (driverHomeomorph b).1 z.1
+        (z.1.im*Real.exp (-2*(Target.a κ)*(j+1)))
+      τ≤(T:ℝ≥0∞) →
+      let v := actualState (Target.a κ) (driverHomeomorph b).1 τ.toNNReal z.1
+      C≤|X v/Y v|) :
+    ENNReal.ofReal (∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b))≤
+      ENNReal.ofReal (B*C^(-q))*angularCompactMass (Target.a κ) q j K (driverHomeomorph b) := by
+  rw [actualUpperMass_ofReal_integral (Target.a κ) _ ha (Real.exp_pos _)
+      (level_step_lt_one (Target.a κ) ha j).le T (driverHomeomorph b) f hf,
+    angularCompactMass_const_mul (Target.a κ) q ha j (driverHomeomorph b) K hK]
+  apply lintegral_mono
+  intro z
+  by_cases hz : z∈tsupport f
+  · rw [Set.indicator_of_mem (hfK hz)]
+    change timeMassDensity (Target.a κ) (Real.exp (-2*(Target.a κ)*(j+1))) T
+      (driverHomeomorph b) z.1*ENNReal.ofReal (f z)≤_
+    calc
+      _ ≤ (ENNReal.ofReal (C^(-q))*angularDensity (Target.a κ) q j (driverHomeomorph b) z)*
+          ENNReal.ofReal B :=
+        mul_le_mul' (timeMassDensity_le_angular κ q C hq hC b T j z (hangle z hz))
+          (ENNReal.ofReal_le_ofReal (hfB z hz))
+      _ = _ := by rw [ENNReal.ofReal_mul hB]; ac_rfl
+  · simp only [image_eq_zero_of_notMem_tsupport hz,ENNReal.ofReal_zero,
+      mul_zero,zero_le]
+
+ 
+
+theorem actual_limit_integral_zero_away_trace (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (q : ℝ) (hq : 0<q) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀ k j,j∈s k→0≤c k j) (hs : ∀ k j,j∈s k→k≤j)
+    (b : TDriver) (hb : AngularRegular κ b)
+    (hcont : ContinuousOn (Target.trace κ b) (Ici 0)) (T : ℝ≥0)
+    (K : Set UpperPlane) (hK : MeasurableSet K)
+    (f : C_c(UpperPlane,ℝ)) (hf : ∀z,0≤f z) (hfK : tsupport f⊆K)
+    (haway : ∀ z∈tsupport f,z.1∉(Target.trace κ b) '' Icc 0 ((T:ℝ)/κ))
+    (μ : Measure UpperPlane)
+    (hlim : Tendsto (fun k => ∑ j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b)) atTop (𝓝 (∫z,f z∂μ)))
+    (hangular : liminf (fun k => convexAngularMass (Target.a κ) q (s k) (c k) K
+        (driverHomeomorph b)) atTop<∞) : (∫z,f z∂μ)=0 := by
+  obtain ⟨B,hB⟩ := f.hasCompactSupport.bddAbove_image f.continuous.continuousOn
+  let D : ℝ := max 0 B
+  have hD : 0≤D := le_max_left _ _
+  have hDf : ∀ z∈tsupport f,f z≤D := fun z hz => (hB ⟨z,hz,rfl⟩).trans (le_max_right _ _)
+  obtain ⟨L,hL,_⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hangular
+  have hfreq := Filter.frequently_lt_of_liminf_lt (h := hL)
+  have hbound (C : ℝ) (hC : 0<C) : ENNReal.ofReal (∫z,f z∂μ)≤
+      ENNReal.ofReal (D*C^(-q))*(L:ℝ≥0∞) := by
+    obtain ⟨N,hN⟩ := eventually_atTop.mp (compact_away_angle_large κ hκ b hb hcont T
+      (tsupport f) f.hasCompactSupport haway C hC.le)
+    have hev : ∀ᶠ k : ℕ in atTop,
+        ENNReal.ofReal (∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+          (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b))≤
+        ENNReal.ofReal (D*C^(-q))*convexAngularMass (Target.a κ) q (s k) (c k) K
+          (driverHomeomorph b) := by
+      filter_upwards [eventually_ge_atTop N] with k hk
+      rw [ENNReal.ofReal_sum_of_nonneg (fun j hj => mul_nonneg (hc k j hj) (integral_nonneg hf)),
+        convexAngularMass,Finset.mul_sum]
+      apply Finset.sum_le_sum
+      intro j hj
+      rw [ENNReal.ofReal_mul (hc k j hj)]
+      have hbj := actualUpperMass_integral_le_angular κ q C D
+        (TraceMassIntegration.a_gt_quarter hκ hκ8) hq.le hC hD b T j K hK f hf hfK hDf
+        (fun z hz hτ => (hN j (hk.trans (hs k j hj)) z hz hτ).le)
+      exact (mul_le_mul' le_rfl hbj).trans_eq (by ac_rfl)
+    apply le_of_tendsto_of_frequently (ENNReal.tendsto_ofReal hlim)
+    exact (hfreq.and_eventually hev).mono (fun k hk => hk.2.trans (mul_le_mul' le_rfl hk.1.le))
+  have hdecay : Tendsto (fun C : ℝ => ENNReal.ofReal ((D*C^(-q))*(L:ℝ))) atTop (𝓝 0) := by
+    simpa only [mul_zero,zero_mul,ENNReal.ofReal_zero] using
+      ENNReal.tendsto_ofReal (((tendsto_rpow_neg_atTop hq).const_mul D).mul_const (L:ℝ))
+  have hz : ENNReal.ofReal (∫z,f z∂μ)≤0 := by
+    apply ge_of_tendsto hdecay
+    filter_upwards [eventually_gt_atTop (0:ℝ)] with C hC
+    rw [ENNReal.ofReal_mul (mul_nonneg hD (Real.rpow_pos_of_pos hC _).le),ENNReal.ofReal_coe_nnreal]
+    exact hbound C hC
+  exact le_antisymm (ENNReal.ofReal_eq_zero.mp (le_antisymm hz bot_le)) (integral_nonneg hf)
+
+end SLEExactGauge.TraceLevelLocalization
+
+namespace SLEExactGauge.MassVersions
+
+ 
+
+lemma open_null_of_nonneg_compact_tests {X : Type*} [TopologicalSpace X] [T2Space X]
+    [LocallyCompactSpace X] [MeasurableSpace X] [BorelSpace X]
+    (μ : Measure X) [μ.Regular]
+    (U : Set X) (hU : IsOpen U)
+    (htest : ∀ f : C_c(X,ℝ), (∀x,0≤f x) → tsupport f⊆U → (∫x,f x∂μ)=0)
+    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) : μ U=0 := by
+  rw [hU.measure_eq_iSup_isCompact]
+  simp only [ENNReal.iSup_eq_zero]
+  intro K hKU hK
+  obtain ⟨f,hfK,hfc,hfU,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK hU hKU
+  let g : C_c(X,ℝ) := ⟨f,hfc⟩
+  have hz : (fun x => f x)=ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg
+    (fun x => (hf01 x).1) (f.continuous.integrable_of_hasCompactSupport hfc)).mp
+      (htest g (fun x => (hf01 x).1) hfU)
+  rw [measure_eq_zero_iff_ae_notMem]
+  filter_upwards [hz] with x hx hKx
+  have he := hfK hKx
+  change f x=0 at hx
+  change f x=1 at he
+  linarith
+
+end SLEExactGauge.MassVersions
+
+namespace SLEExactGauge.TraceLevelLocalization
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
+
+ 
+
+theorem actual_limit_past_trace_support (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8)
+    (q : ℝ) (hq : 0<q) (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ)
+    (hc : ∀ k j,j∈s k→0≤c k j) (hs : ∀ k j,j∈s k→k≤j)
+    (b : TDriver) (hb : AngularRegular κ b)
+    (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
+    (K : CompactExhaustion UpperPlane)
+    (hangular : ∀ n,liminf (fun k => convexAngularMass (Target.a κ) q (s k) (c k) (K n)
+        (driverHomeomorph b)) atTop<∞)
+    (T : ℝ≥0) (μ : Measure UpperPlane) [μ.Regular]
+    (hlim : ∀ f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) T (driverHomeomorph b)) atTop (𝓝 (∫z,f z∂μ)))
+    (_hfinite : IsFiniteMeasureOnCompacts μ := by infer_instance) :
+    μ {z : UpperPlane | z.1∉(Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)}=0 := by
+  have htr : IsCompact ((Target.trace κ b) '' Icc 0 ((T:ℝ)/κ)) :=
+    isCompact_Icc.image_of_continuousOn (hcont.mono (fun _ hx => hx.1))
+  apply MassVersions.open_null_of_nonneg_compact_tests μ _
+    (htr.isClosed.isOpen_compl.preimage continuous_subtype_val)
+  intro f hf hfU
+  obtain ⟨n,hn⟩ := K.exists_superset_of_isCompact f.hasCompactSupport
+  exact actual_limit_integral_zero_away_trace κ hκ hκ8 q hq s c hc hs b hb hcont T
+    (K n) (K.isCompact n).measurableSet f hf hn hfU μ (hlim f) (hangular n)
+
+end SLEExactGauge.TraceLevelLocalization
+end
+end
+ 
+
+ 
+section
+ 
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open ExactSLEGreen
+
+ 
+def upperDilation (c : ℝ≥0) (hc : 0<c) : UpperPlane ≃ₜ UpperPlane where
+  toFun z := ⟨((c:ℝ):ℂ)*z.1,by
+    simpa [Complex.mul_im] using mul_pos (show 0<(c:ℝ) from hc) z.2⟩
+  invFun z := ⟨((c:ℝ):ℂ)⁻¹*z.1,by
+    simpa [←Complex.ofReal_inv,Complex.mul_im] using
+      mul_pos (inv_pos.mpr (show 0<(c:ℝ) from hc)) z.2⟩
+  left_inv z := by
+    apply Subtype.ext
+    simp only [←mul_assoc,inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr
+      (NNReal.coe_ne_zero.mpr hc.ne')),one_mul]
+  right_inv z := by
+    apply Subtype.ext
+    simp only [←mul_assoc,mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr
+      (NNReal.coe_ne_zero.mpr hc.ne')),one_mul]
+  continuous_toFun := (continuous_const.mul continuous_subtype_val).subtype_mk _
+  continuous_invFun := (continuous_const.mul continuous_subtype_val).subtype_mk _
+
+@[simp] lemma upperDilation_apply (c : ℝ≥0) (hc : 0<c) (z : UpperPlane) :
+    (upperDilation c hc z).1=((c:ℝ):ℂ)*z.1 := rfl
+
+@[simp] lemma upperDilation_symm_apply (c : ℝ≥0) (hc : 0<c) (z : UpperPlane) :
+    ((upperDilation c hc).symm z).1=((c:ℝ):ℂ)⁻¹*z.1 := rfl
+
+lemma upperDilation_symm (c : ℝ≥0) (hc : 0<c) :
+    (upperDilation c hc).symm=upperDilation c⁻¹ (inv_pos.mpr hc) := by
+  ext z
+  simp
+
+lemma map_upperArea (c : ℝ≥0) (hc : 0<c) :
+    upperArea.map (upperDilation c hc)=(c:ℝ≥0∞)⁻¹^2 • upperArea := by
+  let f : ℂ → ℂ := fun z => ((c:ℝ):ℂ)*z
+  have hf : Measurable f := (continuous_const.mul continuous_id).measurable
+  have hpre : f ⁻¹' {z : ℂ | 0<z.im}={z : ℂ | 0<z.im} := by
+    ext z
+    simp only [mem_preimage,mem_ofPred_eq,f,Complex.mul_im,Complex.ofReal_re,
+      Complex.ofReal_im,zero_mul,add_zero]
+    exact mul_pos_iff_of_pos_left (show 0<(c:ℝ) from hc)
+  have hmap : (volume : Measure ℂ).map f=(c:ℝ≥0∞)⁻¹^2 • volume := by
+    have hm := Measure.map_addHaar_smul (volume : Measure ℂ)
+      (show (c:ℝ)≠0 from NNReal.coe_ne_zero.mpr hc.ne')
+    simpa only [Complex.real_smul,Complex.finrank_real_complex,abs_of_nonneg
+      (inv_nonneg.mpr (sq_nonneg (c:ℝ))),
+      ENNReal.ofReal_inv_of_pos (sq_pos_of_pos (show 0<(c:ℝ) from hc)),
+      ENNReal.ofReal_pow c.coe_nonneg,ENNReal.ofReal_coe_nnreal,ENNReal.inv_pow,f] using hm
+  apply upper_embedding.map_injective
+  calc
+    (upperArea.map (upperDilation c hc)).map Subtype.val =
+        (upperArea.map (Subtype.val : UpperPlane → ℂ)).map f := by
+      rw [Measure.map_map measurable_subtype_coe (upperDilation c hc).measurable,
+        Measure.map_map hf measurable_subtype_coe]
+      rfl
+    _ = ((volume : Measure ℂ).restrict {z | 0<z.im}).map f := by
+      rw [upperArea,upper_embedding.map_comap,Subtype.range_val_subtype]
+    _ = ((volume : Measure ℂ).map f).restrict {z | 0<z.im} := by
+      rw [Measure.restrict_map hf upperPlane_open.measurableSet,hpre]
+    _ = ((c:ℝ≥0∞)⁻¹^2 • upperArea).map Subtype.val := by
+      rw [hmap,Measure.map_smul _ measurable_subtype_coe.aemeasurable,
+        upperArea,upper_embedding.map_comap,Measure.restrict_smul,
+        Subtype.range_val_subtype]
+
+ 
+lemma map_withDensity_comp {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    (e : X ≃ᵐ Y) (μ : Measure X) (f : Y → ℝ≥0∞) (hf : Measurable f) :
+    (μ.withDensity (fun x => f (e x))).map e=(μ.map e).withDensity f := by
+  ext A hA
+  rw [Measure.map_apply e.measurable hA,withDensity_apply _ (e.measurable hA),
+    withDensity_apply _ hA,Measure.restrict_map e.measurable hA,
+    lintegral_map hf e.measurable]
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open ExactSLEMain ExactSLEGreen
+
+lemma scaling_factor (a : ℝ) {c : ℝ≥0} (hc : 0<c) :
+    ENNReal.ofReal ((c:ℝ)^(alpha a))*(c:ℝ≥0∞)⁻¹^2 =
+      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) := by
+  rw [Real.rpow_sub (show 0<(c:ℝ) from hc),Real.rpow_two,
+    ENNReal.ofReal_div_of_pos (sq_pos_of_pos (show 0<(c:ℝ) from hc)),
+    ENNReal.ofReal_pow c.coe_nonneg,ENNReal.ofReal_coe_nnreal,div_eq_mul_inv,ENNReal.inv_pow]
+
+lemma map_actualUpperMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (q : ℝ) (hq : 0<q) (hq1 : q≤1) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    (actualUpperMass a q t (scaleDriver c b)).map (upperDilation c hc) =
+      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) • actualUpperMass a q ((c:ℝ≥0∞)^2*t) b := by
+  let e := upperDilation c hc
+  let f : UpperPlane → ℝ≥0∞ := fun z => timeMassDensity a q ((c:ℝ≥0∞)^2*t) b z.1
+  have hf : Measurable f := (timeMassDensity_measurable a ha q hq hq1 _).comp
+    (measurable_const.prodMk measurable_subtype_coe)
+  have he : (fun z : UpperPlane => timeMassDensity a q t (scaleDriver c b) z.1) =
+      fun z => ENNReal.ofReal ((c:ℝ)^(alpha a))*f (e z) := by
+    funext z
+    exact timeMassDensity_scaled_upper a (by linarith) hc q hq hq1 t b z.1 z.2
+  simp only [actualUpperMass,he]
+  rw [show (fun z => ENNReal.ofReal ((c:ℝ)^(alpha a))*f (e z)) =
+    ENNReal.ofReal ((c:ℝ)^(alpha a)) • (fun z => f (e z)) from rfl,
+    withDensity_smul' _ _ ENNReal.ofReal_ne_top,
+    Measure.map_smul _ (upperDilation c hc).measurable.aemeasurable]
+  have hmf : (upperArea.withDensity (fun z => f (e z))).map e =
+      (upperArea.map e).withDensity f := map_withDensity_comp e.toMeasurableEquiv upperArea f hf
+  rw [hmf,map_upperArea c hc,withDensity_smul_measure,smul_smul,scaling_factor a hc]
+
+lemma map_actualConvexMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (s : Finset ℕ) (v : ℕ → ℝ) (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    (actualConvexMass a s v t (scaleDriver c b)).map (upperDilation c hc) =
+      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) • actualConvexMass a s v ((c:ℝ≥0∞)^2*t) b := by
+  simp only [actualConvexMass,Measure.map_finset_sum (upperDilation c hc).measurable.aemeasurable,
+    Measure.map_smul _ (upperDilation c hc).measurable.aemeasurable,
+    map_actualUpperMass_scaled a ha hc _ (Real.exp_pos _) (level_step_le_one a ha _),
+    Finset.smul_sum]
+  apply Finset.sum_congr rfl
+  intro j hj
+  exact smul_comm _ _ _
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.MassVersions
+
+variable {X Y : Type*}
+    [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    [TopologicalSpace Y] [T2Space Y] [LocallyCompactSpace Y]
+    [SecondCountableTopology Y] [MeasurableSpace Y] [BorelSpace Y]
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] [BorelSpace X] [T2Space Y] [LocallyCompactSpace Y]
+    [SecondCountableTopology Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma compact_comp_symm_comp [T2Space X] [LocallyCompactSpace X]
+    [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X]
+    [T2Space Y] [LocallyCompactSpace Y] [SecondCountableTopology Y]
+    [MeasurableSpace Y] [BorelSpace Y] (e : X ≃ₜ Y) (f : C_c(X,ℝ)) :
+    (f.comp e.symm.toCocompactMap).comp e.toCocompactMap=f := by
+  ext x
+  exact congrArg f (e.symm_apply_apply x)
+
+lemma testConverges_equiv (e : X ≃ₜ Y) {r : ℝ} (hr : 0<r)
+    (L : ℕ → (C_c(X,ℝ) →ₚ[ℝ] ℝ)) (R : ℕ → (C_c(Y,ℝ) →ₚ[ℝ] ℝ))
+    (h : ∀ k (f : C_c(Y,ℝ)), L k (f.comp e.toCocompactMap)=r*R k f) :
+    TestConverges L ↔ TestConverges R := by
+  constructor
+  · intro hL i
+    have ht := (tendsto_radonLimit L hL
+      ((determiningTests i).comp e.toCocompactMap)).const_mul r⁻¹
+    simp only [h,←mul_assoc,inv_mul_cancel₀ hr.ne',one_mul] at ht
+    exact ⟨_,ht⟩
+  · intro hR i
+    have hb (k : ℕ) : L k (determiningTests i)=
+        r*R k ((determiningTests i).comp e.symm.toCocompactMap) := by
+      simpa only [compact_comp_symm_comp] using h k
+        ((determiningTests i).comp e.symm.toCocompactMap)
+    refine ⟨r * ∫ y, ((determiningTests i).comp e.symm.toCocompactMap) y ∂radonLimit R,?_⟩
+    simpa only [hb] using (tendsto_radonLimit R hR
+      ((determiningTests i).comp e.symm.toCocompactMap)).const_mul r
+
+ 
+
+theorem radonLimit_equiv (e : X ≃ₜ Y) {r : ℝ} (hr : 0<r)
+    (L : ℕ → (C_c(X,ℝ) →ₚ[ℝ] ℝ)) (R : ℕ → (C_c(Y,ℝ) →ₚ[ℝ] ℝ))
+    (h : ∀ k (f : C_c(Y,ℝ)), L k (f.comp e.toCocompactMap)=r*R k f) :
+    (radonLimit L).map e=ENNReal.ofReal r • radonLimit R := by
+  classical
+  by_cases hR : TestConverges R
+  · have hL := (testConverges_equiv e hr L R h).mpr hR
+    let := radonLimit_regular L
+    let := radonLimit_regular R
+    let : ((radonLimit L).map e).Regular := Measure.Regular.map e
+    let : (ENNReal.ofReal r • radonLimit R).Regular :=
+      Measure.Regular.smul ENNReal.ofReal_ne_top
+    apply Measure.ext_of_integral_eq_on_compactlySupported
+    intro f
+    rw [integral_smul_measure,ENNReal.toReal_ofReal hr.le,smul_eq_mul]
+    rw [show (∫ y,f y ∂(radonLimit L).map e) =
+      ∫ x,(f.comp e.toCocompactMap) x ∂radonLimit L from
+      integral_map_equiv e.toMeasurableEquiv f]
+    apply tendsto_nhds_unique (tendsto_radonLimit L hL (f.comp e.toCocompactMap))
+    simpa only [h] using (tendsto_radonLimit R hR f).const_mul r
+  · have hL := (testConverges_equiv e hr L R h).not.mpr hR
+    simp only [radonLimit,dite_eq_right hL,dite_eq_right hR,Measure.map_zero,smul_zero]
+
+end SLEExactGauge.MassVersions
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Set Filter Topology CompactlySupported
+open scoped NNReal ENNReal
+namespace SLEExactGauge.LoewnerScaling
+open ExactSLEMain ExactSLEGreen SLEExactGauge.MassVersions
+
+lemma actualMassFunctionals_integral (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (k : ℕ) (f : C_c(UpperPlane,ℝ)) :
+    actualMassFunctionals a ha s v hv t b k f = ∫ z,f z ∂actualConvexMass a (s k) (v k) t b := by
+  rw [actualMassFunctionals_apply,actualConvexMass_integral a ha _ _ (hv k)]
+
+lemma actualMassFunctionals_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (k : ℕ) (f : C_c(UpperPlane,ℝ)) :
+    actualMassFunctionals a ha s v hv t (scaleDriver c b) k
+      (f.comp (upperDilation c hc).toCocompactMap) =
+    (c:ℝ)^(alpha a-2)* actualMassFunctionals a ha s v hv ((c:ℝ≥0∞)^2*t) b k f := by
+  simp only [actualMassFunctionals_integral]
+  calc
+    _ = ∫ z,f z ∂(actualConvexMass a (s k) (v k) t (scaleDriver c b)).map
+        (upperDilation c hc) := (integral_map_equiv (upperDilation c hc).toMeasurableEquiv f).symm
+    _ = _ := by
+      rw [map_actualConvexMass_scaled a ha hc,integral_smul_measure,
+        ENNReal.toReal_ofReal (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _).le,smul_eq_mul]
+
+ 
+theorem map_recoveredMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    (recoveredMass a ha s v hv t (scaleDriver c b)).map (upperDilation c hc) =
+      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) • recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b := by
+  apply radonLimit_equiv (upperDilation c hc)
+    (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _) _ _
+  exact actualMassFunctionals_scaled a ha hc s v hv t b
+
+ 
+
+theorem recoveredMass_scaled (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) (A : Set UpperPlane) :
+    recoveredMass a ha s v hv t (scaleDriver c b) A =
+      ENNReal.ofReal ((c:ℝ)^(alpha a-2)) *
+        recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b ((upperDilation c hc) '' A) := by
+  have h := congrArg (fun μ : Measure UpperPlane => μ ((upperDilation c hc) '' A))
+    (map_recoveredMass_scaled a ha hc s v hv t b)
+  simpa only [(upperDilation c hc).measurableEmbedding.map_apply,
+    Set.preimage_image_eq _ (upperDilation c hc).injective,Measure.smul_apply,smul_eq_mul] using h
+
+lemma recoveredMass_scaled_univ_pos_iff (a : ℝ) (ha : 1/4<a) {c : ℝ≥0} (hc : 0<c)
+    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
+    (t : ℝ≥0∞) (b : ExactSLEGenerator.Driver) :
+    0 < recoveredMass a ha s v hv t (scaleDriver c b) univ ↔
+      0 < recoveredMass a ha s v hv ((c:ℝ≥0∞)^2*t) b univ := by
+  rw [recoveredMass_scaled a ha hc s v hv t b univ,
+    image_univ,range_eq_univ.mpr (upperDilation c hc).surjective]
+  have hr : 0<ENNReal.ofReal ((c:ℝ)^(alpha a-2)) :=
+    ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (show 0<(c:ℝ) from hc) _)
+  rw [ENNReal.mul_pos_iff]
+  exact and_iff_right hr
+
+ 
+
+theorem recoveredMass_positivity_probability (a : ℝ) (ha : 1/4<a)
+    (s : ℕ → Finset ℕ) (v : ℕ → ℕ → ℝ) (hv : ∀ k j, j∈s k → 0≤v k j)
+    (W : Measure ExactSLEGenerator.Driver)
+    (hW : IsBrownianReal (fun t (b : ExactSLEGenerator.Driver) => b.1 t) W)
+    (u t : ℝ≥0) (hu : 0<u) (ht : 0<t) :
+    W {b | 0<recoveredMass a ha s v hv u b univ}=
+      W {b | 0<recoveredMass a ha s v hv t b univ} := by
+  let c := NNReal.sqrt (t/u)
+  have hc : 0<c := NNReal.sqrt_pos.mpr (div_pos ht hu)
+  have hcu : c^2*u=t := by
+    dsimp only [c]
+    rw [NNReal.sq_sqrt,div_mul_cancel₀ _ hu.ne']
+  have hcu' : (c:ℝ≥0∞)^2*(u:ℝ≥0∞)=(t:ℝ≥0∞) := by exact_mod_cast hcu
+  let E := {b | 0<recoveredMass a ha s v hv u b univ}
+  have hE : MeasurableSet E := measurableSet_lt measurable_const
+    ((Measure.measurable_coe MeasurableSet.univ).comp (recoveredMass_measurable a ha s v hv u))
+  calc
+    _ = W ((scaleDriver c) ⁻¹' E) := by
+      exact ((measurePreserving_scaleDriver hW hc).measure_preimage hE.nullMeasurableSet).symm
+    _ = _ := by
+      apply congrArg (fun A : Set ExactSLEGenerator.Driver => W A)
+      apply Set.ext
+      intro b
+      change (0<recoveredMass a ha s v hv u (scaleDriver c b) univ) ↔ _
+      rw [recoveredMass_scaled_univ_pos_iff a ha hc s v hv u b,hcu']
+      rfl
+
+end SLEExactGauge.LoewnerScaling
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TraceRestriction.lean b/lean/OAI/Probability/SLEGauge/TraceRestriction.lean
index 07ea843a62e6712d0d28f3830ee9d824cdfe67ef..93e54004be652a90d1e8d9914899db78bc67a83d 100644
--- a/lean/OAI/Probability/SLEGauge/TraceRestriction.lean
+++ b/lean/OAI/Probability/SLEGauge/TraceRestriction.lean
@@ -1,538 +1,538 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.LoewnerScaling
-import OAI.Probability.SLEGauge.FiniteProductLimit
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology
-open scoped Classical NNReal ENNReal symmDiff
-namespace SLEExactGauge.TraceRestriction
-
-variable {X Y : Type*} [MeasurableSpace X] [MetricSpace Y]
-    [MeasurableSpace Y] [BorelSpace Y]
-
- 
-def visited (ι : X → Y) (γ : ℝ≥0 → Y) (t : ℝ≥0) : Set X :=
-  ι ⁻¹' (γ '' Icc 0 t)
-
-omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma visited_mono (ι : X → Y) (γ : ℝ≥0 → Y) : Monotone (visited ι γ) := by
-  intro s t hst
-  exact preimage_mono (image_mono (Icc_subset_Icc_right hst))
-
-lemma measurableSet_visited (ι : X → Y) (hι : Measurable ι)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (t : ℝ≥0) :
-    MeasurableSet (visited ι γ t) :=
-  ((isCompact_Icc.image hγ).isClosed.measurableSet).preimage hι
-
- 
-
-omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma visited_symmDiff_subset (ι : X → Y) (γ : ℝ≥0 → Y) (s t : ℝ≥0) :
-    visited ι γ s ∆ visited ι γ t ⊆ ι ⁻¹' (γ '' uIcc s t) := by
-  intro x hx
-  rcases hx with ⟨hxs,hxt⟩ | ⟨hxt,hxs⟩
-  · rcases hxs with ⟨u,hu,he⟩
-    have htu : t<u := lt_of_not_ge (fun hut => hxt ⟨u,⟨hu.1,hut⟩,he⟩)
-    exact ⟨u,⟨(min_le_right s t).trans htu.le,hu.2.trans (le_max_left s t)⟩,he⟩
-  · rcases hxt with ⟨u,hu,he⟩
-    have hsu : s<u := lt_of_not_ge (fun hus => hxs ⟨u,⟨hu.1,hus⟩,he⟩)
-    exact ⟨u,⟨(min_le_left s t).trans hsu.le,hu.2.trans (le_max_right s t)⟩,he⟩
-
-lemma nnreal_dist_right_le {s t u : ℝ≥0} (hu : u∈uIcc s t) :
-    dist u t≤dist s t := by
-  apply Real.dist_right_le_of_mem_uIcc
-  change min (s : ℝ) (t : ℝ)≤(u : ℝ) ∧ (u : ℝ) ≤ max (s : ℝ) (t : ℝ)
-  exact ⟨hu.1,hu.2⟩
-
- 
-
-omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
-lemma eventually_not_mem_visited_symmDiff (ι : X → Y)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (t : ℝ≥0) (x : X)
-    (hx : ι x≠γ t) : ∀ᶠ s in 𝓝 t, x∉visited ι γ s ∆ visited ι γ t := by
-  have hr : 0<dist (ι x) (γ t) := dist_pos.mpr hx
-  obtain ⟨δ,hδ,hδγ⟩ := Metric.continuousAt_iff.mp (hγ.continuousAt (x := t)) _ hr
-  filter_upwards [Metric.ball_mem_nhds t hδ] with s hs hxs
-  obtain ⟨u,hu,he⟩ := visited_symmDiff_subset ι γ s t hxs
-  have hud : dist u t<δ := (nnreal_dist_right_le hu).trans_lt hs
-  have hlt := hδγ hud
-  rw [he] at hlt
-  exact lt_irrefl _ hlt
-
-omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
-lemma eventually_visited_iff (ι : X → Y)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (t : ℝ≥0) (x : X)
-    (hx : ι x≠γ t) : ∀ᶠ s in 𝓝 t, (x∈visited ι γ s ↔ x∈visited ι γ t) := by
-  filter_upwards [eventually_not_mem_visited_symmDiff ι γ hγ t x hx] with s hs
-  constructor
-  · intro hx
-    by_contra ht
-    exact hs (Or.inl ⟨hx,ht⟩)
-  · intro hx
-    by_contra ht
-    exact hs (Or.inr ⟨hx,ht⟩)
-
-omit [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma preimage_singleton_null_of_injective (ν : Measure X) [NullSingletonClass ν]
-    (ι : X → Y) (hinj : Function.Injective ι) (y : Y) : ν (ι ⁻¹' {y})=0 := by
-  by_cases h : ∃ x, ι x=y
-  · obtain ⟨x,hx⟩ := h
-    have he : ι ⁻¹' {y}={x} := by
-      ext z
-      simp only [mem_preimage,mem_singleton_iff,← hx,hinj.eq_iff]
-    rw [he,measure_singleton]
-  · have he : ι ⁻¹' {y}=∅ := by
-      ext x
-      simp only [mem_preimage,mem_singleton_iff,mem_empty_iff_false,iff_false]
-      exact fun hx => h ⟨x,hx⟩
-    rw [he,measure_empty]
-
- 
-
-lemma tendsto_measure_visited_symmDiff (ν : Measure X)
-    (ι : X → Y) (hι : Measurable ι) (γ : ℝ≥0 → Y) (hγ : Continuous γ)
-    (t : ℝ≥0) (hpoint : ν (ι ⁻¹' {γ t})=0)
-    (K : Set X) (_hK : MeasurableSet K) (hfin : ν K≠∞) :
-    Tendsto (fun s => ν (K∩(visited ι γ s ∆ visited ι γ t))) (𝓝 t) (𝓝 0) := by
-  let μ := ν.restrict K
-  have : IsFiniteMeasure μ := ⟨by simpa only [μ,Measure.restrict_apply_univ] using hfin.lt_top⟩
-  let F : ℝ≥0 → X → ℝ≥0∞ := fun s => (visited ι γ s ∆ visited ι γ t).indicator 1
-  have hF (s : ℝ≥0) : Measurable (F s) :=
-    measurable_const.indicator
-      ((measurableSet_visited ι hι γ hγ s).symmDiff (measurableSet_visited ι hι γ hγ t))
-  have hbound (s : ℝ≥0) : ∀ᵐ x ∂μ, F s x≤1 := Eventually.of_forall (fun x => by
-    by_cases hx : x∈visited ι γ s ∆ visited ι γ t <;> simp [F,hx])
-  have hzero : μ (ι ⁻¹' {γ t})=0 := by
-    change (ν.restrict K) (ι ⁻¹' {γ t})=0
-    rw [Measure.restrict_apply ((measurableSet_singleton (γ t)).preimage hι)]
-    exact measure_mono_null inter_subset_left hpoint
-  have hlim : ∀ᵐ x ∂μ, Tendsto (fun s => F s x) (𝓝 t) (𝓝 0) := by
-    filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with x hx
-    have hxt : ι x≠γ t := hx
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_not_mem_visited_symmDiff ι γ hγ t x hxt] with s hs
-    simp only [F,indicator_of_notMem hs]
-  have ht := tendsto_lintegral_filter_of_dominated_convergence (μ := μ) (fun _ => (1 : ℝ≥0∞))
-    (Eventually.of_forall hF) (Eventually.of_forall hbound)
-    (by simp : (∫⁻ x, (1 : ℝ≥0∞) ∂μ)≠∞) hlim
-  have he (s : ℝ≥0) : (∫⁻x, F s x ∂μ)=ν (K∩(visited ι γ s ∆ visited ι γ t)) := by
-    have hs := (measurableSet_visited ι hι γ hγ s).symmDiff (measurableSet_visited ι hι γ hγ t)
-    change (∫⁻x, (visited ι γ s ∆ visited ι γ t).indicator 1 x ∂(ν.restrict K))=_
-    rw [lintegral_indicator_one hs,Measure.restrict_apply hs,inter_comm]
-  simpa only [he,lintegral_zero] using ht
-
- 
-lemma continuous_integral_visited (ν : Measure X) [NullSingletonClass ν]
-    (ι : X → Y) (hι : Measurable ι) (hinj : Function.Injective ι)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (f : X → ℝ) (hf : Integrable f ν) :
-    Continuous (fun t => ∫x in visited ι γ t, f x ∂ν) := by
-  rw [continuous_iff_continuousAt]
-  intro t
-  simp_rw [← integral_indicator (measurableSet_visited ι hι γ hγ _)]
-  apply continuousAt_of_dominated (bound := fun x => ‖f x‖)
-  · exact Eventually.of_forall (fun s => hf.aestronglyMeasurable.indicator
-      (measurableSet_visited ι hι γ hγ s))
-  · exact Eventually.of_forall (fun s => Eventually.of_forall (fun x => norm_indicator_le_norm_self _ _))
-  · exact hf.norm
-  · filter_upwards [measure_eq_zero_iff_ae_notMem.mp
-      (preimage_singleton_null_of_injective ν ι hinj (γ t))] with x hx
-    have hxt : ι x≠γ t := hx
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_visited_iff ι γ hγ t x hxt] with s hs
-    by_cases ht : x∈visited ι γ t
-    · simp only [indicator_of_mem ht,indicator_of_mem (hs.mpr ht)]
-    · simp only [indicator_of_notMem ht,indicator_of_notMem (mt hs.mp ht)]
-
- 
-
-def firstVisit (ι : X → Y) (γ : ℝ≥0 → Y) (x : X) : ℝ≥0∞ :=
-  if _h : ({u : ℝ≥0 | γ u=ι x}).Nonempty then
-    (sInf {u : ℝ≥0 | γ u=ι x} : ℝ≥0) else ∞
-
-omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
-lemma firstVisit_le_iff (ι : X → Y) (γ : ℝ≥0 → Y) (hγ : Continuous γ)
-    (x : X) (t : ℝ≥0) : firstVisit ι γ x≤(t : ℝ≥0∞) ↔ x∈visited ι γ t := by
-  classical
-  by_cases h : ({u : ℝ≥0 | γ u=ι x}).Nonempty
-  · have hclosed : IsClosed {u : ℝ≥0 | γ u=ι x} := isClosed_eq hγ continuous_const
-    have hmem := hclosed.csInf_mem h ⟨0,fun _ _ => bot_le⟩
-    simp only [firstVisit,dite_eq_left h,ENNReal.coe_le_coe]
-    constructor
-    · intro ht
-      exact ⟨_,⟨bot_le,ht⟩,hmem⟩
-    · rintro ⟨u,hu,he⟩
-      exact (csInf_le (s := {u : ℝ≥0 | γ u=ι x}) ⟨0,fun _ _ => bot_le⟩ he).trans hu.2
-  · simp only [firstVisit,dite_eq_right h,top_le_iff,ENNReal.coe_ne_top,false_iff]
-    rintro ⟨u,hu,he⟩
-    exact h ⟨u,he⟩
-
-omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma firstVisit_lt_top_iff (ι : X → Y) (γ : ℝ≥0 → Y) (x : X) :
-    firstVisit ι γ x<∞ ↔ ι x∈range γ := by
-  classical
-  by_cases h : ({u : ℝ≥0 | γ u=ι x}).Nonempty
-  · simp only [firstVisit,dite_eq_left h,ENNReal.coe_lt_top,true_iff]
-    exact h
-  · simp only [firstVisit,dite_eq_right h,lt_self_iff_false,false_iff]
-    exact h
-
-lemma measurable_firstVisit (ι : X → Y) (hι : Measurable ι)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ) : Measurable (firstVisit ι γ) := by
-  apply measurable_of_Iic
-  intro t
-  induction t using ENNReal.recTopCoe with
-  | top => simp only [Iic_top,preimage_univ,MeasurableSet.univ]
-  | coe t =>
-    have he : (firstVisit ι γ) ⁻¹' Iic (t : ℝ≥0∞)=visited ι γ t := by
-      ext x
-      exact firstVisit_le_iff ι γ hγ x t
-    rw [he]
-    exact measurableSet_visited ι hι γ hγ t
-
- 
-
-omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma iUnion_visited_nat (ι : X → Y) (γ : ℝ≥0 → Y) :
-    (⋃ n : ℕ, visited ι γ n)=ι ⁻¹' range γ := by
-  ext x
-  constructor
-  · rintro ⟨_,⟨n,rfl⟩,u,hu,he⟩
-    exact ⟨u,he⟩
-  · rintro ⟨u,he⟩
-    obtain ⟨n,hn⟩ := exists_nat_ge u
-    exact mem_iUnion.mpr ⟨n,u,⟨bot_le,hn⟩,he⟩
-
- 
-
-lemma restrict_sub_restrict (ν : Measure X) (S T : Set X) (hS : MeasurableSet S) :
-    ν.restrict T-ν.restrict S=ν.restrict (T\S) := by
-  have hp : (ν.restrict T-ν.restrict S).restrict S=0 := by
-    rw [Measure.restrict_sub_eq_restrict_sub_restrict hS,
-      Measure.restrict_restrict hS,Measure.restrict_restrict hS,inter_self]
-    exact Measure.sub_eq_zero_of_le (Measure.restrict_mono inter_subset_left le_rfl)
-  have hf : (ν.restrict T-ν.restrict S).restrict Sᶜ=ν.restrict (T\S) := by
-    rw [Measure.restrict_sub_eq_restrict_sub_restrict hS.compl,
-      Measure.restrict_restrict hS.compl,Measure.restrict_restrict hS.compl,
-      compl_inter_self,Measure.restrict_empty,Measure.sub_zero]
-    rw [sdiff_eq,inter_comm]
-  have hi := Measure.restrict_add_restrict_compl (μ := ν.restrict T-ν.restrict S) hS
-  rw [hp,hf,zero_add] at hi
-  exact hi.symm
-
-omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
-lemma visited_diff_subset (ι : X → Y) (γ : ℝ≥0 → Y) {s t : ℝ≥0} (_hst : s≤t) :
-    visited ι γ t\visited ι γ s⊆ι ⁻¹' (γ '' Icc s t) := by
-  rintro x ⟨⟨u,hu,he⟩,hn⟩
-  have hsu : s<u := lt_of_not_ge (fun hus => hn ⟨u,⟨hu.1,hus⟩,he⟩)
-  exact ⟨u,⟨hsu.le,hu.2⟩,he⟩
-
- 
-
-lemma increment_carried_by_segment (ν : Measure X) (ι : X → Y) (hι : Measurable ι)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ) {s t : ℝ≥0} (hst : s≤t) :
-    (ν.restrict (visited ι γ t)-ν.restrict (visited ι γ s))
-        (ι ⁻¹' (γ '' Icc s t))ᶜ=0 := by
-  rw [restrict_sub_restrict ν _ _ (measurableSet_visited ι hι γ hγ s),
-    Measure.restrict_apply (((isCompact_Icc.image hγ).isClosed.measurableSet).preimage hι).compl]
-  have he : (ι ⁻¹' (γ '' Icc s t))ᶜ∩(visited ι γ t\visited ι γ s)=∅ := by
-    apply eq_empty_iff_forall_notMem.mpr
-    rintro x ⟨hn,hx⟩
-    exact hn (visited_diff_subset ι γ hst hx)
-  rw [he,measure_empty]
-
-end SLEExactGauge.TraceRestriction
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.DenseTimeSqueeze
-
- 
-
-lemma tendsto_of_dense {I : Type*} {l : Filter I}
-    (f : I → ℝ≥0 → ℝ) (F : ℝ≥0 → ℝ) (D : Set ℝ≥0)
-    (hD : Dense D) (h0 : 0 ∈ D) (hmono : ∀ i, Monotone (f i))
-    (hF : Continuous F)
-    (hc : ∀ q ∈ D, Tendsto (fun i => f i q) l (𝓝 (F q)))
-    (t : ℝ≥0) : Tendsto (fun i => f i t) l (𝓝 (F t)) := by
-  by_cases ht : t = 0
-  · subst t
-    exact hc 0 h0
-  rw [tendsto_order]
-  constructor
-  · intro a ha
-    have he : ∀ᶠ q in 𝓝 t, a < F q := hF.continuousAt.tendsto.eventually (lt_mem_nhds ha)
-    obtain ⟨u, hut, hu⟩ := exists_Ioc_subset_of_mem_nhds he (⟨0, bot_lt_iff_ne_bot.mpr ht⟩)
-    obtain ⟨q, hq, huq, hqt⟩ := hD.exists_between hut
-    have haq : a < F q := hu ⟨huq, hqt.le⟩
-    filter_upwards [(hc q hq).eventually (lt_mem_nhds haq)] with i hi
-    exact hi.trans_le (hmono i hqt.le)
-  · intro a ha
-    have he : ∀ᶠ q in 𝓝 t, F q < a := hF.continuousAt.tendsto.eventually (gt_mem_nhds ha)
-    obtain ⟨u, htu, hu⟩ := exists_Ico_subset_of_mem_nhds he (exists_gt t)
-    obtain ⟨q, hq, htq, hqu⟩ := hD.exists_between htu
-    have hqa : F q < a := hu ⟨htq.le, hqu⟩
-    filter_upwards [(hc q hq).eventually (gt_mem_nhds hqa)] with i hi
-    exact (hmono i htq.le).trans_lt hi
-
-end SLEExactGauge.DenseTimeSqueeze
-
-namespace SLEExactGauge.DenseTimeSqueeze
-
-variable {X : Type*} [TopologicalSpace X] [T2Space X]
-    [MeasurableSpace X] [BorelSpace X]
-
- 
-
-omit [T2Space X] in
-lemma tendsto_integral_of_dense {I : Type*} {l : Filter I}
-    (μ : I → ℝ≥0 → Measure X) (ν : ℝ≥0 → Measure X)
-    [∀ i t, IsFiniteMeasureOnCompacts (μ i t)]
-    [∀ t, IsFiniteMeasureOnCompacts (ν t)]
-    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0 ∈ D)
-    (hmono : ∀ i, Monotone (μ i))
-    (hν : ∀ f : C_c(X, ℝ), Continuous (fun t => ∫ x, f x ∂ν t))
-    (hc : ∀ q ∈ D, ∀ f : C_c(X, ℝ),
-      Tendsto (fun i => ∫ x, f x ∂μ i q) l (𝓝 (∫ x, f x ∂ν q)))
-    (t : ℝ≥0) (f : C_c(X, ℝ)) :
-    Tendsto (fun i => ∫ x, f x ∂μ i t) l (𝓝 (∫ x, f x ∂ν t)) := by
-  have hp (g : C_c(X, ℝ)) (hg : 0 ≤ g) :
-      Tendsto (fun i => ∫ x, g x ∂μ i t) l (𝓝 (∫ x, g x ∂ν t)) := by
-    apply tendsto_of_dense (fun i q => ∫ x, g x ∂μ i q)
-      (fun q => ∫ x, g x ∂ν q) D hD h0 _ _ _ t
-    · intro i s t hst
-      exact integral_mono_measure (hmono i hst) (Eventually.of_forall hg) g.integrable
-    · exact hν g
-    · exact fun q hq => hc q hq g
-  have hplus := hp (f ⊔ 0) le_sup_right
-  have hminus := hp ((-f) ⊔ 0) le_sup_right
-  have he : ∀ x, f x = (f ⊔ 0) x - ((-f) ⊔ 0) x := by
-    intro x
-    change f x = max (f x) 0 - max (-f x) 0
-    rcases le_total (f x) 0 with h | h
-    · rw [max_eq_right h, max_eq_left (neg_nonneg.mpr h)]
-      ring
-    · rw [max_eq_left h, max_eq_right (neg_nonpos.mpr h)]
-      ring
-  have hi (ρ : Measure X) [IsFiniteMeasureOnCompacts ρ] :
-      (∫ x, f x ∂ρ) = (∫ x, (f ⊔ 0) x ∂ρ) - ∫ x, ((-f) ⊔ 0) x ∂ρ := by
-    simp_rw [he]
-    exact integral_sub (f ⊔ 0).integrable ((-f) ⊔ 0).integrable
-  simpa only [hi] using hplus.sub hminus
-
-end SLEExactGauge.DenseTimeSqueeze
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal
-
-namespace SLEExactGauge.FirstVisitReconstruction
-open TraceRestriction DenseTimeSqueeze MassVersions
-
-variable {X Y : Type*} [TopologicalSpace X] [T2Space X]
-    [LocallyCompactSpace X] [SecondCountableTopology X]
-    [MeasurableSpace X] [BorelSpace X]
-    [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y]
-
-omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
-lemma tendsto_all_times (μ : ℕ → ℝ≥0 → Measure X)
-    [∀ n t, IsFiniteMeasureOnCompacts (μ n t)]
-    (ν : Measure X) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
-    (ι : X → Y) (hι : Measurable ι) (hinj : Function.Injective ι)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ)
-    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0 ∈ D)
-    (hmono : ∀ n, Monotone (μ n))
-    (hc : ∀ q ∈ D, ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n q)
-      atTop (𝓝 (∫ x in visited ι γ q, f x ∂ν))) :
-    ∀ t : ℝ≥0, ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n t)
-      atTop (𝓝 (∫ x in visited ι γ t, f x ∂ν)) := by
-  apply tendsto_integral_of_dense μ (fun t => ν.restrict (visited ι γ t)) D hD h0 hmono
-  · intro f
-    exact continuous_integral_visited ν ι hι hinj γ hγ f f.integrable
-  · exact hc
-
- 
-
-lemma radonLimit_all_times (μ : ℕ → ℝ≥0 → Measure X)
-    [∀ n t, IsFiniteMeasureOnCompacts (μ n t)]
-    (ν : Measure X) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
-    (ι : X → Y) (hι : Measurable ι) (hinj : Function.Injective ι)
-    (γ : ℝ≥0 → Y) (hγ : Continuous γ)
-    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0 ∈ D)
-    (hmono : ∀ n, Monotone (μ n))
-    (hc : ∀ q ∈ D, ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n q)
-      atTop (𝓝 (∫ x in visited ι γ q, f x ∂ν))) :
-    ∀ t : ℝ≥0,
-      radonLimit (fun n => CompactlySupportedContinuousMap.integralPositiveLinearMap (μ n t)) =
-        ν.restrict (visited ι γ t) := by
-  intro t
-  apply radonLimit_eq
-  exact tendsto_all_times μ ν ι hι hinj γ hγ D hD h0 hmono hc t
-
-end SLEExactGauge.FirstVisitReconstruction
-end
-end
- 
-
- 
-section
- 
-
-noncomputable section
-open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
-open scoped NNReal ENNReal symmDiff
-namespace ExactSLEGreen
-open ExactSLEGenerator ExactSLEMain
-
-lemma actualConvexMass_mono_time (a : ℝ) (s : Finset ℕ) (c : ℕ→ℝ)
-    (b : ExactSLEGenerator.Driver) : Monotone (fun t => actualConvexMass a s c t b) := by
-  intro u v huv
-  apply Measure.le_iff'.mpr
-  intro A
-  simp only [actualConvexMass,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul]
-  exact Finset.sum_le_sum (fun j _ => mul_le_mul' le_rfl
-    (Measure.le_iff'.mp (actualUpperMass_mono_time a _ b huv) A))
-
-end ExactSLEGreen
-
-namespace SLEExactGauge.TraceMassContinuity
-open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
-open TraceRestriction MassVersions
-
- 
-def internalTrace (κ : ℝ) (b : TDriver) (u : ℝ≥0) : ℂ :=
-  Target.trace κ b ((u:ℝ)/κ)
-
-lemma continuous_internalTrace {κ : ℝ} (hκ : 0<κ) (b : TDriver)
-    (hb : ContinuousOn (Target.trace κ b) (Ici 0)) : Continuous (internalTrace κ b) :=
-  hb.comp_continuous (continuous_subtype_val.div_const κ) (fun u => div_nonneg u.2 hκ.le)
-
-lemma internalTrace_image {κ : ℝ} (hκ : 0<κ) (b : TDriver) (u v : ℝ≥0) :
-    internalTrace κ b '' Icc u v = Target.trace κ b '' Icc ((u:ℝ)/κ) ((v:ℝ)/κ) := by
-  ext z
-  constructor
-  · rintro ⟨t,ht,rfl⟩
-    exact ⟨_,⟨div_le_div_of_nonneg_right ht.1 hκ.le,
-      div_le_div_of_nonneg_right ht.2 hκ.le⟩,rfl⟩
-  · rintro ⟨t,ht,rfl⟩
-    have ht0 : 0≤t := (div_nonneg u.2 hκ.le).trans ht.1
-    refine ⟨⟨κ*t,mul_nonneg hκ.le ht0⟩,⟨?_,?_⟩,?_⟩
-    · change (u:ℝ)≤κ*t
-      nlinarith [(div_le_iff₀ hκ).mp ht.1]
-    · change κ*t≤v
-      nlinarith [(le_div_iff₀ hκ).mp ht.2]
-    · change Target.trace κ b ((κ*t)/κ)=Target.trace κ b t
-      rw [mul_div_cancel_left₀ _ hκ.ne']
-
-lemma visited_internalTrace {κ : ℝ} (hκ : 0<κ) (b : TDriver) (t : ℝ≥0) :
-    visited (fun z : UpperPlane => z.1) (internalTrace κ b) t =
-      {z : UpperPlane | z.1∈Target.trace κ b '' Icc 0 ((t:ℝ)/κ)} := by
-  simp only [visited,internalTrace_image hκ,NNReal.coe_zero,zero_div]
-  rfl
-
- 
-lemma actualMass_tendsto_all_times (κ : ℝ) (hκ : 0<κ) (ha : 1/4<Target.a κ)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (b : TDriver) (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
-    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
-    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0∈D)
-    (hconv : ∀q∈D, ∀ f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) q (driverHomeomorph b))
-      atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) q,f z∂ν))) :
-    ∀t : ℝ≥0, ∀ f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) t (driverHomeomorph b))
-      atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) t,f z∂ν)) := by
-  let M (k : ℕ) (t : ℝ≥0) :=
-    actualConvexMass (Target.a κ) (s k) (c k) t (driverHomeomorph b)
-  let (k : ℕ) (t : ℝ≥0) : IsFiniteMeasureOnCompacts (M k t) :=
-    actualConvexMass_finiteOnCompacts (Target.a κ) ha (s k) (c k) t (driverHomeomorph b)
-  have hv := FirstVisitReconstruction.tendsto_all_times M ν
-    (fun z : UpperPlane => z.1) measurable_subtype_coe Subtype.val_injective
-    (internalTrace κ b) (continuous_internalTrace hκ b hcont) D hD h0
-    (fun k u v huv => actualConvexMass_mono_time (Target.a κ) (s k) (c k)
-      (driverHomeomorph b) (ENNReal.coe_le_coe.mpr huv))
-    (by
-      intro q hq f
-      simpa only [M,actualConvexMass_integral (Target.a κ) ha _ _ (hc _)] using hconv q hq f)
-  intro t f
-  simpa only [M,actualConvexMass_integral (Target.a κ) ha _ _ (hc _)] using hv t f
-
- 
-
-lemma recoveredMass_all_times (κ : ℝ) (hκ : 0<κ) (ha : 1/4<Target.a κ)
-    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
-    (b : TDriver) (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
-    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
-    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0∈D)
-    (hconv : ∀q∈D, ∀ f : C_c(UpperPlane,ℝ), Tendsto
-      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
-        (Real.exp (-2*(Target.a κ)*(j+1))) q (driverHomeomorph b))
-      atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) q,f z∂ν))) :
-    ∀t : ℝ≥0,
-      recoveredMass (Target.a κ) ha s c hc t (driverHomeomorph b)=
-        ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t) := by
-  intro t
-  apply radonLimit_eq
-  intro f
-  simpa only [actualMassFunctionals_apply] using
-    actualMass_tendsto_all_times κ hκ ha s c hc b hcont ν D hD h0 hconv t f
-
- 
-
-lemma local_variation_visited (ν : Measure UpperPlane) [NullSingletonClass ν]
-    {κ : ℝ} (hκ : 0<κ) (b : TDriver)
-    (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
-    (K : Set UpperPlane) (hK : MeasurableSet K) (hfin : ν K≠∞) (t : ℝ≥0) :
-    Tendsto (fun u : ℝ≥0 =>
-      ((ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) u)-
-        ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t))+
-       (ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t)-
-        ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) u))) K)
-        (𝓝 t) (𝓝 0) := by
-  let V := visited (fun z : UpperPlane => z.1) (internalTrace κ b)
-  have hV (u : ℝ≥0) : MeasurableSet (V u) :=
-    measurableSet_visited _ measurable_subtype_coe _ (continuous_internalTrace hκ b hcont) u
-  have he (u : ℝ≥0) :
-      ((ν.restrict (V u)-ν.restrict (V t))+(ν.restrict (V t)-ν.restrict (V u))) K=
-        ν (K∩(V u ∆ V t)) := by
-    rw [restrict_sub_restrict ν _ _ (hV t),restrict_sub_restrict ν _ _ (hV u),
-      ←Measure.restrict_union (disjoint_sdiff_sdiff (x := V u) (y := V t)) ((hV t).diff (hV u)),
-      Measure.restrict_apply hK]
-    rfl
-  change Tendsto (fun u : ℝ≥0 =>
-    ((ν.restrict (V u)-ν.restrict (V t))+(ν.restrict (V t)-ν.restrict (V u))) K) (𝓝 t) (𝓝 0)
-  simp_rw [he]
-  exact tendsto_measure_visited_symmDiff ν _ measurable_subtype_coe _
-    (continuous_internalTrace hκ b hcont) t
-    (preimage_singleton_null_of_injective ν _ Subtype.val_injective _) K hK hfin
-
-end SLEExactGauge.TraceMassContinuity
-end
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.LoewnerScaling
+import OAI.Probability.SLEGauge.FiniteProductLimit
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology
+open scoped Classical NNReal ENNReal symmDiff
+namespace SLEExactGauge.TraceRestriction
+
+variable {X Y : Type*} [MeasurableSpace X] [MetricSpace Y]
+    [MeasurableSpace Y] [BorelSpace Y]
+
+ 
+def visited (ι : X → Y) (γ : ℝ≥0 → Y) (t : ℝ≥0) : Set X :=
+  ι ⁻¹' (γ '' Icc 0 t)
+
+omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma visited_mono (ι : X → Y) (γ : ℝ≥0 → Y) : Monotone (visited ι γ) := by
+  intro s t hst
+  exact preimage_mono (image_mono (Icc_subset_Icc_right hst))
+
+lemma measurableSet_visited (ι : X → Y) (hι : Measurable ι)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (t : ℝ≥0) :
+    MeasurableSet (visited ι γ t) :=
+  ((isCompact_Icc.image hγ).isClosed.measurableSet).preimage hι
+
+ 
+
+omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma visited_symmDiff_subset (ι : X → Y) (γ : ℝ≥0 → Y) (s t : ℝ≥0) :
+    visited ι γ s ∆ visited ι γ t ⊆ ι ⁻¹' (γ '' uIcc s t) := by
+  intro x hx
+  rcases hx with ⟨hxs,hxt⟩ | ⟨hxt,hxs⟩
+  · rcases hxs with ⟨u,hu,he⟩
+    have htu : t<u := lt_of_not_ge (fun hut => hxt ⟨u,⟨hu.1,hut⟩,he⟩)
+    exact ⟨u,⟨(min_le_right s t).trans htu.le,hu.2.trans (le_max_left s t)⟩,he⟩
+  · rcases hxt with ⟨u,hu,he⟩
+    have hsu : s<u := lt_of_not_ge (fun hus => hxs ⟨u,⟨hu.1,hus⟩,he⟩)
+    exact ⟨u,⟨(min_le_left s t).trans hsu.le,hu.2.trans (le_max_right s t)⟩,he⟩
+
+lemma nnreal_dist_right_le {s t u : ℝ≥0} (hu : u∈uIcc s t) :
+    dist u t≤dist s t := by
+  apply Real.dist_right_le_of_mem_uIcc
+  change min (s : ℝ) (t : ℝ)≤(u : ℝ) ∧ (u : ℝ) ≤ max (s : ℝ) (t : ℝ)
+  exact ⟨hu.1,hu.2⟩
+
+ 
+
+omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
+lemma eventually_not_mem_visited_symmDiff (ι : X → Y)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (t : ℝ≥0) (x : X)
+    (hx : ι x≠γ t) : ∀ᶠ s in 𝓝 t, x∉visited ι γ s ∆ visited ι γ t := by
+  have hr : 0<dist (ι x) (γ t) := dist_pos.mpr hx
+  obtain ⟨δ,hδ,hδγ⟩ := Metric.continuousAt_iff.mp (hγ.continuousAt (x := t)) _ hr
+  filter_upwards [Metric.ball_mem_nhds t hδ] with s hs hxs
+  obtain ⟨u,hu,he⟩ := visited_symmDiff_subset ι γ s t hxs
+  have hud : dist u t<δ := (nnreal_dist_right_le hu).trans_lt hs
+  have hlt := hδγ hud
+  rw [he] at hlt
+  exact lt_irrefl _ hlt
+
+omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
+lemma eventually_visited_iff (ι : X → Y)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (t : ℝ≥0) (x : X)
+    (hx : ι x≠γ t) : ∀ᶠ s in 𝓝 t, (x∈visited ι γ s ↔ x∈visited ι γ t) := by
+  filter_upwards [eventually_not_mem_visited_symmDiff ι γ hγ t x hx] with s hs
+  constructor
+  · intro hx
+    by_contra ht
+    exact hs (Or.inl ⟨hx,ht⟩)
+  · intro hx
+    by_contra ht
+    exact hs (Or.inr ⟨hx,ht⟩)
+
+omit [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma preimage_singleton_null_of_injective (ν : Measure X) [NullSingletonClass ν]
+    (ι : X → Y) (hinj : Function.Injective ι) (y : Y) : ν (ι ⁻¹' {y})=0 := by
+  by_cases h : ∃ x, ι x=y
+  · obtain ⟨x,hx⟩ := h
+    have he : ι ⁻¹' {y}={x} := by
+      ext z
+      simp only [mem_preimage,mem_singleton_iff,← hx,hinj.eq_iff]
+    rw [he,measure_singleton]
+  · have he : ι ⁻¹' {y}=∅ := by
+      ext x
+      simp only [mem_preimage,mem_singleton_iff,mem_empty_iff_false,iff_false]
+      exact fun hx => h ⟨x,hx⟩
+    rw [he,measure_empty]
+
+ 
+
+lemma tendsto_measure_visited_symmDiff (ν : Measure X)
+    (ι : X → Y) (hι : Measurable ι) (γ : ℝ≥0 → Y) (hγ : Continuous γ)
+    (t : ℝ≥0) (hpoint : ν (ι ⁻¹' {γ t})=0)
+    (K : Set X) (_hK : MeasurableSet K) (hfin : ν K≠∞) :
+    Tendsto (fun s => ν (K∩(visited ι γ s ∆ visited ι γ t))) (𝓝 t) (𝓝 0) := by
+  let μ := ν.restrict K
+  have : IsFiniteMeasure μ := ⟨by simpa only [μ,Measure.restrict_apply_univ] using hfin.lt_top⟩
+  let F : ℝ≥0 → X → ℝ≥0∞ := fun s => (visited ι γ s ∆ visited ι γ t).indicator 1
+  have hF (s : ℝ≥0) : Measurable (F s) :=
+    measurable_const.indicator
+      ((measurableSet_visited ι hι γ hγ s).symmDiff (measurableSet_visited ι hι γ hγ t))
+  have hbound (s : ℝ≥0) : ∀ᵐ x ∂μ, F s x≤1 := Eventually.of_forall (fun x => by
+    by_cases hx : x∈visited ι γ s ∆ visited ι γ t <;> simp [F,hx])
+  have hzero : μ (ι ⁻¹' {γ t})=0 := by
+    change (ν.restrict K) (ι ⁻¹' {γ t})=0
+    rw [Measure.restrict_apply ((measurableSet_singleton (γ t)).preimage hι)]
+    exact measure_mono_null inter_subset_left hpoint
+  have hlim : ∀ᵐ x ∂μ, Tendsto (fun s => F s x) (𝓝 t) (𝓝 0) := by
+    filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with x hx
+    have hxt : ι x≠γ t := hx
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_not_mem_visited_symmDiff ι γ hγ t x hxt] with s hs
+    simp only [F,indicator_of_notMem hs]
+  have ht := tendsto_lintegral_filter_of_dominated_convergence (μ := μ) (fun _ => (1 : ℝ≥0∞))
+    (Eventually.of_forall hF) (Eventually.of_forall hbound)
+    (by simp : (∫⁻ x, (1 : ℝ≥0∞) ∂μ)≠∞) hlim
+  have he (s : ℝ≥0) : (∫⁻x, F s x ∂μ)=ν (K∩(visited ι γ s ∆ visited ι γ t)) := by
+    have hs := (measurableSet_visited ι hι γ hγ s).symmDiff (measurableSet_visited ι hι γ hγ t)
+    change (∫⁻x, (visited ι γ s ∆ visited ι γ t).indicator 1 x ∂(ν.restrict K))=_
+    rw [lintegral_indicator_one hs,Measure.restrict_apply hs,inter_comm]
+  simpa only [he,lintegral_zero] using ht
+
+ 
+lemma continuous_integral_visited (ν : Measure X) [NullSingletonClass ν]
+    (ι : X → Y) (hι : Measurable ι) (hinj : Function.Injective ι)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ) (f : X → ℝ) (hf : Integrable f ν) :
+    Continuous (fun t => ∫x in visited ι γ t, f x ∂ν) := by
+  rw [continuous_iff_continuousAt]
+  intro t
+  simp_rw [← integral_indicator (measurableSet_visited ι hι γ hγ _)]
+  apply continuousAt_of_dominated (bound := fun x => ‖f x‖)
+  · exact Eventually.of_forall (fun s => hf.aestronglyMeasurable.indicator
+      (measurableSet_visited ι hι γ hγ s))
+  · exact Eventually.of_forall (fun s => Eventually.of_forall (fun x => norm_indicator_le_norm_self _ _))
+  · exact hf.norm
+  · filter_upwards [measure_eq_zero_iff_ae_notMem.mp
+      (preimage_singleton_null_of_injective ν ι hinj (γ t))] with x hx
+    have hxt : ι x≠γ t := hx
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_visited_iff ι γ hγ t x hxt] with s hs
+    by_cases ht : x∈visited ι γ t
+    · simp only [indicator_of_mem ht,indicator_of_mem (hs.mpr ht)]
+    · simp only [indicator_of_notMem ht,indicator_of_notMem (mt hs.mp ht)]
+
+ 
+
+def firstVisit (ι : X → Y) (γ : ℝ≥0 → Y) (x : X) : ℝ≥0∞ :=
+  if _h : ({u : ℝ≥0 | γ u=ι x}).Nonempty then
+    (sInf {u : ℝ≥0 | γ u=ι x} : ℝ≥0) else ∞
+
+omit [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] in
+lemma firstVisit_le_iff (ι : X → Y) (γ : ℝ≥0 → Y) (hγ : Continuous γ)
+    (x : X) (t : ℝ≥0) : firstVisit ι γ x≤(t : ℝ≥0∞) ↔ x∈visited ι γ t := by
+  classical
+  by_cases h : ({u : ℝ≥0 | γ u=ι x}).Nonempty
+  · have hclosed : IsClosed {u : ℝ≥0 | γ u=ι x} := isClosed_eq hγ continuous_const
+    have hmem := hclosed.csInf_mem h ⟨0,fun _ _ => bot_le⟩
+    simp only [firstVisit,dite_eq_left h,ENNReal.coe_le_coe]
+    constructor
+    · intro ht
+      exact ⟨_,⟨bot_le,ht⟩,hmem⟩
+    · rintro ⟨u,hu,he⟩
+      exact (csInf_le (s := {u : ℝ≥0 | γ u=ι x}) ⟨0,fun _ _ => bot_le⟩ he).trans hu.2
+  · simp only [firstVisit,dite_eq_right h,top_le_iff,ENNReal.coe_ne_top,false_iff]
+    rintro ⟨u,hu,he⟩
+    exact h ⟨u,he⟩
+
+omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma firstVisit_lt_top_iff (ι : X → Y) (γ : ℝ≥0 → Y) (x : X) :
+    firstVisit ι γ x<∞ ↔ ι x∈range γ := by
+  classical
+  by_cases h : ({u : ℝ≥0 | γ u=ι x}).Nonempty
+  · simp only [firstVisit,dite_eq_left h,ENNReal.coe_lt_top,true_iff]
+    exact h
+  · simp only [firstVisit,dite_eq_right h,lt_self_iff_false,false_iff]
+    exact h
+
+lemma measurable_firstVisit (ι : X → Y) (hι : Measurable ι)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ) : Measurable (firstVisit ι γ) := by
+  apply measurable_of_Iic
+  intro t
+  induction t using ENNReal.recTopCoe with
+  | top => simp only [Iic_top,preimage_univ,MeasurableSet.univ]
+  | coe t =>
+    have he : (firstVisit ι γ) ⁻¹' Iic (t : ℝ≥0∞)=visited ι γ t := by
+      ext x
+      exact firstVisit_le_iff ι γ hγ x t
+    rw [he]
+    exact measurableSet_visited ι hι γ hγ t
+
+ 
+
+omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma iUnion_visited_nat (ι : X → Y) (γ : ℝ≥0 → Y) :
+    (⋃ n : ℕ, visited ι γ n)=ι ⁻¹' range γ := by
+  ext x
+  constructor
+  · rintro ⟨_,⟨n,rfl⟩,u,hu,he⟩
+    exact ⟨u,he⟩
+  · rintro ⟨u,he⟩
+    obtain ⟨n,hn⟩ := exists_nat_ge u
+    exact mem_iUnion.mpr ⟨n,u,⟨bot_le,hn⟩,he⟩
+
+ 
+
+lemma restrict_sub_restrict (ν : Measure X) (S T : Set X) (hS : MeasurableSet S) :
+    ν.restrict T-ν.restrict S=ν.restrict (T\S) := by
+  have hp : (ν.restrict T-ν.restrict S).restrict S=0 := by
+    rw [Measure.restrict_sub_eq_restrict_sub_restrict hS,
+      Measure.restrict_restrict hS,Measure.restrict_restrict hS,inter_self]
+    exact Measure.sub_eq_zero_of_le (Measure.restrict_mono inter_subset_left le_rfl)
+  have hf : (ν.restrict T-ν.restrict S).restrict Sᶜ=ν.restrict (T\S) := by
+    rw [Measure.restrict_sub_eq_restrict_sub_restrict hS.compl,
+      Measure.restrict_restrict hS.compl,Measure.restrict_restrict hS.compl,
+      compl_inter_self,Measure.restrict_empty,Measure.sub_zero]
+    rw [sdiff_eq,inter_comm]
+  have hi := Measure.restrict_add_restrict_compl (μ := ν.restrict T-ν.restrict S) hS
+  rw [hp,hf,zero_add] at hi
+  exact hi.symm
+
+omit [MeasurableSpace X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] in
+lemma visited_diff_subset (ι : X → Y) (γ : ℝ≥0 → Y) {s t : ℝ≥0} (_hst : s≤t) :
+    visited ι γ t\visited ι γ s⊆ι ⁻¹' (γ '' Icc s t) := by
+  rintro x ⟨⟨u,hu,he⟩,hn⟩
+  have hsu : s<u := lt_of_not_ge (fun hus => hn ⟨u,⟨hu.1,hus⟩,he⟩)
+  exact ⟨u,⟨hsu.le,hu.2⟩,he⟩
+
+ 
+
+lemma increment_carried_by_segment (ν : Measure X) (ι : X → Y) (hι : Measurable ι)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ) {s t : ℝ≥0} (hst : s≤t) :
+    (ν.restrict (visited ι γ t)-ν.restrict (visited ι γ s))
+        (ι ⁻¹' (γ '' Icc s t))ᶜ=0 := by
+  rw [restrict_sub_restrict ν _ _ (measurableSet_visited ι hι γ hγ s),
+    Measure.restrict_apply (((isCompact_Icc.image hγ).isClosed.measurableSet).preimage hι).compl]
+  have he : (ι ⁻¹' (γ '' Icc s t))ᶜ∩(visited ι γ t\visited ι γ s)=∅ := by
+    apply eq_empty_iff_forall_notMem.mpr
+    rintro x ⟨hn,hx⟩
+    exact hn (visited_diff_subset ι γ hst hx)
+  rw [he,measure_empty]
+
+end SLEExactGauge.TraceRestriction
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.DenseTimeSqueeze
+
+ 
+
+lemma tendsto_of_dense {I : Type*} {l : Filter I}
+    (f : I → ℝ≥0 → ℝ) (F : ℝ≥0 → ℝ) (D : Set ℝ≥0)
+    (hD : Dense D) (h0 : 0 ∈ D) (hmono : ∀ i, Monotone (f i))
+    (hF : Continuous F)
+    (hc : ∀ q ∈ D, Tendsto (fun i => f i q) l (𝓝 (F q)))
+    (t : ℝ≥0) : Tendsto (fun i => f i t) l (𝓝 (F t)) := by
+  by_cases ht : t = 0
+  · subst t
+    exact hc 0 h0
+  rw [tendsto_order]
+  constructor
+  · intro a ha
+    have he : ∀ᶠ q in 𝓝 t, a < F q := hF.continuousAt.tendsto.eventually (lt_mem_nhds ha)
+    obtain ⟨u, hut, hu⟩ := exists_Ioc_subset_of_mem_nhds he (⟨0, bot_lt_iff_ne_bot.mpr ht⟩)
+    obtain ⟨q, hq, huq, hqt⟩ := hD.exists_between hut
+    have haq : a < F q := hu ⟨huq, hqt.le⟩
+    filter_upwards [(hc q hq).eventually (lt_mem_nhds haq)] with i hi
+    exact hi.trans_le (hmono i hqt.le)
+  · intro a ha
+    have he : ∀ᶠ q in 𝓝 t, F q < a := hF.continuousAt.tendsto.eventually (gt_mem_nhds ha)
+    obtain ⟨u, htu, hu⟩ := exists_Ico_subset_of_mem_nhds he (exists_gt t)
+    obtain ⟨q, hq, htq, hqu⟩ := hD.exists_between htu
+    have hqa : F q < a := hu ⟨htq.le, hqu⟩
+    filter_upwards [(hc q hq).eventually (gt_mem_nhds hqa)] with i hi
+    exact (hmono i htq.le).trans_lt hi
+
+end SLEExactGauge.DenseTimeSqueeze
+
+namespace SLEExactGauge.DenseTimeSqueeze
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X]
+    [MeasurableSpace X] [BorelSpace X]
+
+ 
+
+omit [T2Space X] in
+lemma tendsto_integral_of_dense {I : Type*} {l : Filter I}
+    (μ : I → ℝ≥0 → Measure X) (ν : ℝ≥0 → Measure X)
+    [∀ i t, IsFiniteMeasureOnCompacts (μ i t)]
+    [∀ t, IsFiniteMeasureOnCompacts (ν t)]
+    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0 ∈ D)
+    (hmono : ∀ i, Monotone (μ i))
+    (hν : ∀ f : C_c(X, ℝ), Continuous (fun t => ∫ x, f x ∂ν t))
+    (hc : ∀ q ∈ D, ∀ f : C_c(X, ℝ),
+      Tendsto (fun i => ∫ x, f x ∂μ i q) l (𝓝 (∫ x, f x ∂ν q)))
+    (t : ℝ≥0) (f : C_c(X, ℝ)) :
+    Tendsto (fun i => ∫ x, f x ∂μ i t) l (𝓝 (∫ x, f x ∂ν t)) := by
+  have hp (g : C_c(X, ℝ)) (hg : 0 ≤ g) :
+      Tendsto (fun i => ∫ x, g x ∂μ i t) l (𝓝 (∫ x, g x ∂ν t)) := by
+    apply tendsto_of_dense (fun i q => ∫ x, g x ∂μ i q)
+      (fun q => ∫ x, g x ∂ν q) D hD h0 _ _ _ t
+    · intro i s t hst
+      exact integral_mono_measure (hmono i hst) (Eventually.of_forall hg) g.integrable
+    · exact hν g
+    · exact fun q hq => hc q hq g
+  have hplus := hp (f ⊔ 0) le_sup_right
+  have hminus := hp ((-f) ⊔ 0) le_sup_right
+  have he : ∀ x, f x = (f ⊔ 0) x - ((-f) ⊔ 0) x := by
+    intro x
+    change f x = max (f x) 0 - max (-f x) 0
+    rcases le_total (f x) 0 with h | h
+    · rw [max_eq_right h, max_eq_left (neg_nonneg.mpr h)]
+      ring
+    · rw [max_eq_left h, max_eq_right (neg_nonpos.mpr h)]
+      ring
+  have hi (ρ : Measure X) [IsFiniteMeasureOnCompacts ρ] :
+      (∫ x, f x ∂ρ) = (∫ x, (f ⊔ 0) x ∂ρ) - ∫ x, ((-f) ⊔ 0) x ∂ρ := by
+    simp_rw [he]
+    exact integral_sub (f ⊔ 0).integrable ((-f) ⊔ 0).integrable
+  simpa only [hi] using hplus.sub hminus
+
+end SLEExactGauge.DenseTimeSqueeze
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal
+
+namespace SLEExactGauge.FirstVisitReconstruction
+open TraceRestriction DenseTimeSqueeze MassVersions
+
+variable {X Y : Type*} [TopologicalSpace X] [T2Space X]
+    [LocallyCompactSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] [BorelSpace X]
+    [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y]
+
+omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] in
+lemma tendsto_all_times (μ : ℕ → ℝ≥0 → Measure X)
+    [∀ n t, IsFiniteMeasureOnCompacts (μ n t)]
+    (ν : Measure X) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
+    (ι : X → Y) (hι : Measurable ι) (hinj : Function.Injective ι)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ)
+    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0 ∈ D)
+    (hmono : ∀ n, Monotone (μ n))
+    (hc : ∀ q ∈ D, ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n q)
+      atTop (𝓝 (∫ x in visited ι γ q, f x ∂ν))) :
+    ∀ t : ℝ≥0, ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n t)
+      atTop (𝓝 (∫ x in visited ι γ t, f x ∂ν)) := by
+  apply tendsto_integral_of_dense μ (fun t => ν.restrict (visited ι γ t)) D hD h0 hmono
+  · intro f
+    exact continuous_integral_visited ν ι hι hinj γ hγ f f.integrable
+  · exact hc
+
+ 
+
+lemma radonLimit_all_times (μ : ℕ → ℝ≥0 → Measure X)
+    [∀ n t, IsFiniteMeasureOnCompacts (μ n t)]
+    (ν : Measure X) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
+    (ι : X → Y) (hι : Measurable ι) (hinj : Function.Injective ι)
+    (γ : ℝ≥0 → Y) (hγ : Continuous γ)
+    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0 ∈ D)
+    (hmono : ∀ n, Monotone (μ n))
+    (hc : ∀ q ∈ D, ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n q)
+      atTop (𝓝 (∫ x in visited ι γ q, f x ∂ν))) :
+    ∀ t : ℝ≥0,
+      radonLimit (fun n => CompactlySupportedContinuousMap.integralPositiveLinearMap (μ n t)) =
+        ν.restrict (visited ι γ t) := by
+  intro t
+  apply radonLimit_eq
+  exact tendsto_all_times μ ν ι hι hinj γ hγ D hD h0 hmono hc t
+
+end SLEExactGauge.FirstVisitReconstruction
+end
+end
+ 
+
+ 
+section
+ 
+
+noncomputable section
+open MeasureTheory ProbabilityTheory Filter Set Topology CompactlySupported
+open scoped NNReal ENNReal symmDiff
+namespace ExactSLEGreen
+open ExactSLEGenerator ExactSLEMain
+
+lemma actualConvexMass_mono_time (a : ℝ) (s : Finset ℕ) (c : ℕ→ℝ)
+    (b : ExactSLEGenerator.Driver) : Monotone (fun t => actualConvexMass a s c t b) := by
+  intro u v huv
+  apply Measure.le_iff'.mpr
+  intro A
+  simp only [actualConvexMass,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul]
+  exact Finset.sum_le_sum (fun j _ => mul_le_mul' le_rfl
+    (Measure.le_iff'.mp (actualUpperMass_mono_time a _ b huv) A))
+
+end ExactSLEGreen
+
+namespace SLEExactGauge.TraceMassContinuity
+open FlowIdentification ExactSLEMain ExactSLEGenerator ExactSLEGreen
+open TraceRestriction MassVersions
+
+ 
+def internalTrace (κ : ℝ) (b : TDriver) (u : ℝ≥0) : ℂ :=
+  Target.trace κ b ((u:ℝ)/κ)
+
+lemma continuous_internalTrace {κ : ℝ} (hκ : 0<κ) (b : TDriver)
+    (hb : ContinuousOn (Target.trace κ b) (Ici 0)) : Continuous (internalTrace κ b) :=
+  hb.comp_continuous (continuous_subtype_val.div_const κ) (fun u => div_nonneg u.2 hκ.le)
+
+lemma internalTrace_image {κ : ℝ} (hκ : 0<κ) (b : TDriver) (u v : ℝ≥0) :
+    internalTrace κ b '' Icc u v = Target.trace κ b '' Icc ((u:ℝ)/κ) ((v:ℝ)/κ) := by
+  ext z
+  constructor
+  · rintro ⟨t,ht,rfl⟩
+    exact ⟨_,⟨div_le_div_of_nonneg_right ht.1 hκ.le,
+      div_le_div_of_nonneg_right ht.2 hκ.le⟩,rfl⟩
+  · rintro ⟨t,ht,rfl⟩
+    have ht0 : 0≤t := (div_nonneg u.2 hκ.le).trans ht.1
+    refine ⟨⟨κ*t,mul_nonneg hκ.le ht0⟩,⟨?_,?_⟩,?_⟩
+    · change (u:ℝ)≤κ*t
+      nlinarith [(div_le_iff₀ hκ).mp ht.1]
+    · change κ*t≤v
+      nlinarith [(le_div_iff₀ hκ).mp ht.2]
+    · change Target.trace κ b ((κ*t)/κ)=Target.trace κ b t
+      rw [mul_div_cancel_left₀ _ hκ.ne']
+
+lemma visited_internalTrace {κ : ℝ} (hκ : 0<κ) (b : TDriver) (t : ℝ≥0) :
+    visited (fun z : UpperPlane => z.1) (internalTrace κ b) t =
+      {z : UpperPlane | z.1∈Target.trace κ b '' Icc 0 ((t:ℝ)/κ)} := by
+  simp only [visited,internalTrace_image hκ,NNReal.coe_zero,zero_div]
+  rfl
+
+ 
+lemma actualMass_tendsto_all_times (κ : ℝ) (hκ : 0<κ) (ha : 1/4<Target.a κ)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (b : TDriver) (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
+    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
+    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0∈D)
+    (hconv : ∀q∈D, ∀ f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) q (driverHomeomorph b))
+      atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) q,f z∂ν))) :
+    ∀t : ℝ≥0, ∀ f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) t (driverHomeomorph b))
+      atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) t,f z∂ν)) := by
+  let M (k : ℕ) (t : ℝ≥0) :=
+    actualConvexMass (Target.a κ) (s k) (c k) t (driverHomeomorph b)
+  let (k : ℕ) (t : ℝ≥0) : IsFiniteMeasureOnCompacts (M k t) :=
+    actualConvexMass_finiteOnCompacts (Target.a κ) ha (s k) (c k) t (driverHomeomorph b)
+  have hv := FirstVisitReconstruction.tendsto_all_times M ν
+    (fun z : UpperPlane => z.1) measurable_subtype_coe Subtype.val_injective
+    (internalTrace κ b) (continuous_internalTrace hκ b hcont) D hD h0
+    (fun k u v huv => actualConvexMass_mono_time (Target.a κ) (s k) (c k)
+      (driverHomeomorph b) (ENNReal.coe_le_coe.mpr huv))
+    (by
+      intro q hq f
+      simpa only [M,actualConvexMass_integral (Target.a κ) ha _ _ (hc _)] using hconv q hq f)
+  intro t f
+  simpa only [M,actualConvexMass_integral (Target.a κ) ha _ _ (hc _)] using hv t f
+
+ 
+
+lemma recoveredMass_all_times (κ : ℝ) (hκ : 0<κ) (ha : 1/4<Target.a κ)
+    (s : ℕ→Finset ℕ) (c : ℕ→ℕ→ℝ) (hc : ∀k j,j∈s k→0≤c k j)
+    (b : TDriver) (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
+    (ν : Measure UpperPlane) [IsFiniteMeasureOnCompacts ν] [NullSingletonClass ν]
+    (D : Set ℝ≥0) (hD : Dense D) (h0 : 0∈D)
+    (hconv : ∀q∈D, ∀ f : C_c(UpperPlane,ℝ), Tendsto
+      (fun k => ∑j∈s k,c k j*∫z,f z∂actualUpperMass (Target.a κ)
+        (Real.exp (-2*(Target.a κ)*(j+1))) q (driverHomeomorph b))
+      atTop (𝓝 (∫z in visited (fun z : UpperPlane => z.1) (internalTrace κ b) q,f z∂ν))) :
+    ∀t : ℝ≥0,
+      recoveredMass (Target.a κ) ha s c hc t (driverHomeomorph b)=
+        ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t) := by
+  intro t
+  apply radonLimit_eq
+  intro f
+  simpa only [actualMassFunctionals_apply] using
+    actualMass_tendsto_all_times κ hκ ha s c hc b hcont ν D hD h0 hconv t f
+
+ 
+
+lemma local_variation_visited (ν : Measure UpperPlane) [NullSingletonClass ν]
+    {κ : ℝ} (hκ : 0<κ) (b : TDriver)
+    (hcont : ContinuousOn (Target.trace κ b) (Ici 0))
+    (K : Set UpperPlane) (hK : MeasurableSet K) (hfin : ν K≠∞) (t : ℝ≥0) :
+    Tendsto (fun u : ℝ≥0 =>
+      ((ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) u)-
+        ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t))+
+       (ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) t)-
+        ν.restrict (visited (fun z : UpperPlane => z.1) (internalTrace κ b) u))) K)
+        (𝓝 t) (𝓝 0) := by
+  let V := visited (fun z : UpperPlane => z.1) (internalTrace κ b)
+  have hV (u : ℝ≥0) : MeasurableSet (V u) :=
+    measurableSet_visited _ measurable_subtype_coe _ (continuous_internalTrace hκ b hcont) u
+  have he (u : ℝ≥0) :
+      ((ν.restrict (V u)-ν.restrict (V t))+(ν.restrict (V t)-ν.restrict (V u))) K=
+        ν (K∩(V u ∆ V t)) := by
+    rw [restrict_sub_restrict ν _ _ (hV t),restrict_sub_restrict ν _ _ (hV u),
+      ←Measure.restrict_union (disjoint_sdiff_sdiff (x := V u) (y := V t)) ((hV t).diff (hV u)),
+      Measure.restrict_apply hK]
+    rfl
+  change Tendsto (fun u : ℝ≥0 =>
+    ((ν.restrict (V u)-ν.restrict (V t))+(ν.restrict (V t)-ν.restrict (V u))) K) (𝓝 t) (𝓝 0)
+  simp_rw [he]
+  exact tendsto_measure_visited_symmDiff ν _ measurable_subtype_coe _
+    (continuous_internalTrace hκ b hcont) t
+    (preimage_singleton_null_of_injective ν _ Subtype.val_injective _) K hK hfin
+
+end SLEExactGauge.TraceMassContinuity
+end
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLEGauge/TwoPointGrowth.lean b/lean/OAI/Probability/SLEGauge/TwoPointGrowth.lean
index 3a741c020c16d7d1762266e3887d99b61aa35d0f..24460a5b7081e079008ca20cc54acf0dfc455aff 100644
--- a/lean/OAI/Probability/SLEGauge/TwoPointGrowth.lean
+++ b/lean/OAI/Probability/SLEGauge/TwoPointGrowth.lean
@@ -1,454 +1,454 @@
-import Mathlib
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLEGauge.StateBounds
-
-namespace OAI
-
-/-! Hausdorff gauges, Loewner evolution and trace measures. -/
-
-section
-open MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-open scoped Topology
-open Set
-open MeasureTheory Filter
-open scoped NNReal Topology
-open MeasureTheory Filter Set Metric Function ODE
-open MeasureTheory ProbabilityTheory Filter Set
-open Filter
-open Filter Set
-namespace ExactSLEMain
-open Set
-theorem loewner_two_point_growth {A : ℝ} (hA : 0 < A)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {T : ℝ≥0} {z w : ℂ}
-    (hz : alive A b T z) (hw : alive A b T w) (hzw : z ≠ w) :
-    ‖z-w‖ ≤ (2 * (loewnerMap A b T z).im / ‖deriv (loewnerMap A b T) z‖) *
-      (‖loewnerMap A b T z-loewnerMap A b T w‖ /
-       ‖loewnerMap A b T z-star (loewnerMap A b T w)‖) /
-      (1-‖loewnerMap A b T z-loewnerMap A b T w‖ /
-       ‖loewnerMap A b T z-star (loewnerMap A b T w)‖)^2 := by
-  obtain ⟨f,hf⟩ := hz
-  obtain ⟨g,hg⟩ := hw
-  have hz : alive A b T z := ⟨f,hf⟩
-  have hw : alive A b T w := ⟨g,hg⟩
-  have hyz : 0 < z.im := by simpa only [hf.1] using (hf.2 0 (by simp)).1
-  have hyw : 0 < w.im := by simpa only [hg.1] using (hg.2 0 (by simp)).1
-  let V : ℝ → ℝ := fun u => (f u).im/Complex.normSq (f u-(b u.toNNReal : ℂ))
-  let W : ℝ → ℝ := fun u => (g u).im/Complex.normSq (g u-(b u.toNNReal : ℂ))
-  have hVc : ContinuousOn V (Icc (0 : ℝ) T) :=
-    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)).div
-      (Complex.continuous_normSq.comp_continuousOn (odeUntil_center_continuousOn hb hf))
-      (fun u hu => (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf hu)).ne')
-  let v : ℝ → ℝ := fun u => V (projIcc (0 : ℝ) T T.coe_nonneg u)
-  have hv : Continuous v := hVc.domRestrict.comp continuous_projIcc
-  have hve (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : v u = V u := by
-    dsimp [v]; rw [projIcc_of_mem _ hu]
-  let R : ℝ → ℝ := fun u => 2*z.im*Real.exp (-2*A*(∫ s in (0 : ℝ)..u, v s^2))
-  let r : ℝ → ℝ := fun u => ‖f u-g u‖/‖f u-star (g u)‖
-  have hR (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt R (-2*A*(V u)^2*R u) (Icc (0 : ℝ) T) u := by
-    have hi := intervalIntegral.integral_hasDerivAt_right ((hv.pow 2).intervalIntegrable 0 u)
-      (hv.pow 2).aestronglyMeasurable.stronglyMeasurableAtFilter (hv.pow 2).continuousAt
-    have hh := (((hi.const_mul (-2*A)).exp).const_mul (2*z.im)).hasDerivWithinAt (s := Icc (0 : ℝ) T)
-    convert hh using 1
-    try rfl
-    dsimp only [R,Pi.pow_apply]
-    rw [hve u hu]
-    ring
-  have hr (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt r (2*A*V u*W u*r u) (Icc (0 : ℝ) T) u :=
-    odeUntil_pair_ratio_deriv hb hf hg hzw hu
-  have hl (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : r u < 1 := by
-    have hlt := upper_pair_norm_lt (hf.2 u hu).1 (hg.2 u hu).1
-    exact (div_lt_one ((norm_nonneg _).trans_lt hlt)).mpr hlt
-  have hH (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      V u*(1-r u) ≤ W u*(1+r u) := by
-    have hh := upper_velocity_harnack
-      (z := f u-(b u.toNNReal : ℂ)) (w := g u-(b u.toNNReal : ℂ))
-      (by simpa using (hf.2 u hu).1) (by simpa using (hg.2 u hu).1)
-    have he : f u-(b u.toNNReal : ℂ)-(g u-(b u.toNNReal : ℂ)) = f u-g u := by ring
-    have he' : f u-(b u.toNNReal : ℂ)-star (g u-(b u.toNNReal : ℂ)) = f u-star (g u) := by simp
-    simp only [he,he',Complex.sub_im,Complex.ofReal_im,sub_zero] at hh
-    have hD := (norm_nonneg _).trans_lt (upper_pair_norm_lt (hf.2 u hu).1 (hg.2 u hu).1)
-    have hdiv := (div_le_div_iff_of_pos_right hD).mpr hh
-    change V u*(‖f u-star (g u)‖-‖f u-g u‖)/‖f u-star (g u)‖ ≤
-      W u*(‖f u-star (g u)‖+‖f u-g u‖)/‖f u-star (g u)‖ at hdiv
-    calc
-      V u*(1-r u) = V u*(‖f u-star (g u)‖-‖f u-g u‖)/‖f u-star (g u)‖ := by
-        change _*(1-‖f u-g u‖/‖f u-star (g u)‖) = _
-        simp only [mul_sub,sub_div,mul_div_assoc,div_self hD.ne',mul_one]
-      _ ≤ _ := hdiv
-      _ = W u*(1+r u) := by
-        change _ = W u*(1+‖f u-g u‖/‖f u-star (g u)‖)
-        simp only [mul_add,add_div,mul_div_assoc,div_self hD.ne',mul_one]
-  let K : ℝ → ℝ := fun u => R u*r u/(1-r u)^2
-  let K' : ℝ → ℝ := fun u => 2*A*R u*r u*V u/(1-r u)^3 *
-    (W u*(1+r u)-V u*(1-r u))
-  have hK (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
-      HasDerivWithinAt K (K' u) (Icc (0 : ℝ) T) u :=
-    growth_factor_deriv (hR u hu) (hr u hu) (ne_of_lt (hl u hu))
-  have hpos (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : 0 ≤ K' u := by
-    have hRp : 0 < R u := by dsimp [R]; positivity
-    have hrp : 0 ≤ r u := div_nonneg (norm_nonneg _) (norm_nonneg _)
-    have hVp : 0 < V u := div_pos (hf.2 u hu).1
-      (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf hu))
-    have hlp : 0 < 1-r u := sub_pos.mpr (hl u hu)
-    have hHp : 0 ≤ W u*(1+r u)-V u*(1-r u) := sub_nonneg.mpr (hH u hu)
-    dsimp [K']; positivity
-  have hmono : MonotoneOn K (Icc (0 : ℝ) T) :=
-    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _) (HasDerivWithinAt.continuousOn hK)
-      (fun u hu => (hK u (interior_subset hu)).mono interior_subset)
-      (fun u hu => hpos u (interior_subset hu))
-  have hstart : ‖z-w‖ ≤ K 0 := by
-    simpa [K,R,r,hf.1,hg.1] using upper_initial_growth hyz hyw
-  have hstop := hstart.trans (hmono (by simp) (by simp) T.coe_nonneg)
-  have heR : R T = 2*(loewnerMap A b T z).im/‖deriv (loewnerMap A b T) z‖ := by
-    have hi : (∫ u in (0 : ℝ)..(T : ℝ), v u^2) = ∫ u in (0 : ℝ)..(T : ℝ),
-      ((loewnerMap A b u.toNNReal z-(b u.toNNReal : ℂ)).im/
-        Complex.normSq (loewnerMap A b u.toNNReal z-(b u.toNNReal : ℂ)))^2 := by
-      apply intervalIntegral.integral_congr
-      intro u hu
-      have hu' : u ∈ Icc (0 : ℝ) T := by simpa only [uIcc_of_le T.coe_nonneg] using hu
-      dsimp only
-      rw [hve u hu',loewnerMap_eq_solution_time hf hu']
-      simp [V]
-    dsimp only [R]
-    rw [hi]
-    have hh := loewner_radius_ratio hb hz
-    rw [show 2*(loewnerMap A b T z).im/‖deriv (loewnerMap A b T) z‖ =
-      2*((loewnerMap A b T z).im/‖deriv (loewnerMap A b T) z‖) by ring,hh]
-    ring
-  dsimp only [K,r] at hstop
-  rw [heR,← loewnerMap_eq hf,← loewnerMap_eq hg] at hstop
-  exact hstop
-
-end ExactSLEMain
-open MeasureTheory Set
-namespace ExactSLEMain
-
-lemma interaction_integral_strong {a c S T : ℝ}
-    (ha : 0 < a) (hc : 0 ≤ c) (hac : 2*c < a) (hST : S ≤ T)
-    {L s V W : ℝ → ℝ}
-    (hV : ContinuousOn V (Icc S T)) (hW : ContinuousOn W (Icc S T))
-    (hL : ∀ t ∈ Icc S T, HasDerivWithinAt L (-2*a*V t*W t) (Icc S T) t)
-    (hs : ∀ t ∈ Icc S T, HasDerivWithinAt s ((V t)^2+(W t)^2) (Icc S T) t)
-    (hs0 : ∀ t ∈ Icc S T, 0 ≤ s t)
-    (hL0 : ∀ t ∈ Icc S T, 0 ≤ L t)
-    (hdecay : ∀ t ∈ Icc S T, L t ≤ Real.exp (-a*s t/2)) :
-    (∫ t in S..T, Real.exp (c*s t)*V t*W t) ≤ 1/(2*(a-2*c)) := by
-  let k := a/2-c
-  have hk : 0 < k := by dsimp [k]; linarith
-  have hsc := HasDerivWithinAt.continuousOn hs
-  let q : ℝ → ℝ := fun t => Real.exp (c*s t)*V t*W t
-  have hq : ContinuousOn q (Icc S T) :=
-    ((Real.continuous_exp.comp_continuousOn (hsc.const_mul c)).mul hV).mul hW
-  let q₀ : ℝ → ℝ := fun t => q (projIcc S T hST t)
-  have hq₀ : Continuous q₀ := hq.domRestrict.comp continuous_projIcc
-  have hqe (t : ℝ) (ht : t ∈ Icc S T) : q₀ t = q t := by
-    dsimp [q₀]; rw [projIcc_of_mem _ ht]
-  let F : ℝ → ℝ := fun t => Real.exp (c*s t)*L t + c/k*Real.exp (-k*s t) +
-    2*a*(∫ v in S..t, q₀ v)
-  let F' : ℝ → ℝ := fun t => c*((V t)^2+(W t)^2)*
-    (Real.exp (c*s t)*L t-Real.exp (-k*s t))
-  have hd (t : ℝ) (ht : t ∈ Icc S T) :
-      HasDerivWithinAt F (F' t) (Icc S T) t := by
-    have hi := intervalIntegral.integral_hasDerivAt_right (hq₀.intervalIntegrable S t)
-      hq₀.aestronglyMeasurable.stronglyMeasurableAtFilter hq₀.continuousAt
-    have hh := ((((hs t ht).const_mul c).exp.mul (hL t ht)).add
-      ((((hs t ht).const_mul (-k)).exp).const_mul (c/k))).add
-      (hi.const_mul (2*a)).hasDerivWithinAt
-    convert hh using 1
-    try rfl
-    dsimp only [F',q]
-    rw [hqe t ht]
-    dsimp only [q]
-    field_simp
-    ring
-  have hbound (t : ℝ) (ht : t ∈ Icc S T) :
-      Real.exp (c*s t)*L t ≤ Real.exp (-k*s t) := by
-    calc
-      _ ≤ Real.exp (c*s t)*Real.exp (-a*s t/2) :=
-        mul_le_mul_of_nonneg_left (hdecay t ht) (Real.exp_pos _).le
-      _ = Real.exp (-k*s t) := by rw [←Real.exp_add]; congr 1; dsimp [k]; ring
-  have hn (t : ℝ) (ht : t ∈ Icc S T) : F' t ≤ 0 := by
-    exact mul_nonpos_of_nonneg_of_nonpos (mul_nonneg hc (add_nonneg (sq_nonneg _) (sq_nonneg _)))
-      (sub_nonpos.mpr (hbound t ht))
-  have hanti : AntitoneOn F (Icc S T) :=
-    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _) (HasDerivWithinAt.continuousOn hd)
-      (fun t ht => (hd t (interior_subset ht)).mono interior_subset)
-      (fun t ht => hn t (interior_subset ht))
-  have htS : S ∈ Icc S T := ⟨le_rfl,hST⟩
-  have htT : T ∈ Icc S T := ⟨hST,le_rfl⟩
-  have hS : F S ≤ 1+c/k := by
-    have hex : Real.exp (-k*s S) ≤ 1 :=
-      Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hk.le) (hs0 S htS))
-    have hb := hbound S htS
-    have hc' := mul_le_mul_of_nonneg_left hex (div_nonneg hc hk.le)
-    dsimp only [F]
-    simp only [intervalIntegral.integral_same,mul_zero,add_zero]
-    linarith
-  have hT : 2*a*(∫ v in S..T, q₀ v) ≤ F T := by
-    have h1 := mul_nonneg (Real.exp_pos (c*s T)).le (hL0 T htT)
-    have h2 := mul_nonneg (div_nonneg hc hk.le) (Real.exp_pos (-k*s T)).le
-    dsimp only [F]; linarith
-  have hall := hT.trans ((hanti htS htT hST).trans hS)
-  have hie : (∫ v in S..T, q₀ v) = ∫ v in S..T, Real.exp (c*s v)*V v*W v := by
-    apply intervalIntegral.integral_congr
-    intro t ht
-    exact hqe t (by simpa only [uIcc_of_le hST] using ht)
-  rw [hie] at hall
-  have hdiv : (∫ v in S..T, Real.exp (c*s v)*V v*W v) ≤ (1+c/k)/(2*a) :=
-    (le_div_iff₀ (show 0 < 2*a by positivity)).mpr (by convert hall using 1; ring)
-  have he : (1+c/k)/(2*a) = 1/(2*(a-2*c)) := by
-    have hh : a-2*c = 2*k := by dsimp [k]; ring
-    rw [hh]
-    field_simp
-    dsimp [k]
-    ring
-  rwa [he] at hdiv
-
-lemma interaction_constant_comparison {a c : ℝ} (ha : 0 < a) (hc : 0 ≤ c)
-    (hac : 2*c < a) :
-    1/(2*(a-2*c)) ≤ Real.exp (2*c)/(2*a*(1-Real.exp (2*c-a))) := by
-  have hδ : 0 < a-2*c := sub_pos.mpr hac
-  have he : 0 < 1-Real.exp (2*c-a) := sub_pos.mpr
-    (Real.exp_lt_one_iff.mpr (sub_neg.mpr hac))
-  have hp := Real.exp_pos (2*c-a)
-  have h1 := Real.add_one_le_exp (2*c-a)
-  have h2 := Real.add_one_le_exp (2*c)
-  have h3 : (a-2*c)*(1-Real.exp (2*c-a)) ≤ a-2*c := by
-    nlinarith [mul_pos hδ hp]
-  have h4 : 2*c*(1-Real.exp (2*c-a)) ≤ 2*c*(a-2*c) := by
-    exact mul_le_mul_of_nonneg_left (by linarith) (by positivity)
-  have h5 := mul_le_mul_of_nonneg_left h2 hδ.le
-  apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr
-  nlinarith
-
-theorem weighted_interaction_integral {a c S T : ℝ}
-    (ha : 0 < a) (hc : 0 ≤ c) (hac : 2*c < a) (hST : S ≤ T)
-    {L s V W : ℝ → ℝ}
-    (hV : ContinuousOn V (Icc S T)) (hW : ContinuousOn W (Icc S T))
-    (hL : ∀ t ∈ Icc S T, HasDerivWithinAt L (-2*a*V t*W t) (Icc S T) t)
-    (hs : ∀ t ∈ Icc S T, HasDerivWithinAt s ((V t)^2+(W t)^2) (Icc S T) t)
-    (hs0 : ∀ t ∈ Icc S T, 0 ≤ s t)
-    (hL0 : ∀ t ∈ Icc S T, 0 ≤ L t)
-    (hdecay : ∀ t ∈ Icc S T, L t ≤ Real.exp (-a*s t/2)) :
-    (∫ t in S..T, Real.exp (c*s t)*V t*W t) ≤
-      Real.exp (2*c)/(2*a*(1-Real.exp (2*c-a))) :=
-  (interaction_integral_strong ha hc hac hST hV hW hL hs hs0 hL0 hdecay).trans
-    (interaction_constant_comparison ha hc hac)
-
-end ExactSLEMain
-
-namespace ExactSLEMain
- 
-
-noncomputable def fullLoewnerRadius (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ≥0) : ℝ :=
-  2*(loewnerMap A b t z).im/‖deriv (loewnerMap A b t) z‖
- 
-
-noncomputable def odeSpeed (b : ℝ≥0 → ℝ) (T : ℝ≥0) (f : ℝ → ℂ) (t : ℝ) : ℝ :=
-  let u : ℝ := projIcc (0:ℝ) T T.coe_nonneg t
-  (f u).im/Complex.normSq (f u-(b u.toNNReal : ℂ))
-noncomputable def activeODEClock (b : ℝ≥0 → ℝ) (T σ : ℝ≥0) (f : ℝ → ℂ) (t : ℝ) : ℝ :=
-  ∫ u in (σ:ℝ)..t, odeSpeed b T f u^2
-end ExactSLEMain
-namespace ExactSLEMain
-lemma odeSpeed_eq {b : ℝ≥0 → ℝ} {T : ℝ≥0} (f : ℝ → ℂ) {t : ℝ}
-    (ht : t∈Icc (0:ℝ) T) :
-    odeSpeed b T f t=(f t).im/Complex.normSq (f t-(b t.toNNReal : ℂ)) := by
-  simp only [odeSpeed,projIcc_of_mem _ ht]
-lemma odeSpeed_continuous {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f) :
-    Continuous (odeSpeed b T f) := by
-  have hv : ContinuousOn (fun t => (f t).im/Complex.normSq (f t-(b t.toNNReal : ℂ))) (Icc (0:ℝ) T) :=
-    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)).div
-      (Complex.continuous_normSq.comp_continuousOn (odeUntil_center_continuousOn hb hf))
-      (fun t ht => (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf ht)).ne')
-  exact hv.domRestrict.comp continuous_projIcc
-lemma fullRadius_primitive {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
-    {t : ℝ≥0} (ht : t≤T) :
-    fullLoewnerRadius a b z t=2*z.im*Real.exp
-      (-2*a*(∫ u in (0:ℝ)..(t:ℝ), odeSpeed b T f u^2)) := by
-  have hz : alive a b t z := ⟨f,odeUntil_mono hf ht⟩
-  have he : (∫ u in (0:ℝ)..(t:ℝ), odeSpeed b T f u^2) =
-      ∫ u in (0:ℝ)..(t:ℝ),
-        ((loewnerMap a b u.toNNReal z-(b u.toNNReal : ℂ)).im/
-          Complex.normSq (loewnerMap a b u.toNNReal z-(b u.toNNReal : ℂ)))^2 := by
-    apply intervalIntegral.integral_congr
-    intro u hu
-    have hu0 : u∈Icc (0:ℝ) t := by simpa only [uIcc_of_le t.coe_nonneg] using hu
-    have hu' : u∈Icc (0:ℝ) T := ⟨hu0.1,hu0.2.trans (by exact_mod_cast ht)⟩
-    dsimp only
-    rw [odeSpeed_eq f hu',loewnerMap_eq_solution_time hf hu']
-    simp
-  unfold fullLoewnerRadius
-  rw [show 2*(loewnerMap a b t z).im/‖deriv (loewnerMap a b t) z‖=
-    2*((loewnerMap a b t z).im/‖deriv (loewnerMap a b t) z‖) by ring,
-    loewner_radius_ratio hb hz,he]
-  ring
-lemma fullRadius_pos {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
-    {t : ℝ≥0} (ht : t≤T) : 0<fullLoewnerRadius a b z t := by
-  have hz : 0<z.im := by simpa only [hf.1] using (hf.2 0 (by simp)).1
-  rw [fullRadius_primitive hb hf ht]
-  positivity
-lemma fullRadius_active_clock {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
-    {σ t : ℝ≥0} (hσ : σ≤T) (ht : t≤T) :
-    fullLoewnerRadius a b z t=fullLoewnerRadius a b z σ*
-      Real.exp (-2*a*activeODEClock b T σ f t) := by
-  have hc := (odeSpeed_continuous hb hf).pow 2
-  have he := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
-    (hc.intervalIntegrable 0 (σ:ℝ)) (hc.intervalIntegrable (σ:ℝ) (t:ℝ))
-  simp only [Pi.pow_apply] at he
-  rw [fullRadius_primitive hb hf ht,fullRadius_primitive hb hf hσ]
-  unfold activeODEClock
-  rw [mul_assoc (2*z.im),← Real.exp_add,← mul_add,he]
-lemma activeODEClock_deriv {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
-    (σ : ℝ≥0) (t : ℝ) :
-    HasDerivAt (activeODEClock b T σ f) (odeSpeed b T f t^2) t := by
-  have hc := (odeSpeed_continuous hb hf).pow 2
-  exact intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable (σ:ℝ) t)
-    hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-lemma activeODEClock_nonneg {b : ℝ≥0 → ℝ} {T σ : ℝ≥0} {f : ℝ → ℂ} {t : ℝ}
-    (hσ : (σ:ℝ)≤t) : 0≤activeODEClock b T σ f t := by
-  exact intervalIntegral.integral_nonneg hσ (fun _ _ => sq_nonneg _)
-end ExactSLEMain
-namespace ExactSLEMain
-lemma neg_log_le_sqrt_ratio {r : ℝ} (hr : 0<r) (hr1 : r≤1) :
-    -Real.log r≤(1-r)/Real.sqrt r := by
-  have hl : 0≤-Real.log r/2 := by linarith [Real.log_nonpos hr.le hr1]
-  have hs := Real.self_le_sinh_iff.mpr hl
-  have he : Real.exp (Real.log r/2)=Real.sqrt r := by rw [Real.exp_half,Real.exp_log hr]
-  have he' : Real.exp (-Real.log r/2)=(Real.sqrt r)⁻¹ := by
-    rw [neg_div,Real.exp_neg,he]
-  rw [Real.sinh_eq,he',show -(-Real.log r/2)=Real.log r/2 by ring,he] at hs
-  have hsq := Real.sq_sqrt hr.le
-  exact (le_div_iff₀ (Real.sqrt_pos.mpr hr)).mpr (by
-    have hh := mul_le_mul_of_nonneg_right hs (Real.sqrt_nonneg r)
-    field_simp at hh
-    nlinarith)
-lemma log_sq_mul_le {r : ℝ} (hr : 0<r) (hr1 : r≤1) :
-    (Real.log r)^2*r≤(1-r)^2 := by
-  have h := (le_div_iff₀ (Real.sqrt_pos.mpr hr)).mp (neg_log_le_sqrt_ratio hr hr1)
-  have hl : 0≤-Real.log r := neg_nonneg.mpr (Real.log_nonpos hr.le hr1)
-  have hprod := mul_nonneg hl (Real.sqrt_nonneg r)
-  have hsq := mul_self_le_mul_self hprod h
-  nlinarith [Real.sq_sqrt hr.le]
-end ExactSLEMain
-namespace ExactSLEMain
-lemma ode_pair_ratio_pos_lt_one {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
-    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
-    {t : ℝ} (ht : t∈Icc (0:ℝ) T) :
-    0<‖f t-g t‖/‖f t-star (g t)‖ ∧ ‖f t-g t‖/‖f t-star (g t)‖<1 := by
-  have hlt := upper_pair_norm_lt (hf.2 t ht).1 (hg.2 t ht).1
-  have hden : 0<‖f t-star (g t)‖ := (norm_nonneg _).trans_lt hlt
-  exact ⟨div_pos (norm_pos_iff.mpr (odeUntil_pair_ne hb hf hg hzw ht)) hden,
-    (div_lt_one hden).mpr hlt⟩
-lemma actual_pair_clock_decay {a : ℝ} (ha : 0<a)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z w : ℂ} {f g : ℝ → ℂ}
-    {T σ t : ℝ≥0} (hσ : σ≤T) (ht : t≤T)
-    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
-    (hcap : fullLoewnerRadius a b z σ≤‖z-w‖) :
-    -Real.log (‖f t-g t‖/‖f t-star (g t)‖)≤
-      Real.exp (-a*activeODEClock b T σ f t) := by
-  let r := ‖f t-g t‖/‖f t-star (g t)‖
-  have ht' : (t:ℝ)∈Icc (0:ℝ) T := ⟨t.coe_nonneg,by exact_mod_cast ht⟩
-  have hr := ode_pair_ratio_pos_lt_one hb hf hg hzw ht'
-  have hgrowth := loewner_two_point_growth ha hb
-    (show alive a b t z from ⟨f,odeUntil_mono hf ht⟩)
-    (show alive a b t w from ⟨g,odeUntil_mono hg ht⟩) hzw
-  change ‖z-w‖ ≤ fullLoewnerRadius a b z t * _ / _ at hgrowth
-  rw [loewnerMap_eq (odeUntil_mono hf ht),loewnerMap_eq (odeUntil_mono hg ht)] at hgrowth
-  change ‖z-w‖≤fullLoewnerRadius a b z t*r/(1-r)^2 at hgrowth
-  have hden : 0<(1-r)^2 := sq_pos_of_pos (sub_pos.mpr hr.2)
-  have hgrowth' := (le_div_iff₀ hden).mp hgrowth
-  rw [fullRadius_active_clock hb hf hσ ht] at hgrowth'
-  have hR := fullRadius_pos hb hf hσ
-  have hmul : fullLoewnerRadius a b z σ*(1-r)^2 ≤
-      fullLoewnerRadius a b z σ*(Real.exp (-2*a*activeODEClock b T σ f t)*r) :=
-    (mul_le_mul_of_nonneg_right hcap hden.le).trans (by simpa only [mul_assoc] using hgrowth')
-  have hbound := (mul_le_mul_iff_right₀ hR).mp hmul
-  have hsq : (Real.log r)^2≤Real.exp (-2*a*activeODEClock b T σ f t) :=
-    (mul_le_mul_iff_left₀ hr.1).mp ((log_sq_mul_le hr.1 hr.2.le).trans hbound)
-  have he : Real.exp (-2*a*activeODEClock b T σ f t) =
-      (Real.exp (-a*activeODEClock b T σ f t))^2 := by
-    rw [← Real.exp_nat_mul]
-    congr 1
-    ring
-  rw [he] at hsq
-  change -Real.log r≤_
-  nlinarith [Real.exp_pos (-a*activeODEClock b T σ f t)]
-lemma ode_pair_log_deriv {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
-    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
-    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
-    {t : ℝ} (ht : t∈Icc (0:ℝ) T) :
-    HasDerivWithinAt (fun u => -Real.log (‖f u-g u‖/‖f u-star (g u)‖))
-      (-2*a*odeSpeed b T f t*odeSpeed b T g t) (Icc (0:ℝ) T) t := by
-  have hp := (ode_pair_ratio_pos_lt_one hb hf hg hzw ht).1
-  have hd := ((odeUntil_pair_ratio_deriv hb hf hg hzw ht).log hp.ne').neg
-  rw [mul_div_cancel_right₀ _ hp.ne'] at hd
-  convert hd using 1
-  try rfl
-  rw [odeSpeed_eq f ht,odeSpeed_eq g ht]
-  ring
-end ExactSLEMain
-namespace ExactSLEMain
- 
-
-theorem actual_pair_interaction {a c : ℝ} (ha : 0<a) (hc : 0≤c) (hac : 2*c<a)
-    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z w : ℂ} {f g : ℝ → ℂ}
-    {T S σ τ : ℝ≥0} (hST : S≤T) (hσ : σ≤S) (hτ : τ≤S)
-    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
-    (hcapz : fullLoewnerRadius a b z σ≤‖z-w‖)
-    (hcapw : fullLoewnerRadius a b w τ≤‖z-w‖) :
-    (∫ t in (S:ℝ)..(T:ℝ), Real.exp (c*(activeODEClock b T σ f t+
-      activeODEClock b T τ g t))*odeSpeed b T f t*odeSpeed b T g t) ≤
-      Real.exp (2*c)/(2*a*(1-Real.exp (2*c-a))) := by
-  let L := fun t : ℝ => -Real.log (‖f t-g t‖/‖f t-star (g t)‖)
-  let u := activeODEClock b T σ f
-  let v := activeODEClock b T τ g
-  have hST' : (S:ℝ)≤T := by exact_mod_cast hST
-  have hsub : Icc (S:ℝ) T ⊆ Icc (0:ℝ) T := Icc_subset_Icc_left S.coe_nonneg
-  have hnon (t : ℝ) (ht : t∈Icc (S:ℝ) T) : 0≤L t := by
-    have hr := ode_pair_ratio_pos_lt_one hb hf hg hzw (hsub ht)
-    exact neg_nonneg.mpr (Real.log_nonpos hr.1.le hr.2.le)
-  have hdecay (t : ℝ) (ht : t∈Icc (S:ℝ) T) :
-      L t≤Real.exp (-a*(u t+v t)/2) := by
-    have ht0 : 0≤t := (hsub ht).1
-    have htT : t.toNNReal≤T := by
-      exact_mod_cast (show (t.toNNReal:ℝ)≤T by simpa only [Real.coe_toNNReal _ ht0] using ht.2)
-    have hz := actual_pair_clock_decay ha hb (hσ.trans hST) htT hf hg hzw hcapz
-    have hcapw' : fullLoewnerRadius a b w τ≤‖w-z‖ := by simpa only [norm_sub_rev] using hcapw
-    have hw := actual_pair_clock_decay ha hb (hτ.trans hST) htT hg hf hzw.symm hcapw'
-    rw [Real.coe_toNNReal _ ht0] at hz hw
-    have hnorm : ‖g t-star (f t)‖=‖f t-star (g t)‖ := by
-      calc
-        _ = ‖star (g t-star (f t))‖ := (norm_star _).symm
-        _ = ‖star (g t)-f t‖ := by rw [star_sub,star_star]
-        _ = _ := norm_sub_rev _ _
-    rw [hnorm,norm_sub_rev (g t) (f t)] at hw
-    change L t≤Real.exp (-a*u t) at hz
-    change L t≤Real.exp (-a*v t) at hw
-    rcases le_total (u t) (v t) with h|h
-    · exact hw.trans (Real.exp_le_exp.mpr (by nlinarith))
-    · exact hz.trans (Real.exp_le_exp.mpr (by nlinarith))
-  exact weighted_interaction_integral ha hc hac hST'
-    (L:=L) (s:=fun t => u t+v t) (V:=odeSpeed b T f) (W:=odeSpeed b T g)
-    (odeSpeed_continuous hb hf).continuousOn (odeSpeed_continuous hb hg).continuousOn
-    (fun t ht => (ode_pair_log_deriv hb hf hg hzw (hsub ht)).mono hsub)
-    (fun t _ => ((activeODEClock_deriv hb hf σ t).add
-      (activeODEClock_deriv hb hg τ t)).hasDerivWithinAt)
-    (fun t ht => add_nonneg (activeODEClock_nonneg ((by exact_mod_cast hσ : (σ:ℝ)≤S).trans ht.1))
-      (activeODEClock_nonneg ((by exact_mod_cast hτ : (τ:ℝ)≤S).trans ht.1))) hnon hdecay
-end ExactSLEMain
-namespace ExactSLEGreen
-open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
-end ExactSLEGreen
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLEGauge.StateBounds
+
+namespace OAI
+
+/-! Hausdorff gauges, Loewner evolution and trace measures. -/
+
+section
+open MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+open scoped Topology
+open Set
+open MeasureTheory Filter
+open scoped NNReal Topology
+open MeasureTheory Filter Set Metric Function ODE
+open MeasureTheory ProbabilityTheory Filter Set
+open Filter
+open Filter Set
+namespace ExactSLEMain
+open Set
+theorem loewner_two_point_growth {A : ℝ} (hA : 0 < A)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {T : ℝ≥0} {z w : ℂ}
+    (hz : alive A b T z) (hw : alive A b T w) (hzw : z ≠ w) :
+    ‖z-w‖ ≤ (2 * (loewnerMap A b T z).im / ‖deriv (loewnerMap A b T) z‖) *
+      (‖loewnerMap A b T z-loewnerMap A b T w‖ /
+       ‖loewnerMap A b T z-star (loewnerMap A b T w)‖) /
+      (1-‖loewnerMap A b T z-loewnerMap A b T w‖ /
+       ‖loewnerMap A b T z-star (loewnerMap A b T w)‖)^2 := by
+  obtain ⟨f,hf⟩ := hz
+  obtain ⟨g,hg⟩ := hw
+  have hz : alive A b T z := ⟨f,hf⟩
+  have hw : alive A b T w := ⟨g,hg⟩
+  have hyz : 0 < z.im := by simpa only [hf.1] using (hf.2 0 (by simp)).1
+  have hyw : 0 < w.im := by simpa only [hg.1] using (hg.2 0 (by simp)).1
+  let V : ℝ → ℝ := fun u => (f u).im/Complex.normSq (f u-(b u.toNNReal : ℂ))
+  let W : ℝ → ℝ := fun u => (g u).im/Complex.normSq (g u-(b u.toNNReal : ℂ))
+  have hVc : ContinuousOn V (Icc (0 : ℝ) T) :=
+    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)).div
+      (Complex.continuous_normSq.comp_continuousOn (odeUntil_center_continuousOn hb hf))
+      (fun u hu => (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf hu)).ne')
+  let v : ℝ → ℝ := fun u => V (projIcc (0 : ℝ) T T.coe_nonneg u)
+  have hv : Continuous v := hVc.domRestrict.comp continuous_projIcc
+  have hve (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : v u = V u := by
+    dsimp [v]; rw [projIcc_of_mem _ hu]
+  let R : ℝ → ℝ := fun u => 2*z.im*Real.exp (-2*A*(∫ s in (0 : ℝ)..u, v s^2))
+  let r : ℝ → ℝ := fun u => ‖f u-g u‖/‖f u-star (g u)‖
+  have hR (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt R (-2*A*(V u)^2*R u) (Icc (0 : ℝ) T) u := by
+    have hi := intervalIntegral.integral_hasDerivAt_right ((hv.pow 2).intervalIntegrable 0 u)
+      (hv.pow 2).aestronglyMeasurable.stronglyMeasurableAtFilter (hv.pow 2).continuousAt
+    have hh := (((hi.const_mul (-2*A)).exp).const_mul (2*z.im)).hasDerivWithinAt (s := Icc (0 : ℝ) T)
+    convert hh using 1
+    try rfl
+    dsimp only [R,Pi.pow_apply]
+    rw [hve u hu]
+    ring
+  have hr (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt r (2*A*V u*W u*r u) (Icc (0 : ℝ) T) u :=
+    odeUntil_pair_ratio_deriv hb hf hg hzw hu
+  have hl (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : r u < 1 := by
+    have hlt := upper_pair_norm_lt (hf.2 u hu).1 (hg.2 u hu).1
+    exact (div_lt_one ((norm_nonneg _).trans_lt hlt)).mpr hlt
+  have hH (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      V u*(1-r u) ≤ W u*(1+r u) := by
+    have hh := upper_velocity_harnack
+      (z := f u-(b u.toNNReal : ℂ)) (w := g u-(b u.toNNReal : ℂ))
+      (by simpa using (hf.2 u hu).1) (by simpa using (hg.2 u hu).1)
+    have he : f u-(b u.toNNReal : ℂ)-(g u-(b u.toNNReal : ℂ)) = f u-g u := by ring
+    have he' : f u-(b u.toNNReal : ℂ)-star (g u-(b u.toNNReal : ℂ)) = f u-star (g u) := by simp
+    simp only [he,he',Complex.sub_im,Complex.ofReal_im,sub_zero] at hh
+    have hD := (norm_nonneg _).trans_lt (upper_pair_norm_lt (hf.2 u hu).1 (hg.2 u hu).1)
+    have hdiv := (div_le_div_iff_of_pos_right hD).mpr hh
+    change V u*(‖f u-star (g u)‖-‖f u-g u‖)/‖f u-star (g u)‖ ≤
+      W u*(‖f u-star (g u)‖+‖f u-g u‖)/‖f u-star (g u)‖ at hdiv
+    calc
+      V u*(1-r u) = V u*(‖f u-star (g u)‖-‖f u-g u‖)/‖f u-star (g u)‖ := by
+        change _*(1-‖f u-g u‖/‖f u-star (g u)‖) = _
+        simp only [mul_sub,sub_div,mul_div_assoc,div_self hD.ne',mul_one]
+      _ ≤ _ := hdiv
+      _ = W u*(1+r u) := by
+        change _ = W u*(1+‖f u-g u‖/‖f u-star (g u)‖)
+        simp only [mul_add,add_div,mul_div_assoc,div_self hD.ne',mul_one]
+  let K : ℝ → ℝ := fun u => R u*r u/(1-r u)^2
+  let K' : ℝ → ℝ := fun u => 2*A*R u*r u*V u/(1-r u)^3 *
+    (W u*(1+r u)-V u*(1-r u))
+  have hK (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) :
+      HasDerivWithinAt K (K' u) (Icc (0 : ℝ) T) u :=
+    growth_factor_deriv (hR u hu) (hr u hu) (ne_of_lt (hl u hu))
+  have hpos (u : ℝ) (hu : u ∈ Icc (0 : ℝ) T) : 0 ≤ K' u := by
+    have hRp : 0 < R u := by dsimp [R]; positivity
+    have hrp : 0 ≤ r u := div_nonneg (norm_nonneg _) (norm_nonneg _)
+    have hVp : 0 < V u := div_pos (hf.2 u hu).1
+      (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf hu))
+    have hlp : 0 < 1-r u := sub_pos.mpr (hl u hu)
+    have hHp : 0 ≤ W u*(1+r u)-V u*(1-r u) := sub_nonneg.mpr (hH u hu)
+    dsimp [K']; positivity
+  have hmono : MonotoneOn K (Icc (0 : ℝ) T) :=
+    monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _) (HasDerivWithinAt.continuousOn hK)
+      (fun u hu => (hK u (interior_subset hu)).mono interior_subset)
+      (fun u hu => hpos u (interior_subset hu))
+  have hstart : ‖z-w‖ ≤ K 0 := by
+    simpa [K,R,r,hf.1,hg.1] using upper_initial_growth hyz hyw
+  have hstop := hstart.trans (hmono (by simp) (by simp) T.coe_nonneg)
+  have heR : R T = 2*(loewnerMap A b T z).im/‖deriv (loewnerMap A b T) z‖ := by
+    have hi : (∫ u in (0 : ℝ)..(T : ℝ), v u^2) = ∫ u in (0 : ℝ)..(T : ℝ),
+      ((loewnerMap A b u.toNNReal z-(b u.toNNReal : ℂ)).im/
+        Complex.normSq (loewnerMap A b u.toNNReal z-(b u.toNNReal : ℂ)))^2 := by
+      apply intervalIntegral.integral_congr
+      intro u hu
+      have hu' : u ∈ Icc (0 : ℝ) T := by simpa only [uIcc_of_le T.coe_nonneg] using hu
+      dsimp only
+      rw [hve u hu',loewnerMap_eq_solution_time hf hu']
+      simp [V]
+    dsimp only [R]
+    rw [hi]
+    have hh := loewner_radius_ratio hb hz
+    rw [show 2*(loewnerMap A b T z).im/‖deriv (loewnerMap A b T) z‖ =
+      2*((loewnerMap A b T z).im/‖deriv (loewnerMap A b T) z‖) by ring,hh]
+    ring
+  dsimp only [K,r] at hstop
+  rw [heR,← loewnerMap_eq hf,← loewnerMap_eq hg] at hstop
+  exact hstop
+
+end ExactSLEMain
+open MeasureTheory Set
+namespace ExactSLEMain
+
+lemma interaction_integral_strong {a c S T : ℝ}
+    (ha : 0 < a) (hc : 0 ≤ c) (hac : 2*c < a) (hST : S ≤ T)
+    {L s V W : ℝ → ℝ}
+    (hV : ContinuousOn V (Icc S T)) (hW : ContinuousOn W (Icc S T))
+    (hL : ∀ t ∈ Icc S T, HasDerivWithinAt L (-2*a*V t*W t) (Icc S T) t)
+    (hs : ∀ t ∈ Icc S T, HasDerivWithinAt s ((V t)^2+(W t)^2) (Icc S T) t)
+    (hs0 : ∀ t ∈ Icc S T, 0 ≤ s t)
+    (hL0 : ∀ t ∈ Icc S T, 0 ≤ L t)
+    (hdecay : ∀ t ∈ Icc S T, L t ≤ Real.exp (-a*s t/2)) :
+    (∫ t in S..T, Real.exp (c*s t)*V t*W t) ≤ 1/(2*(a-2*c)) := by
+  let k := a/2-c
+  have hk : 0 < k := by dsimp [k]; linarith
+  have hsc := HasDerivWithinAt.continuousOn hs
+  let q : ℝ → ℝ := fun t => Real.exp (c*s t)*V t*W t
+  have hq : ContinuousOn q (Icc S T) :=
+    ((Real.continuous_exp.comp_continuousOn (hsc.const_mul c)).mul hV).mul hW
+  let q₀ : ℝ → ℝ := fun t => q (projIcc S T hST t)
+  have hq₀ : Continuous q₀ := hq.domRestrict.comp continuous_projIcc
+  have hqe (t : ℝ) (ht : t ∈ Icc S T) : q₀ t = q t := by
+    dsimp [q₀]; rw [projIcc_of_mem _ ht]
+  let F : ℝ → ℝ := fun t => Real.exp (c*s t)*L t + c/k*Real.exp (-k*s t) +
+    2*a*(∫ v in S..t, q₀ v)
+  let F' : ℝ → ℝ := fun t => c*((V t)^2+(W t)^2)*
+    (Real.exp (c*s t)*L t-Real.exp (-k*s t))
+  have hd (t : ℝ) (ht : t ∈ Icc S T) :
+      HasDerivWithinAt F (F' t) (Icc S T) t := by
+    have hi := intervalIntegral.integral_hasDerivAt_right (hq₀.intervalIntegrable S t)
+      hq₀.aestronglyMeasurable.stronglyMeasurableAtFilter hq₀.continuousAt
+    have hh := ((((hs t ht).const_mul c).exp.mul (hL t ht)).add
+      ((((hs t ht).const_mul (-k)).exp).const_mul (c/k))).add
+      (hi.const_mul (2*a)).hasDerivWithinAt
+    convert hh using 1
+    try rfl
+    dsimp only [F',q]
+    rw [hqe t ht]
+    dsimp only [q]
+    field_simp
+    ring
+  have hbound (t : ℝ) (ht : t ∈ Icc S T) :
+      Real.exp (c*s t)*L t ≤ Real.exp (-k*s t) := by
+    calc
+      _ ≤ Real.exp (c*s t)*Real.exp (-a*s t/2) :=
+        mul_le_mul_of_nonneg_left (hdecay t ht) (Real.exp_pos _).le
+      _ = Real.exp (-k*s t) := by rw [←Real.exp_add]; congr 1; dsimp [k]; ring
+  have hn (t : ℝ) (ht : t ∈ Icc S T) : F' t ≤ 0 := by
+    exact mul_nonpos_of_nonneg_of_nonpos (mul_nonneg hc (add_nonneg (sq_nonneg _) (sq_nonneg _)))
+      (sub_nonpos.mpr (hbound t ht))
+  have hanti : AntitoneOn F (Icc S T) :=
+    antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _) (HasDerivWithinAt.continuousOn hd)
+      (fun t ht => (hd t (interior_subset ht)).mono interior_subset)
+      (fun t ht => hn t (interior_subset ht))
+  have htS : S ∈ Icc S T := ⟨le_rfl,hST⟩
+  have htT : T ∈ Icc S T := ⟨hST,le_rfl⟩
+  have hS : F S ≤ 1+c/k := by
+    have hex : Real.exp (-k*s S) ≤ 1 :=
+      Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hk.le) (hs0 S htS))
+    have hb := hbound S htS
+    have hc' := mul_le_mul_of_nonneg_left hex (div_nonneg hc hk.le)
+    dsimp only [F]
+    simp only [intervalIntegral.integral_same,mul_zero,add_zero]
+    linarith
+  have hT : 2*a*(∫ v in S..T, q₀ v) ≤ F T := by
+    have h1 := mul_nonneg (Real.exp_pos (c*s T)).le (hL0 T htT)
+    have h2 := mul_nonneg (div_nonneg hc hk.le) (Real.exp_pos (-k*s T)).le
+    dsimp only [F]; linarith
+  have hall := hT.trans ((hanti htS htT hST).trans hS)
+  have hie : (∫ v in S..T, q₀ v) = ∫ v in S..T, Real.exp (c*s v)*V v*W v := by
+    apply intervalIntegral.integral_congr
+    intro t ht
+    exact hqe t (by simpa only [uIcc_of_le hST] using ht)
+  rw [hie] at hall
+  have hdiv : (∫ v in S..T, Real.exp (c*s v)*V v*W v) ≤ (1+c/k)/(2*a) :=
+    (le_div_iff₀ (show 0 < 2*a by positivity)).mpr (by convert hall using 1; ring)
+  have he : (1+c/k)/(2*a) = 1/(2*(a-2*c)) := by
+    have hh : a-2*c = 2*k := by dsimp [k]; ring
+    rw [hh]
+    field_simp
+    dsimp [k]
+    ring
+  rwa [he] at hdiv
+
+lemma interaction_constant_comparison {a c : ℝ} (ha : 0 < a) (hc : 0 ≤ c)
+    (hac : 2*c < a) :
+    1/(2*(a-2*c)) ≤ Real.exp (2*c)/(2*a*(1-Real.exp (2*c-a))) := by
+  have hδ : 0 < a-2*c := sub_pos.mpr hac
+  have he : 0 < 1-Real.exp (2*c-a) := sub_pos.mpr
+    (Real.exp_lt_one_iff.mpr (sub_neg.mpr hac))
+  have hp := Real.exp_pos (2*c-a)
+  have h1 := Real.add_one_le_exp (2*c-a)
+  have h2 := Real.add_one_le_exp (2*c)
+  have h3 : (a-2*c)*(1-Real.exp (2*c-a)) ≤ a-2*c := by
+    nlinarith [mul_pos hδ hp]
+  have h4 : 2*c*(1-Real.exp (2*c-a)) ≤ 2*c*(a-2*c) := by
+    exact mul_le_mul_of_nonneg_left (by linarith) (by positivity)
+  have h5 := mul_le_mul_of_nonneg_left h2 hδ.le
+  apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr
+  nlinarith
+
+theorem weighted_interaction_integral {a c S T : ℝ}
+    (ha : 0 < a) (hc : 0 ≤ c) (hac : 2*c < a) (hST : S ≤ T)
+    {L s V W : ℝ → ℝ}
+    (hV : ContinuousOn V (Icc S T)) (hW : ContinuousOn W (Icc S T))
+    (hL : ∀ t ∈ Icc S T, HasDerivWithinAt L (-2*a*V t*W t) (Icc S T) t)
+    (hs : ∀ t ∈ Icc S T, HasDerivWithinAt s ((V t)^2+(W t)^2) (Icc S T) t)
+    (hs0 : ∀ t ∈ Icc S T, 0 ≤ s t)
+    (hL0 : ∀ t ∈ Icc S T, 0 ≤ L t)
+    (hdecay : ∀ t ∈ Icc S T, L t ≤ Real.exp (-a*s t/2)) :
+    (∫ t in S..T, Real.exp (c*s t)*V t*W t) ≤
+      Real.exp (2*c)/(2*a*(1-Real.exp (2*c-a))) :=
+  (interaction_integral_strong ha hc hac hST hV hW hL hs hs0 hL0 hdecay).trans
+    (interaction_constant_comparison ha hc hac)
+
+end ExactSLEMain
+
+namespace ExactSLEMain
+ 
+
+noncomputable def fullLoewnerRadius (A : ℝ) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ≥0) : ℝ :=
+  2*(loewnerMap A b t z).im/‖deriv (loewnerMap A b t) z‖
+ 
+
+noncomputable def odeSpeed (b : ℝ≥0 → ℝ) (T : ℝ≥0) (f : ℝ → ℂ) (t : ℝ) : ℝ :=
+  let u : ℝ := projIcc (0:ℝ) T T.coe_nonneg t
+  (f u).im/Complex.normSq (f u-(b u.toNNReal : ℂ))
+noncomputable def activeODEClock (b : ℝ≥0 → ℝ) (T σ : ℝ≥0) (f : ℝ → ℂ) (t : ℝ) : ℝ :=
+  ∫ u in (σ:ℝ)..t, odeSpeed b T f u^2
+end ExactSLEMain
+namespace ExactSLEMain
+lemma odeSpeed_eq {b : ℝ≥0 → ℝ} {T : ℝ≥0} (f : ℝ → ℂ) {t : ℝ}
+    (ht : t∈Icc (0:ℝ) T) :
+    odeSpeed b T f t=(f t).im/Complex.normSq (f t-(b t.toNNReal : ℂ)) := by
+  simp only [odeSpeed,projIcc_of_mem _ ht]
+lemma odeSpeed_continuous {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f) :
+    Continuous (odeSpeed b T f) := by
+  have hv : ContinuousOn (fun t => (f t).im/Complex.normSq (f t-(b t.toNNReal : ℂ))) (Icc (0:ℝ) T) :=
+    (Complex.continuous_im.comp_continuousOn (odeUntil_continuousOn hf)).div
+      (Complex.continuous_normSq.comp_continuousOn (odeUntil_center_continuousOn hb hf))
+      (fun t ht => (Complex.normSq_pos.mpr (odeUntil_center_ne_zero hf ht)).ne')
+  exact hv.domRestrict.comp continuous_projIcc
+lemma fullRadius_primitive {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
+    {t : ℝ≥0} (ht : t≤T) :
+    fullLoewnerRadius a b z t=2*z.im*Real.exp
+      (-2*a*(∫ u in (0:ℝ)..(t:ℝ), odeSpeed b T f u^2)) := by
+  have hz : alive a b t z := ⟨f,odeUntil_mono hf ht⟩
+  have he : (∫ u in (0:ℝ)..(t:ℝ), odeSpeed b T f u^2) =
+      ∫ u in (0:ℝ)..(t:ℝ),
+        ((loewnerMap a b u.toNNReal z-(b u.toNNReal : ℂ)).im/
+          Complex.normSq (loewnerMap a b u.toNNReal z-(b u.toNNReal : ℂ)))^2 := by
+    apply intervalIntegral.integral_congr
+    intro u hu
+    have hu0 : u∈Icc (0:ℝ) t := by simpa only [uIcc_of_le t.coe_nonneg] using hu
+    have hu' : u∈Icc (0:ℝ) T := ⟨hu0.1,hu0.2.trans (by exact_mod_cast ht)⟩
+    dsimp only
+    rw [odeSpeed_eq f hu',loewnerMap_eq_solution_time hf hu']
+    simp
+  unfold fullLoewnerRadius
+  rw [show 2*(loewnerMap a b t z).im/‖deriv (loewnerMap a b t) z‖=
+    2*((loewnerMap a b t z).im/‖deriv (loewnerMap a b t) z‖) by ring,
+    loewner_radius_ratio hb hz,he]
+  ring
+lemma fullRadius_pos {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
+    {t : ℝ≥0} (ht : t≤T) : 0<fullLoewnerRadius a b z t := by
+  have hz : 0<z.im := by simpa only [hf.1] using (hf.2 0 (by simp)).1
+  rw [fullRadius_primitive hb hf ht]
+  positivity
+lemma fullRadius_active_clock {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
+    {σ t : ℝ≥0} (hσ : σ≤T) (ht : t≤T) :
+    fullLoewnerRadius a b z t=fullLoewnerRadius a b z σ*
+      Real.exp (-2*a*activeODEClock b T σ f t) := by
+  have hc := (odeSpeed_continuous hb hf).pow 2
+  have he := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
+    (hc.intervalIntegrable 0 (σ:ℝ)) (hc.intervalIntegrable (σ:ℝ) (t:ℝ))
+  simp only [Pi.pow_apply] at he
+  rw [fullRadius_primitive hb hf ht,fullRadius_primitive hb hf hσ]
+  unfold activeODEClock
+  rw [mul_assoc (2*z.im),← Real.exp_add,← mul_add,he]
+lemma activeODEClock_deriv {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z : ℂ} {T : ℝ≥0} {f : ℝ → ℂ} (hf : odeUntil a b z T f)
+    (σ : ℝ≥0) (t : ℝ) :
+    HasDerivAt (activeODEClock b T σ f) (odeSpeed b T f t^2) t := by
+  have hc := (odeSpeed_continuous hb hf).pow 2
+  exact intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable (σ:ℝ) t)
+    hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+lemma activeODEClock_nonneg {b : ℝ≥0 → ℝ} {T σ : ℝ≥0} {f : ℝ → ℂ} {t : ℝ}
+    (hσ : (σ:ℝ)≤t) : 0≤activeODEClock b T σ f t := by
+  exact intervalIntegral.integral_nonneg hσ (fun _ _ => sq_nonneg _)
+end ExactSLEMain
+namespace ExactSLEMain
+lemma neg_log_le_sqrt_ratio {r : ℝ} (hr : 0<r) (hr1 : r≤1) :
+    -Real.log r≤(1-r)/Real.sqrt r := by
+  have hl : 0≤-Real.log r/2 := by linarith [Real.log_nonpos hr.le hr1]
+  have hs := Real.self_le_sinh_iff.mpr hl
+  have he : Real.exp (Real.log r/2)=Real.sqrt r := by rw [Real.exp_half,Real.exp_log hr]
+  have he' : Real.exp (-Real.log r/2)=(Real.sqrt r)⁻¹ := by
+    rw [neg_div,Real.exp_neg,he]
+  rw [Real.sinh_eq,he',show -(-Real.log r/2)=Real.log r/2 by ring,he] at hs
+  have hsq := Real.sq_sqrt hr.le
+  exact (le_div_iff₀ (Real.sqrt_pos.mpr hr)).mpr (by
+    have hh := mul_le_mul_of_nonneg_right hs (Real.sqrt_nonneg r)
+    field_simp at hh
+    nlinarith)
+lemma log_sq_mul_le {r : ℝ} (hr : 0<r) (hr1 : r≤1) :
+    (Real.log r)^2*r≤(1-r)^2 := by
+  have h := (le_div_iff₀ (Real.sqrt_pos.mpr hr)).mp (neg_log_le_sqrt_ratio hr hr1)
+  have hl : 0≤-Real.log r := neg_nonneg.mpr (Real.log_nonpos hr.le hr1)
+  have hprod := mul_nonneg hl (Real.sqrt_nonneg r)
+  have hsq := mul_self_le_mul_self hprod h
+  nlinarith [Real.sq_sqrt hr.le]
+end ExactSLEMain
+namespace ExactSLEMain
+lemma ode_pair_ratio_pos_lt_one {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
+    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
+    {t : ℝ} (ht : t∈Icc (0:ℝ) T) :
+    0<‖f t-g t‖/‖f t-star (g t)‖ ∧ ‖f t-g t‖/‖f t-star (g t)‖<1 := by
+  have hlt := upper_pair_norm_lt (hf.2 t ht).1 (hg.2 t ht).1
+  have hden : 0<‖f t-star (g t)‖ := (norm_nonneg _).trans_lt hlt
+  exact ⟨div_pos (norm_pos_iff.mpr (odeUntil_pair_ne hb hf hg hzw ht)) hden,
+    (div_lt_one hden).mpr hlt⟩
+lemma actual_pair_clock_decay {a : ℝ} (ha : 0<a)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z w : ℂ} {f g : ℝ → ℂ}
+    {T σ t : ℝ≥0} (hσ : σ≤T) (ht : t≤T)
+    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
+    (hcap : fullLoewnerRadius a b z σ≤‖z-w‖) :
+    -Real.log (‖f t-g t‖/‖f t-star (g t)‖)≤
+      Real.exp (-a*activeODEClock b T σ f t) := by
+  let r := ‖f t-g t‖/‖f t-star (g t)‖
+  have ht' : (t:ℝ)∈Icc (0:ℝ) T := ⟨t.coe_nonneg,by exact_mod_cast ht⟩
+  have hr := ode_pair_ratio_pos_lt_one hb hf hg hzw ht'
+  have hgrowth := loewner_two_point_growth ha hb
+    (show alive a b t z from ⟨f,odeUntil_mono hf ht⟩)
+    (show alive a b t w from ⟨g,odeUntil_mono hg ht⟩) hzw
+  change ‖z-w‖ ≤ fullLoewnerRadius a b z t * _ / _ at hgrowth
+  rw [loewnerMap_eq (odeUntil_mono hf ht),loewnerMap_eq (odeUntil_mono hg ht)] at hgrowth
+  change ‖z-w‖≤fullLoewnerRadius a b z t*r/(1-r)^2 at hgrowth
+  have hden : 0<(1-r)^2 := sq_pos_of_pos (sub_pos.mpr hr.2)
+  have hgrowth' := (le_div_iff₀ hden).mp hgrowth
+  rw [fullRadius_active_clock hb hf hσ ht] at hgrowth'
+  have hR := fullRadius_pos hb hf hσ
+  have hmul : fullLoewnerRadius a b z σ*(1-r)^2 ≤
+      fullLoewnerRadius a b z σ*(Real.exp (-2*a*activeODEClock b T σ f t)*r) :=
+    (mul_le_mul_of_nonneg_right hcap hden.le).trans (by simpa only [mul_assoc] using hgrowth')
+  have hbound := (mul_le_mul_iff_right₀ hR).mp hmul
+  have hsq : (Real.log r)^2≤Real.exp (-2*a*activeODEClock b T σ f t) :=
+    (mul_le_mul_iff_left₀ hr.1).mp ((log_sq_mul_le hr.1 hr.2.le).trans hbound)
+  have he : Real.exp (-2*a*activeODEClock b T σ f t) =
+      (Real.exp (-a*activeODEClock b T σ f t))^2 := by
+    rw [← Real.exp_nat_mul]
+    congr 1
+    ring
+  rw [he] at hsq
+  change -Real.log r≤_
+  nlinarith [Real.exp_pos (-a*activeODEClock b T σ f t)]
+lemma ode_pair_log_deriv {a : ℝ} {b : ℝ≥0 → ℝ} (hb : Continuous b)
+    {z w : ℂ} {T : ℝ≥0} {f g : ℝ → ℂ}
+    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
+    {t : ℝ} (ht : t∈Icc (0:ℝ) T) :
+    HasDerivWithinAt (fun u => -Real.log (‖f u-g u‖/‖f u-star (g u)‖))
+      (-2*a*odeSpeed b T f t*odeSpeed b T g t) (Icc (0:ℝ) T) t := by
+  have hp := (ode_pair_ratio_pos_lt_one hb hf hg hzw ht).1
+  have hd := ((odeUntil_pair_ratio_deriv hb hf hg hzw ht).log hp.ne').neg
+  rw [mul_div_cancel_right₀ _ hp.ne'] at hd
+  convert hd using 1
+  try rfl
+  rw [odeSpeed_eq f ht,odeSpeed_eq g ht]
+  ring
+end ExactSLEMain
+namespace ExactSLEMain
+ 
+
+theorem actual_pair_interaction {a c : ℝ} (ha : 0<a) (hc : 0≤c) (hac : 2*c<a)
+    {b : ℝ≥0 → ℝ} (hb : Continuous b) {z w : ℂ} {f g : ℝ → ℂ}
+    {T S σ τ : ℝ≥0} (hST : S≤T) (hσ : σ≤S) (hτ : τ≤S)
+    (hf : odeUntil a b z T f) (hg : odeUntil a b w T g) (hzw : z≠w)
+    (hcapz : fullLoewnerRadius a b z σ≤‖z-w‖)
+    (hcapw : fullLoewnerRadius a b w τ≤‖z-w‖) :
+    (∫ t in (S:ℝ)..(T:ℝ), Real.exp (c*(activeODEClock b T σ f t+
+      activeODEClock b T τ g t))*odeSpeed b T f t*odeSpeed b T g t) ≤
+      Real.exp (2*c)/(2*a*(1-Real.exp (2*c-a))) := by
+  let L := fun t : ℝ => -Real.log (‖f t-g t‖/‖f t-star (g t)‖)
+  let u := activeODEClock b T σ f
+  let v := activeODEClock b T τ g
+  have hST' : (S:ℝ)≤T := by exact_mod_cast hST
+  have hsub : Icc (S:ℝ) T ⊆ Icc (0:ℝ) T := Icc_subset_Icc_left S.coe_nonneg
+  have hnon (t : ℝ) (ht : t∈Icc (S:ℝ) T) : 0≤L t := by
+    have hr := ode_pair_ratio_pos_lt_one hb hf hg hzw (hsub ht)
+    exact neg_nonneg.mpr (Real.log_nonpos hr.1.le hr.2.le)
+  have hdecay (t : ℝ) (ht : t∈Icc (S:ℝ) T) :
+      L t≤Real.exp (-a*(u t+v t)/2) := by
+    have ht0 : 0≤t := (hsub ht).1
+    have htT : t.toNNReal≤T := by
+      exact_mod_cast (show (t.toNNReal:ℝ)≤T by simpa only [Real.coe_toNNReal _ ht0] using ht.2)
+    have hz := actual_pair_clock_decay ha hb (hσ.trans hST) htT hf hg hzw hcapz
+    have hcapw' : fullLoewnerRadius a b w τ≤‖w-z‖ := by simpa only [norm_sub_rev] using hcapw
+    have hw := actual_pair_clock_decay ha hb (hτ.trans hST) htT hg hf hzw.symm hcapw'
+    rw [Real.coe_toNNReal _ ht0] at hz hw
+    have hnorm : ‖g t-star (f t)‖=‖f t-star (g t)‖ := by
+      calc
+        _ = ‖star (g t-star (f t))‖ := (norm_star _).symm
+        _ = ‖star (g t)-f t‖ := by rw [star_sub,star_star]
+        _ = _ := norm_sub_rev _ _
+    rw [hnorm,norm_sub_rev (g t) (f t)] at hw
+    change L t≤Real.exp (-a*u t) at hz
+    change L t≤Real.exp (-a*v t) at hw
+    rcases le_total (u t) (v t) with h|h
+    · exact hw.trans (Real.exp_le_exp.mpr (by nlinarith))
+    · exact hz.trans (Real.exp_le_exp.mpr (by nlinarith))
+  exact weighted_interaction_integral ha hc hac hST'
+    (L:=L) (s:=fun t => u t+v t) (V:=odeSpeed b T f) (W:=odeSpeed b T g)
+    (odeSpeed_continuous hb hf).continuousOn (odeSpeed_continuous hb hg).continuousOn
+    (fun t ht => (ode_pair_log_deriv hb hf hg hzw (hsub ht)).mono hsub)
+    (fun t _ => ((activeODEClock_deriv hb hf σ t).add
+      (activeODEClock_deriv hb hg τ t)).hasDerivWithinAt)
+    (fun t ht => add_nonneg (activeODEClock_nonneg ((by exact_mod_cast hσ : (σ:ℝ)≤S).trans ht.1))
+      (activeODEClock_nonneg ((by exact_mod_cast hτ : (τ:ℝ)≤S).trans ht.1))) hnon hdecay
+end ExactSLEMain
+namespace ExactSLEGreen
+open ExactSLEMain ExactSLEGenerator ExactSLEFiniteTilt
+end ExactSLEGreen
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/AngularClock.lean b/lean/OAI/Probability/SLELocalization/AngularClock.lean
index 80b37bf890addb654dbe538b608960834d313ece..d20565bc1a3cce2e9a4ed5868c6c774c7e3d6336 100644
--- a/lean/OAI/Probability/SLELocalization/AngularClock.lean
+++ b/lean/OAI/Probability/SLELocalization/AngularClock.lean
@@ -1,982 +1,982 @@
-import Mathlib
-import OAI.Probability.SLELocalization.AngularMoments
-import OAI.Probability.SLE.ClippedCoordinates
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Gamma Field estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.GammaField
-open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-lemma field_nonneg {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) :
-    0 ≤ field hε hT l b η ρ sdev U z t x :=
-  AngularBarrier.value_nonneg hl hb (remaining_pos hε hT hη ρ U z t)
-    (squaredSine_pos _).le
-
-lemma field_initial {z : ℂ} (hy : ε ≤ z.im) {u : ℝ} (hu : 0 ≤ u)
-    (l b η sdev : ℝ) (U : C(I, ℝ)) :
-    field hε hT l b η ((2*z.im)*Real.exp (-u)) sdev U z 0 0 = profile l b (η+u) (z.re/z.im) := by
-  have hz : 0 < z.im := hε.trans_le hy
-  simp only [field, remaining, rawRemaining, angleRatio, height, position_initial,
-    logDerivative_initial, mul_zero, sub_zero, max_eq_right hy]
-  rw [Real.log_mul (by positivity) (Real.exp_pos _).ne', Real.log_exp]
-  have he : η + Real.log (2*z.im) - (Real.log (2*z.im) + -u) = η+u := by ring
-  rw [he, max_eq_right (by linarith)]
-
-lemma measurable_stop (hTp : 0 < T) (z : ℂ) (ρ δ : ℝ) :
-    @Measurable C(I, ℝ) ℝ (borel _) _ (fun U => stop hε hT U z ρ δ) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  apply measurable_of_Ioi
-  intro a
-  by_cases ha : a < 0
-  · have he : (fun U => stop hε hT U z ρ δ) ⁻¹' Ioi a = univ := by
-      ext U; simp only [mem_preimage, mem_Ioi, mem_univ, iff_true]
-      exact ha.trans_le (stop_bounds hε hT U z ρ δ).1
-    rw [he]; exact MeasurableSet.univ
-  · by_cases hat : a < T
-    · have he : (fun U => stop hε hT U z ρ δ) ⁻¹' Ioi a =
-        {U | ∀ s ∈ Icc 0 a, 0 < deficit hε hT U z ρ δ s} := by
-        ext U; exact lt_stop_iff hε hT hTp U z ρ δ ⟨not_lt.1 ha,hat⟩
-      rw [he]; exact (isOpen_positive_on hε hT z ρ δ a).measurableSet
-    · have he : (fun U => stop hε hT U z ρ δ) ⁻¹' Ioi a = ∅ := by
-        ext U; simp only [mem_preimage, mem_Ioi, mem_empty_iff_false, iff_false]
-        exact not_lt_of_ge ((stop_bounds hε hT U z ρ δ).2.trans (not_lt.1 hat))
-      rw [he]; exact MeasurableSet.empty
-
-lemma integral_drift_nonpos {κ η ρ : ℝ} (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (_hρ : 0 < ρ)
-    (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (δ : ℝ) (B : ℝ → ℝ) (hB : Continuous B)
-    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ*B s) :
-    (∫ t in (0:ℝ)..(stop hε hT U z ρ δ), drift hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t (B t)) +
-    (1/2:ℝ)*∫ t in (0:ℝ)..(stop hε hT U z ρ δ),
-      iteratedDeriv 2 (field hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t) (B t) ≤ 0 := by
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have htau := (stop_bounds hε hT U z ρ δ).1
-  have hm : Continuous (fun t : ℝ => ((U,t),B t)) :=
-    (continuous_const.prodMk continuous_id).prodMk hB
-  have hv := ((continuous_drift hε hT hl hb hη ρ (Real.sqrt κ) z).comp hm).continuousOn (s := Icc 0 (stop hε hT U z ρ δ))
-  have hc := ((continuous_derivative hε hT hl hb hη ρ (Real.sqrt κ) z 2).comp hm).continuousOn (s := Icc 0 (stop hε hT U z ρ δ))
-  have hiV := hv.intervalIntegrable_of_Icc (μ:=volume) htau
-  have hiC := hc.intervalIntegrable_of_Icc (μ:=volume) htau
-  dsimp only [Function.comp_def] at hiV hiC
-  rw [← intervalIntegral.integral_const_mul, ← intervalIntegral.integral_add hiV (hiC.const_mul (1/2:ℝ))]
-  rw [intervalIntegral.integral_of_le htau]
-  apply integral_nonpos_of_ae
-  have hne : ∀ᵐ t : ℝ ∂volume, t ≠ stop hε hT U z ρ δ := by simp [ae_iff]
-  filter_upwards [ae_restrict_mem measurableSet_Ioc,
-    ae_restrict_of_ae hne] with t ht he
-  have hlt : t < stop hε hT U z ρ δ := lt_of_le_of_ne ht.2 he
-  have hh := (deficit_pos_iff hε hT U z ρ δ t).1
-    (positive_before hε hT hTp U z ρ δ ht.1.le hlt)
-  exact drift_nonpos hε hT hκ hη ρ U z t (B t) hh.2.1.le
-    (hU t ⟨ht.1.le,ht.2.trans (stop_bounds hε hT U z ρ δ).2⟩)
-
-end SLEExactGauge.GammaField
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GammaField AngularBarrier
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable (hz : z ∈ halfPlane) (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-include h hz hu hc hb0
-
-lemma remaining_upper (η : ℝ) {t : ℝ} (ht : t ∈ Icc 0 (C.stopTime κ B sample)) :
-    remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-u)) (C.paths κ B sample) z t ≤ η+u := by
-  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle t).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)).2.1
-  have hrad := radius_eq_source C.height_pos C.horizon_pos h
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) _ z _ C.angle
-    (C.path_eq κ B sample hc) h0 ht
-  have ha := (h.log_radius_strictAnti (C.bounds κ B sample).1
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (C.source_live κ B sample h hz hu hc hb0)).antitoneOn
-    (show (0:ℝ) ∈ Icc 0 (C.stopTime κ B sample) from ⟨le_rfl,(C.bounds κ B sample).1⟩) ht ht.1
-  rw [h.radius_initial hz] at ha
-  unfold remaining
-  apply max_le (by linarith)
-  rw [rawRemaining_eq C.height_pos C.horizon.coe_nonneg η _ _ z t hh, hrad,
-    Real.log_mul (by have hy : 0 < z.im := hz; positivity) (Real.exp_pos _).ne', Real.log_exp]
-  linarith
-
-omit h in
-lemma ratio_upper {t : ℝ} (ht : t ∈ Icc 0 (C.stopTime κ B sample)) :
-    |angleRatio C.height_pos C.horizon.coe_nonneg (Real.sqrt κ) (C.paths κ B sample) z t (B t.toNNReal sample)| ≤ 1/C.angle := by
-  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hh := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle t).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)
-  exact ratio_bound C.height_pos C.horizon.coe_nonneg C.angle_pos _ _ z t _ hh.2.1 hh.2.2
-    (C.path_eq κ B sample hc ⟨ht.1,ht.2.trans (C.bounds κ B sample).2⟩)
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.GammaField
-open LoewnerCutoff LoewnerExit AngularBarrier
-
-def pathTerminal {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0)
-    (l b η ρ δ sdev : ℝ) (z : ℂ) (U : C(I, ℝ)) : ℝ :=
-  let tau := stop hε T.coe_nonneg U z ρ δ
-  field hε T.coe_nonneg l b η ρ sdev U z tau (extendDriver T U tau/sdev)
-
-lemma continuous_actualField {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0)
-    {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (ρ sdev : ℝ) (z : ℂ) :
-    Continuous (fun v : C(I, ℝ) × ℝ =>
-      field hε T.coe_nonneg l b η ρ sdev v.1 z v.2 (extendDriver T v.1 v.2/sdev)) := by
-  have hp : Continuous (fun v : C(I, ℝ) × ℝ => (v,extendDriver T v.1 v.2/sdev)) :=
-    continuous_id.prodMk ((continuous_joint_extendDriver T).div_const sdev)
-  have hh := Continuous.comp (g := fun v : (C(I, ℝ) × ℝ) × ℝ =>
-    field hε T.coe_nonneg l b η ρ sdev v.1.1 z v.1.2 v.2)
-    (f := fun v : C(I, ℝ) × ℝ => (v,extendDriver T v.1 v.2/sdev))
-    (continuous_field hε T.coe_nonneg hl hb hη ρ sdev z) hp
-  dsimp only [Function.comp_def] at hh
-  exact hh
-
-lemma pathTerminal_measurable {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ))
-    {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (ρ δ sdev : ℝ) (z : ℂ) :
-    @Measurable C(I, ℝ) ℝ (borel _) _ (pathTerminal hε T l b η ρ δ sdev z) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hτ : Measurable (fun U : C(I, ℝ) => stop hε T.coe_nonneg U z ρ δ) :=
-    measurable_stop hε T.coe_nonneg hT z ρ δ
-  have hm : Measurable (fun U : C(I, ℝ) => (U,stop hε T.coe_nonneg U z ρ δ)) :=
-    measurable_id.prodMk hτ
-  have hh := (continuous_actualField hε T hl hb hη ρ sdev z).measurable.comp hm
-  dsimp only [Function.comp_def] at hh
-  exact hh
-
-lemma pathTerminal_eq {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0)
-    (l b η ρ δ sdev : ℝ) (hsdev : sdev ≠ 0) (z : ℂ) :
-    let U := BrownianDriver.path B sdev T sample
-    let tau := stop hε T.coe_nonneg U z ρ δ
-    pathTerminal hε T l b η ρ δ sdev z U =
-      field hε T.coe_nonneg l b η ρ sdev U z tau (B tau.toNNReal sample) := by
-  dsimp only [pathTerminal]
-  congr 1
-  rw [BrownianDriver.extend_eq B sdev T sample hc (stop_bounds hε T.coe_nonneg _ z ρ δ),
-    mul_div_cancel_left₀ _ hsdev]
-
-lemma terminal_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ))
-    {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (ρ δ sdev : ℝ) (hsdev : sdev ≠ 0) (z : ℂ) :
-    let U := BrownianDriver.path B sdev T
-    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-    AEMeasurable (fun sample => field hε T.coe_nonneg l b η ρ sdev (U sample) z (τ sample)
-      (B (τ sample).toNNReal sample)) P := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  have hm := (pathTerminal_measurable hε T hT hl hb hη ρ δ sdev z).comp_aemeasurable
-    (BrownianDriver.path_aemeasurable hB sdev T)
-  apply hm.congr
-  filter_upwards [hB.cont] with sample hc
-  exact pathTerminal_eq B sample hc hε T l b η ρ δ sdev hsdev z
-
-end SLEExactGauge.GammaField
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open GammaField
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-def barrier {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
-  field C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η ((2*z.im)*Real.exp (-u)) (Real.sqrt κ)
-    (C.paths κ B sample) z (C.stopTime κ B sample) (B (C.stopTime κ B sample).toNNReal sample)
-
-lemma barrier_nonneg {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) : 0 ≤ C.barrier η κ B sample := by
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  exact field_nonneg C.height_pos C.horizon.coe_nonneg hl hb hη _ _ _ z _ _
-
-lemma barrier_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (η κ : ℝ)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) : AEMeasurable (C.barrier η κ B) P := by
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  exact terminal_aemeasurable hB C.height_pos C.horizon C.horizon_pos hl hb hη _ _ _
-    (Real.sqrt_pos.2 hκ.1).ne' z
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open GammaField GammaPredictable BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-lemma barrier_expectation {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (η κ : ℝ)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (hz : z ∈ halfPlane) (hu : 0 < u)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample)) :
-    Integrable (C.barrier η κ B) P ∧
-      (∫ sample, C.barrier η κ B sample ∂P) ≤ AngularBarrier.profile (κ/2) ((8/κ-1)/2) (η+u) (z.re/z.im) := by
-  let l := κ/2
-  let b := (8/κ-1)/2
-  let ρ := (2*z.im)*Real.exp (-u)
-  have hl : 0 < l := div_pos hκ.1 (by norm_num)
-  have hb : 0 < b := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    dsimp [b]; linarith
-  have hρ : 0 < ρ := by have hy : 0 < z.im := hz; dsimp [ρ]; positivity
-  let U := C.paths κ B
-  let τ := C.stopTime κ B
-  let F := fun sample => field C.height_pos C.horizon.coe_nonneg l b η ρ (Real.sqrt κ) (U sample) z
-  let V := fun sample => drift C.height_pos C.horizon.coe_nonneg l b η ρ (Real.sqrt κ) (U sample) z
-  let G := AngularBarrier.profile l b (η+u) (z.re/z.im)
-  have hreg : ∀ᵐ sample ∂P, BrownianLocal.LocalRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact regular C.height_pos C.horizon.coe_nonneg hl hb hη hρ C.horizon_pos
-      (Real.sqrt κ) (U sample) z C.angle _ (hc.comp continuous_real_toNNReal)
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ C.horizon :=
-    Filter.Eventually.of_forall (fun sample => C.bounds κ B sample)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coefficient_bound C.height_pos C.horizon.coe_nonneg hl hb hη
-    ρ (Real.sqrt κ) (η+u) (1/C.angle) z 1
-  obtain ⟨K₂,hK₂,hb₂⟩ := coefficient_bound C.height_pos C.horizon.coe_nonneg hl hb hη
-    ρ (Real.sqrt κ) (η+u) (1/C.angle) z 2
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t x : ℝ), remaining C.height_pos C.horizon.coe_nonneg η ρ W z t ≤ η+u →
-        |angleRatio C.height_pos C.horizon.coe_nonneg (Real.sqrt κ) W z t x| ≤ 1/C.angle →
-        |iteratedDeriv k (field C.height_pos C.horizon.coe_nonneg l b η ρ (Real.sqrt κ) W z t) x| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B C.horizon τ F n i sample| ≤ K := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 hm
-    unfold stoppedCoeff
-    split_ifs with hh
-    · have ht : mesh C.horizon n i ∈ Icc 0 (C.stopTime κ B sample) :=
-        ⟨by unfold mesh; positivity,hh.le⟩
-      exact hb (U sample) _ _ (C.remaining_upper κ B sample hm hz hu hc hb0 η ht)
-        (by simpa only [mesh_toNNReal] using C.ratio_upper κ B sample hz hu hc hb0 ht)
-    · simpa only [abs_zero] using hK
-  let A := fun n i => rawCoeff C.height_pos 1 B C.horizon (dyadicTime C.horizon n i) l b η (Real.sqrt κ) z ρ C.angle
-  let D := fun n i => rawCoeff C.height_pos 2 B C.horizon (dyadicTime C.horizon n i) l b η (Real.sqrt κ) z ρ C.angle
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B C.horizon τ F n i :=
-    rawCoeff_stoppedCoeff_ae C.height_pos hB 1 C.horizon C.horizon_pos l b η (Real.sqrt κ) z ρ C.angle n i
-  have hDe (n i : ℕ) : D n i =ᵐ[P] stoppedCoeff 2 B C.horizon τ F n i :=
-    rawCoeff_stoppedCoeff_ae C.height_pos hB 2 C.horizon C.horizon_pos l b η (Real.sqrt κ) z ρ C.angle n i
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (A n i) :=
-    rawCoeff_measurable_past C.height_pos 1 (Or.inl rfl) B C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle
-  have hDm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (D n i) :=
-    rawCoeff_measurable_past C.height_pos 2 (Or.inr rfl) B C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 1 (Or.inl rfl) C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle).aestronglyMeasurable K₁
-    filter_upwards [hAe n i,hsbound 1 K₁ hK₁ hb₁ n i] with sample he hh
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hh
-  have hD2 (n i : ℕ) : MemLp (D n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 2 (Or.inr rfl) C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle).aestronglyMeasurable K₂
-    filter_upwards [hDe n i,hsbound 2 K₂ hK₂ hb₂ n i] with sample he hh
-    change |D n i sample| ≤ K₂
-    rw [he]
-    exact hh
-  have hi := BrownianLocal.stopped_field_ito_centered_ae hB C.horizon τ F V htau hreg
-    A D hAe hDe hDm hD2 K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-    (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-    (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
-  change Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 at hi
-  have hmaj : ∀ᵐ sample ∂P, C.barrier η κ B sample ≤ Y sample+G := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
-    have hd := integral_drift_nonpos C.height_pos C.horizon.coe_nonneg hκ hη hρ C.horizon_pos
-      (U sample) z C.angle (fun t => B t.toNNReal sample) (hc.comp continuous_real_toNNReal)
-      (C.path_eq κ B sample hc)
-    have hin : F sample 0 (B 0 sample) = G := by
-      rw [hb0]
-      exact field_initial C.height_pos C.horizon.coe_nonneg (C.height_initial hz hu.le).le hu.le l b η (Real.sqrt κ) (U sample)
-    change _ ≤ Y sample+G
-    dsimp only [Y]
-    rw [hin]
-    change C.barrier η κ B sample ≤ C.barrier η κ B sample - G - _ - _ + G
-    change (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) +
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample) ≤ 0 at hd
-    linarith only [hd]
-  have hInt : Integrable (fun sample => Y sample+G) P := hi.1.add (integrable_const G)
-  have hI : Integrable (C.barrier η κ B) P := by
-    apply hInt.mono' (C.barrier_aemeasurable hB η κ hκ hη).aestronglyMeasurable
-    filter_upwards [hmaj] with sample hh
-    simpa only [Real.norm_eq_abs, abs_of_nonneg (C.barrier_nonneg η κ B sample hκ hη)] using hh
-  refine ⟨hI, ?_⟩
-  calc
-    (∫ sample, C.barrier η κ B sample ∂P) ≤ ∫ sample, Y sample+G ∂P := integral_mono_ae hI hInt hmaj
-    _ = G := by rw [integral_add hi.1 (integrable_const G), hi.2]; simp
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GammaField AngularBarrier
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable (hz : z ∈ halfPlane) (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-include h hz hu hc hb0
-
-lemma barrier_at_radius (η : ℝ) {t : ℝ≥0} (ht : C.stopTime κ B sample = t)
-    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
-    C.barrier η κ B sample = profile (κ/2) ((8/κ-1)/2) η
-      (((g t z).re-Real.sqrt κ*B t sample)/(g t z).im) := by
-  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hs := source_until_stop C.height_pos C.horizon_pos h _ z _ C.angle
-    (C.path_eq κ B sample hc) h0
-  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0
-      ⟨(C.bounds κ B sample).1,le_rfl⟩)).2.1
-  have hpos := hs.2 ⟨(C.bounds κ B sample).1,le_rfl⟩
-  have hr := C.source_radius κ B sample h hz hu hc hb0
-  rw [ht,hrt] at hr
-  unfold barrier GammaField.field remaining
-  rw [rawRemaining_eq C.height_pos C.horizon.coe_nonneg η _ _ z _ hh, ht, hr]
-  simp only [add_sub_cancel_right, max_self]
-  unfold angleRatio LoewnerField.height
-  rw [← ht, max_eq_right hh, ← hpos, ht, Real.toNNReal_coe]
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt
-open AngularBarrier
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-include h hz hu hc hb0
-
-lemma eventually_stop_at_radius {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
-    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
-    ∀ᶠ n : ℕ in atTop, (cutoffs hz u hu.le n).stopTime κ B sample = t := by
-  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
-  have hδ : ∀ᶠ n : ℕ in atTop, shrinkingAngle z n < η :=
-    (shrinkingAngle_tendsto z).eventually (gt_mem_nhds hη)
-  filter_upwards [hδ] with n hn
-  let C := cutoffs hz u hu.le n
-  have hh : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
-    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · exact hr
-    · have htm := hm (C.stopTime κ B sample)
-        ⟨(C.bounds κ B sample).1,C.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
-      rw [hs] at htm
-      exact False.elim ((not_lt_of_ge htm) hn)
-  exact h.live_radius_injective
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (C.bounds κ B sample).1 t.coe_nonneg (C.source_live κ B sample h hz hu hc hb0) hzt
-    (hh.trans hrt.symm)
-
-omit h hz hu hc hb0 in
-def timeShift (n : ℕ) : ℝ := 1/((n:ℝ)+1)
-
-omit h hz hu hc hb0 in
-lemma timeShift_pos (n : ℕ) : 0 < timeShift n := by unfold timeShift; positivity
-
-omit h hz hu hc hb0 in
-lemma timeShift_tendsto : Tendsto timeShift atTop (𝓝[>] (0:ℝ)) := by
-  apply tendsto_nhdsWithin_iff.2
-  constructor
-  · change Tendsto (fun n : ℕ => (1:ℝ)/((n:ℝ)+1)) atTop (𝓝 0)
-    simpa only [Function.comp_def, Nat.cast_add, Nat.cast_one] using
-      (tendsto_const_div_atTop_nhds_zero_nat (1:ℝ)).comp (tendsto_add_atTop_nat 1)
-  · exact Filter.Eventually.of_forall timeShift_pos
-
-lemma barrier_tendsto_on_hit (hκ : 0 < κ ∧ κ < 8)
-    (hhit : firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞)) :
-    Tendsto (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).barrier (timeShift n) κ B sample)) atTop (𝓝 1) := by
-  have hex : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧ loewnerRadius g t z = (2*z.im)*Real.exp (-u) := by
-    by_contra hn
-    have he : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧ loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ := by
-      exact Set.eq_empty_iff_forall_notMem.2 (fun t ht => hn ⟨t,ht⟩)
-    exact hhit (by simp only [firstLiveRadiusStop, he, Set.image_empty, sInf_empty])
-  obtain ⟨t,hzt,hrt⟩ := hex
-  have he : (fun n : ℕ => (cutoffs hz u hu.le n).barrier (timeShift n) κ B sample) =ᶠ[atTop]
-      (fun n => profile (κ/2) ((8/κ-1)/2) (timeShift n)
-        (((g t z).re-Real.sqrt κ*B t sample)/(g t z).im)) := by
-    filter_upwards [eventually_stop_at_radius κ B sample h hz hu hc hb0 hzt hrt] with n hn
-    exact (cutoffs hz u hu.le n).barrier_at_radius κ B sample h hz hu hc hb0 (timeShift n) hn hrt
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have ht := (value_tendsto_zero_time hl hb (squaredSine_pos
-    (((g t z).re-Real.sqrt κ*B t sample)/(g t z).im))).comp timeShift_tendsto
-  have ht' := ht.congr' he.symm
-  simpa only [ENNReal.ofReal_one, Function.comp_def] using
-    ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht'
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteTilt AngularBarrier
-
-lemma radiusApproachProfile {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u) :
-    P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
-      ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) u (z.re/z.im)) := by
-  let C := cutoffs hz u hu.le
-  let F := fun n sample => ENNReal.ofReal ((C n).barrier (timeShift n) κ B sample)
-  let Y := fun sample => liminf (fun n => F n sample) atTop
-  have hFm (n : ℕ) : AEMeasurable (F n) P :=
-    ENNReal.measurable_ofReal.comp_aemeasurable
-      ((C n).barrier_aemeasurable hB (timeShift n) κ hκ (timeShift_pos n))
-  have hYm : AEMeasurable Y P := by
-    dsimp only [Y]
-    simp_rw [liminf_eq_iSup_iInf_of_nat]
-    exact AEMeasurable.iSup (fun n => AEMeasurable.iInf (fun i => AEMeasurable.iInf (fun (_ : n ≤ i) => hFm i)))
-  have hsub : {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤ᵐ[P]
-      {sample | (1:ℝ≥0∞) ≤ Y sample} := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0 hhit
-    have hh := barrier_tendsto_on_hit κ B sample hm hz hu hc hb0 hκ hhit
-    have he : Y sample = 1 := hh.liminf_eq
-    exact he.ge
-  have hexp (n : ℕ) : (∫⁻ sample, F n sample ∂P) ≤
-      ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) (timeShift n+u) (z.re/z.im)) := by
-    have hi := (C n).barrier_expectation hB (timeShift n) κ hκ (timeShift_pos n) hz hu γ g f hmaps
-    rw [← ofReal_integral_eq_lintegral_ofReal hi.1 (Filter.Eventually.of_forall
-      (fun sample => (C n).barrier_nonneg (timeShift n) κ B sample hκ (timeShift_pos n)))]
-    exact ENNReal.ofReal_le_ofReal hi.2
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have ht : Tendsto (fun n : ℕ => ENNReal.ofReal
-      (profile (κ/2) ((8/κ-1)/2) (timeShift n+u) (z.re/z.im))) atTop
-      (𝓝 (ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) u (z.re/z.im)))) := by
-    have hs : Tendsto (fun n => (timeShift n+u,z.re/z.im)) atTop (𝓝 (u,z.re/z.im)) := by
-      have ht0 : Tendsto timeShift atTop (𝓝 (0:ℝ)) := timeShift_tendsto.mono_right nhdsWithin_le_nhds
-      simpa only [zero_add] using (ht0.add_const u).prodMk_nhds tendsto_const_nhds
-    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-      ((contDiffAt_profile (v := (u,z.re/z.im)) hl hb 0 hu).continuousAt.tendsto.comp hs)
-  calc
-    P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
-        P {sample | (1:ℝ≥0∞) ≤ Y sample} := measure_mono_ae hsub
-    _ ≤ ∫⁻ sample, Y sample ∂P := by
-      simpa only [div_one] using meas_ge_le_lintegral_div hYm (by norm_num : (1:ℝ≥0∞) ≠ 0) (by norm_num : (1:ℝ≥0∞) ≠ ⊤)
-    _ ≤ liminf (fun n => ∫⁻ sample, F n sample ∂P) atTop := lintegral_liminf_le' hFm
-    _ ≤ liminf (fun n => ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) (timeShift n+u) (z.re/z.im))) atTop :=
-      liminf_le_liminf (Filter.Eventually.of_forall hexp)
-    _ = _ := ht.liminf_eq
-
-lemma squaredSine_initial {z : ℂ} (hz : z ∈ halfPlane) :
-    squaredSine (z.re/z.im) = (z.im/‖z‖)^2 := by
-  have hy : 0 < z.im := hz
-  have hn : 0 < ‖z‖ := lt_of_lt_of_le hy (le_trans (le_abs_self _) (Complex.abs_im_le_norm z))
-  have he : ‖z‖^2 = z.re^2+z.im^2 := by
-    rw [← Complex.normSq_eq_norm_sq, Complex.normSq_apply]
-    ring
-  unfold squaredSine
-  field_simp
-  nlinarith
-
-theorem radiusApproachBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C : ℝ, 0 < C ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (_hz : z ∈ halfPlane) (u : ℝ) (_hu : 1 ≤ u),
-      P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
-        ENNReal.ofReal (C * Real.exp (-codimension κ * u) *
-          (z.im / ‖z‖) ^ (8 / κ - 1)) := by
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  obtain ⟨C,hC,hdec⟩ := value_decay hl hb
-  refine ⟨C,hC,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz u hu
-  have hu0 : 0 < u := by linarith
-  apply (radiusApproachProfile κ hκ B hB γ g f hmaps z hz u hu0).trans
-  apply ENNReal.ofReal_le_ofReal
-  have hp : (κ/2)*((8/κ-1)/2)/2 = codimension κ := by
-    unfold codimension dimension
-    field_simp [hκ.1.ne']
-    ring
-  have hq : (squaredSine (z.re/z.im))^((8/κ-1)/2) = (z.im/‖z‖)^(8/κ-1) := by
-    rw [squaredSine_initial hz, ← Real.rpow_natCast (z.im/‖z‖) 2,
-      ← Real.rpow_mul (initial_sine_pos hz).le]
-    congr 1
-    ring
-  have hh := hdec u (squaredSine (z.re/z.im)) hu (squaredSine_pos _).le
-  simpa only [profile,hp,hq] using hh
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.UnivalentDistance
-
-lemma derivative_variation {f : ℂ → ℂ} {a : ℂ} {r A : ℝ}
-    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball a (2*r)))
-    (hb : ∀ z ∈ closedBall a r, ‖deriv f z‖ ≤ 2*A)
-    {z : ℂ} (hz : z ∈ closedBall a (r/2)) :
-    ‖deriv f z - deriv f a‖ ≤ (4*A/r)*‖z-a‖ := by
-  have hd' := hd.deriv isOpen_ball
-  have hsub : closedBall a (r/2) ⊆ ball a (2*r) := closedBall_subset_ball (by linarith)
-  have hbound (w : ℂ) (hw : w ∈ closedBall a (r/2)) :
-      ‖deriv (deriv f) w‖ ≤ 4*A/r := by
-    have hs : closedBall w (r/2) ⊆ closedBall a r := by
-      apply closedBall_subset_closedBall'
-      have hh := mem_closedBall.1 hw
-      linarith
-    have hsc : closure (ball w (r/2)) ⊆ ball a (2*r) := by
-      rw [closure_ball _ (by linarith : r/2 ≠ 0)]
-      exact hs.trans (closedBall_subset_ball (by linarith))
-    have hh := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le
-      (f := deriv f) (by linarith : 0 < r/2) (hd'.mono hsc).diffContOnCl
-      (fun v hv => hb v (hs (sphere_subset_closedBall hv)))
-    convert hh using 1; field_simp; ring
-  exact (convex_closedBall a (r/2)).norm_image_sub_le_of_norm_deriv_le
-    (fun w hw => hd'.differentiableAt (isOpen_ball.mem_nhds (hsub hw))) hbound
-    (mem_closedBall_self (by linarith)) hz
-
-lemma local_image_disk {f : ℂ → ℂ} {a : ℂ} {r : ℝ}
-    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball a (2*r)))
-    (ha : deriv f a ≠ 0)
-    (hb : ∀ z ∈ closedBall a r, ‖deriv f z‖ ≤ 2*‖deriv f a‖) :
-    closedBall (f a) (‖deriv f a‖*r/32) ⊆ f '' closedBall a (r/16) := by
-  let F : ℂ → ℂ := fun z => (f z-f a)/deriv f a
-  have hA : 0 < ‖deriv f a‖ := norm_pos_iff.2 ha
-  have hs : closedBall a (r/16) ⊆ ball a (2*r) := closedBall_subset_ball (by linarith)
-  have hdF (z : ℂ) (hz : z ∈ closedBall a (r/16)) :
-      HasDerivAt F (deriv f z/deriv f a) z :=
-    ((hd.differentiableAt (isOpen_ball.mem_nhds (hs hz))).hasDerivAt.sub_const _).div_const _
-  have hbound (z : ℂ) (hz : z ∈ closedBall a (r/16)) :
-      ‖deriv f z/deriv f a-1‖ ≤ (1/2:ℝ) := by
-    have hv := derivative_variation hr hd hb
-      (closedBall_subset_closedBall (by linarith : r/16 ≤ r/2) hz)
-    have he : deriv f z/deriv f a-1 = (deriv f z-deriv f a)/deriv f a := by
-      field_simp
-    rw [he,norm_div,div_le_iff₀ hA]
-    have hn := mem_closedBall_iff_norm.1 hz
-    have hn' : (4*‖deriv f a‖/r)*‖z-a‖ ≤ (4*‖deriv f a‖/r)*(r/16) :=
-      mul_le_mul_of_nonneg_left hn (by positivity)
-    have he' : (4*‖deriv f a‖/r)*(r/16) = ‖deriv f a‖/4 := by field_simp; ring
-    rw [he'] at hn'
-    linarith
-  have happ : ApproximatesLinearOn F (ContinuousLinearMap.id ℂ ℂ)
-      (closedBall a (r/16)) (1/2:ℝ≥0) := by
-    intro x hx y hy
-    have hh := (convex_closedBall a (r/16)).norm_image_sub_le_of_norm_hasDerivWithin_le
-      (fun z hz => ((hdF z hz).sub (hasDerivAt_id z)).hasDerivWithinAt) hbound hy hx
-    change ‖(F x-x)-(F y-y)‖ ≤ (1/2:ℝ)*‖x-y‖ at hh
-    have he : (F x-x)-(F y-y) = F x-F y-(x-y) := by ring
-    rw [he] at hh
-    exact hh
-  have hsurj := happ.surjOn_closedBall_of_nonlinearRightInverse
-    (ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse (by linarith : 0 ≤ r/16) Subset.rfl
-  have hsurj' : closedBall 0 (r/32) ⊆ F '' closedBall a (r/16) := by
-    change closedBall (F a) (((↑(ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse.nnnorm : ℝ)⁻¹ - (1/2)) * (r/16)) ⊆ _ at hsurj
-    have hn : (↑(ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse.nnnorm : ℝ) = 1 := by
-      simp [ContinuousLinearEquiv.toNonlinearRightInverse]
-    rw [hn,show ((1:ℝ)⁻¹-(1/2))*(r/16) = r/32 by norm_num; ring] at hsurj
-    simpa only [F,sub_self,zero_div] using hsurj
-  intro w hw
-  have hw' : (w-f a)/deriv f a ∈ closedBall (0:ℂ) (r/32) := by
-    rw [mem_closedBall_iff_norm,sub_zero,norm_div,div_le_iff₀ hA]
-    have hh := mem_closedBall_iff_norm.1 hw
-    nlinarith
-  obtain ⟨v,hv,he⟩ := hsurj' hw'
-  refine ⟨v,hv,?_⟩
-  dsimp only [F] at he
-  have he' : f v-f a = w-f a := (div_left_inj' ha).mp he
-  exact sub_left_injective he'
-
-lemma image_disk {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1))
-    (h0 : deriv f 0 ≠ 0) :
-    ∃ b : ℂ, closedBall b (‖deriv f 0‖/128) ⊆ f '' ball 0 1 := by
-  have hs : closedBall (0:ℂ) (1/2:ℝ) ⊆ ball 0 1 := closedBall_subset_ball (by norm_num)
-  have hc : ContinuousOn (fun z : ℂ => (1/2-‖z‖)*‖deriv f z‖)
-      (closedBall 0 (1/2:ℝ)) :=
-    (continuous_const.sub continuous_norm).continuousOn.mul
-      ((hd.deriv isOpen_ball).continuousOn.mono hs).norm
-  obtain ⟨a,ha,hm⟩ := (isCompact_closedBall (0:ℂ) (1/2:ℝ)).exists_isMaxOn
-    (nonempty_closedBall.mpr (by norm_num)) hc
-  have ham : ‖a‖ ≤ (1/2:ℝ) := by simpa using ha
-  have hmax0 := hm (mem_closedBall_self (by norm_num : (0:ℝ) ≤ 1/2))
-  change (1/2-‖(0:ℂ)‖)*‖deriv f 0‖ ≤ (1/2-‖a‖)*‖deriv f a‖ at hmax0
-  simp only [norm_zero,sub_zero] at hmax0
-  have h0p := norm_pos_iff.mpr h0
-  have hAp : 0 < ‖deriv f a‖ := by
-    by_contra hh
-    have he : ‖deriv f a‖ = 0 := le_antisymm (le_of_not_gt hh) (norm_nonneg _)
-    rw [he,mul_zero] at hmax0
-    linarith
-  have halt : ‖a‖ < (1/2:ℝ) := by nlinarith
-  let r : ℝ := (1/2-‖a‖)/2
-  have hr : 0 < r := by dsimp [r]; linarith
-  have he : 2*r = 1/2-‖a‖ := by dsimp [r]; ring
-  have hsub : ball a (2*r) ⊆ ball (0:ℂ) 1 := by
-    apply ball_subset_ball'
-    rw [dist_zero_right,he]
-    linarith
-  have hb (z : ℂ) (hz : z ∈ closedBall a r) : ‖deriv f z‖ ≤ 2*‖deriv f a‖ := by
-    have hzn : ‖z‖ ≤ ‖a‖+r := by
-      have hh := norm_le_norm_add_norm_sub a z
-      rw [norm_sub_rev a z] at hh
-      have hz' := mem_closedBall_iff_norm.1 hz
-      linarith
-    have hz0 : z ∈ closedBall (0:ℂ) (1/2:ℝ) := by
-      simp only [mem_closedBall_iff_norm,sub_zero]
-      linarith
-    have hmz := hm hz0
-    change (1/2-‖z‖)*‖deriv f z‖ ≤ (1/2-‖a‖)*‖deriv f a‖ at hmz
-    have hwel : r ≤ 1/2-‖z‖ := by linarith
-    have hh := mul_le_mul_of_nonneg_right hwel (norm_nonneg (deriv f z))
-    nlinarith
-  refine ⟨f a,?_⟩
-  have hrad : ‖deriv f 0‖/128 ≤ ‖deriv f a‖*r/32 := by nlinarith
-  apply (closedBall_subset_closedBall hrad).trans
-  apply (local_image_disk hr (hd.mono hsub) (norm_pos_iff.mp hAp) hb).trans
-  apply image_mono
-  apply closedBall_subset_ball'
-  rw [dist_zero_right]
-  linarith
-
-lemma exp_injective_derivative_bound {h : ℂ → ℂ}
-    (hd : DifferentiableOn ℂ h (ball 0 1)) (hi : InjOn (Complex.exp ∘ h) (ball 0 1)) :
-    ‖deriv h 0‖ ≤ 256*Real.pi := by
-  by_contra hb
-  have hp := Real.pi_pos
-  have hn : 0 < ‖deriv h 0‖ := by linarith
-  obtain ⟨b,hbim⟩ := image_disk hd (norm_pos_iff.mp hn)
-  have hb0 : b ∈ closedBall b (‖deriv h 0‖/128) := mem_closedBall_self (by positivity)
-  have hb1 : b+2*Real.pi*Complex.I ∈ closedBall b (‖deriv h 0‖/128) := by
-    rw [mem_closedBall_iff_norm,add_sub_cancel_left]
-    norm_num [norm_mul,Complex.norm_real,abs_of_pos hp]
-    linarith
-  obtain ⟨z0,hz0,hz0'⟩ := hbim hb0
-  obtain ⟨z1,hz1,hz1'⟩ := hbim hb1
-  have he : z1 = z0 := hi hz1 hz0 (by
-    simp only [Function.comp_apply,hz0',hz1']
-    exact Complex.exp_periodic b)
-  have he' : (2:ℂ)*Real.pi*Complex.I = 0 := by
-    rw [he,hz0'] at hz1'
-    linear_combination -hz1'
-  have hh : (Real.pi:ℂ) = 0 := (mul_eq_zero.mp he').resolve_right Complex.I_ne_zero |>
-    (fun h => (mul_eq_zero.mp h).resolve_left (by norm_num))
-  exact Real.pi_ne_zero (Complex.ofReal_eq_zero.mp hh)
-
-lemma omitted_value_bound {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1))
-    (hi : InjOn f (ball 0 1)) {w : ℂ} (hw : w ∉ f '' ball 0 1) :
-    ‖deriv f 0‖ ≤ (256*Real.pi)*‖f 0-w‖ := by
-  have hzero : (0:ℂ) ∈ ball 0 1 := mem_ball_self zero_lt_one
-  have hn (z : ℂ) (hz : z ∈ ball (0:ℂ) 1) : f z-w ≠ 0 := by
-    intro he
-    exact hw ⟨z,hz,sub_eq_zero.mp he⟩
-  have hc : IsSimplyConnected (ball (0:ℂ) 1) := by
-    let := (convex_ball (0:ℂ) (1:ℝ)).contractibleSpace (nonempty_ball.mpr zero_lt_one)
-    change SimplyConnectedSpace (ball (0:ℂ) 1)
-    infer_instance
-  obtain ⟨h,hhc,hhe⟩ := Complex.exists_continuousOn_eqOn_exp_comp hc isOpen_ball
-    (hd.continuousOn.sub continuousOn_const) (by
-      rintro ⟨z,hz,he⟩
-      exact hn z hz he)
-  have hhd (z : ℂ) (hz : z ∈ ball (0:ℂ) 1) :
-      HasDerivAt h (deriv f z/(f z-w)) z := by
-    have hh := HasDerivAt.of_comp_left
-      (hhc.continuousAt (isOpen_ball.mem_nhds hz))
-      (Complex.hasDerivAt_exp (h z))
-      ((hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const w)
-      (Complex.exp_ne_zero _) (hhe.eventuallyEq_of_mem (isOpen_ball.mem_nhds hz))
-    have he : Complex.exp (h z) = f z-w := hhe hz
-    simpa only [he] using hh
-  have hhd' : DifferentiableOn ℂ h (ball 0 1) := fun z hz => (hhd z hz).differentiableAt.differentiableWithinAt
-  have hhi : InjOn (Complex.exp ∘ h) (ball 0 1) := by
-    intro z hz v hv he
-    apply hi hz hv
-    rw [hhe hz,hhe hv] at he
-    exact sub_left_injective he
-  have hh := exp_injective_derivative_bound hhd' hhi
-  rw [(hhd 0 hzero).deriv,norm_div,div_le_iff₀ (norm_pos_iff.mpr (hn 0 hzero))] at hh
-  exact hh
-
-end SLEExactGauge.UnivalentDistance
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma radius_le_complement_distance (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t)
-    {w : ℂ} (hw : w ∉ localizationDomain γ t) :
-    loewnerRadius g t z ≤ (512*Real.pi)*dist z w := by
-  have hpos := h.im_pos ht hz
-  have hdn := h.deriv_ne_zero ht hz
-  rcases h with ⟨hγ,hzero,him,htrans,hinit,hg,hf,hgmap,hfmap,hfg,hgf,hc,hd,htip⟩
-  let φ : ℂ → ℂ := fun q => g t z + ((g t z).im:ℂ)*q
-  let F : ℂ → ℂ := fun q => f t (φ q)
-  have hφ0 : φ 0 = g t z := by simp [φ]
-  have hφ (q : ℂ) (hq : q ∈ ball (0:ℂ) 1) : φ q ∈ halfPlane := by
-    have hqn : ‖q‖ < 1 := by simpa using hq
-    have hqi := Complex.abs_im_le_norm q
-    have hql := neg_abs_le q.im
-    change 0 < (φ q).im
-    simp only [φ,Complex.add_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,
-      zero_mul,add_zero]
-    nlinarith
-  have hdφ (q : ℂ) : HasDerivAt φ ((g t z).im:ℂ) q := by
-    simpa only [mul_one,id_eq] using! ((hasDerivAt_id q).const_mul ((g t z).im:ℂ)).const_add (g t z)
-  have hdF (q : ℂ) (hq : q ∈ ball (0:ℂ) 1) : DifferentiableAt ℂ F q :=
-    (hf t ht _ (hφ q hq)).differentiableAt.comp q (hdφ q).differentiableAt
-  have hiF : InjOn F (ball 0 1) := by
-    intro q hq v hv he
-    have hh := congrArg (g t) he
-    change g t (f t (φ q)) = g t (f t (φ v)) at hh
-    rw [hgf t ht _ (hφ q hq),hgf t ht _ (hφ v hv)] at hh
-    dsimp only [φ] at hh
-    exact mul_left_cancel₀ (Complex.ofReal_ne_zero.mpr hpos.ne') (add_left_cancel hh)
-  have hFw : w ∉ F '' ball 0 1 := by
-    rintro ⟨q,hq,he⟩
-    exact hw (he ▸ hfmap t ht (hφ q hq))
-  have hgd := (hg t ht z hz).differentiableAt.hasDerivAt
-  have hfd := (hf t ht _ (hgmap t ht hz)).differentiableAt.hasDerivAt
-  have heq : (fun v => f t (g t v)) =ᶠ[𝓝 z] id := by
-    filter_upwards [(survivingDomain_open hγ ht).mem_nhds hz] with v hv
-    exact hfg t ht v hv
-  have he := ((hfd.comp z hgd).congr_of_eventuallyEq heq.symm).unique (hasDerivAt_id z)
-  have hen : ‖deriv (f t) (g t z)‖*‖deriv (g t) z‖ = 1 := by
-    simpa only [norm_mul,norm_one] using congrArg norm he
-  have hFd : deriv F 0 = deriv (f t) (g t z)*((g t z).im:ℂ) := by
-    have hfd' : HasDerivAt (f t) (deriv (f t) (g t z)) (φ 0) := hφ0.symm ▸ hfd
-    exact (hfd'.comp 0 (hdφ 0)).deriv
-  have hF0 : F 0 = z := by simp only [F,hφ0,hfg t ht z hz]
-  have hR : loewnerRadius g t z = 2*‖deriv F 0‖ := by
-    rw [loewnerRadius,hFd,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hpos,
-      div_eq_iff (norm_ne_zero_iff.mpr hdn)]
-    nlinarith
-  have hh := UnivalentDistance.omitted_value_bound
-    (fun q hq => (hdF q hq).differentiableWithinAt) hiF hFw
-  rw [hF0] at hh
-  rw [hR,dist_eq_norm]
-  nlinarith
-
-lemma radius_le_boundary_distance (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    loewnerRadius g t z ≤ (512*Real.pi)*infDist z (localizationDomain γ t)ᶜ := by
-  have hp : 0 < 512*Real.pi := by positivity
-  have hn : ((localizationDomain γ t)ᶜ).Nonempty := by
-    refine ⟨0,?_⟩
-    intro h0
-    exact (lt_irrefl (0:ℝ)) (by simpa [halfPlane] using h0.1.1)
-  rw [mul_comm (512*Real.pi),← div_le_iff₀ hp]
-  apply (le_infDist hn).2
-  intro w hw
-  rw [div_le_iff₀ hp]
-  exact (h.radius_le_complement_distance ht hz hw).trans_eq (mul_comm _ _)
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-def liveDistanceApproach (γ : ℝ → ℂ) (z : ℂ) (ε : ℝ) : Prop :=
-  ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧ infDist z (localizationDomain γ t)ᶜ < ε
-
-lemma liveDistanceApproach_radius_hit {driver : ℝ → ℝ} {γ : ℝ → ℂ}
-    {g f : ℝ → ℂ → ℂ} (h : IsLoewnerTraceUsing driver γ g f)
-    (hd : ContinuousOn driver (Ici 0)) {z : ℂ} (hz : z ∈ halfPlane)
-    {ε u : ℝ} (hr : (2*z.im)*Real.exp (-u) = (512*Real.pi)*ε)
-    (hsmall : (512*Real.pi)*ε ≤ 2*z.im)
-    (hhit : liveDistanceApproach γ z ε) :
-    firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞) := by
-  obtain ⟨t,hzt,hclose⟩ := hhit
-  have hrt : loewnerRadius g t z < (2*z.im)*Real.exp (-u) := by
-    rw [hr]
-    exact (h.radius_le_boundary_distance t.coe_nonneg hzt).trans_lt
-      (mul_lt_mul_of_pos_left hclose (by positivity))
-  obtain ⟨s,hs,hseq⟩ := intermediate_value_Icc' t.coe_nonneg
-    (h.continuous_radius t.coe_nonneg (hd.mono Icc_subset_Ici_self) hzt)
-    ⟨hrt.le,by rw [h.radius_initial hz,hr]; exact hsmall⟩
-  apply ne_top_of_le_ne_top (ENNReal.coe_ne_top (r := ⟨s,hs.1⟩))
-  apply sInf_le
-  exact ⟨⟨s,hs.1⟩,⟨survivingDomain_mono hs.2 hzt,hseq⟩,rfl⟩
-
-theorem euclideanApproachBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ c < 1 ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (_hz : z ∈ halfPlane) (ε : ℝ) (_hε : 0 < ε) (_hsmall : ε < c*z.im),
-      P {sample | liveDistanceApproach (γ sample) z ε} ≤
-        ENNReal.ofReal (C * ((2*z.im)^(-codimension κ) *
-          (z.im/‖z‖)^(8/κ-1)) * ε^(codimension κ)) := by
-  obtain ⟨C,hC,hbound⟩ := radiusApproachBound κ hκ
-  let K : ℝ := 512*Real.pi
-  have hK : 0 < K := by dsimp [K]; positivity
-  have hK1 : 1 < K := by dsimp [K]; nlinarith [Real.pi_gt_three]
-  have he : 1 ≤ Real.exp 1 := Real.one_le_exp (by norm_num)
-  have heK : 1 < K*Real.exp 1 := by nlinarith
-  refine ⟨C*K^(codimension κ),1/(K*Real.exp 1),mul_pos hC (Real.rpow_pos_of_pos hK _),
-    by positivity,(div_lt_one (by positivity)).2 heK,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz ε hε hsmall
-  have hy : 0 < z.im := hz
-  have hsmall' : ε*(K*Real.exp 1) < z.im := by
-    rw [one_div_mul_eq_div] at hsmall
-    exact (lt_div_iff₀ (by positivity)).mp hsmall
-  have hKε : 0 < K*ε := mul_pos hK hε
-  let u : ℝ := Real.log ((2*z.im)/(K*ε))
-  have hu : 1 ≤ u := by
-    have hh : Real.exp 1 ≤ (2*z.im)/(K*ε) := by
-      rw [le_div_iff₀ hKε]
-      nlinarith
-    simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos 1) hh
-  have hr : (2*z.im)*Real.exp (-u) = K*ε := by
-    rw [Real.exp_neg,Real.exp_log (by positivity : 0 < (2*z.im)/(K*ε))]
-    field_simp
-  have hstart : K*ε ≤ 2*z.im := by
-    have hh : K*ε ≤ ε*(K*Real.exp 1) := by nlinarith
-    linarith
-  have hsub : {sample | liveDistanceApproach (γ sample) z ε} ≤ᵐ[P]
-      {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} := by
-    filter_upwards [hmaps,hB.cont] with sample hm hc hhit
-    exact liveDistanceApproach_radius_hit hm
-      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-      hz hr hstart hhit
-  apply (measure_mono_ae hsub).trans
-  apply (hbound P B hB γ g f hγ hmaps z hz u hu).trans
-  apply ENNReal.ofReal_le_ofReal
-  have hexp : Real.exp (-codimension κ*u) =
-      (K*ε)^(codimension κ)*(2*z.im)^(-codimension κ) := by
-    rw [Real.rpow_def_of_pos hKε,Real.rpow_def_of_pos (by positivity : 0 < 2*z.im),
-      ← Real.exp_add]
-    congr 1
-    dsimp [u]
-    rw [Real.log_div (by positivity : 2*z.im ≠ 0) hKε.ne']
-    ring
-  rw [hexp,Real.mul_rpow hK.le hε.le]
-  exact le_of_eq (by ring)
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma future_tip_mem_closure (h : IsLoewnerTraceUsing driver γ g f)
-    {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) : γ t ∈ closure (localizationDomain γ s) := by
-  rcases h with ⟨hγ,hzero,him,htrans,hinit,hg,hf,hgmap,hfmap,hfg,hgf,hc,hd,htip⟩
-  apply isClosed_closure.mem_of_tendsto (htip t (hs.trans hst))
-  filter_upwards [self_mem_nhdsWithin] with y hy
-  have hh : (driver t : ℂ) + (y:ℂ)*Complex.I ∈ halfPlane := by
-    simpa only [halfPlane,mem_ofPred_eq,mem_Ioi,Complex.add_im,Complex.ofReal_im,Complex.mul_im,
-      Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_mul,add_zero,zero_add] using hy
-  exact subset_closure (survivingDomain_mono hst (hfmap t (hs.trans hst) hh))
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.AngularMoments
+import OAI.Probability.SLE.ClippedCoordinates
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Gamma Field estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.GammaField
+open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+lemma field_nonneg {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) :
+    0 ≤ field hε hT l b η ρ sdev U z t x :=
+  AngularBarrier.value_nonneg hl hb (remaining_pos hε hT hη ρ U z t)
+    (squaredSine_pos _).le
+
+lemma field_initial {z : ℂ} (hy : ε ≤ z.im) {u : ℝ} (hu : 0 ≤ u)
+    (l b η sdev : ℝ) (U : C(I, ℝ)) :
+    field hε hT l b η ((2*z.im)*Real.exp (-u)) sdev U z 0 0 = profile l b (η+u) (z.re/z.im) := by
+  have hz : 0 < z.im := hε.trans_le hy
+  simp only [field, remaining, rawRemaining, angleRatio, height, position_initial,
+    logDerivative_initial, mul_zero, sub_zero, max_eq_right hy]
+  rw [Real.log_mul (by positivity) (Real.exp_pos _).ne', Real.log_exp]
+  have he : η + Real.log (2*z.im) - (Real.log (2*z.im) + -u) = η+u := by ring
+  rw [he, max_eq_right (by linarith)]
+
+lemma measurable_stop (hTp : 0 < T) (z : ℂ) (ρ δ : ℝ) :
+    @Measurable C(I, ℝ) ℝ (borel _) _ (fun U => stop hε hT U z ρ δ) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  apply measurable_of_Ioi
+  intro a
+  by_cases ha : a < 0
+  · have he : (fun U => stop hε hT U z ρ δ) ⁻¹' Ioi a = univ := by
+      ext U; simp only [mem_preimage, mem_Ioi, mem_univ, iff_true]
+      exact ha.trans_le (stop_bounds hε hT U z ρ δ).1
+    rw [he]; exact MeasurableSet.univ
+  · by_cases hat : a < T
+    · have he : (fun U => stop hε hT U z ρ δ) ⁻¹' Ioi a =
+        {U | ∀ s ∈ Icc 0 a, 0 < deficit hε hT U z ρ δ s} := by
+        ext U; exact lt_stop_iff hε hT hTp U z ρ δ ⟨not_lt.1 ha,hat⟩
+      rw [he]; exact (isOpen_positive_on hε hT z ρ δ a).measurableSet
+    · have he : (fun U => stop hε hT U z ρ δ) ⁻¹' Ioi a = ∅ := by
+        ext U; simp only [mem_preimage, mem_Ioi, mem_empty_iff_false, iff_false]
+        exact not_lt_of_ge ((stop_bounds hε hT U z ρ δ).2.trans (not_lt.1 hat))
+      rw [he]; exact MeasurableSet.empty
+
+lemma integral_drift_nonpos {κ η ρ : ℝ} (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (_hρ : 0 < ρ)
+    (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (δ : ℝ) (B : ℝ → ℝ) (hB : Continuous B)
+    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ*B s) :
+    (∫ t in (0:ℝ)..(stop hε hT U z ρ δ), drift hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t (B t)) +
+    (1/2:ℝ)*∫ t in (0:ℝ)..(stop hε hT U z ρ δ),
+      iteratedDeriv 2 (field hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t) (B t) ≤ 0 := by
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have htau := (stop_bounds hε hT U z ρ δ).1
+  have hm : Continuous (fun t : ℝ => ((U,t),B t)) :=
+    (continuous_const.prodMk continuous_id).prodMk hB
+  have hv := ((continuous_drift hε hT hl hb hη ρ (Real.sqrt κ) z).comp hm).continuousOn (s := Icc 0 (stop hε hT U z ρ δ))
+  have hc := ((continuous_derivative hε hT hl hb hη ρ (Real.sqrt κ) z 2).comp hm).continuousOn (s := Icc 0 (stop hε hT U z ρ δ))
+  have hiV := hv.intervalIntegrable_of_Icc (μ:=volume) htau
+  have hiC := hc.intervalIntegrable_of_Icc (μ:=volume) htau
+  dsimp only [Function.comp_def] at hiV hiC
+  rw [← intervalIntegral.integral_const_mul, ← intervalIntegral.integral_add hiV (hiC.const_mul (1/2:ℝ))]
+  rw [intervalIntegral.integral_of_le htau]
+  apply integral_nonpos_of_ae
+  have hne : ∀ᵐ t : ℝ ∂volume, t ≠ stop hε hT U z ρ δ := by simp [ae_iff]
+  filter_upwards [ae_restrict_mem measurableSet_Ioc,
+    ae_restrict_of_ae hne] with t ht he
+  have hlt : t < stop hε hT U z ρ δ := lt_of_le_of_ne ht.2 he
+  have hh := (deficit_pos_iff hε hT U z ρ δ t).1
+    (positive_before hε hT hTp U z ρ δ ht.1.le hlt)
+  exact drift_nonpos hε hT hκ hη ρ U z t (B t) hh.2.1.le
+    (hU t ⟨ht.1.le,ht.2.trans (stop_bounds hε hT U z ρ δ).2⟩)
+
+end SLEExactGauge.GammaField
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GammaField AngularBarrier
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable (hz : z ∈ halfPlane) (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+include h hz hu hc hb0
+
+lemma remaining_upper (η : ℝ) {t : ℝ} (ht : t ∈ Icc 0 (C.stopTime κ B sample)) :
+    remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-u)) (C.paths κ B sample) z t ≤ η+u := by
+  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle t).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)).2.1
+  have hrad := radius_eq_source C.height_pos C.horizon_pos h
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) _ z _ C.angle
+    (C.path_eq κ B sample hc) h0 ht
+  have ha := (h.log_radius_strictAnti (C.bounds κ B sample).1
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (C.source_live κ B sample h hz hu hc hb0)).antitoneOn
+    (show (0:ℝ) ∈ Icc 0 (C.stopTime κ B sample) from ⟨le_rfl,(C.bounds κ B sample).1⟩) ht ht.1
+  rw [h.radius_initial hz] at ha
+  unfold remaining
+  apply max_le (by linarith)
+  rw [rawRemaining_eq C.height_pos C.horizon.coe_nonneg η _ _ z t hh, hrad,
+    Real.log_mul (by have hy : 0 < z.im := hz; positivity) (Real.exp_pos _).ne', Real.log_exp]
+  linarith
+
+omit h in
+lemma ratio_upper {t : ℝ} (ht : t ∈ Icc 0 (C.stopTime κ B sample)) :
+    |angleRatio C.height_pos C.horizon.coe_nonneg (Real.sqrt κ) (C.paths κ B sample) z t (B t.toNNReal sample)| ≤ 1/C.angle := by
+  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hh := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle t).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)
+  exact ratio_bound C.height_pos C.horizon.coe_nonneg C.angle_pos _ _ z t _ hh.2.1 hh.2.2
+    (C.path_eq κ B sample hc ⟨ht.1,ht.2.trans (C.bounds κ B sample).2⟩)
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.GammaField
+open LoewnerCutoff LoewnerExit AngularBarrier
+
+def pathTerminal {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0)
+    (l b η ρ δ sdev : ℝ) (z : ℂ) (U : C(I, ℝ)) : ℝ :=
+  let tau := stop hε T.coe_nonneg U z ρ δ
+  field hε T.coe_nonneg l b η ρ sdev U z tau (extendDriver T U tau/sdev)
+
+lemma continuous_actualField {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0)
+    {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (ρ sdev : ℝ) (z : ℂ) :
+    Continuous (fun v : C(I, ℝ) × ℝ =>
+      field hε T.coe_nonneg l b η ρ sdev v.1 z v.2 (extendDriver T v.1 v.2/sdev)) := by
+  have hp : Continuous (fun v : C(I, ℝ) × ℝ => (v,extendDriver T v.1 v.2/sdev)) :=
+    continuous_id.prodMk ((continuous_joint_extendDriver T).div_const sdev)
+  have hh := Continuous.comp (g := fun v : (C(I, ℝ) × ℝ) × ℝ =>
+    field hε T.coe_nonneg l b η ρ sdev v.1.1 z v.1.2 v.2)
+    (f := fun v : C(I, ℝ) × ℝ => (v,extendDriver T v.1 v.2/sdev))
+    (continuous_field hε T.coe_nonneg hl hb hη ρ sdev z) hp
+  dsimp only [Function.comp_def] at hh
+  exact hh
+
+lemma pathTerminal_measurable {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ))
+    {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (ρ δ sdev : ℝ) (z : ℂ) :
+    @Measurable C(I, ℝ) ℝ (borel _) _ (pathTerminal hε T l b η ρ δ sdev z) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hτ : Measurable (fun U : C(I, ℝ) => stop hε T.coe_nonneg U z ρ δ) :=
+    measurable_stop hε T.coe_nonneg hT z ρ δ
+  have hm : Measurable (fun U : C(I, ℝ) => (U,stop hε T.coe_nonneg U z ρ δ)) :=
+    measurable_id.prodMk hτ
+  have hh := (continuous_actualField hε T hl hb hη ρ sdev z).measurable.comp hm
+  dsimp only [Function.comp_def] at hh
+  exact hh
+
+lemma pathTerminal_eq {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0)
+    (l b η ρ δ sdev : ℝ) (hsdev : sdev ≠ 0) (z : ℂ) :
+    let U := BrownianDriver.path B sdev T sample
+    let tau := stop hε T.coe_nonneg U z ρ δ
+    pathTerminal hε T l b η ρ δ sdev z U =
+      field hε T.coe_nonneg l b η ρ sdev U z tau (B tau.toNNReal sample) := by
+  dsimp only [pathTerminal]
+  congr 1
+  rw [BrownianDriver.extend_eq B sdev T sample hc (stop_bounds hε T.coe_nonneg _ z ρ δ),
+    mul_div_cancel_left₀ _ hsdev]
+
+lemma terminal_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ))
+    {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (ρ δ sdev : ℝ) (hsdev : sdev ≠ 0) (z : ℂ) :
+    let U := BrownianDriver.path B sdev T
+    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+    AEMeasurable (fun sample => field hε T.coe_nonneg l b η ρ sdev (U sample) z (τ sample)
+      (B (τ sample).toNNReal sample)) P := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  have hm := (pathTerminal_measurable hε T hT hl hb hη ρ δ sdev z).comp_aemeasurable
+    (BrownianDriver.path_aemeasurable hB sdev T)
+  apply hm.congr
+  filter_upwards [hB.cont] with sample hc
+  exact pathTerminal_eq B sample hc hε T l b η ρ δ sdev hsdev z
+
+end SLEExactGauge.GammaField
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open GammaField
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+def barrier {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
+  field C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η ((2*z.im)*Real.exp (-u)) (Real.sqrt κ)
+    (C.paths κ B sample) z (C.stopTime κ B sample) (B (C.stopTime κ B sample).toNNReal sample)
+
+lemma barrier_nonneg {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) : 0 ≤ C.barrier η κ B sample := by
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  exact field_nonneg C.height_pos C.horizon.coe_nonneg hl hb hη _ _ _ z _ _
+
+lemma barrier_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (η κ : ℝ)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) : AEMeasurable (C.barrier η κ B) P := by
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  exact terminal_aemeasurable hB C.height_pos C.horizon C.horizon_pos hl hb hη _ _ _
+    (Real.sqrt_pos.2 hκ.1).ne' z
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open GammaField GammaPredictable BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+lemma barrier_expectation {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (η κ : ℝ)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (hz : z ∈ halfPlane) (hu : 0 < u)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample)) :
+    Integrable (C.barrier η κ B) P ∧
+      (∫ sample, C.barrier η κ B sample ∂P) ≤ AngularBarrier.profile (κ/2) ((8/κ-1)/2) (η+u) (z.re/z.im) := by
+  let l := κ/2
+  let b := (8/κ-1)/2
+  let ρ := (2*z.im)*Real.exp (-u)
+  have hl : 0 < l := div_pos hκ.1 (by norm_num)
+  have hb : 0 < b := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    dsimp [b]; linarith
+  have hρ : 0 < ρ := by have hy : 0 < z.im := hz; dsimp [ρ]; positivity
+  let U := C.paths κ B
+  let τ := C.stopTime κ B
+  let F := fun sample => field C.height_pos C.horizon.coe_nonneg l b η ρ (Real.sqrt κ) (U sample) z
+  let V := fun sample => drift C.height_pos C.horizon.coe_nonneg l b η ρ (Real.sqrt κ) (U sample) z
+  let G := AngularBarrier.profile l b (η+u) (z.re/z.im)
+  have hreg : ∀ᵐ sample ∂P, BrownianLocal.LocalRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact regular C.height_pos C.horizon.coe_nonneg hl hb hη hρ C.horizon_pos
+      (Real.sqrt κ) (U sample) z C.angle _ (hc.comp continuous_real_toNNReal)
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ C.horizon :=
+    Filter.Eventually.of_forall (fun sample => C.bounds κ B sample)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coefficient_bound C.height_pos C.horizon.coe_nonneg hl hb hη
+    ρ (Real.sqrt κ) (η+u) (1/C.angle) z 1
+  obtain ⟨K₂,hK₂,hb₂⟩ := coefficient_bound C.height_pos C.horizon.coe_nonneg hl hb hη
+    ρ (Real.sqrt κ) (η+u) (1/C.angle) z 2
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t x : ℝ), remaining C.height_pos C.horizon.coe_nonneg η ρ W z t ≤ η+u →
+        |angleRatio C.height_pos C.horizon.coe_nonneg (Real.sqrt κ) W z t x| ≤ 1/C.angle →
+        |iteratedDeriv k (field C.height_pos C.horizon.coe_nonneg l b η ρ (Real.sqrt κ) W z t) x| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B C.horizon τ F n i sample| ≤ K := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 hm
+    unfold stoppedCoeff
+    split_ifs with hh
+    · have ht : mesh C.horizon n i ∈ Icc 0 (C.stopTime κ B sample) :=
+        ⟨by unfold mesh; positivity,hh.le⟩
+      exact hb (U sample) _ _ (C.remaining_upper κ B sample hm hz hu hc hb0 η ht)
+        (by simpa only [mesh_toNNReal] using C.ratio_upper κ B sample hz hu hc hb0 ht)
+    · simpa only [abs_zero] using hK
+  let A := fun n i => rawCoeff C.height_pos 1 B C.horizon (dyadicTime C.horizon n i) l b η (Real.sqrt κ) z ρ C.angle
+  let D := fun n i => rawCoeff C.height_pos 2 B C.horizon (dyadicTime C.horizon n i) l b η (Real.sqrt κ) z ρ C.angle
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B C.horizon τ F n i :=
+    rawCoeff_stoppedCoeff_ae C.height_pos hB 1 C.horizon C.horizon_pos l b η (Real.sqrt κ) z ρ C.angle n i
+  have hDe (n i : ℕ) : D n i =ᵐ[P] stoppedCoeff 2 B C.horizon τ F n i :=
+    rawCoeff_stoppedCoeff_ae C.height_pos hB 2 C.horizon C.horizon_pos l b η (Real.sqrt κ) z ρ C.angle n i
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (A n i) :=
+    rawCoeff_measurable_past C.height_pos 1 (Or.inl rfl) B C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle
+  have hDm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (D n i) :=
+    rawCoeff_measurable_past C.height_pos 2 (Or.inr rfl) B C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 1 (Or.inl rfl) C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle).aestronglyMeasurable K₁
+    filter_upwards [hAe n i,hsbound 1 K₁ hK₁ hb₁ n i] with sample he hh
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hh
+  have hD2 (n i : ℕ) : MemLp (D n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 2 (Or.inr rfl) C.horizon _ l b η (Real.sqrt κ) hl hb hη z ρ C.angle).aestronglyMeasurable K₂
+    filter_upwards [hDe n i,hsbound 2 K₂ hK₂ hb₂ n i] with sample he hh
+    change |D n i sample| ≤ K₂
+    rw [he]
+    exact hh
+  have hi := BrownianLocal.stopped_field_ito_centered_ae hB C.horizon τ F V htau hreg
+    A D hAe hDe hDm hD2 K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+    (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+    (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
+  change Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 at hi
+  have hmaj : ∀ᵐ sample ∂P, C.barrier η κ B sample ≤ Y sample+G := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
+    have hd := integral_drift_nonpos C.height_pos C.horizon.coe_nonneg hκ hη hρ C.horizon_pos
+      (U sample) z C.angle (fun t => B t.toNNReal sample) (hc.comp continuous_real_toNNReal)
+      (C.path_eq κ B sample hc)
+    have hin : F sample 0 (B 0 sample) = G := by
+      rw [hb0]
+      exact field_initial C.height_pos C.horizon.coe_nonneg (C.height_initial hz hu.le).le hu.le l b η (Real.sqrt κ) (U sample)
+    change _ ≤ Y sample+G
+    dsimp only [Y]
+    rw [hin]
+    change C.barrier η κ B sample ≤ C.barrier η κ B sample - G - _ - _ + G
+    change (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) +
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample) ≤ 0 at hd
+    linarith only [hd]
+  have hInt : Integrable (fun sample => Y sample+G) P := hi.1.add (integrable_const G)
+  have hI : Integrable (C.barrier η κ B) P := by
+    apply hInt.mono' (C.barrier_aemeasurable hB η κ hκ hη).aestronglyMeasurable
+    filter_upwards [hmaj] with sample hh
+    simpa only [Real.norm_eq_abs, abs_of_nonneg (C.barrier_nonneg η κ B sample hκ hη)] using hh
+  refine ⟨hI, ?_⟩
+  calc
+    (∫ sample, C.barrier η κ B sample ∂P) ≤ ∫ sample, Y sample+G ∂P := integral_mono_ae hI hInt hmaj
+    _ = G := by rw [integral_add hi.1 (integrable_const G), hi.2]; simp
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GammaField AngularBarrier
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable (hz : z ∈ halfPlane) (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+include h hz hu hc hb0
+
+lemma barrier_at_radius (η : ℝ) {t : ℝ≥0} (ht : C.stopTime κ B sample = t)
+    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
+    C.barrier η κ B sample = profile (κ/2) ((8/κ-1)/2) η
+      (((g t z).re-Real.sqrt κ*B t sample)/(g t z).im) := by
+  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hs := source_until_stop C.height_pos C.horizon_pos h _ z _ C.angle
+    (C.path_eq κ B sample hc) h0
+  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0
+      ⟨(C.bounds κ B sample).1,le_rfl⟩)).2.1
+  have hpos := hs.2 ⟨(C.bounds κ B sample).1,le_rfl⟩
+  have hr := C.source_radius κ B sample h hz hu hc hb0
+  rw [ht,hrt] at hr
+  unfold barrier GammaField.field remaining
+  rw [rawRemaining_eq C.height_pos C.horizon.coe_nonneg η _ _ z _ hh, ht, hr]
+  simp only [add_sub_cancel_right, max_self]
+  unfold angleRatio LoewnerField.height
+  rw [← ht, max_eq_right hh, ← hpos, ht, Real.toNNReal_coe]
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt
+open AngularBarrier
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+include h hz hu hc hb0
+
+lemma eventually_stop_at_radius {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
+    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
+    ∀ᶠ n : ℕ in atTop, (cutoffs hz u hu.le n).stopTime κ B sample = t := by
+  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
+  have hδ : ∀ᶠ n : ℕ in atTop, shrinkingAngle z n < η :=
+    (shrinkingAngle_tendsto z).eventually (gt_mem_nhds hη)
+  filter_upwards [hδ] with n hn
+  let C := cutoffs hz u hu.le n
+  have hh : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
+    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · exact hr
+    · have htm := hm (C.stopTime κ B sample)
+        ⟨(C.bounds κ B sample).1,C.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
+      rw [hs] at htm
+      exact False.elim ((not_lt_of_ge htm) hn)
+  exact h.live_radius_injective
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (C.bounds κ B sample).1 t.coe_nonneg (C.source_live κ B sample h hz hu hc hb0) hzt
+    (hh.trans hrt.symm)
+
+omit h hz hu hc hb0 in
+def timeShift (n : ℕ) : ℝ := 1/((n:ℝ)+1)
+
+omit h hz hu hc hb0 in
+lemma timeShift_pos (n : ℕ) : 0 < timeShift n := by unfold timeShift; positivity
+
+omit h hz hu hc hb0 in
+lemma timeShift_tendsto : Tendsto timeShift atTop (𝓝[>] (0:ℝ)) := by
+  apply tendsto_nhdsWithin_iff.2
+  constructor
+  · change Tendsto (fun n : ℕ => (1:ℝ)/((n:ℝ)+1)) atTop (𝓝 0)
+    simpa only [Function.comp_def, Nat.cast_add, Nat.cast_one] using
+      (tendsto_const_div_atTop_nhds_zero_nat (1:ℝ)).comp (tendsto_add_atTop_nat 1)
+  · exact Filter.Eventually.of_forall timeShift_pos
+
+lemma barrier_tendsto_on_hit (hκ : 0 < κ ∧ κ < 8)
+    (hhit : firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞)) :
+    Tendsto (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).barrier (timeShift n) κ B sample)) atTop (𝓝 1) := by
+  have hex : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧ loewnerRadius g t z = (2*z.im)*Real.exp (-u) := by
+    by_contra hn
+    have he : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧ loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ := by
+      exact Set.eq_empty_iff_forall_notMem.2 (fun t ht => hn ⟨t,ht⟩)
+    exact hhit (by simp only [firstLiveRadiusStop, he, Set.image_empty, sInf_empty])
+  obtain ⟨t,hzt,hrt⟩ := hex
+  have he : (fun n : ℕ => (cutoffs hz u hu.le n).barrier (timeShift n) κ B sample) =ᶠ[atTop]
+      (fun n => profile (κ/2) ((8/κ-1)/2) (timeShift n)
+        (((g t z).re-Real.sqrt κ*B t sample)/(g t z).im)) := by
+    filter_upwards [eventually_stop_at_radius κ B sample h hz hu hc hb0 hzt hrt] with n hn
+    exact (cutoffs hz u hu.le n).barrier_at_radius κ B sample h hz hu hc hb0 (timeShift n) hn hrt
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have ht := (value_tendsto_zero_time hl hb (squaredSine_pos
+    (((g t z).re-Real.sqrt κ*B t sample)/(g t z).im))).comp timeShift_tendsto
+  have ht' := ht.congr' he.symm
+  simpa only [ENNReal.ofReal_one, Function.comp_def] using
+    ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht'
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteTilt AngularBarrier
+
+lemma radiusApproachProfile {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u) :
+    P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
+      ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) u (z.re/z.im)) := by
+  let C := cutoffs hz u hu.le
+  let F := fun n sample => ENNReal.ofReal ((C n).barrier (timeShift n) κ B sample)
+  let Y := fun sample => liminf (fun n => F n sample) atTop
+  have hFm (n : ℕ) : AEMeasurable (F n) P :=
+    ENNReal.measurable_ofReal.comp_aemeasurable
+      ((C n).barrier_aemeasurable hB (timeShift n) κ hκ (timeShift_pos n))
+  have hYm : AEMeasurable Y P := by
+    dsimp only [Y]
+    simp_rw [liminf_eq_iSup_iInf_of_nat]
+    exact AEMeasurable.iSup (fun n => AEMeasurable.iInf (fun i => AEMeasurable.iInf (fun (_ : n ≤ i) => hFm i)))
+  have hsub : {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤ᵐ[P]
+      {sample | (1:ℝ≥0∞) ≤ Y sample} := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0 hhit
+    have hh := barrier_tendsto_on_hit κ B sample hm hz hu hc hb0 hκ hhit
+    have he : Y sample = 1 := hh.liminf_eq
+    exact he.ge
+  have hexp (n : ℕ) : (∫⁻ sample, F n sample ∂P) ≤
+      ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) (timeShift n+u) (z.re/z.im)) := by
+    have hi := (C n).barrier_expectation hB (timeShift n) κ hκ (timeShift_pos n) hz hu γ g f hmaps
+    rw [← ofReal_integral_eq_lintegral_ofReal hi.1 (Filter.Eventually.of_forall
+      (fun sample => (C n).barrier_nonneg (timeShift n) κ B sample hκ (timeShift_pos n)))]
+    exact ENNReal.ofReal_le_ofReal hi.2
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have ht : Tendsto (fun n : ℕ => ENNReal.ofReal
+      (profile (κ/2) ((8/κ-1)/2) (timeShift n+u) (z.re/z.im))) atTop
+      (𝓝 (ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) u (z.re/z.im)))) := by
+    have hs : Tendsto (fun n => (timeShift n+u,z.re/z.im)) atTop (𝓝 (u,z.re/z.im)) := by
+      have ht0 : Tendsto timeShift atTop (𝓝 (0:ℝ)) := timeShift_tendsto.mono_right nhdsWithin_le_nhds
+      simpa only [zero_add] using (ht0.add_const u).prodMk_nhds tendsto_const_nhds
+    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+      ((contDiffAt_profile (v := (u,z.re/z.im)) hl hb 0 hu).continuousAt.tendsto.comp hs)
+  calc
+    P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
+        P {sample | (1:ℝ≥0∞) ≤ Y sample} := measure_mono_ae hsub
+    _ ≤ ∫⁻ sample, Y sample ∂P := by
+      simpa only [div_one] using meas_ge_le_lintegral_div hYm (by norm_num : (1:ℝ≥0∞) ≠ 0) (by norm_num : (1:ℝ≥0∞) ≠ ⊤)
+    _ ≤ liminf (fun n => ∫⁻ sample, F n sample ∂P) atTop := lintegral_liminf_le' hFm
+    _ ≤ liminf (fun n => ENNReal.ofReal (profile (κ/2) ((8/κ-1)/2) (timeShift n+u) (z.re/z.im))) atTop :=
+      liminf_le_liminf (Filter.Eventually.of_forall hexp)
+    _ = _ := ht.liminf_eq
+
+lemma squaredSine_initial {z : ℂ} (hz : z ∈ halfPlane) :
+    squaredSine (z.re/z.im) = (z.im/‖z‖)^2 := by
+  have hy : 0 < z.im := hz
+  have hn : 0 < ‖z‖ := lt_of_lt_of_le hy (le_trans (le_abs_self _) (Complex.abs_im_le_norm z))
+  have he : ‖z‖^2 = z.re^2+z.im^2 := by
+    rw [← Complex.normSq_eq_norm_sq, Complex.normSq_apply]
+    ring
+  unfold squaredSine
+  field_simp
+  nlinarith
+
+theorem radiusApproachBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C : ℝ, 0 < C ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (_hz : z ∈ halfPlane) (u : ℝ) (_hu : 1 ≤ u),
+      P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
+        ENNReal.ofReal (C * Real.exp (-codimension κ * u) *
+          (z.im / ‖z‖) ^ (8 / κ - 1)) := by
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  obtain ⟨C,hC,hdec⟩ := value_decay hl hb
+  refine ⟨C,hC,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz u hu
+  have hu0 : 0 < u := by linarith
+  apply (radiusApproachProfile κ hκ B hB γ g f hmaps z hz u hu0).trans
+  apply ENNReal.ofReal_le_ofReal
+  have hp : (κ/2)*((8/κ-1)/2)/2 = codimension κ := by
+    unfold codimension dimension
+    field_simp [hκ.1.ne']
+    ring
+  have hq : (squaredSine (z.re/z.im))^((8/κ-1)/2) = (z.im/‖z‖)^(8/κ-1) := by
+    rw [squaredSine_initial hz, ← Real.rpow_natCast (z.im/‖z‖) 2,
+      ← Real.rpow_mul (initial_sine_pos hz).le]
+    congr 1
+    ring
+  have hh := hdec u (squaredSine (z.re/z.im)) hu (squaredSine_pos _).le
+  simpa only [profile,hp,hq] using hh
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.UnivalentDistance
+
+lemma derivative_variation {f : ℂ → ℂ} {a : ℂ} {r A : ℝ}
+    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball a (2*r)))
+    (hb : ∀ z ∈ closedBall a r, ‖deriv f z‖ ≤ 2*A)
+    {z : ℂ} (hz : z ∈ closedBall a (r/2)) :
+    ‖deriv f z - deriv f a‖ ≤ (4*A/r)*‖z-a‖ := by
+  have hd' := hd.deriv isOpen_ball
+  have hsub : closedBall a (r/2) ⊆ ball a (2*r) := closedBall_subset_ball (by linarith)
+  have hbound (w : ℂ) (hw : w ∈ closedBall a (r/2)) :
+      ‖deriv (deriv f) w‖ ≤ 4*A/r := by
+    have hs : closedBall w (r/2) ⊆ closedBall a r := by
+      apply closedBall_subset_closedBall'
+      have hh := mem_closedBall.1 hw
+      linarith
+    have hsc : closure (ball w (r/2)) ⊆ ball a (2*r) := by
+      rw [closure_ball _ (by linarith : r/2 ≠ 0)]
+      exact hs.trans (closedBall_subset_ball (by linarith))
+    have hh := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le
+      (f := deriv f) (by linarith : 0 < r/2) (hd'.mono hsc).diffContOnCl
+      (fun v hv => hb v (hs (sphere_subset_closedBall hv)))
+    convert hh using 1; field_simp; ring
+  exact (convex_closedBall a (r/2)).norm_image_sub_le_of_norm_deriv_le
+    (fun w hw => hd'.differentiableAt (isOpen_ball.mem_nhds (hsub hw))) hbound
+    (mem_closedBall_self (by linarith)) hz
+
+lemma local_image_disk {f : ℂ → ℂ} {a : ℂ} {r : ℝ}
+    (hr : 0 < r) (hd : DifferentiableOn ℂ f (ball a (2*r)))
+    (ha : deriv f a ≠ 0)
+    (hb : ∀ z ∈ closedBall a r, ‖deriv f z‖ ≤ 2*‖deriv f a‖) :
+    closedBall (f a) (‖deriv f a‖*r/32) ⊆ f '' closedBall a (r/16) := by
+  let F : ℂ → ℂ := fun z => (f z-f a)/deriv f a
+  have hA : 0 < ‖deriv f a‖ := norm_pos_iff.2 ha
+  have hs : closedBall a (r/16) ⊆ ball a (2*r) := closedBall_subset_ball (by linarith)
+  have hdF (z : ℂ) (hz : z ∈ closedBall a (r/16)) :
+      HasDerivAt F (deriv f z/deriv f a) z :=
+    ((hd.differentiableAt (isOpen_ball.mem_nhds (hs hz))).hasDerivAt.sub_const _).div_const _
+  have hbound (z : ℂ) (hz : z ∈ closedBall a (r/16)) :
+      ‖deriv f z/deriv f a-1‖ ≤ (1/2:ℝ) := by
+    have hv := derivative_variation hr hd hb
+      (closedBall_subset_closedBall (by linarith : r/16 ≤ r/2) hz)
+    have he : deriv f z/deriv f a-1 = (deriv f z-deriv f a)/deriv f a := by
+      field_simp
+    rw [he,norm_div,div_le_iff₀ hA]
+    have hn := mem_closedBall_iff_norm.1 hz
+    have hn' : (4*‖deriv f a‖/r)*‖z-a‖ ≤ (4*‖deriv f a‖/r)*(r/16) :=
+      mul_le_mul_of_nonneg_left hn (by positivity)
+    have he' : (4*‖deriv f a‖/r)*(r/16) = ‖deriv f a‖/4 := by field_simp; ring
+    rw [he'] at hn'
+    linarith
+  have happ : ApproximatesLinearOn F (ContinuousLinearMap.id ℂ ℂ)
+      (closedBall a (r/16)) (1/2:ℝ≥0) := by
+    intro x hx y hy
+    have hh := (convex_closedBall a (r/16)).norm_image_sub_le_of_norm_hasDerivWithin_le
+      (fun z hz => ((hdF z hz).sub (hasDerivAt_id z)).hasDerivWithinAt) hbound hy hx
+    change ‖(F x-x)-(F y-y)‖ ≤ (1/2:ℝ)*‖x-y‖ at hh
+    have he : (F x-x)-(F y-y) = F x-F y-(x-y) := by ring
+    rw [he] at hh
+    exact hh
+  have hsurj := happ.surjOn_closedBall_of_nonlinearRightInverse
+    (ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse (by linarith : 0 ≤ r/16) Subset.rfl
+  have hsurj' : closedBall 0 (r/32) ⊆ F '' closedBall a (r/16) := by
+    change closedBall (F a) (((↑(ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse.nnnorm : ℝ)⁻¹ - (1/2)) * (r/16)) ⊆ _ at hsurj
+    have hn : (↑(ContinuousLinearEquiv.refl ℂ ℂ).toNonlinearRightInverse.nnnorm : ℝ) = 1 := by
+      simp [ContinuousLinearEquiv.toNonlinearRightInverse]
+    rw [hn,show ((1:ℝ)⁻¹-(1/2))*(r/16) = r/32 by norm_num; ring] at hsurj
+    simpa only [F,sub_self,zero_div] using hsurj
+  intro w hw
+  have hw' : (w-f a)/deriv f a ∈ closedBall (0:ℂ) (r/32) := by
+    rw [mem_closedBall_iff_norm,sub_zero,norm_div,div_le_iff₀ hA]
+    have hh := mem_closedBall_iff_norm.1 hw
+    nlinarith
+  obtain ⟨v,hv,he⟩ := hsurj' hw'
+  refine ⟨v,hv,?_⟩
+  dsimp only [F] at he
+  have he' : f v-f a = w-f a := (div_left_inj' ha).mp he
+  exact sub_left_injective he'
+
+lemma image_disk {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1))
+    (h0 : deriv f 0 ≠ 0) :
+    ∃ b : ℂ, closedBall b (‖deriv f 0‖/128) ⊆ f '' ball 0 1 := by
+  have hs : closedBall (0:ℂ) (1/2:ℝ) ⊆ ball 0 1 := closedBall_subset_ball (by norm_num)
+  have hc : ContinuousOn (fun z : ℂ => (1/2-‖z‖)*‖deriv f z‖)
+      (closedBall 0 (1/2:ℝ)) :=
+    (continuous_const.sub continuous_norm).continuousOn.mul
+      ((hd.deriv isOpen_ball).continuousOn.mono hs).norm
+  obtain ⟨a,ha,hm⟩ := (isCompact_closedBall (0:ℂ) (1/2:ℝ)).exists_isMaxOn
+    (nonempty_closedBall.mpr (by norm_num)) hc
+  have ham : ‖a‖ ≤ (1/2:ℝ) := by simpa using ha
+  have hmax0 := hm (mem_closedBall_self (by norm_num : (0:ℝ) ≤ 1/2))
+  change (1/2-‖(0:ℂ)‖)*‖deriv f 0‖ ≤ (1/2-‖a‖)*‖deriv f a‖ at hmax0
+  simp only [norm_zero,sub_zero] at hmax0
+  have h0p := norm_pos_iff.mpr h0
+  have hAp : 0 < ‖deriv f a‖ := by
+    by_contra hh
+    have he : ‖deriv f a‖ = 0 := le_antisymm (le_of_not_gt hh) (norm_nonneg _)
+    rw [he,mul_zero] at hmax0
+    linarith
+  have halt : ‖a‖ < (1/2:ℝ) := by nlinarith
+  let r : ℝ := (1/2-‖a‖)/2
+  have hr : 0 < r := by dsimp [r]; linarith
+  have he : 2*r = 1/2-‖a‖ := by dsimp [r]; ring
+  have hsub : ball a (2*r) ⊆ ball (0:ℂ) 1 := by
+    apply ball_subset_ball'
+    rw [dist_zero_right,he]
+    linarith
+  have hb (z : ℂ) (hz : z ∈ closedBall a r) : ‖deriv f z‖ ≤ 2*‖deriv f a‖ := by
+    have hzn : ‖z‖ ≤ ‖a‖+r := by
+      have hh := norm_le_norm_add_norm_sub a z
+      rw [norm_sub_rev a z] at hh
+      have hz' := mem_closedBall_iff_norm.1 hz
+      linarith
+    have hz0 : z ∈ closedBall (0:ℂ) (1/2:ℝ) := by
+      simp only [mem_closedBall_iff_norm,sub_zero]
+      linarith
+    have hmz := hm hz0
+    change (1/2-‖z‖)*‖deriv f z‖ ≤ (1/2-‖a‖)*‖deriv f a‖ at hmz
+    have hwel : r ≤ 1/2-‖z‖ := by linarith
+    have hh := mul_le_mul_of_nonneg_right hwel (norm_nonneg (deriv f z))
+    nlinarith
+  refine ⟨f a,?_⟩
+  have hrad : ‖deriv f 0‖/128 ≤ ‖deriv f a‖*r/32 := by nlinarith
+  apply (closedBall_subset_closedBall hrad).trans
+  apply (local_image_disk hr (hd.mono hsub) (norm_pos_iff.mp hAp) hb).trans
+  apply image_mono
+  apply closedBall_subset_ball'
+  rw [dist_zero_right]
+  linarith
+
+lemma exp_injective_derivative_bound {h : ℂ → ℂ}
+    (hd : DifferentiableOn ℂ h (ball 0 1)) (hi : InjOn (Complex.exp ∘ h) (ball 0 1)) :
+    ‖deriv h 0‖ ≤ 256*Real.pi := by
+  by_contra hb
+  have hp := Real.pi_pos
+  have hn : 0 < ‖deriv h 0‖ := by linarith
+  obtain ⟨b,hbim⟩ := image_disk hd (norm_pos_iff.mp hn)
+  have hb0 : b ∈ closedBall b (‖deriv h 0‖/128) := mem_closedBall_self (by positivity)
+  have hb1 : b+2*Real.pi*Complex.I ∈ closedBall b (‖deriv h 0‖/128) := by
+    rw [mem_closedBall_iff_norm,add_sub_cancel_left]
+    norm_num [norm_mul,Complex.norm_real,abs_of_pos hp]
+    linarith
+  obtain ⟨z0,hz0,hz0'⟩ := hbim hb0
+  obtain ⟨z1,hz1,hz1'⟩ := hbim hb1
+  have he : z1 = z0 := hi hz1 hz0 (by
+    simp only [Function.comp_apply,hz0',hz1']
+    exact Complex.exp_periodic b)
+  have he' : (2:ℂ)*Real.pi*Complex.I = 0 := by
+    rw [he,hz0'] at hz1'
+    linear_combination -hz1'
+  have hh : (Real.pi:ℂ) = 0 := (mul_eq_zero.mp he').resolve_right Complex.I_ne_zero |>
+    (fun h => (mul_eq_zero.mp h).resolve_left (by norm_num))
+  exact Real.pi_ne_zero (Complex.ofReal_eq_zero.mp hh)
+
+lemma omitted_value_bound {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball 0 1))
+    (hi : InjOn f (ball 0 1)) {w : ℂ} (hw : w ∉ f '' ball 0 1) :
+    ‖deriv f 0‖ ≤ (256*Real.pi)*‖f 0-w‖ := by
+  have hzero : (0:ℂ) ∈ ball 0 1 := mem_ball_self zero_lt_one
+  have hn (z : ℂ) (hz : z ∈ ball (0:ℂ) 1) : f z-w ≠ 0 := by
+    intro he
+    exact hw ⟨z,hz,sub_eq_zero.mp he⟩
+  have hc : IsSimplyConnected (ball (0:ℂ) 1) := by
+    let := (convex_ball (0:ℂ) (1:ℝ)).contractibleSpace (nonempty_ball.mpr zero_lt_one)
+    change SimplyConnectedSpace (ball (0:ℂ) 1)
+    infer_instance
+  obtain ⟨h,hhc,hhe⟩ := Complex.exists_continuousOn_eqOn_exp_comp hc isOpen_ball
+    (hd.continuousOn.sub continuousOn_const) (by
+      rintro ⟨z,hz,he⟩
+      exact hn z hz he)
+  have hhd (z : ℂ) (hz : z ∈ ball (0:ℂ) 1) :
+      HasDerivAt h (deriv f z/(f z-w)) z := by
+    have hh := HasDerivAt.of_comp_left
+      (hhc.continuousAt (isOpen_ball.mem_nhds hz))
+      (Complex.hasDerivAt_exp (h z))
+      ((hd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const w)
+      (Complex.exp_ne_zero _) (hhe.eventuallyEq_of_mem (isOpen_ball.mem_nhds hz))
+    have he : Complex.exp (h z) = f z-w := hhe hz
+    simpa only [he] using hh
+  have hhd' : DifferentiableOn ℂ h (ball 0 1) := fun z hz => (hhd z hz).differentiableAt.differentiableWithinAt
+  have hhi : InjOn (Complex.exp ∘ h) (ball 0 1) := by
+    intro z hz v hv he
+    apply hi hz hv
+    rw [hhe hz,hhe hv] at he
+    exact sub_left_injective he
+  have hh := exp_injective_derivative_bound hhd' hhi
+  rw [(hhd 0 hzero).deriv,norm_div,div_le_iff₀ (norm_pos_iff.mpr (hn 0 hzero))] at hh
+  exact hh
+
+end SLEExactGauge.UnivalentDistance
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma radius_le_complement_distance (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t)
+    {w : ℂ} (hw : w ∉ localizationDomain γ t) :
+    loewnerRadius g t z ≤ (512*Real.pi)*dist z w := by
+  have hpos := h.im_pos ht hz
+  have hdn := h.deriv_ne_zero ht hz
+  rcases h with ⟨hγ,hzero,him,htrans,hinit,hg,hf,hgmap,hfmap,hfg,hgf,hc,hd,htip⟩
+  let φ : ℂ → ℂ := fun q => g t z + ((g t z).im:ℂ)*q
+  let F : ℂ → ℂ := fun q => f t (φ q)
+  have hφ0 : φ 0 = g t z := by simp [φ]
+  have hφ (q : ℂ) (hq : q ∈ ball (0:ℂ) 1) : φ q ∈ halfPlane := by
+    have hqn : ‖q‖ < 1 := by simpa using hq
+    have hqi := Complex.abs_im_le_norm q
+    have hql := neg_abs_le q.im
+    change 0 < (φ q).im
+    simp only [φ,Complex.add_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,
+      zero_mul,add_zero]
+    nlinarith
+  have hdφ (q : ℂ) : HasDerivAt φ ((g t z).im:ℂ) q := by
+    simpa only [mul_one,id_eq] using! ((hasDerivAt_id q).const_mul ((g t z).im:ℂ)).const_add (g t z)
+  have hdF (q : ℂ) (hq : q ∈ ball (0:ℂ) 1) : DifferentiableAt ℂ F q :=
+    (hf t ht _ (hφ q hq)).differentiableAt.comp q (hdφ q).differentiableAt
+  have hiF : InjOn F (ball 0 1) := by
+    intro q hq v hv he
+    have hh := congrArg (g t) he
+    change g t (f t (φ q)) = g t (f t (φ v)) at hh
+    rw [hgf t ht _ (hφ q hq),hgf t ht _ (hφ v hv)] at hh
+    dsimp only [φ] at hh
+    exact mul_left_cancel₀ (Complex.ofReal_ne_zero.mpr hpos.ne') (add_left_cancel hh)
+  have hFw : w ∉ F '' ball 0 1 := by
+    rintro ⟨q,hq,he⟩
+    exact hw (he ▸ hfmap t ht (hφ q hq))
+  have hgd := (hg t ht z hz).differentiableAt.hasDerivAt
+  have hfd := (hf t ht _ (hgmap t ht hz)).differentiableAt.hasDerivAt
+  have heq : (fun v => f t (g t v)) =ᶠ[𝓝 z] id := by
+    filter_upwards [(survivingDomain_open hγ ht).mem_nhds hz] with v hv
+    exact hfg t ht v hv
+  have he := ((hfd.comp z hgd).congr_of_eventuallyEq heq.symm).unique (hasDerivAt_id z)
+  have hen : ‖deriv (f t) (g t z)‖*‖deriv (g t) z‖ = 1 := by
+    simpa only [norm_mul,norm_one] using congrArg norm he
+  have hFd : deriv F 0 = deriv (f t) (g t z)*((g t z).im:ℂ) := by
+    have hfd' : HasDerivAt (f t) (deriv (f t) (g t z)) (φ 0) := hφ0.symm ▸ hfd
+    exact (hfd'.comp 0 (hdφ 0)).deriv
+  have hF0 : F 0 = z := by simp only [F,hφ0,hfg t ht z hz]
+  have hR : loewnerRadius g t z = 2*‖deriv F 0‖ := by
+    rw [loewnerRadius,hFd,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hpos,
+      div_eq_iff (norm_ne_zero_iff.mpr hdn)]
+    nlinarith
+  have hh := UnivalentDistance.omitted_value_bound
+    (fun q hq => (hdF q hq).differentiableWithinAt) hiF hFw
+  rw [hF0] at hh
+  rw [hR,dist_eq_norm]
+  nlinarith
+
+lemma radius_le_boundary_distance (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    loewnerRadius g t z ≤ (512*Real.pi)*infDist z (localizationDomain γ t)ᶜ := by
+  have hp : 0 < 512*Real.pi := by positivity
+  have hn : ((localizationDomain γ t)ᶜ).Nonempty := by
+    refine ⟨0,?_⟩
+    intro h0
+    exact (lt_irrefl (0:ℝ)) (by simpa [halfPlane] using h0.1.1)
+  rw [mul_comm (512*Real.pi),← div_le_iff₀ hp]
+  apply (le_infDist hn).2
+  intro w hw
+  rw [div_le_iff₀ hp]
+  exact (h.radius_le_complement_distance ht hz hw).trans_eq (mul_comm _ _)
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+def liveDistanceApproach (γ : ℝ → ℂ) (z : ℂ) (ε : ℝ) : Prop :=
+  ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧ infDist z (localizationDomain γ t)ᶜ < ε
+
+lemma liveDistanceApproach_radius_hit {driver : ℝ → ℝ} {γ : ℝ → ℂ}
+    {g f : ℝ → ℂ → ℂ} (h : IsLoewnerTraceUsing driver γ g f)
+    (hd : ContinuousOn driver (Ici 0)) {z : ℂ} (hz : z ∈ halfPlane)
+    {ε u : ℝ} (hr : (2*z.im)*Real.exp (-u) = (512*Real.pi)*ε)
+    (hsmall : (512*Real.pi)*ε ≤ 2*z.im)
+    (hhit : liveDistanceApproach γ z ε) :
+    firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞) := by
+  obtain ⟨t,hzt,hclose⟩ := hhit
+  have hrt : loewnerRadius g t z < (2*z.im)*Real.exp (-u) := by
+    rw [hr]
+    exact (h.radius_le_boundary_distance t.coe_nonneg hzt).trans_lt
+      (mul_lt_mul_of_pos_left hclose (by positivity))
+  obtain ⟨s,hs,hseq⟩ := intermediate_value_Icc' t.coe_nonneg
+    (h.continuous_radius t.coe_nonneg (hd.mono Icc_subset_Ici_self) hzt)
+    ⟨hrt.le,by rw [h.radius_initial hz,hr]; exact hsmall⟩
+  apply ne_top_of_le_ne_top (ENNReal.coe_ne_top (r := ⟨s,hs.1⟩))
+  apply sInf_le
+  exact ⟨⟨s,hs.1⟩,⟨survivingDomain_mono hs.2 hzt,hseq⟩,rfl⟩
+
+theorem euclideanApproachBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ c < 1 ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (_hz : z ∈ halfPlane) (ε : ℝ) (_hε : 0 < ε) (_hsmall : ε < c*z.im),
+      P {sample | liveDistanceApproach (γ sample) z ε} ≤
+        ENNReal.ofReal (C * ((2*z.im)^(-codimension κ) *
+          (z.im/‖z‖)^(8/κ-1)) * ε^(codimension κ)) := by
+  obtain ⟨C,hC,hbound⟩ := radiusApproachBound κ hκ
+  let K : ℝ := 512*Real.pi
+  have hK : 0 < K := by dsimp [K]; positivity
+  have hK1 : 1 < K := by dsimp [K]; nlinarith [Real.pi_gt_three]
+  have he : 1 ≤ Real.exp 1 := Real.one_le_exp (by norm_num)
+  have heK : 1 < K*Real.exp 1 := by nlinarith
+  refine ⟨C*K^(codimension κ),1/(K*Real.exp 1),mul_pos hC (Real.rpow_pos_of_pos hK _),
+    by positivity,(div_lt_one (by positivity)).2 heK,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz ε hε hsmall
+  have hy : 0 < z.im := hz
+  have hsmall' : ε*(K*Real.exp 1) < z.im := by
+    rw [one_div_mul_eq_div] at hsmall
+    exact (lt_div_iff₀ (by positivity)).mp hsmall
+  have hKε : 0 < K*ε := mul_pos hK hε
+  let u : ℝ := Real.log ((2*z.im)/(K*ε))
+  have hu : 1 ≤ u := by
+    have hh : Real.exp 1 ≤ (2*z.im)/(K*ε) := by
+      rw [le_div_iff₀ hKε]
+      nlinarith
+    simpa only [Real.log_exp] using Real.log_le_log (Real.exp_pos 1) hh
+  have hr : (2*z.im)*Real.exp (-u) = K*ε := by
+    rw [Real.exp_neg,Real.exp_log (by positivity : 0 < (2*z.im)/(K*ε))]
+    field_simp
+  have hstart : K*ε ≤ 2*z.im := by
+    have hh : K*ε ≤ ε*(K*Real.exp 1) := by nlinarith
+    linarith
+  have hsub : {sample | liveDistanceApproach (γ sample) z ε} ≤ᵐ[P]
+      {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} := by
+    filter_upwards [hmaps,hB.cont] with sample hm hc hhit
+    exact liveDistanceApproach_radius_hit hm
+      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+      hz hr hstart hhit
+  apply (measure_mono_ae hsub).trans
+  apply (hbound P B hB γ g f hγ hmaps z hz u hu).trans
+  apply ENNReal.ofReal_le_ofReal
+  have hexp : Real.exp (-codimension κ*u) =
+      (K*ε)^(codimension κ)*(2*z.im)^(-codimension κ) := by
+    rw [Real.rpow_def_of_pos hKε,Real.rpow_def_of_pos (by positivity : 0 < 2*z.im),
+      ← Real.exp_add]
+    congr 1
+    dsimp [u]
+    rw [Real.log_div (by positivity : 2*z.im ≠ 0) hKε.ne']
+    ring
+  rw [hexp,Real.mul_rpow hK.le hε.le]
+  exact le_of_eq (by ring)
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma future_tip_mem_closure (h : IsLoewnerTraceUsing driver γ g f)
+    {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t) : γ t ∈ closure (localizationDomain γ s) := by
+  rcases h with ⟨hγ,hzero,him,htrans,hinit,hg,hf,hgmap,hfmap,hfg,hgf,hc,hd,htip⟩
+  apply isClosed_closure.mem_of_tendsto (htip t (hs.trans hst))
+  filter_upwards [self_mem_nhdsWithin] with y hy
+  have hh : (driver t : ℂ) + (y:ℂ)*Complex.I ∈ halfPlane := by
+    simpa only [halfPlane,mem_ofPred_eq,mem_Ioi,Complex.add_im,Complex.ofReal_im,Complex.mul_im,
+      Complex.ofReal_re,Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_mul,add_zero,zero_add] using hy
+  exact subset_closure (survivingDomain_mono hst (hfmap t (hs.trans hst) hh))
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/AngularMoments.lean b/lean/OAI/Probability/SLELocalization/AngularMoments.lean
index 4f5d84ce890c01e799789430e93378f49a87130c..5e3e3b8c6187d4824529e76d7e298e7581d375d4 100644
--- a/lean/OAI/Probability/SLELocalization/AngularMoments.lean
+++ b/lean/OAI/Probability/SLELocalization/AngularMoments.lean
@@ -1,941 +1,941 @@
-import Mathlib
-import OAI.Probability.SLELocalization.ReverseFlow
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Angular Barrier estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.AngularBarrier
-
-lemma value_decay {l b : ℝ} (hl : 0 < l) (hb : 0 < b) :
-    ∃ K : ℝ, 0 < K ∧ ∀ t q : ℝ, 1 ≤ t → 0 ≤ q →
-      value l b t q ≤ K*Real.exp (-(l*b/2)*t)*q^b := by
-  let c := (1-Real.exp (-l))/(2*(b+1))
-  have hc : 0 < c := div_pos (sub_pos.2 (Real.exp_lt_one_iff.2 (by linarith))) (by positivity)
-  have hg : 0 < Real.Gamma b := Real.Gamma_pos_of_pos hb
-  refine ⟨1/(b*Real.Gamma b*c^b), by positivity, ?_⟩
-  intro t q ht hq
-  have ht0 : 0 < t := by linarith
-  have ha := scale_pos hl hb ht0
-  have hlower : c*Real.exp (l*t) ≤ scale l b t := by
-    have hh : 1 ≤ Real.exp (l*t)*Real.exp (-l) := by
-      rw [← Real.exp_add]
-      exact Real.one_le_exp_iff.2 (by nlinarith)
-    dsimp only [c,scale]
-    apply (le_div_iff₀ (by positivity : 0 < 2*(b+1))).2
-    have heq : ((1-Real.exp (-l))/(2*(b+1))*Real.exp (l*t))*(2*(b+1)) =
-        (1-Real.exp (-l))*Real.exp (l*t) := by field_simp
-    rw [heq]
-    nlinarith
-  have hratio : q/scale l b t ≤ q/(c*Real.exp (l*t)) :=
-    div_le_div_of_nonneg_left hq (mul_pos hc (Real.exp_pos _)) hlower
-  have hpow := Real.rpow_le_rpow (div_nonneg hq ha.le) hratio hb.le
-  have hbound := (cdf_le_power hb (div_nonneg hq ha.le)).trans
-    (div_le_div_of_nonneg_right hpow (mul_pos hb hg).le)
-  have he : Real.exp (l*b/2*t)*(q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b) =
-      1/(b*Real.Gamma b*c^b)*Real.exp (-(l*b/2)*t)*q^b := by
-    rw [Real.div_rpow hq (mul_pos hc (Real.exp_pos _)).le,
-      Real.mul_rpow hc.le (Real.exp_pos _).le, ← Real.exp_mul]
-    have hratioexp : Real.exp (l*b/2*t)/Real.exp (l*t*b) = Real.exp (-(l*b/2)*t) := by
-      rw [← Real.exp_sub]; congr 1; ring
-    calc
-      _ = (Real.exp (l*b/2*t)/Real.exp (l*t*b))*q^b/(b*Real.Gamma b*c^b) := by ring
-      _ = _ := by rw [hratioexp]; ring
-  calc
-    _ ≤ Real.exp (l*b/2*t)*((q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b)) :=
-      mul_le_mul_of_nonneg_left hbound (Real.exp_pos _).le
-    _ = _ := by rw [← mul_div_assoc, he]
-
-def squaredSine (w : ℝ) := 1/(1+w^2)
-
-def profile (l b t w : ℝ) := value l b t (squaredSine w)
-
-def profileT (l b t w : ℝ) := valueT l b t (squaredSine w)
-
-def profileW (l b t w : ℝ) := (-2*w/(1+w^2)^2)*valueQ l b t (squaredSine w)
-
-def profileWW (l b t w : ℝ) := (6*w^2-2)/(1+w^2)^3*valueQ l b t (squaredSine w) +
-  4*w^2/(1+w^2)^4*valueQQ l b t (squaredSine w)
-
-lemma squaredSine_pos (w : ℝ) : 0 < squaredSine w := by unfold squaredSine; positivity
-
-lemma squaredSine_le_one (w : ℝ) : squaredSine w ≤ 1 := by
-  unfold squaredSine; exact (div_le_one (by positivity)).2 (by nlinarith [sq_nonneg w])
-
-lemma hasDerivAt_squaredSine (w : ℝ) :
-    HasDerivAt squaredSine (-2*w/(1+w^2)^2) w := by
-  apply ((hasDerivAt_const w 1).div (((hasDerivAt_id w).pow 2).const_add 1) (by simp only [Pi.pow_apply, id_eq]; positivity)).congr_deriv
-  simp only [Pi.pow_apply, id_eq]; ring
-
-lemma contDiff_squaredSine (n : WithTop ℕ∞) : ContDiff ℝ n squaredSine := by
-  unfold squaredSine
-  exact contDiff_const.div (contDiff_const.add (contDiff_id.pow 2)) (fun w => by positivity)
-
-lemma contDiffAt_profile {l b : ℝ} (hl : 0 < l) (hb : 0 < b) (n : ℕ)
-    {v : ℝ × ℝ} (ht : 0 < v.1) :
-    ContDiffAt ℝ n (fun v : ℝ × ℝ => profile l b v.1 v.2) v := by
-  have hq := (contDiff_squaredSine n).contDiffAt.comp v contDiffAt_snd
-  have hm : ContDiffAt ℝ n (fun v : ℝ × ℝ => (v.1,squaredSine v.2)) v :=
-    contDiffAt_fst.prodMk hq
-  have hh := (contDiffAt_value (v:=(v.1,squaredSine v.2)) hl hb n ht (squaredSine_pos v.2)).comp v hm
-  dsimp only [Function.comp_def] at hh
-  exact hh
-
-lemma hasDerivAt_profile_w {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (w : ℝ) :
-    HasDerivAt (profile l b t) (profileW l b t w) w := by
-  have hh := (hasDerivAt_value_q hl hb ht (squaredSine_pos w)).comp w (hasDerivAt_squaredSine w)
-  apply hh.congr_deriv
-  dsimp only [profileW]; ring
-
-lemma hasDerivAt_profile_t {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (w : ℝ) :
-    HasDerivAt (fun s => profile l b s w) (profileT l b t w) t :=
-  hasDerivAt_value_t hl hb ht (squaredSine_pos w)
-
-lemma deriv_squaredSine_hasDerivAt (w : ℝ) :
-    HasDerivAt (fun w : ℝ => -2*w/(1+w^2)^2) ((6*w^2-2)/(1+w^2)^3) w := by
-  have hd := ((hasDerivAt_id w).const_mul (-2)).div
-    ((((hasDerivAt_id w).pow 2).const_add 1).pow 2) (by simp only [Pi.pow_apply, id_eq]; positivity)
-  apply hd.congr_deriv
-  simp only [Pi.pow_apply, id_eq]
-  field_simp
-  ring
-
-lemma hasDerivAt_profileW {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (w : ℝ) :
-    HasDerivAt (profileW l b t) (profileWW l b t w) w := by
-  have hh := (deriv_squaredSine_hasDerivAt w).mul
-    ((hasDerivAt_valueQ_q hl hb ht (squaredSine_pos w)).comp w (hasDerivAt_squaredSine w))
-  apply hh.congr_deriv
-  dsimp only [profileWW, Function.comp_def]
-  field_simp
-  ring
-
-lemma deriv_profile {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) :
-    deriv (profile l b t) = profileW l b t := funext fun w => (hasDerivAt_profile_w hl hb ht w).deriv
-
-lemma second_profile {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) :
-    iteratedDeriv 2 (profile l b t) = profileWW l b t := by
-  simp only [iteratedDeriv_succ, iteratedDeriv_zero, deriv_profile hl hb ht]
-  exact funext fun w => (hasDerivAt_profileW hl hb ht w).deriv
-
-lemma parameter_deriv_smooth (f : ℝ → ℝ → ℝ) (n : ℕ) {v : ℝ × ℝ}
-    (hf : ContDiffAt ℝ (n+1) (Function.uncurry f) v) :
-    ContDiffAt ℝ n (fun v : ℝ × ℝ => deriv (f v.1) v.2) v := by
-  let g : (ℝ × ℝ) → ℝ → ℝ := fun v w => f v.1 w
-  have hh : ContDiffAt ℝ (n+1) (Function.uncurry g) (v,v.2) := by
-    have hm : ContDiffAt ℝ (n+1) (fun r : (ℝ × ℝ) × ℝ => (r.1.1,r.2)) (v,v.2) := by fun_prop
-    exact hf.comp (v,v.2) hm
-  have hd := hh.fderiv (m:=n) contDiffAt_snd (by simp)
-  exact hd.clm_apply contDiffAt_const
-
-lemma parameter_iteratedDeriv_smooth (f : ℝ → ℝ → ℝ)
-    (hf : ∀ n : ℕ, ∀ v : ℝ × ℝ, 0 < v.1 → ContDiffAt ℝ n (Function.uncurry f) v)
-    (k n : ℕ) (v : ℝ × ℝ) (hv : 0 < v.1) :
-    ContDiffAt ℝ n (fun v : ℝ × ℝ => iteratedDeriv k (f v.1) v.2) v := by
-  induction k generalizing n with
-  | zero => exact hf n v hv
-  | succ k ih =>
-    simpa only [iteratedDeriv_succ] using parameter_deriv_smooth (fun t => iteratedDeriv k (f t)) n (ih (n+1))
-
-lemma profile_derivative_continuousAt {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
-    (k : ℕ) {v : ℝ × ℝ} (hv : 0 < v.1) :
-    ContinuousAt (fun v : ℝ × ℝ => iteratedDeriv k (profile l b v.1) v.2) v :=
-  (parameter_iteratedDeriv_smooth (profile l b) (fun n _v hv => contDiffAt_profile hl hb n hv)
-    k 0 v hv).continuousAt
-
-lemma profile_super {κ t : ℝ} (hκ : 0 < κ ∧ κ < 8) (ht : 0 < t) (w : ℝ) :
-    w*(1+w^2)*profileW (κ/2) ((8/κ-1)/2) t w +
-      κ/8*(1+w^2)^2*profileWW (κ/2) ((8/κ-1)/2) t w ≤
-        profileT (κ/2) ((8/κ-1)/2) t w := by
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have hi := supersolution (l:=κ/2) (b:=(8/κ-1)/2) (t:=t) (q:=squaredSine w) (div_pos hκ.1 (by norm_num)) hb ht (squaredSine_pos w) (squaredSine_le_one w)
-  change _ ≤ valueT (κ/2) ((8/κ-1)/2) t (squaredSine w)
-  convert hi using 1
-  dsimp only [profileW, profileWW, squaredSine]
-  field_simp [hκ.1.ne']
-  ring
-
-lemma hasDerivAt_value_path {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
-    {T Q : ℝ → ℝ} {s T' Q' : ℝ} (hT : HasDerivAt T T' s) (hQ : HasDerivAt Q Q' s)
-    (ht : 0 < T s) (hq : 0 < Q s) :
-    HasDerivAt (fun v => value l b (T v) (Q v))
-      (valueT l b (T s) (Q s)*T' + valueQ l b (T s) (Q s)*Q') s := by
-  have ha := (hasDerivAt_scale l b (T s)).comp s hT
-  have hs := scale_pos hl hb ht
-  have hh := ((hT.const_mul (l*b/2)).exp).mul
-    ((hasDerivAt_cdf hb (div_pos hq hs)).comp s (hQ.div ha hs.ne'))
-  apply hh.congr_deriv
-  simp only [Function.comp_def, Pi.div_apply]
-  dsimp only [valueT, valueQ, value]
-  have he : Real.exp (l*T s) = 2*(b+1)*scale l b (T s)+1 := by unfold scale; field_simp; ring
-  rw [he]
-  field_simp
-  ring
-
-lemma hasDerivAt_profile_path {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
-    {T W : ℝ → ℝ} {s T' W' : ℝ} (hT : HasDerivAt T T' s) (hW : HasDerivAt W W' s)
-    (ht : 0 < T s) :
-    HasDerivAt (fun v => profile l b (T v) (W v))
-      (profileT l b (T s) (W s)*T' + profileW l b (T s) (W s)*W') s := by
-  have hq := (hasDerivAt_squaredSine (W s)).comp s hW
-  have hh := hasDerivAt_value_path hl hb hT hq ht (squaredSine_pos (W s))
-  apply hh.congr_deriv
-  simp only [profileT,profileW,Function.comp_def]
-  ring
-
-lemma profile_smooth {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (n : ℕ) :
-    ContDiff ℝ n (profile l b t) := by
-  apply contDiff_iff_contDiffAt.2
-  intro w
-  have hh := (contDiffAt_profile (v:=(t,w)) hl hb n ht).comp w (contDiffAt_const.prodMk contDiffAt_id)
-  dsimp only [Function.comp_def] at hh
-  exact hh
-
-lemma iteratedDeriv_affine (f : ℝ → ℝ) (hf : ∀ n : ℕ, ContDiff ℝ n f) (k : ℕ) (a c x : ℝ) :
-    iteratedDeriv k (fun x => f (a*x+c)) x = a^k*iteratedDeriv k f (a*x+c) := by
-  induction k generalizing x with
-  | zero => simp
-  | succ k ih =>
-    have he : iteratedDeriv k (fun x => f (a*x+c)) = fun x => a^k*iteratedDeriv k f (a*x+c) := funext ih
-    rw [iteratedDeriv_succ, he]
-    have hd : Differentiable ℝ (iteratedDeriv k f) :=
-      ContDiff.differentiable_iteratedDeriv k (hf (k+1)) (by exact_mod_cast Nat.lt_succ_self k)
-    have hh := ((hd (a*x+c)).hasDerivAt.comp x
-      (((hasDerivAt_id x).const_mul a).add_const c)).const_mul (a^k)
-    dsimp only [Function.comp_def, id_eq] at hh
-    rw [hh.deriv]
-    simp only [iteratedDeriv_succ,pow_succ,mul_one]
-    ring
-
-end SLEExactGauge.AngularBarrier
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianLocal
-open BrownianPredictable PredictableLimit BrownianIto TaylorMeshes TimeField
-open BrownianField BrownianStopped PartitionMeshes StoppedMeshes
-
-structure LocalRegular (F V : ℝ → ℝ → ℝ) (b : ℝ → ℝ) (T : ℝ) : Prop where
-  smooth : ∀ t, ContDiff ℝ 3 (F t)
-  third_cont : ContinuousOn (fun p : ℝ × ℝ => iteratedDeriv 3 (F p.1) p.2) ((Icc 0 T) ×ˢ univ)
-  time_cont : ∀ x, ContinuousOn (fun t => F t x) (Icc 0 T)
-  time_deriv : ∀ t ∈ Ioo 0 T, ∀ x, HasDerivAt (fun u => F u x) (V t x) t
-  drift_cont : ContinuousOn (fun p : ℝ × ℝ => V p.1 (b p.2)) ((Icc 0 T) ×ˢ (Icc 0 T))
-  second_cont : ContinuousOn (fun t => iteratedDeriv 2 (F t) (b t)) (Icc 0 T)
-
-lemma family_remainder_tendsto_of_eventually_bounded {T : ℝ} (p : PartitionSequence T)
-    {f : ℕ → ℕ → ℝ → ℝ} {b : ℝ → ℝ}
-    (hf : ∀ n i, ContDiff ℝ 3 (f n i)) {V K : ℝ}
-    (hb : ContinuousOn b (Icc 0 T)) (hV : 0 ≤ V) (hK : 0 ≤ K)
-    (h3 : ∀ n i, i < p.count n → ∀ x ∈ uIcc (b (p.time n i)) (b (p.time n (i+1))),
-      |iteratedDeriv 3 (f n i) x| ≤ K)
-    (hQ : ∀ᶠ n in atTop, (∑ i ∈ Finset.range (p.count n),
-      (b (p.time n (i+1))-b (p.time n i))^2) ≤ V) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      remainder₂ (f n i) (b (p.time n i)) (b (p.time n (i+1)))) atTop (𝓝 0) := by
-  apply Metric.tendsto_atTop.2
-  intro ε hε
-  let δ := ε / ((K+1)*(V+1))
-  have hδ : 0 < δ := div_pos hε (mul_pos (by linarith) (by linarith))
-  have hs := p.increments_small hb hδ
-  obtain ⟨N, hN⟩ := eventually_atTop.1 (hs.and hQ)
-  refine ⟨N, fun n hn => ?_⟩
-  rcases hN n hn with ⟨hs, hq⟩
-  have hr := sum_family_remainder₂_bound (hf n) (fun i => b (p.time n i))
-    (p.count n) K δ hK hδ.le (fun i hi z hz => h3 n i hi z hz) (fun i hi => (hs i hi).le)
-  simp only [Real.dist_eq, sub_zero]
-  calc
-    _ ≤ K * δ / 6 * ∑ i ∈ Finset.range (p.count n),
-        (b (p.time n (i+1))-b (p.time n i))^2 := hr
-    _ ≤ K * δ / 6 * (V+1) :=
-      mul_le_mul_of_nonneg_left (hq.trans (by linarith)) (by positivity)
-    _ < ε := by
-      dsimp [δ]
-      have hden : 0 < (K+1)*(V+1) := mul_pos (by linarith) (by linarith)
-      have hid : K * (ε / ((K+1)*(V+1))) / 6 * (V+1) = ε * (K / (6*(K+1))) := by
-        field_simp
-      rw [hid]
-      have hk : K / (6*(K+1)) < 1 := (div_lt_one (by positivity)).2 (by linarith)
-      simpa using mul_lt_mul_of_pos_left hk hε
-
-lemma stopped_field_taylor_limit {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T)
-    {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ}
-    (hb : ContinuousOn b (Icc 0 T)) (hreg : LocalRegular F V b τ)
-    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1)) - b (mesh T n i))^2) atTop (𝓝 T))
-    {K₁ K₂ : ℝ} (hK₁ : 0 ≤ K₁) (hK₂ : 0 ≤ K₂)
-    (hA : ∀ n i, mesh T n i < τ →
-      |deriv (F (mesh T n i)) (b (mesh T n i))| ≤ K₁)
-    (hC : ∀ n i, mesh T n i < τ →
-      |iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i))| ≤ K₂)
-    (hE : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-      iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i)) *
-        ((b (mesh T n (i+1)) - b (mesh T n i))^2 - T/(2 : ℝ)^n) else 0) atTop (𝓝 0)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-      deriv (F (mesh T n i)) (b (mesh T n i)) *
-        (b (mesh T n (i+1)) - b (mesh T n i)) else 0) atTop
-      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, iteratedDeriv 2 (F t) (b t))) := by
-  let p := clippedDyadic T τ hT hτ hτT
-  let C : ℝ → ℝ := fun t => iteratedDeriv 2 (F t) (b t)
-  let A : ℝ → ℝ := fun t => deriv (F t) (b t)
-  have hbτ := hb.mono (Icc_subset_Icc le_rfl hτT)
-  have hQc := clipped_quadratic_eventually_bounded hT hτ hτT hb hQ
-  obtain ⟨R,hR⟩ := isCompact_Icc.exists_bound_of_continuousOn hbτ
-  have hc := hreg.third_cont.mono (Set.prod_mono Subset.rfl (subset_univ (Icc (-R) R)))
-  obtain ⟨K,hK⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hc
-  have hr := family_remainder_tendsto_of_eventually_bounded p
-    (f := fun n i => F (p.time n i)) (fun _ _ => hreg.smooth _)
-    hbτ (show 0 ≤ T+2 by linarith) (le_max_left 0 K) (fun n i hi x hx => by
-      have hleft : b (p.time n i) ∈ Icc (-R) R := abs_le.1 (by simpa only [Real.norm_eq_abs] using hR _ (p.mem n i hi.le))
-      have hright : b (p.time n (i+1)) ∈ Icc (-R) R := abs_le.1 (by simpa only [Real.norm_eq_abs] using hR _ (p.mem n (i+1) hi))
-      have hkx := hK (p.time n i,x) ⟨p.mem n i hi.le, uIcc_subset_Icc hleft hright hx⟩
-      rw [Real.norm_eq_abs] at hkx
-      exact hkx.trans (le_max_right 0 K)) hQc
-  have htime := p.time_increment_tendsto hreg.time_cont hreg.time_deriv hreg.drift_cont
-  have hriem := prestop_riemann_tendsto hT hτ hτT hreg.second_cont
-  let Qpred : ℕ → ℝ := fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-    C (mesh T n i) * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0
-  have hQpred : Tendsto Qpred atTop (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
-    have heq (n : ℕ) : Qpred n =
-      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-        C (mesh T n i) * ((b (mesh T n (i+1))-b (mesh T n i))^2-T/(2 : ℝ)^n) else 0) +
-      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then C (mesh T n i)*(T/(2 : ℝ)^n) else 0) := by
-      unfold Qpred
-      rw [← Finset.sum_add_distrib]
-      apply Finset.sum_congr rfl
-      intro i hi
-      split_ifs <;> ring
-    dsimp only [C] at heq
-    simpa only [← heq, zero_add] using hE.add hriem
-  have hQdiff := clipped_quadraticSum_difference_tendsto hT hτ hτT hb
-    (fun n i => C (p.time n i))
-    (K := max K₂ |C τ|) (hK₂.trans (le_max_left _ _))
-    (coefficient_clipped_bound C hC)
-  have hQeq (n i : ℕ) :
-      (if mesh T n i < τ then C (p.time n i) *
-        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) =
-      (if mesh T n i < τ then C (mesh T n i) *
-        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) := by
-    split_ifs with h
-    · simp only [p, clippedDyadic, min_eq_left h.le]
-    · rfl
-  simp only [hQeq] at hQdiff
-  have hQweighted : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      C (p.time n i) * (b (p.time n (i+1)) - b (p.time n i))^2) atTop
-      (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
-    simpa only [Qpred, sub_add_cancel, zero_add, p, clippedDyadic] using hQdiff.add hQpred
-  have hTel (n : ℕ) : ∑ i ∈ Finset.range (p.count n),
-      (F (p.time n (i+1)) (b (p.time n (i+1))) -
-        F (p.time n i) (b (p.time n i))) = F τ (b τ) - F 0 (b 0) := by
-    rw [Finset.sum_range_sub (fun i => F (p.time n i) (b (p.time n i))), p.start, p.finish]
-  have hIdentity (n : ℕ) : (∑ i ∈ Finset.range (p.count n),
-      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) =
-      F τ (b τ) - F 0 (b 0) -
-      (∑ i ∈ Finset.range (p.count n),
-        (F (p.time n (i+1)) (b (p.time n (i+1))) -
-          F (p.time n i) (b (p.time n (i+1))))) - (1/2 : ℝ)*
-      (∑ i ∈ Finset.range (p.count n),
-        C (p.time n i)*(b (p.time n (i+1))-b (p.time n i))^2) -
-      (∑ i ∈ Finset.range (p.count n), remainder₂ (F (p.time n i))
-        (b (p.time n i)) (b (p.time n (i+1)))) := by
-    rw [← hTel n]
-    simp only [Finset.mul_sum, ← Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl
-    intro i hi
-    simp only [A, C, remainder₂, iteratedDeriv_succ, iteratedDeriv_zero]
-    ring
-  have hAclip : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) atTop
-      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, C t)) := by
-    have hh := (((tendsto_const_nhds (x := F τ (b τ) - F 0 (b 0))).sub
-      htime).sub (hQweighted.const_mul (1/2 : ℝ))).sub hr
-    simpa only [← hIdentity, sub_zero] using hh
-  have hAdiff := clipped_linearSum_difference_tendsto hT hτ hτT hb
-    (fun n i => A (p.time n i))
-    (K := max K₁ |A τ|) (hK₁.trans (le_max_left _ _))
-    (coefficient_clipped_bound A hA)
-  have hAeq (n i : ℕ) :
-      (if mesh T n i < τ then A (p.time n i) *
-        (b (mesh T n (i+1))-b (mesh T n i)) else 0) =
-      (if mesh T n i < τ then A (mesh T n i) *
-        (b (mesh T n (i+1))-b (mesh T n i)) else 0) := by
-    split_ifs with h
-    · simp only [p, clippedDyadic, min_eq_left h.le]
-    · rfl
-  simp only [hAeq] at hAdiff
-  have hh := hAclip.sub hAdiff
-  simpa only [p, clippedDyadic, A, C, sub_sub_cancel, sub_zero] using hh
-
-theorem stopped_fieldSum_tendsto_ae {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
-    (hτ : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T)
-    (hreg : ∀ᵐ sample ∂P, LocalRegular (F sample) (V sample) (fun t => B t.toNNReal sample) (τ sample))
-    (C : ℕ → ℕ → Ω → ℝ)
-    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
-    (hC2 : ∀ n i, MemLp (C n i) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 2 B T τ F n i sample| ≤ K₂)
-    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
-    (hAbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 1 B T τ F n i sample| ≤ K₁) :
-    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n sample) atTop
-      (𝓝 (F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-        (∫ t in (0 : ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ sample),
-          iteratedDeriv 2 (F sample t) (B t.toNNReal sample))) := by
-  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T
-    C hC hC2 K₂ hK₂ (fun n i => by
-      filter_upwards [hCeq n i, hbound n i] with sample he hb
-      simpa only [he] using hb)
-  have hCeq_all : ∀ᵐ sample ∂P, ∀ n i, C n i sample = stoppedCoeff 2 B T τ F n i sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hCeq n))
-  have hC_all : ∀ᵐ sample ∂P, ∀ n i, |stoppedCoeff 2 B T τ F n i sample| ≤ K₂ :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hbound n))
-  have hA_all : ∀ᵐ sample ∂P, ∀ n i, |stoppedCoeff 1 B T τ F n i sample| ≤ K₁ :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAbound n))
-  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hτ, hC_all, hA_all, hCeq_all]
-    with sample hb hQ hE hr hτsample hCsample hAsample hCeqsample
-  have hEs : Tendsto (fun n => dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n sample) atTop (𝓝 0) := by
-    convert hE using 1
-    funext n
-    unfold dyadicQuadraticError
-    simp only [hCeqsample]
-  let b : ℝ → ℝ := fun t => B t.toNNReal sample
-  have hbc : Continuous b := hb.comp continuous_real_toNNReal
-  have hQ' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
-    simpa only [quadraticSum, b, mesh_toNNReal] using hQ
-  have hE' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ sample then
-      iteratedDeriv 2 (F sample (mesh T n i)) (b (mesh T n i)) *
-        ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0)
-        atTop (𝓝 0) := by
-    have heq (n : ℕ) : dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n sample =
-        ∑ i ∈ Finset.range (2^n), if mesh T n i < τ sample then
-          iteratedDeriv 2 (F sample (mesh T n i)) (b (mesh T n i)) *
-            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0 := by
-      unfold dyadicQuadraticError
-      apply Finset.sum_congr rfl
-      intro i hi
-      simp only [stoppedCoeff, b, mesh_toNNReal]
-      split_ifs <;> simp
-    simpa only [heq] using hEs
-  have hA' (n i : ℕ) (hi : mesh T n i < τ sample) :
-      |deriv (F sample (mesh T n i)) (b (mesh T n i))| ≤ K₁ := by
-    simpa only [stoppedCoeff, ite_eq_left hi, iteratedDeriv_succ, iteratedDeriv_zero,
-      b, mesh_toNNReal] using hAsample n i
-  have hC' (n i : ℕ) (hi : mesh T n i < τ sample) :
-      |iteratedDeriv 2 (F sample (mesh T n i)) (b (mesh T n i))| ≤ K₂ := by
-    simpa only [stoppedCoeff, ite_eq_left hi, b, mesh_toNNReal] using hCsample n i
-  have ht := stopped_field_taylor_limit T.coe_nonneg hτsample.1 hτsample.2 hbc.continuousOn
-    hr hQ' hK₁ hK₂ hA' hC' hE'
-  have heq (n : ℕ) : dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n sample =
-      ∑ i ∈ Finset.range (2^n), if mesh T n i < τ sample then
-        deriv (F sample (mesh T n i)) (b (mesh T n i)) *
-          (b (mesh T n (i+1))-b (mesh T n i)) else 0 := by
-    unfold dyadicDriverSum
-    apply Finset.sum_congr rfl
-    intro i hi
-    simp only [stoppedCoeff, b, mesh_toNNReal, iteratedDeriv_succ, iteratedDeriv_zero]
-    split_ifs <;> simp
-  simpa only [← heq, b, Real.toNNReal_zero] using ht
-
-theorem stopped_field_ito_centered_ae {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
-    (hτ : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T)
-    (hreg : ∀ᵐ sample ∂P, LocalRegular (F sample) (V sample) (fun t => B t.toNNReal sample) (τ sample))
-    (A C : ℕ → ℕ → Ω → ℝ)
-    (hAeq : ∀ n i, A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i)
-    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
-    (hC2 : ∀ n i, MemLp (C n i) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 2 B T τ F n i sample| ≤ K₂)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
-    (hA2 : ∀ n i, MemLp (A n i) 2 P)
-    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
-    (hAbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 1 B T τ F n i sample| ≤ K₁) :
-    let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0 : ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
-    Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 := by
-  have ht := stopped_fieldSum_tendsto_ae hB T τ F V hτ hreg C hCeq hC hC2
-    K₂ hK₂ hbound K₁ hK₁ hAbound
-  have hAeq_all : ∀ᵐ sample ∂P, ∀ n i, A n i sample = stoppedCoeff 1 B T τ F n i sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAeq n))
-  apply limit_centered hB T A hA hA2 K₁ hK₁ ?_ _ ?_
-  · intro n i
-    filter_upwards [hAeq n i, hAbound n i] with sample he hb
-    simpa only [he] using hb
-  · filter_upwards [ht, hAeq_all] with sample hsample he
-    convert hsample using 1
-    funext n
-    unfold dyadicDriverSum
-    simp only [he]
-
-end SLEExactGauge.BrownianLocal
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.AngularBarrier
-
-lemma profileT_continuousAt {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
-    {v : ℝ × ℝ} (hv : 0 < v.1) :
-    ContinuousAt (fun v : ℝ × ℝ => profileT l b v.1 v.2) v := by
-  have hs : ContinuousAt (fun v : ℝ × ℝ => scale l b v.1) v := by
-    unfold scale; fun_prop
-  have hq := (contDiff_squaredSine 0).continuous.continuousAt.comp (continuousAt_snd : ContinuousAt (fun v : ℝ × ℝ => v.2) v)
-  have hz : ContinuousAt (fun v : ℝ × ℝ => squaredSine v.2 / scale l b v.1) v := hq.div hs (scale_pos hl hb hv).ne'
-  have hp : ContinuousAt (fun v : ℝ × ℝ => pdf b (squaredSine v.2 / scale l b v.1)) v := (hasDerivAt_pdf hb (div_pos (squaredSine_pos v.2) (scale_pos hl hb hv))).continuousAt.comp_of_eq hz rfl
-  have hf := (contDiffAt_profile hl hb 0 hv).continuousAt
-  have he : ContinuousAt (fun v : ℝ × ℝ => Real.exp (l*b/2*v.1)) v := by fun_prop
-  have hc : ContinuousAt (fun v : ℝ × ℝ => l+l/(2*(b+1)*scale l b v.1)) v :=
-    continuousAt_const.add (continuousAt_const.div (continuousAt_const.mul hs)
-      (mul_ne_zero (by positivity) (scale_pos hl hb hv).ne'))
-  exact (continuousAt_const.mul hf).sub (((he.mul hp).mul hz).mul hc)
-
-end SLEExactGauge.AngularBarrier
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.GammaField
-open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-def rawRemaining (η ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  η + Real.log (2*height hε hT U z t) - logDerivative hε hT U z t - Real.log ρ
-
-def remaining (η ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  max η (rawRemaining hε hT η ρ U z t)
-
-def angleRatio (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) : ℝ :=
-  ((position hε hT U z t).re-sdev*x)/height hε hT U z t
-
-def field (l b η ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) : ℝ :=
-  profile l b (remaining hε hT η ρ U z t) (angleRatio hε hT sdev U z t x)
-
-def drift (l b η ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) : ℝ :=
-  let y := height hε hT U z t
-  let w := angleRatio hε hT sdev U z t x
-  let v := velocity ε (extendDriver T U t) (position hε hT U z t)
-  profileT l b (remaining hε hT η ρ U z t) w *
-    (v.im/y - variation hε hT U z t) +
-  profileW l b (remaining hε hT η ρ U z t) w * (v.re/y-w*v.im/y)
-
-lemma remaining_pos {η : ℝ} (hη : 0 < η) (ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    0 < remaining hε hT η ρ U z t := hη.trans_le (le_max_left _ _)
-
-lemma continuous_height (z : ℂ) :
-    Continuous (fun v : C(I, ℝ) × ℝ => height hε hT v.1 z v.2) :=
-  continuous_const.max (Complex.continuous_im.comp (continuous_position hε hT z))
-
-lemma continuous_rawRemaining (η ρ : ℝ) (z : ℂ) :
-    Continuous (fun v : C(I, ℝ) × ℝ => rawRemaining hε hT η ρ v.1 z v.2) := by
-  exact ((continuous_const.add ((continuous_const.mul (continuous_height hε hT z)).log
-    (fun v => (mul_pos (by norm_num) (height_pos hε hT v.1 z v.2)).ne'))).sub
-    (continuous_logDerivative hε hT z)).sub continuous_const
-
-lemma continuous_remaining (η ρ : ℝ) (z : ℂ) :
-    Continuous (fun v : C(I, ℝ) × ℝ => remaining hε hT η ρ v.1 z v.2) :=
-  continuous_const.max (continuous_rawRemaining hε hT η ρ z)
-
-lemma continuous_angleRatio (sdev : ℝ) (z : ℂ) :
-    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => angleRatio hε hT sdev v.1.1 z v.1.2 v.2) := by
-  have hp := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
-  have hy := (continuous_height hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
-  exact ((Complex.continuous_re.comp hp).sub (continuous_const.mul continuous_snd)).div hy
-    (fun v => (height_pos hε hT v.1.1 z v.1.2).ne')
-
-lemma smooth {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) (n : ℕ) :
-    ContDiff ℝ n (field hε hT l b η ρ sdev U z t) := by
-  apply (profile_smooth hl hb (remaining_pos hε hT hη ρ U z t) n).comp
-  unfold angleRatio
-  fun_prop
-
-lemma iteratedDeriv_eq {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) (k : ℕ) :
-    iteratedDeriv k (field hε hT l b η ρ sdev U z t) x =
-      (-sdev/height hε hT U z t)^k *
-        iteratedDeriv k (profile l b (remaining hε hT η ρ U z t)) (angleRatio hε hT sdev U z t x) := by
-  have he : field hε hT l b η ρ sdev U z t = fun x =>
-      profile l b (remaining hε hT η ρ U z t)
-        ((-sdev/height hε hT U z t)*x+(position hε hT U z t).re/height hε hT U z t) := by
-    funext x; unfold field angleRatio; congr 1; ring
-  rw [he, iteratedDeriv_affine _ (profile_smooth hl hb (remaining_pos hε hT hη ρ U z t))]
-  congr 2
-  unfold angleRatio; ring
-
-lemma continuous_derivative {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev : ℝ) (z : ℂ) (k : ℕ) :
-    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv k (field hε hT l b η ρ sdev v.1.1 z v.1.2) v.2) := by
-  simp only [iteratedDeriv_eq hε hT hl hb hη]
-  apply continuous_iff_continuousAt.2
-  intro v
-  have ht := ((continuous_remaining hε hT η ρ z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v)
-  have hw := (continuous_angleRatio hε hT sdev z).continuousAt (x := v)
-  have hh := (profile_derivative_continuousAt hl hb k (remaining_pos hε hT hη ρ v.1.1 z v.1.2)).comp (ht.prodMk hw)
-  exact ((continuousAt_const.div ((continuous_height hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt
-    (height_pos hε hT v.1.1 z v.1.2).ne').pow k).mul hh
-
-lemma continuous_field {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev : ℝ) (z : ℂ) :
-    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => field hε hT l b η ρ sdev v.1.1 z v.1.2 v.2) := by
-  simpa using continuous_derivative hε hT hl hb hη ρ sdev z 0
-
-lemma continuous_drift {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev : ℝ) (z : ℂ) :
-    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => drift hε hT l b η ρ sdev v.1.1 z v.1.2 v.2) := by
-  apply continuous_iff_continuousAt.2
-  intro v
-  have hR := ((continuous_remaining hε hT η ρ z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v)
-  have hW := (continuous_angleRatio hε hT sdev z).continuousAt (x := v)
-  have htime := (profileT_continuousAt hl hb (remaining_pos hε hT hη ρ v.1.1 z v.1.2)).comp (hR.prodMk hW)
-  have hspace := (profile_derivative_continuousAt hl hb 1 (remaining_pos hε hT hη ρ v.1.1 z v.1.2)).comp (hR.prodMk hW)
-  have hspace' : ContinuousAt (fun v : (C(I, ℝ) × ℝ) × ℝ => profileW l b
-      (remaining hε hT η ρ v.1.1 z v.1.2) (angleRatio hε hT sdev v.1.1 z v.1.2 v.2)) v := by
-    simpa only [Function.comp_def, iteratedDeriv_succ, iteratedDeriv_zero,
-      deriv_profile hl hb (remaining_pos hε hT hη ρ _ _ _)] using hspace
-  have hp := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
-  have hd := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
-  have hv := ((continuous_velocity hε).comp (hd.prodMk hp)).continuousAt (x := v)
-  have hy := ((continuous_height hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v)
-  have hn := (height_pos hε hT v.1.1 z v.1.2).ne'
-  exact (htime.mul (((Complex.continuous_im.continuousAt.comp hv).div hy hn).sub
-    ((continuous_variation hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v))).add
-      (hspace'.mul (((Complex.continuous_re.continuousAt.comp hv).div hy hn).sub
-        ((hW.mul (Complex.continuous_im.continuousAt.comp hv)).div hy hn)))
-
-lemma rawRemaining_eq (η ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im) :
-    rawRemaining hε hT η ρ U z t = η + Real.log (radius hε hT U z t) - Real.log ρ := by
-  have hz := hε.trans_le hy
-  unfold rawRemaining radius height
-  rw [max_eq_right hy, Real.log_div (by positivity) (Real.exp_pos _).ne', Real.log_exp]
-  ring
-
-lemma rawRemaining_gt (η : ℝ) {ρ : ℝ} (hρ : 0 < ρ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im) (hr : ρ < radius hε hT U z t) :
-    η < rawRemaining hε hT η ρ U z t := by
-  rw [rawRemaining_eq hε hT η ρ U z t hy]
-  have hh := Real.strictMonoOn_log hρ (hρ.trans hr) hr
-  linarith
-
-lemma time_deriv {l b η ρ : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (hρ : 0 < ρ)
-    (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) {t : ℝ}
-    (ht : t ∈ Ioo 0 T) (hy : ε < (position hε hT U z t).im)
-    (hr : ρ < radius hε hT U z t) (x : ℝ) :
-    HasDerivAt (fun s => field hε hT l b η ρ sdev U z s x) (drift hε hT l b η ρ sdev U z t x) t := by
-  have hH := position_deriv hε hT U z ht
-  have hL := hasDerivAt_logDerivative hε hT U z ht
-  have hIm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hH
-  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
-  dsimp only [Function.comp_def, Complex.imCLM_apply, Complex.reCLM_apply] at hIm hRe
-  have hY : HasDerivAt (height hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)).im t := by
-    apply hIm.congr_of_eventuallyEq
-    filter_upwards [hIm.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy)] with s hs
-    exact max_eq_right hs.le
-  have hraw : HasDerivAt (rawRemaining hε hT η ρ U z)
-      ((velocity ε (extendDriver T U t) (position hε hT U z t)).im / height hε hT U z t -
-        variation hε hT U z t) t := by
-    have hh := (((hY.const_mul 2).log (by have := height_pos hε hT U z t; positivity)).const_add η).sub hL
-    apply (hh.sub_const (Real.log ρ)).congr_deriv
-    ring
-  have hR : HasDerivAt (remaining hε hT η ρ U z)
-      ((velocity ε (extendDriver T U t) (position hε hT U z t)).im / height hε hT U z t -
-        variation hε hT U z t) t := by
-    apply hraw.congr_of_eventuallyEq
-    have hh := rawRemaining_gt hε hT η hρ U z t hy.le hr
-    filter_upwards [hraw.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hh)] with s hs
-    exact max_eq_right hs.le
-  have hW : HasDerivAt (fun s => angleRatio hε hT sdev U z s x)
-      ((velocity ε (extendDriver T U t) (position hε hT U z t)).re / height hε hT U z t -
-        angleRatio hε hT sdev U z t x * (velocity ε (extendDriver T U t) (position hε hT U z t)).im / height hε hT U z t) t := by
-    apply ((hRe.sub_const (sdev*x)).div hY (height_pos hε hT U z t).ne').congr_deriv
-    dsimp only [angleRatio]
-    field_simp
-  exact hasDerivAt_profile_path hl hb hR hW (remaining_pos hε hT hη ρ U z t)
-
-lemma field_prefix_eq {ε S T : ℝ} (hε : 0 < ε) (hS : 0 ≤ S) (hT : 0 ≤ T)
-    (hST : S ≤ T) (U V : C(I, ℝ)) (z : ℂ) (l b η ρ sdev : ℝ)
-    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
-    {t : ℝ} (ht : t ∈ Icc 0 S) :
-    field hε hS l b η ρ sdev U z t = field hε hT l b η ρ sdev V z t := by
-  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
-  have hL := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
-  funext x
-  simp only [field, remaining, rawRemaining, angleRatio, height, hH, hL]
-
-lemma regular {l b η ρ : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (hρ : 0 < ρ)
-    (hTp : 0 < T) (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (δ : ℝ)
-    (B : ℝ → ℝ) (hB : Continuous B) :
-    BrownianLocal.LocalRegular (field hε hT l b η ρ sdev U z)
-      (drift hε hT l b η ρ sdev U z) B (stop hε hT U z ρ δ) := by
-  constructor
-  · intro t; exact smooth hε hT hl hb hη ρ sdev U z t 3
-  · have hm : Continuous (fun v : ℝ × ℝ => ((U,v.1),v.2)) :=
-      (continuous_const.prodMk continuous_fst).prodMk continuous_snd
-    exact ((continuous_derivative hε hT hl hb hη ρ sdev z 3).comp hm).continuousOn
-  · intro x
-    have hm : Continuous (fun t : ℝ => ((U,t),x)) :=
-      (continuous_const.prodMk continuous_id).prodMk continuous_const
-    exact ((continuous_field hε hT hl hb hη ρ sdev z).comp hm).continuousOn
-  · intro t ht x
-    have htT : t ∈ Ioo 0 T := ⟨ht.1,ht.2.trans_le (stop_bounds hε hT U z ρ δ).2⟩
-    have hh := (deficit_pos_iff hε hT U z ρ δ t).1
-      (positive_before hε hT hTp U z ρ δ ht.1.le ht.2)
-    exact time_deriv hε hT hl hb hη hρ sdev U z htT hh.2.1 hh.1 x
-  · have hm : Continuous (fun v : ℝ × ℝ => ((U,v.1),B v.2)) :=
-      (continuous_const.prodMk continuous_fst).prodMk (hB.comp continuous_snd)
-    exact ((continuous_drift hε hT hl hb hη ρ sdev z).comp hm).continuousOn
-  · have hm : Continuous (fun t : ℝ => ((U,t),B t)) :=
-      (continuous_const.prodMk continuous_id).prodMk hB
-    exact ((continuous_derivative hε hT hl hb hη ρ sdev z 2).comp hm).continuousOn
-
-lemma coefficient_bound {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
-    (ρ sdev M D : ℝ) (z : ℂ) (k : ℕ) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t x : ℝ),
-      remaining hε hT η ρ U z t ≤ M → |angleRatio hε hT sdev U z t x| ≤ D →
-      |iteratedDeriv k (field hε hT l b η ρ sdev U z t) x| ≤ K := by
-  have hc : ContinuousOn (fun v : ℝ × ℝ => iteratedDeriv k (profile l b v.1) v.2)
-      ((Icc η M) ×ˢ (Icc (-D) D)) := by
-    intro v hv
-    exact (profile_derivative_continuousAt hl hb k (hη.trans_le hv.1.1)).continuousWithinAt
-  obtain ⟨K,hK⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hc
-  refine ⟨(|sdev|/ε)^k * max 0 K, by positivity, ?_⟩
-  intro U t x ht hw
-  rw [iteratedDeriv_eq hε hT hl hb hη, abs_mul, abs_pow, abs_div, abs_neg,
-    abs_of_pos (height_pos hε hT U z t)]
-  apply mul_le_mul
-  · exact pow_le_pow_left₀ (by positivity)
-      (div_le_div_of_nonneg_left (abs_nonneg _) hε (le_max_left _ _)) k
-  · apply le_trans _ (le_max_right 0 K)
-    simpa only [Real.norm_eq_abs] using hK
-      (remaining hε hT η ρ U z t, angleRatio hε hT sdev U z t x)
-      ⟨⟨le_max_left _ _,ht⟩,abs_le.1 hw⟩
-  · exact abs_nonneg _
-  · positivity
-
-lemma physical_algebra (κ x y A B C : ℝ) (hy : 0 < y) :
-    A*((-2*y/(x^2+y^2))/y - (-2*(x^2-y^2)/(x^2+y^2)^2)) +
-      B*((2*x/(x^2+y^2))/y - (x/y)*(-2*y/(x^2+y^2))/y) +
-      (1/2:ℝ)*(κ/y^2*C) =
-    (4*y^2/(x^2+y^2)^2)*(-A+(x/y)*(1+(x/y)^2)*B+κ/8*(1+(x/y)^2)^2*C) := by
-  have hn : x^2+y^2 ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma drift_nonpos {κ η : ℝ} (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η)
-    (ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im)
-    (hU : extendDriver T U t = Real.sqrt κ*x) :
-    drift hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t x +
-      (1/2:ℝ)*iteratedDeriv 2 (field hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t) x ≤ 0 := by
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have ht := remaining_pos hε hT hη ρ U z t
-  have hh : height hε hT U z t = (position hε hT U z t).im := max_eq_right hy
-  rw [iteratedDeriv_eq hε hT hl hb hη, second_profile hl hb ht]
-  dsimp only [drift]
-  rw [variation_of_unclipped hε hT U z t hy, velocity_eq hy, hh, hU]
-  simp only [Complex.div_re, Complex.div_im, Complex.re_ofNat, Complex.im_ofNat,
-    Complex.sub_re, Complex.ofReal_re, Complex.sub_im, Complex.ofReal_im,
-    sub_zero, zero_mul, Complex.normSq_apply]
-  simp only [neg_div, neg_sq, div_pow, Real.sq_sqrt hκ.1.le]
-  have he := physical_algebra κ ((position hε hT U z t).re-Real.sqrt κ*x)
-    (position hε hT U z t).im
-    (profileT (κ/2) ((8/κ-1)/2) (remaining hε hT η ρ U z t) (angleRatio hε hT (Real.sqrt κ) U z t x))
-    (profileW (κ/2) ((8/κ-1)/2) (remaining hε hT η ρ U z t) (angleRatio hε hT (Real.sqrt κ) U z t x))
-    (profileWW (κ/2) ((8/κ-1)/2) (remaining hε hT η ρ U z t) (angleRatio hε hT (Real.sqrt κ) U z t x)) (hε.trans_le hy)
-  have ha : angleRatio hε hT (Real.sqrt κ) U z t x =
-      ((position hε hT U z t).re-Real.sqrt κ*x)/(position hε hT U z t).im := by
-    unfold angleRatio; rw [hh]
-  rw [← ha] at he
-  convert (he.trans_le (mul_nonpos_of_nonneg_of_nonpos (by positivity) (by
-    have hs := profile_super hκ ht (angleRatio hε hT (Real.sqrt κ) U z t x)
-    linarith))) using 1; ring
-
-end SLEExactGauge.GammaField
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.GammaPredictable
-open LoewnerCutoff LoewnerExit GammaField BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-open scoped unitInterval
-variable {ε : ℝ} (hε : 0 < ε)
-
-def coeffAt (k : ℕ) (s : ℝ≥0) (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ)
-    (q : C(I, ℝ) × ℝ) : ℝ := by
-  classical
-  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
-    iteratedDeriv k (field hε s.coe_nonneg l b η ρ sdev q.1 z s) q.2 else 0
-
-lemma coeffAt_measurable (k : ℕ) (_hk : k = 1 ∨ k = 2) (s : ℝ≥0)
-    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (ρ δ : ℝ) :
-    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s l b η sdev z ρ δ) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
-      iteratedDeriv k (field hε s.coe_nonneg l b η ρ sdev q.1 z s) q.2) := by
-    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
-      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
-    have hc := (continuous_derivative hε s.coe_nonneg hl hb hη ρ sdev z k).comp hp
-    dsimp only [Function.comp_def] at hc
-    exact hc
-  apply hc.measurable.ite _ measurable_const
-  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
-
-def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω) : ℝ :=
-  if s < T then coeffAt hε k s l b η sdev z ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
-
-lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (ρ δ : ℝ) :
-    Measurable[pastSigma B s] (rawCoeff hε k B T s l b η sdev z ρ δ) := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  let : MeasurableSpace Ω := pastSigma B s
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z ρ δ).comp
-      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
-  · exact measurable_const
-
-lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
-    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (ρ δ : ℝ) :
-    AEMeasurable (rawCoeff hε k B T s l b η sdev z ρ δ) P := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z ρ δ).comp_aemeasurable
-      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
-  · exact aemeasurable_const
-
-lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    rawCoeff hε k B T s l b η sdev z ρ δ sample =
-      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
-        iteratedDeriv k (field hε T.coe_nonneg l b η ρ sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
-  classical
-  by_cases hsT : s < T
-  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
-    have hf := field_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z l b η ρ sdev he
-      ⟨s.coe_nonneg, le_rfl⟩
-    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
-    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
-        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
-      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
-      exact forall₂_congr (fun t ht => by rw [hd ht])
-    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
-  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
-      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
-    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
-
-lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (n i : ℕ) :
-    rawCoeff hε k B T (dyadicTime T n i) l b η sdev z ρ δ =ᵐ[P]
-      stoppedCoeff k B T
-        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
-        (fun sample => field hε T.coe_nonneg l b η ρ sdev (BrownianDriver.path B sdev T sample) z) n i := by
-  filter_upwards [hB.cont] with sample hc
-  rw [rawCoeff_eq hε k B T _ hT l b η sdev z ρ δ sample hc]
-  simp only [stoppedCoeff, coe_dyadicTime]
-
-end SLEExactGauge.GammaPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.GammaField
-open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-lemma ratio_bound {δ : ℝ} (hδ : 0 < δ) (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im) (hs : δ ≤ sine hε hT U z t)
-    (hU : extendDriver T U t = sdev*x) :
-    |angleRatio hε hT sdev U z t x| ≤ 1/δ := by
-  have hyp := hε.trans_le hy
-  have hn : 0 < ‖position hε hT U z t - (extendDriver T U t : ℂ)‖ := by
-    apply norm_pos_iff.2
-    intro he
-    have hi := congrArg Complex.im he
-    simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hi
-    linarith
-  have hh : δ * ‖position hε hT U z t - (extendDriver T U t : ℂ)‖ ≤ (position hε hT U z t).im := by
-    apply (le_div_iff₀ hn).1
-    unfold sine at hs
-    rw [clampIm_eq (show ε ≤ (position hε hT U z t - (extendDriver T U t : ℂ)).im by simpa using hy)] at hs
-    exact hs
-  have hx := mul_le_mul_of_nonneg_left (Complex.abs_re_le_norm
-    (position hε hT U z t - (extendDriver T U t : ℂ))) hδ.le
-  have hxx := hx.trans hh
-  simp only [Complex.sub_re, Complex.ofReal_re, hU] at hxx
-  rw [angleRatio, height, max_eq_right hy, abs_div, abs_of_pos hyp]
-  apply (div_le_iff₀ hyp).2
-  rw [div_mul_eq_mul_div, one_mul]
-  apply (le_div_iff₀ hδ).2
-  nlinarith [hxx]
-
-end SLEExactGauge.GammaField
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.ReverseFlow
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Angular Barrier estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.AngularBarrier
+
+lemma value_decay {l b : ℝ} (hl : 0 < l) (hb : 0 < b) :
+    ∃ K : ℝ, 0 < K ∧ ∀ t q : ℝ, 1 ≤ t → 0 ≤ q →
+      value l b t q ≤ K*Real.exp (-(l*b/2)*t)*q^b := by
+  let c := (1-Real.exp (-l))/(2*(b+1))
+  have hc : 0 < c := div_pos (sub_pos.2 (Real.exp_lt_one_iff.2 (by linarith))) (by positivity)
+  have hg : 0 < Real.Gamma b := Real.Gamma_pos_of_pos hb
+  refine ⟨1/(b*Real.Gamma b*c^b), by positivity, ?_⟩
+  intro t q ht hq
+  have ht0 : 0 < t := by linarith
+  have ha := scale_pos hl hb ht0
+  have hlower : c*Real.exp (l*t) ≤ scale l b t := by
+    have hh : 1 ≤ Real.exp (l*t)*Real.exp (-l) := by
+      rw [← Real.exp_add]
+      exact Real.one_le_exp_iff.2 (by nlinarith)
+    dsimp only [c,scale]
+    apply (le_div_iff₀ (by positivity : 0 < 2*(b+1))).2
+    have heq : ((1-Real.exp (-l))/(2*(b+1))*Real.exp (l*t))*(2*(b+1)) =
+        (1-Real.exp (-l))*Real.exp (l*t) := by field_simp
+    rw [heq]
+    nlinarith
+  have hratio : q/scale l b t ≤ q/(c*Real.exp (l*t)) :=
+    div_le_div_of_nonneg_left hq (mul_pos hc (Real.exp_pos _)) hlower
+  have hpow := Real.rpow_le_rpow (div_nonneg hq ha.le) hratio hb.le
+  have hbound := (cdf_le_power hb (div_nonneg hq ha.le)).trans
+    (div_le_div_of_nonneg_right hpow (mul_pos hb hg).le)
+  have he : Real.exp (l*b/2*t)*(q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b) =
+      1/(b*Real.Gamma b*c^b)*Real.exp (-(l*b/2)*t)*q^b := by
+    rw [Real.div_rpow hq (mul_pos hc (Real.exp_pos _)).le,
+      Real.mul_rpow hc.le (Real.exp_pos _).le, ← Real.exp_mul]
+    have hratioexp : Real.exp (l*b/2*t)/Real.exp (l*t*b) = Real.exp (-(l*b/2)*t) := by
+      rw [← Real.exp_sub]; congr 1; ring
+    calc
+      _ = (Real.exp (l*b/2*t)/Real.exp (l*t*b))*q^b/(b*Real.Gamma b*c^b) := by ring
+      _ = _ := by rw [hratioexp]; ring
+  calc
+    _ ≤ Real.exp (l*b/2*t)*((q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b)) :=
+      mul_le_mul_of_nonneg_left hbound (Real.exp_pos _).le
+    _ = _ := by rw [← mul_div_assoc, he]
+
+def squaredSine (w : ℝ) := 1/(1+w^2)
+
+def profile (l b t w : ℝ) := value l b t (squaredSine w)
+
+def profileT (l b t w : ℝ) := valueT l b t (squaredSine w)
+
+def profileW (l b t w : ℝ) := (-2*w/(1+w^2)^2)*valueQ l b t (squaredSine w)
+
+def profileWW (l b t w : ℝ) := (6*w^2-2)/(1+w^2)^3*valueQ l b t (squaredSine w) +
+  4*w^2/(1+w^2)^4*valueQQ l b t (squaredSine w)
+
+lemma squaredSine_pos (w : ℝ) : 0 < squaredSine w := by unfold squaredSine; positivity
+
+lemma squaredSine_le_one (w : ℝ) : squaredSine w ≤ 1 := by
+  unfold squaredSine; exact (div_le_one (by positivity)).2 (by nlinarith [sq_nonneg w])
+
+lemma hasDerivAt_squaredSine (w : ℝ) :
+    HasDerivAt squaredSine (-2*w/(1+w^2)^2) w := by
+  apply ((hasDerivAt_const w 1).div (((hasDerivAt_id w).pow 2).const_add 1) (by simp only [Pi.pow_apply, id_eq]; positivity)).congr_deriv
+  simp only [Pi.pow_apply, id_eq]; ring
+
+lemma contDiff_squaredSine (n : WithTop ℕ∞) : ContDiff ℝ n squaredSine := by
+  unfold squaredSine
+  exact contDiff_const.div (contDiff_const.add (contDiff_id.pow 2)) (fun w => by positivity)
+
+lemma contDiffAt_profile {l b : ℝ} (hl : 0 < l) (hb : 0 < b) (n : ℕ)
+    {v : ℝ × ℝ} (ht : 0 < v.1) :
+    ContDiffAt ℝ n (fun v : ℝ × ℝ => profile l b v.1 v.2) v := by
+  have hq := (contDiff_squaredSine n).contDiffAt.comp v contDiffAt_snd
+  have hm : ContDiffAt ℝ n (fun v : ℝ × ℝ => (v.1,squaredSine v.2)) v :=
+    contDiffAt_fst.prodMk hq
+  have hh := (contDiffAt_value (v:=(v.1,squaredSine v.2)) hl hb n ht (squaredSine_pos v.2)).comp v hm
+  dsimp only [Function.comp_def] at hh
+  exact hh
+
+lemma hasDerivAt_profile_w {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (w : ℝ) :
+    HasDerivAt (profile l b t) (profileW l b t w) w := by
+  have hh := (hasDerivAt_value_q hl hb ht (squaredSine_pos w)).comp w (hasDerivAt_squaredSine w)
+  apply hh.congr_deriv
+  dsimp only [profileW]; ring
+
+lemma hasDerivAt_profile_t {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (w : ℝ) :
+    HasDerivAt (fun s => profile l b s w) (profileT l b t w) t :=
+  hasDerivAt_value_t hl hb ht (squaredSine_pos w)
+
+lemma deriv_squaredSine_hasDerivAt (w : ℝ) :
+    HasDerivAt (fun w : ℝ => -2*w/(1+w^2)^2) ((6*w^2-2)/(1+w^2)^3) w := by
+  have hd := ((hasDerivAt_id w).const_mul (-2)).div
+    ((((hasDerivAt_id w).pow 2).const_add 1).pow 2) (by simp only [Pi.pow_apply, id_eq]; positivity)
+  apply hd.congr_deriv
+  simp only [Pi.pow_apply, id_eq]
+  field_simp
+  ring
+
+lemma hasDerivAt_profileW {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (w : ℝ) :
+    HasDerivAt (profileW l b t) (profileWW l b t w) w := by
+  have hh := (deriv_squaredSine_hasDerivAt w).mul
+    ((hasDerivAt_valueQ_q hl hb ht (squaredSine_pos w)).comp w (hasDerivAt_squaredSine w))
+  apply hh.congr_deriv
+  dsimp only [profileWW, Function.comp_def]
+  field_simp
+  ring
+
+lemma deriv_profile {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) :
+    deriv (profile l b t) = profileW l b t := funext fun w => (hasDerivAt_profile_w hl hb ht w).deriv
+
+lemma second_profile {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) :
+    iteratedDeriv 2 (profile l b t) = profileWW l b t := by
+  simp only [iteratedDeriv_succ, iteratedDeriv_zero, deriv_profile hl hb ht]
+  exact funext fun w => (hasDerivAt_profileW hl hb ht w).deriv
+
+lemma parameter_deriv_smooth (f : ℝ → ℝ → ℝ) (n : ℕ) {v : ℝ × ℝ}
+    (hf : ContDiffAt ℝ (n+1) (Function.uncurry f) v) :
+    ContDiffAt ℝ n (fun v : ℝ × ℝ => deriv (f v.1) v.2) v := by
+  let g : (ℝ × ℝ) → ℝ → ℝ := fun v w => f v.1 w
+  have hh : ContDiffAt ℝ (n+1) (Function.uncurry g) (v,v.2) := by
+    have hm : ContDiffAt ℝ (n+1) (fun r : (ℝ × ℝ) × ℝ => (r.1.1,r.2)) (v,v.2) := by fun_prop
+    exact hf.comp (v,v.2) hm
+  have hd := hh.fderiv (m:=n) contDiffAt_snd (by simp)
+  exact hd.clm_apply contDiffAt_const
+
+lemma parameter_iteratedDeriv_smooth (f : ℝ → ℝ → ℝ)
+    (hf : ∀ n : ℕ, ∀ v : ℝ × ℝ, 0 < v.1 → ContDiffAt ℝ n (Function.uncurry f) v)
+    (k n : ℕ) (v : ℝ × ℝ) (hv : 0 < v.1) :
+    ContDiffAt ℝ n (fun v : ℝ × ℝ => iteratedDeriv k (f v.1) v.2) v := by
+  induction k generalizing n with
+  | zero => exact hf n v hv
+  | succ k ih =>
+    simpa only [iteratedDeriv_succ] using parameter_deriv_smooth (fun t => iteratedDeriv k (f t)) n (ih (n+1))
+
+lemma profile_derivative_continuousAt {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
+    (k : ℕ) {v : ℝ × ℝ} (hv : 0 < v.1) :
+    ContinuousAt (fun v : ℝ × ℝ => iteratedDeriv k (profile l b v.1) v.2) v :=
+  (parameter_iteratedDeriv_smooth (profile l b) (fun n _v hv => contDiffAt_profile hl hb n hv)
+    k 0 v hv).continuousAt
+
+lemma profile_super {κ t : ℝ} (hκ : 0 < κ ∧ κ < 8) (ht : 0 < t) (w : ℝ) :
+    w*(1+w^2)*profileW (κ/2) ((8/κ-1)/2) t w +
+      κ/8*(1+w^2)^2*profileWW (κ/2) ((8/κ-1)/2) t w ≤
+        profileT (κ/2) ((8/κ-1)/2) t w := by
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have hi := supersolution (l:=κ/2) (b:=(8/κ-1)/2) (t:=t) (q:=squaredSine w) (div_pos hκ.1 (by norm_num)) hb ht (squaredSine_pos w) (squaredSine_le_one w)
+  change _ ≤ valueT (κ/2) ((8/κ-1)/2) t (squaredSine w)
+  convert hi using 1
+  dsimp only [profileW, profileWW, squaredSine]
+  field_simp [hκ.1.ne']
+  ring
+
+lemma hasDerivAt_value_path {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
+    {T Q : ℝ → ℝ} {s T' Q' : ℝ} (hT : HasDerivAt T T' s) (hQ : HasDerivAt Q Q' s)
+    (ht : 0 < T s) (hq : 0 < Q s) :
+    HasDerivAt (fun v => value l b (T v) (Q v))
+      (valueT l b (T s) (Q s)*T' + valueQ l b (T s) (Q s)*Q') s := by
+  have ha := (hasDerivAt_scale l b (T s)).comp s hT
+  have hs := scale_pos hl hb ht
+  have hh := ((hT.const_mul (l*b/2)).exp).mul
+    ((hasDerivAt_cdf hb (div_pos hq hs)).comp s (hQ.div ha hs.ne'))
+  apply hh.congr_deriv
+  simp only [Function.comp_def, Pi.div_apply]
+  dsimp only [valueT, valueQ, value]
+  have he : Real.exp (l*T s) = 2*(b+1)*scale l b (T s)+1 := by unfold scale; field_simp; ring
+  rw [he]
+  field_simp
+  ring
+
+lemma hasDerivAt_profile_path {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
+    {T W : ℝ → ℝ} {s T' W' : ℝ} (hT : HasDerivAt T T' s) (hW : HasDerivAt W W' s)
+    (ht : 0 < T s) :
+    HasDerivAt (fun v => profile l b (T v) (W v))
+      (profileT l b (T s) (W s)*T' + profileW l b (T s) (W s)*W') s := by
+  have hq := (hasDerivAt_squaredSine (W s)).comp s hW
+  have hh := hasDerivAt_value_path hl hb hT hq ht (squaredSine_pos (W s))
+  apply hh.congr_deriv
+  simp only [profileT,profileW,Function.comp_def]
+  ring
+
+lemma profile_smooth {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) (n : ℕ) :
+    ContDiff ℝ n (profile l b t) := by
+  apply contDiff_iff_contDiffAt.2
+  intro w
+  have hh := (contDiffAt_profile (v:=(t,w)) hl hb n ht).comp w (contDiffAt_const.prodMk contDiffAt_id)
+  dsimp only [Function.comp_def] at hh
+  exact hh
+
+lemma iteratedDeriv_affine (f : ℝ → ℝ) (hf : ∀ n : ℕ, ContDiff ℝ n f) (k : ℕ) (a c x : ℝ) :
+    iteratedDeriv k (fun x => f (a*x+c)) x = a^k*iteratedDeriv k f (a*x+c) := by
+  induction k generalizing x with
+  | zero => simp
+  | succ k ih =>
+    have he : iteratedDeriv k (fun x => f (a*x+c)) = fun x => a^k*iteratedDeriv k f (a*x+c) := funext ih
+    rw [iteratedDeriv_succ, he]
+    have hd : Differentiable ℝ (iteratedDeriv k f) :=
+      ContDiff.differentiable_iteratedDeriv k (hf (k+1)) (by exact_mod_cast Nat.lt_succ_self k)
+    have hh := ((hd (a*x+c)).hasDerivAt.comp x
+      (((hasDerivAt_id x).const_mul a).add_const c)).const_mul (a^k)
+    dsimp only [Function.comp_def, id_eq] at hh
+    rw [hh.deriv]
+    simp only [iteratedDeriv_succ,pow_succ,mul_one]
+    ring
+
+end SLEExactGauge.AngularBarrier
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianLocal
+open BrownianPredictable PredictableLimit BrownianIto TaylorMeshes TimeField
+open BrownianField BrownianStopped PartitionMeshes StoppedMeshes
+
+structure LocalRegular (F V : ℝ → ℝ → ℝ) (b : ℝ → ℝ) (T : ℝ) : Prop where
+  smooth : ∀ t, ContDiff ℝ 3 (F t)
+  third_cont : ContinuousOn (fun p : ℝ × ℝ => iteratedDeriv 3 (F p.1) p.2) ((Icc 0 T) ×ˢ univ)
+  time_cont : ∀ x, ContinuousOn (fun t => F t x) (Icc 0 T)
+  time_deriv : ∀ t ∈ Ioo 0 T, ∀ x, HasDerivAt (fun u => F u x) (V t x) t
+  drift_cont : ContinuousOn (fun p : ℝ × ℝ => V p.1 (b p.2)) ((Icc 0 T) ×ˢ (Icc 0 T))
+  second_cont : ContinuousOn (fun t => iteratedDeriv 2 (F t) (b t)) (Icc 0 T)
+
+lemma family_remainder_tendsto_of_eventually_bounded {T : ℝ} (p : PartitionSequence T)
+    {f : ℕ → ℕ → ℝ → ℝ} {b : ℝ → ℝ}
+    (hf : ∀ n i, ContDiff ℝ 3 (f n i)) {V K : ℝ}
+    (hb : ContinuousOn b (Icc 0 T)) (hV : 0 ≤ V) (hK : 0 ≤ K)
+    (h3 : ∀ n i, i < p.count n → ∀ x ∈ uIcc (b (p.time n i)) (b (p.time n (i+1))),
+      |iteratedDeriv 3 (f n i) x| ≤ K)
+    (hQ : ∀ᶠ n in atTop, (∑ i ∈ Finset.range (p.count n),
+      (b (p.time n (i+1))-b (p.time n i))^2) ≤ V) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      remainder₂ (f n i) (b (p.time n i)) (b (p.time n (i+1)))) atTop (𝓝 0) := by
+  apply Metric.tendsto_atTop.2
+  intro ε hε
+  let δ := ε / ((K+1)*(V+1))
+  have hδ : 0 < δ := div_pos hε (mul_pos (by linarith) (by linarith))
+  have hs := p.increments_small hb hδ
+  obtain ⟨N, hN⟩ := eventually_atTop.1 (hs.and hQ)
+  refine ⟨N, fun n hn => ?_⟩
+  rcases hN n hn with ⟨hs, hq⟩
+  have hr := sum_family_remainder₂_bound (hf n) (fun i => b (p.time n i))
+    (p.count n) K δ hK hδ.le (fun i hi z hz => h3 n i hi z hz) (fun i hi => (hs i hi).le)
+  simp only [Real.dist_eq, sub_zero]
+  calc
+    _ ≤ K * δ / 6 * ∑ i ∈ Finset.range (p.count n),
+        (b (p.time n (i+1))-b (p.time n i))^2 := hr
+    _ ≤ K * δ / 6 * (V+1) :=
+      mul_le_mul_of_nonneg_left (hq.trans (by linarith)) (by positivity)
+    _ < ε := by
+      dsimp [δ]
+      have hden : 0 < (K+1)*(V+1) := mul_pos (by linarith) (by linarith)
+      have hid : K * (ε / ((K+1)*(V+1))) / 6 * (V+1) = ε * (K / (6*(K+1))) := by
+        field_simp
+      rw [hid]
+      have hk : K / (6*(K+1)) < 1 := (div_lt_one (by positivity)).2 (by linarith)
+      simpa using mul_lt_mul_of_pos_left hk hε
+
+lemma stopped_field_taylor_limit {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T)
+    {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ}
+    (hb : ContinuousOn b (Icc 0 T)) (hreg : LocalRegular F V b τ)
+    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1)) - b (mesh T n i))^2) atTop (𝓝 T))
+    {K₁ K₂ : ℝ} (hK₁ : 0 ≤ K₁) (hK₂ : 0 ≤ K₂)
+    (hA : ∀ n i, mesh T n i < τ →
+      |deriv (F (mesh T n i)) (b (mesh T n i))| ≤ K₁)
+    (hC : ∀ n i, mesh T n i < τ →
+      |iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i))| ≤ K₂)
+    (hE : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+      iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i)) *
+        ((b (mesh T n (i+1)) - b (mesh T n i))^2 - T/(2 : ℝ)^n) else 0) atTop (𝓝 0)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+      deriv (F (mesh T n i)) (b (mesh T n i)) *
+        (b (mesh T n (i+1)) - b (mesh T n i)) else 0) atTop
+      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, iteratedDeriv 2 (F t) (b t))) := by
+  let p := clippedDyadic T τ hT hτ hτT
+  let C : ℝ → ℝ := fun t => iteratedDeriv 2 (F t) (b t)
+  let A : ℝ → ℝ := fun t => deriv (F t) (b t)
+  have hbτ := hb.mono (Icc_subset_Icc le_rfl hτT)
+  have hQc := clipped_quadratic_eventually_bounded hT hτ hτT hb hQ
+  obtain ⟨R,hR⟩ := isCompact_Icc.exists_bound_of_continuousOn hbτ
+  have hc := hreg.third_cont.mono (Set.prod_mono Subset.rfl (subset_univ (Icc (-R) R)))
+  obtain ⟨K,hK⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hc
+  have hr := family_remainder_tendsto_of_eventually_bounded p
+    (f := fun n i => F (p.time n i)) (fun _ _ => hreg.smooth _)
+    hbτ (show 0 ≤ T+2 by linarith) (le_max_left 0 K) (fun n i hi x hx => by
+      have hleft : b (p.time n i) ∈ Icc (-R) R := abs_le.1 (by simpa only [Real.norm_eq_abs] using hR _ (p.mem n i hi.le))
+      have hright : b (p.time n (i+1)) ∈ Icc (-R) R := abs_le.1 (by simpa only [Real.norm_eq_abs] using hR _ (p.mem n (i+1) hi))
+      have hkx := hK (p.time n i,x) ⟨p.mem n i hi.le, uIcc_subset_Icc hleft hright hx⟩
+      rw [Real.norm_eq_abs] at hkx
+      exact hkx.trans (le_max_right 0 K)) hQc
+  have htime := p.time_increment_tendsto hreg.time_cont hreg.time_deriv hreg.drift_cont
+  have hriem := prestop_riemann_tendsto hT hτ hτT hreg.second_cont
+  let Qpred : ℕ → ℝ := fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+    C (mesh T n i) * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0
+  have hQpred : Tendsto Qpred atTop (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
+    have heq (n : ℕ) : Qpred n =
+      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+        C (mesh T n i) * ((b (mesh T n (i+1))-b (mesh T n i))^2-T/(2 : ℝ)^n) else 0) +
+      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then C (mesh T n i)*(T/(2 : ℝ)^n) else 0) := by
+      unfold Qpred
+      rw [← Finset.sum_add_distrib]
+      apply Finset.sum_congr rfl
+      intro i hi
+      split_ifs <;> ring
+    dsimp only [C] at heq
+    simpa only [← heq, zero_add] using hE.add hriem
+  have hQdiff := clipped_quadraticSum_difference_tendsto hT hτ hτT hb
+    (fun n i => C (p.time n i))
+    (K := max K₂ |C τ|) (hK₂.trans (le_max_left _ _))
+    (coefficient_clipped_bound C hC)
+  have hQeq (n i : ℕ) :
+      (if mesh T n i < τ then C (p.time n i) *
+        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) =
+      (if mesh T n i < τ then C (mesh T n i) *
+        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) := by
+    split_ifs with h
+    · simp only [p, clippedDyadic, min_eq_left h.le]
+    · rfl
+  simp only [hQeq] at hQdiff
+  have hQweighted : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      C (p.time n i) * (b (p.time n (i+1)) - b (p.time n i))^2) atTop
+      (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
+    simpa only [Qpred, sub_add_cancel, zero_add, p, clippedDyadic] using hQdiff.add hQpred
+  have hTel (n : ℕ) : ∑ i ∈ Finset.range (p.count n),
+      (F (p.time n (i+1)) (b (p.time n (i+1))) -
+        F (p.time n i) (b (p.time n i))) = F τ (b τ) - F 0 (b 0) := by
+    rw [Finset.sum_range_sub (fun i => F (p.time n i) (b (p.time n i))), p.start, p.finish]
+  have hIdentity (n : ℕ) : (∑ i ∈ Finset.range (p.count n),
+      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) =
+      F τ (b τ) - F 0 (b 0) -
+      (∑ i ∈ Finset.range (p.count n),
+        (F (p.time n (i+1)) (b (p.time n (i+1))) -
+          F (p.time n i) (b (p.time n (i+1))))) - (1/2 : ℝ)*
+      (∑ i ∈ Finset.range (p.count n),
+        C (p.time n i)*(b (p.time n (i+1))-b (p.time n i))^2) -
+      (∑ i ∈ Finset.range (p.count n), remainder₂ (F (p.time n i))
+        (b (p.time n i)) (b (p.time n (i+1)))) := by
+    rw [← hTel n]
+    simp only [Finset.mul_sum, ← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro i hi
+    simp only [A, C, remainder₂, iteratedDeriv_succ, iteratedDeriv_zero]
+    ring
+  have hAclip : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) atTop
+      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, C t)) := by
+    have hh := (((tendsto_const_nhds (x := F τ (b τ) - F 0 (b 0))).sub
+      htime).sub (hQweighted.const_mul (1/2 : ℝ))).sub hr
+    simpa only [← hIdentity, sub_zero] using hh
+  have hAdiff := clipped_linearSum_difference_tendsto hT hτ hτT hb
+    (fun n i => A (p.time n i))
+    (K := max K₁ |A τ|) (hK₁.trans (le_max_left _ _))
+    (coefficient_clipped_bound A hA)
+  have hAeq (n i : ℕ) :
+      (if mesh T n i < τ then A (p.time n i) *
+        (b (mesh T n (i+1))-b (mesh T n i)) else 0) =
+      (if mesh T n i < τ then A (mesh T n i) *
+        (b (mesh T n (i+1))-b (mesh T n i)) else 0) := by
+    split_ifs with h
+    · simp only [p, clippedDyadic, min_eq_left h.le]
+    · rfl
+  simp only [hAeq] at hAdiff
+  have hh := hAclip.sub hAdiff
+  simpa only [p, clippedDyadic, A, C, sub_sub_cancel, sub_zero] using hh
+
+theorem stopped_fieldSum_tendsto_ae {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
+    (hτ : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T)
+    (hreg : ∀ᵐ sample ∂P, LocalRegular (F sample) (V sample) (fun t => B t.toNNReal sample) (τ sample))
+    (C : ℕ → ℕ → Ω → ℝ)
+    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
+    (hC2 : ∀ n i, MemLp (C n i) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 2 B T τ F n i sample| ≤ K₂)
+    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
+    (hAbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 1 B T τ F n i sample| ≤ K₁) :
+    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n sample) atTop
+      (𝓝 (F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+        (∫ t in (0 : ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ sample),
+          iteratedDeriv 2 (F sample t) (B t.toNNReal sample))) := by
+  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T
+    C hC hC2 K₂ hK₂ (fun n i => by
+      filter_upwards [hCeq n i, hbound n i] with sample he hb
+      simpa only [he] using hb)
+  have hCeq_all : ∀ᵐ sample ∂P, ∀ n i, C n i sample = stoppedCoeff 2 B T τ F n i sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hCeq n))
+  have hC_all : ∀ᵐ sample ∂P, ∀ n i, |stoppedCoeff 2 B T τ F n i sample| ≤ K₂ :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hbound n))
+  have hA_all : ∀ᵐ sample ∂P, ∀ n i, |stoppedCoeff 1 B T τ F n i sample| ≤ K₁ :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAbound n))
+  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hτ, hC_all, hA_all, hCeq_all]
+    with sample hb hQ hE hr hτsample hCsample hAsample hCeqsample
+  have hEs : Tendsto (fun n => dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n sample) atTop (𝓝 0) := by
+    convert hE using 1
+    funext n
+    unfold dyadicQuadraticError
+    simp only [hCeqsample]
+  let b : ℝ → ℝ := fun t => B t.toNNReal sample
+  have hbc : Continuous b := hb.comp continuous_real_toNNReal
+  have hQ' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
+    simpa only [quadraticSum, b, mesh_toNNReal] using hQ
+  have hE' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ sample then
+      iteratedDeriv 2 (F sample (mesh T n i)) (b (mesh T n i)) *
+        ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0)
+        atTop (𝓝 0) := by
+    have heq (n : ℕ) : dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n sample =
+        ∑ i ∈ Finset.range (2^n), if mesh T n i < τ sample then
+          iteratedDeriv 2 (F sample (mesh T n i)) (b (mesh T n i)) *
+            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0 := by
+      unfold dyadicQuadraticError
+      apply Finset.sum_congr rfl
+      intro i hi
+      simp only [stoppedCoeff, b, mesh_toNNReal]
+      split_ifs <;> simp
+    simpa only [heq] using hEs
+  have hA' (n i : ℕ) (hi : mesh T n i < τ sample) :
+      |deriv (F sample (mesh T n i)) (b (mesh T n i))| ≤ K₁ := by
+    simpa only [stoppedCoeff, ite_eq_left hi, iteratedDeriv_succ, iteratedDeriv_zero,
+      b, mesh_toNNReal] using hAsample n i
+  have hC' (n i : ℕ) (hi : mesh T n i < τ sample) :
+      |iteratedDeriv 2 (F sample (mesh T n i)) (b (mesh T n i))| ≤ K₂ := by
+    simpa only [stoppedCoeff, ite_eq_left hi, b, mesh_toNNReal] using hCsample n i
+  have ht := stopped_field_taylor_limit T.coe_nonneg hτsample.1 hτsample.2 hbc.continuousOn
+    hr hQ' hK₁ hK₂ hA' hC' hE'
+  have heq (n : ℕ) : dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n sample =
+      ∑ i ∈ Finset.range (2^n), if mesh T n i < τ sample then
+        deriv (F sample (mesh T n i)) (b (mesh T n i)) *
+          (b (mesh T n (i+1))-b (mesh T n i)) else 0 := by
+    unfold dyadicDriverSum
+    apply Finset.sum_congr rfl
+    intro i hi
+    simp only [stoppedCoeff, b, mesh_toNNReal, iteratedDeriv_succ, iteratedDeriv_zero]
+    split_ifs <;> simp
+  simpa only [← heq, b, Real.toNNReal_zero] using ht
+
+theorem stopped_field_ito_centered_ae {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
+    (hτ : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T)
+    (hreg : ∀ᵐ sample ∂P, LocalRegular (F sample) (V sample) (fun t => B t.toNNReal sample) (τ sample))
+    (A C : ℕ → ℕ → Ω → ℝ)
+    (hAeq : ∀ n i, A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i)
+    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
+    (hC2 : ∀ n i, MemLp (C n i) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 2 B T τ F n i sample| ≤ K₂)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
+    (hA2 : ∀ n i, MemLp (A n i) 2 P)
+    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
+    (hAbound : ∀ n i, ∀ᵐ sample ∂P, |stoppedCoeff 1 B T τ F n i sample| ≤ K₁) :
+    let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0 : ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
+    Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 := by
+  have ht := stopped_fieldSum_tendsto_ae hB T τ F V hτ hreg C hCeq hC hC2
+    K₂ hK₂ hbound K₁ hK₁ hAbound
+  have hAeq_all : ∀ᵐ sample ∂P, ∀ n i, A n i sample = stoppedCoeff 1 B T τ F n i sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAeq n))
+  apply limit_centered hB T A hA hA2 K₁ hK₁ ?_ _ ?_
+  · intro n i
+    filter_upwards [hAeq n i, hAbound n i] with sample he hb
+    simpa only [he] using hb
+  · filter_upwards [ht, hAeq_all] with sample hsample he
+    convert hsample using 1
+    funext n
+    unfold dyadicDriverSum
+    simp only [he]
+
+end SLEExactGauge.BrownianLocal
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.AngularBarrier
+
+lemma profileT_continuousAt {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
+    {v : ℝ × ℝ} (hv : 0 < v.1) :
+    ContinuousAt (fun v : ℝ × ℝ => profileT l b v.1 v.2) v := by
+  have hs : ContinuousAt (fun v : ℝ × ℝ => scale l b v.1) v := by
+    unfold scale; fun_prop
+  have hq := (contDiff_squaredSine 0).continuous.continuousAt.comp (continuousAt_snd : ContinuousAt (fun v : ℝ × ℝ => v.2) v)
+  have hz : ContinuousAt (fun v : ℝ × ℝ => squaredSine v.2 / scale l b v.1) v := hq.div hs (scale_pos hl hb hv).ne'
+  have hp : ContinuousAt (fun v : ℝ × ℝ => pdf b (squaredSine v.2 / scale l b v.1)) v := (hasDerivAt_pdf hb (div_pos (squaredSine_pos v.2) (scale_pos hl hb hv))).continuousAt.comp_of_eq hz rfl
+  have hf := (contDiffAt_profile hl hb 0 hv).continuousAt
+  have he : ContinuousAt (fun v : ℝ × ℝ => Real.exp (l*b/2*v.1)) v := by fun_prop
+  have hc : ContinuousAt (fun v : ℝ × ℝ => l+l/(2*(b+1)*scale l b v.1)) v :=
+    continuousAt_const.add (continuousAt_const.div (continuousAt_const.mul hs)
+      (mul_ne_zero (by positivity) (scale_pos hl hb hv).ne'))
+  exact (continuousAt_const.mul hf).sub (((he.mul hp).mul hz).mul hc)
+
+end SLEExactGauge.AngularBarrier
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.GammaField
+open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+def rawRemaining (η ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  η + Real.log (2*height hε hT U z t) - logDerivative hε hT U z t - Real.log ρ
+
+def remaining (η ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  max η (rawRemaining hε hT η ρ U z t)
+
+def angleRatio (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) : ℝ :=
+  ((position hε hT U z t).re-sdev*x)/height hε hT U z t
+
+def field (l b η ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) : ℝ :=
+  profile l b (remaining hε hT η ρ U z t) (angleRatio hε hT sdev U z t x)
+
+def drift (l b η ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) : ℝ :=
+  let y := height hε hT U z t
+  let w := angleRatio hε hT sdev U z t x
+  let v := velocity ε (extendDriver T U t) (position hε hT U z t)
+  profileT l b (remaining hε hT η ρ U z t) w *
+    (v.im/y - variation hε hT U z t) +
+  profileW l b (remaining hε hT η ρ U z t) w * (v.re/y-w*v.im/y)
+
+lemma remaining_pos {η : ℝ} (hη : 0 < η) (ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    0 < remaining hε hT η ρ U z t := hη.trans_le (le_max_left _ _)
+
+lemma continuous_height (z : ℂ) :
+    Continuous (fun v : C(I, ℝ) × ℝ => height hε hT v.1 z v.2) :=
+  continuous_const.max (Complex.continuous_im.comp (continuous_position hε hT z))
+
+lemma continuous_rawRemaining (η ρ : ℝ) (z : ℂ) :
+    Continuous (fun v : C(I, ℝ) × ℝ => rawRemaining hε hT η ρ v.1 z v.2) := by
+  exact ((continuous_const.add ((continuous_const.mul (continuous_height hε hT z)).log
+    (fun v => (mul_pos (by norm_num) (height_pos hε hT v.1 z v.2)).ne'))).sub
+    (continuous_logDerivative hε hT z)).sub continuous_const
+
+lemma continuous_remaining (η ρ : ℝ) (z : ℂ) :
+    Continuous (fun v : C(I, ℝ) × ℝ => remaining hε hT η ρ v.1 z v.2) :=
+  continuous_const.max (continuous_rawRemaining hε hT η ρ z)
+
+lemma continuous_angleRatio (sdev : ℝ) (z : ℂ) :
+    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => angleRatio hε hT sdev v.1.1 z v.1.2 v.2) := by
+  have hp := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
+  have hy := (continuous_height hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
+  exact ((Complex.continuous_re.comp hp).sub (continuous_const.mul continuous_snd)).div hy
+    (fun v => (height_pos hε hT v.1.1 z v.1.2).ne')
+
+lemma smooth {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) (n : ℕ) :
+    ContDiff ℝ n (field hε hT l b η ρ sdev U z t) := by
+  apply (profile_smooth hl hb (remaining_pos hε hT hη ρ U z t) n).comp
+  unfold angleRatio
+  fun_prop
+
+lemma iteratedDeriv_eq {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ) (k : ℕ) :
+    iteratedDeriv k (field hε hT l b η ρ sdev U z t) x =
+      (-sdev/height hε hT U z t)^k *
+        iteratedDeriv k (profile l b (remaining hε hT η ρ U z t)) (angleRatio hε hT sdev U z t x) := by
+  have he : field hε hT l b η ρ sdev U z t = fun x =>
+      profile l b (remaining hε hT η ρ U z t)
+        ((-sdev/height hε hT U z t)*x+(position hε hT U z t).re/height hε hT U z t) := by
+    funext x; unfold field angleRatio; congr 1; ring
+  rw [he, iteratedDeriv_affine _ (profile_smooth hl hb (remaining_pos hε hT hη ρ U z t))]
+  congr 2
+  unfold angleRatio; ring
+
+lemma continuous_derivative {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev : ℝ) (z : ℂ) (k : ℕ) :
+    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv k (field hε hT l b η ρ sdev v.1.1 z v.1.2) v.2) := by
+  simp only [iteratedDeriv_eq hε hT hl hb hη]
+  apply continuous_iff_continuousAt.2
+  intro v
+  have ht := ((continuous_remaining hε hT η ρ z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v)
+  have hw := (continuous_angleRatio hε hT sdev z).continuousAt (x := v)
+  have hh := (profile_derivative_continuousAt hl hb k (remaining_pos hε hT hη ρ v.1.1 z v.1.2)).comp (ht.prodMk hw)
+  exact ((continuousAt_const.div ((continuous_height hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt
+    (height_pos hε hT v.1.1 z v.1.2).ne').pow k).mul hh
+
+lemma continuous_field {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev : ℝ) (z : ℂ) :
+    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => field hε hT l b η ρ sdev v.1.1 z v.1.2 v.2) := by
+  simpa using continuous_derivative hε hT hl hb hη ρ sdev z 0
+
+lemma continuous_drift {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev : ℝ) (z : ℂ) :
+    Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => drift hε hT l b η ρ sdev v.1.1 z v.1.2 v.2) := by
+  apply continuous_iff_continuousAt.2
+  intro v
+  have hR := ((continuous_remaining hε hT η ρ z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v)
+  have hW := (continuous_angleRatio hε hT sdev z).continuousAt (x := v)
+  have htime := (profileT_continuousAt hl hb (remaining_pos hε hT hη ρ v.1.1 z v.1.2)).comp (hR.prodMk hW)
+  have hspace := (profile_derivative_continuousAt hl hb 1 (remaining_pos hε hT hη ρ v.1.1 z v.1.2)).comp (hR.prodMk hW)
+  have hspace' : ContinuousAt (fun v : (C(I, ℝ) × ℝ) × ℝ => profileW l b
+      (remaining hε hT η ρ v.1.1 z v.1.2) (angleRatio hε hT sdev v.1.1 z v.1.2 v.2)) v := by
+    simpa only [Function.comp_def, iteratedDeriv_succ, iteratedDeriv_zero,
+      deriv_profile hl hb (remaining_pos hε hT hη ρ _ _ _)] using hspace
+  have hp := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
+  have hd := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))
+  have hv := ((continuous_velocity hε).comp (hd.prodMk hp)).continuousAt (x := v)
+  have hy := ((continuous_height hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v)
+  have hn := (height_pos hε hT v.1.1 z v.1.2).ne'
+  exact (htime.mul (((Complex.continuous_im.continuousAt.comp hv).div hy hn).sub
+    ((continuous_variation hε hT z).comp (continuous_fst : Continuous (fun v : (C(I, ℝ) × ℝ) × ℝ => v.1))).continuousAt (x := v))).add
+      (hspace'.mul (((Complex.continuous_re.continuousAt.comp hv).div hy hn).sub
+        ((hW.mul (Complex.continuous_im.continuousAt.comp hv)).div hy hn)))
+
+lemma rawRemaining_eq (η ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im) :
+    rawRemaining hε hT η ρ U z t = η + Real.log (radius hε hT U z t) - Real.log ρ := by
+  have hz := hε.trans_le hy
+  unfold rawRemaining radius height
+  rw [max_eq_right hy, Real.log_div (by positivity) (Real.exp_pos _).ne', Real.log_exp]
+  ring
+
+lemma rawRemaining_gt (η : ℝ) {ρ : ℝ} (hρ : 0 < ρ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im) (hr : ρ < radius hε hT U z t) :
+    η < rawRemaining hε hT η ρ U z t := by
+  rw [rawRemaining_eq hε hT η ρ U z t hy]
+  have hh := Real.strictMonoOn_log hρ (hρ.trans hr) hr
+  linarith
+
+lemma time_deriv {l b η ρ : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (hρ : 0 < ρ)
+    (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) {t : ℝ}
+    (ht : t ∈ Ioo 0 T) (hy : ε < (position hε hT U z t).im)
+    (hr : ρ < radius hε hT U z t) (x : ℝ) :
+    HasDerivAt (fun s => field hε hT l b η ρ sdev U z s x) (drift hε hT l b η ρ sdev U z t x) t := by
+  have hH := position_deriv hε hT U z ht
+  have hL := hasDerivAt_logDerivative hε hT U z ht
+  have hIm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hH
+  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
+  dsimp only [Function.comp_def, Complex.imCLM_apply, Complex.reCLM_apply] at hIm hRe
+  have hY : HasDerivAt (height hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)).im t := by
+    apply hIm.congr_of_eventuallyEq
+    filter_upwards [hIm.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy)] with s hs
+    exact max_eq_right hs.le
+  have hraw : HasDerivAt (rawRemaining hε hT η ρ U z)
+      ((velocity ε (extendDriver T U t) (position hε hT U z t)).im / height hε hT U z t -
+        variation hε hT U z t) t := by
+    have hh := (((hY.const_mul 2).log (by have := height_pos hε hT U z t; positivity)).const_add η).sub hL
+    apply (hh.sub_const (Real.log ρ)).congr_deriv
+    ring
+  have hR : HasDerivAt (remaining hε hT η ρ U z)
+      ((velocity ε (extendDriver T U t) (position hε hT U z t)).im / height hε hT U z t -
+        variation hε hT U z t) t := by
+    apply hraw.congr_of_eventuallyEq
+    have hh := rawRemaining_gt hε hT η hρ U z t hy.le hr
+    filter_upwards [hraw.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hh)] with s hs
+    exact max_eq_right hs.le
+  have hW : HasDerivAt (fun s => angleRatio hε hT sdev U z s x)
+      ((velocity ε (extendDriver T U t) (position hε hT U z t)).re / height hε hT U z t -
+        angleRatio hε hT sdev U z t x * (velocity ε (extendDriver T U t) (position hε hT U z t)).im / height hε hT U z t) t := by
+    apply ((hRe.sub_const (sdev*x)).div hY (height_pos hε hT U z t).ne').congr_deriv
+    dsimp only [angleRatio]
+    field_simp
+  exact hasDerivAt_profile_path hl hb hR hW (remaining_pos hε hT hη ρ U z t)
+
+lemma field_prefix_eq {ε S T : ℝ} (hε : 0 < ε) (hS : 0 ≤ S) (hT : 0 ≤ T)
+    (hST : S ≤ T) (U V : C(I, ℝ)) (z : ℂ) (l b η ρ sdev : ℝ)
+    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
+    {t : ℝ} (ht : t ∈ Icc 0 S) :
+    field hε hS l b η ρ sdev U z t = field hε hT l b η ρ sdev V z t := by
+  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
+  have hL := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
+  funext x
+  simp only [field, remaining, rawRemaining, angleRatio, height, hH, hL]
+
+lemma regular {l b η ρ : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (hρ : 0 < ρ)
+    (hTp : 0 < T) (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (δ : ℝ)
+    (B : ℝ → ℝ) (hB : Continuous B) :
+    BrownianLocal.LocalRegular (field hε hT l b η ρ sdev U z)
+      (drift hε hT l b η ρ sdev U z) B (stop hε hT U z ρ δ) := by
+  constructor
+  · intro t; exact smooth hε hT hl hb hη ρ sdev U z t 3
+  · have hm : Continuous (fun v : ℝ × ℝ => ((U,v.1),v.2)) :=
+      (continuous_const.prodMk continuous_fst).prodMk continuous_snd
+    exact ((continuous_derivative hε hT hl hb hη ρ sdev z 3).comp hm).continuousOn
+  · intro x
+    have hm : Continuous (fun t : ℝ => ((U,t),x)) :=
+      (continuous_const.prodMk continuous_id).prodMk continuous_const
+    exact ((continuous_field hε hT hl hb hη ρ sdev z).comp hm).continuousOn
+  · intro t ht x
+    have htT : t ∈ Ioo 0 T := ⟨ht.1,ht.2.trans_le (stop_bounds hε hT U z ρ δ).2⟩
+    have hh := (deficit_pos_iff hε hT U z ρ δ t).1
+      (positive_before hε hT hTp U z ρ δ ht.1.le ht.2)
+    exact time_deriv hε hT hl hb hη hρ sdev U z htT hh.2.1 hh.1 x
+  · have hm : Continuous (fun v : ℝ × ℝ => ((U,v.1),B v.2)) :=
+      (continuous_const.prodMk continuous_fst).prodMk (hB.comp continuous_snd)
+    exact ((continuous_drift hε hT hl hb hη ρ sdev z).comp hm).continuousOn
+  · have hm : Continuous (fun t : ℝ => ((U,t),B t)) :=
+      (continuous_const.prodMk continuous_id).prodMk hB
+    exact ((continuous_derivative hε hT hl hb hη ρ sdev z 2).comp hm).continuousOn
+
+lemma coefficient_bound {l b η : ℝ} (hl : 0 < l) (hb : 0 < b) (hη : 0 < η)
+    (ρ sdev M D : ℝ) (z : ℂ) (k : ℕ) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t x : ℝ),
+      remaining hε hT η ρ U z t ≤ M → |angleRatio hε hT sdev U z t x| ≤ D →
+      |iteratedDeriv k (field hε hT l b η ρ sdev U z t) x| ≤ K := by
+  have hc : ContinuousOn (fun v : ℝ × ℝ => iteratedDeriv k (profile l b v.1) v.2)
+      ((Icc η M) ×ˢ (Icc (-D) D)) := by
+    intro v hv
+    exact (profile_derivative_continuousAt hl hb k (hη.trans_le hv.1.1)).continuousWithinAt
+  obtain ⟨K,hK⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hc
+  refine ⟨(|sdev|/ε)^k * max 0 K, by positivity, ?_⟩
+  intro U t x ht hw
+  rw [iteratedDeriv_eq hε hT hl hb hη, abs_mul, abs_pow, abs_div, abs_neg,
+    abs_of_pos (height_pos hε hT U z t)]
+  apply mul_le_mul
+  · exact pow_le_pow_left₀ (by positivity)
+      (div_le_div_of_nonneg_left (abs_nonneg _) hε (le_max_left _ _)) k
+  · apply le_trans _ (le_max_right 0 K)
+    simpa only [Real.norm_eq_abs] using hK
+      (remaining hε hT η ρ U z t, angleRatio hε hT sdev U z t x)
+      ⟨⟨le_max_left _ _,ht⟩,abs_le.1 hw⟩
+  · exact abs_nonneg _
+  · positivity
+
+lemma physical_algebra (κ x y A B C : ℝ) (hy : 0 < y) :
+    A*((-2*y/(x^2+y^2))/y - (-2*(x^2-y^2)/(x^2+y^2)^2)) +
+      B*((2*x/(x^2+y^2))/y - (x/y)*(-2*y/(x^2+y^2))/y) +
+      (1/2:ℝ)*(κ/y^2*C) =
+    (4*y^2/(x^2+y^2)^2)*(-A+(x/y)*(1+(x/y)^2)*B+κ/8*(1+(x/y)^2)^2*C) := by
+  have hn : x^2+y^2 ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma drift_nonpos {κ η : ℝ} (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η)
+    (ρ : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im)
+    (hU : extendDriver T U t = Real.sqrt κ*x) :
+    drift hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t x +
+      (1/2:ℝ)*iteratedDeriv 2 (field hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t) x ≤ 0 := by
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have ht := remaining_pos hε hT hη ρ U z t
+  have hh : height hε hT U z t = (position hε hT U z t).im := max_eq_right hy
+  rw [iteratedDeriv_eq hε hT hl hb hη, second_profile hl hb ht]
+  dsimp only [drift]
+  rw [variation_of_unclipped hε hT U z t hy, velocity_eq hy, hh, hU]
+  simp only [Complex.div_re, Complex.div_im, Complex.re_ofNat, Complex.im_ofNat,
+    Complex.sub_re, Complex.ofReal_re, Complex.sub_im, Complex.ofReal_im,
+    sub_zero, zero_mul, Complex.normSq_apply]
+  simp only [neg_div, neg_sq, div_pow, Real.sq_sqrt hκ.1.le]
+  have he := physical_algebra κ ((position hε hT U z t).re-Real.sqrt κ*x)
+    (position hε hT U z t).im
+    (profileT (κ/2) ((8/κ-1)/2) (remaining hε hT η ρ U z t) (angleRatio hε hT (Real.sqrt κ) U z t x))
+    (profileW (κ/2) ((8/κ-1)/2) (remaining hε hT η ρ U z t) (angleRatio hε hT (Real.sqrt κ) U z t x))
+    (profileWW (κ/2) ((8/κ-1)/2) (remaining hε hT η ρ U z t) (angleRatio hε hT (Real.sqrt κ) U z t x)) (hε.trans_le hy)
+  have ha : angleRatio hε hT (Real.sqrt κ) U z t x =
+      ((position hε hT U z t).re-Real.sqrt κ*x)/(position hε hT U z t).im := by
+    unfold angleRatio; rw [hh]
+  rw [← ha] at he
+  convert (he.trans_le (mul_nonpos_of_nonneg_of_nonpos (by positivity) (by
+    have hs := profile_super hκ ht (angleRatio hε hT (Real.sqrt κ) U z t x)
+    linarith))) using 1; ring
+
+end SLEExactGauge.GammaField
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.GammaPredictable
+open LoewnerCutoff LoewnerExit GammaField BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+open scoped unitInterval
+variable {ε : ℝ} (hε : 0 < ε)
+
+def coeffAt (k : ℕ) (s : ℝ≥0) (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ)
+    (q : C(I, ℝ) × ℝ) : ℝ := by
+  classical
+  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
+    iteratedDeriv k (field hε s.coe_nonneg l b η ρ sdev q.1 z s) q.2 else 0
+
+lemma coeffAt_measurable (k : ℕ) (_hk : k = 1 ∨ k = 2) (s : ℝ≥0)
+    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (ρ δ : ℝ) :
+    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s l b η sdev z ρ δ) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
+      iteratedDeriv k (field hε s.coe_nonneg l b η ρ sdev q.1 z s) q.2) := by
+    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
+      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
+    have hc := (continuous_derivative hε s.coe_nonneg hl hb hη ρ sdev z k).comp hp
+    dsimp only [Function.comp_def] at hc
+    exact hc
+  apply hc.measurable.ite _ measurable_const
+  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
+
+def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω) : ℝ :=
+  if s < T then coeffAt hε k s l b η sdev z ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
+
+lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (ρ δ : ℝ) :
+    Measurable[pastSigma B s] (rawCoeff hε k B T s l b η sdev z ρ δ) := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  let : MeasurableSpace Ω := pastSigma B s
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z ρ δ).comp
+      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
+  · exact measurable_const
+
+lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
+    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (ρ δ : ℝ) :
+    AEMeasurable (rawCoeff hε k B T s l b η sdev z ρ δ) P := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z ρ δ).comp_aemeasurable
+      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
+  · exact aemeasurable_const
+
+lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    rawCoeff hε k B T s l b η sdev z ρ δ sample =
+      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
+        iteratedDeriv k (field hε T.coe_nonneg l b η ρ sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
+  classical
+  by_cases hsT : s < T
+  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
+    have hf := field_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z l b η ρ sdev he
+      ⟨s.coe_nonneg, le_rfl⟩
+    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
+    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
+        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
+      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
+      exact forall₂_congr (fun t ht => by rw [hd ht])
+    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
+  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
+      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
+    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
+
+lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (n i : ℕ) :
+    rawCoeff hε k B T (dyadicTime T n i) l b η sdev z ρ δ =ᵐ[P]
+      stoppedCoeff k B T
+        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
+        (fun sample => field hε T.coe_nonneg l b η ρ sdev (BrownianDriver.path B sdev T sample) z) n i := by
+  filter_upwards [hB.cont] with sample hc
+  rw [rawCoeff_eq hε k B T _ hT l b η sdev z ρ δ sample hc]
+  simp only [stoppedCoeff, coe_dyadicTime]
+
+end SLEExactGauge.GammaPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.GammaField
+open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+lemma ratio_bound {δ : ℝ} (hδ : 0 < δ) (sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t x : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im) (hs : δ ≤ sine hε hT U z t)
+    (hU : extendDriver T U t = sdev*x) :
+    |angleRatio hε hT sdev U z t x| ≤ 1/δ := by
+  have hyp := hε.trans_le hy
+  have hn : 0 < ‖position hε hT U z t - (extendDriver T U t : ℂ)‖ := by
+    apply norm_pos_iff.2
+    intro he
+    have hi := congrArg Complex.im he
+    simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hi
+    linarith
+  have hh : δ * ‖position hε hT U z t - (extendDriver T U t : ℂ)‖ ≤ (position hε hT U z t).im := by
+    apply (le_div_iff₀ hn).1
+    unfold sine at hs
+    rw [clampIm_eq (show ε ≤ (position hε hT U z t - (extendDriver T U t : ℂ)).im by simpa using hy)] at hs
+    exact hs
+  have hx := mul_le_mul_of_nonneg_left (Complex.abs_re_le_norm
+    (position hε hT U z t - (extendDriver T U t : ℂ))) hδ.le
+  have hxx := hx.trans hh
+  simp only [Complex.sub_re, Complex.ofReal_re, hU] at hxx
+  rw [angleRatio, height, max_eq_right hy, abs_div, abs_of_pos hyp]
+  apply (div_le_iff₀ hyp).2
+  rw [div_mul_eq_mul_div, one_mul]
+  apply (le_div_iff₀ hδ).2
+  nlinarith [hxx]
+
+end SLEExactGauge.GammaField
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/BoundaryDiffusion.lean b/lean/OAI/Probability/SLELocalization/BoundaryDiffusion.lean
index 5e6f7fbaac03c381b1ae1312328c9613fd49fe3d..f7bf4df4dd432f5da3f79288590badafe6347f6a 100644
--- a/lean/OAI/Probability/SLELocalization/BoundaryDiffusion.lean
+++ b/lean/OAI/Probability/SLELocalization/BoundaryDiffusion.lean
@@ -1,1028 +1,1028 @@
-import Mathlib
-import OAI.Probability.SLELocalization.SwallowingLaws
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Boundary Green estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing GreenPhysical
-
-lemma deterministic_budget {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (κ β a : ℝ) (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (U : C(I, ℝ)) (x L : ℝ) (b : ℝ → ℝ) (hb : Continuous b)
-    (hb0 : b 0=0) (hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*b t) (Icc 0 T)) :
-    let τ := stop hε hT.le U x β a L
-    let F := field hε hT.le β a (Real.sqrt κ) U x
-    let V := drift hε hT.le β a (Real.sqrt κ) U x
-    F τ (b τ) ≤
-        (F τ (b τ)-F 0 (b 0)-(∫ t in (0:ℝ)..τ,V t (b t))-
-          (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (b t))+radial (-β) x ε := by
-  dsimp only
-  let τ := stop hε hT.le U x β a L
-  let F := field hε hT.le β a (Real.sqrt κ) U x
-  let V := drift hε hT.le β a (Real.sqrt κ) U x
-  have htau := stop_bounds hε hT.le U x β a L
-  have hmap : Continuous (fun t => ((U,t),b t)) := (continuous_const.prodMk continuous_id).prodMk hb
-  have hV : Continuous (fun t => V t (b t)) := by
-    have hh := (continuous_drift hε hT.le β a (Real.sqrt κ) x).comp hmap
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  have hQ : Continuous (fun t => iteratedDeriv 2 (F t) (b t)) := by
-    have hh := (continuous_coeff hε hT.le β a (Real.sqrt κ) x 2 (Or.inr rfl)).comp hmap
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  have hneg : (∫ t in (0:ℝ)..τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) ≤ 0 := by
-    have hh : (∫ t in (0:ℝ)..τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) ≤
-        ∫ _t in (0:ℝ)..τ, (0:ℝ) := by
-      apply intervalIntegral.integral_mono_on_of_le_Ioo htau.1
-        ((hV.add (continuous_const.mul hQ)).intervalIntegrable _ _)
-        (intervalIntegrable_const)
-      intro t ht
-      have hh := (deficit_pos_iff hε hT.le U x β a L t).1
-        (positive_before hε hT.le hT U x β a L ht.1.le ht.2)
-      change V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) ≤ 0
-      have he := generator_field hε hT.le κ β a hκ U x t (b t) hh.1.le
-        (hU ⟨ht.1.le,ht.2.le.trans htau.2⟩)
-      change V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) = _ at he
-      rw [he]
-      exact mul_nonpos_of_nonneg_of_nonpos (factor_pos hε hT.le β a U x t).le
-        (generator_nonpos hκ hβ ha hba (hε.trans hh.1) hε)
-    simpa using hh
-  have hhalf : IntervalIntegrable (fun t => (1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) volume 0 τ :=
-    (hQ.intervalIntegrable _ _).const_mul _
-  rw [intervalIntegral.integral_add (hV.intervalIntegrable _ _) hhalf,
-    intervalIntegral.integral_const_mul] at hneg
-  have hini : F 0 (b 0) = radial (-β) x ε := by rw [hb0]; exact field_initial hε hT.le β a _ U x
-  change F τ (b τ) ≤ (F τ (b τ)-F 0 (b 0)-(∫ t in (0:ℝ)..τ,V t (b t))-
-    (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (b t))+radial (-β) x ε
-  rw [hini]
-  linarith
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.Swallowing
-open LoewnerCutoff LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-variable {ε : ℝ} (hε : 0 < ε)
-
-def coeffAt (k : ℕ) (s : ℝ≥0) (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ)
-    (q : C(I, ℝ) × ℝ) : ℝ := by
-  classical
-  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
-    iteratedDeriv k (field hε s.coe_nonneg β C sdev q.1 z s) q.2 else 0
-
-lemma coeffAt_measurable (k : ℕ) (hk : k = 1 ∨ k = 2) (s : ℝ≥0)
-    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s β C sdev z ρ δ) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
-      iteratedDeriv k (field hε s.coe_nonneg β C sdev q.1 z s) q.2) := by
-    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
-      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
-    have hh := (continuous_coeff hε s.coe_nonneg β C sdev z k hk).comp hp
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  apply hc.measurable.ite _ measurable_const
-  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
-
-def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω) : ℝ :=
-  if s < T then coeffAt hε k s β C sdev z ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
-
-lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    Measurable[pastSigma B s] (rawCoeff hε k B T s β C sdev z ρ δ) := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  let : MeasurableSpace Ω := pastSigma B s
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s β C sdev z ρ δ).comp
-      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
-  · exact measurable_const
-
-lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
-    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    AEMeasurable (rawCoeff hε k B T s β C sdev z ρ δ) P := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s β C sdev z ρ δ).comp_aemeasurable
-      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
-  · exact aemeasurable_const
-
-lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    rawCoeff hε k B T s β C sdev z ρ δ sample =
-      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
-        iteratedDeriv k (field hε T.coe_nonneg β C sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
-  classical
-  by_cases hsT : s < T
-  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
-    have hf := field_prefix_eq hε T.coe_nonneg s.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z β C sdev he
-      ⟨s.coe_nonneg, le_rfl⟩
-    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
-    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
-        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
-      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
-      exact forall₂_congr (fun t ht => by rw [hd ht])
-    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
-  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
-      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
-    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
-
-lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (n i : ℕ) :
-    rawCoeff hε k B T (dyadicTime T n i) β C sdev z ρ δ =ᵐ[P]
-      stoppedCoeff k B T
-        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
-        (fun sample => field hε T.coe_nonneg β C sdev (BrownianDriver.path B sdev T sample) z) n i := by
-  filter_upwards [hB.cont] with sample hc
-  rw [rawCoeff_eq hε k B T _ hT β C sdev z ρ δ sample hc]
-  simp only [stoppedCoeff, coe_dyadicTime]
-
-theorem localized_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (β C sdev : ℝ) (hβ : β ≤ 1)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (ρ δ : ℝ) :
-    let U := BrownianDriver.path B sdev T
-    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-    let F := fun sample => field hε T.coe_nonneg β C sdev (U sample) z
-    let V := fun sample => drift hε T.coe_nonneg β C sdev (U sample) z
-    let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
-    Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 := by
-  dsimp only
-  let U := BrownianDriver.path B sdev T
-  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-  let F := fun sample => field hε T.coe_nonneg β C sdev (U sample) z
-  let V := fun sample => drift hε T.coe_nonneg β C sdev (U sample) z
-  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact regular hε T.coe_nonneg hT β C sdev hβ (U sample) z ρ δ _
-      (hc.comp continuous_real_toNNReal)
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
-    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) z ρ δ)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg β C sdev hβ z 1 (Or.inl rfl)
-  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg β C sdev hβ z 2 (Or.inr (Or.inl rfl))
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
-        |iteratedDeriv k (field hε T.coe_nonneg β C sdev W z t) b| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
-    apply Filter.Eventually.of_forall
-    intro sample
-    unfold stoppedCoeff
-    split_ifs
-    · exact hb _ _ _
-    · simpa only [abs_zero] using hK
-  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) β C sdev z ρ δ
-  let Q := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) β C sdev z ρ δ
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 1 T hT β C sdev z ρ δ n i
-  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 2 T hT β C sdev z ρ δ n i
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
-    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ z ρ δ
-  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
-    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ z ρ δ
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₁
-    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hb
-  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₂
-    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |Q n i sample| ≤ K₂
-    rw [he]
-    exact hb
-  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A Q hAe hCe hCm hC2
-    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  exact hi
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.Swallowing
-open GreenPhysical LoewnerCutoff LoewnerExit
-
-abbrev correction {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ :=
-  let U := BrownianDriver.path B (Real.sqrt κ) T ω
-  let τ := stop hε T.coe_nonneg U z 0 0
-  let F := field hε T.coe_nonneg β C (Real.sqrt κ) U z
-  let V := drift hε T.coe_nonneg β C (Real.sqrt κ) U z
-  F τ (B τ.toNNReal ω)-F 0 (B 0 ω)-
-    (∫ t in (0:ℝ)..τ, V t (B t.toNNReal ω))-
-      (1/2:ℝ)*∫ t in (0:ℝ)..τ, iteratedDeriv 2 (F t) (B t.toNNReal ω)
-
-def budget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ :=
-  if T=0 then lyapunov β C z.re z.im else
-    max (correction B κ β C hε T z ω+lyapunov β C z.re z.im) 0
-
-lemma field_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (hy : ε ≤ z.im) :
-    field hε hT β C s U z 0 0 = lyapunov β C z.re z.im := by
-  simp only [field,LoewnerField.height,position_initial,mul_zero,sub_zero,max_eq_right hy]
-
-lemma generator_field {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (κ β C : ℝ) (hκ : 0 ≤ κ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im)
-    (hU : extendDriver T U t = Real.sqrt κ*b) :
-    drift hε hT β C (Real.sqrt κ) U z t b+
-      (1/2:ℝ)*iteratedDeriv 2 (field hε hT β C (Real.sqrt κ) U z t) b =
-    generator κ β C ((position hε hT U z t).re-Real.sqrt κ*b) (LoewnerField.height hε hT U z t) := by
-  have hh : LoewnerField.height hε hT U z t = (position hε hT U z t).im := max_eq_right hy
-  rw [second_eq,Real.sq_sqrt hκ]
-  dsimp only [drift]
-  rw [velocity_eq hy,hh,hU]
-  simp only [generator,Complex.div_re,Complex.div_im,Complex.re_ofNat,Complex.im_ofNat,
-    Complex.sub_re,Complex.ofReal_re,Complex.sub_im,Complex.ofReal_im,sub_zero,zero_mul,
-    Complex.normSq_apply]
-  ring
-
-lemma deterministic_budget {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (κ β : ℝ) (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (U : C(I, ℝ)) (z : ℂ) (hz : ε ≤ z.im) (b : ℝ → ℝ) (hb : Continuous b)
-    (hb0 : b 0=0) (hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*b t) (Icc 0 T)) :
-    let C := (2*κ-8-κ*β)/4
-    let d := β*(κ-4-κ*β)/2
-    let τ := stop hε hT.le U z 0 0
-    let F := field hε hT.le β C (Real.sqrt κ) U z
-    let V := drift hε hT.le β C (Real.sqrt κ) U z
-    let E := ∫ t in (0:ℝ)..τ,
-      (((position hε hT.le U z t).re-Real.sqrt κ*b t)^2+(LoewnerField.height hε hT.le U z t)^2)^(β/2-1)
-    0 ≤ E ∧
-      F τ (b τ)+d*E ≤
-        (F τ (b τ)-F 0 (b 0)- (∫ t in (0:ℝ)..τ,V t (b t))-
-          (1/2:ℝ)*∫ t in (0:ℝ)..τ, iteratedDeriv 2 (F t) (b t))+lyapunov β C z.re z.im := by
-  dsimp only
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  let τ := stop hε hT.le U z 0 0
-  let F := field hε hT.le β C (Real.sqrt κ) U z
-  let V := drift hε hT.le β C (Real.sqrt κ) U z
-  let H := fun t => (((position hε hT.le U z t).re-Real.sqrt κ*b t)^2+
-    (LoewnerField.height hε hT.le U z t)^2)^(β/2-1)
-  have htau := stop_bounds hε hT.le U z 0 0
-  have hmap : Continuous (fun t => ((U,t),b t)) := (continuous_const.prodMk continuous_id).prodMk hb
-  have hV : Continuous (fun t => V t (b t)) := by
-    have hh := (continuous_drift hε hT.le β C (Real.sqrt κ) z).comp hmap
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  have hQ : Continuous (fun t => iteratedDeriv 2 (F t) (b t)) := by
-    have hh := (continuous_coeff hε hT.le β C (Real.sqrt κ) z 2 (Or.inr rfl)).comp hmap
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  have hHc : Continuous H := by
-    have hP := (continuous_position hε hT.le z).comp (continuous_const.prodMk continuous_id : Continuous (fun t : ℝ => (U,t)))
-    have hx := (Complex.continuous_re.comp hP).sub (continuous_const.mul hb : Continuous (fun t => Real.sqrt κ*b t))
-    have hy : Continuous (LoewnerField.height hε hT.le U z) := continuous_const.max (Complex.continuous_im.comp hP)
-    exact ((hx.pow 2).add (hy.pow 2)).rpow_const (fun t => Or.inl (q_pos _ (LoewnerField.height_pos hε hT.le U z t)).ne')
-  have hH0 : 0 ≤ ∫ t in (0:ℝ)..τ,H t := by
-    apply intervalIntegral.integral_nonneg htau.1
-    intro t ht
-    exact Real.rpow_nonneg (by positivity) _
-  refine ⟨hH0,?_⟩
-  have hpoint : ∀ t ∈ Ioo 0 τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) ≤ -d*H t := by
-    intro t ht
-    have hh := (deficit_pos_iff hε hT.le U z 0 0 t).1
-      (positive_before hε hT.le hT U z 0 0 ht.1.le ht.2)
-    have he := generator_field hε hT.le κ β C (by linarith) U z t (b t) hh.2.1.le
-      (hU ⟨ht.1.le,ht.2.le.trans htau.2⟩)
-    change V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) = _ at he
-    rw [he]
-    exact (generator_negative hκ hβ hβκ _ (LoewnerField.height_pos hε hT.le U z t)).2.2
-  have hm : (∫ t in (0:ℝ)..τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) ≤
-      ∫ t in (0:ℝ)..τ, -d*H t := by
-    exact intervalIntegral.integral_mono_on_of_le_Ioo htau.1
-      ((hV.add (continuous_const.mul hQ)).intervalIntegrable _ _)
-      ((continuous_const.mul hHc).intervalIntegrable _ _) hpoint
-  have hhalf : IntervalIntegrable (fun t => (1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) volume 0 τ :=
-    (hQ.intervalIntegrable _ _).const_mul _
-  rw [intervalIntegral.integral_add (hV.intervalIntegrable _ _) hhalf,
-    intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul] at hm
-  have hinit : F 0 (b 0) = lyapunov β C z.re z.im := by rw [hb0]; exact field_initial hε hT.le β C _ U z hz
-  change F τ (b τ)+d*(∫ t in (0:ℝ)..τ,H t) ≤
-    (F τ (b τ)-F 0 (b 0)-(∫ t in (0:ℝ)..τ,V t (b t))-
-      (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (b t))+lyapunov β C z.re z.im
-  rw [hinit]
-  linarith
-
-lemma budget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ β : ℝ) (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ) (hz : ε ≤ z.im) :
-    let C := (2*κ-8-κ*β)/4
-    Integrable (budget B κ β C hε T z) P ∧
-      (∫ ω,budget B κ β C hε T z ω ∂P) = lyapunov β C z.re z.im ∧
-      ∀ ω, 0 ≤ budget B κ β C hε T z ω := by
-  dsimp only
-  let C := (2*κ-8-κ*β)/4
-  have hgen := generator_negative hκ hβ hβκ z.re (hε.trans_le hz)
-  have hC : 0 ≤ C := hgen.1.le
-  have hφ : 0 ≤ lyapunov β C z.re z.im := lyapunov_nonneg hC _ (hε.trans_le hz)
-  by_cases hT0 : T=0
-  · subst T
-    have he0 : budget B κ β ((2*κ-8-κ*β)/4) hε 0 z =
-        fun _ : Ω => lyapunov β C z.re z.im := by
-      funext ω
-      exact ite_eq_left rfl
-    rw [he0]
-    exact ⟨integrable_const _,by simp [C],fun _ => hφ⟩
-  have hT : 0 < (T:ℝ) := NNReal.coe_pos.2 (pos_iff_ne_zero.2 hT0)
-  have hb1 : β ≤ 1 := by have hh := div_pos (by norm_num : (0:ℝ)<4) (by linarith : 0<κ); linarith
-  have hcenter := localized_centered hB β C (Real.sqrt κ) hb1 hε T hT z 0 0
-  change Integrable (correction B κ β C hε T z) P ∧ (∫ ω,correction B κ β C hε T z ω ∂P)=0 at hcenter
-  have he : budget B κ β C hε T z =ᵐ[P]
-      (fun ω => correction B κ β C hε T z ω+lyapunov β C z.re z.im) := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hc hb0
-    have hh := deterministic_budget hε hT κ β hκ hβ hβκ (BrownianDriver.path B (Real.sqrt κ) T ω) z hz
-      (fun t => B t.toNNReal ω) (hc.comp continuous_real_toNNReal) (by simpa using hb0)
-      (fun t ht => BrownianDriver.extend_eq B _ T ω hc ht)
-    dsimp only at hh
-    have hn := lyapunov_nonneg (β := β) hC
-      ((position hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z
-        (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z 0 0)).re-
-        Real.sqrt κ*B (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z 0 0).toNNReal ω)
-      (LoewnerField.height_pos hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z
-        (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z 0 0))
-    have hh2 := hh.2
-    simp only [Real.toNNReal_zero] at hh2
-    have hpos : 0 ≤ correction B κ β C hε T z ω+lyapunov β C z.re z.im :=
-      (add_nonneg hn (mul_nonneg hgen.2.1.le hh.1)).trans hh2
-    simp only [budget,ite_eq_right hT0,max_eq_left hpos]
-  have hi := hcenter.1.add (integrable_const (lyapunov β C z.re z.im))
-  refine ⟨hi.congr he.symm,?_,fun ω => ?_⟩
-  · rw [integral_congr_ae he,integral_add hcenter.1 (integrable_const _),hcenter.2,zero_add]
-    simp [C]
-  · simp only [budget,ite_eq_right hT0]
-    exact le_max_right _ _
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-
-lemma measure_eventually_budget {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (W : ℕ → Ω → ℝ) (M a : ℝ) (ha : 0 < a)
-    (hi : ∀ n, Integrable (W n) P) (hn : ∀ n ω, 0 ≤ W n ω)
-    (hmean : ∀ n, (∫ ω,W n ω ∂P)=M) (E : Set Ω)
-    (hE : ∀ᵐ ω ∂P, ω∈E → ∀ᶠ n in atTop, a ≤ W n ω) :
-    P E ≤ ENNReal.ofReal (M/a) := by
-  let L := fun ω => liminf (fun n => ENNReal.ofReal (W n ω)) atTop
-  have hm (n : ℕ) : AEMeasurable (fun ω => ENNReal.ofReal (W n ω)) P :=
-    ENNReal.measurable_ofReal.comp_aemeasurable (hi n).aemeasurable
-  have hLm : AEMeasurable L P := by
-    simp only [L,liminf_eq_iSup_iInf_of_nat]
-    exact AEMeasurable.iSup (fun n => AEMeasurable.iInf (fun i => AEMeasurable.iInf (fun _ => hm i)))
-  have hLi : (∫⁻ ω,L ω ∂P) ≤ ENNReal.ofReal M := by
-    have hf := lintegral_liminf_le' (u := (atTop : Filter ℕ)) hm
-    have he (n : ℕ) : (∫⁻ ω, ENNReal.ofReal (W n ω) ∂P)=ENNReal.ofReal M := by
-      rw [← ofReal_integral_eq_lintegral_ofReal (hi n) (Filter.Eventually.of_forall (hn n)),hmean n]
-    simpa only [he,liminf_const] using hf
-  have hsub : E ≤ᵐ[P] {ω | ENNReal.ofReal a ≤ L ω} := by
-    filter_upwards [hE] with ω hω
-    intro hωE
-    apply le_liminf_of_le (by isBoundedDefault)
-    exact (hω hωE).mono (fun n hh => ENNReal.ofReal_le_ofReal hh)
-  calc
-    P E ≤ P {ω | ENNReal.ofReal a ≤ L ω} := measure_mono_ae hsub
-    _ ≤ (∫⁻ ω,L ω ∂P)/ENNReal.ofReal a :=
-      meas_ge_le_lintegral_div hLm (ENNReal.ofReal_ne_zero_iff.2 ha) ENNReal.ofReal_ne_top
-    _ ≤ ENNReal.ofReal M/ENNReal.ofReal a := by gcongr
-    _ = ENNReal.ofReal (M/a) := (ENNReal.ofReal_div_of_pos ha).symm
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing
-
-abbrev correction {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (x L : ℝ) (ω : Ω) : ℝ :=
-  let U := BrownianDriver.path B (Real.sqrt κ) T ω
-  let τ := stop hε T.coe_nonneg U x β a L
-  let F := field hε T.coe_nonneg β a (Real.sqrt κ) U x
-  let V := drift hε T.coe_nonneg β a (Real.sqrt κ) U x
-  F τ (B τ.toNNReal ω)-F 0 (B 0 ω)-
-    (∫ t in (0:ℝ)..τ,V t (B t.toNNReal ω))-
-    (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (B t.toNNReal ω)
-
-def budget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (x L : ℝ) (ω : Ω) : ℝ :=
-  max (correction B κ β a hε T x L ω+radial (-β) x ε) 0
-
-lemma budget_nonneg {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (x L : ℝ) (ω : Ω) :
-    0 ≤ budget B κ β a hε T x L ω := le_max_right _ _
-
-lemma budget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x L : ℝ) :
-    Integrable (budget B κ β a hε T x L) P ∧
-      (∫ ω,budget B κ β a hε T x L ω ∂P) = radial (-β) x ε := by
-  have hc := localized_centered hB β a (Real.sqrt κ) hβ hε T hT x L
-  change Integrable (correction B κ β a hε T x L) P ∧
-    (∫ ω,correction B κ β a hε T x L ω ∂P)=0 at hc
-  have he : budget B κ β a hε T x L =ᵐ[P] fun ω => correction B κ β a hε T x L ω+radial (-β) x ε := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hcont hb0
-    have hh := deterministic_budget hε hT κ β a hκ hβ ha hba
-      (BrownianDriver.path B (Real.sqrt κ) T ω) x L (fun t => B t.toNNReal ω)
-      (hcont.comp continuous_real_toNNReal) (by simpa using hb0)
-      (fun t ht => BrownianDriver.extend_eq B _ T ω hcont ht)
-    dsimp only at hh
-    simp only [Real.toNNReal_zero] at hh
-    have hn := (field_pos hε T.coe_nonneg β a (Real.sqrt κ)
-      (BrownianDriver.path B (Real.sqrt κ) T ω) x
-      (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) x β a L)
-      (B (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) x β a L).toNNReal ω)).le
-    exact max_eq_left (hn.trans hh)
-  have hi := hc.1.add (integrable_const (radial (-β) x ε))
-  refine ⟨hi.congr he.symm,?_⟩
-  rw [integral_congr_ae he,integral_add hc.1 (integrable_const _),hc.2,zero_add]
-  simp
-
-def reach (U : C(I, ℝ)) {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (x β a L : ℝ) : Prop :=
-  ∃ t ∈ Icc 0 T, (∀ s ∈ Icc 0 t, ε < gap hε hT U x s) ∧ L ≤ weight hε hT U x β a t
-
-lemma reached_terminal {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (U : C(I, ℝ))
-    (x β a L : ℝ) (hreach : reach U hε hT.le x β a L) :
-    L ≤ weight hε hT.le U x β a (stop hε hT.le U x β a L) := by
-  obtain ⟨t,ht,hgap,hweight⟩ := hreach
-  have hst : stop hε hT.le U x β a L ≤ t := by
-    by_contra hn
-    have hp := (deficit_pos_iff hε hT.le U x β a L t).1
-      (positive_before hε hT.le hT U x β a L ht.1 (lt_of_not_ge hn))
-    exact (not_lt_of_ge hweight) hp.2
-  have hb := stop_bounds hε hT.le U x β a L
-  by_cases hend : stop hε hT.le U x β a L < T
-  · have hh := nonpos_at_of_lt hε hT.le hT U x β a L hend
-    have hg := hgap _ ⟨hb.1,hst⟩
-    simp only [deficit,min_le_iff,sub_nonpos] at hh
-    rcases hh with hh | hh
-    · exact (not_le_of_gt hg hh).elim
-    · exact hh
-  · have he : stop hε hT.le U x β a L=t := le_antisymm hst (ht.2.trans (not_lt.1 hend))
-    rw [he]
-    exact hweight
-
-lemma budget_dominates_reach {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x L : ℝ) (ω : Ω)
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    (hr : reach (BrownianDriver.path B (Real.sqrt κ) T ω) hε T.coe_nonneg x β a L) :
-    L ≤ budget B κ β a hε T x L ω := by
-  let U := BrownianDriver.path B (Real.sqrt κ) T ω
-  let τ := stop hε T.coe_nonneg U x β a L
-  have htau := stop_bounds hε T.coe_nonneg U x β a L
-  have hh := deterministic_budget hε hT κ β a hκ hβ ha hba U x L
-    (fun t => B t.toNNReal ω) (hc.comp continuous_real_toNNReal) (by simpa using hb0)
-    (fun t ht => BrownianDriver.extend_eq B _ T ω hc ht)
-  have hw := reached_terminal hε hT U x β a L hr
-  have he : field hε T.coe_nonneg β a (Real.sqrt κ) U x τ (B τ.toNNReal ω) =
-      weight hε T.coe_nonneg U x β a τ := by
-    have hU : extendDriver T U τ = Real.sqrt κ*B τ.toNNReal ω :=
-      BrownianDriver.extend_eq B _ T ω hc htau
-    simp only [field,weight,gap,hU]
-  dsimp only at hh
-  simp only [Real.toNNReal_zero] at hh
-  change field hε T.coe_nonneg β a (Real.sqrt κ) U x τ (B τ.toNNReal ω) ≤
-    correction B κ β a hε T x L ω+radial (-β) x ε at hh
-  rw [he] at hh
-  exact (hw.trans hh).trans (le_max_left _ _)
-
-theorem maximal_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x L : ℝ) (hL : 0 < L) :
-    P {ω | reach (BrownianDriver.path B (Real.sqrt κ) T ω) hε T.coe_nonneg x β a L} ≤
-      ENNReal.ofReal (radial (-β) x ε/L) := by
-  have hm := budget_mean hB κ β a hκ hβ ha hba hε T hT x L
-  apply measure_eventually_budget (fun _ : ℕ => budget B κ β a hε T x L) _ L hL
-    (fun _ => hm.1) (fun _ => budget_nonneg B κ β a hε T x L) (fun _ => hm.2)
-  filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hc hb0
-  intro hr
-  exact Filter.Eventually.of_forall (fun _ => budget_dominates_reach B κ β a hκ hβ ha hba hε T hT x L ω hc hb0 hr)
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryFunctional
-
-lemma continuous_position_time {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) : Continuous (position hε hT U z) := by
-  have hp : Continuous (fun t : ℝ => (U, t)) := continuous_const.prodMk continuous_id
-  have hh := (continuous_position hε hT z).comp hp
-  change Continuous (position hε hT U z) at hh
-  exact hh
-
-lemma continuous_logDerivative_time {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) : Continuous (logDerivative hε hT U z) := by
-  have hp : Continuous (fun t : ℝ => (U, t)) := continuous_const.prodMk continuous_id
-  have hh := (continuous_logDerivative hε hT z).comp hp
-  change Continuous (logDerivative hε hT U z) at hh
-  exact hh
-
-end SLEExactGauge.BoundaryFunctional
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-
-lemma continuous_gap_time {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (x : ℝ) : Continuous (gap hε hT U x) := by
-  have hp : Continuous (fun t : ℝ => (U,t)) := continuous_const.prodMk continuous_id
-  have hh := (continuous_gap hε hT x).comp hp
-  change Continuous (gap hε hT U x) at hh
-  exact hh
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing
-
-lemma norm_sum_lower {X δ : ℝ} {D : ℂ} (hX : 0 < X) (hδ : δ ≤ 1/2)
-    (hD : ‖D‖ ≤ δ*X) : X/2 ≤ ‖(X:ℂ)+D‖ := by
-  have hh := norm_sub_le ((X:ℂ)+D) D
-  rw [add_sub_cancel_right,Complex.norm_real,Real.norm_of_nonneg hX.le] at hh
-  have hm := mul_le_mul_of_nonneg_right hδ hX.le
-  linarith
-
-lemma ratio_real_lower {X δ a : ℝ} {D : ℂ} (hX : 0 < X) (hδ0 : 0 ≤ δ) (hδ : δ ≤ 1/2)
-    (hD : ‖D‖ ≤ δ*X) (haδ : a ≤ 2-4*δ) :
-    a ≤ 2*((X:ℂ)/((X:ℂ)+D)).re := by
-  have hlo := norm_sum_lower hX hδ hD
-  have hn : (X:ℂ)+D ≠ 0 := norm_pos_iff.mp (lt_of_lt_of_le (by positivity) hlo)
-  have hquot : ‖D/((X:ℂ)+D)‖ ≤ 2*δ := by
-    rw [norm_div]
-    apply (div_le_iff₀ (norm_pos_iff.mpr hn)).2
-    calc
-      ‖D‖ ≤ δ*X := hD
-      _ ≤ (2*δ)*‖(X:ℂ)+D‖ := by nlinarith
-  have he : (X:ℂ)/((X:ℂ)+D) = 1-D/((X:ℂ)+D) := by field_simp; ring
-  rw [he,Complex.sub_re,Complex.one_re]
-  have hre := (Complex.re_le_norm (D/((X:ℂ)+D))).trans hquot
-  linarith
-
-lemma normSq_slope (X : ℝ) (D : ℂ) (hX : X ≠ 0) (hD : (X:ℂ)+D ≠ 0) :
-    2*D.re*(2/((X:ℂ)+D)-2/(X:ℂ)).re +
-      2*D.im*(2/((X:ℂ)+D)-2/(X:ℂ)).im =
-    -4*Complex.normSq D / X^2 * ((X:ℂ)/((X:ℂ)+D)).re := by
-  have hQ : (X+D.re)^2+D.im^2 ≠ 0 := by
-    have hh := (Complex.normSq_pos.mpr hD).ne'
-    simpa only [Complex.normSq_apply,Complex.add_re,Complex.ofReal_re,Complex.add_im,
-      Complex.ofReal_im,zero_add,pow_two] using hh
-  simp only [Complex.sub_re,Complex.sub_im,Complex.div_re,Complex.div_im,Complex.re_ofNat,
-    Complex.im_ofNat,Complex.add_re,Complex.add_im,Complex.ofReal_re,Complex.ofReal_im,
-    Complex.normSq_apply,zero_add,zero_mul,mul_zero,add_zero]
-  simp only [← pow_two]
-  field_simp
-  ring
-
-lemma energy_deriv {D : ℝ → ℂ} {l : ℝ → ℝ} {t X a : ℝ}
-    (hX : X ≠ 0) (hD : (X:ℂ)+D t ≠ 0)
-    (hd : HasDerivAt D (2/((X:ℂ)+D t)-2/(X:ℂ)) t)
-    (hl : HasDerivAt l (-2/X^2) t) :
-    HasDerivAt (fun s => Complex.normSq (D s)*Real.exp (-a*l s))
-      (Real.exp (-a*l t)*(-4*Complex.normSq (D t)/X^2*((X:ℂ)/((X:ℂ)+D t)).re+
-        2*a/X^2*Complex.normSq (D t))) t := by
-  have hre := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hd
-  have him := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hd
-  dsimp only [Function.comp_def,Complex.reCLM_apply,Complex.imCLM_apply] at hre him
-  have hq := (hre.pow 2).add (him.pow 2)
-  have hq' : HasDerivAt (fun s => Complex.normSq (D s))
-      (2*(D t).re*(2/((X:ℂ)+D t)-2/(X:ℂ)).re+2*(D t).im*(2/((X:ℂ)+D t)-2/(X:ℂ)).im) t := by
-    convert hq using 1 <;> (try simp only [Complex.normSq_apply,pow_two]) <;> first | rfl | ring
-  rw [normSq_slope _ _ hX hD] at hq'
-  apply (hq'.mul ((hl.const_mul (-a)).exp)).congr_deriv
-  ring
-
-lemma energy_antitone {S δ a : ℝ} (hδ0 : 0 ≤ δ) (hδ : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (D : ℝ → ℂ) (l X : ℝ → ℝ) (hDc : ContinuousOn D (Icc 0 S)) (hlc : Continuous l)
-    (hX : ∀ t ∈ Ioo 0 S, 0 < X t)
-    (hDd : ∀ t ∈ Ioo 0 S, HasDerivAt D (2/((X t:ℂ)+D t)-2/(X t:ℂ)) t)
-    (hld : ∀ t ∈ Ioo 0 S, HasDerivAt l (-2/(X t)^2) t)
-    (hratio : ∀ t ∈ Ioo 0 S, ‖D t‖ ≤ δ*X t) :
-    AntitoneOn (fun t => Complex.normSq (D t)*Real.exp (-a*l t)) (Icc 0 S) := by
-  have hEc : ContinuousOn (fun t => Complex.normSq (D t)*Real.exp (-a*l t)) (Icc 0 S) :=
-    (Complex.continuous_normSq.comp_continuousOn hDc).mul
-      ((Real.continuous_exp.comp (continuous_const.mul hlc)).continuousOn)
-  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 S) hEc
-    (f' := fun t => Real.exp (-a*l t)*(-4*Complex.normSq (D t)/(X t)^2*
-      ((X t:ℂ)/((X t:ℂ)+D t)).re+2*a/(X t)^2*Complex.normSq (D t)))
-  · intro t ht
-    rw [interior_Icc] at ht
-    have hn : (X t:ℂ)+D t ≠ 0 := norm_pos_iff.mp
-      (lt_of_lt_of_le (half_pos (hX t ht)) (norm_sum_lower (hX t ht) hδ (hratio t ht)))
-    exact (energy_deriv (hX t ht).ne' hn (hDd t ht) (hld t ht)).hasDerivWithinAt
-  · intro t ht
-    rw [interior_Icc] at ht
-    have hr := ratio_real_lower (hX t ht) hδ0 hδ (hratio t ht) haδ
-    apply mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le
-    have hq := Complex.normSq_nonneg (D t)
-    have hh := mul_le_mul_of_nonneg_left hr (show 0 ≤ 2*Complex.normSq (D t)/(X t)^2 by positivity)
-    convert sub_nonpos.mpr hh using 1
-    ring
-
-lemma gap_coefficients {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (x t : ℝ) (hx : ε ≤ gap hε hT U x t) :
-    velocity ε (extendDriver T U t) (position hε hT U (x:ℂ) t) = 2/(gap hε hT U x t:ℂ) ∧
-      variation hε hT U (x:ℂ) t = -2/(gap hε hT U x t)^2 := by
-  have hz : position hε hT U (x:ℂ) t = ((position hε hT U (x:ℂ) t).re:ℂ) := by
-    apply Complex.ext <;> simp [position_im_zero]
-  have hc : clampRe ε (position hε hT U (x:ℂ) t-(extendDriver T U t:ℂ)) =
-      (gap hε hT U x t:ℂ) := by
-    change BoundaryCutoff.clamp 1 ε _ = _
-    rw [BoundaryCutoff.clamp_eq (by norm_num : |(1:ℝ)|=1) (by simpa [gap] using hx),hz]
-    simp only [gap,Complex.ofReal_sub]
-  constructor
-  · change 2/clampRe ε _ = _; rw [hc]
-  · unfold variation
-    rw [hc]
-    norm_cast
-
-lemma gap_difference_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (x : ℝ) {t : ℝ} (ht : t ∈ Ioo 0 T)
-    (hx : ε ≤ gap hε hT U x t) (G : ℝ → ℂ)
-    (hG : HasDerivAt G (2/(G t-(extendDriver T U t:ℂ))) t) :
-    HasDerivAt (fun s => G s-position hε hT U (x:ℂ) s)
-      (2/((gap hε hT U x t:ℂ)+(G t-position hε hT U (x:ℂ) t))-
-        2/(gap hε hT U x t:ℂ)) t := by
-  have hh := hG.sub (position_deriv hε hT U (x:ℂ) ht)
-  rw [(gap_coefficients hε hT U x t hx).1] at hh
-  have hz : position hε hT U (x:ℂ) t = ((position hε hT U (x:ℂ) t).re:ℂ) := by
-    apply Complex.ext <;> simp [position_im_zero]
-  have hg : G t-(extendDriver T U t:ℂ) = (gap hε hT U x t:ℂ)+(G t-position hε hT U (x:ℂ) t) := by
-    unfold gap
-    rw [Complex.ofReal_sub,← hz]
-    ring
-  rw [hg] at hh
-  exact hh
-
-lemma ratio_energy {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
-    (U : C(I, ℝ)) (x a δ : ℝ) (hδ0 : 0 ≤ δ) (hδ : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (G : ℝ → ℂ) (hGc : ContinuousOn G (Icc 0 S))
-    (hGd : ∀ t ∈ Ioo 0 S, HasDerivAt G (2/(G t-(extendDriver T U t:ℂ))) t)
-    (hgap : ∀ t ∈ Icc 0 S, ε ≤ gap hε hT U x t)
-    (hratio : ∀ t ∈ Icc 0 S, ‖G t-position hε hT U (x:ℂ) t‖ ≤ δ*gap hε hT U x t) :
-    Complex.normSq (G S-position hε hT U (x:ℂ) S) *
-      Real.exp (-a*logDerivative hε hT U (x:ℂ) S) ≤ Complex.normSq (G 0-(x:ℂ)) := by
-  let D : ℝ → ℂ := fun t => G t-position hε hT U (x:ℂ) t
-  let l : ℝ → ℝ := logDerivative hε hT U (x:ℂ)
-  let X : ℝ → ℝ := gap hε hT U x
-  have hHc : Continuous (fun t => position hε hT U (x:ℂ) t) :=
-    continuous_position_time hε hT U (x:ℂ)
-  have hDc : ContinuousOn D (Icc 0 S) := hGc.sub hHc.continuousOn
-  have hlc : Continuous l :=
-    continuous_logDerivative_time hε hT U (x:ℂ)
-  have hXd (t : ℝ) (ht : t ∈ Ioo 0 S) : 0 < X t := hε.trans_le (hgap t ⟨ht.1.le,ht.2.le⟩)
-  have hDd (t : ℝ) (ht : t ∈ Ioo 0 S) : HasDerivAt D (2/((X t:ℂ)+D t)-2/(X t:ℂ)) t :=
-    gap_difference_deriv hε hT U x ⟨ht.1,ht.2.trans_le hST⟩ (hgap t ⟨ht.1.le,ht.2.le⟩) G (hGd t ht)
-  have hld (t : ℝ) (ht : t ∈ Ioo 0 S) : HasDerivAt l (-2/(X t)^2) t := by
-    have hh := hasDerivAt_logDerivative hε hT U (x:ℂ) ⟨ht.1,ht.2.trans_le hST⟩
-    rw [(gap_coefficients hε hT U x t (hgap t ⟨ht.1.le,ht.2.le⟩)).2] at hh
-    exact hh
-  have ha := energy_antitone hδ0 hδ haδ D l X hDc hlc hXd hDd hld
-    (fun t ht => hratio t ⟨ht.1.le,ht.2.le⟩)
-  have hh := ha ⟨le_rfl,hS⟩ ⟨hS,le_rfl⟩ hS
-  change Complex.normSq (D S)*Real.exp (-a*l S) ≤ Complex.normSq (G 0-(x:ℂ))
-  have he : Complex.normSq (D 0)*Real.exp (-a*l 0) = Complex.normSq (G 0-(x:ℂ)) := by
-    dsimp only [D,l]
-    rw [position_initial,logDerivative_initial,mul_zero,Real.exp_zero,mul_one]
-  exact hh.trans_eq he
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open Swallowing
-
-lemma energy_weight_bound {β a l d δ X ε N : ℝ} (hβ : 0 < β) (hd : 0 < d)
-    (hδ : 0 ≤ δ) (hX : 0 < X) (hε0 : 0 ≤ ε) (hε : ε ≤ X) (_hN : 0 ≤ N)
-    (henergy : N^2*Real.exp (-a*l) ≤ d^2)
-    (hweight : Real.exp ((a*β/2)*l)*radial (-β) X ε ≤ (δ/(4*d))^β) :
-    N ≤ (δ/2)*X := by
-  let E := Real.exp ((a/2)*l)
-  let Q := Real.sqrt (X^2+ε^2)
-  have hE : 0 < E := Real.exp_pos _
-  have hQ2 : 0 < X^2+ε^2 := by positivity
-  have hQ : 0 < Q := Real.sqrt_pos.2 hQ2
-  have hQsq : Q^2=X^2+ε^2 := Real.sq_sqrt hQ2.le
-  have hQbound : Q ≤ 2*X := by nlinarith [sq_nonneg (X-ε)]
-  have hep : Real.exp (-a*l)*E^2 = 1 := by
-    dsimp only [E]
-    rw [sq,← Real.exp_add,← Real.exp_add]
-    convert Real.exp_zero using 1
-    congr 1
-    ring
-  have hNEsq : N^2 ≤ (d*E)^2 := by
-    have hh := mul_le_mul_of_nonneg_right henergy (sq_nonneg E)
-    rw [mul_assoc,hep,mul_one] at hh
-    nlinarith only [hh]
-  have hNE : N ≤ d*E := by nlinarith [mul_pos hd hE]
-  have hEp : E^β=Real.exp ((a*β/2)*l) := by
-    dsimp only [E]
-    rw [← Real.exp_mul]
-    congr 1
-    ring
-  have hQp : Q^β=(X^2+ε^2)^(β/2) := by
-    dsimp only [Q]
-    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hQ2.le]
-    congr 1
-    ring
-  have hwp : Real.exp ((a*β/2)*l)*radial (-β) X ε = (E/Q)^β := by
-    rw [Real.div_rpow hE.le hQ.le,hEp,hQp]
-    unfold radial
-    rw [show (-β)/2 = -(β/2) by ring,Real.rpow_neg hQ2.le]
-    rfl
-  rw [hwp] at hweight
-  have hR : 0 ≤ δ/(4*d) := by positivity
-  have hER : E/Q ≤ δ/(4*d) := (Real.rpow_le_rpow_iff (div_nonneg hE.le hQ.le) hR hβ).1 hweight
-  have heQ : E ≤ (δ/(4*d))*Q := (div_le_iff₀ hQ).1 hER
-  calc
-    N ≤ d*E := hNE
-    _ ≤ d*((δ/(4*d))*Q) := mul_le_mul_of_nonneg_left heQ hd.le
-    _ ≤ d*((δ/(4*d))*(2*X)) := by gcongr
-    _ = (δ/2)*X := by field_simp; ring
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge.BoundaryGreen
-
-lemma first_zero_of_nonpos {F : ℝ → ℝ} {S : ℝ} (hS : 0 ≤ S)
-    (hF : Continuous F) (h0 : 0 < F 0) (hSval : F S ≤ 0) :
-    ∃ σ ∈ Ioc 0 S, F σ = 0 ∧ ∀ s ∈ Icc 0 σ, 0 ≤ F s := by
-  let K : Set ℝ := Icc 0 S ∩ {s | F s ≤ 0}
-  have hK : IsCompact K := isCompact_Icc.inter_right (isClosed_le hF continuous_const)
-  have hne : K.Nonempty := ⟨S,⟨hS,le_rfl⟩,hSval⟩
-  let σ := sInf K
-  have hmem : σ ∈ K := hK.sInf_mem hne
-  have hσ : 0 < σ := by
-    apply lt_of_le_of_ne hmem.1.1
-    intro heq
-    have hh := hmem.2
-    rw [← heq] at hh
-    exact (not_le_of_gt h0) hh
-  have hbefore : ∀ s ∈ Ico 0 σ, 0 < F s := by
-    intro s hs
-    by_contra hn
-    have hm : s ∈ K := ⟨⟨hs.1,hs.2.le.trans hmem.1.2⟩,not_lt.1 hn⟩
-    exact (not_le_of_gt hs.2) (csInf_le hK.bddBelow hm)
-  have hσval : 0 ≤ F σ := by
-    apply ge_of_tendsto (hF.continuousAt.tendsto.mono_left nhdsWithin_le_nhds :
-      Tendsto F (𝓝[<] σ) (𝓝 (F σ)))
-    filter_upwards [Ioo_mem_nhdsLT hσ] with s hs
-    exact (hbefore s ⟨hs.1.le,hs.2⟩).le
-  refine ⟨σ,⟨hσ,hmem.1.2⟩,le_antisymm hmem.2 hσval,?_⟩
-  intro s hs
-  rcases lt_or_eq_of_le hs.2 with hlt | rfl
-  · exact (hbefore s ⟨hs.1,hlt⟩).le
-  · exact hσval
-
-lemma first_zero_of_nonpos_on {F : ℝ → ℝ} {S : ℝ} (hS : 0 ≤ S)
-    (hF : ContinuousOn F (Icc 0 S)) (h0 : 0 < F 0) (hSval : F S ≤ 0) :
-    ∃ σ ∈ Ioc 0 S, F σ = 0 ∧ ∀ s ∈ Icc 0 σ, 0 ≤ F s := by
-  let Fc : ℝ → ℝ := fun s => F (projIcc 0 S hS s)
-  have hc : Continuous Fc := hF.comp_continuous
-    (continuous_subtype_val.comp (continuous_projIcc (h := hS))) (fun s => (projIcc 0 S hS s).property)
-  have he (s : ℝ) (hs : s ∈ Icc 0 S) : Fc s=F s := by
-    simp only [Fc,projIcc,max_eq_right hs.1,min_eq_right hs.2]
-  obtain ⟨σ,hσ,hz,hpre⟩ := first_zero_of_nonpos hS hc (by rwa [he 0 ⟨le_rfl,hS⟩])
-    (by rwa [he S ⟨hS,le_rfl⟩])
-  refine ⟨σ,hσ,?_,?_⟩
-  · rwa [he σ ⟨hσ.1.le,hσ.2⟩] at hz
-  · intro s hs
-    have hh := hpre s hs
-    rwa [he s ⟨hs.1,hs.2.trans hσ.2⟩] at hh
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing
-
-lemma tip_forces_reach {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
-    (U : C(I, ℝ)) (hu0 : extendDriver T U 0=0)
-    (x a β δ d y : ℝ) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
-    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x) (hεx : 2*ε < x) (hy : δ*ε < y)
-    (G : ℝ → ℂ) (hGc : ContinuousOn G (Icc 0 S))
-    (hGd : ∀ t ∈ Ioo 0 S, HasDerivAt G (2/(G t-(extendDriver T U t:ℂ))) t)
-    (hG0 : ‖G 0-(x:ℂ)‖ ≤ d) (hheight : ∀ t ∈ Icc 0 S, y ≤ (G t).im)
-    (hGend : (G S).re=extendDriver T U S) :
-    reach U hε hT x β a ((δ/(4*d))^β) := by
-  let H : ℝ → ℂ := position hε hT U (x:ℂ)
-  let X : ℝ → ℝ := gap hε hT U x
-  let D : ℝ → ℂ := fun t => G t-H t
-  let F : ℝ → ℝ := fun t => min (X t-2*ε) (δ*X t-‖D t‖)
-  have hXc : Continuous X := continuous_gap_time hε hT U x
-  have hDc : ContinuousOn D (Icc 0 S) :=
-    hGc.sub (continuous_position_time hε hT U (x:ℂ)).continuousOn
-  have hFc : ContinuousOn F (Icc 0 S) := continuous_min.comp_continuousOn
-    ((hXc.continuousOn.sub continuousOn_const).prodMk
-      ((continuousOn_const.mul hXc.continuousOn).sub hDc.norm))
-  have hX0 : X 0=x := by simp only [X,gap,position_initial,Complex.ofReal_re,hu0,sub_zero]
-  have hF0 : 0 < F 0 := by
-    change 0 < min (X 0-2*ε) (δ*X 0-‖G 0-H 0‖)
-    rw [hX0,show H 0=(x:ℂ) from position_initial hε hT U (x:ℂ)]
-    exact lt_min (sub_pos.mpr hεx) (sub_pos.mpr (hG0.trans_lt hdx))
-  have hRe : (D S).re = -X S := by
-    change (G S-position hε hT U (x:ℂ) S).re = -gap hε hT U x S
-    simp only [Complex.sub_re,gap,hGend]
-    ring
-  have hnorm : X S ≤ ‖D S‖ := by
-    have hh := Complex.abs_re_le_norm (D S)
-    rw [hRe,abs_neg] at hh
-    exact (le_abs_self _).trans hh
-  have hFS : F S ≤ 0 := by
-    by_contra hn
-    have hh := lt_min_iff.mp (lt_of_not_ge hn)
-    have hXp : 0 < X S := by linarith [hh.1]
-    have hm := mul_le_mul_of_nonneg_right hδhalf hXp.le
-    linarith [hh.2]
-  obtain ⟨q,hq,hzero,hpre⟩ := first_zero_of_nonpos_on hS hFc hF0 hFS
-  have hgap : ∀ t ∈ Icc 0 q, 2*ε ≤ X t := by
-    intro t ht
-    exact sub_nonneg.mp (le_min_iff.mp (hpre t ht)).1
-  have hratio : ∀ t ∈ Icc 0 q, ‖D t‖ ≤ δ*X t := by
-    intro t ht
-    exact sub_nonneg.mp (le_min_iff.mp (hpre t ht)).2
-  have hXp : 0 < X q := hε.trans_le (by linarith [hgap q ⟨hq.1.le,le_rfl⟩])
-  have hqT : q ≤ T := hq.2.trans hST
-  have henergy := ratio_energy hε hT hq.1.le hqT U x a δ hδ.le hδhalf haδ G
-    (hGc.mono (Icc_subset_Icc_right hq.2)) (fun t ht => hGd t ⟨ht.1,ht.2.trans_le hq.2⟩)
-    (fun t ht => by change ε ≤ X t; linarith [hgap t ht]) hratio
-  have hEn : ‖D q‖^2*Real.exp (-a*logDerivative hε hT U (x:ℂ) q) ≤ d^2 := by
-    rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at henergy
-    exact henergy.trans (sq_le_sq₀ (norm_nonneg _) hd.le |>.2 hG0)
-  by_contra hreach
-  have hweight : weight hε hT U x β a q ≤ (δ/(4*d))^β := by
-    apply (not_le.mp ?_).le
-    intro hh
-    exact hreach ⟨q,⟨hq.1.le,hqT⟩,(fun t ht => by change ε < X t; linarith [hgap t ht]),hh⟩
-  have hW : Real.exp ((a*β/2)*logDerivative hε hT U (x:ℂ) q)*radial (-β) (X q) ε ≤ (δ/(4*d))^β := hweight
-  have hsmall := energy_weight_bound hβ hd hδ.le hXp hε.le
-    (show ε ≤ X q by linarith [hgap q ⟨hq.1.le,le_rfl⟩]) (norm_nonneg (D q)) hEn hW
-  have hym : y ≤ ‖D q‖ := by
-    have hh := hheight q ⟨hq.1.le,hq.2⟩
-    have him : (D q).im=(G q).im := by simp [D,H,position_im_zero]
-    rw [← him] at hh
-    exact hh.trans (Complex.im_le_norm _)
-  have hgt : 2*ε < X q := by nlinarith
-  have hrat : ‖D q‖ < δ*X q := by nlinarith [mul_pos hδ hXp]
-  have hpos : 0 < F q := lt_min (sub_pos.mpr hgt) (sub_pos.mpr hrat)
-  exact (ne_of_gt hpos) hzero
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-
-lemma inverse_tip_point {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {t : ℝ} (ht : 0 ≤ t)
-    {x : ℂ} {d : ℝ} (hclose : dist (γ t) x < d) :
-    ∃ (w : ℂ) (y : ℝ), 0 < y ∧ dist w x < d ∧
-      w ∈ localizationDomain γ t ∧ g t w=(driver t:ℂ)+(y:ℂ)*Complex.I := by
-  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2.2 t ht
-  have he : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), dist (f t ((driver t:ℂ)+(y:ℂ)*Complex.I)) x < d :=
-    htip.eventually (isOpen_ball.mem_nhds hclose)
-  have hp : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), 0 < y := self_mem_nhdsWithin
-  obtain ⟨y,hy,hball⟩ := (hp.and he).exists
-  have hz : (driver t:ℂ)+(y:ℂ)*Complex.I ∈ halfPlane := by
-    change 0 < ((driver t:ℂ)+(y:ℂ)*Complex.I).im
-    simpa using hy
-  refine ⟨f t ((driver t:ℂ)+(y:ℂ)*Complex.I),y,hy,hball,?_,?_⟩
-  · exact h.2.2.2.2.2.2.2.2.1 t ht hz
-  · exact h.2.2.2.2.2.2.2.2.2.2.1 t ht _ hz
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ReverseLoewner
-open LoewnerCutoff
-
-def velocity (ε u : ℝ) (w : ℂ) : ℂ := -LoewnerCutoff.velocity ε u w
-
-lemma velocity_lipschitz {ε : ℝ} (hε : 0 < ε) (u : ℝ) :
-    LipschitzWith ⟨4 / ε^2, by positivity⟩ (velocity ε u) :=
-  (LoewnerCutoff.velocity_lipschitz hε u).neg
-
-end SLEExactGauge.ReverseLoewner
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.SwallowingLaws
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Boundary Green estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing GreenPhysical
+
+lemma deterministic_budget {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (κ β a : ℝ) (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (U : C(I, ℝ)) (x L : ℝ) (b : ℝ → ℝ) (hb : Continuous b)
+    (hb0 : b 0=0) (hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*b t) (Icc 0 T)) :
+    let τ := stop hε hT.le U x β a L
+    let F := field hε hT.le β a (Real.sqrt κ) U x
+    let V := drift hε hT.le β a (Real.sqrt κ) U x
+    F τ (b τ) ≤
+        (F τ (b τ)-F 0 (b 0)-(∫ t in (0:ℝ)..τ,V t (b t))-
+          (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (b t))+radial (-β) x ε := by
+  dsimp only
+  let τ := stop hε hT.le U x β a L
+  let F := field hε hT.le β a (Real.sqrt κ) U x
+  let V := drift hε hT.le β a (Real.sqrt κ) U x
+  have htau := stop_bounds hε hT.le U x β a L
+  have hmap : Continuous (fun t => ((U,t),b t)) := (continuous_const.prodMk continuous_id).prodMk hb
+  have hV : Continuous (fun t => V t (b t)) := by
+    have hh := (continuous_drift hε hT.le β a (Real.sqrt κ) x).comp hmap
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  have hQ : Continuous (fun t => iteratedDeriv 2 (F t) (b t)) := by
+    have hh := (continuous_coeff hε hT.le β a (Real.sqrt κ) x 2 (Or.inr rfl)).comp hmap
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  have hneg : (∫ t in (0:ℝ)..τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) ≤ 0 := by
+    have hh : (∫ t in (0:ℝ)..τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) ≤
+        ∫ _t in (0:ℝ)..τ, (0:ℝ) := by
+      apply intervalIntegral.integral_mono_on_of_le_Ioo htau.1
+        ((hV.add (continuous_const.mul hQ)).intervalIntegrable _ _)
+        (intervalIntegrable_const)
+      intro t ht
+      have hh := (deficit_pos_iff hε hT.le U x β a L t).1
+        (positive_before hε hT.le hT U x β a L ht.1.le ht.2)
+      change V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) ≤ 0
+      have he := generator_field hε hT.le κ β a hκ U x t (b t) hh.1.le
+        (hU ⟨ht.1.le,ht.2.le.trans htau.2⟩)
+      change V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) = _ at he
+      rw [he]
+      exact mul_nonpos_of_nonneg_of_nonpos (factor_pos hε hT.le β a U x t).le
+        (generator_nonpos hκ hβ ha hba (hε.trans hh.1) hε)
+    simpa using hh
+  have hhalf : IntervalIntegrable (fun t => (1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) volume 0 τ :=
+    (hQ.intervalIntegrable _ _).const_mul _
+  rw [intervalIntegral.integral_add (hV.intervalIntegrable _ _) hhalf,
+    intervalIntegral.integral_const_mul] at hneg
+  have hini : F 0 (b 0) = radial (-β) x ε := by rw [hb0]; exact field_initial hε hT.le β a _ U x
+  change F τ (b τ) ≤ (F τ (b τ)-F 0 (b 0)-(∫ t in (0:ℝ)..τ,V t (b t))-
+    (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (b t))+radial (-β) x ε
+  rw [hini]
+  linarith
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.Swallowing
+open LoewnerCutoff LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+variable {ε : ℝ} (hε : 0 < ε)
+
+def coeffAt (k : ℕ) (s : ℝ≥0) (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ)
+    (q : C(I, ℝ) × ℝ) : ℝ := by
+  classical
+  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
+    iteratedDeriv k (field hε s.coe_nonneg β C sdev q.1 z s) q.2 else 0
+
+lemma coeffAt_measurable (k : ℕ) (hk : k = 1 ∨ k = 2) (s : ℝ≥0)
+    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s β C sdev z ρ δ) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
+      iteratedDeriv k (field hε s.coe_nonneg β C sdev q.1 z s) q.2) := by
+    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
+      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
+    have hh := (continuous_coeff hε s.coe_nonneg β C sdev z k hk).comp hp
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  apply hc.measurable.ite _ measurable_const
+  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
+
+def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω) : ℝ :=
+  if s < T then coeffAt hε k s β C sdev z ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
+
+lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    Measurable[pastSigma B s] (rawCoeff hε k B T s β C sdev z ρ δ) := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  let : MeasurableSpace Ω := pastSigma B s
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s β C sdev z ρ δ).comp
+      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
+  · exact measurable_const
+
+lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
+    (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    AEMeasurable (rawCoeff hε k B T s β C sdev z ρ δ) P := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s β C sdev z ρ δ).comp_aemeasurable
+      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
+  · exact aemeasurable_const
+
+lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    rawCoeff hε k B T s β C sdev z ρ δ sample =
+      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
+        iteratedDeriv k (field hε T.coe_nonneg β C sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
+  classical
+  by_cases hsT : s < T
+  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
+    have hf := field_prefix_eq hε T.coe_nonneg s.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z β C sdev he
+      ⟨s.coe_nonneg, le_rfl⟩
+    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
+    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
+        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
+      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
+      exact forall₂_congr (fun t ht => by rw [hd ht])
+    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
+  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
+      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
+    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
+
+lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (β C sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (n i : ℕ) :
+    rawCoeff hε k B T (dyadicTime T n i) β C sdev z ρ δ =ᵐ[P]
+      stoppedCoeff k B T
+        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
+        (fun sample => field hε T.coe_nonneg β C sdev (BrownianDriver.path B sdev T sample) z) n i := by
+  filter_upwards [hB.cont] with sample hc
+  rw [rawCoeff_eq hε k B T _ hT β C sdev z ρ δ sample hc]
+  simp only [stoppedCoeff, coe_dyadicTime]
+
+theorem localized_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (β C sdev : ℝ) (hβ : β ≤ 1)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (ρ δ : ℝ) :
+    let U := BrownianDriver.path B sdev T
+    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+    let F := fun sample => field hε T.coe_nonneg β C sdev (U sample) z
+    let V := fun sample => drift hε T.coe_nonneg β C sdev (U sample) z
+    let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
+    Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 := by
+  dsimp only
+  let U := BrownianDriver.path B sdev T
+  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+  let F := fun sample => field hε T.coe_nonneg β C sdev (U sample) z
+  let V := fun sample => drift hε T.coe_nonneg β C sdev (U sample) z
+  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact regular hε T.coe_nonneg hT β C sdev hβ (U sample) z ρ δ _
+      (hc.comp continuous_real_toNNReal)
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
+    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) z ρ δ)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg β C sdev hβ z 1 (Or.inl rfl)
+  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg β C sdev hβ z 2 (Or.inr (Or.inl rfl))
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
+        |iteratedDeriv k (field hε T.coe_nonneg β C sdev W z t) b| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
+    apply Filter.Eventually.of_forall
+    intro sample
+    unfold stoppedCoeff
+    split_ifs
+    · exact hb _ _ _
+    · simpa only [abs_zero] using hK
+  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) β C sdev z ρ δ
+  let Q := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) β C sdev z ρ δ
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 1 T hT β C sdev z ρ δ n i
+  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 2 T hT β C sdev z ρ δ n i
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
+    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ z ρ δ
+  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
+    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ z ρ δ
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₁
+    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hb
+  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₂
+    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |Q n i sample| ≤ K₂
+    rw [he]
+    exact hb
+  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A Q hAe hCe hCm hC2
+    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  exact hi
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.Swallowing
+open GreenPhysical LoewnerCutoff LoewnerExit
+
+abbrev correction {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ :=
+  let U := BrownianDriver.path B (Real.sqrt κ) T ω
+  let τ := stop hε T.coe_nonneg U z 0 0
+  let F := field hε T.coe_nonneg β C (Real.sqrt κ) U z
+  let V := drift hε T.coe_nonneg β C (Real.sqrt κ) U z
+  F τ (B τ.toNNReal ω)-F 0 (B 0 ω)-
+    (∫ t in (0:ℝ)..τ, V t (B t.toNNReal ω))-
+      (1/2:ℝ)*∫ t in (0:ℝ)..τ, iteratedDeriv 2 (F t) (B t.toNNReal ω)
+
+def budget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ :=
+  if T=0 then lyapunov β C z.re z.im else
+    max (correction B κ β C hε T z ω+lyapunov β C z.re z.im) 0
+
+lemma field_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (hy : ε ≤ z.im) :
+    field hε hT β C s U z 0 0 = lyapunov β C z.re z.im := by
+  simp only [field,LoewnerField.height,position_initial,mul_zero,sub_zero,max_eq_right hy]
+
+lemma generator_field {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (κ β C : ℝ) (hκ : 0 ≤ κ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im)
+    (hU : extendDriver T U t = Real.sqrt κ*b) :
+    drift hε hT β C (Real.sqrt κ) U z t b+
+      (1/2:ℝ)*iteratedDeriv 2 (field hε hT β C (Real.sqrt κ) U z t) b =
+    generator κ β C ((position hε hT U z t).re-Real.sqrt κ*b) (LoewnerField.height hε hT U z t) := by
+  have hh : LoewnerField.height hε hT U z t = (position hε hT U z t).im := max_eq_right hy
+  rw [second_eq,Real.sq_sqrt hκ]
+  dsimp only [drift]
+  rw [velocity_eq hy,hh,hU]
+  simp only [generator,Complex.div_re,Complex.div_im,Complex.re_ofNat,Complex.im_ofNat,
+    Complex.sub_re,Complex.ofReal_re,Complex.sub_im,Complex.ofReal_im,sub_zero,zero_mul,
+    Complex.normSq_apply]
+  ring
+
+lemma deterministic_budget {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (κ β : ℝ) (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (U : C(I, ℝ)) (z : ℂ) (hz : ε ≤ z.im) (b : ℝ → ℝ) (hb : Continuous b)
+    (hb0 : b 0=0) (hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*b t) (Icc 0 T)) :
+    let C := (2*κ-8-κ*β)/4
+    let d := β*(κ-4-κ*β)/2
+    let τ := stop hε hT.le U z 0 0
+    let F := field hε hT.le β C (Real.sqrt κ) U z
+    let V := drift hε hT.le β C (Real.sqrt κ) U z
+    let E := ∫ t in (0:ℝ)..τ,
+      (((position hε hT.le U z t).re-Real.sqrt κ*b t)^2+(LoewnerField.height hε hT.le U z t)^2)^(β/2-1)
+    0 ≤ E ∧
+      F τ (b τ)+d*E ≤
+        (F τ (b τ)-F 0 (b 0)- (∫ t in (0:ℝ)..τ,V t (b t))-
+          (1/2:ℝ)*∫ t in (0:ℝ)..τ, iteratedDeriv 2 (F t) (b t))+lyapunov β C z.re z.im := by
+  dsimp only
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  let τ := stop hε hT.le U z 0 0
+  let F := field hε hT.le β C (Real.sqrt κ) U z
+  let V := drift hε hT.le β C (Real.sqrt κ) U z
+  let H := fun t => (((position hε hT.le U z t).re-Real.sqrt κ*b t)^2+
+    (LoewnerField.height hε hT.le U z t)^2)^(β/2-1)
+  have htau := stop_bounds hε hT.le U z 0 0
+  have hmap : Continuous (fun t => ((U,t),b t)) := (continuous_const.prodMk continuous_id).prodMk hb
+  have hV : Continuous (fun t => V t (b t)) := by
+    have hh := (continuous_drift hε hT.le β C (Real.sqrt κ) z).comp hmap
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  have hQ : Continuous (fun t => iteratedDeriv 2 (F t) (b t)) := by
+    have hh := (continuous_coeff hε hT.le β C (Real.sqrt κ) z 2 (Or.inr rfl)).comp hmap
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  have hHc : Continuous H := by
+    have hP := (continuous_position hε hT.le z).comp (continuous_const.prodMk continuous_id : Continuous (fun t : ℝ => (U,t)))
+    have hx := (Complex.continuous_re.comp hP).sub (continuous_const.mul hb : Continuous (fun t => Real.sqrt κ*b t))
+    have hy : Continuous (LoewnerField.height hε hT.le U z) := continuous_const.max (Complex.continuous_im.comp hP)
+    exact ((hx.pow 2).add (hy.pow 2)).rpow_const (fun t => Or.inl (q_pos _ (LoewnerField.height_pos hε hT.le U z t)).ne')
+  have hH0 : 0 ≤ ∫ t in (0:ℝ)..τ,H t := by
+    apply intervalIntegral.integral_nonneg htau.1
+    intro t ht
+    exact Real.rpow_nonneg (by positivity) _
+  refine ⟨hH0,?_⟩
+  have hpoint : ∀ t ∈ Ioo 0 τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) ≤ -d*H t := by
+    intro t ht
+    have hh := (deficit_pos_iff hε hT.le U z 0 0 t).1
+      (positive_before hε hT.le hT U z 0 0 ht.1.le ht.2)
+    have he := generator_field hε hT.le κ β C (by linarith) U z t (b t) hh.2.1.le
+      (hU ⟨ht.1.le,ht.2.le.trans htau.2⟩)
+    change V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t) = _ at he
+    rw [he]
+    exact (generator_negative hκ hβ hβκ _ (LoewnerField.height_pos hε hT.le U z t)).2.2
+  have hm : (∫ t in (0:ℝ)..τ, V t (b t)+(1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) ≤
+      ∫ t in (0:ℝ)..τ, -d*H t := by
+    exact intervalIntegral.integral_mono_on_of_le_Ioo htau.1
+      ((hV.add (continuous_const.mul hQ)).intervalIntegrable _ _)
+      ((continuous_const.mul hHc).intervalIntegrable _ _) hpoint
+  have hhalf : IntervalIntegrable (fun t => (1/2:ℝ)*iteratedDeriv 2 (F t) (b t)) volume 0 τ :=
+    (hQ.intervalIntegrable _ _).const_mul _
+  rw [intervalIntegral.integral_add (hV.intervalIntegrable _ _) hhalf,
+    intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul] at hm
+  have hinit : F 0 (b 0) = lyapunov β C z.re z.im := by rw [hb0]; exact field_initial hε hT.le β C _ U z hz
+  change F τ (b τ)+d*(∫ t in (0:ℝ)..τ,H t) ≤
+    (F τ (b τ)-F 0 (b 0)-(∫ t in (0:ℝ)..τ,V t (b t))-
+      (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (b t))+lyapunov β C z.re z.im
+  rw [hinit]
+  linarith
+
+lemma budget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ β : ℝ) (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ) (hz : ε ≤ z.im) :
+    let C := (2*κ-8-κ*β)/4
+    Integrable (budget B κ β C hε T z) P ∧
+      (∫ ω,budget B κ β C hε T z ω ∂P) = lyapunov β C z.re z.im ∧
+      ∀ ω, 0 ≤ budget B κ β C hε T z ω := by
+  dsimp only
+  let C := (2*κ-8-κ*β)/4
+  have hgen := generator_negative hκ hβ hβκ z.re (hε.trans_le hz)
+  have hC : 0 ≤ C := hgen.1.le
+  have hφ : 0 ≤ lyapunov β C z.re z.im := lyapunov_nonneg hC _ (hε.trans_le hz)
+  by_cases hT0 : T=0
+  · subst T
+    have he0 : budget B κ β ((2*κ-8-κ*β)/4) hε 0 z =
+        fun _ : Ω => lyapunov β C z.re z.im := by
+      funext ω
+      exact ite_eq_left rfl
+    rw [he0]
+    exact ⟨integrable_const _,by simp [C],fun _ => hφ⟩
+  have hT : 0 < (T:ℝ) := NNReal.coe_pos.2 (pos_iff_ne_zero.2 hT0)
+  have hb1 : β ≤ 1 := by have hh := div_pos (by norm_num : (0:ℝ)<4) (by linarith : 0<κ); linarith
+  have hcenter := localized_centered hB β C (Real.sqrt κ) hb1 hε T hT z 0 0
+  change Integrable (correction B κ β C hε T z) P ∧ (∫ ω,correction B κ β C hε T z ω ∂P)=0 at hcenter
+  have he : budget B κ β C hε T z =ᵐ[P]
+      (fun ω => correction B κ β C hε T z ω+lyapunov β C z.re z.im) := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hc hb0
+    have hh := deterministic_budget hε hT κ β hκ hβ hβκ (BrownianDriver.path B (Real.sqrt κ) T ω) z hz
+      (fun t => B t.toNNReal ω) (hc.comp continuous_real_toNNReal) (by simpa using hb0)
+      (fun t ht => BrownianDriver.extend_eq B _ T ω hc ht)
+    dsimp only at hh
+    have hn := lyapunov_nonneg (β := β) hC
+      ((position hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z
+        (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z 0 0)).re-
+        Real.sqrt κ*B (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z 0 0).toNNReal ω)
+      (LoewnerField.height_pos hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z
+        (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) z 0 0))
+    have hh2 := hh.2
+    simp only [Real.toNNReal_zero] at hh2
+    have hpos : 0 ≤ correction B κ β C hε T z ω+lyapunov β C z.re z.im :=
+      (add_nonneg hn (mul_nonneg hgen.2.1.le hh.1)).trans hh2
+    simp only [budget,ite_eq_right hT0,max_eq_left hpos]
+  have hi := hcenter.1.add (integrable_const (lyapunov β C z.re z.im))
+  refine ⟨hi.congr he.symm,?_,fun ω => ?_⟩
+  · rw [integral_congr_ae he,integral_add hcenter.1 (integrable_const _),hcenter.2,zero_add]
+    simp [C]
+  · simp only [budget,ite_eq_right hT0]
+    exact le_max_right _ _
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+
+lemma measure_eventually_budget {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (W : ℕ → Ω → ℝ) (M a : ℝ) (ha : 0 < a)
+    (hi : ∀ n, Integrable (W n) P) (hn : ∀ n ω, 0 ≤ W n ω)
+    (hmean : ∀ n, (∫ ω,W n ω ∂P)=M) (E : Set Ω)
+    (hE : ∀ᵐ ω ∂P, ω∈E → ∀ᶠ n in atTop, a ≤ W n ω) :
+    P E ≤ ENNReal.ofReal (M/a) := by
+  let L := fun ω => liminf (fun n => ENNReal.ofReal (W n ω)) atTop
+  have hm (n : ℕ) : AEMeasurable (fun ω => ENNReal.ofReal (W n ω)) P :=
+    ENNReal.measurable_ofReal.comp_aemeasurable (hi n).aemeasurable
+  have hLm : AEMeasurable L P := by
+    simp only [L,liminf_eq_iSup_iInf_of_nat]
+    exact AEMeasurable.iSup (fun n => AEMeasurable.iInf (fun i => AEMeasurable.iInf (fun _ => hm i)))
+  have hLi : (∫⁻ ω,L ω ∂P) ≤ ENNReal.ofReal M := by
+    have hf := lintegral_liminf_le' (u := (atTop : Filter ℕ)) hm
+    have he (n : ℕ) : (∫⁻ ω, ENNReal.ofReal (W n ω) ∂P)=ENNReal.ofReal M := by
+      rw [← ofReal_integral_eq_lintegral_ofReal (hi n) (Filter.Eventually.of_forall (hn n)),hmean n]
+    simpa only [he,liminf_const] using hf
+  have hsub : E ≤ᵐ[P] {ω | ENNReal.ofReal a ≤ L ω} := by
+    filter_upwards [hE] with ω hω
+    intro hωE
+    apply le_liminf_of_le (by isBoundedDefault)
+    exact (hω hωE).mono (fun n hh => ENNReal.ofReal_le_ofReal hh)
+  calc
+    P E ≤ P {ω | ENNReal.ofReal a ≤ L ω} := measure_mono_ae hsub
+    _ ≤ (∫⁻ ω,L ω ∂P)/ENNReal.ofReal a :=
+      meas_ge_le_lintegral_div hLm (ENNReal.ofReal_ne_zero_iff.2 ha) ENNReal.ofReal_ne_top
+    _ ≤ ENNReal.ofReal M/ENNReal.ofReal a := by gcongr
+    _ = ENNReal.ofReal (M/a) := (ENNReal.ofReal_div_of_pos ha).symm
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing
+
+abbrev correction {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (x L : ℝ) (ω : Ω) : ℝ :=
+  let U := BrownianDriver.path B (Real.sqrt κ) T ω
+  let τ := stop hε T.coe_nonneg U x β a L
+  let F := field hε T.coe_nonneg β a (Real.sqrt κ) U x
+  let V := drift hε T.coe_nonneg β a (Real.sqrt κ) U x
+  F τ (B τ.toNNReal ω)-F 0 (B 0 ω)-
+    (∫ t in (0:ℝ)..τ,V t (B t.toNNReal ω))-
+    (1/2:ℝ)*∫ t in (0:ℝ)..τ,iteratedDeriv 2 (F t) (B t.toNNReal ω)
+
+def budget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (x L : ℝ) (ω : Ω) : ℝ :=
+  max (correction B κ β a hε T x L ω+radial (-β) x ε) 0
+
+lemma budget_nonneg {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (x L : ℝ) (ω : Ω) :
+    0 ≤ budget B κ β a hε T x L ω := le_max_right _ _
+
+lemma budget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x L : ℝ) :
+    Integrable (budget B κ β a hε T x L) P ∧
+      (∫ ω,budget B κ β a hε T x L ω ∂P) = radial (-β) x ε := by
+  have hc := localized_centered hB β a (Real.sqrt κ) hβ hε T hT x L
+  change Integrable (correction B κ β a hε T x L) P ∧
+    (∫ ω,correction B κ β a hε T x L ω ∂P)=0 at hc
+  have he : budget B κ β a hε T x L =ᵐ[P] fun ω => correction B κ β a hε T x L ω+radial (-β) x ε := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hcont hb0
+    have hh := deterministic_budget hε hT κ β a hκ hβ ha hba
+      (BrownianDriver.path B (Real.sqrt κ) T ω) x L (fun t => B t.toNNReal ω)
+      (hcont.comp continuous_real_toNNReal) (by simpa using hb0)
+      (fun t ht => BrownianDriver.extend_eq B _ T ω hcont ht)
+    dsimp only at hh
+    simp only [Real.toNNReal_zero] at hh
+    have hn := (field_pos hε T.coe_nonneg β a (Real.sqrt κ)
+      (BrownianDriver.path B (Real.sqrt κ) T ω) x
+      (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) x β a L)
+      (B (stop hε T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T ω) x β a L).toNNReal ω)).le
+    exact max_eq_left (hn.trans hh)
+  have hi := hc.1.add (integrable_const (radial (-β) x ε))
+  refine ⟨hi.congr he.symm,?_⟩
+  rw [integral_congr_ae he,integral_add hc.1 (integrable_const _),hc.2,zero_add]
+  simp
+
+def reach (U : C(I, ℝ)) {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (x β a L : ℝ) : Prop :=
+  ∃ t ∈ Icc 0 T, (∀ s ∈ Icc 0 t, ε < gap hε hT U x s) ∧ L ≤ weight hε hT U x β a t
+
+lemma reached_terminal {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (U : C(I, ℝ))
+    (x β a L : ℝ) (hreach : reach U hε hT.le x β a L) :
+    L ≤ weight hε hT.le U x β a (stop hε hT.le U x β a L) := by
+  obtain ⟨t,ht,hgap,hweight⟩ := hreach
+  have hst : stop hε hT.le U x β a L ≤ t := by
+    by_contra hn
+    have hp := (deficit_pos_iff hε hT.le U x β a L t).1
+      (positive_before hε hT.le hT U x β a L ht.1 (lt_of_not_ge hn))
+    exact (not_lt_of_ge hweight) hp.2
+  have hb := stop_bounds hε hT.le U x β a L
+  by_cases hend : stop hε hT.le U x β a L < T
+  · have hh := nonpos_at_of_lt hε hT.le hT U x β a L hend
+    have hg := hgap _ ⟨hb.1,hst⟩
+    simp only [deficit,min_le_iff,sub_nonpos] at hh
+    rcases hh with hh | hh
+    · exact (not_le_of_gt hg hh).elim
+    · exact hh
+  · have he : stop hε hT.le U x β a L=t := le_antisymm hst (ht.2.trans (not_lt.1 hend))
+    rw [he]
+    exact hweight
+
+lemma budget_dominates_reach {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x L : ℝ) (ω : Ω)
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    (hr : reach (BrownianDriver.path B (Real.sqrt κ) T ω) hε T.coe_nonneg x β a L) :
+    L ≤ budget B κ β a hε T x L ω := by
+  let U := BrownianDriver.path B (Real.sqrt κ) T ω
+  let τ := stop hε T.coe_nonneg U x β a L
+  have htau := stop_bounds hε T.coe_nonneg U x β a L
+  have hh := deterministic_budget hε hT κ β a hκ hβ ha hba U x L
+    (fun t => B t.toNNReal ω) (hc.comp continuous_real_toNNReal) (by simpa using hb0)
+    (fun t ht => BrownianDriver.extend_eq B _ T ω hc ht)
+  have hw := reached_terminal hε hT U x β a L hr
+  have he : field hε T.coe_nonneg β a (Real.sqrt κ) U x τ (B τ.toNNReal ω) =
+      weight hε T.coe_nonneg U x β a τ := by
+    have hU : extendDriver T U τ = Real.sqrt κ*B τ.toNNReal ω :=
+      BrownianDriver.extend_eq B _ T ω hc htau
+    simp only [field,weight,gap,hU]
+  dsimp only at hh
+  simp only [Real.toNNReal_zero] at hh
+  change field hε T.coe_nonneg β a (Real.sqrt κ) U x τ (B τ.toNNReal ω) ≤
+    correction B κ β a hε T x L ω+radial (-β) x ε at hh
+  rw [he] at hh
+  exact (hw.trans hh).trans (le_max_left _ _)
+
+theorem maximal_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x L : ℝ) (hL : 0 < L) :
+    P {ω | reach (BrownianDriver.path B (Real.sqrt κ) T ω) hε T.coe_nonneg x β a L} ≤
+      ENNReal.ofReal (radial (-β) x ε/L) := by
+  have hm := budget_mean hB κ β a hκ hβ ha hba hε T hT x L
+  apply measure_eventually_budget (fun _ : ℕ => budget B κ β a hε T x L) _ L hL
+    (fun _ => hm.1) (fun _ => budget_nonneg B κ β a hε T x L) (fun _ => hm.2)
+  filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hc hb0
+  intro hr
+  exact Filter.Eventually.of_forall (fun _ => budget_dominates_reach B κ β a hκ hβ ha hba hε T hT x L ω hc hb0 hr)
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryFunctional
+
+lemma continuous_position_time {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) : Continuous (position hε hT U z) := by
+  have hp : Continuous (fun t : ℝ => (U, t)) := continuous_const.prodMk continuous_id
+  have hh := (continuous_position hε hT z).comp hp
+  change Continuous (position hε hT U z) at hh
+  exact hh
+
+lemma continuous_logDerivative_time {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) : Continuous (logDerivative hε hT U z) := by
+  have hp : Continuous (fun t : ℝ => (U, t)) := continuous_const.prodMk continuous_id
+  have hh := (continuous_logDerivative hε hT z).comp hp
+  change Continuous (logDerivative hε hT U z) at hh
+  exact hh
+
+end SLEExactGauge.BoundaryFunctional
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+
+lemma continuous_gap_time {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (x : ℝ) : Continuous (gap hε hT U x) := by
+  have hp : Continuous (fun t : ℝ => (U,t)) := continuous_const.prodMk continuous_id
+  have hh := (continuous_gap hε hT x).comp hp
+  change Continuous (gap hε hT U x) at hh
+  exact hh
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing
+
+lemma norm_sum_lower {X δ : ℝ} {D : ℂ} (hX : 0 < X) (hδ : δ ≤ 1/2)
+    (hD : ‖D‖ ≤ δ*X) : X/2 ≤ ‖(X:ℂ)+D‖ := by
+  have hh := norm_sub_le ((X:ℂ)+D) D
+  rw [add_sub_cancel_right,Complex.norm_real,Real.norm_of_nonneg hX.le] at hh
+  have hm := mul_le_mul_of_nonneg_right hδ hX.le
+  linarith
+
+lemma ratio_real_lower {X δ a : ℝ} {D : ℂ} (hX : 0 < X) (hδ0 : 0 ≤ δ) (hδ : δ ≤ 1/2)
+    (hD : ‖D‖ ≤ δ*X) (haδ : a ≤ 2-4*δ) :
+    a ≤ 2*((X:ℂ)/((X:ℂ)+D)).re := by
+  have hlo := norm_sum_lower hX hδ hD
+  have hn : (X:ℂ)+D ≠ 0 := norm_pos_iff.mp (lt_of_lt_of_le (by positivity) hlo)
+  have hquot : ‖D/((X:ℂ)+D)‖ ≤ 2*δ := by
+    rw [norm_div]
+    apply (div_le_iff₀ (norm_pos_iff.mpr hn)).2
+    calc
+      ‖D‖ ≤ δ*X := hD
+      _ ≤ (2*δ)*‖(X:ℂ)+D‖ := by nlinarith
+  have he : (X:ℂ)/((X:ℂ)+D) = 1-D/((X:ℂ)+D) := by field_simp; ring
+  rw [he,Complex.sub_re,Complex.one_re]
+  have hre := (Complex.re_le_norm (D/((X:ℂ)+D))).trans hquot
+  linarith
+
+lemma normSq_slope (X : ℝ) (D : ℂ) (hX : X ≠ 0) (hD : (X:ℂ)+D ≠ 0) :
+    2*D.re*(2/((X:ℂ)+D)-2/(X:ℂ)).re +
+      2*D.im*(2/((X:ℂ)+D)-2/(X:ℂ)).im =
+    -4*Complex.normSq D / X^2 * ((X:ℂ)/((X:ℂ)+D)).re := by
+  have hQ : (X+D.re)^2+D.im^2 ≠ 0 := by
+    have hh := (Complex.normSq_pos.mpr hD).ne'
+    simpa only [Complex.normSq_apply,Complex.add_re,Complex.ofReal_re,Complex.add_im,
+      Complex.ofReal_im,zero_add,pow_two] using hh
+  simp only [Complex.sub_re,Complex.sub_im,Complex.div_re,Complex.div_im,Complex.re_ofNat,
+    Complex.im_ofNat,Complex.add_re,Complex.add_im,Complex.ofReal_re,Complex.ofReal_im,
+    Complex.normSq_apply,zero_add,zero_mul,mul_zero,add_zero]
+  simp only [← pow_two]
+  field_simp
+  ring
+
+lemma energy_deriv {D : ℝ → ℂ} {l : ℝ → ℝ} {t X a : ℝ}
+    (hX : X ≠ 0) (hD : (X:ℂ)+D t ≠ 0)
+    (hd : HasDerivAt D (2/((X:ℂ)+D t)-2/(X:ℂ)) t)
+    (hl : HasDerivAt l (-2/X^2) t) :
+    HasDerivAt (fun s => Complex.normSq (D s)*Real.exp (-a*l s))
+      (Real.exp (-a*l t)*(-4*Complex.normSq (D t)/X^2*((X:ℂ)/((X:ℂ)+D t)).re+
+        2*a/X^2*Complex.normSq (D t))) t := by
+  have hre := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hd
+  have him := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hd
+  dsimp only [Function.comp_def,Complex.reCLM_apply,Complex.imCLM_apply] at hre him
+  have hq := (hre.pow 2).add (him.pow 2)
+  have hq' : HasDerivAt (fun s => Complex.normSq (D s))
+      (2*(D t).re*(2/((X:ℂ)+D t)-2/(X:ℂ)).re+2*(D t).im*(2/((X:ℂ)+D t)-2/(X:ℂ)).im) t := by
+    convert hq using 1 <;> (try simp only [Complex.normSq_apply,pow_two]) <;> first | rfl | ring
+  rw [normSq_slope _ _ hX hD] at hq'
+  apply (hq'.mul ((hl.const_mul (-a)).exp)).congr_deriv
+  ring
+
+lemma energy_antitone {S δ a : ℝ} (hδ0 : 0 ≤ δ) (hδ : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (D : ℝ → ℂ) (l X : ℝ → ℝ) (hDc : ContinuousOn D (Icc 0 S)) (hlc : Continuous l)
+    (hX : ∀ t ∈ Ioo 0 S, 0 < X t)
+    (hDd : ∀ t ∈ Ioo 0 S, HasDerivAt D (2/((X t:ℂ)+D t)-2/(X t:ℂ)) t)
+    (hld : ∀ t ∈ Ioo 0 S, HasDerivAt l (-2/(X t)^2) t)
+    (hratio : ∀ t ∈ Ioo 0 S, ‖D t‖ ≤ δ*X t) :
+    AntitoneOn (fun t => Complex.normSq (D t)*Real.exp (-a*l t)) (Icc 0 S) := by
+  have hEc : ContinuousOn (fun t => Complex.normSq (D t)*Real.exp (-a*l t)) (Icc 0 S) :=
+    (Complex.continuous_normSq.comp_continuousOn hDc).mul
+      ((Real.continuous_exp.comp (continuous_const.mul hlc)).continuousOn)
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 S) hEc
+    (f' := fun t => Real.exp (-a*l t)*(-4*Complex.normSq (D t)/(X t)^2*
+      ((X t:ℂ)/((X t:ℂ)+D t)).re+2*a/(X t)^2*Complex.normSq (D t)))
+  · intro t ht
+    rw [interior_Icc] at ht
+    have hn : (X t:ℂ)+D t ≠ 0 := norm_pos_iff.mp
+      (lt_of_lt_of_le (half_pos (hX t ht)) (norm_sum_lower (hX t ht) hδ (hratio t ht)))
+    exact (energy_deriv (hX t ht).ne' hn (hDd t ht) (hld t ht)).hasDerivWithinAt
+  · intro t ht
+    rw [interior_Icc] at ht
+    have hr := ratio_real_lower (hX t ht) hδ0 hδ (hratio t ht) haδ
+    apply mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le
+    have hq := Complex.normSq_nonneg (D t)
+    have hh := mul_le_mul_of_nonneg_left hr (show 0 ≤ 2*Complex.normSq (D t)/(X t)^2 by positivity)
+    convert sub_nonpos.mpr hh using 1
+    ring
+
+lemma gap_coefficients {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (x t : ℝ) (hx : ε ≤ gap hε hT U x t) :
+    velocity ε (extendDriver T U t) (position hε hT U (x:ℂ) t) = 2/(gap hε hT U x t:ℂ) ∧
+      variation hε hT U (x:ℂ) t = -2/(gap hε hT U x t)^2 := by
+  have hz : position hε hT U (x:ℂ) t = ((position hε hT U (x:ℂ) t).re:ℂ) := by
+    apply Complex.ext <;> simp [position_im_zero]
+  have hc : clampRe ε (position hε hT U (x:ℂ) t-(extendDriver T U t:ℂ)) =
+      (gap hε hT U x t:ℂ) := by
+    change BoundaryCutoff.clamp 1 ε _ = _
+    rw [BoundaryCutoff.clamp_eq (by norm_num : |(1:ℝ)|=1) (by simpa [gap] using hx),hz]
+    simp only [gap,Complex.ofReal_sub]
+  constructor
+  · change 2/clampRe ε _ = _; rw [hc]
+  · unfold variation
+    rw [hc]
+    norm_cast
+
+lemma gap_difference_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (x : ℝ) {t : ℝ} (ht : t ∈ Ioo 0 T)
+    (hx : ε ≤ gap hε hT U x t) (G : ℝ → ℂ)
+    (hG : HasDerivAt G (2/(G t-(extendDriver T U t:ℂ))) t) :
+    HasDerivAt (fun s => G s-position hε hT U (x:ℂ) s)
+      (2/((gap hε hT U x t:ℂ)+(G t-position hε hT U (x:ℂ) t))-
+        2/(gap hε hT U x t:ℂ)) t := by
+  have hh := hG.sub (position_deriv hε hT U (x:ℂ) ht)
+  rw [(gap_coefficients hε hT U x t hx).1] at hh
+  have hz : position hε hT U (x:ℂ) t = ((position hε hT U (x:ℂ) t).re:ℂ) := by
+    apply Complex.ext <;> simp [position_im_zero]
+  have hg : G t-(extendDriver T U t:ℂ) = (gap hε hT U x t:ℂ)+(G t-position hε hT U (x:ℂ) t) := by
+    unfold gap
+    rw [Complex.ofReal_sub,← hz]
+    ring
+  rw [hg] at hh
+  exact hh
+
+lemma ratio_energy {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
+    (U : C(I, ℝ)) (x a δ : ℝ) (hδ0 : 0 ≤ δ) (hδ : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (G : ℝ → ℂ) (hGc : ContinuousOn G (Icc 0 S))
+    (hGd : ∀ t ∈ Ioo 0 S, HasDerivAt G (2/(G t-(extendDriver T U t:ℂ))) t)
+    (hgap : ∀ t ∈ Icc 0 S, ε ≤ gap hε hT U x t)
+    (hratio : ∀ t ∈ Icc 0 S, ‖G t-position hε hT U (x:ℂ) t‖ ≤ δ*gap hε hT U x t) :
+    Complex.normSq (G S-position hε hT U (x:ℂ) S) *
+      Real.exp (-a*logDerivative hε hT U (x:ℂ) S) ≤ Complex.normSq (G 0-(x:ℂ)) := by
+  let D : ℝ → ℂ := fun t => G t-position hε hT U (x:ℂ) t
+  let l : ℝ → ℝ := logDerivative hε hT U (x:ℂ)
+  let X : ℝ → ℝ := gap hε hT U x
+  have hHc : Continuous (fun t => position hε hT U (x:ℂ) t) :=
+    continuous_position_time hε hT U (x:ℂ)
+  have hDc : ContinuousOn D (Icc 0 S) := hGc.sub hHc.continuousOn
+  have hlc : Continuous l :=
+    continuous_logDerivative_time hε hT U (x:ℂ)
+  have hXd (t : ℝ) (ht : t ∈ Ioo 0 S) : 0 < X t := hε.trans_le (hgap t ⟨ht.1.le,ht.2.le⟩)
+  have hDd (t : ℝ) (ht : t ∈ Ioo 0 S) : HasDerivAt D (2/((X t:ℂ)+D t)-2/(X t:ℂ)) t :=
+    gap_difference_deriv hε hT U x ⟨ht.1,ht.2.trans_le hST⟩ (hgap t ⟨ht.1.le,ht.2.le⟩) G (hGd t ht)
+  have hld (t : ℝ) (ht : t ∈ Ioo 0 S) : HasDerivAt l (-2/(X t)^2) t := by
+    have hh := hasDerivAt_logDerivative hε hT U (x:ℂ) ⟨ht.1,ht.2.trans_le hST⟩
+    rw [(gap_coefficients hε hT U x t (hgap t ⟨ht.1.le,ht.2.le⟩)).2] at hh
+    exact hh
+  have ha := energy_antitone hδ0 hδ haδ D l X hDc hlc hXd hDd hld
+    (fun t ht => hratio t ⟨ht.1.le,ht.2.le⟩)
+  have hh := ha ⟨le_rfl,hS⟩ ⟨hS,le_rfl⟩ hS
+  change Complex.normSq (D S)*Real.exp (-a*l S) ≤ Complex.normSq (G 0-(x:ℂ))
+  have he : Complex.normSq (D 0)*Real.exp (-a*l 0) = Complex.normSq (G 0-(x:ℂ)) := by
+    dsimp only [D,l]
+    rw [position_initial,logDerivative_initial,mul_zero,Real.exp_zero,mul_one]
+  exact hh.trans_eq he
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open Swallowing
+
+lemma energy_weight_bound {β a l d δ X ε N : ℝ} (hβ : 0 < β) (hd : 0 < d)
+    (hδ : 0 ≤ δ) (hX : 0 < X) (hε0 : 0 ≤ ε) (hε : ε ≤ X) (_hN : 0 ≤ N)
+    (henergy : N^2*Real.exp (-a*l) ≤ d^2)
+    (hweight : Real.exp ((a*β/2)*l)*radial (-β) X ε ≤ (δ/(4*d))^β) :
+    N ≤ (δ/2)*X := by
+  let E := Real.exp ((a/2)*l)
+  let Q := Real.sqrt (X^2+ε^2)
+  have hE : 0 < E := Real.exp_pos _
+  have hQ2 : 0 < X^2+ε^2 := by positivity
+  have hQ : 0 < Q := Real.sqrt_pos.2 hQ2
+  have hQsq : Q^2=X^2+ε^2 := Real.sq_sqrt hQ2.le
+  have hQbound : Q ≤ 2*X := by nlinarith [sq_nonneg (X-ε)]
+  have hep : Real.exp (-a*l)*E^2 = 1 := by
+    dsimp only [E]
+    rw [sq,← Real.exp_add,← Real.exp_add]
+    convert Real.exp_zero using 1
+    congr 1
+    ring
+  have hNEsq : N^2 ≤ (d*E)^2 := by
+    have hh := mul_le_mul_of_nonneg_right henergy (sq_nonneg E)
+    rw [mul_assoc,hep,mul_one] at hh
+    nlinarith only [hh]
+  have hNE : N ≤ d*E := by nlinarith [mul_pos hd hE]
+  have hEp : E^β=Real.exp ((a*β/2)*l) := by
+    dsimp only [E]
+    rw [← Real.exp_mul]
+    congr 1
+    ring
+  have hQp : Q^β=(X^2+ε^2)^(β/2) := by
+    dsimp only [Q]
+    rw [Real.sqrt_eq_rpow,← Real.rpow_mul hQ2.le]
+    congr 1
+    ring
+  have hwp : Real.exp ((a*β/2)*l)*radial (-β) X ε = (E/Q)^β := by
+    rw [Real.div_rpow hE.le hQ.le,hEp,hQp]
+    unfold radial
+    rw [show (-β)/2 = -(β/2) by ring,Real.rpow_neg hQ2.le]
+    rfl
+  rw [hwp] at hweight
+  have hR : 0 ≤ δ/(4*d) := by positivity
+  have hER : E/Q ≤ δ/(4*d) := (Real.rpow_le_rpow_iff (div_nonneg hE.le hQ.le) hR hβ).1 hweight
+  have heQ : E ≤ (δ/(4*d))*Q := (div_le_iff₀ hQ).1 hER
+  calc
+    N ≤ d*E := hNE
+    _ ≤ d*((δ/(4*d))*Q) := mul_le_mul_of_nonneg_left heQ hd.le
+    _ ≤ d*((δ/(4*d))*(2*X)) := by gcongr
+    _ = (δ/2)*X := by field_simp; ring
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge.BoundaryGreen
+
+lemma first_zero_of_nonpos {F : ℝ → ℝ} {S : ℝ} (hS : 0 ≤ S)
+    (hF : Continuous F) (h0 : 0 < F 0) (hSval : F S ≤ 0) :
+    ∃ σ ∈ Ioc 0 S, F σ = 0 ∧ ∀ s ∈ Icc 0 σ, 0 ≤ F s := by
+  let K : Set ℝ := Icc 0 S ∩ {s | F s ≤ 0}
+  have hK : IsCompact K := isCompact_Icc.inter_right (isClosed_le hF continuous_const)
+  have hne : K.Nonempty := ⟨S,⟨hS,le_rfl⟩,hSval⟩
+  let σ := sInf K
+  have hmem : σ ∈ K := hK.sInf_mem hne
+  have hσ : 0 < σ := by
+    apply lt_of_le_of_ne hmem.1.1
+    intro heq
+    have hh := hmem.2
+    rw [← heq] at hh
+    exact (not_le_of_gt h0) hh
+  have hbefore : ∀ s ∈ Ico 0 σ, 0 < F s := by
+    intro s hs
+    by_contra hn
+    have hm : s ∈ K := ⟨⟨hs.1,hs.2.le.trans hmem.1.2⟩,not_lt.1 hn⟩
+    exact (not_le_of_gt hs.2) (csInf_le hK.bddBelow hm)
+  have hσval : 0 ≤ F σ := by
+    apply ge_of_tendsto (hF.continuousAt.tendsto.mono_left nhdsWithin_le_nhds :
+      Tendsto F (𝓝[<] σ) (𝓝 (F σ)))
+    filter_upwards [Ioo_mem_nhdsLT hσ] with s hs
+    exact (hbefore s ⟨hs.1.le,hs.2⟩).le
+  refine ⟨σ,⟨hσ,hmem.1.2⟩,le_antisymm hmem.2 hσval,?_⟩
+  intro s hs
+  rcases lt_or_eq_of_le hs.2 with hlt | rfl
+  · exact (hbefore s ⟨hs.1,hlt⟩).le
+  · exact hσval
+
+lemma first_zero_of_nonpos_on {F : ℝ → ℝ} {S : ℝ} (hS : 0 ≤ S)
+    (hF : ContinuousOn F (Icc 0 S)) (h0 : 0 < F 0) (hSval : F S ≤ 0) :
+    ∃ σ ∈ Ioc 0 S, F σ = 0 ∧ ∀ s ∈ Icc 0 σ, 0 ≤ F s := by
+  let Fc : ℝ → ℝ := fun s => F (projIcc 0 S hS s)
+  have hc : Continuous Fc := hF.comp_continuous
+    (continuous_subtype_val.comp (continuous_projIcc (h := hS))) (fun s => (projIcc 0 S hS s).property)
+  have he (s : ℝ) (hs : s ∈ Icc 0 S) : Fc s=F s := by
+    simp only [Fc,projIcc,max_eq_right hs.1,min_eq_right hs.2]
+  obtain ⟨σ,hσ,hz,hpre⟩ := first_zero_of_nonpos hS hc (by rwa [he 0 ⟨le_rfl,hS⟩])
+    (by rwa [he S ⟨hS,le_rfl⟩])
+  refine ⟨σ,hσ,?_,?_⟩
+  · rwa [he σ ⟨hσ.1.le,hσ.2⟩] at hz
+  · intro s hs
+    have hh := hpre s hs
+    rwa [he s ⟨hs.1,hs.2.trans hσ.2⟩] at hh
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing
+
+lemma tip_forces_reach {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
+    (U : C(I, ℝ)) (hu0 : extendDriver T U 0=0)
+    (x a β δ d y : ℝ) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
+    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x) (hεx : 2*ε < x) (hy : δ*ε < y)
+    (G : ℝ → ℂ) (hGc : ContinuousOn G (Icc 0 S))
+    (hGd : ∀ t ∈ Ioo 0 S, HasDerivAt G (2/(G t-(extendDriver T U t:ℂ))) t)
+    (hG0 : ‖G 0-(x:ℂ)‖ ≤ d) (hheight : ∀ t ∈ Icc 0 S, y ≤ (G t).im)
+    (hGend : (G S).re=extendDriver T U S) :
+    reach U hε hT x β a ((δ/(4*d))^β) := by
+  let H : ℝ → ℂ := position hε hT U (x:ℂ)
+  let X : ℝ → ℝ := gap hε hT U x
+  let D : ℝ → ℂ := fun t => G t-H t
+  let F : ℝ → ℝ := fun t => min (X t-2*ε) (δ*X t-‖D t‖)
+  have hXc : Continuous X := continuous_gap_time hε hT U x
+  have hDc : ContinuousOn D (Icc 0 S) :=
+    hGc.sub (continuous_position_time hε hT U (x:ℂ)).continuousOn
+  have hFc : ContinuousOn F (Icc 0 S) := continuous_min.comp_continuousOn
+    ((hXc.continuousOn.sub continuousOn_const).prodMk
+      ((continuousOn_const.mul hXc.continuousOn).sub hDc.norm))
+  have hX0 : X 0=x := by simp only [X,gap,position_initial,Complex.ofReal_re,hu0,sub_zero]
+  have hF0 : 0 < F 0 := by
+    change 0 < min (X 0-2*ε) (δ*X 0-‖G 0-H 0‖)
+    rw [hX0,show H 0=(x:ℂ) from position_initial hε hT U (x:ℂ)]
+    exact lt_min (sub_pos.mpr hεx) (sub_pos.mpr (hG0.trans_lt hdx))
+  have hRe : (D S).re = -X S := by
+    change (G S-position hε hT U (x:ℂ) S).re = -gap hε hT U x S
+    simp only [Complex.sub_re,gap,hGend]
+    ring
+  have hnorm : X S ≤ ‖D S‖ := by
+    have hh := Complex.abs_re_le_norm (D S)
+    rw [hRe,abs_neg] at hh
+    exact (le_abs_self _).trans hh
+  have hFS : F S ≤ 0 := by
+    by_contra hn
+    have hh := lt_min_iff.mp (lt_of_not_ge hn)
+    have hXp : 0 < X S := by linarith [hh.1]
+    have hm := mul_le_mul_of_nonneg_right hδhalf hXp.le
+    linarith [hh.2]
+  obtain ⟨q,hq,hzero,hpre⟩ := first_zero_of_nonpos_on hS hFc hF0 hFS
+  have hgap : ∀ t ∈ Icc 0 q, 2*ε ≤ X t := by
+    intro t ht
+    exact sub_nonneg.mp (le_min_iff.mp (hpre t ht)).1
+  have hratio : ∀ t ∈ Icc 0 q, ‖D t‖ ≤ δ*X t := by
+    intro t ht
+    exact sub_nonneg.mp (le_min_iff.mp (hpre t ht)).2
+  have hXp : 0 < X q := hε.trans_le (by linarith [hgap q ⟨hq.1.le,le_rfl⟩])
+  have hqT : q ≤ T := hq.2.trans hST
+  have henergy := ratio_energy hε hT hq.1.le hqT U x a δ hδ.le hδhalf haδ G
+    (hGc.mono (Icc_subset_Icc_right hq.2)) (fun t ht => hGd t ⟨ht.1,ht.2.trans_le hq.2⟩)
+    (fun t ht => by change ε ≤ X t; linarith [hgap t ht]) hratio
+  have hEn : ‖D q‖^2*Real.exp (-a*logDerivative hε hT U (x:ℂ) q) ≤ d^2 := by
+    rw [Complex.normSq_eq_norm_sq,Complex.normSq_eq_norm_sq] at henergy
+    exact henergy.trans (sq_le_sq₀ (norm_nonneg _) hd.le |>.2 hG0)
+  by_contra hreach
+  have hweight : weight hε hT U x β a q ≤ (δ/(4*d))^β := by
+    apply (not_le.mp ?_).le
+    intro hh
+    exact hreach ⟨q,⟨hq.1.le,hqT⟩,(fun t ht => by change ε < X t; linarith [hgap t ht]),hh⟩
+  have hW : Real.exp ((a*β/2)*logDerivative hε hT U (x:ℂ) q)*radial (-β) (X q) ε ≤ (δ/(4*d))^β := hweight
+  have hsmall := energy_weight_bound hβ hd hδ.le hXp hε.le
+    (show ε ≤ X q by linarith [hgap q ⟨hq.1.le,le_rfl⟩]) (norm_nonneg (D q)) hEn hW
+  have hym : y ≤ ‖D q‖ := by
+    have hh := hheight q ⟨hq.1.le,hq.2⟩
+    have him : (D q).im=(G q).im := by simp [D,H,position_im_zero]
+    rw [← him] at hh
+    exact hh.trans (Complex.im_le_norm _)
+  have hgt : 2*ε < X q := by nlinarith
+  have hrat : ‖D q‖ < δ*X q := by nlinarith [mul_pos hδ hXp]
+  have hpos : 0 < F q := lt_min (sub_pos.mpr hgt) (sub_pos.mpr hrat)
+  exact (ne_of_gt hpos) hzero
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+
+lemma inverse_tip_point {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {t : ℝ} (ht : 0 ≤ t)
+    {x : ℂ} {d : ℝ} (hclose : dist (γ t) x < d) :
+    ∃ (w : ℂ) (y : ℝ), 0 < y ∧ dist w x < d ∧
+      w ∈ localizationDomain γ t ∧ g t w=(driver t:ℂ)+(y:ℂ)*Complex.I := by
+  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2.2 t ht
+  have he : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), dist (f t ((driver t:ℂ)+(y:ℂ)*Complex.I)) x < d :=
+    htip.eventually (isOpen_ball.mem_nhds hclose)
+  have hp : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), 0 < y := self_mem_nhdsWithin
+  obtain ⟨y,hy,hball⟩ := (hp.and he).exists
+  have hz : (driver t:ℂ)+(y:ℂ)*Complex.I ∈ halfPlane := by
+    change 0 < ((driver t:ℂ)+(y:ℂ)*Complex.I).im
+    simpa using hy
+  refine ⟨f t ((driver t:ℂ)+(y:ℂ)*Complex.I),y,hy,hball,?_,?_⟩
+  · exact h.2.2.2.2.2.2.2.2.1 t ht hz
+  · exact h.2.2.2.2.2.2.2.2.2.2.1 t ht _ hz
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ReverseLoewner
+open LoewnerCutoff
+
+def velocity (ε u : ℝ) (w : ℂ) : ℂ := -LoewnerCutoff.velocity ε u w
+
+lemma velocity_lipschitz {ε : ℝ} (hε : 0 < ε) (u : ℝ) :
+    LipschitzWith ⟨4 / ε^2, by positivity⟩ (velocity ε u) :=
+  (LoewnerCutoff.velocity_lipschitz hε u).neg
+
+end SLEExactGauge.ReverseLoewner
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/BrownianSupport.lean b/lean/OAI/Probability/SLELocalization/BrownianSupport.lean
index e892a7d59a678e7df6ab523a1b9d8d412c34c4cb..45e4cd8dd61ea9325be3dba4602ef369910c68d0 100644
--- a/lean/OAI/Probability/SLELocalization/BrownianSupport.lean
+++ b/lean/OAI/Probability/SLELocalization/BrownianSupport.lean
@@ -1,731 +1,731 @@
-import Mathlib
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Brownian Support estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianSupport
-open BrownianPredictable
-
-lemma preBrownian_past_square_le {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsPreBrownianReal B P) (hBm : ∀ t, Measurable (B t))
-    {s t : ℝ≥0} (hst : s ≤ t) {A : Set Ω} (hA : MeasurableSet[pastSigma B s] A) :
-    (∫ sample in A, (B s sample)^2 ∂P) ≤ ∫ sample in A, (B t sample)^2 ∂P := by
-  have hraw : pastSigma B s ≤ ‹MeasurableSpace Ω› :=
-    (Measurable.of_eval (fun time : Set.Iic s => hBm time)).comap_le
-  have hAm : MeasurableSet A := hraw _ hA
-  have hs := (hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two
-  have ht := (hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two
-  have hi : IndepFun (A.indicator (B s)) (fun sample => B t sample-B s sample) P := by
-    have hall := (hB.indepFun_shift s).symm
-    have hX : Measurable[pastSigma B s] (A.indicator (B s)) :=
-      (past_measurable_eval B le_rfl).indicator hA
-    have hf := hall.comp measurable_id (measurable_pi_apply (t-s))
-    simp only [Function.comp_def,id_eq,add_tsub_cancel_of_le hst] at hf
-    exact indep_of_indep_of_le_left hf hX.comap_le
-  have hz : (∫ sample in A, B s sample*(B t sample-B s sample) ∂P) = 0 := by
-    have he := hi.integral_fun_mul_eq_mul_integral
-      ((hBm s).indicator hAm).aestronglyMeasurable ((hBm t).sub (hBm s)).aestronglyMeasurable
-    have hind : (fun sample => A.indicator (B s) sample*(B t sample-B s sample)) =
-        A.indicator (fun sample => B s sample*(B t sample-B s sample)) := by
-      funext sample
-      by_cases ha : sample ∈ A <;> simp [ha]
-    rw [hind,integral_indicator hAm,integral_sub (hB.integrable_eval t) (hB.integrable_eval s),
-      hB.integral_eval t,hB.integral_eval s,sub_self,mul_zero] at he
-    exact he
-  have hcross : Integrable (fun sample => B s sample*(B t sample-B s sample)) P :=
-    hs.integrable_mul (ht.sub hs)
-  have his : Integrable (fun sample => (B s sample)^2) P := hs.integrable_sq
-  have hid : Integrable (fun sample => (B t sample-B s sample)^2) P := (ht.sub hs).integrable_sq
-  have he : (∫ sample in A, (B t sample)^2 ∂P) =
-      (∫ sample in A, (B s sample)^2 ∂P) + ∫ sample in A, (B t sample-B s sample)^2 ∂P := by
-    calc
-      _ = ∫ sample in A, ((B s sample)^2+2*(B s sample*(B t sample-B s sample)))+
-          (B t sample-B s sample)^2 ∂P := integral_congr_ae (Eventually.of_forall (fun _ => by ring))
-      _ = (∫ sample in A, (B s sample)^2+2*(B s sample*(B t sample-B s sample)) ∂P) +
-          ∫ sample in A, (B t sample-B s sample)^2 ∂P :=
-        integral_add ((his.restrict (s := A)).add ((hcross.restrict (s := A)).const_mul 2)) (hid.restrict (s := A))
-      _ = ((∫ sample in A, (B s sample)^2 ∂P) +
-          2*(∫ sample in A, B s sample*(B t sample-B s sample) ∂P)) +
-          ∫ sample in A, (B t sample-B s sample)^2 ∂P := by
-        congr 1
-        calc
-          _ = (∫ sample in A, (B s sample)^2 ∂P) +
-              ∫ sample in A, 2*(B s sample*(B t sample-B s sample)) ∂P :=
-            integral_add (his.restrict (s := A)) ((hcross.restrict (s := A)).const_mul 2)
-          _ = _ := by rw [integral_const_mul]
-      _ = _ := by rw [hz,mul_zero,add_zero]
-  rw [he]
-  exact le_add_of_nonneg_right (integral_nonneg (fun _ => sq_nonneg _))
-
-lemma preBrownian_square_submartingale {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsPreBrownianReal B P) (hBm : ∀ t, Measurable (B t))
-    (t : ℕ → ℝ≥0) (ht : Monotone t) :
-    let F : Filtration ℕ ‹MeasurableSpace Ω› :=
-      ⟨fun n => pastSigma B (t n),fun _ _ hij => pastSigma_mono B (ht hij),
-        fun n => (Measurable.of_eval (fun time : Set.Iic (t n) => hBm time)).comap_le⟩
-    Submartingale (fun n sample => (B (t n) sample)^2) F P := by
-  intro F
-  apply submartingale_of_setIntegral_le
-  · intro n
-    exact ((past_measurable_eval B le_rfl).pow_const 2).stronglyMeasurable
-  · intro n
-    exact (hB.isGaussianProcess.hasGaussianLaw_eval (t n)).memLp_two.integrable_sq
-  · intro i j hij A hA
-    exact preBrownian_past_square_le hB hBm (ht hij) hA
-
-lemma preBrownian_finite_max_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsPreBrownianReal B P) (hBm : ∀ t, Measurable (B t))
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (n : ℕ) {a : ℝ} (ha : 0 < a) :
-    P {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} ≤ ENNReal.ofReal ((t n:ℝ)/a^2) := by
-  let f : ℕ → Ω → ℝ := fun n sample => (B (t n) sample)^2
-  have hm := maximal_ineq (preBrownian_square_submartingale hB hBm t ht)
-    (show 0 ≤ f from fun _ _ => sq_nonneg _) (ε := (a^2).toNNReal) n
-  simp only [Real.coe_toNNReal _ (sq_nonneg a)] at hm
-  have he : {sample | (a^2) ≤ (Finset.range (n+1)).sup' Finset.nonempty_range_add_one (fun k => f k sample)} =
-      {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} := by
-    ext sample
-    simp only [mem_ofPred_eq,Finset.le_sup'_iff,Finset.mem_range,Nat.lt_succ_iff,f]
-    constructor
-    · rintro ⟨i,hi,hv⟩
-      exact ⟨i,hi,(sq_le_sq₀ ha.le (abs_nonneg _)).1 (by simpa using hv)⟩
-    · rintro ⟨i,hi,hv⟩
-      exact ⟨i,hi,by simpa using (sq_le_sq₀ ha.le (abs_nonneg _)).2 hv⟩
-  change ENNReal.ofReal (a^2) * P _ ≤ _ at hm
-  rw [he] at hm
-  have hi : (∫ sample in {sample | ∃ i ≤ n, a ≤ |B (t i) sample|}, (B (t n) sample)^2 ∂P) ≤ (t n:ℝ) := by
-    calc
-      _ ≤ ∫ sample, (B (t n) sample)^2 ∂P := integral_mono_measure Measure.restrict_le_self
-        (Eventually.of_forall (fun _ => sq_nonneg _))
-        (hB.isGaussianProcess.hasGaussianLaw_eval (t n)).memLp_two.integrable_sq
-      _ = _ := ((hB.hasLaw_eval (t n)).integral_comp
-        (show AEStronglyMeasurable (fun x : ℝ => x^2) _ from by fun_prop)).trans
-          (BrownianPredictable.gaussian_second_moment _)
-  have hh := hm.trans (ENNReal.ofReal_le_ofReal hi)
-  rw [ENNReal.ofReal_div_of_pos (sq_pos_of_pos ha)]
-  exact (ENNReal.le_div_iff_mul_le (Or.inl (ne_of_gt (ENNReal.ofReal_pos.2 (sq_pos_of_pos ha)))) (Or.inl ENNReal.ofReal_ne_top)).2 (by simpa [mul_comm] using hh)
-
-end SLEExactGauge.BrownianSupport
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-
-def restartIndex {Ω : Type*} (τ : Ω → ℝ≥0) (n : ℕ) (sample : Ω) : ℕ :=
-  Nat.ceil (τ sample*((n:ℝ≥0)+1))
-
-def restartGrid (n k : ℕ) : ℝ≥0 := (k:ℝ≥0)/((n:ℝ≥0)+1)
-
-def restartTime {Ω : Type*} (τ : Ω → ℝ≥0) (n : ℕ) (sample : Ω) : ℝ≥0 :=
-  restartGrid n (restartIndex τ n sample)
-
-lemma restartIndex_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {τ : Ω → ℝ≥0} (hτ : AEMeasurable τ P) (n : ℕ) : AEMeasurable (restartIndex τ n) P :=
-  Nat.measurable_ceil.comp_aemeasurable (hτ.mul_const _)
-
-lemma restartTime_bounds {Ω : Type*} (τ : Ω → ℝ≥0) (n : ℕ) (sample : Ω) :
-    τ sample ≤ restartTime τ n sample ∧
-      restartTime τ n sample ≤ τ sample+1/((n:ℝ≥0)+1) := by
-  have ha : 0 < (n:ℝ≥0)+1 := by positivity
-  constructor
-  · exact (le_div_iff₀ ha).2 (Nat.le_ceil _)
-  · change (Nat.ceil (τ sample*((n:ℝ≥0)+1)):ℝ≥0)/((n:ℝ≥0)+1) ≤ _
-    apply le_of_lt
-    calc
-      _ < (τ sample*((n:ℝ≥0)+1)+1)/((n:ℝ≥0)+1) :=
-        (div_lt_div_iff_of_pos_right ha).2 (Nat.ceil_lt_add_one (by positivity))
-      _ = _ := by rw [add_div,mul_div_cancel_right₀ _ ha.ne']
-
-lemma restartTime_tendsto {Ω : Type*} (τ : Ω → ℝ≥0) (sample : Ω) :
-    Tendsto (fun n => restartTime τ n sample) atTop (𝓝 (τ sample)) := by
-  have hh : Tendsto (fun n : ℕ => τ sample+1/((n:ℝ≥0)+1)) atTop (𝓝 (τ sample)) := by
-    simpa only [add_zero] using tendsto_const_nhds.add
-      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ≥0))
-  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hh
-    (fun n => (restartTime_bounds τ n sample).1) (fun n => (restartTime_bounds τ n sample).2)
-
-lemma restartIndex_past {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (τ : Ω → ℝ≥0) (A : Set Ω)
-    (hstop : ∀ t : ℝ≥0, MeasurableSet[pastSigma B t] (A ∩ {sample | τ sample ≤ t}))
-    (n k : ℕ) : MeasurableSet[pastSigma B (restartGrid n k)]
-      (A ∩ {sample | restartIndex τ n sample = k}) := by
-  have ha : 0 < (n:ℝ≥0)+1 := by positivity
-  by_cases hk : k=0
-  · subst k
-    have he : A ∩ {sample | restartIndex τ n sample = 0} = A ∩ {sample | τ sample ≤ 0} := by
-      ext sample
-      simp only [mem_inter_iff,mem_ofPred_eq,restartIndex,Nat.ceil_eq_zero]
-      constructor
-      · rintro ⟨hA,ht⟩
-        exact ⟨hA,(mul_le_mul_iff_left₀ ha).1 (by simpa using ht)⟩
-      · rintro ⟨hA,ht⟩
-        exact ⟨hA,mul_nonpos_of_nonpos_of_nonneg ht ha.le⟩
-    rw [he]
-    have hz : restartGrid n 0 = 0 := by simp [restartGrid]
-    rw [hz]
-    exact hstop 0
-  · have hle : restartGrid n (k-1) ≤ restartGrid n k := by
-      apply div_le_div_of_nonneg_right _ ha.le
-      exact_mod_cast Nat.sub_le k 1
-    have hprev : MeasurableSet[pastSigma B (restartGrid n k)]
-        (A ∩ {sample | τ sample ≤ restartGrid n (k-1)}) :=
-      (pastSigma_mono B hle) _ (hstop _)
-    have he : A ∩ {sample | restartIndex τ n sample = k} =
-        (A ∩ {sample | τ sample ≤ restartGrid n k}) \
-          (A ∩ {sample | τ sample ≤ restartGrid n (k-1)}) := by
-      ext sample
-      simp only [mem_inter_iff,mem_ofPred_eq,Set.mem_sdiff,restartIndex,Nat.ceil_eq_iff hk,
-        restartGrid,le_div_iff₀ ha,not_and,not_le]
-      constructor
-      · rintro ⟨hA,hl,hu⟩
-        exact ⟨⟨hA,hu⟩,fun _ => hl⟩
-      · rintro ⟨⟨hA,hu⟩,hl⟩
-        exact ⟨hA,hl hA,hu⟩
-    rw [he]
-    exact (hstop _).diff hprev
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianSupport
-open BrownianPredictable
-
-lemma brownian_finite_max_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsPreBrownianReal B P)
-    (t : ℕ → ℝ≥0) (ht : Monotone t) (n : ℕ) {a : ℝ} (ha : 0 < a) :
-    P {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} ≤ ENNReal.ofReal ((t n:ℝ)/a^2) := by
-  let C : ℝ≥0 → Ω → ℝ := fun t => (hB.aemeasurable t).mk (B t)
-  have hm (t : ℝ≥0) : Measurable (C t) := (hB.aemeasurable t).measurable_mk
-  have he (t : ℝ≥0) : B t =ᵐ[P] C t := (hB.aemeasurable t).ae_eq_mk
-  have hC : IsPreBrownianReal C P := hB.congr he
-  have hmeasure : P {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} =
-      P {sample | ∃ i ≤ n, a ≤ |C (t i) sample|} := by
-    apply measure_congr
-    filter_upwards [ae_all_iff.2 (fun i : ℕ => he (t i))] with sample hh
-    apply propext
-    simp only [hh]
-  rw [hmeasure]
-  exact preBrownian_finite_max_bound hC hm t ht n ha
-
-lemma brownian_oscillation_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) :
-    P {sample | ∃ t ∈ Icc 0 T, a < |B t sample|} ≤ ENNReal.ofReal ((T:ℝ)/a^2) := by
-  let grid (n i : ℕ) : ℝ≥0 := min T (restartGrid n i)
-  let N (n : ℕ) := Nat.ceil (T*((n:ℝ≥0)+1))
-  let E (n : ℕ) : Set Ω := {sample | ∃ i ≤ N n, a ≤ |B (grid n i) sample|}
-  have hgrid (n : ℕ) : Monotone (grid n) := by
-    intro i j hij
-    apply min_le_min le_rfl
-    exact div_le_div_of_nonneg_right (by exact_mod_cast hij) (by positivity)
-  have hend (n : ℕ) : grid n (N n) = T := by
-    apply min_eq_left
-    exact (restartTime_bounds (fun _ : Unit => T) n ()).1
-  have hE (n : ℕ) : P (E n) ≤ ENNReal.ofReal ((T:ℝ)/a^2) := by
-    have hh := brownian_finite_max_bound hB.toIsPreBrownianReal (grid n) (hgrid n) (N n) ha
-    simpa only [hend] using hh
-  let F (k : ℕ) : Set Ω := ⋂ n ≥ k, E n
-  have hd : Monotone F := by
-    intro k l hkl sample hs
-    simp only [F,mem_iInter] at hs ⊢
-    exact fun n hn => hs n (hkl.trans hn)
-  have hsub : {sample | ∃ t ∈ Icc 0 T, a < |B t sample|} ≤ᵐ[P] ⋃ k, F k := by
-    filter_upwards [hB.cont] with sample hc hs
-    rcases hs with ⟨t,ht,haB⟩
-    have htend : Tendsto (fun n => min T (restartTime (fun _ : Unit => t) n ())) atTop (𝓝 t) := by
-      have hh := (tendsto_const_nhds (x := T)).min (restartTime_tendsto (fun _ : Unit => t) ())
-      simpa only [min_eq_right ht.2] using hh
-    have hev := ((hc.continuousAt.tendsto.comp htend).abs).eventually (lt_mem_nhds haB)
-    obtain ⟨k,hk⟩ := eventually_atTop.1 hev
-    apply mem_iUnion.2
-    refine ⟨k,?_⟩
-    simp only [F,mem_iInter]
-    intro n hn
-    refine ⟨restartIndex (fun _ : Unit => t) n (),?_,(hk n hn).le⟩
-    apply Nat.ceil_mono
-    exact mul_le_mul_of_nonneg_right ht.2 (by positivity)
-  calc
-    _ ≤ P (⋃ k, F k) := measure_mono_ae hsub
-    _ = ⨆ k, P (F k) := hd.directed_le.measure_iUnion
-    _ ≤ _ := iSup_le (fun k => (measure_mono (by
-      intro sample hs
-      exact mem_iInter.1 (mem_iInter.1 hs k) le_rfl)).trans (hE k))
-
-end SLEExactGauge.BrownianSupport
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianSupport
-
-private def densityFloor : ℝ := (Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-(1/2:ℝ))
-
-private lemma densityFloor_pos : 0 < densityFloor := by
-  dsimp [densityFloor]
-  positivity
-
-private lemma gaussian_pdf_lower {v : ℝ≥0} (hv : 0 < v) {x : ℝ} (hx : |x| ≤ Real.sqrt v) :
-    densityFloor / Real.sqrt v ≤ gaussianPDFReal 0 v x := by
-  have hvR : 0 < (v:ℝ) := hv
-  have hs : 0 < Real.sqrt (v:ℝ) := Real.sqrt_pos.2 hvR
-  have hx2 : x^2 ≤ (v:ℝ) := by
-    have hh := (sq_le_sq₀ (abs_nonneg x) hs.le).2 hx
-    simpa only [sq_abs,Real.sq_sqrt hvR.le] using hh
-  have he : -(1/2:ℝ) ≤ -x^2/(2*(v:ℝ)) := by
-    apply (le_div_iff₀ (by positivity)).2
-    nlinarith
-  calc
-    _ = (Real.sqrt (2*Real.pi*(v:ℝ)))⁻¹ * Real.exp (-(1/2:ℝ)) := by
-      rw [Real.sqrt_mul (by positivity)]
-      dsimp [densityFloor]
-      field_simp
-    _ ≤ _ := by
-      change _ ≤ (Real.sqrt (2*Real.pi*(v:ℝ)))⁻¹ * Real.exp (-(x-0)^2/(2*(v:ℝ)))
-      simp only [sub_zero]
-      exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.2 he) (by positivity)
-
-lemma gaussian_short_interval_lower {v : ℝ≥0} (hv : 0 < v) {c α L : ℝ}
-    (hα : 0 < α) (hc : |c| ≤ L*(v:ℝ))
-    (hsmall : (L+α)*(v:ℝ) ≤ Real.sqrt v) :
-    ENNReal.ofReal (2*densityFloor*α*Real.sqrt v) ≤
-      gaussianReal 0 v {x : ℝ | |x-c| < α*(v:ℝ)} := by
-  have hvR : 0 < (v:ℝ) := hv
-  have hs : 0 < Real.sqrt (v:ℝ) := Real.sqrt_pos.2 hvR
-  let A : Set ℝ := Ioo (c-α*(v:ℝ)) (c+α*(v:ℝ))
-  have hA : {x : ℝ | |x-c| < α*(v:ℝ)} = A := by
-    ext x
-    simp only [mem_ofPred_eq,A,mem_Ioo,abs_lt]
-    constructor <;> intro hh <;> constructor <;> linarith [hh.1,hh.2]
-  rw [hA,gaussianReal_apply_eq_integral 0 hv.ne']
-  apply ENNReal.ofReal_le_ofReal
-  calc
-    _ = ∫ x in A, densityFloor/Real.sqrt v := by
-      rw [setIntegral_const]
-      simp only [smul_eq_mul,A,Real.volume_real_Ioo]
-      rw [max_eq_left (by nlinarith [mul_pos hα hvR] : 0 ≤ c+α*(v:ℝ)-(c-α*(v:ℝ)))]
-      field_simp
-      rw [Real.sq_sqrt hvR.le]
-      ring
-    _ ≤ _ := by
-      apply setIntegral_mono_on (integrable_const _) (integrable_gaussianPDFReal 0 v).integrableOn measurableSet_Ioo
-      intro x hx
-      apply gaussian_pdf_lower hv
-      have hxc : |x-c| < α*(v:ℝ) := by
-        have hh : x ∈ {x : ℝ | |x-c| < α*(v:ℝ)} := hA.symm ▸ hx
-        exact hh
-      calc
-        |x| ≤ |x-c|+|c| := by simpa using abs_add_le (x-c) c
-        _ ≤ (L+α)*(v:ℝ) := by nlinarith
-        _ ≤ _ := hsmall
-
-lemma short_endpoint_dominates {L α b : ℝ} (hL : 0 ≤ L) (hα : 0 < α) (hb : 0 < b) :
-    ∃ δ : ℝ, 0 < δ ∧ ∀ (v : ℝ≥0), 0 < v → (v:ℝ) ≤ δ → ∀ c : ℝ,
-      |c| ≤ L*(v:ℝ) → ENNReal.ofReal ((v:ℝ)/b^2) <
-        gaussianReal 0 v {x : ℝ | |x-c| < α*(v:ℝ)} := by
-  let r : ℝ := min (1/(L+α+1)) (densityFloor*α*b^2)
-  have hr : 0 < r := lt_min (by positivity) (by positivity [densityFloor_pos])
-  refine ⟨r^2,by positivity,fun v hv hvr c hc => ?_⟩
-  have hvR : 0 < (v:ℝ) := hv
-  have hs : 0 < Real.sqrt (v:ℝ) := Real.sqrt_pos.2 hvR
-  have hsr : Real.sqrt (v:ℝ) ≤ r := (Real.sqrt_le_iff).2 ⟨hr.le,hvr⟩
-  have hsmall : (L+α)*(v:ℝ) ≤ Real.sqrt v := by
-    have hr1 : r ≤ 1/(L+α+1) := min_le_left _ _
-    have hh : (L+α+1)*Real.sqrt (v:ℝ) ≤ 1 := by
-      have hc' := (le_div_iff₀ (by positivity : 0 < L+α+1)).1 (hsr.trans hr1)
-      nlinarith
-    nlinarith [Real.sq_sqrt hvR.le]
-  apply lt_of_lt_of_le _ (gaussian_short_interval_lower hv hα hc hsmall)
-  apply (ENNReal.ofReal_lt_ofReal_iff (by positivity [densityFloor_pos])).2
-  apply (div_lt_iff₀ (sq_pos_of_pos hb)).2
-  have hr2 : r ≤ densityFloor*α*b^2 := min_le_right _ _
-  have hh := hsr.trans hr2
-  have hmul := mul_le_mul_of_nonneg_left hh hs.le
-  nlinarith [Real.sq_sqrt hvR.le,densityFloor_pos]
-
-end SLEExactGauge.BrownianSupport
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BrownianSupport
-open BrownianPredictable
-
-def probeTime (T : ℝ≥0) (q : I) : ℝ≥0 := T * ⟨q,q.property.1⟩
-
-lemma probeTime_le (T : ℝ≥0) (q : I) : probeTime T q ≤ T := by
-  change (T:ℝ)*(q:ℝ) ≤ T
-  simpa using mul_le_mul_of_nonneg_left q.property.2 T.coe_nonneg
-
-lemma continuous_probeTime (T : ℝ≥0) : Continuous (probeTime T) := by
-  apply Continuous.const_mul
-  exact continuous_subtype_val.subtype_mk (fun q => q.property.1)
-
-def stepSet (h : ℝ≥0) (c α b : ℝ) : Set (ℝ≥0 → ℝ) :=
-  {f | |f h-c| < α*(h:ℝ) ∧ ∀ n : ℕ, |f (probeTime h (denseSeq I n))| ≤ b}
-
-lemma measurableSet_stepSet (h : ℝ≥0) (c α b : ℝ) : MeasurableSet (stepSet h c α b) := by
-  change MeasurableSet ({f : ℝ≥0 → ℝ | |f h-c| < α*(h:ℝ)} ∩
-    {f : ℝ≥0 → ℝ | ∀ n : ℕ, |f (probeTime h (denseSeq I n))| ≤ b})
-  apply MeasurableSet.inter
-  · exact measurableSet_lt (((measurable_pi_apply h : Measurable (fun f : ℝ≥0 → ℝ => f h))).sub_const c).abs measurable_const
-  · simp only [ofPred_forall]
-    exact MeasurableSet.iInter fun n => measurableSet_le (measurable_pi_apply _).abs measurable_const
-
-lemma probe_bound_all {f : ℝ≥0 → ℝ} (hf : Continuous f) (T : ℝ≥0) {b : ℝ}
-    (hb : ∀ n : ℕ, |f (probeTime T (denseSeq I n))| ≤ b) :
-    ∀ t ≤ T, |f t| ≤ b := by
-  have hall (q : I) : |f (probeTime T q)| ≤ b :=
-    (denseRange_denseSeq I).induction_on q
-      (isClosed_le ((hf.comp (continuous_probeTime T)).abs) continuous_const) hb
-  intro t ht
-  by_cases hT : T=0
-  · have ht0 : t=0 := le_antisymm (ht.trans hT.le) (bot_le)
-    have hz : probeTime T (0:I) = 0 := by
-      apply NNReal.coe_injective
-      change (T:ℝ)*(0:ℝ) = (0:ℝ)
-      exact mul_zero _
-    simpa only [ht0,hz] using hall (0:I)
-  · have hTp : 0 < (T:ℝ) := by exact_mod_cast lt_of_le_of_ne (bot_le : 0 ≤ T) (Ne.symm hT)
-    let q : I := ⟨(t:ℝ)/(T:ℝ),⟨div_nonneg t.coe_nonneg T.coe_nonneg,(div_le_one hTp).2 ht⟩⟩
-    have he : probeTime T q = t := by
-      apply NNReal.coe_injective
-      change (T:ℝ)*((t:ℝ)/(T:ℝ)) = t
-      field_simp
-    simpa only [he] using hall q
-
-def stepEvent {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (t h : ℝ≥0) (c α b : ℝ) : Set Ω :=
-  (fun sample s => B (t+s) sample-B t sample) ⁻¹' stepSet h c α b
-
-lemma stepEvent_past {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (t h : ℝ≥0) (c α b : ℝ) : MeasurableSet[pastSigma B (t+h)] (stepEvent B t h c α b) := by
-  let : MeasurableSpace Ω := pastSigma B (t+h)
-  change MeasurableSet[pastSigma B (t+h)]
-    ({sample | |B (t+h) sample-B t sample-c| < α*(h:ℝ)} ∩
-     {sample | ∀ n : ℕ, |B (t+probeTime h (denseSeq I n)) sample-B t sample| ≤ b})
-  apply MeasurableSet.inter
-  · apply measurableSet_lt _ measurable_const
-    exact (((past_measurable_eval B (s:=t+h) (t:=t+h) le_rfl).sub
-      (past_measurable_eval B (s:=t) (t:=t+h) (le_add_right le_rfl))).sub_const c).abs
-  · simp only [ofPred_forall]
-    apply MeasurableSet.iInter
-    intro n
-    apply measurableSet_le _ measurable_const
-    exact ((past_measurable_eval B (s:=t+probeTime h _) (t:=t+h) (by gcongr; exact probeTime_le h _)).sub
-      (past_measurable_eval B (s:=t) (t:=t+h) (le_add_right le_rfl))).abs
-
-lemma small_step_pos {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) {L α b : ℝ} (hL : 0 ≤ L) (hα : 0 < α) (hb : 0 < b) :
-    ∃ δ : ℝ, 0 < δ ∧ ∀ (h : ℝ≥0), 0 < h → (h:ℝ) ≤ δ → ∀ c : ℝ,
-      |c| ≤ L*(h:ℝ) → ∀ t : ℝ≥0, 0 < P (stepEvent B t h c α b) := by
-  obtain ⟨δ,hδ,hdom⟩ := short_endpoint_dominates hL hα hb
-  refine ⟨δ,hδ,fun h hh hhδ c hc t => ?_⟩
-  let C : ℝ≥0 → Ω → ℝ := fun s sample => B (t+s) sample-B t sample
-  have hC : IsBrownianReal C P := hB.shift t
-  let E : Set Ω := {sample | |C h sample-c| < α*(h:ℝ)}
-  let Bad : Set Ω := {sample | ∃ s ∈ Icc 0 h, b < |C s sample|}
-  have hE : P E = gaussianReal 0 h {x : ℝ | |x-c| < α*(h:ℝ)} :=
-    (hC.toIsPreBrownianReal.hasLaw_eval h).measure_eq
-      (p := fun x : ℝ => |x-c| < α*(h:ℝ))
-      (measurableSet_lt (measurable_id.sub_const c).abs measurable_const)
-  have hbad : P Bad < P E := by
-    rw [hE]
-    exact (brownian_oscillation_bound hC h hb).trans_lt (hdom h hh hhδ c hc)
-  have hsub : E ⊆ stepEvent B t h c α b ∪ Bad := by
-    intro sample hs
-    by_cases hbad : sample ∈ Bad
-    · exact Or.inr hbad
-    · apply Or.inl
-      refine ⟨hs,fun n => ?_⟩
-      apply le_of_not_gt
-      intro hn
-      exact hbad ⟨probeTime h (denseSeq I n),⟨bot_le,probeTime_le h _⟩,hn⟩
-  apply pos_iff_ne_zero.2
-  intro hz
-  have hm := (measure_mono (μ:=P) hsub).trans (measure_union_le _ _)
-  rw [hz,zero_add] at hm
-  exact (not_le_of_gt hbad) hm
-
-lemma past_inter_step {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (t h : ℝ≥0) (c α b : ℝ) (A : Set Ω) (hA : MeasurableSet[pastSigma B t] A) :
-    P (A ∩ stepEvent B t h c α b) = P A * P (stepEvent B t h c α b) := by
-  have hi := (hB.toIsPreBrownianReal.indepFun_shift t).symm
-  apply hi.meas_inter hA
-  exact (measurableSet_stepSet h c α b).preimage (comap_measurable _)
-
-def stepsEvent {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (h : ℝ≥0) (c : ℕ → ℝ)
-    (α b : ℝ) (n : ℕ) : Set Ω :=
-  {sample | ∀ i < n, sample ∈ stepEvent B ((i:ℝ≥0)*h) h (c i) α b}
-
-lemma stepsEvent_zero {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (h : ℝ≥0) (c : ℕ → ℝ)
-    (α b : ℝ) : stepsEvent B h c α b 0 = univ := by
-  ext sample
-  simp [stepsEvent]
-
-lemma stepsEvent_succ {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (h : ℝ≥0) (c : ℕ → ℝ)
-    (α b : ℝ) (n : ℕ) : stepsEvent B h c α b (n+1) =
-      stepsEvent B h c α b n ∩ stepEvent B ((n:ℝ≥0)*h) h (c n) α b := by
-  ext sample
-  simp only [stepsEvent,mem_ofPred_eq,mem_inter_iff,Nat.lt_succ_iff]
-  exact ⟨fun hh => ⟨fun i hi => hh i (Nat.le_of_lt hi),hh n le_rfl⟩,
-    fun hh i hi => (lt_or_eq_of_le hi).elim (hh.1 i) (fun he => he ▸ hh.2)⟩
-
-lemma stepsEvent_past {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
-    (h : ℝ≥0) (c : ℕ → ℝ) (α b : ℝ) (n : ℕ) :
-    MeasurableSet[pastSigma B ((n:ℝ≥0)*h)] (stepsEvent B h c α b n) := by
-  induction n with
-  | zero => rw [stepsEvent_zero]; exact .univ
-  | succ n ih =>
-    rw [stepsEvent_succ]
-    have he : ((n+1:ℕ):ℝ≥0)*h = (n:ℝ≥0)*h+h := by push_cast; ring
-    rw [he]
-    exact ((pastSigma_mono B (s := (n:ℝ≥0)*h) (t := (n:ℝ≥0)*h+h) (le_add_right le_rfl)) _ ih).inter
-      (stepEvent_past B _ _ _ _ _)
-
-lemma stepsEvent_pos {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (h : ℝ≥0) (c : ℕ → ℝ) (α b : ℝ) (n : ℕ)
-    (hpos : ∀ i < n, 0 < P (stepEvent B ((i:ℝ≥0)*h) h (c i) α b)) :
-    0 < P (stepsEvent B h c α b n) := by
-  induction n with
-  | zero => rw [stepsEvent_zero,measure_univ]; exact zero_lt_one
-  | succ n ih =>
-    rw [stepsEvent_succ,past_inter_step hB _ _ _ _ _ _ (stepsEvent_past B h c α b n)]
-    exact ENNReal.mul_pos (ih (fun i hi => hpos i (Nat.lt_succ_of_lt hi))).ne' (hpos n (Nat.lt_succ_self n)).ne'
-
-lemma exists_grid_interval (h : ℝ≥0) {n : ℕ} (hn : 0 < n) {t : ℝ≥0}
-    (ht : t ≤ (n:ℝ≥0)*h) :
-    ∃ i < n, (i:ℝ≥0)*h ≤ t ∧ t ≤ ((i+1:ℕ):ℝ≥0)*h := by
-  induction n with
-  | zero => omega
-  | succ n ih =>
-    by_cases hn0 : n=0
-    · subst n
-      exact ⟨0,by omega,by simp,by simpa using ht⟩
-    · by_cases hh : t ≤ (n:ℝ≥0)*h
-      · obtain ⟨i,hi,hi1,hi2⟩ := ih (Nat.pos_of_ne_zero hn0) hh
-        exact ⟨i,Nat.lt_succ_of_lt hi,hi1,hi2⟩
-      · exact ⟨n,Nat.lt_succ_self n,(lt_of_not_ge hh).le,ht⟩
-
-lemma endpoint_error_bound (f u : ℝ≥0 → ℝ) (h : ℝ≥0) (α : ℝ) (n : ℕ)
-    (hz : f 0 = u 0)
-    (herr : ∀ i < n, |f ((i:ℝ≥0)*h+h)-f ((i:ℝ≥0)*h)-
-      (u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h))| ≤ α*(h:ℝ)) :
-    ∀ i ≤ n, |f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h)| ≤ (i:ℝ)*α*(h:ℝ) := by
-  intro i hi
-  induction i with
-  | zero => simp [hz]
-  | succ i ih =>
-    have hei : ((i+1:ℕ):ℝ≥0)*h = (i:ℝ≥0)*h+h := by push_cast; ring
-    rw [hei]
-    have hh := herr i (by omega)
-    have hh2 := ih (by omega)
-    calc
-      _ = |(f ((i:ℝ≥0)*h+h)-f ((i:ℝ≥0)*h)-
-            (u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h)))+
-            (f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h))| := by congr 1; ring
-      _ ≤ _ := abs_add_le _ _
-      _ ≤ _ := add_le_add hh hh2
-      _ = _ := by push_cast; ring
-
-lemma tube_of_steps (f u : ℝ≥0 → ℝ) (hf : Continuous f) (h : ℝ≥0) (n : ℕ)
-    (hn : 0 < n) {α b L : ℝ} (hα : 0 ≤ α) (hL : 0 ≤ L) (hz : f 0=u 0)
-    (hu : ∀ s ∈ Icc 0 ((n:ℝ≥0)*h), ∀ t ∈ Icc 0 ((n:ℝ≥0)*h),
-      |u s-u t| ≤ L*|((s:ℝ)-(t:ℝ))|)
-    (hs : ∀ i < n, (fun s => f ((i:ℝ≥0)*h+s)-f ((i:ℝ≥0)*h)) ∈
-      stepSet h (u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h)) α b) :
-    ∀ t ∈ Icc 0 ((n:ℝ≥0)*h), |f t-u t| ≤ b+(n:ℝ)*α*(h:ℝ)+L*(h:ℝ) := by
-  have he := endpoint_error_bound f u h α n hz (fun i hi => (hs i hi).1.le)
-  intro t ht
-  obtain ⟨i,hi,hit,hti⟩ := exists_grid_interval h hn ht.2
-  have hiadd : ((i+1:ℕ):ℝ≥0)*h = (i:ℝ≥0)*h+h := by push_cast; ring
-  have hsub : t-(i:ℝ≥0)*h ≤ h := by
-    rw [tsub_le_iff_right]
-    rw [hiadd] at hti
-    simpa only [add_comm] using hti
-  have hosc : |f t-f ((i:ℝ≥0)*h)| ≤ b := by
-    have hc : Continuous (fun s => f ((i:ℝ≥0)*h+s)-f ((i:ℝ≥0)*h)) :=
-      (hf.comp (continuous_const.add continuous_id)).sub continuous_const
-    have hh := probe_bound_all hc h (hs i hi).2 (t-(i:ℝ≥0)*h) hsub
-    simpa only [add_tsub_cancel_of_le hit] using hh
-  have hig : (i:ℝ≥0)*h ≤ (n:ℝ≥0)*h := by gcongr
-  have hul : |u ((i:ℝ≥0)*h)-u t| ≤ L*(h:ℝ) := by
-    apply (hu _ ⟨bot_le,hig⟩ _ ht).trans
-    have habs : |((i:ℝ≥0)*h:ℝ)-(t:ℝ)| ≤ (h:ℝ) := by
-      simp only [NNReal.coe_natCast]
-      rw [abs_of_nonpos (sub_nonpos.2 (show (i:ℝ)*(h:ℝ) ≤ (t:ℝ) by exact_mod_cast hit))]
-      have htR : (t:ℝ) ≤ (i:ℝ)*(h:ℝ)+(h:ℝ) := by exact_mod_cast (hiadd ▸ hti)
-      linarith
-    exact mul_le_mul_of_nonneg_left habs hL
-  calc
-    _ = |(f t-f ((i:ℝ≥0)*h))+(f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h))+
-          (u ((i:ℝ≥0)*h)-u t)| := by congr 1; ring
-    _ ≤ (|f t-f ((i:ℝ≥0)*h)|+|f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h)|)+
-          |u ((i:ℝ≥0)*h)-u t| :=
-        (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
-    _ ≤ b+(i:ℝ)*α*(h:ℝ)+L*(h:ℝ) := add_le_add (add_le_add hosc (he i hi.le)) hul
-    _ ≤ _ := by gcongr
-
-end SLEExactGauge.BrownianSupport
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianSupport
-
-lemma lipschitzTube_pos {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T : ℝ≥0) (u : ℝ≥0 → ℝ)
-    (hu0 : u 0=0) {L : ℝ} (hL : 0 ≤ L)
-    (hu : ∀ s ∈ Icc 0 T, ∀ t ∈ Icc 0 T, |u s-u t| ≤ L*|(s:ℝ)-(t:ℝ)|)
-    {ε : ℝ} (hε : 0 < ε) :
-    0 < P {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
-  by_cases hT : T=0
-  · have hsub : (univ : Set Ω) ≤ᵐ[P] {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
-      filter_upwards [hB.eval_zero_ae_eq_zero] with sample hs _
-      intro t ht
-      have ht0 : t=0 := le_antisymm (ht.2.trans hT.le) bot_le
-      simpa [ht0,hs,hu0] using hε
-    have hm := measure_mono_ae hsub
-    rw [measure_univ] at hm
-    exact zero_lt_one.trans_le hm
-  have hTp : 0 < T := lt_of_le_of_ne bot_le (Ne.symm hT)
-  have hTR : 0 < (T:ℝ) := hTp
-  let α : ℝ := ε/(4*((T:ℝ)+1))
-  let b : ℝ := ε/4
-  have hα : 0 < α := by dsimp [α]; positivity
-  have hb : 0 < b := by dsimp [b]; positivity
-  obtain ⟨δ,hδ,hd⟩ := small_step_pos hB hL hα hb
-  let H : ℝ := min δ ((ε/4)/(L+1))
-  have hH : 0 < H := lt_min hδ (div_pos (by positivity) (by linarith))
-  obtain ⟨n,hn⟩ := exists_nat_gt (max 1 ((T:ℝ)/H))
-  have hnR : 0 < (n:ℝ) := lt_trans (by norm_num) (lt_of_le_of_lt (le_max_left _ _) hn)
-  have hnN : 0 < n := by exact_mod_cast hnR
-  have hnNN : 0 < (n:ℝ≥0) := by exact_mod_cast hnN
-  let h : ℝ≥0 := T/(n:ℝ≥0)
-  have hh : 0 < h := div_pos hTp hnNN
-  have hhH : (h:ℝ) ≤ H := by
-    change (T:ℝ)/(n:ℝ) ≤ H
-    apply (div_le_iff₀ hnR).2
-    have hx := (div_lt_iff₀ hH).1 (lt_of_le_of_lt (le_max_right _ _) hn)
-    linarith
-  have hhδ : (h:ℝ) ≤ δ := hhH.trans (min_le_left _ _)
-  have hLtune : L*(h:ℝ) ≤ ε/4 := by
-    have h1 := (le_div_iff₀ (by linarith : 0 < L+1)).1 (hhH.trans (min_le_right _ _))
-    nlinarith [h.coe_nonneg]
-  have hαtune : α*(T:ℝ) ≤ ε/4 := by
-    have heq : α*((T:ℝ)+1) = ε/4 := by dsimp [α]; field_simp
-    nlinarith
-  have hend : (n:ℝ≥0)*h = T := by dsimp [h]; field_simp
-  have hendR : (n:ℝ)*(h:ℝ) = T := by exact_mod_cast hend
-  let c (i : ℕ) : ℝ := u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h)
-  have hgrid (i : ℕ) (hi : i<n) : (i:ℝ≥0)*h+h ≤ T := by
-    calc
-      _ = ((i+1:ℕ):ℝ≥0)*h := by push_cast; ring
-      _ ≤ (n:ℝ≥0)*h := by gcongr; omega
-      _ = _ := hend
-  have hc (i : ℕ) (hi : i<n) : |c i| ≤ L*(h:ℝ) := by
-    have hs := hu _ ⟨bot_le,hgrid i hi⟩ _ ⟨bot_le,(le_add_right le_rfl).trans (hgrid i hi)⟩
-    simpa [c,NNReal.coe_add,abs_of_nonneg h.coe_nonneg] using hs
-  have hp : 0 < P (stepsEvent B h c α b n) :=
-    stepsEvent_pos hB h c α b n (fun i hi => hd h hh hhδ (c i) (hc i hi) _)
-  have hsub : stepsEvent B h c α b n ≤ᵐ[P]
-      {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
-    filter_upwards [hB.cont,hB.eval_zero_ae_eq_zero] with sample hcont hzero hs
-    have hhbound := tube_of_steps (fun t => B t sample) u hcont h n hnN hα.le hL
-      (hzero.trans hu0.symm) (by simpa only [hend] using hu) hs
-    intro t ht
-    have hbt := hhbound t (by simpa only [hend] using ht)
-    have hcst : b+(n:ℝ)*α*(h:ℝ)+L*(h:ℝ) < ε := by
-      have heq : (n:ℝ)*α*(h:ℝ) = α*(T:ℝ) := by rw [← hendR]; ring
-      rw [heq]
-      dsimp [b]
-      linarith
-    exact hbt.trans_lt hcst
-  exact hp.trans_le (measure_mono_ae hsub)
-
-theorem uniformTube_pos {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T : ℝ≥0) (u : ℝ≥0 → ℝ)
-    (hu : ContinuousOn u (Icc 0 T)) (hu0 : u 0 = 0) {ε : ℝ} (hε : 0 < ε) :
-    0 < P {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
-  let v : ℝ → ℝ := fun x => u x.toNNReal
-  have hv : ContinuousOn v (Icc 0 (T:ℝ)) :=
-    hu.comp continuous_real_toNNReal.continuousOn
-      (fun x hx => ⟨bot_le,Real.toNNReal_le_iff_le_coe.2 hx.2⟩)
-  obtain ⟨p,hp⟩ := exists_polynomial_near_of_continuousOn 0 (T:ℝ) v hv (ε/8) (by positivity)
-  let w : ℝ≥0 → ℝ := fun t => p.eval (t:ℝ)-p.eval 0
-  have hw0 : w 0=0 := by simp [w]
-  have hp0 : |p.eval 0| < ε/8 := by
-    have hh := hp 0 ⟨le_rfl,T.coe_nonneg⟩
-    simpa [v,hu0] using hh
-  have hwu (t : ℝ≥0) (ht : t ∈ Icc 0 T) : |w t-u t| < ε/4 := by
-    have hpt : |p.eval (t:ℝ)-u t| < ε/8 := by
-      simpa [v] using hp (t:ℝ) ⟨t.coe_nonneg,ht.2⟩
-    calc
-      _ = |(p.eval (t:ℝ)-u t)-p.eval 0| := by dsimp [w]; congr 1; ring
-      _ ≤ |p.eval (t:ℝ)-u t|+|p.eval 0| := by
-        simpa only [sub_eq_add_neg,abs_neg] using abs_add_le (p.eval (t:ℝ)-u t) (-p.eval 0)
-      _ < ε/4 := by linarith
-  have hpc : ContDiff ℝ 1 (fun x : ℝ => p.eval x) := by
-    simpa using p.contDiff_aeval (𝕜:=ℝ) 1
-  obtain ⟨L,hL⟩ := hpc.contDiffOn.exists_lipschitzOnWith (by norm_num) (convex_Icc 0 (T:ℝ)) isCompact_Icc
-  have hw : ∀ s ∈ Icc 0 T, ∀ t ∈ Icc 0 T, |w s-w t| ≤ (L:ℝ)*|(s:ℝ)-(t:ℝ)| := by
-    intro s hs t ht
-    have hh := hL.dist_le_mul (s:ℝ) ⟨s.coe_nonneg,hs.2⟩ (t:ℝ) ⟨t.coe_nonneg,ht.2⟩
-    simpa [w,Real.dist_eq] using hh
-  have hpw := lipschitzTube_pos hB T w hw0 L.coe_nonneg hw (show 0 < ε/2 by positivity)
-  apply hpw.trans_le
-  apply measure_mono
-  intro sample hs t ht
-  calc
-    _ = |(B t sample-w t)+(w t-u t)| := by congr 1; ring
-    _ ≤ |B t sample-w t|+|w t-u t| := abs_add_le _ _
-    _ < ε := by linarith [hs t ht,hwu t ht]
-
-end SLEExactGauge.BrownianSupport
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Brownian Support estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianSupport
+open BrownianPredictable
+
+lemma preBrownian_past_square_le {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsPreBrownianReal B P) (hBm : ∀ t, Measurable (B t))
+    {s t : ℝ≥0} (hst : s ≤ t) {A : Set Ω} (hA : MeasurableSet[pastSigma B s] A) :
+    (∫ sample in A, (B s sample)^2 ∂P) ≤ ∫ sample in A, (B t sample)^2 ∂P := by
+  have hraw : pastSigma B s ≤ ‹MeasurableSpace Ω› :=
+    (Measurable.of_eval (fun time : Set.Iic s => hBm time)).comap_le
+  have hAm : MeasurableSet A := hraw _ hA
+  have hs := (hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two
+  have ht := (hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two
+  have hi : IndepFun (A.indicator (B s)) (fun sample => B t sample-B s sample) P := by
+    have hall := (hB.indepFun_shift s).symm
+    have hX : Measurable[pastSigma B s] (A.indicator (B s)) :=
+      (past_measurable_eval B le_rfl).indicator hA
+    have hf := hall.comp measurable_id (measurable_pi_apply (t-s))
+    simp only [Function.comp_def,id_eq,add_tsub_cancel_of_le hst] at hf
+    exact indep_of_indep_of_le_left hf hX.comap_le
+  have hz : (∫ sample in A, B s sample*(B t sample-B s sample) ∂P) = 0 := by
+    have he := hi.integral_fun_mul_eq_mul_integral
+      ((hBm s).indicator hAm).aestronglyMeasurable ((hBm t).sub (hBm s)).aestronglyMeasurable
+    have hind : (fun sample => A.indicator (B s) sample*(B t sample-B s sample)) =
+        A.indicator (fun sample => B s sample*(B t sample-B s sample)) := by
+      funext sample
+      by_cases ha : sample ∈ A <;> simp [ha]
+    rw [hind,integral_indicator hAm,integral_sub (hB.integrable_eval t) (hB.integrable_eval s),
+      hB.integral_eval t,hB.integral_eval s,sub_self,mul_zero] at he
+    exact he
+  have hcross : Integrable (fun sample => B s sample*(B t sample-B s sample)) P :=
+    hs.integrable_mul (ht.sub hs)
+  have his : Integrable (fun sample => (B s sample)^2) P := hs.integrable_sq
+  have hid : Integrable (fun sample => (B t sample-B s sample)^2) P := (ht.sub hs).integrable_sq
+  have he : (∫ sample in A, (B t sample)^2 ∂P) =
+      (∫ sample in A, (B s sample)^2 ∂P) + ∫ sample in A, (B t sample-B s sample)^2 ∂P := by
+    calc
+      _ = ∫ sample in A, ((B s sample)^2+2*(B s sample*(B t sample-B s sample)))+
+          (B t sample-B s sample)^2 ∂P := integral_congr_ae (Eventually.of_forall (fun _ => by ring))
+      _ = (∫ sample in A, (B s sample)^2+2*(B s sample*(B t sample-B s sample)) ∂P) +
+          ∫ sample in A, (B t sample-B s sample)^2 ∂P :=
+        integral_add ((his.restrict (s := A)).add ((hcross.restrict (s := A)).const_mul 2)) (hid.restrict (s := A))
+      _ = ((∫ sample in A, (B s sample)^2 ∂P) +
+          2*(∫ sample in A, B s sample*(B t sample-B s sample) ∂P)) +
+          ∫ sample in A, (B t sample-B s sample)^2 ∂P := by
+        congr 1
+        calc
+          _ = (∫ sample in A, (B s sample)^2 ∂P) +
+              ∫ sample in A, 2*(B s sample*(B t sample-B s sample)) ∂P :=
+            integral_add (his.restrict (s := A)) ((hcross.restrict (s := A)).const_mul 2)
+          _ = _ := by rw [integral_const_mul]
+      _ = _ := by rw [hz,mul_zero,add_zero]
+  rw [he]
+  exact le_add_of_nonneg_right (integral_nonneg (fun _ => sq_nonneg _))
+
+lemma preBrownian_square_submartingale {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsPreBrownianReal B P) (hBm : ∀ t, Measurable (B t))
+    (t : ℕ → ℝ≥0) (ht : Monotone t) :
+    let F : Filtration ℕ ‹MeasurableSpace Ω› :=
+      ⟨fun n => pastSigma B (t n),fun _ _ hij => pastSigma_mono B (ht hij),
+        fun n => (Measurable.of_eval (fun time : Set.Iic (t n) => hBm time)).comap_le⟩
+    Submartingale (fun n sample => (B (t n) sample)^2) F P := by
+  intro F
+  apply submartingale_of_setIntegral_le
+  · intro n
+    exact ((past_measurable_eval B le_rfl).pow_const 2).stronglyMeasurable
+  · intro n
+    exact (hB.isGaussianProcess.hasGaussianLaw_eval (t n)).memLp_two.integrable_sq
+  · intro i j hij A hA
+    exact preBrownian_past_square_le hB hBm (ht hij) hA
+
+lemma preBrownian_finite_max_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsPreBrownianReal B P) (hBm : ∀ t, Measurable (B t))
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (n : ℕ) {a : ℝ} (ha : 0 < a) :
+    P {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} ≤ ENNReal.ofReal ((t n:ℝ)/a^2) := by
+  let f : ℕ → Ω → ℝ := fun n sample => (B (t n) sample)^2
+  have hm := maximal_ineq (preBrownian_square_submartingale hB hBm t ht)
+    (show 0 ≤ f from fun _ _ => sq_nonneg _) (ε := (a^2).toNNReal) n
+  simp only [Real.coe_toNNReal _ (sq_nonneg a)] at hm
+  have he : {sample | (a^2) ≤ (Finset.range (n+1)).sup' Finset.nonempty_range_add_one (fun k => f k sample)} =
+      {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} := by
+    ext sample
+    simp only [mem_ofPred_eq,Finset.le_sup'_iff,Finset.mem_range,Nat.lt_succ_iff,f]
+    constructor
+    · rintro ⟨i,hi,hv⟩
+      exact ⟨i,hi,(sq_le_sq₀ ha.le (abs_nonneg _)).1 (by simpa using hv)⟩
+    · rintro ⟨i,hi,hv⟩
+      exact ⟨i,hi,by simpa using (sq_le_sq₀ ha.le (abs_nonneg _)).2 hv⟩
+  change ENNReal.ofReal (a^2) * P _ ≤ _ at hm
+  rw [he] at hm
+  have hi : (∫ sample in {sample | ∃ i ≤ n, a ≤ |B (t i) sample|}, (B (t n) sample)^2 ∂P) ≤ (t n:ℝ) := by
+    calc
+      _ ≤ ∫ sample, (B (t n) sample)^2 ∂P := integral_mono_measure Measure.restrict_le_self
+        (Eventually.of_forall (fun _ => sq_nonneg _))
+        (hB.isGaussianProcess.hasGaussianLaw_eval (t n)).memLp_two.integrable_sq
+      _ = _ := ((hB.hasLaw_eval (t n)).integral_comp
+        (show AEStronglyMeasurable (fun x : ℝ => x^2) _ from by fun_prop)).trans
+          (BrownianPredictable.gaussian_second_moment _)
+  have hh := hm.trans (ENNReal.ofReal_le_ofReal hi)
+  rw [ENNReal.ofReal_div_of_pos (sq_pos_of_pos ha)]
+  exact (ENNReal.le_div_iff_mul_le (Or.inl (ne_of_gt (ENNReal.ofReal_pos.2 (sq_pos_of_pos ha)))) (Or.inl ENNReal.ofReal_ne_top)).2 (by simpa [mul_comm] using hh)
+
+end SLEExactGauge.BrownianSupport
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+
+def restartIndex {Ω : Type*} (τ : Ω → ℝ≥0) (n : ℕ) (sample : Ω) : ℕ :=
+  Nat.ceil (τ sample*((n:ℝ≥0)+1))
+
+def restartGrid (n k : ℕ) : ℝ≥0 := (k:ℝ≥0)/((n:ℝ≥0)+1)
+
+def restartTime {Ω : Type*} (τ : Ω → ℝ≥0) (n : ℕ) (sample : Ω) : ℝ≥0 :=
+  restartGrid n (restartIndex τ n sample)
+
+lemma restartIndex_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {τ : Ω → ℝ≥0} (hτ : AEMeasurable τ P) (n : ℕ) : AEMeasurable (restartIndex τ n) P :=
+  Nat.measurable_ceil.comp_aemeasurable (hτ.mul_const _)
+
+lemma restartTime_bounds {Ω : Type*} (τ : Ω → ℝ≥0) (n : ℕ) (sample : Ω) :
+    τ sample ≤ restartTime τ n sample ∧
+      restartTime τ n sample ≤ τ sample+1/((n:ℝ≥0)+1) := by
+  have ha : 0 < (n:ℝ≥0)+1 := by positivity
+  constructor
+  · exact (le_div_iff₀ ha).2 (Nat.le_ceil _)
+  · change (Nat.ceil (τ sample*((n:ℝ≥0)+1)):ℝ≥0)/((n:ℝ≥0)+1) ≤ _
+    apply le_of_lt
+    calc
+      _ < (τ sample*((n:ℝ≥0)+1)+1)/((n:ℝ≥0)+1) :=
+        (div_lt_div_iff_of_pos_right ha).2 (Nat.ceil_lt_add_one (by positivity))
+      _ = _ := by rw [add_div,mul_div_cancel_right₀ _ ha.ne']
+
+lemma restartTime_tendsto {Ω : Type*} (τ : Ω → ℝ≥0) (sample : Ω) :
+    Tendsto (fun n => restartTime τ n sample) atTop (𝓝 (τ sample)) := by
+  have hh : Tendsto (fun n : ℕ => τ sample+1/((n:ℝ≥0)+1)) atTop (𝓝 (τ sample)) := by
+    simpa only [add_zero] using tendsto_const_nhds.add
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ≥0))
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hh
+    (fun n => (restartTime_bounds τ n sample).1) (fun n => (restartTime_bounds τ n sample).2)
+
+lemma restartIndex_past {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (τ : Ω → ℝ≥0) (A : Set Ω)
+    (hstop : ∀ t : ℝ≥0, MeasurableSet[pastSigma B t] (A ∩ {sample | τ sample ≤ t}))
+    (n k : ℕ) : MeasurableSet[pastSigma B (restartGrid n k)]
+      (A ∩ {sample | restartIndex τ n sample = k}) := by
+  have ha : 0 < (n:ℝ≥0)+1 := by positivity
+  by_cases hk : k=0
+  · subst k
+    have he : A ∩ {sample | restartIndex τ n sample = 0} = A ∩ {sample | τ sample ≤ 0} := by
+      ext sample
+      simp only [mem_inter_iff,mem_ofPred_eq,restartIndex,Nat.ceil_eq_zero]
+      constructor
+      · rintro ⟨hA,ht⟩
+        exact ⟨hA,(mul_le_mul_iff_left₀ ha).1 (by simpa using ht)⟩
+      · rintro ⟨hA,ht⟩
+        exact ⟨hA,mul_nonpos_of_nonpos_of_nonneg ht ha.le⟩
+    rw [he]
+    have hz : restartGrid n 0 = 0 := by simp [restartGrid]
+    rw [hz]
+    exact hstop 0
+  · have hle : restartGrid n (k-1) ≤ restartGrid n k := by
+      apply div_le_div_of_nonneg_right _ ha.le
+      exact_mod_cast Nat.sub_le k 1
+    have hprev : MeasurableSet[pastSigma B (restartGrid n k)]
+        (A ∩ {sample | τ sample ≤ restartGrid n (k-1)}) :=
+      (pastSigma_mono B hle) _ (hstop _)
+    have he : A ∩ {sample | restartIndex τ n sample = k} =
+        (A ∩ {sample | τ sample ≤ restartGrid n k}) \
+          (A ∩ {sample | τ sample ≤ restartGrid n (k-1)}) := by
+      ext sample
+      simp only [mem_inter_iff,mem_ofPred_eq,Set.mem_sdiff,restartIndex,Nat.ceil_eq_iff hk,
+        restartGrid,le_div_iff₀ ha,not_and,not_le]
+      constructor
+      · rintro ⟨hA,hl,hu⟩
+        exact ⟨⟨hA,hu⟩,fun _ => hl⟩
+      · rintro ⟨⟨hA,hu⟩,hl⟩
+        exact ⟨hA,hl hA,hu⟩
+    rw [he]
+    exact (hstop _).diff hprev
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianSupport
+open BrownianPredictable
+
+lemma brownian_finite_max_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsPreBrownianReal B P)
+    (t : ℕ → ℝ≥0) (ht : Monotone t) (n : ℕ) {a : ℝ} (ha : 0 < a) :
+    P {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} ≤ ENNReal.ofReal ((t n:ℝ)/a^2) := by
+  let C : ℝ≥0 → Ω → ℝ := fun t => (hB.aemeasurable t).mk (B t)
+  have hm (t : ℝ≥0) : Measurable (C t) := (hB.aemeasurable t).measurable_mk
+  have he (t : ℝ≥0) : B t =ᵐ[P] C t := (hB.aemeasurable t).ae_eq_mk
+  have hC : IsPreBrownianReal C P := hB.congr he
+  have hmeasure : P {sample | ∃ i ≤ n, a ≤ |B (t i) sample|} =
+      P {sample | ∃ i ≤ n, a ≤ |C (t i) sample|} := by
+    apply measure_congr
+    filter_upwards [ae_all_iff.2 (fun i : ℕ => he (t i))] with sample hh
+    apply propext
+    simp only [hh]
+  rw [hmeasure]
+  exact preBrownian_finite_max_bound hC hm t ht n ha
+
+lemma brownian_oscillation_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) :
+    P {sample | ∃ t ∈ Icc 0 T, a < |B t sample|} ≤ ENNReal.ofReal ((T:ℝ)/a^2) := by
+  let grid (n i : ℕ) : ℝ≥0 := min T (restartGrid n i)
+  let N (n : ℕ) := Nat.ceil (T*((n:ℝ≥0)+1))
+  let E (n : ℕ) : Set Ω := {sample | ∃ i ≤ N n, a ≤ |B (grid n i) sample|}
+  have hgrid (n : ℕ) : Monotone (grid n) := by
+    intro i j hij
+    apply min_le_min le_rfl
+    exact div_le_div_of_nonneg_right (by exact_mod_cast hij) (by positivity)
+  have hend (n : ℕ) : grid n (N n) = T := by
+    apply min_eq_left
+    exact (restartTime_bounds (fun _ : Unit => T) n ()).1
+  have hE (n : ℕ) : P (E n) ≤ ENNReal.ofReal ((T:ℝ)/a^2) := by
+    have hh := brownian_finite_max_bound hB.toIsPreBrownianReal (grid n) (hgrid n) (N n) ha
+    simpa only [hend] using hh
+  let F (k : ℕ) : Set Ω := ⋂ n ≥ k, E n
+  have hd : Monotone F := by
+    intro k l hkl sample hs
+    simp only [F,mem_iInter] at hs ⊢
+    exact fun n hn => hs n (hkl.trans hn)
+  have hsub : {sample | ∃ t ∈ Icc 0 T, a < |B t sample|} ≤ᵐ[P] ⋃ k, F k := by
+    filter_upwards [hB.cont] with sample hc hs
+    rcases hs with ⟨t,ht,haB⟩
+    have htend : Tendsto (fun n => min T (restartTime (fun _ : Unit => t) n ())) atTop (𝓝 t) := by
+      have hh := (tendsto_const_nhds (x := T)).min (restartTime_tendsto (fun _ : Unit => t) ())
+      simpa only [min_eq_right ht.2] using hh
+    have hev := ((hc.continuousAt.tendsto.comp htend).abs).eventually (lt_mem_nhds haB)
+    obtain ⟨k,hk⟩ := eventually_atTop.1 hev
+    apply mem_iUnion.2
+    refine ⟨k,?_⟩
+    simp only [F,mem_iInter]
+    intro n hn
+    refine ⟨restartIndex (fun _ : Unit => t) n (),?_,(hk n hn).le⟩
+    apply Nat.ceil_mono
+    exact mul_le_mul_of_nonneg_right ht.2 (by positivity)
+  calc
+    _ ≤ P (⋃ k, F k) := measure_mono_ae hsub
+    _ = ⨆ k, P (F k) := hd.directed_le.measure_iUnion
+    _ ≤ _ := iSup_le (fun k => (measure_mono (by
+      intro sample hs
+      exact mem_iInter.1 (mem_iInter.1 hs k) le_rfl)).trans (hE k))
+
+end SLEExactGauge.BrownianSupport
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianSupport
+
+private def densityFloor : ℝ := (Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-(1/2:ℝ))
+
+private lemma densityFloor_pos : 0 < densityFloor := by
+  dsimp [densityFloor]
+  positivity
+
+private lemma gaussian_pdf_lower {v : ℝ≥0} (hv : 0 < v) {x : ℝ} (hx : |x| ≤ Real.sqrt v) :
+    densityFloor / Real.sqrt v ≤ gaussianPDFReal 0 v x := by
+  have hvR : 0 < (v:ℝ) := hv
+  have hs : 0 < Real.sqrt (v:ℝ) := Real.sqrt_pos.2 hvR
+  have hx2 : x^2 ≤ (v:ℝ) := by
+    have hh := (sq_le_sq₀ (abs_nonneg x) hs.le).2 hx
+    simpa only [sq_abs,Real.sq_sqrt hvR.le] using hh
+  have he : -(1/2:ℝ) ≤ -x^2/(2*(v:ℝ)) := by
+    apply (le_div_iff₀ (by positivity)).2
+    nlinarith
+  calc
+    _ = (Real.sqrt (2*Real.pi*(v:ℝ)))⁻¹ * Real.exp (-(1/2:ℝ)) := by
+      rw [Real.sqrt_mul (by positivity)]
+      dsimp [densityFloor]
+      field_simp
+    _ ≤ _ := by
+      change _ ≤ (Real.sqrt (2*Real.pi*(v:ℝ)))⁻¹ * Real.exp (-(x-0)^2/(2*(v:ℝ)))
+      simp only [sub_zero]
+      exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.2 he) (by positivity)
+
+lemma gaussian_short_interval_lower {v : ℝ≥0} (hv : 0 < v) {c α L : ℝ}
+    (hα : 0 < α) (hc : |c| ≤ L*(v:ℝ))
+    (hsmall : (L+α)*(v:ℝ) ≤ Real.sqrt v) :
+    ENNReal.ofReal (2*densityFloor*α*Real.sqrt v) ≤
+      gaussianReal 0 v {x : ℝ | |x-c| < α*(v:ℝ)} := by
+  have hvR : 0 < (v:ℝ) := hv
+  have hs : 0 < Real.sqrt (v:ℝ) := Real.sqrt_pos.2 hvR
+  let A : Set ℝ := Ioo (c-α*(v:ℝ)) (c+α*(v:ℝ))
+  have hA : {x : ℝ | |x-c| < α*(v:ℝ)} = A := by
+    ext x
+    simp only [mem_ofPred_eq,A,mem_Ioo,abs_lt]
+    constructor <;> intro hh <;> constructor <;> linarith [hh.1,hh.2]
+  rw [hA,gaussianReal_apply_eq_integral 0 hv.ne']
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ = ∫ x in A, densityFloor/Real.sqrt v := by
+      rw [setIntegral_const]
+      simp only [smul_eq_mul,A,Real.volume_real_Ioo]
+      rw [max_eq_left (by nlinarith [mul_pos hα hvR] : 0 ≤ c+α*(v:ℝ)-(c-α*(v:ℝ)))]
+      field_simp
+      rw [Real.sq_sqrt hvR.le]
+      ring
+    _ ≤ _ := by
+      apply setIntegral_mono_on (integrable_const _) (integrable_gaussianPDFReal 0 v).integrableOn measurableSet_Ioo
+      intro x hx
+      apply gaussian_pdf_lower hv
+      have hxc : |x-c| < α*(v:ℝ) := by
+        have hh : x ∈ {x : ℝ | |x-c| < α*(v:ℝ)} := hA.symm ▸ hx
+        exact hh
+      calc
+        |x| ≤ |x-c|+|c| := by simpa using abs_add_le (x-c) c
+        _ ≤ (L+α)*(v:ℝ) := by nlinarith
+        _ ≤ _ := hsmall
+
+lemma short_endpoint_dominates {L α b : ℝ} (hL : 0 ≤ L) (hα : 0 < α) (hb : 0 < b) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ (v : ℝ≥0), 0 < v → (v:ℝ) ≤ δ → ∀ c : ℝ,
+      |c| ≤ L*(v:ℝ) → ENNReal.ofReal ((v:ℝ)/b^2) <
+        gaussianReal 0 v {x : ℝ | |x-c| < α*(v:ℝ)} := by
+  let r : ℝ := min (1/(L+α+1)) (densityFloor*α*b^2)
+  have hr : 0 < r := lt_min (by positivity) (by positivity [densityFloor_pos])
+  refine ⟨r^2,by positivity,fun v hv hvr c hc => ?_⟩
+  have hvR : 0 < (v:ℝ) := hv
+  have hs : 0 < Real.sqrt (v:ℝ) := Real.sqrt_pos.2 hvR
+  have hsr : Real.sqrt (v:ℝ) ≤ r := (Real.sqrt_le_iff).2 ⟨hr.le,hvr⟩
+  have hsmall : (L+α)*(v:ℝ) ≤ Real.sqrt v := by
+    have hr1 : r ≤ 1/(L+α+1) := min_le_left _ _
+    have hh : (L+α+1)*Real.sqrt (v:ℝ) ≤ 1 := by
+      have hc' := (le_div_iff₀ (by positivity : 0 < L+α+1)).1 (hsr.trans hr1)
+      nlinarith
+    nlinarith [Real.sq_sqrt hvR.le]
+  apply lt_of_lt_of_le _ (gaussian_short_interval_lower hv hα hc hsmall)
+  apply (ENNReal.ofReal_lt_ofReal_iff (by positivity [densityFloor_pos])).2
+  apply (div_lt_iff₀ (sq_pos_of_pos hb)).2
+  have hr2 : r ≤ densityFloor*α*b^2 := min_le_right _ _
+  have hh := hsr.trans hr2
+  have hmul := mul_le_mul_of_nonneg_left hh hs.le
+  nlinarith [Real.sq_sqrt hvR.le,densityFloor_pos]
+
+end SLEExactGauge.BrownianSupport
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BrownianSupport
+open BrownianPredictable
+
+def probeTime (T : ℝ≥0) (q : I) : ℝ≥0 := T * ⟨q,q.property.1⟩
+
+lemma probeTime_le (T : ℝ≥0) (q : I) : probeTime T q ≤ T := by
+  change (T:ℝ)*(q:ℝ) ≤ T
+  simpa using mul_le_mul_of_nonneg_left q.property.2 T.coe_nonneg
+
+lemma continuous_probeTime (T : ℝ≥0) : Continuous (probeTime T) := by
+  apply Continuous.const_mul
+  exact continuous_subtype_val.subtype_mk (fun q => q.property.1)
+
+def stepSet (h : ℝ≥0) (c α b : ℝ) : Set (ℝ≥0 → ℝ) :=
+  {f | |f h-c| < α*(h:ℝ) ∧ ∀ n : ℕ, |f (probeTime h (denseSeq I n))| ≤ b}
+
+lemma measurableSet_stepSet (h : ℝ≥0) (c α b : ℝ) : MeasurableSet (stepSet h c α b) := by
+  change MeasurableSet ({f : ℝ≥0 → ℝ | |f h-c| < α*(h:ℝ)} ∩
+    {f : ℝ≥0 → ℝ | ∀ n : ℕ, |f (probeTime h (denseSeq I n))| ≤ b})
+  apply MeasurableSet.inter
+  · exact measurableSet_lt (((measurable_pi_apply h : Measurable (fun f : ℝ≥0 → ℝ => f h))).sub_const c).abs measurable_const
+  · simp only [ofPred_forall]
+    exact MeasurableSet.iInter fun n => measurableSet_le (measurable_pi_apply _).abs measurable_const
+
+lemma probe_bound_all {f : ℝ≥0 → ℝ} (hf : Continuous f) (T : ℝ≥0) {b : ℝ}
+    (hb : ∀ n : ℕ, |f (probeTime T (denseSeq I n))| ≤ b) :
+    ∀ t ≤ T, |f t| ≤ b := by
+  have hall (q : I) : |f (probeTime T q)| ≤ b :=
+    (denseRange_denseSeq I).induction_on q
+      (isClosed_le ((hf.comp (continuous_probeTime T)).abs) continuous_const) hb
+  intro t ht
+  by_cases hT : T=0
+  · have ht0 : t=0 := le_antisymm (ht.trans hT.le) (bot_le)
+    have hz : probeTime T (0:I) = 0 := by
+      apply NNReal.coe_injective
+      change (T:ℝ)*(0:ℝ) = (0:ℝ)
+      exact mul_zero _
+    simpa only [ht0,hz] using hall (0:I)
+  · have hTp : 0 < (T:ℝ) := by exact_mod_cast lt_of_le_of_ne (bot_le : 0 ≤ T) (Ne.symm hT)
+    let q : I := ⟨(t:ℝ)/(T:ℝ),⟨div_nonneg t.coe_nonneg T.coe_nonneg,(div_le_one hTp).2 ht⟩⟩
+    have he : probeTime T q = t := by
+      apply NNReal.coe_injective
+      change (T:ℝ)*((t:ℝ)/(T:ℝ)) = t
+      field_simp
+    simpa only [he] using hall q
+
+def stepEvent {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (t h : ℝ≥0) (c α b : ℝ) : Set Ω :=
+  (fun sample s => B (t+s) sample-B t sample) ⁻¹' stepSet h c α b
+
+lemma stepEvent_past {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (t h : ℝ≥0) (c α b : ℝ) : MeasurableSet[pastSigma B (t+h)] (stepEvent B t h c α b) := by
+  let : MeasurableSpace Ω := pastSigma B (t+h)
+  change MeasurableSet[pastSigma B (t+h)]
+    ({sample | |B (t+h) sample-B t sample-c| < α*(h:ℝ)} ∩
+     {sample | ∀ n : ℕ, |B (t+probeTime h (denseSeq I n)) sample-B t sample| ≤ b})
+  apply MeasurableSet.inter
+  · apply measurableSet_lt _ measurable_const
+    exact (((past_measurable_eval B (s:=t+h) (t:=t+h) le_rfl).sub
+      (past_measurable_eval B (s:=t) (t:=t+h) (le_add_right le_rfl))).sub_const c).abs
+  · simp only [ofPred_forall]
+    apply MeasurableSet.iInter
+    intro n
+    apply measurableSet_le _ measurable_const
+    exact ((past_measurable_eval B (s:=t+probeTime h _) (t:=t+h) (by gcongr; exact probeTime_le h _)).sub
+      (past_measurable_eval B (s:=t) (t:=t+h) (le_add_right le_rfl))).abs
+
+lemma small_step_pos {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) {L α b : ℝ} (hL : 0 ≤ L) (hα : 0 < α) (hb : 0 < b) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ (h : ℝ≥0), 0 < h → (h:ℝ) ≤ δ → ∀ c : ℝ,
+      |c| ≤ L*(h:ℝ) → ∀ t : ℝ≥0, 0 < P (stepEvent B t h c α b) := by
+  obtain ⟨δ,hδ,hdom⟩ := short_endpoint_dominates hL hα hb
+  refine ⟨δ,hδ,fun h hh hhδ c hc t => ?_⟩
+  let C : ℝ≥0 → Ω → ℝ := fun s sample => B (t+s) sample-B t sample
+  have hC : IsBrownianReal C P := hB.shift t
+  let E : Set Ω := {sample | |C h sample-c| < α*(h:ℝ)}
+  let Bad : Set Ω := {sample | ∃ s ∈ Icc 0 h, b < |C s sample|}
+  have hE : P E = gaussianReal 0 h {x : ℝ | |x-c| < α*(h:ℝ)} :=
+    (hC.toIsPreBrownianReal.hasLaw_eval h).measure_eq
+      (p := fun x : ℝ => |x-c| < α*(h:ℝ))
+      (measurableSet_lt (measurable_id.sub_const c).abs measurable_const)
+  have hbad : P Bad < P E := by
+    rw [hE]
+    exact (brownian_oscillation_bound hC h hb).trans_lt (hdom h hh hhδ c hc)
+  have hsub : E ⊆ stepEvent B t h c α b ∪ Bad := by
+    intro sample hs
+    by_cases hbad : sample ∈ Bad
+    · exact Or.inr hbad
+    · apply Or.inl
+      refine ⟨hs,fun n => ?_⟩
+      apply le_of_not_gt
+      intro hn
+      exact hbad ⟨probeTime h (denseSeq I n),⟨bot_le,probeTime_le h _⟩,hn⟩
+  apply pos_iff_ne_zero.2
+  intro hz
+  have hm := (measure_mono (μ:=P) hsub).trans (measure_union_le _ _)
+  rw [hz,zero_add] at hm
+  exact (not_le_of_gt hbad) hm
+
+lemma past_inter_step {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (t h : ℝ≥0) (c α b : ℝ) (A : Set Ω) (hA : MeasurableSet[pastSigma B t] A) :
+    P (A ∩ stepEvent B t h c α b) = P A * P (stepEvent B t h c α b) := by
+  have hi := (hB.toIsPreBrownianReal.indepFun_shift t).symm
+  apply hi.meas_inter hA
+  exact (measurableSet_stepSet h c α b).preimage (comap_measurable _)
+
+def stepsEvent {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (h : ℝ≥0) (c : ℕ → ℝ)
+    (α b : ℝ) (n : ℕ) : Set Ω :=
+  {sample | ∀ i < n, sample ∈ stepEvent B ((i:ℝ≥0)*h) h (c i) α b}
+
+lemma stepsEvent_zero {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (h : ℝ≥0) (c : ℕ → ℝ)
+    (α b : ℝ) : stepsEvent B h c α b 0 = univ := by
+  ext sample
+  simp [stepsEvent]
+
+lemma stepsEvent_succ {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (h : ℝ≥0) (c : ℕ → ℝ)
+    (α b : ℝ) (n : ℕ) : stepsEvent B h c α b (n+1) =
+      stepsEvent B h c α b n ∩ stepEvent B ((n:ℝ≥0)*h) h (c n) α b := by
+  ext sample
+  simp only [stepsEvent,mem_ofPred_eq,mem_inter_iff,Nat.lt_succ_iff]
+  exact ⟨fun hh => ⟨fun i hi => hh i (Nat.le_of_lt hi),hh n le_rfl⟩,
+    fun hh i hi => (lt_or_eq_of_le hi).elim (hh.1 i) (fun he => he ▸ hh.2)⟩
+
+lemma stepsEvent_past {Ω : Type*} [MeasurableSpace Ω] (B : ℝ≥0 → Ω → ℝ)
+    (h : ℝ≥0) (c : ℕ → ℝ) (α b : ℝ) (n : ℕ) :
+    MeasurableSet[pastSigma B ((n:ℝ≥0)*h)] (stepsEvent B h c α b n) := by
+  induction n with
+  | zero => rw [stepsEvent_zero]; exact .univ
+  | succ n ih =>
+    rw [stepsEvent_succ]
+    have he : ((n+1:ℕ):ℝ≥0)*h = (n:ℝ≥0)*h+h := by push_cast; ring
+    rw [he]
+    exact ((pastSigma_mono B (s := (n:ℝ≥0)*h) (t := (n:ℝ≥0)*h+h) (le_add_right le_rfl)) _ ih).inter
+      (stepEvent_past B _ _ _ _ _)
+
+lemma stepsEvent_pos {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (h : ℝ≥0) (c : ℕ → ℝ) (α b : ℝ) (n : ℕ)
+    (hpos : ∀ i < n, 0 < P (stepEvent B ((i:ℝ≥0)*h) h (c i) α b)) :
+    0 < P (stepsEvent B h c α b n) := by
+  induction n with
+  | zero => rw [stepsEvent_zero,measure_univ]; exact zero_lt_one
+  | succ n ih =>
+    rw [stepsEvent_succ,past_inter_step hB _ _ _ _ _ _ (stepsEvent_past B h c α b n)]
+    exact ENNReal.mul_pos (ih (fun i hi => hpos i (Nat.lt_succ_of_lt hi))).ne' (hpos n (Nat.lt_succ_self n)).ne'
+
+lemma exists_grid_interval (h : ℝ≥0) {n : ℕ} (hn : 0 < n) {t : ℝ≥0}
+    (ht : t ≤ (n:ℝ≥0)*h) :
+    ∃ i < n, (i:ℝ≥0)*h ≤ t ∧ t ≤ ((i+1:ℕ):ℝ≥0)*h := by
+  induction n with
+  | zero => omega
+  | succ n ih =>
+    by_cases hn0 : n=0
+    · subst n
+      exact ⟨0,by omega,by simp,by simpa using ht⟩
+    · by_cases hh : t ≤ (n:ℝ≥0)*h
+      · obtain ⟨i,hi,hi1,hi2⟩ := ih (Nat.pos_of_ne_zero hn0) hh
+        exact ⟨i,Nat.lt_succ_of_lt hi,hi1,hi2⟩
+      · exact ⟨n,Nat.lt_succ_self n,(lt_of_not_ge hh).le,ht⟩
+
+lemma endpoint_error_bound (f u : ℝ≥0 → ℝ) (h : ℝ≥0) (α : ℝ) (n : ℕ)
+    (hz : f 0 = u 0)
+    (herr : ∀ i < n, |f ((i:ℝ≥0)*h+h)-f ((i:ℝ≥0)*h)-
+      (u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h))| ≤ α*(h:ℝ)) :
+    ∀ i ≤ n, |f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h)| ≤ (i:ℝ)*α*(h:ℝ) := by
+  intro i hi
+  induction i with
+  | zero => simp [hz]
+  | succ i ih =>
+    have hei : ((i+1:ℕ):ℝ≥0)*h = (i:ℝ≥0)*h+h := by push_cast; ring
+    rw [hei]
+    have hh := herr i (by omega)
+    have hh2 := ih (by omega)
+    calc
+      _ = |(f ((i:ℝ≥0)*h+h)-f ((i:ℝ≥0)*h)-
+            (u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h)))+
+            (f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h))| := by congr 1; ring
+      _ ≤ _ := abs_add_le _ _
+      _ ≤ _ := add_le_add hh hh2
+      _ = _ := by push_cast; ring
+
+lemma tube_of_steps (f u : ℝ≥0 → ℝ) (hf : Continuous f) (h : ℝ≥0) (n : ℕ)
+    (hn : 0 < n) {α b L : ℝ} (hα : 0 ≤ α) (hL : 0 ≤ L) (hz : f 0=u 0)
+    (hu : ∀ s ∈ Icc 0 ((n:ℝ≥0)*h), ∀ t ∈ Icc 0 ((n:ℝ≥0)*h),
+      |u s-u t| ≤ L*|((s:ℝ)-(t:ℝ))|)
+    (hs : ∀ i < n, (fun s => f ((i:ℝ≥0)*h+s)-f ((i:ℝ≥0)*h)) ∈
+      stepSet h (u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h)) α b) :
+    ∀ t ∈ Icc 0 ((n:ℝ≥0)*h), |f t-u t| ≤ b+(n:ℝ)*α*(h:ℝ)+L*(h:ℝ) := by
+  have he := endpoint_error_bound f u h α n hz (fun i hi => (hs i hi).1.le)
+  intro t ht
+  obtain ⟨i,hi,hit,hti⟩ := exists_grid_interval h hn ht.2
+  have hiadd : ((i+1:ℕ):ℝ≥0)*h = (i:ℝ≥0)*h+h := by push_cast; ring
+  have hsub : t-(i:ℝ≥0)*h ≤ h := by
+    rw [tsub_le_iff_right]
+    rw [hiadd] at hti
+    simpa only [add_comm] using hti
+  have hosc : |f t-f ((i:ℝ≥0)*h)| ≤ b := by
+    have hc : Continuous (fun s => f ((i:ℝ≥0)*h+s)-f ((i:ℝ≥0)*h)) :=
+      (hf.comp (continuous_const.add continuous_id)).sub continuous_const
+    have hh := probe_bound_all hc h (hs i hi).2 (t-(i:ℝ≥0)*h) hsub
+    simpa only [add_tsub_cancel_of_le hit] using hh
+  have hig : (i:ℝ≥0)*h ≤ (n:ℝ≥0)*h := by gcongr
+  have hul : |u ((i:ℝ≥0)*h)-u t| ≤ L*(h:ℝ) := by
+    apply (hu _ ⟨bot_le,hig⟩ _ ht).trans
+    have habs : |((i:ℝ≥0)*h:ℝ)-(t:ℝ)| ≤ (h:ℝ) := by
+      simp only [NNReal.coe_natCast]
+      rw [abs_of_nonpos (sub_nonpos.2 (show (i:ℝ)*(h:ℝ) ≤ (t:ℝ) by exact_mod_cast hit))]
+      have htR : (t:ℝ) ≤ (i:ℝ)*(h:ℝ)+(h:ℝ) := by exact_mod_cast (hiadd ▸ hti)
+      linarith
+    exact mul_le_mul_of_nonneg_left habs hL
+  calc
+    _ = |(f t-f ((i:ℝ≥0)*h))+(f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h))+
+          (u ((i:ℝ≥0)*h)-u t)| := by congr 1; ring
+    _ ≤ (|f t-f ((i:ℝ≥0)*h)|+|f ((i:ℝ≥0)*h)-u ((i:ℝ≥0)*h)|)+
+          |u ((i:ℝ≥0)*h)-u t| :=
+        (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)
+    _ ≤ b+(i:ℝ)*α*(h:ℝ)+L*(h:ℝ) := add_le_add (add_le_add hosc (he i hi.le)) hul
+    _ ≤ _ := by gcongr
+
+end SLEExactGauge.BrownianSupport
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianSupport
+
+lemma lipschitzTube_pos {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T : ℝ≥0) (u : ℝ≥0 → ℝ)
+    (hu0 : u 0=0) {L : ℝ} (hL : 0 ≤ L)
+    (hu : ∀ s ∈ Icc 0 T, ∀ t ∈ Icc 0 T, |u s-u t| ≤ L*|(s:ℝ)-(t:ℝ)|)
+    {ε : ℝ} (hε : 0 < ε) :
+    0 < P {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
+  by_cases hT : T=0
+  · have hsub : (univ : Set Ω) ≤ᵐ[P] {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
+      filter_upwards [hB.eval_zero_ae_eq_zero] with sample hs _
+      intro t ht
+      have ht0 : t=0 := le_antisymm (ht.2.trans hT.le) bot_le
+      simpa [ht0,hs,hu0] using hε
+    have hm := measure_mono_ae hsub
+    rw [measure_univ] at hm
+    exact zero_lt_one.trans_le hm
+  have hTp : 0 < T := lt_of_le_of_ne bot_le (Ne.symm hT)
+  have hTR : 0 < (T:ℝ) := hTp
+  let α : ℝ := ε/(4*((T:ℝ)+1))
+  let b : ℝ := ε/4
+  have hα : 0 < α := by dsimp [α]; positivity
+  have hb : 0 < b := by dsimp [b]; positivity
+  obtain ⟨δ,hδ,hd⟩ := small_step_pos hB hL hα hb
+  let H : ℝ := min δ ((ε/4)/(L+1))
+  have hH : 0 < H := lt_min hδ (div_pos (by positivity) (by linarith))
+  obtain ⟨n,hn⟩ := exists_nat_gt (max 1 ((T:ℝ)/H))
+  have hnR : 0 < (n:ℝ) := lt_trans (by norm_num) (lt_of_le_of_lt (le_max_left _ _) hn)
+  have hnN : 0 < n := by exact_mod_cast hnR
+  have hnNN : 0 < (n:ℝ≥0) := by exact_mod_cast hnN
+  let h : ℝ≥0 := T/(n:ℝ≥0)
+  have hh : 0 < h := div_pos hTp hnNN
+  have hhH : (h:ℝ) ≤ H := by
+    change (T:ℝ)/(n:ℝ) ≤ H
+    apply (div_le_iff₀ hnR).2
+    have hx := (div_lt_iff₀ hH).1 (lt_of_le_of_lt (le_max_right _ _) hn)
+    linarith
+  have hhδ : (h:ℝ) ≤ δ := hhH.trans (min_le_left _ _)
+  have hLtune : L*(h:ℝ) ≤ ε/4 := by
+    have h1 := (le_div_iff₀ (by linarith : 0 < L+1)).1 (hhH.trans (min_le_right _ _))
+    nlinarith [h.coe_nonneg]
+  have hαtune : α*(T:ℝ) ≤ ε/4 := by
+    have heq : α*((T:ℝ)+1) = ε/4 := by dsimp [α]; field_simp
+    nlinarith
+  have hend : (n:ℝ≥0)*h = T := by dsimp [h]; field_simp
+  have hendR : (n:ℝ)*(h:ℝ) = T := by exact_mod_cast hend
+  let c (i : ℕ) : ℝ := u ((i:ℝ≥0)*h+h)-u ((i:ℝ≥0)*h)
+  have hgrid (i : ℕ) (hi : i<n) : (i:ℝ≥0)*h+h ≤ T := by
+    calc
+      _ = ((i+1:ℕ):ℝ≥0)*h := by push_cast; ring
+      _ ≤ (n:ℝ≥0)*h := by gcongr; omega
+      _ = _ := hend
+  have hc (i : ℕ) (hi : i<n) : |c i| ≤ L*(h:ℝ) := by
+    have hs := hu _ ⟨bot_le,hgrid i hi⟩ _ ⟨bot_le,(le_add_right le_rfl).trans (hgrid i hi)⟩
+    simpa [c,NNReal.coe_add,abs_of_nonneg h.coe_nonneg] using hs
+  have hp : 0 < P (stepsEvent B h c α b n) :=
+    stepsEvent_pos hB h c α b n (fun i hi => hd h hh hhδ (c i) (hc i hi) _)
+  have hsub : stepsEvent B h c α b n ≤ᵐ[P]
+      {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
+    filter_upwards [hB.cont,hB.eval_zero_ae_eq_zero] with sample hcont hzero hs
+    have hhbound := tube_of_steps (fun t => B t sample) u hcont h n hnN hα.le hL
+      (hzero.trans hu0.symm) (by simpa only [hend] using hu) hs
+    intro t ht
+    have hbt := hhbound t (by simpa only [hend] using ht)
+    have hcst : b+(n:ℝ)*α*(h:ℝ)+L*(h:ℝ) < ε := by
+      have heq : (n:ℝ)*α*(h:ℝ) = α*(T:ℝ) := by rw [← hendR]; ring
+      rw [heq]
+      dsimp [b]
+      linarith
+    exact hbt.trans_lt hcst
+  exact hp.trans_le (measure_mono_ae hsub)
+
+theorem uniformTube_pos {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T : ℝ≥0) (u : ℝ≥0 → ℝ)
+    (hu : ContinuousOn u (Icc 0 T)) (hu0 : u 0 = 0) {ε : ℝ} (hε : 0 < ε) :
+    0 < P {sample | ∀ t ∈ Icc 0 T, |B t sample-u t| < ε} := by
+  let v : ℝ → ℝ := fun x => u x.toNNReal
+  have hv : ContinuousOn v (Icc 0 (T:ℝ)) :=
+    hu.comp continuous_real_toNNReal.continuousOn
+      (fun x hx => ⟨bot_le,Real.toNNReal_le_iff_le_coe.2 hx.2⟩)
+  obtain ⟨p,hp⟩ := exists_polynomial_near_of_continuousOn 0 (T:ℝ) v hv (ε/8) (by positivity)
+  let w : ℝ≥0 → ℝ := fun t => p.eval (t:ℝ)-p.eval 0
+  have hw0 : w 0=0 := by simp [w]
+  have hp0 : |p.eval 0| < ε/8 := by
+    have hh := hp 0 ⟨le_rfl,T.coe_nonneg⟩
+    simpa [v,hu0] using hh
+  have hwu (t : ℝ≥0) (ht : t ∈ Icc 0 T) : |w t-u t| < ε/4 := by
+    have hpt : |p.eval (t:ℝ)-u t| < ε/8 := by
+      simpa [v] using hp (t:ℝ) ⟨t.coe_nonneg,ht.2⟩
+    calc
+      _ = |(p.eval (t:ℝ)-u t)-p.eval 0| := by dsimp [w]; congr 1; ring
+      _ ≤ |p.eval (t:ℝ)-u t|+|p.eval 0| := by
+        simpa only [sub_eq_add_neg,abs_neg] using abs_add_le (p.eval (t:ℝ)-u t) (-p.eval 0)
+      _ < ε/4 := by linarith
+  have hpc : ContDiff ℝ 1 (fun x : ℝ => p.eval x) := by
+    simpa using p.contDiff_aeval (𝕜:=ℝ) 1
+  obtain ⟨L,hL⟩ := hpc.contDiffOn.exists_lipschitzOnWith (by norm_num) (convex_Icc 0 (T:ℝ)) isCompact_Icc
+  have hw : ∀ s ∈ Icc 0 T, ∀ t ∈ Icc 0 T, |w s-w t| ≤ (L:ℝ)*|(s:ℝ)-(t:ℝ)| := by
+    intro s hs t ht
+    have hh := hL.dist_le_mul (s:ℝ) ⟨s.coe_nonneg,hs.2⟩ (t:ℝ) ⟨t.coe_nonneg,ht.2⟩
+    simpa [w,Real.dist_eq] using hh
+  have hpw := lipschitzTube_pos hB T w hw0 L.coe_nonneg hw (show 0 < ε/2 by positivity)
+  apply hpw.trans_le
+  apply measure_mono
+  intro sample hs t ht
+  calc
+    _ = |(B t sample-w t)+(w t-u t)| := by congr 1; ring
+    _ ≤ |B t sample-w t|+|w t-u t| := abs_add_le _ _
+    _ < ε := by linarith [hs t ht,hwu t ht]
+
+end SLEExactGauge.BrownianSupport
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/Charge.lean b/lean/OAI/Probability/SLELocalization/Charge.lean
index 4998d50215421945758f91ac4102d8e289bed067..70d48daee61b315bf846ad30038cce2c841563ed 100644
--- a/lean/OAI/Probability/SLELocalization/Charge.lean
+++ b/lean/OAI/Probability/SLELocalization/Charge.lean
@@ -1,874 +1,874 @@
-import Mathlib
-import OAI.Probability.SLELocalization.AngularClock
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Charge estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-def sausage (γ : ℝ → ℂ) (v : ℝ) : Set ℂ :=
-  {w | ∃ t : ℝ, 0 ≤ t ∧ dist w (γ t) < v}
-
-def sausageCharge (κ : ℝ) (γ : ℝ → ℂ) (v : ℝ) (A : Set ℂ) : ℝ≥0∞ :=
-  ENNReal.ofReal (v^(-codimension κ)) * volume (A ∩ sausage γ v)
-
-lemma sausage_open (γ : ℝ → ℂ) (v : ℝ) : IsOpen (sausage γ v) := by
-  have he : sausage γ v = ⋃ t : ℝ, ⋃ (_ : 0 ≤ t), ball (γ t) v := by
-    ext w
-    simp [sausage,mem_ball]
-  rw [he]
-  exact isOpen_iUnion (fun t => isOpen_iUnion (fun _ => isOpen_ball))
-
-lemma sausage_eq_dense {γ : ℝ → ℂ} (hγ : ContinuousOn γ (Ici 0)) (v : ℝ) :
-    sausage γ v = ⋃ n : ℕ, ball (γ (denseSeq ℝ≥0 n)) v := by
-  ext w
-  simp only [sausage,mem_ofPred_eq,mem_iUnion,mem_ball]
-  constructor
-  · rintro ⟨t,ht,hvisit⟩
-    by_contra hn
-    push Not at hn
-    have hc : Continuous (fun q : ℝ≥0 => dist w (γ q)) :=
-      continuous_const.dist (hγ.comp_continuous continuous_subtype_val (fun q => q.coe_nonneg))
-    have hh : v ≤ dist w (γ t) := (denseRange_denseSeq ℝ≥0).induction_on (p := fun q : ℝ≥0 => v ≤ dist w (γ q)) (⟨t,ht⟩ : ℝ≥0)
-      (isClosed_le continuous_const hc) hn
-    exact (not_lt_of_ge hh) hvisit
-  · rintro ⟨n,hn⟩
-    exact ⟨denseSeq ℝ≥0 n,(denseSeq ℝ≥0 n).coe_nonneg,hn⟩
-
-lemma measurable_sausage_representative {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (γ : Ω → ℝ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hc : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0)) (v : ℝ) :
-    ∃ E : Set (Ω × ℂ), MeasurableSet E ∧
-      ∀ᵐ sample ∂P, ∀ w : ℂ, (sample,w) ∈ E ↔ w ∈ sausage (γ sample) v := by
-  let Y (n : ℕ) := (hγ (denseSeq ℝ≥0 n) (denseSeq ℝ≥0 n).coe_nonneg).mk (fun sample => γ sample (denseSeq ℝ≥0 n))
-  have hY (n : ℕ) : Measurable (Y n) := (hγ _ (denseSeq ℝ≥0 n).coe_nonneg).measurable_mk
-  have he : ∀ᵐ sample ∂P, ∀ n : ℕ, γ sample (denseSeq ℝ≥0 n) = Y n sample :=
-    ae_all_iff.2 (fun n => (hγ _ (denseSeq ℝ≥0 n).coe_nonneg).ae_eq_mk)
-  let E : Set (Ω × ℂ) := ⋃ n : ℕ, {x | dist x.2 (Y n x.1) < v}
-  refine ⟨E,MeasurableSet.iUnion (fun n => measurableSet_lt
-    (measurable_snd.dist ((hY n).comp measurable_fst)) measurable_const),?_⟩
-  filter_upwards [hc,he] with sample hcont heq
-  intro w
-  rw [sausage_eq_dense hcont v]
-  simp only [E,mem_iUnion,mem_ofPred_eq,mem_ball,heq]
-
-lemma expected_sausage_area {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [SFinite P]
-    (γ : Ω → ℝ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hc : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0)) (v : ℝ)
-    (A : Set ℂ) (_hA : MeasurableSet A) :
-    (∫⁻ sample, volume (A ∩ sausage (γ sample) v) ∂P) =
-      ∫⁻ w in A, P {sample | w ∈ sausage (γ sample) v} := by
-  classical
-  obtain ⟨E,hE,he⟩ := measurable_sausage_representative γ hγ hc v
-  let F : Ω → ℂ → ℝ≥0∞ := fun sample w => E.indicator (fun _ => 1) (sample,w)
-  have hFm : Measurable (Function.uncurry F) := measurable_const.indicator hE
-  have hleft : (fun sample => volume (A ∩ sausage (γ sample) v)) =ᵐ[P]
-      (fun sample => ∫⁻ w in A, F sample w) := by
-    filter_upwards [he] with sample heq
-    have hf : F sample = (sausage (γ sample) v).indicator (fun _ => (1:ℝ≥0∞)) := by
-      funext w
-      simp only [F,indicator_apply,heq]
-    rw [hf,lintegral_indicator_fun_one (sausage_open (γ sample) v).measurableSet,
-      Measure.restrict_apply (sausage_open (γ sample) v).measurableSet,inter_comm]
-  rw [lintegral_congr_ae hleft,lintegral_lintegral_swap hFm.aemeasurable]
-  apply lintegral_congr
-  intro w
-  have hset : MeasurableSet {sample | (sample,w) ∈ E} := hE.preimage (measurable_id.prodMk measurable_const)
-  have hFe : (fun sample => F sample w) = ({sample | (sample,w) ∈ E}).indicator (fun _ => (1:ℝ≥0∞)) := rfl
-  rw [hFe,lintegral_indicator_fun_one hset]
-  apply measure_congr
-  filter_upwards [he] with sample heq
-  exact propext (heq w)
-
-end SLEExactGauge
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma IsChordalSLE.maps {Ω : Type*} [MeasurableSpace Ω] {κ : ℝ} {P : Measure Ω}
-    {γ : Ω → ℝ → ℂ} (h : IsChordalSLE κ P γ) :
-    ∃ B : ℝ≥0 → Ω → ℝ, ∃ g f : Ω → ℝ → ℂ → ℂ,
-       IsBrownianReal B P ∧ ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-         (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample) := by
-  classical
-  obtain ⟨hm,B,hB,htrace⟩ := h
-  have hex (sample : Ω) : ∃ g f : ℝ → ℂ → ℂ,
-      IsLoewnerTrace (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) →
-      IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) g f := by
-    by_cases hh : IsLoewnerTrace (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample)
-    · obtain ⟨hc,hz,hi,ht,g,f,hr⟩ := hh
-      exact ⟨g,f,fun _ => ⟨hc,hz,hi,ht,hr⟩⟩
-    · exact ⟨0,0,fun hh' => (hh hh').elim⟩
-  choose g f hgf using hex
-  exact ⟨B,g,f,hB,htrace.mono (fun sample hs => hgf sample hs)⟩
-
-lemma IsChordalSLE.continuous {Ω : Type*} [MeasurableSpace Ω] {κ : ℝ} {P : Measure Ω}
-    {γ : Ω → ℝ → ℂ} (h : IsChordalSLE κ P γ) :
-    ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0) := by
-  obtain ⟨B,hB,ht⟩ := h.2
-  exact ht.mono (fun _ hh => hh.1)
-
-lemma IsChordalSLE.no_sausage_below {Ω : Type*} [MeasurableSpace Ω] {κ : ℝ} {P : Measure Ω}
-    {γ : Ω → ℝ → ℂ} (h : IsChordalSLE κ P γ) {z : ℂ} {v : ℝ} (hz : z.im ≤ -v) :
-    P {sample | z ∈ sausage (γ sample) v} = 0 := by
-  obtain ⟨B,hB,ht⟩ := h.2
-  apply measure_mono_null (t := {sample | ¬ IsLoewnerTrace
-    (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample)})
-  · intro sample hs hh
-    obtain ⟨t,ht0,hv⟩ := hs
-    have hi := hh.2.2.1 t ht0
-    have hab := Complex.abs_im_le_norm (z-γ sample t)
-    have hn := neg_abs_le (z-γ sample t).im
-    rw [dist_eq_norm] at hv
-    simp only [Complex.sub_im] at hab hn
-    linarith
-  · exact ae_iff.1 ht
-
-theorem sausage_point_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ A K : ℝ, 1 < A ∧ 0 < K ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (γ : Ω → ℝ → ℂ) (_hγ : IsChordalSLE κ P γ) (z : ℂ) (v : ℝ)
-      (_hv : 0 < v) (_hhigh : A*v < z.im),
-      P {sample | z ∈ sausage (γ sample) v} ≤
-        ENNReal.ofReal (K*v^(codimension κ)*z.im^(-codimension κ)) := by
-  obtain ⟨C,c,hC,hc,hc1,hbound⟩ := euclideanApproachBound κ hκ
-  let p := codimension κ
-  refine ⟨6/c,C*3^p*2^(-p),by rw [lt_div_iff₀ hc]; linarith,
-    by positivity,?_⟩
-  intro Ω _ P _ γ hγ z v hv hhigh
-  obtain ⟨B,g,f,hB,hmaps⟩ := hγ.maps
-  have hy : 0 < z.im := lt_trans (by positivity : 0 < (6/c)*v) hhigh
-  have hcy : 6*v < c*z.im := by
-    have hx := mul_lt_mul_of_pos_left hhigh hc
-    field_simp at hx
-    nlinarith
-  have hsmall : 2*v < z.im := by nlinarith
-  have hsub : {sample | z ∈ sausage (γ sample) v} ≤ᵐ[P]
-      {sample | liveDistanceApproach (γ sample) z (3*v)} := by
-    filter_upwards [hmaps] with sample hm hs
-    obtain ⟨visitTime,hvisitTime,hvisit⟩ := hs
-    let visitNonneg : ℝ≥0 := ⟨visitTime,hvisitTime⟩
-    have hcont : Continuous (fun time : ℝ≥0 => dist z (γ sample time)) :=
-      continuous_const.dist (hm.1.comp_continuous continuous_subtype_val
-        (fun time => time.coe_nonneg))
-    let approachTimes : Set ℝ≥0 := Icc 0 visitNonneg ∩
-      {time | dist z (γ sample time) ≤ 2*v}
-    have hcompact : IsCompact approachTimes :=
-      isCompact_Icc.inter_right (isClosed_le hcont continuous_const)
-    obtain ⟨firstTime,hfirst,hmin⟩ := hcompact.exists_isLeast
-      ⟨visitNonneg,⟨⟨zero_le,le_rfl⟩,by change dist z (γ sample visitTime) ≤ 2*v; linarith⟩⟩
-    have hfirstTime : (firstTime:ℝ) ≤ visitTime := hfirst.1.2
-    have hstrict (time : ℝ≥0) (htime : time < firstTime) :
-        2*v < dist z (γ sample time) := by
-      by_contra hn
-      exact (not_le_of_gt htime)
-        (hmin ⟨⟨zero_le,htime.le.trans hfirst.1.2⟩,not_lt.1 hn⟩)
-    have hinit : 2*v < dist z (γ sample 0) := by
-      rw [hm.2.1,dist_zero_right]
-      exact hsmall.trans_le (Complex.im_le_norm z)
-    have hdist : dist z (γ sample firstTime) = 2*v := by
-      apply le_antisymm hfirst.2
-      by_contra hn
-      have hlt : dist z (γ sample firstTime) < 2*v := not_le.1 hn
-      have hcc : ContinuousOn (fun time : ℝ => dist z (γ sample time))
-          (Icc 0 (firstTime:ℝ)) :=
-        (continuous_const.dist continuous_id).comp_continuousOn
-          (hm.1.mono Icc_subset_Ici_self)
-      obtain ⟨time,htime,heq⟩ := intermediate_value_Icc' firstTime.coe_nonneg hcc
-        ⟨hlt.le,hinit.le⟩
-      have htimeFirst : time < (firstTime:ℝ) :=
-        lt_of_le_of_ne htime.2 (by intro heq; subst time; linarith)
-      have hh := hstrict ⟨time,htime.1⟩ htimeFirst
-      change 2*v < dist z (γ sample time) at hh
-      exact (not_lt_of_ge heq.le) hh
-    have hbefore (time : ℝ) (htime : time ∈ Icc 0 (firstTime:ℝ)) :
-        2*v ≤ dist z (γ sample time) := by
-      rcases lt_or_eq_of_le htime.2 with hlt | heq
-      · exact (hstrict ⟨time,htime.1⟩ hlt).le
-      · rw [heq,hdist]
-    have hfree : ball z (2*v) ⊆ halfPlane \ localizationSegment (γ sample) 0 firstTime := by
-      intro point hpoint
-      have hnorm : ‖point-z‖ < 2*v := by
-        simpa only [mem_ball,dist_eq_norm] using hpoint
-      constructor
-      · have hh := Complex.abs_im_le_norm (point-z)
-        have hn := neg_abs_le (point-z).im
-        change 0 < point.im
-        simp only [Complex.sub_im] at hh hn
-        linarith
-      · rintro ⟨time,htime,rfl⟩
-        have hh := hbefore time htime
-        have hh' := mem_ball.1 hpoint
-        rw [dist_comm] at hh'
-        linarith
-    have hclosure := hm.future_tip_mem_closure firstTime.coe_nonneg hfirstTime
-    have hball : γ sample visitTime ∈ ball z (2*v) := by
-      rw [mem_ball,dist_comm]
-      linarith
-    obtain ⟨point,hpointBall,hpointDomain⟩ :=
-      mem_closure_iff.1 hclosure (ball z (2*v)) isOpen_ball hball
-    have hcomponent :=
-      (convex_ball z (2*v)).isPreconnected.subset_connectedComponentIn hpointBall hfree
-    have hzball : z ∈ ball z (2*v) := mem_ball_self (by positivity)
-    have hzlive : z ∈ localizationDomain (γ sample) firstTime := by
-      refine ⟨hfree hzball,?_⟩
-      rw [← connectedComponentIn_eq (hcomponent hzball)]
-      exact hpointDomain.2
-    refine ⟨firstTime,hzlive,?_⟩
-    have hnot : γ sample firstTime ∈ (localizationDomain (γ sample) firstTime)ᶜ := by
-      intro hh
-      exact hh.1.2 ⟨firstTime,⟨firstTime.coe_nonneg,le_rfl⟩,rfl⟩
-    have hh := infDist_le_dist_of_mem (x := z) hnot
-    rw [hdist] at hh
-    linarith
-  apply (measure_mono_ae hsub).trans
-  apply (hbound P B hB γ g f hγ.1 hmaps z hy (3*v) (by positivity) (by linarith)).trans
-  apply ENNReal.ofReal_le_ofReal
-  have hn : 0 < ‖z‖ := lt_of_lt_of_le hy (Complex.im_le_norm z)
-  have ha : 0 ≤ 8/κ-1 := by rw [sub_nonneg,le_div_iff₀ hκ.1]; linarith [hκ.2]
-  have hs : (z.im/‖z‖)^(8/κ-1) ≤ 1 :=
-    Real.rpow_le_one (by positivity) ((div_le_one hn).2 (Complex.im_le_norm z)) ha
-  have hnon : 0 ≤ C*(2*z.im)^(-p)*(3*v)^p := by positivity
-  calc
-    C*((2*z.im)^(-p)*(z.im/‖z‖)^(8/κ-1))*(3*v)^p
-        = (C*(2*z.im)^(-p)*(3*v)^p)*((z.im/‖z‖)^(8/κ-1)) := by ring
-    _ ≤ (C*(2*z.im)^(-p)*(3*v)^p)*1 := mul_le_mul_of_nonneg_left hs hnon
-    _ = (C*3^p*2^(-p))*v^p*z.im^(-p) := by
-      rw [Real.mul_rpow (by norm_num : (0:ℝ) ≤ 2) hy.le,
-        Real.mul_rpow (by norm_num : (0:ℝ) ≤ 3) hv.le]
-      ring
-
-end SLEExactGauge
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-def regionBand (R a b : ℝ) : Set ℂ :=
-  {w | w.re ∈ Icc (-R) R ∧ w.im ∈ Icc a b}
-
-lemma regionBand_measurable (R a b : ℝ) : MeasurableSet (regionBand R a b) :=
-  (Complex.measurable_re measurableSet_Icc).inter (Complex.measurable_im measurableSet_Icc)
-
-lemma lintegral_regionBand (R a b : ℝ) (F : ℝ → ℝ≥0∞) (hF : Measurable F) :
-    (∫⁻ w in regionBand R a b, F w.im) =
-      ENNReal.ofReal (2*R) * ∫⁻ y in Icc a b, F y := by
-  have he := Complex.volume_preserving_equiv_real_prod.setLIntegral_comp_preimage_emb
-    Complex.measurableEquivRealProd.measurableEmbedding (fun x : ℝ × ℝ => F x.2)
-      (Icc (-R) R ×ˢ Icc a b)
-  change (∫⁻ w in regionBand R a b, F w.im) = _ at he
-  rw [he,Measure.volume_eq_prod,← Measure.prod_restrict,
-    lintegral_prod (fun x : ℝ × ℝ => F x.2) (hF.comp measurable_snd).aemeasurable]
-  simp only [lintegral_const,Measure.restrict_apply_univ,Real.volume_Icc,sub_neg_eq_add,
-    ← two_mul R,mul_comm]
-
-lemma volume_regionBand (R a b : ℝ) :
-    volume (regionBand R a b) = ENNReal.ofReal (2*R)*ENNReal.ofReal (b-a) := by
-  simpa only [lintegral_one,Measure.restrict_apply_univ,Real.volume_Icc] using
-    lintegral_regionBand R a b (fun _ => 1) measurable_const
-
-lemma height_kernel_finite (R p : ℝ) (hR : 0 ≤ R) (hp : p < 1) :
-    (∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-p))) < ⊤ := by
-  rw [lintegral_regionBand R 0 R (fun y : ℝ => ENNReal.ofReal (y^(-p))) (measurable_id.pow_const (-p)).ennreal_ofReal]
-  apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top
-  have hi : IntegrableOn (fun y : ℝ => y^(-p)) (Icc 0 R) :=
-    (intervalIntegrable_iff_integrableOn_Icc_of_le hR).1 (intervalIntegral.intervalIntegrable_rpow' (by linarith))
-  rw [← ofReal_integral_eq_lintegral_ofReal hi]
-  · exact ENNReal.ofReal_lt_top
-  · filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy
-    exact Real.rpow_nonneg hy.1 _
-
-end SLEExactGauge
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-universe u
-
-lemma expected_sausage_area_bound
-    {κ R A K : ℝ} (_hR : 0 < R) (hA : 1 < A) (hK : 0 < K)
-    (hb : ∀ {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (γ : Ω → ℝ → ℂ) (_hγ : IsChordalSLE κ P γ) (z : ℂ) (v : ℝ)
-      (_hv : 0 < v) (_hhigh : A*v < z.im),
-      P {sample | z ∈ sausage (γ sample) v} ≤
-        ENNReal.ofReal (K*v^(codimension κ)*z.im^(-codimension κ)))
-    {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-    (γ : Ω → ℝ → ℂ) (hγ : IsChordalSLE κ P γ) (v : ℝ) (hv : 0 < v) :
-    (∫⁻ sample, volume (ball 0 R ∩ sausage (γ sample) v) ∂P) ≤
-      ENNReal.ofReal (2*R)*ENNReal.ofReal ((A+1)*v) +
-        ENNReal.ofReal (K*v^(codimension κ)) *
-          ∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-codimension κ)) := by
-  rw [expected_sausage_area γ hγ.1 hγ.continuous v (ball 0 R) measurableSet_ball]
-  let F := (regionBand R (-v) (A*v)).indicator (fun _ => (1 : ℝ≥0∞))
-  let G := (regionBand R 0 R).indicator (fun w => ENNReal.ofReal (w.im^(-codimension κ)))
-  have hFm : Measurable F := measurable_const.indicator (regionBand_measurable _ _ _)
-  have hGm : Measurable G :=
-    (Complex.measurable_im.pow_const _).ennreal_ofReal.indicator (regionBand_measurable _ _ _)
-  have hmono : (∫⁻ w in ball 0 R, P {sample | w ∈ sausage (γ sample) v}) ≤
-      ∫⁻ w in ball 0 R, F w + ENNReal.ofReal (K*v^(codimension κ))*G w := by
-    apply lintegral_mono_ae
-    filter_upwards [ae_restrict_mem measurableSet_ball] with w hw
-    have hnorm : ‖w‖ < R := by simpa only [mem_ball,dist_zero_right] using hw
-    have hre : w.re ∈ Icc (-R) R := by
-      have hh := Complex.abs_re_le_norm w
-      exact ⟨by linarith [neg_abs_le w.re],by linarith [le_abs_self w.re]⟩
-    by_cases hhigh : A*v < w.im
-    · have hwG : w ∈ regionBand R 0 R := ⟨hre,by constructor; nlinarith; exact (Complex.im_le_norm w).trans hnorm.le⟩
-      have hh := hb P γ hγ w v hv hhigh
-      have hmul : ENNReal.ofReal (K*v^(codimension κ)*w.im^(-codimension κ)) =
-          ENNReal.ofReal (K*v^(codimension κ))*G w := by
-        rw [show G w = ENNReal.ofReal (w.im^(-codimension κ)) from indicator_of_mem hwG _]
-        exact ENNReal.ofReal_mul (by positivity)
-      rw [hmul] at hh
-      exact hh.trans (le_add_left le_rfl)
-    · by_cases hlow : w.im ≤ -v
-      · rw [hγ.no_sausage_below hlow]
-        exact zero_le
-      · have hwF : w ∈ regionBand R (-v) (A*v) := ⟨hre,le_of_lt (not_le.1 hlow),not_lt.1 hhigh⟩
-        have hh : P {sample | w ∈ sausage (γ sample) v} ≤ 1 := prob_le_one
-        rw [show F w = 1 from indicator_of_mem hwF _]
-        exact hh.trans (le_add_right le_rfl)
-  apply hmono.trans
-  apply (setLIntegral_le_lintegral (ball 0 R) (fun w => F w + ENNReal.ofReal (K*v^(codimension κ))*G w)).trans
-  rw [lintegral_add_left hFm,lintegral_const_mul _ hGm]
-  rw [show (∫⁻ w, F w) = volume (regionBand R (-v) (A*v)) from
-      lintegral_indicator_fun_one (regionBand_measurable _ _ _),
-    show (∫⁻ w, G w) = ∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-codimension κ)) from
-      lintegral_indicator (regionBand_measurable _ _ _) _,volume_regionBand]
-  have he : A*v-(-v) = (A+1)*v := by ring
-  rw [he]
-
-theorem uniformBoundedRegionCharge (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (R : ℝ) (hR : 0 < R) :
-    ∃ C : ℝ≥0, ∀ {Ω : Type u} [MeasurableSpace Ω]
-      (P : Measure Ω) [IsProbabilityMeasure P] (γ : Ω → ℝ → ℂ)
-      (_hγ : IsChordalSLE κ P γ) (v : ℝ) (_hv : 0 < v) (_hv1 : v ≤ 1),
-      (∫⁻ sample, sausageCharge κ (γ sample) v (ball 0 R) ∂P) ≤ C := by
-  obtain ⟨A,K,hA,hK,hbound⟩ := sausage_point_bound κ hκ
-  let p := codimension κ
-  have hp : p < 1 := by dsimp [p,codimension,dimension]; linarith [hκ.1]
-  let D := ∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-p))
-  have hD : D < ⊤ := height_kernel_finite R p hR.le hp
-  let Z := ENNReal.ofReal (2*R*(A+1))+ENNReal.ofReal K*D
-  have hZ : Z ≠ ⊤ := (ENNReal.add_lt_top.2
-    ⟨ENNReal.ofReal_lt_top,ENNReal.mul_lt_top ENNReal.ofReal_lt_top hD⟩).ne
-  refine ⟨Z.toNNReal,?_⟩
-  intro Ω _ P _ γ hγ v hv hv1
-  rw [ENNReal.coe_toNNReal hZ]
-  change (∫⁻ sample, ENNReal.ofReal (v^(-p))*volume (ball 0 R ∩ sausage (γ sample) v) ∂P) ≤ Z
-  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
-  apply (mul_le_mul_right (expected_sausage_area_bound hR hA hK hbound P γ hγ v hv) _).trans
-  change ENNReal.ofReal (v^(-p)) *
-    (ENNReal.ofReal (2*R)*ENNReal.ofReal ((A+1)*v)+ENNReal.ofReal (K*v^p)*D) ≤ Z
-  rw [mul_add]
-  apply add_le_add
-  · rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ 2*R),← ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _)]
-    apply ENNReal.ofReal_le_ofReal
-    have hh : v^(-p)*v ≤ 1 := by
-      have he : v^(-p)*v = v^(-p+1) := by
-        simpa only [Real.rpow_one] using (Real.rpow_add hv (-p) 1).symm
-      rw [he]
-      exact Real.rpow_le_one hv.le hv1 (by linarith)
-    calc
-      v^(-p)*(2*R*((A+1)*v)) = (2*R*(A+1))*(v^(-p)*v) := by ring
-      _ ≤ (2*R*(A+1))*1 := mul_le_mul_of_nonneg_left hh (by positivity)
-      _ = 2*R*(A+1) := mul_one _
-  · have he : v^(-p)*(K*v^p) = K := by
-      calc
-        v^(-p)*(K*v^p) = K*(v^(-p)*v^p) := by ring
-        _ = K := by rw [← Real.rpow_add hv,neg_add_cancel,Real.rpow_zero,mul_one]
-    rw [← mul_assoc,← ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _),he]
-
-end SLEExactGauge
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.AngularClock
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Charge estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+def sausage (γ : ℝ → ℂ) (v : ℝ) : Set ℂ :=
+  {w | ∃ t : ℝ, 0 ≤ t ∧ dist w (γ t) < v}
+
+def sausageCharge (κ : ℝ) (γ : ℝ → ℂ) (v : ℝ) (A : Set ℂ) : ℝ≥0∞ :=
+  ENNReal.ofReal (v^(-codimension κ)) * volume (A ∩ sausage γ v)
+
+lemma sausage_open (γ : ℝ → ℂ) (v : ℝ) : IsOpen (sausage γ v) := by
+  have he : sausage γ v = ⋃ t : ℝ, ⋃ (_ : 0 ≤ t), ball (γ t) v := by
+    ext w
+    simp [sausage,mem_ball]
+  rw [he]
+  exact isOpen_iUnion (fun t => isOpen_iUnion (fun _ => isOpen_ball))
+
+lemma sausage_eq_dense {γ : ℝ → ℂ} (hγ : ContinuousOn γ (Ici 0)) (v : ℝ) :
+    sausage γ v = ⋃ n : ℕ, ball (γ (denseSeq ℝ≥0 n)) v := by
+  ext w
+  simp only [sausage,mem_ofPred_eq,mem_iUnion,mem_ball]
+  constructor
+  · rintro ⟨t,ht,hvisit⟩
+    by_contra hn
+    push Not at hn
+    have hc : Continuous (fun q : ℝ≥0 => dist w (γ q)) :=
+      continuous_const.dist (hγ.comp_continuous continuous_subtype_val (fun q => q.coe_nonneg))
+    have hh : v ≤ dist w (γ t) := (denseRange_denseSeq ℝ≥0).induction_on (p := fun q : ℝ≥0 => v ≤ dist w (γ q)) (⟨t,ht⟩ : ℝ≥0)
+      (isClosed_le continuous_const hc) hn
+    exact (not_lt_of_ge hh) hvisit
+  · rintro ⟨n,hn⟩
+    exact ⟨denseSeq ℝ≥0 n,(denseSeq ℝ≥0 n).coe_nonneg,hn⟩
+
+lemma measurable_sausage_representative {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (γ : Ω → ℝ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hc : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0)) (v : ℝ) :
+    ∃ E : Set (Ω × ℂ), MeasurableSet E ∧
+      ∀ᵐ sample ∂P, ∀ w : ℂ, (sample,w) ∈ E ↔ w ∈ sausage (γ sample) v := by
+  let Y (n : ℕ) := (hγ (denseSeq ℝ≥0 n) (denseSeq ℝ≥0 n).coe_nonneg).mk (fun sample => γ sample (denseSeq ℝ≥0 n))
+  have hY (n : ℕ) : Measurable (Y n) := (hγ _ (denseSeq ℝ≥0 n).coe_nonneg).measurable_mk
+  have he : ∀ᵐ sample ∂P, ∀ n : ℕ, γ sample (denseSeq ℝ≥0 n) = Y n sample :=
+    ae_all_iff.2 (fun n => (hγ _ (denseSeq ℝ≥0 n).coe_nonneg).ae_eq_mk)
+  let E : Set (Ω × ℂ) := ⋃ n : ℕ, {x | dist x.2 (Y n x.1) < v}
+  refine ⟨E,MeasurableSet.iUnion (fun n => measurableSet_lt
+    (measurable_snd.dist ((hY n).comp measurable_fst)) measurable_const),?_⟩
+  filter_upwards [hc,he] with sample hcont heq
+  intro w
+  rw [sausage_eq_dense hcont v]
+  simp only [E,mem_iUnion,mem_ofPred_eq,mem_ball,heq]
+
+lemma expected_sausage_area {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [SFinite P]
+    (γ : Ω → ℝ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hc : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0)) (v : ℝ)
+    (A : Set ℂ) (_hA : MeasurableSet A) :
+    (∫⁻ sample, volume (A ∩ sausage (γ sample) v) ∂P) =
+      ∫⁻ w in A, P {sample | w ∈ sausage (γ sample) v} := by
+  classical
+  obtain ⟨E,hE,he⟩ := measurable_sausage_representative γ hγ hc v
+  let F : Ω → ℂ → ℝ≥0∞ := fun sample w => E.indicator (fun _ => 1) (sample,w)
+  have hFm : Measurable (Function.uncurry F) := measurable_const.indicator hE
+  have hleft : (fun sample => volume (A ∩ sausage (γ sample) v)) =ᵐ[P]
+      (fun sample => ∫⁻ w in A, F sample w) := by
+    filter_upwards [he] with sample heq
+    have hf : F sample = (sausage (γ sample) v).indicator (fun _ => (1:ℝ≥0∞)) := by
+      funext w
+      simp only [F,indicator_apply,heq]
+    rw [hf,lintegral_indicator_fun_one (sausage_open (γ sample) v).measurableSet,
+      Measure.restrict_apply (sausage_open (γ sample) v).measurableSet,inter_comm]
+  rw [lintegral_congr_ae hleft,lintegral_lintegral_swap hFm.aemeasurable]
+  apply lintegral_congr
+  intro w
+  have hset : MeasurableSet {sample | (sample,w) ∈ E} := hE.preimage (measurable_id.prodMk measurable_const)
+  have hFe : (fun sample => F sample w) = ({sample | (sample,w) ∈ E}).indicator (fun _ => (1:ℝ≥0∞)) := rfl
+  rw [hFe,lintegral_indicator_fun_one hset]
+  apply measure_congr
+  filter_upwards [he] with sample heq
+  exact propext (heq w)
+
+end SLEExactGauge
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma IsChordalSLE.maps {Ω : Type*} [MeasurableSpace Ω] {κ : ℝ} {P : Measure Ω}
+    {γ : Ω → ℝ → ℂ} (h : IsChordalSLE κ P γ) :
+    ∃ B : ℝ≥0 → Ω → ℝ, ∃ g f : Ω → ℝ → ℂ → ℂ,
+       IsBrownianReal B P ∧ ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+         (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample) := by
+  classical
+  obtain ⟨hm,B,hB,htrace⟩ := h
+  have hex (sample : Ω) : ∃ g f : ℝ → ℂ → ℂ,
+      IsLoewnerTrace (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) →
+      IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) g f := by
+    by_cases hh : IsLoewnerTrace (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample)
+    · obtain ⟨hc,hz,hi,ht,g,f,hr⟩ := hh
+      exact ⟨g,f,fun _ => ⟨hc,hz,hi,ht,hr⟩⟩
+    · exact ⟨0,0,fun hh' => (hh hh').elim⟩
+  choose g f hgf using hex
+  exact ⟨B,g,f,hB,htrace.mono (fun sample hs => hgf sample hs)⟩
+
+lemma IsChordalSLE.continuous {Ω : Type*} [MeasurableSpace Ω] {κ : ℝ} {P : Measure Ω}
+    {γ : Ω → ℝ → ℂ} (h : IsChordalSLE κ P γ) :
+    ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0) := by
+  obtain ⟨B,hB,ht⟩ := h.2
+  exact ht.mono (fun _ hh => hh.1)
+
+lemma IsChordalSLE.no_sausage_below {Ω : Type*} [MeasurableSpace Ω] {κ : ℝ} {P : Measure Ω}
+    {γ : Ω → ℝ → ℂ} (h : IsChordalSLE κ P γ) {z : ℂ} {v : ℝ} (hz : z.im ≤ -v) :
+    P {sample | z ∈ sausage (γ sample) v} = 0 := by
+  obtain ⟨B,hB,ht⟩ := h.2
+  apply measure_mono_null (t := {sample | ¬ IsLoewnerTrace
+    (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample)})
+  · intro sample hs hh
+    obtain ⟨t,ht0,hv⟩ := hs
+    have hi := hh.2.2.1 t ht0
+    have hab := Complex.abs_im_le_norm (z-γ sample t)
+    have hn := neg_abs_le (z-γ sample t).im
+    rw [dist_eq_norm] at hv
+    simp only [Complex.sub_im] at hab hn
+    linarith
+  · exact ae_iff.1 ht
+
+theorem sausage_point_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ A K : ℝ, 1 < A ∧ 0 < K ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (γ : Ω → ℝ → ℂ) (_hγ : IsChordalSLE κ P γ) (z : ℂ) (v : ℝ)
+      (_hv : 0 < v) (_hhigh : A*v < z.im),
+      P {sample | z ∈ sausage (γ sample) v} ≤
+        ENNReal.ofReal (K*v^(codimension κ)*z.im^(-codimension κ)) := by
+  obtain ⟨C,c,hC,hc,hc1,hbound⟩ := euclideanApproachBound κ hκ
+  let p := codimension κ
+  refine ⟨6/c,C*3^p*2^(-p),by rw [lt_div_iff₀ hc]; linarith,
+    by positivity,?_⟩
+  intro Ω _ P _ γ hγ z v hv hhigh
+  obtain ⟨B,g,f,hB,hmaps⟩ := hγ.maps
+  have hy : 0 < z.im := lt_trans (by positivity : 0 < (6/c)*v) hhigh
+  have hcy : 6*v < c*z.im := by
+    have hx := mul_lt_mul_of_pos_left hhigh hc
+    field_simp at hx
+    nlinarith
+  have hsmall : 2*v < z.im := by nlinarith
+  have hsub : {sample | z ∈ sausage (γ sample) v} ≤ᵐ[P]
+      {sample | liveDistanceApproach (γ sample) z (3*v)} := by
+    filter_upwards [hmaps] with sample hm hs
+    obtain ⟨visitTime,hvisitTime,hvisit⟩ := hs
+    let visitNonneg : ℝ≥0 := ⟨visitTime,hvisitTime⟩
+    have hcont : Continuous (fun time : ℝ≥0 => dist z (γ sample time)) :=
+      continuous_const.dist (hm.1.comp_continuous continuous_subtype_val
+        (fun time => time.coe_nonneg))
+    let approachTimes : Set ℝ≥0 := Icc 0 visitNonneg ∩
+      {time | dist z (γ sample time) ≤ 2*v}
+    have hcompact : IsCompact approachTimes :=
+      isCompact_Icc.inter_right (isClosed_le hcont continuous_const)
+    obtain ⟨firstTime,hfirst,hmin⟩ := hcompact.exists_isLeast
+      ⟨visitNonneg,⟨⟨zero_le,le_rfl⟩,by change dist z (γ sample visitTime) ≤ 2*v; linarith⟩⟩
+    have hfirstTime : (firstTime:ℝ) ≤ visitTime := hfirst.1.2
+    have hstrict (time : ℝ≥0) (htime : time < firstTime) :
+        2*v < dist z (γ sample time) := by
+      by_contra hn
+      exact (not_le_of_gt htime)
+        (hmin ⟨⟨zero_le,htime.le.trans hfirst.1.2⟩,not_lt.1 hn⟩)
+    have hinit : 2*v < dist z (γ sample 0) := by
+      rw [hm.2.1,dist_zero_right]
+      exact hsmall.trans_le (Complex.im_le_norm z)
+    have hdist : dist z (γ sample firstTime) = 2*v := by
+      apply le_antisymm hfirst.2
+      by_contra hn
+      have hlt : dist z (γ sample firstTime) < 2*v := not_le.1 hn
+      have hcc : ContinuousOn (fun time : ℝ => dist z (γ sample time))
+          (Icc 0 (firstTime:ℝ)) :=
+        (continuous_const.dist continuous_id).comp_continuousOn
+          (hm.1.mono Icc_subset_Ici_self)
+      obtain ⟨time,htime,heq⟩ := intermediate_value_Icc' firstTime.coe_nonneg hcc
+        ⟨hlt.le,hinit.le⟩
+      have htimeFirst : time < (firstTime:ℝ) :=
+        lt_of_le_of_ne htime.2 (by intro heq; subst time; linarith)
+      have hh := hstrict ⟨time,htime.1⟩ htimeFirst
+      change 2*v < dist z (γ sample time) at hh
+      exact (not_lt_of_ge heq.le) hh
+    have hbefore (time : ℝ) (htime : time ∈ Icc 0 (firstTime:ℝ)) :
+        2*v ≤ dist z (γ sample time) := by
+      rcases lt_or_eq_of_le htime.2 with hlt | heq
+      · exact (hstrict ⟨time,htime.1⟩ hlt).le
+      · rw [heq,hdist]
+    have hfree : ball z (2*v) ⊆ halfPlane \ localizationSegment (γ sample) 0 firstTime := by
+      intro point hpoint
+      have hnorm : ‖point-z‖ < 2*v := by
+        simpa only [mem_ball,dist_eq_norm] using hpoint
+      constructor
+      · have hh := Complex.abs_im_le_norm (point-z)
+        have hn := neg_abs_le (point-z).im
+        change 0 < point.im
+        simp only [Complex.sub_im] at hh hn
+        linarith
+      · rintro ⟨time,htime,rfl⟩
+        have hh := hbefore time htime
+        have hh' := mem_ball.1 hpoint
+        rw [dist_comm] at hh'
+        linarith
+    have hclosure := hm.future_tip_mem_closure firstTime.coe_nonneg hfirstTime
+    have hball : γ sample visitTime ∈ ball z (2*v) := by
+      rw [mem_ball,dist_comm]
+      linarith
+    obtain ⟨point,hpointBall,hpointDomain⟩ :=
+      mem_closure_iff.1 hclosure (ball z (2*v)) isOpen_ball hball
+    have hcomponent :=
+      (convex_ball z (2*v)).isPreconnected.subset_connectedComponentIn hpointBall hfree
+    have hzball : z ∈ ball z (2*v) := mem_ball_self (by positivity)
+    have hzlive : z ∈ localizationDomain (γ sample) firstTime := by
+      refine ⟨hfree hzball,?_⟩
+      rw [← connectedComponentIn_eq (hcomponent hzball)]
+      exact hpointDomain.2
+    refine ⟨firstTime,hzlive,?_⟩
+    have hnot : γ sample firstTime ∈ (localizationDomain (γ sample) firstTime)ᶜ := by
+      intro hh
+      exact hh.1.2 ⟨firstTime,⟨firstTime.coe_nonneg,le_rfl⟩,rfl⟩
+    have hh := infDist_le_dist_of_mem (x := z) hnot
+    rw [hdist] at hh
+    linarith
+  apply (measure_mono_ae hsub).trans
+  apply (hbound P B hB γ g f hγ.1 hmaps z hy (3*v) (by positivity) (by linarith)).trans
+  apply ENNReal.ofReal_le_ofReal
+  have hn : 0 < ‖z‖ := lt_of_lt_of_le hy (Complex.im_le_norm z)
+  have ha : 0 ≤ 8/κ-1 := by rw [sub_nonneg,le_div_iff₀ hκ.1]; linarith [hκ.2]
+  have hs : (z.im/‖z‖)^(8/κ-1) ≤ 1 :=
+    Real.rpow_le_one (by positivity) ((div_le_one hn).2 (Complex.im_le_norm z)) ha
+  have hnon : 0 ≤ C*(2*z.im)^(-p)*(3*v)^p := by positivity
+  calc
+    C*((2*z.im)^(-p)*(z.im/‖z‖)^(8/κ-1))*(3*v)^p
+        = (C*(2*z.im)^(-p)*(3*v)^p)*((z.im/‖z‖)^(8/κ-1)) := by ring
+    _ ≤ (C*(2*z.im)^(-p)*(3*v)^p)*1 := mul_le_mul_of_nonneg_left hs hnon
+    _ = (C*3^p*2^(-p))*v^p*z.im^(-p) := by
+      rw [Real.mul_rpow (by norm_num : (0:ℝ) ≤ 2) hy.le,
+        Real.mul_rpow (by norm_num : (0:ℝ) ≤ 3) hv.le]
+      ring
+
+end SLEExactGauge
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+def regionBand (R a b : ℝ) : Set ℂ :=
+  {w | w.re ∈ Icc (-R) R ∧ w.im ∈ Icc a b}
+
+lemma regionBand_measurable (R a b : ℝ) : MeasurableSet (regionBand R a b) :=
+  (Complex.measurable_re measurableSet_Icc).inter (Complex.measurable_im measurableSet_Icc)
+
+lemma lintegral_regionBand (R a b : ℝ) (F : ℝ → ℝ≥0∞) (hF : Measurable F) :
+    (∫⁻ w in regionBand R a b, F w.im) =
+      ENNReal.ofReal (2*R) * ∫⁻ y in Icc a b, F y := by
+  have he := Complex.volume_preserving_equiv_real_prod.setLIntegral_comp_preimage_emb
+    Complex.measurableEquivRealProd.measurableEmbedding (fun x : ℝ × ℝ => F x.2)
+      (Icc (-R) R ×ˢ Icc a b)
+  change (∫⁻ w in regionBand R a b, F w.im) = _ at he
+  rw [he,Measure.volume_eq_prod,← Measure.prod_restrict,
+    lintegral_prod (fun x : ℝ × ℝ => F x.2) (hF.comp measurable_snd).aemeasurable]
+  simp only [lintegral_const,Measure.restrict_apply_univ,Real.volume_Icc,sub_neg_eq_add,
+    ← two_mul R,mul_comm]
+
+lemma volume_regionBand (R a b : ℝ) :
+    volume (regionBand R a b) = ENNReal.ofReal (2*R)*ENNReal.ofReal (b-a) := by
+  simpa only [lintegral_one,Measure.restrict_apply_univ,Real.volume_Icc] using
+    lintegral_regionBand R a b (fun _ => 1) measurable_const
+
+lemma height_kernel_finite (R p : ℝ) (hR : 0 ≤ R) (hp : p < 1) :
+    (∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-p))) < ⊤ := by
+  rw [lintegral_regionBand R 0 R (fun y : ℝ => ENNReal.ofReal (y^(-p))) (measurable_id.pow_const (-p)).ennreal_ofReal]
+  apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top
+  have hi : IntegrableOn (fun y : ℝ => y^(-p)) (Icc 0 R) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hR).1 (intervalIntegral.intervalIntegrable_rpow' (by linarith))
+  rw [← ofReal_integral_eq_lintegral_ofReal hi]
+  · exact ENNReal.ofReal_lt_top
+  · filter_upwards [ae_restrict_mem measurableSet_Icc] with y hy
+    exact Real.rpow_nonneg hy.1 _
+
+end SLEExactGauge
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric TopologicalSpace
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+universe u
+
+lemma expected_sausage_area_bound
+    {κ R A K : ℝ} (_hR : 0 < R) (hA : 1 < A) (hK : 0 < K)
+    (hb : ∀ {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (γ : Ω → ℝ → ℂ) (_hγ : IsChordalSLE κ P γ) (z : ℂ) (v : ℝ)
+      (_hv : 0 < v) (_hhigh : A*v < z.im),
+      P {sample | z ∈ sausage (γ sample) v} ≤
+        ENNReal.ofReal (K*v^(codimension κ)*z.im^(-codimension κ)))
+    {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+    (γ : Ω → ℝ → ℂ) (hγ : IsChordalSLE κ P γ) (v : ℝ) (hv : 0 < v) :
+    (∫⁻ sample, volume (ball 0 R ∩ sausage (γ sample) v) ∂P) ≤
+      ENNReal.ofReal (2*R)*ENNReal.ofReal ((A+1)*v) +
+        ENNReal.ofReal (K*v^(codimension κ)) *
+          ∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-codimension κ)) := by
+  rw [expected_sausage_area γ hγ.1 hγ.continuous v (ball 0 R) measurableSet_ball]
+  let F := (regionBand R (-v) (A*v)).indicator (fun _ => (1 : ℝ≥0∞))
+  let G := (regionBand R 0 R).indicator (fun w => ENNReal.ofReal (w.im^(-codimension κ)))
+  have hFm : Measurable F := measurable_const.indicator (regionBand_measurable _ _ _)
+  have hGm : Measurable G :=
+    (Complex.measurable_im.pow_const _).ennreal_ofReal.indicator (regionBand_measurable _ _ _)
+  have hmono : (∫⁻ w in ball 0 R, P {sample | w ∈ sausage (γ sample) v}) ≤
+      ∫⁻ w in ball 0 R, F w + ENNReal.ofReal (K*v^(codimension κ))*G w := by
+    apply lintegral_mono_ae
+    filter_upwards [ae_restrict_mem measurableSet_ball] with w hw
+    have hnorm : ‖w‖ < R := by simpa only [mem_ball,dist_zero_right] using hw
+    have hre : w.re ∈ Icc (-R) R := by
+      have hh := Complex.abs_re_le_norm w
+      exact ⟨by linarith [neg_abs_le w.re],by linarith [le_abs_self w.re]⟩
+    by_cases hhigh : A*v < w.im
+    · have hwG : w ∈ regionBand R 0 R := ⟨hre,by constructor; nlinarith; exact (Complex.im_le_norm w).trans hnorm.le⟩
+      have hh := hb P γ hγ w v hv hhigh
+      have hmul : ENNReal.ofReal (K*v^(codimension κ)*w.im^(-codimension κ)) =
+          ENNReal.ofReal (K*v^(codimension κ))*G w := by
+        rw [show G w = ENNReal.ofReal (w.im^(-codimension κ)) from indicator_of_mem hwG _]
+        exact ENNReal.ofReal_mul (by positivity)
+      rw [hmul] at hh
+      exact hh.trans (le_add_left le_rfl)
+    · by_cases hlow : w.im ≤ -v
+      · rw [hγ.no_sausage_below hlow]
+        exact zero_le
+      · have hwF : w ∈ regionBand R (-v) (A*v) := ⟨hre,le_of_lt (not_le.1 hlow),not_lt.1 hhigh⟩
+        have hh : P {sample | w ∈ sausage (γ sample) v} ≤ 1 := prob_le_one
+        rw [show F w = 1 from indicator_of_mem hwF _]
+        exact hh.trans (le_add_right le_rfl)
+  apply hmono.trans
+  apply (setLIntegral_le_lintegral (ball 0 R) (fun w => F w + ENNReal.ofReal (K*v^(codimension κ))*G w)).trans
+  rw [lintegral_add_left hFm,lintegral_const_mul _ hGm]
+  rw [show (∫⁻ w, F w) = volume (regionBand R (-v) (A*v)) from
+      lintegral_indicator_fun_one (regionBand_measurable _ _ _),
+    show (∫⁻ w, G w) = ∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-codimension κ)) from
+      lintegral_indicator (regionBand_measurable _ _ _) _,volume_regionBand]
+  have he : A*v-(-v) = (A+1)*v := by ring
+  rw [he]
+
+theorem uniformBoundedRegionCharge (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (R : ℝ) (hR : 0 < R) :
+    ∃ C : ℝ≥0, ∀ {Ω : Type u} [MeasurableSpace Ω]
+      (P : Measure Ω) [IsProbabilityMeasure P] (γ : Ω → ℝ → ℂ)
+      (_hγ : IsChordalSLE κ P γ) (v : ℝ) (_hv : 0 < v) (_hv1 : v ≤ 1),
+      (∫⁻ sample, sausageCharge κ (γ sample) v (ball 0 R) ∂P) ≤ C := by
+  obtain ⟨A,K,hA,hK,hbound⟩ := sausage_point_bound κ hκ
+  let p := codimension κ
+  have hp : p < 1 := by dsimp [p,codimension,dimension]; linarith [hκ.1]
+  let D := ∫⁻ w in regionBand R 0 R, ENNReal.ofReal (w.im^(-p))
+  have hD : D < ⊤ := height_kernel_finite R p hR.le hp
+  let Z := ENNReal.ofReal (2*R*(A+1))+ENNReal.ofReal K*D
+  have hZ : Z ≠ ⊤ := (ENNReal.add_lt_top.2
+    ⟨ENNReal.ofReal_lt_top,ENNReal.mul_lt_top ENNReal.ofReal_lt_top hD⟩).ne
+  refine ⟨Z.toNNReal,?_⟩
+  intro Ω _ P _ γ hγ v hv hv1
+  rw [ENNReal.coe_toNNReal hZ]
+  change (∫⁻ sample, ENNReal.ofReal (v^(-p))*volume (ball 0 R ∩ sausage (γ sample) v) ∂P) ≤ Z
+  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+  apply (mul_le_mul_right (expected_sausage_area_bound hR hA hK hbound P γ hγ v hv) _).trans
+  change ENNReal.ofReal (v^(-p)) *
+    (ENNReal.ofReal (2*R)*ENNReal.ofReal ((A+1)*v)+ENNReal.ofReal (K*v^p)*D) ≤ Z
+  rw [mul_add]
+  apply add_le_add
+  · rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ 2*R),← ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _)]
+    apply ENNReal.ofReal_le_ofReal
+    have hh : v^(-p)*v ≤ 1 := by
+      have he : v^(-p)*v = v^(-p+1) := by
+        simpa only [Real.rpow_one] using (Real.rpow_add hv (-p) 1).symm
+      rw [he]
+      exact Real.rpow_le_one hv.le hv1 (by linarith)
+    calc
+      v^(-p)*(2*R*((A+1)*v)) = (2*R*(A+1))*(v^(-p)*v) := by ring
+      _ ≤ (2*R*(A+1))*1 := mul_le_mul_of_nonneg_left hh (by positivity)
+      _ = 2*R*(A+1) := mul_one _
+  · have he : v^(-p)*(K*v^p) = K := by
+      calc
+        v^(-p)*(K*v^p) = K*(v^(-p)*v^p) := by ring
+        _ = K := by rw [← Real.rpow_add hv,neg_add_cancel,Real.rpow_zero,mul_one]
+    rw [← mul_assoc,← ENNReal.ofReal_mul (Real.rpow_nonneg hv.le _),he]
+
+end SLEExactGauge
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/CircleMeans.lean b/lean/OAI/Probability/SLELocalization/CircleMeans.lean
index 1e5e81f2711837e3d4cf469ee3e3c1f1a1f55be1..2174a4a8850acaa26c73bfdb36956bc220cf3abf 100644
--- a/lean/OAI/Probability/SLELocalization/CircleMeans.lean
+++ b/lean/OAI/Probability/SLELocalization/CircleMeans.lean
@@ -1,96 +1,96 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-
-namespace OAI
-
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-open Layer
-
-lemma circle_derivative_square {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1))
-    {r : ℝ} (hr : |r| < 1) :
-    (circleDMean u r)^2 ≤ (2*Real.pi)*(∫ θ in -Real.pi..Real.pi,
-      ‖fderiv ℝ u (r • circleMap 0 1 θ)‖^2) := by
-  let D := fun θ : ℝ => fderiv ℝ u (r • circleMap 0 1 θ) (circleMap 0 1 θ)
-  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
-    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
-  have hgrad : Continuous (fun θ : ℝ => fderiv ℝ u (r • circleMap 0 1 θ)) := by
-    apply continuousOn_univ.1
-    exact hdu.continuousOn.comp ((continuous_circleMap 0 1).const_smul r).continuousOn
-      (fun θ _ => radial_mem hr θ)
-  have hD : Continuous D := hgrad.clm_apply (continuous_circleMap 0 1)
-  have hp : -Real.pi ≤ Real.pi := by linarith [Real.pi_pos]
-  have hc := integral_mul_square_le (continuousOn_memLp_two_Ioc hp hD.continuousOn)
-    (continuousOn_memLp_two_Ioc hp (continuous_const (y := (1:ℝ))).continuousOn)
-  have heD : (∫ θ in Ioc (-Real.pi) Real.pi, D θ*1) = circleDMean u r := by
-    simp only [mul_one,circleDMean,intervalIntegral.integral_of_le hp,D]
-  have he1 : (∫ θ in Ioc (-Real.pi) Real.pi, (1:ℝ)^2) = 2*Real.pi := by
-    rw [← intervalIntegral.integral_of_le hp]
-    simp
-    ring
-  rw [heD,he1] at hc
-  have hle : (∫ θ in Ioc (-Real.pi) Real.pi, D θ^2) ≤
-      ∫ θ in -Real.pi..Real.pi, ‖fderiv ℝ u (r • circleMap 0 1 θ)‖^2 := by
-    rw [intervalIntegral.integral_of_le hp]
-    apply integral_mono_ae ((hD.pow 2).intervalIntegrable _ _).1
-      ((hgrad.norm.pow 2).intervalIntegrable _ _).1
-    filter_upwards with θ
-    have hb := (fderiv ℝ u (r • circleMap 0 1 θ)).le_opNorm (circleMap 0 1 θ)
-    norm_num only [norm_circleMap_zero,abs_one,mul_one] at hb
-    have hh := pow_le_pow_left₀ (norm_nonneg _) hb 2
-    simpa only [Real.norm_eq_abs,sq_abs,D,Pi.pow_apply] using hh
-  exact hc.trans (by nlinarith [Real.pi_pos])
-
-theorem critical_circle_means {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1))
-    {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) (hb : b < 1) :
-    (circleMean u b-circleMean u a)^2 ≤ 2*Real.pi*Real.log (b/a)*
-      (∫ s in a..b, s*(∫ θ in -Real.pi..Real.pi, ‖fderiv ℝ u (s • circleMap 0 1 θ)‖^2)) := by
-  have hs (s : ℝ) (hs : s ∈ Icc a b) : |s| < 1 :=
-    abs_lt.2 ⟨by linarith [hs.1],hs.2.trans_lt hb⟩
-  have hD : ContinuousOn (circleDMean u) (Icc a b) := by
-    intro s hs'
-    exact ((circle_means_derivatives hu (hs s hs')).2.continuousAt).continuousWithinAt
-  have ht := weighted_interval_energy ha hab hD
-    (fun s hs' => (circle_means_derivatives hu (hs s hs')).1)
-  let E := fun s : ℝ => ∫ θ in -Real.pi..Real.pi, ‖fderiv ℝ u (s • circleMap 0 1 θ)‖^2
-  let h := (b+1)/2
-  have hh : h < 1 := by dsimp [h]; linarith
-  have hbh : b < h := by dsimp [h]; linarith
-  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
-    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
-  have henergy : ContinuousOn (fun p : ℝ × ℝ => ‖fderiv ℝ u (p.1 • circleMap 0 1 p.2)‖^2)
-      (Icc 0 h ×ˢ Icc (-Real.pi) Real.pi) :=
-    (hdu.continuousOn.comp radial_continuous.continuousOn
-      (radial_interval_maps (by norm_num) hh)).norm.pow 2
-  have hE : ContinuousOn E (Icc a b) := by
-    intro s hs'
-    exact (compact_parameter_continuousAt (F := fun s θ => ‖fderiv ℝ u (s • circleMap 0 1 θ)‖^2)
-      (by linarith [Real.pi_pos])
-      (ha.trans_le hs'.1) (hs'.2.trans_lt hbh) henergy).continuousWithinAt
-  have hle : (∫ s in a..b, s*(circleDMean u s)^2) ≤
-      ∫ s in a..b, (2*Real.pi)*(s*E s) := by
-    apply intervalIntegral.integral_mono_on hab
-      ((continuousOn_id.mul (hD.pow 2)).intervalIntegrable_of_Icc hab)
-      ((continuousOn_const.mul (continuousOn_id.mul hE)).intervalIntegrable_of_Icc hab)
-    intro s hs'
-    have hc := mul_le_mul_of_nonneg_left (circle_derivative_square hu (hs s hs'))
-      (ha.trans_le hs'.1).le
-    dsimp only [E]
-    exact hc.trans_eq (by simp only [Pi.mul_apply,id_eq]; ring)
-  have hl : 0 ≤ Real.log (b/a) := Real.log_nonneg ((one_le_div ha).2 hab)
-  have hh' := mul_le_mul_of_nonneg_left hle hl
-  rw [intervalIntegral.integral_const_mul] at hh'
-  apply ht.trans
-  convert hh' using 1
-  ring
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+
+namespace OAI
+
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+open Layer
+
+lemma circle_derivative_square {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1))
+    {r : ℝ} (hr : |r| < 1) :
+    (circleDMean u r)^2 ≤ (2*Real.pi)*(∫ θ in -Real.pi..Real.pi,
+      ‖fderiv ℝ u (r • circleMap 0 1 θ)‖^2) := by
+  let D := fun θ : ℝ => fderiv ℝ u (r • circleMap 0 1 θ) (circleMap 0 1 θ)
+  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
+    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
+  have hgrad : Continuous (fun θ : ℝ => fderiv ℝ u (r • circleMap 0 1 θ)) := by
+    apply continuousOn_univ.1
+    exact hdu.continuousOn.comp ((continuous_circleMap 0 1).const_smul r).continuousOn
+      (fun θ _ => radial_mem hr θ)
+  have hD : Continuous D := hgrad.clm_apply (continuous_circleMap 0 1)
+  have hp : -Real.pi ≤ Real.pi := by linarith [Real.pi_pos]
+  have hc := integral_mul_square_le (continuousOn_memLp_two_Ioc hp hD.continuousOn)
+    (continuousOn_memLp_two_Ioc hp (continuous_const (y := (1:ℝ))).continuousOn)
+  have heD : (∫ θ in Ioc (-Real.pi) Real.pi, D θ*1) = circleDMean u r := by
+    simp only [mul_one,circleDMean,intervalIntegral.integral_of_le hp,D]
+  have he1 : (∫ θ in Ioc (-Real.pi) Real.pi, (1:ℝ)^2) = 2*Real.pi := by
+    rw [← intervalIntegral.integral_of_le hp]
+    simp
+    ring
+  rw [heD,he1] at hc
+  have hle : (∫ θ in Ioc (-Real.pi) Real.pi, D θ^2) ≤
+      ∫ θ in -Real.pi..Real.pi, ‖fderiv ℝ u (r • circleMap 0 1 θ)‖^2 := by
+    rw [intervalIntegral.integral_of_le hp]
+    apply integral_mono_ae ((hD.pow 2).intervalIntegrable _ _).1
+      ((hgrad.norm.pow 2).intervalIntegrable _ _).1
+    filter_upwards with θ
+    have hb := (fderiv ℝ u (r • circleMap 0 1 θ)).le_opNorm (circleMap 0 1 θ)
+    norm_num only [norm_circleMap_zero,abs_one,mul_one] at hb
+    have hh := pow_le_pow_left₀ (norm_nonneg _) hb 2
+    simpa only [Real.norm_eq_abs,sq_abs,D,Pi.pow_apply] using hh
+  exact hc.trans (by nlinarith [Real.pi_pos])
+
+theorem critical_circle_means {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1))
+    {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) (hb : b < 1) :
+    (circleMean u b-circleMean u a)^2 ≤ 2*Real.pi*Real.log (b/a)*
+      (∫ s in a..b, s*(∫ θ in -Real.pi..Real.pi, ‖fderiv ℝ u (s • circleMap 0 1 θ)‖^2)) := by
+  have hs (s : ℝ) (hs : s ∈ Icc a b) : |s| < 1 :=
+    abs_lt.2 ⟨by linarith [hs.1],hs.2.trans_lt hb⟩
+  have hD : ContinuousOn (circleDMean u) (Icc a b) := by
+    intro s hs'
+    exact ((circle_means_derivatives hu (hs s hs')).2.continuousAt).continuousWithinAt
+  have ht := weighted_interval_energy ha hab hD
+    (fun s hs' => (circle_means_derivatives hu (hs s hs')).1)
+  let E := fun s : ℝ => ∫ θ in -Real.pi..Real.pi, ‖fderiv ℝ u (s • circleMap 0 1 θ)‖^2
+  let h := (b+1)/2
+  have hh : h < 1 := by dsimp [h]; linarith
+  have hbh : b < h := by dsimp [h]; linarith
+  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
+    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
+  have henergy : ContinuousOn (fun p : ℝ × ℝ => ‖fderiv ℝ u (p.1 • circleMap 0 1 p.2)‖^2)
+      (Icc 0 h ×ˢ Icc (-Real.pi) Real.pi) :=
+    (hdu.continuousOn.comp radial_continuous.continuousOn
+      (radial_interval_maps (by norm_num) hh)).norm.pow 2
+  have hE : ContinuousOn E (Icc a b) := by
+    intro s hs'
+    exact (compact_parameter_continuousAt (F := fun s θ => ‖fderiv ℝ u (s • circleMap 0 1 θ)‖^2)
+      (by linarith [Real.pi_pos])
+      (ha.trans_le hs'.1) (hs'.2.trans_lt hbh) henergy).continuousWithinAt
+  have hle : (∫ s in a..b, s*(circleDMean u s)^2) ≤
+      ∫ s in a..b, (2*Real.pi)*(s*E s) := by
+    apply intervalIntegral.integral_mono_on hab
+      ((continuousOn_id.mul (hD.pow 2)).intervalIntegrable_of_Icc hab)
+      ((continuousOn_const.mul (continuousOn_id.mul hE)).intervalIntegrable_of_Icc hab)
+    intro s hs'
+    have hc := mul_le_mul_of_nonneg_left (circle_derivative_square hu (hs s hs'))
+      (ha.trans_le hs'.1).le
+    dsimp only [E]
+    exact hc.trans_eq (by simp only [Pi.mul_apply,id_eq]; ring)
+  have hl : 0 ≤ Real.log (b/a) := Real.log_nonneg ((one_le_div ha).2 hab)
+  have hh' := mul_le_mul_of_nonneg_left hle hl
+  rw [intervalIntegral.integral_const_mul] at hh'
+  apply ht.trans
+  convert hh' using 1
+  ring
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/CommonClocks.lean b/lean/OAI/Probability/SLELocalization/CommonClocks.lean
index 1d586d5ad4d53bdddaa02afc3b6f86160c80069a..69995c8c05675ca981024d9d18c84843e69db626 100644
--- a/lean/OAI/Probability/SLELocalization/CommonClocks.lean
+++ b/lean/OAI/Probability/SLELocalization/CommonClocks.lean
@@ -1,1084 +1,1084 @@
-import Mathlib
-import OAI.Probability.SLELocalization.Charge
-import OAI.Probability.SLELocalization.PhysicalPast
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Finite Tilt estimates. -/
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology NNReal
-namespace SLEExactGauge.FiniteTilt
-
-structure CommonCutoff (A : Set ℂ) (u : ℂ → ℝ) where
-  angle : ℝ
-  angle_pos : 0 < angle
-  height : ℝ
-  height_pos : 0 < height
-  horizon : ℝ≥0
-  horizon_pos : 0 < (horizon:ℝ)
-  angle_initial : ∀ z ∈ A, angle < z.im/‖z‖
-  height_small : ∀ z ∈ A, height < z.im*Real.exp (-u z/(2*angle^2))
-  time_large : ∀ z ∈ A, u z*z.im^2/(4*angle^4) < horizon
-
-def CommonCutoff.at {A : Set ℂ} {u : ℂ → ℝ} (C : CommonCutoff A u)
-    (z : ℂ) (hz : z ∈ A) : Cutoff z (u z) where
-  angle := C.angle
-  angle_pos := C.angle_pos
-  angle_initial := C.angle_initial z hz
-  height := C.height
-  height_pos := C.height_pos
-  horizon := C.horizon
-  horizon_pos := C.horizon_pos
-  height_small := C.height_small z hz
-  time_large := C.time_large z hz
-
-lemma commonCutoffs {A : Set ℂ} {u : ℂ → ℝ} {H U α : ℝ}
-    (hH : 1 ≤ H) (hU : 0 ≤ U) (hα : 0 < α)
-    (hy : ∀ z ∈ A, 1 ≤ z.im ∧ z.im ≤ H)
-    (hu : ∀ z ∈ A, 0 ≤ u z ∧ u z ≤ U)
-    (hs : ∀ z ∈ A, α ≤ z.im/‖z‖) :
-    ∃ Cs : ℕ → CommonCutoff A u, Tendsto (fun n => (Cs n).angle) atTop (𝓝 0) := by
-  let δ : ℕ → ℝ := fun n => (α/2)/((n:ℝ)+1)
-  have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity
-  have hδα (n : ℕ) : δ n < α := by
-    have hh : 1 ≤ (n:ℝ)+1 := by linarith [Nat.cast_nonneg (α := ℝ) n]
-    have he := div_le_self (by positivity : 0 ≤ α/2) hh
-    exact he.trans_lt (by linarith)
-  let ε : ℕ → ℝ := fun n => Real.exp (-U/(2*(δ n)^2))/2
-  let T : ℕ → ℝ≥0 := fun n => ⟨U*H^2/(4*(δ n)^4)+1,by positivity⟩
-  let Cs : ℕ → CommonCutoff A u := fun n => {
-    angle := δ n
-    angle_pos := hδ n
-    height := ε n
-    height_pos := by dsimp [ε]; positivity
-    horizon := T n
-    horizon_pos := by
-      have hn : 0 ≤ U*H^2/(4*(δ n)^4) := by positivity
-      change 0 < U*H^2/(4*(δ n)^4)+1
-      linarith
-    angle_initial := fun z hz => (hδα n).trans_le (hs z hz)
-    height_small := by
-      intro z hz
-      have hu' := (hu z hz).2
-      have he : Real.exp (-U/(2*(δ n)^2)) ≤ Real.exp (-u z/(2*(δ n)^2)) := by
-        apply Real.exp_le_exp.2
-        exact div_le_div_of_nonneg_right (neg_le_neg hu') (by positivity)
-      have hz' := (hy z hz).1
-      dsimp only [ε]
-      have hp := Real.exp_pos (-U/(2*(δ n)^2))
-      nlinarith [Real.exp_pos (-u z/(2*(δ n)^2))]
-    time_large := by
-      intro z hz
-      have hz0 : 0 ≤ z.im := (by norm_num : (0:ℝ) ≤ 1).trans (hy z hz).1
-      have hz2 : z.im^2 ≤ H^2 := pow_le_pow_left₀ hz0 (hy z hz).2 2
-      have hm := mul_le_mul (hu z hz).2 hz2 (sq_nonneg _) hU
-      have hd := div_le_div_of_nonneg_right hm (by positivity : 0 ≤ 4*(δ n)^4)
-      change u z*z.im^2/(4*(δ n)^4) < U*H^2/(4*(δ n)^4)+1
-      linarith }
-  refine ⟨Cs,?_⟩
-  have ht : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
-    tendsto_atTop_add_const_right _ 1 tendsto_natCast_atTop_atTop
-  have hh := (tendsto_const_nhds (x := α/2) (f := atTop)).div_atTop ht
-  simpa only [Cs,δ] using hh
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerCutoff
-
-lemma solution_mark_dist_le {ε T D δ : ℝ} (hε : 0 < ε)
-    (hD : 0 ≤ D) (hδ : 0 ≤ δ) {U V : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
-    (h0 : dist (H 0) (G 0) ≤ δ) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    dist (H t) (G t) ≤ (δ+D)*Real.exp ((4/ε^2)*T) := by
-  let K : ℝ≥0 := ⟨4/ε^2,by positivity⟩
-  have hK : (0:ℝ) < K := by change 0 < 4/ε^2; positivity
-  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
-      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K:ℝ)*D := by
-    rw [dist_eq_norm]
-    calc
-      _ ≤ (4/ε^2)*|V s-U s| := norm_velocity_driver_sub_le hε _ _ _
-      _ ≤ (K:ℝ)*D := by
-        rw [abs_sub_comm]
-        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1,hs.2.le⟩) hK.le
-  have hh := dist_le_of_approx_trajectories_ODE
-    (fun t => velocity_lipschitz hε (U t))
-    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
-    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
-    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
-    hb h0 t ht
-  have he : gronwallBound δ K (0+(K:ℝ)*D) (t-0) =
-      δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) := by
-    simp only [gronwallBound,zero_add,sub_zero,ite_eq_right (ne_of_gt hK)]
-    field_simp
-  rw [he] at hh
-  have hexp : Real.exp ((K:ℝ)*t) ≤ Real.exp ((K:ℝ)*T) :=
-    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
-  apply hh.trans
-  change δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) ≤ (δ+D)*Real.exp ((K:ℝ)*T)
-  have h1 := mul_le_mul_of_nonneg_left hexp hδ
-  have h2 := mul_le_mul_of_nonneg_left hexp hD
-  nlinarith
-
-lemma solutionPath_joint_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
-    LipschitzWith ⟨2*Real.exp ((4/ε^2)*T),by positivity⟩
-      (fun p : C(I, ℝ) × ℂ => solutionPath hε hT p.1 p.2) := by
-  apply LipschitzWith.of_dist_le_mul
-  rintro ⟨U,z⟩ ⟨V,w⟩
-  apply (ContinuousMap.dist_le (show 0 ≤ (2*Real.exp ((4/ε^2)*T))*dist (U,z) (V,w) by positivity)).2
-  intro t
-  have hh := solution_mark_dist_le hε (D := dist U V) (δ := dist z w)
-    dist_nonneg dist_nonneg (solution_deriv hε hT U z) (solution_deriv hε hT V w)
-    (by rw [solution_initial,solution_initial])
-    (fun s _ => ContinuousMap.dist_apply_le_dist (f := U) (g := V) (projIcc 0 1 zero_le_one (s/T))) t.property
-  apply hh.trans
-  change (dist z w+dist U V)*Real.exp ((4/ε^2)*T) ≤
-    (2*Real.exp ((4/ε^2)*T))*max (dist U V) (dist z w)
-  have hsum : dist z w+dist U V ≤ 2*max (dist U V) (dist z w) := by
-    linarith [le_max_left (dist U V) (dist z w),le_max_right (dist U V) (dist z w)]
-  nlinarith [mul_le_mul_of_nonneg_right hsum (Real.exp_pos ((4/ε^2)*T)).le]
-
-lemma continuous_mark_position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
-    Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => position hε hT p.1.1 p.1.2 p.2) := by
-  exact continuous_eval.comp (((solutionPath_joint_lipschitz hε hT).continuous.comp continuous_fst).prodMk
-    ((continuous_projIcc (h := hT)).comp continuous_snd))
-
-lemma continuous_mark_variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
-    Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => variation hε hT p.1.1 p.1.2 p.2) := by
-  have hd : Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => extendDriver T p.1.1 p.2) :=
-    (continuous_joint_extendDriver T).comp ((continuous_fst.fst).prodMk continuous_snd)
-  have hw := (continuous_mark_position hε hT).sub (Complex.continuous_ofReal.comp hd)
-  exact Complex.continuous_re.comp (continuous_const.div
-    (((continuous_clampIm ε).comp hw).pow 2) (fun _ => pow_ne_zero 2 (clampIm_ne_zero hε _)))
-
-lemma continuous_mark_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
-    Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => logDerivative hε hT p.1.1 p.1.2 p.2) := by
-  have hc := intervalIntegral.continuous_parametric_primitive_of_continuous
-    (μ := volume) (a₀ := 0) (f := fun p : C(I,ℝ) × ℂ => variation hε hT p.1 p.2)
-    (continuous_mark_variation hε hT)
-  exact hc.comp (continuous_fst.prodMk
-    (continuous_subtype_val.comp ((continuous_projIcc (h := hT)).comp continuous_snd)))
-
-lemma continuous_mark_green {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (p a : ℝ) :
-    Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ =>
-      LoewnerField.field hε hT p a 1 q.1.1 q.1.2 q.2 (extendDriver T q.1.1 q.2)) := by
-  have hH := continuous_mark_position hε hT
-  have hD : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => extendDriver T q.1.1 q.2) :=
-    (continuous_joint_extendDriver T).comp (continuous_fst.fst.prodMk continuous_snd)
-  apply GreenPhysical.continuous_green_comp p a
-  · exact (Complex.continuous_re.comp hH).sub (continuous_const.mul hD)
-  · exact continuous_const.max (Complex.continuous_im.comp hH)
-  · exact continuous_mark_logDerivative hε hT
-  · intro q
-    exact LoewnerField.height_pos hε hT q.1.1 q.1.2 q.2
-
-end SLEExactGauge.LoewnerCutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerCutoff
-open LoewnerField LoewnerExit
-local instance commonClocksMarkRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance commonClocksMarkRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-
-lemma continuous_mark_field {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (p a sdev : ℝ) :
-    Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ =>
-      field hε hT p a sdev q.1.1.1 q.1.1.2 q.1.2 q.2) := by
-  have hH := (continuous_mark_position hε hT).comp
-    (continuous_fst : Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => q.1))
-  apply GreenPhysical.continuous_green_comp p a
-  · exact (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd)
-  · exact continuous_const.max (Complex.continuous_im.comp hH)
-  · exact (continuous_mark_logDerivative hε hT).comp continuous_fst
-  · intro q
-    exact height_pos hε hT q.1.1.1 q.1.1.2 q.1.2
-
-lemma continuous_mark_first {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (p a sdev : ℝ) :
-    Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ =>
-      iteratedDeriv 1 (field hε hT p a sdev q.1.1.1 q.1.1.2 q.1.2) q.2) := by
-  simp only [first_eq]
-  have hH := (continuous_mark_position hε hT).comp
-    (continuous_fst : Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => sdev*q.2))
-  have hy : Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => height hε hT q.1.1.1 q.1.1.2 q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  exact ((continuous_const.mul hx).div ((hx.pow 2).add (hy.pow 2))
-    (fun q => (GreenPhysical.q_pos _ (height_pos hε hT q.1.1.1 q.1.1.2 q.1.2)).ne')).mul
-      (continuous_mark_field hε hT p a sdev)
-
-lemma continuous_mark_deficit {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ) :
-    Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => deficit hε hT q.1.1 q.1.2 ρ δ q.2) := by
-  have hH := continuous_mark_position hε hT
-  have hD : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => extendDriver T q.1.1 q.2) :=
-    (continuous_joint_extendDriver T).comp (continuous_fst.fst.prodMk continuous_snd)
-  have hr : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => radius hε hT q.1.1 q.1.2 q.2) :=
-    (continuous_const.mul (Complex.continuous_im.comp hH)).div
-      (Real.continuous_exp.comp (continuous_mark_logDerivative hε hT)) (fun _ => Real.exp_ne_zero _)
-  have hs : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => sine hε hT q.1.1 q.1.2 q.2) :=
-    (Complex.continuous_im.comp hH).div
-      (((continuous_clampIm ε).comp (hH.sub (Complex.continuous_ofReal.comp hD))).norm)
-      (fun _ => norm_ne_zero_iff.2 (clampIm_ne_zero hε _))
-  exact (hr.sub continuous_const).min
-    (((Complex.continuous_im.comp hH).sub continuous_const).min (hs.sub continuous_const))
-
-lemma continuous_mark_exitPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ) :
-    Continuous (fun q : C(I,ℝ) × ℂ => exitPath hε hT q.1 q.2 ρ δ) := by
-  apply ContinuousMap.continuous_of_continuous_uncurry _
-  exact (continuous_mark_deficit hε hT ρ δ).comp
-    (continuous_fst.prodMk (continuous_const.mul (continuous_subtype_val.comp continuous_snd)))
-
-lemma measurable_mark_stop {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ) :
-    Measurable (fun q : C(I,ℝ) × ℂ => stop hε hT q.1 q.2 ρ δ) :=
-  measurable_const.mul (ContinuousExit.measurable_time.comp (continuous_mark_exitPath hε hT ρ δ).measurable)
-
-lemma isOpen_mark_positive_on {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ t : ℝ) :
-    IsOpen {q : C(I,ℝ) × ℂ | ∀ s ∈ Icc 0 t, 0 < deficit hε hT q.1 q.2 ρ δ s} := by
-  let S : Set ((C(I,ℝ) × ℂ) × Icc 0 t) := {q | deficit hε hT q.1.1 q.1.2 ρ δ q.2 ≤ 0}
-  have hc : IsClosed S := isClosed_le ((continuous_mark_deficit hε hT ρ δ).comp
-    (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))) continuous_const
-  have hh := isClosedMap_fst_of_compactSpace S hc
-  have he : {q : C(I,ℝ) × ℂ | ∀ s ∈ Icc 0 t, 0 < deficit hε hT q.1 q.2 ρ δ s} = (Prod.fst '' S)ᶜ := by
-    ext q
-    simp only [mem_ofPred_eq,mem_compl_iff,mem_image,S,not_exists,not_and]
-    constructor
-    · intro hp v hv hvq
-      subst q
-      exact (not_lt_of_ge hv) (hp v.2 v.2.property)
-    · intro hp s hs
-      by_contra hn
-      exact hp (q,⟨s,hs⟩) (not_lt.1 hn) rfl
-  rw [he]
-  exact hh.isOpen_compl
-
-lemma measurable_mark_coeffAt {ε : ℝ} (hε : 0 < ε) (s : ℝ≥0) (p a sdev ρ δ : ℝ) :
-    Measurable (fun q : (C(I,ℝ) × ℝ) × ℂ =>
-      LoewnerLocalized.coeffAt hε 1 s p a sdev q.2 ρ δ q.1) := by
-  have hp : Continuous (fun q : (C(I,ℝ) × ℝ) × ℂ => (((q.1.1,q.2),(s:ℝ)),q.1.2)) :=
-    ((continuous_fst.fst.prodMk continuous_snd).prodMk continuous_const).prodMk continuous_fst.snd
-  have hc := (continuous_mark_first hε s.coe_nonneg p a sdev).comp hp
-  dsimp only [Function.comp_def] at hc
-  exact hc.measurable.ite (((isOpen_mark_positive_on hε s.coe_nonneg ρ δ s).preimage
-    (continuous_fst.fst.prodMk continuous_snd)).measurableSet) measurable_const
-
-end SLEExactGauge.LoewnerCutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.PhysicalPast
-
-lemma joint_avoids_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (ε : ℝ) (T : ℝ≥0) :
-    @MeasurableSet (Ω × ℂ) (Prod.instMeasurableSpace (α := Ω) (β := ℂ)
-      (m₁ := BrownianPredictable.pastSigma B T) (m₂ := inferInstance))
-      {q | q.1 ∈ avoids κ B q.2 ε T} := by
-  let : MeasurableSpace Ω := BrownianPredictable.pastSigma B T
-  change MeasurableSet {q : Ω × ℂ | ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤
-    dist q.2 (CanonicalTrace.value κ B (sampleTime T (TopologicalSpace.denseSeq I n)) q.1)}
-  simp only [ofPred_exists,ofPred_forall]
-  apply MeasurableSet.iUnion
-  intro m
-  apply MeasurableSet.iInter
-  intro n
-  exact measurableSet_le measurable_const (measurable_snd.dist
-    (((CanonicalTrace.measurable_past κ B _).mono
-      (BrownianPredictable.pastSigma_mono B (sampleTime_le T (TopologicalSpace.denseSeq I n))) le_rfl).comp measurable_fst))
-
-end SLEExactGauge.PhysicalPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open BrownianPredictable
-local instance commonClocksPastRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance commonClocksPastRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-
-lemma joint_rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    {ε : ℝ} (hε : 0 < ε) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0) (p a sdev ρ δ : ℝ) :
-    @Measurable (Ω × ℂ) ℝ (Prod.instMeasurableSpace (α := Ω) (β := ℂ)
-      (m₁ := pastSigma B s) (m₂ := inferInstance)) _
-      (fun q => rawCoeff hε 1 B T s p a sdev q.2 ρ δ q.1) := by
-  let : MeasurableSpace Ω := pastSigma B s
-  unfold rawCoeff
-  split_ifs
-  · have hp : Measurable (fun q : Ω × ℂ =>
-        ((BrownianDriver.path B sdev s q.1,B s q.1),q.2)) :=
-      (((BrownianDriver.path_measurable_past B sdev s).comp measurable_fst).prodMk
-        ((past_measurable_eval B (s := s) le_rfl).comp measurable_fst)).prodMk measurable_snd
-    have hh := (LoewnerCutoff.measurable_mark_coeffAt hε s p a sdev ρ δ).comp hp
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  · exact measurable_const
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.PhysicalPast
-
-lemma joint_avoids_nullMeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (ν : Measure ℂ) [SFinite ν] (κ ε : ℝ) (T : ℝ≥0) :
-    NullMeasurableSet {q : Ω × ℂ | q.1 ∈ avoids κ B q.2 ε T} (P.prod ν) := by
-  change NullMeasurableSet {q : Ω × ℂ | ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤
-    dist q.2 (CanonicalTrace.value κ B (sampleTime T (TopologicalSpace.denseSeq I n)) q.1)} (P.prod ν)
-  simp only [ofPred_exists,ofPred_forall]
-  apply NullMeasurableSet.iUnion
-  intro m
-  apply NullMeasurableSet.iInter
-  intro n
-  exact nullMeasurableSet_le aemeasurable_const
-    (measurable_snd.aemeasurable.dist ((CanonicalTrace.aemeasurable κ hB _).comp_fst))
-
-end SLEExactGauge.PhysicalPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-local instance commonClocksJointRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance commonClocksJointRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-
-lemma joint_rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (ν : Measure ℂ) [SFinite ν] {ε : ℝ} (hε : 0 < ε)
-    (T s : ℝ≥0) (p a sdev ρ δ : ℝ) :
-    AEMeasurable (fun q : Ω × ℂ => rawCoeff hε 1 B T s p a sdev q.2 ρ δ q.1) (P.prod ν) := by
-  unfold rawCoeff
-  split_ifs
-  · have hp : AEMeasurable (fun q : Ω × ℂ =>
-        ((BrownianDriver.path B sdev s q.1,B s q.1),q.2)) (P.prod ν) :=
-      (((BrownianDriver.path_aemeasurable hB sdev s).comp_fst).prodMk
-        ((hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable.comp_fst)).prodMk measurable_snd.aemeasurable
-    have hh := (LoewnerCutoff.measurable_mark_coeffAt hε s p a sdev ρ δ).comp_aemeasurable hp
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  · exact aemeasurable_const
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerLocalized BrownianPredictable
-
-def coeff {Ω : Type*} {ε : ℝ} (hε : 0 < ε) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (T s : ℝ≥0) (ρ δ e : ℝ) (q : Ω × ℂ) : ℝ :=
-  if q.1 ∈ PhysicalPast.avoids κ B q.2 e s then
-    rawCoeff hε 1 B T s (1-κ/8) (8/κ-1) (Real.sqrt κ) q.2 ρ δ q.1 else 0
-
-lemma coeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    {ε : ℝ} (hε : 0 < ε) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0) (ρ δ e : ℝ) :
-    @Measurable (Ω × ℂ) ℝ (Prod.instMeasurableSpace (m₁ := pastSigma B s) (m₂ := inferInstance)) _
-      (coeff hε κ B T s ρ δ e) := by
-  exact (joint_rawCoeff_measurable_past hε B T s _ _ _ ρ δ).ite
-    (PhysicalPast.joint_avoids_measurable_past κ B e s) measurable_const
-
-lemma coeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (ν : Measure ℂ) [SFinite ν] {ε : ℝ} (hε : 0 < ε)
-    (κ : ℝ) (T s : ℝ≥0) (ρ δ e : ℝ) :
-    AEMeasurable (coeff hε κ B T s ρ δ e) (P.prod ν) := by
-  exact (joint_rawCoeff_aemeasurable hB ν hε T s _ _ _ ρ δ).indicator₀
-    (PhysicalPast.joint_avoids_nullMeasurable hB ν κ e s)
-
-lemma coeff_uniform_bound {Ω : Type*} {ε : ℝ} (hε : 0 < ε)
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (ρ δ : ℝ) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ e s q, |coeff hε κ B T s ρ δ e q| ≤ K := by
-  let p := 1-κ/8
-  let a := 8/κ-1
-  have hp : 0 ≤ p := by dsimp [p]; linarith [hκ.2]
-  have ha : 0 ≤ a := by dsimp [a]; have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by linarith [hκ.2]); linarith
-  let K := max 0 (|Real.sqrt κ| *((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε))
-  refine ⟨K,le_max_left _ _,?_⟩
-  intro e s q
-  unfold coeff rawCoeff coeffAt
-  split_ifs with h1 h2 h3
-  · have hh := first_bound hε s.coe_nonneg p a (Real.sqrt κ) hp ha
-      (BrownianDriver.path B (Real.sqrt κ) s q.1) q.2 s (B s q.1)
-    apply hh.trans
-    have hs : (s:ℝ) ≤ T := h2.le
-    have he : Real.exp ((2/ε^2)*s) ≤ Real.exp ((2/ε^2)*T) :=
-      Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left hs (by positivity))
-    have hb : (2*ε/Real.exp ((2/ε^2)*T)) ≤ (2*ε/Real.exp ((2/ε^2)*s)) :=
-      div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) he
-    have hr := Real.rpow_le_rpow_of_nonpos (by positivity : 0 < 2*ε/Real.exp ((2/ε^2)*T)) hb (neg_nonpos.2 hp)
-    apply le_trans _ (le_max_right _ _)
-    gcongr
-  all_goals simpa only [abs_zero] using (le_max_left 0 (|Real.sqrt κ| *((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε)))
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.CompactHit
-local instance commonClocksHitComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance commonClocksHitComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-def hits (e : ℝ) (p : C(I,ℂ) × ℂ) : Prop := ∃ q : I, dist p.2 (p.1 q) ≤ e
-
-def time (e : ℝ) (p : C(I,ℂ) × ℂ) : I := sInf {q : I | dist p.2 (p.1 q) ≤ e}
-
-lemma hits_closed (e : ℝ) : IsClosed {p : C(I,ℂ) × ℂ | hits e p} := by
-  have hc : IsClosed {q : (C(I,ℂ) × ℂ) × I | dist q.1.2 (q.1.1 q.2) ≤ e} :=
-    isClosed_le (continuous_fst.snd.dist (continuous_eval.comp (continuous_fst.fst.prodMk continuous_snd))) continuous_const
-  have he : {p : C(I,ℂ) × ℂ | hits e p} = Prod.fst ''
-      {q : (C(I,ℂ) × ℂ) × I | dist q.1.2 (q.1.1 q.2) ≤ e} := by
-    ext p
-    simp [hits]
-  rw [he]
-  exact isClosedMap_fst_of_compactSpace _ hc
-
-lemma time_mem {e : ℝ} {p : C(I,ℂ) × ℂ} (hp : hits e p) :
-    dist p.2 (p.1 (time e p)) ≤ e := by
-  exact (isClosed_le (continuous_const.dist p.1.continuous) continuous_const).isCompact.sInf_mem hp
-
-lemma time_le {e : ℝ} {p : C(I,ℂ) × ℂ} {q : I} (hq : dist p.2 (p.1 q) ≤ e) :
-    time e p ≤ q := sInf_le hq
-
-lemma before {e : ℝ} {p : C(I,ℂ) × ℂ} {q : I} (hq : q < time e p) :
-    e < dist p.2 (p.1 q) := by
-  by_contra hm
-  exact not_le_of_gt hq (time_le (not_lt.1 hm))
-
-lemma time_eq_one_of_not_hits {e : ℝ} {p : C(I,ℂ) × ℂ} (hp : ¬ hits e p) : time e p = 1 := by
-  have he : {q : I | dist p.2 (p.1 q) ≤ e} = ∅ := by
-    ext q
-    simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]
-    exact fun hq => hp ⟨q,hq⟩
-  simp only [time,he,sInf_empty]
-  rfl
-
-lemma time_le_iff {e : ℝ} {p : C(I,ℂ) × ℂ} {q : I} (hq : q < 1) :
-    time e p ≤ q ↔ ∃ r : I, r ≤ q ∧ dist p.2 (p.1 r) ≤ e := by
-  constructor
-  · intro ht
-    have hp : hits e p := by
-      by_contra hn
-      rw [time_eq_one_of_not_hits hn] at ht
-      exact not_le_of_gt hq ht
-    exact ⟨time e p,ht,time_mem hp⟩
-  · rintro ⟨r,hr,hmem⟩
-    exact (time_le hmem).trans hr
-
-lemma measurable_time (e : ℝ) : Measurable (time e) := by
-  apply measurable_of_Iic
-  intro q
-  by_cases hq : q = 1
-  · have he : time e ⁻¹' Iic q = univ := by
-      subst q
-      ext p
-      simp only [mem_preimage,mem_Iic,mem_univ,iff_true]
-      exact (time e p).property.2
-    rw [he]
-    exact MeasurableSet.univ
-  · have hq1 : q < 1 := lt_of_le_of_ne q.property.2 hq
-    have hc : IsClosed {r : (C(I,ℂ) × ℂ) × I | r.2 ≤ q ∧ dist r.1.2 (r.1.1 r.2) ≤ e} :=
-      (isClosed_le continuous_snd continuous_const).inter
-        (isClosed_le (continuous_fst.snd.dist (continuous_eval.comp (continuous_fst.fst.prodMk continuous_snd))) continuous_const)
-    have he : time e ⁻¹' Iic q = Prod.fst ''
-        {r : (C(I,ℂ) × ℂ) × I | r.2 ≤ q ∧ dist r.1.2 (r.1.1 r.2) ≤ e} := by
-      ext p
-      change (time e p ≤ q) ↔ ∃ r, r ∈ {r : (C(I,ℂ) × ℂ) × I | r.2 ≤ q ∧ dist r.1.2 (r.1.1 r.2) ≤ e} ∧ r.1 = p
-      rw [time_le_iff hq1]
-      constructor
-      · rintro ⟨r,hr,hd⟩
-        exact ⟨(p,r),⟨hr,hd⟩,rfl⟩
-      · rintro ⟨⟨p',r⟩,⟨hr,hd⟩,he⟩
-        change p' = p at he
-        subst p'
-        exact ⟨r,hr,hd⟩
-    rw [he]
-    exact (isClosedMap_fst_of_compactSpace _ hc).measurableSet
-
-end SLEExactGauge.CompactHit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.CompactHit
-
-def realTime (T e : ℝ) (V : C(I,ℂ)) (z : ℂ) : ℝ := T*time e (V,z)
-
-lemma realTime_bounds {T : ℝ} (hT : 0 ≤ T) (e : ℝ) (V : C(I,ℂ)) (z : ℂ) :
-    realTime T e V z ∈ Icc 0 T :=
-  ⟨mul_nonneg hT (time e (V,z)).property.1,by
-    simpa only [realTime,mul_one] using mul_le_mul_of_nonneg_left (time e (V,z)).property.2 hT⟩
-
-lemma realTime_le {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
-    (hV : ∀ q : I, V q = γ (T*q)) {t : ℝ} (ht : t ∈ Icc 0 T)
-    (hd : dist z (γ t) ≤ e) : realTime T e V z ≤ t := by
-  let q : I := ⟨t/T,⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩⟩
-  have he : T*(q:ℝ) = t := mul_div_cancel₀ _ hT.ne'
-  have hq : dist z (V q) ≤ e := by rw [hV,he]; exact hd
-  have hm : (time e (V,z):ℝ) ≤ (q:ℝ) := time_le hq
-  have hx := mul_le_mul_of_nonneg_left hm hT.le
-  rwa [he] at hx
-
-lemma realTime_hits {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
-    (hV : ∀ q : I, V q = γ (T*q)) {t : ℝ} (ht : t ∈ Icc 0 T)
-    (hd : dist z (γ t) ≤ e) : hits e (V,z) := by
-  let q : I := ⟨t/T,⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩⟩
-  refine ⟨q,?_⟩
-  change dist z (V q) ≤ e
-  rw [hV,mul_div_cancel₀ _ hT.ne']
-  exact hd
-
-lemma realTime_stop {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
-    (hV : ∀ q : I, V q = γ (T*q)) (hh : hits e (V,z))
-    (hl : z ∈ localizationDomain γ (realTime T e V z)) :
-    firstLiveDistanceStop γ z e = ENNReal.ofReal (realTime T e V z) := by
-  have hb := realTime_bounds hT.le e V z
-  have hd : dist z (γ (realTime T e V z)) ≤ e := by
-    simpa only [hV,realTime] using time_mem hh
-  let t₀ : ℝ≥0 := ⟨realTime T e V z,hb.1⟩
-  have hle : firstLiveDistanceStop γ z e ≤ (t₀ : ℝ≥0∞) := PhysicalStop.le_of_hit hl hd
-  have hc : (t₀ : ℝ≥0∞) = ENNReal.ofReal (realTime T e V z) :=
-    ENNReal.ofReal_coe_nnreal.symm
-  rw [hc] at hle
-  apply le_antisymm hle
-  apply le_sInf
-  rintro r ⟨s,⟨hsl,hsd⟩,rfl⟩
-  have ht : realTime T e V z ≤ s := by
-    by_cases hsT : (s:ℝ) ≤ T
-    · exact realTime_le hT V γ z hV ⟨s.coe_nonneg,hsT⟩ hsd
-    · exact hb.2.trans (not_le.1 hsT).le
-  exact (ENNReal.ofReal_le_ofReal ht).trans_eq (ENNReal.ofReal_coe_nnreal (p := s))
-
-lemma live_byTime_iff {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
-    (hc : ContinuousOn γ (Ici 0)) (hV : ∀ q : I, V q = γ (T*q)) :
-    firstLiveDistanceStop γ z e ≤ ENNReal.ofReal T ↔
-      hits e (V,z) ∧ z ∈ localizationDomain γ (realTime T e V z) := by
-  constructor
-  · intro hh
-    obtain ⟨t,he,hl,hd,hmin⟩ := PhysicalStop.finite_spec hc
-      (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hh)
-    have ht : (t:ℝ) ≤ T := by
-      rw [he,← ENNReal.ofReal_coe_nnreal (p := t)] at hh
-      exact (ENNReal.ofReal_le_ofReal_iff hT.le).1 hh
-    exact ⟨realTime_hits hT V γ z hV ⟨t.coe_nonneg,ht⟩ hd,
-      survivingDomain_mono (realTime_le hT V γ z hV ⟨t.coe_nonneg,ht⟩ hd) hl⟩
-  · rintro ⟨hh,hl⟩
-    rw [realTime_stop hT V γ z hV hh hl]
-    exact ENNReal.ofReal_le_ofReal (realTime_bounds hT.le e V z).2
-
-lemma cap_eq_realTime {T e τ : ℝ} (hT : 0 < T) (hτ : τ ∈ Icc 0 T)
-    (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ) (hγ : ContinuousOn γ (Ici 0))
-    (hV : ∀ q : I, V q = γ (T*q)) (hl : z ∈ localizationDomain γ τ) :
-    PhysicalStop.cap (firstLiveDistanceStop γ z e) τ = min (realTime T e V z) τ := by
-  let r := realTime T e V z
-  have hr := realTime_bounds hT.le e V z
-  have hsmall (hrt : r < τ) : firstLiveDistanceStop γ z e = ENNReal.ofReal r := by
-    have hh : hits e (V,z) := by
-      by_contra hn
-      have he : r = T := by simp only [r,realTime,time_eq_one_of_not_hits hn]; simp
-      linarith [hτ.2]
-    exact realTime_stop hT V γ z hV hh (survivingDomain_mono hrt.le hl)
-  by_cases hfin : firstLiveDistanceStop γ z e = ⊤
-  · rw [PhysicalStop.cap,ite_eq_left hfin]
-    apply (min_eq_right _).symm
-    by_contra hn
-    have he := hsmall (lt_of_not_ge hn)
-    rw [hfin] at he
-    exact ENNReal.ofReal_ne_top he.symm
-  · obtain ⟨s,hs,hsl,hsd,hmin⟩ := PhysicalStop.finite_spec hγ hfin
-    rw [PhysicalStop.cap,ite_eq_right hfin,hs,ENNReal.coe_toReal]
-    apply le_antisymm
-    · apply le_min _ (min_le_right _ _)
-      by_cases hrt : r < τ
-      · have he := hsmall hrt
-        rw [hs,← ENNReal.ofReal_coe_nnreal (p := s)] at he
-        have hsre : (s:ℝ) = r := (ENNReal.ofReal_eq_ofReal_iff s.coe_nonneg (show 0 ≤ r from hr.1)).1 he
-        rw [hsre]
-        exact min_le_left _ _
-      · exact (min_le_right _ _).trans (not_lt.1 hrt)
-    · apply le_min _ (min_le_right _ _)
-      by_cases hsT : (s:ℝ) ≤ T
-      · exact (min_le_left _ _).trans (realTime_le hT V γ z hV ⟨s.coe_nonneg,hsT⟩ hsd)
-      · exact (min_le_right _ _).trans (hτ.2.trans (not_le.1 hsT).le)
-
-lemma realTime_antitone (T : ℝ) (hT : 0 ≤ T) (V : C(I,ℂ)) (z : ℂ) {e e' : ℝ} (he : e ≤ e') :
-    realTime T e' V z ≤ realTime T e V z := by
-  apply mul_le_mul_of_nonneg_left _ hT
-  by_cases hh : hits e (V,z)
-  · exact time_le ((time_mem hh).trans he)
-  · rw [time_eq_one_of_not_hits hh]
-    exact (time e' (V,z)).property.2
-
-end SLEExactGauge.CompactHit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized
-local instance commonClocksObservedRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance commonClocksObservedRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance commonClocksObservedComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance commonClocksObservedComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-def observedTime {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e : ℝ)
-    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ :=
-  min (CompactHit.realTime T e q.1.2 q.2) (stop hε hT q.1.1 q.2 ρ δ)
-
-lemma measurable_observedTime {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e : ℝ) :
-    Measurable (observedTime hε hT ρ δ e) := by
-  have hv : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => (q.1.2,q.2)) :=
-    measurable_fst.snd.prodMk measurable_snd
-  have hv' := (measurable_subtype_coe.comp ((CompactHit.measurable_time e).comp hv)).const_mul T
-  have hu : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => (q.1.1,q.2)) :=
-    measurable_fst.fst.prodMk measurable_snd
-  have hs := (measurable_mark_stop hε hT ρ δ).comp hu
-  have hh := hv'.min hs
-  exact hh
-
-def observedValue {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e p a : ℝ)
-    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ :=
-  let t := observedTime hε hT ρ δ e q
-  field hε hT p a 1 q.1.1 q.2 t (extendDriver T q.1.1 t)
-
-lemma measurable_observedValue {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e p a : ℝ) :
-    Measurable (observedValue hε hT ρ δ e p a) := by
-  have ht := measurable_observedTime hε hT ρ δ e
-  have hx : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ =>
-      (((q.1.1,q.2),observedTime hε hT ρ δ e q),extendDriver T q.1.1 (observedTime hε hT ρ δ e q))) :=
-    ((measurable_fst.fst.prodMk measurable_snd).prodMk ht).prodMk
-      ((continuous_joint_extendDriver T).measurable.comp (measurable_fst.fst.prodMk ht))
-  have hh := (continuous_mark_field hε hT p a 1).measurable.comp hx
-  exact hh
-
-lemma coeff_common {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
-    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (z : ℂ) (hz : z ∈ A) {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ) (e : ℝ) (s : ℝ≥0) (ω : Ω) :
-    (C.at z hz).physicalCoeff 1 κ B e s ω =
-      coeff C.height_pos κ B C.horizon s ρ C.angle e (ω,z) := by
-  simp only [FiniteTilt.Cutoff.physicalCoeff,FiniteTilt.CommonCutoff.at,coeff,hr]
-
-lemma observedValue_eq {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
-    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
-    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
-    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e : ℝ) :
-    observedValue C.height_pos C.horizon.coe_nonneg ρ C.angle e (1-κ/8) (8/κ-1)
-      ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V),z) =
-        (C.at z hz).physicalValue κ B γ e ω := by
-  let Cz := C.at z hz
-  have hs := Cz.source_live κ B ω hm hhalf hu hc hb0
-  have hb := Cz.bounds κ B ω
-  have ht := CompactHit.cap_eq_realTime (e := e) C.horizon_pos hb V (γ ω) z hm.1 hV hs
-  have he : observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e
-      ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V),z) = Cz.physicalTime κ B γ e ω := by
-    rw [FiniteTilt.Cutoff.physicalTime,ht]
-    simp only [observedTime,FiniteTilt.Cutoff.stopTime,FiniteTilt.Cutoff.paths,Cz,FiniteTilt.CommonCutoff.at,hr]
-  have hb' := Cz.physicalTime_bounds κ B γ e ω
-  have hU := Cz.path_eq κ B ω hc ⟨hb'.1,hb'.2.trans hb.2⟩
-  dsimp only [observedValue]
-  rw [he]
-  change field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) 1 _ z _ _ =
-    field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) _ z _ _
-  dsimp only [field]
-  rw [one_mul]
-  change extendDriver (C.horizon:ℝ) (BrownianDriver.path B (Real.sqrt κ) C.horizon ω)
-    (Cz.physicalTime κ B γ e ω) = _ at hU
-  rw [hU]
-  rfl
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Mesh
-open TaylorMeshes
-
-def slab (D : ℝ → ℝ) (a b t : ℝ) : ℝ :=
-  (if t < b then D t else 0) - (if t < a then D t else 0)
-
-lemma prestop_product (D E : ℝ → ℝ) (a b t : ℝ) :
-    (if t < a then D t else 0)*(if t < b then E t else 0) =
-      (if t < min a b then D t*E t else 0) := by
-  simp only [lt_min_iff]
-  split_ifs <;> simp_all
-
-lemma slab_product (D E : ℝ → ℝ) (a b c d t : ℝ) :
-    slab D a b t * slab E c d t =
-      (if t < min b d then D t*E t else 0) -
-      (if t < min b c then D t*E t else 0) -
-      (if t < min a d then D t*E t else 0) +
-      (if t < min a c then D t*E t else 0) := by
-  unfold slab
-  rw [sub_mul,mul_sub,mul_sub]
-  simp only [prestop_product]
-  ring
-
-lemma slab_riemann_tendsto {T a b c d : ℝ} (hT : 0 ≤ T)
-    (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T)
-    (hc : c ∈ Icc 0 T) (hd : d ∈ Icc 0 T)
-    {D E : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T))
-    (hE : ContinuousOn E (Icc 0 T)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (slab D a b (mesh T n i)*slab E c d (mesh T n i))*(T/(2:ℝ)^n))
-      atTop (𝓝 ((∫ t in 0..min b d, D t*E t) - (∫ t in 0..min b c, D t*E t) -
-        (∫ t in 0..min a d, D t*E t) + (∫ t in 0..min a c, D t*E t))) := by
-  have hDE : ContinuousOn (fun t => D t*E t) (Icc 0 T) := hD.mul hE
-  have hlim (s t : ℝ) (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T) :=
-    StoppedMeshes.prestop_riemann_tendsto hT (le_min hs.1 ht.1)
-      ((min_le_left s t).trans hs.2) (hDE.mono (Icc_subset_Icc_right ((min_le_left s t).trans hs.2)))
-  have hh := ((hlim b d hb hd).sub (hlim b c hb hc)).sub (hlim a d ha hd) |>.add (hlim a c ha hc)
-  convert hh using 1
-  funext n
-  simp [slab_product,sub_mul,add_mul,Finset.sum_sub_distrib,Finset.sum_add_distrib,ite_mul]
-
-lemma integral_prestop {T a : ℝ} (ha : a ∈ Icc 0 T) (D : ℝ → ℝ) :
-    (∫ t in (0:ℝ)..T, if t < a then D t else 0) = ∫ t in (0:ℝ)..a, D t := by
-  calc
-    _ = ∫ t in (0:ℝ)..T, Set.indicator {t : ℝ | t ≤ a} D t := by
-      apply intervalIntegral.integral_congr_ae
-      filter_upwards [show ∀ᵐ t : ℝ ∂volume, t ≠ a from by rw [ae_iff]; simp] with t ht htI
-      simp only [Set.indicator_apply,mem_ofPred_eq]
-      congr 1
-      exact propext (lt_iff_le_and_ne.trans (and_iff_left ht))
-    _ = _ := intervalIntegral.integral_indicator ha
-
-lemma prestop_intervalIntegrable {T a : ℝ} (hT : 0 ≤ T)
-    {D : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T)) :
-    IntervalIntegrable (fun t => if t < a then D t else 0) volume 0 T := by
-  have hh : IntegrableOn D (Ioc 0 T) volume :=
-    (hD.integrableOn_Icc).mono_set Ioc_subset_Icc_self
-  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hT]
-  exact hh.indicator measurableSet_Iio
-
-lemma slab_product_integral {T a b c d : ℝ} (hT : 0 ≤ T)
-    (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T)
-    (hc : c ∈ Icc 0 T) (hd : d ∈ Icc 0 T)
-    {D E : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T))
-    (hE : ContinuousOn E (Icc 0 T)) :
-    (∫ t in (0:ℝ)..T, slab D a b t*slab E c d t) =
-      (∫ t in 0..min b d, D t*E t) - (∫ t in 0..min b c, D t*E t) -
-      (∫ t in 0..min a d, D t*E t) + (∫ t in 0..min a c, D t*E t) := by
-  simp_rw [slab_product]
-  have hDE : ContinuousOn (fun t => D t*E t) (Icc 0 T) := hD.mul hE
-  have hi (s : ℝ) := prestop_intervalIntegrable (a := s) hT hDE
-  rw [intervalIntegral.integral_add ((hi _ |>.sub (hi _)).sub (hi _)) (hi _),
-    intervalIntegral.integral_sub (hi _ |>.sub (hi _)) (hi _),
-    intervalIntegral.integral_sub (hi _) (hi _)]
-  rw [integral_prestop ⟨le_min hb.1 hd.1,(min_le_left _ _).trans hb.2⟩,
-    integral_prestop ⟨le_min hb.1 hc.1,(min_le_left _ _).trans hb.2⟩,
-    integral_prestop ⟨le_min ha.1 hd.1,(min_le_left _ _).trans ha.2⟩,
-    integral_prestop ⟨le_min ha.1 hc.1,(min_le_left _ _).trans ha.2⟩]
-
-lemma slab_riemann_tendsto_integral {T a b c d : ℝ} (hT : 0 ≤ T)
-    (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T)
-    (hc : c ∈ Icc 0 T) (hd : d ∈ Icc 0 T)
-    {D E : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T))
-    (hE : ContinuousOn E (Icc 0 T)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (slab D a b (mesh T n i)*slab E c d (mesh T n i))*(T/(2:ℝ)^n))
-      atTop (𝓝 (∫ t in (0:ℝ)..T, slab D a b t*slab E c d t)) := by
-  rw [slab_product_integral hT ha hb hc hd hD hE]
-  exact slab_riemann_tendsto hT ha hb hc hd hD hE
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge.Mesh
-
-lemma slab_eq_zero_of_not_mem {D : ℝ → ℝ} {a b t : ℝ} (hab : a ≤ b) (ht : t ∉ Icc a b) :
-    slab D a b t = 0 := by
-  unfold slab
-  by_cases hta : t < a
-  · simp [hta,hta.trans_le hab]
-  · have htb : ¬t < b := by intro h; exact ht ⟨not_lt.1 hta,h.le⟩
-    simp [hta,htb]
-
-lemma slab_abs_le {D : ℝ → ℝ} {a b t M : ℝ} (hab : a ≤ b) (hM : 0 ≤ M) (h : |D t| ≤ M) :
-    |slab D a b t| ≤ M := by
-  unfold slab
-  by_cases hta : t < a
-  · simpa [hta,hta.trans_le hab] using hM
-  · by_cases htb : t < b
-    · simpa [hta,htb] using h
-    · simpa [hta,htb] using hM
-
-lemma measurable_slab {Z : Type*} [MeasurableSpace Z] {D : Z → ℝ → ℝ} {a b : Z → ℝ}
-    (hD : Measurable (fun q : ℝ × Z => D q.2 q.1)) (ha : Measurable a) (hb : Measurable b) :
-    Measurable (fun q : ℝ × Z => slab (D q.2) (a q.2) (b q.2) q.1) := by
-  exact (hD.ite (measurableSet_lt measurable_fst (hb.comp measurable_snd)) measurable_const).sub
-    (hD.ite (measurableSet_lt measurable_fst (ha.comp measurable_snd)) measurable_const)
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerExit
-local instance commonClocksBoundsRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance commonClocksBoundsRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance commonClocksBoundsComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance commonClocksBoundsComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-lemma observedTime_bounds {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e : ℝ)
-    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : observedTime hε hT ρ δ e q ∈ Icc 0 T := by
-  have hr := CompactHit.realTime_bounds hT e q.1.2 q.2
-  have hs := stop_bounds hε hT q.1.1 q.2 ρ δ
-  exact ⟨le_min hr.1 hs.1,(min_le_left _ _).trans hr.2⟩
-
-lemma observedTime_antitone {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ)
-    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) {e r : ℝ} (her : e ≤ r) :
-    observedTime hε hT ρ δ r q ≤ observedTime hε hT ρ δ e q :=
-  min_le_min_right _ (CompactHit.realTime_antitone T hT q.1.2 q.2 her)
-
-lemma observedTime_eq {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
-    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
-    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
-    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e : ℝ) :
-    observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e
-      ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V),z) =
-        (C.at z hz).physicalTime κ B γ e ω := by
-  let Cz := C.at z hz
-  have hs := Cz.source_live κ B ω hm hhalf hu hc hb0
-  have hb := Cz.bounds κ B ω
-  have ht := CompactHit.cap_eq_realTime (e := e) C.horizon_pos hb V (γ ω) z hm.1 hV hs
-  rw [FiniteTilt.Cutoff.physicalTime,ht]
-  simp only [observedTime,FiniteTilt.Cutoff.stopTime,FiniteTilt.Cutoff.paths,Cz,FiniteTilt.CommonCutoff.at,hr]
-
-lemma observedTime_approach {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (ρ δ e r : ℝ)
-    (U : C(I,ℝ)) (V : C(I,ℂ)) (γ : ℝ → ℂ) (hV : ∀ q : I, V q = γ (T*q)) (z : ℂ)
-    (hstrict : observedTime hε hT.le ρ δ r ((U,V),z) < observedTime hε hT.le ρ δ e ((U,V),z)) :
-    dist z (γ (observedTime hε hT.le ρ δ r ((U,V),z))) ≤ r := by
-  have hr : CompactHit.realTime T r V z < stop hε hT.le U z ρ δ := by
-    have hh : min (CompactHit.realTime T r V z) (stop hε hT.le U z ρ δ) <
-        stop hε hT.le U z ρ δ := hstrict.trans_le (min_le_right _ _)
-    exact (min_lt_iff.1 hh).resolve_right (lt_irrefl _)
-  have ht := (stop_bounds hε hT.le U z ρ δ).2
-  have hh : CompactHit.hits r (V,z) := by
-    by_contra hn
-    have he : CompactHit.realTime T r V z = T := by
-      simp only [CompactHit.realTime,CompactHit.time_eq_one_of_not_hits hn]; simp
-    rw [he] at hr
-    exact (not_lt_of_ge ht) hr
-  rw [observedTime,min_eq_left hr.le]
-  simpa only [hV,CompactHit.realTime] using CompactHit.time_mem hh
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge.Mesh
-
-lemma coefficient_le_poisson {z : ℂ} (hy : 0 < z.im)
-    {a p s R R₀ : ℝ} (ha : 1 ≤ a) (hp : 0 ≤ p) (hs : 0 ≤ s)
-    (hR₀ : 0 < R₀) (hR : R₀ ≤ R) :
-    |s*a*z.re/‖z‖^2 * (R^(-p)*(z.im/‖z‖)^a)| ≤
-      (s*a*R₀^(-p)) * (z.im/‖z‖^2) := by
-  have hz : 0 < ‖z‖ := norm_pos_iff.2 (by intro h; simp [h] at hy)
-  have hsin : 0 ≤ z.im/‖z‖ := by positivity
-  have hsinle : z.im/‖z‖ ≤ 1 := (div_le_one hz).2 (Complex.im_le_norm z)
-  have hpower := Real.rpow_le_self_of_le_one hsin hsinle ha
-  have hr := Real.rpow_le_rpow_of_nonpos hR₀ hR (neg_nonpos.2 hp)
-  have ha0 : 0 ≤ a := le_trans zero_le_one ha
-  have hpwr : 0 ≤ R^(-p) := Real.rpow_nonneg (hR₀.le.trans hR) _
-  simp only [abs_mul,abs_div,abs_of_nonneg hs,abs_of_nonneg ha0,abs_pow,abs_norm,
-    abs_of_nonneg hpwr,abs_of_nonneg (Real.rpow_nonneg hsin a)]
-  calc
-    s*a*|z.re|/‖z‖^2*(R^(-p)*(z.im/‖z‖)^a)
-        ≤ s*a*‖z‖/‖z‖^2*(R₀^(-p)*(z.im/‖z‖)) := by
-          gcongr
-          exact Complex.abs_re_le_norm z
-    _ = (s*a*R₀^(-p))*(z.im/‖z‖^2) := by field_simp
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-lemma hessian_scalar_comp {u : ℂ → ℝ} {φ : ℝ → ℝ} {z v : ℂ}
-    (hu : ContDiffAt ℝ 2 u z) (hφ : ContDiffAt ℝ 2 φ (u z)) :
-    fderiv ℝ (fderiv ℝ (φ ∘ u)) z v v =
-      deriv (deriv φ) (u z)*(fderiv ℝ u z v)^2 +
-        deriv φ (u z)*fderiv ℝ (fderiv ℝ u) z v v := by
-  have hdu : ContDiffAt ℝ 1 (fderiv ℝ u) z := hu.fderiv_right (by norm_num)
-  have hdφ : ContDiffAt ℝ 1 (deriv φ) (u z) := hφ.derivWithin (by norm_num)
-  have h₁ := ((hdφ.differentiableAt (by norm_num)).hasDerivAt.comp_hasFDerivAt z
-    (hu.differentiableAt (by norm_num)).hasFDerivAt)
-  have h₂ := (hdu.differentiableAt (by norm_num)).hasFDerivAt
-  have hh := h₁.smul h₂
-  have he : fderiv ℝ (φ ∘ u) =ᶠ[𝓝 z] fun y => deriv φ (u y) • fderiv ℝ u y := by
-    have heu : ∀ᶠ y in 𝓝 z, ContDiffAt ℝ 2 u y := hu.eventually (by norm_num)
-    have heφ : ∀ᶠ y in 𝓝 z, ContDiffAt ℝ 2 φ (u y) :=
-      hu.continuousAt.tendsto.eventually (hφ.eventually (by norm_num))
-    filter_upwards [heu,heφ] with y hy hyφ
-    exact ((hyφ.differentiableAt (by norm_num)).hasDerivAt.comp_hasFDerivAt y
-      (hy.differentiableAt (by norm_num)).hasFDerivAt).fderiv
-  have hh' := (hh.congr_of_eventuallyEq he).fderiv
-  rw [hh']
-  simp only [add_apply,ContinuousLinearMap.smulRight_apply,
-    smul_apply,smul_eq_mul,Function.comp_apply]
-  ring
-
-lemma laplacian_scalar_comp {u : ℂ → ℝ} {φ : ℝ → ℝ} {z : ℂ}
-    (hu : ContDiffAt ℝ 2 u z) (hφ : ContDiffAt ℝ 2 φ (u z)) :
-    Laplacian.laplacian (φ ∘ u) z =
-      deriv (deriv φ) (u z)*((fderiv ℝ u z 1)^2+(fderiv ℝ u z Complex.I)^2) +
-        deriv φ (u z)*Laplacian.laplacian u z := by
-  rw [laplacian_eq_hessian,hessian_scalar_comp hu hφ,hessian_scalar_comp hu hφ,
-    laplacian_eq_hessian]
-  ring
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology RealInnerProductSpace
-namespace SLEExactGauge.Layer
-
-def diskFactor (z : ℂ) : ℝ := 1-‖z‖^2
-
-lemma diskFactor_pos {z : ℂ} (hz : ‖z‖ < 1) : 0 < diskFactor z := by
-  dsimp [diskFactor]
-  nlinarith [norm_nonneg z]
-
-lemma diskFactor_smooth : ContDiff ℝ 2 diskFactor :=
-  contDiff_const.sub (contDiff_norm_sq ℝ)
-
-lemma diskFactor_fderiv (z : ℂ) : fderiv ℝ diskFactor z = -((2:ℝ) • innerSL ℝ z) := by
-  simpa only [two_smul, diskFactor] using!
-    ((hasStrictFDerivAt_norm_sq z).hasFDerivAt.const_sub (1:ℝ)).fderiv
-
-lemma diskFactor_hessian (z : ℂ) :
-    fderiv ℝ (fderiv ℝ diskFactor) z = -((2:ℝ) • innerSL ℝ (E := ℂ)) := by
-  have he : fderiv ℝ diskFactor = fun z => -((2:ℝ) • innerSL ℝ z) := funext diskFactor_fderiv
-  rw [he]
-  convert! ((innerSL ℝ (E := ℂ)).hasFDerivAt (x := z) |>.const_smul (2:ℝ)).neg.fderiv using 1
-
-lemma diskFactor_gradient (z : ℂ) :
-    fderiv ℝ diskFactor z 1 = -2*z.re ∧
-      fderiv ℝ diskFactor z Complex.I = -2*z.im := by
-  rw [diskFactor_fderiv]
-  simp [Complex.inner,innerSL_apply_apply]
-
-lemma diskFactor_laplacian (z : ℂ) : Laplacian.laplacian diskFactor z = -4 := by
-  rw [laplacian_eq_hessian,diskFactor_hessian]
-  simp only [neg_apply,smul_apply,smul_eq_mul]
-  norm_num [innerSL,innerₛₗ,Complex.inner]
-
-lemma logFactor_smooth {z : ℂ} (hz : ‖z‖ < 1) :
-    ContDiffAt ℝ 2 (fun z => Real.log (diskFactor z)) z :=
-  diskFactor_smooth.contDiffAt.log (ne_of_gt (diskFactor_pos hz))
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.Charge
+import OAI.Probability.SLELocalization.PhysicalPast
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Finite Tilt estimates. -/
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology NNReal
+namespace SLEExactGauge.FiniteTilt
+
+structure CommonCutoff (A : Set ℂ) (u : ℂ → ℝ) where
+  angle : ℝ
+  angle_pos : 0 < angle
+  height : ℝ
+  height_pos : 0 < height
+  horizon : ℝ≥0
+  horizon_pos : 0 < (horizon:ℝ)
+  angle_initial : ∀ z ∈ A, angle < z.im/‖z‖
+  height_small : ∀ z ∈ A, height < z.im*Real.exp (-u z/(2*angle^2))
+  time_large : ∀ z ∈ A, u z*z.im^2/(4*angle^4) < horizon
+
+def CommonCutoff.at {A : Set ℂ} {u : ℂ → ℝ} (C : CommonCutoff A u)
+    (z : ℂ) (hz : z ∈ A) : Cutoff z (u z) where
+  angle := C.angle
+  angle_pos := C.angle_pos
+  angle_initial := C.angle_initial z hz
+  height := C.height
+  height_pos := C.height_pos
+  horizon := C.horizon
+  horizon_pos := C.horizon_pos
+  height_small := C.height_small z hz
+  time_large := C.time_large z hz
+
+lemma commonCutoffs {A : Set ℂ} {u : ℂ → ℝ} {H U α : ℝ}
+    (hH : 1 ≤ H) (hU : 0 ≤ U) (hα : 0 < α)
+    (hy : ∀ z ∈ A, 1 ≤ z.im ∧ z.im ≤ H)
+    (hu : ∀ z ∈ A, 0 ≤ u z ∧ u z ≤ U)
+    (hs : ∀ z ∈ A, α ≤ z.im/‖z‖) :
+    ∃ Cs : ℕ → CommonCutoff A u, Tendsto (fun n => (Cs n).angle) atTop (𝓝 0) := by
+  let δ : ℕ → ℝ := fun n => (α/2)/((n:ℝ)+1)
+  have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity
+  have hδα (n : ℕ) : δ n < α := by
+    have hh : 1 ≤ (n:ℝ)+1 := by linarith [Nat.cast_nonneg (α := ℝ) n]
+    have he := div_le_self (by positivity : 0 ≤ α/2) hh
+    exact he.trans_lt (by linarith)
+  let ε : ℕ → ℝ := fun n => Real.exp (-U/(2*(δ n)^2))/2
+  let T : ℕ → ℝ≥0 := fun n => ⟨U*H^2/(4*(δ n)^4)+1,by positivity⟩
+  let Cs : ℕ → CommonCutoff A u := fun n => {
+    angle := δ n
+    angle_pos := hδ n
+    height := ε n
+    height_pos := by dsimp [ε]; positivity
+    horizon := T n
+    horizon_pos := by
+      have hn : 0 ≤ U*H^2/(4*(δ n)^4) := by positivity
+      change 0 < U*H^2/(4*(δ n)^4)+1
+      linarith
+    angle_initial := fun z hz => (hδα n).trans_le (hs z hz)
+    height_small := by
+      intro z hz
+      have hu' := (hu z hz).2
+      have he : Real.exp (-U/(2*(δ n)^2)) ≤ Real.exp (-u z/(2*(δ n)^2)) := by
+        apply Real.exp_le_exp.2
+        exact div_le_div_of_nonneg_right (neg_le_neg hu') (by positivity)
+      have hz' := (hy z hz).1
+      dsimp only [ε]
+      have hp := Real.exp_pos (-U/(2*(δ n)^2))
+      nlinarith [Real.exp_pos (-u z/(2*(δ n)^2))]
+    time_large := by
+      intro z hz
+      have hz0 : 0 ≤ z.im := (by norm_num : (0:ℝ) ≤ 1).trans (hy z hz).1
+      have hz2 : z.im^2 ≤ H^2 := pow_le_pow_left₀ hz0 (hy z hz).2 2
+      have hm := mul_le_mul (hu z hz).2 hz2 (sq_nonneg _) hU
+      have hd := div_le_div_of_nonneg_right hm (by positivity : 0 ≤ 4*(δ n)^4)
+      change u z*z.im^2/(4*(δ n)^4) < U*H^2/(4*(δ n)^4)+1
+      linarith }
+  refine ⟨Cs,?_⟩
+  have ht : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
+    tendsto_atTop_add_const_right _ 1 tendsto_natCast_atTop_atTop
+  have hh := (tendsto_const_nhds (x := α/2) (f := atTop)).div_atTop ht
+  simpa only [Cs,δ] using hh
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerCutoff
+
+lemma solution_mark_dist_le {ε T D δ : ℝ} (hε : 0 < ε)
+    (hD : 0 ≤ D) (hδ : 0 ≤ δ) {U V : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
+    (h0 : dist (H 0) (G 0) ≤ δ) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    dist (H t) (G t) ≤ (δ+D)*Real.exp ((4/ε^2)*T) := by
+  let K : ℝ≥0 := ⟨4/ε^2,by positivity⟩
+  have hK : (0:ℝ) < K := by change 0 < 4/ε^2; positivity
+  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
+      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K:ℝ)*D := by
+    rw [dist_eq_norm]
+    calc
+      _ ≤ (4/ε^2)*|V s-U s| := norm_velocity_driver_sub_le hε _ _ _
+      _ ≤ (K:ℝ)*D := by
+        rw [abs_sub_comm]
+        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1,hs.2.le⟩) hK.le
+  have hh := dist_le_of_approx_trajectories_ODE
+    (fun t => velocity_lipschitz hε (U t))
+    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
+    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
+    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
+    hb h0 t ht
+  have he : gronwallBound δ K (0+(K:ℝ)*D) (t-0) =
+      δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) := by
+    simp only [gronwallBound,zero_add,sub_zero,ite_eq_right (ne_of_gt hK)]
+    field_simp
+  rw [he] at hh
+  have hexp : Real.exp ((K:ℝ)*t) ≤ Real.exp ((K:ℝ)*T) :=
+    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
+  apply hh.trans
+  change δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) ≤ (δ+D)*Real.exp ((K:ℝ)*T)
+  have h1 := mul_le_mul_of_nonneg_left hexp hδ
+  have h2 := mul_le_mul_of_nonneg_left hexp hD
+  nlinarith
+
+lemma solutionPath_joint_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
+    LipschitzWith ⟨2*Real.exp ((4/ε^2)*T),by positivity⟩
+      (fun p : C(I, ℝ) × ℂ => solutionPath hε hT p.1 p.2) := by
+  apply LipschitzWith.of_dist_le_mul
+  rintro ⟨U,z⟩ ⟨V,w⟩
+  apply (ContinuousMap.dist_le (show 0 ≤ (2*Real.exp ((4/ε^2)*T))*dist (U,z) (V,w) by positivity)).2
+  intro t
+  have hh := solution_mark_dist_le hε (D := dist U V) (δ := dist z w)
+    dist_nonneg dist_nonneg (solution_deriv hε hT U z) (solution_deriv hε hT V w)
+    (by rw [solution_initial,solution_initial])
+    (fun s _ => ContinuousMap.dist_apply_le_dist (f := U) (g := V) (projIcc 0 1 zero_le_one (s/T))) t.property
+  apply hh.trans
+  change (dist z w+dist U V)*Real.exp ((4/ε^2)*T) ≤
+    (2*Real.exp ((4/ε^2)*T))*max (dist U V) (dist z w)
+  have hsum : dist z w+dist U V ≤ 2*max (dist U V) (dist z w) := by
+    linarith [le_max_left (dist U V) (dist z w),le_max_right (dist U V) (dist z w)]
+  nlinarith [mul_le_mul_of_nonneg_right hsum (Real.exp_pos ((4/ε^2)*T)).le]
+
+lemma continuous_mark_position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
+    Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => position hε hT p.1.1 p.1.2 p.2) := by
+  exact continuous_eval.comp (((solutionPath_joint_lipschitz hε hT).continuous.comp continuous_fst).prodMk
+    ((continuous_projIcc (h := hT)).comp continuous_snd))
+
+lemma continuous_mark_variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
+    Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => variation hε hT p.1.1 p.1.2 p.2) := by
+  have hd : Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => extendDriver T p.1.1 p.2) :=
+    (continuous_joint_extendDriver T).comp ((continuous_fst.fst).prodMk continuous_snd)
+  have hw := (continuous_mark_position hε hT).sub (Complex.continuous_ofReal.comp hd)
+  exact Complex.continuous_re.comp (continuous_const.div
+    (((continuous_clampIm ε).comp hw).pow 2) (fun _ => pow_ne_zero 2 (clampIm_ne_zero hε _)))
+
+lemma continuous_mark_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
+    Continuous (fun p : (C(I,ℝ) × ℂ) × ℝ => logDerivative hε hT p.1.1 p.1.2 p.2) := by
+  have hc := intervalIntegral.continuous_parametric_primitive_of_continuous
+    (μ := volume) (a₀ := 0) (f := fun p : C(I,ℝ) × ℂ => variation hε hT p.1 p.2)
+    (continuous_mark_variation hε hT)
+  exact hc.comp (continuous_fst.prodMk
+    (continuous_subtype_val.comp ((continuous_projIcc (h := hT)).comp continuous_snd)))
+
+lemma continuous_mark_green {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (p a : ℝ) :
+    Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ =>
+      LoewnerField.field hε hT p a 1 q.1.1 q.1.2 q.2 (extendDriver T q.1.1 q.2)) := by
+  have hH := continuous_mark_position hε hT
+  have hD : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => extendDriver T q.1.1 q.2) :=
+    (continuous_joint_extendDriver T).comp (continuous_fst.fst.prodMk continuous_snd)
+  apply GreenPhysical.continuous_green_comp p a
+  · exact (Complex.continuous_re.comp hH).sub (continuous_const.mul hD)
+  · exact continuous_const.max (Complex.continuous_im.comp hH)
+  · exact continuous_mark_logDerivative hε hT
+  · intro q
+    exact LoewnerField.height_pos hε hT q.1.1 q.1.2 q.2
+
+end SLEExactGauge.LoewnerCutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerCutoff
+open LoewnerField LoewnerExit
+local instance commonClocksMarkRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance commonClocksMarkRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+
+lemma continuous_mark_field {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (p a sdev : ℝ) :
+    Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ =>
+      field hε hT p a sdev q.1.1.1 q.1.1.2 q.1.2 q.2) := by
+  have hH := (continuous_mark_position hε hT).comp
+    (continuous_fst : Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => q.1))
+  apply GreenPhysical.continuous_green_comp p a
+  · exact (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd)
+  · exact continuous_const.max (Complex.continuous_im.comp hH)
+  · exact (continuous_mark_logDerivative hε hT).comp continuous_fst
+  · intro q
+    exact height_pos hε hT q.1.1.1 q.1.1.2 q.1.2
+
+lemma continuous_mark_first {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (p a sdev : ℝ) :
+    Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ =>
+      iteratedDeriv 1 (field hε hT p a sdev q.1.1.1 q.1.1.2 q.1.2) q.2) := by
+  simp only [first_eq]
+  have hH := (continuous_mark_position hε hT).comp
+    (continuous_fst : Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => sdev*q.2))
+  have hy : Continuous (fun q : ((C(I,ℝ) × ℂ) × ℝ) × ℝ => height hε hT q.1.1.1 q.1.1.2 q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  exact ((continuous_const.mul hx).div ((hx.pow 2).add (hy.pow 2))
+    (fun q => (GreenPhysical.q_pos _ (height_pos hε hT q.1.1.1 q.1.1.2 q.1.2)).ne')).mul
+      (continuous_mark_field hε hT p a sdev)
+
+lemma continuous_mark_deficit {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ) :
+    Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => deficit hε hT q.1.1 q.1.2 ρ δ q.2) := by
+  have hH := continuous_mark_position hε hT
+  have hD : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => extendDriver T q.1.1 q.2) :=
+    (continuous_joint_extendDriver T).comp (continuous_fst.fst.prodMk continuous_snd)
+  have hr : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => radius hε hT q.1.1 q.1.2 q.2) :=
+    (continuous_const.mul (Complex.continuous_im.comp hH)).div
+      (Real.continuous_exp.comp (continuous_mark_logDerivative hε hT)) (fun _ => Real.exp_ne_zero _)
+  have hs : Continuous (fun q : (C(I,ℝ) × ℂ) × ℝ => sine hε hT q.1.1 q.1.2 q.2) :=
+    (Complex.continuous_im.comp hH).div
+      (((continuous_clampIm ε).comp (hH.sub (Complex.continuous_ofReal.comp hD))).norm)
+      (fun _ => norm_ne_zero_iff.2 (clampIm_ne_zero hε _))
+  exact (hr.sub continuous_const).min
+    (((Complex.continuous_im.comp hH).sub continuous_const).min (hs.sub continuous_const))
+
+lemma continuous_mark_exitPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ) :
+    Continuous (fun q : C(I,ℝ) × ℂ => exitPath hε hT q.1 q.2 ρ δ) := by
+  apply ContinuousMap.continuous_of_continuous_uncurry _
+  exact (continuous_mark_deficit hε hT ρ δ).comp
+    (continuous_fst.prodMk (continuous_const.mul (continuous_subtype_val.comp continuous_snd)))
+
+lemma measurable_mark_stop {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ) :
+    Measurable (fun q : C(I,ℝ) × ℂ => stop hε hT q.1 q.2 ρ δ) :=
+  measurable_const.mul (ContinuousExit.measurable_time.comp (continuous_mark_exitPath hε hT ρ δ).measurable)
+
+lemma isOpen_mark_positive_on {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ t : ℝ) :
+    IsOpen {q : C(I,ℝ) × ℂ | ∀ s ∈ Icc 0 t, 0 < deficit hε hT q.1 q.2 ρ δ s} := by
+  let S : Set ((C(I,ℝ) × ℂ) × Icc 0 t) := {q | deficit hε hT q.1.1 q.1.2 ρ δ q.2 ≤ 0}
+  have hc : IsClosed S := isClosed_le ((continuous_mark_deficit hε hT ρ δ).comp
+    (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))) continuous_const
+  have hh := isClosedMap_fst_of_compactSpace S hc
+  have he : {q : C(I,ℝ) × ℂ | ∀ s ∈ Icc 0 t, 0 < deficit hε hT q.1 q.2 ρ δ s} = (Prod.fst '' S)ᶜ := by
+    ext q
+    simp only [mem_ofPred_eq,mem_compl_iff,mem_image,S,not_exists,not_and]
+    constructor
+    · intro hp v hv hvq
+      subst q
+      exact (not_lt_of_ge hv) (hp v.2 v.2.property)
+    · intro hp s hs
+      by_contra hn
+      exact hp (q,⟨s,hs⟩) (not_lt.1 hn) rfl
+  rw [he]
+  exact hh.isOpen_compl
+
+lemma measurable_mark_coeffAt {ε : ℝ} (hε : 0 < ε) (s : ℝ≥0) (p a sdev ρ δ : ℝ) :
+    Measurable (fun q : (C(I,ℝ) × ℝ) × ℂ =>
+      LoewnerLocalized.coeffAt hε 1 s p a sdev q.2 ρ δ q.1) := by
+  have hp : Continuous (fun q : (C(I,ℝ) × ℝ) × ℂ => (((q.1.1,q.2),(s:ℝ)),q.1.2)) :=
+    ((continuous_fst.fst.prodMk continuous_snd).prodMk continuous_const).prodMk continuous_fst.snd
+  have hc := (continuous_mark_first hε s.coe_nonneg p a sdev).comp hp
+  dsimp only [Function.comp_def] at hc
+  exact hc.measurable.ite (((isOpen_mark_positive_on hε s.coe_nonneg ρ δ s).preimage
+    (continuous_fst.fst.prodMk continuous_snd)).measurableSet) measurable_const
+
+end SLEExactGauge.LoewnerCutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.PhysicalPast
+
+lemma joint_avoids_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (ε : ℝ) (T : ℝ≥0) :
+    @MeasurableSet (Ω × ℂ) (Prod.instMeasurableSpace (α := Ω) (β := ℂ)
+      (m₁ := BrownianPredictable.pastSigma B T) (m₂ := inferInstance))
+      {q | q.1 ∈ avoids κ B q.2 ε T} := by
+  let : MeasurableSpace Ω := BrownianPredictable.pastSigma B T
+  change MeasurableSet {q : Ω × ℂ | ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤
+    dist q.2 (CanonicalTrace.value κ B (sampleTime T (TopologicalSpace.denseSeq I n)) q.1)}
+  simp only [ofPred_exists,ofPred_forall]
+  apply MeasurableSet.iUnion
+  intro m
+  apply MeasurableSet.iInter
+  intro n
+  exact measurableSet_le measurable_const (measurable_snd.dist
+    (((CanonicalTrace.measurable_past κ B _).mono
+      (BrownianPredictable.pastSigma_mono B (sampleTime_le T (TopologicalSpace.denseSeq I n))) le_rfl).comp measurable_fst))
+
+end SLEExactGauge.PhysicalPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open BrownianPredictable
+local instance commonClocksPastRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance commonClocksPastRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+
+lemma joint_rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    {ε : ℝ} (hε : 0 < ε) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0) (p a sdev ρ δ : ℝ) :
+    @Measurable (Ω × ℂ) ℝ (Prod.instMeasurableSpace (α := Ω) (β := ℂ)
+      (m₁ := pastSigma B s) (m₂ := inferInstance)) _
+      (fun q => rawCoeff hε 1 B T s p a sdev q.2 ρ δ q.1) := by
+  let : MeasurableSpace Ω := pastSigma B s
+  unfold rawCoeff
+  split_ifs
+  · have hp : Measurable (fun q : Ω × ℂ =>
+        ((BrownianDriver.path B sdev s q.1,B s q.1),q.2)) :=
+      (((BrownianDriver.path_measurable_past B sdev s).comp measurable_fst).prodMk
+        ((past_measurable_eval B (s := s) le_rfl).comp measurable_fst)).prodMk measurable_snd
+    have hh := (LoewnerCutoff.measurable_mark_coeffAt hε s p a sdev ρ δ).comp hp
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  · exact measurable_const
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.PhysicalPast
+
+lemma joint_avoids_nullMeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (ν : Measure ℂ) [SFinite ν] (κ ε : ℝ) (T : ℝ≥0) :
+    NullMeasurableSet {q : Ω × ℂ | q.1 ∈ avoids κ B q.2 ε T} (P.prod ν) := by
+  change NullMeasurableSet {q : Ω × ℂ | ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤
+    dist q.2 (CanonicalTrace.value κ B (sampleTime T (TopologicalSpace.denseSeq I n)) q.1)} (P.prod ν)
+  simp only [ofPred_exists,ofPred_forall]
+  apply NullMeasurableSet.iUnion
+  intro m
+  apply NullMeasurableSet.iInter
+  intro n
+  exact nullMeasurableSet_le aemeasurable_const
+    (measurable_snd.aemeasurable.dist ((CanonicalTrace.aemeasurable κ hB _).comp_fst))
+
+end SLEExactGauge.PhysicalPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+local instance commonClocksJointRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance commonClocksJointRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+
+lemma joint_rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (ν : Measure ℂ) [SFinite ν] {ε : ℝ} (hε : 0 < ε)
+    (T s : ℝ≥0) (p a sdev ρ δ : ℝ) :
+    AEMeasurable (fun q : Ω × ℂ => rawCoeff hε 1 B T s p a sdev q.2 ρ δ q.1) (P.prod ν) := by
+  unfold rawCoeff
+  split_ifs
+  · have hp : AEMeasurable (fun q : Ω × ℂ =>
+        ((BrownianDriver.path B sdev s q.1,B s q.1),q.2)) (P.prod ν) :=
+      (((BrownianDriver.path_aemeasurable hB sdev s).comp_fst).prodMk
+        ((hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable.comp_fst)).prodMk measurable_snd.aemeasurable
+    have hh := (LoewnerCutoff.measurable_mark_coeffAt hε s p a sdev ρ δ).comp_aemeasurable hp
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  · exact aemeasurable_const
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerLocalized BrownianPredictable
+
+def coeff {Ω : Type*} {ε : ℝ} (hε : 0 < ε) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (T s : ℝ≥0) (ρ δ e : ℝ) (q : Ω × ℂ) : ℝ :=
+  if q.1 ∈ PhysicalPast.avoids κ B q.2 e s then
+    rawCoeff hε 1 B T s (1-κ/8) (8/κ-1) (Real.sqrt κ) q.2 ρ δ q.1 else 0
+
+lemma coeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    {ε : ℝ} (hε : 0 < ε) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0) (ρ δ e : ℝ) :
+    @Measurable (Ω × ℂ) ℝ (Prod.instMeasurableSpace (m₁ := pastSigma B s) (m₂ := inferInstance)) _
+      (coeff hε κ B T s ρ δ e) := by
+  exact (joint_rawCoeff_measurable_past hε B T s _ _ _ ρ δ).ite
+    (PhysicalPast.joint_avoids_measurable_past κ B e s) measurable_const
+
+lemma coeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (ν : Measure ℂ) [SFinite ν] {ε : ℝ} (hε : 0 < ε)
+    (κ : ℝ) (T s : ℝ≥0) (ρ δ e : ℝ) :
+    AEMeasurable (coeff hε κ B T s ρ δ e) (P.prod ν) := by
+  exact (joint_rawCoeff_aemeasurable hB ν hε T s _ _ _ ρ δ).indicator₀
+    (PhysicalPast.joint_avoids_nullMeasurable hB ν κ e s)
+
+lemma coeff_uniform_bound {Ω : Type*} {ε : ℝ} (hε : 0 < ε)
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (ρ δ : ℝ) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ e s q, |coeff hε κ B T s ρ δ e q| ≤ K := by
+  let p := 1-κ/8
+  let a := 8/κ-1
+  have hp : 0 ≤ p := by dsimp [p]; linarith [hκ.2]
+  have ha : 0 ≤ a := by dsimp [a]; have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by linarith [hκ.2]); linarith
+  let K := max 0 (|Real.sqrt κ| *((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε))
+  refine ⟨K,le_max_left _ _,?_⟩
+  intro e s q
+  unfold coeff rawCoeff coeffAt
+  split_ifs with h1 h2 h3
+  · have hh := first_bound hε s.coe_nonneg p a (Real.sqrt κ) hp ha
+      (BrownianDriver.path B (Real.sqrt κ) s q.1) q.2 s (B s q.1)
+    apply hh.trans
+    have hs : (s:ℝ) ≤ T := h2.le
+    have he : Real.exp ((2/ε^2)*s) ≤ Real.exp ((2/ε^2)*T) :=
+      Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left hs (by positivity))
+    have hb : (2*ε/Real.exp ((2/ε^2)*T)) ≤ (2*ε/Real.exp ((2/ε^2)*s)) :=
+      div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) he
+    have hr := Real.rpow_le_rpow_of_nonpos (by positivity : 0 < 2*ε/Real.exp ((2/ε^2)*T)) hb (neg_nonpos.2 hp)
+    apply le_trans _ (le_max_right _ _)
+    gcongr
+  all_goals simpa only [abs_zero] using (le_max_left 0 (|Real.sqrt κ| *((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε)))
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.CompactHit
+local instance commonClocksHitComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance commonClocksHitComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+def hits (e : ℝ) (p : C(I,ℂ) × ℂ) : Prop := ∃ q : I, dist p.2 (p.1 q) ≤ e
+
+def time (e : ℝ) (p : C(I,ℂ) × ℂ) : I := sInf {q : I | dist p.2 (p.1 q) ≤ e}
+
+lemma hits_closed (e : ℝ) : IsClosed {p : C(I,ℂ) × ℂ | hits e p} := by
+  have hc : IsClosed {q : (C(I,ℂ) × ℂ) × I | dist q.1.2 (q.1.1 q.2) ≤ e} :=
+    isClosed_le (continuous_fst.snd.dist (continuous_eval.comp (continuous_fst.fst.prodMk continuous_snd))) continuous_const
+  have he : {p : C(I,ℂ) × ℂ | hits e p} = Prod.fst ''
+      {q : (C(I,ℂ) × ℂ) × I | dist q.1.2 (q.1.1 q.2) ≤ e} := by
+    ext p
+    simp [hits]
+  rw [he]
+  exact isClosedMap_fst_of_compactSpace _ hc
+
+lemma time_mem {e : ℝ} {p : C(I,ℂ) × ℂ} (hp : hits e p) :
+    dist p.2 (p.1 (time e p)) ≤ e := by
+  exact (isClosed_le (continuous_const.dist p.1.continuous) continuous_const).isCompact.sInf_mem hp
+
+lemma time_le {e : ℝ} {p : C(I,ℂ) × ℂ} {q : I} (hq : dist p.2 (p.1 q) ≤ e) :
+    time e p ≤ q := sInf_le hq
+
+lemma before {e : ℝ} {p : C(I,ℂ) × ℂ} {q : I} (hq : q < time e p) :
+    e < dist p.2 (p.1 q) := by
+  by_contra hm
+  exact not_le_of_gt hq (time_le (not_lt.1 hm))
+
+lemma time_eq_one_of_not_hits {e : ℝ} {p : C(I,ℂ) × ℂ} (hp : ¬ hits e p) : time e p = 1 := by
+  have he : {q : I | dist p.2 (p.1 q) ≤ e} = ∅ := by
+    ext q
+    simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false]
+    exact fun hq => hp ⟨q,hq⟩
+  simp only [time,he,sInf_empty]
+  rfl
+
+lemma time_le_iff {e : ℝ} {p : C(I,ℂ) × ℂ} {q : I} (hq : q < 1) :
+    time e p ≤ q ↔ ∃ r : I, r ≤ q ∧ dist p.2 (p.1 r) ≤ e := by
+  constructor
+  · intro ht
+    have hp : hits e p := by
+      by_contra hn
+      rw [time_eq_one_of_not_hits hn] at ht
+      exact not_le_of_gt hq ht
+    exact ⟨time e p,ht,time_mem hp⟩
+  · rintro ⟨r,hr,hmem⟩
+    exact (time_le hmem).trans hr
+
+lemma measurable_time (e : ℝ) : Measurable (time e) := by
+  apply measurable_of_Iic
+  intro q
+  by_cases hq : q = 1
+  · have he : time e ⁻¹' Iic q = univ := by
+      subst q
+      ext p
+      simp only [mem_preimage,mem_Iic,mem_univ,iff_true]
+      exact (time e p).property.2
+    rw [he]
+    exact MeasurableSet.univ
+  · have hq1 : q < 1 := lt_of_le_of_ne q.property.2 hq
+    have hc : IsClosed {r : (C(I,ℂ) × ℂ) × I | r.2 ≤ q ∧ dist r.1.2 (r.1.1 r.2) ≤ e} :=
+      (isClosed_le continuous_snd continuous_const).inter
+        (isClosed_le (continuous_fst.snd.dist (continuous_eval.comp (continuous_fst.fst.prodMk continuous_snd))) continuous_const)
+    have he : time e ⁻¹' Iic q = Prod.fst ''
+        {r : (C(I,ℂ) × ℂ) × I | r.2 ≤ q ∧ dist r.1.2 (r.1.1 r.2) ≤ e} := by
+      ext p
+      change (time e p ≤ q) ↔ ∃ r, r ∈ {r : (C(I,ℂ) × ℂ) × I | r.2 ≤ q ∧ dist r.1.2 (r.1.1 r.2) ≤ e} ∧ r.1 = p
+      rw [time_le_iff hq1]
+      constructor
+      · rintro ⟨r,hr,hd⟩
+        exact ⟨(p,r),⟨hr,hd⟩,rfl⟩
+      · rintro ⟨⟨p',r⟩,⟨hr,hd⟩,he⟩
+        change p' = p at he
+        subst p'
+        exact ⟨r,hr,hd⟩
+    rw [he]
+    exact (isClosedMap_fst_of_compactSpace _ hc).measurableSet
+
+end SLEExactGauge.CompactHit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.CompactHit
+
+def realTime (T e : ℝ) (V : C(I,ℂ)) (z : ℂ) : ℝ := T*time e (V,z)
+
+lemma realTime_bounds {T : ℝ} (hT : 0 ≤ T) (e : ℝ) (V : C(I,ℂ)) (z : ℂ) :
+    realTime T e V z ∈ Icc 0 T :=
+  ⟨mul_nonneg hT (time e (V,z)).property.1,by
+    simpa only [realTime,mul_one] using mul_le_mul_of_nonneg_left (time e (V,z)).property.2 hT⟩
+
+lemma realTime_le {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
+    (hV : ∀ q : I, V q = γ (T*q)) {t : ℝ} (ht : t ∈ Icc 0 T)
+    (hd : dist z (γ t) ≤ e) : realTime T e V z ≤ t := by
+  let q : I := ⟨t/T,⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩⟩
+  have he : T*(q:ℝ) = t := mul_div_cancel₀ _ hT.ne'
+  have hq : dist z (V q) ≤ e := by rw [hV,he]; exact hd
+  have hm : (time e (V,z):ℝ) ≤ (q:ℝ) := time_le hq
+  have hx := mul_le_mul_of_nonneg_left hm hT.le
+  rwa [he] at hx
+
+lemma realTime_hits {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
+    (hV : ∀ q : I, V q = γ (T*q)) {t : ℝ} (ht : t ∈ Icc 0 T)
+    (hd : dist z (γ t) ≤ e) : hits e (V,z) := by
+  let q : I := ⟨t/T,⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩⟩
+  refine ⟨q,?_⟩
+  change dist z (V q) ≤ e
+  rw [hV,mul_div_cancel₀ _ hT.ne']
+  exact hd
+
+lemma realTime_stop {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
+    (hV : ∀ q : I, V q = γ (T*q)) (hh : hits e (V,z))
+    (hl : z ∈ localizationDomain γ (realTime T e V z)) :
+    firstLiveDistanceStop γ z e = ENNReal.ofReal (realTime T e V z) := by
+  have hb := realTime_bounds hT.le e V z
+  have hd : dist z (γ (realTime T e V z)) ≤ e := by
+    simpa only [hV,realTime] using time_mem hh
+  let t₀ : ℝ≥0 := ⟨realTime T e V z,hb.1⟩
+  have hle : firstLiveDistanceStop γ z e ≤ (t₀ : ℝ≥0∞) := PhysicalStop.le_of_hit hl hd
+  have hc : (t₀ : ℝ≥0∞) = ENNReal.ofReal (realTime T e V z) :=
+    ENNReal.ofReal_coe_nnreal.symm
+  rw [hc] at hle
+  apply le_antisymm hle
+  apply le_sInf
+  rintro r ⟨s,⟨hsl,hsd⟩,rfl⟩
+  have ht : realTime T e V z ≤ s := by
+    by_cases hsT : (s:ℝ) ≤ T
+    · exact realTime_le hT V γ z hV ⟨s.coe_nonneg,hsT⟩ hsd
+    · exact hb.2.trans (not_le.1 hsT).le
+  exact (ENNReal.ofReal_le_ofReal ht).trans_eq (ENNReal.ofReal_coe_nnreal (p := s))
+
+lemma live_byTime_iff {T e : ℝ} (hT : 0 < T) (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ)
+    (hc : ContinuousOn γ (Ici 0)) (hV : ∀ q : I, V q = γ (T*q)) :
+    firstLiveDistanceStop γ z e ≤ ENNReal.ofReal T ↔
+      hits e (V,z) ∧ z ∈ localizationDomain γ (realTime T e V z) := by
+  constructor
+  · intro hh
+    obtain ⟨t,he,hl,hd,hmin⟩ := PhysicalStop.finite_spec hc
+      (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hh)
+    have ht : (t:ℝ) ≤ T := by
+      rw [he,← ENNReal.ofReal_coe_nnreal (p := t)] at hh
+      exact (ENNReal.ofReal_le_ofReal_iff hT.le).1 hh
+    exact ⟨realTime_hits hT V γ z hV ⟨t.coe_nonneg,ht⟩ hd,
+      survivingDomain_mono (realTime_le hT V γ z hV ⟨t.coe_nonneg,ht⟩ hd) hl⟩
+  · rintro ⟨hh,hl⟩
+    rw [realTime_stop hT V γ z hV hh hl]
+    exact ENNReal.ofReal_le_ofReal (realTime_bounds hT.le e V z).2
+
+lemma cap_eq_realTime {T e τ : ℝ} (hT : 0 < T) (hτ : τ ∈ Icc 0 T)
+    (V : C(I,ℂ)) (γ : ℝ → ℂ) (z : ℂ) (hγ : ContinuousOn γ (Ici 0))
+    (hV : ∀ q : I, V q = γ (T*q)) (hl : z ∈ localizationDomain γ τ) :
+    PhysicalStop.cap (firstLiveDistanceStop γ z e) τ = min (realTime T e V z) τ := by
+  let r := realTime T e V z
+  have hr := realTime_bounds hT.le e V z
+  have hsmall (hrt : r < τ) : firstLiveDistanceStop γ z e = ENNReal.ofReal r := by
+    have hh : hits e (V,z) := by
+      by_contra hn
+      have he : r = T := by simp only [r,realTime,time_eq_one_of_not_hits hn]; simp
+      linarith [hτ.2]
+    exact realTime_stop hT V γ z hV hh (survivingDomain_mono hrt.le hl)
+  by_cases hfin : firstLiveDistanceStop γ z e = ⊤
+  · rw [PhysicalStop.cap,ite_eq_left hfin]
+    apply (min_eq_right _).symm
+    by_contra hn
+    have he := hsmall (lt_of_not_ge hn)
+    rw [hfin] at he
+    exact ENNReal.ofReal_ne_top he.symm
+  · obtain ⟨s,hs,hsl,hsd,hmin⟩ := PhysicalStop.finite_spec hγ hfin
+    rw [PhysicalStop.cap,ite_eq_right hfin,hs,ENNReal.coe_toReal]
+    apply le_antisymm
+    · apply le_min _ (min_le_right _ _)
+      by_cases hrt : r < τ
+      · have he := hsmall hrt
+        rw [hs,← ENNReal.ofReal_coe_nnreal (p := s)] at he
+        have hsre : (s:ℝ) = r := (ENNReal.ofReal_eq_ofReal_iff s.coe_nonneg (show 0 ≤ r from hr.1)).1 he
+        rw [hsre]
+        exact min_le_left _ _
+      · exact (min_le_right _ _).trans (not_lt.1 hrt)
+    · apply le_min _ (min_le_right _ _)
+      by_cases hsT : (s:ℝ) ≤ T
+      · exact (min_le_left _ _).trans (realTime_le hT V γ z hV ⟨s.coe_nonneg,hsT⟩ hsd)
+      · exact (min_le_right _ _).trans (hτ.2.trans (not_le.1 hsT).le)
+
+lemma realTime_antitone (T : ℝ) (hT : 0 ≤ T) (V : C(I,ℂ)) (z : ℂ) {e e' : ℝ} (he : e ≤ e') :
+    realTime T e' V z ≤ realTime T e V z := by
+  apply mul_le_mul_of_nonneg_left _ hT
+  by_cases hh : hits e (V,z)
+  · exact time_le ((time_mem hh).trans he)
+  · rw [time_eq_one_of_not_hits hh]
+    exact (time e' (V,z)).property.2
+
+end SLEExactGauge.CompactHit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized
+local instance commonClocksObservedRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance commonClocksObservedRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance commonClocksObservedComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance commonClocksObservedComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+def observedTime {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e : ℝ)
+    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ :=
+  min (CompactHit.realTime T e q.1.2 q.2) (stop hε hT q.1.1 q.2 ρ δ)
+
+lemma measurable_observedTime {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e : ℝ) :
+    Measurable (observedTime hε hT ρ δ e) := by
+  have hv : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => (q.1.2,q.2)) :=
+    measurable_fst.snd.prodMk measurable_snd
+  have hv' := (measurable_subtype_coe.comp ((CompactHit.measurable_time e).comp hv)).const_mul T
+  have hu : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => (q.1.1,q.2)) :=
+    measurable_fst.fst.prodMk measurable_snd
+  have hs := (measurable_mark_stop hε hT ρ δ).comp hu
+  have hh := hv'.min hs
+  exact hh
+
+def observedValue {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e p a : ℝ)
+    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ :=
+  let t := observedTime hε hT ρ δ e q
+  field hε hT p a 1 q.1.1 q.2 t (extendDriver T q.1.1 t)
+
+lemma measurable_observedValue {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e p a : ℝ) :
+    Measurable (observedValue hε hT ρ δ e p a) := by
+  have ht := measurable_observedTime hε hT ρ δ e
+  have hx : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ =>
+      (((q.1.1,q.2),observedTime hε hT ρ δ e q),extendDriver T q.1.1 (observedTime hε hT ρ δ e q))) :=
+    ((measurable_fst.fst.prodMk measurable_snd).prodMk ht).prodMk
+      ((continuous_joint_extendDriver T).measurable.comp (measurable_fst.fst.prodMk ht))
+  have hh := (continuous_mark_field hε hT p a 1).measurable.comp hx
+  exact hh
+
+lemma coeff_common {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
+    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (z : ℂ) (hz : z ∈ A) {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ) (e : ℝ) (s : ℝ≥0) (ω : Ω) :
+    (C.at z hz).physicalCoeff 1 κ B e s ω =
+      coeff C.height_pos κ B C.horizon s ρ C.angle e (ω,z) := by
+  simp only [FiniteTilt.Cutoff.physicalCoeff,FiniteTilt.CommonCutoff.at,coeff,hr]
+
+lemma observedValue_eq {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
+    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
+    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
+    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e : ℝ) :
+    observedValue C.height_pos C.horizon.coe_nonneg ρ C.angle e (1-κ/8) (8/κ-1)
+      ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V),z) =
+        (C.at z hz).physicalValue κ B γ e ω := by
+  let Cz := C.at z hz
+  have hs := Cz.source_live κ B ω hm hhalf hu hc hb0
+  have hb := Cz.bounds κ B ω
+  have ht := CompactHit.cap_eq_realTime (e := e) C.horizon_pos hb V (γ ω) z hm.1 hV hs
+  have he : observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e
+      ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V),z) = Cz.physicalTime κ B γ e ω := by
+    rw [FiniteTilt.Cutoff.physicalTime,ht]
+    simp only [observedTime,FiniteTilt.Cutoff.stopTime,FiniteTilt.Cutoff.paths,Cz,FiniteTilt.CommonCutoff.at,hr]
+  have hb' := Cz.physicalTime_bounds κ B γ e ω
+  have hU := Cz.path_eq κ B ω hc ⟨hb'.1,hb'.2.trans hb.2⟩
+  dsimp only [observedValue]
+  rw [he]
+  change field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) 1 _ z _ _ =
+    field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) _ z _ _
+  dsimp only [field]
+  rw [one_mul]
+  change extendDriver (C.horizon:ℝ) (BrownianDriver.path B (Real.sqrt κ) C.horizon ω)
+    (Cz.physicalTime κ B γ e ω) = _ at hU
+  rw [hU]
+  rfl
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Mesh
+open TaylorMeshes
+
+def slab (D : ℝ → ℝ) (a b t : ℝ) : ℝ :=
+  (if t < b then D t else 0) - (if t < a then D t else 0)
+
+lemma prestop_product (D E : ℝ → ℝ) (a b t : ℝ) :
+    (if t < a then D t else 0)*(if t < b then E t else 0) =
+      (if t < min a b then D t*E t else 0) := by
+  simp only [lt_min_iff]
+  split_ifs <;> simp_all
+
+lemma slab_product (D E : ℝ → ℝ) (a b c d t : ℝ) :
+    slab D a b t * slab E c d t =
+      (if t < min b d then D t*E t else 0) -
+      (if t < min b c then D t*E t else 0) -
+      (if t < min a d then D t*E t else 0) +
+      (if t < min a c then D t*E t else 0) := by
+  unfold slab
+  rw [sub_mul,mul_sub,mul_sub]
+  simp only [prestop_product]
+  ring
+
+lemma slab_riemann_tendsto {T a b c d : ℝ} (hT : 0 ≤ T)
+    (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T)
+    (hc : c ∈ Icc 0 T) (hd : d ∈ Icc 0 T)
+    {D E : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T))
+    (hE : ContinuousOn E (Icc 0 T)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (slab D a b (mesh T n i)*slab E c d (mesh T n i))*(T/(2:ℝ)^n))
+      atTop (𝓝 ((∫ t in 0..min b d, D t*E t) - (∫ t in 0..min b c, D t*E t) -
+        (∫ t in 0..min a d, D t*E t) + (∫ t in 0..min a c, D t*E t))) := by
+  have hDE : ContinuousOn (fun t => D t*E t) (Icc 0 T) := hD.mul hE
+  have hlim (s t : ℝ) (hs : s ∈ Icc 0 T) (ht : t ∈ Icc 0 T) :=
+    StoppedMeshes.prestop_riemann_tendsto hT (le_min hs.1 ht.1)
+      ((min_le_left s t).trans hs.2) (hDE.mono (Icc_subset_Icc_right ((min_le_left s t).trans hs.2)))
+  have hh := ((hlim b d hb hd).sub (hlim b c hb hc)).sub (hlim a d ha hd) |>.add (hlim a c ha hc)
+  convert hh using 1
+  funext n
+  simp [slab_product,sub_mul,add_mul,Finset.sum_sub_distrib,Finset.sum_add_distrib,ite_mul]
+
+lemma integral_prestop {T a : ℝ} (ha : a ∈ Icc 0 T) (D : ℝ → ℝ) :
+    (∫ t in (0:ℝ)..T, if t < a then D t else 0) = ∫ t in (0:ℝ)..a, D t := by
+  calc
+    _ = ∫ t in (0:ℝ)..T, Set.indicator {t : ℝ | t ≤ a} D t := by
+      apply intervalIntegral.integral_congr_ae
+      filter_upwards [show ∀ᵐ t : ℝ ∂volume, t ≠ a from by rw [ae_iff]; simp] with t ht htI
+      simp only [Set.indicator_apply,mem_ofPred_eq]
+      congr 1
+      exact propext (lt_iff_le_and_ne.trans (and_iff_left ht))
+    _ = _ := intervalIntegral.integral_indicator ha
+
+lemma prestop_intervalIntegrable {T a : ℝ} (hT : 0 ≤ T)
+    {D : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T)) :
+    IntervalIntegrable (fun t => if t < a then D t else 0) volume 0 T := by
+  have hh : IntegrableOn D (Ioc 0 T) volume :=
+    (hD.integrableOn_Icc).mono_set Ioc_subset_Icc_self
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hT]
+  exact hh.indicator measurableSet_Iio
+
+lemma slab_product_integral {T a b c d : ℝ} (hT : 0 ≤ T)
+    (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T)
+    (hc : c ∈ Icc 0 T) (hd : d ∈ Icc 0 T)
+    {D E : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T))
+    (hE : ContinuousOn E (Icc 0 T)) :
+    (∫ t in (0:ℝ)..T, slab D a b t*slab E c d t) =
+      (∫ t in 0..min b d, D t*E t) - (∫ t in 0..min b c, D t*E t) -
+      (∫ t in 0..min a d, D t*E t) + (∫ t in 0..min a c, D t*E t) := by
+  simp_rw [slab_product]
+  have hDE : ContinuousOn (fun t => D t*E t) (Icc 0 T) := hD.mul hE
+  have hi (s : ℝ) := prestop_intervalIntegrable (a := s) hT hDE
+  rw [intervalIntegral.integral_add ((hi _ |>.sub (hi _)).sub (hi _)) (hi _),
+    intervalIntegral.integral_sub (hi _ |>.sub (hi _)) (hi _),
+    intervalIntegral.integral_sub (hi _) (hi _)]
+  rw [integral_prestop ⟨le_min hb.1 hd.1,(min_le_left _ _).trans hb.2⟩,
+    integral_prestop ⟨le_min hb.1 hc.1,(min_le_left _ _).trans hb.2⟩,
+    integral_prestop ⟨le_min ha.1 hd.1,(min_le_left _ _).trans ha.2⟩,
+    integral_prestop ⟨le_min ha.1 hc.1,(min_le_left _ _).trans ha.2⟩]
+
+lemma slab_riemann_tendsto_integral {T a b c d : ℝ} (hT : 0 ≤ T)
+    (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T)
+    (hc : c ∈ Icc 0 T) (hd : d ∈ Icc 0 T)
+    {D E : ℝ → ℝ} (hD : ContinuousOn D (Icc 0 T))
+    (hE : ContinuousOn E (Icc 0 T)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (slab D a b (mesh T n i)*slab E c d (mesh T n i))*(T/(2:ℝ)^n))
+      atTop (𝓝 (∫ t in (0:ℝ)..T, slab D a b t*slab E c d t)) := by
+  rw [slab_product_integral hT ha hb hc hd hD hE]
+  exact slab_riemann_tendsto hT ha hb hc hd hD hE
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge.Mesh
+
+lemma slab_eq_zero_of_not_mem {D : ℝ → ℝ} {a b t : ℝ} (hab : a ≤ b) (ht : t ∉ Icc a b) :
+    slab D a b t = 0 := by
+  unfold slab
+  by_cases hta : t < a
+  · simp [hta,hta.trans_le hab]
+  · have htb : ¬t < b := by intro h; exact ht ⟨not_lt.1 hta,h.le⟩
+    simp [hta,htb]
+
+lemma slab_abs_le {D : ℝ → ℝ} {a b t M : ℝ} (hab : a ≤ b) (hM : 0 ≤ M) (h : |D t| ≤ M) :
+    |slab D a b t| ≤ M := by
+  unfold slab
+  by_cases hta : t < a
+  · simpa [hta,hta.trans_le hab] using hM
+  · by_cases htb : t < b
+    · simpa [hta,htb] using h
+    · simpa [hta,htb] using hM
+
+lemma measurable_slab {Z : Type*} [MeasurableSpace Z] {D : Z → ℝ → ℝ} {a b : Z → ℝ}
+    (hD : Measurable (fun q : ℝ × Z => D q.2 q.1)) (ha : Measurable a) (hb : Measurable b) :
+    Measurable (fun q : ℝ × Z => slab (D q.2) (a q.2) (b q.2) q.1) := by
+  exact (hD.ite (measurableSet_lt measurable_fst (hb.comp measurable_snd)) measurable_const).sub
+    (hD.ite (measurableSet_lt measurable_fst (ha.comp measurable_snd)) measurable_const)
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerExit
+local instance commonClocksBoundsRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance commonClocksBoundsRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance commonClocksBoundsComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance commonClocksBoundsComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+lemma observedTime_bounds {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ e : ℝ)
+    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : observedTime hε hT ρ δ e q ∈ Icc 0 T := by
+  have hr := CompactHit.realTime_bounds hT e q.1.2 q.2
+  have hs := stop_bounds hε hT q.1.1 q.2 ρ δ
+  exact ⟨le_min hr.1 hs.1,(min_le_left _ _).trans hr.2⟩
+
+lemma observedTime_antitone {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (ρ δ : ℝ)
+    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) {e r : ℝ} (her : e ≤ r) :
+    observedTime hε hT ρ δ r q ≤ observedTime hε hT ρ δ e q :=
+  min_le_min_right _ (CompactHit.realTime_antitone T hT q.1.2 q.2 her)
+
+lemma observedTime_eq {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
+    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
+    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
+    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e : ℝ) :
+    observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e
+      ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V),z) =
+        (C.at z hz).physicalTime κ B γ e ω := by
+  let Cz := C.at z hz
+  have hs := Cz.source_live κ B ω hm hhalf hu hc hb0
+  have hb := Cz.bounds κ B ω
+  have ht := CompactHit.cap_eq_realTime (e := e) C.horizon_pos hb V (γ ω) z hm.1 hV hs
+  rw [FiniteTilt.Cutoff.physicalTime,ht]
+  simp only [observedTime,FiniteTilt.Cutoff.stopTime,FiniteTilt.Cutoff.paths,Cz,FiniteTilt.CommonCutoff.at,hr]
+
+lemma observedTime_approach {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (ρ δ e r : ℝ)
+    (U : C(I,ℝ)) (V : C(I,ℂ)) (γ : ℝ → ℂ) (hV : ∀ q : I, V q = γ (T*q)) (z : ℂ)
+    (hstrict : observedTime hε hT.le ρ δ r ((U,V),z) < observedTime hε hT.le ρ δ e ((U,V),z)) :
+    dist z (γ (observedTime hε hT.le ρ δ r ((U,V),z))) ≤ r := by
+  have hr : CompactHit.realTime T r V z < stop hε hT.le U z ρ δ := by
+    have hh : min (CompactHit.realTime T r V z) (stop hε hT.le U z ρ δ) <
+        stop hε hT.le U z ρ δ := hstrict.trans_le (min_le_right _ _)
+    exact (min_lt_iff.1 hh).resolve_right (lt_irrefl _)
+  have ht := (stop_bounds hε hT.le U z ρ δ).2
+  have hh : CompactHit.hits r (V,z) := by
+    by_contra hn
+    have he : CompactHit.realTime T r V z = T := by
+      simp only [CompactHit.realTime,CompactHit.time_eq_one_of_not_hits hn]; simp
+    rw [he] at hr
+    exact (not_lt_of_ge ht) hr
+  rw [observedTime,min_eq_left hr.le]
+  simpa only [hV,CompactHit.realTime] using CompactHit.time_mem hh
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge.Mesh
+
+lemma coefficient_le_poisson {z : ℂ} (hy : 0 < z.im)
+    {a p s R R₀ : ℝ} (ha : 1 ≤ a) (hp : 0 ≤ p) (hs : 0 ≤ s)
+    (hR₀ : 0 < R₀) (hR : R₀ ≤ R) :
+    |s*a*z.re/‖z‖^2 * (R^(-p)*(z.im/‖z‖)^a)| ≤
+      (s*a*R₀^(-p)) * (z.im/‖z‖^2) := by
+  have hz : 0 < ‖z‖ := norm_pos_iff.2 (by intro h; simp [h] at hy)
+  have hsin : 0 ≤ z.im/‖z‖ := by positivity
+  have hsinle : z.im/‖z‖ ≤ 1 := (div_le_one hz).2 (Complex.im_le_norm z)
+  have hpower := Real.rpow_le_self_of_le_one hsin hsinle ha
+  have hr := Real.rpow_le_rpow_of_nonpos hR₀ hR (neg_nonpos.2 hp)
+  have ha0 : 0 ≤ a := le_trans zero_le_one ha
+  have hpwr : 0 ≤ R^(-p) := Real.rpow_nonneg (hR₀.le.trans hR) _
+  simp only [abs_mul,abs_div,abs_of_nonneg hs,abs_of_nonneg ha0,abs_pow,abs_norm,
+    abs_of_nonneg hpwr,abs_of_nonneg (Real.rpow_nonneg hsin a)]
+  calc
+    s*a*|z.re|/‖z‖^2*(R^(-p)*(z.im/‖z‖)^a)
+        ≤ s*a*‖z‖/‖z‖^2*(R₀^(-p)*(z.im/‖z‖)) := by
+          gcongr
+          exact Complex.abs_re_le_norm z
+    _ = (s*a*R₀^(-p))*(z.im/‖z‖^2) := by field_simp
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+lemma hessian_scalar_comp {u : ℂ → ℝ} {φ : ℝ → ℝ} {z v : ℂ}
+    (hu : ContDiffAt ℝ 2 u z) (hφ : ContDiffAt ℝ 2 φ (u z)) :
+    fderiv ℝ (fderiv ℝ (φ ∘ u)) z v v =
+      deriv (deriv φ) (u z)*(fderiv ℝ u z v)^2 +
+        deriv φ (u z)*fderiv ℝ (fderiv ℝ u) z v v := by
+  have hdu : ContDiffAt ℝ 1 (fderiv ℝ u) z := hu.fderiv_right (by norm_num)
+  have hdφ : ContDiffAt ℝ 1 (deriv φ) (u z) := hφ.derivWithin (by norm_num)
+  have h₁ := ((hdφ.differentiableAt (by norm_num)).hasDerivAt.comp_hasFDerivAt z
+    (hu.differentiableAt (by norm_num)).hasFDerivAt)
+  have h₂ := (hdu.differentiableAt (by norm_num)).hasFDerivAt
+  have hh := h₁.smul h₂
+  have he : fderiv ℝ (φ ∘ u) =ᶠ[𝓝 z] fun y => deriv φ (u y) • fderiv ℝ u y := by
+    have heu : ∀ᶠ y in 𝓝 z, ContDiffAt ℝ 2 u y := hu.eventually (by norm_num)
+    have heφ : ∀ᶠ y in 𝓝 z, ContDiffAt ℝ 2 φ (u y) :=
+      hu.continuousAt.tendsto.eventually (hφ.eventually (by norm_num))
+    filter_upwards [heu,heφ] with y hy hyφ
+    exact ((hyφ.differentiableAt (by norm_num)).hasDerivAt.comp_hasFDerivAt y
+      (hy.differentiableAt (by norm_num)).hasFDerivAt).fderiv
+  have hh' := (hh.congr_of_eventuallyEq he).fderiv
+  rw [hh']
+  simp only [add_apply,ContinuousLinearMap.smulRight_apply,
+    smul_apply,smul_eq_mul,Function.comp_apply]
+  ring
+
+lemma laplacian_scalar_comp {u : ℂ → ℝ} {φ : ℝ → ℝ} {z : ℂ}
+    (hu : ContDiffAt ℝ 2 u z) (hφ : ContDiffAt ℝ 2 φ (u z)) :
+    Laplacian.laplacian (φ ∘ u) z =
+      deriv (deriv φ) (u z)*((fderiv ℝ u z 1)^2+(fderiv ℝ u z Complex.I)^2) +
+        deriv φ (u z)*Laplacian.laplacian u z := by
+  rw [laplacian_eq_hessian,hessian_scalar_comp hu hφ,hessian_scalar_comp hu hφ,
+    laplacian_eq_hessian]
+  ring
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology RealInnerProductSpace
+namespace SLEExactGauge.Layer
+
+def diskFactor (z : ℂ) : ℝ := 1-‖z‖^2
+
+lemma diskFactor_pos {z : ℂ} (hz : ‖z‖ < 1) : 0 < diskFactor z := by
+  dsimp [diskFactor]
+  nlinarith [norm_nonneg z]
+
+lemma diskFactor_smooth : ContDiff ℝ 2 diskFactor :=
+  contDiff_const.sub (contDiff_norm_sq ℝ)
+
+lemma diskFactor_fderiv (z : ℂ) : fderiv ℝ diskFactor z = -((2:ℝ) • innerSL ℝ z) := by
+  simpa only [two_smul, diskFactor] using!
+    ((hasStrictFDerivAt_norm_sq z).hasFDerivAt.const_sub (1:ℝ)).fderiv
+
+lemma diskFactor_hessian (z : ℂ) :
+    fderiv ℝ (fderiv ℝ diskFactor) z = -((2:ℝ) • innerSL ℝ (E := ℂ)) := by
+  have he : fderiv ℝ diskFactor = fun z => -((2:ℝ) • innerSL ℝ z) := funext diskFactor_fderiv
+  rw [he]
+  convert! ((innerSL ℝ (E := ℂ)).hasFDerivAt (x := z) |>.const_smul (2:ℝ)).neg.fderiv using 1
+
+lemma diskFactor_gradient (z : ℂ) :
+    fderiv ℝ diskFactor z 1 = -2*z.re ∧
+      fderiv ℝ diskFactor z Complex.I = -2*z.im := by
+  rw [diskFactor_fderiv]
+  simp [Complex.inner,innerSL_apply_apply]
+
+lemma diskFactor_laplacian (z : ℂ) : Laplacian.laplacian diskFactor z = -4 := by
+  rw [laplacian_eq_hessian,diskFactor_hessian]
+  simp only [neg_apply,smul_apply,smul_eq_mul]
+  norm_num [innerSL,innerₛₗ,Complex.inner]
+
+lemma logFactor_smooth {z : ℂ} (hz : ‖z‖ < 1) :
+    ContDiffAt ℝ 2 (fun z => Real.log (diskFactor z)) z :=
+  diskFactor_smooth.contDiffAt.log (ne_of_gt (diskFactor_pos hz))
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/ConditionalAngles.lean b/lean/OAI/Probability/SLELocalization/ConditionalAngles.lean
index 652a2587e14a3916593292211a13df78c06718c4..99bf72e6c6855e4e1c9e5518156d164fd18c2333 100644
--- a/lean/OAI/Probability/SLELocalization/ConditionalAngles.lean
+++ b/lean/OAI/Probability/SLELocalization/ConditionalAngles.lean
@@ -1,657 +1,657 @@
-import Mathlib
-import OAI.Probability.SLELocalization.ConditionalTilts
-import OAI.Probability.SLE.ClippedCoordinates
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Finite Tilt estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized BrownianPredictable
-  BrownianStopped BrownianIto TaylorMeshes PredictableLimit
-
-lemma target_setIntegral_le {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η)
-    {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ} (hu : 0 ≤ u) (huv1 : u+1 ≤ v)
-    (C : Cutoff z v) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hsle : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (E : Set Ω) (hE : NullMeasurableSet E P)
-    (hstop : ∀ t : ℝ≥0, ∃ S : Set Ω, MeasurableSet[pastSigma B t] S ∧
-      S =ᵐ[P] (E ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)) :
-    (∫ sample in E, C.barrier η κ B sample ∂P) ≤
-      ∫ sample in E, C.targetValue hz (by linarith : u ≤ v) η κ B sample ∂P := by
-  have huv : u ≤ v := by linarith
-  have hv : 0 < v := by linarith
-  let D := C.retarget hz huv
-  let Q := C.targetCoeff 1 η κ B
-  have hQe (w : ℝ) (hwv : w ≤ v) (n i : ℕ) := C.targetCoeff_eq hB 1 η κ hz w hwv n i
-  obtain ⟨K,hK,hQb⟩ := C.targetCoeff_bound hB 1 η κ hκ hη hz hv γ g f hsle
-  have hQm (w : ℝ) (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (Q w n i) :=
-    C.targetCoeff_measurable B 1 (Or.inl rfl) η κ hκ hη w n i
-  have hQ2 (w : ℝ) (hwv : w ≤ v) (n i : ℕ) : MemLp (Q w n i) 2 P := by
-    apply MemLp.of_bound (C.targetCoeff_aemeasurable hB 1 (Or.inl rfl) η κ hκ hη w n i).aestronglyMeasurable K
-    simpa only [Real.norm_eq_abs] using hQb w hwv n i
-  have hsupport (n i : ℕ) : ∀ᵐ sample ∂P,
-      (dyadicTime C.horizon n i : ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u →
-        Q v n i sample = Q u n i sample := by
-    filter_upwards [hQe v le_rfl n i,hQe u huv n i,hsle,hB.cont,
-      hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hev heu hs hc hb0
-    intro ht
-    rcases eq_or_lt_of_le hu with rfl | hu'
-    · rw [firstLiveRadiusStop_zero hs hz] at ht
-      exact False.elim (not_lt_of_ge bot_le ht)
-    dsimp only [Q]
-    rw [hev,heu]
-    change (if mesh C.horizon n i < C.stopTime κ B sample then _ else 0) =
-      (if mesh C.horizon n i < D.stopTime κ B sample then _ else 0)
-    have hle := C.retarget_stop_le hz huv κ B sample
-    by_cases hD : mesh C.horizon n i < D.stopTime κ B sample
-    · rw [ite_eq_left hD,ite_eq_left (hD.trans_le hle)]
-    · by_cases hC : mesh C.horizon n i < C.stopTime κ B sample
-      · have hh := C.retarget_has_hit_of_lt hz hu' huv κ B sample hs hc hb0
-          ((le_of_not_gt hD).trans_lt hC)
-        have hbad : firstLiveRadiusStop (γ sample) (g sample) z u ≤
-            (dyadicTime C.horizon n i : ℝ≥0∞) := by
-          rw [hh,← ENNReal.ofReal_coe_nnreal,coe_dyadicTime]
-          exact ENNReal.ofReal_le_ofReal (le_of_not_gt hD)
-        exact False.elim ((not_lt_of_ge hbad) ht)
-      · rw [ite_eq_right hC,ite_eq_right hD]
-  have hlv := C.target_sum_tendsto hB η κ hκ hη hz hv γ g f hsle v le_rfl
-  have hlu := C.target_sum_tendsto hB η κ hκ hη hz hv γ g f hsle u huv
-  have hcenter := indicator_difference_centered hB C.horizon (Q v) (Q u)
-    (hQm v) (hQm u) (hQ2 v le_rfl) (hQ2 u huv) K hK (hQb v le_rfl) (hQb u huv)
-    (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u) hsupport E hE hstop
-    (C.targetRemainder η κ B (C.stopTime κ B))
-    (C.targetRemainder η κ B (D.stopTime κ B)) hlv hlu
-  have hdiff : (∫ sample in E, C.targetRemainder η κ B (C.stopTime κ B) sample -
-      C.targetRemainder η κ B (D.stopTime κ B) sample ∂P) = 0 := by
-    simpa only [integral_indicator₀ hE] using hcenter.2
-  have hdiffint : IntegrableOn (fun sample => C.targetRemainder η κ B (C.stopTime κ B) sample -
-      C.targetRemainder η κ B (D.stopTime κ B) sample) E P :=
-    hcenter.1.integrableOn.congr (by
-      filter_upwards [ae_restrict_mem₀ hE] with sample hs
-      exact Set.indicator_of_mem hs _)
-  have hbi := (C.barrier_expectation hB η κ hκ hη hz hv γ g f hsle).1
-  obtain ⟨L,hL,hLb⟩ := targetValue_bound κ hκ
-  have hint : Integrable (C.targetValue hz huv η κ B) P := by
-    apply ((D.expectation hB κ hκ hz hu).1.const_mul
-      (L*((2*z.im)*Real.exp (-v))^(codimension κ))).mono'
-      (C.targetValue_aemeasurable hB hz huv η κ hκ hη).aestronglyMeasurable
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
-    rw [Real.norm_eq_abs,abs_of_nonneg (C.targetValue_nonneg hz huv η κ hκ hη B sample)]
-    exact hLb hz hu huv1 C B sample η hη hc hb0
-  have hle : (∫ sample in E, C.barrier η κ B sample-C.targetValue hz huv η κ B sample ∂P) ≤ 0 := by
-    rw [← hdiff]
-    apply integral_mono_ae (hbi.integrableOn.sub hint.integrableOn) hdiffint
-    filter_upwards [ae_restrict_of_ae hB.cont] with sample hc
-    exact C.target_difference_le_remainder hz huv η κ hκ hη B sample hc
-  rw [integral_sub hbi.integrableOn hint.integrableOn] at hle
-  exact sub_nonpos.mp hle
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerField LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes PredictableLimit
-
-theorem localized_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (ρ δ : ℝ) :
-    let U := BrownianDriver.path B (Real.sqrt κ) T
-    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-    let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T
-      (fun n i => rawCoeff hε 1 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ)
-      n sample) atTop
-      (𝓝 (F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample))) := by
-  dsimp only
-  let U := BrownianDriver.path B (Real.sqrt κ) T
-  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by
-    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
-    linarith
-  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
-      (hc.comp continuous_real_toNNReal)
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
-    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) z ρ δ)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
-        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
-    apply Filter.Eventually.of_forall
-    intro sample
-    unfold stoppedCoeff
-    split_ifs
-    · exact hb _ _ _
-    · simpa only [abs_zero] using hK
-  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
-  let C := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 1 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
-  have hCe (n i : ℕ) : C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 2 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
-    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ z ρ δ
-  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) :=
-    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ z ρ δ
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₁
-    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hb
-  have hC2 (n i : ℕ) : MemLp (C n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₂
-    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |C n i sample| ≤ K₂
-    rw [he]
-    exact hb
-  have hi := stopped_fieldSum_tendsto_ae hB T τ F V htau hreg C hCe hCm hC2
-    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
-      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
-    filter_upwards [hB.cont] with sample hc
-    have hcanc := integral_drift_cancel hε hT κ hκ.1 (U sample) z ρ δ
-      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
-    dsimp only [F,V,τ]
-    linarith
-  have hAeall : ∀ᵐ sample ∂P, ∀ n i, A n i sample = stoppedCoeff 1 B T τ F n i sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAe n))
-  filter_upwards [hi,he,hAeall] with sample hi he ha
-  rw [he] at hi
-  convert hi using 1
-  funext n
-  unfold dyadicDriverSum
-  simp only [← ha]
-  rfl
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized BrownianPredictable
-  BrownianStopped BrownianIto TaylorMeshes PredictableLimit
-
-lemma retarget_setIntegral {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v)
-    (C : Cutoff z v) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hsle : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (E : Set Ω) (hE : NullMeasurableSet E P)
-    (hstop : ∀ t : ℝ≥0, ∃ S : Set Ω, MeasurableSet[pastSigma B t] S ∧
-      S =ᵐ[P] (E ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)) :
-    (∫ sample in E, C.value κ B sample ∂P) =
-      ∫ sample in E, (C.retarget hz huv).value κ B sample ∂P := by
-  let D := C.retarget hz huv
-  let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
-    (Real.sqrt κ) (C.paths κ B sample) z
-  let Q := fun w n i => rawCoeff C.height_pos 1 B C.horizon (dyadicTime C.horizon n i)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-w)) C.angle
-  have hQe (w : ℝ) (n i : ℕ) : Q w n i =ᵐ[P]
-      stoppedCoeff 1 B C.horizon
-        (fun sample => stop C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
-          ((2*z.im)*Real.exp (-w)) C.angle) F n i :=
-    rawCoeff_stoppedCoeff_ae C.height_pos hB 1 C.horizon C.horizon_pos
-      (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-w)) C.angle n i
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := sub_nonneg.2 ((le_div_iff₀ hκ.1).2 (by linarith [hκ.2]))
-  obtain ⟨K,hK,hKb⟩ := coeff_uniform_bound C.height_pos C.horizon.coe_nonneg 1 (Or.inl rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  have hQb (w : ℝ) (n i : ℕ) : ∀ᵐ sample ∂P, |Q w n i sample| ≤ K := by
-    filter_upwards [hQe w n i] with sample he
-    rw [he]
-    unfold stoppedCoeff
-    split_ifs
-    · exact hKb _ _ _
-    · simpa only [abs_zero] using hK
-  have hQm (w : ℝ) (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (Q w n i) :=
-    rawCoeff_measurable_past C.height_pos 1 (Or.inl rfl) B C.horizon _ _ _ _ z _ C.angle
-  have hQ2 (w : ℝ) (n i : ℕ) : MemLp (Q w n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 1 (Or.inl rfl) C.horizon
-      (dyadicTime C.horizon n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z _ C.angle).aestronglyMeasurable K
-    simpa only [Real.norm_eq_abs] using hQb w n i
-  have hsupport (n i : ℕ) : ∀ᵐ sample ∂P,
-      (dyadicTime C.horizon n i : ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u →
-        Q v n i sample = Q u n i sample := by
-    filter_upwards [hQe v n i,hQe u n i,hsle,hB.cont,
-      hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hev heu hs hc hb0
-    intro ht
-    rcases eq_or_lt_of_le hu with rfl | hu'
-    · rw [firstLiveRadiusStop_zero hs hz] at ht
-      exact False.elim (not_lt_of_ge bot_le ht)
-    rw [hev,heu]
-    change (if mesh C.horizon n i < C.stopTime κ B sample then _ else 0) =
-      (if mesh C.horizon n i < D.stopTime κ B sample then _ else 0)
-    have hle := C.retarget_stop_le hz huv κ B sample
-    by_cases hD : mesh C.horizon n i < D.stopTime κ B sample
-    · rw [ite_eq_left hD,ite_eq_left (hD.trans_le hle)]
-    · by_cases hC : mesh C.horizon n i < C.stopTime κ B sample
-      · have hh := C.retarget_has_hit_of_lt hz hu' huv κ B sample hs hc hb0
-          ((le_of_not_gt hD).trans_lt hC)
-        have hbad : firstLiveRadiusStop (γ sample) (g sample) z u ≤
-            (dyadicTime C.horizon n i : ℝ≥0∞) := by
-          rw [hh,← ENNReal.ofReal_coe_nnreal,coe_dyadicTime]
-          exact ENNReal.ofReal_le_ofReal (le_of_not_gt hD)
-        exact False.elim ((not_lt_of_ge hbad) ht)
-      · rw [ite_eq_right hC,ite_eq_right hD]
-  have hlv := localized_sum_tendsto hB κ hκ C.height_pos C.horizon C.horizon_pos z
-    ((2*z.im)*Real.exp (-v)) C.angle
-  have hlu := localized_sum_tendsto hB κ hκ C.height_pos C.horizon C.horizon_pos z
-    ((2*z.im)*Real.exp (-u)) C.angle
-  have hcenter := indicator_difference_centered hB C.horizon (Q v) (Q u)
-    (hQm v) (hQm u) (hQ2 v) (hQ2 u) K hK (hQb v) (hQb u)
-    (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u) hsupport E hE hstop
-    (fun sample => C.value κ B sample-F sample 0 (B 0 sample))
-    (fun sample => D.value κ B sample-F sample 0 (B 0 sample)) hlv hlu
-  have hdiff : (∫ sample in E, C.value κ B sample-D.value κ B sample ∂P) = 0 := by
-    simpa only [sub_sub_sub_cancel_right,integral_indicator₀ hE] using hcenter.2
-  have hv : Integrable (C.value κ B) P := (C.expectation hB κ hκ hz (hu.trans huv)).1
-  have hu' : Integrable (D.value κ B) P := (D.expectation hB κ hκ hz hu).1
-  rw [integral_sub hv.integrableOn hu'.integrableOn] at hdiff
-  exact sub_eq_zero.mp hdiff
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.FiniteTilt
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-variable (C : ℕ → Cutoff z u) (hC : Tendsto (fun n => (C n).angle) atTop (𝓝 0))
-include h hz hu hc hb0 hC
-
-lemma eventually_value_at_radius_general {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
-    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
-    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((C n).value κ B sample) =
-      finiteStoppedDensity κ (fun t => B t sample) γ g z u := by
-  obtain ⟨η, hη, hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
-  have hδ : ∀ᶠ n : ℕ in atTop, (C n).angle < η :=
-    hC.eventually (gt_mem_nhds hη)
-  filter_upwards [hδ] with n hn
-  let D := C n
-  have hh : loewnerRadius g (D.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
-    rcases D.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · exact hr
-    · have htm := hm (D.stopTime κ B sample)
-        ⟨(D.bounds κ B sample).1,D.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
-      rw [hs] at htm
-      exact False.elim ((not_lt_of_ge htm) hn)
-  have he : D.stopTime κ B sample = t := h.live_radius_injective
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (D.bounds κ B sample).1 t.coe_nonneg (D.source_live κ B sample h hz hu hc hb0) hzt
-    (hh.trans hrt.symm)
-  have htau := h.firstLiveRadiusStop_eq
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hzt hrt
-  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  rw [finiteStoppedDensity, htau, ite_eq_right (ENNReal.coe_ne_top), ENNReal.coe_toReal, hp,
-    ← he, ← D.value_eq_source κ B sample h hz hu hc hb0]
-
-lemma tendsto_value_killed_general (hκ : 0 < κ ∧ κ < 8) :
-    Tendsto (fun n : ℕ => ENNReal.ofReal ((C n).value κ B sample)) atTop
-      (𝓝 (finiteStoppedDensity κ (fun t => B t sample) γ g z u)) := by
-  by_cases he : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧
-      loewnerRadius g t z = (2*z.im)*Real.exp (-u)
-  · obtain ⟨t, hzt, hrt⟩ := he
-    have hh : (fun n : ℕ => ENNReal.ofReal ((C n).value κ B sample)) =ᶠ[atTop]
-        (fun _ => finiteStoppedDensity κ (fun t => B t sample) γ g z u) :=
-      eventually_value_at_radius_general κ B sample h hz hu hc hb0 C hC hzt hrt
-    exact tendsto_const_nhds.congr' hh.symm
-  · have hset : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
-        loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ := by
-      apply Set.eq_empty_iff_forall_notMem.2
-      intro t ht
-      exact he ⟨t,ht⟩
-    have htau : firstLiveRadiusStop γ g z u = (⊤ : ℝ≥0∞) := by
-      simp only [firstLiveRadiusStop, hset, Set.image_empty, sInf_empty]
-    rw [finiteStoppedDensity, htau, ite_eq_left rfl]
-    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
-    have hlim := hC.rpow_const_nhds_zero ha
-    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
-        ((C n).angle)^(8/κ-1)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
-    have hl : Tendsto (fun n : ℕ => (C n).value κ B sample) atTop (𝓝 0) := by
-      apply squeeze_zero (fun n => (C n).value_nonneg κ B sample) _ hlim'
-      intro n
-      let D := C n
-      have hs : loewnerAngleSine κ (fun t => B t sample) g (D.stopTime κ B sample) z = D.angle := by
-        rcases D.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-        · exact False.elim (he ⟨⟨D.stopTime κ B sample,(D.bounds κ B sample).1⟩,
-            D.source_live κ B sample h hz hu hc hb0, hr⟩)
-        · exact hs
-      exact D.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
-    simpa only [ENNReal.ofReal_zero, Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.FiniteTilt
-
-lemma tendsto_setLIntegral_value {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 ≤ u)
-    (C : ℕ → Cutoff z u) (hC : Tendsto (fun n => (C n).angle) atTop (𝓝 0)) (E : Set Ω) :
-    Tendsto (fun n => ∫⁻ sample in E, ENNReal.ofReal ((C n).value κ B sample) ∂P) atTop
-      (𝓝 (∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
-        (γ sample) (g sample) z u ∂P)) := by
-  rcases eq_or_lt_of_le hu with rfl | hu
-  · have he (n : ℕ) : (∫⁻ sample in E, ENNReal.ofReal ((C n).value κ B sample) ∂P) =
-        ∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
-          (γ sample) (g sample) z 0 ∂P := by
-      apply lintegral_congr_ae
-      apply ae_restrict_of_ae
-      filter_upwards [hmaps,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hb0
-      rw [(C n).value_zero hz κ B sample hb0,finiteStoppedDensity_zero (κ := κ) (B := fun t => B t sample) hm hb0 hz]
-    simp only [he]
-    exact tendsto_const_nhds
-  · let F := fun n sample => ENNReal.ofReal ((C n).value κ B sample)
-    have hFm (n : ℕ) : AEMeasurable (F n) (P.restrict E) :=
-      (ENNReal.measurable_ofReal.comp_aemeasurable
-        ((C n).expectation hB κ hκ hz hu.le).1.aemeasurable).mono_measure Measure.restrict_le_self
-    have hFb (n : ℕ) : ∀ᵐ sample ∂P.restrict E,
-        F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
-      apply ae_restrict_of_ae
-      filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
-      exact ENNReal.ofReal_le_ofReal ((C n).value_bound κ hκ B sample hz hu hc hb0)
-    have hFl : ∀ᵐ sample ∂P.restrict E, Tendsto (fun n => F n sample) atTop
-        (𝓝 (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u)) := by
-      apply ae_restrict_of_ae
-      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0
-      exact tendsto_value_killed_general κ B sample hm hz hu hc hb0 C hC hκ
-    have hfin : (∫⁻ _ : Ω in E, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ ⊤ := by
-      rw [lintegral_const]
-      exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)
-    exact tendsto_lintegral_of_dominated_convergence'
-      (fun _ : Ω => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8)))) hFm hFb hfin hFl
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteTilt BrownianPredictable
-
-theorem finiteStoppedDensity_consistent {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u v : ℝ) (hu : 0 ≤ u) (huv : u ≤ v)
-    (A : Set Ω) (hA : NullMeasurableSet A P)
-    (hstop : ∀ t : ℝ≥0, ∃ E : Set Ω, MeasurableSet[pastSigma B t] E ∧
-      E =ᵐ[P] (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)) :
-    (∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z v ∂P) =
-      ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P := by
-  let C := cutoffs hz v (hu.trans huv)
-  let D := fun n => (C n).retarget hz huv
-  have hv := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz (hu.trans huv) C
-    (shrinkingAngle_tendsto z) A
-  have hu' := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz hu D
-    (shrinkingAngle_tendsto z) A
-  have he (n : ℕ) : (∫⁻ sample in A, ENNReal.ofReal ((C n).value κ B sample) ∂P) =
-      ∫⁻ sample in A, ENNReal.ofReal ((D n).value κ B sample) ∂P := by
-    rw [← ofReal_integral_eq_lintegral_ofReal ((C n).expectation hB κ hκ hz (hu.trans huv)).1.integrableOn
-      (Filter.Eventually.of_forall (fun sample => (C n).value_nonneg κ B sample)),
-      ← ofReal_integral_eq_lintegral_ofReal ((D n).expectation hB κ hκ hz hu).1.integrableOn
-      (Filter.Eventually.of_forall (fun sample => (D n).value_nonneg κ B sample))]
-    congr 1
-    exact (C n).retarget_setIntegral hB κ hκ hz hu huv γ g f hmaps A hA hstop
-  exact tendsto_nhds_unique (hv.congr' (Filter.Eventually.of_forall he)) hu'
-
-theorem conditionalRadiusApproachBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C : ℝ, 0 < C ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (_hz : z ∈ halfPlane) (u v : ℝ) (_hu : 0 ≤ u) (_huv : u+1 ≤ v)
-      (A : Set Ω) (_hA : NullMeasurableSet A P)
-      (_hstop : ∀ t : ℝ≥0, ∃ E : Set Ω,
-        MeasurableSet[pastSigma B t] E ∧
-        E =ᵐ[P] (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)),
-      P (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z v ≠ (⊤ : ℝ≥0∞)}) ≤
-        ENNReal.ofReal (C * ((2*z.im)*Real.exp (-v))^(codimension κ)) *
-          ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P := by
-  obtain ⟨K,hK,hKb⟩ := Cutoff.targetValue_bound κ hκ
-  refine ⟨K,hK,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz u v hu huv A hA hstop
-  have hz0 : 0 < z.im := hz
-  have hv : 0 < v := by linarith
-  have huv' : u ≤ v := by linarith
-  let C := cutoffs hz v hv.le
-  let D := fun n => (C n).retarget hz huv'
-  let F := fun n => A.indicator (fun sample => ENNReal.ofReal ((C n).barrier (timeShift n) κ B sample))
-  let Y := fun sample => liminf (fun n => F n sample) atTop
-  have hFm (n : ℕ) : AEMeasurable (F n) P :=
-    (ENNReal.measurable_ofReal.comp_aemeasurable
-      ((C n).barrier_aemeasurable hB (timeShift n) κ hκ (timeShift_pos n))).indicator₀ hA
-  have hYm : AEMeasurable Y P := by
-    dsimp only [Y]
-    simp_rw [liminf_eq_iSup_iInf_of_nat]
-    exact AEMeasurable.iSup (fun n => AEMeasurable.iInf (fun i => AEMeasurable.iInf (fun (_ : n ≤ i) => hFm i)))
-  have hsub : (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z v ≠ (⊤ : ℝ≥0∞)} : Set Ω) ≤ᵐ[P]
-      {sample | (1:ℝ≥0∞) ≤ Y sample} := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0 hs
-    have hh := barrier_tendsto_on_hit κ B sample hm hz hv hc hb0 hκ hs.2
-    have he : Y sample = 1 := by
-      dsimp only [Y,F]
-      simp only [Set.indicator_of_mem hs.1]
-      exact hh.liminf_eq
-    exact he.ge
-  have hexp (n : ℕ) : (∫⁻ sample, F n sample ∂P) ≤
-      ENNReal.ofReal (K*((2*z.im)*Real.exp (-v))^(codimension κ)) *
-        ∫⁻ sample in A, ENNReal.ofReal ((D n).value κ B sample) ∂P := by
-    have hbi := ((C n).barrier_expectation hB (timeShift n) κ hκ (timeShift_pos n) hz hv γ g f hmaps).1
-    have hti : Integrable ((C n).targetValue hz huv' (timeShift n) κ B) P := by
-      apply (((D n).expectation hB κ hκ hz hu).1.const_mul
-        (K*((2*z.im)*Real.exp (-v))^(codimension κ))).mono'
-        ((C n).targetValue_aemeasurable hB hz huv' (timeShift n) κ hκ (timeShift_pos n)).aestronglyMeasurable
-      filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
-      rw [Real.norm_eq_abs,abs_of_nonneg ((C n).targetValue_nonneg hz huv' (timeShift n) κ hκ (timeShift_pos n) B sample)]
-      exact hKb hz hu huv (C n) B sample (timeShift n) (timeShift_pos n) hc hb0
-    have hle := (C n).target_setIntegral_le hB (timeShift n) κ hκ (timeShift_pos n) hz hu huv
-      γ g f hmaps A hA hstop
-    rw [lintegral_indicator₀ hA]
-    calc
-      _ = ENNReal.ofReal (∫ sample in A, (C n).barrier (timeShift n) κ B sample ∂P) := by
-        symm
-        exact ofReal_integral_eq_lintegral_ofReal hbi.integrableOn (Eventually.of_forall
-          (fun sample => (C n).barrier_nonneg (timeShift n) κ B sample hκ (timeShift_pos n)))
-      _ ≤ ENNReal.ofReal (∫ sample in A, (C n).targetValue hz huv' (timeShift n) κ B sample ∂P) :=
-        ENNReal.ofReal_le_ofReal hle
-      _ = ∫⁻ sample in A, ENNReal.ofReal ((C n).targetValue hz huv' (timeShift n) κ B sample) ∂P :=
-        ofReal_integral_eq_lintegral_ofReal hti.integrableOn (Eventually.of_forall
-          (fun sample => (C n).targetValue_nonneg hz huv' (timeShift n) κ hκ (timeShift_pos n) B sample))
-      _ ≤ ∫⁻ sample in A, ENNReal.ofReal (K*((2*z.im)*Real.exp (-v))^(codimension κ)) *
-          ENNReal.ofReal ((D n).value κ B sample) ∂P := by
-        apply lintegral_mono_ae
-        apply ae_restrict_of_ae
-        filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
-        rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ K*((2*z.im)*Real.exp (-v))^(codimension κ))]
-        exact ENNReal.ofReal_le_ofReal (hKb hz hu huv (C n) B sample (timeShift n) (timeShift_pos n) hc hb0)
-      _ = _ := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-  have ht := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz hu D (shrinkingAngle_tendsto z) A
-  have ht' := ENNReal.Tendsto.const_mul ht (Or.inr (ENNReal.ofReal_ne_top (r := K*((2*z.im)*Real.exp (-v))^(codimension κ))))
-  calc
-    _ ≤ P {sample | (1:ℝ≥0∞) ≤ Y sample} := measure_mono_ae hsub
-    _ ≤ ∫⁻ sample, Y sample ∂P := by
-      simpa only [div_one] using meas_ge_le_lintegral_div hYm (by norm_num : (1:ℝ≥0∞) ≠ 0) (by norm_num : (1:ℝ≥0∞) ≠ ⊤)
-    _ ≤ liminf (fun n => ∫⁻ sample, F n sample ∂P) atTop := lintegral_liminf_le' hFm
-    _ ≤ _ := (liminf_le_liminf (Eventually.of_forall hexp)).trans_eq ht'.liminf_eq
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable FiniteTilt
-
-theorem conditionalCenteredInverseAngleBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C : ℝ, 0 < C ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (_hz : z ∈ halfPlane) (u v : ℝ) (_hu : 0 ≤ u) (_huv : u+1 ≤ v)
-      (A : Set Ω) (_hA : NullMeasurableSet A P)
-      (_hstop : ∀ t : ℝ≥0, ∃ E : Set Ω,
-        MeasurableSet[pastSigma B t] E ∧
-        E =ᵐ[P] (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)),
-      (∫⁻ sample in A, radiusStoppedInverseAngle κ B γ g z v sample
-          ∂finiteCenteredMeasure κ P B γ g z v) ≤
-        ENNReal.ofReal C * (finiteCenteredMeasure κ P B γ g z u) A := by
-  obtain ⟨C,hC,hbound⟩ := conditionalRadiusApproachBound κ hκ
-  refine ⟨C,hC,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz u v hu huv A hA hstop
-  have hv : 0 < v := by linarith
-  have hz0 : 0 < z.im := hz
-  let M₀ := ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))
-  let K := ENNReal.ofReal (((2*z.im)*Real.exp (-v))^(-codimension κ)) / M₀
-  let H : Set Ω := {sample | firstLiveRadiusStop (γ sample) (g sample) z v ≠ (⊤ : ℝ≥0∞)}
-  have hmpos := localizationInitialGreen_pos κ hz
-  have hm0 : M₀ ≠ 0 := (ENNReal.ofReal_pos.2 hmpos).ne'
-  have hd := finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz v hv
-  have hdm : AEMeasurable (fun sample => finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z v / M₀) P := hd.div_const M₀
-  have hdf : ∀ᵐ sample ∂P, finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z v / M₀ < (⊤ : ℝ≥0∞) := Filter.Eventually.of_forall
-    (fun sample => ENNReal.div_lt_top (finiteStoppedDensity_lt_top κ _ _ _ z v).ne hm0)
-  have he : (fun sample => (finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z v / M₀) * radiusStoppedInverseAngle κ B γ g z v sample) =ᵐ[P]
-      H.indicator (fun _ => K) := by
-    filter_upwards [hmaps,hB.cont] with sample hm hc
-    have hh := finiteStoppedDensity_mul_inverseAngle κ B γ g f z hz v sample hm hc
-    rw [show (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z v / M₀) *
-      radiusStoppedInverseAngle κ B γ g z v sample = (finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z v * radiusStoppedInverseAngle κ B γ g z v sample) / M₀ by
-        simp only [div_eq_mul_inv]; ac_rfl, hh]
-    classical
-    by_cases hhit : sample ∈ H
-    · simp only [H] at hhit
-      simp only [H, Set.indicator_of_mem hhit, K]
-    · simp only [H] at hhit
-      simp only [H, Set.indicator_of_notMem hhit, ENNReal.zero_div]
-  have heleft : (∫⁻ sample in A, radiusStoppedInverseAngle κ B γ g z v sample
-      ∂finiteCenteredMeasure κ P B γ g z v) ≤ K*P (A∩H) := by
-    change (∫⁻ sample in A, _ ∂P.withDensity _) ≤ _
-    rw [setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀' P A hdm.restrict _
-      (ae_restrict_of_ae hdf)]
-    simp only [Pi.mul_apply]
-    rw [lintegral_congr_ae (ae_restrict_of_ae he)]
-    calc
-      _ ≤ K*(P.restrict A) H := lintegral_indicator_const_le H K
-      _ = _ := by rw [Measure.restrict_apply₀' hA,inter_comm]
-  have hemeasure : (finiteCenteredMeasure κ P B γ g z u) A =
-      M₀⁻¹ * ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample)
-        (γ sample) (g sample) z u ∂P := by
-    rw [finiteCenteredMeasure,withDensity_apply₀ _ hA]
-    simp_rw [ENNReal.div_eq_inv_mul]
-    exact lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hm0)
-  have hcancel : ENNReal.ofReal (((2*z.im)*Real.exp (-v))^(-codimension κ)) *
-      ENNReal.ofReal (C*((2*z.im)*Real.exp (-v))^(codimension κ)) = ENNReal.ofReal C := by
-    rw [← ENNReal.ofReal_mul (Real.rpow_nonneg (by positivity) _)]
-    congr 1
-    calc
-      _ = C*(((2*z.im)*Real.exp (-v))^(-codimension κ)*((2*z.im)*Real.exp (-v))^(codimension κ)) := by ring
-      _ = C := by rw [← Real.rpow_add (by positivity),neg_add_cancel,Real.rpow_zero,mul_one]
-  calc
-    _ ≤ K*P (A∩H) := heleft
-    _ ≤ K*(ENNReal.ofReal (C*((2*z.im)*Real.exp (-v))^(codimension κ)) *
-        ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P) :=
-      mul_le_mul' le_rfl (hbound P B hB γ g f hγ hmaps z hz u v hu huv A hA hstop)
-    _ = _ := by
-      rw [hemeasure]
-      dsimp only [K]
-      rw [div_eq_mul_inv]
-      calc
-        _ = (ENNReal.ofReal (((2*z.im)*Real.exp (-v))^(-codimension κ)) *
-            ENNReal.ofReal (C*((2*z.im)*Real.exp (-v))^(codimension κ))) *
-            (M₀⁻¹ * ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P) := by ac_rfl
-        _ = _ := by rw [hcancel]
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.ConditionalTilts
+import OAI.Probability.SLE.ClippedCoordinates
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Finite Tilt estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized BrownianPredictable
+  BrownianStopped BrownianIto TaylorMeshes PredictableLimit
+
+lemma target_setIntegral_le {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η)
+    {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ} (hu : 0 ≤ u) (huv1 : u+1 ≤ v)
+    (C : Cutoff z v) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hsle : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (E : Set Ω) (hE : NullMeasurableSet E P)
+    (hstop : ∀ t : ℝ≥0, ∃ S : Set Ω, MeasurableSet[pastSigma B t] S ∧
+      S =ᵐ[P] (E ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)) :
+    (∫ sample in E, C.barrier η κ B sample ∂P) ≤
+      ∫ sample in E, C.targetValue hz (by linarith : u ≤ v) η κ B sample ∂P := by
+  have huv : u ≤ v := by linarith
+  have hv : 0 < v := by linarith
+  let D := C.retarget hz huv
+  let Q := C.targetCoeff 1 η κ B
+  have hQe (w : ℝ) (hwv : w ≤ v) (n i : ℕ) := C.targetCoeff_eq hB 1 η κ hz w hwv n i
+  obtain ⟨K,hK,hQb⟩ := C.targetCoeff_bound hB 1 η κ hκ hη hz hv γ g f hsle
+  have hQm (w : ℝ) (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (Q w n i) :=
+    C.targetCoeff_measurable B 1 (Or.inl rfl) η κ hκ hη w n i
+  have hQ2 (w : ℝ) (hwv : w ≤ v) (n i : ℕ) : MemLp (Q w n i) 2 P := by
+    apply MemLp.of_bound (C.targetCoeff_aemeasurable hB 1 (Or.inl rfl) η κ hκ hη w n i).aestronglyMeasurable K
+    simpa only [Real.norm_eq_abs] using hQb w hwv n i
+  have hsupport (n i : ℕ) : ∀ᵐ sample ∂P,
+      (dyadicTime C.horizon n i : ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u →
+        Q v n i sample = Q u n i sample := by
+    filter_upwards [hQe v le_rfl n i,hQe u huv n i,hsle,hB.cont,
+      hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hev heu hs hc hb0
+    intro ht
+    rcases eq_or_lt_of_le hu with rfl | hu'
+    · rw [firstLiveRadiusStop_zero hs hz] at ht
+      exact False.elim (not_lt_of_ge bot_le ht)
+    dsimp only [Q]
+    rw [hev,heu]
+    change (if mesh C.horizon n i < C.stopTime κ B sample then _ else 0) =
+      (if mesh C.horizon n i < D.stopTime κ B sample then _ else 0)
+    have hle := C.retarget_stop_le hz huv κ B sample
+    by_cases hD : mesh C.horizon n i < D.stopTime κ B sample
+    · rw [ite_eq_left hD,ite_eq_left (hD.trans_le hle)]
+    · by_cases hC : mesh C.horizon n i < C.stopTime κ B sample
+      · have hh := C.retarget_has_hit_of_lt hz hu' huv κ B sample hs hc hb0
+          ((le_of_not_gt hD).trans_lt hC)
+        have hbad : firstLiveRadiusStop (γ sample) (g sample) z u ≤
+            (dyadicTime C.horizon n i : ℝ≥0∞) := by
+          rw [hh,← ENNReal.ofReal_coe_nnreal,coe_dyadicTime]
+          exact ENNReal.ofReal_le_ofReal (le_of_not_gt hD)
+        exact False.elim ((not_lt_of_ge hbad) ht)
+      · rw [ite_eq_right hC,ite_eq_right hD]
+  have hlv := C.target_sum_tendsto hB η κ hκ hη hz hv γ g f hsle v le_rfl
+  have hlu := C.target_sum_tendsto hB η κ hκ hη hz hv γ g f hsle u huv
+  have hcenter := indicator_difference_centered hB C.horizon (Q v) (Q u)
+    (hQm v) (hQm u) (hQ2 v le_rfl) (hQ2 u huv) K hK (hQb v le_rfl) (hQb u huv)
+    (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u) hsupport E hE hstop
+    (C.targetRemainder η κ B (C.stopTime κ B))
+    (C.targetRemainder η κ B (D.stopTime κ B)) hlv hlu
+  have hdiff : (∫ sample in E, C.targetRemainder η κ B (C.stopTime κ B) sample -
+      C.targetRemainder η κ B (D.stopTime κ B) sample ∂P) = 0 := by
+    simpa only [integral_indicator₀ hE] using hcenter.2
+  have hdiffint : IntegrableOn (fun sample => C.targetRemainder η κ B (C.stopTime κ B) sample -
+      C.targetRemainder η κ B (D.stopTime κ B) sample) E P :=
+    hcenter.1.integrableOn.congr (by
+      filter_upwards [ae_restrict_mem₀ hE] with sample hs
+      exact Set.indicator_of_mem hs _)
+  have hbi := (C.barrier_expectation hB η κ hκ hη hz hv γ g f hsle).1
+  obtain ⟨L,hL,hLb⟩ := targetValue_bound κ hκ
+  have hint : Integrable (C.targetValue hz huv η κ B) P := by
+    apply ((D.expectation hB κ hκ hz hu).1.const_mul
+      (L*((2*z.im)*Real.exp (-v))^(codimension κ))).mono'
+      (C.targetValue_aemeasurable hB hz huv η κ hκ hη).aestronglyMeasurable
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
+    rw [Real.norm_eq_abs,abs_of_nonneg (C.targetValue_nonneg hz huv η κ hκ hη B sample)]
+    exact hLb hz hu huv1 C B sample η hη hc hb0
+  have hle : (∫ sample in E, C.barrier η κ B sample-C.targetValue hz huv η κ B sample ∂P) ≤ 0 := by
+    rw [← hdiff]
+    apply integral_mono_ae (hbi.integrableOn.sub hint.integrableOn) hdiffint
+    filter_upwards [ae_restrict_of_ae hB.cont] with sample hc
+    exact C.target_difference_le_remainder hz huv η κ hκ hη B sample hc
+  rw [integral_sub hbi.integrableOn hint.integrableOn] at hle
+  exact sub_nonpos.mp hle
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerField LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes PredictableLimit
+
+theorem localized_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (ρ δ : ℝ) :
+    let U := BrownianDriver.path B (Real.sqrt κ) T
+    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+    let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T
+      (fun n i => rawCoeff hε 1 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ)
+      n sample) atTop
+      (𝓝 (F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample))) := by
+  dsimp only
+  let U := BrownianDriver.path B (Real.sqrt κ) T
+  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by
+    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
+    linarith
+  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
+      (hc.comp continuous_real_toNNReal)
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
+    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) z ρ δ)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
+        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
+    apply Filter.Eventually.of_forall
+    intro sample
+    unfold stoppedCoeff
+    split_ifs
+    · exact hb _ _ _
+    · simpa only [abs_zero] using hK
+  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
+  let C := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 1 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
+  have hCe (n i : ℕ) : C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 2 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
+    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ z ρ δ
+  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) :=
+    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ z ρ δ
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₁
+    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hb
+  have hC2 (n i : ℕ) : MemLp (C n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₂
+    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |C n i sample| ≤ K₂
+    rw [he]
+    exact hb
+  have hi := stopped_fieldSum_tendsto_ae hB T τ F V htau hreg C hCe hCm hC2
+    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
+      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
+    filter_upwards [hB.cont] with sample hc
+    have hcanc := integral_drift_cancel hε hT κ hκ.1 (U sample) z ρ δ
+      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
+    dsimp only [F,V,τ]
+    linarith
+  have hAeall : ∀ᵐ sample ∂P, ∀ n i, A n i sample = stoppedCoeff 1 B T τ F n i sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAe n))
+  filter_upwards [hi,he,hAeall] with sample hi he ha
+  rw [he] at hi
+  convert hi using 1
+  funext n
+  unfold dyadicDriverSum
+  simp only [← ha]
+  rfl
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized BrownianPredictable
+  BrownianStopped BrownianIto TaylorMeshes PredictableLimit
+
+lemma retarget_setIntegral {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v)
+    (C : Cutoff z v) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hsle : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (E : Set Ω) (hE : NullMeasurableSet E P)
+    (hstop : ∀ t : ℝ≥0, ∃ S : Set Ω, MeasurableSet[pastSigma B t] S ∧
+      S =ᵐ[P] (E ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)) :
+    (∫ sample in E, C.value κ B sample ∂P) =
+      ∫ sample in E, (C.retarget hz huv).value κ B sample ∂P := by
+  let D := C.retarget hz huv
+  let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
+    (Real.sqrt κ) (C.paths κ B sample) z
+  let Q := fun w n i => rawCoeff C.height_pos 1 B C.horizon (dyadicTime C.horizon n i)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-w)) C.angle
+  have hQe (w : ℝ) (n i : ℕ) : Q w n i =ᵐ[P]
+      stoppedCoeff 1 B C.horizon
+        (fun sample => stop C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
+          ((2*z.im)*Real.exp (-w)) C.angle) F n i :=
+    rawCoeff_stoppedCoeff_ae C.height_pos hB 1 C.horizon C.horizon_pos
+      (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-w)) C.angle n i
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := sub_nonneg.2 ((le_div_iff₀ hκ.1).2 (by linarith [hκ.2]))
+  obtain ⟨K,hK,hKb⟩ := coeff_uniform_bound C.height_pos C.horizon.coe_nonneg 1 (Or.inl rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  have hQb (w : ℝ) (n i : ℕ) : ∀ᵐ sample ∂P, |Q w n i sample| ≤ K := by
+    filter_upwards [hQe w n i] with sample he
+    rw [he]
+    unfold stoppedCoeff
+    split_ifs
+    · exact hKb _ _ _
+    · simpa only [abs_zero] using hK
+  have hQm (w : ℝ) (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (Q w n i) :=
+    rawCoeff_measurable_past C.height_pos 1 (Or.inl rfl) B C.horizon _ _ _ _ z _ C.angle
+  have hQ2 (w : ℝ) (n i : ℕ) : MemLp (Q w n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 1 (Or.inl rfl) C.horizon
+      (dyadicTime C.horizon n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z _ C.angle).aestronglyMeasurable K
+    simpa only [Real.norm_eq_abs] using hQb w n i
+  have hsupport (n i : ℕ) : ∀ᵐ sample ∂P,
+      (dyadicTime C.horizon n i : ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u →
+        Q v n i sample = Q u n i sample := by
+    filter_upwards [hQe v n i,hQe u n i,hsle,hB.cont,
+      hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hev heu hs hc hb0
+    intro ht
+    rcases eq_or_lt_of_le hu with rfl | hu'
+    · rw [firstLiveRadiusStop_zero hs hz] at ht
+      exact False.elim (not_lt_of_ge bot_le ht)
+    rw [hev,heu]
+    change (if mesh C.horizon n i < C.stopTime κ B sample then _ else 0) =
+      (if mesh C.horizon n i < D.stopTime κ B sample then _ else 0)
+    have hle := C.retarget_stop_le hz huv κ B sample
+    by_cases hD : mesh C.horizon n i < D.stopTime κ B sample
+    · rw [ite_eq_left hD,ite_eq_left (hD.trans_le hle)]
+    · by_cases hC : mesh C.horizon n i < C.stopTime κ B sample
+      · have hh := C.retarget_has_hit_of_lt hz hu' huv κ B sample hs hc hb0
+          ((le_of_not_gt hD).trans_lt hC)
+        have hbad : firstLiveRadiusStop (γ sample) (g sample) z u ≤
+            (dyadicTime C.horizon n i : ℝ≥0∞) := by
+          rw [hh,← ENNReal.ofReal_coe_nnreal,coe_dyadicTime]
+          exact ENNReal.ofReal_le_ofReal (le_of_not_gt hD)
+        exact False.elim ((not_lt_of_ge hbad) ht)
+      · rw [ite_eq_right hC,ite_eq_right hD]
+  have hlv := localized_sum_tendsto hB κ hκ C.height_pos C.horizon C.horizon_pos z
+    ((2*z.im)*Real.exp (-v)) C.angle
+  have hlu := localized_sum_tendsto hB κ hκ C.height_pos C.horizon C.horizon_pos z
+    ((2*z.im)*Real.exp (-u)) C.angle
+  have hcenter := indicator_difference_centered hB C.horizon (Q v) (Q u)
+    (hQm v) (hQm u) (hQ2 v) (hQ2 u) K hK (hQb v) (hQb u)
+    (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u) hsupport E hE hstop
+    (fun sample => C.value κ B sample-F sample 0 (B 0 sample))
+    (fun sample => D.value κ B sample-F sample 0 (B 0 sample)) hlv hlu
+  have hdiff : (∫ sample in E, C.value κ B sample-D.value κ B sample ∂P) = 0 := by
+    simpa only [sub_sub_sub_cancel_right,integral_indicator₀ hE] using hcenter.2
+  have hv : Integrable (C.value κ B) P := (C.expectation hB κ hκ hz (hu.trans huv)).1
+  have hu' : Integrable (D.value κ B) P := (D.expectation hB κ hκ hz hu).1
+  rw [integral_sub hv.integrableOn hu'.integrableOn] at hdiff
+  exact sub_eq_zero.mp hdiff
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.FiniteTilt
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+variable (C : ℕ → Cutoff z u) (hC : Tendsto (fun n => (C n).angle) atTop (𝓝 0))
+include h hz hu hc hb0 hC
+
+lemma eventually_value_at_radius_general {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
+    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
+    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((C n).value κ B sample) =
+      finiteStoppedDensity κ (fun t => B t sample) γ g z u := by
+  obtain ⟨η, hη, hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
+  have hδ : ∀ᶠ n : ℕ in atTop, (C n).angle < η :=
+    hC.eventually (gt_mem_nhds hη)
+  filter_upwards [hδ] with n hn
+  let D := C n
+  have hh : loewnerRadius g (D.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
+    rcases D.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · exact hr
+    · have htm := hm (D.stopTime κ B sample)
+        ⟨(D.bounds κ B sample).1,D.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
+      rw [hs] at htm
+      exact False.elim ((not_lt_of_ge htm) hn)
+  have he : D.stopTime κ B sample = t := h.live_radius_injective
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (D.bounds κ B sample).1 t.coe_nonneg (D.source_live κ B sample h hz hu hc hb0) hzt
+    (hh.trans hrt.symm)
+  have htau := h.firstLiveRadiusStop_eq
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hzt hrt
+  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  rw [finiteStoppedDensity, htau, ite_eq_right (ENNReal.coe_ne_top), ENNReal.coe_toReal, hp,
+    ← he, ← D.value_eq_source κ B sample h hz hu hc hb0]
+
+lemma tendsto_value_killed_general (hκ : 0 < κ ∧ κ < 8) :
+    Tendsto (fun n : ℕ => ENNReal.ofReal ((C n).value κ B sample)) atTop
+      (𝓝 (finiteStoppedDensity κ (fun t => B t sample) γ g z u)) := by
+  by_cases he : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧
+      loewnerRadius g t z = (2*z.im)*Real.exp (-u)
+  · obtain ⟨t, hzt, hrt⟩ := he
+    have hh : (fun n : ℕ => ENNReal.ofReal ((C n).value κ B sample)) =ᶠ[atTop]
+        (fun _ => finiteStoppedDensity κ (fun t => B t sample) γ g z u) :=
+      eventually_value_at_radius_general κ B sample h hz hu hc hb0 C hC hzt hrt
+    exact tendsto_const_nhds.congr' hh.symm
+  · have hset : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
+        loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ := by
+      apply Set.eq_empty_iff_forall_notMem.2
+      intro t ht
+      exact he ⟨t,ht⟩
+    have htau : firstLiveRadiusStop γ g z u = (⊤ : ℝ≥0∞) := by
+      simp only [firstLiveRadiusStop, hset, Set.image_empty, sInf_empty]
+    rw [finiteStoppedDensity, htau, ite_eq_left rfl]
+    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
+    have hlim := hC.rpow_const_nhds_zero ha
+    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
+        ((C n).angle)^(8/κ-1)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
+    have hl : Tendsto (fun n : ℕ => (C n).value κ B sample) atTop (𝓝 0) := by
+      apply squeeze_zero (fun n => (C n).value_nonneg κ B sample) _ hlim'
+      intro n
+      let D := C n
+      have hs : loewnerAngleSine κ (fun t => B t sample) g (D.stopTime κ B sample) z = D.angle := by
+        rcases D.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+        · exact False.elim (he ⟨⟨D.stopTime κ B sample,(D.bounds κ B sample).1⟩,
+            D.source_live κ B sample h hz hu hc hb0, hr⟩)
+        · exact hs
+      exact D.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
+    simpa only [ENNReal.ofReal_zero, Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.FiniteTilt
+
+lemma tendsto_setLIntegral_value {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 ≤ u)
+    (C : ℕ → Cutoff z u) (hC : Tendsto (fun n => (C n).angle) atTop (𝓝 0)) (E : Set Ω) :
+    Tendsto (fun n => ∫⁻ sample in E, ENNReal.ofReal ((C n).value κ B sample) ∂P) atTop
+      (𝓝 (∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
+        (γ sample) (g sample) z u ∂P)) := by
+  rcases eq_or_lt_of_le hu with rfl | hu
+  · have he (n : ℕ) : (∫⁻ sample in E, ENNReal.ofReal ((C n).value κ B sample) ∂P) =
+        ∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
+          (γ sample) (g sample) z 0 ∂P := by
+      apply lintegral_congr_ae
+      apply ae_restrict_of_ae
+      filter_upwards [hmaps,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hb0
+      rw [(C n).value_zero hz κ B sample hb0,finiteStoppedDensity_zero (κ := κ) (B := fun t => B t sample) hm hb0 hz]
+    simp only [he]
+    exact tendsto_const_nhds
+  · let F := fun n sample => ENNReal.ofReal ((C n).value κ B sample)
+    have hFm (n : ℕ) : AEMeasurable (F n) (P.restrict E) :=
+      (ENNReal.measurable_ofReal.comp_aemeasurable
+        ((C n).expectation hB κ hκ hz hu.le).1.aemeasurable).mono_measure Measure.restrict_le_self
+    have hFb (n : ℕ) : ∀ᵐ sample ∂P.restrict E,
+        F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
+      apply ae_restrict_of_ae
+      filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
+      exact ENNReal.ofReal_le_ofReal ((C n).value_bound κ hκ B sample hz hu hc hb0)
+    have hFl : ∀ᵐ sample ∂P.restrict E, Tendsto (fun n => F n sample) atTop
+        (𝓝 (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u)) := by
+      apply ae_restrict_of_ae
+      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0
+      exact tendsto_value_killed_general κ B sample hm hz hu hc hb0 C hC hκ
+    have hfin : (∫⁻ _ : Ω in E, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ ⊤ := by
+      rw [lintegral_const]
+      exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)
+    exact tendsto_lintegral_of_dominated_convergence'
+      (fun _ : Ω => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8)))) hFm hFb hfin hFl
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteTilt BrownianPredictable
+
+theorem finiteStoppedDensity_consistent {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u v : ℝ) (hu : 0 ≤ u) (huv : u ≤ v)
+    (A : Set Ω) (hA : NullMeasurableSet A P)
+    (hstop : ∀ t : ℝ≥0, ∃ E : Set Ω, MeasurableSet[pastSigma B t] E ∧
+      E =ᵐ[P] (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)) :
+    (∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z v ∂P) =
+      ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P := by
+  let C := cutoffs hz v (hu.trans huv)
+  let D := fun n => (C n).retarget hz huv
+  have hv := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz (hu.trans huv) C
+    (shrinkingAngle_tendsto z) A
+  have hu' := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz hu D
+    (shrinkingAngle_tendsto z) A
+  have he (n : ℕ) : (∫⁻ sample in A, ENNReal.ofReal ((C n).value κ B sample) ∂P) =
+      ∫⁻ sample in A, ENNReal.ofReal ((D n).value κ B sample) ∂P := by
+    rw [← ofReal_integral_eq_lintegral_ofReal ((C n).expectation hB κ hκ hz (hu.trans huv)).1.integrableOn
+      (Filter.Eventually.of_forall (fun sample => (C n).value_nonneg κ B sample)),
+      ← ofReal_integral_eq_lintegral_ofReal ((D n).expectation hB κ hκ hz hu).1.integrableOn
+      (Filter.Eventually.of_forall (fun sample => (D n).value_nonneg κ B sample))]
+    congr 1
+    exact (C n).retarget_setIntegral hB κ hκ hz hu huv γ g f hmaps A hA hstop
+  exact tendsto_nhds_unique (hv.congr' (Filter.Eventually.of_forall he)) hu'
+
+theorem conditionalRadiusApproachBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C : ℝ, 0 < C ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (_hz : z ∈ halfPlane) (u v : ℝ) (_hu : 0 ≤ u) (_huv : u+1 ≤ v)
+      (A : Set Ω) (_hA : NullMeasurableSet A P)
+      (_hstop : ∀ t : ℝ≥0, ∃ E : Set Ω,
+        MeasurableSet[pastSigma B t] E ∧
+        E =ᵐ[P] (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)),
+      P (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z v ≠ (⊤ : ℝ≥0∞)}) ≤
+        ENNReal.ofReal (C * ((2*z.im)*Real.exp (-v))^(codimension κ)) *
+          ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P := by
+  obtain ⟨K,hK,hKb⟩ := Cutoff.targetValue_bound κ hκ
+  refine ⟨K,hK,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz u v hu huv A hA hstop
+  have hz0 : 0 < z.im := hz
+  have hv : 0 < v := by linarith
+  have huv' : u ≤ v := by linarith
+  let C := cutoffs hz v hv.le
+  let D := fun n => (C n).retarget hz huv'
+  let F := fun n => A.indicator (fun sample => ENNReal.ofReal ((C n).barrier (timeShift n) κ B sample))
+  let Y := fun sample => liminf (fun n => F n sample) atTop
+  have hFm (n : ℕ) : AEMeasurable (F n) P :=
+    (ENNReal.measurable_ofReal.comp_aemeasurable
+      ((C n).barrier_aemeasurable hB (timeShift n) κ hκ (timeShift_pos n))).indicator₀ hA
+  have hYm : AEMeasurable Y P := by
+    dsimp only [Y]
+    simp_rw [liminf_eq_iSup_iInf_of_nat]
+    exact AEMeasurable.iSup (fun n => AEMeasurable.iInf (fun i => AEMeasurable.iInf (fun (_ : n ≤ i) => hFm i)))
+  have hsub : (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z v ≠ (⊤ : ℝ≥0∞)} : Set Ω) ≤ᵐ[P]
+      {sample | (1:ℝ≥0∞) ≤ Y sample} := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0 hs
+    have hh := barrier_tendsto_on_hit κ B sample hm hz hv hc hb0 hκ hs.2
+    have he : Y sample = 1 := by
+      dsimp only [Y,F]
+      simp only [Set.indicator_of_mem hs.1]
+      exact hh.liminf_eq
+    exact he.ge
+  have hexp (n : ℕ) : (∫⁻ sample, F n sample ∂P) ≤
+      ENNReal.ofReal (K*((2*z.im)*Real.exp (-v))^(codimension κ)) *
+        ∫⁻ sample in A, ENNReal.ofReal ((D n).value κ B sample) ∂P := by
+    have hbi := ((C n).barrier_expectation hB (timeShift n) κ hκ (timeShift_pos n) hz hv γ g f hmaps).1
+    have hti : Integrable ((C n).targetValue hz huv' (timeShift n) κ B) P := by
+      apply (((D n).expectation hB κ hκ hz hu).1.const_mul
+        (K*((2*z.im)*Real.exp (-v))^(codimension κ))).mono'
+        ((C n).targetValue_aemeasurable hB hz huv' (timeShift n) κ hκ (timeShift_pos n)).aestronglyMeasurable
+      filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
+      rw [Real.norm_eq_abs,abs_of_nonneg ((C n).targetValue_nonneg hz huv' (timeShift n) κ hκ (timeShift_pos n) B sample)]
+      exact hKb hz hu huv (C n) B sample (timeShift n) (timeShift_pos n) hc hb0
+    have hle := (C n).target_setIntegral_le hB (timeShift n) κ hκ (timeShift_pos n) hz hu huv
+      γ g f hmaps A hA hstop
+    rw [lintegral_indicator₀ hA]
+    calc
+      _ = ENNReal.ofReal (∫ sample in A, (C n).barrier (timeShift n) κ B sample ∂P) := by
+        symm
+        exact ofReal_integral_eq_lintegral_ofReal hbi.integrableOn (Eventually.of_forall
+          (fun sample => (C n).barrier_nonneg (timeShift n) κ B sample hκ (timeShift_pos n)))
+      _ ≤ ENNReal.ofReal (∫ sample in A, (C n).targetValue hz huv' (timeShift n) κ B sample ∂P) :=
+        ENNReal.ofReal_le_ofReal hle
+      _ = ∫⁻ sample in A, ENNReal.ofReal ((C n).targetValue hz huv' (timeShift n) κ B sample) ∂P :=
+        ofReal_integral_eq_lintegral_ofReal hti.integrableOn (Eventually.of_forall
+          (fun sample => (C n).targetValue_nonneg hz huv' (timeShift n) κ hκ (timeShift_pos n) B sample))
+      _ ≤ ∫⁻ sample in A, ENNReal.ofReal (K*((2*z.im)*Real.exp (-v))^(codimension κ)) *
+          ENNReal.ofReal ((D n).value κ B sample) ∂P := by
+        apply lintegral_mono_ae
+        apply ae_restrict_of_ae
+        filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb0
+        rw [← ENNReal.ofReal_mul (by positivity : 0 ≤ K*((2*z.im)*Real.exp (-v))^(codimension κ))]
+        exact ENNReal.ofReal_le_ofReal (hKb hz hu huv (C n) B sample (timeShift n) (timeShift_pos n) hc hb0)
+      _ = _ := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+  have ht := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz hu D (shrinkingAngle_tendsto z) A
+  have ht' := ENNReal.Tendsto.const_mul ht (Or.inr (ENNReal.ofReal_ne_top (r := K*((2*z.im)*Real.exp (-v))^(codimension κ))))
+  calc
+    _ ≤ P {sample | (1:ℝ≥0∞) ≤ Y sample} := measure_mono_ae hsub
+    _ ≤ ∫⁻ sample, Y sample ∂P := by
+      simpa only [div_one] using meas_ge_le_lintegral_div hYm (by norm_num : (1:ℝ≥0∞) ≠ 0) (by norm_num : (1:ℝ≥0∞) ≠ ⊤)
+    _ ≤ liminf (fun n => ∫⁻ sample, F n sample ∂P) atTop := lintegral_liminf_le' hFm
+    _ ≤ _ := (liminf_le_liminf (Eventually.of_forall hexp)).trans_eq ht'.liminf_eq
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable FiniteTilt
+
+theorem conditionalCenteredInverseAngleBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C : ℝ, 0 < C ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (_hz : z ∈ halfPlane) (u v : ℝ) (_hu : 0 ≤ u) (_huv : u+1 ≤ v)
+      (A : Set Ω) (_hA : NullMeasurableSet A P)
+      (_hstop : ∀ t : ℝ≥0, ∃ E : Set Ω,
+        MeasurableSet[pastSigma B t] E ∧
+        E =ᵐ[P] (A ∩ {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≤ (t : ℝ≥0∞)} : Set Ω)),
+      (∫⁻ sample in A, radiusStoppedInverseAngle κ B γ g z v sample
+          ∂finiteCenteredMeasure κ P B γ g z v) ≤
+        ENNReal.ofReal C * (finiteCenteredMeasure κ P B γ g z u) A := by
+  obtain ⟨C,hC,hbound⟩ := conditionalRadiusApproachBound κ hκ
+  refine ⟨C,hC,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz u v hu huv A hA hstop
+  have hv : 0 < v := by linarith
+  have hz0 : 0 < z.im := hz
+  let M₀ := ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))
+  let K := ENNReal.ofReal (((2*z.im)*Real.exp (-v))^(-codimension κ)) / M₀
+  let H : Set Ω := {sample | firstLiveRadiusStop (γ sample) (g sample) z v ≠ (⊤ : ℝ≥0∞)}
+  have hmpos := localizationInitialGreen_pos κ hz
+  have hm0 : M₀ ≠ 0 := (ENNReal.ofReal_pos.2 hmpos).ne'
+  have hd := finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz v hv
+  have hdm : AEMeasurable (fun sample => finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z v / M₀) P := hd.div_const M₀
+  have hdf : ∀ᵐ sample ∂P, finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z v / M₀ < (⊤ : ℝ≥0∞) := Filter.Eventually.of_forall
+    (fun sample => ENNReal.div_lt_top (finiteStoppedDensity_lt_top κ _ _ _ z v).ne hm0)
+  have he : (fun sample => (finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z v / M₀) * radiusStoppedInverseAngle κ B γ g z v sample) =ᵐ[P]
+      H.indicator (fun _ => K) := by
+    filter_upwards [hmaps,hB.cont] with sample hm hc
+    have hh := finiteStoppedDensity_mul_inverseAngle κ B γ g f z hz v sample hm hc
+    rw [show (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z v / M₀) *
+      radiusStoppedInverseAngle κ B γ g z v sample = (finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z v * radiusStoppedInverseAngle κ B γ g z v sample) / M₀ by
+        simp only [div_eq_mul_inv]; ac_rfl, hh]
+    classical
+    by_cases hhit : sample ∈ H
+    · simp only [H] at hhit
+      simp only [H, Set.indicator_of_mem hhit, K]
+    · simp only [H] at hhit
+      simp only [H, Set.indicator_of_notMem hhit, ENNReal.zero_div]
+  have heleft : (∫⁻ sample in A, radiusStoppedInverseAngle κ B γ g z v sample
+      ∂finiteCenteredMeasure κ P B γ g z v) ≤ K*P (A∩H) := by
+    change (∫⁻ sample in A, _ ∂P.withDensity _) ≤ _
+    rw [setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀' P A hdm.restrict _
+      (ae_restrict_of_ae hdf)]
+    simp only [Pi.mul_apply]
+    rw [lintegral_congr_ae (ae_restrict_of_ae he)]
+    calc
+      _ ≤ K*(P.restrict A) H := lintegral_indicator_const_le H K
+      _ = _ := by rw [Measure.restrict_apply₀' hA,inter_comm]
+  have hemeasure : (finiteCenteredMeasure κ P B γ g z u) A =
+      M₀⁻¹ * ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample)
+        (γ sample) (g sample) z u ∂P := by
+    rw [finiteCenteredMeasure,withDensity_apply₀ _ hA]
+    simp_rw [ENNReal.div_eq_inv_mul]
+    exact lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hm0)
+  have hcancel : ENNReal.ofReal (((2*z.im)*Real.exp (-v))^(-codimension κ)) *
+      ENNReal.ofReal (C*((2*z.im)*Real.exp (-v))^(codimension κ)) = ENNReal.ofReal C := by
+    rw [← ENNReal.ofReal_mul (Real.rpow_nonneg (by positivity) _)]
+    congr 1
+    calc
+      _ = C*(((2*z.im)*Real.exp (-v))^(-codimension κ)*((2*z.im)*Real.exp (-v))^(codimension κ)) := by ring
+      _ = C := by rw [← Real.rpow_add (by positivity),neg_add_cancel,Real.rpow_zero,mul_one]
+  calc
+    _ ≤ K*P (A∩H) := heleft
+    _ ≤ K*(ENNReal.ofReal (C*((2*z.im)*Real.exp (-v))^(codimension κ)) *
+        ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P) :=
+      mul_le_mul' le_rfl (hbound P B hB γ g f hγ hmaps z hz u v hu huv A hA hstop)
+    _ = _ := by
+      rw [hemeasure]
+      dsimp only [K]
+      rw [div_eq_mul_inv]
+      calc
+        _ = (ENNReal.ofReal (((2*z.im)*Real.exp (-v))^(-codimension κ)) *
+            ENNReal.ofReal (C*((2*z.im)*Real.exp (-v))^(codimension κ))) *
+            (M₀⁻¹ * ∫⁻ sample in A, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P) := by ac_rfl
+        _ = _ := by rw [hcancel]
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/ConditionalTilts.lean b/lean/OAI/Probability/SLELocalization/ConditionalTilts.lean
index 91f7079f938315f617a41bced05a87bf46c0b8f6..7dfae7e9cf02e027a906d5b7d0fdbe825aae7d4b 100644
--- a/lean/OAI/Probability/SLELocalization/ConditionalTilts.lean
+++ b/lean/OAI/Probability/SLELocalization/ConditionalTilts.lean
@@ -1,977 +1,977 @@
-import Mathlib
-import OAI.Probability.SLELocalization.AngularClock
-import OAI.Probability.SLE.ClippedCoordinates
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Brownian Predictable estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-open PredictableLimit
-
-lemma indicator_difference_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (A D : ℕ → ℕ → Ω → ℝ)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
-    (hD : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (D n i))
-    (hA2 : ∀ n i, MemLp (A n i) 2 P) (hD2 : ∀ n i, MemLp (D n i) 2 P)
-    (K : ℝ) (hK : 0 ≤ K)
-    (hbA : ∀ n i, ∀ᵐ sample ∂P, |A n i sample| ≤ K)
-    (hbD : ∀ n i, ∀ᵐ sample ∂P, |D n i sample| ≤ K)
-    (θ : Ω → ℝ≥0∞)
-    (hsupport : ∀ n i, ∀ᵐ sample ∂P,
-      (dyadicTime T n i : ℝ≥0∞) < θ sample → A n i sample = D n i sample)
-    (E : Set Ω) (hE : NullMeasurableSet E P)
-    (hstop : ∀ t : ℝ≥0, ∃ S : Set Ω, MeasurableSet[pastSigma B t] S ∧
-      S =ᵐ[P] (E ∩ {sample | θ sample ≤ (t : ℝ≥0∞)} : Set Ω))
-    (YA YD : Ω → ℝ)
-    (hlA : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A n sample) atTop (𝓝 (YA sample)))
-    (hlD : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T D n sample) atTop (𝓝 (YD sample))) :
-    Integrable (E.indicator (fun sample => YA sample-YD sample)) P ∧
-      (∫ sample, E.indicator (fun sample => YA sample-YD sample) sample ∂P) = 0 := by
-  classical
-  choose S hSm hSeq using hstop
-  let Q := fun n i => (S (dyadicTime T n i)).indicator (fun sample => A n i sample-D n i sample)
-  have hQm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
-    ((hA n i).sub (hD n i)).indicator (hSm _)
-  have heq (n i : ℕ) : Q n i =ᵐ[P] E.indicator (fun sample => A n i sample-D n i sample) := by
-    filter_upwards [hSeq (dyadicTime T n i), hsupport n i] with sample he hs
-    have he : sample ∈ S (dyadicTime T n i) ↔ sample ∈ E ∧ θ sample ≤ (dyadicTime T n i : ℝ≥0∞) := Iff.of_eq he
-    by_cases hz : θ sample ≤ (dyadicTime T n i : ℝ≥0∞)
-    · simp only [Q,Set.indicator_apply,he,hz,and_true]
-    · have hh := hs (lt_of_not_ge hz)
-      simp only [Q,Set.indicator_apply,hh,sub_self,ite_self]
-  have hQ2 (n i : ℕ) : MemLp (Q n i) 2 P := by
-    have hd := (hA2 n i).sub (hD2 n i)
-    have hm : MemLp (E.indicator (fun sample => A n i sample-D n i sample)) 2 P :=
-      hd.of_le (hd.aestronglyMeasurable.indicator₀ hE)
-        (Filter.Eventually.of_forall (fun sample => norm_indicator_le_norm_self _ sample))
-    exact hm.ae_eq (heq n i).symm
-  have hQb (n i : ℕ) : ∀ᵐ sample ∂P, |Q n i sample| ≤ 2*K := by
-    filter_upwards [hbA n i,hbD n i] with sample ha hd
-    dsimp only [Q]
-    rw [Set.indicator_apply]
-    split_ifs
-    · exact (abs_sub _ _).trans (by linarith)
-    · simpa only [abs_zero] using (show 0 ≤ 2*K by positivity)
-  have heall : ∀ᵐ sample ∂P, ∀ n i, Q n i sample = E.indicator (fun sample => A n i sample-D n i sample) sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (heq n))
-  apply limit_centered hB T Q hQm hQ2 (2*K) (by positivity) hQb _
-  filter_upwards [heall,hlA,hlD] with sample he ha hd
-  have heqsum (n : ℕ) : dyadicDriverSum B T Q n sample =
-      E.indicator (fun sample => dyadicDriverSum B T A n sample-dyadicDriverSum B T D n sample) sample := by
-    unfold dyadicDriverSum
-    simp only [he,Set.indicator_apply]
-    split_ifs <;> simp [sub_mul,Finset.sum_sub_distrib]
-  simp only [heqsum,Set.indicator_apply]
-  split_ifs
-  · exact ha.sub hd
-  · exact tendsto_const_nhds
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.GammaTwoRadius
-open LoewnerCutoff LoewnerExit GammaField BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-open scoped unitInterval
-variable {ε : ℝ} (hε : 0 < ε)
-
-def coeffAt (k : ℕ) (s : ℝ≥0) (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ)
-    (q : C(I, ℝ) × ℝ) : ℝ := by
-  classical
-  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
-    iteratedDeriv k (field hε s.coe_nonneg l b η r sdev q.1 z s) q.2 else 0
-
-lemma coeffAt_measurable (k : ℕ) (_hk : k = 1 ∨ k = 2) (s : ℝ≥0)
-    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (r ρ δ : ℝ) :
-    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s l b η sdev z r ρ δ) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
-      iteratedDeriv k (field hε s.coe_nonneg l b η r sdev q.1 z s) q.2) := by
-    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
-      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
-    have hc := (continuous_derivative hε s.coe_nonneg hl hb hη r sdev z k).comp hp
-    dsimp only [Function.comp_def] at hc
-    exact hc
-  apply hc.measurable.ite _ measurable_const
-  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
-
-def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ) (sample : Ω) : ℝ :=
-  if s < T then coeffAt hε k s l b η sdev z r ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
-
-lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (r ρ δ : ℝ) :
-    Measurable[pastSigma B s] (rawCoeff hε k B T s l b η sdev z r ρ δ) := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  let : MeasurableSpace Ω := pastSigma B s
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z r ρ δ).comp
-      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
-  · exact measurable_const
-
-lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
-    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (r ρ δ : ℝ) :
-    AEMeasurable (rawCoeff hε k B T s l b η sdev z r ρ δ) P := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z r ρ δ).comp_aemeasurable
-      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
-  · exact aemeasurable_const
-
-lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    rawCoeff hε k B T s l b η sdev z r ρ δ sample =
-      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
-        iteratedDeriv k (field hε T.coe_nonneg l b η r sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
-  classical
-  by_cases hsT : s < T
-  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
-    have hf := field_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z l b η r sdev he
-      ⟨s.coe_nonneg, le_rfl⟩
-    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
-    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
-        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
-      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
-      exact forall₂_congr (fun t ht => by rw [hd ht])
-    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
-  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
-      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
-    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
-
-lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ) (n i : ℕ) :
-    rawCoeff hε k B T (dyadicTime T n i) l b η sdev z r ρ δ =ᵐ[P]
-      stoppedCoeff k B T
-        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
-        (fun sample => field hε T.coe_nonneg l b η r sdev (BrownianDriver.path B sdev T sample) z) n i := by
-  filter_upwards [hB.cont] with sample hc
-  rw [rawCoeff_eq hε k B T _ hT l b η sdev z r ρ δ sample hc]
-  simp only [stoppedCoeff, coe_dyadicTime]
-
-end SLEExactGauge.GammaTwoRadius
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerExit
-open LoewnerCutoff
-
-lemma stop_antitone_radius {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (U : C(I, ℝ)) (z : ℂ) (ρ ρ₂ δ : ℝ) (hρρ₂ : ρ ≤ ρ₂) :
-    stop hε hT.le U z ρ₂ δ ≤ stop hε hT.le U z ρ δ := by
-  by_contra hn
-  have hlt : stop hε hT.le U z ρ δ < stop hε hT.le U z ρ₂ δ := lt_of_not_ge hn
-  have hp := (deficit_pos_iff hε hT.le U z ρ₂ δ _).1
-    (positive_before hε hT.le hT U z ρ₂ δ (stop_bounds hε hT.le U z ρ δ).1 hlt)
-  have hpos : 0 < deficit hε hT.le U z ρ δ (stop hε hT.le U z ρ δ) :=
-    (deficit_pos_iff hε hT.le U z ρ δ _).2 ⟨hρρ₂.trans_lt hp.1,hp.2⟩
-  exact (not_lt_of_ge (nonpos_at_of_lt hε hT.le hT U z ρ δ
-    (hlt.trans_le (stop_bounds hε hT.le U z ρ₂ δ).2))) hpos
-
-end SLEExactGauge.LoewnerExit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized
-variable {z : ℂ} {u v : ℝ}
-
-def retarget (C : Cutoff z v) (hz : z ∈ halfPlane) (huv : u ≤ v) : Cutoff z u where
-  angle := C.angle
-  angle_pos := C.angle_pos
-  angle_initial := C.angle_initial
-  height := C.height
-  height_pos := C.height_pos
-  horizon := C.horizon
-  horizon_pos := C.horizon_pos
-  height_small := C.height_small.trans_le (by
-    apply mul_le_mul_of_nonneg_left _ hz.le
-    apply Real.exp_le_exp.mpr
-    exact div_le_div_of_nonneg_right (neg_le_neg huv) (by positivity))
-  time_large := lt_of_le_of_lt (by
-    apply div_le_div_of_nonneg_right _ (by positivity)
-    exact mul_le_mul_of_nonneg_right huv (sq_nonneg _)) C.time_large
-
-lemma retarget_stop_le (C : Cutoff z v) (hz : z ∈ halfPlane) (huv : u ≤ v)
-    {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
-    (C.retarget hz huv).stopTime κ B sample ≤ C.stopTime κ B sample := by
-  apply stop_antitone_radius C.height_pos C.horizon_pos
-  exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (neg_le_neg huv)) (by exact mul_nonneg (by norm_num) hz.le)
-
-lemma retarget_has_hit_of_lt (C : Cutoff z v) (hz : z ∈ halfPlane)
-    (hu : 0 < u) (huv : u ≤ v) {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-    (hlt : (C.retarget hz huv).stopTime κ B sample < C.stopTime κ B sample) :
-    firstLiveRadiusStop γ g z u = ENNReal.ofReal ((C.retarget hz huv).stopTime κ B sample) := by
-  let D := C.retarget hz huv
-  have hd := D.initial_deficit_brownian κ B sample hz hu hc hb0
-  have ht := D.bounds κ B sample
-  have hR : loewnerRadius g (D.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
-    rcases D.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · exact hr
-    · have hp := (deficit_pos_iff C.height_pos C.horizon.coe_nonneg (C.paths κ B sample)
-        z ((2*z.im)*Real.exp (-v)) C.angle (D.stopTime κ B sample)).1
-        (positive_before C.height_pos C.horizon.coe_nonneg C.horizon_pos (C.paths κ B sample)
-          z ((2*z.im)*Real.exp (-v)) C.angle ht.1 hlt)
-      have he := D.source_sine κ B sample h hz hu hc hb0
-      have hi : C.angle < loewnerAngleSine κ (fun t => B t sample) g (D.stopTime κ B sample) z := by
-        rw [← he]
-        exact hp.2.2
-      exact False.elim ((ne_of_gt hi) hs)
-  have hh := h.firstLiveRadiusStop_eq
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (t := ⟨D.stopTime κ B sample,ht.1⟩) (D.source_live κ B sample h hz hu hc hb0) hR
-  rw [ENNReal.ofReal_eq_coe_nnreal ht.1]
-  exact hh
-
-lemma stopTime_zero (C : Cutoff z 0) {Ω : Type*} (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : C.stopTime κ B sample = 0 := by
-  apply le_antisymm _ (C.bounds κ B sample).1
-  by_contra hn
-  have hp := (deficit_pos_iff C.height_pos C.horizon.coe_nonneg (C.paths κ B sample)
-    z ((2*z.im)*Real.exp (-0)) C.angle 0).1
-    (positive_before C.height_pos C.horizon.coe_nonneg C.horizon_pos (C.paths κ B sample)
-      z ((2*z.im)*Real.exp (-0)) C.angle le_rfl (lt_of_not_ge hn))
-  simpa only [LoewnerExit.radius,position_initial,logDerivative_initial,neg_zero,
-    Real.exp_zero,mul_one,div_one,lt_self_iff_false] using hp.1
-
-lemma value_zero (C : Cutoff z 0) (hz : z ∈ halfPlane) {Ω : Type*} (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (sample : Ω) (hb0 : B 0 sample = 0) :
-    ENNReal.ofReal (C.value κ B sample) =
-      ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) := by
-  unfold value
-  rw [C.stopTime_zero,Real.toNNReal_zero,hb0,field_initial C.height_pos C.horizon.coe_nonneg
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z (C.height_initial hz le_rfl).le,
-    green_initial κ hz]
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BrownianLocal.LocalRegular
-
-lemma mono {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ} {S T : ℝ}
-    (h : BrownianLocal.LocalRegular F V b T) (hST : S ≤ T) :
-    BrownianLocal.LocalRegular F V b S where
-  smooth := h.smooth
-  third_cont := h.third_cont.mono (Set.prod_mono (Icc_subset_Icc_right hST) Subset.rfl)
-  time_cont := fun x => (h.time_cont x).mono (Icc_subset_Icc_right hST)
-  time_deriv := fun t ht => h.time_deriv t ⟨ht.1,ht.2.trans_le hST⟩
-  drift_cont := h.drift_cont.mono (Set.prod_mono (Icc_subset_Icc_right hST) (Icc_subset_Icc_right hST))
-  second_cont := h.second_cont.mono (Icc_subset_Icc_right hST)
-
-end SLEExactGauge.BrownianLocal.LocalRegular
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open GammaField GammaTwoRadius BrownianPredictable BrownianStopped BrownianIto TaylorMeshes PredictableLimit
-variable {z : ℂ} {v : ℝ} (C : Cutoff z v)
-
-def targetField {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ → ℝ → ℝ :=
-  GammaField.field C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (C.paths κ B sample) z
-
-def targetDrift {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ → ℝ → ℝ :=
-  GammaField.drift C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (C.paths κ B sample) z
-
-def targetCoeff {Ω : Type*} (k : ℕ) (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (w : ℝ) (n i : ℕ) : Ω → ℝ :=
-  GammaTwoRadius.rawCoeff C.height_pos k B C.horizon (dyadicTime C.horizon n i)
-    (κ/2) ((8/κ-1)/2) η (Real.sqrt κ) z ((2*z.im)*Real.exp (-v)) ((2*z.im)*Real.exp (-w)) C.angle
-
-def targetRemainder {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (t : Ω → ℝ) (sample : Ω) : ℝ :=
-  C.targetField η κ B sample (t sample) (B (t sample).toNNReal sample) -
-    C.targetField η κ B sample 0 (B 0 sample) -
-    (∫ s in (0:ℝ)..(t sample), C.targetDrift η κ B sample s (B s.toNNReal sample)) -
-    (1/2:ℝ)*∫ s in (0:ℝ)..(t sample),
-      iteratedDeriv 2 (C.targetField η κ B sample s) (B s.toNNReal sample)
-
-lemma targetCoeff_eq {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (η κ : ℝ)
-    (hz : z ∈ halfPlane) (w : ℝ) (hwv : w ≤ v) (n i : ℕ) :
-    C.targetCoeff k η κ B w n i =ᵐ[P]
-      stoppedCoeff k B C.horizon ((C.retarget hz hwv).stopTime κ B) (C.targetField η κ B) n i :=
-  rawCoeff_stoppedCoeff_ae C.height_pos hB k C.horizon C.horizon_pos (κ/2) ((8/κ-1)/2)
-    η (Real.sqrt κ) z _ _ C.angle n i
-
-lemma targetCoeff_measurable {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (k : ℕ) (hk : k = 1 ∨ k = 2) (η κ : ℝ)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (w : ℝ) (n i : ℕ) :
-    Measurable[pastSigma B (dyadicTime C.horizon n i)] (C.targetCoeff k η κ B w n i) := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  exact rawCoeff_measurable_past C.height_pos k hk B C.horizon _ _ _ _ _ hl hb hη z _ _ C.angle
-
-lemma targetCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (hk : k = 1 ∨ k = 2) (η κ : ℝ)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (w : ℝ) (n i : ℕ) :
-    AEMeasurable (C.targetCoeff k η κ B w n i) P := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  exact rawCoeff_aemeasurable C.height_pos hB k hk C.horizon _ _ _ _ _ hl hb hη z _ _ C.angle
-
-lemma targetCoeff_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (η κ : ℝ)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (hz : z ∈ halfPlane) (hv : 0 < v)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample)) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ w, w ≤ v → ∀ n i,
-      ∀ᵐ sample ∂P, |C.targetCoeff k η κ B w n i sample| ≤ K := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  obtain ⟨K,hK,hKb⟩ := coefficient_bound C.height_pos C.horizon.coe_nonneg hl hb hη
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (η+v) (1/C.angle) z k
-  refine ⟨K,hK,?_⟩
-  intro w hwv n i
-  filter_upwards [C.targetCoeff_eq hB k η κ hz w hwv n i,hmaps,hB.cont,
-    hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample he hm hc hb0
-  rw [he]
-  unfold stoppedCoeff
-  split_ifs with hh
-  · have ht : mesh C.horizon n i ∈ Icc 0 (C.stopTime κ B sample) :=
-      ⟨by unfold mesh; positivity,hh.le.trans (C.retarget_stop_le hz hwv κ B sample)⟩
-    exact hKb (C.paths κ B sample) _ _ (C.remaining_upper κ B sample hm hz hv hc hb0 η ht)
-      (by simpa only [mesh_toNNReal] using C.ratio_upper κ B sample hz hv hc hb0 ht)
-  · simpa only [abs_zero] using hK
-
-lemma target_sum_tendsto {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (η κ : ℝ)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (hz : z ∈ halfPlane) (hv : 0 < v)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (w : ℝ) (hwv : w ≤ v) :
-    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B C.horizon (C.targetCoeff 1 η κ B w) n sample)
-      atTop (𝓝 (C.targetRemainder η κ B ((C.retarget hz hwv).stopTime κ B) sample)) := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  let D := C.retarget hz hwv
-  have hreg : ∀ᵐ sample ∂P, BrownianLocal.LocalRegular (C.targetField η κ B sample)
-      (C.targetDrift η κ B sample) (fun t => B t.toNNReal sample) (D.stopTime κ B sample) := by
-    filter_upwards [hB.cont] with sample hc
-    have hρ : 0 < (2*z.im)*Real.exp (-v) := by have hy : 0 < z.im := hz; positivity
-    exact (regular C.height_pos C.horizon.coe_nonneg hl hb hη hρ C.horizon_pos
-      (Real.sqrt κ) (C.paths κ B sample) z C.angle _ (hc.comp continuous_real_toNNReal)).mono
-      (C.retarget_stop_le hz hwv κ B sample)
-  obtain ⟨K₁,hK₁,hb₁⟩ := C.targetCoeff_bound hB 1 η κ hκ hη hz hv γ g f hmaps
-  obtain ⟨K₂,hK₂,hb₂⟩ := C.targetCoeff_bound hB 2 η κ hκ hη hz hv γ g f hmaps
-  have hMem : ∀ n i, MemLp (C.targetCoeff 2 η κ B w n i) 2 P := by
-    intro n i
-    apply MemLp.of_bound (C.targetCoeff_aemeasurable hB 2 (Or.inr rfl) η κ hκ hη w n i).aestronglyMeasurable K₂
-    simpa only [Real.norm_eq_abs] using hb₂ w hwv n i
-  have hi := BrownianLocal.stopped_fieldSum_tendsto_ae hB C.horizon (D.stopTime κ B)
-    (C.targetField η κ B) (C.targetDrift η κ B)
-    (Filter.Eventually.of_forall (fun sample => D.bounds κ B sample)) hreg
-    (C.targetCoeff 2 η κ B w) (C.targetCoeff_eq hB 2 η κ hz w hwv)
-    (C.targetCoeff_measurable B 2 (Or.inr rfl) η κ hκ hη w) hMem
-    K₂ hK₂ (fun n i => by
-      filter_upwards [C.targetCoeff_eq hB 2 η κ hz w hwv n i,hb₂ w hwv n i] with sample he hh
-      rwa [← he])
-    K₁ hK₁ (fun n i => by
-      filter_upwards [C.targetCoeff_eq hB 1 η κ hz w hwv n i,hb₁ w hwv n i] with sample he hh
-      rwa [← he])
-  have heall : ∀ᵐ sample ∂P, ∀ n i, C.targetCoeff 1 η κ B w n i sample =
-      stoppedCoeff 1 B C.horizon (D.stopTime κ B) (C.targetField η κ B) n i sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (C.targetCoeff_eq hB 1 η κ hz w hwv n))
-  filter_upwards [hi,heall] with sample hi he
-  simpa only [targetRemainder,dyadicDriverSum,he] using hi
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.GammaField
-open LoewnerCutoff LoewnerExit
-
-lemma integral_drift_nonpos_from {ε T κ η : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (ρ : ℝ)
-    (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (δ : ℝ) (B : ℝ → ℝ) (hB : Continuous B)
-    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ*B s)
-    {a : ℝ} (ha : 0 ≤ a) (haτ : a ≤ stop hε hT U z ρ δ) :
-    (∫ t in a..(stop hε hT U z ρ δ), drift hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t (B t)) +
-    (1/2:ℝ)*∫ t in a..(stop hε hT U z ρ δ),
-      iteratedDeriv 2 (field hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t) (B t) ≤ 0 := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have hm : Continuous (fun t : ℝ => ((U,t),B t)) :=
-    (continuous_const.prodMk continuous_id).prodMk hB
-  have hv := (continuous_drift hε hT hl hb hη ρ (Real.sqrt κ) z).comp hm
-  have hc := (continuous_derivative hε hT hl hb hη ρ (Real.sqrt κ) z 2).comp hm
-  have hiV := hv.intervalIntegrable (μ := volume) a (stop hε hT U z ρ δ)
-  have hiC := hc.intervalIntegrable (μ := volume) a (stop hε hT U z ρ δ)
-  dsimp only [Function.comp_def] at hiV hiC
-  rw [← intervalIntegral.integral_const_mul, ← intervalIntegral.integral_add hiV (hiC.const_mul (1/2:ℝ))]
-  rw [intervalIntegral.integral_of_le haτ]
-  apply integral_nonpos_of_ae
-  have hne : ∀ᵐ t : ℝ ∂volume, t ≠ stop hε hT U z ρ δ := by simp [ae_iff]
-  filter_upwards [ae_restrict_mem measurableSet_Ioc,ae_restrict_of_ae hne] with t ht he
-  have hlt : t < stop hε hT U z ρ δ := lt_of_le_of_ne ht.2 he
-  have hh := (deficit_pos_iff hε hT U z ρ δ t).1
-    (positive_before hε hT hTp U z ρ δ (ha.trans ht.1.le) hlt)
-  exact drift_nonpos hε hT hκ hη ρ U z t (B t) hh.2.1.le
-    (hU t ⟨ha.trans ht.1.le,ht.2.trans (stop_bounds hε hT U z ρ δ).2⟩)
-
-end SLEExactGauge.GammaField
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open GammaField LoewnerCutoff LoewnerExit
-variable {z : ℂ} {v : ℝ} (C : Cutoff z v)
-
-def targetValue {Ω : Type*} (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
-    (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
-  C.targetField η κ B sample ((C.retarget hz hwv).stopTime κ B sample)
-    (B ((C.retarget hz hwv).stopTime κ B sample).toNNReal sample)
-
-lemma targetValue_nonneg {Ω : Type*} (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
-    (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
-    0 ≤ C.targetValue hz hwv η κ B sample := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  exact field_nonneg C.height_pos C.horizon.coe_nonneg hl hb hη _ _ _ _ _ _
-
-lemma targetValue_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
-    (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) :
-    AEMeasurable (C.targetValue hz hwv η κ B) P := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  let τ := fun U : C(I, ℝ) => stop C.height_pos C.horizon.coe_nonneg U z ((2*z.im)*Real.exp (-w)) C.angle
-  have hτ : Measurable τ := measurable_stop C.height_pos C.horizon.coe_nonneg C.horizon_pos z _ C.angle
-  have hm := ((continuous_actualField C.height_pos C.horizon hl hb hη
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) z).measurable.comp (measurable_id.prodMk hτ)).comp_aemeasurable
-    (BrownianDriver.path_aemeasurable hB (Real.sqrt κ) C.horizon)
-  apply hm.congr
-  filter_upwards [hB.cont] with sample hc
-  change GammaField.field C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (C.paths κ B sample) z
-    ((C.retarget hz hwv).stopTime κ B sample)
-    (LoewnerCutoff.extendDriver C.horizon (C.paths κ B sample)
-      ((C.retarget hz hwv).stopTime κ B sample) / Real.sqrt κ) = _
-  rw [C.path_eq κ B sample hc ((C.retarget hz hwv).bounds κ B sample),mul_div_cancel_left₀ _ (Real.sqrt_pos.2 hκ.1).ne']
-  rfl
-
-lemma target_difference_le_remainder {Ω : Type*} (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
-    (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    C.barrier η κ B sample-C.targetValue hz hwv η κ B sample ≤
-      C.targetRemainder η κ B (C.stopTime κ B) sample -
-      C.targetRemainder η κ B ((C.retarget hz hwv).stopTime κ B) sample := by
-  let D := C.retarget hz hwv
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  have hm : Continuous (fun t : ℝ => ((C.paths κ B sample,t),B t.toNNReal sample)) :=
-    (continuous_const.prodMk continuous_id).prodMk (hc.comp continuous_real_toNNReal)
-  have hd := (continuous_drift C.height_pos C.horizon.coe_nonneg hl hb hη
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) z).comp hm
-  have h2 := (continuous_derivative C.height_pos C.horizon.coe_nonneg hl hb hη
-    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) z 2).comp hm
-  have hid := intervalIntegral.integral_add_adjacent_intervals
-    (hd.intervalIntegrable (μ := volume) 0 (D.stopTime κ B sample))
-    (hd.intervalIntegrable (μ := volume) (D.stopTime κ B sample) (C.stopTime κ B sample))
-  have hi2 := intervalIntegral.integral_add_adjacent_intervals
-    (h2.intervalIntegrable (μ := volume) 0 (D.stopTime κ B sample))
-    (h2.intervalIntegrable (μ := volume) (D.stopTime κ B sample) (C.stopTime κ B sample))
-  have hn := integral_drift_nonpos_from C.height_pos C.horizon.coe_nonneg hκ hη
-    ((2*z.im)*Real.exp (-v)) C.horizon_pos (C.paths κ B sample) z C.angle
-    (fun t => B t.toNNReal sample) (hc.comp continuous_real_toNNReal)
-    (C.path_eq κ B sample hc) (D.bounds κ B sample).1 (C.retarget_stop_le hz hwv κ B sample)
-  dsimp only [Function.comp_def] at hid hi2
-  unfold targetRemainder targetValue targetField targetDrift barrier
-  dsimp only [D, stopTime] at hid hi2 hn ⊢
-  linarith only [hn,hid,hi2]
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.AngularBarrier
-
-lemma value_decay_from {l b t₀ : ℝ} (hl : 0 < l) (hb : 0 < b) (ht₀ : 0 < t₀) :
-    ∃ K : ℝ, 0 < K ∧ ∀ t q : ℝ, t₀ ≤ t → 0 ≤ q →
-      value l b t q ≤ K*Real.exp (-(l*b/2)*t)*q^b := by
-  let c := (1-Real.exp (-l*t₀))/(2*(b+1))
-  have hc : 0 < c := div_pos (sub_pos.2 (Real.exp_lt_one_iff.2 (by nlinarith))) (by positivity)
-  have hg : 0 < Real.Gamma b := Real.Gamma_pos_of_pos hb
-  refine ⟨1/(b*Real.Gamma b*c^b), by positivity, ?_⟩
-  intro t q ht hq
-  have ht0 : 0 < t := ht₀.trans_le ht
-  have ha := scale_pos hl hb ht0
-  have hlower : c*Real.exp (l*t) ≤ scale l b t := by
-    have hh : 1 ≤ Real.exp (l*t)*Real.exp (-l*t₀) := by
-      rw [← Real.exp_add]
-      exact Real.one_le_exp_iff.2 (by nlinarith)
-    dsimp only [c,scale]
-    apply (le_div_iff₀ (by positivity : 0 < 2*(b+1))).2
-    have heq : ((1-Real.exp (-l*t₀))/(2*(b+1))*Real.exp (l*t))*(2*(b+1)) =
-        (1-Real.exp (-l*t₀))*Real.exp (l*t) := by field_simp
-    rw [heq]
-    nlinarith
-  have hratio : q/scale l b t ≤ q/(c*Real.exp (l*t)) :=
-    div_le_div_of_nonneg_left hq (mul_pos hc (Real.exp_pos _)) hlower
-  have hpow := Real.rpow_le_rpow (div_nonneg hq ha.le) hratio hb.le
-  have hbound := (cdf_le_power hb (div_nonneg hq ha.le)).trans
-    (div_le_div_of_nonneg_right hpow (mul_pos hb hg).le)
-  have he : Real.exp (l*b/2*t)*(q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b) =
-      1/(b*Real.Gamma b*c^b)*Real.exp (-(l*b/2)*t)*q^b := by
-    rw [Real.div_rpow hq (mul_pos hc (Real.exp_pos _)).le,
-      Real.mul_rpow hc.le (Real.exp_pos _).le, ← Real.exp_mul]
-    have hratioexp : Real.exp (l*b/2*t)/Real.exp (l*t*b) = Real.exp (-(l*b/2)*t) := by
-      rw [← Real.exp_sub]; congr 1; ring
-    calc
-      _ = (Real.exp (l*b/2*t)/Real.exp (l*t*b))*q^b/(b*Real.Gamma b*c^b) := by ring
-      _ = _ := by rw [hratioexp]; ring
-  calc
-    _ ≤ Real.exp (l*b/2*t)*((q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b)) :=
-      mul_le_mul_of_nonneg_left hbound (Real.exp_pos _).le
-    _ = _ := by rw [← mul_div_assoc, he]
-
-end SLEExactGauge.AngularBarrier
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteTilt AngularBarrier
-
-theorem radiusApproachBoundFrom (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (u₀ : ℝ) (hu₀ : 0 < u₀) :
-    ∃ C : ℝ, 0 < C ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (_hz : z ∈ halfPlane) (u : ℝ) (_hu : u₀ ≤ u),
-      P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
-        ENNReal.ofReal (C * Real.exp (-codimension κ * u) *
-          (z.im / ‖z‖) ^ (8 / κ - 1)) := by
-  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  obtain ⟨C,hC,hdec⟩ := value_decay_from hl hb hu₀
-  refine ⟨C,hC,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz u hu
-  have hu0 : 0 < u := hu₀.trans_le hu
-  apply (radiusApproachProfile κ hκ B hB γ g f hmaps z hz u hu0).trans
-  apply ENNReal.ofReal_le_ofReal
-  have hp : (κ/2)*((8/κ-1)/2)/2 = codimension κ := by
-    unfold codimension dimension
-    field_simp [hκ.1.ne']
-    ring
-  have hq : (squaredSine (z.re/z.im))^((8/κ-1)/2) = (z.im/‖z‖)^(8/κ-1) := by
-    rw [squaredSine_initial hz, ← Real.rpow_natCast (z.im/‖z‖) 2,
-      ← Real.rpow_mul (initial_sine_pos hz).le]
-    congr 1
-    ring
-  have hh := hdec u (squaredSine (z.re/z.im)) hu (squaredSine_pos _).le
-  simpa only [profile,hp,hq] using hh
-
-def finiteCenteredMeasure {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (P : Measure Ω) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) : Measure Ω :=
-  P.withDensity (fun sample =>
-    finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u /
-      ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))
-
-def radiusStoppedInverseAngle {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) (sample : Ω) : ℝ≥0∞ :=
-  let τ := firstLiveRadiusStop (γ sample) (g sample) z u
-  if τ = (⊤ : ℝ≥0∞) then 1 else ENNReal.ofReal
-    ((loewnerAngleSine κ (fun t => B t sample) (g sample) τ.toReal z)^(-(8/κ-1)))
-
-lemma finiteStoppedDensity_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u) :
-    AEMeasurable (fun sample => finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z u) P := by
-  let C := cutoffs hz u hu.le
-  let F := fun n sample => ENNReal.ofReal ((C n).value κ B sample)
-  have hexp (n : ℕ) := (C n).expectation hB κ hκ hz hu.le
-  have hFm (n : ℕ) : AEMeasurable (F n) P :=
-    ENNReal.measurable_ofReal.comp_aemeasurable (hexp n).1.aemeasurable
-  have hFl : ∀ᵐ sample ∂P, Tendsto (fun n => F n sample) atTop
-      (𝓝 (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u)) := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
-    exact tendsto_value_killed κ B sample hm hz hu hc hb hκ
-  exact aemeasurable_of_tendsto_metrizable_ae' hFm hFl
-
-lemma finiteStoppedDensity_lt_top (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ → ℂ)
-    (g : ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) : finiteStoppedDensity κ B γ g z u < (⊤ : ℝ≥0∞) := by
-  dsimp only [finiteStoppedDensity]
-  split_ifs <;> simp
-
-lemma localizationInitialGreen_pos (κ : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
-    0 < (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1) := by
-  have hy : 0 < z.im := hz
-  exact mul_pos (Real.rpow_pos_of_pos (by positivity) _) (Real.rpow_pos_of_pos (initial_sine_pos hz) _)
-
-lemma finiteCenteredMeasure_probability {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (P : Measure Ω) [IsProbabilityMeasure P]
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u) :
-    IsProbabilityMeasure (finiteCenteredMeasure κ P B γ g z u) := by
-  have hm0 : ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) ≠ 0 :=
-    (ENNReal.ofReal_pos.2 (localizationInitialGreen_pos κ hz)).ne'
-  constructor
-  rw [finiteCenteredMeasure, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ]
-  simp_rw [ENNReal.div_eq_inv_mul]
-  rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hm0),
-    finiteStoppedCenteredLaw κ hκ B hB γ g f hγ hmaps z hz u hu.le,
-    ENNReal.inv_mul_cancel hm0 ENNReal.ofReal_ne_top]
-
-lemma finiteStoppedDensity_mul_inverseAngle {Ω : Type*} (κ : ℝ)
-    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (sample : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample)
-      (γ sample) (g sample) (f sample))
-    (hc : Continuous (fun t => B t sample)) :
-    finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u *
-      radiusStoppedInverseAngle κ B γ g z u sample =
-    {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)}.indicator
-      (fun _ => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-codimension κ))) sample := by
-  classical
-  by_cases hhit : firstLiveRadiusStop (γ sample) (g sample) z u = (⊤ : ℝ≥0∞)
-  · simp [finiteStoppedDensity,radiusStoppedInverseAngle,hhit]
-  have hex : ∃ t : ℝ≥0, z ∈ localizationDomain (γ sample) t ∧
-      loewnerRadius (g sample) t z = (2*z.im)*Real.exp (-u) := by
-    by_contra hn
-    have he : {t : ℝ≥0 | z ∈ localizationDomain (γ sample) t ∧
-        loewnerRadius (g sample) t z = (2*z.im)*Real.exp (-u)} = ∅ :=
-      Set.eq_empty_iff_forall_notMem.2 (fun t ht => hn ⟨t,ht⟩)
-    exact hhit (by simp only [firstLiveRadiusStop,he,Set.image_empty,sInf_empty])
-  obtain ⟨t,hzt,hrt⟩ := hex
-  have htau := hm.firstLiveRadiusStop_eq ((continuous_const.mul
-    (hc.comp continuous_real_toNNReal)).continuousOn) hzt hrt
-  have hs : 0 < loewnerAngleSine κ (fun t => B t sample) (g sample) t z := by
-    have hh := div_pos (hm.im_pos t.coe_nonneg hzt) (norm_pos_iff.2 (hm.centered_ne_zero t.coe_nonneg hzt))
-    simpa only [loewnerAngleSine, Complex.ofReal_mul] using hh
-  have hr : 0 < ((2*z.im)*Real.exp (-u))^(-codimension κ) := by
-    apply Real.rpow_pos_of_pos
-    have hy : 0 < z.im := hz
-    positivity
-  simp only [finiteStoppedDensity,radiusStoppedInverseAngle,htau,ENNReal.coe_ne_top,
-    ite_false,ENNReal.coe_toReal,hrt,Set.mem_ofPred_eq,Ne,not_false_eq_true,
-    Set.indicator_of_mem]
-  rw [← ENNReal.ofReal_mul (mul_pos hr (Real.rpow_pos_of_pos hs _)).le,
-    mul_assoc, ← Real.rpow_add hs, add_neg_cancel, Real.rpow_zero,mul_one]
-
-lemma radius_green_cancel (κ u : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
-    (((2*z.im)*Real.exp (-u))^(-codimension κ) /
-      ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))) *
-      (Real.exp (-codimension κ*u)*(z.im/‖z‖)^(8/κ-1)) = 1 := by
-  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
-  have hs := Real.rpow_pos_of_pos (initial_sine_pos hz) (8/κ-1)
-  have hp := Real.rpow_pos_of_pos hy (-codimension κ)
-  rw [Real.mul_rpow hy.le (Real.exp_pos _).le, ← Real.exp_mul]
-  have he : Real.exp (-u * -codimension κ) * Real.exp (-codimension κ*u) = 1 := by
-    rw [← Real.exp_add, show -u * -codimension κ + -codimension κ*u = 0 by ring, Real.exp_zero]
-  field_simp
-  convert he using 1
-  congr 2 <;> ring
-
-theorem finiteCenteredInverseAngleBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (u₀ : ℝ) (hu₀ : 0 < u₀) :
-    ∃ C : ℝ, 0 < C ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (_hz : z ∈ halfPlane) (u : ℝ) (_hu : u₀ ≤ u),
-      IsProbabilityMeasure (finiteCenteredMeasure κ P B γ g z u) ∧
-      (∫⁻ sample, radiusStoppedInverseAngle κ B γ g z u sample
-        ∂finiteCenteredMeasure κ P B γ g z u) ≤ ENNReal.ofReal C := by
-  obtain ⟨C,hC,hbound⟩ := radiusApproachBoundFrom κ hκ u₀ hu₀
-  refine ⟨C,hC,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz u hu
-  have hu0 : 0 < u := hu₀.trans_le hu
-  refine ⟨finiteCenteredMeasure_probability κ hκ P B hB γ g f hγ hmaps z hz u hu0, ?_⟩
-  let M₀ := ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))
-  let K := ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-codimension κ)) / M₀
-  let H : Set Ω := {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)}
-  have hmpos := localizationInitialGreen_pos κ hz
-  have hm0 : M₀ ≠ 0 := (ENNReal.ofReal_pos.2 hmpos).ne'
-  have hd := finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0
-  have hdm : AEMeasurable (fun sample => finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z u / M₀) P := hd.div_const M₀
-  have hdf : ∀ᵐ sample ∂P, finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z u / M₀ < (⊤ : ℝ≥0∞) := Filter.Eventually.of_forall
-    (fun sample => ENNReal.div_lt_top (finiteStoppedDensity_lt_top κ _ _ _ z u).ne hm0)
-  have he : (fun sample => (finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z u / M₀) * radiusStoppedInverseAngle κ B γ g z u sample) =ᵐ[P]
-      H.indicator (fun _ => K) := by
-    filter_upwards [hmaps,hB.cont] with sample hm hc
-    have hh := finiteStoppedDensity_mul_inverseAngle κ B γ g f z hz u sample hm hc
-    rw [show (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u / M₀) *
-      radiusStoppedInverseAngle κ B γ g z u sample = (finiteStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z u * radiusStoppedInverseAngle κ B γ g z u sample) / M₀ by
-        simp only [div_eq_mul_inv]; ac_rfl, hh]
-    classical
-    by_cases hhit : sample ∈ H
-    · simp only [H] at hhit
-      simp only [H, Set.indicator_of_mem hhit, K]
-    · simp only [H] at hhit
-      simp only [H, Set.indicator_of_notMem hhit, ENNReal.zero_div]
-  change (∫⁻ sample, radiusStoppedInverseAngle κ B γ g z u sample ∂P.withDensity
-    (fun sample => finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u / M₀)) ≤ _
-  rw [lintegral_withDensity_eq_lintegral_mul_non_measurable₀ P hdm hdf]
-  simp only [Pi.mul_apply]
-  rw [lintegral_congr_ae he]
-  calc
-    (∫⁻ sample, H.indicator (fun _ => K) sample ∂P) ≤ K*P H := lintegral_indicator_const_le H K
-    _ ≤ K * ENNReal.ofReal (C * Real.exp (-codimension κ*u)*(z.im/‖z‖)^(8/κ-1)) :=
-      mul_le_mul' le_rfl (hbound P B hB γ g f hγ hmaps z hz u hu)
-    _ = ENNReal.ofReal C := by
-      dsimp only [K,M₀]
-      have hy : 0 < z.im := hz
-      have hratio : 0 ≤ ((2*z.im)*Real.exp (-u))^(-codimension κ) /
-          ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) :=
-        div_nonneg (Real.rpow_nonneg (by positivity) _) hmpos.le
-      rw [← ENNReal.ofReal_div_of_pos hmpos, ← ENNReal.ofReal_mul hratio]
-      congr 1
-      calc
-        _ = C * (((2*z.im)*Real.exp (-u))^(-codimension κ) /
-            ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) *
-            (Real.exp (-codimension κ*u)*(z.im/‖z‖)^(8/κ-1))) := by ring
-        _ = C := by rw [radius_green_cancel κ u hz,mul_one]
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.GammaField
-open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier GreenPhysical
-
-lemma profile_green_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ K : ℝ, 0 < K ∧ ∀ x y l t ρ : ℝ, 0 < y → 0 < ρ → 1 ≤ t →
-      Real.log (2*y)-l-Real.log ρ ≤ t →
-      profile (κ/2) ((8/κ-1)/2) t (x/y) ≤
-        K*ρ^(codimension κ)*green (1-κ/8) (8/κ-1) x y l := by
-  have hl : 0 < κ/2 := by linarith [hκ.1]
-  have hb : 0 < (8/κ-1)/2 := by
-    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
-    linarith
-  obtain ⟨K,hK,hdec⟩ := value_decay_from hl hb (by norm_num : (0:ℝ)<1)
-  refine ⟨K,hK,?_⟩
-  intro x y l t ρ hy hρ ht hr
-  have hp : (κ/2)*((8/κ-1)/2)/2 = codimension κ := by
-    unfold codimension dimension
-    field_simp [hκ.1.ne']
-    ring
-  have hp' : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  have hp0 : 0 ≤ codimension κ := by rw [hp']; linarith [hκ.2]
-  have hs : 0 < Real.sqrt (x^2+y^2) := Real.sqrt_pos.2 (by positivity)
-  have hq0 : squaredSine (x/y) = (y/Real.sqrt (x^2+y^2))^2 := by
-    have hsq := Real.sq_sqrt (by positivity : 0 ≤ x^2+y^2)
-    unfold squaredSine
-    field_simp
-    nlinarith
-  have hq : (squaredSine (x/y))^((8/κ-1)/2) = (y/Real.sqrt (x^2+y^2))^(8/κ-1) := by
-    rw [hq0, ← Real.rpow_natCast (y/Real.sqrt (x^2+y^2)) 2,
-      ← Real.rpow_mul (div_pos hy hs).le]
-    congr 1
-    ring
-  have he : Real.exp (-(codimension κ)*t) ≤
-      ρ^(codimension κ)*(2*y/Real.exp l)^(-(1-κ/8)) := by
-    rw [Real.rpow_def_of_pos hρ,Real.rpow_def_of_pos (div_pos (by positivity) (Real.exp_pos _)),
-      ← Real.exp_add]
-    apply Real.exp_le_exp.2
-    rw [Real.log_div (by positivity) (Real.exp_ne_zero _),Real.log_exp,← hp']
-    nlinarith [mul_le_mul_of_nonneg_left hr hp0]
-  have hh := hdec t (squaredSine (x/y)) ht (squaredSine_pos _).le
-  rw [hp,hq] at hh
-  calc
-    _ ≤ K*Real.exp (-(codimension κ)*t)*(y/Real.sqrt (x^2+y^2))^(8/κ-1) := hh
-    _ ≤ K*(ρ^(codimension κ)*(2*y/Real.exp l)^(-(1-κ/8)))*(y/Real.sqrt (x^2+y^2))^(8/κ-1) :=
-      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left he hK.le) (Real.rpow_nonneg (div_pos hy hs).le _)
-    _ = _ := by rw [green_eq_source _ _ _ _ _ hy]; ring
-
-end SLEExactGauge.GammaField
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit GammaField
-
-lemma height_radius_at_stop {z : ℂ} {u : ℝ} (C : Cutoff z u) (hz : z ∈ halfPlane)
-    (hu : 0 ≤ u) {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
-    C.height ≤ (position C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z (C.stopTime κ B sample)).im ∧
-    (2*z.im)*Real.exp (-u) ≤ radius C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z (C.stopTime κ B sample) := by
-  rcases eq_or_lt_of_le hu with rfl | hu'
-  · rw [C.stopTime_zero,position_initial]
-    constructor
-    · exact (C.height_initial hz le_rfl).le
-    · simp only [radius,position_initial,logDerivative_initial,neg_zero,Real.exp_zero,mul_one,div_one,le_refl]
-  · have hd := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-      (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle
-        (C.initial_deficit_brownian κ B sample hz hu' hc hb0) ⟨(C.bounds κ B sample).1,le_rfl⟩)
-    exact ⟨hd.2.1,hd.1⟩
-
-lemma targetValue_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ K : ℝ, 0 < K ∧ ∀ {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ}
-      (_hu : 0 ≤ u) (huv : u+1 ≤ v) (C : Cutoff z v) {Ω : Type*}
-      (B : ℝ≥0 → Ω → ℝ) (sample : Ω) (η : ℝ), 0 < η →
-      Continuous (fun t => B t sample) → B 0 sample = 0 →
-      C.targetValue hz (by linarith : u ≤ v) η κ B sample ≤
-        K*((2*z.im)*Real.exp (-v))^(codimension κ) *
-          (C.retarget hz (by linarith : u ≤ v)).value κ B sample := by
-  obtain ⟨K,hK,hbound⟩ := profile_green_bound κ hκ
-  refine ⟨K,hK,?_⟩
-  intro z hz u v hu huv C Ω B sample η hη hc hb0
-  let D := C.retarget hz (by linarith : u ≤ v)
-  let τ := D.stopTime κ B sample
-  let U := C.paths κ B sample
-  have hd := D.height_radius_at_stop hz hu κ B sample hc hb0
-  have hy : C.height ≤ (position C.height_pos C.horizon.coe_nonneg U z τ).im := hd.1
-  have hR : (2*z.im)*Real.exp (-u) ≤ radius C.height_pos C.horizon.coe_nonneg U z τ := hd.2
-  have hz0 : 0 < z.im := hz
-  have hρu : 0 < (2*z.im)*Real.exp (-u) := by positivity
-  have hρv : 0 < (2*z.im)*Real.exp (-v) := by positivity
-  have hlog := Real.log_le_log hρu hR
-  have hlρ (w : ℝ) : Real.log ((2*z.im)*Real.exp (-w)) = Real.log (2*z.im)-w := by
-    rw [Real.log_mul (by positivity) (Real.exp_ne_zero _),Real.log_exp]; ring
-  have hr0 := le_max_right η (rawRemaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ)
-  change rawRemaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ ≤
-    remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ at hr0
-  have hr1 : 1 ≤ remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ := by
-    rw [rawRemaining_eq C.height_pos C.horizon.coe_nonneg _ _ _ _ _ hy,hlρ] at hr0
-    rw [hlρ] at hlog
-    exact le_trans (by linarith : 1 ≤ η+Real.log (radius C.height_pos C.horizon.coe_nonneg U z τ)-(Real.log (2*z.im)-v)) hr0
-  have hr2 : Real.log (2*LoewnerField.height C.height_pos C.horizon.coe_nonneg U z τ)-
-      logDerivative C.height_pos C.horizon.coe_nonneg U z τ-Real.log ((2*z.im)*Real.exp (-v)) ≤
-      remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ := by
-    unfold rawRemaining at hr0
-    linarith
-  exact hbound _ _ _ _ _ (LoewnerField.height_pos _ _ _ _ _) hρv hr1 hr2
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.AngularClock
+import OAI.Probability.SLE.ClippedCoordinates
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Brownian Predictable estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+open PredictableLimit
+
+lemma indicator_difference_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (A D : ℕ → ℕ → Ω → ℝ)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
+    (hD : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (D n i))
+    (hA2 : ∀ n i, MemLp (A n i) 2 P) (hD2 : ∀ n i, MemLp (D n i) 2 P)
+    (K : ℝ) (hK : 0 ≤ K)
+    (hbA : ∀ n i, ∀ᵐ sample ∂P, |A n i sample| ≤ K)
+    (hbD : ∀ n i, ∀ᵐ sample ∂P, |D n i sample| ≤ K)
+    (θ : Ω → ℝ≥0∞)
+    (hsupport : ∀ n i, ∀ᵐ sample ∂P,
+      (dyadicTime T n i : ℝ≥0∞) < θ sample → A n i sample = D n i sample)
+    (E : Set Ω) (hE : NullMeasurableSet E P)
+    (hstop : ∀ t : ℝ≥0, ∃ S : Set Ω, MeasurableSet[pastSigma B t] S ∧
+      S =ᵐ[P] (E ∩ {sample | θ sample ≤ (t : ℝ≥0∞)} : Set Ω))
+    (YA YD : Ω → ℝ)
+    (hlA : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A n sample) atTop (𝓝 (YA sample)))
+    (hlD : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T D n sample) atTop (𝓝 (YD sample))) :
+    Integrable (E.indicator (fun sample => YA sample-YD sample)) P ∧
+      (∫ sample, E.indicator (fun sample => YA sample-YD sample) sample ∂P) = 0 := by
+  classical
+  choose S hSm hSeq using hstop
+  let Q := fun n i => (S (dyadicTime T n i)).indicator (fun sample => A n i sample-D n i sample)
+  have hQm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
+    ((hA n i).sub (hD n i)).indicator (hSm _)
+  have heq (n i : ℕ) : Q n i =ᵐ[P] E.indicator (fun sample => A n i sample-D n i sample) := by
+    filter_upwards [hSeq (dyadicTime T n i), hsupport n i] with sample he hs
+    have he : sample ∈ S (dyadicTime T n i) ↔ sample ∈ E ∧ θ sample ≤ (dyadicTime T n i : ℝ≥0∞) := Iff.of_eq he
+    by_cases hz : θ sample ≤ (dyadicTime T n i : ℝ≥0∞)
+    · simp only [Q,Set.indicator_apply,he,hz,and_true]
+    · have hh := hs (lt_of_not_ge hz)
+      simp only [Q,Set.indicator_apply,hh,sub_self,ite_self]
+  have hQ2 (n i : ℕ) : MemLp (Q n i) 2 P := by
+    have hd := (hA2 n i).sub (hD2 n i)
+    have hm : MemLp (E.indicator (fun sample => A n i sample-D n i sample)) 2 P :=
+      hd.of_le (hd.aestronglyMeasurable.indicator₀ hE)
+        (Filter.Eventually.of_forall (fun sample => norm_indicator_le_norm_self _ sample))
+    exact hm.ae_eq (heq n i).symm
+  have hQb (n i : ℕ) : ∀ᵐ sample ∂P, |Q n i sample| ≤ 2*K := by
+    filter_upwards [hbA n i,hbD n i] with sample ha hd
+    dsimp only [Q]
+    rw [Set.indicator_apply]
+    split_ifs
+    · exact (abs_sub _ _).trans (by linarith)
+    · simpa only [abs_zero] using (show 0 ≤ 2*K by positivity)
+  have heall : ∀ᵐ sample ∂P, ∀ n i, Q n i sample = E.indicator (fun sample => A n i sample-D n i sample) sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (heq n))
+  apply limit_centered hB T Q hQm hQ2 (2*K) (by positivity) hQb _
+  filter_upwards [heall,hlA,hlD] with sample he ha hd
+  have heqsum (n : ℕ) : dyadicDriverSum B T Q n sample =
+      E.indicator (fun sample => dyadicDriverSum B T A n sample-dyadicDriverSum B T D n sample) sample := by
+    unfold dyadicDriverSum
+    simp only [he,Set.indicator_apply]
+    split_ifs <;> simp [sub_mul,Finset.sum_sub_distrib]
+  simp only [heqsum,Set.indicator_apply]
+  split_ifs
+  · exact ha.sub hd
+  · exact tendsto_const_nhds
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.GammaTwoRadius
+open LoewnerCutoff LoewnerExit GammaField BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+open scoped unitInterval
+variable {ε : ℝ} (hε : 0 < ε)
+
+def coeffAt (k : ℕ) (s : ℝ≥0) (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ)
+    (q : C(I, ℝ) × ℝ) : ℝ := by
+  classical
+  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
+    iteratedDeriv k (field hε s.coe_nonneg l b η r sdev q.1 z s) q.2 else 0
+
+lemma coeffAt_measurable (k : ℕ) (_hk : k = 1 ∨ k = 2) (s : ℝ≥0)
+    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (r ρ δ : ℝ) :
+    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s l b η sdev z r ρ δ) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
+      iteratedDeriv k (field hε s.coe_nonneg l b η r sdev q.1 z s) q.2) := by
+    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
+      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
+    have hc := (continuous_derivative hε s.coe_nonneg hl hb hη r sdev z k).comp hp
+    dsimp only [Function.comp_def] at hc
+    exact hc
+  apply hc.measurable.ite _ measurable_const
+  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
+
+def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ) (sample : Ω) : ℝ :=
+  if s < T then coeffAt hε k s l b η sdev z r ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
+
+lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (r ρ δ : ℝ) :
+    Measurable[pastSigma B s] (rawCoeff hε k B T s l b η sdev z r ρ δ) := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  let : MeasurableSpace Ω := pastSigma B s
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z r ρ δ).comp
+      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
+  · exact measurable_const
+
+lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
+    (l b η sdev : ℝ) (hl : 0 < l) (hb : 0 < b) (hη : 0 < η) (z : ℂ) (r ρ δ : ℝ) :
+    AEMeasurable (rawCoeff hε k B T s l b η sdev z r ρ δ) P := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s l b η sdev hl hb hη z r ρ δ).comp_aemeasurable
+      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
+  · exact aemeasurable_const
+
+lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    rawCoeff hε k B T s l b η sdev z r ρ δ sample =
+      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
+        iteratedDeriv k (field hε T.coe_nonneg l b η r sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
+  classical
+  by_cases hsT : s < T
+  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
+    have hf := field_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z l b η r sdev he
+      ⟨s.coe_nonneg, le_rfl⟩
+    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
+    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
+        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
+      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
+      exact forall₂_congr (fun t ht => by rw [hd ht])
+    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
+  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
+      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
+    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
+
+lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (l b η sdev : ℝ) (z : ℂ) (r ρ δ : ℝ) (n i : ℕ) :
+    rawCoeff hε k B T (dyadicTime T n i) l b η sdev z r ρ δ =ᵐ[P]
+      stoppedCoeff k B T
+        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
+        (fun sample => field hε T.coe_nonneg l b η r sdev (BrownianDriver.path B sdev T sample) z) n i := by
+  filter_upwards [hB.cont] with sample hc
+  rw [rawCoeff_eq hε k B T _ hT l b η sdev z r ρ δ sample hc]
+  simp only [stoppedCoeff, coe_dyadicTime]
+
+end SLEExactGauge.GammaTwoRadius
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerExit
+open LoewnerCutoff
+
+lemma stop_antitone_radius {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (U : C(I, ℝ)) (z : ℂ) (ρ ρ₂ δ : ℝ) (hρρ₂ : ρ ≤ ρ₂) :
+    stop hε hT.le U z ρ₂ δ ≤ stop hε hT.le U z ρ δ := by
+  by_contra hn
+  have hlt : stop hε hT.le U z ρ δ < stop hε hT.le U z ρ₂ δ := lt_of_not_ge hn
+  have hp := (deficit_pos_iff hε hT.le U z ρ₂ δ _).1
+    (positive_before hε hT.le hT U z ρ₂ δ (stop_bounds hε hT.le U z ρ δ).1 hlt)
+  have hpos : 0 < deficit hε hT.le U z ρ δ (stop hε hT.le U z ρ δ) :=
+    (deficit_pos_iff hε hT.le U z ρ δ _).2 ⟨hρρ₂.trans_lt hp.1,hp.2⟩
+  exact (not_lt_of_ge (nonpos_at_of_lt hε hT.le hT U z ρ δ
+    (hlt.trans_le (stop_bounds hε hT.le U z ρ₂ δ).2))) hpos
+
+end SLEExactGauge.LoewnerExit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized
+variable {z : ℂ} {u v : ℝ}
+
+def retarget (C : Cutoff z v) (hz : z ∈ halfPlane) (huv : u ≤ v) : Cutoff z u where
+  angle := C.angle
+  angle_pos := C.angle_pos
+  angle_initial := C.angle_initial
+  height := C.height
+  height_pos := C.height_pos
+  horizon := C.horizon
+  horizon_pos := C.horizon_pos
+  height_small := C.height_small.trans_le (by
+    apply mul_le_mul_of_nonneg_left _ hz.le
+    apply Real.exp_le_exp.mpr
+    exact div_le_div_of_nonneg_right (neg_le_neg huv) (by positivity))
+  time_large := lt_of_le_of_lt (by
+    apply div_le_div_of_nonneg_right _ (by positivity)
+    exact mul_le_mul_of_nonneg_right huv (sq_nonneg _)) C.time_large
+
+lemma retarget_stop_le (C : Cutoff z v) (hz : z ∈ halfPlane) (huv : u ≤ v)
+    {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
+    (C.retarget hz huv).stopTime κ B sample ≤ C.stopTime κ B sample := by
+  apply stop_antitone_radius C.height_pos C.horizon_pos
+  exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (neg_le_neg huv)) (by exact mul_nonneg (by norm_num) hz.le)
+
+lemma retarget_has_hit_of_lt (C : Cutoff z v) (hz : z ∈ halfPlane)
+    (hu : 0 < u) (huv : u ≤ v) {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+    (hlt : (C.retarget hz huv).stopTime κ B sample < C.stopTime κ B sample) :
+    firstLiveRadiusStop γ g z u = ENNReal.ofReal ((C.retarget hz huv).stopTime κ B sample) := by
+  let D := C.retarget hz huv
+  have hd := D.initial_deficit_brownian κ B sample hz hu hc hb0
+  have ht := D.bounds κ B sample
+  have hR : loewnerRadius g (D.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
+    rcases D.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · exact hr
+    · have hp := (deficit_pos_iff C.height_pos C.horizon.coe_nonneg (C.paths κ B sample)
+        z ((2*z.im)*Real.exp (-v)) C.angle (D.stopTime κ B sample)).1
+        (positive_before C.height_pos C.horizon.coe_nonneg C.horizon_pos (C.paths κ B sample)
+          z ((2*z.im)*Real.exp (-v)) C.angle ht.1 hlt)
+      have he := D.source_sine κ B sample h hz hu hc hb0
+      have hi : C.angle < loewnerAngleSine κ (fun t => B t sample) g (D.stopTime κ B sample) z := by
+        rw [← he]
+        exact hp.2.2
+      exact False.elim ((ne_of_gt hi) hs)
+  have hh := h.firstLiveRadiusStop_eq
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (t := ⟨D.stopTime κ B sample,ht.1⟩) (D.source_live κ B sample h hz hu hc hb0) hR
+  rw [ENNReal.ofReal_eq_coe_nnreal ht.1]
+  exact hh
+
+lemma stopTime_zero (C : Cutoff z 0) {Ω : Type*} (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : C.stopTime κ B sample = 0 := by
+  apply le_antisymm _ (C.bounds κ B sample).1
+  by_contra hn
+  have hp := (deficit_pos_iff C.height_pos C.horizon.coe_nonneg (C.paths κ B sample)
+    z ((2*z.im)*Real.exp (-0)) C.angle 0).1
+    (positive_before C.height_pos C.horizon.coe_nonneg C.horizon_pos (C.paths κ B sample)
+      z ((2*z.im)*Real.exp (-0)) C.angle le_rfl (lt_of_not_ge hn))
+  simpa only [LoewnerExit.radius,position_initial,logDerivative_initial,neg_zero,
+    Real.exp_zero,mul_one,div_one,lt_self_iff_false] using hp.1
+
+lemma value_zero (C : Cutoff z 0) (hz : z ∈ halfPlane) {Ω : Type*} (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (sample : Ω) (hb0 : B 0 sample = 0) :
+    ENNReal.ofReal (C.value κ B sample) =
+      ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) := by
+  unfold value
+  rw [C.stopTime_zero,Real.toNNReal_zero,hb0,field_initial C.height_pos C.horizon.coe_nonneg
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z (C.height_initial hz le_rfl).le,
+    green_initial κ hz]
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BrownianLocal.LocalRegular
+
+lemma mono {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ} {S T : ℝ}
+    (h : BrownianLocal.LocalRegular F V b T) (hST : S ≤ T) :
+    BrownianLocal.LocalRegular F V b S where
+  smooth := h.smooth
+  third_cont := h.third_cont.mono (Set.prod_mono (Icc_subset_Icc_right hST) Subset.rfl)
+  time_cont := fun x => (h.time_cont x).mono (Icc_subset_Icc_right hST)
+  time_deriv := fun t ht => h.time_deriv t ⟨ht.1,ht.2.trans_le hST⟩
+  drift_cont := h.drift_cont.mono (Set.prod_mono (Icc_subset_Icc_right hST) (Icc_subset_Icc_right hST))
+  second_cont := h.second_cont.mono (Icc_subset_Icc_right hST)
+
+end SLEExactGauge.BrownianLocal.LocalRegular
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open GammaField GammaTwoRadius BrownianPredictable BrownianStopped BrownianIto TaylorMeshes PredictableLimit
+variable {z : ℂ} {v : ℝ} (C : Cutoff z v)
+
+def targetField {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ → ℝ → ℝ :=
+  GammaField.field C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (C.paths κ B sample) z
+
+def targetDrift {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ → ℝ → ℝ :=
+  GammaField.drift C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (C.paths κ B sample) z
+
+def targetCoeff {Ω : Type*} (k : ℕ) (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (w : ℝ) (n i : ℕ) : Ω → ℝ :=
+  GammaTwoRadius.rawCoeff C.height_pos k B C.horizon (dyadicTime C.horizon n i)
+    (κ/2) ((8/κ-1)/2) η (Real.sqrt κ) z ((2*z.im)*Real.exp (-v)) ((2*z.im)*Real.exp (-w)) C.angle
+
+def targetRemainder {Ω : Type*} (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (t : Ω → ℝ) (sample : Ω) : ℝ :=
+  C.targetField η κ B sample (t sample) (B (t sample).toNNReal sample) -
+    C.targetField η κ B sample 0 (B 0 sample) -
+    (∫ s in (0:ℝ)..(t sample), C.targetDrift η κ B sample s (B s.toNNReal sample)) -
+    (1/2:ℝ)*∫ s in (0:ℝ)..(t sample),
+      iteratedDeriv 2 (C.targetField η κ B sample s) (B s.toNNReal sample)
+
+lemma targetCoeff_eq {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (η κ : ℝ)
+    (hz : z ∈ halfPlane) (w : ℝ) (hwv : w ≤ v) (n i : ℕ) :
+    C.targetCoeff k η κ B w n i =ᵐ[P]
+      stoppedCoeff k B C.horizon ((C.retarget hz hwv).stopTime κ B) (C.targetField η κ B) n i :=
+  rawCoeff_stoppedCoeff_ae C.height_pos hB k C.horizon C.horizon_pos (κ/2) ((8/κ-1)/2)
+    η (Real.sqrt κ) z _ _ C.angle n i
+
+lemma targetCoeff_measurable {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (k : ℕ) (hk : k = 1 ∨ k = 2) (η κ : ℝ)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (w : ℝ) (n i : ℕ) :
+    Measurable[pastSigma B (dyadicTime C.horizon n i)] (C.targetCoeff k η κ B w n i) := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  exact rawCoeff_measurable_past C.height_pos k hk B C.horizon _ _ _ _ _ hl hb hη z _ _ C.angle
+
+lemma targetCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (hk : k = 1 ∨ k = 2) (η κ : ℝ)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (w : ℝ) (n i : ℕ) :
+    AEMeasurable (C.targetCoeff k η κ B w n i) P := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  exact rawCoeff_aemeasurable C.height_pos hB k hk C.horizon _ _ _ _ _ hl hb hη z _ _ C.angle
+
+lemma targetCoeff_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (η κ : ℝ)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (hz : z ∈ halfPlane) (hv : 0 < v)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample)) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ w, w ≤ v → ∀ n i,
+      ∀ᵐ sample ∂P, |C.targetCoeff k η κ B w n i sample| ≤ K := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  obtain ⟨K,hK,hKb⟩ := coefficient_bound C.height_pos C.horizon.coe_nonneg hl hb hη
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (η+v) (1/C.angle) z k
+  refine ⟨K,hK,?_⟩
+  intro w hwv n i
+  filter_upwards [C.targetCoeff_eq hB k η κ hz w hwv n i,hmaps,hB.cont,
+    hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample he hm hc hb0
+  rw [he]
+  unfold stoppedCoeff
+  split_ifs with hh
+  · have ht : mesh C.horizon n i ∈ Icc 0 (C.stopTime κ B sample) :=
+      ⟨by unfold mesh; positivity,hh.le.trans (C.retarget_stop_le hz hwv κ B sample)⟩
+    exact hKb (C.paths κ B sample) _ _ (C.remaining_upper κ B sample hm hz hv hc hb0 η ht)
+      (by simpa only [mesh_toNNReal] using C.ratio_upper κ B sample hz hv hc hb0 ht)
+  · simpa only [abs_zero] using hK
+
+lemma target_sum_tendsto {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (η κ : ℝ)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (hz : z ∈ halfPlane) (hv : 0 < v)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (w : ℝ) (hwv : w ≤ v) :
+    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B C.horizon (C.targetCoeff 1 η κ B w) n sample)
+      atTop (𝓝 (C.targetRemainder η κ B ((C.retarget hz hwv).stopTime κ B) sample)) := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  let D := C.retarget hz hwv
+  have hreg : ∀ᵐ sample ∂P, BrownianLocal.LocalRegular (C.targetField η κ B sample)
+      (C.targetDrift η κ B sample) (fun t => B t.toNNReal sample) (D.stopTime κ B sample) := by
+    filter_upwards [hB.cont] with sample hc
+    have hρ : 0 < (2*z.im)*Real.exp (-v) := by have hy : 0 < z.im := hz; positivity
+    exact (regular C.height_pos C.horizon.coe_nonneg hl hb hη hρ C.horizon_pos
+      (Real.sqrt κ) (C.paths κ B sample) z C.angle _ (hc.comp continuous_real_toNNReal)).mono
+      (C.retarget_stop_le hz hwv κ B sample)
+  obtain ⟨K₁,hK₁,hb₁⟩ := C.targetCoeff_bound hB 1 η κ hκ hη hz hv γ g f hmaps
+  obtain ⟨K₂,hK₂,hb₂⟩ := C.targetCoeff_bound hB 2 η κ hκ hη hz hv γ g f hmaps
+  have hMem : ∀ n i, MemLp (C.targetCoeff 2 η κ B w n i) 2 P := by
+    intro n i
+    apply MemLp.of_bound (C.targetCoeff_aemeasurable hB 2 (Or.inr rfl) η κ hκ hη w n i).aestronglyMeasurable K₂
+    simpa only [Real.norm_eq_abs] using hb₂ w hwv n i
+  have hi := BrownianLocal.stopped_fieldSum_tendsto_ae hB C.horizon (D.stopTime κ B)
+    (C.targetField η κ B) (C.targetDrift η κ B)
+    (Filter.Eventually.of_forall (fun sample => D.bounds κ B sample)) hreg
+    (C.targetCoeff 2 η κ B w) (C.targetCoeff_eq hB 2 η κ hz w hwv)
+    (C.targetCoeff_measurable B 2 (Or.inr rfl) η κ hκ hη w) hMem
+    K₂ hK₂ (fun n i => by
+      filter_upwards [C.targetCoeff_eq hB 2 η κ hz w hwv n i,hb₂ w hwv n i] with sample he hh
+      rwa [← he])
+    K₁ hK₁ (fun n i => by
+      filter_upwards [C.targetCoeff_eq hB 1 η κ hz w hwv n i,hb₁ w hwv n i] with sample he hh
+      rwa [← he])
+  have heall : ∀ᵐ sample ∂P, ∀ n i, C.targetCoeff 1 η κ B w n i sample =
+      stoppedCoeff 1 B C.horizon (D.stopTime κ B) (C.targetField η κ B) n i sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (C.targetCoeff_eq hB 1 η κ hz w hwv n))
+  filter_upwards [hi,heall] with sample hi he
+  simpa only [targetRemainder,dyadicDriverSum,he] using hi
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.GammaField
+open LoewnerCutoff LoewnerExit
+
+lemma integral_drift_nonpos_from {ε T κ η : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (ρ : ℝ)
+    (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (δ : ℝ) (B : ℝ → ℝ) (hB : Continuous B)
+    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ*B s)
+    {a : ℝ} (ha : 0 ≤ a) (haτ : a ≤ stop hε hT U z ρ δ) :
+    (∫ t in a..(stop hε hT U z ρ δ), drift hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t (B t)) +
+    (1/2:ℝ)*∫ t in a..(stop hε hT U z ρ δ),
+      iteratedDeriv 2 (field hε hT (κ/2) ((8/κ-1)/2) η ρ (Real.sqrt κ) U z t) (B t) ≤ 0 := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have hm : Continuous (fun t : ℝ => ((U,t),B t)) :=
+    (continuous_const.prodMk continuous_id).prodMk hB
+  have hv := (continuous_drift hε hT hl hb hη ρ (Real.sqrt κ) z).comp hm
+  have hc := (continuous_derivative hε hT hl hb hη ρ (Real.sqrt κ) z 2).comp hm
+  have hiV := hv.intervalIntegrable (μ := volume) a (stop hε hT U z ρ δ)
+  have hiC := hc.intervalIntegrable (μ := volume) a (stop hε hT U z ρ δ)
+  dsimp only [Function.comp_def] at hiV hiC
+  rw [← intervalIntegral.integral_const_mul, ← intervalIntegral.integral_add hiV (hiC.const_mul (1/2:ℝ))]
+  rw [intervalIntegral.integral_of_le haτ]
+  apply integral_nonpos_of_ae
+  have hne : ∀ᵐ t : ℝ ∂volume, t ≠ stop hε hT U z ρ δ := by simp [ae_iff]
+  filter_upwards [ae_restrict_mem measurableSet_Ioc,ae_restrict_of_ae hne] with t ht he
+  have hlt : t < stop hε hT U z ρ δ := lt_of_le_of_ne ht.2 he
+  have hh := (deficit_pos_iff hε hT U z ρ δ t).1
+    (positive_before hε hT hTp U z ρ δ (ha.trans ht.1.le) hlt)
+  exact drift_nonpos hε hT hκ hη ρ U z t (B t) hh.2.1.le
+    (hU t ⟨ha.trans ht.1.le,ht.2.trans (stop_bounds hε hT U z ρ δ).2⟩)
+
+end SLEExactGauge.GammaField
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open GammaField LoewnerCutoff LoewnerExit
+variable {z : ℂ} {v : ℝ} (C : Cutoff z v)
+
+def targetValue {Ω : Type*} (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
+    (η κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
+  C.targetField η κ B sample ((C.retarget hz hwv).stopTime κ B sample)
+    (B ((C.retarget hz hwv).stopTime κ B sample).toNNReal sample)
+
+lemma targetValue_nonneg {Ω : Type*} (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
+    (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
+    0 ≤ C.targetValue hz hwv η κ B sample := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  exact field_nonneg C.height_pos C.horizon.coe_nonneg hl hb hη _ _ _ _ _ _
+
+lemma targetValue_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
+    (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) :
+    AEMeasurable (C.targetValue hz hwv η κ B) P := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  let τ := fun U : C(I, ℝ) => stop C.height_pos C.horizon.coe_nonneg U z ((2*z.im)*Real.exp (-w)) C.angle
+  have hτ : Measurable τ := measurable_stop C.height_pos C.horizon.coe_nonneg C.horizon_pos z _ C.angle
+  have hm := ((continuous_actualField C.height_pos C.horizon hl hb hη
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) z).measurable.comp (measurable_id.prodMk hτ)).comp_aemeasurable
+    (BrownianDriver.path_aemeasurable hB (Real.sqrt κ) C.horizon)
+  apply hm.congr
+  filter_upwards [hB.cont] with sample hc
+  change GammaField.field C.height_pos C.horizon.coe_nonneg (κ/2) ((8/κ-1)/2) η
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) (C.paths κ B sample) z
+    ((C.retarget hz hwv).stopTime κ B sample)
+    (LoewnerCutoff.extendDriver C.horizon (C.paths κ B sample)
+      ((C.retarget hz hwv).stopTime κ B sample) / Real.sqrt κ) = _
+  rw [C.path_eq κ B sample hc ((C.retarget hz hwv).bounds κ B sample),mul_div_cancel_left₀ _ (Real.sqrt_pos.2 hκ.1).ne']
+  rfl
+
+lemma target_difference_le_remainder {Ω : Type*} (hz : z ∈ halfPlane) {w : ℝ} (hwv : w ≤ v)
+    (η κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hη : 0 < η) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    C.barrier η κ B sample-C.targetValue hz hwv η κ B sample ≤
+      C.targetRemainder η κ B (C.stopTime κ B) sample -
+      C.targetRemainder η κ B ((C.retarget hz hwv).stopTime κ B) sample := by
+  let D := C.retarget hz hwv
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  have hm : Continuous (fun t : ℝ => ((C.paths κ B sample,t),B t.toNNReal sample)) :=
+    (continuous_const.prodMk continuous_id).prodMk (hc.comp continuous_real_toNNReal)
+  have hd := (continuous_drift C.height_pos C.horizon.coe_nonneg hl hb hη
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) z).comp hm
+  have h2 := (continuous_derivative C.height_pos C.horizon.coe_nonneg hl hb hη
+    ((2*z.im)*Real.exp (-v)) (Real.sqrt κ) z 2).comp hm
+  have hid := intervalIntegral.integral_add_adjacent_intervals
+    (hd.intervalIntegrable (μ := volume) 0 (D.stopTime κ B sample))
+    (hd.intervalIntegrable (μ := volume) (D.stopTime κ B sample) (C.stopTime κ B sample))
+  have hi2 := intervalIntegral.integral_add_adjacent_intervals
+    (h2.intervalIntegrable (μ := volume) 0 (D.stopTime κ B sample))
+    (h2.intervalIntegrable (μ := volume) (D.stopTime κ B sample) (C.stopTime κ B sample))
+  have hn := integral_drift_nonpos_from C.height_pos C.horizon.coe_nonneg hκ hη
+    ((2*z.im)*Real.exp (-v)) C.horizon_pos (C.paths κ B sample) z C.angle
+    (fun t => B t.toNNReal sample) (hc.comp continuous_real_toNNReal)
+    (C.path_eq κ B sample hc) (D.bounds κ B sample).1 (C.retarget_stop_le hz hwv κ B sample)
+  dsimp only [Function.comp_def] at hid hi2
+  unfold targetRemainder targetValue targetField targetDrift barrier
+  dsimp only [D, stopTime] at hid hi2 hn ⊢
+  linarith only [hn,hid,hi2]
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.AngularBarrier
+
+lemma value_decay_from {l b t₀ : ℝ} (hl : 0 < l) (hb : 0 < b) (ht₀ : 0 < t₀) :
+    ∃ K : ℝ, 0 < K ∧ ∀ t q : ℝ, t₀ ≤ t → 0 ≤ q →
+      value l b t q ≤ K*Real.exp (-(l*b/2)*t)*q^b := by
+  let c := (1-Real.exp (-l*t₀))/(2*(b+1))
+  have hc : 0 < c := div_pos (sub_pos.2 (Real.exp_lt_one_iff.2 (by nlinarith))) (by positivity)
+  have hg : 0 < Real.Gamma b := Real.Gamma_pos_of_pos hb
+  refine ⟨1/(b*Real.Gamma b*c^b), by positivity, ?_⟩
+  intro t q ht hq
+  have ht0 : 0 < t := ht₀.trans_le ht
+  have ha := scale_pos hl hb ht0
+  have hlower : c*Real.exp (l*t) ≤ scale l b t := by
+    have hh : 1 ≤ Real.exp (l*t)*Real.exp (-l*t₀) := by
+      rw [← Real.exp_add]
+      exact Real.one_le_exp_iff.2 (by nlinarith)
+    dsimp only [c,scale]
+    apply (le_div_iff₀ (by positivity : 0 < 2*(b+1))).2
+    have heq : ((1-Real.exp (-l*t₀))/(2*(b+1))*Real.exp (l*t))*(2*(b+1)) =
+        (1-Real.exp (-l*t₀))*Real.exp (l*t) := by field_simp
+    rw [heq]
+    nlinarith
+  have hratio : q/scale l b t ≤ q/(c*Real.exp (l*t)) :=
+    div_le_div_of_nonneg_left hq (mul_pos hc (Real.exp_pos _)) hlower
+  have hpow := Real.rpow_le_rpow (div_nonneg hq ha.le) hratio hb.le
+  have hbound := (cdf_le_power hb (div_nonneg hq ha.le)).trans
+    (div_le_div_of_nonneg_right hpow (mul_pos hb hg).le)
+  have he : Real.exp (l*b/2*t)*(q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b) =
+      1/(b*Real.Gamma b*c^b)*Real.exp (-(l*b/2)*t)*q^b := by
+    rw [Real.div_rpow hq (mul_pos hc (Real.exp_pos _)).le,
+      Real.mul_rpow hc.le (Real.exp_pos _).le, ← Real.exp_mul]
+    have hratioexp : Real.exp (l*b/2*t)/Real.exp (l*t*b) = Real.exp (-(l*b/2)*t) := by
+      rw [← Real.exp_sub]; congr 1; ring
+    calc
+      _ = (Real.exp (l*b/2*t)/Real.exp (l*t*b))*q^b/(b*Real.Gamma b*c^b) := by ring
+      _ = _ := by rw [hratioexp]; ring
+  calc
+    _ ≤ Real.exp (l*b/2*t)*((q/(c*Real.exp (l*t)))^b/(b*Real.Gamma b)) :=
+      mul_le_mul_of_nonneg_left hbound (Real.exp_pos _).le
+    _ = _ := by rw [← mul_div_assoc, he]
+
+end SLEExactGauge.AngularBarrier
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteTilt AngularBarrier
+
+theorem radiusApproachBoundFrom (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (u₀ : ℝ) (hu₀ : 0 < u₀) :
+    ∃ C : ℝ, 0 < C ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (_hz : z ∈ halfPlane) (u : ℝ) (_hu : u₀ ≤ u),
+      P {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)} ≤
+        ENNReal.ofReal (C * Real.exp (-codimension κ * u) *
+          (z.im / ‖z‖) ^ (8 / κ - 1)) := by
+  have hl : 0 < κ/2 := div_pos hκ.1 (by norm_num)
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  obtain ⟨C,hC,hdec⟩ := value_decay_from hl hb hu₀
+  refine ⟨C,hC,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz u hu
+  have hu0 : 0 < u := hu₀.trans_le hu
+  apply (radiusApproachProfile κ hκ B hB γ g f hmaps z hz u hu0).trans
+  apply ENNReal.ofReal_le_ofReal
+  have hp : (κ/2)*((8/κ-1)/2)/2 = codimension κ := by
+    unfold codimension dimension
+    field_simp [hκ.1.ne']
+    ring
+  have hq : (squaredSine (z.re/z.im))^((8/κ-1)/2) = (z.im/‖z‖)^(8/κ-1) := by
+    rw [squaredSine_initial hz, ← Real.rpow_natCast (z.im/‖z‖) 2,
+      ← Real.rpow_mul (initial_sine_pos hz).le]
+    congr 1
+    ring
+  have hh := hdec u (squaredSine (z.re/z.im)) hu (squaredSine_pos _).le
+  simpa only [profile,hp,hq] using hh
+
+def finiteCenteredMeasure {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (P : Measure Ω) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) : Measure Ω :=
+  P.withDensity (fun sample =>
+    finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u /
+      ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))
+
+def radiusStoppedInverseAngle {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) (sample : Ω) : ℝ≥0∞ :=
+  let τ := firstLiveRadiusStop (γ sample) (g sample) z u
+  if τ = (⊤ : ℝ≥0∞) then 1 else ENNReal.ofReal
+    ((loewnerAngleSine κ (fun t => B t sample) (g sample) τ.toReal z)^(-(8/κ-1)))
+
+lemma finiteStoppedDensity_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u) :
+    AEMeasurable (fun sample => finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z u) P := by
+  let C := cutoffs hz u hu.le
+  let F := fun n sample => ENNReal.ofReal ((C n).value κ B sample)
+  have hexp (n : ℕ) := (C n).expectation hB κ hκ hz hu.le
+  have hFm (n : ℕ) : AEMeasurable (F n) P :=
+    ENNReal.measurable_ofReal.comp_aemeasurable (hexp n).1.aemeasurable
+  have hFl : ∀ᵐ sample ∂P, Tendsto (fun n => F n sample) atTop
+      (𝓝 (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u)) := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
+    exact tendsto_value_killed κ B sample hm hz hu hc hb hκ
+  exact aemeasurable_of_tendsto_metrizable_ae' hFm hFl
+
+lemma finiteStoppedDensity_lt_top (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ → ℂ)
+    (g : ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) : finiteStoppedDensity κ B γ g z u < (⊤ : ℝ≥0∞) := by
+  dsimp only [finiteStoppedDensity]
+  split_ifs <;> simp
+
+lemma localizationInitialGreen_pos (κ : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
+    0 < (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1) := by
+  have hy : 0 < z.im := hz
+  exact mul_pos (Real.rpow_pos_of_pos (by positivity) _) (Real.rpow_pos_of_pos (initial_sine_pos hz) _)
+
+lemma finiteCenteredMeasure_probability {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (P : Measure Ω) [IsProbabilityMeasure P]
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u) :
+    IsProbabilityMeasure (finiteCenteredMeasure κ P B γ g z u) := by
+  have hm0 : ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) ≠ 0 :=
+    (ENNReal.ofReal_pos.2 (localizationInitialGreen_pos κ hz)).ne'
+  constructor
+  rw [finiteCenteredMeasure, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ]
+  simp_rw [ENNReal.div_eq_inv_mul]
+  rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hm0),
+    finiteStoppedCenteredLaw κ hκ B hB γ g f hγ hmaps z hz u hu.le,
+    ENNReal.inv_mul_cancel hm0 ENNReal.ofReal_ne_top]
+
+lemma finiteStoppedDensity_mul_inverseAngle {Ω : Type*} (κ : ℝ)
+    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (sample : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample)
+      (γ sample) (g sample) (f sample))
+    (hc : Continuous (fun t => B t sample)) :
+    finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u *
+      radiusStoppedInverseAngle κ B γ g z u sample =
+    {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)}.indicator
+      (fun _ => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-codimension κ))) sample := by
+  classical
+  by_cases hhit : firstLiveRadiusStop (γ sample) (g sample) z u = (⊤ : ℝ≥0∞)
+  · simp [finiteStoppedDensity,radiusStoppedInverseAngle,hhit]
+  have hex : ∃ t : ℝ≥0, z ∈ localizationDomain (γ sample) t ∧
+      loewnerRadius (g sample) t z = (2*z.im)*Real.exp (-u) := by
+    by_contra hn
+    have he : {t : ℝ≥0 | z ∈ localizationDomain (γ sample) t ∧
+        loewnerRadius (g sample) t z = (2*z.im)*Real.exp (-u)} = ∅ :=
+      Set.eq_empty_iff_forall_notMem.2 (fun t ht => hn ⟨t,ht⟩)
+    exact hhit (by simp only [firstLiveRadiusStop,he,Set.image_empty,sInf_empty])
+  obtain ⟨t,hzt,hrt⟩ := hex
+  have htau := hm.firstLiveRadiusStop_eq ((continuous_const.mul
+    (hc.comp continuous_real_toNNReal)).continuousOn) hzt hrt
+  have hs : 0 < loewnerAngleSine κ (fun t => B t sample) (g sample) t z := by
+    have hh := div_pos (hm.im_pos t.coe_nonneg hzt) (norm_pos_iff.2 (hm.centered_ne_zero t.coe_nonneg hzt))
+    simpa only [loewnerAngleSine, Complex.ofReal_mul] using hh
+  have hr : 0 < ((2*z.im)*Real.exp (-u))^(-codimension κ) := by
+    apply Real.rpow_pos_of_pos
+    have hy : 0 < z.im := hz
+    positivity
+  simp only [finiteStoppedDensity,radiusStoppedInverseAngle,htau,ENNReal.coe_ne_top,
+    ite_false,ENNReal.coe_toReal,hrt,Set.mem_ofPred_eq,Ne,not_false_eq_true,
+    Set.indicator_of_mem]
+  rw [← ENNReal.ofReal_mul (mul_pos hr (Real.rpow_pos_of_pos hs _)).le,
+    mul_assoc, ← Real.rpow_add hs, add_neg_cancel, Real.rpow_zero,mul_one]
+
+lemma radius_green_cancel (κ u : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
+    (((2*z.im)*Real.exp (-u))^(-codimension κ) /
+      ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))) *
+      (Real.exp (-codimension κ*u)*(z.im/‖z‖)^(8/κ-1)) = 1 := by
+  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
+  have hs := Real.rpow_pos_of_pos (initial_sine_pos hz) (8/κ-1)
+  have hp := Real.rpow_pos_of_pos hy (-codimension κ)
+  rw [Real.mul_rpow hy.le (Real.exp_pos _).le, ← Real.exp_mul]
+  have he : Real.exp (-u * -codimension κ) * Real.exp (-codimension κ*u) = 1 := by
+    rw [← Real.exp_add, show -u * -codimension κ + -codimension κ*u = 0 by ring, Real.exp_zero]
+  field_simp
+  convert he using 1
+  congr 2 <;> ring
+
+theorem finiteCenteredInverseAngleBound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (u₀ : ℝ) (hu₀ : 0 < u₀) :
+    ∃ C : ℝ, 0 < C ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (_hz : z ∈ halfPlane) (u : ℝ) (_hu : u₀ ≤ u),
+      IsProbabilityMeasure (finiteCenteredMeasure κ P B γ g z u) ∧
+      (∫⁻ sample, radiusStoppedInverseAngle κ B γ g z u sample
+        ∂finiteCenteredMeasure κ P B γ g z u) ≤ ENNReal.ofReal C := by
+  obtain ⟨C,hC,hbound⟩ := radiusApproachBoundFrom κ hκ u₀ hu₀
+  refine ⟨C,hC,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz u hu
+  have hu0 : 0 < u := hu₀.trans_le hu
+  refine ⟨finiteCenteredMeasure_probability κ hκ P B hB γ g f hγ hmaps z hz u hu0, ?_⟩
+  let M₀ := ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))
+  let K := ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-codimension κ)) / M₀
+  let H : Set Ω := {sample | firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞)}
+  have hmpos := localizationInitialGreen_pos κ hz
+  have hm0 : M₀ ≠ 0 := (ENNReal.ofReal_pos.2 hmpos).ne'
+  have hd := finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0
+  have hdm : AEMeasurable (fun sample => finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z u / M₀) P := hd.div_const M₀
+  have hdf : ∀ᵐ sample ∂P, finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z u / M₀ < (⊤ : ℝ≥0∞) := Filter.Eventually.of_forall
+    (fun sample => ENNReal.div_lt_top (finiteStoppedDensity_lt_top κ _ _ _ z u).ne hm0)
+  have he : (fun sample => (finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z u / M₀) * radiusStoppedInverseAngle κ B γ g z u sample) =ᵐ[P]
+      H.indicator (fun _ => K) := by
+    filter_upwards [hmaps,hB.cont] with sample hm hc
+    have hh := finiteStoppedDensity_mul_inverseAngle κ B γ g f z hz u sample hm hc
+    rw [show (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u / M₀) *
+      radiusStoppedInverseAngle κ B γ g z u sample = (finiteStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z u * radiusStoppedInverseAngle κ B γ g z u sample) / M₀ by
+        simp only [div_eq_mul_inv]; ac_rfl, hh]
+    classical
+    by_cases hhit : sample ∈ H
+    · simp only [H] at hhit
+      simp only [H, Set.indicator_of_mem hhit, K]
+    · simp only [H] at hhit
+      simp only [H, Set.indicator_of_notMem hhit, ENNReal.zero_div]
+  change (∫⁻ sample, radiusStoppedInverseAngle κ B γ g z u sample ∂P.withDensity
+    (fun sample => finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u / M₀)) ≤ _
+  rw [lintegral_withDensity_eq_lintegral_mul_non_measurable₀ P hdm hdf]
+  simp only [Pi.mul_apply]
+  rw [lintegral_congr_ae he]
+  calc
+    (∫⁻ sample, H.indicator (fun _ => K) sample ∂P) ≤ K*P H := lintegral_indicator_const_le H K
+    _ ≤ K * ENNReal.ofReal (C * Real.exp (-codimension κ*u)*(z.im/‖z‖)^(8/κ-1)) :=
+      mul_le_mul' le_rfl (hbound P B hB γ g f hγ hmaps z hz u hu)
+    _ = ENNReal.ofReal C := by
+      dsimp only [K,M₀]
+      have hy : 0 < z.im := hz
+      have hratio : 0 ≤ ((2*z.im)*Real.exp (-u))^(-codimension κ) /
+          ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) :=
+        div_nonneg (Real.rpow_nonneg (by positivity) _) hmpos.le
+      rw [← ENNReal.ofReal_div_of_pos hmpos, ← ENNReal.ofReal_mul hratio]
+      congr 1
+      calc
+        _ = C * (((2*z.im)*Real.exp (-u))^(-codimension κ) /
+            ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) *
+            (Real.exp (-codimension κ*u)*(z.im/‖z‖)^(8/κ-1))) := by ring
+        _ = C := by rw [radius_green_cancel κ u hz,mul_one]
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.GammaField
+open LoewnerCutoff LoewnerField LoewnerExit AngularBarrier GreenPhysical
+
+lemma profile_green_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ K : ℝ, 0 < K ∧ ∀ x y l t ρ : ℝ, 0 < y → 0 < ρ → 1 ≤ t →
+      Real.log (2*y)-l-Real.log ρ ≤ t →
+      profile (κ/2) ((8/κ-1)/2) t (x/y) ≤
+        K*ρ^(codimension κ)*green (1-κ/8) (8/κ-1) x y l := by
+  have hl : 0 < κ/2 := by linarith [hκ.1]
+  have hb : 0 < (8/κ-1)/2 := by
+    have hh : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by simpa using hκ.2)
+    linarith
+  obtain ⟨K,hK,hdec⟩ := value_decay_from hl hb (by norm_num : (0:ℝ)<1)
+  refine ⟨K,hK,?_⟩
+  intro x y l t ρ hy hρ ht hr
+  have hp : (κ/2)*((8/κ-1)/2)/2 = codimension κ := by
+    unfold codimension dimension
+    field_simp [hκ.1.ne']
+    ring
+  have hp' : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  have hp0 : 0 ≤ codimension κ := by rw [hp']; linarith [hκ.2]
+  have hs : 0 < Real.sqrt (x^2+y^2) := Real.sqrt_pos.2 (by positivity)
+  have hq0 : squaredSine (x/y) = (y/Real.sqrt (x^2+y^2))^2 := by
+    have hsq := Real.sq_sqrt (by positivity : 0 ≤ x^2+y^2)
+    unfold squaredSine
+    field_simp
+    nlinarith
+  have hq : (squaredSine (x/y))^((8/κ-1)/2) = (y/Real.sqrt (x^2+y^2))^(8/κ-1) := by
+    rw [hq0, ← Real.rpow_natCast (y/Real.sqrt (x^2+y^2)) 2,
+      ← Real.rpow_mul (div_pos hy hs).le]
+    congr 1
+    ring
+  have he : Real.exp (-(codimension κ)*t) ≤
+      ρ^(codimension κ)*(2*y/Real.exp l)^(-(1-κ/8)) := by
+    rw [Real.rpow_def_of_pos hρ,Real.rpow_def_of_pos (div_pos (by positivity) (Real.exp_pos _)),
+      ← Real.exp_add]
+    apply Real.exp_le_exp.2
+    rw [Real.log_div (by positivity) (Real.exp_ne_zero _),Real.log_exp,← hp']
+    nlinarith [mul_le_mul_of_nonneg_left hr hp0]
+  have hh := hdec t (squaredSine (x/y)) ht (squaredSine_pos _).le
+  rw [hp,hq] at hh
+  calc
+    _ ≤ K*Real.exp (-(codimension κ)*t)*(y/Real.sqrt (x^2+y^2))^(8/κ-1) := hh
+    _ ≤ K*(ρ^(codimension κ)*(2*y/Real.exp l)^(-(1-κ/8)))*(y/Real.sqrt (x^2+y^2))^(8/κ-1) :=
+      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left he hK.le) (Real.rpow_nonneg (div_pos hy hs).le _)
+    _ = _ := by rw [green_eq_source _ _ _ _ _ hy]; ring
+
+end SLEExactGauge.GammaField
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit GammaField
+
+lemma height_radius_at_stop {z : ℂ} {u : ℝ} (C : Cutoff z u) (hz : z ∈ halfPlane)
+    (hu : 0 ≤ u) {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
+    C.height ≤ (position C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z (C.stopTime κ B sample)).im ∧
+    (2*z.im)*Real.exp (-u) ≤ radius C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z (C.stopTime κ B sample) := by
+  rcases eq_or_lt_of_le hu with rfl | hu'
+  · rw [C.stopTime_zero,position_initial]
+    constructor
+    · exact (C.height_initial hz le_rfl).le
+    · simp only [radius,position_initial,logDerivative_initial,neg_zero,Real.exp_zero,mul_one,div_one,le_refl]
+  · have hd := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+      (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle
+        (C.initial_deficit_brownian κ B sample hz hu' hc hb0) ⟨(C.bounds κ B sample).1,le_rfl⟩)
+    exact ⟨hd.2.1,hd.1⟩
+
+lemma targetValue_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ K : ℝ, 0 < K ∧ ∀ {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ}
+      (_hu : 0 ≤ u) (huv : u+1 ≤ v) (C : Cutoff z v) {Ω : Type*}
+      (B : ℝ≥0 → Ω → ℝ) (sample : Ω) (η : ℝ), 0 < η →
+      Continuous (fun t => B t sample) → B 0 sample = 0 →
+      C.targetValue hz (by linarith : u ≤ v) η κ B sample ≤
+        K*((2*z.im)*Real.exp (-v))^(codimension κ) *
+          (C.retarget hz (by linarith : u ≤ v)).value κ B sample := by
+  obtain ⟨K,hK,hbound⟩ := profile_green_bound κ hκ
+  refine ⟨K,hK,?_⟩
+  intro z hz u v hu huv C Ω B sample η hη hc hb0
+  let D := C.retarget hz (by linarith : u ≤ v)
+  let τ := D.stopTime κ B sample
+  let U := C.paths κ B sample
+  have hd := D.height_radius_at_stop hz hu κ B sample hc hb0
+  have hy : C.height ≤ (position C.height_pos C.horizon.coe_nonneg U z τ).im := hd.1
+  have hR : (2*z.im)*Real.exp (-u) ≤ radius C.height_pos C.horizon.coe_nonneg U z τ := hd.2
+  have hz0 : 0 < z.im := hz
+  have hρu : 0 < (2*z.im)*Real.exp (-u) := by positivity
+  have hρv : 0 < (2*z.im)*Real.exp (-v) := by positivity
+  have hlog := Real.log_le_log hρu hR
+  have hlρ (w : ℝ) : Real.log ((2*z.im)*Real.exp (-w)) = Real.log (2*z.im)-w := by
+    rw [Real.log_mul (by positivity) (Real.exp_ne_zero _),Real.log_exp]; ring
+  have hr0 := le_max_right η (rawRemaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ)
+  change rawRemaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ ≤
+    remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ at hr0
+  have hr1 : 1 ≤ remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ := by
+    rw [rawRemaining_eq C.height_pos C.horizon.coe_nonneg _ _ _ _ _ hy,hlρ] at hr0
+    rw [hlρ] at hlog
+    exact le_trans (by linarith : 1 ≤ η+Real.log (radius C.height_pos C.horizon.coe_nonneg U z τ)-(Real.log (2*z.im)-v)) hr0
+  have hr2 : Real.log (2*LoewnerField.height C.height_pos C.horizon.coe_nonneg U z τ)-
+      logDerivative C.height_pos C.horizon.coe_nonneg U z τ-Real.log ((2*z.im)*Real.exp (-v)) ≤
+      remaining C.height_pos C.horizon.coe_nonneg η ((2*z.im)*Real.exp (-v)) U z τ := by
+    unfold rawRemaining at hr0
+    linarith
+  exact hbound _ _ _ _ _ (LoewnerField.height_pos _ _ _ _ _) hρv hr1 hr2
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/ConformalKernels.lean b/lean/OAI/Probability/SLELocalization/ConformalKernels.lean
index ca3650d3fcbd4f2a2a5c0d776fcbd018beb2277f..d1c964968f745dde3ba4be14cebe2e3b767e437d 100644
--- a/lean/OAI/Probability/SLELocalization/ConformalKernels.lean
+++ b/lean/OAI/Probability/SLELocalization/ConformalKernels.lean
@@ -1,973 +1,973 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-
-namespace OAI
-
-
-/-! Conformal localization and Layer estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology RealInnerProductSpace
-namespace SLEExactGauge.Layer
-
-lemma logFactor_laplacian {z : ℂ} (hz : ‖z‖ < 1) :
-    Laplacian.laplacian (fun z => Real.log (diskFactor z)) z = -4/(diskFactor z)^2 := by
-  have ha := diskFactor_pos hz
-  have hl : ContDiffAt ℝ 2 Real.log (diskFactor z) := Real.contDiffAt_log.mpr (ne_of_gt ha)
-  have hh := laplacian_scalar_comp diskFactor_smooth.contDiffAt hl
-  have hlogd : deriv Real.log = fun t => t⁻¹ := funext Real.deriv_log
-  rw [diskFactor_laplacian,(diskFactor_gradient z).1,(diskFactor_gradient z).2,
-    hlogd] at hh
-  have he : deriv (fun t : ℝ => t⁻¹) (diskFactor z) = -((diskFactor z)^2)⁻¹ := by
-    simp
-  rw [he] at hh
-  change Laplacian.laplacian (fun z => Real.log (diskFactor z)) z = _ at hh
-  rw [hh]
-  have hn : ‖z‖^2 = z.re^2+z.im^2 := by rw [Complex.sq_norm,Complex.normSq_apply]; ring
-  dsimp [diskFactor] at ha ⊢
-  rw [hn] at ha ⊢
-  field_simp [ne_of_gt ha]
-  ring
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology RealInnerProductSpace
-namespace SLEExactGauge.Layer
-
-lemma logNorm_fderiv {H : ℂ → ℂ} {z D : ℂ} (hd : HasDerivAt H D z)
-    (hn : H z ≠ 0) (v : ℂ) :
-    fderiv ℝ (fun w => Real.log ‖H w‖) z v = (D*v/H z).re := by
-  have hn2 : ‖H z‖^2 ≠ 0 := pow_ne_zero 2 (norm_ne_zero_iff.2 hn)
-  have hh := ((hd.hasFDerivAt.restrictScalars ℝ).norm_sq.log hn2).const_smul (1/2:ℝ)
-  have he : (fun w => (1/2:ℝ) • Real.log (‖H w‖^2)) = (fun w => Real.log ‖H w‖) := by
-    funext w
-    rw [Real.log_pow]
-    simp only [smul_eq_mul]
-    ring
-  change HasFDerivAt (fun w => (1/2:ℝ) • Real.log (‖H w‖^2)) _ z at hh
-  rw [he] at hh
-  rw [hh.fderiv]
-  simp only [smul_apply,smul_eq_mul,two_smul]
-  change 1/2*((‖H z‖^2)⁻¹*(inner ℝ (H z) (v*D)+inner ℝ (H z) (v*D))) = (D*v/H z).re
-  rw [Complex.inner,Complex.div_re]
-  have heq : Complex.normSq (H z) = ‖H z‖^2 := (Complex.sq_norm _).symm
-  rw [heq]
-  simp only [Complex.mul_re,Complex.mul_im,Complex.conj_re,Complex.conj_im]
-  field_simp
-  ring
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-structure DiskPair (U : Set ℂ) (F G : ℂ → ℂ) : Prop where
-  isOpen : IsOpen U
-  toDomain : MapsTo F (ball 0 1) U
-  toDisk : MapsTo G U (ball 0 1)
-  left_inv : ∀ z ∈ ball (0:ℂ) 1, G (F z) = z
-  right_inv : ∀ z ∈ U, F (G z) = z
-  forward : DifferentiableOn ℂ F (ball 0 1)
-  inverse : DifferentiableOn ℂ G U
-
-lemma DiskPair.injOn {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G) :
-    InjOn F (ball 0 1) := by
-  intro z hz w hw he
-  simpa only [h.left_inv z hz,h.left_inv w hw] using congrArg G he
-
-lemma DiskPair.inverse_derivative {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
-    {z : ℂ} (hz : z ∈ ball (0:ℂ) 1) : deriv G (F z)*deriv F z = 1 := by
-  have he : (G ∘ F) =ᶠ[𝓝 z] id := by
-    filter_upwards [isOpen_ball.mem_nhds hz] with w hw
-    exact h.left_inv w hw
-  exact (((h.inverse.differentiableAt (h.isOpen.mem_nhds (h.toDomain hz))).hasDerivAt.comp z
-    (h.forward.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt).congr_of_eventuallyEq he.symm).unique
-      (hasDerivAt_id z)
-
-lemma DiskPair.deriv_ne_zero {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
-    {z : ℂ} (hz : z ∈ ball (0:ℂ) 1) : deriv F z ≠ 0 := by
-  intro he
-  simpa [he] using h.inverse_derivative hz
-
-lemma DiskPair.second_derivative_zero {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G) :
-    ‖deriv (deriv F) 0‖ ≤ (2*(512*Real.pi)^2)*‖deriv F 0‖ := by
-  have hzero : (0:ℂ) ∈ ball 0 1 := mem_ball_self zero_lt_one
-  have hA : 0 < ‖deriv F 0‖ := norm_pos_iff.2 (h.deriv_ne_zero hzero)
-  let R : ℝ := ‖deriv F 0‖/(512*Real.pi)
-  have hR : 0 < R := div_pos hA (by positivity)
-  have hsub : closedBall (F 0) R ⊆ U := by
-    intro w hw
-    by_contra hwu
-    have hwi : w ∉ F '' ball 0 1 := by
-      rintro ⟨z,hz,rfl⟩
-      exact hwu (h.toDomain hz)
-    have hh := UnivalentDistance.omitted_value_bound h.forward h.injOn hwi
-    have hnorm : ‖F 0-w‖ ≤ R := by
-      rw [norm_sub_rev,← dist_eq_norm]
-      exact mem_closedBall.1 hw
-    have hbound := hh.trans (mul_le_mul_of_nonneg_left hnorm (by positivity : 0 ≤ 256*Real.pi))
-    have he : (256*Real.pi)*R = ‖deriv F 0‖/2 := by
-      dsimp [R]; field_simp; ring
-    rw [he] at hbound
-    linarith
-  have hb : ‖deriv (deriv G) (F 0)‖ ≤ 2/R^2 := by
-    have hh := Complex.norm_iteratedDeriv_le_of_forall_mem_sphere_norm_le 2 hR
-      ((h.inverse.mono hsub).diffContOnCl_ball Subset.rfl)
-      (fun z hz => (mem_ball_zero_iff.1 (h.toDisk (hsub (sphere_subset_closedBall hz)))).le)
-    simpa [iteratedDeriv_succ,iteratedDeriv_zero] using hh
-  have hFd := h.forward.differentiableAt (isOpen_ball.mem_nhds hzero)
-  have hFdd := (h.forward.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hzero)
-  have hGdd := (h.inverse.deriv h.isOpen).differentiableAt (h.isOpen.mem_nhds (h.toDomain hzero))
-  have he : (fun z => deriv G (F z)*deriv F z) =ᶠ[𝓝 (0:ℂ)] fun _ => 1 := by
-    filter_upwards [isOpen_ball.mem_nhds hzero] with z hz
-    exact h.inverse_derivative hz
-  have hh := (((hGdd.hasDerivAt.comp 0 hFd.hasDerivAt).mul hFdd.hasDerivAt).congr_of_eventuallyEq
-    he.symm).unique (hasDerivAt_const (0:ℂ) (1:ℂ))
-  simp only [Function.comp_apply] at hh
-  have hh0 := h.inverse_derivative hzero
-  have hrel : deriv (deriv F) 0 = -deriv (deriv G) (F 0)*(deriv F 0)^3 := by
-    linear_combination (deriv F 0)*hh - (deriv (deriv F) 0)*hh0
-  rw [hrel,norm_mul,norm_neg,norm_pow]
-  calc
-    _ ≤ (2/R^2)*‖deriv F 0‖^3 := mul_le_mul_of_nonneg_right hb (by positivity)
-    _ = (2*(512*Real.pi)^2)*‖deriv F 0‖ := by
-      dsimp [R]
-      field_simp
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.SlitMaps
-
-def diskAuto (c z : ℂ) : ℂ := (z-c)/(1-starRingEnd ℂ c*z)
-
-lemma diskAuto_den_ne {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
-    1-starRingEnd ℂ c*z ≠ 0 := by
-  have hn : ‖starRingEnd ℂ c*z‖ < 1 := by
-    rw [norm_mul, Complex.norm_conj]
-    calc
-      ‖c‖*‖z‖ ≤ ‖c‖*1 := mul_le_mul_of_nonneg_left hz.le (norm_nonneg _)
-      _ < 1 := by simpa using hc
-  intro he
-  have hh : starRingEnd ℂ c*z = 1 := (sub_eq_zero.1 he).symm
-  simp [hh] at hn
-
-lemma disk_norm_identity (c z : ℂ) :
-    ‖1-starRingEnd ℂ c*z‖^2-‖z-c‖^2 = (1-‖c‖^2)*(1-‖z‖^2) := by
-  simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im,
-    Complex.one_re, Complex.one_im, Complex.mul_re, Complex.mul_im,
-    Complex.conj_re, Complex.conj_im]
-  ring
-
-lemma diskAuto_mapsTo {c : ℂ} (hc : ‖c‖ < 1) : MapsTo (diskAuto c) (ball 0 1) (ball 0 1) := by
-  intro z hz
-  rw [mem_ball_zero_iff] at hz ⊢
-  have hn := diskAuto_den_ne hc hz
-  rw [diskAuto,norm_div,div_lt_one₀ (norm_pos_iff.2 hn)]
-  have hp : 0 < (1-‖c‖^2)*(1-‖z‖^2) := mul_pos (by nlinarith [norm_nonneg c]) (by nlinarith [norm_nonneg z])
-  have he := disk_norm_identity c z
-  nlinarith [norm_nonneg (z-c), norm_nonneg (1-starRingEnd ℂ c*z)]
-
-lemma diskAuto_hasDerivAt {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
-    HasDerivAt (diskAuto c) ((1-starRingEnd ℂ c*c)/(1-starRingEnd ℂ c*z)^2) z := by
-  have hd := ((hasDerivAt_id z).sub_const c).div
-    ((hasDerivAt_const z (1:ℂ)).sub ((hasDerivAt_id z).const_mul (starRingEnd ℂ c)))
-    (diskAuto_den_ne hc hz)
-  change HasDerivAt (diskAuto c)
-    ((1*(1-starRingEnd ℂ c*z)-(z-c)*(0-starRingEnd ℂ c*1))/(1-starRingEnd ℂ c*z)^2) z at hd
-  have he : 1*(1-starRingEnd ℂ c*z)-(z-c)*(0-starRingEnd ℂ c*1) = 1-starRingEnd ℂ c*c := by ring
-  rwa [he] at hd
-
-lemma diskAuto_deriv {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
-    deriv (diskAuto c) z = (1-starRingEnd ℂ c*c)/(1-starRingEnd ℂ c*z)^2 :=
-  (diskAuto_hasDerivAt hc hz).deriv
-
-lemma diskAuto_deriv_ne {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
-    deriv (diskAuto c) z ≠ 0 := by
-  rw [diskAuto_deriv hc hz]
-  exact div_ne_zero (diskAuto_den_ne hc hc) (pow_ne_zero _ (diskAuto_den_ne hc hz))
-
-lemma diskAuto_injOn {c : ℂ} (hc : ‖c‖ < 1) : InjOn (diskAuto c) (ball 0 1) := by
-  intro z hz w hw he
-  have hzn := diskAuto_den_ne hc (mem_ball_zero_iff.1 hz)
-  have hwn := diskAuto_den_ne hc (mem_ball_zero_iff.1 hw)
-  have he' := (div_eq_div_iff hzn hwn).1 he
-  have hn := diskAuto_den_ne hc hc
-  have hh : (z-w)*(1-starRingEnd ℂ c*c) = 0 := by linear_combination he'
-  exact sub_eq_zero.1 ((mul_eq_zero.1 hh).resolve_right hn)
-
-@[simp] lemma diskAuto_self (c : ℂ) : diskAuto c c = 0 := by simp [diskAuto]
-
-@[simp] lemma diskAuto_zero (c : ℂ) : diskAuto c 0 = -c := by simp [diskAuto]
-
-lemma diskAuto_norm_deriv_zero {c : ℂ} (hc : ‖c‖ < 1) :
-    ‖deriv (diskAuto c) 0‖ = 1-‖c‖^2 := by
-  rw [diskAuto_deriv hc (by simp)]
-  simp only [mul_zero,sub_zero,one_pow,div_one]
-  rw [mul_comm (starRingEnd ℂ c),Complex.mul_conj']
-  have he : (1:ℂ)-(‖c‖:ℂ)^2 = ((1-‖c‖^2:ℝ):ℂ) := by push_cast; rfl
-  rw [he,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (by nlinarith [norm_nonneg c])]
-
-lemma diskAuto_norm_deriv_self {c : ℂ} (hc : ‖c‖ < 1) :
-    ‖deriv (diskAuto c) c‖ = 1/(1-‖c‖^2) := by
-  have hh : (1-starRingEnd ℂ c*c)/(1-starRingEnd ℂ c*c)^2 = (1-starRingEnd ℂ c*c)⁻¹ := by
-    field_simp [diskAuto_den_ne hc hc]
-  rw [diskAuto_deriv hc hc,hh]
-  rw [norm_inv]
-  have he : ‖1-starRingEnd ℂ c*c‖ = 1-‖c‖^2 := by
-    have hh := diskAuto_norm_deriv_zero hc
-    rw [diskAuto_deriv hc (by simp)] at hh
-    simpa using hh
-  simp [he,one_div]
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-namespace SLEExactGauge.Layer
-open SlitMaps
-
-lemma diskAuto_neg_inverse {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
-    diskAuto c (diskAuto (-c) z) = z := by
-  have hnc : ‖-c‖ < 1 := by simpa using hc
-  have hφ := diskAuto_mapsTo hnc (mem_ball_zero_iff.2 hz)
-  have hd1 := diskAuto_den_ne hnc hz
-  have hd2 := diskAuto_den_ne hc (mem_ball_zero_iff.1 hφ)
-  have ha := diskAuto_den_ne hc hc
-  dsimp [diskAuto] at hd1 hd2 ⊢
-  simp only [map_neg,neg_mul,sub_neg_eq_add] at hd1 hd2 ⊢
-  apply (div_eq_iff hd2).2
-  have he := div_mul_cancel₀ (z+c) hd1
-  linear_combination he
-
-lemma DiskPair.recenter {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
-    {c : ℂ} (hc : ‖c‖ < 1) :
-    DiskPair U (F ∘ diskAuto (-c)) (diskAuto c ∘ G) := by
-  have hnc : ‖-c‖ < 1 := by simpa using hc
-  refine ⟨h.isOpen,h.toDomain.comp (diskAuto_mapsTo hnc),(diskAuto_mapsTo hc).comp h.toDisk,?_,?_,?_,?_⟩
-  · intro z hz
-    dsimp only [Function.comp_apply]
-    rw [h.left_inv _ (diskAuto_mapsTo hnc hz)]
-    exact diskAuto_neg_inverse hc (mem_ball_zero_iff.1 hz)
-  · intro z hz
-    dsimp only [Function.comp_apply]
-    have he := diskAuto_neg_inverse hnc (mem_ball_zero_iff.1 (h.toDisk hz))
-    simp only [neg_neg] at he
-    rw [he,h.right_inv z hz]
-  · exact h.forward.comp (fun z hz => (diskAuto_hasDerivAt hnc (mem_ball_zero_iff.1 hz)).differentiableAt.differentiableWithinAt)
-      (diskAuto_mapsTo hnc)
-  · have ha : DifferentiableOn ℂ (diskAuto c) (ball 0 1) := fun z hz =>
-      (diskAuto_hasDerivAt hc (mem_ball_zero_iff.1 hz)).differentiableAt.differentiableWithinAt
-    exact ha.comp h.inverse h.toDisk
-
-lemma diskAuto_deriv_zero_exact {c : ℂ} (hc : ‖c‖ < 1) :
-    deriv (diskAuto (-c)) 0 = 1-(starRingEnd ℂ c)*c := by
-  rw [diskAuto_deriv (by simpa using hc) (by simp)]
-  simp
-
-lemma diskAuto_second_zero {c : ℂ} (hc : ‖c‖ < 1) :
-    deriv (deriv (diskAuto (-c))) 0 = -2*(starRingEnd ℂ c)*(1-(starRingEnd ℂ c)*c) := by
-  have hnc : ‖-c‖ < 1 := by simpa using hc
-  have he : deriv (diskAuto (-c)) =ᶠ[𝓝 (0:ℂ)]
-      fun z => (1-(starRingEnd ℂ c)*c)/(1+(starRingEnd ℂ c)*z)^2 := by
-    filter_upwards [Metric.ball_mem_nhds (0:ℂ) zero_lt_one] with z hz
-    rw [diskAuto_deriv hnc (mem_ball_zero_iff.1 hz)]
-    simp
-  rw [he.deriv_eq]
-  have hd := (hasDerivAt_const (0:ℂ) (1-(starRingEnd ℂ c)*c)).div
-    ((((hasDerivAt_id (0:ℂ)).const_mul (starRingEnd ℂ c)).const_add 1).pow 2) (by simp)
-  have hh := hd.deriv
-  dsimp only [Pi.div_apply,Pi.pow_apply,id_eq] at hh
-  change deriv ((fun x : ℂ => 1-(starRingEnd ℂ c)*c) / (fun x : ℂ => 1+(starRingEnd ℂ c)*x)^2) 0 = _
-  rw [hh]
-  simp only [mul_zero,add_zero,one_pow,zero_sub,div_one,mul_one]
-  norm_num
-  ring
-
-lemma DiskPair.preSchwarzian {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
-    {c : ℂ} (hc : ‖c‖ < 1) :
-    ‖((1-‖c‖^2:ℝ):ℂ)*deriv (deriv F) c - 2*(starRingEnd ℂ c)*deriv F c‖ ≤
-      (2*(512*Real.pi)^2)*‖deriv F c‖ := by
-  let φ := diskAuto (-c)
-  let A : ℂ := 1-(starRingEnd ℂ c)*c
-  have hnc : ‖-c‖ < 1 := by simpa using hc
-  have hcD : c ∈ ball (0:ℂ) 1 := mem_ball_zero_iff.2 hc
-  have hφ0 : φ 0 = c := by simp [φ]
-  have hdφ : deriv φ 0 = A := diskAuto_deriv_zero_exact hc
-  have hddφ : deriv (deriv φ) 0 = -2*(starRingEnd ℂ c)*A := diskAuto_second_zero hc
-  have hφd : DifferentiableOn ℂ φ (ball 0 1) := fun z hz =>
-    (diskAuto_hasDerivAt hnc (mem_ball_zero_iff.1 hz)).differentiableAt.differentiableWithinAt
-  have hFd := h.forward.differentiableAt (isOpen_ball.mem_nhds hcD)
-  have hFdd := (h.forward.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hcD)
-  have hφd0 := hφd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self zero_lt_one))
-  have hφdd0 := (hφd.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds (mem_ball_self zero_lt_one))
-  have hd : deriv (F ∘ φ) 0 = deriv F c*A := by
-    have hFd' : HasDerivAt F (deriv F c) (φ 0) := hφ0.symm ▸ hFd.hasDerivAt
-    have hh := hFd'.comp (0:ℂ) hφd0.hasDerivAt
-    simpa only [hdφ] using hh.deriv
-  have he : deriv (F ∘ φ) =ᶠ[𝓝 (0:ℂ)] fun z => deriv F (φ z)*deriv φ z := by
-    filter_upwards [Metric.ball_mem_nhds (0:ℂ) zero_lt_one] with z hz
-    exact ((h.forward.differentiableAt (isOpen_ball.mem_nhds (diskAuto_mapsTo hnc hz))).hasDerivAt.comp z
-      (hφd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt).deriv
-  have hdd : deriv (deriv (F ∘ φ)) 0 = A*(A*deriv (deriv F) c-2*(starRingEnd ℂ c)*deriv F c) := by
-    rw [he.deriv_eq]
-    have hFdd' : HasDerivAt (deriv F) (deriv (deriv F) c) (φ 0) := hφ0.symm ▸ hFdd.hasDerivAt
-    have hh := ((hFdd'.comp 0 hφd0.hasDerivAt).mul hφdd0.hasDerivAt).deriv
-    dsimp only [Pi.mul_apply,Function.comp_apply] at hh
-    simp only [hφ0,hdφ,hddφ] at hh
-    change deriv ((deriv F ∘ φ)*deriv φ) 0 = _
-    rw [hh]
-    ring
-  have hAeq : A = ((1-‖c‖^2:ℝ):ℂ) := by
-    dsimp only [A]
-    rw [mul_comm (starRingEnd ℂ c),Complex.mul_conj']
-    push_cast
-    rfl
-  have hAp : 0 < ‖A‖ := by
-    rw [hAeq,Complex.norm_real,Real.norm_eq_abs]
-    exact abs_pos.2 (by nlinarith [norm_nonneg c] : 1-‖c‖^2 ≠ 0)
-  have hb := (h.recenter hc).second_derivative_zero
-  change ‖deriv (deriv (F ∘ φ)) 0‖ ≤ (2*(512*Real.pi)^2)*‖deriv (F ∘ φ) 0‖ at hb
-  rw [hdd,hd,norm_mul,norm_mul] at hb
-  have hh : ‖A*deriv (deriv F) c-2*(starRingEnd ℂ c)*deriv F c‖ ≤ (2*(512*Real.pi)^2)*‖deriv F c‖ := by
-    nlinarith
-  rwa [hAeq] at hh
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-def logRadius (F : ℂ → ℂ) (z : ℂ) : ℝ :=
-  Real.log (diskFactor z) + Real.log ‖deriv F z‖
-
-lemma DiskPair.logRadius_smooth {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
-    ContDiffAt ℝ 2 (logRadius F) z := by
-  have hh := ((h.forward.deriv isOpen_ball).analyticOnNhd isOpen_ball z
-    (mem_ball_zero_iff.2 hz)).harmonicAt_log_norm (h.deriv_ne_zero (mem_ball_zero_iff.2 hz))
-  exact (logFactor_smooth hz).add hh.1
-
-lemma DiskPair.logRadius_laplacian {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
-    Laplacian.laplacian (logRadius F) z = -4/(diskFactor z)^2 := by
-  have hh := ((h.forward.deriv isOpen_ball).analyticOnNhd isOpen_ball z
-    (mem_ball_zero_iff.2 hz)).harmonicAt_log_norm (h.deriv_ne_zero (mem_ball_zero_iff.2 hz))
-  have he := (logFactor_smooth hz).laplacian_add hh.1
-  change Laplacian.laplacian (logRadius F) z = _ at he
-  rw [he,logFactor_laplacian hz,hh.2.eq_of_nhds]
-  simp
-
-lemma DiskPair.logRadius_gradient {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
-    fderiv ℝ (logRadius F) z 1 = (deriv (deriv F) z/deriv F z).re - 2*z.re/diskFactor z ∧
-    fderiv ℝ (logRadius F) z Complex.I = -(deriv (deriv F) z/deriv F z).im - 2*z.im/diskFactor z := by
-  have ha := diskFactor_pos hz
-  have hdz := (h.forward.deriv isOpen_ball).differentiableAt
-    (isOpen_ball.mem_nhds (mem_ball_zero_iff.2 hz))
-  have hn := h.deriv_ne_zero (mem_ball_zero_iff.2 hz)
-  have hl := (diskFactor_smooth.differentiable (by norm_num) z).hasFDerivAt.log (ne_of_gt ha)
-  have hd : DifferentiableAt ℝ (fun z => Real.log ‖deriv F z‖) z :=
-    (((h.forward.deriv isOpen_ball).analyticOnNhd isOpen_ball z
-      (mem_ball_zero_iff.2 hz)).harmonicAt_log_norm hn).1.differentiableAt (by norm_num)
-  have he := (hl.add hd.hasFDerivAt).fderiv
-  change fderiv ℝ (logRadius F) z = _ at he
-  rw [he]
-  simp only [add_apply,smul_apply,smul_eq_mul,
-    (diskFactor_gradient z).1,(diskFactor_gradient z).2,
-    logNorm_fderiv hdz.hasDerivAt hn,mul_one]
-  constructor
-  · ring
-  · rw [mul_div_right_comm,Complex.mul_re]
-    simp
-    ring
-
-lemma DiskPair.logRadius_gradient_sq {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
-    (fderiv ℝ (logRadius F) z 1)^2 + (fderiv ℝ (logRadius F) z Complex.I)^2 ≤
-      (2*(512*Real.pi)^2)^2/(diskFactor z)^2 := by
-  let a := diskFactor z
-  let c := deriv (deriv F) z/deriv F z
-  let b : ℂ := (a:ℂ)*c - 2*(starRingEnd ℂ z)
-  have ha : 0 < a := diskFactor_pos hz
-  have hn := h.deriv_ne_zero (mem_ball_zero_iff.2 hz)
-  have he : b = ((a:ℂ)*deriv (deriv F) z - 2*(starRingEnd ℂ z)*deriv F z)/deriv F z := by
-    dsimp [b,c]
-    field_simp
-  have hb : ‖b‖ ≤ 2*(512*Real.pi)^2 := by
-    rw [he,norm_div]
-    apply (div_le_iff₀ (norm_pos_iff.2 hn)).2
-    exact h.preSchwarzian hz
-  have hi : ((fderiv ℝ (logRadius F) z 1)^2 +
-      (fderiv ℝ (logRadius F) z Complex.I)^2)*a^2 = ‖b‖^2 := by
-    rw [(h.logRadius_gradient hz).1,(h.logRadius_gradient hz).2,
-      Complex.sq_norm,Complex.normSq_apply]
-    change ((c.re-2*z.re/a)^2+(-c.im-2*z.im/a)^2)*a^2 = _
-    dsimp [b]
-    simp only [Complex.mul_re,Complex.mul_im,
-      Complex.ofReal_re,Complex.ofReal_im,
-      Complex.conj_re,Complex.conj_im,zero_mul,sub_zero,add_zero]
-    norm_num
-    field_simp
-    ring
-  apply (le_div_iff₀ (pow_pos ha 2)).2
-  rw [hi]
-  exact pow_le_pow_left₀ (norm_nonneg b) hb 2
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-def layerProfile (α t : ℝ) : ℝ := Real.exp (α*t)/(1+Real.exp (α*t))
-
-def layerProfileD (α t : ℝ) : ℝ := α*Real.exp (α*t)/(1+Real.exp (α*t))^2
-
-def layerProfileDD (α t : ℝ) : ℝ :=
-  α^2*Real.exp (α*t)*(1-Real.exp (α*t))/(1+Real.exp (α*t))^3
-
-lemma profile_smooth (α : ℝ) : ContDiff ℝ 2 (layerProfile α) := by
-  unfold layerProfile
-  apply ContDiff.div (by fun_prop) (by fun_prop)
-  intro x
-  positivity
-
-lemma profile_derivative (α t : ℝ) : HasDerivAt (layerProfile α) (layerProfileD α t) t := by
-  have he := ((hasDerivAt_id t).const_mul α).exp
-  have hh := he.div (he.const_add 1) (by positivity : 1+Real.exp (α*t) ≠ 0)
-  convert! hh using 1
-  dsimp [layerProfile,layerProfileD]
-  ring
-
-lemma profile_second (α t : ℝ) : HasDerivAt (layerProfileD α) (layerProfileDD α t) t := by
-  have he := ((hasDerivAt_id t).const_mul α).exp
-  have hh := (he.const_mul α).div ((he.const_add 1).pow 2)
-    (by positivity : (1+Real.exp (α*t))^2 ≠ 0)
-  convert! hh using 1
-  dsimp [layerProfileD,layerProfileDD]
-  field_simp
-  ring
-
-lemma profile_bounds (α t : ℝ) : 0 ≤ layerProfile α t ∧ layerProfile α t ≤ 1 := by
-  have he := Real.exp_pos (α*t)
-  dsimp [layerProfile]
-  exact ⟨by positivity,(div_le_one (by positivity)).2 (by linarith)⟩
-
-lemma profileD_nonneg {α : ℝ} (hα : 0 ≤ α) (t : ℝ) : 0 ≤ layerProfileD α t := by
-  dsimp [layerProfileD]
-  positivity
-
-lemma profileDD_le {α : ℝ} (_hα : 0 ≤ α) (t : ℝ) :
-    layerProfileDD α t ≤ α*layerProfileD α t := by
-  have he := Real.exp_pos (α*t)
-  have hd : 0 < 1+Real.exp (α*t) := by positivity
-  rw [layerProfileDD,layerProfileD,div_le_iff₀ (pow_pos hd 3)]
-  have hi : α*(α*Real.exp (α*t)/(1+Real.exp (α*t))^2)*(1+Real.exp (α*t))^3 =
-      α^2*Real.exp (α*t)*(1+Real.exp (α*t)) := by field_simp
-  rw [hi]
-  exact mul_le_mul_of_nonneg_left (by linarith : 1-Real.exp (α*t) ≤ 1+Real.exp (α*t)) (by positivity)
-
-lemma profileD_lower {α t : ℝ} (hα : 0 ≤ α) (hα₂ : α ≤ 2) (ht : t ≤ 0) :
-    α*Real.exp (2*t)/4 ≤ layerProfileD α t := by
-  have he := Real.exp_pos (α*t)
-  have he1 : Real.exp (α*t) ≤ 1 := Real.exp_le_one_iff.2 (mul_nonpos_of_nonneg_of_nonpos hα ht)
-  have he2 : Real.exp (2*t) ≤ Real.exp (α*t) := Real.exp_le_exp.2 (mul_le_mul_of_nonpos_right hα₂ ht)
-  have hden : (1+Real.exp (α*t))^2 ≤ 4 := by nlinarith
-  calc
-    _ ≤ α*Real.exp (α*t)/4 := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left he2 hα) (by norm_num)
-    _ ≤ _ := div_le_div_of_nonneg_left (by positivity) (by positivity) hden
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-def layerExponent : ℝ := 1/(1+(2*(512*Real.pi)^2)^2)
-
-lemma layerExponent_bounds : 0 < layerExponent ∧ layerExponent ≤ 2 ∧
-    layerExponent*(2*(512*Real.pi)^2)^2 ≤ 2 := by
-  have hL : 0 ≤ (2*(512*Real.pi)^2)^2 := sq_nonneg _
-  have hd : 0 < 1+(2*(512*Real.pi)^2)^2 := by positivity
-  dsimp [layerExponent]
-  refine ⟨by positivity,?_,?_⟩
-  · apply (div_le_iff₀ hd).2
-    nlinarith
-  · rw [div_mul_eq_mul_div,mul_comm (1:ℝ),mul_one]
-    apply (div_le_iff₀ hd).2
-    linarith
-
-def shiftedRadius (F : ℂ → ℂ) (R : ℝ) (z : ℂ) : ℝ := logRadius F z - Real.log R
-
-def layerBarrier (F : ℂ → ℂ) (R : ℝ) (z : ℂ) : ℝ :=
-  R^2 * layerProfile layerExponent (shiftedRadius F R z)
-
-lemma DiskPair.shifted_smooth {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
-    ContDiffAt ℝ 2 (shiftedRadius F R) z := (h.logRadius_smooth hz).sub contDiffAt_const
-
-lemma DiskPair.shifted_laplacian {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
-    Laplacian.laplacian (shiftedRadius F R) z = -4/(diskFactor z)^2 := by
-  have hh := (h.logRadius_smooth hz).laplacian_sub (contDiffAt_const (c := Real.log R))
-  change Laplacian.laplacian (shiftedRadius F R) z = _ at hh
-  rw [hh,h.logRadius_laplacian hz]
-  simp
-
-lemma shifted_fderiv (F : ℂ → ℂ) (R : ℝ) (z : ℂ) :
-    fderiv ℝ (shiftedRadius F R) z = fderiv ℝ (logRadius F) z := fderiv_sub_const _
-
-lemma DiskPair.barrier_smooth {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) (R : ℝ) : ContDiffOn ℝ 2 (layerBarrier F R) (ball 0 1) := by
-  intro z hz
-  exact (((profile_smooth layerExponent).contDiffAt.comp z
-    (h.shifted_smooth (mem_ball_zero_iff.1 hz) R)).const_smul (R^2)).contDiffWithinAt
-
-lemma barrier_bounds (F : ℂ → ℂ) (R : ℝ) (z : ℂ) :
-    0 ≤ layerBarrier F R z ∧ layerBarrier F R z ≤ R^2 := by
-  have hh := profile_bounds layerExponent (shiftedRadius F R z)
-  dsimp [layerBarrier]
-  exact ⟨mul_nonneg (sq_nonneg _) hh.1,by simpa using mul_le_mul_of_nonneg_left hh.2 (sq_nonneg R)⟩
-
-lemma profile_laplacian_estimate {u : ℂ → ℝ} {z : ℂ} {a α L : ℝ}
-    (hu : ContDiffAt ℝ 2 u z) (_ha : 0 < a) (hα : 0 ≤ α) (hαL : α*L ≤ 2)
-    (hΔ : Laplacian.laplacian u z = -4/a^2)
-    (hgrad : (fderiv ℝ u z 1)^2+(fderiv ℝ u z Complex.I)^2 ≤ L/a^2) :
-    Laplacian.laplacian (layerProfile α ∘ u) z ≤ -2*layerProfileD α (u z)/a^2 := by
-  rw [laplacian_scalar_comp hu (profile_smooth α).contDiffAt]
-  have hd : deriv (layerProfile α) = layerProfileD α := funext (fun t => (profile_derivative α t).deriv)
-  rw [hd,(profile_second α (u z)).deriv,hΔ]
-  have hD := profileD_nonneg hα (u z)
-  have hDD := profileDD_le hα (u z)
-  calc
-    _ ≤ (α*layerProfileD α (u z))*(L/a^2) + layerProfileD α (u z)*(-4/a^2) := by
-      apply add_le_add_left
-      exact (mul_le_mul_of_nonneg_right hDD (add_nonneg (sq_nonneg _) (sq_nonneg _))).trans
-        (mul_le_mul_of_nonneg_left hgrad (mul_nonneg hα hD))
-    _ = layerProfileD α (u z)*(α*L-4)/a^2 := by ring
-    _ ≤ _ := by
-      apply div_le_div_of_nonneg_right _ (sq_nonneg a)
-      nlinarith [mul_le_mul_of_nonneg_left hαL hD]
-
-lemma DiskPair.barrier_drop {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
-    Laplacian.laplacian (layerBarrier F R) z ≤
-      -2*R^2*layerProfileD layerExponent (shiftedRadius F R z)/(diskFactor z)^2 := by
-  have hs := h.shifted_smooth hz R
-  have hc := (profile_smooth layerExponent).contDiffAt.comp z hs
-  have hh := InnerProductSpace.laplacian_smul (R^2) hc
-  change Laplacian.laplacian (layerBarrier F R) z = R^2*Laplacian.laplacian (layerProfile layerExponent ∘ shiftedRadius F R) z at hh
-  rw [hh]
-  have hb := profile_laplacian_estimate hs (diskFactor_pos hz)
-    layerExponent_bounds.1.le layerExponent_bounds.2.2 (h.shifted_laplacian hz R)
-    (by rw [shifted_fderiv]; exact h.logRadius_gradient_sq hz)
-  convert! mul_le_mul_of_nonneg_left hb (sq_nonneg R) using 1
-  ring
-
-lemma DiskPair.barrier_superharmonic {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
-    Laplacian.laplacian (layerBarrier F R) z ≤ 0 := by
-  apply (h.barrier_drop hz R).trans
-  apply div_nonpos_of_nonpos_of_nonneg _ (sq_nonneg _)
-  have hh := profileD_nonneg layerExponent_bounds.1.le (shiftedRadius F R z)
-  nlinarith [sq_nonneg R]
-
-lemma DiskPair.shifted_exp {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) {R : ℝ} (hR : 0 < R) :
-    Real.exp (shiftedRadius F R z) = diskFactor z * ‖deriv F z‖/R := by
-  have ha := diskFactor_pos hz
-  have hn := norm_pos_iff.2 (h.deriv_ne_zero (mem_ball_zero_iff.2 hz))
-  dsimp [shiftedRadius,logRadius]
-  rw [Real.exp_sub,Real.exp_add,Real.exp_log ha,Real.exp_log hn,Real.exp_log hR]
-
-lemma DiskPair.barrier_layer {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) {R : ℝ} (hR : 0 < R)
-    (hl : diskFactor z * ‖deriv F z‖ ≤ R) :
-    layerExponent/2*‖deriv F z‖^2 ≤ -Laplacian.laplacian (layerBarrier F R) z := by
-  have ha := diskFactor_pos hz
-  have ht : shiftedRadius F R z ≤ 0 := by
-    apply Real.exp_le_one_iff.1
-    rw [h.shifted_exp hz hR]
-    exact (div_le_one hR).2 hl
-  have he : Real.exp (2*shiftedRadius F R z) = (diskFactor z * ‖deriv F z‖/R)^2 := by
-    rw [two_mul,Real.exp_add,h.shifted_exp hz hR,pow_two]
-  have hp := profileD_lower layerExponent_bounds.1.le layerExponent_bounds.2.1 ht
-  rw [he] at hp
-  have hm := mul_le_mul_of_nonneg_left hp (show 0 ≤ 2*R^2/(diskFactor z)^2 by positivity)
-  have hid : (2*R^2/(diskFactor z)^2)*(layerExponent*(diskFactor z*‖deriv F z‖/R)^2/4) =
-      layerExponent/2*‖deriv F z‖^2 := by
-    field_simp
-    ring
-  rw [hid] at hm
-  have hb := h.barrier_drop hz R
-  nlinarith [show (2*R^2/(diskFactor z)^2)*layerProfileD layerExponent (shiftedRadius F R z) =
-      -(-2*R^2*layerProfileD layerExponent (shiftedRadius F R z)/(diskFactor z)^2) by ring]
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-lemma radial_green_annulus {m d L : ℝ → ℝ} {M e R : ℝ}
-    (he : 0 < e) (heR : e ≤ R) (hR : R < 1)
-    (hd : ∀ r ∈ Ico (0:ℝ) 1, HasDerivAt m (d r) r)
-    (hc : ContinuousOn d (Ico (0:ℝ) 1))
-    (hA : ∀ r ∈ Ioo (0:ℝ) 1, HasDerivAt (fun r => r*d r) (r*L r) r)
-    (hLc : ContinuousOn L (Ioo (0:ℝ) 1))
-    (hL : ∀ r ∈ Ioo (0:ℝ) 1, L r ≤ 0)
-    (hm : ∀ r ∈ Ico (0:ℝ) 1, 0 ≤ m r ∧ m r ≤ M) :
-    (∫ r in e..R, Real.log (R/r)*(-r*L r)) ≤ M := by
-  have hR0 : 0 < R := he.trans_le heR
-  have hsub : Icc (0:ℝ) R ⊆ Ico (0:ℝ) 1 := fun r hr => ⟨hr.1,hr.2.trans_lt hR⟩
-  have hant : AntitoneOn (fun r => r*d r) (Icc (0:ℝ) R) := by
-    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
-      (continuousOn_id.mul (hc.mono hsub))
-    · intro r hr
-      rw [interior_Icc] at hr
-      exact (hA r ⟨hr.1,hr.2.trans hR⟩).hasDerivWithinAt
-    · intro r hr
-      rw [interior_Icc] at hr
-      exact mul_nonpos_of_nonneg_of_nonpos hr.1.le (hL r ⟨hr.1,hr.2.trans hR⟩)
-  have hAe : e*d e ≤ 0 := by
-    simpa using hant ⟨le_rfl,hR0.le⟩ ⟨he.le,heR⟩ he.le
-  have hue : uIcc e R = Icc e R := uIcc_of_le heR
-  have hs (r : ℝ) (hr : r ∈ uIcc e R) : r ∈ Ioo (0:ℝ) 1 := by
-    rw [hue] at hr
-    exact ⟨he.trans_le hr.1,hr.2.trans_lt hR⟩
-  have hlog (r : ℝ) (hr : r ∈ uIcc e R) :
-      HasDerivAt (fun r => Real.log (R/r)) (-r⁻¹) r := by
-    have hh := (Real.hasDerivAt_log (ne_of_gt (hs r hr).1)).const_sub (Real.log R)
-    convert hh.congr_of_eventuallyEq ?_ using 1
-    filter_upwards [Ioi_mem_nhds (hs r hr).1] with t ht
-    exact Real.log_div (ne_of_gt hR0) (ne_of_gt ht)
-  have hprim (r : ℝ) (hr : r ∈ uIcc e R) :
-      HasDerivAt (fun r => -Real.log (R/r)*(r*d r)-m r)
-        (Real.log (R/r)*(-r*L r)) r := by
-    have hh := ((hlog r hr).neg.mul (hA r (hs r hr))).sub (hd r ⟨(hs r hr).1.le,(hs r hr).2⟩)
-    convert! hh using 1
-    simp only [Pi.neg_apply,neg_neg]
-    field_simp [ne_of_gt (hs r hr).1]
-    ring
-  have hi : IntervalIntegrable (fun r => Real.log (R/r)*(-r*L r)) volume e R := by
-    apply ContinuousOn.intervalIntegrable
-    apply ContinuousOn.mul
-    · exact fun r hr => (hlog r hr).continuousAt.continuousWithinAt
-    · exact (continuousOn_id.neg.mul (hLc.mono (fun r hr => hs r hr)))
-  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt hprim hi
-  have hRR : Real.log (R/R) = 0 := by simp [ne_of_gt hR0]
-  rw [hRR,neg_zero,zero_mul,zero_sub] at hftc
-  rw [hftc]
-  have hlg : 0 ≤ Real.log (R/e) := Real.log_nonneg ((one_le_div he).2 heR)
-  have hmul := mul_nonpos_of_nonneg_of_nonpos hlg hAe
-  have hme := hm e ⟨he.le,heR.trans_lt hR⟩
-  have hmR := hm R ⟨hR0.le,hR⟩
-  nlinarith
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal
-namespace SLEExactGauge.Layer
-
-lemma radial_green_integral {m d L : ℝ → ℝ} {M : ℝ}
-    (hd : ∀ r ∈ Ico (0:ℝ) 1, HasDerivAt m (d r) r)
-    (hc : ContinuousOn d (Ico (0:ℝ) 1))
-    (hA : ∀ r ∈ Ioo (0:ℝ) 1, HasDerivAt (fun r => r*d r) (r*L r) r)
-    (hLc : ContinuousOn L (Ioo (0:ℝ) 1))
-    (hL : ∀ r ∈ Ioo (0:ℝ) 1, L r ≤ 0)
-    (hm : ∀ r ∈ Ico (0:ℝ) 1, 0 ≤ m r ∧ m r ≤ M) :
-    (∫⁻ r in Ioo (0:ℝ) 1, ENNReal.ofReal (Real.log (1/r)*(-r*L r))) ≤ ENNReal.ofReal M := by
-  let e : ℕ → ℝ := fun n => (1/((n:ℝ)+1))/2
-  let R : ℕ → ℝ := fun n => 1-e n
-  have he (n : ℕ) : 0 < e n := by dsimp [e]; positivity
-  have hehalf (n : ℕ) : e n ≤ 1/2 := by
-    dsimp [e]
-    apply div_le_div_of_nonneg_right _ (by norm_num)
-    exact (div_le_one (by positivity)).2 (by have hn := Nat.cast_nonneg (α := ℝ) n; linarith)
-  have heR (n : ℕ) : e n ≤ R n := by dsimp [R]; linarith [hehalf n]
-  have hR (n : ℕ) : R n < 1 := by dsimp [R]; linarith [he n]
-  have hR0 (n : ℕ) : 0 < R n := (he n).trans_le (heR n)
-  have hsub (n : ℕ) : Icc (e n) (R n) ⊆ Ioo (0:ℝ) 1 :=
-    fun r hr => ⟨(he n).trans_le hr.1,hr.2.trans_lt (hR n)⟩
-  let f : ℕ → ℝ → ℝ≥0∞ := fun n => (Icc (e n) (R n)).indicator
-    (fun r => ENNReal.ofReal (Real.log (R n/r)*(-r*L r)))
-  let μ := volume.restrict (Ioo (0:ℝ) 1)
-  have hfm (n : ℕ) : AEMeasurable (f n) μ := by
-    apply AEMeasurable.indicator _ measurableSet_Icc
-    apply AEMeasurable.ennreal_ofReal
-    exact ((measurable_const.div measurable_id).log.aemeasurable).mul
-      (measurable_id.neg.aemeasurable.mul (hLc.aemeasurable measurableSet_Ioo))
-  have hbound (n : ℕ) : (∫⁻ r, f n r ∂μ) ≤ ENNReal.ofReal M := by
-    let q : ℝ → ℝ := fun r => Real.log (R n/r)*(-r*L r)
-    have hqc : ContinuousOn q (Icc (e n) (R n)) := by
-      apply ContinuousOn.mul
-      · exact (continuousOn_const.div continuousOn_id (fun r hr => ne_of_gt (hsub n hr).1)).log
-          (fun r hr => ne_of_gt (div_pos (hR0 n) (hsub n hr).1))
-      · exact continuousOn_id.neg.mul (hLc.mono (hsub n))
-    have hqi : IntegrableOn q (Icc (e n) (R n)) := hqc.integrableOn_Icc
-    have hqnn : 0 ≤ᵐ[volume.restrict (Icc (e n) (R n))] q := by
-      filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr
-      exact mul_nonneg (Real.log_nonneg ((one_le_div (hsub n hr).1).2 hr.2))
-        (mul_nonneg_of_nonpos_of_nonpos (neg_nonpos.2 (hsub n hr).1.le) (hL r (hsub n hr)))
-    change (∫⁻ r, (Icc (e n) (R n)).indicator (fun r => ENNReal.ofReal (q r)) r ∂μ) ≤ _
-    rw [lintegral_indicator measurableSet_Icc]
-    change (∫⁻ r, ENNReal.ofReal (q r) ∂(volume.restrict (Ioo (0:ℝ) 1)).restrict (Icc (e n) (R n))) ≤ _
-    rw [Measure.restrict_restrict measurableSet_Icc,inter_eq_left.2 (hsub n),← ofReal_integral_eq_lintegral_ofReal hqi hqnn]
-    apply ENNReal.ofReal_le_ofReal
-    rw [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le (heR n)]
-    exact radial_green_annulus (he n) (heR n) (hR n) hd hc hA hLc hL hm
-  have helim : Tendsto e atTop (𝓝 0) := by
-    simpa only [zero_div,e] using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).div_const 2
-  have hRlim : Tendsto R atTop (𝓝 1) := by
-    simpa only [sub_zero,R] using tendsto_const_nhds.sub helim
-  have hlim (r : ℝ) (hr : r ∈ Ioo (0:ℝ) 1) :
-      Tendsto (fun n => f n r) atTop (𝓝 (ENNReal.ofReal (Real.log (1/r)*(-r*L r)))) := by
-    have hevent : ∀ᶠ n in atTop, r ∈ Icc (e n) (R n) := by
-      filter_upwards [helim.eventually_lt_const hr.1,hRlim.eventually_const_lt hr.2] with n hn hn'
-      exact ⟨hn.le,hn'.le⟩
-    have ht := ENNReal.tendsto_ofReal (((hRlim.div_const r).log (div_ne_zero one_ne_zero (ne_of_gt hr.1))).mul_const (-r*L r))
-    apply ht.congr'
-    filter_upwards [hevent] with n hn
-    exact (indicator_of_mem hn (fun r => ENNReal.ofReal (Real.log (R n/r)*(-r*L r)))).symm
-  calc
-    _ = ∫⁻ r, liminf (fun n => f n r) atTop ∂μ := by
-      apply lintegral_congr_ae
-      filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr
-      exact (hlim r hr).liminf_eq.symm
-    _ ≤ liminf (fun n => ∫⁻ r, f n r ∂μ) atTop := lintegral_liminf_le' hfm
-    _ ≤ _ := liminf_le_of_frequently_le (Eventually.frequently (Eventually.of_forall hbound))
-
-lemma superharmonic_radial_green {u : ℂ → ℝ} {M : ℝ}
-    (hu : ContDiffOn ℝ 2 u (ball 0 1))
-    (hb : ∀ z ∈ ball (0:ℂ) 1, 0 ≤ u z ∧ u z ≤ M)
-    (hΔ : ∀ z ∈ ball (0:ℂ) 1, Laplacian.laplacian u z ≤ 0) :
-    (∫⁻ r in Ioo (0:ℝ) 1, ENNReal.ofReal (Real.log (1/r)*(-r*circleLMean u r))) ≤
-      ENNReal.ofReal (2*Real.pi*M) := by
-  have hrad (r : ℝ) (hr : r ∈ Ico (0:ℝ) 1) : |r| < 1 := by rw [abs_of_nonneg hr.1]; exact hr.2
-  have hθc (r : ℝ) (hr : r ∈ Ico (0:ℝ) 1) : Continuous (fun θ => u (r • circleMap 0 1 θ)) :=
-    continuousOn_univ.1 (hu.continuousOn.comp ((continuous_circleMap 0 1).const_smul r).continuousOn
-      (fun θ _ => radial_mem (hrad r hr) θ))
-  apply radial_green_integral (m := circleMean u) (d := circleDMean u)
-  · intro r hr
-    exact (circle_means_derivatives hu (hrad r hr)).1
-  · intro r hr
-    exact (circle_means_derivatives hu (hrad r hr)).2.continuousAt.continuousWithinAt
-  · intro r hr
-    exact weighted_circle_derivative hu hr.1 hr.2
-  · exact (circle_lmean_continuous hu).mono (fun r hr => ⟨by linarith [hr.1],hr.2⟩)
-  · intro r hr
-    have hh := intervalIntegral.integral_nonneg (μ := volume) (neg_le_self Real.pi_pos.le)
-      (f := fun θ => -Laplacian.laplacian u (r • circleMap 0 1 θ))
-      (fun θ _ => neg_nonneg.2 (hΔ _ (radial_mem (by rw [abs_of_pos hr.1]; exact hr.2) θ)))
-    rw [intervalIntegral.integral_neg] at hh
-    exact neg_nonneg.1 hh
-  · intro r hr
-    constructor
-    · apply intervalIntegral.integral_nonneg (neg_le_self Real.pi_pos.le)
-      intro θ _
-      exact (hb _ (radial_mem (hrad r hr) θ)).1
-    · have hi := intervalIntegral.integral_mono_on (μ := volume) (neg_le_self Real.pi_pos.le)
-        ((hθc r hr).intervalIntegrable _ _) (intervalIntegrable_const (c := M))
-        (fun θ _ => (hb _ (radial_mem (hrad r hr) θ)).2)
-      change circleMean u r ≤ _ at hi
-      exact hi.trans_eq (by simp only [intervalIntegral.integral_const,smul_eq_mul]; ring)
-
-theorem superharmonic_disk_green {u : ℂ → ℝ} {M : ℝ}
-    (hu : ContDiffOn ℝ 2 u (ball 0 1))
-    (hb : ∀ z ∈ ball (0:ℂ) 1, 0 ≤ u z ∧ u z ≤ M)
-    (hΔ : ∀ z ∈ ball (0:ℂ) 1, Laplacian.laplacian u z ≤ 0) :
-    (∫⁻ z in ball (0:ℂ) 1, ENNReal.ofReal (Real.log (1/‖z‖)*(-Laplacian.laplacian u z))) ≤
-      ENNReal.ofReal (2*Real.pi*M) := by
-  apply (polar_disk_lintegral_le _).trans
-  have he : (∫⁻ r in Ioo (0:ℝ) 1, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-      ENNReal.ofReal r * ENNReal.ofReal (Real.log (1/‖r • circleMap 0 1 θ‖)*
-        (-Laplacian.laplacian u (r • circleMap 0 1 θ)))) =
-      ∫⁻ r in Ioo (0:ℝ) 1, ENNReal.ofReal (Real.log (1/r)*(-r*circleLMean u r)) := by
-    apply lintegral_congr_ae
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr
-    have hrabs : |r| < 1 := by rw [abs_of_pos hr.1]; exact hr.2
-    let q : ℝ → ℝ := fun θ => Real.log (1/r)*(-r*Laplacian.laplacian u (r • circleMap 0 1 θ))
-    have hq : Continuous q := ((circle_laplacian_continuous hu hrabs).const_mul (-r)).const_mul _
-    have hqi : IntegrableOn q (Ioc (-Real.pi) Real.pi) := (hq.intervalIntegrable _ _).1
-    have hq0 : 0 ≤ᵐ[volume.restrict (Ioc (-Real.pi) Real.pi)] q := by
-      filter_upwards with θ
-      apply mul_nonneg
-      · exact Real.log_nonneg ((one_le_div hr.1).2 hr.2.le)
-      · exact mul_nonneg_of_nonpos_of_nonpos (neg_nonpos.2 hr.1.le) (hΔ _ (radial_mem hrabs θ))
-    have hqint : (∫ θ in -Real.pi..Real.pi, q θ) = Real.log (1/r)*(-r*circleLMean u r) := by
-      dsimp only [q,circleLMean]
-      rw [intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul]
-    calc
-      _ = ∫⁻ θ in Ioo (-Real.pi) Real.pi, ENNReal.ofReal (q θ) := by
-        apply lintegral_congr
-        intro θ
-        rw [radial_norm,abs_of_pos hr.1,← ENNReal.ofReal_mul hr.1.le]
-        congr 1
-        dsimp only [q]
-        ring
-      _ = ENNReal.ofReal (∫ θ in -Real.pi..Real.pi, q θ) := by
-        rw [restrict_Ioo_eq_restrict_Ioc,intervalIntegral.integral_of_le (neg_le_self Real.pi_pos.le)]
-        exact (ofReal_integral_eq_lintegral_ofReal hqi hq0).symm
-      _ = _ := congrArg ENNReal.ofReal hqint
-  rw [he]
-  exact superharmonic_radial_green hu hb hΔ
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-lemma univalent_not_locally_constant {F : ℂ → ℂ} (hi : InjOn F (ball 0 1))
-    {c : ℂ} (hc : c ∈ ball (0:ℂ) 1) : ¬ ∀ᶠ z in 𝓝 c, F z = F c := by
-  intro hh
-  have he : ∀ᶠ z in 𝓝 c, z = c := by
-    filter_upwards [hh,isOpen_ball.mem_nhds hc] with z hz hzD
-    exact hi hzD hc hz
-  have hb : ∀ᶠ z in 𝓝[≠] c, False := by
-    filter_upwards [he.filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with z hz hne
-    exact hne hz
-  obtain ⟨z,hz⟩ := hb.exists
-  exact hz
-
-lemma univalent_punctured_derivative {F : ℂ → ℂ}
-    (hf : DifferentiableOn ℂ F (ball 0 1)) (hi : InjOn F (ball 0 1))
-    {c : ℂ} (hc : c ∈ ball (0:ℂ) 1) : ∀ᶠ z in 𝓝[≠] c, deriv F z ≠ 0 := by
-  have ha := hf.analyticOnNhd isOpen_ball c hc
-  apply ha.deriv.eventually_eq_zero_or_eventually_ne_zero.resolve_left
-  intro hz
-  obtain ⟨r,hr,hsub⟩ := Metric.mem_nhds_iff.1 (hz.and (isOpen_ball.mem_nhds hc))
-  have hbd : ball c r ⊆ ball (0:ℂ) 1 := fun z hz => (hsub hz).2
-  have hconst : ∀ᶠ z in 𝓝 c, F z = F c := by
-    filter_upwards [ball_mem_nhds c hr] with z hz
-    exact isOpen_ball.is_const_of_deriv_eq_zero (convex_ball c r).isPreconnected
-      (hf.mono hbd) (fun z hz => (hsub hz).1) hz (mem_ball_self hr)
-  exact univalent_not_locally_constant hi hc hconst
-
-theorem univalent_diskPair {F : ℂ → ℂ}
-    (hf : DifferentiableOn ℂ F (ball 0 1)) (hi : InjOn F (ball 0 1)) :
-    ∃ G : ℂ → ℂ, DiskPair (F '' ball 0 1) F G := by
-  let U := F '' ball (0:ℂ) 1
-  let G := invFunOn F (ball (0:ℂ) 1)
-  have hl : ∀ z ∈ ball (0:ℂ) 1, G (F z) = z := hi.leftInvOn_invFunOn
-  have hr : ∀ w ∈ U, F (G w) = w := (surjOn_image F (ball (0:ℂ) 1)).rightInvOn_invFunOn
-  have hgmap : MapsTo G U (ball (0:ℂ) 1) := (surjOn_image F (ball (0:ℂ) 1)).mapsTo_invFunOn
-  have hopen (c : ℂ) (hc : c ∈ ball (0:ℂ) 1) : 𝓝 (F c) ≤ map F (𝓝 c) :=
-    ((hf.analyticOnNhd isOpen_ball c hc).eventually_constant_or_nhds_le_map_nhds).resolve_left
-      (univalent_not_locally_constant hi hc)
-  have hU : IsOpen U := by
-    apply isOpen_iff_mem_nhds.2
-    rintro w ⟨c,hc,rfl⟩
-    apply hopen c hc
-    change ∀ᶠ z in 𝓝 c, F z ∈ U
-    filter_upwards [isOpen_ball.mem_nhds hc] with z hz
-    exact mem_image_of_mem F hz
-  have hGc (w : ℂ) (hw : w ∈ U) : ContinuousAt G w := by
-    rcases hw with ⟨c,hc,rfl⟩
-    rw [ContinuousAt,hl c hc]
-    calc
-      map G (𝓝 (F c)) ≤ map G (map F (𝓝 c)) := map_mono (hopen c hc)
-      _ = map id (𝓝 c) := by
-        rw [map_map]
-        apply Filter.map_congr
-        filter_upwards [isOpen_ball.mem_nhds hc] with z hz
-        exact hl z hz
-      _ = 𝓝 c := map_id
-  have hgood (w : ℂ) (hw : w ∈ U) (hn : deriv F (G w) ≠ 0) : DifferentiableAt ℂ G w := by
-    apply HasDerivAt.differentiableAt (f' := (deriv F (G w))⁻¹)
-    apply HasDerivAt.of_local_left_inverse (hGc w hw)
-      (hf.differentiableAt (isOpen_ball.mem_nhds (hgmap hw))).hasDerivAt hn
-    filter_upwards [hU.mem_nhds hw] with y hy
-    exact hr y hy
-  have hGd (w : ℂ) (hw : w ∈ U) : DifferentiableAt ℂ G w := by
-    have hc := hgmap hw
-    have htn : Tendsto G (𝓝[≠] w) (𝓝[≠] G w) := by
-      apply tendsto_nhdsWithin_iff.2
-      refine ⟨(hGc w hw).tendsto.mono_left nhdsWithin_le_nhds,?_⟩
-      filter_upwards [(show ∀ᶠ y in 𝓝 w, y ∈ U from hU.mem_nhds hw).filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with y hy hne
-      change G y ≠ G w
-      intro hgy
-      apply hne
-      change y = w
-      rw [← hr y hy,hgy,hr w hw]
-    have hne := htn.eventually (univalent_punctured_derivative hf hi hc)
-    apply (Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt ?_ (hGc w hw)).differentiableAt
-    filter_upwards [hne,(show ∀ᶠ y in 𝓝 w, y ∈ U from hU.mem_nhds hw).filter_mono nhdsWithin_le_nhds] with y hn hy
-    exact hgood y hy hn
-  exact ⟨G,hU,(fun z hz => mem_image_of_mem F hz),hgmap,hl,hr,hf,fun w hw => (hGd w hw).differentiableWithinAt⟩
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+
+namespace OAI
+
+
+/-! Conformal localization and Layer estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology RealInnerProductSpace
+namespace SLEExactGauge.Layer
+
+lemma logFactor_laplacian {z : ℂ} (hz : ‖z‖ < 1) :
+    Laplacian.laplacian (fun z => Real.log (diskFactor z)) z = -4/(diskFactor z)^2 := by
+  have ha := diskFactor_pos hz
+  have hl : ContDiffAt ℝ 2 Real.log (diskFactor z) := Real.contDiffAt_log.mpr (ne_of_gt ha)
+  have hh := laplacian_scalar_comp diskFactor_smooth.contDiffAt hl
+  have hlogd : deriv Real.log = fun t => t⁻¹ := funext Real.deriv_log
+  rw [diskFactor_laplacian,(diskFactor_gradient z).1,(diskFactor_gradient z).2,
+    hlogd] at hh
+  have he : deriv (fun t : ℝ => t⁻¹) (diskFactor z) = -((diskFactor z)^2)⁻¹ := by
+    simp
+  rw [he] at hh
+  change Laplacian.laplacian (fun z => Real.log (diskFactor z)) z = _ at hh
+  rw [hh]
+  have hn : ‖z‖^2 = z.re^2+z.im^2 := by rw [Complex.sq_norm,Complex.normSq_apply]; ring
+  dsimp [diskFactor] at ha ⊢
+  rw [hn] at ha ⊢
+  field_simp [ne_of_gt ha]
+  ring
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology RealInnerProductSpace
+namespace SLEExactGauge.Layer
+
+lemma logNorm_fderiv {H : ℂ → ℂ} {z D : ℂ} (hd : HasDerivAt H D z)
+    (hn : H z ≠ 0) (v : ℂ) :
+    fderiv ℝ (fun w => Real.log ‖H w‖) z v = (D*v/H z).re := by
+  have hn2 : ‖H z‖^2 ≠ 0 := pow_ne_zero 2 (norm_ne_zero_iff.2 hn)
+  have hh := ((hd.hasFDerivAt.restrictScalars ℝ).norm_sq.log hn2).const_smul (1/2:ℝ)
+  have he : (fun w => (1/2:ℝ) • Real.log (‖H w‖^2)) = (fun w => Real.log ‖H w‖) := by
+    funext w
+    rw [Real.log_pow]
+    simp only [smul_eq_mul]
+    ring
+  change HasFDerivAt (fun w => (1/2:ℝ) • Real.log (‖H w‖^2)) _ z at hh
+  rw [he] at hh
+  rw [hh.fderiv]
+  simp only [smul_apply,smul_eq_mul,two_smul]
+  change 1/2*((‖H z‖^2)⁻¹*(inner ℝ (H z) (v*D)+inner ℝ (H z) (v*D))) = (D*v/H z).re
+  rw [Complex.inner,Complex.div_re]
+  have heq : Complex.normSq (H z) = ‖H z‖^2 := (Complex.sq_norm _).symm
+  rw [heq]
+  simp only [Complex.mul_re,Complex.mul_im,Complex.conj_re,Complex.conj_im]
+  field_simp
+  ring
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+structure DiskPair (U : Set ℂ) (F G : ℂ → ℂ) : Prop where
+  isOpen : IsOpen U
+  toDomain : MapsTo F (ball 0 1) U
+  toDisk : MapsTo G U (ball 0 1)
+  left_inv : ∀ z ∈ ball (0:ℂ) 1, G (F z) = z
+  right_inv : ∀ z ∈ U, F (G z) = z
+  forward : DifferentiableOn ℂ F (ball 0 1)
+  inverse : DifferentiableOn ℂ G U
+
+lemma DiskPair.injOn {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G) :
+    InjOn F (ball 0 1) := by
+  intro z hz w hw he
+  simpa only [h.left_inv z hz,h.left_inv w hw] using congrArg G he
+
+lemma DiskPair.inverse_derivative {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
+    {z : ℂ} (hz : z ∈ ball (0:ℂ) 1) : deriv G (F z)*deriv F z = 1 := by
+  have he : (G ∘ F) =ᶠ[𝓝 z] id := by
+    filter_upwards [isOpen_ball.mem_nhds hz] with w hw
+    exact h.left_inv w hw
+  exact (((h.inverse.differentiableAt (h.isOpen.mem_nhds (h.toDomain hz))).hasDerivAt.comp z
+    (h.forward.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt).congr_of_eventuallyEq he.symm).unique
+      (hasDerivAt_id z)
+
+lemma DiskPair.deriv_ne_zero {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
+    {z : ℂ} (hz : z ∈ ball (0:ℂ) 1) : deriv F z ≠ 0 := by
+  intro he
+  simpa [he] using h.inverse_derivative hz
+
+lemma DiskPair.second_derivative_zero {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G) :
+    ‖deriv (deriv F) 0‖ ≤ (2*(512*Real.pi)^2)*‖deriv F 0‖ := by
+  have hzero : (0:ℂ) ∈ ball 0 1 := mem_ball_self zero_lt_one
+  have hA : 0 < ‖deriv F 0‖ := norm_pos_iff.2 (h.deriv_ne_zero hzero)
+  let R : ℝ := ‖deriv F 0‖/(512*Real.pi)
+  have hR : 0 < R := div_pos hA (by positivity)
+  have hsub : closedBall (F 0) R ⊆ U := by
+    intro w hw
+    by_contra hwu
+    have hwi : w ∉ F '' ball 0 1 := by
+      rintro ⟨z,hz,rfl⟩
+      exact hwu (h.toDomain hz)
+    have hh := UnivalentDistance.omitted_value_bound h.forward h.injOn hwi
+    have hnorm : ‖F 0-w‖ ≤ R := by
+      rw [norm_sub_rev,← dist_eq_norm]
+      exact mem_closedBall.1 hw
+    have hbound := hh.trans (mul_le_mul_of_nonneg_left hnorm (by positivity : 0 ≤ 256*Real.pi))
+    have he : (256*Real.pi)*R = ‖deriv F 0‖/2 := by
+      dsimp [R]; field_simp; ring
+    rw [he] at hbound
+    linarith
+  have hb : ‖deriv (deriv G) (F 0)‖ ≤ 2/R^2 := by
+    have hh := Complex.norm_iteratedDeriv_le_of_forall_mem_sphere_norm_le 2 hR
+      ((h.inverse.mono hsub).diffContOnCl_ball Subset.rfl)
+      (fun z hz => (mem_ball_zero_iff.1 (h.toDisk (hsub (sphere_subset_closedBall hz)))).le)
+    simpa [iteratedDeriv_succ,iteratedDeriv_zero] using hh
+  have hFd := h.forward.differentiableAt (isOpen_ball.mem_nhds hzero)
+  have hFdd := (h.forward.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hzero)
+  have hGdd := (h.inverse.deriv h.isOpen).differentiableAt (h.isOpen.mem_nhds (h.toDomain hzero))
+  have he : (fun z => deriv G (F z)*deriv F z) =ᶠ[𝓝 (0:ℂ)] fun _ => 1 := by
+    filter_upwards [isOpen_ball.mem_nhds hzero] with z hz
+    exact h.inverse_derivative hz
+  have hh := (((hGdd.hasDerivAt.comp 0 hFd.hasDerivAt).mul hFdd.hasDerivAt).congr_of_eventuallyEq
+    he.symm).unique (hasDerivAt_const (0:ℂ) (1:ℂ))
+  simp only [Function.comp_apply] at hh
+  have hh0 := h.inverse_derivative hzero
+  have hrel : deriv (deriv F) 0 = -deriv (deriv G) (F 0)*(deriv F 0)^3 := by
+    linear_combination (deriv F 0)*hh - (deriv (deriv F) 0)*hh0
+  rw [hrel,norm_mul,norm_neg,norm_pow]
+  calc
+    _ ≤ (2/R^2)*‖deriv F 0‖^3 := mul_le_mul_of_nonneg_right hb (by positivity)
+    _ = (2*(512*Real.pi)^2)*‖deriv F 0‖ := by
+      dsimp [R]
+      field_simp
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.SlitMaps
+
+def diskAuto (c z : ℂ) : ℂ := (z-c)/(1-starRingEnd ℂ c*z)
+
+lemma diskAuto_den_ne {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
+    1-starRingEnd ℂ c*z ≠ 0 := by
+  have hn : ‖starRingEnd ℂ c*z‖ < 1 := by
+    rw [norm_mul, Complex.norm_conj]
+    calc
+      ‖c‖*‖z‖ ≤ ‖c‖*1 := mul_le_mul_of_nonneg_left hz.le (norm_nonneg _)
+      _ < 1 := by simpa using hc
+  intro he
+  have hh : starRingEnd ℂ c*z = 1 := (sub_eq_zero.1 he).symm
+  simp [hh] at hn
+
+lemma disk_norm_identity (c z : ℂ) :
+    ‖1-starRingEnd ℂ c*z‖^2-‖z-c‖^2 = (1-‖c‖^2)*(1-‖z‖^2) := by
+  simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im,
+    Complex.one_re, Complex.one_im, Complex.mul_re, Complex.mul_im,
+    Complex.conj_re, Complex.conj_im]
+  ring
+
+lemma diskAuto_mapsTo {c : ℂ} (hc : ‖c‖ < 1) : MapsTo (diskAuto c) (ball 0 1) (ball 0 1) := by
+  intro z hz
+  rw [mem_ball_zero_iff] at hz ⊢
+  have hn := diskAuto_den_ne hc hz
+  rw [diskAuto,norm_div,div_lt_one₀ (norm_pos_iff.2 hn)]
+  have hp : 0 < (1-‖c‖^2)*(1-‖z‖^2) := mul_pos (by nlinarith [norm_nonneg c]) (by nlinarith [norm_nonneg z])
+  have he := disk_norm_identity c z
+  nlinarith [norm_nonneg (z-c), norm_nonneg (1-starRingEnd ℂ c*z)]
+
+lemma diskAuto_hasDerivAt {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
+    HasDerivAt (diskAuto c) ((1-starRingEnd ℂ c*c)/(1-starRingEnd ℂ c*z)^2) z := by
+  have hd := ((hasDerivAt_id z).sub_const c).div
+    ((hasDerivAt_const z (1:ℂ)).sub ((hasDerivAt_id z).const_mul (starRingEnd ℂ c)))
+    (diskAuto_den_ne hc hz)
+  change HasDerivAt (diskAuto c)
+    ((1*(1-starRingEnd ℂ c*z)-(z-c)*(0-starRingEnd ℂ c*1))/(1-starRingEnd ℂ c*z)^2) z at hd
+  have he : 1*(1-starRingEnd ℂ c*z)-(z-c)*(0-starRingEnd ℂ c*1) = 1-starRingEnd ℂ c*c := by ring
+  rwa [he] at hd
+
+lemma diskAuto_deriv {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
+    deriv (diskAuto c) z = (1-starRingEnd ℂ c*c)/(1-starRingEnd ℂ c*z)^2 :=
+  (diskAuto_hasDerivAt hc hz).deriv
+
+lemma diskAuto_deriv_ne {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
+    deriv (diskAuto c) z ≠ 0 := by
+  rw [diskAuto_deriv hc hz]
+  exact div_ne_zero (diskAuto_den_ne hc hc) (pow_ne_zero _ (diskAuto_den_ne hc hz))
+
+lemma diskAuto_injOn {c : ℂ} (hc : ‖c‖ < 1) : InjOn (diskAuto c) (ball 0 1) := by
+  intro z hz w hw he
+  have hzn := diskAuto_den_ne hc (mem_ball_zero_iff.1 hz)
+  have hwn := diskAuto_den_ne hc (mem_ball_zero_iff.1 hw)
+  have he' := (div_eq_div_iff hzn hwn).1 he
+  have hn := diskAuto_den_ne hc hc
+  have hh : (z-w)*(1-starRingEnd ℂ c*c) = 0 := by linear_combination he'
+  exact sub_eq_zero.1 ((mul_eq_zero.1 hh).resolve_right hn)
+
+@[simp] lemma diskAuto_self (c : ℂ) : diskAuto c c = 0 := by simp [diskAuto]
+
+@[simp] lemma diskAuto_zero (c : ℂ) : diskAuto c 0 = -c := by simp [diskAuto]
+
+lemma diskAuto_norm_deriv_zero {c : ℂ} (hc : ‖c‖ < 1) :
+    ‖deriv (diskAuto c) 0‖ = 1-‖c‖^2 := by
+  rw [diskAuto_deriv hc (by simp)]
+  simp only [mul_zero,sub_zero,one_pow,div_one]
+  rw [mul_comm (starRingEnd ℂ c),Complex.mul_conj']
+  have he : (1:ℂ)-(‖c‖:ℂ)^2 = ((1-‖c‖^2:ℝ):ℂ) := by push_cast; rfl
+  rw [he,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (by nlinarith [norm_nonneg c])]
+
+lemma diskAuto_norm_deriv_self {c : ℂ} (hc : ‖c‖ < 1) :
+    ‖deriv (diskAuto c) c‖ = 1/(1-‖c‖^2) := by
+  have hh : (1-starRingEnd ℂ c*c)/(1-starRingEnd ℂ c*c)^2 = (1-starRingEnd ℂ c*c)⁻¹ := by
+    field_simp [diskAuto_den_ne hc hc]
+  rw [diskAuto_deriv hc hc,hh]
+  rw [norm_inv]
+  have he : ‖1-starRingEnd ℂ c*c‖ = 1-‖c‖^2 := by
+    have hh := diskAuto_norm_deriv_zero hc
+    rw [diskAuto_deriv hc (by simp)] at hh
+    simpa using hh
+  simp [he,one_div]
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+namespace SLEExactGauge.Layer
+open SlitMaps
+
+lemma diskAuto_neg_inverse {c z : ℂ} (hc : ‖c‖ < 1) (hz : ‖z‖ < 1) :
+    diskAuto c (diskAuto (-c) z) = z := by
+  have hnc : ‖-c‖ < 1 := by simpa using hc
+  have hφ := diskAuto_mapsTo hnc (mem_ball_zero_iff.2 hz)
+  have hd1 := diskAuto_den_ne hnc hz
+  have hd2 := diskAuto_den_ne hc (mem_ball_zero_iff.1 hφ)
+  have ha := diskAuto_den_ne hc hc
+  dsimp [diskAuto] at hd1 hd2 ⊢
+  simp only [map_neg,neg_mul,sub_neg_eq_add] at hd1 hd2 ⊢
+  apply (div_eq_iff hd2).2
+  have he := div_mul_cancel₀ (z+c) hd1
+  linear_combination he
+
+lemma DiskPair.recenter {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
+    {c : ℂ} (hc : ‖c‖ < 1) :
+    DiskPair U (F ∘ diskAuto (-c)) (diskAuto c ∘ G) := by
+  have hnc : ‖-c‖ < 1 := by simpa using hc
+  refine ⟨h.isOpen,h.toDomain.comp (diskAuto_mapsTo hnc),(diskAuto_mapsTo hc).comp h.toDisk,?_,?_,?_,?_⟩
+  · intro z hz
+    dsimp only [Function.comp_apply]
+    rw [h.left_inv _ (diskAuto_mapsTo hnc hz)]
+    exact diskAuto_neg_inverse hc (mem_ball_zero_iff.1 hz)
+  · intro z hz
+    dsimp only [Function.comp_apply]
+    have he := diskAuto_neg_inverse hnc (mem_ball_zero_iff.1 (h.toDisk hz))
+    simp only [neg_neg] at he
+    rw [he,h.right_inv z hz]
+  · exact h.forward.comp (fun z hz => (diskAuto_hasDerivAt hnc (mem_ball_zero_iff.1 hz)).differentiableAt.differentiableWithinAt)
+      (diskAuto_mapsTo hnc)
+  · have ha : DifferentiableOn ℂ (diskAuto c) (ball 0 1) := fun z hz =>
+      (diskAuto_hasDerivAt hc (mem_ball_zero_iff.1 hz)).differentiableAt.differentiableWithinAt
+    exact ha.comp h.inverse h.toDisk
+
+lemma diskAuto_deriv_zero_exact {c : ℂ} (hc : ‖c‖ < 1) :
+    deriv (diskAuto (-c)) 0 = 1-(starRingEnd ℂ c)*c := by
+  rw [diskAuto_deriv (by simpa using hc) (by simp)]
+  simp
+
+lemma diskAuto_second_zero {c : ℂ} (hc : ‖c‖ < 1) :
+    deriv (deriv (diskAuto (-c))) 0 = -2*(starRingEnd ℂ c)*(1-(starRingEnd ℂ c)*c) := by
+  have hnc : ‖-c‖ < 1 := by simpa using hc
+  have he : deriv (diskAuto (-c)) =ᶠ[𝓝 (0:ℂ)]
+      fun z => (1-(starRingEnd ℂ c)*c)/(1+(starRingEnd ℂ c)*z)^2 := by
+    filter_upwards [Metric.ball_mem_nhds (0:ℂ) zero_lt_one] with z hz
+    rw [diskAuto_deriv hnc (mem_ball_zero_iff.1 hz)]
+    simp
+  rw [he.deriv_eq]
+  have hd := (hasDerivAt_const (0:ℂ) (1-(starRingEnd ℂ c)*c)).div
+    ((((hasDerivAt_id (0:ℂ)).const_mul (starRingEnd ℂ c)).const_add 1).pow 2) (by simp)
+  have hh := hd.deriv
+  dsimp only [Pi.div_apply,Pi.pow_apply,id_eq] at hh
+  change deriv ((fun x : ℂ => 1-(starRingEnd ℂ c)*c) / (fun x : ℂ => 1+(starRingEnd ℂ c)*x)^2) 0 = _
+  rw [hh]
+  simp only [mul_zero,add_zero,one_pow,zero_sub,div_one,mul_one]
+  norm_num
+  ring
+
+lemma DiskPair.preSchwarzian {U : Set ℂ} {F G : ℂ → ℂ} (h : DiskPair U F G)
+    {c : ℂ} (hc : ‖c‖ < 1) :
+    ‖((1-‖c‖^2:ℝ):ℂ)*deriv (deriv F) c - 2*(starRingEnd ℂ c)*deriv F c‖ ≤
+      (2*(512*Real.pi)^2)*‖deriv F c‖ := by
+  let φ := diskAuto (-c)
+  let A : ℂ := 1-(starRingEnd ℂ c)*c
+  have hnc : ‖-c‖ < 1 := by simpa using hc
+  have hcD : c ∈ ball (0:ℂ) 1 := mem_ball_zero_iff.2 hc
+  have hφ0 : φ 0 = c := by simp [φ]
+  have hdφ : deriv φ 0 = A := diskAuto_deriv_zero_exact hc
+  have hddφ : deriv (deriv φ) 0 = -2*(starRingEnd ℂ c)*A := diskAuto_second_zero hc
+  have hφd : DifferentiableOn ℂ φ (ball 0 1) := fun z hz =>
+    (diskAuto_hasDerivAt hnc (mem_ball_zero_iff.1 hz)).differentiableAt.differentiableWithinAt
+  have hFd := h.forward.differentiableAt (isOpen_ball.mem_nhds hcD)
+  have hFdd := (h.forward.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds hcD)
+  have hφd0 := hφd.differentiableAt (isOpen_ball.mem_nhds (mem_ball_self zero_lt_one))
+  have hφdd0 := (hφd.deriv isOpen_ball).differentiableAt (isOpen_ball.mem_nhds (mem_ball_self zero_lt_one))
+  have hd : deriv (F ∘ φ) 0 = deriv F c*A := by
+    have hFd' : HasDerivAt F (deriv F c) (φ 0) := hφ0.symm ▸ hFd.hasDerivAt
+    have hh := hFd'.comp (0:ℂ) hφd0.hasDerivAt
+    simpa only [hdφ] using hh.deriv
+  have he : deriv (F ∘ φ) =ᶠ[𝓝 (0:ℂ)] fun z => deriv F (φ z)*deriv φ z := by
+    filter_upwards [Metric.ball_mem_nhds (0:ℂ) zero_lt_one] with z hz
+    exact ((h.forward.differentiableAt (isOpen_ball.mem_nhds (diskAuto_mapsTo hnc hz))).hasDerivAt.comp z
+      (hφd.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt).deriv
+  have hdd : deriv (deriv (F ∘ φ)) 0 = A*(A*deriv (deriv F) c-2*(starRingEnd ℂ c)*deriv F c) := by
+    rw [he.deriv_eq]
+    have hFdd' : HasDerivAt (deriv F) (deriv (deriv F) c) (φ 0) := hφ0.symm ▸ hFdd.hasDerivAt
+    have hh := ((hFdd'.comp 0 hφd0.hasDerivAt).mul hφdd0.hasDerivAt).deriv
+    dsimp only [Pi.mul_apply,Function.comp_apply] at hh
+    simp only [hφ0,hdφ,hddφ] at hh
+    change deriv ((deriv F ∘ φ)*deriv φ) 0 = _
+    rw [hh]
+    ring
+  have hAeq : A = ((1-‖c‖^2:ℝ):ℂ) := by
+    dsimp only [A]
+    rw [mul_comm (starRingEnd ℂ c),Complex.mul_conj']
+    push_cast
+    rfl
+  have hAp : 0 < ‖A‖ := by
+    rw [hAeq,Complex.norm_real,Real.norm_eq_abs]
+    exact abs_pos.2 (by nlinarith [norm_nonneg c] : 1-‖c‖^2 ≠ 0)
+  have hb := (h.recenter hc).second_derivative_zero
+  change ‖deriv (deriv (F ∘ φ)) 0‖ ≤ (2*(512*Real.pi)^2)*‖deriv (F ∘ φ) 0‖ at hb
+  rw [hdd,hd,norm_mul,norm_mul] at hb
+  have hh : ‖A*deriv (deriv F) c-2*(starRingEnd ℂ c)*deriv F c‖ ≤ (2*(512*Real.pi)^2)*‖deriv F c‖ := by
+    nlinarith
+  rwa [hAeq] at hh
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+def logRadius (F : ℂ → ℂ) (z : ℂ) : ℝ :=
+  Real.log (diskFactor z) + Real.log ‖deriv F z‖
+
+lemma DiskPair.logRadius_smooth {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
+    ContDiffAt ℝ 2 (logRadius F) z := by
+  have hh := ((h.forward.deriv isOpen_ball).analyticOnNhd isOpen_ball z
+    (mem_ball_zero_iff.2 hz)).harmonicAt_log_norm (h.deriv_ne_zero (mem_ball_zero_iff.2 hz))
+  exact (logFactor_smooth hz).add hh.1
+
+lemma DiskPair.logRadius_laplacian {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
+    Laplacian.laplacian (logRadius F) z = -4/(diskFactor z)^2 := by
+  have hh := ((h.forward.deriv isOpen_ball).analyticOnNhd isOpen_ball z
+    (mem_ball_zero_iff.2 hz)).harmonicAt_log_norm (h.deriv_ne_zero (mem_ball_zero_iff.2 hz))
+  have he := (logFactor_smooth hz).laplacian_add hh.1
+  change Laplacian.laplacian (logRadius F) z = _ at he
+  rw [he,logFactor_laplacian hz,hh.2.eq_of_nhds]
+  simp
+
+lemma DiskPair.logRadius_gradient {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
+    fderiv ℝ (logRadius F) z 1 = (deriv (deriv F) z/deriv F z).re - 2*z.re/diskFactor z ∧
+    fderiv ℝ (logRadius F) z Complex.I = -(deriv (deriv F) z/deriv F z).im - 2*z.im/diskFactor z := by
+  have ha := diskFactor_pos hz
+  have hdz := (h.forward.deriv isOpen_ball).differentiableAt
+    (isOpen_ball.mem_nhds (mem_ball_zero_iff.2 hz))
+  have hn := h.deriv_ne_zero (mem_ball_zero_iff.2 hz)
+  have hl := (diskFactor_smooth.differentiable (by norm_num) z).hasFDerivAt.log (ne_of_gt ha)
+  have hd : DifferentiableAt ℝ (fun z => Real.log ‖deriv F z‖) z :=
+    (((h.forward.deriv isOpen_ball).analyticOnNhd isOpen_ball z
+      (mem_ball_zero_iff.2 hz)).harmonicAt_log_norm hn).1.differentiableAt (by norm_num)
+  have he := (hl.add hd.hasFDerivAt).fderiv
+  change fderiv ℝ (logRadius F) z = _ at he
+  rw [he]
+  simp only [add_apply,smul_apply,smul_eq_mul,
+    (diskFactor_gradient z).1,(diskFactor_gradient z).2,
+    logNorm_fderiv hdz.hasDerivAt hn,mul_one]
+  constructor
+  · ring
+  · rw [mul_div_right_comm,Complex.mul_re]
+    simp
+    ring
+
+lemma DiskPair.logRadius_gradient_sq {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) :
+    (fderiv ℝ (logRadius F) z 1)^2 + (fderiv ℝ (logRadius F) z Complex.I)^2 ≤
+      (2*(512*Real.pi)^2)^2/(diskFactor z)^2 := by
+  let a := diskFactor z
+  let c := deriv (deriv F) z/deriv F z
+  let b : ℂ := (a:ℂ)*c - 2*(starRingEnd ℂ z)
+  have ha : 0 < a := diskFactor_pos hz
+  have hn := h.deriv_ne_zero (mem_ball_zero_iff.2 hz)
+  have he : b = ((a:ℂ)*deriv (deriv F) z - 2*(starRingEnd ℂ z)*deriv F z)/deriv F z := by
+    dsimp [b,c]
+    field_simp
+  have hb : ‖b‖ ≤ 2*(512*Real.pi)^2 := by
+    rw [he,norm_div]
+    apply (div_le_iff₀ (norm_pos_iff.2 hn)).2
+    exact h.preSchwarzian hz
+  have hi : ((fderiv ℝ (logRadius F) z 1)^2 +
+      (fderiv ℝ (logRadius F) z Complex.I)^2)*a^2 = ‖b‖^2 := by
+    rw [(h.logRadius_gradient hz).1,(h.logRadius_gradient hz).2,
+      Complex.sq_norm,Complex.normSq_apply]
+    change ((c.re-2*z.re/a)^2+(-c.im-2*z.im/a)^2)*a^2 = _
+    dsimp [b]
+    simp only [Complex.mul_re,Complex.mul_im,
+      Complex.ofReal_re,Complex.ofReal_im,
+      Complex.conj_re,Complex.conj_im,zero_mul,sub_zero,add_zero]
+    norm_num
+    field_simp
+    ring
+  apply (le_div_iff₀ (pow_pos ha 2)).2
+  rw [hi]
+  exact pow_le_pow_left₀ (norm_nonneg b) hb 2
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+def layerProfile (α t : ℝ) : ℝ := Real.exp (α*t)/(1+Real.exp (α*t))
+
+def layerProfileD (α t : ℝ) : ℝ := α*Real.exp (α*t)/(1+Real.exp (α*t))^2
+
+def layerProfileDD (α t : ℝ) : ℝ :=
+  α^2*Real.exp (α*t)*(1-Real.exp (α*t))/(1+Real.exp (α*t))^3
+
+lemma profile_smooth (α : ℝ) : ContDiff ℝ 2 (layerProfile α) := by
+  unfold layerProfile
+  apply ContDiff.div (by fun_prop) (by fun_prop)
+  intro x
+  positivity
+
+lemma profile_derivative (α t : ℝ) : HasDerivAt (layerProfile α) (layerProfileD α t) t := by
+  have he := ((hasDerivAt_id t).const_mul α).exp
+  have hh := he.div (he.const_add 1) (by positivity : 1+Real.exp (α*t) ≠ 0)
+  convert! hh using 1
+  dsimp [layerProfile,layerProfileD]
+  ring
+
+lemma profile_second (α t : ℝ) : HasDerivAt (layerProfileD α) (layerProfileDD α t) t := by
+  have he := ((hasDerivAt_id t).const_mul α).exp
+  have hh := (he.const_mul α).div ((he.const_add 1).pow 2)
+    (by positivity : (1+Real.exp (α*t))^2 ≠ 0)
+  convert! hh using 1
+  dsimp [layerProfileD,layerProfileDD]
+  field_simp
+  ring
+
+lemma profile_bounds (α t : ℝ) : 0 ≤ layerProfile α t ∧ layerProfile α t ≤ 1 := by
+  have he := Real.exp_pos (α*t)
+  dsimp [layerProfile]
+  exact ⟨by positivity,(div_le_one (by positivity)).2 (by linarith)⟩
+
+lemma profileD_nonneg {α : ℝ} (hα : 0 ≤ α) (t : ℝ) : 0 ≤ layerProfileD α t := by
+  dsimp [layerProfileD]
+  positivity
+
+lemma profileDD_le {α : ℝ} (_hα : 0 ≤ α) (t : ℝ) :
+    layerProfileDD α t ≤ α*layerProfileD α t := by
+  have he := Real.exp_pos (α*t)
+  have hd : 0 < 1+Real.exp (α*t) := by positivity
+  rw [layerProfileDD,layerProfileD,div_le_iff₀ (pow_pos hd 3)]
+  have hi : α*(α*Real.exp (α*t)/(1+Real.exp (α*t))^2)*(1+Real.exp (α*t))^3 =
+      α^2*Real.exp (α*t)*(1+Real.exp (α*t)) := by field_simp
+  rw [hi]
+  exact mul_le_mul_of_nonneg_left (by linarith : 1-Real.exp (α*t) ≤ 1+Real.exp (α*t)) (by positivity)
+
+lemma profileD_lower {α t : ℝ} (hα : 0 ≤ α) (hα₂ : α ≤ 2) (ht : t ≤ 0) :
+    α*Real.exp (2*t)/4 ≤ layerProfileD α t := by
+  have he := Real.exp_pos (α*t)
+  have he1 : Real.exp (α*t) ≤ 1 := Real.exp_le_one_iff.2 (mul_nonpos_of_nonneg_of_nonpos hα ht)
+  have he2 : Real.exp (2*t) ≤ Real.exp (α*t) := Real.exp_le_exp.2 (mul_le_mul_of_nonpos_right hα₂ ht)
+  have hden : (1+Real.exp (α*t))^2 ≤ 4 := by nlinarith
+  calc
+    _ ≤ α*Real.exp (α*t)/4 := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left he2 hα) (by norm_num)
+    _ ≤ _ := div_le_div_of_nonneg_left (by positivity) (by positivity) hden
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+def layerExponent : ℝ := 1/(1+(2*(512*Real.pi)^2)^2)
+
+lemma layerExponent_bounds : 0 < layerExponent ∧ layerExponent ≤ 2 ∧
+    layerExponent*(2*(512*Real.pi)^2)^2 ≤ 2 := by
+  have hL : 0 ≤ (2*(512*Real.pi)^2)^2 := sq_nonneg _
+  have hd : 0 < 1+(2*(512*Real.pi)^2)^2 := by positivity
+  dsimp [layerExponent]
+  refine ⟨by positivity,?_,?_⟩
+  · apply (div_le_iff₀ hd).2
+    nlinarith
+  · rw [div_mul_eq_mul_div,mul_comm (1:ℝ),mul_one]
+    apply (div_le_iff₀ hd).2
+    linarith
+
+def shiftedRadius (F : ℂ → ℂ) (R : ℝ) (z : ℂ) : ℝ := logRadius F z - Real.log R
+
+def layerBarrier (F : ℂ → ℂ) (R : ℝ) (z : ℂ) : ℝ :=
+  R^2 * layerProfile layerExponent (shiftedRadius F R z)
+
+lemma DiskPair.shifted_smooth {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
+    ContDiffAt ℝ 2 (shiftedRadius F R) z := (h.logRadius_smooth hz).sub contDiffAt_const
+
+lemma DiskPair.shifted_laplacian {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
+    Laplacian.laplacian (shiftedRadius F R) z = -4/(diskFactor z)^2 := by
+  have hh := (h.logRadius_smooth hz).laplacian_sub (contDiffAt_const (c := Real.log R))
+  change Laplacian.laplacian (shiftedRadius F R) z = _ at hh
+  rw [hh,h.logRadius_laplacian hz]
+  simp
+
+lemma shifted_fderiv (F : ℂ → ℂ) (R : ℝ) (z : ℂ) :
+    fderiv ℝ (shiftedRadius F R) z = fderiv ℝ (logRadius F) z := fderiv_sub_const _
+
+lemma DiskPair.barrier_smooth {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) (R : ℝ) : ContDiffOn ℝ 2 (layerBarrier F R) (ball 0 1) := by
+  intro z hz
+  exact (((profile_smooth layerExponent).contDiffAt.comp z
+    (h.shifted_smooth (mem_ball_zero_iff.1 hz) R)).const_smul (R^2)).contDiffWithinAt
+
+lemma barrier_bounds (F : ℂ → ℂ) (R : ℝ) (z : ℂ) :
+    0 ≤ layerBarrier F R z ∧ layerBarrier F R z ≤ R^2 := by
+  have hh := profile_bounds layerExponent (shiftedRadius F R z)
+  dsimp [layerBarrier]
+  exact ⟨mul_nonneg (sq_nonneg _) hh.1,by simpa using mul_le_mul_of_nonneg_left hh.2 (sq_nonneg R)⟩
+
+lemma profile_laplacian_estimate {u : ℂ → ℝ} {z : ℂ} {a α L : ℝ}
+    (hu : ContDiffAt ℝ 2 u z) (_ha : 0 < a) (hα : 0 ≤ α) (hαL : α*L ≤ 2)
+    (hΔ : Laplacian.laplacian u z = -4/a^2)
+    (hgrad : (fderiv ℝ u z 1)^2+(fderiv ℝ u z Complex.I)^2 ≤ L/a^2) :
+    Laplacian.laplacian (layerProfile α ∘ u) z ≤ -2*layerProfileD α (u z)/a^2 := by
+  rw [laplacian_scalar_comp hu (profile_smooth α).contDiffAt]
+  have hd : deriv (layerProfile α) = layerProfileD α := funext (fun t => (profile_derivative α t).deriv)
+  rw [hd,(profile_second α (u z)).deriv,hΔ]
+  have hD := profileD_nonneg hα (u z)
+  have hDD := profileDD_le hα (u z)
+  calc
+    _ ≤ (α*layerProfileD α (u z))*(L/a^2) + layerProfileD α (u z)*(-4/a^2) := by
+      apply add_le_add_left
+      exact (mul_le_mul_of_nonneg_right hDD (add_nonneg (sq_nonneg _) (sq_nonneg _))).trans
+        (mul_le_mul_of_nonneg_left hgrad (mul_nonneg hα hD))
+    _ = layerProfileD α (u z)*(α*L-4)/a^2 := by ring
+    _ ≤ _ := by
+      apply div_le_div_of_nonneg_right _ (sq_nonneg a)
+      nlinarith [mul_le_mul_of_nonneg_left hαL hD]
+
+lemma DiskPair.barrier_drop {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
+    Laplacian.laplacian (layerBarrier F R) z ≤
+      -2*R^2*layerProfileD layerExponent (shiftedRadius F R z)/(diskFactor z)^2 := by
+  have hs := h.shifted_smooth hz R
+  have hc := (profile_smooth layerExponent).contDiffAt.comp z hs
+  have hh := InnerProductSpace.laplacian_smul (R^2) hc
+  change Laplacian.laplacian (layerBarrier F R) z = R^2*Laplacian.laplacian (layerProfile layerExponent ∘ shiftedRadius F R) z at hh
+  rw [hh]
+  have hb := profile_laplacian_estimate hs (diskFactor_pos hz)
+    layerExponent_bounds.1.le layerExponent_bounds.2.2 (h.shifted_laplacian hz R)
+    (by rw [shifted_fderiv]; exact h.logRadius_gradient_sq hz)
+  convert! mul_le_mul_of_nonneg_left hb (sq_nonneg R) using 1
+  ring
+
+lemma DiskPair.barrier_superharmonic {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) (R : ℝ) :
+    Laplacian.laplacian (layerBarrier F R) z ≤ 0 := by
+  apply (h.barrier_drop hz R).trans
+  apply div_nonpos_of_nonpos_of_nonneg _ (sq_nonneg _)
+  have hh := profileD_nonneg layerExponent_bounds.1.le (shiftedRadius F R z)
+  nlinarith [sq_nonneg R]
+
+lemma DiskPair.shifted_exp {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) {R : ℝ} (hR : 0 < R) :
+    Real.exp (shiftedRadius F R z) = diskFactor z * ‖deriv F z‖/R := by
+  have ha := diskFactor_pos hz
+  have hn := norm_pos_iff.2 (h.deriv_ne_zero (mem_ball_zero_iff.2 hz))
+  dsimp [shiftedRadius,logRadius]
+  rw [Real.exp_sub,Real.exp_add,Real.exp_log ha,Real.exp_log hn,Real.exp_log hR]
+
+lemma DiskPair.barrier_layer {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z : ℂ} (hz : ‖z‖ < 1) {R : ℝ} (hR : 0 < R)
+    (hl : diskFactor z * ‖deriv F z‖ ≤ R) :
+    layerExponent/2*‖deriv F z‖^2 ≤ -Laplacian.laplacian (layerBarrier F R) z := by
+  have ha := diskFactor_pos hz
+  have ht : shiftedRadius F R z ≤ 0 := by
+    apply Real.exp_le_one_iff.1
+    rw [h.shifted_exp hz hR]
+    exact (div_le_one hR).2 hl
+  have he : Real.exp (2*shiftedRadius F R z) = (diskFactor z * ‖deriv F z‖/R)^2 := by
+    rw [two_mul,Real.exp_add,h.shifted_exp hz hR,pow_two]
+  have hp := profileD_lower layerExponent_bounds.1.le layerExponent_bounds.2.1 ht
+  rw [he] at hp
+  have hm := mul_le_mul_of_nonneg_left hp (show 0 ≤ 2*R^2/(diskFactor z)^2 by positivity)
+  have hid : (2*R^2/(diskFactor z)^2)*(layerExponent*(diskFactor z*‖deriv F z‖/R)^2/4) =
+      layerExponent/2*‖deriv F z‖^2 := by
+    field_simp
+    ring
+  rw [hid] at hm
+  have hb := h.barrier_drop hz R
+  nlinarith [show (2*R^2/(diskFactor z)^2)*layerProfileD layerExponent (shiftedRadius F R z) =
+      -(-2*R^2*layerProfileD layerExponent (shiftedRadius F R z)/(diskFactor z)^2) by ring]
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+lemma radial_green_annulus {m d L : ℝ → ℝ} {M e R : ℝ}
+    (he : 0 < e) (heR : e ≤ R) (hR : R < 1)
+    (hd : ∀ r ∈ Ico (0:ℝ) 1, HasDerivAt m (d r) r)
+    (hc : ContinuousOn d (Ico (0:ℝ) 1))
+    (hA : ∀ r ∈ Ioo (0:ℝ) 1, HasDerivAt (fun r => r*d r) (r*L r) r)
+    (hLc : ContinuousOn L (Ioo (0:ℝ) 1))
+    (hL : ∀ r ∈ Ioo (0:ℝ) 1, L r ≤ 0)
+    (hm : ∀ r ∈ Ico (0:ℝ) 1, 0 ≤ m r ∧ m r ≤ M) :
+    (∫ r in e..R, Real.log (R/r)*(-r*L r)) ≤ M := by
+  have hR0 : 0 < R := he.trans_le heR
+  have hsub : Icc (0:ℝ) R ⊆ Ico (0:ℝ) 1 := fun r hr => ⟨hr.1,hr.2.trans_lt hR⟩
+  have hant : AntitoneOn (fun r => r*d r) (Icc (0:ℝ) R) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
+      (continuousOn_id.mul (hc.mono hsub))
+    · intro r hr
+      rw [interior_Icc] at hr
+      exact (hA r ⟨hr.1,hr.2.trans hR⟩).hasDerivWithinAt
+    · intro r hr
+      rw [interior_Icc] at hr
+      exact mul_nonpos_of_nonneg_of_nonpos hr.1.le (hL r ⟨hr.1,hr.2.trans hR⟩)
+  have hAe : e*d e ≤ 0 := by
+    simpa using hant ⟨le_rfl,hR0.le⟩ ⟨he.le,heR⟩ he.le
+  have hue : uIcc e R = Icc e R := uIcc_of_le heR
+  have hs (r : ℝ) (hr : r ∈ uIcc e R) : r ∈ Ioo (0:ℝ) 1 := by
+    rw [hue] at hr
+    exact ⟨he.trans_le hr.1,hr.2.trans_lt hR⟩
+  have hlog (r : ℝ) (hr : r ∈ uIcc e R) :
+      HasDerivAt (fun r => Real.log (R/r)) (-r⁻¹) r := by
+    have hh := (Real.hasDerivAt_log (ne_of_gt (hs r hr).1)).const_sub (Real.log R)
+    convert hh.congr_of_eventuallyEq ?_ using 1
+    filter_upwards [Ioi_mem_nhds (hs r hr).1] with t ht
+    exact Real.log_div (ne_of_gt hR0) (ne_of_gt ht)
+  have hprim (r : ℝ) (hr : r ∈ uIcc e R) :
+      HasDerivAt (fun r => -Real.log (R/r)*(r*d r)-m r)
+        (Real.log (R/r)*(-r*L r)) r := by
+    have hh := ((hlog r hr).neg.mul (hA r (hs r hr))).sub (hd r ⟨(hs r hr).1.le,(hs r hr).2⟩)
+    convert! hh using 1
+    simp only [Pi.neg_apply,neg_neg]
+    field_simp [ne_of_gt (hs r hr).1]
+    ring
+  have hi : IntervalIntegrable (fun r => Real.log (R/r)*(-r*L r)) volume e R := by
+    apply ContinuousOn.intervalIntegrable
+    apply ContinuousOn.mul
+    · exact fun r hr => (hlog r hr).continuousAt.continuousWithinAt
+    · exact (continuousOn_id.neg.mul (hLc.mono (fun r hr => hs r hr)))
+  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt hprim hi
+  have hRR : Real.log (R/R) = 0 := by simp [ne_of_gt hR0]
+  rw [hRR,neg_zero,zero_mul,zero_sub] at hftc
+  rw [hftc]
+  have hlg : 0 ≤ Real.log (R/e) := Real.log_nonneg ((one_le_div he).2 heR)
+  have hmul := mul_nonpos_of_nonneg_of_nonpos hlg hAe
+  have hme := hm e ⟨he.le,heR.trans_lt hR⟩
+  have hmR := hm R ⟨hR0.le,hR⟩
+  nlinarith
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal
+namespace SLEExactGauge.Layer
+
+lemma radial_green_integral {m d L : ℝ → ℝ} {M : ℝ}
+    (hd : ∀ r ∈ Ico (0:ℝ) 1, HasDerivAt m (d r) r)
+    (hc : ContinuousOn d (Ico (0:ℝ) 1))
+    (hA : ∀ r ∈ Ioo (0:ℝ) 1, HasDerivAt (fun r => r*d r) (r*L r) r)
+    (hLc : ContinuousOn L (Ioo (0:ℝ) 1))
+    (hL : ∀ r ∈ Ioo (0:ℝ) 1, L r ≤ 0)
+    (hm : ∀ r ∈ Ico (0:ℝ) 1, 0 ≤ m r ∧ m r ≤ M) :
+    (∫⁻ r in Ioo (0:ℝ) 1, ENNReal.ofReal (Real.log (1/r)*(-r*L r))) ≤ ENNReal.ofReal M := by
+  let e : ℕ → ℝ := fun n => (1/((n:ℝ)+1))/2
+  let R : ℕ → ℝ := fun n => 1-e n
+  have he (n : ℕ) : 0 < e n := by dsimp [e]; positivity
+  have hehalf (n : ℕ) : e n ≤ 1/2 := by
+    dsimp [e]
+    apply div_le_div_of_nonneg_right _ (by norm_num)
+    exact (div_le_one (by positivity)).2 (by have hn := Nat.cast_nonneg (α := ℝ) n; linarith)
+  have heR (n : ℕ) : e n ≤ R n := by dsimp [R]; linarith [hehalf n]
+  have hR (n : ℕ) : R n < 1 := by dsimp [R]; linarith [he n]
+  have hR0 (n : ℕ) : 0 < R n := (he n).trans_le (heR n)
+  have hsub (n : ℕ) : Icc (e n) (R n) ⊆ Ioo (0:ℝ) 1 :=
+    fun r hr => ⟨(he n).trans_le hr.1,hr.2.trans_lt (hR n)⟩
+  let f : ℕ → ℝ → ℝ≥0∞ := fun n => (Icc (e n) (R n)).indicator
+    (fun r => ENNReal.ofReal (Real.log (R n/r)*(-r*L r)))
+  let μ := volume.restrict (Ioo (0:ℝ) 1)
+  have hfm (n : ℕ) : AEMeasurable (f n) μ := by
+    apply AEMeasurable.indicator _ measurableSet_Icc
+    apply AEMeasurable.ennreal_ofReal
+    exact ((measurable_const.div measurable_id).log.aemeasurable).mul
+      (measurable_id.neg.aemeasurable.mul (hLc.aemeasurable measurableSet_Ioo))
+  have hbound (n : ℕ) : (∫⁻ r, f n r ∂μ) ≤ ENNReal.ofReal M := by
+    let q : ℝ → ℝ := fun r => Real.log (R n/r)*(-r*L r)
+    have hqc : ContinuousOn q (Icc (e n) (R n)) := by
+      apply ContinuousOn.mul
+      · exact (continuousOn_const.div continuousOn_id (fun r hr => ne_of_gt (hsub n hr).1)).log
+          (fun r hr => ne_of_gt (div_pos (hR0 n) (hsub n hr).1))
+      · exact continuousOn_id.neg.mul (hLc.mono (hsub n))
+    have hqi : IntegrableOn q (Icc (e n) (R n)) := hqc.integrableOn_Icc
+    have hqnn : 0 ≤ᵐ[volume.restrict (Icc (e n) (R n))] q := by
+      filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr
+      exact mul_nonneg (Real.log_nonneg ((one_le_div (hsub n hr).1).2 hr.2))
+        (mul_nonneg_of_nonpos_of_nonpos (neg_nonpos.2 (hsub n hr).1.le) (hL r (hsub n hr)))
+    change (∫⁻ r, (Icc (e n) (R n)).indicator (fun r => ENNReal.ofReal (q r)) r ∂μ) ≤ _
+    rw [lintegral_indicator measurableSet_Icc]
+    change (∫⁻ r, ENNReal.ofReal (q r) ∂(volume.restrict (Ioo (0:ℝ) 1)).restrict (Icc (e n) (R n))) ≤ _
+    rw [Measure.restrict_restrict measurableSet_Icc,inter_eq_left.2 (hsub n),← ofReal_integral_eq_lintegral_ofReal hqi hqnn]
+    apply ENNReal.ofReal_le_ofReal
+    rw [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le (heR n)]
+    exact radial_green_annulus (he n) (heR n) (hR n) hd hc hA hLc hL hm
+  have helim : Tendsto e atTop (𝓝 0) := by
+    simpa only [zero_div,e] using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).div_const 2
+  have hRlim : Tendsto R atTop (𝓝 1) := by
+    simpa only [sub_zero,R] using tendsto_const_nhds.sub helim
+  have hlim (r : ℝ) (hr : r ∈ Ioo (0:ℝ) 1) :
+      Tendsto (fun n => f n r) atTop (𝓝 (ENNReal.ofReal (Real.log (1/r)*(-r*L r)))) := by
+    have hevent : ∀ᶠ n in atTop, r ∈ Icc (e n) (R n) := by
+      filter_upwards [helim.eventually_lt_const hr.1,hRlim.eventually_const_lt hr.2] with n hn hn'
+      exact ⟨hn.le,hn'.le⟩
+    have ht := ENNReal.tendsto_ofReal (((hRlim.div_const r).log (div_ne_zero one_ne_zero (ne_of_gt hr.1))).mul_const (-r*L r))
+    apply ht.congr'
+    filter_upwards [hevent] with n hn
+    exact (indicator_of_mem hn (fun r => ENNReal.ofReal (Real.log (R n/r)*(-r*L r)))).symm
+  calc
+    _ = ∫⁻ r, liminf (fun n => f n r) atTop ∂μ := by
+      apply lintegral_congr_ae
+      filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr
+      exact (hlim r hr).liminf_eq.symm
+    _ ≤ liminf (fun n => ∫⁻ r, f n r ∂μ) atTop := lintegral_liminf_le' hfm
+    _ ≤ _ := liminf_le_of_frequently_le (Eventually.frequently (Eventually.of_forall hbound))
+
+lemma superharmonic_radial_green {u : ℂ → ℝ} {M : ℝ}
+    (hu : ContDiffOn ℝ 2 u (ball 0 1))
+    (hb : ∀ z ∈ ball (0:ℂ) 1, 0 ≤ u z ∧ u z ≤ M)
+    (hΔ : ∀ z ∈ ball (0:ℂ) 1, Laplacian.laplacian u z ≤ 0) :
+    (∫⁻ r in Ioo (0:ℝ) 1, ENNReal.ofReal (Real.log (1/r)*(-r*circleLMean u r))) ≤
+      ENNReal.ofReal (2*Real.pi*M) := by
+  have hrad (r : ℝ) (hr : r ∈ Ico (0:ℝ) 1) : |r| < 1 := by rw [abs_of_nonneg hr.1]; exact hr.2
+  have hθc (r : ℝ) (hr : r ∈ Ico (0:ℝ) 1) : Continuous (fun θ => u (r • circleMap 0 1 θ)) :=
+    continuousOn_univ.1 (hu.continuousOn.comp ((continuous_circleMap 0 1).const_smul r).continuousOn
+      (fun θ _ => radial_mem (hrad r hr) θ))
+  apply radial_green_integral (m := circleMean u) (d := circleDMean u)
+  · intro r hr
+    exact (circle_means_derivatives hu (hrad r hr)).1
+  · intro r hr
+    exact (circle_means_derivatives hu (hrad r hr)).2.continuousAt.continuousWithinAt
+  · intro r hr
+    exact weighted_circle_derivative hu hr.1 hr.2
+  · exact (circle_lmean_continuous hu).mono (fun r hr => ⟨by linarith [hr.1],hr.2⟩)
+  · intro r hr
+    have hh := intervalIntegral.integral_nonneg (μ := volume) (neg_le_self Real.pi_pos.le)
+      (f := fun θ => -Laplacian.laplacian u (r • circleMap 0 1 θ))
+      (fun θ _ => neg_nonneg.2 (hΔ _ (radial_mem (by rw [abs_of_pos hr.1]; exact hr.2) θ)))
+    rw [intervalIntegral.integral_neg] at hh
+    exact neg_nonneg.1 hh
+  · intro r hr
+    constructor
+    · apply intervalIntegral.integral_nonneg (neg_le_self Real.pi_pos.le)
+      intro θ _
+      exact (hb _ (radial_mem (hrad r hr) θ)).1
+    · have hi := intervalIntegral.integral_mono_on (μ := volume) (neg_le_self Real.pi_pos.le)
+        ((hθc r hr).intervalIntegrable _ _) (intervalIntegrable_const (c := M))
+        (fun θ _ => (hb _ (radial_mem (hrad r hr) θ)).2)
+      change circleMean u r ≤ _ at hi
+      exact hi.trans_eq (by simp only [intervalIntegral.integral_const,smul_eq_mul]; ring)
+
+theorem superharmonic_disk_green {u : ℂ → ℝ} {M : ℝ}
+    (hu : ContDiffOn ℝ 2 u (ball 0 1))
+    (hb : ∀ z ∈ ball (0:ℂ) 1, 0 ≤ u z ∧ u z ≤ M)
+    (hΔ : ∀ z ∈ ball (0:ℂ) 1, Laplacian.laplacian u z ≤ 0) :
+    (∫⁻ z in ball (0:ℂ) 1, ENNReal.ofReal (Real.log (1/‖z‖)*(-Laplacian.laplacian u z))) ≤
+      ENNReal.ofReal (2*Real.pi*M) := by
+  apply (polar_disk_lintegral_le _).trans
+  have he : (∫⁻ r in Ioo (0:ℝ) 1, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+      ENNReal.ofReal r * ENNReal.ofReal (Real.log (1/‖r • circleMap 0 1 θ‖)*
+        (-Laplacian.laplacian u (r • circleMap 0 1 θ)))) =
+      ∫⁻ r in Ioo (0:ℝ) 1, ENNReal.ofReal (Real.log (1/r)*(-r*circleLMean u r)) := by
+    apply lintegral_congr_ae
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr
+    have hrabs : |r| < 1 := by rw [abs_of_pos hr.1]; exact hr.2
+    let q : ℝ → ℝ := fun θ => Real.log (1/r)*(-r*Laplacian.laplacian u (r • circleMap 0 1 θ))
+    have hq : Continuous q := ((circle_laplacian_continuous hu hrabs).const_mul (-r)).const_mul _
+    have hqi : IntegrableOn q (Ioc (-Real.pi) Real.pi) := (hq.intervalIntegrable _ _).1
+    have hq0 : 0 ≤ᵐ[volume.restrict (Ioc (-Real.pi) Real.pi)] q := by
+      filter_upwards with θ
+      apply mul_nonneg
+      · exact Real.log_nonneg ((one_le_div hr.1).2 hr.2.le)
+      · exact mul_nonneg_of_nonpos_of_nonpos (neg_nonpos.2 hr.1.le) (hΔ _ (radial_mem hrabs θ))
+    have hqint : (∫ θ in -Real.pi..Real.pi, q θ) = Real.log (1/r)*(-r*circleLMean u r) := by
+      dsimp only [q,circleLMean]
+      rw [intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul]
+    calc
+      _ = ∫⁻ θ in Ioo (-Real.pi) Real.pi, ENNReal.ofReal (q θ) := by
+        apply lintegral_congr
+        intro θ
+        rw [radial_norm,abs_of_pos hr.1,← ENNReal.ofReal_mul hr.1.le]
+        congr 1
+        dsimp only [q]
+        ring
+      _ = ENNReal.ofReal (∫ θ in -Real.pi..Real.pi, q θ) := by
+        rw [restrict_Ioo_eq_restrict_Ioc,intervalIntegral.integral_of_le (neg_le_self Real.pi_pos.le)]
+        exact (ofReal_integral_eq_lintegral_ofReal hqi hq0).symm
+      _ = _ := congrArg ENNReal.ofReal hqint
+  rw [he]
+  exact superharmonic_radial_green hu hb hΔ
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+lemma univalent_not_locally_constant {F : ℂ → ℂ} (hi : InjOn F (ball 0 1))
+    {c : ℂ} (hc : c ∈ ball (0:ℂ) 1) : ¬ ∀ᶠ z in 𝓝 c, F z = F c := by
+  intro hh
+  have he : ∀ᶠ z in 𝓝 c, z = c := by
+    filter_upwards [hh,isOpen_ball.mem_nhds hc] with z hz hzD
+    exact hi hzD hc hz
+  have hb : ∀ᶠ z in 𝓝[≠] c, False := by
+    filter_upwards [he.filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with z hz hne
+    exact hne hz
+  obtain ⟨z,hz⟩ := hb.exists
+  exact hz
+
+lemma univalent_punctured_derivative {F : ℂ → ℂ}
+    (hf : DifferentiableOn ℂ F (ball 0 1)) (hi : InjOn F (ball 0 1))
+    {c : ℂ} (hc : c ∈ ball (0:ℂ) 1) : ∀ᶠ z in 𝓝[≠] c, deriv F z ≠ 0 := by
+  have ha := hf.analyticOnNhd isOpen_ball c hc
+  apply ha.deriv.eventually_eq_zero_or_eventually_ne_zero.resolve_left
+  intro hz
+  obtain ⟨r,hr,hsub⟩ := Metric.mem_nhds_iff.1 (hz.and (isOpen_ball.mem_nhds hc))
+  have hbd : ball c r ⊆ ball (0:ℂ) 1 := fun z hz => (hsub hz).2
+  have hconst : ∀ᶠ z in 𝓝 c, F z = F c := by
+    filter_upwards [ball_mem_nhds c hr] with z hz
+    exact isOpen_ball.is_const_of_deriv_eq_zero (convex_ball c r).isPreconnected
+      (hf.mono hbd) (fun z hz => (hsub hz).1) hz (mem_ball_self hr)
+  exact univalent_not_locally_constant hi hc hconst
+
+theorem univalent_diskPair {F : ℂ → ℂ}
+    (hf : DifferentiableOn ℂ F (ball 0 1)) (hi : InjOn F (ball 0 1)) :
+    ∃ G : ℂ → ℂ, DiskPair (F '' ball 0 1) F G := by
+  let U := F '' ball (0:ℂ) 1
+  let G := invFunOn F (ball (0:ℂ) 1)
+  have hl : ∀ z ∈ ball (0:ℂ) 1, G (F z) = z := hi.leftInvOn_invFunOn
+  have hr : ∀ w ∈ U, F (G w) = w := (surjOn_image F (ball (0:ℂ) 1)).rightInvOn_invFunOn
+  have hgmap : MapsTo G U (ball (0:ℂ) 1) := (surjOn_image F (ball (0:ℂ) 1)).mapsTo_invFunOn
+  have hopen (c : ℂ) (hc : c ∈ ball (0:ℂ) 1) : 𝓝 (F c) ≤ map F (𝓝 c) :=
+    ((hf.analyticOnNhd isOpen_ball c hc).eventually_constant_or_nhds_le_map_nhds).resolve_left
+      (univalent_not_locally_constant hi hc)
+  have hU : IsOpen U := by
+    apply isOpen_iff_mem_nhds.2
+    rintro w ⟨c,hc,rfl⟩
+    apply hopen c hc
+    change ∀ᶠ z in 𝓝 c, F z ∈ U
+    filter_upwards [isOpen_ball.mem_nhds hc] with z hz
+    exact mem_image_of_mem F hz
+  have hGc (w : ℂ) (hw : w ∈ U) : ContinuousAt G w := by
+    rcases hw with ⟨c,hc,rfl⟩
+    rw [ContinuousAt,hl c hc]
+    calc
+      map G (𝓝 (F c)) ≤ map G (map F (𝓝 c)) := map_mono (hopen c hc)
+      _ = map id (𝓝 c) := by
+        rw [map_map]
+        apply Filter.map_congr
+        filter_upwards [isOpen_ball.mem_nhds hc] with z hz
+        exact hl z hz
+      _ = 𝓝 c := map_id
+  have hgood (w : ℂ) (hw : w ∈ U) (hn : deriv F (G w) ≠ 0) : DifferentiableAt ℂ G w := by
+    apply HasDerivAt.differentiableAt (f' := (deriv F (G w))⁻¹)
+    apply HasDerivAt.of_local_left_inverse (hGc w hw)
+      (hf.differentiableAt (isOpen_ball.mem_nhds (hgmap hw))).hasDerivAt hn
+    filter_upwards [hU.mem_nhds hw] with y hy
+    exact hr y hy
+  have hGd (w : ℂ) (hw : w ∈ U) : DifferentiableAt ℂ G w := by
+    have hc := hgmap hw
+    have htn : Tendsto G (𝓝[≠] w) (𝓝[≠] G w) := by
+      apply tendsto_nhdsWithin_iff.2
+      refine ⟨(hGc w hw).tendsto.mono_left nhdsWithin_le_nhds,?_⟩
+      filter_upwards [(show ∀ᶠ y in 𝓝 w, y ∈ U from hU.mem_nhds hw).filter_mono nhdsWithin_le_nhds,self_mem_nhdsWithin] with y hy hne
+      change G y ≠ G w
+      intro hgy
+      apply hne
+      change y = w
+      rw [← hr y hy,hgy,hr w hw]
+    have hne := htn.eventually (univalent_punctured_derivative hf hi hc)
+    apply (Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt ?_ (hGc w hw)).differentiableAt
+    filter_upwards [hne,(show ∀ᶠ y in 𝓝 w, y ∈ U from hU.mem_nhds hw).filter_mono nhdsWithin_le_nhds] with y hn hy
+    exact hgood y hy hn
+  exact ⟨G,hU,(fun z hz => mem_image_of_mem F hz),hgmap,hl,hr,hf,fun w hw => (hGd w hw).differentiableWithinAt⟩
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/ConformalLocalization.lean b/lean/OAI/Probability/SLELocalization/ConformalLocalization.lean
index 8422afae57664f2fdd416ce5feaba2694aa9a9a0..77b53cc7a85ffcb778193f7b26188c65ae98724e 100644
--- a/lean/OAI/Probability/SLELocalization/ConformalLocalization.lean
+++ b/lean/OAI/Probability/SLELocalization/ConformalLocalization.lean
@@ -1,375 +1,375 @@
-import Mathlib
-import OAI.Probability.SLELocalization.EuclideanApproach
-
-namespace OAI
-
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-theorem strip_slab_half_mean {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
-    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-    (hlarge : 32*Real.pi*localizationMeanConstant ≤ H)
-    (j : ℤ) (hj : (j:ℝ) ≤ localizationStripWidth*H) :
-    stripMean H q r Fbar (unitSlab j) ≤ H/2 := by
-  let K := localizationMeanConstant
-  let E := totalStripEnergy H q r Fbar
-  have hK : 0 < K := localizationMeanConstant_pos
-  have hE : 0 ≤ E := totalStripEnergy_nonneg _ _ _ _
-  have hEt := totalStripEnergy_bound hF hinj hext hH hr q
-  change E ≤ 2*Real.pi*H at hEt
-  have hKc : slabMeanCoefficient ≤ K := by dsimp [K,localizationMeanConstant]; linarith [Real.pi_pos]
-  have hKπ : Real.pi ≤ K := by dsimp [K,localizationMeanConstant]; linarith [slabMeanCoefficient_pos]
-  have hm : (stripMean H q r Fbar (unitSlab j))^2 ≤ K*(max (j:ℝ) 0+2)*E := by
-    by_cases hj0 : 0 ≤ j
-    · have hjr : (0:ℝ) ≤ j := by exact_mod_cast hj0
-      rw [max_eq_left hjr]
-      exact (positive_unitSlab_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess j hjr).trans
-        (by gcongr)
-    · have hjr : (j:ℝ) ≤ -1 := by exact_mod_cast (show j ≤ -1 by omega)
-      rw [max_eq_right (by linarith : (j:ℝ) ≤ 0),zero_add]
-      exact (negative_unitSlab_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess j hjr).trans
-        (mul_le_mul_of_nonneg_right (by linarith) hE)
-  have hmax : max (j:ℝ) 0 ≤ localizationStripWidth*H :=
-    max_le hj (mul_nonneg localizationStripWidth_pos.le hH.le)
-  have hbpos := localizationStripWidth_pos
-  have hfinal : K*(max (j:ℝ) 0+2)*E ≤ (H/2)^2 := by
-    calc
-      _ ≤ K*(localizationStripWidth*H+2)*(2*Real.pi*H) := by gcongr
-      _ = H^2/16+4*Real.pi*K*H := by
-        unfold localizationStripWidth
-        change K*((1/(32*Real.pi*K))*H+2)*(2*Real.pi*H) = _
-        field_simp
-        ring
-      _ ≤ (H/2)^2 := by
-        have ht := mul_le_mul_of_nonneg_right hlarge hH.le
-        change (32*Real.pi*K)*H ≤ H*H at ht
-        nlinarith
-  have hh := hm.trans hfinal
-  nlinarith [sq_nonneg (stripMean H q r Fbar (unitSlab j)+H/2)]
-
-lemma unitSlab_cells_disjoint : Pairwise (fun j k : ℤ => Disjoint (unitSlab j).cells (unitSlab k).cells) := by
-  intro j k hjk
-  have hdis (j k : ℤ) (hjk : j < k) : Disjoint (unitSlab j).cells (unitSlab k).cells := by
-    apply disjoint_left.mpr
-    intro x hx hy
-    change x.1 ∈ Ioc (j:ℝ) (j+1) ∧ _ at hx
-    change x.1 ∈ Ioc (k:ℝ) (k+1) ∧ _ at hy
-    have hjk' : (j:ℝ)+1 ≤ k := by exact_mod_cast (show j+1 ≤ k by omega)
-    linarith [hx.1.2,hy.1.1]
-  rcases lt_or_gt_of_ne hjk with h | h
-  · exact hdis j k h
-  · exact (hdis k j h).symm
-
-lemma DRect.measure_eq_restrict (Q : DRect) : Q.measure = volume.restrict Q.cells := by
-  rw [DRect.measure,DRect.cells,Measure.volume_eq_prod]
-  change _ = (volume.prod volume).restrict ((Ioc Q.a Q.b) ×ˢ (Ioc Q.c Q.d))
-  rw [← Measure.prod_restrict]
-
-theorem strip_slab_energy_sum {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    (∑' j : ℤ, ENNReal.ofReal (stripEnergy H q r Fbar (unitSlab j))) ≤
-      ENNReal.ofReal (2*Real.pi*H) := by
-  let G := fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
-  let ν := (volume : Measure (ℝ × ℝ)).withDensity (fun p => ENNReal.ofReal (G p))
-  let S := (univ : Set ℝ).prod (Ioc (0:ℝ) Real.pi)
-  have hmeas (Q : DRect) : MeasurableSet Q.cells := measurableSet_Ioc.prod measurableSet_Ioc
-  have he (j : ℤ) : ENNReal.ofReal (stripEnergy H q r Fbar (unitSlab j)) = ν (unitSlab j).cells := by
-    rw [withDensity_apply _ (hmeas _)]
-    have hi := (strip_gradient_rect_memLp hF hinj hext hH hr q (j:ℝ) (j+1) le_rfl le_rfl).integrable_sq
-    change Integrable G (unitSlab j).measure at hi
-    rw [DRect.measure_eq_restrict] at hi
-    rw [← ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun p => sq_nonneg _))]
-    congr 1
-    exact congrArg (fun μ => ∫ p, G p ∂μ) (DRect.measure_eq_restrict _)
-  have hS : ν S = ENNReal.ofReal (totalStripEnergy H q r Fbar) := by
-    change ((volume : Measure (ℝ × ℝ)).withDensity (fun p => ENNReal.ofReal (G p)))
-      ((univ : Set ℝ) ×ˢ Ioc (0:ℝ) Real.pi) = _
-    rw [withDensity_apply _ (MeasurableSet.univ.prod measurableSet_Ioc)]
-    change (∫⁻ p, ENNReal.ofReal (G p) ∂(volume.prod volume).restrict ((univ : Set ℝ) ×ˢ Ioc (0:ℝ) Real.pi)) = _
-    rw [← Measure.prod_restrict,Measure.restrict_univ]
-    have hi := strip_energy_integrable_prod hF hinj hext hH hr q
-    rw [restrict_Ioo_eq_restrict_Ioc] at hi
-    exact (ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun p => sq_nonneg _))).symm
-  rw [tsum_congr he,← measure_iUnion unitSlab_cells_disjoint (fun j => hmeas (unitSlab j))]
-  calc
-    _ ≤ ν S := measure_mono (by
-      intro p hp
-      obtain ⟨j,hj⟩ := mem_iUnion.mp hp
-      exact ⟨mem_univ _,hj.2⟩)
-    _ = ENNReal.ofReal (totalStripEnergy H q r Fbar) := hS
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (totalStripEnergy_bound hF hinj hext hH hr q)
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma energy_exp_linearize {A a H E : ℝ} (hA : 0 ≤ A) (ha : 0 ≤ a)
-    (hH : 0 < H) (hE : 0 ≤ E) (hEt : E ≤ 2*Real.pi*H) :
-    A*(E/H^2)*Real.exp (-a*H^2/E) ≤
-      ((A/H^2)*Real.exp (-(a/(2*Real.pi))*H))*E := by
-  by_cases h0 : E = 0
-  · simp [h0]
-  have hEp : 0 < E := lt_of_le_of_ne hE (Ne.symm h0)
-  have hexp : -a*H^2/E ≤ -(a/(2*Real.pi))*H := by
-    rw [div_le_iff₀ hEp]
-    have hmul := mul_le_mul_of_nonneg_left hEt
-      (by positivity : 0 ≤ a*H/(2*Real.pi))
-    have hh : (a*H/(2*Real.pi))*(2*Real.pi*H) = a*H^2 := by
-      field_simp
-    rw [hh] at hmul
-    convert neg_le_neg hmul using 1 <;> ring
-  calc
-    _ ≤ A*(E/H^2)*Real.exp (-(a/(2*Real.pi))*H) := by gcongr
-    _ = _ := by ring
-
-theorem energy_event_union {ι Ω : Type*} [Countable ι] [MeasurableSpace Ω]
-    (P : Measure Ω) (events : ι → Set Ω) (E : ι → ℝ)
-    {A a H : ℝ} (hA : 0 ≤ A) (ha : 0 ≤ a) (hH : 1 ≤ H)
-    (hE : ∀ j, 0 ≤ E j) (hEt : ∀ j, E j ≤ 2*Real.pi*H)
-    (hsum : (∑' j, ENNReal.ofReal (E j)) ≤ ENNReal.ofReal (2*Real.pi*H))
-    (hhit : ∀ j, P (events j) ≤ ENNReal.ofReal (A*(E j/H^2)*Real.exp (-a*H^2/E j))) :
-    P (⋃ j, events j) ≤ ENNReal.ofReal ((2*Real.pi*A)*Real.exp (-(a/(2*Real.pi))*H)) := by
-  have hHp : 0 < H := lt_of_lt_of_le zero_lt_one hH
-  let k := (A/H^2)*Real.exp (-(a/(2*Real.pi))*H)
-  have hk : 0 ≤ k := by dsimp [k]; positivity
-  calc
-    _ ≤ ∑' j, P (events j) := measure_iUnion_le _
-    _ ≤ ∑' j, ENNReal.ofReal (k*E j) := ENNReal.tsum_le_tsum (fun j =>
-      (hhit j).trans (ENNReal.ofReal_le_ofReal (energy_exp_linearize hA ha hHp (hE j) (hEt j))))
-    _ = ENNReal.ofReal k*(∑' j, ENNReal.ofReal (E j)) := by
-      simp_rw [ENNReal.ofReal_mul hk]
-      exact ENNReal.tsum_mul_left
-    _ ≤ ENNReal.ofReal k*ENNReal.ofReal (2*Real.pi*H) := by gcongr
-    _ = ENNReal.ofReal (k*(2*Real.pi*H)) := (ENNReal.ofReal_mul hk).symm
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (by
-      have hid : k*(2*Real.pi*H) = (2*Real.pi*A/H)*Real.exp (-(a/(2*Real.pi))*H) := by
-        dsimp [k]
-        field_simp
-      rw [hid]
-      apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
-      exact div_le_self (by positivity) hH)
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-def logEnergyExit {Ω : Type*} (γ : Ω → ℝ → ℂ) (Fbar : ℂ → OnePoint ℂ)
-    (H : ℝ) (q : ℂ) (r : ℝ) (ω : Ω) : Prop :=
-  ∃ x ∈ logTrace γ ω, x.1 ≤ localizationStripWidth*H ∧ 0 ≤ x.2 ∧ x.2 ≤ Real.pi ∧
-    stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H
-
-lemma rectCoefficient_unitSlab (j : ℝ) :
-    rectCoefficient (unitSlab j) = rectCoefficient (unitSlab 0) := by
-  unfold rectCoefficient unitSlab
-  dsimp only
-  rw [show j+1-j=1 by ring]
-  norm_num
-
-theorem logTrace_localization (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ c C H₀ : ℝ, 0 < c ∧ 0 < C ∧ 0 < H₀ ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (F : ℂ → ℂ) (_hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-        (_hinj : InjOn F {z : ℂ | 0 < z.im}) (Fbar : ℂ → OnePoint ℂ)
-        (_hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-        (_hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-        (q : ℂ) (r : ℝ) (_hr : 0 < r)
-        (η : ℝ → ℂ) (_hη : ContinuousOn η (Icc 0 1)) (_hη0 : η 0 = 0)
-        (_hη1 : ‖η 1‖ = 1) (_hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-        (_haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-          (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-        (H : ℝ), H₀ ≤ H → P {ω | logEnergyExit γ Fbar H q r ω} ≤
-          ENNReal.ofReal (C*Real.exp (-c*H)) := by
-  obtain ⟨C₀,ρ,hC₀,hρ,hρ1,hhit⟩ := logTrace_dyadic_hitting κ hκ
-  let D := rectCoefficient (unitSlab 0)
-  let T := ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j
-  let A := 64*D*(C₀+1)*(T+1)/ρ
-  let a := (-Real.log ρ)/(48*D)
-  let H₀ := max 1 (32*Real.pi*localizationMeanConstant)
-  have hD : 0 < D := rectCoefficient_pos _
-  have hT : 0 ≤ T := tsum_nonneg (fun j => mul_nonneg (sq_nonneg _) (pow_nonneg hρ.le j))
-  have hA : 0 < A := by dsimp [A]; positivity
-  have ha : 0 < a := div_pos (neg_pos.2 (Real.log_neg hρ hρ1)) (by positivity)
-  have hH₀ : 0 < H₀ := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
-  refine ⟨a/(2*Real.pi),2*Real.pi*A,H₀,by positivity,by positivity,hH₀,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps F hF hinj Fbar hext hcont q r hr η hη hη0 hη1 hηH haccess H hlarge
-  have hH1 : 1 ≤ H := (le_max_left _ _).trans hlarge
-  have hH : 0 < H := lt_of_lt_of_le zero_lt_one hH1
-  have hHH : 32*Real.pi*localizationMeanConstant ≤ H := (le_max_right _ _).trans hlarge
-  let E := fun j : ℤ => stripEnergy H q r Fbar (unitSlab j)
-  let events := fun j : ℤ => {ω | (j:ℝ) ≤ localizationStripWidth*H ∧
-    ∃ x ∈ logTrace γ ω, x ∈ (unitSlab j).closed ∧
-      stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H}
-  have hsub : {ω | logEnergyExit γ Fbar H q r ω} ⊆ ⋃ j, events j := by
-    rintro ω ⟨x,hx,hxv,hx0,hxπ,hxH⟩
-    apply mem_iUnion.mpr
-    refine ⟨Int.floor x.1,?_,x,hx,?_,hxH⟩
-    · exact (Int.floor_le x.1).trans hxv
-    · exact ⟨⟨Int.floor_le x.1,(Int.lt_floor_add_one x.1).le⟩,hx0,hxπ⟩
-  have hE : ∀ j, 0 ≤ E j := fun j => integral_nonneg (fun _ => sq_nonneg _)
-  have hEt : ∀ j, E j ≤ 2*Real.pi*H := fun j =>
-    (stripEnergy_le_total hF hinj hext hH hr q (unitSlab j) le_rfl le_rfl).trans
-      (totalStripEnergy_bound hF hinj hext hH hr q)
-  have hsum := strip_slab_energy_sum hF hinj hext hH hr q
-  have hev (j : ℤ) : P (events j) ≤ ENNReal.ofReal (A*(E j/H^2)*Real.exp (-a*H^2/E j)) := by
-    by_cases hj : (j:ℝ) ≤ localizationStripWidth*H
-    · have hm := strip_slab_half_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess hHH j hj
-      have hb := strip_rectangle_hitting hF hinj hext hcont hH hr q (unitSlab j) le_rfl le_rfl hm
-        P (logTrace γ) hC₀.le hρ hρ1 (hhit P B hB γ g f hγ hmaps j)
-      dsimp only at hb
-      rw [rectCoefficient_unitSlab] at hb
-      change P _ ≤ ENNReal.ofReal (A*(E j/H^2)*Real.exp (-a*H^2/E j)) at hb
-      exact (measure_mono (fun ω hω => hω.2)).trans hb
-    · have he : events j = ∅ := by ext ω; simp [events,hj]
-      rw [he,measure_empty]
-      exact bot_le
-  exact (measure_mono hsub).trans (energy_event_union P events E hA.le ha.le hH1 hE hEt hsum hev)
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology ENNReal NNReal OnePoint
-namespace SLEExactGauge
-open RectangleEnergy
-
-def conformalExit (Fbar : ℂ → OnePoint ℂ) (γ : ℝ → ℂ)
-    (q : ℂ) (radius A : ℝ) : Prop :=
-  ∃ t : ℝ, 0 ≤ t ∧ (∀ s : ℝ, 0 ≤ s → s < t → ‖γ s‖ < A) ∧
-    Fbar (γ t) ∉ (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.ball q radius
-
-lemma norm_le_of_pre_modulus {γ : ℝ → ℂ} (hc : ContinuousOn γ (Ici 0))
-    (h0 : γ 0 = 0) {t A : ℝ} (ht : 0 ≤ t) (hA : 0 < A)
-    (hpre : ∀ s : ℝ, 0 ≤ s → s < t → ‖γ s‖ < A) : ‖γ t‖ ≤ A := by
-  by_contra hn
-  have hgt : A < ‖γ t‖ := lt_of_not_ge hn
-  have hm : A ∈ Icc ‖γ 0‖ ‖γ t‖ := by simpa [h0] using And.intro hA.le hgt.le
-  obtain ⟨s,hs,he⟩ := intermediate_value_Icc ht (hc.mono Icc_subset_Ici_self).norm hm
-  have hst : s < t := lt_of_le_of_ne hs.2 (by intro heq; subst s; linarith)
-  exact (ne_of_lt (hpre s hs.1 hst)) he
-
-lemma conformalExit_logEnergyExit {Ω : Type*} {γ : Ω → ℝ → ℂ}
-    {ω : Ω} {Fbar : ℂ → OnePoint ℂ} {q : ℂ} {r L : ℝ}
-    (hr : 0 < r) (hL : 1 < L) (hc : ContinuousOn (γ ω) (Ici 0))
-    (h0 : γ ω 0 = 0) (hhalf : ∀ t, 0 ≤ t → 0 ≤ (γ ω t).im)
-    (hstart : Fbar 0 ∈ (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-    (hexit : conformalExit Fbar (γ ω) q (L*r) (L^localizationStripWidth)) :
-      logEnergyExit γ Fbar (Real.log L) q r ω := by
-  obtain ⟨t,ht,hpre,hout⟩ := hexit
-  have hLp : 0 < L := zero_lt_one.trans hL
-  have hz : γ ω t ≠ 0 := by
-    intro he
-    apply hout
-    rw [he]
-    obtain ⟨w,hw,heq⟩ := hstart
-    refine ⟨w,?_,heq⟩
-    exact mem_ball.2 (lt_of_le_of_lt (mem_closedBall.1 hw) (by nlinarith))
-  have hn := norm_le_of_pre_modulus hc h0 ht (Real.rpow_pos_of_pos hLp _) hpre
-  let x : ℝ × ℝ := (Real.log ‖γ ω t‖, Complex.arg (γ ω t))
-  have hexp : Complex.exp ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = γ ω t := by
-    change Complex.exp (Complex.log (γ ω t)) = γ ω t
-    exact Complex.exp_log hz
-  refine ⟨x,⟨t,ht,hexp.symm⟩,?_,Complex.arg_nonneg_iff.2 (hhalf t ht),Complex.arg_le_pi _,?_⟩
-  · have hb := Real.log_le_log (norm_pos_iff.2 hz) hn
-    rw [Real.log_rpow hLp] at hb
-    exact hb
-  · unfold stripPotential
-    rw [hexp]
-    apply sphereTruncation_top_of_outside (Real.log_pos hL).le hr q
-    simpa [Real.exp_log hLp] using hout
-
-theorem uniform_conformal_localization (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ b c C L₀ : ℝ, 0 < b ∧ 0 < c ∧ 0 < C ∧ 1 < L₀ ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ)
-        (_hF : AnalyticOnNhd ℂ F halfPlane) (_hinj : Set.InjOn F halfPlane)
-        (_hext : ∀ z ∈ halfPlane, Fbar z = (F z : OnePoint ℂ))
-        (_hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-        (η : ℝ → ℂ) (_hη : ContinuousOn η (Icc 0 1)) (_hη0 : η 0 = 0)
-        (_hη1 : ‖η 1‖ = 1) (_hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-        (q : ℂ) (r : ℝ) (_hr : 0 < r)
-        (_haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-          (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.closedBall q r)
-        (L : ℝ), L₀ ≤ L →
-        P {sample | conformalExit Fbar (γ sample) q (L*r) (L^b)} ≤
-          ENNReal.ofReal (C*L^(-c)) := by
-  obtain ⟨c,C,H₀,hc,hC,hH₀,hlog⟩ := logTrace_localization κ hκ
-  refine ⟨localizationStripWidth,c,C,Real.exp (H₀+1),localizationStripWidth_pos,hc,hC,?_,?_⟩
-  · exact Real.one_lt_exp_iff.2 (by linarith)
-  intro Ω _ P _ B hB γ g f hγ hmaps F Fbar hF hinj hext hcont η hη hη0 hη1 hηH q r hr haccess L hlarge
-  have hL : 1 < L := (Real.one_lt_exp_iff.2 (by linarith : 0 < H₀+1)).trans_le hlarge
-  have hLp : 0 < L := zero_lt_one.trans hL
-  have hH : H₀ ≤ Real.log L := by
-    have hh := Real.log_le_log (Real.exp_pos (H₀+1)) hlarge
-    rw [Real.log_exp] at hh
-    linarith
-  have hstart : Fbar 0 ∈ (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r := by
-    simpa [hη0] using haccess 0 (by norm_num : (0:ℝ) ∈ Icc 0 1)
-  have hm : P {ω | conformalExit Fbar (γ ω) q (L*r) (L^localizationStripWidth)} ≤
-      P {ω | logEnergyExit γ Fbar (Real.log L) q r ω} := by
-    apply measure_mono_ae
-    filter_upwards [hmaps] with ω hw
-    intro hexit
-    exact conformalExit_logEnergyExit hr hL hw.1 hw.2.1 hw.2.2.1 hstart hexit
-  have hb := hlog P B hB γ g f hγ hmaps F hF hinj Fbar hext hcont q r hr η hη hη0 hη1 hηH haccess (Real.log L) hH
-  have he : Real.exp (-c*Real.log L) = L^(-c) := by
-    rw [Real.rpow_def_of_pos hLp]
-    congr 1
-    ring
-  rw [he] at hb
-  exact hm.trans hb
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.EuclideanApproach
+
+namespace OAI
+
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+theorem strip_slab_half_mean {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
+    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+    (hlarge : 32*Real.pi*localizationMeanConstant ≤ H)
+    (j : ℤ) (hj : (j:ℝ) ≤ localizationStripWidth*H) :
+    stripMean H q r Fbar (unitSlab j) ≤ H/2 := by
+  let K := localizationMeanConstant
+  let E := totalStripEnergy H q r Fbar
+  have hK : 0 < K := localizationMeanConstant_pos
+  have hE : 0 ≤ E := totalStripEnergy_nonneg _ _ _ _
+  have hEt := totalStripEnergy_bound hF hinj hext hH hr q
+  change E ≤ 2*Real.pi*H at hEt
+  have hKc : slabMeanCoefficient ≤ K := by dsimp [K,localizationMeanConstant]; linarith [Real.pi_pos]
+  have hKπ : Real.pi ≤ K := by dsimp [K,localizationMeanConstant]; linarith [slabMeanCoefficient_pos]
+  have hm : (stripMean H q r Fbar (unitSlab j))^2 ≤ K*(max (j:ℝ) 0+2)*E := by
+    by_cases hj0 : 0 ≤ j
+    · have hjr : (0:ℝ) ≤ j := by exact_mod_cast hj0
+      rw [max_eq_left hjr]
+      exact (positive_unitSlab_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess j hjr).trans
+        (by gcongr)
+    · have hjr : (j:ℝ) ≤ -1 := by exact_mod_cast (show j ≤ -1 by omega)
+      rw [max_eq_right (by linarith : (j:ℝ) ≤ 0),zero_add]
+      exact (negative_unitSlab_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess j hjr).trans
+        (mul_le_mul_of_nonneg_right (by linarith) hE)
+  have hmax : max (j:ℝ) 0 ≤ localizationStripWidth*H :=
+    max_le hj (mul_nonneg localizationStripWidth_pos.le hH.le)
+  have hbpos := localizationStripWidth_pos
+  have hfinal : K*(max (j:ℝ) 0+2)*E ≤ (H/2)^2 := by
+    calc
+      _ ≤ K*(localizationStripWidth*H+2)*(2*Real.pi*H) := by gcongr
+      _ = H^2/16+4*Real.pi*K*H := by
+        unfold localizationStripWidth
+        change K*((1/(32*Real.pi*K))*H+2)*(2*Real.pi*H) = _
+        field_simp
+        ring
+      _ ≤ (H/2)^2 := by
+        have ht := mul_le_mul_of_nonneg_right hlarge hH.le
+        change (32*Real.pi*K)*H ≤ H*H at ht
+        nlinarith
+  have hh := hm.trans hfinal
+  nlinarith [sq_nonneg (stripMean H q r Fbar (unitSlab j)+H/2)]
+
+lemma unitSlab_cells_disjoint : Pairwise (fun j k : ℤ => Disjoint (unitSlab j).cells (unitSlab k).cells) := by
+  intro j k hjk
+  have hdis (j k : ℤ) (hjk : j < k) : Disjoint (unitSlab j).cells (unitSlab k).cells := by
+    apply disjoint_left.mpr
+    intro x hx hy
+    change x.1 ∈ Ioc (j:ℝ) (j+1) ∧ _ at hx
+    change x.1 ∈ Ioc (k:ℝ) (k+1) ∧ _ at hy
+    have hjk' : (j:ℝ)+1 ≤ k := by exact_mod_cast (show j+1 ≤ k by omega)
+    linarith [hx.1.2,hy.1.1]
+  rcases lt_or_gt_of_ne hjk with h | h
+  · exact hdis j k h
+  · exact (hdis k j h).symm
+
+lemma DRect.measure_eq_restrict (Q : DRect) : Q.measure = volume.restrict Q.cells := by
+  rw [DRect.measure,DRect.cells,Measure.volume_eq_prod]
+  change _ = (volume.prod volume).restrict ((Ioc Q.a Q.b) ×ˢ (Ioc Q.c Q.d))
+  rw [← Measure.prod_restrict]
+
+theorem strip_slab_energy_sum {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    (∑' j : ℤ, ENNReal.ofReal (stripEnergy H q r Fbar (unitSlab j))) ≤
+      ENNReal.ofReal (2*Real.pi*H) := by
+  let G := fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
+  let ν := (volume : Measure (ℝ × ℝ)).withDensity (fun p => ENNReal.ofReal (G p))
+  let S := (univ : Set ℝ).prod (Ioc (0:ℝ) Real.pi)
+  have hmeas (Q : DRect) : MeasurableSet Q.cells := measurableSet_Ioc.prod measurableSet_Ioc
+  have he (j : ℤ) : ENNReal.ofReal (stripEnergy H q r Fbar (unitSlab j)) = ν (unitSlab j).cells := by
+    rw [withDensity_apply _ (hmeas _)]
+    have hi := (strip_gradient_rect_memLp hF hinj hext hH hr q (j:ℝ) (j+1) le_rfl le_rfl).integrable_sq
+    change Integrable G (unitSlab j).measure at hi
+    rw [DRect.measure_eq_restrict] at hi
+    rw [← ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun p => sq_nonneg _))]
+    congr 1
+    exact congrArg (fun μ => ∫ p, G p ∂μ) (DRect.measure_eq_restrict _)
+  have hS : ν S = ENNReal.ofReal (totalStripEnergy H q r Fbar) := by
+    change ((volume : Measure (ℝ × ℝ)).withDensity (fun p => ENNReal.ofReal (G p)))
+      ((univ : Set ℝ) ×ˢ Ioc (0:ℝ) Real.pi) = _
+    rw [withDensity_apply _ (MeasurableSet.univ.prod measurableSet_Ioc)]
+    change (∫⁻ p, ENNReal.ofReal (G p) ∂(volume.prod volume).restrict ((univ : Set ℝ) ×ˢ Ioc (0:ℝ) Real.pi)) = _
+    rw [← Measure.prod_restrict,Measure.restrict_univ]
+    have hi := strip_energy_integrable_prod hF hinj hext hH hr q
+    rw [restrict_Ioo_eq_restrict_Ioc] at hi
+    exact (ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun p => sq_nonneg _))).symm
+  rw [tsum_congr he,← measure_iUnion unitSlab_cells_disjoint (fun j => hmeas (unitSlab j))]
+  calc
+    _ ≤ ν S := measure_mono (by
+      intro p hp
+      obtain ⟨j,hj⟩ := mem_iUnion.mp hp
+      exact ⟨mem_univ _,hj.2⟩)
+    _ = ENNReal.ofReal (totalStripEnergy H q r Fbar) := hS
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (totalStripEnergy_bound hF hinj hext hH hr q)
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma energy_exp_linearize {A a H E : ℝ} (hA : 0 ≤ A) (ha : 0 ≤ a)
+    (hH : 0 < H) (hE : 0 ≤ E) (hEt : E ≤ 2*Real.pi*H) :
+    A*(E/H^2)*Real.exp (-a*H^2/E) ≤
+      ((A/H^2)*Real.exp (-(a/(2*Real.pi))*H))*E := by
+  by_cases h0 : E = 0
+  · simp [h0]
+  have hEp : 0 < E := lt_of_le_of_ne hE (Ne.symm h0)
+  have hexp : -a*H^2/E ≤ -(a/(2*Real.pi))*H := by
+    rw [div_le_iff₀ hEp]
+    have hmul := mul_le_mul_of_nonneg_left hEt
+      (by positivity : 0 ≤ a*H/(2*Real.pi))
+    have hh : (a*H/(2*Real.pi))*(2*Real.pi*H) = a*H^2 := by
+      field_simp
+    rw [hh] at hmul
+    convert neg_le_neg hmul using 1 <;> ring
+  calc
+    _ ≤ A*(E/H^2)*Real.exp (-(a/(2*Real.pi))*H) := by gcongr
+    _ = _ := by ring
+
+theorem energy_event_union {ι Ω : Type*} [Countable ι] [MeasurableSpace Ω]
+    (P : Measure Ω) (events : ι → Set Ω) (E : ι → ℝ)
+    {A a H : ℝ} (hA : 0 ≤ A) (ha : 0 ≤ a) (hH : 1 ≤ H)
+    (hE : ∀ j, 0 ≤ E j) (hEt : ∀ j, E j ≤ 2*Real.pi*H)
+    (hsum : (∑' j, ENNReal.ofReal (E j)) ≤ ENNReal.ofReal (2*Real.pi*H))
+    (hhit : ∀ j, P (events j) ≤ ENNReal.ofReal (A*(E j/H^2)*Real.exp (-a*H^2/E j))) :
+    P (⋃ j, events j) ≤ ENNReal.ofReal ((2*Real.pi*A)*Real.exp (-(a/(2*Real.pi))*H)) := by
+  have hHp : 0 < H := lt_of_lt_of_le zero_lt_one hH
+  let k := (A/H^2)*Real.exp (-(a/(2*Real.pi))*H)
+  have hk : 0 ≤ k := by dsimp [k]; positivity
+  calc
+    _ ≤ ∑' j, P (events j) := measure_iUnion_le _
+    _ ≤ ∑' j, ENNReal.ofReal (k*E j) := ENNReal.tsum_le_tsum (fun j =>
+      (hhit j).trans (ENNReal.ofReal_le_ofReal (energy_exp_linearize hA ha hHp (hE j) (hEt j))))
+    _ = ENNReal.ofReal k*(∑' j, ENNReal.ofReal (E j)) := by
+      simp_rw [ENNReal.ofReal_mul hk]
+      exact ENNReal.tsum_mul_left
+    _ ≤ ENNReal.ofReal k*ENNReal.ofReal (2*Real.pi*H) := by gcongr
+    _ = ENNReal.ofReal (k*(2*Real.pi*H)) := (ENNReal.ofReal_mul hk).symm
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (by
+      have hid : k*(2*Real.pi*H) = (2*Real.pi*A/H)*Real.exp (-(a/(2*Real.pi))*H) := by
+        dsimp [k]
+        field_simp
+      rw [hid]
+      apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
+      exact div_le_self (by positivity) hH)
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+def logEnergyExit {Ω : Type*} (γ : Ω → ℝ → ℂ) (Fbar : ℂ → OnePoint ℂ)
+    (H : ℝ) (q : ℂ) (r : ℝ) (ω : Ω) : Prop :=
+  ∃ x ∈ logTrace γ ω, x.1 ≤ localizationStripWidth*H ∧ 0 ≤ x.2 ∧ x.2 ≤ Real.pi ∧
+    stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H
+
+lemma rectCoefficient_unitSlab (j : ℝ) :
+    rectCoefficient (unitSlab j) = rectCoefficient (unitSlab 0) := by
+  unfold rectCoefficient unitSlab
+  dsimp only
+  rw [show j+1-j=1 by ring]
+  norm_num
+
+theorem logTrace_localization (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ c C H₀ : ℝ, 0 < c ∧ 0 < C ∧ 0 < H₀ ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (F : ℂ → ℂ) (_hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+        (_hinj : InjOn F {z : ℂ | 0 < z.im}) (Fbar : ℂ → OnePoint ℂ)
+        (_hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+        (_hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+        (q : ℂ) (r : ℝ) (_hr : 0 < r)
+        (η : ℝ → ℂ) (_hη : ContinuousOn η (Icc 0 1)) (_hη0 : η 0 = 0)
+        (_hη1 : ‖η 1‖ = 1) (_hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+        (_haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+          (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+        (H : ℝ), H₀ ≤ H → P {ω | logEnergyExit γ Fbar H q r ω} ≤
+          ENNReal.ofReal (C*Real.exp (-c*H)) := by
+  obtain ⟨C₀,ρ,hC₀,hρ,hρ1,hhit⟩ := logTrace_dyadic_hitting κ hκ
+  let D := rectCoefficient (unitSlab 0)
+  let T := ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j
+  let A := 64*D*(C₀+1)*(T+1)/ρ
+  let a := (-Real.log ρ)/(48*D)
+  let H₀ := max 1 (32*Real.pi*localizationMeanConstant)
+  have hD : 0 < D := rectCoefficient_pos _
+  have hT : 0 ≤ T := tsum_nonneg (fun j => mul_nonneg (sq_nonneg _) (pow_nonneg hρ.le j))
+  have hA : 0 < A := by dsimp [A]; positivity
+  have ha : 0 < a := div_pos (neg_pos.2 (Real.log_neg hρ hρ1)) (by positivity)
+  have hH₀ : 0 < H₀ := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
+  refine ⟨a/(2*Real.pi),2*Real.pi*A,H₀,by positivity,by positivity,hH₀,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps F hF hinj Fbar hext hcont q r hr η hη hη0 hη1 hηH haccess H hlarge
+  have hH1 : 1 ≤ H := (le_max_left _ _).trans hlarge
+  have hH : 0 < H := lt_of_lt_of_le zero_lt_one hH1
+  have hHH : 32*Real.pi*localizationMeanConstant ≤ H := (le_max_right _ _).trans hlarge
+  let E := fun j : ℤ => stripEnergy H q r Fbar (unitSlab j)
+  let events := fun j : ℤ => {ω | (j:ℝ) ≤ localizationStripWidth*H ∧
+    ∃ x ∈ logTrace γ ω, x ∈ (unitSlab j).closed ∧
+      stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H}
+  have hsub : {ω | logEnergyExit γ Fbar H q r ω} ⊆ ⋃ j, events j := by
+    rintro ω ⟨x,hx,hxv,hx0,hxπ,hxH⟩
+    apply mem_iUnion.mpr
+    refine ⟨Int.floor x.1,?_,x,hx,?_,hxH⟩
+    · exact (Int.floor_le x.1).trans hxv
+    · exact ⟨⟨Int.floor_le x.1,(Int.lt_floor_add_one x.1).le⟩,hx0,hxπ⟩
+  have hE : ∀ j, 0 ≤ E j := fun j => integral_nonneg (fun _ => sq_nonneg _)
+  have hEt : ∀ j, E j ≤ 2*Real.pi*H := fun j =>
+    (stripEnergy_le_total hF hinj hext hH hr q (unitSlab j) le_rfl le_rfl).trans
+      (totalStripEnergy_bound hF hinj hext hH hr q)
+  have hsum := strip_slab_energy_sum hF hinj hext hH hr q
+  have hev (j : ℤ) : P (events j) ≤ ENNReal.ofReal (A*(E j/H^2)*Real.exp (-a*H^2/E j)) := by
+    by_cases hj : (j:ℝ) ≤ localizationStripWidth*H
+    · have hm := strip_slab_half_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess hHH j hj
+      have hb := strip_rectangle_hitting hF hinj hext hcont hH hr q (unitSlab j) le_rfl le_rfl hm
+        P (logTrace γ) hC₀.le hρ hρ1 (hhit P B hB γ g f hγ hmaps j)
+      dsimp only at hb
+      rw [rectCoefficient_unitSlab] at hb
+      change P _ ≤ ENNReal.ofReal (A*(E j/H^2)*Real.exp (-a*H^2/E j)) at hb
+      exact (measure_mono (fun ω hω => hω.2)).trans hb
+    · have he : events j = ∅ := by ext ω; simp [events,hj]
+      rw [he,measure_empty]
+      exact bot_le
+  exact (measure_mono hsub).trans (energy_event_union P events E hA.le ha.le hH1 hE hEt hsum hev)
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology ENNReal NNReal OnePoint
+namespace SLEExactGauge
+open RectangleEnergy
+
+def conformalExit (Fbar : ℂ → OnePoint ℂ) (γ : ℝ → ℂ)
+    (q : ℂ) (radius A : ℝ) : Prop :=
+  ∃ t : ℝ, 0 ≤ t ∧ (∀ s : ℝ, 0 ≤ s → s < t → ‖γ s‖ < A) ∧
+    Fbar (γ t) ∉ (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.ball q radius
+
+lemma norm_le_of_pre_modulus {γ : ℝ → ℂ} (hc : ContinuousOn γ (Ici 0))
+    (h0 : γ 0 = 0) {t A : ℝ} (ht : 0 ≤ t) (hA : 0 < A)
+    (hpre : ∀ s : ℝ, 0 ≤ s → s < t → ‖γ s‖ < A) : ‖γ t‖ ≤ A := by
+  by_contra hn
+  have hgt : A < ‖γ t‖ := lt_of_not_ge hn
+  have hm : A ∈ Icc ‖γ 0‖ ‖γ t‖ := by simpa [h0] using And.intro hA.le hgt.le
+  obtain ⟨s,hs,he⟩ := intermediate_value_Icc ht (hc.mono Icc_subset_Ici_self).norm hm
+  have hst : s < t := lt_of_le_of_ne hs.2 (by intro heq; subst s; linarith)
+  exact (ne_of_lt (hpre s hs.1 hst)) he
+
+lemma conformalExit_logEnergyExit {Ω : Type*} {γ : Ω → ℝ → ℂ}
+    {ω : Ω} {Fbar : ℂ → OnePoint ℂ} {q : ℂ} {r L : ℝ}
+    (hr : 0 < r) (hL : 1 < L) (hc : ContinuousOn (γ ω) (Ici 0))
+    (h0 : γ ω 0 = 0) (hhalf : ∀ t, 0 ≤ t → 0 ≤ (γ ω t).im)
+    (hstart : Fbar 0 ∈ (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+    (hexit : conformalExit Fbar (γ ω) q (L*r) (L^localizationStripWidth)) :
+      logEnergyExit γ Fbar (Real.log L) q r ω := by
+  obtain ⟨t,ht,hpre,hout⟩ := hexit
+  have hLp : 0 < L := zero_lt_one.trans hL
+  have hz : γ ω t ≠ 0 := by
+    intro he
+    apply hout
+    rw [he]
+    obtain ⟨w,hw,heq⟩ := hstart
+    refine ⟨w,?_,heq⟩
+    exact mem_ball.2 (lt_of_le_of_lt (mem_closedBall.1 hw) (by nlinarith))
+  have hn := norm_le_of_pre_modulus hc h0 ht (Real.rpow_pos_of_pos hLp _) hpre
+  let x : ℝ × ℝ := (Real.log ‖γ ω t‖, Complex.arg (γ ω t))
+  have hexp : Complex.exp ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = γ ω t := by
+    change Complex.exp (Complex.log (γ ω t)) = γ ω t
+    exact Complex.exp_log hz
+  refine ⟨x,⟨t,ht,hexp.symm⟩,?_,Complex.arg_nonneg_iff.2 (hhalf t ht),Complex.arg_le_pi _,?_⟩
+  · have hb := Real.log_le_log (norm_pos_iff.2 hz) hn
+    rw [Real.log_rpow hLp] at hb
+    exact hb
+  · unfold stripPotential
+    rw [hexp]
+    apply sphereTruncation_top_of_outside (Real.log_pos hL).le hr q
+    simpa [Real.exp_log hLp] using hout
+
+theorem uniform_conformal_localization (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ b c C L₀ : ℝ, 0 < b ∧ 0 < c ∧ 0 < C ∧ 1 < L₀ ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ)
+        (_hF : AnalyticOnNhd ℂ F halfPlane) (_hinj : Set.InjOn F halfPlane)
+        (_hext : ∀ z ∈ halfPlane, Fbar z = (F z : OnePoint ℂ))
+        (_hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+        (η : ℝ → ℂ) (_hη : ContinuousOn η (Icc 0 1)) (_hη0 : η 0 = 0)
+        (_hη1 : ‖η 1‖ = 1) (_hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+        (q : ℂ) (r : ℝ) (_hr : 0 < r)
+        (_haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+          (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.closedBall q r)
+        (L : ℝ), L₀ ≤ L →
+        P {sample | conformalExit Fbar (γ sample) q (L*r) (L^b)} ≤
+          ENNReal.ofReal (C*L^(-c)) := by
+  obtain ⟨c,C,H₀,hc,hC,hH₀,hlog⟩ := logTrace_localization κ hκ
+  refine ⟨localizationStripWidth,c,C,Real.exp (H₀+1),localizationStripWidth_pos,hc,hC,?_,?_⟩
+  · exact Real.one_lt_exp_iff.2 (by linarith)
+  intro Ω _ P _ B hB γ g f hγ hmaps F Fbar hF hinj hext hcont η hη hη0 hη1 hηH q r hr haccess L hlarge
+  have hL : 1 < L := (Real.one_lt_exp_iff.2 (by linarith : 0 < H₀+1)).trans_le hlarge
+  have hLp : 0 < L := zero_lt_one.trans hL
+  have hH : H₀ ≤ Real.log L := by
+    have hh := Real.log_le_log (Real.exp_pos (H₀+1)) hlarge
+    rw [Real.log_exp] at hh
+    linarith
+  have hstart : Fbar 0 ∈ (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r := by
+    simpa [hη0] using haccess 0 (by norm_num : (0:ℝ) ∈ Icc 0 1)
+  have hm : P {ω | conformalExit Fbar (γ ω) q (L*r) (L^localizationStripWidth)} ≤
+      P {ω | logEnergyExit γ Fbar (Real.log L) q r ω} := by
+    apply measure_mono_ae
+    filter_upwards [hmaps] with ω hw
+    intro hexit
+    exact conformalExit_logEnergyExit hr hL hw.1 hw.2.1 hw.2.2.1 hstart hexit
+  have hb := hlog P B hB γ g f hγ hmaps F hF hinj Fbar hext hcont q r hr η hη hη0 hη1 hηH haccess (Real.log L) hH
+  have he : Real.exp (-c*Real.log L) = L^(-c) := by
+    rw [Real.rpow_def_of_pos hLp]
+    congr 1
+    ring
+  rw [he] at hb
+  exact hm.trans hb
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/DriverStability.lean b/lean/OAI/Probability/SLELocalization/DriverStability.lean
index bf2ef2bc7dad37bd01382a8652514f8cb5aa2740..f9be040e849aecc99bab53b221233995609b7c20 100644
--- a/lean/OAI/Probability/SLELocalization/DriverStability.lean
+++ b/lean/OAI/Probability/SLELocalization/DriverStability.lean
@@ -1,193 +1,193 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-
-namespace OAI
-
-
-/-! Conformal localization and Driver Stability estimates. -/
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal unitInterval
-namespace SLEExactGauge.DriverStability
-open LoewnerCutoff
-
-def timePath {T : ℝ} (hT : 0 ≤ T) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) : C(I,ℝ) :=
-  ⟨fun q => U (T*(q:ℝ)), hU.comp_continuous (continuous_const.mul continuous_subtype_val)
-    (fun q => ⟨mul_nonneg hT q.property.1,by nlinarith [q.property.2]⟩)⟩
-
-lemma extend_timePath {T : ℝ} (hT : 0 < T) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) :
-    EqOn (extendDriver T (timePath hT.le U hU)) U (Icc 0 T) := by
-  intro t ht
-  have hq : t/T ∈ Icc (0:ℝ) 1 := ⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩
-  change U (T*(projIcc 0 1 zero_le_one (t/T):ℝ)) = U t
-  rw [projIcc_of_mem _ hq]
-  congr 1
-  field_simp
-
-lemma timePath_dist_le {T η : ℝ} (hT : 0 ≤ T) (hη : 0 ≤ η)
-    (U V : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (hV : ContinuousOn V (Icc 0 T))
-    (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ η) :
-    dist (timePath hT U hU) (timePath hT V hV) ≤ η := by
-  apply (ContinuousMap.dist_le hη).2
-  intro q
-  exact hUV _ ⟨mul_nonneg hT q.property.1,by nlinarith [q.property.2]⟩
-
-lemma radius_of_positive_path
-    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing U γ g f) {T ε : ℝ} (hT : 0 ≤ T) (hε : 0 < ε)
-    (hU : ContinuousOn U (Icc 0 T)) (V : C(I,ℝ)) (z : ℂ)
-    (hVU : EqOn (extendDriver T V) U (Icc 0 T))
-    (hheight : ∀ t ∈ Icc 0 T, ε ≤ (position hε hT V z t).im) :
-    z ∈ localizationDomain γ T ∧ LoewnerExit.radius hε hT V z T = loewnerRadius g T z := by
-  have hh := h.cutoff_solution_live hε hT
-    (H:=position hε hT V z)
-    (fun t ht => by rw [← hVU ht]; exact position_derivWithin hε hT V z ht) hheight
-  rw [position_initial] at hh
-  have hl : logDerivative hε hT V z T = Real.log ‖deriv (g T) z‖ := by
-    rw [logDerivative_eq_integral hε hT V z ⟨hT,le_rfl⟩,
-      h.log_deriv_norm_eq hT hU hh.1]
-    have hi := (h.continuous_spatialCoefficient hT hU hh.1).intervalIntegrable_of_Icc (μ:=MeasureTheory.volume) hT
-    rw [← Complex.reCLM_apply,← Complex.reCLM.intervalIntegral_comp_comm hi]
-    apply intervalIntegral.integral_congr
-    intro s hs
-    rw [uIcc_of_le hT] at hs
-    unfold variation
-    rw [clampIm_eq (by simpa using hheight s hs)]
-    simp only [← hh.2 hs,hVU hs,Complex.reCLM_apply]
-  refine ⟨hh.1,?_⟩
-  rw [LoewnerExit.radius,hl,Real.exp_log (norm_pos_iff.2 (h.deriv_ne_zero hT hh.1)),← hh.2 ⟨hT,le_rfl⟩]
-  rfl
-
-lemma live_radius_stable
-    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing U γ g f) {T ρ : ℝ} (hT : 0 < T)
-    (hU : ContinuousOn U (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hρ : loewnerRadius g T z < ρ) :
-    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
-      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
-      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
-      z ∈ localizationDomain ξ T ∧ loewnerRadius G T z < ρ := by
-  let ε : ℝ := (g T z).im/4
-  have hε : 0 < ε := div_pos (h.im_pos hT.le hz) (by norm_num)
-  let u : C(I,ℝ) := timePath hT.le U hU
-  have heu : EqOn (extendDriver T u) U (Icc 0 T) := extend_timePath hT U hU
-  have hbase : EqOn (fun t => g t z) (position hε hT.le u z) (Icc 0 T) :=
-    h.eq_cutoff_solution hT.le hε hU hz (by dsimp [ε]; linarith [h.im_pos hT.le hz])
-      (position_initial hε hT.le u z)
-      (fun t ht => by rw [← heu ht]; exact position_derivWithin hε hT.le u z ht)
-  have hbaseheight (t : ℝ) (ht : t ∈ Icc 0 T) : 4*ε ≤ (position hε hT.le u z t).im := by
-    rw [← hbase ht]
-    have hh := h.height_antitone hT.le hz ht ⟨hT.le,le_rfl⟩ ht.2
-    dsimp [ε]
-    linarith
-  have hver : LoewnerExit.radius hε hT.le u z T = loewnerRadius g T z :=
-    (radius_of_positive_path h hT.le hε hU u z heu (fun t ht => by linarith [hbaseheight t ht])).2
-  have hp : ContinuousAt (fun v : C(I,ℝ) => solutionPath hε hT.le v z) u :=
-    (solutionPath_lipschitz hε hT.le z).continuous.continuousAt
-  have hr : ContinuousAt (fun v : C(I,ℝ) => LoewnerExit.radius hε hT.le v z T) u :=
-    ((LoewnerExit.continuous_radius hε hT.le z).comp (continuous_id.prodMk continuous_const)).continuousAt
-  have hn1 : ∀ᶠ v in 𝓝 u, dist (solutionPath hε hT.le v z) (solutionPath hε hT.le u z) < ε :=
-    hp.eventually (ball_mem_nhds _ hε)
-  have hn2 : ∀ᶠ v in 𝓝 u, LoewnerExit.radius hε hT.le v z T < ρ :=
-    hr.eventually (gt_mem_nhds (by change LoewnerExit.radius hε hT.le u z T < ρ; rw [hver]; exact hρ))
-  obtain ⟨δ,hδ,hd⟩ := Metric.eventually_nhds_iff.1 (hn1.and hn2)
-  refine ⟨δ/2,by positivity,fun V ξ G F hξ hV hclose => ?_⟩
-  let v : C(I,ℝ) := timePath hT.le V hV
-  have hev : EqOn (extendDriver T v) V (Icc 0 T) := extend_timePath hT V hV
-  have hdist : dist v u < δ :=
-    (timePath_dist_le hT.le (by positivity) V U hV hU (fun t ht => (hclose t ht).le)).trans_lt (by linarith)
-  have hdv := hd hdist
-  have hh (t : ℝ) (ht : t ∈ Icc 0 T) : ε ≤ (position hε hT.le v z t).im := by
-    have hdiff : dist (position hε hT.le v z t) (position hε hT.le u z t) < ε := by
-      rw [position_eq hε hT.le v z ht,position_eq hε hT.le u z ht]
-      exact (ContinuousMap.dist_apply_le_dist (f:=solutionPath hε hT.le v z)
-        (g:=solutionPath hε hT.le u z) (⟨t,ht⟩:Icc 0 T)).trans_lt hdv.1
-    have him := Complex.abs_im_le_norm (position hε hT.le v z t-position hε hT.le u z t)
-    rw [dist_eq_norm] at hdiff
-    simp only [Complex.sub_im] at him
-    have hlo := neg_abs_le ((position hε hT.le v z t).im-(position hε hT.le u z t).im)
-    linarith [hbaseheight t ht]
-  have hs := radius_of_positive_path hξ hT.le hε hV v z hev hh
-  exact ⟨hs.1,hs.2 ▸ hdv.2⟩
-
-end SLEExactGauge.DriverStability
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-theorem uniformDriver_forces_tipVisit
-    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing U γ g f) {T r : ℝ} (hT : 0 < T)
-    (hU : ContinuousOn U (Icc 0 T)) (hqr : 0 < (γ T).im) (hr : 0 < r) :
-    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
-      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
-      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
-      ∃ t ∈ Icc 0 T, ξ t ∈ Metric.ball (γ T) r := by
-  let q : ℂ := γ T
-  let a : ℝ := min (r/4) (q.im/4)
-  have ha : 0 < a := lt_min (by positivity) (by dsimp [q]; positivity)
-  have har : a ≤ r/4 := min_le_left _ _
-  have haq : a ≤ q.im/4 := min_le_right _ _
-  let C : ℝ := 512*Real.pi
-  have hC : 0 < C := by dsimp [C]; positivity
-  let δ : ℝ := min (a/2) (a/(4*C))
-  have hδ : 0 < δ := lt_min (by positivity) (by positivity)
-  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2.2 T hT.le
-  have hev : ∀ᶠ (y : ℝ) in 𝓝[>] (0:ℝ), dist (f T ((U T:ℂ)+(y:ℂ)*Complex.I)) q < δ :=
-    htip.eventually (ball_mem_nhds _ hδ)
-  obtain ⟨y,hy,hyδ⟩ := (Filter.Eventually.and (self_mem_nhdsWithin : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), 0 < y) hev).exists
-  have hypos : 0 < y := hy
-  let x : ℂ := f T ((U T:ℂ)+(y:ℂ)*Complex.I)
-  have hx : x ∈ localizationDomain γ T :=
-    h.2.2.2.2.2.2.2.2.1 T hT.le (by simpa [halfPlane] using hypos)
-  have hxq : dist x q < δ := hyδ
-  have hxsmall : dist x q < a/2 := hxq.trans_le (min_le_left _ _)
-  have hnq : q ∉ localizationDomain γ T := by
-    intro hq
-    exact hq.1.2 ⟨T,⟨hT.le,le_rfl⟩,rfl⟩
-  have hR : loewnerRadius g T x < a/2 := by
-    calc
-      _ ≤ C*dist x q := h.radius_le_complement_distance hT.le hx hnq
-      _ < C*(a/(4*C)) := mul_lt_mul_of_pos_left (hxq.trans_le (min_le_right _ _)) hC
-      _ = a/4 := by field_simp
-      _ < a/2 := by linarith
-  obtain ⟨η,hη,hstable⟩ := DriverStability.live_radius_stable h hT hU hx hR
-  refine ⟨η,hη,fun V ξ G F hξ hV hclose => ?_⟩
-  have hs := hstable V ξ G F hξ hV hclose
-  by_contra hn
-  push Not at hn
-  have hfree : closedBall x a ⊆ halfPlane \ localizationSegment ξ 0 T := by
-    intro w hw
-    have hwx : dist w x ≤ a := hw
-    have hwq : dist w q ≤ dist w x+dist x q := dist_triangle _ _ _
-    have hwr : dist w q < r := by linarith
-    have hwi : dist w q < q.im := by linarith
-    have him := Complex.abs_im_le_norm (w-q)
-    have hlo := neg_abs_le (w.im-q.im)
-    rw [← dist_eq_norm] at him
-    simp only [Complex.sub_im] at him
-    refine ⟨by change 0 < w.im; linarith,?_⟩
-    rintro ⟨t,ht,he⟩
-    apply hn t ht
-    change dist (ξ t) q < r
-    simpa only [he] using hwr
-  have hball := closedBall_subset_survivingDomain ha.le hs.1 hfree
-  have hl := hξ.radius_ge_half_free_ball hT.le ha hs.1 hball
-  exact (not_le_of_gt hs.2) hl
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+
+namespace OAI
+
+
+/-! Conformal localization and Driver Stability estimates. -/
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal unitInterval
+namespace SLEExactGauge.DriverStability
+open LoewnerCutoff
+
+def timePath {T : ℝ} (hT : 0 ≤ T) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) : C(I,ℝ) :=
+  ⟨fun q => U (T*(q:ℝ)), hU.comp_continuous (continuous_const.mul continuous_subtype_val)
+    (fun q => ⟨mul_nonneg hT q.property.1,by nlinarith [q.property.2]⟩)⟩
+
+lemma extend_timePath {T : ℝ} (hT : 0 < T) (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) :
+    EqOn (extendDriver T (timePath hT.le U hU)) U (Icc 0 T) := by
+  intro t ht
+  have hq : t/T ∈ Icc (0:ℝ) 1 := ⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩
+  change U (T*(projIcc 0 1 zero_le_one (t/T):ℝ)) = U t
+  rw [projIcc_of_mem _ hq]
+  congr 1
+  field_simp
+
+lemma timePath_dist_le {T η : ℝ} (hT : 0 ≤ T) (hη : 0 ≤ η)
+    (U V : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (hV : ContinuousOn V (Icc 0 T))
+    (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ η) :
+    dist (timePath hT U hU) (timePath hT V hV) ≤ η := by
+  apply (ContinuousMap.dist_le hη).2
+  intro q
+  exact hUV _ ⟨mul_nonneg hT q.property.1,by nlinarith [q.property.2]⟩
+
+lemma radius_of_positive_path
+    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing U γ g f) {T ε : ℝ} (hT : 0 ≤ T) (hε : 0 < ε)
+    (hU : ContinuousOn U (Icc 0 T)) (V : C(I,ℝ)) (z : ℂ)
+    (hVU : EqOn (extendDriver T V) U (Icc 0 T))
+    (hheight : ∀ t ∈ Icc 0 T, ε ≤ (position hε hT V z t).im) :
+    z ∈ localizationDomain γ T ∧ LoewnerExit.radius hε hT V z T = loewnerRadius g T z := by
+  have hh := h.cutoff_solution_live hε hT
+    (H:=position hε hT V z)
+    (fun t ht => by rw [← hVU ht]; exact position_derivWithin hε hT V z ht) hheight
+  rw [position_initial] at hh
+  have hl : logDerivative hε hT V z T = Real.log ‖deriv (g T) z‖ := by
+    rw [logDerivative_eq_integral hε hT V z ⟨hT,le_rfl⟩,
+      h.log_deriv_norm_eq hT hU hh.1]
+    have hi := (h.continuous_spatialCoefficient hT hU hh.1).intervalIntegrable_of_Icc (μ:=MeasureTheory.volume) hT
+    rw [← Complex.reCLM_apply,← Complex.reCLM.intervalIntegral_comp_comm hi]
+    apply intervalIntegral.integral_congr
+    intro s hs
+    rw [uIcc_of_le hT] at hs
+    unfold variation
+    rw [clampIm_eq (by simpa using hheight s hs)]
+    simp only [← hh.2 hs,hVU hs,Complex.reCLM_apply]
+  refine ⟨hh.1,?_⟩
+  rw [LoewnerExit.radius,hl,Real.exp_log (norm_pos_iff.2 (h.deriv_ne_zero hT hh.1)),← hh.2 ⟨hT,le_rfl⟩]
+  rfl
+
+lemma live_radius_stable
+    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing U γ g f) {T ρ : ℝ} (hT : 0 < T)
+    (hU : ContinuousOn U (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hρ : loewnerRadius g T z < ρ) :
+    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
+      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
+      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
+      z ∈ localizationDomain ξ T ∧ loewnerRadius G T z < ρ := by
+  let ε : ℝ := (g T z).im/4
+  have hε : 0 < ε := div_pos (h.im_pos hT.le hz) (by norm_num)
+  let u : C(I,ℝ) := timePath hT.le U hU
+  have heu : EqOn (extendDriver T u) U (Icc 0 T) := extend_timePath hT U hU
+  have hbase : EqOn (fun t => g t z) (position hε hT.le u z) (Icc 0 T) :=
+    h.eq_cutoff_solution hT.le hε hU hz (by dsimp [ε]; linarith [h.im_pos hT.le hz])
+      (position_initial hε hT.le u z)
+      (fun t ht => by rw [← heu ht]; exact position_derivWithin hε hT.le u z ht)
+  have hbaseheight (t : ℝ) (ht : t ∈ Icc 0 T) : 4*ε ≤ (position hε hT.le u z t).im := by
+    rw [← hbase ht]
+    have hh := h.height_antitone hT.le hz ht ⟨hT.le,le_rfl⟩ ht.2
+    dsimp [ε]
+    linarith
+  have hver : LoewnerExit.radius hε hT.le u z T = loewnerRadius g T z :=
+    (radius_of_positive_path h hT.le hε hU u z heu (fun t ht => by linarith [hbaseheight t ht])).2
+  have hp : ContinuousAt (fun v : C(I,ℝ) => solutionPath hε hT.le v z) u :=
+    (solutionPath_lipschitz hε hT.le z).continuous.continuousAt
+  have hr : ContinuousAt (fun v : C(I,ℝ) => LoewnerExit.radius hε hT.le v z T) u :=
+    ((LoewnerExit.continuous_radius hε hT.le z).comp (continuous_id.prodMk continuous_const)).continuousAt
+  have hn1 : ∀ᶠ v in 𝓝 u, dist (solutionPath hε hT.le v z) (solutionPath hε hT.le u z) < ε :=
+    hp.eventually (ball_mem_nhds _ hε)
+  have hn2 : ∀ᶠ v in 𝓝 u, LoewnerExit.radius hε hT.le v z T < ρ :=
+    hr.eventually (gt_mem_nhds (by change LoewnerExit.radius hε hT.le u z T < ρ; rw [hver]; exact hρ))
+  obtain ⟨δ,hδ,hd⟩ := Metric.eventually_nhds_iff.1 (hn1.and hn2)
+  refine ⟨δ/2,by positivity,fun V ξ G F hξ hV hclose => ?_⟩
+  let v : C(I,ℝ) := timePath hT.le V hV
+  have hev : EqOn (extendDriver T v) V (Icc 0 T) := extend_timePath hT V hV
+  have hdist : dist v u < δ :=
+    (timePath_dist_le hT.le (by positivity) V U hV hU (fun t ht => (hclose t ht).le)).trans_lt (by linarith)
+  have hdv := hd hdist
+  have hh (t : ℝ) (ht : t ∈ Icc 0 T) : ε ≤ (position hε hT.le v z t).im := by
+    have hdiff : dist (position hε hT.le v z t) (position hε hT.le u z t) < ε := by
+      rw [position_eq hε hT.le v z ht,position_eq hε hT.le u z ht]
+      exact (ContinuousMap.dist_apply_le_dist (f:=solutionPath hε hT.le v z)
+        (g:=solutionPath hε hT.le u z) (⟨t,ht⟩:Icc 0 T)).trans_lt hdv.1
+    have him := Complex.abs_im_le_norm (position hε hT.le v z t-position hε hT.le u z t)
+    rw [dist_eq_norm] at hdiff
+    simp only [Complex.sub_im] at him
+    have hlo := neg_abs_le ((position hε hT.le v z t).im-(position hε hT.le u z t).im)
+    linarith [hbaseheight t ht]
+  have hs := radius_of_positive_path hξ hT.le hε hV v z hev hh
+  exact ⟨hs.1,hs.2 ▸ hdv.2⟩
+
+end SLEExactGauge.DriverStability
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+theorem uniformDriver_forces_tipVisit
+    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing U γ g f) {T r : ℝ} (hT : 0 < T)
+    (hU : ContinuousOn U (Icc 0 T)) (hqr : 0 < (γ T).im) (hr : 0 < r) :
+    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
+      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
+      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
+      ∃ t ∈ Icc 0 T, ξ t ∈ Metric.ball (γ T) r := by
+  let q : ℂ := γ T
+  let a : ℝ := min (r/4) (q.im/4)
+  have ha : 0 < a := lt_min (by positivity) (by dsimp [q]; positivity)
+  have har : a ≤ r/4 := min_le_left _ _
+  have haq : a ≤ q.im/4 := min_le_right _ _
+  let C : ℝ := 512*Real.pi
+  have hC : 0 < C := by dsimp [C]; positivity
+  let δ : ℝ := min (a/2) (a/(4*C))
+  have hδ : 0 < δ := lt_min (by positivity) (by positivity)
+  have htip := h.2.2.2.2.2.2.2.2.2.2.2.2.2 T hT.le
+  have hev : ∀ᶠ (y : ℝ) in 𝓝[>] (0:ℝ), dist (f T ((U T:ℂ)+(y:ℂ)*Complex.I)) q < δ :=
+    htip.eventually (ball_mem_nhds _ hδ)
+  obtain ⟨y,hy,hyδ⟩ := (Filter.Eventually.and (self_mem_nhdsWithin : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), 0 < y) hev).exists
+  have hypos : 0 < y := hy
+  let x : ℂ := f T ((U T:ℂ)+(y:ℂ)*Complex.I)
+  have hx : x ∈ localizationDomain γ T :=
+    h.2.2.2.2.2.2.2.2.1 T hT.le (by simpa [halfPlane] using hypos)
+  have hxq : dist x q < δ := hyδ
+  have hxsmall : dist x q < a/2 := hxq.trans_le (min_le_left _ _)
+  have hnq : q ∉ localizationDomain γ T := by
+    intro hq
+    exact hq.1.2 ⟨T,⟨hT.le,le_rfl⟩,rfl⟩
+  have hR : loewnerRadius g T x < a/2 := by
+    calc
+      _ ≤ C*dist x q := h.radius_le_complement_distance hT.le hx hnq
+      _ < C*(a/(4*C)) := mul_lt_mul_of_pos_left (hxq.trans_le (min_le_right _ _)) hC
+      _ = a/4 := by field_simp
+      _ < a/2 := by linarith
+  obtain ⟨η,hη,hstable⟩ := DriverStability.live_radius_stable h hT hU hx hR
+  refine ⟨η,hη,fun V ξ G F hξ hV hclose => ?_⟩
+  have hs := hstable V ξ G F hξ hV hclose
+  by_contra hn
+  push Not at hn
+  have hfree : closedBall x a ⊆ halfPlane \ localizationSegment ξ 0 T := by
+    intro w hw
+    have hwx : dist w x ≤ a := hw
+    have hwq : dist w q ≤ dist w x+dist x q := dist_triangle _ _ _
+    have hwr : dist w q < r := by linarith
+    have hwi : dist w q < q.im := by linarith
+    have him := Complex.abs_im_le_norm (w-q)
+    have hlo := neg_abs_le (w.im-q.im)
+    rw [← dist_eq_norm] at him
+    simp only [Complex.sub_im] at him
+    refine ⟨by change 0 < w.im; linarith,?_⟩
+    rintro ⟨t,ht,he⟩
+    apply hn t ht
+    change dist (ξ t) q < r
+    simpa only [he] using hwr
+  have hball := closedBall_subset_survivingDomain ha.le hs.1 hfree
+  have hl := hξ.radius_ge_half_free_ball hT.le ha hs.1 hball
+  exact (not_le_of_gt hs.2) hl
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/DyadicKernels.lean b/lean/OAI/Probability/SLELocalization/DyadicKernels.lean
index 479f8c68a9298bccc26c1c254e7ec5f18ebb8814..9c9596c8dd7dfd3dabb1c65da354212142391a30 100644
--- a/lean/OAI/Probability/SLELocalization/DyadicKernels.lean
+++ b/lean/OAI/Probability/SLELocalization/DyadicKernels.lean
@@ -1,898 +1,898 @@
-import Mathlib
-import OAI.Probability.SLELocalization.WeightedEnergy
-
-namespace OAI
-
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-open Layer
-
-lemma annular_invSquare_integral {r R : ℝ} (hr : 0 < r) (hR : r < R) :
-    (∫⁻ z in {z : ℂ | r < ‖z‖ ∧ ‖z‖ < R}, ENNReal.ofReal (1/‖z‖^2)) ≤
-      ENNReal.ofReal (2*Real.pi*Real.log (R/r)) := by
-  apply (polar_annulus_lintegral_le _ hr).trans
-  have he : (∫⁻ s in Ioo r R, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-      ENNReal.ofReal s*ENNReal.ofReal (1/‖s • circleMap 0 1 θ‖^2)) =
-      ENNReal.ofReal (2*Real.pi)*(∫⁻ s in Ioo r R, ENNReal.ofReal s⁻¹) := by
-    calc
-      _ = ∫⁻ s in Ioo r R, ENNReal.ofReal (2*Real.pi)*ENNReal.ofReal s⁻¹ := by
-        apply lintegral_congr_ae
-        filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs
-        have hs0 := hr.trans hs.1
-        simp only [radial_norm,abs_of_pos hs0]
-        have hmul : ENNReal.ofReal s*ENNReal.ofReal (1/s^2) = ENNReal.ofReal s⁻¹ := by
-          rw [← ENNReal.ofReal_mul hs0.le]
-          congr 1
-          field_simp
-        simp only [hmul,lintegral_const,Measure.restrict_apply_univ,Real.volume_Ioo,
-          show Real.pi - -Real.pi = 2*Real.pi by ring]
-        exact mul_comm _ _
-      _ = _ := by rw [lintegral_const_mul']; exact ENNReal.ofReal_ne_top
-  rw [he]
-  have hc : ContinuousOn (fun s : ℝ => s⁻¹) (Icc r R) :=
-    continuousOn_id.inv₀ (fun s hs => (hr.trans_le hs.1).ne')
-  have hi : IntegrableOn (fun s : ℝ => s⁻¹) (Ioc r R) :=
-    (hc.intervalIntegrable_of_Icc hR.le).1
-  have hnon : 0 ≤ᵐ[volume.restrict (Ioc r R)] (fun s : ℝ => s⁻¹) := by
-    filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
-    exact inv_nonneg.2 (hr.trans hs.1).le
-  have hint : (∫ s in r..R, s⁻¹) = Real.log (R/r) := by
-    apply integral_inv
-    rw [uIcc_of_le hR.le]
-    intro hz
-    exact (not_le_of_gt hr) hz.1
-  have heq : (∫⁻ s in Ioo r R, ENNReal.ofReal s⁻¹) = ENNReal.ofReal (Real.log (R/r)) := by
-    rw [restrict_Ioo_eq_restrict_Ioc,← ofReal_integral_eq_lintegral_ofReal hi hnon,
-      ← intervalIntegral.integral_of_le hR.le,hint]
-  rw [heq,← ENNReal.ofReal_mul (by positivity)]
-
-def annularKernel (r R : ℝ) (z : ℂ) : ℝ≥0∞ :=
-  if r < ‖z‖ ∧ ‖z‖ < R then ENNReal.ofReal (1/‖z‖^2) else 0
-
-lemma annularKernel_integral {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    (∫⁻ z : ℂ, annularKernel r (Real.exp H*r) (z-q)) ≤ ENNReal.ofReal (2*Real.pi*H) := by
-  rw [lintegral_sub_right_eq_self]
-  have he : annularKernel r (Real.exp H*r) =
-      {z : ℂ | r < ‖z‖ ∧ ‖z‖ < Real.exp H*r}.indicator (fun z => ENNReal.ofReal (1/‖z‖^2)) := by
-    ext z
-    simp [annularKernel,Set.indicator]
-  have hA : MeasurableSet {z : ℂ | r < ‖z‖ ∧ ‖z‖ < Real.exp H*r} :=
-    (measurableSet_lt measurable_const measurable_norm).inter
-      (measurableSet_lt measurable_norm measurable_const)
-  rw [he,lintegral_indicator hA]
-  have hR : r < Real.exp H*r := (lt_mul_iff_one_lt_left hr).2 (Real.one_lt_exp_iff.2 hH)
-  have hh := annular_invSquare_integral hr hR
-  have hlog : Real.log (Real.exp H*r/r) = H := by
-    rw [mul_div_cancel_right₀ _ hr.ne',Real.log_exp]
-  simpa only [hlog] using hh
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma complex_real_jacobian (c : ℂ) :
-    ((ContinuousLinearMap.toSpanSingleton ℂ c).restrictScalars ℝ).det = ‖c‖^2 := by
-  simp [ContinuousLinearMap.det,LinearMap.det_restrictScalars,
-    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
-
-theorem conformal_truncation_energy {U : Set ℂ} (hU : IsOpen U)
-    {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F U) (hinj : InjOn F U)
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    (∫⁻ z in U, ENNReal.ofReal (‖fderiv ℝ (radialTruncation H q r ∘ F) z‖^2)) ≤
-      ENNReal.ofReal (2*Real.pi*H) := by
-  have hchange := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hU.measurableSet
-    (fun z hz => ((hF.differentiableAt (hU.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt)
-    hinj (fun w => annularKernel r (Real.exp H*r) (w-q))
-  calc
-    _ ≤ ∫⁻ z in U, ENNReal.ofReal (‖deriv F z‖^2)*annularKernel r (Real.exp H*r) (F z-q) := by
-      apply setLIntegral_mono' hU.measurableSet
-      intro z hz
-      have hb := radialTruncation_fderiv_bound hH.le hr (q := q)
-        (hF.differentiableAt (hU.mem_nhds hz))
-      apply (ENNReal.ofReal_le_ofReal hb).trans
-      by_cases hh : r < ‖F z-q‖ ∧ ‖F z-q‖ < Real.exp H*r
-      · rw [ite_eq_left hh,annularKernel,ite_eq_left hh,← ENNReal.ofReal_mul (sq_nonneg _)]
-        apply le_of_eq
-        congr 1
-        ring
-      · simp only [ite_eq_right hh,annularKernel,ENNReal.ofReal_zero,mul_zero,le_refl]
-    _ = ∫⁻ w in F '' U, annularKernel r (Real.exp H*r) (w-q) := by
-      rw [hchange]
-      apply setLIntegral_congr_fun hU.measurableSet
-      intro z hz
-      dsimp only
-      rw [complex_real_jacobian,abs_of_nonneg (sq_nonneg _)]
-    _ ≤ ∫⁻ w : ℂ, annularKernel r (Real.exp H*r) (w-q) := lintegral_mono_set (subset_univ _) |> fun h => by simpa using h
-    _ ≤ _ := annularKernel_integral hH hr q
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal OnePoint
-namespace SLEExactGauge.RectangleEnergy
-
-def sphereTruncation (H : ℝ) (q : ℂ) (r : ℝ) : OnePoint ℂ → ℝ :=
-  OnePoint.rec H (radialTruncation H q r)
-
-@[simp] lemma sphereTruncation_coe (H : ℝ) (q : ℂ) (r : ℝ) (z : ℂ) :
-    sphereTruncation H q r (z : OnePoint ℂ) = radialTruncation H q r z := rfl
-
-@[simp] lemma sphereTruncation_infty (H : ℝ) (q : ℂ) (r : ℝ) :
-    sphereTruncation H q r ∞ = H := rfl
-
-lemma sphereTruncation_continuous {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
-    Continuous (sphereTruncation H q r) := by
-  rw [continuous_iff_continuousAt]
-  intro z
-  induction z using OnePoint.rec with
-  | infty =>
-    apply OnePoint.continuousAt_infty.2
-    intro s hs
-    refine ⟨closedBall q (Real.exp H*r),isClosed_closedBall,isCompact_closedBall _ _,?_⟩
-    intro w hw
-    have hd : Real.exp H*r ≤ ‖w-q‖ := by
-      have hn : Real.exp H*r < dist w q := lt_of_not_ge hw
-      exact (show Real.exp H*r < ‖w-q‖ by simpa [dist_eq_norm] using hn).le
-    change radialTruncation H q r w ∈ s
-    rw [radialTruncation_top hH hr hd]
-    exact mem_of_mem_nhds hs
-  | coe w =>
-    exact OnePoint.continuousAt_coe.2 (radialTruncation_continuous hr H q).continuousAt
-
-lemma sphereTruncation_bounds {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    (z : OnePoint ℂ) : 0 ≤ sphereTruncation H q r z ∧ sphereTruncation H q r z ≤ H := by
-  induction z using OnePoint.rec with
-  | infty => exact ⟨hH,le_rfl⟩
-  | coe w => exact radialTruncation_bounds hH hr q w
-
-lemma sphereTruncation_top_of_outside {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    {z : OnePoint ℂ} (hz : z ∉ (fun w : ℂ => (w : OnePoint ℂ)) '' ball q (Real.exp H*r)) :
-    sphereTruncation H q r z = H := by
-  induction z using OnePoint.rec with
-  | infty => rfl
-  | coe w =>
-    apply radialTruncation_top hH hr
-    have hn : w ∉ ball q (Real.exp H*r) := fun hw => hz ⟨w,hw,rfl⟩
-    simpa [mem_ball,dist_eq_norm] using (le_of_not_gt hn : Real.exp H*r ≤ dist w q)
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-def openStrip : Set ℂ := {z | 0 < z.im ∧ z.im < Real.pi}
-
-def closedStrip : Set ℂ := {z | 0 ≤ z.im ∧ z.im ≤ Real.pi}
-
-def stripPotential (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) (z : ℂ) : ℝ :=
-  sphereTruncation H q r (Fbar (Complex.exp z))
-
-lemma openStrip_isOpen : IsOpen openStrip :=
-  (isOpen_lt continuous_const Complex.continuous_im).inter
-    (isOpen_lt Complex.continuous_im continuous_const)
-
-lemma exp_mem_halfPlane {z : ℂ} (hz : z ∈ openStrip) : 0 < (Complex.exp z).im := by
-  rw [Complex.exp_im]
-  exact mul_pos (Real.exp_pos _) (Real.sin_pos_of_pos_of_lt_pi hz.1 hz.2)
-
-lemma exp_mem_closedHalfPlane {z : ℂ} (hz : z ∈ closedStrip) : 0 ≤ (Complex.exp z).im := by
-  rw [Complex.exp_im]
-  exact mul_nonneg (Real.exp_pos _).le (Real.sin_nonneg_of_mem_Icc hz)
-
-lemma exp_injOn_strip : InjOn Complex.exp openStrip := by
-  intro z hz w hw hzw
-  exact Complex.exp_inj_of_neg_pi_lt_of_le_pi
-    ((neg_lt_zero.2 Real.pi_pos).trans hz.1) hz.2.le
-    ((neg_lt_zero.2 Real.pi_pos).trans hw.1) hw.2.le hzw
-
-lemma stripPotential_continuous {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    {Fbar : ℂ → OnePoint ℂ} (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im}) :
-    ContinuousOn (stripPotential H q r Fbar) closedStrip :=
-  (sphereTruncation_continuous hH hr q).continuousOn.comp
-    (hcont.comp Complex.continuous_exp.continuousOn (fun z hz => exp_mem_closedHalfPlane (z := z) hz))
-    (fun _ _ => mem_univ _)
-
-theorem strip_potential_energy {F : ℂ → ℂ} (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    (∫⁻ z in openStrip, ENNReal.ofReal (‖fderiv ℝ (stripPotential H q r Fbar) z‖^2)) ≤
-      ENNReal.ofReal (2*Real.pi*H) := by
-  have hdiff : DifferentiableOn ℂ (F ∘ Complex.exp) openStrip := by
-    intro z hz
-    exact ((hF _ (exp_mem_halfPlane hz)).differentiableAt.comp z
-      (Complex.differentiable_exp z)).differentiableWithinAt
-  have hi : InjOn (F ∘ Complex.exp) openStrip :=
-    hinj.comp exp_injOn_strip (fun z hz => exp_mem_halfPlane hz)
-  have hb := conformal_truncation_energy openStrip_isOpen hdiff hi hH hr q
-  have he : (∫⁻ z in openStrip, ENNReal.ofReal (‖fderiv ℝ (stripPotential H q r Fbar) z‖^2)) =
-      ∫⁻ z in openStrip, ENNReal.ofReal (‖fderiv ℝ (radialTruncation H q r ∘ (F ∘ Complex.exp)) z‖^2) := by
-    apply setLIntegral_congr_fun openStrip_isOpen.measurableSet
-    intro z hz
-    have heq : stripPotential H q r Fbar =ᶠ[𝓝 z]
-        (radialTruncation H q r ∘ (F ∘ Complex.exp)) := by
-      filter_upwards [openStrip_isOpen.mem_nhds hz] with w hw
-      simp only [stripPotential,hext _ (exp_mem_halfPlane hw),sphereTruncation_coe,Function.comp_apply]
-    dsimp only
-    rw [heq.fderiv_eq]
-  rw [he]
-  exact hb
-
-theorem access_section_zero {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    {Fbar : ℂ → OnePoint ℂ} {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1))
-    (hη0 : η 0 = 0) (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-    {v : ℝ} (hv : v < 0) : ∃ φ ∈ Icc (0:ℝ) Real.pi,
-      stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I) = 0 := by
-  have hm : Real.exp v ∈ Icc ‖η 0‖ ‖η 1‖ := by
-    rw [hη0,hη1,norm_zero]
-    exact ⟨(Real.exp_pos _).le,(Real.exp_lt_one_iff.2 hv).le⟩
-  obtain ⟨t,ht,he⟩ := intermediate_value_Icc (by norm_num : (0:ℝ) ≤ 1) hη.norm hm
-  dsimp only at he
-  have henz : η t ≠ 0 := norm_ne_zero_iff.1 (by rw [he]; exact (Real.exp_pos _).ne')
-  have hlogRe : (Complex.log (η t)).re = v := by rw [Complex.log_re,he,Real.log_exp]
-  have hlog : (v:ℂ)+(Complex.arg (η t):ℂ)*Complex.I = Complex.log (η t) := by
-    rw [← hlogRe,← Complex.log_im]
-    exact Complex.re_add_im _
-  refine ⟨Complex.arg (η t),⟨Complex.arg_nonneg_iff.2 (hηH t ht),Complex.arg_le_pi _⟩,?_⟩
-  obtain ⟨w,hw,hew⟩ := haccess t ht
-  simp only [stripPotential,hlog,Complex.exp_log henz,← hew,sphereTruncation_coe]
-  apply radialTruncation_zero hH hr
-  simpa only [mem_closedBall,dist_eq_norm] using hw
-
-lemma stripPotential_interior_eq {F : ℂ → ℂ} {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (H r : ℝ) (q : ℂ) {z : ℂ} (hz : z ∈ openStrip) :
-    stripPotential H q r Fbar =ᶠ[𝓝 z] radialTruncation H q r ∘ (F ∘ Complex.exp) := by
-  filter_upwards [openStrip_isOpen.mem_nhds hz] with w hw
-  simp only [stripPotential,hext _ (exp_mem_halfPlane hw),sphereTruncation_coe,Function.comp_apply]
-
-lemma stripPotential_vertical_AC_interior {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hr : 0 < r) (q : ℂ) (v : ℝ) {a b : ℝ}
-    (ha : a ∈ Ioo (0:ℝ) Real.pi) (hb : b ∈ Ioo (0:ℝ) Real.pi) :
-    AbsolutelyContinuousOnInterval
-      (fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)) a b := by
-  let c := fun φ : ℝ => (v:ℂ)+(φ:ℂ)*Complex.I
-  have hc : ContDiff ℝ 1 c := contDiff_const.add (Complex.ofRealCLM.contDiff.mul contDiff_const)
-  have hmap : MapsTo c (uIcc a b) openStrip := by
-    intro φ hφ
-    have hlo : 0 < φ := lt_of_lt_of_le (lt_min ha.1 hb.1) hφ.1
-    have hhi : φ < Real.pi := lt_of_le_of_lt hφ.2 (max_lt ha.2 hb.2)
-    simpa [c,openStrip] using And.intro hlo hhi
-  have hdiff : DifferentiableOn ℂ (F ∘ Complex.exp) openStrip := by
-    intro z hz
-    exact ((hF _ (exp_mem_halfPlane hz)).differentiableAt.comp z
-      (Complex.differentiable_exp z)).differentiableWithinAt
-  have hG : ContDiffOn ℝ 1 ((F ∘ Complex.exp) ∘ c) (uIcc a b) :=
-    ((hdiff.contDiffOn openStrip_isOpen).restrict_scalars ℝ).comp hc.contDiffOn hmap
-  apply acl_congr (radialTruncation_along_C1_AC hr H q hG)
-  intro φ hφ
-  exact ((stripPotential_interior_eq hext H r q (hmap hφ)).self_of_nhds).symm
-
-lemma stripPotential_vertical_deriv {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ) {φ : ℝ}
-    (hφ : φ ∈ Ioo (0:ℝ) Real.pi) :
-    deriv (fun ψ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(ψ:ℂ)*Complex.I)) φ =
-      fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(φ:ℂ)*Complex.I) Complex.I := by
-  let c := fun ψ : ℝ => (v:ℂ)+(ψ:ℂ)*Complex.I
-  have hmem : c φ ∈ openStrip := by simpa [c,openStrip] using hφ
-  have he := stripPotential_interior_eq hext H r q hmem
-  have hc : HasDerivAt c Complex.I φ := by
-    simpa [c,Complex.ofRealCLM] using ((Complex.ofRealCLM.hasDerivAt).mul_const Complex.I).const_add (v:ℂ)
-  have hd := radialTruncation_directional_deriv hH hr q (c := c) (t := φ) (d := Complex.I)
-    ((hF _ (exp_mem_halfPlane hmem)).differentiableAt.comp _ (Complex.differentiable_exp _)) hc
-  rw [← he.fderiv_eq] at hd
-  refine (he.comp_tendsto hc.continuousAt).deriv_eq.trans hd
-
-theorem stripPotential_vertical_zero_energy {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ)
-    (hE : MemLp (fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
-      ((v:ℂ)+(φ:ℂ)*Complex.I)‖) 2 (volume.restrict (Ioc (0:ℝ) Real.pi)))
-    {z : ℝ} (hz : z ∈ Icc (0:ℝ) Real.pi)
-    (hzero : stripPotential H q r Fbar ((v:ℂ)+(z:ℂ)*Complex.I) = 0)
-    {φ : ℝ} (hφ : φ ∈ Icc (0:ℝ) Real.pi) :
-    (stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I))^2 ≤
-      Real.pi*(∫ ψ in (0:ℝ)..Real.pi, ‖fderiv ℝ (stripPotential H q r Fbar)
-        ((v:ℂ)+(ψ:ℂ)*Complex.I)‖^2) := by
-  let f := fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)
-  let e := fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(φ:ℂ)*Complex.I)‖
-  have hb : ∀ᵐ ψ ∂volume.restrict (Ioc (0:ℝ) Real.pi), ‖deriv f ψ‖ ≤ e ψ := by
-    rw [← restrict_Ioo_eq_restrict_Ioc]
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with ψ hψ
-    rw [stripPotential_vertical_deriv hF hext hH hr q v hψ]
-    simpa only [e,Complex.norm_I,mul_one] using
-      (fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(ψ:ℂ)*Complex.I)).le_opNorm Complex.I
-  have hD := hE.mono' (aestronglyMeasurable_deriv f _) hb
-  have hc : ContinuousOn f (Icc 0 Real.pi) :=
-    (stripPotential_continuous hH hr q hcont).comp
-      (continuous_const.add (Complex.continuous_ofReal.mul_const _)).continuousOn
-      (by intro ψ hψ; simpa [closedStrip] using hψ)
-  have ht := boundary_zero_energy Real.pi_pos hc hD
-    (fun a ha b hb => stripPotential_vertical_AC_interior hF hext hr q v ha hb) hz hzero hφ
-  have hm : (∫ ψ in (0:ℝ)..Real.pi, (deriv f ψ)^2) ≤
-      ∫ ψ in (0:ℝ)..Real.pi, (e ψ)^2 := by
-    simp only [intervalIntegral.integral_of_le Real.pi_pos.le]
-    apply integral_mono_ae hD.integrable_sq hE.integrable_sq
-    filter_upwards [hb] with ψ hψ
-    simpa only [Real.norm_eq_abs,sq_abs] using pow_le_pow_left₀ (norm_nonneg _) hψ 2
-  simp only [sub_zero] at ht
-  exact ht.trans (mul_le_mul_of_nonneg_left hm Real.pi_pos.le)
-
-lemma strip_energy_integrable {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    IntegrableOn (fun z => ‖fderiv ℝ (stripPotential H q r Fbar) z‖^2) openStrip := by
-  refine ⟨((measurable_fderiv ℝ _).norm.pow_const 2).aestronglyMeasurable,?_⟩
-  rw [hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun z => sq_nonneg _))]
-  exact (strip_potential_energy hF hinj hext hH hr q).trans_lt (ENNReal.ofReal_lt_top)
-
-lemma strip_energy_integrable_prod {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    Integrable (fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
-      ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2)
-      (volume.prod (volume.restrict (Ioo (0:ℝ) Real.pi))) := by
-  have hi := (Complex.volume_preserving_equiv_real_prod.symm _).integrableOn_comp_preimage
-    Complex.measurableEquivRealProd.symm.measurableEmbedding |>.2
-      (strip_energy_integrable hF hinj hext hH hr q)
-  have hs : Complex.measurableEquivRealProd.symm ⁻¹' openStrip =
-      (univ : Set ℝ).prod (Ioo (0:ℝ) Real.pi) := by
-    ext p
-    change (0 < p.2 ∧ p.2 < Real.pi) ↔ True ∧ (0 < p.2 ∧ p.2 < Real.pi)
-    simp
-  rw [hs] at hi
-  change Integrable _ ((volume.prod volume).restrict ((univ : Set ℝ) ×ˢ Ioo (0:ℝ) Real.pi)) at hi
-  rw [← Measure.prod_restrict,Measure.restrict_univ] at hi
-  convert hi using 1
-  funext p
-  have hp : (p.1:ℂ)+(p.2:ℂ)*Complex.I = Complex.measurableEquivRealProd.symm p := by
-    apply Complex.ext <;> simp
-  rw [hp]
-  rfl
-
-theorem strip_vertical_memLp_ae {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    ∀ᵐ v : ℝ, MemLp (fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
-      ((v:ℂ)+(φ:ℂ)*Complex.I)‖) 2 (volume.restrict (Ioc (0:ℝ) Real.pi)) := by
-  have hi := (strip_energy_integrable_prod hF hinj hext hH hr q).prod_right_ae
-  filter_upwards [hi] with v hv
-  rw [← restrict_Ioo_eq_restrict_Ioc]
-  apply (memLp_two_iff_integrable_sq ?_).2 hv
-  exact ((measurable_fderiv ℝ _).comp
-    ((measurable_const.add (Complex.measurable_ofReal.mul_const _)))).norm.aestronglyMeasurable
-
-theorem strip_negative_section_energy {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
-    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r) :
-    ∀ᵐ v : ℝ, v < 0 → ∀ φ ∈ Icc (0:ℝ) Real.pi,
-      (stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I))^2 ≤
-        Real.pi*(∫ ψ in (0:ℝ)..Real.pi, ‖fderiv ℝ (stripPotential H q r Fbar)
-          ((v:ℂ)+(ψ:ℂ)*Complex.I)‖^2) := by
-  filter_upwards [strip_vertical_memLp_ae hF hinj hext hH hr q] with v hv
-  intro hv0 φ hφ
-  obtain ⟨z,hz,hzero⟩ := access_section_zero hH.le hr q hη hη0 hη1 hηH haccess hv0
-  exact stripPotential_vertical_zero_energy hF hext hcont hH.le hr q v hv hz hzero hφ
-
-lemma mean_difference_abs_le {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
-    {μ : Measure α} {ν : Measure β} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
-    {u : α → ℝ} {v : β → ℝ} (hu : Integrable u μ) (hv : Integrable v ν) :
-    |(∫ x, u x ∂μ)-(∫ y, v y ∂ν)| ≤ ∫ x, ∫ y, |u x-v y| ∂ν ∂μ := by
-  have hD : Integrable (fun p : α × β => u p.1-v p.2) (μ.prod ν) :=
-    (hu.comp_fst ν).sub (hv.comp_snd μ)
-  have he : (∫ p : α × β, u p.1-v p.2 ∂μ.prod ν) =
-      (∫ x, u x ∂μ)-(∫ y, v y ∂ν) := by
-    rw [integral_prod _ hD]
-    simp_rw [integral_sub (integrable_const _) hv]
-    simp only [integral_const,probReal_univ,one_smul]
-    rw [integral_sub hu (integrable_const _)]
-    simp
-  have hh := norm_integral_le_integral_norm (μ := μ.prod ν) (fun p : α × β => u p.1-v p.2)
-  rw [he,integral_prod _ hD.norm] at hh
-  simpa only [Real.norm_eq_abs] using hh
-
-theorem rectangle_mean_L1_bound {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
-    {μ μ' : Measure X} {ν ν' : Measure Y}
-    [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
-    [IsProbabilityMeasure ν] [IsProbabilityMeasure ν']
-    {u : X × Y → ℝ} (hu : Integrable u (μ.prod ν)) (hu' : Integrable u (μ'.prod ν'))
-    {R : Y → ℝ} {C : X → ℝ} (hR : Integrable R ν') (hC : Integrable C μ)
-    (hbound : ∀ᵐ p ∂μ'.prod ν', ∀ᵐ q ∂μ.prod ν, |u p-u q| ≤ R p.2+C q.1) :
-    |(∫ p, u p ∂μ'.prod ν')-(∫ p, u p ∂μ.prod ν)| ≤
-      (∫ y, R y ∂ν')+(∫ x, C x ∂μ) := by
-  let A := μ'.prod ν'
-  let B := μ.prod ν
-  have hD : Integrable (fun p : (X × Y) × (X × Y) => |u p.1-u p.2|) (A.prod B) :=
-    ((hu'.comp_fst B).sub (hu.comp_snd A)).abs
-  have hRi : Integrable (fun p : X × Y => R p.2) A := hR.comp_snd μ'
-  have hCi : Integrable (fun p : X × Y => C p.1) B := hC.comp_fst ν
-  have heC : (∫ p : X × Y, C p.1 ∂B) = ∫ x, C x ∂μ := by
-    rw [integral_prod _ hCi]
-    simp only [integral_const,probReal_univ,one_smul]
-  have heR : (∫ p : X × Y, R p.2 ∂A) = ∫ y, R y ∂ν' := by
-    rw [integral_prod _ hRi]
-    simp only [integral_const,probReal_univ,one_smul]
-  calc
-    _ ≤ ∫ p, ∫ q, |u p-u q| ∂B ∂A := mean_difference_abs_le hu' hu
-    _ ≤ ∫ p, R p.2+(∫ q, C q.1 ∂B) ∂A := by
-      apply integral_mono_ae hD.integral_prod_left (hRi.add (integrable_const _))
-      filter_upwards [hD.prod_right_ae,hbound] with p hp hb
-      calc
-        _ ≤ ∫ q, R p.2+C q.1 ∂B := integral_mono_ae hp ((integrable_const _).add hCi) hb
-        _ = _ := by rw [integral_add (integrable_const _) hCi]; simp
-    _ = _ := by
-      rw [integral_add hRi (integrable_const _),heC,heR]
-      simp
-
-def intervalProb (a b : ℝ) : Measure ℝ :=
-  (ENNReal.ofReal (b-a))⁻¹ • volume.restrict (Ioc a b)
-
-lemma intervalProb_isProbability {a b : ℝ} (hab : a < b) : IsProbabilityMeasure (intervalProb a b) := by
-  constructor
-  simp only [intervalProb,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul,Real.volume_Ioc]
-  exact ENNReal.inv_mul_cancel (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hab)) ENNReal.ofReal_ne_top
-
-lemma integral_intervalProb {a b : ℝ} (hab : a ≤ b) (f : ℝ → ℝ) :
-    (∫ x, f x ∂intervalProb a b) = (b-a)⁻¹*(∫ x in a..b, f x) := by
-  rw [intervalProb,integral_smul_measure,ENNReal.toReal_inv,ENNReal.toReal_ofReal (sub_nonneg.2 hab),
-    smul_eq_mul,intervalIntegral.integral_of_le hab]
-
-lemma intervalProb_ae {a b : ℝ} {p : ℝ → Prop} :
-    (∀ᵐ x ∂intervalProb a b, p x) ↔ ∀ᵐ x ∂volume.restrict (Ioc a b), p x := by
-  exact Measure.ae_ennreal_smul_measure_iff (ENNReal.inv_ne_zero.2 ENNReal.ofReal_ne_top)
-
-lemma integrable_intervalProb {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
-    (hf : IntegrableOn f (Ioc a b)) : Integrable f (intervalProb a b) := by
-  exact hf.smul_measure (ENNReal.inv_ne_top.2 (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hab)))
-
-lemma intervalProb_integral_subinterval {a b c d K : ℝ} (hab : a < b) (hcd : c < d)
-    (hac : a ≤ c) (hdb : d ≤ b) (hK : 0 ≤ K) (hlen : b-a ≤ K*(d-c))
-    {f : ℝ → ℝ} (hf : IntegrableOn f (Ioc a b))
-    (hf0 : ∀ᵐ x ∂volume.restrict (Ioc a b), 0 ≤ f x) :
-    (∫ x, f x ∂intervalProb c d) ≤ K*(∫ x, f x ∂intervalProb a b) := by
-  have hsub : Ioc c d ⊆ Ioc a b := fun x hx => ⟨hac.trans_lt hx.1,hx.2.trans hdb⟩
-  have him : (∫ x in c..d, f x) ≤ ∫ x in a..b, f x := by
-    rw [intervalIntegral.integral_of_le hcd.le,intervalIntegral.integral_of_le hab.le]
-    exact setIntegral_mono_set hf hf0 (ae_of_all _ (fun x hx => hsub hx))
-  have hnon : 0 ≤ ∫ x in a..b, f x := by
-    rw [intervalIntegral.integral_of_le hab.le]
-    exact integral_nonneg_of_ae hf0
-  have hcoef : (d-c)⁻¹ ≤ K*(b-a)⁻¹ := by
-    rw [← one_div,← div_eq_mul_inv]
-    exact (div_le_div_iff₀ (sub_pos.2 hcd) (sub_pos.2 hab)).2 (by simpa using hlen)
-  rw [integral_intervalProb hcd.le,integral_intervalProb hab.le]
-  calc
-    _ ≤ (d-c)⁻¹*(∫ x in a..b, f x) := mul_le_mul_of_nonneg_left him (inv_nonneg.2 (sub_nonneg.2 hcd.le))
-    _ ≤ (K*(b-a)⁻¹)*(∫ x in a..b, f x) :=
-      mul_le_mul hcoef le_rfl hnon (mul_nonneg hK (inv_nonneg.2 (sub_nonneg.2 hab.le)))
-    _ = _ := by ring
-
-lemma acl_line_L1 {a b : ℝ} {f g : ℝ → ℝ}
-    (hf : AbsolutelyContinuousOnInterval f a b)
-    (hg : IntegrableOn g (Ioc a b))
-    (hbound : ∀ᵐ s ∂volume.restrict (Ioc a b), ‖deriv f s‖ ≤ g s)
-    {x y : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b) :
-    |f y-f x| ≤ ∫ s in a..b, g s := by
-  have hab := hx.1.trans hx.2
-  have hD : IntegrableOn (deriv f) (Ioc a b) :=
-    hg.mono' (aestronglyMeasurable_deriv f _) hbound
-  suffices ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, x ≤ y →
-      |f y-f x| ≤ ∫ s in a..b, g s by
-    rcases le_total x y with h | h
-    · exact this x hx y hy h
-    · simpa only [abs_sub_comm] using this y hy x hx h
-  intro x hx y hy hxy
-  have hsub : Ioc x y ⊆ Ioc a b := fun s hs => ⟨hx.1.trans_lt hs.1,hs.2.trans hy.2⟩
-  have hfac : AbsolutelyContinuousOnInterval f x y := hf.mono
-    (by simpa only [uIcc_of_le hab,uIcc_of_le hxy] using Icc_subset_Icc hx.1 hy.2)
-  calc
-    _ = ‖∫ s in x..y, deriv f s‖ := by rw [hfac.integral_deriv_eq_sub,Real.norm_eq_abs]
-    _ ≤ ∫ s in x..y, ‖deriv f s‖ := intervalIntegral.norm_integral_le_integral_norm hxy
-    _ ≤ ∫ s in a..b, ‖deriv f s‖ := by
-      rw [intervalIntegral.integral_of_le hxy,intervalIntegral.integral_of_le hab]
-      exact setIntegral_mono_set hD.norm (ae_of_all _ (fun s => norm_nonneg _))
-        (ae_of_all _ (fun s hs => hsub hs))
-    _ ≤ ∫ s in a..b, g s := by
-      rw [intervalIntegral.integral_of_le hab,intervalIntegral.integral_of_le hab]
-      exact integral_mono_ae hD.norm hg hbound
-
-theorem rectangle_poincare_L1 {a b c d a' b' c' d' K : ℝ}
-    (hab : a < b) (hcd : c < d) (hab' : a' < b') (hcd' : c' < d')
-    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d)
-    (hK : 0 ≤ K) (hlen : d-c ≤ K*(d'-c'))
-    {u g : ℝ × ℝ → ℝ}
-    (hu : Integrable u ((intervalProb a b).prod (intervalProb c d)))
-    (hu' : Integrable u ((intervalProb a' b').prod (intervalProb c' d')))
-    (hg : Integrable g ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))))
-    (hg0 : ∀ p, 0 ≤ g p)
-    (hhor : ∀ᵐ y ∂volume.restrict (Ioc c d),
-      AbsolutelyContinuousOnInterval (fun x => u (x,y)) a b ∧
-      ∀ᵐ x ∂volume.restrict (Ioc a b), ‖deriv (fun s => u (s,y)) x‖ ≤ g (x,y))
-    (hver : ∀ᵐ x ∂volume.restrict (Ioc a b),
-      AbsolutelyContinuousOnInterval (fun y => u (x,y)) c d ∧
-      ∀ᵐ y ∂volume.restrict (Ioc c d), ‖deriv (fun s => u (x,s)) y‖ ≤ g (x,y)) :
-    |(∫ p, u p ∂(intervalProb a' b').prod (intervalProb c' d'))-
-      (∫ p, u p ∂(intervalProb a b).prod (intervalProb c d))| ≤
-      (K/(d-c)+1/(b-a))*(∫ p, g p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-  let := intervalProb_isProbability hab
-  let := intervalProb_isProbability hcd
-  let := intervalProb_isProbability hab'
-  let := intervalProb_isProbability hcd'
-  let R := fun y => ∫ x in a..b, g (x,y)
-  let C := fun x => ∫ y in c..d, g (x,y)
-  have hRi : IntegrableOn R (Ioc c d) := by
-    simpa only [IntegrableOn,R,intervalIntegral.integral_of_le hab.le] using hg.integral_prod_right
-  have hCi : IntegrableOn C (Ioc a b) := by
-    simpa only [IntegrableOn,C,intervalIntegral.integral_of_le hcd.le] using hg.integral_prod_left
-  have subX : Ioc a' b' ⊆ Ioc a b := fun x hx => ⟨haa.trans_lt hx.1,hx.2.trans hbb⟩
-  have subY : Ioc c' d' ⊆ Ioc c d := fun y hy => ⟨hcc.trans_lt hy.1,hy.2.trans hdd⟩
-  have hH : ∀ᵐ y ∂intervalProb c' d', ∀ x₁ ∈ Icc a b, ∀ x₂ ∈ Icc a b,
-      |u (x₁,y)-u (x₂,y)| ≤ R y := by
-    apply intervalProb_ae.2
-    apply (ae_mono (Measure.restrict_mono_set volume subY))
-    filter_upwards [hhor,hg.prod_left_ae] with y hy hgy
-    intro x₁ hx₁ x₂ hx₂
-    exact acl_line_L1 hy.1 hgy hy.2 hx₂ hx₁
-  have hV : ∀ᵐ x ∂intervalProb a b, ∀ y₁ ∈ Icc c d, ∀ y₂ ∈ Icc c d,
-      |u (x,y₁)-u (x,y₂)| ≤ C x := by
-    apply intervalProb_ae.2
-    filter_upwards [hver,hg.prod_right_ae] with x hx hgx
-    intro y₁ hy₁ y₂ hy₂
-    exact acl_line_L1 hx.1 hgx hx.2 hy₂ hy₁
-  have hmem {l r : ℝ} : ∀ᵐ x ∂intervalProb l r, x ∈ Icc l r :=
-    intervalProb_ae.2 ((ae_restrict_mem measurableSet_Ioc).mono (fun x hx => Ioc_subset_Icc_self hx))
-  have hb : ∀ᵐ p ∂(intervalProb a' b').prod (intervalProb c' d'),
-      ∀ᵐ q ∂(intervalProb a b).prod (intervalProb c d), |u p-u q| ≤ R p.2+C q.1 := by
-    have hpX := (Measure.quasiMeasurePreserving_fst (μ := intervalProb a' b') (ν := intervalProb c' d')).tendsto_ae.eventually (hmem (l := a') (r := b'))
-    have hpY := (Measure.quasiMeasurePreserving_snd (μ := intervalProb a' b') (ν := intervalProb c' d')).tendsto_ae.eventually (hmem (l := c') (r := d'))
-    have hpH := (Measure.quasiMeasurePreserving_snd (μ := intervalProb a' b') (ν := intervalProb c' d')).tendsto_ae.eventually hH
-    have hqX := (Measure.quasiMeasurePreserving_fst (μ := intervalProb a b) (ν := intervalProb c d)).tendsto_ae.eventually (hmem (l := a) (r := b))
-    have hqY := (Measure.quasiMeasurePreserving_snd (μ := intervalProb a b) (ν := intervalProb c d)).tendsto_ae.eventually (hmem (l := c) (r := d))
-    have hqV := (Measure.quasiMeasurePreserving_fst (μ := intervalProb a b) (ν := intervalProb c d)).tendsto_ae.eventually hV
-    filter_upwards [hpX,hpY,hpH] with p hpx hpy hph
-    filter_upwards [hqX,hqY,hqV] with q hqx hqy hqv
-    have hpx' : p.1 ∈ Icc a b := ⟨haa.trans hpx.1,hpx.2.trans hbb⟩
-    have hpy' : p.2 ∈ Icc c d := ⟨hcc.trans hpy.1,hpy.2.trans hdd⟩
-    exact (abs_sub_le (u p) (u (q.1,p.2)) (u q)).trans
-      (add_le_add (hph p.1 hpx' q.1 hqx) (hqv p.2 hpy' q.2 hqy))
-  have hmain := rectangle_mean_L1_bound hu hu'
-    (integrable_intervalProb hcd' (hRi.mono_set subY)) (integrable_intervalProb hab hCi) hb
-  have hR0 : ∀ y, 0 ≤ R y := fun y => intervalIntegral.integral_nonneg hab.le (fun x hx => hg0 (x,y))
-  have hRbound := intervalProb_integral_subinterval hcd hcd' hcc hdd hK hlen hRi (ae_of_all _ hR0)
-  have hRE : (∫ y, R y ∂intervalProb c d) =
-      (d-c)⁻¹*(∫ p, g p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-    rw [integral_intervalProb hcd.le]
-    congr 1
-    rw [intervalIntegral.integral_of_le hcd.le,integral_prod_symm _ hg]
-    simp only [R,intervalIntegral.integral_of_le hab.le]
-  have hCE : (∫ x, C x ∂intervalProb a b) =
-      (b-a)⁻¹*(∫ p, g p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-    rw [integral_intervalProb hab.le]
-    congr 1
-    rw [intervalIntegral.integral_of_le hab.le,integral_prod _ hg]
-    simp only [C,intervalIntegral.integral_of_le hcd.le]
-  calc
-    _ ≤ (∫ y, R y ∂intervalProb c' d')+(∫ x, C x ∂intervalProb a b) := hmain
-    _ ≤ K*(∫ y, R y ∂intervalProb c d)+(∫ x, C x ∂intervalProb a b) := add_le_add hRbound le_rfl
-    _ = _ := by rw [hRE,hCE]; ring
-
-lemma stripPotential_horizontal_AC {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hr : 0 < r) (q : ℂ) {φ : ℝ}
-    (hφ : φ ∈ Ioo (0:ℝ) Real.pi) (a b : ℝ) :
-    AbsolutelyContinuousOnInterval
-      (fun v : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)) a b := by
-  let c := fun v : ℝ => (v:ℂ)+(φ:ℂ)*Complex.I
-  have hc : ContDiff ℝ 1 c := Complex.ofRealCLM.contDiff.add contDiff_const
-  have hmap : MapsTo c (uIcc a b) openStrip := by
-    intro v hv
-    simpa [c,openStrip] using hφ
-  have hdiff : DifferentiableOn ℂ (F ∘ Complex.exp) openStrip := by
-    intro z hz
-    exact ((hF _ (exp_mem_halfPlane hz)).differentiableAt.comp z
-      (Complex.differentiable_exp z)).differentiableWithinAt
-  have hG : ContDiffOn ℝ 1 ((F ∘ Complex.exp) ∘ c) (uIcc a b) :=
-    ((hdiff.contDiffOn openStrip_isOpen).restrict_scalars ℝ).comp hc.contDiffOn hmap
-  apply acl_congr (radialTruncation_along_C1_AC hr H q hG)
-  intro v hv
-  exact ((stripPotential_interior_eq hext H r q (hmap hv)).self_of_nhds).symm
-
-lemma stripPotential_horizontal_deriv {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ) {φ : ℝ}
-    (hφ : φ ∈ Ioo (0:ℝ) Real.pi) :
-    deriv (fun s : ℝ => stripPotential H q r Fbar ((s:ℂ)+(φ:ℂ)*Complex.I)) v =
-      fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(φ:ℂ)*Complex.I) 1 := by
-  let c := fun s : ℝ => (s:ℂ)+(φ:ℂ)*Complex.I
-  have hmem : c v ∈ openStrip := by simpa [c,openStrip] using hφ
-  have he := stripPotential_interior_eq hext H r q hmem
-  have hc : HasDerivAt c 1 v := by
-    simpa [c,Complex.ofRealCLM] using Complex.ofRealCLM.hasDerivAt.add_const ((φ:ℂ)*Complex.I)
-  have hd := radialTruncation_directional_deriv hH hr q (c := c) (t := v) (d := 1)
-    ((hF _ (exp_mem_halfPlane hmem)).differentiableAt.comp _ (Complex.differentiable_exp _)) hc
-  rw [← he.fderiv_eq] at hd
-  exact (he.comp_tendsto hc.continuousAt).deriv_eq.trans hd
-
-lemma stripPotential_vertical_AC {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ)
-    (hE : MemLp (fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
-      ((v:ℂ)+(φ:ℂ)*Complex.I)‖) 2 (volume.restrict (Ioc (0:ℝ) Real.pi))) :
-    AbsolutelyContinuousOnInterval
-      (fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)) 0 Real.pi := by
-  let f := fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)
-  have hb : ∀ᵐ ψ ∂volume.restrict (Ioc (0:ℝ) Real.pi), ‖deriv f ψ‖ ≤
-      ‖fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(ψ:ℂ)*Complex.I)‖ := by
-    rw [← restrict_Ioo_eq_restrict_Ioc]
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with ψ hψ
-    rw [stripPotential_vertical_deriv hF hext hH hr q v hψ]
-    simpa only [Complex.norm_I,mul_one] using
-      (fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(ψ:ℂ)*Complex.I)).le_opNorm Complex.I
-  have hD := hE.mono' (aestronglyMeasurable_deriv f _) hb
-  have hc : ContinuousOn f (Icc 0 Real.pi) :=
-    (stripPotential_continuous hH hr q hcont).comp
-      (continuous_const.add (Complex.continuous_ofReal.mul_const _)).continuousOn
-      (by intro ψ hψ; simpa [closedStrip] using hψ)
-  exact acl_of_interior Real.pi_pos hc
-    ((intervalIntegrable_iff_integrableOn_Ioc_of_le Real.pi_pos.le).2 (hD.integrable (by norm_num)))
-    (fun a ha b hb => stripPotential_vertical_AC_interior hF hext hr q v ha hb)
-
-theorem strip_coordinate_ACL {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {a b c d : ℝ} (hcd : c < d) (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
-    (∀ᵐ y : ℝ ∂volume.restrict (Ioc c d),
-      AbsolutelyContinuousOnInterval
-        (fun x : ℝ => stripPotential H q r Fbar ((x:ℂ)+(y:ℂ)*Complex.I)) a b ∧
-      ∀ᵐ x : ℝ ∂volume.restrict (Ioc a b),
-        ‖deriv (fun s : ℝ => stripPotential H q r Fbar ((s:ℂ)+(y:ℂ)*Complex.I)) x‖ ≤
-          ‖fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)‖) ∧
-    (∀ᵐ x : ℝ ∂volume.restrict (Ioc a b),
-      AbsolutelyContinuousOnInterval
-        (fun y : ℝ => stripPotential H q r Fbar ((x:ℂ)+(y:ℂ)*Complex.I)) c d ∧
-      ∀ᵐ y : ℝ ∂volume.restrict (Ioc c d),
-        ‖deriv (fun s : ℝ => stripPotential H q r Fbar ((x:ℂ)+(s:ℂ)*Complex.I)) y‖ ≤
-          ‖fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)‖) := by
-  have hy : ∀ᵐ y : ℝ ∂volume.restrict (Ioc c d), y ∈ Ioo (0:ℝ) Real.pi := by
-    rw [← restrict_Ioo_eq_restrict_Ioc]
-    filter_upwards [ae_restrict_mem measurableSet_Ioo] with y hy
-    exact ⟨hc.trans_lt hy.1,hy.2.trans_le hd⟩
-  constructor
-  · filter_upwards [hy] with y hy
-    refine ⟨stripPotential_horizontal_AC hF hext hr q hy a b,ae_of_all _ (fun x => ?_)⟩
-    rw [stripPotential_horizontal_deriv hF hext hH.le hr q x hy]
-    simpa only [norm_one,mul_one] using
-      (fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)).le_opNorm 1
-  · filter_upwards [(strip_vertical_memLp_ae hF hinj hext hH hr q).filter_mono
-      (ae_mono Measure.restrict_le_self)] with x hx
-    refine ⟨(stripPotential_vertical_AC hF hext hcont hH.le hr q x hx).mono ?_,?_⟩
-    · rw [uIcc_of_le hcd.le,uIcc_of_le Real.pi_pos.le]
-      exact fun y hy => ⟨hc.trans hy.1,hy.2.trans hd⟩
-    · filter_upwards [hy] with y hy
-      rw [stripPotential_vertical_deriv hF hext hH.le hr q x hy]
-      simpa only [Complex.norm_I,mul_one] using
-        (fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)).le_opNorm Complex.I
-
-lemma strip_gradient_rect_memLp {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    (a b : ℝ) {c d : ℝ} (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
-    MemLp (fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
-      ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖) 2
-      ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-  have hi := strip_energy_integrable_prod hF hinj hext hH hr q
-  rw [restrict_Ioo_eq_restrict_Ioc] at hi
-  have hm : (volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d)) ≤
-      volume.prod (volume.restrict (Ioc (0:ℝ) Real.pi)) :=
-    Measure.prod_mono Measure.restrict_le_self (Measure.restrict_mono_set _
-      (fun y hy => ⟨hc.trans_lt hy.1,hy.2.trans hd⟩))
-  apply (memLp_two_iff_integrable_sq ?_).2 (hi.mono_measure hm)
-  exact ((measurable_fderiv ℝ _).comp
-    ((Complex.measurable_ofReal.comp measurable_fst).add
-      ((Complex.measurable_ofReal.comp measurable_snd).mul_const _))).norm.aestronglyMeasurable
-
-lemma strip_potential_rect_integrable {Fbar : ℂ → OnePoint ℂ}
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    (a b : ℝ) {c d : ℝ} (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
-    Integrable (fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I))
-      ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-  have hh : ContinuousOn (fun p : ℝ × ℝ => stripPotential H q r Fbar
-      ((p.1:ℂ)+(p.2:ℂ)*Complex.I)) ((Icc a b).prod (Icc c d)) :=
-    (stripPotential_continuous hH hr q hcont).comp
-      ((Complex.continuous_ofReal.comp continuous_fst).add
-        ((Complex.continuous_ofReal.comp continuous_snd).mul_const _)).continuousOn
-      (by intro p hp; simpa [closedStrip] using And.intro (hc.trans hp.2.1) (hp.2.2.trans hd))
-  have hi := hh.integrableOn_compact (μ := volume) (isCompact_Icc.prod isCompact_Icc)
-  have hj := hi.mono_set (prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)
-  rw [IntegrableOn,Measure.volume_eq_prod,← Measure.prod_restrict] at hj
-  exact hj
-
-lemma strip_potential_mean_integrable {Fbar : ℂ → OnePoint ℂ}
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    {a b c d : ℝ} (hab : a < b) (hcd : c < d) (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
-    Integrable (fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I))
-      ((intervalProb a b).prod (intervalProb c d)) := by
-  rw [intervalProb,intervalProb,Measure.prod_smul_left,Measure.prod_smul_right]
-  exact ((strip_potential_rect_integrable hcont hH hr q a b hc hd).smul_measure
-    (ENNReal.inv_ne_top.2 (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hcd)))).smul_measure
-      (ENNReal.inv_ne_top.2 (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hab)))
-
-theorem strip_rectangle_poincare {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {a b c d a' b' c' d' K : ℝ}
-    (hab : a < b) (hcd : c < d) (hab' : a' < b') (hcd' : c' < d')
-    (hlo : 0 ≤ c) (hhi : d ≤ Real.pi)
-    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d)
-    (hK : 0 ≤ K) (hlen : d-c ≤ K*(d'-c')) :
-    |(∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb a' b').prod (intervalProb c' d'))-
-      (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb a b).prod (intervalProb c d))| ≤
-      (K/(d-c)+1/(b-a))*(∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar)
-        ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖
-          ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-  have hh := strip_coordinate_ACL hF hinj hext hcont hH hr q (a := a) (b := b) hcd hlo hhi
-  exact rectangle_poincare_L1 hab hcd hab' hcd' haa hbb hcc hdd hK hlen
-    (strip_potential_mean_integrable hcont hH.le hr q hab hcd hlo hhi)
-    (strip_potential_mean_integrable hcont hH.le hr q hab' hcd' (hlo.trans hcc) (hdd.trans hhi))
-    ((strip_gradient_rect_memLp hF hinj hext hH hr q a b hlo hhi).integrable (by norm_num))
-    (fun _ => norm_nonneg _) hh.1 hh.2
-
-lemma dyadic_subsum_le {a : ℕ → ℝ} (ha : ∀ j, a j = 0 ∨ a j = (2:ℝ)^j) (n : ℕ) :
-    ∑ j ∈ Finset.range n, a j ≤ (2:ℝ)^n-1 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ,pow_succ]
-    rcases ha n with h | h
-    · rw [h,add_zero]
-      linarith [pow_pos (by norm_num : (0:ℝ) < 2) n]
-    · rw [h]
-      linarith
-
-lemma dyadic_subsum_square {a : ℕ → ℝ} (ha : ∀ j, a j = 0 ∨ a j = (2:ℝ)^j) (n : ℕ) :
-    (∑ j ∈ Finset.range n, a j)^2 ≤ 3*∑ j ∈ Finset.range n, (a j)^2 := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    rw [Finset.sum_range_succ,Finset.sum_range_succ]
-    rcases ha n with h | h
-    · simpa only [h,add_zero,zero_pow (by norm_num : (2:ℕ) ≠ 0)] using ih
-    · rw [h]
-      have hb := dyadic_subsum_le ha n
-      have hp : 0 ≤ (2:ℝ)^n := by positivity
-      have hm := mul_nonneg hp (show 0 ≤ (2:ℝ)^n-∑ j ∈ Finset.range n, a j by linarith)
-      nlinarith
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-section Measure
-variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [IsFiniteMeasure μ]
-
-def dyadicKernel (Q : ℕ → Set α) (n : ℕ) (x : α) : ℝ :=
-  ∑ j ∈ Finset.range n, (Q j).indicator (fun _ => (2:ℝ)^j) x
-
-lemma dyadicKernel_memLp {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j)) (n : ℕ) :
-    MemLp (dyadicKernel Q n) 2 μ := by
-  have hh := memLp_finsetSum' (μ := μ) (p := (2:ℝ≥0∞)) (Finset.range n)
-    (fun j hj => (memLp_const ((2:ℝ)^j)).indicator (hQ j))
-  have he : dyadicKernel Q n = ∑ j ∈ Finset.range n, (Q j).indicator (fun _ => (2:ℝ)^j) := by
-    funext x
-    simp only [dyadicKernel,Finset.sum_apply]
-  rw [he]
-  exact hh
-
-lemma dyadicKernel_square_energy {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
-    {V : ℝ} (n : ℕ) (harea : ∀ j < n, ((2:ℝ)^j)^2*μ.real (Q j) ≤ V) :
-    (∫ x, (dyadicKernel Q n x)^2 ∂μ) ≤ 3*(n:ℝ)*V := by
-  have hpoint (x : α) : (dyadicKernel Q n x)^2 ≤
-      3*∑ j ∈ Finset.range n, ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2 := by
-    apply dyadic_subsum_square
-    intro j
-    by_cases hx : x ∈ Q j
-    · right; exact indicator_of_mem hx _
-    · left; exact indicator_of_notMem hx _
-  have hi (j : ℕ) : Integrable (fun x => ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2) μ :=
-    ((memLp_const ((2:ℝ)^j) : MemLp _ 2 μ).indicator (hQ j)).integrable_sq
-  calc
-    _ ≤ ∫ x, 3*∑ j ∈ Finset.range n, ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2 ∂μ :=
-      integral_mono (dyadicKernel_memLp hQ n).integrable_sq
-        ((integrable_finsetSum _ (fun j hj => hi j)).const_mul 3) hpoint
-    _ = 3*∑ j ∈ Finset.range n, ((2:ℝ)^j)^2*μ.real (Q j) := by
-      rw [integral_const_mul,integral_finsetSum _ (fun j hj => hi j)]
-      congr 1
-      apply Finset.sum_congr rfl
-      intro j hj
-      have he : (fun x => ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2) =
-          (Q j).indicator (fun _ => ((2:ℝ)^j)^2) := by
-        funext x
-        by_cases hx : x ∈ Q j <;> simp [hx]
-      rw [he,integral_indicator (hQ j),integral_const]
-      simp [Measure.real,smul_eq_mul,mul_comm]
-    _ ≤ 3*(n:ℝ)*V := by
-      have hh := Finset.sum_le_sum (s := Finset.range n)
-        (fun j hj => harea j (Finset.mem_range.1 hj))
-      simpa only [Finset.sum_const,Finset.card_range,nsmul_eq_mul,mul_assoc] using
-        mul_le_mul_of_nonneg_left hh (by norm_num : (0:ℝ) ≤ 3)
-
-end Measure
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.WeightedEnergy
+
+namespace OAI
+
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+open Layer
+
+lemma annular_invSquare_integral {r R : ℝ} (hr : 0 < r) (hR : r < R) :
+    (∫⁻ z in {z : ℂ | r < ‖z‖ ∧ ‖z‖ < R}, ENNReal.ofReal (1/‖z‖^2)) ≤
+      ENNReal.ofReal (2*Real.pi*Real.log (R/r)) := by
+  apply (polar_annulus_lintegral_le _ hr).trans
+  have he : (∫⁻ s in Ioo r R, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+      ENNReal.ofReal s*ENNReal.ofReal (1/‖s • circleMap 0 1 θ‖^2)) =
+      ENNReal.ofReal (2*Real.pi)*(∫⁻ s in Ioo r R, ENNReal.ofReal s⁻¹) := by
+    calc
+      _ = ∫⁻ s in Ioo r R, ENNReal.ofReal (2*Real.pi)*ENNReal.ofReal s⁻¹ := by
+        apply lintegral_congr_ae
+        filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs
+        have hs0 := hr.trans hs.1
+        simp only [radial_norm,abs_of_pos hs0]
+        have hmul : ENNReal.ofReal s*ENNReal.ofReal (1/s^2) = ENNReal.ofReal s⁻¹ := by
+          rw [← ENNReal.ofReal_mul hs0.le]
+          congr 1
+          field_simp
+        simp only [hmul,lintegral_const,Measure.restrict_apply_univ,Real.volume_Ioo,
+          show Real.pi - -Real.pi = 2*Real.pi by ring]
+        exact mul_comm _ _
+      _ = _ := by rw [lintegral_const_mul']; exact ENNReal.ofReal_ne_top
+  rw [he]
+  have hc : ContinuousOn (fun s : ℝ => s⁻¹) (Icc r R) :=
+    continuousOn_id.inv₀ (fun s hs => (hr.trans_le hs.1).ne')
+  have hi : IntegrableOn (fun s : ℝ => s⁻¹) (Ioc r R) :=
+    (hc.intervalIntegrable_of_Icc hR.le).1
+  have hnon : 0 ≤ᵐ[volume.restrict (Ioc r R)] (fun s : ℝ => s⁻¹) := by
+    filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
+    exact inv_nonneg.2 (hr.trans hs.1).le
+  have hint : (∫ s in r..R, s⁻¹) = Real.log (R/r) := by
+    apply integral_inv
+    rw [uIcc_of_le hR.le]
+    intro hz
+    exact (not_le_of_gt hr) hz.1
+  have heq : (∫⁻ s in Ioo r R, ENNReal.ofReal s⁻¹) = ENNReal.ofReal (Real.log (R/r)) := by
+    rw [restrict_Ioo_eq_restrict_Ioc,← ofReal_integral_eq_lintegral_ofReal hi hnon,
+      ← intervalIntegral.integral_of_le hR.le,hint]
+  rw [heq,← ENNReal.ofReal_mul (by positivity)]
+
+def annularKernel (r R : ℝ) (z : ℂ) : ℝ≥0∞ :=
+  if r < ‖z‖ ∧ ‖z‖ < R then ENNReal.ofReal (1/‖z‖^2) else 0
+
+lemma annularKernel_integral {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    (∫⁻ z : ℂ, annularKernel r (Real.exp H*r) (z-q)) ≤ ENNReal.ofReal (2*Real.pi*H) := by
+  rw [lintegral_sub_right_eq_self]
+  have he : annularKernel r (Real.exp H*r) =
+      {z : ℂ | r < ‖z‖ ∧ ‖z‖ < Real.exp H*r}.indicator (fun z => ENNReal.ofReal (1/‖z‖^2)) := by
+    ext z
+    simp [annularKernel,Set.indicator]
+  have hA : MeasurableSet {z : ℂ | r < ‖z‖ ∧ ‖z‖ < Real.exp H*r} :=
+    (measurableSet_lt measurable_const measurable_norm).inter
+      (measurableSet_lt measurable_norm measurable_const)
+  rw [he,lintegral_indicator hA]
+  have hR : r < Real.exp H*r := (lt_mul_iff_one_lt_left hr).2 (Real.one_lt_exp_iff.2 hH)
+  have hh := annular_invSquare_integral hr hR
+  have hlog : Real.log (Real.exp H*r/r) = H := by
+    rw [mul_div_cancel_right₀ _ hr.ne',Real.log_exp]
+  simpa only [hlog] using hh
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma complex_real_jacobian (c : ℂ) :
+    ((ContinuousLinearMap.toSpanSingleton ℂ c).restrictScalars ℝ).det = ‖c‖^2 := by
+  simp [ContinuousLinearMap.det,LinearMap.det_restrictScalars,
+    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
+
+theorem conformal_truncation_energy {U : Set ℂ} (hU : IsOpen U)
+    {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F U) (hinj : InjOn F U)
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    (∫⁻ z in U, ENNReal.ofReal (‖fderiv ℝ (radialTruncation H q r ∘ F) z‖^2)) ≤
+      ENNReal.ofReal (2*Real.pi*H) := by
+  have hchange := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hU.measurableSet
+    (fun z hz => ((hF.differentiableAt (hU.mem_nhds hz)).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt)
+    hinj (fun w => annularKernel r (Real.exp H*r) (w-q))
+  calc
+    _ ≤ ∫⁻ z in U, ENNReal.ofReal (‖deriv F z‖^2)*annularKernel r (Real.exp H*r) (F z-q) := by
+      apply setLIntegral_mono' hU.measurableSet
+      intro z hz
+      have hb := radialTruncation_fderiv_bound hH.le hr (q := q)
+        (hF.differentiableAt (hU.mem_nhds hz))
+      apply (ENNReal.ofReal_le_ofReal hb).trans
+      by_cases hh : r < ‖F z-q‖ ∧ ‖F z-q‖ < Real.exp H*r
+      · rw [ite_eq_left hh,annularKernel,ite_eq_left hh,← ENNReal.ofReal_mul (sq_nonneg _)]
+        apply le_of_eq
+        congr 1
+        ring
+      · simp only [ite_eq_right hh,annularKernel,ENNReal.ofReal_zero,mul_zero,le_refl]
+    _ = ∫⁻ w in F '' U, annularKernel r (Real.exp H*r) (w-q) := by
+      rw [hchange]
+      apply setLIntegral_congr_fun hU.measurableSet
+      intro z hz
+      dsimp only
+      rw [complex_real_jacobian,abs_of_nonneg (sq_nonneg _)]
+    _ ≤ ∫⁻ w : ℂ, annularKernel r (Real.exp H*r) (w-q) := lintegral_mono_set (subset_univ _) |> fun h => by simpa using h
+    _ ≤ _ := annularKernel_integral hH hr q
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal OnePoint
+namespace SLEExactGauge.RectangleEnergy
+
+def sphereTruncation (H : ℝ) (q : ℂ) (r : ℝ) : OnePoint ℂ → ℝ :=
+  OnePoint.rec H (radialTruncation H q r)
+
+@[simp] lemma sphereTruncation_coe (H : ℝ) (q : ℂ) (r : ℝ) (z : ℂ) :
+    sphereTruncation H q r (z : OnePoint ℂ) = radialTruncation H q r z := rfl
+
+@[simp] lemma sphereTruncation_infty (H : ℝ) (q : ℂ) (r : ℝ) :
+    sphereTruncation H q r ∞ = H := rfl
+
+lemma sphereTruncation_continuous {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
+    Continuous (sphereTruncation H q r) := by
+  rw [continuous_iff_continuousAt]
+  intro z
+  induction z using OnePoint.rec with
+  | infty =>
+    apply OnePoint.continuousAt_infty.2
+    intro s hs
+    refine ⟨closedBall q (Real.exp H*r),isClosed_closedBall,isCompact_closedBall _ _,?_⟩
+    intro w hw
+    have hd : Real.exp H*r ≤ ‖w-q‖ := by
+      have hn : Real.exp H*r < dist w q := lt_of_not_ge hw
+      exact (show Real.exp H*r < ‖w-q‖ by simpa [dist_eq_norm] using hn).le
+    change radialTruncation H q r w ∈ s
+    rw [radialTruncation_top hH hr hd]
+    exact mem_of_mem_nhds hs
+  | coe w =>
+    exact OnePoint.continuousAt_coe.2 (radialTruncation_continuous hr H q).continuousAt
+
+lemma sphereTruncation_bounds {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    (z : OnePoint ℂ) : 0 ≤ sphereTruncation H q r z ∧ sphereTruncation H q r z ≤ H := by
+  induction z using OnePoint.rec with
+  | infty => exact ⟨hH,le_rfl⟩
+  | coe w => exact radialTruncation_bounds hH hr q w
+
+lemma sphereTruncation_top_of_outside {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    {z : OnePoint ℂ} (hz : z ∉ (fun w : ℂ => (w : OnePoint ℂ)) '' ball q (Real.exp H*r)) :
+    sphereTruncation H q r z = H := by
+  induction z using OnePoint.rec with
+  | infty => rfl
+  | coe w =>
+    apply radialTruncation_top hH hr
+    have hn : w ∉ ball q (Real.exp H*r) := fun hw => hz ⟨w,hw,rfl⟩
+    simpa [mem_ball,dist_eq_norm] using (le_of_not_gt hn : Real.exp H*r ≤ dist w q)
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+def openStrip : Set ℂ := {z | 0 < z.im ∧ z.im < Real.pi}
+
+def closedStrip : Set ℂ := {z | 0 ≤ z.im ∧ z.im ≤ Real.pi}
+
+def stripPotential (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) (z : ℂ) : ℝ :=
+  sphereTruncation H q r (Fbar (Complex.exp z))
+
+lemma openStrip_isOpen : IsOpen openStrip :=
+  (isOpen_lt continuous_const Complex.continuous_im).inter
+    (isOpen_lt Complex.continuous_im continuous_const)
+
+lemma exp_mem_halfPlane {z : ℂ} (hz : z ∈ openStrip) : 0 < (Complex.exp z).im := by
+  rw [Complex.exp_im]
+  exact mul_pos (Real.exp_pos _) (Real.sin_pos_of_pos_of_lt_pi hz.1 hz.2)
+
+lemma exp_mem_closedHalfPlane {z : ℂ} (hz : z ∈ closedStrip) : 0 ≤ (Complex.exp z).im := by
+  rw [Complex.exp_im]
+  exact mul_nonneg (Real.exp_pos _).le (Real.sin_nonneg_of_mem_Icc hz)
+
+lemma exp_injOn_strip : InjOn Complex.exp openStrip := by
+  intro z hz w hw hzw
+  exact Complex.exp_inj_of_neg_pi_lt_of_le_pi
+    ((neg_lt_zero.2 Real.pi_pos).trans hz.1) hz.2.le
+    ((neg_lt_zero.2 Real.pi_pos).trans hw.1) hw.2.le hzw
+
+lemma stripPotential_continuous {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    {Fbar : ℂ → OnePoint ℂ} (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im}) :
+    ContinuousOn (stripPotential H q r Fbar) closedStrip :=
+  (sphereTruncation_continuous hH hr q).continuousOn.comp
+    (hcont.comp Complex.continuous_exp.continuousOn (fun z hz => exp_mem_closedHalfPlane (z := z) hz))
+    (fun _ _ => mem_univ _)
+
+theorem strip_potential_energy {F : ℂ → ℂ} (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    (∫⁻ z in openStrip, ENNReal.ofReal (‖fderiv ℝ (stripPotential H q r Fbar) z‖^2)) ≤
+      ENNReal.ofReal (2*Real.pi*H) := by
+  have hdiff : DifferentiableOn ℂ (F ∘ Complex.exp) openStrip := by
+    intro z hz
+    exact ((hF _ (exp_mem_halfPlane hz)).differentiableAt.comp z
+      (Complex.differentiable_exp z)).differentiableWithinAt
+  have hi : InjOn (F ∘ Complex.exp) openStrip :=
+    hinj.comp exp_injOn_strip (fun z hz => exp_mem_halfPlane hz)
+  have hb := conformal_truncation_energy openStrip_isOpen hdiff hi hH hr q
+  have he : (∫⁻ z in openStrip, ENNReal.ofReal (‖fderiv ℝ (stripPotential H q r Fbar) z‖^2)) =
+      ∫⁻ z in openStrip, ENNReal.ofReal (‖fderiv ℝ (radialTruncation H q r ∘ (F ∘ Complex.exp)) z‖^2) := by
+    apply setLIntegral_congr_fun openStrip_isOpen.measurableSet
+    intro z hz
+    have heq : stripPotential H q r Fbar =ᶠ[𝓝 z]
+        (radialTruncation H q r ∘ (F ∘ Complex.exp)) := by
+      filter_upwards [openStrip_isOpen.mem_nhds hz] with w hw
+      simp only [stripPotential,hext _ (exp_mem_halfPlane hw),sphereTruncation_coe,Function.comp_apply]
+    dsimp only
+    rw [heq.fderiv_eq]
+  rw [he]
+  exact hb
+
+theorem access_section_zero {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    {Fbar : ℂ → OnePoint ℂ} {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1))
+    (hη0 : η 0 = 0) (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+    {v : ℝ} (hv : v < 0) : ∃ φ ∈ Icc (0:ℝ) Real.pi,
+      stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I) = 0 := by
+  have hm : Real.exp v ∈ Icc ‖η 0‖ ‖η 1‖ := by
+    rw [hη0,hη1,norm_zero]
+    exact ⟨(Real.exp_pos _).le,(Real.exp_lt_one_iff.2 hv).le⟩
+  obtain ⟨t,ht,he⟩ := intermediate_value_Icc (by norm_num : (0:ℝ) ≤ 1) hη.norm hm
+  dsimp only at he
+  have henz : η t ≠ 0 := norm_ne_zero_iff.1 (by rw [he]; exact (Real.exp_pos _).ne')
+  have hlogRe : (Complex.log (η t)).re = v := by rw [Complex.log_re,he,Real.log_exp]
+  have hlog : (v:ℂ)+(Complex.arg (η t):ℂ)*Complex.I = Complex.log (η t) := by
+    rw [← hlogRe,← Complex.log_im]
+    exact Complex.re_add_im _
+  refine ⟨Complex.arg (η t),⟨Complex.arg_nonneg_iff.2 (hηH t ht),Complex.arg_le_pi _⟩,?_⟩
+  obtain ⟨w,hw,hew⟩ := haccess t ht
+  simp only [stripPotential,hlog,Complex.exp_log henz,← hew,sphereTruncation_coe]
+  apply radialTruncation_zero hH hr
+  simpa only [mem_closedBall,dist_eq_norm] using hw
+
+lemma stripPotential_interior_eq {F : ℂ → ℂ} {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (H r : ℝ) (q : ℂ) {z : ℂ} (hz : z ∈ openStrip) :
+    stripPotential H q r Fbar =ᶠ[𝓝 z] radialTruncation H q r ∘ (F ∘ Complex.exp) := by
+  filter_upwards [openStrip_isOpen.mem_nhds hz] with w hw
+  simp only [stripPotential,hext _ (exp_mem_halfPlane hw),sphereTruncation_coe,Function.comp_apply]
+
+lemma stripPotential_vertical_AC_interior {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hr : 0 < r) (q : ℂ) (v : ℝ) {a b : ℝ}
+    (ha : a ∈ Ioo (0:ℝ) Real.pi) (hb : b ∈ Ioo (0:ℝ) Real.pi) :
+    AbsolutelyContinuousOnInterval
+      (fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)) a b := by
+  let c := fun φ : ℝ => (v:ℂ)+(φ:ℂ)*Complex.I
+  have hc : ContDiff ℝ 1 c := contDiff_const.add (Complex.ofRealCLM.contDiff.mul contDiff_const)
+  have hmap : MapsTo c (uIcc a b) openStrip := by
+    intro φ hφ
+    have hlo : 0 < φ := lt_of_lt_of_le (lt_min ha.1 hb.1) hφ.1
+    have hhi : φ < Real.pi := lt_of_le_of_lt hφ.2 (max_lt ha.2 hb.2)
+    simpa [c,openStrip] using And.intro hlo hhi
+  have hdiff : DifferentiableOn ℂ (F ∘ Complex.exp) openStrip := by
+    intro z hz
+    exact ((hF _ (exp_mem_halfPlane hz)).differentiableAt.comp z
+      (Complex.differentiable_exp z)).differentiableWithinAt
+  have hG : ContDiffOn ℝ 1 ((F ∘ Complex.exp) ∘ c) (uIcc a b) :=
+    ((hdiff.contDiffOn openStrip_isOpen).restrict_scalars ℝ).comp hc.contDiffOn hmap
+  apply acl_congr (radialTruncation_along_C1_AC hr H q hG)
+  intro φ hφ
+  exact ((stripPotential_interior_eq hext H r q (hmap hφ)).self_of_nhds).symm
+
+lemma stripPotential_vertical_deriv {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ) {φ : ℝ}
+    (hφ : φ ∈ Ioo (0:ℝ) Real.pi) :
+    deriv (fun ψ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(ψ:ℂ)*Complex.I)) φ =
+      fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(φ:ℂ)*Complex.I) Complex.I := by
+  let c := fun ψ : ℝ => (v:ℂ)+(ψ:ℂ)*Complex.I
+  have hmem : c φ ∈ openStrip := by simpa [c,openStrip] using hφ
+  have he := stripPotential_interior_eq hext H r q hmem
+  have hc : HasDerivAt c Complex.I φ := by
+    simpa [c,Complex.ofRealCLM] using ((Complex.ofRealCLM.hasDerivAt).mul_const Complex.I).const_add (v:ℂ)
+  have hd := radialTruncation_directional_deriv hH hr q (c := c) (t := φ) (d := Complex.I)
+    ((hF _ (exp_mem_halfPlane hmem)).differentiableAt.comp _ (Complex.differentiable_exp _)) hc
+  rw [← he.fderiv_eq] at hd
+  refine (he.comp_tendsto hc.continuousAt).deriv_eq.trans hd
+
+theorem stripPotential_vertical_zero_energy {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ)
+    (hE : MemLp (fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
+      ((v:ℂ)+(φ:ℂ)*Complex.I)‖) 2 (volume.restrict (Ioc (0:ℝ) Real.pi)))
+    {z : ℝ} (hz : z ∈ Icc (0:ℝ) Real.pi)
+    (hzero : stripPotential H q r Fbar ((v:ℂ)+(z:ℂ)*Complex.I) = 0)
+    {φ : ℝ} (hφ : φ ∈ Icc (0:ℝ) Real.pi) :
+    (stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I))^2 ≤
+      Real.pi*(∫ ψ in (0:ℝ)..Real.pi, ‖fderiv ℝ (stripPotential H q r Fbar)
+        ((v:ℂ)+(ψ:ℂ)*Complex.I)‖^2) := by
+  let f := fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)
+  let e := fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(φ:ℂ)*Complex.I)‖
+  have hb : ∀ᵐ ψ ∂volume.restrict (Ioc (0:ℝ) Real.pi), ‖deriv f ψ‖ ≤ e ψ := by
+    rw [← restrict_Ioo_eq_restrict_Ioc]
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with ψ hψ
+    rw [stripPotential_vertical_deriv hF hext hH hr q v hψ]
+    simpa only [e,Complex.norm_I,mul_one] using
+      (fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(ψ:ℂ)*Complex.I)).le_opNorm Complex.I
+  have hD := hE.mono' (aestronglyMeasurable_deriv f _) hb
+  have hc : ContinuousOn f (Icc 0 Real.pi) :=
+    (stripPotential_continuous hH hr q hcont).comp
+      (continuous_const.add (Complex.continuous_ofReal.mul_const _)).continuousOn
+      (by intro ψ hψ; simpa [closedStrip] using hψ)
+  have ht := boundary_zero_energy Real.pi_pos hc hD
+    (fun a ha b hb => stripPotential_vertical_AC_interior hF hext hr q v ha hb) hz hzero hφ
+  have hm : (∫ ψ in (0:ℝ)..Real.pi, (deriv f ψ)^2) ≤
+      ∫ ψ in (0:ℝ)..Real.pi, (e ψ)^2 := by
+    simp only [intervalIntegral.integral_of_le Real.pi_pos.le]
+    apply integral_mono_ae hD.integrable_sq hE.integrable_sq
+    filter_upwards [hb] with ψ hψ
+    simpa only [Real.norm_eq_abs,sq_abs] using pow_le_pow_left₀ (norm_nonneg _) hψ 2
+  simp only [sub_zero] at ht
+  exact ht.trans (mul_le_mul_of_nonneg_left hm Real.pi_pos.le)
+
+lemma strip_energy_integrable {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    IntegrableOn (fun z => ‖fderiv ℝ (stripPotential H q r Fbar) z‖^2) openStrip := by
+  refine ⟨((measurable_fderiv ℝ _).norm.pow_const 2).aestronglyMeasurable,?_⟩
+  rw [hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun z => sq_nonneg _))]
+  exact (strip_potential_energy hF hinj hext hH hr q).trans_lt (ENNReal.ofReal_lt_top)
+
+lemma strip_energy_integrable_prod {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    Integrable (fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
+      ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2)
+      (volume.prod (volume.restrict (Ioo (0:ℝ) Real.pi))) := by
+  have hi := (Complex.volume_preserving_equiv_real_prod.symm _).integrableOn_comp_preimage
+    Complex.measurableEquivRealProd.symm.measurableEmbedding |>.2
+      (strip_energy_integrable hF hinj hext hH hr q)
+  have hs : Complex.measurableEquivRealProd.symm ⁻¹' openStrip =
+      (univ : Set ℝ).prod (Ioo (0:ℝ) Real.pi) := by
+    ext p
+    change (0 < p.2 ∧ p.2 < Real.pi) ↔ True ∧ (0 < p.2 ∧ p.2 < Real.pi)
+    simp
+  rw [hs] at hi
+  change Integrable _ ((volume.prod volume).restrict ((univ : Set ℝ) ×ˢ Ioo (0:ℝ) Real.pi)) at hi
+  rw [← Measure.prod_restrict,Measure.restrict_univ] at hi
+  convert hi using 1
+  funext p
+  have hp : (p.1:ℂ)+(p.2:ℂ)*Complex.I = Complex.measurableEquivRealProd.symm p := by
+    apply Complex.ext <;> simp
+  rw [hp]
+  rfl
+
+theorem strip_vertical_memLp_ae {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    ∀ᵐ v : ℝ, MemLp (fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
+      ((v:ℂ)+(φ:ℂ)*Complex.I)‖) 2 (volume.restrict (Ioc (0:ℝ) Real.pi)) := by
+  have hi := (strip_energy_integrable_prod hF hinj hext hH hr q).prod_right_ae
+  filter_upwards [hi] with v hv
+  rw [← restrict_Ioo_eq_restrict_Ioc]
+  apply (memLp_two_iff_integrable_sq ?_).2 hv
+  exact ((measurable_fderiv ℝ _).comp
+    ((measurable_const.add (Complex.measurable_ofReal.mul_const _)))).norm.aestronglyMeasurable
+
+theorem strip_negative_section_energy {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
+    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r) :
+    ∀ᵐ v : ℝ, v < 0 → ∀ φ ∈ Icc (0:ℝ) Real.pi,
+      (stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I))^2 ≤
+        Real.pi*(∫ ψ in (0:ℝ)..Real.pi, ‖fderiv ℝ (stripPotential H q r Fbar)
+          ((v:ℂ)+(ψ:ℂ)*Complex.I)‖^2) := by
+  filter_upwards [strip_vertical_memLp_ae hF hinj hext hH hr q] with v hv
+  intro hv0 φ hφ
+  obtain ⟨z,hz,hzero⟩ := access_section_zero hH.le hr q hη hη0 hη1 hηH haccess hv0
+  exact stripPotential_vertical_zero_energy hF hext hcont hH.le hr q v hv hz hzero hφ
+
+lemma mean_difference_abs_le {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
+    {μ : Measure α} {ν : Measure β} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    {u : α → ℝ} {v : β → ℝ} (hu : Integrable u μ) (hv : Integrable v ν) :
+    |(∫ x, u x ∂μ)-(∫ y, v y ∂ν)| ≤ ∫ x, ∫ y, |u x-v y| ∂ν ∂μ := by
+  have hD : Integrable (fun p : α × β => u p.1-v p.2) (μ.prod ν) :=
+    (hu.comp_fst ν).sub (hv.comp_snd μ)
+  have he : (∫ p : α × β, u p.1-v p.2 ∂μ.prod ν) =
+      (∫ x, u x ∂μ)-(∫ y, v y ∂ν) := by
+    rw [integral_prod _ hD]
+    simp_rw [integral_sub (integrable_const _) hv]
+    simp only [integral_const,probReal_univ,one_smul]
+    rw [integral_sub hu (integrable_const _)]
+    simp
+  have hh := norm_integral_le_integral_norm (μ := μ.prod ν) (fun p : α × β => u p.1-v p.2)
+  rw [he,integral_prod _ hD.norm] at hh
+  simpa only [Real.norm_eq_abs] using hh
+
+theorem rectangle_mean_L1_bound {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    {μ μ' : Measure X} {ν ν' : Measure Y}
+    [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']
+    [IsProbabilityMeasure ν] [IsProbabilityMeasure ν']
+    {u : X × Y → ℝ} (hu : Integrable u (μ.prod ν)) (hu' : Integrable u (μ'.prod ν'))
+    {R : Y → ℝ} {C : X → ℝ} (hR : Integrable R ν') (hC : Integrable C μ)
+    (hbound : ∀ᵐ p ∂μ'.prod ν', ∀ᵐ q ∂μ.prod ν, |u p-u q| ≤ R p.2+C q.1) :
+    |(∫ p, u p ∂μ'.prod ν')-(∫ p, u p ∂μ.prod ν)| ≤
+      (∫ y, R y ∂ν')+(∫ x, C x ∂μ) := by
+  let A := μ'.prod ν'
+  let B := μ.prod ν
+  have hD : Integrable (fun p : (X × Y) × (X × Y) => |u p.1-u p.2|) (A.prod B) :=
+    ((hu'.comp_fst B).sub (hu.comp_snd A)).abs
+  have hRi : Integrable (fun p : X × Y => R p.2) A := hR.comp_snd μ'
+  have hCi : Integrable (fun p : X × Y => C p.1) B := hC.comp_fst ν
+  have heC : (∫ p : X × Y, C p.1 ∂B) = ∫ x, C x ∂μ := by
+    rw [integral_prod _ hCi]
+    simp only [integral_const,probReal_univ,one_smul]
+  have heR : (∫ p : X × Y, R p.2 ∂A) = ∫ y, R y ∂ν' := by
+    rw [integral_prod _ hRi]
+    simp only [integral_const,probReal_univ,one_smul]
+  calc
+    _ ≤ ∫ p, ∫ q, |u p-u q| ∂B ∂A := mean_difference_abs_le hu' hu
+    _ ≤ ∫ p, R p.2+(∫ q, C q.1 ∂B) ∂A := by
+      apply integral_mono_ae hD.integral_prod_left (hRi.add (integrable_const _))
+      filter_upwards [hD.prod_right_ae,hbound] with p hp hb
+      calc
+        _ ≤ ∫ q, R p.2+C q.1 ∂B := integral_mono_ae hp ((integrable_const _).add hCi) hb
+        _ = _ := by rw [integral_add (integrable_const _) hCi]; simp
+    _ = _ := by
+      rw [integral_add hRi (integrable_const _),heC,heR]
+      simp
+
+def intervalProb (a b : ℝ) : Measure ℝ :=
+  (ENNReal.ofReal (b-a))⁻¹ • volume.restrict (Ioc a b)
+
+lemma intervalProb_isProbability {a b : ℝ} (hab : a < b) : IsProbabilityMeasure (intervalProb a b) := by
+  constructor
+  simp only [intervalProb,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul,Real.volume_Ioc]
+  exact ENNReal.inv_mul_cancel (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hab)) ENNReal.ofReal_ne_top
+
+lemma integral_intervalProb {a b : ℝ} (hab : a ≤ b) (f : ℝ → ℝ) :
+    (∫ x, f x ∂intervalProb a b) = (b-a)⁻¹*(∫ x in a..b, f x) := by
+  rw [intervalProb,integral_smul_measure,ENNReal.toReal_inv,ENNReal.toReal_ofReal (sub_nonneg.2 hab),
+    smul_eq_mul,intervalIntegral.integral_of_le hab]
+
+lemma intervalProb_ae {a b : ℝ} {p : ℝ → Prop} :
+    (∀ᵐ x ∂intervalProb a b, p x) ↔ ∀ᵐ x ∂volume.restrict (Ioc a b), p x := by
+  exact Measure.ae_ennreal_smul_measure_iff (ENNReal.inv_ne_zero.2 ENNReal.ofReal_ne_top)
+
+lemma integrable_intervalProb {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
+    (hf : IntegrableOn f (Ioc a b)) : Integrable f (intervalProb a b) := by
+  exact hf.smul_measure (ENNReal.inv_ne_top.2 (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hab)))
+
+lemma intervalProb_integral_subinterval {a b c d K : ℝ} (hab : a < b) (hcd : c < d)
+    (hac : a ≤ c) (hdb : d ≤ b) (hK : 0 ≤ K) (hlen : b-a ≤ K*(d-c))
+    {f : ℝ → ℝ} (hf : IntegrableOn f (Ioc a b))
+    (hf0 : ∀ᵐ x ∂volume.restrict (Ioc a b), 0 ≤ f x) :
+    (∫ x, f x ∂intervalProb c d) ≤ K*(∫ x, f x ∂intervalProb a b) := by
+  have hsub : Ioc c d ⊆ Ioc a b := fun x hx => ⟨hac.trans_lt hx.1,hx.2.trans hdb⟩
+  have him : (∫ x in c..d, f x) ≤ ∫ x in a..b, f x := by
+    rw [intervalIntegral.integral_of_le hcd.le,intervalIntegral.integral_of_le hab.le]
+    exact setIntegral_mono_set hf hf0 (ae_of_all _ (fun x hx => hsub hx))
+  have hnon : 0 ≤ ∫ x in a..b, f x := by
+    rw [intervalIntegral.integral_of_le hab.le]
+    exact integral_nonneg_of_ae hf0
+  have hcoef : (d-c)⁻¹ ≤ K*(b-a)⁻¹ := by
+    rw [← one_div,← div_eq_mul_inv]
+    exact (div_le_div_iff₀ (sub_pos.2 hcd) (sub_pos.2 hab)).2 (by simpa using hlen)
+  rw [integral_intervalProb hcd.le,integral_intervalProb hab.le]
+  calc
+    _ ≤ (d-c)⁻¹*(∫ x in a..b, f x) := mul_le_mul_of_nonneg_left him (inv_nonneg.2 (sub_nonneg.2 hcd.le))
+    _ ≤ (K*(b-a)⁻¹)*(∫ x in a..b, f x) :=
+      mul_le_mul hcoef le_rfl hnon (mul_nonneg hK (inv_nonneg.2 (sub_nonneg.2 hab.le)))
+    _ = _ := by ring
+
+lemma acl_line_L1 {a b : ℝ} {f g : ℝ → ℝ}
+    (hf : AbsolutelyContinuousOnInterval f a b)
+    (hg : IntegrableOn g (Ioc a b))
+    (hbound : ∀ᵐ s ∂volume.restrict (Ioc a b), ‖deriv f s‖ ≤ g s)
+    {x y : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b) :
+    |f y-f x| ≤ ∫ s in a..b, g s := by
+  have hab := hx.1.trans hx.2
+  have hD : IntegrableOn (deriv f) (Ioc a b) :=
+    hg.mono' (aestronglyMeasurable_deriv f _) hbound
+  suffices ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, x ≤ y →
+      |f y-f x| ≤ ∫ s in a..b, g s by
+    rcases le_total x y with h | h
+    · exact this x hx y hy h
+    · simpa only [abs_sub_comm] using this y hy x hx h
+  intro x hx y hy hxy
+  have hsub : Ioc x y ⊆ Ioc a b := fun s hs => ⟨hx.1.trans_lt hs.1,hs.2.trans hy.2⟩
+  have hfac : AbsolutelyContinuousOnInterval f x y := hf.mono
+    (by simpa only [uIcc_of_le hab,uIcc_of_le hxy] using Icc_subset_Icc hx.1 hy.2)
+  calc
+    _ = ‖∫ s in x..y, deriv f s‖ := by rw [hfac.integral_deriv_eq_sub,Real.norm_eq_abs]
+    _ ≤ ∫ s in x..y, ‖deriv f s‖ := intervalIntegral.norm_integral_le_integral_norm hxy
+    _ ≤ ∫ s in a..b, ‖deriv f s‖ := by
+      rw [intervalIntegral.integral_of_le hxy,intervalIntegral.integral_of_le hab]
+      exact setIntegral_mono_set hD.norm (ae_of_all _ (fun s => norm_nonneg _))
+        (ae_of_all _ (fun s hs => hsub hs))
+    _ ≤ ∫ s in a..b, g s := by
+      rw [intervalIntegral.integral_of_le hab,intervalIntegral.integral_of_le hab]
+      exact integral_mono_ae hD.norm hg hbound
+
+theorem rectangle_poincare_L1 {a b c d a' b' c' d' K : ℝ}
+    (hab : a < b) (hcd : c < d) (hab' : a' < b') (hcd' : c' < d')
+    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d)
+    (hK : 0 ≤ K) (hlen : d-c ≤ K*(d'-c'))
+    {u g : ℝ × ℝ → ℝ}
+    (hu : Integrable u ((intervalProb a b).prod (intervalProb c d)))
+    (hu' : Integrable u ((intervalProb a' b').prod (intervalProb c' d')))
+    (hg : Integrable g ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))))
+    (hg0 : ∀ p, 0 ≤ g p)
+    (hhor : ∀ᵐ y ∂volume.restrict (Ioc c d),
+      AbsolutelyContinuousOnInterval (fun x => u (x,y)) a b ∧
+      ∀ᵐ x ∂volume.restrict (Ioc a b), ‖deriv (fun s => u (s,y)) x‖ ≤ g (x,y))
+    (hver : ∀ᵐ x ∂volume.restrict (Ioc a b),
+      AbsolutelyContinuousOnInterval (fun y => u (x,y)) c d ∧
+      ∀ᵐ y ∂volume.restrict (Ioc c d), ‖deriv (fun s => u (x,s)) y‖ ≤ g (x,y)) :
+    |(∫ p, u p ∂(intervalProb a' b').prod (intervalProb c' d'))-
+      (∫ p, u p ∂(intervalProb a b).prod (intervalProb c d))| ≤
+      (K/(d-c)+1/(b-a))*(∫ p, g p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+  let := intervalProb_isProbability hab
+  let := intervalProb_isProbability hcd
+  let := intervalProb_isProbability hab'
+  let := intervalProb_isProbability hcd'
+  let R := fun y => ∫ x in a..b, g (x,y)
+  let C := fun x => ∫ y in c..d, g (x,y)
+  have hRi : IntegrableOn R (Ioc c d) := by
+    simpa only [IntegrableOn,R,intervalIntegral.integral_of_le hab.le] using hg.integral_prod_right
+  have hCi : IntegrableOn C (Ioc a b) := by
+    simpa only [IntegrableOn,C,intervalIntegral.integral_of_le hcd.le] using hg.integral_prod_left
+  have subX : Ioc a' b' ⊆ Ioc a b := fun x hx => ⟨haa.trans_lt hx.1,hx.2.trans hbb⟩
+  have subY : Ioc c' d' ⊆ Ioc c d := fun y hy => ⟨hcc.trans_lt hy.1,hy.2.trans hdd⟩
+  have hH : ∀ᵐ y ∂intervalProb c' d', ∀ x₁ ∈ Icc a b, ∀ x₂ ∈ Icc a b,
+      |u (x₁,y)-u (x₂,y)| ≤ R y := by
+    apply intervalProb_ae.2
+    apply (ae_mono (Measure.restrict_mono_set volume subY))
+    filter_upwards [hhor,hg.prod_left_ae] with y hy hgy
+    intro x₁ hx₁ x₂ hx₂
+    exact acl_line_L1 hy.1 hgy hy.2 hx₂ hx₁
+  have hV : ∀ᵐ x ∂intervalProb a b, ∀ y₁ ∈ Icc c d, ∀ y₂ ∈ Icc c d,
+      |u (x,y₁)-u (x,y₂)| ≤ C x := by
+    apply intervalProb_ae.2
+    filter_upwards [hver,hg.prod_right_ae] with x hx hgx
+    intro y₁ hy₁ y₂ hy₂
+    exact acl_line_L1 hx.1 hgx hx.2 hy₂ hy₁
+  have hmem {l r : ℝ} : ∀ᵐ x ∂intervalProb l r, x ∈ Icc l r :=
+    intervalProb_ae.2 ((ae_restrict_mem measurableSet_Ioc).mono (fun x hx => Ioc_subset_Icc_self hx))
+  have hb : ∀ᵐ p ∂(intervalProb a' b').prod (intervalProb c' d'),
+      ∀ᵐ q ∂(intervalProb a b).prod (intervalProb c d), |u p-u q| ≤ R p.2+C q.1 := by
+    have hpX := (Measure.quasiMeasurePreserving_fst (μ := intervalProb a' b') (ν := intervalProb c' d')).tendsto_ae.eventually (hmem (l := a') (r := b'))
+    have hpY := (Measure.quasiMeasurePreserving_snd (μ := intervalProb a' b') (ν := intervalProb c' d')).tendsto_ae.eventually (hmem (l := c') (r := d'))
+    have hpH := (Measure.quasiMeasurePreserving_snd (μ := intervalProb a' b') (ν := intervalProb c' d')).tendsto_ae.eventually hH
+    have hqX := (Measure.quasiMeasurePreserving_fst (μ := intervalProb a b) (ν := intervalProb c d)).tendsto_ae.eventually (hmem (l := a) (r := b))
+    have hqY := (Measure.quasiMeasurePreserving_snd (μ := intervalProb a b) (ν := intervalProb c d)).tendsto_ae.eventually (hmem (l := c) (r := d))
+    have hqV := (Measure.quasiMeasurePreserving_fst (μ := intervalProb a b) (ν := intervalProb c d)).tendsto_ae.eventually hV
+    filter_upwards [hpX,hpY,hpH] with p hpx hpy hph
+    filter_upwards [hqX,hqY,hqV] with q hqx hqy hqv
+    have hpx' : p.1 ∈ Icc a b := ⟨haa.trans hpx.1,hpx.2.trans hbb⟩
+    have hpy' : p.2 ∈ Icc c d := ⟨hcc.trans hpy.1,hpy.2.trans hdd⟩
+    exact (abs_sub_le (u p) (u (q.1,p.2)) (u q)).trans
+      (add_le_add (hph p.1 hpx' q.1 hqx) (hqv p.2 hpy' q.2 hqy))
+  have hmain := rectangle_mean_L1_bound hu hu'
+    (integrable_intervalProb hcd' (hRi.mono_set subY)) (integrable_intervalProb hab hCi) hb
+  have hR0 : ∀ y, 0 ≤ R y := fun y => intervalIntegral.integral_nonneg hab.le (fun x hx => hg0 (x,y))
+  have hRbound := intervalProb_integral_subinterval hcd hcd' hcc hdd hK hlen hRi (ae_of_all _ hR0)
+  have hRE : (∫ y, R y ∂intervalProb c d) =
+      (d-c)⁻¹*(∫ p, g p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+    rw [integral_intervalProb hcd.le]
+    congr 1
+    rw [intervalIntegral.integral_of_le hcd.le,integral_prod_symm _ hg]
+    simp only [R,intervalIntegral.integral_of_le hab.le]
+  have hCE : (∫ x, C x ∂intervalProb a b) =
+      (b-a)⁻¹*(∫ p, g p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+    rw [integral_intervalProb hab.le]
+    congr 1
+    rw [intervalIntegral.integral_of_le hab.le,integral_prod _ hg]
+    simp only [C,intervalIntegral.integral_of_le hcd.le]
+  calc
+    _ ≤ (∫ y, R y ∂intervalProb c' d')+(∫ x, C x ∂intervalProb a b) := hmain
+    _ ≤ K*(∫ y, R y ∂intervalProb c d)+(∫ x, C x ∂intervalProb a b) := add_le_add hRbound le_rfl
+    _ = _ := by rw [hRE,hCE]; ring
+
+lemma stripPotential_horizontal_AC {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hr : 0 < r) (q : ℂ) {φ : ℝ}
+    (hφ : φ ∈ Ioo (0:ℝ) Real.pi) (a b : ℝ) :
+    AbsolutelyContinuousOnInterval
+      (fun v : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)) a b := by
+  let c := fun v : ℝ => (v:ℂ)+(φ:ℂ)*Complex.I
+  have hc : ContDiff ℝ 1 c := Complex.ofRealCLM.contDiff.add contDiff_const
+  have hmap : MapsTo c (uIcc a b) openStrip := by
+    intro v hv
+    simpa [c,openStrip] using hφ
+  have hdiff : DifferentiableOn ℂ (F ∘ Complex.exp) openStrip := by
+    intro z hz
+    exact ((hF _ (exp_mem_halfPlane hz)).differentiableAt.comp z
+      (Complex.differentiable_exp z)).differentiableWithinAt
+  have hG : ContDiffOn ℝ 1 ((F ∘ Complex.exp) ∘ c) (uIcc a b) :=
+    ((hdiff.contDiffOn openStrip_isOpen).restrict_scalars ℝ).comp hc.contDiffOn hmap
+  apply acl_congr (radialTruncation_along_C1_AC hr H q hG)
+  intro v hv
+  exact ((stripPotential_interior_eq hext H r q (hmap hv)).self_of_nhds).symm
+
+lemma stripPotential_horizontal_deriv {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ) {φ : ℝ}
+    (hφ : φ ∈ Ioo (0:ℝ) Real.pi) :
+    deriv (fun s : ℝ => stripPotential H q r Fbar ((s:ℂ)+(φ:ℂ)*Complex.I)) v =
+      fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(φ:ℂ)*Complex.I) 1 := by
+  let c := fun s : ℝ => (s:ℂ)+(φ:ℂ)*Complex.I
+  have hmem : c v ∈ openStrip := by simpa [c,openStrip] using hφ
+  have he := stripPotential_interior_eq hext H r q hmem
+  have hc : HasDerivAt c 1 v := by
+    simpa [c,Complex.ofRealCLM] using Complex.ofRealCLM.hasDerivAt.add_const ((φ:ℂ)*Complex.I)
+  have hd := radialTruncation_directional_deriv hH hr q (c := c) (t := v) (d := 1)
+    ((hF _ (exp_mem_halfPlane hmem)).differentiableAt.comp _ (Complex.differentiable_exp _)) hc
+  rw [← he.fderiv_eq] at hd
+  exact (he.comp_tendsto hc.continuousAt).deriv_eq.trans hd
+
+lemma stripPotential_vertical_AC {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (v : ℝ)
+    (hE : MemLp (fun φ : ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
+      ((v:ℂ)+(φ:ℂ)*Complex.I)‖) 2 (volume.restrict (Ioc (0:ℝ) Real.pi))) :
+    AbsolutelyContinuousOnInterval
+      (fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)) 0 Real.pi := by
+  let f := fun φ : ℝ => stripPotential H q r Fbar ((v:ℂ)+(φ:ℂ)*Complex.I)
+  have hb : ∀ᵐ ψ ∂volume.restrict (Ioc (0:ℝ) Real.pi), ‖deriv f ψ‖ ≤
+      ‖fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(ψ:ℂ)*Complex.I)‖ := by
+    rw [← restrict_Ioo_eq_restrict_Ioc]
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with ψ hψ
+    rw [stripPotential_vertical_deriv hF hext hH hr q v hψ]
+    simpa only [Complex.norm_I,mul_one] using
+      (fderiv ℝ (stripPotential H q r Fbar) ((v:ℂ)+(ψ:ℂ)*Complex.I)).le_opNorm Complex.I
+  have hD := hE.mono' (aestronglyMeasurable_deriv f _) hb
+  have hc : ContinuousOn f (Icc 0 Real.pi) :=
+    (stripPotential_continuous hH hr q hcont).comp
+      (continuous_const.add (Complex.continuous_ofReal.mul_const _)).continuousOn
+      (by intro ψ hψ; simpa [closedStrip] using hψ)
+  exact acl_of_interior Real.pi_pos hc
+    ((intervalIntegrable_iff_integrableOn_Ioc_of_le Real.pi_pos.le).2 (hD.integrable (by norm_num)))
+    (fun a ha b hb => stripPotential_vertical_AC_interior hF hext hr q v ha hb)
+
+theorem strip_coordinate_ACL {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {a b c d : ℝ} (hcd : c < d) (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
+    (∀ᵐ y : ℝ ∂volume.restrict (Ioc c d),
+      AbsolutelyContinuousOnInterval
+        (fun x : ℝ => stripPotential H q r Fbar ((x:ℂ)+(y:ℂ)*Complex.I)) a b ∧
+      ∀ᵐ x : ℝ ∂volume.restrict (Ioc a b),
+        ‖deriv (fun s : ℝ => stripPotential H q r Fbar ((s:ℂ)+(y:ℂ)*Complex.I)) x‖ ≤
+          ‖fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)‖) ∧
+    (∀ᵐ x : ℝ ∂volume.restrict (Ioc a b),
+      AbsolutelyContinuousOnInterval
+        (fun y : ℝ => stripPotential H q r Fbar ((x:ℂ)+(y:ℂ)*Complex.I)) c d ∧
+      ∀ᵐ y : ℝ ∂volume.restrict (Ioc c d),
+        ‖deriv (fun s : ℝ => stripPotential H q r Fbar ((x:ℂ)+(s:ℂ)*Complex.I)) y‖ ≤
+          ‖fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)‖) := by
+  have hy : ∀ᵐ y : ℝ ∂volume.restrict (Ioc c d), y ∈ Ioo (0:ℝ) Real.pi := by
+    rw [← restrict_Ioo_eq_restrict_Ioc]
+    filter_upwards [ae_restrict_mem measurableSet_Ioo] with y hy
+    exact ⟨hc.trans_lt hy.1,hy.2.trans_le hd⟩
+  constructor
+  · filter_upwards [hy] with y hy
+    refine ⟨stripPotential_horizontal_AC hF hext hr q hy a b,ae_of_all _ (fun x => ?_)⟩
+    rw [stripPotential_horizontal_deriv hF hext hH.le hr q x hy]
+    simpa only [norm_one,mul_one] using
+      (fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)).le_opNorm 1
+  · filter_upwards [(strip_vertical_memLp_ae hF hinj hext hH hr q).filter_mono
+      (ae_mono Measure.restrict_le_self)] with x hx
+    refine ⟨(stripPotential_vertical_AC hF hext hcont hH.le hr q x hx).mono ?_,?_⟩
+    · rw [uIcc_of_le hcd.le,uIcc_of_le Real.pi_pos.le]
+      exact fun y hy => ⟨hc.trans hy.1,hy.2.trans hd⟩
+    · filter_upwards [hy] with y hy
+      rw [stripPotential_vertical_deriv hF hext hH.le hr q x hy]
+      simpa only [Complex.norm_I,mul_one] using
+        (fderiv ℝ (stripPotential H q r Fbar) ((x:ℂ)+(y:ℂ)*Complex.I)).le_opNorm Complex.I
+
+lemma strip_gradient_rect_memLp {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    (a b : ℝ) {c d : ℝ} (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
+    MemLp (fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar)
+      ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖) 2
+      ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+  have hi := strip_energy_integrable_prod hF hinj hext hH hr q
+  rw [restrict_Ioo_eq_restrict_Ioc] at hi
+  have hm : (volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d)) ≤
+      volume.prod (volume.restrict (Ioc (0:ℝ) Real.pi)) :=
+    Measure.prod_mono Measure.restrict_le_self (Measure.restrict_mono_set _
+      (fun y hy => ⟨hc.trans_lt hy.1,hy.2.trans hd⟩))
+  apply (memLp_two_iff_integrable_sq ?_).2 (hi.mono_measure hm)
+  exact ((measurable_fderiv ℝ _).comp
+    ((Complex.measurable_ofReal.comp measurable_fst).add
+      ((Complex.measurable_ofReal.comp measurable_snd).mul_const _))).norm.aestronglyMeasurable
+
+lemma strip_potential_rect_integrable {Fbar : ℂ → OnePoint ℂ}
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    (a b : ℝ) {c d : ℝ} (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
+    Integrable (fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I))
+      ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+  have hh : ContinuousOn (fun p : ℝ × ℝ => stripPotential H q r Fbar
+      ((p.1:ℂ)+(p.2:ℂ)*Complex.I)) ((Icc a b).prod (Icc c d)) :=
+    (stripPotential_continuous hH hr q hcont).comp
+      ((Complex.continuous_ofReal.comp continuous_fst).add
+        ((Complex.continuous_ofReal.comp continuous_snd).mul_const _)).continuousOn
+      (by intro p hp; simpa [closedStrip] using And.intro (hc.trans hp.2.1) (hp.2.2.trans hd))
+  have hi := hh.integrableOn_compact (μ := volume) (isCompact_Icc.prod isCompact_Icc)
+  have hj := hi.mono_set (prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)
+  rw [IntegrableOn,Measure.volume_eq_prod,← Measure.prod_restrict] at hj
+  exact hj
+
+lemma strip_potential_mean_integrable {Fbar : ℂ → OnePoint ℂ}
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    {a b c d : ℝ} (hab : a < b) (hcd : c < d) (hc : 0 ≤ c) (hd : d ≤ Real.pi) :
+    Integrable (fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I))
+      ((intervalProb a b).prod (intervalProb c d)) := by
+  rw [intervalProb,intervalProb,Measure.prod_smul_left,Measure.prod_smul_right]
+  exact ((strip_potential_rect_integrable hcont hH hr q a b hc hd).smul_measure
+    (ENNReal.inv_ne_top.2 (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hcd)))).smul_measure
+      (ENNReal.inv_ne_top.2 (ENNReal.ofReal_ne_zero_iff.2 (sub_pos.2 hab)))
+
+theorem strip_rectangle_poincare {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {a b c d a' b' c' d' K : ℝ}
+    (hab : a < b) (hcd : c < d) (hab' : a' < b') (hcd' : c' < d')
+    (hlo : 0 ≤ c) (hhi : d ≤ Real.pi)
+    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d)
+    (hK : 0 ≤ K) (hlen : d-c ≤ K*(d'-c')) :
+    |(∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb a' b').prod (intervalProb c' d'))-
+      (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb a b).prod (intervalProb c d))| ≤
+      (K/(d-c)+1/(b-a))*(∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar)
+        ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖
+          ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+  have hh := strip_coordinate_ACL hF hinj hext hcont hH hr q (a := a) (b := b) hcd hlo hhi
+  exact rectangle_poincare_L1 hab hcd hab' hcd' haa hbb hcc hdd hK hlen
+    (strip_potential_mean_integrable hcont hH.le hr q hab hcd hlo hhi)
+    (strip_potential_mean_integrable hcont hH.le hr q hab' hcd' (hlo.trans hcc) (hdd.trans hhi))
+    ((strip_gradient_rect_memLp hF hinj hext hH hr q a b hlo hhi).integrable (by norm_num))
+    (fun _ => norm_nonneg _) hh.1 hh.2
+
+lemma dyadic_subsum_le {a : ℕ → ℝ} (ha : ∀ j, a j = 0 ∨ a j = (2:ℝ)^j) (n : ℕ) :
+    ∑ j ∈ Finset.range n, a j ≤ (2:ℝ)^n-1 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ,pow_succ]
+    rcases ha n with h | h
+    · rw [h,add_zero]
+      linarith [pow_pos (by norm_num : (0:ℝ) < 2) n]
+    · rw [h]
+      linarith
+
+lemma dyadic_subsum_square {a : ℕ → ℝ} (ha : ∀ j, a j = 0 ∨ a j = (2:ℝ)^j) (n : ℕ) :
+    (∑ j ∈ Finset.range n, a j)^2 ≤ 3*∑ j ∈ Finset.range n, (a j)^2 := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [Finset.sum_range_succ,Finset.sum_range_succ]
+    rcases ha n with h | h
+    · simpa only [h,add_zero,zero_pow (by norm_num : (2:ℕ) ≠ 0)] using ih
+    · rw [h]
+      have hb := dyadic_subsum_le ha n
+      have hp : 0 ≤ (2:ℝ)^n := by positivity
+      have hm := mul_nonneg hp (show 0 ≤ (2:ℝ)^n-∑ j ∈ Finset.range n, a j by linarith)
+      nlinarith
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+section Measure
+variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [IsFiniteMeasure μ]
+
+def dyadicKernel (Q : ℕ → Set α) (n : ℕ) (x : α) : ℝ :=
+  ∑ j ∈ Finset.range n, (Q j).indicator (fun _ => (2:ℝ)^j) x
+
+lemma dyadicKernel_memLp {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j)) (n : ℕ) :
+    MemLp (dyadicKernel Q n) 2 μ := by
+  have hh := memLp_finsetSum' (μ := μ) (p := (2:ℝ≥0∞)) (Finset.range n)
+    (fun j hj => (memLp_const ((2:ℝ)^j)).indicator (hQ j))
+  have he : dyadicKernel Q n = ∑ j ∈ Finset.range n, (Q j).indicator (fun _ => (2:ℝ)^j) := by
+    funext x
+    simp only [dyadicKernel,Finset.sum_apply]
+  rw [he]
+  exact hh
+
+lemma dyadicKernel_square_energy {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
+    {V : ℝ} (n : ℕ) (harea : ∀ j < n, ((2:ℝ)^j)^2*μ.real (Q j) ≤ V) :
+    (∫ x, (dyadicKernel Q n x)^2 ∂μ) ≤ 3*(n:ℝ)*V := by
+  have hpoint (x : α) : (dyadicKernel Q n x)^2 ≤
+      3*∑ j ∈ Finset.range n, ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2 := by
+    apply dyadic_subsum_square
+    intro j
+    by_cases hx : x ∈ Q j
+    · right; exact indicator_of_mem hx _
+    · left; exact indicator_of_notMem hx _
+  have hi (j : ℕ) : Integrable (fun x => ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2) μ :=
+    ((memLp_const ((2:ℝ)^j) : MemLp _ 2 μ).indicator (hQ j)).integrable_sq
+  calc
+    _ ≤ ∫ x, 3*∑ j ∈ Finset.range n, ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2 ∂μ :=
+      integral_mono (dyadicKernel_memLp hQ n).integrable_sq
+        ((integrable_finsetSum _ (fun j hj => hi j)).const_mul 3) hpoint
+    _ = 3*∑ j ∈ Finset.range n, ((2:ℝ)^j)^2*μ.real (Q j) := by
+      rw [integral_const_mul,integral_finsetSum _ (fun j hj => hi j)]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro j hj
+      have he : (fun x => ((Q j).indicator (fun _ => (2:ℝ)^j) x)^2) =
+          (Q j).indicator (fun _ => ((2:ℝ)^j)^2) := by
+        funext x
+        by_cases hx : x ∈ Q j <;> simp [hx]
+      rw [he,integral_indicator (hQ j),integral_const]
+      simp [Measure.real,smul_eq_mul,mul_comm]
+    _ ≤ 3*(n:ℝ)*V := by
+      have hh := Finset.sum_le_sum (s := Finset.range n)
+        (fun j hj => harea j (Finset.mem_range.1 hj))
+      simpa only [Finset.sum_const,Finset.card_range,nsmul_eq_mul,mul_assoc] using
+        mul_le_mul_of_nonneg_left hh (by norm_num : (0:ℝ) ≤ 3)
+
+end Measure
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/EuclideanApproach.lean b/lean/OAI/Probability/SLELocalization/EuclideanApproach.lean
index 37b2ecbdd729e946a402bb2eebb8e567e492622d..dd7a6fd6909cd83acc3655fa82a26a954f5a2fcc 100644
--- a/lean/OAI/Probability/SLELocalization/EuclideanApproach.lean
+++ b/lean/OAI/Probability/SLELocalization/EuclideanApproach.lean
@@ -1,991 +1,991 @@
-import Mathlib
-import OAI.Probability.SLELocalization.AngularClock
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Is Loewner Trace Using estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma visit_liveDistanceApproach (h : IsLoewnerTraceUsing driver γ g f)
-    {z : ℂ} {v : ℝ} (hv : 0 < v) (hsmall : 2*v < z.im)
-    (hvisit : ∃ T : ℝ, 0 ≤ T ∧ dist z (γ T) < v) :
-    liveDistanceApproach γ z (3*v) := by
-  obtain ⟨T,hT,hvisit⟩ := hvisit
-  let TN : ℝ≥0 := ⟨T,hT⟩
-  have hc : Continuous (fun t : ℝ≥0 => dist z (γ t)) :=
-    continuous_const.dist (h.1.comp_continuous continuous_subtype_val (fun t => t.coe_nonneg))
-  let A : Set ℝ≥0 := Icc 0 TN ∩ {t | dist z (γ t) ≤ 2*v}
-  have hA : IsCompact A := isCompact_Icc.inter_right (isClosed_le hc continuous_const)
-  obtain ⟨s,hs,hmin⟩ := hA.exists_isLeast ⟨TN,⟨⟨zero_le,le_rfl⟩,by change dist z (γ T) ≤ 2*v; linarith⟩⟩
-  have hsT : (s:ℝ) ≤ T := hs.1.2
-  have hstrict (q : ℝ≥0) (hqs : q < s) : 2*v < dist z (γ q) := by
-    by_contra hn
-    exact (not_le_of_gt hqs) (hmin ⟨⟨zero_le,hqs.le.trans hs.1.2⟩,not_lt.1 hn⟩)
-  have hinit : 2*v < dist z (γ 0) := by
-    rw [h.2.1,dist_zero_right]
-    exact hsmall.trans_le (Complex.im_le_norm z)
-  have hdist : dist z (γ s) = 2*v := by
-    apply le_antisymm hs.2
-    by_contra hn
-    have hl : dist z (γ s) < 2*v := not_le.1 hn
-    have hcc : ContinuousOn (fun q : ℝ => dist z (γ q)) (Icc 0 (s:ℝ)) :=
-      (continuous_const.dist continuous_id).comp_continuousOn (h.1.mono Icc_subset_Ici_self)
-    obtain ⟨q,hq,he⟩ := intermediate_value_Icc' s.coe_nonneg hcc ⟨hl.le,hinit.le⟩
-    have hqs : q < (s:ℝ) := lt_of_le_of_ne hq.2 (by intro heq; subst q; linarith)
-    have hh := hstrict ⟨q,hq.1⟩ hqs
-    change 2*v < dist z (γ q) at hh
-    exact (not_lt_of_ge he.le) hh
-  have hbefore (q : ℝ) (hq : q ∈ Icc 0 (s:ℝ)) : 2*v ≤ dist z (γ q) := by
-    rcases lt_or_eq_of_le hq.2 with hh | hh
-    · exact (hstrict ⟨q,hq.1⟩ hh).le
-    · rw [hh,hdist]
-  have hfree : ball z (2*v) ⊆ halfPlane \ localizationSegment γ 0 s := by
-    intro w hw
-    have hnorm : ‖w-z‖ < 2*v := by simpa only [mem_ball,dist_eq_norm] using hw
-    constructor
-    · have hh := Complex.abs_im_le_norm (w-z)
-      have hn := neg_abs_le (w-z).im
-      change 0 < w.im
-      simp only [Complex.sub_im] at hh hn
-      linarith
-    · rintro ⟨q,hq,rfl⟩
-      have hh := hbefore q hq
-      have hh' := mem_ball.1 hw
-      rw [dist_comm] at hh'
-      linarith
-  have hcl := h.future_tip_mem_closure s.coe_nonneg hsT
-  have hb : γ T ∈ ball z (2*v) := by rw [mem_ball,dist_comm]; linarith
-  obtain ⟨w,hwb,hwD⟩ := mem_closure_iff.1 hcl (ball z (2*v)) isOpen_ball hb
-  have hcomp := (convex_ball z (2*v)).isPreconnected.subset_connectedComponentIn hwb hfree
-  have hzball : z ∈ ball z (2*v) := mem_ball_self (by positivity)
-  have hzlive : z ∈ localizationDomain γ s := by
-    refine ⟨hfree hzball,?_⟩
-    rw [← connectedComponentIn_eq (hcomp hzball)]
-    exact hwD.2
-  refine ⟨s,hzlive,?_⟩
-  have hnot : γ s ∈ (localizationDomain γ s)ᶜ := by
-    intro hh
-    exact hh.1.2 ⟨s,⟨s.coe_nonneg,le_rfl⟩,rfl⟩
-  have hh := infDist_le_dist_of_mem (x := z) hnot
-  rw [hdist] at hh
-  linarith
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.SmallTarget
-
-lemma sqrt_rpow {ε : ℝ} (hε : 0 ≤ ε) (β : ℝ) : (Real.sqrt ε)^β=ε^(β/2) := by
-  rw [Real.sqrt_eq_rpow,← Real.rpow_mul hε]
-  congr 1
-  ring
-
-lemma interior_weight {p b ε : ℝ} {z : ℂ} (hp : 0 ≤ p) (hb : 0 ≤ b)
-    (hε : 0 < ε) (hy : Real.sqrt ε ≤ z.im) :
-    (2*z.im)^(-p)*(z.im/‖z‖)^b ≤ ε^(-(p/2)) := by
-  have hyr : 0 < z.im := (Real.sqrt_pos.2 hε).trans_le hy
-  have hn : 0 < ‖z‖ := hyr.trans_le (Complex.im_le_norm z)
-  have hsin0 : 0 ≤ z.im/‖z‖ := div_nonneg hyr.le hn.le
-  have hsin1 : z.im/‖z‖ ≤ 1 := (div_le_one hn).2 (Complex.im_le_norm z)
-  have hsin := Real.rpow_le_one hsin0 hsin1 hb
-  have hrad : (2*z.im)^(-p) ≤ (Real.sqrt ε)^(-p) :=
-    Real.rpow_le_rpow_of_nonpos (Real.sqrt_pos.2 hε) (by linarith) (neg_nonpos.mpr hp)
-  calc
-    _ ≤ (2*z.im)^(-p)*1 := mul_le_mul_of_nonneg_left hsin (Real.rpow_nonneg (by positivity) _)
-    _ ≤ (Real.sqrt ε)^(-p) := by simpa using hrad
-    _ = ε^(-(p/2)) := by rw [sqrt_rpow hε.le]; congr 1; ring
-
-lemma interior_power_product {ε C p : ℝ} (hε : 0 < ε) :
-    C*ε^(-(p/2))*(3*ε)^p = C*3^p*ε^(p/2) := by
-  rw [Real.mul_rpow (by norm_num : (0:ℝ)≤3) hε.le]
-  calc
-    _ = C*3^p*(ε^(-(p/2))*ε^p) := by ring
-    _ = _ := by rw [← Real.rpow_add hε]; congr 2; ring
-
-lemma radius_power {A ε β : ℝ} (hA : 0 ≤ A) (hε : 0 ≤ ε) :
-    (A*Real.sqrt ε)^β=A^β*ε^(β/2) := by
-  rw [Real.mul_rpow hA (Real.sqrt_nonneg _),sqrt_rpow hε]
-
-theorem interior_trace_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ c < 1 ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (ε : ℝ) (_hε : 0 < ε) (_hy : Real.sqrt ε ≤ z.im)
-      (_hsmall : 3*ε < c*z.im),
-      P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < ε} ≤
-        ENNReal.ofReal (C*ε^(codimension κ/2)) := by
-  obtain ⟨C,c,hC,hc,hc1,hbound⟩ := euclideanApproachBound κ hκ
-  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
-  have hb : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
-  refine ⟨C*3^(codimension κ),c,mul_pos hC (Real.rpow_pos_of_pos (by norm_num) _),hc,hc1,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z ε hε hy hsmall
-  have hz : z ∈ halfPlane := (Real.sqrt_pos.2 hε).trans_le hy
-  have hsm : 2*ε < z.im := by
-    have hh := mul_lt_mul_of_pos_right hc1 (show 0 < z.im from hz)
-    linarith
-  have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < ε} ≤ᵐ[P]
-      {sample | liveDistanceApproach (γ sample) z (3*ε)} := by
-    filter_upwards [hmaps] with sample hm hh
-    exact hm.visit_liveDistanceApproach hε hsm hh
-  refine (measure_mono_ae hsub).trans ((hbound P B hB γ g f hγ hmaps z hz (3*ε) (by positivity) hsmall).trans
-    (ENNReal.ofReal_le_ofReal ?_))
-  have hm := interior_weight hp.le hb.le hε hy
-  have hpε : 0 ≤ (3*ε)^(codimension κ) := Real.rpow_nonneg (by positivity) _
-  calc
-    C*((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))*(3*ε)^(codimension κ) ≤
-        C*ε^(-(codimension κ/2))*(3*ε)^(codimension κ) := by gcongr
-    _ = C*3^(codimension κ)*ε^(codimension κ/2) := interior_power_product hε
-
-end SLEExactGauge.SmallTarget
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.SmallTarget
-
-lemma epsilon_le_sqrt {ε : ℝ} (hε : 0 ≤ ε) (hε1 : ε ≤ 1) : ε ≤ Real.sqrt ε := by
-  have hs := Real.sq_sqrt hε
-  have hp := Real.sqrt_nonneg ε
-  nlinarith [mul_nonneg hε (sub_nonneg.mpr hε1)]
-
-lemma near_real_geometry {v : ℂ} {ε K : ℝ} (hv : 0 ≤ v.im) (hn : 1/2 ≤ ‖v‖)
-    (hε : 0 < ε) (hε1 : ε ≤ 1) (_hK : 0 ≤ K)
-    (hy : v.im ≤ K*Real.sqrt ε) (hsmall : (K+1)*Real.sqrt ε < 1/4) :
-    1/4 ≤ |v.re| ∧ ∀ w : ℂ, dist v w < ε → dist w (v.re:ℂ) < (K+1)*Real.sqrt ε := by
-  have hs := Real.sqrt_nonneg ε
-  have hy4 : v.im < 1/4 := by nlinarith
-  have hab := Complex.norm_le_abs_re_add_abs_im v
-  rw [abs_of_nonneg hv] at hab
-  refine ⟨by linarith,?_⟩
-  intro w hw
-  have hvx : dist v (v.re:ℂ) ≤ v.im := by
-    rw [dist_eq_norm]
-    have hh := Complex.norm_le_abs_re_add_abs_im (v-(v.re:ℂ))
-    simpa only [Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,
-      sub_self,sub_zero,abs_zero,zero_add,abs_of_nonneg hv] using hh
-  have he := epsilon_le_sqrt hε.le hε1
-  calc
-    dist w (v.re:ℂ) ≤ dist w v+dist v (v.re:ℂ) := dist_triangle _ _ _
-    _ < ε+v.im := add_lt_add_of_lt_of_le (by simpa only [dist_comm] using hw) hvx
-    _ ≤ (K+1)*Real.sqrt ε := by nlinarith
-
-lemma cutoff_sqrt_small {A ε : ℝ} (hA : 0 < A) (hε : 0 < ε)
-    (hsmall : ε < 1/(16*A^2)) : A*Real.sqrt ε < 1/4 := by
-  have hh := (lt_div_iff₀ (show 0 < 16*A^2 by positivity)).mp hsmall
-  have hs := Real.sq_sqrt hε.le
-  have hp : 0 ≤ A*Real.sqrt ε := by positivity
-  nlinarith [sq_nonneg (A*Real.sqrt ε-1/4)]
-
-end SLEExactGauge.SmallTarget
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open SmallTarget BoundaryGreen
-
-theorem uniform_small_target (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (v : ℂ), 0 ≤ v.im → 1/2 ≤ ‖v‖ → ‖v‖ ≤ 2 →
-        ∀ ε : ℝ, 0 < ε → ε < 1/16 →
-        P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < ε} ≤
-          ENNReal.ofReal (C*ε^c) := by
-  let β := (8/κ-1)/2
-  let a := 1+κ/8
-  obtain ⟨hβ,ha,ha2,hba⟩ := exponent_choice hκ.1 hκ.2
-  change 0 < β at hβ
-  change 0 < a at ha
-  change a < 2 at ha2
-  change κ*(β+1) < 4+2*a at hba
-  let δ := (2-a)/8
-  have hδ : 0 < δ := by dsimp [δ]; linarith
-  have hδhalf : δ ≤ 1/2 := by dsimp [δ]; linarith
-  have haδ : a ≤ 2-4*δ := by dsimp [δ]; linarith
-  obtain ⟨Ci,e,hi,he,he1,hinter⟩ := interior_trace_bound κ hκ
-  let K := max 1 (3/e)
-  have hK1 : 1 ≤ K := le_max_left _ _
-  have hKe : 3/e ≤ K := le_max_right _ _
-  have hK : 0 < K := lt_of_lt_of_le zero_lt_one hK1
-  let A := K+1
-  have hA : 0 < A := by dsimp [A]; positivity
-  let ρ := 1/(16*A^2)
-  have hρ : 0 < ρ := by dsimp [ρ]; positivity
-  let c := min (β/2) (codimension κ/2)
-  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
-  have hc : 0 < c := lt_min (half_pos hβ) (half_pos hp)
-  let Cb := (16/δ)^β*A^β
-  have hCb : 0 < Cb := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (Real.rpow_pos_of_pos hA _)
-  let C := Cb+Ci+ρ^(-c)
-  have hR : 0 < ρ^(-c) := Real.rpow_pos_of_pos hρ _
-  have hC : 0 < C := by dsimp only [C]; positivity
-  have hbC : Cb ≤ C := by dsimp only [C]; linarith
-  have hiC : Ci ≤ C := by dsimp only [C]; linarith
-  have hrC : ρ^(-c) ≤ C := by dsimp only [C]; linarith
-  refine ⟨C,c,hC,hc,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps v hv hn hlo ε hε hε16
-  have hε1 : ε ≤ 1 := by linarith
-  have hs : 0 < Real.sqrt ε := Real.sqrt_pos.2 hε
-  have heps := epsilon_le_sqrt hε.le hε1
-  by_cases hsmall : ε < ρ
-  · by_cases hnear : v.im ≤ K*Real.sqrt ε
-    · have hgeom := near_real_geometry hv hn hε hε1 hK.le hnear (cutoff_sqrt_small hA hε hsmall)
-      have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < ε} ⊆
-          {sample | ∃ t : ℝ, 0 ≤ t ∧ dist (γ sample t) (v.re:ℂ) < A*Real.sqrt ε} := by
-        rintro sample ⟨t,ht,hclose⟩
-        exact ⟨t,ht,hgeom.2 _ hclose⟩
-      have hh := annulus_real_bound hB κ β a δ hκ.1.le hβ ha.le hba.le hδ hδhalf haδ
-        γ g f hmaps hgeom.1 (mul_pos hA hs)
-      refine (measure_mono hsub).trans (hh.trans (ENNReal.ofReal_le_ofReal ?_))
-      rw [radius_power hA.le hε.le]
-      have hepow : ε^(β/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_left _ _)
-      calc
-        (16/δ)^β*(A^β*ε^(β/2)) = Cb*ε^(β/2) := by dsimp only [Cb]; ring
-        _ ≤ Cb*ε^c := mul_le_mul_of_nonneg_left hepow hCb.le
-        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hbC (Real.rpow_nonneg hε.le _)
-    · have hy : Real.sqrt ε ≤ v.im := by
-        have hh := mul_le_mul_of_nonneg_right hK1 hs.le
-        linarith [not_le.mp hnear]
-      have hm : 3*ε < e*v.im := by
-        have hh := (div_le_iff₀ he).mp hKe
-        have hm := mul_le_mul_of_nonneg_right hh hs.le
-        have hlt := mul_lt_mul_of_pos_left (not_le.mp hnear) he
-        nlinarith
-      refine (hinter P B hB γ g f hγ hmaps v ε hε hy hm).trans (ENNReal.ofReal_le_ofReal ?_)
-      have hepow : ε^(codimension κ/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_right _ _)
-      exact (mul_le_mul_of_nonneg_left hepow hi.le).trans
-        (mul_le_mul_of_nonneg_right hiC (Real.rpow_nonneg hε.le _))
-  · have hrε := Real.rpow_le_rpow hρ.le (not_lt.1 hsmall) hc.le
-    have hprod : ρ^(-c)*ρ^c = 1 := by
-      rw [Real.rpow_neg hρ.le,inv_mul_cancel₀ (Real.rpow_pos_of_pos hρ _).ne']
-    have h1 : 1 ≤ C*ε^c := by
-      calc
-        1 = ρ^(-c)*ρ^c := hprod.symm
-        _ ≤ ρ^(-c)*ε^c := mul_le_mul_of_nonneg_left hrε hR.le
-        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hrC (Real.rpow_nonneg hε.le _)
-    exact prob_le_one.trans (by simpa using ENNReal.ofReal_le_ofReal h1)
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.SmallTarget
-
-lemma scaled_near_real_geometry {v : ℂ} {R ε K : ℝ} (hR : 0 < R)
-    (hv : 0 ≤ v.im) (hn : R/2 ≤ ‖v‖) (hε : 0 < ε) (hε1 : ε ≤ 1) (_hK : 0 ≤ K)
-    (hy : v.im ≤ R*(K*Real.sqrt ε)) (hsmall : (K+1)*Real.sqrt ε < 1/4) :
-    R/4 ≤ |v.re| ∧ ∀ w : ℂ, dist v w < R*ε →
-      dist w (v.re:ℂ) < R*((K+1)*Real.sqrt ε) := by
-  have hs := Real.sqrt_nonneg ε
-  have hmul := mul_lt_mul_of_pos_left hsmall hR
-  have hy4 : v.im < R/4 := by nlinarith [mul_nonneg hR.le hs]
-  have hab := Complex.norm_le_abs_re_add_abs_im v
-  rw [abs_of_nonneg hv] at hab
-  refine ⟨by linarith,?_⟩
-  intro w hw
-  have hvx : dist v (v.re:ℂ) ≤ v.im := by
-    rw [dist_eq_norm]
-    have hh := Complex.norm_le_abs_re_add_abs_im (v-(v.re:ℂ))
-    simpa only [Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,
-      sub_self,sub_zero,abs_zero,zero_add,abs_of_nonneg hv] using hh
-  have he := mul_le_mul_of_nonneg_left (epsilon_le_sqrt hε.le hε1) hR.le
-  calc
-    dist w (v.re:ℂ) ≤ dist w v+dist v (v.re:ℂ) := dist_triangle _ _ _
-    _ < R*ε+v.im := add_lt_add_of_lt_of_le (by simpa only [dist_comm] using hw) hvx
-    _ ≤ R*((K+1)*Real.sqrt ε) := by nlinarith
-
-lemma scaled_interior_power {R ε p : ℝ} (hR : 0 < R) (hε : 0 < ε) :
-    (R*Real.sqrt ε)^(-p)*(3*(R*ε))^p = 3^p*ε^(p/2) := by
-  rw [Real.mul_rpow hR.le (Real.sqrt_nonneg _),sqrt_rpow hε.le,
-    Real.mul_rpow (by norm_num : (0:ℝ)≤3) (mul_nonneg hR.le hε.le),Real.mul_rpow hR.le hε.le]
-  calc
-    _ = 3^p*(R^(-p)*R^p)*(ε^(-p/2)*ε^p) := by ring
-    _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hε,neg_add_cancel,Real.rpow_zero,mul_one,show -p/2+p=p/2 by ring]
-
- theorem interior_trace_scaled_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ c < 1 ∧
-    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-      (z : ℂ) (R ε : ℝ) (_hR : 0 < R) (_hε : 0 < ε) (_hy : R*Real.sqrt ε ≤ z.im)
-      (_hsmall : 3*(R*ε) < c*z.im),
-      P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < R*ε} ≤
-        ENNReal.ofReal (C*ε^(codimension κ/2)) := by
-  obtain ⟨C,c,hC,hc,hc1,hbound⟩ := euclideanApproachBound κ hκ
-  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
-  have hb : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
-  refine ⟨C*3^(codimension κ),c,mul_pos hC (Real.rpow_pos_of_pos (by norm_num) _),hc,hc1,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z R ε hR hε hy hsmall
-  have hs := Real.sqrt_pos.2 hε
-  have hz : z ∈ halfPlane := (mul_pos hR hs).trans_le hy
-  have hzim : 0 < z.im := hz
-  have hsm : 2*(R*ε) < z.im := by
-    have hh := mul_lt_mul_of_pos_right hc1 (show 0 < z.im from hz)
-    linarith [mul_pos hR hε]
-  have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < R*ε} ≤ᵐ[P]
-      {sample | liveDistanceApproach (γ sample) z (3*(R*ε))} := by
-    filter_upwards [hmaps] with sample hm hh
-    exact hm.visit_liveDistanceApproach (mul_pos hR hε) hsm hh
-  refine (measure_mono_ae hsub).trans ((hbound P B hB γ g f hγ hmaps z hz _ (by positivity) hsmall).trans
-    (ENNReal.ofReal_le_ofReal ?_))
-  have hn : 0 < ‖z‖ := hz.trans_le (Complex.im_le_norm z)
-  have hsin := Real.rpow_le_one (div_nonneg hz.le hn.le)
-    ((div_le_one hn).2 (Complex.im_le_norm z)) hb.le
-  have hrad : (2*z.im)^(-codimension κ) ≤ (R*Real.sqrt ε)^(-codimension κ) :=
-    Real.rpow_le_rpow_of_nonpos (mul_pos hR hs) (by linarith [hzim]) (by linarith)
-  calc
-    _ ≤ C*((2*z.im)^(-codimension κ)*1)*(3*(R*ε))^(codimension κ)  := by
-      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left
-        (mul_le_mul_of_nonneg_left hsin (Real.rpow_nonneg (by positivity) _)) hC.le)
-        (Real.rpow_nonneg (by positivity) _)
-    _ ≤ C*((R*Real.sqrt ε)^(-codimension κ))*(3*(R*ε))^(codimension κ) := by
-      simp only [mul_one]; gcongr
-    _ = C*3^(codimension κ)*ε^(codimension κ/2) := by
-      rw [mul_assoc,scaled_interior_power hR hε]; ring
-
-end SLEExactGauge.SmallTarget
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open SmallTarget BoundaryGreen
-
-lemma scaled_annulus_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    {x R d : ℝ} (hR : 0 < R) (hx : R/4 ≤ |x|) (hd : 0 < d) :
-    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < R*d} ≤ ENNReal.ofReal ((16/δ)^β*d^β) := by
-  have hxp : 0 < |x| := (by positivity : 0 < R/4).trans_le hx
-  refine (real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps hxp (mul_pos hR hd)).trans
-    (ENNReal.ofReal_le_ofReal ?_)
-  have hr : 4*(R*d)/(δ*|x|) ≤ (16/δ)*d := by
-    rw [div_le_iff₀ (mul_pos hδ hxp)]
-    have hh := mul_le_mul_of_nonneg_left hx (show 0 ≤ 16*d by positivity)
-    field_simp
-    nlinarith
-  have hh := Real.rpow_le_rpow (by positivity : 0 ≤ 4*(R*d)/(δ*|x|)) hr hβ.le
-  rwa [Real.mul_rpow (by positivity : 0 ≤ 16/δ) hd.le] at hh
-
-theorem uniform_small_target_scaled (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (v : ℂ) (R : ℝ), 0 < R → 0 ≤ v.im → R/2 ≤ ‖v‖ →
-        ∀ ε : ℝ, 0 < ε → ε < 1/16 →
-        P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < R*ε} ≤
-          ENNReal.ofReal (C*ε^c) := by
-  let β := (8/κ-1)/2
-  let a := 1+κ/8
-  obtain ⟨hβ,ha,ha2,hba⟩ := exponent_choice hκ.1 hκ.2
-  change 0 < β at hβ
-  change 0 < a at ha
-  change a < 2 at ha2
-  change κ*(β+1) < 4+2*a at hba
-  let δ := (2-a)/8
-  have hδ : 0 < δ := by dsimp [δ]; linarith
-  have hδhalf : δ ≤ 1/2 := by dsimp [δ]; linarith
-  have haδ : a ≤ 2-4*δ := by dsimp [δ]; linarith
-  obtain ⟨Ci,e,hi,he,he1,hinter⟩ := interior_trace_scaled_bound κ hκ
-  let K := max 1 (3/e)
-  have hK1 : 1 ≤ K := le_max_left _ _
-  have hKe : 3/e ≤ K := le_max_right _ _
-  have hK : 0 < K := lt_of_lt_of_le zero_lt_one hK1
-  let A := K+1
-  have hA : 0 < A := by dsimp [A]; positivity
-  let ρ := 1/(16*A^2)
-  have hρ : 0 < ρ := by dsimp [ρ]; positivity
-  let c := min (β/2) (codimension κ/2)
-  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
-  have hc : 0 < c := lt_min (half_pos hβ) (half_pos hp)
-  let Cb := (16/δ)^β*A^β
-  have hCb : 0 < Cb := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (Real.rpow_pos_of_pos hA _)
-  let C := Cb+Ci+ρ^(-c)
-  have hρpow : 0 < ρ^(-c) := Real.rpow_pos_of_pos hρ _
-  have hC : 0 < C := by dsimp only [C]; positivity
-  have hbC : Cb ≤ C := by dsimp only [C]; linarith
-  have hiC : Ci ≤ C := by dsimp only [C]; linarith
-  have hrC : ρ^(-c) ≤ C := by dsimp only [C]; linarith
-  refine ⟨C,c,hC,hc,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps v R hR hv hn ε hε hε16
-  have hε1 : ε ≤ 1 := by linarith
-  have hs : 0 < Real.sqrt ε := Real.sqrt_pos.2 hε
-  have heps := epsilon_le_sqrt hε.le hε1
-  by_cases hsmall : ε < ρ
-  · by_cases hnear : v.im ≤ R*(K*Real.sqrt ε)
-    · have hgeom := scaled_near_real_geometry hR hv hn hε hε1 hK.le hnear (cutoff_sqrt_small hA hε hsmall)
-      have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < R*ε} ⊆
-          {sample | ∃ t : ℝ, 0 ≤ t ∧ dist (γ sample t) (v.re:ℂ) < R*(A*Real.sqrt ε)} := by
-        rintro sample ⟨t,ht,hclose⟩
-        exact ⟨t,ht,hgeom.2 _ hclose⟩
-      have hh := scaled_annulus_real_bound hB κ β a δ hκ.1.le hβ ha.le hba.le hδ hδhalf haδ
-        γ g f hmaps hR hgeom.1 (mul_pos hA hs)
-      refine (measure_mono hsub).trans (hh.trans (ENNReal.ofReal_le_ofReal ?_))
-      rw [radius_power hA.le hε.le]
-      have hepow : ε^(β/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_left _ _)
-      calc
-        (16/δ)^β*(A^β*ε^(β/2)) = Cb*ε^(β/2) := by dsimp only [Cb]; ring
-        _ ≤ Cb*ε^c := mul_le_mul_of_nonneg_left hepow hCb.le
-        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hbC (Real.rpow_nonneg hε.le _)
-    · have hy : R*Real.sqrt ε ≤ v.im := by
-        have hh := mul_le_mul_of_nonneg_left
-          (mul_le_mul_of_nonneg_right hK1 hs.le) hR.le
-        nlinarith [not_le.mp hnear]
-      have hm : 3*(R*ε) < e*v.im := by
-        have hh := (div_le_iff₀ he).mp hKe
-        calc
-          3*(R*ε) ≤ 3*(R*Real.sqrt ε) := by gcongr
-          _ ≤ e*(R*(K*Real.sqrt ε)) := by
-            have hm := mul_le_mul_of_nonneg_right hh (mul_nonneg hR.le hs.le)
-            nlinarith
-          _ < e*v.im := mul_lt_mul_of_pos_left (not_le.mp hnear) he
-      refine (hinter P B hB γ g f hγ hmaps v R ε hR hε hy hm).trans (ENNReal.ofReal_le_ofReal ?_)
-      have hepow : ε^(codimension κ/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_right _ _)
-      exact (mul_le_mul_of_nonneg_left hepow hi.le).trans
-        (mul_le_mul_of_nonneg_right hiC (Real.rpow_nonneg hε.le _))
-  · have hrε := Real.rpow_le_rpow hρ.le (not_lt.1 hsmall) hc.le
-    have hprod : ρ^(-c)*ρ^c = 1 := by
-      rw [Real.rpow_neg hρ.le,inv_mul_cancel₀ (Real.rpow_pos_of_pos hρ _).ne']
-    have h1 : 1 ≤ C*ε^c := by
-      calc
-        1 = ρ^(-c)*ρ^c := hprod.symm
-        _ ≤ ρ^(-c)*ε^c := mul_le_mul_of_nonneg_left hrε hρpow.le
-        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hrC (Real.rpow_nonneg hε.le _)
-    exact prob_le_one.trans (by simpa using ENNReal.ofReal_le_ofReal h1)
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-def logTrace {Ω : Type*} (γ : Ω → ℝ → ℂ) (ω : Ω) : Set (ℝ × ℝ) :=
-  {x | ∃ t : ℝ, 0 ≤ t ∧ γ ω t = Complex.exp ((x.1:ℂ)+(x.2:ℂ)*Complex.I)}
-
-lemma dyadic_radius_power (K c : ℝ) (hK : 0 ≤ K) (n : ℕ) :
-    (K/(2:ℝ)^n)^c = K^c*((1/2:ℝ)^c)^n := by
-  rw [div_eq_mul_inv,Real.mul_rpow hK (by positivity),← inv_pow,
-    ← Real.rpow_pow_comm (by norm_num : (0:ℝ) ≤ 2⁻¹)]
-  simp only [one_div]
-
-theorem logTrace_dyadic_hitting (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
-    ∃ C₀ ρ : ℝ, 0 < C₀ ∧ 0 < ρ ∧ ρ < 1 ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (j : ℝ) (n : ℕ) (i : DIndex n),
-          P {ω | ∃ x ∈ logTrace γ ω, x ∈ (dyadicRect (unitSlab j) n i).closed} ≤
-            ENNReal.ofReal (C₀*ρ^n) := by
-  obtain ⟨C,c,hC,hc,hsmall⟩ := uniform_small_target_scaled κ hκ
-  let K := expCellScale
-  let ρ := (1/2:ℝ)^c
-  let C₀ := (C+16^c)*K^c
-  have hK : 0 < K := expCellScale_pos
-  have hρ : 0 < ρ := Real.rpow_pos_of_pos (by norm_num) _
-  have hρ1 : ρ < 1 := Real.rpow_lt_one (by norm_num) (by norm_num) hc
-  have h16 : 0 < (16:ℝ)^c := Real.rpow_pos_of_pos (by norm_num) _
-  have hC₀ : 0 < C₀ := mul_pos (add_pos hC h16) (Real.rpow_pos_of_pos hK _)
-  refine ⟨C₀,ρ,hC₀,hρ,hρ1,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps j n i
-  let ε := K/(2:ℝ)^n
-  have hε : 0 < ε := by dsimp [ε]; positivity
-  have heq : ε^c = K^c*ρ^n := dyadic_radius_power K c hK.le n
-  by_cases hεsmall : ε < 1/16
-  · let Q := dyadicRect (unitSlab j) n i
-    let v := Complex.exp ((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)
-    have hg := dyadicCell_exp_geometry j n i hεsmall
-    have hb := hsmall P B hB γ g f hγ hmaps v (Real.exp j) (Real.exp_pos _) hg.1 hg.2.1 ε hε hεsmall
-    have hs : {ω | ∃ x ∈ logTrace γ ω, x ∈ Q.closed} ⊆
-        {ω | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ ω t) < Real.exp j*ε} := by
-      rintro ω ⟨x,⟨t,ht,he⟩,hx⟩
-      refine ⟨t,ht,?_⟩
-      rw [he]
-      exact hg.2.2 x hx
-    refine (measure_mono hs).trans (hb.trans (ENNReal.ofReal_le_ofReal ?_))
-    rw [heq]
-    change C*(K^c*ρ^n) ≤ ((C+16^c)*K^c)*ρ^n
-    nlinarith [mul_nonneg h16.le (mul_nonneg (Real.rpow_nonneg hK.le c) (pow_nonneg hρ.le n))]
-  · have hbase : 1 ≤ 16*ε := by linarith [not_lt.mp hεsmall]
-    have hp := Real.one_le_rpow hbase hc.le
-    rw [Real.mul_rpow (by norm_num : (0:ℝ)≤16) hε.le,heq] at hp
-    have h1 : 1 ≤ C₀*ρ^n := hp.trans (by
-      change 16^c*(K^c*ρ^n) ≤ ((C+16^c)*K^c)*ρ^n
-      nlinarith [mul_nonneg hC.le (mul_nonneg (Real.rpow_nonneg hK.le c) (pow_nonneg hρ.le n))])
-    exact prob_le_one.trans (by simpa using ENNReal.ofReal_le_ofReal h1)
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-namespace DRect
-
-instance (Q : DRect) : IsFiniteMeasure Q.measure := by unfold measure; infer_instance
-
-lemma measure_subset {Q R : DRect} (ha : Q.a ≤ R.a) (hb : R.b ≤ Q.b)
-    (hc : Q.c ≤ R.c) (hd : R.d ≤ Q.d) : R.measure ≤ Q.measure :=
-  Measure.prod_mono (Measure.restrict_mono_set _ (fun _x hx => ⟨ha.trans_lt hx.1,hx.2.trans hb⟩))
-    (Measure.restrict_mono_set _ (fun _y hy => ⟨hc.trans_lt hy.1,hy.2.trans hd⟩))
-
-lemma child_measure_le (Q : DRect) (s : Bool × Bool) : (Q.child s).measure ≤ Q.measure := by
-  obtain ⟨ha,hb,hc,hd⟩ := Q.child_bounds s
-  exact measure_subset ha hb hc hd
-
-lemma measure_interval_split {l r : ℝ} (h : l < r) :
-    volume.restrict (Ioc l r) = volume.restrict (Ioc l ((l+r)/2))+
-      volume.restrict (Ioc ((l+r)/2) r) := by
-  rw [← Ioc_union_Ioc_eq_Ioc (show l ≤ (l+r)/2 by linarith) (show (l+r)/2 ≤ r by linarith)]
-  apply Measure.restrict_union _ measurableSet_Ioc
-  exact disjoint_left.2 (fun x hx hy => (not_lt_of_ge hx.2) hy.1)
-
-lemma measure_child_sum (Q : DRect) : Q.measure = ∑ s : Bool × Bool, (Q.child s).measure := by
-  simp only [Fintype.sum_prod_type,Fintype.sum_bool,measure,child,halfLo,halfHi,↓reduceIte]
-  rw [measure_interval_split Q.hab,measure_interval_split Q.hcd,Measure.add_prod,
-    Measure.prod_add,Measure.prod_add]
-  ac_rfl
-
-end DRect
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma dyadicRect_measure_le (Q : DRect) (n : ℕ) (i : DIndex n) :
-    (dyadicRect Q n i).measure ≤ Q.measure := by
-  obtain ⟨ha,hb,hc,hd⟩ := dyadicRect_bounds Q n i
-  exact DRect.measure_subset ha hb hc hd
-
-lemma dyadic_measure_sum (Q : DRect) (n : ℕ) :
-    Q.measure = ∑ i : DIndex n, (dyadicRect Q n i).measure := by
-  induction n with
-  | zero =>
-    let : Unique (DIndex 0) := inferInstanceAs (Unique Unit)
-    rw [Fintype.sum_unique]
-    rfl
-  | succ n ih =>
-    change Q.measure = ∑ i : DIndex n × (Bool × Bool), ((dyadicRect Q n i.1).child i.2).measure
-    rw [Fintype.sum_prod_type]
-    simp_rw [← DRect.measure_child_sum]
-    exact ih
-
-lemma dyadic_integral_sum (Q : DRect) {g : ℝ × ℝ → ℝ} (hg : Integrable g Q.measure) (n : ℕ) :
-    ∑ i : DIndex n, (∫ p, g p ∂(dyadicRect Q n i).measure) = ∫ p, g p ∂Q.measure := by
-  conv_rhs => rw [dyadic_measure_sum Q n]
-  symm
-  exact integral_finsetSum_measure (fun i hi => hg.mono_measure (dyadicRect_measure_le Q n i))
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-def jumpSize (H : ℝ) (j : ℕ) : ℝ := H/(8*((j:ℝ)+1)*((j:ℝ)+2))
-
-lemma jumpSize_pos {H : ℝ} (hH : 0 < H) (j : ℕ) : 0 < jumpSize H j := by
-  unfold jumpSize
-  positivity
-
-lemma jumpSize_telescope (H : ℝ) (j : ℕ) :
-    H/8*(1-1/((j:ℝ)+1)) + jumpSize H j = H/8*(1-1/((j:ℝ)+2)) := by
-  unfold jumpSize
-  have h1 : (j:ℝ)+1 ≠ 0 := by positivity
-  have h2 : (j:ℝ)+2 ≠ 0 := by positivity
-  field_simp
-  ring
-
-lemma exists_large_increment {H : ℝ} (hH : 0 < H) {v : ℕ → ℝ}
-    (hstart : v 0 ≤ 3*H/4) (hlim : Tendsto v atTop (𝓝 H)) :
-    ∃ j : ℕ, jumpSize H j < |v (j+1)-v j| := by
-  by_contra hn
-  push Not at hn
-  have hb (n : ℕ) : v n ≤ v 0 + H/8*(1-1/((n:ℝ)+1)) := by
-    induction n with
-    | zero => simp
-    | succ n ih =>
-      have hd := le_trans (le_abs_self (v (n+1)-v n)) (hn n)
-      have hh := jumpSize_telescope H n
-      have hc : ((n+1:ℕ):ℝ)+1 = (n:ℝ)+2 := by norm_num; ring
-      rw [hc]
-      linarith
-  have hb' (n : ℕ) : v n ≤ 7*H/8 := by
-    have hinv : 0 ≤ 1/((n:ℝ)+1) := by positivity
-    have hp : 0 ≤ H/8 := by positivity
-    have hm := mul_nonneg hp hinv
-    linarith [hb n]
-  have hh := le_of_tendsto hlim (Filter.Eventually.of_forall hb')
-  linarith
-
-lemma geometric_moment_summable {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
-    Summable (fun j : ℕ => (((j:ℝ)+1)*((j:ℝ)+2))^2*q^j) := by
-  have hn : ‖q‖ < 1 := by simpa [Real.norm_eq_abs,abs_of_nonneg hq] using hq1
-  have h0 := summable_pow_mul_geometric_of_norm_lt_one 0 hn
-  have h1 := summable_pow_mul_geometric_of_norm_lt_one 1 hn
-  have h2 := summable_pow_mul_geometric_of_norm_lt_one 2 hn
-  have h3 := summable_pow_mul_geometric_of_norm_lt_one 3 hn
-  have h4 := summable_pow_mul_geometric_of_norm_lt_one 4 hn
-  have hh := (((h4.add (h3.mul_left 6)).add (h2.mul_left 13)).add
-    (h1.mul_left 12)).add (h0.mul_left 4)
-  apply hh.congr
-  intro j
-  ring
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-section Tree
-variable {X : Type} {I : ℕ → Type} [finiteIndex : ∀ n, Fintype (I n)]
-  (parent : ∀ n, I (n+1) → I n) (branch : X → ∀ n, I n)
-  (means energy : ∀ n, I n → ℝ) (value : X → ℝ)
-
-omit finiteIndex in
-lemma branch_bad_parent [∀ level, Fintype (I level)] {H D : ℝ} (hH : 0 < H) (N : ℕ)
-    (hbranch : ∀ x n, parent n (branch x (n+1)) = branch x n)
-    (hroot : ∀ i, means N i ≤ 3*H/4)
-    (hstep : ∀ n i, |means (n+1) i - means n (parent n i)|^2 ≤ D*energy n (parent n i))
-    (hlim : ∀ x, Tendsto (fun n => means n (branch x n)) atTop (𝓝 (value x)))
-    {x : X} (hx : value x = H) :
-    ∃ j : ℕ, (jumpSize H j)^2 < D*energy (N+j) (branch x (N+j)) := by
-  have hl : Tendsto (fun n => means (N+n) (branch x (N+n))) atTop (𝓝 H) := by
-    have ht : Tendsto (fun n : ℕ => N+n) atTop atTop := by
-      simpa only [Nat.add_comm] using tendsto_add_atTop_nat N
-    simpa only [Function.comp_def,hx] using (hlim x).comp ht
-  obtain ⟨j,hj⟩ := exists_large_increment hH (by simpa using hroot (branch x N)) hl
-  refine ⟨j,?_⟩
-  have hs := hstep (N+j) (branch x ((N+j)+1))
-  rw [hbranch] at hs
-  have hj' : jumpSize H j < |means ((N+j)+1) (branch x ((N+j)+1)) - means (N+j) (branch x (N+j))| := by
-    have he : N+(j+1) = (N+j)+1 := (Nat.add_assoc N j 1).symm
-    rw [he] at hj
-    exact hj
-  exact ((sq_lt_sq₀ (jumpSize_pos hH j).le (abs_nonneg _)).2 hj').trans_le hs
-
-end Tree
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma jump_energy_scale {H : ℝ} (hH : H ≠ 0) (D E C q : ℝ) (N j : ℕ) :
-    D*E/(jumpSize H j)^2*(C*q^(N+j)) =
-      (64*D*C*E/H^2*q^N)*((((j:ℝ)+1)*((j:ℝ)+2))^2*q^j) := by
-  unfold jumpSize
-  rw [pow_add]
-  field_simp
-  ring
-
-theorem dyadic_energy_hitting
-    {X : Type} {I : ℕ → Type} [∀ n, Fintype (I n)]
-    (parent : ∀ n, I (n+1) → I n) (branch : X → ∀ n, I n)
-    (means energy : ∀ n, I n → ℝ) (value : X → ℝ)
-    {H D E C₀ q : ℝ} (hH : 0 < H) (hD : 0 ≤ D) (hE : 0 ≤ E)
-    (hC : 0 ≤ C₀) (hq : 0 ≤ q) (hq1 : q < 1) (N : ℕ)
-    (hbranch : ∀ x n, parent n (branch x (n+1)) = branch x n)
-    (hroot : ∀ i, means N i ≤ 3*H/4)
-    (hstep : ∀ n i, |means (n+1) i - means n (parent n i)|^2 ≤ D*energy n (parent n i))
-    (hlim : ∀ x, Tendsto (fun n => means n (branch x n)) atTop (𝓝 (value x)))
-    (hnon : ∀ n i, 0 ≤ energy n i) (hsum : ∀ n, ∑ i, energy n i ≤ E)
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) (Ξ : Ω → Set X)
-    (hhit : ∀ n i, P {ω | ∃ x ∈ Ξ ω, branch x n = i} ≤ ENNReal.ofReal (C₀*q^n)) :
-    P {ω | ∃ x ∈ Ξ ω, value x = H} ≤
-      ENNReal.ofReal ((64*D*C₀*E/H^2*q^N)*
-        ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*q^j) := by
-  classical
-  let event (j : ℕ) (i : I (N+j)) : Set Ω :=
-    {ω | (jumpSize H j)^2 < D*energy (N+j) i ∧ ∃ x ∈ Ξ ω, branch x (N+j) = i}
-  have hcover : {ω | ∃ x ∈ Ξ ω, value x = H} ⊆ ⋃ j : ℕ, ⋃ i : I (N+j), event j i := by
-    rintro ω ⟨x,hx,hv⟩
-    obtain ⟨j,hj⟩ := branch_bad_parent parent branch means energy value hH N hbranch hroot hstep hlim hv
-    exact mem_iUnion.2 ⟨j,mem_iUnion.2 ⟨branch x (N+j),hj,x,hx,rfl⟩⟩
-  have hb (j : ℕ) (i : I (N+j)) :
-      P (event j i) ≤ ENNReal.ofReal (D*energy (N+j) i/(jumpSize H j)^2*(C₀*q^(N+j))) := by
-    by_cases hh : (jumpSize H j)^2 < D*energy (N+j) i
-    · have hs : event j i ⊆ {ω | ∃ x ∈ Ξ ω, branch x (N+j) = i} := fun _ h => h.2
-      refine (measure_mono hs).trans ((hhit _ _).trans (ENNReal.ofReal_le_ofReal ?_))
-      have hratio : 1 ≤ D*energy (N+j) i/(jumpSize H j)^2 :=
-        (le_div_iff₀ (sq_pos_of_pos (jumpSize_pos hH j))).2 (by simpa using hh.le)
-      exact le_mul_of_one_le_left (mul_nonneg hC (pow_nonneg hq _)) hratio
-    · have he : event j i = ∅ := by ext ω; simp [event,hh]
-      rw [he,measure_empty]
-      exact bot_le
-  have hlevel (j : ℕ) : P (⋃ i : I (N+j), event j i) ≤
-      ENNReal.ofReal (D*E/(jumpSize H j)^2*(C₀*q^(N+j))) := by
-    calc
-      _ ≤ ∑ i : I (N+j), P (event j i) := measure_iUnion_fintype_le _ _
-      _ ≤ ∑ i : I (N+j), ENNReal.ofReal (D*energy (N+j) i/(jumpSize H j)^2*(C₀*q^(N+j))) :=
-        Finset.sum_le_sum (fun i _ => hb j i)
-      _ = ENNReal.ofReal (∑ i : I (N+j), D*energy (N+j) i/(jumpSize H j)^2*(C₀*q^(N+j))) := by
-        symm
-        apply ENNReal.ofReal_sum_of_nonneg
-        intro i hi
-        exact mul_nonneg (div_nonneg (mul_nonneg hD (hnon _ _)) (sq_nonneg _))
-          (mul_nonneg hC (pow_nonneg hq _))
-      _ ≤ _ := by
-        apply ENNReal.ofReal_le_ofReal
-        simp only [← Finset.sum_mul,← Finset.sum_div,← Finset.mul_sum]
-        exact mul_le_mul_of_nonneg_right
-          (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left (hsum _) hD) (sq_nonneg _))
-          (mul_nonneg hC (pow_nonneg hq _))
-  let K := 64*D*C₀*E/H^2*q^N
-  have hK : 0 ≤ K := by dsimp [K]; positivity
-  have hs := (geometric_moment_summable hq hq1).mul_left K
-  calc
-    _ ≤ P (⋃ j : ℕ, ⋃ i : I (N+j), event j i) := measure_mono hcover
-    _ ≤ ∑' j : ℕ, P (⋃ i : I (N+j), event j i) := measure_iUnion_le _
-    _ ≤ ∑' j : ℕ, ENNReal.ofReal (D*E/(jumpSize H j)^2*(C₀*q^(N+j))) := ENNReal.tsum_le_tsum hlevel
-    _ = ∑' j : ℕ, ENNReal.ofReal (K*((((j:ℝ)+1)*((j:ℝ)+2))^2*q^j)) := by
-      apply tsum_congr
-      intro j
-      rw [jump_energy_scale hH.ne']
-    _ = ENNReal.ofReal (∑' j : ℕ, K*((((j:ℝ)+1)*((j:ℝ)+2))^2*q^j)) := by
-      symm
-      exact ENNReal.ofReal_tsum_of_nonneg (fun j => by positivity) hs
-    _ = _ := by rw [tsum_mul_left]
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-theorem strip_rectangle_hitting_discrete {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
-    (hmean : stripMean H q r Fbar Q ≤ H/2) (N : ℕ)
-    (hN : 3*rectCoefficient Q*(N:ℝ)*stripEnergy H q r Fbar Q ≤ (H/4)^2)
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) (Ξ : Ω → Set (ℝ × ℝ))
-    {C₀ ρ : ℝ} (hC : 0 ≤ C₀) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1)
-    (hhit : ∀ n i, P {ω | ∃ x ∈ Ξ ω, x ∈ (dyadicRect Q n i).closed} ≤
-      ENNReal.ofReal (C₀*ρ^n)) :
-    P {ω | ∃ x ∈ Ξ ω, x ∈ Q.closed ∧
-      stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H} ≤
-      ENNReal.ofReal ((64*rectCoefficient Q*C₀*stripEnergy H q r Fbar Q/H^2*ρ^N)*
-        ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j) := by
-  let means := fun n i => stripMean H q r Fbar (dyadicRect Q n i)
-  let energy := fun n i => stripEnergy H q r Fbar (dyadicRect Q n i)
-  let value := fun x : Q.closed => stripPotential H q r Fbar ((x.val.1:ℂ)+(x.val.2:ℂ)*Complex.I)
-  let branch := fun x : Q.closed => dPointBranch Q x.val
-  let Ξ' := fun ω => {x : Q.closed | x.val ∈ Ξ ω}
-  have hroot (i : DIndex N) : means N i ≤ 3*H/4 := by
-    have hb := (strip_dyadic_mean_critical hF hinj hext hcont hH hr Q hlo hhi N i).trans hN
-    have hd := le_of_sq_le_sq hb (show 0 ≤ H/4 by positivity)
-    dsimp only [means]
-    linarith
-  have hstep (n : ℕ) (i : DIndex (n+1)) :
-      |means (n+1) i-means n (dParent n i)|^2 ≤ rectCoefficient Q*energy n (dParent n i) := by
-    rw [sq_abs]
-    exact strip_dyadic_mean_step hF hinj hext hcont hH hr Q hlo hhi n i
-  have hlim (x : Q.closed) : Tendsto (fun n => means n (branch x n)) atTop (𝓝 (value x)) :=
-    strip_dyadic_mean_limit hcont hH.le hr Q hlo hhi x.property
-  have hnon (n : ℕ) (i : DIndex n) : 0 ≤ energy n i := integral_nonneg (fun _ => sq_nonneg _)
-  have hE : 0 ≤ stripEnergy H q r Fbar Q := integral_nonneg (fun _ => sq_nonneg _)
-  have hsum (n : ℕ) : ∑ i, energy n i ≤ stripEnergy H q r Fbar Q := by
-    apply le_of_eq
-    exact dyadic_integral_sum Q (strip_gradient_rect_memLp hF hinj hext hH hr q Q.a Q.b hlo hhi).integrable_sq n
-  have hh (n : ℕ) (i : DIndex n) : P {ω | ∃ x ∈ Ξ' ω, branch x n = i} ≤ ENNReal.ofReal (C₀*ρ^n) := by
-    refine (measure_mono ?_).trans (hhit n i)
-    rintro ω ⟨x,hx,hi⟩
-    have hxmem := dPointBranch_mem x.property n
-    change x.val ∈ (dyadicRect Q n (branch x n)).closed at hxmem
-    rw [hi] at hxmem
-    exact ⟨x.val,hx,hxmem⟩
-  have ht := dyadic_energy_hitting dParent branch means energy value hH (rectCoefficient_pos Q).le hE
-    hC hρ hρ1 N (fun x n => dPointBranch_parent Q x.val n) hroot hstep hlim hnon hsum P Ξ' hh
-  refine (measure_mono ?_).trans ht
-  rintro ω ⟨x,hx,hxQ,hvalue⟩
-  exact ⟨⟨x,hxQ⟩,hx,hvalue⟩
-
-lemma geometric_floor_bound {ρ : ℝ} (hρ : 0 < ρ) (hρ1 : ρ < 1) (x : ℝ) :
-    ρ^Nat.floor x ≤ Real.exp (Real.log ρ*x)/ρ := by
-  have hh := Real.rpow_le_rpow_of_exponent_ge hρ hρ1.le (Nat.sub_one_lt_floor x).le
-  rw [Real.rpow_natCast] at hh
-  refine hh.trans_eq ?_
-  rw [Real.rpow_sub hρ,Real.rpow_one,Real.rpow_def_of_pos hρ]
-
-lemma critical_floor {D E H : ℝ} (hD : 0 < D) (hE : 0 < E) :
-    3*D*(Nat.floor (H^2/(48*D*E)):ℝ)*E ≤ (H/4)^2 := by
-  have hf := Nat.floor_le (show 0 ≤ H^2/(48*D*E) by positivity)
-  calc
-    _ ≤ 3*D*(H^2/(48*D*E))*E := by gcongr
-    _ = _ := by field_simp; ring
-
-theorem strip_rectangle_hitting {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
-    (hmean : stripMean H q r Fbar Q ≤ H/2)
-    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) (Ξ : Ω → Set (ℝ × ℝ))
-    {C₀ ρ : ℝ} (hC : 0 ≤ C₀) (hρ : 0 < ρ) (hρ1 : ρ < 1)
-    (hhit : ∀ n i, P {ω | ∃ x ∈ Ξ ω, x ∈ (dyadicRect Q n i).closed} ≤
-      ENNReal.ofReal (C₀*ρ^n)) :
-    let D := rectCoefficient Q
-    let E := stripEnergy H q r Fbar Q
-    let T := ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j
-    P {ω | ∃ x ∈ Ξ ω, x ∈ Q.closed ∧
-      stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H} ≤
-      ENNReal.ofReal ((64*D*(C₀+1)*(T+1)/ρ)*(E/H^2)*
-        Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)) := by
-  dsimp only
-  let D := rectCoefficient Q
-  let E := stripEnergy H q r Fbar Q
-  let T := ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j
-  have hD : 0 < D := rectCoefficient_pos Q
-  have hE0 : 0 ≤ E := integral_nonneg (fun _ => sq_nonneg _)
-  have hT : 0 ≤ T := tsum_nonneg (fun j => mul_nonneg (sq_nonneg _) (pow_nonneg hρ.le _))
-  by_cases hE : E = 0
-  · have ht := strip_rectangle_hitting_discrete hF hinj hext hcont hH hr q Q hlo hhi hmean 0
-      (by simp; positivity) P Ξ hC hρ.le hρ1 hhit
-    change P _ ≤ ENNReal.ofReal ((64*D*C₀*E/H^2*ρ^0)*T) at ht
-    change P _ ≤ ENNReal.ofReal ((64*D*(C₀+1)*(T+1)/ρ)*(E/H^2)*_)
-    simpa only [hE,mul_zero,zero_div,zero_mul,ENNReal.ofReal_zero] using ht
-  · have hEpos : 0 < E := lt_of_le_of_ne hE0 (Ne.symm hE)
-    let x := H^2/(48*D*E)
-    have ht := strip_rectangle_hitting_discrete hF hinj hext hcont hH hr q Q hlo hhi hmean (Nat.floor x)
-      (critical_floor hD hEpos) P Ξ hC hρ.le hρ1 hhit
-    refine ht.trans (ENNReal.ofReal_le_ofReal ?_)
-    change (64*D*C₀*E/H^2*ρ^Nat.floor x)*T ≤
-      (64*D*(C₀+1)*(T+1)/ρ)*(E/H^2)*Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)
-    have hexp : Real.log ρ*x = -((-Real.log ρ)/(48*D))*H^2/E := by dsimp [x]; ring
-    have hg := geometric_floor_bound hρ hρ1 x
-    rw [hexp] at hg
-    have hcoef : C₀*T ≤ (C₀+1)*(T+1) := by nlinarith
-    calc
-      _ ≤ (64*D*C₀*E/H^2*(Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)/ρ))*T := by gcongr
-      _ = (64*D/ρ)*(E/H^2)*Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)*(C₀*T) := by ring
-      _ ≤ (64*D/ρ)*(E/H^2)*Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)*((C₀+1)*(T+1)) := by
-        exact mul_le_mul_of_nonneg_left hcoef (by positivity)
-      _ = _ := by ring
-
-def localizationMeanConstant : ℝ := slabMeanCoefficient+Real.pi+1
-
-lemma localizationMeanConstant_pos : 0 < localizationMeanConstant := by
-  unfold localizationMeanConstant
-  linarith [slabMeanCoefficient_pos,Real.pi_pos]
-
-def localizationStripWidth : ℝ := 1/(32*Real.pi*localizationMeanConstant)
-
-lemma localizationStripWidth_pos : 0 < localizationStripWidth := by
-  unfold localizationStripWidth
-  have hk := localizationMeanConstant_pos
-  positivity
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.AngularClock
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Is Loewner Trace Using estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma visit_liveDistanceApproach (h : IsLoewnerTraceUsing driver γ g f)
+    {z : ℂ} {v : ℝ} (hv : 0 < v) (hsmall : 2*v < z.im)
+    (hvisit : ∃ T : ℝ, 0 ≤ T ∧ dist z (γ T) < v) :
+    liveDistanceApproach γ z (3*v) := by
+  obtain ⟨T,hT,hvisit⟩ := hvisit
+  let TN : ℝ≥0 := ⟨T,hT⟩
+  have hc : Continuous (fun t : ℝ≥0 => dist z (γ t)) :=
+    continuous_const.dist (h.1.comp_continuous continuous_subtype_val (fun t => t.coe_nonneg))
+  let A : Set ℝ≥0 := Icc 0 TN ∩ {t | dist z (γ t) ≤ 2*v}
+  have hA : IsCompact A := isCompact_Icc.inter_right (isClosed_le hc continuous_const)
+  obtain ⟨s,hs,hmin⟩ := hA.exists_isLeast ⟨TN,⟨⟨zero_le,le_rfl⟩,by change dist z (γ T) ≤ 2*v; linarith⟩⟩
+  have hsT : (s:ℝ) ≤ T := hs.1.2
+  have hstrict (q : ℝ≥0) (hqs : q < s) : 2*v < dist z (γ q) := by
+    by_contra hn
+    exact (not_le_of_gt hqs) (hmin ⟨⟨zero_le,hqs.le.trans hs.1.2⟩,not_lt.1 hn⟩)
+  have hinit : 2*v < dist z (γ 0) := by
+    rw [h.2.1,dist_zero_right]
+    exact hsmall.trans_le (Complex.im_le_norm z)
+  have hdist : dist z (γ s) = 2*v := by
+    apply le_antisymm hs.2
+    by_contra hn
+    have hl : dist z (γ s) < 2*v := not_le.1 hn
+    have hcc : ContinuousOn (fun q : ℝ => dist z (γ q)) (Icc 0 (s:ℝ)) :=
+      (continuous_const.dist continuous_id).comp_continuousOn (h.1.mono Icc_subset_Ici_self)
+    obtain ⟨q,hq,he⟩ := intermediate_value_Icc' s.coe_nonneg hcc ⟨hl.le,hinit.le⟩
+    have hqs : q < (s:ℝ) := lt_of_le_of_ne hq.2 (by intro heq; subst q; linarith)
+    have hh := hstrict ⟨q,hq.1⟩ hqs
+    change 2*v < dist z (γ q) at hh
+    exact (not_lt_of_ge he.le) hh
+  have hbefore (q : ℝ) (hq : q ∈ Icc 0 (s:ℝ)) : 2*v ≤ dist z (γ q) := by
+    rcases lt_or_eq_of_le hq.2 with hh | hh
+    · exact (hstrict ⟨q,hq.1⟩ hh).le
+    · rw [hh,hdist]
+  have hfree : ball z (2*v) ⊆ halfPlane \ localizationSegment γ 0 s := by
+    intro w hw
+    have hnorm : ‖w-z‖ < 2*v := by simpa only [mem_ball,dist_eq_norm] using hw
+    constructor
+    · have hh := Complex.abs_im_le_norm (w-z)
+      have hn := neg_abs_le (w-z).im
+      change 0 < w.im
+      simp only [Complex.sub_im] at hh hn
+      linarith
+    · rintro ⟨q,hq,rfl⟩
+      have hh := hbefore q hq
+      have hh' := mem_ball.1 hw
+      rw [dist_comm] at hh'
+      linarith
+  have hcl := h.future_tip_mem_closure s.coe_nonneg hsT
+  have hb : γ T ∈ ball z (2*v) := by rw [mem_ball,dist_comm]; linarith
+  obtain ⟨w,hwb,hwD⟩ := mem_closure_iff.1 hcl (ball z (2*v)) isOpen_ball hb
+  have hcomp := (convex_ball z (2*v)).isPreconnected.subset_connectedComponentIn hwb hfree
+  have hzball : z ∈ ball z (2*v) := mem_ball_self (by positivity)
+  have hzlive : z ∈ localizationDomain γ s := by
+    refine ⟨hfree hzball,?_⟩
+    rw [← connectedComponentIn_eq (hcomp hzball)]
+    exact hwD.2
+  refine ⟨s,hzlive,?_⟩
+  have hnot : γ s ∈ (localizationDomain γ s)ᶜ := by
+    intro hh
+    exact hh.1.2 ⟨s,⟨s.coe_nonneg,le_rfl⟩,rfl⟩
+  have hh := infDist_le_dist_of_mem (x := z) hnot
+  rw [hdist] at hh
+  linarith
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.SmallTarget
+
+lemma sqrt_rpow {ε : ℝ} (hε : 0 ≤ ε) (β : ℝ) : (Real.sqrt ε)^β=ε^(β/2) := by
+  rw [Real.sqrt_eq_rpow,← Real.rpow_mul hε]
+  congr 1
+  ring
+
+lemma interior_weight {p b ε : ℝ} {z : ℂ} (hp : 0 ≤ p) (hb : 0 ≤ b)
+    (hε : 0 < ε) (hy : Real.sqrt ε ≤ z.im) :
+    (2*z.im)^(-p)*(z.im/‖z‖)^b ≤ ε^(-(p/2)) := by
+  have hyr : 0 < z.im := (Real.sqrt_pos.2 hε).trans_le hy
+  have hn : 0 < ‖z‖ := hyr.trans_le (Complex.im_le_norm z)
+  have hsin0 : 0 ≤ z.im/‖z‖ := div_nonneg hyr.le hn.le
+  have hsin1 : z.im/‖z‖ ≤ 1 := (div_le_one hn).2 (Complex.im_le_norm z)
+  have hsin := Real.rpow_le_one hsin0 hsin1 hb
+  have hrad : (2*z.im)^(-p) ≤ (Real.sqrt ε)^(-p) :=
+    Real.rpow_le_rpow_of_nonpos (Real.sqrt_pos.2 hε) (by linarith) (neg_nonpos.mpr hp)
+  calc
+    _ ≤ (2*z.im)^(-p)*1 := mul_le_mul_of_nonneg_left hsin (Real.rpow_nonneg (by positivity) _)
+    _ ≤ (Real.sqrt ε)^(-p) := by simpa using hrad
+    _ = ε^(-(p/2)) := by rw [sqrt_rpow hε.le]; congr 1; ring
+
+lemma interior_power_product {ε C p : ℝ} (hε : 0 < ε) :
+    C*ε^(-(p/2))*(3*ε)^p = C*3^p*ε^(p/2) := by
+  rw [Real.mul_rpow (by norm_num : (0:ℝ)≤3) hε.le]
+  calc
+    _ = C*3^p*(ε^(-(p/2))*ε^p) := by ring
+    _ = _ := by rw [← Real.rpow_add hε]; congr 2; ring
+
+lemma radius_power {A ε β : ℝ} (hA : 0 ≤ A) (hε : 0 ≤ ε) :
+    (A*Real.sqrt ε)^β=A^β*ε^(β/2) := by
+  rw [Real.mul_rpow hA (Real.sqrt_nonneg _),sqrt_rpow hε]
+
+theorem interior_trace_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ c < 1 ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (ε : ℝ) (_hε : 0 < ε) (_hy : Real.sqrt ε ≤ z.im)
+      (_hsmall : 3*ε < c*z.im),
+      P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < ε} ≤
+        ENNReal.ofReal (C*ε^(codimension κ/2)) := by
+  obtain ⟨C,c,hC,hc,hc1,hbound⟩ := euclideanApproachBound κ hκ
+  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
+  have hb : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
+  refine ⟨C*3^(codimension κ),c,mul_pos hC (Real.rpow_pos_of_pos (by norm_num) _),hc,hc1,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z ε hε hy hsmall
+  have hz : z ∈ halfPlane := (Real.sqrt_pos.2 hε).trans_le hy
+  have hsm : 2*ε < z.im := by
+    have hh := mul_lt_mul_of_pos_right hc1 (show 0 < z.im from hz)
+    linarith
+  have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < ε} ≤ᵐ[P]
+      {sample | liveDistanceApproach (γ sample) z (3*ε)} := by
+    filter_upwards [hmaps] with sample hm hh
+    exact hm.visit_liveDistanceApproach hε hsm hh
+  refine (measure_mono_ae hsub).trans ((hbound P B hB γ g f hγ hmaps z hz (3*ε) (by positivity) hsmall).trans
+    (ENNReal.ofReal_le_ofReal ?_))
+  have hm := interior_weight hp.le hb.le hε hy
+  have hpε : 0 ≤ (3*ε)^(codimension κ) := Real.rpow_nonneg (by positivity) _
+  calc
+    C*((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))*(3*ε)^(codimension κ) ≤
+        C*ε^(-(codimension κ/2))*(3*ε)^(codimension κ) := by gcongr
+    _ = C*3^(codimension κ)*ε^(codimension κ/2) := interior_power_product hε
+
+end SLEExactGauge.SmallTarget
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.SmallTarget
+
+lemma epsilon_le_sqrt {ε : ℝ} (hε : 0 ≤ ε) (hε1 : ε ≤ 1) : ε ≤ Real.sqrt ε := by
+  have hs := Real.sq_sqrt hε
+  have hp := Real.sqrt_nonneg ε
+  nlinarith [mul_nonneg hε (sub_nonneg.mpr hε1)]
+
+lemma near_real_geometry {v : ℂ} {ε K : ℝ} (hv : 0 ≤ v.im) (hn : 1/2 ≤ ‖v‖)
+    (hε : 0 < ε) (hε1 : ε ≤ 1) (_hK : 0 ≤ K)
+    (hy : v.im ≤ K*Real.sqrt ε) (hsmall : (K+1)*Real.sqrt ε < 1/4) :
+    1/4 ≤ |v.re| ∧ ∀ w : ℂ, dist v w < ε → dist w (v.re:ℂ) < (K+1)*Real.sqrt ε := by
+  have hs := Real.sqrt_nonneg ε
+  have hy4 : v.im < 1/4 := by nlinarith
+  have hab := Complex.norm_le_abs_re_add_abs_im v
+  rw [abs_of_nonneg hv] at hab
+  refine ⟨by linarith,?_⟩
+  intro w hw
+  have hvx : dist v (v.re:ℂ) ≤ v.im := by
+    rw [dist_eq_norm]
+    have hh := Complex.norm_le_abs_re_add_abs_im (v-(v.re:ℂ))
+    simpa only [Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,
+      sub_self,sub_zero,abs_zero,zero_add,abs_of_nonneg hv] using hh
+  have he := epsilon_le_sqrt hε.le hε1
+  calc
+    dist w (v.re:ℂ) ≤ dist w v+dist v (v.re:ℂ) := dist_triangle _ _ _
+    _ < ε+v.im := add_lt_add_of_lt_of_le (by simpa only [dist_comm] using hw) hvx
+    _ ≤ (K+1)*Real.sqrt ε := by nlinarith
+
+lemma cutoff_sqrt_small {A ε : ℝ} (hA : 0 < A) (hε : 0 < ε)
+    (hsmall : ε < 1/(16*A^2)) : A*Real.sqrt ε < 1/4 := by
+  have hh := (lt_div_iff₀ (show 0 < 16*A^2 by positivity)).mp hsmall
+  have hs := Real.sq_sqrt hε.le
+  have hp : 0 ≤ A*Real.sqrt ε := by positivity
+  nlinarith [sq_nonneg (A*Real.sqrt ε-1/4)]
+
+end SLEExactGauge.SmallTarget
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open SmallTarget BoundaryGreen
+
+theorem uniform_small_target (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (v : ℂ), 0 ≤ v.im → 1/2 ≤ ‖v‖ → ‖v‖ ≤ 2 →
+        ∀ ε : ℝ, 0 < ε → ε < 1/16 →
+        P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < ε} ≤
+          ENNReal.ofReal (C*ε^c) := by
+  let β := (8/κ-1)/2
+  let a := 1+κ/8
+  obtain ⟨hβ,ha,ha2,hba⟩ := exponent_choice hκ.1 hκ.2
+  change 0 < β at hβ
+  change 0 < a at ha
+  change a < 2 at ha2
+  change κ*(β+1) < 4+2*a at hba
+  let δ := (2-a)/8
+  have hδ : 0 < δ := by dsimp [δ]; linarith
+  have hδhalf : δ ≤ 1/2 := by dsimp [δ]; linarith
+  have haδ : a ≤ 2-4*δ := by dsimp [δ]; linarith
+  obtain ⟨Ci,e,hi,he,he1,hinter⟩ := interior_trace_bound κ hκ
+  let K := max 1 (3/e)
+  have hK1 : 1 ≤ K := le_max_left _ _
+  have hKe : 3/e ≤ K := le_max_right _ _
+  have hK : 0 < K := lt_of_lt_of_le zero_lt_one hK1
+  let A := K+1
+  have hA : 0 < A := by dsimp [A]; positivity
+  let ρ := 1/(16*A^2)
+  have hρ : 0 < ρ := by dsimp [ρ]; positivity
+  let c := min (β/2) (codimension κ/2)
+  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
+  have hc : 0 < c := lt_min (half_pos hβ) (half_pos hp)
+  let Cb := (16/δ)^β*A^β
+  have hCb : 0 < Cb := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (Real.rpow_pos_of_pos hA _)
+  let C := Cb+Ci+ρ^(-c)
+  have hR : 0 < ρ^(-c) := Real.rpow_pos_of_pos hρ _
+  have hC : 0 < C := by dsimp only [C]; positivity
+  have hbC : Cb ≤ C := by dsimp only [C]; linarith
+  have hiC : Ci ≤ C := by dsimp only [C]; linarith
+  have hrC : ρ^(-c) ≤ C := by dsimp only [C]; linarith
+  refine ⟨C,c,hC,hc,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps v hv hn hlo ε hε hε16
+  have hε1 : ε ≤ 1 := by linarith
+  have hs : 0 < Real.sqrt ε := Real.sqrt_pos.2 hε
+  have heps := epsilon_le_sqrt hε.le hε1
+  by_cases hsmall : ε < ρ
+  · by_cases hnear : v.im ≤ K*Real.sqrt ε
+    · have hgeom := near_real_geometry hv hn hε hε1 hK.le hnear (cutoff_sqrt_small hA hε hsmall)
+      have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < ε} ⊆
+          {sample | ∃ t : ℝ, 0 ≤ t ∧ dist (γ sample t) (v.re:ℂ) < A*Real.sqrt ε} := by
+        rintro sample ⟨t,ht,hclose⟩
+        exact ⟨t,ht,hgeom.2 _ hclose⟩
+      have hh := annulus_real_bound hB κ β a δ hκ.1.le hβ ha.le hba.le hδ hδhalf haδ
+        γ g f hmaps hgeom.1 (mul_pos hA hs)
+      refine (measure_mono hsub).trans (hh.trans (ENNReal.ofReal_le_ofReal ?_))
+      rw [radius_power hA.le hε.le]
+      have hepow : ε^(β/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_left _ _)
+      calc
+        (16/δ)^β*(A^β*ε^(β/2)) = Cb*ε^(β/2) := by dsimp only [Cb]; ring
+        _ ≤ Cb*ε^c := mul_le_mul_of_nonneg_left hepow hCb.le
+        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hbC (Real.rpow_nonneg hε.le _)
+    · have hy : Real.sqrt ε ≤ v.im := by
+        have hh := mul_le_mul_of_nonneg_right hK1 hs.le
+        linarith [not_le.mp hnear]
+      have hm : 3*ε < e*v.im := by
+        have hh := (div_le_iff₀ he).mp hKe
+        have hm := mul_le_mul_of_nonneg_right hh hs.le
+        have hlt := mul_lt_mul_of_pos_left (not_le.mp hnear) he
+        nlinarith
+      refine (hinter P B hB γ g f hγ hmaps v ε hε hy hm).trans (ENNReal.ofReal_le_ofReal ?_)
+      have hepow : ε^(codimension κ/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_right _ _)
+      exact (mul_le_mul_of_nonneg_left hepow hi.le).trans
+        (mul_le_mul_of_nonneg_right hiC (Real.rpow_nonneg hε.le _))
+  · have hrε := Real.rpow_le_rpow hρ.le (not_lt.1 hsmall) hc.le
+    have hprod : ρ^(-c)*ρ^c = 1 := by
+      rw [Real.rpow_neg hρ.le,inv_mul_cancel₀ (Real.rpow_pos_of_pos hρ _).ne']
+    have h1 : 1 ≤ C*ε^c := by
+      calc
+        1 = ρ^(-c)*ρ^c := hprod.symm
+        _ ≤ ρ^(-c)*ε^c := mul_le_mul_of_nonneg_left hrε hR.le
+        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hrC (Real.rpow_nonneg hε.le _)
+    exact prob_le_one.trans (by simpa using ENNReal.ofReal_le_ofReal h1)
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.SmallTarget
+
+lemma scaled_near_real_geometry {v : ℂ} {R ε K : ℝ} (hR : 0 < R)
+    (hv : 0 ≤ v.im) (hn : R/2 ≤ ‖v‖) (hε : 0 < ε) (hε1 : ε ≤ 1) (_hK : 0 ≤ K)
+    (hy : v.im ≤ R*(K*Real.sqrt ε)) (hsmall : (K+1)*Real.sqrt ε < 1/4) :
+    R/4 ≤ |v.re| ∧ ∀ w : ℂ, dist v w < R*ε →
+      dist w (v.re:ℂ) < R*((K+1)*Real.sqrt ε) := by
+  have hs := Real.sqrt_nonneg ε
+  have hmul := mul_lt_mul_of_pos_left hsmall hR
+  have hy4 : v.im < R/4 := by nlinarith [mul_nonneg hR.le hs]
+  have hab := Complex.norm_le_abs_re_add_abs_im v
+  rw [abs_of_nonneg hv] at hab
+  refine ⟨by linarith,?_⟩
+  intro w hw
+  have hvx : dist v (v.re:ℂ) ≤ v.im := by
+    rw [dist_eq_norm]
+    have hh := Complex.norm_le_abs_re_add_abs_im (v-(v.re:ℂ))
+    simpa only [Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,
+      sub_self,sub_zero,abs_zero,zero_add,abs_of_nonneg hv] using hh
+  have he := mul_le_mul_of_nonneg_left (epsilon_le_sqrt hε.le hε1) hR.le
+  calc
+    dist w (v.re:ℂ) ≤ dist w v+dist v (v.re:ℂ) := dist_triangle _ _ _
+    _ < R*ε+v.im := add_lt_add_of_lt_of_le (by simpa only [dist_comm] using hw) hvx
+    _ ≤ R*((K+1)*Real.sqrt ε) := by nlinarith
+
+lemma scaled_interior_power {R ε p : ℝ} (hR : 0 < R) (hε : 0 < ε) :
+    (R*Real.sqrt ε)^(-p)*(3*(R*ε))^p = 3^p*ε^(p/2) := by
+  rw [Real.mul_rpow hR.le (Real.sqrt_nonneg _),sqrt_rpow hε.le,
+    Real.mul_rpow (by norm_num : (0:ℝ)≤3) (mul_nonneg hR.le hε.le),Real.mul_rpow hR.le hε.le]
+  calc
+    _ = 3^p*(R^(-p)*R^p)*(ε^(-p/2)*ε^p) := by ring
+    _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hε,neg_add_cancel,Real.rpow_zero,mul_one,show -p/2+p=p/2 by ring]
+
+ theorem interior_trace_scaled_bound (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ c < 1 ∧
+    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+      (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+      (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+      (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+        (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+      (z : ℂ) (R ε : ℝ) (_hR : 0 < R) (_hε : 0 < ε) (_hy : R*Real.sqrt ε ≤ z.im)
+      (_hsmall : 3*(R*ε) < c*z.im),
+      P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < R*ε} ≤
+        ENNReal.ofReal (C*ε^(codimension κ/2)) := by
+  obtain ⟨C,c,hC,hc,hc1,hbound⟩ := euclideanApproachBound κ hκ
+  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
+  have hb : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
+  refine ⟨C*3^(codimension κ),c,mul_pos hC (Real.rpow_pos_of_pos (by norm_num) _),hc,hc1,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z R ε hR hε hy hsmall
+  have hs := Real.sqrt_pos.2 hε
+  have hz : z ∈ halfPlane := (mul_pos hR hs).trans_le hy
+  have hzim : 0 < z.im := hz
+  have hsm : 2*(R*ε) < z.im := by
+    have hh := mul_lt_mul_of_pos_right hc1 (show 0 < z.im from hz)
+    linarith [mul_pos hR hε]
+  have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist z (γ sample t) < R*ε} ≤ᵐ[P]
+      {sample | liveDistanceApproach (γ sample) z (3*(R*ε))} := by
+    filter_upwards [hmaps] with sample hm hh
+    exact hm.visit_liveDistanceApproach (mul_pos hR hε) hsm hh
+  refine (measure_mono_ae hsub).trans ((hbound P B hB γ g f hγ hmaps z hz _ (by positivity) hsmall).trans
+    (ENNReal.ofReal_le_ofReal ?_))
+  have hn : 0 < ‖z‖ := hz.trans_le (Complex.im_le_norm z)
+  have hsin := Real.rpow_le_one (div_nonneg hz.le hn.le)
+    ((div_le_one hn).2 (Complex.im_le_norm z)) hb.le
+  have hrad : (2*z.im)^(-codimension κ) ≤ (R*Real.sqrt ε)^(-codimension κ) :=
+    Real.rpow_le_rpow_of_nonpos (mul_pos hR hs) (by linarith [hzim]) (by linarith)
+  calc
+    _ ≤ C*((2*z.im)^(-codimension κ)*1)*(3*(R*ε))^(codimension κ)  := by
+      exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_left hsin (Real.rpow_nonneg (by positivity) _)) hC.le)
+        (Real.rpow_nonneg (by positivity) _)
+    _ ≤ C*((R*Real.sqrt ε)^(-codimension κ))*(3*(R*ε))^(codimension κ) := by
+      simp only [mul_one]; gcongr
+    _ = C*3^(codimension κ)*ε^(codimension κ/2) := by
+      rw [mul_assoc,scaled_interior_power hR hε]; ring
+
+end SLEExactGauge.SmallTarget
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open SmallTarget BoundaryGreen
+
+lemma scaled_annulus_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    {x R d : ℝ} (hR : 0 < R) (hx : R/4 ≤ |x|) (hd : 0 < d) :
+    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < R*d} ≤ ENNReal.ofReal ((16/δ)^β*d^β) := by
+  have hxp : 0 < |x| := (by positivity : 0 < R/4).trans_le hx
+  refine (real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps hxp (mul_pos hR hd)).trans
+    (ENNReal.ofReal_le_ofReal ?_)
+  have hr : 4*(R*d)/(δ*|x|) ≤ (16/δ)*d := by
+    rw [div_le_iff₀ (mul_pos hδ hxp)]
+    have hh := mul_le_mul_of_nonneg_left hx (show 0 ≤ 16*d by positivity)
+    field_simp
+    nlinarith
+  have hh := Real.rpow_le_rpow (by positivity : 0 ≤ 4*(R*d)/(δ*|x|)) hr hβ.le
+  rwa [Real.mul_rpow (by positivity : 0 ≤ 16/δ) hd.le] at hh
+
+theorem uniform_small_target_scaled (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (v : ℂ) (R : ℝ), 0 < R → 0 ≤ v.im → R/2 ≤ ‖v‖ →
+        ∀ ε : ℝ, 0 < ε → ε < 1/16 →
+        P {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < R*ε} ≤
+          ENNReal.ofReal (C*ε^c) := by
+  let β := (8/κ-1)/2
+  let a := 1+κ/8
+  obtain ⟨hβ,ha,ha2,hba⟩ := exponent_choice hκ.1 hκ.2
+  change 0 < β at hβ
+  change 0 < a at ha
+  change a < 2 at ha2
+  change κ*(β+1) < 4+2*a at hba
+  let δ := (2-a)/8
+  have hδ : 0 < δ := by dsimp [δ]; linarith
+  have hδhalf : δ ≤ 1/2 := by dsimp [δ]; linarith
+  have haδ : a ≤ 2-4*δ := by dsimp [δ]; linarith
+  obtain ⟨Ci,e,hi,he,he1,hinter⟩ := interior_trace_scaled_bound κ hκ
+  let K := max 1 (3/e)
+  have hK1 : 1 ≤ K := le_max_left _ _
+  have hKe : 3/e ≤ K := le_max_right _ _
+  have hK : 0 < K := lt_of_lt_of_le zero_lt_one hK1
+  let A := K+1
+  have hA : 0 < A := by dsimp [A]; positivity
+  let ρ := 1/(16*A^2)
+  have hρ : 0 < ρ := by dsimp [ρ]; positivity
+  let c := min (β/2) (codimension κ/2)
+  have hp : 0 < codimension κ := by unfold codimension dimension; linarith [hκ.2]
+  have hc : 0 < c := lt_min (half_pos hβ) (half_pos hp)
+  let Cb := (16/δ)^β*A^β
+  have hCb : 0 < Cb := mul_pos (Real.rpow_pos_of_pos (by positivity) _) (Real.rpow_pos_of_pos hA _)
+  let C := Cb+Ci+ρ^(-c)
+  have hρpow : 0 < ρ^(-c) := Real.rpow_pos_of_pos hρ _
+  have hC : 0 < C := by dsimp only [C]; positivity
+  have hbC : Cb ≤ C := by dsimp only [C]; linarith
+  have hiC : Ci ≤ C := by dsimp only [C]; linarith
+  have hrC : ρ^(-c) ≤ C := by dsimp only [C]; linarith
+  refine ⟨C,c,hC,hc,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps v R hR hv hn ε hε hε16
+  have hε1 : ε ≤ 1 := by linarith
+  have hs : 0 < Real.sqrt ε := Real.sqrt_pos.2 hε
+  have heps := epsilon_le_sqrt hε.le hε1
+  by_cases hsmall : ε < ρ
+  · by_cases hnear : v.im ≤ R*(K*Real.sqrt ε)
+    · have hgeom := scaled_near_real_geometry hR hv hn hε hε1 hK.le hnear (cutoff_sqrt_small hA hε hsmall)
+      have hsub : {sample | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ sample t) < R*ε} ⊆
+          {sample | ∃ t : ℝ, 0 ≤ t ∧ dist (γ sample t) (v.re:ℂ) < R*(A*Real.sqrt ε)} := by
+        rintro sample ⟨t,ht,hclose⟩
+        exact ⟨t,ht,hgeom.2 _ hclose⟩
+      have hh := scaled_annulus_real_bound hB κ β a δ hκ.1.le hβ ha.le hba.le hδ hδhalf haδ
+        γ g f hmaps hR hgeom.1 (mul_pos hA hs)
+      refine (measure_mono hsub).trans (hh.trans (ENNReal.ofReal_le_ofReal ?_))
+      rw [radius_power hA.le hε.le]
+      have hepow : ε^(β/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_left _ _)
+      calc
+        (16/δ)^β*(A^β*ε^(β/2)) = Cb*ε^(β/2) := by dsimp only [Cb]; ring
+        _ ≤ Cb*ε^c := mul_le_mul_of_nonneg_left hepow hCb.le
+        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hbC (Real.rpow_nonneg hε.le _)
+    · have hy : R*Real.sqrt ε ≤ v.im := by
+        have hh := mul_le_mul_of_nonneg_left
+          (mul_le_mul_of_nonneg_right hK1 hs.le) hR.le
+        nlinarith [not_le.mp hnear]
+      have hm : 3*(R*ε) < e*v.im := by
+        have hh := (div_le_iff₀ he).mp hKe
+        calc
+          3*(R*ε) ≤ 3*(R*Real.sqrt ε) := by gcongr
+          _ ≤ e*(R*(K*Real.sqrt ε)) := by
+            have hm := mul_le_mul_of_nonneg_right hh (mul_nonneg hR.le hs.le)
+            nlinarith
+          _ < e*v.im := mul_lt_mul_of_pos_left (not_le.mp hnear) he
+      refine (hinter P B hB γ g f hγ hmaps v R ε hR hε hy hm).trans (ENNReal.ofReal_le_ofReal ?_)
+      have hepow : ε^(codimension κ/2) ≤ ε^c := Real.rpow_le_rpow_of_exponent_ge hε hε1 (min_le_right _ _)
+      exact (mul_le_mul_of_nonneg_left hepow hi.le).trans
+        (mul_le_mul_of_nonneg_right hiC (Real.rpow_nonneg hε.le _))
+  · have hrε := Real.rpow_le_rpow hρ.le (not_lt.1 hsmall) hc.le
+    have hprod : ρ^(-c)*ρ^c = 1 := by
+      rw [Real.rpow_neg hρ.le,inv_mul_cancel₀ (Real.rpow_pos_of_pos hρ _).ne']
+    have h1 : 1 ≤ C*ε^c := by
+      calc
+        1 = ρ^(-c)*ρ^c := hprod.symm
+        _ ≤ ρ^(-c)*ε^c := mul_le_mul_of_nonneg_left hrε hρpow.le
+        _ ≤ C*ε^c := mul_le_mul_of_nonneg_right hrC (Real.rpow_nonneg hε.le _)
+    exact prob_le_one.trans (by simpa using ENNReal.ofReal_le_ofReal h1)
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+def logTrace {Ω : Type*} (γ : Ω → ℝ → ℂ) (ω : Ω) : Set (ℝ × ℝ) :=
+  {x | ∃ t : ℝ, 0 ≤ t ∧ γ ω t = Complex.exp ((x.1:ℂ)+(x.2:ℂ)*Complex.I)}
+
+lemma dyadic_radius_power (K c : ℝ) (hK : 0 ≤ K) (n : ℕ) :
+    (K/(2:ℝ)^n)^c = K^c*((1/2:ℝ)^c)^n := by
+  rw [div_eq_mul_inv,Real.mul_rpow hK (by positivity),← inv_pow,
+    ← Real.rpow_pow_comm (by norm_num : (0:ℝ) ≤ 2⁻¹)]
+  simp only [one_div]
+
+theorem logTrace_dyadic_hitting (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) :
+    ∃ C₀ ρ : ℝ, 0 < C₀ ∧ 0 < ρ ∧ ρ < 1 ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (j : ℝ) (n : ℕ) (i : DIndex n),
+          P {ω | ∃ x ∈ logTrace γ ω, x ∈ (dyadicRect (unitSlab j) n i).closed} ≤
+            ENNReal.ofReal (C₀*ρ^n) := by
+  obtain ⟨C,c,hC,hc,hsmall⟩ := uniform_small_target_scaled κ hκ
+  let K := expCellScale
+  let ρ := (1/2:ℝ)^c
+  let C₀ := (C+16^c)*K^c
+  have hK : 0 < K := expCellScale_pos
+  have hρ : 0 < ρ := Real.rpow_pos_of_pos (by norm_num) _
+  have hρ1 : ρ < 1 := Real.rpow_lt_one (by norm_num) (by norm_num) hc
+  have h16 : 0 < (16:ℝ)^c := Real.rpow_pos_of_pos (by norm_num) _
+  have hC₀ : 0 < C₀ := mul_pos (add_pos hC h16) (Real.rpow_pos_of_pos hK _)
+  refine ⟨C₀,ρ,hC₀,hρ,hρ1,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps j n i
+  let ε := K/(2:ℝ)^n
+  have hε : 0 < ε := by dsimp [ε]; positivity
+  have heq : ε^c = K^c*ρ^n := dyadic_radius_power K c hK.le n
+  by_cases hεsmall : ε < 1/16
+  · let Q := dyadicRect (unitSlab j) n i
+    let v := Complex.exp ((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)
+    have hg := dyadicCell_exp_geometry j n i hεsmall
+    have hb := hsmall P B hB γ g f hγ hmaps v (Real.exp j) (Real.exp_pos _) hg.1 hg.2.1 ε hε hεsmall
+    have hs : {ω | ∃ x ∈ logTrace γ ω, x ∈ Q.closed} ⊆
+        {ω | ∃ t : ℝ, 0 ≤ t ∧ dist v (γ ω t) < Real.exp j*ε} := by
+      rintro ω ⟨x,⟨t,ht,he⟩,hx⟩
+      refine ⟨t,ht,?_⟩
+      rw [he]
+      exact hg.2.2 x hx
+    refine (measure_mono hs).trans (hb.trans (ENNReal.ofReal_le_ofReal ?_))
+    rw [heq]
+    change C*(K^c*ρ^n) ≤ ((C+16^c)*K^c)*ρ^n
+    nlinarith [mul_nonneg h16.le (mul_nonneg (Real.rpow_nonneg hK.le c) (pow_nonneg hρ.le n))]
+  · have hbase : 1 ≤ 16*ε := by linarith [not_lt.mp hεsmall]
+    have hp := Real.one_le_rpow hbase hc.le
+    rw [Real.mul_rpow (by norm_num : (0:ℝ)≤16) hε.le,heq] at hp
+    have h1 : 1 ≤ C₀*ρ^n := hp.trans (by
+      change 16^c*(K^c*ρ^n) ≤ ((C+16^c)*K^c)*ρ^n
+      nlinarith [mul_nonneg hC.le (mul_nonneg (Real.rpow_nonneg hK.le c) (pow_nonneg hρ.le n))])
+    exact prob_le_one.trans (by simpa using ENNReal.ofReal_le_ofReal h1)
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+namespace DRect
+
+instance (Q : DRect) : IsFiniteMeasure Q.measure := by unfold measure; infer_instance
+
+lemma measure_subset {Q R : DRect} (ha : Q.a ≤ R.a) (hb : R.b ≤ Q.b)
+    (hc : Q.c ≤ R.c) (hd : R.d ≤ Q.d) : R.measure ≤ Q.measure :=
+  Measure.prod_mono (Measure.restrict_mono_set _ (fun _x hx => ⟨ha.trans_lt hx.1,hx.2.trans hb⟩))
+    (Measure.restrict_mono_set _ (fun _y hy => ⟨hc.trans_lt hy.1,hy.2.trans hd⟩))
+
+lemma child_measure_le (Q : DRect) (s : Bool × Bool) : (Q.child s).measure ≤ Q.measure := by
+  obtain ⟨ha,hb,hc,hd⟩ := Q.child_bounds s
+  exact measure_subset ha hb hc hd
+
+lemma measure_interval_split {l r : ℝ} (h : l < r) :
+    volume.restrict (Ioc l r) = volume.restrict (Ioc l ((l+r)/2))+
+      volume.restrict (Ioc ((l+r)/2) r) := by
+  rw [← Ioc_union_Ioc_eq_Ioc (show l ≤ (l+r)/2 by linarith) (show (l+r)/2 ≤ r by linarith)]
+  apply Measure.restrict_union _ measurableSet_Ioc
+  exact disjoint_left.2 (fun x hx hy => (not_lt_of_ge hx.2) hy.1)
+
+lemma measure_child_sum (Q : DRect) : Q.measure = ∑ s : Bool × Bool, (Q.child s).measure := by
+  simp only [Fintype.sum_prod_type,Fintype.sum_bool,measure,child,halfLo,halfHi,↓reduceIte]
+  rw [measure_interval_split Q.hab,measure_interval_split Q.hcd,Measure.add_prod,
+    Measure.prod_add,Measure.prod_add]
+  ac_rfl
+
+end DRect
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma dyadicRect_measure_le (Q : DRect) (n : ℕ) (i : DIndex n) :
+    (dyadicRect Q n i).measure ≤ Q.measure := by
+  obtain ⟨ha,hb,hc,hd⟩ := dyadicRect_bounds Q n i
+  exact DRect.measure_subset ha hb hc hd
+
+lemma dyadic_measure_sum (Q : DRect) (n : ℕ) :
+    Q.measure = ∑ i : DIndex n, (dyadicRect Q n i).measure := by
+  induction n with
+  | zero =>
+    let : Unique (DIndex 0) := inferInstanceAs (Unique Unit)
+    rw [Fintype.sum_unique]
+    rfl
+  | succ n ih =>
+    change Q.measure = ∑ i : DIndex n × (Bool × Bool), ((dyadicRect Q n i.1).child i.2).measure
+    rw [Fintype.sum_prod_type]
+    simp_rw [← DRect.measure_child_sum]
+    exact ih
+
+lemma dyadic_integral_sum (Q : DRect) {g : ℝ × ℝ → ℝ} (hg : Integrable g Q.measure) (n : ℕ) :
+    ∑ i : DIndex n, (∫ p, g p ∂(dyadicRect Q n i).measure) = ∫ p, g p ∂Q.measure := by
+  conv_rhs => rw [dyadic_measure_sum Q n]
+  symm
+  exact integral_finsetSum_measure (fun i hi => hg.mono_measure (dyadicRect_measure_le Q n i))
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+def jumpSize (H : ℝ) (j : ℕ) : ℝ := H/(8*((j:ℝ)+1)*((j:ℝ)+2))
+
+lemma jumpSize_pos {H : ℝ} (hH : 0 < H) (j : ℕ) : 0 < jumpSize H j := by
+  unfold jumpSize
+  positivity
+
+lemma jumpSize_telescope (H : ℝ) (j : ℕ) :
+    H/8*(1-1/((j:ℝ)+1)) + jumpSize H j = H/8*(1-1/((j:ℝ)+2)) := by
+  unfold jumpSize
+  have h1 : (j:ℝ)+1 ≠ 0 := by positivity
+  have h2 : (j:ℝ)+2 ≠ 0 := by positivity
+  field_simp
+  ring
+
+lemma exists_large_increment {H : ℝ} (hH : 0 < H) {v : ℕ → ℝ}
+    (hstart : v 0 ≤ 3*H/4) (hlim : Tendsto v atTop (𝓝 H)) :
+    ∃ j : ℕ, jumpSize H j < |v (j+1)-v j| := by
+  by_contra hn
+  push Not at hn
+  have hb (n : ℕ) : v n ≤ v 0 + H/8*(1-1/((n:ℝ)+1)) := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      have hd := le_trans (le_abs_self (v (n+1)-v n)) (hn n)
+      have hh := jumpSize_telescope H n
+      have hc : ((n+1:ℕ):ℝ)+1 = (n:ℝ)+2 := by norm_num; ring
+      rw [hc]
+      linarith
+  have hb' (n : ℕ) : v n ≤ 7*H/8 := by
+    have hinv : 0 ≤ 1/((n:ℝ)+1) := by positivity
+    have hp : 0 ≤ H/8 := by positivity
+    have hm := mul_nonneg hp hinv
+    linarith [hb n]
+  have hh := le_of_tendsto hlim (Filter.Eventually.of_forall hb')
+  linarith
+
+lemma geometric_moment_summable {q : ℝ} (hq : 0 ≤ q) (hq1 : q < 1) :
+    Summable (fun j : ℕ => (((j:ℝ)+1)*((j:ℝ)+2))^2*q^j) := by
+  have hn : ‖q‖ < 1 := by simpa [Real.norm_eq_abs,abs_of_nonneg hq] using hq1
+  have h0 := summable_pow_mul_geometric_of_norm_lt_one 0 hn
+  have h1 := summable_pow_mul_geometric_of_norm_lt_one 1 hn
+  have h2 := summable_pow_mul_geometric_of_norm_lt_one 2 hn
+  have h3 := summable_pow_mul_geometric_of_norm_lt_one 3 hn
+  have h4 := summable_pow_mul_geometric_of_norm_lt_one 4 hn
+  have hh := (((h4.add (h3.mul_left 6)).add (h2.mul_left 13)).add
+    (h1.mul_left 12)).add (h0.mul_left 4)
+  apply hh.congr
+  intro j
+  ring
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+section Tree
+variable {X : Type} {I : ℕ → Type} [finiteIndex : ∀ n, Fintype (I n)]
+  (parent : ∀ n, I (n+1) → I n) (branch : X → ∀ n, I n)
+  (means energy : ∀ n, I n → ℝ) (value : X → ℝ)
+
+omit finiteIndex in
+lemma branch_bad_parent [∀ level, Fintype (I level)] {H D : ℝ} (hH : 0 < H) (N : ℕ)
+    (hbranch : ∀ x n, parent n (branch x (n+1)) = branch x n)
+    (hroot : ∀ i, means N i ≤ 3*H/4)
+    (hstep : ∀ n i, |means (n+1) i - means n (parent n i)|^2 ≤ D*energy n (parent n i))
+    (hlim : ∀ x, Tendsto (fun n => means n (branch x n)) atTop (𝓝 (value x)))
+    {x : X} (hx : value x = H) :
+    ∃ j : ℕ, (jumpSize H j)^2 < D*energy (N+j) (branch x (N+j)) := by
+  have hl : Tendsto (fun n => means (N+n) (branch x (N+n))) atTop (𝓝 H) := by
+    have ht : Tendsto (fun n : ℕ => N+n) atTop atTop := by
+      simpa only [Nat.add_comm] using tendsto_add_atTop_nat N
+    simpa only [Function.comp_def,hx] using (hlim x).comp ht
+  obtain ⟨j,hj⟩ := exists_large_increment hH (by simpa using hroot (branch x N)) hl
+  refine ⟨j,?_⟩
+  have hs := hstep (N+j) (branch x ((N+j)+1))
+  rw [hbranch] at hs
+  have hj' : jumpSize H j < |means ((N+j)+1) (branch x ((N+j)+1)) - means (N+j) (branch x (N+j))| := by
+    have he : N+(j+1) = (N+j)+1 := (Nat.add_assoc N j 1).symm
+    rw [he] at hj
+    exact hj
+  exact ((sq_lt_sq₀ (jumpSize_pos hH j).le (abs_nonneg _)).2 hj').trans_le hs
+
+end Tree
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma jump_energy_scale {H : ℝ} (hH : H ≠ 0) (D E C q : ℝ) (N j : ℕ) :
+    D*E/(jumpSize H j)^2*(C*q^(N+j)) =
+      (64*D*C*E/H^2*q^N)*((((j:ℝ)+1)*((j:ℝ)+2))^2*q^j) := by
+  unfold jumpSize
+  rw [pow_add]
+  field_simp
+  ring
+
+theorem dyadic_energy_hitting
+    {X : Type} {I : ℕ → Type} [∀ n, Fintype (I n)]
+    (parent : ∀ n, I (n+1) → I n) (branch : X → ∀ n, I n)
+    (means energy : ∀ n, I n → ℝ) (value : X → ℝ)
+    {H D E C₀ q : ℝ} (hH : 0 < H) (hD : 0 ≤ D) (hE : 0 ≤ E)
+    (hC : 0 ≤ C₀) (hq : 0 ≤ q) (hq1 : q < 1) (N : ℕ)
+    (hbranch : ∀ x n, parent n (branch x (n+1)) = branch x n)
+    (hroot : ∀ i, means N i ≤ 3*H/4)
+    (hstep : ∀ n i, |means (n+1) i - means n (parent n i)|^2 ≤ D*energy n (parent n i))
+    (hlim : ∀ x, Tendsto (fun n => means n (branch x n)) atTop (𝓝 (value x)))
+    (hnon : ∀ n i, 0 ≤ energy n i) (hsum : ∀ n, ∑ i, energy n i ≤ E)
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) (Ξ : Ω → Set X)
+    (hhit : ∀ n i, P {ω | ∃ x ∈ Ξ ω, branch x n = i} ≤ ENNReal.ofReal (C₀*q^n)) :
+    P {ω | ∃ x ∈ Ξ ω, value x = H} ≤
+      ENNReal.ofReal ((64*D*C₀*E/H^2*q^N)*
+        ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*q^j) := by
+  classical
+  let event (j : ℕ) (i : I (N+j)) : Set Ω :=
+    {ω | (jumpSize H j)^2 < D*energy (N+j) i ∧ ∃ x ∈ Ξ ω, branch x (N+j) = i}
+  have hcover : {ω | ∃ x ∈ Ξ ω, value x = H} ⊆ ⋃ j : ℕ, ⋃ i : I (N+j), event j i := by
+    rintro ω ⟨x,hx,hv⟩
+    obtain ⟨j,hj⟩ := branch_bad_parent parent branch means energy value hH N hbranch hroot hstep hlim hv
+    exact mem_iUnion.2 ⟨j,mem_iUnion.2 ⟨branch x (N+j),hj,x,hx,rfl⟩⟩
+  have hb (j : ℕ) (i : I (N+j)) :
+      P (event j i) ≤ ENNReal.ofReal (D*energy (N+j) i/(jumpSize H j)^2*(C₀*q^(N+j))) := by
+    by_cases hh : (jumpSize H j)^2 < D*energy (N+j) i
+    · have hs : event j i ⊆ {ω | ∃ x ∈ Ξ ω, branch x (N+j) = i} := fun _ h => h.2
+      refine (measure_mono hs).trans ((hhit _ _).trans (ENNReal.ofReal_le_ofReal ?_))
+      have hratio : 1 ≤ D*energy (N+j) i/(jumpSize H j)^2 :=
+        (le_div_iff₀ (sq_pos_of_pos (jumpSize_pos hH j))).2 (by simpa using hh.le)
+      exact le_mul_of_one_le_left (mul_nonneg hC (pow_nonneg hq _)) hratio
+    · have he : event j i = ∅ := by ext ω; simp [event,hh]
+      rw [he,measure_empty]
+      exact bot_le
+  have hlevel (j : ℕ) : P (⋃ i : I (N+j), event j i) ≤
+      ENNReal.ofReal (D*E/(jumpSize H j)^2*(C₀*q^(N+j))) := by
+    calc
+      _ ≤ ∑ i : I (N+j), P (event j i) := measure_iUnion_fintype_le _ _
+      _ ≤ ∑ i : I (N+j), ENNReal.ofReal (D*energy (N+j) i/(jumpSize H j)^2*(C₀*q^(N+j))) :=
+        Finset.sum_le_sum (fun i _ => hb j i)
+      _ = ENNReal.ofReal (∑ i : I (N+j), D*energy (N+j) i/(jumpSize H j)^2*(C₀*q^(N+j))) := by
+        symm
+        apply ENNReal.ofReal_sum_of_nonneg
+        intro i hi
+        exact mul_nonneg (div_nonneg (mul_nonneg hD (hnon _ _)) (sq_nonneg _))
+          (mul_nonneg hC (pow_nonneg hq _))
+      _ ≤ _ := by
+        apply ENNReal.ofReal_le_ofReal
+        simp only [← Finset.sum_mul,← Finset.sum_div,← Finset.mul_sum]
+        exact mul_le_mul_of_nonneg_right
+          (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left (hsum _) hD) (sq_nonneg _))
+          (mul_nonneg hC (pow_nonneg hq _))
+  let K := 64*D*C₀*E/H^2*q^N
+  have hK : 0 ≤ K := by dsimp [K]; positivity
+  have hs := (geometric_moment_summable hq hq1).mul_left K
+  calc
+    _ ≤ P (⋃ j : ℕ, ⋃ i : I (N+j), event j i) := measure_mono hcover
+    _ ≤ ∑' j : ℕ, P (⋃ i : I (N+j), event j i) := measure_iUnion_le _
+    _ ≤ ∑' j : ℕ, ENNReal.ofReal (D*E/(jumpSize H j)^2*(C₀*q^(N+j))) := ENNReal.tsum_le_tsum hlevel
+    _ = ∑' j : ℕ, ENNReal.ofReal (K*((((j:ℝ)+1)*((j:ℝ)+2))^2*q^j)) := by
+      apply tsum_congr
+      intro j
+      rw [jump_energy_scale hH.ne']
+    _ = ENNReal.ofReal (∑' j : ℕ, K*((((j:ℝ)+1)*((j:ℝ)+2))^2*q^j)) := by
+      symm
+      exact ENNReal.ofReal_tsum_of_nonneg (fun j => by positivity) hs
+    _ = _ := by rw [tsum_mul_left]
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+theorem strip_rectangle_hitting_discrete {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
+    (hmean : stripMean H q r Fbar Q ≤ H/2) (N : ℕ)
+    (hN : 3*rectCoefficient Q*(N:ℝ)*stripEnergy H q r Fbar Q ≤ (H/4)^2)
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) (Ξ : Ω → Set (ℝ × ℝ))
+    {C₀ ρ : ℝ} (hC : 0 ≤ C₀) (hρ : 0 ≤ ρ) (hρ1 : ρ < 1)
+    (hhit : ∀ n i, P {ω | ∃ x ∈ Ξ ω, x ∈ (dyadicRect Q n i).closed} ≤
+      ENNReal.ofReal (C₀*ρ^n)) :
+    P {ω | ∃ x ∈ Ξ ω, x ∈ Q.closed ∧
+      stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H} ≤
+      ENNReal.ofReal ((64*rectCoefficient Q*C₀*stripEnergy H q r Fbar Q/H^2*ρ^N)*
+        ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j) := by
+  let means := fun n i => stripMean H q r Fbar (dyadicRect Q n i)
+  let energy := fun n i => stripEnergy H q r Fbar (dyadicRect Q n i)
+  let value := fun x : Q.closed => stripPotential H q r Fbar ((x.val.1:ℂ)+(x.val.2:ℂ)*Complex.I)
+  let branch := fun x : Q.closed => dPointBranch Q x.val
+  let Ξ' := fun ω => {x : Q.closed | x.val ∈ Ξ ω}
+  have hroot (i : DIndex N) : means N i ≤ 3*H/4 := by
+    have hb := (strip_dyadic_mean_critical hF hinj hext hcont hH hr Q hlo hhi N i).trans hN
+    have hd := le_of_sq_le_sq hb (show 0 ≤ H/4 by positivity)
+    dsimp only [means]
+    linarith
+  have hstep (n : ℕ) (i : DIndex (n+1)) :
+      |means (n+1) i-means n (dParent n i)|^2 ≤ rectCoefficient Q*energy n (dParent n i) := by
+    rw [sq_abs]
+    exact strip_dyadic_mean_step hF hinj hext hcont hH hr Q hlo hhi n i
+  have hlim (x : Q.closed) : Tendsto (fun n => means n (branch x n)) atTop (𝓝 (value x)) :=
+    strip_dyadic_mean_limit hcont hH.le hr Q hlo hhi x.property
+  have hnon (n : ℕ) (i : DIndex n) : 0 ≤ energy n i := integral_nonneg (fun _ => sq_nonneg _)
+  have hE : 0 ≤ stripEnergy H q r Fbar Q := integral_nonneg (fun _ => sq_nonneg _)
+  have hsum (n : ℕ) : ∑ i, energy n i ≤ stripEnergy H q r Fbar Q := by
+    apply le_of_eq
+    exact dyadic_integral_sum Q (strip_gradient_rect_memLp hF hinj hext hH hr q Q.a Q.b hlo hhi).integrable_sq n
+  have hh (n : ℕ) (i : DIndex n) : P {ω | ∃ x ∈ Ξ' ω, branch x n = i} ≤ ENNReal.ofReal (C₀*ρ^n) := by
+    refine (measure_mono ?_).trans (hhit n i)
+    rintro ω ⟨x,hx,hi⟩
+    have hxmem := dPointBranch_mem x.property n
+    change x.val ∈ (dyadicRect Q n (branch x n)).closed at hxmem
+    rw [hi] at hxmem
+    exact ⟨x.val,hx,hxmem⟩
+  have ht := dyadic_energy_hitting dParent branch means energy value hH (rectCoefficient_pos Q).le hE
+    hC hρ hρ1 N (fun x n => dPointBranch_parent Q x.val n) hroot hstep hlim hnon hsum P Ξ' hh
+  refine (measure_mono ?_).trans ht
+  rintro ω ⟨x,hx,hxQ,hvalue⟩
+  exact ⟨⟨x,hxQ⟩,hx,hvalue⟩
+
+lemma geometric_floor_bound {ρ : ℝ} (hρ : 0 < ρ) (hρ1 : ρ < 1) (x : ℝ) :
+    ρ^Nat.floor x ≤ Real.exp (Real.log ρ*x)/ρ := by
+  have hh := Real.rpow_le_rpow_of_exponent_ge hρ hρ1.le (Nat.sub_one_lt_floor x).le
+  rw [Real.rpow_natCast] at hh
+  refine hh.trans_eq ?_
+  rw [Real.rpow_sub hρ,Real.rpow_one,Real.rpow_def_of_pos hρ]
+
+lemma critical_floor {D E H : ℝ} (hD : 0 < D) (hE : 0 < E) :
+    3*D*(Nat.floor (H^2/(48*D*E)):ℝ)*E ≤ (H/4)^2 := by
+  have hf := Nat.floor_le (show 0 ≤ H^2/(48*D*E) by positivity)
+  calc
+    _ ≤ 3*D*(H^2/(48*D*E))*E := by gcongr
+    _ = _ := by field_simp; ring
+
+theorem strip_rectangle_hitting {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
+    (hmean : stripMean H q r Fbar Q ≤ H/2)
+    {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) (Ξ : Ω → Set (ℝ × ℝ))
+    {C₀ ρ : ℝ} (hC : 0 ≤ C₀) (hρ : 0 < ρ) (hρ1 : ρ < 1)
+    (hhit : ∀ n i, P {ω | ∃ x ∈ Ξ ω, x ∈ (dyadicRect Q n i).closed} ≤
+      ENNReal.ofReal (C₀*ρ^n)) :
+    let D := rectCoefficient Q
+    let E := stripEnergy H q r Fbar Q
+    let T := ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j
+    P {ω | ∃ x ∈ Ξ ω, x ∈ Q.closed ∧
+      stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I) = H} ≤
+      ENNReal.ofReal ((64*D*(C₀+1)*(T+1)/ρ)*(E/H^2)*
+        Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)) := by
+  dsimp only
+  let D := rectCoefficient Q
+  let E := stripEnergy H q r Fbar Q
+  let T := ∑' j : ℕ, (((j:ℝ)+1)*((j:ℝ)+2))^2*ρ^j
+  have hD : 0 < D := rectCoefficient_pos Q
+  have hE0 : 0 ≤ E := integral_nonneg (fun _ => sq_nonneg _)
+  have hT : 0 ≤ T := tsum_nonneg (fun j => mul_nonneg (sq_nonneg _) (pow_nonneg hρ.le _))
+  by_cases hE : E = 0
+  · have ht := strip_rectangle_hitting_discrete hF hinj hext hcont hH hr q Q hlo hhi hmean 0
+      (by simp; positivity) P Ξ hC hρ.le hρ1 hhit
+    change P _ ≤ ENNReal.ofReal ((64*D*C₀*E/H^2*ρ^0)*T) at ht
+    change P _ ≤ ENNReal.ofReal ((64*D*(C₀+1)*(T+1)/ρ)*(E/H^2)*_)
+    simpa only [hE,mul_zero,zero_div,zero_mul,ENNReal.ofReal_zero] using ht
+  · have hEpos : 0 < E := lt_of_le_of_ne hE0 (Ne.symm hE)
+    let x := H^2/(48*D*E)
+    have ht := strip_rectangle_hitting_discrete hF hinj hext hcont hH hr q Q hlo hhi hmean (Nat.floor x)
+      (critical_floor hD hEpos) P Ξ hC hρ.le hρ1 hhit
+    refine ht.trans (ENNReal.ofReal_le_ofReal ?_)
+    change (64*D*C₀*E/H^2*ρ^Nat.floor x)*T ≤
+      (64*D*(C₀+1)*(T+1)/ρ)*(E/H^2)*Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)
+    have hexp : Real.log ρ*x = -((-Real.log ρ)/(48*D))*H^2/E := by dsimp [x]; ring
+    have hg := geometric_floor_bound hρ hρ1 x
+    rw [hexp] at hg
+    have hcoef : C₀*T ≤ (C₀+1)*(T+1) := by nlinarith
+    calc
+      _ ≤ (64*D*C₀*E/H^2*(Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)/ρ))*T := by gcongr
+      _ = (64*D/ρ)*(E/H^2)*Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)*(C₀*T) := by ring
+      _ ≤ (64*D/ρ)*(E/H^2)*Real.exp (-((-Real.log ρ)/(48*D))*H^2/E)*((C₀+1)*(T+1)) := by
+        exact mul_le_mul_of_nonneg_left hcoef (by positivity)
+      _ = _ := by ring
+
+def localizationMeanConstant : ℝ := slabMeanCoefficient+Real.pi+1
+
+lemma localizationMeanConstant_pos : 0 < localizationMeanConstant := by
+  unfold localizationMeanConstant
+  linarith [slabMeanCoefficient_pos,Real.pi_pos]
+
+def localizationStripWidth : ℝ := 1/(32*Real.pi*localizationMeanConstant)
+
+lemma localizationStripWidth_pos : 0 < localizationStripWidth := by
+  unfold localizationStripWidth
+  have hk := localizationMeanConstant_pos
+  positivity
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/FilledObstacle.lean b/lean/OAI/Probability/SLELocalization/FilledObstacle.lean
index 64352d34ed9d532a570e64170e17ae39de27768a..32e11a67f19c1aacbd4d289fb3ed8f09cee74cbd 100644
--- a/lean/OAI/Probability/SLELocalization/FilledObstacle.lean
+++ b/lean/OAI/Probability/SLELocalization/FilledObstacle.lean
@@ -1,149 +1,149 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-import OAI.Probability.SLELocalization.PastObstacle
-
-namespace OAI
-
-
-/-! Conformal localization and Filled Obstacle estimates. -/
-
-noncomputable section
-open Set Filter Topology
-open scoped Topology
-namespace SLEExactGauge
-
-lemma nearest_segment_subset_filling {A K : Set ℂ} (hAK : A ⊆ K)
-    (hfill : ∀ x ∈ K, connectedComponentIn Aᶜ x ⊆ K)
-    {x a : ℂ} (hx : x ∈ K) (ha : a ∈ A)
-    (hnear : ∀ b ∈ A, dist x a ≤ dist x b) :
-    _root_.segment ℝ x a ⊆ K := by
-  by_cases he : x = a
-  · subst a
-    simpa using singleton_subset_iff.mpr hx
-  have hd : 0 < dist x a := dist_pos.mpr he
-  let L : ℝ → ℂ := AffineMap.lineMap x a
-  have hc : Continuous L := (lipschitzWith_lineMap x a).continuous
-  have hLA : L '' Ico 0 1 ⊆ Aᶜ := by
-    rintro _ ⟨t,ht,rfl⟩ hb
-    have hh := hnear _ hb
-    rw [dist_left_lineMap,Real.norm_eq_abs,abs_of_nonneg ht.1] at hh
-    have ht1 := ht.2
-    nlinarith
-  have hxL : x ∈ L '' Ico 0 1 := ⟨0,by norm_num,by simp [L]⟩
-  have hLK : L '' Ico 0 1 ⊆ K := (isPreconnected_Ico.image _ hc.continuousOn
-    |>.subset_connectedComponentIn hxL hLA).trans (hfill x hx)
-  rw [segment_eq_image_lineMap]
-  rintro _ ⟨t,ht,rfl⟩
-  rcases lt_or_eq_of_le ht.2 with ht1 | rfl
-  · exact hLK ⟨t,⟨ht.1,ht1⟩,rfl⟩
-  · simpa using hAK ha
-
-lemma past_obstacle_subset_filled {γ : ℝ → ℂ} {T : ℝ} :
-    halfPlaneᶜ ∪ localizationSegment γ 0 T ⊆ (localizationDomain γ T)ᶜ := by
-  intro z hz hd
-  rcases hz with h | h
-  · exact h hd.1.1
-  · exact hd.1.2 h
-
-lemma past_filled_component_closed {γ : ℝ → ℂ} {T : ℝ} :
-    ∀ x ∈ (localizationDomain γ T)ᶜ,
-      connectedComponentIn (halfPlaneᶜ ∪ localizationSegment γ 0 T)ᶜ x ⊆
-        (localizationDomain γ T)ᶜ := by
-  intro x hx y hy hys
-  have he : (halfPlaneᶜ ∪ localizationSegment γ 0 T)ᶜ = halfPlane \ localizationSegment γ 0 T := by
-    ext z; simp
-  rw [he] at hy
-  have hxx : x ∈ halfPlane \ localizationSegment γ 0 T :=
-    (connectedComponentIn_nonempty_iff).mp ⟨y,hy⟩
-  apply hx
-  refine ⟨hxx,?_⟩
-  simpa only [connectedComponentIn_eq hy] using hys.2
-
-lemma locallyConnected_of_component_balls {K : Set ℂ}
-    (h : ∀ x ∈ K, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
-      ∀ y ∈ K, dist y x < δ → y ∈ connectedComponentIn (K ∩ Metric.ball x ε) x) :
-    LocallyConnectedSpace K := by
-  rw [locallyConnectedSpace_iff_connected_subsets]
-  intro x U hU
-  obtain ⟨ε,hε,hεU⟩ := Metric.mem_nhds_iff.mp hU
-  obtain ⟨δ,hδ,hh⟩ := h x x.property ε hε
-  let C := connectedComponentIn (K ∩ Metric.ball (x:ℂ) ε) (x:ℂ)
-  have hCK : C ⊆ K := (connectedComponentIn_subset _ _).trans inter_subset_left
-  refine ⟨Subtype.val ⁻¹' C,?_,?_,?_⟩
-  · apply Metric.mem_nhds_iff.mpr
-    exact ⟨δ,hδ,fun y hy => hh y y.property hy⟩
-  · apply IsInducing.subtypeVal.isPreconnected_image.mp
-    simpa only [Subtype.image_preimage_coe,inter_eq_right.mpr hCK] using
-      (isPreconnected_connectedComponentIn : IsPreconnected C)
-  · intro y hy
-    exact hεU ((connectedComponentIn_subset _ _ hy).2)
-
-lemma locallyConnected_filling {A K : Set ℂ} (hA : IsClosed A)
-    [LocallyConnectedSpace A] (hAK : A ⊆ K)
-    (hfill : ∀ x ∈ K, connectedComponentIn Aᶜ x ⊆ K) :
-    LocallyConnectedSpace K := by
-  apply locallyConnected_of_component_balls
-  intro x hx ε hε
-  by_cases hxA : x ∈ A
-  · let xA : A := ⟨x,hxA⟩
-    obtain ⟨V,hV,hVc,hVε⟩ := locallyConnectedSpace_iff_connected_subsets.1
-      (inferInstance : LocallyConnectedSpace A) xA (Metric.ball xA ε)
-      (Metric.ball_mem_nhds _ hε)
-    obtain ⟨r,hr,hrV⟩ := Metric.mem_nhds_iff.mp hV
-    let C : Set ℂ := Subtype.val '' V
-    have hCA : C ⊆ A := Subtype.coe_image_subset _ _
-    have hCε : C ⊆ Metric.ball x ε := by
-      rintro _ ⟨v,hv,rfl⟩
-      exact hVε hv
-    have hxC : x ∈ C := ⟨xA,mem_of_mem_nhds hV,rfl⟩
-    have hCc : IsPreconnected C := hVc.image _ continuous_subtype_val.continuousOn
-    refine ⟨min r ε / 4,by positivity,?_⟩
-    intro y hy hyx
-    obtain ⟨a,ha,hea⟩ := hA.exists_infDist_eq_dist ⟨x,hxA⟩ y
-    have hnear : ∀ b ∈ A, dist y a ≤ dist y b := by
-      intro b hb
-      rw [← hea]
-      exact Metric.infDist_le_dist_of_mem hb
-    have hya : dist y a ≤ dist y x := hnear x hxA
-    have hax : dist a x < r := by
-      have hh := dist_triangle a y x
-      rw [dist_comm a y] at hh
-      have hm := min_le_left r ε
-      linarith
-    have haC : a ∈ C := ⟨⟨a,ha⟩,hrV hax,rfl⟩
-    have hyε : y ∈ Metric.ball x ε := by
-      have hm := min_le_right r ε
-      change dist y x < ε
-      linarith
-    let S := _root_.segment ℝ y a
-    have hSK : S ⊆ K := nearest_segment_subset_filling hAK hfill hy ha hnear
-    have hSε : S ⊆ Metric.ball x ε := (convex_ball x ε).segment_subset hyε (hCε haC)
-    have hUc : IsPreconnected (C ∪ S) := hCc.union a haC (right_mem_segment ℝ y a) (convex_segment y a).isPreconnected
-    exact hUc.subset_connectedComponentIn (F := K ∩ Metric.ball x ε) (Or.inl hxC)
-      (union_subset (fun _ hz => ⟨hAK (hCA hz),hCε hz⟩)
-        (fun _ hz => ⟨hSK hz,hSε hz⟩)) (Or.inr (left_mem_segment ℝ y a))
-  · obtain ⟨r,hr,hrA⟩ := Metric.isOpen_iff.mp hA.isOpen_compl x hxA
-    refine ⟨min r ε,lt_min hr hε,?_⟩
-    intro y hy hyx
-    let B := Metric.ball x (min r ε)
-    have hxB : x ∈ B := Metric.mem_ball_self (lt_min hr hε)
-    have hBA : B ⊆ Aᶜ := (Metric.ball_subset_ball (min_le_left _ _)).trans hrA
-    have hBK : B ⊆ K := ((convex_ball x (min r ε)).isPreconnected.subset_connectedComponentIn
-      hxB hBA).trans (hfill x hx)
-    exact (convex_ball x (min r ε)).isPreconnected.subset_connectedComponentIn
-      (F := K ∩ Metric.ball x ε) hxB
-      (fun _ hz => ⟨hBK hz,(Metric.ball_subset_ball (min_le_right _ _)) hz⟩) hyx
-
-theorem past_filled_locallyConnected {γ : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
-    (hc : ContinuousOn γ (Icc 0 T)) :
-    LocallyConnectedSpace (↑((localizationDomain γ T)ᶜ) : Type) := by
-  have := past_obstacle_locallyConnected hT hc
-  exact locallyConnected_filling
-    (halfPlane_open.isClosed_compl.union (past_trace_continuum hT hc).1.isClosed)
-    past_obstacle_subset_filled past_filled_component_closed
-
-end SLEExactGauge
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+import OAI.Probability.SLELocalization.PastObstacle
+
+namespace OAI
+
+
+/-! Conformal localization and Filled Obstacle estimates. -/
+
+noncomputable section
+open Set Filter Topology
+open scoped Topology
+namespace SLEExactGauge
+
+lemma nearest_segment_subset_filling {A K : Set ℂ} (hAK : A ⊆ K)
+    (hfill : ∀ x ∈ K, connectedComponentIn Aᶜ x ⊆ K)
+    {x a : ℂ} (hx : x ∈ K) (ha : a ∈ A)
+    (hnear : ∀ b ∈ A, dist x a ≤ dist x b) :
+    _root_.segment ℝ x a ⊆ K := by
+  by_cases he : x = a
+  · subst a
+    simpa using singleton_subset_iff.mpr hx
+  have hd : 0 < dist x a := dist_pos.mpr he
+  let L : ℝ → ℂ := AffineMap.lineMap x a
+  have hc : Continuous L := (lipschitzWith_lineMap x a).continuous
+  have hLA : L '' Ico 0 1 ⊆ Aᶜ := by
+    rintro _ ⟨t,ht,rfl⟩ hb
+    have hh := hnear _ hb
+    rw [dist_left_lineMap,Real.norm_eq_abs,abs_of_nonneg ht.1] at hh
+    have ht1 := ht.2
+    nlinarith
+  have hxL : x ∈ L '' Ico 0 1 := ⟨0,by norm_num,by simp [L]⟩
+  have hLK : L '' Ico 0 1 ⊆ K := (isPreconnected_Ico.image _ hc.continuousOn
+    |>.subset_connectedComponentIn hxL hLA).trans (hfill x hx)
+  rw [segment_eq_image_lineMap]
+  rintro _ ⟨t,ht,rfl⟩
+  rcases lt_or_eq_of_le ht.2 with ht1 | rfl
+  · exact hLK ⟨t,⟨ht.1,ht1⟩,rfl⟩
+  · simpa using hAK ha
+
+lemma past_obstacle_subset_filled {γ : ℝ → ℂ} {T : ℝ} :
+    halfPlaneᶜ ∪ localizationSegment γ 0 T ⊆ (localizationDomain γ T)ᶜ := by
+  intro z hz hd
+  rcases hz with h | h
+  · exact h hd.1.1
+  · exact hd.1.2 h
+
+lemma past_filled_component_closed {γ : ℝ → ℂ} {T : ℝ} :
+    ∀ x ∈ (localizationDomain γ T)ᶜ,
+      connectedComponentIn (halfPlaneᶜ ∪ localizationSegment γ 0 T)ᶜ x ⊆
+        (localizationDomain γ T)ᶜ := by
+  intro x hx y hy hys
+  have he : (halfPlaneᶜ ∪ localizationSegment γ 0 T)ᶜ = halfPlane \ localizationSegment γ 0 T := by
+    ext z; simp
+  rw [he] at hy
+  have hxx : x ∈ halfPlane \ localizationSegment γ 0 T :=
+    (connectedComponentIn_nonempty_iff).mp ⟨y,hy⟩
+  apply hx
+  refine ⟨hxx,?_⟩
+  simpa only [connectedComponentIn_eq hy] using hys.2
+
+lemma locallyConnected_of_component_balls {K : Set ℂ}
+    (h : ∀ x ∈ K, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧
+      ∀ y ∈ K, dist y x < δ → y ∈ connectedComponentIn (K ∩ Metric.ball x ε) x) :
+    LocallyConnectedSpace K := by
+  rw [locallyConnectedSpace_iff_connected_subsets]
+  intro x U hU
+  obtain ⟨ε,hε,hεU⟩ := Metric.mem_nhds_iff.mp hU
+  obtain ⟨δ,hδ,hh⟩ := h x x.property ε hε
+  let C := connectedComponentIn (K ∩ Metric.ball (x:ℂ) ε) (x:ℂ)
+  have hCK : C ⊆ K := (connectedComponentIn_subset _ _).trans inter_subset_left
+  refine ⟨Subtype.val ⁻¹' C,?_,?_,?_⟩
+  · apply Metric.mem_nhds_iff.mpr
+    exact ⟨δ,hδ,fun y hy => hh y y.property hy⟩
+  · apply IsInducing.subtypeVal.isPreconnected_image.mp
+    simpa only [Subtype.image_preimage_coe,inter_eq_right.mpr hCK] using
+      (isPreconnected_connectedComponentIn : IsPreconnected C)
+  · intro y hy
+    exact hεU ((connectedComponentIn_subset _ _ hy).2)
+
+lemma locallyConnected_filling {A K : Set ℂ} (hA : IsClosed A)
+    [LocallyConnectedSpace A] (hAK : A ⊆ K)
+    (hfill : ∀ x ∈ K, connectedComponentIn Aᶜ x ⊆ K) :
+    LocallyConnectedSpace K := by
+  apply locallyConnected_of_component_balls
+  intro x hx ε hε
+  by_cases hxA : x ∈ A
+  · let xA : A := ⟨x,hxA⟩
+    obtain ⟨V,hV,hVc,hVε⟩ := locallyConnectedSpace_iff_connected_subsets.1
+      (inferInstance : LocallyConnectedSpace A) xA (Metric.ball xA ε)
+      (Metric.ball_mem_nhds _ hε)
+    obtain ⟨r,hr,hrV⟩ := Metric.mem_nhds_iff.mp hV
+    let C : Set ℂ := Subtype.val '' V
+    have hCA : C ⊆ A := Subtype.coe_image_subset _ _
+    have hCε : C ⊆ Metric.ball x ε := by
+      rintro _ ⟨v,hv,rfl⟩
+      exact hVε hv
+    have hxC : x ∈ C := ⟨xA,mem_of_mem_nhds hV,rfl⟩
+    have hCc : IsPreconnected C := hVc.image _ continuous_subtype_val.continuousOn
+    refine ⟨min r ε / 4,by positivity,?_⟩
+    intro y hy hyx
+    obtain ⟨a,ha,hea⟩ := hA.exists_infDist_eq_dist ⟨x,hxA⟩ y
+    have hnear : ∀ b ∈ A, dist y a ≤ dist y b := by
+      intro b hb
+      rw [← hea]
+      exact Metric.infDist_le_dist_of_mem hb
+    have hya : dist y a ≤ dist y x := hnear x hxA
+    have hax : dist a x < r := by
+      have hh := dist_triangle a y x
+      rw [dist_comm a y] at hh
+      have hm := min_le_left r ε
+      linarith
+    have haC : a ∈ C := ⟨⟨a,ha⟩,hrV hax,rfl⟩
+    have hyε : y ∈ Metric.ball x ε := by
+      have hm := min_le_right r ε
+      change dist y x < ε
+      linarith
+    let S := _root_.segment ℝ y a
+    have hSK : S ⊆ K := nearest_segment_subset_filling hAK hfill hy ha hnear
+    have hSε : S ⊆ Metric.ball x ε := (convex_ball x ε).segment_subset hyε (hCε haC)
+    have hUc : IsPreconnected (C ∪ S) := hCc.union a haC (right_mem_segment ℝ y a) (convex_segment y a).isPreconnected
+    exact hUc.subset_connectedComponentIn (F := K ∩ Metric.ball x ε) (Or.inl hxC)
+      (union_subset (fun _ hz => ⟨hAK (hCA hz),hCε hz⟩)
+        (fun _ hz => ⟨hSK hz,hSε hz⟩)) (Or.inr (left_mem_segment ℝ y a))
+  · obtain ⟨r,hr,hrA⟩ := Metric.isOpen_iff.mp hA.isOpen_compl x hxA
+    refine ⟨min r ε,lt_min hr hε,?_⟩
+    intro y hy hyx
+    let B := Metric.ball x (min r ε)
+    have hxB : x ∈ B := Metric.mem_ball_self (lt_min hr hε)
+    have hBA : B ⊆ Aᶜ := (Metric.ball_subset_ball (min_le_left _ _)).trans hrA
+    have hBK : B ⊆ K := ((convex_ball x (min r ε)).isPreconnected.subset_connectedComponentIn
+      hxB hBA).trans (hfill x hx)
+    exact (convex_ball x (min r ε)).isPreconnected.subset_connectedComponentIn
+      (F := K ∩ Metric.ball x ε) hxB
+      (fun _ hz => ⟨hBK hz,(Metric.ball_subset_ball (min_le_right _ _)) hz⟩) hyx
+
+theorem past_filled_locallyConnected {γ : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
+    (hc : ContinuousOn γ (Icc 0 T)) :
+    LocallyConnectedSpace (↑((localizationDomain γ T)ᶜ) : Type) := by
+  have := past_obstacle_locallyConnected hT hc
+  exact locallyConnected_filling
+    (halfPlane_open.isClosed_compl.union (past_trace_continuum hT hc).1.isClosed)
+    past_obstacle_subset_filled past_filled_component_closed
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/FiniteClock.lean b/lean/OAI/Probability/SLELocalization/FiniteClock.lean
index 0bee8f499ce822b5b13b87d406856bbb46516eab..b1ea39ac6d663a2f301ac3ad77d7195c50f18e83 100644
--- a/lean/OAI/Probability/SLELocalization/FiniteClock.lean
+++ b/lean/OAI/Probability/SLELocalization/FiniteClock.lean
@@ -1,254 +1,254 @@
-import Mathlib
-import OAI.Probability.SLELocalization.LoewnerExit
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Finite Clock estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.FiniteClock
-
-lemma live_density_below {κ : ℝ} (hκ : 0 < κ ∧ κ < 8)
-    {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
-    (hc : Continuous B) {z : ℂ} {s u δ : ℝ}
-    (hs : 0 ≤ s) (hz : z ∈ localizationDomain γ s)
-    (hclock : Real.log (2*z.im)-Real.log (loewnerRadius g s z) = u)
-    (hangle : loewnerAngleSine κ B g s z < angleFloor δ u) :
-    finiteStoppedDensity κ B γ g z u ∈ Ioo 0 (threshold κ z δ u) := by
-  have hU : ContinuousOn (fun t : ℝ => Real.sqrt κ*B t.toNNReal) (Ici 0) := (continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn
-  have hy : 0 < z.im := hz.1.1
-  have hr := hm.radius_pos hs hz
-  have hrad : loewnerRadius g s z = (2*z.im)*Real.exp (-u) := by
-    have he : Real.log (loewnerRadius g s z) = Real.log (2*z.im)+(-u) := by linarith
-    have hh := congrArg Real.exp he
-    simpa only [Real.exp_log hr,Real.exp_add,Real.exp_log (show 0<2*z.im by positivity)] using hh
-  have hstop := hm.firstLiveRadiusStop_eq hU (t := ⟨s,hs⟩) hz hrad
-  have hsin : 0 < loewnerAngleSine κ B g s z := by
-    apply div_pos (hm.im_pos hs hz)
-    simpa only [Complex.ofReal_mul] using (norm_pos_iff.2 (hm.centered_ne_zero hs hz))
-  have hRp := Real.rpow_pos_of_pos hr (-codimension κ)
-  have hval : 0 < (loewnerRadius g s z)^(-codimension κ)*(loewnerAngleSine κ B g s z)^(8/κ-1) :=
-    mul_pos hRp (Real.rpow_pos_of_pos hsin _)
-  have hlt : (loewnerRadius g s z)^(-codimension κ)*(loewnerAngleSine κ B g s z)^(8/κ-1) <
-      ((2*z.im)*Real.exp (-u))^(-codimension κ)*(angleFloor δ u)^(8/κ-1) := by
-    rw [← hrad]
-    exact mul_lt_mul_of_pos_left (Real.rpow_lt_rpow hsin.le hangle (exponent_pos hκ)) hRp
-  simp only [finiteStoppedDensity,hstop]
-  exact ⟨ENNReal.ofReal_pos.2 hval,ENNReal.ofReal_lt_ofReal_iff (hval.trans hlt) |>.2 hlt⟩
-
-lemma exponential_clock_bound {T δ Y₀ : ℝ} {Y C Q : ℝ → ℝ}
-    (hT : 0 ≤ T) (hδ : 0 < δ) (hY₀ : 0 < Y₀)
-    (hYinit : Y 0 = Y₀) (hCinit : C 0 = 0)
-    (hY : ContinuousOn Y (Icc 0 T)) (hC : ContinuousOn C (Icc 0 T))
-    (_hYpos : ∀ t ∈ Icc 0 T, 0 < Y t)
-    (hQpos : ∀ t ∈ Ioo 0 T, 0 < Q t)
-    (hYQ : ∀ t ∈ Ioo 0 T, (Y t)^2 ≤ Q t)
-    (hangle : ∀ t ∈ Ioo 0 T, (angleFloor δ (C t))^2 * Q t ≤ (Y t)^2)
-    (hdY : ∀ t ∈ Ioo 0 T, HasDerivAt Y (-2*Y t/Q t) t)
-    (hdC : ∀ t ∈ Ioo 0 T, HasDerivAt C (4*(Y t)^2/(Q t)^2) t) :
-    T ≤ Y₀^2/(2*δ^4) := by
-  have hh : AntitoneOn (fun t => (Y t)^2 * Real.exp (C t)) (Icc 0 T) := by
-    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T)
-      ((hY.pow 2).mul (Real.continuous_exp.comp_continuousOn hC))
-      (f' := fun t => (2*Y t*(-2*Y t/Q t))*Real.exp (C t) +
-        (Y t)^2*(Real.exp (C t)*(4*(Y t)^2/(Q t)^2)))
-    · intro t ht
-      rw [interior_Icc] at ht
-      convert (((hdY t ht).pow 2).mul (hdC t ht).exp).hasDerivWithinAt
-        (s := interior (Icc 0 T)) using 1 <;> first | rfl | (simp only [Pi.pow_apply]; ring)
-    · intro t ht
-      rw [interior_Icc] at ht
-      have hq := hQpos t ht
-      apply (mul_le_mul_iff_of_pos_right (sq_pos_of_pos hq)).1
-      have hd := hYQ t ht
-      have hb := mul_nonpos_of_nonneg_of_nonpos
-        (show 0 ≤ 4*(Y t)^2*Real.exp (C t) by positivity) (sub_nonpos.2 hd)
-      field_simp
-      nlinarith [hb]
-  have height (t : ℝ) (ht : t ∈ Icc 0 T) :
-      (Y t)^2 ≤ Y₀^2*(Real.exp (-C t/2))^2 := by
-    have hh' := hh ⟨le_rfl,hT⟩ ht ht.1
-    simp only [hYinit,hCinit,Real.exp_zero,mul_one] at hh'
-    have he : Real.exp (C t)*(Real.exp (-C t/2))^2 = 1 := by
-      rw [pow_two,← Real.exp_add,← Real.exp_add]
-      convert Real.exp_zero using 1; congr 1; ring
-    have hb := mul_le_mul_of_nonneg_right hh' (sq_nonneg (Real.exp (-C t/2)))
-    rwa [mul_assoc,he,mul_one] at hb
-  have rate (t : ℝ) (ht : t ∈ Ioo 0 T) :
-      2*δ^4/Y₀^2 ≤ (4*(Y t)^2/(Q t)^2)/2*Real.exp (-C t/2) := by
-    have hq := hQpos t ht
-    have hD := Real.exp_pos (-C t/2)
-    have ha := pow_le_pow_left₀
-      (mul_nonneg (sq_nonneg _) hq.le) (hangle t ht) 2
-    have hE : (Real.exp (-C t/8))^4 = Real.exp (-C t/2) := by
-      rw [← Real.exp_nat_mul]
-      congr 1
-      ring
-    have ha' : δ^4*Real.exp (-C t/2)*(Q t)^2 ≤ (Y t)^4 := by
-      simpa only [angleFloor,mul_pow,← pow_mul,show 2*2=4 by norm_num,hE] using ha
-    have hb := mul_le_mul_of_nonneg_right (height t ⟨ht.1.le,ht.2.le⟩) (sq_nonneg (Y t))
-    have hc : δ^4*(Q t)^2 ≤ Y₀^2*Real.exp (-C t/2)*(Y t)^2 := by
-      apply (mul_le_mul_iff_of_pos_left hD).1
-      nlinarith [ha',hb]
-    apply (div_le_iff₀ (sq_pos_of_pos hY₀)).2
-    apply (mul_le_mul_iff_of_pos_right (sq_pos_of_pos hq)).1
-    field_simp
-    simp only [neg_div] at hc
-    nlinarith [hc]
-  have hK : 0 < 2*δ^4/Y₀^2 := by positivity
-  have hm : AntitoneOn (fun t => Real.exp (-C t/2)+(2*δ^4/Y₀^2)*t) (Icc 0 T) := by
-    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T)
-      ((Real.continuous_exp.comp_continuousOn (hC.neg.div_const 2)).add (continuousOn_id.const_mul _))
-      (f' := fun t => Real.exp (-C t/2)*(-(4*(Y t)^2/(Q t)^2)/2)+2*δ^4/Y₀^2)
-    · intro t ht
-      rw [interior_Icc] at ht
-      convert (((hdC t ht).neg.div_const 2).exp.add
-          ((hasDerivAt_id t).const_mul (2*δ^4/Y₀^2))).hasDerivWithinAt
-        (s := interior (Icc 0 T)) using 1 <;> first | rfl | (simp only [Pi.neg_apply]; ring)
-    · intro t ht
-      rw [interior_Icc] at ht
-      nlinarith [rate t ht]
-  have hb := hm ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
-  simp only [hCinit,neg_zero,zero_div,Real.exp_zero,mul_zero,add_zero] at hb
-  have ht : T ≤ 1/(2*δ^4/Y₀^2) := (le_div_iff₀ hK).2 (by nlinarith [Real.exp_pos (-C T/2)])
-  convert ht using 1; field_simp
-
-lemma good_physical_clock {κ : ℝ} (hκ : 0 < κ ∧ κ < 8)
-    {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
-    (hc : Continuous B) {z : ℂ} (hz : z ∈ halfPlane) {δ v : ℝ}
-    (hδ : 0 < δ) (hv : 0 < v)
-    (hhit : firstLiveRadiusStop γ g z v ≠ (⊤ : ℝ≥0∞))
-    (hgood : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ v →
-      finiteStoppedDensity κ B γ g z q ∉ Ioo 0 (threshold κ z δ q)) :
-    firstLiveRadiusStop γ g z v ≤ ENNReal.ofReal (z.im^2/(2*δ^4)) := by
-  have hUc : Continuous (fun t : ℝ => Real.sqrt κ*B t.toNNReal) :=
-    continuous_const.mul (hc.comp continuous_real_toNNReal)
-  have hU := hUc.continuousOn (s := Ici 0)
-  obtain ⟨T,hstop,hzT,hrT⟩ := RadiusPast.live_radius_witness hm hU hhit
-  have hT : 0 < (T:ℝ) := by
-    have hp := hm.firstLiveRadiusStop_pos hU hz hv
-    rw [hstop] at hp
-    exact ENNReal.coe_pos.1 hp
-  let C := fun t => Real.log (2*z.im)-Real.log (loewnerRadius g t z)
-  have hC : ContinuousOn C (Icc 0 T) :=
-    continuousOn_const.sub (hm.continuous_log_radius T.coe_nonneg (hU.mono Icc_subset_Ici_self) hzT)
-  have hCs : StrictMonoOn C (Icc 0 T) := by
-    intro a ha b hb hab
-    have hh := hm.log_radius_strictAnti T.coe_nonneg (hU.mono Icc_subset_Ici_self) hzT ha hb hab
-    dsimp [C]
-    linarith
-  have hC0 : C 0 = 0 := by simp only [C,hm.radius_initial hz,sub_self]
-  have hCT : C T = v := by
-    have hy : 0 < 2*z.im := by have : 0<z.im := hz; positivity
-    dsimp [C]
-    rw [hrT,Real.log_mul hy.ne' (Real.exp_ne_zero _),Real.log_exp]
-    ring
-  have hg := hm.continuous_eval T.coe_nonneg hzT
-  have hsine : ContinuousOn (fun t => loewnerAngleSine κ B g t z) (Icc 0 T) := by
-    have hh := (Complex.continuous_im.comp_continuousOn hg).div
-      (hg.sub (Complex.continuous_ofReal.comp hUc).continuousOn).norm
-      (fun t ht => norm_ne_zero_iff.2 (hm.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hzT)))
-    convert hh using 1; first | rfl | (funext t; simp only [loewnerAngleSine,Complex.ofReal_mul,Function.comp_def,Pi.div_apply,Pi.sub_apply])
-  have hangle : ∀ t : ℝ, t ∈ Ioo 0 (T:ℝ) → angleFloor δ (C t) ≤ loewnerAngleSine κ B g t z := by
-    have hF : ContinuousOn (fun t => loewnerAngleSine κ B g t z-angleFloor δ (C t)) (Icc 0 T) :=
-      hsine.sub (continuousOn_const.mul (Real.continuous_exp.comp_continuousOn (hC.neg.div_const 8)))
-    have hn := rational_clock_nonneg hT hC hCs hC0 hCT hF ?_
-    · intro t ht
-      exact sub_nonneg.1 (hn t ht)
-    · intro q hq hqv s hs hseq
-      apply sub_nonneg.2
-      by_contra hn
-      exact hgood q hq hqv (live_density_below hκ hm hc hs.1
-        (survivingDomain_mono hs.2 hzT) hseq (by rw [← hseq]; exact lt_of_not_ge hn))
-  have hbound : (T:ℝ) ≤ z.im^2/(2*δ^4) := by
-    apply exponential_clock_bound (Y := fun t => (g t z).im) (C := C)
-      (Q := fun t => Complex.normSq (g t z-((Real.sqrt κ*B t.toNNReal : ℝ) : ℂ)))
-      T.coe_nonneg hδ (show 0<z.im from hz) (hm.initial z hz |>.symm ▸ rfl) hC0
-      (Complex.continuous_im.comp_continuousOn hg) hC
-    · intro t ht
-      exact hm.im_pos ht.1 (survivingDomain_mono ht.2 hzT)
-    · intro t ht
-      exact Complex.normSq_pos.2 (hm.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hzT))
-    · intro t ht
-      simp only [Complex.normSq_apply,Complex.sub_im,Complex.ofReal_im,sub_zero]
-      nlinarith [sq_nonneg ((g t z-((Real.sqrt κ*B t.toNNReal : ℝ) : ℂ)).re)]
-    · intro t ht
-      have hp := norm_pos_iff.2 (hm.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hzT))
-      have hs := hangle t ht
-      simp only [loewnerAngleSine,← Complex.ofReal_mul] at hs
-      have ha := (le_div_iff₀ hp).1 hs
-      have hb := pow_le_pow_left₀ (mul_nonneg (angleFloor_pos hδ _).le hp.le) ha 2
-      simpa only [mul_pow,Complex.normSq_eq_norm_sq] using hb
-    · intro t ht
-      exact hm.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hzT)
-    · intro t ht
-      apply ((hm.hasDerivAt_log_radius T.coe_nonneg (hU.mono Icc_subset_Ici_self) hzT ht).const_sub
-        (Real.log (2*z.im))).congr_deriv
-      ring
-  rw [hstop]
-  simpa only [ENNReal.ofReal_coe_nnreal] using (ENNReal.ofReal_le_ofReal hbound)
-
-end SLEExactGauge.FiniteClock
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteClock
-
-theorem finiteCenteredPhysicalTime_tight {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (ε : ℝ) (hε : 0 < ε) :
-    ∃ T : ℝ, 0 < T ∧ ∀ u : ℝ, 1 ≤ u →
-      (finiteCenteredMeasure κ P B γ g z u)
-        {sample | ENNReal.ofReal T < firstLiveRadiusStop (γ sample) (g sample) z u}
-        < ENNReal.ofReal ε := by
-  obtain ⟨D,hD,hbound⟩ := bad_bound κ hκ B hB γ g f hγ hmaps z hz
-  obtain ⟨δ,hδ,hsmall⟩ := exists_small_floor (exponent_pos hκ) hD hε
-  refine ⟨z.im^2/(2*δ^4),by have hy : 0<z.im := hz; positivity,?_⟩
-  intro u hu
-  have hu0 : 0 < u := by linarith
-  let Q := finiteCenteredMeasure κ P B γ g z u
-  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
-  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
-    change ∀ᵐ sample ∂P.withDensity _,_
-    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
-    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
-  have hsub : {sample | ENNReal.ofReal (z.im^2/(2*δ^4)) <
-        firstLiveRadiusStop (γ sample) (g sample) z u} ≤ᵐ[Q] bad κ B γ g z δ u := by
-    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hm hc hh hx
-    by_contra hn
-    have hg : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ u →
-        finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z q ∉ Ioo 0 (threshold κ z δ q) := by
-      intro q hq hqu hbad
-      exact hn (mem_bad_of_rational κ B γ g z δ u q hq hqu hbad)
-    exact not_lt_of_ge (good_physical_clock (κ := κ) (B := fun t => B t sample) (γ := γ sample)
-      (g := g sample) (f := f sample) hκ hm hc hz hδ hu0 hh hg) hx
-  exact (measure_mono_ae hsub).trans_lt
-    ((hbound δ hδ u).trans_lt (ENNReal.ofReal_lt_ofReal_iff hε |>.2 hsmall))
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.LoewnerExit
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Finite Clock estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.FiniteClock
+
+lemma live_density_below {κ : ℝ} (hκ : 0 < κ ∧ κ < 8)
+    {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
+    (hc : Continuous B) {z : ℂ} {s u δ : ℝ}
+    (hs : 0 ≤ s) (hz : z ∈ localizationDomain γ s)
+    (hclock : Real.log (2*z.im)-Real.log (loewnerRadius g s z) = u)
+    (hangle : loewnerAngleSine κ B g s z < angleFloor δ u) :
+    finiteStoppedDensity κ B γ g z u ∈ Ioo 0 (threshold κ z δ u) := by
+  have hU : ContinuousOn (fun t : ℝ => Real.sqrt κ*B t.toNNReal) (Ici 0) := (continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn
+  have hy : 0 < z.im := hz.1.1
+  have hr := hm.radius_pos hs hz
+  have hrad : loewnerRadius g s z = (2*z.im)*Real.exp (-u) := by
+    have he : Real.log (loewnerRadius g s z) = Real.log (2*z.im)+(-u) := by linarith
+    have hh := congrArg Real.exp he
+    simpa only [Real.exp_log hr,Real.exp_add,Real.exp_log (show 0<2*z.im by positivity)] using hh
+  have hstop := hm.firstLiveRadiusStop_eq hU (t := ⟨s,hs⟩) hz hrad
+  have hsin : 0 < loewnerAngleSine κ B g s z := by
+    apply div_pos (hm.im_pos hs hz)
+    simpa only [Complex.ofReal_mul] using (norm_pos_iff.2 (hm.centered_ne_zero hs hz))
+  have hRp := Real.rpow_pos_of_pos hr (-codimension κ)
+  have hval : 0 < (loewnerRadius g s z)^(-codimension κ)*(loewnerAngleSine κ B g s z)^(8/κ-1) :=
+    mul_pos hRp (Real.rpow_pos_of_pos hsin _)
+  have hlt : (loewnerRadius g s z)^(-codimension κ)*(loewnerAngleSine κ B g s z)^(8/κ-1) <
+      ((2*z.im)*Real.exp (-u))^(-codimension κ)*(angleFloor δ u)^(8/κ-1) := by
+    rw [← hrad]
+    exact mul_lt_mul_of_pos_left (Real.rpow_lt_rpow hsin.le hangle (exponent_pos hκ)) hRp
+  simp only [finiteStoppedDensity,hstop]
+  exact ⟨ENNReal.ofReal_pos.2 hval,ENNReal.ofReal_lt_ofReal_iff (hval.trans hlt) |>.2 hlt⟩
+
+lemma exponential_clock_bound {T δ Y₀ : ℝ} {Y C Q : ℝ → ℝ}
+    (hT : 0 ≤ T) (hδ : 0 < δ) (hY₀ : 0 < Y₀)
+    (hYinit : Y 0 = Y₀) (hCinit : C 0 = 0)
+    (hY : ContinuousOn Y (Icc 0 T)) (hC : ContinuousOn C (Icc 0 T))
+    (_hYpos : ∀ t ∈ Icc 0 T, 0 < Y t)
+    (hQpos : ∀ t ∈ Ioo 0 T, 0 < Q t)
+    (hYQ : ∀ t ∈ Ioo 0 T, (Y t)^2 ≤ Q t)
+    (hangle : ∀ t ∈ Ioo 0 T, (angleFloor δ (C t))^2 * Q t ≤ (Y t)^2)
+    (hdY : ∀ t ∈ Ioo 0 T, HasDerivAt Y (-2*Y t/Q t) t)
+    (hdC : ∀ t ∈ Ioo 0 T, HasDerivAt C (4*(Y t)^2/(Q t)^2) t) :
+    T ≤ Y₀^2/(2*δ^4) := by
+  have hh : AntitoneOn (fun t => (Y t)^2 * Real.exp (C t)) (Icc 0 T) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T)
+      ((hY.pow 2).mul (Real.continuous_exp.comp_continuousOn hC))
+      (f' := fun t => (2*Y t*(-2*Y t/Q t))*Real.exp (C t) +
+        (Y t)^2*(Real.exp (C t)*(4*(Y t)^2/(Q t)^2)))
+    · intro t ht
+      rw [interior_Icc] at ht
+      convert (((hdY t ht).pow 2).mul (hdC t ht).exp).hasDerivWithinAt
+        (s := interior (Icc 0 T)) using 1 <;> first | rfl | (simp only [Pi.pow_apply]; ring)
+    · intro t ht
+      rw [interior_Icc] at ht
+      have hq := hQpos t ht
+      apply (mul_le_mul_iff_of_pos_right (sq_pos_of_pos hq)).1
+      have hd := hYQ t ht
+      have hb := mul_nonpos_of_nonneg_of_nonpos
+        (show 0 ≤ 4*(Y t)^2*Real.exp (C t) by positivity) (sub_nonpos.2 hd)
+      field_simp
+      nlinarith [hb]
+  have height (t : ℝ) (ht : t ∈ Icc 0 T) :
+      (Y t)^2 ≤ Y₀^2*(Real.exp (-C t/2))^2 := by
+    have hh' := hh ⟨le_rfl,hT⟩ ht ht.1
+    simp only [hYinit,hCinit,Real.exp_zero,mul_one] at hh'
+    have he : Real.exp (C t)*(Real.exp (-C t/2))^2 = 1 := by
+      rw [pow_two,← Real.exp_add,← Real.exp_add]
+      convert Real.exp_zero using 1; congr 1; ring
+    have hb := mul_le_mul_of_nonneg_right hh' (sq_nonneg (Real.exp (-C t/2)))
+    rwa [mul_assoc,he,mul_one] at hb
+  have rate (t : ℝ) (ht : t ∈ Ioo 0 T) :
+      2*δ^4/Y₀^2 ≤ (4*(Y t)^2/(Q t)^2)/2*Real.exp (-C t/2) := by
+    have hq := hQpos t ht
+    have hD := Real.exp_pos (-C t/2)
+    have ha := pow_le_pow_left₀
+      (mul_nonneg (sq_nonneg _) hq.le) (hangle t ht) 2
+    have hE : (Real.exp (-C t/8))^4 = Real.exp (-C t/2) := by
+      rw [← Real.exp_nat_mul]
+      congr 1
+      ring
+    have ha' : δ^4*Real.exp (-C t/2)*(Q t)^2 ≤ (Y t)^4 := by
+      simpa only [angleFloor,mul_pow,← pow_mul,show 2*2=4 by norm_num,hE] using ha
+    have hb := mul_le_mul_of_nonneg_right (height t ⟨ht.1.le,ht.2.le⟩) (sq_nonneg (Y t))
+    have hc : δ^4*(Q t)^2 ≤ Y₀^2*Real.exp (-C t/2)*(Y t)^2 := by
+      apply (mul_le_mul_iff_of_pos_left hD).1
+      nlinarith [ha',hb]
+    apply (div_le_iff₀ (sq_pos_of_pos hY₀)).2
+    apply (mul_le_mul_iff_of_pos_right (sq_pos_of_pos hq)).1
+    field_simp
+    simp only [neg_div] at hc
+    nlinarith [hc]
+  have hK : 0 < 2*δ^4/Y₀^2 := by positivity
+  have hm : AntitoneOn (fun t => Real.exp (-C t/2)+(2*δ^4/Y₀^2)*t) (Icc 0 T) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T)
+      ((Real.continuous_exp.comp_continuousOn (hC.neg.div_const 2)).add (continuousOn_id.const_mul _))
+      (f' := fun t => Real.exp (-C t/2)*(-(4*(Y t)^2/(Q t)^2)/2)+2*δ^4/Y₀^2)
+    · intro t ht
+      rw [interior_Icc] at ht
+      convert (((hdC t ht).neg.div_const 2).exp.add
+          ((hasDerivAt_id t).const_mul (2*δ^4/Y₀^2))).hasDerivWithinAt
+        (s := interior (Icc 0 T)) using 1 <;> first | rfl | (simp only [Pi.neg_apply]; ring)
+    · intro t ht
+      rw [interior_Icc] at ht
+      nlinarith [rate t ht]
+  have hb := hm ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
+  simp only [hCinit,neg_zero,zero_div,Real.exp_zero,mul_zero,add_zero] at hb
+  have ht : T ≤ 1/(2*δ^4/Y₀^2) := (le_div_iff₀ hK).2 (by nlinarith [Real.exp_pos (-C T/2)])
+  convert ht using 1; field_simp
+
+lemma good_physical_clock {κ : ℝ} (hκ : 0 < κ ∧ κ < 8)
+    {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
+    (hc : Continuous B) {z : ℂ} (hz : z ∈ halfPlane) {δ v : ℝ}
+    (hδ : 0 < δ) (hv : 0 < v)
+    (hhit : firstLiveRadiusStop γ g z v ≠ (⊤ : ℝ≥0∞))
+    (hgood : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ v →
+      finiteStoppedDensity κ B γ g z q ∉ Ioo 0 (threshold κ z δ q)) :
+    firstLiveRadiusStop γ g z v ≤ ENNReal.ofReal (z.im^2/(2*δ^4)) := by
+  have hUc : Continuous (fun t : ℝ => Real.sqrt κ*B t.toNNReal) :=
+    continuous_const.mul (hc.comp continuous_real_toNNReal)
+  have hU := hUc.continuousOn (s := Ici 0)
+  obtain ⟨T,hstop,hzT,hrT⟩ := RadiusPast.live_radius_witness hm hU hhit
+  have hT : 0 < (T:ℝ) := by
+    have hp := hm.firstLiveRadiusStop_pos hU hz hv
+    rw [hstop] at hp
+    exact ENNReal.coe_pos.1 hp
+  let C := fun t => Real.log (2*z.im)-Real.log (loewnerRadius g t z)
+  have hC : ContinuousOn C (Icc 0 T) :=
+    continuousOn_const.sub (hm.continuous_log_radius T.coe_nonneg (hU.mono Icc_subset_Ici_self) hzT)
+  have hCs : StrictMonoOn C (Icc 0 T) := by
+    intro a ha b hb hab
+    have hh := hm.log_radius_strictAnti T.coe_nonneg (hU.mono Icc_subset_Ici_self) hzT ha hb hab
+    dsimp [C]
+    linarith
+  have hC0 : C 0 = 0 := by simp only [C,hm.radius_initial hz,sub_self]
+  have hCT : C T = v := by
+    have hy : 0 < 2*z.im := by have : 0<z.im := hz; positivity
+    dsimp [C]
+    rw [hrT,Real.log_mul hy.ne' (Real.exp_ne_zero _),Real.log_exp]
+    ring
+  have hg := hm.continuous_eval T.coe_nonneg hzT
+  have hsine : ContinuousOn (fun t => loewnerAngleSine κ B g t z) (Icc 0 T) := by
+    have hh := (Complex.continuous_im.comp_continuousOn hg).div
+      (hg.sub (Complex.continuous_ofReal.comp hUc).continuousOn).norm
+      (fun t ht => norm_ne_zero_iff.2 (hm.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hzT)))
+    convert hh using 1; first | rfl | (funext t; simp only [loewnerAngleSine,Complex.ofReal_mul,Function.comp_def,Pi.div_apply,Pi.sub_apply])
+  have hangle : ∀ t : ℝ, t ∈ Ioo 0 (T:ℝ) → angleFloor δ (C t) ≤ loewnerAngleSine κ B g t z := by
+    have hF : ContinuousOn (fun t => loewnerAngleSine κ B g t z-angleFloor δ (C t)) (Icc 0 T) :=
+      hsine.sub (continuousOn_const.mul (Real.continuous_exp.comp_continuousOn (hC.neg.div_const 8)))
+    have hn := rational_clock_nonneg hT hC hCs hC0 hCT hF ?_
+    · intro t ht
+      exact sub_nonneg.1 (hn t ht)
+    · intro q hq hqv s hs hseq
+      apply sub_nonneg.2
+      by_contra hn
+      exact hgood q hq hqv (live_density_below hκ hm hc hs.1
+        (survivingDomain_mono hs.2 hzT) hseq (by rw [← hseq]; exact lt_of_not_ge hn))
+  have hbound : (T:ℝ) ≤ z.im^2/(2*δ^4) := by
+    apply exponential_clock_bound (Y := fun t => (g t z).im) (C := C)
+      (Q := fun t => Complex.normSq (g t z-((Real.sqrt κ*B t.toNNReal : ℝ) : ℂ)))
+      T.coe_nonneg hδ (show 0<z.im from hz) (hm.initial z hz |>.symm ▸ rfl) hC0
+      (Complex.continuous_im.comp_continuousOn hg) hC
+    · intro t ht
+      exact hm.im_pos ht.1 (survivingDomain_mono ht.2 hzT)
+    · intro t ht
+      exact Complex.normSq_pos.2 (hm.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hzT))
+    · intro t ht
+      simp only [Complex.normSq_apply,Complex.sub_im,Complex.ofReal_im,sub_zero]
+      nlinarith [sq_nonneg ((g t z-((Real.sqrt κ*B t.toNNReal : ℝ) : ℂ)).re)]
+    · intro t ht
+      have hp := norm_pos_iff.2 (hm.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hzT))
+      have hs := hangle t ht
+      simp only [loewnerAngleSine,← Complex.ofReal_mul] at hs
+      have ha := (le_div_iff₀ hp).1 hs
+      have hb := pow_le_pow_left₀ (mul_nonneg (angleFloor_pos hδ _).le hp.le) ha 2
+      simpa only [mul_pow,Complex.normSq_eq_norm_sq] using hb
+    · intro t ht
+      exact hm.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hzT)
+    · intro t ht
+      apply ((hm.hasDerivAt_log_radius T.coe_nonneg (hU.mono Icc_subset_Ici_self) hzT ht).const_sub
+        (Real.log (2*z.im))).congr_deriv
+      ring
+  rw [hstop]
+  simpa only [ENNReal.ofReal_coe_nnreal] using (ENNReal.ofReal_le_ofReal hbound)
+
+end SLEExactGauge.FiniteClock
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteClock
+
+theorem finiteCenteredPhysicalTime_tight {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (ε : ℝ) (hε : 0 < ε) :
+    ∃ T : ℝ, 0 < T ∧ ∀ u : ℝ, 1 ≤ u →
+      (finiteCenteredMeasure κ P B γ g z u)
+        {sample | ENNReal.ofReal T < firstLiveRadiusStop (γ sample) (g sample) z u}
+        < ENNReal.ofReal ε := by
+  obtain ⟨D,hD,hbound⟩ := bad_bound κ hκ B hB γ g f hγ hmaps z hz
+  obtain ⟨δ,hδ,hsmall⟩ := exists_small_floor (exponent_pos hκ) hD hε
+  refine ⟨z.im^2/(2*δ^4),by have hy : 0<z.im := hz; positivity,?_⟩
+  intro u hu
+  have hu0 : 0 < u := by linarith
+  let Q := finiteCenteredMeasure κ P B γ g z u
+  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
+  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
+    change ∀ᵐ sample ∂P.withDensity _,_
+    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
+    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
+  have hsub : {sample | ENNReal.ofReal (z.im^2/(2*δ^4)) <
+        firstLiveRadiusStop (γ sample) (g sample) z u} ≤ᵐ[Q] bad κ B γ g z δ u := by
+    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hm hc hh hx
+    by_contra hn
+    have hg : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ u →
+        finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z q ∉ Ioo 0 (threshold κ z δ q) := by
+      intro q hq hqu hbad
+      exact hn (mem_bad_of_rational κ B γ g z δ u q hq hqu hbad)
+    exact not_lt_of_ge (good_physical_clock (κ := κ) (B := fun t => B t sample) (γ := γ sample)
+      (g := g sample) (f := f sample) hκ hm hc hz hδ hu0 hh hg) hx
+  exact (measure_mono_ae hsub).trans_lt
+    ((hbound δ hδ u).trans_lt (ENNReal.ofReal_lt_ofReal_iff hε |>.2 hsmall))
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/FiniteTimeMass.lean b/lean/OAI/Probability/SLELocalization/FiniteTimeMass.lean
index 78161d6abebaa952c38606609ce5bad2c50562eb..8655f313171424eb0dabba55ddd79236ae0bf23a 100644
--- a/lean/OAI/Probability/SLELocalization/FiniteTimeMass.lean
+++ b/lean/OAI/Probability/SLELocalization/FiniteTimeMass.lean
@@ -1,362 +1,362 @@
-import Mathlib
-import OAI.Probability.SLELocalization.FiniteClock
-import OAI.Probability.SLELocalization.PhysicalPast
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Finite Time Mass estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteTilt GreenPhysical
-
-theorem physicalDistanceStoppedDensity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (ε : ℝ) (hε : 0 < ε) (hsmall : ε < z.im) :
-    (∫⁻ sample, physicalStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z ε ∂P) =
-        ENNReal.ofReal ((2*z.im)^(-codimension κ) * (z.im/‖z‖)^(8/κ-1)) := by
-  obtain ⟨u,hu,hρ⟩ : ∃ u : ℝ, 0 < u ∧ (2*z.im)*Real.exp (-u) < ε/4 := by
-    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
-    exact ((eventually_gt_atTop 0).and (hl.eventually (gt_mem_nhds (by linarith : 0 < ε/4)))).exists
-  let C := cutoffs hz u hu.le
-  let F := fun n sample => ENNReal.ofReal ((C n).physicalValue κ B γ ε sample)
-  have hexp (n : ℕ) := (C n).physical_expectation hB κ hκ γ g f hmaps hz hu ε
-  have hFm (n : ℕ) : AEMeasurable (F n) P :=
-    ENNReal.measurable_ofReal.comp_aemeasurable (hexp n).1.aemeasurable
-  have hFb (n : ℕ) : ∀ᵐ sample ∂P, F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb
-    exact ENNReal.ofReal_le_ofReal ((C n).physicalValue_bound κ hκ B γ ε sample hz hu hc hb)
-  have hFl : ∀ᵐ sample ∂P, Tendsto (fun n => F n sample) atTop
-      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε)) := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
-    exact tendsto_physicalValue_killed κ B γ g f sample hm hz hu hc hb hε hsmall hρ hκ
-  have hfin : (∫⁻ _ : Ω, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ (⊤ : ℝ≥0∞) := by
-    rw [lintegral_const, measure_univ, mul_one]
-    exact ENNReal.ofReal_ne_top
-  have hdct := tendsto_lintegral_of_dominated_convergence'
-    (fun _ : Ω => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8)))) hFm hFb hfin hFl
-  have hint (n : ℕ) : (∫⁻ sample, F n sample ∂P) = ENNReal.ofReal (green (1-κ/8) (8/κ-1) z.re z.im 0) := by
-    rw [← ofReal_integral_eq_lintegral_ofReal (hexp n).1
-      (Filter.Eventually.of_forall (fun sample => (C n).physicalValue_nonneg κ B γ ε sample)), (hexp n).2]
-  simp only [hint] at hdct
-  have he := tendsto_nhds_unique hdct tendsto_const_nhds
-  rw [green_initial κ hz] at he
-  exact he
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.FiniteTilt
-
-lemma physicalValue_aemeasurable_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-    {e : ℝ} (he : 0 < e) (hes : e < z.im) (hr : (2*z.im)*Real.exp (-u) < e/4) :
-    (∀ n : ℕ, AEMeasurable (fun sample => ENNReal.ofReal
-      ((cutoffs hz u hu.le n).physicalValue κ B γ e sample)) P) ∧
-    ∀ᵐ sample ∂P, Tendsto (fun n => ENNReal.ofReal
-      ((cutoffs hz u hu.le n).physicalValue κ B γ e sample)) atTop
-      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e)) := by
-  constructor
-  · intro n
-    exact ENNReal.measurable_ofReal.comp_aemeasurable
-      ((cutoffs hz u hu.le n).physical_expectation hB κ hκ γ g f hmaps hz hu e).1.aemeasurable
-  · filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
-    exact tendsto_physicalValue_killed κ B γ g f sample hm hz hu hc hb he hes hr hκ
-
-lemma tendsto_setLIntegral_physicalValue {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-    {e : ℝ} (he : 0 < e) (hes : e < z.im) (hr : (2*z.im)*Real.exp (-u) < e/4) (E : Set Ω) :
-    Tendsto (fun n => ∫⁻ sample in E, ENNReal.ofReal
-      ((cutoffs hz u hu.le n).physicalValue κ B γ e sample) ∂P) atTop
-      (𝓝 (∫⁻ sample in E, physicalStoppedDensity κ (fun t => B t sample)
-        (γ sample) (g sample) z e ∂P)) := by
-  let C := cutoffs hz u hu.le
-  let F := fun n sample => ENNReal.ofReal ((C n).physicalValue κ B γ e sample)
-  obtain ⟨hFm,hFl⟩ := physicalValue_aemeasurable_tendsto hB κ hκ γ g f hmaps hz hu he hes hr
-  have hFb (n : ℕ) : ∀ᵐ sample ∂P.restrict E,
-      F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
-    apply ae_restrict_of_ae
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb
-    exact ENNReal.ofReal_le_ofReal ((C n).physicalValue_bound κ hκ B γ e sample hz hu hc hb)
-  have hfin : (∫⁻ _ : Ω in E, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ ⊤ := by
-    rw [lintegral_const]
-    exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)
-  exact tendsto_lintegral_of_dominated_convergence' _
-    (fun n => (hFm n).mono_measure Measure.restrict_le_self) hFb hfin (ae_restrict_of_ae hFl)
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteTilt
-
-lemma physical_radial_avoidance_identity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u)
-    (e : ℝ) (he : 0 < e) (hes : e < z.im) (hr : (2*z.im)*Real.exp (-u) < e/4) (S : ℝ≥0) :
-    (∫⁻ sample in PhysicalPast.avoids κ B z e S,
-      physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P) =
-    ∫⁻ sample in PhysicalPast.avoids κ B z e S,
-      finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P := by
-  let C := cutoffs hz u hu.le
-  let E := PhysicalPast.avoids κ B z e S
-  have hp := tendsto_setLIntegral_physicalValue hB κ hκ γ g f hmaps hz hu he hes hr E
-  have hr := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz hu.le C
-    (shrinkingAngle_tendsto z) E
-  have hh (n : ℕ) : (∫⁻ sample in E, ENNReal.ofReal ((C n).physicalValue κ B γ e sample) ∂P) =
-      ∫⁻ sample in E, ENNReal.ofReal ((C n).value κ B sample) ∂P := by
-    rw [← ofReal_integral_eq_lintegral_ofReal
-      ((C n).physical_expectation hB κ hκ γ g f hmaps hz hu e).1.integrableOn
-      (Filter.Eventually.of_forall (fun sample => (C n).physicalValue_nonneg κ B γ e sample)),
-      ← ofReal_integral_eq_lintegral_ofReal ((C n).expectation hB κ hκ hz hu.le).1.integrableOn
-      (Filter.Eventually.of_forall (fun sample => (C n).value_nonneg κ B sample))]
-    exact congrArg ENNReal.ofReal ((C n).physical_avoidance_setIntegral hB κ hκ hz hu γ g f hmaps e S).symm
-  exact tendsto_nhds_unique (hp.congr' (Filter.Eventually.of_forall hh)) hr
-
-lemma physicalStoppedDensity_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (e : ℝ) (he : 0 < e) (hes : e < z.im) :
-    AEMeasurable (fun sample => physicalStoppedDensity κ (fun t => B t sample)
-      (γ sample) (g sample) z e) P := by
-  obtain ⟨u,hu,hr⟩ : ∃ u : ℝ, 0 < u ∧ (2*z.im)*Real.exp (-u) < e/4 := by
-    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
-    exact ((eventually_gt_atTop 0).and (hl.eventually (gt_mem_nhds (by linarith : 0 < e/4)))).exists
-  obtain ⟨hFm,hFl⟩ := physicalValue_aemeasurable_tendsto hB κ hκ γ g f hmaps hz hu he hes hr
-  exact aemeasurable_of_tendsto_metrizable_ae' hFm hFl
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma physicalStoppedDensity_pos_iff {κ : ℝ} {B : ℝ≥0 → ℝ}
-    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
-    (z : ℂ) (e : ℝ) :
-    0 < physicalStoppedDensity κ B γ g z e ↔ firstLiveDistanceStop γ z e ≠ (⊤ : ℝ≥0∞) := by
-  by_cases hh : firstLiveDistanceStop γ z e = (⊤ : ℝ≥0∞)
-  · simp only [physicalStoppedDensity,hh,ite_true,lt_self_iff_false,ne_eq,not_true_eq_false]
-  · rw [physicalStoppedDensity,ite_eq_right hh]
-    constructor
-    · exact fun _ => hh
-    · intro _
-      obtain ⟨t,ht,hlive,_,_⟩ := PhysicalStop.finite_spec hm.1 hh
-      rw [ht,ENNReal.coe_toReal]
-      apply ENNReal.ofReal_pos.2
-      have hs : 0 < loewnerAngleSine κ B g t z := by
-        apply div_pos (hm.im_pos t.coe_nonneg hlive)
-        apply norm_pos_iff.2
-        simpa only [Complex.ofReal_mul] using (hm.centered_ne_zero t.coe_nonneg hlive)
-      exact mul_pos (Real.rpow_pos_of_pos (hm.radius_pos t.coe_nonneg hlive) _)
-        (Real.rpow_pos_of_pos hs _)
-
-lemma physical_avoidance_at_finite_hit {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-    (hc : Continuous (fun t => B t sample))
-    (z : ℂ) (e : ℝ) (S : ℝ≥0) (hh : firstLiveDistanceStop γ z e ≠ (⊤ : ℝ≥0∞)) :
-    sample ∈ PhysicalPast.avoids κ B z e S ↔ (S:ℝ≥0∞) < firstLiveDistanceStop γ z e := by
-  rw [PhysicalPast.avoids_eq_source κ B sample hm hc z e S]
-  obtain ⟨t,ht,hlive,hd,hmin⟩ := PhysicalStop.finite_spec hm.1 hh
-  rw [ht,ENNReal.coe_lt_coe]
-  constructor
-  · intro h
-    by_contra hn
-    exact (not_lt_of_ge hd) (h t (not_lt.1 hn))
-  · intro h s hs
-    exact hmin s (hs.trans_lt h)
-
-lemma physical_before_deep_radius {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing U γ g f) (hU : ContinuousOn U (Ici 0))
-    {z : ℂ} {e u : ℝ} (he : 0 < e) (hes : e < z.im)
-    (hr : (2*z.im)*Real.exp (-u) < e/4)
-    (hh : firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞)) :
-    firstLiveDistanceStop γ z e < firstLiveRadiusStop γ g z u := by
-  obtain ⟨t,ht,hlive,hradius⟩ := RadiusPast.live_radius_witness hm hU hh
-  rw [ht]
-  by_contra hn
-  have hb := hm.radius_ge_before_distance_stop he hes (not_lt.1 hn) hlive
-  rw [hradius] at hb
-  exact (not_lt_of_ge hb) hr
-
-lemma physical_early_nullMeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (e : ℝ) (he : 0 < e) (hes : e < z.im) (S : ℝ≥0) :
-    NullMeasurableSet {sample | firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞)} P := by
-  have hd := physicalStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz e he hes
-  have hA := (nullMeasurableSet_lt (aemeasurable_const (b := (0 : ℝ≥0∞))) hd).inter
-    (PhysicalPast.avoids_nullMeasurable κ hB z e S).compl
-  apply hA.congr
-  filter_upwards [hmaps,hB.cont] with sample hm hc
-  change (0 < physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∧
-    sample ∉ PhysicalPast.avoids κ B z e S) = (firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞))
-  apply propext
-  rw [physicalStoppedDensity_pos_iff (κ := κ) (B := fun t => B t sample) hm]
-  constructor
-  · rintro ⟨hh,hno⟩
-    exact not_lt.1 (hno ∘ (physical_avoidance_at_finite_hit κ B sample hm hc z e S hh).2)
-  · intro hle
-    have hh := ne_top_of_le_ne_top ENNReal.coe_ne_top hle
-    exact ⟨hh,fun hs => not_lt_of_ge hle ((physical_avoidance_at_finite_hit κ B sample hm hc z e S hh).1 hs)⟩
-
-theorem finiteTimePhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) :
-    ∃ T : ℝ, 0 < T ∧ ∀ e : ℝ, 0 < e → e < z.im →
-      ENNReal.ofReal (((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))/2) ≤
-        ∫⁻ sample in {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T},
-          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P := by
-  classical
-  let m := (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)
-  let M := ENNReal.ofReal m
-  have hm : 0 < m := localizationInitialGreen_pos κ hz
-  have hM0 : M ≠ 0 := (ENNReal.ofReal_pos.2 hm).ne'
-  have hMtop : M ≠ ⊤ := ENNReal.ofReal_ne_top
-  obtain ⟨T,hT,hclock⟩ := finiteCenteredPhysicalTime_tight κ hκ B hB γ g f hγ hmaps z hz
-    (1/2) (by norm_num)
-  refine ⟨T,hT,?_⟩
-  intro e he hes
-  let S : ℝ≥0 := ⟨T,hT.le⟩
-  have hS : (S : ℝ≥0∞) = ENNReal.ofReal T := by
-    change (S : ℝ≥0∞) = ENNReal.ofReal (S : ℝ)
-    exact ENNReal.ofReal_coe_nnreal.symm
-  let E := PhysicalPast.avoids κ B z e S
-  let A := {sample | firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞)}
-  let D := fun sample => physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e
-  have hE : NullMeasurableSet E P := PhysicalPast.avoids_nullMeasurable κ hB z e S
-  have hA : NullMeasurableSet A P := physical_early_nullMeasurable κ hκ B hB γ g f hmaps z hz e he hes S
-  have hD : AEMeasurable D P := physicalStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz e he hes
-  obtain ⟨u,hu,hr⟩ : ∃ u : ℝ, 1 < u ∧ (2*z.im)*Real.exp (-u) < e/4 := by
-    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
-    exact ((eventually_gt_atTop 1).and (hl.eventually (gt_mem_nhds (by linarith : 0 < e/4)))).exists
-  have hu0 : 0 < u := by linarith
-  let Q := finiteCenteredMeasure κ P B γ g z u
-  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
-  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
-    change ∀ᵐ sample ∂P.withDensity _,_
-    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
-    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
-  have hsub : E ≤ᵐ[Q] {sample | (S:ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u} := by
-    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hmaps hc hhit hmem
-    have hpr := physical_before_deep_radius hmaps
-      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) he hes hr hhit
-    have hfp : firstLiveDistanceStop (γ sample) z e ≠ (⊤ : ℝ≥0∞) := ne_top_of_lt hpr
-    exact ((physical_avoidance_at_finite_hit κ B sample hmaps hc z e S hfp).1 hmem).trans hpr
-  have hQE : Q E ≤ ENNReal.ofReal (1/2) := by
-    apply (measure_mono_ae hsub).trans
-    simpa only [hS] using (hclock u hu.le).le
-  have hid : (∫⁻ sample in E, D sample ∂P) = M*Q E := by
-    rw [physical_radial_avoidance_identity κ hκ B hB γ g f hmaps z hz u hu0 e he hes hr S]
-    rw [show Q E = ∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
-        (γ sample) (g sample) z u / M ∂P from withDensity_apply₀ _ hE]
-    simp_rw [ENNReal.div_eq_inv_mul]
-    rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hM0),← mul_assoc,ENNReal.mul_inv_cancel hM0 hMtop,one_mul]
-  have hhalf : M*ENNReal.ofReal (1/2) = ENNReal.ofReal (m/2) := by
-    rw [← ENNReal.ofReal_mul hm.le]
-    congr 1
-    ring
-  have htail : (∫⁻ sample in E, D sample ∂P) ≤ ENNReal.ofReal (m/2) := by
-    rw [hid,← hhalf]
-    exact mul_le_mul' le_rfl hQE
-  have hi : Eᶜ.indicator D =ᵐ[P] A.indicator D := by
-    filter_upwards [hmaps,hB.cont] with sample hm hc
-    by_cases hh : firstLiveDistanceStop (γ sample) z e = (⊤ : ℝ≥0∞)
-    · have hd : D sample = 0 := by simp only [D,physicalStoppedDensity,hh,ite_true]
-      simp only [Set.indicator_apply,hd,ite_self]
-    · have hv := physical_avoidance_at_finite_hit κ B sample hm hc z e S hh
-      have hae : sample ∈ Eᶜ ↔ sample ∈ A := by
-        simpa only [E,A,mem_compl_iff,mem_ofPred_eq,not_lt] using hv.not
-      simp only [Set.indicator_apply,hae]
-  have hall : (∫⁻ sample in E, D sample ∂P)+(∫⁻ sample in A, D sample ∂P) = M := by
-    rw [← lintegral_indicator₀ hA,← lintegral_congr_ae hi,← lintegral_indicator₀ hE,
-      ← lintegral_add_left' (hD.indicator₀ hE)]
-    have hp : (fun sample => E.indicator D sample+Eᶜ.indicator D sample) = D := by
-      funext sample
-      by_cases hs : sample∈E <;> simp [hs]
-    rw [hp]
-    exact physicalDistanceStoppedDensity κ hκ B hB γ g f hγ hmaps z hz e he hes
-  have hdouble : M = ENNReal.ofReal (m/2)+ENNReal.ofReal (m/2) := by
-    change ENNReal.ofReal m = _
-    rw [← ENNReal.ofReal_add (by positivity : 0 ≤ m/2) (by positivity : 0 ≤ m/2)]
-    congr 1
-    ring
-  have hlow : ENNReal.ofReal (m/2) ≤ ∫⁻ sample in A, D sample ∂P := by
-    apply ENNReal.le_of_add_le_add_left ENNReal.ofReal_ne_top
-    rw [← hdouble,← hall]
-    exact add_le_add htail le_rfl
-  simpa only [A,D,m,hS] using hlow
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.FiniteClock
+import OAI.Probability.SLELocalization.PhysicalPast
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Finite Time Mass estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteTilt GreenPhysical
+
+theorem physicalDistanceStoppedDensity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (ε : ℝ) (hε : 0 < ε) (hsmall : ε < z.im) :
+    (∫⁻ sample, physicalStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z ε ∂P) =
+        ENNReal.ofReal ((2*z.im)^(-codimension κ) * (z.im/‖z‖)^(8/κ-1)) := by
+  obtain ⟨u,hu,hρ⟩ : ∃ u : ℝ, 0 < u ∧ (2*z.im)*Real.exp (-u) < ε/4 := by
+    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
+    exact ((eventually_gt_atTop 0).and (hl.eventually (gt_mem_nhds (by linarith : 0 < ε/4)))).exists
+  let C := cutoffs hz u hu.le
+  let F := fun n sample => ENNReal.ofReal ((C n).physicalValue κ B γ ε sample)
+  have hexp (n : ℕ) := (C n).physical_expectation hB κ hκ γ g f hmaps hz hu ε
+  have hFm (n : ℕ) : AEMeasurable (F n) P :=
+    ENNReal.measurable_ofReal.comp_aemeasurable (hexp n).1.aemeasurable
+  have hFb (n : ℕ) : ∀ᵐ sample ∂P, F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb
+    exact ENNReal.ofReal_le_ofReal ((C n).physicalValue_bound κ hκ B γ ε sample hz hu hc hb)
+  have hFl : ∀ᵐ sample ∂P, Tendsto (fun n => F n sample) atTop
+      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε)) := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
+    exact tendsto_physicalValue_killed κ B γ g f sample hm hz hu hc hb hε hsmall hρ hκ
+  have hfin : (∫⁻ _ : Ω, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ (⊤ : ℝ≥0∞) := by
+    rw [lintegral_const, measure_univ, mul_one]
+    exact ENNReal.ofReal_ne_top
+  have hdct := tendsto_lintegral_of_dominated_convergence'
+    (fun _ : Ω => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8)))) hFm hFb hfin hFl
+  have hint (n : ℕ) : (∫⁻ sample, F n sample ∂P) = ENNReal.ofReal (green (1-κ/8) (8/κ-1) z.re z.im 0) := by
+    rw [← ofReal_integral_eq_lintegral_ofReal (hexp n).1
+      (Filter.Eventually.of_forall (fun sample => (C n).physicalValue_nonneg κ B γ ε sample)), (hexp n).2]
+  simp only [hint] at hdct
+  have he := tendsto_nhds_unique hdct tendsto_const_nhds
+  rw [green_initial κ hz] at he
+  exact he
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.FiniteTilt
+
+lemma physicalValue_aemeasurable_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+    {e : ℝ} (he : 0 < e) (hes : e < z.im) (hr : (2*z.im)*Real.exp (-u) < e/4) :
+    (∀ n : ℕ, AEMeasurable (fun sample => ENNReal.ofReal
+      ((cutoffs hz u hu.le n).physicalValue κ B γ e sample)) P) ∧
+    ∀ᵐ sample ∂P, Tendsto (fun n => ENNReal.ofReal
+      ((cutoffs hz u hu.le n).physicalValue κ B γ e sample)) atTop
+      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e)) := by
+  constructor
+  · intro n
+    exact ENNReal.measurable_ofReal.comp_aemeasurable
+      ((cutoffs hz u hu.le n).physical_expectation hB κ hκ γ g f hmaps hz hu e).1.aemeasurable
+  · filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
+    exact tendsto_physicalValue_killed κ B γ g f sample hm hz hu hc hb he hes hr hκ
+
+lemma tendsto_setLIntegral_physicalValue {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+    {e : ℝ} (he : 0 < e) (hes : e < z.im) (hr : (2*z.im)*Real.exp (-u) < e/4) (E : Set Ω) :
+    Tendsto (fun n => ∫⁻ sample in E, ENNReal.ofReal
+      ((cutoffs hz u hu.le n).physicalValue κ B γ e sample) ∂P) atTop
+      (𝓝 (∫⁻ sample in E, physicalStoppedDensity κ (fun t => B t sample)
+        (γ sample) (g sample) z e ∂P)) := by
+  let C := cutoffs hz u hu.le
+  let F := fun n sample => ENNReal.ofReal ((C n).physicalValue κ B γ e sample)
+  obtain ⟨hFm,hFl⟩ := physicalValue_aemeasurable_tendsto hB κ hκ γ g f hmaps hz hu he hes hr
+  have hFb (n : ℕ) : ∀ᵐ sample ∂P.restrict E,
+      F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
+    apply ae_restrict_of_ae
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb
+    exact ENNReal.ofReal_le_ofReal ((C n).physicalValue_bound κ hκ B γ e sample hz hu hc hb)
+  have hfin : (∫⁻ _ : Ω in E, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ ⊤ := by
+    rw [lintegral_const]
+    exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)
+  exact tendsto_lintegral_of_dominated_convergence' _
+    (fun n => (hFm n).mono_measure Measure.restrict_le_self) hFb hfin (ae_restrict_of_ae hFl)
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteTilt
+
+lemma physical_radial_avoidance_identity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u)
+    (e : ℝ) (he : 0 < e) (hes : e < z.im) (hr : (2*z.im)*Real.exp (-u) < e/4) (S : ℝ≥0) :
+    (∫⁻ sample in PhysicalPast.avoids κ B z e S,
+      physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P) =
+    ∫⁻ sample in PhysicalPast.avoids κ B z e S,
+      finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P := by
+  let C := cutoffs hz u hu.le
+  let E := PhysicalPast.avoids κ B z e S
+  have hp := tendsto_setLIntegral_physicalValue hB κ hκ γ g f hmaps hz hu he hes hr E
+  have hr := tendsto_setLIntegral_value hB κ hκ γ g f hmaps hz hu.le C
+    (shrinkingAngle_tendsto z) E
+  have hh (n : ℕ) : (∫⁻ sample in E, ENNReal.ofReal ((C n).physicalValue κ B γ e sample) ∂P) =
+      ∫⁻ sample in E, ENNReal.ofReal ((C n).value κ B sample) ∂P := by
+    rw [← ofReal_integral_eq_lintegral_ofReal
+      ((C n).physical_expectation hB κ hκ γ g f hmaps hz hu e).1.integrableOn
+      (Filter.Eventually.of_forall (fun sample => (C n).physicalValue_nonneg κ B γ e sample)),
+      ← ofReal_integral_eq_lintegral_ofReal ((C n).expectation hB κ hκ hz hu.le).1.integrableOn
+      (Filter.Eventually.of_forall (fun sample => (C n).value_nonneg κ B sample))]
+    exact congrArg ENNReal.ofReal ((C n).physical_avoidance_setIntegral hB κ hκ hz hu γ g f hmaps e S).symm
+  exact tendsto_nhds_unique (hp.congr' (Filter.Eventually.of_forall hh)) hr
+
+lemma physicalStoppedDensity_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (e : ℝ) (he : 0 < e) (hes : e < z.im) :
+    AEMeasurable (fun sample => physicalStoppedDensity κ (fun t => B t sample)
+      (γ sample) (g sample) z e) P := by
+  obtain ⟨u,hu,hr⟩ : ∃ u : ℝ, 0 < u ∧ (2*z.im)*Real.exp (-u) < e/4 := by
+    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
+    exact ((eventually_gt_atTop 0).and (hl.eventually (gt_mem_nhds (by linarith : 0 < e/4)))).exists
+  obtain ⟨hFm,hFl⟩ := physicalValue_aemeasurable_tendsto hB κ hκ γ g f hmaps hz hu he hes hr
+  exact aemeasurable_of_tendsto_metrizable_ae' hFm hFl
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma physicalStoppedDensity_pos_iff {κ : ℝ} {B : ℝ≥0 → ℝ}
+    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
+    (z : ℂ) (e : ℝ) :
+    0 < physicalStoppedDensity κ B γ g z e ↔ firstLiveDistanceStop γ z e ≠ (⊤ : ℝ≥0∞) := by
+  by_cases hh : firstLiveDistanceStop γ z e = (⊤ : ℝ≥0∞)
+  · simp only [physicalStoppedDensity,hh,ite_true,lt_self_iff_false,ne_eq,not_true_eq_false]
+  · rw [physicalStoppedDensity,ite_eq_right hh]
+    constructor
+    · exact fun _ => hh
+    · intro _
+      obtain ⟨t,ht,hlive,_,_⟩ := PhysicalStop.finite_spec hm.1 hh
+      rw [ht,ENNReal.coe_toReal]
+      apply ENNReal.ofReal_pos.2
+      have hs : 0 < loewnerAngleSine κ B g t z := by
+        apply div_pos (hm.im_pos t.coe_nonneg hlive)
+        apply norm_pos_iff.2
+        simpa only [Complex.ofReal_mul] using (hm.centered_ne_zero t.coe_nonneg hlive)
+      exact mul_pos (Real.rpow_pos_of_pos (hm.radius_pos t.coe_nonneg hlive) _)
+        (Real.rpow_pos_of_pos hs _)
+
+lemma physical_avoidance_at_finite_hit {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+    (hc : Continuous (fun t => B t sample))
+    (z : ℂ) (e : ℝ) (S : ℝ≥0) (hh : firstLiveDistanceStop γ z e ≠ (⊤ : ℝ≥0∞)) :
+    sample ∈ PhysicalPast.avoids κ B z e S ↔ (S:ℝ≥0∞) < firstLiveDistanceStop γ z e := by
+  rw [PhysicalPast.avoids_eq_source κ B sample hm hc z e S]
+  obtain ⟨t,ht,hlive,hd,hmin⟩ := PhysicalStop.finite_spec hm.1 hh
+  rw [ht,ENNReal.coe_lt_coe]
+  constructor
+  · intro h
+    by_contra hn
+    exact (not_lt_of_ge hd) (h t (not_lt.1 hn))
+  · intro h s hs
+    exact hmin s (hs.trans_lt h)
+
+lemma physical_before_deep_radius {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing U γ g f) (hU : ContinuousOn U (Ici 0))
+    {z : ℂ} {e u : ℝ} (he : 0 < e) (hes : e < z.im)
+    (hr : (2*z.im)*Real.exp (-u) < e/4)
+    (hh : firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞)) :
+    firstLiveDistanceStop γ z e < firstLiveRadiusStop γ g z u := by
+  obtain ⟨t,ht,hlive,hradius⟩ := RadiusPast.live_radius_witness hm hU hh
+  rw [ht]
+  by_contra hn
+  have hb := hm.radius_ge_before_distance_stop he hes (not_lt.1 hn) hlive
+  rw [hradius] at hb
+  exact (not_lt_of_ge hb) hr
+
+lemma physical_early_nullMeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (e : ℝ) (he : 0 < e) (hes : e < z.im) (S : ℝ≥0) :
+    NullMeasurableSet {sample | firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞)} P := by
+  have hd := physicalStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz e he hes
+  have hA := (nullMeasurableSet_lt (aemeasurable_const (b := (0 : ℝ≥0∞))) hd).inter
+    (PhysicalPast.avoids_nullMeasurable κ hB z e S).compl
+  apply hA.congr
+  filter_upwards [hmaps,hB.cont] with sample hm hc
+  change (0 < physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∧
+    sample ∉ PhysicalPast.avoids κ B z e S) = (firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞))
+  apply propext
+  rw [physicalStoppedDensity_pos_iff (κ := κ) (B := fun t => B t sample) hm]
+  constructor
+  · rintro ⟨hh,hno⟩
+    exact not_lt.1 (hno ∘ (physical_avoidance_at_finite_hit κ B sample hm hc z e S hh).2)
+  · intro hle
+    have hh := ne_top_of_le_ne_top ENNReal.coe_ne_top hle
+    exact ⟨hh,fun hs => not_lt_of_ge hle ((physical_avoidance_at_finite_hit κ B sample hm hc z e S hh).1 hs)⟩
+
+theorem finiteTimePhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) :
+    ∃ T : ℝ, 0 < T ∧ ∀ e : ℝ, 0 < e → e < z.im →
+      ENNReal.ofReal (((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))/2) ≤
+        ∫⁻ sample in {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T},
+          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P := by
+  classical
+  let m := (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)
+  let M := ENNReal.ofReal m
+  have hm : 0 < m := localizationInitialGreen_pos κ hz
+  have hM0 : M ≠ 0 := (ENNReal.ofReal_pos.2 hm).ne'
+  have hMtop : M ≠ ⊤ := ENNReal.ofReal_ne_top
+  obtain ⟨T,hT,hclock⟩ := finiteCenteredPhysicalTime_tight κ hκ B hB γ g f hγ hmaps z hz
+    (1/2) (by norm_num)
+  refine ⟨T,hT,?_⟩
+  intro e he hes
+  let S : ℝ≥0 := ⟨T,hT.le⟩
+  have hS : (S : ℝ≥0∞) = ENNReal.ofReal T := by
+    change (S : ℝ≥0∞) = ENNReal.ofReal (S : ℝ)
+    exact ENNReal.ofReal_coe_nnreal.symm
+  let E := PhysicalPast.avoids κ B z e S
+  let A := {sample | firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞)}
+  let D := fun sample => physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e
+  have hE : NullMeasurableSet E P := PhysicalPast.avoids_nullMeasurable κ hB z e S
+  have hA : NullMeasurableSet A P := physical_early_nullMeasurable κ hκ B hB γ g f hmaps z hz e he hes S
+  have hD : AEMeasurable D P := physicalStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz e he hes
+  obtain ⟨u,hu,hr⟩ : ∃ u : ℝ, 1 < u ∧ (2*z.im)*Real.exp (-u) < e/4 := by
+    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
+    exact ((eventually_gt_atTop 1).and (hl.eventually (gt_mem_nhds (by linarith : 0 < e/4)))).exists
+  have hu0 : 0 < u := by linarith
+  let Q := finiteCenteredMeasure κ P B γ g z u
+  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
+  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
+    change ∀ᵐ sample ∂P.withDensity _,_
+    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
+    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
+  have hsub : E ≤ᵐ[Q] {sample | (S:ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u} := by
+    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hmaps hc hhit hmem
+    have hpr := physical_before_deep_radius hmaps
+      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) he hes hr hhit
+    have hfp : firstLiveDistanceStop (γ sample) z e ≠ (⊤ : ℝ≥0∞) := ne_top_of_lt hpr
+    exact ((physical_avoidance_at_finite_hit κ B sample hmaps hc z e S hfp).1 hmem).trans hpr
+  have hQE : Q E ≤ ENNReal.ofReal (1/2) := by
+    apply (measure_mono_ae hsub).trans
+    simpa only [hS] using (hclock u hu.le).le
+  have hid : (∫⁻ sample in E, D sample ∂P) = M*Q E := by
+    rw [physical_radial_avoidance_identity κ hκ B hB γ g f hmaps z hz u hu0 e he hes hr S]
+    rw [show Q E = ∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
+        (γ sample) (g sample) z u / M ∂P from withDensity_apply₀ _ hE]
+    simp_rw [ENNReal.div_eq_inv_mul]
+    rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hM0),← mul_assoc,ENNReal.mul_inv_cancel hM0 hMtop,one_mul]
+  have hhalf : M*ENNReal.ofReal (1/2) = ENNReal.ofReal (m/2) := by
+    rw [← ENNReal.ofReal_mul hm.le]
+    congr 1
+    ring
+  have htail : (∫⁻ sample in E, D sample ∂P) ≤ ENNReal.ofReal (m/2) := by
+    rw [hid,← hhalf]
+    exact mul_le_mul' le_rfl hQE
+  have hi : Eᶜ.indicator D =ᵐ[P] A.indicator D := by
+    filter_upwards [hmaps,hB.cont] with sample hm hc
+    by_cases hh : firstLiveDistanceStop (γ sample) z e = (⊤ : ℝ≥0∞)
+    · have hd : D sample = 0 := by simp only [D,physicalStoppedDensity,hh,ite_true]
+      simp only [Set.indicator_apply,hd,ite_self]
+    · have hv := physical_avoidance_at_finite_hit κ B sample hm hc z e S hh
+      have hae : sample ∈ Eᶜ ↔ sample ∈ A := by
+        simpa only [E,A,mem_compl_iff,mem_ofPred_eq,not_lt] using hv.not
+      simp only [Set.indicator_apply,hae]
+  have hall : (∫⁻ sample in E, D sample ∂P)+(∫⁻ sample in A, D sample ∂P) = M := by
+    rw [← lintegral_indicator₀ hA,← lintegral_congr_ae hi,← lintegral_indicator₀ hE,
+      ← lintegral_add_left' (hD.indicator₀ hE)]
+    have hp : (fun sample => E.indicator D sample+Eᶜ.indicator D sample) = D := by
+      funext sample
+      by_cases hs : sample∈E <;> simp [hs]
+    rw [hp]
+    exact physicalDistanceStoppedDensity κ hκ B hB γ g f hγ hmaps z hz e he hes
+  have hdouble : M = ENNReal.ofReal (m/2)+ENNReal.ofReal (m/2) := by
+    change ENNReal.ofReal m = _
+    rw [← ENNReal.ofReal_add (by positivity : 0 ≤ m/2) (by positivity : 0 ≤ m/2)]
+    congr 1
+    ring
+  have hlow : ENNReal.ofReal (m/2) ≤ ∫⁻ sample in A, D sample ∂P := by
+    apply ENNReal.le_of_add_le_add_left ENNReal.ofReal_ne_top
+    rw [← hdouble,← hall]
+    exact add_le_add htail le_rfl
+  simpa only [A,D,m,hS] using hlow
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/FirstDisk.lean b/lean/OAI/Probability/SLELocalization/FirstDisk.lean
index 4166f4f6b921797f008650c5b2344f5eedf8a9a2..f9e3fadc0b61d9a07696b32a53c788ede692fd38 100644
--- a/lean/OAI/Probability/SLELocalization/FirstDisk.lean
+++ b/lean/OAI/Probability/SLELocalization/FirstDisk.lean
@@ -1,577 +1,577 @@
-import Mathlib
-import OAI.Probability.SLELocalization.LoewnerExit
-import OAI.Probability.SLELocalization.PhysicalPast
-
-namespace OAI
-
-
-/-! Conformal localization and Is Loewner Trace Using estimates. -/
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge
-namespace IsLoewnerTraceUsing
-variable {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma hasDerivAt_real (h : IsLoewnerTraceUsing U γ g f)
-    {t : ℝ} (ht : 0 < t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    HasDerivAt (fun s => (g s z).re)
-      (2 * ((g t z).re-U t) / Complex.normSq (g t z-(U t:ℂ))) t := by
-  have hh := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t (h.hasDerivAt_eval ht hz)
-  simpa [Complex.div_re,Function.comp_def] using hh
-
-lemma initial_real_le (h : IsLoewnerTraceUsing U γ g f)
-    {T M : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hU : ∀ t ∈ Icc 0 T, U t ≤ M) (hend : (g T z).re ≤ M) : z.re ≤ M := by
-  let R : ℝ → ℝ := fun s => (g (T-s) z).re
-  have hc : ContinuousOn R (Icc 0 T) :=
-    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).comp
-      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
-  have hd (s : ℝ) (hs : s ∈ Ico 0 T) : HasDerivAt R
-      (-2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
-    have he := (h.hasDerivAt_real (sub_pos.2 hs.2)
-      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
-      ((hasDerivAt_id s).const_sub T)
-    simpa [R,neg_div,Function.comp_def] using he
-  apply le_of_forall_pos_le_add
-  intro ε hε
-  have hb := image_le_of_deriv_right_lt_deriv_boundary hc
-    (fun s hs => (hd s hs).hasDerivWithinAt)
-    (B:=fun _ => M+ε) (B':=fun _ => 0)
-    (by dsimp [R]; simpa using hend.trans (by linarith : M ≤ M+ε))
-    (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T) ⟨hT,le_rfl⟩
-  · simpa [R,h.initial z hz.1.1] using hb
-  · have hden := Complex.normSq_pos.2
-      (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
-    have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
-    dsimp [R] at he
-    exact div_neg_of_neg_of_pos (by nlinarith) hden
-
-lemma initial_real_ge (h : IsLoewnerTraceUsing U γ g f)
-    {T m : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hU : ∀ t ∈ Icc 0 T, m ≤ U t) (hend : m ≤ (g T z).re) : m ≤ z.re := by
-  let R : ℝ → ℝ := fun s => -(g (T-s) z).re
-  have hc : ContinuousOn R (Icc 0 T) :=
-    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).neg.comp
-      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
-  have hd (s : ℝ) (hs : s ∈ Ico 0 T) : HasDerivAt R
-      (2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
-    have he := ((h.hasDerivAt_real (sub_pos.2 hs.2)
-      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
-      ((hasDerivAt_id s).const_sub T)).neg
-    convert he using 1
-    all_goals first | rfl | norm_num
-  have hn : -z.re ≤ -m := by
-    apply le_of_forall_pos_le_add
-    intro ε hε
-    have hb := image_le_of_deriv_right_lt_deriv_boundary hc
-      (fun s hs => (hd s hs).hasDerivWithinAt)
-      (B:=fun _ => -m+ε) (B':=fun _ => 0)
-      (by dsimp [R]; simpa using (show -(g T z).re ≤ -m+ε by linarith))
-      (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T) ⟨hT,le_rfl⟩
-    · simpa [R,h.initial z hz.1.1] using hb
-    · have hden := Complex.normSq_pos.2
-        (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
-      have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
-      dsimp [R] at he
-      exact div_neg_of_neg_of_pos (by nlinarith) hden
-  linarith
-
-lemma initial_height_sq_le (h : IsLoewnerTraceUsing U γ g f)
-    {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    z.im^2 ≤ (g T z).im^2+4*T := by
-  have hm : MonotoneOn (fun s => (g s z).im^2+4*s) (Icc 0 T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 T)
-      (((Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)).pow 2).add
-        (continuous_const.mul continuous_id).continuousOn)
-      (f':=fun t => -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ))+4)
-    · intro t ht
-      rw [interior_Icc] at ht
-      convert (((h.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hz)).pow 2).add
-        ((hasDerivAt_id t).const_mul 4)).hasDerivWithinAt using 1
-      all_goals first | rfl | (norm_num; ring)
-    · intro t ht
-      rw [interior_Icc] at ht
-      have hd := Complex.normSq_pos.2 (h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hz))
-      have he : (g t z).im^2 ≤ Complex.normSq (g t z-(U t:ℂ)) := by
-        simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
-        nlinarith [sq_nonneg ((g t z).re-U t)]
-      have hb : -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ)) ≥ -4 := by
-        exact (le_div_iff₀ hd).2 (by nlinarith)
-      linarith
-  simpa [h.initial z hz.1.1] using hm ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
-
-end IsLoewnerTraceUsing
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge
-
-theorem trace_driver_rectangle
-    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing U γ g f) {T m M : ℝ} (_hT : 0 ≤ T)
-    (_hU : ContinuousOn U (Icc 0 T))
-    (hb : ∀ t ∈ Icc 0 T, m ≤ U t ∧ U t ≤ M) :
-    ∀ t ∈ Icc 0 T, m ≤ (γ t).re ∧ (γ t).re ≤ M ∧ (γ t).im ^ 2 ≤ 4*t := by
-  intro t ht
-  let w : ℝ → ℂ := fun y => (U t:ℂ)+(y:ℂ)*Complex.I
-  let z : ℝ → ℂ := fun y => f t (w y)
-  have hw (y : ℝ) (hy : 0 < y) : w y ∈ halfPlane := by simpa [w,halfPlane] using hy
-  have hz (y : ℝ) (hy : 0 < y) : z y ∈ localizationDomain γ t :=
-    h.2.2.2.2.2.2.2.2.1 t ht.1 (hw y hy)
-  have he (y : ℝ) (hy : 0 < y) : g t (z y) = w y :=
-    h.2.2.2.2.2.2.2.2.2.2.1 t ht.1 _ (hw y hy)
-  have hlim : Tendsto z (𝓝[>] (0:ℝ)) (𝓝 (γ t)) :=
-    h.2.2.2.2.2.2.2.2.2.2.2.2.2 t ht.1
-  have hleft (y : ℝ) (hy : 0 < y) : m ≤ (z y).re :=
-    h.initial_real_ge ht.1 (hz y hy)
-      (fun s hs => (hb s ⟨hs.1,hs.2.trans ht.2⟩).1)
-      (by simpa [he y hy,w] using (hb t ht).1)
-  have hright (y : ℝ) (hy : 0 < y) : (z y).re ≤ M :=
-    h.initial_real_le ht.1 (hz y hy)
-      (fun s hs => (hb s ⟨hs.1,hs.2.trans ht.2⟩).2)
-      (by simpa [he y hy,w] using (hb t ht).2)
-  have hheight (y : ℝ) (hy : 0 < y) : (z y).im^2 ≤ y^2+4*t := by
-    simpa [he y hy,w] using h.initial_height_sq_le ht.1 (hz y hy)
-  refine ⟨ge_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim) ?_,
-    le_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim) ?_,?_⟩
-  · filter_upwards [self_mem_nhdsWithin] with y hy using hleft y hy
-  · filter_upwards [self_mem_nhdsWithin] with y hy using hright y hy
-  · have hr : Tendsto (fun y : ℝ => y^2+4*t) (𝓝[>] (0:ℝ)) (𝓝 (4*t)) := by
-      simpa using (show ContinuousAt (fun y : ℝ => y^2+4*t) 0 by fun_prop).tendsto.mono_left nhdsWithin_le_nhds
-    apply le_of_tendsto_of_tendsto ((Complex.continuous_im.continuousAt.tendsto.comp hlim).pow 2) hr
-    filter_upwards [self_mem_nhdsWithin] with y hy using hheight y hy
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge
-namespace IsLoewnerTraceUsing
-variable {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma earlier_real_le (h : IsLoewnerTraceUsing U γ g f)
-    {S T M : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hU : ∀ t ∈ Icc S T, U t ≤ M) (hend : (g T z).re ≤ M) : (g S z).re ≤ M := by
-  have hT : 0 ≤ T := hS.trans hST
-  let R : ℝ → ℝ := fun s => (g (T-s) z).re
-  have hc : ContinuousOn R (Icc 0 (T-S)) :=
-    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).comp
-      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
-  have hd (s : ℝ) (hs : s ∈ Ico 0 (T-S)) : HasDerivAt R
-      (-2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
-    have he := (h.hasDerivAt_real (by linarith [hs.2] : 0 < T-s)
-      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
-      ((hasDerivAt_id s).const_sub T)
-    simpa [R,neg_div,Function.comp_def] using he
-  apply le_of_forall_pos_le_add
-  intro ε hε
-  have hb := image_le_of_deriv_right_lt_deriv_boundary hc
-    (fun s hs => (hd s hs).hasDerivWithinAt)
-    (B:=fun _ => M+ε) (B':=fun _ => 0)
-    (by dsimp [R]; simpa using hend.trans (by linarith : M ≤ M+ε))
-    (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T-S) ⟨sub_nonneg.2 hST,le_rfl⟩
-  · simpa [R,sub_sub_cancel] using hb
-  · have hden := Complex.normSq_pos.2
-      (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
-    have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
-    dsimp [R] at he
-    exact div_neg_of_neg_of_pos (by nlinarith) hden
-
-lemma earlier_real_ge (h : IsLoewnerTraceUsing U γ g f)
-    {S T m : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hU : ∀ t ∈ Icc S T, m ≤ U t) (hend : m ≤ (g T z).re) : m ≤ (g S z).re := by
-  have hT : 0 ≤ T := hS.trans hST
-  let R : ℝ → ℝ := fun s => -(g (T-s) z).re
-  have hc : ContinuousOn R (Icc 0 (T-S)) :=
-    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).neg.comp
-      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
-  have hd (s : ℝ) (hs : s ∈ Ico 0 (T-S)) : HasDerivAt R
-      (2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
-    have he := ((h.hasDerivAt_real (by linarith [hs.2] : 0 < T-s)
-      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
-      ((hasDerivAt_id s).const_sub T)).neg
-    convert he using 1
-    all_goals first | rfl | norm_num
-  have hn : -(g S z).re ≤ -m := by
-    apply le_of_forall_pos_le_add
-    intro ε hε
-    have hb := image_le_of_deriv_right_lt_deriv_boundary hc
-      (fun s hs => (hd s hs).hasDerivWithinAt)
-      (B:=fun _ => -m+ε) (B':=fun _ => 0)
-      (by dsimp [R]; simpa using (show -(g T z).re ≤ -m+ε by linarith))
-      (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T-S) ⟨sub_nonneg.2 hST,le_rfl⟩
-    · simpa [R,sub_sub_cancel] using hb
-    · have hden := Complex.normSq_pos.2
-        (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
-      have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
-      dsimp [R] at he
-      exact div_neg_of_neg_of_pos (by nlinarith) hden
-  linarith
-
-lemma earlier_height_sq_le (h : IsLoewnerTraceUsing U γ g f)
-    {S T : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    (g S z).im^2 ≤ (g T z).im^2+4*(T-S) := by
-  have hT : 0 ≤ T := hS.trans hST
-  have hcont := (h.continuous_eval hT hz).mono (Icc_subset_Icc hS le_rfl)
-  have hm : MonotoneOn (fun s => (g s z).im^2+4*s) (Icc S T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc S T)
-      (((Complex.continuous_im.comp_continuousOn hcont).pow 2).add
-        (continuous_const.mul continuous_id).continuousOn)
-      (f':=fun t => -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ))+4)
-    · intro t ht
-      rw [interior_Icc] at ht
-      convert (((h.hasDerivAt_height (hS.trans_lt ht.1) (survivingDomain_mono ht.2.le hz)).pow 2).add
-        ((hasDerivAt_id t).const_mul 4)).hasDerivWithinAt using 1
-      all_goals first | rfl | (norm_num; ring)
-    · intro t ht
-      rw [interior_Icc] at ht
-      have hd := Complex.normSq_pos.2 (h.centered_ne_zero (hS.trans ht.1.le) (survivingDomain_mono ht.2.le hz))
-      have he : (g t z).im^2 ≤ Complex.normSq (g t z-(U t:ℂ)) := by
-        simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
-        nlinarith [sq_nonneg ((g t z).re-U t)]
-      have hb : -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ)) ≥ -4 := by
-        exact (le_div_iff₀ hd).2 (by nlinarith)
-      linarith
-  have hh := hm ⟨le_rfl,hST⟩ ⟨hST,le_rfl⟩ hST
-  linarith
-
-lemma later_real_le (h : IsLoewnerTraceUsing U γ g f)
-    {S T M ε : ℝ} (hS : 0 < S) (hST : S ≤ T) (hε : 0 < ε)
-    {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hU : ∀ t ∈ Icc S T, U t ≤ M) (hstart : (g S z).re ≤ M) :
-    (g T z).re ≤ M+ε+(2/ε)*(T-S) := by
-  have hc := Complex.continuous_re.comp_continuousOn
-    ((h.continuous_eval (hS.le.trans hST) hz).mono (Icc_subset_Icc hS.le le_rfl))
-  have hd (t : ℝ) (ht : t ∈ Ico S T) :=
-    h.hasDerivAt_real (hS.trans_le ht.1) (survivingDomain_mono ht.2.le hz)
-  apply image_le_of_deriv_right_lt_deriv_boundary hc
-    (fun t ht => (hd t ht).hasDerivWithinAt)
-    (B:=fun t => M+ε+(2/ε)*(t-S)) (B':=fun _ => 2/ε)
-    (by simpa using hstart.trans (show M ≤ M+ε by linarith))
-    (fun t => by simpa using (((hasDerivAt_id t).sub_const S).const_mul (2/ε)).const_add (M+ε))
-    (fun t ht he => ?_) ⟨hST,le_rfl⟩
-  have hden := Complex.normSq_pos.2 (h.centered_ne_zero (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz))
-  have hu := hU t ⟨ht.1,ht.2.le⟩
-  have hx : ε ≤ (g t z).re-U t := by
-    have hh : 0 ≤ (2/ε)*(t-S) := mul_nonneg (by positivity) (sub_nonneg.2 ht.1)
-    dsimp only [Function.comp_apply] at he
-    linarith
-  have hy := h.im_pos (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz)
-  rw [div_lt_div_iff₀ hden hε]
-  simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
-  nlinarith [sq_pos_of_pos hy, mul_nonneg (sub_nonneg.2 hx) (show 0 ≤ (g t z).re-U t by linarith)]
-
-lemma later_real_ge (h : IsLoewnerTraceUsing U γ g f)
-    {S T m ε : ℝ} (hS : 0 < S) (hST : S ≤ T) (hε : 0 < ε)
-    {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hU : ∀ t ∈ Icc S T, m ≤ U t) (hstart : m ≤ (g S z).re) :
-    m-ε-(2/ε)*(T-S) ≤ (g T z).re := by
-  have hc := (Complex.continuous_re.comp_continuousOn
-    ((h.continuous_eval (hS.le.trans hST) hz).mono (Icc_subset_Icc hS.le le_rfl))).neg
-  have hd (t : ℝ) (ht : t ∈ Ico S T) :=
-    (h.hasDerivAt_real (hS.trans_le ht.1) (survivingDomain_mono ht.2.le hz)).neg
-  have hh := image_le_of_deriv_right_lt_deriv_boundary hc
-    (fun t ht => (hd t ht).hasDerivWithinAt)
-    (B:=fun t => -m+ε+(2/ε)*(t-S)) (B':=fun _ => 2/ε)
-    (by simpa using (show -(g S z).re ≤ -m+ε by linarith))
-    (fun t => by simpa using (((hasDerivAt_id t).sub_const S).const_mul (2/ε)).const_add (-m+ε))
-    (fun t ht he => ?_) (x:=T) ⟨hST,le_rfl⟩
-  · dsimp only [Function.comp_apply, Pi.neg_apply] at hh
-    linarith
-  · have hden := Complex.normSq_pos.2 (h.centered_ne_zero (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz))
-    have hu := hU t ⟨ht.1,ht.2.le⟩
-    have hx : ε ≤ U t-(g t z).re := by
-      have hh : 0 ≤ (2/ε)*(t-S) := mul_nonneg (by positivity) (sub_nonneg.2 ht.1)
-      dsimp only [Function.comp_apply, Pi.neg_apply] at he
-      linarith
-    have hy := h.im_pos (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz)
-    rw [← neg_div,div_lt_div_iff₀ hden hε]
-    simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
-    nlinarith [sq_pos_of_pos hy, mul_nonneg (sub_nonneg.2 hx) (show 0 ≤ U t-(g t z).re by linarith)]
-
-theorem earlier_tip_rectangle (h : IsLoewnerTraceUsing U γ g f)
-    {S T m M : ℝ} (hS : 0 ≤ S) (hST : S ≤ T)
-    (hlive : γ T ∈ localizationDomain γ S)
-    (hb : ∀ t ∈ Icc S T, m ≤ U t ∧ U t ≤ M) :
-    m ≤ (g S (γ T)).re ∧ (g S (γ T)).re ≤ M ∧ (g S (γ T)).im ^ 2 ≤ 4*(T-S) := by
-  have hT : 0 ≤ T := hS.trans hST
-  let w : ℝ → ℂ := fun y => (U T:ℂ)+(y:ℂ)*Complex.I
-  let z : ℝ → ℂ := fun y => f T (w y)
-  have hw (y : ℝ) (hy : 0 < y) : w y ∈ halfPlane := by simpa [w,halfPlane] using hy
-  have hz (y : ℝ) (hy : 0 < y) : z y ∈ localizationDomain γ T :=
-    h.2.2.2.2.2.2.2.2.1 T hT (hw y hy)
-  have he (y : ℝ) (hy : 0 < y) : g T (z y) = w y :=
-    h.2.2.2.2.2.2.2.2.2.2.1 T hT _ (hw y hy)
-  have hlim : Tendsto z (𝓝[>] (0:ℝ)) (𝓝 (γ T)) :=
-    h.2.2.2.2.2.2.2.2.2.2.2.2.2 T hT
-  have hlim' : Tendsto (fun y => g S (z y)) (𝓝[>] (0:ℝ)) (𝓝 (g S (γ T))) :=
-    (h.analytic_eval hS hlive).continuousAt.tendsto.comp hlim
-  have hleft (y : ℝ) (hy : 0 < y) : m ≤ (g S (z y)).re :=
-    h.earlier_real_ge hS hST (hz y hy) (fun s hs => (hb s hs).1)
-      (by simpa [he y hy,w] using (hb T ⟨hST,le_rfl⟩).1)
-  have hright (y : ℝ) (hy : 0 < y) : (g S (z y)).re ≤ M :=
-    h.earlier_real_le hS hST (hz y hy) (fun s hs => (hb s hs).2)
-      (by simpa [he y hy,w] using (hb T ⟨hST,le_rfl⟩).2)
-  have hheight (y : ℝ) (hy : 0 < y) : (g S (z y)).im^2 ≤ y^2+4*(T-S) := by
-    simpa [he y hy,w] using h.earlier_height_sq_le hS hST (hz y hy)
-  refine ⟨ge_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim') ?_,
-    le_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim') ?_,?_⟩
-  · filter_upwards [self_mem_nhdsWithin] with y hy using hleft y hy
-  · filter_upwards [self_mem_nhdsWithin] with y hy using hright y hy
-  · have hr : Tendsto (fun y : ℝ => y^2+4*(T-S)) (𝓝[>] (0:ℝ)) (𝓝 (4*(T-S))) := by
-      simpa using (show ContinuousAt (fun y : ℝ => y^2+4*(T-S)) 0 by fun_prop).tendsto.mono_left nhdsWithin_le_nhds
-    apply le_of_tendsto_of_tendsto ((Complex.continuous_im.continuousAt.tendsto.comp hlim').pow 2) hr
-    filter_upwards [self_mem_nhdsWithin] with y hy using hheight y hy
-
-end IsLoewnerTraceUsing
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge
-
-lemma ball_subset_localizationDomain {γ : ℝ → ℂ} {t : ℝ} {z : ℂ} {r : ℝ}
-    (hr : 0 < r) (hz : z ∈ localizationDomain γ t)
-    (hfree : ball z r ⊆ halfPlane \ localizationSegment γ 0 t) :
-    ball z r ⊆ localizationDomain γ t := by
-  have hc := (convex_ball z r).isPreconnected.subset_connectedComponentIn
-    (mem_ball_self hr) hfree
-  intro w hw
-  refine ⟨hfree hw,?_⟩
-  rw [← connectedComponentIn_eq (hc hw)]
-  exact hz.2
-
-lemma first_disk_hit_free {γ : ℝ → ℂ} {T r : ℝ} {z : ℂ}
-    (hr : 0 < r) (hball : closedBall z r ⊆ halfPlane)
-    (hlive : z ∈ localizationDomain γ T) (hhit : dist (γ T) z = r)
-    (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
-    ball z r ⊆ localizationDomain γ T ∧ ∀ s ∈ Ico 0 T,
-      closedBall z r ⊆ localizationDomain γ s := by
-  constructor
-  · apply ball_subset_localizationDomain hr hlive
-    intro w hw
-    refine ⟨hball (ball_subset_closedBall hw),?_⟩
-    rintro ⟨t,ht,he⟩
-    subst w
-    rcases lt_or_eq_of_le ht.2 with hlt | rfl
-    · exact (not_le_of_gt (hpre t ⟨ht.1,hlt⟩)) (mem_ball.1 hw).le
-    · exact (ne_of_lt (mem_ball.1 hw)) hhit
-  · intro s hs
-    apply closedBall_subset_survivingDomain hr.le (survivingDomain_mono hs.2.le hlive)
-    intro w hw
-    refine ⟨hball hw,?_⟩
-    rintro ⟨t,ht,he⟩
-    subst w
-    exact (not_le_of_gt (hpre t ⟨ht.1,ht.2.trans_lt hs.2⟩)) (mem_closedBall.1 hw)
-
-lemma first_disk_hit_live_before {γ : ℝ → ℂ} {T r : ℝ} {z : ℂ}
-    (hr : 0 < r) (hball : closedBall z r ⊆ halfPlane)
-    (hlive : z ∈ localizationDomain γ T) (hhit : dist (γ T) z = r)
-    (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
-    ∀ s ∈ Ico 0 T, γ T ∈ localizationDomain γ s := by
-  intro s hs
-  exact (first_disk_hit_free hr hball hlive hhit hpre).2 s hs (mem_closedBall.2 hhit.le)
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge
-namespace IsLoewnerTraceUsing
-variable {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-theorem fresh_tip_access (h : IsLoewnerTraceUsing U γ g f) {T : ℝ}
-    (hT : 0 < T) (hU : ContinuousOn U (Icc 0 T))
-    (hlive : ∀ s ∈ Ico 0 T, γ T ∈ localizationDomain γ s) :
-    Tendsto (g T) (𝓝[localizationDomain γ T] (γ T)) (𝓝 (U T : ℂ)) := by
-  apply Metric.tendsto_nhds.2
-  intro ε hε
-  let e := ε/10
-  have he : 0 < e := by dsimp [e]; positivity
-  obtain ⟨η,hη,hηU⟩ := Metric.continuousWithinAt_iff.1
-    (hU.continuousWithinAt ⟨hT.le,le_rfl⟩) e he
-  let δ := min η (e^2/8)
-  have hδ : 0 < δ := lt_min hη (by positivity)
-  obtain ⟨S,hSlo,hST⟩ := exists_between (max_lt hT (sub_lt_self T hδ))
-  have hS : 0 < S := (le_max_left _ _).trans_lt hSlo
-  have hSδ : T-δ < S := (le_max_right _ _).trans_lt hSlo
-  have htimeη : T-S < η := (by linarith : T-S < δ).trans_le (min_le_left _ _)
-  have htimee : T-S < e^2/8 := (by linarith : T-S < δ).trans_le (min_le_right _ _)
-  have hUb (s : ℝ) (hs : s ∈ Icc S T) : U T-e ≤ U s ∧ U s ≤ U T+e := by
-    have hh := hηU ⟨hS.le.trans hs.1,hs.2⟩ (show dist s T < η by
-      rw [Real.dist_eq,abs_of_nonpos (sub_nonpos.2 hs.2)]
-      linarith [hs.1])
-    rw [Real.dist_eq] at hh
-    have ha := abs_lt.1 hh
-    constructor <;> linarith [ha.1,ha.2]
-  have htip := h.earlier_tip_rectangle hS.le hST.le (hlive S ⟨hS.le,hST⟩) hUb
-  have htipim : (g S (γ T)).im < e := by
-    have hy := h.im_pos hS.le (hlive S ⟨hS.le,hST⟩)
-    nlinarith [htip.2.2]
-  have hevent : ∀ᶠ w in 𝓝[localizationDomain γ T] (γ T),
-      dist (g S w) (g S (γ T)) < e :=
-    (Metric.tendsto_nhds.1 (h.analytic_eval hS.le (hlive S ⟨hS.le,hST⟩)).continuousAt.tendsto e he).filter_mono nhdsWithin_le_nhds
-  filter_upwards [hevent,self_mem_nhdsWithin] with w hnear hw
-  rw [dist_eq_norm] at hnear
-  have hrnear : |(g S w).re-(g S (γ T)).re| < e := by
-    simpa only [Complex.sub_re] using (Complex.abs_re_le_norm (g S w-g S (γ T))).trans_lt hnear
-  have hinear : |(g S w).im-(g S (γ T)).im| < e := by
-    simpa only [Complex.sub_im] using (Complex.abs_im_le_norm (g S w-g S (γ T))).trans_lt hnear
-  have hstartlo : U T-2*e ≤ (g S w).re := by nlinarith [(abs_lt.1 hrnear).1,htip.1]
-  have hstarthi : (g S w).re ≤ U T+2*e := by nlinarith [(abs_lt.1 hrnear).2,htip.2.1]
-  have hlo := h.later_real_ge hS hST.le he hw
-    (fun s hs => (show U T-2*e ≤ U T-e by linarith).trans (hUb s hs).1) hstartlo
-  have hhi := h.later_real_le hS hST.le he hw
-    (fun s hs => (hUb s hs).2.trans (show U T+e ≤ U T+2*e by linarith)) hstarthi
-  have hdt : (2/e)*(T-S) < e := by
-    calc
-      _ = 2*(T-S)/e := by ring
-      _ < e := (div_lt_iff₀ he).2 (by nlinarith)
-  have hre : |(g T w).re-U T| < 4*e := by
-    rw [abs_lt]
-    constructor <;> linarith
-  have him : (g T w).im < 2*e := by
-    have hm := h.height_antitone hT.le hw ⟨hS.le,hST.le⟩ ⟨hT.le,le_rfl⟩ hST.le
-    linarith [(abs_lt.1 hinear).2]
-  have himabs : |(g T w).im| < 2*e := by rw [abs_of_pos (h.im_pos hT.le hw)]; exact him
-  rw [dist_eq_norm]
-  calc
-    _ ≤ |(g T w).re-U T|+|(g T w).im| := by
-      simpa only [Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero] using
-        Complex.norm_le_abs_re_add_abs_im (g T w-(U T:ℂ))
-    _ < 6*e := by linarith
-    _ < ε := by dsimp [e]; linarith
-
-end IsLoewnerTraceUsing
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge
-
-theorem first_disk_hit_access {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing U γ g f) {T r : ℝ} {z : ℂ}
-    (hT : 0 < T) (hr : 0 < r) (hU : ContinuousOn U (Icc 0 T))
-    (hball : Metric.closedBall z r ⊆ halfPlane)
-    (hlive : z ∈ localizationDomain γ T)
-    (hhit : dist (γ T) z = r)
-    (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
-    Tendsto (g T) (𝓝[localizationDomain γ T] (γ T)) (𝓝 (U T : ℂ)) :=
-  h.fresh_tip_access hT hU (first_disk_hit_live_before hr hball hlive hhit hpre)
-
-theorem first_disk_hit_access_path {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing U γ g f) {T r : ℝ} {z : ℂ}
-    (hT : 0 < T) (hr : 0 < r) (hU : ContinuousOn U (Icc 0 T))
-    (hball : closedBall z r ⊆ halfPlane) (hlive : z ∈ localizationDomain γ T)
-    (hhit : dist (γ T) z = r) (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
-    ∃ η : ℝ → ℂ, ContinuousOn η (Icc 0 1) ∧ η 0 = 0 ∧ ‖η 1‖ = 1 ∧
-      (∀ u ∈ Ioc 0 1, 0 < (η u).im ∧
-        f T ((U T:ℂ)+(‖g T z-(U T:ℂ)‖:ℂ)*η u) =
-          z+((1-u:ℝ):ℂ)*(γ T-z)) := by
-  classical
-  let R : ℝ → ℂ := fun u => z+((1-u:ℝ):ℂ)*(γ T-z)
-  let N := ‖g T z-(U T:ℂ)‖
-  let H : ℝ → ℂ := fun u => ((N⁻¹:ℝ):ℂ)*(g T (R u)-(U T:ℂ))
-  let η := Function.update H 0 0
-  have hN : 0 < N := norm_pos_iff.2 (h.centered_ne_zero hT.le hlive)
-  have hNc : (N:ℂ) ≠ 0 := by exact_mod_cast hN.ne'
-  have hR0 : R 0 = γ T := by simp [R]
-  have hR1 : R 1 = z := by simp [R]
-  have hRc : Continuous R := by dsimp [R]; fun_prop
-  have hRball (u : ℝ) (hu : u ∈ Ioc 0 1) : R u ∈ ball z r := by
-    have hdist : dist (R u) z = (1-u)*r := by
-      rw [dist_eq_norm]
-      dsimp [R]
-      rw [add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,
-        abs_of_nonneg (sub_nonneg.2 hu.2)]
-      rw [← dist_eq_norm,hhit]
-    rw [mem_ball,hdist]
-    nlinarith [mul_pos hu.1 hr]
-  have hRl (u : ℝ) (hu : u ∈ Ioc 0 1) : R u ∈ localizationDomain γ T :=
-    (first_disk_hit_free hr hball hlive hhit hpre).1 (hRball u hu)
-  have hlimR : Tendsto R (𝓝[Icc (0:ℝ) 1 \ {0}] 0) (𝓝[localizationDomain γ T] (γ T)) := by
-    apply tendsto_nhdsWithin_iff.2
-    refine ⟨?_,?_⟩
-    · have hh : Tendsto R (𝓝[Icc (0:ℝ) 1 \ {0}] 0) (𝓝 (R 0)) := hRc.continuousAt.tendsto.mono_left nhdsWithin_le_nhds
-      rw [hR0] at hh
-      exact hh
-    · filter_upwards [self_mem_nhdsWithin] with u hu
-      exact hRl u ⟨lt_of_le_of_ne hu.1.1 (Ne.symm hu.2),hu.1.2⟩
-  have hlimG := (first_disk_hit_access h hT hr hU hball hlive hhit hpre).comp hlimR
-  have hηcont : ContinuousOn η (Icc 0 1) := by
-    intro u hu
-    by_cases hu0 : u = 0
-    · subst u
-      rw [show η = Function.update H 0 0 from rfl,continuousWithinAt_update_same]
-      simpa only [H,Function.comp_def,sub_self,mul_zero] using (hlimG.sub (tendsto_const_nhds (x := (U T:ℂ)))).const_mul ((N⁻¹:ℝ):ℂ)
-    · have hcH : ContinuousAt H u := by
-        exact continuousAt_const.mul (((h.analytic_eval hT.le (hRl u ⟨lt_of_le_of_ne hu.1 (Ne.symm hu0),hu.2⟩)).continuousAt.comp hRc.continuousAt).sub continuousAt_const)
-      apply hcH.continuousWithinAt.congr_of_eventuallyEq_of_mem _ hu
-      have hne : ∀ᶠ v in 𝓝 u, v ≠ (0:ℝ) := isOpen_ne.mem_nhds hu0
-      filter_upwards [hne.filter_mono nhdsWithin_le_nhds] with v hv
-      exact Function.update_of_ne hv _ _
-  refine ⟨η,hηcont,by simp [η],?_,?_⟩
-  · rw [show η 1 = H 1 by simp [η]]
-    dsimp [H]
-    rw [hR1,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (inv_pos.2 hN)]
-    exact inv_mul_cancel₀ hN.ne'
-  · intro u hu
-    have hη : η u = H u := Function.update_of_ne (ne_of_gt hu.1) _ _
-    rw [hη]
-    constructor
-    · have hy := h.im_pos hT.le (hRl u hu)
-      dsimp [H]
-      simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,mul_zero,
-        add_zero,Complex.sub_im,sub_zero] using mul_pos (inv_pos.2 hN) hy
-    · have heq : (U T:ℂ)+(N:ℂ)*H u = g T (R u) := by
-        dsimp [H]
-        rw [Complex.ofReal_inv,← mul_assoc,mul_inv_cancel₀ hNc,one_mul]
-        ring
-      change f T ((U T:ℂ)+(N:ℂ)*H u) = R u
-      rw [heq]
-      exact h.2.2.2.2.2.2.2.2.2.1 T hT.le _ (hRl u hu)
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.LoewnerExit
+import OAI.Probability.SLELocalization.PhysicalPast
+
+namespace OAI
+
+
+/-! Conformal localization and Is Loewner Trace Using estimates. -/
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge
+namespace IsLoewnerTraceUsing
+variable {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma hasDerivAt_real (h : IsLoewnerTraceUsing U γ g f)
+    {t : ℝ} (ht : 0 < t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    HasDerivAt (fun s => (g s z).re)
+      (2 * ((g t z).re-U t) / Complex.normSq (g t z-(U t:ℂ))) t := by
+  have hh := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t (h.hasDerivAt_eval ht hz)
+  simpa [Complex.div_re,Function.comp_def] using hh
+
+lemma initial_real_le (h : IsLoewnerTraceUsing U γ g f)
+    {T M : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hU : ∀ t ∈ Icc 0 T, U t ≤ M) (hend : (g T z).re ≤ M) : z.re ≤ M := by
+  let R : ℝ → ℝ := fun s => (g (T-s) z).re
+  have hc : ContinuousOn R (Icc 0 T) :=
+    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).comp
+      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
+  have hd (s : ℝ) (hs : s ∈ Ico 0 T) : HasDerivAt R
+      (-2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
+    have he := (h.hasDerivAt_real (sub_pos.2 hs.2)
+      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
+      ((hasDerivAt_id s).const_sub T)
+    simpa [R,neg_div,Function.comp_def] using he
+  apply le_of_forall_pos_le_add
+  intro ε hε
+  have hb := image_le_of_deriv_right_lt_deriv_boundary hc
+    (fun s hs => (hd s hs).hasDerivWithinAt)
+    (B:=fun _ => M+ε) (B':=fun _ => 0)
+    (by dsimp [R]; simpa using hend.trans (by linarith : M ≤ M+ε))
+    (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T) ⟨hT,le_rfl⟩
+  · simpa [R,h.initial z hz.1.1] using hb
+  · have hden := Complex.normSq_pos.2
+      (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
+    have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
+    dsimp [R] at he
+    exact div_neg_of_neg_of_pos (by nlinarith) hden
+
+lemma initial_real_ge (h : IsLoewnerTraceUsing U γ g f)
+    {T m : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hU : ∀ t ∈ Icc 0 T, m ≤ U t) (hend : m ≤ (g T z).re) : m ≤ z.re := by
+  let R : ℝ → ℝ := fun s => -(g (T-s) z).re
+  have hc : ContinuousOn R (Icc 0 T) :=
+    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).neg.comp
+      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
+  have hd (s : ℝ) (hs : s ∈ Ico 0 T) : HasDerivAt R
+      (2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
+    have he := ((h.hasDerivAt_real (sub_pos.2 hs.2)
+      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
+      ((hasDerivAt_id s).const_sub T)).neg
+    convert he using 1
+    all_goals first | rfl | norm_num
+  have hn : -z.re ≤ -m := by
+    apply le_of_forall_pos_le_add
+    intro ε hε
+    have hb := image_le_of_deriv_right_lt_deriv_boundary hc
+      (fun s hs => (hd s hs).hasDerivWithinAt)
+      (B:=fun _ => -m+ε) (B':=fun _ => 0)
+      (by dsimp [R]; simpa using (show -(g T z).re ≤ -m+ε by linarith))
+      (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T) ⟨hT,le_rfl⟩
+    · simpa [R,h.initial z hz.1.1] using hb
+    · have hden := Complex.normSq_pos.2
+        (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
+      have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
+      dsimp [R] at he
+      exact div_neg_of_neg_of_pos (by nlinarith) hden
+  linarith
+
+lemma initial_height_sq_le (h : IsLoewnerTraceUsing U γ g f)
+    {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    z.im^2 ≤ (g T z).im^2+4*T := by
+  have hm : MonotoneOn (fun s => (g s z).im^2+4*s) (Icc 0 T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 T)
+      (((Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)).pow 2).add
+        (continuous_const.mul continuous_id).continuousOn)
+      (f':=fun t => -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ))+4)
+    · intro t ht
+      rw [interior_Icc] at ht
+      convert (((h.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hz)).pow 2).add
+        ((hasDerivAt_id t).const_mul 4)).hasDerivWithinAt using 1
+      all_goals first | rfl | (norm_num; ring)
+    · intro t ht
+      rw [interior_Icc] at ht
+      have hd := Complex.normSq_pos.2 (h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hz))
+      have he : (g t z).im^2 ≤ Complex.normSq (g t z-(U t:ℂ)) := by
+        simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
+        nlinarith [sq_nonneg ((g t z).re-U t)]
+      have hb : -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ)) ≥ -4 := by
+        exact (le_div_iff₀ hd).2 (by nlinarith)
+      linarith
+  simpa [h.initial z hz.1.1] using hm ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
+
+end IsLoewnerTraceUsing
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge
+
+theorem trace_driver_rectangle
+    {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing U γ g f) {T m M : ℝ} (_hT : 0 ≤ T)
+    (_hU : ContinuousOn U (Icc 0 T))
+    (hb : ∀ t ∈ Icc 0 T, m ≤ U t ∧ U t ≤ M) :
+    ∀ t ∈ Icc 0 T, m ≤ (γ t).re ∧ (γ t).re ≤ M ∧ (γ t).im ^ 2 ≤ 4*t := by
+  intro t ht
+  let w : ℝ → ℂ := fun y => (U t:ℂ)+(y:ℂ)*Complex.I
+  let z : ℝ → ℂ := fun y => f t (w y)
+  have hw (y : ℝ) (hy : 0 < y) : w y ∈ halfPlane := by simpa [w,halfPlane] using hy
+  have hz (y : ℝ) (hy : 0 < y) : z y ∈ localizationDomain γ t :=
+    h.2.2.2.2.2.2.2.2.1 t ht.1 (hw y hy)
+  have he (y : ℝ) (hy : 0 < y) : g t (z y) = w y :=
+    h.2.2.2.2.2.2.2.2.2.2.1 t ht.1 _ (hw y hy)
+  have hlim : Tendsto z (𝓝[>] (0:ℝ)) (𝓝 (γ t)) :=
+    h.2.2.2.2.2.2.2.2.2.2.2.2.2 t ht.1
+  have hleft (y : ℝ) (hy : 0 < y) : m ≤ (z y).re :=
+    h.initial_real_ge ht.1 (hz y hy)
+      (fun s hs => (hb s ⟨hs.1,hs.2.trans ht.2⟩).1)
+      (by simpa [he y hy,w] using (hb t ht).1)
+  have hright (y : ℝ) (hy : 0 < y) : (z y).re ≤ M :=
+    h.initial_real_le ht.1 (hz y hy)
+      (fun s hs => (hb s ⟨hs.1,hs.2.trans ht.2⟩).2)
+      (by simpa [he y hy,w] using (hb t ht).2)
+  have hheight (y : ℝ) (hy : 0 < y) : (z y).im^2 ≤ y^2+4*t := by
+    simpa [he y hy,w] using h.initial_height_sq_le ht.1 (hz y hy)
+  refine ⟨ge_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim) ?_,
+    le_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim) ?_,?_⟩
+  · filter_upwards [self_mem_nhdsWithin] with y hy using hleft y hy
+  · filter_upwards [self_mem_nhdsWithin] with y hy using hright y hy
+  · have hr : Tendsto (fun y : ℝ => y^2+4*t) (𝓝[>] (0:ℝ)) (𝓝 (4*t)) := by
+      simpa using (show ContinuousAt (fun y : ℝ => y^2+4*t) 0 by fun_prop).tendsto.mono_left nhdsWithin_le_nhds
+    apply le_of_tendsto_of_tendsto ((Complex.continuous_im.continuousAt.tendsto.comp hlim).pow 2) hr
+    filter_upwards [self_mem_nhdsWithin] with y hy using hheight y hy
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge
+namespace IsLoewnerTraceUsing
+variable {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma earlier_real_le (h : IsLoewnerTraceUsing U γ g f)
+    {S T M : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hU : ∀ t ∈ Icc S T, U t ≤ M) (hend : (g T z).re ≤ M) : (g S z).re ≤ M := by
+  have hT : 0 ≤ T := hS.trans hST
+  let R : ℝ → ℝ := fun s => (g (T-s) z).re
+  have hc : ContinuousOn R (Icc 0 (T-S)) :=
+    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).comp
+      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
+  have hd (s : ℝ) (hs : s ∈ Ico 0 (T-S)) : HasDerivAt R
+      (-2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
+    have he := (h.hasDerivAt_real (by linarith [hs.2] : 0 < T-s)
+      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
+      ((hasDerivAt_id s).const_sub T)
+    simpa [R,neg_div,Function.comp_def] using he
+  apply le_of_forall_pos_le_add
+  intro ε hε
+  have hb := image_le_of_deriv_right_lt_deriv_boundary hc
+    (fun s hs => (hd s hs).hasDerivWithinAt)
+    (B:=fun _ => M+ε) (B':=fun _ => 0)
+    (by dsimp [R]; simpa using hend.trans (by linarith : M ≤ M+ε))
+    (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T-S) ⟨sub_nonneg.2 hST,le_rfl⟩
+  · simpa [R,sub_sub_cancel] using hb
+  · have hden := Complex.normSq_pos.2
+      (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
+    have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
+    dsimp [R] at he
+    exact div_neg_of_neg_of_pos (by nlinarith) hden
+
+lemma earlier_real_ge (h : IsLoewnerTraceUsing U γ g f)
+    {S T m : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hU : ∀ t ∈ Icc S T, m ≤ U t) (hend : m ≤ (g T z).re) : m ≤ (g S z).re := by
+  have hT : 0 ≤ T := hS.trans hST
+  let R : ℝ → ℝ := fun s => -(g (T-s) z).re
+  have hc : ContinuousOn R (Icc 0 (T-S)) :=
+    (Complex.continuous_re.comp_continuousOn (h.continuous_eval hT hz)).neg.comp
+      (continuous_const.sub continuous_id).continuousOn (fun s hs => by constructor <;> linarith [hs.1,hs.2])
+  have hd (s : ℝ) (hs : s ∈ Ico 0 (T-S)) : HasDerivAt R
+      (2*((g (T-s) z).re-U (T-s))/Complex.normSq (g (T-s) z-(U (T-s):ℂ))) s := by
+    have he := ((h.hasDerivAt_real (by linarith [hs.2] : 0 < T-s)
+      (survivingDomain_mono (by linarith [hs.1]) hz)).comp s
+      ((hasDerivAt_id s).const_sub T)).neg
+    convert he using 1
+    all_goals first | rfl | norm_num
+  have hn : -(g S z).re ≤ -m := by
+    apply le_of_forall_pos_le_add
+    intro ε hε
+    have hb := image_le_of_deriv_right_lt_deriv_boundary hc
+      (fun s hs => (hd s hs).hasDerivWithinAt)
+      (B:=fun _ => -m+ε) (B':=fun _ => 0)
+      (by dsimp [R]; simpa using (show -(g T z).re ≤ -m+ε by linarith))
+      (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) (x:=T-S) ⟨sub_nonneg.2 hST,le_rfl⟩
+    · simpa [R,sub_sub_cancel] using hb
+    · have hden := Complex.normSq_pos.2
+        (h.centered_ne_zero (t:=T-s) (z:=z) (by linarith [hs.2]) (survivingDomain_mono (by linarith [hs.1]) hz))
+      have hh := hU (T-s) (by constructor <;> linarith [hs.1,hs.2])
+      dsimp [R] at he
+      exact div_neg_of_neg_of_pos (by nlinarith) hden
+  linarith
+
+lemma earlier_height_sq_le (h : IsLoewnerTraceUsing U γ g f)
+    {S T : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    (g S z).im^2 ≤ (g T z).im^2+4*(T-S) := by
+  have hT : 0 ≤ T := hS.trans hST
+  have hcont := (h.continuous_eval hT hz).mono (Icc_subset_Icc hS le_rfl)
+  have hm : MonotoneOn (fun s => (g s z).im^2+4*s) (Icc S T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc S T)
+      (((Complex.continuous_im.comp_continuousOn hcont).pow 2).add
+        (continuous_const.mul continuous_id).continuousOn)
+      (f':=fun t => -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ))+4)
+    · intro t ht
+      rw [interior_Icc] at ht
+      convert (((h.hasDerivAt_height (hS.trans_lt ht.1) (survivingDomain_mono ht.2.le hz)).pow 2).add
+        ((hasDerivAt_id t).const_mul 4)).hasDerivWithinAt using 1
+      all_goals first | rfl | (norm_num; ring)
+    · intro t ht
+      rw [interior_Icc] at ht
+      have hd := Complex.normSq_pos.2 (h.centered_ne_zero (hS.trans ht.1.le) (survivingDomain_mono ht.2.le hz))
+      have he : (g t z).im^2 ≤ Complex.normSq (g t z-(U t:ℂ)) := by
+        simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
+        nlinarith [sq_nonneg ((g t z).re-U t)]
+      have hb : -4*(g t z).im^2/Complex.normSq (g t z-(U t:ℂ)) ≥ -4 := by
+        exact (le_div_iff₀ hd).2 (by nlinarith)
+      linarith
+  have hh := hm ⟨le_rfl,hST⟩ ⟨hST,le_rfl⟩ hST
+  linarith
+
+lemma later_real_le (h : IsLoewnerTraceUsing U γ g f)
+    {S T M ε : ℝ} (hS : 0 < S) (hST : S ≤ T) (hε : 0 < ε)
+    {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hU : ∀ t ∈ Icc S T, U t ≤ M) (hstart : (g S z).re ≤ M) :
+    (g T z).re ≤ M+ε+(2/ε)*(T-S) := by
+  have hc := Complex.continuous_re.comp_continuousOn
+    ((h.continuous_eval (hS.le.trans hST) hz).mono (Icc_subset_Icc hS.le le_rfl))
+  have hd (t : ℝ) (ht : t ∈ Ico S T) :=
+    h.hasDerivAt_real (hS.trans_le ht.1) (survivingDomain_mono ht.2.le hz)
+  apply image_le_of_deriv_right_lt_deriv_boundary hc
+    (fun t ht => (hd t ht).hasDerivWithinAt)
+    (B:=fun t => M+ε+(2/ε)*(t-S)) (B':=fun _ => 2/ε)
+    (by simpa using hstart.trans (show M ≤ M+ε by linarith))
+    (fun t => by simpa using (((hasDerivAt_id t).sub_const S).const_mul (2/ε)).const_add (M+ε))
+    (fun t ht he => ?_) ⟨hST,le_rfl⟩
+  have hden := Complex.normSq_pos.2 (h.centered_ne_zero (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz))
+  have hu := hU t ⟨ht.1,ht.2.le⟩
+  have hx : ε ≤ (g t z).re-U t := by
+    have hh : 0 ≤ (2/ε)*(t-S) := mul_nonneg (by positivity) (sub_nonneg.2 ht.1)
+    dsimp only [Function.comp_apply] at he
+    linarith
+  have hy := h.im_pos (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz)
+  rw [div_lt_div_iff₀ hden hε]
+  simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
+  nlinarith [sq_pos_of_pos hy, mul_nonneg (sub_nonneg.2 hx) (show 0 ≤ (g t z).re-U t by linarith)]
+
+lemma later_real_ge (h : IsLoewnerTraceUsing U γ g f)
+    {S T m ε : ℝ} (hS : 0 < S) (hST : S ≤ T) (hε : 0 < ε)
+    {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hU : ∀ t ∈ Icc S T, m ≤ U t) (hstart : m ≤ (g S z).re) :
+    m-ε-(2/ε)*(T-S) ≤ (g T z).re := by
+  have hc := (Complex.continuous_re.comp_continuousOn
+    ((h.continuous_eval (hS.le.trans hST) hz).mono (Icc_subset_Icc hS.le le_rfl))).neg
+  have hd (t : ℝ) (ht : t ∈ Ico S T) :=
+    (h.hasDerivAt_real (hS.trans_le ht.1) (survivingDomain_mono ht.2.le hz)).neg
+  have hh := image_le_of_deriv_right_lt_deriv_boundary hc
+    (fun t ht => (hd t ht).hasDerivWithinAt)
+    (B:=fun t => -m+ε+(2/ε)*(t-S)) (B':=fun _ => 2/ε)
+    (by simpa using (show -(g S z).re ≤ -m+ε by linarith))
+    (fun t => by simpa using (((hasDerivAt_id t).sub_const S).const_mul (2/ε)).const_add (-m+ε))
+    (fun t ht he => ?_) (x:=T) ⟨hST,le_rfl⟩
+  · dsimp only [Function.comp_apply, Pi.neg_apply] at hh
+    linarith
+  · have hden := Complex.normSq_pos.2 (h.centered_ne_zero (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz))
+    have hu := hU t ⟨ht.1,ht.2.le⟩
+    have hx : ε ≤ U t-(g t z).re := by
+      have hh : 0 ≤ (2/ε)*(t-S) := mul_nonneg (by positivity) (sub_nonneg.2 ht.1)
+      dsimp only [Function.comp_apply, Pi.neg_apply] at he
+      linarith
+    have hy := h.im_pos (hS.le.trans ht.1) (survivingDomain_mono ht.2.le hz)
+    rw [← neg_div,div_lt_div_iff₀ hden hε]
+    simp only [Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero]
+    nlinarith [sq_pos_of_pos hy, mul_nonneg (sub_nonneg.2 hx) (show 0 ≤ U t-(g t z).re by linarith)]
+
+theorem earlier_tip_rectangle (h : IsLoewnerTraceUsing U γ g f)
+    {S T m M : ℝ} (hS : 0 ≤ S) (hST : S ≤ T)
+    (hlive : γ T ∈ localizationDomain γ S)
+    (hb : ∀ t ∈ Icc S T, m ≤ U t ∧ U t ≤ M) :
+    m ≤ (g S (γ T)).re ∧ (g S (γ T)).re ≤ M ∧ (g S (γ T)).im ^ 2 ≤ 4*(T-S) := by
+  have hT : 0 ≤ T := hS.trans hST
+  let w : ℝ → ℂ := fun y => (U T:ℂ)+(y:ℂ)*Complex.I
+  let z : ℝ → ℂ := fun y => f T (w y)
+  have hw (y : ℝ) (hy : 0 < y) : w y ∈ halfPlane := by simpa [w,halfPlane] using hy
+  have hz (y : ℝ) (hy : 0 < y) : z y ∈ localizationDomain γ T :=
+    h.2.2.2.2.2.2.2.2.1 T hT (hw y hy)
+  have he (y : ℝ) (hy : 0 < y) : g T (z y) = w y :=
+    h.2.2.2.2.2.2.2.2.2.2.1 T hT _ (hw y hy)
+  have hlim : Tendsto z (𝓝[>] (0:ℝ)) (𝓝 (γ T)) :=
+    h.2.2.2.2.2.2.2.2.2.2.2.2.2 T hT
+  have hlim' : Tendsto (fun y => g S (z y)) (𝓝[>] (0:ℝ)) (𝓝 (g S (γ T))) :=
+    (h.analytic_eval hS hlive).continuousAt.tendsto.comp hlim
+  have hleft (y : ℝ) (hy : 0 < y) : m ≤ (g S (z y)).re :=
+    h.earlier_real_ge hS hST (hz y hy) (fun s hs => (hb s hs).1)
+      (by simpa [he y hy,w] using (hb T ⟨hST,le_rfl⟩).1)
+  have hright (y : ℝ) (hy : 0 < y) : (g S (z y)).re ≤ M :=
+    h.earlier_real_le hS hST (hz y hy) (fun s hs => (hb s hs).2)
+      (by simpa [he y hy,w] using (hb T ⟨hST,le_rfl⟩).2)
+  have hheight (y : ℝ) (hy : 0 < y) : (g S (z y)).im^2 ≤ y^2+4*(T-S) := by
+    simpa [he y hy,w] using h.earlier_height_sq_le hS hST (hz y hy)
+  refine ⟨ge_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim') ?_,
+    le_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp hlim') ?_,?_⟩
+  · filter_upwards [self_mem_nhdsWithin] with y hy using hleft y hy
+  · filter_upwards [self_mem_nhdsWithin] with y hy using hright y hy
+  · have hr : Tendsto (fun y : ℝ => y^2+4*(T-S)) (𝓝[>] (0:ℝ)) (𝓝 (4*(T-S))) := by
+      simpa using (show ContinuousAt (fun y : ℝ => y^2+4*(T-S)) 0 by fun_prop).tendsto.mono_left nhdsWithin_le_nhds
+    apply le_of_tendsto_of_tendsto ((Complex.continuous_im.continuousAt.tendsto.comp hlim').pow 2) hr
+    filter_upwards [self_mem_nhdsWithin] with y hy using hheight y hy
+
+end IsLoewnerTraceUsing
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge
+
+lemma ball_subset_localizationDomain {γ : ℝ → ℂ} {t : ℝ} {z : ℂ} {r : ℝ}
+    (hr : 0 < r) (hz : z ∈ localizationDomain γ t)
+    (hfree : ball z r ⊆ halfPlane \ localizationSegment γ 0 t) :
+    ball z r ⊆ localizationDomain γ t := by
+  have hc := (convex_ball z r).isPreconnected.subset_connectedComponentIn
+    (mem_ball_self hr) hfree
+  intro w hw
+  refine ⟨hfree hw,?_⟩
+  rw [← connectedComponentIn_eq (hc hw)]
+  exact hz.2
+
+lemma first_disk_hit_free {γ : ℝ → ℂ} {T r : ℝ} {z : ℂ}
+    (hr : 0 < r) (hball : closedBall z r ⊆ halfPlane)
+    (hlive : z ∈ localizationDomain γ T) (hhit : dist (γ T) z = r)
+    (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
+    ball z r ⊆ localizationDomain γ T ∧ ∀ s ∈ Ico 0 T,
+      closedBall z r ⊆ localizationDomain γ s := by
+  constructor
+  · apply ball_subset_localizationDomain hr hlive
+    intro w hw
+    refine ⟨hball (ball_subset_closedBall hw),?_⟩
+    rintro ⟨t,ht,he⟩
+    subst w
+    rcases lt_or_eq_of_le ht.2 with hlt | rfl
+    · exact (not_le_of_gt (hpre t ⟨ht.1,hlt⟩)) (mem_ball.1 hw).le
+    · exact (ne_of_lt (mem_ball.1 hw)) hhit
+  · intro s hs
+    apply closedBall_subset_survivingDomain hr.le (survivingDomain_mono hs.2.le hlive)
+    intro w hw
+    refine ⟨hball hw,?_⟩
+    rintro ⟨t,ht,he⟩
+    subst w
+    exact (not_le_of_gt (hpre t ⟨ht.1,ht.2.trans_lt hs.2⟩)) (mem_closedBall.1 hw)
+
+lemma first_disk_hit_live_before {γ : ℝ → ℂ} {T r : ℝ} {z : ℂ}
+    (hr : 0 < r) (hball : closedBall z r ⊆ halfPlane)
+    (hlive : z ∈ localizationDomain γ T) (hhit : dist (γ T) z = r)
+    (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
+    ∀ s ∈ Ico 0 T, γ T ∈ localizationDomain γ s := by
+  intro s hs
+  exact (first_disk_hit_free hr hball hlive hhit hpre).2 s hs (mem_closedBall.2 hhit.le)
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge
+namespace IsLoewnerTraceUsing
+variable {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+theorem fresh_tip_access (h : IsLoewnerTraceUsing U γ g f) {T : ℝ}
+    (hT : 0 < T) (hU : ContinuousOn U (Icc 0 T))
+    (hlive : ∀ s ∈ Ico 0 T, γ T ∈ localizationDomain γ s) :
+    Tendsto (g T) (𝓝[localizationDomain γ T] (γ T)) (𝓝 (U T : ℂ)) := by
+  apply Metric.tendsto_nhds.2
+  intro ε hε
+  let e := ε/10
+  have he : 0 < e := by dsimp [e]; positivity
+  obtain ⟨η,hη,hηU⟩ := Metric.continuousWithinAt_iff.1
+    (hU.continuousWithinAt ⟨hT.le,le_rfl⟩) e he
+  let δ := min η (e^2/8)
+  have hδ : 0 < δ := lt_min hη (by positivity)
+  obtain ⟨S,hSlo,hST⟩ := exists_between (max_lt hT (sub_lt_self T hδ))
+  have hS : 0 < S := (le_max_left _ _).trans_lt hSlo
+  have hSδ : T-δ < S := (le_max_right _ _).trans_lt hSlo
+  have htimeη : T-S < η := (by linarith : T-S < δ).trans_le (min_le_left _ _)
+  have htimee : T-S < e^2/8 := (by linarith : T-S < δ).trans_le (min_le_right _ _)
+  have hUb (s : ℝ) (hs : s ∈ Icc S T) : U T-e ≤ U s ∧ U s ≤ U T+e := by
+    have hh := hηU ⟨hS.le.trans hs.1,hs.2⟩ (show dist s T < η by
+      rw [Real.dist_eq,abs_of_nonpos (sub_nonpos.2 hs.2)]
+      linarith [hs.1])
+    rw [Real.dist_eq] at hh
+    have ha := abs_lt.1 hh
+    constructor <;> linarith [ha.1,ha.2]
+  have htip := h.earlier_tip_rectangle hS.le hST.le (hlive S ⟨hS.le,hST⟩) hUb
+  have htipim : (g S (γ T)).im < e := by
+    have hy := h.im_pos hS.le (hlive S ⟨hS.le,hST⟩)
+    nlinarith [htip.2.2]
+  have hevent : ∀ᶠ w in 𝓝[localizationDomain γ T] (γ T),
+      dist (g S w) (g S (γ T)) < e :=
+    (Metric.tendsto_nhds.1 (h.analytic_eval hS.le (hlive S ⟨hS.le,hST⟩)).continuousAt.tendsto e he).filter_mono nhdsWithin_le_nhds
+  filter_upwards [hevent,self_mem_nhdsWithin] with w hnear hw
+  rw [dist_eq_norm] at hnear
+  have hrnear : |(g S w).re-(g S (γ T)).re| < e := by
+    simpa only [Complex.sub_re] using (Complex.abs_re_le_norm (g S w-g S (γ T))).trans_lt hnear
+  have hinear : |(g S w).im-(g S (γ T)).im| < e := by
+    simpa only [Complex.sub_im] using (Complex.abs_im_le_norm (g S w-g S (γ T))).trans_lt hnear
+  have hstartlo : U T-2*e ≤ (g S w).re := by nlinarith [(abs_lt.1 hrnear).1,htip.1]
+  have hstarthi : (g S w).re ≤ U T+2*e := by nlinarith [(abs_lt.1 hrnear).2,htip.2.1]
+  have hlo := h.later_real_ge hS hST.le he hw
+    (fun s hs => (show U T-2*e ≤ U T-e by linarith).trans (hUb s hs).1) hstartlo
+  have hhi := h.later_real_le hS hST.le he hw
+    (fun s hs => (hUb s hs).2.trans (show U T+e ≤ U T+2*e by linarith)) hstarthi
+  have hdt : (2/e)*(T-S) < e := by
+    calc
+      _ = 2*(T-S)/e := by ring
+      _ < e := (div_lt_iff₀ he).2 (by nlinarith)
+  have hre : |(g T w).re-U T| < 4*e := by
+    rw [abs_lt]
+    constructor <;> linarith
+  have him : (g T w).im < 2*e := by
+    have hm := h.height_antitone hT.le hw ⟨hS.le,hST.le⟩ ⟨hT.le,le_rfl⟩ hST.le
+    linarith [(abs_lt.1 hinear).2]
+  have himabs : |(g T w).im| < 2*e := by rw [abs_of_pos (h.im_pos hT.le hw)]; exact him
+  rw [dist_eq_norm]
+  calc
+    _ ≤ |(g T w).re-U T|+|(g T w).im| := by
+      simpa only [Complex.sub_re,Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero] using
+        Complex.norm_le_abs_re_add_abs_im (g T w-(U T:ℂ))
+    _ < 6*e := by linarith
+    _ < ε := by dsimp [e]; linarith
+
+end IsLoewnerTraceUsing
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge
+
+theorem first_disk_hit_access {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing U γ g f) {T r : ℝ} {z : ℂ}
+    (hT : 0 < T) (hr : 0 < r) (hU : ContinuousOn U (Icc 0 T))
+    (hball : Metric.closedBall z r ⊆ halfPlane)
+    (hlive : z ∈ localizationDomain γ T)
+    (hhit : dist (γ T) z = r)
+    (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
+    Tendsto (g T) (𝓝[localizationDomain γ T] (γ T)) (𝓝 (U T : ℂ)) :=
+  h.fresh_tip_access hT hU (first_disk_hit_live_before hr hball hlive hhit hpre)
+
+theorem first_disk_hit_access_path {U : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing U γ g f) {T r : ℝ} {z : ℂ}
+    (hT : 0 < T) (hr : 0 < r) (hU : ContinuousOn U (Icc 0 T))
+    (hball : closedBall z r ⊆ halfPlane) (hlive : z ∈ localizationDomain γ T)
+    (hhit : dist (γ T) z = r) (hpre : ∀ t ∈ Ico 0 T, r < dist (γ t) z) :
+    ∃ η : ℝ → ℂ, ContinuousOn η (Icc 0 1) ∧ η 0 = 0 ∧ ‖η 1‖ = 1 ∧
+      (∀ u ∈ Ioc 0 1, 0 < (η u).im ∧
+        f T ((U T:ℂ)+(‖g T z-(U T:ℂ)‖:ℂ)*η u) =
+          z+((1-u:ℝ):ℂ)*(γ T-z)) := by
+  classical
+  let R : ℝ → ℂ := fun u => z+((1-u:ℝ):ℂ)*(γ T-z)
+  let N := ‖g T z-(U T:ℂ)‖
+  let H : ℝ → ℂ := fun u => ((N⁻¹:ℝ):ℂ)*(g T (R u)-(U T:ℂ))
+  let η := Function.update H 0 0
+  have hN : 0 < N := norm_pos_iff.2 (h.centered_ne_zero hT.le hlive)
+  have hNc : (N:ℂ) ≠ 0 := by exact_mod_cast hN.ne'
+  have hR0 : R 0 = γ T := by simp [R]
+  have hR1 : R 1 = z := by simp [R]
+  have hRc : Continuous R := by dsimp [R]; fun_prop
+  have hRball (u : ℝ) (hu : u ∈ Ioc 0 1) : R u ∈ ball z r := by
+    have hdist : dist (R u) z = (1-u)*r := by
+      rw [dist_eq_norm]
+      dsimp [R]
+      rw [add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+        abs_of_nonneg (sub_nonneg.2 hu.2)]
+      rw [← dist_eq_norm,hhit]
+    rw [mem_ball,hdist]
+    nlinarith [mul_pos hu.1 hr]
+  have hRl (u : ℝ) (hu : u ∈ Ioc 0 1) : R u ∈ localizationDomain γ T :=
+    (first_disk_hit_free hr hball hlive hhit hpre).1 (hRball u hu)
+  have hlimR : Tendsto R (𝓝[Icc (0:ℝ) 1 \ {0}] 0) (𝓝[localizationDomain γ T] (γ T)) := by
+    apply tendsto_nhdsWithin_iff.2
+    refine ⟨?_,?_⟩
+    · have hh : Tendsto R (𝓝[Icc (0:ℝ) 1 \ {0}] 0) (𝓝 (R 0)) := hRc.continuousAt.tendsto.mono_left nhdsWithin_le_nhds
+      rw [hR0] at hh
+      exact hh
+    · filter_upwards [self_mem_nhdsWithin] with u hu
+      exact hRl u ⟨lt_of_le_of_ne hu.1.1 (Ne.symm hu.2),hu.1.2⟩
+  have hlimG := (first_disk_hit_access h hT hr hU hball hlive hhit hpre).comp hlimR
+  have hηcont : ContinuousOn η (Icc 0 1) := by
+    intro u hu
+    by_cases hu0 : u = 0
+    · subst u
+      rw [show η = Function.update H 0 0 from rfl,continuousWithinAt_update_same]
+      simpa only [H,Function.comp_def,sub_self,mul_zero] using (hlimG.sub (tendsto_const_nhds (x := (U T:ℂ)))).const_mul ((N⁻¹:ℝ):ℂ)
+    · have hcH : ContinuousAt H u := by
+        exact continuousAt_const.mul (((h.analytic_eval hT.le (hRl u ⟨lt_of_le_of_ne hu.1 (Ne.symm hu0),hu.2⟩)).continuousAt.comp hRc.continuousAt).sub continuousAt_const)
+      apply hcH.continuousWithinAt.congr_of_eventuallyEq_of_mem _ hu
+      have hne : ∀ᶠ v in 𝓝 u, v ≠ (0:ℝ) := isOpen_ne.mem_nhds hu0
+      filter_upwards [hne.filter_mono nhdsWithin_le_nhds] with v hv
+      exact Function.update_of_ne hv _ _
+  refine ⟨η,hηcont,by simp [η],?_,?_⟩
+  · rw [show η 1 = H 1 by simp [η]]
+    dsimp [H]
+    rw [hR1,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (inv_pos.2 hN)]
+    exact inv_mul_cancel₀ hN.ne'
+  · intro u hu
+    have hη : η u = H u := Function.update_of_ne (ne_of_gt hu.1) _ _
+    rw [hη]
+    constructor
+    · have hy := h.im_pos hT.le (hRl u hu)
+      dsimp [H]
+      simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,mul_zero,
+        add_zero,Complex.sub_im,sub_zero] using mul_pos (inv_pos.2 hN) hy
+    · have heq : (U T:ℂ)+(N:ℂ)*H u = g T (R u) := by
+        dsimp [H]
+        rw [Complex.ofReal_inv,← mul_assoc,mul_inv_cancel₀ hNc,one_mul]
+        ring
+      change f T ((U T:ℂ)+(N:ℂ)*H u) = R u
+      rw [heq]
+      exact h.2.2.2.2.2.2.2.2.2.1 T hT.le _ (hRl u hu)
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/GreenTilts.lean b/lean/OAI/Probability/SLELocalization/GreenTilts.lean
index 696caa6dbd5dbe6980b529029406feb11feea31a..b7067d18fb880f547c07e6a9a68ce8ac4ab7b06b 100644
--- a/lean/OAI/Probability/SLELocalization/GreenTilts.lean
+++ b/lean/OAI/Probability/SLELocalization/GreenTilts.lean
@@ -1,1336 +1,1336 @@
-import Mathlib
-import OAI.Probability.SLELocalization.LoewnerFlow
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Loewner Localized estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerExit LoewnerField
-open scoped unitInterval
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-
-lemma source_until_stop (h : IsLoewnerTraceUsing driver γ g f)
-    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (h0 : 0 < deficit hε hT.le U z ρ δ 0) :
-    z ∈ localizationDomain γ (stop hε hT.le U z ρ δ) ∧
-      EqOn (fun s => g s z) (position hε hT.le U z) (Icc 0 (stop hε hT.le U z ρ δ)) := by
-  have hb := stop_bounds hε hT.le U z ρ δ
-  have hh := h.cutoff_solution_live hε hb.1 (H := position hε hT.le U z) ?_ ?_
-  · simpa only [position_initial] using hh
-  · intro t ht
-    rw [← hU ⟨ht.1,ht.2.trans hb.2⟩]
-    exact (position_derivWithin hε hT.le U z ⟨ht.1,ht.2.trans hb.2⟩).mono
-      (Icc_subset_Icc le_rfl hb.2)
-  · intro t ht
-    exact ((deficit_nonneg_iff hε hT.le U z ρ δ t).1
-      (nonneg_until hε hT.le hT U z ρ δ h0 ht)).2.1
-
-lemma logDerivative_eq_source (h : IsLoewnerTraceUsing driver γ g f)
-    (hcont : ContinuousOn driver (Icc 0 T))
-    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (h0 : 0 < deficit hε hT.le U z ρ δ 0) {t : ℝ}
-    (ht : t ∈ Icc 0 (stop hε hT.le U z ρ δ)) :
-    logDerivative hε hT.le U z t = Real.log ‖deriv (g t) z‖ := by
-  have hb := stop_bounds hε hT.le U z ρ δ
-  have hs := source_until_stop hε hT h U z ρ δ hU h0
-  have hz := survivingDomain_mono ht.2 hs.1
-  have hu := hcont.mono (Icc_subset_Icc le_rfl (ht.2.trans hb.2))
-  rw [logDerivative_eq_integral hε hT.le U z ⟨ht.1,ht.2.trans hb.2⟩,
-    h.log_deriv_norm_eq ht.1 hu hz]
-  have hi := (h.continuous_spatialCoefficient ht.1 hu hz).intervalIntegrable_of_Icc (μ := volume) ht.1
-  rw [← Complex.reCLM_apply, ← Complex.reCLM.intervalIntegral_comp_comm hi]
-  apply intervalIntegral.integral_congr
-  intro s hst
-  rw [uIcc_of_le ht.1] at hst
-  have hss : s ∈ Icc 0 (stop hε hT.le U z ρ δ) := ⟨hst.1,hst.2.trans ht.2⟩
-  have hh := ((deficit_nonneg_iff hε hT.le U z ρ δ s).1
-    (nonneg_until hε hT.le hT U z ρ δ h0 hss)).2.1
-  unfold variation
-  rw [clampIm_eq (by simpa using hh)]
-  simp only [← hs.2 hss, hU ⟨hst.1,hss.2.trans hb.2⟩, Complex.reCLM_apply]
-
-lemma radius_eq_source (h : IsLoewnerTraceUsing driver γ g f)
-    (hcont : ContinuousOn driver (Icc 0 T))
-    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (h0 : 0 < deficit hε hT.le U z ρ δ 0) {t : ℝ}
-    (ht : t ∈ Icc 0 (stop hε hT.le U z ρ δ)) :
-    radius hε hT.le U z t = loewnerRadius g t z := by
-  have hs := source_until_stop hε hT h U z ρ δ hU h0
-  rw [radius, logDerivative_eq_source hε hT h hcont U z ρ δ hU h0 ht,
-    Real.exp_log (norm_pos_iff.2 (h.deriv_ne_zero ht.1 (survivingDomain_mono ht.2 hs.1))),
-    ← hs.2 ht]
-  rfl
-
-lemma sine_eq_source (h : IsLoewnerTraceUsing driver γ g f)
-    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (h0 : 0 < deficit hε hT.le U z ρ δ 0) {t : ℝ}
-    (ht : t ∈ Icc 0 (stop hε hT.le U z ρ δ)) :
-    sine hε hT.le U z t = (g t z).im / ‖g t z - (driver t : ℂ)‖ := by
-  have hs := source_until_stop hε hT h U z ρ δ hU h0
-  have hh := ((deficit_nonneg_iff hε hT.le U z ρ δ t).1
-    (nonneg_until hε hT.le hT U z ρ δ h0 ht)).2.1
-  rw [sine, clampIm_eq (by simpa using hh), ← hs.2 ht,
-    hU ⟨ht.1,ht.2.trans (stop_bounds hε hT.le U z ρ δ).2⟩]
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerExit LoewnerField GreenPhysical
-open scoped unitInterval
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-
-lemma field_eq_radius_sine (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ)
-    (hy : ε ≤ (position hε hT.le U z t).im)
-    (hU : extendDriver T U t = sdev*b) :
-    field hε hT.le p a sdev U z t b =
-      (radius hε hT.le U z t)^(-p) * (sine hε hT.le U z t)^a := by
-  rw [field, green_eq_source _ _ _ _ _ (height_pos hε hT.le U z t)]
-  have hh : height hε hT.le U z t = (position hε hT.le U z t).im := max_eq_right hy
-  rw [hh, radius, sine, clampIm_eq (by simpa using hy), hU]
-  simp only [Complex.norm_def, Complex.normSq_apply, Complex.sub_re, Complex.ofReal_re,
-    Complex.sub_im, Complex.ofReal_im, sub_zero, pow_two]
-
-lemma clock_bounds_stop (h : IsLoewnerTraceUsing driver γ g f)
-    (hcont : ContinuousOn driver (Icc 0 T))
-    (U : C(I, ℝ)) (z : ℂ) (u δ : ℝ) (hδ : 0 < δ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (h0 : 0 < deficit hε hT.le U z ((2*z.im)*Real.exp (-u)) δ 0) :
-    z.im*Real.exp (-u/(2*δ^2)) ≤ (position hε hT.le U z
-      (stop hε hT.le U z ((2*z.im)*Real.exp (-u)) δ)).im ∧
-      stop hε hT.le U z ((2*z.im)*Real.exp (-u)) δ ≤ u*z.im^2/(4*δ^4) := by
-  let ρ := (2*z.im)*Real.exp (-u)
-  let τ := stop hε hT.le U z ρ δ
-  have hb : τ ∈ Icc 0 T := stop_bounds hε hT.le U z ρ δ
-  have hs := source_until_stop hε hT h U z ρ δ hU h0
-  have ha (t : ℝ) (ht : t ∈ Icc 0 τ) : δ ≤ (g t z).im / ‖g t z - (driver t : ℂ)‖ := by
-    rw [← sine_eq_source hε hT h U z ρ δ hU h0 ht]
-    exact ((deficit_nonneg_iff hε hT.le U z ρ δ t).1
-      (nonneg_until hε hT.le hT U z ρ δ h0 ht)).2.2
-  have hc := h.finite_clock_bounds (u := u) hb.1 hδ (hcont.mono (Icc_subset_Icc le_rfl hb.2)) hs.1 ?_ ?_
-  · have heq : g τ z = position hε hT.le U z τ := hs.2 ⟨hb.1,le_rfl⟩
-    rw [heq] at hc
-    exact hc
-  · intro t ht
-    have hz := survivingDomain_mono ht.2.le hs.1
-    have hnorm : 0 < ‖g t z - (driver t : ℂ)‖ := norm_pos_iff.2 (h.centered_ne_zero ht.1.le hz)
-    have hq : δ * ‖g t z - (driver t : ℂ)‖ ≤ (g t z).im :=
-      (le_div_iff₀ hnorm).1 (ha t ⟨ht.1.le,ht.2.le⟩)
-    simpa only [mul_pow, Complex.normSq_eq_norm_sq] using
-      pow_le_pow_left₀ (mul_nonneg hδ.le hnorm.le) hq 2
-  · rw [← radius_eq_source hε hT h hcont U z ρ δ hU h0 ⟨hb.1,le_rfl⟩]
-    exact ((deficit_nonneg_iff hε hT.le U z ρ δ τ).1
-      (nonneg_until hε hT.le hT U z ρ δ h0 ⟨hb.1,le_rfl⟩)).1
-
-lemma terminal_cases (h : IsLoewnerTraceUsing driver γ g f)
-    (hcont : ContinuousOn driver (Icc 0 T))
-    (U : C(I, ℝ)) (z : ℂ) (u δ : ℝ) (hδ : 0 < δ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (h0 : 0 < deficit hε hT.le U z ((2*z.im)*Real.exp (-u)) δ 0)
-    (hsmall : ε < z.im*Real.exp (-u/(2*δ^2)))
-    (hlarge : u*z.im^2/(4*δ^4) < T) :
-    let ρ := (2*z.im)*Real.exp (-u)
-    let τ := stop hε hT.le U z ρ δ
-    radius hε hT.le U z τ = ρ ∨ sine hε hT.le U z τ = δ := by
-  dsimp only
-  let ρ := (2*z.im)*Real.exp (-u)
-  let τ := stop hε hT.le U z ρ δ
-  have hb : τ ∈ Icc 0 T := stop_bounds hε hT.le U z ρ δ
-  have hc := clock_bounds_stop hε hT h hcont U z u δ hδ hU h0
-  have hn := nonpos_at_of_lt hε hT.le hT U z ρ δ (hc.2.trans_lt hlarge)
-  have hp := (deficit_nonneg_iff hε hT.le U z ρ δ τ).1
-    (nonneg_until hε hT.le hT U z ρ δ h0 ⟨hb.1,le_rfl⟩)
-  by_contra he
-  push Not at he
-  have hh : 0 < deficit hε hT.le U z ρ δ τ :=
-    (deficit_pos_iff hε hT.le U z ρ δ τ).2
-      ⟨lt_of_le_of_ne hp.1 (Ne.symm he.1),hsmall.trans_le hc.1,
-        lt_of_le_of_ne hp.2.2 (Ne.symm he.2)⟩
-  exact (not_lt_of_ge hn) hh
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.FiniteTilt
-open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GreenPhysical
-open scoped unitInterval
-
-structure Cutoff (z : ℂ) (u : ℝ) where
-  angle : ℝ
-  angle_pos : 0 < angle
-  angle_initial : angle < z.im / ‖z‖
-  height : ℝ
-  height_pos : 0 < height
-  horizon : ℝ≥0
-  horizon_pos : 0 < (horizon:ℝ)
-  height_small : height < z.im * Real.exp (-u/(2*angle^2))
-  time_large : u*z.im^2/(4*angle^4) < horizon
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.FiniteTilt
-open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GreenPhysical
-open scoped unitInterval
-namespace Cutoff
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-lemma height_initial (hz : z ∈ halfPlane) (hu : 0 ≤ u) : C.height < z.im := by
-  have he : Real.exp (-u/(2*C.angle^2)) ≤ 1 := Real.exp_le_one_iff.2
-    (div_nonpos_of_nonpos_of_nonneg (neg_nonpos.2 hu) (by positivity))
-  exact C.height_small.trans_le (by simpa only [mul_one] using mul_le_mul_of_nonneg_left he hz.le)
-
-lemma initial_deficit (hz : z ∈ halfPlane) (hu : 0 < u) (V : C(I, ℝ))
-    (hV0 : extendDriver C.horizon V 0 = 0) :
-    0 < deficit C.height_pos C.horizon.coe_nonneg V z ((2*z.im)*Real.exp (-u)) C.angle 0 := by
-  apply (deficit_pos_iff _ _ _ _ _ _ _).2
-  have hy0 : 0 < z.im := hz
-  have hy := C.height_initial hz hu.le
-  have hr : (2*z.im)*Real.exp (-u) < 2*z.im := by
-    simpa only [mul_one] using mul_lt_mul_of_pos_left
-      (Real.exp_lt_one_iff.2 (neg_neg_of_pos hu)) (show 0 < 2*z.im by positivity)
-  refine ⟨?_, ?_, ?_⟩
-  · simpa only [LoewnerExit.radius, position_initial, logDerivative_initial, Real.exp_zero, div_one] using hr
-  · simpa only [position_initial] using hy
-  · simpa only [LoewnerExit.sine, position_initial, hV0, Complex.ofReal_zero, sub_zero,
-      clampIm_eq hy.le] using C.angle_initial
-
-def paths {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) : Ω → C(I, ℝ) :=
-  BrownianDriver.path B (Real.sqrt κ) C.horizon
-
-def stopTime {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
-  stop C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z ((2*z.im)*Real.exp (-u)) C.angle
-
-def value {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
-  field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
-    (C.stopTime κ B sample) (B (C.stopTime κ B sample).toNNReal sample)
-
-lemma path_eq {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    EqOn (extendDriver C.horizon (C.paths κ B sample)) (fun t => Real.sqrt κ*B t.toNNReal sample)
-      (Icc 0 (C.horizon:ℝ)) := by
-  intro t ht
-  exact BrownianDriver.extend_eq B (Real.sqrt κ) C.horizon sample hc ht
-
-lemma initial_deficit_brownian {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    (hz : z ∈ halfPlane) (hu : 0 < u)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
-    0 < deficit C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
-      ((2*z.im)*Real.exp (-u)) C.angle 0 := by
-  apply C.initial_deficit hz hu
-  simpa only [Real.toNNReal_zero, hb0, mul_zero] using
-    C.path_eq κ B sample hc ⟨le_rfl,C.horizon.coe_nonneg⟩
-
-lemma bounds {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
-    C.stopTime κ B sample ∈ Icc 0 (C.horizon:ℝ) :=
-  stop_bounds C.height_pos C.horizon.coe_nonneg _ z _ C.angle
-
-lemma value_nonneg {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
-    0 ≤ C.value κ B sample := (green_pos _ _ _ _ _).le
-
-lemma value_bound {Ω : Type*} (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (sample : Ω) (hz : z ∈ halfPlane) (hu : 0 < u)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
-    C.value κ B sample ≤ ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) := by
-  have hy0 : 0 < z.im := hz
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := sub_nonneg.2 ((le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le))
-  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hd := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0
-      ⟨(C.bounds κ B sample).1,le_rfl⟩)
-  apply green_le_fixed_radius _ _ _ _ _ _ (LoewnerField.height_pos _ _ _ _ _) ha hp (by positivity)
-  simpa only [LoewnerExit.radius, LoewnerField.height, max_eq_right hd.2.1] using hd.1
-
-lemma expectation {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (hz : z ∈ halfPlane) (hu : 0 ≤ u) :
-    Integrable (C.value κ B) P ∧ (∫ sample, C.value κ B sample ∂P) =
-      green (1-κ/8) (8/κ-1) z.re z.im 0 :=
-  localized_expectation hB κ hκ C.height_pos C.horizon C.horizon_pos z (C.height_initial hz hu).le _ C.angle
-
-end Cutoff
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.FiniteTilt
-open scoped unitInterval
-
-lemma initial_sine_pos {z : ℂ} (hz : z ∈ halfPlane) : 0 < z.im/‖z‖ := by
-  have hy : 0 < z.im := hz
-  have hne : z ≠ 0 := by
-    intro he
-    simp only [he, Complex.zero_im, lt_self_iff_false] at hy
-  exact div_pos hy (norm_pos_iff.2 hne)
-
-def shrinkingAngle (z : ℂ) (n : ℕ) : ℝ := (z.im/‖z‖)/((n:ℝ)+2)
-
-lemma shrinkingAngle_pos {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) :
-    0 < shrinkingAngle z n := div_pos (initial_sine_pos hz) (by positivity)
-
-lemma shrinkingAngle_initial {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) :
-    shrinkingAngle z n < z.im/‖z‖ := by
-  apply div_lt_self (initial_sine_pos hz)
-  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n
-  linarith
-
-lemma shrinkingAngle_tendsto (z : ℂ) : Tendsto (shrinkingAngle z) atTop (𝓝 0) := by
-  have ht := (tendsto_const_div_atTop_nhds_zero_nat (z.im/‖z‖)).comp (tendsto_add_atTop_nat 2)
-  change Tendsto (fun n : ℕ => (z.im/‖z‖)/((n:ℝ)+2)) atTop (𝓝 0)
-  simpa only [Function.comp_def, Nat.cast_add, Nat.cast_ofNat] using ht
-
-def cutoffs {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (n : ℕ) : Cutoff z u where
-  angle := shrinkingAngle z n
-  angle_pos := shrinkingAngle_pos hz n
-  angle_initial := shrinkingAngle_initial hz n
-  height := z.im*Real.exp (-u/(2*(shrinkingAngle z n)^2))/2
-  height_pos := by have hy : 0 < z.im := hz; positivity
-  horizon := ⟨u*z.im^2/(4*(shrinkingAngle z n)^4)+1, by positivity⟩
-  horizon_pos := by change 0 < u*z.im^2/(4*(shrinkingAngle z n)^4)+1; positivity
-  height_small := by
-    have he := mul_pos (show 0 < z.im from hz) (Real.exp_pos (-u/(2*(shrinkingAngle z n)^2)))
-    linarith
-  time_large := by
-    change u*z.im^2/(4*(shrinkingAngle z n)^4) < u*z.im^2/(4*(shrinkingAngle z n)^4)+1
-    linarith
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GreenPhysical
-open scoped unitInterval
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable (hz : z ∈ halfPlane) (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-include h hz hu hc hb0
-
-lemma source_live : z ∈ localizationDomain γ (C.stopTime κ B sample) := by
-  exact (source_until_stop C.height_pos C.horizon_pos h _ z _ C.angle
-    (C.path_eq κ B sample hc) (C.initial_deficit_brownian κ B sample hz hu hc hb0)).1
-
-lemma source_radius : radius C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
-    (C.stopTime κ B sample) = loewnerRadius g (C.stopTime κ B sample) z := by
-  exact radius_eq_source C.height_pos C.horizon_pos h
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) _ z _ C.angle
-    (C.path_eq κ B sample hc) (C.initial_deficit_brownian κ B sample hz hu hc hb0)
-    ⟨(C.bounds κ B sample).1,le_rfl⟩
-
-lemma source_sine : sine C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
-    (C.stopTime κ B sample) = loewnerAngleSine κ (fun t => B t sample) g
-      (C.stopTime κ B sample) z := by
-  unfold loewnerAngleSine
-  simpa only [Complex.ofReal_mul] using sine_eq_source C.height_pos C.horizon_pos h _ z _ C.angle
-    (C.path_eq κ B sample hc) (C.initial_deficit_brownian κ B sample hz hu hc hb0)
-    ⟨(C.bounds κ B sample).1,le_rfl⟩
-
-lemma radius_lower : (2*z.im)*Real.exp (-u) ≤ loewnerRadius g (C.stopTime κ B sample) z := by
-  rw [← C.source_radius κ B sample h hz hu hc hb0]
-  exact ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle
-      (C.initial_deficit_brownian κ B sample hz hu hc hb0) ⟨(C.bounds κ B sample).1,le_rfl⟩)).1
-
-lemma value_eq_source : C.value κ B sample =
-    (loewnerRadius g (C.stopTime κ B sample) z)^(-(1-κ/8)) *
-      (loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z)^(8/κ-1) := by
-  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle
-      (C.initial_deficit_brownian κ B sample hz hu hc hb0) ⟨(C.bounds κ B sample).1,le_rfl⟩)).2.1
-  rw [Cutoff.value, field_eq_radius_sine C.height_pos C.horizon_pos _ _ _ _ z _ _ hh
-    (C.path_eq κ B sample hc (C.bounds κ B sample)),
-    C.source_radius κ B sample h hz hu hc hb0, C.source_sine κ B sample h hz hu hc hb0]
-
-lemma source_terminal : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) ∨
-    loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z = C.angle := by
-  have hh := terminal_cases C.height_pos C.horizon_pos h
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (C.paths κ B sample) z u C.angle C.angle_pos (C.path_eq κ B sample hc)
-    (C.initial_deficit_brownian κ B sample hz hu hc hb0) C.height_small C.time_large
-  change radius _ _ _ _ (C.stopTime κ B sample) = _ ∨
-    sine _ _ _ _ (C.stopTime κ B sample) = _ at hh
-  rwa [C.source_radius κ B sample h hz hu hc hb0, C.source_sine κ B sample h hz hu hc hb0] at hh
-
-lemma stopTime_le_radius {t : ℝ} (ht : 0 ≤ t) (_hzt : z ∈ localizationDomain γ t)
-    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) : C.stopTime κ B sample ≤ t := by
-  by_contra hn
-  have hlt : t < C.stopTime κ B sample := lt_of_not_ge hn
-  have ha := h.radius_strictAnti (C.bounds κ B sample).1
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (C.source_live κ B sample h hz hu hc hb0) ⟨ht,hlt.le⟩ ⟨(C.bounds κ B sample).1,le_rfl⟩ hlt
-  dsimp only at ha
-  rw [hrt] at ha
-  exact (not_lt_of_ge (C.radius_lower κ B sample h hz hu hc hb0)) ha
-
-lemma angle_exit_bound (hκ : 0 < κ ∧ κ < 8)
-    (he : loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z = C.angle) :
-    C.value κ B sample ≤ ((2*z.im)*Real.exp (-u))^(-(1-κ/8))*C.angle^(8/κ-1) := by
-  rw [C.value_eq_source κ B sample h hz hu hc hb0, he]
-  apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg C.angle_pos.le _)
-  exact Real.rpow_le_rpow_of_nonpos (by have hy : 0 < z.im := hz; positivity)
-    (C.radius_lower κ B sample h hz hu hc hb0) (by linarith [hκ.2])
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {κ : ℝ} {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma sine_positive_minimum (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal) γ g f)
-    (hc : Continuous B) {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ∃ η : ℝ, 0 < η ∧ ∀ t ∈ Icc 0 T, η ≤ loewnerAngleSine κ B g t z := by
-  have hgc := h.continuous_eval hT hz
-  have hdc : Continuous (fun t : ℝ => ((Real.sqrt κ * B t.toNNReal : ℝ) : ℂ)) :=
-    Complex.continuous_ofReal.comp (continuous_const.mul (hc.comp continuous_real_toNNReal))
-  have hnum := Complex.continuous_im.comp_continuousOn hgc
-  have hden := (hgc.sub hdc.continuousOn).norm
-  have hn (t : ℝ) (ht : t ∈ Icc 0 T) :
-      ‖g t z - ((Real.sqrt κ * B t.toNNReal : ℝ) : ℂ)‖ ≠ 0 :=
-    norm_ne_zero_iff.2 (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz))
-  have hx := hnum.div hden hn
-  have hct : ContinuousOn (fun t => loewnerAngleSine κ B g t z) (Icc 0 T) := by
-    apply hx.congr
-    intro t ht
-    simp only [loewnerAngleSine, Function.comp_def, Complex.ofReal_mul, Pi.div_apply, Pi.sub_apply]
-  obtain ⟨t, ht, hm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.2 hT) hct
-  refine ⟨loewnerAngleSine κ B g t z, ?_, hm⟩
-  have hh := div_pos (h.im_pos ht.1 (survivingDomain_mono ht.2 hz))
-    (norm_pos_iff.2 (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz)))
-  simpa only [loewnerAngleSine, Complex.ofReal_mul] using hh
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.FiniteTilt
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-include h hz hu hc hb0
-
-lemma eventually_value_at_radius {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
-    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
-    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((cutoffs hz u hu.le n).value κ B sample) =
-      finiteStoppedDensity κ (fun t => B t sample) γ g z u := by
-  obtain ⟨η, hη, hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
-  have hδ : ∀ᶠ n : ℕ in atTop, shrinkingAngle z n < η :=
-    (shrinkingAngle_tendsto z).eventually (gt_mem_nhds hη)
-  filter_upwards [hδ] with n hn
-  let C := cutoffs hz u hu.le n
-  have hh : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
-    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · exact hr
-    · have htm := hm (C.stopTime κ B sample)
-        ⟨(C.bounds κ B sample).1,C.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
-      rw [hs] at htm
-      exact False.elim ((not_lt_of_ge htm) hn)
-  have he : C.stopTime κ B sample = t := h.live_radius_injective
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (C.bounds κ B sample).1 t.coe_nonneg (C.source_live κ B sample h hz hu hc hb0) hzt
-    (hh.trans hrt.symm)
-  have htau := h.firstLiveRadiusStop_eq
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hzt hrt
-  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  rw [finiteStoppedDensity, htau, ite_eq_right (ENNReal.coe_ne_top), ENNReal.coe_toReal, hp,
-    ← he, ← C.value_eq_source κ B sample h hz hu hc hb0]
-
-lemma tendsto_value_killed (hκ : 0 < κ ∧ κ < 8) :
-    Tendsto (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).value κ B sample)) atTop
-      (𝓝 (finiteStoppedDensity κ (fun t => B t sample) γ g z u)) := by
-  by_cases he : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧
-      loewnerRadius g t z = (2*z.im)*Real.exp (-u)
-  · obtain ⟨t, hzt, hrt⟩ := he
-    have hh : (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).value κ B sample)) =ᶠ[atTop]
-        (fun _ => finiteStoppedDensity κ (fun t => B t sample) γ g z u) :=
-      eventually_value_at_radius κ B sample h hz hu hc hb0 hzt hrt
-    exact tendsto_const_nhds.congr' hh.symm
-  · have hset : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
-        loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ := by
-      apply Set.eq_empty_iff_forall_notMem.2
-      intro t ht
-      exact he ⟨t,ht⟩
-    have htau : firstLiveRadiusStop γ g z u = (⊤ : ℝ≥0∞) := by
-      simp only [firstLiveRadiusStop, hset, Set.image_empty, sInf_empty]
-    rw [finiteStoppedDensity, htau, ite_eq_left rfl]
-    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
-    have hlim := (shrinkingAngle_tendsto z).rpow_const_nhds_zero ha
-    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
-        (shrinkingAngle z n)^(8/κ-1)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
-    have hl : Tendsto (fun n : ℕ => (cutoffs hz u hu.le n).value κ B sample) atTop (𝓝 0) := by
-      apply squeeze_zero (fun n => (cutoffs hz u hu.le n).value_nonneg κ B sample) _ hlim'
-      intro n
-      let C := cutoffs hz u hu.le n
-      have hs : loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z = C.angle := by
-        rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-        · exact False.elim (he ⟨⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩,
-            C.source_live κ B sample h hz hu hc hb0, hr⟩)
-        · exact hs
-      exact C.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
-    simpa only [ENNReal.ofReal_zero, Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge
-open FiniteTilt GreenPhysical
-
-lemma finiteStoppedDensity_zero {κ : ℝ} {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
-    (hb0 : B 0 = 0) {z : ℂ} (hz : z ∈ halfPlane) :
-    finiteStoppedDensity κ B γ g z 0 =
-      ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) := by
-  rw [finiteStoppedDensity, firstLiveRadiusStop_zero h hz, ite_eq_right ENNReal.zero_ne_top,
-    ENNReal.toReal_zero, h.radius_initial hz]
-  simp only [loewnerAngleSine, h.initial z hz, Real.toNNReal_zero, hb0, mul_zero,
-    Complex.ofReal_zero, sub_zero]
-
-lemma green_initial (κ : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
-    green (1-κ/8) (8/κ-1) z.re z.im 0 =
-      (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1) := by
-  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  rw [green_eq_source _ _ _ _ _ hz, Real.exp_zero, div_one, hp]
-  congr 3
-  rw [Complex.norm_eq_sqrt_sq_add_sq]
-
-theorem finiteStoppedCenteredLaw {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) :
-    (∫⁻ sample, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P) =
-      ENNReal.ofReal ((2 * z.im) ^ (-codimension κ) *
-        (z.im / ‖z‖) ^ (8 / κ - 1)) := by
-  rcases eq_or_lt_of_le hu with rfl | hu
-  · have he : (fun sample => finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z 0) =ᵐ[P]
-        (fun _ => ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))) := by
-      filter_upwards [hmaps,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hb
-      exact finiteStoppedDensity_zero (κ := κ) (B := fun t => B t sample) hm hb hz
-    rw [lintegral_congr_ae he, lintegral_const, measure_univ, mul_one]
-  · let C := cutoffs hz u hu.le
-    let F := fun n sample => ENNReal.ofReal ((C n).value κ B sample)
-    have hexp (n : ℕ) := (C n).expectation hB κ hκ hz hu.le
-    have hFm (n : ℕ) : AEMeasurable (F n) P :=
-      ENNReal.measurable_ofReal.comp_aemeasurable (hexp n).1.aemeasurable
-    have hFb (n : ℕ) : ∀ᵐ sample ∂P, F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
-      filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb
-      exact ENNReal.ofReal_le_ofReal ((C n).value_bound κ hκ B sample hz hu hc hb)
-    have hFl : ∀ᵐ sample ∂P, Tendsto (fun n => F n sample) atTop
-        (𝓝 (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u)) := by
-      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
-      exact tendsto_value_killed κ B sample hm hz hu hc hb hκ
-    have hfin : (∫⁻ _ : Ω, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ (⊤ : ℝ≥0∞) := by
-      rw [lintegral_const, measure_univ, mul_one]
-      exact ENNReal.ofReal_ne_top
-    have hdct := tendsto_lintegral_of_dominated_convergence'
-      (fun _ : Ω => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8)))) hFm hFb hfin hFl
-    have hint (n : ℕ) : (∫⁻ sample, F n sample ∂P) = ENNReal.ofReal (green (1-κ/8) (8/κ-1) z.re z.im 0) := by
-      rw [← ofReal_integral_eq_lintegral_ofReal (hexp n).1
-        (Filter.Eventually.of_forall (fun sample => (C n).value_nonneg κ B sample)), (hexp n).2]
-    simp only [hint] at hdct
-    have he := tendsto_nhds_unique hdct tendsto_const_nhds
-    rw [green_initial κ hz] at he
-    exact he
-
-end SLEExactGauge
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BoundaryFunctional
-
-def clampRe (ε : ℝ) (w : ℂ) : ℂ := BoundaryCutoff.clamp 1 ε w
-
-def velocity (ε u : ℝ) (w : ℂ) : ℂ := BoundaryCutoff.velocity 1 ε u w
-
-lemma clampRe_ne_zero {ε : ℝ} (hε : 0 < ε) (w : ℂ) : clampRe ε w ≠ 0 :=
-  BoundaryCutoff.clamp_ne_zero (by norm_num) hε w
-
-lemma clampRe_norm {ε : ℝ} (hε : 0 < ε) (w : ℂ) : ε ≤ ‖clampRe ε w‖ :=
-  BoundaryCutoff.clamp_norm (by norm_num) hε w
-
-lemma clampRe_norm_sub (ε : ℝ) (z w : ℂ) : ‖clampRe ε z-clampRe ε w‖ ≤ 2*‖z-w‖ :=
-  BoundaryCutoff.clamp_norm_sub (by norm_num) z w
-
-lemma velocity_lipschitz {ε : ℝ} (hε : 0 < ε) (u : ℝ) :
-    LipschitzWith ⟨4/ε^2,by positivity⟩ (velocity ε u) :=
-  BoundaryCutoff.velocity_lipschitz (by norm_num) hε u
-
-lemma norm_velocity_sub_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (z w : ℂ) :
-    ‖velocity ε u z-velocity ε u w‖ ≤ (4/ε^2)*‖z-w‖ :=
-  BoundaryCutoff.norm_velocity_sub_le (by norm_num) hε u z w
-
-lemma continuous_velocity {ε : ℝ} (hε : 0 < ε) :
-    Continuous (fun p : ℝ × ℂ => velocity ε p.1 p.2) :=
-  BoundaryCutoff.continuous_velocity (by norm_num) hε
-
-lemma exists_solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
-    ∃ H : ℝ → ℂ, H 0=z ∧ ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t :=
-  BoundaryCutoff.exists_solution (by norm_num) hε hT U hU z
-
-lemma norm_velocity_driver_sub_le {ε : ℝ} (hε : 0 < ε) (u v : ℝ) (z : ℂ) :
-    ‖velocity ε u z - velocity ε v z‖ ≤ (4 / ε^2) * |u-v| := by
-  have he (a : ℝ) : velocity ε 0 (z - (a : ℂ)) = velocity ε a z := by
-    simp [velocity,BoundaryCutoff.velocity]
-  have hh := norm_velocity_sub_le hε 0 (z-(u:ℂ)) (z-(v:ℂ))
-  rw [he, he] at hh
-  convert hh using 1
-  rw [sub_sub_sub_cancel_left, ← Complex.ofReal_sub, Complex.norm_real, abs_sub_comm]
-  simp [Real.norm_eq_abs]
-
-lemma derivWithin_right {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ico 0 T) :
-    HasDerivWithinAt H (velocity ε (U t) (H t)) (Ici t) t :=
-  (hH t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
-
-lemma derivWithin_left {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ioc 0 T) :
-    HasDerivWithinAt H (velocity ε (U t) (H t)) (Iic t) t :=
-  (hH t ⟨ht.1.le, ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
-
-lemma solution_eq {ε T : ℝ} (hε : 0 < ε) {U : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (U t) (G t)) (Icc 0 T) t)
-    (h0 : H 0 = G 0) : EqOn H G (Icc 0 T) := by
-  apply ODE_solution_unique_of_mem_Icc_right
-    (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith)
-    (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_right hH ht)
-    (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hG)
-    (fun _ ht => derivWithin_right hG ht) (fun _ _ => mem_univ _) h0
-
-lemma solution_dist_le {ε T D : ℝ} (hε : 0 < ε) (hD : 0 ≤ D)
-    {U V : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
-    (h0 : H 0 = G 0) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    dist (H t) (G t) ≤ D * Real.exp ((4 / ε^2) * T) := by
-  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
-  have hK : (0 : ℝ) < K := by change 0 < 4 / ε^2; positivity
-  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
-      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K : ℝ) * D := by
-    rw [dist_eq_norm]
-    calc
-      _ ≤ (4 / ε^2) * |V s-U s| := norm_velocity_driver_sub_le hε _ _ _
-      _ ≤ (K : ℝ) * D := by
-        rw [abs_sub_comm]
-        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1, hs.2.le⟩) hK.le
-  have hh := dist_le_of_approx_trajectories_ODE
-    (fun t => velocity_lipschitz hε (U t))
-    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
-    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
-    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
-    hb (le_of_eq (dist_eq_zero.2 h0)) t ht
-  have he : gronwallBound 0 K (0 + (K : ℝ)*D) (t-0) =
-      D * (Real.exp ((K : ℝ)*t)-1) := by
-    simp only [gronwallBound, zero_add, sub_zero, ite_eq_right (ne_of_gt hK)]
-    field_simp
-    ring
-  rw [he] at hh
-  have hexp : Real.exp ((K : ℝ)*t) ≤ Real.exp ((K : ℝ)*T) :=
-    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
-  apply hh.trans
-  change D * (Real.exp ((4 / ε^2)*t)-1) ≤ D * Real.exp ((4 / ε^2)*T)
-  change Real.exp ((4 / ε^2)*t) ≤ Real.exp ((4 / ε^2)*T) at hexp
-  exact mul_le_mul_of_nonneg_left (by linarith) hD
-
-end SLEExactGauge.BoundaryFunctional
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BoundaryFunctional
-open scoped unitInterval
-
-def extendDriver (T : ℝ) (U : C(I, ℝ)) (t : ℝ) : ℝ :=
-  U (projIcc 0 1 zero_le_one (t / T))
-
-lemma continuous_extendDriver (T : ℝ) (U : C(I, ℝ)) : Continuous (extendDriver T U) :=
-  U.continuous.comp (continuous_projIcc.comp (continuous_id.div_const T))
-
-def solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) : ℝ → ℂ :=
-  (exists_solution hε hT (extendDriver T U) (continuous_extendDriver T U).continuousOn z).choose
-
-lemma solution_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    solution hε hT U z 0 = z :=
-  (exists_solution hε hT (extendDriver T U)
-    (continuous_extendDriver T U).continuousOn z).choose_spec.1
-
-lemma solution_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (solution hε hT U z)
-      (velocity ε (extendDriver T U t) (solution hε hT U z t)) (Icc 0 T) t :=
-  (exists_solution hε hT (extendDriver T U)
-    (continuous_extendDriver T U).continuousOn z).choose_spec.2
-
-def solutionPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    C(Icc 0 T, ℂ) :=
-  ⟨fun t => solution hε hT U z t,
-    continuousOn_iff_continuous_domRestrict.1 (HasDerivWithinAt.continuousOn (solution_deriv hε hT U z))⟩
-
-lemma solutionPath_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    LipschitzWith ⟨Real.exp ((4 / ε^2) * T), (Real.exp_pos _).le⟩
-      (fun U : C(I, ℝ) => solutionPath hε hT U z) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro U V
-  apply (ContinuousMap.dist_le (mul_nonneg (Real.exp_pos _).le dist_nonneg)).2
-  intro t
-  have hUV (s : ℝ) (_hs : s ∈ Icc 0 T) :
-      |extendDriver T U s-extendDriver T V s| ≤ dist U V := by
-    exact (ContinuousMap.dist_apply_le_dist (f := U) (g := V)
-      (projIcc 0 1 zero_le_one (s/T)))
-  have hh := solution_dist_le hε dist_nonneg (solution_deriv hε hT U z)
-    (solution_deriv hε hT V z)
-    ((solution_initial hε hT U z).trans (solution_initial hε hT V z).symm)
-    hUV t.property
-  change dist (solution hε hT U z t) (solution hε hT V z t) ≤
-    Real.exp ((4 / ε^2)*T) * dist U V
-  simpa only [mul_comm] using hh
-
-def position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℂ :=
-  solutionPath hε hT U z (projIcc 0 T hT t)
-
-lemma position_eq {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    position hε hT U z t = solution hε hT U z t := by
-  simp only [position, solutionPath, projIcc, max_eq_right ht.1,
-    min_eq_right ht.2]
-  rfl
-
-lemma continuous_position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => position hε hT p.1 z p.2) := by
-  exact continuous_eval.comp (((solutionPath_lipschitz hε hT z).continuous.comp continuous_fst).prodMk
-    ((continuous_projIcc (h := hT)).comp continuous_snd))
-
-lemma continuous_joint_extendDriver (T : ℝ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => extendDriver T p.1 p.2) := by
-  exact continuous_eval.comp (continuous_fst.prodMk
-    ((continuous_projIcc (h := zero_le_one)).comp (continuous_snd.div_const T)))
-
-def variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  (-2 / (clampRe ε (position hε hT U z t - (extendDriver T U t : ℂ)))^2).re
-
-lemma continuous_clampRe (ε : ℝ) : Continuous (clampRe ε) := by
-  change Continuous (fun w : ℂ => Complex.equivRealProdCLM.symm (1*max (1*w.re) ε,w.im))
-  apply Complex.equivRealProdCLM.symm.continuous.comp
-  apply Continuous.prodMk <;> fun_prop
-
-lemma continuous_variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => variation hε hT p.1 z p.2) := by
-  have hw := (continuous_position hε hT z).sub
-    (Complex.continuous_ofReal.comp (continuous_joint_extendDriver T))
-  exact Complex.continuous_re.comp (continuous_const.div
-    (((continuous_clampRe ε).comp hw).pow 2)
-    (fun p => pow_ne_zero 2 (clampRe_ne_zero hε _)))
-
-def logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  ∫ s in (0 : ℝ)..(projIcc 0 T hT t : ℝ), variation hε hT U z s
-
-lemma continuous_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => logDerivative hε hT p.1 z p.2) := by
-  have hc := intervalIntegral.continuous_parametric_primitive_of_continuous
-    (μ := volume) (a₀ := 0) (f := fun U t => variation hε hT U z t)
-    (continuous_variation hε hT z)
-  exact hc.comp (continuous_fst.prodMk
-    (continuous_subtype_val.comp ((continuous_projIcc (h := hT)).comp continuous_snd)))
-
-lemma abs_variation_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    |variation hε hT U z t| ≤ 2/ε^2 := by
-  apply (Complex.abs_re_le_norm _).trans
-  rw [norm_div, norm_neg, Complex.norm_ofNat, norm_pow]
-  apply div_le_div_of_nonneg_left (by norm_num) (sq_pos_of_pos hε)
-  exact pow_le_pow_left₀ hε.le (clampRe_norm hε _) 2
-
-lemma logDerivative_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) : logDerivative hε hT U z 0 = 0 := by
-  simp only [logDerivative, projIcc_of_mem hT (left_mem_Icc.mpr hT), intervalIntegral.integral_same]
-
-lemma abs_logDerivative_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    |logDerivative hε hT U z t| ≤ (2/ε^2)*T := by
-  have hb := (projIcc 0 T hT t).property
-  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (0:ℝ)) (b := (projIcc 0 T hT t : ℝ))
-    (fun s (_hs : s ∈ uIoc 0 (projIcc 0 T hT t : ℝ)) =>
-      show ‖variation hε hT U z s‖ ≤ 2/ε^2 from abs_variation_le hε hT U z s)
-  change |logDerivative hε hT U z t| ≤ _
-  apply hi.trans
-  rw [sub_zero, abs_of_nonneg hb.1]
-  exact mul_le_mul_of_nonneg_left hb.2 (by positivity)
-
-lemma position_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) : position hε hT U z 0 = z := by
-  rw [position_eq hε hT U z ⟨le_rfl,hT⟩, solution_initial]
-
-lemma position_derivWithin {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    HasDerivWithinAt (position hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)) (Icc 0 T) t := by
-  rw [position_eq hε hT U z ht]
-  exact (solution_deriv hε hT U z t ht).congr
-    (fun s hs => position_eq hε hT U z hs) (position_eq hε hT U z ht)
-
-lemma position_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
-    HasDerivAt (position hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)) t :=
-  (position_derivWithin hε hT U z ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2)
-
-lemma logDerivative_eq_integral {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    logDerivative hε hT U z t = ∫ s in (0:ℝ)..t, variation hε hT U z s := by
-  simp only [logDerivative, projIcc_of_mem hT ht]
-
-lemma hasDerivAt_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
-    HasDerivAt (logDerivative hε hT U z) (variation hε hT U z t) t := by
-  have hp : Continuous (fun t : ℝ => (U,t)) := continuous_const.prodMk continuous_id
-  have hc := (continuous_variation hε hT z).comp hp
-  dsimp only [Function.comp_def] at hc
-  have hd : HasDerivAt (fun v : ℝ => ∫ s in (0:ℝ)..v, variation hε hT U z s)
-      (variation hε hT U z t) t :=
-    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
-      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-  apply hd.congr_of_eventuallyEq
-  filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
-  exact logDerivative_eq_integral hε hT U z hs
-
-lemma position_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
-    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (_hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
-    (U V : C(I, ℝ)) (z : ℂ)
-    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
-    EqOn (position hε hT₁ U z) (position hε hT₂ V z) (Icc 0 S) := by
-  apply solution_eq hε (U := extendDriver T₁ U) ?_ ?_ ((position_initial hε hT₁ U z).trans (position_initial hε hT₂ V z).symm)
-  · intro t ht
-    exact (position_derivWithin hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩).mono
-      (Icc_subset_Icc le_rfl hS₁)
-  · intro t ht
-    rw [hUV ht]
-    exact (position_derivWithin hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩).mono
-      (Icc_subset_Icc le_rfl hS₂)
-
-lemma logDerivative_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
-    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
-    (U V : C(I, ℝ)) (z : ℂ)
-    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
-    EqOn (logDerivative hε hT₁ U z) (logDerivative hε hT₂ V z) (Icc 0 S) := by
-  have hH := position_prefix_eq hε hT₁ hT₂ hS hS₁ hS₂ U V z hUV
-  intro t ht
-  rw [logDerivative_eq_integral hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩,
-    logDerivative_eq_integral hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩]
-  apply intervalIntegral.integral_congr
-  intro s hs
-  rw [uIcc_of_le ht.1] at hs
-  have hsS : s ∈ Icc 0 S := ⟨hs.1,hs.2.trans ht.2⟩
-  simp only [variation, hH hsS, hUV hsS]
-
-lemma position_im_zero {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (x t : ℝ) : (position hε hT U (x:ℂ) t).im = 0 := by
-  have hc := BoundaryCutoff.conjugate_deriv (solution_deriv hε hT U (x:ℂ))
-  have he := solution_eq hε (solution_deriv hε hT U (x:ℂ)) hc (by
-    rw [solution_initial]; simp)
-  let v := projIcc 0 T hT t
-  have hv := he v.property
-  change solution hε hT U (x:ℂ) v = star (solution hε hT U (x:ℂ) v) at hv
-  have hh := congrArg Complex.im hv
-  change (solution hε hT U (x:ℂ) v).im = 0
-  change (solution hε hT U (x:ℂ) v).im = -(solution hε hT U (x:ℂ) v).im at hh
-  linarith
-
-end SLEExactGauge.BoundaryFunctional
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-open GreenPhysical
-
-def radial (β x y : ℝ) : ℝ := (x^2+y^2)^(β/2)
-
-def lyapunov (β C x y : ℝ) : ℝ := radial β x y+C*y^β
-
-def radialX (β x y : ℝ) : ℝ := β*x*(x^2+y^2)^(β/2-1)
-
-def radialXX (β x y : ℝ) : ℝ :=
-  β*(x^2+y^2)^(β/2-1)+β*(β-2)*x^2*(x^2+y^2)^(β/2-2)
-
-def radialXXX (β x y : ℝ) : ℝ :=
-  3*β*(β-2)*x*(x^2+y^2)^(β/2-2)+
-  β*(β-2)*(β-4)*x^3*(x^2+y^2)^(β/2-3)
-
-lemma radial_pos (β x : ℝ) {y : ℝ} (hy : 0 < y) : 0 < radial β x y :=
-  Real.rpow_pos_of_pos (q_pos x hy) _
-
-lemma hasDerivAt_radial_path (β : ℝ) {X Y : ℝ → ℝ} {t X' Y' : ℝ}
-    (hX : HasDerivAt X X' t) (hY : HasDerivAt Y Y' t) (hy : 0 < Y t) :
-    HasDerivAt (fun t => radial β (X t) (Y t))
-      (β*(X t*X'+Y t*Y')*(X t^2+Y t^2)^(β/2-1)) t := by
-  have h := (((hX.pow 2).add (hY.pow 2)).rpow_const (p := β/2) (Or.inl (q_pos (X t) hy).ne'))
-  simp only [Pi.add_apply,Pi.pow_apply] at h
-  apply h.congr_deriv
-  ring
-
-lemma hasDerivAt_radial_x (β x y : ℝ) (hy : 0 < y) :
-    HasDerivAt (radial β · y) (radialX β x y) x := by
-  convert hasDerivAt_radial_path β (hasDerivAt_id x) (hasDerivAt_const x y) hy using 1 <;>
-    simp [radialX]
-
-lemma hasDerivAt_radialX_x (β x y : ℝ) (hy : 0 < y) :
-    HasDerivAt (radialX β · y) (radialXX β x y) x := by
-  have hq := ((hasDerivAt_id x).pow 2).add_const (y^2)
-  have h := ((hasDerivAt_id x).const_mul β).mul
-    (hq.rpow_const (p := β/2-1) (Or.inl (q_pos x hy).ne'))
-  apply h.congr_deriv
-  dsimp only [id_eq,Pi.pow_apply,radialXX]
-  convert_to β*(x^2+y^2)^(β/2-1)+β*x*((2*x)*((β/2-1)*(x^2+y^2)^(β/2-1-1))) = _ using 1
-  · ring
-  rw [show β/2-1-1 = β/2-2 by ring]
-  ring
-
-lemma hasDerivAt_radialXX_x (β x y : ℝ) (hy : 0 < y) :
-    HasDerivAt (radialXX β · y) (radialXXX β x y) x := by
-  have hq := ((hasDerivAt_id x).pow 2).add_const (y^2)
-  have h1 := (hq.rpow_const (p := β/2-1) (Or.inl (q_pos x hy).ne')).const_mul β
-  have h2 := (((hasDerivAt_id x).pow 2).const_mul (β*(β-2))).mul
-    (hq.rpow_const (p := β/2-2) (Or.inl (q_pos x hy).ne'))
-  apply (h1.add h2).congr_deriv
-  dsimp only [id_eq,Pi.pow_apply,radialXXX]
-  rw [show β/2-1-1 = β/2-2 by ring,show β/2-2-1 = β/2-3 by ring]
-  ring
-
-lemma contDiff_radial_driver (β c y s : ℝ) (hy : 0 < y) :
-    ContDiff ℝ ⊤ (fun b => radial β (c-s*b) y) := by
-  unfold radial
-  apply ContDiff.rpow_const_of_ne
-  · fun_prop
-  · intro x; exact (q_pos _ hy).ne'
-
-lemma lyapunov_nonneg {β C : ℝ} (hC : 0 ≤ C) (x : ℝ) {y : ℝ} (hy : 0 < y) :
-    0 ≤ lyapunov β C x y := by
-  exact add_nonneg (radial_pos β x hy).le (mul_nonneg hC (Real.rpow_nonneg hy.le _))
-
-def generator (κ β C x y : ℝ) : ℝ :=
-  β*(x*(2*x/(x^2+y^2))+y*(-2*y/(x^2+y^2)))*(x^2+y^2)^(β/2-1) +
-  C*(β*y^(β-1)*(-2*y/(x^2+y^2))) + κ/2*radialXX β x y
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.LoewnerFlow
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Loewner Localized estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerExit LoewnerField
+open scoped unitInterval
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+
+lemma source_until_stop (h : IsLoewnerTraceUsing driver γ g f)
+    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (h0 : 0 < deficit hε hT.le U z ρ δ 0) :
+    z ∈ localizationDomain γ (stop hε hT.le U z ρ δ) ∧
+      EqOn (fun s => g s z) (position hε hT.le U z) (Icc 0 (stop hε hT.le U z ρ δ)) := by
+  have hb := stop_bounds hε hT.le U z ρ δ
+  have hh := h.cutoff_solution_live hε hb.1 (H := position hε hT.le U z) ?_ ?_
+  · simpa only [position_initial] using hh
+  · intro t ht
+    rw [← hU ⟨ht.1,ht.2.trans hb.2⟩]
+    exact (position_derivWithin hε hT.le U z ⟨ht.1,ht.2.trans hb.2⟩).mono
+      (Icc_subset_Icc le_rfl hb.2)
+  · intro t ht
+    exact ((deficit_nonneg_iff hε hT.le U z ρ δ t).1
+      (nonneg_until hε hT.le hT U z ρ δ h0 ht)).2.1
+
+lemma logDerivative_eq_source (h : IsLoewnerTraceUsing driver γ g f)
+    (hcont : ContinuousOn driver (Icc 0 T))
+    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (h0 : 0 < deficit hε hT.le U z ρ δ 0) {t : ℝ}
+    (ht : t ∈ Icc 0 (stop hε hT.le U z ρ δ)) :
+    logDerivative hε hT.le U z t = Real.log ‖deriv (g t) z‖ := by
+  have hb := stop_bounds hε hT.le U z ρ δ
+  have hs := source_until_stop hε hT h U z ρ δ hU h0
+  have hz := survivingDomain_mono ht.2 hs.1
+  have hu := hcont.mono (Icc_subset_Icc le_rfl (ht.2.trans hb.2))
+  rw [logDerivative_eq_integral hε hT.le U z ⟨ht.1,ht.2.trans hb.2⟩,
+    h.log_deriv_norm_eq ht.1 hu hz]
+  have hi := (h.continuous_spatialCoefficient ht.1 hu hz).intervalIntegrable_of_Icc (μ := volume) ht.1
+  rw [← Complex.reCLM_apply, ← Complex.reCLM.intervalIntegral_comp_comm hi]
+  apply intervalIntegral.integral_congr
+  intro s hst
+  rw [uIcc_of_le ht.1] at hst
+  have hss : s ∈ Icc 0 (stop hε hT.le U z ρ δ) := ⟨hst.1,hst.2.trans ht.2⟩
+  have hh := ((deficit_nonneg_iff hε hT.le U z ρ δ s).1
+    (nonneg_until hε hT.le hT U z ρ δ h0 hss)).2.1
+  unfold variation
+  rw [clampIm_eq (by simpa using hh)]
+  simp only [← hs.2 hss, hU ⟨hst.1,hss.2.trans hb.2⟩, Complex.reCLM_apply]
+
+lemma radius_eq_source (h : IsLoewnerTraceUsing driver γ g f)
+    (hcont : ContinuousOn driver (Icc 0 T))
+    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (h0 : 0 < deficit hε hT.le U z ρ δ 0) {t : ℝ}
+    (ht : t ∈ Icc 0 (stop hε hT.le U z ρ δ)) :
+    radius hε hT.le U z t = loewnerRadius g t z := by
+  have hs := source_until_stop hε hT h U z ρ δ hU h0
+  rw [radius, logDerivative_eq_source hε hT h hcont U z ρ δ hU h0 ht,
+    Real.exp_log (norm_pos_iff.2 (h.deriv_ne_zero ht.1 (survivingDomain_mono ht.2 hs.1))),
+    ← hs.2 ht]
+  rfl
+
+lemma sine_eq_source (h : IsLoewnerTraceUsing driver γ g f)
+    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (h0 : 0 < deficit hε hT.le U z ρ δ 0) {t : ℝ}
+    (ht : t ∈ Icc 0 (stop hε hT.le U z ρ δ)) :
+    sine hε hT.le U z t = (g t z).im / ‖g t z - (driver t : ℂ)‖ := by
+  have hs := source_until_stop hε hT h U z ρ δ hU h0
+  have hh := ((deficit_nonneg_iff hε hT.le U z ρ δ t).1
+    (nonneg_until hε hT.le hT U z ρ δ h0 ht)).2.1
+  rw [sine, clampIm_eq (by simpa using hh), ← hs.2 ht,
+    hU ⟨ht.1,ht.2.trans (stop_bounds hε hT.le U z ρ δ).2⟩]
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerExit LoewnerField GreenPhysical
+open scoped unitInterval
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+
+lemma field_eq_radius_sine (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ)
+    (hy : ε ≤ (position hε hT.le U z t).im)
+    (hU : extendDriver T U t = sdev*b) :
+    field hε hT.le p a sdev U z t b =
+      (radius hε hT.le U z t)^(-p) * (sine hε hT.le U z t)^a := by
+  rw [field, green_eq_source _ _ _ _ _ (height_pos hε hT.le U z t)]
+  have hh : height hε hT.le U z t = (position hε hT.le U z t).im := max_eq_right hy
+  rw [hh, radius, sine, clampIm_eq (by simpa using hy), hU]
+  simp only [Complex.norm_def, Complex.normSq_apply, Complex.sub_re, Complex.ofReal_re,
+    Complex.sub_im, Complex.ofReal_im, sub_zero, pow_two]
+
+lemma clock_bounds_stop (h : IsLoewnerTraceUsing driver γ g f)
+    (hcont : ContinuousOn driver (Icc 0 T))
+    (U : C(I, ℝ)) (z : ℂ) (u δ : ℝ) (hδ : 0 < δ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (h0 : 0 < deficit hε hT.le U z ((2*z.im)*Real.exp (-u)) δ 0) :
+    z.im*Real.exp (-u/(2*δ^2)) ≤ (position hε hT.le U z
+      (stop hε hT.le U z ((2*z.im)*Real.exp (-u)) δ)).im ∧
+      stop hε hT.le U z ((2*z.im)*Real.exp (-u)) δ ≤ u*z.im^2/(4*δ^4) := by
+  let ρ := (2*z.im)*Real.exp (-u)
+  let τ := stop hε hT.le U z ρ δ
+  have hb : τ ∈ Icc 0 T := stop_bounds hε hT.le U z ρ δ
+  have hs := source_until_stop hε hT h U z ρ δ hU h0
+  have ha (t : ℝ) (ht : t ∈ Icc 0 τ) : δ ≤ (g t z).im / ‖g t z - (driver t : ℂ)‖ := by
+    rw [← sine_eq_source hε hT h U z ρ δ hU h0 ht]
+    exact ((deficit_nonneg_iff hε hT.le U z ρ δ t).1
+      (nonneg_until hε hT.le hT U z ρ δ h0 ht)).2.2
+  have hc := h.finite_clock_bounds (u := u) hb.1 hδ (hcont.mono (Icc_subset_Icc le_rfl hb.2)) hs.1 ?_ ?_
+  · have heq : g τ z = position hε hT.le U z τ := hs.2 ⟨hb.1,le_rfl⟩
+    rw [heq] at hc
+    exact hc
+  · intro t ht
+    have hz := survivingDomain_mono ht.2.le hs.1
+    have hnorm : 0 < ‖g t z - (driver t : ℂ)‖ := norm_pos_iff.2 (h.centered_ne_zero ht.1.le hz)
+    have hq : δ * ‖g t z - (driver t : ℂ)‖ ≤ (g t z).im :=
+      (le_div_iff₀ hnorm).1 (ha t ⟨ht.1.le,ht.2.le⟩)
+    simpa only [mul_pow, Complex.normSq_eq_norm_sq] using
+      pow_le_pow_left₀ (mul_nonneg hδ.le hnorm.le) hq 2
+  · rw [← radius_eq_source hε hT h hcont U z ρ δ hU h0 ⟨hb.1,le_rfl⟩]
+    exact ((deficit_nonneg_iff hε hT.le U z ρ δ τ).1
+      (nonneg_until hε hT.le hT U z ρ δ h0 ⟨hb.1,le_rfl⟩)).1
+
+lemma terminal_cases (h : IsLoewnerTraceUsing driver γ g f)
+    (hcont : ContinuousOn driver (Icc 0 T))
+    (U : C(I, ℝ)) (z : ℂ) (u δ : ℝ) (hδ : 0 < δ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (h0 : 0 < deficit hε hT.le U z ((2*z.im)*Real.exp (-u)) δ 0)
+    (hsmall : ε < z.im*Real.exp (-u/(2*δ^2)))
+    (hlarge : u*z.im^2/(4*δ^4) < T) :
+    let ρ := (2*z.im)*Real.exp (-u)
+    let τ := stop hε hT.le U z ρ δ
+    radius hε hT.le U z τ = ρ ∨ sine hε hT.le U z τ = δ := by
+  dsimp only
+  let ρ := (2*z.im)*Real.exp (-u)
+  let τ := stop hε hT.le U z ρ δ
+  have hb : τ ∈ Icc 0 T := stop_bounds hε hT.le U z ρ δ
+  have hc := clock_bounds_stop hε hT h hcont U z u δ hδ hU h0
+  have hn := nonpos_at_of_lt hε hT.le hT U z ρ δ (hc.2.trans_lt hlarge)
+  have hp := (deficit_nonneg_iff hε hT.le U z ρ δ τ).1
+    (nonneg_until hε hT.le hT U z ρ δ h0 ⟨hb.1,le_rfl⟩)
+  by_contra he
+  push Not at he
+  have hh : 0 < deficit hε hT.le U z ρ δ τ :=
+    (deficit_pos_iff hε hT.le U z ρ δ τ).2
+      ⟨lt_of_le_of_ne hp.1 (Ne.symm he.1),hsmall.trans_le hc.1,
+        lt_of_le_of_ne hp.2.2 (Ne.symm he.2)⟩
+  exact (not_lt_of_ge hn) hh
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.FiniteTilt
+open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GreenPhysical
+open scoped unitInterval
+
+structure Cutoff (z : ℂ) (u : ℝ) where
+  angle : ℝ
+  angle_pos : 0 < angle
+  angle_initial : angle < z.im / ‖z‖
+  height : ℝ
+  height_pos : 0 < height
+  horizon : ℝ≥0
+  horizon_pos : 0 < (horizon:ℝ)
+  height_small : height < z.im * Real.exp (-u/(2*angle^2))
+  time_large : u*z.im^2/(4*angle^4) < horizon
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.FiniteTilt
+open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GreenPhysical
+open scoped unitInterval
+namespace Cutoff
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+lemma height_initial (hz : z ∈ halfPlane) (hu : 0 ≤ u) : C.height < z.im := by
+  have he : Real.exp (-u/(2*C.angle^2)) ≤ 1 := Real.exp_le_one_iff.2
+    (div_nonpos_of_nonpos_of_nonneg (neg_nonpos.2 hu) (by positivity))
+  exact C.height_small.trans_le (by simpa only [mul_one] using mul_le_mul_of_nonneg_left he hz.le)
+
+lemma initial_deficit (hz : z ∈ halfPlane) (hu : 0 < u) (V : C(I, ℝ))
+    (hV0 : extendDriver C.horizon V 0 = 0) :
+    0 < deficit C.height_pos C.horizon.coe_nonneg V z ((2*z.im)*Real.exp (-u)) C.angle 0 := by
+  apply (deficit_pos_iff _ _ _ _ _ _ _).2
+  have hy0 : 0 < z.im := hz
+  have hy := C.height_initial hz hu.le
+  have hr : (2*z.im)*Real.exp (-u) < 2*z.im := by
+    simpa only [mul_one] using mul_lt_mul_of_pos_left
+      (Real.exp_lt_one_iff.2 (neg_neg_of_pos hu)) (show 0 < 2*z.im by positivity)
+  refine ⟨?_, ?_, ?_⟩
+  · simpa only [LoewnerExit.radius, position_initial, logDerivative_initial, Real.exp_zero, div_one] using hr
+  · simpa only [position_initial] using hy
+  · simpa only [LoewnerExit.sine, position_initial, hV0, Complex.ofReal_zero, sub_zero,
+      clampIm_eq hy.le] using C.angle_initial
+
+def paths {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) : Ω → C(I, ℝ) :=
+  BrownianDriver.path B (Real.sqrt κ) C.horizon
+
+def stopTime {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
+  stop C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z ((2*z.im)*Real.exp (-u)) C.angle
+
+def value {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) : ℝ :=
+  field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
+    (C.stopTime κ B sample) (B (C.stopTime κ B sample).toNNReal sample)
+
+lemma path_eq {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    EqOn (extendDriver C.horizon (C.paths κ B sample)) (fun t => Real.sqrt κ*B t.toNNReal sample)
+      (Icc 0 (C.horizon:ℝ)) := by
+  intro t ht
+  exact BrownianDriver.extend_eq B (Real.sqrt κ) C.horizon sample hc ht
+
+lemma initial_deficit_brownian {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    (hz : z ∈ halfPlane) (hu : 0 < u)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
+    0 < deficit C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
+      ((2*z.im)*Real.exp (-u)) C.angle 0 := by
+  apply C.initial_deficit hz hu
+  simpa only [Real.toNNReal_zero, hb0, mul_zero] using
+    C.path_eq κ B sample hc ⟨le_rfl,C.horizon.coe_nonneg⟩
+
+lemma bounds {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
+    C.stopTime κ B sample ∈ Icc 0 (C.horizon:ℝ) :=
+  stop_bounds C.height_pos C.horizon.coe_nonneg _ z _ C.angle
+
+lemma value_nonneg {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω) :
+    0 ≤ C.value κ B sample := (green_pos _ _ _ _ _).le
+
+lemma value_bound {Ω : Type*} (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (sample : Ω) (hz : z ∈ halfPlane) (hu : 0 < u)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
+    C.value κ B sample ≤ ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) := by
+  have hy0 : 0 < z.im := hz
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := sub_nonneg.2 ((le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le))
+  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hd := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0
+      ⟨(C.bounds κ B sample).1,le_rfl⟩)
+  apply green_le_fixed_radius _ _ _ _ _ _ (LoewnerField.height_pos _ _ _ _ _) ha hp (by positivity)
+  simpa only [LoewnerExit.radius, LoewnerField.height, max_eq_right hd.2.1] using hd.1
+
+lemma expectation {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (hz : z ∈ halfPlane) (hu : 0 ≤ u) :
+    Integrable (C.value κ B) P ∧ (∫ sample, C.value κ B sample ∂P) =
+      green (1-κ/8) (8/κ-1) z.re z.im 0 :=
+  localized_expectation hB κ hκ C.height_pos C.horizon C.horizon_pos z (C.height_initial hz hu).le _ C.angle
+
+end Cutoff
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.FiniteTilt
+open scoped unitInterval
+
+lemma initial_sine_pos {z : ℂ} (hz : z ∈ halfPlane) : 0 < z.im/‖z‖ := by
+  have hy : 0 < z.im := hz
+  have hne : z ≠ 0 := by
+    intro he
+    simp only [he, Complex.zero_im, lt_self_iff_false] at hy
+  exact div_pos hy (norm_pos_iff.2 hne)
+
+def shrinkingAngle (z : ℂ) (n : ℕ) : ℝ := (z.im/‖z‖)/((n:ℝ)+2)
+
+lemma shrinkingAngle_pos {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) :
+    0 < shrinkingAngle z n := div_pos (initial_sine_pos hz) (by positivity)
+
+lemma shrinkingAngle_initial {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) :
+    shrinkingAngle z n < z.im/‖z‖ := by
+  apply div_lt_self (initial_sine_pos hz)
+  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n
+  linarith
+
+lemma shrinkingAngle_tendsto (z : ℂ) : Tendsto (shrinkingAngle z) atTop (𝓝 0) := by
+  have ht := (tendsto_const_div_atTop_nhds_zero_nat (z.im/‖z‖)).comp (tendsto_add_atTop_nat 2)
+  change Tendsto (fun n : ℕ => (z.im/‖z‖)/((n:ℝ)+2)) atTop (𝓝 0)
+  simpa only [Function.comp_def, Nat.cast_add, Nat.cast_ofNat] using ht
+
+def cutoffs {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (n : ℕ) : Cutoff z u where
+  angle := shrinkingAngle z n
+  angle_pos := shrinkingAngle_pos hz n
+  angle_initial := shrinkingAngle_initial hz n
+  height := z.im*Real.exp (-u/(2*(shrinkingAngle z n)^2))/2
+  height_pos := by have hy : 0 < z.im := hz; positivity
+  horizon := ⟨u*z.im^2/(4*(shrinkingAngle z n)^4)+1, by positivity⟩
+  horizon_pos := by change 0 < u*z.im^2/(4*(shrinkingAngle z n)^4)+1; positivity
+  height_small := by
+    have he := mul_pos (show 0 < z.im from hz) (Real.exp_pos (-u/(2*(shrinkingAngle z n)^2)))
+    linarith
+  time_large := by
+    change u*z.im^2/(4*(shrinkingAngle z n)^4) < u*z.im^2/(4*(shrinkingAngle z n)^4)+1
+    linarith
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerExit LoewnerField LoewnerLocalized GreenPhysical
+open scoped unitInterval
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable (hz : z ∈ halfPlane) (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+include h hz hu hc hb0
+
+lemma source_live : z ∈ localizationDomain γ (C.stopTime κ B sample) := by
+  exact (source_until_stop C.height_pos C.horizon_pos h _ z _ C.angle
+    (C.path_eq κ B sample hc) (C.initial_deficit_brownian κ B sample hz hu hc hb0)).1
+
+lemma source_radius : radius C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
+    (C.stopTime κ B sample) = loewnerRadius g (C.stopTime κ B sample) z := by
+  exact radius_eq_source C.height_pos C.horizon_pos h
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) _ z _ C.angle
+    (C.path_eq κ B sample hc) (C.initial_deficit_brownian κ B sample hz hu hc hb0)
+    ⟨(C.bounds κ B sample).1,le_rfl⟩
+
+lemma source_sine : sine C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
+    (C.stopTime κ B sample) = loewnerAngleSine κ (fun t => B t sample) g
+      (C.stopTime κ B sample) z := by
+  unfold loewnerAngleSine
+  simpa only [Complex.ofReal_mul] using sine_eq_source C.height_pos C.horizon_pos h _ z _ C.angle
+    (C.path_eq κ B sample hc) (C.initial_deficit_brownian κ B sample hz hu hc hb0)
+    ⟨(C.bounds κ B sample).1,le_rfl⟩
+
+lemma radius_lower : (2*z.im)*Real.exp (-u) ≤ loewnerRadius g (C.stopTime κ B sample) z := by
+  rw [← C.source_radius κ B sample h hz hu hc hb0]
+  exact ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle
+      (C.initial_deficit_brownian κ B sample hz hu hc hb0) ⟨(C.bounds κ B sample).1,le_rfl⟩)).1
+
+lemma value_eq_source : C.value κ B sample =
+    (loewnerRadius g (C.stopTime κ B sample) z)^(-(1-κ/8)) *
+      (loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z)^(8/κ-1) := by
+  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle
+      (C.initial_deficit_brownian κ B sample hz hu hc hb0) ⟨(C.bounds κ B sample).1,le_rfl⟩)).2.1
+  rw [Cutoff.value, field_eq_radius_sine C.height_pos C.horizon_pos _ _ _ _ z _ _ hh
+    (C.path_eq κ B sample hc (C.bounds κ B sample)),
+    C.source_radius κ B sample h hz hu hc hb0, C.source_sine κ B sample h hz hu hc hb0]
+
+lemma source_terminal : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) ∨
+    loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z = C.angle := by
+  have hh := terminal_cases C.height_pos C.horizon_pos h
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (C.paths κ B sample) z u C.angle C.angle_pos (C.path_eq κ B sample hc)
+    (C.initial_deficit_brownian κ B sample hz hu hc hb0) C.height_small C.time_large
+  change radius _ _ _ _ (C.stopTime κ B sample) = _ ∨
+    sine _ _ _ _ (C.stopTime κ B sample) = _ at hh
+  rwa [C.source_radius κ B sample h hz hu hc hb0, C.source_sine κ B sample h hz hu hc hb0] at hh
+
+lemma stopTime_le_radius {t : ℝ} (ht : 0 ≤ t) (_hzt : z ∈ localizationDomain γ t)
+    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) : C.stopTime κ B sample ≤ t := by
+  by_contra hn
+  have hlt : t < C.stopTime κ B sample := lt_of_not_ge hn
+  have ha := h.radius_strictAnti (C.bounds κ B sample).1
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (C.source_live κ B sample h hz hu hc hb0) ⟨ht,hlt.le⟩ ⟨(C.bounds κ B sample).1,le_rfl⟩ hlt
+  dsimp only at ha
+  rw [hrt] at ha
+  exact (not_lt_of_ge (C.radius_lower κ B sample h hz hu hc hb0)) ha
+
+lemma angle_exit_bound (hκ : 0 < κ ∧ κ < 8)
+    (he : loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z = C.angle) :
+    C.value κ B sample ≤ ((2*z.im)*Real.exp (-u))^(-(1-κ/8))*C.angle^(8/κ-1) := by
+  rw [C.value_eq_source κ B sample h hz hu hc hb0, he]
+  apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg C.angle_pos.le _)
+  exact Real.rpow_le_rpow_of_nonpos (by have hy : 0 < z.im := hz; positivity)
+    (C.radius_lower κ B sample h hz hu hc hb0) (by linarith [hκ.2])
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {κ : ℝ} {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma sine_positive_minimum (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal) γ g f)
+    (hc : Continuous B) {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ∃ η : ℝ, 0 < η ∧ ∀ t ∈ Icc 0 T, η ≤ loewnerAngleSine κ B g t z := by
+  have hgc := h.continuous_eval hT hz
+  have hdc : Continuous (fun t : ℝ => ((Real.sqrt κ * B t.toNNReal : ℝ) : ℂ)) :=
+    Complex.continuous_ofReal.comp (continuous_const.mul (hc.comp continuous_real_toNNReal))
+  have hnum := Complex.continuous_im.comp_continuousOn hgc
+  have hden := (hgc.sub hdc.continuousOn).norm
+  have hn (t : ℝ) (ht : t ∈ Icc 0 T) :
+      ‖g t z - ((Real.sqrt κ * B t.toNNReal : ℝ) : ℂ)‖ ≠ 0 :=
+    norm_ne_zero_iff.2 (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz))
+  have hx := hnum.div hden hn
+  have hct : ContinuousOn (fun t => loewnerAngleSine κ B g t z) (Icc 0 T) := by
+    apply hx.congr
+    intro t ht
+    simp only [loewnerAngleSine, Function.comp_def, Complex.ofReal_mul, Pi.div_apply, Pi.sub_apply]
+  obtain ⟨t, ht, hm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.2 hT) hct
+  refine ⟨loewnerAngleSine κ B g t z, ?_, hm⟩
+  have hh := div_pos (h.im_pos ht.1 (survivingDomain_mono ht.2 hz))
+    (norm_pos_iff.2 (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz)))
+  simpa only [loewnerAngleSine, Complex.ofReal_mul] using hh
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.FiniteTilt
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+include h hz hu hc hb0
+
+lemma eventually_value_at_radius {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
+    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
+    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((cutoffs hz u hu.le n).value κ B sample) =
+      finiteStoppedDensity κ (fun t => B t sample) γ g z u := by
+  obtain ⟨η, hη, hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
+  have hδ : ∀ᶠ n : ℕ in atTop, shrinkingAngle z n < η :=
+    (shrinkingAngle_tendsto z).eventually (gt_mem_nhds hη)
+  filter_upwards [hδ] with n hn
+  let C := cutoffs hz u hu.le n
+  have hh : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
+    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · exact hr
+    · have htm := hm (C.stopTime κ B sample)
+        ⟨(C.bounds κ B sample).1,C.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
+      rw [hs] at htm
+      exact False.elim ((not_lt_of_ge htm) hn)
+  have he : C.stopTime κ B sample = t := h.live_radius_injective
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (C.bounds κ B sample).1 t.coe_nonneg (C.source_live κ B sample h hz hu hc hb0) hzt
+    (hh.trans hrt.symm)
+  have htau := h.firstLiveRadiusStop_eq
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hzt hrt
+  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  rw [finiteStoppedDensity, htau, ite_eq_right (ENNReal.coe_ne_top), ENNReal.coe_toReal, hp,
+    ← he, ← C.value_eq_source κ B sample h hz hu hc hb0]
+
+lemma tendsto_value_killed (hκ : 0 < κ ∧ κ < 8) :
+    Tendsto (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).value κ B sample)) atTop
+      (𝓝 (finiteStoppedDensity κ (fun t => B t sample) γ g z u)) := by
+  by_cases he : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧
+      loewnerRadius g t z = (2*z.im)*Real.exp (-u)
+  · obtain ⟨t, hzt, hrt⟩ := he
+    have hh : (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).value κ B sample)) =ᶠ[atTop]
+        (fun _ => finiteStoppedDensity κ (fun t => B t sample) γ g z u) :=
+      eventually_value_at_radius κ B sample h hz hu hc hb0 hzt hrt
+    exact tendsto_const_nhds.congr' hh.symm
+  · have hset : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
+        loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ := by
+      apply Set.eq_empty_iff_forall_notMem.2
+      intro t ht
+      exact he ⟨t,ht⟩
+    have htau : firstLiveRadiusStop γ g z u = (⊤ : ℝ≥0∞) := by
+      simp only [firstLiveRadiusStop, hset, Set.image_empty, sInf_empty]
+    rw [finiteStoppedDensity, htau, ite_eq_left rfl]
+    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
+    have hlim := (shrinkingAngle_tendsto z).rpow_const_nhds_zero ha
+    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
+        (shrinkingAngle z n)^(8/κ-1)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
+    have hl : Tendsto (fun n : ℕ => (cutoffs hz u hu.le n).value κ B sample) atTop (𝓝 0) := by
+      apply squeeze_zero (fun n => (cutoffs hz u hu.le n).value_nonneg κ B sample) _ hlim'
+      intro n
+      let C := cutoffs hz u hu.le n
+      have hs : loewnerAngleSine κ (fun t => B t sample) g (C.stopTime κ B sample) z = C.angle := by
+        rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+        · exact False.elim (he ⟨⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩,
+            C.source_live κ B sample h hz hu hc hb0, hr⟩)
+        · exact hs
+      exact C.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
+    simpa only [ENNReal.ofReal_zero, Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge
+open FiniteTilt GreenPhysical
+
+lemma finiteStoppedDensity_zero {κ : ℝ} {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
+    (hb0 : B 0 = 0) {z : ℂ} (hz : z ∈ halfPlane) :
+    finiteStoppedDensity κ B γ g z 0 =
+      ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) := by
+  rw [finiteStoppedDensity, firstLiveRadiusStop_zero h hz, ite_eq_right ENNReal.zero_ne_top,
+    ENNReal.toReal_zero, h.radius_initial hz]
+  simp only [loewnerAngleSine, h.initial z hz, Real.toNNReal_zero, hb0, mul_zero,
+    Complex.ofReal_zero, sub_zero]
+
+lemma green_initial (κ : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
+    green (1-κ/8) (8/κ-1) z.re z.im 0 =
+      (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1) := by
+  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  rw [green_eq_source _ _ _ _ _ hz, Real.exp_zero, div_one, hp]
+  congr 3
+  rw [Complex.norm_eq_sqrt_sq_add_sq]
+
+theorem finiteStoppedCenteredLaw {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) :
+    (∫⁻ sample, finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∂P) =
+      ENNReal.ofReal ((2 * z.im) ^ (-codimension κ) *
+        (z.im / ‖z‖) ^ (8 / κ - 1)) := by
+  rcases eq_or_lt_of_le hu with rfl | hu
+  · have he : (fun sample => finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z 0) =ᵐ[P]
+        (fun _ => ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))) := by
+      filter_upwards [hmaps,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hb
+      exact finiteStoppedDensity_zero (κ := κ) (B := fun t => B t sample) hm hb hz
+    rw [lintegral_congr_ae he, lintegral_const, measure_univ, mul_one]
+  · let C := cutoffs hz u hu.le
+    let F := fun n sample => ENNReal.ofReal ((C n).value κ B sample)
+    have hexp (n : ℕ) := (C n).expectation hB κ hκ hz hu.le
+    have hFm (n : ℕ) : AEMeasurable (F n) P :=
+      ENNReal.measurable_ofReal.comp_aemeasurable (hexp n).1.aemeasurable
+    have hFb (n : ℕ) : ∀ᵐ sample ∂P, F n sample ≤ ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) := by
+      filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hc hb
+      exact ENNReal.ofReal_le_ofReal ((C n).value_bound κ hκ B sample hz hu hc hb)
+    have hFl : ∀ᵐ sample ∂P, Tendsto (fun n => F n sample) atTop
+        (𝓝 (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u)) := by
+      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb
+      exact tendsto_value_killed κ B sample hm hz hu hc hb hκ
+    have hfin : (∫⁻ _ : Ω, ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8))) ∂P) ≠ (⊤ : ℝ≥0∞) := by
+      rw [lintegral_const, measure_univ, mul_one]
+      exact ENNReal.ofReal_ne_top
+    have hdct := tendsto_lintegral_of_dominated_convergence'
+      (fun _ : Ω => ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-(1-κ/8)))) hFm hFb hfin hFl
+    have hint (n : ℕ) : (∫⁻ sample, F n sample ∂P) = ENNReal.ofReal (green (1-κ/8) (8/κ-1) z.re z.im 0) := by
+      rw [← ofReal_integral_eq_lintegral_ofReal (hexp n).1
+        (Filter.Eventually.of_forall (fun sample => (C n).value_nonneg κ B sample)), (hexp n).2]
+    simp only [hint] at hdct
+    have he := tendsto_nhds_unique hdct tendsto_const_nhds
+    rw [green_initial κ hz] at he
+    exact he
+
+end SLEExactGauge
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BoundaryFunctional
+
+def clampRe (ε : ℝ) (w : ℂ) : ℂ := BoundaryCutoff.clamp 1 ε w
+
+def velocity (ε u : ℝ) (w : ℂ) : ℂ := BoundaryCutoff.velocity 1 ε u w
+
+lemma clampRe_ne_zero {ε : ℝ} (hε : 0 < ε) (w : ℂ) : clampRe ε w ≠ 0 :=
+  BoundaryCutoff.clamp_ne_zero (by norm_num) hε w
+
+lemma clampRe_norm {ε : ℝ} (hε : 0 < ε) (w : ℂ) : ε ≤ ‖clampRe ε w‖ :=
+  BoundaryCutoff.clamp_norm (by norm_num) hε w
+
+lemma clampRe_norm_sub (ε : ℝ) (z w : ℂ) : ‖clampRe ε z-clampRe ε w‖ ≤ 2*‖z-w‖ :=
+  BoundaryCutoff.clamp_norm_sub (by norm_num) z w
+
+lemma velocity_lipschitz {ε : ℝ} (hε : 0 < ε) (u : ℝ) :
+    LipschitzWith ⟨4/ε^2,by positivity⟩ (velocity ε u) :=
+  BoundaryCutoff.velocity_lipschitz (by norm_num) hε u
+
+lemma norm_velocity_sub_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (z w : ℂ) :
+    ‖velocity ε u z-velocity ε u w‖ ≤ (4/ε^2)*‖z-w‖ :=
+  BoundaryCutoff.norm_velocity_sub_le (by norm_num) hε u z w
+
+lemma continuous_velocity {ε : ℝ} (hε : 0 < ε) :
+    Continuous (fun p : ℝ × ℂ => velocity ε p.1 p.2) :=
+  BoundaryCutoff.continuous_velocity (by norm_num) hε
+
+lemma exists_solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
+    ∃ H : ℝ → ℂ, H 0=z ∧ ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t :=
+  BoundaryCutoff.exists_solution (by norm_num) hε hT U hU z
+
+lemma norm_velocity_driver_sub_le {ε : ℝ} (hε : 0 < ε) (u v : ℝ) (z : ℂ) :
+    ‖velocity ε u z - velocity ε v z‖ ≤ (4 / ε^2) * |u-v| := by
+  have he (a : ℝ) : velocity ε 0 (z - (a : ℂ)) = velocity ε a z := by
+    simp [velocity,BoundaryCutoff.velocity]
+  have hh := norm_velocity_sub_le hε 0 (z-(u:ℂ)) (z-(v:ℂ))
+  rw [he, he] at hh
+  convert hh using 1
+  rw [sub_sub_sub_cancel_left, ← Complex.ofReal_sub, Complex.norm_real, abs_sub_comm]
+  simp [Real.norm_eq_abs]
+
+lemma derivWithin_right {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ico 0 T) :
+    HasDerivWithinAt H (velocity ε (U t) (H t)) (Ici t) t :=
+  (hH t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
+
+lemma derivWithin_left {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ioc 0 T) :
+    HasDerivWithinAt H (velocity ε (U t) (H t)) (Iic t) t :=
+  (hH t ⟨ht.1.le, ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
+
+lemma solution_eq {ε T : ℝ} (hε : 0 < ε) {U : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (U t) (G t)) (Icc 0 T) t)
+    (h0 : H 0 = G 0) : EqOn H G (Icc 0 T) := by
+  apply ODE_solution_unique_of_mem_Icc_right
+    (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith)
+    (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_right hH ht)
+    (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hG)
+    (fun _ ht => derivWithin_right hG ht) (fun _ _ => mem_univ _) h0
+
+lemma solution_dist_le {ε T D : ℝ} (hε : 0 < ε) (hD : 0 ≤ D)
+    {U V : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
+    (h0 : H 0 = G 0) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    dist (H t) (G t) ≤ D * Real.exp ((4 / ε^2) * T) := by
+  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
+  have hK : (0 : ℝ) < K := by change 0 < 4 / ε^2; positivity
+  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
+      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K : ℝ) * D := by
+    rw [dist_eq_norm]
+    calc
+      _ ≤ (4 / ε^2) * |V s-U s| := norm_velocity_driver_sub_le hε _ _ _
+      _ ≤ (K : ℝ) * D := by
+        rw [abs_sub_comm]
+        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1, hs.2.le⟩) hK.le
+  have hh := dist_le_of_approx_trajectories_ODE
+    (fun t => velocity_lipschitz hε (U t))
+    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
+    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
+    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
+    hb (le_of_eq (dist_eq_zero.2 h0)) t ht
+  have he : gronwallBound 0 K (0 + (K : ℝ)*D) (t-0) =
+      D * (Real.exp ((K : ℝ)*t)-1) := by
+    simp only [gronwallBound, zero_add, sub_zero, ite_eq_right (ne_of_gt hK)]
+    field_simp
+    ring
+  rw [he] at hh
+  have hexp : Real.exp ((K : ℝ)*t) ≤ Real.exp ((K : ℝ)*T) :=
+    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
+  apply hh.trans
+  change D * (Real.exp ((4 / ε^2)*t)-1) ≤ D * Real.exp ((4 / ε^2)*T)
+  change Real.exp ((4 / ε^2)*t) ≤ Real.exp ((4 / ε^2)*T) at hexp
+  exact mul_le_mul_of_nonneg_left (by linarith) hD
+
+end SLEExactGauge.BoundaryFunctional
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BoundaryFunctional
+open scoped unitInterval
+
+def extendDriver (T : ℝ) (U : C(I, ℝ)) (t : ℝ) : ℝ :=
+  U (projIcc 0 1 zero_le_one (t / T))
+
+lemma continuous_extendDriver (T : ℝ) (U : C(I, ℝ)) : Continuous (extendDriver T U) :=
+  U.continuous.comp (continuous_projIcc.comp (continuous_id.div_const T))
+
+def solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) : ℝ → ℂ :=
+  (exists_solution hε hT (extendDriver T U) (continuous_extendDriver T U).continuousOn z).choose
+
+lemma solution_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    solution hε hT U z 0 = z :=
+  (exists_solution hε hT (extendDriver T U)
+    (continuous_extendDriver T U).continuousOn z).choose_spec.1
+
+lemma solution_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (solution hε hT U z)
+      (velocity ε (extendDriver T U t) (solution hε hT U z t)) (Icc 0 T) t :=
+  (exists_solution hε hT (extendDriver T U)
+    (continuous_extendDriver T U).continuousOn z).choose_spec.2
+
+def solutionPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    C(Icc 0 T, ℂ) :=
+  ⟨fun t => solution hε hT U z t,
+    continuousOn_iff_continuous_domRestrict.1 (HasDerivWithinAt.continuousOn (solution_deriv hε hT U z))⟩
+
+lemma solutionPath_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    LipschitzWith ⟨Real.exp ((4 / ε^2) * T), (Real.exp_pos _).le⟩
+      (fun U : C(I, ℝ) => solutionPath hε hT U z) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro U V
+  apply (ContinuousMap.dist_le (mul_nonneg (Real.exp_pos _).le dist_nonneg)).2
+  intro t
+  have hUV (s : ℝ) (_hs : s ∈ Icc 0 T) :
+      |extendDriver T U s-extendDriver T V s| ≤ dist U V := by
+    exact (ContinuousMap.dist_apply_le_dist (f := U) (g := V)
+      (projIcc 0 1 zero_le_one (s/T)))
+  have hh := solution_dist_le hε dist_nonneg (solution_deriv hε hT U z)
+    (solution_deriv hε hT V z)
+    ((solution_initial hε hT U z).trans (solution_initial hε hT V z).symm)
+    hUV t.property
+  change dist (solution hε hT U z t) (solution hε hT V z t) ≤
+    Real.exp ((4 / ε^2)*T) * dist U V
+  simpa only [mul_comm] using hh
+
+def position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℂ :=
+  solutionPath hε hT U z (projIcc 0 T hT t)
+
+lemma position_eq {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    position hε hT U z t = solution hε hT U z t := by
+  simp only [position, solutionPath, projIcc, max_eq_right ht.1,
+    min_eq_right ht.2]
+  rfl
+
+lemma continuous_position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => position hε hT p.1 z p.2) := by
+  exact continuous_eval.comp (((solutionPath_lipschitz hε hT z).continuous.comp continuous_fst).prodMk
+    ((continuous_projIcc (h := hT)).comp continuous_snd))
+
+lemma continuous_joint_extendDriver (T : ℝ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => extendDriver T p.1 p.2) := by
+  exact continuous_eval.comp (continuous_fst.prodMk
+    ((continuous_projIcc (h := zero_le_one)).comp (continuous_snd.div_const T)))
+
+def variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  (-2 / (clampRe ε (position hε hT U z t - (extendDriver T U t : ℂ)))^2).re
+
+lemma continuous_clampRe (ε : ℝ) : Continuous (clampRe ε) := by
+  change Continuous (fun w : ℂ => Complex.equivRealProdCLM.symm (1*max (1*w.re) ε,w.im))
+  apply Complex.equivRealProdCLM.symm.continuous.comp
+  apply Continuous.prodMk <;> fun_prop
+
+lemma continuous_variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => variation hε hT p.1 z p.2) := by
+  have hw := (continuous_position hε hT z).sub
+    (Complex.continuous_ofReal.comp (continuous_joint_extendDriver T))
+  exact Complex.continuous_re.comp (continuous_const.div
+    (((continuous_clampRe ε).comp hw).pow 2)
+    (fun p => pow_ne_zero 2 (clampRe_ne_zero hε _)))
+
+def logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  ∫ s in (0 : ℝ)..(projIcc 0 T hT t : ℝ), variation hε hT U z s
+
+lemma continuous_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => logDerivative hε hT p.1 z p.2) := by
+  have hc := intervalIntegral.continuous_parametric_primitive_of_continuous
+    (μ := volume) (a₀ := 0) (f := fun U t => variation hε hT U z t)
+    (continuous_variation hε hT z)
+  exact hc.comp (continuous_fst.prodMk
+    (continuous_subtype_val.comp ((continuous_projIcc (h := hT)).comp continuous_snd)))
+
+lemma abs_variation_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    |variation hε hT U z t| ≤ 2/ε^2 := by
+  apply (Complex.abs_re_le_norm _).trans
+  rw [norm_div, norm_neg, Complex.norm_ofNat, norm_pow]
+  apply div_le_div_of_nonneg_left (by norm_num) (sq_pos_of_pos hε)
+  exact pow_le_pow_left₀ hε.le (clampRe_norm hε _) 2
+
+lemma logDerivative_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) : logDerivative hε hT U z 0 = 0 := by
+  simp only [logDerivative, projIcc_of_mem hT (left_mem_Icc.mpr hT), intervalIntegral.integral_same]
+
+lemma abs_logDerivative_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    |logDerivative hε hT U z t| ≤ (2/ε^2)*T := by
+  have hb := (projIcc 0 T hT t).property
+  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (0:ℝ)) (b := (projIcc 0 T hT t : ℝ))
+    (fun s (_hs : s ∈ uIoc 0 (projIcc 0 T hT t : ℝ)) =>
+      show ‖variation hε hT U z s‖ ≤ 2/ε^2 from abs_variation_le hε hT U z s)
+  change |logDerivative hε hT U z t| ≤ _
+  apply hi.trans
+  rw [sub_zero, abs_of_nonneg hb.1]
+  exact mul_le_mul_of_nonneg_left hb.2 (by positivity)
+
+lemma position_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) : position hε hT U z 0 = z := by
+  rw [position_eq hε hT U z ⟨le_rfl,hT⟩, solution_initial]
+
+lemma position_derivWithin {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    HasDerivWithinAt (position hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)) (Icc 0 T) t := by
+  rw [position_eq hε hT U z ht]
+  exact (solution_deriv hε hT U z t ht).congr
+    (fun s hs => position_eq hε hT U z hs) (position_eq hε hT U z ht)
+
+lemma position_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
+    HasDerivAt (position hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)) t :=
+  (position_derivWithin hε hT U z ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2)
+
+lemma logDerivative_eq_integral {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    logDerivative hε hT U z t = ∫ s in (0:ℝ)..t, variation hε hT U z s := by
+  simp only [logDerivative, projIcc_of_mem hT ht]
+
+lemma hasDerivAt_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
+    HasDerivAt (logDerivative hε hT U z) (variation hε hT U z t) t := by
+  have hp : Continuous (fun t : ℝ => (U,t)) := continuous_const.prodMk continuous_id
+  have hc := (continuous_variation hε hT z).comp hp
+  dsimp only [Function.comp_def] at hc
+  have hd : HasDerivAt (fun v : ℝ => ∫ s in (0:ℝ)..v, variation hε hT U z s)
+      (variation hε hT U z t) t :=
+    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
+      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
+  exact logDerivative_eq_integral hε hT U z hs
+
+lemma position_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
+    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (_hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
+    (U V : C(I, ℝ)) (z : ℂ)
+    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
+    EqOn (position hε hT₁ U z) (position hε hT₂ V z) (Icc 0 S) := by
+  apply solution_eq hε (U := extendDriver T₁ U) ?_ ?_ ((position_initial hε hT₁ U z).trans (position_initial hε hT₂ V z).symm)
+  · intro t ht
+    exact (position_derivWithin hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩).mono
+      (Icc_subset_Icc le_rfl hS₁)
+  · intro t ht
+    rw [hUV ht]
+    exact (position_derivWithin hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩).mono
+      (Icc_subset_Icc le_rfl hS₂)
+
+lemma logDerivative_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
+    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
+    (U V : C(I, ℝ)) (z : ℂ)
+    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
+    EqOn (logDerivative hε hT₁ U z) (logDerivative hε hT₂ V z) (Icc 0 S) := by
+  have hH := position_prefix_eq hε hT₁ hT₂ hS hS₁ hS₂ U V z hUV
+  intro t ht
+  rw [logDerivative_eq_integral hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩,
+    logDerivative_eq_integral hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩]
+  apply intervalIntegral.integral_congr
+  intro s hs
+  rw [uIcc_of_le ht.1] at hs
+  have hsS : s ∈ Icc 0 S := ⟨hs.1,hs.2.trans ht.2⟩
+  simp only [variation, hH hsS, hUV hsS]
+
+lemma position_im_zero {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (x t : ℝ) : (position hε hT U (x:ℂ) t).im = 0 := by
+  have hc := BoundaryCutoff.conjugate_deriv (solution_deriv hε hT U (x:ℂ))
+  have he := solution_eq hε (solution_deriv hε hT U (x:ℂ)) hc (by
+    rw [solution_initial]; simp)
+  let v := projIcc 0 T hT t
+  have hv := he v.property
+  change solution hε hT U (x:ℂ) v = star (solution hε hT U (x:ℂ) v) at hv
+  have hh := congrArg Complex.im hv
+  change (solution hε hT U (x:ℂ) v).im = 0
+  change (solution hε hT U (x:ℂ) v).im = -(solution hε hT U (x:ℂ) v).im at hh
+  linarith
+
+end SLEExactGauge.BoundaryFunctional
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+open GreenPhysical
+
+def radial (β x y : ℝ) : ℝ := (x^2+y^2)^(β/2)
+
+def lyapunov (β C x y : ℝ) : ℝ := radial β x y+C*y^β
+
+def radialX (β x y : ℝ) : ℝ := β*x*(x^2+y^2)^(β/2-1)
+
+def radialXX (β x y : ℝ) : ℝ :=
+  β*(x^2+y^2)^(β/2-1)+β*(β-2)*x^2*(x^2+y^2)^(β/2-2)
+
+def radialXXX (β x y : ℝ) : ℝ :=
+  3*β*(β-2)*x*(x^2+y^2)^(β/2-2)+
+  β*(β-2)*(β-4)*x^3*(x^2+y^2)^(β/2-3)
+
+lemma radial_pos (β x : ℝ) {y : ℝ} (hy : 0 < y) : 0 < radial β x y :=
+  Real.rpow_pos_of_pos (q_pos x hy) _
+
+lemma hasDerivAt_radial_path (β : ℝ) {X Y : ℝ → ℝ} {t X' Y' : ℝ}
+    (hX : HasDerivAt X X' t) (hY : HasDerivAt Y Y' t) (hy : 0 < Y t) :
+    HasDerivAt (fun t => radial β (X t) (Y t))
+      (β*(X t*X'+Y t*Y')*(X t^2+Y t^2)^(β/2-1)) t := by
+  have h := (((hX.pow 2).add (hY.pow 2)).rpow_const (p := β/2) (Or.inl (q_pos (X t) hy).ne'))
+  simp only [Pi.add_apply,Pi.pow_apply] at h
+  apply h.congr_deriv
+  ring
+
+lemma hasDerivAt_radial_x (β x y : ℝ) (hy : 0 < y) :
+    HasDerivAt (radial β · y) (radialX β x y) x := by
+  convert hasDerivAt_radial_path β (hasDerivAt_id x) (hasDerivAt_const x y) hy using 1 <;>
+    simp [radialX]
+
+lemma hasDerivAt_radialX_x (β x y : ℝ) (hy : 0 < y) :
+    HasDerivAt (radialX β · y) (radialXX β x y) x := by
+  have hq := ((hasDerivAt_id x).pow 2).add_const (y^2)
+  have h := ((hasDerivAt_id x).const_mul β).mul
+    (hq.rpow_const (p := β/2-1) (Or.inl (q_pos x hy).ne'))
+  apply h.congr_deriv
+  dsimp only [id_eq,Pi.pow_apply,radialXX]
+  convert_to β*(x^2+y^2)^(β/2-1)+β*x*((2*x)*((β/2-1)*(x^2+y^2)^(β/2-1-1))) = _ using 1
+  · ring
+  rw [show β/2-1-1 = β/2-2 by ring]
+  ring
+
+lemma hasDerivAt_radialXX_x (β x y : ℝ) (hy : 0 < y) :
+    HasDerivAt (radialXX β · y) (radialXXX β x y) x := by
+  have hq := ((hasDerivAt_id x).pow 2).add_const (y^2)
+  have h1 := (hq.rpow_const (p := β/2-1) (Or.inl (q_pos x hy).ne')).const_mul β
+  have h2 := (((hasDerivAt_id x).pow 2).const_mul (β*(β-2))).mul
+    (hq.rpow_const (p := β/2-2) (Or.inl (q_pos x hy).ne'))
+  apply (h1.add h2).congr_deriv
+  dsimp only [id_eq,Pi.pow_apply,radialXXX]
+  rw [show β/2-1-1 = β/2-2 by ring,show β/2-2-1 = β/2-3 by ring]
+  ring
+
+lemma contDiff_radial_driver (β c y s : ℝ) (hy : 0 < y) :
+    ContDiff ℝ ⊤ (fun b => radial β (c-s*b) y) := by
+  unfold radial
+  apply ContDiff.rpow_const_of_ne
+  · fun_prop
+  · intro x; exact (q_pos _ hy).ne'
+
+lemma lyapunov_nonneg {β C : ℝ} (hC : 0 ≤ C) (x : ℝ) {y : ℝ} (hy : 0 < y) :
+    0 ≤ lyapunov β C x y := by
+  exact add_nonneg (radial_pos β x hy).le (mul_nonneg hC (Real.rpow_nonneg hy.le _))
+
+def generator (κ β C x y : ℝ) : ℝ :=
+  β*(x*(2*x/(x^2+y^2))+y*(-2*y/(x^2+y^2)))*(x^2+y^2)^(β/2-1) +
+  C*(β*y^(β-1)*(-2*y/(x^2+y^2))) + κ/2*radialXX β x y
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/IntegratedMeshSimple.lean b/lean/OAI/Probability/SLELocalization/IntegratedMeshSimple.lean
index 26de42d72e1af77ea34f3c185ee046f731df69f6..31184cf43ad54faf113605842759b07ac133217d 100644
--- a/lean/OAI/Probability/SLELocalization/IntegratedMeshSimple.lean
+++ b/lean/OAI/Probability/SLELocalization/IntegratedMeshSimple.lean
@@ -1,908 +1,908 @@
-import Mathlib
-import OAI.Probability.SLELocalization.ConformalKernels
-
-namespace OAI
-
-
-/-! Conformal localization and Integrated Mesh Simple estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal
-namespace SLEExactGauge
-open Layer
-
-lemma layer_set_measurable {F : ℂ → ℂ} (hf : DifferentiableOn ℂ F (ball 0 1)) (R : ℝ) :
-    MeasurableSet {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R} := by
-  classical
-  let f : ℂ → ℝ := fun w => diskFactor w*‖deriv F w‖
-  have hc : ContinuousOn f (ball 0 1) :=
-    diskFactor_smooth.continuous.continuousOn.mul (hf.deriv isOpen_ball).continuousOn.norm
-  have hm : Measurable ((ball (0:ℂ) 1).piecewise f (fun _ => 0)) :=
-    hc.measurable_piecewise continuousOn_const measurableSet_ball
-  have he : {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R} =
-      ball (0:ℂ) 1 ∩ {w | (ball (0:ℂ) 1).piecewise f (fun _ => 0) w ≤ R} := by
-    ext w
-    by_cases hw : w ∈ ball (0:ℂ) 1
-    · have hw1 := mem_ball_zero_iff.1 hw
-      simp [hw,hw1,f,diskFactor]
-    · have hn : ¬‖w‖ < 1 := by simpa only [mem_ball_zero_iff] using hw
-      simp [hn,hw]
-  rw [he]
-  exact measurableSet_ball.inter (measurableSet_le hm measurable_const)
-
-theorem conformalRadiusLayer :
-    ∃ C : ℝ, 0 < C ∧ ∀ F : ℂ → ℂ,
-      DifferentiableOn ℂ F (Metric.ball 0 1) → Set.InjOn F (Metric.ball 0 1) →
-      ∀ R : ℝ, 0 < R →
-      (∫⁻ w in {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R},
-        ENNReal.ofReal (Real.log (1/‖w‖)*‖deriv F w‖^2) ∂volume) ≤
-      ENNReal.ofReal (C*R^2) := by
-  refine ⟨4*Real.pi/Layer.layerExponent,div_pos (by positivity) layerExponent_bounds.1,?_⟩
-  intro F hf hi R hR
-  obtain ⟨G,h⟩ := univalent_diskPair hf hi
-  let S := {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R}
-  let u := layerBarrier F R
-  have hα := layerExponent_bounds.1
-  have hC : 0 ≤ 2/Layer.layerExponent := by positivity
-  have hpoint (w : ℂ) (hw : w ∈ S) :
-      ENNReal.ofReal (Real.log (1/‖w‖)*‖deriv F w‖^2) ≤
-      ENNReal.ofReal (2/Layer.layerExponent) * ENNReal.ofReal
-        (Real.log (1/‖w‖)*(-Laplacian.laplacian u w)) := by
-    have hb := h.barrier_layer hw.1 hR hw.2
-    have hlog : 0 ≤ Real.log (1/‖w‖) := by
-      by_cases hz : w = 0
-      · simp [hz]
-      · exact Real.log_nonneg ((one_le_div (norm_pos_iff.2 hz)).2 hw.1.le)
-    rw [← ENNReal.ofReal_mul hC]
-    apply ENNReal.ofReal_le_ofReal
-    have hb' : ‖deriv F w‖^2 ≤ (2/Layer.layerExponent)*(-Laplacian.laplacian u w) := by
-      have hm := mul_le_mul_of_nonneg_left hb hC
-      have he : (2/Layer.layerExponent)*(Layer.layerExponent/2*‖deriv F w‖^2) = ‖deriv F w‖^2 := by
-        field_simp
-      rw [he] at hm
-      exact hm
-    nlinarith [mul_le_mul_of_nonneg_left hb' hlog]
-  have hS : MeasurableSet S := layer_set_measurable hf R
-  have hsub : S ⊆ ball (0:ℂ) 1 := fun w hw => mem_ball_zero_iff.2 hw.1
-  have hb := superharmonic_disk_green (h.barrier_smooth R)
-    (fun z _ => barrier_bounds F R z)
-    (fun z hz => h.barrier_superharmonic (mem_ball_zero_iff.1 hz) R)
-  calc
-    _ ≤ ∫⁻ w in S, ENNReal.ofReal (2/Layer.layerExponent) *
-        ENNReal.ofReal (Real.log (1/‖w‖)*(-Laplacian.laplacian u w)) :=
-      setLIntegral_mono' hS hpoint
-    _ ≤ ∫⁻ w in ball (0:ℂ) 1, ENNReal.ofReal (2/Layer.layerExponent) *
-        ENNReal.ofReal (Real.log (1/‖w‖)*(-Laplacian.laplacian u w)) := lintegral_mono_set hsub
-    _ = ENNReal.ofReal (2/Layer.layerExponent) * (∫⁻ w in ball (0:ℂ) 1,
-        ENNReal.ofReal (Real.log (1/‖w‖)*(-Laplacian.laplacian u w))) := by
-      rw [lintegral_const_mul']
-      exact ENNReal.ofReal_ne_top
-    _ ≤ ENNReal.ofReal (2/Layer.layerExponent) * ENNReal.ofReal (2*Real.pi*R^2) :=
-      mul_le_mul_right hb _
-    _ = _ := by
-      rw [← ENNReal.ofReal_mul hC]
-      congr 1
-      ring
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal
-namespace SLEExactGauge.Layer
-open SlitMaps
-
-lemma DiskPair.radius_le_omitted {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) {z w : ℂ} (hz : ‖z‖ < 1) (hw : w ∉ U) :
-    diskFactor z*‖deriv F z‖ ≤ (256*Real.pi)*dist (F z) w := by
-  have hr := h.recenter hz
-  have homit : w ∉ (F ∘ diskAuto (-z)) '' ball (0:ℂ) 1 := by
-    rintro ⟨q,hq,rfl⟩
-    exact hw (hr.toDomain hq)
-  have hb := UnivalentDistance.omitted_value_bound hr.forward hr.injOn homit
-  have hF := h.forward.differentiableAt (isOpen_ball.mem_nhds (mem_ball_zero_iff.2 hz))
-  have hφ := diskAuto_hasDerivAt (show ‖-z‖ < 1 by simpa using hz) (z := (0:ℂ)) (by simp)
-  have hφ0 : diskAuto (-z) 0 = z := by simp
-  have he := (hφ0.symm ▸ hF.hasDerivAt).comp (0:ℂ) hφ
-  have hd : deriv (F ∘ diskAuto (-z)) 0 = deriv F z*((diskFactor z:ℝ):ℂ) := by
-    have hdφ := diskAuto_deriv_zero_exact hz
-    have ha : (1-(starRingEnd ℂ z)*z) = ((diskFactor z:ℝ):ℂ) := by
-      rw [mul_comm (starRingEnd ℂ z),Complex.mul_conj']
-      simp only [diskFactor,Complex.ofReal_sub,Complex.ofReal_one,Complex.ofReal_pow]
-    rw [ha] at hdφ
-    simpa only [hφ0,map_neg,neg_mul_neg,mul_zero,sub_zero,one_pow,div_one,ha] using he.deriv
-  rw [hd,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (diskFactor_pos hz)] at hb
-  simpa only [Function.comp_apply,hφ0,dist_eq_norm,mul_comm (diskFactor z)] using hb
-
-lemma DiskPair.radius_le_infDist {U : Set ℂ} {F G : ℂ → ℂ}
-    (h : DiskPair U F G) (hne : Uᶜ.Nonempty) {z : ℂ} (hz : ‖z‖ < 1) :
-    diskFactor z*‖deriv F z‖ ≤ (256*Real.pi)*infDist (F z) Uᶜ := by
-  have hp : 0 < 256*Real.pi := by positivity
-  rw [mul_comm (256*Real.pi),← div_le_iff₀ hp]
-  apply (le_infDist hne).2
-  intro w hw
-  rw [div_le_iff₀ hp]
-  exact (h.radius_le_omitted hz hw).trans_eq (mul_comm _ _)
-
-lemma complexJacobian (c : ℂ) :
-    ((ContinuousLinearMap.toSpanSingleton ℂ c).restrictScalars ℝ).det = ‖c‖^2 := by
-  simp [ContinuousLinearMap.det,LinearMap.det_restrictScalars,
-    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
-
-theorem physical_boundary_layer :
-    ∃ C : ℝ, 0 < C ∧ ∀ (U : Set ℂ) (F G : ℂ → ℂ), DiskPair U F G → Uᶜ.Nonempty →
-      ∀ R : ℝ, 0 < R →
-      (∫⁻ w in {w : ℂ | w ∈ U ∧ infDist w Uᶜ ≤ R},
-        ENNReal.ofReal (Real.log (1/‖G w‖))) ≤ ENNReal.ofReal (C*R^2) := by
-  obtain ⟨C,hC,hbound⟩ := conformalRadiusLayer
-  refine ⟨C*(256*Real.pi)^2,by positivity,?_⟩
-  intro U F G h hne R hR
-  let S := {z : ℂ | ‖z‖ < 1 ∧ (1-‖z‖^2)*‖deriv F z‖ ≤ (256*Real.pi)*R}
-  have hS : MeasurableSet S := layer_set_measurable h.forward _
-  have hsub : S ⊆ ball (0:ℂ) 1 := fun z hz => mem_ball_zero_iff.2 hz.1
-  have himage : {w : ℂ | w ∈ U ∧ infDist w Uᶜ ≤ R} ⊆ F '' S := by
-    intro w hw
-    refine ⟨G w,⟨mem_ball_zero_iff.1 (h.toDisk hw.1),?_⟩,h.right_inv w hw.1⟩
-    have hh := h.radius_le_infDist hne (mem_ball_zero_iff.1 (h.toDisk hw.1))
-    rw [h.right_inv w hw.1] at hh
-    exact hh.trans (mul_le_mul_of_nonneg_left hw.2 (by positivity))
-  have hchange := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hS
-    (fun z hz => ((h.forward.differentiableAt (isOpen_ball.mem_nhds (hsub hz))).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt)
-    (h.injOn.mono hsub) (fun w => ENNReal.ofReal (Real.log (1/‖G w‖)))
-  calc
-    _ ≤ ∫⁻ w in F '' S, ENNReal.ofReal (Real.log (1/‖G w‖)) := lintegral_mono_set himage
-    _ = ∫⁻ z in S, ENNReal.ofReal (Real.log (1/‖z‖)*‖deriv F z‖^2) := by
-      rw [hchange]
-      apply setLIntegral_congr_fun hS
-      intro z hz
-      dsimp only
-      rw [complexJacobian,abs_of_nonneg (sq_nonneg _),h.left_inv z (hsub hz),
-        ← ENNReal.ofReal_mul (sq_nonneg _)]
-      congr 1
-      ring
-    _ ≤ ENNReal.ofReal (C*((256*Real.pi)*R)^2) := hbound F h.forward h.injOn _ (by positivity)
-    _ = _ := by congr 1; ring
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.Mesh
-
-def halfPlaneGreen (z w : ℂ) : ℝ := Real.log ‖(z-(starRingEnd ℂ w))/(z-w)‖
-
-lemma hasDerivAt_log_norm {f : ℝ → ℂ} {D : ℂ} {t : ℝ}
-    (hd : HasDerivAt f D t) (hn : f t ≠ 0) :
-    HasDerivAt (fun u => Real.log ‖f u‖) (D/f t).re t := by
-  have hh : ContDiffAt ℝ 2 (fun z : ℂ => Real.log ‖z‖) (f t) :=
-    (analyticAt_id.harmonicAt_log_norm hn).1
-  have hc := (hh.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt t hd
-  have he := Layer.logNorm_fderiv (hasDerivAt_id (f t)) hn D
-  apply hc.congr_deriv
-  simpa only [id_eq,one_mul] using he
-
-lemma green_eq_difference {z w : ℂ} (hn : z ≠ w) (hp : z ≠ starRingEnd ℂ w) :
-    halfPlaneGreen z w = Real.log ‖z-(starRingEnd ℂ w)‖ - Real.log ‖z-w‖ := by
-  rw [halfPlaneGreen,norm_div,Real.log_div (norm_ne_zero_iff.2 (sub_ne_zero.2 hp))
-    (norm_ne_zero_iff.2 (sub_ne_zero.2 hn))]
-
-lemma difference_velocity {z w : ℂ} (hz : z ≠ 0) (hw : w ≠ 0) (hd : z ≠ w) :
-    (2/z-2/w)/(z-w) = -2/(z*w) := by
-  field_simp
-  ring
-
-lemma green_velocity {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    (-2/(z*(starRingEnd ℂ w))).re - (-2/(z*w)).re =
-      -4*z.im/(‖z‖^2) * (w.im/(‖w‖^2)) := by
-  have hzn : Complex.normSq z ≠ 0 := by
-    rw [← Complex.sq_norm]
-    exact pow_ne_zero 2 (norm_ne_zero_iff.2 (by intro he; simp [he] at hz))
-  have hwn : Complex.normSq w ≠ 0 := by
-    rw [← Complex.sq_norm]
-    exact pow_ne_zero 2 (norm_ne_zero_iff.2 (by intro he; simp [he] at hw))
-  rw [Complex.div_re,Complex.div_re,Complex.normSq_mul,Complex.normSq_mul,Complex.normSq_conj,
-    Complex.sq_norm,Complex.sq_norm]
-  simp only [Complex.mul_re,Complex.conj_re,Complex.conj_im,Complex.neg_re,Complex.neg_im]
-  norm_num
-  field_simp
-  ring
-
-theorem green_decrease {g h : ℝ → ℂ} {t u : ℝ}
-    (hz : 0 < (g t).im) (hw : 0 < (h t).im) (hne : g t ≠ h t)
-    (hg : HasDerivAt g (2/(g t-(u:ℂ))) t)
-    (hh : HasDerivAt h (2/(h t-(u:ℂ))) t) :
-    HasDerivAt (fun s => halfPlaneGreen (g s) (h s))
-      (-4*(g t).im/‖g t-(u:ℂ)‖^2 * ((h t).im/‖h t-(u:ℂ)‖^2)) t := by
-  have hp : g t ≠ starRingEnd ℂ (h t) := by
-    intro he
-    have hi := congrArg Complex.im he
-    simp only [Complex.conj_im] at hi
-    linarith
-  have hc := hg.sub hh
-  have hcc := hg.sub (Complex.conjCLE.hasFDerivAt.comp_hasDerivAt t hh)
-  have h₁ := hasDerivAt_log_norm hcc (sub_ne_zero.2 hp)
-  have h₂ := hasDerivAt_log_norm hc (sub_ne_zero.2 hne)
-  have he : (fun s => halfPlaneGreen (g s) (h s)) =ᶠ[𝓝 t]
-      (fun s => Real.log ‖g s-starRingEnd ℂ (h s)‖-Real.log ‖g s-h s‖) := by
-    filter_upwards [(hg.continuousAt.sub hh.continuousAt).eventually_ne (sub_ne_zero.2 hne),
-      hcc.continuousAt.eventually_ne (sub_ne_zero.2 hp)] with s hs hs'
-    exact green_eq_difference (sub_ne_zero.1 hs) (sub_ne_zero.1 hs')
-  apply ((h₁.sub h₂).congr_of_eventuallyEq he).congr_deriv
-  let z := g t-(u:ℂ)
-  let w := h t-(u:ℂ)
-  have hzi : 0 < z.im := by simpa [z] using hz
-  have hwi : 0 < w.im := by simpa [w] using hw
-  have hzn : z ≠ 0 := by intro he; simp [he] at hzi
-  have hwn : w ≠ 0 := by intro he; simp [he] at hwi
-  have hzw : z ≠ w := by simpa [z,w] using hne
-  have hzwc : z ≠ starRingEnd ℂ w := by
-    intro he
-    have hi := congrArg Complex.im he
-    simp only [Complex.conj_im] at hi
-    linarith
-  have hdiff : g t-h t = z-w := by dsimp [z,w]; ring
-  have hdiffc : g t-starRingEnd ℂ (h t) = z-starRingEnd ℂ w := by
-    simp only [z,w,map_sub,Complex.conj_ofReal]
-    ring
-  have hconj : starRingEnd ℂ (2/(h t-(u:ℂ))) = 2/(starRingEnd ℂ w) := by simp [w,Complex.conj_ofNat]
-  change ((2/z-starRingEnd ℂ (2/w))/(g t-starRingEnd ℂ (h t))).re -
-    ((2/z-2/w)/(g t-h t)).re = _
-  rw [hdiff,hdiffc]
-  have hconj2 : starRingEnd ℂ (2/w) = 2/(starRingEnd ℂ w) := by
-    simpa only [Complex.conj_ofNat] using (map_div₀ (starRingEnd ℂ) (2:ℂ) w)
-  rw [hconj2,difference_velocity hzn (by simpa using hwn) hzwc,
-    difference_velocity hzn hwn hzw,green_velocity hzi hwi]
-  simp [z,w]
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Function MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge.Layer
-
-def halfToDisk (a w : ℂ) : ℂ := (w-a)/(w-starRingEnd ℂ a)
-
-def diskToHalf (a v : ℂ) : ℂ := (a-starRingEnd ℂ a*v)/(1-v)
-
-lemma half_den_ne {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
-    w-starRingEnd ℂ a ≠ 0 := by
-  intro he
-  have hi := congrArg Complex.im he
-  simp only [Complex.sub_im,Complex.conj_im,Complex.zero_im] at hi
-  linarith
-
-lemma disk_den_ne {v : ℂ} (hv : ‖v‖ < 1) : (1:ℂ)-v ≠ 0 := by
-  intro he
-  have he' : v = 1 := (sub_eq_zero.1 he).symm
-  simp [he'] at hv
-
-lemma halfToDisk_maps {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
-    ‖halfToDisk a w‖ < 1 := by
-  have hd := half_den_ne ha hw
-  have he : ‖w-starRingEnd ℂ a‖^2 - ‖w-a‖^2 = 4*w.im*a.im := by
-    simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
-      Complex.conj_re,Complex.conj_im]
-    ring
-  rw [halfToDisk,norm_div,div_lt_one (norm_pos_iff.2 hd)]
-  nlinarith [norm_nonneg (w-a),norm_nonneg (w-starRingEnd ℂ a),mul_pos hw ha]
-
-lemma diskToHalf_im (a v : ℂ) :
-    (diskToHalf a v).im = a.im*(1-‖v‖^2)/‖1-v‖^2 := by
-  simp only [diskToHalf,Complex.div_im,Complex.sq_norm,Complex.normSq_apply,
-    Complex.sub_re,Complex.sub_im,Complex.mul_re,Complex.mul_im,Complex.conj_re,
-    Complex.conj_im,Complex.one_re,Complex.one_im]
-  ring
-
-lemma diskToHalf_maps {a v : ℂ} (ha : 0 < a.im) (hv : ‖v‖ < 1) :
-    0 < (diskToHalf a v).im := by
-  rw [diskToHalf_im]
-  apply div_pos (mul_pos ha _) (sq_pos_of_ne_zero (norm_ne_zero_iff.2 (disk_den_ne hv)))
-  nlinarith [norm_nonneg v]
-
-lemma disk_half_inverse {a v : ℂ} (ha : 0 < a.im) (hv : ‖v‖ < 1) :
-    halfToDisk a (diskToHalf a v) = v := by
-  have h₁ := disk_den_ne hv
-  have h₂ := half_den_ne ha (diskToHalf_maps ha hv)
-  dsimp [halfToDisk,diskToHalf] at h₂ ⊢
-  apply (div_eq_iff h₂).2
-  field_simp
-  ring
-
-lemma half_disk_inverse {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
-    diskToHalf a (halfToDisk a w) = w := by
-  have h₁ := half_den_ne ha hw
-  have h₂ := disk_den_ne (halfToDisk_maps ha hw)
-  dsimp [halfToDisk,diskToHalf] at h₂ ⊢
-  apply (div_eq_iff h₂).2
-  field_simp
-  ring
-
-lemma halfToDisk_differentiable {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
-    DifferentiableAt ℂ (halfToDisk a) w := by
-  exact (differentiableAt_id.sub_const a).div (differentiableAt_id.sub_const _) (half_den_ne ha hw)
-
-lemma diskToHalf_differentiable {a v : ℂ} (_ha : 0 < a.im) (hv : ‖v‖ < 1) :
-    DifferentiableAt ℂ (diskToHalf a) v := by
-  exact ((differentiableAt_id.const_mul _).const_sub a).div
-    (differentiableAt_id.const_sub 1) (disk_den_ne hv)
-
-lemma green_disk_eq (a w : ℂ) :
-    Real.log (1/‖halfToDisk a w‖) = Mesh.halfPlaneGreen a w := by
-  have hn : ‖w-starRingEnd ℂ a‖ = ‖a-starRingEnd ℂ w‖ := by
-    rw [← norm_neg (w-starRingEnd ℂ a),neg_sub]
-    rw [← Complex.norm_conj (starRingEnd ℂ a-w)]
-    simp only [map_sub,starRingEnd_self_apply]
-  rw [halfToDisk,norm_div,one_div_div,hn,norm_sub_rev w,Mesh.halfPlaneGreen,norm_div]
-
-theorem halfPlane_boundary_layer :
-    ∃ C : ℝ, 0 < C ∧ ∀ (U : Set ℂ) (g f : ℂ → ℂ),
-      IsOpen U → Uᶜ.Nonempty →
-      MapsTo g U {z : ℂ | 0 < z.im} → MapsTo f {z : ℂ | 0 < z.im} U →
-      (∀ z ∈ U, f (g z) = z) → (∀ z : ℂ, 0 < z.im → g (f z) = z) →
-      DifferentiableOn ℂ g U → DifferentiableOn ℂ f {z : ℂ | 0 < z.im} →
-      ∀ a : ℂ, 0 < a.im → ∀ R : ℝ, 0 < R →
-      (∫⁻ w in {w : ℂ | w ∈ U ∧ infDist w Uᶜ ≤ R},
-        ENNReal.ofReal (Mesh.halfPlaneGreen a (g w))) ≤ ENNReal.ofReal (C*R^2) := by
-  obtain ⟨C,hC,hbound⟩ := physical_boundary_layer
-  refine ⟨C,hC,?_⟩
-  intro U g f hU hne hg hf hfg hgf hgd hfd a ha R hR
-  have hH : IsOpen {z : ℂ | 0 < z.im} := isOpen_lt continuous_const Complex.continuous_im
-  have hpair : DiskPair U (f ∘ diskToHalf a) (halfToDisk a ∘ g) := by
-    refine ⟨hU,?_,?_,?_,?_,?_,?_⟩
-    · intro v hv
-      exact hf (diskToHalf_maps ha (mem_ball_zero_iff.1 hv))
-    · intro w hw
-      exact mem_ball_zero_iff.2 (halfToDisk_maps ha (hg hw))
-    · intro v hv
-      dsimp only [Function.comp_apply]
-      rw [hgf _ (diskToHalf_maps ha (mem_ball_zero_iff.1 hv))]
-      exact disk_half_inverse ha (mem_ball_zero_iff.1 hv)
-    · intro w hw
-      dsimp only [Function.comp_apply]
-      rw [half_disk_inverse ha (hg hw),hfg w hw]
-    · intro v hv
-      exact ((hfd.differentiableAt (hH.mem_nhds (diskToHalf_maps ha (mem_ball_zero_iff.1 hv)))).comp v
-        (diskToHalf_differentiable ha (mem_ball_zero_iff.1 hv))).differentiableWithinAt
-    · intro w hw
-      exact ((halfToDisk_differentiable ha (hg hw)).comp w
-        (hgd.differentiableAt (hU.mem_nhds hw))).differentiableWithinAt
-  have hh := hbound U _ _ hpair hne R hR
-  simpa only [Function.comp_apply,green_disk_eq] using hh
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge.Mesh
-
-lemma green_nonneg {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
-    0 ≤ halfPlaneGreen z w := by
-  by_cases he : z = w
-  · subst w
-    simp [halfPlaneGreen]
-  have hn : 0 < ‖z-w‖ := norm_pos_iff.2 (sub_ne_zero.2 he)
-  have hd : ‖z-starRingEnd ℂ w‖^2 - ‖z-w‖^2 = 4*z.im*w.im := by
-    simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
-      Complex.conj_re,Complex.conj_im]
-    ring
-  apply Real.log_nonneg
-  rw [norm_div,le_div_iff₀ hn,one_mul]
-  nlinarith [norm_nonneg (z-starRingEnd ℂ w),mul_pos hz hw]
-
-lemma green_continuousAt {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) (hne : z ≠ w) :
-    ContinuousAt (fun q : ℂ × ℂ => halfPlaneGreen q.1 q.2) (z,w) := by
-  apply ContinuousAt.log
-  · exact (((continuous_fst.sub (Complex.continuous_conj.comp continuous_snd)).continuousAt).div
-      (continuous_fst.sub continuous_snd).continuousAt (sub_ne_zero.2 hne)).norm
-  · exact norm_ne_zero_iff.2 (div_ne_zero (Layer.half_den_ne hw hz) (sub_ne_zero.2 hne))
-
-def poissonFactor (driver : ℝ → ℝ) (g : ℝ → ℂ → ℂ) (t : ℝ) (z : ℂ) : ℝ :=
-  (g t z).im/‖g t z-(driver t:ℂ)‖^2
-
-lemma poissonFactor_continuousOn {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {T : ℝ} (hT : 0 ≤ T)
-    (hU : ContinuousOn driver (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun t => poissonFactor driver g t z) (Icc 0 T) := by
-  apply (Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)).div
-    (((h.continuous_eval hT hz).sub (Complex.continuous_ofReal.comp_continuousOn hU)).norm.pow 2)
-  intro t ht
-  exact pow_ne_zero 2 (norm_ne_zero_iff.2 (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz)))
-
-lemma uniformization_injective {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {t : ℝ} (ht : 0 ≤ t) :
-    InjOn (g t) (localizationDomain γ t) := by
-  rcases h with ⟨_,_,_,_,_,_,_,_,_,hfg,_,_,_,_⟩
-  intro z hz w hw he
-  simpa only [hfg t ht z hz,hfg t ht w hw] using congrArg (f t) he
-
-lemma pair_integral {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
-    (hU : ContinuousOn driver (Icc 0 t)) {z w : ℂ}
-    (hz : z ∈ localizationDomain γ t) (hw : w ∈ localizationDomain γ t) (hne : z ≠ w) :
-    (∫ u in s..t, 4*poissonFactor driver g u z*poissonFactor driver g u w) =
-      halfPlaneGreen (g s z) (g s w)-halfPlaneGreen (g t z) (g t w) := by
-  have ht : 0 ≤ t := hs.trans hst
-  have hsub : Icc s t ⊆ Icc 0 t := Icc_subset_Icc hs le_rfl
-  have hgcont : ContinuousOn (fun u => halfPlaneGreen (g u z) (g u w)) (Icc s t) := by
-    intro u hu
-    have hc := (green_continuousAt (h.im_pos (hs.trans hu.1) (survivingDomain_mono hu.2 hz))
-      (h.im_pos (hs.trans hu.1) (survivingDomain_mono hu.2 hw))
-      (fun he => hne (uniformization_injective h (hs.trans hu.1)
-        (survivingDomain_mono hu.2 hz) (survivingDomain_mono hu.2 hw) he)))
-    have hp := (((h.continuous_eval ht hz).mono hsub) u hu).prodMk
-      (((h.continuous_eval ht hw).mono hsub) u hu)
-    exact ContinuousAt.comp_continuousWithinAt (f := fun v => (g v z,g v w)) hc hp
-  have hi : IntervalIntegrable
-      (fun u => -4*poissonFactor driver g u z*poissonFactor driver g u w) volume s t := by
-    apply ContinuousOn.intervalIntegrable
-    rw [uIcc_of_le hst]
-    exact ((((poissonFactor_continuousOn h ht hU hz).const_mul (-4)).mul
-      (poissonFactor_continuousOn h ht hU hw)).mono hsub)
-  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hst hgcont
-    (f' := fun u => -4*poissonFactor driver g u z*poissonFactor driver g u w)
-    (fun u hu => by
-      have hu0 := hs.trans_lt hu.1
-      have hz' := survivingDomain_mono hu.2.le hz
-      have hw' := survivingDomain_mono hu.2.le hw
-      have hd := green_decrease (h.im_pos hu0.le hz') (h.im_pos hu0.le hw')
-        (fun he => hne (uniformization_injective h hu0.le hz' hw' he))
-        (h.hasDerivAt_eval hu0 hz') (h.hasDerivAt_eval hu0 hw')
-      exact hd.congr_deriv (by dsimp [poissonFactor]; ring)) hi
-  have he' : (fun u => -4*poissonFactor driver g u z*poissonFactor driver g u w) =
-      (fun u => -(4*poissonFactor driver g u z*poissonFactor driver g u w)) := by funext u; ring
-  rw [he',intervalIntegral.integral_neg] at he
-  linarith
-
-lemma pair_integral_le {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
-    (hU : ContinuousOn driver (Icc 0 t)) {z w : ℂ}
-    (hz : z ∈ localizationDomain γ t) (hw : w ∈ localizationDomain γ t) (hne : z ≠ w) :
-    (∫ u in s..t, 4*poissonFactor driver g u z*poissonFactor driver g u w) ≤
-      halfPlaneGreen (g s z) (g s w) := by
-  rw [pair_integral h hs hst hU hz hw hne]
-  exact sub_le_self _ (green_nonneg (h.im_pos (hs.trans hst) hz) (h.im_pos (hs.trans hst) hw))
-
-theorem loewner_boundary_layer :
-    ∃ C : ℝ, 0 < C ∧ ∀ (driver : ℝ → ℝ) (γ : ℝ → ℂ) (g f : ℝ → ℂ → ℂ),
-      IsLoewnerTraceUsing driver γ g f → ∀ t : ℝ, 0 ≤ t →
-      ∀ z ∈ localizationDomain γ t, ∀ R : ℝ, 0 < R →
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
-        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2) := by
-  obtain ⟨C,hC,hbound⟩ := Layer.halfPlane_boundary_layer
-  refine ⟨C,hC,?_⟩
-  intro driver γ g f h t ht z hz R hR
-  rcases h with ⟨hγ,_,_,_,_,hg,hf,hgm,hfm,hfg,hgf,_,_,_⟩
-  apply hbound _ (g t) (f t) (survivingDomain_open hγ ht) _ (hgm t ht) (hfm t ht)
-    (hfg t ht) (hgf t ht) (hg t ht).differentiableOn (hf t ht).differentiableOn
-    (g t z) (hgm t ht hz) R hR
-  refine ⟨0,?_⟩
-  intro hzero
-  exact (lt_irrefl (0:ℝ)) (by simpa [halfPlane] using hzero.1.1)
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.PhysicalStop
-
-lemma ofReal_cap_le (s : ℝ≥0∞) (a : ℝ) : ENNReal.ofReal (cap s a) ≤ s := by
-  by_cases hs : s = ⊤
-  · simp [hs]
-  · rw [cap,ite_eq_right hs]
-    exact (ENNReal.ofReal_le_ofReal (min_le_left _ _)).trans_eq (ENNReal.ofReal_toReal hs)
-
-end SLEExactGauge.PhysicalStop
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-lemma first_eq_source {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (hz : z ∈ halfPlane) (hu : 0 < u) {t : ℝ} (ht : t ∈ Icc 0 (C.stopTime κ B ω)) :
-    iteratedDeriv 1 (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
-      (Real.sqrt κ) (C.paths κ B ω) z t) (B t.toNNReal ω) =
-        Real.sqrt κ*(8/κ-1)*(g ω t z-(Real.sqrt κ*B t.toNNReal ω:ℂ)).re /
-          ‖g ω t z-(Real.sqrt κ*B t.toNNReal ω:ℂ)‖^2 *
-            ((loewnerRadius (g ω) t z)^(-(1-κ/8)) *
-              (loewnerAngleSine κ (fun s => B s ω) (g ω) t z)^(8/κ-1)) := by
-  have hU := C.path_eq κ B ω hc
-  have h0 := C.initial_deficit_brownian κ B ω hz hu hc hb0
-  have hs := source_until_stop C.height_pos C.horizon_pos hm _ z _ C.angle hU h0
-  have hhi := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle t).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)).2.1
-  have hts := ht.2.trans (C.bounds κ B ω).2
-  rw [first_eq,field_eq_radius_sine C.height_pos C.horizon_pos _ _ _ _ z t _ hhi (hU ⟨ht.1,hts⟩),
-    radius_eq_source C.height_pos C.horizon_pos hm
-      (continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn _ z _ C.angle hU h0 ht,
-    sine_eq_source C.height_pos C.horizon_pos hm _ z _ C.angle hU h0 ht]
-  rw [LoewnerField.height,max_eq_right hhi,← hs.2 ht]
-  simp only [loewnerAngleSine,Complex.sub_re,Complex.sq_norm,Complex.normSq_apply,
-    Complex.sub_im,Complex.ofReal_mul]
-  simp only [Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,mul_zero,zero_mul,sub_zero,add_zero,pow_two]
-
-lemma first_le_poisson {Ω : Type*} (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4)
-    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (hz : z ∈ halfPlane) (hu : 0 < u) {e t : ℝ} (he : 0 < e) (hes : e < z.im)
-    (ht : t ∈ Icc 0 (C.physicalTime κ B γ e ω)) :
-    |iteratedDeriv 1 (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
-      (Real.sqrt κ) (C.paths κ B ω) z t) (B t.toNNReal ω)| ≤
-      (Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8))) *
-        Mesh.poissonFactor (fun t => Real.sqrt κ*B t.toNNReal ω) (g ω) t z := by
-  have hstop := ht.2.trans (C.physicalTime_bounds κ B γ e ω).2
-  have hl := survivingDomain_mono hstop (C.source_live κ B ω hm hz hu hc hb0)
-  let tn : ℝ≥0 := ⟨t,ht.1⟩
-  have hh : (tn:ℝ≥0∞) ≤ firstLiveDistanceStop (γ ω) z e := by
-    rw [← ENNReal.ofReal_coe_nnreal (p := tn)]
-    exact (ENNReal.ofReal_le_ofReal ht.2).trans (PhysicalStop.ofReal_cap_le _ _)
-  have hR : e/4 ≤ loewnerRadius (g ω) t z := hm.radius_ge_before_distance_stop he hes hh hl
-  have ha : 1 ≤ 8/κ-1 := by
-    have hh : 2 ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by linarith [hκ.2])
-    linarith
-  rw [C.first_eq_source κ B γ g f ω hm hc hb0 hz hu ⟨ht.1,hstop⟩]
-  have hz' : 0 < (g ω t z-(Real.sqrt κ*B t.toNNReal ω:ℂ)).im := by
-    simpa only [Complex.sub_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,mul_zero,zero_mul,add_zero,sub_zero] using hm.im_pos ht.1 hl
-  have hb := Mesh.coefficient_le_poisson hz' ha (show 0 ≤ 1-κ/8 by linarith [hκ.2])
-    (Real.sqrt_nonneg κ) (show 0 < e/4 by positivity) hR
-  simpa only [Mesh.poissonFactor,loewnerAngleSine,Complex.sub_im,Complex.mul_im,Complex.ofReal_im,
-    mul_zero,zero_mul,add_zero,sub_zero,Complex.ofReal_mul,tn,NNReal.coe_mk] using hb
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge.Mesh
-
-lemma poissonFactor_pos {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {t : ℝ} (ht : 0 ≤ t)
-    {z : ℂ} (hz : z ∈ localizationDomain γ t) : 0 < poissonFactor driver g t z := by
-  apply div_pos (h.im_pos ht hz)
-  apply sq_pos_of_pos
-  apply norm_pos_iff.2
-  intro he
-  have hi := congrArg Complex.im he
-  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
-  exact (h.im_pos ht hz).ne' hi
-
-lemma overlap_lintegral_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
-    (hU : ContinuousOn driver (Icc 0 t)) {z w : ℂ}
-    (hz : z ∈ localizationDomain γ t) (hw : w ∈ localizationDomain γ t) (hne : z ≠ w)
-    (b c : ℝ → ℝ) {K : ℝ} (hK : 0 ≤ K)
-    (hb : ∀ v ∈ Icc s t, |b v| ≤ K*poissonFactor driver g v z)
-    (hc : ∀ v ∈ Icc s t, |c v| ≤ K*poissonFactor driver g v w) :
-    (∫⁻ v in Icc s t, ENNReal.ofReal |b v*c v|) ≤
-      ENNReal.ofReal (K^2 * halfPlaneGreen (g s z) (g s w)) := by
-  let F : ℝ → ℝ := fun v => 4*poissonFactor driver g v z*poissonFactor driver g v w
-  have hF : ContinuousOn F (Icc s t) :=
-    (((poissonFactor_continuousOn h (hs.trans hst) hU hz).const_mul 4).mul
-      (poissonFactor_continuousOn h (hs.trans hst) hU hw)).mono
-        (Icc_subset_Icc hs le_rfl)
-  have hFp (v : ℝ) (hv : v ∈ Icc s t) : 0 ≤ F v := by
-    dsimp [F]
-    exact mul_nonneg (mul_nonneg (by norm_num)
-      (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hz)).le)
-      (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hw)).le
-  calc
-    _ ≤ ∫⁻ v in Icc s t, ENNReal.ofReal (K^2*F v) := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem measurableSet_Icc] with v hv
-      apply ENNReal.ofReal_le_ofReal
-      rw [abs_mul]
-      have hp := mul_nonneg
-        (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hz)).le
-        (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hw)).le
-      calc
-        _ ≤ (K*poissonFactor driver g v z)*(K*poissonFactor driver g v w) :=
-          mul_le_mul (hb v hv) (hc v hv) (abs_nonneg _) (mul_nonneg hK
-            (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hz)).le)
-        _ ≤ K^2*F v := by dsimp [F]; nlinarith [mul_nonneg (sq_nonneg K) hp]
-    _ = ENNReal.ofReal (∫ v in Icc s t, K^2*F v) := by
-      symm
-      apply ofReal_integral_eq_lintegral_ofReal
-        ((hF.const_mul (K^2)).integrableOn_compact isCompact_Icc)
-      filter_upwards [ae_restrict_mem measurableSet_Icc] with v hv
-      exact mul_nonneg (sq_nonneg K) (hFp v hv)
-    _ = ENNReal.ofReal (K^2*(∫ v in s..t, F v)) := by
-      rw [integral_const_mul,integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le hst]
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left
-      (pair_integral_le h hs hst hU hz hw hne) (sq_nonneg K))
-
-lemma approached_in_boundary_layer {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (_h : IsLoewnerTraceUsing driver γ g f) {s t R : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
-    {w : ℂ} (hd : dist w (γ s) ≤ R) :
-    infDist w (localizationDomain γ t)ᶜ ≤ R := by
-  apply (infDist_le_dist_of_mem ?_).trans hd
-  intro hm
-  exact hm.1.2 ⟨s,⟨hs,hst⟩,rfl⟩
-
-lemma green_symm (z w : ℂ) : halfPlaneGreen z w = halfPlaneGreen w z := by
-  have hn : ‖z-starRingEnd ℂ w‖ = ‖w-starRingEnd ℂ z‖ := by
-    calc
-      _ = ‖starRingEnd ℂ (z-starRingEnd ℂ w)‖ := (Complex.norm_conj _).symm
-      _ = ‖starRingEnd ℂ z-w‖ := by simp only [map_sub,starRingEnd_self_apply]
-      _ = _ := norm_sub_rev _ _
-  simp only [halfPlaneGreen,norm_div,hn,norm_sub_rev z w]
-
-def laterGreen (g : ℝ → ℂ → ℂ) (a b : ℂ → ℝ) (z w : ℂ) : ℝ≥0∞ :=
-  if a w ≤ a z ∧ a z ≤ b w then
-    ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w)) else 0
-
-lemma laterGreen_row {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {A : Set ℂ} (hA : MeasurableSet A)
-    {a b : ℂ → ℝ} {R C : ℝ} (_hR : 0 < R)
-    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
-    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
-    (happroach : ∀ z ∈ A, dist z (γ (a z)) ≤ R)
-    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
-        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2))
-    {z : ℂ} (hz : z ∈ A) :
-    (∫⁻ w in A, laterGreen g a b z w) ≤ ENNReal.ofReal (C*R^2) := by
-  let L : Set ℂ := {w | w ∈ localizationDomain γ (a z) ∧
-    infDist w (localizationDomain γ (a z))ᶜ ≤ R}
-  have hLm : MeasurableSet L := (survivingDomain_open h.1 (hstart z hz)).measurableSet.inter
-    (isClosed_le (continuous_infDist_pt _) continuous_const).measurableSet
-  calc
-    _ ≤ ∫⁻ w in A, L.indicator
-        (fun w => ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w))) w := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem hA] with w hw
-      unfold laterGreen
-      split_ifs with horder
-      · have hwL : w ∈ L := ⟨survivingDomain_mono horder.2 (hlive w hw),
-          approached_in_boundary_layer h (hstart w hw) horder.1 (happroach w hw)⟩
-        rw [indicator_of_mem hwL]
-      · exact zero_le
-    _ ≤ ∫⁻ w, L.indicator
-        (fun w => ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w))) w :=
-      lintegral_mono' Measure.restrict_le_self le_rfl
-    _ = ∫⁻ w in L, ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w)) :=
-      lintegral_indicator hLm _
-    _ ≤ _ := hlayer (a z) (hstart z hz) z (survivingDomain_mono (hab z hz) (hlive z hz))
-
-lemma supported_pair_lintegral {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ} {z w : ℂ}
-    (hz : ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
-    (hw : ∀ t, t ∉ Icc (a w) (b w) → B t w = 0) :
-    (∫⁻ t, ENNReal.ofReal |B t z*B t w|) =
-      ∫⁻ t in Icc (max (a z) (a w)) (min (b z) (b w)), ENNReal.ofReal |B t z*B t w| := by
-  rw [← lintegral_indicator measurableSet_Icc]
-  apply lintegral_congr
-  intro t
-  by_cases ht : t ∈ Icc (max (a z) (a w)) (min (b z) (b w))
-  · rw [indicator_of_mem ht]
-  · rw [indicator_of_notMem ht]
-    have hh : t ∉ Icc (a z) (b z) ∨ t ∉ Icc (a w) (b w) := by
-      by_contra hn
-      push Not at hn
-      exact ht ⟨max_le hn.1.1 hn.2.1,le_min hn.1.2 hn.2.2⟩
-    rcases hh with hh | hh <;> simp [hz t,hw t,hh]
-
-lemma asynchronous_pair_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
-    {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ} {z w : ℂ} (hne : z ≠ w)
-    (hz0 : 0 ≤ a z) (_hw0 : 0 ≤ a w) (_hzab : a z ≤ b z) (_hwab : a w ≤ b w)
-    (hzlive : z ∈ localizationDomain γ (b z)) (hwlive : w ∈ localizationDomain γ (b w))
-    (hzsup : ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
-    (hwsup : ∀ t, t ∉ Icc (a w) (b w) → B t w = 0)
-    {K : ℝ} (hK : 0 ≤ K)
-    (hzb : ∀ t ∈ Icc (a z) (b z), |B t z| ≤ K*poissonFactor driver g t z)
-    (hwb : ∀ t ∈ Icc (a w) (b w), |B t w| ≤ K*poissonFactor driver g t w) :
-    (∫⁻ t, ENNReal.ofReal |B t z*B t w|) ≤
-      ENNReal.ofReal (K^2) * (laterGreen g a b z w+laterGreen g a b w z) := by
-  rw [supported_pair_lintegral hzsup hwsup]
-  by_cases hover : max (a z) (a w) ≤ min (b z) (b w)
-  · have hi := overlap_lintegral_bound h (hz0.trans (le_max_left _ _)) hover hU.continuousOn
-      (survivingDomain_mono (min_le_left _ _) hzlive)
-      (survivingDomain_mono (min_le_right _ _) hwlive) hne (fun t => B t z) (fun t => B t w) hK
-      (fun t ht => hzb t ⟨(le_max_left _ _).trans ht.1,ht.2.trans (min_le_left _ _)⟩)
-      (fun t ht => hwb t ⟨(le_max_right _ _).trans ht.1,ht.2.trans (min_le_right _ _)⟩)
-    rw [ENNReal.ofReal_mul (sq_nonneg K)] at hi
-    apply hi.trans
-    apply mul_le_mul_right
-    by_cases horder : a w ≤ a z
-    · rw [max_eq_left horder]
-      have hzw : a z ≤ b w := (le_max_left _ _).trans (hover.trans (min_le_right _ _))
-      rw [laterGreen,ite_eq_left ⟨horder,hzw⟩]
-      exact le_add_right le_rfl
-    · have hwz : a z ≤ a w := (lt_of_not_ge horder).le
-      rw [max_eq_right hwz]
-      have hwzb : a w ≤ b z := (le_max_right _ _).trans (hover.trans (min_le_left _ _))
-      have hk : halfPlaneGreen (g (a w) z) (g (a w) w) =
-          halfPlaneGreen (g (a w) w) (g (a w) z) := green_symm _ _
-      rw [hk]
-      have hright : laterGreen g a b w z = ENNReal.ofReal (halfPlaneGreen (g (a w) w) (g (a w) z)) := by
-        rw [laterGreen,ite_eq_left ⟨hwz,hwzb⟩]
-      rw [← hright]
-      exact le_add_left le_rfl
-  · rw [Icc_eq_empty hover,Measure.restrict_empty,lintegral_zero_measure]
-    exact zero_le
-
-lemma norm_integral_sq_le_pairs {E : Type*} [MeasurableSpace E] {μ : Measure E}
-    (v : E → ℝ) (hv : Measurable v) :
-    ENNReal.ofReal (|∫ z, v z ∂μ|^2) ≤
-      ∫⁻ z, ∫⁻ w, ENNReal.ofReal |v z*v w| ∂μ ∂μ := by
-  have hn : ENNReal.ofReal |∫ z, v z ∂μ| ≤ ∫⁻ z, ENNReal.ofReal |v z| ∂μ := by
-    simpa only [← ofReal_norm,Real.norm_eq_abs] using
-      (enorm_integral_le_lintegral_enorm (μ := μ) v)
-  rw [ENNReal.ofReal_pow (abs_nonneg _)]
-  calc
-    _ ≤ (∫⁻ z, ENNReal.ofReal |v z| ∂μ)^2 := pow_le_pow_left₀ zero_le hn 2
-    _ = ∫⁻ z, ∫⁻ w, ENNReal.ofReal |v z*v w| ∂μ ∂μ := by
-      simp_rw [abs_mul,ENNReal.ofReal_mul (abs_nonneg _)]
-      rw [pow_two,lintegral_lintegral_mul hv.abs.ennreal_ofReal.aemeasurable hv.abs.ennreal_ofReal.aemeasurable]
-
-lemma bracket_le_pair_integral {A : Set ℂ} (B : ℝ → ℂ → ℝ)
-    (hB : Measurable (Function.uncurry B)) :
-    (∫⁻ t, ENNReal.ofReal (|∫ z in A, B t z|^2)) ≤
-      ∫⁻ z in A, ∫⁻ w in A, ∫⁻ t, ENNReal.ofReal |B t z*B t w| := by
-  have hb (t : ℝ) : Measurable (B t) := hB.comp (measurable_const.prodMk measurable_id)
-  have hF : Measurable (fun q : (ℝ × ℂ) × ℂ => ENNReal.ofReal |B q.1.1 q.1.2*B q.1.1 q.2|) := by
-    exact ((hB.comp measurable_fst).mul
-      (hB.comp (measurable_fst.fst.prodMk measurable_snd))).abs.ennreal_ofReal
-  calc
-    _ ≤ ∫⁻ t, ∫⁻ z in A, ∫⁻ w in A, ENNReal.ofReal |B t z*B t w| :=
-      lintegral_mono (fun t => norm_integral_sq_le_pairs (B t) (hb t))
-    _ = ∫⁻ z in A, ∫⁻ t, ∫⁻ w in A, ENNReal.ofReal |B t z*B t w| :=
-      lintegral_lintegral_swap hF.lintegral_prod_right.aemeasurable
-    _ = _ := by
-      apply lintegral_congr
-      intro z
-      apply lintegral_lintegral_swap
-      exact (((hB.comp (measurable_fst.prodMk measurable_const)).mul hB).abs.ennreal_ofReal).aemeasurable
-
-lemma symmetric_kernel_integral {A : Set ℂ} (L : ℂ → ℂ → ℝ≥0∞)
-    (hL : Measurable (Function.uncurry L)) {D : ℝ≥0∞}
-    (hrow : ∀ z ∈ A, (∫⁻ w in A, L z w) ≤ D) (hA : MeasurableSet A) :
-    (∫⁻ z in A, ∫⁻ w in A, L z w+L w z) ≤ 2*D*volume A := by
-  have hswap := lintegral_lintegral_swap (μ := volume.restrict A) (ν := volume.restrict A) hL.aemeasurable
-  have hr : (∫⁻ z in A, ∫⁻ w in A, L z w) ≤ D*volume A := by
-    calc
-      _ ≤ ∫⁻ z in A, D := by
-        apply lintegral_mono_ae
-        filter_upwards [ae_restrict_mem hA] with z hz
-        exact hrow z hz
-      _ = _ := by simp
-  have he (z : ℂ) : (∫⁻ w in A, L z w+L w z) =
-      (∫⁻ w in A, L z w)+(∫⁻ w in A, L w z) :=
-    lintegral_add_left (hL.comp (measurable_const.prodMk measurable_id)) _
-  simp_rw [he]
-  rw [lintegral_add_left hL.lintegral_prod_right,← hswap]
-  calc
-    _ ≤ D*volume A+D*volume A := add_le_add hr hr
-    _ = _ := by ring
-
-theorem asynchronous_bracket_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
-    {A : Set ℂ} (hA : MeasurableSet A) {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ}
-    (hB : Measurable (Function.uncurry B))
-    (hL : Measurable (Function.uncurry (laterGreen g a b)))
-    {K R C : ℝ} (hK : 0 ≤ K) (hR : 0 < R)
-    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
-    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
-    (happroach : ∀ z ∈ A, dist z (γ (a z)) ≤ R)
-    (hsup : ∀ z ∈ A, ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
-    (hcoeff : ∀ z ∈ A, ∀ t ∈ Icc (a z) (b z), |B t z| ≤ K*poissonFactor driver g t z)
-    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
-        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2)) :
-    (∫⁻ t, ENNReal.ofReal (|∫ z in A, B t z|^2)) ≤
-      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (C*R^2)*volume A := by
-  apply (bracket_le_pair_integral B hB).trans
-  calc
-    _ ≤ ∫⁻ z in A, ∫⁻ w in A,
-        ENNReal.ofReal (K^2)*(laterGreen g a b z w+laterGreen g a b w z) := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem hA] with z hz
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem hA,ae_restrict_of_ae (show ∀ᵐ w : ℂ ∂volume, w ≠ z from by rw [ae_iff]; simp)] with w hw hne
-      exact asynchronous_pair_bound h hU hne.symm (hstart z hz) (hstart w hw)
-        (hab z hz) (hab w hw) (hlive z hz) (hlive w hw) (hsup z hz) (hsup w hw) hK
-        (hcoeff z hz) (hcoeff w hw)
-    _ = ENNReal.ofReal (K^2)*(∫⁻ z in A, ∫⁻ w in A, laterGreen g a b z w+laterGreen g a b w z) := by
-      simp only [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
-    _ ≤ ENNReal.ofReal (K^2)*(2*ENNReal.ofReal (C*R^2)*volume A) := by
-      apply mul_le_mul_right
-      exact symmetric_kernel_integral _ hL
-        (fun z hz => laterGreen_row h hA hR hstart hab hlive happroach hlayer hz) hA
-    _ = _ := by ring
-
-theorem asynchronous_bracket_bound_with_kernel {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
-    {A : Set ℂ} (hA : MeasurableSet A) {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ}
-    (hB : Measurable (Function.uncurry B))
-    (L : ℂ → ℂ → ℝ≥0∞) (hL : Measurable (Function.uncurry L))
-    (hLeq : ∀ z ∈ A, ∀ w ∈ A, L z w = laterGreen g a b z w)
-    {K R C : ℝ} (hK : 0 ≤ K) (hR : 0 < R)
-    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
-    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
-    (happroach : ∀ z ∈ A, dist z (γ (a z)) ≤ R)
-    (hsup : ∀ z ∈ A, ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
-    (hcoeff : ∀ z ∈ A, ∀ t ∈ Icc (a z) (b z), |B t z| ≤ K*poissonFactor driver g t z)
-    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
-        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2)) :
-    (∫⁻ t, ENNReal.ofReal (|∫ z in A, B t z|^2)) ≤
-      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (C*R^2)*volume A := by
-  apply (bracket_le_pair_integral B hB).trans
-  calc
-    _ ≤ ∫⁻ z in A, ∫⁻ w in A,
-        ENNReal.ofReal (K^2)*(L z w+L w z) := by
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem hA] with z hz
-      apply lintegral_mono_ae
-      filter_upwards [ae_restrict_mem hA,ae_restrict_of_ae (show ∀ᵐ w : ℂ ∂volume, w ≠ z from by rw [ae_iff]; simp)] with w hw hne
-      rw [hLeq z hz w hw,hLeq w hw z hz]
-      exact asynchronous_pair_bound h hU hne.symm (hstart z hz) (hstart w hw)
-        (hab z hz) (hab w hw) (hlive z hz) (hlive w hw) (hsup z hz) (hsup w hw) hK
-        (hcoeff z hz) (hcoeff w hw)
-    _ = ENNReal.ofReal (K^2)*(∫⁻ z in A, ∫⁻ w in A, L z w+L w z) := by
-      simp only [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
-    _ ≤ ENNReal.ofReal (K^2)*(2*ENNReal.ofReal (C*R^2)*volume A) := by
-      apply mul_le_mul_right
-      exact symmetric_kernel_integral _ hL
-        (fun z hz => by
-          have he : (∫⁻ w in A, L z w) = ∫⁻ w in A, laterGreen g a b z w := by
-            apply lintegral_congr_ae
-            filter_upwards [ae_restrict_mem hA] with w hw
-            exact hLeq z hz w hw
-          rw [he]
-          exact laterGreen_row h hA hR hstart hab hlive happroach hlayer hz) hA
-    _ = _ := by ring
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.ConformalKernels
+
+namespace OAI
+
+
+/-! Conformal localization and Integrated Mesh Simple estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal
+namespace SLEExactGauge
+open Layer
+
+lemma layer_set_measurable {F : ℂ → ℂ} (hf : DifferentiableOn ℂ F (ball 0 1)) (R : ℝ) :
+    MeasurableSet {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R} := by
+  classical
+  let f : ℂ → ℝ := fun w => diskFactor w*‖deriv F w‖
+  have hc : ContinuousOn f (ball 0 1) :=
+    diskFactor_smooth.continuous.continuousOn.mul (hf.deriv isOpen_ball).continuousOn.norm
+  have hm : Measurable ((ball (0:ℂ) 1).piecewise f (fun _ => 0)) :=
+    hc.measurable_piecewise continuousOn_const measurableSet_ball
+  have he : {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R} =
+      ball (0:ℂ) 1 ∩ {w | (ball (0:ℂ) 1).piecewise f (fun _ => 0) w ≤ R} := by
+    ext w
+    by_cases hw : w ∈ ball (0:ℂ) 1
+    · have hw1 := mem_ball_zero_iff.1 hw
+      simp [hw,hw1,f,diskFactor]
+    · have hn : ¬‖w‖ < 1 := by simpa only [mem_ball_zero_iff] using hw
+      simp [hn,hw]
+  rw [he]
+  exact measurableSet_ball.inter (measurableSet_le hm measurable_const)
+
+theorem conformalRadiusLayer :
+    ∃ C : ℝ, 0 < C ∧ ∀ F : ℂ → ℂ,
+      DifferentiableOn ℂ F (Metric.ball 0 1) → Set.InjOn F (Metric.ball 0 1) →
+      ∀ R : ℝ, 0 < R →
+      (∫⁻ w in {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R},
+        ENNReal.ofReal (Real.log (1/‖w‖)*‖deriv F w‖^2) ∂volume) ≤
+      ENNReal.ofReal (C*R^2) := by
+  refine ⟨4*Real.pi/Layer.layerExponent,div_pos (by positivity) layerExponent_bounds.1,?_⟩
+  intro F hf hi R hR
+  obtain ⟨G,h⟩ := univalent_diskPair hf hi
+  let S := {w : ℂ | ‖w‖ < 1 ∧ (1-‖w‖^2)*‖deriv F w‖ ≤ R}
+  let u := layerBarrier F R
+  have hα := layerExponent_bounds.1
+  have hC : 0 ≤ 2/Layer.layerExponent := by positivity
+  have hpoint (w : ℂ) (hw : w ∈ S) :
+      ENNReal.ofReal (Real.log (1/‖w‖)*‖deriv F w‖^2) ≤
+      ENNReal.ofReal (2/Layer.layerExponent) * ENNReal.ofReal
+        (Real.log (1/‖w‖)*(-Laplacian.laplacian u w)) := by
+    have hb := h.barrier_layer hw.1 hR hw.2
+    have hlog : 0 ≤ Real.log (1/‖w‖) := by
+      by_cases hz : w = 0
+      · simp [hz]
+      · exact Real.log_nonneg ((one_le_div (norm_pos_iff.2 hz)).2 hw.1.le)
+    rw [← ENNReal.ofReal_mul hC]
+    apply ENNReal.ofReal_le_ofReal
+    have hb' : ‖deriv F w‖^2 ≤ (2/Layer.layerExponent)*(-Laplacian.laplacian u w) := by
+      have hm := mul_le_mul_of_nonneg_left hb hC
+      have he : (2/Layer.layerExponent)*(Layer.layerExponent/2*‖deriv F w‖^2) = ‖deriv F w‖^2 := by
+        field_simp
+      rw [he] at hm
+      exact hm
+    nlinarith [mul_le_mul_of_nonneg_left hb' hlog]
+  have hS : MeasurableSet S := layer_set_measurable hf R
+  have hsub : S ⊆ ball (0:ℂ) 1 := fun w hw => mem_ball_zero_iff.2 hw.1
+  have hb := superharmonic_disk_green (h.barrier_smooth R)
+    (fun z _ => barrier_bounds F R z)
+    (fun z hz => h.barrier_superharmonic (mem_ball_zero_iff.1 hz) R)
+  calc
+    _ ≤ ∫⁻ w in S, ENNReal.ofReal (2/Layer.layerExponent) *
+        ENNReal.ofReal (Real.log (1/‖w‖)*(-Laplacian.laplacian u w)) :=
+      setLIntegral_mono' hS hpoint
+    _ ≤ ∫⁻ w in ball (0:ℂ) 1, ENNReal.ofReal (2/Layer.layerExponent) *
+        ENNReal.ofReal (Real.log (1/‖w‖)*(-Laplacian.laplacian u w)) := lintegral_mono_set hsub
+    _ = ENNReal.ofReal (2/Layer.layerExponent) * (∫⁻ w in ball (0:ℂ) 1,
+        ENNReal.ofReal (Real.log (1/‖w‖)*(-Laplacian.laplacian u w))) := by
+      rw [lintegral_const_mul']
+      exact ENNReal.ofReal_ne_top
+    _ ≤ ENNReal.ofReal (2/Layer.layerExponent) * ENNReal.ofReal (2*Real.pi*R^2) :=
+      mul_le_mul_right hb _
+    _ = _ := by
+      rw [← ENNReal.ofReal_mul hC]
+      congr 1
+      ring
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal
+namespace SLEExactGauge.Layer
+open SlitMaps
+
+lemma DiskPair.radius_le_omitted {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) {z w : ℂ} (hz : ‖z‖ < 1) (hw : w ∉ U) :
+    diskFactor z*‖deriv F z‖ ≤ (256*Real.pi)*dist (F z) w := by
+  have hr := h.recenter hz
+  have homit : w ∉ (F ∘ diskAuto (-z)) '' ball (0:ℂ) 1 := by
+    rintro ⟨q,hq,rfl⟩
+    exact hw (hr.toDomain hq)
+  have hb := UnivalentDistance.omitted_value_bound hr.forward hr.injOn homit
+  have hF := h.forward.differentiableAt (isOpen_ball.mem_nhds (mem_ball_zero_iff.2 hz))
+  have hφ := diskAuto_hasDerivAt (show ‖-z‖ < 1 by simpa using hz) (z := (0:ℂ)) (by simp)
+  have hφ0 : diskAuto (-z) 0 = z := by simp
+  have he := (hφ0.symm ▸ hF.hasDerivAt).comp (0:ℂ) hφ
+  have hd : deriv (F ∘ diskAuto (-z)) 0 = deriv F z*((diskFactor z:ℝ):ℂ) := by
+    have hdφ := diskAuto_deriv_zero_exact hz
+    have ha : (1-(starRingEnd ℂ z)*z) = ((diskFactor z:ℝ):ℂ) := by
+      rw [mul_comm (starRingEnd ℂ z),Complex.mul_conj']
+      simp only [diskFactor,Complex.ofReal_sub,Complex.ofReal_one,Complex.ofReal_pow]
+    rw [ha] at hdφ
+    simpa only [hφ0,map_neg,neg_mul_neg,mul_zero,sub_zero,one_pow,div_one,ha] using he.deriv
+  rw [hd,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (diskFactor_pos hz)] at hb
+  simpa only [Function.comp_apply,hφ0,dist_eq_norm,mul_comm (diskFactor z)] using hb
+
+lemma DiskPair.radius_le_infDist {U : Set ℂ} {F G : ℂ → ℂ}
+    (h : DiskPair U F G) (hne : Uᶜ.Nonempty) {z : ℂ} (hz : ‖z‖ < 1) :
+    diskFactor z*‖deriv F z‖ ≤ (256*Real.pi)*infDist (F z) Uᶜ := by
+  have hp : 0 < 256*Real.pi := by positivity
+  rw [mul_comm (256*Real.pi),← div_le_iff₀ hp]
+  apply (le_infDist hne).2
+  intro w hw
+  rw [div_le_iff₀ hp]
+  exact (h.radius_le_omitted hz hw).trans_eq (mul_comm _ _)
+
+lemma complexJacobian (c : ℂ) :
+    ((ContinuousLinearMap.toSpanSingleton ℂ c).restrictScalars ℝ).det = ‖c‖^2 := by
+  simp [ContinuousLinearMap.det,LinearMap.det_restrictScalars,
+    Algebra.norm_complex_eq,Complex.normSq_eq_norm_sq]
+
+theorem physical_boundary_layer :
+    ∃ C : ℝ, 0 < C ∧ ∀ (U : Set ℂ) (F G : ℂ → ℂ), DiskPair U F G → Uᶜ.Nonempty →
+      ∀ R : ℝ, 0 < R →
+      (∫⁻ w in {w : ℂ | w ∈ U ∧ infDist w Uᶜ ≤ R},
+        ENNReal.ofReal (Real.log (1/‖G w‖))) ≤ ENNReal.ofReal (C*R^2) := by
+  obtain ⟨C,hC,hbound⟩ := conformalRadiusLayer
+  refine ⟨C*(256*Real.pi)^2,by positivity,?_⟩
+  intro U F G h hne R hR
+  let S := {z : ℂ | ‖z‖ < 1 ∧ (1-‖z‖^2)*‖deriv F z‖ ≤ (256*Real.pi)*R}
+  have hS : MeasurableSet S := layer_set_measurable h.forward _
+  have hsub : S ⊆ ball (0:ℂ) 1 := fun z hz => mem_ball_zero_iff.2 hz.1
+  have himage : {w : ℂ | w ∈ U ∧ infDist w Uᶜ ≤ R} ⊆ F '' S := by
+    intro w hw
+    refine ⟨G w,⟨mem_ball_zero_iff.1 (h.toDisk hw.1),?_⟩,h.right_inv w hw.1⟩
+    have hh := h.radius_le_infDist hne (mem_ball_zero_iff.1 (h.toDisk hw.1))
+    rw [h.right_inv w hw.1] at hh
+    exact hh.trans (mul_le_mul_of_nonneg_left hw.2 (by positivity))
+  have hchange := lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hS
+    (fun z hz => ((h.forward.differentiableAt (isOpen_ball.mem_nhds (hsub hz))).hasDerivAt.hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt)
+    (h.injOn.mono hsub) (fun w => ENNReal.ofReal (Real.log (1/‖G w‖)))
+  calc
+    _ ≤ ∫⁻ w in F '' S, ENNReal.ofReal (Real.log (1/‖G w‖)) := lintegral_mono_set himage
+    _ = ∫⁻ z in S, ENNReal.ofReal (Real.log (1/‖z‖)*‖deriv F z‖^2) := by
+      rw [hchange]
+      apply setLIntegral_congr_fun hS
+      intro z hz
+      dsimp only
+      rw [complexJacobian,abs_of_nonneg (sq_nonneg _),h.left_inv z (hsub hz),
+        ← ENNReal.ofReal_mul (sq_nonneg _)]
+      congr 1
+      ring
+    _ ≤ ENNReal.ofReal (C*((256*Real.pi)*R)^2) := hbound F h.forward h.injOn _ (by positivity)
+    _ = _ := by congr 1; ring
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.Mesh
+
+def halfPlaneGreen (z w : ℂ) : ℝ := Real.log ‖(z-(starRingEnd ℂ w))/(z-w)‖
+
+lemma hasDerivAt_log_norm {f : ℝ → ℂ} {D : ℂ} {t : ℝ}
+    (hd : HasDerivAt f D t) (hn : f t ≠ 0) :
+    HasDerivAt (fun u => Real.log ‖f u‖) (D/f t).re t := by
+  have hh : ContDiffAt ℝ 2 (fun z : ℂ => Real.log ‖z‖) (f t) :=
+    (analyticAt_id.harmonicAt_log_norm hn).1
+  have hc := (hh.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt t hd
+  have he := Layer.logNorm_fderiv (hasDerivAt_id (f t)) hn D
+  apply hc.congr_deriv
+  simpa only [id_eq,one_mul] using he
+
+lemma green_eq_difference {z w : ℂ} (hn : z ≠ w) (hp : z ≠ starRingEnd ℂ w) :
+    halfPlaneGreen z w = Real.log ‖z-(starRingEnd ℂ w)‖ - Real.log ‖z-w‖ := by
+  rw [halfPlaneGreen,norm_div,Real.log_div (norm_ne_zero_iff.2 (sub_ne_zero.2 hp))
+    (norm_ne_zero_iff.2 (sub_ne_zero.2 hn))]
+
+lemma difference_velocity {z w : ℂ} (hz : z ≠ 0) (hw : w ≠ 0) (hd : z ≠ w) :
+    (2/z-2/w)/(z-w) = -2/(z*w) := by
+  field_simp
+  ring
+
+lemma green_velocity {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    (-2/(z*(starRingEnd ℂ w))).re - (-2/(z*w)).re =
+      -4*z.im/(‖z‖^2) * (w.im/(‖w‖^2)) := by
+  have hzn : Complex.normSq z ≠ 0 := by
+    rw [← Complex.sq_norm]
+    exact pow_ne_zero 2 (norm_ne_zero_iff.2 (by intro he; simp [he] at hz))
+  have hwn : Complex.normSq w ≠ 0 := by
+    rw [← Complex.sq_norm]
+    exact pow_ne_zero 2 (norm_ne_zero_iff.2 (by intro he; simp [he] at hw))
+  rw [Complex.div_re,Complex.div_re,Complex.normSq_mul,Complex.normSq_mul,Complex.normSq_conj,
+    Complex.sq_norm,Complex.sq_norm]
+  simp only [Complex.mul_re,Complex.conj_re,Complex.conj_im,Complex.neg_re,Complex.neg_im]
+  norm_num
+  field_simp
+  ring
+
+theorem green_decrease {g h : ℝ → ℂ} {t u : ℝ}
+    (hz : 0 < (g t).im) (hw : 0 < (h t).im) (hne : g t ≠ h t)
+    (hg : HasDerivAt g (2/(g t-(u:ℂ))) t)
+    (hh : HasDerivAt h (2/(h t-(u:ℂ))) t) :
+    HasDerivAt (fun s => halfPlaneGreen (g s) (h s))
+      (-4*(g t).im/‖g t-(u:ℂ)‖^2 * ((h t).im/‖h t-(u:ℂ)‖^2)) t := by
+  have hp : g t ≠ starRingEnd ℂ (h t) := by
+    intro he
+    have hi := congrArg Complex.im he
+    simp only [Complex.conj_im] at hi
+    linarith
+  have hc := hg.sub hh
+  have hcc := hg.sub (Complex.conjCLE.hasFDerivAt.comp_hasDerivAt t hh)
+  have h₁ := hasDerivAt_log_norm hcc (sub_ne_zero.2 hp)
+  have h₂ := hasDerivAt_log_norm hc (sub_ne_zero.2 hne)
+  have he : (fun s => halfPlaneGreen (g s) (h s)) =ᶠ[𝓝 t]
+      (fun s => Real.log ‖g s-starRingEnd ℂ (h s)‖-Real.log ‖g s-h s‖) := by
+    filter_upwards [(hg.continuousAt.sub hh.continuousAt).eventually_ne (sub_ne_zero.2 hne),
+      hcc.continuousAt.eventually_ne (sub_ne_zero.2 hp)] with s hs hs'
+    exact green_eq_difference (sub_ne_zero.1 hs) (sub_ne_zero.1 hs')
+  apply ((h₁.sub h₂).congr_of_eventuallyEq he).congr_deriv
+  let z := g t-(u:ℂ)
+  let w := h t-(u:ℂ)
+  have hzi : 0 < z.im := by simpa [z] using hz
+  have hwi : 0 < w.im := by simpa [w] using hw
+  have hzn : z ≠ 0 := by intro he; simp [he] at hzi
+  have hwn : w ≠ 0 := by intro he; simp [he] at hwi
+  have hzw : z ≠ w := by simpa [z,w] using hne
+  have hzwc : z ≠ starRingEnd ℂ w := by
+    intro he
+    have hi := congrArg Complex.im he
+    simp only [Complex.conj_im] at hi
+    linarith
+  have hdiff : g t-h t = z-w := by dsimp [z,w]; ring
+  have hdiffc : g t-starRingEnd ℂ (h t) = z-starRingEnd ℂ w := by
+    simp only [z,w,map_sub,Complex.conj_ofReal]
+    ring
+  have hconj : starRingEnd ℂ (2/(h t-(u:ℂ))) = 2/(starRingEnd ℂ w) := by simp [w,Complex.conj_ofNat]
+  change ((2/z-starRingEnd ℂ (2/w))/(g t-starRingEnd ℂ (h t))).re -
+    ((2/z-2/w)/(g t-h t)).re = _
+  rw [hdiff,hdiffc]
+  have hconj2 : starRingEnd ℂ (2/w) = 2/(starRingEnd ℂ w) := by
+    simpa only [Complex.conj_ofNat] using (map_div₀ (starRingEnd ℂ) (2:ℂ) w)
+  rw [hconj2,difference_velocity hzn (by simpa using hwn) hzwc,
+    difference_velocity hzn hwn hzw,green_velocity hzi hwi]
+  simp [z,w]
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Function MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge.Layer
+
+def halfToDisk (a w : ℂ) : ℂ := (w-a)/(w-starRingEnd ℂ a)
+
+def diskToHalf (a v : ℂ) : ℂ := (a-starRingEnd ℂ a*v)/(1-v)
+
+lemma half_den_ne {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
+    w-starRingEnd ℂ a ≠ 0 := by
+  intro he
+  have hi := congrArg Complex.im he
+  simp only [Complex.sub_im,Complex.conj_im,Complex.zero_im] at hi
+  linarith
+
+lemma disk_den_ne {v : ℂ} (hv : ‖v‖ < 1) : (1:ℂ)-v ≠ 0 := by
+  intro he
+  have he' : v = 1 := (sub_eq_zero.1 he).symm
+  simp [he'] at hv
+
+lemma halfToDisk_maps {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
+    ‖halfToDisk a w‖ < 1 := by
+  have hd := half_den_ne ha hw
+  have he : ‖w-starRingEnd ℂ a‖^2 - ‖w-a‖^2 = 4*w.im*a.im := by
+    simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
+      Complex.conj_re,Complex.conj_im]
+    ring
+  rw [halfToDisk,norm_div,div_lt_one (norm_pos_iff.2 hd)]
+  nlinarith [norm_nonneg (w-a),norm_nonneg (w-starRingEnd ℂ a),mul_pos hw ha]
+
+lemma diskToHalf_im (a v : ℂ) :
+    (diskToHalf a v).im = a.im*(1-‖v‖^2)/‖1-v‖^2 := by
+  simp only [diskToHalf,Complex.div_im,Complex.sq_norm,Complex.normSq_apply,
+    Complex.sub_re,Complex.sub_im,Complex.mul_re,Complex.mul_im,Complex.conj_re,
+    Complex.conj_im,Complex.one_re,Complex.one_im]
+  ring
+
+lemma diskToHalf_maps {a v : ℂ} (ha : 0 < a.im) (hv : ‖v‖ < 1) :
+    0 < (diskToHalf a v).im := by
+  rw [diskToHalf_im]
+  apply div_pos (mul_pos ha _) (sq_pos_of_ne_zero (norm_ne_zero_iff.2 (disk_den_ne hv)))
+  nlinarith [norm_nonneg v]
+
+lemma disk_half_inverse {a v : ℂ} (ha : 0 < a.im) (hv : ‖v‖ < 1) :
+    halfToDisk a (diskToHalf a v) = v := by
+  have h₁ := disk_den_ne hv
+  have h₂ := half_den_ne ha (diskToHalf_maps ha hv)
+  dsimp [halfToDisk,diskToHalf] at h₂ ⊢
+  apply (div_eq_iff h₂).2
+  field_simp
+  ring
+
+lemma half_disk_inverse {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
+    diskToHalf a (halfToDisk a w) = w := by
+  have h₁ := half_den_ne ha hw
+  have h₂ := disk_den_ne (halfToDisk_maps ha hw)
+  dsimp [halfToDisk,diskToHalf] at h₂ ⊢
+  apply (div_eq_iff h₂).2
+  field_simp
+  ring
+
+lemma halfToDisk_differentiable {a w : ℂ} (ha : 0 < a.im) (hw : 0 < w.im) :
+    DifferentiableAt ℂ (halfToDisk a) w := by
+  exact (differentiableAt_id.sub_const a).div (differentiableAt_id.sub_const _) (half_den_ne ha hw)
+
+lemma diskToHalf_differentiable {a v : ℂ} (_ha : 0 < a.im) (hv : ‖v‖ < 1) :
+    DifferentiableAt ℂ (diskToHalf a) v := by
+  exact ((differentiableAt_id.const_mul _).const_sub a).div
+    (differentiableAt_id.const_sub 1) (disk_den_ne hv)
+
+lemma green_disk_eq (a w : ℂ) :
+    Real.log (1/‖halfToDisk a w‖) = Mesh.halfPlaneGreen a w := by
+  have hn : ‖w-starRingEnd ℂ a‖ = ‖a-starRingEnd ℂ w‖ := by
+    rw [← norm_neg (w-starRingEnd ℂ a),neg_sub]
+    rw [← Complex.norm_conj (starRingEnd ℂ a-w)]
+    simp only [map_sub,starRingEnd_self_apply]
+  rw [halfToDisk,norm_div,one_div_div,hn,norm_sub_rev w,Mesh.halfPlaneGreen,norm_div]
+
+theorem halfPlane_boundary_layer :
+    ∃ C : ℝ, 0 < C ∧ ∀ (U : Set ℂ) (g f : ℂ → ℂ),
+      IsOpen U → Uᶜ.Nonempty →
+      MapsTo g U {z : ℂ | 0 < z.im} → MapsTo f {z : ℂ | 0 < z.im} U →
+      (∀ z ∈ U, f (g z) = z) → (∀ z : ℂ, 0 < z.im → g (f z) = z) →
+      DifferentiableOn ℂ g U → DifferentiableOn ℂ f {z : ℂ | 0 < z.im} →
+      ∀ a : ℂ, 0 < a.im → ∀ R : ℝ, 0 < R →
+      (∫⁻ w in {w : ℂ | w ∈ U ∧ infDist w Uᶜ ≤ R},
+        ENNReal.ofReal (Mesh.halfPlaneGreen a (g w))) ≤ ENNReal.ofReal (C*R^2) := by
+  obtain ⟨C,hC,hbound⟩ := physical_boundary_layer
+  refine ⟨C,hC,?_⟩
+  intro U g f hU hne hg hf hfg hgf hgd hfd a ha R hR
+  have hH : IsOpen {z : ℂ | 0 < z.im} := isOpen_lt continuous_const Complex.continuous_im
+  have hpair : DiskPair U (f ∘ diskToHalf a) (halfToDisk a ∘ g) := by
+    refine ⟨hU,?_,?_,?_,?_,?_,?_⟩
+    · intro v hv
+      exact hf (diskToHalf_maps ha (mem_ball_zero_iff.1 hv))
+    · intro w hw
+      exact mem_ball_zero_iff.2 (halfToDisk_maps ha (hg hw))
+    · intro v hv
+      dsimp only [Function.comp_apply]
+      rw [hgf _ (diskToHalf_maps ha (mem_ball_zero_iff.1 hv))]
+      exact disk_half_inverse ha (mem_ball_zero_iff.1 hv)
+    · intro w hw
+      dsimp only [Function.comp_apply]
+      rw [half_disk_inverse ha (hg hw),hfg w hw]
+    · intro v hv
+      exact ((hfd.differentiableAt (hH.mem_nhds (diskToHalf_maps ha (mem_ball_zero_iff.1 hv)))).comp v
+        (diskToHalf_differentiable ha (mem_ball_zero_iff.1 hv))).differentiableWithinAt
+    · intro w hw
+      exact ((halfToDisk_differentiable ha (hg hw)).comp w
+        (hgd.differentiableAt (hU.mem_nhds hw))).differentiableWithinAt
+  have hh := hbound U _ _ hpair hne R hR
+  simpa only [Function.comp_apply,green_disk_eq] using hh
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge.Mesh
+
+lemma green_nonneg {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) :
+    0 ≤ halfPlaneGreen z w := by
+  by_cases he : z = w
+  · subst w
+    simp [halfPlaneGreen]
+  have hn : 0 < ‖z-w‖ := norm_pos_iff.2 (sub_ne_zero.2 he)
+  have hd : ‖z-starRingEnd ℂ w‖^2 - ‖z-w‖^2 = 4*z.im*w.im := by
+    simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,
+      Complex.conj_re,Complex.conj_im]
+    ring
+  apply Real.log_nonneg
+  rw [norm_div,le_div_iff₀ hn,one_mul]
+  nlinarith [norm_nonneg (z-starRingEnd ℂ w),mul_pos hz hw]
+
+lemma green_continuousAt {z w : ℂ} (hz : 0 < z.im) (hw : 0 < w.im) (hne : z ≠ w) :
+    ContinuousAt (fun q : ℂ × ℂ => halfPlaneGreen q.1 q.2) (z,w) := by
+  apply ContinuousAt.log
+  · exact (((continuous_fst.sub (Complex.continuous_conj.comp continuous_snd)).continuousAt).div
+      (continuous_fst.sub continuous_snd).continuousAt (sub_ne_zero.2 hne)).norm
+  · exact norm_ne_zero_iff.2 (div_ne_zero (Layer.half_den_ne hw hz) (sub_ne_zero.2 hne))
+
+def poissonFactor (driver : ℝ → ℝ) (g : ℝ → ℂ → ℂ) (t : ℝ) (z : ℂ) : ℝ :=
+  (g t z).im/‖g t z-(driver t:ℂ)‖^2
+
+lemma poissonFactor_continuousOn {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {T : ℝ} (hT : 0 ≤ T)
+    (hU : ContinuousOn driver (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun t => poissonFactor driver g t z) (Icc 0 T) := by
+  apply (Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)).div
+    (((h.continuous_eval hT hz).sub (Complex.continuous_ofReal.comp_continuousOn hU)).norm.pow 2)
+  intro t ht
+  exact pow_ne_zero 2 (norm_ne_zero_iff.2 (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz)))
+
+lemma uniformization_injective {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {t : ℝ} (ht : 0 ≤ t) :
+    InjOn (g t) (localizationDomain γ t) := by
+  rcases h with ⟨_,_,_,_,_,_,_,_,_,hfg,_,_,_,_⟩
+  intro z hz w hw he
+  simpa only [hfg t ht z hz,hfg t ht w hw] using congrArg (f t) he
+
+lemma pair_integral {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
+    (hU : ContinuousOn driver (Icc 0 t)) {z w : ℂ}
+    (hz : z ∈ localizationDomain γ t) (hw : w ∈ localizationDomain γ t) (hne : z ≠ w) :
+    (∫ u in s..t, 4*poissonFactor driver g u z*poissonFactor driver g u w) =
+      halfPlaneGreen (g s z) (g s w)-halfPlaneGreen (g t z) (g t w) := by
+  have ht : 0 ≤ t := hs.trans hst
+  have hsub : Icc s t ⊆ Icc 0 t := Icc_subset_Icc hs le_rfl
+  have hgcont : ContinuousOn (fun u => halfPlaneGreen (g u z) (g u w)) (Icc s t) := by
+    intro u hu
+    have hc := (green_continuousAt (h.im_pos (hs.trans hu.1) (survivingDomain_mono hu.2 hz))
+      (h.im_pos (hs.trans hu.1) (survivingDomain_mono hu.2 hw))
+      (fun he => hne (uniformization_injective h (hs.trans hu.1)
+        (survivingDomain_mono hu.2 hz) (survivingDomain_mono hu.2 hw) he)))
+    have hp := (((h.continuous_eval ht hz).mono hsub) u hu).prodMk
+      (((h.continuous_eval ht hw).mono hsub) u hu)
+    exact ContinuousAt.comp_continuousWithinAt (f := fun v => (g v z,g v w)) hc hp
+  have hi : IntervalIntegrable
+      (fun u => -4*poissonFactor driver g u z*poissonFactor driver g u w) volume s t := by
+    apply ContinuousOn.intervalIntegrable
+    rw [uIcc_of_le hst]
+    exact ((((poissonFactor_continuousOn h ht hU hz).const_mul (-4)).mul
+      (poissonFactor_continuousOn h ht hU hw)).mono hsub)
+  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hst hgcont
+    (f' := fun u => -4*poissonFactor driver g u z*poissonFactor driver g u w)
+    (fun u hu => by
+      have hu0 := hs.trans_lt hu.1
+      have hz' := survivingDomain_mono hu.2.le hz
+      have hw' := survivingDomain_mono hu.2.le hw
+      have hd := green_decrease (h.im_pos hu0.le hz') (h.im_pos hu0.le hw')
+        (fun he => hne (uniformization_injective h hu0.le hz' hw' he))
+        (h.hasDerivAt_eval hu0 hz') (h.hasDerivAt_eval hu0 hw')
+      exact hd.congr_deriv (by dsimp [poissonFactor]; ring)) hi
+  have he' : (fun u => -4*poissonFactor driver g u z*poissonFactor driver g u w) =
+      (fun u => -(4*poissonFactor driver g u z*poissonFactor driver g u w)) := by funext u; ring
+  rw [he',intervalIntegral.integral_neg] at he
+  linarith
+
+lemma pair_integral_le {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
+    (hU : ContinuousOn driver (Icc 0 t)) {z w : ℂ}
+    (hz : z ∈ localizationDomain γ t) (hw : w ∈ localizationDomain γ t) (hne : z ≠ w) :
+    (∫ u in s..t, 4*poissonFactor driver g u z*poissonFactor driver g u w) ≤
+      halfPlaneGreen (g s z) (g s w) := by
+  rw [pair_integral h hs hst hU hz hw hne]
+  exact sub_le_self _ (green_nonneg (h.im_pos (hs.trans hst) hz) (h.im_pos (hs.trans hst) hw))
+
+theorem loewner_boundary_layer :
+    ∃ C : ℝ, 0 < C ∧ ∀ (driver : ℝ → ℝ) (γ : ℝ → ℂ) (g f : ℝ → ℂ → ℂ),
+      IsLoewnerTraceUsing driver γ g f → ∀ t : ℝ, 0 ≤ t →
+      ∀ z ∈ localizationDomain γ t, ∀ R : ℝ, 0 < R →
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
+        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2) := by
+  obtain ⟨C,hC,hbound⟩ := Layer.halfPlane_boundary_layer
+  refine ⟨C,hC,?_⟩
+  intro driver γ g f h t ht z hz R hR
+  rcases h with ⟨hγ,_,_,_,_,hg,hf,hgm,hfm,hfg,hgf,_,_,_⟩
+  apply hbound _ (g t) (f t) (survivingDomain_open hγ ht) _ (hgm t ht) (hfm t ht)
+    (hfg t ht) (hgf t ht) (hg t ht).differentiableOn (hf t ht).differentiableOn
+    (g t z) (hgm t ht hz) R hR
+  refine ⟨0,?_⟩
+  intro hzero
+  exact (lt_irrefl (0:ℝ)) (by simpa [halfPlane] using hzero.1.1)
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.PhysicalStop
+
+lemma ofReal_cap_le (s : ℝ≥0∞) (a : ℝ) : ENNReal.ofReal (cap s a) ≤ s := by
+  by_cases hs : s = ⊤
+  · simp [hs]
+  · rw [cap,ite_eq_right hs]
+    exact (ENNReal.ofReal_le_ofReal (min_le_left _ _)).trans_eq (ENNReal.ofReal_toReal hs)
+
+end SLEExactGauge.PhysicalStop
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+lemma first_eq_source {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (hz : z ∈ halfPlane) (hu : 0 < u) {t : ℝ} (ht : t ∈ Icc 0 (C.stopTime κ B ω)) :
+    iteratedDeriv 1 (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
+      (Real.sqrt κ) (C.paths κ B ω) z t) (B t.toNNReal ω) =
+        Real.sqrt κ*(8/κ-1)*(g ω t z-(Real.sqrt κ*B t.toNNReal ω:ℂ)).re /
+          ‖g ω t z-(Real.sqrt κ*B t.toNNReal ω:ℂ)‖^2 *
+            ((loewnerRadius (g ω) t z)^(-(1-κ/8)) *
+              (loewnerAngleSine κ (fun s => B s ω) (g ω) t z)^(8/κ-1)) := by
+  have hU := C.path_eq κ B ω hc
+  have h0 := C.initial_deficit_brownian κ B ω hz hu hc hb0
+  have hs := source_until_stop C.height_pos C.horizon_pos hm _ z _ C.angle hU h0
+  have hhi := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle t).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)).2.1
+  have hts := ht.2.trans (C.bounds κ B ω).2
+  rw [first_eq,field_eq_radius_sine C.height_pos C.horizon_pos _ _ _ _ z t _ hhi (hU ⟨ht.1,hts⟩),
+    radius_eq_source C.height_pos C.horizon_pos hm
+      (continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn _ z _ C.angle hU h0 ht,
+    sine_eq_source C.height_pos C.horizon_pos hm _ z _ C.angle hU h0 ht]
+  rw [LoewnerField.height,max_eq_right hhi,← hs.2 ht]
+  simp only [loewnerAngleSine,Complex.sub_re,Complex.sq_norm,Complex.normSq_apply,
+    Complex.sub_im,Complex.ofReal_mul]
+  simp only [Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,mul_zero,zero_mul,sub_zero,add_zero,pow_two]
+
+lemma first_le_poisson {Ω : Type*} (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4)
+    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (hz : z ∈ halfPlane) (hu : 0 < u) {e t : ℝ} (he : 0 < e) (hes : e < z.im)
+    (ht : t ∈ Icc 0 (C.physicalTime κ B γ e ω)) :
+    |iteratedDeriv 1 (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
+      (Real.sqrt κ) (C.paths κ B ω) z t) (B t.toNNReal ω)| ≤
+      (Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8))) *
+        Mesh.poissonFactor (fun t => Real.sqrt κ*B t.toNNReal ω) (g ω) t z := by
+  have hstop := ht.2.trans (C.physicalTime_bounds κ B γ e ω).2
+  have hl := survivingDomain_mono hstop (C.source_live κ B ω hm hz hu hc hb0)
+  let tn : ℝ≥0 := ⟨t,ht.1⟩
+  have hh : (tn:ℝ≥0∞) ≤ firstLiveDistanceStop (γ ω) z e := by
+    rw [← ENNReal.ofReal_coe_nnreal (p := tn)]
+    exact (ENNReal.ofReal_le_ofReal ht.2).trans (PhysicalStop.ofReal_cap_le _ _)
+  have hR : e/4 ≤ loewnerRadius (g ω) t z := hm.radius_ge_before_distance_stop he hes hh hl
+  have ha : 1 ≤ 8/κ-1 := by
+    have hh : 2 ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by linarith [hκ.2])
+    linarith
+  rw [C.first_eq_source κ B γ g f ω hm hc hb0 hz hu ⟨ht.1,hstop⟩]
+  have hz' : 0 < (g ω t z-(Real.sqrt κ*B t.toNNReal ω:ℂ)).im := by
+    simpa only [Complex.sub_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,mul_zero,zero_mul,add_zero,sub_zero] using hm.im_pos ht.1 hl
+  have hb := Mesh.coefficient_le_poisson hz' ha (show 0 ≤ 1-κ/8 by linarith [hκ.2])
+    (Real.sqrt_nonneg κ) (show 0 < e/4 by positivity) hR
+  simpa only [Mesh.poissonFactor,loewnerAngleSine,Complex.sub_im,Complex.mul_im,Complex.ofReal_im,
+    mul_zero,zero_mul,add_zero,sub_zero,Complex.ofReal_mul,tn,NNReal.coe_mk] using hb
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge.Mesh
+
+lemma poissonFactor_pos {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {t : ℝ} (ht : 0 ≤ t)
+    {z : ℂ} (hz : z ∈ localizationDomain γ t) : 0 < poissonFactor driver g t z := by
+  apply div_pos (h.im_pos ht hz)
+  apply sq_pos_of_pos
+  apply norm_pos_iff.2
+  intro he
+  have hi := congrArg Complex.im he
+  simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi
+  exact (h.im_pos ht hz).ne' hi
+
+lemma overlap_lintegral_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {s t : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
+    (hU : ContinuousOn driver (Icc 0 t)) {z w : ℂ}
+    (hz : z ∈ localizationDomain γ t) (hw : w ∈ localizationDomain γ t) (hne : z ≠ w)
+    (b c : ℝ → ℝ) {K : ℝ} (hK : 0 ≤ K)
+    (hb : ∀ v ∈ Icc s t, |b v| ≤ K*poissonFactor driver g v z)
+    (hc : ∀ v ∈ Icc s t, |c v| ≤ K*poissonFactor driver g v w) :
+    (∫⁻ v in Icc s t, ENNReal.ofReal |b v*c v|) ≤
+      ENNReal.ofReal (K^2 * halfPlaneGreen (g s z) (g s w)) := by
+  let F : ℝ → ℝ := fun v => 4*poissonFactor driver g v z*poissonFactor driver g v w
+  have hF : ContinuousOn F (Icc s t) :=
+    (((poissonFactor_continuousOn h (hs.trans hst) hU hz).const_mul 4).mul
+      (poissonFactor_continuousOn h (hs.trans hst) hU hw)).mono
+        (Icc_subset_Icc hs le_rfl)
+  have hFp (v : ℝ) (hv : v ∈ Icc s t) : 0 ≤ F v := by
+    dsimp [F]
+    exact mul_nonneg (mul_nonneg (by norm_num)
+      (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hz)).le)
+      (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hw)).le
+  calc
+    _ ≤ ∫⁻ v in Icc s t, ENNReal.ofReal (K^2*F v) := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem measurableSet_Icc] with v hv
+      apply ENNReal.ofReal_le_ofReal
+      rw [abs_mul]
+      have hp := mul_nonneg
+        (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hz)).le
+        (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hw)).le
+      calc
+        _ ≤ (K*poissonFactor driver g v z)*(K*poissonFactor driver g v w) :=
+          mul_le_mul (hb v hv) (hc v hv) (abs_nonneg _) (mul_nonneg hK
+            (poissonFactor_pos h (hs.trans hv.1) (survivingDomain_mono hv.2 hz)).le)
+        _ ≤ K^2*F v := by dsimp [F]; nlinarith [mul_nonneg (sq_nonneg K) hp]
+    _ = ENNReal.ofReal (∫ v in Icc s t, K^2*F v) := by
+      symm
+      apply ofReal_integral_eq_lintegral_ofReal
+        ((hF.const_mul (K^2)).integrableOn_compact isCompact_Icc)
+      filter_upwards [ae_restrict_mem measurableSet_Icc] with v hv
+      exact mul_nonneg (sq_nonneg K) (hFp v hv)
+    _ = ENNReal.ofReal (K^2*(∫ v in s..t, F v)) := by
+      rw [integral_const_mul,integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le hst]
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left
+      (pair_integral_le h hs hst hU hz hw hne) (sq_nonneg K))
+
+lemma approached_in_boundary_layer {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (_h : IsLoewnerTraceUsing driver γ g f) {s t R : ℝ} (hs : 0 ≤ s) (hst : s ≤ t)
+    {w : ℂ} (hd : dist w (γ s) ≤ R) :
+    infDist w (localizationDomain γ t)ᶜ ≤ R := by
+  apply (infDist_le_dist_of_mem ?_).trans hd
+  intro hm
+  exact hm.1.2 ⟨s,⟨hs,hst⟩,rfl⟩
+
+lemma green_symm (z w : ℂ) : halfPlaneGreen z w = halfPlaneGreen w z := by
+  have hn : ‖z-starRingEnd ℂ w‖ = ‖w-starRingEnd ℂ z‖ := by
+    calc
+      _ = ‖starRingEnd ℂ (z-starRingEnd ℂ w)‖ := (Complex.norm_conj _).symm
+      _ = ‖starRingEnd ℂ z-w‖ := by simp only [map_sub,starRingEnd_self_apply]
+      _ = _ := norm_sub_rev _ _
+  simp only [halfPlaneGreen,norm_div,hn,norm_sub_rev z w]
+
+def laterGreen (g : ℝ → ℂ → ℂ) (a b : ℂ → ℝ) (z w : ℂ) : ℝ≥0∞ :=
+  if a w ≤ a z ∧ a z ≤ b w then
+    ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w)) else 0
+
+lemma laterGreen_row {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {A : Set ℂ} (hA : MeasurableSet A)
+    {a b : ℂ → ℝ} {R C : ℝ} (_hR : 0 < R)
+    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
+    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
+    (happroach : ∀ z ∈ A, dist z (γ (a z)) ≤ R)
+    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
+        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2))
+    {z : ℂ} (hz : z ∈ A) :
+    (∫⁻ w in A, laterGreen g a b z w) ≤ ENNReal.ofReal (C*R^2) := by
+  let L : Set ℂ := {w | w ∈ localizationDomain γ (a z) ∧
+    infDist w (localizationDomain γ (a z))ᶜ ≤ R}
+  have hLm : MeasurableSet L := (survivingDomain_open h.1 (hstart z hz)).measurableSet.inter
+    (isClosed_le (continuous_infDist_pt _) continuous_const).measurableSet
+  calc
+    _ ≤ ∫⁻ w in A, L.indicator
+        (fun w => ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w))) w := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem hA] with w hw
+      unfold laterGreen
+      split_ifs with horder
+      · have hwL : w ∈ L := ⟨survivingDomain_mono horder.2 (hlive w hw),
+          approached_in_boundary_layer h (hstart w hw) horder.1 (happroach w hw)⟩
+        rw [indicator_of_mem hwL]
+      · exact zero_le
+    _ ≤ ∫⁻ w, L.indicator
+        (fun w => ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w))) w :=
+      lintegral_mono' Measure.restrict_le_self le_rfl
+    _ = ∫⁻ w in L, ENNReal.ofReal (halfPlaneGreen (g (a z) z) (g (a z) w)) :=
+      lintegral_indicator hLm _
+    _ ≤ _ := hlayer (a z) (hstart z hz) z (survivingDomain_mono (hab z hz) (hlive z hz))
+
+lemma supported_pair_lintegral {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ} {z w : ℂ}
+    (hz : ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
+    (hw : ∀ t, t ∉ Icc (a w) (b w) → B t w = 0) :
+    (∫⁻ t, ENNReal.ofReal |B t z*B t w|) =
+      ∫⁻ t in Icc (max (a z) (a w)) (min (b z) (b w)), ENNReal.ofReal |B t z*B t w| := by
+  rw [← lintegral_indicator measurableSet_Icc]
+  apply lintegral_congr
+  intro t
+  by_cases ht : t ∈ Icc (max (a z) (a w)) (min (b z) (b w))
+  · rw [indicator_of_mem ht]
+  · rw [indicator_of_notMem ht]
+    have hh : t ∉ Icc (a z) (b z) ∨ t ∉ Icc (a w) (b w) := by
+      by_contra hn
+      push Not at hn
+      exact ht ⟨max_le hn.1.1 hn.2.1,le_min hn.1.2 hn.2.2⟩
+    rcases hh with hh | hh <;> simp [hz t,hw t,hh]
+
+lemma asynchronous_pair_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
+    {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ} {z w : ℂ} (hne : z ≠ w)
+    (hz0 : 0 ≤ a z) (_hw0 : 0 ≤ a w) (_hzab : a z ≤ b z) (_hwab : a w ≤ b w)
+    (hzlive : z ∈ localizationDomain γ (b z)) (hwlive : w ∈ localizationDomain γ (b w))
+    (hzsup : ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
+    (hwsup : ∀ t, t ∉ Icc (a w) (b w) → B t w = 0)
+    {K : ℝ} (hK : 0 ≤ K)
+    (hzb : ∀ t ∈ Icc (a z) (b z), |B t z| ≤ K*poissonFactor driver g t z)
+    (hwb : ∀ t ∈ Icc (a w) (b w), |B t w| ≤ K*poissonFactor driver g t w) :
+    (∫⁻ t, ENNReal.ofReal |B t z*B t w|) ≤
+      ENNReal.ofReal (K^2) * (laterGreen g a b z w+laterGreen g a b w z) := by
+  rw [supported_pair_lintegral hzsup hwsup]
+  by_cases hover : max (a z) (a w) ≤ min (b z) (b w)
+  · have hi := overlap_lintegral_bound h (hz0.trans (le_max_left _ _)) hover hU.continuousOn
+      (survivingDomain_mono (min_le_left _ _) hzlive)
+      (survivingDomain_mono (min_le_right _ _) hwlive) hne (fun t => B t z) (fun t => B t w) hK
+      (fun t ht => hzb t ⟨(le_max_left _ _).trans ht.1,ht.2.trans (min_le_left _ _)⟩)
+      (fun t ht => hwb t ⟨(le_max_right _ _).trans ht.1,ht.2.trans (min_le_right _ _)⟩)
+    rw [ENNReal.ofReal_mul (sq_nonneg K)] at hi
+    apply hi.trans
+    apply mul_le_mul_right
+    by_cases horder : a w ≤ a z
+    · rw [max_eq_left horder]
+      have hzw : a z ≤ b w := (le_max_left _ _).trans (hover.trans (min_le_right _ _))
+      rw [laterGreen,ite_eq_left ⟨horder,hzw⟩]
+      exact le_add_right le_rfl
+    · have hwz : a z ≤ a w := (lt_of_not_ge horder).le
+      rw [max_eq_right hwz]
+      have hwzb : a w ≤ b z := (le_max_right _ _).trans (hover.trans (min_le_left _ _))
+      have hk : halfPlaneGreen (g (a w) z) (g (a w) w) =
+          halfPlaneGreen (g (a w) w) (g (a w) z) := green_symm _ _
+      rw [hk]
+      have hright : laterGreen g a b w z = ENNReal.ofReal (halfPlaneGreen (g (a w) w) (g (a w) z)) := by
+        rw [laterGreen,ite_eq_left ⟨hwz,hwzb⟩]
+      rw [← hright]
+      exact le_add_left le_rfl
+  · rw [Icc_eq_empty hover,Measure.restrict_empty,lintegral_zero_measure]
+    exact zero_le
+
+lemma norm_integral_sq_le_pairs {E : Type*} [MeasurableSpace E] {μ : Measure E}
+    (v : E → ℝ) (hv : Measurable v) :
+    ENNReal.ofReal (|∫ z, v z ∂μ|^2) ≤
+      ∫⁻ z, ∫⁻ w, ENNReal.ofReal |v z*v w| ∂μ ∂μ := by
+  have hn : ENNReal.ofReal |∫ z, v z ∂μ| ≤ ∫⁻ z, ENNReal.ofReal |v z| ∂μ := by
+    simpa only [← ofReal_norm,Real.norm_eq_abs] using
+      (enorm_integral_le_lintegral_enorm (μ := μ) v)
+  rw [ENNReal.ofReal_pow (abs_nonneg _)]
+  calc
+    _ ≤ (∫⁻ z, ENNReal.ofReal |v z| ∂μ)^2 := pow_le_pow_left₀ zero_le hn 2
+    _ = ∫⁻ z, ∫⁻ w, ENNReal.ofReal |v z*v w| ∂μ ∂μ := by
+      simp_rw [abs_mul,ENNReal.ofReal_mul (abs_nonneg _)]
+      rw [pow_two,lintegral_lintegral_mul hv.abs.ennreal_ofReal.aemeasurable hv.abs.ennreal_ofReal.aemeasurable]
+
+lemma bracket_le_pair_integral {A : Set ℂ} (B : ℝ → ℂ → ℝ)
+    (hB : Measurable (Function.uncurry B)) :
+    (∫⁻ t, ENNReal.ofReal (|∫ z in A, B t z|^2)) ≤
+      ∫⁻ z in A, ∫⁻ w in A, ∫⁻ t, ENNReal.ofReal |B t z*B t w| := by
+  have hb (t : ℝ) : Measurable (B t) := hB.comp (measurable_const.prodMk measurable_id)
+  have hF : Measurable (fun q : (ℝ × ℂ) × ℂ => ENNReal.ofReal |B q.1.1 q.1.2*B q.1.1 q.2|) := by
+    exact ((hB.comp measurable_fst).mul
+      (hB.comp (measurable_fst.fst.prodMk measurable_snd))).abs.ennreal_ofReal
+  calc
+    _ ≤ ∫⁻ t, ∫⁻ z in A, ∫⁻ w in A, ENNReal.ofReal |B t z*B t w| :=
+      lintegral_mono (fun t => norm_integral_sq_le_pairs (B t) (hb t))
+    _ = ∫⁻ z in A, ∫⁻ t, ∫⁻ w in A, ENNReal.ofReal |B t z*B t w| :=
+      lintegral_lintegral_swap hF.lintegral_prod_right.aemeasurable
+    _ = _ := by
+      apply lintegral_congr
+      intro z
+      apply lintegral_lintegral_swap
+      exact (((hB.comp (measurable_fst.prodMk measurable_const)).mul hB).abs.ennreal_ofReal).aemeasurable
+
+lemma symmetric_kernel_integral {A : Set ℂ} (L : ℂ → ℂ → ℝ≥0∞)
+    (hL : Measurable (Function.uncurry L)) {D : ℝ≥0∞}
+    (hrow : ∀ z ∈ A, (∫⁻ w in A, L z w) ≤ D) (hA : MeasurableSet A) :
+    (∫⁻ z in A, ∫⁻ w in A, L z w+L w z) ≤ 2*D*volume A := by
+  have hswap := lintegral_lintegral_swap (μ := volume.restrict A) (ν := volume.restrict A) hL.aemeasurable
+  have hr : (∫⁻ z in A, ∫⁻ w in A, L z w) ≤ D*volume A := by
+    calc
+      _ ≤ ∫⁻ z in A, D := by
+        apply lintegral_mono_ae
+        filter_upwards [ae_restrict_mem hA] with z hz
+        exact hrow z hz
+      _ = _ := by simp
+  have he (z : ℂ) : (∫⁻ w in A, L z w+L w z) =
+      (∫⁻ w in A, L z w)+(∫⁻ w in A, L w z) :=
+    lintegral_add_left (hL.comp (measurable_const.prodMk measurable_id)) _
+  simp_rw [he]
+  rw [lintegral_add_left hL.lintegral_prod_right,← hswap]
+  calc
+    _ ≤ D*volume A+D*volume A := add_le_add hr hr
+    _ = _ := by ring
+
+theorem asynchronous_bracket_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
+    {A : Set ℂ} (hA : MeasurableSet A) {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ}
+    (hB : Measurable (Function.uncurry B))
+    (hL : Measurable (Function.uncurry (laterGreen g a b)))
+    {K R C : ℝ} (hK : 0 ≤ K) (hR : 0 < R)
+    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
+    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
+    (happroach : ∀ z ∈ A, dist z (γ (a z)) ≤ R)
+    (hsup : ∀ z ∈ A, ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
+    (hcoeff : ∀ z ∈ A, ∀ t ∈ Icc (a z) (b z), |B t z| ≤ K*poissonFactor driver g t z)
+    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
+        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2)) :
+    (∫⁻ t, ENNReal.ofReal (|∫ z in A, B t z|^2)) ≤
+      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (C*R^2)*volume A := by
+  apply (bracket_le_pair_integral B hB).trans
+  calc
+    _ ≤ ∫⁻ z in A, ∫⁻ w in A,
+        ENNReal.ofReal (K^2)*(laterGreen g a b z w+laterGreen g a b w z) := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem hA] with z hz
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem hA,ae_restrict_of_ae (show ∀ᵐ w : ℂ ∂volume, w ≠ z from by rw [ae_iff]; simp)] with w hw hne
+      exact asynchronous_pair_bound h hU hne.symm (hstart z hz) (hstart w hw)
+        (hab z hz) (hab w hw) (hlive z hz) (hlive w hw) (hsup z hz) (hsup w hw) hK
+        (hcoeff z hz) (hcoeff w hw)
+    _ = ENNReal.ofReal (K^2)*(∫⁻ z in A, ∫⁻ w in A, laterGreen g a b z w+laterGreen g a b w z) := by
+      simp only [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+    _ ≤ ENNReal.ofReal (K^2)*(2*ENNReal.ofReal (C*R^2)*volume A) := by
+      apply mul_le_mul_right
+      exact symmetric_kernel_integral _ hL
+        (fun z hz => laterGreen_row h hA hR hstart hab hlive happroach hlayer hz) hA
+    _ = _ := by ring
+
+theorem asynchronous_bracket_bound_with_kernel {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
+    {A : Set ℂ} (hA : MeasurableSet A) {a b : ℂ → ℝ} {B : ℝ → ℂ → ℝ}
+    (hB : Measurable (Function.uncurry B))
+    (L : ℂ → ℂ → ℝ≥0∞) (hL : Measurable (Function.uncurry L))
+    (hLeq : ∀ z ∈ A, ∀ w ∈ A, L z w = laterGreen g a b z w)
+    {K R C : ℝ} (hK : 0 ≤ K) (hR : 0 < R)
+    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
+    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
+    (happroach : ∀ z ∈ A, dist z (γ (a z)) ≤ R)
+    (hsup : ∀ z ∈ A, ∀ t, t ∉ Icc (a z) (b z) → B t z = 0)
+    (hcoeff : ∀ z ∈ A, ∀ t ∈ Icc (a z) (b z), |B t z| ≤ K*poissonFactor driver g t z)
+    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
+        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2)) :
+    (∫⁻ t, ENNReal.ofReal (|∫ z in A, B t z|^2)) ≤
+      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (C*R^2)*volume A := by
+  apply (bracket_le_pair_integral B hB).trans
+  calc
+    _ ≤ ∫⁻ z in A, ∫⁻ w in A,
+        ENNReal.ofReal (K^2)*(L z w+L w z) := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem hA] with z hz
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem hA,ae_restrict_of_ae (show ∀ᵐ w : ℂ ∂volume, w ≠ z from by rw [ae_iff]; simp)] with w hw hne
+      rw [hLeq z hz w hw,hLeq w hw z hz]
+      exact asynchronous_pair_bound h hU hne.symm (hstart z hz) (hstart w hw)
+        (hab z hz) (hab w hw) (hlive z hz) (hlive w hw) (hsup z hz) (hsup w hw) hK
+        (hcoeff z hz) (hcoeff w hw)
+    _ = ENNReal.ofReal (K^2)*(∫⁻ z in A, ∫⁻ w in A, L z w+L w z) := by
+      simp only [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+    _ ≤ ENNReal.ofReal (K^2)*(2*ENNReal.ofReal (C*R^2)*volume A) := by
+      apply mul_le_mul_right
+      exact symmetric_kernel_integral _ hL
+        (fun z hz => by
+          have he : (∫⁻ w in A, L z w) = ∫⁻ w in A, laterGreen g a b z w := by
+            apply lintegral_congr_ae
+            filter_upwards [ae_restrict_mem hA] with w hw
+            exact hLeq z hz w hw
+          rw [he]
+          exact laterGreen_row h hA hR hstart hab hlive happroach hlayer hz) hA
+    _ = _ := by ring
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/LoewnerCocycle.lean b/lean/OAI/Probability/SLELocalization/LoewnerCocycle.lean
index 11dc04dc99917d9bd0bec35a89778711f91c39d3..5a27d7aa71f51b526c51eee63d19732a3786b8c7 100644
--- a/lean/OAI/Probability/SLELocalization/LoewnerCocycle.lean
+++ b/lean/OAI/Probability/SLELocalization/LoewnerCocycle.lean
@@ -1,130 +1,130 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-
-namespace OAI
-
-
-/-! Conformal localization and Loewner Cocycle estimates. -/
-
-section
-noncomputable section
-open Set Filter
-open scoped Topology
-namespace SLEExactGauge
-open LoewnerCutoff
-
-theorem loewner_inverse_cocycle {U : ℝ → ℝ} {γ η : ℝ → ℂ}
-    {g f G F : ℝ → ℂ → ℂ} (h : IsLoewnerTraceUsing U γ g f)
-    {T : ℝ} (hT : 0 ≤ T)
-    (hnew : IsLoewnerTraceUsing (fun t => U (T+t)-U T) η G F)
-    {s : ℝ} (hs : 0 ≤ s) {w : ℂ} (hw : w ∈ halfPlane) :
-    f (T+s) (w+(U T:ℂ)) = f T (F s w+(U T:ℂ)) := by
-  let x := f (T+s) (w+(U T:ℂ))
-  let y := F s w
-  have hsum : 0 ≤ T+s := add_nonneg hT hs
-  have hw' : w+(U T:ℂ) ∈ halfPlane := by simpa [halfPlane] using hw
-  have hx : x ∈ localizationDomain γ (T+s) := h.2.2.2.2.2.2.2.2.1 (T+s) hsum hw'
-  have hy : y ∈ localizationDomain η s := hnew.2.2.2.2.2.2.2.2.1 s hs hw
-  have hfin : g (T+s) x = w+(U T:ℂ) := h.2.2.2.2.2.2.2.2.2.2.1 (T+s) hsum _ hw'
-  have hnewfin : G s y = w := hnew.2.2.2.2.2.2.2.2.2.2.1 s hs _ hw
-  have hxl (t : ℝ) (ht : t ∈ Icc 0 s) : w.im ≤ (g (T+t) x).im := by
-    have HH := h.height_antitone hsum hx
-      ⟨add_nonneg hT ht.1,by linarith [ht.2]⟩ ⟨hsum,le_rfl⟩
-      (by linarith [ht.2])
-    simpa only [hfin,Complex.add_im,Complex.ofReal_im,add_zero] using HH
-  have hyl (t : ℝ) (ht : t ∈ Icc 0 s) : w.im ≤ (G t y).im := by
-    simpa only [hnewfin] using hnew.height_antitone hs hy ht ⟨hs,le_rfl⟩ ht.2
-  have hc : ContinuousOn (fun t => g (T+t) x-(U T:ℂ)) (Icc 0 s) :=
-    ((h.continuous_eval hsum hx).comp (continuous_const.add continuous_id).continuousOn
-      (fun t ht => ⟨add_nonneg hT ht.1,by change T+t ≤ T+s; linarith [ht.2]⟩)).sub continuousOn_const
-  have hd (t : ℝ) (ht : t ∈ Ioc 0 s) :
-      HasDerivWithinAt (fun t => g (T+t) x-(U T:ℂ))
-        (velocity w.im (U (T+t)-U T) (g (T+t) x-(U T:ℂ))) (Iic t) t := by
-    rw [velocity_eq (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using hxl t ⟨ht.1.le,ht.2⟩)]
-    have HH := ((h.2.2.2.2.2.2.2.2.2.2.2.2.1 (T+t) (by linarith [ht.1]) x
-      (survivingDomain_mono (by linarith [ht.2]) hx)).scomp t
-      ((hasDerivAt_id t).const_add T)).sub_const (U T:ℂ)
-    have heq : g (T+t) x-(U T:ℂ)-((U (T+t)-U T:ℝ):ℂ) =
-        g (T+t) x-(U (T+t):ℂ) := by rw [Complex.ofReal_sub]; ring
-    rw [heq]
-    simpa only [one_smul,Function.comp_def,id_eq] using HH.hasDerivWithinAt
-  have hdnew (t : ℝ) (ht : t ∈ Ioc 0 s) :
-      HasDerivWithinAt (fun t => G t y)
-        (velocity w.im (U (T+t)-U T) (G t y)) (Iic t) t := by
-    rw [velocity_eq (hyl t ⟨ht.1.le,ht.2⟩)]
-    exact (hnew.2.2.2.2.2.2.2.2.2.2.2.2.1 t ht.1 y
-      (survivingDomain_mono ht.2 hy)).hasDerivWithinAt
-  have he : EqOn (fun t => g (T+t) x-(U T:ℂ)) (fun t => G t y) (Icc 0 s) :=
-    ODE_solution_unique_of_mem_Icc_left (s := fun _ => univ)
-      (fun t _ => (velocity_lipschitz hw (U (T+t)-U T)).lipschitzOnWith)
-      hc hd (fun _ _ => mem_univ _)
-      (hnew.continuous_eval hs hy) hdnew (fun _ _ => mem_univ _)
-      (by simp only [hfin,hnewfin,add_sub_cancel_right])
-  have he0 := he ⟨le_rfl,hs⟩
-  dsimp only at he0
-  rw [add_zero,hnew.initial y hy.1.1] at he0
-  have hearg : g T x = y+(U T:ℂ) := by rw [← he0]; ring
-  change x = f T (y+(U T:ℂ))
-  rw [← hearg]
-  exact (h.2.2.2.2.2.2.2.2.2.1 T hT x
-    (survivingDomain_mono (le_add_of_nonneg_right hs) hx)).symm
-
-end SLEExactGauge
-
-end
-
-end
-
-/-! Conformal localization and Loewner Trace Composition estimates. -/
-
-noncomputable section
-open Set Filter
-open scoped Topology OnePoint
-namespace SLEExactGauge
-
-theorem loewner_trace_composition_of_extension {U : ℝ → ℝ} {γ η : ℝ → ℂ}
-    {g f G F : ℝ → ℂ → ℂ} (h : IsLoewnerTraceUsing U γ g f)
-    {T : ℝ} (hT : 0 ≤ T)
-    (hnew : IsLoewnerTraceUsing (fun t => U (T+t)-U T) η G F)
-    (Fbar : ℂ → OnePoint ℂ)
-    (hext : ∀ w ∈ halfPlane, Fbar w = (f T w : OnePoint ℂ))
-    (hc : ContinuousOn Fbar {w : ℂ | 0 ≤ w.im})
-    {s : ℝ} (hs : 0 ≤ s) :
-    Fbar (η s+(U T:ℂ)) = (γ (T+s) : OnePoint ℂ) := by
-  let V := U (T+s)-U T
-  let a : ℝ → ℂ := fun y => (V:ℂ)+(y:ℂ)*Complex.I
-  have ha (y : ℝ) (hy : 0 < y) : a y ∈ halfPlane := by
-    simpa [a,halfPlane] using hy
-  have hfa (y : ℝ) (hy : 0 < y) : F s (a y)+(U T:ℂ) ∈ halfPlane := by
-    have hh := (hnew.2.2.2.2.2.2.2.2.1 s hs (ha y hy)).1.1
-    simpa only [halfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero] using hh
-  have hlim : Tendsto (fun y => F s (a y)+(U T:ℂ)) (𝓝[>] (0:ℝ))
-      (𝓝[{w : ℂ | 0 ≤ w.im}] (η s+(U T:ℂ))) := by
-    apply tendsto_nhdsWithin_iff.2
-    refine ⟨?_,?_⟩
-    · exact (hnew.2.2.2.2.2.2.2.2.2.2.2.2.2 s hs).add_const (U T:ℂ)
-    · filter_upwards [self_mem_nhdsWithin] with y hy
-      exact (hfa y hy).le
-  have hp : 0 ≤ (η s+(U T:ℂ)).im := by
-    simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using hnew.2.2.1 s hs
-  have hleft := ((hc _ hp).tendsto).comp hlim
-  have hright : Tendsto (fun y : ℝ => (f (T+s)
-      ((U (T+s):ℂ)+(y:ℂ)*Complex.I) : OnePoint ℂ)) (𝓝[>] (0:ℝ))
-      (𝓝 (γ (T+s) : OnePoint ℂ)) :=
-    OnePoint.continuous_coe.continuousAt.tendsto.comp
-      (h.2.2.2.2.2.2.2.2.2.2.2.2.2 (T+s) (add_nonneg hT hs))
-  have heq : (fun y : ℝ => Fbar (F s (a y)+(U T:ℂ))) =ᶠ[𝓝[>] (0:ℝ)]
-      (fun y : ℝ => (f (T+s) ((U (T+s):ℂ)+(y:ℂ)*Complex.I) : OnePoint ℂ)) := by
-    filter_upwards [self_mem_nhdsWithin] with y hy
-    rw [hext _ (hfa y hy),← loewner_inverse_cocycle h hT hnew hs (ha y hy)]
-    congr 2
-    dsimp [a,V]
-    rw [Complex.ofReal_sub]
-    ring
-  exact tendsto_nhds_unique hleft (hright.congr' heq.symm)
-
-end SLEExactGauge
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+
+namespace OAI
+
+
+/-! Conformal localization and Loewner Cocycle estimates. -/
+
+section
+noncomputable section
+open Set Filter
+open scoped Topology
+namespace SLEExactGauge
+open LoewnerCutoff
+
+theorem loewner_inverse_cocycle {U : ℝ → ℝ} {γ η : ℝ → ℂ}
+    {g f G F : ℝ → ℂ → ℂ} (h : IsLoewnerTraceUsing U γ g f)
+    {T : ℝ} (hT : 0 ≤ T)
+    (hnew : IsLoewnerTraceUsing (fun t => U (T+t)-U T) η G F)
+    {s : ℝ} (hs : 0 ≤ s) {w : ℂ} (hw : w ∈ halfPlane) :
+    f (T+s) (w+(U T:ℂ)) = f T (F s w+(U T:ℂ)) := by
+  let x := f (T+s) (w+(U T:ℂ))
+  let y := F s w
+  have hsum : 0 ≤ T+s := add_nonneg hT hs
+  have hw' : w+(U T:ℂ) ∈ halfPlane := by simpa [halfPlane] using hw
+  have hx : x ∈ localizationDomain γ (T+s) := h.2.2.2.2.2.2.2.2.1 (T+s) hsum hw'
+  have hy : y ∈ localizationDomain η s := hnew.2.2.2.2.2.2.2.2.1 s hs hw
+  have hfin : g (T+s) x = w+(U T:ℂ) := h.2.2.2.2.2.2.2.2.2.2.1 (T+s) hsum _ hw'
+  have hnewfin : G s y = w := hnew.2.2.2.2.2.2.2.2.2.2.1 s hs _ hw
+  have hxl (t : ℝ) (ht : t ∈ Icc 0 s) : w.im ≤ (g (T+t) x).im := by
+    have HH := h.height_antitone hsum hx
+      ⟨add_nonneg hT ht.1,by linarith [ht.2]⟩ ⟨hsum,le_rfl⟩
+      (by linarith [ht.2])
+    simpa only [hfin,Complex.add_im,Complex.ofReal_im,add_zero] using HH
+  have hyl (t : ℝ) (ht : t ∈ Icc 0 s) : w.im ≤ (G t y).im := by
+    simpa only [hnewfin] using hnew.height_antitone hs hy ht ⟨hs,le_rfl⟩ ht.2
+  have hc : ContinuousOn (fun t => g (T+t) x-(U T:ℂ)) (Icc 0 s) :=
+    ((h.continuous_eval hsum hx).comp (continuous_const.add continuous_id).continuousOn
+      (fun t ht => ⟨add_nonneg hT ht.1,by change T+t ≤ T+s; linarith [ht.2]⟩)).sub continuousOn_const
+  have hd (t : ℝ) (ht : t ∈ Ioc 0 s) :
+      HasDerivWithinAt (fun t => g (T+t) x-(U T:ℂ))
+        (velocity w.im (U (T+t)-U T) (g (T+t) x-(U T:ℂ))) (Iic t) t := by
+    rw [velocity_eq (by simpa only [Complex.sub_im,Complex.ofReal_im,sub_zero] using hxl t ⟨ht.1.le,ht.2⟩)]
+    have HH := ((h.2.2.2.2.2.2.2.2.2.2.2.2.1 (T+t) (by linarith [ht.1]) x
+      (survivingDomain_mono (by linarith [ht.2]) hx)).scomp t
+      ((hasDerivAt_id t).const_add T)).sub_const (U T:ℂ)
+    have heq : g (T+t) x-(U T:ℂ)-((U (T+t)-U T:ℝ):ℂ) =
+        g (T+t) x-(U (T+t):ℂ) := by rw [Complex.ofReal_sub]; ring
+    rw [heq]
+    simpa only [one_smul,Function.comp_def,id_eq] using HH.hasDerivWithinAt
+  have hdnew (t : ℝ) (ht : t ∈ Ioc 0 s) :
+      HasDerivWithinAt (fun t => G t y)
+        (velocity w.im (U (T+t)-U T) (G t y)) (Iic t) t := by
+    rw [velocity_eq (hyl t ⟨ht.1.le,ht.2⟩)]
+    exact (hnew.2.2.2.2.2.2.2.2.2.2.2.2.1 t ht.1 y
+      (survivingDomain_mono ht.2 hy)).hasDerivWithinAt
+  have he : EqOn (fun t => g (T+t) x-(U T:ℂ)) (fun t => G t y) (Icc 0 s) :=
+    ODE_solution_unique_of_mem_Icc_left (s := fun _ => univ)
+      (fun t _ => (velocity_lipschitz hw (U (T+t)-U T)).lipschitzOnWith)
+      hc hd (fun _ _ => mem_univ _)
+      (hnew.continuous_eval hs hy) hdnew (fun _ _ => mem_univ _)
+      (by simp only [hfin,hnewfin,add_sub_cancel_right])
+  have he0 := he ⟨le_rfl,hs⟩
+  dsimp only at he0
+  rw [add_zero,hnew.initial y hy.1.1] at he0
+  have hearg : g T x = y+(U T:ℂ) := by rw [← he0]; ring
+  change x = f T (y+(U T:ℂ))
+  rw [← hearg]
+  exact (h.2.2.2.2.2.2.2.2.2.1 T hT x
+    (survivingDomain_mono (le_add_of_nonneg_right hs) hx)).symm
+
+end SLEExactGauge
+
+end
+
+end
+
+/-! Conformal localization and Loewner Trace Composition estimates. -/
+
+noncomputable section
+open Set Filter
+open scoped Topology OnePoint
+namespace SLEExactGauge
+
+theorem loewner_trace_composition_of_extension {U : ℝ → ℝ} {γ η : ℝ → ℂ}
+    {g f G F : ℝ → ℂ → ℂ} (h : IsLoewnerTraceUsing U γ g f)
+    {T : ℝ} (hT : 0 ≤ T)
+    (hnew : IsLoewnerTraceUsing (fun t => U (T+t)-U T) η G F)
+    (Fbar : ℂ → OnePoint ℂ)
+    (hext : ∀ w ∈ halfPlane, Fbar w = (f T w : OnePoint ℂ))
+    (hc : ContinuousOn Fbar {w : ℂ | 0 ≤ w.im})
+    {s : ℝ} (hs : 0 ≤ s) :
+    Fbar (η s+(U T:ℂ)) = (γ (T+s) : OnePoint ℂ) := by
+  let V := U (T+s)-U T
+  let a : ℝ → ℂ := fun y => (V:ℂ)+(y:ℂ)*Complex.I
+  have ha (y : ℝ) (hy : 0 < y) : a y ∈ halfPlane := by
+    simpa [a,halfPlane] using hy
+  have hfa (y : ℝ) (hy : 0 < y) : F s (a y)+(U T:ℂ) ∈ halfPlane := by
+    have hh := (hnew.2.2.2.2.2.2.2.2.1 s hs (ha y hy)).1.1
+    simpa only [halfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero] using hh
+  have hlim : Tendsto (fun y => F s (a y)+(U T:ℂ)) (𝓝[>] (0:ℝ))
+      (𝓝[{w : ℂ | 0 ≤ w.im}] (η s+(U T:ℂ))) := by
+    apply tendsto_nhdsWithin_iff.2
+    refine ⟨?_,?_⟩
+    · exact (hnew.2.2.2.2.2.2.2.2.2.2.2.2.2 s hs).add_const (U T:ℂ)
+    · filter_upwards [self_mem_nhdsWithin] with y hy
+      exact (hfa y hy).le
+  have hp : 0 ≤ (η s+(U T:ℂ)).im := by
+    simpa only [Complex.add_im,Complex.ofReal_im,add_zero] using hnew.2.2.1 s hs
+  have hleft := ((hc _ hp).tendsto).comp hlim
+  have hright : Tendsto (fun y : ℝ => (f (T+s)
+      ((U (T+s):ℂ)+(y:ℂ)*Complex.I) : OnePoint ℂ)) (𝓝[>] (0:ℝ))
+      (𝓝 (γ (T+s) : OnePoint ℂ)) :=
+    OnePoint.continuous_coe.continuousAt.tendsto.comp
+      (h.2.2.2.2.2.2.2.2.2.2.2.2.2 (T+s) (add_nonneg hT hs))
+  have heq : (fun y : ℝ => Fbar (F s (a y)+(U T:ℂ))) =ᶠ[𝓝[>] (0:ℝ)]
+      (fun y : ℝ => (f (T+s) ((U (T+s):ℂ)+(y:ℂ)*Complex.I) : OnePoint ℂ)) := by
+    filter_upwards [self_mem_nhdsWithin] with y hy
+    rw [hext _ (hfa y hy),← loewner_inverse_cocycle h hT hnew hs (ha y hy)]
+    congr 2
+    dsimp [a,V]
+    rw [Complex.ofReal_sub]
+    ring
+  exact tendsto_nhds_unique hleft (hright.congr' heq.symm)
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/LoewnerExit.lean b/lean/OAI/Probability/SLELocalization/LoewnerExit.lean
index cd7535562828af7c87488b0d160bc0983c1b354b..ca3653407893dce1cc4460d25e20e53673fc934e 100644
--- a/lean/OAI/Probability/SLELocalization/LoewnerExit.lean
+++ b/lean/OAI/Probability/SLELocalization/LoewnerExit.lean
@@ -1,1093 +1,1093 @@
-import Mathlib
-import OAI.Probability.SLELocalization.ConditionalAngles
-import OAI.Probability.SLE.ClippedCoordinates
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Loewner Exit estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerExit
-open LoewnerCutoff
-
-lemma stop_le_of_nonpos {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (U : C(I,ℝ)) (z : ℂ) (ρ δ : ℝ) {t : ℝ} (ht : t ∈ Icc 0 T)
-    (hn : deficit hε hT.le U z ρ δ t ≤ 0) :
-    stop hε hT.le U z ρ δ ≤ t := by
-  have hq : t/T ∈ Icc (0:ℝ) 1 := ⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩
-  have hh := ContinuousExit.time_le_of_nonpos hq
-    (show ContinuousExit.extend (exitPath hε hT.le U z ρ δ) (t/T) ≤ 0 by
-      rw [exitPath_extend_div hε hT.le hT U z ρ δ ht]; exact hn)
-  change T*ContinuousExit.time (exitPath hε hT.le U z ρ δ) ≤ t
-  exact (mul_le_mul_of_nonneg_left hh hT.le).trans_eq (mul_div_cancel₀ _ hT.ne')
-
-lemma stop_prefix_min {ε S T : ℝ} (hε : 0 < ε) (hS : 0 < S) (hT : 0 < T)
-    (hST : S ≤ T) (U V : C(I,ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S)) :
-    stop hε hS.le U z ρ δ = min S (stop hε hT.le V z ρ δ) := by
-  let a := stop hε hS.le U z ρ δ
-  let b := stop hε hT.le V z ρ δ
-  have ha : 0 ≤ a ∧ a ≤ S := stop_bounds hε hS.le U z ρ δ
-  have hb : 0 ≤ b ∧ b ≤ T := stop_bounds hε hT.le V z ρ δ
-  have hd := deficit_prefix_eq hε hS.le hT.le hST U V z ρ δ hUV
-  apply le_antisymm
-  · apply le_min ha.2
-    by_contra hn
-    have hba : b < a := lt_of_not_ge hn
-    have hbs : b < S := hba.trans_le ha.2
-    have hbT : b < T := hbs.trans_le hST
-    have hn := nonpos_at_of_lt hε hT.le hT V z ρ δ hbT
-    have hp := positive_before hε hS.le hS U z ρ δ hb.1 hba
-    have he := hd ⟨hb.1,hbs.le⟩
-    change deficit hε hS.le U z ρ δ b = deficit hε hT.le V z ρ δ b at he
-    rw [he] at hp
-    exact (not_lt_of_ge hn) hp
-  · by_contra hn
-    have haa : a < min S b := lt_of_not_ge hn
-    have haS : a < S := (lt_min_iff.1 haa).1
-    have hab : a < b := (lt_min_iff.1 haa).2
-    have hn := nonpos_at_of_lt hε hS.le hS U z ρ δ haS
-    have hp := positive_before hε hT.le hT V z ρ δ ha.1 hab
-    have he := hd ⟨ha.1,haS.le⟩
-    change deficit hε hS.le U z ρ δ a = deficit hε hT.le V z ρ δ a at he
-    rw [← he] at hp
-    exact (not_lt_of_ge hn) hp
-
-end SLEExactGauge.LoewnerExit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-
-def withHorizon {z : ℂ} {u : ℝ} (C : Cutoff z u) (T : ℝ≥0) : Cutoff z u where
-  angle := C.angle
-  angle_pos := C.angle_pos
-  angle_initial := C.angle_initial
-  height := C.height
-  height_pos := C.height_pos
-  horizon := max C.horizon (T+1)
-  horizon_pos := C.horizon_pos.trans_le (le_max_left _ _)
-  height_small := C.height_small
-  time_large := C.time_large.trans_le (le_max_left _ _)
-
-lemma withHorizon_gt {z : ℂ} {u : ℝ} (C : Cutoff z u) (T : ℝ≥0) :
-    (T:ℝ) < (C.withHorizon T).horizon := by
-  change (T:ℝ) < max (C.horizon:ℝ) (T+1)
-  exact (lt_add_one _).trans_le (le_max_right _ _)
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.RadiusPast
-open LoewnerCutoff LoewnerExit LoewnerLocalized FiniteTilt
-
-def hitData {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (κ : ℝ) (U : C(I,ℝ)) (z : ℂ) (ρ δ : ℝ) : ℝ × ℝ :=
-  let τ := stop hε hT U z ρ δ
-  (τ, if radius hε hT U z τ ≤ ρ then
-    (radius hε hT U z τ)^(-codimension κ)*(sine hε hT U z τ)^(8/κ-1) else 0)
-
-lemma measurable_hitData {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (κ : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    @Measurable C(I,ℝ) (ℝ × ℝ) (borel _) _ (fun U => hitData hε hT.le κ U z ρ δ) := by
-  let : MeasurableSpace C(I,ℝ) := borel _
-  let : BorelSpace C(I,ℝ) := ⟨rfl⟩
-  have ht := GammaField.measurable_stop hε hT.le hT z ρ δ
-  have hr := (continuous_radius hε hT.le z).measurable.comp (measurable_id.prodMk ht)
-  have hs := (continuous_sine hε hT.le z).measurable.comp (measurable_id.prodMk ht)
-  exact ht.prodMk (Measurable.ite (measurableSet_le hr measurable_const)
-    ((hr.pow_const _).mul (hs.pow_const _)) measurable_const)
-
-end SLEExactGauge.RadiusPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.RadiusPast
-open LoewnerCutoff LoewnerExit LoewnerLocalized FiniteTilt
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
-variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-variable (C : Cutoff z u) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (hTC : T ≤ C.horizon)
-
-include h hz hu hc hb0 hT hTC in
-lemma raw_data_source :
-    let U := BrownianDriver.path B (Real.sqrt κ) T sample
-    let τ := stop C.height_pos T.coe_nonneg U z ((2*z.im)*Real.exp (-u)) C.angle
-    τ = min (T:ℝ) (C.stopTime κ B sample) ∧ z ∈ localizationDomain γ τ ∧
-    hitData C.height_pos T.coe_nonneg κ U z ((2*z.im)*Real.exp (-u)) C.angle =
-      (τ, if loewnerRadius g τ z = (2*z.im)*Real.exp (-u) then
-        (loewnerRadius g τ z)^(-codimension κ) *
-          (loewnerAngleSine κ (fun t => B t sample) g τ z)^(8/κ-1) else 0) := by
-  classical
-  dsimp only
-  let U := BrownianDriver.path B (Real.sqrt κ) T sample
-  let ρ := (2*z.im)*Real.exp (-u)
-  let τ := stop C.height_pos T.coe_nonneg U z ρ C.angle
-  have hUV := BrownianDriver.prefix_eq B (Real.sqrt κ) T C.horizon sample hc hTC
-  have h0 : 0 < deficit C.height_pos T.coe_nonneg U z ρ C.angle 0 := by
-    rw [deficit_prefix_eq C.height_pos T.coe_nonneg C.horizon.coe_nonneg hTC
-      U (C.paths κ B sample) z ρ C.angle hUV ⟨le_rfl,T.coe_nonneg⟩]
-    exact C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*B t.toNNReal sample) (Icc 0 (T:ℝ)) :=
-    fun t ht => BrownianDriver.extend_eq B (Real.sqrt κ) T sample hc ht
-  have hsrc := source_until_stop C.height_pos hT h U z ρ C.angle hU h0
-  have hb := stop_bounds C.height_pos T.coe_nonneg U z ρ C.angle
-  have ht : τ ∈ Icc 0 τ := ⟨hb.1,le_rfl⟩
-  have hr := radius_eq_source C.height_pos hT h
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) U z ρ C.angle hU h0 ht
-  have hs : sine C.height_pos T.coe_nonneg U z τ =
-      loewnerAngleSine κ (fun t => B t sample) g τ z := by
-    simpa only [loewnerAngleSine,Complex.ofReal_mul] using
-      sine_eq_source C.height_pos hT h U z ρ C.angle hU h0 ht
-  have hlo : ρ ≤ radius C.height_pos T.coe_nonneg U z τ :=
-    ((deficit_nonneg_iff C.height_pos T.coe_nonneg U z ρ C.angle τ).1
-      (nonneg_until C.height_pos T.coe_nonneg hT U z ρ C.angle h0 ht)).1
-  refine ⟨stop_prefix_min C.height_pos hT C.horizon_pos hTC U (C.paths κ B sample) z ρ C.angle hUV,hsrc.1,?_⟩
-  change (τ, if radius C.height_pos T.coe_nonneg U z τ ≤ ρ then
-    (radius C.height_pos T.coe_nonneg U z τ)^(-codimension κ) *
-    (sine C.height_pos T.coe_nonneg U z τ)^(8/κ-1) else 0) =
-    (τ, if loewnerRadius g τ z = ρ then (loewnerRadius g τ z)^(-codimension κ) *
-      (loewnerAngleSine κ (fun t => B t sample) g τ z)^(8/κ-1) else 0)
-  by_cases he : loewnerRadius g τ z = ρ
-  · have hle : radius C.height_pos T.coe_nonneg U z τ ≤ ρ := by rw [hr,he]
-    simp only [hr,hs,he,le_refl,ite_true]
-  · have hn : ¬ radius C.height_pos T.coe_nonneg U z τ ≤ ρ := by
-      intro hl
-      exact he (hr.symm.trans (le_antisymm hl hlo))
-    simp only [hn,he,ite_false]
-
-include h hz hu hc hb0 in
-lemma eventually_stop_general (D : ℕ → Cutoff z u)
-    (hD : Tendsto (fun n => (D n).angle) atTop (𝓝 0))
-    {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
-    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
-    ∀ᶠ n : ℕ in atTop, (D n).stopTime κ B sample = t := by
-  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
-  filter_upwards [hD.eventually (gt_mem_nhds hη)] with n hn
-  let C := D n
-  have hh : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
-    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · exact hr
-    · have htm := hm (C.stopTime κ B sample)
-        ⟨(C.bounds κ B sample).1,C.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
-      rw [hs] at htm
-      exact False.elim ((not_lt_of_ge htm) hn)
-  exact h.live_radius_injective
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-    (C.bounds κ B sample).1 t.coe_nonneg (C.source_live κ B sample h hz hu hc hb0) hzt
-    (hh.trans hrt.symm)
-
-end SLEExactGauge.RadiusPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.RadiusPast
-open LoewnerCutoff LoewnerExit LoewnerLocalized FiniteTilt
-
-lemma live_radius_witness {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hU : ContinuousOn driver (Ici 0))
-    {z : ℂ} {u : ℝ} (hh : firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞)) :
-    ∃ t : ℝ≥0, firstLiveRadiusStop γ g z u = (t : ℝ≥0∞) ∧
-      z ∈ localizationDomain γ t ∧ loewnerRadius g t z = (2*z.im)*Real.exp (-u) := by
-  have he : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧
-      loewnerRadius g t z = (2*z.im)*Real.exp (-u) := by
-    by_contra hn
-    have he : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
-        loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ :=
-      Set.eq_empty_iff_forall_notMem.2 (fun t ht => hn ⟨t,ht⟩)
-    exact hh (by simp only [firstLiveRadiusStop,he,Set.image_empty,sInf_empty])
-  obtain ⟨t,hzt,hrt⟩ := he
-  exact ⟨t,h.firstLiveRadiusStop_eq hU hzt hrt,hzt,hrt⟩
-
-def byTime {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) (T : ℝ≥0) (sample : Ω) : ℝ≥0∞ :=
-  if firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T : ℝ≥0∞) then
-    finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u else 0
-
-def approx {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (n : ℕ) (sample : Ω) : ℝ≥0∞ :=
-  let C := (cutoffs hz u hu n).withHorizon T
-  ENNReal.ofReal (hitData C.height_pos T.coe_nonneg κ
-    (BrownianDriver.path B (Real.sqrt κ) T sample) z ((2*z.im)*Real.exp (-u)) C.angle).2
-
-lemma eventually_approx {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-    (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
-    ∀ᶠ n : ℕ in atTop, approx κ B hz u hu.le T n sample = byTime κ B γ g z u T sample := by
-  classical
-  let D := fun n => (cutoffs hz u hu.le n).withHorizon T
-  have hU : ContinuousOn (fun t : ℝ => Real.sqrt κ*B t.toNNReal sample) (Ici 0) := (continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn
-  by_cases hh : firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T : ℝ≥0∞)
-  · obtain ⟨t,htau,hzt,hrt⟩ := live_radius_witness h hU
-      (ne_top_of_le_ne_top ENNReal.coe_ne_top hh)
-    have htT : t ≤ T := by simpa only [htau,ENNReal.coe_le_coe] using hh
-    have hD : Tendsto (fun n => (D n).angle) atTop (𝓝 0) := shrinkingAngle_tendsto z
-    filter_upwards [eventually_stop_general κ B sample h hz hu hc hb0 D hD hzt hrt] with n hn
-    have hs := raw_data_source κ B sample h hz hu hc hb0 (D n) T hT
-      ((cutoffs hz u hu.le n).withHorizon_gt T).le
-    dsimp only at hs
-    rw [hn,min_eq_right (show (t:ℝ) ≤ T from htT)] at hs
-    dsimp only [approx]
-    unfold byTime
-    rw [ite_eq_left hh, hs.2.2, Prod.snd, hs.1, ite_eq_left hrt,
-      finiteStoppedDensity,htau,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal]
-  · apply Filter.Eventually.of_forall
-    intro n
-    have hs := raw_data_source κ B sample h hz hu hc hb0 (D n) T hT
-      ((cutoffs hz u hu.le n).withHorizon_gt T).le
-    dsimp only at hs
-    let U := BrownianDriver.path B (Real.sqrt κ) T sample
-    let τ := stop (D n).height_pos T.coe_nonneg U z ((2*z.im)*Real.exp (-u)) (D n).angle
-    have hn : loewnerRadius (g sample) τ z ≠ (2*z.im)*Real.exp (-u) := by
-      intro he
-      have ht := stop_bounds (D n).height_pos T.coe_nonneg U z ((2*z.im)*Real.exp (-u)) (D n).angle
-      have heq := h.firstLiveRadiusStop_eq hU (t := ⟨τ,ht.1⟩) hs.2.1 he
-      apply hh
-      rw [heq]
-      exact ENNReal.coe_le_coe.2 (show (⟨τ,ht.1⟩ : ℝ≥0) ≤ T from ht.2)
-    dsimp only [approx]
-    unfold byTime
-    rw [ite_eq_right hh,hs.2.2,Prod.snd,ite_eq_right hn,ENNReal.ofReal_zero]
-
-end SLEExactGauge.RadiusPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.RadiusPast
-open FiniteTilt BrownianPredictable
-local instance : MeasurableSpace C(I, ℝ) := borel _
-local instance : BorelSpace C(I, ℝ) := ⟨rfl⟩
-
-def observed {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (sample : Ω) : ℝ≥0∞ :=
-  ENNReal.ofReal (ContinuousDriver.selectedLimit 0 (fun n => (approx κ B hz u hu T n sample).toReal))
-
-lemma measurable_observed {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
-    Measurable[pastSigma B T] (observed κ B hz u hu T) := by
-  let : MeasurableSpace Ω := pastSigma B T
-  apply ENNReal.measurable_ofReal.comp
-  apply ContinuousDriver.measurable_selectedLimit
-  intro n
-  let C := (cutoffs hz u hu n).withHorizon T
-  exact ENNReal.measurable_toReal.comp (ENNReal.measurable_ofReal.comp
-    (((measurable_hitData C.height_pos hT κ z ((2*z.im)*Real.exp (-u)) C.angle).snd).comp
-      (BrownianDriver.path_measurable_past B (Real.sqrt κ) T)))
-
-lemma aemeasurable_observed {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
-    AEMeasurable (observed κ B hz u hu T) P := by
-  apply ENNReal.measurable_ofReal.comp_aemeasurable
-  apply ContinuousDriver.aemeasurable_selectedLimit
-  intro n
-  let C := (cutoffs hz u hu n).withHorizon T
-  exact ENNReal.measurable_toReal.comp_aemeasurable (ENNReal.measurable_ofReal.comp_aemeasurable
-    (((measurable_hitData C.height_pos hT κ z ((2*z.im)*Real.exp (-u)) C.angle).snd).comp_aemeasurable
-      (BrownianDriver.path_aemeasurable hB (Real.sqrt κ) T)))
-
-lemma observed_eq {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-    (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
-    observed κ B hz u hu.le T sample = byTime κ B γ g z u T sample := by
-  have hf : byTime κ B γ g z u T sample ≠ ⊤ := by
-    simp only [byTime,finiteStoppedDensity]
-    split_ifs <;> simp
-  unfold observed
-  rw [ContinuousDriver.selectedLimit_eq (y := (byTime κ B γ g z u T sample).toReal) 0 (by
-    apply tendsto_const_nhds.congr'
-    filter_upwards [eventually_approx κ B γ g f sample h hz hu hc hb0 T hT] with n hn
-    exact congrArg ENNReal.toReal hn.symm), ENNReal.ofReal_toReal hf]
-
-lemma byTime_pos_iff {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    {z : ℂ} (_hz : z ∈ halfPlane) (u : ℝ)
-    (hc : Continuous (fun t => B t sample)) (T : ℝ≥0) :
-    0 < byTime κ B γ g z u T sample ↔ firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T : ℝ≥0∞) := by
-  unfold byTime
-  split_ifs with hh
-  · obtain ⟨t,htau,hzt,hrt⟩ := live_radius_witness h
-      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-      (ne_top_of_le_ne_top ENNReal.coe_ne_top hh)
-    have hs : 0 < loewnerAngleSine κ (fun t => B t sample) (g sample) t z := by
-      simpa only [loewnerAngleSine,Complex.ofReal_mul] using
-        div_pos (h.im_pos t.coe_nonneg hzt) (norm_pos_iff.2 (h.centered_ne_zero t.coe_nonneg hzt))
-    have hr : 0 < loewnerRadius (g sample) t z := h.radius_pos t.coe_nonneg hzt
-    simp only [finiteStoppedDensity,htau,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal]
-    exact iff_of_true (ENNReal.ofReal_pos.2
-      (mul_pos (Real.rpow_pos_of_pos hr _) (Real.rpow_pos_of_pos hs _))) (htau ▸ hh)
-  · exact iff_of_false (lt_irrefl _) hh
-
-end SLEExactGauge.RadiusPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-variable (h : IsLoewnerTraceUsing driver γ g f) (hU : ContinuousOn driver (Ici 0))
-
-include h hU in
-lemma firstLiveRadiusStop_mono {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v) :
-    firstLiveRadiusStop γ g z u ≤ firstLiveRadiusStop γ g z v := by
-  by_cases hv : firstLiveRadiusStop γ g z v = (⊤ : ℝ≥0∞)
-  · simp only [hv,le_top]
-  obtain ⟨t,htau,hzt,hrt⟩ := RadiusPast.live_radius_witness h hU hv
-  have hρ : (2*z.im)*Real.exp (-v) ≤ (2*z.im)*Real.exp (-u) :=
-    mul_le_mul_of_nonneg_left (Real.exp_le_exp.2 (neg_le_neg huv)) (by have hy : 0 < z.im := hz; positivity)
-  have hρ0 : (2*z.im)*Real.exp (-u) ≤ loewnerRadius g 0 z := by
-    rw [h.radius_initial hz]
-    exact mul_le_of_le_one_right (by have hy : 0 < z.im := hz; positivity)
-      (Real.exp_le_one_iff.2 (neg_nonpos.2 hu))
-  obtain ⟨s,hs,hseq⟩ := intermediate_value_Icc' t.coe_nonneg
-    (h.continuous_radius t.coe_nonneg (hU.mono Icc_subset_Ici_self) hzt) ⟨hrt ▸ hρ,hρ0⟩
-  have hstop := h.firstLiveRadiusStop_eq hU (t := ⟨s,hs.1⟩) (survivingDomain_mono hs.2 hzt) hseq
-  rw [hstop,htau]
-  exact ENNReal.coe_le_coe.2 (show (⟨s,hs.1⟩ : ℝ≥0) ≤ t from hs.2)
-
-include h hU in
-lemma firstLiveRadiusStop_pos {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u) :
-    0 < firstLiveRadiusStop γ g z u := by
-  by_cases hv : firstLiveRadiusStop γ g z u = (⊤ : ℝ≥0∞)
-  · simp only [hv,ENNReal.zero_lt_top]
-  obtain ⟨t,htau,hzt,hrt⟩ := RadiusPast.live_radius_witness h hU hv
-  have ht : 0 < t := by
-    by_contra hn
-    have he : t = 0 := le_antisymm (not_lt.1 hn) zero_le
-    rw [he,NNReal.coe_zero,h.radius_initial hz] at hrt
-    have hr : (2*z.im)*Real.exp (-u) < 2*z.im :=
-      mul_lt_of_lt_one_right (by have hy : 0 < z.im := hz; positivity)
-        (Real.exp_lt_one_iff.2 (neg_neg_of_pos hu))
-    exact (ne_of_lt hr) hrt.symm
-  rw [htau]
-  exact ENNReal.coe_pos.2 ht
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable
-
-def CompletedStoppedPast {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (P : Measure Ω) (τ : Ω → ℝ≥0∞) (A : Set Ω) : Prop :=
-  NullMeasurableSet A P ∧ ∀ t : ℝ≥0, ∃ E : Set Ω, MeasurableSet[pastSigma B t] E ∧
-    E =ᵐ[P] (A ∩ {sample | τ sample ≤ (t : ℝ≥0∞)} : Set Ω)
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable
-namespace CompletedStoppedPast
-variable {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} {P : Measure Ω}
-variable {τ σ : Ω → ℝ≥0∞} {A C : Set Ω}
-
-lemma empty : CompletedStoppedPast B P τ ∅ := by
-  refine ⟨nullMeasurableSet_empty,fun t => ⟨∅,@MeasurableSet.empty Ω (pastSigma B t),?_⟩⟩
-  simp only [empty_inter]
-  exact Filter.EventuallyEq.rfl
-
-lemma union (hA : CompletedStoppedPast B P τ A) (hC : CompletedStoppedPast B P τ C) :
-    CompletedStoppedPast B P τ (A ∪ C) := by
-  refine ⟨hA.1.union hC.1,?_⟩
-  intro t
-  obtain ⟨E,hE,he⟩ := hA.2 t
-  obtain ⟨F,hF,hf⟩ := hC.2 t
-  refine ⟨E ∪ F,hE.union hF,?_⟩
-  filter_upwards [he,hf] with x he hf
-  change (x∈E ∨ x∈F) = ((x∈A ∨ x∈C) ∧ τ x ≤ (t:ℝ≥0∞))
-  change (x∈E) = (x∈A ∧ τ x ≤ (t:ℝ≥0∞)) at he
-  change (x∈F) = (x∈C ∧ τ x ≤ (t:ℝ≥0∞)) at hf
-  rw [he,hf]
-  exact propext (by tauto)
-
-lemma diff (hA : CompletedStoppedPast B P τ A) (hC : CompletedStoppedPast B P τ C) :
-    CompletedStoppedPast B P τ (A \ C) := by
-  refine ⟨hA.1.diff hC.1,?_⟩
-  intro t
-  obtain ⟨E,hE,he⟩ := hA.2 t
-  obtain ⟨F,hF,hf⟩ := hC.2 t
-  refine ⟨E \ F,hE.diff hF,?_⟩
-  filter_upwards [he,hf] with x he hf
-  change (x∈E ∧ x∉F) = ((x∈A ∧ x∉C) ∧ τ x ≤ (t:ℝ≥0∞))
-  change (x∈E) = (x∈A ∧ τ x ≤ (t:ℝ≥0∞)) at he
-  change (x∈F) = (x∈C ∧ τ x ≤ (t:ℝ≥0∞)) at hf
-  rw [he,hf]
-  exact propext (by tauto)
-
-lemma mono (hA : CompletedStoppedPast B P τ A)
-    (hu : CompletedStoppedPast B P σ univ) (hτσ : ∀ᵐ x ∂P, τ x ≤ σ x) :
-    CompletedStoppedPast B P σ A := by
-  refine ⟨hA.1,?_⟩
-  intro t
-  obtain ⟨E,hE,he⟩ := hA.2 t
-  obtain ⟨F,hF,hf⟩ := hu.2 t
-  refine ⟨E ∩ F,hE.inter hF,?_⟩
-  filter_upwards [he,hf,hτσ] with x he hf hle
-  change (x∈E ∧ x∈F) = (x∈A ∧ σ x ≤ (t:ℝ≥0∞))
-  change (x∈E) = (x∈A ∧ τ x ≤ (t:ℝ≥0∞)) at he
-  change (x∈F) = (True ∧ σ x ≤ (t:ℝ≥0∞)) at hf
-  rw [he,hf]
-  apply propext
-  constructor
-  · exact fun hh => ⟨hh.1.1,hh.2.2⟩
-  · exact fun hh => ⟨⟨hh.1,hle.trans hh.2⟩,⟨trivial,hh.2⟩⟩
-
-end CompletedStoppedPast
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable
-
-lemma finiteStoppedDensity_stoppedPast {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u)
-    (S : Set ℝ≥0∞) (hS : MeasurableSet S) :
-    CompletedStoppedPast B P (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u)
-      {sample | finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∈ S} := by
-  refine ⟨?_,?_⟩
-  · exact (finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu).nullMeasurableSet_preimage hS
-  intro T
-  by_cases ht : T = 0
-  · refine ⟨∅,@MeasurableSet.empty Ω (pastSigma B T),?_⟩
-    filter_upwards [hmaps,hB.cont] with sample hm hc
-    have hp := hm.firstLiveRadiusStop_pos
-      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hz hu
-    apply propext
-    simp only [mem_empty_iff_false,mem_inter_iff,mem_ofPred_eq,ht,ENNReal.coe_zero]
-    constructor
-    · exact False.elim
-    · exact fun hh => (not_le_of_gt hp) hh.2
-  have hTp : 0 < (T:ℝ) := by exact_mod_cast pos_iff_ne_zero.2 ht
-  let F := RadiusPast.observed κ B hz u hu.le T
-  have hFm := RadiusPast.measurable_observed κ B hz u hu.le T hTp
-  refine ⟨{sample | 0 < F sample} ∩ F ⁻¹' S,
-    (measurableSet_lt measurable_const hFm).inter (hS.preimage hFm),?_⟩
-  filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0
-  have he := RadiusPast.observed_eq κ B γ g f sample hm hz hu hc hb0 T hTp
-  have hp := RadiusPast.byTime_pos_iff κ B γ g f sample hm hz u hc T
-  apply propext
-  change (0 < F sample ∧ F sample ∈ S) ↔
-    (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∈ S ∧
-      firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T:ℝ≥0∞))
-  change F sample = _ at he
-  rw [he,hp]
-  by_cases hh : firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T:ℝ≥0∞)
-  · simp only [RadiusPast.byTime,hh,ite_true,true_and,and_true]
-  · simp only [hh,false_and,and_false]
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace CompletedStoppedPast
-variable {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} {P : Measure Ω}
-variable {τ : Ω → ℝ≥0∞}
-
-lemma finsetUnion {ι : Type*} (s : Finset ι) (A : ι → Set Ω)
-    (hA : ∀ i ∈ s, CompletedStoppedPast B P τ (A i)) :
-    CompletedStoppedPast B P τ (⋃ i ∈ s, A i) := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => simpa only [Finset.notMem_empty, iUnion_of_empty, iUnion_empty] using (empty (B:=B) (P:=P) (τ:=τ))
-  | @insert a s ha ih =>
-    simpa only [Finset.set_biUnion_insert] using
-      (hA a (Finset.mem_insert_self _ _)).union (ih (fun i hi => hA i (Finset.mem_insert_of_mem hi)))
-
-end CompletedStoppedPast
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace FiniteFirstEvents
-variable {Ω ι : Type*} [LinearOrder ι]
-
-def first (s : Finset ι) (A : ι → Set Ω) (i : ι) : Set Ω :=
-  A i \ ⋃ j ∈ s.filter (fun j => j < i), A j
-
-lemma first_subset (s : Finset ι) (A : ι → Set Ω) (i : ι) : first s A i ⊆ A i :=
-  sdiff_subset
-
-lemma first_disjoint (s : Finset ι) (A : ι → Set Ω) {i j : ι}
-    (hi : i ∈ s) (hj : j ∈ s) (hij : i ≠ j) : Disjoint (first s A i) (first s A j) := by
-  classical
-  apply Set.disjoint_left.2
-  intro x hxi hxj
-  rcases lt_or_gt_of_ne hij with hij | hji
-  · exact hxj.2 (mem_iUnion.2 ⟨i,mem_iUnion.2 ⟨Finset.mem_filter.2 ⟨hi,hij⟩,hxi.1⟩⟩)
-  · exact hxi.2 (mem_iUnion.2 ⟨j,mem_iUnion.2 ⟨Finset.mem_filter.2 ⟨hj,hji⟩,hxj.1⟩⟩)
-
-lemma union_first (s : Finset ι) (A : ι → Set Ω) :
-    (⋃ i ∈ s, first s A i) = ⋃ i ∈ s, A i := by
-  classical
-  apply Subset.antisymm
-  · exact iUnion₂_mono (fun i _ => first_subset s A i)
-  · intro x hx
-    obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.1 hx
-    let t := s.filter (fun i => x ∈ A i)
-    have ht : t.Nonempty := ⟨i,Finset.mem_filter.2 ⟨hi,hxi⟩⟩
-    let j := t.min' ht
-    have hj := Finset.mem_filter.1 (Finset.min'_mem t ht)
-    apply mem_iUnion₂.2 ⟨j,hj.1,?_⟩
-    refine ⟨hj.2,?_⟩
-    intro hbad
-    obtain ⟨k,hk,hxk⟩ := mem_iUnion₂.1 hbad
-    have hkj := (Finset.mem_filter.1 hk).2
-    have hjk := Finset.min'_le t k (Finset.mem_filter.2 ⟨(Finset.mem_filter.1 hk).1,hxk⟩)
-    exact (not_lt_of_ge hjk) hkj
-
-lemma first_stoppedPast [MeasurableSpace Ω]
-    {B : ℝ≥0 → Ω → ℝ} {P : Measure Ω} {τ : Ω → ℝ≥0∞}
-    (s : Finset ι) (A : ι → Set Ω) (i : ι)
-    (hi : CompletedStoppedPast B P τ (A i))
-    (hj : ∀ j ∈ s, j < i → CompletedStoppedPast B P τ (A j)) :
-    CompletedStoppedPast B P τ (first s A i) := by
-  classical
-  exact hi.diff (CompletedStoppedPast.finsetUnion (s.filter (fun j => j < i)) A
-    (fun j hj' => hj j (Finset.mem_filter.1 hj').1 (Finset.mem_filter.1 hj').2))
-
-end FiniteFirstEvents
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open BrownianPredictable
-
-lemma finiteCenteredMeasure_event_le {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (u v : ℝ) (hu : 0 ≤ u) (huv : u ≤ v)
-    (A : Set Ω)
-    (hA : CompletedStoppedPast B P (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u) A)
-    (K : ℝ≥0∞)
-    (hb : ∀ᵐ sample ∂P.restrict A,
-      finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ≤ K) :
-    finiteCenteredMeasure κ P B γ g z v A ≤
-      (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K * P A := by
-  have he := finiteStoppedDensity_consistent κ hκ B hB γ g f hγ hmaps z hz u v hu huv A hA.1 hA.2
-  have hm0 : ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) ≠ 0 :=
-    (ENNReal.ofReal_pos.2 (localizationInitialGreen_pos κ hz)).ne'
-  rw [finiteCenteredMeasure,withDensity_apply₀ _ hA.1]
-  simp_rw [ENNReal.div_eq_inv_mul]
-  rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hm0),he,mul_assoc]
-  apply mul_le_mul' le_rfl
-  exact (lintegral_mono_ae hb).trans_eq (by rw [lintegral_const,Measure.restrict_apply_univ])
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteFirstEvents
-
-def densityBelow {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) (K : ℝ≥0∞) : Set Ω :=
-  {sample | finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∈ Ioo 0 K}
-
-lemma finiteCentered_grid_le {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (v : ℝ) (s : Finset ℚ)
-    (hs : ∀ q ∈ s, 0 < (q:ℝ) ∧ (q:ℝ) ≤ v)
-    (a : ℚ → ℝ≥0∞) (K : ℝ≥0∞) (ha : ∀ q ∈ s, a q ≤ K) :
-    finiteCenteredMeasure κ P B γ g z v (⋃ q ∈ s, densityBelow κ B γ g z q (a q)) ≤
-      (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K *
-        P (⋃ q ∈ s, densityBelow κ B γ g z q (a q)) := by
-  classical
-  let A := fun q : ℚ => densityBelow κ B γ g z q (a q)
-  let Q := finiteCenteredMeasure κ P B γ g z v
-  let k := (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K
-  have hpast (q : ℚ) (hq : q ∈ s) : CompletedStoppedPast B P
-      (fun sample => firstLiveRadiusStop (γ sample) (g sample) z q) (A q) :=
-    finiteStoppedDensity_stoppedPast κ hκ B hB γ g f hmaps z hz q (hs q hq).1 (Ioo 0 (a q)) measurableSet_Ioo
-  have huniv (q : ℚ) (hq : q ∈ s) : CompletedStoppedPast B P
-      (fun sample => firstLiveRadiusStop (γ sample) (g sample) z q) univ := by
-    simpa only [mem_univ,ofPred_true] using
-      finiteStoppedDensity_stoppedPast κ hκ B hB γ g f hmaps z hz q (hs q hq).1 univ MeasurableSet.univ
-  have hfirst (q : ℚ) (hq : q∈s) : CompletedStoppedPast B P
-      (fun sample => firstLiveRadiusStop (γ sample) (g sample) z q) (first s A q) := by
-    apply first_stoppedPast s A q (hpast q hq)
-    intro j hj hjq
-    apply (hpast j hj).mono (huniv q hq)
-    filter_upwards [hmaps,hB.cont] with sample hm hc
-    exact hm.firstLiveRadiusStop_mono
-      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hz (hs j hj).1.le
-      (by exact_mod_cast hjq.le)
-  have hle (q : ℚ) (hq : q∈s) : Q (first s A q) ≤ k * P (first s A q) := by
-    apply finiteCenteredMeasure_event_le κ hκ B hB γ g f hγ hmaps z hz q v
-      (hs q hq).1.le (hs q hq).2 (first s A q) (hfirst q hq) K
-    filter_upwards [ae_restrict_mem₀ (hfirst q hq).1] with sample hx
-    exact hx.1.2.le.trans (ha q hq)
-  have hdis : PairwiseDisjoint (↑s : Set ℚ) (first s A) :=
-    fun i hi j hj hij => first_disjoint s A hi hj hij
-  have hPQ : Q ≪ P := withDensity_absolutelyContinuous _ _
-  have hsumQ : Q (⋃ q∈s,first s A q) = ∑ q∈s,Q (first s A q) :=
-    measure_biUnion_finset₀ hdis.aedisjoint (fun q hq => (hfirst q hq).1.mono_ac hPQ)
-  have hsumP : P (⋃ q∈s,first s A q) = ∑ q∈s,P (first s A q) :=
-    measure_biUnion_finset₀ hdis.aedisjoint (fun q hq => (hfirst q hq).1)
-  change Q (⋃ q∈s,A q) ≤ k * P (⋃ q∈s,A q)
-  rw [← union_first s A,hsumQ,hsumP]
-  calc
-    _ ≤ ∑ q∈s,k*P (first s A q) := Finset.sum_le_sum hle
-    _ = _ := by rw [Finset.mul_sum]
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma finiteCentered_countableGrid_le {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (v : ℝ) (J : Set ℚ)
-    (hJ : ∀ q ∈ J, 0 < (q:ℝ) ∧ (q:ℝ) ≤ v)
-    (a : ℚ → ℝ≥0∞) (K : ℝ≥0∞) (ha : ∀ q ∈ J, a q ≤ K) :
-    finiteCenteredMeasure κ P B γ g z v (⋃ q ∈ J, densityBelow κ B γ g z q (a q)) ≤
-      (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K *
-        P (⋃ q ∈ J, densityBelow κ B γ g z q (a q)) := by
-  classical
-  let A := fun q : ℚ => densityBelow κ B γ g z q (a q)
-  let F := fun s : Finset ℚ => ⋃ q ∈ s.filter (fun q => q∈J), A q
-  have hm : Monotone F := by
-    intro s t hst
-    exact iUnion₂_mono' (fun q hq => ⟨q, Finset.mem_filter.2
-      ⟨hst (Finset.mem_filter.1 hq).1,(Finset.mem_filter.1 hq).2⟩,Subset.rfl⟩)
-  have he : (⋃ s, F s) = ⋃ q ∈ J, A q := by
-    ext sample
-    simp only [F,mem_iUnion,Finset.mem_filter]
-    constructor
-    · rintro ⟨s,q,⟨hq,hJq⟩,hx⟩
-      exact ⟨q,hJq,hx⟩
-    · rintro ⟨q,hq,hx⟩
-      exact ⟨{q},q,⟨Finset.mem_singleton_self q,hq⟩,hx⟩
-  have hsub (s : Finset ℚ) : F s ⊆ ⋃ q ∈ J,A q := by
-    rw [← he]
-    exact subset_iUnion F s
-  change finiteCenteredMeasure κ P B γ g z v (⋃ q∈J,A q) ≤ _
-  rw [← he,hm.measure_iUnion]
-  apply iSup_le
-  intro s
-  apply (finiteCentered_grid_le κ hκ B hB γ g f hγ hmaps z hz v
-    (s.filter (fun q => q∈J))
-    (fun q hq => hJ q (Finset.mem_filter.1 hq).2) a K
-    (fun q hq => ha q (Finset.mem_filter.1 hq).2)).trans
-  exact mul_le_mul' le_rfl (measure_mono (he ▸ hsub s))
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace FiniteClock
-
-def angleFloor (δ u : ℝ) : ℝ := δ * Real.exp (-u/8)
-
-def threshold (κ : ℝ) (z : ℂ) (δ u : ℝ) : ℝ≥0∞ :=
-  ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-codimension κ) * (angleFloor δ u)^(8/κ-1))
-
-def blockThreshold (κ : ℝ) (z : ℂ) (δ n : ℝ) : ℝ :=
-  (2*z.im)^(-codimension κ) * Real.exp (codimension κ*(n+1)) *
-    δ^(8/κ-1) * Real.exp (-(8/κ-1)*n/8)
-
-lemma exponent_pos {κ : ℝ} (hκ : 0 < κ ∧ κ < 8) : 0 < 8/κ-1 := by
-  have h : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by linarith)
-  linarith
-
-lemma codimension_pos {κ : ℝ} (hκ : 0 < κ ∧ κ < 8) : 0 < codimension κ := by
-  unfold codimension dimension
-  linarith
-
-lemma angleFloor_pos {δ : ℝ} (hδ : 0 < δ) (u : ℝ) : 0 < angleFloor δ u := by
-  unfold angleFloor
-  positivity
-
-lemma threshold_formula (κ : ℝ) {z : ℂ} (hz : z ∈ halfPlane) {δ : ℝ} (hδ : 0 < δ) (u : ℝ) :
-    threshold κ z δ u = ENNReal.ofReal
-      ((2*z.im)^(-codimension κ) * Real.exp (codimension κ*u) *
-        δ^(8/κ-1) * Real.exp (-(8/κ-1)*u/8)) := by
-  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
-  unfold threshold angleFloor
-  rw [Real.mul_rpow hy.le (Real.exp_pos _).le,Real.mul_rpow hδ.le (Real.exp_pos _).le,
-    ← Real.exp_mul, ← Real.exp_mul]
-  congr 1
-  rw [show -u * -codimension κ = codimension κ*u by ring,
-    show (-u/8)*(8/κ-1) = -(8/κ-1)*u/8 by ring]
-  ring
-
-lemma threshold_le_block (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    {z : ℂ} (hz : z ∈ halfPlane) {δ : ℝ} (hδ : 0 < δ)
-    {n u : ℝ} (hnu : n ≤ u) (hun : u ≤ n+1) :
-    threshold κ z δ u ≤ ENNReal.ofReal (blockThreshold κ z δ n) := by
-  rw [threshold_formula κ hz hδ]
-  apply ENNReal.ofReal_le_ofReal
-  unfold blockThreshold
-  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
-  have hp := codimension_pos hκ
-  have ha := exponent_pos hκ
-  apply mul_le_mul
-  · apply mul_le_mul_of_nonneg_right _ (Real.rpow_pos_of_pos hδ _).le
-    exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left hun hp.le))
-      (Real.rpow_pos_of_pos hy _).le
-  · apply Real.exp_le_exp.2
-    nlinarith
-  · positivity
-  · positivity
-
-lemma block_cancel (κ C δ n : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
-    ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))⁻¹ *
-      blockThreshold κ z δ n *
-      (C*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)) =
-      C * Real.exp (codimension κ) * δ^(8/κ-1) * Real.exp (-(8/κ-1)*n/8) := by
-  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
-  have hs := Real.rpow_pos_of_pos (FiniteTilt.initial_sine_pos hz) (8/κ-1)
-  have hp := Real.rpow_pos_of_pos hy (-codimension κ)
-  have he : Real.exp (codimension κ*(n+1))*Real.exp (-codimension κ*n) = Real.exp (codimension κ) := by
-    rw [← Real.exp_add]
-    congr 1
-    ring
-  unfold blockThreshold
-  field_simp
-  simp only [neg_mul] at he
-  linear_combination δ^(8/κ-1)*C * he
-
-def blockIndex (v : ℝ) (n : ℕ) : Set ℚ :=
-  {q | 0 < (q:ℝ) ∧ (q:ℝ) ≤ v ∧ (n:ℝ) ≤ (q:ℝ) ∧ (q:ℝ) ≤ (n:ℝ)+1}
-
-def blockBad {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (δ v : ℝ) (n : ℕ) : Set Ω :=
-  ⋃ q ∈ blockIndex v n, densityBelow κ B γ g z q (threshold κ z δ q)
-
-lemma blockBad_subset_hit {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (δ v : ℝ) (n : ℕ) :
-    blockBad κ B γ g z δ v n ≤ᵐ[P]
-      {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} := by
-  filter_upwards [hmaps,hB.cont] with sample hm hc hx
-  rcases mem_iUnion.1 hx with ⟨q,hq⟩
-  rcases mem_iUnion.1 hq with ⟨hq,hx⟩
-  have hhit : firstLiveRadiusStop (γ sample) (g sample) z q ≠ (⊤ : ℝ≥0∞) := by
-    intro he
-    have hh := hx.1
-    simp only [finiteStoppedDensity,he,ite_true,lt_self_iff_false] at hh
-  have hh := hm.firstLiveRadiusStop_mono
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hz
-    (Nat.cast_nonneg n) hq.2.2.1
-  exact ne_top_of_le_ne_top hhit hh
-
-lemma ordinary_natural_radius_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) :
-    ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ,
-      P {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} ≤
-        ENNReal.ofReal (C*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)) := by
-  obtain ⟨C,hC,hbound⟩ := radiusApproachBound κ hκ
-  let S := (z.im/‖z‖)^(8/κ-1)
-  have hS : 0 < S := Real.rpow_pos_of_pos (FiniteTilt.initial_sine_pos hz) _
-  refine ⟨C+S⁻¹,by positivity,?_⟩
-  intro n
-  cases n with
-  | zero =>
-    simp only [Nat.cast_zero,mul_zero,Real.exp_zero,mul_one]
-    apply (prob_le_one).trans
-    apply ENNReal.one_le_ofReal.2
-    change 1 ≤ (C+S⁻¹)*S
-    rw [add_mul,inv_mul_cancel₀ hS.ne']
-    linarith [mul_pos hC hS]
-  | succ n =>
-    simp only [Nat.cast_add,Nat.cast_one]
-    apply (hbound P B hB γ g f hγ hmaps z hz (n+1)
-      (by norm_num)).trans
-    apply ENNReal.ofReal_le_ofReal
-    exact mul_le_mul_of_nonneg_right
-      (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (inv_nonneg.2 hS.le))
-        (Real.exp_pos _).le) hS.le
-
-lemma block_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) (C : ℝ) (_hC : 0 < C)
-    (hbound : ∀ n : ℕ,
-      P {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} ≤
-        ENNReal.ofReal (C*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)))
-    (δ : ℝ) (hδ : 0 < δ) (v : ℝ) (n : ℕ) :
-    finiteCenteredMeasure κ P B γ g z v (blockBad κ B γ g z δ v n) ≤
-      ENNReal.ofReal (C * Real.exp (codimension κ) * δ^(8/κ-1) *
-        Real.exp (-(8/κ-1)*n/8)) := by
-  have hm0 := localizationInitialGreen_pos κ hz
-  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
-  have hK : 0 < blockThreshold κ z δ n := by
-    unfold blockThreshold
-    positivity
-  have hp := (measure_mono_ae (blockBad_subset_hit κ B hB γ g f hmaps z hz δ v n)).trans (hbound n)
-  have hg := finiteCentered_countableGrid_le κ hκ B hB γ g f hγ hmaps z hz v
-    (blockIndex v n) (fun q hq => ⟨hq.1,hq.2.1⟩)
-    (fun q => threshold κ z δ q) (ENNReal.ofReal (blockThreshold κ z δ n))
-    (fun q hq => threshold_le_block κ hκ hz hδ hq.2.2.1 hq.2.2.2)
-  apply hg.trans
-  apply (mul_le_mul' le_rfl hp).trans_eq
-  rw [← ENNReal.ofReal_inv_of_pos hm0, ← ENNReal.ofReal_mul (inv_nonneg.2 hm0.le),
-    ← ENNReal.ofReal_mul (mul_nonneg (inv_nonneg.2 hm0.le) hK.le)]
-  rw [block_cancel κ C δ n hz]
-
-end FiniteClock
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.FiniteClock
-
-def bad {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (δ v : ℝ) : Set Ω :=
-  ⋃ n : ℕ, blockBad κ B γ g z δ v n
-
-lemma mem_bad_of_rational {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (δ v : ℝ)
-    (q : ℚ) (hq : 0 < (q:ℝ)) (hqv : (q:ℝ) ≤ v) {sample : Ω}
-    (hx : sample ∈ densityBelow κ B γ g z q (threshold κ z δ q)) :
-    sample ∈ bad κ B γ g z δ v := by
-  apply mem_iUnion.2 ⟨⌊(q:ℝ)⌋₊,?_⟩
-  apply mem_iUnion.2 ⟨q,?_⟩
-  apply mem_iUnion.2 ⟨⟨hq,hqv,Nat.floor_le hq.le,(Nat.lt_floor_add_one (q:ℝ)).le⟩,hx⟩
-
-lemma geometric_blocks (κ C δ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hC : 0 < C) (hδ : 0 < δ) :
-    (∑' n : ℕ, ENNReal.ofReal (C*Real.exp (codimension κ)*δ^(8/κ-1)*Real.exp (-(8/κ-1)*n/8))) =
-      ENNReal.ofReal ((C*Real.exp (codimension κ)/(1-Real.exp (-(8/κ-1)/8)))*δ^(8/κ-1)) := by
-  have ha := exponent_pos hκ
-  have hr : Real.exp (-(8/κ-1)/8) < 1 := Real.exp_lt_one_iff.2 (by linarith)
-  have he (n : ℕ) : Real.exp (-(8/κ-1)*n/8) = (Real.exp (-(8/κ-1)/8))^n := by
-    rw [← Real.exp_nat_mul]
-    congr 1
-    ring
-  simp_rw [he,ENNReal.ofReal_mul (show 0≤C*Real.exp (codimension κ)*δ^(8/κ-1) by positivity),
-    ENNReal.ofReal_pow (Real.exp_pos _).le]
-  rw [ENNReal.tsum_mul_left,ENNReal.tsum_geometric,← ENNReal.ofReal_one,
-    ← ENNReal.ofReal_sub 1 (Real.exp_pos _).le,← ENNReal.ofReal_inv_of_pos (sub_pos.2 hr),
-    ← ENNReal.ofReal_mul (show 0≤C*Real.exp (codimension κ)*δ^(8/κ-1) by positivity)]
-  congr 1
-  ring
-
-lemma bad_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (z : ℂ) (hz : z ∈ halfPlane) :
-    ∃ D : ℝ, 0 < D ∧ ∀ δ : ℝ, 0 < δ → ∀ v : ℝ,
-      finiteCenteredMeasure κ P B γ g z v (bad κ B γ g z δ v) ≤ ENNReal.ofReal (D*δ^(8/κ-1)) := by
-  obtain ⟨C,hC,hbound⟩ := ordinary_natural_radius_bound κ hκ B hB γ g f hγ hmaps z hz
-  have hr : Real.exp (-(8/κ-1)/8) < 1 := Real.exp_lt_one_iff.2 (by have := exponent_pos hκ; linarith)
-  refine ⟨C*Real.exp (codimension κ)/(1-Real.exp (-(8/κ-1)/8)),by positivity,?_⟩
-  intro δ hδ v
-  apply (measure_iUnion_le _).trans
-  apply (ENNReal.tsum_le_tsum (fun n => block_bound κ hκ B hB γ g f hγ hmaps z hz C hC hbound δ hδ v n)).trans_eq
-  exact geometric_blocks κ C δ hκ hC hδ
-
-lemma exists_small_floor {a D ε : ℝ} (ha : 0 < a) (hD : 0 < D) (hε : 0 < ε) :
-    ∃ δ : ℝ, 0 < δ ∧ D*δ^a < ε := by
-  have hp : 0 < ε/(2*D) := by positivity
-  refine ⟨(ε/(2*D))^(1/a),Real.rpow_pos_of_pos hp _,?_⟩
-  rw [← Real.rpow_mul hp.le,one_div_mul_cancel ha.ne',Real.rpow_one]
-  have he : D*(ε/(2*D)) = ε/2 := by field_simp
-  rw [he]
-  linarith
-
-lemma rational_clock_nonneg {T v : ℝ} {C F : ℝ → ℝ}
-    (hT : 0 < T) (hC : ContinuousOn C (Icc 0 T)) (hCs : StrictMonoOn C (Icc 0 T))
-    (hC0 : C 0 = 0) (hCT : C T = v) (hF : ContinuousOn F (Icc 0 T))
-    (hrat : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ v →
-      ∀ s ∈ Icc 0 T, C s = (q:ℝ) → 0 ≤ F s) :
-    ∀ t ∈ Ioo 0 T, 0 ≤ F t := by
-  intro t ht
-  by_contra hn
-  have hneg : F t < 0 := lt_of_not_ge hn
-  have hFc : ContinuousAt F t := (hF t ⟨ht.1.le,ht.2.le⟩).continuousAt (Icc_mem_nhds ht.1 ht.2)
-  have he : ∀ᶠ s in 𝓝 t, F s < 0 := hFc.tendsto.eventually (gt_mem_nhds hneg)
-  obtain ⟨e,he0,he⟩ := Metric.eventually_nhds_iff.1 he
-  have hlow : max 0 (t-e) < t := max_lt ht.1 (by linarith)
-  obtain ⟨a,ha,hat⟩ := exists_between hlow
-  have ha0 : 0 < a := (le_max_left 0 (t-e)).trans_lt ha
-  have hae : t-e < a := (le_max_right 0 (t-e)).trans_lt ha
-  have haq : a ∈ Icc 0 T := ⟨ha0.le,(hat.trans ht.2).le⟩
-  obtain ⟨q,hqa,hqt⟩ := exists_rat_btwn (hCs haq ⟨ht.1.le,ht.2.le⟩ hat)
-  have hq0 : 0 < (q:ℝ) := by
-    have hc := hCs ⟨le_rfl,hT.le⟩ haq ha0
-    rw [hC0] at hc
-    exact hc.trans hqa
-  have hqv : (q:ℝ) ≤ v := by
-    have hc := hCs.monotoneOn ⟨ht.1.le,ht.2.le⟩ ⟨hT.le,le_rfl⟩ ht.2.le
-    rw [hCT] at hc
-    exact hqt.le.trans hc
-  obtain ⟨s,hs,hsq⟩ := intermediate_value_Icc hat.le
-    (hC.mono (Icc_subset_Icc haq.1 ht.2.le)) ⟨hqa.le,hqt.le⟩
-  have hst : s ∈ Icc 0 T := ⟨ha0.le.trans hs.1,hs.2.trans ht.2.le⟩
-  have hsn : F s < 0 := he (by
-    rw [Real.dist_eq,abs_of_nonpos (sub_nonpos.2 hs.2)]
-    linarith [hs.1])
-  exact (not_lt_of_ge (hrat q hq0 hqv s hst hsq)) hsn
-
-end SLEExactGauge.FiniteClock
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.ConditionalAngles
+import OAI.Probability.SLE.ClippedCoordinates
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Loewner Exit estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerExit
+open LoewnerCutoff
+
+lemma stop_le_of_nonpos {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (U : C(I,ℝ)) (z : ℂ) (ρ δ : ℝ) {t : ℝ} (ht : t ∈ Icc 0 T)
+    (hn : deficit hε hT.le U z ρ δ t ≤ 0) :
+    stop hε hT.le U z ρ δ ≤ t := by
+  have hq : t/T ∈ Icc (0:ℝ) 1 := ⟨div_nonneg ht.1 hT.le,(div_le_one hT).2 ht.2⟩
+  have hh := ContinuousExit.time_le_of_nonpos hq
+    (show ContinuousExit.extend (exitPath hε hT.le U z ρ δ) (t/T) ≤ 0 by
+      rw [exitPath_extend_div hε hT.le hT U z ρ δ ht]; exact hn)
+  change T*ContinuousExit.time (exitPath hε hT.le U z ρ δ) ≤ t
+  exact (mul_le_mul_of_nonneg_left hh hT.le).trans_eq (mul_div_cancel₀ _ hT.ne')
+
+lemma stop_prefix_min {ε S T : ℝ} (hε : 0 < ε) (hS : 0 < S) (hT : 0 < T)
+    (hST : S ≤ T) (U V : C(I,ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S)) :
+    stop hε hS.le U z ρ δ = min S (stop hε hT.le V z ρ δ) := by
+  let a := stop hε hS.le U z ρ δ
+  let b := stop hε hT.le V z ρ δ
+  have ha : 0 ≤ a ∧ a ≤ S := stop_bounds hε hS.le U z ρ δ
+  have hb : 0 ≤ b ∧ b ≤ T := stop_bounds hε hT.le V z ρ δ
+  have hd := deficit_prefix_eq hε hS.le hT.le hST U V z ρ δ hUV
+  apply le_antisymm
+  · apply le_min ha.2
+    by_contra hn
+    have hba : b < a := lt_of_not_ge hn
+    have hbs : b < S := hba.trans_le ha.2
+    have hbT : b < T := hbs.trans_le hST
+    have hn := nonpos_at_of_lt hε hT.le hT V z ρ δ hbT
+    have hp := positive_before hε hS.le hS U z ρ δ hb.1 hba
+    have he := hd ⟨hb.1,hbs.le⟩
+    change deficit hε hS.le U z ρ δ b = deficit hε hT.le V z ρ δ b at he
+    rw [he] at hp
+    exact (not_lt_of_ge hn) hp
+  · by_contra hn
+    have haa : a < min S b := lt_of_not_ge hn
+    have haS : a < S := (lt_min_iff.1 haa).1
+    have hab : a < b := (lt_min_iff.1 haa).2
+    have hn := nonpos_at_of_lt hε hS.le hS U z ρ δ haS
+    have hp := positive_before hε hT.le hT V z ρ δ ha.1 hab
+    have he := hd ⟨ha.1,haS.le⟩
+    change deficit hε hS.le U z ρ δ a = deficit hε hT.le V z ρ δ a at he
+    rw [← he] at hp
+    exact (not_lt_of_ge hn) hp
+
+end SLEExactGauge.LoewnerExit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+
+def withHorizon {z : ℂ} {u : ℝ} (C : Cutoff z u) (T : ℝ≥0) : Cutoff z u where
+  angle := C.angle
+  angle_pos := C.angle_pos
+  angle_initial := C.angle_initial
+  height := C.height
+  height_pos := C.height_pos
+  horizon := max C.horizon (T+1)
+  horizon_pos := C.horizon_pos.trans_le (le_max_left _ _)
+  height_small := C.height_small
+  time_large := C.time_large.trans_le (le_max_left _ _)
+
+lemma withHorizon_gt {z : ℂ} {u : ℝ} (C : Cutoff z u) (T : ℝ≥0) :
+    (T:ℝ) < (C.withHorizon T).horizon := by
+  change (T:ℝ) < max (C.horizon:ℝ) (T+1)
+  exact (lt_add_one _).trans_le (le_max_right _ _)
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.RadiusPast
+open LoewnerCutoff LoewnerExit LoewnerLocalized FiniteTilt
+
+def hitData {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (κ : ℝ) (U : C(I,ℝ)) (z : ℂ) (ρ δ : ℝ) : ℝ × ℝ :=
+  let τ := stop hε hT U z ρ δ
+  (τ, if radius hε hT U z τ ≤ ρ then
+    (radius hε hT U z τ)^(-codimension κ)*(sine hε hT U z τ)^(8/κ-1) else 0)
+
+lemma measurable_hitData {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (κ : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    @Measurable C(I,ℝ) (ℝ × ℝ) (borel _) _ (fun U => hitData hε hT.le κ U z ρ δ) := by
+  let : MeasurableSpace C(I,ℝ) := borel _
+  let : BorelSpace C(I,ℝ) := ⟨rfl⟩
+  have ht := GammaField.measurable_stop hε hT.le hT z ρ δ
+  have hr := (continuous_radius hε hT.le z).measurable.comp (measurable_id.prodMk ht)
+  have hs := (continuous_sine hε hT.le z).measurable.comp (measurable_id.prodMk ht)
+  exact ht.prodMk (Measurable.ite (measurableSet_le hr measurable_const)
+    ((hr.pow_const _).mul (hs.pow_const _)) measurable_const)
+
+end SLEExactGauge.RadiusPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.RadiusPast
+open LoewnerCutoff LoewnerExit LoewnerLocalized FiniteTilt
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+variable {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) γ g f)
+variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+variable (C : Cutoff z u) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (hTC : T ≤ C.horizon)
+
+include h hz hu hc hb0 hT hTC in
+lemma raw_data_source :
+    let U := BrownianDriver.path B (Real.sqrt κ) T sample
+    let τ := stop C.height_pos T.coe_nonneg U z ((2*z.im)*Real.exp (-u)) C.angle
+    τ = min (T:ℝ) (C.stopTime κ B sample) ∧ z ∈ localizationDomain γ τ ∧
+    hitData C.height_pos T.coe_nonneg κ U z ((2*z.im)*Real.exp (-u)) C.angle =
+      (τ, if loewnerRadius g τ z = (2*z.im)*Real.exp (-u) then
+        (loewnerRadius g τ z)^(-codimension κ) *
+          (loewnerAngleSine κ (fun t => B t sample) g τ z)^(8/κ-1) else 0) := by
+  classical
+  dsimp only
+  let U := BrownianDriver.path B (Real.sqrt κ) T sample
+  let ρ := (2*z.im)*Real.exp (-u)
+  let τ := stop C.height_pos T.coe_nonneg U z ρ C.angle
+  have hUV := BrownianDriver.prefix_eq B (Real.sqrt κ) T C.horizon sample hc hTC
+  have h0 : 0 < deficit C.height_pos T.coe_nonneg U z ρ C.angle 0 := by
+    rw [deficit_prefix_eq C.height_pos T.coe_nonneg C.horizon.coe_nonneg hTC
+      U (C.paths κ B sample) z ρ C.angle hUV ⟨le_rfl,T.coe_nonneg⟩]
+    exact C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*B t.toNNReal sample) (Icc 0 (T:ℝ)) :=
+    fun t ht => BrownianDriver.extend_eq B (Real.sqrt κ) T sample hc ht
+  have hsrc := source_until_stop C.height_pos hT h U z ρ C.angle hU h0
+  have hb := stop_bounds C.height_pos T.coe_nonneg U z ρ C.angle
+  have ht : τ ∈ Icc 0 τ := ⟨hb.1,le_rfl⟩
+  have hr := radius_eq_source C.height_pos hT h
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) U z ρ C.angle hU h0 ht
+  have hs : sine C.height_pos T.coe_nonneg U z τ =
+      loewnerAngleSine κ (fun t => B t sample) g τ z := by
+    simpa only [loewnerAngleSine,Complex.ofReal_mul] using
+      sine_eq_source C.height_pos hT h U z ρ C.angle hU h0 ht
+  have hlo : ρ ≤ radius C.height_pos T.coe_nonneg U z τ :=
+    ((deficit_nonneg_iff C.height_pos T.coe_nonneg U z ρ C.angle τ).1
+      (nonneg_until C.height_pos T.coe_nonneg hT U z ρ C.angle h0 ht)).1
+  refine ⟨stop_prefix_min C.height_pos hT C.horizon_pos hTC U (C.paths κ B sample) z ρ C.angle hUV,hsrc.1,?_⟩
+  change (τ, if radius C.height_pos T.coe_nonneg U z τ ≤ ρ then
+    (radius C.height_pos T.coe_nonneg U z τ)^(-codimension κ) *
+    (sine C.height_pos T.coe_nonneg U z τ)^(8/κ-1) else 0) =
+    (τ, if loewnerRadius g τ z = ρ then (loewnerRadius g τ z)^(-codimension κ) *
+      (loewnerAngleSine κ (fun t => B t sample) g τ z)^(8/κ-1) else 0)
+  by_cases he : loewnerRadius g τ z = ρ
+  · have hle : radius C.height_pos T.coe_nonneg U z τ ≤ ρ := by rw [hr,he]
+    simp only [hr,hs,he,le_refl,ite_true]
+  · have hn : ¬ radius C.height_pos T.coe_nonneg U z τ ≤ ρ := by
+      intro hl
+      exact he (hr.symm.trans (le_antisymm hl hlo))
+    simp only [hn,he,ite_false]
+
+include h hz hu hc hb0 in
+lemma eventually_stop_general (D : ℕ → Cutoff z u)
+    (hD : Tendsto (fun n => (D n).angle) atTop (𝓝 0))
+    {t : ℝ≥0} (hzt : z ∈ localizationDomain γ t)
+    (hrt : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
+    ∀ᶠ n : ℕ in atTop, (D n).stopTime κ B sample = t := by
+  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
+  filter_upwards [hD.eventually (gt_mem_nhds hη)] with n hn
+  let C := D n
+  have hh : loewnerRadius g (C.stopTime κ B sample) z = (2*z.im)*Real.exp (-u) := by
+    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · exact hr
+    · have htm := hm (C.stopTime κ B sample)
+        ⟨(C.bounds κ B sample).1,C.stopTime_le_radius κ B sample h hz hu hc hb0 t.coe_nonneg hzt hrt⟩
+      rw [hs] at htm
+      exact False.elim ((not_lt_of_ge htm) hn)
+  exact h.live_radius_injective
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+    (C.bounds κ B sample).1 t.coe_nonneg (C.source_live κ B sample h hz hu hc hb0) hzt
+    (hh.trans hrt.symm)
+
+end SLEExactGauge.RadiusPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.RadiusPast
+open LoewnerCutoff LoewnerExit LoewnerLocalized FiniteTilt
+
+lemma live_radius_witness {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hU : ContinuousOn driver (Ici 0))
+    {z : ℂ} {u : ℝ} (hh : firstLiveRadiusStop γ g z u ≠ (⊤ : ℝ≥0∞)) :
+    ∃ t : ℝ≥0, firstLiveRadiusStop γ g z u = (t : ℝ≥0∞) ∧
+      z ∈ localizationDomain γ t ∧ loewnerRadius g t z = (2*z.im)*Real.exp (-u) := by
+  have he : ∃ t : ℝ≥0, z ∈ localizationDomain γ t ∧
+      loewnerRadius g t z = (2*z.im)*Real.exp (-u) := by
+    by_contra hn
+    have he : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
+        loewnerRadius g t z = (2*z.im)*Real.exp (-u)} = ∅ :=
+      Set.eq_empty_iff_forall_notMem.2 (fun t ht => hn ⟨t,ht⟩)
+    exact hh (by simp only [firstLiveRadiusStop,he,Set.image_empty,sInf_empty])
+  obtain ⟨t,hzt,hrt⟩ := he
+  exact ⟨t,h.firstLiveRadiusStop_eq hU hzt hrt,hzt,hrt⟩
+
+def byTime {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) (T : ℝ≥0) (sample : Ω) : ℝ≥0∞ :=
+  if firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T : ℝ≥0∞) then
+    finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u else 0
+
+def approx {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (n : ℕ) (sample : Ω) : ℝ≥0∞ :=
+  let C := (cutoffs hz u hu n).withHorizon T
+  ENNReal.ofReal (hitData C.height_pos T.coe_nonneg κ
+    (BrownianDriver.path B (Real.sqrt κ) T sample) z ((2*z.im)*Real.exp (-u)) C.angle).2
+
+lemma eventually_approx {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+    (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
+    ∀ᶠ n : ℕ in atTop, approx κ B hz u hu.le T n sample = byTime κ B γ g z u T sample := by
+  classical
+  let D := fun n => (cutoffs hz u hu.le n).withHorizon T
+  have hU : ContinuousOn (fun t : ℝ => Real.sqrt κ*B t.toNNReal sample) (Ici 0) := (continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn
+  by_cases hh : firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T : ℝ≥0∞)
+  · obtain ⟨t,htau,hzt,hrt⟩ := live_radius_witness h hU
+      (ne_top_of_le_ne_top ENNReal.coe_ne_top hh)
+    have htT : t ≤ T := by simpa only [htau,ENNReal.coe_le_coe] using hh
+    have hD : Tendsto (fun n => (D n).angle) atTop (𝓝 0) := shrinkingAngle_tendsto z
+    filter_upwards [eventually_stop_general κ B sample h hz hu hc hb0 D hD hzt hrt] with n hn
+    have hs := raw_data_source κ B sample h hz hu hc hb0 (D n) T hT
+      ((cutoffs hz u hu.le n).withHorizon_gt T).le
+    dsimp only at hs
+    rw [hn,min_eq_right (show (t:ℝ) ≤ T from htT)] at hs
+    dsimp only [approx]
+    unfold byTime
+    rw [ite_eq_left hh, hs.2.2, Prod.snd, hs.1, ite_eq_left hrt,
+      finiteStoppedDensity,htau,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal]
+  · apply Filter.Eventually.of_forall
+    intro n
+    have hs := raw_data_source κ B sample h hz hu hc hb0 (D n) T hT
+      ((cutoffs hz u hu.le n).withHorizon_gt T).le
+    dsimp only at hs
+    let U := BrownianDriver.path B (Real.sqrt κ) T sample
+    let τ := stop (D n).height_pos T.coe_nonneg U z ((2*z.im)*Real.exp (-u)) (D n).angle
+    have hn : loewnerRadius (g sample) τ z ≠ (2*z.im)*Real.exp (-u) := by
+      intro he
+      have ht := stop_bounds (D n).height_pos T.coe_nonneg U z ((2*z.im)*Real.exp (-u)) (D n).angle
+      have heq := h.firstLiveRadiusStop_eq hU (t := ⟨τ,ht.1⟩) hs.2.1 he
+      apply hh
+      rw [heq]
+      exact ENNReal.coe_le_coe.2 (show (⟨τ,ht.1⟩ : ℝ≥0) ≤ T from ht.2)
+    dsimp only [approx]
+    unfold byTime
+    rw [ite_eq_right hh,hs.2.2,Prod.snd,ite_eq_right hn,ENNReal.ofReal_zero]
+
+end SLEExactGauge.RadiusPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.RadiusPast
+open FiniteTilt BrownianPredictable
+local instance : MeasurableSpace C(I, ℝ) := borel _
+local instance : BorelSpace C(I, ℝ) := ⟨rfl⟩
+
+def observed {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (sample : Ω) : ℝ≥0∞ :=
+  ENNReal.ofReal (ContinuousDriver.selectedLimit 0 (fun n => (approx κ B hz u hu T n sample).toReal))
+
+lemma measurable_observed {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
+    Measurable[pastSigma B T] (observed κ B hz u hu T) := by
+  let : MeasurableSpace Ω := pastSigma B T
+  apply ENNReal.measurable_ofReal.comp
+  apply ContinuousDriver.measurable_selectedLimit
+  intro n
+  let C := (cutoffs hz u hu n).withHorizon T
+  exact ENNReal.measurable_toReal.comp (ENNReal.measurable_ofReal.comp
+    (((measurable_hitData C.height_pos hT κ z ((2*z.im)*Real.exp (-u)) C.angle).snd).comp
+      (BrownianDriver.path_measurable_past B (Real.sqrt κ) T)))
+
+lemma aemeasurable_observed {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    {z : ℂ} (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
+    AEMeasurable (observed κ B hz u hu T) P := by
+  apply ENNReal.measurable_ofReal.comp_aemeasurable
+  apply ContinuousDriver.aemeasurable_selectedLimit
+  intro n
+  let C := (cutoffs hz u hu n).withHorizon T
+  exact ENNReal.measurable_toReal.comp_aemeasurable (ENNReal.measurable_ofReal.comp_aemeasurable
+    (((measurable_hitData C.height_pos hT κ z ((2*z.im)*Real.exp (-u)) C.angle).snd).comp_aemeasurable
+      (BrownianDriver.path_aemeasurable hB (Real.sqrt κ) T)))
+
+lemma observed_eq {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+    (T : ℝ≥0) (hT : 0 < (T:ℝ)) :
+    observed κ B hz u hu.le T sample = byTime κ B γ g z u T sample := by
+  have hf : byTime κ B γ g z u T sample ≠ ⊤ := by
+    simp only [byTime,finiteStoppedDensity]
+    split_ifs <;> simp
+  unfold observed
+  rw [ContinuousDriver.selectedLimit_eq (y := (byTime κ B γ g z u T sample).toReal) 0 (by
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_approx κ B γ g f sample h hz hu hc hb0 T hT] with n hn
+    exact congrArg ENNReal.toReal hn.symm), ENNReal.ofReal_toReal hf]
+
+lemma byTime_pos_iff {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    {z : ℂ} (_hz : z ∈ halfPlane) (u : ℝ)
+    (hc : Continuous (fun t => B t sample)) (T : ℝ≥0) :
+    0 < byTime κ B γ g z u T sample ↔ firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T : ℝ≥0∞) := by
+  unfold byTime
+  split_ifs with hh
+  · obtain ⟨t,htau,hzt,hrt⟩ := live_radius_witness h
+      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+      (ne_top_of_le_ne_top ENNReal.coe_ne_top hh)
+    have hs : 0 < loewnerAngleSine κ (fun t => B t sample) (g sample) t z := by
+      simpa only [loewnerAngleSine,Complex.ofReal_mul] using
+        div_pos (h.im_pos t.coe_nonneg hzt) (norm_pos_iff.2 (h.centered_ne_zero t.coe_nonneg hzt))
+    have hr : 0 < loewnerRadius (g sample) t z := h.radius_pos t.coe_nonneg hzt
+    simp only [finiteStoppedDensity,htau,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal]
+    exact iff_of_true (ENNReal.ofReal_pos.2
+      (mul_pos (Real.rpow_pos_of_pos hr _) (Real.rpow_pos_of_pos hs _))) (htau ▸ hh)
+  · exact iff_of_false (lt_irrefl _) hh
+
+end SLEExactGauge.RadiusPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+variable (h : IsLoewnerTraceUsing driver γ g f) (hU : ContinuousOn driver (Ici 0))
+
+include h hU in
+lemma firstLiveRadiusStop_mono {z : ℂ} (hz : z ∈ halfPlane) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v) :
+    firstLiveRadiusStop γ g z u ≤ firstLiveRadiusStop γ g z v := by
+  by_cases hv : firstLiveRadiusStop γ g z v = (⊤ : ℝ≥0∞)
+  · simp only [hv,le_top]
+  obtain ⟨t,htau,hzt,hrt⟩ := RadiusPast.live_radius_witness h hU hv
+  have hρ : (2*z.im)*Real.exp (-v) ≤ (2*z.im)*Real.exp (-u) :=
+    mul_le_mul_of_nonneg_left (Real.exp_le_exp.2 (neg_le_neg huv)) (by have hy : 0 < z.im := hz; positivity)
+  have hρ0 : (2*z.im)*Real.exp (-u) ≤ loewnerRadius g 0 z := by
+    rw [h.radius_initial hz]
+    exact mul_le_of_le_one_right (by have hy : 0 < z.im := hz; positivity)
+      (Real.exp_le_one_iff.2 (neg_nonpos.2 hu))
+  obtain ⟨s,hs,hseq⟩ := intermediate_value_Icc' t.coe_nonneg
+    (h.continuous_radius t.coe_nonneg (hU.mono Icc_subset_Ici_self) hzt) ⟨hrt ▸ hρ,hρ0⟩
+  have hstop := h.firstLiveRadiusStop_eq hU (t := ⟨s,hs.1⟩) (survivingDomain_mono hs.2 hzt) hseq
+  rw [hstop,htau]
+  exact ENNReal.coe_le_coe.2 (show (⟨s,hs.1⟩ : ℝ≥0) ≤ t from hs.2)
+
+include h hU in
+lemma firstLiveRadiusStop_pos {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u) :
+    0 < firstLiveRadiusStop γ g z u := by
+  by_cases hv : firstLiveRadiusStop γ g z u = (⊤ : ℝ≥0∞)
+  · simp only [hv,ENNReal.zero_lt_top]
+  obtain ⟨t,htau,hzt,hrt⟩ := RadiusPast.live_radius_witness h hU hv
+  have ht : 0 < t := by
+    by_contra hn
+    have he : t = 0 := le_antisymm (not_lt.1 hn) zero_le
+    rw [he,NNReal.coe_zero,h.radius_initial hz] at hrt
+    have hr : (2*z.im)*Real.exp (-u) < 2*z.im :=
+      mul_lt_of_lt_one_right (by have hy : 0 < z.im := hz; positivity)
+        (Real.exp_lt_one_iff.2 (neg_neg_of_pos hu))
+    exact (ne_of_lt hr) hrt.symm
+  rw [htau]
+  exact ENNReal.coe_pos.2 ht
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable
+
+def CompletedStoppedPast {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (P : Measure Ω) (τ : Ω → ℝ≥0∞) (A : Set Ω) : Prop :=
+  NullMeasurableSet A P ∧ ∀ t : ℝ≥0, ∃ E : Set Ω, MeasurableSet[pastSigma B t] E ∧
+    E =ᵐ[P] (A ∩ {sample | τ sample ≤ (t : ℝ≥0∞)} : Set Ω)
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable
+namespace CompletedStoppedPast
+variable {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} {P : Measure Ω}
+variable {τ σ : Ω → ℝ≥0∞} {A C : Set Ω}
+
+lemma empty : CompletedStoppedPast B P τ ∅ := by
+  refine ⟨nullMeasurableSet_empty,fun t => ⟨∅,@MeasurableSet.empty Ω (pastSigma B t),?_⟩⟩
+  simp only [empty_inter]
+  exact Filter.EventuallyEq.rfl
+
+lemma union (hA : CompletedStoppedPast B P τ A) (hC : CompletedStoppedPast B P τ C) :
+    CompletedStoppedPast B P τ (A ∪ C) := by
+  refine ⟨hA.1.union hC.1,?_⟩
+  intro t
+  obtain ⟨E,hE,he⟩ := hA.2 t
+  obtain ⟨F,hF,hf⟩ := hC.2 t
+  refine ⟨E ∪ F,hE.union hF,?_⟩
+  filter_upwards [he,hf] with x he hf
+  change (x∈E ∨ x∈F) = ((x∈A ∨ x∈C) ∧ τ x ≤ (t:ℝ≥0∞))
+  change (x∈E) = (x∈A ∧ τ x ≤ (t:ℝ≥0∞)) at he
+  change (x∈F) = (x∈C ∧ τ x ≤ (t:ℝ≥0∞)) at hf
+  rw [he,hf]
+  exact propext (by tauto)
+
+lemma diff (hA : CompletedStoppedPast B P τ A) (hC : CompletedStoppedPast B P τ C) :
+    CompletedStoppedPast B P τ (A \ C) := by
+  refine ⟨hA.1.diff hC.1,?_⟩
+  intro t
+  obtain ⟨E,hE,he⟩ := hA.2 t
+  obtain ⟨F,hF,hf⟩ := hC.2 t
+  refine ⟨E \ F,hE.diff hF,?_⟩
+  filter_upwards [he,hf] with x he hf
+  change (x∈E ∧ x∉F) = ((x∈A ∧ x∉C) ∧ τ x ≤ (t:ℝ≥0∞))
+  change (x∈E) = (x∈A ∧ τ x ≤ (t:ℝ≥0∞)) at he
+  change (x∈F) = (x∈C ∧ τ x ≤ (t:ℝ≥0∞)) at hf
+  rw [he,hf]
+  exact propext (by tauto)
+
+lemma mono (hA : CompletedStoppedPast B P τ A)
+    (hu : CompletedStoppedPast B P σ univ) (hτσ : ∀ᵐ x ∂P, τ x ≤ σ x) :
+    CompletedStoppedPast B P σ A := by
+  refine ⟨hA.1,?_⟩
+  intro t
+  obtain ⟨E,hE,he⟩ := hA.2 t
+  obtain ⟨F,hF,hf⟩ := hu.2 t
+  refine ⟨E ∩ F,hE.inter hF,?_⟩
+  filter_upwards [he,hf,hτσ] with x he hf hle
+  change (x∈E ∧ x∈F) = (x∈A ∧ σ x ≤ (t:ℝ≥0∞))
+  change (x∈E) = (x∈A ∧ τ x ≤ (t:ℝ≥0∞)) at he
+  change (x∈F) = (True ∧ σ x ≤ (t:ℝ≥0∞)) at hf
+  rw [he,hf]
+  apply propext
+  constructor
+  · exact fun hh => ⟨hh.1.1,hh.2.2⟩
+  · exact fun hh => ⟨⟨hh.1,hle.trans hh.2⟩,⟨trivial,hh.2⟩⟩
+
+end CompletedStoppedPast
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable
+
+lemma finiteStoppedDensity_stoppedPast {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u : ℝ) (hu : 0 < u)
+    (S : Set ℝ≥0∞) (hS : MeasurableSet S) :
+    CompletedStoppedPast B P (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u)
+      {sample | finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∈ S} := by
+  refine ⟨?_,?_⟩
+  · exact (finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu).nullMeasurableSet_preimage hS
+  intro T
+  by_cases ht : T = 0
+  · refine ⟨∅,@MeasurableSet.empty Ω (pastSigma B T),?_⟩
+    filter_upwards [hmaps,hB.cont] with sample hm hc
+    have hp := hm.firstLiveRadiusStop_pos
+      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hz hu
+    apply propext
+    simp only [mem_empty_iff_false,mem_inter_iff,mem_ofPred_eq,ht,ENNReal.coe_zero]
+    constructor
+    · exact False.elim
+    · exact fun hh => (not_le_of_gt hp) hh.2
+  have hTp : 0 < (T:ℝ) := by exact_mod_cast pos_iff_ne_zero.2 ht
+  let F := RadiusPast.observed κ B hz u hu.le T
+  have hFm := RadiusPast.measurable_observed κ B hz u hu.le T hTp
+  refine ⟨{sample | 0 < F sample} ∩ F ⁻¹' S,
+    (measurableSet_lt measurable_const hFm).inter (hS.preimage hFm),?_⟩
+  filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hm hc hb0
+  have he := RadiusPast.observed_eq κ B γ g f sample hm hz hu hc hb0 T hTp
+  have hp := RadiusPast.byTime_pos_iff κ B γ g f sample hm hz u hc T
+  apply propext
+  change (0 < F sample ∧ F sample ∈ S) ↔
+    (finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∈ S ∧
+      firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T:ℝ≥0∞))
+  change F sample = _ at he
+  rw [he,hp]
+  by_cases hh : firstLiveRadiusStop (γ sample) (g sample) z u ≤ (T:ℝ≥0∞)
+  · simp only [RadiusPast.byTime,hh,ite_true,true_and,and_true]
+  · simp only [hh,false_and,and_false]
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace CompletedStoppedPast
+variable {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} {P : Measure Ω}
+variable {τ : Ω → ℝ≥0∞}
+
+lemma finsetUnion {ι : Type*} (s : Finset ι) (A : ι → Set Ω)
+    (hA : ∀ i ∈ s, CompletedStoppedPast B P τ (A i)) :
+    CompletedStoppedPast B P τ (⋃ i ∈ s, A i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simpa only [Finset.notMem_empty, iUnion_of_empty, iUnion_empty] using (empty (B:=B) (P:=P) (τ:=τ))
+  | @insert a s ha ih =>
+    simpa only [Finset.set_biUnion_insert] using
+      (hA a (Finset.mem_insert_self _ _)).union (ih (fun i hi => hA i (Finset.mem_insert_of_mem hi)))
+
+end CompletedStoppedPast
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace FiniteFirstEvents
+variable {Ω ι : Type*} [LinearOrder ι]
+
+def first (s : Finset ι) (A : ι → Set Ω) (i : ι) : Set Ω :=
+  A i \ ⋃ j ∈ s.filter (fun j => j < i), A j
+
+lemma first_subset (s : Finset ι) (A : ι → Set Ω) (i : ι) : first s A i ⊆ A i :=
+  sdiff_subset
+
+lemma first_disjoint (s : Finset ι) (A : ι → Set Ω) {i j : ι}
+    (hi : i ∈ s) (hj : j ∈ s) (hij : i ≠ j) : Disjoint (first s A i) (first s A j) := by
+  classical
+  apply Set.disjoint_left.2
+  intro x hxi hxj
+  rcases lt_or_gt_of_ne hij with hij | hji
+  · exact hxj.2 (mem_iUnion.2 ⟨i,mem_iUnion.2 ⟨Finset.mem_filter.2 ⟨hi,hij⟩,hxi.1⟩⟩)
+  · exact hxi.2 (mem_iUnion.2 ⟨j,mem_iUnion.2 ⟨Finset.mem_filter.2 ⟨hj,hji⟩,hxj.1⟩⟩)
+
+lemma union_first (s : Finset ι) (A : ι → Set Ω) :
+    (⋃ i ∈ s, first s A i) = ⋃ i ∈ s, A i := by
+  classical
+  apply Subset.antisymm
+  · exact iUnion₂_mono (fun i _ => first_subset s A i)
+  · intro x hx
+    obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.1 hx
+    let t := s.filter (fun i => x ∈ A i)
+    have ht : t.Nonempty := ⟨i,Finset.mem_filter.2 ⟨hi,hxi⟩⟩
+    let j := t.min' ht
+    have hj := Finset.mem_filter.1 (Finset.min'_mem t ht)
+    apply mem_iUnion₂.2 ⟨j,hj.1,?_⟩
+    refine ⟨hj.2,?_⟩
+    intro hbad
+    obtain ⟨k,hk,hxk⟩ := mem_iUnion₂.1 hbad
+    have hkj := (Finset.mem_filter.1 hk).2
+    have hjk := Finset.min'_le t k (Finset.mem_filter.2 ⟨(Finset.mem_filter.1 hk).1,hxk⟩)
+    exact (not_lt_of_ge hjk) hkj
+
+lemma first_stoppedPast [MeasurableSpace Ω]
+    {B : ℝ≥0 → Ω → ℝ} {P : Measure Ω} {τ : Ω → ℝ≥0∞}
+    (s : Finset ι) (A : ι → Set Ω) (i : ι)
+    (hi : CompletedStoppedPast B P τ (A i))
+    (hj : ∀ j ∈ s, j < i → CompletedStoppedPast B P τ (A j)) :
+    CompletedStoppedPast B P τ (first s A i) := by
+  classical
+  exact hi.diff (CompletedStoppedPast.finsetUnion (s.filter (fun j => j < i)) A
+    (fun j hj' => hj j (Finset.mem_filter.1 hj').1 (Finset.mem_filter.1 hj').2))
+
+end FiniteFirstEvents
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open BrownianPredictable
+
+lemma finiteCenteredMeasure_event_le {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (u v : ℝ) (hu : 0 ≤ u) (huv : u ≤ v)
+    (A : Set Ω)
+    (hA : CompletedStoppedPast B P (fun sample => firstLiveRadiusStop (γ sample) (g sample) z u) A)
+    (K : ℝ≥0∞)
+    (hb : ∀ᵐ sample ∂P.restrict A,
+      finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ≤ K) :
+    finiteCenteredMeasure κ P B γ g z v A ≤
+      (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K * P A := by
+  have he := finiteStoppedDensity_consistent κ hκ B hB γ g f hγ hmaps z hz u v hu huv A hA.1 hA.2
+  have hm0 : ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)) ≠ 0 :=
+    (ENNReal.ofReal_pos.2 (localizationInitialGreen_pos κ hz)).ne'
+  rw [finiteCenteredMeasure,withDensity_apply₀ _ hA.1]
+  simp_rw [ENNReal.div_eq_inv_mul]
+  rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hm0),he,mul_assoc]
+  apply mul_le_mul' le_rfl
+  exact (lintegral_mono_ae hb).trans_eq (by rw [lintegral_const,Measure.restrict_apply_univ])
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteFirstEvents
+
+def densityBelow {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) (K : ℝ≥0∞) : Set Ω :=
+  {sample | finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z u ∈ Ioo 0 K}
+
+lemma finiteCentered_grid_le {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (v : ℝ) (s : Finset ℚ)
+    (hs : ∀ q ∈ s, 0 < (q:ℝ) ∧ (q:ℝ) ≤ v)
+    (a : ℚ → ℝ≥0∞) (K : ℝ≥0∞) (ha : ∀ q ∈ s, a q ≤ K) :
+    finiteCenteredMeasure κ P B γ g z v (⋃ q ∈ s, densityBelow κ B γ g z q (a q)) ≤
+      (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K *
+        P (⋃ q ∈ s, densityBelow κ B γ g z q (a q)) := by
+  classical
+  let A := fun q : ℚ => densityBelow κ B γ g z q (a q)
+  let Q := finiteCenteredMeasure κ P B γ g z v
+  let k := (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K
+  have hpast (q : ℚ) (hq : q ∈ s) : CompletedStoppedPast B P
+      (fun sample => firstLiveRadiusStop (γ sample) (g sample) z q) (A q) :=
+    finiteStoppedDensity_stoppedPast κ hκ B hB γ g f hmaps z hz q (hs q hq).1 (Ioo 0 (a q)) measurableSet_Ioo
+  have huniv (q : ℚ) (hq : q ∈ s) : CompletedStoppedPast B P
+      (fun sample => firstLiveRadiusStop (γ sample) (g sample) z q) univ := by
+    simpa only [mem_univ,ofPred_true] using
+      finiteStoppedDensity_stoppedPast κ hκ B hB γ g f hmaps z hz q (hs q hq).1 univ MeasurableSet.univ
+  have hfirst (q : ℚ) (hq : q∈s) : CompletedStoppedPast B P
+      (fun sample => firstLiveRadiusStop (γ sample) (g sample) z q) (first s A q) := by
+    apply first_stoppedPast s A q (hpast q hq)
+    intro j hj hjq
+    apply (hpast j hj).mono (huniv q hq)
+    filter_upwards [hmaps,hB.cont] with sample hm hc
+    exact hm.firstLiveRadiusStop_mono
+      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hz (hs j hj).1.le
+      (by exact_mod_cast hjq.le)
+  have hle (q : ℚ) (hq : q∈s) : Q (first s A q) ≤ k * P (first s A q) := by
+    apply finiteCenteredMeasure_event_le κ hκ B hB γ g f hγ hmaps z hz q v
+      (hs q hq).1.le (hs q hq).2 (first s A q) (hfirst q hq) K
+    filter_upwards [ae_restrict_mem₀ (hfirst q hq).1] with sample hx
+    exact hx.1.2.le.trans (ha q hq)
+  have hdis : PairwiseDisjoint (↑s : Set ℚ) (first s A) :=
+    fun i hi j hj hij => first_disjoint s A hi hj hij
+  have hPQ : Q ≪ P := withDensity_absolutelyContinuous _ _
+  have hsumQ : Q (⋃ q∈s,first s A q) = ∑ q∈s,Q (first s A q) :=
+    measure_biUnion_finset₀ hdis.aedisjoint (fun q hq => (hfirst q hq).1.mono_ac hPQ)
+  have hsumP : P (⋃ q∈s,first s A q) = ∑ q∈s,P (first s A q) :=
+    measure_biUnion_finset₀ hdis.aedisjoint (fun q hq => (hfirst q hq).1)
+  change Q (⋃ q∈s,A q) ≤ k * P (⋃ q∈s,A q)
+  rw [← union_first s A,hsumQ,hsumP]
+  calc
+    _ ≤ ∑ q∈s,k*P (first s A q) := Finset.sum_le_sum hle
+    _ = _ := by rw [Finset.mul_sum]
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma finiteCentered_countableGrid_le {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (v : ℝ) (J : Set ℚ)
+    (hJ : ∀ q ∈ J, 0 < (q:ℝ) ∧ (q:ℝ) ≤ v)
+    (a : ℚ → ℝ≥0∞) (K : ℝ≥0∞) (ha : ∀ q ∈ J, a q ≤ K) :
+    finiteCenteredMeasure κ P B γ g z v (⋃ q ∈ J, densityBelow κ B γ g z q (a q)) ≤
+      (ENNReal.ofReal ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)))⁻¹ * K *
+        P (⋃ q ∈ J, densityBelow κ B γ g z q (a q)) := by
+  classical
+  let A := fun q : ℚ => densityBelow κ B γ g z q (a q)
+  let F := fun s : Finset ℚ => ⋃ q ∈ s.filter (fun q => q∈J), A q
+  have hm : Monotone F := by
+    intro s t hst
+    exact iUnion₂_mono' (fun q hq => ⟨q, Finset.mem_filter.2
+      ⟨hst (Finset.mem_filter.1 hq).1,(Finset.mem_filter.1 hq).2⟩,Subset.rfl⟩)
+  have he : (⋃ s, F s) = ⋃ q ∈ J, A q := by
+    ext sample
+    simp only [F,mem_iUnion,Finset.mem_filter]
+    constructor
+    · rintro ⟨s,q,⟨hq,hJq⟩,hx⟩
+      exact ⟨q,hJq,hx⟩
+    · rintro ⟨q,hq,hx⟩
+      exact ⟨{q},q,⟨Finset.mem_singleton_self q,hq⟩,hx⟩
+  have hsub (s : Finset ℚ) : F s ⊆ ⋃ q ∈ J,A q := by
+    rw [← he]
+    exact subset_iUnion F s
+  change finiteCenteredMeasure κ P B γ g z v (⋃ q∈J,A q) ≤ _
+  rw [← he,hm.measure_iUnion]
+  apply iSup_le
+  intro s
+  apply (finiteCentered_grid_le κ hκ B hB γ g f hγ hmaps z hz v
+    (s.filter (fun q => q∈J))
+    (fun q hq => hJ q (Finset.mem_filter.1 hq).2) a K
+    (fun q hq => ha q (Finset.mem_filter.1 hq).2)).trans
+  exact mul_le_mul' le_rfl (measure_mono (he ▸ hsub s))
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace FiniteClock
+
+def angleFloor (δ u : ℝ) : ℝ := δ * Real.exp (-u/8)
+
+def threshold (κ : ℝ) (z : ℂ) (δ u : ℝ) : ℝ≥0∞ :=
+  ENNReal.ofReal (((2*z.im)*Real.exp (-u))^(-codimension κ) * (angleFloor δ u)^(8/κ-1))
+
+def blockThreshold (κ : ℝ) (z : ℂ) (δ n : ℝ) : ℝ :=
+  (2*z.im)^(-codimension κ) * Real.exp (codimension κ*(n+1)) *
+    δ^(8/κ-1) * Real.exp (-(8/κ-1)*n/8)
+
+lemma exponent_pos {κ : ℝ} (hκ : 0 < κ ∧ κ < 8) : 0 < 8/κ-1 := by
+  have h : 1 < 8/κ := (lt_div_iff₀ hκ.1).2 (by linarith)
+  linarith
+
+lemma codimension_pos {κ : ℝ} (hκ : 0 < κ ∧ κ < 8) : 0 < codimension κ := by
+  unfold codimension dimension
+  linarith
+
+lemma angleFloor_pos {δ : ℝ} (hδ : 0 < δ) (u : ℝ) : 0 < angleFloor δ u := by
+  unfold angleFloor
+  positivity
+
+lemma threshold_formula (κ : ℝ) {z : ℂ} (hz : z ∈ halfPlane) {δ : ℝ} (hδ : 0 < δ) (u : ℝ) :
+    threshold κ z δ u = ENNReal.ofReal
+      ((2*z.im)^(-codimension κ) * Real.exp (codimension κ*u) *
+        δ^(8/κ-1) * Real.exp (-(8/κ-1)*u/8)) := by
+  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
+  unfold threshold angleFloor
+  rw [Real.mul_rpow hy.le (Real.exp_pos _).le,Real.mul_rpow hδ.le (Real.exp_pos _).le,
+    ← Real.exp_mul, ← Real.exp_mul]
+  congr 1
+  rw [show -u * -codimension κ = codimension κ*u by ring,
+    show (-u/8)*(8/κ-1) = -(8/κ-1)*u/8 by ring]
+  ring
+
+lemma threshold_le_block (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    {z : ℂ} (hz : z ∈ halfPlane) {δ : ℝ} (hδ : 0 < δ)
+    {n u : ℝ} (hnu : n ≤ u) (hun : u ≤ n+1) :
+    threshold κ z δ u ≤ ENNReal.ofReal (blockThreshold κ z δ n) := by
+  rw [threshold_formula κ hz hδ]
+  apply ENNReal.ofReal_le_ofReal
+  unfold blockThreshold
+  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
+  have hp := codimension_pos hκ
+  have ha := exponent_pos hκ
+  apply mul_le_mul
+  · apply mul_le_mul_of_nonneg_right _ (Real.rpow_pos_of_pos hδ _).le
+    exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left hun hp.le))
+      (Real.rpow_pos_of_pos hy _).le
+  · apply Real.exp_le_exp.2
+    nlinarith
+  · positivity
+  · positivity
+
+lemma block_cancel (κ C δ n : ℝ) {z : ℂ} (hz : z ∈ halfPlane) :
+    ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))⁻¹ *
+      blockThreshold κ z δ n *
+      (C*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)) =
+      C * Real.exp (codimension κ) * δ^(8/κ-1) * Real.exp (-(8/κ-1)*n/8) := by
+  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
+  have hs := Real.rpow_pos_of_pos (FiniteTilt.initial_sine_pos hz) (8/κ-1)
+  have hp := Real.rpow_pos_of_pos hy (-codimension κ)
+  have he : Real.exp (codimension κ*(n+1))*Real.exp (-codimension κ*n) = Real.exp (codimension κ) := by
+    rw [← Real.exp_add]
+    congr 1
+    ring
+  unfold blockThreshold
+  field_simp
+  simp only [neg_mul] at he
+  linear_combination δ^(8/κ-1)*C * he
+
+def blockIndex (v : ℝ) (n : ℕ) : Set ℚ :=
+  {q | 0 < (q:ℝ) ∧ (q:ℝ) ≤ v ∧ (n:ℝ) ≤ (q:ℝ) ∧ (q:ℝ) ≤ (n:ℝ)+1}
+
+def blockBad {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (δ v : ℝ) (n : ℕ) : Set Ω :=
+  ⋃ q ∈ blockIndex v n, densityBelow κ B γ g z q (threshold κ z δ q)
+
+lemma blockBad_subset_hit {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (δ v : ℝ) (n : ℕ) :
+    blockBad κ B γ g z δ v n ≤ᵐ[P]
+      {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} := by
+  filter_upwards [hmaps,hB.cont] with sample hm hc hx
+  rcases mem_iUnion.1 hx with ⟨q,hq⟩
+  rcases mem_iUnion.1 hq with ⟨hq,hx⟩
+  have hhit : firstLiveRadiusStop (γ sample) (g sample) z q ≠ (⊤ : ℝ≥0∞) := by
+    intro he
+    have hh := hx.1
+    simp only [finiteStoppedDensity,he,ite_true,lt_self_iff_false] at hh
+  have hh := hm.firstLiveRadiusStop_mono
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) hz
+    (Nat.cast_nonneg n) hq.2.2.1
+  exact ne_top_of_le_ne_top hhit hh
+
+lemma ordinary_natural_radius_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) :
+    ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ,
+      P {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} ≤
+        ENNReal.ofReal (C*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)) := by
+  obtain ⟨C,hC,hbound⟩ := radiusApproachBound κ hκ
+  let S := (z.im/‖z‖)^(8/κ-1)
+  have hS : 0 < S := Real.rpow_pos_of_pos (FiniteTilt.initial_sine_pos hz) _
+  refine ⟨C+S⁻¹,by positivity,?_⟩
+  intro n
+  cases n with
+  | zero =>
+    simp only [Nat.cast_zero,mul_zero,Real.exp_zero,mul_one]
+    apply (prob_le_one).trans
+    apply ENNReal.one_le_ofReal.2
+    change 1 ≤ (C+S⁻¹)*S
+    rw [add_mul,inv_mul_cancel₀ hS.ne']
+    linarith [mul_pos hC hS]
+  | succ n =>
+    simp only [Nat.cast_add,Nat.cast_one]
+    apply (hbound P B hB γ g f hγ hmaps z hz (n+1)
+      (by norm_num)).trans
+    apply ENNReal.ofReal_le_ofReal
+    exact mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (inv_nonneg.2 hS.le))
+        (Real.exp_pos _).le) hS.le
+
+lemma block_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) (C : ℝ) (_hC : 0 < C)
+    (hbound : ∀ n : ℕ,
+      P {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} ≤
+        ENNReal.ofReal (C*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)))
+    (δ : ℝ) (hδ : 0 < δ) (v : ℝ) (n : ℕ) :
+    finiteCenteredMeasure κ P B γ g z v (blockBad κ B γ g z δ v n) ≤
+      ENNReal.ofReal (C * Real.exp (codimension κ) * δ^(8/κ-1) *
+        Real.exp (-(8/κ-1)*n/8)) := by
+  have hm0 := localizationInitialGreen_pos κ hz
+  have hy : 0 < 2*z.im := by have : 0 < z.im := hz; positivity
+  have hK : 0 < blockThreshold κ z δ n := by
+    unfold blockThreshold
+    positivity
+  have hp := (measure_mono_ae (blockBad_subset_hit κ B hB γ g f hmaps z hz δ v n)).trans (hbound n)
+  have hg := finiteCentered_countableGrid_le κ hκ B hB γ g f hγ hmaps z hz v
+    (blockIndex v n) (fun q hq => ⟨hq.1,hq.2.1⟩)
+    (fun q => threshold κ z δ q) (ENNReal.ofReal (blockThreshold κ z δ n))
+    (fun q hq => threshold_le_block κ hκ hz hδ hq.2.2.1 hq.2.2.2)
+  apply hg.trans
+  apply (mul_le_mul' le_rfl hp).trans_eq
+  rw [← ENNReal.ofReal_inv_of_pos hm0, ← ENNReal.ofReal_mul (inv_nonneg.2 hm0.le),
+    ← ENNReal.ofReal_mul (mul_nonneg (inv_nonneg.2 hm0.le) hK.le)]
+  rw [block_cancel κ C δ n hz]
+
+end FiniteClock
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.FiniteClock
+
+def bad {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (δ v : ℝ) : Set Ω :=
+  ⋃ n : ℕ, blockBad κ B γ g z δ v n
+
+lemma mem_bad_of_rational {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (δ v : ℝ)
+    (q : ℚ) (hq : 0 < (q:ℝ)) (hqv : (q:ℝ) ≤ v) {sample : Ω}
+    (hx : sample ∈ densityBelow κ B γ g z q (threshold κ z δ q)) :
+    sample ∈ bad κ B γ g z δ v := by
+  apply mem_iUnion.2 ⟨⌊(q:ℝ)⌋₊,?_⟩
+  apply mem_iUnion.2 ⟨q,?_⟩
+  apply mem_iUnion.2 ⟨⟨hq,hqv,Nat.floor_le hq.le,(Nat.lt_floor_add_one (q:ℝ)).le⟩,hx⟩
+
+lemma geometric_blocks (κ C δ : ℝ) (hκ : 0 < κ ∧ κ < 8) (hC : 0 < C) (hδ : 0 < δ) :
+    (∑' n : ℕ, ENNReal.ofReal (C*Real.exp (codimension κ)*δ^(8/κ-1)*Real.exp (-(8/κ-1)*n/8))) =
+      ENNReal.ofReal ((C*Real.exp (codimension κ)/(1-Real.exp (-(8/κ-1)/8)))*δ^(8/κ-1)) := by
+  have ha := exponent_pos hκ
+  have hr : Real.exp (-(8/κ-1)/8) < 1 := Real.exp_lt_one_iff.2 (by linarith)
+  have he (n : ℕ) : Real.exp (-(8/κ-1)*n/8) = (Real.exp (-(8/κ-1)/8))^n := by
+    rw [← Real.exp_nat_mul]
+    congr 1
+    ring
+  simp_rw [he,ENNReal.ofReal_mul (show 0≤C*Real.exp (codimension κ)*δ^(8/κ-1) by positivity),
+    ENNReal.ofReal_pow (Real.exp_pos _).le]
+  rw [ENNReal.tsum_mul_left,ENNReal.tsum_geometric,← ENNReal.ofReal_one,
+    ← ENNReal.ofReal_sub 1 (Real.exp_pos _).le,← ENNReal.ofReal_inv_of_pos (sub_pos.2 hr),
+    ← ENNReal.ofReal_mul (show 0≤C*Real.exp (codimension κ)*δ^(8/κ-1) by positivity)]
+  congr 1
+  ring
+
+lemma bad_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (z : ℂ) (hz : z ∈ halfPlane) :
+    ∃ D : ℝ, 0 < D ∧ ∀ δ : ℝ, 0 < δ → ∀ v : ℝ,
+      finiteCenteredMeasure κ P B γ g z v (bad κ B γ g z δ v) ≤ ENNReal.ofReal (D*δ^(8/κ-1)) := by
+  obtain ⟨C,hC,hbound⟩ := ordinary_natural_radius_bound κ hκ B hB γ g f hγ hmaps z hz
+  have hr : Real.exp (-(8/κ-1)/8) < 1 := Real.exp_lt_one_iff.2 (by have := exponent_pos hκ; linarith)
+  refine ⟨C*Real.exp (codimension κ)/(1-Real.exp (-(8/κ-1)/8)),by positivity,?_⟩
+  intro δ hδ v
+  apply (measure_iUnion_le _).trans
+  apply (ENNReal.tsum_le_tsum (fun n => block_bound κ hκ B hB γ g f hγ hmaps z hz C hC hbound δ hδ v n)).trans_eq
+  exact geometric_blocks κ C δ hκ hC hδ
+
+lemma exists_small_floor {a D ε : ℝ} (ha : 0 < a) (hD : 0 < D) (hε : 0 < ε) :
+    ∃ δ : ℝ, 0 < δ ∧ D*δ^a < ε := by
+  have hp : 0 < ε/(2*D) := by positivity
+  refine ⟨(ε/(2*D))^(1/a),Real.rpow_pos_of_pos hp _,?_⟩
+  rw [← Real.rpow_mul hp.le,one_div_mul_cancel ha.ne',Real.rpow_one]
+  have he : D*(ε/(2*D)) = ε/2 := by field_simp
+  rw [he]
+  linarith
+
+lemma rational_clock_nonneg {T v : ℝ} {C F : ℝ → ℝ}
+    (hT : 0 < T) (hC : ContinuousOn C (Icc 0 T)) (hCs : StrictMonoOn C (Icc 0 T))
+    (hC0 : C 0 = 0) (hCT : C T = v) (hF : ContinuousOn F (Icc 0 T))
+    (hrat : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ v →
+      ∀ s ∈ Icc 0 T, C s = (q:ℝ) → 0 ≤ F s) :
+    ∀ t ∈ Ioo 0 T, 0 ≤ F t := by
+  intro t ht
+  by_contra hn
+  have hneg : F t < 0 := lt_of_not_ge hn
+  have hFc : ContinuousAt F t := (hF t ⟨ht.1.le,ht.2.le⟩).continuousAt (Icc_mem_nhds ht.1 ht.2)
+  have he : ∀ᶠ s in 𝓝 t, F s < 0 := hFc.tendsto.eventually (gt_mem_nhds hneg)
+  obtain ⟨e,he0,he⟩ := Metric.eventually_nhds_iff.1 he
+  have hlow : max 0 (t-e) < t := max_lt ht.1 (by linarith)
+  obtain ⟨a,ha,hat⟩ := exists_between hlow
+  have ha0 : 0 < a := (le_max_left 0 (t-e)).trans_lt ha
+  have hae : t-e < a := (le_max_right 0 (t-e)).trans_lt ha
+  have haq : a ∈ Icc 0 T := ⟨ha0.le,(hat.trans ht.2).le⟩
+  obtain ⟨q,hqa,hqt⟩ := exists_rat_btwn (hCs haq ⟨ht.1.le,ht.2.le⟩ hat)
+  have hq0 : 0 < (q:ℝ) := by
+    have hc := hCs ⟨le_rfl,hT.le⟩ haq ha0
+    rw [hC0] at hc
+    exact hc.trans hqa
+  have hqv : (q:ℝ) ≤ v := by
+    have hc := hCs.monotoneOn ⟨ht.1.le,ht.2.le⟩ ⟨hT.le,le_rfl⟩ ht.2.le
+    rw [hCT] at hc
+    exact hqt.le.trans hc
+  obtain ⟨s,hs,hsq⟩ := intermediate_value_Icc hat.le
+    (hC.mono (Icc_subset_Icc haq.1 ht.2.le)) ⟨hqa.le,hqt.le⟩
+  have hst : s ∈ Icc 0 T := ⟨ha0.le.trans hs.1,hs.2.trans ht.2.le⟩
+  have hsn : F s < 0 := he (by
+    rw [Real.dist_eq,abs_of_nonpos (sub_nonpos.2 hs.2)]
+    linarith [hs.1])
+  exact (not_lt_of_ge (hrat q hq0 hqv s hst hsq)) hsn
+
+end SLEExactGauge.FiniteClock
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/LoewnerFlow.lean b/lean/OAI/Probability/SLELocalization/LoewnerFlow.lean
index 7d11f9a718bf97c96f45df84daa6e99c10ff6a5f..0a8022d9bdaad1563d91e37a4577a1c6f248bd2a 100644
--- a/lean/OAI/Probability/SLELocalization/LoewnerFlow.lean
+++ b/lean/OAI/Probability/SLELocalization/LoewnerFlow.lean
@@ -1,1440 +1,1440 @@
-import Mathlib
-import OAI.Probability.SLELocalization.StoppedIto
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Loewner Field estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-namespace SLEExactGauge.LoewnerField
-open LoewnerCutoff GreenPhysical
-open scoped unitInterval
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-lemma height_pos (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : 0 < height hε hT U z t :=
-  hε.trans_le (le_max_left _ _)
-
-lemma smooth (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    ContDiff ℝ 3 (field hε hT p a sdev U z t) :=
-  contDiff_green_driver p a _ _ _ sdev (height_pos hε hT U z t)
-
-lemma continuous_field (p a sdev : ℝ) (z : ℂ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => field hε hT p a sdev q.1.1 z q.1.2 q.2) := by
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hL := (continuous_logDerivative hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  apply continuous_green_comp p a
-  · exact (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd)
-  · exact continuous_const.max (Complex.continuous_im.comp hH)
-  · exact hL
-  · intro q
-    exact height_pos hε hT q.1.1 z q.1.2
-
-lemma continuous_drift (p a sdev : ℝ) (z : ℂ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => drift hε hT p a sdev q.1.1 z q.1.2 q.2) := by
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hD := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hV := (continuous_velocity hε).comp (hD.prodMk hH)
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => sdev*q.2))
-  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => height hε hT q.1.1 z q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  have hv := (continuous_variation hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  unfold drift
-  apply (continuous_field hε hT p a sdev z).mul
-  apply Continuous.sub
-  · exact (continuous_const.mul hv).add
-      ((continuous_const.mul (Complex.continuous_im.comp hV)).div hy
-        (fun q => (height_pos hε hT q.1.1 z q.1.2).ne'))
-  · exact (continuous_const.mul ((hx.mul (Complex.continuous_re.comp hV)).add
-      (hy.mul (Complex.continuous_im.comp hV)))).div ((hx.pow 2).add (hy.pow 2))
-        (fun q => (q_pos _ (height_pos hε hT q.1.1 z q.1.2)).ne')
-
-lemma field_upper (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
-    (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    field hε hT p a sdev U z t b ≤ (2*ε/Real.exp ((2/ε^2)*T))^(-p) := by
-  apply green_le_fixed_radius p a _ _ _ _ (height_pos hε hT U z t) ha hp (by positivity)
-  have hL : Real.exp (logDerivative hε hT U z t) ≤ Real.exp ((2/ε^2)*T) :=
-    Real.exp_le_exp.mpr ((le_abs_self _).trans (abs_logDerivative_le hε hT U z t))
-  calc
-    2*ε / Real.exp ((2/ε^2)*T) ≤ 2*ε / Real.exp (logDerivative hε hT U z t) :=
-      div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) hL
-    _ ≤ _ := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left (le_max_left _ _) (by norm_num))
-      (Real.exp_pos _).le
-
-lemma field_third_bound (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
-    (U : C(I, ℝ)) (z : ℂ) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ t b, |iteratedDeriv 3 (field hε hT p a sdev U z t) b| ≤ K := by
-  let M := (2*ε/Real.exp ((2/ε^2)*T))^(-p)
-  refine ⟨|sdev|^3 * (M*(a*(a+2)*(a+4))/ε^3), by positivity, ?_⟩
-  intro t b
-  change |iteratedDeriv 3 (fun b => green p a ((position hε hT U z t).re-sdev*b)
-    (height hε hT U z t) (logDerivative hε hT U z t)) b| ≤ _
-  rw [iteratedDeriv_green_driver p a _ _ _ sdev (height_pos hε hT U z t) 3 b,
-    abs_mul, abs_pow, abs_neg]
-  apply mul_le_mul_of_nonneg_left _ (by positivity)
-  exact third_green_bound hε (le_max_left _ _) ha (field_upper hε hT p a sdev hp ha U z t b)
-
-lemma time_deriv (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) {t : ℝ}
-    (ht : t ∈ Ioo 0 T) (hy : ε < (position hε hT U z t).im) (b : ℝ) :
-    HasDerivAt (fun s => field hε hT p a sdev U z s b) (drift hε hT p a sdev U z t b) t := by
-  have hH := position_deriv hε hT U z ht
-  have hL := hasDerivAt_logDerivative hε hT U z ht
-  have hIm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hH
-  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
-  dsimp only [Function.comp_def, Complex.imCLM_apply, Complex.reCLM_apply] at hIm hRe
-  have hY : HasDerivAt (height hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)).im t := by
-    apply hIm.congr_of_eventuallyEq
-    have hc := hIm.continuousAt
-    filter_upwards [hc.preimage_mem_nhds (Ioi_mem_nhds hy)] with s hs
-    exact max_eq_right hs.le
-  exact hasDerivAt_green_path p a (hRe.sub_const (sdev*b)) hY hL (height_pos hε hT U z t)
-
-lemma second_eq (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    iteratedDeriv 2 (field hε hT p a sdev U z t) b =
-      sdev^2 * (field hε hT p a sdev U z t b *
-        (a*(a+1)*((position hε hT U z t).re-sdev*b)^2 - a*(height hε hT U z t)^2) /
-        (((position hε hT U z t).re-sdev*b)^2+(height hε hT U z t)^2)^2) := by
-  simp only [iteratedDeriv_succ, iteratedDeriv_zero, field]
-  exact deriv_deriv_green_driver p a _ _ _ sdev b (height_pos hε hT U z t)
-
-lemma continuous_second (p a sdev : ℝ) (z : ℂ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv 2 (field hε hT p a sdev q.1.1 z q.1.2) q.2) := by
-  simp only [second_eq]
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => sdev*q.2))
-  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => height hε hT q.1.1 z q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  apply continuous_const.mul
-  apply Continuous.div
-  · exact (continuous_field hε hT p a sdev z).mul
-      ((continuous_const.mul (hx.pow 2)).sub (continuous_const.mul (hy.pow 2)))
-  · exact ((hx.pow 2).add (hy.pow 2)).pow 2
-  · exact fun q => pow_ne_zero 2 (q_pos _ (height_pos hε hT q.1.1 z q.1.2)).ne'
-
-lemma first_eq (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    iteratedDeriv 1 (field hε hT p a sdev U z t) b =
-      sdev*a*((position hε hT U z t).re-sdev*b) /
-        (((position hε hT U z t).re-sdev*b)^2+(height hε hT U z t)^2) *
-        field hε hT p a sdev U z t b := by
-  simp only [iteratedDeriv_succ, iteratedDeriv_zero, field]
-  exact deriv_green_driver p a _ _ _ sdev b (height_pos hε hT U z t)
-
-lemma continuous_first (p a sdev : ℝ) (z : ℂ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv 1 (field hε hT p a sdev q.1.1 z q.1.2) q.2) := by
-  simp only [first_eq]
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => sdev*q.2))
-  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => height hε hT q.1.1 z q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  exact ((continuous_const.mul hx).div ((hx.pow 2).add (hy.pow 2))
-    (fun q => (q_pos _ (height_pos hε hT q.1.1 z q.1.2)).ne')).mul
-      (continuous_field hε hT p a sdev z)
-
-lemma first_bound (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
-    (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    |iteratedDeriv 1 (field hε hT p a sdev U z t) b| ≤
-      |sdev| * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε) := by
-  change |iteratedDeriv 1 (fun b => green p a ((position hε hT U z t).re-sdev*b)
-    (height hε hT U z t) (logDerivative hε hT U z t)) b| ≤ _
-  rw [iteratedDeriv_green_driver p a _ _ _ sdev (height_pos hε hT U z t) 1 b,
-    abs_mul, abs_pow, abs_neg, pow_one]
-  simp only [iteratedDeriv_succ, iteratedDeriv_zero]
-  exact mul_le_mul_of_nonneg_left
-    (first_green_bound hε (le_max_left _ _) ha (field_upper hε hT p a sdev hp ha U z t b))
-    (abs_nonneg _)
-
-lemma second_bound (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
-    (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    |iteratedDeriv 2 (field hε hT p a sdev U z t) b| ≤
-      |sdev|^2 * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*(a*(a+2))/ε^2) := by
-  change |iteratedDeriv 2 (fun b => green p a ((position hε hT U z t).re-sdev*b)
-    (height hε hT U z t) (logDerivative hε hT U z t)) b| ≤ _
-  rw [iteratedDeriv_green_driver p a _ _ _ sdev (height_pos hε hT U z t) 2 b,
-    abs_mul, abs_pow, abs_neg]
-  exact mul_le_mul_of_nonneg_left
-    (second_green_bound hε (le_max_left _ _) ha (field_upper hε hT p a sdev hp ha U z t b))
-    (by positivity)
-
-lemma variation_of_unclipped (U : C(I, ℝ)) (z : ℂ) (t : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im) :
-    variation hε hT U z t =
-      -2 * (((position hε hT U z t).re - extendDriver T U t)^2 - (position hε hT U z t).im^2) /
-        (((position hε hT U z t).re - extendDriver T U t)^2 + (position hε hT U z t).im^2)^2 := by
-  unfold variation
-  rw [clampIm_eq (by simpa using hy)]
-  simp only [Complex.div_re, Complex.neg_re, Complex.re_ofNat, Complex.neg_im,
-    Complex.im_ofNat, neg_zero, zero_mul, map_pow]
-  simp only [pow_two, Complex.mul_re, Complex.sub_re, Complex.ofReal_re,
-    Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.normSq_apply]
-  ring
-
-lemma drift_cancellation (κ : ℝ) (hκ : 0 < κ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ)
-    (hy : ε ≤ (position hε hT U z t).im)
-    (hU : extendDriver T U t = Real.sqrt κ * b) :
-    drift hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t b +
-      (1/2 : ℝ) * iteratedDeriv 2 (field hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) b = 0 := by
-  have hypos := hε.trans_le hy
-  have hh : height hε hT U z t = (position hε hT U z t).im := max_eq_right hy
-  rw [second_eq, Real.sq_sqrt hκ.le]
-  dsimp only [drift]
-  rw [variation_of_unclipped hε hT U z t hy, velocity_eq hy,
-    hh, hU]
-  have hc := green_drift_cancellation κ
-    ((position hε hT U z t).re-Real.sqrt κ*b) (position hε hT U z t).im
-    (field hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t b) hκ.ne' hypos
-  dsimp only at hc
-  simp only [Complex.div_re, Complex.div_im, Complex.re_ofNat, Complex.im_ofNat,
-    Complex.sub_re, Complex.ofReal_re, Complex.sub_im, Complex.ofReal_im,
-    sub_zero, zero_mul, Complex.normSq_apply] at *
-  convert hc using 1
-  ring
-
-end SLEExactGauge.LoewnerField
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-namespace SLEExactGauge.LoewnerField
-open LoewnerCutoff
-open scoped unitInterval
-
-lemma field_prefix_eq {ε S T : ℝ} (hε : 0 < ε) (hS : 0 ≤ S) (hT : 0 ≤ T)
-    (hST : S ≤ T) (U V : C(I, ℝ)) (z : ℂ) (p a sdev : ℝ)
-    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
-    {t : ℝ} (ht : t ∈ Icc 0 S) :
-    field hε hS p a sdev U z t = field hε hT p a sdev V z t := by
-  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
-  have hL := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
-  funext b
-  simp only [field, height, hH, hL]
-
-end SLEExactGauge.LoewnerField
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.ContinuousExit
-
-def extend (p : C(I, ℝ)) (t : ℝ) : ℝ := p (projIcc 0 1 zero_le_one t)
-
-lemma continuous_extend (p : C(I, ℝ)) : Continuous (extend p) :=
-  p.continuous.comp (continuous_projIcc (a := (0:ℝ)) (b := 1) (h := zero_le_one))
-
-lemma extend_coe (p : C(I, ℝ)) (t : I) : extend p t = p t := by
-  simp only [extend, projIcc, max_eq_right t.property.1, min_eq_right t.property.2]
-
-def exitSet (p : C(I, ℝ)) : Set ℝ := insert 1 (Icc 0 1 ∩ {t | extend p t ≤ 0})
-
-def time (p : C(I, ℝ)) : ℝ := sInf (exitSet p)
-
-lemma exitSet_nonempty (p : C(I, ℝ)) : (exitSet p).Nonempty := ⟨1, mem_insert _ _⟩
-
-lemma exitSet_subset (p : C(I, ℝ)) : exitSet p ⊆ Icc 0 1 := by
-  rintro t (rfl | ht)
-  · exact ⟨zero_le_one,le_rfl⟩
-  · exact ht.1
-
-lemma compact_exitSet (p : C(I, ℝ)) : IsCompact (exitSet p) := by
-  exact (isCompact_Icc.inter_right (isClosed_le (continuous_extend p) continuous_const)).insert 1
-
-lemma time_mem (p : C(I, ℝ)) : time p ∈ exitSet p :=
-  (compact_exitSet p).sInf_mem (exitSet_nonempty p)
-
-lemma time_bounds (p : C(I, ℝ)) : time p ∈ Icc 0 1 := exitSet_subset p (time_mem p)
-
-lemma time_le_of_nonpos {p : C(I, ℝ)} {t : ℝ} (ht : t ∈ Icc 0 1) (hp : extend p t ≤ 0) :
-    time p ≤ t := csInf_le ((compact_exitSet p).bddBelow) (mem_insert_of_mem 1 ⟨ht,hp⟩)
-
-lemma positive_before {p : C(I, ℝ)} {t : ℝ} (ht : 0 ≤ t) (hτ : t < time p) :
-    0 < extend p t := by
-  by_contra hn
-  exact (not_le_of_gt hτ) (time_le_of_nonpos ⟨ht,hτ.le.trans (time_bounds p).2⟩ (not_lt.1 hn))
-
-lemma nonpos_at_of_lt_one {p : C(I, ℝ)} (hτ : time p < 1) : extend p (time p) ≤ 0 := by
-  rcases time_mem p with h | h
-  · exact (ne_of_lt hτ h).elim
-  · exact h.2
-
-lemma lt_time_iff {p : C(I, ℝ)} {t : ℝ} (ht : t ∈ Ico 0 1) :
-    t < time p ↔ ∀ s ∈ Icc 0 t, 0 < extend p s := by
-  constructor
-  · intro h s hs
-    exact positive_before hs.1 (hs.2.trans_lt h)
-  · intro h
-    by_contra hn
-    have hτ : time p ≤ t := not_lt.1 hn
-    have hp := h (time p) ⟨(time_bounds p).1,hτ⟩
-    exact (not_lt_of_ge (nonpos_at_of_lt_one (hτ.trans_lt ht.2))) hp
-
-lemma isOpen_positive_on (t : ℝ) :
-    IsOpen {p : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < extend p s} := by
-  let S : Set (C(I, ℝ) × Icc 0 t) := {q | extend q.1 q.2 ≤ 0}
-  have hc : IsClosed S := by
-    apply isClosed_le ?_ continuous_const
-    exact continuous_eval.comp (continuous_fst.prodMk
-      ((continuous_projIcc (a := (0:ℝ)) (b := 1) (h := zero_le_one)).comp (continuous_subtype_val.comp continuous_snd)))
-  have hh := isClosedMap_fst_of_compactSpace S hc
-  have he : {p : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < extend p s} = (Prod.fst '' S)ᶜ := by
-    ext p
-    simp only [mem_ofPred_eq, mem_compl_iff, mem_image, S, not_exists, not_and]
-    constructor
-    · intro hp q hq hqp
-      subst p
-      exact (not_lt_of_ge hq) (hp q.2 q.2.property)
-    · intro hp s hs
-      by_contra hn
-      exact hp (p,⟨s,hs⟩) (not_lt.1 hn) rfl
-  rw [he]
-  exact hh.isOpen_compl
-
-lemma measurable_time : @Measurable C(I, ℝ) ℝ (borel _) _ time := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  apply measurable_of_Ioi
-  intro t
-  by_cases h0 : t < 0
-  · have he : time ⁻¹' Ioi t = univ := by
-      ext p
-      simp only [mem_preimage, mem_Ioi, mem_univ, iff_true]
-      exact h0.trans_le (time_bounds p).1
-    rw [he]
-    exact MeasurableSet.univ
-  by_cases h1 : 1 ≤ t
-  · have he : time ⁻¹' Ioi t = ∅ := by
-      ext p
-      simp only [mem_preimage, mem_Ioi, mem_empty_iff_false, iff_false]
-      exact not_lt_of_ge ((time_bounds p).2.trans h1)
-    rw [he]
-    exact MeasurableSet.empty
-  have he : time ⁻¹' Ioi t = {p : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < extend p s} := by
-    ext p
-    exact lt_time_iff ⟨not_lt.1 h0,not_le.1 h1⟩
-  rw [he]
-  exact (isOpen_positive_on t).measurableSet
-
-lemma time_pos {p : C(I, ℝ)} (h0 : 0 < extend p 0) : 0 < time p := by
-  apply (lt_time_iff (p := p) (t := 0) ⟨le_rfl, zero_lt_one⟩).2
-  intro s hs
-  have hs0 : s = 0 := le_antisymm hs.2 hs.1
-  simpa only [hs0] using h0
-
-lemma nonneg_until {p : C(I, ℝ)} (h0 : 0 < extend p 0) {t : ℝ}
-    (ht : t ∈ Icc 0 (time p)) : 0 ≤ extend p t := by
-  rcases lt_or_eq_of_le ht.2 with hlt | rfl
-  · exact (positive_before ht.1 hlt).le
-  have htend : Tendsto (extend p) (𝓝[<] time p) (𝓝 (extend p (time p))) :=
-    (continuous_extend p).continuousAt.tendsto.mono_left nhdsWithin_le_nhds
-  apply ge_of_tendsto htend
-  filter_upwards [Ioo_mem_nhdsLT (time_pos h0)] with s hs
-  exact (positive_before hs.1.le hs.2).le
-
-lemma eq_zero_at_of_lt_one {p : C(I, ℝ)} (h0 : 0 < extend p 0)
-    (hτ : time p < 1) : extend p (time p) = 0 :=
-  le_antisymm (nonpos_at_of_lt_one hτ) (nonneg_until h0 ⟨(time_bounds p).1, le_rfl⟩)
-
-end SLEExactGauge.ContinuousExit
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerExit
-open LoewnerCutoff
-open scoped unitInterval
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-def radius (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  2*(position hε hT U z t).im / Real.exp (logDerivative hε hT U z t)
-
-def sine (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  (position hε hT U z t).im /
-    ‖clampIm ε (position hε hT U z t - (extendDriver T U t : ℂ))‖
-
-def deficit (U : C(I, ℝ)) (z : ℂ) (ρ δ t : ℝ) : ℝ :=
-  min (radius hε hT U z t - ρ)
-    (min ((position hε hT U z t).im-ε) (sine hε hT U z t-δ))
-
-lemma continuous_radius (z : ℂ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => radius hε hT q.1 z q.2) := by
-  exact (continuous_const.mul (Complex.continuous_im.comp (continuous_position hε hT z))).div
-    (Real.continuous_exp.comp (continuous_logDerivative hε hT z)) (fun _ => Real.exp_ne_zero _)
-
-lemma continuous_sine (z : ℂ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => sine hε hT q.1 z q.2) := by
-  exact (Complex.continuous_im.comp (continuous_position hε hT z)).div
-    (((continuous_clampIm ε).comp ((continuous_position hε hT z).sub
-      (Complex.continuous_ofReal.comp (continuous_joint_extendDriver T)))).norm)
-    (fun _ => norm_ne_zero_iff.2 (clampIm_ne_zero hε _))
-
-lemma continuous_deficit (z : ℂ) (ρ δ : ℝ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => deficit hε hT q.1 z ρ δ q.2) :=
-  ((continuous_radius hε hT z).sub continuous_const).min
-    (((Complex.continuous_im.comp (continuous_position hε hT z)).sub continuous_const).min
-      ((continuous_sine hε hT z).sub continuous_const))
-
-lemma deficit_pos_iff (U : C(I, ℝ)) (z : ℂ) (ρ δ t : ℝ) :
-    0 < deficit hε hT U z ρ δ t ↔
-      ρ < radius hε hT U z t ∧ ε < (position hε hT U z t).im ∧ δ < sine hε hT U z t := by
-  simp only [deficit, lt_min_iff, sub_pos]
-
-lemma deficit_nonneg_iff (U : C(I, ℝ)) (z : ℂ) (ρ δ t : ℝ) :
-    0 ≤ deficit hε hT U z ρ δ t ↔
-      ρ ≤ radius hε hT U z t ∧ ε ≤ (position hε hT U z t).im ∧ δ ≤ sine hε hT U z t := by
-  simp only [deficit, le_min_iff, sub_nonneg]
-
-def exitPath (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) : C(I, ℝ) := by
-  refine ⟨fun q => deficit hε hT U z ρ δ (T*q), ?_⟩
-  have hp : Continuous (fun q : I => (U,T*q)) :=
-    continuous_const.prodMk (continuous_const.mul continuous_subtype_val)
-  have hc := (continuous_deficit hε hT z ρ δ).comp hp
-  dsimp only [Function.comp_def] at hc
-  exact hc
-
-def stop (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) : ℝ :=
-  T * ContinuousExit.time (exitPath hε hT U z ρ δ)
-
-lemma stop_bounds (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) :
-    stop hε hT U z ρ δ ∈ Icc 0 T := by
-  have hh := ContinuousExit.time_bounds (exitPath hε hT U z ρ δ)
-  exact ⟨mul_nonneg hT hh.1, by simpa only [stop, mul_one] using mul_le_mul_of_nonneg_left hh.2 hT⟩
-
-lemma exitPath_extend (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) {t : ℝ} (ht : t ∈ Icc 0 1) :
-    ContinuousExit.extend (exitPath hε hT U z ρ δ) t = deficit hε hT U z ρ δ (T*t) := by
-  simpa only [exitPath, ContinuousMap.coe_mk] using ContinuousExit.extend_coe
-    (exitPath hε hT U z ρ δ) (⟨t,ht⟩ : I)
-
-lemma exitPath_extend_div (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    ContinuousExit.extend (exitPath hε hT U z ρ δ) (t/T) = deficit hε hT U z ρ δ t := by
-  rw [exitPath_extend hε hT U z ρ δ
-    ⟨div_nonneg ht.1 hT, (div_le_one hTp).2 ht.2⟩, mul_div_cancel₀ _ hTp.ne']
-
-lemma positive_before (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    {t : ℝ} (ht : 0 ≤ t) (htτ : t < stop hε hT U z ρ δ) :
-    0 < deficit hε hT U z ρ δ t := by
-  rw [← exitPath_extend_div hε hT hTp U z ρ δ ⟨ht,htτ.le.trans (stop_bounds hε hT U z ρ δ).2⟩]
-  apply ContinuousExit.positive_before (div_nonneg ht hT)
-  exact (div_lt_iff₀ hTp).2 (by simpa only [stop, mul_comm] using htτ)
-
-lemma stop_pos (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (h0 : 0 < deficit hε hT U z ρ δ 0) : 0 < stop hε hT U z ρ δ := by
-  apply mul_pos hTp (ContinuousExit.time_pos _)
-  simpa only [exitPath_extend hε hT U z ρ δ ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
-
-lemma nonneg_until (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (h0 : 0 < deficit hε hT U z ρ δ 0) {t : ℝ}
-    (ht : t ∈ Icc 0 (stop hε hT U z ρ δ)) : 0 ≤ deficit hε hT U z ρ δ t := by
-  rw [← exitPath_extend_div hε hT hTp U z ρ δ ⟨ht.1,ht.2.trans (stop_bounds hε hT U z ρ δ).2⟩]
-  apply ContinuousExit.nonneg_until
-  · simpa only [exitPath_extend hε hT U z ρ δ ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
-  · exact ⟨div_nonneg ht.1 hT, (div_le_iff₀ hTp).2 (by simpa only [stop, mul_comm] using ht.2)⟩
-
-lemma nonpos_at_of_lt (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hτ : stop hε hT U z ρ δ < T) :
-    deficit hε hT U z ρ δ (stop hε hT U z ρ δ) ≤ 0 := by
-  have hs : ContinuousExit.time (exitPath hε hT U z ρ δ) < 1 := by
-    apply (mul_lt_mul_iff_right₀ hTp).mp
-    simpa only [mul_one, stop] using hτ
-  exact (exitPath_extend hε hT U z ρ δ (ContinuousExit.time_bounds _)) ▸
-    ContinuousExit.nonpos_at_of_lt_one hs
-
-lemma lt_stop_iff (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    {t : ℝ} (ht : t ∈ Ico 0 T) :
-    t < stop hε hT U z ρ δ ↔ ∀ s ∈ Icc 0 t, 0 < deficit hε hT U z ρ δ s := by
-  constructor
-  · intro hh s hs
-    exact positive_before hε hT hTp U z ρ δ hs.1 (hs.2.trans_lt hh)
-  · intro hh
-    change t < T * ContinuousExit.time (exitPath hε hT U z ρ δ)
-    rw [mul_comm]
-    apply (div_lt_iff₀ hTp).1
-    apply (ContinuousExit.lt_time_iff (p := exitPath hε hT U z ρ δ)
-      ⟨div_nonneg ht.1 hT, (div_lt_one hTp).2 ht.2⟩).2
-    intro s hs
-    rw [exitPath_extend hε hT U z ρ δ
-      ⟨hs.1, hs.2.trans ((div_le_one hTp).2 ht.2.le)⟩]
-    apply hh
-    exact ⟨mul_nonneg hT hs.1, (mul_le_mul_of_nonneg_left hs.2 hT).trans_eq
-      (mul_div_cancel₀ _ hTp.ne')⟩
-
-lemma isOpen_positive_on (z : ℂ) (ρ δ t : ℝ) :
-    IsOpen {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U z ρ δ s} := by
-  let S : Set (C(I, ℝ) × Icc 0 t) := {q | deficit hε hT q.1 z ρ δ q.2 ≤ 0}
-  have hmap : Continuous (fun q : C(I, ℝ) × Icc 0 t => (q.1,(q.2:ℝ))) :=
-    continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)
-  have hd := (continuous_deficit hε hT z ρ δ).comp hmap
-  dsimp only [Function.comp_def] at hd
-  have hc : IsClosed S := isClosed_le hd continuous_const
-  have hh := isClosedMap_fst_of_compactSpace S hc
-  have he : {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U z ρ δ s} = (Prod.fst '' S)ᶜ := by
-    ext U
-    simp only [mem_ofPred_eq, mem_compl_iff, mem_image, S, not_exists, not_and]
-    constructor
-    · intro hp q hq hqp
-      subst U
-      exact (not_lt_of_ge hq) (hp q.2 q.2.property)
-    · intro hp s hs
-      by_contra hn
-      exact hp (U,⟨s,hs⟩) (not_lt.1 hn) rfl
-  rw [he]
-  exact hh.isOpen_compl
-
-end SLEExactGauge.LoewnerExit
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerExit
-open LoewnerCutoff
-open scoped unitInterval
-
-lemma deficit_prefix_eq {ε S T : ℝ} (hε : 0 < ε) (hS : 0 ≤ S) (hT : 0 ≤ T)
-    (hST : S ≤ T) (U V : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S)) :
-    EqOn (deficit hε hS U z ρ δ) (deficit hε hT V z ρ δ) (Icc 0 S) := by
-  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV
-  have hL := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V z hUV
-  intro t ht
-  simp only [deficit, radius, sine, hH ht, hL ht, hUV ht]
-
-end SLEExactGauge.LoewnerExit
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerField LoewnerExit BrownianField
-open scoped unitInterval
-
-lemma regular {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (p a sdev : ℝ)
-    (hp : 0 ≤ p) (ha : 0 ≤ a) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
-    (b : ℝ → ℝ) (hb : Continuous b) :
-    FieldRegular (field hε hT.le p a sdev U z) (drift hε hT.le p a sdev U z) b
-      (stop hε hT.le U z ρ δ) := by
-  constructor
-  · exact smooth hε hT.le p a sdev U z
-  · exact field_third_bound hε hT.le p a sdev hp ha U z
-  · intro x
-    have hm : Continuous (fun t : ℝ => ((U,t),x)) :=
-      (continuous_const.prodMk continuous_id).prodMk continuous_const
-    have hc := (continuous_field hε hT.le p a sdev z).comp hm
-    exact hc.continuousOn
-  · intro t ht x
-    have htT : t ∈ Ioo 0 T := ⟨ht.1,ht.2.trans_le (stop_bounds hε hT.le U z ρ δ).2⟩
-    have hh := (deficit_pos_iff hε hT.le U z ρ δ t).1
-      (positive_before hε hT.le hT U z ρ δ ht.1.le ht.2)
-    exact time_deriv hε hT.le p a sdev U z htT hh.2.1 x
-  · have hm : Continuous (fun q : ℝ × ℝ => ((U,q.1),b q.2)) :=
-      (continuous_const.prodMk continuous_fst).prodMk (hb.comp continuous_snd)
-    have hc := (continuous_drift hε hT.le p a sdev z).comp hm
-    exact hc.continuousOn
-  · have hm : Continuous (fun t : ℝ => ((U,t),b t)) :=
-      (continuous_const.prodMk continuous_id).prodMk hb
-    have hc := (continuous_second hε hT.le p a sdev z).comp hm
-    exact hc.continuousOn
-
-lemma integral_drift_cancel {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (κ : ℝ) (hκ : 0 < κ)
-    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) (b : ℝ → ℝ)
-    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ * b s) :
-    (∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ), drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) +
-      (1/2 : ℝ)*∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ),
-        iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) = 0 := by
-  have he : (∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ), drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) =
-      ∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ),
-        -(1/2 : ℝ)*iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) := by
-    apply intervalIntegral.integral_congr_Ioo_of_le (stop_bounds hε hT.le U z ρ δ).1
-    intro t ht
-    have htT : t ∈ Icc 0 T := ⟨ht.1.le,ht.2.le.trans (stop_bounds hε hT.le U z ρ δ).2⟩
-    have hh := (deficit_pos_iff hε hT.le U z ρ δ t).1
-      (positive_before hε hT.le hT U z ρ δ ht.1.le ht.2)
-    have hc := drift_cancellation hε hT.le κ hκ U z t (b t) hh.2.1.le (hU t htT)
-    linarith
-  rw [he, intervalIntegral.integral_const_mul]
-  ring
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.ContinuousDriver
-
-lemma aemeasurable_selectedLimit {Ω Y : Type*} [MeasurableSpace Ω]
-    [MetricSpace Y] [CompleteSpace Y] [MeasurableSpace Y] [BorelSpace Y]
-    [SecondCountableTopology Y] {P : Measure Ω}
-    (y₀ : Y) (a : ℕ → Ω → Y) (ha : ∀ n, AEMeasurable (a n) P) :
-    AEMeasurable (fun sample => selectedLimit y₀ (fun n => a n sample)) P := by
-  have hm : Measurable (fun a : ℕ → Y => selectedLimit y₀ a) :=
-    measurable_selectedLimit y₀ (fun (n : ℕ) (a : ℕ → Y) => a n) (fun n => measurable_pi_apply n)
-  exact hm.comp_aemeasurable (AEMeasurable.of_eval ha)
-
-lemma aemeasurable_path {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (x : Ω → I → ℝ) (hx : ∀ t, AEMeasurable (fun sample => x sample t) P) :
-    @AEMeasurable Ω C(I, ℝ) (borel C(I, ℝ)) _ (fun sample => path (x sample)) P := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  apply aemeasurable_selectedLimit
-  intro n
-  apply Finset.aemeasurable_fun_sum
-  intro k _
-  exact (hx (bernstein.z k)).smul aemeasurable_const
-
-end SLEExactGauge.ContinuousDriver
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.BrownianDriver
-open BrownianPredictable LoewnerCutoff
-open scoped unitInterval
-
-def path {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) (sample : Ω) : C(I, ℝ) :=
-  ContinuousDriver.path (fun q : I => sdev * B ((T:ℝ)*q).toNNReal sample)
-
-lemma path_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) :
-    @Measurable Ω C(I, ℝ) (pastSigma B T) (borel C(I, ℝ)) (path B sdev T) := by
-  let : MeasurableSpace Ω := pastSigma B T
-  apply ContinuousDriver.measurable_path
-  intro q
-  apply Measurable.const_mul
-  apply past_measurable_eval
-  have hh : (T:ℝ)*q ≤ T := by
-    simpa only [mul_one] using mul_le_mul_of_nonneg_left q.property.2 T.coe_nonneg
-  exact (Real.toNNReal_le_toNNReal hh).trans_eq Real.toNNReal_coe
-
-lemma path_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (sdev : ℝ) (T : ℝ≥0) :
-    @AEMeasurable Ω C(I, ℝ) (borel C(I, ℝ)) _ (path B sdev T) P := by
-  apply ContinuousDriver.aemeasurable_path
-  intro q
-  exact ((hB.toIsPreBrownianReal.hasLaw_eval _).aemeasurable).const_mul sdev
-
-lemma path_eval {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) (q : I) :
-    path B sdev T sample q = sdev * B ((T:ℝ)*q).toNNReal sample := by
-  apply ContinuousDriver.path_eval_of_continuous
-  exact continuous_const.mul (hc.comp (continuous_real_toNNReal.comp
-    (continuous_const.mul continuous_subtype_val)))
-
-lemma extend_eq {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) {t : ℝ} (ht : t ∈ Icc 0 (T:ℝ)) :
-    extendDriver T (path B sdev T sample) t = sdev * B t.toNNReal sample := by
-  rw [extendDriver, path_eval B sdev T sample hc]
-  by_cases hT : (T:ℝ) = 0
-  · have ht0 : t = 0 := le_antisymm (hT ▸ ht.2) ht.1
-    simp only [hT, ht0, zero_mul]
-  · have hTp : 0 < (T:ℝ) := lt_of_le_of_ne T.coe_nonneg (Ne.symm hT)
-    rw [projIcc_of_mem zero_le_one ⟨div_nonneg ht.1 T.coe_nonneg, (div_le_one hTp).2 ht.2⟩,
-      mul_div_cancel₀ _ hT]
-
-lemma prefix_eq {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (S T : ℝ≥0) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) (hST : S ≤ T) :
-    EqOn (extendDriver S (path B sdev S sample)) (extendDriver T (path B sdev T sample)) (Icc 0 (S:ℝ)) := by
-  intro t ht
-  rw [extend_eq B sdev S sample hc ht, extend_eq B sdev T sample hc ⟨ht.1,ht.2.trans hST⟩]
-
-end SLEExactGauge.BrownianDriver
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerField LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-open scoped unitInterval
-variable {ε : ℝ} (hε : 0 < ε)
-
-def coeffAt (k : ℕ) (s : ℝ≥0) (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ)
-    (q : C(I, ℝ) × ℝ) : ℝ := by
-  classical
-  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
-    iteratedDeriv k (field hε s.coe_nonneg p a sdev q.1 z s) q.2 else 0
-
-lemma coeffAt_measurable (k : ℕ) (hk : k = 1 ∨ k = 2) (s : ℝ≥0)
-    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s p a sdev z ρ δ) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
-      iteratedDeriv k (field hε s.coe_nonneg p a sdev q.1 z s) q.2) := by
-    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
-      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
-    rcases hk with rfl | rfl
-    · have hc := (continuous_first hε s.coe_nonneg p a sdev z).comp hp
-      dsimp only [Function.comp_def] at hc
-      exact hc
-    · have hc := (continuous_second hε s.coe_nonneg p a sdev z).comp hp
-      dsimp only [Function.comp_def] at hc
-      exact hc
-  apply hc.measurable.ite _ measurable_const
-  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
-
-def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω) : ℝ :=
-  if s < T then coeffAt hε k s p a sdev z ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
-
-lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    Measurable[pastSigma B s] (rawCoeff hε k B T s p a sdev z ρ δ) := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  let : MeasurableSpace Ω := pastSigma B s
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s p a sdev z ρ δ).comp
-      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
-  · exact measurable_const
-
-lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
-    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
-    AEMeasurable (rawCoeff hε k B T s p a sdev z ρ δ) P := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s p a sdev z ρ δ).comp_aemeasurable
-      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
-  · exact aemeasurable_const
-
-lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    rawCoeff hε k B T s p a sdev z ρ δ sample =
-      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
-        iteratedDeriv k (field hε T.coe_nonneg p a sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
-  classical
-  by_cases hsT : s < T
-  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
-    have hf := field_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z p a sdev he
-      ⟨s.coe_nonneg, le_rfl⟩
-    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
-    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
-        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
-      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
-      exact forall₂_congr (fun t ht => by rw [hd ht])
-    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
-  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
-      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
-    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
-
-lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (n i : ℕ) :
-    rawCoeff hε k B T (dyadicTime T n i) p a sdev z ρ δ =ᵐ[P]
-      stoppedCoeff k B T
-        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
-        (fun sample => field hε T.coe_nonneg p a sdev (BrownianDriver.path B sdev T sample) z) n i := by
-  filter_upwards [hB.cont] with sample hc
-  rw [rawCoeff_eq hε k B T _ hT p a sdev z ρ δ sample hc]
-  simp only [stoppedCoeff, coe_dyadicTime]
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerField LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-open scoped unitInterval
-
-lemma coeff_uniform_bound {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
-    (z : ℂ) : ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t b : ℝ),
-      |iteratedDeriv k (field hε hT p a sdev U z t) b| ≤ K := by
-  rcases hk with rfl | rfl
-  · refine ⟨max 0 (|sdev| * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε)), le_max_left _ _, ?_⟩
-    intro U t b
-    exact (first_bound hε hT p a sdev hp ha U z t b).trans (le_max_right _ _)
-  · refine ⟨max 0 (|sdev|^2 * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*(a*(a+2))/ε^2)), le_max_left _ _, ?_⟩
-    intro U t b
-    exact (second_bound hε hT p a sdev hp ha U z t b).trans (le_max_right _ _)
-
-theorem localized_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (ρ δ : ℝ) :
-    let U := BrownianDriver.path B (Real.sqrt κ) T
-    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-    let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-    Integrable (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) P ∧
-      (∫ sample, F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) ∂P) = 0 := by
-  dsimp only
-  let U := BrownianDriver.path B (Real.sqrt κ) T
-  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by
-    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
-    linarith
-  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
-      (hc.comp continuous_real_toNNReal)
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
-    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) z ρ δ)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
-        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
-    apply Filter.Eventually.of_forall
-    intro sample
-    unfold stoppedCoeff
-    split_ifs
-    · exact hb _ _ _
-    · simpa only [abs_zero] using hK
-  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
-  let C := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 1 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
-  have hCe (n i : ℕ) : C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 2 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
-    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ z ρ δ
-  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) :=
-    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ z ρ δ
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₁
-    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hb
-  have hC2 (n i : ℕ) : MemLp (C n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₂
-    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |C n i sample| ≤ K₂
-    rw [he]
-    exact hb
-  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A C hAe hCe hCm hC2
-    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
-      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
-    filter_upwards [hB.cont] with sample hc
-    have hcanc := integral_drift_cancel hε hT κ hκ.1 (U sample) z ρ δ
-      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
-    dsimp only [F,V,τ]
-    linarith
-  exact ⟨hi.1.congr he, (integral_congr_ae he).symm.trans hi.2⟩
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerField LoewnerExit GreenPhysical
-open scoped unitInterval
-
-lemma field_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (hy : ε ≤ z.im) :
-    field hε hT p a sdev U z 0 0 = green p a z.re z.im 0 := by
-  simp only [field, height, position_initial, logDerivative_initial, mul_zero, sub_zero,
-    max_eq_right hy]
-
-theorem localized_expectation {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hy : ε ≤ z.im) (ρ δ : ℝ) :
-    let U := BrownianDriver.path B (Real.sqrt κ) T
-    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-    let M := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-      (τ sample) (B (τ sample).toNNReal sample)
-    Integrable M P ∧ (∫ sample, M sample ∂P) = green (1-κ/8) (8/κ-1) z.re z.im 0 := by
-  dsimp only
-  let U := BrownianDriver.path B (Real.sqrt κ) T
-  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
-  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  let M := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample)
-  let G := green (1-κ/8) (8/κ-1) z.re z.im 0
-  have hi := localized_centered hB κ hκ hε T hT z ρ δ
-  have he : (fun sample => M sample - F sample 0 (B 0 sample)) =ᵐ[P]
-      (fun sample => M sample - G) := by
-    filter_upwards [hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hb
-    rw [hb]
-    congr 1
-    exact field_initial hε T.coe_nonneg _ _ _ _ z hy
-  have hd : Integrable (fun sample => M sample-G) P := hi.1.congr he
-  have hv : (∫ sample, M sample-G ∂P) = 0 := (integral_congr_ae he).symm.trans hi.2
-  have hm : Integrable M P := by
-    exact (hd.add (integrable_const G)).congr
-      (Filter.Eventually.of_forall (fun sample => sub_add_cancel (M sample) G))
-  refine ⟨hm, ?_⟩
-  have hh := integral_sub hm (integrable_const (μ := P) G)
-  simp only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul] at hh
-  rw [hh] at hv
-  change (∫ sample, M sample ∂P) = G
-  linarith
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology unitInterval
-namespace SLEExactGauge.IsLoewnerTraceUsing
-open LoewnerCutoff
-
-lemma cutoff_solution_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt H (velocity ε (driver t) (H t)) (Icc 0 T) t)
-    (hheight : ∀ t ∈ Icc 0 T, ε ≤ (H t).im) :
-    H 0 ∈ localizationDomain γ T ∧ EqOn (fun t => g t (H 0)) H (Icc 0 T) := by
-  have hp : H T ∈ halfPlane := hε.trans_le (hheight T ⟨hT, le_rfl⟩)
-  let z' := f T (H T)
-  have hz' : z' ∈ localizationDomain γ T := h.2.2.2.2.2.2.2.2.1 T hT hp
-  have hfinal : g T z' = H T := h.2.2.2.2.2.2.2.2.2.2.1 T hT _ hp
-  have hht (t : ℝ) (ht : t ∈ Icc 0 T) : ε ≤ (g t z').im := by
-    have ha := h.height_antitone hT hz' ht ⟨hT, le_rfl⟩ ht.2
-    dsimp only at ha
-    rw [hfinal] at ha
-    exact (hheight T ⟨hT, le_rfl⟩).trans ha
-  have hd (t : ℝ) (ht : t ∈ Ioc 0 T) :
-      HasDerivWithinAt (fun s => g s z')
-        (velocity ε (driver t) (g t z')) (Iic t) t := by
-    rw [velocity_eq (hht t ⟨ht.1.le, ht.2⟩)]
-    exact (h.2.2.2.2.2.2.2.2.2.2.2.2.1 t ht.1 z'
-      (survivingDomain_mono ht.2 hz')).hasDerivWithinAt
-  have he : EqOn (fun t => g t z') H (Icc 0 T) :=
-    ODE_solution_unique_of_mem_Icc_left
-      (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (driver t)).lipschitzOnWith)
-      (h.continuous_eval hT hz') hd (fun _ _ => mem_univ _)
-      (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_left hH ht)
-      (fun _ _ => mem_univ _) hfinal
-  have hz0 : z' = H 0 := by
-    rw [← h.initial z' hz'.1.1]
-    exact he ⟨le_rfl, hT⟩
-  exact ⟨hz0 ▸ hz', hz0 ▸ he⟩
-
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma hasDerivWithinAt_eval (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    HasDerivWithinAt (fun s => g s z)
-      (2 / (g t z - (driver t : ℂ))) (Icc 0 T) t := by
-  have hc := h.continuous_velocity hT hU hz
-  have hi : IntervalIntegrable (fun s => 2 / (g s z - (driver s : ℂ))) volume 0 t :=
-    (hc.intervalIntegrable_of_Icc hT).mono_set (by
-      rw [uIcc_of_le ht.1, uIcc_of_le hT]
-      exact Icc_subset_Icc le_rfl ht.2)
-  have : Fact (t ∈ Icc 0 T) := ⟨ht⟩
-  have hd := intervalIntegral.integral_hasDerivWithinAt_right (s := Icc 0 T) (t := Icc 0 T)
-    hi (hc.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc t) (hc t ht)
-  have he : EqOn (fun s => g 0 z + ∫ v in (0 : ℝ)..s, 2 / (g v z - (driver v : ℂ)))
-      (fun s => g s z) (Icc 0 T) := by
-    intro s hs
-    dsimp only
-    rw [h.integral_velocity hT hU hz ⟨le_rfl, hT⟩ hs hs.1]
-    abel
-  exact (hd.const_add (g 0 z)).congr (fun s hs => (he hs).symm) (he ht).symm
-
-lemma eq_cutoff_solution (h : IsLoewnerTraceUsing driver γ g f)
-    {T ε : ℝ} (hT : 0 ≤ T) (hε : 0 < ε) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) (hheight : ε ≤ (g T z).im)
-    {H : ℝ → ℂ} (hH0 : H 0 = z)
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
-      (velocity ε (driver t) (H t)) (Icc 0 T) t) :
-    EqOn (fun t => g t z) H (Icc 0 T) := by
-  apply solution_eq hε ?_ hH ((h.initial z hz.1.1).trans hH0.symm)
-  intro t ht
-  rw [velocity_eq (hheight.trans (h.height_antitone hT hz ht ⟨hT, le_rfl⟩ ht.2))]
-  exact h.hasDerivWithinAt_eval hT hU hz ht
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.StoppedIto
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Loewner Field estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+namespace SLEExactGauge.LoewnerField
+open LoewnerCutoff GreenPhysical
+open scoped unitInterval
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+lemma height_pos (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : 0 < height hε hT U z t :=
+  hε.trans_le (le_max_left _ _)
+
+lemma smooth (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    ContDiff ℝ 3 (field hε hT p a sdev U z t) :=
+  contDiff_green_driver p a _ _ _ sdev (height_pos hε hT U z t)
+
+lemma continuous_field (p a sdev : ℝ) (z : ℂ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => field hε hT p a sdev q.1.1 z q.1.2 q.2) := by
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hL := (continuous_logDerivative hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  apply continuous_green_comp p a
+  · exact (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd)
+  · exact continuous_const.max (Complex.continuous_im.comp hH)
+  · exact hL
+  · intro q
+    exact height_pos hε hT q.1.1 z q.1.2
+
+lemma continuous_drift (p a sdev : ℝ) (z : ℂ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => drift hε hT p a sdev q.1.1 z q.1.2 q.2) := by
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hD := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hV := (continuous_velocity hε).comp (hD.prodMk hH)
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => sdev*q.2))
+  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => height hε hT q.1.1 z q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  have hv := (continuous_variation hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  unfold drift
+  apply (continuous_field hε hT p a sdev z).mul
+  apply Continuous.sub
+  · exact (continuous_const.mul hv).add
+      ((continuous_const.mul (Complex.continuous_im.comp hV)).div hy
+        (fun q => (height_pos hε hT q.1.1 z q.1.2).ne'))
+  · exact (continuous_const.mul ((hx.mul (Complex.continuous_re.comp hV)).add
+      (hy.mul (Complex.continuous_im.comp hV)))).div ((hx.pow 2).add (hy.pow 2))
+        (fun q => (q_pos _ (height_pos hε hT q.1.1 z q.1.2)).ne')
+
+lemma field_upper (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
+    (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    field hε hT p a sdev U z t b ≤ (2*ε/Real.exp ((2/ε^2)*T))^(-p) := by
+  apply green_le_fixed_radius p a _ _ _ _ (height_pos hε hT U z t) ha hp (by positivity)
+  have hL : Real.exp (logDerivative hε hT U z t) ≤ Real.exp ((2/ε^2)*T) :=
+    Real.exp_le_exp.mpr ((le_abs_self _).trans (abs_logDerivative_le hε hT U z t))
+  calc
+    2*ε / Real.exp ((2/ε^2)*T) ≤ 2*ε / Real.exp (logDerivative hε hT U z t) :=
+      div_le_div_of_nonneg_left (by positivity) (Real.exp_pos _) hL
+    _ ≤ _ := div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left (le_max_left _ _) (by norm_num))
+      (Real.exp_pos _).le
+
+lemma field_third_bound (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
+    (U : C(I, ℝ)) (z : ℂ) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ t b, |iteratedDeriv 3 (field hε hT p a sdev U z t) b| ≤ K := by
+  let M := (2*ε/Real.exp ((2/ε^2)*T))^(-p)
+  refine ⟨|sdev|^3 * (M*(a*(a+2)*(a+4))/ε^3), by positivity, ?_⟩
+  intro t b
+  change |iteratedDeriv 3 (fun b => green p a ((position hε hT U z t).re-sdev*b)
+    (height hε hT U z t) (logDerivative hε hT U z t)) b| ≤ _
+  rw [iteratedDeriv_green_driver p a _ _ _ sdev (height_pos hε hT U z t) 3 b,
+    abs_mul, abs_pow, abs_neg]
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  exact third_green_bound hε (le_max_left _ _) ha (field_upper hε hT p a sdev hp ha U z t b)
+
+lemma time_deriv (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) {t : ℝ}
+    (ht : t ∈ Ioo 0 T) (hy : ε < (position hε hT U z t).im) (b : ℝ) :
+    HasDerivAt (fun s => field hε hT p a sdev U z s b) (drift hε hT p a sdev U z t b) t := by
+  have hH := position_deriv hε hT U z ht
+  have hL := hasDerivAt_logDerivative hε hT U z ht
+  have hIm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hH
+  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
+  dsimp only [Function.comp_def, Complex.imCLM_apply, Complex.reCLM_apply] at hIm hRe
+  have hY : HasDerivAt (height hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)).im t := by
+    apply hIm.congr_of_eventuallyEq
+    have hc := hIm.continuousAt
+    filter_upwards [hc.preimage_mem_nhds (Ioi_mem_nhds hy)] with s hs
+    exact max_eq_right hs.le
+  exact hasDerivAt_green_path p a (hRe.sub_const (sdev*b)) hY hL (height_pos hε hT U z t)
+
+lemma second_eq (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    iteratedDeriv 2 (field hε hT p a sdev U z t) b =
+      sdev^2 * (field hε hT p a sdev U z t b *
+        (a*(a+1)*((position hε hT U z t).re-sdev*b)^2 - a*(height hε hT U z t)^2) /
+        (((position hε hT U z t).re-sdev*b)^2+(height hε hT U z t)^2)^2) := by
+  simp only [iteratedDeriv_succ, iteratedDeriv_zero, field]
+  exact deriv_deriv_green_driver p a _ _ _ sdev b (height_pos hε hT U z t)
+
+lemma continuous_second (p a sdev : ℝ) (z : ℂ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv 2 (field hε hT p a sdev q.1.1 z q.1.2) q.2) := by
+  simp only [second_eq]
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => sdev*q.2))
+  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => height hε hT q.1.1 z q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  apply continuous_const.mul
+  apply Continuous.div
+  · exact (continuous_field hε hT p a sdev z).mul
+      ((continuous_const.mul (hx.pow 2)).sub (continuous_const.mul (hy.pow 2)))
+  · exact ((hx.pow 2).add (hy.pow 2)).pow 2
+  · exact fun q => pow_ne_zero 2 (q_pos _ (height_pos hε hT q.1.1 z q.1.2)).ne'
+
+lemma first_eq (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    iteratedDeriv 1 (field hε hT p a sdev U z t) b =
+      sdev*a*((position hε hT U z t).re-sdev*b) /
+        (((position hε hT U z t).re-sdev*b)^2+(height hε hT U z t)^2) *
+        field hε hT p a sdev U z t b := by
+  simp only [iteratedDeriv_succ, iteratedDeriv_zero, field]
+  exact deriv_green_driver p a _ _ _ sdev b (height_pos hε hT U z t)
+
+lemma continuous_first (p a sdev : ℝ) (z : ℂ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv 1 (field hε hT p a sdev q.1.1 z q.1.2) q.2) := by
+  simp only [first_eq]
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => sdev*q.2))
+  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => height hε hT q.1.1 z q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  exact ((continuous_const.mul hx).div ((hx.pow 2).add (hy.pow 2))
+    (fun q => (q_pos _ (height_pos hε hT q.1.1 z q.1.2)).ne')).mul
+      (continuous_field hε hT p a sdev z)
+
+lemma first_bound (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
+    (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    |iteratedDeriv 1 (field hε hT p a sdev U z t) b| ≤
+      |sdev| * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε) := by
+  change |iteratedDeriv 1 (fun b => green p a ((position hε hT U z t).re-sdev*b)
+    (height hε hT U z t) (logDerivative hε hT U z t)) b| ≤ _
+  rw [iteratedDeriv_green_driver p a _ _ _ sdev (height_pos hε hT U z t) 1 b,
+    abs_mul, abs_pow, abs_neg, pow_one]
+  simp only [iteratedDeriv_succ, iteratedDeriv_zero]
+  exact mul_le_mul_of_nonneg_left
+    (first_green_bound hε (le_max_left _ _) ha (field_upper hε hT p a sdev hp ha U z t b))
+    (abs_nonneg _)
+
+lemma second_bound (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
+    (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    |iteratedDeriv 2 (field hε hT p a sdev U z t) b| ≤
+      |sdev|^2 * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*(a*(a+2))/ε^2) := by
+  change |iteratedDeriv 2 (fun b => green p a ((position hε hT U z t).re-sdev*b)
+    (height hε hT U z t) (logDerivative hε hT U z t)) b| ≤ _
+  rw [iteratedDeriv_green_driver p a _ _ _ sdev (height_pos hε hT U z t) 2 b,
+    abs_mul, abs_pow, abs_neg]
+  exact mul_le_mul_of_nonneg_left
+    (second_green_bound hε (le_max_left _ _) ha (field_upper hε hT p a sdev hp ha U z t b))
+    (by positivity)
+
+lemma variation_of_unclipped (U : C(I, ℝ)) (z : ℂ) (t : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im) :
+    variation hε hT U z t =
+      -2 * (((position hε hT U z t).re - extendDriver T U t)^2 - (position hε hT U z t).im^2) /
+        (((position hε hT U z t).re - extendDriver T U t)^2 + (position hε hT U z t).im^2)^2 := by
+  unfold variation
+  rw [clampIm_eq (by simpa using hy)]
+  simp only [Complex.div_re, Complex.neg_re, Complex.re_ofNat, Complex.neg_im,
+    Complex.im_ofNat, neg_zero, zero_mul, map_pow]
+  simp only [pow_two, Complex.mul_re, Complex.sub_re, Complex.ofReal_re,
+    Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.normSq_apply]
+  ring
+
+lemma drift_cancellation (κ : ℝ) (hκ : 0 < κ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ)
+    (hy : ε ≤ (position hε hT U z t).im)
+    (hU : extendDriver T U t = Real.sqrt κ * b) :
+    drift hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t b +
+      (1/2 : ℝ) * iteratedDeriv 2 (field hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) b = 0 := by
+  have hypos := hε.trans_le hy
+  have hh : height hε hT U z t = (position hε hT U z t).im := max_eq_right hy
+  rw [second_eq, Real.sq_sqrt hκ.le]
+  dsimp only [drift]
+  rw [variation_of_unclipped hε hT U z t hy, velocity_eq hy,
+    hh, hU]
+  have hc := green_drift_cancellation κ
+    ((position hε hT U z t).re-Real.sqrt κ*b) (position hε hT U z t).im
+    (field hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t b) hκ.ne' hypos
+  dsimp only at hc
+  simp only [Complex.div_re, Complex.div_im, Complex.re_ofNat, Complex.im_ofNat,
+    Complex.sub_re, Complex.ofReal_re, Complex.sub_im, Complex.ofReal_im,
+    sub_zero, zero_mul, Complex.normSq_apply] at *
+  convert hc using 1
+  ring
+
+end SLEExactGauge.LoewnerField
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+namespace SLEExactGauge.LoewnerField
+open LoewnerCutoff
+open scoped unitInterval
+
+lemma field_prefix_eq {ε S T : ℝ} (hε : 0 < ε) (hS : 0 ≤ S) (hT : 0 ≤ T)
+    (hST : S ≤ T) (U V : C(I, ℝ)) (z : ℂ) (p a sdev : ℝ)
+    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
+    {t : ℝ} (ht : t ∈ Icc 0 S) :
+    field hε hS p a sdev U z t = field hε hT p a sdev V z t := by
+  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
+  have hL := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
+  funext b
+  simp only [field, height, hH, hL]
+
+end SLEExactGauge.LoewnerField
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.ContinuousExit
+
+def extend (p : C(I, ℝ)) (t : ℝ) : ℝ := p (projIcc 0 1 zero_le_one t)
+
+lemma continuous_extend (p : C(I, ℝ)) : Continuous (extend p) :=
+  p.continuous.comp (continuous_projIcc (a := (0:ℝ)) (b := 1) (h := zero_le_one))
+
+lemma extend_coe (p : C(I, ℝ)) (t : I) : extend p t = p t := by
+  simp only [extend, projIcc, max_eq_right t.property.1, min_eq_right t.property.2]
+
+def exitSet (p : C(I, ℝ)) : Set ℝ := insert 1 (Icc 0 1 ∩ {t | extend p t ≤ 0})
+
+def time (p : C(I, ℝ)) : ℝ := sInf (exitSet p)
+
+lemma exitSet_nonempty (p : C(I, ℝ)) : (exitSet p).Nonempty := ⟨1, mem_insert _ _⟩
+
+lemma exitSet_subset (p : C(I, ℝ)) : exitSet p ⊆ Icc 0 1 := by
+  rintro t (rfl | ht)
+  · exact ⟨zero_le_one,le_rfl⟩
+  · exact ht.1
+
+lemma compact_exitSet (p : C(I, ℝ)) : IsCompact (exitSet p) := by
+  exact (isCompact_Icc.inter_right (isClosed_le (continuous_extend p) continuous_const)).insert 1
+
+lemma time_mem (p : C(I, ℝ)) : time p ∈ exitSet p :=
+  (compact_exitSet p).sInf_mem (exitSet_nonempty p)
+
+lemma time_bounds (p : C(I, ℝ)) : time p ∈ Icc 0 1 := exitSet_subset p (time_mem p)
+
+lemma time_le_of_nonpos {p : C(I, ℝ)} {t : ℝ} (ht : t ∈ Icc 0 1) (hp : extend p t ≤ 0) :
+    time p ≤ t := csInf_le ((compact_exitSet p).bddBelow) (mem_insert_of_mem 1 ⟨ht,hp⟩)
+
+lemma positive_before {p : C(I, ℝ)} {t : ℝ} (ht : 0 ≤ t) (hτ : t < time p) :
+    0 < extend p t := by
+  by_contra hn
+  exact (not_le_of_gt hτ) (time_le_of_nonpos ⟨ht,hτ.le.trans (time_bounds p).2⟩ (not_lt.1 hn))
+
+lemma nonpos_at_of_lt_one {p : C(I, ℝ)} (hτ : time p < 1) : extend p (time p) ≤ 0 := by
+  rcases time_mem p with h | h
+  · exact (ne_of_lt hτ h).elim
+  · exact h.2
+
+lemma lt_time_iff {p : C(I, ℝ)} {t : ℝ} (ht : t ∈ Ico 0 1) :
+    t < time p ↔ ∀ s ∈ Icc 0 t, 0 < extend p s := by
+  constructor
+  · intro h s hs
+    exact positive_before hs.1 (hs.2.trans_lt h)
+  · intro h
+    by_contra hn
+    have hτ : time p ≤ t := not_lt.1 hn
+    have hp := h (time p) ⟨(time_bounds p).1,hτ⟩
+    exact (not_lt_of_ge (nonpos_at_of_lt_one (hτ.trans_lt ht.2))) hp
+
+lemma isOpen_positive_on (t : ℝ) :
+    IsOpen {p : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < extend p s} := by
+  let S : Set (C(I, ℝ) × Icc 0 t) := {q | extend q.1 q.2 ≤ 0}
+  have hc : IsClosed S := by
+    apply isClosed_le ?_ continuous_const
+    exact continuous_eval.comp (continuous_fst.prodMk
+      ((continuous_projIcc (a := (0:ℝ)) (b := 1) (h := zero_le_one)).comp (continuous_subtype_val.comp continuous_snd)))
+  have hh := isClosedMap_fst_of_compactSpace S hc
+  have he : {p : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < extend p s} = (Prod.fst '' S)ᶜ := by
+    ext p
+    simp only [mem_ofPred_eq, mem_compl_iff, mem_image, S, not_exists, not_and]
+    constructor
+    · intro hp q hq hqp
+      subst p
+      exact (not_lt_of_ge hq) (hp q.2 q.2.property)
+    · intro hp s hs
+      by_contra hn
+      exact hp (p,⟨s,hs⟩) (not_lt.1 hn) rfl
+  rw [he]
+  exact hh.isOpen_compl
+
+lemma measurable_time : @Measurable C(I, ℝ) ℝ (borel _) _ time := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  apply measurable_of_Ioi
+  intro t
+  by_cases h0 : t < 0
+  · have he : time ⁻¹' Ioi t = univ := by
+      ext p
+      simp only [mem_preimage, mem_Ioi, mem_univ, iff_true]
+      exact h0.trans_le (time_bounds p).1
+    rw [he]
+    exact MeasurableSet.univ
+  by_cases h1 : 1 ≤ t
+  · have he : time ⁻¹' Ioi t = ∅ := by
+      ext p
+      simp only [mem_preimage, mem_Ioi, mem_empty_iff_false, iff_false]
+      exact not_lt_of_ge ((time_bounds p).2.trans h1)
+    rw [he]
+    exact MeasurableSet.empty
+  have he : time ⁻¹' Ioi t = {p : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < extend p s} := by
+    ext p
+    exact lt_time_iff ⟨not_lt.1 h0,not_le.1 h1⟩
+  rw [he]
+  exact (isOpen_positive_on t).measurableSet
+
+lemma time_pos {p : C(I, ℝ)} (h0 : 0 < extend p 0) : 0 < time p := by
+  apply (lt_time_iff (p := p) (t := 0) ⟨le_rfl, zero_lt_one⟩).2
+  intro s hs
+  have hs0 : s = 0 := le_antisymm hs.2 hs.1
+  simpa only [hs0] using h0
+
+lemma nonneg_until {p : C(I, ℝ)} (h0 : 0 < extend p 0) {t : ℝ}
+    (ht : t ∈ Icc 0 (time p)) : 0 ≤ extend p t := by
+  rcases lt_or_eq_of_le ht.2 with hlt | rfl
+  · exact (positive_before ht.1 hlt).le
+  have htend : Tendsto (extend p) (𝓝[<] time p) (𝓝 (extend p (time p))) :=
+    (continuous_extend p).continuousAt.tendsto.mono_left nhdsWithin_le_nhds
+  apply ge_of_tendsto htend
+  filter_upwards [Ioo_mem_nhdsLT (time_pos h0)] with s hs
+  exact (positive_before hs.1.le hs.2).le
+
+lemma eq_zero_at_of_lt_one {p : C(I, ℝ)} (h0 : 0 < extend p 0)
+    (hτ : time p < 1) : extend p (time p) = 0 :=
+  le_antisymm (nonpos_at_of_lt_one hτ) (nonneg_until h0 ⟨(time_bounds p).1, le_rfl⟩)
+
+end SLEExactGauge.ContinuousExit
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerExit
+open LoewnerCutoff
+open scoped unitInterval
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+def radius (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  2*(position hε hT U z t).im / Real.exp (logDerivative hε hT U z t)
+
+def sine (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  (position hε hT U z t).im /
+    ‖clampIm ε (position hε hT U z t - (extendDriver T U t : ℂ))‖
+
+def deficit (U : C(I, ℝ)) (z : ℂ) (ρ δ t : ℝ) : ℝ :=
+  min (radius hε hT U z t - ρ)
+    (min ((position hε hT U z t).im-ε) (sine hε hT U z t-δ))
+
+lemma continuous_radius (z : ℂ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => radius hε hT q.1 z q.2) := by
+  exact (continuous_const.mul (Complex.continuous_im.comp (continuous_position hε hT z))).div
+    (Real.continuous_exp.comp (continuous_logDerivative hε hT z)) (fun _ => Real.exp_ne_zero _)
+
+lemma continuous_sine (z : ℂ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => sine hε hT q.1 z q.2) := by
+  exact (Complex.continuous_im.comp (continuous_position hε hT z)).div
+    (((continuous_clampIm ε).comp ((continuous_position hε hT z).sub
+      (Complex.continuous_ofReal.comp (continuous_joint_extendDriver T)))).norm)
+    (fun _ => norm_ne_zero_iff.2 (clampIm_ne_zero hε _))
+
+lemma continuous_deficit (z : ℂ) (ρ δ : ℝ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => deficit hε hT q.1 z ρ δ q.2) :=
+  ((continuous_radius hε hT z).sub continuous_const).min
+    (((Complex.continuous_im.comp (continuous_position hε hT z)).sub continuous_const).min
+      ((continuous_sine hε hT z).sub continuous_const))
+
+lemma deficit_pos_iff (U : C(I, ℝ)) (z : ℂ) (ρ δ t : ℝ) :
+    0 < deficit hε hT U z ρ δ t ↔
+      ρ < radius hε hT U z t ∧ ε < (position hε hT U z t).im ∧ δ < sine hε hT U z t := by
+  simp only [deficit, lt_min_iff, sub_pos]
+
+lemma deficit_nonneg_iff (U : C(I, ℝ)) (z : ℂ) (ρ δ t : ℝ) :
+    0 ≤ deficit hε hT U z ρ δ t ↔
+      ρ ≤ radius hε hT U z t ∧ ε ≤ (position hε hT U z t).im ∧ δ ≤ sine hε hT U z t := by
+  simp only [deficit, le_min_iff, sub_nonneg]
+
+def exitPath (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) : C(I, ℝ) := by
+  refine ⟨fun q => deficit hε hT U z ρ δ (T*q), ?_⟩
+  have hp : Continuous (fun q : I => (U,T*q)) :=
+    continuous_const.prodMk (continuous_const.mul continuous_subtype_val)
+  have hc := (continuous_deficit hε hT z ρ δ).comp hp
+  dsimp only [Function.comp_def] at hc
+  exact hc
+
+def stop (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) : ℝ :=
+  T * ContinuousExit.time (exitPath hε hT U z ρ δ)
+
+lemma stop_bounds (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) :
+    stop hε hT U z ρ δ ∈ Icc 0 T := by
+  have hh := ContinuousExit.time_bounds (exitPath hε hT U z ρ δ)
+  exact ⟨mul_nonneg hT hh.1, by simpa only [stop, mul_one] using mul_le_mul_of_nonneg_left hh.2 hT⟩
+
+lemma exitPath_extend (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) {t : ℝ} (ht : t ∈ Icc 0 1) :
+    ContinuousExit.extend (exitPath hε hT U z ρ δ) t = deficit hε hT U z ρ δ (T*t) := by
+  simpa only [exitPath, ContinuousMap.coe_mk] using ContinuousExit.extend_coe
+    (exitPath hε hT U z ρ δ) (⟨t,ht⟩ : I)
+
+lemma exitPath_extend_div (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    ContinuousExit.extend (exitPath hε hT U z ρ δ) (t/T) = deficit hε hT U z ρ δ t := by
+  rw [exitPath_extend hε hT U z ρ δ
+    ⟨div_nonneg ht.1 hT, (div_le_one hTp).2 ht.2⟩, mul_div_cancel₀ _ hTp.ne']
+
+lemma positive_before (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    {t : ℝ} (ht : 0 ≤ t) (htτ : t < stop hε hT U z ρ δ) :
+    0 < deficit hε hT U z ρ δ t := by
+  rw [← exitPath_extend_div hε hT hTp U z ρ δ ⟨ht,htτ.le.trans (stop_bounds hε hT U z ρ δ).2⟩]
+  apply ContinuousExit.positive_before (div_nonneg ht hT)
+  exact (div_lt_iff₀ hTp).2 (by simpa only [stop, mul_comm] using htτ)
+
+lemma stop_pos (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (h0 : 0 < deficit hε hT U z ρ δ 0) : 0 < stop hε hT U z ρ δ := by
+  apply mul_pos hTp (ContinuousExit.time_pos _)
+  simpa only [exitPath_extend hε hT U z ρ δ ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
+
+lemma nonneg_until (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (h0 : 0 < deficit hε hT U z ρ δ 0) {t : ℝ}
+    (ht : t ∈ Icc 0 (stop hε hT U z ρ δ)) : 0 ≤ deficit hε hT U z ρ δ t := by
+  rw [← exitPath_extend_div hε hT hTp U z ρ δ ⟨ht.1,ht.2.trans (stop_bounds hε hT U z ρ δ).2⟩]
+  apply ContinuousExit.nonneg_until
+  · simpa only [exitPath_extend hε hT U z ρ δ ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
+  · exact ⟨div_nonneg ht.1 hT, (div_le_iff₀ hTp).2 (by simpa only [stop, mul_comm] using ht.2)⟩
+
+lemma nonpos_at_of_lt (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hτ : stop hε hT U z ρ δ < T) :
+    deficit hε hT U z ρ δ (stop hε hT U z ρ δ) ≤ 0 := by
+  have hs : ContinuousExit.time (exitPath hε hT U z ρ δ) < 1 := by
+    apply (mul_lt_mul_iff_right₀ hTp).mp
+    simpa only [mul_one, stop] using hτ
+  exact (exitPath_extend hε hT U z ρ δ (ContinuousExit.time_bounds _)) ▸
+    ContinuousExit.nonpos_at_of_lt_one hs
+
+lemma lt_stop_iff (hTp : 0 < T) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    {t : ℝ} (ht : t ∈ Ico 0 T) :
+    t < stop hε hT U z ρ δ ↔ ∀ s ∈ Icc 0 t, 0 < deficit hε hT U z ρ δ s := by
+  constructor
+  · intro hh s hs
+    exact positive_before hε hT hTp U z ρ δ hs.1 (hs.2.trans_lt hh)
+  · intro hh
+    change t < T * ContinuousExit.time (exitPath hε hT U z ρ δ)
+    rw [mul_comm]
+    apply (div_lt_iff₀ hTp).1
+    apply (ContinuousExit.lt_time_iff (p := exitPath hε hT U z ρ δ)
+      ⟨div_nonneg ht.1 hT, (div_lt_one hTp).2 ht.2⟩).2
+    intro s hs
+    rw [exitPath_extend hε hT U z ρ δ
+      ⟨hs.1, hs.2.trans ((div_le_one hTp).2 ht.2.le)⟩]
+    apply hh
+    exact ⟨mul_nonneg hT hs.1, (mul_le_mul_of_nonneg_left hs.2 hT).trans_eq
+      (mul_div_cancel₀ _ hTp.ne')⟩
+
+lemma isOpen_positive_on (z : ℂ) (ρ δ t : ℝ) :
+    IsOpen {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U z ρ δ s} := by
+  let S : Set (C(I, ℝ) × Icc 0 t) := {q | deficit hε hT q.1 z ρ δ q.2 ≤ 0}
+  have hmap : Continuous (fun q : C(I, ℝ) × Icc 0 t => (q.1,(q.2:ℝ))) :=
+    continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)
+  have hd := (continuous_deficit hε hT z ρ δ).comp hmap
+  dsimp only [Function.comp_def] at hd
+  have hc : IsClosed S := isClosed_le hd continuous_const
+  have hh := isClosedMap_fst_of_compactSpace S hc
+  have he : {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U z ρ δ s} = (Prod.fst '' S)ᶜ := by
+    ext U
+    simp only [mem_ofPred_eq, mem_compl_iff, mem_image, S, not_exists, not_and]
+    constructor
+    · intro hp q hq hqp
+      subst U
+      exact (not_lt_of_ge hq) (hp q.2 q.2.property)
+    · intro hp s hs
+      by_contra hn
+      exact hp (U,⟨s,hs⟩) (not_lt.1 hn) rfl
+  rw [he]
+  exact hh.isOpen_compl
+
+end SLEExactGauge.LoewnerExit
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerExit
+open LoewnerCutoff
+open scoped unitInterval
+
+lemma deficit_prefix_eq {ε S T : ℝ} (hε : 0 < ε) (hS : 0 ≤ S) (hT : 0 ≤ T)
+    (hST : S ≤ T) (U V : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S)) :
+    EqOn (deficit hε hS U z ρ δ) (deficit hε hT V z ρ δ) (Icc 0 S) := by
+  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV
+  have hL := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V z hUV
+  intro t ht
+  simp only [deficit, radius, sine, hH ht, hL ht, hUV ht]
+
+end SLEExactGauge.LoewnerExit
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerField LoewnerExit BrownianField
+open scoped unitInterval
+
+lemma regular {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (p a sdev : ℝ)
+    (hp : 0 ≤ p) (ha : 0 ≤ a) (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ)
+    (b : ℝ → ℝ) (hb : Continuous b) :
+    FieldRegular (field hε hT.le p a sdev U z) (drift hε hT.le p a sdev U z) b
+      (stop hε hT.le U z ρ δ) := by
+  constructor
+  · exact smooth hε hT.le p a sdev U z
+  · exact field_third_bound hε hT.le p a sdev hp ha U z
+  · intro x
+    have hm : Continuous (fun t : ℝ => ((U,t),x)) :=
+      (continuous_const.prodMk continuous_id).prodMk continuous_const
+    have hc := (continuous_field hε hT.le p a sdev z).comp hm
+    exact hc.continuousOn
+  · intro t ht x
+    have htT : t ∈ Ioo 0 T := ⟨ht.1,ht.2.trans_le (stop_bounds hε hT.le U z ρ δ).2⟩
+    have hh := (deficit_pos_iff hε hT.le U z ρ δ t).1
+      (positive_before hε hT.le hT U z ρ δ ht.1.le ht.2)
+    exact time_deriv hε hT.le p a sdev U z htT hh.2.1 x
+  · have hm : Continuous (fun q : ℝ × ℝ => ((U,q.1),b q.2)) :=
+      (continuous_const.prodMk continuous_fst).prodMk (hb.comp continuous_snd)
+    have hc := (continuous_drift hε hT.le p a sdev z).comp hm
+    exact hc.continuousOn
+  · have hm : Continuous (fun t : ℝ => ((U,t),b t)) :=
+      (continuous_const.prodMk continuous_id).prodMk hb
+    have hc := (continuous_second hε hT.le p a sdev z).comp hm
+    exact hc.continuousOn
+
+lemma integral_drift_cancel {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (κ : ℝ) (hκ : 0 < κ)
+    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) (b : ℝ → ℝ)
+    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ * b s) :
+    (∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ), drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) +
+      (1/2 : ℝ)*∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ),
+        iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) = 0 := by
+  have he : (∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ), drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) =
+      ∫ t in (0:ℝ)..(stop hε hT.le U z ρ δ),
+        -(1/2 : ℝ)*iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) := by
+    apply intervalIntegral.integral_congr_Ioo_of_le (stop_bounds hε hT.le U z ρ δ).1
+    intro t ht
+    have htT : t ∈ Icc 0 T := ⟨ht.1.le,ht.2.le.trans (stop_bounds hε hT.le U z ρ δ).2⟩
+    have hh := (deficit_pos_iff hε hT.le U z ρ δ t).1
+      (positive_before hε hT.le hT U z ρ δ ht.1.le ht.2)
+    have hc := drift_cancellation hε hT.le κ hκ U z t (b t) hh.2.1.le (hU t htT)
+    linarith
+  rw [he, intervalIntegral.integral_const_mul]
+  ring
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.ContinuousDriver
+
+lemma aemeasurable_selectedLimit {Ω Y : Type*} [MeasurableSpace Ω]
+    [MetricSpace Y] [CompleteSpace Y] [MeasurableSpace Y] [BorelSpace Y]
+    [SecondCountableTopology Y] {P : Measure Ω}
+    (y₀ : Y) (a : ℕ → Ω → Y) (ha : ∀ n, AEMeasurable (a n) P) :
+    AEMeasurable (fun sample => selectedLimit y₀ (fun n => a n sample)) P := by
+  have hm : Measurable (fun a : ℕ → Y => selectedLimit y₀ a) :=
+    measurable_selectedLimit y₀ (fun (n : ℕ) (a : ℕ → Y) => a n) (fun n => measurable_pi_apply n)
+  exact hm.comp_aemeasurable (AEMeasurable.of_eval ha)
+
+lemma aemeasurable_path {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (x : Ω → I → ℝ) (hx : ∀ t, AEMeasurable (fun sample => x sample t) P) :
+    @AEMeasurable Ω C(I, ℝ) (borel C(I, ℝ)) _ (fun sample => path (x sample)) P := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  apply aemeasurable_selectedLimit
+  intro n
+  apply Finset.aemeasurable_fun_sum
+  intro k _
+  exact (hx (bernstein.z k)).smul aemeasurable_const
+
+end SLEExactGauge.ContinuousDriver
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.BrownianDriver
+open BrownianPredictable LoewnerCutoff
+open scoped unitInterval
+
+def path {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) (sample : Ω) : C(I, ℝ) :=
+  ContinuousDriver.path (fun q : I => sdev * B ((T:ℝ)*q).toNNReal sample)
+
+lemma path_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) :
+    @Measurable Ω C(I, ℝ) (pastSigma B T) (borel C(I, ℝ)) (path B sdev T) := by
+  let : MeasurableSpace Ω := pastSigma B T
+  apply ContinuousDriver.measurable_path
+  intro q
+  apply Measurable.const_mul
+  apply past_measurable_eval
+  have hh : (T:ℝ)*q ≤ T := by
+    simpa only [mul_one] using mul_le_mul_of_nonneg_left q.property.2 T.coe_nonneg
+  exact (Real.toNNReal_le_toNNReal hh).trans_eq Real.toNNReal_coe
+
+lemma path_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (sdev : ℝ) (T : ℝ≥0) :
+    @AEMeasurable Ω C(I, ℝ) (borel C(I, ℝ)) _ (path B sdev T) P := by
+  apply ContinuousDriver.aemeasurable_path
+  intro q
+  exact ((hB.toIsPreBrownianReal.hasLaw_eval _).aemeasurable).const_mul sdev
+
+lemma path_eval {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) (q : I) :
+    path B sdev T sample q = sdev * B ((T:ℝ)*q).toNNReal sample := by
+  apply ContinuousDriver.path_eval_of_continuous
+  exact continuous_const.mul (hc.comp (continuous_real_toNNReal.comp
+    (continuous_const.mul continuous_subtype_val)))
+
+lemma extend_eq {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (T : ℝ≥0) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) {t : ℝ} (ht : t ∈ Icc 0 (T:ℝ)) :
+    extendDriver T (path B sdev T sample) t = sdev * B t.toNNReal sample := by
+  rw [extendDriver, path_eval B sdev T sample hc]
+  by_cases hT : (T:ℝ) = 0
+  · have ht0 : t = 0 := le_antisymm (hT ▸ ht.2) ht.1
+    simp only [hT, ht0, zero_mul]
+  · have hTp : 0 < (T:ℝ) := lt_of_le_of_ne T.coe_nonneg (Ne.symm hT)
+    rw [projIcc_of_mem zero_le_one ⟨div_nonneg ht.1 T.coe_nonneg, (div_le_one hTp).2 ht.2⟩,
+      mul_div_cancel₀ _ hT]
+
+lemma prefix_eq {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (sdev : ℝ) (S T : ℝ≥0) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) (hST : S ≤ T) :
+    EqOn (extendDriver S (path B sdev S sample)) (extendDriver T (path B sdev T sample)) (Icc 0 (S:ℝ)) := by
+  intro t ht
+  rw [extend_eq B sdev S sample hc ht, extend_eq B sdev T sample hc ⟨ht.1,ht.2.trans hST⟩]
+
+end SLEExactGauge.BrownianDriver
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerField LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+open scoped unitInterval
+variable {ε : ℝ} (hε : 0 < ε)
+
+def coeffAt (k : ℕ) (s : ℝ≥0) (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ)
+    (q : C(I, ℝ) × ℝ) : ℝ := by
+  classical
+  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 z ρ δ t then
+    iteratedDeriv k (field hε s.coe_nonneg p a sdev q.1 z s) q.2 else 0
+
+lemma coeffAt_measurable (k : ℕ) (hk : k = 1 ∨ k = 2) (s : ℝ≥0)
+    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s p a sdev z ρ δ) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
+      iteratedDeriv k (field hε s.coe_nonneg p a sdev q.1 z s) q.2) := by
+    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
+      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
+    rcases hk with rfl | rfl
+    · have hc := (continuous_first hε s.coe_nonneg p a sdev z).comp hp
+      dsimp only [Function.comp_def] at hc
+      exact hc
+    · have hc := (continuous_second hε s.coe_nonneg p a sdev z).comp hp
+      dsimp only [Function.comp_def] at hc
+      exact hc
+  apply hc.measurable.ite _ measurable_const
+  exact ((isOpen_positive_on hε s.coe_nonneg z ρ δ s).preimage continuous_fst).measurableSet
+
+def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω) : ℝ :=
+  if s < T then coeffAt hε k s p a sdev z ρ δ (BrownianDriver.path B sdev s sample, B s sample) else 0
+
+lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    Measurable[pastSigma B s] (rawCoeff hε k B T s p a sdev z ρ δ) := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  let : MeasurableSpace Ω := pastSigma B s
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s p a sdev z ρ δ).comp
+      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
+  · exact measurable_const
+
+lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
+    (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) :
+    AEMeasurable (rawCoeff hε k B T s p a sdev z ρ δ) P := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s p a sdev z ρ δ).comp_aemeasurable
+      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
+  · exact aemeasurable_const
+
+lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    rawCoeff hε k B T s p a sdev z ρ δ sample =
+      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ then
+        iteratedDeriv k (field hε T.coe_nonneg p a sdev (BrownianDriver.path B sdev T sample) z s) (B s sample) else 0 := by
+  classical
+  by_cases hsT : s < T
+  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
+    have hf := field_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z p a sdev he
+      ⟨s.coe_nonneg, le_rfl⟩
+    have hd := deficit_prefix_eq hε s.coe_nonneg T.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) z ρ δ he
+    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) z ρ δ t) ↔
+        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ := by
+      rw [lt_stop_iff hε T.coe_nonneg hT _ z ρ δ ⟨s.coe_nonneg,hsT⟩]
+      exact forall₂_congr (fun t ht => by rw [hd ht])
+    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
+  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ :=
+      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ z ρ δ).2.trans (not_lt.1 hsT))
+    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
+
+lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (p a sdev : ℝ) (z : ℂ) (ρ δ : ℝ) (n i : ℕ) :
+    rawCoeff hε k B T (dyadicTime T n i) p a sdev z ρ δ =ᵐ[P]
+      stoppedCoeff k B T
+        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) z ρ δ)
+        (fun sample => field hε T.coe_nonneg p a sdev (BrownianDriver.path B sdev T sample) z) n i := by
+  filter_upwards [hB.cont] with sample hc
+  rw [rawCoeff_eq hε k B T _ hT p a sdev z ρ δ sample hc]
+  simp only [stoppedCoeff, coe_dyadicTime]
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerField LoewnerExit BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+open scoped unitInterval
+
+lemma coeff_uniform_bound {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (p a sdev : ℝ) (hp : 0 ≤ p) (ha : 0 ≤ a)
+    (z : ℂ) : ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t b : ℝ),
+      |iteratedDeriv k (field hε hT p a sdev U z t) b| ≤ K := by
+  rcases hk with rfl | rfl
+  · refine ⟨max 0 (|sdev| * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*a/ε)), le_max_left _ _, ?_⟩
+    intro U t b
+    exact (first_bound hε hT p a sdev hp ha U z t b).trans (le_max_right _ _)
+  · refine ⟨max 0 (|sdev|^2 * ((2*ε/Real.exp ((2/ε^2)*T))^(-p)*(a*(a+2))/ε^2)), le_max_left _ _, ?_⟩
+    intro U t b
+    exact (second_bound hε hT p a sdev hp ha U z t b).trans (le_max_right _ _)
+
+theorem localized_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (ρ δ : ℝ) :
+    let U := BrownianDriver.path B (Real.sqrt κ) T
+    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+    let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+    Integrable (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) P ∧
+      (∫ sample, F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) ∂P) = 0 := by
+  dsimp only
+  let U := BrownianDriver.path B (Real.sqrt κ) T
+  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by
+    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
+    linarith
+  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
+      (hc.comp continuous_real_toNNReal)
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
+    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) z ρ δ)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
+        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
+    apply Filter.Eventually.of_forall
+    intro sample
+    unfold stoppedCoeff
+    split_ifs
+    · exact hb _ _ _
+    · simpa only [abs_zero] using hK
+  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
+  let C := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 1 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
+  have hCe (n i : ℕ) : C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 2 T hT (1-κ/8) (8/κ-1) (Real.sqrt κ) z ρ δ n i
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
+    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ z ρ δ
+  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) :=
+    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ z ρ δ
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₁
+    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hb
+  have hC2 (n i : ℕ) : MemLp (C n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ z ρ δ).aestronglyMeasurable K₂
+    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |C n i sample| ≤ K₂
+    rw [he]
+    exact hb
+  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A C hAe hCe hCm hC2
+    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
+      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
+    filter_upwards [hB.cont] with sample hc
+    have hcanc := integral_drift_cancel hε hT κ hκ.1 (U sample) z ρ δ
+      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
+    dsimp only [F,V,τ]
+    linarith
+  exact ⟨hi.1.congr he, (integral_congr_ae he).symm.trans hi.2⟩
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerField LoewnerExit GreenPhysical
+open scoped unitInterval
+
+lemma field_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (hy : ε ≤ z.im) :
+    field hε hT p a sdev U z 0 0 = green p a z.re z.im 0 := by
+  simp only [field, height, position_initial, logDerivative_initial, mul_zero, sub_zero,
+    max_eq_right hy]
+
+theorem localized_expectation {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hy : ε ≤ z.im) (ρ δ : ℝ) :
+    let U := BrownianDriver.path B (Real.sqrt κ) T
+    let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+    let M := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+      (τ sample) (B (τ sample).toNNReal sample)
+    Integrable M P ∧ (∫ sample, M sample ∂P) = green (1-κ/8) (8/κ-1) z.re z.im 0 := by
+  dsimp only
+  let U := BrownianDriver.path B (Real.sqrt κ) T
+  let τ := fun sample => stop hε T.coe_nonneg (U sample) z ρ δ
+  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  let M := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample)
+  let G := green (1-κ/8) (8/κ-1) z.re z.im 0
+  have hi := localized_centered hB κ hκ hε T hT z ρ δ
+  have he : (fun sample => M sample - F sample 0 (B 0 sample)) =ᵐ[P]
+      (fun sample => M sample - G) := by
+    filter_upwards [hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hb
+    rw [hb]
+    congr 1
+    exact field_initial hε T.coe_nonneg _ _ _ _ z hy
+  have hd : Integrable (fun sample => M sample-G) P := hi.1.congr he
+  have hv : (∫ sample, M sample-G ∂P) = 0 := (integral_congr_ae he).symm.trans hi.2
+  have hm : Integrable M P := by
+    exact (hd.add (integrable_const G)).congr
+      (Filter.Eventually.of_forall (fun sample => sub_add_cancel (M sample) G))
+  refine ⟨hm, ?_⟩
+  have hh := integral_sub hm (integrable_const (μ := P) G)
+  simp only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul] at hh
+  rw [hh] at hv
+  change (∫ sample, M sample ∂P) = G
+  linarith
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology unitInterval
+namespace SLEExactGauge.IsLoewnerTraceUsing
+open LoewnerCutoff
+
+lemma cutoff_solution_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt H (velocity ε (driver t) (H t)) (Icc 0 T) t)
+    (hheight : ∀ t ∈ Icc 0 T, ε ≤ (H t).im) :
+    H 0 ∈ localizationDomain γ T ∧ EqOn (fun t => g t (H 0)) H (Icc 0 T) := by
+  have hp : H T ∈ halfPlane := hε.trans_le (hheight T ⟨hT, le_rfl⟩)
+  let z' := f T (H T)
+  have hz' : z' ∈ localizationDomain γ T := h.2.2.2.2.2.2.2.2.1 T hT hp
+  have hfinal : g T z' = H T := h.2.2.2.2.2.2.2.2.2.2.1 T hT _ hp
+  have hht (t : ℝ) (ht : t ∈ Icc 0 T) : ε ≤ (g t z').im := by
+    have ha := h.height_antitone hT hz' ht ⟨hT, le_rfl⟩ ht.2
+    dsimp only at ha
+    rw [hfinal] at ha
+    exact (hheight T ⟨hT, le_rfl⟩).trans ha
+  have hd (t : ℝ) (ht : t ∈ Ioc 0 T) :
+      HasDerivWithinAt (fun s => g s z')
+        (velocity ε (driver t) (g t z')) (Iic t) t := by
+    rw [velocity_eq (hht t ⟨ht.1.le, ht.2⟩)]
+    exact (h.2.2.2.2.2.2.2.2.2.2.2.2.1 t ht.1 z'
+      (survivingDomain_mono ht.2 hz')).hasDerivWithinAt
+  have he : EqOn (fun t => g t z') H (Icc 0 T) :=
+    ODE_solution_unique_of_mem_Icc_left
+      (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (driver t)).lipschitzOnWith)
+      (h.continuous_eval hT hz') hd (fun _ _ => mem_univ _)
+      (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_left hH ht)
+      (fun _ _ => mem_univ _) hfinal
+  have hz0 : z' = H 0 := by
+    rw [← h.initial z' hz'.1.1]
+    exact he ⟨le_rfl, hT⟩
+  exact ⟨hz0 ▸ hz', hz0 ▸ he⟩
+
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma hasDerivWithinAt_eval (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    HasDerivWithinAt (fun s => g s z)
+      (2 / (g t z - (driver t : ℂ))) (Icc 0 T) t := by
+  have hc := h.continuous_velocity hT hU hz
+  have hi : IntervalIntegrable (fun s => 2 / (g s z - (driver s : ℂ))) volume 0 t :=
+    (hc.intervalIntegrable_of_Icc hT).mono_set (by
+      rw [uIcc_of_le ht.1, uIcc_of_le hT]
+      exact Icc_subset_Icc le_rfl ht.2)
+  have : Fact (t ∈ Icc 0 T) := ⟨ht⟩
+  have hd := intervalIntegral.integral_hasDerivWithinAt_right (s := Icc 0 T) (t := Icc 0 T)
+    hi (hc.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc t) (hc t ht)
+  have he : EqOn (fun s => g 0 z + ∫ v in (0 : ℝ)..s, 2 / (g v z - (driver v : ℂ)))
+      (fun s => g s z) (Icc 0 T) := by
+    intro s hs
+    dsimp only
+    rw [h.integral_velocity hT hU hz ⟨le_rfl, hT⟩ hs hs.1]
+    abel
+  exact (hd.const_add (g 0 z)).congr (fun s hs => (he hs).symm) (he ht).symm
+
+lemma eq_cutoff_solution (h : IsLoewnerTraceUsing driver γ g f)
+    {T ε : ℝ} (hT : 0 ≤ T) (hε : 0 < ε) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) (hheight : ε ≤ (g T z).im)
+    {H : ℝ → ℂ} (hH0 : H 0 = z)
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
+      (velocity ε (driver t) (H t)) (Icc 0 T) t) :
+    EqOn (fun t => g t z) H (Icc 0 T) := by
+  apply solution_eq hε ?_ hH ((h.initial z hz.1.1).trans hH0.symm)
+  intro t ht
+  rw [velocity_eq (hheight.trans (h.height_antitone hT hz ht ⟨hT, le_rfl⟩ ht.2))]
+  exact h.hasDerivWithinAt_eval hT hU hz ht
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/MappedExit.lean b/lean/OAI/Probability/SLELocalization/MappedExit.lean
index 40fa2c35e76ac7cc441c62810c4cb87839aaf64f..59c6eac94091d7ac536a7e28f13e5a2e4a4f9b1f 100644
--- a/lean/OAI/Probability/SLELocalization/MappedExit.lean
+++ b/lean/OAI/Probability/SLELocalization/MappedExit.lean
@@ -1,694 +1,694 @@
-import Mathlib
-import OAI.Probability.SLELocalization.BrownianSupport
-import OAI.Probability.SLELocalization.CommonClocks
-import OAI.Probability.SLELocalization.ConformalLocalization
-import OAI.Probability.SLELocalization.FirstDisk
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-
-namespace OAI
-
-
-/-! Conformal localization and Swallowing estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.Swallowing
-open GreenPhysical LoewnerCutoff LoewnerExit
-
-lemma cutoff_eq_until {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (U : C(I,ℝ)) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (hc : ContinuousOn driver (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hy : ε < (g T z).im) :
-    EqOn (position hε hT.le U z) (fun t => g t z) (Icc 0 T) ∧
-      stop hε hT.le U z 0 0 = T := by
-  have he := h.eq_cutoff_solution hT.le hε hc hz hy.le (position_initial hε hT.le U z) (by
-    intro t ht
-    rw [← hU ht]
-    exact position_derivWithin hε hT.le U z ht)
-  have hpos (t : ℝ) (ht : t ∈ Icc 0 T) : ε < (position hε hT.le U z t).im := by
-    rw [← he ht]
-    exact hy.trans_le (h.height_antitone hT.le hz ht ⟨hT.le,le_rfl⟩ ht.2)
-  refine ⟨he.symm,?_⟩
-  apply le_antisymm (stop_bounds hε hT.le U z 0 0).2
-  by_contra hn
-  have hlt := lt_of_not_ge hn
-  have hn0 := nonpos_at_of_lt hε hT.le hT U z 0 0 hlt
-  have ht := stop_bounds hε hT.le U z 0 0
-  have hh := hpos _ ht
-  have hp : 0 < deficit hε hT.le U z 0 0 (stop hε hT.le U z 0 0) := by
-    apply (deficit_pos_iff hε hT.le U z 0 0 _).2
-    refine ⟨?_,hh,?_⟩
-    · exact div_pos (mul_pos (by norm_num) (hε.trans hh)) (Real.exp_pos _)
-    · exact div_pos (hε.trans hh) (norm_pos_iff.2 (clampIm_ne_zero hε _))
-  exact (not_lt_of_ge hn0) hp
-
-def physicalQ {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ : ℝ)
-    (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (ω : Ω) (t : ℝ) : ℝ :=
-  ((g ω t z).re-Real.sqrt κ*B t.toNNReal ω)^2+(g ω t z).im^2
-
-def physicalPhi {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
-    (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (ω : Ω) (t : ℝ) : ℝ :=
-  lyapunov β C ((g ω t z).re-Real.sqrt κ*B t.toNNReal ω) (g ω t z).im
-
-lemma budget_dominates {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
-    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ)
-    (hz : z ∈ localizationDomain (γ ω) T) (hy : ε < (g ω T z).im) :
-    let C := (2*κ-8-κ*β)/4
-    let d := β*(κ-4-κ*β)/2
-    physicalPhi B κ β C g z ω T+d*(∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
-      budget B κ β C hε T z ω := by
-  dsimp only
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  let U := BrownianDriver.path B (Real.sqrt κ) T ω
-  have hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*B t.toNNReal ω) (Icc 0 (T:ℝ)) :=
-    fun t ht => BrownianDriver.extend_eq B _ T ω hc ht
-  have hb := hc.comp continuous_real_toNNReal
-  have hd : Continuous (fun t : ℝ => Real.sqrt κ*B t.toNNReal ω) := continuous_const.mul hb
-  obtain ⟨he,hs⟩ := cutoff_eq_until h hε hT U hU hd.continuousOn hz hy
-  have hh0 := h.height_antitone T.coe_nonneg hz ⟨le_rfl,T.coe_nonneg⟩ ⟨T.coe_nonneg,le_rfl⟩ T.coe_nonneg
-  dsimp only at hh0
-  rw [h.initial z hz.1.1] at hh0
-  have hin := deterministic_budget hε hT κ β hκ hβ hβκ U z (hy.le.trans hh0)
-    (fun t => B t.toNNReal ω) hb (by simpa using hb0) hU
-  dsimp only at hin
-  have heheight (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : LoewnerField.height hε T.coe_nonneg U z t = (g ω t z).im := by
-    rw [LoewnerField.height,he ht,max_eq_right]
-    exact hy.le.trans (h.height_antitone T.coe_nonneg hz ht ⟨T.coe_nonneg,le_rfl⟩ ht.2)
-  have hE : (∫ t in (0:ℝ)..stop hε T.coe_nonneg U z 0 0,
-      (((position hε T.coe_nonneg U z t).re-Real.sqrt κ*B t.toNNReal ω)^2+
-        (LoewnerField.height hε T.coe_nonneg U z t)^2)^(β/2-1)) =
-      ∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1) := by
-    rw [hs]
-    apply intervalIntegral.integral_congr
-    intro t ht
-    rw [uIcc_of_le T.coe_nonneg] at ht
-    dsimp only
-    rw [he ht,heheight t ht]
-    rfl
-  have hF : field hε T.coe_nonneg β C (Real.sqrt κ) U z (stop hε T.coe_nonneg U z 0 0)
-      (B (stop hε T.coe_nonneg U z 0 0).toNNReal ω) = physicalPhi B κ β C g z ω T := by
-    rw [hs]
-    simp only [field,physicalPhi]
-    rw [he ⟨T.coe_nonneg,le_rfl⟩,heheight _ ⟨T.coe_nonneg,le_rfl⟩]
-  have hbound : physicalPhi B κ β C g z ω T+d*(∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
-      correction B κ β C hε T z ω+lyapunov β C z.re z.im := by
-    have hh := hin.2
-    simp only [Real.toNNReal_zero] at hh
-    change field hε T.coe_nonneg β C (Real.sqrt κ) U z (stop hε T.coe_nonneg U z 0 0)
-        (B (stop hε T.coe_nonneg U z 0 0).toNNReal ω) + d*_ ≤
-      correction B κ β C hε T z ω+lyapunov β C z.re z.im at hh
-    rw [hF,hE] at hh
-    exact hh
-  exact hbound.trans (by
-    rw [budget,ite_eq_right (by exact ne_of_gt (show (0:ℝ≥0)<T from hT))]
-    exact le_max_left _ _)
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-
-lemma radial_occupation_point {q T α : ℝ} (hq : 0 < q) (hT : 0 < T)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1) : T^α ≤ q^α+T*q^(α-1) := by
-  by_cases hqT : T ≤ q
-  · exact (Real.rpow_le_rpow hT.le hqT hα).trans (le_add_of_nonneg_right (by positivity))
-  · have hh := Real.rpow_le_rpow_of_nonpos hq (not_le.1 hqT).le (by linarith : α-1≤0)
-    have hid : T*T^(α-1)=T^α := by
-      rw [Real.rpow_sub hT,Real.rpow_one]
-      field_simp
-    calc
-      T^α = T*T^(α-1) := hid.symm
-      _ ≤ T*q^(α-1) := mul_le_mul_of_nonneg_left hh hT.le
-      _ ≤ q^α+T*q^(α-1) := le_add_of_nonneg_left (by positivity)
-
-lemma occupation_lower {q φ : ℝ → ℝ} {T α : ℝ} (hT : 0 < T)
-    (hα : 0 ≤ α) (hα1 : α ≤ 1)
-    (hq : ContinuousOn q (Icc 0 T)) (hφ : ContinuousOn φ (Icc 0 T))
-    (hp : ∀ t ∈ Icc 0 T, 0 < q t) (hφq : ∀ t ∈ Icc 0 T, (q t)^α ≤ φ t) :
-    T^α ≤ (∫ t in (0:ℝ)..T, (q t)^(α-1))+(∫ t in (0:ℝ)..T,φ t)/T := by
-  have hH : ContinuousOn (fun t => (q t)^(α-1)) (Icc 0 T) :=
-    hq.rpow_const (fun t ht => Or.inl (hp t ht).ne')
-  have hiφ := hφ.intervalIntegrable_of_Icc (μ := volume) hT.le
-  have hiH := hH.intervalIntegrable_of_Icc (μ := volume) hT.le
-  have hm : (∫ _ in (0:ℝ)..T,T^α) ≤ ∫ t in (0:ℝ)..T,φ t+T*(q t)^(α-1) := by
-    apply intervalIntegral.integral_mono_on hT.le (intervalIntegrable_const) (hiφ.add (hiH.const_mul T))
-    intro t ht
-    exact (radial_occupation_point (hp t ht) hT hα hα1).trans (add_le_add (hφq t ht) le_rfl)
-  rw [intervalIntegral.integral_const,sub_zero,smul_eq_mul,
-    intervalIntegral.integral_add hiφ (hiH.const_mul T),intervalIntegral.integral_const_mul] at hm
-  have hm := (le_div_iff₀ hT).2 (by simpa only [mul_comm] using hm)
-  calc
-    T^α ≤ ((∫ t in (0:ℝ)..T,φ t)+T*(∫ t in (0:ℝ)..T,(q t)^(α-1)))/T := hm
-    _ = _ := by field_simp; ring
-
-lemma average_riemann_tendsto {φ : ℝ → ℝ} {T : ℝ} (hT : 0 < T)
-    (hφ : ContinuousOn φ (Icc 0 T)) :
-    Tendsto (fun n => (∑ i ∈ Finset.range (2^n),φ (TaylorMeshes.mesh T n i))/(2:ℝ)^n)
-      atTop (𝓝 ((∫ t in (0:ℝ)..T,φ t)/T)) := by
-  have hh := (TaylorMeshes.mesh_riemann_tendsto hT.le hφ).div_const T
-  convert hh using 1
-  funext n
-  rw [← Finset.sum_mul]
-  field_simp
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-open BrownianPredictable
-
-def heightCut (z : ℂ) (n : ℕ) : ℝ := z.im / ((n:ℝ)+1)
-
-lemma heightCut_pos {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) : 0 < heightCut z n :=
-  div_pos hz (by positivity)
-
-lemma heightCut_le {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) : heightCut z n ≤ z.im := by
-  apply div_le_self hz.le
-  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n
-  linarith
-
-lemma heightCut_tendsto (z : ℂ) : Tendsto (heightCut z) atTop (𝓝 0) := by
-  change Tendsto (fun n : ℕ => z.im/((n:ℝ)+1)) atTop (𝓝 0)
-  simpa only [mul_one,mul_zero,div_eq_mul_inv,one_mul] using
-    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul z.im
-
-def averagedBudget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
-    (T : ℝ≥0) (z : ℂ) (hz : z∈halfPlane) (n : ℕ) (ω : Ω) : ℝ :=
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  budget B κ β C (heightCut_pos hz n) T z ω/d+
-    (∑ i ∈ Finset.range (2^n), budget B κ β C (heightCut_pos hz n) (dyadicTime T n i) z ω)/(2:ℝ)^n
-
-lemma averagedBudget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β : ℝ)
-    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (T : ℝ≥0) (z : ℂ) (hz : z∈halfPlane) (n : ℕ) :
-    let C := (2*κ-8-κ*β)/4
-    let d := β*(κ-4-κ*β)/2
-    Integrable (averagedBudget B κ β T z hz n) P ∧
-      (∫ ω,averagedBudget B κ β T z hz n ω ∂P) =
-        lyapunov β C z.re z.im/d+lyapunov β C z.re z.im ∧
-      ∀ ω, 0 ≤ averagedBudget B κ β T z hz n ω := by
-  dsimp only
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  have hd : 0 < d := (generator_negative hκ hβ hβκ z.re hz).2.1
-  have hb (S : ℝ≥0) := budget_mean hB κ β hκ hβ hβκ (heightCut_pos hz n) S z (heightCut_le hz n)
-  have hs : Integrable (fun ω => ∑ i ∈ Finset.range (2^n), budget B κ β C (heightCut_pos hz n) (dyadicTime T n i) z ω) P :=
-    integrable_finsetSum _ (fun i hi => (hb _).1)
-  refine ⟨((hb T).1.div_const d).add (hs.div_const _),?_,?_⟩
-  · change (∫ ω,budget B κ β C (heightCut_pos hz n) T z ω/d+
-      (∑ i ∈ Finset.range (2^n), budget B κ β C (heightCut_pos hz n) (dyadicTime T n i) z ω)/(2:ℝ)^n ∂P)=_
-    rw [integral_add ((hb T).1.div_const _) (hs.div_const _),integral_div,integral_div,
-      (hb T).2.1,integral_finsetSum _ (fun i hi => (hb _).1)]
-    simp only [(hb _).2.1,Finset.sum_const,Finset.card_range,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat]
-    change lyapunov β C z.re z.im/d+((2:ℝ)^n*lyapunov β C z.re z.im)/(2:ℝ)^n =
-      lyapunov β C z.re z.im/d+lyapunov β C z.re z.im
-    congr 1
-    field_simp
-  · intro ω
-    apply add_nonneg
-    · exact div_nonneg ((hb T).2.2 ω) hd.le
-    · exact div_nonneg (Finset.sum_nonneg fun i hi => (hb _).2.2 ω) (by positivity)
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-open BrownianPredictable BrownianField BrownianIto TaylorMeshes GreenPhysical
-
-lemma budget_terminal_bound {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
-    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ)
-    (hz : z ∈ localizationDomain (γ ω) T) (hy : ε < (g ω T z).im) :
-    let C := (2*κ-8-κ*β)/4
-    physicalPhi B κ β C g z ω T ≤ budget B κ β C hε T z ω := by
-  dsimp only
-  by_cases ht : T=0
-  · subst T
-    simp [physicalPhi,h.initial z hz.1.1,hb0,budget]
-  have hT : 0 < (T:ℝ) := NNReal.coe_pos.2 (pos_iff_ne_zero.2 ht)
-  have hh := budget_dominates B κ β hκ hβ hβκ γ g f ω h hc hb0 hε T hT z hz hy
-  have hd := (generator_negative hκ hβ hβκ z.re hz.1.1).2.1
-  have hE : 0 ≤ ∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1) := by
-    apply intervalIntegral.integral_nonneg T.coe_nonneg
-    intro t ht
-    exact Real.rpow_nonneg (by dsimp [physicalQ]; positivity) _
-  exact (le_add_of_nonneg_right (mul_nonneg hd.le hE)).trans hh
-
-lemma physical_continuous {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) {T : ℝ} (hT : 0 ≤ T) (z : ℂ)
-    (hz : z ∈ localizationDomain (γ ω) T) :
-    ContinuousOn (physicalQ B κ g z ω) (Icc 0 T) ∧
-      ContinuousOn (physicalPhi B κ β C g z ω) (Icc 0 T) ∧
-      ∀ t ∈ Icc 0 T, 0 < physicalQ B κ g z ω t := by
-  have hg := h.continuous_eval hT hz
-  have hx := (Complex.continuous_re.comp_continuousOn hg).sub
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal) : Continuous (fun t : ℝ => Real.sqrt κ*B t.toNNReal ω)).continuousOn)
-  have hy := Complex.continuous_im.comp_continuousOn hg
-  have hyp (t : ℝ) (ht : t∈Icc 0 T) := h.im_pos ht.1 (survivingDomain_mono ht.2 hz)
-  have hq : ContinuousOn (physicalQ B κ g z ω) (Icc 0 T) := (hx.pow 2).add (hy.pow 2)
-  have hqp (t : ℝ) (ht : t∈Icc 0 T) : 0 < physicalQ B κ g z ω t := q_pos _ (hyp t ht)
-  refine ⟨hq,?_,hqp⟩
-  exact (hq.rpow_const (fun t ht => Or.inl (hqp t ht).ne')).add
-    (continuousOn_const.mul (hy.rpow_const (fun t ht => Or.inl (hyp t ht).ne')))
-
-lemma averagedBudget_lower {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
-    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hz : z ∈ halfPlane)
-    (halive : z ∈ localizationDomain (γ ω) T) (n : ℕ) (hy : heightCut z n < (g ω T z).im) :
-    let C := (2*κ-8-κ*β)/4
-    (∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1))+
-      (∑ i ∈ Finset.range (2^n),physicalPhi B κ β C g z ω (mesh T n i))/(2:ℝ)^n ≤
-        averagedBudget B κ β T z hz n ω := by
-  dsimp only
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  have hgen := generator_negative hκ hβ hβκ z.re hz
-  have hd : 0 < d := hgen.2.1
-  have hphi : 0 ≤ physicalPhi B κ β C g z ω T :=
-    lyapunov_nonneg hgen.1.le _ (h.im_pos T.coe_nonneg halive)
-  have hh := budget_dominates B κ β hκ hβ hβκ γ g f ω h hc hb0 (heightCut_pos hz n) T hT z halive hy
-  have hEc : (∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
-      budget B κ β C (heightCut_pos hz n) T z ω/d := by
-    apply (le_div_iff₀ hd).2
-    change physicalPhi B κ β C g z ω T+d*(∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
-      budget B κ β C (heightCut_pos hz n) T z ω at hh
-    nlinarith only [hh,hphi]
-  apply add_le_add hEc
-  apply div_le_div_of_nonneg_right _ (by positivity)
-  apply Finset.sum_le_sum
-  intro i hi
-  have hti : (dyadicTime T n i : ℝ) ∈ Icc 0 (T:ℝ) := by
-    rw [coe_dyadicTime]
-    exact mesh_mem T T.coe_nonneg n i (Finset.mem_range.1 hi).le
-  have hiY := hy.trans_le (h.height_antitone T.coe_nonneg halive hti ⟨T.coe_nonneg,le_rfl⟩ hti.2)
-  have hbound := budget_terminal_bound B κ β hκ hβ hβκ γ g f ω h hc hb0 (heightCut_pos hz n)
-    (dyadicTime T n i) z (survivingDomain_mono hti.2 halive) hiY
-  simpa only [coe_dyadicTime] using hbound
-
-lemma eventually_budget_large {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
-    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hz : z ∈ halfPlane)
-    (halive : z ∈ localizationDomain (γ ω) T) :
-    ∀ᶠ n in atTop, (T:ℝ)^(β/2)/2 ≤ averagedBudget B κ β T z hz n ω := by
-  let C := (2*κ-8-κ*β)/4
-  let E := ∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)
-  let φ := physicalPhi B κ β C g z ω
-  have hh := physical_continuous B κ β C γ g f ω h hc T.coe_nonneg z halive
-  have hg := generator_negative hκ hβ hβκ z.re hz
-  have hβ1 : β/2 ≤ 1 := by have h4 := div_pos (by norm_num : (0:ℝ)<4) (by linarith : 0<κ); linarith
-  have hφq (t : ℝ) (ht : t∈Icc 0 (T:ℝ)) : (physicalQ B κ g z ω t)^(β/2) ≤ φ t := by
-    apply le_add_of_nonneg_right
-    exact mul_nonneg hg.1.le (Real.rpow_nonneg (h.im_pos ht.1 (survivingDomain_mono ht.2 halive)).le _)
-  have hbound := occupation_lower hT (by positivity : (0:ℝ)≤β/2) hβ1 hh.1 hh.2.1 hh.2.2 hφq
-  have hlim := (average_riemann_tendsto hT hh.2.1).const_add E
-  have hlt : (T:ℝ)^(β/2)/2 < E+(∫ t in (0:ℝ)..(T:ℝ),φ t)/(T:ℝ) := by
-    have hp := Real.rpow_pos_of_pos hT (β/2)
-    linarith
-  have hlower := hlim.eventually (lt_mem_nhds hlt)
-  have hcut := (heightCut_tendsto z).eventually (gt_mem_nhds (h.im_pos T.coe_nonneg halive))
-  filter_upwards [hcut,hlower] with n hcut hn
-  exact hn.le.trans (averagedBudget_lower B κ β hκ hβ hβκ γ g f ω h hc hb0 T hT z hz halive n hcut)
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-
-lemma initial_uniform {β C : ℝ} (hβ : 0 ≤ β) (hC : 0 ≤ C) {z : ℂ}
-    (hz : z ∈ halfPlane) (hn : ‖z‖=1) : lyapunov β C z.re z.im ≤ 1+C := by
-  have hq : z.re^2+z.im^2=1 := by
-    have hh : ‖z‖^2=1 := by rw [hn]; norm_num
-    rw [Complex.sq_norm] at hh
-    simpa only [Complex.normSq_apply,pow_two] using hh
-  have hy1 : z.im ≤ 1 := by simpa only [hn] using Complex.im_le_norm z
-  have hpow := Real.rpow_le_rpow hz.le hy1 hβ
-  simp only [Real.one_rpow] at hpow
-  simp only [lyapunov,radial,hq,Real.one_rpow]
-  exact add_le_add le_rfl (by simpa only [mul_one] using mul_le_mul_of_nonneg_left hpow hC)
-
-lemma survival_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β : ℝ)
-    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hz : z∈halfPlane) (hn : ‖z‖=1) :
-    let C := (2*κ-8-κ*β)/4
-    let d := β*(κ-4-κ*β)/2
-    P {ω | z ∈ localizationDomain (γ ω) T} ≤
-      ENNReal.ofReal (2*((1+C)/d+(1+C))*(T:ℝ)^(-(β/2))) := by
-  dsimp only
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  let M := lyapunov β C z.re z.im/d+lyapunov β C z.re z.im
-  have hgen := generator_negative hκ hβ hβκ z.re hz
-  have hd : 0 < d := hgen.2.1
-  have hW (n : ℕ) := averagedBudget_mean hB κ β hκ hβ hβκ T z hz n
-  have he : ∀ᵐ ω ∂P, z ∈ localizationDomain (γ ω) T →
-      ∀ᶠ n in atTop, (T:ℝ)^(β/2)/2 ≤ averagedBudget B κ β T z hz n ω := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hm hc hb0
-    intro halive
-    exact eventually_budget_large B κ β hκ hβ hβκ γ g f ω hm hc hb0 T hT z hz halive
-  have hbound := measure_eventually_budget (averagedBudget B κ β T z hz) M ((T:ℝ)^(β/2)/2)
-    (by positivity) (fun n => (hW n).1) (fun n ω => (hW n).2.2 ω) (fun n => (hW n).2.1)
-    {ω | z ∈ localizationDomain (γ ω) T} he
-  have hinit := initial_uniform hβ.le hgen.1.le hz hn
-  have hM : M ≤ (1+C)/d+(1+C) := add_le_add (div_le_div_of_nonneg_right hinit hd.le) hinit
-  have hp : 0 < (T:ℝ)^(β/2) := Real.rpow_pos_of_pos hT _
-  refine hbound.trans (ENNReal.ofReal_le_ofReal ?_)
-  calc
-    M/((T:ℝ)^(β/2)/2) ≤ ((1+C)/d+(1+C))/((T:ℝ)^(β/2)/2) := div_le_div_of_nonneg_right hM (by positivity)
-    _ = _ := by
-      change ((1+C)/d+(1+C))/((T:ℝ)^(β/2)/2) = 2*((1+C)/d+(1+C))*(T:ℝ)^(-(β/2))
-      rw [Real.rpow_neg hT.le]
-      field_simp
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open Swallowing
-
-theorem swallowingTime_tail (κ : ℝ) (hκ : 4 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (z : ℂ), z ∈ halfPlane → ‖z‖ = 1 → ∀ t : ℝ, 0 < t →
-        P {sample | z ∈ localizationDomain (γ sample) t} ≤ ENNReal.ofReal (C*t^(-c)) := by
-  let β := (1-4/κ)/2
-  have hκ0 : 0 < κ := by linarith [hκ.1]
-  have ha : 0 < 1-4/κ := sub_pos.2 ((div_lt_one hκ0).2 hκ.1)
-  have hβ : 0 < β := half_pos ha
-  have hβκ : β < 1-4/κ := half_lt_self ha
-  let C := (2*κ-8-κ*β)/4
-  let d := β*(κ-4-κ*β)/2
-  have hgen := generator_negative hκ.1 hβ hβκ 0 (by norm_num : (0:ℝ)<1)
-  have hC : 0 < C := hgen.1
-  have hd : 0 < d := hgen.2.1
-  refine ⟨2*((1+C)/d+(1+C)),β/2,by positivity,by positivity,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz hn t ht
-  exact survival_bound hB κ β hκ.1 hβ hβκ γ g f hmaps ⟨t,ht.le⟩ ht z hz hn
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-def excursionBeforeSwallowing (γ : ℝ → ℂ) (z : ℂ) (A : ℝ) : Prop :=
-  ∃ t : ℝ, 0 ≤ t ∧ A ≤ ‖γ t‖ ∧
-    ∀ s ∈ Ico 0 t, z ∈ localizationDomain γ s
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-namespace Swallowing
-
-lemma early_excursion_forces_driver {κ A : ℝ} (hκ : 0 < κ) (hA : 8 < A)
-    {B : ℝ≥0 → ℝ} (hB : Continuous B) {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal) γ g f)
-    {t : ℝ} (ht : t ∈ Icc 0 A) (he : A ≤ ‖γ t‖) :
-    ∃ s ∈ Icc 0 (A.toNNReal), A/(2*Real.sqrt κ) < |B s| := by
-  by_contra! hmax
-  have hsqrt : 0 < Real.sqrt κ := Real.sqrt_pos.2 hκ
-  have hbound : ∀ s ∈ Icc 0 A,
-      -A/2 ≤ Real.sqrt κ * B s.toNNReal ∧ Real.sqrt κ * B s.toNNReal ≤ A/2 := by
-    intro s hs
-    have hb := hmax s.toNNReal ⟨by positivity, Real.toNNReal_le_toNNReal hs.2⟩
-    have ha : |Real.sqrt κ * B s.toNNReal| ≤ A/2 := by
-      rw [abs_mul,abs_of_pos hsqrt]
-      calc
-        Real.sqrt κ * |B s.toNNReal| ≤ Real.sqrt κ*(A/(2*Real.sqrt κ)) :=
-          mul_le_mul_of_nonneg_left hb hsqrt.le
-        _ = A/2 := by field_simp
-    exact ⟨by linarith [(abs_le.1 ha).1], (abs_le.1 ha).2⟩
-  have hr := trace_driver_rectangle hm (by linarith : 0 ≤ A)
-    ((continuous_const.mul (hB.comp continuous_real_toNNReal)).continuousOn) hbound t ht
-  have hx : (γ t).re^2 ≤ A^2/4 := by nlinarith [sq_nonneg ((γ t).re+A/2),sq_nonneg ((γ t).re-A/2)]
-  have hn : A^2 ≤ (γ t).re^2+(γ t).im^2 := by
-    have hh := sq_le_sq₀ (by linarith : 0 ≤ A) (norm_nonneg (γ t)) |>.2 he
-    rw [Complex.sq_norm,Complex.normSq_apply] at hh
-    simpa only [pow_two] using hh
-  nlinarith [sq_nonneg (A-8),hr.2.2,ht.2]
-
-lemma excursion_split {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (z : ℂ) (A : ℝ) (hA : 8 < A) :
-    P {ω | excursionBeforeSwallowing (γ ω) z A} ≤
-      P {ω | z ∈ localizationDomain (γ ω) A}+ ENNReal.ofReal (4*κ/A) := by
-  have hsub : ∀ᵐ ω ∂P, excursionBeforeSwallowing (γ ω) z A →
-      z ∈ localizationDomain (γ ω) A ∨
-      ∃ s ∈ Icc 0 A.toNNReal, A/(2*Real.sqrt κ) < |B s ω| := by
-    filter_upwards [hmaps,hB.cont] with ω hm hc he
-    obtain ⟨t,ht,hn,hlive⟩ := he
-    by_cases hta : t ≤ A
-    · exact Or.inr (early_excursion_forces_driver hκ hA hc hm ⟨ht,hta⟩ hn)
-    · exact Or.inl (hlive A ⟨by linarith,lt_of_not_ge hta⟩)
-  have hb := BrownianSupport.brownian_oscillation_bound hB A.toNNReal
-    (a := A/(2*Real.sqrt κ)) (by positivity)
-  have hcalc : (A.toNNReal:ℝ)/(A/(2*Real.sqrt κ))^2 = 4*κ/A := by
-    rw [Real.coe_toNNReal A (by linarith)]
-    have ha0 : A ≠ 0 := by linarith
-    have hs0 : Real.sqrt κ ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hκ)
-    field_simp
-    nlinarith [Real.sq_sqrt hκ.le]
-  rw [hcalc] at hb
-  have hm : P {ω | excursionBeforeSwallowing (γ ω) z A} ≤
-      P ({ω | z ∈ localizationDomain (γ ω) A} ∪
-        {ω | ∃ s ∈ Icc 0 A.toNNReal, A/(2*Real.sqrt κ) < |B s ω|}) := measure_mono_ae hsub
-  exact hm.trans ((measure_union_le _ _).trans (add_le_add le_rfl hb))
-
-end Swallowing
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-theorem swallowing_excursion_tail (κ : ℝ) (hκ : 4 < κ ∧ κ < 8) :
-    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (z : ℂ), z ∈ halfPlane → ‖z‖ = 1 → ∀ A : ℝ, 2 ≤ A →
-        P {sample | excursionBeforeSwallowing (γ sample) z A} ≤
-          ENNReal.ofReal (C*A^(-c)) := by
-  obtain ⟨C₀,c₀,hC₀,hc₀,htail⟩ := swallowingTime_tail κ hκ
-  let c := min c₀ 1
-  let C := C₀+4*κ+8
-  have hκ0 : 0 < κ := by linarith [hκ.1]
-  have hc : 0 < c := lt_min hc₀ zero_lt_one
-  have hc1 : c ≤ 1 := min_le_right _ _
-  have hcc : c ≤ c₀ := min_le_left _ _
-  have hC8 : 8 ≤ C := by dsimp [C]; linarith
-  refine ⟨C,c,by linarith,hc,?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps z hz hn A hA
-  have hA0 : 0 < A := by linarith
-  have hA1 : 1 ≤ A := by linarith
-  have hp : 0 < A^(-c) := Real.rpow_pos_of_pos hA0 _
-  by_cases hlarge : 8 < A
-  · have hsplit := Swallowing.excursion_split hB κ hκ0 γ g f hmaps z A hlarge
-    have htime := htail P B hB γ g f hγ hmaps z hz hn A hA0
-    refine (hsplit.trans (add_le_add htime le_rfl)).trans ?_
-    rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
-    apply ENNReal.ofReal_le_ofReal
-    have hpow := Real.rpow_le_rpow_of_exponent_le hA1 (neg_le_neg hcc)
-    have hinv : (1:ℝ)/A ≤ A^(-c) := by
-      have hh := Real.rpow_le_rpow_of_exponent_le hA1 (neg_le_neg hc1)
-      simpa only [Real.rpow_neg_one,one_div] using hh
-    calc
-      C₀*A^(-c₀)+4*κ/A ≤ C₀*A^(-c)+4*κ*A^(-c) :=
-        add_le_add (mul_le_mul_of_nonneg_left hpow hC₀.le) (by
-          simpa only [div_eq_mul_inv,mul_one,one_mul,mul_assoc] using
-            mul_le_mul_of_nonneg_left hinv (show 0 ≤ 4*κ by positivity))
-      _ ≤ C*A^(-c) := by dsimp [C]; nlinarith
-  · have hsmall : A ≤ 8 := le_of_not_gt hlarge
-    have hac : A^c ≤ 8 := by
-      have hh := Real.rpow_le_rpow_of_exponent_le hA1 hc1
-      have hh' : A^c ≤ A := by simpa only [Real.rpow_one] using hh
-      exact hh'.trans hsmall
-    have h8 : (1:ℝ) ≤ 8*A^(-c) := by
-      rw [Real.rpow_neg hA0.le,← div_eq_mul_inv]
-      exact (le_div_iff₀ (Real.rpow_pos_of_pos hA0 c)).2 (by simpa using hac)
-    have hone : (1:ℝ) ≤ C*A^(-c) := h8.trans (mul_le_mul_of_nonneg_right hC8 hp.le)
-    exact prob_le_one.trans (by simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal hone)
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology ENNReal NNReal OnePoint
-namespace SLEExactGauge
-
-def mappedLiveExit (Fbar : ℂ → OnePoint ℂ) (γ : ℝ → ℂ)
-    (z q : ℂ) (radius : ℝ) : Prop :=
-  ∃ t : ℝ, 0 ≤ t ∧ z ∈ localizationDomain γ t ∧
-    Fbar (γ t) ∉ (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.ball q radius
-
-lemma mappedLiveExit_split {Fbar : ℂ → OnePoint ℂ} {γ : ℝ → ℂ}
-    {z q : ℂ} {radius A : ℝ} (he : mappedLiveExit Fbar γ z q radius) :
-    conformalExit Fbar γ q radius A ∨ excursionBeforeSwallowing γ z A := by
-  obtain ⟨t,ht,hz,hout⟩ := he
-  by_cases hpre : ∀ s : ℝ, 0 ≤ s → s < t → ‖γ s‖ < A
-  · exact Or.inl ⟨t,ht,hpre,hout⟩
-  · push Not at hpre
-    obtain ⟨s,hs,hst,hn⟩ := hpre
-    exact Or.inr ⟨s,hs,hn,fun u hu => survivingDomain_mono (hu.2.trans hst).le hz⟩
-
-theorem mapped_live_exit_tail (κ : ℝ) (hκ : 4 < κ ∧ κ < 8) :
-    ∃ c C L₀ : ℝ, 0 < c ∧ 0 < C ∧ 1 < L₀ ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ)
-        (_hF : AnalyticOnNhd ℂ F halfPlane) (_hinj : Set.InjOn F halfPlane)
-        (_hext : ∀ z ∈ halfPlane, Fbar z = (F z : OnePoint ℂ))
-        (_hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-        (η : ℝ → ℂ) (_hη : ContinuousOn η (Icc 0 1)) (_hη0 : η 0 = 0)
-        (_hη1 : ‖η 1‖ = 1) (_hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-        (z q : ℂ) (_hz : z ∈ halfPlane) (_hznorm : ‖z‖ = 1) (_hzq : F z = q)
-        (r : ℝ) (_hr : 0 < r)
-        (_haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-          (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.closedBall q r)
-        (L : ℝ), L₀ ≤ L →
-        P {sample | mappedLiveExit Fbar (γ sample) z q (L*r)} ≤
-          ENNReal.ofReal (C*L^(-c)) := by
-  have hκ0 : 0 < κ := by linarith [hκ.1]
-  obtain ⟨b,c₁,C₁,L₁,hb,hc₁,hC₁,hL₁,hloc⟩ := uniform_conformal_localization κ ⟨hκ0,hκ.2⟩
-  obtain ⟨C₂,c₂,hC₂,hc₂,hswallow⟩ := swallowing_excursion_tail κ hκ
-  let c := min c₁ (b*c₂)
-  let L₀ := max L₁ ((2:ℝ)^(1/b))
-  have hc : 0 < c := lt_min hc₁ (mul_pos hb hc₂)
-  refine ⟨c,C₁+C₂,L₀,hc,add_pos hC₁ hC₂,hL₁.trans_le (le_max_left _ _),?_⟩
-  intro Ω _ P _ B hB γ g f hγ hmaps F Fbar hF hinj hext hcont η hη hη0 hη1 hηH z q hz hznorm hzq r hr haccess L hlarge
-  have hLL : L₁ ≤ L := (le_max_left _ _).trans hlarge
-  have hL : 1 < L := hL₁.trans_le hLL
-  have hLp : 0 < L := zero_lt_one.trans hL
-  have hA : (2:ℝ) ≤ L^b := by
-    calc
-      2 = ((2:ℝ)^(1/b))^b := by rw [← Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),one_div_mul_cancel hb.ne',Real.rpow_one]
-      _ ≤ L^b := Real.rpow_le_rpow (Real.rpow_nonneg (by norm_num) _) ((le_max_right _ _).trans hlarge) hb.le
-  have hp₁ := hloc P B hB γ g f hγ hmaps F Fbar hF hinj hext hcont η hη hη0 hη1 hηH q r hr haccess L hLL
-  have hp₂ := hswallow P B hB γ g f hγ hmaps z hz hznorm (L^b) hA
-  have hsub : {ω | mappedLiveExit Fbar (γ ω) z q (L*r)} ⊆
-      {ω | conformalExit Fbar (γ ω) q (L*r) (L^b)} ∪
-        {ω | excursionBeforeSwallowing (γ ω) z (L^b)} := fun ω he => mappedLiveExit_split he
-  have hpow₁ : L^(-c₁) ≤ L^(-c) := Real.rpow_le_rpow_of_exponent_le hL.le (neg_le_neg (min_le_left _ _))
-  have hpow₂ : (L^b)^(-c₂) ≤ L^(-c) := by
-    rw [← Real.rpow_mul hLp.le]
-    apply Real.rpow_le_rpow_of_exponent_le hL.le
-    nlinarith [show c ≤ b*c₂ from min_le_right _ _]
-  calc
-    _ ≤ P ({ω | conformalExit Fbar (γ ω) q (L*r) (L^b)} ∪
-        {ω | excursionBeforeSwallowing (γ ω) z (L^b)}) := measure_mono hsub
-    _ ≤ P {ω | conformalExit Fbar (γ ω) q (L*r) (L^b)}+
-        P {ω | excursionBeforeSwallowing (γ ω) z (L^b)} := measure_union_le _ _
-    _ ≤ ENNReal.ofReal (C₁*L^(-c₁))+ENNReal.ofReal (C₂*(L^b)^(-c₂)) := add_le_add hp₁ hp₂
-    _ = ENNReal.ofReal (C₁*L^(-c₁)+C₂*(L^b)^(-c₂)) := (ENNReal.ofReal_add (by positivity) (by positivity)).symm
-    _ ≤ _ := ENNReal.ofReal_le_ofReal (by
-      calc
-        _ ≤ C₁*L^(-c)+C₂*L^(-c) := add_le_add (mul_le_mul_of_nonneg_left hpow₁ hC₁.le) (mul_le_mul_of_nonneg_left hpow₂ hC₂.le)
-        _ = _ := by ring)
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.BrownianSupport
+import OAI.Probability.SLELocalization.CommonClocks
+import OAI.Probability.SLELocalization.ConformalLocalization
+import OAI.Probability.SLELocalization.FirstDisk
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+
+namespace OAI
+
+
+/-! Conformal localization and Swallowing estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.Swallowing
+open GreenPhysical LoewnerCutoff LoewnerExit
+
+lemma cutoff_eq_until {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (U : C(I,ℝ)) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (hc : ContinuousOn driver (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hy : ε < (g T z).im) :
+    EqOn (position hε hT.le U z) (fun t => g t z) (Icc 0 T) ∧
+      stop hε hT.le U z 0 0 = T := by
+  have he := h.eq_cutoff_solution hT.le hε hc hz hy.le (position_initial hε hT.le U z) (by
+    intro t ht
+    rw [← hU ht]
+    exact position_derivWithin hε hT.le U z ht)
+  have hpos (t : ℝ) (ht : t ∈ Icc 0 T) : ε < (position hε hT.le U z t).im := by
+    rw [← he ht]
+    exact hy.trans_le (h.height_antitone hT.le hz ht ⟨hT.le,le_rfl⟩ ht.2)
+  refine ⟨he.symm,?_⟩
+  apply le_antisymm (stop_bounds hε hT.le U z 0 0).2
+  by_contra hn
+  have hlt := lt_of_not_ge hn
+  have hn0 := nonpos_at_of_lt hε hT.le hT U z 0 0 hlt
+  have ht := stop_bounds hε hT.le U z 0 0
+  have hh := hpos _ ht
+  have hp : 0 < deficit hε hT.le U z 0 0 (stop hε hT.le U z 0 0) := by
+    apply (deficit_pos_iff hε hT.le U z 0 0 _).2
+    refine ⟨?_,hh,?_⟩
+    · exact div_pos (mul_pos (by norm_num) (hε.trans hh)) (Real.exp_pos _)
+    · exact div_pos (hε.trans hh) (norm_pos_iff.2 (clampIm_ne_zero hε _))
+  exact (not_lt_of_ge hn0) hp
+
+def physicalQ {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ : ℝ)
+    (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (ω : Ω) (t : ℝ) : ℝ :=
+  ((g ω t z).re-Real.sqrt κ*B t.toNNReal ω)^2+(g ω t z).im^2
+
+def physicalPhi {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
+    (g : Ω → ℝ → ℂ → ℂ) (z : ℂ) (ω : Ω) (t : ℝ) : ℝ :=
+  lyapunov β C ((g ω t z).re-Real.sqrt κ*B t.toNNReal ω) (g ω t z).im
+
+lemma budget_dominates {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
+    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ)
+    (hz : z ∈ localizationDomain (γ ω) T) (hy : ε < (g ω T z).im) :
+    let C := (2*κ-8-κ*β)/4
+    let d := β*(κ-4-κ*β)/2
+    physicalPhi B κ β C g z ω T+d*(∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
+      budget B κ β C hε T z ω := by
+  dsimp only
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  let U := BrownianDriver.path B (Real.sqrt κ) T ω
+  have hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*B t.toNNReal ω) (Icc 0 (T:ℝ)) :=
+    fun t ht => BrownianDriver.extend_eq B _ T ω hc ht
+  have hb := hc.comp continuous_real_toNNReal
+  have hd : Continuous (fun t : ℝ => Real.sqrt κ*B t.toNNReal ω) := continuous_const.mul hb
+  obtain ⟨he,hs⟩ := cutoff_eq_until h hε hT U hU hd.continuousOn hz hy
+  have hh0 := h.height_antitone T.coe_nonneg hz ⟨le_rfl,T.coe_nonneg⟩ ⟨T.coe_nonneg,le_rfl⟩ T.coe_nonneg
+  dsimp only at hh0
+  rw [h.initial z hz.1.1] at hh0
+  have hin := deterministic_budget hε hT κ β hκ hβ hβκ U z (hy.le.trans hh0)
+    (fun t => B t.toNNReal ω) hb (by simpa using hb0) hU
+  dsimp only at hin
+  have heheight (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : LoewnerField.height hε T.coe_nonneg U z t = (g ω t z).im := by
+    rw [LoewnerField.height,he ht,max_eq_right]
+    exact hy.le.trans (h.height_antitone T.coe_nonneg hz ht ⟨T.coe_nonneg,le_rfl⟩ ht.2)
+  have hE : (∫ t in (0:ℝ)..stop hε T.coe_nonneg U z 0 0,
+      (((position hε T.coe_nonneg U z t).re-Real.sqrt κ*B t.toNNReal ω)^2+
+        (LoewnerField.height hε T.coe_nonneg U z t)^2)^(β/2-1)) =
+      ∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1) := by
+    rw [hs]
+    apply intervalIntegral.integral_congr
+    intro t ht
+    rw [uIcc_of_le T.coe_nonneg] at ht
+    dsimp only
+    rw [he ht,heheight t ht]
+    rfl
+  have hF : field hε T.coe_nonneg β C (Real.sqrt κ) U z (stop hε T.coe_nonneg U z 0 0)
+      (B (stop hε T.coe_nonneg U z 0 0).toNNReal ω) = physicalPhi B κ β C g z ω T := by
+    rw [hs]
+    simp only [field,physicalPhi]
+    rw [he ⟨T.coe_nonneg,le_rfl⟩,heheight _ ⟨T.coe_nonneg,le_rfl⟩]
+  have hbound : physicalPhi B κ β C g z ω T+d*(∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
+      correction B κ β C hε T z ω+lyapunov β C z.re z.im := by
+    have hh := hin.2
+    simp only [Real.toNNReal_zero] at hh
+    change field hε T.coe_nonneg β C (Real.sqrt κ) U z (stop hε T.coe_nonneg U z 0 0)
+        (B (stop hε T.coe_nonneg U z 0 0).toNNReal ω) + d*_ ≤
+      correction B κ β C hε T z ω+lyapunov β C z.re z.im at hh
+    rw [hF,hE] at hh
+    exact hh
+  exact hbound.trans (by
+    rw [budget,ite_eq_right (by exact ne_of_gt (show (0:ℝ≥0)<T from hT))]
+    exact le_max_left _ _)
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+
+lemma radial_occupation_point {q T α : ℝ} (hq : 0 < q) (hT : 0 < T)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1) : T^α ≤ q^α+T*q^(α-1) := by
+  by_cases hqT : T ≤ q
+  · exact (Real.rpow_le_rpow hT.le hqT hα).trans (le_add_of_nonneg_right (by positivity))
+  · have hh := Real.rpow_le_rpow_of_nonpos hq (not_le.1 hqT).le (by linarith : α-1≤0)
+    have hid : T*T^(α-1)=T^α := by
+      rw [Real.rpow_sub hT,Real.rpow_one]
+      field_simp
+    calc
+      T^α = T*T^(α-1) := hid.symm
+      _ ≤ T*q^(α-1) := mul_le_mul_of_nonneg_left hh hT.le
+      _ ≤ q^α+T*q^(α-1) := le_add_of_nonneg_left (by positivity)
+
+lemma occupation_lower {q φ : ℝ → ℝ} {T α : ℝ} (hT : 0 < T)
+    (hα : 0 ≤ α) (hα1 : α ≤ 1)
+    (hq : ContinuousOn q (Icc 0 T)) (hφ : ContinuousOn φ (Icc 0 T))
+    (hp : ∀ t ∈ Icc 0 T, 0 < q t) (hφq : ∀ t ∈ Icc 0 T, (q t)^α ≤ φ t) :
+    T^α ≤ (∫ t in (0:ℝ)..T, (q t)^(α-1))+(∫ t in (0:ℝ)..T,φ t)/T := by
+  have hH : ContinuousOn (fun t => (q t)^(α-1)) (Icc 0 T) :=
+    hq.rpow_const (fun t ht => Or.inl (hp t ht).ne')
+  have hiφ := hφ.intervalIntegrable_of_Icc (μ := volume) hT.le
+  have hiH := hH.intervalIntegrable_of_Icc (μ := volume) hT.le
+  have hm : (∫ _ in (0:ℝ)..T,T^α) ≤ ∫ t in (0:ℝ)..T,φ t+T*(q t)^(α-1) := by
+    apply intervalIntegral.integral_mono_on hT.le (intervalIntegrable_const) (hiφ.add (hiH.const_mul T))
+    intro t ht
+    exact (radial_occupation_point (hp t ht) hT hα hα1).trans (add_le_add (hφq t ht) le_rfl)
+  rw [intervalIntegral.integral_const,sub_zero,smul_eq_mul,
+    intervalIntegral.integral_add hiφ (hiH.const_mul T),intervalIntegral.integral_const_mul] at hm
+  have hm := (le_div_iff₀ hT).2 (by simpa only [mul_comm] using hm)
+  calc
+    T^α ≤ ((∫ t in (0:ℝ)..T,φ t)+T*(∫ t in (0:ℝ)..T,(q t)^(α-1)))/T := hm
+    _ = _ := by field_simp; ring
+
+lemma average_riemann_tendsto {φ : ℝ → ℝ} {T : ℝ} (hT : 0 < T)
+    (hφ : ContinuousOn φ (Icc 0 T)) :
+    Tendsto (fun n => (∑ i ∈ Finset.range (2^n),φ (TaylorMeshes.mesh T n i))/(2:ℝ)^n)
+      atTop (𝓝 ((∫ t in (0:ℝ)..T,φ t)/T)) := by
+  have hh := (TaylorMeshes.mesh_riemann_tendsto hT.le hφ).div_const T
+  convert hh using 1
+  funext n
+  rw [← Finset.sum_mul]
+  field_simp
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+open BrownianPredictable
+
+def heightCut (z : ℂ) (n : ℕ) : ℝ := z.im / ((n:ℝ)+1)
+
+lemma heightCut_pos {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) : 0 < heightCut z n :=
+  div_pos hz (by positivity)
+
+lemma heightCut_le {z : ℂ} (hz : z ∈ halfPlane) (n : ℕ) : heightCut z n ≤ z.im := by
+  apply div_le_self hz.le
+  have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n
+  linarith
+
+lemma heightCut_tendsto (z : ℂ) : Tendsto (heightCut z) atTop (𝓝 0) := by
+  change Tendsto (fun n : ℕ => z.im/((n:ℝ)+1)) atTop (𝓝 0)
+  simpa only [mul_one,mul_zero,div_eq_mul_inv,one_mul] using
+    (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul z.im
+
+def averagedBudget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
+    (T : ℝ≥0) (z : ℂ) (hz : z∈halfPlane) (n : ℕ) (ω : Ω) : ℝ :=
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  budget B κ β C (heightCut_pos hz n) T z ω/d+
+    (∑ i ∈ Finset.range (2^n), budget B κ β C (heightCut_pos hz n) (dyadicTime T n i) z ω)/(2:ℝ)^n
+
+lemma averagedBudget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β : ℝ)
+    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (T : ℝ≥0) (z : ℂ) (hz : z∈halfPlane) (n : ℕ) :
+    let C := (2*κ-8-κ*β)/4
+    let d := β*(κ-4-κ*β)/2
+    Integrable (averagedBudget B κ β T z hz n) P ∧
+      (∫ ω,averagedBudget B κ β T z hz n ω ∂P) =
+        lyapunov β C z.re z.im/d+lyapunov β C z.re z.im ∧
+      ∀ ω, 0 ≤ averagedBudget B κ β T z hz n ω := by
+  dsimp only
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  have hd : 0 < d := (generator_negative hκ hβ hβκ z.re hz).2.1
+  have hb (S : ℝ≥0) := budget_mean hB κ β hκ hβ hβκ (heightCut_pos hz n) S z (heightCut_le hz n)
+  have hs : Integrable (fun ω => ∑ i ∈ Finset.range (2^n), budget B κ β C (heightCut_pos hz n) (dyadicTime T n i) z ω) P :=
+    integrable_finsetSum _ (fun i hi => (hb _).1)
+  refine ⟨((hb T).1.div_const d).add (hs.div_const _),?_,?_⟩
+  · change (∫ ω,budget B κ β C (heightCut_pos hz n) T z ω/d+
+      (∑ i ∈ Finset.range (2^n), budget B κ β C (heightCut_pos hz n) (dyadicTime T n i) z ω)/(2:ℝ)^n ∂P)=_
+    rw [integral_add ((hb T).1.div_const _) (hs.div_const _),integral_div,integral_div,
+      (hb T).2.1,integral_finsetSum _ (fun i hi => (hb _).1)]
+    simp only [(hb _).2.1,Finset.sum_const,Finset.card_range,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat]
+    change lyapunov β C z.re z.im/d+((2:ℝ)^n*lyapunov β C z.re z.im)/(2:ℝ)^n =
+      lyapunov β C z.re z.im/d+lyapunov β C z.re z.im
+    congr 1
+    field_simp
+  · intro ω
+    apply add_nonneg
+    · exact div_nonneg ((hb T).2.2 ω) hd.le
+    · exact div_nonneg (Finset.sum_nonneg fun i hi => (hb _).2.2 ω) (by positivity)
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+open BrownianPredictable BrownianField BrownianIto TaylorMeshes GreenPhysical
+
+lemma budget_terminal_bound {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
+    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (z : ℂ)
+    (hz : z ∈ localizationDomain (γ ω) T) (hy : ε < (g ω T z).im) :
+    let C := (2*κ-8-κ*β)/4
+    physicalPhi B κ β C g z ω T ≤ budget B κ β C hε T z ω := by
+  dsimp only
+  by_cases ht : T=0
+  · subst T
+    simp [physicalPhi,h.initial z hz.1.1,hb0,budget]
+  have hT : 0 < (T:ℝ) := NNReal.coe_pos.2 (pos_iff_ne_zero.2 ht)
+  have hh := budget_dominates B κ β hκ hβ hβκ γ g f ω h hc hb0 hε T hT z hz hy
+  have hd := (generator_negative hκ hβ hβκ z.re hz.1.1).2.1
+  have hE : 0 ≤ ∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1) := by
+    apply intervalIntegral.integral_nonneg T.coe_nonneg
+    intro t ht
+    exact Real.rpow_nonneg (by dsimp [physicalQ]; positivity) _
+  exact (le_add_of_nonneg_right (mul_nonneg hd.le hE)).trans hh
+
+lemma physical_continuous {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β C : ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) {T : ℝ} (hT : 0 ≤ T) (z : ℂ)
+    (hz : z ∈ localizationDomain (γ ω) T) :
+    ContinuousOn (physicalQ B κ g z ω) (Icc 0 T) ∧
+      ContinuousOn (physicalPhi B κ β C g z ω) (Icc 0 T) ∧
+      ∀ t ∈ Icc 0 T, 0 < physicalQ B κ g z ω t := by
+  have hg := h.continuous_eval hT hz
+  have hx := (Complex.continuous_re.comp_continuousOn hg).sub
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal) : Continuous (fun t : ℝ => Real.sqrt κ*B t.toNNReal ω)).continuousOn)
+  have hy := Complex.continuous_im.comp_continuousOn hg
+  have hyp (t : ℝ) (ht : t∈Icc 0 T) := h.im_pos ht.1 (survivingDomain_mono ht.2 hz)
+  have hq : ContinuousOn (physicalQ B κ g z ω) (Icc 0 T) := (hx.pow 2).add (hy.pow 2)
+  have hqp (t : ℝ) (ht : t∈Icc 0 T) : 0 < physicalQ B κ g z ω t := q_pos _ (hyp t ht)
+  refine ⟨hq,?_,hqp⟩
+  exact (hq.rpow_const (fun t ht => Or.inl (hqp t ht).ne')).add
+    (continuousOn_const.mul (hy.rpow_const (fun t ht => Or.inl (hyp t ht).ne')))
+
+lemma averagedBudget_lower {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
+    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hz : z ∈ halfPlane)
+    (halive : z ∈ localizationDomain (γ ω) T) (n : ℕ) (hy : heightCut z n < (g ω T z).im) :
+    let C := (2*κ-8-κ*β)/4
+    (∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1))+
+      (∑ i ∈ Finset.range (2^n),physicalPhi B κ β C g z ω (mesh T n i))/(2:ℝ)^n ≤
+        averagedBudget B κ β T z hz n ω := by
+  dsimp only
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  have hgen := generator_negative hκ hβ hβκ z.re hz
+  have hd : 0 < d := hgen.2.1
+  have hphi : 0 ≤ physicalPhi B κ β C g z ω T :=
+    lyapunov_nonneg hgen.1.le _ (h.im_pos T.coe_nonneg halive)
+  have hh := budget_dominates B κ β hκ hβ hβκ γ g f ω h hc hb0 (heightCut_pos hz n) T hT z halive hy
+  have hEc : (∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
+      budget B κ β C (heightCut_pos hz n) T z ω/d := by
+    apply (le_div_iff₀ hd).2
+    change physicalPhi B κ β C g z ω T+d*(∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)) ≤
+      budget B κ β C (heightCut_pos hz n) T z ω at hh
+    nlinarith only [hh,hphi]
+  apply add_le_add hEc
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  apply Finset.sum_le_sum
+  intro i hi
+  have hti : (dyadicTime T n i : ℝ) ∈ Icc 0 (T:ℝ) := by
+    rw [coe_dyadicTime]
+    exact mesh_mem T T.coe_nonneg n i (Finset.mem_range.1 hi).le
+  have hiY := hy.trans_le (h.height_antitone T.coe_nonneg halive hti ⟨T.coe_nonneg,le_rfl⟩ hti.2)
+  have hbound := budget_terminal_bound B κ β hκ hβ hβκ γ g f ω h hc hb0 (heightCut_pos hz n)
+    (dyadicTime T n i) z (survivingDomain_mono hti.2 halive) hiY
+  simpa only [coe_dyadicTime] using hbound
+
+lemma eventually_budget_large {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β : ℝ)
+    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hz : z ∈ halfPlane)
+    (halive : z ∈ localizationDomain (γ ω) T) :
+    ∀ᶠ n in atTop, (T:ℝ)^(β/2)/2 ≤ averagedBudget B κ β T z hz n ω := by
+  let C := (2*κ-8-κ*β)/4
+  let E := ∫ t in (0:ℝ)..(T:ℝ), (physicalQ B κ g z ω t)^(β/2-1)
+  let φ := physicalPhi B κ β C g z ω
+  have hh := physical_continuous B κ β C γ g f ω h hc T.coe_nonneg z halive
+  have hg := generator_negative hκ hβ hβκ z.re hz
+  have hβ1 : β/2 ≤ 1 := by have h4 := div_pos (by norm_num : (0:ℝ)<4) (by linarith : 0<κ); linarith
+  have hφq (t : ℝ) (ht : t∈Icc 0 (T:ℝ)) : (physicalQ B κ g z ω t)^(β/2) ≤ φ t := by
+    apply le_add_of_nonneg_right
+    exact mul_nonneg hg.1.le (Real.rpow_nonneg (h.im_pos ht.1 (survivingDomain_mono ht.2 halive)).le _)
+  have hbound := occupation_lower hT (by positivity : (0:ℝ)≤β/2) hβ1 hh.1 hh.2.1 hh.2.2 hφq
+  have hlim := (average_riemann_tendsto hT hh.2.1).const_add E
+  have hlt : (T:ℝ)^(β/2)/2 < E+(∫ t in (0:ℝ)..(T:ℝ),φ t)/(T:ℝ) := by
+    have hp := Real.rpow_pos_of_pos hT (β/2)
+    linarith
+  have hlower := hlim.eventually (lt_mem_nhds hlt)
+  have hcut := (heightCut_tendsto z).eventually (gt_mem_nhds (h.im_pos T.coe_nonneg halive))
+  filter_upwards [hcut,hlower] with n hcut hn
+  exact hn.le.trans (averagedBudget_lower B κ β hκ hβ hβκ γ g f ω h hc hb0 T hT z hz halive n hcut)
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+
+lemma initial_uniform {β C : ℝ} (hβ : 0 ≤ β) (hC : 0 ≤ C) {z : ℂ}
+    (hz : z ∈ halfPlane) (hn : ‖z‖=1) : lyapunov β C z.re z.im ≤ 1+C := by
+  have hq : z.re^2+z.im^2=1 := by
+    have hh : ‖z‖^2=1 := by rw [hn]; norm_num
+    rw [Complex.sq_norm] at hh
+    simpa only [Complex.normSq_apply,pow_two] using hh
+  have hy1 : z.im ≤ 1 := by simpa only [hn] using Complex.im_le_norm z
+  have hpow := Real.rpow_le_rpow hz.le hy1 hβ
+  simp only [Real.one_rpow] at hpow
+  simp only [lyapunov,radial,hq,Real.one_rpow]
+  exact add_le_add le_rfl (by simpa only [mul_one] using mul_le_mul_of_nonneg_left hpow hC)
+
+lemma survival_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β : ℝ)
+    (hκ : 4 < κ) (hβ : 0 < β) (hβκ : β < 1-4/κ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (T : ℝ≥0) (hT : 0 < (T:ℝ)) (z : ℂ) (hz : z∈halfPlane) (hn : ‖z‖=1) :
+    let C := (2*κ-8-κ*β)/4
+    let d := β*(κ-4-κ*β)/2
+    P {ω | z ∈ localizationDomain (γ ω) T} ≤
+      ENNReal.ofReal (2*((1+C)/d+(1+C))*(T:ℝ)^(-(β/2))) := by
+  dsimp only
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  let M := lyapunov β C z.re z.im/d+lyapunov β C z.re z.im
+  have hgen := generator_negative hκ hβ hβκ z.re hz
+  have hd : 0 < d := hgen.2.1
+  have hW (n : ℕ) := averagedBudget_mean hB κ β hκ hβ hβκ T z hz n
+  have he : ∀ᵐ ω ∂P, z ∈ localizationDomain (γ ω) T →
+      ∀ᶠ n in atTop, (T:ℝ)^(β/2)/2 ≤ averagedBudget B κ β T z hz n ω := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hm hc hb0
+    intro halive
+    exact eventually_budget_large B κ β hκ hβ hβκ γ g f ω hm hc hb0 T hT z hz halive
+  have hbound := measure_eventually_budget (averagedBudget B κ β T z hz) M ((T:ℝ)^(β/2)/2)
+    (by positivity) (fun n => (hW n).1) (fun n ω => (hW n).2.2 ω) (fun n => (hW n).2.1)
+    {ω | z ∈ localizationDomain (γ ω) T} he
+  have hinit := initial_uniform hβ.le hgen.1.le hz hn
+  have hM : M ≤ (1+C)/d+(1+C) := add_le_add (div_le_div_of_nonneg_right hinit hd.le) hinit
+  have hp : 0 < (T:ℝ)^(β/2) := Real.rpow_pos_of_pos hT _
+  refine hbound.trans (ENNReal.ofReal_le_ofReal ?_)
+  calc
+    M/((T:ℝ)^(β/2)/2) ≤ ((1+C)/d+(1+C))/((T:ℝ)^(β/2)/2) := div_le_div_of_nonneg_right hM (by positivity)
+    _ = _ := by
+      change ((1+C)/d+(1+C))/((T:ℝ)^(β/2)/2) = 2*((1+C)/d+(1+C))*(T:ℝ)^(-(β/2))
+      rw [Real.rpow_neg hT.le]
+      field_simp
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open Swallowing
+
+theorem swallowingTime_tail (κ : ℝ) (hκ : 4 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (z : ℂ), z ∈ halfPlane → ‖z‖ = 1 → ∀ t : ℝ, 0 < t →
+        P {sample | z ∈ localizationDomain (γ sample) t} ≤ ENNReal.ofReal (C*t^(-c)) := by
+  let β := (1-4/κ)/2
+  have hκ0 : 0 < κ := by linarith [hκ.1]
+  have ha : 0 < 1-4/κ := sub_pos.2 ((div_lt_one hκ0).2 hκ.1)
+  have hβ : 0 < β := half_pos ha
+  have hβκ : β < 1-4/κ := half_lt_self ha
+  let C := (2*κ-8-κ*β)/4
+  let d := β*(κ-4-κ*β)/2
+  have hgen := generator_negative hκ.1 hβ hβκ 0 (by norm_num : (0:ℝ)<1)
+  have hC : 0 < C := hgen.1
+  have hd : 0 < d := hgen.2.1
+  refine ⟨2*((1+C)/d+(1+C)),β/2,by positivity,by positivity,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz hn t ht
+  exact survival_bound hB κ β hκ.1 hβ hβκ γ g f hmaps ⟨t,ht.le⟩ ht z hz hn
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+def excursionBeforeSwallowing (γ : ℝ → ℂ) (z : ℂ) (A : ℝ) : Prop :=
+  ∃ t : ℝ, 0 ≤ t ∧ A ≤ ‖γ t‖ ∧
+    ∀ s ∈ Ico 0 t, z ∈ localizationDomain γ s
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+namespace Swallowing
+
+lemma early_excursion_forces_driver {κ A : ℝ} (hκ : 0 < κ) (hA : 8 < A)
+    {B : ℝ≥0 → ℝ} (hB : Continuous B) {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal) γ g f)
+    {t : ℝ} (ht : t ∈ Icc 0 A) (he : A ≤ ‖γ t‖) :
+    ∃ s ∈ Icc 0 (A.toNNReal), A/(2*Real.sqrt κ) < |B s| := by
+  by_contra! hmax
+  have hsqrt : 0 < Real.sqrt κ := Real.sqrt_pos.2 hκ
+  have hbound : ∀ s ∈ Icc 0 A,
+      -A/2 ≤ Real.sqrt κ * B s.toNNReal ∧ Real.sqrt κ * B s.toNNReal ≤ A/2 := by
+    intro s hs
+    have hb := hmax s.toNNReal ⟨by positivity, Real.toNNReal_le_toNNReal hs.2⟩
+    have ha : |Real.sqrt κ * B s.toNNReal| ≤ A/2 := by
+      rw [abs_mul,abs_of_pos hsqrt]
+      calc
+        Real.sqrt κ * |B s.toNNReal| ≤ Real.sqrt κ*(A/(2*Real.sqrt κ)) :=
+          mul_le_mul_of_nonneg_left hb hsqrt.le
+        _ = A/2 := by field_simp
+    exact ⟨by linarith [(abs_le.1 ha).1], (abs_le.1 ha).2⟩
+  have hr := trace_driver_rectangle hm (by linarith : 0 ≤ A)
+    ((continuous_const.mul (hB.comp continuous_real_toNNReal)).continuousOn) hbound t ht
+  have hx : (γ t).re^2 ≤ A^2/4 := by nlinarith [sq_nonneg ((γ t).re+A/2),sq_nonneg ((γ t).re-A/2)]
+  have hn : A^2 ≤ (γ t).re^2+(γ t).im^2 := by
+    have hh := sq_le_sq₀ (by linarith : 0 ≤ A) (norm_nonneg (γ t)) |>.2 he
+    rw [Complex.sq_norm,Complex.normSq_apply] at hh
+    simpa only [pow_two] using hh
+  nlinarith [sq_nonneg (A-8),hr.2.2,ht.2]
+
+lemma excursion_split {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (κ : ℝ) (hκ : 0 < κ) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (z : ℂ) (A : ℝ) (hA : 8 < A) :
+    P {ω | excursionBeforeSwallowing (γ ω) z A} ≤
+      P {ω | z ∈ localizationDomain (γ ω) A}+ ENNReal.ofReal (4*κ/A) := by
+  have hsub : ∀ᵐ ω ∂P, excursionBeforeSwallowing (γ ω) z A →
+      z ∈ localizationDomain (γ ω) A ∨
+      ∃ s ∈ Icc 0 A.toNNReal, A/(2*Real.sqrt κ) < |B s ω| := by
+    filter_upwards [hmaps,hB.cont] with ω hm hc he
+    obtain ⟨t,ht,hn,hlive⟩ := he
+    by_cases hta : t ≤ A
+    · exact Or.inr (early_excursion_forces_driver hκ hA hc hm ⟨ht,hta⟩ hn)
+    · exact Or.inl (hlive A ⟨by linarith,lt_of_not_ge hta⟩)
+  have hb := BrownianSupport.brownian_oscillation_bound hB A.toNNReal
+    (a := A/(2*Real.sqrt κ)) (by positivity)
+  have hcalc : (A.toNNReal:ℝ)/(A/(2*Real.sqrt κ))^2 = 4*κ/A := by
+    rw [Real.coe_toNNReal A (by linarith)]
+    have ha0 : A ≠ 0 := by linarith
+    have hs0 : Real.sqrt κ ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hκ)
+    field_simp
+    nlinarith [Real.sq_sqrt hκ.le]
+  rw [hcalc] at hb
+  have hm : P {ω | excursionBeforeSwallowing (γ ω) z A} ≤
+      P ({ω | z ∈ localizationDomain (γ ω) A} ∪
+        {ω | ∃ s ∈ Icc 0 A.toNNReal, A/(2*Real.sqrt κ) < |B s ω|}) := measure_mono_ae hsub
+  exact hm.trans ((measure_union_le _ _).trans (add_le_add le_rfl hb))
+
+end Swallowing
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+theorem swallowing_excursion_tail (κ : ℝ) (hκ : 4 < κ ∧ κ < 8) :
+    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (z : ℂ), z ∈ halfPlane → ‖z‖ = 1 → ∀ A : ℝ, 2 ≤ A →
+        P {sample | excursionBeforeSwallowing (γ sample) z A} ≤
+          ENNReal.ofReal (C*A^(-c)) := by
+  obtain ⟨C₀,c₀,hC₀,hc₀,htail⟩ := swallowingTime_tail κ hκ
+  let c := min c₀ 1
+  let C := C₀+4*κ+8
+  have hκ0 : 0 < κ := by linarith [hκ.1]
+  have hc : 0 < c := lt_min hc₀ zero_lt_one
+  have hc1 : c ≤ 1 := min_le_right _ _
+  have hcc : c ≤ c₀ := min_le_left _ _
+  have hC8 : 8 ≤ C := by dsimp [C]; linarith
+  refine ⟨C,c,by linarith,hc,?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps z hz hn A hA
+  have hA0 : 0 < A := by linarith
+  have hA1 : 1 ≤ A := by linarith
+  have hp : 0 < A^(-c) := Real.rpow_pos_of_pos hA0 _
+  by_cases hlarge : 8 < A
+  · have hsplit := Swallowing.excursion_split hB κ hκ0 γ g f hmaps z A hlarge
+    have htime := htail P B hB γ g f hγ hmaps z hz hn A hA0
+    refine (hsplit.trans (add_le_add htime le_rfl)).trans ?_
+    rw [← ENNReal.ofReal_add (by positivity) (by positivity)]
+    apply ENNReal.ofReal_le_ofReal
+    have hpow := Real.rpow_le_rpow_of_exponent_le hA1 (neg_le_neg hcc)
+    have hinv : (1:ℝ)/A ≤ A^(-c) := by
+      have hh := Real.rpow_le_rpow_of_exponent_le hA1 (neg_le_neg hc1)
+      simpa only [Real.rpow_neg_one,one_div] using hh
+    calc
+      C₀*A^(-c₀)+4*κ/A ≤ C₀*A^(-c)+4*κ*A^(-c) :=
+        add_le_add (mul_le_mul_of_nonneg_left hpow hC₀.le) (by
+          simpa only [div_eq_mul_inv,mul_one,one_mul,mul_assoc] using
+            mul_le_mul_of_nonneg_left hinv (show 0 ≤ 4*κ by positivity))
+      _ ≤ C*A^(-c) := by dsimp [C]; nlinarith
+  · have hsmall : A ≤ 8 := le_of_not_gt hlarge
+    have hac : A^c ≤ 8 := by
+      have hh := Real.rpow_le_rpow_of_exponent_le hA1 hc1
+      have hh' : A^c ≤ A := by simpa only [Real.rpow_one] using hh
+      exact hh'.trans hsmall
+    have h8 : (1:ℝ) ≤ 8*A^(-c) := by
+      rw [Real.rpow_neg hA0.le,← div_eq_mul_inv]
+      exact (le_div_iff₀ (Real.rpow_pos_of_pos hA0 c)).2 (by simpa using hac)
+    have hone : (1:ℝ) ≤ C*A^(-c) := h8.trans (mul_le_mul_of_nonneg_right hC8 hp.le)
+    exact prob_le_one.trans (by simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal hone)
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology ENNReal NNReal OnePoint
+namespace SLEExactGauge
+
+def mappedLiveExit (Fbar : ℂ → OnePoint ℂ) (γ : ℝ → ℂ)
+    (z q : ℂ) (radius : ℝ) : Prop :=
+  ∃ t : ℝ, 0 ≤ t ∧ z ∈ localizationDomain γ t ∧
+    Fbar (γ t) ∉ (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.ball q radius
+
+lemma mappedLiveExit_split {Fbar : ℂ → OnePoint ℂ} {γ : ℝ → ℂ}
+    {z q : ℂ} {radius A : ℝ} (he : mappedLiveExit Fbar γ z q radius) :
+    conformalExit Fbar γ q radius A ∨ excursionBeforeSwallowing γ z A := by
+  obtain ⟨t,ht,hz,hout⟩ := he
+  by_cases hpre : ∀ s : ℝ, 0 ≤ s → s < t → ‖γ s‖ < A
+  · exact Or.inl ⟨t,ht,hpre,hout⟩
+  · push Not at hpre
+    obtain ⟨s,hs,hst,hn⟩ := hpre
+    exact Or.inr ⟨s,hs,hn,fun u hu => survivingDomain_mono (hu.2.trans hst).le hz⟩
+
+theorem mapped_live_exit_tail (κ : ℝ) (hκ : 4 < κ ∧ κ < 8) :
+    ∃ c C L₀ : ℝ, 0 < c ∧ 0 < C ∧ 1 < L₀ ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (F : ℂ → ℂ) (Fbar : ℂ → OnePoint ℂ)
+        (_hF : AnalyticOnNhd ℂ F halfPlane) (_hinj : Set.InjOn F halfPlane)
+        (_hext : ∀ z ∈ halfPlane, Fbar z = (F z : OnePoint ℂ))
+        (_hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+        (η : ℝ → ℂ) (_hη : ContinuousOn η (Icc 0 1)) (_hη0 : η 0 = 0)
+        (_hη1 : ‖η 1‖ = 1) (_hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+        (z q : ℂ) (_hz : z ∈ halfPlane) (_hznorm : ‖z‖ = 1) (_hzq : F z = q)
+        (r : ℝ) (_hr : 0 < r)
+        (_haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+          (fun w : ℂ => (w : OnePoint ℂ)) '' Metric.closedBall q r)
+        (L : ℝ), L₀ ≤ L →
+        P {sample | mappedLiveExit Fbar (γ sample) z q (L*r)} ≤
+          ENNReal.ofReal (C*L^(-c)) := by
+  have hκ0 : 0 < κ := by linarith [hκ.1]
+  obtain ⟨b,c₁,C₁,L₁,hb,hc₁,hC₁,hL₁,hloc⟩ := uniform_conformal_localization κ ⟨hκ0,hκ.2⟩
+  obtain ⟨C₂,c₂,hC₂,hc₂,hswallow⟩ := swallowing_excursion_tail κ hκ
+  let c := min c₁ (b*c₂)
+  let L₀ := max L₁ ((2:ℝ)^(1/b))
+  have hc : 0 < c := lt_min hc₁ (mul_pos hb hc₂)
+  refine ⟨c,C₁+C₂,L₀,hc,add_pos hC₁ hC₂,hL₁.trans_le (le_max_left _ _),?_⟩
+  intro Ω _ P _ B hB γ g f hγ hmaps F Fbar hF hinj hext hcont η hη hη0 hη1 hηH z q hz hznorm hzq r hr haccess L hlarge
+  have hLL : L₁ ≤ L := (le_max_left _ _).trans hlarge
+  have hL : 1 < L := hL₁.trans_le hLL
+  have hLp : 0 < L := zero_lt_one.trans hL
+  have hA : (2:ℝ) ≤ L^b := by
+    calc
+      2 = ((2:ℝ)^(1/b))^b := by rw [← Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),one_div_mul_cancel hb.ne',Real.rpow_one]
+      _ ≤ L^b := Real.rpow_le_rpow (Real.rpow_nonneg (by norm_num) _) ((le_max_right _ _).trans hlarge) hb.le
+  have hp₁ := hloc P B hB γ g f hγ hmaps F Fbar hF hinj hext hcont η hη hη0 hη1 hηH q r hr haccess L hLL
+  have hp₂ := hswallow P B hB γ g f hγ hmaps z hz hznorm (L^b) hA
+  have hsub : {ω | mappedLiveExit Fbar (γ ω) z q (L*r)} ⊆
+      {ω | conformalExit Fbar (γ ω) q (L*r) (L^b)} ∪
+        {ω | excursionBeforeSwallowing (γ ω) z (L^b)} := fun ω he => mappedLiveExit_split he
+  have hpow₁ : L^(-c₁) ≤ L^(-c) := Real.rpow_le_rpow_of_exponent_le hL.le (neg_le_neg (min_le_left _ _))
+  have hpow₂ : (L^b)^(-c₂) ≤ L^(-c) := by
+    rw [← Real.rpow_mul hLp.le]
+    apply Real.rpow_le_rpow_of_exponent_le hL.le
+    nlinarith [show c ≤ b*c₂ from min_le_right _ _]
+  calc
+    _ ≤ P ({ω | conformalExit Fbar (γ ω) q (L*r) (L^b)} ∪
+        {ω | excursionBeforeSwallowing (γ ω) z (L^b)}) := measure_mono hsub
+    _ ≤ P {ω | conformalExit Fbar (γ ω) q (L*r) (L^b)}+
+        P {ω | excursionBeforeSwallowing (γ ω) z (L^b)} := measure_union_le _ _
+    _ ≤ ENNReal.ofReal (C₁*L^(-c₁))+ENNReal.ofReal (C₂*(L^b)^(-c₂)) := add_le_add hp₁ hp₂
+    _ = ENNReal.ofReal (C₁*L^(-c₁)+C₂*(L^b)^(-c₂)) := (ENNReal.ofReal_add (by positivity) (by positivity)).symm
+    _ ≤ _ := ENNReal.ofReal_le_ofReal (by
+      calc
+        _ ≤ C₁*L^(-c)+C₂*L^(-c) := add_le_add (mul_le_mul_of_nonneg_left hpow₁ hC₁.le) (mul_le_mul_of_nonneg_left hpow₂ hC₂.le)
+        _ = _ := by ring)
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/MeshExponents.lean b/lean/OAI/Probability/SLELocalization/MeshExponents.lean
index 00287b2718ec5c3d28dabe0b68ab4cff67fad03f..b6f80805cdc6cb8a23b9f9dd76004a3885130525 100644
--- a/lean/OAI/Probability/SLELocalization/MeshExponents.lean
+++ b/lean/OAI/Probability/SLELocalization/MeshExponents.lean
@@ -1,1082 +1,1082 @@
-import Mathlib
-import OAI.Probability.SLELocalization.SlabMoments
-import OAI.Probability.SLE.ClippedCoordinates
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Mesh estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Mesh
-
-lemma interval_square_bound {A : Set ℂ} (hA : volume A ≠ ⊤) (C : ℝ → ℂ → ℝ)
-    {T K J R : ℝ} (hT : 0 ≤ T) (hJ : 0 ≤ J)
-    (hb : (∫⁻ t, ENNReal.ofReal (|∫ z in A, C t z|^2)) ≤
-      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (J*R^2)*volume A) :
-    (∫ t in (0:ℝ)..T, (∫ z in A, C t z)^2) ≤ 2*K^2*(J*R^2)*(volume A).toReal := by
-  rw [intervalIntegral.integral_of_le hT]
-  have hh : ENNReal.ofReal (∫ t in Ioc (0:ℝ) T, (∫ z in A, C t z)^2) ≤
-      ∫⁻ t in Ioc (0:ℝ) T, ENNReal.ofReal ((∫ z in A, C t z)^2) := by
-    calc
-      _ ≤ ‖∫ t in Ioc (0:ℝ) T, (∫ z in A, C t z)^2‖ₑ := by
-        rw [← ofReal_norm]
-        exact ENNReal.ofReal_le_ofReal (le_abs_self _)
-      _ ≤ ∫⁻ t in Ioc (0:ℝ) T, ‖(∫ z in A, C t z)^2‖ₑ :=
-        enorm_integral_le_lintegral_enorm _
-      _ = _ := by
-        apply lintegral_congr
-        intro t
-        rw [← ofReal_norm,Real.norm_of_nonneg (sq_nonneg _)]
-  have hh' := hh.trans (lintegral_mono' (μ := volume.restrict (Ioc (0:ℝ) T))
-    (ν := volume) Measure.restrict_le_self le_rfl)
-  simp only [sq_abs] at hb
-  have hrhs : 2*ENNReal.ofReal (K^2)*ENNReal.ofReal (J*R^2)*volume A =
-      ENNReal.ofReal (2*K^2*(J*R^2)*(volume A).toReal) := by
-    rw [ENNReal.ofReal_mul (mul_nonneg (mul_nonneg (by norm_num) (sq_nonneg _)) (mul_nonneg hJ (sq_nonneg _))),
-      ENNReal.ofReal_toReal hA,ENNReal.ofReal_mul (mul_nonneg (by norm_num) (sq_nonneg _)),
-      ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2)]
-    norm_num
-  apply (ENNReal.ofReal_le_ofReal_iff (by positivity : 0 ≤ 2*K^2*(J*R^2)*(volume A).toReal)).1
-  exact (hh'.trans hb).trans_eq hrhs
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
-open BrownianPredictable BrownianIto TaylorMeshes PredictableLimit
-local instance meshExponentsCommonRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsCommonRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance meshExponentsCommonComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance meshExponentsCommonComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-theorem common_slab_mgf {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {A : Set ℂ} {u : ℂ → ℝ} (C : FiniteTilt.CommonCutoff A u)
-    (hA : MeasurableSet A) [IsFiniteMeasure (volume.restrict A)]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
-    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
-    (V : Ω → C(I,ℂ)) (hVm : Measurable V)
-    (hV : ∀ᵐ ω ∂P, ∀ q : I, V ω q = γ ω ((C.horizon:ℝ)*q))
-    (e r : ℝ) (he : 0 < e) (her : e ≤ r) (hes : ∀ z ∈ A, e < z.im)
-    (J : ℝ) (hJ : 0 ≤ J)
-    (hlayer : ∀ ω, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω) →
-      ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain (γ ω) t,
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain (γ ω) t ∧ infDist w (localizationDomain (γ ω) t)ᶜ ≤ r},
-        ENNReal.ofReal (halfPlaneGreen (g ω t z) (g ω t w))) ≤ ENNReal.ofReal (J*r^2)) (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z in A,
-      observedDifference C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
-        ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V ω),z)))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(2*(Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2*
-        (J*r^2)*(volume A).toReal)/2)) := by
-  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
-  let ν := volume.restrict A
-  let X := fun n i => slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r
-  let Y := fun q : Ω × ℂ => observedDifference C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
-    ((BrownianDriver.path B (Real.sqrt κ) C.horizon q.1,V q.1),q.2)
-  have hXm (n i : ℕ) : AEMeasurable (X n i) (P.prod ν) :=
-    (coeff_aemeasurable hB ν C.height_pos κ _ _ ρ C.angle e).sub
-      (coeff_aemeasurable hB ν C.height_pos κ _ _ ρ C.angle r)
-  have hXp (n i : ℕ) : @Measurable (Ω × ℂ) ℝ
-      (Prod.instMeasurableSpace (m₁ := pastSigma B (dyadicTime C.horizon n i)) (m₂ := inferInstance)) _ (X n i) :=
-    (coeff_measurable_past C.height_pos κ B _ _ ρ C.angle e).sub
-      (coeff_measurable_past C.height_pos κ B _ _ ρ C.angle r)
-  obtain ⟨K,hK,hbound⟩ := coeff_uniform_bound C.height_pos κ hκ' B C.horizon ρ C.angle
-  have hXbound (n i : ℕ) (q : Ω × ℂ) : |X n i q| ≤ 2*K := by
-    exact (abs_sub _ _).trans ((add_le_add (hbound e _ q) (hbound r _ q)).trans_eq (by ring))
-  have hsum := SpatialLimit.bounded_sums ν hB C.horizon X hXp hXm (2*K) (by positivity) hXbound
-  have hY := observedDifference_aemeasurable hB ν C.height_pos κ ρ C.angle e r C.horizon V hVm
-  have hlim : ∀ᵐ q ∂P.prod ν, Tendsto
-      (fun n => dyadicDriverSum B C.horizon (fun n i ω => X n i (ω,q.2)) n q.1) atTop (𝓝 (Y q)) := by
-    apply SpatialLimit.ae_prod_tendsto_of_fibers
-      (fun n => SpatialLimit.sums_aemeasurable hB C.horizon X hXm n) hY
-    filter_upwards [ae_restrict_mem hA] with z hz
-    exact slab_sum_tendsto C κ hκ' B hB γ g f hmaps z hz (hhalf z hz) (hu z hz) (hr z hz) V hV e r
-  let Q := fun ω => ∫ t in (0:ℝ)..C.horizon, (∫ z in A,
-    pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
-      (BrownianDriver.path B (Real.sqrt κ) C.horizon ω) (V ω) (fun t => B t ω) t z)^2
-  have hQ : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicVariance C.horizon
-      (fun n i ω => ∫ z, X n i (ω,z) ∂ν) n ω) atTop (𝓝 (Q ω)) := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hV] with ω hm hc hb0 hv
-    exact slabCoeff_variance_tendsto C hA κ hκ' B γ g f ω hm hc hb0 hhalf hu hr (V ω) hv e r her
-  let J' := 2*(Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2*(J*r^2)*(volume A).toReal
-  have hJ' : ∀ᵐ ω ∂P, Q ω ≤ J' := by
-    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hV] with ω hm hc hb0 hv
-    have hbr := observed_bracket C hA κ hκ B γ g f ω hm hc hb0 hhalf hu hr (V ω) hv e r he her hes J (hlayer ω hm)
-    have hAf : volume A ≠ ⊤ := by
-      simpa only [Measure.restrict_apply_univ] using measure_ne_top (volume.restrict A) univ
-    exact interval_square_bound hAf _ C.horizon.coe_nonneg hJ hbr
-  exact SpatialLimit.spatial_exponential_bound ν hB C.horizon X hsum.1 hXp Y
-    (fun n => (hsum.2 n).1) (∫ _ : ℂ, (2*K)^2*(C.horizon:ℝ) ∂ν)
-    (fun n => (hsum.2 n).2) hlim Q hQ J' θ hJ'
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
-variable (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-variable {ε : ℝ} (hε : 0 < ε) (hsmall : ε < z.im)
-variable (hρ : (2*z.im)*Real.exp (-u) < ε/4)
-variable (Cs : ℕ → Cutoff z u) (hCs : Tendsto (fun n => (Cs n).angle) atTop (𝓝 0))
-include h hz hu hc hb0 hε hsmall hρ hCs
-
-lemma eventually_physicalValue_at_hit_general (hfin : firstLiveDistanceStop (γ sample) z ε ≠ ⊤) :
-    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((Cs n).physicalValue κ B γ ε sample) =
-      physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε := by
-  obtain ⟨t,ht,hzt,hd,hmin⟩ := PhysicalStop.finite_spec h.1 hfin
-  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
-  have hδ : ∀ᶠ n : ℕ in atTop, (Cs n).angle < η :=
-    hCs.eventually (gt_mem_nhds hη)
-  filter_upwards [hδ] with n hn
-  let C := Cs n
-  have htc : (t:ℝ) ≤ C.stopTime κ B sample := by
-    by_contra hnot
-    have hlt : C.stopTime κ B sample < t := lt_of_not_ge hnot
-    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · have hh := h.radius_ge_before_distance_stop hε hsmall
-        (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
-        (by rw [ht]; exact ENNReal.coe_le_coe.2 hlt.le)
-        (C.source_live κ B sample h hz hu hc hb0)
-      change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
-      rw [hr] at hh
-      exact (not_lt_of_ge hh) hρ
-    · have hh := hm (C.stopTime κ B sample) ⟨(C.bounds κ B sample).1,hlt.le⟩
-      rw [hs] at hh
-      exact (not_lt_of_ge hh) hn
-  have he : C.physicalTime κ B γ ε sample = t := by
-    rw [Cutoff.physicalTime,ht,PhysicalStop.cap_coe htc]
-  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  rw [physicalStoppedDensity,ht,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,hp,
-    C.physicalValue_eq_source κ B γ g f sample h hz hu hc hb0 ε,he]
-
-lemma tendsto_physicalValue_killed_general (hκ : 0 < κ ∧ κ < 8) :
-    Tendsto (fun n : ℕ => ENNReal.ofReal ((Cs n).physicalValue κ B γ ε sample)) atTop
-      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε)) := by
-  by_cases he : firstLiveDistanceStop (γ sample) z ε = ⊤
-  · rw [physicalStoppedDensity,he,ite_eq_left rfl]
-    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
-    have hlim := hCs.rpow_const_nhds_zero ha
-    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
-        ((Cs n).angle)^(8/κ-1)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
-    have hl : Tendsto (fun n : ℕ => (Cs n).physicalValue κ B γ ε sample) atTop (𝓝 0) := by
-      apply squeeze_zero (fun n => (Cs n).physicalValue_nonneg κ B γ ε sample) _ hlim'
-      intro n
-      let C := Cs n
-      have ht : C.physicalTime κ B γ ε sample = C.stopTime κ B sample := by
-        simp only [Cutoff.physicalTime,PhysicalStop.cap,he,ite_true]
-      have hv : C.physicalValue κ B γ ε sample = C.value κ B sample := by
-        simp only [Cutoff.physicalValue,ht,Cutoff.value]
-      rw [hv]
-      have hs : loewnerAngleSine κ (fun t => B t sample) (g sample) (C.stopTime κ B sample) z = C.angle := by
-        rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-        · have hh := h.radius_ge_before_distance_stop hε hsmall
-            (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
-            (by rw [he]; exact le_top) (C.source_live κ B sample h hz hu hc hb0)
-          change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
-          rw [hr] at hh
-          exact False.elim ((not_lt_of_ge hh) hρ)
-        · exact hs
-      exact C.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
-    simpa only [ENNReal.ofReal_zero,Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
-  · have hh := eventually_physicalValue_at_hit_general κ B γ g f sample h hz hu hc hb0 hε hsmall hρ Cs hCs he
-    exact tendsto_const_nhds.congr' (hh.mono (fun _ heq => heq.symm))
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.JointStopped
-open LoewnerCutoff
-
-def good {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I,ℝ)) (z : ℂ) (t : ℝ) : Prop :=
-  t ∈ Icc 0 T ∧ ∀ q : I, ε ≤ (position hε hT U z (t*q)).im
-
-lemma good_height {ε T t : ℝ} {hε : 0 < ε} {hT : 0 ≤ T}
-    {U : C(I,ℝ)} {z : ℂ} (hg : good hε hT U z t) :
-    ∀ s ∈ Icc 0 t, ε ≤ (position hε hT U z s).im := by
-  intro s hs
-  by_cases ht : t = 0
-  · have hz : s=0 := le_antisymm (ht ▸ hs.2) hs.1
-    simpa only [ht,hz,zero_mul] using hg.2 (0:I)
-  · have htp : 0 < t := lt_of_le_of_ne hg.1.1 (Ne.symm ht)
-    let q : I := ⟨s/t,⟨div_nonneg hs.1 hg.1.1,(div_le_one htp).2 hs.2⟩⟩
-    have he : t*(q:ℝ) = s := mul_div_cancel₀ _ ht
-    simpa only [he] using hg.2 q
-
-lemma good_closed {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
-    IsClosed {p : (C(I,ℝ) × ℂ) × ℝ | good hε hT p.1.1 p.1.2 p.2} := by
-  have ht : IsClosed {p : (C(I,ℝ) × ℂ) × ℝ | p.2 ∈ Icc 0 T} := isClosed_Icc.preimage continuous_snd
-  simp only [good,ofPred_and,ofPred_forall]
-  apply ht.inter
-  apply isClosed_iInter
-  intro q
-  apply isClosed_le continuous_const
-  exact Complex.continuous_im.comp ((continuous_mark_position hε hT).comp
-    (continuous_fst.prodMk (continuous_snd.mul continuous_const)))
-
-lemma good_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing driver γ g f)
-    {ε T t : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (hg : good hε hT U z t) :
-    z ∈ localizationDomain γ t ∧ EqOn (fun s => g s z) (position hε hT U z) (Icc 0 t) := by
-  have hd (s : ℝ) (hs : s ∈ Icc 0 t) :
-      HasDerivWithinAt (position hε hT U z)
-        (velocity ε (driver s) (position hε hT U z s)) (Icc 0 t) s := by
-    have hsT : s ∈ Icc 0 T := ⟨hs.1,hs.2.trans hg.1.2⟩
-    rw [← hU hsT]
-    exact (position_derivWithin hε hT U z hsT).mono (Icc_subset_Icc le_rfl hg.1.2)
-  have hh := hm.cutoff_solution_live hε hg.1.1 hd (good_height hg)
-  simpa only [position_initial] using hh
-
-lemma good_of_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    {ε T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
-    (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (ht : t ∈ Icc 0 T) (hz : z ∈ localizationDomain γ t) (hheight : ε ≤ (g t z).im) :
-    good hε hT U z t := by
-  have hh (s : ℝ) (hs : s ∈ Icc 0 t) : ε ≤ (g s z).im :=
-    hheight.trans (hm.height_antitone ht.1 hz hs ⟨ht.1,le_rfl⟩ hs.2)
-  have hpos (s : ℝ) (hs : s ∈ Icc 0 t) :
-      HasDerivWithinAt (position hε hT U z)
-        (velocity ε (extendDriver T U s) (position hε hT U z s)) (Icc 0 t) s :=
-    (position_derivWithin hε hT U z ⟨hs.1,hs.2.trans ht.2⟩).mono (Icc_subset_Icc le_rfl ht.2)
-  have hsource (s : ℝ) (hs : s ∈ Icc 0 t) :
-      HasDerivWithinAt (fun r => g r z) (velocity ε (extendDriver T U s) (g s z)) (Icc 0 t) s := by
-    rw [velocity_eq (hh s hs),hU ⟨hs.1,hs.2.trans ht.2⟩]
-    exact hm.hasDerivWithinAt_eval ht.1 (hc.mono (Icc_subset_Icc le_rfl ht.2)) hz hs
-  have he := solution_eq hε hpos hsource (by rw [position_initial,hm.initial z hz.1.1])
-  refine ⟨ht,fun q => ?_⟩
-  have hs : t*(q:ℝ) ∈ Icc 0 t := ⟨mul_nonneg ht.1 q.property.1,by simpa only [mul_one] using mul_le_mul_of_nonneg_left q.property.2 ht.1⟩
-  rw [he hs]
-  exact hh _ hs
-
-lemma log_of_good {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    {ε T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
-    (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (hg : good hε hT U z t) :
-    logDerivative hε hT U z t = Real.log ‖deriv (g t) z‖ := by
-  have hl := good_live hm hε hT U z hU hg
-  have hct := hc.mono (Icc_subset_Icc le_rfl hg.1.2)
-  rw [logDerivative_eq_integral hε hT U z hg.1,hm.log_deriv_norm_eq hg.1.1 hct hl.1]
-  have hi := (hm.continuous_spatialCoefficient hg.1.1 hct hl.1).intervalIntegrable_of_Icc (μ := volume) hg.1.1
-  rw [← Complex.reCLM_apply,← Complex.reCLM.intervalIntegral_comp_comm hi]
-  apply intervalIntegral.integral_congr
-  intro s hs
-  rw [uIcc_of_le hg.1.1] at hs
-  unfold variation
-  rw [clampIm_eq (by simpa using good_height hg s hs),← hl.2 hs,hU ⟨hs.1,hs.2.trans hg.1.2⟩]
-  rfl
-
-end SLEExactGauge.JointStopped
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.JointStopped
-open LoewnerCutoff
-local instance meshExponentsGreenRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsGreenRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-
-lemma field_good {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    {ε T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
-    (hε : 0 < ε) (hT : 0 ≤ T) (p a : ℝ)
-    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (hg : good hε hT U z t) :
-    LoewnerField.field hε hT p a 1 U z t (extendDriver T U t) =
-      (loewnerRadius g t z)^(-p)*((g t z).im/‖g t z-(driver t:ℂ)‖)^a := by
-  have hl := good_live hm hε hT U z hU hg
-  have hy : ε ≤ (position hε hT U z t).im := good_height hg t ⟨hg.1.1,le_rfl⟩
-  rw [LoewnerField.field,GreenPhysical.green_eq_source _ _ _ _ _ (LoewnerField.height_pos hε hT U z t),
-    LoewnerField.height,max_eq_right hy,log_of_good hm hc hε hT U z hU hg,
-    Real.exp_log (norm_pos_iff.2 (hm.deriv_ne_zero hg.1.1 hl.1)),← hl.2 ⟨hg.1.1,le_rfl⟩,
-    hU hg.1,one_mul]
-  simp only [loewnerRadius,Complex.norm_def,Complex.normSq_apply,Complex.sub_re,
-    Complex.ofReal_re,Complex.sub_im,Complex.ofReal_im,sub_zero,pow_two]
-
-def killedGreen (driver : ℝ → ℝ) (γ : ℝ → ℂ) (g : ℝ → ℂ → ℂ)
-    (p a : ℝ) (z : ℂ) (t : ℝ) : ℝ :=
-  if z ∈ localizationDomain γ t then
-    (loewnerRadius g t z)^(-p)*((g t z).im/‖g t z-(driver t:ℂ)‖)^a else 0
-
-def greenApprox {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) (n : ℕ)
-    (q : (C(I,ℝ) × ℂ) × ℝ) : ℝ :=
-  if good (show 0 < 1/((n:ℝ)+1) by positivity) hT q.1.1 q.1.2 q.2 then
-    LoewnerField.field (show 0 < 1/((n:ℝ)+1) by positivity) hT p a 1 q.1.1 q.1.2 q.2
-      (extendDriver T q.1.1 q.2) else 0
-
-lemma measurable_greenApprox {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) (n : ℕ) :
-    Measurable (greenApprox hT p a n) := by
-  unfold greenApprox
-  exact Measurable.ite (good_closed (show 0 < 1/((n:ℝ)+1) by positivity) hT).measurableSet
-    (continuous_mark_green (show 0 < 1/((n:ℝ)+1) by positivity) hT p a).measurable measurable_const
-
-lemma eventually_greenApprox {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    {T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
-    (hT : 0 ≤ T) (p a : ℝ) (U : C(I,ℝ)) (z : ℂ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T)) (ht : t ∈ Icc 0 T) :
-    ∀ᶠ n : ℕ in atTop, greenApprox hT p a n ((U,z),t) = killedGreen driver γ g p a z t := by
-  classical
-  by_cases hz : z ∈ localizationDomain γ t
-  · have hp := hm.im_pos ht.1 hz
-    have hlim : Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
-    filter_upwards [hlim.eventually (gt_mem_nhds hp)] with n hn
-    have hg := good_of_live hm hc (show 0 < 1/((n:ℝ)+1) by positivity) hT U z hU ht hz hn.le
-    rw [greenApprox,ite_eq_left hg,killedGreen,ite_eq_left hz]
-    exact field_good hm hc _ hT p a U z hU hg
-  · apply Filter.Eventually.of_forall
-    intro n
-    have hn : ¬ good (show 0 < 1/((n:ℝ)+1) by positivity) hT U z t :=
-      fun hg => hz (good_live hm _ hT U z hU hg).1
-    simp only [greenApprox,hn,ite_false,killedGreen,hz]
-
-def observedGreen {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) (q : (C(I,ℝ) × ℂ) × ℝ) : ℝ :=
-  ContinuousDriver.selectedLimit 0 (fun n => greenApprox hT p a n q)
-
-lemma measurable_observedGreen {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) :
-    Measurable (observedGreen hT p a) :=
-  ContinuousDriver.measurable_selectedLimit 0 _ (fun n => measurable_greenApprox hT p a n)
-
-lemma observedGreen_eq {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    {T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
-    (hT : 0 ≤ T) (p a : ℝ) (U : C(I,ℝ)) (z : ℂ)
-    (hU : EqOn (extendDriver T U) driver (Icc 0 T)) (ht : t ∈ Icc 0 T) :
-    observedGreen hT p a ((U,z),t) = killedGreen driver γ g p a z t := by
-  apply ContinuousDriver.selectedLimit_eq
-  apply tendsto_const_nhds.congr'
-  filter_upwards [eventually_greenApprox hm hc hT p a U z hU ht] with n hn
-  exact hn.symm
-
-end SLEExactGauge.JointStopped
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Classical
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.JointStopped
-open LoewnerCutoff
-local instance meshExponentsObservedRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsObservedRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance meshExponentsObservedComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance meshExponentsObservedComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-def observedPhysical {T : ℝ} (hT : 0 ≤ T) (p a e : ℝ)
-    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ≥0∞ :=
-  if CompactHit.hits e (q.1.2,q.2) then
-    ENNReal.ofReal (observedGreen hT p a ((q.1.1,q.2),CompactHit.realTime T e q.1.2 q.2)) else 0
-
-lemma measurable_observedPhysical {T : ℝ} (hT : 0 ≤ T) (p a e : ℝ) :
-    Measurable (observedPhysical hT p a e) := by
-  have hv : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => (q.1.2,q.2)) :=
-    measurable_fst.snd.prodMk measurable_snd
-  have ht : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => CompactHit.realTime T e q.1.2 q.2) :=
-    measurable_const.mul (measurable_subtype_coe.comp ((CompactHit.measurable_time e).comp hv))
-  apply Measurable.ite ((CompactHit.hits_closed e).measurableSet.preimage hv)
-  · exact ENNReal.measurable_ofReal.comp ((measurable_observedGreen hT p a).comp
-      ((measurable_fst.fst.prodMk measurable_snd).prodMk ht))
-  · exact measurable_const
-
-lemma observedPhysical_eq {κ : ℝ} {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    {T : ℝ} (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
-    (hc : ContinuousOn (fun t => Real.sqrt κ*B t.toNNReal) (Icc 0 T))
-    (hT : 0 < T) (e : ℝ) (U : C(I,ℝ)) (V : C(I,ℂ)) (z : ℂ)
-    (hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*B t.toNNReal) (Icc 0 T))
-    (hV : ∀ q : I, V q = γ (T*q)) :
-    observedPhysical hT.le (codimension κ) (8/κ-1) e ((U,V),z) =
-      if firstLiveDistanceStop γ z e ≤ ENNReal.ofReal T then
-        physicalStoppedDensity κ B γ g z e else 0 := by
-  classical
-  have heq := CompactHit.live_byTime_iff hT V γ z hm.1 hV (e := e)
-  have hg := observedGreen_eq hm hc hT.le (codimension κ) (8/κ-1) U z hU
-    (CompactHit.realTime_bounds hT.le e V z)
-  unfold observedPhysical
-  by_cases hh : CompactHit.hits e (V,z)
-  · rw [ite_eq_left hh,hg]
-    by_cases hl : z ∈ localizationDomain γ (CompactHit.realTime T e V z)
-    · have hstop := CompactHit.realTime_stop hT V γ z hV hh hl
-      rw [ite_eq_left (heq.2 ⟨hh,hl⟩),physicalStoppedDensity,hstop,ite_eq_right ENNReal.ofReal_ne_top,
-        ENNReal.toReal_ofReal (CompactHit.realTime_bounds hT.le e V z).1,
-        killedGreen,ite_eq_left hl]
-      simp only [loewnerAngleSine,Complex.ofReal_mul]
-    · rw [ite_eq_right (fun h => hl (heq.1 h).2),killedGreen,ite_eq_right hl,ENNReal.ofReal_zero]
-  · rw [ite_eq_right hh,ite_eq_right (fun h => hh (heq.1 h).1)]
-
-end SLEExactGauge.JointStopped
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ContinuousDriver
-local instance meshExponentsDriverRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsDriverRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance meshExponentsDriverComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance meshExponentsDriverComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-def complexParts (p : C(I,ℝ) × C(I,ℝ)) : C(I,ℂ) :=
-  ⟨fun q => (p.1 q:ℂ)+(p.2 q:ℂ)*Complex.I,
-    (Complex.continuous_ofReal.comp p.1.continuous).add
-      ((Complex.continuous_ofReal.comp p.2.continuous).mul continuous_const)⟩
-
-lemma continuous_complexParts : Continuous complexParts := by
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  exact (Complex.continuous_ofReal.comp (continuous_eval.comp (continuous_fst.fst.prodMk continuous_snd))).add
-    ((Complex.continuous_ofReal.comp (continuous_eval.comp (continuous_fst.snd.prodMk continuous_snd))).mul continuous_const)
-
-def complexPath (x : I → ℂ) : C(I,ℂ) :=
-  complexParts (path (fun q => (x q).re),path (fun q => (x q).im))
-
-lemma complexPath_eval {x : I → ℂ} (hx : Continuous x) (q : I) : complexPath x q = x q := by
-  change (path (fun q => (x q).re) q:ℂ)+(path (fun q => (x q).im) q:ℂ)*Complex.I = x q
-  rw [path_eval_of_continuous (show Continuous (fun q => (x q).re) from Complex.continuous_re.comp hx),
-    path_eval_of_continuous (show Continuous (fun q => (x q).im) from Complex.continuous_im.comp hx)]
-  exact Complex.re_add_im _
-
-lemma aemeasurable_complexPath {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (x : Ω → I → ℂ) (hx : ∀ q, AEMeasurable (fun sample => x sample q) P) :
-    AEMeasurable (fun sample => complexPath (x sample)) P := by
-  exact continuous_complexParts.measurable.comp_aemeasurable
-    ((aemeasurable_path (fun sample q => (x sample q).re) (fun q => Complex.measurable_re.comp_aemeasurable (hx q))).prodMk
-    (aemeasurable_path (fun sample q => (x sample q).im) (fun q => Complex.measurable_im.comp_aemeasurable (hx q))))
-
-lemma exists_measurable_trace_path {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (γ : Ω → ℝ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hc : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0))
-    (T : ℝ) (hT : 0 ≤ T) :
-    ∃ V : Ω → C(I,ℂ), Measurable V ∧ ∀ᵐ sample ∂P, ∀ q : I, V sample q = γ sample (T*q) := by
-  let V₀ := fun sample => complexPath (fun q : I => γ sample (T*q))
-  have hV : AEMeasurable V₀ P := aemeasurable_complexPath _ (fun q => hγ _ (mul_nonneg hT q.property.1))
-  refine ⟨hV.mk V₀,hV.measurable_mk,?_⟩
-  filter_upwards [hV.ae_eq_mk,hc] with sample he hcont
-  intro q
-  rw [← he]
-  apply complexPath_eval
-  exact hcont.comp_continuous (continuous_const.mul continuous_subtype_val) (fun q => mul_nonneg hT q.property.1)
-
-end SLEExactGauge.ContinuousDriver
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge
-local instance meshExponentsJointRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsJointRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance meshExponentsJointComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance meshExponentsJointComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-theorem jointFiniteTimePhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (_hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (T : ℝ) (hT : 0 < T) (e : ℝ) (_he : 0 < e) :
-    ∃ W : Ω × ℂ → ℝ≥0∞, Measurable W ∧ ∀ᵐ sample ∂P, ∀ z : ℂ,
-      W (sample,z) = if firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T
-        then physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e
-        else 0 := by
-  have hcont : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0) := hmaps.mono (fun _ hm => hm.1)
-  obtain ⟨V,hVm,hVe⟩ := ContinuousDriver.exists_measurable_trace_path γ hγ hcont T hT.le
-  let S : ℝ≥0 := ⟨T,hT.le⟩
-  have hu := BrownianDriver.path_aemeasurable hB (Real.sqrt κ) S
-  let U := hu.mk (BrownianDriver.path B (Real.sqrt κ) S)
-  have hUm : Measurable U := hu.measurable_mk
-  let W := fun q : Ω × ℂ => JointStopped.observedPhysical hT.le (codimension κ) (8/κ-1) e
-    ((U q.1,V q.1),q.2)
-  refine ⟨W,?_,?_⟩
-  · exact (JointStopped.measurable_observedPhysical hT.le (codimension κ) (8/κ-1) e).comp
-      (((hUm.comp measurable_fst).prodMk (hVm.comp measurable_fst)).prodMk measurable_snd)
-  · filter_upwards [hmaps,hB.cont,hu.ae_eq_mk,hVe] with sample hm hb huEq hvEq
-    change BrownianDriver.path B (Real.sqrt κ) S sample = U sample at huEq
-    intro z
-    apply JointStopped.observedPhysical_eq (κ := κ) (B := fun t => B t sample) hm
-      (continuous_const.mul (hb.comp continuous_real_toNNReal)).continuousOn hT e (U sample) (V sample) z _ hvEq
-    intro t ht
-    rw [← huEq]
-    exact BrownianDriver.extend_eq B (Real.sqrt κ) S sample hb ht
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-theorem jointPhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (e : ℝ) (he : 0 < e) :
-    ∃ W : Ω × ℂ → ℝ≥0∞, Measurable W ∧ ∀ᵐ ω ∂P, ∀ z,
-      W (ω,z) = physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e := by
-  choose W hW hWe using (fun n : ℕ => jointFiniteTimePhysicalDensity κ hκ B hB γ g f hγ hmaps
-    ((n:ℝ)+1) (by positivity) e he)
-  refine ⟨fun q => ⨆ n, W n q,Measurable.iSup hW,?_⟩
-  filter_upwards [ae_all_iff.2 hWe] with ω hω z
-  simp only [hω]
-  by_cases ht : firstLiveDistanceStop (γ ω) z e = ⊤
-  · simp only [physicalStoppedDensity,ht,ite_true,ite_self,iSup_const]
-  · have hf := ENNReal.ofReal_toReal ht
-    obtain ⟨n,hn⟩ := exists_nat_gt (firstLiveDistanceStop (γ ω) z e).toReal
-    have hb : firstLiveDistanceStop (γ ω) z e ≤ ENNReal.ofReal ((n:ℝ)+1) := by
-      rw [← hf]
-      exact ENNReal.ofReal_le_ofReal (by linarith)
-    apply le_antisymm
-    · apply iSup_le
-      intro m
-      split_ifs
-      · exact le_rfl
-      · exact zero_le
-    · exact le_iSup_of_le n (by rw [ite_eq_left hb])
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open FiniteTilt
-local instance meshExponentsDensityRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsDensityRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance meshExponentsDensityComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance meshExponentsDensityComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-lemma physicalStoppedDensity_ne_top (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ → ℂ)
-    (g : ℝ → ℂ → ℂ) (z : ℂ) (e : ℝ) : physicalStoppedDensity κ B γ g z e ≠ ⊤ := by
-  simp only [physicalStoppedDensity]
-  split_ifs <;> simp
-
-lemma physicalValue_real_tendsto {Ω : Type*}
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
-    (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    {e : ℝ} (he : 0 < e) (hes : e < z.im)
-    (hr : (2*z.im)*Real.exp (-u) < e/4)
-    (Cs : ℕ → Cutoff z u) (hCs : Tendsto (fun n => (Cs n).angle) atTop (𝓝 0)) :
-    Tendsto (fun n => (Cs n).physicalValue κ B γ e ω) atTop
-      (𝓝 (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal) := by
-  have hh := tendsto_physicalValue_killed_general κ B γ g f ω hm hz hu hc hb0 he hes hr Cs hCs hκ
-  have ht := (ENNReal.continuousAt_toReal (physicalStoppedDensity_ne_top κ (fun t => B t ω)
-    (γ ω) (g ω) z e)).tendsto.comp hh
-  simpa only [Function.comp_def,ENNReal.toReal_ofReal ((Cs _).physicalValue_nonneg κ B γ e ω)] using ht
-
-lemma exponential_bound_of_ae_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (X : ℕ → Ω → ℝ) (Y : Ω → ℝ)
-    (hX : ∀ n, AEMeasurable (X n) P)
-    (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
-    (K θ : ℝ)
-    (hb : ∀ n, (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*X n ω)) ∂P) ≤ ENNReal.ofReal (Real.exp K)) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*Y ω)) ∂P) ≤ ENNReal.ofReal (Real.exp K) := by
-  have hm (n : ℕ) := ((hX n).const_mul θ).exp.ennreal_ofReal
-  have hl : ∀ᵐ ω ∂P, Tendsto (fun n => ENNReal.ofReal (Real.exp (θ*X n ω))) atTop
-      (𝓝 (ENNReal.ofReal (Real.exp (θ*Y ω)))) := by
-    filter_upwards [hlim] with ω hω
-    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-      (Real.continuous_exp.continuousAt.tendsto.comp (hω.const_mul θ))
-  calc
-    _ = ∫⁻ ω, liminf (fun n => ENNReal.ofReal (Real.exp (θ*X n ω))) atTop ∂P := by
-      apply lintegral_congr_ae
-      filter_upwards [hl] with ω hω
-      exact hω.liminf_eq.symm
-    _ ≤ liminf (fun n => ∫⁻ ω, ENNReal.ofReal (Real.exp (θ*X n ω)) ∂P) atTop := lintegral_liminf_le' hm
-    _ ≤ _ := liminf_le_of_frequently_le (Filter.Eventually.frequently (Filter.Eventually.of_forall hb))
-
-lemma common_slab_limit {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {A : Set ℂ} {u : ℂ → ℝ} (Cs : ℕ → CommonCutoff A u)
-    (hCs : Tendsto (fun n => (Cs n).angle) atTop (𝓝 0))
-    (hA : MeasurableSet A) [IsFiniteMeasure (volume.restrict A)]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
-    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
-    (V : ℕ → Ω → C(I,ℂ)) (hVm : ∀ n, Measurable (V n))
-    (hV : ∀ n, ∀ᵐ ω ∂P, ∀ q : I, V n ω q = γ ω (((Cs n).horizon:ℝ)*q))
-    (e r : ℝ) (he : 0 < e) (her : e ≤ r) (hrs : ∀ z ∈ A, r < z.im) (hρ : ρ < e/4) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => ∫ z in A,
-      observedDifference (Cs n).height_pos (Cs n).horizon.coe_nonneg κ ρ (Cs n).angle e r
-        ((BrownianDriver.path B (Real.sqrt κ) (Cs n).horizon ω,V n ω),z)) atTop
-      (𝓝 (∫ z in A, (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal -
-        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z r).toReal)) := by
-  let F := fun n (q : Ω × ℂ) => observedDifference (Cs n).height_pos (Cs n).horizon.coe_nonneg κ ρ (Cs n).angle e r
-    ((BrownianDriver.path B (Real.sqrt κ) (Cs n).horizon q.1,V n q.1),q.2)
-  have hm (n : ℕ) : AEStronglyMeasurable (F n) (P.prod (volume.restrict A)) :=
-    (observedDifference_aemeasurable hB _ (Cs n).height_pos κ ρ (Cs n).angle e r (Cs n).horizon (V n) (hVm n)).aestronglyMeasurable
-  have hf (n : ℕ) : ∀ᵐ ω ∂P, AEStronglyMeasurable (fun z => F n (ω,z)) (volume.restrict A) := (hm n).prodMk_left
-  filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,ae_all_iff.2 hV,ae_all_iff.2 hf]
-    with ω hw hc hb0 hv hfω
-  have heq (n : ℕ) (z : ℂ) (hz : z ∈ A) : F n (ω,z) =
-      ((Cs n).at z hz).physicalValue κ B γ e ω - ((Cs n).at z hz).physicalValue κ B γ r ω := by
-    dsimp only [F,observedDifference]
-    rw [observedValue_eq (Cs n) κ B γ g f ω hw hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) (V n ω) (hv n) e,
-      observedValue_eq (Cs n) κ B γ g f ω hw hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) (V n ω) (hv n) r]
-  apply tendsto_integral_of_dominated_convergence (fun _ : ℂ => 2*ρ^(-(1-κ/8))) hfω (integrable_const _)
-  · intro n
-    filter_upwards [ae_restrict_mem hA] with z hz
-    rw [heq n z hz,Real.norm_eq_abs]
-    have hn (s : ℝ) := ((Cs n).at z hz).physicalValue_nonneg κ B γ s ω
-    have hb (s : ℝ) := ((Cs n).at z hz).physicalValue_bound κ hκ B γ s ω (hhalf z hz) (hu z hz) hc hb0
-    rw [hr z hz] at hb
-    calc
-      _ ≤ |((Cs n).at z hz).physicalValue κ B γ e ω| + |((Cs n).at z hz).physicalValue κ B γ r ω| := abs_sub _ _
-      _ ≤ 2*ρ^(-(1-κ/8)) := by rw [abs_of_nonneg (hn e),abs_of_nonneg (hn r)]; linarith [hb e,hb r]
-  · filter_upwards [ae_restrict_mem hA] with z hz
-    have hl (s : ℝ) (hs : 0 < s) (hse : s < z.im) (hρs : ρ < s/4) := physicalValue_real_tendsto κ hκ B γ g f ω hw
-      (hhalf z hz) (hu z hz) hc hb0 hs hse (by rw [hr z hz]; exact hρs) (fun n => (Cs n).at z hz) hCs
-    exact ((hl e he (her.trans_lt (hrs z hz)) hρ).sub (hl r (he.trans_le her) (hrs z hz) (by linarith))).congr
-      (fun n => (heq n z hz).symm)
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.FiniteTilt
-
-lemma compact_log_cutoffs {A : Set ℂ} (hA : IsCompact A) (hne : A.Nonempty)
-    (hy : ∀ z ∈ A, 1 ≤ z.im) {ρ : ℝ} (hρ : 0 < ρ) (hρs : ρ < 1) :
-    ∃ Cs : ℕ → CommonCutoff A (fun z => Real.log (2*z.im/ρ)),
-      Tendsto (fun n => (Cs n).angle) atTop (𝓝 0) := by
-  have hy0 (z : ℂ) (hz : z ∈ A) : 0 < z.im := lt_of_lt_of_le (by norm_num) (hy z hz)
-  have hR (z : ℂ) (hz : z ∈ A) : 1 < 2*z.im/ρ := (lt_div_iff₀ hρ).2 (by linarith [hy z hz])
-  have hu (z : ℂ) (hz : z ∈ A) : 0 < Real.log (2*z.im/ρ) := Real.log_pos (hR z hz)
-  have huc : ContinuousOn (fun z : ℂ => Real.log (2*z.im/ρ)) A :=
-    ((continuous_const.mul Complex.continuous_im).div_const ρ).continuousOn.log (fun z hz => (lt_trans zero_lt_one (hR z hz)).ne')
-  have hsc : ContinuousOn (fun z : ℂ => z.im/‖z‖) A := Complex.continuous_im.continuousOn.div
-    continuous_norm.continuousOn (fun z hz => norm_ne_zero_iff.2 (by intro he; simpa [he] using hy0 z hz))
-  obtain ⟨zH,hzH,hH⟩ := hA.exists_isMaxOn hne Complex.continuous_im.continuousOn
-  obtain ⟨zU,hzU,hU⟩ := hA.exists_isMaxOn hne huc
-  obtain ⟨zS,hzS,hS⟩ := hA.exists_isMinOn hne hsc
-  exact commonCutoffs (hy zH hzH) (hu zU hzU).le (initial_sine_pos (hy0 zS hzS))
-    (fun z hz => ⟨hy z hz,hH hz⟩) (fun z hz => ⟨(hu z hz).le,hU hz⟩) (fun z hz => hS hz)
-
-lemma logRadius_pos {z : ℂ} (hz : 1 ≤ z.im) {ρ : ℝ} (hρ : 0 < ρ) (hρs : ρ < 1) :
-    0 < Real.log (2*z.im/ρ) := Real.log_pos ((lt_div_iff₀ hρ).2 (by linarith))
-
-lemma logRadius_eq {z : ℂ} (hz : 0 < z.im) {ρ : ℝ} (hρ : 0 < ρ) :
-    (2*z.im)*Real.exp (-Real.log (2*z.im/ρ)) = ρ := by
-  rw [Real.exp_neg,Real.exp_log (by positivity)]
-  field_simp
-
-lemma collar_height {z₀ : ℂ} (hcollar : closedBall z₀ 2 ⊆ halfPlane) :
-    ∀ z ∈ closedBall z₀ 1, 1 ≤ z.im := by
-  have hbase : 2 < z₀.im := by
-    have hm : z₀-2*Complex.I ∈ closedBall z₀ 2 := by
-      rw [mem_closedBall,dist_eq_norm]
-      simp
-    have hh := hcollar hm
-    change 0 < (z₀-2*Complex.I).im at hh
-    norm_num at hh
-    linarith
-  intro z hz
-  have hd := mem_closedBall.1 hz
-  have hi := Complex.abs_im_le_norm (z-z₀)
-  rw [Complex.sub_im,← dist_eq_norm] at hi
-  have hlo := neg_le_abs (z.im-z₀.im)
-  linarith
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open FiniteTilt
-local instance meshExponentsMgfRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance meshExponentsMgfRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance meshExponentsMgfComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance meshExponentsMgfComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-theorem physical_slab_mgf :
-    ∃ J : ℝ, 0 < J ∧ ∀ {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-      {A : Set ℂ}, IsCompact A → A.Nonempty → (∀ z ∈ A, 1 ≤ z.im) →
-      ∀ (κ : ℝ), 0 < κ ∧ κ ≤ 4 → ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
-      ∀ (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ),
-      (∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P) →
-      (∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω)) →
-      ∀ e r : ℝ, 0 < e → e ≤ r → r < 1 → ∀ θ : ℝ,
-      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z in A,
-        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal -
-        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z r).toReal))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(2*(Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2*
-        (J*r^2)*(volume A).toReal)/2)) := by
-  obtain ⟨J,hJ,hJb⟩ := Mesh.loewner_boundary_layer
-  refine ⟨J,hJ,?_⟩
-  intro Ω m P hP A hA hne hy κ hκ B hB γ g f hγ hmaps e r he her hr θ
-  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
-  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.2 hA.measure_lt_top.ne
-  let ρ := e/8
-  have hρ : 0 < ρ := by dsimp [ρ]; positivity
-  have hρs : ρ < 1 := by dsimp [ρ]; linarith
-  obtain ⟨Cs,hCs⟩ := compact_log_cutoffs hA hne hy hρ hρs
-  have hcont : ∀ᵐ ω ∂P, ContinuousOn (γ ω) (Ici 0) := hmaps.mono (fun _ hw => hw.1)
-  choose V hVm hV using (fun n => ContinuousDriver.exists_measurable_trace_path γ hγ hcont
-    ((Cs n).horizon:ℝ) (Cs n).horizon.coe_nonneg)
-  have hh (z : ℂ) (hz : z ∈ A) : z ∈ halfPlane := lt_of_lt_of_le zero_lt_one (hy z hz)
-  have hu (z : ℂ) (hz : z ∈ A) : 0 < Real.log (2*z.im/ρ) := logRadius_pos (hy z hz) hρ hρs
-  have hrad (z : ℂ) (hz : z ∈ A) : (2*z.im)*Real.exp (-Real.log (2*z.im/ρ)) = ρ := logRadius_eq (hh z hz) hρ
-  let X := fun n ω => ∫ z in A, observedDifference (Cs n).height_pos (Cs n).horizon.coe_nonneg κ ρ (Cs n).angle e r
-    ((BrownianDriver.path B (Real.sqrt κ) (Cs n).horizon ω,V n ω),z)
-  have hXm (n : ℕ) : AEMeasurable (X n) P :=
-    (observedDifference_aemeasurable hB (volume.restrict A) (Cs n).height_pos κ ρ (Cs n).angle e r
-      (Cs n).horizon (V n) (hVm n)).aestronglyMeasurable.integral_prod_right'.aemeasurable
-  have hlim := common_slab_limit Cs hCs hA.measurableSet κ hκ' B hB γ g f hmaps hh hu hrad
-    V hVm hV e r he her (fun z hz => hr.trans_le (hy z hz)) (by dsimp [ρ]; linarith)
-  apply exponential_bound_of_ae_tendsto P X _ hXm hlim _ θ
-  intro n
-  exact common_slab_mgf (Cs n) hA.measurableSet κ hκ B hB γ g f hmaps hh hu hrad (V n) (hVm n) (hV n)
-    e r he her (fun z hz => her.trans_lt (hr.trans_le (hy z hz))) J hJ.le
-    (fun ω hw t ht z hz => hJb _ _ _ _ hw t ht z hz r (he.trans_le her)) θ
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.MeshSlab
-
-lemma physicalDensity_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (hm : IsLoewnerTraceUsing driver γ g f) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → ℝ) {z : ℂ} {e : ℝ} (he : 0 < e) (hes : e < z.im) :
-    (physicalStoppedDensity κ B γ g z e).toReal ≤ (e/4)^(-codimension κ) := by
-  by_cases hf : firstLiveDistanceStop γ z e = ⊤
-  · simp only [physicalStoppedDensity,hf,ite_true,ENNReal.toReal_zero]
-    positivity
-  obtain ⟨t,ht,hzt,hd,hmin⟩ := PhysicalStop.finite_spec hm.1 hf
-  have hR := hm.radius_ge_before_distance_stop he hes (t := t) (by rw [ht]) hzt
-  have hp : 0 ≤ codimension κ := by unfold codimension dimension; linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le); linarith
-  have hy := hm.im_pos t.coe_nonneg hzt
-  let w := g t z - (Real.sqrt κ*B t : ℂ)
-  have hwi : w.im = (g t z).im := by simp [w]
-  have hwn : 0 < ‖w‖ := norm_pos_iff.2 (by intro hh; have hi := congrArg Complex.im hh; rw [hwi] at hi; simpa using hy.ne' hi)
-  have hs0 : 0 ≤ loewnerAngleSine κ B g t z := by unfold loewnerAngleSine; positivity
-  have hs1 : loewnerAngleSine κ B g t z ≤ 1 := by
-    simp only [loewnerAngleSine,Real.toNNReal_coe]
-    change (g t z).im / ‖w‖ ≤ 1
-    apply (div_le_one hwn).2
-    rw [← hwi]
-    exact Complex.im_le_norm w
-  have hs := Real.rpow_le_one hs0 hs1 ha
-  have hR0 : 0 < loewnerRadius g t z := (by positivity : 0 < e/4).trans_le hR
-  have hrpow := Real.rpow_le_rpow_of_nonpos (by positivity : 0 < e/4) hR (neg_nonpos.2 hp)
-  have hv0 : 0 ≤ loewnerRadius g t z ^ (-codimension κ) * loewnerAngleSine κ B g t z ^ (8/κ-1) := by positivity
-  rw [physicalStoppedDensity,ht,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,ENNReal.toReal_ofReal hv0]
-  exact (mul_le_of_le_one_right (Real.rpow_nonneg hR0.le _) hs).trans hrpow
-
-def physicalMass {Ω : Type*} (A : Set ℂ) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (e : ℝ) (ω : Ω) : ℝ :=
-  ∫ z in A, (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal
-
-lemma physicalMass_nonneg {Ω : Type*} (A : Set ℂ) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (e : ℝ) (ω : Ω) :
-    0 ≤ physicalMass A κ B γ g e ω := integral_nonneg (fun _ => ENNReal.toReal_nonneg)
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.MeshSlab
-variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-variable (A : Set ℂ) [sfiniteVolume : SFinite (volume.restrict A)]
-variable (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-variable (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-variable (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P)
-variable (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-variable (e : ℝ) (he : 0 < e)
-include hκ hB hγ hmaps he
-
-lemma physicalMass_aemeasurable : AEMeasurable (physicalMass A κ B γ g e) P := by
-  obtain ⟨W,hW,hWe⟩ := jointPhysicalDensity κ hκ B hB γ g f hγ hmaps e he
-  apply (hW.ennreal_toReal.stronglyMeasurable.integral_prod_right' (ν := volume.restrict A)).aemeasurable.congr
-  filter_upwards [hWe] with ω hw
-  exact integral_congr_ae (Filter.Eventually.of_forall (fun z => congrArg ENNReal.toReal (hw z)))
-
-omit sfiniteVolume in
-lemma physicalDensity_integrable [SFinite (volume.restrict A)] [IsFiniteMeasure (volume.restrict A)] (hA : MeasurableSet A)
-    (hes : ∀ z ∈ A, e < z.im) :
-    ∀ᵐ ω ∂P, Integrable (fun z => (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal) (volume.restrict A) := by
-  obtain ⟨W,hW,hWe⟩ := jointPhysicalDensity κ hκ B hB γ g f hγ hmaps e he
-  filter_upwards [hWe,hmaps] with ω hw hm
-  have hwm : Measurable (fun z => (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal) := by
-    have hh : Measurable (fun z : ℂ => (W (ω,z)).toReal) := hW.ennreal_toReal.comp (measurable_const.prodMk measurable_id)
-    convert hh using 1
-    funext z
-    rw [hw z]
-  apply Integrable.of_bound hwm.aestronglyMeasurable ((e/4)^(-codimension κ))
-  filter_upwards [ae_restrict_mem hA] with z hz
-  rw [Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg]
-  exact physicalDensity_bound hm κ hκ (fun t => B t ω) he (hes z hz)
-
-lemma physicalMass_lintegral [IsFiniteMeasure (volume.restrict A)] (hA : MeasurableSet A)
-    (hes : ∀ z ∈ A, e < z.im) :
-    ∀ᵐ ω ∂P, ENNReal.ofReal (physicalMass A κ B γ g e ω) =
-      ∫⁻ z in A, physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e := by
-  filter_upwards [physicalDensity_integrable A κ hκ B hB γ g f hγ hmaps e he hA hes] with ω hi
-  rw [physicalMass,ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall (fun _ => ENNReal.toReal_nonneg))]
-  apply lintegral_congr
-  intro z
-  exact ENNReal.ofReal_toReal (physicalStoppedDensity_ne_top _ _ _ _ _ _)
-
-lemma physicalMass_bound [IsFiniteMeasure (volume.restrict A)] (hA : MeasurableSet A)
-    (hes : ∀ z ∈ A, e < z.im) :
-    ∀ᵐ ω ∂P, physicalMass A κ B γ g e ω ≤ (volume A).toReal*(e/4)^(-codimension κ) := by
-  filter_upwards [physicalDensity_integrable A κ hκ B hB γ g f hγ hmaps e he hA hes,hmaps] with ω hi hm
-  have hh := integral_mono_ae hi (integrable_const ((e/4)^(-codimension κ))) (μ := volume.restrict A) (by
-    filter_upwards [ae_restrict_mem hA] with z hz
-    exact physicalDensity_bound hm κ hκ (fun t => B t ω) he (hes z hz))
-  simpa only [integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul,physicalMass] using hh
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.MeshSlab
-
-lemma slab_radius_scale (p : ℝ) {r : ℝ} (hr : 0 < r) :
-    (r/8)^(-p)*r = (8:ℝ)^p*r^(1-p) := by
-  rw [Real.div_rpow hr.le (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 8),div_inv_eq_mul,
-    show 1-p = 1+(-p) by ring,Real.rpow_add hr,Real.rpow_one]
-  ring
-
-theorem slab_mass_mgf (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) :
-    ∃ K : ℝ, 0 < K ∧ ∀ {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-      (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
-      ∀ (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ),
-      (∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P) →
-      (∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω)) →
-      ∀ (z₀ : ℂ), closedBall z₀ 2 ⊆ halfPlane → ∀ e r : ℝ,
-      0 < e → e ≤ r → r/2 ≤ e → r < 1 → ∀ θ : ℝ,
-      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(physicalMass (closedBall z₀ 1) κ B γ g e ω -
-        physicalMass (closedBall z₀ 1) κ B γ g r ω))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(K*r^(1-codimension κ))^2/2)) := by
-  obtain ⟨J,hJ,hmgf⟩ := physical_slab_mgf
-  let L := Real.sqrt κ*(8/κ-1)*(8:ℝ)^(codimension κ)
-  have hp : 0 ≤ codimension κ := by unfold codimension dimension; linarith [hκ.2]
-  have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by linarith [hκ.2]))
-  have hL : 0 < L := mul_pos (mul_pos (Real.sqrt_pos.2 hκ.1) ha) (Real.rpow_pos_of_pos (by norm_num) _)
-  let K := Real.sqrt (2*J*Real.pi)*L
-  have hJπ : 0 < 2*J*Real.pi := by positivity
-  refine ⟨K,mul_pos (Real.sqrt_pos.2 hJπ) hL,?_⟩
-  intro Ω m P hP B hB γ g f hγ hmaps z₀ hcollar e r he her hel hr θ
-  let A := closedBall z₀ (1:ℝ)
-  have hAc : IsCompact A := isCompact_closedBall _ _
-  have hAm : MeasurableSet A := hAc.measurableSet
-  have hy := FiniteTilt.collar_height hcollar
-  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.2 hAc.measure_lt_top.ne
-  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
-  have hh := hmgf hAc (nonempty_closedBall.2 (by norm_num)) hy κ hκ B hB γ g f hγ hmaps e r he her hr θ
-  have hIe := physicalDensity_integrable A κ hκ' B hB γ g f hγ hmaps e he hAm
-    (fun z hz => her.trans_lt (hr.trans_le (hy z hz)))
-  have hIr := physicalDensity_integrable A κ hκ' B hB γ g f hγ hmaps r (he.trans_le her) hAm
-    (fun z hz => hr.trans_le (hy z hz))
-  have heq : (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(physicalMass A κ B γ g e ω - physicalMass A κ B γ g r ω))) ∂P) =
-      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z in A,
-        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal -
-        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z r).toReal))) ∂P) := by
-    apply lintegral_congr_ae
-    filter_upwards [hIe,hIr] with ω hie hir
-    rw [integral_sub hie hir]
-    rfl
-  rw [heq]
-  apply hh.trans
-  apply ENNReal.ofReal_le_ofReal
-  apply Real.exp_le_exp.2
-  apply div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left _ (sq_nonneg θ)) (by norm_num)
-  have hr0 := he.trans_le her
-  have hrad := Real.rpow_le_rpow_of_nonpos (by positivity : 0 < r/8) (by linarith : r/8 ≤ e/4) (neg_nonpos.2 hp)
-  have hmul : (Real.sqrt κ*(8/κ-1)*(e/4)^(-codimension κ))*r ≤ L*r^(1-codimension κ) := by
-    calc
-      _ ≤ (Real.sqrt κ*(8/κ-1)*(r/8)^(-codimension κ))*r := by gcongr
-      _ = _ := by rw [mul_assoc,slab_radius_scale _ hr0]; dsimp [L]; ring
-  have hleft : 0 ≤ (Real.sqrt κ*(8/κ-1)*(e/4)^(-codimension κ))*r := by positivity
-  have hsq := pow_le_pow_left₀ hleft hmul 2
-  have hv : (volume A).toReal = Real.pi := by
-    dsimp [A]
-    rw [Complex.volume_closedBall]
-    norm_num [ENNReal.toReal_ofReal Real.pi_pos.le]
-  rw [hv]
-  have hcode : 1-κ/8 = codimension κ := by unfold codimension dimension; ring
-  rw [hcode]
-  have hmul' := mul_le_mul_of_nonneg_left hsq hJπ.le
-  dsimp [K] at *
-  have hsqrt := Real.sq_sqrt hJπ.le
-  nlinarith
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.MeshSlab
-
-lemma finite_mgf_sum {Ω ι : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (t : Finset ι) (Y : ι → Ω → ℝ) (σ : ι → ℝ)
-    (hσ : ∀ i ∈ t, 0 < σ i) (hY : ∀ i ∈ t, AEMeasurable (Y i) P)
-    (hmgf : ∀ i ∈ t, ∀ θ : ℝ, (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*Y i ω)) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(σ i)^2/2))) (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∑ i ∈ t, Y i ω))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(∑ i ∈ t, σ i)^2/2)) := by
-  classical
-  by_cases ht : t.Nonempty
-  · let S := ∑ i ∈ t, σ i
-    have hS : 0 < S := Finset.sum_pos (fun i hi => hσ i hi) ht
-    let w := fun i => σ i/S
-    have hw (i : ι) (hi : i ∈ t) : 0 ≤ w i := (div_pos (hσ i hi) hS).le
-    have hw1 : ∑ i ∈ t, w i = 1 := by simp only [w,← Finset.sum_div,S,div_self hS.ne']
-    have hpoint (ω : Ω) : Real.exp (θ*(∑ i ∈ t, Y i ω)) ≤
-        ∑ i ∈ t, w i * Real.exp ((θ*S/σ i)*Y i ω) := by
-      have hh := convexOn_exp.map_sum_le (t := t) (w := w) (p := fun i => (θ*S/σ i)*Y i ω)
-        hw hw1 (fun _ _ => mem_univ _)
-      have he : (∑ i ∈ t, w i • ((θ*S/σ i)*Y i ω)) = θ*(∑ i ∈ t, Y i ω) := by
-        conv_rhs => rw [Finset.mul_sum]
-        apply Finset.sum_congr rfl
-        intro i hi
-        dsimp [w]
-        field_simp [hS.ne', (hσ i hi).ne']
-      rw [he] at hh
-      simpa only [smul_eq_mul] using hh
-    calc
-      _ ≤ ∫⁻ ω, ∑ i ∈ t, ENNReal.ofReal (w i) * ENNReal.ofReal (Real.exp ((θ*S/σ i)*Y i ω)) ∂P := by
-        apply lintegral_mono
-        intro ω
-        calc
-          _ ≤ ENNReal.ofReal (∑ i ∈ t, w i * Real.exp ((θ*S/σ i)*Y i ω)) := ENNReal.ofReal_le_ofReal (hpoint ω)
-          _ = _ := by
-            rw [ENNReal.ofReal_sum_of_nonneg (fun i hi => mul_nonneg (hw i hi) (Real.exp_pos _).le)]
-            apply Finset.sum_congr rfl
-            intro i hi
-            exact ENNReal.ofReal_mul (hw i hi)
-      _ = ∑ i ∈ t, ENNReal.ofReal (w i) * (∫⁻ ω, ENNReal.ofReal (Real.exp ((θ*S/σ i)*Y i ω)) ∂P) := by
-        rw [lintegral_finsetSum' _ (fun i hi => ((hY i hi).const_mul _).exp.ennreal_ofReal.const_mul _)]
-        apply Finset.sum_congr rfl
-        intro i hi
-        exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-      _ ≤ ∑ i ∈ t, ENNReal.ofReal (w i) * ENNReal.ofReal (Real.exp (θ^2*S^2/2)) := by
-        apply Finset.sum_le_sum
-        intro i hi
-        have hh := hmgf i hi (θ*S/σ i)
-        have he : (θ*S/σ i)^2*(σ i)^2/2 = θ^2*S^2/2 := by field_simp [(hσ i hi).ne']
-        rw [he] at hh
-        exact mul_le_mul_right hh _
-      _ = _ := by
-        rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg hw,hw1,ENNReal.ofReal_one,one_mul]
-  · have he : t = ∅ := Finset.not_nonempty_iff_eq_empty.1 ht
-    simp [he]
-
-lemma geo_radius_power {δ : ℝ} (hδ : 0 < δ) (a : ℝ) (n : ℕ) :
-    (δ*(1/2:ℝ)^n)^a = δ^a*((1/2:ℝ)^a)^n := by
-  rw [Real.mul_rpow hδ.le (by positivity),← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 1/2),
-    mul_comm (n:ℝ) a,Real.rpow_mul_natCast (by norm_num : (0:ℝ) ≤ 1/2)]
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.SlabMoments
+import OAI.Probability.SLE.ClippedCoordinates
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Mesh estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Mesh
+
+lemma interval_square_bound {A : Set ℂ} (hA : volume A ≠ ⊤) (C : ℝ → ℂ → ℝ)
+    {T K J R : ℝ} (hT : 0 ≤ T) (hJ : 0 ≤ J)
+    (hb : (∫⁻ t, ENNReal.ofReal (|∫ z in A, C t z|^2)) ≤
+      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (J*R^2)*volume A) :
+    (∫ t in (0:ℝ)..T, (∫ z in A, C t z)^2) ≤ 2*K^2*(J*R^2)*(volume A).toReal := by
+  rw [intervalIntegral.integral_of_le hT]
+  have hh : ENNReal.ofReal (∫ t in Ioc (0:ℝ) T, (∫ z in A, C t z)^2) ≤
+      ∫⁻ t in Ioc (0:ℝ) T, ENNReal.ofReal ((∫ z in A, C t z)^2) := by
+    calc
+      _ ≤ ‖∫ t in Ioc (0:ℝ) T, (∫ z in A, C t z)^2‖ₑ := by
+        rw [← ofReal_norm]
+        exact ENNReal.ofReal_le_ofReal (le_abs_self _)
+      _ ≤ ∫⁻ t in Ioc (0:ℝ) T, ‖(∫ z in A, C t z)^2‖ₑ :=
+        enorm_integral_le_lintegral_enorm _
+      _ = _ := by
+        apply lintegral_congr
+        intro t
+        rw [← ofReal_norm,Real.norm_of_nonneg (sq_nonneg _)]
+  have hh' := hh.trans (lintegral_mono' (μ := volume.restrict (Ioc (0:ℝ) T))
+    (ν := volume) Measure.restrict_le_self le_rfl)
+  simp only [sq_abs] at hb
+  have hrhs : 2*ENNReal.ofReal (K^2)*ENNReal.ofReal (J*R^2)*volume A =
+      ENNReal.ofReal (2*K^2*(J*R^2)*(volume A).toReal) := by
+    rw [ENNReal.ofReal_mul (mul_nonneg (mul_nonneg (by norm_num) (sq_nonneg _)) (mul_nonneg hJ (sq_nonneg _))),
+      ENNReal.ofReal_toReal hA,ENNReal.ofReal_mul (mul_nonneg (by norm_num) (sq_nonneg _)),
+      ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2)]
+    norm_num
+  apply (ENNReal.ofReal_le_ofReal_iff (by positivity : 0 ≤ 2*K^2*(J*R^2)*(volume A).toReal)).1
+  exact (hh'.trans hb).trans_eq hrhs
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
+open BrownianPredictable BrownianIto TaylorMeshes PredictableLimit
+local instance meshExponentsCommonRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsCommonRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance meshExponentsCommonComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance meshExponentsCommonComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+theorem common_slab_mgf {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {A : Set ℂ} {u : ℂ → ℝ} (C : FiniteTilt.CommonCutoff A u)
+    (hA : MeasurableSet A) [IsFiniteMeasure (volume.restrict A)]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
+    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
+    (V : Ω → C(I,ℂ)) (hVm : Measurable V)
+    (hV : ∀ᵐ ω ∂P, ∀ q : I, V ω q = γ ω ((C.horizon:ℝ)*q))
+    (e r : ℝ) (he : 0 < e) (her : e ≤ r) (hes : ∀ z ∈ A, e < z.im)
+    (J : ℝ) (hJ : 0 ≤ J)
+    (hlayer : ∀ ω, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω) →
+      ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain (γ ω) t,
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain (γ ω) t ∧ infDist w (localizationDomain (γ ω) t)ᶜ ≤ r},
+        ENNReal.ofReal (halfPlaneGreen (g ω t z) (g ω t w))) ≤ ENNReal.ofReal (J*r^2)) (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z in A,
+      observedDifference C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
+        ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V ω),z)))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(2*(Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2*
+        (J*r^2)*(volume A).toReal)/2)) := by
+  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
+  let ν := volume.restrict A
+  let X := fun n i => slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r
+  let Y := fun q : Ω × ℂ => observedDifference C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
+    ((BrownianDriver.path B (Real.sqrt κ) C.horizon q.1,V q.1),q.2)
+  have hXm (n i : ℕ) : AEMeasurable (X n i) (P.prod ν) :=
+    (coeff_aemeasurable hB ν C.height_pos κ _ _ ρ C.angle e).sub
+      (coeff_aemeasurable hB ν C.height_pos κ _ _ ρ C.angle r)
+  have hXp (n i : ℕ) : @Measurable (Ω × ℂ) ℝ
+      (Prod.instMeasurableSpace (m₁ := pastSigma B (dyadicTime C.horizon n i)) (m₂ := inferInstance)) _ (X n i) :=
+    (coeff_measurable_past C.height_pos κ B _ _ ρ C.angle e).sub
+      (coeff_measurable_past C.height_pos κ B _ _ ρ C.angle r)
+  obtain ⟨K,hK,hbound⟩ := coeff_uniform_bound C.height_pos κ hκ' B C.horizon ρ C.angle
+  have hXbound (n i : ℕ) (q : Ω × ℂ) : |X n i q| ≤ 2*K := by
+    exact (abs_sub _ _).trans ((add_le_add (hbound e _ q) (hbound r _ q)).trans_eq (by ring))
+  have hsum := SpatialLimit.bounded_sums ν hB C.horizon X hXp hXm (2*K) (by positivity) hXbound
+  have hY := observedDifference_aemeasurable hB ν C.height_pos κ ρ C.angle e r C.horizon V hVm
+  have hlim : ∀ᵐ q ∂P.prod ν, Tendsto
+      (fun n => dyadicDriverSum B C.horizon (fun n i ω => X n i (ω,q.2)) n q.1) atTop (𝓝 (Y q)) := by
+    apply SpatialLimit.ae_prod_tendsto_of_fibers
+      (fun n => SpatialLimit.sums_aemeasurable hB C.horizon X hXm n) hY
+    filter_upwards [ae_restrict_mem hA] with z hz
+    exact slab_sum_tendsto C κ hκ' B hB γ g f hmaps z hz (hhalf z hz) (hu z hz) (hr z hz) V hV e r
+  let Q := fun ω => ∫ t in (0:ℝ)..C.horizon, (∫ z in A,
+    pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
+      (BrownianDriver.path B (Real.sqrt κ) C.horizon ω) (V ω) (fun t => B t ω) t z)^2
+  have hQ : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicVariance C.horizon
+      (fun n i ω => ∫ z, X n i (ω,z) ∂ν) n ω) atTop (𝓝 (Q ω)) := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hV] with ω hm hc hb0 hv
+    exact slabCoeff_variance_tendsto C hA κ hκ' B γ g f ω hm hc hb0 hhalf hu hr (V ω) hv e r her
+  let J' := 2*(Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2*(J*r^2)*(volume A).toReal
+  have hJ' : ∀ᵐ ω ∂P, Q ω ≤ J' := by
+    filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hV] with ω hm hc hb0 hv
+    have hbr := observed_bracket C hA κ hκ B γ g f ω hm hc hb0 hhalf hu hr (V ω) hv e r he her hes J (hlayer ω hm)
+    have hAf : volume A ≠ ⊤ := by
+      simpa only [Measure.restrict_apply_univ] using measure_ne_top (volume.restrict A) univ
+    exact interval_square_bound hAf _ C.horizon.coe_nonneg hJ hbr
+  exact SpatialLimit.spatial_exponential_bound ν hB C.horizon X hsum.1 hXp Y
+    (fun n => (hsum.2 n).1) (∫ _ : ℂ, (2*K)^2*(C.horizon:ℝ) ∂ν)
+    (fun n => (hsum.2 n).2) hlim Q hQ J' θ hJ'
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
+variable (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+variable {ε : ℝ} (hε : 0 < ε) (hsmall : ε < z.im)
+variable (hρ : (2*z.im)*Real.exp (-u) < ε/4)
+variable (Cs : ℕ → Cutoff z u) (hCs : Tendsto (fun n => (Cs n).angle) atTop (𝓝 0))
+include h hz hu hc hb0 hε hsmall hρ hCs
+
+lemma eventually_physicalValue_at_hit_general (hfin : firstLiveDistanceStop (γ sample) z ε ≠ ⊤) :
+    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((Cs n).physicalValue κ B γ ε sample) =
+      physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε := by
+  obtain ⟨t,ht,hzt,hd,hmin⟩ := PhysicalStop.finite_spec h.1 hfin
+  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
+  have hδ : ∀ᶠ n : ℕ in atTop, (Cs n).angle < η :=
+    hCs.eventually (gt_mem_nhds hη)
+  filter_upwards [hδ] with n hn
+  let C := Cs n
+  have htc : (t:ℝ) ≤ C.stopTime κ B sample := by
+    by_contra hnot
+    have hlt : C.stopTime κ B sample < t := lt_of_not_ge hnot
+    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · have hh := h.radius_ge_before_distance_stop hε hsmall
+        (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
+        (by rw [ht]; exact ENNReal.coe_le_coe.2 hlt.le)
+        (C.source_live κ B sample h hz hu hc hb0)
+      change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
+      rw [hr] at hh
+      exact (not_lt_of_ge hh) hρ
+    · have hh := hm (C.stopTime κ B sample) ⟨(C.bounds κ B sample).1,hlt.le⟩
+      rw [hs] at hh
+      exact (not_lt_of_ge hh) hn
+  have he : C.physicalTime κ B γ ε sample = t := by
+    rw [Cutoff.physicalTime,ht,PhysicalStop.cap_coe htc]
+  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  rw [physicalStoppedDensity,ht,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,hp,
+    C.physicalValue_eq_source κ B γ g f sample h hz hu hc hb0 ε,he]
+
+lemma tendsto_physicalValue_killed_general (hκ : 0 < κ ∧ κ < 8) :
+    Tendsto (fun n : ℕ => ENNReal.ofReal ((Cs n).physicalValue κ B γ ε sample)) atTop
+      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε)) := by
+  by_cases he : firstLiveDistanceStop (γ sample) z ε = ⊤
+  · rw [physicalStoppedDensity,he,ite_eq_left rfl]
+    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
+    have hlim := hCs.rpow_const_nhds_zero ha
+    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
+        ((Cs n).angle)^(8/κ-1)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
+    have hl : Tendsto (fun n : ℕ => (Cs n).physicalValue κ B γ ε sample) atTop (𝓝 0) := by
+      apply squeeze_zero (fun n => (Cs n).physicalValue_nonneg κ B γ ε sample) _ hlim'
+      intro n
+      let C := Cs n
+      have ht : C.physicalTime κ B γ ε sample = C.stopTime κ B sample := by
+        simp only [Cutoff.physicalTime,PhysicalStop.cap,he,ite_true]
+      have hv : C.physicalValue κ B γ ε sample = C.value κ B sample := by
+        simp only [Cutoff.physicalValue,ht,Cutoff.value]
+      rw [hv]
+      have hs : loewnerAngleSine κ (fun t => B t sample) (g sample) (C.stopTime κ B sample) z = C.angle := by
+        rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+        · have hh := h.radius_ge_before_distance_stop hε hsmall
+            (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
+            (by rw [he]; exact le_top) (C.source_live κ B sample h hz hu hc hb0)
+          change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
+          rw [hr] at hh
+          exact False.elim ((not_lt_of_ge hh) hρ)
+        · exact hs
+      exact C.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
+    simpa only [ENNReal.ofReal_zero,Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
+  · have hh := eventually_physicalValue_at_hit_general κ B γ g f sample h hz hu hc hb0 hε hsmall hρ Cs hCs he
+    exact tendsto_const_nhds.congr' (hh.mono (fun _ heq => heq.symm))
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.JointStopped
+open LoewnerCutoff
+
+def good {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I,ℝ)) (z : ℂ) (t : ℝ) : Prop :=
+  t ∈ Icc 0 T ∧ ∀ q : I, ε ≤ (position hε hT U z (t*q)).im
+
+lemma good_height {ε T t : ℝ} {hε : 0 < ε} {hT : 0 ≤ T}
+    {U : C(I,ℝ)} {z : ℂ} (hg : good hε hT U z t) :
+    ∀ s ∈ Icc 0 t, ε ≤ (position hε hT U z s).im := by
+  intro s hs
+  by_cases ht : t = 0
+  · have hz : s=0 := le_antisymm (ht ▸ hs.2) hs.1
+    simpa only [ht,hz,zero_mul] using hg.2 (0:I)
+  · have htp : 0 < t := lt_of_le_of_ne hg.1.1 (Ne.symm ht)
+    let q : I := ⟨s/t,⟨div_nonneg hs.1 hg.1.1,(div_le_one htp).2 hs.2⟩⟩
+    have he : t*(q:ℝ) = s := mul_div_cancel₀ _ ht
+    simpa only [he] using hg.2 q
+
+lemma good_closed {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
+    IsClosed {p : (C(I,ℝ) × ℂ) × ℝ | good hε hT p.1.1 p.1.2 p.2} := by
+  have ht : IsClosed {p : (C(I,ℝ) × ℂ) × ℝ | p.2 ∈ Icc 0 T} := isClosed_Icc.preimage continuous_snd
+  simp only [good,ofPred_and,ofPred_forall]
+  apply ht.inter
+  apply isClosed_iInter
+  intro q
+  apply isClosed_le continuous_const
+  exact Complex.continuous_im.comp ((continuous_mark_position hε hT).comp
+    (continuous_fst.prodMk (continuous_snd.mul continuous_const)))
+
+lemma good_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing driver γ g f)
+    {ε T t : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (hg : good hε hT U z t) :
+    z ∈ localizationDomain γ t ∧ EqOn (fun s => g s z) (position hε hT U z) (Icc 0 t) := by
+  have hd (s : ℝ) (hs : s ∈ Icc 0 t) :
+      HasDerivWithinAt (position hε hT U z)
+        (velocity ε (driver s) (position hε hT U z s)) (Icc 0 t) s := by
+    have hsT : s ∈ Icc 0 T := ⟨hs.1,hs.2.trans hg.1.2⟩
+    rw [← hU hsT]
+    exact (position_derivWithin hε hT U z hsT).mono (Icc_subset_Icc le_rfl hg.1.2)
+  have hh := hm.cutoff_solution_live hε hg.1.1 hd (good_height hg)
+  simpa only [position_initial] using hh
+
+lemma good_of_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    {ε T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
+    (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (ht : t ∈ Icc 0 T) (hz : z ∈ localizationDomain γ t) (hheight : ε ≤ (g t z).im) :
+    good hε hT U z t := by
+  have hh (s : ℝ) (hs : s ∈ Icc 0 t) : ε ≤ (g s z).im :=
+    hheight.trans (hm.height_antitone ht.1 hz hs ⟨ht.1,le_rfl⟩ hs.2)
+  have hpos (s : ℝ) (hs : s ∈ Icc 0 t) :
+      HasDerivWithinAt (position hε hT U z)
+        (velocity ε (extendDriver T U s) (position hε hT U z s)) (Icc 0 t) s :=
+    (position_derivWithin hε hT U z ⟨hs.1,hs.2.trans ht.2⟩).mono (Icc_subset_Icc le_rfl ht.2)
+  have hsource (s : ℝ) (hs : s ∈ Icc 0 t) :
+      HasDerivWithinAt (fun r => g r z) (velocity ε (extendDriver T U s) (g s z)) (Icc 0 t) s := by
+    rw [velocity_eq (hh s hs),hU ⟨hs.1,hs.2.trans ht.2⟩]
+    exact hm.hasDerivWithinAt_eval ht.1 (hc.mono (Icc_subset_Icc le_rfl ht.2)) hz hs
+  have he := solution_eq hε hpos hsource (by rw [position_initial,hm.initial z hz.1.1])
+  refine ⟨ht,fun q => ?_⟩
+  have hs : t*(q:ℝ) ∈ Icc 0 t := ⟨mul_nonneg ht.1 q.property.1,by simpa only [mul_one] using mul_le_mul_of_nonneg_left q.property.2 ht.1⟩
+  rw [he hs]
+  exact hh _ hs
+
+lemma log_of_good {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    {ε T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
+    (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (hg : good hε hT U z t) :
+    logDerivative hε hT U z t = Real.log ‖deriv (g t) z‖ := by
+  have hl := good_live hm hε hT U z hU hg
+  have hct := hc.mono (Icc_subset_Icc le_rfl hg.1.2)
+  rw [logDerivative_eq_integral hε hT U z hg.1,hm.log_deriv_norm_eq hg.1.1 hct hl.1]
+  have hi := (hm.continuous_spatialCoefficient hg.1.1 hct hl.1).intervalIntegrable_of_Icc (μ := volume) hg.1.1
+  rw [← Complex.reCLM_apply,← Complex.reCLM.intervalIntegral_comp_comm hi]
+  apply intervalIntegral.integral_congr
+  intro s hs
+  rw [uIcc_of_le hg.1.1] at hs
+  unfold variation
+  rw [clampIm_eq (by simpa using good_height hg s hs),← hl.2 hs,hU ⟨hs.1,hs.2.trans hg.1.2⟩]
+  rfl
+
+end SLEExactGauge.JointStopped
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.JointStopped
+open LoewnerCutoff
+local instance meshExponentsGreenRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsGreenRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+
+lemma field_good {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    {ε T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
+    (hε : 0 < ε) (hT : 0 ≤ T) (p a : ℝ)
+    (U : C(I,ℝ)) (z : ℂ) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (hg : good hε hT U z t) :
+    LoewnerField.field hε hT p a 1 U z t (extendDriver T U t) =
+      (loewnerRadius g t z)^(-p)*((g t z).im/‖g t z-(driver t:ℂ)‖)^a := by
+  have hl := good_live hm hε hT U z hU hg
+  have hy : ε ≤ (position hε hT U z t).im := good_height hg t ⟨hg.1.1,le_rfl⟩
+  rw [LoewnerField.field,GreenPhysical.green_eq_source _ _ _ _ _ (LoewnerField.height_pos hε hT U z t),
+    LoewnerField.height,max_eq_right hy,log_of_good hm hc hε hT U z hU hg,
+    Real.exp_log (norm_pos_iff.2 (hm.deriv_ne_zero hg.1.1 hl.1)),← hl.2 ⟨hg.1.1,le_rfl⟩,
+    hU hg.1,one_mul]
+  simp only [loewnerRadius,Complex.norm_def,Complex.normSq_apply,Complex.sub_re,
+    Complex.ofReal_re,Complex.sub_im,Complex.ofReal_im,sub_zero,pow_two]
+
+def killedGreen (driver : ℝ → ℝ) (γ : ℝ → ℂ) (g : ℝ → ℂ → ℂ)
+    (p a : ℝ) (z : ℂ) (t : ℝ) : ℝ :=
+  if z ∈ localizationDomain γ t then
+    (loewnerRadius g t z)^(-p)*((g t z).im/‖g t z-(driver t:ℂ)‖)^a else 0
+
+def greenApprox {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) (n : ℕ)
+    (q : (C(I,ℝ) × ℂ) × ℝ) : ℝ :=
+  if good (show 0 < 1/((n:ℝ)+1) by positivity) hT q.1.1 q.1.2 q.2 then
+    LoewnerField.field (show 0 < 1/((n:ℝ)+1) by positivity) hT p a 1 q.1.1 q.1.2 q.2
+      (extendDriver T q.1.1 q.2) else 0
+
+lemma measurable_greenApprox {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) (n : ℕ) :
+    Measurable (greenApprox hT p a n) := by
+  unfold greenApprox
+  exact Measurable.ite (good_closed (show 0 < 1/((n:ℝ)+1) by positivity) hT).measurableSet
+    (continuous_mark_green (show 0 < 1/((n:ℝ)+1) by positivity) hT p a).measurable measurable_const
+
+lemma eventually_greenApprox {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    {T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
+    (hT : 0 ≤ T) (p a : ℝ) (U : C(I,ℝ)) (z : ℂ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T)) (ht : t ∈ Icc 0 T) :
+    ∀ᶠ n : ℕ in atTop, greenApprox hT p a n ((U,z),t) = killedGreen driver γ g p a z t := by
+  classical
+  by_cases hz : z ∈ localizationDomain γ t
+  · have hp := hm.im_pos ht.1 hz
+    have hlim : Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+    filter_upwards [hlim.eventually (gt_mem_nhds hp)] with n hn
+    have hg := good_of_live hm hc (show 0 < 1/((n:ℝ)+1) by positivity) hT U z hU ht hz hn.le
+    rw [greenApprox,ite_eq_left hg,killedGreen,ite_eq_left hz]
+    exact field_good hm hc _ hT p a U z hU hg
+  · apply Filter.Eventually.of_forall
+    intro n
+    have hn : ¬ good (show 0 < 1/((n:ℝ)+1) by positivity) hT U z t :=
+      fun hg => hz (good_live hm _ hT U z hU hg).1
+    simp only [greenApprox,hn,ite_false,killedGreen,hz]
+
+def observedGreen {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) (q : (C(I,ℝ) × ℂ) × ℝ) : ℝ :=
+  ContinuousDriver.selectedLimit 0 (fun n => greenApprox hT p a n q)
+
+lemma measurable_observedGreen {T : ℝ} (hT : 0 ≤ T) (p a : ℝ) :
+    Measurable (observedGreen hT p a) :=
+  ContinuousDriver.measurable_selectedLimit 0 _ (fun n => measurable_greenApprox hT p a n)
+
+lemma observedGreen_eq {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    {T t : ℝ} (hm : IsLoewnerTraceUsing driver γ g f) (hc : ContinuousOn driver (Icc 0 T))
+    (hT : 0 ≤ T) (p a : ℝ) (U : C(I,ℝ)) (z : ℂ)
+    (hU : EqOn (extendDriver T U) driver (Icc 0 T)) (ht : t ∈ Icc 0 T) :
+    observedGreen hT p a ((U,z),t) = killedGreen driver γ g p a z t := by
+  apply ContinuousDriver.selectedLimit_eq
+  apply tendsto_const_nhds.congr'
+  filter_upwards [eventually_greenApprox hm hc hT p a U z hU ht] with n hn
+  exact hn.symm
+
+end SLEExactGauge.JointStopped
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Classical
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.JointStopped
+open LoewnerCutoff
+local instance meshExponentsObservedRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsObservedRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance meshExponentsObservedComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance meshExponentsObservedComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+def observedPhysical {T : ℝ} (hT : 0 ≤ T) (p a e : ℝ)
+    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ≥0∞ :=
+  if CompactHit.hits e (q.1.2,q.2) then
+    ENNReal.ofReal (observedGreen hT p a ((q.1.1,q.2),CompactHit.realTime T e q.1.2 q.2)) else 0
+
+lemma measurable_observedPhysical {T : ℝ} (hT : 0 ≤ T) (p a e : ℝ) :
+    Measurable (observedPhysical hT p a e) := by
+  have hv : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => (q.1.2,q.2)) :=
+    measurable_fst.snd.prodMk measurable_snd
+  have ht : Measurable (fun q : (C(I,ℝ) × C(I,ℂ)) × ℂ => CompactHit.realTime T e q.1.2 q.2) :=
+    measurable_const.mul (measurable_subtype_coe.comp ((CompactHit.measurable_time e).comp hv))
+  apply Measurable.ite ((CompactHit.hits_closed e).measurableSet.preimage hv)
+  · exact ENNReal.measurable_ofReal.comp ((measurable_observedGreen hT p a).comp
+      ((measurable_fst.fst.prodMk measurable_snd).prodMk ht))
+  · exact measurable_const
+
+lemma observedPhysical_eq {κ : ℝ} {B : ℝ≥0 → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    {T : ℝ} (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal) γ g f)
+    (hc : ContinuousOn (fun t => Real.sqrt κ*B t.toNNReal) (Icc 0 T))
+    (hT : 0 < T) (e : ℝ) (U : C(I,ℝ)) (V : C(I,ℂ)) (z : ℂ)
+    (hU : EqOn (extendDriver T U) (fun t => Real.sqrt κ*B t.toNNReal) (Icc 0 T))
+    (hV : ∀ q : I, V q = γ (T*q)) :
+    observedPhysical hT.le (codimension κ) (8/κ-1) e ((U,V),z) =
+      if firstLiveDistanceStop γ z e ≤ ENNReal.ofReal T then
+        physicalStoppedDensity κ B γ g z e else 0 := by
+  classical
+  have heq := CompactHit.live_byTime_iff hT V γ z hm.1 hV (e := e)
+  have hg := observedGreen_eq hm hc hT.le (codimension κ) (8/κ-1) U z hU
+    (CompactHit.realTime_bounds hT.le e V z)
+  unfold observedPhysical
+  by_cases hh : CompactHit.hits e (V,z)
+  · rw [ite_eq_left hh,hg]
+    by_cases hl : z ∈ localizationDomain γ (CompactHit.realTime T e V z)
+    · have hstop := CompactHit.realTime_stop hT V γ z hV hh hl
+      rw [ite_eq_left (heq.2 ⟨hh,hl⟩),physicalStoppedDensity,hstop,ite_eq_right ENNReal.ofReal_ne_top,
+        ENNReal.toReal_ofReal (CompactHit.realTime_bounds hT.le e V z).1,
+        killedGreen,ite_eq_left hl]
+      simp only [loewnerAngleSine,Complex.ofReal_mul]
+    · rw [ite_eq_right (fun h => hl (heq.1 h).2),killedGreen,ite_eq_right hl,ENNReal.ofReal_zero]
+  · rw [ite_eq_right hh,ite_eq_right (fun h => hh (heq.1 h).1)]
+
+end SLEExactGauge.JointStopped
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ContinuousDriver
+local instance meshExponentsDriverRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsDriverRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance meshExponentsDriverComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance meshExponentsDriverComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+def complexParts (p : C(I,ℝ) × C(I,ℝ)) : C(I,ℂ) :=
+  ⟨fun q => (p.1 q:ℂ)+(p.2 q:ℂ)*Complex.I,
+    (Complex.continuous_ofReal.comp p.1.continuous).add
+      ((Complex.continuous_ofReal.comp p.2.continuous).mul continuous_const)⟩
+
+lemma continuous_complexParts : Continuous complexParts := by
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  exact (Complex.continuous_ofReal.comp (continuous_eval.comp (continuous_fst.fst.prodMk continuous_snd))).add
+    ((Complex.continuous_ofReal.comp (continuous_eval.comp (continuous_fst.snd.prodMk continuous_snd))).mul continuous_const)
+
+def complexPath (x : I → ℂ) : C(I,ℂ) :=
+  complexParts (path (fun q => (x q).re),path (fun q => (x q).im))
+
+lemma complexPath_eval {x : I → ℂ} (hx : Continuous x) (q : I) : complexPath x q = x q := by
+  change (path (fun q => (x q).re) q:ℂ)+(path (fun q => (x q).im) q:ℂ)*Complex.I = x q
+  rw [path_eval_of_continuous (show Continuous (fun q => (x q).re) from Complex.continuous_re.comp hx),
+    path_eval_of_continuous (show Continuous (fun q => (x q).im) from Complex.continuous_im.comp hx)]
+  exact Complex.re_add_im _
+
+lemma aemeasurable_complexPath {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (x : Ω → I → ℂ) (hx : ∀ q, AEMeasurable (fun sample => x sample q) P) :
+    AEMeasurable (fun sample => complexPath (x sample)) P := by
+  exact continuous_complexParts.measurable.comp_aemeasurable
+    ((aemeasurable_path (fun sample q => (x sample q).re) (fun q => Complex.measurable_re.comp_aemeasurable (hx q))).prodMk
+    (aemeasurable_path (fun sample q => (x sample q).im) (fun q => Complex.measurable_im.comp_aemeasurable (hx q))))
+
+lemma exists_measurable_trace_path {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (γ : Ω → ℝ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hc : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0))
+    (T : ℝ) (hT : 0 ≤ T) :
+    ∃ V : Ω → C(I,ℂ), Measurable V ∧ ∀ᵐ sample ∂P, ∀ q : I, V sample q = γ sample (T*q) := by
+  let V₀ := fun sample => complexPath (fun q : I => γ sample (T*q))
+  have hV : AEMeasurable V₀ P := aemeasurable_complexPath _ (fun q => hγ _ (mul_nonneg hT q.property.1))
+  refine ⟨hV.mk V₀,hV.measurable_mk,?_⟩
+  filter_upwards [hV.ae_eq_mk,hc] with sample he hcont
+  intro q
+  rw [← he]
+  apply complexPath_eval
+  exact hcont.comp_continuous (continuous_const.mul continuous_subtype_val) (fun q => mul_nonneg hT q.property.1)
+
+end SLEExactGauge.ContinuousDriver
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge
+local instance meshExponentsJointRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsJointRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance meshExponentsJointComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance meshExponentsJointComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+theorem jointFiniteTimePhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (_hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (T : ℝ) (hT : 0 < T) (e : ℝ) (_he : 0 < e) :
+    ∃ W : Ω × ℂ → ℝ≥0∞, Measurable W ∧ ∀ᵐ sample ∂P, ∀ z : ℂ,
+      W (sample,z) = if firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T
+        then physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e
+        else 0 := by
+  have hcont : ∀ᵐ sample ∂P, ContinuousOn (γ sample) (Ici 0) := hmaps.mono (fun _ hm => hm.1)
+  obtain ⟨V,hVm,hVe⟩ := ContinuousDriver.exists_measurable_trace_path γ hγ hcont T hT.le
+  let S : ℝ≥0 := ⟨T,hT.le⟩
+  have hu := BrownianDriver.path_aemeasurable hB (Real.sqrt κ) S
+  let U := hu.mk (BrownianDriver.path B (Real.sqrt κ) S)
+  have hUm : Measurable U := hu.measurable_mk
+  let W := fun q : Ω × ℂ => JointStopped.observedPhysical hT.le (codimension κ) (8/κ-1) e
+    ((U q.1,V q.1),q.2)
+  refine ⟨W,?_,?_⟩
+  · exact (JointStopped.measurable_observedPhysical hT.le (codimension κ) (8/κ-1) e).comp
+      (((hUm.comp measurable_fst).prodMk (hVm.comp measurable_fst)).prodMk measurable_snd)
+  · filter_upwards [hmaps,hB.cont,hu.ae_eq_mk,hVe] with sample hm hb huEq hvEq
+    change BrownianDriver.path B (Real.sqrt κ) S sample = U sample at huEq
+    intro z
+    apply JointStopped.observedPhysical_eq (κ := κ) (B := fun t => B t sample) hm
+      (continuous_const.mul (hb.comp continuous_real_toNNReal)).continuousOn hT e (U sample) (V sample) z _ hvEq
+    intro t ht
+    rw [← huEq]
+    exact BrownianDriver.extend_eq B (Real.sqrt κ) S sample hb ht
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+theorem jointPhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (e : ℝ) (he : 0 < e) :
+    ∃ W : Ω × ℂ → ℝ≥0∞, Measurable W ∧ ∀ᵐ ω ∂P, ∀ z,
+      W (ω,z) = physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e := by
+  choose W hW hWe using (fun n : ℕ => jointFiniteTimePhysicalDensity κ hκ B hB γ g f hγ hmaps
+    ((n:ℝ)+1) (by positivity) e he)
+  refine ⟨fun q => ⨆ n, W n q,Measurable.iSup hW,?_⟩
+  filter_upwards [ae_all_iff.2 hWe] with ω hω z
+  simp only [hω]
+  by_cases ht : firstLiveDistanceStop (γ ω) z e = ⊤
+  · simp only [physicalStoppedDensity,ht,ite_true,ite_self,iSup_const]
+  · have hf := ENNReal.ofReal_toReal ht
+    obtain ⟨n,hn⟩ := exists_nat_gt (firstLiveDistanceStop (γ ω) z e).toReal
+    have hb : firstLiveDistanceStop (γ ω) z e ≤ ENNReal.ofReal ((n:ℝ)+1) := by
+      rw [← hf]
+      exact ENNReal.ofReal_le_ofReal (by linarith)
+    apply le_antisymm
+    · apply iSup_le
+      intro m
+      split_ifs
+      · exact le_rfl
+      · exact zero_le
+    · exact le_iSup_of_le n (by rw [ite_eq_left hb])
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open FiniteTilt
+local instance meshExponentsDensityRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsDensityRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance meshExponentsDensityComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance meshExponentsDensityComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+lemma physicalStoppedDensity_ne_top (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ → ℂ)
+    (g : ℝ → ℂ → ℂ) (z : ℂ) (e : ℝ) : physicalStoppedDensity κ B γ g z e ≠ ⊤ := by
+  simp only [physicalStoppedDensity]
+  split_ifs <;> simp
+
+lemma physicalValue_real_tendsto {Ω : Type*}
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
+    (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    {e : ℝ} (he : 0 < e) (hes : e < z.im)
+    (hr : (2*z.im)*Real.exp (-u) < e/4)
+    (Cs : ℕ → Cutoff z u) (hCs : Tendsto (fun n => (Cs n).angle) atTop (𝓝 0)) :
+    Tendsto (fun n => (Cs n).physicalValue κ B γ e ω) atTop
+      (𝓝 (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal) := by
+  have hh := tendsto_physicalValue_killed_general κ B γ g f ω hm hz hu hc hb0 he hes hr Cs hCs hκ
+  have ht := (ENNReal.continuousAt_toReal (physicalStoppedDensity_ne_top κ (fun t => B t ω)
+    (γ ω) (g ω) z e)).tendsto.comp hh
+  simpa only [Function.comp_def,ENNReal.toReal_ofReal ((Cs _).physicalValue_nonneg κ B γ e ω)] using ht
+
+lemma exponential_bound_of_ae_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (X : ℕ → Ω → ℝ) (Y : Ω → ℝ)
+    (hX : ∀ n, AEMeasurable (X n) P)
+    (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
+    (K θ : ℝ)
+    (hb : ∀ n, (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*X n ω)) ∂P) ≤ ENNReal.ofReal (Real.exp K)) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*Y ω)) ∂P) ≤ ENNReal.ofReal (Real.exp K) := by
+  have hm (n : ℕ) := ((hX n).const_mul θ).exp.ennreal_ofReal
+  have hl : ∀ᵐ ω ∂P, Tendsto (fun n => ENNReal.ofReal (Real.exp (θ*X n ω))) atTop
+      (𝓝 (ENNReal.ofReal (Real.exp (θ*Y ω)))) := by
+    filter_upwards [hlim] with ω hω
+    exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+      (Real.continuous_exp.continuousAt.tendsto.comp (hω.const_mul θ))
+  calc
+    _ = ∫⁻ ω, liminf (fun n => ENNReal.ofReal (Real.exp (θ*X n ω))) atTop ∂P := by
+      apply lintegral_congr_ae
+      filter_upwards [hl] with ω hω
+      exact hω.liminf_eq.symm
+    _ ≤ liminf (fun n => ∫⁻ ω, ENNReal.ofReal (Real.exp (θ*X n ω)) ∂P) atTop := lintegral_liminf_le' hm
+    _ ≤ _ := liminf_le_of_frequently_le (Filter.Eventually.frequently (Filter.Eventually.of_forall hb))
+
+lemma common_slab_limit {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {A : Set ℂ} {u : ℂ → ℝ} (Cs : ℕ → CommonCutoff A u)
+    (hCs : Tendsto (fun n => (Cs n).angle) atTop (𝓝 0))
+    (hA : MeasurableSet A) [IsFiniteMeasure (volume.restrict A)]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
+    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
+    (V : ℕ → Ω → C(I,ℂ)) (hVm : ∀ n, Measurable (V n))
+    (hV : ∀ n, ∀ᵐ ω ∂P, ∀ q : I, V n ω q = γ ω (((Cs n).horizon:ℝ)*q))
+    (e r : ℝ) (he : 0 < e) (her : e ≤ r) (hrs : ∀ z ∈ A, r < z.im) (hρ : ρ < e/4) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => ∫ z in A,
+      observedDifference (Cs n).height_pos (Cs n).horizon.coe_nonneg κ ρ (Cs n).angle e r
+        ((BrownianDriver.path B (Real.sqrt κ) (Cs n).horizon ω,V n ω),z)) atTop
+      (𝓝 (∫ z in A, (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal -
+        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z r).toReal)) := by
+  let F := fun n (q : Ω × ℂ) => observedDifference (Cs n).height_pos (Cs n).horizon.coe_nonneg κ ρ (Cs n).angle e r
+    ((BrownianDriver.path B (Real.sqrt κ) (Cs n).horizon q.1,V n q.1),q.2)
+  have hm (n : ℕ) : AEStronglyMeasurable (F n) (P.prod (volume.restrict A)) :=
+    (observedDifference_aemeasurable hB _ (Cs n).height_pos κ ρ (Cs n).angle e r (Cs n).horizon (V n) (hVm n)).aestronglyMeasurable
+  have hf (n : ℕ) : ∀ᵐ ω ∂P, AEStronglyMeasurable (fun z => F n (ω,z)) (volume.restrict A) := (hm n).prodMk_left
+  filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,ae_all_iff.2 hV,ae_all_iff.2 hf]
+    with ω hw hc hb0 hv hfω
+  have heq (n : ℕ) (z : ℂ) (hz : z ∈ A) : F n (ω,z) =
+      ((Cs n).at z hz).physicalValue κ B γ e ω - ((Cs n).at z hz).physicalValue κ B γ r ω := by
+    dsimp only [F,observedDifference]
+    rw [observedValue_eq (Cs n) κ B γ g f ω hw hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) (V n ω) (hv n) e,
+      observedValue_eq (Cs n) κ B γ g f ω hw hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) (V n ω) (hv n) r]
+  apply tendsto_integral_of_dominated_convergence (fun _ : ℂ => 2*ρ^(-(1-κ/8))) hfω (integrable_const _)
+  · intro n
+    filter_upwards [ae_restrict_mem hA] with z hz
+    rw [heq n z hz,Real.norm_eq_abs]
+    have hn (s : ℝ) := ((Cs n).at z hz).physicalValue_nonneg κ B γ s ω
+    have hb (s : ℝ) := ((Cs n).at z hz).physicalValue_bound κ hκ B γ s ω (hhalf z hz) (hu z hz) hc hb0
+    rw [hr z hz] at hb
+    calc
+      _ ≤ |((Cs n).at z hz).physicalValue κ B γ e ω| + |((Cs n).at z hz).physicalValue κ B γ r ω| := abs_sub _ _
+      _ ≤ 2*ρ^(-(1-κ/8)) := by rw [abs_of_nonneg (hn e),abs_of_nonneg (hn r)]; linarith [hb e,hb r]
+  · filter_upwards [ae_restrict_mem hA] with z hz
+    have hl (s : ℝ) (hs : 0 < s) (hse : s < z.im) (hρs : ρ < s/4) := physicalValue_real_tendsto κ hκ B γ g f ω hw
+      (hhalf z hz) (hu z hz) hc hb0 hs hse (by rw [hr z hz]; exact hρs) (fun n => (Cs n).at z hz) hCs
+    exact ((hl e he (her.trans_lt (hrs z hz)) hρ).sub (hl r (he.trans_le her) (hrs z hz) (by linarith))).congr
+      (fun n => (heq n z hz).symm)
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.FiniteTilt
+
+lemma compact_log_cutoffs {A : Set ℂ} (hA : IsCompact A) (hne : A.Nonempty)
+    (hy : ∀ z ∈ A, 1 ≤ z.im) {ρ : ℝ} (hρ : 0 < ρ) (hρs : ρ < 1) :
+    ∃ Cs : ℕ → CommonCutoff A (fun z => Real.log (2*z.im/ρ)),
+      Tendsto (fun n => (Cs n).angle) atTop (𝓝 0) := by
+  have hy0 (z : ℂ) (hz : z ∈ A) : 0 < z.im := lt_of_lt_of_le (by norm_num) (hy z hz)
+  have hR (z : ℂ) (hz : z ∈ A) : 1 < 2*z.im/ρ := (lt_div_iff₀ hρ).2 (by linarith [hy z hz])
+  have hu (z : ℂ) (hz : z ∈ A) : 0 < Real.log (2*z.im/ρ) := Real.log_pos (hR z hz)
+  have huc : ContinuousOn (fun z : ℂ => Real.log (2*z.im/ρ)) A :=
+    ((continuous_const.mul Complex.continuous_im).div_const ρ).continuousOn.log (fun z hz => (lt_trans zero_lt_one (hR z hz)).ne')
+  have hsc : ContinuousOn (fun z : ℂ => z.im/‖z‖) A := Complex.continuous_im.continuousOn.div
+    continuous_norm.continuousOn (fun z hz => norm_ne_zero_iff.2 (by intro he; simpa [he] using hy0 z hz))
+  obtain ⟨zH,hzH,hH⟩ := hA.exists_isMaxOn hne Complex.continuous_im.continuousOn
+  obtain ⟨zU,hzU,hU⟩ := hA.exists_isMaxOn hne huc
+  obtain ⟨zS,hzS,hS⟩ := hA.exists_isMinOn hne hsc
+  exact commonCutoffs (hy zH hzH) (hu zU hzU).le (initial_sine_pos (hy0 zS hzS))
+    (fun z hz => ⟨hy z hz,hH hz⟩) (fun z hz => ⟨(hu z hz).le,hU hz⟩) (fun z hz => hS hz)
+
+lemma logRadius_pos {z : ℂ} (hz : 1 ≤ z.im) {ρ : ℝ} (hρ : 0 < ρ) (hρs : ρ < 1) :
+    0 < Real.log (2*z.im/ρ) := Real.log_pos ((lt_div_iff₀ hρ).2 (by linarith))
+
+lemma logRadius_eq {z : ℂ} (hz : 0 < z.im) {ρ : ℝ} (hρ : 0 < ρ) :
+    (2*z.im)*Real.exp (-Real.log (2*z.im/ρ)) = ρ := by
+  rw [Real.exp_neg,Real.exp_log (by positivity)]
+  field_simp
+
+lemma collar_height {z₀ : ℂ} (hcollar : closedBall z₀ 2 ⊆ halfPlane) :
+    ∀ z ∈ closedBall z₀ 1, 1 ≤ z.im := by
+  have hbase : 2 < z₀.im := by
+    have hm : z₀-2*Complex.I ∈ closedBall z₀ 2 := by
+      rw [mem_closedBall,dist_eq_norm]
+      simp
+    have hh := hcollar hm
+    change 0 < (z₀-2*Complex.I).im at hh
+    norm_num at hh
+    linarith
+  intro z hz
+  have hd := mem_closedBall.1 hz
+  have hi := Complex.abs_im_le_norm (z-z₀)
+  rw [Complex.sub_im,← dist_eq_norm] at hi
+  have hlo := neg_le_abs (z.im-z₀.im)
+  linarith
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open FiniteTilt
+local instance meshExponentsMgfRealMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance meshExponentsMgfRealBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance meshExponentsMgfComplexMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance meshExponentsMgfComplexBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+theorem physical_slab_mgf :
+    ∃ J : ℝ, 0 < J ∧ ∀ {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+      {A : Set ℂ}, IsCompact A → A.Nonempty → (∀ z ∈ A, 1 ≤ z.im) →
+      ∀ (κ : ℝ), 0 < κ ∧ κ ≤ 4 → ∀ (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
+      ∀ (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ),
+      (∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P) →
+      (∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω)) →
+      ∀ e r : ℝ, 0 < e → e ≤ r → r < 1 → ∀ θ : ℝ,
+      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z in A,
+        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal -
+        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z r).toReal))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(2*(Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2*
+        (J*r^2)*(volume A).toReal)/2)) := by
+  obtain ⟨J,hJ,hJb⟩ := Mesh.loewner_boundary_layer
+  refine ⟨J,hJ,?_⟩
+  intro Ω m P hP A hA hne hy κ hκ B hB γ g f hγ hmaps e r he her hr θ
+  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
+  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.2 hA.measure_lt_top.ne
+  let ρ := e/8
+  have hρ : 0 < ρ := by dsimp [ρ]; positivity
+  have hρs : ρ < 1 := by dsimp [ρ]; linarith
+  obtain ⟨Cs,hCs⟩ := compact_log_cutoffs hA hne hy hρ hρs
+  have hcont : ∀ᵐ ω ∂P, ContinuousOn (γ ω) (Ici 0) := hmaps.mono (fun _ hw => hw.1)
+  choose V hVm hV using (fun n => ContinuousDriver.exists_measurable_trace_path γ hγ hcont
+    ((Cs n).horizon:ℝ) (Cs n).horizon.coe_nonneg)
+  have hh (z : ℂ) (hz : z ∈ A) : z ∈ halfPlane := lt_of_lt_of_le zero_lt_one (hy z hz)
+  have hu (z : ℂ) (hz : z ∈ A) : 0 < Real.log (2*z.im/ρ) := logRadius_pos (hy z hz) hρ hρs
+  have hrad (z : ℂ) (hz : z ∈ A) : (2*z.im)*Real.exp (-Real.log (2*z.im/ρ)) = ρ := logRadius_eq (hh z hz) hρ
+  let X := fun n ω => ∫ z in A, observedDifference (Cs n).height_pos (Cs n).horizon.coe_nonneg κ ρ (Cs n).angle e r
+    ((BrownianDriver.path B (Real.sqrt κ) (Cs n).horizon ω,V n ω),z)
+  have hXm (n : ℕ) : AEMeasurable (X n) P :=
+    (observedDifference_aemeasurable hB (volume.restrict A) (Cs n).height_pos κ ρ (Cs n).angle e r
+      (Cs n).horizon (V n) (hVm n)).aestronglyMeasurable.integral_prod_right'.aemeasurable
+  have hlim := common_slab_limit Cs hCs hA.measurableSet κ hκ' B hB γ g f hmaps hh hu hrad
+    V hVm hV e r he her (fun z hz => hr.trans_le (hy z hz)) (by dsimp [ρ]; linarith)
+  apply exponential_bound_of_ae_tendsto P X _ hXm hlim _ θ
+  intro n
+  exact common_slab_mgf (Cs n) hA.measurableSet κ hκ B hB γ g f hmaps hh hu hrad (V n) (hVm n) (hV n)
+    e r he her (fun z hz => her.trans_lt (hr.trans_le (hy z hz))) J hJ.le
+    (fun ω hw t ht z hz => hJb _ _ _ _ hw t ht z hz r (he.trans_le her)) θ
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.MeshSlab
+
+lemma physicalDensity_bound {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (hm : IsLoewnerTraceUsing driver γ g f) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → ℝ) {z : ℂ} {e : ℝ} (he : 0 < e) (hes : e < z.im) :
+    (physicalStoppedDensity κ B γ g z e).toReal ≤ (e/4)^(-codimension κ) := by
+  by_cases hf : firstLiveDistanceStop γ z e = ⊤
+  · simp only [physicalStoppedDensity,hf,ite_true,ENNReal.toReal_zero]
+    positivity
+  obtain ⟨t,ht,hzt,hd,hmin⟩ := PhysicalStop.finite_spec hm.1 hf
+  have hR := hm.radius_ge_before_distance_stop he hes (t := t) (by rw [ht]) hzt
+  have hp : 0 ≤ codimension κ := by unfold codimension dimension; linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le); linarith
+  have hy := hm.im_pos t.coe_nonneg hzt
+  let w := g t z - (Real.sqrt κ*B t : ℂ)
+  have hwi : w.im = (g t z).im := by simp [w]
+  have hwn : 0 < ‖w‖ := norm_pos_iff.2 (by intro hh; have hi := congrArg Complex.im hh; rw [hwi] at hi; simpa using hy.ne' hi)
+  have hs0 : 0 ≤ loewnerAngleSine κ B g t z := by unfold loewnerAngleSine; positivity
+  have hs1 : loewnerAngleSine κ B g t z ≤ 1 := by
+    simp only [loewnerAngleSine,Real.toNNReal_coe]
+    change (g t z).im / ‖w‖ ≤ 1
+    apply (div_le_one hwn).2
+    rw [← hwi]
+    exact Complex.im_le_norm w
+  have hs := Real.rpow_le_one hs0 hs1 ha
+  have hR0 : 0 < loewnerRadius g t z := (by positivity : 0 < e/4).trans_le hR
+  have hrpow := Real.rpow_le_rpow_of_nonpos (by positivity : 0 < e/4) hR (neg_nonpos.2 hp)
+  have hv0 : 0 ≤ loewnerRadius g t z ^ (-codimension κ) * loewnerAngleSine κ B g t z ^ (8/κ-1) := by positivity
+  rw [physicalStoppedDensity,ht,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,ENNReal.toReal_ofReal hv0]
+  exact (mul_le_of_le_one_right (Real.rpow_nonneg hR0.le _) hs).trans hrpow
+
+def physicalMass {Ω : Type*} (A : Set ℂ) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (e : ℝ) (ω : Ω) : ℝ :=
+  ∫ z in A, (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal
+
+lemma physicalMass_nonneg {Ω : Type*} (A : Set ℂ) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g : Ω → ℝ → ℂ → ℂ) (e : ℝ) (ω : Ω) :
+    0 ≤ physicalMass A κ B γ g e ω := integral_nonneg (fun _ => ENNReal.toReal_nonneg)
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.MeshSlab
+variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+variable (A : Set ℂ) [sfiniteVolume : SFinite (volume.restrict A)]
+variable (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+variable (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+variable (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P)
+variable (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+variable (e : ℝ) (he : 0 < e)
+include hκ hB hγ hmaps he
+
+lemma physicalMass_aemeasurable : AEMeasurable (physicalMass A κ B γ g e) P := by
+  obtain ⟨W,hW,hWe⟩ := jointPhysicalDensity κ hκ B hB γ g f hγ hmaps e he
+  apply (hW.ennreal_toReal.stronglyMeasurable.integral_prod_right' (ν := volume.restrict A)).aemeasurable.congr
+  filter_upwards [hWe] with ω hw
+  exact integral_congr_ae (Filter.Eventually.of_forall (fun z => congrArg ENNReal.toReal (hw z)))
+
+omit sfiniteVolume in
+lemma physicalDensity_integrable [SFinite (volume.restrict A)] [IsFiniteMeasure (volume.restrict A)] (hA : MeasurableSet A)
+    (hes : ∀ z ∈ A, e < z.im) :
+    ∀ᵐ ω ∂P, Integrable (fun z => (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal) (volume.restrict A) := by
+  obtain ⟨W,hW,hWe⟩ := jointPhysicalDensity κ hκ B hB γ g f hγ hmaps e he
+  filter_upwards [hWe,hmaps] with ω hw hm
+  have hwm : Measurable (fun z => (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal) := by
+    have hh : Measurable (fun z : ℂ => (W (ω,z)).toReal) := hW.ennreal_toReal.comp (measurable_const.prodMk measurable_id)
+    convert hh using 1
+    funext z
+    rw [hw z]
+  apply Integrable.of_bound hwm.aestronglyMeasurable ((e/4)^(-codimension κ))
+  filter_upwards [ae_restrict_mem hA] with z hz
+  rw [Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg]
+  exact physicalDensity_bound hm κ hκ (fun t => B t ω) he (hes z hz)
+
+lemma physicalMass_lintegral [IsFiniteMeasure (volume.restrict A)] (hA : MeasurableSet A)
+    (hes : ∀ z ∈ A, e < z.im) :
+    ∀ᵐ ω ∂P, ENNReal.ofReal (physicalMass A κ B γ g e ω) =
+      ∫⁻ z in A, physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e := by
+  filter_upwards [physicalDensity_integrable A κ hκ B hB γ g f hγ hmaps e he hA hes] with ω hi
+  rw [physicalMass,ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall (fun _ => ENNReal.toReal_nonneg))]
+  apply lintegral_congr
+  intro z
+  exact ENNReal.ofReal_toReal (physicalStoppedDensity_ne_top _ _ _ _ _ _)
+
+lemma physicalMass_bound [IsFiniteMeasure (volume.restrict A)] (hA : MeasurableSet A)
+    (hes : ∀ z ∈ A, e < z.im) :
+    ∀ᵐ ω ∂P, physicalMass A κ B γ g e ω ≤ (volume A).toReal*(e/4)^(-codimension κ) := by
+  filter_upwards [physicalDensity_integrable A κ hκ B hB γ g f hγ hmaps e he hA hes,hmaps] with ω hi hm
+  have hh := integral_mono_ae hi (integrable_const ((e/4)^(-codimension κ))) (μ := volume.restrict A) (by
+    filter_upwards [ae_restrict_mem hA] with z hz
+    exact physicalDensity_bound hm κ hκ (fun t => B t ω) he (hes z hz))
+  simpa only [integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul,physicalMass] using hh
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.MeshSlab
+
+lemma slab_radius_scale (p : ℝ) {r : ℝ} (hr : 0 < r) :
+    (r/8)^(-p)*r = (8:ℝ)^p*r^(1-p) := by
+  rw [Real.div_rpow hr.le (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 8),div_inv_eq_mul,
+    show 1-p = 1+(-p) by ring,Real.rpow_add hr,Real.rpow_one]
+  ring
+
+theorem slab_mass_mgf (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) :
+    ∃ K : ℝ, 0 < K ∧ ∀ {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+      (B : ℝ≥0 → Ω → ℝ), IsBrownianReal B P →
+      ∀ (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ),
+      (∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P) →
+      (∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω)) →
+      ∀ (z₀ : ℂ), closedBall z₀ 2 ⊆ halfPlane → ∀ e r : ℝ,
+      0 < e → e ≤ r → r/2 ≤ e → r < 1 → ∀ θ : ℝ,
+      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(physicalMass (closedBall z₀ 1) κ B γ g e ω -
+        physicalMass (closedBall z₀ 1) κ B γ g r ω))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(K*r^(1-codimension κ))^2/2)) := by
+  obtain ⟨J,hJ,hmgf⟩ := physical_slab_mgf
+  let L := Real.sqrt κ*(8/κ-1)*(8:ℝ)^(codimension κ)
+  have hp : 0 ≤ codimension κ := by unfold codimension dimension; linarith [hκ.2]
+  have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by linarith [hκ.2]))
+  have hL : 0 < L := mul_pos (mul_pos (Real.sqrt_pos.2 hκ.1) ha) (Real.rpow_pos_of_pos (by norm_num) _)
+  let K := Real.sqrt (2*J*Real.pi)*L
+  have hJπ : 0 < 2*J*Real.pi := by positivity
+  refine ⟨K,mul_pos (Real.sqrt_pos.2 hJπ) hL,?_⟩
+  intro Ω m P hP B hB γ g f hγ hmaps z₀ hcollar e r he her hel hr θ
+  let A := closedBall z₀ (1:ℝ)
+  have hAc : IsCompact A := isCompact_closedBall _ _
+  have hAm : MeasurableSet A := hAc.measurableSet
+  have hy := FiniteTilt.collar_height hcollar
+  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.2 hAc.measure_lt_top.ne
+  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
+  have hh := hmgf hAc (nonempty_closedBall.2 (by norm_num)) hy κ hκ B hB γ g f hγ hmaps e r he her hr θ
+  have hIe := physicalDensity_integrable A κ hκ' B hB γ g f hγ hmaps e he hAm
+    (fun z hz => her.trans_lt (hr.trans_le (hy z hz)))
+  have hIr := physicalDensity_integrable A κ hκ' B hB γ g f hγ hmaps r (he.trans_le her) hAm
+    (fun z hz => hr.trans_le (hy z hz))
+  have heq : (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(physicalMass A κ B γ g e ω - physicalMass A κ B γ g r ω))) ∂P) =
+      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z in A,
+        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z e).toReal -
+        (physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z r).toReal))) ∂P) := by
+    apply lintegral_congr_ae
+    filter_upwards [hIe,hIr] with ω hie hir
+    rw [integral_sub hie hir]
+    rfl
+  rw [heq]
+  apply hh.trans
+  apply ENNReal.ofReal_le_ofReal
+  apply Real.exp_le_exp.2
+  apply div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left _ (sq_nonneg θ)) (by norm_num)
+  have hr0 := he.trans_le her
+  have hrad := Real.rpow_le_rpow_of_nonpos (by positivity : 0 < r/8) (by linarith : r/8 ≤ e/4) (neg_nonpos.2 hp)
+  have hmul : (Real.sqrt κ*(8/κ-1)*(e/4)^(-codimension κ))*r ≤ L*r^(1-codimension κ) := by
+    calc
+      _ ≤ (Real.sqrt κ*(8/κ-1)*(r/8)^(-codimension κ))*r := by gcongr
+      _ = _ := by rw [mul_assoc,slab_radius_scale _ hr0]; dsimp [L]; ring
+  have hleft : 0 ≤ (Real.sqrt κ*(8/κ-1)*(e/4)^(-codimension κ))*r := by positivity
+  have hsq := pow_le_pow_left₀ hleft hmul 2
+  have hv : (volume A).toReal = Real.pi := by
+    dsimp [A]
+    rw [Complex.volume_closedBall]
+    norm_num [ENNReal.toReal_ofReal Real.pi_pos.le]
+  rw [hv]
+  have hcode : 1-κ/8 = codimension κ := by unfold codimension dimension; ring
+  rw [hcode]
+  have hmul' := mul_le_mul_of_nonneg_left hsq hJπ.le
+  dsimp [K] at *
+  have hsqrt := Real.sq_sqrt hJπ.le
+  nlinarith
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.MeshSlab
+
+lemma finite_mgf_sum {Ω ι : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (t : Finset ι) (Y : ι → Ω → ℝ) (σ : ι → ℝ)
+    (hσ : ∀ i ∈ t, 0 < σ i) (hY : ∀ i ∈ t, AEMeasurable (Y i) P)
+    (hmgf : ∀ i ∈ t, ∀ θ : ℝ, (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*Y i ω)) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(σ i)^2/2))) (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∑ i ∈ t, Y i ω))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(∑ i ∈ t, σ i)^2/2)) := by
+  classical
+  by_cases ht : t.Nonempty
+  · let S := ∑ i ∈ t, σ i
+    have hS : 0 < S := Finset.sum_pos (fun i hi => hσ i hi) ht
+    let w := fun i => σ i/S
+    have hw (i : ι) (hi : i ∈ t) : 0 ≤ w i := (div_pos (hσ i hi) hS).le
+    have hw1 : ∑ i ∈ t, w i = 1 := by simp only [w,← Finset.sum_div,S,div_self hS.ne']
+    have hpoint (ω : Ω) : Real.exp (θ*(∑ i ∈ t, Y i ω)) ≤
+        ∑ i ∈ t, w i * Real.exp ((θ*S/σ i)*Y i ω) := by
+      have hh := convexOn_exp.map_sum_le (t := t) (w := w) (p := fun i => (θ*S/σ i)*Y i ω)
+        hw hw1 (fun _ _ => mem_univ _)
+      have he : (∑ i ∈ t, w i • ((θ*S/σ i)*Y i ω)) = θ*(∑ i ∈ t, Y i ω) := by
+        conv_rhs => rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro i hi
+        dsimp [w]
+        field_simp [hS.ne', (hσ i hi).ne']
+      rw [he] at hh
+      simpa only [smul_eq_mul] using hh
+    calc
+      _ ≤ ∫⁻ ω, ∑ i ∈ t, ENNReal.ofReal (w i) * ENNReal.ofReal (Real.exp ((θ*S/σ i)*Y i ω)) ∂P := by
+        apply lintegral_mono
+        intro ω
+        calc
+          _ ≤ ENNReal.ofReal (∑ i ∈ t, w i * Real.exp ((θ*S/σ i)*Y i ω)) := ENNReal.ofReal_le_ofReal (hpoint ω)
+          _ = _ := by
+            rw [ENNReal.ofReal_sum_of_nonneg (fun i hi => mul_nonneg (hw i hi) (Real.exp_pos _).le)]
+            apply Finset.sum_congr rfl
+            intro i hi
+            exact ENNReal.ofReal_mul (hw i hi)
+      _ = ∑ i ∈ t, ENNReal.ofReal (w i) * (∫⁻ ω, ENNReal.ofReal (Real.exp ((θ*S/σ i)*Y i ω)) ∂P) := by
+        rw [lintegral_finsetSum' _ (fun i hi => ((hY i hi).const_mul _).exp.ennreal_ofReal.const_mul _)]
+        apply Finset.sum_congr rfl
+        intro i hi
+        exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+      _ ≤ ∑ i ∈ t, ENNReal.ofReal (w i) * ENNReal.ofReal (Real.exp (θ^2*S^2/2)) := by
+        apply Finset.sum_le_sum
+        intro i hi
+        have hh := hmgf i hi (θ*S/σ i)
+        have he : (θ*S/σ i)^2*(σ i)^2/2 = θ^2*S^2/2 := by field_simp [(hσ i hi).ne']
+        rw [he] at hh
+        exact mul_le_mul_right hh _
+      _ = _ := by
+        rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg hw,hw1,ENNReal.ofReal_one,one_mul]
+  · have he : t = ∅ := Finset.not_nonempty_iff_eq_empty.1 ht
+    simp [he]
+
+lemma geo_radius_power {δ : ℝ} (hδ : 0 < δ) (a : ℝ) (n : ℕ) :
+    (δ*(1/2:ℝ)^n)^a = δ^a*((1/2:ℝ)^a)^n := by
+  rw [Real.mul_rpow hδ.le (by positivity),← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 1/2),
+    mul_comm (n:ℝ) a,Real.rpow_mul_natCast (by norm_num : (0:ℝ) ≤ 1/2)]
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/MeshSlab.lean b/lean/OAI/Probability/SLELocalization/MeshSlab.lean
index 0696f82fcbdbb3298083e22b0b168906d5dd60c6..70539344147ef0b53009aa8ed4eda1217eca01ca 100644
--- a/lean/OAI/Probability/SLELocalization/MeshSlab.lean
+++ b/lean/OAI/Probability/SLELocalization/MeshSlab.lean
@@ -1,213 +1,213 @@
-import Mathlib
-import OAI.Probability.SLELocalization.MeshExponents
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Mesh Slab estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.MeshSlab
-
-lemma geometric_mesh_mgf {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    (X : ℝ → Ω → ℝ) {p K δ rf : ℝ} (hp : p < 1) (hK : 0 < K)
-    (hrf : 0 < rf) (hrδ : rf ≤ δ)
-    (hX : ∀ r, 0 < r → r ≤ δ → AEMeasurable (X r) P)
-    (hslab : ∀ e r, 0 < e → e ≤ r → r/2 ≤ e → r ≤ δ → ∀ θ : ℝ,
-      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(X e ω-X r ω))) ∂P) ≤
-        ENNReal.ofReal (Real.exp (θ^2*(K*r^(1-p))^2/2))) (θ : ℝ) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(X rf ω-X δ ω))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(K*δ^(1-p)/(1-(1/2:ℝ)^(1-p)))^2/2)) := by
-  let q : ℝ := (1/2:ℝ)^(1-p)
-  have hq : 0 < q := Real.rpow_pos_of_pos (by norm_num) _
-  have hq1 : q < 1 := Real.rpow_lt_one (by norm_num) (by norm_num) (by linarith)
-  have hδ : 0 < δ := hrf.trans_le hrδ
-  let R : ℕ → ℝ := fun n => max rf (δ*(1/2:ℝ)^n)
-  have hRpos (n : ℕ) : 0 < R n := hrf.trans_le (le_max_left _ _)
-  have hRle (n : ℕ) : R n ≤ δ := max_le hrδ (by
-    have hh : (1/2:ℝ)^n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
-    simpa using mul_le_mul_of_nonneg_left hh hδ.le)
-  have hRm (n : ℕ) : R (n+1) ≤ R n := by
-    apply max_le_max le_rfl
-    rw [pow_succ]
-    nlinarith [pow_pos (by norm_num : (0:ℝ) < 1/2) n]
-  have hR0 : R 0 = δ := by dsimp [R]; simp only [pow_zero,mul_one,max_eq_right hrδ]
-  have ht : Tendsto (fun n : ℕ => δ*(1/2:ℝ)^n) atTop (𝓝 0) := by
-    simpa only [mul_zero] using (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2)
-      (by norm_num : (1/2:ℝ) < 1)).const_mul δ
-  obtain ⟨N,hN⟩ := (ht.eventually (gt_mem_nhds hrf)).exists
-  have hRN : R N = rf := max_eq_left hN.le
-  let Y := fun n ω => X (R (n+1)) ω-X (R n) ω
-  let σ := fun n => K*δ^(1-p)*q^n
-  have hσ (n : ℕ) : 0 < σ n := by dsimp [σ]; positivity
-  have hYm (n : ℕ) : AEMeasurable (Y n) P :=
-    (hX _ (hRpos (n+1)) (hRle (n+1))).sub (hX _ (hRpos n) (hRle n))
-  have hmgf (n : ℕ) (s : ℝ) : (∫⁻ ω, ENNReal.ofReal (Real.exp (s*Y n ω)) ∂P) ≤
-      ENNReal.ofReal (Real.exp (s^2*(σ n)^2/2)) := by
-    by_cases hn : rf ≤ δ*(1/2:ℝ)^n
-    · have hRn : R n = δ*(1/2:ℝ)^n := max_eq_right hn
-      have hhalf : R n/2 ≤ R (n+1) := by
-        rw [hRn]
-        apply le_trans _ (le_max_right _ _)
-        rw [pow_succ]
-        ring_nf
-        exact le_rfl
-      have hh := hslab _ _ (hRpos (n+1)) (hRm n) hhalf (hRle n) s
-      have he : K*(R n)^(1-p) = σ n := by
-        rw [hRn,geo_radius_power hδ]
-        dsimp [σ,q]
-        ring
-      simpa only [Y,he] using hh
-    · have he : R n = rf := max_eq_left (le_of_not_ge hn)
-      have hne : R (n+1) = rf := le_antisymm (by rw [← he]; exact hRm n) (le_max_left _ _)
-      have hY0 : ∀ ω, Y n ω = 0 := fun ω => by dsimp [Y]; rw [he,hne]; ring
-      simp only [hY0,mul_zero,Real.exp_zero,ENNReal.ofReal_one,lintegral_const,measure_univ,mul_one]
-      have hn0 : 0 ≤ s^2*(σ n)^2/2 := by positivity
-      have he1 : 1 ≤ Real.exp (s^2*(σ n)^2/2) := Real.one_le_exp hn0
-      simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal he1
-  have hsum := finite_mgf_sum (Finset.range N) Y σ (fun n _ => hσ n) (fun n _ => hYm n)
-    (fun n _ s => hmgf n s) θ
-  have htel (ω : Ω) : (∑ n ∈ Finset.range N, Y n ω) = X rf ω-X δ ω := by
-    rw [show (∑ n ∈ Finset.range N, Y n ω) = X (R N) ω-X (R 0) ω from Finset.sum_range_sub (fun n => X (R n) ω) N,
-      hRN,hR0]
-  simp only [htel] at hsum
-  apply hsum.trans
-  apply ENNReal.ofReal_le_ofReal
-  apply Real.exp_le_exp.2
-  apply div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left _ (sq_nonneg θ)) (by norm_num)
-  apply pow_le_pow_left₀ (Finset.sum_nonneg (fun n _ => (hσ n).le))
-  rw [show (∑ n ∈ Finset.range N, σ n) = K*δ^(1-p)*(∑ n ∈ Finset.range N, q^n) by simp only [σ,Finset.mul_sum]]
-  have hg : (∑ n ∈ Finset.range N, q^n) ≤ 1/(1-q) := by
-    apply (le_div_iff₀ (sub_pos.2 hq1)).2
-    rw [geom_sum_mul_neg]
-    exact sub_le_self _ (pow_nonneg hq.le N)
-  have hh := mul_le_mul_of_nonneg_left hg (show 0 ≤ K*δ^(1-p) by positivity)
-  simpa only [mul_one_div,q] using hh
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.MeshSlab
-
-lemma tail_scale (p : ℝ) {δ K : ℝ} (hδ : 0 < δ) (hK : 0 < K) :
-    (δ^(-p))^2/(2*(K*δ^(1-p))^2) = (1/(2*K^2))*δ^(-2:ℝ) := by
-  have hp : (δ^(-p))^2 = (δ^(1-p))^2*δ^(-2:ℝ) := by
-    rw [← Real.rpow_mul_natCast hδ.le,← Real.rpow_mul_natCast hδ.le,← Real.rpow_add hδ]
-    congr 1
-    norm_num
-    ring
-  rw [hp,mul_pow]
-  field_simp [(Real.rpow_pos_of_pos hδ (1-p)).ne',hK.ne']
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open MeshSlab
-
-theorem integratedMeshTail_halfPlane_simple (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) :
-    ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧
-      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
-        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
-        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-        (z₀ : ℂ) (_hcollar : Metric.closedBall z₀ 2 ⊆ halfPlane)
-        (r_f δ : ℝ), 0 < r_f → r_f ≤ δ → δ ≤ δ₀ →
-        P {sample | ENNReal.ofReal (C*δ^(-codimension κ)) <
-          ∫⁻ z in Metric.closedBall z₀ 1,
-            physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z r_f ∂volume} ≤
-            ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
-  obtain ⟨K,hK,hslab⟩ := slab_mass_mgf κ hκ
-  let p := codimension κ
-  have hp : p < 1 := by dsimp [p,codimension,dimension]; linarith [hκ.1]
-  let q : ℝ := (1/2:ℝ)^(1-p)
-  have hq : q < 1 := Real.rpow_lt_one (by norm_num) (by norm_num) (by linarith)
-  let K' := K/(1-q)
-  have hK' : 0 < K' := div_pos hK (sub_pos.2 hq)
-  let C := max 1 (Real.pi*(4:ℝ)^p+1)
-  have hC1 : 1 ≤ C := le_max_left _ _
-  have hCbase : Real.pi*(4:ℝ)^p+1 ≤ C := le_max_right _ _
-  let c := 1/(2*K'^2)
-  have hc : 0 < c := by dsimp [c]; positivity
-  refine ⟨C,c,1/2,lt_of_lt_of_le zero_lt_one hC1,hc,by norm_num,?_⟩
-  intro Ω m P hP B hB γ g f hγ hmaps z₀ hcollar rf δ hrf hrδ hδsmall
-  let A := closedBall z₀ (1:ℝ)
-  let X := physicalMass A κ B γ g
-  have hAc : IsCompact A := isCompact_closedBall _ _
-  have hA : MeasurableSet A := hAc.measurableSet
-  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.2 hAc.measure_lt_top.ne
-  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
-  have hδ : 0 < δ := hrf.trans_le hrδ
-  have hδ1 : δ < 1 := by linarith
-  have hy := FiniteTilt.collar_height hcollar
-  have hXm (r : ℝ) (hr : 0 < r) : AEMeasurable (X r) P := by
-    exact physicalMass_aemeasurable A κ hκ' B hB γ g f hγ hmaps r hr
-  have hmgf (θ : ℝ) : (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(X rf ω-X δ ω))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*(K'*δ^(1-p))^2/2)) := by
-    have hh := geometric_mesh_mgf X hp hK hrf hrδ (fun r hr _ => hXm r hr)
-      (fun e r he her hel hrδ θ => hslab B hB γ g f hγ hmaps z₀ hcollar e r he her hel (hrδ.trans_lt hδ1) θ) θ
-    convert hh using 1
-    congr 3
-    dsimp [K',q]
-    ring
-  have htail := BrownianPredictable.tail_of_exponential_bound (fun ω => X rf ω-X δ ω)
-    ((hXm rf hrf).sub (hXm δ hδ)) (sq_pos_of_pos (mul_pos hK' (Real.rpow_pos_of_pos hδ _)))
-    hmgf (Real.rpow_pos_of_pos hδ (-p))
-  have hbound := physicalMass_bound A κ hκ' B hB γ g f hγ hmaps δ hδ hA
-    (fun z hz => hδ1.trans_le (hy z hz))
-  have hlint := physicalMass_lintegral A κ hκ' B hB γ g f hγ hmaps rf hrf hA
-    (fun z hz => hrδ.trans_lt (hδ1.trans_le (hy z hz)))
-  have hv : (volume A).toReal = Real.pi := by dsimp [A]; rw [Complex.volume_closedBall]; norm_num
-  have hinit : (volume A).toReal*(δ/4)^(-codimension κ) = Real.pi*(4:ℝ)^p*δ^(-p) := by
-    rw [hv,Real.div_rpow hδ.le (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 4),div_inv_eq_mul]
-    dsimp [p]
-    ring
-  have hsub : {ω | ENNReal.ofReal (C*δ^(-codimension κ)) <
-      ∫⁻ z in A, physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z rf} ≤ᵐ[P]
-      {ω | δ^(-p) < X rf ω-X δ ω} := by
-    filter_upwards [hbound,hlint] with ω hb hl hs
-    rw [hinit] at hb
-    change X δ ω ≤ Real.pi*(4:ℝ)^p*δ^(-p) at hb
-    change ENNReal.ofReal (C*δ^(-codimension κ)) <
-      (∫⁻ z in A, physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z rf) at hs
-    rw [← hl] at hs
-    by_contra hn
-    have hd := le_of_not_gt hn
-    have hscale := mul_le_mul_of_nonneg_right hCbase (Real.rpow_nonneg hδ.le (-p))
-    have he : X rf ω ≤ C*δ^(-codimension κ) := by change X rf ω ≤ C*δ^(-p); nlinarith
-    exact (not_lt_of_ge (ENNReal.ofReal_le_ofReal he)) hs
-  apply (measure_mono_ae hsub).trans
-  apply htail.trans
-  rw [neg_div,tail_scale p hδ hK']
-  apply ENNReal.ofReal_le_ofReal
-  have he : -(1/(2*K'^2)*δ^(-2:ℝ)) = -c*δ^(-2:ℝ) := by dsimp [c]; ring
-  rw [he]
-  exact le_mul_of_one_le_left (Real.exp_pos _).le hC1
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.MeshExponents
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Mesh Slab estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.MeshSlab
+
+lemma geometric_mesh_mgf {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    (X : ℝ → Ω → ℝ) {p K δ rf : ℝ} (hp : p < 1) (hK : 0 < K)
+    (hrf : 0 < rf) (hrδ : rf ≤ δ)
+    (hX : ∀ r, 0 < r → r ≤ δ → AEMeasurable (X r) P)
+    (hslab : ∀ e r, 0 < e → e ≤ r → r/2 ≤ e → r ≤ δ → ∀ θ : ℝ,
+      (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(X e ω-X r ω))) ∂P) ≤
+        ENNReal.ofReal (Real.exp (θ^2*(K*r^(1-p))^2/2))) (θ : ℝ) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(X rf ω-X δ ω))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(K*δ^(1-p)/(1-(1/2:ℝ)^(1-p)))^2/2)) := by
+  let q : ℝ := (1/2:ℝ)^(1-p)
+  have hq : 0 < q := Real.rpow_pos_of_pos (by norm_num) _
+  have hq1 : q < 1 := Real.rpow_lt_one (by norm_num) (by norm_num) (by linarith)
+  have hδ : 0 < δ := hrf.trans_le hrδ
+  let R : ℕ → ℝ := fun n => max rf (δ*(1/2:ℝ)^n)
+  have hRpos (n : ℕ) : 0 < R n := hrf.trans_le (le_max_left _ _)
+  have hRle (n : ℕ) : R n ≤ δ := max_le hrδ (by
+    have hh : (1/2:ℝ)^n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
+    simpa using mul_le_mul_of_nonneg_left hh hδ.le)
+  have hRm (n : ℕ) : R (n+1) ≤ R n := by
+    apply max_le_max le_rfl
+    rw [pow_succ]
+    nlinarith [pow_pos (by norm_num : (0:ℝ) < 1/2) n]
+  have hR0 : R 0 = δ := by dsimp [R]; simp only [pow_zero,mul_one,max_eq_right hrδ]
+  have ht : Tendsto (fun n : ℕ => δ*(1/2:ℝ)^n) atTop (𝓝 0) := by
+    simpa only [mul_zero] using (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2)
+      (by norm_num : (1/2:ℝ) < 1)).const_mul δ
+  obtain ⟨N,hN⟩ := (ht.eventually (gt_mem_nhds hrf)).exists
+  have hRN : R N = rf := max_eq_left hN.le
+  let Y := fun n ω => X (R (n+1)) ω-X (R n) ω
+  let σ := fun n => K*δ^(1-p)*q^n
+  have hσ (n : ℕ) : 0 < σ n := by dsimp [σ]; positivity
+  have hYm (n : ℕ) : AEMeasurable (Y n) P :=
+    (hX _ (hRpos (n+1)) (hRle (n+1))).sub (hX _ (hRpos n) (hRle n))
+  have hmgf (n : ℕ) (s : ℝ) : (∫⁻ ω, ENNReal.ofReal (Real.exp (s*Y n ω)) ∂P) ≤
+      ENNReal.ofReal (Real.exp (s^2*(σ n)^2/2)) := by
+    by_cases hn : rf ≤ δ*(1/2:ℝ)^n
+    · have hRn : R n = δ*(1/2:ℝ)^n := max_eq_right hn
+      have hhalf : R n/2 ≤ R (n+1) := by
+        rw [hRn]
+        apply le_trans _ (le_max_right _ _)
+        rw [pow_succ]
+        ring_nf
+        exact le_rfl
+      have hh := hslab _ _ (hRpos (n+1)) (hRm n) hhalf (hRle n) s
+      have he : K*(R n)^(1-p) = σ n := by
+        rw [hRn,geo_radius_power hδ]
+        dsimp [σ,q]
+        ring
+      simpa only [Y,he] using hh
+    · have he : R n = rf := max_eq_left (le_of_not_ge hn)
+      have hne : R (n+1) = rf := le_antisymm (by rw [← he]; exact hRm n) (le_max_left _ _)
+      have hY0 : ∀ ω, Y n ω = 0 := fun ω => by dsimp [Y]; rw [he,hne]; ring
+      simp only [hY0,mul_zero,Real.exp_zero,ENNReal.ofReal_one,lintegral_const,measure_univ,mul_one]
+      have hn0 : 0 ≤ s^2*(σ n)^2/2 := by positivity
+      have he1 : 1 ≤ Real.exp (s^2*(σ n)^2/2) := Real.one_le_exp hn0
+      simpa only [ENNReal.ofReal_one] using ENNReal.ofReal_le_ofReal he1
+  have hsum := finite_mgf_sum (Finset.range N) Y σ (fun n _ => hσ n) (fun n _ => hYm n)
+    (fun n _ s => hmgf n s) θ
+  have htel (ω : Ω) : (∑ n ∈ Finset.range N, Y n ω) = X rf ω-X δ ω := by
+    rw [show (∑ n ∈ Finset.range N, Y n ω) = X (R N) ω-X (R 0) ω from Finset.sum_range_sub (fun n => X (R n) ω) N,
+      hRN,hR0]
+  simp only [htel] at hsum
+  apply hsum.trans
+  apply ENNReal.ofReal_le_ofReal
+  apply Real.exp_le_exp.2
+  apply div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left _ (sq_nonneg θ)) (by norm_num)
+  apply pow_le_pow_left₀ (Finset.sum_nonneg (fun n _ => (hσ n).le))
+  rw [show (∑ n ∈ Finset.range N, σ n) = K*δ^(1-p)*(∑ n ∈ Finset.range N, q^n) by simp only [σ,Finset.mul_sum]]
+  have hg : (∑ n ∈ Finset.range N, q^n) ≤ 1/(1-q) := by
+    apply (le_div_iff₀ (sub_pos.2 hq1)).2
+    rw [geom_sum_mul_neg]
+    exact sub_le_self _ (pow_nonneg hq.le N)
+  have hh := mul_le_mul_of_nonneg_left hg (show 0 ≤ K*δ^(1-p) by positivity)
+  simpa only [mul_one_div,q] using hh
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.MeshSlab
+
+lemma tail_scale (p : ℝ) {δ K : ℝ} (hδ : 0 < δ) (hK : 0 < K) :
+    (δ^(-p))^2/(2*(K*δ^(1-p))^2) = (1/(2*K^2))*δ^(-2:ℝ) := by
+  have hp : (δ^(-p))^2 = (δ^(1-p))^2*δ^(-2:ℝ) := by
+    rw [← Real.rpow_mul_natCast hδ.le,← Real.rpow_mul_natCast hδ.le,← Real.rpow_add hδ]
+    congr 1
+    norm_num
+    ring
+  rw [hp,mul_pow]
+  field_simp [(Real.rpow_pos_of_pos hδ (1-p)).ne',hK.ne']
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open MeshSlab
+
+theorem integratedMeshTail_halfPlane_simple (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) :
+    ∃ C c δ₀ : ℝ, 0 < C ∧ 0 < c ∧ 0 < δ₀ ∧
+      ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
+        (B : ℝ≥0 → Ω → ℝ) (_hB : IsBrownianReal B P)
+        (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+        (_hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+        (_hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+          (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+        (z₀ : ℂ) (_hcollar : Metric.closedBall z₀ 2 ⊆ halfPlane)
+        (r_f δ : ℝ), 0 < r_f → r_f ≤ δ → δ ≤ δ₀ →
+        P {sample | ENNReal.ofReal (C*δ^(-codimension κ)) <
+          ∫⁻ z in Metric.closedBall z₀ 1,
+            physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z r_f ∂volume} ≤
+            ENNReal.ofReal (C*Real.exp (-c*δ^(-2:ℝ))) := by
+  obtain ⟨K,hK,hslab⟩ := slab_mass_mgf κ hκ
+  let p := codimension κ
+  have hp : p < 1 := by dsimp [p,codimension,dimension]; linarith [hκ.1]
+  let q : ℝ := (1/2:ℝ)^(1-p)
+  have hq : q < 1 := Real.rpow_lt_one (by norm_num) (by norm_num) (by linarith)
+  let K' := K/(1-q)
+  have hK' : 0 < K' := div_pos hK (sub_pos.2 hq)
+  let C := max 1 (Real.pi*(4:ℝ)^p+1)
+  have hC1 : 1 ≤ C := le_max_left _ _
+  have hCbase : Real.pi*(4:ℝ)^p+1 ≤ C := le_max_right _ _
+  let c := 1/(2*K'^2)
+  have hc : 0 < c := by dsimp [c]; positivity
+  refine ⟨C,c,1/2,lt_of_lt_of_le zero_lt_one hC1,hc,by norm_num,?_⟩
+  intro Ω m P hP B hB γ g f hγ hmaps z₀ hcollar rf δ hrf hrδ hδsmall
+  let A := closedBall z₀ (1:ℝ)
+  let X := physicalMass A κ B γ g
+  have hAc : IsCompact A := isCompact_closedBall _ _
+  have hA : MeasurableSet A := hAc.measurableSet
+  let : IsFiniteMeasure (volume.restrict A) := isFiniteMeasure_restrict.2 hAc.measure_lt_top.ne
+  have hκ' : 0 < κ ∧ κ < 8 := ⟨hκ.1,by linarith [hκ.2]⟩
+  have hδ : 0 < δ := hrf.trans_le hrδ
+  have hδ1 : δ < 1 := by linarith
+  have hy := FiniteTilt.collar_height hcollar
+  have hXm (r : ℝ) (hr : 0 < r) : AEMeasurable (X r) P := by
+    exact physicalMass_aemeasurable A κ hκ' B hB γ g f hγ hmaps r hr
+  have hmgf (θ : ℝ) : (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(X rf ω-X δ ω))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*(K'*δ^(1-p))^2/2)) := by
+    have hh := geometric_mesh_mgf X hp hK hrf hrδ (fun r hr _ => hXm r hr)
+      (fun e r he her hel hrδ θ => hslab B hB γ g f hγ hmaps z₀ hcollar e r he her hel (hrδ.trans_lt hδ1) θ) θ
+    convert hh using 1
+    congr 3
+    dsimp [K',q]
+    ring
+  have htail := BrownianPredictable.tail_of_exponential_bound (fun ω => X rf ω-X δ ω)
+    ((hXm rf hrf).sub (hXm δ hδ)) (sq_pos_of_pos (mul_pos hK' (Real.rpow_pos_of_pos hδ _)))
+    hmgf (Real.rpow_pos_of_pos hδ (-p))
+  have hbound := physicalMass_bound A κ hκ' B hB γ g f hγ hmaps δ hδ hA
+    (fun z hz => hδ1.trans_le (hy z hz))
+  have hlint := physicalMass_lintegral A κ hκ' B hB γ g f hγ hmaps rf hrf hA
+    (fun z hz => hrδ.trans_lt (hδ1.trans_le (hy z hz)))
+  have hv : (volume A).toReal = Real.pi := by dsimp [A]; rw [Complex.volume_closedBall]; norm_num
+  have hinit : (volume A).toReal*(δ/4)^(-codimension κ) = Real.pi*(4:ℝ)^p*δ^(-p) := by
+    rw [hv,Real.div_rpow hδ.le (by norm_num),Real.rpow_neg (by norm_num : (0:ℝ) ≤ 4),div_inv_eq_mul]
+    dsimp [p]
+    ring
+  have hsub : {ω | ENNReal.ofReal (C*δ^(-codimension κ)) <
+      ∫⁻ z in A, physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z rf} ≤ᵐ[P]
+      {ω | δ^(-p) < X rf ω-X δ ω} := by
+    filter_upwards [hbound,hlint] with ω hb hl hs
+    rw [hinit] at hb
+    change X δ ω ≤ Real.pi*(4:ℝ)^p*δ^(-p) at hb
+    change ENNReal.ofReal (C*δ^(-codimension κ)) <
+      (∫⁻ z in A, physicalStoppedDensity κ (fun t => B t ω) (γ ω) (g ω) z rf) at hs
+    rw [← hl] at hs
+    by_contra hn
+    have hd := le_of_not_gt hn
+    have hscale := mul_le_mul_of_nonneg_right hCbase (Real.rpow_nonneg hδ.le (-p))
+    have he : X rf ω ≤ C*δ^(-codimension κ) := by change X rf ω ≤ C*δ^(-p); nlinarith
+    exact (not_lt_of_ge (ENNReal.ofReal_le_ofReal he)) hs
+  apply (measure_mono_ae hsub).trans
+  apply htail.trans
+  rw [neg_div,tail_scale p hδ hK']
+  apply ENNReal.ofReal_le_ofReal
+  have he : -(1/(2*K'^2)*δ^(-2:ℝ)) = -c*δ^(-2:ℝ) := by dsimp [c]; ring
+  rw [he]
+  exact le_mul_of_one_le_left (Real.exp_pos _).le hC1
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/PastObstacle.lean b/lean/OAI/Probability/SLELocalization/PastObstacle.lean
index 2b6b93789dfa8a18d96a1987c696ab43bed3f5e7..30b261c196f0748faba40da2fb53a33bceee844d 100644
--- a/lean/OAI/Probability/SLELocalization/PastObstacle.lean
+++ b/lean/OAI/Probability/SLELocalization/PastObstacle.lean
@@ -1,135 +1,135 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-
-namespace OAI
-
-
-/-! Conformal localization and Past Obstacle estimates. -/
-
-noncomputable section
-open Set Filter Topology
-open scoped Topology
-namespace SLEExactGauge
-
-lemma locallyConnected_sum {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-    [LocallyConnectedSpace X] [LocallyConnectedSpace Y] : LocallyConnectedSpace (X ⊕ Y) := by
-  rw [locallyConnectedSpace_iff_subsets_isOpen_isConnected]
-  rintro (x | y) U hU
-  · have hp : Sum.inl ⁻¹' U ∈ 𝓝 x := continuous_inl.continuousAt.preimage_mem_nhds hU
-    obtain ⟨V,hVU,hVo,hxV,hVc⟩ := locallyConnectedSpace_iff_subsets_isOpen_isConnected.1
-      (inferInstance : LocallyConnectedSpace X) x _ hp
-    refine ⟨Sum.inl '' V,?_,isOpenMap_inl _ hVo,mem_image_of_mem _ hxV,
-      hVc.image _ continuous_inl.continuousOn⟩
-    rintro _ ⟨v,hv,rfl⟩
-    exact hVU hv
-  · have hp : Sum.inr ⁻¹' U ∈ 𝓝 y := continuous_inr.continuousAt.preimage_mem_nhds hU
-    obtain ⟨V,hVU,hVo,hyV,hVc⟩ := locallyConnectedSpace_iff_subsets_isOpen_isConnected.1
-      (inferInstance : LocallyConnectedSpace Y) y _ hp
-    refine ⟨Sum.inr '' V,?_,isOpenMap_inr _ hVo,mem_image_of_mem _ hyV,
-      hVc.image _ continuous_inr.continuousOn⟩
-    rintro _ ⟨v,hv,rfl⟩
-    exact hVU hv
-
-lemma locallyConnected_closed_union {X : Type*} [TopologicalSpace X]
-    {A B : Set X} (hA : IsClosed A) (hB : IsClosed B)
-    [LocallyConnectedSpace A] [LocallyConnectedSpace B] :
-    LocallyConnectedSpace (↑(A ∪ B) : Type _) := by
-  have := locallyConnected_sum (X := A) (Y := B)
-  let a : A → ↑(A ∪ B) := fun x => ⟨x,Or.inl x.property⟩
-  let b : B → ↑(A ∪ B) := fun x => ⟨x,Or.inr x.property⟩
-  have hca : Continuous a := continuous_subtype_val.subtype_mk _
-  have hcb : Continuous b := continuous_subtype_val.subtype_mk _
-  have hcla : IsClosedMap a := by
-    intro S hS
-    have he : a '' S = Subtype.val ⁻¹' (Subtype.val '' S) := by
-      ext x
-      constructor
-      · rintro ⟨y,hy,rfl⟩; exact ⟨y,hy,rfl⟩
-      · rintro ⟨y,hy,he⟩; exact ⟨y,hy,Subtype.ext he⟩
-    rw [he]
-    exact (hA.isClosedEmbedding_subtypeVal.isClosedMap S hS).preimage continuous_subtype_val
-  have hclb : IsClosedMap b := by
-    intro S hS
-    have he : b '' S = Subtype.val ⁻¹' (Subtype.val '' S) := by
-      ext x
-      constructor
-      · rintro ⟨y,hy,rfl⟩; exact ⟨y,hy,rfl⟩
-      · rintro ⟨y,hy,he⟩; exact ⟨y,hy,Subtype.ext he⟩
-    rw [he]
-    exact (hB.isClosedEmbedding_subtypeVal.isClosedMap S hS).preimage continuous_subtype_val
-  have hsurj : Function.Surjective (Sum.elim a b) := by
-    intro x
-    rcases x.property with ha | hb
-    · exact ⟨Sum.inl ⟨x,ha⟩,rfl⟩
-    · exact ⟨Sum.inr ⟨x,hb⟩,rfl⟩
-  exact ((hcla.sumElim hclb).isQuotientMap (hca.sumElim hcb) hsurj).isCoinducing.locallyConnectedSpace
-
-lemma lower_halfPlane_locallyConnected : LocallyConnectedSpace (↑(halfPlaneᶜ) : Type) := by
-  let π : ℂ → ↑(halfPlaneᶜ) := fun z => ⟨⟨z.re,min z.im 0⟩,by
-    change ¬ 0 < min z.im 0
-    exact not_lt_of_ge (min_le_right _ _)⟩
-  have hc : Continuous π := by
-    apply Continuous.subtype_mk
-    change Continuous (fun z : ℂ => Complex.mk z.re (min z.im 0))
-    have hh : Continuous (fun z : ℂ => (z.re:ℂ)+((min z.im 0:ℝ):ℂ)*Complex.I) := by fun_prop
-    convert hh using 1
-    funext z
-    apply Complex.ext <;> simp
-  have hr (z : ↑(halfPlaneᶜ)) : π z = z := by
-    have hz : z.val.im ≤ 0 := le_of_not_gt z.property
-    apply Subtype.ext
-    apply Complex.ext <;> simp [π,min_eq_left hz]
-  have hq : IsQuotientMap π := by
-    refine ⟨.of_isOpen_preimage_iff_isOpen (fun S => ?_),?_⟩
-    · constructor
-      · intro hS
-        have hh := hS.preimage (continuous_subtype_val : Continuous (Subtype.val : ↑(halfPlaneᶜ) → ℂ))
-        have he : Subtype.val ⁻¹' (π ⁻¹' S) = S := by ext z; simp only [mem_preimage,hr]
-        rwa [he] at hh
-      · intro hS
-        exact hS.preimage hc
-    · intro z
-      exact ⟨z,hr z⟩
-  exact hq.isCoinducing.locallyConnectedSpace
-
-end SLEExactGauge
-
-end
-
-/-! Conformal localization and Past Trace Continuum estimates. -/
-
-noncomputable section
-open Set Filter Topology
-open scoped Topology
-namespace SLEExactGauge
-
-theorem past_trace_continuum {γ : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
-    (hc : ContinuousOn γ (Icc 0 T)) :
-    IsCompact (localizationSegment γ 0 T) ∧ IsConnected (localizationSegment γ 0 T) ∧
-      LocallyConnectedSpace (localizationSegment γ 0 T) := by
-  have : LocallyConnectedSpace (Icc (0:ℝ) T) :=
-    (isQuotientMap_projIcc (a := (0:ℝ)) (b := T) (h := hT)).isCoinducing.locallyConnectedSpace
-  let F : Icc (0:ℝ) T → localizationSegment γ 0 T :=
-    fun t => ⟨γ t, ⟨t,t.property,rfl⟩⟩
-  have hFc : Continuous F :=
-    (continuousOn_iff_continuous_domRestrict.1 hc).subtype_mk _
-  have hFs : Function.Surjective F := by
-    rintro ⟨z,t,ht,rfl⟩
-    exact ⟨⟨t,ht⟩,rfl⟩
-  refine ⟨isCompact_Icc.image_of_continuousOn hc,
-    (isConnected_Icc hT).image γ hc,?_⟩
-  exact (IsQuotientMap.of_surjective_continuous hFs hFc).isCoinducing.locallyConnectedSpace
-
-theorem past_obstacle_locallyConnected {γ : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
-    (hc : ContinuousOn γ (Icc 0 T)) :
-    LocallyConnectedSpace (↑(halfPlaneᶜ ∪ localizationSegment γ 0 T) : Type) := by
-  have := lower_halfPlane_locallyConnected
-  have := (past_trace_continuum hT hc).2.2
-  exact locallyConnected_closed_union halfPlane_open.isClosed_compl
-    (past_trace_continuum hT hc).1.isClosed
-
-end SLEExactGauge
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+
+namespace OAI
+
+
+/-! Conformal localization and Past Obstacle estimates. -/
+
+noncomputable section
+open Set Filter Topology
+open scoped Topology
+namespace SLEExactGauge
+
+lemma locallyConnected_sum {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    [LocallyConnectedSpace X] [LocallyConnectedSpace Y] : LocallyConnectedSpace (X ⊕ Y) := by
+  rw [locallyConnectedSpace_iff_subsets_isOpen_isConnected]
+  rintro (x | y) U hU
+  · have hp : Sum.inl ⁻¹' U ∈ 𝓝 x := continuous_inl.continuousAt.preimage_mem_nhds hU
+    obtain ⟨V,hVU,hVo,hxV,hVc⟩ := locallyConnectedSpace_iff_subsets_isOpen_isConnected.1
+      (inferInstance : LocallyConnectedSpace X) x _ hp
+    refine ⟨Sum.inl '' V,?_,isOpenMap_inl _ hVo,mem_image_of_mem _ hxV,
+      hVc.image _ continuous_inl.continuousOn⟩
+    rintro _ ⟨v,hv,rfl⟩
+    exact hVU hv
+  · have hp : Sum.inr ⁻¹' U ∈ 𝓝 y := continuous_inr.continuousAt.preimage_mem_nhds hU
+    obtain ⟨V,hVU,hVo,hyV,hVc⟩ := locallyConnectedSpace_iff_subsets_isOpen_isConnected.1
+      (inferInstance : LocallyConnectedSpace Y) y _ hp
+    refine ⟨Sum.inr '' V,?_,isOpenMap_inr _ hVo,mem_image_of_mem _ hyV,
+      hVc.image _ continuous_inr.continuousOn⟩
+    rintro _ ⟨v,hv,rfl⟩
+    exact hVU hv
+
+lemma locallyConnected_closed_union {X : Type*} [TopologicalSpace X]
+    {A B : Set X} (hA : IsClosed A) (hB : IsClosed B)
+    [LocallyConnectedSpace A] [LocallyConnectedSpace B] :
+    LocallyConnectedSpace (↑(A ∪ B) : Type _) := by
+  have := locallyConnected_sum (X := A) (Y := B)
+  let a : A → ↑(A ∪ B) := fun x => ⟨x,Or.inl x.property⟩
+  let b : B → ↑(A ∪ B) := fun x => ⟨x,Or.inr x.property⟩
+  have hca : Continuous a := continuous_subtype_val.subtype_mk _
+  have hcb : Continuous b := continuous_subtype_val.subtype_mk _
+  have hcla : IsClosedMap a := by
+    intro S hS
+    have he : a '' S = Subtype.val ⁻¹' (Subtype.val '' S) := by
+      ext x
+      constructor
+      · rintro ⟨y,hy,rfl⟩; exact ⟨y,hy,rfl⟩
+      · rintro ⟨y,hy,he⟩; exact ⟨y,hy,Subtype.ext he⟩
+    rw [he]
+    exact (hA.isClosedEmbedding_subtypeVal.isClosedMap S hS).preimage continuous_subtype_val
+  have hclb : IsClosedMap b := by
+    intro S hS
+    have he : b '' S = Subtype.val ⁻¹' (Subtype.val '' S) := by
+      ext x
+      constructor
+      · rintro ⟨y,hy,rfl⟩; exact ⟨y,hy,rfl⟩
+      · rintro ⟨y,hy,he⟩; exact ⟨y,hy,Subtype.ext he⟩
+    rw [he]
+    exact (hB.isClosedEmbedding_subtypeVal.isClosedMap S hS).preimage continuous_subtype_val
+  have hsurj : Function.Surjective (Sum.elim a b) := by
+    intro x
+    rcases x.property with ha | hb
+    · exact ⟨Sum.inl ⟨x,ha⟩,rfl⟩
+    · exact ⟨Sum.inr ⟨x,hb⟩,rfl⟩
+  exact ((hcla.sumElim hclb).isQuotientMap (hca.sumElim hcb) hsurj).isCoinducing.locallyConnectedSpace
+
+lemma lower_halfPlane_locallyConnected : LocallyConnectedSpace (↑(halfPlaneᶜ) : Type) := by
+  let π : ℂ → ↑(halfPlaneᶜ) := fun z => ⟨⟨z.re,min z.im 0⟩,by
+    change ¬ 0 < min z.im 0
+    exact not_lt_of_ge (min_le_right _ _)⟩
+  have hc : Continuous π := by
+    apply Continuous.subtype_mk
+    change Continuous (fun z : ℂ => Complex.mk z.re (min z.im 0))
+    have hh : Continuous (fun z : ℂ => (z.re:ℂ)+((min z.im 0:ℝ):ℂ)*Complex.I) := by fun_prop
+    convert hh using 1
+    funext z
+    apply Complex.ext <;> simp
+  have hr (z : ↑(halfPlaneᶜ)) : π z = z := by
+    have hz : z.val.im ≤ 0 := le_of_not_gt z.property
+    apply Subtype.ext
+    apply Complex.ext <;> simp [π,min_eq_left hz]
+  have hq : IsQuotientMap π := by
+    refine ⟨.of_isOpen_preimage_iff_isOpen (fun S => ?_),?_⟩
+    · constructor
+      · intro hS
+        have hh := hS.preimage (continuous_subtype_val : Continuous (Subtype.val : ↑(halfPlaneᶜ) → ℂ))
+        have he : Subtype.val ⁻¹' (π ⁻¹' S) = S := by ext z; simp only [mem_preimage,hr]
+        rwa [he] at hh
+      · intro hS
+        exact hS.preimage hc
+    · intro z
+      exact ⟨z,hr z⟩
+  exact hq.isCoinducing.locallyConnectedSpace
+
+end SLEExactGauge
+
+end
+
+/-! Conformal localization and Past Trace Continuum estimates. -/
+
+noncomputable section
+open Set Filter Topology
+open scoped Topology
+namespace SLEExactGauge
+
+theorem past_trace_continuum {γ : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
+    (hc : ContinuousOn γ (Icc 0 T)) :
+    IsCompact (localizationSegment γ 0 T) ∧ IsConnected (localizationSegment γ 0 T) ∧
+      LocallyConnectedSpace (localizationSegment γ 0 T) := by
+  have : LocallyConnectedSpace (Icc (0:ℝ) T) :=
+    (isQuotientMap_projIcc (a := (0:ℝ)) (b := T) (h := hT)).isCoinducing.locallyConnectedSpace
+  let F : Icc (0:ℝ) T → localizationSegment γ 0 T :=
+    fun t => ⟨γ t, ⟨t,t.property,rfl⟩⟩
+  have hFc : Continuous F :=
+    (continuousOn_iff_continuous_domRestrict.1 hc).subtype_mk _
+  have hFs : Function.Surjective F := by
+    rintro ⟨z,t,ht,rfl⟩
+    exact ⟨⟨t,ht⟩,rfl⟩
+  refine ⟨isCompact_Icc.image_of_continuousOn hc,
+    (isConnected_Icc hT).image γ hc,?_⟩
+  exact (IsQuotientMap.of_surjective_continuous hFs hFc).isCoinducing.locallyConnectedSpace
+
+theorem past_obstacle_locallyConnected {γ : ℝ → ℂ} {T : ℝ} (hT : 0 ≤ T)
+    (hc : ContinuousOn γ (Icc 0 T)) :
+    LocallyConnectedSpace (↑(halfPlaneᶜ ∪ localizationSegment γ 0 T) : Type) := by
+  have := lower_halfPlane_locallyConnected
+  have := (past_trace_continuum hT hc).2.2
+  exact locallyConnected_closed_union halfPlane_open.isClosed_compl
+    (past_trace_continuum hT hc).1.isClosed
+
+end SLEExactGauge
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/PhysicalPast.lean b/lean/OAI/Probability/SLELocalization/PhysicalPast.lean
index 9e059797176db4447f6fbefa1740f19d2c3d4997..143209fafcc32beaed606799d911df6c9c313361 100644
--- a/lean/OAI/Probability/SLELocalization/PhysicalPast.lean
+++ b/lean/OAI/Probability/SLELocalization/PhysicalPast.lean
@@ -1,1070 +1,1070 @@
-import Mathlib
-import OAI.Probability.SLELocalization.ConditionalAngles
-import OAI.Probability.SLE.ClippedCoordinates
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Physical Past estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.PhysicalPast
-
-def sampleTime (T : ℝ≥0) (q : I) : ℝ≥0 := T * ⟨q,q.property.1⟩
-
-lemma sampleTime_le (T : ℝ≥0) (q : I) : sampleTime T q ≤ T := by
-  change (T:ℝ)*(q:ℝ) ≤ T
-  simpa using mul_le_mul_of_nonneg_left q.property.2 T.coe_nonneg
-
-lemma uniform_margin_iff {F : I → ℝ} (hF : Continuous F) (ε : ℝ) :
-    (∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤ F (denseSeq I n)) ↔
-      ∀ q : I, ε < F q := by
-  constructor
-  · rintro ⟨m,hm⟩ q
-    have hh : ε + CanonicalTrace.height m ≤ F q :=
-      (denseRange_denseSeq I).induction_on q (isClosed_le continuous_const hF) hm
-    linarith [CanonicalTrace.height_pos m]
-  · intro hh
-    obtain ⟨q,hq,hmin⟩ := isCompact_univ.exists_isMinOn (univ_nonempty : (univ : Set I).Nonempty) hF.continuousOn
-    obtain ⟨m,hm⟩ := exists_nat_one_div_lt (sub_pos.2 (hh q))
-    refine ⟨m,fun n => ?_⟩
-    have hb := hmin (mem_univ (denseSeq I n))
-    change F q ≤ F (denseSeq I n) at hb
-    dsimp [CanonicalTrace.height]
-    linarith
-
-def avoids {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (z : ℂ) (ε : ℝ)
-    (T : ℝ≥0) : Set Ω :=
-  {sample | ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤
-    dist z (CanonicalTrace.value κ B (sampleTime T (denseSeq I n)) sample)}
-
-lemma avoids_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (z : ℂ) (ε : ℝ) (T : ℝ≥0) :
-    @MeasurableSet Ω (BrownianPredictable.pastSigma B T) (avoids κ B z ε T) := by
-  let : MeasurableSpace Ω := BrownianPredictable.pastSigma B T
-  simp only [avoids,ofPred_exists,ofPred_forall]
-  apply MeasurableSet.iUnion
-  intro m
-  apply MeasurableSet.iInter
-  intro n
-  apply measurableSet_le measurable_const
-  exact measurable_const.dist ((CanonicalTrace.measurable_past κ B _).mono
-    (BrownianPredictable.pastSigma_mono B (sampleTime_le T (denseSeq I n))) le_rfl)
-
-lemma avoids_nullMeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (z : ℂ) (ε : ℝ) (T : ℝ≥0) : NullMeasurableSet (avoids κ B z ε T) P := by
-  simp only [avoids,ofPred_exists,ofPred_forall]
-  apply NullMeasurableSet.iUnion
-  intro m
-  apply NullMeasurableSet.iInter
-  intro n
-  exact nullMeasurableSet_le aemeasurable_const
-    (aemeasurable_const.dist (CanonicalTrace.aemeasurable κ hB _))
-
-lemma avoids_eq_source {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal sample) γ g f)
-    (hc : Continuous (fun t => B t sample)) (z : ℂ) (ε : ℝ) (T : ℝ≥0) :
-    sample ∈ avoids κ B z ε T ↔ ∀ s : ℝ≥0, s ≤ T → ε < dist z (γ s) := by
-  have he : (sample ∈ avoids κ B z ε T) ↔
-      ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤ dist z (γ (sampleTime T (denseSeq I n))) := by
-    simp only [avoids,mem_ofPred_eq,CanonicalTrace.eq_source κ B sample h hc]
-  rw [he]
-  have hcont : Continuous (fun q : I => dist z (γ (sampleTime T q))) :=
-    continuous_const.dist (h.1.comp_continuous (continuous_const.mul continuous_subtype_val)
-      (fun q => mul_nonneg T.coe_nonneg q.property.1))
-  rw [uniform_margin_iff hcont ε]
-  constructor
-  · intro hh s hs
-    by_cases hT : T = 0
-    · have hs0 : s=0 := by simpa only [hT,le_zero_iff] using hs
-      have hzero : sampleTime T (0:I) = 0 := by
-        apply NNReal.coe_injective
-        change (T:ℝ) * (0:ℝ) = 0
-        exact mul_zero _
-      simpa only [hzero,hs0,NNReal.coe_zero] using hh (0:I)
-    · have hTp : 0 < (T:ℝ) := lt_of_le_of_ne T.coe_nonneg (by exact_mod_cast Ne.symm hT)
-      let q : I := ⟨(s:ℝ)/(T:ℝ),⟨div_nonneg s.coe_nonneg T.coe_nonneg,
-        (div_le_one hTp).2 hs⟩⟩
-      have hsq : sampleTime T q = s := by
-        apply NNReal.coe_injective
-        change (T:ℝ)*((s:ℝ)/(T:ℝ)) = s
-        field_simp
-      simpa only [hsq] using hh q
-  · intro hh q
-    exact hh _ (sampleTime_le T q)
-
-end SLEExactGauge.PhysicalPast
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-lemma closedBall_subset_survivingDomain {γ : ℝ → ℂ} {t : ℝ} {z : ℂ} {r : ℝ}
-    (hr : 0 ≤ r) (hz : z ∈ localizationDomain γ t)
-    (hfree : closedBall z r ⊆ halfPlane \ localizationSegment γ 0 t) :
-    closedBall z r ⊆ localizationDomain γ t := by
-  have hc := (convex_closedBall z r).isPreconnected.subset_connectedComponentIn
-    (mem_closedBall_self hr) hfree
-  intro w hw
-  refine ⟨hfree hw,?_⟩
-  rw [← connectedComponentIn_eq (hc hw)]
-  exact hz.2
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge
-namespace IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma radius_ge_half_free_ball (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} {r : ℝ} (hr : 0 < r)
-    (hz : z ∈ localizationDomain γ t) (hball : closedBall z r ⊆ localizationDomain γ t) :
-    r/2 ≤ loewnerRadius g t z := by
-  let y := (g t z).im
-  have hy : 0 < y := h.im_pos ht hz
-  let F : ℂ → ℂ := fun w => (g t w - (starRingEnd ℂ) (g t z))⁻¹
-  have hden (w : ℂ) (hw : w ∈ localizationDomain γ t) :
-      y ≤ ‖g t w - (starRingEnd ℂ) (g t z)‖ := by
-    have hp := h.im_pos ht hw
-    have hh := Complex.im_le_norm (g t w - (starRingEnd ℂ) (g t z))
-    simp only [Complex.sub_im,Complex.conj_im] at hh
-    dsimp [y]
-    linarith
-  have hn (w : ℂ) (hw : w ∈ localizationDomain γ t) : g t w-(starRingEnd ℂ) (g t z) ≠ 0 :=
-    norm_pos_iff.1 (hy.trans_le (hden w hw))
-  have hd (w : ℂ) (hw : w ∈ localizationDomain γ t) : DifferentiableAt ℂ F w :=
-    ((h.analytic_eval ht hw).differentiableAt.sub_const _).inv (hn w hw)
-  have hF : DiffContOnCl ℂ F (ball z r) := by
-    apply DifferentiableOn.diffContOnCl
-    rw [closure_ball _ hr.ne']
-    exact fun w hw => (hd w (hball hw)).differentiableWithinAt
-  have hb : ∀ w ∈ sphere z r, ‖F w‖ ≤ 1/y := by
-    intro w hw
-    change ‖(g t w - (starRingEnd ℂ) (g t z))⁻¹‖ ≤ 1/y
-    rw [norm_inv,one_div]
-    exact inv_anti₀ hy (hden w (hball (sphere_subset_closedBall hw)))
-  have hc := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le hr hF hb
-  have hdF : HasDerivAt F (-deriv (g t) z / (g t z-(starRingEnd ℂ) (g t z))^2) z := by
-    exact ((h.analytic_eval ht hz).differentiableAt.hasDerivAt.sub_const _).inv (hn z hz)
-  have hnorm : ‖g t z-(starRingEnd ℂ) (g t z)‖ = 2*y := by
-    rw [Complex.sub_conj,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one]
-    exact abs_of_pos (by dsimp [y] at hy ⊢; positivity)
-  rw [hdF.deriv,norm_div,norm_neg,norm_pow,hnorm] at hc
-  have hdn := norm_pos_iff.2 (h.deriv_ne_zero ht hz)
-  change r/2 ≤ 2*y/‖deriv (g t) z‖
-  rw [le_div_iff₀ hdn]
-  have hh := (div_le_iff₀ (sq_pos_of_pos (by positivity : 0 < 2*y))).1 hc
-  have hh' : ‖deriv (g t) z‖ * r ≤ 4*y := by
-    have hrr := (mul_le_mul_of_nonneg_right hh hr.le)
-    field_simp at hrr
-    nlinarith
-  nlinarith
-
-end IsLoewnerTraceUsing
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-def firstLiveDistanceStop (γ : ℝ → ℂ) (z : ℂ) (ε : ℝ) : ℝ≥0∞ :=
-  sInf ((fun t : ℝ≥0 => (t : ℝ≥0∞)) ''
-    {t : ℝ≥0 | z ∈ localizationDomain γ t ∧ dist z (γ t) ≤ ε})
-
-def physicalStoppedDensity (κ : ℝ) (B : ℝ≥0 → ℝ)
-    (γ : ℝ → ℂ) (g : ℝ → ℂ → ℂ) (z : ℂ) (ε : ℝ) : ℝ≥0∞ :=
-  let τ := firstLiveDistanceStop γ z ε
-  if τ = ⊤ then 0 else
-    ENNReal.ofReal ((loewnerRadius g τ.toReal z)^(-codimension κ) *
-      (loewnerAngleSine κ B g τ.toReal z)^(8/κ-1))
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-namespace PhysicalStop
-variable {γ : ℝ → ℂ} {z : ℂ} {ε : ℝ}
-
-lemma le_of_hit {t : ℝ≥0} (hz : z ∈ localizationDomain γ t) (hd : dist z (γ t) ≤ ε) :
-    firstLiveDistanceStop γ z ε ≤ (t:ℝ≥0∞) :=
-  sInf_le ⟨t,⟨hz,hd⟩,rfl⟩
-
-lemma eq_top_iff : firstLiveDistanceStop γ z ε = ⊤ ↔
-    ∀ t : ℝ≥0, z ∈ localizationDomain γ t → ε < dist z (γ t) := by
-  constructor
-  · intro he t hz
-    by_contra hn
-    have hh := le_of_hit hz (not_lt.1 hn)
-    rw [he] at hh
-    exact ENNReal.coe_ne_top (top_le_iff.1 hh)
-  · intro hn
-    have he : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧ dist z (γ t) ≤ ε} = ∅ := by
-      exact Set.eq_empty_iff_forall_notMem.2 (fun t ht => (not_lt_of_ge ht.2) (hn t ht.1))
-    simp only [firstLiveDistanceStop,he,image_empty,sInf_empty]
-
-lemma finite_spec (hγ : ContinuousOn γ (Ici 0))
-    (hfin : firstLiveDistanceStop γ z ε ≠ ⊤) :
-    ∃ t : ℝ≥0, firstLiveDistanceStop γ z ε = (t:ℝ≥0∞) ∧
-      z ∈ localizationDomain γ t ∧ dist z (γ t) ≤ ε ∧
-      ∀ s : ℝ≥0, s < t → ε < dist z (γ s) := by
-  have hex : ∃ T : ℝ≥0, z ∈ localizationDomain γ T ∧ dist z (γ T) ≤ ε := by
-    by_contra hn
-    push Not at hn
-    exact hfin (eq_top_iff.2 hn)
-  obtain ⟨T,hTlive,hTdist⟩ := hex
-  have hc : Continuous (fun s : ℝ≥0 => dist z (γ s)) :=
-    continuous_const.dist (hγ.comp_continuous continuous_subtype_val (fun s => s.coe_nonneg))
-  let A : Set ℝ≥0 := Icc 0 T ∩ {s | dist z (γ s) ≤ ε}
-  have hA : IsCompact A := isCompact_Icc.inter_right (isClosed_le hc continuous_const)
-  obtain ⟨t,ht,hmin⟩ := hA.exists_isLeast ⟨T,⟨⟨zero_le,le_rfl⟩,hTdist⟩⟩
-  have htlive : z ∈ localizationDomain γ t := survivingDomain_mono ht.1.2 hTlive
-  have hleast (s : ℝ≥0) (hs : z ∈ localizationDomain γ s ∧ dist z (γ s) ≤ ε) : t ≤ s := by
-    by_cases hsT : s ≤ T
-    · exact hmin ⟨⟨zero_le,hsT⟩,hs.2⟩
-    · exact ht.1.2.trans (le_of_not_ge hsT)
-  have he : firstLiveDistanceStop γ z ε = (t:ℝ≥0∞) := by
-    apply le_antisymm (le_of_hit htlive ht.2)
-    apply le_sInf
-    rintro _ ⟨s,hs,rfl⟩
-    exact ENNReal.coe_le_coe.2 (hleast s hs)
-  refine ⟨t,he,htlive,ht.2,?_⟩
-  intro s hs
-  by_contra hn
-  exact (not_le_of_gt hs) (hleast s ⟨survivingDomain_mono hs.le htlive,not_lt.1 hn⟩)
-
-lemma lt_stop_distance {t : ℝ≥0} (ht : (t:ℝ≥0∞) < firstLiveDistanceStop γ z ε)
-    (hlive : z ∈ localizationDomain γ t) : ε < dist z (γ t) := by
-  by_contra hn
-  exact (not_le_of_gt ht) (le_of_hit hlive (not_lt.1 hn))
-
-lemma stopped_survival_iff (hγ : ContinuousOn γ (Ici 0)) {t : ℝ≥0}
-    (hlive : z ∈ localizationDomain γ t) :
-    (t:ℝ≥0∞) < firstLiveDistanceStop γ z ε ↔
-      ∀ s : ℝ≥0, s ≤ t → ε < dist z (γ s) := by
-  constructor
-  · intro ht s hs
-    exact lt_stop_distance ((ENNReal.coe_le_coe.2 hs).trans_lt ht) (survivingDomain_mono hs hlive)
-  · intro hh
-    by_contra hn
-    have he : firstLiveDistanceStop γ z ε ≠ ⊤ :=
-      ne_top_of_le_ne_top ENNReal.coe_ne_top (not_lt.1 hn)
-    obtain ⟨s,hs,hslive,hsdist,hmin⟩ := finite_spec hγ he
-    rw [hs] at hn
-    have hst : s ≤ t := ENNReal.coe_le_coe.1 (not_lt.1 hn)
-    exact (not_lt_of_ge hsdist) (hh s hst)
-
-lemma finite_dist_eq (hγ : ContinuousOn γ (Ici 0))
-    (hinit : ε < dist z (γ 0)) (hfin : firstLiveDistanceStop γ z ε ≠ ⊤) :
-    dist z (γ (firstLiveDistanceStop γ z ε).toReal) = ε := by
-  obtain ⟨t,he,hlive,hdist,hmin⟩ := finite_spec hγ hfin
-  rw [he,ENNReal.coe_toReal]
-  apply le_antisymm hdist
-  by_contra hn
-  have hlt : dist z (γ t) < ε := not_le.1 hn
-  have hc : ContinuousOn (fun s : ℝ => dist z (γ s)) (Icc 0 (t:ℝ)) :=
-    (continuous_const.dist continuous_id).comp_continuousOn (hγ.mono Icc_subset_Ici_self)
-  obtain ⟨s,hs,hseq⟩ := intermediate_value_Icc' t.coe_nonneg hc ⟨hlt.le,hinit.le⟩
-  have hst : s < (t:ℝ) := lt_of_le_of_ne hs.2 (by intro hst; subst s; linarith)
-  have hh := hmin ⟨s,hs.1⟩ hst
-  exact (not_lt_of_ge hseq.le) hh
-
-lemma trace_distance_ge_before (hγ : ContinuousOn γ (Ici 0))
-    (hinit : ε < dist z (γ 0)) {t : ℝ≥0}
-    (ht : (t:ℝ≥0∞) ≤ firstLiveDistanceStop γ z ε)
-    (hlive : z ∈ localizationDomain γ t) {s : ℝ} (hs : s ∈ Icc 0 (t:ℝ)) :
-    ε ≤ dist z (γ s) := by
-  rcases lt_or_eq_of_le ((ENNReal.coe_le_coe.2 (show (⟨s,hs.1⟩:ℝ≥0) ≤ t from hs.2)).trans ht) with hlt | he
-  · exact (lt_stop_distance hlt (survivingDomain_mono hs.2 hlive)).le
-  · have hfin : firstLiveDistanceStop γ z ε ≠ ⊤ := he ▸ ENNReal.coe_ne_top
-    have hh := finite_dist_eq hγ hinit hfin
-    rw [← he] at hh
-    change dist z (γ s) = ε at hh
-    exact hh.symm.le
-
-end PhysicalStop
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-open Set Metric
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma radius_ge_before_distance_stop (h : IsLoewnerTraceUsing driver γ g f)
-    {z : ℂ} {ε : ℝ} (hε : 0 < ε) (hsmall : ε < z.im) {t : ℝ≥0}
-    (ht : (t:ℝ≥0∞) ≤ firstLiveDistanceStop γ z ε) (hlive : z ∈ localizationDomain γ t) :
-    ε/4 ≤ loewnerRadius g t z := by
-  have hinit : ε < dist z (γ 0) := by
-    rw [h.2.1,dist_zero_right]
-    exact hsmall.trans_le (Complex.im_le_norm z)
-  have hdist (s : ℝ) (hs : s ∈ Icc 0 (t:ℝ)) : ε ≤ dist z (γ s) :=
-    PhysicalStop.trace_distance_ge_before h.1 hinit ht hlive hs
-  have hfree : closedBall z (ε/2) ⊆ halfPlane \ localizationSegment γ 0 t := by
-    intro w hw
-    have hnorm : ‖w-z‖ ≤ ε/2 := by simpa only [mem_closedBall,dist_eq_norm] using hw
-    constructor
-    · have hh := Complex.abs_im_le_norm (w-z)
-      have hn := neg_abs_le (w-z).im
-      change 0 < w.im
-      simp only [Complex.sub_im] at hh hn
-      linarith
-    · rintro ⟨s,hs,rfl⟩
-      have hh := hdist s hs
-      have hh' := mem_closedBall.1 hw
-      rw [dist_comm] at hh'
-      linarith
-  have hb := closedBall_subset_survivingDomain (by linarith : 0 ≤ ε/2) hlive hfree
-  have hh := h.radius_ge_half_free_ball t.coe_nonneg (by linarith : 0 < ε/2) hlive hb
-  convert hh using 1
-  ring
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.PhysicalStop
-
-def cap (tauE : ℝ≥0∞) (a : ℝ) : ℝ := if tauE = ⊤ then a else min tauE.toReal a
-
-lemma cap_bounds (tauE : ℝ≥0∞) {a : ℝ} (ha : 0 ≤ a) : 0 ≤ cap tauE a ∧ cap tauE a ≤ a := by
-  unfold cap
-  split_ifs
-  · exact ⟨ha,le_rfl⟩
-  · exact ⟨le_min ENNReal.toReal_nonneg ha,min_le_right _ _⟩
-
-lemma lt_cap (tauE : ℝ≥0∞) (a : ℝ) (s : ℝ≥0) :
-    (s:ℝ) < cap tauE a ↔ (s:ℝ) < a ∧ (s:ℝ≥0∞) < tauE := by
-  by_cases htauE : tauE = ⊤
-  · simp [cap,htauE]
-  · rw [cap,ite_eq_right htauE,lt_min_iff,and_comm]
-    apply and_congr_right
-    intro _
-    simpa only [ENNReal.coe_toReal] using ENNReal.toReal_lt_toReal (a := (s:ℝ≥0∞)) ENNReal.coe_ne_top htauE
-
-lemma cap_coe {t : ℝ≥0} {a : ℝ} (hta : (t:ℝ) ≤ a) : cap (t:ℝ≥0∞) a = t := by
-  simp only [cap,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,min_eq_left hta]
-
-end SLEExactGauge.PhysicalStop
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BrownianField.FieldRegular
-
-lemma restrict {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ} {s t : ℝ}
-    (h : FieldRegular F V b t) (hst : s ≤ t) : FieldRegular F V b s := by
-  refine ⟨h.smooth,h.third_bound,?_,?_,?_,?_⟩
-  · exact fun x => (h.time_cont x).mono (Icc_subset_Icc le_rfl hst)
-  · exact fun r hr => h.time_deriv r ⟨hr.1,hr.2.trans_le hst⟩
-  · exact h.drift_cont.mono (prod_mono (Icc_subset_Icc le_rfl hst) (Icc_subset_Icc le_rfl hst))
-  · exact h.second_cont.mono (Icc_subset_Icc le_rfl hst)
-
-end SLEExactGauge.BrownianField.FieldRegular
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerLocalized
-open LoewnerCutoff LoewnerField LoewnerExit
-
-lemma integral_drift_cancel_before {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (κ : ℝ) (hκ : 0 < κ)
-    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) (b : ℝ → ℝ)
-    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ * b s)
-    {τ : ℝ} (hτ0 : 0 ≤ τ) (hτ : τ ≤ stop hε hT.le U z ρ δ) :
-    (∫ t in (0:ℝ)..τ, drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) +
-      (1/2 : ℝ)*∫ t in (0:ℝ)..τ,
-        iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) = 0 := by
-  have he : (∫ t in (0:ℝ)..τ, drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) =
-      ∫ t in (0:ℝ)..τ,
-        -(1/2 : ℝ)*iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) := by
-    apply intervalIntegral.integral_congr_Ioo_of_le hτ0
-    intro t ht
-    have htT : t ∈ Icc 0 T := ⟨ht.1.le,ht.2.le.trans (hτ.trans (stop_bounds hε hT.le U z ρ δ).2)⟩
-    have hh := (deficit_pos_iff hε hT.le U z ρ δ t).1
-      (positive_before hε hT.le hT U z ρ δ ht.1.le (ht.2.trans_le hτ))
-    have hc := drift_cancellation hε hT.le κ hκ U z t (b t) hh.2.1.le (hU t htT)
-    linarith
-  rw [he, intervalIntegral.integral_const_mul]
-  ring
-
-end SLEExactGauge.LoewnerLocalized
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-def physicalTime {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) : ℝ :=
-  PhysicalStop.cap (firstLiveDistanceStop (γ sample) z ε) (C.stopTime κ B sample)
-
-def physicalValue {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) : ℝ :=
-  field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
-    (C.physicalTime κ B γ ε sample) (B (C.physicalTime κ B γ ε sample).toNNReal sample)
-
-lemma physicalTime_bounds {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) :
-    0 ≤ C.physicalTime κ B γ ε sample ∧ C.physicalTime κ B γ ε sample ≤ C.stopTime κ B sample :=
-  PhysicalStop.cap_bounds _ (C.bounds κ B sample).1
-
-lemma physicalValue_nonneg {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) :
-    0 ≤ C.physicalValue κ B γ ε sample := (green_pos _ _ _ _ _).le
-
-lemma physicalValue_bound {Ω : Type*} (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) (hz : z ∈ halfPlane) (hu : 0 < u)
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
-    C.physicalValue κ B γ ε sample ≤ ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) := by
-  have hy0 : 0 < z.im := hz
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by
-    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
-    linarith
-  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hd := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0
-      (C.physicalTime_bounds κ B γ ε sample))
-  apply green_le_fixed_radius _ _ _ _ _ _ (LoewnerField.height_pos _ _ _ _ _) ha hp (by positivity)
-  simpa only [LoewnerExit.radius,LoewnerField.height,max_eq_right hd.2.1] using hd.1
-
-def physicalCoeff {Ω : Type*} (k : ℕ) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (ε : ℝ) (s : ℝ≥0) (sample : Ω) : ℝ :=
-  by
-  classical
-  exact if sample ∈ PhysicalPast.avoids κ B z ε s then
-    rawCoeff C.height_pos k B C.horizon s (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle sample
-  else 0
-
-lemma physicalCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (k : ℕ) (hk : k=1 ∨ k=2) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (ε : ℝ) (s : ℝ≥0) :
-    Measurable[BrownianPredictable.pastSigma B s] (C.physicalCoeff k κ B ε s) := by
-  classical
-  exact (rawCoeff_measurable_past C.height_pos k hk B C.horizon s _ _ _ _ _ _).ite
-    (PhysicalPast.avoids_measurable_past κ B z ε s) measurable_const
-
-lemma physicalCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (k : ℕ) (hk : k=1 ∨ k=2) (κ : ℝ) (ε : ℝ) (s : ℝ≥0) :
-    AEMeasurable (C.physicalCoeff k κ B ε s) P := by
-  classical
-  exact (rawCoeff_aemeasurable C.height_pos hB k hk C.horizon s _ _ _ _ _ _).indicator₀
-    (PhysicalPast.avoids_nullMeasurable κ hB z ε s)
-
-lemma physicalCoeff_eq {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
-    (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) (k : ℕ) (s : ℝ≥0) :
-    C.physicalCoeff k κ B ε s sample =
-      if (s:ℝ) < C.physicalTime κ B γ ε sample then
-        iteratedDeriv k (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ)
-          (C.paths κ B sample) z s) (B s sample) else 0 := by
-  classical
-  have he := rawCoeff_eq C.height_pos k B C.horizon s C.horizon_pos (1-κ/8) (8/κ-1)
-    (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle sample hc
-  change rawCoeff _ _ _ _ _ _ _ _ _ _ _ _ =
-    (if (s:ℝ) < C.stopTime κ B sample then _ else 0) at he
-  have ht := PhysicalStop.lt_cap (firstLiveDistanceStop (γ sample) z ε) (C.stopTime κ B sample) s
-  change ((s:ℝ) < C.physicalTime κ B γ ε sample) ↔ _ at ht
-  by_cases hs : (s:ℝ) < C.stopTime κ B sample
-  · have hlive := survivingDomain_mono hs.le (C.source_live κ B sample h hz hu hc hb0)
-    have hav := (PhysicalPast.avoids_eq_source κ B sample h hc z ε s).trans
-      (PhysicalStop.stopped_survival_iff h.1 hlive).symm
-    simp only [physicalCoeff,he,hav,ht,hs,ite_true,true_and,paths]
-  · have hcap : ¬ (s:ℝ) < C.physicalTime κ B γ ε sample := fun hh => hs (ht.1 hh).1
-    simp only [physicalCoeff,he,ite_eq_right hs,ite_eq_right hcap,ite_self]
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
-open BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-theorem physical_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) :
-    let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
-    Integrable (fun sample => C.physicalValue κ B γ ε sample - F sample 0 (B 0 sample)) P ∧
-      (∫ sample, C.physicalValue κ B γ ε sample - F sample 0 (B 0 sample) ∂P) = 0 := by
-  let T := C.horizon
-  let hT := C.horizon_pos
-  let hε := C.height_pos
-  let ρ := (2*z.im)*Real.exp (-u)
-  let δ := C.angle
-  let U := C.paths κ B
-  let τ := C.physicalTime κ B γ ε
-  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by
-    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
-    linarith
-  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact (regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
-      (hc.comp continuous_real_toNNReal)).restrict (C.physicalTime_bounds κ B γ ε sample).2
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
-    Filter.Eventually.of_forall (fun sample => ⟨(C.physicalTime_bounds κ B γ ε sample).1,
-      (C.physicalTime_bounds κ B γ ε sample).2.trans (C.bounds κ B sample).2⟩)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
-        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
-    apply Filter.Eventually.of_forall
-    intro sample
-    unfold stoppedCoeff
-    split_ifs
-    · exact hb _ _ _
-    · simpa only [abs_zero] using hK
-  let A := fun n i => C.physicalCoeff 1 κ B ε (dyadicTime T n i)
-  let Q := fun n i => C.physicalCoeff 2 κ B ε (dyadicTime T n i)
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
-    simpa only [A,stoppedCoeff,coe_dyadicTime] using
-      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 1 (dyadicTime T n i)
-  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
-    simpa only [Q,stoppedCoeff,coe_dyadicTime] using
-      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 2 (dyadicTime T n i)
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
-    C.physicalCoeff_measurable_past 1 (Or.inl rfl) κ B ε _
-  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
-    C.physicalCoeff_measurable_past 2 (Or.inr rfl) κ B ε _
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 1 (Or.inl rfl) κ ε _).aestronglyMeasurable K₁
-    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hb
-  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
-    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 2 (Or.inr rfl) κ ε _).aestronglyMeasurable K₂
-    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |Q n i sample| ≤ K₂
-    rw [he]
-    exact hb
-  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A Q hAe hCe hCm hC2
-    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
-      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
-    filter_upwards [hB.cont] with sample hc
-    have hcanc := integral_drift_cancel_before hε hT κ hκ.1 (U sample) z ρ δ
-      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
-      (C.physicalTime_bounds κ B γ ε sample).1 (C.physicalTime_bounds κ B γ ε sample).2
-    dsimp only [F,V,τ]
-    linarith
-  exact ⟨hi.1.congr he, (integral_congr_ae he).symm.trans hi.2⟩
-
-theorem physical_expectation {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) :
-    Integrable (C.physicalValue κ B γ ε) P ∧
-      (∫ sample, C.physicalValue κ B γ ε sample ∂P) = green (1-κ/8) (8/κ-1) z.re z.im 0 := by
-  let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
-  let M := C.physicalValue κ B γ ε
-  let G := green (1-κ/8) (8/κ-1) z.re z.im 0
-  have hi := C.physical_centered hB κ hκ γ g f hmaps hz hu ε
-  have he : (fun sample => M sample - F sample 0 (B 0 sample)) =ᵐ[P]
-      (fun sample => M sample - G) := by
-    filter_upwards [hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hb
-    rw [hb]
-    congr 1
-    exact field_initial C.height_pos C.horizon.coe_nonneg _ _ _ _ z (C.height_initial hz hu.le).le
-  have hd : Integrable (fun sample => M sample-G) P := hi.1.congr he
-  have hv : (∫ sample, M sample-G ∂P) = 0 := (integral_congr_ae he).symm.trans hi.2
-  have hm : Integrable M P := by
-    exact (hd.add (integrable_const G)).congr
-      (Filter.Eventually.of_forall (fun sample => sub_add_cancel (M sample) G))
-  refine ⟨hm, ?_⟩
-  have hh := integral_sub hm (integrable_const (μ := P) G)
-  simp only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul] at hh
-  rw [hh] at hv
-  change (∫ sample, M sample ∂P) = G
-  linarith
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
-open BrownianPredictable BrownianStopped BrownianIto TaylorMeshes PredictableLimit
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-
-theorem physical_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) :
-    let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
-      (Real.sqrt κ) (C.paths κ B sample) z
-    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B C.horizon
-      (fun n i => C.physicalCoeff 1 κ B ε (dyadicTime C.horizon n i)) n sample) atTop
-      (𝓝 (C.physicalValue κ B γ ε sample-F sample 0 (B 0 sample))) := by
-  let T := C.horizon
-  let hT := C.horizon_pos
-  let hε := C.height_pos
-  let ρ := (2*z.im)*Real.exp (-u)
-  let δ := C.angle
-  let U := C.paths κ B
-  let τ := C.physicalTime κ B γ ε
-  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by
-    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
-    linarith
-  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact (regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
-      (hc.comp continuous_real_toNNReal)).restrict (C.physicalTime_bounds κ B γ ε sample).2
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
-    Filter.Eventually.of_forall (fun sample => ⟨(C.physicalTime_bounds κ B γ ε sample).1,
-      (C.physicalTime_bounds κ B γ ε sample).2.trans (C.bounds κ B sample).2⟩)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
-        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
-    apply Filter.Eventually.of_forall
-    intro sample
-    unfold stoppedCoeff
-    split_ifs
-    · exact hb _ _ _
-    · simpa only [abs_zero] using hK
-  let A := fun n i => C.physicalCoeff 1 κ B ε (dyadicTime T n i)
-  let Q := fun n i => C.physicalCoeff 2 κ B ε (dyadicTime T n i)
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
-    simpa only [A,stoppedCoeff,coe_dyadicTime] using
-      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 1 (dyadicTime T n i)
-  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i := by
-    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
-    simpa only [Q,stoppedCoeff,coe_dyadicTime] using
-      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 2 (dyadicTime T n i)
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
-    C.physicalCoeff_measurable_past 1 (Or.inl rfl) κ B ε _
-  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
-    C.physicalCoeff_measurable_past 2 (Or.inr rfl) κ B ε _
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 1 (Or.inl rfl) κ ε _).aestronglyMeasurable K₁
-    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hb
-  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
-    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 2 (Or.inr rfl) κ ε _).aestronglyMeasurable K₂
-    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |Q n i sample| ≤ K₂
-    rw [he]
-    exact hb
-  have hi := stopped_fieldSum_tendsto_ae hB T τ F V htau hreg Q hCe hCm hC2
-    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
-      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
-    filter_upwards [hB.cont] with sample hc
-    have hcanc := integral_drift_cancel_before hε hT κ hκ.1 (U sample) z ρ δ
-      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
-      (C.physicalTime_bounds κ B γ ε sample).1 (C.physicalTime_bounds κ B γ ε sample).2
-    dsimp only [F,V,τ]
-    linarith
-  have hAeall : ∀ᵐ sample ∂P, ∀ n i, A n i sample = stoppedCoeff 1 B T τ F n i sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAe n))
-  filter_upwards [hi,he,hAeall] with sample hi he ha
-  rw [he] at hi
-  convert hi using 1
-  · funext n
-    unfold dyadicDriverSum
-    simp only [← ha]
-    rfl
-  · rfl
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-open PredictableLimit
-
-lemma delayed_indicator_difference_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T S : ℝ≥0)
-    (A D : ℕ → ℕ → Ω → ℝ)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
-    (hD : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (D n i))
-    (hA2 : ∀ n i, MemLp (A n i) 2 P) (hD2 : ∀ n i, MemLp (D n i) 2 P)
-    (K : ℝ) (hK : 0 ≤ K)
-    (hbA : ∀ n i, ∀ᵐ sample ∂P, |A n i sample| ≤ K)
-    (hbD : ∀ n i, ∀ᵐ sample ∂P, |D n i sample| ≤ K)
-    (E : Set Ω) (hE : NullMeasurableSet E P)
-    (hEm : MeasurableSet[pastSigma B S] E)
-    (hsupport : ∀ n i, dyadicTime T n i < S → ∀ᵐ sample ∂P,
-      sample ∈ E → A n i sample = D n i sample)
-    (YA YD : Ω → ℝ)
-    (hlA : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A n sample) atTop (𝓝 (YA sample)))
-    (hlD : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T D n sample) atTop (𝓝 (YD sample))) :
-    Integrable (E.indicator (fun sample => YA sample-YD sample)) P ∧
-      (∫ sample, E.indicator (fun sample => YA sample-YD sample) sample ∂P) = 0 := by
-  classical
-  let Q := fun n i => if S ≤ dyadicTime T n i then
-    E.indicator (fun sample => A n i sample-D n i sample) else (fun _ => 0)
-  have hQm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) := by
-    dsimp only [Q]
-    split_ifs with ht
-    · exact ((hA n i).sub (hD n i)).indicator (pastSigma_mono B ht _ hEm)
-    · exact measurable_const
-  have heq (n i : ℕ) : Q n i =ᵐ[P] E.indicator (fun sample => A n i sample-D n i sample) := by
-    by_cases ht : S ≤ dyadicTime T n i
-    · exact Filter.Eventually.of_forall (fun _ => by simp only [Q,ite_eq_left ht])
-    · filter_upwards [hsupport n i (lt_of_not_ge ht)] with sample hs
-      simp only [Q,ite_eq_right ht,Set.indicator_apply]
-      split_ifs with hmem
-      · simp only [hs hmem,sub_self]
-      · rfl
-  have hQ2 (n i : ℕ) : MemLp (Q n i) 2 P := by
-    have hd := (hA2 n i).sub (hD2 n i)
-    have hm : MemLp (E.indicator (fun sample => A n i sample-D n i sample)) 2 P :=
-      hd.of_le (hd.aestronglyMeasurable.indicator₀ hE)
-        (Filter.Eventually.of_forall (fun sample => norm_indicator_le_norm_self _ sample))
-    exact hm.ae_eq (heq n i).symm
-  have hQb (n i : ℕ) : ∀ᵐ sample ∂P, |Q n i sample| ≤ 2*K := by
-    filter_upwards [hbA n i,hbD n i,heq n i] with sample ha hd he
-    rw [he,Set.indicator_apply]
-    split_ifs
-    · exact (abs_sub _ _).trans (by linarith)
-    · simpa only [abs_zero] using (show 0 ≤ 2*K by positivity)
-  have heall : ∀ᵐ sample ∂P, ∀ n i, Q n i sample = E.indicator (fun sample => A n i sample-D n i sample) sample :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (heq n))
-  apply limit_centered hB T Q hQm hQ2 (2*K) (by positivity) hQb _
-  filter_upwards [heall,hlA,hlD] with sample he ha hd
-  have heqsum (n : ℕ) : dyadicDriverSum B T Q n sample =
-      E.indicator (fun sample => dyadicDriverSum B T A n sample-dyadicDriverSum B T D n sample) sample := by
-    unfold dyadicDriverSum
-    simp only [he,Set.indicator_apply]
-    split_ifs <;> simp [sub_mul,Finset.sum_sub_distrib]
-  simp only [heqsum,Set.indicator_apply]
-  split_ifs
-  · exact ha.sub hd
-  · exact tendsto_const_nhds
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized BrownianPredictable
-  BrownianStopped BrownianIto TaylorMeshes PredictableLimit
-
-lemma physical_avoidance_setIntegral {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-    (C : Cutoff z u) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (e : ℝ) (S : ℝ≥0) :
-    (∫ sample in PhysicalPast.avoids κ B z e S, C.value κ B sample ∂P) =
-      ∫ sample in PhysicalPast.avoids κ B z e S, C.physicalValue κ B γ e sample ∂P := by
-  let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
-    (Real.sqrt κ) (C.paths κ B sample) z
-  let A := fun n i => rawCoeff C.height_pos 1 B C.horizon (dyadicTime C.horizon n i)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle
-  let D := fun n i => C.physicalCoeff 1 κ B e (dyadicTime C.horizon n i)
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B C.horizon (C.stopTime κ B) F n i :=
-    rawCoeff_stoppedCoeff_ae C.height_pos hB 1 C.horizon C.horizon_pos
-      (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle n i
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := sub_nonneg.2 ((le_div_iff₀ hκ.1).2 (by linarith [hκ.2]))
-  obtain ⟨K,hK,hKb⟩ := coeff_uniform_bound C.height_pos C.horizon.coe_nonneg 1 (Or.inl rfl)
-    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
-  have hAb (n i : ℕ) : ∀ᵐ sample ∂P, |A n i sample| ≤ K := by
-    filter_upwards [hAe n i] with sample he
-    rw [he]
-    unfold stoppedCoeff
-    split_ifs
-    · exact hKb _ _ _
-    · simpa only [abs_zero] using hK
-  have hDb (n i : ℕ) : ∀ᵐ sample ∂P, |D n i sample| ≤ K := by
-    filter_upwards [hAb n i] with sample ha
-    dsimp only [D,physicalCoeff]
-    split_ifs
-    · exact ha
-    · simpa only [abs_zero] using hK
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (A n i) :=
-    rawCoeff_measurable_past C.height_pos 1 (Or.inl rfl) B C.horizon _ _ _ _ z _ C.angle
-  have hDm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (D n i) :=
-    C.physicalCoeff_measurable_past 1 (Or.inl rfl) κ B e _
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 1 (Or.inl rfl) C.horizon
-      (dyadicTime C.horizon n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z _ C.angle).aestronglyMeasurable K
-    simpa only [Real.norm_eq_abs] using hAb n i
-  have hD2 (n i : ℕ) : MemLp (D n i) 2 P := by
-    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 1 (Or.inl rfl) κ e _).aestronglyMeasurable K
-    simpa only [Real.norm_eq_abs] using hDb n i
-  have hsupport (n i : ℕ) (ht : dyadicTime C.horizon n i < S) : ∀ᵐ sample ∂P,
-      sample ∈ PhysicalPast.avoids κ B z e S → A n i sample = D n i sample := by
-    filter_upwards [hmaps,hB.cont] with sample hm hc he
-    have hh := (PhysicalPast.avoids_eq_source κ B sample hm hc z e S).1 he
-    have hs : sample ∈ PhysicalPast.avoids κ B z e (dyadicTime C.horizon n i) :=
-      (PhysicalPast.avoids_eq_source κ B sample hm hc z e _).2 (fun s hs => hh s (hs.trans ht.le))
-    simp only [D,physicalCoeff,ite_eq_left hs]
-    rfl
-  have hlA := localized_sum_tendsto hB κ hκ C.height_pos C.horizon C.horizon_pos z
-    ((2*z.im)*Real.exp (-u)) C.angle
-  have hlD := C.physical_sum_tendsto hB κ hκ γ g f hmaps hz hu e
-  have hE := PhysicalPast.avoids_nullMeasurable κ hB z e S
-  have hh := delayed_indicator_difference_centered hB C.horizon S A D hAm hDm hA2 hD2 K hK hAb hDb
-    (PhysicalPast.avoids κ B z e S) hE (PhysicalPast.avoids_measurable_past κ B z e S) hsupport
-    (fun sample => C.value κ B sample-F sample 0 (B 0 sample))
-    (fun sample => C.physicalValue κ B γ e sample-F sample 0 (B 0 sample)) hlA hlD
-  have hdiff : (∫ sample in PhysicalPast.avoids κ B z e S,
-      C.value κ B sample-C.physicalValue κ B γ e sample ∂P) = 0 := by
-    simpa only [sub_sub_sub_cancel_right,integral_indicator₀ hE] using hh.2
-  rw [integral_sub ((C.expectation hB κ hκ hz hu.le).1.integrableOn)
-    ((C.physical_expectation hB κ hκ γ g f hmaps hz hu e).1.integrableOn)] at hdiff
-  exact sub_eq_zero.1 hdiff
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt.Cutoff
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
-variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
-variable (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-variable (hz : z ∈ halfPlane) (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-include h hz hu hc hb0
-
-lemma physicalValue_eq_source (ε : ℝ) : C.physicalValue κ B γ ε sample =
-    (loewnerRadius (g sample) (C.physicalTime κ B γ ε sample) z)^(-(1-κ/8)) *
-      (loewnerAngleSine κ (fun t => B t sample) (g sample) (C.physicalTime κ B γ ε sample) z)^(8/κ-1) := by
-  have ht := C.physicalTime_bounds κ B γ ε sample
-  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
-  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
-    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)).2.1
-  have hr := radius_eq_source C.height_pos C.horizon_pos h
-    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) _ z _ C.angle
-    (C.path_eq κ B sample hc) h0 ht
-  have hs := sine_eq_source C.height_pos C.horizon_pos h _ z _ C.angle
-    (C.path_eq κ B sample hc) h0 ht
-  change _ = loewnerRadius (g sample) (C.physicalTime κ B γ ε sample) z at hr
-  have hs' : sine C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
-      (C.physicalTime κ B γ ε sample) =
-      loewnerAngleSine κ (fun t => B t sample) (g sample) (C.physicalTime κ B γ ε sample) z := by
-    simpa only [loewnerAngleSine,Complex.ofReal_mul] using hs
-  rw [physicalValue,field_eq_radius_sine C.height_pos C.horizon_pos _ _ _ _ z _ _ hh
-    (C.path_eq κ B sample hc ⟨ht.1,ht.2.trans (C.bounds κ B sample).2⟩),hr,hs']
-
-end SLEExactGauge.FiniteTilt.Cutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.FiniteTilt
-variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
-variable (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
-variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
-variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
-variable {ε : ℝ} (hε : 0 < ε) (hsmall : ε < z.im)
-variable (hρ : (2*z.im)*Real.exp (-u) < ε/4)
-include h hz hu hc hb0 hε hsmall hρ
-
-lemma eventually_physicalValue_at_hit (hfin : firstLiveDistanceStop (γ sample) z ε ≠ ⊤) :
-    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((cutoffs hz u hu.le n).physicalValue κ B γ ε sample) =
-      physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε := by
-  obtain ⟨t,ht,hzt,hd,hmin⟩ := PhysicalStop.finite_spec h.1 hfin
-  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
-  have hδ : ∀ᶠ n : ℕ in atTop, shrinkingAngle z n < η :=
-    (shrinkingAngle_tendsto z).eventually (gt_mem_nhds hη)
-  filter_upwards [hδ] with n hn
-  let C := cutoffs hz u hu.le n
-  have htc : (t:ℝ) ≤ C.stopTime κ B sample := by
-    by_contra hnot
-    have hlt : C.stopTime κ B sample < t := lt_of_not_ge hnot
-    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-    · have hh := h.radius_ge_before_distance_stop hε hsmall
-        (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
-        (by rw [ht]; exact ENNReal.coe_le_coe.2 hlt.le)
-        (C.source_live κ B sample h hz hu hc hb0)
-      change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
-      rw [hr] at hh
-      exact (not_lt_of_ge hh) hρ
-    · have hh := hm (C.stopTime κ B sample) ⟨(C.bounds κ B sample).1,hlt.le⟩
-      rw [hs] at hh
-      exact (not_lt_of_ge hh) hn
-  have he : C.physicalTime κ B γ ε sample = t := by
-    rw [Cutoff.physicalTime,ht,PhysicalStop.cap_coe htc]
-  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
-  rw [physicalStoppedDensity,ht,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,hp,
-    C.physicalValue_eq_source κ B γ g f sample h hz hu hc hb0 ε,he]
-
-lemma tendsto_physicalValue_killed (hκ : 0 < κ ∧ κ < 8) :
-    Tendsto (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).physicalValue κ B γ ε sample)) atTop
-      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε)) := by
-  by_cases he : firstLiveDistanceStop (γ sample) z ε = ⊤
-  · rw [physicalStoppedDensity,he,ite_eq_left rfl]
-    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
-    have hlim := (shrinkingAngle_tendsto z).rpow_const_nhds_zero ha
-    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
-        (shrinkingAngle z n)^(8/κ-1)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
-    have hl : Tendsto (fun n : ℕ => (cutoffs hz u hu.le n).physicalValue κ B γ ε sample) atTop (𝓝 0) := by
-      apply squeeze_zero (fun n => (cutoffs hz u hu.le n).physicalValue_nonneg κ B γ ε sample) _ hlim'
-      intro n
-      let C := cutoffs hz u hu.le n
-      have ht : C.physicalTime κ B γ ε sample = C.stopTime κ B sample := by
-        simp only [Cutoff.physicalTime,PhysicalStop.cap,he,ite_true]
-      have hv : C.physicalValue κ B γ ε sample = C.value κ B sample := by
-        simp only [Cutoff.physicalValue,ht,Cutoff.value]
-      rw [hv]
-      have hs : loewnerAngleSine κ (fun t => B t sample) (g sample) (C.stopTime κ B sample) z = C.angle := by
-        rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
-        · have hh := h.radius_ge_before_distance_stop hε hsmall
-            (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
-            (by rw [he]; exact le_top) (C.source_live κ B sample h hz hu hc hb0)
-          change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
-          rw [hr] at hh
-          exact False.elim ((not_lt_of_ge hh) hρ)
-        · exact hs
-      exact C.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
-    simpa only [ENNReal.ofReal_zero,Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
-  · have hh := eventually_physicalValue_at_hit κ B γ g f sample h hz hu hc hb0 hε hsmall hρ he
-    exact tendsto_const_nhds.congr' (hh.mono (fun _ heq => heq.symm))
-
-end SLEExactGauge.FiniteTilt
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.ConditionalAngles
+import OAI.Probability.SLE.ClippedCoordinates
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Physical Past estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.PhysicalPast
+
+def sampleTime (T : ℝ≥0) (q : I) : ℝ≥0 := T * ⟨q,q.property.1⟩
+
+lemma sampleTime_le (T : ℝ≥0) (q : I) : sampleTime T q ≤ T := by
+  change (T:ℝ)*(q:ℝ) ≤ T
+  simpa using mul_le_mul_of_nonneg_left q.property.2 T.coe_nonneg
+
+lemma uniform_margin_iff {F : I → ℝ} (hF : Continuous F) (ε : ℝ) :
+    (∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤ F (denseSeq I n)) ↔
+      ∀ q : I, ε < F q := by
+  constructor
+  · rintro ⟨m,hm⟩ q
+    have hh : ε + CanonicalTrace.height m ≤ F q :=
+      (denseRange_denseSeq I).induction_on q (isClosed_le continuous_const hF) hm
+    linarith [CanonicalTrace.height_pos m]
+  · intro hh
+    obtain ⟨q,hq,hmin⟩ := isCompact_univ.exists_isMinOn (univ_nonempty : (univ : Set I).Nonempty) hF.continuousOn
+    obtain ⟨m,hm⟩ := exists_nat_one_div_lt (sub_pos.2 (hh q))
+    refine ⟨m,fun n => ?_⟩
+    have hb := hmin (mem_univ (denseSeq I n))
+    change F q ≤ F (denseSeq I n) at hb
+    dsimp [CanonicalTrace.height]
+    linarith
+
+def avoids {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (z : ℂ) (ε : ℝ)
+    (T : ℝ≥0) : Set Ω :=
+  {sample | ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤
+    dist z (CanonicalTrace.value κ B (sampleTime T (denseSeq I n)) sample)}
+
+lemma avoids_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (z : ℂ) (ε : ℝ) (T : ℝ≥0) :
+    @MeasurableSet Ω (BrownianPredictable.pastSigma B T) (avoids κ B z ε T) := by
+  let : MeasurableSpace Ω := BrownianPredictable.pastSigma B T
+  simp only [avoids,ofPred_exists,ofPred_forall]
+  apply MeasurableSet.iUnion
+  intro m
+  apply MeasurableSet.iInter
+  intro n
+  apply measurableSet_le measurable_const
+  exact measurable_const.dist ((CanonicalTrace.measurable_past κ B _).mono
+    (BrownianPredictable.pastSigma_mono B (sampleTime_le T (denseSeq I n))) le_rfl)
+
+lemma avoids_nullMeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (z : ℂ) (ε : ℝ) (T : ℝ≥0) : NullMeasurableSet (avoids κ B z ε T) P := by
+  simp only [avoids,ofPred_exists,ofPred_forall]
+  apply NullMeasurableSet.iUnion
+  intro m
+  apply NullMeasurableSet.iInter
+  intro n
+  exact nullMeasurableSet_le aemeasurable_const
+    (aemeasurable_const.dist (CanonicalTrace.aemeasurable κ hB _))
+
+lemma avoids_eq_source {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal sample) γ g f)
+    (hc : Continuous (fun t => B t sample)) (z : ℂ) (ε : ℝ) (T : ℝ≥0) :
+    sample ∈ avoids κ B z ε T ↔ ∀ s : ℝ≥0, s ≤ T → ε < dist z (γ s) := by
+  have he : (sample ∈ avoids κ B z ε T) ↔
+      ∃ m : ℕ, ∀ n : ℕ, ε + CanonicalTrace.height m ≤ dist z (γ (sampleTime T (denseSeq I n))) := by
+    simp only [avoids,mem_ofPred_eq,CanonicalTrace.eq_source κ B sample h hc]
+  rw [he]
+  have hcont : Continuous (fun q : I => dist z (γ (sampleTime T q))) :=
+    continuous_const.dist (h.1.comp_continuous (continuous_const.mul continuous_subtype_val)
+      (fun q => mul_nonneg T.coe_nonneg q.property.1))
+  rw [uniform_margin_iff hcont ε]
+  constructor
+  · intro hh s hs
+    by_cases hT : T = 0
+    · have hs0 : s=0 := by simpa only [hT,le_zero_iff] using hs
+      have hzero : sampleTime T (0:I) = 0 := by
+        apply NNReal.coe_injective
+        change (T:ℝ) * (0:ℝ) = 0
+        exact mul_zero _
+      simpa only [hzero,hs0,NNReal.coe_zero] using hh (0:I)
+    · have hTp : 0 < (T:ℝ) := lt_of_le_of_ne T.coe_nonneg (by exact_mod_cast Ne.symm hT)
+      let q : I := ⟨(s:ℝ)/(T:ℝ),⟨div_nonneg s.coe_nonneg T.coe_nonneg,
+        (div_le_one hTp).2 hs⟩⟩
+      have hsq : sampleTime T q = s := by
+        apply NNReal.coe_injective
+        change (T:ℝ)*((s:ℝ)/(T:ℝ)) = s
+        field_simp
+      simpa only [hsq] using hh q
+  · intro hh q
+    exact hh _ (sampleTime_le T q)
+
+end SLEExactGauge.PhysicalPast
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+lemma closedBall_subset_survivingDomain {γ : ℝ → ℂ} {t : ℝ} {z : ℂ} {r : ℝ}
+    (hr : 0 ≤ r) (hz : z ∈ localizationDomain γ t)
+    (hfree : closedBall z r ⊆ halfPlane \ localizationSegment γ 0 t) :
+    closedBall z r ⊆ localizationDomain γ t := by
+  have hc := (convex_closedBall z r).isPreconnected.subset_connectedComponentIn
+    (mem_closedBall_self hr) hfree
+  intro w hw
+  refine ⟨hfree hw,?_⟩
+  rw [← connectedComponentIn_eq (hc hw)]
+  exact hz.2
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge
+namespace IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma radius_ge_half_free_ball (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} {r : ℝ} (hr : 0 < r)
+    (hz : z ∈ localizationDomain γ t) (hball : closedBall z r ⊆ localizationDomain γ t) :
+    r/2 ≤ loewnerRadius g t z := by
+  let y := (g t z).im
+  have hy : 0 < y := h.im_pos ht hz
+  let F : ℂ → ℂ := fun w => (g t w - (starRingEnd ℂ) (g t z))⁻¹
+  have hden (w : ℂ) (hw : w ∈ localizationDomain γ t) :
+      y ≤ ‖g t w - (starRingEnd ℂ) (g t z)‖ := by
+    have hp := h.im_pos ht hw
+    have hh := Complex.im_le_norm (g t w - (starRingEnd ℂ) (g t z))
+    simp only [Complex.sub_im,Complex.conj_im] at hh
+    dsimp [y]
+    linarith
+  have hn (w : ℂ) (hw : w ∈ localizationDomain γ t) : g t w-(starRingEnd ℂ) (g t z) ≠ 0 :=
+    norm_pos_iff.1 (hy.trans_le (hden w hw))
+  have hd (w : ℂ) (hw : w ∈ localizationDomain γ t) : DifferentiableAt ℂ F w :=
+    ((h.analytic_eval ht hw).differentiableAt.sub_const _).inv (hn w hw)
+  have hF : DiffContOnCl ℂ F (ball z r) := by
+    apply DifferentiableOn.diffContOnCl
+    rw [closure_ball _ hr.ne']
+    exact fun w hw => (hd w (hball hw)).differentiableWithinAt
+  have hb : ∀ w ∈ sphere z r, ‖F w‖ ≤ 1/y := by
+    intro w hw
+    change ‖(g t w - (starRingEnd ℂ) (g t z))⁻¹‖ ≤ 1/y
+    rw [norm_inv,one_div]
+    exact inv_anti₀ hy (hden w (hball (sphere_subset_closedBall hw)))
+  have hc := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le hr hF hb
+  have hdF : HasDerivAt F (-deriv (g t) z / (g t z-(starRingEnd ℂ) (g t z))^2) z := by
+    exact ((h.analytic_eval ht hz).differentiableAt.hasDerivAt.sub_const _).inv (hn z hz)
+  have hnorm : ‖g t z-(starRingEnd ℂ) (g t z)‖ = 2*y := by
+    rw [Complex.sub_conj,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one]
+    exact abs_of_pos (by dsimp [y] at hy ⊢; positivity)
+  rw [hdF.deriv,norm_div,norm_neg,norm_pow,hnorm] at hc
+  have hdn := norm_pos_iff.2 (h.deriv_ne_zero ht hz)
+  change r/2 ≤ 2*y/‖deriv (g t) z‖
+  rw [le_div_iff₀ hdn]
+  have hh := (div_le_iff₀ (sq_pos_of_pos (by positivity : 0 < 2*y))).1 hc
+  have hh' : ‖deriv (g t) z‖ * r ≤ 4*y := by
+    have hrr := (mul_le_mul_of_nonneg_right hh hr.le)
+    field_simp at hrr
+    nlinarith
+  nlinarith
+
+end IsLoewnerTraceUsing
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+def firstLiveDistanceStop (γ : ℝ → ℂ) (z : ℂ) (ε : ℝ) : ℝ≥0∞ :=
+  sInf ((fun t : ℝ≥0 => (t : ℝ≥0∞)) ''
+    {t : ℝ≥0 | z ∈ localizationDomain γ t ∧ dist z (γ t) ≤ ε})
+
+def physicalStoppedDensity (κ : ℝ) (B : ℝ≥0 → ℝ)
+    (γ : ℝ → ℂ) (g : ℝ → ℂ → ℂ) (z : ℂ) (ε : ℝ) : ℝ≥0∞ :=
+  let τ := firstLiveDistanceStop γ z ε
+  if τ = ⊤ then 0 else
+    ENNReal.ofReal ((loewnerRadius g τ.toReal z)^(-codimension κ) *
+      (loewnerAngleSine κ B g τ.toReal z)^(8/κ-1))
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+namespace PhysicalStop
+variable {γ : ℝ → ℂ} {z : ℂ} {ε : ℝ}
+
+lemma le_of_hit {t : ℝ≥0} (hz : z ∈ localizationDomain γ t) (hd : dist z (γ t) ≤ ε) :
+    firstLiveDistanceStop γ z ε ≤ (t:ℝ≥0∞) :=
+  sInf_le ⟨t,⟨hz,hd⟩,rfl⟩
+
+lemma eq_top_iff : firstLiveDistanceStop γ z ε = ⊤ ↔
+    ∀ t : ℝ≥0, z ∈ localizationDomain γ t → ε < dist z (γ t) := by
+  constructor
+  · intro he t hz
+    by_contra hn
+    have hh := le_of_hit hz (not_lt.1 hn)
+    rw [he] at hh
+    exact ENNReal.coe_ne_top (top_le_iff.1 hh)
+  · intro hn
+    have he : {t : ℝ≥0 | z ∈ localizationDomain γ t ∧ dist z (γ t) ≤ ε} = ∅ := by
+      exact Set.eq_empty_iff_forall_notMem.2 (fun t ht => (not_lt_of_ge ht.2) (hn t ht.1))
+    simp only [firstLiveDistanceStop,he,image_empty,sInf_empty]
+
+lemma finite_spec (hγ : ContinuousOn γ (Ici 0))
+    (hfin : firstLiveDistanceStop γ z ε ≠ ⊤) :
+    ∃ t : ℝ≥0, firstLiveDistanceStop γ z ε = (t:ℝ≥0∞) ∧
+      z ∈ localizationDomain γ t ∧ dist z (γ t) ≤ ε ∧
+      ∀ s : ℝ≥0, s < t → ε < dist z (γ s) := by
+  have hex : ∃ T : ℝ≥0, z ∈ localizationDomain γ T ∧ dist z (γ T) ≤ ε := by
+    by_contra hn
+    push Not at hn
+    exact hfin (eq_top_iff.2 hn)
+  obtain ⟨T,hTlive,hTdist⟩ := hex
+  have hc : Continuous (fun s : ℝ≥0 => dist z (γ s)) :=
+    continuous_const.dist (hγ.comp_continuous continuous_subtype_val (fun s => s.coe_nonneg))
+  let A : Set ℝ≥0 := Icc 0 T ∩ {s | dist z (γ s) ≤ ε}
+  have hA : IsCompact A := isCompact_Icc.inter_right (isClosed_le hc continuous_const)
+  obtain ⟨t,ht,hmin⟩ := hA.exists_isLeast ⟨T,⟨⟨zero_le,le_rfl⟩,hTdist⟩⟩
+  have htlive : z ∈ localizationDomain γ t := survivingDomain_mono ht.1.2 hTlive
+  have hleast (s : ℝ≥0) (hs : z ∈ localizationDomain γ s ∧ dist z (γ s) ≤ ε) : t ≤ s := by
+    by_cases hsT : s ≤ T
+    · exact hmin ⟨⟨zero_le,hsT⟩,hs.2⟩
+    · exact ht.1.2.trans (le_of_not_ge hsT)
+  have he : firstLiveDistanceStop γ z ε = (t:ℝ≥0∞) := by
+    apply le_antisymm (le_of_hit htlive ht.2)
+    apply le_sInf
+    rintro _ ⟨s,hs,rfl⟩
+    exact ENNReal.coe_le_coe.2 (hleast s hs)
+  refine ⟨t,he,htlive,ht.2,?_⟩
+  intro s hs
+  by_contra hn
+  exact (not_le_of_gt hs) (hleast s ⟨survivingDomain_mono hs.le htlive,not_lt.1 hn⟩)
+
+lemma lt_stop_distance {t : ℝ≥0} (ht : (t:ℝ≥0∞) < firstLiveDistanceStop γ z ε)
+    (hlive : z ∈ localizationDomain γ t) : ε < dist z (γ t) := by
+  by_contra hn
+  exact (not_le_of_gt ht) (le_of_hit hlive (not_lt.1 hn))
+
+lemma stopped_survival_iff (hγ : ContinuousOn γ (Ici 0)) {t : ℝ≥0}
+    (hlive : z ∈ localizationDomain γ t) :
+    (t:ℝ≥0∞) < firstLiveDistanceStop γ z ε ↔
+      ∀ s : ℝ≥0, s ≤ t → ε < dist z (γ s) := by
+  constructor
+  · intro ht s hs
+    exact lt_stop_distance ((ENNReal.coe_le_coe.2 hs).trans_lt ht) (survivingDomain_mono hs hlive)
+  · intro hh
+    by_contra hn
+    have he : firstLiveDistanceStop γ z ε ≠ ⊤ :=
+      ne_top_of_le_ne_top ENNReal.coe_ne_top (not_lt.1 hn)
+    obtain ⟨s,hs,hslive,hsdist,hmin⟩ := finite_spec hγ he
+    rw [hs] at hn
+    have hst : s ≤ t := ENNReal.coe_le_coe.1 (not_lt.1 hn)
+    exact (not_lt_of_ge hsdist) (hh s hst)
+
+lemma finite_dist_eq (hγ : ContinuousOn γ (Ici 0))
+    (hinit : ε < dist z (γ 0)) (hfin : firstLiveDistanceStop γ z ε ≠ ⊤) :
+    dist z (γ (firstLiveDistanceStop γ z ε).toReal) = ε := by
+  obtain ⟨t,he,hlive,hdist,hmin⟩ := finite_spec hγ hfin
+  rw [he,ENNReal.coe_toReal]
+  apply le_antisymm hdist
+  by_contra hn
+  have hlt : dist z (γ t) < ε := not_le.1 hn
+  have hc : ContinuousOn (fun s : ℝ => dist z (γ s)) (Icc 0 (t:ℝ)) :=
+    (continuous_const.dist continuous_id).comp_continuousOn (hγ.mono Icc_subset_Ici_self)
+  obtain ⟨s,hs,hseq⟩ := intermediate_value_Icc' t.coe_nonneg hc ⟨hlt.le,hinit.le⟩
+  have hst : s < (t:ℝ) := lt_of_le_of_ne hs.2 (by intro hst; subst s; linarith)
+  have hh := hmin ⟨s,hs.1⟩ hst
+  exact (not_lt_of_ge hseq.le) hh
+
+lemma trace_distance_ge_before (hγ : ContinuousOn γ (Ici 0))
+    (hinit : ε < dist z (γ 0)) {t : ℝ≥0}
+    (ht : (t:ℝ≥0∞) ≤ firstLiveDistanceStop γ z ε)
+    (hlive : z ∈ localizationDomain γ t) {s : ℝ} (hs : s ∈ Icc 0 (t:ℝ)) :
+    ε ≤ dist z (γ s) := by
+  rcases lt_or_eq_of_le ((ENNReal.coe_le_coe.2 (show (⟨s,hs.1⟩:ℝ≥0) ≤ t from hs.2)).trans ht) with hlt | he
+  · exact (lt_stop_distance hlt (survivingDomain_mono hs.2 hlive)).le
+  · have hfin : firstLiveDistanceStop γ z ε ≠ ⊤ := he ▸ ENNReal.coe_ne_top
+    have hh := finite_dist_eq hγ hinit hfin
+    rw [← he] at hh
+    change dist z (γ s) = ε at hh
+    exact hh.symm.le
+
+end PhysicalStop
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+open Set Metric
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma radius_ge_before_distance_stop (h : IsLoewnerTraceUsing driver γ g f)
+    {z : ℂ} {ε : ℝ} (hε : 0 < ε) (hsmall : ε < z.im) {t : ℝ≥0}
+    (ht : (t:ℝ≥0∞) ≤ firstLiveDistanceStop γ z ε) (hlive : z ∈ localizationDomain γ t) :
+    ε/4 ≤ loewnerRadius g t z := by
+  have hinit : ε < dist z (γ 0) := by
+    rw [h.2.1,dist_zero_right]
+    exact hsmall.trans_le (Complex.im_le_norm z)
+  have hdist (s : ℝ) (hs : s ∈ Icc 0 (t:ℝ)) : ε ≤ dist z (γ s) :=
+    PhysicalStop.trace_distance_ge_before h.1 hinit ht hlive hs
+  have hfree : closedBall z (ε/2) ⊆ halfPlane \ localizationSegment γ 0 t := by
+    intro w hw
+    have hnorm : ‖w-z‖ ≤ ε/2 := by simpa only [mem_closedBall,dist_eq_norm] using hw
+    constructor
+    · have hh := Complex.abs_im_le_norm (w-z)
+      have hn := neg_abs_le (w-z).im
+      change 0 < w.im
+      simp only [Complex.sub_im] at hh hn
+      linarith
+    · rintro ⟨s,hs,rfl⟩
+      have hh := hdist s hs
+      have hh' := mem_closedBall.1 hw
+      rw [dist_comm] at hh'
+      linarith
+  have hb := closedBall_subset_survivingDomain (by linarith : 0 ≤ ε/2) hlive hfree
+  have hh := h.radius_ge_half_free_ball t.coe_nonneg (by linarith : 0 < ε/2) hlive hb
+  convert hh using 1
+  ring
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.PhysicalStop
+
+def cap (tauE : ℝ≥0∞) (a : ℝ) : ℝ := if tauE = ⊤ then a else min tauE.toReal a
+
+lemma cap_bounds (tauE : ℝ≥0∞) {a : ℝ} (ha : 0 ≤ a) : 0 ≤ cap tauE a ∧ cap tauE a ≤ a := by
+  unfold cap
+  split_ifs
+  · exact ⟨ha,le_rfl⟩
+  · exact ⟨le_min ENNReal.toReal_nonneg ha,min_le_right _ _⟩
+
+lemma lt_cap (tauE : ℝ≥0∞) (a : ℝ) (s : ℝ≥0) :
+    (s:ℝ) < cap tauE a ↔ (s:ℝ) < a ∧ (s:ℝ≥0∞) < tauE := by
+  by_cases htauE : tauE = ⊤
+  · simp [cap,htauE]
+  · rw [cap,ite_eq_right htauE,lt_min_iff,and_comm]
+    apply and_congr_right
+    intro _
+    simpa only [ENNReal.coe_toReal] using ENNReal.toReal_lt_toReal (a := (s:ℝ≥0∞)) ENNReal.coe_ne_top htauE
+
+lemma cap_coe {t : ℝ≥0} {a : ℝ} (hta : (t:ℝ) ≤ a) : cap (t:ℝ≥0∞) a = t := by
+  simp only [cap,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,min_eq_left hta]
+
+end SLEExactGauge.PhysicalStop
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BrownianField.FieldRegular
+
+lemma restrict {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ} {s t : ℝ}
+    (h : FieldRegular F V b t) (hst : s ≤ t) : FieldRegular F V b s := by
+  refine ⟨h.smooth,h.third_bound,?_,?_,?_,?_⟩
+  · exact fun x => (h.time_cont x).mono (Icc_subset_Icc le_rfl hst)
+  · exact fun r hr => h.time_deriv r ⟨hr.1,hr.2.trans_le hst⟩
+  · exact h.drift_cont.mono (prod_mono (Icc_subset_Icc le_rfl hst) (Icc_subset_Icc le_rfl hst))
+  · exact h.second_cont.mono (Icc_subset_Icc le_rfl hst)
+
+end SLEExactGauge.BrownianField.FieldRegular
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerLocalized
+open LoewnerCutoff LoewnerField LoewnerExit
+
+lemma integral_drift_cancel_before {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T) (κ : ℝ) (hκ : 0 < κ)
+    (U : C(I, ℝ)) (z : ℂ) (ρ δ : ℝ) (b : ℝ → ℝ)
+    (hU : ∀ s ∈ Icc 0 T, extendDriver T U s = Real.sqrt κ * b s)
+    {τ : ℝ} (hτ0 : 0 ≤ τ) (hτ : τ ≤ stop hε hT.le U z ρ δ) :
+    (∫ t in (0:ℝ)..τ, drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) +
+      (1/2 : ℝ)*∫ t in (0:ℝ)..τ,
+        iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) = 0 := by
+  have he : (∫ t in (0:ℝ)..τ, drift hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t (b t)) =
+      ∫ t in (0:ℝ)..τ,
+        -(1/2 : ℝ)*iteratedDeriv 2 (field hε hT.le (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t) := by
+    apply intervalIntegral.integral_congr_Ioo_of_le hτ0
+    intro t ht
+    have htT : t ∈ Icc 0 T := ⟨ht.1.le,ht.2.le.trans (hτ.trans (stop_bounds hε hT.le U z ρ δ).2)⟩
+    have hh := (deficit_pos_iff hε hT.le U z ρ δ t).1
+      (positive_before hε hT.le hT U z ρ δ ht.1.le (ht.2.trans_le hτ))
+    have hc := drift_cancellation hε hT.le κ hκ U z t (b t) hh.2.1.le (hU t htT)
+    linarith
+  rw [he, intervalIntegral.integral_const_mul]
+  ring
+
+end SLEExactGauge.LoewnerLocalized
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+def physicalTime {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) : ℝ :=
+  PhysicalStop.cap (firstLiveDistanceStop (γ sample) z ε) (C.stopTime κ B sample)
+
+def physicalValue {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) : ℝ :=
+  field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
+    (C.physicalTime κ B γ ε sample) (B (C.physicalTime κ B γ ε sample).toNNReal sample)
+
+lemma physicalTime_bounds {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) :
+    0 ≤ C.physicalTime κ B γ ε sample ∧ C.physicalTime κ B γ ε sample ≤ C.stopTime κ B sample :=
+  PhysicalStop.cap_bounds _ (C.bounds κ B sample).1
+
+lemma physicalValue_nonneg {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) :
+    0 ≤ C.physicalValue κ B γ ε sample := (green_pos _ _ _ _ _).le
+
+lemma physicalValue_bound {Ω : Type*} (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ) (ε : ℝ) (sample : Ω) (hz : z ∈ halfPlane) (hu : 0 < u)
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0) :
+    C.physicalValue κ B γ ε sample ≤ ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) := by
+  have hy0 : 0 < z.im := hz
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by
+    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
+    linarith
+  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hd := (deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0
+      (C.physicalTime_bounds κ B γ ε sample))
+  apply green_le_fixed_radius _ _ _ _ _ _ (LoewnerField.height_pos _ _ _ _ _) ha hp (by positivity)
+  simpa only [LoewnerExit.radius,LoewnerField.height,max_eq_right hd.2.1] using hd.1
+
+def physicalCoeff {Ω : Type*} (k : ℕ) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (ε : ℝ) (s : ℝ≥0) (sample : Ω) : ℝ :=
+  by
+  classical
+  exact if sample ∈ PhysicalPast.avoids κ B z ε s then
+    rawCoeff C.height_pos k B C.horizon s (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle sample
+  else 0
+
+lemma physicalCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (k : ℕ) (hk : k=1 ∨ k=2) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (ε : ℝ) (s : ℝ≥0) :
+    Measurable[BrownianPredictable.pastSigma B s] (C.physicalCoeff k κ B ε s) := by
+  classical
+  exact (rawCoeff_measurable_past C.height_pos k hk B C.horizon s _ _ _ _ _ _).ite
+    (PhysicalPast.avoids_measurable_past κ B z ε s) measurable_const
+
+lemma physicalCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (k : ℕ) (hk : k=1 ∨ k=2) (κ : ℝ) (ε : ℝ) (s : ℝ≥0) :
+    AEMeasurable (C.physicalCoeff k κ B ε s) P := by
+  classical
+  exact (rawCoeff_aemeasurable C.height_pos hB k hk C.horizon s _ _ _ _ _ _).indicator₀
+    (PhysicalPast.avoids_nullMeasurable κ hB z ε s)
+
+lemma physicalCoeff_eq {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
+    (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) (k : ℕ) (s : ℝ≥0) :
+    C.physicalCoeff k κ B ε s sample =
+      if (s:ℝ) < C.physicalTime κ B γ ε sample then
+        iteratedDeriv k (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ)
+          (C.paths κ B sample) z s) (B s sample) else 0 := by
+  classical
+  have he := rawCoeff_eq C.height_pos k B C.horizon s C.horizon_pos (1-κ/8) (8/κ-1)
+    (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle sample hc
+  change rawCoeff _ _ _ _ _ _ _ _ _ _ _ _ =
+    (if (s:ℝ) < C.stopTime κ B sample then _ else 0) at he
+  have ht := PhysicalStop.lt_cap (firstLiveDistanceStop (γ sample) z ε) (C.stopTime κ B sample) s
+  change ((s:ℝ) < C.physicalTime κ B γ ε sample) ↔ _ at ht
+  by_cases hs : (s:ℝ) < C.stopTime κ B sample
+  · have hlive := survivingDomain_mono hs.le (C.source_live κ B sample h hz hu hc hb0)
+    have hav := (PhysicalPast.avoids_eq_source κ B sample h hc z ε s).trans
+      (PhysicalStop.stopped_survival_iff h.1 hlive).symm
+    simp only [physicalCoeff,he,hav,ht,hs,ite_true,true_and,paths]
+  · have hcap : ¬ (s:ℝ) < C.physicalTime κ B γ ε sample := fun hh => hs (ht.1 hh).1
+    simp only [physicalCoeff,he,ite_eq_right hs,ite_eq_right hcap,ite_self]
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
+open BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+theorem physical_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) :
+    let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
+    Integrable (fun sample => C.physicalValue κ B γ ε sample - F sample 0 (B 0 sample)) P ∧
+      (∫ sample, C.physicalValue κ B γ ε sample - F sample 0 (B 0 sample) ∂P) = 0 := by
+  let T := C.horizon
+  let hT := C.horizon_pos
+  let hε := C.height_pos
+  let ρ := (2*z.im)*Real.exp (-u)
+  let δ := C.angle
+  let U := C.paths κ B
+  let τ := C.physicalTime κ B γ ε
+  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by
+    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
+    linarith
+  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact (regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
+      (hc.comp continuous_real_toNNReal)).restrict (C.physicalTime_bounds κ B γ ε sample).2
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
+    Filter.Eventually.of_forall (fun sample => ⟨(C.physicalTime_bounds κ B γ ε sample).1,
+      (C.physicalTime_bounds κ B γ ε sample).2.trans (C.bounds κ B sample).2⟩)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
+        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
+    apply Filter.Eventually.of_forall
+    intro sample
+    unfold stoppedCoeff
+    split_ifs
+    · exact hb _ _ _
+    · simpa only [abs_zero] using hK
+  let A := fun n i => C.physicalCoeff 1 κ B ε (dyadicTime T n i)
+  let Q := fun n i => C.physicalCoeff 2 κ B ε (dyadicTime T n i)
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
+    simpa only [A,stoppedCoeff,coe_dyadicTime] using
+      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 1 (dyadicTime T n i)
+  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
+    simpa only [Q,stoppedCoeff,coe_dyadicTime] using
+      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 2 (dyadicTime T n i)
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
+    C.physicalCoeff_measurable_past 1 (Or.inl rfl) κ B ε _
+  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
+    C.physicalCoeff_measurable_past 2 (Or.inr rfl) κ B ε _
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 1 (Or.inl rfl) κ ε _).aestronglyMeasurable K₁
+    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hb
+  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
+    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 2 (Or.inr rfl) κ ε _).aestronglyMeasurable K₂
+    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |Q n i sample| ≤ K₂
+    rw [he]
+    exact hb
+  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A Q hAe hCe hCm hC2
+    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
+      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
+    filter_upwards [hB.cont] with sample hc
+    have hcanc := integral_drift_cancel_before hε hT κ hκ.1 (U sample) z ρ δ
+      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
+      (C.physicalTime_bounds κ B γ ε sample).1 (C.physicalTime_bounds κ B γ ε sample).2
+    dsimp only [F,V,τ]
+    linarith
+  exact ⟨hi.1.congr he, (integral_congr_ae he).symm.trans hi.2⟩
+
+theorem physical_expectation {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) :
+    Integrable (C.physicalValue κ B γ ε) P ∧
+      (∫ sample, C.physicalValue κ B γ ε sample ∂P) = green (1-κ/8) (8/κ-1) z.re z.im 0 := by
+  let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (C.paths κ B sample) z
+  let M := C.physicalValue κ B γ ε
+  let G := green (1-κ/8) (8/κ-1) z.re z.im 0
+  have hi := C.physical_centered hB κ hκ γ g f hmaps hz hu ε
+  have he : (fun sample => M sample - F sample 0 (B 0 sample)) =ᵐ[P]
+      (fun sample => M sample - G) := by
+    filter_upwards [hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with sample hb
+    rw [hb]
+    congr 1
+    exact field_initial C.height_pos C.horizon.coe_nonneg _ _ _ _ z (C.height_initial hz hu.le).le
+  have hd : Integrable (fun sample => M sample-G) P := hi.1.congr he
+  have hv : (∫ sample, M sample-G ∂P) = 0 := (integral_congr_ae he).symm.trans hi.2
+  have hm : Integrable M P := by
+    exact (hd.add (integrable_const G)).congr
+      (Filter.Eventually.of_forall (fun sample => sub_add_cancel (M sample) G))
+  refine ⟨hm, ?_⟩
+  have hh := integral_sub hm (integrable_const (μ := P) G)
+  simp only [integral_const, Measure.real, measure_univ, ENNReal.toReal_one, one_smul] at hh
+  rw [hh] at hv
+  change (∫ sample, M sample ∂P) = G
+  linarith
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
+open BrownianPredictable BrownianStopped BrownianIto TaylorMeshes PredictableLimit
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+
+theorem physical_sum_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (hz : z ∈ halfPlane) (hu : 0 < u) (ε : ℝ) :
+    let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
+      (Real.sqrt κ) (C.paths κ B sample) z
+    ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B C.horizon
+      (fun n i => C.physicalCoeff 1 κ B ε (dyadicTime C.horizon n i)) n sample) atTop
+      (𝓝 (C.physicalValue κ B γ ε sample-F sample 0 (B 0 sample))) := by
+  let T := C.horizon
+  let hT := C.horizon_pos
+  let hε := C.height_pos
+  let ρ := (2*z.im)*Real.exp (-u)
+  let δ := C.angle
+  let U := C.paths κ B
+  let τ := C.physicalTime κ B γ ε
+  let F := fun sample => field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  let V := fun sample => drift hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) (U sample) z
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by
+    have hh : (1:ℝ) ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
+    linarith
+  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact (regular hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha (U sample) z ρ δ _
+      (hc.comp continuous_real_toNNReal)).restrict (C.physicalTime_bounds κ B γ ε sample).2
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
+    Filter.Eventually.of_forall (fun sample => ⟨(C.physicalTime_bounds κ B γ ε sample).1,
+      (C.physicalTime_bounds κ B γ ε sample).2.trans (C.bounds κ B sample).2⟩)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg 1 (Or.inl rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg 2 (Or.inr rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
+        |iteratedDeriv k (field hε T.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) W z t) b| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
+    apply Filter.Eventually.of_forall
+    intro sample
+    unfold stoppedCoeff
+    split_ifs
+    · exact hb _ _ _
+    · simpa only [abs_zero] using hK
+  let A := fun n i => C.physicalCoeff 1 κ B ε (dyadicTime T n i)
+  let Q := fun n i => C.physicalCoeff 2 κ B ε (dyadicTime T n i)
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
+    simpa only [A,stoppedCoeff,coe_dyadicTime] using
+      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 1 (dyadicTime T n i)
+  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i := by
+    filter_upwards [hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hmaps] with sample hc hb0 h
+    simpa only [Q,stoppedCoeff,coe_dyadicTime] using
+      C.physicalCoeff_eq κ B γ g f sample h hc hb0 hz hu ε 2 (dyadicTime T n i)
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
+    C.physicalCoeff_measurable_past 1 (Or.inl rfl) κ B ε _
+  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
+    C.physicalCoeff_measurable_past 2 (Or.inr rfl) κ B ε _
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 1 (Or.inl rfl) κ ε _).aestronglyMeasurable K₁
+    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hb
+  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
+    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 2 (Or.inr rfl) κ ε _).aestronglyMeasurable K₂
+    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |Q n i sample| ≤ K₂
+    rw [he]
+    exact hb
+  have hi := stopped_fieldSum_tendsto_ae hB T τ F V htau hreg Q hCe hCm hC2
+    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  have he : (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)) =ᵐ[P]
+      (fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample)) := by
+    filter_upwards [hB.cont] with sample hc
+    have hcanc := integral_drift_cancel_before hε hT κ hκ.1 (U sample) z ρ δ
+      (fun t => B t.toNNReal sample) (fun t ht => BrownianDriver.extend_eq B _ T sample hc ht)
+      (C.physicalTime_bounds κ B γ ε sample).1 (C.physicalTime_bounds κ B γ ε sample).2
+    dsimp only [F,V,τ]
+    linarith
+  have hAeall : ∀ᵐ sample ∂P, ∀ n i, A n i sample = stoppedCoeff 1 B T τ F n i sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAe n))
+  filter_upwards [hi,he,hAeall] with sample hi he ha
+  rw [he] at hi
+  convert hi using 1
+  · funext n
+    unfold dyadicDriverSum
+    simp only [← ha]
+    rfl
+  · rfl
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+open PredictableLimit
+
+lemma delayed_indicator_difference_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T S : ℝ≥0)
+    (A D : ℕ → ℕ → Ω → ℝ)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
+    (hD : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (D n i))
+    (hA2 : ∀ n i, MemLp (A n i) 2 P) (hD2 : ∀ n i, MemLp (D n i) 2 P)
+    (K : ℝ) (hK : 0 ≤ K)
+    (hbA : ∀ n i, ∀ᵐ sample ∂P, |A n i sample| ≤ K)
+    (hbD : ∀ n i, ∀ᵐ sample ∂P, |D n i sample| ≤ K)
+    (E : Set Ω) (hE : NullMeasurableSet E P)
+    (hEm : MeasurableSet[pastSigma B S] E)
+    (hsupport : ∀ n i, dyadicTime T n i < S → ∀ᵐ sample ∂P,
+      sample ∈ E → A n i sample = D n i sample)
+    (YA YD : Ω → ℝ)
+    (hlA : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A n sample) atTop (𝓝 (YA sample)))
+    (hlD : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T D n sample) atTop (𝓝 (YD sample))) :
+    Integrable (E.indicator (fun sample => YA sample-YD sample)) P ∧
+      (∫ sample, E.indicator (fun sample => YA sample-YD sample) sample ∂P) = 0 := by
+  classical
+  let Q := fun n i => if S ≤ dyadicTime T n i then
+    E.indicator (fun sample => A n i sample-D n i sample) else (fun _ => 0)
+  have hQm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) := by
+    dsimp only [Q]
+    split_ifs with ht
+    · exact ((hA n i).sub (hD n i)).indicator (pastSigma_mono B ht _ hEm)
+    · exact measurable_const
+  have heq (n i : ℕ) : Q n i =ᵐ[P] E.indicator (fun sample => A n i sample-D n i sample) := by
+    by_cases ht : S ≤ dyadicTime T n i
+    · exact Filter.Eventually.of_forall (fun _ => by simp only [Q,ite_eq_left ht])
+    · filter_upwards [hsupport n i (lt_of_not_ge ht)] with sample hs
+      simp only [Q,ite_eq_right ht,Set.indicator_apply]
+      split_ifs with hmem
+      · simp only [hs hmem,sub_self]
+      · rfl
+  have hQ2 (n i : ℕ) : MemLp (Q n i) 2 P := by
+    have hd := (hA2 n i).sub (hD2 n i)
+    have hm : MemLp (E.indicator (fun sample => A n i sample-D n i sample)) 2 P :=
+      hd.of_le (hd.aestronglyMeasurable.indicator₀ hE)
+        (Filter.Eventually.of_forall (fun sample => norm_indicator_le_norm_self _ sample))
+    exact hm.ae_eq (heq n i).symm
+  have hQb (n i : ℕ) : ∀ᵐ sample ∂P, |Q n i sample| ≤ 2*K := by
+    filter_upwards [hbA n i,hbD n i,heq n i] with sample ha hd he
+    rw [he,Set.indicator_apply]
+    split_ifs
+    · exact (abs_sub _ _).trans (by linarith)
+    · simpa only [abs_zero] using (show 0 ≤ 2*K by positivity)
+  have heall : ∀ᵐ sample ∂P, ∀ n i, Q n i sample = E.indicator (fun sample => A n i sample-D n i sample) sample :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (heq n))
+  apply limit_centered hB T Q hQm hQ2 (2*K) (by positivity) hQb _
+  filter_upwards [heall,hlA,hlD] with sample he ha hd
+  have heqsum (n : ℕ) : dyadicDriverSum B T Q n sample =
+      E.indicator (fun sample => dyadicDriverSum B T A n sample-dyadicDriverSum B T D n sample) sample := by
+    unfold dyadicDriverSum
+    simp only [he,Set.indicator_apply]
+    split_ifs <;> simp [sub_mul,Finset.sum_sub_distrib]
+  simp only [heqsum,Set.indicator_apply]
+  split_ifs
+  · exact ha.sub hd
+  · exact tendsto_const_nhds
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized BrownianPredictable
+  BrownianStopped BrownianIto TaylorMeshes PredictableLimit
+
+lemma physical_avoidance_setIntegral {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+    (C : Cutoff z u) (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (e : ℝ) (S : ℝ≥0) :
+    (∫ sample in PhysicalPast.avoids κ B z e S, C.value κ B sample ∂P) =
+      ∫ sample in PhysicalPast.avoids κ B z e S, C.physicalValue κ B γ e sample ∂P := by
+  let F := fun sample => field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1)
+    (Real.sqrt κ) (C.paths κ B sample) z
+  let A := fun n i => rawCoeff C.height_pos 1 B C.horizon (dyadicTime C.horizon n i)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle
+  let D := fun n i => C.physicalCoeff 1 κ B e (dyadicTime C.horizon n i)
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B C.horizon (C.stopTime κ B) F n i :=
+    rawCoeff_stoppedCoeff_ae C.height_pos hB 1 C.horizon C.horizon_pos
+      (1-κ/8) (8/κ-1) (Real.sqrt κ) z ((2*z.im)*Real.exp (-u)) C.angle n i
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := sub_nonneg.2 ((le_div_iff₀ hκ.1).2 (by linarith [hκ.2]))
+  obtain ⟨K,hK,hKb⟩ := coeff_uniform_bound C.height_pos C.horizon.coe_nonneg 1 (Or.inl rfl)
+    (1-κ/8) (8/κ-1) (Real.sqrt κ) hp ha z
+  have hAb (n i : ℕ) : ∀ᵐ sample ∂P, |A n i sample| ≤ K := by
+    filter_upwards [hAe n i] with sample he
+    rw [he]
+    unfold stoppedCoeff
+    split_ifs
+    · exact hKb _ _ _
+    · simpa only [abs_zero] using hK
+  have hDb (n i : ℕ) : ∀ᵐ sample ∂P, |D n i sample| ≤ K := by
+    filter_upwards [hAb n i] with sample ha
+    dsimp only [D,physicalCoeff]
+    split_ifs
+    · exact ha
+    · simpa only [abs_zero] using hK
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (A n i) :=
+    rawCoeff_measurable_past C.height_pos 1 (Or.inl rfl) B C.horizon _ _ _ _ z _ C.angle
+  have hDm (n i : ℕ) : Measurable[pastSigma B (dyadicTime C.horizon n i)] (D n i) :=
+    C.physicalCoeff_measurable_past 1 (Or.inl rfl) κ B e _
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable C.height_pos hB 1 (Or.inl rfl) C.horizon
+      (dyadicTime C.horizon n i) (1-κ/8) (8/κ-1) (Real.sqrt κ) z _ C.angle).aestronglyMeasurable K
+    simpa only [Real.norm_eq_abs] using hAb n i
+  have hD2 (n i : ℕ) : MemLp (D n i) 2 P := by
+    apply MemLp.of_bound (C.physicalCoeff_aemeasurable hB 1 (Or.inl rfl) κ e _).aestronglyMeasurable K
+    simpa only [Real.norm_eq_abs] using hDb n i
+  have hsupport (n i : ℕ) (ht : dyadicTime C.horizon n i < S) : ∀ᵐ sample ∂P,
+      sample ∈ PhysicalPast.avoids κ B z e S → A n i sample = D n i sample := by
+    filter_upwards [hmaps,hB.cont] with sample hm hc he
+    have hh := (PhysicalPast.avoids_eq_source κ B sample hm hc z e S).1 he
+    have hs : sample ∈ PhysicalPast.avoids κ B z e (dyadicTime C.horizon n i) :=
+      (PhysicalPast.avoids_eq_source κ B sample hm hc z e _).2 (fun s hs => hh s (hs.trans ht.le))
+    simp only [D,physicalCoeff,ite_eq_left hs]
+    rfl
+  have hlA := localized_sum_tendsto hB κ hκ C.height_pos C.horizon C.horizon_pos z
+    ((2*z.im)*Real.exp (-u)) C.angle
+  have hlD := C.physical_sum_tendsto hB κ hκ γ g f hmaps hz hu e
+  have hE := PhysicalPast.avoids_nullMeasurable κ hB z e S
+  have hh := delayed_indicator_difference_centered hB C.horizon S A D hAm hDm hA2 hD2 K hK hAb hDb
+    (PhysicalPast.avoids κ B z e S) hE (PhysicalPast.avoids_measurable_past κ B z e S) hsupport
+    (fun sample => C.value κ B sample-F sample 0 (B 0 sample))
+    (fun sample => C.physicalValue κ B γ e sample-F sample 0 (B 0 sample)) hlA hlD
+  have hdiff : (∫ sample in PhysicalPast.avoids κ B z e S,
+      C.value κ B sample-C.physicalValue κ B γ e sample ∂P) = 0 := by
+    simpa only [sub_sub_sub_cancel_right,integral_indicator₀ hE] using hh.2
+  rw [integral_sub ((C.expectation hB κ hκ hz hu.le).1.integrableOn)
+    ((C.physical_expectation hB κ hκ γ g f hmaps hz hu e).1.integrableOn)] at hdiff
+  exact sub_eq_zero.1 hdiff
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt.Cutoff
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized GreenPhysical
+variable {z : ℂ} {u : ℝ} (C : Cutoff z u)
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
+variable (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+variable (hz : z ∈ halfPlane) (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+include h hz hu hc hb0
+
+lemma physicalValue_eq_source (ε : ℝ) : C.physicalValue κ B γ ε sample =
+    (loewnerRadius (g sample) (C.physicalTime κ B γ ε sample) z)^(-(1-κ/8)) *
+      (loewnerAngleSine κ (fun t => B t sample) (g sample) (C.physicalTime κ B γ ε sample) z)^(8/κ-1) := by
+  have ht := C.physicalTime_bounds κ B γ ε sample
+  have h0 := C.initial_deficit_brownian κ B sample hz hu hc hb0
+  have hh := ((deficit_nonneg_iff C.height_pos C.horizon.coe_nonneg _ z _ C.angle _).1
+    (nonneg_until C.height_pos C.horizon.coe_nonneg C.horizon_pos _ z _ C.angle h0 ht)).2.1
+  have hr := radius_eq_source C.height_pos C.horizon_pos h
+    ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) _ z _ C.angle
+    (C.path_eq κ B sample hc) h0 ht
+  have hs := sine_eq_source C.height_pos C.horizon_pos h _ z _ C.angle
+    (C.path_eq κ B sample hc) h0 ht
+  change _ = loewnerRadius (g sample) (C.physicalTime κ B γ ε sample) z at hr
+  have hs' : sine C.height_pos C.horizon.coe_nonneg (C.paths κ B sample) z
+      (C.physicalTime κ B γ ε sample) =
+      loewnerAngleSine κ (fun t => B t sample) (g sample) (C.physicalTime κ B γ ε sample) z := by
+    simpa only [loewnerAngleSine,Complex.ofReal_mul] using hs
+  rw [physicalValue,field_eq_radius_sine C.height_pos C.horizon_pos _ _ _ _ z _ _ hh
+    (C.path_eq κ B sample hc ⟨ht.1,ht.2.trans (C.bounds κ B sample).2⟩),hr,hs']
+
+end SLEExactGauge.FiniteTilt.Cutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.FiniteTilt
+variable {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ → ℂ)
+variable (g f : Ω → ℝ → ℂ → ℂ) (sample : Ω)
+variable (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+variable {z : ℂ} (hz : z ∈ halfPlane) {u : ℝ} (hu : 0 < u)
+variable (hc : Continuous (fun t => B t sample)) (hb0 : B 0 sample = 0)
+variable {ε : ℝ} (hε : 0 < ε) (hsmall : ε < z.im)
+variable (hρ : (2*z.im)*Real.exp (-u) < ε/4)
+include h hz hu hc hb0 hε hsmall hρ
+
+lemma eventually_physicalValue_at_hit (hfin : firstLiveDistanceStop (γ sample) z ε ≠ ⊤) :
+    ∀ᶠ n : ℕ in atTop, ENNReal.ofReal ((cutoffs hz u hu.le n).physicalValue κ B γ ε sample) =
+      physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε := by
+  obtain ⟨t,ht,hzt,hd,hmin⟩ := PhysicalStop.finite_spec h.1 hfin
+  obtain ⟨η,hη,hm⟩ := h.sine_positive_minimum (κ := κ) (B := fun t => B t sample) hc t.coe_nonneg hzt
+  have hδ : ∀ᶠ n : ℕ in atTop, shrinkingAngle z n < η :=
+    (shrinkingAngle_tendsto z).eventually (gt_mem_nhds hη)
+  filter_upwards [hδ] with n hn
+  let C := cutoffs hz u hu.le n
+  have htc : (t:ℝ) ≤ C.stopTime κ B sample := by
+    by_contra hnot
+    have hlt : C.stopTime κ B sample < t := lt_of_not_ge hnot
+    rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+    · have hh := h.radius_ge_before_distance_stop hε hsmall
+        (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
+        (by rw [ht]; exact ENNReal.coe_le_coe.2 hlt.le)
+        (C.source_live κ B sample h hz hu hc hb0)
+      change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
+      rw [hr] at hh
+      exact (not_lt_of_ge hh) hρ
+    · have hh := hm (C.stopTime κ B sample) ⟨(C.bounds κ B sample).1,hlt.le⟩
+      rw [hs] at hh
+      exact (not_lt_of_ge hh) hn
+  have he : C.physicalTime κ B γ ε sample = t := by
+    rw [Cutoff.physicalTime,ht,PhysicalStop.cap_coe htc]
+  have hp : codimension κ = 1-κ/8 := by unfold codimension dimension; ring
+  rw [physicalStoppedDensity,ht,ite_eq_right ENNReal.coe_ne_top,ENNReal.coe_toReal,hp,
+    C.physicalValue_eq_source κ B γ g f sample h hz hu hc hb0 ε,he]
+
+lemma tendsto_physicalValue_killed (hκ : 0 < κ ∧ κ < 8) :
+    Tendsto (fun n : ℕ => ENNReal.ofReal ((cutoffs hz u hu.le n).physicalValue κ B γ ε sample)) atTop
+      (𝓝 (physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z ε)) := by
+  by_cases he : firstLiveDistanceStop (γ sample) z ε = ⊤
+  · rw [physicalStoppedDensity,he,ite_eq_left rfl]
+    have ha : 0 < 8/κ-1 := sub_pos.2 ((lt_div_iff₀ hκ.1).2 (by simpa using hκ.2))
+    have hlim := (shrinkingAngle_tendsto z).rpow_const_nhds_zero ha
+    have hlim' : Tendsto (fun n : ℕ => ((2*z.im)*Real.exp (-u))^(-(1-κ/8)) *
+        (shrinkingAngle z n)^(8/κ-1)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul hlim
+    have hl : Tendsto (fun n : ℕ => (cutoffs hz u hu.le n).physicalValue κ B γ ε sample) atTop (𝓝 0) := by
+      apply squeeze_zero (fun n => (cutoffs hz u hu.le n).physicalValue_nonneg κ B γ ε sample) _ hlim'
+      intro n
+      let C := cutoffs hz u hu.le n
+      have ht : C.physicalTime κ B γ ε sample = C.stopTime κ B sample := by
+        simp only [Cutoff.physicalTime,PhysicalStop.cap,he,ite_true]
+      have hv : C.physicalValue κ B γ ε sample = C.value κ B sample := by
+        simp only [Cutoff.physicalValue,ht,Cutoff.value]
+      rw [hv]
+      have hs : loewnerAngleSine κ (fun t => B t sample) (g sample) (C.stopTime κ B sample) z = C.angle := by
+        rcases C.source_terminal κ B sample h hz hu hc hb0 with hr | hs
+        · have hh := h.radius_ge_before_distance_stop hε hsmall
+            (t := ⟨C.stopTime κ B sample,(C.bounds κ B sample).1⟩)
+            (by rw [he]; exact le_top) (C.source_live κ B sample h hz hu hc hb0)
+          change ε/4 ≤ loewnerRadius (g sample) (C.stopTime κ B sample) z at hh
+          rw [hr] at hh
+          exact False.elim ((not_lt_of_ge hh) hρ)
+        · exact hs
+      exact C.angle_exit_bound κ B sample h hz hu hc hb0 hκ hs
+    simpa only [ENNReal.ofReal_zero,Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hl
+  · have hh := eventually_physicalValue_at_hit κ B γ g f sample h hz hu hc hb0 hε hsmall hρ he
+    exact tendsto_const_nhds.congr' (hh.mono (fun _ heq => heq.symm))
+
+end SLEExactGauge.FiniteTilt
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/RetainedFirstMoment.lean b/lean/OAI/Probability/SLELocalization/RetainedFirstMoment.lean
index 16e076f40bef709e9b681e57cb981c840ff3e04e..c282e8db178680db54db62a3069de2a20cd32703 100644
--- a/lean/OAI/Probability/SLELocalization/RetainedFirstMoment.lean
+++ b/lean/OAI/Probability/SLELocalization/RetainedFirstMoment.lean
@@ -1,289 +1,289 @@
-import Mathlib
-import OAI.Probability.SLELocalization.FiniteTimeMass
-import OAI.Probability.SLELocalization.MeshExponents
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Retained First Moment estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-open FiniteClock
-
-lemma finiteCenteredPhysicalTime_uniform {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (Y s ε : ℝ) (hY : 0 < Y) (hs : 0 < s) (hε : 0 < ε) :
-    ∃ T : ℝ, 0 < T ∧ ∀ z : ℂ, z ∈ halfPlane → z.im ≤ Y →
-      s ≤ (z.im/‖z‖)^(8/κ-1) → ∀ u : ℝ, 1 ≤ u →
-      finiteCenteredMeasure κ P B γ g z u
-        {sample | ENNReal.ofReal T < firstLiveRadiusStop (γ sample) (g sample) z u} <
-        ENNReal.ofReal ε := by
-  obtain ⟨C,hC,hbound⟩ := radiusApproachBound κ hκ
-  let C' := C+s⁻¹
-  have hC' : 0 < C' := by dsimp only [C']; positivity
-  have hr : Real.exp (-(8/κ-1)/8) < 1 := Real.exp_lt_one_iff.2 (by have := exponent_pos hκ; linarith)
-  let D := C'*Real.exp (codimension κ)/(1-Real.exp (-(8/κ-1)/8))
-  have hD : 0 < D := by dsimp only [D]; positivity
-  obtain ⟨δ,hδ,hsmall⟩ := exists_small_floor (exponent_pos hκ) hD hε
-  refine ⟨Y^2/(2*δ^4),by positivity,?_⟩
-  intro z hz hzy hzs u hu
-  have hS : 0 < (z.im/‖z‖)^(8/κ-1) := hs.trans_le hzs
-  have hrad (n : ℕ) : P {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} ≤
-      ENNReal.ofReal (C'*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)) := by
-    cases n with
-    | zero =>
-      simp only [Nat.cast_zero,mul_zero,Real.exp_zero,mul_one]
-      apply (prob_le_one).trans
-      apply ENNReal.one_le_ofReal.2
-      have hmul := mul_le_mul_of_nonneg_left hzs (inv_nonneg.2 hs.le)
-      rw [inv_mul_cancel₀ hs.ne'] at hmul
-      dsimp only [C']
-      rw [add_mul]
-      linarith [mul_pos hC hS]
-    | succ n =>
-      simp only [Nat.cast_add,Nat.cast_one]
-      apply (hbound P B hB γ g f hγ hmaps z hz (n+1) (by norm_num)).trans
-      apply ENNReal.ofReal_le_ofReal
-      exact mul_le_mul_of_nonneg_right
-        (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (inv_nonneg.2 hs.le))
-          (Real.exp_pos _).le) hS.le
-  have hbad : finiteCenteredMeasure κ P B γ g z u (bad κ B γ g z δ u) ≤
-      ENNReal.ofReal (D*δ^(8/κ-1)) := by
-    apply (measure_iUnion_le _).trans
-    apply (ENNReal.tsum_le_tsum (fun n => block_bound κ hκ B hB γ g f hγ hmaps z hz C' hC' hrad δ hδ u n)).trans_eq
-    exact geometric_blocks κ C' δ hκ hC' hδ
-  have hu0 : 0 < u := by linarith
-  let Q := finiteCenteredMeasure κ P B γ g z u
-  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
-  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
-    change ∀ᵐ sample ∂P.withDensity _,_
-    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
-    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
-  have hsub : {sample | ENNReal.ofReal (Y^2/(2*δ^4)) <
-      firstLiveRadiusStop (γ sample) (g sample) z u} ≤ᵐ[Q] bad κ B γ g z δ u := by
-    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hm hc hh hx
-    by_contra hn
-    have hg : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ u →
-        finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z q ∉ Ioo 0 (threshold κ z δ q) := by
-      intro q hq hqu hbad
-      exact hn (mem_bad_of_rational κ B γ g z δ u q hq hqu hbad)
-    have ht := good_physical_clock (κ := κ) (B := fun t => B t sample) (γ := γ sample)
-      (g := g sample) (f := f sample) hκ hm hc hz hδ hu0 hh hg
-    have hy : 0 < z.im := hz
-    have htY : z.im^2/(2*δ^4) ≤ Y^2/(2*δ^4) :=
-      div_le_div_of_nonneg_right (sq_le_sq₀ hy.le hY.le |>.2 hzy) (by positivity)
-    exact not_lt_of_ge (ht.trans (ENNReal.ofReal_le_ofReal htY)) hx
-  exact (measure_mono_ae hsub).trans_lt
-    (hbad.trans_lt (ENNReal.ofReal_lt_ofReal_iff hε |>.2 hsmall))
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge
-
-lemma uniformFiniteTimePhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (Y s : ℝ) (hY : 0 < Y) (hs : 0 < s) :
-    ∃ T : ℝ, 0 < T ∧ ∀ z : ℂ, z ∈ halfPlane → z.im ≤ Y →
-      s ≤ (z.im/‖z‖)^(8/κ-1) → ∀ e : ℝ, 0 < e → e < z.im →
-      ENNReal.ofReal (((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))/2) ≤
-        ∫⁻ sample in {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T},
-          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P := by
-  classical
-  obtain ⟨T,hT,hclock⟩ := finiteCenteredPhysicalTime_uniform κ hκ B hB γ g f hγ hmaps
-    Y s (1/2) hY hs (by norm_num)
-  refine ⟨T,hT,?_⟩
-  intro z hz hzy hzs e he hes
-  let m := (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)
-  let M := ENNReal.ofReal m
-  have hm : 0 < m := localizationInitialGreen_pos κ hz
-  have hM0 : M ≠ 0 := (ENNReal.ofReal_pos.2 hm).ne'
-  have hMtop : M ≠ ⊤ := ENNReal.ofReal_ne_top
-  let S : ℝ≥0 := ⟨T,hT.le⟩
-  have hS : (S : ℝ≥0∞) = ENNReal.ofReal T := by
-    change (S : ℝ≥0∞) = ENNReal.ofReal (S : ℝ)
-    exact ENNReal.ofReal_coe_nnreal.symm
-  let E := PhysicalPast.avoids κ B z e S
-  let A := {sample | firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞)}
-  let D := fun sample => physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e
-  have hE : NullMeasurableSet E P := PhysicalPast.avoids_nullMeasurable κ hB z e S
-  have hA : NullMeasurableSet A P := physical_early_nullMeasurable κ hκ B hB γ g f hmaps z hz e he hes S
-  have hD : AEMeasurable D P := physicalStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz e he hes
-  obtain ⟨u,hu,hr⟩ : ∃ u : ℝ, 1 < u ∧ (2*z.im)*Real.exp (-u) < e/4 := by
-    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
-      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
-    exact ((eventually_gt_atTop 1).and (hl.eventually (gt_mem_nhds (by linarith : 0 < e/4)))).exists
-  have hu0 : 0 < u := by linarith
-  let Q := finiteCenteredMeasure κ P B γ g z u
-  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
-  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
-    change ∀ᵐ sample ∂P.withDensity _,_
-    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
-    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
-  have hsub : E ≤ᵐ[Q] {sample | (S:ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u} := by
-    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hmaps hc hhit hmem
-    have hpr := physical_before_deep_radius hmaps
-      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) he hes hr hhit
-    have hfp : firstLiveDistanceStop (γ sample) z e ≠ (⊤ : ℝ≥0∞) := ne_top_of_lt hpr
-    exact ((physical_avoidance_at_finite_hit κ B sample hmaps hc z e S hfp).1 hmem).trans hpr
-  have hQE : Q E ≤ ENNReal.ofReal (1/2) := by
-    apply (measure_mono_ae hsub).trans
-    simpa only [hS] using (hclock z hz hzy hzs u hu.le).le
-  have hid : (∫⁻ sample in E, D sample ∂P) = M*Q E := by
-    rw [physical_radial_avoidance_identity κ hκ B hB γ g f hmaps z hz u hu0 e he hes hr S]
-    rw [show Q E = ∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
-        (γ sample) (g sample) z u / M ∂P from withDensity_apply₀ _ hE]
-    simp_rw [ENNReal.div_eq_inv_mul]
-    rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hM0),← mul_assoc,ENNReal.mul_inv_cancel hM0 hMtop,one_mul]
-  have hhalf : M*ENNReal.ofReal (1/2) = ENNReal.ofReal (m/2) := by
-    rw [← ENNReal.ofReal_mul hm.le]
-    congr 1
-    ring
-  have htail : (∫⁻ sample in E, D sample ∂P) ≤ ENNReal.ofReal (m/2) := by
-    rw [hid,← hhalf]
-    exact mul_le_mul' le_rfl hQE
-  have hi : Eᶜ.indicator D =ᵐ[P] A.indicator D := by
-    filter_upwards [hmaps,hB.cont] with sample hm hc
-    by_cases hh : firstLiveDistanceStop (γ sample) z e = (⊤ : ℝ≥0∞)
-    · have hd : D sample = 0 := by simp only [D,physicalStoppedDensity,hh,ite_true]
-      simp only [Set.indicator_apply,hd,ite_self]
-    · have hv := physical_avoidance_at_finite_hit κ B sample hm hc z e S hh
-      have hae : sample ∈ Eᶜ ↔ sample ∈ A := by
-        simpa only [E,A,mem_compl_iff,mem_ofPred_eq,not_lt] using hv.not
-      simp only [Set.indicator_apply,hae]
-  have hall : (∫⁻ sample in E, D sample ∂P)+(∫⁻ sample in A, D sample ∂P) = M := by
-    rw [← lintegral_indicator₀ hA,← lintegral_congr_ae hi,← lintegral_indicator₀ hE,
-      ← lintegral_add_left' (hD.indicator₀ hE)]
-    have hp : (fun sample => E.indicator D sample+Eᶜ.indicator D sample) = D := by
-      funext sample
-      by_cases hs : sample∈E <;> simp [hs]
-    rw [hp]
-    exact physicalDistanceStoppedDensity κ hκ B hB γ g f hγ hmaps z hz e he hes
-  have hdouble : M = ENNReal.ofReal (m/2)+ENNReal.ofReal (m/2) := by
-    change ENNReal.ofReal m = _
-    rw [← ENNReal.ofReal_add (by positivity : 0 ≤ m/2) (by positivity : 0 ≤ m/2)]
-    congr 1
-    ring
-  have hlow : ENNReal.ofReal (m/2) ≤ ∫⁻ sample in A, D sample ∂P := by
-    apply ENNReal.le_of_add_le_add_left ENNReal.ofReal_ne_top
-    rw [← hdouble,← hall]
-    exact add_le_add htail le_rfl
-  simpa only [A,D,m,hS] using hlow
-
-lemma compact_initial_bounds (κ : ℝ) {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty)
-    (hhalf : K ⊆ halfPlane) :
-    ∃ e₀ Y s m : ℝ, 0 < e₀ ∧ 0 < Y ∧ 0 < s ∧ 0 < m ∧ ∀ z ∈ K,
-      e₀ ≤ z.im ∧ z.im ≤ Y ∧ s ≤ (z.im/‖z‖)^(8/κ-1) ∧
-        m ≤ ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))/2 := by
-  have hy (z : ℂ) (hz : z ∈ K) : 0 < z.im := hhalf hz
-  have hn (z : ℂ) (hz : z ∈ K) : ‖z‖ ≠ 0 := norm_ne_zero_iff.2 (by
-    intro he
-    simpa only [he,Complex.zero_im,lt_self_iff_false] using hy z hz)
-  have hsin : ContinuousOn (fun z : ℂ => z.im/‖z‖) K :=
-    Complex.continuous_im.continuousOn.div continuous_norm.continuousOn hn
-  have hsp := hsin.rpow_const (p := 8/κ-1) (fun z hz => Or.inl (FiniteTilt.initial_sine_pos (hhalf hz)).ne')
-  have hr : ContinuousOn (fun z : ℂ => (2*z.im)^(-codimension κ)) K :=
-    (continuous_const.mul Complex.continuous_im).continuousOn.rpow_const
-      (fun z hz => Or.inl (mul_pos (by norm_num) (hy z hz)).ne')
-  have hm := (hr.mul hsp).div_const 2
-  obtain ⟨z₀,hz₀,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn
-  obtain ⟨z₁,hz₁,hmax⟩ := hK.exists_isMaxOn hne Complex.continuous_im.continuousOn
-  obtain ⟨z₂,hz₂,hsmin⟩ := hK.exists_isMinOn hne hsp
-  obtain ⟨z₃,hz₃,hmmin⟩ := hK.exists_isMinOn hne hm
-  refine ⟨z₀.im,z₁.im,(z₂.im/‖z₂‖)^(8/κ-1),
-    ((2*z₃.im)^(-codimension κ)*(z₃.im/‖z₃‖)^(8/κ-1))/2,
-    hy z₀ hz₀,hy z₁ hz₁,Real.rpow_pos_of_pos (FiniteTilt.initial_sine_pos (hhalf hz₂)) _,
-    div_pos (localizationInitialGreen_pos κ (hhalf hz₃)) (by norm_num),?_⟩
-  intro z hz
-  exact ⟨hmin hz,hmax hz,hsmin hz,hmmin hz⟩
-
-theorem retainedPositiveFirstMoment {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
-    (c : ℂ) (r : ℝ) (hr : 0 < r) (hK : Metric.closedBall c r ⊆ halfPlane) :
-    ∃ T e₀ m₀ : ℝ, 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
-      ENNReal.ofReal m₀ ≤ ∫⁻ sample, ∫⁻ z in Metric.closedBall c r,
-        (if firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T then
-          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e else 0)
-        ∂volume ∂P := by
-  classical
-  let K := Metric.closedBall c r
-  have hKm : MeasurableSet K := Metric.isClosed_closedBall.measurableSet
-  obtain ⟨e₀,Y,s,m,he₀,hY,hs,hm,hbd⟩ := compact_initial_bounds κ
-    (isCompact_closedBall c r) ⟨c,Metric.mem_closedBall_self hr.le⟩ hK
-  obtain ⟨T,hT,hfirst⟩ := uniformFiniteTimePhysicalDensity κ hκ B hB γ g f hγ hmaps Y s hY hs
-  have hvol : 0 < (volume K).toReal := ENNReal.toReal_pos
-    (Metric.measure_closedBall_pos volume c hr).ne' measure_closedBall_lt_top.ne
-  refine ⟨T,e₀,m*(volume K).toReal,hT,he₀,mul_pos hm hvol,?_⟩
-  intro e he hee
-  obtain ⟨W,hW,hWe⟩ := jointFiniteTimePhysicalDensity κ hκ B hB γ g f hγ hmaps T hT e he
-  have hwfirst (z : ℂ) (hz : z ∈ K) : ENNReal.ofReal m ≤ ∫⁻ sample, W (sample,z) ∂P := by
-    have hes : e < z.im := hee.trans_le (hbd z hz).1
-    let S : ℝ≥0 := ⟨T,hT.le⟩
-    have hS : (S : ℝ≥0∞) = ENNReal.ofReal T := ENNReal.ofReal_coe_nnreal.symm
-    have hA : NullMeasurableSet {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T} P := by
-      simpa only [hS] using
-        physical_early_nullMeasurable κ hκ B hB γ g f hmaps z (hK hz) e he hes S
-    have hideq : (∫⁻ sample, W (sample,z) ∂P) =
-        ∫⁻ sample in {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T},
-          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P := by
-      rw [← lintegral_indicator₀ hA]
-      apply lintegral_congr_ae
-      filter_upwards [hWe] with sample hw
-      simpa only [Set.indicator_apply,Set.mem_ofPred_eq] using hw z
-    rw [hideq]
-    exact (ENNReal.ofReal_le_ofReal (hbd z hz).2.2.2).trans
-      (hfirst z (hK hz) (hbd z hz).2.1 (hbd z hz).2.2.1 e he hes)
-  have hmass : ENNReal.ofReal (m*(volume K).toReal) ≤
-      ∫⁻ z in K, ∫⁻ sample, W (sample,z) ∂P ∂volume := by
-    calc
-      _ = ∫⁻ z in K, ENNReal.ofReal m ∂volume := by
-        rw [lintegral_const,Measure.restrict_apply_univ,ENNReal.ofReal_mul hm.le,
-          ENNReal.ofReal_toReal measure_closedBall_lt_top.ne]
-      _ ≤ _ := lintegral_mono_ae (ae_restrict_of_forall_mem hKm hwfirst)
-  have hswap : (∫⁻ z in K, ∫⁻ sample, W (sample,z) ∂P ∂volume) =
-      ∫⁻ sample, ∫⁻ z in K, W (sample,z) ∂volume ∂P := by
-    exact (lintegral_lintegral_swap (μ := P) (ν := volume.restrict K)
-      (f := fun sample z => W (sample,z)) hW.aemeasurable).symm
-  rw [hswap] at hmass
-  apply hmass.trans_eq
-  apply lintegral_congr_ae
-  filter_upwards [hWe] with sample hw
-  exact lintegral_congr hw
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.FiniteTimeMass
+import OAI.Probability.SLELocalization.MeshExponents
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Retained First Moment estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+open FiniteClock
+
+lemma finiteCenteredPhysicalTime_uniform {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (Y s ε : ℝ) (hY : 0 < Y) (hs : 0 < s) (hε : 0 < ε) :
+    ∃ T : ℝ, 0 < T ∧ ∀ z : ℂ, z ∈ halfPlane → z.im ≤ Y →
+      s ≤ (z.im/‖z‖)^(8/κ-1) → ∀ u : ℝ, 1 ≤ u →
+      finiteCenteredMeasure κ P B γ g z u
+        {sample | ENNReal.ofReal T < firstLiveRadiusStop (γ sample) (g sample) z u} <
+        ENNReal.ofReal ε := by
+  obtain ⟨C,hC,hbound⟩ := radiusApproachBound κ hκ
+  let C' := C+s⁻¹
+  have hC' : 0 < C' := by dsimp only [C']; positivity
+  have hr : Real.exp (-(8/κ-1)/8) < 1 := Real.exp_lt_one_iff.2 (by have := exponent_pos hκ; linarith)
+  let D := C'*Real.exp (codimension κ)/(1-Real.exp (-(8/κ-1)/8))
+  have hD : 0 < D := by dsimp only [D]; positivity
+  obtain ⟨δ,hδ,hsmall⟩ := exists_small_floor (exponent_pos hκ) hD hε
+  refine ⟨Y^2/(2*δ^4),by positivity,?_⟩
+  intro z hz hzy hzs u hu
+  have hS : 0 < (z.im/‖z‖)^(8/κ-1) := hs.trans_le hzs
+  have hrad (n : ℕ) : P {sample | firstLiveRadiusStop (γ sample) (g sample) z n ≠ (⊤ : ℝ≥0∞)} ≤
+      ENNReal.ofReal (C'*Real.exp (-codimension κ*n)*(z.im/‖z‖)^(8/κ-1)) := by
+    cases n with
+    | zero =>
+      simp only [Nat.cast_zero,mul_zero,Real.exp_zero,mul_one]
+      apply (prob_le_one).trans
+      apply ENNReal.one_le_ofReal.2
+      have hmul := mul_le_mul_of_nonneg_left hzs (inv_nonneg.2 hs.le)
+      rw [inv_mul_cancel₀ hs.ne'] at hmul
+      dsimp only [C']
+      rw [add_mul]
+      linarith [mul_pos hC hS]
+    | succ n =>
+      simp only [Nat.cast_add,Nat.cast_one]
+      apply (hbound P B hB γ g f hγ hmaps z hz (n+1) (by norm_num)).trans
+      apply ENNReal.ofReal_le_ofReal
+      exact mul_le_mul_of_nonneg_right
+        (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (inv_nonneg.2 hs.le))
+          (Real.exp_pos _).le) hS.le
+  have hbad : finiteCenteredMeasure κ P B γ g z u (bad κ B γ g z δ u) ≤
+      ENNReal.ofReal (D*δ^(8/κ-1)) := by
+    apply (measure_iUnion_le _).trans
+    apply (ENNReal.tsum_le_tsum (fun n => block_bound κ hκ B hB γ g f hγ hmaps z hz C' hC' hrad δ hδ u n)).trans_eq
+    exact geometric_blocks κ C' δ hκ hC' hδ
+  have hu0 : 0 < u := by linarith
+  let Q := finiteCenteredMeasure κ P B γ g z u
+  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
+  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
+    change ∀ᵐ sample ∂P.withDensity _,_
+    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
+    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
+  have hsub : {sample | ENNReal.ofReal (Y^2/(2*δ^4)) <
+      firstLiveRadiusStop (γ sample) (g sample) z u} ≤ᵐ[Q] bad κ B γ g z δ u := by
+    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hm hc hh hx
+    by_contra hn
+    have hg : ∀ q : ℚ, 0 < (q:ℝ) → (q:ℝ) ≤ u →
+        finiteStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z q ∉ Ioo 0 (threshold κ z δ q) := by
+      intro q hq hqu hbad
+      exact hn (mem_bad_of_rational κ B γ g z δ u q hq hqu hbad)
+    have ht := good_physical_clock (κ := κ) (B := fun t => B t sample) (γ := γ sample)
+      (g := g sample) (f := f sample) hκ hm hc hz hδ hu0 hh hg
+    have hy : 0 < z.im := hz
+    have htY : z.im^2/(2*δ^4) ≤ Y^2/(2*δ^4) :=
+      div_le_div_of_nonneg_right (sq_le_sq₀ hy.le hY.le |>.2 hzy) (by positivity)
+    exact not_lt_of_ge (ht.trans (ENNReal.ofReal_le_ofReal htY)) hx
+  exact (measure_mono_ae hsub).trans_lt
+    (hbad.trans_lt (ENNReal.ofReal_lt_ofReal_iff hε |>.2 hsmall))
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge
+
+lemma uniformFiniteTimePhysicalDensity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (Y s : ℝ) (hY : 0 < Y) (hs : 0 < s) :
+    ∃ T : ℝ, 0 < T ∧ ∀ z : ℂ, z ∈ halfPlane → z.im ≤ Y →
+      s ≤ (z.im/‖z‖)^(8/κ-1) → ∀ e : ℝ, 0 < e → e < z.im →
+      ENNReal.ofReal (((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))/2) ≤
+        ∫⁻ sample in {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T},
+          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P := by
+  classical
+  obtain ⟨T,hT,hclock⟩ := finiteCenteredPhysicalTime_uniform κ hκ B hB γ g f hγ hmaps
+    Y s (1/2) hY hs (by norm_num)
+  refine ⟨T,hT,?_⟩
+  intro z hz hzy hzs e he hes
+  let m := (2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1)
+  let M := ENNReal.ofReal m
+  have hm : 0 < m := localizationInitialGreen_pos κ hz
+  have hM0 : M ≠ 0 := (ENNReal.ofReal_pos.2 hm).ne'
+  have hMtop : M ≠ ⊤ := ENNReal.ofReal_ne_top
+  let S : ℝ≥0 := ⟨T,hT.le⟩
+  have hS : (S : ℝ≥0∞) = ENNReal.ofReal T := by
+    change (S : ℝ≥0∞) = ENNReal.ofReal (S : ℝ)
+    exact ENNReal.ofReal_coe_nnreal.symm
+  let E := PhysicalPast.avoids κ B z e S
+  let A := {sample | firstLiveDistanceStop (γ sample) z e ≤ (S:ℝ≥0∞)}
+  let D := fun sample => physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e
+  have hE : NullMeasurableSet E P := PhysicalPast.avoids_nullMeasurable κ hB z e S
+  have hA : NullMeasurableSet A P := physical_early_nullMeasurable κ hκ B hB γ g f hmaps z hz e he hes S
+  have hD : AEMeasurable D P := physicalStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz e he hes
+  obtain ⟨u,hu,hr⟩ : ∃ u : ℝ, 1 < u ∧ (2*z.im)*Real.exp (-u) < e/4 := by
+    have hl : Tendsto (fun u : ℝ => (2*z.im)*Real.exp (-u)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using tendsto_const_nhds.mul Real.tendsto_exp_neg_atTop_nhds_zero
+    exact ((eventually_gt_atTop 1).and (hl.eventually (gt_mem_nhds (by linarith : 0 < e/4)))).exists
+  have hu0 : 0 < u := by linarith
+  let Q := finiteCenteredMeasure κ P B γ g z u
+  have hQP : Q ≪ P := withDensity_absolutelyContinuous _ _
+  have hhit : ∀ᵐ sample ∂Q, firstLiveRadiusStop (γ sample) (g sample) z u ≠ (⊤ : ℝ≥0∞) := by
+    change ∀ᵐ sample ∂P.withDensity _,_
+    rw [ae_withDensity_iff' ((finiteStoppedDensity_aemeasurable κ hκ B hB γ g f hmaps z hz u hu0).div_const _)]
+    exact Filter.Eventually.of_forall (fun sample hn ht => hn (by simp only [finiteStoppedDensity,ht,ite_true,ENNReal.zero_div]))
+  have hsub : E ≤ᵐ[Q] {sample | (S:ℝ≥0∞) < firstLiveRadiusStop (γ sample) (g sample) z u} := by
+    filter_upwards [hQP.ae_le hmaps,hQP.ae_le hB.cont,hhit] with sample hmaps hc hhit hmem
+    have hpr := physical_before_deep_radius hmaps
+      ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn) he hes hr hhit
+    have hfp : firstLiveDistanceStop (γ sample) z e ≠ (⊤ : ℝ≥0∞) := ne_top_of_lt hpr
+    exact ((physical_avoidance_at_finite_hit κ B sample hmaps hc z e S hfp).1 hmem).trans hpr
+  have hQE : Q E ≤ ENNReal.ofReal (1/2) := by
+    apply (measure_mono_ae hsub).trans
+    simpa only [hS] using (hclock z hz hzy hzs u hu.le).le
+  have hid : (∫⁻ sample in E, D sample ∂P) = M*Q E := by
+    rw [physical_radial_avoidance_identity κ hκ B hB γ g f hmaps z hz u hu0 e he hes hr S]
+    rw [show Q E = ∫⁻ sample in E, finiteStoppedDensity κ (fun t => B t sample)
+        (γ sample) (g sample) z u / M ∂P from withDensity_apply₀ _ hE]
+    simp_rw [ENNReal.div_eq_inv_mul]
+    rw [lintegral_const_mul' _ _ (ENNReal.inv_ne_top.2 hM0),← mul_assoc,ENNReal.mul_inv_cancel hM0 hMtop,one_mul]
+  have hhalf : M*ENNReal.ofReal (1/2) = ENNReal.ofReal (m/2) := by
+    rw [← ENNReal.ofReal_mul hm.le]
+    congr 1
+    ring
+  have htail : (∫⁻ sample in E, D sample ∂P) ≤ ENNReal.ofReal (m/2) := by
+    rw [hid,← hhalf]
+    exact mul_le_mul' le_rfl hQE
+  have hi : Eᶜ.indicator D =ᵐ[P] A.indicator D := by
+    filter_upwards [hmaps,hB.cont] with sample hm hc
+    by_cases hh : firstLiveDistanceStop (γ sample) z e = (⊤ : ℝ≥0∞)
+    · have hd : D sample = 0 := by simp only [D,physicalStoppedDensity,hh,ite_true]
+      simp only [Set.indicator_apply,hd,ite_self]
+    · have hv := physical_avoidance_at_finite_hit κ B sample hm hc z e S hh
+      have hae : sample ∈ Eᶜ ↔ sample ∈ A := by
+        simpa only [E,A,mem_compl_iff,mem_ofPred_eq,not_lt] using hv.not
+      simp only [Set.indicator_apply,hae]
+  have hall : (∫⁻ sample in E, D sample ∂P)+(∫⁻ sample in A, D sample ∂P) = M := by
+    rw [← lintegral_indicator₀ hA,← lintegral_congr_ae hi,← lintegral_indicator₀ hE,
+      ← lintegral_add_left' (hD.indicator₀ hE)]
+    have hp : (fun sample => E.indicator D sample+Eᶜ.indicator D sample) = D := by
+      funext sample
+      by_cases hs : sample∈E <;> simp [hs]
+    rw [hp]
+    exact physicalDistanceStoppedDensity κ hκ B hB γ g f hγ hmaps z hz e he hes
+  have hdouble : M = ENNReal.ofReal (m/2)+ENNReal.ofReal (m/2) := by
+    change ENNReal.ofReal m = _
+    rw [← ENNReal.ofReal_add (by positivity : 0 ≤ m/2) (by positivity : 0 ≤ m/2)]
+    congr 1
+    ring
+  have hlow : ENNReal.ofReal (m/2) ≤ ∫⁻ sample in A, D sample ∂P := by
+    apply ENNReal.le_of_add_le_add_left ENNReal.ofReal_ne_top
+    rw [← hdouble,← hall]
+    exact add_le_add htail le_rfl
+  simpa only [A,D,m,hS] using hlow
+
+lemma compact_initial_bounds (κ : ℝ) {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty)
+    (hhalf : K ⊆ halfPlane) :
+    ∃ e₀ Y s m : ℝ, 0 < e₀ ∧ 0 < Y ∧ 0 < s ∧ 0 < m ∧ ∀ z ∈ K,
+      e₀ ≤ z.im ∧ z.im ≤ Y ∧ s ≤ (z.im/‖z‖)^(8/κ-1) ∧
+        m ≤ ((2*z.im)^(-codimension κ)*(z.im/‖z‖)^(8/κ-1))/2 := by
+  have hy (z : ℂ) (hz : z ∈ K) : 0 < z.im := hhalf hz
+  have hn (z : ℂ) (hz : z ∈ K) : ‖z‖ ≠ 0 := norm_ne_zero_iff.2 (by
+    intro he
+    simpa only [he,Complex.zero_im,lt_self_iff_false] using hy z hz)
+  have hsin : ContinuousOn (fun z : ℂ => z.im/‖z‖) K :=
+    Complex.continuous_im.continuousOn.div continuous_norm.continuousOn hn
+  have hsp := hsin.rpow_const (p := 8/κ-1) (fun z hz => Or.inl (FiniteTilt.initial_sine_pos (hhalf hz)).ne')
+  have hr : ContinuousOn (fun z : ℂ => (2*z.im)^(-codimension κ)) K :=
+    (continuous_const.mul Complex.continuous_im).continuousOn.rpow_const
+      (fun z hz => Or.inl (mul_pos (by norm_num) (hy z hz)).ne')
+  have hm := (hr.mul hsp).div_const 2
+  obtain ⟨z₀,hz₀,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn
+  obtain ⟨z₁,hz₁,hmax⟩ := hK.exists_isMaxOn hne Complex.continuous_im.continuousOn
+  obtain ⟨z₂,hz₂,hsmin⟩ := hK.exists_isMinOn hne hsp
+  obtain ⟨z₃,hz₃,hmmin⟩ := hK.exists_isMinOn hne hm
+  refine ⟨z₀.im,z₁.im,(z₂.im/‖z₂‖)^(8/κ-1),
+    ((2*z₃.im)^(-codimension κ)*(z₃.im/‖z₃‖)^(8/κ-1))/2,
+    hy z₀ hz₀,hy z₁ hz₁,Real.rpow_pos_of_pos (FiniteTilt.initial_sine_pos (hhalf hz₂)) _,
+    div_pos (localizationInitialGreen_pos κ (hhalf hz₃)) (by norm_num),?_⟩
+  intro z hz
+  exact ⟨hmin hz,hmax hz,hsmin hz,hmmin hz⟩
+
+theorem retainedPositiveFirstMoment {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hγ : ∀ t, 0 ≤ t → AEMeasurable (fun sample => γ sample t) P)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ*B t.toNNReal sample) (γ sample) (g sample) (f sample))
+    (c : ℂ) (r : ℝ) (hr : 0 < r) (hK : Metric.closedBall c r ⊆ halfPlane) :
+    ∃ T e₀ m₀ : ℝ, 0 < T ∧ 0 < e₀ ∧ 0 < m₀ ∧ ∀ e : ℝ, 0 < e → e < e₀ →
+      ENNReal.ofReal m₀ ≤ ∫⁻ sample, ∫⁻ z in Metric.closedBall c r,
+        (if firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T then
+          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e else 0)
+        ∂volume ∂P := by
+  classical
+  let K := Metric.closedBall c r
+  have hKm : MeasurableSet K := Metric.isClosed_closedBall.measurableSet
+  obtain ⟨e₀,Y,s,m,he₀,hY,hs,hm,hbd⟩ := compact_initial_bounds κ
+    (isCompact_closedBall c r) ⟨c,Metric.mem_closedBall_self hr.le⟩ hK
+  obtain ⟨T,hT,hfirst⟩ := uniformFiniteTimePhysicalDensity κ hκ B hB γ g f hγ hmaps Y s hY hs
+  have hvol : 0 < (volume K).toReal := ENNReal.toReal_pos
+    (Metric.measure_closedBall_pos volume c hr).ne' measure_closedBall_lt_top.ne
+  refine ⟨T,e₀,m*(volume K).toReal,hT,he₀,mul_pos hm hvol,?_⟩
+  intro e he hee
+  obtain ⟨W,hW,hWe⟩ := jointFiniteTimePhysicalDensity κ hκ B hB γ g f hγ hmaps T hT e he
+  have hwfirst (z : ℂ) (hz : z ∈ K) : ENNReal.ofReal m ≤ ∫⁻ sample, W (sample,z) ∂P := by
+    have hes : e < z.im := hee.trans_le (hbd z hz).1
+    let S : ℝ≥0 := ⟨T,hT.le⟩
+    have hS : (S : ℝ≥0∞) = ENNReal.ofReal T := ENNReal.ofReal_coe_nnreal.symm
+    have hA : NullMeasurableSet {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T} P := by
+      simpa only [hS] using
+        physical_early_nullMeasurable κ hκ B hB γ g f hmaps z (hK hz) e he hes S
+    have hideq : (∫⁻ sample, W (sample,z) ∂P) =
+        ∫⁻ sample in {sample | firstLiveDistanceStop (γ sample) z e ≤ ENNReal.ofReal T},
+          physicalStoppedDensity κ (fun t => B t sample) (γ sample) (g sample) z e ∂P := by
+      rw [← lintegral_indicator₀ hA]
+      apply lintegral_congr_ae
+      filter_upwards [hWe] with sample hw
+      simpa only [Set.indicator_apply,Set.mem_ofPred_eq] using hw z
+    rw [hideq]
+    exact (ENNReal.ofReal_le_ofReal (hbd z hz).2.2.2).trans
+      (hfirst z (hK hz) (hbd z hz).2.1 (hbd z hz).2.2.1 e he hes)
+  have hmass : ENNReal.ofReal (m*(volume K).toReal) ≤
+      ∫⁻ z in K, ∫⁻ sample, W (sample,z) ∂P ∂volume := by
+    calc
+      _ = ∫⁻ z in K, ENNReal.ofReal m ∂volume := by
+        rw [lintegral_const,Measure.restrict_apply_univ,ENNReal.ofReal_mul hm.le,
+          ENNReal.ofReal_toReal measure_closedBall_lt_top.ne]
+      _ ≤ _ := lintegral_mono_ae (ae_restrict_of_forall_mem hKm hwfirst)
+  have hswap : (∫⁻ z in K, ∫⁻ sample, W (sample,z) ∂P ∂volume) =
+      ∫⁻ sample, ∫⁻ z in K, W (sample,z) ∂volume ∂P := by
+    exact (lintegral_lintegral_swap (μ := P) (ν := volume.restrict K)
+      (f := fun sample z => W (sample,z)) hW.aemeasurable).symm
+  rw [hswap] at hmass
+  apply hmass.trans_eq
+  apply lintegral_congr_ae
+  filter_upwards [hWe] with sample hw
+  exact lintegral_congr hw
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/ReverseFlow.lean b/lean/OAI/Probability/SLELocalization/ReverseFlow.lean
index 260dec6a90ad642501bf2d1b1cc092ea134e0859..fbf17377984490582c0ae82a8f0e27c91c9d8d8c 100644
--- a/lean/OAI/Probability/SLELocalization/ReverseFlow.lean
+++ b/lean/OAI/Probability/SLELocalization/ReverseFlow.lean
@@ -1,1014 +1,1014 @@
-import Mathlib
-import OAI.Probability.SLELocalization.BoundaryDiffusion
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Reverse Loewner estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ReverseLoewner
-open LoewnerCutoff
-
-lemma continuous_velocity {ε : ℝ} (hε : 0 < ε) :
-    Continuous (fun p : ℝ × ℂ => velocity ε p.1 p.2) :=
-  (LoewnerCutoff.continuous_velocity hε).neg
-
-lemma norm_velocity_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (w : ℂ) :
-    ‖velocity ε u w‖ ≤ 2 / ε := by
-  simpa only [velocity,norm_neg] using LoewnerCutoff.norm_velocity_le hε u w
-
-lemma exists_solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
-    ∃ H : ℝ → ℂ, H 0 = z ∧ ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t := by
-  let L : ℝ≥0 := ⟨2 / ε, by positivity⟩
-  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
-  let a : ℝ≥0 := ⟨(2 / ε) * T, mul_nonneg (by positivity) hT⟩
-  have hp : IsPicardLindelof (fun t => velocity ε (U t))
-      ⟨0, ⟨le_rfl, hT⟩⟩ z a 0 L K := by
-    constructor
-    · exact fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith
-    · intro w _
-      have hc : ContinuousOn (fun t : ℝ => (U t, w)) (Icc 0 T) :=
-        hU.prodMk continuousOn_const
-      simpa only [Function.comp_def] using
-        (continuous_velocity hε).comp_continuousOn hc
-    · exact fun t _ w _ => norm_velocity_le hε (U t) w
-    · dsimp [L, a]
-      simp only [sub_zero, sub_self, max_eq_left hT]
-      rfl
-  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
-
-end SLEExactGauge.ReverseLoewner
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ReverseLoewner
-open LoewnerCutoff
-
-lemma norm_velocity_driver_sub_le {ε : ℝ} (hε : 0 < ε) (u v : ℝ) (z : ℂ) :
-    ‖velocity ε u z - velocity ε v z‖ ≤ (4 / ε^2) * |u-v| := by
-  simpa only [velocity, neg_sub_neg, norm_sub_rev] using
-    LoewnerCutoff.norm_velocity_driver_sub_le hε u v z
-
-lemma derivWithin_right {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ico 0 T) :
-    HasDerivWithinAt H (velocity ε (U t) (H t)) (Ici t) t :=
-  (hH t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
-
-lemma derivWithin_left {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ioc 0 T) :
-    HasDerivWithinAt H (velocity ε (U t) (H t)) (Iic t) t :=
-  (hH t ⟨ht.1.le, ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
-
-lemma solution_eq {ε T : ℝ} (hε : 0 < ε) {U : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (U t) (G t)) (Icc 0 T) t)
-    (h0 : H 0 = G 0) : EqOn H G (Icc 0 T) := by
-  apply ODE_solution_unique_of_mem_Icc_right
-    (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith)
-    (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_right hH ht)
-    (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hG)
-    (fun _ ht => derivWithin_right hG ht) (fun _ _ => mem_univ _) h0
-
-lemma solution_dist_le {ε T D : ℝ} (hε : 0 < ε) (hD : 0 ≤ D)
-    {U V : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
-    (h0 : H 0 = G 0) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    dist (H t) (G t) ≤ D * Real.exp ((4 / ε^2) * T) := by
-  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
-  have hK : (0 : ℝ) < K := by change 0 < 4 / ε^2; positivity
-  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
-      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K : ℝ) * D := by
-    rw [dist_eq_norm]
-    calc
-      _ ≤ (4 / ε^2) * |V s-U s| := norm_velocity_driver_sub_le hε _ _ _
-      _ ≤ (K : ℝ) * D := by
-        rw [abs_sub_comm]
-        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1, hs.2.le⟩) hK.le
-  have hh := dist_le_of_approx_trajectories_ODE
-    (fun t => velocity_lipschitz hε (U t))
-    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
-    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
-    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
-    hb (le_of_eq (dist_eq_zero.2 h0)) t ht
-  have he : gronwallBound 0 K (0 + (K : ℝ)*D) (t-0) =
-      D * (Real.exp ((K : ℝ)*t)-1) := by
-    simp only [gronwallBound, zero_add, sub_zero, ite_eq_right (ne_of_gt hK)]
-    field_simp
-    ring
-  rw [he] at hh
-  have hexp : Real.exp ((K : ℝ)*t) ≤ Real.exp ((K : ℝ)*T) :=
-    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
-  apply hh.trans
-  change D * (Real.exp ((4 / ε^2)*t)-1) ≤ D * Real.exp ((4 / ε^2)*T)
-  change Real.exp ((4 / ε^2)*t) ≤ Real.exp ((4 / ε^2)*T) at hexp
-  exact mul_le_mul_of_nonneg_left (by linarith) hD
-
-end SLEExactGauge.ReverseLoewner
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ReverseLoewner
-open LoewnerCutoff
-open scoped unitInterval
-
-def solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) : ℝ → ℂ :=
-  (exists_solution hε hT (extendDriver T U) (continuous_extendDriver T U).continuousOn z).choose
-
-lemma solution_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    solution hε hT U z 0 = z :=
-  (exists_solution hε hT (extendDriver T U)
-    (continuous_extendDriver T U).continuousOn z).choose_spec.1
-
-lemma solution_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (solution hε hT U z)
-      (velocity ε (extendDriver T U t) (solution hε hT U z t)) (Icc 0 T) t :=
-  (exists_solution hε hT (extendDriver T U)
-    (continuous_extendDriver T U).continuousOn z).choose_spec.2
-
-def solutionPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    C(Icc 0 T, ℂ) :=
-  ⟨fun t => solution hε hT U z t,
-    continuousOn_iff_continuous_domRestrict.1 (HasDerivWithinAt.continuousOn (solution_deriv hε hT U z))⟩
-
-lemma solutionPath_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    LipschitzWith ⟨Real.exp ((4 / ε^2) * T), (Real.exp_pos _).le⟩
-      (fun U : C(I, ℝ) => solutionPath hε hT U z) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro U V
-  apply (ContinuousMap.dist_le (mul_nonneg (Real.exp_pos _).le dist_nonneg)).2
-  intro t
-  have hUV (s : ℝ) (_hs : s ∈ Icc 0 T) :
-      |extendDriver T U s-extendDriver T V s| ≤ dist U V := by
-    exact (ContinuousMap.dist_apply_le_dist (f := U) (g := V)
-      (projIcc 0 1 zero_le_one (s/T)))
-  have hh := solution_dist_le hε dist_nonneg (solution_deriv hε hT U z)
-    (solution_deriv hε hT V z)
-    ((solution_initial hε hT U z).trans (solution_initial hε hT V z).symm)
-    hUV t.property
-  change dist (solution hε hT U z t) (solution hε hT V z t) ≤
-    Real.exp ((4 / ε^2)*T) * dist U V
-  simpa only [mul_comm] using hh
-
-end SLEExactGauge.ReverseLoewner
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ReverseLoewner
-open LoewnerCutoff
-
-def reverseCentered (U : C(I, ℝ)) : C(I, ℝ) :=
-  ⟨fun q => U (unitInterval.symm q) - U 1,
-    (U.continuous.comp unitInterval.continuous_symm).sub continuous_const⟩
-
-lemma reverseCentered_lipschitz : LipschitzWith 2 reverseCentered := by
-  apply LipschitzWith.of_dist_le_mul
-  intro U V
-  apply (ContinuousMap.dist_le (by positivity)).2
-  intro q
-  change dist (U (unitInterval.symm q)-U 1) (V (unitInterval.symm q)-V 1) ≤ (2:ℝ)*dist U V
-  calc
-    _ ≤ dist (U (unitInterval.symm q)) (V (unitInterval.symm q)) + dist (U 1) (V 1) := dist_sub_sub_le _ _ _ _
-    _ ≤ dist U V + dist U V := add_le_add (ContinuousMap.dist_apply_le_dist _) (ContinuousMap.dist_apply_le_dist _)
-    _ = _ := by ring
-
-lemma extend_reverseCentered {T : ℝ} (hT : T ≠ 0) (U : C(I, ℝ)) (s : ℝ) :
-    extendDriver T (reverseCentered U) s = extendDriver T U (T-s) - U 1 := by
-  dsimp only [extendDriver,reverseCentered,ContinuousMap.coe_mk]
-  rw [unitInterval.symm_projIcc]
-  congr 2
-  field_simp
-
-lemma extend_end {T : ℝ} (hT : T ≠ 0) (U : C(I, ℝ)) : extendDriver T U T = U 1 := by
-  simp only [extendDriver,div_self hT]
-  congr 1
-  exact Subtype.ext (by simp [projIcc])
-
-def tipApprox {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) : ℂ :=
-  solution hε hT (reverseCentered U) ((ε:ℂ)*Complex.I) T + (U 1 : ℂ)
-
-lemma continuous_tipApprox {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
-    Continuous (tipApprox hε hT) := by
-  have hc : Continuous (fun U : C(I, ℝ) =>
-      solutionPath hε hT (reverseCentered U) ((ε:ℂ)*Complex.I)) :=
-    (solutionPath_lipschitz hε hT _).continuous.comp reverseCentered_lipschitz.continuous
-  have he := (continuous_eval_const (⟨T,⟨hT,le_rfl⟩⟩ : Icc 0 T)).comp hc
-  exact he.add (Complex.continuous_ofReal.comp (continuous_eval_const (1:I)))
-
-lemma tipApprox_eq_inverse {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
-    (hU : ContinuousOn driver (Icc 0 T)) (U : C(I, ℝ))
-    (he : EqOn (extendDriver T U) driver (Icc 0 T)) :
-    tipApprox hε hT.le U = f T ((driver T:ℂ)+(ε:ℂ)*Complex.I) := by
-  let w : ℂ := (driver T:ℂ)+(ε:ℂ)*Complex.I
-  have hw : w ∈ halfPlane := by simpa [w,halfPlane] using hε
-  let z := f T w
-  have hz : z ∈ localizationDomain γ T := h.2.2.2.2.2.2.2.2.1 T hT.le hw
-  have hfinal : g T z = w := h.2.2.2.2.2.2.2.2.2.2.1 T hT.le _ hw
-  have hend : (U 1:ℝ) = driver T := (extend_end hT.ne' U).symm.trans (he ⟨hT.le,le_rfl⟩)
-  let H : ℝ → ℂ := fun s => g (T-s) z - (driver T:ℂ)
-  have hH0 : H 0 = (ε:ℂ)*Complex.I := by simp [H,hfinal,w]
-  have hheight (s : ℝ) (hs : s ∈ Icc 0 T) : ε ≤ (H s).im := by
-    have hh := h.height_antitone hT.le hz (show T-s ∈ Icc 0 T by constructor <;> linarith [hs.1,hs.2])
-      ⟨hT.le,le_rfl⟩ (by linarith [hs.1] : T-s ≤ T)
-    simp only [hfinal,w,Complex.add_im,Complex.ofReal_im,Complex.mul_I_im,Complex.ofReal_re,zero_add] at hh
-    simpa only [H,Complex.sub_im,Complex.ofReal_im,sub_zero] using hh
-  have hHd (s : ℝ) (hs : s ∈ Icc 0 T) :
-      HasDerivWithinAt H (velocity ε (extendDriver T (reverseCentered U) s) (H s)) (Icc 0 T) s := by
-    have hst : T-s ∈ Icc 0 T := by constructor <;> linarith [hs.1,hs.2]
-    have htder : HasDerivWithinAt (fun r : ℝ => T-r) (-1) (Icc 0 T) s := by
-      simpa only [id_eq] using ((hasDerivAt_id s).const_sub T).hasDerivWithinAt
-    have hd := (h.hasDerivWithinAt_eval hT.le hU hz hst).scomp s htder
-      (show MapsTo (fun r : ℝ => T-r) (Icc 0 T) (Icc 0 T) from fun r hr => by constructor <;> linarith [hr.1,hr.2])
-    have hd' := hd.sub_const (driver T : ℂ)
-    apply hd'.congr_deriv
-    rw [velocity,LoewnerCutoff.velocity_eq (hheight s hs),extend_reverseCentered hT.ne' U s,he hst,hend]
-    dsimp only [H,Function.comp_def]
-    simp only [neg_smul,one_smul]
-    push_cast
-    ring
-  have hsol := solution_eq hε hHd (solution_deriv hε hT.le (reverseCentered U) ((ε:ℂ)*Complex.I))
-    (hH0.trans (solution_initial hε hT.le (reverseCentered U) ((ε:ℂ)*Complex.I)).symm)
-  have hendpoint := hsol ⟨hT.le,le_rfl⟩
-  simp only [H,sub_self,h.initial z hz.1.1] at hendpoint
-  dsimp only [tipApprox]
-  rw [← hendpoint,hend]
-  simp only [sub_add_cancel,z,w]
-
-end SLEExactGauge.ReverseLoewner
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.CanonicalTrace
-open ReverseLoewner
-
-def height (n : ℕ) : ℝ := 1 / ((n:ℝ)+1)
-
-lemma height_pos (n : ℕ) : 0 < height n := by unfold height; positivity
-
-lemma height_tendsto : Tendsto height atTop (𝓝[>] (0:ℝ)) := by
-  apply tendsto_nhdsWithin_iff.2
-  constructor
-  · change Tendsto (fun n : ℕ => (1:ℝ)/((n:ℝ)+1)) atTop (𝓝 0)
-    simpa only [Function.comp_def, Nat.cast_add, Nat.cast_one] using
-      (tendsto_const_div_atTop_nhds_zero_nat (1:ℝ)).comp (tendsto_add_atTop_nat 1)
-  · exact Filter.Eventually.of_forall height_pos
-
-def value {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (sample : Ω) : ℂ :=
-  if T = 0 then 0 else ContinuousDriver.selectedLimit 0
-    (fun n => tipApprox (height_pos n) T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T sample))
-
-lemma measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) :
-    @Measurable Ω ℂ (BrownianPredictable.pastSigma B T) (borel ℂ) (value κ B T) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  let : MeasurableSpace Ω := BrownianPredictable.pastSigma B T
-  change Measurable (fun sample => value κ B T sample)
-  by_cases hT : T = 0
-  · simp only [value,ite_eq_left hT]
-    exact measurable_const
-  · simp only [value,ite_eq_right hT]
-    apply ContinuousDriver.measurable_selectedLimit
-    intro n
-    exact (continuous_tipApprox (height_pos n) T.coe_nonneg).measurable.comp
-      (BrownianDriver.path_measurable_past B (Real.sqrt κ) T)
-
-lemma aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    (κ : ℝ) {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) :
-    AEMeasurable (value κ B T) P := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  change AEMeasurable (fun sample => value κ B T sample) P
-  by_cases hT : T = 0
-  · simp only [value,ite_eq_left hT]
-    exact aemeasurable_const
-  · simp only [value,ite_eq_right hT]
-    apply ContinuousDriver.aemeasurable_selectedLimit
-    intro n
-    exact (continuous_tipApprox (height_pos n) T.coe_nonneg).measurable.comp_aemeasurable
-      (BrownianDriver.path_aemeasurable hB (Real.sqrt κ) T)
-
-lemma eq_source {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
-    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal sample) γ g f)
-    (hc : Continuous (fun t => B t sample)) (T : ℝ≥0) : value κ B T sample = γ T := by
-  by_cases hT : T = 0
-  · simpa [value,hT] using h.2.1.symm
-  · simp only [value,ite_eq_right hT]
-    apply ContinuousDriver.selectedLimit_eq
-    have hTp : 0 < (T:ℝ) := lt_of_le_of_ne T.coe_nonneg (by exact_mod_cast Ne.symm hT)
-    have he (n : ℕ) :
-        tipApprox (height_pos n) T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T sample) =
-          f T ((Real.sqrt κ * B T sample:ℝ) + (height n:ℂ)*Complex.I) := by
-      simpa only [Real.toNNReal_coe] using
-        tipApprox_eq_inverse h (height_pos n) hTp
-          ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
-          (BrownianDriver.path B (Real.sqrt κ) T sample)
-          (fun _ ht => BrownianDriver.extend_eq B (Real.sqrt κ) T sample hc ht)
-    have ht := (h.2.2.2.2.2.2.2.2.2.2.2.2.2 T T.coe_nonneg).comp height_tendsto
-    simpa only [Function.comp_def,he,Real.toNNReal_coe] using ht
-
-end SLEExactGauge.CanonicalTrace
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge
-
-theorem canonicalTraceFromBrownianPast {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    (κ : ℝ) (_hκ : 0 < κ ∧ κ < 8)
-    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
-      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample)) :
-    ∃ Γ : ℝ≥0 → Ω → ℂ,
-      (∀ t, @Measurable Ω ℂ (BrownianPredictable.pastSigma B t) (borel ℂ) (Γ t)) ∧
-      (∀ t, AEMeasurable (Γ t) P) ∧
-      ∀ᵐ sample ∂P, ∀ t : ℝ≥0, Γ t sample = γ sample t := by
-  refine ⟨CanonicalTrace.value κ B, CanonicalTrace.measurable_past κ B,
-    CanonicalTrace.aemeasurable κ hB, ?_⟩
-  filter_upwards [hmaps,hB.cont] with sample hm hc
-  exact CanonicalTrace.eq_source κ B sample hm hc
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional
-
-lemma source_point_reach {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hzero : driver 0=0)
-    {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
-    (U : C(I, ℝ)) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
-    (x a β δ d y : ℝ) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
-    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x) (hεx : 2*ε < x) (hy : δ*ε < y)
-    (w : ℂ) (hw : w ∈ localizationDomain γ S) (hclose : dist w (x:ℂ) < d)
-    (hwS : g S w=(driver S:ℂ)+(y:ℂ)*Complex.I) :
-    reach U hε hT x β a ((δ/(4*d))^β) := by
-  apply tip_forces_reach hε hT hS hST U (by rw [hU ⟨le_rfl,hT⟩,hzero]) x a β δ d y
-    hβ hδ hδhalf haδ hd hdx hεx hy (fun t => g t w) (h.continuous_eval hS hw)
-  · intro t ht
-    rw [hU ⟨ht.1.le,ht.2.le.trans hST⟩]
-    exact h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hw)
-  · rw [h.initial w hw.1.1]
-    exact (dist_eq_norm w (x:ℂ) ▸ hclose).le
-  · intro t ht
-    have hh := h.height_antitone hS hw ht ⟨hS,le_rfl⟩ ht.2
-    dsimp only at hh
-    rw [hwS] at hh
-    simpa using hh
-  · rw [hwS,hU ⟨hS,hST⟩]
-    simp
-
-lemma source_hit_eventually_reach {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ : ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    (x a β δ d : ℝ) (hx : 0 < x) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
-    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x)
-    (hhit : ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d) :
-    ∀ᶠ n : ℕ in atTop, reach (BrownianDriver.path B (Real.sqrt κ) (n+1) ω)
-      (CanonicalTrace.height_pos n) (n+1 : ℝ≥0).coe_nonneg x β a ((δ/(4*d))^β) := by
-  obtain ⟨t,ht,hclose⟩ := hhit
-  obtain ⟨w,y,hy,hwd,hw,hwS⟩ := inverse_tip_point h ht hclose
-  have hsmall : ∀ᶠ n : ℕ in atTop, CanonicalTrace.height n < min (x/2) (y/δ) :=
-    (CanonicalTrace.height_tendsto.mono_right nhdsWithin_le_nhds).eventually
-      (gt_mem_nhds (lt_min (half_pos hx) (div_pos hy hδ)))
-  have hlarge : ∀ᶠ n : ℕ in atTop, t ≤ ((n+1:ℝ≥0):ℝ) := by
-    have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
-    simpa only [NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using hh.eventually (eventually_ge_atTop t)
-  filter_upwards [hsmall,hlarge] with n hn htn
-  have hn' := lt_min_iff.mp hn
-  have hxcut : 2*CanonicalTrace.height n < x := by linarith [hn'.1]
-  have hycut : δ*CanonicalTrace.height n < y := by
-    have hh := (lt_div_iff₀ hδ).mp hn'.2
-    simpa only [mul_comm] using hh
-  apply source_point_reach h (by simp [hb0]) (CanonicalTrace.height_pos n)
-    (n+1:ℝ≥0).coe_nonneg ht htn (BrownianDriver.path B (Real.sqrt κ) (n+1) ω)
-    (fun s hs => BrownianDriver.extend_eq B _ (n+1) ω hc hs)
-    x a β δ d y hβ hδ hδhalf haδ hd hdx hxcut hycut w hw hwd hwS
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing
-
-lemma initial_bound {β x ε : ℝ} (hβ : 0 ≤ β) (hx : 0 < x) :
-    radial (-β) x ε ≤ x^(-β) := by
-  have hh := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hx)
-    (le_add_of_nonneg_right (sq_nonneg ε)) (show (-β)/2 ≤ 0 by linarith)
-  change (x^2+ε^2)^((-β)/2) ≤ x^(-β)
-  calc
-    _ ≤ (x^2)^((-β)/2) := hh
-    _ = x^(-β) := by
-      rw [← Real.rpow_natCast_mul hx.le]
-      congr 1
-      ring
-
-def uniformBudget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a x L : ℝ) (n : ℕ) (ω : Ω) : ℝ :=
-  budget B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω +
-    (x^(-β)-radial (-β) x (CanonicalTrace.height n))
-
-lemma uniformBudget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (x L : ℝ) (hx : 0 < x) (n : ℕ) :
-    Integrable (uniformBudget B κ β a x L n) P ∧
-      (∫ ω,uniformBudget B κ β a x L n ω ∂P)=x^(-β) ∧
-      ∀ ω, 0 ≤ uniformBudget B κ β a x L n ω := by
-  have hT : (0:ℝ)<((n+1:ℝ≥0):ℝ) := by positivity
-  have hh := budget_mean hB κ β a hκ hβ ha hba (CanonicalTrace.height_pos n) (n+1) hT x L
-  refine ⟨hh.1.add (integrable_const _),?_,?_⟩
-  · change (∫ ω,budget B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω+
-      (x^(-β)-radial (-β) x (CanonicalTrace.height n)) ∂P)=_
-    rw [integral_add hh.1 (integrable_const _),hh.2]
-    simp
-  · intro ω
-    exact add_nonneg (budget_nonneg B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω)
-      (sub_nonneg.mpr (initial_bound hβ hx))
-
-lemma uniformBudget_dominates {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a x L : ℝ)
-    (hβ : 0 ≤ β) (hx : 0 < x) (n : ℕ) (ω : Ω) :
-    budget B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω ≤ uniformBudget B κ β a x L n ω :=
-  le_add_of_nonneg_right (sub_nonneg.mpr (initial_bound hβ hx))
-
-lemma ratio_power_eq {x d δ β : ℝ} (hx : 0 < x) (hd : 0 < d) (hδ : 0 < δ) :
-    x^(-β)/(δ/(4*d))^β = (4*d/(δ*x))^β := by
-  rw [Real.rpow_neg hx.le,← Real.inv_rpow hx.le,← Real.div_rpow (inv_nonneg.mpr hx.le) (by positivity)]
-  congr 1
-  field_simp
-
-theorem positive_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    {x d : ℝ} (hx : 0 < x) (hd : 0 < d) :
-    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d} ≤ ENNReal.ofReal ((4*d/(δ*x))^β) := by
-  by_cases hdx : d < δ*x
-  · let L : ℝ := (δ/(4*d))^β
-    have hL : 0 < L := Real.rpow_pos_of_pos (by positivity) _
-    have hm (n : ℕ) := uniformBudget_mean hB κ β a hκ hβ.le ha hba x L hx n
-    have hE : ∀ᵐ ω ∂P, (∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d) →
-        ∀ᶠ n : ℕ in atTop, L ≤ uniformBudget B κ β a x L n ω := by
-      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hh hc hb0
-      intro hhit
-      filter_upwards [source_hit_eventually_reach B κ γ g f ω hh hc hb0 x a β δ d hx hβ hδ hδhalf haδ hd hdx hhit] with n hn
-      have hb := budget_dominates_reach B κ β a hκ hβ.le ha hba (CanonicalTrace.height_pos n)
-        (n+1) (by positivity) x L ω hc hb0 hn
-      exact hb.trans (uniformBudget_dominates B κ β a x L hβ.le hx n ω)
-    have hh := measure_eventually_budget (uniformBudget B κ β a x L) (x^(-β)) L hL
-      (fun n => (hm n).1) (fun n => (hm n).2.2) (fun n => (hm n).2.1) _ hE
-    rwa [ratio_power_eq hx hd hδ] at hh
-  · have hratio : 1 ≤ 4*d/(δ*x) := by
-      rw [le_div_iff₀ (mul_pos hδ hx)]
-      have hh : δ*x ≤ d := not_lt.1 hdx
-      linarith
-    have hp : 1 ≤ (4*d/(δ*x))^β := Real.one_le_rpow hratio hβ.le
-    exact (prob_le_one).trans (by simpa using ENNReal.ofReal_le_ofReal hp)
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory Metric
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional
-
-def mirror (z : ℂ) : ℂ := -(starRingEnd ℂ) z
-
-lemma mirror_im (z : ℂ) : (mirror z).im=z.im := by simp [mirror]
-
-lemma mirror_re (z : ℂ) : (mirror z).re= -z.re := by simp [mirror]
-
-lemma continuous_mirror : Continuous mirror := Complex.continuous_conj.neg
-
-lemma mirror_dist_real (w : ℂ) (x : ℝ) : ‖mirror w-(x:ℂ)‖ = dist w ((-x:ℝ):ℂ) := by
-  rw [dist_eq_norm]
-  have he : mirror w-(x:ℂ) = -(starRingEnd ℂ) (w-((-x:ℝ):ℂ)) := by simp [mirror]; ring
-  rw [he,norm_neg,Complex.norm_conj]
-
-lemma mirror_ode {G : ℝ → ℂ} {t u : ℝ} (hG : HasDerivAt G (2/(G t-(u:ℂ))) t) :
-    HasDerivAt (fun s => mirror (G s)) (2/(mirror (G t)-((-u:ℝ):ℂ))) t := by
-  have hh := (Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivAt t hG).neg
-  change HasDerivAt (fun s => -(starRingEnd ℂ) (G s)) (-(starRingEnd ℂ) (2/(G t-(u:ℂ)))) t at hh
-  apply hh.congr_deriv
-  simp only [map_div₀,map_ofNat,map_sub,Complex.conj_ofReal,Complex.ofReal_neg,mirror]
-  rw [show -(starRingEnd ℂ) (G t)- -(u:ℂ) = -((starRingEnd ℂ) (G t)-(u:ℂ)) by ring,div_neg]
-
-lemma negative_source_point_reach {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hzero : driver 0=0)
-    {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
-    (U : C(I, ℝ)) (hU : EqOn (extendDriver T U) (fun s => -driver s) (Icc 0 T))
-    (x a β δ d y : ℝ) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
-    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x) (hεx : 2*ε < x) (hy : δ*ε < y)
-    (w : ℂ) (hw : w ∈ localizationDomain γ S) (hclose : dist w ((-x:ℝ):ℂ) < d)
-    (hwS : g S w=(driver S:ℂ)+(y:ℂ)*Complex.I) :
-    reach U hε hT x β a ((δ/(4*d))^β) := by
-  apply tip_forces_reach hε hT hS hST U (by rw [hU ⟨le_rfl,hT⟩]; simp [hzero]) x a β δ d y
-    hβ hδ hδhalf haδ hd hdx hεx hy (fun t => mirror (g t w))
-    (continuous_mirror.comp_continuousOn (h.continuous_eval hS hw))
-  · intro t ht
-    rw [hU ⟨ht.1.le,ht.2.le.trans hST⟩]
-    exact mirror_ode (h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hw))
-  · rw [h.initial w hw.1.1,mirror_dist_real]
-    exact hclose.le
-  · intro t ht
-    rw [mirror_im]
-    have hh := h.height_antitone hS hw ht ⟨hS,le_rfl⟩ ht.2
-    dsimp only at hh
-    rw [hwS] at hh
-    simpa using hh
-  · rw [mirror_re,hwS,hU ⟨hS,hST⟩]
-    simp
-
-lemma negative_source_hit_eventually_reach {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ : ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
-    (x a β δ d : ℝ) (hx : 0 < x) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
-    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x)
-    (hhit : ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) ((-x:ℝ):ℂ) < d) :
-    ∀ᶠ n : ℕ in atTop, reach (BrownianDriver.path (-B) (Real.sqrt κ) (n+1) ω)
-      (CanonicalTrace.height_pos n) (n+1 : ℝ≥0).coe_nonneg x β a ((δ/(4*d))^β) := by
-  obtain ⟨t,ht,hclose⟩ := hhit
-  obtain ⟨w,y,hy,hwd,hw,hwS⟩ := inverse_tip_point h ht hclose
-  have hsmall : ∀ᶠ n : ℕ in atTop, CanonicalTrace.height n < min (x/2) (y/δ) :=
-    (CanonicalTrace.height_tendsto.mono_right nhdsWithin_le_nhds).eventually
-      (gt_mem_nhds (lt_min (half_pos hx) (div_pos hy hδ)))
-  have hlarge : ∀ᶠ n : ℕ in atTop, t ≤ ((n+1:ℝ≥0):ℝ) := by
-    have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
-      tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
-    simpa only [NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using hh.eventually (eventually_ge_atTop t)
-  filter_upwards [hsmall,hlarge] with n hn htn
-  have hn' := lt_min_iff.mp hn
-  have hxcut : 2*CanonicalTrace.height n < x := by linarith [hn'.1]
-  have hycut : δ*CanonicalTrace.height n < y := by
-    have hh := (lt_div_iff₀ hδ).mp hn'.2
-    simpa only [mul_comm] using hh
-  apply negative_source_point_reach h (by simp [hb0]) (CanonicalTrace.height_pos n)
-    (n+1:ℝ≥0).coe_nonneg ht htn (BrownianDriver.path (-B) (Real.sqrt κ) (n+1) ω)
-    (fun s hs => by
-      have he := BrownianDriver.extend_eq (-B) (Real.sqrt κ) (n+1) ω hc.neg hs
-      change extendDriver ((n+1:ℝ≥0):ℝ) (BrownianDriver.path (-B) (Real.sqrt κ) (n+1) ω) s =
-        Real.sqrt κ*(-B s.toNNReal ω) at he
-      simpa only [mul_neg] using he)
-    x a β δ d y hβ hδ hδhalf haδ hd hdx hxcut hycut w hw hwd hwS
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing
-
-theorem negative_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    {x d : ℝ} (hx : 0 < x) (hd : 0 < d) :
-    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) ((-x:ℝ):ℂ) < d} ≤ ENNReal.ofReal ((4*d/(δ*x))^β) := by
-  by_cases hdx : d < δ*x
-  · let L : ℝ := (δ/(4*d))^β
-    have hL : 0 < L := Real.rpow_pos_of_pos (by positivity) _
-    have hm (n : ℕ) := uniformBudget_mean hB.neg κ β a hκ hβ.le ha hba x L hx n
-    have hE : ∀ᵐ ω ∂P, (∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) ((-x:ℝ):ℂ) < d) →
-        ∀ᶠ n : ℕ in atTop, L ≤ uniformBudget (-B) κ β a x L n ω := by
-      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hh hc hb0
-      intro hhit
-      filter_upwards [negative_source_hit_eventually_reach B κ γ g f ω hh hc hb0 x a β δ d hx hβ hδ hδhalf haδ hd hdx hhit] with n hn
-      have hb := budget_dominates_reach (-B) κ β a hκ hβ.le ha hba (CanonicalTrace.height_pos n)
-        (n+1) (by positivity) x L ω hc.neg (by simpa using congrArg Neg.neg hb0) hn
-      exact hb.trans (uniformBudget_dominates (-B) κ β a x L hβ.le hx n ω)
-    have hh := measure_eventually_budget (uniformBudget (-B) κ β a x L) (x^(-β)) L hL
-      (fun n => (hm n).1) (fun n => (hm n).2.2) (fun n => (hm n).2.1) _ hE
-    rwa [ratio_power_eq hx hd hδ] at hh
-  · have hratio : 1 ≤ 4*d/(δ*x) := by
-      rw [le_div_iff₀ (mul_pos hδ hx)]
-      have hh : δ*x ≤ d := not_lt.1 hdx
-      linarith
-    have hp : 1 ≤ (4*d/(δ*x))^β := Real.one_le_rpow hratio hβ.le
-    exact (prob_le_one).trans (by simpa using ENNReal.ofReal_le_ofReal hp)
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-
-theorem real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    {x d : ℝ} (hx : 0 < |x|) (hd : 0 < d) :
-    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d} ≤ ENNReal.ofReal ((4*d/(δ*|x|))^β) := by
-  rcases le_or_gt 0 x with hpos | hneg
-  · rw [abs_of_nonneg hpos] at hx ⊢
-    exact positive_real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps hx hd
-  · have hh := negative_real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps (neg_pos.2 hneg) hd
-    simpa only [neg_neg,abs_of_neg hneg] using hh
-
-lemma annulus_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
-    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
-    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    {x d : ℝ} (hx : 1/4 ≤ |x|) (hd : 0 < d) :
-    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d} ≤ ENNReal.ofReal ((16/δ)^β*d^β) := by
-  have hxp : 0 < |x| := lt_of_lt_of_le (by norm_num) hx
-  refine (real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps hxp hd).trans
-    (ENNReal.ofReal_le_ofReal ?_)
-  have hr : 4*d/(δ*|x|) ≤ (16/δ)*d := by
-    rw [div_le_iff₀ (mul_pos hδ hxp)]
-    have hh := mul_le_mul_of_nonneg_left hx (show 0 ≤ 16*d by positivity)
-    field_simp
-    nlinarith
-  have hh := Real.rpow_le_rpow (by positivity : 0 ≤ 4*d/(δ*|x|)) hr hβ.le
-  rwa [Real.mul_rpow (by positivity : 0 ≤ 16/δ) hd.le] at hh
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.AngularBarrier
-
-def lowerGamma (b s : ℝ) : ℝ := ∫ x in (0:ℝ)..s, Real.exp (-x)*x^(b-1)
-
-def cdf (b s : ℝ) : ℝ := lowerGamma b s / Real.Gamma b
-
-def pdf (b s : ℝ) : ℝ := Real.exp (-s)*s^(b-1)/Real.Gamma b
-
-lemma integrable_kernel {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) :
-    IntervalIntegrable (fun x : ℝ => Real.exp (-x)*x^(b-1)) volume 0 s := by
-  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hs]
-  exact (Real.GammaIntegral_convergent hb).mono_set Ioc_subset_Ioi_self
-
-lemma kernel_continuousAt (b : ℝ) {s : ℝ} (hs : 0 < s) :
-    ContinuousAt (fun x : ℝ => Real.exp (-x)*x^(b-1)) s :=
-  Real.continuous_exp.continuousAt.comp continuousAt_neg |>.mul
-    (continuousAt_id.rpow_const (Or.inl hs.ne'))
-
-lemma lowerGamma_nonneg {b : ℝ} (_hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) : 0 ≤ lowerGamma b s := by
-  apply intervalIntegral.integral_nonneg hs
-  intro x hx
-  exact mul_nonneg (Real.exp_pos _).le (Real.rpow_nonneg hx.1 _)
-
-lemma cdf_nonneg {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) : 0 ≤ cdf b s :=
-  div_nonneg (lowerGamma_nonneg hb hs) (Real.Gamma_pos_of_pos hb).le
-
-lemma pdf_pos {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) : 0 < pdf b s :=
-  div_pos (mul_pos (Real.exp_pos _) (Real.rpow_pos_of_pos hs _)) (Real.Gamma_pos_of_pos hb)
-
-lemma power_integral {b : ℝ} (hb : 0 < b) {s : ℝ} (_hs : 0 ≤ s) :
-    (∫ x in (0:ℝ)..s, x^(b-1)) = s^b/b := by
-  rw [integral_rpow (Or.inl (by linarith : -1 < b-1))]
-  rw [show b-1+1 = b by ring, Real.zero_rpow hb.ne', sub_zero]
-
-lemma cdf_le_power {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) :
-    cdf b s ≤ s^b/(b*Real.Gamma b) := by
-  have hmono : lowerGamma b s ≤ ∫ x in (0:ℝ)..s, x^(b-1) := by
-    apply intervalIntegral.integral_mono_on hs (integrable_kernel hb hs)
-      (intervalIntegral.intervalIntegrable_rpow' (by linarith : -1 < b-1))
-    intro x hx
-    simpa only [one_mul] using mul_le_mul_of_nonneg_right
-      (Real.exp_le_one_iff.2 (neg_nonpos.2 hx.1)) (Real.rpow_nonneg hx.1 (b-1))
-  rw [power_integral hb hs] at hmono
-  have hh := div_le_div_of_nonneg_right hmono (Real.Gamma_pos_of_pos hb).le
-  simpa only [cdf, div_div] using hh
-
-lemma hasDerivAt_lowerGamma {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
-    HasDerivAt (lowerGamma b) (Real.exp (-s)*s^(b-1)) s := by
-  apply intervalIntegral.integral_hasDerivAt_right (integrable_kernel hb hs.le)
-  · exact (Real.measurable_exp.comp measurable_neg |>.mul
-      (measurable_id.pow_const (b-1))).aestronglyMeasurable.stronglyMeasurableAtFilter
-  · exact kernel_continuousAt b hs
-
-lemma hasDerivAt_cdf {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
-    HasDerivAt (cdf b) (pdf b s) s :=
-  (hasDerivAt_lowerGamma hb hs).div_const (Real.Gamma b)
-
-lemma deriv_cdf {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
-    deriv (cdf b) s = pdf b s := (hasDerivAt_cdf hb hs).deriv
-
-lemma hasDerivAt_pdf {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
-    HasDerivAt (pdf b) (pdf b s*((b-1)/s-1)) s := by
-  have hd := (((hasDerivAt_neg s).exp).mul
-    ((hasDerivAt_id s).rpow_const (p:=b-1) (Or.inl hs.ne'))).div_const (Real.Gamma b)
-  apply hd.congr_deriv
-  unfold pdf
-  simp only [id_eq]
-  rw [Real.rpow_sub_one hs.ne' (b-1)]
-  field_simp
-  ring
-
-lemma lowerGamma_tendsto {b : ℝ} (hb : 0 < b) :
-    Tendsto (lowerGamma b) atTop (𝓝 (Real.Gamma b)) := by
-  rw [Real.Gamma_eq_integral hb]
-  exact intervalIntegral_tendsto_integral_Ioi 0 (Real.GammaIntegral_convergent hb) tendsto_id
-
-lemma cdf_tendsto {b : ℝ} (hb : 0 < b) : Tendsto (cdf b) atTop (𝓝 1) := by
-  change Tendsto (fun s => lowerGamma b s / Real.Gamma b) atTop (𝓝 1)
-  have hh := (lowerGamma_tendsto hb).div_const (Real.Gamma b)
-  convert hh using 1
-  rw [div_self (Real.Gamma_pos_of_pos hb).ne']
-
-lemma power_mul {b x : ℝ} (hb : 0 < b) (hx : 0 ≤ x) :
-    x^(b-1)*x = x^b := by
-  rcases eq_or_lt_of_le hx with h | h
-  · subst x; simp [Real.zero_rpow hb.ne']
-  · rw [← Real.rpow_add_one h.ne', sub_add_cancel]
-
-lemma kernel_lower {b s x : ℝ} (hb : 0 < b) (hx : 0 ≤ x) :
-    Real.exp (-s)*((1+s)*x^(b-1)-x^b) ≤ Real.exp (-x)*x^(b-1) := by
-  have hh := Real.add_one_le_exp (s-x)
-  have he : Real.exp (-s)*Real.exp (s-x) = Real.exp (-x) := by
-    rw [← Real.exp_add]; congr 1; ring
-  have hi := mul_le_mul_of_nonneg_right
-    (mul_le_mul_of_nonneg_left hh (Real.exp_pos (-s)).le) (Real.rpow_nonneg hx (b-1))
-  rw [he] at hi
-  calc
-    _ = Real.exp (-s)*(s-x+1)*x^(b-1) := by rw [← power_mul hb hx]; ring
-    _ ≤ _ := hi
-
-lemma lowerGamma_lower {b s : ℝ} (hb : 0 < b) (hs : 0 ≤ s) :
-    Real.exp (-s)*((1+s)*s^b/b-s^(b+1)/(b+1)) ≤ lowerGamma b s := by
-  have hi₁ : IntervalIntegrable (fun x : ℝ => x^(b-1)) volume 0 s :=
-    intervalIntegral.intervalIntegrable_rpow' (by linarith)
-  have hi₂ : IntervalIntegrable (fun x : ℝ => x^b) volume 0 s :=
-    intervalIntegral.intervalIntegrable_rpow' (by linarith)
-  have hh := intervalIntegral.integral_mono_on hs
-    (((hi₁.const_mul (1+s)).sub hi₂).const_mul (Real.exp (-s)))
-    (integrable_kernel hb hs) (fun x hx => kernel_lower hb hx.1 (s:=s))
-  rw [intervalIntegral.integral_const_mul,
-    intervalIntegral.integral_sub (hi₁.const_mul _) hi₂,
-    intervalIntegral.integral_const_mul, power_integral hb hs] at hh
-  have hi₂eq : (∫ x in (0:ℝ)..s, x^b) = s^(b+1)/(b+1) := by
-    simpa only [add_sub_cancel_right] using power_integral (by linarith : 0 < b+1) hs
-  rw [hi₂eq] at hh
-  simpa only [lowerGamma, mul_div_assoc] using hh
-
-lemma cdf_pdf_inequality {b s : ℝ} (hb : 0 < b) (hs : 0 < s) :
-    s^2*pdf b s/(b+1) ≤ b*cdf b s-s*pdf b s := by
-  have hl := lowerGamma_lower hb hs.le
-  have hg : 0 < Real.Gamma b := Real.Gamma_pos_of_pos hb
-  have hbp : 0 < b+1 := by linarith
-  rw [Real.rpow_add_one hs.ne' b] at hl
-  have heq : s^b = s^(b-1)*s := (power_mul hb hs.le).symm
-  rw [heq] at hl
-  unfold cdf pdf
-  apply (div_le_div_iff_of_pos_right hg).1
-  have hh := mul_le_mul_of_nonneg_left hl hb.le
-  field_simp [hb.ne', hbp.ne'] at hh ⊢
-  nlinarith
-
-lemma contDiffAt_pdf (n : WithTop ℕ∞) (b : ℝ) {s : ℝ} (hs : 0 < s) :
-    ContDiffAt ℝ n (pdf b) s := by
-  have he : ContDiffAt ℝ n (fun x : ℝ => Real.exp (-x)) s :=
-    (contDiffAt_id.neg).exp
-  have hp : ContDiffAt ℝ n (fun x : ℝ => x^(b-1)) s :=
-    contDiffAt_id.rpow contDiffAt_const hs.ne'
-  exact (he.mul hp).div_const _
-
-lemma contDiffOn_cdf {b : ℝ} (hb : 0 < b) (n : ℕ) :
-    ContDiffOn ℝ (n+1) (cdf b) (Ioi 0) := by
-  rw [contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioi]
-  refine ⟨fun s hs => (hasDerivAt_cdf hb hs).differentiableAt.differentiableWithinAt,
-    ?_, ?_⟩
-  · intro hn; simp at hn
-  · have hh : ContDiffOn ℝ n (pdf b) (Ioi 0) :=
-      fun s hs => (contDiffAt_pdf n b hs).contDiffWithinAt
-    exact hh.congr (fun s hs => deriv_cdf hb hs)
-
-lemma contDiffAt_cdf {b : ℝ} (hb : 0 < b) (n : ℕ) {s : ℝ} (hs : 0 < s) :
-    ContDiffAt ℝ n (cdf b) s := by
-  have hh := (contDiffOn_cdf hb n).contDiffAt (isOpen_Ioi.mem_nhds hs)
-  exact hh.of_le (by exact_mod_cast Nat.le_succ n)
-
-def scale (l b t : ℝ) := (Real.exp (l*t)-1)/(2*(b+1))
-
-def value (l b t q : ℝ) := Real.exp (l*b/2*t)*cdf b (q/scale l b t)
-
-def valueQ (l b t q : ℝ) := Real.exp (l*b/2*t)*pdf b (q/scale l b t)/scale l b t
-
-def valueQQ (l b t q : ℝ) := Real.exp (l*b/2*t)*pdf b (q/scale l b t)*
-  ((b-1)/(q/scale l b t)-1)/(scale l b t)^2
-
-def valueT (l b t q : ℝ) := l*b/2*value l b t q -
-  Real.exp (l*b/2*t)*pdf b (q/scale l b t)*(q/scale l b t)*
-    (l+l/(2*(b+1)*scale l b t))
-
-lemma scale_pos {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) :
-    0 < scale l b t := by
-  apply div_pos
-  · exact sub_pos.2 (Real.one_lt_exp_iff.2 (mul_pos hl ht))
-  · positivity
-
-lemma hasDerivAt_scale (l b t : ℝ) :
-    HasDerivAt (scale l b) (l*Real.exp (l*t)/(2*(b+1))) t := by
-  apply (((((hasDerivAt_id t).const_mul l).exp).sub_const 1).div_const (2*(b+1))).congr_deriv
-  simp only [id_eq]
-  ring
-
-lemma hasDerivAt_value_q {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
-    (ht : 0 < t) (hq : 0 < q) : HasDerivAt (value l b t) (valueQ l b t q) q := by
-  have hs := scale_pos hl hb ht
-  have hh := ((hasDerivAt_cdf hb (div_pos hq hs)).comp q
-    ((hasDerivAt_id q).div_const (scale l b t))).const_mul (Real.exp (l*b/2*t))
-  apply hh.congr_deriv
-  dsimp only [valueQ]
-  ring
-
-lemma hasDerivAt_valueQ_q {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
-    (ht : 0 < t) (hq : 0 < q) : HasDerivAt (valueQ l b t) (valueQQ l b t q) q := by
-  have hs := scale_pos hl hb ht
-  have hh := (((hasDerivAt_pdf hb (div_pos hq hs)).comp q
-    ((hasDerivAt_id q).div_const (scale l b t))).const_mul (Real.exp (l*b/2*t))).div_const (scale l b t)
-  apply hh.congr_deriv
-  dsimp only [valueQQ]
-  ring
-
-lemma hasDerivAt_value_t {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
-    (ht : 0 < t) (hq : 0 < q) :
-    HasDerivAt (fun s => value l b s q) (valueT l b t q) t := by
-  have hs := scale_pos hl hb ht
-  have hbp : b+1 ≠ 0 := by positivity
-  have hh := ((((hasDerivAt_id t).const_mul (l*b/2)).exp).mul
-    ((hasDerivAt_cdf hb (div_pos hq hs)).comp t ((hasDerivAt_const t q).div (hasDerivAt_scale l b t) hs.ne')))
-  apply hh.congr_deriv
-  simp only [id_eq, Function.comp_def, Pi.div_apply]
-  dsimp only [valueT, value]
-  have he : Real.exp (l*t) = 2*(b+1)*scale l b t+1 := by unfold scale; field_simp; ring
-  rw [he]
-  field_simp
-  ring
-
-theorem supersolution {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
-    (ht : 0 < t) (hq : 0 < q) (hq1 : q ≤ 1) :
-    l*q*(1-q)*valueQQ l b t q +
-      l*((1-b)-(3/2-b)*q)*valueQ l b t q ≤ valueT l b t q := by
-  let α := scale l b t
-  let s := q/α
-  let W := pdf b s
-  let F := cdf b s
-  have hα : 0 < α := scale_pos hl hb ht
-  have hs : 0 < s := div_pos hq hα
-  have hW : 0 ≤ W := (pdf_pos hb hs).le
-  have hbp : 0 < b+1 := by linarith
-  have hi : s^2*W/(b+1) ≤ b*F-s*W := cdf_pdf_inequality hb hs
-  have hαs : α*s = q := by dsimp [s]; field_simp
-  have hA : 0 ≤ 1-1/(2*(b+1)) := by
-    rw [sub_nonneg, div_le_one (by positivity)]
-    linarith
-  have hrest : 0 ≤ l*(1-1/(2*(b+1)))*(1-q)*s*W/α := by
-    exact div_nonneg (mul_nonneg (mul_nonneg (mul_nonneg (mul_nonneg hl.le hA) (sub_nonneg.2 hq1)) hs.le) hW) hα.le
-  have hmain : 0 ≤ l/2*(b*F-s*W-s^2*W/(b+1)) := mul_nonneg (by positivity) (sub_nonneg.2 hi)
-  have he : valueT l b t q - (l*q*(1-q)*valueQQ l b t q +
-        l*((1-b)-(3/2-b)*q)*valueQ l b t q) =
-      Real.exp (l*b/2*t) *
-        (l/2*(b*F-s*W-s^2*W/(b+1)) + l*(1-1/(2*(b+1)))*(1-q)*s*W/α) := by
-    dsimp only [valueT, value, valueQQ, valueQ]
-    change l*b/2*(Real.exp (l*b/2*t)*F) - Real.exp (l*b/2*t)*W*s*
-        (l+l/(2*(b+1)*α)) - (l*q*(1-q)*(Real.exp (l*b/2*t)*W*((b-1)/s-1)/α^2) +
-        l*((1-b)-(3/2-b)*q)*(Real.exp (l*b/2*t)*W/α)) = _
-    dsimp only [s]
-    field_simp
-    ring
-  have hh : 0 ≤ valueT l b t q - (l*q*(1-q)*valueQQ l b t q +
-      l*((1-b)-(3/2-b)*q)*valueQ l b t q) := by
-    rw [he]
-    exact mul_nonneg (Real.exp_pos _).le (add_nonneg hmain hrest)
-  exact sub_nonneg.1 hh
-
-lemma value_nonneg {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
-    (ht : 0 < t) (hq : 0 ≤ q) : 0 ≤ value l b t q :=
-  mul_nonneg (Real.exp_pos _).le (cdf_nonneg hb (div_nonneg hq (scale_pos hl hb ht).le))
-
-lemma contDiffAt_value {l b : ℝ} (hl : 0 < l) (hb : 0 < b) (n : ℕ)
-    {v : ℝ × ℝ} (ht : 0 < v.1) (hq : 0 < v.2) :
-    ContDiffAt ℝ n (fun v : ℝ × ℝ => value l b v.1 v.2) v := by
-  have he : ContDiffAt ℝ n (fun v : ℝ × ℝ => Real.exp (l*b/2*v.1)) v := by fun_prop
-  have ha : ContDiffAt ℝ n (fun v : ℝ × ℝ => scale l b v.1) v := by
-    unfold scale; fun_prop
-  have hs := scale_pos hl hb ht
-  have hqf : ContDiffAt ℝ n (fun v : ℝ × ℝ => v.2/scale l b v.1) v :=
-    contDiffAt_snd.div ha hs.ne'
-  have hh := (contDiffAt_cdf hb n (div_pos hq hs)).comp v hqf
-  dsimp only [Function.comp_def] at hh
-  exact he.mul hh
-
-lemma continuousAt_value {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
-    {v : ℝ × ℝ} (ht : 0 < v.1) (hq : 0 < v.2) :
-    ContinuousAt (fun v : ℝ × ℝ => value l b v.1 v.2) v :=
-  (contDiffAt_value hl hb 0 ht hq).continuousAt
-
-lemma scale_tendsto {l b : ℝ} (hl : 0 < l) (hb : 0 < b) :
-    Tendsto (scale l b) (𝓝[>] (0:ℝ)) (𝓝[>] (0:ℝ)) := by
-  apply tendsto_nhdsWithin_iff.2
-  constructor
-  · have hh : Continuous (scale l b) := by unfold scale; fun_prop
-    simpa [scale] using (hh.continuousAt (x:=0)).tendsto.mono_left nhdsWithin_le_nhds
-  · filter_upwards [self_mem_nhdsWithin] with t ht
-    exact scale_pos hl hb ht
-
-lemma value_tendsto_zero_time {l b q : ℝ} (hl : 0 < l) (hb : 0 < b) (hq : 0 < q) :
-    Tendsto (fun t => value l b t q) (𝓝[>] (0:ℝ)) (𝓝 1) := by
-  have hs : Tendsto (fun t => q/scale l b t) (𝓝[>] (0:ℝ)) atTop := by
-    simpa only [div_eq_mul_inv, Function.comp_def] using
-      (tendsto_inv_nhdsGT_zero.comp (scale_tendsto hl hb)).const_mul_atTop hq
-  have he : Tendsto (fun t : ℝ => Real.exp (l*b/2*t)) (𝓝[>] (0:ℝ)) (𝓝 1) := by
-    have hh : Continuous (fun t : ℝ => Real.exp (l*b/2*t)) := by fun_prop
-    simpa using (hh.continuousAt (x:=0)).tendsto.mono_left nhdsWithin_le_nhds
-  simpa only [value, one_mul, Function.comp_def] using he.mul ((cdf_tendsto hb).comp hs)
-
-end SLEExactGauge.AngularBarrier
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.BoundaryDiffusion
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Reverse Loewner estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ReverseLoewner
+open LoewnerCutoff
+
+lemma continuous_velocity {ε : ℝ} (hε : 0 < ε) :
+    Continuous (fun p : ℝ × ℂ => velocity ε p.1 p.2) :=
+  (LoewnerCutoff.continuous_velocity hε).neg
+
+lemma norm_velocity_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (w : ℂ) :
+    ‖velocity ε u w‖ ≤ 2 / ε := by
+  simpa only [velocity,norm_neg] using LoewnerCutoff.norm_velocity_le hε u w
+
+lemma exists_solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
+    ∃ H : ℝ → ℂ, H 0 = z ∧ ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t := by
+  let L : ℝ≥0 := ⟨2 / ε, by positivity⟩
+  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
+  let a : ℝ≥0 := ⟨(2 / ε) * T, mul_nonneg (by positivity) hT⟩
+  have hp : IsPicardLindelof (fun t => velocity ε (U t))
+      ⟨0, ⟨le_rfl, hT⟩⟩ z a 0 L K := by
+    constructor
+    · exact fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith
+    · intro w _
+      have hc : ContinuousOn (fun t : ℝ => (U t, w)) (Icc 0 T) :=
+        hU.prodMk continuousOn_const
+      simpa only [Function.comp_def] using
+        (continuous_velocity hε).comp_continuousOn hc
+    · exact fun t _ w _ => norm_velocity_le hε (U t) w
+    · dsimp [L, a]
+      simp only [sub_zero, sub_self, max_eq_left hT]
+      rfl
+  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
+
+end SLEExactGauge.ReverseLoewner
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ReverseLoewner
+open LoewnerCutoff
+
+lemma norm_velocity_driver_sub_le {ε : ℝ} (hε : 0 < ε) (u v : ℝ) (z : ℂ) :
+    ‖velocity ε u z - velocity ε v z‖ ≤ (4 / ε^2) * |u-v| := by
+  simpa only [velocity, neg_sub_neg, norm_sub_rev] using
+    LoewnerCutoff.norm_velocity_driver_sub_le hε u v z
+
+lemma derivWithin_right {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ico 0 T) :
+    HasDerivWithinAt H (velocity ε (U t) (H t)) (Ici t) t :=
+  (hH t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
+
+lemma derivWithin_left {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ioc 0 T) :
+    HasDerivWithinAt H (velocity ε (U t) (H t)) (Iic t) t :=
+  (hH t ⟨ht.1.le, ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
+
+lemma solution_eq {ε T : ℝ} (hε : 0 < ε) {U : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (U t) (G t)) (Icc 0 T) t)
+    (h0 : H 0 = G 0) : EqOn H G (Icc 0 T) := by
+  apply ODE_solution_unique_of_mem_Icc_right
+    (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith)
+    (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_right hH ht)
+    (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hG)
+    (fun _ ht => derivWithin_right hG ht) (fun _ _ => mem_univ _) h0
+
+lemma solution_dist_le {ε T D : ℝ} (hε : 0 < ε) (hD : 0 ≤ D)
+    {U V : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
+    (h0 : H 0 = G 0) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    dist (H t) (G t) ≤ D * Real.exp ((4 / ε^2) * T) := by
+  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
+  have hK : (0 : ℝ) < K := by change 0 < 4 / ε^2; positivity
+  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
+      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K : ℝ) * D := by
+    rw [dist_eq_norm]
+    calc
+      _ ≤ (4 / ε^2) * |V s-U s| := norm_velocity_driver_sub_le hε _ _ _
+      _ ≤ (K : ℝ) * D := by
+        rw [abs_sub_comm]
+        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1, hs.2.le⟩) hK.le
+  have hh := dist_le_of_approx_trajectories_ODE
+    (fun t => velocity_lipschitz hε (U t))
+    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
+    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
+    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
+    hb (le_of_eq (dist_eq_zero.2 h0)) t ht
+  have he : gronwallBound 0 K (0 + (K : ℝ)*D) (t-0) =
+      D * (Real.exp ((K : ℝ)*t)-1) := by
+    simp only [gronwallBound, zero_add, sub_zero, ite_eq_right (ne_of_gt hK)]
+    field_simp
+    ring
+  rw [he] at hh
+  have hexp : Real.exp ((K : ℝ)*t) ≤ Real.exp ((K : ℝ)*T) :=
+    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
+  apply hh.trans
+  change D * (Real.exp ((4 / ε^2)*t)-1) ≤ D * Real.exp ((4 / ε^2)*T)
+  change Real.exp ((4 / ε^2)*t) ≤ Real.exp ((4 / ε^2)*T) at hexp
+  exact mul_le_mul_of_nonneg_left (by linarith) hD
+
+end SLEExactGauge.ReverseLoewner
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ReverseLoewner
+open LoewnerCutoff
+open scoped unitInterval
+
+def solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) : ℝ → ℂ :=
+  (exists_solution hε hT (extendDriver T U) (continuous_extendDriver T U).continuousOn z).choose
+
+lemma solution_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    solution hε hT U z 0 = z :=
+  (exists_solution hε hT (extendDriver T U)
+    (continuous_extendDriver T U).continuousOn z).choose_spec.1
+
+lemma solution_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (solution hε hT U z)
+      (velocity ε (extendDriver T U t) (solution hε hT U z t)) (Icc 0 T) t :=
+  (exists_solution hε hT (extendDriver T U)
+    (continuous_extendDriver T U).continuousOn z).choose_spec.2
+
+def solutionPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    C(Icc 0 T, ℂ) :=
+  ⟨fun t => solution hε hT U z t,
+    continuousOn_iff_continuous_domRestrict.1 (HasDerivWithinAt.continuousOn (solution_deriv hε hT U z))⟩
+
+lemma solutionPath_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    LipschitzWith ⟨Real.exp ((4 / ε^2) * T), (Real.exp_pos _).le⟩
+      (fun U : C(I, ℝ) => solutionPath hε hT U z) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro U V
+  apply (ContinuousMap.dist_le (mul_nonneg (Real.exp_pos _).le dist_nonneg)).2
+  intro t
+  have hUV (s : ℝ) (_hs : s ∈ Icc 0 T) :
+      |extendDriver T U s-extendDriver T V s| ≤ dist U V := by
+    exact (ContinuousMap.dist_apply_le_dist (f := U) (g := V)
+      (projIcc 0 1 zero_le_one (s/T)))
+  have hh := solution_dist_le hε dist_nonneg (solution_deriv hε hT U z)
+    (solution_deriv hε hT V z)
+    ((solution_initial hε hT U z).trans (solution_initial hε hT V z).symm)
+    hUV t.property
+  change dist (solution hε hT U z t) (solution hε hT V z t) ≤
+    Real.exp ((4 / ε^2)*T) * dist U V
+  simpa only [mul_comm] using hh
+
+end SLEExactGauge.ReverseLoewner
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ReverseLoewner
+open LoewnerCutoff
+
+def reverseCentered (U : C(I, ℝ)) : C(I, ℝ) :=
+  ⟨fun q => U (unitInterval.symm q) - U 1,
+    (U.continuous.comp unitInterval.continuous_symm).sub continuous_const⟩
+
+lemma reverseCentered_lipschitz : LipschitzWith 2 reverseCentered := by
+  apply LipschitzWith.of_dist_le_mul
+  intro U V
+  apply (ContinuousMap.dist_le (by positivity)).2
+  intro q
+  change dist (U (unitInterval.symm q)-U 1) (V (unitInterval.symm q)-V 1) ≤ (2:ℝ)*dist U V
+  calc
+    _ ≤ dist (U (unitInterval.symm q)) (V (unitInterval.symm q)) + dist (U 1) (V 1) := dist_sub_sub_le _ _ _ _
+    _ ≤ dist U V + dist U V := add_le_add (ContinuousMap.dist_apply_le_dist _) (ContinuousMap.dist_apply_le_dist _)
+    _ = _ := by ring
+
+lemma extend_reverseCentered {T : ℝ} (hT : T ≠ 0) (U : C(I, ℝ)) (s : ℝ) :
+    extendDriver T (reverseCentered U) s = extendDriver T U (T-s) - U 1 := by
+  dsimp only [extendDriver,reverseCentered,ContinuousMap.coe_mk]
+  rw [unitInterval.symm_projIcc]
+  congr 2
+  field_simp
+
+lemma extend_end {T : ℝ} (hT : T ≠ 0) (U : C(I, ℝ)) : extendDriver T U T = U 1 := by
+  simp only [extendDriver,div_self hT]
+  congr 1
+  exact Subtype.ext (by simp [projIcc])
+
+def tipApprox {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) : ℂ :=
+  solution hε hT (reverseCentered U) ((ε:ℂ)*Complex.I) T + (U 1 : ℂ)
+
+lemma continuous_tipApprox {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) :
+    Continuous (tipApprox hε hT) := by
+  have hc : Continuous (fun U : C(I, ℝ) =>
+      solutionPath hε hT (reverseCentered U) ((ε:ℂ)*Complex.I)) :=
+    (solutionPath_lipschitz hε hT _).continuous.comp reverseCentered_lipschitz.continuous
+  have he := (continuous_eval_const (⟨T,⟨hT,le_rfl⟩⟩ : Icc 0 T)).comp hc
+  exact he.add (Complex.continuous_ofReal.comp (continuous_eval_const (1:I)))
+
+lemma tipApprox_eq_inverse {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {ε T : ℝ} (hε : 0 < ε) (hT : 0 < T)
+    (hU : ContinuousOn driver (Icc 0 T)) (U : C(I, ℝ))
+    (he : EqOn (extendDriver T U) driver (Icc 0 T)) :
+    tipApprox hε hT.le U = f T ((driver T:ℂ)+(ε:ℂ)*Complex.I) := by
+  let w : ℂ := (driver T:ℂ)+(ε:ℂ)*Complex.I
+  have hw : w ∈ halfPlane := by simpa [w,halfPlane] using hε
+  let z := f T w
+  have hz : z ∈ localizationDomain γ T := h.2.2.2.2.2.2.2.2.1 T hT.le hw
+  have hfinal : g T z = w := h.2.2.2.2.2.2.2.2.2.2.1 T hT.le _ hw
+  have hend : (U 1:ℝ) = driver T := (extend_end hT.ne' U).symm.trans (he ⟨hT.le,le_rfl⟩)
+  let H : ℝ → ℂ := fun s => g (T-s) z - (driver T:ℂ)
+  have hH0 : H 0 = (ε:ℂ)*Complex.I := by simp [H,hfinal,w]
+  have hheight (s : ℝ) (hs : s ∈ Icc 0 T) : ε ≤ (H s).im := by
+    have hh := h.height_antitone hT.le hz (show T-s ∈ Icc 0 T by constructor <;> linarith [hs.1,hs.2])
+      ⟨hT.le,le_rfl⟩ (by linarith [hs.1] : T-s ≤ T)
+    simp only [hfinal,w,Complex.add_im,Complex.ofReal_im,Complex.mul_I_im,Complex.ofReal_re,zero_add] at hh
+    simpa only [H,Complex.sub_im,Complex.ofReal_im,sub_zero] using hh
+  have hHd (s : ℝ) (hs : s ∈ Icc 0 T) :
+      HasDerivWithinAt H (velocity ε (extendDriver T (reverseCentered U) s) (H s)) (Icc 0 T) s := by
+    have hst : T-s ∈ Icc 0 T := by constructor <;> linarith [hs.1,hs.2]
+    have htder : HasDerivWithinAt (fun r : ℝ => T-r) (-1) (Icc 0 T) s := by
+      simpa only [id_eq] using ((hasDerivAt_id s).const_sub T).hasDerivWithinAt
+    have hd := (h.hasDerivWithinAt_eval hT.le hU hz hst).scomp s htder
+      (show MapsTo (fun r : ℝ => T-r) (Icc 0 T) (Icc 0 T) from fun r hr => by constructor <;> linarith [hr.1,hr.2])
+    have hd' := hd.sub_const (driver T : ℂ)
+    apply hd'.congr_deriv
+    rw [velocity,LoewnerCutoff.velocity_eq (hheight s hs),extend_reverseCentered hT.ne' U s,he hst,hend]
+    dsimp only [H,Function.comp_def]
+    simp only [neg_smul,one_smul]
+    push_cast
+    ring
+  have hsol := solution_eq hε hHd (solution_deriv hε hT.le (reverseCentered U) ((ε:ℂ)*Complex.I))
+    (hH0.trans (solution_initial hε hT.le (reverseCentered U) ((ε:ℂ)*Complex.I)).symm)
+  have hendpoint := hsol ⟨hT.le,le_rfl⟩
+  simp only [H,sub_self,h.initial z hz.1.1] at hendpoint
+  dsimp only [tipApprox]
+  rw [← hendpoint,hend]
+  simp only [sub_add_cancel,z,w]
+
+end SLEExactGauge.ReverseLoewner
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.CanonicalTrace
+open ReverseLoewner
+
+def height (n : ℕ) : ℝ := 1 / ((n:ℝ)+1)
+
+lemma height_pos (n : ℕ) : 0 < height n := by unfold height; positivity
+
+lemma height_tendsto : Tendsto height atTop (𝓝[>] (0:ℝ)) := by
+  apply tendsto_nhdsWithin_iff.2
+  constructor
+  · change Tendsto (fun n : ℕ => (1:ℝ)/((n:ℝ)+1)) atTop (𝓝 0)
+    simpa only [Function.comp_def, Nat.cast_add, Nat.cast_one] using
+      (tendsto_const_div_atTop_nhds_zero_nat (1:ℝ)).comp (tendsto_add_atTop_nat 1)
+  · exact Filter.Eventually.of_forall height_pos
+
+def value {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (sample : Ω) : ℂ :=
+  if T = 0 then 0 else ContinuousDriver.selectedLimit 0
+    (fun n => tipApprox (height_pos n) T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T sample))
+
+lemma measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) :
+    @Measurable Ω ℂ (BrownianPredictable.pastSigma B T) (borel ℂ) (value κ B T) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  let : MeasurableSpace Ω := BrownianPredictable.pastSigma B T
+  change Measurable (fun sample => value κ B T sample)
+  by_cases hT : T = 0
+  · simp only [value,ite_eq_left hT]
+    exact measurable_const
+  · simp only [value,ite_eq_right hT]
+    apply ContinuousDriver.measurable_selectedLimit
+    intro n
+    exact (continuous_tipApprox (height_pos n) T.coe_nonneg).measurable.comp
+      (BrownianDriver.path_measurable_past B (Real.sqrt κ) T)
+
+lemma aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    (κ : ℝ) {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0) :
+    AEMeasurable (value κ B T) P := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  change AEMeasurable (fun sample => value κ B T sample) P
+  by_cases hT : T = 0
+  · simp only [value,ite_eq_left hT]
+    exact aemeasurable_const
+  · simp only [value,ite_eq_right hT]
+    apply ContinuousDriver.aemeasurable_selectedLimit
+    intro n
+    exact (continuous_tipApprox (height_pos n) T.coe_nonneg).measurable.comp_aemeasurable
+      (BrownianDriver.path_aemeasurable hB (Real.sqrt κ) T)
+
+lemma eq_source {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (sample : Ω)
+    {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ * B t.toNNReal sample) γ g f)
+    (hc : Continuous (fun t => B t sample)) (T : ℝ≥0) : value κ B T sample = γ T := by
+  by_cases hT : T = 0
+  · simpa [value,hT] using h.2.1.symm
+  · simp only [value,ite_eq_right hT]
+    apply ContinuousDriver.selectedLimit_eq
+    have hTp : 0 < (T:ℝ) := lt_of_le_of_ne T.coe_nonneg (by exact_mod_cast Ne.symm hT)
+    have he (n : ℕ) :
+        tipApprox (height_pos n) T.coe_nonneg (BrownianDriver.path B (Real.sqrt κ) T sample) =
+          f T ((Real.sqrt κ * B T sample:ℝ) + (height n:ℂ)*Complex.I) := by
+      simpa only [Real.toNNReal_coe] using
+        tipApprox_eq_inverse h (height_pos n) hTp
+          ((continuous_const.mul (hc.comp continuous_real_toNNReal)).continuousOn)
+          (BrownianDriver.path B (Real.sqrt κ) T sample)
+          (fun _ ht => BrownianDriver.extend_eq B (Real.sqrt κ) T sample hc ht)
+    have ht := (h.2.2.2.2.2.2.2.2.2.2.2.2.2 T T.coe_nonneg).comp height_tendsto
+    simpa only [Function.comp_def,he,Real.toNNReal_coe] using ht
+
+end SLEExactGauge.CanonicalTrace
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge
+
+theorem canonicalTraceFromBrownianPast {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    (κ : ℝ) (_hκ : 0 < κ ∧ κ < 8)
+    (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ sample ∂P, IsLoewnerTraceUsing
+      (fun t => Real.sqrt κ * B t.toNNReal sample) (γ sample) (g sample) (f sample)) :
+    ∃ Γ : ℝ≥0 → Ω → ℂ,
+      (∀ t, @Measurable Ω ℂ (BrownianPredictable.pastSigma B t) (borel ℂ) (Γ t)) ∧
+      (∀ t, AEMeasurable (Γ t) P) ∧
+      ∀ᵐ sample ∂P, ∀ t : ℝ≥0, Γ t sample = γ sample t := by
+  refine ⟨CanonicalTrace.value κ B, CanonicalTrace.measurable_past κ B,
+    CanonicalTrace.aemeasurable κ hB, ?_⟩
+  filter_upwards [hmaps,hB.cont] with sample hm hc
+  exact CanonicalTrace.eq_source κ B sample hm hc
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional
+
+lemma source_point_reach {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hzero : driver 0=0)
+    {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
+    (U : C(I, ℝ)) (hU : EqOn (extendDriver T U) driver (Icc 0 T))
+    (x a β δ d y : ℝ) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
+    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x) (hεx : 2*ε < x) (hy : δ*ε < y)
+    (w : ℂ) (hw : w ∈ localizationDomain γ S) (hclose : dist w (x:ℂ) < d)
+    (hwS : g S w=(driver S:ℂ)+(y:ℂ)*Complex.I) :
+    reach U hε hT x β a ((δ/(4*d))^β) := by
+  apply tip_forces_reach hε hT hS hST U (by rw [hU ⟨le_rfl,hT⟩,hzero]) x a β δ d y
+    hβ hδ hδhalf haδ hd hdx hεx hy (fun t => g t w) (h.continuous_eval hS hw)
+  · intro t ht
+    rw [hU ⟨ht.1.le,ht.2.le.trans hST⟩]
+    exact h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hw)
+  · rw [h.initial w hw.1.1]
+    exact (dist_eq_norm w (x:ℂ) ▸ hclose).le
+  · intro t ht
+    have hh := h.height_antitone hS hw ht ⟨hS,le_rfl⟩ ht.2
+    dsimp only at hh
+    rw [hwS] at hh
+    simpa using hh
+  · rw [hwS,hU ⟨hS,hST⟩]
+    simp
+
+lemma source_hit_eventually_reach {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ : ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    (x a β δ d : ℝ) (hx : 0 < x) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
+    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x)
+    (hhit : ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d) :
+    ∀ᶠ n : ℕ in atTop, reach (BrownianDriver.path B (Real.sqrt κ) (n+1) ω)
+      (CanonicalTrace.height_pos n) (n+1 : ℝ≥0).coe_nonneg x β a ((δ/(4*d))^β) := by
+  obtain ⟨t,ht,hclose⟩ := hhit
+  obtain ⟨w,y,hy,hwd,hw,hwS⟩ := inverse_tip_point h ht hclose
+  have hsmall : ∀ᶠ n : ℕ in atTop, CanonicalTrace.height n < min (x/2) (y/δ) :=
+    (CanonicalTrace.height_tendsto.mono_right nhdsWithin_le_nhds).eventually
+      (gt_mem_nhds (lt_min (half_pos hx) (div_pos hy hδ)))
+  have hlarge : ∀ᶠ n : ℕ in atTop, t ≤ ((n+1:ℝ≥0):ℝ) := by
+    have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
+    simpa only [NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using hh.eventually (eventually_ge_atTop t)
+  filter_upwards [hsmall,hlarge] with n hn htn
+  have hn' := lt_min_iff.mp hn
+  have hxcut : 2*CanonicalTrace.height n < x := by linarith [hn'.1]
+  have hycut : δ*CanonicalTrace.height n < y := by
+    have hh := (lt_div_iff₀ hδ).mp hn'.2
+    simpa only [mul_comm] using hh
+  apply source_point_reach h (by simp [hb0]) (CanonicalTrace.height_pos n)
+    (n+1:ℝ≥0).coe_nonneg ht htn (BrownianDriver.path B (Real.sqrt κ) (n+1) ω)
+    (fun s hs => BrownianDriver.extend_eq B _ (n+1) ω hc hs)
+    x a β δ d y hβ hδ hδhalf haδ hd hdx hxcut hycut w hw hwd hwS
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing
+
+lemma initial_bound {β x ε : ℝ} (hβ : 0 ≤ β) (hx : 0 < x) :
+    radial (-β) x ε ≤ x^(-β) := by
+  have hh := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hx)
+    (le_add_of_nonneg_right (sq_nonneg ε)) (show (-β)/2 ≤ 0 by linarith)
+  change (x^2+ε^2)^((-β)/2) ≤ x^(-β)
+  calc
+    _ ≤ (x^2)^((-β)/2) := hh
+    _ = x^(-β) := by
+      rw [← Real.rpow_natCast_mul hx.le]
+      congr 1
+      ring
+
+def uniformBudget {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a x L : ℝ) (n : ℕ) (ω : Ω) : ℝ :=
+  budget B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω +
+    (x^(-β)-radial (-β) x (CanonicalTrace.height n))
+
+lemma uniformBudget_mean {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (x L : ℝ) (hx : 0 < x) (n : ℕ) :
+    Integrable (uniformBudget B κ β a x L n) P ∧
+      (∫ ω,uniformBudget B κ β a x L n ω ∂P)=x^(-β) ∧
+      ∀ ω, 0 ≤ uniformBudget B κ β a x L n ω := by
+  have hT : (0:ℝ)<((n+1:ℝ≥0):ℝ) := by positivity
+  have hh := budget_mean hB κ β a hκ hβ ha hba (CanonicalTrace.height_pos n) (n+1) hT x L
+  refine ⟨hh.1.add (integrable_const _),?_,?_⟩
+  · change (∫ ω,budget B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω+
+      (x^(-β)-radial (-β) x (CanonicalTrace.height n)) ∂P)=_
+    rw [integral_add hh.1 (integrable_const _),hh.2]
+    simp
+  · intro ω
+    exact add_nonneg (budget_nonneg B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω)
+      (sub_nonneg.mpr (initial_bound hβ hx))
+
+lemma uniformBudget_dominates {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ β a x L : ℝ)
+    (hβ : 0 ≤ β) (hx : 0 < x) (n : ℕ) (ω : Ω) :
+    budget B κ β a (CanonicalTrace.height_pos n) (n+1) x L ω ≤ uniformBudget B κ β a x L n ω :=
+  le_add_of_nonneg_right (sub_nonneg.mpr (initial_bound hβ hx))
+
+lemma ratio_power_eq {x d δ β : ℝ} (hx : 0 < x) (hd : 0 < d) (hδ : 0 < δ) :
+    x^(-β)/(δ/(4*d))^β = (4*d/(δ*x))^β := by
+  rw [Real.rpow_neg hx.le,← Real.inv_rpow hx.le,← Real.div_rpow (inv_nonneg.mpr hx.le) (by positivity)]
+  congr 1
+  field_simp
+
+theorem positive_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    {x d : ℝ} (hx : 0 < x) (hd : 0 < d) :
+    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d} ≤ ENNReal.ofReal ((4*d/(δ*x))^β) := by
+  by_cases hdx : d < δ*x
+  · let L : ℝ := (δ/(4*d))^β
+    have hL : 0 < L := Real.rpow_pos_of_pos (by positivity) _
+    have hm (n : ℕ) := uniformBudget_mean hB κ β a hκ hβ.le ha hba x L hx n
+    have hE : ∀ᵐ ω ∂P, (∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d) →
+        ∀ᶠ n : ℕ in atTop, L ≤ uniformBudget B κ β a x L n ω := by
+      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hh hc hb0
+      intro hhit
+      filter_upwards [source_hit_eventually_reach B κ γ g f ω hh hc hb0 x a β δ d hx hβ hδ hδhalf haδ hd hdx hhit] with n hn
+      have hb := budget_dominates_reach B κ β a hκ hβ.le ha hba (CanonicalTrace.height_pos n)
+        (n+1) (by positivity) x L ω hc hb0 hn
+      exact hb.trans (uniformBudget_dominates B κ β a x L hβ.le hx n ω)
+    have hh := measure_eventually_budget (uniformBudget B κ β a x L) (x^(-β)) L hL
+      (fun n => (hm n).1) (fun n => (hm n).2.2) (fun n => (hm n).2.1) _ hE
+    rwa [ratio_power_eq hx hd hδ] at hh
+  · have hratio : 1 ≤ 4*d/(δ*x) := by
+      rw [le_div_iff₀ (mul_pos hδ hx)]
+      have hh : δ*x ≤ d := not_lt.1 hdx
+      linarith
+    have hp : 1 ≤ (4*d/(δ*x))^β := Real.one_le_rpow hratio hβ.le
+    exact (prob_le_one).trans (by simpa using ENNReal.ofReal_le_ofReal hp)
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory Metric
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional
+
+def mirror (z : ℂ) : ℂ := -(starRingEnd ℂ) z
+
+lemma mirror_im (z : ℂ) : (mirror z).im=z.im := by simp [mirror]
+
+lemma mirror_re (z : ℂ) : (mirror z).re= -z.re := by simp [mirror]
+
+lemma continuous_mirror : Continuous mirror := Complex.continuous_conj.neg
+
+lemma mirror_dist_real (w : ℂ) (x : ℝ) : ‖mirror w-(x:ℂ)‖ = dist w ((-x:ℝ):ℂ) := by
+  rw [dist_eq_norm]
+  have he : mirror w-(x:ℂ) = -(starRingEnd ℂ) (w-((-x:ℝ):ℂ)) := by simp [mirror]; ring
+  rw [he,norm_neg,Complex.norm_conj]
+
+lemma mirror_ode {G : ℝ → ℂ} {t u : ℝ} (hG : HasDerivAt G (2/(G t-(u:ℂ))) t) :
+    HasDerivAt (fun s => mirror (G s)) (2/(mirror (G t)-((-u:ℝ):ℂ))) t := by
+  have hh := (Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivAt t hG).neg
+  change HasDerivAt (fun s => -(starRingEnd ℂ) (G s)) (-(starRingEnd ℂ) (2/(G t-(u:ℂ)))) t at hh
+  apply hh.congr_deriv
+  simp only [map_div₀,map_ofNat,map_sub,Complex.conj_ofReal,Complex.ofReal_neg,mirror]
+  rw [show -(starRingEnd ℂ) (G t)- -(u:ℂ) = -((starRingEnd ℂ) (G t)-(u:ℂ)) by ring,div_neg]
+
+lemma negative_source_point_reach {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hzero : driver 0=0)
+    {ε T S : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (hS : 0 ≤ S) (hST : S ≤ T)
+    (U : C(I, ℝ)) (hU : EqOn (extendDriver T U) (fun s => -driver s) (Icc 0 T))
+    (x a β δ d y : ℝ) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
+    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x) (hεx : 2*ε < x) (hy : δ*ε < y)
+    (w : ℂ) (hw : w ∈ localizationDomain γ S) (hclose : dist w ((-x:ℝ):ℂ) < d)
+    (hwS : g S w=(driver S:ℂ)+(y:ℂ)*Complex.I) :
+    reach U hε hT x β a ((δ/(4*d))^β) := by
+  apply tip_forces_reach hε hT hS hST U (by rw [hU ⟨le_rfl,hT⟩]; simp [hzero]) x a β δ d y
+    hβ hδ hδhalf haδ hd hdx hεx hy (fun t => mirror (g t w))
+    (continuous_mirror.comp_continuousOn (h.continuous_eval hS hw))
+  · intro t ht
+    rw [hU ⟨ht.1.le,ht.2.le.trans hST⟩]
+    exact mirror_ode (h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hw))
+  · rw [h.initial w hw.1.1,mirror_dist_real]
+    exact hclose.le
+  · intro t ht
+    rw [mirror_im]
+    have hh := h.height_antitone hS hw ht ⟨hS,le_rfl⟩ ht.2
+    dsimp only at hh
+    rw [hwS] at hh
+    simpa using hh
+  · rw [mirror_re,hwS,hU ⟨hS,hST⟩]
+    simp
+
+lemma negative_source_hit_eventually_reach {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (κ : ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (h : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω=0)
+    (x a β δ d : ℝ) (hx : 0 < x) (hβ : 0 < β) (hδ : 0 < δ) (hδhalf : δ ≤ 1/2)
+    (haδ : a ≤ 2-4*δ) (hd : 0 < d) (hdx : d < δ*x)
+    (hhit : ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) ((-x:ℝ):ℂ) < d) :
+    ∀ᶠ n : ℕ in atTop, reach (BrownianDriver.path (-B) (Real.sqrt κ) (n+1) ω)
+      (CanonicalTrace.height_pos n) (n+1 : ℝ≥0).coe_nonneg x β a ((δ/(4*d))^β) := by
+  obtain ⟨t,ht,hclose⟩ := hhit
+  obtain ⟨w,y,hy,hwd,hw,hwS⟩ := inverse_tip_point h ht hclose
+  have hsmall : ∀ᶠ n : ℕ in atTop, CanonicalTrace.height n < min (x/2) (y/δ) :=
+    (CanonicalTrace.height_tendsto.mono_right nhdsWithin_le_nhds).eventually
+      (gt_mem_nhds (lt_min (half_pos hx) (div_pos hy hδ)))
+  have hlarge : ∀ᶠ n : ℕ in atTop, t ≤ ((n+1:ℝ≥0):ℝ) := by
+    have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
+      tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
+    simpa only [NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using hh.eventually (eventually_ge_atTop t)
+  filter_upwards [hsmall,hlarge] with n hn htn
+  have hn' := lt_min_iff.mp hn
+  have hxcut : 2*CanonicalTrace.height n < x := by linarith [hn'.1]
+  have hycut : δ*CanonicalTrace.height n < y := by
+    have hh := (lt_div_iff₀ hδ).mp hn'.2
+    simpa only [mul_comm] using hh
+  apply negative_source_point_reach h (by simp [hb0]) (CanonicalTrace.height_pos n)
+    (n+1:ℝ≥0).coe_nonneg ht htn (BrownianDriver.path (-B) (Real.sqrt κ) (n+1) ω)
+    (fun s hs => by
+      have he := BrownianDriver.extend_eq (-B) (Real.sqrt κ) (n+1) ω hc.neg hs
+      change extendDriver ((n+1:ℝ≥0):ℝ) (BrownianDriver.path (-B) (Real.sqrt κ) (n+1) ω) s =
+        Real.sqrt κ*(-B s.toNNReal ω) at he
+      simpa only [mul_neg] using he)
+    x a β δ d y hβ hδ hδhalf haδ hd hdx hxcut hycut w hw hwd hwS
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing
+
+theorem negative_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    {x d : ℝ} (hx : 0 < x) (hd : 0 < d) :
+    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) ((-x:ℝ):ℂ) < d} ≤ ENNReal.ofReal ((4*d/(δ*x))^β) := by
+  by_cases hdx : d < δ*x
+  · let L : ℝ := (δ/(4*d))^β
+    have hL : 0 < L := Real.rpow_pos_of_pos (by positivity) _
+    have hm (n : ℕ) := uniformBudget_mean hB.neg κ β a hκ hβ.le ha hba x L hx n
+    have hE : ∀ᵐ ω ∂P, (∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) ((-x:ℝ):ℂ) < d) →
+        ∀ᶠ n : ℕ in atTop, L ≤ uniformBudget (-B) κ β a x L n ω := by
+      filter_upwards [hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero] with ω hh hc hb0
+      intro hhit
+      filter_upwards [negative_source_hit_eventually_reach B κ γ g f ω hh hc hb0 x a β δ d hx hβ hδ hδhalf haδ hd hdx hhit] with n hn
+      have hb := budget_dominates_reach (-B) κ β a hκ hβ.le ha hba (CanonicalTrace.height_pos n)
+        (n+1) (by positivity) x L ω hc.neg (by simpa using congrArg Neg.neg hb0) hn
+      exact hb.trans (uniformBudget_dominates (-B) κ β a x L hβ.le hx n ω)
+    have hh := measure_eventually_budget (uniformBudget (-B) κ β a x L) (x^(-β)) L hL
+      (fun n => (hm n).1) (fun n => (hm n).2.2) (fun n => (hm n).2.1) _ hE
+    rwa [ratio_power_eq hx hd hδ] at hh
+  · have hratio : 1 ≤ 4*d/(δ*x) := by
+      rw [le_div_iff₀ (mul_pos hδ hx)]
+      have hh : δ*x ≤ d := not_lt.1 hdx
+      linarith
+    have hp : 1 ≤ (4*d/(δ*x))^β := Real.one_le_rpow hratio hβ.le
+    exact (prob_le_one).trans (by simpa using ENNReal.ofReal_le_ofReal hp)
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+
+theorem real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    {x d : ℝ} (hx : 0 < |x|) (hd : 0 < d) :
+    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d} ≤ ENNReal.ofReal ((4*d/(δ*|x|))^β) := by
+  rcases le_or_gt 0 x with hpos | hneg
+  · rw [abs_of_nonneg hpos] at hx ⊢
+    exact positive_real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps hx hd
+  · have hh := negative_real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps (neg_pos.2 hneg) hd
+    simpa only [neg_neg,abs_of_neg hneg] using hh
+
+lemma annulus_real_bound {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (κ β a δ : ℝ)
+    (hκ : 0 ≤ κ) (hβ : 0 < β) (ha : 0 ≤ a) (hba : κ*(β+1) ≤ 4+2*a)
+    (hδ : 0 < δ) (hδhalf : δ ≤ 1/2) (haδ : a ≤ 2-4*δ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    {x d : ℝ} (hx : 1/4 ≤ |x|) (hd : 0 < d) :
+    P {ω | ∃ t : ℝ, 0 ≤ t ∧ dist (γ ω t) (x:ℂ) < d} ≤ ENNReal.ofReal ((16/δ)^β*d^β) := by
+  have hxp : 0 < |x| := lt_of_lt_of_le (by norm_num) hx
+  refine (real_bound hB κ β a δ hκ hβ ha hba hδ hδhalf haδ γ g f hmaps hxp hd).trans
+    (ENNReal.ofReal_le_ofReal ?_)
+  have hr : 4*d/(δ*|x|) ≤ (16/δ)*d := by
+    rw [div_le_iff₀ (mul_pos hδ hxp)]
+    have hh := mul_le_mul_of_nonneg_left hx (show 0 ≤ 16*d by positivity)
+    field_simp
+    nlinarith
+  have hh := Real.rpow_le_rpow (by positivity : 0 ≤ 4*d/(δ*|x|)) hr hβ.le
+  rwa [Real.mul_rpow (by positivity : 0 ≤ 16/δ) hd.le] at hh
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.AngularBarrier
+
+def lowerGamma (b s : ℝ) : ℝ := ∫ x in (0:ℝ)..s, Real.exp (-x)*x^(b-1)
+
+def cdf (b s : ℝ) : ℝ := lowerGamma b s / Real.Gamma b
+
+def pdf (b s : ℝ) : ℝ := Real.exp (-s)*s^(b-1)/Real.Gamma b
+
+lemma integrable_kernel {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) :
+    IntervalIntegrable (fun x : ℝ => Real.exp (-x)*x^(b-1)) volume 0 s := by
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hs]
+  exact (Real.GammaIntegral_convergent hb).mono_set Ioc_subset_Ioi_self
+
+lemma kernel_continuousAt (b : ℝ) {s : ℝ} (hs : 0 < s) :
+    ContinuousAt (fun x : ℝ => Real.exp (-x)*x^(b-1)) s :=
+  Real.continuous_exp.continuousAt.comp continuousAt_neg |>.mul
+    (continuousAt_id.rpow_const (Or.inl hs.ne'))
+
+lemma lowerGamma_nonneg {b : ℝ} (_hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) : 0 ≤ lowerGamma b s := by
+  apply intervalIntegral.integral_nonneg hs
+  intro x hx
+  exact mul_nonneg (Real.exp_pos _).le (Real.rpow_nonneg hx.1 _)
+
+lemma cdf_nonneg {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) : 0 ≤ cdf b s :=
+  div_nonneg (lowerGamma_nonneg hb hs) (Real.Gamma_pos_of_pos hb).le
+
+lemma pdf_pos {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) : 0 < pdf b s :=
+  div_pos (mul_pos (Real.exp_pos _) (Real.rpow_pos_of_pos hs _)) (Real.Gamma_pos_of_pos hb)
+
+lemma power_integral {b : ℝ} (hb : 0 < b) {s : ℝ} (_hs : 0 ≤ s) :
+    (∫ x in (0:ℝ)..s, x^(b-1)) = s^b/b := by
+  rw [integral_rpow (Or.inl (by linarith : -1 < b-1))]
+  rw [show b-1+1 = b by ring, Real.zero_rpow hb.ne', sub_zero]
+
+lemma cdf_le_power {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 ≤ s) :
+    cdf b s ≤ s^b/(b*Real.Gamma b) := by
+  have hmono : lowerGamma b s ≤ ∫ x in (0:ℝ)..s, x^(b-1) := by
+    apply intervalIntegral.integral_mono_on hs (integrable_kernel hb hs)
+      (intervalIntegral.intervalIntegrable_rpow' (by linarith : -1 < b-1))
+    intro x hx
+    simpa only [one_mul] using mul_le_mul_of_nonneg_right
+      (Real.exp_le_one_iff.2 (neg_nonpos.2 hx.1)) (Real.rpow_nonneg hx.1 (b-1))
+  rw [power_integral hb hs] at hmono
+  have hh := div_le_div_of_nonneg_right hmono (Real.Gamma_pos_of_pos hb).le
+  simpa only [cdf, div_div] using hh
+
+lemma hasDerivAt_lowerGamma {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
+    HasDerivAt (lowerGamma b) (Real.exp (-s)*s^(b-1)) s := by
+  apply intervalIntegral.integral_hasDerivAt_right (integrable_kernel hb hs.le)
+  · exact (Real.measurable_exp.comp measurable_neg |>.mul
+      (measurable_id.pow_const (b-1))).aestronglyMeasurable.stronglyMeasurableAtFilter
+  · exact kernel_continuousAt b hs
+
+lemma hasDerivAt_cdf {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
+    HasDerivAt (cdf b) (pdf b s) s :=
+  (hasDerivAt_lowerGamma hb hs).div_const (Real.Gamma b)
+
+lemma deriv_cdf {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
+    deriv (cdf b) s = pdf b s := (hasDerivAt_cdf hb hs).deriv
+
+lemma hasDerivAt_pdf {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : 0 < s) :
+    HasDerivAt (pdf b) (pdf b s*((b-1)/s-1)) s := by
+  have hd := (((hasDerivAt_neg s).exp).mul
+    ((hasDerivAt_id s).rpow_const (p:=b-1) (Or.inl hs.ne'))).div_const (Real.Gamma b)
+  apply hd.congr_deriv
+  unfold pdf
+  simp only [id_eq]
+  rw [Real.rpow_sub_one hs.ne' (b-1)]
+  field_simp
+  ring
+
+lemma lowerGamma_tendsto {b : ℝ} (hb : 0 < b) :
+    Tendsto (lowerGamma b) atTop (𝓝 (Real.Gamma b)) := by
+  rw [Real.Gamma_eq_integral hb]
+  exact intervalIntegral_tendsto_integral_Ioi 0 (Real.GammaIntegral_convergent hb) tendsto_id
+
+lemma cdf_tendsto {b : ℝ} (hb : 0 < b) : Tendsto (cdf b) atTop (𝓝 1) := by
+  change Tendsto (fun s => lowerGamma b s / Real.Gamma b) atTop (𝓝 1)
+  have hh := (lowerGamma_tendsto hb).div_const (Real.Gamma b)
+  convert hh using 1
+  rw [div_self (Real.Gamma_pos_of_pos hb).ne']
+
+lemma power_mul {b x : ℝ} (hb : 0 < b) (hx : 0 ≤ x) :
+    x^(b-1)*x = x^b := by
+  rcases eq_or_lt_of_le hx with h | h
+  · subst x; simp [Real.zero_rpow hb.ne']
+  · rw [← Real.rpow_add_one h.ne', sub_add_cancel]
+
+lemma kernel_lower {b s x : ℝ} (hb : 0 < b) (hx : 0 ≤ x) :
+    Real.exp (-s)*((1+s)*x^(b-1)-x^b) ≤ Real.exp (-x)*x^(b-1) := by
+  have hh := Real.add_one_le_exp (s-x)
+  have he : Real.exp (-s)*Real.exp (s-x) = Real.exp (-x) := by
+    rw [← Real.exp_add]; congr 1; ring
+  have hi := mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_left hh (Real.exp_pos (-s)).le) (Real.rpow_nonneg hx (b-1))
+  rw [he] at hi
+  calc
+    _ = Real.exp (-s)*(s-x+1)*x^(b-1) := by rw [← power_mul hb hx]; ring
+    _ ≤ _ := hi
+
+lemma lowerGamma_lower {b s : ℝ} (hb : 0 < b) (hs : 0 ≤ s) :
+    Real.exp (-s)*((1+s)*s^b/b-s^(b+1)/(b+1)) ≤ lowerGamma b s := by
+  have hi₁ : IntervalIntegrable (fun x : ℝ => x^(b-1)) volume 0 s :=
+    intervalIntegral.intervalIntegrable_rpow' (by linarith)
+  have hi₂ : IntervalIntegrable (fun x : ℝ => x^b) volume 0 s :=
+    intervalIntegral.intervalIntegrable_rpow' (by linarith)
+  have hh := intervalIntegral.integral_mono_on hs
+    (((hi₁.const_mul (1+s)).sub hi₂).const_mul (Real.exp (-s)))
+    (integrable_kernel hb hs) (fun x hx => kernel_lower hb hx.1 (s:=s))
+  rw [intervalIntegral.integral_const_mul,
+    intervalIntegral.integral_sub (hi₁.const_mul _) hi₂,
+    intervalIntegral.integral_const_mul, power_integral hb hs] at hh
+  have hi₂eq : (∫ x in (0:ℝ)..s, x^b) = s^(b+1)/(b+1) := by
+    simpa only [add_sub_cancel_right] using power_integral (by linarith : 0 < b+1) hs
+  rw [hi₂eq] at hh
+  simpa only [lowerGamma, mul_div_assoc] using hh
+
+lemma cdf_pdf_inequality {b s : ℝ} (hb : 0 < b) (hs : 0 < s) :
+    s^2*pdf b s/(b+1) ≤ b*cdf b s-s*pdf b s := by
+  have hl := lowerGamma_lower hb hs.le
+  have hg : 0 < Real.Gamma b := Real.Gamma_pos_of_pos hb
+  have hbp : 0 < b+1 := by linarith
+  rw [Real.rpow_add_one hs.ne' b] at hl
+  have heq : s^b = s^(b-1)*s := (power_mul hb hs.le).symm
+  rw [heq] at hl
+  unfold cdf pdf
+  apply (div_le_div_iff_of_pos_right hg).1
+  have hh := mul_le_mul_of_nonneg_left hl hb.le
+  field_simp [hb.ne', hbp.ne'] at hh ⊢
+  nlinarith
+
+lemma contDiffAt_pdf (n : WithTop ℕ∞) (b : ℝ) {s : ℝ} (hs : 0 < s) :
+    ContDiffAt ℝ n (pdf b) s := by
+  have he : ContDiffAt ℝ n (fun x : ℝ => Real.exp (-x)) s :=
+    (contDiffAt_id.neg).exp
+  have hp : ContDiffAt ℝ n (fun x : ℝ => x^(b-1)) s :=
+    contDiffAt_id.rpow contDiffAt_const hs.ne'
+  exact (he.mul hp).div_const _
+
+lemma contDiffOn_cdf {b : ℝ} (hb : 0 < b) (n : ℕ) :
+    ContDiffOn ℝ (n+1) (cdf b) (Ioi 0) := by
+  rw [contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioi]
+  refine ⟨fun s hs => (hasDerivAt_cdf hb hs).differentiableAt.differentiableWithinAt,
+    ?_, ?_⟩
+  · intro hn; simp at hn
+  · have hh : ContDiffOn ℝ n (pdf b) (Ioi 0) :=
+      fun s hs => (contDiffAt_pdf n b hs).contDiffWithinAt
+    exact hh.congr (fun s hs => deriv_cdf hb hs)
+
+lemma contDiffAt_cdf {b : ℝ} (hb : 0 < b) (n : ℕ) {s : ℝ} (hs : 0 < s) :
+    ContDiffAt ℝ n (cdf b) s := by
+  have hh := (contDiffOn_cdf hb n).contDiffAt (isOpen_Ioi.mem_nhds hs)
+  exact hh.of_le (by exact_mod_cast Nat.le_succ n)
+
+def scale (l b t : ℝ) := (Real.exp (l*t)-1)/(2*(b+1))
+
+def value (l b t q : ℝ) := Real.exp (l*b/2*t)*cdf b (q/scale l b t)
+
+def valueQ (l b t q : ℝ) := Real.exp (l*b/2*t)*pdf b (q/scale l b t)/scale l b t
+
+def valueQQ (l b t q : ℝ) := Real.exp (l*b/2*t)*pdf b (q/scale l b t)*
+  ((b-1)/(q/scale l b t)-1)/(scale l b t)^2
+
+def valueT (l b t q : ℝ) := l*b/2*value l b t q -
+  Real.exp (l*b/2*t)*pdf b (q/scale l b t)*(q/scale l b t)*
+    (l+l/(2*(b+1)*scale l b t))
+
+lemma scale_pos {l b t : ℝ} (hl : 0 < l) (hb : 0 < b) (ht : 0 < t) :
+    0 < scale l b t := by
+  apply div_pos
+  · exact sub_pos.2 (Real.one_lt_exp_iff.2 (mul_pos hl ht))
+  · positivity
+
+lemma hasDerivAt_scale (l b t : ℝ) :
+    HasDerivAt (scale l b) (l*Real.exp (l*t)/(2*(b+1))) t := by
+  apply (((((hasDerivAt_id t).const_mul l).exp).sub_const 1).div_const (2*(b+1))).congr_deriv
+  simp only [id_eq]
+  ring
+
+lemma hasDerivAt_value_q {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
+    (ht : 0 < t) (hq : 0 < q) : HasDerivAt (value l b t) (valueQ l b t q) q := by
+  have hs := scale_pos hl hb ht
+  have hh := ((hasDerivAt_cdf hb (div_pos hq hs)).comp q
+    ((hasDerivAt_id q).div_const (scale l b t))).const_mul (Real.exp (l*b/2*t))
+  apply hh.congr_deriv
+  dsimp only [valueQ]
+  ring
+
+lemma hasDerivAt_valueQ_q {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
+    (ht : 0 < t) (hq : 0 < q) : HasDerivAt (valueQ l b t) (valueQQ l b t q) q := by
+  have hs := scale_pos hl hb ht
+  have hh := (((hasDerivAt_pdf hb (div_pos hq hs)).comp q
+    ((hasDerivAt_id q).div_const (scale l b t))).const_mul (Real.exp (l*b/2*t))).div_const (scale l b t)
+  apply hh.congr_deriv
+  dsimp only [valueQQ]
+  ring
+
+lemma hasDerivAt_value_t {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
+    (ht : 0 < t) (hq : 0 < q) :
+    HasDerivAt (fun s => value l b s q) (valueT l b t q) t := by
+  have hs := scale_pos hl hb ht
+  have hbp : b+1 ≠ 0 := by positivity
+  have hh := ((((hasDerivAt_id t).const_mul (l*b/2)).exp).mul
+    ((hasDerivAt_cdf hb (div_pos hq hs)).comp t ((hasDerivAt_const t q).div (hasDerivAt_scale l b t) hs.ne')))
+  apply hh.congr_deriv
+  simp only [id_eq, Function.comp_def, Pi.div_apply]
+  dsimp only [valueT, value]
+  have he : Real.exp (l*t) = 2*(b+1)*scale l b t+1 := by unfold scale; field_simp; ring
+  rw [he]
+  field_simp
+  ring
+
+theorem supersolution {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
+    (ht : 0 < t) (hq : 0 < q) (hq1 : q ≤ 1) :
+    l*q*(1-q)*valueQQ l b t q +
+      l*((1-b)-(3/2-b)*q)*valueQ l b t q ≤ valueT l b t q := by
+  let α := scale l b t
+  let s := q/α
+  let W := pdf b s
+  let F := cdf b s
+  have hα : 0 < α := scale_pos hl hb ht
+  have hs : 0 < s := div_pos hq hα
+  have hW : 0 ≤ W := (pdf_pos hb hs).le
+  have hbp : 0 < b+1 := by linarith
+  have hi : s^2*W/(b+1) ≤ b*F-s*W := cdf_pdf_inequality hb hs
+  have hαs : α*s = q := by dsimp [s]; field_simp
+  have hA : 0 ≤ 1-1/(2*(b+1)) := by
+    rw [sub_nonneg, div_le_one (by positivity)]
+    linarith
+  have hrest : 0 ≤ l*(1-1/(2*(b+1)))*(1-q)*s*W/α := by
+    exact div_nonneg (mul_nonneg (mul_nonneg (mul_nonneg (mul_nonneg hl.le hA) (sub_nonneg.2 hq1)) hs.le) hW) hα.le
+  have hmain : 0 ≤ l/2*(b*F-s*W-s^2*W/(b+1)) := mul_nonneg (by positivity) (sub_nonneg.2 hi)
+  have he : valueT l b t q - (l*q*(1-q)*valueQQ l b t q +
+        l*((1-b)-(3/2-b)*q)*valueQ l b t q) =
+      Real.exp (l*b/2*t) *
+        (l/2*(b*F-s*W-s^2*W/(b+1)) + l*(1-1/(2*(b+1)))*(1-q)*s*W/α) := by
+    dsimp only [valueT, value, valueQQ, valueQ]
+    change l*b/2*(Real.exp (l*b/2*t)*F) - Real.exp (l*b/2*t)*W*s*
+        (l+l/(2*(b+1)*α)) - (l*q*(1-q)*(Real.exp (l*b/2*t)*W*((b-1)/s-1)/α^2) +
+        l*((1-b)-(3/2-b)*q)*(Real.exp (l*b/2*t)*W/α)) = _
+    dsimp only [s]
+    field_simp
+    ring
+  have hh : 0 ≤ valueT l b t q - (l*q*(1-q)*valueQQ l b t q +
+      l*((1-b)-(3/2-b)*q)*valueQ l b t q) := by
+    rw [he]
+    exact mul_nonneg (Real.exp_pos _).le (add_nonneg hmain hrest)
+  exact sub_nonneg.1 hh
+
+lemma value_nonneg {l b t q : ℝ} (hl : 0 < l) (hb : 0 < b)
+    (ht : 0 < t) (hq : 0 ≤ q) : 0 ≤ value l b t q :=
+  mul_nonneg (Real.exp_pos _).le (cdf_nonneg hb (div_nonneg hq (scale_pos hl hb ht).le))
+
+lemma contDiffAt_value {l b : ℝ} (hl : 0 < l) (hb : 0 < b) (n : ℕ)
+    {v : ℝ × ℝ} (ht : 0 < v.1) (hq : 0 < v.2) :
+    ContDiffAt ℝ n (fun v : ℝ × ℝ => value l b v.1 v.2) v := by
+  have he : ContDiffAt ℝ n (fun v : ℝ × ℝ => Real.exp (l*b/2*v.1)) v := by fun_prop
+  have ha : ContDiffAt ℝ n (fun v : ℝ × ℝ => scale l b v.1) v := by
+    unfold scale; fun_prop
+  have hs := scale_pos hl hb ht
+  have hqf : ContDiffAt ℝ n (fun v : ℝ × ℝ => v.2/scale l b v.1) v :=
+    contDiffAt_snd.div ha hs.ne'
+  have hh := (contDiffAt_cdf hb n (div_pos hq hs)).comp v hqf
+  dsimp only [Function.comp_def] at hh
+  exact he.mul hh
+
+lemma continuousAt_value {l b : ℝ} (hl : 0 < l) (hb : 0 < b)
+    {v : ℝ × ℝ} (ht : 0 < v.1) (hq : 0 < v.2) :
+    ContinuousAt (fun v : ℝ × ℝ => value l b v.1 v.2) v :=
+  (contDiffAt_value hl hb 0 ht hq).continuousAt
+
+lemma scale_tendsto {l b : ℝ} (hl : 0 < l) (hb : 0 < b) :
+    Tendsto (scale l b) (𝓝[>] (0:ℝ)) (𝓝[>] (0:ℝ)) := by
+  apply tendsto_nhdsWithin_iff.2
+  constructor
+  · have hh : Continuous (scale l b) := by unfold scale; fun_prop
+    simpa [scale] using (hh.continuousAt (x:=0)).tendsto.mono_left nhdsWithin_le_nhds
+  · filter_upwards [self_mem_nhdsWithin] with t ht
+    exact scale_pos hl hb ht
+
+lemma value_tendsto_zero_time {l b q : ℝ} (hl : 0 < l) (hb : 0 < b) (hq : 0 < q) :
+    Tendsto (fun t => value l b t q) (𝓝[>] (0:ℝ)) (𝓝 1) := by
+  have hs : Tendsto (fun t => q/scale l b t) (𝓝[>] (0:ℝ)) atTop := by
+    simpa only [div_eq_mul_inv, Function.comp_def] using
+      (tendsto_inv_nhdsGT_zero.comp (scale_tendsto hl hb)).const_mul_atTop hq
+  have he : Tendsto (fun t : ℝ => Real.exp (l*b/2*t)) (𝓝[>] (0:ℝ)) (𝓝 1) := by
+    have hh : Continuous (fun t : ℝ => Real.exp (l*b/2*t)) := by fun_prop
+    simpa using (hh.continuousAt (x:=0)).tendsto.mono_left nhdsWithin_le_nhds
+  simpa only [value, one_mul, Function.comp_def] using he.mul ((cdf_tendsto hb).comp hs)
+
+end SLEExactGauge.AngularBarrier
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/SlabMoments.lean b/lean/OAI/Probability/SLELocalization/SlabMoments.lean
index 8f56fa9f81ec891d7e9d935f9fc10848814ebd1a..e8ba84f0f32a4f0282a6f08684cc48502e65bd94 100644
--- a/lean/OAI/Probability/SLELocalization/SlabMoments.lean
+++ b/lean/OAI/Probability/SLELocalization/SlabMoments.lean
@@ -1,1013 +1,1013 @@
-import Mathlib
-import OAI.Probability.SLELocalization.IntegratedMeshSimple
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Mesh estimates. -/
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory
-open scoped Topology ENNReal
-namespace SLEExactGauge.Mesh
-
-lemma asynchronous_bracket_bound_active {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
-    {A : Set ℂ} (hA : MeasurableSet A) {a b : ℂ → ℝ} {D : ℂ → ℝ → ℝ}
-    (ha : Measurable a) (hb : Measurable b)
-    (hB : Measurable (fun q : ℝ × ℂ => slab (D q.2) (a q.2) (b q.2) q.1))
-    (L : ℂ → ℂ → ℝ≥0∞) (hL : Measurable (Function.uncurry L))
-    (hLeq : ∀ z ∈ A, ∀ w ∈ A, L z w = laterGreen g a b z w)
-    {K R C : ℝ} (hK : 0 ≤ K) (hR : 0 < R)
-    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
-    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
-    (happroach : ∀ z ∈ A, a z < b z → dist z (γ (a z)) ≤ R)
-    (hcoeff : ∀ z ∈ A, ∀ t ∈ Icc (a z) (b z),
-      |slab (D z) (a z) (b z) t| ≤ K*poissonFactor driver g t z)
-    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
-        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2)) :
-    (∫⁻ t, ENNReal.ofReal (|∫ z in A, slab (D z) (a z) (b z) t|^2)) ≤
-      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (C*R^2)*volume A := by
-  let A' := A ∩ {z | a z < b z}
-  have hA' : MeasurableSet A' := hA.inter (measurableSet_lt ha hb)
-  have hs : A' ⊆ A := inter_subset_left
-  have he (t : ℝ) : (∫ z in A, slab (D z) (a z) (b z) t) =
-      ∫ z in A', slab (D z) (a z) (b z) t := by
-    apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero hA hs
-    intro z hz
-    have hab' : a z = b z := le_antisymm (hab z hz.1) (not_lt.1 (fun hh => hz.2 ⟨hz.1,hh⟩))
-    simp only [slab,hab',sub_self]
-  simp only [he]
-  apply (asynchronous_bracket_bound_with_kernel (B := fun t z => slab (D z) (a z) (b z) t) h hU hA' hB L hL
-    (fun z hz w hw => hLeq z hz.1 w hw.1) hK hR
-    (fun z hz => hstart z hz.1) (fun z hz => hab z hz.1)
-    (fun z hz => hlive z hz.1) (fun z hz => happroach z hz.1 hz.2)
-    (fun z hz t ht => slab_eq_zero_of_not_mem (hab z hz.1) ht)
-    (fun z hz t ht => hcoeff z hz.1 t ht) hlayer).trans
-  have hv : volume A' ≤ volume A := measure_mono hs
-  exact mul_le_mul_right hv _
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric MeasureTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
-local instance slabMomentsBracketRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance slabMomentsBracketRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance slabMomentsBracketComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance slabMomentsBracketComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-lemma measurable_halfPlaneGreen : Measurable (fun q : ℂ × ℂ => halfPlaneGreen q.1 q.2) := by
-  unfold halfPlaneGreen
-  fun_prop
-
-lemma observed_bracket {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
-    (C : FiniteTilt.CommonCutoff A u) (hA : MeasurableSet A)
-    (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
-    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
-    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q))
-    (e r : ℝ) (he : 0 < e) (her : e ≤ r) (hes : ∀ z ∈ A, e < z.im)
-    (J : ℝ) (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain (γ ω) t,
-      (∫⁻ w in {w : ℂ | w ∈ localizationDomain (γ ω) t ∧ infDist w (localizationDomain (γ ω) t)ᶜ ≤ r},
-        ENNReal.ofReal (halfPlaneGreen (g ω t z) (g ω t w))) ≤ ENNReal.ofReal (J*r^2)) :
-    let U := BrownianDriver.path B (Real.sqrt κ) C.horizon ω
-    let a := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle r ((U,V),z)
-    let b := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e ((U,V),z)
-    let D := fun z t => iteratedDeriv 1
-      (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (B t.toNNReal ω)
-    (∫⁻ t, ENNReal.ofReal (|∫ z in A, slab (D z) (a z) (b z) t|^2)) ≤
-      2*ENNReal.ofReal ((Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2)*ENNReal.ofReal (J*r^2)*volume A := by
-  intro U a b D
-  have ham : Measurable a := by
-    have hh := (measurable_observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle r).comp
-      (measurable_const.prodMk measurable_id : Measurable (fun z : ℂ => ((U,V),z)))
-    exact hh
-  have hbm : Measurable b := by
-    have hh := (measurable_observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e).comp
-      (measurable_const.prodMk measurable_id : Measurable (fun z : ℂ => ((U,V),z)))
-    exact hh
-  have hDc : Continuous (fun q : ℝ × ℂ => D q.2 q.1) := by
-    have hp : Continuous (fun q : ℝ × ℂ => (((U,q.2),q.1),B q.1.toNNReal ω)) :=
-      ((continuous_const.prodMk continuous_snd).prodMk continuous_fst).prodMk
-        ((hc.comp continuous_real_toNNReal).comp continuous_fst)
-    have hh := (continuous_mark_first C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ)).comp hp
-    exact hh
-  have hpos (z : ℂ) (hz : z ∈ A) (t : ℝ) (ht : t ∈ Icc 0 ((C.at z hz).stopTime κ B ω)) :
-      g ω t z = position C.height_pos C.horizon.coe_nonneg U z t := by
-    let Cz := C.at z hz
-    have h0 := Cz.initial_deficit_brownian κ B ω (hhalf z hz) (hu z hz) hc hb0
-    have hs := source_until_stop Cz.height_pos Cz.horizon_pos hm _ z _ Cz.angle (Cz.path_eq κ B ω hc) h0
-    exact hs.2 ht
-  have hat (z : ℂ) (hz : z ∈ A) : a z = (C.at z hz).physicalTime κ B γ r ω :=
-    observedTime_eq C κ B γ g f ω hm hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) V hV r
-  have hbt (z : ℂ) (hz : z ∈ A) : b z = (C.at z hz).physicalTime κ B γ e ω :=
-    observedTime_eq C κ B γ g f ω hm hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) V hV e
-  have hab (z : ℂ) : a z ≤ b z := observedTime_antitone C.height_pos C.horizon.coe_nonneg ρ C.angle ((U,V),z) her
-  have ha0 (z : ℂ) : 0 ≤ a z := (observedTime_bounds C.height_pos C.horizon.coe_nonneg ρ C.angle r ((U,V),z)).1
-  have hbz (z : ℂ) (hz : z ∈ A) : b z ≤ (C.at z hz).stopTime κ B ω := by
-    rw [hbt z hz]; exact ((C.at z hz).physicalTime_bounds κ B γ e ω).2
-  let L : ℂ → ℂ → ℝ≥0∞ := fun z w =>
-    if a w ≤ a z ∧ a z ≤ b w then ENNReal.ofReal (halfPlaneGreen
-      (position C.height_pos C.horizon.coe_nonneg U z (a z))
-      (position C.height_pos C.horizon.coe_nonneg U w (a z))) else 0
-  have hL : Measurable (Function.uncurry L) := by
-    have h1 : Measurable (fun q : ℂ × ℂ => ((U,q.1),a q.1)) :=
-      (measurable_const.prodMk measurable_fst).prodMk (ham.comp measurable_fst)
-    have h2 : Measurable (fun q : ℂ × ℂ => ((U,q.2),a q.1)) :=
-      (measurable_const.prodMk measurable_snd).prodMk (ham.comp measurable_fst)
-    have hp1 := (continuous_mark_position C.height_pos C.horizon.coe_nonneg).measurable.comp h1
-    have hp2 := (continuous_mark_position C.height_pos C.horizon.coe_nonneg).measurable.comp h2
-    have hh := (measurable_halfPlaneGreen.comp (hp1.prodMk hp2)).ennreal_ofReal
-    exact hh.ite ((measurableSet_le (ham.comp measurable_snd) (ham.comp measurable_fst)).inter
-      (measurableSet_le (ham.comp measurable_fst) (hbm.comp measurable_snd))) measurable_const
-  have hLe (z : ℂ) (hz : z ∈ A) (w : ℂ) (hw : w ∈ A) : L z w = laterGreen (g ω) a b z w := by
-    dsimp only [L,laterGreen]
-    split_ifs with hh
-    · rw [← hpos z hz (a z) ⟨ha0 z,(hab z).trans (hbz z hz)⟩,
-        ← hpos w hw (a z) ⟨ha0 z,hh.2.trans (hbz w hw)⟩]
-      rfl
-    · rfl
-  have haexp : 0 ≤ 8/κ-1 := by
-    have hh : 1 ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by linarith [hκ.2])
-    linarith
-  have hKn : 0 ≤ Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)) := by positivity
-  apply asynchronous_bracket_bound_active hm
-    (continuous_const.mul (hc.comp continuous_real_toNNReal)) hA ham hbm
-    (measurable_slab hDc.measurable ham hbm) L hL hLe hKn (he.trans_le her)
-    (fun z _ => ha0 z) (fun z _ => hab z)
-  · intro z hz
-    exact survivingDomain_mono (hbz z hz) ((C.at z hz).source_live κ B ω hm (hhalf z hz) (hu z hz) hc hb0)
-  · intro z hz hh
-    exact observedTime_approach C.height_pos C.horizon_pos ρ C.angle e r U V (γ ω) hV z hh
-  · intro z hz t ht
-    have hti : t ∈ Icc 0 ((C.at z hz).physicalTime κ B γ e ω) := by
-      rw [← hbt z hz]; exact ⟨(ha0 z).trans ht.1,ht.2⟩
-    have hb := (C.at z hz).first_le_poisson κ hκ B γ g f ω hm hc hb0 (hhalf z hz) (hu z hz) he (hes z hz) hti
-    have hl := survivingDomain_mono (hti.2.trans ((C.at z hz).physicalTime_bounds κ B γ e ω).2)
-      ((C.at z hz).source_live κ B ω hm (hhalf z hz) (hu z hz) hc hb0)
-    have hn : 0 ≤ poissonFactor (fun t => Real.sqrt κ*B t.toNNReal ω) (g ω) t z :=
-      div_nonneg (hm.im_pos hti.1 hl).le (sq_nonneg _)
-    exact slab_abs_le (hab z) (mul_nonneg hKn hn) hb
-  · exact hlayer
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-
-lemma gaussian_exp_compensated (v : ℝ≥0) (x a θ : ℝ) :
-    (∫⁻ y : ℝ, ENNReal.ofReal (Real.exp (x+θ*a*y-θ^2*a^2*(v:ℝ)/2))
-      ∂gaussianReal 0 v) = ENNReal.ofReal (Real.exp x) := by
-  have he (y : ℝ) : Real.exp (x+θ*a*y-θ^2*a^2*(v:ℝ)/2) =
-      Real.exp (x-θ^2*a^2*(v:ℝ)/2)*Real.exp ((θ*a)*y) := by
-    rw [← Real.exp_add]
-    congr 1
-    ring
-  simp_rw [he,ENNReal.ofReal_mul (Real.exp_pos _).le]
-  rw [lintegral_const_mul _ (by fun_prop),← ofReal_integral_eq_lintegral_ofReal
-    (integrable_exp_mul_gaussianReal (θ*a)) (Filter.Eventually.of_forall (fun y => (Real.exp_pos _).le))]
-  have hm := congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := v)) (θ*a)
-  change (∫ y : ℝ, Real.exp ((θ*a)*y) ∂gaussianReal 0 v) = _ at hm
-  rw [hm,← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add]
-  congr 2
-  ring
-
-lemma past_exp_step {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) {s t : ℝ≥0} (hst : s ≤ t)
-    (X A : Ω → ℝ) (hX : Measurable[pastSigma B s] X)
-    (hA : Measurable[pastSigma B s] A) (hXa : AEMeasurable X P)
-    (hAa : AEMeasurable A P) (θ : ℝ) :
-    (∫⁻ sample, ENNReal.ofReal (Real.exp (X sample+θ*A sample*(B t sample-B s sample)-
-      θ^2*(A sample)^2*((t-s:ℝ≥0):ℝ)/2)) ∂P) =
-      ∫⁻ sample, ENNReal.ofReal (Real.exp (X sample)) ∂P := by
-  let V : Ω → ℝ × ℝ := fun sample => (X sample,A sample)
-  have hVm : Measurable[pastSigma B s] V := hX.prodMk hA
-  have hVa : AEMeasurable V P := hXa.prodMk hAa
-  have hi := brownian_indep_of_past_measurable hB s (t-s) hVm
-  rw [add_tsub_cancel_of_le hst] at hi
-  have hl : HasLaw V (P.map V) P := ⟨hVa,rfl⟩
-  have hprod := IndepFun.hasLaw_prod hl (increment_hasLaw hB hst) hi
-  let F : (ℝ × ℝ) × ℝ → ℝ≥0∞ := fun q =>
-    ENNReal.ofReal (Real.exp (q.1.1+θ*q.1.2*q.2-θ^2*q.1.2^2*((t-s:ℝ≥0):ℝ)/2))
-  have hFm : Measurable F := by fun_prop
-  have he := hprod.lintegral_comp hFm.aemeasurable
-  change (∫⁻ sample, F (V sample,B t sample-B s sample) ∂P) = _ at he
-  rw [he,lintegral_prod _ hFm.aemeasurable]
-  simp only [F,gaussian_exp_compensated]
-  have hmap := hl.lintegral_comp
-    (show AEMeasurable (fun q : ℝ × ℝ => ENNReal.ofReal (Real.exp q.1)) (P.map V) by fun_prop)
-  exact hmap.symm
-
-def exponentialLogSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (t : ℕ → ℝ≥0) (A : ℕ → Ω → ℝ) (θ : ℝ) (N : ℕ) (sample : Ω) : ℝ :=
-  ∑ i ∈ Finset.range N, (θ*A i sample*(B (t (i+1)) sample-B (t i) sample)-
-    θ^2*(A i sample)^2*((t (i+1)-t i:ℝ≥0):ℝ)/2)
-
-lemma exponentialLogSum_past {Ω : Type*} [MeasurableSpace Ω]
-    (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (ht : Monotone t)
-    (A : ℕ → Ω → ℝ) (hA : ∀ i, Measurable[pastSigma B (t i)] (A i))
-    (θ : ℝ) (N : ℕ) : Measurable[pastSigma B (t N)] (exponentialLogSum B t A θ N) := by
-  apply Finset.measurable_fun_sum
-  intro i hi
-  have hAi := (hA i).mono (pastSigma_mono B (ht (Finset.mem_range.1 hi).le)) le_rfl
-  have hBi := past_measurable_eval B (ht (Finset.mem_range.1 hi).le)
-  have hBin := past_measurable_eval B (ht (Nat.succ_le_of_lt (Finset.mem_range.1 hi)))
-  exact (((hAi.const_mul θ).mul (hBin.sub hBi)).sub
-    (((hAi.pow_const 2).const_mul (θ^2)).mul_const _ |>.div_const 2))
-
-lemma exponentialLogSum_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (t : ℕ → ℝ≥0) (A : ℕ → Ω → ℝ) (hA : ∀ i, AEMeasurable (A i) P)
-    (θ : ℝ) (N : ℕ) : AEMeasurable (exponentialLogSum B t A θ N) P := by
-  apply Finset.aemeasurable_fun_sum
-  intro i hi
-  have hBi := hB.toIsPreBrownianReal.aemeasurable (t i)
-  have hBin := hB.toIsPreBrownianReal.aemeasurable (t (i+1))
-  fun_prop
-
-lemma exponentialLogSum_identity {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (t : ℕ → ℝ≥0) (ht : Monotone t)
-    (A : ℕ → Ω → ℝ) (hA : ∀ i, Measurable[pastSigma B (t i)] (A i))
-    (hAa : ∀ i, AEMeasurable (A i) P) (θ : ℝ) (N : ℕ) :
-    (∫⁻ sample, ENNReal.ofReal (Real.exp (exponentialLogSum B t A θ N sample)) ∂P) = 1 := by
-  induction N with
-  | zero => simp [exponentialLogSum]
-  | succ N hN =>
-    simp only [exponentialLogSum,Finset.sum_range_succ]
-    change (∫⁻ sample, ENNReal.ofReal (Real.exp (exponentialLogSum B t A θ N sample+
-      (θ*A N sample*(B (t (N+1)) sample-B (t N) sample)-
-      θ^2*(A N sample)^2*((t (N+1)-t N:ℝ≥0):ℝ)/2))) ∂P) = 1
-    simp_rw [← add_sub_assoc]
-    rw [past_exp_step hB (ht (Nat.le_succ N)) _ _ (exponentialLogSum_past B t ht A hA θ N)
-      (hA N) (exponentialLogSum_aemeasurable hB t A hAa θ N) (hAa N),hN]
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-open PredictableLimit
-
-def dyadicVariance {Ω : Type*} (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ) (n : ℕ) (sample : Ω) : ℝ :=
-  ∑ i ∈ Finset.range (2^n), (A n i sample)^2 *
-    ((dyadicTime T n (i+1)-dyadicTime T n i:ℝ≥0):ℝ)
-
-lemma exponentialLogSum_dyadic {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
-    (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ) (θ : ℝ) (n : ℕ) (sample : Ω) :
-    exponentialLogSum B (dyadicTime T n) (A n) θ (2^n) sample =
-      θ*dyadicDriverSum B T A n sample - θ^2*dyadicVariance T A n sample/2 := by
-  simp only [exponentialLogSum,dyadicDriverSum,dyadicVariance,Finset.mul_sum,
-    Finset.sum_div,← Finset.sum_sub_distrib]
-  apply Finset.sum_congr rfl
-  intro i hi
-  ring
-
-theorem limit_exponential_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
-    (hAa : ∀ n i, AEMeasurable (A n i) P)
-    (Y Q : Ω → ℝ)
-    (hY : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A n sample) atTop (𝓝 (Y sample)))
-    (hQ : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicVariance T A n sample) atTop (𝓝 (Q sample)))
-    (K : ℝ) (hK : ∀ᵐ sample ∂P, Q sample ≤ K) (θ : ℝ) :
-    (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*K/2)) := by
-  let E : ℕ → Ω → ℝ≥0∞ := fun n sample => ENNReal.ofReal
-    (Real.exp (exponentialLogSum B (dyadicTime T n) (A n) θ (2^n) sample))
-  have hEm (n : ℕ) : AEMeasurable (E n) P :=
-    ((exponentialLogSum_aemeasurable hB _ _ (hAa n) θ (2^n)).exp).ennreal_ofReal
-  have hEi (n : ℕ) : (∫⁻ sample, E n sample ∂P) = 1 :=
-    exponentialLogSum_identity hB _ (dyadicTime_mono T n) _ (hA n) (hAa n) θ (2^n)
-  have hElim : ∀ᵐ sample ∂P, Tendsto (fun n => E n sample) atTop
-      (𝓝 (ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)))) := by
-    filter_upwards [hY,hQ] with sample hy hq
-    have hh := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-      (Real.continuous_exp.continuousAt.tendsto.comp
-        ((hy.const_mul θ).sub ((hq.const_mul (θ^2)).div_const 2)))
-    simpa only [E,exponentialLogSum_dyadic,Function.comp_def] using hh
-  have hFatou : (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) ≤ 1 := by
-    calc
-      _ = ∫⁻ sample, liminf (fun n => E n sample) atTop ∂P := by
-        apply lintegral_congr_ae
-        filter_upwards [hElim] with sample hs
-        exact hs.liminf_eq.symm
-      _ ≤ liminf (fun n => ∫⁻ sample, E n sample ∂P) atTop := lintegral_liminf_le' hEm
-      _ = 1 := by simp only [hEi,liminf_const]
-  have hlimMeas : AEMeasurable (fun sample => ENNReal.ofReal
-      (Real.exp (θ*Y sample-θ^2*Q sample/2))) P :=
-    aemeasurable_of_tendsto_metrizable_ae atTop hEm hElim
-  calc
-    _ ≤ ∫⁻ sample, ENNReal.ofReal (Real.exp (θ^2*K/2)) *
-        ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [hK] with sample hk
-      rw [← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add]
-      apply ENNReal.ofReal_le_ofReal
-      apply Real.exp_le_exp.2
-      nlinarith [mul_le_mul_of_nonneg_left hk (sq_nonneg θ)]
-    _ = ENNReal.ofReal (Real.exp (θ^2*K/2)) *
-        (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) :=
-      lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-    _ ≤ _ := by simpa only [mul_one] using mul_le_mul_right hFatou (ENNReal.ofReal (Real.exp (θ^2*K/2)))
-
-lemma tail_of_exponential_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} (Y : Ω → ℝ) (hY : AEMeasurable Y P)
-    {K : ℝ} (hK : 0 < K)
-    (hbound : ∀ θ : ℝ, (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*K/2))) {x : ℝ} (hx : 0 < x) :
-    P {sample | x < Y sample} ≤ ENNReal.ofReal (Real.exp (-x^2/(2*K))) := by
-  let θ := x/K
-  have hθ : 0 < θ := div_pos hx hK
-  have hYm : AEMeasurable (fun sample => ENNReal.ofReal (Real.exp (θ*Y sample))) P :=
-    (hY.const_mul θ).exp.ennreal_ofReal
-  have he := meas_ge_le_lintegral_div hYm
-    (ENNReal.ofReal_ne_zero_iff.2 (Real.exp_pos (θ*x))) ENNReal.ofReal_ne_top
-  have he' : P {sample | x < Y sample} ≤
-      (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) /
-        ENNReal.ofReal (Real.exp (θ*x)) := by
-    apply (measure_mono _).trans he
-    intro sample hs
-    exact ENNReal.ofReal_le_ofReal (Real.exp_le_exp.2
-      (mul_le_mul_of_nonneg_left hs.le hθ.le))
-  refine he'.trans ((ENNReal.div_le_div_right (hbound θ) _).trans_eq ?_)
-  rw [← ENNReal.ofReal_div_of_pos (Real.exp_pos _),← Real.exp_sub]
-  congr 2
-  dsimp [θ]
-  field_simp
-  ring
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Mesh
-open TaylorMeshes BrownianPredictable BrownianIto
-
-lemma spatial_variance_tendsto {Z : Type*} [MeasurableSpace Z]
-    (ν : Measure Z) [IsFiniteMeasure ν] (T : ℝ≥0)
-    (C : ℝ → Z → ℝ) (hC : Measurable (Function.uncurry C))
-    (a b : Z → ℝ) (ha : ∀ z, a z ∈ Icc 0 (T:ℝ)) (hb : ∀ z, b z ∈ Icc 0 (T:ℝ))
-    (D : Z → ℝ → ℝ) (hD : ∀ z, ContinuousOn (D z) (Icc 0 (T:ℝ)))
-    (he : ∀ t z, C t z = slab (D z) (a z) (b z) t)
-    (K : ℝ) (hK : 0 ≤ K) (hbound : ∀ t ∈ Icc 0 (T:ℝ), ∀ z, |C t z| ≤ K) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (∫ z, C (mesh T n i) z ∂ν)^2 * ((T:ℝ)/(2:ℝ)^n)) atTop
-      (𝓝 (∫ t in (0:ℝ)..T, (∫ z, C t z ∂ν)^2)) := by
-  let F : ℕ → Z × Z → ℝ := fun n q => ∑ i ∈ Finset.range (2^n),
-    (C (mesh T n i) q.1*C (mesh T n i) q.2)*((T:ℝ)/(2:ℝ)^n)
-  let G : Z × Z → ℝ := fun q => ∫ t in (0:ℝ)..T, C t q.1*C t q.2
-  have hsec (t : ℝ) : Measurable (C t) := hC.comp (measurable_const.prodMk measurable_id)
-  have hF (n : ℕ) : Measurable (F n) := by
-    apply Finset.measurable_fun_sum
-    intro i hi
-    exact (((hsec _).comp measurable_fst).mul ((hsec _).comp measurable_snd)).mul_const _
-  have hFb (n : ℕ) (q : Z × Z) : ‖F n q‖ ≤ (T:ℝ)*K^2 := by
-    calc
-      _ ≤ ∑ i ∈ Finset.range (2^n), ‖(C (mesh T n i) q.1*C (mesh T n i) q.2)*((T:ℝ)/(2:ℝ)^n)‖ := norm_sum_le _ _
-      _ ≤ ∑ _i ∈ Finset.range (2^n), K^2*((T:ℝ)/(2:ℝ)^n) := by
-        apply Finset.sum_le_sum
-        intro i hi
-        rw [Real.norm_eq_abs,abs_mul,abs_mul,abs_of_nonneg (by positivity : 0 ≤ (T:ℝ)/(2:ℝ)^n)]
-        rw [pow_two]
-        gcongr <;> exact hbound _ (mesh_mem _ T.coe_nonneg n i (Finset.mem_range.1 hi).le) _
-      _ = _ := by simp; field_simp
-  have hlim (q : Z × Z) : Tendsto (fun n => F n q) atTop (𝓝 (G q)) := by
-    simp only [F,G,he]
-    exact slab_riemann_tendsto_integral T.coe_nonneg (ha q.1) (hb q.1) (ha q.2) (hb q.2) (hD q.1) (hD q.2)
-  have hl := tendsto_integral_of_dominated_convergence (μ := ν.prod ν) (fun _ : Z × Z => (T:ℝ)*K^2)
-    (fun n => (hF n).aestronglyMeasurable) (integrable_const _) (fun n => Filter.Eventually.of_forall (hFb n))
-    (Filter.Eventually.of_forall hlim)
-  have hFi (n : ℕ) : Integrable (F n) (ν.prod ν) :=
-    Integrable.of_bound (hF n).aestronglyMeasurable _ (Filter.Eventually.of_forall (hFb n))
-  have hprod (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : Integrable (fun q : Z × Z => C t q.1*C t q.2) (ν.prod ν) := by
-    apply Integrable.of_bound (((hsec t).comp measurable_fst).mul ((hsec t).comp measurable_snd)).aestronglyMeasurable (K^2)
-    apply Filter.Eventually.of_forall
-    intro q
-    change |C t q.1*C t q.2| ≤ K^2
-    simp only [abs_mul,pow_two]
-    exact mul_le_mul (hbound t ht q.1) (hbound t ht q.2) (abs_nonneg _) hK
-  have hFe (n : ℕ) : (∫ q, F n q ∂ν.prod ν) = ∑ i ∈ Finset.range (2^n),
-      (∫ z, C (mesh T n i) z ∂ν)^2 * ((T:ℝ)/(2:ℝ)^n) := by
-    rw [show F n = fun q => ∑ i ∈ Finset.range (2^n), (C (mesh T n i) q.1*C (mesh T n i) q.2)*((T:ℝ)/(2:ℝ)^n) from rfl,
-      integral_finsetSum]
-    · apply Finset.sum_congr rfl
-      intro i hi
-      rw [integral_mul_const,integral_prod_mul,pow_two]
-    · intro i hi
-      exact (hprod _ (mesh_mem _ T.coe_nonneg n i (Finset.mem_range.1 hi).le)).mul_const _
-  let μ := volume.restrict (Ioc (0:ℝ) (T:ℝ))
-  have hCt : Measurable (fun q : (Z × Z) × ℝ => C q.2 q.1.1*C q.2 q.1.2) :=
-    ((hC.comp (measurable_snd.prodMk measurable_fst.fst)).mul
-      (hC.comp (measurable_snd.prodMk measurable_fst.snd)))
-  have hCti : Integrable (fun q : (Z × Z) × ℝ => C q.2 q.1.1*C q.2 q.1.2) ((ν.prod ν).prod μ) := by
-    apply Integrable.of_bound hCt.aestronglyMeasurable (K^2)
-    have hm : ∀ᵐ t ∂μ, t ∈ Ioc (0:ℝ) (T:ℝ) := ae_restrict_mem measurableSet_Ioc
-    filter_upwards [(Measure.ae_prod_iff_ae_ae (measurableSet_Ioc.preimage measurable_snd)).2 (Filter.Eventually.of_forall (fun q : Z × Z => hm))] with q hq
-    simp only [Real.norm_eq_abs,abs_mul,pow_two]
-    exact mul_le_mul (hbound q.2 ⟨hq.1.le,hq.2⟩ q.1.1) (hbound q.2 ⟨hq.1.le,hq.2⟩ q.1.2) (abs_nonneg _) hK
-  have hGe : (∫ q, G q ∂ν.prod ν) = ∫ t in (0:ℝ)..T, (∫ z, C t z ∂ν)^2 := by
-    simp only [G,intervalIntegral.integral_of_le T.coe_nonneg]
-    rw [integral_integral_swap hCti]
-    apply integral_congr_ae
-    exact Filter.Eventually.of_forall (fun t => by
-      dsimp only
-      rw [integral_prod_mul,pow_two])
-  simpa only [hFe,hGe] using hl
-
-lemma spatial_dyadicVariance_tendsto {Z Ω : Type*} [MeasurableSpace Z]
-    (ν : Measure Z) [IsFiniteMeasure ν] (T : ℝ≥0)
-    (A : ℕ → ℕ → Ω × Z → ℝ) (ω : Ω)
-    (C : ℝ → Z → ℝ) (hC : Measurable (Function.uncurry C))
-    (a b : Z → ℝ) (ha : ∀ z, a z ∈ Icc 0 (T:ℝ)) (hb : ∀ z, b z ∈ Icc 0 (T:ℝ))
-    (D : Z → ℝ → ℝ) (hD : ∀ z, ContinuousOn (D z) (Icc 0 (T:ℝ)))
-    (he : ∀ t z, C t z = slab (D z) (a z) (b z) t)
-    (hAe : ∀ n i, ∀ᵐ z ∂ν, A n i (ω,z) = C (mesh T n i) z)
-    (K : ℝ) (hK : 0 ≤ K) (hbound : ∀ t ∈ Icc 0 (T:ℝ), ∀ z, |C t z| ≤ K) :
-    Tendsto (fun n => dyadicVariance T (fun n i ω => ∫ z, A n i (ω,z) ∂ν) n ω) atTop
-      (𝓝 (∫ t in (0:ℝ)..T, (∫ z, C t z ∂ν)^2)) := by
-  have hl := spatial_variance_tendsto ν T C hC a b ha hb D hD he K hK hbound
-  convert hl using 1
-  funext n
-  unfold dyadicVariance
-  apply Finset.sum_congr rfl
-  intro i hi
-  dsimp only
-  rw [integral_congr_ae (hAe n i)]
-  rw [NNReal.coe_sub (dyadicTime_mono T n (Nat.le_succ i)),dyadicTime_step]
-
-end SLEExactGauge.Mesh
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
-open BrownianPredictable BrownianIto TaylorMeshes PredictableLimit
-local instance slabMomentsCoeffRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance slabMomentsCoeffRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance slabMomentsCoeffComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance slabMomentsCoeffComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-def slabCoeff {Ω : Type*} {ε : ℝ} (hε : 0 < ε) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (T s : ℝ≥0) (ρ δ e r : ℝ) (q : Ω × ℂ) : ℝ :=
-  coeff hε κ B T s ρ δ e q - coeff hε κ B T s ρ δ r q
-
-def pathFirst {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ : ℝ)
-    (U : C(I,ℝ)) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ) : ℝ :=
-  iteratedDeriv 1 (field hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t.toNNReal)
-
-def pathSlab {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ ρ δ e r : ℝ)
-    (U : C(I,ℝ)) (V : C(I,ℂ)) (b : ℝ≥0 → ℝ) (t : ℝ) (z : ℂ) : ℝ :=
-  slab (pathFirst hε hT κ U b z)
-    (observedTime hε hT ρ δ r ((U,V),z)) (observedTime hε hT ρ δ e ((U,V),z)) t
-
-lemma continuous_pathFirst {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ : ℝ)
-    (U : C(I,ℝ)) (b : ℝ≥0 → ℝ) (hb : Continuous b) :
-    Continuous (fun q : ℝ × ℂ => pathFirst hε hT κ U b q.2 q.1) := by
-  have hp : Continuous (fun q : ℝ × ℂ => (((U,q.2),q.1),b q.1.toNNReal)) :=
-    ((continuous_const.prodMk continuous_snd).prodMk continuous_fst).prodMk
-      ((hb.comp continuous_real_toNNReal).comp continuous_fst)
-  have hh := (continuous_mark_first hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ)).comp hp
-  exact hh
-
-lemma measurable_pathSlab {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ ρ δ e r : ℝ)
-    (U : C(I,ℝ)) (V : C(I,ℂ)) (b : ℝ≥0 → ℝ) (hb : Continuous b) :
-    Measurable (Function.uncurry (pathSlab hε hT κ ρ δ e r U V b)) := by
-  have hs (s : ℝ) := (measurable_observedTime hε hT ρ δ s).comp
-    (measurable_const.prodMk measurable_id : Measurable (fun z : ℂ => ((U,V),z)))
-  exact measurable_slab (continuous_pathFirst hε hT κ U b hb).measurable (hs r) (hs e)
-
-lemma pathSlab_uniform_bound {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (ρ δ e r : ℝ) (her : e ≤ r) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I,ℝ)) (V : C(I,ℂ)) (b : ℝ≥0 → ℝ) t z,
-      |pathSlab hε hT κ ρ δ e r U V b t z| ≤ K := by
-  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
-  have ha : 0 ≤ 8/κ-1 := by
-    have hh : 1 ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
-    linarith
-  let K := |Real.sqrt κ| *((2*ε/Real.exp ((2/ε^2)*T))^(-(1-κ/8))*(8/κ-1)/ε)
-  have hK : 0 ≤ K := by dsimp only [K]; positivity
-  refine ⟨K,hK,?_⟩
-  intro U V b t z
-  exact slab_abs_le (observedTime_antitone hε hT ρ δ ((U,V),z) her) hK
-    (first_bound hε hT _ _ _ hp ha U z t (b t.toNNReal))
-
-lemma slabCoeff_eq {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
-    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
-    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
-    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e r : ℝ) (s : ℝ≥0) :
-    slabCoeff C.height_pos κ B C.horizon s ρ C.angle e r (ω,z) =
-      pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
-        (BrownianDriver.path B (Real.sqrt κ) C.horizon ω) V (fun t => B t ω) s z := by
-  unfold slabCoeff
-  rw [← coeff_common C κ B z hz hr e s ω,← coeff_common C κ B z hz hr r s ω,
-    (C.at z hz).physicalCoeff_eq κ B γ g f ω hm hc hb0 hhalf hu e 1 s,
-    (C.at z hz).physicalCoeff_eq κ B γ g f ω hm hc hb0 hhalf hu r 1 s]
-  dsimp only [pathSlab,slab,pathFirst]
-  rw [observedTime_eq C κ B γ g f ω hm hc hb0 z hz hhalf hu hr V hV r,
-    observedTime_eq C κ B γ g f ω hm hc hb0 z hz hhalf hu hr V hV e]
-  simp only [Real.toNNReal_coe]
-  rfl
-
-lemma slabCoeff_variance_tendsto {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
-    (C : FiniteTilt.CommonCutoff A u) (hA : MeasurableSet A) [IsFiniteMeasure (volume.restrict A)]
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
-    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
-    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
-    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
-    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e r : ℝ) (her : e ≤ r) :
-    Tendsto (fun n => dyadicVariance C.horizon
-      (fun n i ω => ∫ z in A, slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r (ω,z)) n ω)
-      atTop (𝓝 (∫ t in (0:ℝ)..C.horizon, (∫ z in A,
-        pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
-          (BrownianDriver.path B (Real.sqrt κ) C.horizon ω) V (fun t => B t ω) t z)^2)) := by
-  let U := BrownianDriver.path B (Real.sqrt κ) C.horizon ω
-  let a := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle r ((U,V),z)
-  let b := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e ((U,V),z)
-  let D := pathFirst C.height_pos C.horizon.coe_nonneg κ U (fun t => B t ω)
-  let S := pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r U V (fun t => B t ω)
-  obtain ⟨K,hK,hbound⟩ := pathSlab_uniform_bound C.height_pos C.horizon.coe_nonneg κ hκ ρ C.angle e r her
-  apply spatial_dyadicVariance_tendsto (K := K) (volume.restrict A) C.horizon
-    (fun n i => slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r)
-    ω S (measurable_pathSlab _ _ _ _ _ _ _ _ _ _ hc) a b
-    (fun z => observedTime_bounds _ _ _ _ _ _) (fun z => observedTime_bounds _ _ _ _ _ _) D
-  · intro z
-    have hh := (continuous_pathFirst C.height_pos C.horizon.coe_nonneg κ U (fun t => B t ω) hc).comp
-      (continuous_id.prodMk continuous_const : Continuous (fun t : ℝ => (t,z)))
-    exact hh.continuousOn
-  · intro t z; rfl
-  · intro n i
-    filter_upwards [ae_restrict_mem hA] with z hz
-    simpa only [coe_dyadicTime] using
-      slabCoeff_eq C κ B γ g f ω hm hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) V hV e r (dyadicTime C.horizon n i)
-  · exact hK
-  · intro t ht z
-    exact hbound U V (fun t => B t ω) t z
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.SpatialLimit
-open L2Limit
-
-lemma uniformIntegrable_one_of_eLpNorm_two_bounded {Ω ι : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {X : ι → Ω → ℝ}
-    (hXm : ∀ i, AEStronglyMeasurable (X i) P) (C : ℝ≥0)
-    (hC : ∀ i, eLpNorm (X i) 2 P ≤ C) : UniformIntegrable X 1 P := by
-  let D : ℝ≥0∞ := (C : ℝ≥0∞) * (P univ) ^ (1/2 : ℝ)
-  have hDt : D ≠ ∞ := by dsimp [D]; finiteness
-  refine ⟨?_, D.toNNReal, ?_⟩
-  · apply unifIntegrable_iff.2
-    intro ε hε
-    by_cases hεt : ε = ∞
-    · subst ε
-      exact ⟨1, by norm_num, fun _ _ _ => le_top⟩
-    have hεr : 0 < ε.toReal := ENNReal.toReal_pos hε.ne' hεt
-    let d : ℝ≥0 := ⟨ε.toReal / ((C : ℝ)+1), div_nonneg hεr.le (by positivity)⟩
-    have hd : 0 < (d : ℝ) := div_pos hεr (by positivity)
-    refine ⟨ENNReal.ofReal ((d : ℝ)^2), ENNReal.ofReal_pos.2 (sq_pos_of_pos hd),
-      fun i s hμs => ?_⟩
-    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (μ := P.restrict s)
-      (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i).restrict
-    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
-    rw [Measure.restrict_apply_univ] at hb
-    have hp : (P s) ^ (1/2 : ℝ) ≤ (d : ℝ≥0∞) := by
-      calc
-        _ ≤ (ENNReal.ofReal ((d : ℝ)^2)) ^ (1/2 : ℝ) :=
-          ENNReal.rpow_le_rpow hμs (by norm_num)
-        _ = _ := by
-          rw [ENNReal.ofReal_pow d.coe_nonneg]
-          simp only [ENNReal.ofReal_coe_nnreal]
-          rw [← ENNReal.rpow_natCast, ← ENNReal.rpow_mul]
-          norm_num
-    calc
-      _ ≤ eLpNorm (X i) 2 (P.restrict s) * (P s) ^ (1/2 : ℝ) := hb
-      _ ≤ C * (d : ℝ≥0∞) := mul_le_mul
-        ((eLpNorm_mono_measure _ Measure.restrict_le_self).trans (hC i)) hp (by positivity) (by positivity)
-      _ ≤ ENNReal.ofReal ε.toReal := by
-        rw [← ENNReal.coe_mul, ← ENNReal.ofReal_coe_nnreal]
-        simp only [NNReal.coe_mul]
-        apply ENNReal.ofReal_le_ofReal
-        change (C : ℝ) * (ε.toReal / ((C : ℝ)+1)) ≤ ε.toReal
-        rw [← mul_div_assoc, div_le_iff₀ (by positivity)]
-        nlinarith
-      _ = ε := ENNReal.ofReal_toReal hεt
-  · intro i
-    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i)
-    rw [ENNReal.coe_toNNReal hDt]
-    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
-    exact hb.trans (mul_le_mul_left (hC i) ((P univ) ^ (1/2 : ℝ)))
-
-lemma L1_tendsto_of_square_bound {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Y : Ω → ℝ}
-    (hX : ∀ n, MemLp (X n) 2 P) (A : ℝ)
-    (hA : ∀ n, (∫ ω, X n ω ^ 2 ∂P) ≤ A)
-    (hXY : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) :
-    Integrable Y P ∧ Tendsto (fun n => eLpNorm (X n-Y) 1 P) atTop (𝓝 0) := by
-  let C : ℝ≥0∞ := (ENNReal.ofReal A) ^ (1/2 : ℝ)
-  have hCt : C ≠ ∞ := ENNReal.rpow_ne_top_of_nonneg (by norm_num) ENNReal.ofReal_ne_top
-  have hC (n : ℕ) : eLpNorm (X n) 2 P ≤ C.toNNReal := by
-    rw [ENNReal.coe_toNNReal hCt, eLpNorm_two_eq_integral_sq (hX n)]
-    exact ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal (hA n)) (by norm_num)
-  have hui := uniformIntegrable_one_of_eLpNorm_two_bounded
-    (fun n => (hX n).aestronglyMeasurable) C.toNNReal hC
-  have hY := hui.memLp_of_ae_tendsto hXY
-  exact ⟨memLp_one_iff_integrable.1 hY,
-    tendsto_Lp_finite_of_tendsto_ae (by norm_num : (1 : ℝ≥0∞) ≤ 1)
-      (by norm_num) (fun n => (hX n).aestronglyMeasurable) hY hui.unifIntegrable hXY⟩
-
-lemma integrate_L1_le {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
-    {X Y : Ω × Z → ℝ} (hX : Integrable X (P.prod ν)) (hY : Integrable Y (P.prod ν)) :
-    eLpNorm ((fun ω => ∫ z, X (ω,z) ∂ν)-(fun ω => ∫ z, Y (ω,z) ∂ν)) 1 P ≤
-      eLpNorm (X-Y) 1 (P.prod ν) := by
-  rw [eLpNorm_one_eq_lintegral_enorm
-    (hX.aestronglyMeasurable.integral_prod_right'.sub
-      hY.aestronglyMeasurable.integral_prod_right'),
-    eLpNorm_one_eq_lintegral_enorm (hX.sub hY).aestronglyMeasurable]
-  rw [lintegral_prod _ (hX.sub hY).aestronglyMeasurable.enorm]
-  apply lintegral_mono_ae
-  filter_upwards [hX.prod_right_ae,hY.prod_right_ae] with ω hx hy
-  simp only [Pi.sub_apply]
-  rw [← integral_sub hx hy]
-  change ‖∫ z, X (ω,z)-Y (ω,z) ∂ν‖ₑ ≤ _
-  exact enorm_integral_le_lintegral_enorm _
-
-lemma spatial_L1_limit {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} {ν : Measure Z} [IsFiniteMeasure P] [IsFiniteMeasure ν]
-    {X : ℕ → Ω × Z → ℝ} {Y : Ω × Z → ℝ}
-    (hX : ∀ n, MemLp (X n) 2 (P.prod ν)) (A : ℝ)
-    (hA : ∀ n, (∫ q, X n q ^ 2 ∂P.prod ν) ≤ A)
-    (hXY : ∀ᵐ q ∂P.prod ν, Tendsto (fun n => X n q) atTop (𝓝 (Y q))) :
-    Integrable Y (P.prod ν) ∧
-      ∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ ω ∂P,
-        Tendsto (fun n => ∫ z, X (ns n) (ω,z) ∂ν) atTop (𝓝 (∫ z, Y (ω,z) ∂ν)) := by
-  obtain ⟨hY,hL⟩ := L1_tendsto_of_square_bound hX A hA hXY
-  have hi (n : ℕ) := (hX n).integrable (by norm_num : (1 : ℝ≥0∞) ≤ 2)
-  have hh : Tendsto (fun n => eLpNorm
-      ((fun ω => ∫ z, X n (ω,z) ∂ν)-(fun ω => ∫ z, Y (ω,z) ∂ν)) 1 P) atTop (𝓝 0) :=
-    tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hL
-      (fun _ => bot_le) (fun n => integrate_L1_le (hi n) hY)
-  have hm := tendstoInMeasure_of_tendsto_eLpNorm (by norm_num : (1 : ℝ≥0∞) ≠ 0)
-    hh
-  exact ⟨hY,hm.exists_seq_tendsto_ae⟩
-
-end SLEExactGauge.SpatialLimit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.SpatialLimit
-
-lemma memLp_two_prod_of_fiber_bound {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} {ν : Measure Z} [SFinite P] [IsFiniteMeasure ν]
-    {X : Ω × Z → ℝ} (hX : AEMeasurable X (P.prod ν))
-    (hXi : ∀ᵐ z ∂ν, MemLp (fun ω => X (ω,z)) 2 P) (A : ℝ)
-    (hA : ∀ᵐ z ∂ν, (∫ ω, X (ω,z)^2 ∂P) ≤ A) :
-    MemLp X 2 (P.prod ν) ∧ (∫ q, X q ^ 2 ∂P.prod ν) ≤ ∫ _ : Z, A ∂ν := by
-  have hsq : AEStronglyMeasurable (fun q => X q^2) (P.prod ν) := (hX.pow_const 2).aestronglyMeasurable
-  have hmeas : AEStronglyMeasurable (fun z => ∫ ω, ‖X (ω,z)^2‖ ∂P) ν :=
-    hsq.norm.prod_swap.integral_prod_right'
-  have he (z : Z) : (∫ ω, ‖X (ω,z)^2‖ ∂P) = ∫ ω, X (ω,z)^2 ∂P := by
-    apply integral_congr_ae
-    exact Filter.Eventually.of_forall (fun ω => Real.norm_of_nonneg (sq_nonneg _))
-  have hi : Integrable (fun q => X q^2) (P.prod ν) := by
-    apply (integrable_prod_iff' hsq).2
-    refine ⟨hXi.mono (fun z hz => hz.integrable_sq),?_⟩
-    apply (integrable_const A).mono' hmeas
-    filter_upwards [hA] with z hz
-    rw [he,Real.norm_of_nonneg (integral_nonneg (fun _ => sq_nonneg _))]
-    exact hz
-  refine ⟨(memLp_two_iff_integrable_sq hX.aestronglyMeasurable).2 hi,?_⟩
-  rw [integral_prod_symm _ hi]
-  exact integral_mono_ae hi.integral_prod_right (integrable_const _) hA
-
-lemma ae_prod_tendsto_of_fibers {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
-    {X : ℕ → Ω × Z → ℝ} {Y : Ω × Z → ℝ}
-    (hX : ∀ n, AEMeasurable (X n) (P.prod ν)) (hY : AEMeasurable Y (P.prod ν))
-    (hh : ∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Tendsto (fun n => X n (ω,z)) atTop (𝓝 (Y (ω,z)))) :
-    ∀ᵐ q ∂P.prod ν, Tendsto (fun n => X n q) atTop (𝓝 (Y q)) := by
-  let F : ℕ → Ω × Z → ℝ := fun n => (hX n).mk (X n)
-  let G := hY.mk Y
-  have hF (n : ℕ) : Measurable (F n) := (hX n).measurable_mk
-  have hG : Measurable G := hY.measurable_mk
-  have he : ∀ᵐ q ∂P.prod ν, (∀ n, X n q = F n q) ∧ Y q = G q :=
-    (ae_all_iff.2 (fun n => (hX n).ae_eq_mk)).and hY.ae_eq_mk
-  have hes : ∀ᵐ q ∂ν.prod P, (∀ n, X n q.swap = F n q.swap) ∧ Y q.swap = G q.swap :=
-    Measure.measurePreserving_swap.quasiMeasurePreserving.ae he
-  have hf : ∀ᵐ q ∂ν.prod P, Tendsto (fun n => F n q.swap) atTop (𝓝 (G q.swap)) := by
-    apply (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun
-      (fun n => (hF n).comp measurable_swap) (hG.comp measurable_swap))).2
-    filter_upwards [hh,Measure.ae_ae_of_ae_prod hes] with z hz hez
-    filter_upwards [hz,hez] with ω hω heω
-    change (∀ n, X n (ω,z) = F n (ω,z)) ∧ Y (ω,z) = G (ω,z) at heω
-    change Tendsto (fun n => F n (ω,z)) atTop (𝓝 (G (ω,z)))
-    simpa only [heω.1,heω.2] using hω
-  have hf' := Measure.measurePreserving_swap.quasiMeasurePreserving.ae hf
-  filter_upwards [he,hf'] with q heq hq
-  simpa only [heq.1,heq.2,Prod.swap_swap] using hq
-
-end SLEExactGauge.SpatialLimit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-open PredictableLimit
-
-theorem limit_exponential_bound_subseq {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
-    (hAa : ∀ n i, AEMeasurable (A n i) P)
-    (ns : ℕ → ℕ) (Y Q : Ω → ℝ)
-    (hY : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A (ns n) sample) atTop (𝓝 (Y sample)))
-    (hQ : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicVariance T A (ns n) sample) atTop (𝓝 (Q sample)))
-    (K : ℝ) (hK : ∀ᵐ sample ∂P, Q sample ≤ K) (θ : ℝ) :
-    (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*K/2)) := by
-  let E : ℕ → Ω → ℝ≥0∞ := fun n sample => ENNReal.ofReal
-    (Real.exp (exponentialLogSum B (dyadicTime T (ns n)) (A (ns n)) θ (2^(ns n)) sample))
-  have hEm (n : ℕ) : AEMeasurable (E n) P :=
-    ((exponentialLogSum_aemeasurable hB _ _ (hAa (ns n)) θ (2^(ns n))).exp).ennreal_ofReal
-  have hEi (n : ℕ) : (∫⁻ sample, E n sample ∂P) = 1 :=
-    exponentialLogSum_identity hB _ (dyadicTime_mono T (ns n)) _ (hA (ns n)) (hAa (ns n)) θ (2^(ns n))
-  have hElim : ∀ᵐ sample ∂P, Tendsto (fun n => E n sample) atTop
-      (𝓝 (ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)))) := by
-    filter_upwards [hY,hQ] with sample hy hq
-    have hh := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
-      (Real.continuous_exp.continuousAt.tendsto.comp
-        ((hy.const_mul θ).sub ((hq.const_mul (θ^2)).div_const 2)))
-    simpa only [E,exponentialLogSum_dyadic,Function.comp_def] using hh
-  have hFatou : (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) ≤ 1 := by
-    calc
-      _ = ∫⁻ sample, liminf (fun n => E n sample) atTop ∂P := by
-        apply lintegral_congr_ae
-        filter_upwards [hElim] with sample hs
-        exact hs.liminf_eq.symm
-      _ ≤ liminf (fun n => ∫⁻ sample, E n sample ∂P) atTop := lintegral_liminf_le' hEm
-      _ = 1 := by simp only [hEi,liminf_const]
-  have hlimMeas : AEMeasurable (fun sample => ENNReal.ofReal
-      (Real.exp (θ*Y sample-θ^2*Q sample/2))) P :=
-    aemeasurable_of_tendsto_metrizable_ae atTop hEm hElim
-  calc
-    _ ≤ ∫⁻ sample, ENNReal.ofReal (Real.exp (θ^2*K/2)) *
-        ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P := by
-      apply lintegral_mono_ae
-      filter_upwards [hK] with sample hk
-      rw [← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add]
-      apply ENNReal.ofReal_le_ofReal
-      apply Real.exp_le_exp.2
-      nlinarith [mul_le_mul_of_nonneg_left hk (sq_nonneg θ)]
-    _ = ENNReal.ofReal (Real.exp (θ^2*K/2)) *
-        (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) :=
-      lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
-    _ ≤ _ := by simpa only [mul_one] using mul_le_mul_right hFatou (ENNReal.ofReal (Real.exp (θ^2*K/2)))
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.SpatialLimit
-open BrownianPredictable PredictableLimit
-
-lemma dyadicDriverSum_integrate {Ω Z : Type*} [MeasurableSpace Z]
-    (ν : Measure Z) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
-    (A : ℕ → ℕ → Ω × Z → ℝ) (n : ℕ) (ω : Ω)
-    (hA : ∀ i, Integrable (fun z => A n i (ω,z)) ν) :
-    dyadicDriverSum B T (fun n i ω => ∫ z, A n i (ω,z) ∂ν) n ω =
-      ∫ z, dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω ∂ν := by
-  unfold dyadicDriverSum
-  rw [integral_finsetSum]
-  · apply Finset.sum_congr rfl
-    intro i hi
-    rw [integral_mul_const]
-  · intro i hi
-    exact (hA i).mul_const _
-
-theorem spatial_exponential_bound {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} [IsProbabilityMeasure P] (ν : Measure Z) [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (A : ℕ → ℕ → Ω × Z → ℝ)
-    (hAi : ∀ n i, Integrable (A n i) (P.prod ν))
-    (hA : ∀ n i, @Measurable (Ω × Z) ℝ
-      (Prod.instMeasurableSpace (m₁ := pastSigma B (dyadicTime T n i)) (m₂ := inferInstance)) _ (A n i))
-    (Y : Ω × Z → ℝ)
-    (hX : ∀ n, MemLp (fun q => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) 2 (P.prod ν))
-    (V : ℝ) (hV : ∀ n, (∫ q, (dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1)^2 ∂P.prod ν) ≤ V)
-    (hY : ∀ᵐ q ∂P.prod ν, Tendsto
-      (fun n => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) atTop (𝓝 (Y q)))
-    (Q : Ω → ℝ)
-    (hQ : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicVariance T (fun n i ω => ∫ z, A n i (ω,z) ∂ν) n ω)
-      atTop (𝓝 (Q ω))) (K θ : ℝ) (hK : ∀ᵐ ω ∂P, Q ω ≤ K) :
-    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z, Y (ω,z) ∂ν))) ∂P) ≤
-      ENNReal.ofReal (Real.exp (θ^2*K/2)) := by
-  let C := fun n i ω => ∫ z, A n i (ω,z) ∂ν
-  have hCi (n i : ℕ) : AEMeasurable (C n i) P :=
-    (hAi n i).aestronglyMeasurable.integral_prod_right'.aemeasurable
-  have hCp (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) := by
-    let : MeasurableSpace Ω := pastSigma B (dyadicTime T n i)
-    exact (hA n i).stronglyMeasurable.integral_prod_right'.measurable
-  obtain ⟨hYi,ns,hns,hlim⟩ := spatial_L1_limit hX V hV hY
-  have he : ∀ᵐ ω ∂P, ∀ n, dyadicDriverSum B T C n ω =
-      ∫ z, dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω ∂ν := by
-    have hh : ∀ᵐ ω ∂P, ∀ n i, Integrable (fun z => A n i (ω,z)) ν :=
-      ae_all_iff.2 (fun n => ae_all_iff.2 (fun i => (hAi n i).prod_right_ae))
-    filter_upwards [hh] with ω hω n
-    exact dyadicDriverSum_integrate ν B T A n ω (hω n)
-  have hS : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B T C (ns n) ω) atTop
-      (𝓝 (∫ z, Y (ω,z) ∂ν)) := by
-    filter_upwards [he,hlim] with ω heω hω
-    simpa only [heω] using hω
-  have hQV : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicVariance T C (ns n) ω) atTop (𝓝 (Q ω)) :=
-    hQ.mono (fun ω hω => hω.comp hns.tendsto_atTop)
-  exact limit_exponential_bound_subseq hB T C hCp hCi ns _ Q hS hQV K hK θ
-
-lemma sums_aemeasurable {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (A : ℕ → ℕ → Ω × Z → ℝ)
-    (hA : ∀ n i, AEMeasurable (A n i) (P.prod ν)) (n : ℕ) :
-    AEMeasurable (fun q => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) (P.prod ν) := by
-  apply Finset.aemeasurable_fun_sum
-  intro i hi
-  exact (hA n i).mul
-    (((hB.toIsPreBrownianReal.hasLaw_eval _).aemeasurable.comp_fst).sub
-      ((hB.toIsPreBrownianReal.hasLaw_eval _).aemeasurable.comp_fst))
-
-lemma bounded_sums {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} [IsProbabilityMeasure P] (ν : Measure Z) [IsFiniteMeasure ν]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (A : ℕ → ℕ → Ω × Z → ℝ)
-    (hA : ∀ n i, @Measurable (Ω × Z) ℝ
-      (Prod.instMeasurableSpace (m₁ := pastSigma B (dyadicTime T n i)) (m₂ := inferInstance)) _ (A n i))
-    (hAm : ∀ n i, AEMeasurable (A n i) (P.prod ν))
-    (K : ℝ) (hK : 0 ≤ K) (hbound : ∀ n i q, |A n i q| ≤ K) :
-    (∀ n i, Integrable (A n i) (P.prod ν)) ∧
-    ∀ n, MemLp (fun q => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) 2 (P.prod ν) ∧
-      (∫ q, (dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1)^2 ∂P.prod ν) ≤
-        ∫ _ : Z, K^2*(T:ℝ) ∂ν := by
-  have hi (n i : ℕ) : Integrable (A n i) (P.prod ν) :=
-    Integrable.of_bound (hAm n i).aestronglyMeasurable K (Filter.Eventually.of_forall (hbound n i))
-  refine ⟨hi,?_⟩
-  intro n
-  have hsect : ∀ᵐ z ∂ν, ∀ n i, AEStronglyMeasurable (fun ω => A n i (ω,z)) P :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (fun i => ((hi n i).prod_left_ae.mono (fun z hz => hz.aestronglyMeasurable))))
-  have hcp (z : Z) (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (fun ω => A n i (ω,z)) := by
-    let : MeasurableSpace Ω := pastSigma B (dyadicTime T n i)
-    exact (hA n i).comp (measurable_id.prodMk measurable_const)
-  have hh : ∀ᵐ z ∂ν,
-      MemLp (fun ω => dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω) 2 P ∧
-      (∫ ω, (dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω)^2 ∂P) ≤ K^2*(T:ℝ) := by
-    filter_upwards [hsect] with z hz
-    have h2 (n i : ℕ) : MemLp (fun ω => A n i (ω,z)) 2 P :=
-      MemLp.of_bound (hz n i) K (Filter.Eventually.of_forall (fun ω => hbound n i (ω,z)))
-    exact ⟨dyadicDriverSum_memLp hB T _ (hcp z) h2 n,
-      dyadicDriverSum_second_moment_le hB T _ (hcp z) h2 K hK
-        (fun n i => Filter.Eventually.of_forall (fun ω => hbound n i (ω,z))) n⟩
-  exact memLp_two_prod_of_fiber_bound (sums_aemeasurable hB T A hAm n)
-    (hh.mono (fun z hz => hz.1)) (K^2*(T:ℝ)) (hh.mono (fun z hz => hz.2))
-
-end SLEExactGauge.SpatialLimit
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.MeshSlab
-open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
-open BrownianPredictable BrownianIto TaylorMeshes PredictableLimit
-local instance slabMomentsDifferenceRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
-local instance slabMomentsDifferenceRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
-local instance slabMomentsDifferenceComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
-local instance slabMomentsDifferenceComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
-
-def observedDifference {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ ρ δ e r : ℝ)
-    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ :=
-  observedValue hε hT ρ δ e (1-κ/8) (8/κ-1) q -
-    observedValue hε hT ρ δ r (1-κ/8) (8/κ-1) q
-
-lemma observedDifference_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (ν : Measure ℂ) [SFinite ν] {ε : ℝ} (hε : 0 < ε)
-    (κ ρ δ e r : ℝ) (T : ℝ≥0) (V : Ω → C(I,ℂ)) (hV : Measurable V) :
-    AEMeasurable (fun q : Ω × ℂ => observedDifference hε T.coe_nonneg κ ρ δ e r
-      ((BrownianDriver.path B (Real.sqrt κ) T q.1,V q.1),q.2)) (P.prod ν) := by
-  have hc := (measurable_observedValue hε T.coe_nonneg ρ δ e (1-κ/8) (8/κ-1)).sub
-    (measurable_observedValue hε T.coe_nonneg ρ δ r (1-κ/8) (8/κ-1))
-  have hp : AEMeasurable (fun q : Ω × ℂ => ((BrownianDriver.path B (Real.sqrt κ) T q.1,V q.1),q.2)) (P.prod ν) :=
-    (((BrownianDriver.path_aemeasurable hB (Real.sqrt κ) T).comp_fst).prodMk
-      ((hV.comp measurable_fst).aemeasurable)).prodMk measurable_snd.aemeasurable
-  exact hc.comp_aemeasurable hp
-
-lemma slab_sum_tendsto {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
-    {A : Set ℂ} {u : ℂ → ℝ} (C : FiniteTilt.CommonCutoff A u)
-    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
-    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
-    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
-    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
-    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
-    (V : Ω → C(I,ℂ)) (hV : ∀ᵐ ω ∂P, ∀ q : I, V ω q = γ ω ((C.horizon:ℝ)*q)) (e r : ℝ) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B C.horizon
-      (fun n i ω => slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r (ω,z)) n ω) atTop
-      (𝓝 (observedDifference C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
-        ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V ω),z))) := by
-  have he := (C.at z hz).physical_sum_tendsto hB κ hκ γ g f hmaps hhalf hu e
-  have hr' := (C.at z hz).physical_sum_tendsto hB κ hκ γ g f hmaps hhalf hu r
-  filter_upwards [he,hr',hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hV] with ω heω hrω hm hc hb0 hVω
-  have hv (s : ℝ) := observedValue_eq C κ B γ g f ω hm hc hb0 z hz hhalf hu hr (V ω) hVω s
-  unfold observedDifference
-  rw [hv e,hv r]
-  have hl := heω.sub hrω
-  convert hl using 1
-  · funext n
-    simp only [dyadicDriverSum,slabCoeff,← coeff_common C κ B z hz hr,
-      sub_mul,Finset.sum_sub_distrib]
-    rfl
-  · congr 1
-    ring
-
-end SLEExactGauge.MeshSlab
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.IntegratedMeshSimple
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Mesh estimates. -/
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory
+open scoped Topology ENNReal
+namespace SLEExactGauge.Mesh
+
+lemma asynchronous_bracket_bound_active {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) (hU : Continuous driver)
+    {A : Set ℂ} (hA : MeasurableSet A) {a b : ℂ → ℝ} {D : ℂ → ℝ → ℝ}
+    (ha : Measurable a) (hb : Measurable b)
+    (hB : Measurable (fun q : ℝ × ℂ => slab (D q.2) (a q.2) (b q.2) q.1))
+    (L : ℂ → ℂ → ℝ≥0∞) (hL : Measurable (Function.uncurry L))
+    (hLeq : ∀ z ∈ A, ∀ w ∈ A, L z w = laterGreen g a b z w)
+    {K R C : ℝ} (hK : 0 ≤ K) (hR : 0 < R)
+    (hstart : ∀ z ∈ A, 0 ≤ a z) (hab : ∀ z ∈ A, a z ≤ b z)
+    (hlive : ∀ z ∈ A, z ∈ localizationDomain γ (b z))
+    (happroach : ∀ z ∈ A, a z < b z → dist z (γ (a z)) ≤ R)
+    (hcoeff : ∀ z ∈ A, ∀ t ∈ Icc (a z) (b z),
+      |slab (D z) (a z) (b z) t| ≤ K*poissonFactor driver g t z)
+    (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain γ t,
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain γ t ∧ infDist w (localizationDomain γ t)ᶜ ≤ R},
+        ENNReal.ofReal (halfPlaneGreen (g t z) (g t w))) ≤ ENNReal.ofReal (C*R^2)) :
+    (∫⁻ t, ENNReal.ofReal (|∫ z in A, slab (D z) (a z) (b z) t|^2)) ≤
+      2*ENNReal.ofReal (K^2)*ENNReal.ofReal (C*R^2)*volume A := by
+  let A' := A ∩ {z | a z < b z}
+  have hA' : MeasurableSet A' := hA.inter (measurableSet_lt ha hb)
+  have hs : A' ⊆ A := inter_subset_left
+  have he (t : ℝ) : (∫ z in A, slab (D z) (a z) (b z) t) =
+      ∫ z in A', slab (D z) (a z) (b z) t := by
+    apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero hA hs
+    intro z hz
+    have hab' : a z = b z := le_antisymm (hab z hz.1) (not_lt.1 (fun hh => hz.2 ⟨hz.1,hh⟩))
+    simp only [slab,hab',sub_self]
+  simp only [he]
+  apply (asynchronous_bracket_bound_with_kernel (B := fun t z => slab (D z) (a z) (b z) t) h hU hA' hB L hL
+    (fun z hz w hw => hLeq z hz.1 w hw.1) hK hR
+    (fun z hz => hstart z hz.1) (fun z hz => hab z hz.1)
+    (fun z hz => hlive z hz.1) (fun z hz => happroach z hz.1 hz.2)
+    (fun z hz t ht => slab_eq_zero_of_not_mem (hab z hz.1) ht)
+    (fun z hz t ht => hcoeff z hz.1 t ht) hlayer).trans
+  have hv : volume A' ≤ volume A := measure_mono hs
+  exact mul_le_mul_right hv _
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric MeasureTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
+local instance slabMomentsBracketRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance slabMomentsBracketRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance slabMomentsBracketComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance slabMomentsBracketComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+lemma measurable_halfPlaneGreen : Measurable (fun q : ℂ × ℂ => halfPlaneGreen q.1 q.2) := by
+  unfold halfPlaneGreen
+  fun_prop
+
+lemma observed_bracket {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
+    (C : FiniteTilt.CommonCutoff A u) (hA : MeasurableSet A)
+    (κ : ℝ) (hκ : 0 < κ ∧ κ ≤ 4) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
+    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
+    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q))
+    (e r : ℝ) (he : 0 < e) (her : e ≤ r) (hes : ∀ z ∈ A, e < z.im)
+    (J : ℝ) (hlayer : ∀ t : ℝ, 0 ≤ t → ∀ z ∈ localizationDomain (γ ω) t,
+      (∫⁻ w in {w : ℂ | w ∈ localizationDomain (γ ω) t ∧ infDist w (localizationDomain (γ ω) t)ᶜ ≤ r},
+        ENNReal.ofReal (halfPlaneGreen (g ω t z) (g ω t w))) ≤ ENNReal.ofReal (J*r^2)) :
+    let U := BrownianDriver.path B (Real.sqrt κ) C.horizon ω
+    let a := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle r ((U,V),z)
+    let b := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e ((U,V),z)
+    let D := fun z t => iteratedDeriv 1
+      (field C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (B t.toNNReal ω)
+    (∫⁻ t, ENNReal.ofReal (|∫ z in A, slab (D z) (a z) (b z) t|^2)) ≤
+      2*ENNReal.ofReal ((Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)))^2)*ENNReal.ofReal (J*r^2)*volume A := by
+  intro U a b D
+  have ham : Measurable a := by
+    have hh := (measurable_observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle r).comp
+      (measurable_const.prodMk measurable_id : Measurable (fun z : ℂ => ((U,V),z)))
+    exact hh
+  have hbm : Measurable b := by
+    have hh := (measurable_observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e).comp
+      (measurable_const.prodMk measurable_id : Measurable (fun z : ℂ => ((U,V),z)))
+    exact hh
+  have hDc : Continuous (fun q : ℝ × ℂ => D q.2 q.1) := by
+    have hp : Continuous (fun q : ℝ × ℂ => (((U,q.2),q.1),B q.1.toNNReal ω)) :=
+      ((continuous_const.prodMk continuous_snd).prodMk continuous_fst).prodMk
+        ((hc.comp continuous_real_toNNReal).comp continuous_fst)
+    have hh := (continuous_mark_first C.height_pos C.horizon.coe_nonneg (1-κ/8) (8/κ-1) (Real.sqrt κ)).comp hp
+    exact hh
+  have hpos (z : ℂ) (hz : z ∈ A) (t : ℝ) (ht : t ∈ Icc 0 ((C.at z hz).stopTime κ B ω)) :
+      g ω t z = position C.height_pos C.horizon.coe_nonneg U z t := by
+    let Cz := C.at z hz
+    have h0 := Cz.initial_deficit_brownian κ B ω (hhalf z hz) (hu z hz) hc hb0
+    have hs := source_until_stop Cz.height_pos Cz.horizon_pos hm _ z _ Cz.angle (Cz.path_eq κ B ω hc) h0
+    exact hs.2 ht
+  have hat (z : ℂ) (hz : z ∈ A) : a z = (C.at z hz).physicalTime κ B γ r ω :=
+    observedTime_eq C κ B γ g f ω hm hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) V hV r
+  have hbt (z : ℂ) (hz : z ∈ A) : b z = (C.at z hz).physicalTime κ B γ e ω :=
+    observedTime_eq C κ B γ g f ω hm hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) V hV e
+  have hab (z : ℂ) : a z ≤ b z := observedTime_antitone C.height_pos C.horizon.coe_nonneg ρ C.angle ((U,V),z) her
+  have ha0 (z : ℂ) : 0 ≤ a z := (observedTime_bounds C.height_pos C.horizon.coe_nonneg ρ C.angle r ((U,V),z)).1
+  have hbz (z : ℂ) (hz : z ∈ A) : b z ≤ (C.at z hz).stopTime κ B ω := by
+    rw [hbt z hz]; exact ((C.at z hz).physicalTime_bounds κ B γ e ω).2
+  let L : ℂ → ℂ → ℝ≥0∞ := fun z w =>
+    if a w ≤ a z ∧ a z ≤ b w then ENNReal.ofReal (halfPlaneGreen
+      (position C.height_pos C.horizon.coe_nonneg U z (a z))
+      (position C.height_pos C.horizon.coe_nonneg U w (a z))) else 0
+  have hL : Measurable (Function.uncurry L) := by
+    have h1 : Measurable (fun q : ℂ × ℂ => ((U,q.1),a q.1)) :=
+      (measurable_const.prodMk measurable_fst).prodMk (ham.comp measurable_fst)
+    have h2 : Measurable (fun q : ℂ × ℂ => ((U,q.2),a q.1)) :=
+      (measurable_const.prodMk measurable_snd).prodMk (ham.comp measurable_fst)
+    have hp1 := (continuous_mark_position C.height_pos C.horizon.coe_nonneg).measurable.comp h1
+    have hp2 := (continuous_mark_position C.height_pos C.horizon.coe_nonneg).measurable.comp h2
+    have hh := (measurable_halfPlaneGreen.comp (hp1.prodMk hp2)).ennreal_ofReal
+    exact hh.ite ((measurableSet_le (ham.comp measurable_snd) (ham.comp measurable_fst)).inter
+      (measurableSet_le (ham.comp measurable_fst) (hbm.comp measurable_snd))) measurable_const
+  have hLe (z : ℂ) (hz : z ∈ A) (w : ℂ) (hw : w ∈ A) : L z w = laterGreen (g ω) a b z w := by
+    dsimp only [L,laterGreen]
+    split_ifs with hh
+    · rw [← hpos z hz (a z) ⟨ha0 z,(hab z).trans (hbz z hz)⟩,
+        ← hpos w hw (a z) ⟨ha0 z,hh.2.trans (hbz w hw)⟩]
+      rfl
+    · rfl
+  have haexp : 0 ≤ 8/κ-1 := by
+    have hh : 1 ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by linarith [hκ.2])
+    linarith
+  have hKn : 0 ≤ Real.sqrt κ*(8/κ-1)*(e/4)^(-(1-κ/8)) := by positivity
+  apply asynchronous_bracket_bound_active hm
+    (continuous_const.mul (hc.comp continuous_real_toNNReal)) hA ham hbm
+    (measurable_slab hDc.measurable ham hbm) L hL hLe hKn (he.trans_le her)
+    (fun z _ => ha0 z) (fun z _ => hab z)
+  · intro z hz
+    exact survivingDomain_mono (hbz z hz) ((C.at z hz).source_live κ B ω hm (hhalf z hz) (hu z hz) hc hb0)
+  · intro z hz hh
+    exact observedTime_approach C.height_pos C.horizon_pos ρ C.angle e r U V (γ ω) hV z hh
+  · intro z hz t ht
+    have hti : t ∈ Icc 0 ((C.at z hz).physicalTime κ B γ e ω) := by
+      rw [← hbt z hz]; exact ⟨(ha0 z).trans ht.1,ht.2⟩
+    have hb := (C.at z hz).first_le_poisson κ hκ B γ g f ω hm hc hb0 (hhalf z hz) (hu z hz) he (hes z hz) hti
+    have hl := survivingDomain_mono (hti.2.trans ((C.at z hz).physicalTime_bounds κ B γ e ω).2)
+      ((C.at z hz).source_live κ B ω hm (hhalf z hz) (hu z hz) hc hb0)
+    have hn : 0 ≤ poissonFactor (fun t => Real.sqrt κ*B t.toNNReal ω) (g ω) t z :=
+      div_nonneg (hm.im_pos hti.1 hl).le (sq_nonneg _)
+    exact slab_abs_le (hab z) (mul_nonneg hKn hn) hb
+  · exact hlayer
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+
+lemma gaussian_exp_compensated (v : ℝ≥0) (x a θ : ℝ) :
+    (∫⁻ y : ℝ, ENNReal.ofReal (Real.exp (x+θ*a*y-θ^2*a^2*(v:ℝ)/2))
+      ∂gaussianReal 0 v) = ENNReal.ofReal (Real.exp x) := by
+  have he (y : ℝ) : Real.exp (x+θ*a*y-θ^2*a^2*(v:ℝ)/2) =
+      Real.exp (x-θ^2*a^2*(v:ℝ)/2)*Real.exp ((θ*a)*y) := by
+    rw [← Real.exp_add]
+    congr 1
+    ring
+  simp_rw [he,ENNReal.ofReal_mul (Real.exp_pos _).le]
+  rw [lintegral_const_mul _ (by fun_prop),← ofReal_integral_eq_lintegral_ofReal
+    (integrable_exp_mul_gaussianReal (θ*a)) (Filter.Eventually.of_forall (fun y => (Real.exp_pos _).le))]
+  have hm := congrFun (mgf_fun_id_gaussianReal (μ := 0) (v := v)) (θ*a)
+  change (∫ y : ℝ, Real.exp ((θ*a)*y) ∂gaussianReal 0 v) = _ at hm
+  rw [hm,← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add]
+  congr 2
+  ring
+
+lemma past_exp_step {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) {s t : ℝ≥0} (hst : s ≤ t)
+    (X A : Ω → ℝ) (hX : Measurable[pastSigma B s] X)
+    (hA : Measurable[pastSigma B s] A) (hXa : AEMeasurable X P)
+    (hAa : AEMeasurable A P) (θ : ℝ) :
+    (∫⁻ sample, ENNReal.ofReal (Real.exp (X sample+θ*A sample*(B t sample-B s sample)-
+      θ^2*(A sample)^2*((t-s:ℝ≥0):ℝ)/2)) ∂P) =
+      ∫⁻ sample, ENNReal.ofReal (Real.exp (X sample)) ∂P := by
+  let V : Ω → ℝ × ℝ := fun sample => (X sample,A sample)
+  have hVm : Measurable[pastSigma B s] V := hX.prodMk hA
+  have hVa : AEMeasurable V P := hXa.prodMk hAa
+  have hi := brownian_indep_of_past_measurable hB s (t-s) hVm
+  rw [add_tsub_cancel_of_le hst] at hi
+  have hl : HasLaw V (P.map V) P := ⟨hVa,rfl⟩
+  have hprod := IndepFun.hasLaw_prod hl (increment_hasLaw hB hst) hi
+  let F : (ℝ × ℝ) × ℝ → ℝ≥0∞ := fun q =>
+    ENNReal.ofReal (Real.exp (q.1.1+θ*q.1.2*q.2-θ^2*q.1.2^2*((t-s:ℝ≥0):ℝ)/2))
+  have hFm : Measurable F := by fun_prop
+  have he := hprod.lintegral_comp hFm.aemeasurable
+  change (∫⁻ sample, F (V sample,B t sample-B s sample) ∂P) = _ at he
+  rw [he,lintegral_prod _ hFm.aemeasurable]
+  simp only [F,gaussian_exp_compensated]
+  have hmap := hl.lintegral_comp
+    (show AEMeasurable (fun q : ℝ × ℝ => ENNReal.ofReal (Real.exp q.1)) (P.map V) by fun_prop)
+  exact hmap.symm
+
+def exponentialLogSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (t : ℕ → ℝ≥0) (A : ℕ → Ω → ℝ) (θ : ℝ) (N : ℕ) (sample : Ω) : ℝ :=
+  ∑ i ∈ Finset.range N, (θ*A i sample*(B (t (i+1)) sample-B (t i) sample)-
+    θ^2*(A i sample)^2*((t (i+1)-t i:ℝ≥0):ℝ)/2)
+
+lemma exponentialLogSum_past {Ω : Type*} [MeasurableSpace Ω]
+    (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) (ht : Monotone t)
+    (A : ℕ → Ω → ℝ) (hA : ∀ i, Measurable[pastSigma B (t i)] (A i))
+    (θ : ℝ) (N : ℕ) : Measurable[pastSigma B (t N)] (exponentialLogSum B t A θ N) := by
+  apply Finset.measurable_fun_sum
+  intro i hi
+  have hAi := (hA i).mono (pastSigma_mono B (ht (Finset.mem_range.1 hi).le)) le_rfl
+  have hBi := past_measurable_eval B (ht (Finset.mem_range.1 hi).le)
+  have hBin := past_measurable_eval B (ht (Nat.succ_le_of_lt (Finset.mem_range.1 hi)))
+  exact (((hAi.const_mul θ).mul (hBin.sub hBi)).sub
+    (((hAi.pow_const 2).const_mul (θ^2)).mul_const _ |>.div_const 2))
+
+lemma exponentialLogSum_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (t : ℕ → ℝ≥0) (A : ℕ → Ω → ℝ) (hA : ∀ i, AEMeasurable (A i) P)
+    (θ : ℝ) (N : ℕ) : AEMeasurable (exponentialLogSum B t A θ N) P := by
+  apply Finset.aemeasurable_fun_sum
+  intro i hi
+  have hBi := hB.toIsPreBrownianReal.aemeasurable (t i)
+  have hBin := hB.toIsPreBrownianReal.aemeasurable (t (i+1))
+  fun_prop
+
+lemma exponentialLogSum_identity {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (t : ℕ → ℝ≥0) (ht : Monotone t)
+    (A : ℕ → Ω → ℝ) (hA : ∀ i, Measurable[pastSigma B (t i)] (A i))
+    (hAa : ∀ i, AEMeasurable (A i) P) (θ : ℝ) (N : ℕ) :
+    (∫⁻ sample, ENNReal.ofReal (Real.exp (exponentialLogSum B t A θ N sample)) ∂P) = 1 := by
+  induction N with
+  | zero => simp [exponentialLogSum]
+  | succ N hN =>
+    simp only [exponentialLogSum,Finset.sum_range_succ]
+    change (∫⁻ sample, ENNReal.ofReal (Real.exp (exponentialLogSum B t A θ N sample+
+      (θ*A N sample*(B (t (N+1)) sample-B (t N) sample)-
+      θ^2*(A N sample)^2*((t (N+1)-t N:ℝ≥0):ℝ)/2))) ∂P) = 1
+    simp_rw [← add_sub_assoc]
+    rw [past_exp_step hB (ht (Nat.le_succ N)) _ _ (exponentialLogSum_past B t ht A hA θ N)
+      (hA N) (exponentialLogSum_aemeasurable hB t A hAa θ N) (hAa N),hN]
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+open PredictableLimit
+
+def dyadicVariance {Ω : Type*} (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ) (n : ℕ) (sample : Ω) : ℝ :=
+  ∑ i ∈ Finset.range (2^n), (A n i sample)^2 *
+    ((dyadicTime T n (i+1)-dyadicTime T n i:ℝ≥0):ℝ)
+
+lemma exponentialLogSum_dyadic {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
+    (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ) (θ : ℝ) (n : ℕ) (sample : Ω) :
+    exponentialLogSum B (dyadicTime T n) (A n) θ (2^n) sample =
+      θ*dyadicDriverSum B T A n sample - θ^2*dyadicVariance T A n sample/2 := by
+  simp only [exponentialLogSum,dyadicDriverSum,dyadicVariance,Finset.mul_sum,
+    Finset.sum_div,← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro i hi
+  ring
+
+theorem limit_exponential_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
+    (hAa : ∀ n i, AEMeasurable (A n i) P)
+    (Y Q : Ω → ℝ)
+    (hY : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A n sample) atTop (𝓝 (Y sample)))
+    (hQ : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicVariance T A n sample) atTop (𝓝 (Q sample)))
+    (K : ℝ) (hK : ∀ᵐ sample ∂P, Q sample ≤ K) (θ : ℝ) :
+    (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*K/2)) := by
+  let E : ℕ → Ω → ℝ≥0∞ := fun n sample => ENNReal.ofReal
+    (Real.exp (exponentialLogSum B (dyadicTime T n) (A n) θ (2^n) sample))
+  have hEm (n : ℕ) : AEMeasurable (E n) P :=
+    ((exponentialLogSum_aemeasurable hB _ _ (hAa n) θ (2^n)).exp).ennreal_ofReal
+  have hEi (n : ℕ) : (∫⁻ sample, E n sample ∂P) = 1 :=
+    exponentialLogSum_identity hB _ (dyadicTime_mono T n) _ (hA n) (hAa n) θ (2^n)
+  have hElim : ∀ᵐ sample ∂P, Tendsto (fun n => E n sample) atTop
+      (𝓝 (ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)))) := by
+    filter_upwards [hY,hQ] with sample hy hq
+    have hh := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+      (Real.continuous_exp.continuousAt.tendsto.comp
+        ((hy.const_mul θ).sub ((hq.const_mul (θ^2)).div_const 2)))
+    simpa only [E,exponentialLogSum_dyadic,Function.comp_def] using hh
+  have hFatou : (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) ≤ 1 := by
+    calc
+      _ = ∫⁻ sample, liminf (fun n => E n sample) atTop ∂P := by
+        apply lintegral_congr_ae
+        filter_upwards [hElim] with sample hs
+        exact hs.liminf_eq.symm
+      _ ≤ liminf (fun n => ∫⁻ sample, E n sample ∂P) atTop := lintegral_liminf_le' hEm
+      _ = 1 := by simp only [hEi,liminf_const]
+  have hlimMeas : AEMeasurable (fun sample => ENNReal.ofReal
+      (Real.exp (θ*Y sample-θ^2*Q sample/2))) P :=
+    aemeasurable_of_tendsto_metrizable_ae atTop hEm hElim
+  calc
+    _ ≤ ∫⁻ sample, ENNReal.ofReal (Real.exp (θ^2*K/2)) *
+        ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [hK] with sample hk
+      rw [← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add]
+      apply ENNReal.ofReal_le_ofReal
+      apply Real.exp_le_exp.2
+      nlinarith [mul_le_mul_of_nonneg_left hk (sq_nonneg θ)]
+    _ = ENNReal.ofReal (Real.exp (θ^2*K/2)) *
+        (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) :=
+      lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+    _ ≤ _ := by simpa only [mul_one] using mul_le_mul_right hFatou (ENNReal.ofReal (Real.exp (θ^2*K/2)))
+
+lemma tail_of_exponential_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} (Y : Ω → ℝ) (hY : AEMeasurable Y P)
+    {K : ℝ} (hK : 0 < K)
+    (hbound : ∀ θ : ℝ, (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*K/2))) {x : ℝ} (hx : 0 < x) :
+    P {sample | x < Y sample} ≤ ENNReal.ofReal (Real.exp (-x^2/(2*K))) := by
+  let θ := x/K
+  have hθ : 0 < θ := div_pos hx hK
+  have hYm : AEMeasurable (fun sample => ENNReal.ofReal (Real.exp (θ*Y sample))) P :=
+    (hY.const_mul θ).exp.ennreal_ofReal
+  have he := meas_ge_le_lintegral_div hYm
+    (ENNReal.ofReal_ne_zero_iff.2 (Real.exp_pos (θ*x))) ENNReal.ofReal_ne_top
+  have he' : P {sample | x < Y sample} ≤
+      (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) /
+        ENNReal.ofReal (Real.exp (θ*x)) := by
+    apply (measure_mono _).trans he
+    intro sample hs
+    exact ENNReal.ofReal_le_ofReal (Real.exp_le_exp.2
+      (mul_le_mul_of_nonneg_left hs.le hθ.le))
+  refine he'.trans ((ENNReal.div_le_div_right (hbound θ) _).trans_eq ?_)
+  rw [← ENNReal.ofReal_div_of_pos (Real.exp_pos _),← Real.exp_sub]
+  congr 2
+  dsimp [θ]
+  field_simp
+  ring
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Mesh
+open TaylorMeshes BrownianPredictable BrownianIto
+
+lemma spatial_variance_tendsto {Z : Type*} [MeasurableSpace Z]
+    (ν : Measure Z) [IsFiniteMeasure ν] (T : ℝ≥0)
+    (C : ℝ → Z → ℝ) (hC : Measurable (Function.uncurry C))
+    (a b : Z → ℝ) (ha : ∀ z, a z ∈ Icc 0 (T:ℝ)) (hb : ∀ z, b z ∈ Icc 0 (T:ℝ))
+    (D : Z → ℝ → ℝ) (hD : ∀ z, ContinuousOn (D z) (Icc 0 (T:ℝ)))
+    (he : ∀ t z, C t z = slab (D z) (a z) (b z) t)
+    (K : ℝ) (hK : 0 ≤ K) (hbound : ∀ t ∈ Icc 0 (T:ℝ), ∀ z, |C t z| ≤ K) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (∫ z, C (mesh T n i) z ∂ν)^2 * ((T:ℝ)/(2:ℝ)^n)) atTop
+      (𝓝 (∫ t in (0:ℝ)..T, (∫ z, C t z ∂ν)^2)) := by
+  let F : ℕ → Z × Z → ℝ := fun n q => ∑ i ∈ Finset.range (2^n),
+    (C (mesh T n i) q.1*C (mesh T n i) q.2)*((T:ℝ)/(2:ℝ)^n)
+  let G : Z × Z → ℝ := fun q => ∫ t in (0:ℝ)..T, C t q.1*C t q.2
+  have hsec (t : ℝ) : Measurable (C t) := hC.comp (measurable_const.prodMk measurable_id)
+  have hF (n : ℕ) : Measurable (F n) := by
+    apply Finset.measurable_fun_sum
+    intro i hi
+    exact (((hsec _).comp measurable_fst).mul ((hsec _).comp measurable_snd)).mul_const _
+  have hFb (n : ℕ) (q : Z × Z) : ‖F n q‖ ≤ (T:ℝ)*K^2 := by
+    calc
+      _ ≤ ∑ i ∈ Finset.range (2^n), ‖(C (mesh T n i) q.1*C (mesh T n i) q.2)*((T:ℝ)/(2:ℝ)^n)‖ := norm_sum_le _ _
+      _ ≤ ∑ _i ∈ Finset.range (2^n), K^2*((T:ℝ)/(2:ℝ)^n) := by
+        apply Finset.sum_le_sum
+        intro i hi
+        rw [Real.norm_eq_abs,abs_mul,abs_mul,abs_of_nonneg (by positivity : 0 ≤ (T:ℝ)/(2:ℝ)^n)]
+        rw [pow_two]
+        gcongr <;> exact hbound _ (mesh_mem _ T.coe_nonneg n i (Finset.mem_range.1 hi).le) _
+      _ = _ := by simp; field_simp
+  have hlim (q : Z × Z) : Tendsto (fun n => F n q) atTop (𝓝 (G q)) := by
+    simp only [F,G,he]
+    exact slab_riemann_tendsto_integral T.coe_nonneg (ha q.1) (hb q.1) (ha q.2) (hb q.2) (hD q.1) (hD q.2)
+  have hl := tendsto_integral_of_dominated_convergence (μ := ν.prod ν) (fun _ : Z × Z => (T:ℝ)*K^2)
+    (fun n => (hF n).aestronglyMeasurable) (integrable_const _) (fun n => Filter.Eventually.of_forall (hFb n))
+    (Filter.Eventually.of_forall hlim)
+  have hFi (n : ℕ) : Integrable (F n) (ν.prod ν) :=
+    Integrable.of_bound (hF n).aestronglyMeasurable _ (Filter.Eventually.of_forall (hFb n))
+  have hprod (t : ℝ) (ht : t ∈ Icc 0 (T:ℝ)) : Integrable (fun q : Z × Z => C t q.1*C t q.2) (ν.prod ν) := by
+    apply Integrable.of_bound (((hsec t).comp measurable_fst).mul ((hsec t).comp measurable_snd)).aestronglyMeasurable (K^2)
+    apply Filter.Eventually.of_forall
+    intro q
+    change |C t q.1*C t q.2| ≤ K^2
+    simp only [abs_mul,pow_two]
+    exact mul_le_mul (hbound t ht q.1) (hbound t ht q.2) (abs_nonneg _) hK
+  have hFe (n : ℕ) : (∫ q, F n q ∂ν.prod ν) = ∑ i ∈ Finset.range (2^n),
+      (∫ z, C (mesh T n i) z ∂ν)^2 * ((T:ℝ)/(2:ℝ)^n) := by
+    rw [show F n = fun q => ∑ i ∈ Finset.range (2^n), (C (mesh T n i) q.1*C (mesh T n i) q.2)*((T:ℝ)/(2:ℝ)^n) from rfl,
+      integral_finsetSum]
+    · apply Finset.sum_congr rfl
+      intro i hi
+      rw [integral_mul_const,integral_prod_mul,pow_two]
+    · intro i hi
+      exact (hprod _ (mesh_mem _ T.coe_nonneg n i (Finset.mem_range.1 hi).le)).mul_const _
+  let μ := volume.restrict (Ioc (0:ℝ) (T:ℝ))
+  have hCt : Measurable (fun q : (Z × Z) × ℝ => C q.2 q.1.1*C q.2 q.1.2) :=
+    ((hC.comp (measurable_snd.prodMk measurable_fst.fst)).mul
+      (hC.comp (measurable_snd.prodMk measurable_fst.snd)))
+  have hCti : Integrable (fun q : (Z × Z) × ℝ => C q.2 q.1.1*C q.2 q.1.2) ((ν.prod ν).prod μ) := by
+    apply Integrable.of_bound hCt.aestronglyMeasurable (K^2)
+    have hm : ∀ᵐ t ∂μ, t ∈ Ioc (0:ℝ) (T:ℝ) := ae_restrict_mem measurableSet_Ioc
+    filter_upwards [(Measure.ae_prod_iff_ae_ae (measurableSet_Ioc.preimage measurable_snd)).2 (Filter.Eventually.of_forall (fun q : Z × Z => hm))] with q hq
+    simp only [Real.norm_eq_abs,abs_mul,pow_two]
+    exact mul_le_mul (hbound q.2 ⟨hq.1.le,hq.2⟩ q.1.1) (hbound q.2 ⟨hq.1.le,hq.2⟩ q.1.2) (abs_nonneg _) hK
+  have hGe : (∫ q, G q ∂ν.prod ν) = ∫ t in (0:ℝ)..T, (∫ z, C t z ∂ν)^2 := by
+    simp only [G,intervalIntegral.integral_of_le T.coe_nonneg]
+    rw [integral_integral_swap hCti]
+    apply integral_congr_ae
+    exact Filter.Eventually.of_forall (fun t => by
+      dsimp only
+      rw [integral_prod_mul,pow_two])
+  simpa only [hFe,hGe] using hl
+
+lemma spatial_dyadicVariance_tendsto {Z Ω : Type*} [MeasurableSpace Z]
+    (ν : Measure Z) [IsFiniteMeasure ν] (T : ℝ≥0)
+    (A : ℕ → ℕ → Ω × Z → ℝ) (ω : Ω)
+    (C : ℝ → Z → ℝ) (hC : Measurable (Function.uncurry C))
+    (a b : Z → ℝ) (ha : ∀ z, a z ∈ Icc 0 (T:ℝ)) (hb : ∀ z, b z ∈ Icc 0 (T:ℝ))
+    (D : Z → ℝ → ℝ) (hD : ∀ z, ContinuousOn (D z) (Icc 0 (T:ℝ)))
+    (he : ∀ t z, C t z = slab (D z) (a z) (b z) t)
+    (hAe : ∀ n i, ∀ᵐ z ∂ν, A n i (ω,z) = C (mesh T n i) z)
+    (K : ℝ) (hK : 0 ≤ K) (hbound : ∀ t ∈ Icc 0 (T:ℝ), ∀ z, |C t z| ≤ K) :
+    Tendsto (fun n => dyadicVariance T (fun n i ω => ∫ z, A n i (ω,z) ∂ν) n ω) atTop
+      (𝓝 (∫ t in (0:ℝ)..T, (∫ z, C t z ∂ν)^2)) := by
+  have hl := spatial_variance_tendsto ν T C hC a b ha hb D hD he K hK hbound
+  convert hl using 1
+  funext n
+  unfold dyadicVariance
+  apply Finset.sum_congr rfl
+  intro i hi
+  dsimp only
+  rw [integral_congr_ae (hAe n i)]
+  rw [NNReal.coe_sub (dyadicTime_mono T n (Nat.le_succ i)),dyadicTime_step]
+
+end SLEExactGauge.Mesh
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
+open BrownianPredictable BrownianIto TaylorMeshes PredictableLimit
+local instance slabMomentsCoeffRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance slabMomentsCoeffRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance slabMomentsCoeffComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance slabMomentsCoeffComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+def slabCoeff {Ω : Type*} {ε : ℝ} (hε : 0 < ε) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (T s : ℝ≥0) (ρ δ e r : ℝ) (q : Ω × ℂ) : ℝ :=
+  coeff hε κ B T s ρ δ e q - coeff hε κ B T s ρ δ r q
+
+def pathFirst {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ : ℝ)
+    (U : C(I,ℝ)) (b : ℝ≥0 → ℝ) (z : ℂ) (t : ℝ) : ℝ :=
+  iteratedDeriv 1 (field hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ) U z t) (b t.toNNReal)
+
+def pathSlab {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ ρ δ e r : ℝ)
+    (U : C(I,ℝ)) (V : C(I,ℂ)) (b : ℝ≥0 → ℝ) (t : ℝ) (z : ℂ) : ℝ :=
+  slab (pathFirst hε hT κ U b z)
+    (observedTime hε hT ρ δ r ((U,V),z)) (observedTime hε hT ρ δ e ((U,V),z)) t
+
+lemma continuous_pathFirst {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ : ℝ)
+    (U : C(I,ℝ)) (b : ℝ≥0 → ℝ) (hb : Continuous b) :
+    Continuous (fun q : ℝ × ℂ => pathFirst hε hT κ U b q.2 q.1) := by
+  have hp : Continuous (fun q : ℝ × ℂ => (((U,q.2),q.1),b q.1.toNNReal)) :=
+    ((continuous_const.prodMk continuous_snd).prodMk continuous_fst).prodMk
+      ((hb.comp continuous_real_toNNReal).comp continuous_fst)
+  have hh := (continuous_mark_first hε hT (1-κ/8) (8/κ-1) (Real.sqrt κ)).comp hp
+  exact hh
+
+lemma measurable_pathSlab {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ ρ δ e r : ℝ)
+    (U : C(I,ℝ)) (V : C(I,ℂ)) (b : ℝ≥0 → ℝ) (hb : Continuous b) :
+    Measurable (Function.uncurry (pathSlab hε hT κ ρ δ e r U V b)) := by
+  have hs (s : ℝ) := (measurable_observedTime hε hT ρ δ s).comp
+    (measurable_const.prodMk measurable_id : Measurable (fun z : ℂ => ((U,V),z)))
+  exact measurable_slab (continuous_pathFirst hε hT κ U b hb).measurable (hs r) (hs e)
+
+lemma pathSlab_uniform_bound {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (ρ δ e r : ℝ) (her : e ≤ r) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I,ℝ)) (V : C(I,ℂ)) (b : ℝ≥0 → ℝ) t z,
+      |pathSlab hε hT κ ρ δ e r U V b t z| ≤ K := by
+  have hp : 0 ≤ 1-κ/8 := by linarith [hκ.2]
+  have ha : 0 ≤ 8/κ-1 := by
+    have hh : 1 ≤ 8/κ := (le_div_iff₀ hκ.1).2 (by simpa using hκ.2.le)
+    linarith
+  let K := |Real.sqrt κ| *((2*ε/Real.exp ((2/ε^2)*T))^(-(1-κ/8))*(8/κ-1)/ε)
+  have hK : 0 ≤ K := by dsimp only [K]; positivity
+  refine ⟨K,hK,?_⟩
+  intro U V b t z
+  exact slab_abs_le (observedTime_antitone hε hT ρ δ ((U,V),z) her) hK
+    (first_bound hε hT _ _ _ hp ha U z t (b t.toNNReal))
+
+lemma slabCoeff_eq {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
+    (C : FiniteTilt.CommonCutoff A u) (κ : ℝ) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
+    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
+    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e r : ℝ) (s : ℝ≥0) :
+    slabCoeff C.height_pos κ B C.horizon s ρ C.angle e r (ω,z) =
+      pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
+        (BrownianDriver.path B (Real.sqrt κ) C.horizon ω) V (fun t => B t ω) s z := by
+  unfold slabCoeff
+  rw [← coeff_common C κ B z hz hr e s ω,← coeff_common C κ B z hz hr r s ω,
+    (C.at z hz).physicalCoeff_eq κ B γ g f ω hm hc hb0 hhalf hu e 1 s,
+    (C.at z hz).physicalCoeff_eq κ B γ g f ω hm hc hb0 hhalf hu r 1 s]
+  dsimp only [pathSlab,slab,pathFirst]
+  rw [observedTime_eq C κ B γ g f ω hm hc hb0 z hz hhalf hu hr V hV r,
+    observedTime_eq C κ B γ g f ω hm hc hb0 z hz hhalf hu hr V hV e]
+  simp only [Real.toNNReal_coe]
+  rfl
+
+lemma slabCoeff_variance_tendsto {Ω : Type*} {A : Set ℂ} {u : ℂ → ℝ}
+    (C : FiniteTilt.CommonCutoff A u) (hA : MeasurableSet A) [IsFiniteMeasure (volume.restrict A)]
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ) (ω : Ω)
+    (hm : IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (hc : Continuous (fun t => B t ω)) (hb0 : B 0 ω = 0)
+    (hhalf : ∀ z ∈ A, z ∈ halfPlane) (hu : ∀ z ∈ A, 0 < u z)
+    {ρ : ℝ} (hr : ∀ z ∈ A, (2*z.im)*Real.exp (-u z) = ρ)
+    (V : C(I,ℂ)) (hV : ∀ q : I, V q = γ ω ((C.horizon:ℝ)*q)) (e r : ℝ) (her : e ≤ r) :
+    Tendsto (fun n => dyadicVariance C.horizon
+      (fun n i ω => ∫ z in A, slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r (ω,z)) n ω)
+      atTop (𝓝 (∫ t in (0:ℝ)..C.horizon, (∫ z in A,
+        pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
+          (BrownianDriver.path B (Real.sqrt κ) C.horizon ω) V (fun t => B t ω) t z)^2)) := by
+  let U := BrownianDriver.path B (Real.sqrt κ) C.horizon ω
+  let a := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle r ((U,V),z)
+  let b := fun z => observedTime C.height_pos C.horizon.coe_nonneg ρ C.angle e ((U,V),z)
+  let D := pathFirst C.height_pos C.horizon.coe_nonneg κ U (fun t => B t ω)
+  let S := pathSlab C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r U V (fun t => B t ω)
+  obtain ⟨K,hK,hbound⟩ := pathSlab_uniform_bound C.height_pos C.horizon.coe_nonneg κ hκ ρ C.angle e r her
+  apply spatial_dyadicVariance_tendsto (K := K) (volume.restrict A) C.horizon
+    (fun n i => slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r)
+    ω S (measurable_pathSlab _ _ _ _ _ _ _ _ _ _ hc) a b
+    (fun z => observedTime_bounds _ _ _ _ _ _) (fun z => observedTime_bounds _ _ _ _ _ _) D
+  · intro z
+    have hh := (continuous_pathFirst C.height_pos C.horizon.coe_nonneg κ U (fun t => B t ω) hc).comp
+      (continuous_id.prodMk continuous_const : Continuous (fun t : ℝ => (t,z)))
+    exact hh.continuousOn
+  · intro t z; rfl
+  · intro n i
+    filter_upwards [ae_restrict_mem hA] with z hz
+    simpa only [coe_dyadicTime] using
+      slabCoeff_eq C κ B γ g f ω hm hc hb0 z hz (hhalf z hz) (hu z hz) (hr z hz) V hV e r (dyadicTime C.horizon n i)
+  · exact hK
+  · intro t ht z
+    exact hbound U V (fun t => B t ω) t z
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.SpatialLimit
+open L2Limit
+
+lemma uniformIntegrable_one_of_eLpNorm_two_bounded {Ω ι : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {X : ι → Ω → ℝ}
+    (hXm : ∀ i, AEStronglyMeasurable (X i) P) (C : ℝ≥0)
+    (hC : ∀ i, eLpNorm (X i) 2 P ≤ C) : UniformIntegrable X 1 P := by
+  let D : ℝ≥0∞ := (C : ℝ≥0∞) * (P univ) ^ (1/2 : ℝ)
+  have hDt : D ≠ ∞ := by dsimp [D]; finiteness
+  refine ⟨?_, D.toNNReal, ?_⟩
+  · apply unifIntegrable_iff.2
+    intro ε hε
+    by_cases hεt : ε = ∞
+    · subst ε
+      exact ⟨1, by norm_num, fun _ _ _ => le_top⟩
+    have hεr : 0 < ε.toReal := ENNReal.toReal_pos hε.ne' hεt
+    let d : ℝ≥0 := ⟨ε.toReal / ((C : ℝ)+1), div_nonneg hεr.le (by positivity)⟩
+    have hd : 0 < (d : ℝ) := div_pos hεr (by positivity)
+    refine ⟨ENNReal.ofReal ((d : ℝ)^2), ENNReal.ofReal_pos.2 (sq_pos_of_pos hd),
+      fun i s hμs => ?_⟩
+    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (μ := P.restrict s)
+      (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i).restrict
+    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
+    rw [Measure.restrict_apply_univ] at hb
+    have hp : (P s) ^ (1/2 : ℝ) ≤ (d : ℝ≥0∞) := by
+      calc
+        _ ≤ (ENNReal.ofReal ((d : ℝ)^2)) ^ (1/2 : ℝ) :=
+          ENNReal.rpow_le_rpow hμs (by norm_num)
+        _ = _ := by
+          rw [ENNReal.ofReal_pow d.coe_nonneg]
+          simp only [ENNReal.ofReal_coe_nnreal]
+          rw [← ENNReal.rpow_natCast, ← ENNReal.rpow_mul]
+          norm_num
+    calc
+      _ ≤ eLpNorm (X i) 2 (P.restrict s) * (P s) ^ (1/2 : ℝ) := hb
+      _ ≤ C * (d : ℝ≥0∞) := mul_le_mul
+        ((eLpNorm_mono_measure _ Measure.restrict_le_self).trans (hC i)) hp (by positivity) (by positivity)
+      _ ≤ ENNReal.ofReal ε.toReal := by
+        rw [← ENNReal.coe_mul, ← ENNReal.ofReal_coe_nnreal]
+        simp only [NNReal.coe_mul]
+        apply ENNReal.ofReal_le_ofReal
+        change (C : ℝ) * (ε.toReal / ((C : ℝ)+1)) ≤ ε.toReal
+        rw [← mul_div_assoc, div_le_iff₀ (by positivity)]
+        nlinarith
+      _ = ε := ENNReal.ofReal_toReal hεt
+  · intro i
+    have hb := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (by norm_num : (1 : ℝ≥0∞) ≤ 2) (hXm i)
+    rw [ENNReal.coe_toNNReal hDt]
+    norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one] at hb
+    exact hb.trans (mul_le_mul_left (hC i) ((P univ) ^ (1/2 : ℝ)))
+
+lemma L1_tendsto_of_square_bound {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Y : Ω → ℝ}
+    (hX : ∀ n, MemLp (X n) 2 P) (A : ℝ)
+    (hA : ∀ n, (∫ ω, X n ω ^ 2 ∂P) ≤ A)
+    (hXY : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) :
+    Integrable Y P ∧ Tendsto (fun n => eLpNorm (X n-Y) 1 P) atTop (𝓝 0) := by
+  let C : ℝ≥0∞ := (ENNReal.ofReal A) ^ (1/2 : ℝ)
+  have hCt : C ≠ ∞ := ENNReal.rpow_ne_top_of_nonneg (by norm_num) ENNReal.ofReal_ne_top
+  have hC (n : ℕ) : eLpNorm (X n) 2 P ≤ C.toNNReal := by
+    rw [ENNReal.coe_toNNReal hCt, eLpNorm_two_eq_integral_sq (hX n)]
+    exact ENNReal.rpow_le_rpow (ENNReal.ofReal_le_ofReal (hA n)) (by norm_num)
+  have hui := uniformIntegrable_one_of_eLpNorm_two_bounded
+    (fun n => (hX n).aestronglyMeasurable) C.toNNReal hC
+  have hY := hui.memLp_of_ae_tendsto hXY
+  exact ⟨memLp_one_iff_integrable.1 hY,
+    tendsto_Lp_finite_of_tendsto_ae (by norm_num : (1 : ℝ≥0∞) ≤ 1)
+      (by norm_num) (fun n => (hX n).aestronglyMeasurable) hY hui.unifIntegrable hXY⟩
+
+lemma integrate_L1_le {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
+    {X Y : Ω × Z → ℝ} (hX : Integrable X (P.prod ν)) (hY : Integrable Y (P.prod ν)) :
+    eLpNorm ((fun ω => ∫ z, X (ω,z) ∂ν)-(fun ω => ∫ z, Y (ω,z) ∂ν)) 1 P ≤
+      eLpNorm (X-Y) 1 (P.prod ν) := by
+  rw [eLpNorm_one_eq_lintegral_enorm
+    (hX.aestronglyMeasurable.integral_prod_right'.sub
+      hY.aestronglyMeasurable.integral_prod_right'),
+    eLpNorm_one_eq_lintegral_enorm (hX.sub hY).aestronglyMeasurable]
+  rw [lintegral_prod _ (hX.sub hY).aestronglyMeasurable.enorm]
+  apply lintegral_mono_ae
+  filter_upwards [hX.prod_right_ae,hY.prod_right_ae] with ω hx hy
+  simp only [Pi.sub_apply]
+  rw [← integral_sub hx hy]
+  change ‖∫ z, X (ω,z)-Y (ω,z) ∂ν‖ₑ ≤ _
+  exact enorm_integral_le_lintegral_enorm _
+
+lemma spatial_L1_limit {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} {ν : Measure Z} [IsFiniteMeasure P] [IsFiniteMeasure ν]
+    {X : ℕ → Ω × Z → ℝ} {Y : Ω × Z → ℝ}
+    (hX : ∀ n, MemLp (X n) 2 (P.prod ν)) (A : ℝ)
+    (hA : ∀ n, (∫ q, X n q ^ 2 ∂P.prod ν) ≤ A)
+    (hXY : ∀ᵐ q ∂P.prod ν, Tendsto (fun n => X n q) atTop (𝓝 (Y q))) :
+    Integrable Y (P.prod ν) ∧
+      ∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ ω ∂P,
+        Tendsto (fun n => ∫ z, X (ns n) (ω,z) ∂ν) atTop (𝓝 (∫ z, Y (ω,z) ∂ν)) := by
+  obtain ⟨hY,hL⟩ := L1_tendsto_of_square_bound hX A hA hXY
+  have hi (n : ℕ) := (hX n).integrable (by norm_num : (1 : ℝ≥0∞) ≤ 2)
+  have hh : Tendsto (fun n => eLpNorm
+      ((fun ω => ∫ z, X n (ω,z) ∂ν)-(fun ω => ∫ z, Y (ω,z) ∂ν)) 1 P) atTop (𝓝 0) :=
+    tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hL
+      (fun _ => bot_le) (fun n => integrate_L1_le (hi n) hY)
+  have hm := tendstoInMeasure_of_tendsto_eLpNorm (by norm_num : (1 : ℝ≥0∞) ≠ 0)
+    hh
+  exact ⟨hY,hm.exists_seq_tendsto_ae⟩
+
+end SLEExactGauge.SpatialLimit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.SpatialLimit
+
+lemma memLp_two_prod_of_fiber_bound {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} {ν : Measure Z} [SFinite P] [IsFiniteMeasure ν]
+    {X : Ω × Z → ℝ} (hX : AEMeasurable X (P.prod ν))
+    (hXi : ∀ᵐ z ∂ν, MemLp (fun ω => X (ω,z)) 2 P) (A : ℝ)
+    (hA : ∀ᵐ z ∂ν, (∫ ω, X (ω,z)^2 ∂P) ≤ A) :
+    MemLp X 2 (P.prod ν) ∧ (∫ q, X q ^ 2 ∂P.prod ν) ≤ ∫ _ : Z, A ∂ν := by
+  have hsq : AEStronglyMeasurable (fun q => X q^2) (P.prod ν) := (hX.pow_const 2).aestronglyMeasurable
+  have hmeas : AEStronglyMeasurable (fun z => ∫ ω, ‖X (ω,z)^2‖ ∂P) ν :=
+    hsq.norm.prod_swap.integral_prod_right'
+  have he (z : Z) : (∫ ω, ‖X (ω,z)^2‖ ∂P) = ∫ ω, X (ω,z)^2 ∂P := by
+    apply integral_congr_ae
+    exact Filter.Eventually.of_forall (fun ω => Real.norm_of_nonneg (sq_nonneg _))
+  have hi : Integrable (fun q => X q^2) (P.prod ν) := by
+    apply (integrable_prod_iff' hsq).2
+    refine ⟨hXi.mono (fun z hz => hz.integrable_sq),?_⟩
+    apply (integrable_const A).mono' hmeas
+    filter_upwards [hA] with z hz
+    rw [he,Real.norm_of_nonneg (integral_nonneg (fun _ => sq_nonneg _))]
+    exact hz
+  refine ⟨(memLp_two_iff_integrable_sq hX.aestronglyMeasurable).2 hi,?_⟩
+  rw [integral_prod_symm _ hi]
+  exact integral_mono_ae hi.integral_prod_right (integrable_const _) hA
+
+lemma ae_prod_tendsto_of_fibers {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
+    {X : ℕ → Ω × Z → ℝ} {Y : Ω × Z → ℝ}
+    (hX : ∀ n, AEMeasurable (X n) (P.prod ν)) (hY : AEMeasurable Y (P.prod ν))
+    (hh : ∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Tendsto (fun n => X n (ω,z)) atTop (𝓝 (Y (ω,z)))) :
+    ∀ᵐ q ∂P.prod ν, Tendsto (fun n => X n q) atTop (𝓝 (Y q)) := by
+  let F : ℕ → Ω × Z → ℝ := fun n => (hX n).mk (X n)
+  let G := hY.mk Y
+  have hF (n : ℕ) : Measurable (F n) := (hX n).measurable_mk
+  have hG : Measurable G := hY.measurable_mk
+  have he : ∀ᵐ q ∂P.prod ν, (∀ n, X n q = F n q) ∧ Y q = G q :=
+    (ae_all_iff.2 (fun n => (hX n).ae_eq_mk)).and hY.ae_eq_mk
+  have hes : ∀ᵐ q ∂ν.prod P, (∀ n, X n q.swap = F n q.swap) ∧ Y q.swap = G q.swap :=
+    Measure.measurePreserving_swap.quasiMeasurePreserving.ae he
+  have hf : ∀ᵐ q ∂ν.prod P, Tendsto (fun n => F n q.swap) atTop (𝓝 (G q.swap)) := by
+    apply (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun
+      (fun n => (hF n).comp measurable_swap) (hG.comp measurable_swap))).2
+    filter_upwards [hh,Measure.ae_ae_of_ae_prod hes] with z hz hez
+    filter_upwards [hz,hez] with ω hω heω
+    change (∀ n, X n (ω,z) = F n (ω,z)) ∧ Y (ω,z) = G (ω,z) at heω
+    change Tendsto (fun n => F n (ω,z)) atTop (𝓝 (G (ω,z)))
+    simpa only [heω.1,heω.2] using hω
+  have hf' := Measure.measurePreserving_swap.quasiMeasurePreserving.ae hf
+  filter_upwards [he,hf'] with q heq hq
+  simpa only [heq.1,heq.2,Prod.swap_swap] using hq
+
+end SLEExactGauge.SpatialLimit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+open PredictableLimit
+
+theorem limit_exponential_bound_subseq {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (T : ℝ≥0) (A : ℕ → ℕ → Ω → ℝ)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
+    (hAa : ∀ n i, AEMeasurable (A n i) P)
+    (ns : ℕ → ℕ) (Y Q : Ω → ℝ)
+    (hY : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicDriverSum B T A (ns n) sample) atTop (𝓝 (Y sample)))
+    (hQ : ∀ᵐ sample ∂P, Tendsto (fun n => dyadicVariance T A (ns n) sample) atTop (𝓝 (Q sample)))
+    (K : ℝ) (hK : ∀ᵐ sample ∂P, Q sample ≤ K) (θ : ℝ) :
+    (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample)) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*K/2)) := by
+  let E : ℕ → Ω → ℝ≥0∞ := fun n sample => ENNReal.ofReal
+    (Real.exp (exponentialLogSum B (dyadicTime T (ns n)) (A (ns n)) θ (2^(ns n)) sample))
+  have hEm (n : ℕ) : AEMeasurable (E n) P :=
+    ((exponentialLogSum_aemeasurable hB _ _ (hAa (ns n)) θ (2^(ns n))).exp).ennreal_ofReal
+  have hEi (n : ℕ) : (∫⁻ sample, E n sample ∂P) = 1 :=
+    exponentialLogSum_identity hB _ (dyadicTime_mono T (ns n)) _ (hA (ns n)) (hAa (ns n)) θ (2^(ns n))
+  have hElim : ∀ᵐ sample ∂P, Tendsto (fun n => E n sample) atTop
+      (𝓝 (ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)))) := by
+    filter_upwards [hY,hQ] with sample hy hq
+    have hh := ENNReal.continuous_ofReal.continuousAt.tendsto.comp
+      (Real.continuous_exp.continuousAt.tendsto.comp
+        ((hy.const_mul θ).sub ((hq.const_mul (θ^2)).div_const 2)))
+    simpa only [E,exponentialLogSum_dyadic,Function.comp_def] using hh
+  have hFatou : (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) ≤ 1 := by
+    calc
+      _ = ∫⁻ sample, liminf (fun n => E n sample) atTop ∂P := by
+        apply lintegral_congr_ae
+        filter_upwards [hElim] with sample hs
+        exact hs.liminf_eq.symm
+      _ ≤ liminf (fun n => ∫⁻ sample, E n sample ∂P) atTop := lintegral_liminf_le' hEm
+      _ = 1 := by simp only [hEi,liminf_const]
+  have hlimMeas : AEMeasurable (fun sample => ENNReal.ofReal
+      (Real.exp (θ*Y sample-θ^2*Q sample/2))) P :=
+    aemeasurable_of_tendsto_metrizable_ae atTop hEm hElim
+  calc
+    _ ≤ ∫⁻ sample, ENNReal.ofReal (Real.exp (θ^2*K/2)) *
+        ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P := by
+      apply lintegral_mono_ae
+      filter_upwards [hK] with sample hk
+      rw [← ENNReal.ofReal_mul (Real.exp_pos _).le,← Real.exp_add]
+      apply ENNReal.ofReal_le_ofReal
+      apply Real.exp_le_exp.2
+      nlinarith [mul_le_mul_of_nonneg_left hk (sq_nonneg θ)]
+    _ = ENNReal.ofReal (Real.exp (θ^2*K/2)) *
+        (∫⁻ sample, ENNReal.ofReal (Real.exp (θ*Y sample-θ^2*Q sample/2)) ∂P) :=
+      lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+    _ ≤ _ := by simpa only [mul_one] using mul_le_mul_right hFatou (ENNReal.ofReal (Real.exp (θ^2*K/2)))
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.SpatialLimit
+open BrownianPredictable PredictableLimit
+
+lemma dyadicDriverSum_integrate {Ω Z : Type*} [MeasurableSpace Z]
+    (ν : Measure Z) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
+    (A : ℕ → ℕ → Ω × Z → ℝ) (n : ℕ) (ω : Ω)
+    (hA : ∀ i, Integrable (fun z => A n i (ω,z)) ν) :
+    dyadicDriverSum B T (fun n i ω => ∫ z, A n i (ω,z) ∂ν) n ω =
+      ∫ z, dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω ∂ν := by
+  unfold dyadicDriverSum
+  rw [integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro i hi
+    rw [integral_mul_const]
+  · intro i hi
+    exact (hA i).mul_const _
+
+theorem spatial_exponential_bound {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} [IsProbabilityMeasure P] (ν : Measure Z) [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (A : ℕ → ℕ → Ω × Z → ℝ)
+    (hAi : ∀ n i, Integrable (A n i) (P.prod ν))
+    (hA : ∀ n i, @Measurable (Ω × Z) ℝ
+      (Prod.instMeasurableSpace (m₁ := pastSigma B (dyadicTime T n i)) (m₂ := inferInstance)) _ (A n i))
+    (Y : Ω × Z → ℝ)
+    (hX : ∀ n, MemLp (fun q => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) 2 (P.prod ν))
+    (V : ℝ) (hV : ∀ n, (∫ q, (dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1)^2 ∂P.prod ν) ≤ V)
+    (hY : ∀ᵐ q ∂P.prod ν, Tendsto
+      (fun n => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) atTop (𝓝 (Y q)))
+    (Q : Ω → ℝ)
+    (hQ : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicVariance T (fun n i ω => ∫ z, A n i (ω,z) ∂ν) n ω)
+      atTop (𝓝 (Q ω))) (K θ : ℝ) (hK : ∀ᵐ ω ∂P, Q ω ≤ K) :
+    (∫⁻ ω, ENNReal.ofReal (Real.exp (θ*(∫ z, Y (ω,z) ∂ν))) ∂P) ≤
+      ENNReal.ofReal (Real.exp (θ^2*K/2)) := by
+  let C := fun n i ω => ∫ z, A n i (ω,z) ∂ν
+  have hCi (n i : ℕ) : AEMeasurable (C n i) P :=
+    (hAi n i).aestronglyMeasurable.integral_prod_right'.aemeasurable
+  have hCp (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (C n i) := by
+    let : MeasurableSpace Ω := pastSigma B (dyadicTime T n i)
+    exact (hA n i).stronglyMeasurable.integral_prod_right'.measurable
+  obtain ⟨hYi,ns,hns,hlim⟩ := spatial_L1_limit hX V hV hY
+  have he : ∀ᵐ ω ∂P, ∀ n, dyadicDriverSum B T C n ω =
+      ∫ z, dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω ∂ν := by
+    have hh : ∀ᵐ ω ∂P, ∀ n i, Integrable (fun z => A n i (ω,z)) ν :=
+      ae_all_iff.2 (fun n => ae_all_iff.2 (fun i => (hAi n i).prod_right_ae))
+    filter_upwards [hh] with ω hω n
+    exact dyadicDriverSum_integrate ν B T A n ω (hω n)
+  have hS : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B T C (ns n) ω) atTop
+      (𝓝 (∫ z, Y (ω,z) ∂ν)) := by
+    filter_upwards [he,hlim] with ω heω hω
+    simpa only [heω] using hω
+  have hQV : ∀ᵐ ω ∂P, Tendsto (fun n => dyadicVariance T C (ns n) ω) atTop (𝓝 (Q ω)) :=
+    hQ.mono (fun ω hω => hω.comp hns.tendsto_atTop)
+  exact limit_exponential_bound_subseq hB T C hCp hCi ns _ Q hS hQV K hK θ
+
+lemma sums_aemeasurable {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (A : ℕ → ℕ → Ω × Z → ℝ)
+    (hA : ∀ n i, AEMeasurable (A n i) (P.prod ν)) (n : ℕ) :
+    AEMeasurable (fun q => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) (P.prod ν) := by
+  apply Finset.aemeasurable_fun_sum
+  intro i hi
+  exact (hA n i).mul
+    (((hB.toIsPreBrownianReal.hasLaw_eval _).aemeasurable.comp_fst).sub
+      ((hB.toIsPreBrownianReal.hasLaw_eval _).aemeasurable.comp_fst))
+
+lemma bounded_sums {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} [IsProbabilityMeasure P] (ν : Measure Z) [IsFiniteMeasure ν]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (A : ℕ → ℕ → Ω × Z → ℝ)
+    (hA : ∀ n i, @Measurable (Ω × Z) ℝ
+      (Prod.instMeasurableSpace (m₁ := pastSigma B (dyadicTime T n i)) (m₂ := inferInstance)) _ (A n i))
+    (hAm : ∀ n i, AEMeasurable (A n i) (P.prod ν))
+    (K : ℝ) (hK : 0 ≤ K) (hbound : ∀ n i q, |A n i q| ≤ K) :
+    (∀ n i, Integrable (A n i) (P.prod ν)) ∧
+    ∀ n, MemLp (fun q => dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1) 2 (P.prod ν) ∧
+      (∫ q, (dyadicDriverSum B T (fun n i ω => A n i (ω,q.2)) n q.1)^2 ∂P.prod ν) ≤
+        ∫ _ : Z, K^2*(T:ℝ) ∂ν := by
+  have hi (n i : ℕ) : Integrable (A n i) (P.prod ν) :=
+    Integrable.of_bound (hAm n i).aestronglyMeasurable K (Filter.Eventually.of_forall (hbound n i))
+  refine ⟨hi,?_⟩
+  intro n
+  have hsect : ∀ᵐ z ∂ν, ∀ n i, AEStronglyMeasurable (fun ω => A n i (ω,z)) P :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (fun i => ((hi n i).prod_left_ae.mono (fun z hz => hz.aestronglyMeasurable))))
+  have hcp (z : Z) (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (fun ω => A n i (ω,z)) := by
+    let : MeasurableSpace Ω := pastSigma B (dyadicTime T n i)
+    exact (hA n i).comp (measurable_id.prodMk measurable_const)
+  have hh : ∀ᵐ z ∂ν,
+      MemLp (fun ω => dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω) 2 P ∧
+      (∫ ω, (dyadicDriverSum B T (fun n i ω => A n i (ω,z)) n ω)^2 ∂P) ≤ K^2*(T:ℝ) := by
+    filter_upwards [hsect] with z hz
+    have h2 (n i : ℕ) : MemLp (fun ω => A n i (ω,z)) 2 P :=
+      MemLp.of_bound (hz n i) K (Filter.Eventually.of_forall (fun ω => hbound n i (ω,z)))
+    exact ⟨dyadicDriverSum_memLp hB T _ (hcp z) h2 n,
+      dyadicDriverSum_second_moment_le hB T _ (hcp z) h2 K hK
+        (fun n i => Filter.Eventually.of_forall (fun ω => hbound n i (ω,z))) n⟩
+  exact memLp_two_prod_of_fiber_bound (sums_aemeasurable hB T A hAm n)
+    (hh.mono (fun z hz => hz.1)) (K^2*(T:ℝ)) (hh.mono (fun z hz => hz.2))
+
+end SLEExactGauge.SpatialLimit
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.MeshSlab
+open LoewnerCutoff LoewnerField LoewnerExit LoewnerLocalized Mesh
+open BrownianPredictable BrownianIto TaylorMeshes PredictableLimit
+local instance slabMomentsDifferenceRealPathMeasurableSpace : MeasurableSpace C(I,ℝ) := borel _
+local instance slabMomentsDifferenceRealPathBorelSpace : BorelSpace C(I,ℝ) := ⟨rfl⟩
+local instance slabMomentsDifferenceComplexPathMeasurableSpace : MeasurableSpace C(I,ℂ) := borel _
+local instance slabMomentsDifferenceComplexPathBorelSpace : BorelSpace C(I,ℂ) := ⟨rfl⟩
+
+def observedDifference {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (κ ρ δ e r : ℝ)
+    (q : (C(I,ℝ) × C(I,ℂ)) × ℂ) : ℝ :=
+  observedValue hε hT ρ δ e (1-κ/8) (8/κ-1) q -
+    observedValue hε hT ρ δ r (1-κ/8) (8/κ-1) q
+
+lemma observedDifference_aemeasurable {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (ν : Measure ℂ) [SFinite ν] {ε : ℝ} (hε : 0 < ε)
+    (κ ρ δ e r : ℝ) (T : ℝ≥0) (V : Ω → C(I,ℂ)) (hV : Measurable V) :
+    AEMeasurable (fun q : Ω × ℂ => observedDifference hε T.coe_nonneg κ ρ δ e r
+      ((BrownianDriver.path B (Real.sqrt κ) T q.1,V q.1),q.2)) (P.prod ν) := by
+  have hc := (measurable_observedValue hε T.coe_nonneg ρ δ e (1-κ/8) (8/κ-1)).sub
+    (measurable_observedValue hε T.coe_nonneg ρ δ r (1-κ/8) (8/κ-1))
+  have hp : AEMeasurable (fun q : Ω × ℂ => ((BrownianDriver.path B (Real.sqrt κ) T q.1,V q.1),q.2)) (P.prod ν) :=
+    (((BrownianDriver.path_aemeasurable hB (Real.sqrt κ) T).comp_fst).prodMk
+      ((hV.comp measurable_fst).aemeasurable)).prodMk measurable_snd.aemeasurable
+  exact hc.comp_aemeasurable hp
+
+lemma slab_sum_tendsto {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
+    {A : Set ℂ} {u : ℂ → ℝ} (C : FiniteTilt.CommonCutoff A u)
+    (κ : ℝ) (hκ : 0 < κ ∧ κ < 8) (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P)
+    (γ : Ω → ℝ → ℂ) (g f : Ω → ℝ → ℂ → ℂ)
+    (hmaps : ∀ᵐ ω ∂P, IsLoewnerTraceUsing (fun t => Real.sqrt κ*B t.toNNReal ω) (γ ω) (g ω) (f ω))
+    (z : ℂ) (hz : z ∈ A) (hhalf : z ∈ halfPlane) (hu : 0 < u z)
+    {ρ : ℝ} (hr : (2*z.im)*Real.exp (-u z) = ρ)
+    (V : Ω → C(I,ℂ)) (hV : ∀ᵐ ω ∂P, ∀ q : I, V ω q = γ ω ((C.horizon:ℝ)*q)) (e r : ℝ) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B C.horizon
+      (fun n i ω => slabCoeff C.height_pos κ B C.horizon (dyadicTime C.horizon n i) ρ C.angle e r (ω,z)) n ω) atTop
+      (𝓝 (observedDifference C.height_pos C.horizon.coe_nonneg κ ρ C.angle e r
+        ((BrownianDriver.path B (Real.sqrt κ) C.horizon ω,V ω),z))) := by
+  have he := (C.at z hz).physical_sum_tendsto hB κ hκ γ g f hmaps hhalf hu e
+  have hr' := (C.at z hz).physical_sum_tendsto hB κ hκ γ g f hmaps hhalf hu r
+  filter_upwards [he,hr',hmaps,hB.cont,hB.toIsPreBrownianReal.eval_zero_ae_eq_zero,hV] with ω heω hrω hm hc hb0 hVω
+  have hv (s : ℝ) := observedValue_eq C κ B γ g f ω hm hc hb0 z hz hhalf hu hr (V ω) hVω s
+  unfold observedDifference
+  rw [hv e,hv r]
+  have hl := heω.sub hrω
+  convert hl using 1
+  · funext n
+    simp only [dyadicDriverSum,slabCoeff,← coeff_common C κ B z hz hr,
+      sub_mul,Finset.sum_sub_distrib]
+    rfl
+  · congr 1
+    ring
+
+end SLEExactGauge.MeshSlab
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/SlitMaps.lean b/lean/OAI/Probability/SLELocalization/SlitMaps.lean
index d9fb64ca8670a0631b45c2c0eb16d89504bbf1d2..fe3efc94ff7f741ead063c3f6ea8933286df3ab1 100644
--- a/lean/OAI/Probability/SLELocalization/SlitMaps.lean
+++ b/lean/OAI/Probability/SLELocalization/SlitMaps.lean
@@ -1,655 +1,655 @@
-/-
-Portions of this file have been adapted from an external source.
-The source copyright header and license information are preserved below.
--/
-
-import Mathlib
-import OAI.Probability.SLELocalization.ConformalKernels
-
-namespace OAI
-
-
-/-! Conformal localization and Slit Maps estimates. -/
-
-section
-noncomputable section
-open Set Filter Metric Topology
-open scoped Topology
-namespace SLEExactGauge.SlitMaps
-
-def liftMap {U : Set ℂ} (f : C(U,ℂ)) (z : ℂ) : ℂ := by
-  classical
-  exact if hz : z ∈ U then f ⟨z,hz⟩ else 0
-
-@[simp] lemma liftMap_apply {U : Set ℂ} (f : C(U,ℂ)) (z : U) :
-    liftMap f z = f z := by simp [liftMap,z.property]
-
-lemma tendsto_liftMap {U : Set ℂ} (hU : IsOpen U)
-    {ι : Type*} {l : Filter ι} {F : ι → C(U,ℂ)} {f : C(U,ℂ)}
-    (hc : Tendsto F l (𝓝 f)) :
-    TendstoLocallyUniformlyOn (fun i => liftMap (F i)) (liftMap f) l U := by
-  rw [tendstoLocallyUniformlyOn_iff_forall_isCompact hU]
-  intro K hKU hK
-  let K' : Set U := Subtype.val ⁻¹' K
-  have he : Subtype.val '' K' = K := by
-    ext z
-    constructor
-    · rintro ⟨x,hx,rfl⟩
-      exact hx
-    · intro hz
-      exact ⟨⟨z,hKU hz⟩,hz,rfl⟩
-  have hK' : IsCompact K' := Topology.IsEmbedding.subtypeVal.isCompact_iff.2 (he.symm ▸ hK)
-  have hu := (ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn.1 hc) K' hK'
-  rw [Metric.tendstoUniformlyOn_iff] at hu ⊢
-  intro ε hε
-  filter_upwards [hu ε hε] with i hi z hz
-  simpa only [liftMap, dite_eq_left (hKU hz)] using hi ⟨z,hKU hz⟩ hz
-
-def boundedHolomorphic (U : Set ℂ) : Set C(U,ℂ) :=
-  {f | DifferentiableOn ℂ (liftMap f) U ∧ ∀ z : U, ‖f z‖ ≤ 1}
-
-lemma boundedHolomorphic_equicontinuous {U : Set ℂ} (hU : IsOpen U) :
-    Equicontinuous (fun (f : boundedHolomorphic U) (z : U) => f.val z) := by
-  intro z
-  rw [Metric.equicontinuousAt_iff]
-  intro ε hε
-  obtain ⟨r,hr,hrU⟩ := Metric.nhds_basis_closedBall.mem_iff.1 (hU.mem_nhds z.property)
-  let δ : ℝ := min (r/2) (ε*r/4)
-  have hδ : 0 < δ := lt_min (by positivity) (by positivity)
-  refine ⟨δ,hδ,fun w hw f => ?_⟩
-  have hz : (z:ℂ) ∈ closedBall (z:ℂ) (r/2) := mem_closedBall_self (by positivity)
-  have hw' : (w:ℂ) ∈ closedBall (z:ℂ) (r/2) := by
-    exact (show dist (w:ℂ) (z:ℂ) < δ from hw).le.trans (min_le_left _ _)
-  have hsub : closedBall (z:ℂ) (r/2) ⊆ U :=
-    (closedBall_subset_closedBall (by linarith : r/2 ≤ r)).trans hrU
-  have hd (x : ℂ) (hx : x ∈ closedBall (z:ℂ) (r/2)) :
-      DifferentiableAt ℂ (liftMap f.val) x := f.property.1.differentiableAt (hU.mem_nhds (hsub hx))
-  have hdB (x : ℂ) (hx : x ∈ closedBall (z:ℂ) (r/2)) :
-      ‖deriv (liftMap f.val) x‖ ≤ 2/r := by
-    have hs : closedBall x (r/2) ⊆ U := by
-      have hh : closedBall x (r/2) ⊆ closedBall (z:ℂ) r := by
-        apply closedBall_subset_closedBall'
-        have hn := mem_closedBall.1 hx
-        linarith
-      exact hh.trans hrU
-    have hdc : DiffContOnCl ℂ (liftMap f.val) (ball x (r/2)) :=
-      f.property.1.diffContOnCl_ball hs
-    have hb (y : ℂ) (hy : y ∈ sphere x (r/2)) : ‖liftMap f.val y‖ ≤ 1 := by
-      exact (liftMap_apply f.val ⟨y,hs (sphere_subset_closedBall hy)⟩) ▸ f.property.2 ⟨y,hs (sphere_subset_closedBall hy)⟩
-    have hh := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le (by positivity : 0 < r/2) hdc hb
-    convert hh using 1
-    field_simp
-  have hb := (convex_closedBall (z:ℂ) (r/2)).norm_image_sub_le_of_norm_deriv_le hd hdB hz hw'
-  have he : ‖(w:ℂ)-(z:ℂ)‖ < ε*r/4 := by
-    rw [← dist_eq_norm]
-    exact (show dist (w:ℂ) (z:ℂ) < δ from hw).trans_le (min_le_right _ _)
-  have hrp : 0 < 2/r := by positivity
-  have hprod : (2/r)*‖(w:ℂ)-(z:ℂ)‖ < ε/2 := by
-    calc
-      _ < (2/r)*(ε*r/4) := mul_lt_mul_of_pos_left he hrp
-      _ = ε/2 := by field_simp; ring
-  rw [liftMap_apply,liftMap_apply] at hb
-  rw [dist_comm,dist_eq_norm]
-  exact hb.trans_lt (hprod.trans_le (by linarith))
-
-lemma boundedHolomorphic_closed {U : Set ℂ} (hU : IsOpen U) :
-    IsClosed (boundedHolomorphic U) := by
-  apply isClosed_of_closure_subset
-  intro f hf
-  have : NeBot (𝓝[boundedHolomorphic U] f) := mem_closure_iff_nhdsWithin_neBot.1 hf
-  have hl : Tendsto (fun x : C(U,ℂ) => x) (𝓝[boundedHolomorphic U] f) (𝓝 f) := nhdsWithin_le_nhds
-  have hd : ∀ᶠ x in 𝓝[boundedHolomorphic U] f, DifferentiableOn ℂ (liftMap x) U :=
-    by
-    filter_upwards [self_mem_nhdsWithin] with x hx
-    exact hx.1
-  refine ⟨(tendsto_liftMap hU hl).differentiableOn hd hU,fun z => ?_⟩
-  have ht : Tendsto (fun x : C(U,ℂ) => ‖x z‖) (𝓝[boundedHolomorphic U] f) (𝓝 ‖f z‖) :=
-    ((ContinuousEvalConst.continuous_eval_const z).norm.continuousAt.tendsto).comp hl
-  apply le_of_tendsto ht
-  filter_upwards [self_mem_nhdsWithin] with x hx
-  exact hx.2 z
-
-lemma boundedHolomorphic_compact {U : Set ℂ} (hU : IsOpen U) :
-    IsCompact (boundedHolomorphic U) := by
-  let : LocallyCompactSpace U := hU.locallyCompactSpace
-  let : T2Space (UniformOnFun U ℂ {K : Set U | IsCompact K}) :=
-    UniformOnFun.t2Space_of_covering (by
-      apply eq_univ_iff_forall.mpr
-      intro x
-      exact mem_sUnion_of_mem (mem_singleton x) isCompact_singleton)
-  have hc : IsCompact (closure (boundedHolomorphic U)) := by
-    apply ArzelaAscoli.isCompact_closure_of_isClosedEmbedding
-      (𝔖:={K : Set U | IsCompact K}) (F:=fun f : C(U,ℂ) => (f : U → ℂ))
-      (fun K hK => hK) ContinuousMap.isUniformEmbedding_toUniformOnFunIsCompact.isClosedEmbedding
-    · intro K hK
-      exact (boundedHolomorphic_equicontinuous hU).equicontinuousOn K
-    · intro K hK z hz
-      refine ⟨closedBall (0:ℂ) 1,isCompact_closedBall _ _,fun f hf => ?_⟩
-      simpa using hf.2 z
-  simpa [(boundedHolomorphic_closed hU).closure_eq] using hc
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology
-namespace SLEExactGauge.SlitMaps
-
-lemma lower_norm_of_sphere {f : ℂ → ℂ} {z : ℂ} {r c : ℝ}
-    (hr : 0 < r) (hc : 0 < c)
-    (hd : DifferentiableOn ℂ f (closedBall z r))
-    (hn : ∀ w ∈ closedBall z r, f w ≠ 0)
-    (hb : ∀ w ∈ sphere z r, c ≤ ‖f w‖) : c ≤ ‖f z‖ := by
-  have hi : DifferentiableOn ℂ (fun w => (f w)⁻¹) (closedBall z r) := hd.inv hn
-  have hi' : DiffContOnCl ℂ (fun w => (f w)⁻¹) (ball z r) := by
-    exact hi.diffContOnCl_ball Subset.rfl
-  have hm : ‖(f z)⁻¹‖ ≤ c⁻¹ := by
-    apply Complex.norm_le_of_forall_mem_frontier_norm_le isBounded_ball hi'
-    · intro w hw
-      rw [frontier_ball z hr.ne'] at hw
-      rw [norm_inv]
-      exact inv_anti₀ hc (hb w hw)
-    · rw [closure_ball z hr.ne']
-      exact mem_closedBall_self hr.le
-  rw [norm_inv] at hm
-  exact (inv_le_inv₀ (norm_pos_iff.2 (hn z (mem_closedBall_self hr.le))) hc).1 hm
-
-theorem locallyUniform_nonvanishing {ι : Type*} {l : Filter ι} [l.NeBot]
-    {F : ι → ℂ → ℂ} {f : ℂ → ℂ} {U : Set ℂ}
-    (hU : IsOpen U) (hconn : IsPreconnected U)
-    (hlim : TendstoLocallyUniformlyOn F f l U)
-    (hF : ∀ᶠ i in l, DifferentiableOn ℂ (F i) U)
-    (hn : ∀ᶠ i in l, ∀ z ∈ U, F i z ≠ 0)
-    {a : ℂ} (ha : a ∈ U) (hfa : f a ≠ 0) : ∀ z ∈ U, f z ≠ 0 := by
-  have hf := hlim.differentiableOn hF hU
-  have haf : AnalyticOnNhd ℂ f U := hf.analyticOnNhd hU
-  intro z hz hzero
-  have hnot : ¬∀ᶠ w in 𝓝 z, f w = 0 := by
-    intro he
-    exact hfa (haf.eqOn_zero_of_preconnected_of_eventuallyEq_zero hconn hz he ha)
-  have hiso := ((haf z hz).eventually_eq_or_eventually_ne (analyticAt_const (v:=0))).resolve_left hnot
-  have hmem : ∀ᶠ w in 𝓝 z, w ∈ U := hU.mem_nhds hz
-  have hevent : ∀ᶠ w in 𝓝 z, w ∈ U ∧ (w ≠ z → f w ≠ 0) :=
-    hmem.and (eventually_nhdsWithin_iff.1 hiso)
-  obtain ⟨r,hr,hrU⟩ := Metric.nhds_basis_closedBall.mem_iff.1 hevent
-  have hrsub : closedBall z r ⊆ U := fun w hw => (hrU hw).1
-  have hfnz : ∀ w ∈ sphere z r, f w ≠ 0 := fun w hw =>
-    (hrU (sphere_subset_closedBall hw)).2 (ne_of_mem_sphere hw hr.ne')
-  have hsne : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
-  have hcfn : ContinuousOn (fun w => ‖f w‖) (sphere z r) :=
-    (haf.continuousOn.mono (sphere_subset_closedBall.trans hrsub)).norm
-  obtain ⟨w,hw,hmin⟩ := (isCompact_sphere z r).exists_isMinOn hsne hcfn
-  let c : ℝ := ‖f w‖
-  have hc : 0 < c := norm_pos_iff.2 (hfnz w hw)
-  have huni : TendstoUniformlyOn F f l (sphere z r) :=
-    (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hlim _
-      (sphere_subset_closedBall.trans hrsub) (isCompact_sphere z r)
-  have hclose := Metric.tendstoUniformlyOn_iff.1 huni (c/2) (by positivity)
-  have hlow : ∀ᶠ i in l, c/2 ≤ ‖F i z‖ := by
-    filter_upwards [hF,hn,hclose] with i hdi hni hci
-    apply lower_norm_of_sphere hr (by positivity) (hdi.mono hrsub)
-      (fun v hv => hni v (hrsub hv))
-    intro v hv
-    have hfv := hmin hv
-    have hdv := hci v hv
-    have htri : ‖f v‖ ≤ ‖f v-F i v‖+‖F i v‖ := norm_le_norm_sub_add _ _
-    change c ≤ ‖f v‖ at hfv
-    rw [dist_eq_norm] at hdv
-    linarith
-  have hh := ge_of_tendsto ((hlim.tendsto_at hz).norm) hlow
-  rw [hzero,norm_zero] at hh
-  linarith
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Topology
-open scoped Topology
-namespace SLEExactGauge.SlitMaps
-
-lemma locallyUniform_deriv_nonvanishing {ι : Type*} {l : Filter ι} [l.NeBot]
-    {F : ι → ℂ → ℂ} {f : ℂ → ℂ} {U : Set ℂ}
-    (hU : IsOpen U) (hc : IsPreconnected U)
-    (hlim : TendstoLocallyUniformlyOn F f l U)
-    (hF : ∀ᶠ i in l, DifferentiableOn ℂ (F i) U)
-    (hn : ∀ᶠ i in l, ∀ z ∈ U, deriv (F i) z ≠ 0)
-    {a : ℂ} (ha : a ∈ U) (hfa : deriv f a ≠ 0) :
-    ∀ z ∈ U, deriv f z ≠ 0 := by
-  apply locallyUniform_nonvanishing hU hc (hlim.deriv hF hU) _ hn ha hfa
-  filter_upwards [hF] with i hi
-  exact hi.deriv hU
-
-lemma locallyUniform_injective {ι : Type*} {l : Filter ι} [l.NeBot]
-    {F : ι → ℂ → ℂ} {f : ℂ → ℂ} {U : Set ℂ}
-    (hU : IsOpen U) (hlim : TendstoLocallyUniformlyOn F f l U)
-    (hF : ∀ᶠ i in l, DifferentiableOn ℂ (F i) U)
-    (hi : ∀ᶠ i in l, InjOn (F i) U)
-    (hnd : ∀ z ∈ U, deriv f z ≠ 0) : InjOn f U := by
-  intro x hx y hy heq
-  by_contra hxy
-  have hmem : U \ {x} ∈ 𝓝 y := (hU.inter isClosed_singleton.isOpen_compl).mem_nhds ⟨hy,by simpa [eq_comm] using hxy⟩
-  obtain ⟨r,hr,hrU⟩ := Metric.mem_nhds_iff.1 hmem
-  have hsub : ball y r ⊆ U := fun z hz => (hrU hz).1
-  have hnox : ∀ z ∈ ball y r, z ≠ x := fun z hz => (hrU hz).2
-  have hex : ∃ b ∈ ball y r, f b ≠ f x := by
-    by_contra! hn
-    have he : f =ᶠ[𝓝 y] fun _ => f x := by
-      filter_upwards [ball_mem_nhds y hr] with z hz
-      exact hn z hz
-    exact hnd y hy (he.deriv_eq.trans (deriv_const y (f x)))
-  obtain ⟨b,hb,hfb⟩ := hex
-  have hl : TendstoLocallyUniformlyOn (fun i z => F i z - F i x)
-      (fun z => f z - f x) l (ball y r) :=
-    (hlim.mono hsub).sub ((hlim.tendsto_at hx).tendstoUniformlyOn_const _).tendstoLocallyUniformlyOn
-  have hd : ∀ᶠ i in l, DifferentiableOn ℂ (fun z => F i z - F i x) (ball y r) := by
-    filter_upwards [hF] with i hdi
-    exact (hdi.mono hsub).sub_const _
-  have hne : ∀ᶠ i in l, ∀ z ∈ ball y r, F i z - F i x ≠ 0 := by
-    filter_upwards [hi] with i hii z hz
-    exact sub_ne_zero.2 (fun hh => hnox z hz (hii (hsub hz) hx hh))
-  have hh := locallyUniform_nonvanishing isOpen_ball (convex_ball y r).isPreconnected
-    hl hd hne hb (sub_ne_zero.2 hfb) y (mem_ball_self hr)
-  exact hh (sub_eq_zero.2 heq.symm)
-
-def markedEmbeddings (U : Set ℂ) (a : U) (d₀ : ℝ) : Set C(U,ℂ) :=
-  {f | f ∈ boundedHolomorphic U ∧ f a = 0 ∧ InjOn (liftMap f) U ∧
-    (∀ z ∈ U, deriv (liftMap f) z ≠ 0) ∧ d₀ ≤ ‖deriv (liftMap f) a‖}
-
-lemma derivative_continuousOn {U : Set ℂ} (hU : IsOpen U) (a : U) :
-    ContinuousOn (fun f : C(U,ℂ) => deriv (liftMap f) a) (boundedHolomorphic U) := by
-  intro f hf
-  change Tendsto _ (𝓝[boundedHolomorphic U] f) _
-  have hlim := tendsto_liftMap hU (show Tendsto (fun x : C(U,ℂ) => x)
-    (𝓝[boundedHolomorphic U] f) (𝓝 f) from nhdsWithin_le_nhds)
-  have hh : ∀ᶠ x in 𝓝[boundedHolomorphic U] f, DifferentiableOn ℂ (liftMap x) U := by
-    filter_upwards [self_mem_nhdsWithin] with x hx
-    exact hx.1
-  exact (hlim.deriv hh hU).tendsto_at a.property
-
-lemma markedEmbeddings_closed {U : Set ℂ} (hU : IsOpen U) (hc : IsPreconnected U)
-    (a : U) {d₀ : ℝ} (hd₀ : 0 < d₀) : IsClosed (markedEmbeddings U a d₀) := by
-  apply isClosed_of_closure_subset
-  intro f hf
-  have : NeBot (𝓝[markedEmbeddings U a d₀] f) := mem_closure_iff_nhdsWithin_neBot.1 hf
-  have hl : Tendsto (fun x : C(U,ℂ) => x) (𝓝[markedEmbeddings U a d₀] f) (𝓝 f) := nhdsWithin_le_nhds
-  have hm : ∀ᶠ x in 𝓝[markedEmbeddings U a d₀] f, x ∈ markedEmbeddings U a d₀ := self_mem_nhdsWithin
-  have he : ∀ᶠ x in 𝓝[markedEmbeddings U a d₀] f, x ∈ boundedHolomorphic U := hm.mono (fun x hx => hx.1)
-  have hfb : f ∈ boundedHolomorphic U := (boundedHolomorphic_closed hU).mem_of_tendsto hl he
-  have hF := he.mono (fun x hx => hx.1)
-  have hd := (tendsto_liftMap hU hl).deriv hF hU
-  have ht := hd.tendsto_at a.property
-  have hbound : d₀ ≤ ‖deriv (liftMap f) a‖ :=
-    ge_of_tendsto ht.norm (hm.mono (fun x hx => hx.2.2.2.2))
-  have hnz : deriv (liftMap f) a ≠ 0 := norm_pos_iff.1 (hd₀.trans_le hbound)
-  have hn := locallyUniform_deriv_nonvanishing hU hc (tendsto_liftMap hU hl) hF
-    (hm.mono (fun x hx => hx.2.2.2.1)) a.property hnz
-  refine ⟨hfb,?_,locallyUniform_injective hU (tendsto_liftMap hU hl) hF
-    (hm.mono (fun x hx => hx.2.2.1)) hn,hn,hbound⟩
-  apply tendsto_nhds_unique ((ContinuousEvalConst.continuous_eval_const a).continuousAt.tendsto.comp hl)
-  exact tendsto_const_nhds.congr' (hm.mono (fun x hx => hx.2.1.symm))
-
-lemma markedEmbeddings_compact {U : Set ℂ} (hU : IsOpen U) (hc : IsPreconnected U)
-    (a : U) {d₀ : ℝ} (hd₀ : 0 < d₀) : IsCompact (markedEmbeddings U a d₀) :=
-  (boundedHolomorphic_compact hU).of_isClosed_subset (markedEmbeddings_closed hU hc a hd₀)
-    (fun _ hx => hx.1)
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-section
-noncomputable section
-open Function Filter Metric Set Complex
-open scoped Pointwise Topology
-namespace SLEExactGauge.SlitMaps
-
-/-
-Copyright (c) 2026 Yury Kudryashov. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury Kudryashov
-
-The following conformal embedding declarations are adapted from Mathlib.Analysis.Complex.RiemannMapping.
-Source: https://github.com/leanprover-community/mathlib4/blob/d13f23b723b8a846827a245b89c10fc7d3f11612/Mathlib/Analysis/Complex/RiemannMapping.lean
-
-This code has been modified.
--/
-
-theorem exists_injective_not_dense_image_deriv_ne_zero {U : Set ℂ} (hUo : IsOpen U)
-    (hUc : IsSimplyConnected U) (hU : U ≠ univ) :
-    ∃ f : ℂ → ℂ, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0 := by
-  
-  wlog hU₀ : 0 ∉ U
-  · rw [ne_univ_iff_exists_notMem] at hU
-    rcases hU with ⟨a, ha⟩
-    specialize this (hUo.vadd (-a)) (by simpa) (by simp [hU])
-      (by simpa [mem_vadd_set_iff_neg_vadd_mem])
-    rcases this with ⟨f, hf_inj, hf_dense, hdf⟩
-    refine ⟨f ∘ (-a + ·), hf_inj.comp (add_right_injective (-a)), ?_, fun z hz ↦ ?_⟩
-    · simpa only [← image_vadd, Set.image_image] using! hf_dense
-    · simpa [Function.comp_def, deriv_comp_const_add] using hdf (-a + z) (mapsTo_image _ _ hz)
-  
-  
-  rcases exists_continuousOn_pow_eq hUc hUo continuousOn_id (by rwa [image_id]) two_ne_zero
-    with ⟨f, hfc, hf_inv⟩
-  replace hf_inv : LeftInverse (· ^ 2) f := hf_inv
-  
-  have hf₀ : ∀ z ∈ U, f z ≠ 0 := by
-    intro z hz hfz
-    simpa [hfz, (ne_of_mem_of_not_mem hz hU₀).symm] using hf_inv z
-  
-  
-  have hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z := by
-    intro z hz
-    apply HasStrictDerivAt.of_local_left_inverse
-    · exact hfc.continuousAt <| hUo.mem_nhds hz
-    · simpa using hasStrictDerivAt_pow 2 (f z)
-    · simpa using hf₀ z hz
-    · exact .of_forall hf_inv
-  
-  refine ⟨f, hf_inv.injective, ?_, fun z hz ↦ ?_⟩
-  · 
-    simp only [Dense, not_forall, mem_closure_iff_frequently, not_frequently]
-    rcases hUc.nonempty with ⟨x, hx⟩
-    
-    use -f x
-    
-    
-    have : f '' U ∈ 𝓝 (f x) := by
-      rw [← (hdf x hx).map_nhds_eq (by simpa using hf₀ x hx)]
-      exact Filter.image_mem_map <| hUo.mem_nhds hx
-    
-    rw [nhds_neg, eventually_neg]
-    
-    
-    
-    filter_upwards [this]
-    rintro _ ⟨a, ha, rfl⟩ ⟨b, hb, hab⟩
-    obtain rfl : a = b := by
-      rw [← hf_inv b, hab]
-      simp [hf_inv a]
-    refine hf₀ a ha ?_
-    linear_combination hab / 2
-  · simpa [(hdf z hz).hasDerivAt.deriv] using hf₀ z hz
-
-lemma exists_mapsTo_unitBall_injOn_deriv_ne_zero {U : Set ℂ} (hUo : IsOpen U)
-    (hUc : IsSimplyConnected U) (hU : U ≠ univ) :
-    ∃ f : ℂ → ℂ, MapsTo f U (ball 0 1) ∧ InjOn f U ∧ ∀ z ∈ U, deriv f z ≠ 0 := by
-  
-  
-  rcases exists_injective_not_dense_image_deriv_ne_zero hUo hUc hU with ⟨f, hf_inj, hfd, hdf⟩
-  
-  obtain ⟨x, ε, hε₀, hε⟩ : ∃ (x : ℂ) (ε : ℝ), 0 < ε ∧ ∀ a ∈ U, ε < dist (f a) x := by
-    simpa [Dense, mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall] using hfd
-  have hfx : ∀ z ∈ U, f z ≠ x := fun z hz ↦ by simpa using hε₀.trans (hε z hz)
-  
-  use fun z ↦ ε / (f z - x)
-  refine ⟨?mapsTo, ?injOn, ?deriv⟩
-  case mapsTo =>
-    intro z hz
-    rw [mem_ball_zero_iff, norm_div, norm_real, Real.norm_of_nonneg hε₀.le, div_lt_one₀]
-    · simpa [dist_eq_norm] using hε z hz
-    · simpa [sub_eq_zero] using hfx z hz
-  case injOn =>
-    intro z hz w hw heq
-    simpa [div_eq_mul_inv, hε₀.ne', hf_inj.eq_iff] using heq
-  case deriv =>
-    intro z hz
-    have hdz : DifferentiableAt ℂ f z := differentiableAt_of_deriv_ne_zero (hdf z hz)
-    rw [(hasDerivAt_const _ _).fun_div (hdz.hasDerivAt.sub_const _) _ |>.deriv] <;>
-      simp [*, ne_of_gt, sub_eq_zero]
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Topology
-open scoped Topology
-namespace SLEExactGauge.SlitMaps
-
-def IsDiskEmbedding (U : Set ℂ) (a : ℂ) (f : ℂ → ℂ) : Prop :=
-  MapsTo f U (ball 0 1) ∧ f a = 0 ∧ InjOn f U ∧ ∀ z ∈ U, deriv f z ≠ 0
-
-lemma IsDiskEmbedding.differentiableOn {U : Set ℂ} {a : ℂ} {f : ℂ → ℂ}
-    (hf : IsDiskEmbedding U a f) : DifferentiableOn ℂ f U :=
-  fun z hz => (differentiableAt_of_deriv_ne_zero (hf.2.2.2 z hz)).differentiableWithinAt
-
-lemma exists_diskEmbedding {U : Set ℂ} (hUo : IsOpen U) (hUc : IsSimplyConnected U)
-    (hU : U ≠ univ) (a : U) : ∃ f, IsDiskEmbedding U a f := by
-  obtain ⟨f,hmap,hinj,hnd⟩ := exists_mapsTo_unitBall_injOn_deriv_ne_zero hUo hUc hU
-  have hc : ‖f a‖ < 1 := mem_ball_zero_iff.1 (hmap a.property)
-  refine ⟨diskAuto (f a) ∘ f,(diskAuto_mapsTo hc).comp hmap,?_,
-    (diskAuto_injOn hc).comp hinj hmap,?_⟩
-  · simp
-  · intro z hz
-    have hdz := (diskAuto_hasDerivAt hc (mem_ball_zero_iff.1 (hmap hz))).comp z
-      (differentiableAt_of_deriv_ne_zero (hnd z hz)).hasDerivAt
-    rw [hdz.deriv]
-    exact mul_ne_zero (by
-      rw [← diskAuto_deriv hc (mem_ball_zero_iff.1 (hmap hz))]
-      exact diskAuto_deriv_ne hc (mem_ball_zero_iff.1 (hmap hz))) (hnd z hz)
-
-def continuousRestriction {U : Set ℂ} (f : ℂ → ℂ) (hf : ContinuousOn f U) : C(U,ℂ) :=
-  ⟨fun z => f z,(continuousOn_iff_continuous_domRestrict.1 hf)⟩
-
-@[simp] lemma continuousRestriction_apply {U : Set ℂ} {f : ℂ → ℂ} (hf : ContinuousOn f U) (z : U) :
-    continuousRestriction f hf z = f z := rfl
-
-lemma lift_restriction_eqOn {U : Set ℂ} {f : ℂ → ℂ} (hf : ContinuousOn f U) :
-    EqOn (liftMap (continuousRestriction f hf)) f U := by
-  intro z hz
-  exact liftMap_apply _ ⟨z,hz⟩
-
-lemma deriv_lift_restriction {U : Set ℂ} (hU : IsOpen U) {f : ℂ → ℂ}
-    (hf : ContinuousOn f U) {z : ℂ} (hz : z ∈ U) :
-    deriv (liftMap (continuousRestriction f hf)) z = deriv f z := by
-  apply Filter.EventuallyEq.deriv_eq
-  filter_upwards [hU.mem_nhds hz] with w hw
-  exact lift_restriction_eqOn hf hw
-
-lemma IsDiskEmbedding.restriction_mem {U : Set ℂ} (hU : IsOpen U) (a : U)
-    {f : ℂ → ℂ} (hf : IsDiskEmbedding U a f) {d₀ : ℝ} (hd : d₀ ≤ ‖deriv f a‖) :
-    continuousRestriction f hf.differentiableOn.continuousOn ∈ markedEmbeddings U a d₀ := by
-  let g := continuousRestriction f hf.differentiableOn.continuousOn
-  have he : EqOn (liftMap g) f U := lift_restriction_eqOn _
-  refine ⟨⟨?_,fun z => (mem_ball_zero_iff.1 (hf.1 z.property)).le⟩,hf.2.1,?_,?_,?_⟩
-  · exact hf.differentiableOn.congr he
-  · intro z hz w hw hh
-    exact hf.2.2.1 hz hw ((he hz).symm.trans (hh.trans (he hw)))
-  · intro z hz
-    rw [deriv_lift_restriction hU _ hz]
-    exact hf.2.2.2 z hz
-  · rwa [deriv_lift_restriction hU _ a.property]
-
-lemma markedEmbeddings_isDiskEmbedding {U : Set ℂ} (hU : IsOpen U) (hc : IsPreconnected U)
-    (a : U) {d₀ : ℝ} {f : C(U,ℂ)} (hf : f ∈ markedEmbeddings U a d₀) :
-    IsDiskEmbedding U a (liftMap f) := by
-  refine ⟨?_,by simpa using hf.2.1,hf.2.2.1,hf.2.2.2.1⟩
-  intro z hz
-  rw [mem_ball_zero_iff]
-  have hle : ‖liftMap f z‖ ≤ 1 := (liftMap_apply f ⟨z,hz⟩) ▸ hf.1.2 ⟨z,hz⟩
-  apply lt_of_le_of_ne hle
-  intro he
-  have hm : IsMaxOn (fun w => ‖liftMap f w‖) U z := by
-    intro w hw
-    change ‖liftMap f w‖ ≤ ‖liftMap f z‖
-    rw [he]
-    exact (liftMap_apply f ⟨w,hw⟩) ▸ hf.1.2 ⟨w,hw⟩
-  have hh := Complex.norm_eqOn_of_isPreconnected_of_isMaxOn hc hU hf.1.1 hz hm a.property
-  simp [liftMap_apply,hf.2.1,he] at hh
-
-theorem exists_extremal_diskEmbedding {U : Set ℂ} (hUo : IsOpen U)
-    (hUc : IsSimplyConnected U) (hU : U ≠ univ) (a : U) :
-    ∃ f, IsDiskEmbedding U a f ∧ ∀ g, IsDiskEmbedding U a g → ‖deriv g a‖ ≤ ‖deriv f a‖ := by
-  have hc : IsPreconnected U := hUc.isPathConnected.isConnected.isPreconnected
-  obtain ⟨f₀,hf₀⟩ := exists_diskEmbedding hUo hUc hU a
-  let d₀ := ‖deriv f₀ a‖
-  have hd₀ : 0 < d₀ := norm_pos_iff.2 (hf₀.2.2.2 a a.property)
-  have hnon : (markedEmbeddings U a d₀).Nonempty :=
-    ⟨continuousRestriction f₀ hf₀.differentiableOn.continuousOn,hf₀.restriction_mem hUo a le_rfl⟩
-  obtain ⟨f,hf,hmax⟩ := (markedEmbeddings_compact hUo hc a hd₀).exists_isMaxOn hnon
-    ((derivative_continuousOn hUo a).norm.mono (fun _ hx => hx.1))
-  refine ⟨liftMap f,markedEmbeddings_isDiskEmbedding hUo hc a hf,fun g hg => ?_⟩
-  by_contra! hh
-  have hgd : d₀ ≤ ‖deriv g a‖ := hf.2.2.2.2.trans hh.le
-  have hgmem := hg.restriction_mem hUo a hgd
-  have he := hmax hgmem
-  change ‖deriv (liftMap (continuousRestriction g hg.differentiableOn.continuousOn)) a‖ ≤ ‖deriv (liftMap f) a‖ at he
-  rw [deriv_lift_restriction hUo _ a.property] at he
-  exact (not_le_of_gt hh) he
-
-lemma diskAuto_comp_deriv {c : ℂ} {f : ℂ → ℂ} {z : ℂ}
-    (hc : ‖c‖ < 1) (hf : DifferentiableAt ℂ f z) (hz : ‖f z‖ < 1) :
-    deriv (diskAuto c ∘ f) z = deriv (diskAuto c) (f z)*deriv f z :=
-  ((diskAuto_hasDerivAt hc hz).differentiableAt.hasDerivAt.comp z hf.hasDerivAt).deriv
-
-lemma square_root_boost_real {x t : ℝ} (hx : 0 < x) (hx1 : x < 1) (ht : 0 < t) :
-    t < (1/(1-x^2))*(((1-(x^2)^2)*t)/(2*x)) := by
-  have hden : 0 < (1-x^2)*(2*x) := mul_pos (by nlinarith) (by positivity)
-  have he : (1/(1-x^2))*(((1-(x^2)^2)*t)/(2*x)) =
-      ((1-(x^2)^2)*t)/((1-x^2)*(2*x)) := by
-    field_simp
-  rw [he,lt_div_iff₀ hden]
-  have hpos : 0 < (1-x^2)*(1-x)^2*t := by
-    exact mul_pos (mul_pos (by nlinarith) (sq_pos_of_ne_zero (by linarith))) ht
-  nlinarith
-
-theorem exists_diskEmbedding_improved {U : Set ℂ} (hUo : IsOpen U)
-    (hUc : IsSimplyConnected U) (a : U) {f : ℂ → ℂ}
-    (hf : IsDiskEmbedding U a f) {b : ℂ} (hb : b ∈ ball 0 1) (homit : b ∉ f '' U) :
-    ∃ q, IsDiskEmbedding U a q ∧ ‖deriv f a‖ < ‖deriv q a‖ := by
-  have hbn : ‖b‖ < 1 := mem_ball_zero_iff.1 hb
-  let H := diskAuto b ∘ f
-  have hHmap : MapsTo H U (ball 0 1) := (diskAuto_mapsTo hbn).comp hf.1
-  have hHinj : InjOn H U := (diskAuto_injOn hbn).comp hf.2.2.1 hf.1
-  have hHder (z : ℂ) (hz : z ∈ U) :
-      deriv H z = deriv (diskAuto b) (f z)*deriv f z :=
-    diskAuto_comp_deriv hbn (differentiableAt_of_deriv_ne_zero (hf.2.2.2 z hz))
-      (mem_ball_zero_iff.1 (hf.1 hz))
-  have hHnd (z : ℂ) (hz : z ∈ U) : deriv H z ≠ 0 := by
-    rw [hHder z hz]
-    exact mul_ne_zero (diskAuto_deriv_ne hbn (mem_ball_zero_iff.1 (hf.1 hz))) (hf.2.2.2 z hz)
-  have hHd : DifferentiableOn ℂ H U := fun z hz =>
-    (differentiableAt_of_deriv_ne_zero (hHnd z hz)).differentiableWithinAt
-  have hHnz (z : ℂ) (hz : z ∈ U) : H z ≠ 0 := by
-    intro he
-    have hh : f z = b := (diskAuto_injOn hbn) (hf.1 hz) hb (he.trans (diskAuto_self b).symm)
-    exact homit ⟨z,hz,hh⟩
-  have hnot : 0 ∉ H '' U := by
-    rintro ⟨z,hz,he⟩
-    exact hHnz z hz he
-  obtain ⟨g,hgc,hgsq⟩ := Complex.exists_continuousOn_pow_eq hUc hUo hHd.continuousOn hnot two_ne_zero
-  have hgnz (z : ℂ) (hz : z ∈ U) : g z ≠ 0 := by
-    intro he
-    exact hHnz z hz (by simpa [he] using (hgsq z).symm)
-  have hgmap : MapsTo g U (ball 0 1) := by
-    intro z hz
-    rw [mem_ball_zero_iff]
-    have he := congrArg norm (hgsq z)
-    rw [norm_pow] at he
-    have hh := mem_ball_zero_iff.1 (hHmap hz)
-    nlinarith [norm_nonneg (g z)]
-  have hgder (z : ℂ) (hz : z ∈ U) : HasDerivAt g (deriv H z/(2*g z)) z := by
-    apply HasDerivAt.of_comp_left (hgc.continuousAt (hUo.mem_nhds hz))
-      (show HasDerivAt (fun w : ℂ => w^2) (2*g z) (g z) by simpa using hasDerivAt_pow 2 (g z))
-      (differentiableAt_of_deriv_ne_zero (hHnd z hz)).hasDerivAt
-      (mul_ne_zero two_ne_zero (hgnz z hz))
-    exact .of_forall hgsq
-  have hgnd (z : ℂ) (hz : z ∈ U) : deriv g z ≠ 0 := by
-    rw [(hgder z hz).deriv]
-    exact div_ne_zero (hHnd z hz) (mul_ne_zero two_ne_zero (hgnz z hz))
-  have hginj : InjOn g U := by
-    intro z hz w hw he
-    apply hHinj hz hw
-    rw [← hgsq z,← hgsq w,he]
-  let c := g a
-  have hcn : ‖c‖ < 1 := mem_ball_zero_iff.1 (hgmap a.property)
-  have hcp : 0 < ‖c‖ := norm_pos_iff.2 (hgnz a a.property)
-  let q := diskAuto c ∘ g
-  have hqder (z : ℂ) (hz : z ∈ U) :
-      deriv q z = deriv (diskAuto c) (g z)*deriv g z :=
-    diskAuto_comp_deriv hcn (hgder z hz).differentiableAt (mem_ball_zero_iff.1 (hgmap hz))
-  have hq : IsDiskEmbedding U a q := by
-    refine ⟨(diskAuto_mapsTo hcn).comp hgmap,?_,(diskAuto_injOn hcn).comp hginj hgmap,?_⟩
-    · exact diskAuto_self c
-    · intro z hz
-      rw [hqder z hz]
-      exact mul_ne_zero (diskAuto_deriv_ne hcn (mem_ball_zero_iff.1 (hgmap hz))) (hgnd z hz)
-  refine ⟨q,hq,?_⟩
-  have hcnorm : ‖c‖^2 = ‖b‖ := by
-    have he := congrArg norm (hgsq a)
-    simpa [H,Function.comp_def,hf.2.1,c] using he
-  have hHnorm : ‖deriv H a‖ = (1-‖b‖^2)*‖deriv f a‖ := by
-    rw [hHder a a.property,norm_mul,hf.2.1,diskAuto_norm_deriv_zero hbn]
-  have hgnorm : ‖deriv g a‖ = ((1-‖b‖^2)*‖deriv f a‖)/(2*‖c‖) := by
-    rw [(hgder a a.property).deriv,norm_div,norm_mul,hHnorm]
-    norm_num [c]
-  have hqnorm : ‖deriv q a‖ = (1/(1-‖c‖^2))*(((1-(‖c‖^2)^2)*‖deriv f a‖)/(2*‖c‖)) := by
-    rw [hqder a a.property,norm_mul]
-    change ‖deriv (diskAuto c) c‖*‖deriv g a‖ = _
-    rw [diskAuto_norm_deriv_self hcn,hgnorm,← hcnorm]
-  rw [hqnorm]
-  exact square_root_boost_real hcp hcn (norm_pos_iff.2 (hf.2.2.2 a a.property))
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric Function
-open scoped Topology
-namespace SLEExactGauge.SlitMaps
-
-theorem exists_biholomorphic_unitDisk {U : Set ℂ} (hUo : IsOpen U)
-    (hUc : IsSimplyConnected U) (hU : U ≠ univ) :
-    ∃ f g : ℂ → ℂ,
-      MapsTo f U (ball 0 1) ∧ MapsTo g (ball 0 1) U ∧
-      (∀ z ∈ U, g (f z) = z) ∧ (∀ z ∈ ball (0:ℂ) 1, f (g z) = z) ∧
-      DifferentiableOn ℂ f U ∧ DifferentiableOn ℂ g (ball (0:ℂ) 1) := by
-  obtain ⟨a,ha⟩ := hUc.nonempty
-  obtain ⟨f,hf,hmax⟩ := exists_extremal_diskEmbedding hUo hUc hU ⟨a,ha⟩
-  have hsurj : SurjOn f U (ball 0 1) := by
-    intro b hb
-    by_contra homit
-    obtain ⟨q,hq,hstrict⟩ := exists_diskEmbedding_improved hUo hUc ⟨a,ha⟩ hf hb homit
-    exact (not_le_of_gt hstrict) (hmax q hq)
-  let g := invFunOn f U
-  have hgmap : MapsTo g (ball 0 1) U := hsurj.mapsTo_invFunOn
-  have hleft : ∀ z ∈ U, g (f z) = z := hf.2.2.1.leftInvOn_invFunOn
-  have hright : ∀ z ∈ ball (0:ℂ) 1, f (g z) = z := hsurj.rightInvOn_invFunOn
-  refine ⟨f,g,hf.1,hgmap,hleft,hright,hf.differentiableOn,?_⟩
-  intro z hz
-  have hgU := hgmap hz
-  have hd := (hf.differentiableOn.analyticOnNhd hUo (g z) hgU).hasStrictDerivAt
-  have he : ∀ᶠ w in 𝓝 (g z), g (f w) = w := by
-    filter_upwards [hUo.mem_nhds hgU] with w hw
-    exact hleft w hw
-  have hgd := hd.to_local_left_inverse (hf.2.2.2 (g z) hgU) he
-  rw [hright z hz] at hgd
-  exact hgd.hasDerivAt.differentiableAt.differentiableWithinAt
-
-end SLEExactGauge.SlitMaps
-
-end
-
-end
-
-end OAI
+/-
+Portions of this file have been adapted from an external source.
+The source copyright header and license information are preserved below.
+-/
+
+import Mathlib
+import OAI.Probability.SLELocalization.ConformalKernels
+
+namespace OAI
+
+
+/-! Conformal localization and Slit Maps estimates. -/
+
+section
+noncomputable section
+open Set Filter Metric Topology
+open scoped Topology
+namespace SLEExactGauge.SlitMaps
+
+def liftMap {U : Set ℂ} (f : C(U,ℂ)) (z : ℂ) : ℂ := by
+  classical
+  exact if hz : z ∈ U then f ⟨z,hz⟩ else 0
+
+@[simp] lemma liftMap_apply {U : Set ℂ} (f : C(U,ℂ)) (z : U) :
+    liftMap f z = f z := by simp [liftMap,z.property]
+
+lemma tendsto_liftMap {U : Set ℂ} (hU : IsOpen U)
+    {ι : Type*} {l : Filter ι} {F : ι → C(U,ℂ)} {f : C(U,ℂ)}
+    (hc : Tendsto F l (𝓝 f)) :
+    TendstoLocallyUniformlyOn (fun i => liftMap (F i)) (liftMap f) l U := by
+  rw [tendstoLocallyUniformlyOn_iff_forall_isCompact hU]
+  intro K hKU hK
+  let K' : Set U := Subtype.val ⁻¹' K
+  have he : Subtype.val '' K' = K := by
+    ext z
+    constructor
+    · rintro ⟨x,hx,rfl⟩
+      exact hx
+    · intro hz
+      exact ⟨⟨z,hKU hz⟩,hz,rfl⟩
+  have hK' : IsCompact K' := Topology.IsEmbedding.subtypeVal.isCompact_iff.2 (he.symm ▸ hK)
+  have hu := (ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn.1 hc) K' hK'
+  rw [Metric.tendstoUniformlyOn_iff] at hu ⊢
+  intro ε hε
+  filter_upwards [hu ε hε] with i hi z hz
+  simpa only [liftMap, dite_eq_left (hKU hz)] using hi ⟨z,hKU hz⟩ hz
+
+def boundedHolomorphic (U : Set ℂ) : Set C(U,ℂ) :=
+  {f | DifferentiableOn ℂ (liftMap f) U ∧ ∀ z : U, ‖f z‖ ≤ 1}
+
+lemma boundedHolomorphic_equicontinuous {U : Set ℂ} (hU : IsOpen U) :
+    Equicontinuous (fun (f : boundedHolomorphic U) (z : U) => f.val z) := by
+  intro z
+  rw [Metric.equicontinuousAt_iff]
+  intro ε hε
+  obtain ⟨r,hr,hrU⟩ := Metric.nhds_basis_closedBall.mem_iff.1 (hU.mem_nhds z.property)
+  let δ : ℝ := min (r/2) (ε*r/4)
+  have hδ : 0 < δ := lt_min (by positivity) (by positivity)
+  refine ⟨δ,hδ,fun w hw f => ?_⟩
+  have hz : (z:ℂ) ∈ closedBall (z:ℂ) (r/2) := mem_closedBall_self (by positivity)
+  have hw' : (w:ℂ) ∈ closedBall (z:ℂ) (r/2) := by
+    exact (show dist (w:ℂ) (z:ℂ) < δ from hw).le.trans (min_le_left _ _)
+  have hsub : closedBall (z:ℂ) (r/2) ⊆ U :=
+    (closedBall_subset_closedBall (by linarith : r/2 ≤ r)).trans hrU
+  have hd (x : ℂ) (hx : x ∈ closedBall (z:ℂ) (r/2)) :
+      DifferentiableAt ℂ (liftMap f.val) x := f.property.1.differentiableAt (hU.mem_nhds (hsub hx))
+  have hdB (x : ℂ) (hx : x ∈ closedBall (z:ℂ) (r/2)) :
+      ‖deriv (liftMap f.val) x‖ ≤ 2/r := by
+    have hs : closedBall x (r/2) ⊆ U := by
+      have hh : closedBall x (r/2) ⊆ closedBall (z:ℂ) r := by
+        apply closedBall_subset_closedBall'
+        have hn := mem_closedBall.1 hx
+        linarith
+      exact hh.trans hrU
+    have hdc : DiffContOnCl ℂ (liftMap f.val) (ball x (r/2)) :=
+      f.property.1.diffContOnCl_ball hs
+    have hb (y : ℂ) (hy : y ∈ sphere x (r/2)) : ‖liftMap f.val y‖ ≤ 1 := by
+      exact (liftMap_apply f.val ⟨y,hs (sphere_subset_closedBall hy)⟩) ▸ f.property.2 ⟨y,hs (sphere_subset_closedBall hy)⟩
+    have hh := Complex.norm_deriv_le_of_forall_mem_sphere_norm_le (by positivity : 0 < r/2) hdc hb
+    convert hh using 1
+    field_simp
+  have hb := (convex_closedBall (z:ℂ) (r/2)).norm_image_sub_le_of_norm_deriv_le hd hdB hz hw'
+  have he : ‖(w:ℂ)-(z:ℂ)‖ < ε*r/4 := by
+    rw [← dist_eq_norm]
+    exact (show dist (w:ℂ) (z:ℂ) < δ from hw).trans_le (min_le_right _ _)
+  have hrp : 0 < 2/r := by positivity
+  have hprod : (2/r)*‖(w:ℂ)-(z:ℂ)‖ < ε/2 := by
+    calc
+      _ < (2/r)*(ε*r/4) := mul_lt_mul_of_pos_left he hrp
+      _ = ε/2 := by field_simp; ring
+  rw [liftMap_apply,liftMap_apply] at hb
+  rw [dist_comm,dist_eq_norm]
+  exact hb.trans_lt (hprod.trans_le (by linarith))
+
+lemma boundedHolomorphic_closed {U : Set ℂ} (hU : IsOpen U) :
+    IsClosed (boundedHolomorphic U) := by
+  apply isClosed_of_closure_subset
+  intro f hf
+  have : NeBot (𝓝[boundedHolomorphic U] f) := mem_closure_iff_nhdsWithin_neBot.1 hf
+  have hl : Tendsto (fun x : C(U,ℂ) => x) (𝓝[boundedHolomorphic U] f) (𝓝 f) := nhdsWithin_le_nhds
+  have hd : ∀ᶠ x in 𝓝[boundedHolomorphic U] f, DifferentiableOn ℂ (liftMap x) U :=
+    by
+    filter_upwards [self_mem_nhdsWithin] with x hx
+    exact hx.1
+  refine ⟨(tendsto_liftMap hU hl).differentiableOn hd hU,fun z => ?_⟩
+  have ht : Tendsto (fun x : C(U,ℂ) => ‖x z‖) (𝓝[boundedHolomorphic U] f) (𝓝 ‖f z‖) :=
+    ((ContinuousEvalConst.continuous_eval_const z).norm.continuousAt.tendsto).comp hl
+  apply le_of_tendsto ht
+  filter_upwards [self_mem_nhdsWithin] with x hx
+  exact hx.2 z
+
+lemma boundedHolomorphic_compact {U : Set ℂ} (hU : IsOpen U) :
+    IsCompact (boundedHolomorphic U) := by
+  let : LocallyCompactSpace U := hU.locallyCompactSpace
+  let : T2Space (UniformOnFun U ℂ {K : Set U | IsCompact K}) :=
+    UniformOnFun.t2Space_of_covering (by
+      apply eq_univ_iff_forall.mpr
+      intro x
+      exact mem_sUnion_of_mem (mem_singleton x) isCompact_singleton)
+  have hc : IsCompact (closure (boundedHolomorphic U)) := by
+    apply ArzelaAscoli.isCompact_closure_of_isClosedEmbedding
+      (𝔖:={K : Set U | IsCompact K}) (F:=fun f : C(U,ℂ) => (f : U → ℂ))
+      (fun K hK => hK) ContinuousMap.isUniformEmbedding_toUniformOnFunIsCompact.isClosedEmbedding
+    · intro K hK
+      exact (boundedHolomorphic_equicontinuous hU).equicontinuousOn K
+    · intro K hK z hz
+      refine ⟨closedBall (0:ℂ) 1,isCompact_closedBall _ _,fun f hf => ?_⟩
+      simpa using hf.2 z
+  simpa [(boundedHolomorphic_closed hU).closure_eq] using hc
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology
+namespace SLEExactGauge.SlitMaps
+
+lemma lower_norm_of_sphere {f : ℂ → ℂ} {z : ℂ} {r c : ℝ}
+    (hr : 0 < r) (hc : 0 < c)
+    (hd : DifferentiableOn ℂ f (closedBall z r))
+    (hn : ∀ w ∈ closedBall z r, f w ≠ 0)
+    (hb : ∀ w ∈ sphere z r, c ≤ ‖f w‖) : c ≤ ‖f z‖ := by
+  have hi : DifferentiableOn ℂ (fun w => (f w)⁻¹) (closedBall z r) := hd.inv hn
+  have hi' : DiffContOnCl ℂ (fun w => (f w)⁻¹) (ball z r) := by
+    exact hi.diffContOnCl_ball Subset.rfl
+  have hm : ‖(f z)⁻¹‖ ≤ c⁻¹ := by
+    apply Complex.norm_le_of_forall_mem_frontier_norm_le isBounded_ball hi'
+    · intro w hw
+      rw [frontier_ball z hr.ne'] at hw
+      rw [norm_inv]
+      exact inv_anti₀ hc (hb w hw)
+    · rw [closure_ball z hr.ne']
+      exact mem_closedBall_self hr.le
+  rw [norm_inv] at hm
+  exact (inv_le_inv₀ (norm_pos_iff.2 (hn z (mem_closedBall_self hr.le))) hc).1 hm
+
+theorem locallyUniform_nonvanishing {ι : Type*} {l : Filter ι} [l.NeBot]
+    {F : ι → ℂ → ℂ} {f : ℂ → ℂ} {U : Set ℂ}
+    (hU : IsOpen U) (hconn : IsPreconnected U)
+    (hlim : TendstoLocallyUniformlyOn F f l U)
+    (hF : ∀ᶠ i in l, DifferentiableOn ℂ (F i) U)
+    (hn : ∀ᶠ i in l, ∀ z ∈ U, F i z ≠ 0)
+    {a : ℂ} (ha : a ∈ U) (hfa : f a ≠ 0) : ∀ z ∈ U, f z ≠ 0 := by
+  have hf := hlim.differentiableOn hF hU
+  have haf : AnalyticOnNhd ℂ f U := hf.analyticOnNhd hU
+  intro z hz hzero
+  have hnot : ¬∀ᶠ w in 𝓝 z, f w = 0 := by
+    intro he
+    exact hfa (haf.eqOn_zero_of_preconnected_of_eventuallyEq_zero hconn hz he ha)
+  have hiso := ((haf z hz).eventually_eq_or_eventually_ne (analyticAt_const (v:=0))).resolve_left hnot
+  have hmem : ∀ᶠ w in 𝓝 z, w ∈ U := hU.mem_nhds hz
+  have hevent : ∀ᶠ w in 𝓝 z, w ∈ U ∧ (w ≠ z → f w ≠ 0) :=
+    hmem.and (eventually_nhdsWithin_iff.1 hiso)
+  obtain ⟨r,hr,hrU⟩ := Metric.nhds_basis_closedBall.mem_iff.1 hevent
+  have hrsub : closedBall z r ⊆ U := fun w hw => (hrU hw).1
+  have hfnz : ∀ w ∈ sphere z r, f w ≠ 0 := fun w hw =>
+    (hrU (sphere_subset_closedBall hw)).2 (ne_of_mem_sphere hw hr.ne')
+  have hsne : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
+  have hcfn : ContinuousOn (fun w => ‖f w‖) (sphere z r) :=
+    (haf.continuousOn.mono (sphere_subset_closedBall.trans hrsub)).norm
+  obtain ⟨w,hw,hmin⟩ := (isCompact_sphere z r).exists_isMinOn hsne hcfn
+  let c : ℝ := ‖f w‖
+  have hc : 0 < c := norm_pos_iff.2 (hfnz w hw)
+  have huni : TendstoUniformlyOn F f l (sphere z r) :=
+    (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).1 hlim _
+      (sphere_subset_closedBall.trans hrsub) (isCompact_sphere z r)
+  have hclose := Metric.tendstoUniformlyOn_iff.1 huni (c/2) (by positivity)
+  have hlow : ∀ᶠ i in l, c/2 ≤ ‖F i z‖ := by
+    filter_upwards [hF,hn,hclose] with i hdi hni hci
+    apply lower_norm_of_sphere hr (by positivity) (hdi.mono hrsub)
+      (fun v hv => hni v (hrsub hv))
+    intro v hv
+    have hfv := hmin hv
+    have hdv := hci v hv
+    have htri : ‖f v‖ ≤ ‖f v-F i v‖+‖F i v‖ := norm_le_norm_sub_add _ _
+    change c ≤ ‖f v‖ at hfv
+    rw [dist_eq_norm] at hdv
+    linarith
+  have hh := ge_of_tendsto ((hlim.tendsto_at hz).norm) hlow
+  rw [hzero,norm_zero] at hh
+  linarith
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Topology
+open scoped Topology
+namespace SLEExactGauge.SlitMaps
+
+lemma locallyUniform_deriv_nonvanishing {ι : Type*} {l : Filter ι} [l.NeBot]
+    {F : ι → ℂ → ℂ} {f : ℂ → ℂ} {U : Set ℂ}
+    (hU : IsOpen U) (hc : IsPreconnected U)
+    (hlim : TendstoLocallyUniformlyOn F f l U)
+    (hF : ∀ᶠ i in l, DifferentiableOn ℂ (F i) U)
+    (hn : ∀ᶠ i in l, ∀ z ∈ U, deriv (F i) z ≠ 0)
+    {a : ℂ} (ha : a ∈ U) (hfa : deriv f a ≠ 0) :
+    ∀ z ∈ U, deriv f z ≠ 0 := by
+  apply locallyUniform_nonvanishing hU hc (hlim.deriv hF hU) _ hn ha hfa
+  filter_upwards [hF] with i hi
+  exact hi.deriv hU
+
+lemma locallyUniform_injective {ι : Type*} {l : Filter ι} [l.NeBot]
+    {F : ι → ℂ → ℂ} {f : ℂ → ℂ} {U : Set ℂ}
+    (hU : IsOpen U) (hlim : TendstoLocallyUniformlyOn F f l U)
+    (hF : ∀ᶠ i in l, DifferentiableOn ℂ (F i) U)
+    (hi : ∀ᶠ i in l, InjOn (F i) U)
+    (hnd : ∀ z ∈ U, deriv f z ≠ 0) : InjOn f U := by
+  intro x hx y hy heq
+  by_contra hxy
+  have hmem : U \ {x} ∈ 𝓝 y := (hU.inter isClosed_singleton.isOpen_compl).mem_nhds ⟨hy,by simpa [eq_comm] using hxy⟩
+  obtain ⟨r,hr,hrU⟩ := Metric.mem_nhds_iff.1 hmem
+  have hsub : ball y r ⊆ U := fun z hz => (hrU hz).1
+  have hnox : ∀ z ∈ ball y r, z ≠ x := fun z hz => (hrU hz).2
+  have hex : ∃ b ∈ ball y r, f b ≠ f x := by
+    by_contra! hn
+    have he : f =ᶠ[𝓝 y] fun _ => f x := by
+      filter_upwards [ball_mem_nhds y hr] with z hz
+      exact hn z hz
+    exact hnd y hy (he.deriv_eq.trans (deriv_const y (f x)))
+  obtain ⟨b,hb,hfb⟩ := hex
+  have hl : TendstoLocallyUniformlyOn (fun i z => F i z - F i x)
+      (fun z => f z - f x) l (ball y r) :=
+    (hlim.mono hsub).sub ((hlim.tendsto_at hx).tendstoUniformlyOn_const _).tendstoLocallyUniformlyOn
+  have hd : ∀ᶠ i in l, DifferentiableOn ℂ (fun z => F i z - F i x) (ball y r) := by
+    filter_upwards [hF] with i hdi
+    exact (hdi.mono hsub).sub_const _
+  have hne : ∀ᶠ i in l, ∀ z ∈ ball y r, F i z - F i x ≠ 0 := by
+    filter_upwards [hi] with i hii z hz
+    exact sub_ne_zero.2 (fun hh => hnox z hz (hii (hsub hz) hx hh))
+  have hh := locallyUniform_nonvanishing isOpen_ball (convex_ball y r).isPreconnected
+    hl hd hne hb (sub_ne_zero.2 hfb) y (mem_ball_self hr)
+  exact hh (sub_eq_zero.2 heq.symm)
+
+def markedEmbeddings (U : Set ℂ) (a : U) (d₀ : ℝ) : Set C(U,ℂ) :=
+  {f | f ∈ boundedHolomorphic U ∧ f a = 0 ∧ InjOn (liftMap f) U ∧
+    (∀ z ∈ U, deriv (liftMap f) z ≠ 0) ∧ d₀ ≤ ‖deriv (liftMap f) a‖}
+
+lemma derivative_continuousOn {U : Set ℂ} (hU : IsOpen U) (a : U) :
+    ContinuousOn (fun f : C(U,ℂ) => deriv (liftMap f) a) (boundedHolomorphic U) := by
+  intro f hf
+  change Tendsto _ (𝓝[boundedHolomorphic U] f) _
+  have hlim := tendsto_liftMap hU (show Tendsto (fun x : C(U,ℂ) => x)
+    (𝓝[boundedHolomorphic U] f) (𝓝 f) from nhdsWithin_le_nhds)
+  have hh : ∀ᶠ x in 𝓝[boundedHolomorphic U] f, DifferentiableOn ℂ (liftMap x) U := by
+    filter_upwards [self_mem_nhdsWithin] with x hx
+    exact hx.1
+  exact (hlim.deriv hh hU).tendsto_at a.property
+
+lemma markedEmbeddings_closed {U : Set ℂ} (hU : IsOpen U) (hc : IsPreconnected U)
+    (a : U) {d₀ : ℝ} (hd₀ : 0 < d₀) : IsClosed (markedEmbeddings U a d₀) := by
+  apply isClosed_of_closure_subset
+  intro f hf
+  have : NeBot (𝓝[markedEmbeddings U a d₀] f) := mem_closure_iff_nhdsWithin_neBot.1 hf
+  have hl : Tendsto (fun x : C(U,ℂ) => x) (𝓝[markedEmbeddings U a d₀] f) (𝓝 f) := nhdsWithin_le_nhds
+  have hm : ∀ᶠ x in 𝓝[markedEmbeddings U a d₀] f, x ∈ markedEmbeddings U a d₀ := self_mem_nhdsWithin
+  have he : ∀ᶠ x in 𝓝[markedEmbeddings U a d₀] f, x ∈ boundedHolomorphic U := hm.mono (fun x hx => hx.1)
+  have hfb : f ∈ boundedHolomorphic U := (boundedHolomorphic_closed hU).mem_of_tendsto hl he
+  have hF := he.mono (fun x hx => hx.1)
+  have hd := (tendsto_liftMap hU hl).deriv hF hU
+  have ht := hd.tendsto_at a.property
+  have hbound : d₀ ≤ ‖deriv (liftMap f) a‖ :=
+    ge_of_tendsto ht.norm (hm.mono (fun x hx => hx.2.2.2.2))
+  have hnz : deriv (liftMap f) a ≠ 0 := norm_pos_iff.1 (hd₀.trans_le hbound)
+  have hn := locallyUniform_deriv_nonvanishing hU hc (tendsto_liftMap hU hl) hF
+    (hm.mono (fun x hx => hx.2.2.2.1)) a.property hnz
+  refine ⟨hfb,?_,locallyUniform_injective hU (tendsto_liftMap hU hl) hF
+    (hm.mono (fun x hx => hx.2.2.1)) hn,hn,hbound⟩
+  apply tendsto_nhds_unique ((ContinuousEvalConst.continuous_eval_const a).continuousAt.tendsto.comp hl)
+  exact tendsto_const_nhds.congr' (hm.mono (fun x hx => hx.2.1.symm))
+
+lemma markedEmbeddings_compact {U : Set ℂ} (hU : IsOpen U) (hc : IsPreconnected U)
+    (a : U) {d₀ : ℝ} (hd₀ : 0 < d₀) : IsCompact (markedEmbeddings U a d₀) :=
+  (boundedHolomorphic_compact hU).of_isClosed_subset (markedEmbeddings_closed hU hc a hd₀)
+    (fun _ hx => hx.1)
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+section
+noncomputable section
+open Function Filter Metric Set Complex
+open scoped Pointwise Topology
+namespace SLEExactGauge.SlitMaps
+
+/-
+Copyright (c) 2026 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+
+The following conformal embedding declarations are adapted from Mathlib.Analysis.Complex.RiemannMapping.
+Source: https://github.com/leanprover-community/mathlib4/blob/d13f23b723b8a846827a245b89c10fc7d3f11612/Mathlib/Analysis/Complex/RiemannMapping.lean
+
+This code has been modified.
+-/
+
+theorem exists_injective_not_dense_image_deriv_ne_zero {U : Set ℂ} (hUo : IsOpen U)
+    (hUc : IsSimplyConnected U) (hU : U ≠ univ) :
+    ∃ f : ℂ → ℂ, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0 := by
+  
+  wlog hU₀ : 0 ∉ U
+  · rw [ne_univ_iff_exists_notMem] at hU
+    rcases hU with ⟨a, ha⟩
+    specialize this (hUo.vadd (-a)) (by simpa) (by simp [hU])
+      (by simpa [mem_vadd_set_iff_neg_vadd_mem])
+    rcases this with ⟨f, hf_inj, hf_dense, hdf⟩
+    refine ⟨f ∘ (-a + ·), hf_inj.comp (add_right_injective (-a)), ?_, fun z hz ↦ ?_⟩
+    · simpa only [← image_vadd, Set.image_image] using! hf_dense
+    · simpa [Function.comp_def, deriv_comp_const_add] using hdf (-a + z) (mapsTo_image _ _ hz)
+  
+  
+  rcases exists_continuousOn_pow_eq hUc hUo continuousOn_id (by rwa [image_id]) two_ne_zero
+    with ⟨f, hfc, hf_inv⟩
+  replace hf_inv : LeftInverse (· ^ 2) f := hf_inv
+  
+  have hf₀ : ∀ z ∈ U, f z ≠ 0 := by
+    intro z hz hfz
+    simpa [hfz, (ne_of_mem_of_not_mem hz hU₀).symm] using hf_inv z
+  
+  
+  have hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z := by
+    intro z hz
+    apply HasStrictDerivAt.of_local_left_inverse
+    · exact hfc.continuousAt <| hUo.mem_nhds hz
+    · simpa using hasStrictDerivAt_pow 2 (f z)
+    · simpa using hf₀ z hz
+    · exact .of_forall hf_inv
+  
+  refine ⟨f, hf_inv.injective, ?_, fun z hz ↦ ?_⟩
+  · 
+    simp only [Dense, not_forall, mem_closure_iff_frequently, not_frequently]
+    rcases hUc.nonempty with ⟨x, hx⟩
+    
+    use -f x
+    
+    
+    have : f '' U ∈ 𝓝 (f x) := by
+      rw [← (hdf x hx).map_nhds_eq (by simpa using hf₀ x hx)]
+      exact Filter.image_mem_map <| hUo.mem_nhds hx
+    
+    rw [nhds_neg, eventually_neg]
+    
+    
+    
+    filter_upwards [this]
+    rintro _ ⟨a, ha, rfl⟩ ⟨b, hb, hab⟩
+    obtain rfl : a = b := by
+      rw [← hf_inv b, hab]
+      simp [hf_inv a]
+    refine hf₀ a ha ?_
+    linear_combination hab / 2
+  · simpa [(hdf z hz).hasDerivAt.deriv] using hf₀ z hz
+
+lemma exists_mapsTo_unitBall_injOn_deriv_ne_zero {U : Set ℂ} (hUo : IsOpen U)
+    (hUc : IsSimplyConnected U) (hU : U ≠ univ) :
+    ∃ f : ℂ → ℂ, MapsTo f U (ball 0 1) ∧ InjOn f U ∧ ∀ z ∈ U, deriv f z ≠ 0 := by
+  
+  
+  rcases exists_injective_not_dense_image_deriv_ne_zero hUo hUc hU with ⟨f, hf_inj, hfd, hdf⟩
+  
+  obtain ⟨x, ε, hε₀, hε⟩ : ∃ (x : ℂ) (ε : ℝ), 0 < ε ∧ ∀ a ∈ U, ε < dist (f a) x := by
+    simpa [Dense, mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall] using hfd
+  have hfx : ∀ z ∈ U, f z ≠ x := fun z hz ↦ by simpa using hε₀.trans (hε z hz)
+  
+  use fun z ↦ ε / (f z - x)
+  refine ⟨?mapsTo, ?injOn, ?deriv⟩
+  case mapsTo =>
+    intro z hz
+    rw [mem_ball_zero_iff, norm_div, norm_real, Real.norm_of_nonneg hε₀.le, div_lt_one₀]
+    · simpa [dist_eq_norm] using hε z hz
+    · simpa [sub_eq_zero] using hfx z hz
+  case injOn =>
+    intro z hz w hw heq
+    simpa [div_eq_mul_inv, hε₀.ne', hf_inj.eq_iff] using heq
+  case deriv =>
+    intro z hz
+    have hdz : DifferentiableAt ℂ f z := differentiableAt_of_deriv_ne_zero (hdf z hz)
+    rw [(hasDerivAt_const _ _).fun_div (hdz.hasDerivAt.sub_const _) _ |>.deriv] <;>
+      simp [*, ne_of_gt, sub_eq_zero]
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Topology
+open scoped Topology
+namespace SLEExactGauge.SlitMaps
+
+def IsDiskEmbedding (U : Set ℂ) (a : ℂ) (f : ℂ → ℂ) : Prop :=
+  MapsTo f U (ball 0 1) ∧ f a = 0 ∧ InjOn f U ∧ ∀ z ∈ U, deriv f z ≠ 0
+
+lemma IsDiskEmbedding.differentiableOn {U : Set ℂ} {a : ℂ} {f : ℂ → ℂ}
+    (hf : IsDiskEmbedding U a f) : DifferentiableOn ℂ f U :=
+  fun z hz => (differentiableAt_of_deriv_ne_zero (hf.2.2.2 z hz)).differentiableWithinAt
+
+lemma exists_diskEmbedding {U : Set ℂ} (hUo : IsOpen U) (hUc : IsSimplyConnected U)
+    (hU : U ≠ univ) (a : U) : ∃ f, IsDiskEmbedding U a f := by
+  obtain ⟨f,hmap,hinj,hnd⟩ := exists_mapsTo_unitBall_injOn_deriv_ne_zero hUo hUc hU
+  have hc : ‖f a‖ < 1 := mem_ball_zero_iff.1 (hmap a.property)
+  refine ⟨diskAuto (f a) ∘ f,(diskAuto_mapsTo hc).comp hmap,?_,
+    (diskAuto_injOn hc).comp hinj hmap,?_⟩
+  · simp
+  · intro z hz
+    have hdz := (diskAuto_hasDerivAt hc (mem_ball_zero_iff.1 (hmap hz))).comp z
+      (differentiableAt_of_deriv_ne_zero (hnd z hz)).hasDerivAt
+    rw [hdz.deriv]
+    exact mul_ne_zero (by
+      rw [← diskAuto_deriv hc (mem_ball_zero_iff.1 (hmap hz))]
+      exact diskAuto_deriv_ne hc (mem_ball_zero_iff.1 (hmap hz))) (hnd z hz)
+
+def continuousRestriction {U : Set ℂ} (f : ℂ → ℂ) (hf : ContinuousOn f U) : C(U,ℂ) :=
+  ⟨fun z => f z,(continuousOn_iff_continuous_domRestrict.1 hf)⟩
+
+@[simp] lemma continuousRestriction_apply {U : Set ℂ} {f : ℂ → ℂ} (hf : ContinuousOn f U) (z : U) :
+    continuousRestriction f hf z = f z := rfl
+
+lemma lift_restriction_eqOn {U : Set ℂ} {f : ℂ → ℂ} (hf : ContinuousOn f U) :
+    EqOn (liftMap (continuousRestriction f hf)) f U := by
+  intro z hz
+  exact liftMap_apply _ ⟨z,hz⟩
+
+lemma deriv_lift_restriction {U : Set ℂ} (hU : IsOpen U) {f : ℂ → ℂ}
+    (hf : ContinuousOn f U) {z : ℂ} (hz : z ∈ U) :
+    deriv (liftMap (continuousRestriction f hf)) z = deriv f z := by
+  apply Filter.EventuallyEq.deriv_eq
+  filter_upwards [hU.mem_nhds hz] with w hw
+  exact lift_restriction_eqOn hf hw
+
+lemma IsDiskEmbedding.restriction_mem {U : Set ℂ} (hU : IsOpen U) (a : U)
+    {f : ℂ → ℂ} (hf : IsDiskEmbedding U a f) {d₀ : ℝ} (hd : d₀ ≤ ‖deriv f a‖) :
+    continuousRestriction f hf.differentiableOn.continuousOn ∈ markedEmbeddings U a d₀ := by
+  let g := continuousRestriction f hf.differentiableOn.continuousOn
+  have he : EqOn (liftMap g) f U := lift_restriction_eqOn _
+  refine ⟨⟨?_,fun z => (mem_ball_zero_iff.1 (hf.1 z.property)).le⟩,hf.2.1,?_,?_,?_⟩
+  · exact hf.differentiableOn.congr he
+  · intro z hz w hw hh
+    exact hf.2.2.1 hz hw ((he hz).symm.trans (hh.trans (he hw)))
+  · intro z hz
+    rw [deriv_lift_restriction hU _ hz]
+    exact hf.2.2.2 z hz
+  · rwa [deriv_lift_restriction hU _ a.property]
+
+lemma markedEmbeddings_isDiskEmbedding {U : Set ℂ} (hU : IsOpen U) (hc : IsPreconnected U)
+    (a : U) {d₀ : ℝ} {f : C(U,ℂ)} (hf : f ∈ markedEmbeddings U a d₀) :
+    IsDiskEmbedding U a (liftMap f) := by
+  refine ⟨?_,by simpa using hf.2.1,hf.2.2.1,hf.2.2.2.1⟩
+  intro z hz
+  rw [mem_ball_zero_iff]
+  have hle : ‖liftMap f z‖ ≤ 1 := (liftMap_apply f ⟨z,hz⟩) ▸ hf.1.2 ⟨z,hz⟩
+  apply lt_of_le_of_ne hle
+  intro he
+  have hm : IsMaxOn (fun w => ‖liftMap f w‖) U z := by
+    intro w hw
+    change ‖liftMap f w‖ ≤ ‖liftMap f z‖
+    rw [he]
+    exact (liftMap_apply f ⟨w,hw⟩) ▸ hf.1.2 ⟨w,hw⟩
+  have hh := Complex.norm_eqOn_of_isPreconnected_of_isMaxOn hc hU hf.1.1 hz hm a.property
+  simp [liftMap_apply,hf.2.1,he] at hh
+
+theorem exists_extremal_diskEmbedding {U : Set ℂ} (hUo : IsOpen U)
+    (hUc : IsSimplyConnected U) (hU : U ≠ univ) (a : U) :
+    ∃ f, IsDiskEmbedding U a f ∧ ∀ g, IsDiskEmbedding U a g → ‖deriv g a‖ ≤ ‖deriv f a‖ := by
+  have hc : IsPreconnected U := hUc.isPathConnected.isConnected.isPreconnected
+  obtain ⟨f₀,hf₀⟩ := exists_diskEmbedding hUo hUc hU a
+  let d₀ := ‖deriv f₀ a‖
+  have hd₀ : 0 < d₀ := norm_pos_iff.2 (hf₀.2.2.2 a a.property)
+  have hnon : (markedEmbeddings U a d₀).Nonempty :=
+    ⟨continuousRestriction f₀ hf₀.differentiableOn.continuousOn,hf₀.restriction_mem hUo a le_rfl⟩
+  obtain ⟨f,hf,hmax⟩ := (markedEmbeddings_compact hUo hc a hd₀).exists_isMaxOn hnon
+    ((derivative_continuousOn hUo a).norm.mono (fun _ hx => hx.1))
+  refine ⟨liftMap f,markedEmbeddings_isDiskEmbedding hUo hc a hf,fun g hg => ?_⟩
+  by_contra! hh
+  have hgd : d₀ ≤ ‖deriv g a‖ := hf.2.2.2.2.trans hh.le
+  have hgmem := hg.restriction_mem hUo a hgd
+  have he := hmax hgmem
+  change ‖deriv (liftMap (continuousRestriction g hg.differentiableOn.continuousOn)) a‖ ≤ ‖deriv (liftMap f) a‖ at he
+  rw [deriv_lift_restriction hUo _ a.property] at he
+  exact (not_le_of_gt hh) he
+
+lemma diskAuto_comp_deriv {c : ℂ} {f : ℂ → ℂ} {z : ℂ}
+    (hc : ‖c‖ < 1) (hf : DifferentiableAt ℂ f z) (hz : ‖f z‖ < 1) :
+    deriv (diskAuto c ∘ f) z = deriv (diskAuto c) (f z)*deriv f z :=
+  ((diskAuto_hasDerivAt hc hz).differentiableAt.hasDerivAt.comp z hf.hasDerivAt).deriv
+
+lemma square_root_boost_real {x t : ℝ} (hx : 0 < x) (hx1 : x < 1) (ht : 0 < t) :
+    t < (1/(1-x^2))*(((1-(x^2)^2)*t)/(2*x)) := by
+  have hden : 0 < (1-x^2)*(2*x) := mul_pos (by nlinarith) (by positivity)
+  have he : (1/(1-x^2))*(((1-(x^2)^2)*t)/(2*x)) =
+      ((1-(x^2)^2)*t)/((1-x^2)*(2*x)) := by
+    field_simp
+  rw [he,lt_div_iff₀ hden]
+  have hpos : 0 < (1-x^2)*(1-x)^2*t := by
+    exact mul_pos (mul_pos (by nlinarith) (sq_pos_of_ne_zero (by linarith))) ht
+  nlinarith
+
+theorem exists_diskEmbedding_improved {U : Set ℂ} (hUo : IsOpen U)
+    (hUc : IsSimplyConnected U) (a : U) {f : ℂ → ℂ}
+    (hf : IsDiskEmbedding U a f) {b : ℂ} (hb : b ∈ ball 0 1) (homit : b ∉ f '' U) :
+    ∃ q, IsDiskEmbedding U a q ∧ ‖deriv f a‖ < ‖deriv q a‖ := by
+  have hbn : ‖b‖ < 1 := mem_ball_zero_iff.1 hb
+  let H := diskAuto b ∘ f
+  have hHmap : MapsTo H U (ball 0 1) := (diskAuto_mapsTo hbn).comp hf.1
+  have hHinj : InjOn H U := (diskAuto_injOn hbn).comp hf.2.2.1 hf.1
+  have hHder (z : ℂ) (hz : z ∈ U) :
+      deriv H z = deriv (diskAuto b) (f z)*deriv f z :=
+    diskAuto_comp_deriv hbn (differentiableAt_of_deriv_ne_zero (hf.2.2.2 z hz))
+      (mem_ball_zero_iff.1 (hf.1 hz))
+  have hHnd (z : ℂ) (hz : z ∈ U) : deriv H z ≠ 0 := by
+    rw [hHder z hz]
+    exact mul_ne_zero (diskAuto_deriv_ne hbn (mem_ball_zero_iff.1 (hf.1 hz))) (hf.2.2.2 z hz)
+  have hHd : DifferentiableOn ℂ H U := fun z hz =>
+    (differentiableAt_of_deriv_ne_zero (hHnd z hz)).differentiableWithinAt
+  have hHnz (z : ℂ) (hz : z ∈ U) : H z ≠ 0 := by
+    intro he
+    have hh : f z = b := (diskAuto_injOn hbn) (hf.1 hz) hb (he.trans (diskAuto_self b).symm)
+    exact homit ⟨z,hz,hh⟩
+  have hnot : 0 ∉ H '' U := by
+    rintro ⟨z,hz,he⟩
+    exact hHnz z hz he
+  obtain ⟨g,hgc,hgsq⟩ := Complex.exists_continuousOn_pow_eq hUc hUo hHd.continuousOn hnot two_ne_zero
+  have hgnz (z : ℂ) (hz : z ∈ U) : g z ≠ 0 := by
+    intro he
+    exact hHnz z hz (by simpa [he] using (hgsq z).symm)
+  have hgmap : MapsTo g U (ball 0 1) := by
+    intro z hz
+    rw [mem_ball_zero_iff]
+    have he := congrArg norm (hgsq z)
+    rw [norm_pow] at he
+    have hh := mem_ball_zero_iff.1 (hHmap hz)
+    nlinarith [norm_nonneg (g z)]
+  have hgder (z : ℂ) (hz : z ∈ U) : HasDerivAt g (deriv H z/(2*g z)) z := by
+    apply HasDerivAt.of_comp_left (hgc.continuousAt (hUo.mem_nhds hz))
+      (show HasDerivAt (fun w : ℂ => w^2) (2*g z) (g z) by simpa using hasDerivAt_pow 2 (g z))
+      (differentiableAt_of_deriv_ne_zero (hHnd z hz)).hasDerivAt
+      (mul_ne_zero two_ne_zero (hgnz z hz))
+    exact .of_forall hgsq
+  have hgnd (z : ℂ) (hz : z ∈ U) : deriv g z ≠ 0 := by
+    rw [(hgder z hz).deriv]
+    exact div_ne_zero (hHnd z hz) (mul_ne_zero two_ne_zero (hgnz z hz))
+  have hginj : InjOn g U := by
+    intro z hz w hw he
+    apply hHinj hz hw
+    rw [← hgsq z,← hgsq w,he]
+  let c := g a
+  have hcn : ‖c‖ < 1 := mem_ball_zero_iff.1 (hgmap a.property)
+  have hcp : 0 < ‖c‖ := norm_pos_iff.2 (hgnz a a.property)
+  let q := diskAuto c ∘ g
+  have hqder (z : ℂ) (hz : z ∈ U) :
+      deriv q z = deriv (diskAuto c) (g z)*deriv g z :=
+    diskAuto_comp_deriv hcn (hgder z hz).differentiableAt (mem_ball_zero_iff.1 (hgmap hz))
+  have hq : IsDiskEmbedding U a q := by
+    refine ⟨(diskAuto_mapsTo hcn).comp hgmap,?_,(diskAuto_injOn hcn).comp hginj hgmap,?_⟩
+    · exact diskAuto_self c
+    · intro z hz
+      rw [hqder z hz]
+      exact mul_ne_zero (diskAuto_deriv_ne hcn (mem_ball_zero_iff.1 (hgmap hz))) (hgnd z hz)
+  refine ⟨q,hq,?_⟩
+  have hcnorm : ‖c‖^2 = ‖b‖ := by
+    have he := congrArg norm (hgsq a)
+    simpa [H,Function.comp_def,hf.2.1,c] using he
+  have hHnorm : ‖deriv H a‖ = (1-‖b‖^2)*‖deriv f a‖ := by
+    rw [hHder a a.property,norm_mul,hf.2.1,diskAuto_norm_deriv_zero hbn]
+  have hgnorm : ‖deriv g a‖ = ((1-‖b‖^2)*‖deriv f a‖)/(2*‖c‖) := by
+    rw [(hgder a a.property).deriv,norm_div,norm_mul,hHnorm]
+    norm_num [c]
+  have hqnorm : ‖deriv q a‖ = (1/(1-‖c‖^2))*(((1-(‖c‖^2)^2)*‖deriv f a‖)/(2*‖c‖)) := by
+    rw [hqder a a.property,norm_mul]
+    change ‖deriv (diskAuto c) c‖*‖deriv g a‖ = _
+    rw [diskAuto_norm_deriv_self hcn,hgnorm,← hcnorm]
+  rw [hqnorm]
+  exact square_root_boost_real hcp hcn (norm_pos_iff.2 (hf.2.2.2 a a.property))
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric Function
+open scoped Topology
+namespace SLEExactGauge.SlitMaps
+
+theorem exists_biholomorphic_unitDisk {U : Set ℂ} (hUo : IsOpen U)
+    (hUc : IsSimplyConnected U) (hU : U ≠ univ) :
+    ∃ f g : ℂ → ℂ,
+      MapsTo f U (ball 0 1) ∧ MapsTo g (ball 0 1) U ∧
+      (∀ z ∈ U, g (f z) = z) ∧ (∀ z ∈ ball (0:ℂ) 1, f (g z) = z) ∧
+      DifferentiableOn ℂ f U ∧ DifferentiableOn ℂ g (ball (0:ℂ) 1) := by
+  obtain ⟨a,ha⟩ := hUc.nonempty
+  obtain ⟨f,hf,hmax⟩ := exists_extremal_diskEmbedding hUo hUc hU ⟨a,ha⟩
+  have hsurj : SurjOn f U (ball 0 1) := by
+    intro b hb
+    by_contra homit
+    obtain ⟨q,hq,hstrict⟩ := exists_diskEmbedding_improved hUo hUc ⟨a,ha⟩ hf hb homit
+    exact (not_le_of_gt hstrict) (hmax q hq)
+  let g := invFunOn f U
+  have hgmap : MapsTo g (ball 0 1) U := hsurj.mapsTo_invFunOn
+  have hleft : ∀ z ∈ U, g (f z) = z := hf.2.2.1.leftInvOn_invFunOn
+  have hright : ∀ z ∈ ball (0:ℂ) 1, f (g z) = z := hsurj.rightInvOn_invFunOn
+  refine ⟨f,g,hf.1,hgmap,hleft,hright,hf.differentiableOn,?_⟩
+  intro z hz
+  have hgU := hgmap hz
+  have hd := (hf.differentiableOn.analyticOnNhd hUo (g z) hgU).hasStrictDerivAt
+  have he : ∀ᶠ w in 𝓝 (g z), g (f w) = w := by
+    filter_upwards [hUo.mem_nhds hgU] with w hw
+    exact hleft w hw
+  have hgd := hd.to_local_left_inverse (hf.2.2.2 (g z) hgU) he
+  rw [hright z hz] at hgd
+  exact hgd.hasDerivAt.differentiableAt.differentiableWithinAt
+
+end SLEExactGauge.SlitMaps
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/SobolevMeans.lean b/lean/OAI/Probability/SLELocalization/SobolevMeans.lean
index 2971fc3182bb4bc31d32cf529f76b3cad9535bc7..8896257fa388334ac2ec40c46b45bebddfa7f948 100644
--- a/lean/OAI/Probability/SLELocalization/SobolevMeans.lean
+++ b/lean/OAI/Probability/SLELocalization/SobolevMeans.lean
@@ -1,940 +1,940 @@
-import Mathlib
-import OAI.Probability.SLELocalization.DyadicKernels
-
-namespace OAI
-
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-section Measure
-variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [IsFiniteMeasure μ]
-
-theorem dyadic_weighted_energy {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
-    {g : α → ℝ} (hg : MemLp g 2 μ) {V : ℝ} (n : ℕ)
-    (harea : ∀ j < n, ((2:ℝ)^j)^2*μ.real (Q j) ≤ V) :
-    (∫ x, dyadicKernel Q n x*g x ∂μ)^2 ≤
-      3*(n:ℝ)*V*(∫ x, (g x)^2 ∂μ) := by
-  exact (integral_mul_square_le (dyadicKernel_memLp hQ n) hg).trans
-    (mul_le_mul_of_nonneg_right (dyadicKernel_square_energy hQ n harea)
-      (integral_nonneg (fun x => sq_nonneg _)))
-
-omit [IsFiniteMeasure μ] in
-lemma dyadicKernel_integral_mul {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
-    {g : α → ℝ} (hg : Integrable g μ) (n : ℕ) :
-    (∫ x, dyadicKernel Q n x*g x ∂μ) =
-      ∑ j ∈ Finset.range n, (2:ℝ)^j*(∫ x in Q j, g x ∂μ) := by
-  have he : (fun x => dyadicKernel Q n x*g x) =
-      fun x => ∑ j ∈ Finset.range n, (Q j).indicator (fun x => (2:ℝ)^j*g x) x := by
-    funext x
-    rw [dyadicKernel,Finset.sum_mul]
-    apply Finset.sum_congr rfl
-    intro j hj
-    by_cases hx : x ∈ Q j <;> simp [hx]
-  rw [he,integral_finsetSum _ (fun j hj => (hg.const_mul _).indicator (hQ j))]
-  apply Finset.sum_congr rfl
-  intro j hj
-  rw [integral_indicator (hQ j),integral_const_mul]
-
-omit [IsFiniteMeasure μ] in
-lemma dyadic_L1_telescoping {Q : ℕ → Set α} {g : α → ℝ} {v : ℕ → ℝ}
-    {C : ℝ} (hstep : ∀ j, |v (j+1)-v j| ≤ C*(2:ℝ)^j*(∫ x in Q j, g x ∂μ))
-    (n : ℕ) : |v n-v 0| ≤ C*∑ j ∈ Finset.range n, (2:ℝ)^j*(∫ x in Q j, g x ∂μ) := by
-  induction n with
-  | zero => simp
-  | succ n ih =>
-    calc
-      _ ≤ |v (n+1)-v n|+|v n-v 0| := abs_sub_le _ _ _
-      _ ≤ C*(2:ℝ)^n*(∫ x in Q n, g x ∂μ)+
-          C*∑ j ∈ Finset.range n, (2:ℝ)^j*(∫ x in Q j, g x ∂μ) := add_le_add (hstep n) ih
-      _ = _ := by rw [Finset.sum_range_succ]; ring
-
-theorem critical_means_from_L1 {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
-    {g : α → ℝ} (hg : MemLp g 2 μ) (hg0 : ∀ᵐ x ∂μ, 0 ≤ g x)
-    {v : ℕ → ℝ} {C V : ℝ} (hC : 0 ≤ C)
-    (hstep : ∀ j, |v (j+1)-v j| ≤ C*(2:ℝ)^j*(∫ x in Q j, g x ∂μ))
-    (n : ℕ) (harea : ∀ j < n, ((2:ℝ)^j)^2*μ.real (Q j) ≤ V) :
-    (v n-v 0)^2 ≤ 3*C^2*(n:ℝ)*V*(∫ x, (g x)^2 ∂μ) := by
-  have hi := hg.integrable (by norm_num : (1:ℝ≥0∞) ≤ 2)
-  have hb := dyadic_L1_telescoping hstep n
-  rw [← dyadicKernel_integral_mul hQ hi n] at hb
-  have hn : 0 ≤ ∫ x, dyadicKernel Q n x*g x ∂μ := by
-    apply integral_nonneg_of_ae
-    filter_upwards [hg0] with x hx
-    apply mul_nonneg _ hx
-    apply Finset.sum_nonneg
-    intro j hj
-    exact indicator_nonneg (fun x hx => by positivity) x
-  have hs := (sq_le_sq₀ (abs_nonneg _) (mul_nonneg hC hn)).2 hb
-  rw [sq_abs,mul_pow] at hs
-  calc
-    _ ≤ C^2*(∫ x, dyadicKernel Q n x*g x ∂μ)^2 := hs
-    _ ≤ C^2*(3*(n:ℝ)*V*(∫ x, (g x)^2 ∂μ)) :=
-      mul_le_mul_of_nonneg_left (dyadic_weighted_energy hQ hg n harea) (sq_nonneg _)
-    _ = _ := by ring
-
-end Measure
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma rectangle_measure_restrict {a b c d a' b' c' d' : ℝ}
-    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d) :
-    ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))).restrict
-      ((Ioc a' b').prod (Ioc c' d')) =
-      (volume.restrict (Ioc a' b')).prod (volume.restrict (Ioc c' d')) := by
-  rw [Measure.prod_restrict,Measure.prod_restrict]
-  have hs : (Ioc a' b').prod (Ioc c' d') ⊆ (Ioc a b).prod (Ioc c d) := by
-    rintro p ⟨hx,hy⟩
-    exact ⟨⟨haa.trans_lt hx.1,hx.2.trans hbb⟩,⟨hcc.trans_lt hy.1,hy.2.trans hdd⟩⟩
-  exact Measure.restrict_restrict_of_subset hs
-
-lemma rectangle_measure_real {a b c d a' b' c' d' : ℝ}
-    (hab : a' ≤ b') (hcd : c' ≤ d')
-    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d) :
-    ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))).real
-      ((Ioc a' b').prod (Ioc c' d')) = (b'-a')*(d'-c') := by
-  rw [Measure.real]
-  have he := congrArg (fun μ : Measure (ℝ × ℝ) => μ univ)
-    (rectangle_measure_restrict haa hbb hcc hdd)
-  simp only [Measure.restrict_apply_univ] at he
-  rw [he,← univ_prod_univ,Measure.prod_prod]
-  simp only [Measure.restrict_apply_univ,Real.volume_Ioc,ENNReal.toReal_mul]
-  rw [ENNReal.toReal_ofReal (sub_nonneg.2 hab),ENNReal.toReal_ofReal (sub_nonneg.2 hcd)]
-
-theorem strip_nested_means {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    (a b c d : ℕ → ℝ) (hab : ∀ j, a j < b j) (hcd : ∀ j, c j < d j)
-    (hlo : 0 ≤ c 0) (hhi : d 0 ≤ Real.pi)
-    (haa : ∀ j, a j ≤ a (j+1)) (hbb : ∀ j, b (j+1) ≤ b j)
-    (hcc : ∀ j, c j ≤ c (j+1)) (hdd : ∀ j, d (j+1) ≤ d j)
-    (hw : ∀ j, b j-a j = (b 0-a 0)/(2:ℝ)^j)
-    (hh : ∀ j, d j-c j = (d 0-c 0)/(2:ℝ)^j) (n : ℕ) :
-    ((∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb (a n) (b n)).prod (intervalProb (c n) (d n)))-
-      (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb (a 0) (b 0)).prod (intervalProb (c 0) (d 0))))^2 ≤
-      3*(2/(d 0-c 0)+1/(b 0-a 0))^2*(n:ℝ)*(b 0-a 0)*(d 0-c 0)*
-        (∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar)
-          ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
-          ∂(volume.restrict (Ioc (a 0) (b 0))).prod (volume.restrict (Ioc (c 0) (d 0)))) := by
-  let μ := (volume.restrict (Ioc (a 0) (b 0))).prod (volume.restrict (Ioc (c 0) (d 0)))
-  let Q := fun j => (Ioc (a j) (b j)).prod (Ioc (c j) (d j))
-  let g := fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖
-  let v := fun j => ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-    ∂(intervalProb (a j) (b j)).prod (intervalProb (c j) (d j))
-  let C := 2/(d 0-c 0)+1/(b 0-a 0)
-  have ha0 (j : ℕ) : a 0 ≤ a j := (monotone_nat_of_le_succ haa) (Nat.zero_le j)
-  have hb0 (j : ℕ) : b j ≤ b 0 := (antitone_nat_of_succ_le hbb) (Nat.zero_le j)
-  have hc0 (j : ℕ) : c 0 ≤ c j := (monotone_nat_of_le_succ hcc) (Nat.zero_le j)
-  have hd0 (j : ℕ) : d j ≤ d 0 := (antitone_nat_of_succ_le hdd) (Nat.zero_le j)
-  have hC : 0 ≤ C := by
-    have hw0 := sub_pos.2 (hab 0)
-    have hh0 := sub_pos.2 (hcd 0)
-    dsimp [C]; positivity
-  have hstep (j : ℕ) : |v (j+1)-v j| ≤ C*(2:ℝ)^j*(∫ p in Q j, g p ∂μ) := by
-    have hlen : d j-c j ≤ 2*(d (j+1)-c (j+1)) := by rw [hh j,hh (j+1),pow_succ]; exact le_of_eq (by ring)
-    have ht := strip_rectangle_poincare hF hinj hext hcont hH hr q
-      (hab j) (hcd j) (hab (j+1)) (hcd (j+1)) (hlo.trans (hc0 j)) ((hd0 j).trans hhi)
-      (haa j) (hbb j) (hcc j) (hdd j) (show (0:ℝ) ≤ 2 by norm_num) hlen
-    change |v (j+1)-v j| ≤ _ at ht
-    have he : 2/(d j-c j)+1/(b j-a j) = C*(2:ℝ)^j := by
-      rw [hw j,hh j]
-      dsimp [C]
-      field_simp
-    rw [he] at ht
-    simpa only [Q,μ,rectangle_measure_restrict (ha0 j) (hb0 j) (hc0 j) (hd0 j)] using ht
-  have harea (j : ℕ) (hj : j < n) : ((2:ℝ)^j)^2*μ.real (Q j) ≤ (b 0-a 0)*(d 0-c 0) := by
-    rw [rectangle_measure_real (hab j).le (hcd j).le (ha0 j) (hb0 j) (hc0 j) (hd0 j)]
-    rw [hw j,hh j]
-    exact le_of_eq (by field_simp)
-  have ht := critical_means_from_L1 (μ := μ) (Q := Q)
-    (fun j => measurableSet_Ioc.prod measurableSet_Ioc)
-    (strip_gradient_rect_memLp hF hinj hext hH hr q (a 0) (b 0) hlo hhi)
-    (ae_of_all _ (fun p => norm_nonneg _)) hC hstep n harea
-  dsimp only [v,C,g] at ht
-  convert ht using 1
-  ring
-
-lemma rectangle_measure_total {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) :
-    ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))).real univ =
-      (b-a)*(d-c) := by
-  rw [Measure.real,← univ_prod_univ,Measure.prod_prod]
-  simp only [Measure.restrict_apply_univ,Real.volume_Ioc,ENNReal.toReal_mul]
-  rw [ENNReal.toReal_ofReal (sub_nonneg.2 hab),ENNReal.toReal_ofReal (sub_nonneg.2 hcd)]
-
-theorem strip_parent_energy {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {a b c d a' b' c' d' : ℝ}
-    (hab : a < b) (hcd : c < d) (hab' : a' < b') (hcd' : c' < d')
-    (hlo : 0 ≤ c) (hhi : d ≤ Real.pi)
-    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d)
-    (hlen : d-c ≤ 2*(d'-c')) :
-    ((∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb a' b').prod (intervalProb c' d'))-
-      (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb a b).prod (intervalProb c d)))^2 ≤
-      (2/(d-c)+1/(b-a))^2*(b-a)*(d-c)*
-        (∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
-          ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-  have hp := strip_rectangle_poincare hF hinj hext hcont hH hr q hab hcd hab' hcd'
-    hlo hhi haa hbb hcc hdd (show (0:ℝ) ≤ 2 by norm_num) hlen
-  have hcoef : 0 ≤ 2/(d-c)+1/(b-a) := by
-    have hb := sub_pos.2 hab
-    have hd := sub_pos.2 hcd
-    positivity
-  have hE := strip_gradient_rect_memLp hF hinj hext hH hr q a b hlo hhi
-  have hi : 0 ≤ ∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖
-      ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d)) := integral_nonneg (fun _ => norm_nonneg _)
-  have hs := (sq_le_sq₀ (abs_nonneg _) (mul_nonneg hcoef hi)).2 hp
-  rw [sq_abs,mul_pow] at hs
-  have hc := integral_mul_square_le (memLp_const (1:ℝ)) hE
-  simp only [one_mul,one_pow,integral_const,smul_eq_mul,mul_one,
-    rectangle_measure_total hab.le hcd.le] at hc
-  exact hs.trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hc (sq_nonneg (2/(d-c)+1/(b-a))))
-
-lemma rectangleProb_integral {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d)
-    (f : ℝ × ℝ → ℝ) :
-    (∫ p, f p ∂(intervalProb a b).prod (intervalProb c d)) =
-      ((b-a)*(d-c))⁻¹*(∫ p, f p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
-  rw [intervalProb,intervalProb,Measure.prod_smul_left,Measure.prod_smul_right,
-    integral_smul_measure,integral_smul_measure,ENNReal.toReal_inv,ENNReal.toReal_inv,
-    ENNReal.toReal_ofReal (sub_nonneg.2 hab),ENNReal.toReal_ofReal (sub_nonneg.2 hcd)]
-  simp only [smul_eq_mul,mul_inv_rev]
-  ring
-
-theorem strip_negative_mean {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
-    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-    {a b : ℝ} (hab : a < b) (hb : b ≤ 0) :
-    (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-      ∂(intervalProb a b).prod (intervalProb 0 Real.pi))^2 ≤
-      (Real.pi/(b-a))*(∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar)
-        ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
-          ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc (0:ℝ) Real.pi))) := by
-  let u := fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-  let G := fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
-  let μ := intervalProb a b
-  let ν := intervalProb 0 Real.pi
-  let : IsProbabilityMeasure μ := intervalProb_isProbability hab
-  let : IsProbabilityMeasure ν := intervalProb_isProbability Real.pi_pos
-  have hu := strip_potential_mean_integrable hcont hH.le hr q hab Real.pi_pos le_rfl le_rfl
-  have hu2 : MemLp u 2 (μ.prod ν) := MemLp.of_bound hu.aestronglyMeasurable H (ae_of_all _ (fun p => by
-    have hh := sphereTruncation_bounds hH.le hr q (Fbar (Complex.exp ((p.1:ℂ)+(p.2:ℂ)*Complex.I)))
-    change ‖stripPotential H q r Fbar _‖ ≤ H
-    rw [Real.norm_eq_abs,stripPotential,abs_of_nonneg hh.1]
-    exact hh.2))
-  have hG := (strip_gradient_rect_memLp hF hinj hext hH hr q a b le_rfl le_rfl).integrable_sq
-  have hR := hG.integral_prod_left
-  have hRμ : Integrable (fun v => ∫ φ in Ioc (0:ℝ) Real.pi, G (v,φ)) μ :=
-    integrable_intervalProb hab hR
-  have hbound : ∀ᵐ p ∂μ.prod ν, u p^2 ≤
-      Real.pi*(∫ φ in Ioc (0:ℝ) Real.pi, G (p.1,φ)) := by
-    have hn := strip_negative_section_energy hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess
-    have hv : ∀ᵐ v ∂μ, ∀ φ ∈ Ioc (0:ℝ) Real.pi, u (v,φ)^2 ≤
-        Real.pi*(∫ ψ in Ioc (0:ℝ) Real.pi, G (v,ψ)) := by
-      apply intervalProb_ae.mpr
-      filter_upwards [ae_restrict_of_ae hn,
-        ae_restrict_mem measurableSet_Ioc,show ∀ᵐ v ∂volume.restrict (Ioc a b), v ≠ (0:ℝ) from
-          ae_restrict_of_ae (volume.ae_ne 0)] with v hv hvab hv0
-      intro φ hφ
-      have hvneg : v < 0 := lt_of_le_of_ne (hvab.2.trans hb) hv0
-      have hh := hv hvneg φ ⟨hφ.1.le,hφ.2⟩
-      rw [intervalIntegral.integral_of_le Real.pi_pos.le] at hh
-      exact hh
-    have hφ : ∀ᵐ φ ∂ν, φ ∈ Ioc (0:ℝ) Real.pi := intervalProb_ae.mpr (ae_restrict_mem measurableSet_Ioc)
-    filter_upwards [Measure.quasiMeasurePreserving_fst.ae hv,
-      Measure.quasiMeasurePreserving_snd.ae hφ] with p hp hpφ
-    exact hp p.2 hpφ
-  have hmono := integral_mono_ae hu2.integrable_sq ((hRμ.comp_fst ν).const_mul Real.pi) hbound
-  have hj := integral_mul_square_le (memLp_const (1:ℝ)) hu2
-  simp only [one_mul,one_pow,integral_const,Measure.real,measure_univ,ENNReal.toReal_one,
-    smul_eq_mul,mul_one,one_mul] at hj
-  refine hj.trans (hmono.trans_eq ?_)
-  rw [integral_const_mul]
-  rw [integral_fun_fst (fun v => ∫ φ in Ioc (0:ℝ) Real.pi, G (v,φ))]
-  simp only [Measure.real,measure_univ,ENNReal.toReal_one,one_smul]
-  rw [integral_intervalProb hab.le,intervalIntegral.integral_of_le hab.le]
-  rw [← integral_prod _ hG]
-  simp only [div_eq_mul_inv,mul_assoc]
-
-structure DRect where
-  a : ℝ
-  b : ℝ
-  c : ℝ
-  d : ℝ
-  hab : a < b
-  hcd : c < d
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-namespace DRect
-
-def closed (Q : DRect) : Set (ℝ × ℝ) := (Icc Q.a Q.b).prod (Icc Q.c Q.d)
-
-def cells (Q : DRect) : Set (ℝ × ℝ) := (Ioc Q.a Q.b).prod (Ioc Q.c Q.d)
-
-def measure (Q : DRect) : Measure (ℝ × ℝ) :=
-  (volume.restrict (Ioc Q.a Q.b)).prod (volume.restrict (Ioc Q.c Q.d))
-
-def halfLo (l r : ℝ) (s : Bool) : ℝ := if s then (l+r)/2 else l
-
-def halfHi (l r : ℝ) (s : Bool) : ℝ := if s then r else (l+r)/2
-
-lemma half_bounds {l r : ℝ} (h : l < r) (s : Bool) :
-    l ≤ halfLo l r s ∧ halfLo l r s < halfHi l r s ∧ halfHi l r s ≤ r := by
-  cases s
-  · change l ≤ l ∧ l < (l+r)/2 ∧ (l+r)/2 ≤ r
-    refine ⟨le_rfl,?_,?_⟩ <;> linarith
-  · change l ≤ (l+r)/2 ∧ (l+r)/2 < r ∧ r ≤ r
-    refine ⟨?_,?_,le_rfl⟩ <;> linarith
-
-lemma half_length (l r : ℝ) (s : Bool) : halfHi l r s-halfLo l r s = (r-l)/2 := by
-  cases s <;> simp [halfLo,halfHi] <;> ring
-
-def child (Q : DRect) (s : Bool × Bool) : DRect where
-  a := halfLo Q.a Q.b s.1
-  b := halfHi Q.a Q.b s.1
-  c := halfLo Q.c Q.d s.2
-  d := halfHi Q.c Q.d s.2
-  hab := (half_bounds Q.hab s.1).2.1
-  hcd := (half_bounds Q.hcd s.2).2.1
-
-lemma child_bounds (Q : DRect) (s : Bool × Bool) :
-    Q.a ≤ (Q.child s).a ∧ (Q.child s).b ≤ Q.b ∧
-    Q.c ≤ (Q.child s).c ∧ (Q.child s).d ≤ Q.d :=
-  ⟨(half_bounds Q.hab s.1).1,(half_bounds Q.hab s.1).2.2,
-    (half_bounds Q.hcd s.2).1,(half_bounds Q.hcd s.2).2.2⟩
-
-lemma child_width (Q : DRect) (s : Bool × Bool) : (Q.child s).b-(Q.child s).a = (Q.b-Q.a)/2 := half_length _ _ _
-
-lemma child_height (Q : DRect) (s : Bool × Bool) : (Q.child s).d-(Q.child s).c = (Q.d-Q.c)/2 := half_length _ _ _
-
-lemma child_closed_subset (Q : DRect) (s : Bool × Bool) : (Q.child s).closed ⊆ Q.closed := by
-  rintro p ⟨hx,hy⟩
-  obtain ⟨ha,hb,hc,hd⟩ := Q.child_bounds s
-  exact ⟨⟨ha.trans hx.1,hx.2.trans hb⟩,⟨hc.trans hy.1,hy.2.trans hd⟩⟩
-
-def chooseSide (l r x : ℝ) : Bool := decide ((l+r)/2 < x)
-
-lemma mem_chosen_half {l r x : ℝ} (hx : x ∈ Icc l r) :
-    x ∈ Icc (halfLo l r (chooseSide l r x)) (halfHi l r (chooseSide l r x)) := by
-  by_cases h : (l+r)/2 < x
-  · simp [chooseSide,h,halfLo,halfHi,h.le,hx.2]
-  · have hm : x ≤ (l+r)/2 := le_of_not_gt h
-    simp [chooseSide,h,halfLo,halfHi,hx.1,hm]
-
-def chooseChild (Q : DRect) (x : ℝ × ℝ) : Bool × Bool :=
-  (chooseSide Q.a Q.b x.1,chooseSide Q.c Q.d x.2)
-
-lemma mem_chosen_child {Q : DRect} {x : ℝ × ℝ} (hx : x ∈ Q.closed) :
-    x ∈ (Q.child (Q.chooseChild x)).closed :=
-  ⟨mem_chosen_half hx.1,mem_chosen_half hx.2⟩
-
-end DRect
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-def DIndex : ℕ → Type
-  | 0 => Unit
-  | n+1 => DIndex n × (Bool × Bool)
-
-instance dIndex_fintype (n : ℕ) : Fintype (DIndex n) := by
-  induction n with
-  | zero => exact inferInstanceAs (Fintype Unit)
-  | succ n ih => exact inferInstanceAs (Fintype (DIndex n × (Bool × Bool)))
-
-def dyadicRect (Q : DRect) : (n : ℕ) → DIndex n → DRect
-  | 0, _ => Q
-  | n+1, i => (dyadicRect Q n i.1).child i.2
-
-def dParent (n : ℕ) (i : DIndex (n+1)) : DIndex n := i.1
-
-def dFromBits (β : ℕ → Bool × Bool) : (n : ℕ) → DIndex n
-  | 0 => ()
-  | n+1 => (dFromBits β n,β n)
-
-lemma dFromBits_congr {β β' : ℕ → Bool × Bool} (n : ℕ) (h : ∀ k < n, β k = β' k) :
-    dFromBits β n = dFromBits β' n := by
-  induction n with
-  | zero => rfl
-  | succ n ih =>
-    change (dFromBits β n,β n) = (dFromBits β' n,β' n)
-    rw [ih (fun k hk => h k (Nat.lt_succ_of_lt hk)),h n (Nat.lt_succ_self n)]
-
-def dExtendBits : (n : ℕ) → DIndex n → (ℕ → Bool × Bool)
-  | 0, _ => fun _ => (false,false)
-  | n+1, i => Function.update (dExtendBits n i.1) n i.2
-
-lemma dFromBits_extend (n : ℕ) (i : DIndex n) : dFromBits (dExtendBits n i) n = i := by
-  induction n with
-  | zero => cases i; rfl
-  | succ n ih =>
-    change (dFromBits (Function.update (dExtendBits n i.1) n i.2) n,
-      Function.update (dExtendBits n i.1) n i.2 n) = i
-    rw [Function.update_self]
-    have he := dFromBits_congr n (β := Function.update (dExtendBits n i.1) n i.2)
-      (β' := dExtendBits n i.1) (fun k hk => Function.update_of_ne (Nat.ne_of_lt hk) _ _)
-    rw [he,ih i.1]
-    rfl
-
-lemma dyadicRect_width (Q : DRect) (n : ℕ) (i : DIndex n) :
-    (dyadicRect Q n i).b-(dyadicRect Q n i).a = (Q.b-Q.a)/(2:ℝ)^n := by
-  induction n with
-  | zero => simp [dyadicRect]
-  | succ n ih => rw [dyadicRect,DRect.child_width,ih,pow_succ]; ring
-
-lemma dyadicRect_height (Q : DRect) (n : ℕ) (i : DIndex n) :
-    (dyadicRect Q n i).d-(dyadicRect Q n i).c = (Q.d-Q.c)/(2:ℝ)^n := by
-  induction n with
-  | zero => simp [dyadicRect]
-  | succ n ih => rw [dyadicRect,DRect.child_height,ih,pow_succ]; ring
-
-lemma dyadicRect_bounds (Q : DRect) (n : ℕ) (i : DIndex n) :
-    Q.a ≤ (dyadicRect Q n i).a ∧ (dyadicRect Q n i).b ≤ Q.b ∧
-    Q.c ≤ (dyadicRect Q n i).c ∧ (dyadicRect Q n i).d ≤ Q.d := by
-  induction n with
-  | zero => exact ⟨le_rfl,le_rfl,le_rfl,le_rfl⟩
-  | succ n ih =>
-    obtain ⟨ha,hb,hc,hd⟩ := ih i.1
-    obtain ⟨ha',hb',hc',hd'⟩ := (dyadicRect Q n i.1).child_bounds i.2
-    exact ⟨ha.trans ha',hb'.trans hb,hc.trans hc',hd'.trans hd⟩
-
-def dPointBranch (Q : DRect) (x : ℝ × ℝ) : (n : ℕ) → DIndex n
-  | 0 => ()
-  | n+1 => (dPointBranch Q x n,(dyadicRect Q n (dPointBranch Q x n)).chooseChild x)
-
-lemma dPointBranch_parent (Q : DRect) (x : ℝ × ℝ) (n : ℕ) :
-    dParent n (dPointBranch Q x (n+1)) = dPointBranch Q x n := rfl
-
-lemma dPointBranch_mem {Q : DRect} {x : ℝ × ℝ} (hx : x ∈ Q.closed) (n : ℕ) :
-    x ∈ (dyadicRect Q n (dPointBranch Q x n)).closed := by
-  induction n with
-  | zero => exact hx
-  | succ n ih => exact DRect.mem_chosen_child ih
-
-lemma probability_means_tendsto {α : Type*} [PseudoMetricSpace α] [MeasurableSpace α]
-    {u : α → ℝ} {S : Set α} {x : α} (hc : ContinuousWithinAt u S x)
-    (ν : ℕ → Measure α) [∀ n, IsProbabilityMeasure (ν n)]
-    (hi : ∀ n, Integrable u (ν n)) {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0))
-    (hs : ∀ n, ∀ᵐ p ∂ν n, p ∈ S ∧ dist p x ≤ e n) :
-    Tendsto (fun n => ∫ p, u p ∂ν n) atTop (𝓝 (u x)) := by
-  apply Metric.tendsto_nhds.2
-  intro ε hε
-  obtain ⟨δ,hδ,hδ'⟩ := Metric.continuousWithinAt_iff.1 hc (ε/2) (half_pos hε)
-  filter_upwards [he.eventually (gt_mem_nhds hδ)] with n hn
-  have hb : ∀ᵐ p ∂ν n, ‖u p-u x‖ ≤ ε/2 := by
-    filter_upwards [hs n] with p hp
-    simpa only [dist_eq_norm] using (hδ' hp.1 (hp.2.trans_lt hn)).le
-  have hm := norm_integral_le_of_norm_le_const hb
-  rw [integral_sub (hi n) (integrable_const _)] at hm
-  simp only [integral_const,Measure.real,measure_univ,ENNReal.toReal_one,one_smul,mul_one] at hm
-  have hh : dist (∫ p, u p ∂ν n) (u x) ≤ ε/2 := by simpa only [dist_eq_norm] using hm
-  exact hh.trans_lt (by linarith)
-
-lemma rectangle_point_dist {a b c d : ℝ} {p x : ℝ × ℝ}
-    (hp : p ∈ (Icc a b).prod (Icc c d)) (hx : x ∈ (Icc a b).prod (Icc c d)) :
-    dist p x ≤ max (b-a) (d-c) := by
-  rw [Prod.dist_eq]
-  apply max_le_max
-  · rw [Real.dist_eq,abs_le]
-    constructor <;> linarith [hp.1.1,hp.1.2,hx.1.1,hx.1.2]
-  · rw [Real.dist_eq,abs_le]
-    constructor <;> linarith [hp.2.1,hp.2.2,hx.2.1,hx.2.2]
-
-theorem strip_rectangle_means_tendsto {Fbar : ℂ → OnePoint ℂ}
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
-    (a b c d : ℕ → ℝ) (hab : ∀ n, a n < b n) (hcd : ∀ n, c n < d n)
-    (hlo : ∀ n, 0 ≤ c n) (hhi : ∀ n, d n ≤ Real.pi)
-    {x : ℝ × ℝ} (hx : ∀ n, x ∈ (Icc (a n) (b n)).prod (Icc (c n) (d n)))
-    (he : Tendsto (fun n => max (b n-a n) (d n-c n)) atTop (𝓝 0)) :
-    Tendsto (fun n => ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-      ∂(intervalProb (a n) (b n)).prod (intervalProb (c n) (d n))) atTop
-      (𝓝 (stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I))) := by
-  let ν := fun n => (intervalProb (a n) (b n)).prod (intervalProb (c n) (d n))
-  have (n : ℕ) : IsProbabilityMeasure (intervalProb (a n) (b n)) := intervalProb_isProbability (hab n)
-  have (n : ℕ) : IsProbabilityMeasure (intervalProb (c n) (d n)) := intervalProb_isProbability (hcd n)
-  let S := {p : ℝ × ℝ | 0 ≤ p.2 ∧ p.2 ≤ Real.pi}
-  have hu : ContinuousOn (fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)) S :=
-    (stripPotential_continuous hH hr q hcont).comp
-      ((Complex.continuous_ofReal.comp continuous_fst).add
-        ((Complex.continuous_ofReal.comp continuous_snd).mul_const _)).continuousOn
-      (by intro p hp; simpa [closedStrip,S] using hp)
-  have hxs : x ∈ S := ⟨(hlo 0).trans (hx 0).2.1,(hx 0).2.2.trans (hhi 0)⟩
-  apply probability_means_tendsto (hu x hxs) ν
-    (fun n => strip_potential_mean_integrable hcont hH hr q (hab n) (hcd n) (hlo n) (hhi n)) he
-  intro n
-  have hmem {l r : ℝ} : ∀ᵐ y : ℝ ∂intervalProb l r, y ∈ Icc l r :=
-    intervalProb_ae.2 ((ae_restrict_mem measurableSet_Ioc).mono (fun y hy => Ioc_subset_Icc_self hy))
-  have hX := (Measure.quasiMeasurePreserving_fst (μ := intervalProb (a n) (b n))
-    (ν := intervalProb (c n) (d n))).tendsto_ae.eventually hmem
-  have hY := (Measure.quasiMeasurePreserving_snd (μ := intervalProb (a n) (b n))
-    (ν := intervalProb (c n) (d n))).tendsto_ae.eventually hmem
-  filter_upwards [hX,hY] with p hpx hpy
-  exact ⟨⟨(hlo n).trans hpy.1,hpy.2.trans (hhi n)⟩,rectangle_point_dist ⟨hpx,hpy⟩ (hx n)⟩
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-variable (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ)
-
-def stripMean (Q : DRect) : ℝ :=
-  ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-    ∂(intervalProb Q.a Q.b).prod (intervalProb Q.c Q.d)
-
-def stripEnergy (Q : DRect) : ℝ :=
-  ∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2 ∂Q.measure
-
-def rectCoefficient (Q : DRect) : ℝ := (2/(Q.d-Q.c)+1/(Q.b-Q.a))^2*(Q.b-Q.a)*(Q.d-Q.c)
-
-lemma rectCoefficient_pos (Q : DRect) : 0 < rectCoefficient Q := by
-  have hw := sub_pos.2 Q.hab
-  have hh := sub_pos.2 Q.hcd
-  unfold rectCoefficient
-  positivity
-
-lemma dyadicRect_coefficient (Q : DRect) (n : ℕ) (i : DIndex n) :
-    rectCoefficient (dyadicRect Q n i) = rectCoefficient Q := by
-  unfold rectCoefficient
-  rw [dyadicRect_width,dyadicRect_height]
-  have h2 : (2:ℝ)^n ≠ 0 := pow_ne_zero _ (by norm_num)
-  have hw : Q.b-Q.a ≠ 0 := (sub_pos.2 Q.hab).ne'
-  have hh : Q.d-Q.c ≠ 0 := (sub_pos.2 Q.hcd).ne'
-  field_simp
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-variable (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ)
-variable {H q r Fbar}
-
-theorem strip_dyadic_mean_critical {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im})
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    (hH : 0 < H) (hr : 0 < r) (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
-    (n : ℕ) (i : DIndex n) :
-    (stripMean H q r Fbar (dyadicRect Q n i)-stripMean H q r Fbar Q)^2 ≤
-      3*rectCoefficient Q*(n:ℝ)*stripEnergy H q r Fbar Q := by
-  let β := dExtendBits n i
-  let R := fun j => dyadicRect Q j (dFromBits β j)
-  have hR0 : R 0 = Q := rfl
-  have hRn : R n = dyadicRect Q n i := by dsimp [R,β]; rw [dFromBits_extend]
-  have hRstep (j : ℕ) : R (j+1) = (R j).child (β j) := rfl
-  have hb (j : ℕ) := (R j).child_bounds (β j)
-  have ht := strip_nested_means hF hinj hext hcont hH hr q
-    (fun j => (R j).a) (fun j => (R j).b) (fun j => (R j).c) (fun j => (R j).d)
-    (fun j => (R j).hab) (fun j => (R j).hcd) hlo hhi
-    (fun j => (hb j).1) (fun j => (hb j).2.1) (fun j => (hb j).2.2.1) (fun j => (hb j).2.2.2)
-    (fun j => dyadicRect_width Q j (dFromBits β j))
-    (fun j => dyadicRect_height Q j (dFromBits β j)) n
-  change (stripMean H q r Fbar (R n)-stripMean H q r Fbar (R 0))^2 ≤ _ at ht
-  rw [hR0,hRn] at ht
-  unfold stripEnergy rectCoefficient DRect.measure
-  convert ht using 1
-  ring
-
-theorem strip_dyadic_mean_step {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im})
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    (hH : 0 < H) (hr : 0 < r) (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
-    (n : ℕ) (i : DIndex (n+1)) :
-    (stripMean H q r Fbar (dyadicRect Q (n+1) i)-stripMean H q r Fbar (dyadicRect Q n (dParent n i)))^2 ≤
-      rectCoefficient Q*stripEnergy H q r Fbar (dyadicRect Q n (dParent n i)) := by
-  let R := dyadicRect Q n i.1
-  obtain ⟨ha,hb,hc,hd⟩ := dyadicRect_bounds Q n i.1
-  obtain ⟨ha',hb',hc',hd'⟩ := R.child_bounds i.2
-  have hl : R.d-R.c ≤ 2*((R.child i.2).d-(R.child i.2).c) := by rw [DRect.child_height]; linarith
-  have ht := strip_parent_energy hF hinj hext hcont hH hr q R.hab R.hcd
-    (R.child i.2).hab (R.child i.2).hcd (hlo.trans hc) (hd.trans hhi) ha' hb' hc' hd' hl
-  change (stripMean H q r Fbar (R.child i.2)-stripMean H q r Fbar R)^2 ≤
-    rectCoefficient R*stripEnergy H q r Fbar R at ht
-  rw [dyadicRect_coefficient] at ht
-  exact ht
-
-theorem strip_dyadic_mean_limit
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    (hH : 0 ≤ H) (hr : 0 < r) (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
-    {x : ℝ × ℝ} (hx : x ∈ Q.closed) :
-    Tendsto (fun n => stripMean H q r Fbar (dyadicRect Q n (dPointBranch Q x n))) atTop
-      (𝓝 (stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I))) := by
-  let R := fun n => dyadicRect Q n (dPointBranch Q x n)
-  have hb (n : ℕ) := dyadicRect_bounds Q n (dPointBranch Q x n)
-  apply strip_rectangle_means_tendsto hcont hH hr q
-    (fun n => (R n).a) (fun n => (R n).b) (fun n => (R n).c) (fun n => (R n).d)
-    (fun n => (R n).hab) (fun n => (R n).hcd)
-    (fun n => hlo.trans (hb n).2.2.1) (fun n => (hb n).2.2.2.trans hhi)
-    (fun n => dPointBranch_mem hx n)
-  have h2 := tendsto_inv_atTop_zero.comp (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ) < 2))
-  have hw : Tendsto (fun n => (R n).b-(R n).a) atTop (𝓝 0) := by
-    simpa only [R,dyadicRect_width,div_eq_mul_inv,mul_zero,Function.comp_def] using tendsto_const_nhds.mul h2
-  have hh : Tendsto (fun n => (R n).d-(R n).c) atTop (𝓝 0) := by
-    simpa only [R,dyadicRect_height,div_eq_mul_inv,mul_zero,Function.comp_def] using tendsto_const_nhds.mul h2
-  simpa only [max_self] using hw.max hh
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-def totalStripEnergy (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) : ℝ :=
-  ∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
-    ∂volume.prod (volume.restrict (Ioc (0:ℝ) Real.pi))
-
-lemma totalStripEnergy_nonneg (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) :
-    0 ≤ totalStripEnergy H q r Fbar := integral_nonneg (fun _ => sq_nonneg _)
-
-lemma totalStripEnergy_eq (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) :
-    totalStripEnergy H q r Fbar = ∫ z in openStrip, ‖fderiv ℝ (stripPotential H q r Fbar) z‖^2 := by
-  have he := (Complex.volume_preserving_equiv_real_prod.symm _).setIntegral_preimage_emb
-    Complex.measurableEquivRealProd.symm.measurableEmbedding
-    (fun z => ‖fderiv ℝ (stripPotential H q r Fbar) z‖^2) openStrip
-  have hs : Complex.measurableEquivRealProd.symm ⁻¹' openStrip =
-      (univ : Set ℝ).prod (Ioo (0:ℝ) Real.pi) := by
-    ext p
-    change (0 < p.2 ∧ p.2 < Real.pi) ↔ True ∧ (0 < p.2 ∧ p.2 < Real.pi)
-    simp
-  rw [hs] at he
-  change (∫ p : ℝ × ℝ, _ ∂(volume.prod volume).restrict ((univ : Set ℝ) ×ˢ Ioo (0:ℝ) Real.pi)) = _ at he
-  rw [← Measure.prod_restrict,Measure.restrict_univ,restrict_Ioo_eq_restrict_Ioc] at he
-  refine Eq.trans ?_ he
-  unfold totalStripEnergy
-  apply integral_congr_ae
-  exact ae_of_all _ (fun p => by
-    have hp : (p.1:ℂ)+(p.2:ℂ)*Complex.I = Complex.measurableEquivRealProd.symm p := by
-      apply Complex.ext <;> simp
-    dsimp only
-    rw [hp])
-
- theorem totalStripEnergy_bound {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
-    totalStripEnergy H q r Fbar ≤ 2*Real.pi*H := by
-  have hi := strip_energy_integrable hF hinj hext hH hr q
-  have hb := strip_potential_energy hF hinj hext hH hr q
-  rw [← ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun z => sq_nonneg _))] at hb
-  rw [totalStripEnergy_eq]
-  exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp hb
-
-lemma stripEnergy_le_total {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi) :
-    stripEnergy H q r Fbar Q ≤ totalStripEnergy H q r Fbar := by
-  have hi := strip_energy_integrable_prod hF hinj hext hH hr q
-  rw [restrict_Ioo_eq_restrict_Ioc] at hi
-  apply integral_mono_measure _ (ae_of_all _ (fun _ => sq_nonneg _)) hi
-  exact Measure.prod_mono Measure.restrict_le_self (Measure.restrict_mono_set _
-    (fun y hy => ⟨hlo.trans_lt hy.1,hy.2.trans hhi⟩))
-
-def unitSlab (j : ℝ) : DRect := ⟨j,j+1,0,Real.pi,by linarith,Real.pi_pos⟩
-
-def longSlab (j : ℝ) (hj : 0 ≤ j) : DRect := ⟨-1,j+1,0,Real.pi,by linarith,Real.pi_pos⟩
-
-def slabMeanCoefficient : ℝ := 3*(2*(2/Real.pi+1)^2*Real.pi+Real.pi)
-
-lemma slabMeanCoefficient_pos : 0 < slabMeanCoefficient := by
-  unfold slabMeanCoefficient
-  positivity
-
-lemma longSlab_coefficient (j : ℝ) (hj : 0 ≤ j) :
-    rectCoefficient (longSlab j hj) ≤ (2/Real.pi+1)^2*Real.pi*(j+2) := by
-  have hw : 1 ≤ j+2 := by linarith
-  have hi : 1/(j+2) ≤ 1 := (div_le_one (by linarith)).2 hw
-  unfold rectCoefficient longSlab
-  dsimp only
-  rw [sub_zero,show j+1-(-1)=j+2 by ring]
-  calc
-    _ ≤ (2/Real.pi+1)^2*(j+2)*Real.pi := by gcongr
-    _ = _ := by ring
-
-theorem negative_unitSlab_mean {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
-    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-    (j : ℝ) (hj : j ≤ -1) :
-    (stripMean H q r Fbar (unitSlab j))^2 ≤ Real.pi*totalStripEnergy H q r Fbar := by
-  have hh := strip_negative_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess
-    (a := j) (b := j+1) (by linarith) (by linarith)
-  change (stripMean H q r Fbar (unitSlab j))^2 ≤
-    (Real.pi/(j+1-j))*stripEnergy H q r Fbar (unitSlab j) at hh
-  rw [show j+1-j=1 by ring,div_one] at hh
-  exact hh.trans (mul_le_mul_of_nonneg_left
-    (stripEnergy_le_total hF hinj hext hH hr q (unitSlab j) le_rfl le_rfl) Real.pi_pos.le)
-
-theorem positive_unitSlab_mean {F : ℂ → ℂ}
-    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
-    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
-    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
-    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
-    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
-    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
-    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
-    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
-      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
-    (j : ℝ) (hj : 0 ≤ j) :
-    (stripMean H q r Fbar (unitSlab j))^2 ≤
-      slabMeanCoefficient*(j+2)*totalStripEnergy H q r Fbar := by
-  let A := stripMean H q r Fbar (unitSlab j)
-  let B := stripMean H q r Fbar (unitSlab (-1))
-  let M := stripMean H q r Fbar (longSlab j hj)
-  let E := totalStripEnergy H q r Fbar
-  let K := (2/Real.pi+1)^2*Real.pi
-  have hE : 0 ≤ E := totalStripEnergy_nonneg _ _ _ _
-  have hK : 0 ≤ K := by dsimp [K]; positivity
-  have hEB := stripEnergy_le_total hF hinj hext hH hr q (longSlab j hj) le_rfl le_rfl
-  have hp (a' b' : ℝ) (hab' : a' < b') (haa : -1 ≤ a') (hbb : b' ≤ j+1) :
-      ((∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb a' b').prod (intervalProb 0 Real.pi))-M)^2 ≤ K*(j+2)*E := by
-    have hh := strip_parent_energy hF hinj hext hcont hH hr q
-      (a := -1) (b := j+1) (c := 0) (d := Real.pi) (a' := a') (b' := b') (c' := 0) (d' := Real.pi)
-      (by linarith) Real.pi_pos hab' Real.pi_pos le_rfl le_rfl haa hbb le_rfl le_rfl
-      (by linarith [Real.pi_pos])
-    change (_-M)^2 ≤ rectCoefficient (longSlab j hj)*stripEnergy H q r Fbar (longSlab j hj) at hh
-    exact hh.trans (mul_le_mul (longSlab_coefficient j hj) hEB
-      (integral_nonneg (fun _ => sq_nonneg _)) (by positivity))
-  have hA : (A-M)^2 ≤ K*(j+2)*E := hp j (j+1) (by linarith) (by linarith) le_rfl
-  have hB : (M-B)^2 ≤ K*(j+2)*E := by
-    have hh := hp (-1) 0 (by norm_num) le_rfl (by linarith)
-    have heB : B = ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
-        ∂(intervalProb (-1) 0).prod (intervalProb 0 Real.pi) := by simp [B,stripMean,unitSlab]
-    rw [← heB] at hh
-    nlinarith
-  have hb : B^2 ≤ Real.pi*E := negative_unitSlab_mean hF hinj hext hcont hH hr q
-    hη hη0 hη1 hηH haccess (-1) le_rfl
-  have hb' : B^2 ≤ Real.pi*(j+2)*E := hb.trans (by nlinarith [mul_nonneg hE (mul_nonneg Real.pi_pos.le (show 0 ≤ j+1 by linarith))])
-  have hsum : A^2 ≤ 3*((A-M)^2+(M-B)^2+B^2) := by
-    nlinarith [sq_nonneg ((A-M)-(M-B)),sq_nonneg ((A-M)-B),sq_nonneg ((M-B)-B)]
-  change A^2 ≤ slabMeanCoefficient*(j+2)*E
-  calc
-    _ ≤ 3*((A-M)^2+(M-B)^2+B^2) := hsum
-    _ ≤ 3*(K*(j+2)*E+K*(j+2)*E+Real.pi*(j+2)*E) := by gcongr
-    _ = _ := by dsimp [K,slabMeanCoefficient]; ring
-
-lemma complex_rectangle_diff_norm (Q : DRect) {x : ℝ × ℝ} (hx : x ∈ Q.closed) :
-    ‖((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)‖ ≤
-      (Q.b-Q.a)+(Q.d-Q.c) := by
-  have hh := Complex.norm_le_abs_re_add_abs_im
-    (((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I))
-  change x.1 ∈ Icc Q.a Q.b ∧ x.2 ∈ Icc Q.c Q.d at hx
-  simp only [Complex.sub_re,Complex.sub_im,Complex.add_re,Complex.add_im,
-    Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,
-    Complex.I_re,Complex.I_im,mul_zero,mul_one,sub_zero,zero_add,add_zero] at hh
-  rw [abs_of_nonneg (sub_nonneg.2 hx.1.1),abs_of_nonneg (sub_nonneg.2 hx.2.1)] at hh
-  linarith [hx.1.2,hx.2.2]
-
-lemma exp_difference_small {z w : ℂ} (hzw : ‖z-w‖ ≤ 1) :
-    ‖Complex.exp z-Complex.exp w‖ ≤ 2*‖z-w‖*Real.exp w.re := by
-  have he : Complex.exp z-Complex.exp w = (Complex.exp (z-w)-1)*Complex.exp w := by
-    rw [sub_mul,one_mul,← Complex.exp_add,sub_add_cancel]
-  rw [he,norm_mul,Complex.norm_exp]
-  exact mul_le_mul_of_nonneg_right (Complex.norm_exp_sub_one_le hzw) (Real.exp_pos _).le
-
-def expCellScale : ℝ := 4*(1+Real.pi)*Real.exp 1
-
-lemma expCellScale_pos : 0 < expCellScale := by unfold expCellScale; positivity
-
-lemma expCellScale_ge : 1+Real.pi ≤ expCellScale := by
-  have he : 1 ≤ Real.exp 1 := Real.one_le_exp_iff.2 (by norm_num)
-  unfold expCellScale
-  nlinarith [Real.pi_pos]
-
-lemma dyadicCell_exp_geometry (j : ℝ) (n : ℕ) (i : DIndex n)
-    (hsmall : expCellScale/(2:ℝ)^n < 1/16) :
-    let Q := dyadicRect (unitSlab j) n i
-    let v := Complex.exp ((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)
-    0 ≤ v.im ∧ Real.exp j/2 ≤ ‖v‖ ∧
-    ∀ x ∈ Q.closed, dist v (Complex.exp ((x.1:ℂ)+(x.2:ℂ)*Complex.I)) <
-      Real.exp j*(expCellScale/(2:ℝ)^n) := by
-  let Q := dyadicRect (unitSlab j) n i
-  have hb := dyadicRect_bounds (unitSlab j) n i
-  change j ≤ Q.a ∧ Q.b ≤ j+1 ∧ 0 ≤ Q.c ∧ Q.d ≤ Real.pi at hb
-  have hl : (Q.b-Q.a)+(Q.d-Q.c) = (1+Real.pi)/(2:ℝ)^n := by
-    rw [dyadicRect_width,dyadicRect_height]
-    simp only [unitSlab]
-    ring
-  have hc : Q.c ≤ Real.pi := Q.hcd.le.trans hb.2.2.2
-  have hre : (((Q.a:ℂ)+(Q.c:ℂ)*Complex.I):ℂ).re = Q.a := by simp
-  have him : (((Q.a:ℂ)+(Q.c:ℂ)*Complex.I):ℂ).im = Q.c := by simp
-  dsimp only
-  change 0 ≤ (Complex.exp ((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)).im ∧ _
-  refine ⟨?_,?_,?_⟩
-  · rw [Complex.exp_im,hre,him]
-    exact mul_nonneg (Real.exp_pos _).le (Real.sin_nonneg_of_mem_Icc ⟨hb.2.2.1,hc⟩)
-  · rw [Complex.norm_exp,hre]
-    have hh := Real.exp_le_exp.mpr hb.1
-    linarith [Real.exp_pos j]
-  · intro x hx
-    have hd := complex_rectangle_diff_norm Q hx
-    rw [hl] at hd
-    have hge : (1+Real.pi)/(2:ℝ)^n ≤ expCellScale/(2:ℝ)^n := by
-      exact div_le_div_of_nonneg_right expCellScale_ge (by positivity)
-    have hdn : ‖((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)‖ ≤ 1 := by
-      linarith [hd.trans hge]
-    rw [dist_comm,dist_eq_norm]
-    have he := exp_difference_small hdn
-    rw [hre] at he
-    have hea : Real.exp Q.a ≤ Real.exp (j+1) := Real.exp_le_exp.mpr (Q.hab.le.trans hb.2.1)
-    calc
-      _ ≤ 2*‖((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)‖*Real.exp Q.a := he
-      _ ≤ 2*((1+Real.pi)/(2:ℝ)^n)*Real.exp (j+1) := by gcongr
-      _ < Real.exp j*(expCellScale/(2:ℝ)^n) := by
-        rw [Real.exp_add,expCellScale]
-        have hp : 0 < (1+Real.pi)/(2:ℝ)^n*Real.exp j*Real.exp 1 := by positivity
-        convert (show 2*((1+Real.pi)/(2:ℝ)^n*Real.exp j*Real.exp 1) <
-          4*((1+Real.pi)/(2:ℝ)^n*Real.exp j*Real.exp 1) by linarith) using 1 <;> ring
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.BoundaryCutoff
-
-def clamp (σ ε : ℝ) (w : ℂ) : ℂ := ⟨σ*max (σ*w.re) ε,w.im⟩
-
-def velocity (σ ε u : ℝ) (w : ℂ) : ℂ := 2/clamp σ ε (w-(u:ℂ))
-
-lemma side_sq {σ : ℝ} (hσ : |σ| = 1) : σ*σ = 1 := by
-  have hh := sq_abs σ
-  rw [hσ] at hh
-  nlinarith
-
-lemma clamp_eq {σ ε : ℝ} (hσ : |σ| = 1) {w : ℂ} (hw : ε ≤ σ*w.re) :
-    clamp σ ε w = w := by
-  apply Complex.ext
-  · change σ*max (σ*w.re) ε = w.re
-    rw [max_eq_left hw,← mul_assoc,side_sq hσ,one_mul]
-  · rfl
-
-lemma clamp_norm {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (w : ℂ) : ε ≤ ‖clamp σ ε w‖ := by
-  have hh : ε ≤ |(clamp σ ε w).re| := by
-    change ε ≤ |σ*max (σ*w.re) ε|
-    rw [abs_mul,hσ,one_mul,abs_of_nonneg (hε.le.trans (le_max_right _ _))]
-    exact le_max_right _ _
-  exact hh.trans (Complex.abs_re_le_norm _)
-
-lemma clamp_ne_zero {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (w : ℂ) : clamp σ ε w ≠ 0 :=
-  norm_pos_iff.1 (hε.trans_le (clamp_norm hσ hε w))
-
-lemma clamp_norm_sub {σ ε : ℝ} (hσ : |σ| = 1) (z w : ℂ) :
-    ‖clamp σ ε z-clamp σ ε w‖ ≤ 2*‖z-w‖ := by
-  have hm : |max (σ*z.re) ε-max (σ*w.re) ε| ≤ |σ*z.re-σ*w.re| := by
-    simpa only [sub_self,abs_zero,max_eq_left (abs_nonneg (σ*z.re-σ*w.re))] using
-      abs_max_sub_max_le_max (σ*z.re) ε (σ*w.re) ε
-  have hr : |(clamp σ ε z-clamp σ ε w).re| ≤ ‖z-w‖ := by
-    change |σ*max (σ*z.re) ε-σ*max (σ*w.re) ε| ≤ ‖z-w‖
-    rw [← mul_sub,abs_mul,hσ,one_mul]
-    have he : |σ*z.re-σ*w.re| = |z.re-w.re| := by rw [← mul_sub,abs_mul,hσ,one_mul]
-    rw [he] at hm
-    exact hm.trans (by simpa using Complex.abs_re_le_norm (z-w))
-  have hi : |(clamp σ ε z-clamp σ ε w).im| ≤ ‖z-w‖ := by
-    simpa [clamp] using Complex.abs_im_le_norm (z-w)
-  exact (Complex.norm_le_abs_re_add_abs_im _).trans (by linarith)
-
-lemma velocity_eq {σ ε u : ℝ} (hσ : |σ| = 1) {w : ℂ} (hw : ε ≤ σ*(w.re-u)) :
-    velocity σ ε u w = 2/(w-(u:ℂ)) := by
-  unfold velocity
-  rw [clamp_eq hσ (by simpa using hw)]
-
-lemma norm_velocity_le {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (u : ℝ) (w : ℂ) :
-    ‖velocity σ ε u w‖ ≤ 2/ε := by
-  rw [velocity,norm_div,Complex.norm_ofNat]
-  exact div_le_div_of_nonneg_left (by norm_num) hε (clamp_norm hσ hε _)
-
-end SLEExactGauge.BoundaryCutoff
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.DyadicKernels
+
+namespace OAI
+
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+section Measure
+variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [IsFiniteMeasure μ]
+
+theorem dyadic_weighted_energy {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
+    {g : α → ℝ} (hg : MemLp g 2 μ) {V : ℝ} (n : ℕ)
+    (harea : ∀ j < n, ((2:ℝ)^j)^2*μ.real (Q j) ≤ V) :
+    (∫ x, dyadicKernel Q n x*g x ∂μ)^2 ≤
+      3*(n:ℝ)*V*(∫ x, (g x)^2 ∂μ) := by
+  exact (integral_mul_square_le (dyadicKernel_memLp hQ n) hg).trans
+    (mul_le_mul_of_nonneg_right (dyadicKernel_square_energy hQ n harea)
+      (integral_nonneg (fun x => sq_nonneg _)))
+
+omit [IsFiniteMeasure μ] in
+lemma dyadicKernel_integral_mul {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
+    {g : α → ℝ} (hg : Integrable g μ) (n : ℕ) :
+    (∫ x, dyadicKernel Q n x*g x ∂μ) =
+      ∑ j ∈ Finset.range n, (2:ℝ)^j*(∫ x in Q j, g x ∂μ) := by
+  have he : (fun x => dyadicKernel Q n x*g x) =
+      fun x => ∑ j ∈ Finset.range n, (Q j).indicator (fun x => (2:ℝ)^j*g x) x := by
+    funext x
+    rw [dyadicKernel,Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro j hj
+    by_cases hx : x ∈ Q j <;> simp [hx]
+  rw [he,integral_finsetSum _ (fun j hj => (hg.const_mul _).indicator (hQ j))]
+  apply Finset.sum_congr rfl
+  intro j hj
+  rw [integral_indicator (hQ j),integral_const_mul]
+
+omit [IsFiniteMeasure μ] in
+lemma dyadic_L1_telescoping {Q : ℕ → Set α} {g : α → ℝ} {v : ℕ → ℝ}
+    {C : ℝ} (hstep : ∀ j, |v (j+1)-v j| ≤ C*(2:ℝ)^j*(∫ x in Q j, g x ∂μ))
+    (n : ℕ) : |v n-v 0| ≤ C*∑ j ∈ Finset.range n, (2:ℝ)^j*(∫ x in Q j, g x ∂μ) := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    calc
+      _ ≤ |v (n+1)-v n|+|v n-v 0| := abs_sub_le _ _ _
+      _ ≤ C*(2:ℝ)^n*(∫ x in Q n, g x ∂μ)+
+          C*∑ j ∈ Finset.range n, (2:ℝ)^j*(∫ x in Q j, g x ∂μ) := add_le_add (hstep n) ih
+      _ = _ := by rw [Finset.sum_range_succ]; ring
+
+theorem critical_means_from_L1 {Q : ℕ → Set α} (hQ : ∀ j, MeasurableSet (Q j))
+    {g : α → ℝ} (hg : MemLp g 2 μ) (hg0 : ∀ᵐ x ∂μ, 0 ≤ g x)
+    {v : ℕ → ℝ} {C V : ℝ} (hC : 0 ≤ C)
+    (hstep : ∀ j, |v (j+1)-v j| ≤ C*(2:ℝ)^j*(∫ x in Q j, g x ∂μ))
+    (n : ℕ) (harea : ∀ j < n, ((2:ℝ)^j)^2*μ.real (Q j) ≤ V) :
+    (v n-v 0)^2 ≤ 3*C^2*(n:ℝ)*V*(∫ x, (g x)^2 ∂μ) := by
+  have hi := hg.integrable (by norm_num : (1:ℝ≥0∞) ≤ 2)
+  have hb := dyadic_L1_telescoping hstep n
+  rw [← dyadicKernel_integral_mul hQ hi n] at hb
+  have hn : 0 ≤ ∫ x, dyadicKernel Q n x*g x ∂μ := by
+    apply integral_nonneg_of_ae
+    filter_upwards [hg0] with x hx
+    apply mul_nonneg _ hx
+    apply Finset.sum_nonneg
+    intro j hj
+    exact indicator_nonneg (fun x hx => by positivity) x
+  have hs := (sq_le_sq₀ (abs_nonneg _) (mul_nonneg hC hn)).2 hb
+  rw [sq_abs,mul_pow] at hs
+  calc
+    _ ≤ C^2*(∫ x, dyadicKernel Q n x*g x ∂μ)^2 := hs
+    _ ≤ C^2*(3*(n:ℝ)*V*(∫ x, (g x)^2 ∂μ)) :=
+      mul_le_mul_of_nonneg_left (dyadic_weighted_energy hQ hg n harea) (sq_nonneg _)
+    _ = _ := by ring
+
+end Measure
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma rectangle_measure_restrict {a b c d a' b' c' d' : ℝ}
+    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d) :
+    ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))).restrict
+      ((Ioc a' b').prod (Ioc c' d')) =
+      (volume.restrict (Ioc a' b')).prod (volume.restrict (Ioc c' d')) := by
+  rw [Measure.prod_restrict,Measure.prod_restrict]
+  have hs : (Ioc a' b').prod (Ioc c' d') ⊆ (Ioc a b).prod (Ioc c d) := by
+    rintro p ⟨hx,hy⟩
+    exact ⟨⟨haa.trans_lt hx.1,hx.2.trans hbb⟩,⟨hcc.trans_lt hy.1,hy.2.trans hdd⟩⟩
+  exact Measure.restrict_restrict_of_subset hs
+
+lemma rectangle_measure_real {a b c d a' b' c' d' : ℝ}
+    (hab : a' ≤ b') (hcd : c' ≤ d')
+    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d) :
+    ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))).real
+      ((Ioc a' b').prod (Ioc c' d')) = (b'-a')*(d'-c') := by
+  rw [Measure.real]
+  have he := congrArg (fun μ : Measure (ℝ × ℝ) => μ univ)
+    (rectangle_measure_restrict haa hbb hcc hdd)
+  simp only [Measure.restrict_apply_univ] at he
+  rw [he,← univ_prod_univ,Measure.prod_prod]
+  simp only [Measure.restrict_apply_univ,Real.volume_Ioc,ENNReal.toReal_mul]
+  rw [ENNReal.toReal_ofReal (sub_nonneg.2 hab),ENNReal.toReal_ofReal (sub_nonneg.2 hcd)]
+
+theorem strip_nested_means {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    (a b c d : ℕ → ℝ) (hab : ∀ j, a j < b j) (hcd : ∀ j, c j < d j)
+    (hlo : 0 ≤ c 0) (hhi : d 0 ≤ Real.pi)
+    (haa : ∀ j, a j ≤ a (j+1)) (hbb : ∀ j, b (j+1) ≤ b j)
+    (hcc : ∀ j, c j ≤ c (j+1)) (hdd : ∀ j, d (j+1) ≤ d j)
+    (hw : ∀ j, b j-a j = (b 0-a 0)/(2:ℝ)^j)
+    (hh : ∀ j, d j-c j = (d 0-c 0)/(2:ℝ)^j) (n : ℕ) :
+    ((∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb (a n) (b n)).prod (intervalProb (c n) (d n)))-
+      (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb (a 0) (b 0)).prod (intervalProb (c 0) (d 0))))^2 ≤
+      3*(2/(d 0-c 0)+1/(b 0-a 0))^2*(n:ℝ)*(b 0-a 0)*(d 0-c 0)*
+        (∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar)
+          ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
+          ∂(volume.restrict (Ioc (a 0) (b 0))).prod (volume.restrict (Ioc (c 0) (d 0)))) := by
+  let μ := (volume.restrict (Ioc (a 0) (b 0))).prod (volume.restrict (Ioc (c 0) (d 0)))
+  let Q := fun j => (Ioc (a j) (b j)).prod (Ioc (c j) (d j))
+  let g := fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖
+  let v := fun j => ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+    ∂(intervalProb (a j) (b j)).prod (intervalProb (c j) (d j))
+  let C := 2/(d 0-c 0)+1/(b 0-a 0)
+  have ha0 (j : ℕ) : a 0 ≤ a j := (monotone_nat_of_le_succ haa) (Nat.zero_le j)
+  have hb0 (j : ℕ) : b j ≤ b 0 := (antitone_nat_of_succ_le hbb) (Nat.zero_le j)
+  have hc0 (j : ℕ) : c 0 ≤ c j := (monotone_nat_of_le_succ hcc) (Nat.zero_le j)
+  have hd0 (j : ℕ) : d j ≤ d 0 := (antitone_nat_of_succ_le hdd) (Nat.zero_le j)
+  have hC : 0 ≤ C := by
+    have hw0 := sub_pos.2 (hab 0)
+    have hh0 := sub_pos.2 (hcd 0)
+    dsimp [C]; positivity
+  have hstep (j : ℕ) : |v (j+1)-v j| ≤ C*(2:ℝ)^j*(∫ p in Q j, g p ∂μ) := by
+    have hlen : d j-c j ≤ 2*(d (j+1)-c (j+1)) := by rw [hh j,hh (j+1),pow_succ]; exact le_of_eq (by ring)
+    have ht := strip_rectangle_poincare hF hinj hext hcont hH hr q
+      (hab j) (hcd j) (hab (j+1)) (hcd (j+1)) (hlo.trans (hc0 j)) ((hd0 j).trans hhi)
+      (haa j) (hbb j) (hcc j) (hdd j) (show (0:ℝ) ≤ 2 by norm_num) hlen
+    change |v (j+1)-v j| ≤ _ at ht
+    have he : 2/(d j-c j)+1/(b j-a j) = C*(2:ℝ)^j := by
+      rw [hw j,hh j]
+      dsimp [C]
+      field_simp
+    rw [he] at ht
+    simpa only [Q,μ,rectangle_measure_restrict (ha0 j) (hb0 j) (hc0 j) (hd0 j)] using ht
+  have harea (j : ℕ) (hj : j < n) : ((2:ℝ)^j)^2*μ.real (Q j) ≤ (b 0-a 0)*(d 0-c 0) := by
+    rw [rectangle_measure_real (hab j).le (hcd j).le (ha0 j) (hb0 j) (hc0 j) (hd0 j)]
+    rw [hw j,hh j]
+    exact le_of_eq (by field_simp)
+  have ht := critical_means_from_L1 (μ := μ) (Q := Q)
+    (fun j => measurableSet_Ioc.prod measurableSet_Ioc)
+    (strip_gradient_rect_memLp hF hinj hext hH hr q (a 0) (b 0) hlo hhi)
+    (ae_of_all _ (fun p => norm_nonneg _)) hC hstep n harea
+  dsimp only [v,C,g] at ht
+  convert ht using 1
+  ring
+
+lemma rectangle_measure_total {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) :
+    ((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))).real univ =
+      (b-a)*(d-c) := by
+  rw [Measure.real,← univ_prod_univ,Measure.prod_prod]
+  simp only [Measure.restrict_apply_univ,Real.volume_Ioc,ENNReal.toReal_mul]
+  rw [ENNReal.toReal_ofReal (sub_nonneg.2 hab),ENNReal.toReal_ofReal (sub_nonneg.2 hcd)]
+
+theorem strip_parent_energy {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {a b c d a' b' c' d' : ℝ}
+    (hab : a < b) (hcd : c < d) (hab' : a' < b') (hcd' : c' < d')
+    (hlo : 0 ≤ c) (hhi : d ≤ Real.pi)
+    (haa : a ≤ a') (hbb : b' ≤ b) (hcc : c ≤ c') (hdd : d' ≤ d)
+    (hlen : d-c ≤ 2*(d'-c')) :
+    ((∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb a' b').prod (intervalProb c' d'))-
+      (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb a b).prod (intervalProb c d)))^2 ≤
+      (2/(d-c)+1/(b-a))^2*(b-a)*(d-c)*
+        (∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
+          ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+  have hp := strip_rectangle_poincare hF hinj hext hcont hH hr q hab hcd hab' hcd'
+    hlo hhi haa hbb hcc hdd (show (0:ℝ) ≤ 2 by norm_num) hlen
+  have hcoef : 0 ≤ 2/(d-c)+1/(b-a) := by
+    have hb := sub_pos.2 hab
+    have hd := sub_pos.2 hcd
+    positivity
+  have hE := strip_gradient_rect_memLp hF hinj hext hH hr q a b hlo hhi
+  have hi : 0 ≤ ∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖
+      ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d)) := integral_nonneg (fun _ => norm_nonneg _)
+  have hs := (sq_le_sq₀ (abs_nonneg _) (mul_nonneg hcoef hi)).2 hp
+  rw [sq_abs,mul_pow] at hs
+  have hc := integral_mul_square_le (memLp_const (1:ℝ)) hE
+  simp only [one_mul,one_pow,integral_const,smul_eq_mul,mul_one,
+    rectangle_measure_total hab.le hcd.le] at hc
+  exact hs.trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hc (sq_nonneg (2/(d-c)+1/(b-a))))
+
+lemma rectangleProb_integral {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d)
+    (f : ℝ × ℝ → ℝ) :
+    (∫ p, f p ∂(intervalProb a b).prod (intervalProb c d)) =
+      ((b-a)*(d-c))⁻¹*(∫ p, f p ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc c d))) := by
+  rw [intervalProb,intervalProb,Measure.prod_smul_left,Measure.prod_smul_right,
+    integral_smul_measure,integral_smul_measure,ENNReal.toReal_inv,ENNReal.toReal_inv,
+    ENNReal.toReal_ofReal (sub_nonneg.2 hab),ENNReal.toReal_ofReal (sub_nonneg.2 hcd)]
+  simp only [smul_eq_mul,mul_inv_rev]
+  ring
+
+theorem strip_negative_mean {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
+    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+    {a b : ℝ} (hab : a < b) (hb : b ≤ 0) :
+    (∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+      ∂(intervalProb a b).prod (intervalProb 0 Real.pi))^2 ≤
+      (Real.pi/(b-a))*(∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar)
+        ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
+          ∂(volume.restrict (Ioc a b)).prod (volume.restrict (Ioc (0:ℝ) Real.pi))) := by
+  let u := fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+  let G := fun p : ℝ × ℝ => ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
+  let μ := intervalProb a b
+  let ν := intervalProb 0 Real.pi
+  let : IsProbabilityMeasure μ := intervalProb_isProbability hab
+  let : IsProbabilityMeasure ν := intervalProb_isProbability Real.pi_pos
+  have hu := strip_potential_mean_integrable hcont hH.le hr q hab Real.pi_pos le_rfl le_rfl
+  have hu2 : MemLp u 2 (μ.prod ν) := MemLp.of_bound hu.aestronglyMeasurable H (ae_of_all _ (fun p => by
+    have hh := sphereTruncation_bounds hH.le hr q (Fbar (Complex.exp ((p.1:ℂ)+(p.2:ℂ)*Complex.I)))
+    change ‖stripPotential H q r Fbar _‖ ≤ H
+    rw [Real.norm_eq_abs,stripPotential,abs_of_nonneg hh.1]
+    exact hh.2))
+  have hG := (strip_gradient_rect_memLp hF hinj hext hH hr q a b le_rfl le_rfl).integrable_sq
+  have hR := hG.integral_prod_left
+  have hRμ : Integrable (fun v => ∫ φ in Ioc (0:ℝ) Real.pi, G (v,φ)) μ :=
+    integrable_intervalProb hab hR
+  have hbound : ∀ᵐ p ∂μ.prod ν, u p^2 ≤
+      Real.pi*(∫ φ in Ioc (0:ℝ) Real.pi, G (p.1,φ)) := by
+    have hn := strip_negative_section_energy hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess
+    have hv : ∀ᵐ v ∂μ, ∀ φ ∈ Ioc (0:ℝ) Real.pi, u (v,φ)^2 ≤
+        Real.pi*(∫ ψ in Ioc (0:ℝ) Real.pi, G (v,ψ)) := by
+      apply intervalProb_ae.mpr
+      filter_upwards [ae_restrict_of_ae hn,
+        ae_restrict_mem measurableSet_Ioc,show ∀ᵐ v ∂volume.restrict (Ioc a b), v ≠ (0:ℝ) from
+          ae_restrict_of_ae (volume.ae_ne 0)] with v hv hvab hv0
+      intro φ hφ
+      have hvneg : v < 0 := lt_of_le_of_ne (hvab.2.trans hb) hv0
+      have hh := hv hvneg φ ⟨hφ.1.le,hφ.2⟩
+      rw [intervalIntegral.integral_of_le Real.pi_pos.le] at hh
+      exact hh
+    have hφ : ∀ᵐ φ ∂ν, φ ∈ Ioc (0:ℝ) Real.pi := intervalProb_ae.mpr (ae_restrict_mem measurableSet_Ioc)
+    filter_upwards [Measure.quasiMeasurePreserving_fst.ae hv,
+      Measure.quasiMeasurePreserving_snd.ae hφ] with p hp hpφ
+    exact hp p.2 hpφ
+  have hmono := integral_mono_ae hu2.integrable_sq ((hRμ.comp_fst ν).const_mul Real.pi) hbound
+  have hj := integral_mul_square_le (memLp_const (1:ℝ)) hu2
+  simp only [one_mul,one_pow,integral_const,Measure.real,measure_univ,ENNReal.toReal_one,
+    smul_eq_mul,mul_one,one_mul] at hj
+  refine hj.trans (hmono.trans_eq ?_)
+  rw [integral_const_mul]
+  rw [integral_fun_fst (fun v => ∫ φ in Ioc (0:ℝ) Real.pi, G (v,φ))]
+  simp only [Measure.real,measure_univ,ENNReal.toReal_one,one_smul]
+  rw [integral_intervalProb hab.le,intervalIntegral.integral_of_le hab.le]
+  rw [← integral_prod _ hG]
+  simp only [div_eq_mul_inv,mul_assoc]
+
+structure DRect where
+  a : ℝ
+  b : ℝ
+  c : ℝ
+  d : ℝ
+  hab : a < b
+  hcd : c < d
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+namespace DRect
+
+def closed (Q : DRect) : Set (ℝ × ℝ) := (Icc Q.a Q.b).prod (Icc Q.c Q.d)
+
+def cells (Q : DRect) : Set (ℝ × ℝ) := (Ioc Q.a Q.b).prod (Ioc Q.c Q.d)
+
+def measure (Q : DRect) : Measure (ℝ × ℝ) :=
+  (volume.restrict (Ioc Q.a Q.b)).prod (volume.restrict (Ioc Q.c Q.d))
+
+def halfLo (l r : ℝ) (s : Bool) : ℝ := if s then (l+r)/2 else l
+
+def halfHi (l r : ℝ) (s : Bool) : ℝ := if s then r else (l+r)/2
+
+lemma half_bounds {l r : ℝ} (h : l < r) (s : Bool) :
+    l ≤ halfLo l r s ∧ halfLo l r s < halfHi l r s ∧ halfHi l r s ≤ r := by
+  cases s
+  · change l ≤ l ∧ l < (l+r)/2 ∧ (l+r)/2 ≤ r
+    refine ⟨le_rfl,?_,?_⟩ <;> linarith
+  · change l ≤ (l+r)/2 ∧ (l+r)/2 < r ∧ r ≤ r
+    refine ⟨?_,?_,le_rfl⟩ <;> linarith
+
+lemma half_length (l r : ℝ) (s : Bool) : halfHi l r s-halfLo l r s = (r-l)/2 := by
+  cases s <;> simp [halfLo,halfHi] <;> ring
+
+def child (Q : DRect) (s : Bool × Bool) : DRect where
+  a := halfLo Q.a Q.b s.1
+  b := halfHi Q.a Q.b s.1
+  c := halfLo Q.c Q.d s.2
+  d := halfHi Q.c Q.d s.2
+  hab := (half_bounds Q.hab s.1).2.1
+  hcd := (half_bounds Q.hcd s.2).2.1
+
+lemma child_bounds (Q : DRect) (s : Bool × Bool) :
+    Q.a ≤ (Q.child s).a ∧ (Q.child s).b ≤ Q.b ∧
+    Q.c ≤ (Q.child s).c ∧ (Q.child s).d ≤ Q.d :=
+  ⟨(half_bounds Q.hab s.1).1,(half_bounds Q.hab s.1).2.2,
+    (half_bounds Q.hcd s.2).1,(half_bounds Q.hcd s.2).2.2⟩
+
+lemma child_width (Q : DRect) (s : Bool × Bool) : (Q.child s).b-(Q.child s).a = (Q.b-Q.a)/2 := half_length _ _ _
+
+lemma child_height (Q : DRect) (s : Bool × Bool) : (Q.child s).d-(Q.child s).c = (Q.d-Q.c)/2 := half_length _ _ _
+
+lemma child_closed_subset (Q : DRect) (s : Bool × Bool) : (Q.child s).closed ⊆ Q.closed := by
+  rintro p ⟨hx,hy⟩
+  obtain ⟨ha,hb,hc,hd⟩ := Q.child_bounds s
+  exact ⟨⟨ha.trans hx.1,hx.2.trans hb⟩,⟨hc.trans hy.1,hy.2.trans hd⟩⟩
+
+def chooseSide (l r x : ℝ) : Bool := decide ((l+r)/2 < x)
+
+lemma mem_chosen_half {l r x : ℝ} (hx : x ∈ Icc l r) :
+    x ∈ Icc (halfLo l r (chooseSide l r x)) (halfHi l r (chooseSide l r x)) := by
+  by_cases h : (l+r)/2 < x
+  · simp [chooseSide,h,halfLo,halfHi,h.le,hx.2]
+  · have hm : x ≤ (l+r)/2 := le_of_not_gt h
+    simp [chooseSide,h,halfLo,halfHi,hx.1,hm]
+
+def chooseChild (Q : DRect) (x : ℝ × ℝ) : Bool × Bool :=
+  (chooseSide Q.a Q.b x.1,chooseSide Q.c Q.d x.2)
+
+lemma mem_chosen_child {Q : DRect} {x : ℝ × ℝ} (hx : x ∈ Q.closed) :
+    x ∈ (Q.child (Q.chooseChild x)).closed :=
+  ⟨mem_chosen_half hx.1,mem_chosen_half hx.2⟩
+
+end DRect
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+def DIndex : ℕ → Type
+  | 0 => Unit
+  | n+1 => DIndex n × (Bool × Bool)
+
+instance dIndex_fintype (n : ℕ) : Fintype (DIndex n) := by
+  induction n with
+  | zero => exact inferInstanceAs (Fintype Unit)
+  | succ n ih => exact inferInstanceAs (Fintype (DIndex n × (Bool × Bool)))
+
+def dyadicRect (Q : DRect) : (n : ℕ) → DIndex n → DRect
+  | 0, _ => Q
+  | n+1, i => (dyadicRect Q n i.1).child i.2
+
+def dParent (n : ℕ) (i : DIndex (n+1)) : DIndex n := i.1
+
+def dFromBits (β : ℕ → Bool × Bool) : (n : ℕ) → DIndex n
+  | 0 => ()
+  | n+1 => (dFromBits β n,β n)
+
+lemma dFromBits_congr {β β' : ℕ → Bool × Bool} (n : ℕ) (h : ∀ k < n, β k = β' k) :
+    dFromBits β n = dFromBits β' n := by
+  induction n with
+  | zero => rfl
+  | succ n ih =>
+    change (dFromBits β n,β n) = (dFromBits β' n,β' n)
+    rw [ih (fun k hk => h k (Nat.lt_succ_of_lt hk)),h n (Nat.lt_succ_self n)]
+
+def dExtendBits : (n : ℕ) → DIndex n → (ℕ → Bool × Bool)
+  | 0, _ => fun _ => (false,false)
+  | n+1, i => Function.update (dExtendBits n i.1) n i.2
+
+lemma dFromBits_extend (n : ℕ) (i : DIndex n) : dFromBits (dExtendBits n i) n = i := by
+  induction n with
+  | zero => cases i; rfl
+  | succ n ih =>
+    change (dFromBits (Function.update (dExtendBits n i.1) n i.2) n,
+      Function.update (dExtendBits n i.1) n i.2 n) = i
+    rw [Function.update_self]
+    have he := dFromBits_congr n (β := Function.update (dExtendBits n i.1) n i.2)
+      (β' := dExtendBits n i.1) (fun k hk => Function.update_of_ne (Nat.ne_of_lt hk) _ _)
+    rw [he,ih i.1]
+    rfl
+
+lemma dyadicRect_width (Q : DRect) (n : ℕ) (i : DIndex n) :
+    (dyadicRect Q n i).b-(dyadicRect Q n i).a = (Q.b-Q.a)/(2:ℝ)^n := by
+  induction n with
+  | zero => simp [dyadicRect]
+  | succ n ih => rw [dyadicRect,DRect.child_width,ih,pow_succ]; ring
+
+lemma dyadicRect_height (Q : DRect) (n : ℕ) (i : DIndex n) :
+    (dyadicRect Q n i).d-(dyadicRect Q n i).c = (Q.d-Q.c)/(2:ℝ)^n := by
+  induction n with
+  | zero => simp [dyadicRect]
+  | succ n ih => rw [dyadicRect,DRect.child_height,ih,pow_succ]; ring
+
+lemma dyadicRect_bounds (Q : DRect) (n : ℕ) (i : DIndex n) :
+    Q.a ≤ (dyadicRect Q n i).a ∧ (dyadicRect Q n i).b ≤ Q.b ∧
+    Q.c ≤ (dyadicRect Q n i).c ∧ (dyadicRect Q n i).d ≤ Q.d := by
+  induction n with
+  | zero => exact ⟨le_rfl,le_rfl,le_rfl,le_rfl⟩
+  | succ n ih =>
+    obtain ⟨ha,hb,hc,hd⟩ := ih i.1
+    obtain ⟨ha',hb',hc',hd'⟩ := (dyadicRect Q n i.1).child_bounds i.2
+    exact ⟨ha.trans ha',hb'.trans hb,hc.trans hc',hd'.trans hd⟩
+
+def dPointBranch (Q : DRect) (x : ℝ × ℝ) : (n : ℕ) → DIndex n
+  | 0 => ()
+  | n+1 => (dPointBranch Q x n,(dyadicRect Q n (dPointBranch Q x n)).chooseChild x)
+
+lemma dPointBranch_parent (Q : DRect) (x : ℝ × ℝ) (n : ℕ) :
+    dParent n (dPointBranch Q x (n+1)) = dPointBranch Q x n := rfl
+
+lemma dPointBranch_mem {Q : DRect} {x : ℝ × ℝ} (hx : x ∈ Q.closed) (n : ℕ) :
+    x ∈ (dyadicRect Q n (dPointBranch Q x n)).closed := by
+  induction n with
+  | zero => exact hx
+  | succ n ih => exact DRect.mem_chosen_child ih
+
+lemma probability_means_tendsto {α : Type*} [PseudoMetricSpace α] [MeasurableSpace α]
+    {u : α → ℝ} {S : Set α} {x : α} (hc : ContinuousWithinAt u S x)
+    (ν : ℕ → Measure α) [∀ n, IsProbabilityMeasure (ν n)]
+    (hi : ∀ n, Integrable u (ν n)) {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0))
+    (hs : ∀ n, ∀ᵐ p ∂ν n, p ∈ S ∧ dist p x ≤ e n) :
+    Tendsto (fun n => ∫ p, u p ∂ν n) atTop (𝓝 (u x)) := by
+  apply Metric.tendsto_nhds.2
+  intro ε hε
+  obtain ⟨δ,hδ,hδ'⟩ := Metric.continuousWithinAt_iff.1 hc (ε/2) (half_pos hε)
+  filter_upwards [he.eventually (gt_mem_nhds hδ)] with n hn
+  have hb : ∀ᵐ p ∂ν n, ‖u p-u x‖ ≤ ε/2 := by
+    filter_upwards [hs n] with p hp
+    simpa only [dist_eq_norm] using (hδ' hp.1 (hp.2.trans_lt hn)).le
+  have hm := norm_integral_le_of_norm_le_const hb
+  rw [integral_sub (hi n) (integrable_const _)] at hm
+  simp only [integral_const,Measure.real,measure_univ,ENNReal.toReal_one,one_smul,mul_one] at hm
+  have hh : dist (∫ p, u p ∂ν n) (u x) ≤ ε/2 := by simpa only [dist_eq_norm] using hm
+  exact hh.trans_lt (by linarith)
+
+lemma rectangle_point_dist {a b c d : ℝ} {p x : ℝ × ℝ}
+    (hp : p ∈ (Icc a b).prod (Icc c d)) (hx : x ∈ (Icc a b).prod (Icc c d)) :
+    dist p x ≤ max (b-a) (d-c) := by
+  rw [Prod.dist_eq]
+  apply max_le_max
+  · rw [Real.dist_eq,abs_le]
+    constructor <;> linarith [hp.1.1,hp.1.2,hx.1.1,hx.1.2]
+  · rw [Real.dist_eq,abs_le]
+    constructor <;> linarith [hp.2.1,hp.2.2,hx.2.1,hx.2.2]
+
+theorem strip_rectangle_means_tendsto {Fbar : ℂ → OnePoint ℂ}
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ)
+    (a b c d : ℕ → ℝ) (hab : ∀ n, a n < b n) (hcd : ∀ n, c n < d n)
+    (hlo : ∀ n, 0 ≤ c n) (hhi : ∀ n, d n ≤ Real.pi)
+    {x : ℝ × ℝ} (hx : ∀ n, x ∈ (Icc (a n) (b n)).prod (Icc (c n) (d n)))
+    (he : Tendsto (fun n => max (b n-a n) (d n-c n)) atTop (𝓝 0)) :
+    Tendsto (fun n => ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+      ∂(intervalProb (a n) (b n)).prod (intervalProb (c n) (d n))) atTop
+      (𝓝 (stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I))) := by
+  let ν := fun n => (intervalProb (a n) (b n)).prod (intervalProb (c n) (d n))
+  have (n : ℕ) : IsProbabilityMeasure (intervalProb (a n) (b n)) := intervalProb_isProbability (hab n)
+  have (n : ℕ) : IsProbabilityMeasure (intervalProb (c n) (d n)) := intervalProb_isProbability (hcd n)
+  let S := {p : ℝ × ℝ | 0 ≤ p.2 ∧ p.2 ≤ Real.pi}
+  have hu : ContinuousOn (fun p : ℝ × ℝ => stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)) S :=
+    (stripPotential_continuous hH hr q hcont).comp
+      ((Complex.continuous_ofReal.comp continuous_fst).add
+        ((Complex.continuous_ofReal.comp continuous_snd).mul_const _)).continuousOn
+      (by intro p hp; simpa [closedStrip,S] using hp)
+  have hxs : x ∈ S := ⟨(hlo 0).trans (hx 0).2.1,(hx 0).2.2.trans (hhi 0)⟩
+  apply probability_means_tendsto (hu x hxs) ν
+    (fun n => strip_potential_mean_integrable hcont hH hr q (hab n) (hcd n) (hlo n) (hhi n)) he
+  intro n
+  have hmem {l r : ℝ} : ∀ᵐ y : ℝ ∂intervalProb l r, y ∈ Icc l r :=
+    intervalProb_ae.2 ((ae_restrict_mem measurableSet_Ioc).mono (fun y hy => Ioc_subset_Icc_self hy))
+  have hX := (Measure.quasiMeasurePreserving_fst (μ := intervalProb (a n) (b n))
+    (ν := intervalProb (c n) (d n))).tendsto_ae.eventually hmem
+  have hY := (Measure.quasiMeasurePreserving_snd (μ := intervalProb (a n) (b n))
+    (ν := intervalProb (c n) (d n))).tendsto_ae.eventually hmem
+  filter_upwards [hX,hY] with p hpx hpy
+  exact ⟨⟨(hlo n).trans hpy.1,hpy.2.trans (hhi n)⟩,rectangle_point_dist ⟨hpx,hpy⟩ (hx n)⟩
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+variable (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ)
+
+def stripMean (Q : DRect) : ℝ :=
+  ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+    ∂(intervalProb Q.a Q.b).prod (intervalProb Q.c Q.d)
+
+def stripEnergy (Q : DRect) : ℝ :=
+  ∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2 ∂Q.measure
+
+def rectCoefficient (Q : DRect) : ℝ := (2/(Q.d-Q.c)+1/(Q.b-Q.a))^2*(Q.b-Q.a)*(Q.d-Q.c)
+
+lemma rectCoefficient_pos (Q : DRect) : 0 < rectCoefficient Q := by
+  have hw := sub_pos.2 Q.hab
+  have hh := sub_pos.2 Q.hcd
+  unfold rectCoefficient
+  positivity
+
+lemma dyadicRect_coefficient (Q : DRect) (n : ℕ) (i : DIndex n) :
+    rectCoefficient (dyadicRect Q n i) = rectCoefficient Q := by
+  unfold rectCoefficient
+  rw [dyadicRect_width,dyadicRect_height]
+  have h2 : (2:ℝ)^n ≠ 0 := pow_ne_zero _ (by norm_num)
+  have hw : Q.b-Q.a ≠ 0 := (sub_pos.2 Q.hab).ne'
+  have hh : Q.d-Q.c ≠ 0 := (sub_pos.2 Q.hcd).ne'
+  field_simp
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+variable (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ)
+variable {H q r Fbar}
+
+theorem strip_dyadic_mean_critical {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im})
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    (hH : 0 < H) (hr : 0 < r) (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
+    (n : ℕ) (i : DIndex n) :
+    (stripMean H q r Fbar (dyadicRect Q n i)-stripMean H q r Fbar Q)^2 ≤
+      3*rectCoefficient Q*(n:ℝ)*stripEnergy H q r Fbar Q := by
+  let β := dExtendBits n i
+  let R := fun j => dyadicRect Q j (dFromBits β j)
+  have hR0 : R 0 = Q := rfl
+  have hRn : R n = dyadicRect Q n i := by dsimp [R,β]; rw [dFromBits_extend]
+  have hRstep (j : ℕ) : R (j+1) = (R j).child (β j) := rfl
+  have hb (j : ℕ) := (R j).child_bounds (β j)
+  have ht := strip_nested_means hF hinj hext hcont hH hr q
+    (fun j => (R j).a) (fun j => (R j).b) (fun j => (R j).c) (fun j => (R j).d)
+    (fun j => (R j).hab) (fun j => (R j).hcd) hlo hhi
+    (fun j => (hb j).1) (fun j => (hb j).2.1) (fun j => (hb j).2.2.1) (fun j => (hb j).2.2.2)
+    (fun j => dyadicRect_width Q j (dFromBits β j))
+    (fun j => dyadicRect_height Q j (dFromBits β j)) n
+  change (stripMean H q r Fbar (R n)-stripMean H q r Fbar (R 0))^2 ≤ _ at ht
+  rw [hR0,hRn] at ht
+  unfold stripEnergy rectCoefficient DRect.measure
+  convert ht using 1
+  ring
+
+theorem strip_dyadic_mean_step {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im})
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    (hH : 0 < H) (hr : 0 < r) (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
+    (n : ℕ) (i : DIndex (n+1)) :
+    (stripMean H q r Fbar (dyadicRect Q (n+1) i)-stripMean H q r Fbar (dyadicRect Q n (dParent n i)))^2 ≤
+      rectCoefficient Q*stripEnergy H q r Fbar (dyadicRect Q n (dParent n i)) := by
+  let R := dyadicRect Q n i.1
+  obtain ⟨ha,hb,hc,hd⟩ := dyadicRect_bounds Q n i.1
+  obtain ⟨ha',hb',hc',hd'⟩ := R.child_bounds i.2
+  have hl : R.d-R.c ≤ 2*((R.child i.2).d-(R.child i.2).c) := by rw [DRect.child_height]; linarith
+  have ht := strip_parent_energy hF hinj hext hcont hH hr q R.hab R.hcd
+    (R.child i.2).hab (R.child i.2).hcd (hlo.trans hc) (hd.trans hhi) ha' hb' hc' hd' hl
+  change (stripMean H q r Fbar (R.child i.2)-stripMean H q r Fbar R)^2 ≤
+    rectCoefficient R*stripEnergy H q r Fbar R at ht
+  rw [dyadicRect_coefficient] at ht
+  exact ht
+
+theorem strip_dyadic_mean_limit
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    (hH : 0 ≤ H) (hr : 0 < r) (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi)
+    {x : ℝ × ℝ} (hx : x ∈ Q.closed) :
+    Tendsto (fun n => stripMean H q r Fbar (dyadicRect Q n (dPointBranch Q x n))) atTop
+      (𝓝 (stripPotential H q r Fbar ((x.1:ℂ)+(x.2:ℂ)*Complex.I))) := by
+  let R := fun n => dyadicRect Q n (dPointBranch Q x n)
+  have hb (n : ℕ) := dyadicRect_bounds Q n (dPointBranch Q x n)
+  apply strip_rectangle_means_tendsto hcont hH hr q
+    (fun n => (R n).a) (fun n => (R n).b) (fun n => (R n).c) (fun n => (R n).d)
+    (fun n => (R n).hab) (fun n => (R n).hcd)
+    (fun n => hlo.trans (hb n).2.2.1) (fun n => (hb n).2.2.2.trans hhi)
+    (fun n => dPointBranch_mem hx n)
+  have h2 := tendsto_inv_atTop_zero.comp (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ) < 2))
+  have hw : Tendsto (fun n => (R n).b-(R n).a) atTop (𝓝 0) := by
+    simpa only [R,dyadicRect_width,div_eq_mul_inv,mul_zero,Function.comp_def] using tendsto_const_nhds.mul h2
+  have hh : Tendsto (fun n => (R n).d-(R n).c) atTop (𝓝 0) := by
+    simpa only [R,dyadicRect_height,div_eq_mul_inv,mul_zero,Function.comp_def] using tendsto_const_nhds.mul h2
+  simpa only [max_self] using hw.max hh
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+def totalStripEnergy (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) : ℝ :=
+  ∫ p : ℝ × ℝ, ‖fderiv ℝ (stripPotential H q r Fbar) ((p.1:ℂ)+(p.2:ℂ)*Complex.I)‖^2
+    ∂volume.prod (volume.restrict (Ioc (0:ℝ) Real.pi))
+
+lemma totalStripEnergy_nonneg (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) :
+    0 ≤ totalStripEnergy H q r Fbar := integral_nonneg (fun _ => sq_nonneg _)
+
+lemma totalStripEnergy_eq (H : ℝ) (q : ℂ) (r : ℝ) (Fbar : ℂ → OnePoint ℂ) :
+    totalStripEnergy H q r Fbar = ∫ z in openStrip, ‖fderiv ℝ (stripPotential H q r Fbar) z‖^2 := by
+  have he := (Complex.volume_preserving_equiv_real_prod.symm _).setIntegral_preimage_emb
+    Complex.measurableEquivRealProd.symm.measurableEmbedding
+    (fun z => ‖fderiv ℝ (stripPotential H q r Fbar) z‖^2) openStrip
+  have hs : Complex.measurableEquivRealProd.symm ⁻¹' openStrip =
+      (univ : Set ℝ).prod (Ioo (0:ℝ) Real.pi) := by
+    ext p
+    change (0 < p.2 ∧ p.2 < Real.pi) ↔ True ∧ (0 < p.2 ∧ p.2 < Real.pi)
+    simp
+  rw [hs] at he
+  change (∫ p : ℝ × ℝ, _ ∂(volume.prod volume).restrict ((univ : Set ℝ) ×ˢ Ioo (0:ℝ) Real.pi)) = _ at he
+  rw [← Measure.prod_restrict,Measure.restrict_univ,restrict_Ioo_eq_restrict_Ioc] at he
+  refine Eq.trans ?_ he
+  unfold totalStripEnergy
+  apply integral_congr_ae
+  exact ae_of_all _ (fun p => by
+    have hp : (p.1:ℂ)+(p.2:ℂ)*Complex.I = Complex.measurableEquivRealProd.symm p := by
+      apply Complex.ext <;> simp
+    dsimp only
+    rw [hp])
+
+ theorem totalStripEnergy_bound {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ) :
+    totalStripEnergy H q r Fbar ≤ 2*Real.pi*H := by
+  have hi := strip_energy_integrable hF hinj hext hH hr q
+  have hb := strip_potential_energy hF hinj hext hH hr q
+  rw [← ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun z => sq_nonneg _))] at hb
+  rw [totalStripEnergy_eq]
+  exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp hb
+
+lemma stripEnergy_le_total {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    (Q : DRect) (hlo : 0 ≤ Q.c) (hhi : Q.d ≤ Real.pi) :
+    stripEnergy H q r Fbar Q ≤ totalStripEnergy H q r Fbar := by
+  have hi := strip_energy_integrable_prod hF hinj hext hH hr q
+  rw [restrict_Ioo_eq_restrict_Ioc] at hi
+  apply integral_mono_measure _ (ae_of_all _ (fun _ => sq_nonneg _)) hi
+  exact Measure.prod_mono Measure.restrict_le_self (Measure.restrict_mono_set _
+    (fun y hy => ⟨hlo.trans_lt hy.1,hy.2.trans hhi⟩))
+
+def unitSlab (j : ℝ) : DRect := ⟨j,j+1,0,Real.pi,by linarith,Real.pi_pos⟩
+
+def longSlab (j : ℝ) (hj : 0 ≤ j) : DRect := ⟨-1,j+1,0,Real.pi,by linarith,Real.pi_pos⟩
+
+def slabMeanCoefficient : ℝ := 3*(2*(2/Real.pi+1)^2*Real.pi+Real.pi)
+
+lemma slabMeanCoefficient_pos : 0 < slabMeanCoefficient := by
+  unfold slabMeanCoefficient
+  positivity
+
+lemma longSlab_coefficient (j : ℝ) (hj : 0 ≤ j) :
+    rectCoefficient (longSlab j hj) ≤ (2/Real.pi+1)^2*Real.pi*(j+2) := by
+  have hw : 1 ≤ j+2 := by linarith
+  have hi : 1/(j+2) ≤ 1 := (div_le_one (by linarith)).2 hw
+  unfold rectCoefficient longSlab
+  dsimp only
+  rw [sub_zero,show j+1-(-1)=j+2 by ring]
+  calc
+    _ ≤ (2/Real.pi+1)^2*(j+2)*Real.pi := by gcongr
+    _ = _ := by ring
+
+theorem negative_unitSlab_mean {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
+    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+    (j : ℝ) (hj : j ≤ -1) :
+    (stripMean H q r Fbar (unitSlab j))^2 ≤ Real.pi*totalStripEnergy H q r Fbar := by
+  have hh := strip_negative_mean hF hinj hext hcont hH hr q hη hη0 hη1 hηH haccess
+    (a := j) (b := j+1) (by linarith) (by linarith)
+  change (stripMean H q r Fbar (unitSlab j))^2 ≤
+    (Real.pi/(j+1-j))*stripEnergy H q r Fbar (unitSlab j) at hh
+  rw [show j+1-j=1 by ring,div_one] at hh
+  exact hh.trans (mul_le_mul_of_nonneg_left
+    (stripEnergy_le_total hF hinj hext hH hr q (unitSlab j) le_rfl le_rfl) Real.pi_pos.le)
+
+theorem positive_unitSlab_mean {F : ℂ → ℂ}
+    (hF : AnalyticOnNhd ℂ F {z : ℂ | 0 < z.im})
+    (hinj : InjOn F {z : ℂ | 0 < z.im}) {Fbar : ℂ → OnePoint ℂ}
+    (hext : ∀ z, 0 < z.im → Fbar z = (F z : OnePoint ℂ))
+    (hcont : ContinuousOn Fbar {z : ℂ | 0 ≤ z.im})
+    {H r : ℝ} (hH : 0 < H) (hr : 0 < r) (q : ℂ)
+    {η : ℝ → ℂ} (hη : ContinuousOn η (Icc 0 1)) (hη0 : η 0 = 0)
+    (hη1 : ‖η 1‖ = 1) (hηH : ∀ t ∈ Icc 0 1, 0 ≤ (η t).im)
+    (haccess : ∀ t ∈ Icc 0 1, Fbar (η t) ∈
+      (fun w : ℂ => (w : OnePoint ℂ)) '' closedBall q r)
+    (j : ℝ) (hj : 0 ≤ j) :
+    (stripMean H q r Fbar (unitSlab j))^2 ≤
+      slabMeanCoefficient*(j+2)*totalStripEnergy H q r Fbar := by
+  let A := stripMean H q r Fbar (unitSlab j)
+  let B := stripMean H q r Fbar (unitSlab (-1))
+  let M := stripMean H q r Fbar (longSlab j hj)
+  let E := totalStripEnergy H q r Fbar
+  let K := (2/Real.pi+1)^2*Real.pi
+  have hE : 0 ≤ E := totalStripEnergy_nonneg _ _ _ _
+  have hK : 0 ≤ K := by dsimp [K]; positivity
+  have hEB := stripEnergy_le_total hF hinj hext hH hr q (longSlab j hj) le_rfl le_rfl
+  have hp (a' b' : ℝ) (hab' : a' < b') (haa : -1 ≤ a') (hbb : b' ≤ j+1) :
+      ((∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb a' b').prod (intervalProb 0 Real.pi))-M)^2 ≤ K*(j+2)*E := by
+    have hh := strip_parent_energy hF hinj hext hcont hH hr q
+      (a := -1) (b := j+1) (c := 0) (d := Real.pi) (a' := a') (b' := b') (c' := 0) (d' := Real.pi)
+      (by linarith) Real.pi_pos hab' Real.pi_pos le_rfl le_rfl haa hbb le_rfl le_rfl
+      (by linarith [Real.pi_pos])
+    change (_-M)^2 ≤ rectCoefficient (longSlab j hj)*stripEnergy H q r Fbar (longSlab j hj) at hh
+    exact hh.trans (mul_le_mul (longSlab_coefficient j hj) hEB
+      (integral_nonneg (fun _ => sq_nonneg _)) (by positivity))
+  have hA : (A-M)^2 ≤ K*(j+2)*E := hp j (j+1) (by linarith) (by linarith) le_rfl
+  have hB : (M-B)^2 ≤ K*(j+2)*E := by
+    have hh := hp (-1) 0 (by norm_num) le_rfl (by linarith)
+    have heB : B = ∫ p : ℝ × ℝ, stripPotential H q r Fbar ((p.1:ℂ)+(p.2:ℂ)*Complex.I)
+        ∂(intervalProb (-1) 0).prod (intervalProb 0 Real.pi) := by simp [B,stripMean,unitSlab]
+    rw [← heB] at hh
+    nlinarith
+  have hb : B^2 ≤ Real.pi*E := negative_unitSlab_mean hF hinj hext hcont hH hr q
+    hη hη0 hη1 hηH haccess (-1) le_rfl
+  have hb' : B^2 ≤ Real.pi*(j+2)*E := hb.trans (by nlinarith [mul_nonneg hE (mul_nonneg Real.pi_pos.le (show 0 ≤ j+1 by linarith))])
+  have hsum : A^2 ≤ 3*((A-M)^2+(M-B)^2+B^2) := by
+    nlinarith [sq_nonneg ((A-M)-(M-B)),sq_nonneg ((A-M)-B),sq_nonneg ((M-B)-B)]
+  change A^2 ≤ slabMeanCoefficient*(j+2)*E
+  calc
+    _ ≤ 3*((A-M)^2+(M-B)^2+B^2) := hsum
+    _ ≤ 3*(K*(j+2)*E+K*(j+2)*E+Real.pi*(j+2)*E) := by gcongr
+    _ = _ := by dsimp [K,slabMeanCoefficient]; ring
+
+lemma complex_rectangle_diff_norm (Q : DRect) {x : ℝ × ℝ} (hx : x ∈ Q.closed) :
+    ‖((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)‖ ≤
+      (Q.b-Q.a)+(Q.d-Q.c) := by
+  have hh := Complex.norm_le_abs_re_add_abs_im
+    (((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I))
+  change x.1 ∈ Icc Q.a Q.b ∧ x.2 ∈ Icc Q.c Q.d at hx
+  simp only [Complex.sub_re,Complex.sub_im,Complex.add_re,Complex.add_im,
+    Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,
+    Complex.I_re,Complex.I_im,mul_zero,mul_one,sub_zero,zero_add,add_zero] at hh
+  rw [abs_of_nonneg (sub_nonneg.2 hx.1.1),abs_of_nonneg (sub_nonneg.2 hx.2.1)] at hh
+  linarith [hx.1.2,hx.2.2]
+
+lemma exp_difference_small {z w : ℂ} (hzw : ‖z-w‖ ≤ 1) :
+    ‖Complex.exp z-Complex.exp w‖ ≤ 2*‖z-w‖*Real.exp w.re := by
+  have he : Complex.exp z-Complex.exp w = (Complex.exp (z-w)-1)*Complex.exp w := by
+    rw [sub_mul,one_mul,← Complex.exp_add,sub_add_cancel]
+  rw [he,norm_mul,Complex.norm_exp]
+  exact mul_le_mul_of_nonneg_right (Complex.norm_exp_sub_one_le hzw) (Real.exp_pos _).le
+
+def expCellScale : ℝ := 4*(1+Real.pi)*Real.exp 1
+
+lemma expCellScale_pos : 0 < expCellScale := by unfold expCellScale; positivity
+
+lemma expCellScale_ge : 1+Real.pi ≤ expCellScale := by
+  have he : 1 ≤ Real.exp 1 := Real.one_le_exp_iff.2 (by norm_num)
+  unfold expCellScale
+  nlinarith [Real.pi_pos]
+
+lemma dyadicCell_exp_geometry (j : ℝ) (n : ℕ) (i : DIndex n)
+    (hsmall : expCellScale/(2:ℝ)^n < 1/16) :
+    let Q := dyadicRect (unitSlab j) n i
+    let v := Complex.exp ((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)
+    0 ≤ v.im ∧ Real.exp j/2 ≤ ‖v‖ ∧
+    ∀ x ∈ Q.closed, dist v (Complex.exp ((x.1:ℂ)+(x.2:ℂ)*Complex.I)) <
+      Real.exp j*(expCellScale/(2:ℝ)^n) := by
+  let Q := dyadicRect (unitSlab j) n i
+  have hb := dyadicRect_bounds (unitSlab j) n i
+  change j ≤ Q.a ∧ Q.b ≤ j+1 ∧ 0 ≤ Q.c ∧ Q.d ≤ Real.pi at hb
+  have hl : (Q.b-Q.a)+(Q.d-Q.c) = (1+Real.pi)/(2:ℝ)^n := by
+    rw [dyadicRect_width,dyadicRect_height]
+    simp only [unitSlab]
+    ring
+  have hc : Q.c ≤ Real.pi := Q.hcd.le.trans hb.2.2.2
+  have hre : (((Q.a:ℂ)+(Q.c:ℂ)*Complex.I):ℂ).re = Q.a := by simp
+  have him : (((Q.a:ℂ)+(Q.c:ℂ)*Complex.I):ℂ).im = Q.c := by simp
+  dsimp only
+  change 0 ≤ (Complex.exp ((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)).im ∧ _
+  refine ⟨?_,?_,?_⟩
+  · rw [Complex.exp_im,hre,him]
+    exact mul_nonneg (Real.exp_pos _).le (Real.sin_nonneg_of_mem_Icc ⟨hb.2.2.1,hc⟩)
+  · rw [Complex.norm_exp,hre]
+    have hh := Real.exp_le_exp.mpr hb.1
+    linarith [Real.exp_pos j]
+  · intro x hx
+    have hd := complex_rectangle_diff_norm Q hx
+    rw [hl] at hd
+    have hge : (1+Real.pi)/(2:ℝ)^n ≤ expCellScale/(2:ℝ)^n := by
+      exact div_le_div_of_nonneg_right expCellScale_ge (by positivity)
+    have hdn : ‖((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)‖ ≤ 1 := by
+      linarith [hd.trans hge]
+    rw [dist_comm,dist_eq_norm]
+    have he := exp_difference_small hdn
+    rw [hre] at he
+    have hea : Real.exp Q.a ≤ Real.exp (j+1) := Real.exp_le_exp.mpr (Q.hab.le.trans hb.2.1)
+    calc
+      _ ≤ 2*‖((x.1:ℂ)+(x.2:ℂ)*Complex.I)-((Q.a:ℂ)+(Q.c:ℂ)*Complex.I)‖*Real.exp Q.a := he
+      _ ≤ 2*((1+Real.pi)/(2:ℝ)^n)*Real.exp (j+1) := by gcongr
+      _ < Real.exp j*(expCellScale/(2:ℝ)^n) := by
+        rw [Real.exp_add,expCellScale]
+        have hp : 0 < (1+Real.pi)/(2:ℝ)^n*Real.exp j*Real.exp 1 := by positivity
+        convert (show 2*((1+Real.pi)/(2:ℝ)^n*Real.exp j*Real.exp 1) <
+          4*((1+Real.pi)/(2:ℝ)^n*Real.exp j*Real.exp 1) by linarith) using 1 <;> ring
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.BoundaryCutoff
+
+def clamp (σ ε : ℝ) (w : ℂ) : ℂ := ⟨σ*max (σ*w.re) ε,w.im⟩
+
+def velocity (σ ε u : ℝ) (w : ℂ) : ℂ := 2/clamp σ ε (w-(u:ℂ))
+
+lemma side_sq {σ : ℝ} (hσ : |σ| = 1) : σ*σ = 1 := by
+  have hh := sq_abs σ
+  rw [hσ] at hh
+  nlinarith
+
+lemma clamp_eq {σ ε : ℝ} (hσ : |σ| = 1) {w : ℂ} (hw : ε ≤ σ*w.re) :
+    clamp σ ε w = w := by
+  apply Complex.ext
+  · change σ*max (σ*w.re) ε = w.re
+    rw [max_eq_left hw,← mul_assoc,side_sq hσ,one_mul]
+  · rfl
+
+lemma clamp_norm {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (w : ℂ) : ε ≤ ‖clamp σ ε w‖ := by
+  have hh : ε ≤ |(clamp σ ε w).re| := by
+    change ε ≤ |σ*max (σ*w.re) ε|
+    rw [abs_mul,hσ,one_mul,abs_of_nonneg (hε.le.trans (le_max_right _ _))]
+    exact le_max_right _ _
+  exact hh.trans (Complex.abs_re_le_norm _)
+
+lemma clamp_ne_zero {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (w : ℂ) : clamp σ ε w ≠ 0 :=
+  norm_pos_iff.1 (hε.trans_le (clamp_norm hσ hε w))
+
+lemma clamp_norm_sub {σ ε : ℝ} (hσ : |σ| = 1) (z w : ℂ) :
+    ‖clamp σ ε z-clamp σ ε w‖ ≤ 2*‖z-w‖ := by
+  have hm : |max (σ*z.re) ε-max (σ*w.re) ε| ≤ |σ*z.re-σ*w.re| := by
+    simpa only [sub_self,abs_zero,max_eq_left (abs_nonneg (σ*z.re-σ*w.re))] using
+      abs_max_sub_max_le_max (σ*z.re) ε (σ*w.re) ε
+  have hr : |(clamp σ ε z-clamp σ ε w).re| ≤ ‖z-w‖ := by
+    change |σ*max (σ*z.re) ε-σ*max (σ*w.re) ε| ≤ ‖z-w‖
+    rw [← mul_sub,abs_mul,hσ,one_mul]
+    have he : |σ*z.re-σ*w.re| = |z.re-w.re| := by rw [← mul_sub,abs_mul,hσ,one_mul]
+    rw [he] at hm
+    exact hm.trans (by simpa using Complex.abs_re_le_norm (z-w))
+  have hi : |(clamp σ ε z-clamp σ ε w).im| ≤ ‖z-w‖ := by
+    simpa [clamp] using Complex.abs_im_le_norm (z-w)
+  exact (Complex.norm_le_abs_re_add_abs_im _).trans (by linarith)
+
+lemma velocity_eq {σ ε u : ℝ} (hσ : |σ| = 1) {w : ℂ} (hw : ε ≤ σ*(w.re-u)) :
+    velocity σ ε u w = 2/(w-(u:ℂ)) := by
+  unfold velocity
+  rw [clamp_eq hσ (by simpa using hw)]
+
+lemma norm_velocity_le {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (u : ℝ) (w : ℂ) :
+    ‖velocity σ ε u w‖ ≤ 2/ε := by
+  rw [velocity,norm_div,Complex.norm_ofNat]
+  exact div_le_div_of_nonneg_left (by norm_num) hε (clamp_norm hσ hε _)
+
+end SLEExactGauge.BoundaryCutoff
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/SpatialLimit.lean b/lean/OAI/Probability/SLELocalization/SpatialLimit.lean
index 4dfc159d4465a3ffc09c56141648b420315a8a0d..b835d5e114731f9ed6069d61ebcf87c33df0f7c4 100644
--- a/lean/OAI/Probability/SLELocalization/SpatialLimit.lean
+++ b/lean/OAI/Probability/SLELocalization/SpatialLimit.lean
@@ -1,46 +1,46 @@
-import Mathlib
-import OAI.Probability.SLELocalization.SlabMoments
-
-namespace OAI
-
-
-/-! Conformal localization and Spatial Limit estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.SpatialLimit
-
-lemma canonical_fiber_limit {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
-    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
-    (X : ℕ → Ω × Z → ℝ) (hX : ∀ n, AEMeasurable (X n) (P.prod ν))
-    (Y : Ω × Z → ℝ)
-    (hlim : ∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Tendsto (fun n => X n (ω,z)) atTop (𝓝 (Y (ω,z)))) :
-    ∃ G : Ω × Z → ℝ, Measurable G ∧
-      (∀ᵐ q ∂P.prod ν, Tendsto (fun n => X n q) atTop (𝓝 (G q))) ∧
-      (∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Y (ω,z) = G (ω,z)) := by
-  let F : ℕ → Ω × Z → ℝ := fun n => (hX n).mk (X n)
-  have hF (n : ℕ) : Measurable (F n) := (hX n).measurable_mk
-  let G : Ω × Z → ℝ := fun q => limUnder atTop (fun n => F n q)
-  have hG : Measurable G := (StronglyMeasurable.limUnder (fun n => (hF n).stronglyMeasurable)).measurable
-  have he : ∀ᵐ q ∂P.prod ν, ∀ n, X n q = F n q :=
-    ae_all_iff.2 (fun n => (hX n).ae_eq_mk)
-  have hes := Measure.ae_ae_of_ae_prod (Measure.measurePreserving_swap.quasiMeasurePreserving.ae he)
-  have hGe : ∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Y (ω,z) = G (ω,z) := by
-    filter_upwards [hlim,hes] with z hz hez
-    filter_upwards [hz,hez] with ω hω heω
-    change ∀ n, X n (ω,z) = F n (ω,z) at heω
-    have hf : Tendsto (fun n => F n (ω,z)) atTop (𝓝 (Y (ω,z))) := by
-      simpa only [heω] using hω
-    exact hf.limUnder_eq.symm
-  refine ⟨G,hG,?_,hGe⟩
-  apply ae_prod_tendsto_of_fibers hX hG.aemeasurable
-  filter_upwards [hlim,hGe] with z hz hgz
-  filter_upwards [hz,hgz] with ω hω heω
-  rwa [← heω]
-
-end SLEExactGauge.SpatialLimit
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.SlabMoments
+
+namespace OAI
+
+
+/-! Conformal localization and Spatial Limit estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.SpatialLimit
+
+lemma canonical_fiber_limit {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z]
+    {P : Measure Ω} {ν : Measure Z} [SFinite P] [SFinite ν]
+    (X : ℕ → Ω × Z → ℝ) (hX : ∀ n, AEMeasurable (X n) (P.prod ν))
+    (Y : Ω × Z → ℝ)
+    (hlim : ∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Tendsto (fun n => X n (ω,z)) atTop (𝓝 (Y (ω,z)))) :
+    ∃ G : Ω × Z → ℝ, Measurable G ∧
+      (∀ᵐ q ∂P.prod ν, Tendsto (fun n => X n q) atTop (𝓝 (G q))) ∧
+      (∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Y (ω,z) = G (ω,z)) := by
+  let F : ℕ → Ω × Z → ℝ := fun n => (hX n).mk (X n)
+  have hF (n : ℕ) : Measurable (F n) := (hX n).measurable_mk
+  let G : Ω × Z → ℝ := fun q => limUnder atTop (fun n => F n q)
+  have hG : Measurable G := (StronglyMeasurable.limUnder (fun n => (hF n).stronglyMeasurable)).measurable
+  have he : ∀ᵐ q ∂P.prod ν, ∀ n, X n q = F n q :=
+    ae_all_iff.2 (fun n => (hX n).ae_eq_mk)
+  have hes := Measure.ae_ae_of_ae_prod (Measure.measurePreserving_swap.quasiMeasurePreserving.ae he)
+  have hGe : ∀ᵐ z ∂ν, ∀ᵐ ω ∂P, Y (ω,z) = G (ω,z) := by
+    filter_upwards [hlim,hes] with z hz hez
+    filter_upwards [hz,hez] with ω hω heω
+    change ∀ n, X n (ω,z) = F n (ω,z) at heω
+    have hf : Tendsto (fun n => F n (ω,z)) atTop (𝓝 (Y (ω,z))) := by
+      simpa only [heω] using hω
+    exact hf.limUnder_eq.symm
+  refine ⟨G,hG,?_,hGe⟩
+  apply ae_prod_tendsto_of_fibers hX hG.aemeasurable
+  filter_upwards [hlim,hGe] with z hz hgz
+  filter_upwards [hz,hgz] with ω hω heω
+  rwa [← heω]
+
+end SLEExactGauge.SpatialLimit
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/StoppedIto.lean b/lean/OAI/Probability/SLELocalization/StoppedIto.lean
index a8d74051d4ee324f25d2eef34a0d114540e37198..786e94d9aa67e87be15e0825229bc2914d12313f 100644
--- a/lean/OAI/Probability/SLELocalization/StoppedIto.lean
+++ b/lean/OAI/Probability/SLELocalization/StoppedIto.lean
@@ -1,1187 +1,1187 @@
-import Mathlib
-import OAI.Probability.SLELocalization.StoppedMeshes
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Brownian Stopped estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.BrownianStopped
-open BrownianPredictable PredictableLimit BrownianIto TaylorMeshes BrownianField
-
-theorem stopped_fieldSum_tendsto_ae {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
-    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
-    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
-    (C : ℕ → ℕ → Ω → ℝ)
-    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
-    (hC2 : ∀ n i, MemLp (C n i) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
-    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
-    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω) atTop
-      (𝓝 (F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
-        (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω),
-          iteratedDeriv 2 (F ω t) (B t.toNNReal ω))) := by
-  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T
-    C hC hC2 K₂ hK₂ (fun n i => by
-      filter_upwards [hCeq n i, hbound n i] with ω he hb
-      simpa only [he] using hb)
-  have hCeq_all : ∀ᵐ ω ∂P, ∀ n i, C n i ω = stoppedCoeff 2 B T τ F n i ω :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hCeq n))
-  have hC_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂ :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hbound n))
-  have hA_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁ :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAbound n))
-  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hτ, hC_all, hA_all, hCeq_all]
-    with ω hb hQ hE hr hτω hCω hAω hCeqω
-  have hEs : Tendsto (fun n => dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n ω) atTop (𝓝 0) := by
-    convert hE using 1
-    funext n
-    unfold dyadicQuadraticError
-    simp only [hCeqω]
-  let b : ℝ → ℝ := fun t => B t.toNNReal ω
-  have hbc : Continuous b := hb.comp continuous_real_toNNReal
-  have hQ' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
-    simpa only [quadraticSum, b, mesh_toNNReal] using hQ
-  have hE' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
-      iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-        ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0)
-        atTop (𝓝 0) := by
-    have heq (n : ℕ) : dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n ω =
-        ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
-          iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0 := by
-      unfold dyadicQuadraticError
-      apply Finset.sum_congr rfl
-      intro i hi
-      simp only [stoppedCoeff, b, mesh_toNNReal]
-      split_ifs <;> simp
-    simpa only [heq] using hEs
-  have hA' (n i : ℕ) (hi : mesh T n i < τ ω) :
-      |deriv (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₁ := by
-    simpa only [stoppedCoeff, ite_eq_left hi, iteratedDeriv_succ, iteratedDeriv_zero,
-      b, mesh_toNNReal] using hAω n i
-  have hC' (n i : ℕ) (hi : mesh T n i < τ ω) :
-      |iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₂ := by
-    simpa only [stoppedCoeff, ite_eq_left hi, b, mesh_toNNReal] using hCω n i
-  have ht := stopped_field_taylor_limit T.coe_nonneg hτω.1 hτω.2 hbc.continuousOn
-    hr hQ' hK₁ hK₂ hA' hC' hE'
-  have heq (n : ℕ) : dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω =
-      ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
-        deriv (F ω (mesh T n i)) (b (mesh T n i)) *
-          (b (mesh T n (i+1))-b (mesh T n i)) else 0 := by
-    unfold dyadicDriverSum
-    apply Finset.sum_congr rfl
-    intro i hi
-    simp only [stoppedCoeff, b, mesh_toNNReal, iteratedDeriv_succ, iteratedDeriv_zero]
-    split_ifs <;> simp
-  simpa only [← heq, b, Real.toNNReal_zero] using ht
-
-theorem stopped_field_ito_centered_ae {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
-    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
-    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
-    (A C : ℕ → ℕ → Ω → ℝ)
-    (hAeq : ∀ n i, A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i)
-    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
-    (hC2 : ∀ n i, MemLp (C n i) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
-    (hA2 : ∀ n i, MemLp (A n i) 2 P)
-    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
-    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
-    let Y := fun ω => F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
-      (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
-      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω), iteratedDeriv 2 (F ω t) (B t.toNNReal ω)
-    Integrable Y P ∧ (∫ ω, Y ω ∂P) = 0 := by
-  have ht := stopped_fieldSum_tendsto_ae hB T τ F V hτ hreg C hCeq hC hC2
-    K₂ hK₂ hbound K₁ hK₁ hAbound
-  have hAeq_all : ∀ᵐ ω ∂P, ∀ n i, A n i ω = stoppedCoeff 1 B T τ F n i ω :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAeq n))
-  apply limit_centered hB T A hA hA2 K₁ hK₁ ?_ _ ?_
-  · intro n i
-    filter_upwards [hAeq n i, hAbound n i] with ω he hb
-    simpa only [he] using hb
-  · filter_upwards [ht, hAeq_all] with ω hω he
-    convert hω using 1
-    funext n
-    unfold dyadicDriverSum
-    simp only [he]
-
-end SLEExactGauge.BrownianStopped
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.LoewnerCutoff
-
-def clampIm (ε : ℝ) (w : ℂ) : ℂ := ⟨w.re, max w.im ε⟩
-
-def velocity (ε u : ℝ) (w : ℂ) : ℂ := 2 / clampIm ε (w - (u : ℂ))
-
-lemma clampIm_eq {ε : ℝ} {w : ℂ} (hw : ε ≤ w.im) : clampIm ε w = w := by
-  apply Complex.ext <;> simp [clampIm, max_eq_left hw]
-
-lemma clampIm_height (ε : ℝ) (w : ℂ) : ε ≤ (clampIm ε w).im := le_max_right _ _
-
-lemma clampIm_norm {ε : ℝ} (_hε : 0 < ε) (w : ℂ) : ε ≤ ‖clampIm ε w‖ :=
-  (clampIm_height ε w).trans (Complex.im_le_norm _)
-
-lemma clampIm_ne_zero {ε : ℝ} (hε : 0 < ε) (w : ℂ) : clampIm ε w ≠ 0 :=
-  norm_pos_iff.1 (hε.trans_le (clampIm_norm hε w))
-
-lemma clampIm_norm_sub (ε : ℝ) (z w : ℂ) :
-    ‖clampIm ε z - clampIm ε w‖ ≤ 2 * ‖z - w‖ := by
-  have hm : |max z.im ε - max w.im ε| ≤ |z.im - w.im| := by
-    simpa only [sub_self, abs_zero, max_eq_left (abs_nonneg (z.im - w.im))] using
-      abs_max_sub_max_le_max z.im ε w.im ε
-  calc
-    ‖clampIm ε z - clampIm ε w‖ ≤ |z.re-w.re| + |max z.im ε-max w.im ε| :=
-      Complex.norm_le_abs_re_add_abs_im _
-    _ ≤ |z.re-w.re| + |z.im-w.im| := add_le_add le_rfl hm
-    _ ≤ ‖z-w‖ + ‖z-w‖ := add_le_add (Complex.abs_re_le_norm (z-w))
-      (Complex.abs_im_le_norm (z-w))
-    _ = 2 * ‖z-w‖ := by ring
-
-lemma velocity_eq {ε u : ℝ} {w : ℂ} (hw : ε ≤ w.im) :
-    velocity ε u w = 2 / (w - (u : ℂ)) := by
-  unfold velocity
-  rw [clampIm_eq (by simpa using hw)]
-
-lemma norm_velocity_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (w : ℂ) :
-    ‖velocity ε u w‖ ≤ 2 / ε := by
-  rw [velocity, norm_div, Complex.norm_ofNat]
-  exact div_le_div_of_nonneg_left (by norm_num) hε (clampIm_norm hε _)
-
-lemma norm_velocity_sub_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (z w : ℂ) :
-    ‖velocity ε u z - velocity ε u w‖ ≤ (4 / ε^2) * ‖z-w‖ := by
-  let a := clampIm ε (z - (u : ℂ))
-  let b := clampIm ε (w - (u : ℂ))
-  have ha : a ≠ 0 := clampIm_ne_zero hε _
-  have hb : b ≠ 0 := clampIm_ne_zero hε _
-  have hna := clampIm_norm hε (z - (u : ℂ))
-  have hnb := clampIm_norm hε (w - (u : ℂ))
-  have hd : ε^2 ≤ ‖a*b‖ := by
-    rw [norm_mul, pow_two]
-    exact mul_le_mul hna hnb hε.le (norm_nonneg _)
-  have he : velocity ε u z - velocity ε u w = 2 * (b-a) / (a*b) := by
-    dsimp [velocity, a, b]
-    field_simp [clampIm_ne_zero hε]
-  rw [he, norm_div, norm_mul, Complex.norm_ofNat]
-  have hn : ‖b-a‖ ≤ 2 * ‖z-w‖ := by
-    simpa only [a, b, norm_sub_rev, sub_sub_sub_cancel_right] using
-      clampIm_norm_sub ε (z-(u:ℂ)) (w-(u:ℂ))
-  calc
-    2 * ‖b-a‖ / ‖a*b‖ ≤ (2 * (2 * ‖z-w‖)) / ε^2 :=
-      div_le_div₀ (by positivity) (mul_le_mul_of_nonneg_left hn (by norm_num))
-        (sq_pos_of_pos hε) hd
-    _ = (4 / ε^2) * ‖z-w‖ := by ring
-
-lemma velocity_lipschitz {ε : ℝ} (hε : 0 < ε) (u : ℝ) :
-    LipschitzWith ⟨4 / ε^2, by positivity⟩ (velocity ε u) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  simp only [dist_eq_norm]
-  exact norm_velocity_sub_le hε u z w
-
-lemma continuous_velocity {ε : ℝ} (hε : 0 < ε) :
-    Continuous (fun p : ℝ × ℂ => velocity ε p.1 p.2) := by
-  unfold velocity clampIm
-  refine continuous_const.div ?_ (fun p => clampIm_ne_zero hε _)
-  change Continuous (fun p : ℝ × ℂ => Complex.equivRealProdCLM.symm
-    ((p.2 - (p.1 : ℂ)).re, max (p.2 - (p.1 : ℂ)).im ε))
-  apply Complex.equivRealProdCLM.symm.continuous.comp
-  apply Continuous.prodMk <;> fun_prop
-
-lemma exists_solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
-    ∃ H : ℝ → ℂ, H 0 = z ∧ ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t := by
-  let L : ℝ≥0 := ⟨2 / ε, by positivity⟩
-  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
-  let a : ℝ≥0 := ⟨(2 / ε) * T, mul_nonneg (by positivity) hT⟩
-  have hp : IsPicardLindelof (fun t => velocity ε (U t))
-      ⟨0, ⟨le_rfl, hT⟩⟩ z a 0 L K := by
-    constructor
-    · exact fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith
-    · intro w _
-      have hc : ContinuousOn (fun t : ℝ => (U t, w)) (Icc 0 T) :=
-        hU.prodMk continuousOn_const
-      simpa only [Function.comp_def] using
-        (continuous_velocity hε).comp_continuousOn hc
-    · exact fun t _ w _ => norm_velocity_le hε (U t) w
-    · dsimp [L, a]
-      simp only [sub_zero, sub_self, max_eq_left hT]
-      rfl
-  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
-
-lemma norm_velocity_driver_sub_le {ε : ℝ} (hε : 0 < ε) (u v : ℝ) (z : ℂ) :
-    ‖velocity ε u z - velocity ε v z‖ ≤ (4 / ε^2) * |u-v| := by
-  have he (a : ℝ) : velocity ε 0 (z - (a : ℂ)) = velocity ε a z := by
-    simp [velocity]
-  have hh := norm_velocity_sub_le hε 0 (z-(u:ℂ)) (z-(v:ℂ))
-  rw [he, he] at hh
-  convert hh using 1
-  rw [sub_sub_sub_cancel_left, ← Complex.ofReal_sub, Complex.norm_real, abs_sub_comm]
-  simp [Real.norm_eq_abs]
-
-lemma derivWithin_right {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ico 0 T) :
-    HasDerivWithinAt H (velocity ε (U t) (H t)) (Ici t) t :=
-  (hH t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
-
-lemma derivWithin_left {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ioc 0 T) :
-    HasDerivWithinAt H (velocity ε (U t) (H t)) (Iic t) t :=
-  (hH t ⟨ht.1.le, ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
-
-lemma solution_eq {ε T : ℝ} (hε : 0 < ε) {U : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (U t) (G t)) (Icc 0 T) t)
-    (h0 : H 0 = G 0) : EqOn H G (Icc 0 T) := by
-  apply ODE_solution_unique_of_mem_Icc_right
-    (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith)
-    (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_right hH ht)
-    (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hG)
-    (fun _ ht => derivWithin_right hG ht) (fun _ _ => mem_univ _) h0
-
-lemma solution_dist_le {ε T D : ℝ} (hε : 0 < ε) (hD : 0 ≤ D)
-    {U V : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
-    (h0 : H 0 = G 0) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    dist (H t) (G t) ≤ D * Real.exp ((4 / ε^2) * T) := by
-  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
-  have hK : (0 : ℝ) < K := by change 0 < 4 / ε^2; positivity
-  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
-      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K : ℝ) * D := by
-    rw [dist_eq_norm]
-    calc
-      _ ≤ (4 / ε^2) * |V s-U s| := norm_velocity_driver_sub_le hε _ _ _
-      _ ≤ (K : ℝ) * D := by
-        rw [abs_sub_comm]
-        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1, hs.2.le⟩) hK.le
-  have hh := dist_le_of_approx_trajectories_ODE
-    (fun t => velocity_lipschitz hε (U t))
-    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
-    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
-    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
-    hb (le_of_eq (dist_eq_zero.2 h0)) t ht
-  have he : gronwallBound 0 K (0 + (K : ℝ)*D) (t-0) =
-      D * (Real.exp ((K : ℝ)*t)-1) := by
-    simp only [gronwallBound, zero_add, sub_zero, ite_eq_right (ne_of_gt hK)]
-    field_simp
-    ring
-  rw [he] at hh
-  have hexp : Real.exp ((K : ℝ)*t) ≤ Real.exp ((K : ℝ)*T) :=
-    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
-  apply hh.trans
-  change D * (Real.exp ((4 / ε^2)*t)-1) ≤ D * Real.exp ((4 / ε^2)*T)
-  change Real.exp ((4 / ε^2)*t) ≤ Real.exp ((4 / ε^2)*T) at hexp
-  exact mul_le_mul_of_nonneg_left (by linarith) hD
-
-end SLEExactGauge.LoewnerCutoff
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.LoewnerCutoff
-open scoped unitInterval
-
-def extendDriver (T : ℝ) (U : C(I, ℝ)) (t : ℝ) : ℝ :=
-  U (projIcc 0 1 zero_le_one (t / T))
-
-lemma continuous_extendDriver (T : ℝ) (U : C(I, ℝ)) : Continuous (extendDriver T U) :=
-  U.continuous.comp (continuous_projIcc.comp (continuous_id.div_const T))
-
-def solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) : ℝ → ℂ :=
-  (exists_solution hε hT (extendDriver T U) (continuous_extendDriver T U).continuousOn z).choose
-
-lemma solution_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    solution hε hT U z 0 = z :=
-  (exists_solution hε hT (extendDriver T U)
-    (continuous_extendDriver T U).continuousOn z).choose_spec.1
-
-lemma solution_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (solution hε hT U z)
-      (velocity ε (extendDriver T U t) (solution hε hT U z t)) (Icc 0 T) t :=
-  (exists_solution hε hT (extendDriver T U)
-    (continuous_extendDriver T U).continuousOn z).choose_spec.2
-
-def solutionPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
-    C(Icc 0 T, ℂ) :=
-  ⟨fun t => solution hε hT U z t,
-    continuousOn_iff_continuous_domRestrict.1 (HasDerivWithinAt.continuousOn (solution_deriv hε hT U z))⟩
-
-lemma solutionPath_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    LipschitzWith ⟨Real.exp ((4 / ε^2) * T), (Real.exp_pos _).le⟩
-      (fun U : C(I, ℝ) => solutionPath hε hT U z) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro U V
-  apply (ContinuousMap.dist_le (mul_nonneg (Real.exp_pos _).le dist_nonneg)).2
-  intro t
-  have hUV (s : ℝ) (_hs : s ∈ Icc 0 T) :
-      |extendDriver T U s-extendDriver T V s| ≤ dist U V := by
-    exact (ContinuousMap.dist_apply_le_dist (f := U) (g := V)
-      (projIcc 0 1 zero_le_one (s/T)))
-  have hh := solution_dist_le hε dist_nonneg (solution_deriv hε hT U z)
-    (solution_deriv hε hT V z)
-    ((solution_initial hε hT U z).trans (solution_initial hε hT V z).symm)
-    hUV t.property
-  change dist (solution hε hT U z t) (solution hε hT V z t) ≤
-    Real.exp ((4 / ε^2)*T) * dist U V
-  simpa only [mul_comm] using hh
-
-end SLEExactGauge.LoewnerCutoff
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ContinuousDriver
-section Limit
-variable {Ω Y : Type*} [MeasurableSpace Ω] [MetricSpace Y] [CompleteSpace Y]
-
-def selectedLimit (y₀ : Y) (a : ℕ → Y) : Y := by
-  classical
-  exact if h : CauchySeq a then (cauchySeq_tendsto_of_complete h).choose else y₀
-
-lemma tendsto_selectedLimit (y₀ : Y) {a : ℕ → Y} (h : CauchySeq a) :
-    Tendsto a atTop (𝓝 (selectedLimit y₀ a)) := by
-  rw [selectedLimit, dite_eq_left h]
-  exact (cauchySeq_tendsto_of_complete h).choose_spec
-
-lemma selectedLimit_eq (y₀ : Y) {a : ℕ → Y} {y : Y} (h : Tendsto a atTop (𝓝 y)) :
-    selectedLimit y₀ a = y :=
-  tendsto_nhds_unique (tendsto_selectedLimit y₀ h.cauchySeq) h
-
-end Limit
-
-end SLEExactGauge.ContinuousDriver
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ContinuousDriver
-section Limit
-variable {Ω Y : Type*} [MeasurableSpace Ω] [MetricSpace Y] [completeY : CompleteSpace Y]
-variable [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y]
-
-omit completeY in
-lemma measurableSet_cauchy [CompleteSpace Y] (a : ℕ → Ω → Y) (ha : ∀ n, Measurable (a n)) :
-    MeasurableSet {ω | CauchySeq (fun n => a n ω)} := by
-  simp only [Metric.uniformity_basis_dist_inv_nat_succ.cauchySeq_iff, forall_const,
-    mem_ofPred_eq, ofPred_forall, ofPred_exists]
-  exact MeasurableSet.iInter fun (k : ℕ) => MeasurableSet.iUnion fun (N : ℕ) =>
-    MeasurableSet.iInter fun (m : ℕ) => MeasurableSet.iInter fun (_ : N ≤ m) =>
-      MeasurableSet.iInter fun (n : ℕ) => MeasurableSet.iInter fun (_ : N ≤ n) =>
-        measurableSet_lt ((ha m).dist (ha n)) measurable_const
-
-lemma measurable_selectedLimit (y₀ : Y) (a : ℕ → Ω → Y) (ha : ∀ n, Measurable (a n)) :
-    Measurable (fun ω => selectedLimit y₀ (fun n => a n ω)) := by
-  classical
-  have hc := measurableSet_cauchy a ha
-  apply measurable_of_tendsto_metrizable
-    (f := fun n ω => if CauchySeq (fun m => a m ω) then a n ω else y₀)
-    (fun n => (ha n).ite hc measurable_const)
-  apply tendsto_pi_nhds.2
-  intro ω
-  by_cases hω : CauchySeq (fun n => a n ω)
-  · simpa only [ite_eq_left hω] using tendsto_selectedLimit y₀ hω
-  · simp only [ite_eq_right hω, selectedLimit, dite_eq_right hω]
-    exact tendsto_const_nhds
-
-end Limit
-
-end SLEExactGauge.ContinuousDriver
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.ContinuousDriver
-
-def rawApprox (x : I → ℝ) (n : ℕ) : C(I, ℝ) :=
-  ∑ k : Fin (n+1), (x (bernstein.z k)) • bernstein n k
-
-lemma rawApprox_eq (f : C(I, ℝ)) (n : ℕ) : rawApprox f n = bernsteinApproximation n f := by
-  apply ContinuousMap.ext
-  intro t
-  simp [rawApprox, bernsteinApproximation, smul_eq_mul, mul_comm]
-
-lemma measurable_rawApprox {Ω : Type*} [MeasurableSpace Ω]
-    (x : Ω → I → ℝ) (hx : ∀ t, Measurable (fun ω => x ω t)) (n : ℕ) :
-    @Measurable Ω C(I, ℝ) _ (borel C(I, ℝ)) (fun ω => rawApprox (x ω) n) := by
-  let : MeasurableSpace C(I, ℝ) := borel C(I, ℝ)
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  apply Finset.measurable_sum
-  intro k _
-  exact (hx (bernstein.z k)).smul measurable_const
-
-def path (x : I → ℝ) : C(I, ℝ) := selectedLimit 0 (rawApprox x)
-
-lemma path_eq (f : C(I, ℝ)) : path f = f := by
-  apply selectedLimit_eq
-  have he : rawApprox f = fun n => bernsteinApproximation n f :=
-    funext (rawApprox_eq f)
-  rw [he]
-  exact bernsteinApproximation_uniform f
-
-lemma path_eval_of_continuous {x : I → ℝ} (hx : Continuous x) (t : I) : path x t = x t := by
-  have hh := path_eq (⟨x, hx⟩ : C(I, ℝ))
-  exact congrArg (fun f : C(I, ℝ) => f t) hh
-
-lemma measurable_path {Ω : Type*} [MeasurableSpace Ω]
-    (x : Ω → I → ℝ) (hx : ∀ t, Measurable (fun ω => x ω t)) :
-    @Measurable Ω C(I, ℝ) _ (borel C(I, ℝ)) (fun ω => path (x ω)) := by
-  let : MeasurableSpace C(I, ℝ) := borel C(I, ℝ)
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  exact measurable_selectedLimit 0 _ (measurable_rawApprox x hx)
-
-end SLEExactGauge.ContinuousDriver
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.LoewnerCutoff
-open scoped unitInterval
-
-def position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℂ :=
-  solutionPath hε hT U z (projIcc 0 T hT t)
-
-lemma position_eq {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    position hε hT U z t = solution hε hT U z t := by
-  simp only [position, solutionPath, projIcc, max_eq_right ht.1,
-    min_eq_right ht.2]
-  rfl
-
-lemma continuous_position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => position hε hT p.1 z p.2) := by
-  exact continuous_eval.comp (((solutionPath_lipschitz hε hT z).continuous.comp continuous_fst).prodMk
-    ((continuous_projIcc (h := hT)).comp continuous_snd))
-
-lemma continuous_joint_extendDriver (T : ℝ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => extendDriver T p.1 p.2) := by
-  exact continuous_eval.comp (continuous_fst.prodMk
-    ((continuous_projIcc (h := zero_le_one)).comp (continuous_snd.div_const T)))
-
-def variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  (-2 / (clampIm ε (position hε hT U z t - (extendDriver T U t : ℂ)))^2).re
-
-lemma continuous_clampIm (ε : ℝ) : Continuous (clampIm ε) := by
-  change Continuous (fun w : ℂ => Complex.equivRealProdCLM.symm (w.re,max w.im ε))
-  apply Complex.equivRealProdCLM.symm.continuous.comp
-  apply Continuous.prodMk <;> fun_prop
-
-lemma continuous_variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => variation hε hT p.1 z p.2) := by
-  have hw := (continuous_position hε hT z).sub
-    (Complex.continuous_ofReal.comp (continuous_joint_extendDriver T))
-  exact Complex.continuous_re.comp (continuous_const.div
-    (((continuous_clampIm ε).comp hw).pow 2)
-    (fun p => pow_ne_zero 2 (clampIm_ne_zero hε _)))
-
-def logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  ∫ s in (0 : ℝ)..(projIcc 0 T hT t : ℝ), variation hε hT U z s
-
-lemma continuous_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
-    Continuous (fun p : C(I, ℝ) × ℝ => logDerivative hε hT p.1 z p.2) := by
-  have hc := intervalIntegral.continuous_parametric_primitive_of_continuous
-    (μ := volume) (a₀ := 0) (f := fun U t => variation hε hT U z t)
-    (continuous_variation hε hT z)
-  exact hc.comp (continuous_fst.prodMk
-    (continuous_subtype_val.comp ((continuous_projIcc (h := hT)).comp continuous_snd)))
-
-lemma abs_variation_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    |variation hε hT U z t| ≤ 2/ε^2 := by
-  apply (Complex.abs_re_le_norm _).trans
-  rw [norm_div, norm_neg, Complex.norm_ofNat, norm_pow]
-  apply div_le_div_of_nonneg_left (by norm_num) (sq_pos_of_pos hε)
-  exact pow_le_pow_left₀ hε.le (clampIm_norm hε _) 2
-
-lemma logDerivative_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) : logDerivative hε hT U z 0 = 0 := by
-  simp only [logDerivative, projIcc_of_mem hT (left_mem_Icc.mpr hT), intervalIntegral.integral_same]
-
-lemma abs_logDerivative_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    |logDerivative hε hT U z t| ≤ (2/ε^2)*T := by
-  have hb := (projIcc 0 T hT t).property
-  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
-    (a := (0:ℝ)) (b := (projIcc 0 T hT t : ℝ))
-    (fun s (_hs : s ∈ uIoc 0 (projIcc 0 T hT t : ℝ)) =>
-      show ‖variation hε hT U z s‖ ≤ 2/ε^2 from abs_variation_le hε hT U z s)
-  change |logDerivative hε hT U z t| ≤ _
-  apply hi.trans
-  rw [sub_zero, abs_of_nonneg hb.1]
-  exact mul_le_mul_of_nonneg_left hb.2 (by positivity)
-
-lemma position_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) : position hε hT U z 0 = z := by
-  rw [position_eq hε hT U z ⟨le_rfl,hT⟩, solution_initial]
-
-lemma position_derivWithin {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    HasDerivWithinAt (position hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)) (Icc 0 T) t := by
-  rw [position_eq hε hT U z ht]
-  exact (solution_deriv hε hT U z t ht).congr
-    (fun s hs => position_eq hε hT U z hs) (position_eq hε hT U z ht)
-
-lemma position_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
-    HasDerivAt (position hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)) t :=
-  (position_derivWithin hε hT U z ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2)
-
-lemma logDerivative_eq_integral {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
-    logDerivative hε hT U z t = ∫ s in (0:ℝ)..t, variation hε hT U z s := by
-  simp only [logDerivative, projIcc_of_mem hT ht]
-
-lemma hasDerivAt_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
-    HasDerivAt (logDerivative hε hT U z) (variation hε hT U z t) t := by
-  have hp : Continuous (fun t : ℝ => (U,t)) := continuous_const.prodMk continuous_id
-  have hc := (continuous_variation hε hT z).comp hp
-  dsimp only [Function.comp_def] at hc
-  have hd : HasDerivAt (fun v : ℝ => ∫ s in (0:ℝ)..v, variation hε hT U z s)
-      (variation hε hT U z t) t :=
-    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
-      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
-  apply hd.congr_of_eventuallyEq
-  filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
-  exact logDerivative_eq_integral hε hT U z hs
-
-lemma position_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
-    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (_hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
-    (U V : C(I, ℝ)) (z : ℂ)
-    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
-    EqOn (position hε hT₁ U z) (position hε hT₂ V z) (Icc 0 S) := by
-  apply solution_eq hε (U := extendDriver T₁ U) ?_ ?_ ((position_initial hε hT₁ U z).trans (position_initial hε hT₂ V z).symm)
-  · intro t ht
-    exact (position_derivWithin hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩).mono
-      (Icc_subset_Icc le_rfl hS₁)
-  · intro t ht
-    rw [hUV ht]
-    exact (position_derivWithin hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩).mono
-      (Icc_subset_Icc le_rfl hS₂)
-
-lemma logDerivative_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
-    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
-    (U V : C(I, ℝ)) (z : ℂ)
-    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
-    EqOn (logDerivative hε hT₁ U z) (logDerivative hε hT₂ V z) (Icc 0 S) := by
-  have hH := position_prefix_eq hε hT₁ hT₂ hS hS₁ hS₂ U V z hUV
-  intro t ht
-  rw [logDerivative_eq_integral hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩,
-    logDerivative_eq_integral hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩]
-  apply intervalIntegral.integral_congr
-  intro s hs
-  rw [uIcc_of_le ht.1] at hs
-  have hsS : s ∈ Icc 0 S := ⟨hs.1,hs.2.trans ht.2⟩
-  simp only [variation, hH hsS, hUV hsS]
-
-end SLEExactGauge.LoewnerCutoff
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-namespace SLEExactGauge.GreenPhysical
-
-def green (p a x y l : ℝ) : ℝ :=
-  Real.exp (p*l - p*Real.log 2 + (a-p)*Real.log y - a/2*Real.log (x^2+y^2))
-
-lemma q_pos (x : ℝ) {y : ℝ} (hy : 0 < y) : 0 < x^2+y^2 := by positivity
-
-lemma green_eq_source (p a x y l : ℝ) (hy : 0 < y) :
-    green p a x y l = (2*y / Real.exp l)^(-p) * (y / Real.sqrt (x^2+y^2))^a := by
-  have hQ := q_pos x hy
-  have hR : 0 < 2*y / Real.exp l := div_pos (by positivity) (Real.exp_pos _)
-  have hS : 0 < y / Real.sqrt (x^2+y^2) := div_pos hy (Real.sqrt_pos.2 hQ)
-  rw [Real.rpow_def_of_pos hR, Real.rpow_def_of_pos hS, ← Real.exp_add]
-  dsimp [green]
-  congr 1
-  rw [Real.log_div (by positivity : 2*y ≠ 0) (Real.exp_ne_zero _),
-    Real.log_mul (by norm_num : (2 : ℝ) ≠ 0) hy.ne', Real.log_exp,
-    Real.log_div hy.ne' (by positivity : Real.sqrt (x^2+y^2) ≠ 0),
-    Real.log_sqrt hQ.le]
-  ring
-
-lemma green_pos (p a x y l : ℝ) : 0 < green p a x y l := Real.exp_pos _
-
-lemma hasDerivAt_green_path (p a : ℝ) {X Y L : ℝ → ℝ} {t X' Y' L' : ℝ}
-    (hX : HasDerivAt X X' t) (hY : HasDerivAt Y Y' t) (hL : HasDerivAt L L' t)
-    (hy : 0 < Y t) :
-    HasDerivAt (fun u => green p a (X u) (Y u) (L u))
-      (green p a (X t) (Y t) (L t) *
-        (p*L' + (a-p)*Y'/Y t - a*(X t*X' + Y t*Y')/(X t^2+Y t^2))) t := by
-  have hlogY := hY.log hy.ne'
-  have hlogQ := ((hX.pow 2).add (hY.pow 2)).log (q_pos (X t) hy).ne'
-  have hd := ((((hL.const_mul p).sub_const (p*Real.log 2)).add
-    (hlogY.const_mul (a-p))).sub (hlogQ.const_mul (a/2))).exp
-  apply hd.congr_deriv
-  dsimp [green]
-  ring
-
-lemma hasDerivAt_green_x (p a x y l : ℝ) (hy : 0 < y) :
-    HasDerivAt (fun u => green p a u y l) (green p a x y l * (-a*x/(x^2+y^2))) x := by
-  convert hasDerivAt_green_path p a (hasDerivAt_id x) (hasDerivAt_const x y)
-    (hasDerivAt_const x l) hy using 1 <;> dsimp only [id_eq]; ring
-
-lemma deriv_green_x (p a x y l : ℝ) (hy : 0 < y) :
-    deriv (fun u => green p a u y l) x = green p a x y l * (-a*x/(x^2+y^2)) :=
-  (hasDerivAt_green_x p a x y l hy).deriv
-
-lemma hasDerivAt_neg_ax_div_q (a x y : ℝ) (hy : 0 < y) :
-    HasDerivAt (fun u => -a*u/(u^2+y^2))
-      (a*(x^2-y^2)/(x^2+y^2)^2) x := by
-  have hd := ((hasDerivAt_id x).const_mul (-a)).div
-    (((hasDerivAt_id x).pow 2).add_const (y^2)) (q_pos x hy).ne'
-  apply hd.congr_deriv
-  simp only [id_eq, Pi.pow_apply]
-  norm_num
-  ring
-
-lemma deriv_deriv_green_x (p a x y l : ℝ) (hy : 0 < y) :
-    deriv (deriv (fun u => green p a u y l)) x =
-      green p a x y l * (a*(a+1)*x^2-a*y^2)/(x^2+y^2)^2 := by
-  have heq : deriv (fun u => green p a u y l) =
-      (fun u => green p a u y l * (-a*u/(u^2+y^2))) :=
-    funext (fun u => deriv_green_x p a u y l hy)
-  rw [heq]
-  have hd := (hasDerivAt_green_x p a x y l hy).mul (hasDerivAt_neg_ax_div_q a x y hy)
-  calc
-    _ = _ := hd.deriv
-    _ = _ := by
-      field_simp
-      ring
-
-lemma green_drift_cancellation (κ x y M : ℝ) (hκ : κ ≠ 0) (hy : 0 < y) :
-    let p := 1-κ/8
-    let a := 8/κ-1
-    M * (p * (-2*(x^2-y^2)/(x^2+y^2)^2) +
-      (a-p) * (-2*y/(x^2+y^2))/y -
-      a * (x*(2*x/(x^2+y^2)) + y*(-2*y/(x^2+y^2)))/(x^2+y^2)) +
-    κ/2 * (M*(a*(a+1)*x^2-a*y^2)/(x^2+y^2)^2) = 0 := by
-  dsimp only
-  field_simp
-  ring
-
-lemma loewner_green_drift {κ t : ℝ} (hκ : 0 < κ) {X Y L : ℝ → ℝ}
-    (hy : 0 < Y t)
-    (hX : HasDerivAt X (2*X t/(X t^2+Y t^2)) t)
-    (hY : HasDerivAt Y (-2*Y t/(X t^2+Y t^2)) t)
-    (hL : HasDerivAt L (-2*(X t^2-Y t^2)/(X t^2+Y t^2)^2) t) :
-    HasDerivAt (fun u => green (1-κ/8) (8/κ-1) (X u) (Y u) (L u))
-      (-(κ/2) * deriv (deriv (fun x => green (1-κ/8) (8/κ-1) x (Y t) (L t))) (X t)) t := by
-  have hd := hasDerivAt_green_path (1-κ/8) (8/κ-1) hX hY hL hy
-  apply hd.congr_deriv
-  rw [deriv_deriv_green_x _ _ _ _ _ hy]
-  have hc := green_drift_cancellation κ (X t) (Y t)
-    (green (1-κ/8) (8/κ-1) (X t) (Y t) (L t)) hκ.ne' hy
-  dsimp only at hc
-  linarith
-
-lemma hasDerivAt_green_driver (p a c y l sdev b : ℝ) (hy : 0 < y) :
-    HasDerivAt (fun v => green p a (c-sdev*v) y l)
-      (sdev*a*(c-sdev*b)/((c-sdev*b)^2+y^2) * green p a (c-sdev*b) y l) b := by
-  have hd := (hasDerivAt_green_x p a (c-sdev*b) y l hy).comp b
-    (((hasDerivAt_id b).const_mul sdev).const_sub c)
-  apply hd.congr_deriv
-  ring
-
-lemma deriv_green_driver (p a c y l sdev b : ℝ) (hy : 0 < y) :
-    deriv (fun v => green p a (c-sdev*v) y l) b =
-      sdev*a*(c-sdev*b)/((c-sdev*b)^2+y^2) * green p a (c-sdev*b) y l :=
-  (hasDerivAt_green_driver p a c y l sdev b hy).deriv
-
-lemma hasDerivAt_deriv_green_x (p a x y l : ℝ) (hy : 0 < y) :
-    HasDerivAt (deriv (fun u => green p a u y l))
-      (green p a x y l * (a*(a+1)*x^2-a*y^2)/(x^2+y^2)^2) x := by
-  have heq : deriv (fun u => green p a u y l) =
-      (fun u => green p a u y l * (-a*u/(u^2+y^2))) :=
-    funext (fun u => deriv_green_x p a u y l hy)
-  rw [heq]
-  apply ((hasDerivAt_green_x p a x y l hy).mul (hasDerivAt_neg_ax_div_q a x y hy)).congr_deriv
-  field_simp
-  ring
-
-lemma deriv_deriv_green_driver (p a c y l sdev b : ℝ) (hy : 0 < y) :
-    deriv (deriv (fun v => green p a (c-sdev*v) y l)) b =
-      sdev^2 * (green p a (c-sdev*b) y l *
-        (a*(a+1)*(c-sdev*b)^2-a*y^2)/((c-sdev*b)^2+y^2)^2) := by
-  have heq : deriv (fun v => green p a (c-sdev*v) y l) =
-      fun v => (-sdev) * deriv (fun u => green p a u y l) (c-sdev*v) := by
-    ext v
-    rw [deriv_green_driver _ _ _ _ _ _ _ hy, deriv_green_x _ _ _ _ _ hy]
-    ring
-  rw [heq]
-  have hd := ((hasDerivAt_deriv_green_x p a (c-sdev*b) y l hy).comp b
-    (((hasDerivAt_id b).const_mul sdev).const_sub c)).const_mul (-sdev)
-  calc
-    _ = _ := hd.deriv
-    _ = _ := by ring
-
-lemma green_le_radius (p a x y l : ℝ) (hy : 0 < y) (ha : 0 ≤ a) :
-    green p a x y l ≤ (2*y/Real.exp l)^(-p) := by
-  rw [green_eq_source _ _ _ _ _ hy]
-  have hQ := q_pos x hy
-  have hyS : y ≤ Real.sqrt (x^2+y^2) := by
-    rw [Real.le_sqrt hy.le (by positivity)]
-    nlinarith [sq_nonneg x]
-  have hr : 0 ≤ (2*y/Real.exp l)^(-p) := Real.rpow_nonneg (by positivity) _
-  have hs : (y/Real.sqrt (x^2+y^2))^a ≤ 1 :=
-    Real.rpow_le_one (by positivity) ((div_le_one (Real.sqrt_pos.2 hQ)).2 hyS) ha
-  simpa using mul_le_mul_of_nonneg_left hs hr
-
-lemma green_le_fixed_radius (p a x y l ρ : ℝ) (hy : 0 < y) (ha : 0 ≤ a)
-    (hp : 0 ≤ p) (hρ : 0 < ρ) (hR : ρ ≤ 2*y/Real.exp l) :
-    green p a x y l ≤ ρ^(-p) := by
-  exact (green_le_radius p a x y l hy ha).trans
-    (Real.rpow_le_rpow_of_nonpos hρ hR (neg_nonpos.2 hp))
-
-lemma contDiff_green_x (p a y l : ℝ) (hy : 0 < y) {n : ℕ∞ω} :
-    ContDiff ℝ n (fun x => green p a x y l) := by
-  unfold green
-  apply ContDiff.exp
-  apply ContDiff.sub contDiff_const
-  exact contDiff_const.mul (((contDiff_id.pow 2).add contDiff_const).log
-    (fun x => (q_pos x hy).ne'))
-
-lemma contDiff_green_driver (p a c y l sdev : ℝ) (hy : 0 < y) {n : ℕ∞ω} :
-    ContDiff ℝ n (fun b => green p a (c-sdev*b) y l) :=
-  (contDiff_green_x p a y l hy).comp (contDiff_const.sub (contDiff_const.mul contDiff_id))
-
-lemma hasDerivAt_second_green_x (p a x y l : ℝ) (hy : 0 < y) :
-    HasDerivAt (deriv (deriv (fun v => green p a v y l)))
-      (green p a x y l * (a*(a+2)*x*(3*y^2-(a+1)*x^2))/(x^2+y^2)^3) x := by
-  have he : deriv (deriv (fun v => green p a v y l)) =
-      fun v => green p a v y l * (a*(a+1)*v^2-a*y^2)/(v^2+y^2)^2 :=
-    funext (fun v => deriv_deriv_green_x p a v y l hy)
-  rw [he]
-  have hn := (hasDerivAt_green_x p a x y l hy).mul
-    ((((hasDerivAt_id x).pow 2).const_mul (a*(a+1))).sub_const (a*y^2))
-  have hq := (((hasDerivAt_id x).pow 2).add_const (y^2)).pow 2
-  apply (hn.div hq (pow_ne_zero 2 (q_pos x hy).ne')).congr_deriv
-  simp only [id_eq, Pi.pow_apply, Pi.mul_apply]
-  field_simp
-  ring
-
-lemma third_green_x (p a x y l : ℝ) (hy : 0 < y) :
-    iteratedDeriv 3 (fun v => green p a v y l) x =
-      green p a x y l * (a*(a+2)*x*(3*y^2-(a+1)*x^2))/(x^2+y^2)^3 := by
-  simpa only [iteratedDeriv_succ, iteratedDeriv_zero] using
-    (hasDerivAt_second_green_x p a x y l hy).deriv
-
-lemma iteratedDeriv_green_driver (p a c y l sdev : ℝ) (hy : 0 < y) (k : ℕ) (b : ℝ) :
-    iteratedDeriv k (fun v => green p a (c-sdev*v) y l) b =
-      (-sdev)^k * iteratedDeriv k (fun x => green p a x y l) (c-sdev*b) := by
-  induction k generalizing b with
-  | zero => simp
-  | succ k ih =>
-    have he : iteratedDeriv k (fun v => green p a (c-sdev*v) y l) =
-        fun v => (-sdev)^k * iteratedDeriv k (fun x => green p a x y l) (c-sdev*v) := funext ih
-    rw [iteratedDeriv_succ, he]
-    have hd : Differentiable ℝ (iteratedDeriv k (fun x => green p a x y l)) :=
-      ContDiff.differentiable_iteratedDeriv k (contDiff_green_x p a y l hy (n := ⊤)) (by simp)
-    have hh := ((hd (c-sdev*b)).hasDerivAt.comp b
-      (((hasDerivAt_id b).const_mul sdev).const_sub c)).const_mul ((-sdev)^k)
-    dsimp only [Function.comp_def, id_eq] at hh
-    rw [hh.deriv]
-    simp only [iteratedDeriv_succ, pow_succ, mul_one]
-    ring
-
-lemma abs_x_div_q_le {ε y : ℝ} (hε : 0 < ε) (hy : ε ≤ y) (x : ℝ) :
-    |x| / (x^2+y^2) ≤ 1/ε := by
-  have hq := q_pos x (hε.trans_le hy)
-  apply (div_le_div_iff₀ hq hε).2
-  have hεy : ε^2 ≤ y^2 := pow_le_pow_left₀ hε.le hy 2
-  nlinarith [sq_nonneg (|x|-ε), sq_abs x, sq_nonneg x]
-
-lemma inv_q_le {ε y : ℝ} (hε : 0 < ε) (hy : ε ≤ y) (x : ℝ) :
-    1/(x^2+y^2) ≤ 1/ε^2 := by
-  apply div_le_div_of_nonneg_left zero_le_one (sq_pos_of_pos hε)
-  exact (pow_le_pow_left₀ hε.le hy 2).trans (le_add_of_nonneg_left (sq_nonneg x))
-
-lemma third_green_bound {p a x y l ε M : ℝ} (hε : 0 < ε) (hy : ε ≤ y)
-    (ha : 0 ≤ a) (hM : green p a x y l ≤ M) :
-    |iteratedDeriv 3 (fun v => green p a v y l) x| ≤
-      M * (a*(a+2)*(a+4)) / ε^3 := by
-  have hypos := hε.trans_le hy
-  have hq := q_pos x hypos
-  have hMp : 0 ≤ M := (green_pos p a x y l).le.trans hM
-  have hP : |3*y^2-(a+1)*x^2| ≤ (a+4)*(x^2+y^2) := by
-    calc
-      _ ≤ |3*y^2| + |(a+1)*x^2| := abs_sub _ _
-      _ = 3*y^2+(a+1)*x^2 := by rw [abs_of_nonneg (by positivity), abs_of_nonneg (by positivity)]
-      _ ≤ _ := by nlinarith [mul_nonneg ha (sq_nonneg y), sq_nonneg x]
-  rw [third_green_x _ _ _ _ _ hypos]
-  simp only [abs_div, abs_mul, abs_pow, abs_of_pos hq, abs_of_pos (green_pos p a x y l),
-    abs_of_nonneg ha, abs_of_nonneg (show 0 ≤ a+2 by linarith)]
-  calc
-    _ ≤ M * (a*(a+2)* |x| *((a+4)*(x^2+y^2))) / (x^2+y^2)^3 := by
-      gcongr
-    _ = M*(a*(a+2)*(a+4)) * (|x|/(x^2+y^2)) * (1/(x^2+y^2)) := by
-      field_simp
-    _ ≤ M*(a*(a+2)*(a+4)) * (1/ε) * (1/ε^2) := by
-      exact mul_le_mul
-        (mul_le_mul_of_nonneg_left (abs_x_div_q_le hε hy x) (by positivity))
-        (inv_q_le hε hy x) (by positivity) (by positivity)
-    _ = _ := by ring
-
-lemma continuous_green_comp {X : Type*} [TopologicalSpace X]
-    (p a : ℝ) {x y l : X → ℝ} (hx : Continuous x) (hy : Continuous y)
-    (hl : Continuous l) (hyp : ∀ t, 0 < y t) :
-    Continuous (fun t => green p a (x t) (y t) (l t)) := by
-  unfold green
-  apply Real.continuous_exp.comp
-  exact (((continuous_const.mul hl).sub continuous_const).add
-    (continuous_const.mul (hy.log (fun t => (hyp t).ne')))).sub
-    (continuous_const.mul (((hx.pow 2).add (hy.pow 2)).log
-      (fun t => (q_pos (x t) (hyp t)).ne')))
-
-lemma first_green_bound {p a x y l ε M : ℝ} (hε : 0 < ε) (hy : ε ≤ y)
-    (ha : 0 ≤ a) (hM : green p a x y l ≤ M) :
-    |deriv (fun v => green p a v y l) x| ≤ M*a/ε := by
-  have hp := green_pos p a x y l
-  rw [deriv_green_x p a x y l (hε.trans_le hy)]
-  simp only [abs_mul, abs_div, abs_neg, abs_of_nonneg ha,
-    abs_of_pos (q_pos x (hε.trans_le hy)), abs_of_pos hp]
-  calc
-    _ = green p a x y l * a * (|x|/(x^2+y^2)) := by ring
-    _ ≤ M*a*(1/ε) := mul_le_mul
-      (mul_le_mul_of_nonneg_right hM ha) (abs_x_div_q_le hε hy x)
-      (by positivity) (mul_nonneg (hp.le.trans hM) ha)
-    _ = _ := by ring
-
-lemma second_green_bound {p a x y l ε M : ℝ} (hε : 0 < ε) (hy : ε ≤ y)
-    (ha : 0 ≤ a) (hM : green p a x y l ≤ M) :
-    |iteratedDeriv 2 (fun v => green p a v y l) x| ≤ M*(a*(a+2))/ε^2 := by
-  have hp := green_pos p a x y l
-  have hMp : 0 ≤ M := hp.le.trans hM
-  have hq := q_pos x (hε.trans_le hy)
-  have hP : |a*(a+1)*x^2-a*y^2| ≤ a*(a+2)*(x^2+y^2) := by
-    calc
-      _ ≤ |a*(a+1)*x^2|+|a*y^2| := abs_sub _ _
-      _ = a*(a+1)*x^2+a*y^2 := by rw [abs_of_nonneg (by positivity), abs_of_nonneg (by positivity)]
-      _ ≤ _ := by nlinarith [mul_nonneg ha (sq_nonneg x), mul_nonneg (mul_nonneg ha (by positivity : 0 ≤ a+1)) (sq_nonneg y)]
-  simp only [iteratedDeriv_succ, iteratedDeriv_zero]
-  rw [deriv_deriv_green_x p a x y l (hε.trans_le hy)]
-  simp only [abs_div, abs_mul, abs_pow, abs_of_pos hq, abs_of_pos hp]
-  calc
-    _ ≤ M*(a*(a+2)*(x^2+y^2))/(x^2+y^2)^2 := by gcongr
-    _ = M*(a*(a+2))*(1/(x^2+y^2)) := by field_simp
-    _ ≤ M*(a*(a+2))*(1/ε^2) := mul_le_mul_of_nonneg_left (inv_q_le hε hy x) (by positivity)
-    _ = _ := by ring
-
-end SLEExactGauge.GreenPhysical
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-noncomputable section
-open Set Filter
-open scoped Topology ContDiff
-namespace SLEExactGauge.LoewnerField
-open LoewnerCutoff GreenPhysical
-open scoped unitInterval
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-def height (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
-  max ε (position hε hT U z t).im
-
-def field (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
-  green p a ((position hε hT U z t).re - sdev*b) (height hε hT U z t)
-    (logDerivative hε hT U z t)
-
-def drift (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
-  let x := (position hε hT U z t).re - sdev*b
-  let y := height hε hT U z t
-  let v := velocity ε (extendDriver T U t) (position hε hT U z t)
-  field hε hT p a sdev U z t b *
-    (p*variation hε hT U z t + (a-p)*v.im/y - a*(x*v.re+y*v.im)/(x^2+y^2))
-
-end SLEExactGauge.LoewnerField
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.StoppedMeshes
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Brownian Stopped estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.BrownianStopped
+open BrownianPredictable PredictableLimit BrownianIto TaylorMeshes BrownianField
+
+theorem stopped_fieldSum_tendsto_ae {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
+    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
+    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
+    (C : ℕ → ℕ → Ω → ℝ)
+    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
+    (hC2 : ∀ n i, MemLp (C n i) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
+    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
+    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω) atTop
+      (𝓝 (F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
+        (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω),
+          iteratedDeriv 2 (F ω t) (B t.toNNReal ω))) := by
+  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T
+    C hC hC2 K₂ hK₂ (fun n i => by
+      filter_upwards [hCeq n i, hbound n i] with ω he hb
+      simpa only [he] using hb)
+  have hCeq_all : ∀ᵐ ω ∂P, ∀ n i, C n i ω = stoppedCoeff 2 B T τ F n i ω :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hCeq n))
+  have hC_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂ :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hbound n))
+  have hA_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁ :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAbound n))
+  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hτ, hC_all, hA_all, hCeq_all]
+    with ω hb hQ hE hr hτω hCω hAω hCeqω
+  have hEs : Tendsto (fun n => dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n ω) atTop (𝓝 0) := by
+    convert hE using 1
+    funext n
+    unfold dyadicQuadraticError
+    simp only [hCeqω]
+  let b : ℝ → ℝ := fun t => B t.toNNReal ω
+  have hbc : Continuous b := hb.comp continuous_real_toNNReal
+  have hQ' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
+    simpa only [quadraticSum, b, mesh_toNNReal] using hQ
+  have hE' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
+      iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+        ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0)
+        atTop (𝓝 0) := by
+    have heq (n : ℕ) : dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n ω =
+        ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
+          iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0 := by
+      unfold dyadicQuadraticError
+      apply Finset.sum_congr rfl
+      intro i hi
+      simp only [stoppedCoeff, b, mesh_toNNReal]
+      split_ifs <;> simp
+    simpa only [heq] using hEs
+  have hA' (n i : ℕ) (hi : mesh T n i < τ ω) :
+      |deriv (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₁ := by
+    simpa only [stoppedCoeff, ite_eq_left hi, iteratedDeriv_succ, iteratedDeriv_zero,
+      b, mesh_toNNReal] using hAω n i
+  have hC' (n i : ℕ) (hi : mesh T n i < τ ω) :
+      |iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₂ := by
+    simpa only [stoppedCoeff, ite_eq_left hi, b, mesh_toNNReal] using hCω n i
+  have ht := stopped_field_taylor_limit T.coe_nonneg hτω.1 hτω.2 hbc.continuousOn
+    hr hQ' hK₁ hK₂ hA' hC' hE'
+  have heq (n : ℕ) : dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω =
+      ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
+        deriv (F ω (mesh T n i)) (b (mesh T n i)) *
+          (b (mesh T n (i+1))-b (mesh T n i)) else 0 := by
+    unfold dyadicDriverSum
+    apply Finset.sum_congr rfl
+    intro i hi
+    simp only [stoppedCoeff, b, mesh_toNNReal, iteratedDeriv_succ, iteratedDeriv_zero]
+    split_ifs <;> simp
+  simpa only [← heq, b, Real.toNNReal_zero] using ht
+
+theorem stopped_field_ito_centered_ae {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
+    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
+    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
+    (A C : ℕ → ℕ → Ω → ℝ)
+    (hAeq : ∀ n i, A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i)
+    (hCeq : ∀ n i, C n i =ᵐ[P] stoppedCoeff 2 B T τ F n i)
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (C n i))
+    (hC2 : ∀ n i, MemLp (C n i) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (A n i))
+    (hA2 : ∀ n i, MemLp (A n i) 2 P)
+    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
+    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
+    let Y := fun ω => F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
+      (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
+      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω), iteratedDeriv 2 (F ω t) (B t.toNNReal ω)
+    Integrable Y P ∧ (∫ ω, Y ω ∂P) = 0 := by
+  have ht := stopped_fieldSum_tendsto_ae hB T τ F V hτ hreg C hCeq hC hC2
+    K₂ hK₂ hbound K₁ hK₁ hAbound
+  have hAeq_all : ∀ᵐ ω ∂P, ∀ n i, A n i ω = stoppedCoeff 1 B T τ F n i ω :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAeq n))
+  apply limit_centered hB T A hA hA2 K₁ hK₁ ?_ _ ?_
+  · intro n i
+    filter_upwards [hAeq n i, hAbound n i] with ω he hb
+    simpa only [he] using hb
+  · filter_upwards [ht, hAeq_all] with ω hω he
+    convert hω using 1
+    funext n
+    unfold dyadicDriverSum
+    simp only [he]
+
+end SLEExactGauge.BrownianStopped
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.LoewnerCutoff
+
+def clampIm (ε : ℝ) (w : ℂ) : ℂ := ⟨w.re, max w.im ε⟩
+
+def velocity (ε u : ℝ) (w : ℂ) : ℂ := 2 / clampIm ε (w - (u : ℂ))
+
+lemma clampIm_eq {ε : ℝ} {w : ℂ} (hw : ε ≤ w.im) : clampIm ε w = w := by
+  apply Complex.ext <;> simp [clampIm, max_eq_left hw]
+
+lemma clampIm_height (ε : ℝ) (w : ℂ) : ε ≤ (clampIm ε w).im := le_max_right _ _
+
+lemma clampIm_norm {ε : ℝ} (_hε : 0 < ε) (w : ℂ) : ε ≤ ‖clampIm ε w‖ :=
+  (clampIm_height ε w).trans (Complex.im_le_norm _)
+
+lemma clampIm_ne_zero {ε : ℝ} (hε : 0 < ε) (w : ℂ) : clampIm ε w ≠ 0 :=
+  norm_pos_iff.1 (hε.trans_le (clampIm_norm hε w))
+
+lemma clampIm_norm_sub (ε : ℝ) (z w : ℂ) :
+    ‖clampIm ε z - clampIm ε w‖ ≤ 2 * ‖z - w‖ := by
+  have hm : |max z.im ε - max w.im ε| ≤ |z.im - w.im| := by
+    simpa only [sub_self, abs_zero, max_eq_left (abs_nonneg (z.im - w.im))] using
+      abs_max_sub_max_le_max z.im ε w.im ε
+  calc
+    ‖clampIm ε z - clampIm ε w‖ ≤ |z.re-w.re| + |max z.im ε-max w.im ε| :=
+      Complex.norm_le_abs_re_add_abs_im _
+    _ ≤ |z.re-w.re| + |z.im-w.im| := add_le_add le_rfl hm
+    _ ≤ ‖z-w‖ + ‖z-w‖ := add_le_add (Complex.abs_re_le_norm (z-w))
+      (Complex.abs_im_le_norm (z-w))
+    _ = 2 * ‖z-w‖ := by ring
+
+lemma velocity_eq {ε u : ℝ} {w : ℂ} (hw : ε ≤ w.im) :
+    velocity ε u w = 2 / (w - (u : ℂ)) := by
+  unfold velocity
+  rw [clampIm_eq (by simpa using hw)]
+
+lemma norm_velocity_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (w : ℂ) :
+    ‖velocity ε u w‖ ≤ 2 / ε := by
+  rw [velocity, norm_div, Complex.norm_ofNat]
+  exact div_le_div_of_nonneg_left (by norm_num) hε (clampIm_norm hε _)
+
+lemma norm_velocity_sub_le {ε : ℝ} (hε : 0 < ε) (u : ℝ) (z w : ℂ) :
+    ‖velocity ε u z - velocity ε u w‖ ≤ (4 / ε^2) * ‖z-w‖ := by
+  let a := clampIm ε (z - (u : ℂ))
+  let b := clampIm ε (w - (u : ℂ))
+  have ha : a ≠ 0 := clampIm_ne_zero hε _
+  have hb : b ≠ 0 := clampIm_ne_zero hε _
+  have hna := clampIm_norm hε (z - (u : ℂ))
+  have hnb := clampIm_norm hε (w - (u : ℂ))
+  have hd : ε^2 ≤ ‖a*b‖ := by
+    rw [norm_mul, pow_two]
+    exact mul_le_mul hna hnb hε.le (norm_nonneg _)
+  have he : velocity ε u z - velocity ε u w = 2 * (b-a) / (a*b) := by
+    dsimp [velocity, a, b]
+    field_simp [clampIm_ne_zero hε]
+  rw [he, norm_div, norm_mul, Complex.norm_ofNat]
+  have hn : ‖b-a‖ ≤ 2 * ‖z-w‖ := by
+    simpa only [a, b, norm_sub_rev, sub_sub_sub_cancel_right] using
+      clampIm_norm_sub ε (z-(u:ℂ)) (w-(u:ℂ))
+  calc
+    2 * ‖b-a‖ / ‖a*b‖ ≤ (2 * (2 * ‖z-w‖)) / ε^2 :=
+      div_le_div₀ (by positivity) (mul_le_mul_of_nonneg_left hn (by norm_num))
+        (sq_pos_of_pos hε) hd
+    _ = (4 / ε^2) * ‖z-w‖ := by ring
+
+lemma velocity_lipschitz {ε : ℝ} (hε : 0 < ε) (u : ℝ) :
+    LipschitzWith ⟨4 / ε^2, by positivity⟩ (velocity ε u) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  simp only [dist_eq_norm]
+  exact norm_velocity_sub_le hε u z w
+
+lemma continuous_velocity {ε : ℝ} (hε : 0 < ε) :
+    Continuous (fun p : ℝ × ℂ => velocity ε p.1 p.2) := by
+  unfold velocity clampIm
+  refine continuous_const.div ?_ (fun p => clampIm_ne_zero hε _)
+  change Continuous (fun p : ℝ × ℂ => Complex.equivRealProdCLM.symm
+    ((p.2 - (p.1 : ℂ)).re, max (p.2 - (p.1 : ℂ)).im ε))
+  apply Complex.equivRealProdCLM.symm.continuous.comp
+  apply Continuous.prodMk <;> fun_prop
+
+lemma exists_solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
+    ∃ H : ℝ → ℂ, H 0 = z ∧ ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t := by
+  let L : ℝ≥0 := ⟨2 / ε, by positivity⟩
+  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
+  let a : ℝ≥0 := ⟨(2 / ε) * T, mul_nonneg (by positivity) hT⟩
+  have hp : IsPicardLindelof (fun t => velocity ε (U t))
+      ⟨0, ⟨le_rfl, hT⟩⟩ z a 0 L K := by
+    constructor
+    · exact fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith
+    · intro w _
+      have hc : ContinuousOn (fun t : ℝ => (U t, w)) (Icc 0 T) :=
+        hU.prodMk continuousOn_const
+      simpa only [Function.comp_def] using
+        (continuous_velocity hε).comp_continuousOn hc
+    · exact fun t _ w _ => norm_velocity_le hε (U t) w
+    · dsimp [L, a]
+      simp only [sub_zero, sub_self, max_eq_left hT]
+      rfl
+  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
+
+lemma norm_velocity_driver_sub_le {ε : ℝ} (hε : 0 < ε) (u v : ℝ) (z : ℂ) :
+    ‖velocity ε u z - velocity ε v z‖ ≤ (4 / ε^2) * |u-v| := by
+  have he (a : ℝ) : velocity ε 0 (z - (a : ℂ)) = velocity ε a z := by
+    simp [velocity]
+  have hh := norm_velocity_sub_le hε 0 (z-(u:ℂ)) (z-(v:ℂ))
+  rw [he, he] at hh
+  convert hh using 1
+  rw [sub_sub_sub_cancel_left, ← Complex.ofReal_sub, Complex.norm_real, abs_sub_comm]
+  simp [Real.norm_eq_abs]
+
+lemma derivWithin_right {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ico 0 T) :
+    HasDerivWithinAt H (velocity ε (U t) (H t)) (Ici t) t :=
+  (hH t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
+
+lemma derivWithin_left {ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ioc 0 T) :
+    HasDerivWithinAt H (velocity ε (U t) (H t)) (Iic t) t :=
+  (hH t ⟨ht.1.le, ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
+
+lemma solution_eq {ε T : ℝ} (hε : 0 < ε) {U : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (U t) (G t)) (Icc 0 T) t)
+    (h0 : H 0 = G 0) : EqOn H G (Icc 0 T) := by
+  apply ODE_solution_unique_of_mem_Icc_right
+    (s := fun _ => univ) (fun t _ => (velocity_lipschitz hε (U t)).lipschitzOnWith)
+    (HasDerivWithinAt.continuousOn hH) (fun _ ht => derivWithin_right hH ht)
+    (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hG)
+    (fun _ ht => derivWithin_right hG ht) (fun _ _ => mem_univ _) h0
+
+lemma solution_dist_le {ε T D : ℝ} (hε : 0 < ε) (hD : 0 ≤ D)
+    {U V : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity ε (V t) (G t)) (Icc 0 T) t)
+    (h0 : H 0 = G 0) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    dist (H t) (G t) ≤ D * Real.exp ((4 / ε^2) * T) := by
+  let K : ℝ≥0 := ⟨4 / ε^2, by positivity⟩
+  have hK : (0 : ℝ) < K := by change 0 < 4 / ε^2; positivity
+  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
+      dist (velocity ε (V s) (G s)) (velocity ε (U s) (G s)) ≤ (K : ℝ) * D := by
+    rw [dist_eq_norm]
+    calc
+      _ ≤ (4 / ε^2) * |V s-U s| := norm_velocity_driver_sub_le hε _ _ _
+      _ ≤ (K : ℝ) * D := by
+        rw [abs_sub_comm]
+        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1, hs.2.le⟩) hK.le
+  have hh := dist_le_of_approx_trajectories_ODE
+    (fun t => velocity_lipschitz hε (U t))
+    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
+    (fun t _ => le_of_eq (dist_self (velocity ε (U t) (H t))))
+    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
+    hb (le_of_eq (dist_eq_zero.2 h0)) t ht
+  have he : gronwallBound 0 K (0 + (K : ℝ)*D) (t-0) =
+      D * (Real.exp ((K : ℝ)*t)-1) := by
+    simp only [gronwallBound, zero_add, sub_zero, ite_eq_right (ne_of_gt hK)]
+    field_simp
+    ring
+  rw [he] at hh
+  have hexp : Real.exp ((K : ℝ)*t) ≤ Real.exp ((K : ℝ)*T) :=
+    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
+  apply hh.trans
+  change D * (Real.exp ((4 / ε^2)*t)-1) ≤ D * Real.exp ((4 / ε^2)*T)
+  change Real.exp ((4 / ε^2)*t) ≤ Real.exp ((4 / ε^2)*T) at hexp
+  exact mul_le_mul_of_nonneg_left (by linarith) hD
+
+end SLEExactGauge.LoewnerCutoff
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.LoewnerCutoff
+open scoped unitInterval
+
+def extendDriver (T : ℝ) (U : C(I, ℝ)) (t : ℝ) : ℝ :=
+  U (projIcc 0 1 zero_le_one (t / T))
+
+lemma continuous_extendDriver (T : ℝ) (U : C(I, ℝ)) : Continuous (extendDriver T U) :=
+  U.continuous.comp (continuous_projIcc.comp (continuous_id.div_const T))
+
+def solution {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) : ℝ → ℂ :=
+  (exists_solution hε hT (extendDriver T U) (continuous_extendDriver T U).continuousOn z).choose
+
+lemma solution_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    solution hε hT U z 0 = z :=
+  (exists_solution hε hT (extendDriver T U)
+    (continuous_extendDriver T U).continuousOn z).choose_spec.1
+
+lemma solution_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (solution hε hT U z)
+      (velocity ε (extendDriver T U t) (solution hε hT U z t)) (Icc 0 T) t :=
+  (exists_solution hε hT (extendDriver T U)
+    (continuous_extendDriver T U).continuousOn z).choose_spec.2
+
+def solutionPath {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (U : C(I, ℝ)) (z : ℂ) :
+    C(Icc 0 T, ℂ) :=
+  ⟨fun t => solution hε hT U z t,
+    continuousOn_iff_continuous_domRestrict.1 (HasDerivWithinAt.continuousOn (solution_deriv hε hT U z))⟩
+
+lemma solutionPath_lipschitz {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    LipschitzWith ⟨Real.exp ((4 / ε^2) * T), (Real.exp_pos _).le⟩
+      (fun U : C(I, ℝ) => solutionPath hε hT U z) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro U V
+  apply (ContinuousMap.dist_le (mul_nonneg (Real.exp_pos _).le dist_nonneg)).2
+  intro t
+  have hUV (s : ℝ) (_hs : s ∈ Icc 0 T) :
+      |extendDriver T U s-extendDriver T V s| ≤ dist U V := by
+    exact (ContinuousMap.dist_apply_le_dist (f := U) (g := V)
+      (projIcc 0 1 zero_le_one (s/T)))
+  have hh := solution_dist_le hε dist_nonneg (solution_deriv hε hT U z)
+    (solution_deriv hε hT V z)
+    ((solution_initial hε hT U z).trans (solution_initial hε hT V z).symm)
+    hUV t.property
+  change dist (solution hε hT U z t) (solution hε hT V z t) ≤
+    Real.exp ((4 / ε^2)*T) * dist U V
+  simpa only [mul_comm] using hh
+
+end SLEExactGauge.LoewnerCutoff
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ContinuousDriver
+section Limit
+variable {Ω Y : Type*} [MeasurableSpace Ω] [MetricSpace Y] [CompleteSpace Y]
+
+def selectedLimit (y₀ : Y) (a : ℕ → Y) : Y := by
+  classical
+  exact if h : CauchySeq a then (cauchySeq_tendsto_of_complete h).choose else y₀
+
+lemma tendsto_selectedLimit (y₀ : Y) {a : ℕ → Y} (h : CauchySeq a) :
+    Tendsto a atTop (𝓝 (selectedLimit y₀ a)) := by
+  rw [selectedLimit, dite_eq_left h]
+  exact (cauchySeq_tendsto_of_complete h).choose_spec
+
+lemma selectedLimit_eq (y₀ : Y) {a : ℕ → Y} {y : Y} (h : Tendsto a atTop (𝓝 y)) :
+    selectedLimit y₀ a = y :=
+  tendsto_nhds_unique (tendsto_selectedLimit y₀ h.cauchySeq) h
+
+end Limit
+
+end SLEExactGauge.ContinuousDriver
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ContinuousDriver
+section Limit
+variable {Ω Y : Type*} [MeasurableSpace Ω] [MetricSpace Y] [completeY : CompleteSpace Y]
+variable [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y]
+
+omit completeY in
+lemma measurableSet_cauchy [CompleteSpace Y] (a : ℕ → Ω → Y) (ha : ∀ n, Measurable (a n)) :
+    MeasurableSet {ω | CauchySeq (fun n => a n ω)} := by
+  simp only [Metric.uniformity_basis_dist_inv_nat_succ.cauchySeq_iff, forall_const,
+    mem_ofPred_eq, ofPred_forall, ofPred_exists]
+  exact MeasurableSet.iInter fun (k : ℕ) => MeasurableSet.iUnion fun (N : ℕ) =>
+    MeasurableSet.iInter fun (m : ℕ) => MeasurableSet.iInter fun (_ : N ≤ m) =>
+      MeasurableSet.iInter fun (n : ℕ) => MeasurableSet.iInter fun (_ : N ≤ n) =>
+        measurableSet_lt ((ha m).dist (ha n)) measurable_const
+
+lemma measurable_selectedLimit (y₀ : Y) (a : ℕ → Ω → Y) (ha : ∀ n, Measurable (a n)) :
+    Measurable (fun ω => selectedLimit y₀ (fun n => a n ω)) := by
+  classical
+  have hc := measurableSet_cauchy a ha
+  apply measurable_of_tendsto_metrizable
+    (f := fun n ω => if CauchySeq (fun m => a m ω) then a n ω else y₀)
+    (fun n => (ha n).ite hc measurable_const)
+  apply tendsto_pi_nhds.2
+  intro ω
+  by_cases hω : CauchySeq (fun n => a n ω)
+  · simpa only [ite_eq_left hω] using tendsto_selectedLimit y₀ hω
+  · simp only [ite_eq_right hω, selectedLimit, dite_eq_right hω]
+    exact tendsto_const_nhds
+
+end Limit
+
+end SLEExactGauge.ContinuousDriver
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.ContinuousDriver
+
+def rawApprox (x : I → ℝ) (n : ℕ) : C(I, ℝ) :=
+  ∑ k : Fin (n+1), (x (bernstein.z k)) • bernstein n k
+
+lemma rawApprox_eq (f : C(I, ℝ)) (n : ℕ) : rawApprox f n = bernsteinApproximation n f := by
+  apply ContinuousMap.ext
+  intro t
+  simp [rawApprox, bernsteinApproximation, smul_eq_mul, mul_comm]
+
+lemma measurable_rawApprox {Ω : Type*} [MeasurableSpace Ω]
+    (x : Ω → I → ℝ) (hx : ∀ t, Measurable (fun ω => x ω t)) (n : ℕ) :
+    @Measurable Ω C(I, ℝ) _ (borel C(I, ℝ)) (fun ω => rawApprox (x ω) n) := by
+  let : MeasurableSpace C(I, ℝ) := borel C(I, ℝ)
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  apply Finset.measurable_sum
+  intro k _
+  exact (hx (bernstein.z k)).smul measurable_const
+
+def path (x : I → ℝ) : C(I, ℝ) := selectedLimit 0 (rawApprox x)
+
+lemma path_eq (f : C(I, ℝ)) : path f = f := by
+  apply selectedLimit_eq
+  have he : rawApprox f = fun n => bernsteinApproximation n f :=
+    funext (rawApprox_eq f)
+  rw [he]
+  exact bernsteinApproximation_uniform f
+
+lemma path_eval_of_continuous {x : I → ℝ} (hx : Continuous x) (t : I) : path x t = x t := by
+  have hh := path_eq (⟨x, hx⟩ : C(I, ℝ))
+  exact congrArg (fun f : C(I, ℝ) => f t) hh
+
+lemma measurable_path {Ω : Type*} [MeasurableSpace Ω]
+    (x : Ω → I → ℝ) (hx : ∀ t, Measurable (fun ω => x ω t)) :
+    @Measurable Ω C(I, ℝ) _ (borel C(I, ℝ)) (fun ω => path (x ω)) := by
+  let : MeasurableSpace C(I, ℝ) := borel C(I, ℝ)
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  exact measurable_selectedLimit 0 _ (measurable_rawApprox x hx)
+
+end SLEExactGauge.ContinuousDriver
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.LoewnerCutoff
+open scoped unitInterval
+
+def position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℂ :=
+  solutionPath hε hT U z (projIcc 0 T hT t)
+
+lemma position_eq {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    position hε hT U z t = solution hε hT U z t := by
+  simp only [position, solutionPath, projIcc, max_eq_right ht.1,
+    min_eq_right ht.2]
+  rfl
+
+lemma continuous_position {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => position hε hT p.1 z p.2) := by
+  exact continuous_eval.comp (((solutionPath_lipschitz hε hT z).continuous.comp continuous_fst).prodMk
+    ((continuous_projIcc (h := hT)).comp continuous_snd))
+
+lemma continuous_joint_extendDriver (T : ℝ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => extendDriver T p.1 p.2) := by
+  exact continuous_eval.comp (continuous_fst.prodMk
+    ((continuous_projIcc (h := zero_le_one)).comp (continuous_snd.div_const T)))
+
+def variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  (-2 / (clampIm ε (position hε hT U z t - (extendDriver T U t : ℂ)))^2).re
+
+lemma continuous_clampIm (ε : ℝ) : Continuous (clampIm ε) := by
+  change Continuous (fun w : ℂ => Complex.equivRealProdCLM.symm (w.re,max w.im ε))
+  apply Complex.equivRealProdCLM.symm.continuous.comp
+  apply Continuous.prodMk <;> fun_prop
+
+lemma continuous_variation {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => variation hε hT p.1 z p.2) := by
+  have hw := (continuous_position hε hT z).sub
+    (Complex.continuous_ofReal.comp (continuous_joint_extendDriver T))
+  exact Complex.continuous_re.comp (continuous_const.div
+    (((continuous_clampIm ε).comp hw).pow 2)
+    (fun p => pow_ne_zero 2 (clampIm_ne_zero hε _)))
+
+def logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  ∫ s in (0 : ℝ)..(projIcc 0 T hT t : ℝ), variation hε hT U z s
+
+lemma continuous_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T) (z : ℂ) :
+    Continuous (fun p : C(I, ℝ) × ℝ => logDerivative hε hT p.1 z p.2) := by
+  have hc := intervalIntegral.continuous_parametric_primitive_of_continuous
+    (μ := volume) (a₀ := 0) (f := fun U t => variation hε hT U z t)
+    (continuous_variation hε hT z)
+  exact hc.comp (continuous_fst.prodMk
+    (continuous_subtype_val.comp ((continuous_projIcc (h := hT)).comp continuous_snd)))
+
+lemma abs_variation_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    |variation hε hT U z t| ≤ 2/ε^2 := by
+  apply (Complex.abs_re_le_norm _).trans
+  rw [norm_div, norm_neg, Complex.norm_ofNat, norm_pow]
+  apply div_le_div_of_nonneg_left (by norm_num) (sq_pos_of_pos hε)
+  exact pow_le_pow_left₀ hε.le (clampIm_norm hε _) 2
+
+lemma logDerivative_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) : logDerivative hε hT U z 0 = 0 := by
+  simp only [logDerivative, projIcc_of_mem hT (left_mem_Icc.mpr hT), intervalIntegral.integral_same]
+
+lemma abs_logDerivative_le {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    |logDerivative hε hT U z t| ≤ (2/ε^2)*T := by
+  have hb := (projIcc 0 T hT t).property
+  have hi := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (0:ℝ)) (b := (projIcc 0 T hT t : ℝ))
+    (fun s (_hs : s ∈ uIoc 0 (projIcc 0 T hT t : ℝ)) =>
+      show ‖variation hε hT U z s‖ ≤ 2/ε^2 from abs_variation_le hε hT U z s)
+  change |logDerivative hε hT U z t| ≤ _
+  apply hi.trans
+  rw [sub_zero, abs_of_nonneg hb.1]
+  exact mul_le_mul_of_nonneg_left hb.2 (by positivity)
+
+lemma position_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) : position hε hT U z 0 = z := by
+  rw [position_eq hε hT U z ⟨le_rfl,hT⟩, solution_initial]
+
+lemma position_derivWithin {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    HasDerivWithinAt (position hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)) (Icc 0 T) t := by
+  rw [position_eq hε hT U z ht]
+  exact (solution_deriv hε hT U z t ht).congr
+    (fun s hs => position_eq hε hT U z hs) (position_eq hε hT U z ht)
+
+lemma position_deriv {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
+    HasDerivAt (position hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)) t :=
+  (position_derivWithin hε hT U z ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2)
+
+lemma logDerivative_eq_integral {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Icc 0 T) :
+    logDerivative hε hT U z t = ∫ s in (0:ℝ)..t, variation hε hT U z s := by
+  simp only [logDerivative, projIcc_of_mem hT ht]
+
+lemma hasDerivAt_logDerivative {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : C(I, ℝ)) (z : ℂ) {t : ℝ} (ht : t ∈ Ioo 0 T) :
+    HasDerivAt (logDerivative hε hT U z) (variation hε hT U z t) t := by
+  have hp : Continuous (fun t : ℝ => (U,t)) := continuous_const.prodMk continuous_id
+  have hc := (continuous_variation hε hT z).comp hp
+  dsimp only [Function.comp_def] at hc
+  have hd : HasDerivAt (fun v : ℝ => ∫ s in (0:ℝ)..v, variation hε hT U z s)
+      (variation hε hT U z t) t :=
+    intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 t)
+      hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
+  exact logDerivative_eq_integral hε hT U z hs
+
+lemma position_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
+    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (_hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
+    (U V : C(I, ℝ)) (z : ℂ)
+    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
+    EqOn (position hε hT₁ U z) (position hε hT₂ V z) (Icc 0 S) := by
+  apply solution_eq hε (U := extendDriver T₁ U) ?_ ?_ ((position_initial hε hT₁ U z).trans (position_initial hε hT₂ V z).symm)
+  · intro t ht
+    exact (position_derivWithin hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩).mono
+      (Icc_subset_Icc le_rfl hS₁)
+  · intro t ht
+    rw [hUV ht]
+    exact (position_derivWithin hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩).mono
+      (Icc_subset_Icc le_rfl hS₂)
+
+lemma logDerivative_prefix_eq {ε T₁ T₂ S : ℝ} (hε : 0 < ε)
+    (hT₁ : 0 ≤ T₁) (hT₂ : 0 ≤ T₂) (hS : 0 ≤ S) (hS₁ : S ≤ T₁) (hS₂ : S ≤ T₂)
+    (U V : C(I, ℝ)) (z : ℂ)
+    (hUV : EqOn (extendDriver T₁ U) (extendDriver T₂ V) (Icc 0 S)) :
+    EqOn (logDerivative hε hT₁ U z) (logDerivative hε hT₂ V z) (Icc 0 S) := by
+  have hH := position_prefix_eq hε hT₁ hT₂ hS hS₁ hS₂ U V z hUV
+  intro t ht
+  rw [logDerivative_eq_integral hε hT₁ U z ⟨ht.1,ht.2.trans hS₁⟩,
+    logDerivative_eq_integral hε hT₂ V z ⟨ht.1,ht.2.trans hS₂⟩]
+  apply intervalIntegral.integral_congr
+  intro s hs
+  rw [uIcc_of_le ht.1] at hs
+  have hsS : s ∈ Icc 0 S := ⟨hs.1,hs.2.trans ht.2⟩
+  simp only [variation, hH hsS, hUV hsS]
+
+end SLEExactGauge.LoewnerCutoff
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+namespace SLEExactGauge.GreenPhysical
+
+def green (p a x y l : ℝ) : ℝ :=
+  Real.exp (p*l - p*Real.log 2 + (a-p)*Real.log y - a/2*Real.log (x^2+y^2))
+
+lemma q_pos (x : ℝ) {y : ℝ} (hy : 0 < y) : 0 < x^2+y^2 := by positivity
+
+lemma green_eq_source (p a x y l : ℝ) (hy : 0 < y) :
+    green p a x y l = (2*y / Real.exp l)^(-p) * (y / Real.sqrt (x^2+y^2))^a := by
+  have hQ := q_pos x hy
+  have hR : 0 < 2*y / Real.exp l := div_pos (by positivity) (Real.exp_pos _)
+  have hS : 0 < y / Real.sqrt (x^2+y^2) := div_pos hy (Real.sqrt_pos.2 hQ)
+  rw [Real.rpow_def_of_pos hR, Real.rpow_def_of_pos hS, ← Real.exp_add]
+  dsimp [green]
+  congr 1
+  rw [Real.log_div (by positivity : 2*y ≠ 0) (Real.exp_ne_zero _),
+    Real.log_mul (by norm_num : (2 : ℝ) ≠ 0) hy.ne', Real.log_exp,
+    Real.log_div hy.ne' (by positivity : Real.sqrt (x^2+y^2) ≠ 0),
+    Real.log_sqrt hQ.le]
+  ring
+
+lemma green_pos (p a x y l : ℝ) : 0 < green p a x y l := Real.exp_pos _
+
+lemma hasDerivAt_green_path (p a : ℝ) {X Y L : ℝ → ℝ} {t X' Y' L' : ℝ}
+    (hX : HasDerivAt X X' t) (hY : HasDerivAt Y Y' t) (hL : HasDerivAt L L' t)
+    (hy : 0 < Y t) :
+    HasDerivAt (fun u => green p a (X u) (Y u) (L u))
+      (green p a (X t) (Y t) (L t) *
+        (p*L' + (a-p)*Y'/Y t - a*(X t*X' + Y t*Y')/(X t^2+Y t^2))) t := by
+  have hlogY := hY.log hy.ne'
+  have hlogQ := ((hX.pow 2).add (hY.pow 2)).log (q_pos (X t) hy).ne'
+  have hd := ((((hL.const_mul p).sub_const (p*Real.log 2)).add
+    (hlogY.const_mul (a-p))).sub (hlogQ.const_mul (a/2))).exp
+  apply hd.congr_deriv
+  dsimp [green]
+  ring
+
+lemma hasDerivAt_green_x (p a x y l : ℝ) (hy : 0 < y) :
+    HasDerivAt (fun u => green p a u y l) (green p a x y l * (-a*x/(x^2+y^2))) x := by
+  convert hasDerivAt_green_path p a (hasDerivAt_id x) (hasDerivAt_const x y)
+    (hasDerivAt_const x l) hy using 1 <;> dsimp only [id_eq]; ring
+
+lemma deriv_green_x (p a x y l : ℝ) (hy : 0 < y) :
+    deriv (fun u => green p a u y l) x = green p a x y l * (-a*x/(x^2+y^2)) :=
+  (hasDerivAt_green_x p a x y l hy).deriv
+
+lemma hasDerivAt_neg_ax_div_q (a x y : ℝ) (hy : 0 < y) :
+    HasDerivAt (fun u => -a*u/(u^2+y^2))
+      (a*(x^2-y^2)/(x^2+y^2)^2) x := by
+  have hd := ((hasDerivAt_id x).const_mul (-a)).div
+    (((hasDerivAt_id x).pow 2).add_const (y^2)) (q_pos x hy).ne'
+  apply hd.congr_deriv
+  simp only [id_eq, Pi.pow_apply]
+  norm_num
+  ring
+
+lemma deriv_deriv_green_x (p a x y l : ℝ) (hy : 0 < y) :
+    deriv (deriv (fun u => green p a u y l)) x =
+      green p a x y l * (a*(a+1)*x^2-a*y^2)/(x^2+y^2)^2 := by
+  have heq : deriv (fun u => green p a u y l) =
+      (fun u => green p a u y l * (-a*u/(u^2+y^2))) :=
+    funext (fun u => deriv_green_x p a u y l hy)
+  rw [heq]
+  have hd := (hasDerivAt_green_x p a x y l hy).mul (hasDerivAt_neg_ax_div_q a x y hy)
+  calc
+    _ = _ := hd.deriv
+    _ = _ := by
+      field_simp
+      ring
+
+lemma green_drift_cancellation (κ x y M : ℝ) (hκ : κ ≠ 0) (hy : 0 < y) :
+    let p := 1-κ/8
+    let a := 8/κ-1
+    M * (p * (-2*(x^2-y^2)/(x^2+y^2)^2) +
+      (a-p) * (-2*y/(x^2+y^2))/y -
+      a * (x*(2*x/(x^2+y^2)) + y*(-2*y/(x^2+y^2)))/(x^2+y^2)) +
+    κ/2 * (M*(a*(a+1)*x^2-a*y^2)/(x^2+y^2)^2) = 0 := by
+  dsimp only
+  field_simp
+  ring
+
+lemma loewner_green_drift {κ t : ℝ} (hκ : 0 < κ) {X Y L : ℝ → ℝ}
+    (hy : 0 < Y t)
+    (hX : HasDerivAt X (2*X t/(X t^2+Y t^2)) t)
+    (hY : HasDerivAt Y (-2*Y t/(X t^2+Y t^2)) t)
+    (hL : HasDerivAt L (-2*(X t^2-Y t^2)/(X t^2+Y t^2)^2) t) :
+    HasDerivAt (fun u => green (1-κ/8) (8/κ-1) (X u) (Y u) (L u))
+      (-(κ/2) * deriv (deriv (fun x => green (1-κ/8) (8/κ-1) x (Y t) (L t))) (X t)) t := by
+  have hd := hasDerivAt_green_path (1-κ/8) (8/κ-1) hX hY hL hy
+  apply hd.congr_deriv
+  rw [deriv_deriv_green_x _ _ _ _ _ hy]
+  have hc := green_drift_cancellation κ (X t) (Y t)
+    (green (1-κ/8) (8/κ-1) (X t) (Y t) (L t)) hκ.ne' hy
+  dsimp only at hc
+  linarith
+
+lemma hasDerivAt_green_driver (p a c y l sdev b : ℝ) (hy : 0 < y) :
+    HasDerivAt (fun v => green p a (c-sdev*v) y l)
+      (sdev*a*(c-sdev*b)/((c-sdev*b)^2+y^2) * green p a (c-sdev*b) y l) b := by
+  have hd := (hasDerivAt_green_x p a (c-sdev*b) y l hy).comp b
+    (((hasDerivAt_id b).const_mul sdev).const_sub c)
+  apply hd.congr_deriv
+  ring
+
+lemma deriv_green_driver (p a c y l sdev b : ℝ) (hy : 0 < y) :
+    deriv (fun v => green p a (c-sdev*v) y l) b =
+      sdev*a*(c-sdev*b)/((c-sdev*b)^2+y^2) * green p a (c-sdev*b) y l :=
+  (hasDerivAt_green_driver p a c y l sdev b hy).deriv
+
+lemma hasDerivAt_deriv_green_x (p a x y l : ℝ) (hy : 0 < y) :
+    HasDerivAt (deriv (fun u => green p a u y l))
+      (green p a x y l * (a*(a+1)*x^2-a*y^2)/(x^2+y^2)^2) x := by
+  have heq : deriv (fun u => green p a u y l) =
+      (fun u => green p a u y l * (-a*u/(u^2+y^2))) :=
+    funext (fun u => deriv_green_x p a u y l hy)
+  rw [heq]
+  apply ((hasDerivAt_green_x p a x y l hy).mul (hasDerivAt_neg_ax_div_q a x y hy)).congr_deriv
+  field_simp
+  ring
+
+lemma deriv_deriv_green_driver (p a c y l sdev b : ℝ) (hy : 0 < y) :
+    deriv (deriv (fun v => green p a (c-sdev*v) y l)) b =
+      sdev^2 * (green p a (c-sdev*b) y l *
+        (a*(a+1)*(c-sdev*b)^2-a*y^2)/((c-sdev*b)^2+y^2)^2) := by
+  have heq : deriv (fun v => green p a (c-sdev*v) y l) =
+      fun v => (-sdev) * deriv (fun u => green p a u y l) (c-sdev*v) := by
+    ext v
+    rw [deriv_green_driver _ _ _ _ _ _ _ hy, deriv_green_x _ _ _ _ _ hy]
+    ring
+  rw [heq]
+  have hd := ((hasDerivAt_deriv_green_x p a (c-sdev*b) y l hy).comp b
+    (((hasDerivAt_id b).const_mul sdev).const_sub c)).const_mul (-sdev)
+  calc
+    _ = _ := hd.deriv
+    _ = _ := by ring
+
+lemma green_le_radius (p a x y l : ℝ) (hy : 0 < y) (ha : 0 ≤ a) :
+    green p a x y l ≤ (2*y/Real.exp l)^(-p) := by
+  rw [green_eq_source _ _ _ _ _ hy]
+  have hQ := q_pos x hy
+  have hyS : y ≤ Real.sqrt (x^2+y^2) := by
+    rw [Real.le_sqrt hy.le (by positivity)]
+    nlinarith [sq_nonneg x]
+  have hr : 0 ≤ (2*y/Real.exp l)^(-p) := Real.rpow_nonneg (by positivity) _
+  have hs : (y/Real.sqrt (x^2+y^2))^a ≤ 1 :=
+    Real.rpow_le_one (by positivity) ((div_le_one (Real.sqrt_pos.2 hQ)).2 hyS) ha
+  simpa using mul_le_mul_of_nonneg_left hs hr
+
+lemma green_le_fixed_radius (p a x y l ρ : ℝ) (hy : 0 < y) (ha : 0 ≤ a)
+    (hp : 0 ≤ p) (hρ : 0 < ρ) (hR : ρ ≤ 2*y/Real.exp l) :
+    green p a x y l ≤ ρ^(-p) := by
+  exact (green_le_radius p a x y l hy ha).trans
+    (Real.rpow_le_rpow_of_nonpos hρ hR (neg_nonpos.2 hp))
+
+lemma contDiff_green_x (p a y l : ℝ) (hy : 0 < y) {n : ℕ∞ω} :
+    ContDiff ℝ n (fun x => green p a x y l) := by
+  unfold green
+  apply ContDiff.exp
+  apply ContDiff.sub contDiff_const
+  exact contDiff_const.mul (((contDiff_id.pow 2).add contDiff_const).log
+    (fun x => (q_pos x hy).ne'))
+
+lemma contDiff_green_driver (p a c y l sdev : ℝ) (hy : 0 < y) {n : ℕ∞ω} :
+    ContDiff ℝ n (fun b => green p a (c-sdev*b) y l) :=
+  (contDiff_green_x p a y l hy).comp (contDiff_const.sub (contDiff_const.mul contDiff_id))
+
+lemma hasDerivAt_second_green_x (p a x y l : ℝ) (hy : 0 < y) :
+    HasDerivAt (deriv (deriv (fun v => green p a v y l)))
+      (green p a x y l * (a*(a+2)*x*(3*y^2-(a+1)*x^2))/(x^2+y^2)^3) x := by
+  have he : deriv (deriv (fun v => green p a v y l)) =
+      fun v => green p a v y l * (a*(a+1)*v^2-a*y^2)/(v^2+y^2)^2 :=
+    funext (fun v => deriv_deriv_green_x p a v y l hy)
+  rw [he]
+  have hn := (hasDerivAt_green_x p a x y l hy).mul
+    ((((hasDerivAt_id x).pow 2).const_mul (a*(a+1))).sub_const (a*y^2))
+  have hq := (((hasDerivAt_id x).pow 2).add_const (y^2)).pow 2
+  apply (hn.div hq (pow_ne_zero 2 (q_pos x hy).ne')).congr_deriv
+  simp only [id_eq, Pi.pow_apply, Pi.mul_apply]
+  field_simp
+  ring
+
+lemma third_green_x (p a x y l : ℝ) (hy : 0 < y) :
+    iteratedDeriv 3 (fun v => green p a v y l) x =
+      green p a x y l * (a*(a+2)*x*(3*y^2-(a+1)*x^2))/(x^2+y^2)^3 := by
+  simpa only [iteratedDeriv_succ, iteratedDeriv_zero] using
+    (hasDerivAt_second_green_x p a x y l hy).deriv
+
+lemma iteratedDeriv_green_driver (p a c y l sdev : ℝ) (hy : 0 < y) (k : ℕ) (b : ℝ) :
+    iteratedDeriv k (fun v => green p a (c-sdev*v) y l) b =
+      (-sdev)^k * iteratedDeriv k (fun x => green p a x y l) (c-sdev*b) := by
+  induction k generalizing b with
+  | zero => simp
+  | succ k ih =>
+    have he : iteratedDeriv k (fun v => green p a (c-sdev*v) y l) =
+        fun v => (-sdev)^k * iteratedDeriv k (fun x => green p a x y l) (c-sdev*v) := funext ih
+    rw [iteratedDeriv_succ, he]
+    have hd : Differentiable ℝ (iteratedDeriv k (fun x => green p a x y l)) :=
+      ContDiff.differentiable_iteratedDeriv k (contDiff_green_x p a y l hy (n := ⊤)) (by simp)
+    have hh := ((hd (c-sdev*b)).hasDerivAt.comp b
+      (((hasDerivAt_id b).const_mul sdev).const_sub c)).const_mul ((-sdev)^k)
+    dsimp only [Function.comp_def, id_eq] at hh
+    rw [hh.deriv]
+    simp only [iteratedDeriv_succ, pow_succ, mul_one]
+    ring
+
+lemma abs_x_div_q_le {ε y : ℝ} (hε : 0 < ε) (hy : ε ≤ y) (x : ℝ) :
+    |x| / (x^2+y^2) ≤ 1/ε := by
+  have hq := q_pos x (hε.trans_le hy)
+  apply (div_le_div_iff₀ hq hε).2
+  have hεy : ε^2 ≤ y^2 := pow_le_pow_left₀ hε.le hy 2
+  nlinarith [sq_nonneg (|x|-ε), sq_abs x, sq_nonneg x]
+
+lemma inv_q_le {ε y : ℝ} (hε : 0 < ε) (hy : ε ≤ y) (x : ℝ) :
+    1/(x^2+y^2) ≤ 1/ε^2 := by
+  apply div_le_div_of_nonneg_left zero_le_one (sq_pos_of_pos hε)
+  exact (pow_le_pow_left₀ hε.le hy 2).trans (le_add_of_nonneg_left (sq_nonneg x))
+
+lemma third_green_bound {p a x y l ε M : ℝ} (hε : 0 < ε) (hy : ε ≤ y)
+    (ha : 0 ≤ a) (hM : green p a x y l ≤ M) :
+    |iteratedDeriv 3 (fun v => green p a v y l) x| ≤
+      M * (a*(a+2)*(a+4)) / ε^3 := by
+  have hypos := hε.trans_le hy
+  have hq := q_pos x hypos
+  have hMp : 0 ≤ M := (green_pos p a x y l).le.trans hM
+  have hP : |3*y^2-(a+1)*x^2| ≤ (a+4)*(x^2+y^2) := by
+    calc
+      _ ≤ |3*y^2| + |(a+1)*x^2| := abs_sub _ _
+      _ = 3*y^2+(a+1)*x^2 := by rw [abs_of_nonneg (by positivity), abs_of_nonneg (by positivity)]
+      _ ≤ _ := by nlinarith [mul_nonneg ha (sq_nonneg y), sq_nonneg x]
+  rw [third_green_x _ _ _ _ _ hypos]
+  simp only [abs_div, abs_mul, abs_pow, abs_of_pos hq, abs_of_pos (green_pos p a x y l),
+    abs_of_nonneg ha, abs_of_nonneg (show 0 ≤ a+2 by linarith)]
+  calc
+    _ ≤ M * (a*(a+2)* |x| *((a+4)*(x^2+y^2))) / (x^2+y^2)^3 := by
+      gcongr
+    _ = M*(a*(a+2)*(a+4)) * (|x|/(x^2+y^2)) * (1/(x^2+y^2)) := by
+      field_simp
+    _ ≤ M*(a*(a+2)*(a+4)) * (1/ε) * (1/ε^2) := by
+      exact mul_le_mul
+        (mul_le_mul_of_nonneg_left (abs_x_div_q_le hε hy x) (by positivity))
+        (inv_q_le hε hy x) (by positivity) (by positivity)
+    _ = _ := by ring
+
+lemma continuous_green_comp {X : Type*} [TopologicalSpace X]
+    (p a : ℝ) {x y l : X → ℝ} (hx : Continuous x) (hy : Continuous y)
+    (hl : Continuous l) (hyp : ∀ t, 0 < y t) :
+    Continuous (fun t => green p a (x t) (y t) (l t)) := by
+  unfold green
+  apply Real.continuous_exp.comp
+  exact (((continuous_const.mul hl).sub continuous_const).add
+    (continuous_const.mul (hy.log (fun t => (hyp t).ne')))).sub
+    (continuous_const.mul (((hx.pow 2).add (hy.pow 2)).log
+      (fun t => (q_pos (x t) (hyp t)).ne')))
+
+lemma first_green_bound {p a x y l ε M : ℝ} (hε : 0 < ε) (hy : ε ≤ y)
+    (ha : 0 ≤ a) (hM : green p a x y l ≤ M) :
+    |deriv (fun v => green p a v y l) x| ≤ M*a/ε := by
+  have hp := green_pos p a x y l
+  rw [deriv_green_x p a x y l (hε.trans_le hy)]
+  simp only [abs_mul, abs_div, abs_neg, abs_of_nonneg ha,
+    abs_of_pos (q_pos x (hε.trans_le hy)), abs_of_pos hp]
+  calc
+    _ = green p a x y l * a * (|x|/(x^2+y^2)) := by ring
+    _ ≤ M*a*(1/ε) := mul_le_mul
+      (mul_le_mul_of_nonneg_right hM ha) (abs_x_div_q_le hε hy x)
+      (by positivity) (mul_nonneg (hp.le.trans hM) ha)
+    _ = _ := by ring
+
+lemma second_green_bound {p a x y l ε M : ℝ} (hε : 0 < ε) (hy : ε ≤ y)
+    (ha : 0 ≤ a) (hM : green p a x y l ≤ M) :
+    |iteratedDeriv 2 (fun v => green p a v y l) x| ≤ M*(a*(a+2))/ε^2 := by
+  have hp := green_pos p a x y l
+  have hMp : 0 ≤ M := hp.le.trans hM
+  have hq := q_pos x (hε.trans_le hy)
+  have hP : |a*(a+1)*x^2-a*y^2| ≤ a*(a+2)*(x^2+y^2) := by
+    calc
+      _ ≤ |a*(a+1)*x^2|+|a*y^2| := abs_sub _ _
+      _ = a*(a+1)*x^2+a*y^2 := by rw [abs_of_nonneg (by positivity), abs_of_nonneg (by positivity)]
+      _ ≤ _ := by nlinarith [mul_nonneg ha (sq_nonneg x), mul_nonneg (mul_nonneg ha (by positivity : 0 ≤ a+1)) (sq_nonneg y)]
+  simp only [iteratedDeriv_succ, iteratedDeriv_zero]
+  rw [deriv_deriv_green_x p a x y l (hε.trans_le hy)]
+  simp only [abs_div, abs_mul, abs_pow, abs_of_pos hq, abs_of_pos hp]
+  calc
+    _ ≤ M*(a*(a+2)*(x^2+y^2))/(x^2+y^2)^2 := by gcongr
+    _ = M*(a*(a+2))*(1/(x^2+y^2)) := by field_simp
+    _ ≤ M*(a*(a+2))*(1/ε^2) := mul_le_mul_of_nonneg_left (inv_q_le hε hy x) (by positivity)
+    _ = _ := by ring
+
+end SLEExactGauge.GreenPhysical
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+noncomputable section
+open Set Filter
+open scoped Topology ContDiff
+namespace SLEExactGauge.LoewnerField
+open LoewnerCutoff GreenPhysical
+open scoped unitInterval
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+def height (U : C(I, ℝ)) (z : ℂ) (t : ℝ) : ℝ :=
+  max ε (position hε hT U z t).im
+
+def field (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
+  green p a ((position hε hT U z t).re - sdev*b) (height hε hT U z t)
+    (logDerivative hε hT U z t)
+
+def drift (p a sdev : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
+  let x := (position hε hT U z t).re - sdev*b
+  let y := height hε hT U z t
+  let v := velocity ε (extendDriver T U t) (position hε hT U z t)
+  field hε hT p a sdev U z t b *
+    (p*variation hε hT U z t + (a-p)*v.im/y - a*(x*v.re+y*v.im)/(x^2+y^2))
+
+end SLEExactGauge.LoewnerField
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/StoppedMeshes.lean b/lean/OAI/Probability/SLELocalization/StoppedMeshes.lean
index cbd6071b583144ee026b8f1f8676b5d963a4fe51..c93b84a0597662ee601fbeff13b05563af4e54bd 100644
--- a/lean/OAI/Probability/SLELocalization/StoppedMeshes.lean
+++ b/lean/OAI/Probability/SLELocalization/StoppedMeshes.lean
@@ -1,1069 +1,1069 @@
-import Mathlib
-import OAI.Probability.SLELocalization.TraceModel
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Localization estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-lemma physical_clock_bounds {T δ Y₀ u : ℝ} {Y C Q : ℝ → ℝ}
-    (hT : 0 ≤ T) (hδ : 0 < δ) (hY₀ : 0 < Y₀) (hYinit : Y 0 = Y₀) (hCinit : C 0 = 0)
-    (hY : ContinuousOn Y (Icc 0 T)) (hC : ContinuousOn C (Icc 0 T))
-    (hYpos : ∀ t ∈ Icc 0 T, 0 < Y t)
-    (hYle : ∀ t ∈ Icc 0 T, Y t ≤ Y₀)
-    (hQpos : ∀ t ∈ Ioo 0 T, 0 < Q t)
-    (hangle : ∀ t ∈ Ioo 0 T, δ^2 * Q t ≤ (Y t)^2)
-    (hdY : ∀ t ∈ Ioo 0 T, HasDerivAt (fun s => Real.log (Y s)) (-2 / Q t) t)
-    (hdC : ∀ t ∈ Ioo 0 T, HasDerivAt C (4*(Y t)^2 / (Q t)^2) t)
-    (hCu : C T ≤ u) :
-    Y₀ * Real.exp (-u/(2*δ^2)) ≤ Y T ∧ T ≤ u * Y₀^2 / (4*δ^4) := by
-  have hd2 : 0 < δ^2 := sq_pos_of_pos hδ
-  have hq2 (t : ℝ) (ht : t ∈ Ioo 0 T) : 0 < Q t ^ 2 := sq_pos_of_pos (hQpos t ht)
-  have hlogY := hY.log (fun t ht => (hYpos t ht).ne')
-  have hm : MonotoneOn (fun t => Real.log (Y t) + C t / (2*δ^2)) (Icc 0 T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 T)
-      (hlogY.add (hC.div_const _))
-      (f' := fun t => -2/Q t + (4*(Y t)^2/(Q t)^2)/(2*δ^2))
-    · intro t ht
-      rw [interior_Icc] at ht
-      exact ((hdY t ht).add ((hdC t ht).div_const _)).hasDerivWithinAt
-    · intro t ht
-      rw [interior_Icc] at ht
-      have hq := hQpos t ht
-      have ha := hangle t ht
-      apply (mul_le_mul_iff_of_pos_right (mul_pos (hq2 t ht) (mul_pos (by norm_num : (0:ℝ)<2) hd2))).1
-      field_simp
-      nlinarith
-  have hlogbound := hm ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
-  simp only [hYinit, hCinit, zero_div, add_zero] at hlogbound
-  have hL : Real.log Y₀ + (-u/(2*δ^2)) ≤ Real.log (Y T) := by
-    have hh : C T/(2*δ^2) ≤ u/(2*δ^2) :=
-      div_le_div_of_nonneg_right hCu (mul_pos (by norm_num) hd2).le
-    simp only [neg_div]
-    linarith
-  have hheight : Y₀ * Real.exp (-u/(2*δ^2)) ≤ Y T := by
-    have hh := Real.exp_le_exp.mpr hL
-    simpa only [Real.exp_add, Real.exp_log hY₀, Real.exp_log (hYpos T ⟨hT,le_rfl⟩)] using hh
-  have hK : 0 < 4*δ^4/Y₀^2 := by positivity
-  have hderivbound (t : ℝ) (ht : t ∈ Ioo 0 T) :
-      4*δ^4/Y₀^2 ≤ 4*(Y t)^2/(Q t)^2 := by
-    apply (div_le_div_iff₀ (sq_pos_of_pos hY₀) (hq2 t ht)).2
-    have ha := hangle t ht
-    have hb : (δ^2*Q t)^2 ≤ ((Y t)^2)^2 :=
-      pow_le_pow_left₀ (mul_nonneg hd2.le (hQpos t ht).le) ha 2
-    have hc : (Y t)^2 ≤ Y₀^2 :=
-      pow_le_pow_left₀ (hYpos t ⟨ht.1.le,ht.2.le⟩).le (hYle t ⟨ht.1.le,ht.2.le⟩) 2
-    have he := mul_le_mul_of_nonneg_left hc (sq_nonneg (Y t))
-    nlinarith
-  have hmC : MonotoneOn (fun t => C t - (4*δ^4/Y₀^2)*t) (Icc 0 T) := by
-    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 T)
-      (hC.sub (continuousOn_id.const_mul _))
-      (f' := fun t => 4*(Y t)^2/(Q t)^2 - (4*δ^4/Y₀^2))
-    · intro t ht
-      rw [interior_Icc] at ht
-      simpa only [mul_one] using
-        ((hdC t ht).sub ((hasDerivAt_id t).const_mul (4*δ^4/Y₀^2))).hasDerivWithinAt
-          (s := interior (Icc 0 T))
-    · intro t ht
-      rw [interior_Icc] at ht
-      exact sub_nonneg.2 (hderivbound t ht)
-  have htime := hmC ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
-  simp only [hCinit, mul_zero, sub_zero] at htime
-  refine ⟨hheight, ?_⟩
-  have hh : T ≤ u / (4*δ^4/Y₀^2) := (le_div_iff₀ hK).2 (by linarith)
-  convert hh using 1
-  field_simp
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma finite_clock_bounds (h : IsLoewnerTraceUsing driver γ g f)
-    {T δ u : ℝ} (hT : 0 ≤ T) (hδ : 0 < δ)
-    (hU : ContinuousOn driver (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    (hangle : ∀ t ∈ Ioo 0 T,
-      δ^2 * Complex.normSq (g t z - (driver t : ℂ)) ≤ (g t z).im^2)
-    (hr : (2*z.im)*Real.exp (-u) ≤ loewnerRadius g T z) :
-    z.im * Real.exp (-u/(2*δ^2)) ≤ (g T z).im ∧
-      T ≤ u*z.im^2/(4*δ^4) := by
-  have hz0 : z ∈ halfPlane := hz.1.1
-  have hy0 : 0 < z.im := hz0
-  let C : ℝ → ℝ := fun t => Real.log (2*z.im) - Real.log (loewnerRadius g t z)
-  apply physical_clock_bounds (Y := fun t => (g t z).im) (C := C)
-    (Q := fun t => Complex.normSq (g t z - (driver t : ℂ))) hT hδ hy0
-  · rw [h.initial z hz0]
-  · simp only [C, h.radius_initial hz0, sub_self]
-  · exact Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)
-  · exact continuousOn_const.sub (h.continuous_log_radius hT hU hz)
-  · intro t ht
-    exact h.im_pos ht.1 (survivingDomain_mono ht.2 hz)
-  · intro t ht
-    exact h.height_le_initial ht.1 (survivingDomain_mono ht.2 hz)
-  · intro t ht
-    exact Complex.normSq_pos.2 (h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hz))
-  · exact hangle
-  · intro t ht
-    have hi := h.im_pos ht.1.le (survivingDomain_mono ht.2.le hz)
-    apply ((h.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hz)).log hi.ne').congr_deriv
-    field_simp
-  · intro t ht
-    change HasDerivAt (fun s => Real.log (2*z.im) - Real.log (loewnerRadius g s z)) _ t
-    apply ((h.hasDerivAt_log_radius hT hU hz ht).const_sub (Real.log (2*z.im))).congr_deriv
-    ring
-  · have hh := Real.log_le_log (mul_pos (mul_pos (by norm_num) hy0) (Real.exp_pos (-u))) hr
-    rw [Real.log_mul (mul_ne_zero (by norm_num) hy0.ne') (Real.exp_ne_zero _), Real.log_exp] at hh
-    dsimp [C]
-    linarith
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.PartitionMeshes
-open TaylorMeshes TimeField
-
-structure PartitionSequence (T : ℝ) where
-  count : ℕ → ℕ
-  time : ℕ → ℕ → ℝ
-  step : ℕ → ℝ
-  start : ∀ n, time n 0 = 0
-  finish : ∀ n, time n (count n) = T
-  mono : ∀ n, Monotone (time n)
-  step_le : ∀ n i, i < count n → time n (i+1)-time n i ≤ step n
-  step_zero : Tendsto step atTop (𝓝 0)
-
-end SLEExactGauge.PartitionMeshes
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.PartitionMeshes
-open TaylorMeshes TimeField
-namespace PartitionSequence
-variable {T : ℝ} (p : PartitionSequence T)
-
-include p in
-lemma nonneg : 0 ≤ T := by
-  simpa only [p.start, p.finish] using p.mono 0 (Nat.zero_le (p.count 0))
-
-lemma mem (n i : ℕ) (hi : i ≤ p.count n) : p.time n i ∈ Icc 0 T := by
-  constructor
-  · simpa only [p.start] using p.mono n (Nat.zero_le i)
-  · simpa only [p.finish] using p.mono n hi
-
-lemma increments_small {b : ℝ → ℝ} (hb : ContinuousOn b (Icc 0 T))
-    {δ : ℝ} (hδ : 0 < δ) :
-    ∀ᶠ n in atTop, ∀ i < p.count n, |b (p.time n (i+1))-b (p.time n i)| < δ := by
-  obtain ⟨ε, hε, hmod⟩ := Metric.uniformContinuousOn_iff.1
-    (isCompact_Icc.uniformContinuousOn_of_continuous hb) δ hδ
-  filter_upwards [p.step_zero.eventually (gt_mem_nhds hε)] with n hn
-  intro i hi
-  apply hmod _ (p.mem n (i+1) hi) _ (p.mem n i hi.le)
-  rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.2 (p.mono n (Nat.le_succ i)))]
-  exact (p.step_le n i hi).trans_lt hn
-
-lemma riemann_tendsto {h : ℝ → ℝ} (hh : ContinuousOn h (Icc 0 T)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      h (p.time n i) * (p.time n (i+1)-p.time n i)) atTop
-      (𝓝 (∫ t in (0 : ℝ)..T, h t)) := by
-  have hT := p.nonneg
-  have hint (n i : ℕ) (hi : i < p.count n) :
-      IntervalIntegrable h volume (p.time n i) (p.time n (i+1)) := by
-    apply ContinuousOn.intervalIntegrable
-    rw [uIcc_of_le (p.mono n (Nat.le_succ i))]
-    exact hh.mono (Icc_subset_Icc (p.mem n i hi.le).1 (p.mem n (i+1) hi).2)
-  have hu := isCompact_Icc.uniformContinuousOn_of_continuous hh
-  apply Metric.tendsto_atTop.2
-  intro ε hε
-  let ζ : ℝ := ε / (T+1)
-  have hζ : 0 < ζ := div_pos hε (by linarith)
-  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1 hu ζ hζ
-  obtain ⟨N, hN⟩ := eventually_atTop.1 (p.step_zero.eventually (gt_mem_nhds hδ))
-  refine ⟨N, fun n hn => ?_⟩
-  have hnδ := hN n hn
-  have hsum := intervalIntegral.sum_integral_adjacent_intervals (fun i hi => hint n i hi)
-  rw [p.start, p.finish] at hsum
-  have heach (i : ℕ) (hi : i < p.count n) :
-      |h (p.time n i) * (p.time n (i+1)-p.time n i) -
-        ∫ t in p.time n i..p.time n (i+1), h t| ≤
-          ζ * (p.time n (i+1)-p.time n i) := by
-    have heq : h (p.time n i) * (p.time n (i+1)-p.time n i) =
-        ∫ _ : ℝ in p.time n i..p.time n (i+1), h (p.time n i) := by
-      rw [intervalIntegral.integral_const, smul_eq_mul, mul_comm]
-    rw [heq, ← intervalIntegral.integral_sub intervalIntegrable_const (hint n i hi)]
-    have hbound : ∀ t ∈ uIoc (p.time n i) (p.time n (i+1)),
-        ‖h (p.time n i)-h t‖ ≤ ζ := by
-      intro t ht
-      rw [uIoc_of_le (p.mono n (Nat.le_succ i))] at ht
-      have htT : t ∈ Icc 0 T := ⟨(p.mem n i hi.le).1.trans ht.1.le,
-        ht.2.trans (p.mem n (i+1) hi).2⟩
-      have hd : dist (p.time n i) t < δ := by
-        rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 ht.1.le)]
-        have hs := p.step_le n i hi
-        have ht' := ht.2
-        linarith
-      simpa only [Real.norm_eq_abs, Real.dist_eq] using
-        (hmod _ (p.mem n i hi.le) _ htT hd).le
-    simpa only [Real.norm_eq_abs,
-      abs_of_nonneg (sub_nonneg.2 (p.mono n (Nat.le_succ i)))] using
-      intervalIntegral.norm_integral_le_of_norm_le_const hbound
-  rw [Real.dist_eq, ← hsum, ← Finset.sum_sub_distrib]
-  calc
-    _ ≤ ∑ i ∈ Finset.range (p.count n),
-        |h (p.time n i) * (p.time n (i+1)-p.time n i) -
-          ∫ t in p.time n i..p.time n (i+1), h t| := Finset.abs_sum_le_sum_abs _ _
-    _ ≤ ∑ i ∈ Finset.range (p.count n), ζ * (p.time n (i+1)-p.time n i) :=
-      Finset.sum_le_sum (fun i hi => heach i (Finset.mem_range.1 hi))
-    _ = ζ * T := by
-      rw [← Finset.mul_sum, Finset.sum_range_sub (p.time n), p.start, p.finish, sub_zero]
-    _ < ε := by
-      dsimp only [ζ]
-      rw [div_mul_eq_mul_div, div_lt_iff₀ (by linarith : 0 < T+1)]
-      nlinarith
-
-lemma family_remainder_tendsto {f : ℕ → ℕ → ℝ → ℝ} {b : ℝ → ℝ}
-    (hf : ∀ n i, ContDiff ℝ 3 (f n i))
-    {V K : ℝ} (hb : ContinuousOn b (Icc 0 T)) (hK : 0 ≤ K)
-    (h3 : ∀ n i x, |iteratedDeriv 3 (f n i) x| ≤ K)
-    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      (b (p.time n (i+1))-b (p.time n i))^2) atTop (𝓝 V)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      remainder₂ (f n i) (b (p.time n i)) (b (p.time n (i+1)))) atTop (𝓝 0) := by
-  have hV : 0 ≤ V := ge_of_tendsto hQ
-    (Eventually.of_forall fun _ => Finset.sum_nonneg fun _ _ => sq_nonneg _)
-  apply Metric.tendsto_atTop.2
-  intro ε hε
-  let δ := ε / ((K+1)*(V+1))
-  have hδ : 0 < δ := div_pos hε (mul_pos (by linarith) (by linarith))
-  have hs := p.increments_small hb hδ
-  have hq := hQ.eventually (gt_mem_nhds (show V < V+1 by linarith))
-  obtain ⟨N, hN⟩ := eventually_atTop.1 (hs.and hq)
-  refine ⟨N, fun n hn => ?_⟩
-  rcases hN n hn with ⟨hs, hq⟩
-  have hr := sum_family_remainder₂_bound (hf n) (fun i => b (p.time n i))
-    (p.count n) K δ hK hδ.le (fun i _ z _ => h3 n i z) (fun i hi => (hs i hi).le)
-  simp only [Real.dist_eq, sub_zero]
-  calc
-    _ ≤ K * δ / 6 * ∑ i ∈ Finset.range (p.count n),
-        (b (p.time n (i+1))-b (p.time n i))^2 := hr
-    _ ≤ K * δ / 6 * (V+1) := mul_le_mul_of_nonneg_left hq.le (by positivity)
-    _ < ε := by
-      dsimp [δ]
-      have hden : 0 < (K+1)*(V+1) := mul_pos (by linarith) (by linarith)
-      have hid : K * (ε / ((K+1)*(V+1))) / 6 * (V+1) = ε * (K / (6*(K+1))) := by
-        field_simp
-      rw [hid]
-      have hk : K / (6*(K+1)) < 1 := (div_lt_one (by positivity)).2 (by linarith)
-      simpa using mul_lt_mul_of_pos_left hk hε
-
-end PartitionSequence
-
-end SLEExactGauge.PartitionMeshes
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.PartitionMeshes
-open TaylorMeshes TimeField
-
-def clippedDyadic (T τ : ℝ) (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T) :
-    PartitionSequence τ where
-  count := fun n => 2^n
-  time := fun n i => min (mesh T n i) τ
-  step := fun n => T / (2 : ℝ)^n
-  start := fun n => by rw [mesh_zero, min_eq_left hτ]
-  finish := fun n => by rw [mesh_end, min_eq_right hτT]
-  mono := fun n i j hij => min_le_min_right τ (mesh_mono T hT n hij)
-  step_le := by
-    intro n i hi
-    have hm := mesh_mono T hT n (Nat.le_succ i)
-    have hs := mesh_step T n i
-    by_cases h₁ : mesh T n i ≤ τ
-    · rw [min_eq_left h₁]
-      have hb := min_le_left (mesh T n (i+1)) τ
-      linarith
-    · rw [min_eq_right (le_of_not_ge h₁),
-        min_eq_right ((le_of_not_ge h₁).trans hm)]
-      simp only [sub_self]
-      positivity
-  step_zero := mesh_step_tendsto T
-
-end SLEExactGauge.PartitionMeshes
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.PartitionMeshes.PartitionSequence
-open SLEExactGauge.TaylorMeshes SLEExactGauge.TimeField
-
-lemma time_increment_tendsto {T : ℝ} (p : PartitionSequence T)
-    {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ}
-    (hF : ∀ x, ContinuousOn (fun t => F t x) (Icc 0 T))
-    (hD : ∀ t ∈ Ioo 0 T, ∀ x, HasDerivAt (fun u => F u x) (V t x) t)
-    (hV : ContinuousOn (fun q : ℝ × ℝ => V q.1 (b q.2)) ((Icc 0 T) ×ˢ (Icc 0 T))) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      (F (p.time n (i+1)) (b (p.time n (i+1))) -
-        F (p.time n i) (b (p.time n (i+1))))) atTop
-      (𝓝 (∫ t in (0 : ℝ)..T, V t (b t))) := by
-  have hT := p.nonneg
-  have hVc : ContinuousOn (fun t => V t (b t)) (Icc 0 T) :=
-    hV.comp (continuousOn_id.prodMk continuousOn_id) (fun t ht => ⟨ht, ht⟩)
-  have hriem := p.riemann_tendsto hVc
-  have herror : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      (F (p.time n (i+1)) (b (p.time n (i+1))) -
-        F (p.time n i) (b (p.time n (i+1))) -
-          V (p.time n i) (b (p.time n i)) * (p.time n (i+1)-p.time n i)))
-          atTop (𝓝 0) := by
-    have hu := (isCompact_Icc.prod isCompact_Icc).uniformContinuousOn_of_continuous hV
-    apply Metric.tendsto_atTop.2
-    intro ε hε
-    let ζ : ℝ := ε / (T+1)
-    have hζ : 0 < ζ := div_pos hε (by linarith)
-    obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1 hu ζ hζ
-    obtain ⟨N, hN⟩ := eventually_atTop.1 (p.step_zero.eventually (gt_mem_nhds hδ))
-    refine ⟨N, fun n hn => ?_⟩
-    have hnδ := hN n hn
-    have heach (i : ℕ) (hi : i < p.count n) :
-        |F (p.time n (i+1)) (b (p.time n (i+1))) -
-          F (p.time n i) (b (p.time n (i+1))) -
-            V (p.time n i) (b (p.time n i)) * (p.time n (i+1)-p.time n i)| ≤
-        ζ * (p.time n (i+1)-p.time n i) := by
-      rcases (p.mono n (Nat.le_succ i)).eq_or_lt with heq | hstep_pos
-      · simp [heq]
-      have hleft := p.mem n i hi.le
-      have hright := p.mem n (i+1) hi
-      obtain ⟨c, hc, heq⟩ := exists_hasDerivAt_eq_slope
-        (fun t => F t (b (p.time n (i+1)))) (fun t => V t (b (p.time n (i+1))))
-        hstep_pos ((hF _).mono (Icc_subset_Icc hleft.1 hright.2))
-        (fun t ht => hD t ⟨hleft.1.trans_lt ht.1, ht.2.trans_le hright.2⟩ _)
-      have hcT : c ∈ Icc 0 T := ⟨hleft.1.trans hc.1.le, hc.2.le.trans hright.2⟩
-      have hdist : dist (c, p.time n (i+1)) (p.time n i, p.time n i) < δ := by
-        rw [Prod.dist_eq, max_lt_iff]
-        constructor
-        · rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.2 hc.1.le)]
-          have hcs := hc.2
-          have hstep := p.step_le n i hi
-          linarith
-        · rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.2 hstep_pos.le)]
-          exact (p.step_le n i hi).trans_lt hnδ
-      have hvmod := (hmod (c, p.time n (i+1)) ⟨hcT, hright⟩
-        (p.time n i, p.time n i) ⟨hleft, hleft⟩ hdist).le
-      simp only [Real.dist_eq] at hvmod
-      have hid : F (p.time n (i+1)) (b (p.time n (i+1))) -
-          F (p.time n i) (b (p.time n (i+1))) =
-          V c (b (p.time n (i+1))) * (p.time n (i+1)-p.time n i) :=
-        ((eq_div_iff (sub_pos.2 hstep_pos).ne').1 heq).symm
-      rw [hid, ← sub_mul, abs_mul, abs_of_nonneg (sub_nonneg.2 hstep_pos.le)]
-      exact mul_le_mul_of_nonneg_right hvmod (sub_nonneg.2 hstep_pos.le)
-    rw [Real.dist_eq, sub_zero]
-    calc
-      _ ≤ ∑ i ∈ Finset.range (p.count n),
-          |F (p.time n (i+1)) (b (p.time n (i+1))) -
-            F (p.time n i) (b (p.time n (i+1))) -
-              V (p.time n i) (b (p.time n i)) * (p.time n (i+1)-p.time n i)| :=
-        Finset.abs_sum_le_sum_abs _ _
-      _ ≤ ∑ i ∈ Finset.range (p.count n), ζ * (p.time n (i+1)-p.time n i) :=
-        Finset.sum_le_sum (fun i hi => heach i (Finset.mem_range.1 hi))
-      _ = ζ * T := by
-        rw [← Finset.mul_sum, Finset.sum_range_sub (p.time n), p.start, p.finish, sub_zero]
-      _ < ε := by
-        dsimp [ζ]
-        rw [div_mul_eq_mul_div, div_lt_iff₀ (by linarith : 0 < T+1)]
-        nlinarith
-  have hadd := herror.add hriem
-  simp only [Finset.sum_sub_distrib, sub_add_cancel, zero_add] at hadd
-  simpa only [Finset.sum_sub_distrib] using hadd
-
-end SLEExactGauge.PartitionMeshes.PartitionSequence
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.StoppedMeshes
-open TaylorMeshes PartitionMeshes
-
-lemma one_crossing_sum_bound {N : ℕ} {t : ℕ → ℝ} (ht : Monotone t)
-    {τ E : ℝ} (hE : 0 ≤ E) {f : ℕ → ℝ}
-    (hz : ∀ i < N, ¬ (t i < τ ∧ τ < t (i+1)) → f i = 0)
-    (hb : ∀ i < N, t i < τ ∧ τ < t (i+1) → |f i| ≤ E) :
-    |∑ i ∈ Finset.range N, f i| ≤ E := by
-  classical
-  by_cases he : ∃ i < N, t i < τ ∧ τ < t (i+1)
-  · obtain ⟨i, hi, hcross⟩ := he
-    rw [Finset.sum_eq_single i]
-    · exact hb i hi hcross
-    · intro j hj hji
-      apply hz j (Finset.mem_range.1 hj)
-      intro hcj
-      rcases lt_or_gt_of_ne hji with hji | hij
-      · have hm := ht (show j+1 ≤ i by omega)
-        linarith [hcj.2, hcross.1]
-      · have hm := ht (show i+1 ≤ j by omega)
-        linarith [hcross.2, hcj.1]
-    · intro hni
-      exact (hni (Finset.mem_range.2 hi)).elim
-  · have heq : ∑ i ∈ Finset.range N, f i = 0 := by
-      apply Finset.sum_eq_zero
-      intro i hi
-      apply hz i (Finset.mem_range.1 hi)
-      intro hc
-      exact he ⟨i, Finset.mem_range.1 hi, hc⟩
-    simpa [heq] using hE
-
-lemma clipped_increment_eq {a c τ : ℝ} (hac : a ≤ c) (b : ℝ → ℝ)
-    (hn : ¬ (a < τ ∧ τ < c)) :
-    b (min c τ) - b (min a τ) = if a < τ then b c - b a else 0 := by
-  by_cases ha : a < τ
-  · rw [ite_eq_left ha, min_eq_left ha.le, min_eq_left (le_of_not_gt (fun hc => hn ⟨ha,hc⟩))]
-  · have hτa : τ ≤ a := le_of_not_gt ha
-    rw [ite_eq_right ha, min_eq_right hτa, min_eq_right (hτa.trans hac), sub_self]
-
-lemma clipped_timeSum_difference_tendsto {T τ : ℝ} (hT : 0 ≤ T)
-    (A : ℕ → ℕ → ℝ) {K : ℝ} (hK : 0 ≤ K) (hA : ∀ n i, |A n i| ≤ K) :
-    Tendsto (fun n =>
-      (∑ i ∈ Finset.range (2^n), A n i *
-        (min (mesh T n (i+1)) τ - min (mesh T n i) τ)) -
-      (∑ i ∈ Finset.range (2^n),
-        if mesh T n i < τ then A n i * (T / (2 : ℝ)^n) else 0)) atTop (𝓝 0) := by
-  refine squeeze_zero_norm (a := fun n => K * (T / (2 : ℝ)^n)) (fun n => ?_) ?_
-  · rw [Real.norm_eq_abs, ← Finset.sum_sub_distrib]
-    apply one_crossing_sum_bound (τ := τ) (mesh_mono T hT n) (mul_nonneg hK (by positivity))
-    · intro i hi hn
-      have heq := clipped_increment_eq (mesh_mono T hT n (Nat.le_succ i)) id hn
-      dsimp only [id_eq] at heq
-      rw [heq]
-      split_ifs <;> simp [mesh_step]
-    · intro i hi hc
-      rw [min_eq_right hc.2.le, min_eq_left hc.1.le, ite_eq_left hc.1]
-      rw [← mul_sub, abs_mul]
-      apply le_trans (mul_le_mul_of_nonneg_right (hA n i) (abs_nonneg _))
-      apply mul_le_mul_of_nonneg_left _ hK
-      have hs := mesh_step T n i
-      rw [abs_of_nonpos (by linarith)]
-      linarith [hc.1]
-  · simpa using (mesh_step_tendsto T).const_mul K
-
-lemma clipped_linearSum_difference_tendsto {T τ : ℝ} (hT : 0 ≤ T)
-    (hτ : 0 ≤ τ) (hτT : τ ≤ T) {b : ℝ → ℝ}
-    (hb : ContinuousOn b (Icc 0 T)) (A : ℕ → ℕ → ℝ)
-    {K : ℝ} (hK : 0 ≤ K) (hA : ∀ n i, |A n i| ≤ K) :
-    Tendsto (fun n =>
-      (∑ i ∈ Finset.range (2^n), A n i *
-        (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))) -
-      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-        A n i * (b (mesh T n (i+1)) - b (mesh T n i)) else 0)) atTop (𝓝 0) := by
-  apply Metric.tendsto_atTop.2
-  intro ε hε
-  let η : ℝ := ε / (K+1)
-  have hη : 0 < η := div_pos hε (by linarith)
-  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1
-    (isCompact_Icc.uniformContinuousOn_of_continuous hb) η hη
-  obtain ⟨N, hN⟩ := eventually_atTop.1 ((mesh_step_tendsto T).eventually (gt_mem_nhds hδ))
-  refine ⟨N, fun n hn => ?_⟩
-  rw [Real.dist_eq, sub_zero, ← Finset.sum_sub_distrib]
-  have hbound : |∑ i ∈ Finset.range (2^n),
-      (A n i * (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ)) -
-        if mesh T n i < τ then A n i * (b (mesh T n (i+1)) - b (mesh T n i)) else 0)| ≤ K*η := by
-    apply one_crossing_sum_bound (τ := τ) (mesh_mono T hT n) (mul_nonneg hK hη.le)
-    · intro i hi hni
-      rw [clipped_increment_eq (mesh_mono T hT n (Nat.le_succ i)) b hni]
-      split_ifs <;> simp
-    · intro i hi hc
-      rw [min_eq_right hc.2.le, min_eq_left hc.1.le, ite_eq_left hc.1,
-        ← mul_sub, sub_sub_sub_cancel_right, abs_mul]
-      apply mul_le_mul (hA n i) _ (abs_nonneg _) hK
-      have hd : dist τ (mesh T n (i+1)) < δ := by
-        rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 hc.2.le)]
-        have hs := mesh_step T n i
-        have hnδ := hN n hn
-        linarith [hc.1]
-      exact (hmod τ ⟨hτ, hτT⟩ _ (mesh_mem T hT n (i+1) hi) hd).le
-  apply hbound.trans_lt
-  dsimp [η]
-  rw [← mul_div_assoc, div_lt_iff₀ (by linarith : 0 < K+1)]
-  nlinarith
-
-lemma abs_sq_sub_sq_le {x y η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1)
-    (hη : 0 ≤ η) (hxy : |x-y| ≤ η) : |x^2-y^2| ≤ 2*η := by
-  calc
-    _ = |x-y| * |x+y| := by rw [← abs_mul]; congr 1; ring
-    _ ≤ η * (|x|+|y|) := mul_le_mul hxy (abs_add_le _ _) (abs_nonneg _) hη
-    _ ≤ η*(1+1) := mul_le_mul_of_nonneg_left (add_le_add hx hy) hη
-    _ = _ := by ring
-
-lemma clipped_quadraticSum_difference_tendsto {T τ : ℝ} (hT : 0 ≤ T)
-    (_hτ : 0 ≤ τ) (_hτT : τ ≤ T) {b : ℝ → ℝ}
-    (hb : ContinuousOn b (Icc 0 T)) (A : ℕ → ℕ → ℝ)
-    {K : ℝ} (hK : 0 ≤ K) (hA : ∀ n i, |A n i| ≤ K) :
-    Tendsto (fun n =>
-      (∑ i ∈ Finset.range (2^n), A n i *
-        (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))^2) -
-      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-        A n i * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0)) atTop (𝓝 0) := by
-  apply Metric.tendsto_atTop.2
-  intro ε hε
-  let η : ℝ := ε / (2*K+1)
-  have hη : 0 < η := div_pos hε (by linarith)
-  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1
-    (isCompact_Icc.uniformContinuousOn_of_continuous hb) (min 1 η) (lt_min (by norm_num) hη)
-  obtain ⟨N, hN⟩ := eventually_atTop.1 ((mesh_step_tendsto T).eventually (gt_mem_nhds hδ))
-  refine ⟨N, fun n hn => ?_⟩
-  rw [Real.dist_eq, sub_zero, ← Finset.sum_sub_distrib]
-  have hbound : |∑ i ∈ Finset.range (2^n),
-      (A n i * (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))^2 -
-        if mesh T n i < τ then A n i * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0)| ≤ K*(2*η) := by
-    apply one_crossing_sum_bound (τ := τ) (mesh_mono T hT n) (by positivity)
-    · intro i hi hni
-      rw [clipped_increment_eq (mesh_mono T hT n (Nat.le_succ i)) b hni]
-      split_ifs <;> simp
-    · intro i hi hc
-      have hdist (x y : ℝ) (hx : x ∈ Icc (mesh T n i) (mesh T n (i+1)))
-          (hy : y ∈ Icc (mesh T n i) (mesh T n (i+1))) : dist x y < δ := by
-        rw [Real.dist_eq, abs_lt]
-        have hs := mesh_step T n i
-        have hnδ := hN n hn
-        constructor <;> linarith [hx.1,hx.2,hy.1,hy.2]
-      have hmod' (x y : ℝ) (hx : x ∈ Icc (mesh T n i) (mesh T n (i+1)))
-          (hy : y ∈ Icc (mesh T n i) (mesh T n (i+1))) :
-          |b x-b y| ≤ min 1 η := by
-        have hsub := Icc_subset_Icc (mesh_mem T hT n i hi.le).1 (mesh_mem T hT n (i+1) hi).2
-        exact (hmod x (hsub hx) y (hsub hy) (hdist x y hx hy)).le
-      have hcurr : mesh T n i ≤ mesh T n (i+1) := mesh_mono T hT n (Nat.le_succ i)
-      have hτmem : τ ∈ Icc (mesh T n i) (mesh T n (i+1)) := ⟨hc.1.le,hc.2.le⟩
-      have hx := (hmod' τ _ hτmem (left_mem_Icc.2 hcurr)).trans (min_le_left _ _)
-      have hy := (hmod' _ _ (right_mem_Icc.2 hcurr) (left_mem_Icc.2 hcurr)).trans (min_le_left _ _)
-      have hxy := (hmod' τ _ hτmem (right_mem_Icc.2 hcurr)).trans (min_le_right _ _)
-      rw [min_eq_right hc.2.le, min_eq_left hc.1.le, ite_eq_left hc.1, ← mul_sub, abs_mul]
-      apply mul_le_mul (hA n i) _ (abs_nonneg _) hK
-      apply abs_sq_sub_sq_le hx hy hη.le
-      simpa only [sub_sub_sub_cancel_right] using hxy
-  apply hbound.trans_lt
-  dsimp [η]
-  rw [show K * (2 * (ε / (2*K+1))) = (2*K*ε)/(2*K+1) by ring,
-    div_lt_iff₀ (by linarith : 0 < 2*K+1)]
-  nlinarith
-
-lemma clipped_mem {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (n i : ℕ) :
-    min (mesh T n i) τ ∈ Icc 0 τ := by
-  constructor
-  · exact le_min (by unfold mesh; positivity) hτ
-  · exact min_le_right _ _
-
-lemma prestop_riemann_tendsto {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T)
-    {C : ℝ → ℝ} (hC : ContinuousOn C (Icc 0 τ)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      if mesh T n i < τ then C (mesh T n i) * (T / (2 : ℝ)^n) else 0) atTop
-      (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
-  let p := clippedDyadic T τ hT hτ hτT
-  have hriem := p.riemann_tendsto hC
-  obtain ⟨K, hK⟩ := isCompact_Icc.exists_bound_of_continuousOn hC
-  have hKn : 0 ≤ K := (norm_nonneg _).trans (hK 0 ⟨le_rfl,hτ⟩)
-  have hdiff := clipped_timeSum_difference_tendsto (τ := τ) hT
-    (fun n i => C (min (mesh T n i) τ)) hKn
-    (fun n i => hK _ (clipped_mem hT hτ n i))
-  have heq (n i : ℕ) :
-      (if mesh T n i < τ then C (min (mesh T n i) τ) * (T / (2 : ℝ)^n) else 0) =
-      (if mesh T n i < τ then C (mesh T n i) * (T / (2 : ℝ)^n) else 0) := by
-    split_ifs with h
-    · rw [min_eq_left h.le]
-    · rfl
-  simp only [heq] at hdiff
-  have hh := hriem.sub hdiff
-  simpa only [p, clippedDyadic, sub_sub_cancel, sub_zero] using hh
-
-end SLEExactGauge.StoppedMeshes
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.PartitionMeshes.PartitionSequence
-open SLEExactGauge.TaylorMeshes SLEExactGauge.TimeField
-
-lemma family_remainder_tendsto_of_eventually_bounded {T : ℝ} (p : PartitionSequence T)
-    {f : ℕ → ℕ → ℝ → ℝ} {b : ℝ → ℝ}
-    (hf : ∀ n i, ContDiff ℝ 3 (f n i)) {V K : ℝ}
-    (hb : ContinuousOn b (Icc 0 T)) (hV : 0 ≤ V) (hK : 0 ≤ K)
-    (h3 : ∀ n i x, |iteratedDeriv 3 (f n i) x| ≤ K)
-    (hQ : ∀ᶠ n in atTop, (∑ i ∈ Finset.range (p.count n),
-      (b (p.time n (i+1))-b (p.time n i))^2) ≤ V) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      remainder₂ (f n i) (b (p.time n i)) (b (p.time n (i+1)))) atTop (𝓝 0) := by
-  apply Metric.tendsto_atTop.2
-  intro ε hε
-  let δ := ε / ((K+1)*(V+1))
-  have hδ : 0 < δ := div_pos hε (mul_pos (by linarith) (by linarith))
-  have hs := p.increments_small hb hδ
-  obtain ⟨N, hN⟩ := eventually_atTop.1 (hs.and hQ)
-  refine ⟨N, fun n hn => ?_⟩
-  rcases hN n hn with ⟨hs, hq⟩
-  have hr := sum_family_remainder₂_bound (hf n) (fun i => b (p.time n i))
-    (p.count n) K δ hK hδ.le (fun i _ z _ => h3 n i z) (fun i hi => (hs i hi).le)
-  simp only [Real.dist_eq, sub_zero]
-  calc
-    _ ≤ K * δ / 6 * ∑ i ∈ Finset.range (p.count n),
-        (b (p.time n (i+1))-b (p.time n i))^2 := hr
-    _ ≤ K * δ / 6 * (V+1) :=
-      mul_le_mul_of_nonneg_left (hq.trans (by linarith)) (by positivity)
-    _ < ε := by
-      dsimp [δ]
-      have hden : 0 < (K+1)*(V+1) := mul_pos (by linarith) (by linarith)
-      have hid : K * (ε / ((K+1)*(V+1))) / 6 * (V+1) = ε * (K / (6*(K+1))) := by
-        field_simp
-      rw [hid]
-      have hk : K / (6*(K+1)) < 1 := (div_lt_one (by positivity)).2 (by linarith)
-      simpa using mul_lt_mul_of_pos_left hk hε
-
-end SLEExactGauge.PartitionMeshes.PartitionSequence
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.StoppedMeshes
-open TaylorMeshes PartitionMeshes
-
-lemma clipped_quadratic_eventually_bounded {T τ V : ℝ} (hT : 0 ≤ T)
-    (hτ : 0 ≤ τ) (hτT : τ ≤ T) {b : ℝ → ℝ} (hb : ContinuousOn b (Icc 0 T))
-    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1)) - b (mesh T n i))^2) atTop (𝓝 V)) :
-    ∀ᶠ n in atTop, (∑ i ∈ Finset.range (2^n),
-      (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))^2) ≤ V+2 := by
-  have hd := clipped_quadraticSum_difference_tendsto hT hτ hτT hb
-    (fun _ _ => 1) (by norm_num : (0 : ℝ) ≤ 1) (fun _ _ => by norm_num)
-  simp only [one_mul] at hd
-  filter_upwards [hd.eventually (gt_mem_nhds (show (0 : ℝ) < 1 by norm_num)),
-    hQ.eventually (gt_mem_nhds (show V < V+1 by linarith))] with n hn hqn
-  have hle : (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-      (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0) ≤
-      ∑ i ∈ Finset.range (2^n), (b (mesh T n (i+1)) - b (mesh T n i))^2 := by
-    apply Finset.sum_le_sum
-    intro i hi
-    split_ifs
-    · exact le_rfl
-    · exact sq_nonneg _
-  linarith
-
-end SLEExactGauge.StoppedMeshes
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.BrownianStopped
-open TaylorMeshes TimeField PartitionMeshes StoppedMeshes BrownianField
-
-lemma coefficient_clipped_bound {T τ K : ℝ} (C : ℝ → ℝ)
-    (hC : ∀ n i, mesh T n i < τ → |C (mesh T n i)| ≤ K) (n i : ℕ) :
-    |C (min (mesh T n i) τ)| ≤ max K |C τ| := by
-  by_cases h : mesh T n i < τ
-  · rw [min_eq_left h.le]
-    exact (hC n i h).trans (le_max_left _ _)
-  · rw [min_eq_right (le_of_not_gt h)]
-    exact le_max_right _ _
-
-lemma stopped_field_taylor_limit {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T)
-    {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ}
-    (hb : ContinuousOn b (Icc 0 T)) (hreg : FieldRegular F V b τ)
-    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1)) - b (mesh T n i))^2) atTop (𝓝 T))
-    {K₁ K₂ : ℝ} (hK₁ : 0 ≤ K₁) (hK₂ : 0 ≤ K₂)
-    (hA : ∀ n i, mesh T n i < τ →
-      |deriv (F (mesh T n i)) (b (mesh T n i))| ≤ K₁)
-    (hC : ∀ n i, mesh T n i < τ →
-      |iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i))| ≤ K₂)
-    (hE : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-      iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i)) *
-        ((b (mesh T n (i+1)) - b (mesh T n i))^2 - T/(2 : ℝ)^n) else 0) atTop (𝓝 0)) :
-    Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-      deriv (F (mesh T n i)) (b (mesh T n i)) *
-        (b (mesh T n (i+1)) - b (mesh T n i)) else 0) atTop
-      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, iteratedDeriv 2 (F t) (b t))) := by
-  let p := clippedDyadic T τ hT hτ hτT
-  let C : ℝ → ℝ := fun t => iteratedDeriv 2 (F t) (b t)
-  let A : ℝ → ℝ := fun t => deriv (F t) (b t)
-  have hbτ := hb.mono (Icc_subset_Icc le_rfl hτT)
-  have hQc := clipped_quadratic_eventually_bounded hT hτ hτT hb hQ
-  obtain ⟨K₃, hK₃, h₃⟩ := hreg.third_bound
-  have hr := p.family_remainder_tendsto_of_eventually_bounded
-    (f := fun n i => F (p.time n i)) (fun _ _ => hreg.smooth _)
-    hbτ (show 0 ≤ T+2 by linarith) hK₃ (fun _ _ x => h₃ _ x) hQc
-  have htime := p.time_increment_tendsto hreg.time_cont hreg.time_deriv hreg.drift_cont
-  have hriem := prestop_riemann_tendsto hT hτ hτT hreg.second_cont
-  let Qpred : ℕ → ℝ := fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-    C (mesh T n i) * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0
-  have hQpred : Tendsto Qpred atTop (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
-    have heq (n : ℕ) : Qpred n =
-      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
-        C (mesh T n i) * ((b (mesh T n (i+1))-b (mesh T n i))^2-T/(2 : ℝ)^n) else 0) +
-      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then C (mesh T n i)*(T/(2 : ℝ)^n) else 0) := by
-      unfold Qpred
-      rw [← Finset.sum_add_distrib]
-      apply Finset.sum_congr rfl
-      intro i hi
-      split_ifs <;> ring
-    dsimp only [C] at heq
-    simpa only [← heq, zero_add] using hE.add hriem
-  have hQdiff := clipped_quadraticSum_difference_tendsto hT hτ hτT hb
-    (fun n i => C (p.time n i))
-    (K := max K₂ |C τ|) (hK₂.trans (le_max_left _ _))
-    (coefficient_clipped_bound C hC)
-  have hQeq (n i : ℕ) :
-      (if mesh T n i < τ then C (p.time n i) *
-        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) =
-      (if mesh T n i < τ then C (mesh T n i) *
-        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) := by
-    split_ifs with h
-    · simp only [p, clippedDyadic, min_eq_left h.le]
-    · rfl
-  simp only [hQeq] at hQdiff
-  have hQweighted : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      C (p.time n i) * (b (p.time n (i+1)) - b (p.time n i))^2) atTop
-      (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
-    simpa only [Qpred, sub_add_cancel, zero_add, p, clippedDyadic] using hQdiff.add hQpred
-  have hTel (n : ℕ) : ∑ i ∈ Finset.range (p.count n),
-      (F (p.time n (i+1)) (b (p.time n (i+1))) -
-        F (p.time n i) (b (p.time n i))) = F τ (b τ) - F 0 (b 0) := by
-    rw [Finset.sum_range_sub (fun i => F (p.time n i) (b (p.time n i))), p.start, p.finish]
-  have hIdentity (n : ℕ) : (∑ i ∈ Finset.range (p.count n),
-      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) =
-      F τ (b τ) - F 0 (b 0) -
-      (∑ i ∈ Finset.range (p.count n),
-        (F (p.time n (i+1)) (b (p.time n (i+1))) -
-          F (p.time n i) (b (p.time n (i+1))))) - (1/2 : ℝ)*
-      (∑ i ∈ Finset.range (p.count n),
-        C (p.time n i)*(b (p.time n (i+1))-b (p.time n i))^2) -
-      (∑ i ∈ Finset.range (p.count n), remainder₂ (F (p.time n i))
-        (b (p.time n i)) (b (p.time n (i+1)))) := by
-    rw [← hTel n]
-    simp only [Finset.mul_sum, ← Finset.sum_sub_distrib]
-    apply Finset.sum_congr rfl
-    intro i hi
-    simp only [A, C, remainder₂, iteratedDeriv_succ, iteratedDeriv_zero]
-    ring
-  have hAclip : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
-      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) atTop
-      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, C t)) := by
-    have hh := (((tendsto_const_nhds (x := F τ (b τ) - F 0 (b 0))).sub
-      htime).sub (hQweighted.const_mul (1/2 : ℝ))).sub hr
-    simpa only [← hIdentity, sub_zero] using hh
-  have hAdiff := clipped_linearSum_difference_tendsto hT hτ hτT hb
-    (fun n i => A (p.time n i))
-    (K := max K₁ |A τ|) (hK₁.trans (le_max_left _ _))
-    (coefficient_clipped_bound A hA)
-  have hAeq (n i : ℕ) :
-      (if mesh T n i < τ then A (p.time n i) *
-        (b (mesh T n (i+1))-b (mesh T n i)) else 0) =
-      (if mesh T n i < τ then A (mesh T n i) *
-        (b (mesh T n (i+1))-b (mesh T n i)) else 0) := by
-    split_ifs with h
-    · simp only [p, clippedDyadic, min_eq_left h.le]
-    · rfl
-  simp only [hAeq] at hAdiff
-  have hh := hAclip.sub hAdiff
-  simpa only [p, clippedDyadic, A, C, sub_sub_cancel, sub_zero] using hh
-
-end SLEExactGauge.BrownianStopped
-
-end
-
-end
-
-end
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set MeasureTheory ProbabilityTheory Filter
-open scoped ENNReal NNReal Topology
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-noncomputable section
-open Set Filter MeasureTheory
-open scoped Topology
-namespace SLEExactGauge.BrownianStopped
-open BrownianPredictable PredictableLimit BrownianIto TaylorMeshes BrownianField
-
-def stoppedCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F : Ω → ℝ → ℝ → ℝ) (n i : ℕ) (ω : Ω) : ℝ :=
-  if mesh T n i < τ ω then
-    iteratedDeriv k (F ω (mesh T n i)) (B (dyadicTime T n i) ω) else 0
-
-theorem stopped_fieldSum_tendsto {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
-    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
-    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (stoppedCoeff 2 B T τ F n i))
-    (hC2 : ∀ n i, MemLp (stoppedCoeff 2 B T τ F n i) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
-    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
-    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
-    ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω) atTop
-      (𝓝 (F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
-        (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
-        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω),
-          iteratedDeriv 2 (F ω t) (B t.toNNReal ω))) := by
-  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T
-    (stoppedCoeff 2 B T τ F) hC hC2 K₂ hK₂ hbound
-  have hC_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂ :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hbound n))
-  have hA_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁ :=
-    ae_all_iff.2 (fun n => ae_all_iff.2 (hAbound n))
-  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hτ, hC_all, hA_all]
-    with ω hb hQ hE hr hτω hCω hAω
-  let b : ℝ → ℝ := fun t => B t.toNNReal ω
-  have hbc : Continuous b := hb.comp continuous_real_toNNReal
-  have hQ' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
-      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
-    simpa only [quadraticSum, b, mesh_toNNReal] using hQ
-  have hE' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
-      iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-        ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0)
-        atTop (𝓝 0) := by
-    have heq (n : ℕ) : dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n ω =
-        ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
-          iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
-            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0 := by
-      unfold dyadicQuadraticError
-      apply Finset.sum_congr rfl
-      intro i hi
-      simp only [stoppedCoeff, b, mesh_toNNReal]
-      split_ifs <;> simp
-    simpa only [heq] using hE
-  have hA' (n i : ℕ) (hi : mesh T n i < τ ω) :
-      |deriv (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₁ := by
-    simpa only [stoppedCoeff, ite_eq_left hi, iteratedDeriv_succ, iteratedDeriv_zero,
-      b, mesh_toNNReal] using hAω n i
-  have hC' (n i : ℕ) (hi : mesh T n i < τ ω) :
-      |iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₂ := by
-    simpa only [stoppedCoeff, ite_eq_left hi, b, mesh_toNNReal] using hCω n i
-  have ht := stopped_field_taylor_limit T.coe_nonneg hτω.1 hτω.2 hbc.continuousOn
-    hr hQ' hK₁ hK₂ hA' hC' hE'
-  have heq (n : ℕ) : dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω =
-      ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
-        deriv (F ω (mesh T n i)) (b (mesh T n i)) *
-          (b (mesh T n (i+1))-b (mesh T n i)) else 0 := by
-    unfold dyadicDriverSum
-    apply Finset.sum_congr rfl
-    intro i hi
-    simp only [stoppedCoeff, b, mesh_toNNReal, iteratedDeriv_succ, iteratedDeriv_zero]
-    split_ifs <;> simp
-  simpa only [← heq, b, Real.toNNReal_zero] using ht
-
-theorem stopped_field_ito_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P]
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
-    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
-    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
-    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
-    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (stoppedCoeff 2 B T τ F n i))
-    (hC2 : ∀ n i, MemLp (stoppedCoeff 2 B T τ F n i) 2 P)
-    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
-    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
-    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (stoppedCoeff 1 B T τ F n i))
-    (hA2 : ∀ n i, MemLp (stoppedCoeff 1 B T τ F n i) 2 P)
-    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
-    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
-    let Y := fun ω => F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
-      (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
-      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω), iteratedDeriv 2 (F ω t) (B t.toNNReal ω)
-    Integrable Y P ∧ (∫ ω, Y ω ∂P) = 0 := by
-  dsimp only
-  exact limit_centered hB T (stoppedCoeff 1 B T τ F) hA hA2 K₁ hK₁ hAbound _
-    (stopped_fieldSum_tendsto hB T τ F V hτ hreg hC hC2 K₂ hK₂ hbound K₁ hK₁ hAbound)
-
-end SLEExactGauge.BrownianStopped
-
-end
-
-end
-
-end
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.TraceModel
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Localization estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+lemma physical_clock_bounds {T δ Y₀ u : ℝ} {Y C Q : ℝ → ℝ}
+    (hT : 0 ≤ T) (hδ : 0 < δ) (hY₀ : 0 < Y₀) (hYinit : Y 0 = Y₀) (hCinit : C 0 = 0)
+    (hY : ContinuousOn Y (Icc 0 T)) (hC : ContinuousOn C (Icc 0 T))
+    (hYpos : ∀ t ∈ Icc 0 T, 0 < Y t)
+    (hYle : ∀ t ∈ Icc 0 T, Y t ≤ Y₀)
+    (hQpos : ∀ t ∈ Ioo 0 T, 0 < Q t)
+    (hangle : ∀ t ∈ Ioo 0 T, δ^2 * Q t ≤ (Y t)^2)
+    (hdY : ∀ t ∈ Ioo 0 T, HasDerivAt (fun s => Real.log (Y s)) (-2 / Q t) t)
+    (hdC : ∀ t ∈ Ioo 0 T, HasDerivAt C (4*(Y t)^2 / (Q t)^2) t)
+    (hCu : C T ≤ u) :
+    Y₀ * Real.exp (-u/(2*δ^2)) ≤ Y T ∧ T ≤ u * Y₀^2 / (4*δ^4) := by
+  have hd2 : 0 < δ^2 := sq_pos_of_pos hδ
+  have hq2 (t : ℝ) (ht : t ∈ Ioo 0 T) : 0 < Q t ^ 2 := sq_pos_of_pos (hQpos t ht)
+  have hlogY := hY.log (fun t ht => (hYpos t ht).ne')
+  have hm : MonotoneOn (fun t => Real.log (Y t) + C t / (2*δ^2)) (Icc 0 T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 T)
+      (hlogY.add (hC.div_const _))
+      (f' := fun t => -2/Q t + (4*(Y t)^2/(Q t)^2)/(2*δ^2))
+    · intro t ht
+      rw [interior_Icc] at ht
+      exact ((hdY t ht).add ((hdC t ht).div_const _)).hasDerivWithinAt
+    · intro t ht
+      rw [interior_Icc] at ht
+      have hq := hQpos t ht
+      have ha := hangle t ht
+      apply (mul_le_mul_iff_of_pos_right (mul_pos (hq2 t ht) (mul_pos (by norm_num : (0:ℝ)<2) hd2))).1
+      field_simp
+      nlinarith
+  have hlogbound := hm ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
+  simp only [hYinit, hCinit, zero_div, add_zero] at hlogbound
+  have hL : Real.log Y₀ + (-u/(2*δ^2)) ≤ Real.log (Y T) := by
+    have hh : C T/(2*δ^2) ≤ u/(2*δ^2) :=
+      div_le_div_of_nonneg_right hCu (mul_pos (by norm_num) hd2).le
+    simp only [neg_div]
+    linarith
+  have hheight : Y₀ * Real.exp (-u/(2*δ^2)) ≤ Y T := by
+    have hh := Real.exp_le_exp.mpr hL
+    simpa only [Real.exp_add, Real.exp_log hY₀, Real.exp_log (hYpos T ⟨hT,le_rfl⟩)] using hh
+  have hK : 0 < 4*δ^4/Y₀^2 := by positivity
+  have hderivbound (t : ℝ) (ht : t ∈ Ioo 0 T) :
+      4*δ^4/Y₀^2 ≤ 4*(Y t)^2/(Q t)^2 := by
+    apply (div_le_div_iff₀ (sq_pos_of_pos hY₀) (hq2 t ht)).2
+    have ha := hangle t ht
+    have hb : (δ^2*Q t)^2 ≤ ((Y t)^2)^2 :=
+      pow_le_pow_left₀ (mul_nonneg hd2.le (hQpos t ht).le) ha 2
+    have hc : (Y t)^2 ≤ Y₀^2 :=
+      pow_le_pow_left₀ (hYpos t ⟨ht.1.le,ht.2.le⟩).le (hYle t ⟨ht.1.le,ht.2.le⟩) 2
+    have he := mul_le_mul_of_nonneg_left hc (sq_nonneg (Y t))
+    nlinarith
+  have hmC : MonotoneOn (fun t => C t - (4*δ^4/Y₀^2)*t) (Icc 0 T) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 T)
+      (hC.sub (continuousOn_id.const_mul _))
+      (f' := fun t => 4*(Y t)^2/(Q t)^2 - (4*δ^4/Y₀^2))
+    · intro t ht
+      rw [interior_Icc] at ht
+      simpa only [mul_one] using
+        ((hdC t ht).sub ((hasDerivAt_id t).const_mul (4*δ^4/Y₀^2))).hasDerivWithinAt
+          (s := interior (Icc 0 T))
+    · intro t ht
+      rw [interior_Icc] at ht
+      exact sub_nonneg.2 (hderivbound t ht)
+  have htime := hmC ⟨le_rfl,hT⟩ ⟨hT,le_rfl⟩ hT
+  simp only [hCinit, mul_zero, sub_zero] at htime
+  refine ⟨hheight, ?_⟩
+  have hh : T ≤ u / (4*δ^4/Y₀^2) := (le_div_iff₀ hK).2 (by linarith)
+  convert hh using 1
+  field_simp
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma finite_clock_bounds (h : IsLoewnerTraceUsing driver γ g f)
+    {T δ u : ℝ} (hT : 0 ≤ T) (hδ : 0 < δ)
+    (hU : ContinuousOn driver (Icc 0 T)) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    (hangle : ∀ t ∈ Ioo 0 T,
+      δ^2 * Complex.normSq (g t z - (driver t : ℂ)) ≤ (g t z).im^2)
+    (hr : (2*z.im)*Real.exp (-u) ≤ loewnerRadius g T z) :
+    z.im * Real.exp (-u/(2*δ^2)) ≤ (g T z).im ∧
+      T ≤ u*z.im^2/(4*δ^4) := by
+  have hz0 : z ∈ halfPlane := hz.1.1
+  have hy0 : 0 < z.im := hz0
+  let C : ℝ → ℝ := fun t => Real.log (2*z.im) - Real.log (loewnerRadius g t z)
+  apply physical_clock_bounds (Y := fun t => (g t z).im) (C := C)
+    (Q := fun t => Complex.normSq (g t z - (driver t : ℂ))) hT hδ hy0
+  · rw [h.initial z hz0]
+  · simp only [C, h.radius_initial hz0, sub_self]
+  · exact Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)
+  · exact continuousOn_const.sub (h.continuous_log_radius hT hU hz)
+  · intro t ht
+    exact h.im_pos ht.1 (survivingDomain_mono ht.2 hz)
+  · intro t ht
+    exact h.height_le_initial ht.1 (survivingDomain_mono ht.2 hz)
+  · intro t ht
+    exact Complex.normSq_pos.2 (h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hz))
+  · exact hangle
+  · intro t ht
+    have hi := h.im_pos ht.1.le (survivingDomain_mono ht.2.le hz)
+    apply ((h.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hz)).log hi.ne').congr_deriv
+    field_simp
+  · intro t ht
+    change HasDerivAt (fun s => Real.log (2*z.im) - Real.log (loewnerRadius g s z)) _ t
+    apply ((h.hasDerivAt_log_radius hT hU hz ht).const_sub (Real.log (2*z.im))).congr_deriv
+    ring
+  · have hh := Real.log_le_log (mul_pos (mul_pos (by norm_num) hy0) (Real.exp_pos (-u))) hr
+    rw [Real.log_mul (mul_ne_zero (by norm_num) hy0.ne') (Real.exp_ne_zero _), Real.log_exp] at hh
+    dsimp [C]
+    linarith
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.PartitionMeshes
+open TaylorMeshes TimeField
+
+structure PartitionSequence (T : ℝ) where
+  count : ℕ → ℕ
+  time : ℕ → ℕ → ℝ
+  step : ℕ → ℝ
+  start : ∀ n, time n 0 = 0
+  finish : ∀ n, time n (count n) = T
+  mono : ∀ n, Monotone (time n)
+  step_le : ∀ n i, i < count n → time n (i+1)-time n i ≤ step n
+  step_zero : Tendsto step atTop (𝓝 0)
+
+end SLEExactGauge.PartitionMeshes
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.PartitionMeshes
+open TaylorMeshes TimeField
+namespace PartitionSequence
+variable {T : ℝ} (p : PartitionSequence T)
+
+include p in
+lemma nonneg : 0 ≤ T := by
+  simpa only [p.start, p.finish] using p.mono 0 (Nat.zero_le (p.count 0))
+
+lemma mem (n i : ℕ) (hi : i ≤ p.count n) : p.time n i ∈ Icc 0 T := by
+  constructor
+  · simpa only [p.start] using p.mono n (Nat.zero_le i)
+  · simpa only [p.finish] using p.mono n hi
+
+lemma increments_small {b : ℝ → ℝ} (hb : ContinuousOn b (Icc 0 T))
+    {δ : ℝ} (hδ : 0 < δ) :
+    ∀ᶠ n in atTop, ∀ i < p.count n, |b (p.time n (i+1))-b (p.time n i)| < δ := by
+  obtain ⟨ε, hε, hmod⟩ := Metric.uniformContinuousOn_iff.1
+    (isCompact_Icc.uniformContinuousOn_of_continuous hb) δ hδ
+  filter_upwards [p.step_zero.eventually (gt_mem_nhds hε)] with n hn
+  intro i hi
+  apply hmod _ (p.mem n (i+1) hi) _ (p.mem n i hi.le)
+  rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.2 (p.mono n (Nat.le_succ i)))]
+  exact (p.step_le n i hi).trans_lt hn
+
+lemma riemann_tendsto {h : ℝ → ℝ} (hh : ContinuousOn h (Icc 0 T)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      h (p.time n i) * (p.time n (i+1)-p.time n i)) atTop
+      (𝓝 (∫ t in (0 : ℝ)..T, h t)) := by
+  have hT := p.nonneg
+  have hint (n i : ℕ) (hi : i < p.count n) :
+      IntervalIntegrable h volume (p.time n i) (p.time n (i+1)) := by
+    apply ContinuousOn.intervalIntegrable
+    rw [uIcc_of_le (p.mono n (Nat.le_succ i))]
+    exact hh.mono (Icc_subset_Icc (p.mem n i hi.le).1 (p.mem n (i+1) hi).2)
+  have hu := isCompact_Icc.uniformContinuousOn_of_continuous hh
+  apply Metric.tendsto_atTop.2
+  intro ε hε
+  let ζ : ℝ := ε / (T+1)
+  have hζ : 0 < ζ := div_pos hε (by linarith)
+  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1 hu ζ hζ
+  obtain ⟨N, hN⟩ := eventually_atTop.1 (p.step_zero.eventually (gt_mem_nhds hδ))
+  refine ⟨N, fun n hn => ?_⟩
+  have hnδ := hN n hn
+  have hsum := intervalIntegral.sum_integral_adjacent_intervals (fun i hi => hint n i hi)
+  rw [p.start, p.finish] at hsum
+  have heach (i : ℕ) (hi : i < p.count n) :
+      |h (p.time n i) * (p.time n (i+1)-p.time n i) -
+        ∫ t in p.time n i..p.time n (i+1), h t| ≤
+          ζ * (p.time n (i+1)-p.time n i) := by
+    have heq : h (p.time n i) * (p.time n (i+1)-p.time n i) =
+        ∫ _ : ℝ in p.time n i..p.time n (i+1), h (p.time n i) := by
+      rw [intervalIntegral.integral_const, smul_eq_mul, mul_comm]
+    rw [heq, ← intervalIntegral.integral_sub intervalIntegrable_const (hint n i hi)]
+    have hbound : ∀ t ∈ uIoc (p.time n i) (p.time n (i+1)),
+        ‖h (p.time n i)-h t‖ ≤ ζ := by
+      intro t ht
+      rw [uIoc_of_le (p.mono n (Nat.le_succ i))] at ht
+      have htT : t ∈ Icc 0 T := ⟨(p.mem n i hi.le).1.trans ht.1.le,
+        ht.2.trans (p.mem n (i+1) hi).2⟩
+      have hd : dist (p.time n i) t < δ := by
+        rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 ht.1.le)]
+        have hs := p.step_le n i hi
+        have ht' := ht.2
+        linarith
+      simpa only [Real.norm_eq_abs, Real.dist_eq] using
+        (hmod _ (p.mem n i hi.le) _ htT hd).le
+    simpa only [Real.norm_eq_abs,
+      abs_of_nonneg (sub_nonneg.2 (p.mono n (Nat.le_succ i)))] using
+      intervalIntegral.norm_integral_le_of_norm_le_const hbound
+  rw [Real.dist_eq, ← hsum, ← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑ i ∈ Finset.range (p.count n),
+        |h (p.time n i) * (p.time n (i+1)-p.time n i) -
+          ∫ t in p.time n i..p.time n (i+1), h t| := Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ i ∈ Finset.range (p.count n), ζ * (p.time n (i+1)-p.time n i) :=
+      Finset.sum_le_sum (fun i hi => heach i (Finset.mem_range.1 hi))
+    _ = ζ * T := by
+      rw [← Finset.mul_sum, Finset.sum_range_sub (p.time n), p.start, p.finish, sub_zero]
+    _ < ε := by
+      dsimp only [ζ]
+      rw [div_mul_eq_mul_div, div_lt_iff₀ (by linarith : 0 < T+1)]
+      nlinarith
+
+lemma family_remainder_tendsto {f : ℕ → ℕ → ℝ → ℝ} {b : ℝ → ℝ}
+    (hf : ∀ n i, ContDiff ℝ 3 (f n i))
+    {V K : ℝ} (hb : ContinuousOn b (Icc 0 T)) (hK : 0 ≤ K)
+    (h3 : ∀ n i x, |iteratedDeriv 3 (f n i) x| ≤ K)
+    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      (b (p.time n (i+1))-b (p.time n i))^2) atTop (𝓝 V)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      remainder₂ (f n i) (b (p.time n i)) (b (p.time n (i+1)))) atTop (𝓝 0) := by
+  have hV : 0 ≤ V := ge_of_tendsto hQ
+    (Eventually.of_forall fun _ => Finset.sum_nonneg fun _ _ => sq_nonneg _)
+  apply Metric.tendsto_atTop.2
+  intro ε hε
+  let δ := ε / ((K+1)*(V+1))
+  have hδ : 0 < δ := div_pos hε (mul_pos (by linarith) (by linarith))
+  have hs := p.increments_small hb hδ
+  have hq := hQ.eventually (gt_mem_nhds (show V < V+1 by linarith))
+  obtain ⟨N, hN⟩ := eventually_atTop.1 (hs.and hq)
+  refine ⟨N, fun n hn => ?_⟩
+  rcases hN n hn with ⟨hs, hq⟩
+  have hr := sum_family_remainder₂_bound (hf n) (fun i => b (p.time n i))
+    (p.count n) K δ hK hδ.le (fun i _ z _ => h3 n i z) (fun i hi => (hs i hi).le)
+  simp only [Real.dist_eq, sub_zero]
+  calc
+    _ ≤ K * δ / 6 * ∑ i ∈ Finset.range (p.count n),
+        (b (p.time n (i+1))-b (p.time n i))^2 := hr
+    _ ≤ K * δ / 6 * (V+1) := mul_le_mul_of_nonneg_left hq.le (by positivity)
+    _ < ε := by
+      dsimp [δ]
+      have hden : 0 < (K+1)*(V+1) := mul_pos (by linarith) (by linarith)
+      have hid : K * (ε / ((K+1)*(V+1))) / 6 * (V+1) = ε * (K / (6*(K+1))) := by
+        field_simp
+      rw [hid]
+      have hk : K / (6*(K+1)) < 1 := (div_lt_one (by positivity)).2 (by linarith)
+      simpa using mul_lt_mul_of_pos_left hk hε
+
+end PartitionSequence
+
+end SLEExactGauge.PartitionMeshes
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.PartitionMeshes
+open TaylorMeshes TimeField
+
+def clippedDyadic (T τ : ℝ) (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T) :
+    PartitionSequence τ where
+  count := fun n => 2^n
+  time := fun n i => min (mesh T n i) τ
+  step := fun n => T / (2 : ℝ)^n
+  start := fun n => by rw [mesh_zero, min_eq_left hτ]
+  finish := fun n => by rw [mesh_end, min_eq_right hτT]
+  mono := fun n i j hij => min_le_min_right τ (mesh_mono T hT n hij)
+  step_le := by
+    intro n i hi
+    have hm := mesh_mono T hT n (Nat.le_succ i)
+    have hs := mesh_step T n i
+    by_cases h₁ : mesh T n i ≤ τ
+    · rw [min_eq_left h₁]
+      have hb := min_le_left (mesh T n (i+1)) τ
+      linarith
+    · rw [min_eq_right (le_of_not_ge h₁),
+        min_eq_right ((le_of_not_ge h₁).trans hm)]
+      simp only [sub_self]
+      positivity
+  step_zero := mesh_step_tendsto T
+
+end SLEExactGauge.PartitionMeshes
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.PartitionMeshes.PartitionSequence
+open SLEExactGauge.TaylorMeshes SLEExactGauge.TimeField
+
+lemma time_increment_tendsto {T : ℝ} (p : PartitionSequence T)
+    {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ}
+    (hF : ∀ x, ContinuousOn (fun t => F t x) (Icc 0 T))
+    (hD : ∀ t ∈ Ioo 0 T, ∀ x, HasDerivAt (fun u => F u x) (V t x) t)
+    (hV : ContinuousOn (fun q : ℝ × ℝ => V q.1 (b q.2)) ((Icc 0 T) ×ˢ (Icc 0 T))) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      (F (p.time n (i+1)) (b (p.time n (i+1))) -
+        F (p.time n i) (b (p.time n (i+1))))) atTop
+      (𝓝 (∫ t in (0 : ℝ)..T, V t (b t))) := by
+  have hT := p.nonneg
+  have hVc : ContinuousOn (fun t => V t (b t)) (Icc 0 T) :=
+    hV.comp (continuousOn_id.prodMk continuousOn_id) (fun t ht => ⟨ht, ht⟩)
+  have hriem := p.riemann_tendsto hVc
+  have herror : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      (F (p.time n (i+1)) (b (p.time n (i+1))) -
+        F (p.time n i) (b (p.time n (i+1))) -
+          V (p.time n i) (b (p.time n i)) * (p.time n (i+1)-p.time n i)))
+          atTop (𝓝 0) := by
+    have hu := (isCompact_Icc.prod isCompact_Icc).uniformContinuousOn_of_continuous hV
+    apply Metric.tendsto_atTop.2
+    intro ε hε
+    let ζ : ℝ := ε / (T+1)
+    have hζ : 0 < ζ := div_pos hε (by linarith)
+    obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1 hu ζ hζ
+    obtain ⟨N, hN⟩ := eventually_atTop.1 (p.step_zero.eventually (gt_mem_nhds hδ))
+    refine ⟨N, fun n hn => ?_⟩
+    have hnδ := hN n hn
+    have heach (i : ℕ) (hi : i < p.count n) :
+        |F (p.time n (i+1)) (b (p.time n (i+1))) -
+          F (p.time n i) (b (p.time n (i+1))) -
+            V (p.time n i) (b (p.time n i)) * (p.time n (i+1)-p.time n i)| ≤
+        ζ * (p.time n (i+1)-p.time n i) := by
+      rcases (p.mono n (Nat.le_succ i)).eq_or_lt with heq | hstep_pos
+      · simp [heq]
+      have hleft := p.mem n i hi.le
+      have hright := p.mem n (i+1) hi
+      obtain ⟨c, hc, heq⟩ := exists_hasDerivAt_eq_slope
+        (fun t => F t (b (p.time n (i+1)))) (fun t => V t (b (p.time n (i+1))))
+        hstep_pos ((hF _).mono (Icc_subset_Icc hleft.1 hright.2))
+        (fun t ht => hD t ⟨hleft.1.trans_lt ht.1, ht.2.trans_le hright.2⟩ _)
+      have hcT : c ∈ Icc 0 T := ⟨hleft.1.trans hc.1.le, hc.2.le.trans hright.2⟩
+      have hdist : dist (c, p.time n (i+1)) (p.time n i, p.time n i) < δ := by
+        rw [Prod.dist_eq, max_lt_iff]
+        constructor
+        · rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.2 hc.1.le)]
+          have hcs := hc.2
+          have hstep := p.step_le n i hi
+          linarith
+        · rw [Real.dist_eq, abs_of_nonneg (sub_nonneg.2 hstep_pos.le)]
+          exact (p.step_le n i hi).trans_lt hnδ
+      have hvmod := (hmod (c, p.time n (i+1)) ⟨hcT, hright⟩
+        (p.time n i, p.time n i) ⟨hleft, hleft⟩ hdist).le
+      simp only [Real.dist_eq] at hvmod
+      have hid : F (p.time n (i+1)) (b (p.time n (i+1))) -
+          F (p.time n i) (b (p.time n (i+1))) =
+          V c (b (p.time n (i+1))) * (p.time n (i+1)-p.time n i) :=
+        ((eq_div_iff (sub_pos.2 hstep_pos).ne').1 heq).symm
+      rw [hid, ← sub_mul, abs_mul, abs_of_nonneg (sub_nonneg.2 hstep_pos.le)]
+      exact mul_le_mul_of_nonneg_right hvmod (sub_nonneg.2 hstep_pos.le)
+    rw [Real.dist_eq, sub_zero]
+    calc
+      _ ≤ ∑ i ∈ Finset.range (p.count n),
+          |F (p.time n (i+1)) (b (p.time n (i+1))) -
+            F (p.time n i) (b (p.time n (i+1))) -
+              V (p.time n i) (b (p.time n i)) * (p.time n (i+1)-p.time n i)| :=
+        Finset.abs_sum_le_sum_abs _ _
+      _ ≤ ∑ i ∈ Finset.range (p.count n), ζ * (p.time n (i+1)-p.time n i) :=
+        Finset.sum_le_sum (fun i hi => heach i (Finset.mem_range.1 hi))
+      _ = ζ * T := by
+        rw [← Finset.mul_sum, Finset.sum_range_sub (p.time n), p.start, p.finish, sub_zero]
+      _ < ε := by
+        dsimp [ζ]
+        rw [div_mul_eq_mul_div, div_lt_iff₀ (by linarith : 0 < T+1)]
+        nlinarith
+  have hadd := herror.add hriem
+  simp only [Finset.sum_sub_distrib, sub_add_cancel, zero_add] at hadd
+  simpa only [Finset.sum_sub_distrib] using hadd
+
+end SLEExactGauge.PartitionMeshes.PartitionSequence
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.StoppedMeshes
+open TaylorMeshes PartitionMeshes
+
+lemma one_crossing_sum_bound {N : ℕ} {t : ℕ → ℝ} (ht : Monotone t)
+    {τ E : ℝ} (hE : 0 ≤ E) {f : ℕ → ℝ}
+    (hz : ∀ i < N, ¬ (t i < τ ∧ τ < t (i+1)) → f i = 0)
+    (hb : ∀ i < N, t i < τ ∧ τ < t (i+1) → |f i| ≤ E) :
+    |∑ i ∈ Finset.range N, f i| ≤ E := by
+  classical
+  by_cases he : ∃ i < N, t i < τ ∧ τ < t (i+1)
+  · obtain ⟨i, hi, hcross⟩ := he
+    rw [Finset.sum_eq_single i]
+    · exact hb i hi hcross
+    · intro j hj hji
+      apply hz j (Finset.mem_range.1 hj)
+      intro hcj
+      rcases lt_or_gt_of_ne hji with hji | hij
+      · have hm := ht (show j+1 ≤ i by omega)
+        linarith [hcj.2, hcross.1]
+      · have hm := ht (show i+1 ≤ j by omega)
+        linarith [hcross.2, hcj.1]
+    · intro hni
+      exact (hni (Finset.mem_range.2 hi)).elim
+  · have heq : ∑ i ∈ Finset.range N, f i = 0 := by
+      apply Finset.sum_eq_zero
+      intro i hi
+      apply hz i (Finset.mem_range.1 hi)
+      intro hc
+      exact he ⟨i, Finset.mem_range.1 hi, hc⟩
+    simpa [heq] using hE
+
+lemma clipped_increment_eq {a c τ : ℝ} (hac : a ≤ c) (b : ℝ → ℝ)
+    (hn : ¬ (a < τ ∧ τ < c)) :
+    b (min c τ) - b (min a τ) = if a < τ then b c - b a else 0 := by
+  by_cases ha : a < τ
+  · rw [ite_eq_left ha, min_eq_left ha.le, min_eq_left (le_of_not_gt (fun hc => hn ⟨ha,hc⟩))]
+  · have hτa : τ ≤ a := le_of_not_gt ha
+    rw [ite_eq_right ha, min_eq_right hτa, min_eq_right (hτa.trans hac), sub_self]
+
+lemma clipped_timeSum_difference_tendsto {T τ : ℝ} (hT : 0 ≤ T)
+    (A : ℕ → ℕ → ℝ) {K : ℝ} (hK : 0 ≤ K) (hA : ∀ n i, |A n i| ≤ K) :
+    Tendsto (fun n =>
+      (∑ i ∈ Finset.range (2^n), A n i *
+        (min (mesh T n (i+1)) τ - min (mesh T n i) τ)) -
+      (∑ i ∈ Finset.range (2^n),
+        if mesh T n i < τ then A n i * (T / (2 : ℝ)^n) else 0)) atTop (𝓝 0) := by
+  refine squeeze_zero_norm (a := fun n => K * (T / (2 : ℝ)^n)) (fun n => ?_) ?_
+  · rw [Real.norm_eq_abs, ← Finset.sum_sub_distrib]
+    apply one_crossing_sum_bound (τ := τ) (mesh_mono T hT n) (mul_nonneg hK (by positivity))
+    · intro i hi hn
+      have heq := clipped_increment_eq (mesh_mono T hT n (Nat.le_succ i)) id hn
+      dsimp only [id_eq] at heq
+      rw [heq]
+      split_ifs <;> simp [mesh_step]
+    · intro i hi hc
+      rw [min_eq_right hc.2.le, min_eq_left hc.1.le, ite_eq_left hc.1]
+      rw [← mul_sub, abs_mul]
+      apply le_trans (mul_le_mul_of_nonneg_right (hA n i) (abs_nonneg _))
+      apply mul_le_mul_of_nonneg_left _ hK
+      have hs := mesh_step T n i
+      rw [abs_of_nonpos (by linarith)]
+      linarith [hc.1]
+  · simpa using (mesh_step_tendsto T).const_mul K
+
+lemma clipped_linearSum_difference_tendsto {T τ : ℝ} (hT : 0 ≤ T)
+    (hτ : 0 ≤ τ) (hτT : τ ≤ T) {b : ℝ → ℝ}
+    (hb : ContinuousOn b (Icc 0 T)) (A : ℕ → ℕ → ℝ)
+    {K : ℝ} (hK : 0 ≤ K) (hA : ∀ n i, |A n i| ≤ K) :
+    Tendsto (fun n =>
+      (∑ i ∈ Finset.range (2^n), A n i *
+        (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))) -
+      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+        A n i * (b (mesh T n (i+1)) - b (mesh T n i)) else 0)) atTop (𝓝 0) := by
+  apply Metric.tendsto_atTop.2
+  intro ε hε
+  let η : ℝ := ε / (K+1)
+  have hη : 0 < η := div_pos hε (by linarith)
+  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1
+    (isCompact_Icc.uniformContinuousOn_of_continuous hb) η hη
+  obtain ⟨N, hN⟩ := eventually_atTop.1 ((mesh_step_tendsto T).eventually (gt_mem_nhds hδ))
+  refine ⟨N, fun n hn => ?_⟩
+  rw [Real.dist_eq, sub_zero, ← Finset.sum_sub_distrib]
+  have hbound : |∑ i ∈ Finset.range (2^n),
+      (A n i * (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ)) -
+        if mesh T n i < τ then A n i * (b (mesh T n (i+1)) - b (mesh T n i)) else 0)| ≤ K*η := by
+    apply one_crossing_sum_bound (τ := τ) (mesh_mono T hT n) (mul_nonneg hK hη.le)
+    · intro i hi hni
+      rw [clipped_increment_eq (mesh_mono T hT n (Nat.le_succ i)) b hni]
+      split_ifs <;> simp
+    · intro i hi hc
+      rw [min_eq_right hc.2.le, min_eq_left hc.1.le, ite_eq_left hc.1,
+        ← mul_sub, sub_sub_sub_cancel_right, abs_mul]
+      apply mul_le_mul (hA n i) _ (abs_nonneg _) hK
+      have hd : dist τ (mesh T n (i+1)) < δ := by
+        rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 hc.2.le)]
+        have hs := mesh_step T n i
+        have hnδ := hN n hn
+        linarith [hc.1]
+      exact (hmod τ ⟨hτ, hτT⟩ _ (mesh_mem T hT n (i+1) hi) hd).le
+  apply hbound.trans_lt
+  dsimp [η]
+  rw [← mul_div_assoc, div_lt_iff₀ (by linarith : 0 < K+1)]
+  nlinarith
+
+lemma abs_sq_sub_sq_le {x y η : ℝ} (hx : |x| ≤ 1) (hy : |y| ≤ 1)
+    (hη : 0 ≤ η) (hxy : |x-y| ≤ η) : |x^2-y^2| ≤ 2*η := by
+  calc
+    _ = |x-y| * |x+y| := by rw [← abs_mul]; congr 1; ring
+    _ ≤ η * (|x|+|y|) := mul_le_mul hxy (abs_add_le _ _) (abs_nonneg _) hη
+    _ ≤ η*(1+1) := mul_le_mul_of_nonneg_left (add_le_add hx hy) hη
+    _ = _ := by ring
+
+lemma clipped_quadraticSum_difference_tendsto {T τ : ℝ} (hT : 0 ≤ T)
+    (_hτ : 0 ≤ τ) (_hτT : τ ≤ T) {b : ℝ → ℝ}
+    (hb : ContinuousOn b (Icc 0 T)) (A : ℕ → ℕ → ℝ)
+    {K : ℝ} (hK : 0 ≤ K) (hA : ∀ n i, |A n i| ≤ K) :
+    Tendsto (fun n =>
+      (∑ i ∈ Finset.range (2^n), A n i *
+        (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))^2) -
+      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+        A n i * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0)) atTop (𝓝 0) := by
+  apply Metric.tendsto_atTop.2
+  intro ε hε
+  let η : ℝ := ε / (2*K+1)
+  have hη : 0 < η := div_pos hε (by linarith)
+  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.1
+    (isCompact_Icc.uniformContinuousOn_of_continuous hb) (min 1 η) (lt_min (by norm_num) hη)
+  obtain ⟨N, hN⟩ := eventually_atTop.1 ((mesh_step_tendsto T).eventually (gt_mem_nhds hδ))
+  refine ⟨N, fun n hn => ?_⟩
+  rw [Real.dist_eq, sub_zero, ← Finset.sum_sub_distrib]
+  have hbound : |∑ i ∈ Finset.range (2^n),
+      (A n i * (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))^2 -
+        if mesh T n i < τ then A n i * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0)| ≤ K*(2*η) := by
+    apply one_crossing_sum_bound (τ := τ) (mesh_mono T hT n) (by positivity)
+    · intro i hi hni
+      rw [clipped_increment_eq (mesh_mono T hT n (Nat.le_succ i)) b hni]
+      split_ifs <;> simp
+    · intro i hi hc
+      have hdist (x y : ℝ) (hx : x ∈ Icc (mesh T n i) (mesh T n (i+1)))
+          (hy : y ∈ Icc (mesh T n i) (mesh T n (i+1))) : dist x y < δ := by
+        rw [Real.dist_eq, abs_lt]
+        have hs := mesh_step T n i
+        have hnδ := hN n hn
+        constructor <;> linarith [hx.1,hx.2,hy.1,hy.2]
+      have hmod' (x y : ℝ) (hx : x ∈ Icc (mesh T n i) (mesh T n (i+1)))
+          (hy : y ∈ Icc (mesh T n i) (mesh T n (i+1))) :
+          |b x-b y| ≤ min 1 η := by
+        have hsub := Icc_subset_Icc (mesh_mem T hT n i hi.le).1 (mesh_mem T hT n (i+1) hi).2
+        exact (hmod x (hsub hx) y (hsub hy) (hdist x y hx hy)).le
+      have hcurr : mesh T n i ≤ mesh T n (i+1) := mesh_mono T hT n (Nat.le_succ i)
+      have hτmem : τ ∈ Icc (mesh T n i) (mesh T n (i+1)) := ⟨hc.1.le,hc.2.le⟩
+      have hx := (hmod' τ _ hτmem (left_mem_Icc.2 hcurr)).trans (min_le_left _ _)
+      have hy := (hmod' _ _ (right_mem_Icc.2 hcurr) (left_mem_Icc.2 hcurr)).trans (min_le_left _ _)
+      have hxy := (hmod' τ _ hτmem (right_mem_Icc.2 hcurr)).trans (min_le_right _ _)
+      rw [min_eq_right hc.2.le, min_eq_left hc.1.le, ite_eq_left hc.1, ← mul_sub, abs_mul]
+      apply mul_le_mul (hA n i) _ (abs_nonneg _) hK
+      apply abs_sq_sub_sq_le hx hy hη.le
+      simpa only [sub_sub_sub_cancel_right] using hxy
+  apply hbound.trans_lt
+  dsimp [η]
+  rw [show K * (2 * (ε / (2*K+1))) = (2*K*ε)/(2*K+1) by ring,
+    div_lt_iff₀ (by linarith : 0 < 2*K+1)]
+  nlinarith
+
+lemma clipped_mem {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (n i : ℕ) :
+    min (mesh T n i) τ ∈ Icc 0 τ := by
+  constructor
+  · exact le_min (by unfold mesh; positivity) hτ
+  · exact min_le_right _ _
+
+lemma prestop_riemann_tendsto {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T)
+    {C : ℝ → ℝ} (hC : ContinuousOn C (Icc 0 τ)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      if mesh T n i < τ then C (mesh T n i) * (T / (2 : ℝ)^n) else 0) atTop
+      (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
+  let p := clippedDyadic T τ hT hτ hτT
+  have hriem := p.riemann_tendsto hC
+  obtain ⟨K, hK⟩ := isCompact_Icc.exists_bound_of_continuousOn hC
+  have hKn : 0 ≤ K := (norm_nonneg _).trans (hK 0 ⟨le_rfl,hτ⟩)
+  have hdiff := clipped_timeSum_difference_tendsto (τ := τ) hT
+    (fun n i => C (min (mesh T n i) τ)) hKn
+    (fun n i => hK _ (clipped_mem hT hτ n i))
+  have heq (n i : ℕ) :
+      (if mesh T n i < τ then C (min (mesh T n i) τ) * (T / (2 : ℝ)^n) else 0) =
+      (if mesh T n i < τ then C (mesh T n i) * (T / (2 : ℝ)^n) else 0) := by
+    split_ifs with h
+    · rw [min_eq_left h.le]
+    · rfl
+  simp only [heq] at hdiff
+  have hh := hriem.sub hdiff
+  simpa only [p, clippedDyadic, sub_sub_cancel, sub_zero] using hh
+
+end SLEExactGauge.StoppedMeshes
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.PartitionMeshes.PartitionSequence
+open SLEExactGauge.TaylorMeshes SLEExactGauge.TimeField
+
+lemma family_remainder_tendsto_of_eventually_bounded {T : ℝ} (p : PartitionSequence T)
+    {f : ℕ → ℕ → ℝ → ℝ} {b : ℝ → ℝ}
+    (hf : ∀ n i, ContDiff ℝ 3 (f n i)) {V K : ℝ}
+    (hb : ContinuousOn b (Icc 0 T)) (hV : 0 ≤ V) (hK : 0 ≤ K)
+    (h3 : ∀ n i x, |iteratedDeriv 3 (f n i) x| ≤ K)
+    (hQ : ∀ᶠ n in atTop, (∑ i ∈ Finset.range (p.count n),
+      (b (p.time n (i+1))-b (p.time n i))^2) ≤ V) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      remainder₂ (f n i) (b (p.time n i)) (b (p.time n (i+1)))) atTop (𝓝 0) := by
+  apply Metric.tendsto_atTop.2
+  intro ε hε
+  let δ := ε / ((K+1)*(V+1))
+  have hδ : 0 < δ := div_pos hε (mul_pos (by linarith) (by linarith))
+  have hs := p.increments_small hb hδ
+  obtain ⟨N, hN⟩ := eventually_atTop.1 (hs.and hQ)
+  refine ⟨N, fun n hn => ?_⟩
+  rcases hN n hn with ⟨hs, hq⟩
+  have hr := sum_family_remainder₂_bound (hf n) (fun i => b (p.time n i))
+    (p.count n) K δ hK hδ.le (fun i _ z _ => h3 n i z) (fun i hi => (hs i hi).le)
+  simp only [Real.dist_eq, sub_zero]
+  calc
+    _ ≤ K * δ / 6 * ∑ i ∈ Finset.range (p.count n),
+        (b (p.time n (i+1))-b (p.time n i))^2 := hr
+    _ ≤ K * δ / 6 * (V+1) :=
+      mul_le_mul_of_nonneg_left (hq.trans (by linarith)) (by positivity)
+    _ < ε := by
+      dsimp [δ]
+      have hden : 0 < (K+1)*(V+1) := mul_pos (by linarith) (by linarith)
+      have hid : K * (ε / ((K+1)*(V+1))) / 6 * (V+1) = ε * (K / (6*(K+1))) := by
+        field_simp
+      rw [hid]
+      have hk : K / (6*(K+1)) < 1 := (div_lt_one (by positivity)).2 (by linarith)
+      simpa using mul_lt_mul_of_pos_left hk hε
+
+end SLEExactGauge.PartitionMeshes.PartitionSequence
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.StoppedMeshes
+open TaylorMeshes PartitionMeshes
+
+lemma clipped_quadratic_eventually_bounded {T τ V : ℝ} (hT : 0 ≤ T)
+    (hτ : 0 ≤ τ) (hτT : τ ≤ T) {b : ℝ → ℝ} (hb : ContinuousOn b (Icc 0 T))
+    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1)) - b (mesh T n i))^2) atTop (𝓝 V)) :
+    ∀ᶠ n in atTop, (∑ i ∈ Finset.range (2^n),
+      (b (min (mesh T n (i+1)) τ) - b (min (mesh T n i) τ))^2) ≤ V+2 := by
+  have hd := clipped_quadraticSum_difference_tendsto hT hτ hτT hb
+    (fun _ _ => 1) (by norm_num : (0 : ℝ) ≤ 1) (fun _ _ => by norm_num)
+  simp only [one_mul] at hd
+  filter_upwards [hd.eventually (gt_mem_nhds (show (0 : ℝ) < 1 by norm_num)),
+    hQ.eventually (gt_mem_nhds (show V < V+1 by linarith))] with n hn hqn
+  have hle : (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+      (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0) ≤
+      ∑ i ∈ Finset.range (2^n), (b (mesh T n (i+1)) - b (mesh T n i))^2 := by
+    apply Finset.sum_le_sum
+    intro i hi
+    split_ifs
+    · exact le_rfl
+    · exact sq_nonneg _
+  linarith
+
+end SLEExactGauge.StoppedMeshes
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.BrownianStopped
+open TaylorMeshes TimeField PartitionMeshes StoppedMeshes BrownianField
+
+lemma coefficient_clipped_bound {T τ K : ℝ} (C : ℝ → ℝ)
+    (hC : ∀ n i, mesh T n i < τ → |C (mesh T n i)| ≤ K) (n i : ℕ) :
+    |C (min (mesh T n i) τ)| ≤ max K |C τ| := by
+  by_cases h : mesh T n i < τ
+  · rw [min_eq_left h.le]
+    exact (hC n i h).trans (le_max_left _ _)
+  · rw [min_eq_right (le_of_not_gt h)]
+    exact le_max_right _ _
+
+lemma stopped_field_taylor_limit {T τ : ℝ} (hT : 0 ≤ T) (hτ : 0 ≤ τ) (hτT : τ ≤ T)
+    {F V : ℝ → ℝ → ℝ} {b : ℝ → ℝ}
+    (hb : ContinuousOn b (Icc 0 T)) (hreg : FieldRegular F V b τ)
+    (hQ : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1)) - b (mesh T n i))^2) atTop (𝓝 T))
+    {K₁ K₂ : ℝ} (hK₁ : 0 ≤ K₁) (hK₂ : 0 ≤ K₂)
+    (hA : ∀ n i, mesh T n i < τ →
+      |deriv (F (mesh T n i)) (b (mesh T n i))| ≤ K₁)
+    (hC : ∀ n i, mesh T n i < τ →
+      |iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i))| ≤ K₂)
+    (hE : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+      iteratedDeriv 2 (F (mesh T n i)) (b (mesh T n i)) *
+        ((b (mesh T n (i+1)) - b (mesh T n i))^2 - T/(2 : ℝ)^n) else 0) atTop (𝓝 0)) :
+    Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+      deriv (F (mesh T n i)) (b (mesh T n i)) *
+        (b (mesh T n (i+1)) - b (mesh T n i)) else 0) atTop
+      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, iteratedDeriv 2 (F t) (b t))) := by
+  let p := clippedDyadic T τ hT hτ hτT
+  let C : ℝ → ℝ := fun t => iteratedDeriv 2 (F t) (b t)
+  let A : ℝ → ℝ := fun t => deriv (F t) (b t)
+  have hbτ := hb.mono (Icc_subset_Icc le_rfl hτT)
+  have hQc := clipped_quadratic_eventually_bounded hT hτ hτT hb hQ
+  obtain ⟨K₃, hK₃, h₃⟩ := hreg.third_bound
+  have hr := p.family_remainder_tendsto_of_eventually_bounded
+    (f := fun n i => F (p.time n i)) (fun _ _ => hreg.smooth _)
+    hbτ (show 0 ≤ T+2 by linarith) hK₃ (fun _ _ x => h₃ _ x) hQc
+  have htime := p.time_increment_tendsto hreg.time_cont hreg.time_deriv hreg.drift_cont
+  have hriem := prestop_riemann_tendsto hT hτ hτT hreg.second_cont
+  let Qpred : ℕ → ℝ := fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+    C (mesh T n i) * (b (mesh T n (i+1)) - b (mesh T n i))^2 else 0
+  have hQpred : Tendsto Qpred atTop (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
+    have heq (n : ℕ) : Qpred n =
+      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then
+        C (mesh T n i) * ((b (mesh T n (i+1))-b (mesh T n i))^2-T/(2 : ℝ)^n) else 0) +
+      (∑ i ∈ Finset.range (2^n), if mesh T n i < τ then C (mesh T n i)*(T/(2 : ℝ)^n) else 0) := by
+      unfold Qpred
+      rw [← Finset.sum_add_distrib]
+      apply Finset.sum_congr rfl
+      intro i hi
+      split_ifs <;> ring
+    dsimp only [C] at heq
+    simpa only [← heq, zero_add] using hE.add hriem
+  have hQdiff := clipped_quadraticSum_difference_tendsto hT hτ hτT hb
+    (fun n i => C (p.time n i))
+    (K := max K₂ |C τ|) (hK₂.trans (le_max_left _ _))
+    (coefficient_clipped_bound C hC)
+  have hQeq (n i : ℕ) :
+      (if mesh T n i < τ then C (p.time n i) *
+        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) =
+      (if mesh T n i < τ then C (mesh T n i) *
+        (b (mesh T n (i+1))-b (mesh T n i))^2 else 0) := by
+    split_ifs with h
+    · simp only [p, clippedDyadic, min_eq_left h.le]
+    · rfl
+  simp only [hQeq] at hQdiff
+  have hQweighted : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      C (p.time n i) * (b (p.time n (i+1)) - b (p.time n i))^2) atTop
+      (𝓝 (∫ t in (0 : ℝ)..τ, C t)) := by
+    simpa only [Qpred, sub_add_cancel, zero_add, p, clippedDyadic] using hQdiff.add hQpred
+  have hTel (n : ℕ) : ∑ i ∈ Finset.range (p.count n),
+      (F (p.time n (i+1)) (b (p.time n (i+1))) -
+        F (p.time n i) (b (p.time n i))) = F τ (b τ) - F 0 (b 0) := by
+    rw [Finset.sum_range_sub (fun i => F (p.time n i) (b (p.time n i))), p.start, p.finish]
+  have hIdentity (n : ℕ) : (∑ i ∈ Finset.range (p.count n),
+      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) =
+      F τ (b τ) - F 0 (b 0) -
+      (∑ i ∈ Finset.range (p.count n),
+        (F (p.time n (i+1)) (b (p.time n (i+1))) -
+          F (p.time n i) (b (p.time n (i+1))))) - (1/2 : ℝ)*
+      (∑ i ∈ Finset.range (p.count n),
+        C (p.time n i)*(b (p.time n (i+1))-b (p.time n i))^2) -
+      (∑ i ∈ Finset.range (p.count n), remainder₂ (F (p.time n i))
+        (b (p.time n i)) (b (p.time n (i+1)))) := by
+    rw [← hTel n]
+    simp only [Finset.mul_sum, ← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro i hi
+    simp only [A, C, remainder₂, iteratedDeriv_succ, iteratedDeriv_zero]
+    ring
+  have hAclip : Tendsto (fun n => ∑ i ∈ Finset.range (p.count n),
+      A (p.time n i)*(b (p.time n (i+1))-b (p.time n i))) atTop
+      (𝓝 (F τ (b τ) - F 0 (b 0) - (∫ t in (0 : ℝ)..τ, V t (b t)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..τ, C t)) := by
+    have hh := (((tendsto_const_nhds (x := F τ (b τ) - F 0 (b 0))).sub
+      htime).sub (hQweighted.const_mul (1/2 : ℝ))).sub hr
+    simpa only [← hIdentity, sub_zero] using hh
+  have hAdiff := clipped_linearSum_difference_tendsto hT hτ hτT hb
+    (fun n i => A (p.time n i))
+    (K := max K₁ |A τ|) (hK₁.trans (le_max_left _ _))
+    (coefficient_clipped_bound A hA)
+  have hAeq (n i : ℕ) :
+      (if mesh T n i < τ then A (p.time n i) *
+        (b (mesh T n (i+1))-b (mesh T n i)) else 0) =
+      (if mesh T n i < τ then A (mesh T n i) *
+        (b (mesh T n (i+1))-b (mesh T n i)) else 0) := by
+    split_ifs with h
+    · simp only [p, clippedDyadic, min_eq_left h.le]
+    · rfl
+  simp only [hAeq] at hAdiff
+  have hh := hAclip.sub hAdiff
+  simpa only [p, clippedDyadic, A, C, sub_sub_cancel, sub_zero] using hh
+
+end SLEExactGauge.BrownianStopped
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set MeasureTheory ProbabilityTheory Filter
+open scoped ENNReal NNReal Topology
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology
+namespace SLEExactGauge.BrownianStopped
+open BrownianPredictable PredictableLimit BrownianIto TaylorMeshes BrownianField
+
+def stoppedCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F : Ω → ℝ → ℝ → ℝ) (n i : ℕ) (ω : Ω) : ℝ :=
+  if mesh T n i < τ ω then
+    iteratedDeriv k (F ω (mesh T n i)) (B (dyadicTime T n i) ω) else 0
+
+theorem stopped_fieldSum_tendsto {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
+    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
+    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (stoppedCoeff 2 B T τ F n i))
+    (hC2 : ∀ n i, MemLp (stoppedCoeff 2 B T τ F n i) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
+    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
+    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
+    ∀ᵐ ω ∂P, Tendsto (fun n => dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω) atTop
+      (𝓝 (F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
+        (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
+        (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω),
+          iteratedDeriv 2 (F ω t) (B t.toNNReal ω))) := by
+  have he := brownian_dyadic_weighted_quadraticError_tendsto hB T
+    (stoppedCoeff 2 B T τ F) hC hC2 K₂ hK₂ hbound
+  have hC_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂ :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hbound n))
+  have hA_all : ∀ᵐ ω ∂P, ∀ n i, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁ :=
+    ae_all_iff.2 (fun n => ae_all_iff.2 (hAbound n))
+  filter_upwards [hB.cont, quadraticSum_tendsto hB T, he, hreg, hτ, hC_all, hA_all]
+    with ω hb hQ hE hr hτω hCω hAω
+  let b : ℝ → ℝ := fun t => B t.toNNReal ω
+  have hbc : Continuous b := hb.comp continuous_real_toNNReal
+  have hQ' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n),
+      (b (mesh T n (i+1))-b (mesh T n i))^2) atTop (𝓝 (T : ℝ)) := by
+    simpa only [quadraticSum, b, mesh_toNNReal] using hQ
+  have hE' : Tendsto (fun n => ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
+      iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+        ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0)
+        atTop (𝓝 0) := by
+    have heq (n : ℕ) : dyadicQuadraticError B T (stoppedCoeff 2 B T τ F) n ω =
+        ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
+          iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i)) *
+            ((b (mesh T n (i+1))-b (mesh T n i))^2 - (T : ℝ)/(2 : ℝ)^n) else 0 := by
+      unfold dyadicQuadraticError
+      apply Finset.sum_congr rfl
+      intro i hi
+      simp only [stoppedCoeff, b, mesh_toNNReal]
+      split_ifs <;> simp
+    simpa only [heq] using hE
+  have hA' (n i : ℕ) (hi : mesh T n i < τ ω) :
+      |deriv (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₁ := by
+    simpa only [stoppedCoeff, ite_eq_left hi, iteratedDeriv_succ, iteratedDeriv_zero,
+      b, mesh_toNNReal] using hAω n i
+  have hC' (n i : ℕ) (hi : mesh T n i < τ ω) :
+      |iteratedDeriv 2 (F ω (mesh T n i)) (b (mesh T n i))| ≤ K₂ := by
+    simpa only [stoppedCoeff, ite_eq_left hi, b, mesh_toNNReal] using hCω n i
+  have ht := stopped_field_taylor_limit T.coe_nonneg hτω.1 hτω.2 hbc.continuousOn
+    hr hQ' hK₁ hK₂ hA' hC' hE'
+  have heq (n : ℕ) : dyadicDriverSum B T (stoppedCoeff 1 B T τ F) n ω =
+      ∑ i ∈ Finset.range (2^n), if mesh T n i < τ ω then
+        deriv (F ω (mesh T n i)) (b (mesh T n i)) *
+          (b (mesh T n (i+1))-b (mesh T n i)) else 0 := by
+    unfold dyadicDriverSum
+    apply Finset.sum_congr rfl
+    intro i hi
+    simp only [stoppedCoeff, b, mesh_toNNReal, iteratedDeriv_succ, iteratedDeriv_zero]
+    split_ifs <;> simp
+  simpa only [← heq, b, Real.toNNReal_zero] using ht
+
+theorem stopped_field_ito_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P]
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (T : ℝ≥0)
+    (τ : Ω → ℝ) (F V : Ω → ℝ → ℝ → ℝ)
+    (hτ : ∀ᵐ ω ∂P, 0 ≤ τ ω ∧ τ ω ≤ T)
+    (hreg : ∀ᵐ ω ∂P, FieldRegular (F ω) (V ω) (fun t => B t.toNNReal ω) (τ ω))
+    (hC : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (stoppedCoeff 2 B T τ F n i))
+    (hC2 : ∀ n i, MemLp (stoppedCoeff 2 B T τ F n i) 2 P)
+    (K₂ : ℝ) (hK₂ : 0 ≤ K₂)
+    (hbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 2 B T τ F n i ω| ≤ K₂)
+    (hA : ∀ n i, Measurable[pastSigma B (dyadicTime T n i)] (stoppedCoeff 1 B T τ F n i))
+    (hA2 : ∀ n i, MemLp (stoppedCoeff 1 B T τ F n i) 2 P)
+    (K₁ : ℝ) (hK₁ : 0 ≤ K₁)
+    (hAbound : ∀ n i, ∀ᵐ ω ∂P, |stoppedCoeff 1 B T τ F n i ω| ≤ K₁) :
+    let Y := fun ω => F ω (τ ω) (B (τ ω).toNNReal ω) - F ω 0 (B 0 ω) -
+      (∫ t in (0 : ℝ)..(τ ω), V ω t (B t.toNNReal ω)) -
+      (1/2 : ℝ) * ∫ t in (0 : ℝ)..(τ ω), iteratedDeriv 2 (F ω t) (B t.toNNReal ω)
+    Integrable Y P ∧ (∫ ω, Y ω ∂P) = 0 := by
+  dsimp only
+  exact limit_centered hB T (stoppedCoeff 1 B T τ F) hA hA2 K₁ hK₁ hAbound _
+    (stopped_fieldSum_tendsto hB T τ F V hτ hreg hC hC2 K₂ hK₂ hbound K₁ hK₁ hAbound)
+
+end SLEExactGauge.BrownianStopped
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/StrongRestart.lean b/lean/OAI/Probability/SLELocalization/StrongRestart.lean
index 63f9ded7031dcec958190a2a745158138f130cde..788b205d574ebf3cb8b4a7f591a29353be72614f 100644
--- a/lean/OAI/Probability/SLELocalization/StrongRestart.lean
+++ b/lean/OAI/Probability/SLELocalization/StrongRestart.lean
@@ -1,138 +1,138 @@
-import Mathlib
-import OAI.Probability.SLELocalization.BrownianSupport
-import OAI.Probability.SLELocalization.CommonClocks
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Brownian Predictable estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BrownianPredictable
-
-lemma aemeasurable_select_nat {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E]
-    {P : Measure Ω} {Y : Ω → ℕ} (hY : AEMeasurable Y P) (F : ℕ → Ω → E)
-    (hF : ∀ k, AEMeasurable (F k) P) : AEMeasurable (fun sample => F (Y sample) sample) P := by
-  have hm : Measurable (fun x : ℕ × (ℕ → E) => x.2 x.1) :=
-    measurable_from_prod_countable_right (fun k => measurable_pi_apply k)
-  exact hm.comp_aemeasurable (hY.prodMk (AEMeasurable.of_eval hF))
-
-lemma measure_nat_partition {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {Y : Ω → ℕ} (hY : AEMeasurable Y P) (S : Set Ω) (hS : NullMeasurableSet S P) :
-    P S = ∑' k : ℕ, P (S ∩ {sample | Y sample = k}) := by
-  have hpieces (k : ℕ) : NullMeasurableSet (S ∩ {sample | Y sample = k}) P :=
-    hS.inter (hY.nullMeasurableSet_preimage (measurableSet_singleton k))
-  have hd : Pairwise (fun i j : ℕ => Disjoint (S ∩ {sample | Y sample = i})
-      (S ∩ {sample | Y sample = j})) := by
-    intro i j hij
-    apply Set.disjoint_left.2
-    intro sample hi hj
-    exact hij (hi.2.symm.trans hj.2)
-  have he : (⋃ k : ℕ, S ∩ {sample | Y sample = k}) = S := by
-    ext sample
-    simp
-  calc
-    P S = P (⋃ k : ℕ, S ∩ {sample | Y sample = k}) := congrArg P he.symm
-    _ = _ := measure_iUnion₀ (fun i j hij => (hd hij).aedisjoint) hpieces
-
-lemma countable_restart_law {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (Y : Ω → ℕ) (hY : AEMeasurable Y P)
-    (s : ℕ → ℝ≥0) (A : Set Ω) (hA : NullMeasurableSet A P) (hA0 : P A ≠ 0)
-    (hpieces : ∀ k, MeasurableSet[pastSigma B (s k)] (A ∩ {sample | Y sample = k}))
-    (I : Finset ℝ≥0) :
-    HasLaw (fun sample => I.restrict (fun t => B (s (Y sample)+t) sample-B (s (Y sample)) sample))
-      (BrownianReal.projectiveFamily I) ((P A)⁻¹ • P.restrict A) := by
-  let F (k : ℕ) (sample : Ω) := I.restrict (fun t => B (s k+t) sample-B (s k) sample)
-  have hF (k : ℕ) : HasLaw (F k) (BrownianReal.projectiveFamily I) P :=
-    (hB.shift (s k)).toIsPreBrownianReal.hasLaw I
-  let X := fun sample => F (Y sample) sample
-  have hX : AEMeasurable X P := aemeasurable_select_nat hY F (fun k => (hF k).aemeasurable)
-  refine ⟨hX.restrict.smul_measure _,?_⟩
-  apply Measure.ext
-  intro E hE
-  rw [Measure.map_apply_of_aemeasurable (hX.restrict.smul_measure _) hE,
-    Measure.smul_apply,smul_eq_mul,Measure.restrict_apply₀ (hX.restrict.nullMeasurableSet_preimage hE),inter_comm]
-  have hind (k : ℕ) : P ((A ∩ {sample | Y sample = k}) ∩ (F k ⁻¹' E)) =
-      P (A ∩ {sample | Y sample = k}) * (BrownianReal.projectiveFamily I) E := by
-    have hi := (hB.toIsPreBrownianReal.indepFun_shift (s k)).symm.comp
-      measurable_id (show Measurable (fun f : ℝ≥0 → ℝ => I.restrict f) from
-        Measurable.of_eval (fun t => measurable_pi_apply (t : ℝ≥0)))
-    have hh := hi.meas_inter (hpieces k) (hE.preimage (comap_measurable (F k)))
-    have hll : P (F k ⁻¹' E) = (BrownianReal.projectiveFamily I) E := (hF k).measure_eq hE
-    rw [hll] at hh
-    exact hh
-  have he (k : ℕ) : (A ∩ X ⁻¹' E) ∩ {sample | Y sample = k} =
-      (A ∩ {sample | Y sample = k}) ∩ F k ⁻¹' E := by
-    ext sample
-    simp only [mem_inter_iff,mem_preimage,mem_ofPred_eq,X]
-    constructor
-    · rintro ⟨⟨ha,hf⟩,hy⟩
-      exact ⟨⟨ha,hy⟩,hy ▸ hf⟩
-    · rintro ⟨⟨ha,hy⟩,hf⟩
-      exact ⟨⟨ha,hy ▸ hf⟩,hy⟩
-  rw [measure_nat_partition hY (A ∩ X ⁻¹' E) (hA.inter (hX.nullMeasurableSet_preimage hE))]
-  simp_rw [he,hind]
-  rw [ENNReal.tsum_mul_right,← measure_nat_partition hY A hA,← mul_assoc,
-    ENNReal.inv_mul_cancel hA0 (measure_ne_top P A),one_mul]
-
-lemma hasLaw_of_ae_limit {Ω E : Type*} [MeasurableSpace Ω] [PseudoMetricSpace E]
-    [MeasurableSpace E] [BorelSpace E] {P : Measure Ω} [IsFiniteMeasure P]
-    {μ : Measure E} [IsFiniteMeasure μ] {X : ℕ → Ω → E} {Y : Ω → E}
-    (hX : ∀ n, HasLaw (X n) μ P)
-    (hlim : ∀ᵐ sample ∂P, Tendsto (fun n => X n sample) atTop (𝓝 (Y sample))) : HasLaw Y μ P := by
-  have hY : AEMeasurable Y P := aemeasurable_of_tendsto_metrizable_ae atTop
-    (fun n => (hX n).aemeasurable) hlim
-  refine ⟨hY,?_⟩
-  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
-  intro φ
-  rw [integral_map hY φ.continuous.measurable.aestronglyMeasurable]
-  have ht := tendsto_integral_of_dominated_convergence
-    (F := fun n sample => φ (X n sample)) (f := fun sample => φ (Y sample)) (fun _ : Ω => ‖φ‖)
-    (fun n => (φ.continuous.measurable.comp_aemeasurable (hX n).aemeasurable).aestronglyMeasurable)
-    (integrable_const ‖φ‖) (fun n => Filter.Eventually.of_forall (fun sample => φ.norm_coe_le_norm _))
-    (hlim.mono (fun sample hh => φ.continuous.continuousAt.tendsto.comp hh))
-  have he (n : ℕ) : (∫ sample, φ (X n sample) ∂P) = ∫ x, φ x ∂μ :=
-    (hX n).integral_comp φ.continuous.measurable.aestronglyMeasurable
-  simp only [he] at ht
-  exact tendsto_nhds_unique ht tendsto_const_nhds
-
-theorem brownianStrongRestart {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (τ : Ω → ℝ≥0) (hτ : AEMeasurable τ P)
-    (A : Set Ω) (hA : NullMeasurableSet A P) (hA0 : P A ≠ 0)
-    (hstop : ∀ t : ℝ≥0, MeasurableSet[pastSigma B t] (A ∩ {sample | τ sample ≤ t})) :
-    IsBrownianReal (fun t sample => B (τ sample+t) sample-B (τ sample) sample)
-      ((P A)⁻¹ • P.restrict A) := by
-  let Q : Measure Ω := (P A)⁻¹ • P.restrict A
-  have hQP : Q ≪ P := Measure.smul_absolutelyContinuous.trans
-    Measure.restrict_le_self.absolutelyContinuous
-  have : IsProbabilityMeasure Q := ⟨by
-    simp only [Q,Measure.smul_apply,smul_eq_mul,Measure.restrict_apply_univ]
-    exact ENNReal.inv_mul_cancel hA0 (measure_ne_top P A)⟩
-  have hc : ∀ᵐ sample ∂Q, Continuous (fun t => B t sample) := hQP.ae_le hB.cont
-  refine ⟨⟨fun I => ?_⟩,?_⟩
-  · apply hasLaw_of_ae_limit (X := fun n sample =>
-      I.restrict (fun t => B (restartTime τ n sample+t) sample-B (restartTime τ n sample) sample))
-    · intro n
-      exact countable_restart_law hB (restartIndex τ n) (restartIndex_aemeasurable hτ n)
-        (restartGrid n) A hA hA0 (restartIndex_past B τ A hstop n) I
-    · filter_upwards [hc] with sample hh
-      apply tendsto_pi_nhds.2
-      intro t
-      exact ((hh.continuousAt.tendsto.comp ((restartTime_tendsto τ sample).add tendsto_const_nhds)).sub
-        (hh.continuousAt.tendsto.comp (restartTime_tendsto τ sample)))
-  · filter_upwards [hc] with sample hh
-    exact (hh.comp (continuous_const.add continuous_id)).sub continuous_const
-
-end SLEExactGauge.BrownianPredictable
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.BrownianSupport
+import OAI.Probability.SLELocalization.CommonClocks
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Brownian Predictable estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BrownianPredictable
+
+lemma aemeasurable_select_nat {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E]
+    {P : Measure Ω} {Y : Ω → ℕ} (hY : AEMeasurable Y P) (F : ℕ → Ω → E)
+    (hF : ∀ k, AEMeasurable (F k) P) : AEMeasurable (fun sample => F (Y sample) sample) P := by
+  have hm : Measurable (fun x : ℕ × (ℕ → E) => x.2 x.1) :=
+    measurable_from_prod_countable_right (fun k => measurable_pi_apply k)
+  exact hm.comp_aemeasurable (hY.prodMk (AEMeasurable.of_eval hF))
+
+lemma measure_nat_partition {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {Y : Ω → ℕ} (hY : AEMeasurable Y P) (S : Set Ω) (hS : NullMeasurableSet S P) :
+    P S = ∑' k : ℕ, P (S ∩ {sample | Y sample = k}) := by
+  have hpieces (k : ℕ) : NullMeasurableSet (S ∩ {sample | Y sample = k}) P :=
+    hS.inter (hY.nullMeasurableSet_preimage (measurableSet_singleton k))
+  have hd : Pairwise (fun i j : ℕ => Disjoint (S ∩ {sample | Y sample = i})
+      (S ∩ {sample | Y sample = j})) := by
+    intro i j hij
+    apply Set.disjoint_left.2
+    intro sample hi hj
+    exact hij (hi.2.symm.trans hj.2)
+  have he : (⋃ k : ℕ, S ∩ {sample | Y sample = k}) = S := by
+    ext sample
+    simp
+  calc
+    P S = P (⋃ k : ℕ, S ∩ {sample | Y sample = k}) := congrArg P he.symm
+    _ = _ := measure_iUnion₀ (fun i j hij => (hd hij).aedisjoint) hpieces
+
+lemma countable_restart_law {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (Y : Ω → ℕ) (hY : AEMeasurable Y P)
+    (s : ℕ → ℝ≥0) (A : Set Ω) (hA : NullMeasurableSet A P) (hA0 : P A ≠ 0)
+    (hpieces : ∀ k, MeasurableSet[pastSigma B (s k)] (A ∩ {sample | Y sample = k}))
+    (I : Finset ℝ≥0) :
+    HasLaw (fun sample => I.restrict (fun t => B (s (Y sample)+t) sample-B (s (Y sample)) sample))
+      (BrownianReal.projectiveFamily I) ((P A)⁻¹ • P.restrict A) := by
+  let F (k : ℕ) (sample : Ω) := I.restrict (fun t => B (s k+t) sample-B (s k) sample)
+  have hF (k : ℕ) : HasLaw (F k) (BrownianReal.projectiveFamily I) P :=
+    (hB.shift (s k)).toIsPreBrownianReal.hasLaw I
+  let X := fun sample => F (Y sample) sample
+  have hX : AEMeasurable X P := aemeasurable_select_nat hY F (fun k => (hF k).aemeasurable)
+  refine ⟨hX.restrict.smul_measure _,?_⟩
+  apply Measure.ext
+  intro E hE
+  rw [Measure.map_apply_of_aemeasurable (hX.restrict.smul_measure _) hE,
+    Measure.smul_apply,smul_eq_mul,Measure.restrict_apply₀ (hX.restrict.nullMeasurableSet_preimage hE),inter_comm]
+  have hind (k : ℕ) : P ((A ∩ {sample | Y sample = k}) ∩ (F k ⁻¹' E)) =
+      P (A ∩ {sample | Y sample = k}) * (BrownianReal.projectiveFamily I) E := by
+    have hi := (hB.toIsPreBrownianReal.indepFun_shift (s k)).symm.comp
+      measurable_id (show Measurable (fun f : ℝ≥0 → ℝ => I.restrict f) from
+        Measurable.of_eval (fun t => measurable_pi_apply (t : ℝ≥0)))
+    have hh := hi.meas_inter (hpieces k) (hE.preimage (comap_measurable (F k)))
+    have hll : P (F k ⁻¹' E) = (BrownianReal.projectiveFamily I) E := (hF k).measure_eq hE
+    rw [hll] at hh
+    exact hh
+  have he (k : ℕ) : (A ∩ X ⁻¹' E) ∩ {sample | Y sample = k} =
+      (A ∩ {sample | Y sample = k}) ∩ F k ⁻¹' E := by
+    ext sample
+    simp only [mem_inter_iff,mem_preimage,mem_ofPred_eq,X]
+    constructor
+    · rintro ⟨⟨ha,hf⟩,hy⟩
+      exact ⟨⟨ha,hy⟩,hy ▸ hf⟩
+    · rintro ⟨⟨ha,hy⟩,hf⟩
+      exact ⟨⟨ha,hy ▸ hf⟩,hy⟩
+  rw [measure_nat_partition hY (A ∩ X ⁻¹' E) (hA.inter (hX.nullMeasurableSet_preimage hE))]
+  simp_rw [he,hind]
+  rw [ENNReal.tsum_mul_right,← measure_nat_partition hY A hA,← mul_assoc,
+    ENNReal.inv_mul_cancel hA0 (measure_ne_top P A),one_mul]
+
+lemma hasLaw_of_ae_limit {Ω E : Type*} [MeasurableSpace Ω] [PseudoMetricSpace E]
+    [MeasurableSpace E] [BorelSpace E] {P : Measure Ω} [IsFiniteMeasure P]
+    {μ : Measure E} [IsFiniteMeasure μ] {X : ℕ → Ω → E} {Y : Ω → E}
+    (hX : ∀ n, HasLaw (X n) μ P)
+    (hlim : ∀ᵐ sample ∂P, Tendsto (fun n => X n sample) atTop (𝓝 (Y sample))) : HasLaw Y μ P := by
+  have hY : AEMeasurable Y P := aemeasurable_of_tendsto_metrizable_ae atTop
+    (fun n => (hX n).aemeasurable) hlim
+  refine ⟨hY,?_⟩
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro φ
+  rw [integral_map hY φ.continuous.measurable.aestronglyMeasurable]
+  have ht := tendsto_integral_of_dominated_convergence
+    (F := fun n sample => φ (X n sample)) (f := fun sample => φ (Y sample)) (fun _ : Ω => ‖φ‖)
+    (fun n => (φ.continuous.measurable.comp_aemeasurable (hX n).aemeasurable).aestronglyMeasurable)
+    (integrable_const ‖φ‖) (fun n => Filter.Eventually.of_forall (fun sample => φ.norm_coe_le_norm _))
+    (hlim.mono (fun sample hh => φ.continuous.continuousAt.tendsto.comp hh))
+  have he (n : ℕ) : (∫ sample, φ (X n sample) ∂P) = ∫ x, φ x ∂μ :=
+    (hX n).integral_comp φ.continuous.measurable.aestronglyMeasurable
+  simp only [he] at ht
+  exact tendsto_nhds_unique ht tendsto_const_nhds
+
+theorem brownianStrongRestart {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (τ : Ω → ℝ≥0) (hτ : AEMeasurable τ P)
+    (A : Set Ω) (hA : NullMeasurableSet A P) (hA0 : P A ≠ 0)
+    (hstop : ∀ t : ℝ≥0, MeasurableSet[pastSigma B t] (A ∩ {sample | τ sample ≤ t})) :
+    IsBrownianReal (fun t sample => B (τ sample+t) sample-B (τ sample) sample)
+      ((P A)⁻¹ • P.restrict A) := by
+  let Q : Measure Ω := (P A)⁻¹ • P.restrict A
+  have hQP : Q ≪ P := Measure.smul_absolutelyContinuous.trans
+    Measure.restrict_le_self.absolutelyContinuous
+  have : IsProbabilityMeasure Q := ⟨by
+    simp only [Q,Measure.smul_apply,smul_eq_mul,Measure.restrict_apply_univ]
+    exact ENNReal.inv_mul_cancel hA0 (measure_ne_top P A)⟩
+  have hc : ∀ᵐ sample ∂Q, Continuous (fun t => B t sample) := hQP.ae_le hB.cont
+  refine ⟨⟨fun I => ?_⟩,?_⟩
+  · apply hasLaw_of_ae_limit (X := fun n sample =>
+      I.restrict (fun t => B (restartTime τ n sample+t) sample-B (restartTime τ n sample) sample))
+    · intro n
+      exact countable_restart_law hB (restartIndex τ n) (restartIndex_aemeasurable hτ n)
+        (restartGrid n) A hA hA0 (restartIndex_past B τ A hstop n) I
+    · filter_upwards [hc] with sample hh
+      apply tendsto_pi_nhds.2
+      intro t
+      exact ((hh.continuousAt.tendsto.comp ((restartTime_tendsto τ sample).add tendsto_const_nhds)).sub
+        (hh.continuousAt.tendsto.comp (restartTime_tendsto τ sample)))
+  · filter_upwards [hc] with sample hh
+    exact (hh.comp (continuous_const.add continuous_id)).sub continuous_const
+
+end SLEExactGauge.BrownianPredictable
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/SwallowingLaws.lean b/lean/OAI/Probability/SLELocalization/SwallowingLaws.lean
index 2983acbe63043b48f2a3c4fea87b22ffd0457f86..4dd3848c2d2781be46106695e06c7b4d42cfd202 100644
--- a/lean/OAI/Probability/SLELocalization/SwallowingLaws.lean
+++ b/lean/OAI/Probability/SLELocalization/SwallowingLaws.lean
@@ -1,945 +1,945 @@
-import Mathlib
-import OAI.Probability.SLELocalization.GreenTilts
-import OAI.Probability.SLE.IntegralLimits
-import OAI.Probability.SLE.ItoIsometry
-import OAI.Probability.SLE.StochasticIntegrals
-
-namespace OAI
-
-
-/-! Conformal localization and Swallowing estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.Swallowing
-open GreenPhysical
-
-lemma generator_eq (κ β C x y : ℝ) (hy : 0 < y) :
-    generator κ β C x y =
-      (x^2+y^2)^(β/2-1) *
-        (β*(κ-4)/2 + β*(8-2*κ+κ*β)/2*(x^2/(x^2+y^2))) -
-      (2*C*β)*y^β/(x^2+y^2) := by
-  have hq := (q_pos x hy).ne'
-  have hp : (x^2+y^2)^(β/2-2) = (x^2+y^2)^(β/2-1)/(x^2+y^2) := by
-    rw [← Real.rpow_sub_one hq]
-    congr 1
-    ring
-  have hyp : y^(β-1)*y = y^β := by
-    conv_lhs => rhs; rw [← Real.rpow_one y]
-    rw [← Real.rpow_add hy]
-    congr 1
-    ring
-  unfold generator radialXX
-  rw [hp]
-  calc
-    _ = (x^2+y^2)^(β/2-1)*
-        (β*(κ-4)/2+β*(8-2*κ+κ*β)/2*(x^2/(x^2+y^2))) -
-        2*C*β*(y^(β-1)*y)/(x^2+y^2) := by field_simp; ring
-    _ = _ := by rw [hyp]
-
-lemma radial_height_ineq {β : ℝ} (hβ : β ≤ 2) (x : ℝ) {y : ℝ} (hy : 0 < y) :
-    (x^2+y^2)^(β/2-1)*y^2 ≤ y^β := by
-  have hh := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hy)
-    (show y^2 ≤ x^2+y^2 by nlinarith [sq_nonneg x]) (show β/2-1 ≤ 0 by linarith)
-  calc
-    _ ≤ (y^2)^(β/2-1)*y^2 := mul_le_mul_of_nonneg_right hh (sq_nonneg y)
-    _ = y^β := by
-      rw [← Real.rpow_natCast_mul hy.le]
-      conv_lhs => rhs; rw [← Real.rpow_natCast]
-      rw [← Real.rpow_add hy]
-      norm_num
-      congr 1
-      ring
-
-lemma generator_negative {κ β : ℝ} (hκ : 4 < κ) (hβ : 0 < β)
-    (hβκ : β < 1-4/κ) (x : ℝ) {y : ℝ} (hy : 0 < y) :
-    let C := (2*κ-8-κ*β)/4
-    let d := β*(κ-4-κ*β)/2
-    0 < C ∧ 0 < d ∧ generator κ β C x y ≤ -d*(x^2+y^2)^(β/2-1) := by
-  have hk : 0 < κ := by linarith
-  have hbk : κ*β < κ-4 := by
-    have hh := (mul_lt_mul_of_pos_left hβκ hk)
-    field_simp at hh
-    linarith
-  have hb2 : β ≤ 2 := by
-    have h4 := div_pos (by norm_num : (0:ℝ)<4) hk
-    linarith
-  dsimp only
-  have hC : 0 < (2*κ-8-κ*β)/4 := by linarith
-  have hd : 0 < β*(κ-4-κ*β)/2 := by positivity
-  refine ⟨hC,hd,?_⟩
-  have he := radial_height_ineq hb2 x hy
-  have hq := q_pos x hy
-  have hh := mul_le_mul_of_nonneg_left he (show 0 ≤ β*(2*κ-8-κ*β)/2 from div_nonneg (mul_nonneg hβ.le (by linarith)) (by norm_num))
-  rw [generator_eq _ _ _ _ _ hy]
-  apply (mul_le_mul_iff_left₀ hq).1
-  have heq :
-      ((x^2+y^2)^(β/2-1)*(β*(κ-4)/2+β*(8-2*κ+κ*β)/2*(x^2/(x^2+y^2))) -
-        (2*((2*κ-8-κ*β)/4)*β)*y^β/(x^2+y^2))*(x^2+y^2) =
-      -(β*(κ-4-κ*β)/2)*(x^2+y^2)^(β/2-1)*(x^2+y^2)+
-        (β*(2*κ-8-κ*β)/2)*((x^2+y^2)^(β/2-1)*y^2-y^β) := by
-    field_simp
-    ring
-  rw [heq]
-  nlinarith
-
-lemma monomial_bound {ε : ℝ} (hε : 0 < ε) (x : ℝ) {y a : ℝ} (hy : ε ≤ y)
-    (n : ℕ) (ha : a+(n:ℝ)/2 ≤ 0) :
-    |x|^n*(x^2+y^2)^a ≤ (ε^2)^(a+(n:ℝ)/2) := by
-  have hq := q_pos x (hε.trans_le hy)
-  have hx : |x| ≤ Real.sqrt (x^2+y^2) := Real.abs_le_sqrt (by nlinarith [sq_nonneg y])
-  have hl : ε^2 ≤ x^2+y^2 := by nlinarith [sq_nonneg x]
-  calc
-    _ ≤ (Real.sqrt (x^2+y^2))^n*(x^2+y^2)^a := by gcongr
-    _ = (x^2+y^2)^(a+(n:ℝ)/2) := by
-      rw [Real.sqrt_eq_rpow,← Real.rpow_mul_natCast hq.le,← Real.rpow_add hq]
-      congr 1
-      ring
-    _ ≤ _ := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hε) hl ha
-
-lemma radialX_bound {ε β : ℝ} (hε : 0 < ε) (hβ : β ≤ 1) (x : ℝ) {y : ℝ} (hy : ε ≤ y) :
-    |radialX β x y| ≤ |β| *(ε^2)^(β/2-1/2) := by
-  have hh := monomial_bound (a := β/2-1) hε x hy 1 (by norm_num; linarith)
-  norm_num at hh
-  rw [radialX,abs_mul,abs_mul,abs_of_nonneg (Real.rpow_nonneg (q_pos x (hε.trans_le hy)).le _)]
-  have he : β/2-1+1/2 = β/2-1/2 := by ring
-  rw [he] at hh
-  simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh (abs_nonneg β)
-
-lemma radialXX_bound {ε β : ℝ} (hε : 0 < ε) (hβ : β ≤ 1) (x : ℝ) {y : ℝ} (hy : ε ≤ y) :
-    |radialXX β x y| ≤ (|β|+|β*(β-2)|)*(ε^2)^(β/2-1) := by
-  have h0 := monomial_bound (a := β/2-1) hε x hy 0 (by norm_num; linarith)
-  have h2 := monomial_bound (a := β/2-2) hε x hy 2 (by norm_num; linarith)
-  norm_num at h0 h2
-  rw [show β/2-2+1 = β/2-1 by ring] at h2
-  calc
-    _ ≤ |β| *(x^2+y^2)^(β/2-1)+|β*(β-2)| *(|x|^2*(x^2+y^2)^(β/2-2)) := by
-      unfold radialXX
-      exact (abs_add_le _ _).trans_eq (by simp [abs_mul,abs_pow,abs_of_nonneg (Real.rpow_nonneg (q_pos x (hε.trans_le hy)).le _)];ring)
-    _ ≤ |β| *(ε^2)^(β/2-1)+|β*(β-2)| *(ε^2)^(β/2-1) := by
-      gcongr
-      simpa only [sq_abs] using h2
-    _ = _ := by ring
-
-lemma radialXXX_bound {ε β : ℝ} (hε : 0 < ε) (hβ : β ≤ 1) (x : ℝ) {y : ℝ} (hy : ε ≤ y) :
-    |radialXXX β x y| ≤ (|3*β*(β-2)|+|β*(β-2)*(β-4)|)*(ε^2)^(β/2-3/2) := by
-  have h1 := monomial_bound (a := β/2-2) hε x hy 1 (by norm_num; linarith)
-  have h3 := monomial_bound (a := β/2-3) hε x hy 3 (by norm_num; linarith)
-  norm_num at h1 h3
-  rw [show β/2-2+1/2 = β/2-3/2 by ring] at h1
-  rw [show β/2-3+3/2 = β/2-3/2 by ring] at h3
-  calc
-    _ ≤ |3*β*(β-2)| *(|x| *(x^2+y^2)^(β/2-2))+
-        |β*(β-2)*(β-4)| *(|x|^3*(x^2+y^2)^(β/2-3)) := by
-      unfold radialXXX
-      exact (abs_add_le _ _).trans_eq (by simp [abs_mul,abs_pow,abs_of_nonneg (Real.rpow_nonneg (q_pos x (hε.trans_le hy)).le _)];ring)
-    _ ≤ |3*β*(β-2)| *(ε^2)^(β/2-3/2)+|β*(β-2)*(β-4)| *(ε^2)^(β/2-3/2) := by gcongr
-    _ = _ := by ring
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.Swallowing
-open GreenPhysical LoewnerCutoff LoewnerExit
-
-lemma deriv_lyapunov_driver (β C c y s b : ℝ) (hy : 0 < y) :
-    deriv (fun b => lyapunov β C (c-s*b) y) b = -s*radialX β (c-s*b) y := by
-  have hd := ((hasDerivAt_radial_x β (c-s*b) y hy).comp b
-    (((hasDerivAt_id b).const_mul s).const_sub c)).add_const (C*y^β)
-  convert hd.deriv using 1 <;> simp [lyapunov]; ring
-
-lemma second_lyapunov_driver (β C c y s b : ℝ) (hy : 0 < y) :
-    iteratedDeriv 2 (fun b => lyapunov β C (c-s*b) y) b = s^2*radialXX β (c-s*b) y := by
-  have he : deriv (fun b => lyapunov β C (c-s*b) y) = fun b => -s*radialX β (c-s*b) y :=
-    funext (fun b => deriv_lyapunov_driver β C c y s b hy)
-  simp only [iteratedDeriv_succ,iteratedDeriv_zero]
-  rw [he]
-  have hd := ((hasDerivAt_radialX_x β (c-s*b) y hy).comp b
-    (((hasDerivAt_id b).const_mul s).const_sub c)).const_mul (-s)
-  simp only [Function.comp_def,id_eq] at hd
-  convert hd.deriv using 1
-  ring
-
-lemma third_lyapunov_driver (β C c y s b : ℝ) (hy : 0 < y) :
-    iteratedDeriv 3 (fun b => lyapunov β C (c-s*b) y) b = -s^3*radialXXX β (c-s*b) y := by
-  have he : iteratedDeriv 2 (fun b => lyapunov β C (c-s*b) y) = fun b => s^2*radialXX β (c-s*b) y :=
-    funext (fun b => second_lyapunov_driver β C c y s b hy)
-  rw [show (3:ℕ)=2+1 by rfl,iteratedDeriv_succ,he]
-  have hd := ((hasDerivAt_radialXX_x β (c-s*b) y hy).comp b
-    (((hasDerivAt_id b).const_mul s).const_sub c)).const_mul (s^2)
-  simp only [Function.comp_def,id_eq] at hd
-  convert hd.deriv using 1
-  ring
-
-lemma continuous_radial_comp {α : Type*} [TopologicalSpace α] {X Y : α → ℝ}
-    (hX : Continuous X) (hY : Continuous Y) (hy : ∀ a, 0 < Y a) (β : ℝ) :
-    Continuous (fun a => radial β (X a) (Y a)) := by
-  exact ((hX.pow 2).add (hY.pow 2)).rpow_const (fun a => Or.inl (q_pos _ (hy a)).ne')
-
-lemma continuous_radialX_comp {α : Type*} [TopologicalSpace α] {X Y : α → ℝ}
-    (hX : Continuous X) (hY : Continuous Y) (hy : ∀ a, 0 < Y a) (β : ℝ) :
-    Continuous (fun a => radialX β (X a) (Y a)) := by
-  exact (continuous_const.mul hX).mul (((hX.pow 2).add (hY.pow 2)).rpow_const
-    (fun a => Or.inl (q_pos _ (hy a)).ne'))
-
-lemma continuous_radialXX_comp {α : Type*} [TopologicalSpace α] {X Y : α → ℝ}
-    (hX : Continuous X) (hY : Continuous Y) (hy : ∀ a, 0 < Y a) (β : ℝ) :
-    Continuous (fun a => radialXX β (X a) (Y a)) := by
-  have hp (e : ℝ) := ((hX.pow 2).add (hY.pow 2)).rpow_const (p := e)
-    (fun a => Or.inl (q_pos _ (hy a)).ne')
-  exact (continuous_const.mul (hp _)).add ((continuous_const.mul (hX.pow 2)).mul (hp _))
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.Swallowing
-open GreenPhysical LoewnerCutoff LoewnerExit
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-def field (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
-  lyapunov β C ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t)
-
-def drift (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
-  let x := (position hε hT U z t).re-s*b
-  let y := LoewnerField.height hε hT U z t
-  let v := velocity ε (extendDriver T U t) (position hε hT U z t)
-  β*(x*v.re+y*v.im)*(x^2+y^2)^(β/2-1)+C*(β*y^(β-1)*v.im)
-
-lemma first_eq (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    iteratedDeriv 1 (field hε hT β C s U z t) b =
-      -s*radialX β ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t) := by
-  simp only [iteratedDeriv_succ,iteratedDeriv_zero]
-  exact deriv_lyapunov_driver _ _ _ _ _ _ (LoewnerField.height_pos hε hT U z t)
-
-lemma second_eq (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    iteratedDeriv 2 (field hε hT β C s U z t) b =
-      s^2*radialXX β ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t) :=
-  second_lyapunov_driver _ _ _ _ _ _ (LoewnerField.height_pos hε hT U z t)
-
-lemma third_eq (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
-    iteratedDeriv 3 (field hε hT β C s U z t) b =
-      -s^3*radialXXX β ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t) :=
-  third_lyapunov_driver _ _ _ _ _ _ (LoewnerField.height_pos hε hT U z t)
-
-lemma smooth (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
-    ContDiff ℝ 3 (field hε hT β C s U z t) := by
-  exact ((contDiff_radial_driver β _ _ s (LoewnerField.height_pos hε hT U z t)).of_le (by simp)).add contDiff_const
-
-lemma coeff_uniform_bound (β C s : ℝ) (hβ : β ≤ 1) (z : ℂ) (k : ℕ)
-    (hk : k=1 ∨ k=2 ∨ k=3) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t b : ℝ),
-      |iteratedDeriv k (field hε hT β C s U z t) b| ≤ K := by
-  rcases hk with rfl|rfl|rfl
-  · refine ⟨|s| * (|β| * (ε^2)^(β/2-1/2)),by positivity,?_⟩
-    intro U t b
-    rw [first_eq,abs_mul,abs_neg]
-    exact mul_le_mul_of_nonneg_left (radialX_bound hε hβ _ (le_max_left _ _)) (abs_nonneg _)
-  · refine ⟨|s|^2*((|β|+|β*(β-2)|)*(ε^2)^(β/2-1)),by positivity,?_⟩
-    intro U t b
-    rw [second_eq,abs_mul,abs_pow]
-    exact mul_le_mul_of_nonneg_left (radialXX_bound hε hβ _ (le_max_left _ _)) (by positivity)
-  · refine ⟨|s|^3*((|3*β*(β-2)|+|β*(β-2)*(β-4)|)*(ε^2)^(β/2-3/2)),by positivity,?_⟩
-    intro U t b
-    rw [third_eq,abs_mul,abs_neg,abs_pow]
-    exact mul_le_mul_of_nonneg_left (radialXXX_bound hε hβ _ (le_max_left _ _)) (by positivity)
-
-lemma continuous_field (β C s : ℝ) (z : ℂ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => field hε hT β C s q.1.1 z q.1.2 q.2) := by
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
-  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => LoewnerField.height hε hT q.1.1 z q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  exact (continuous_radial_comp hx hy (fun q => LoewnerField.height_pos hε hT q.1.1 z q.1.2) β).add
-    (continuous_const.mul (hy.rpow_const (fun q => Or.inl (LoewnerField.height_pos hε hT q.1.1 z q.1.2).ne')))
-
-lemma continuous_coeff (β C s : ℝ) (z : ℂ) (k : ℕ) (hk : k=1 ∨ k=2) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv k (field hε hT β C s q.1.1 z q.1.2) q.2) := by
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
-  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => LoewnerField.height hε hT q.1.1 z q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  rcases hk with rfl|rfl
-  · simp only [first_eq]
-    exact continuous_const.mul (continuous_radialX_comp hx hy (fun q => LoewnerField.height_pos hε hT q.1.1 z q.1.2) β)
-  · simp only [second_eq]
-    exact continuous_const.mul (continuous_radialXX_comp hx hy (fun q => LoewnerField.height_pos hε hT q.1.1 z q.1.2) β)
-
-lemma continuous_drift (β C s : ℝ) (z : ℂ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => drift hε hT β C s q.1.1 z q.1.2 q.2) := by
-  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hD := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hV := (continuous_velocity hε).comp (hD.prodMk hH)
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
-  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => LoewnerField.height hε hT q.1.1 z q.1.2) :=
-    continuous_const.max (Complex.continuous_im.comp hH)
-  have hp (e : ℝ) := ((hx.pow 2).add (hy.pow 2)).rpow_const (p := e)
-    (fun q => Or.inl (q_pos _ (LoewnerField.height_pos hε hT q.1.1 z q.1.2)).ne')
-  exact ((continuous_const.mul ((hx.mul (Complex.continuous_re.comp hV)).add
-    (hy.mul (Complex.continuous_im.comp hV)))).mul (hp _)).add
-    (continuous_const.mul ((continuous_const.mul (hy.rpow_const (fun q =>
-      Or.inl (LoewnerField.height_pos hε hT q.1.1 z q.1.2).ne'))).mul (Complex.continuous_im.comp hV)))
-
-lemma time_deriv (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) {t : ℝ}
-    (ht : t ∈ Ioo 0 T) (hy : ε < (position hε hT U z t).im) (b : ℝ) :
-    HasDerivAt (fun v => field hε hT β C s U z v b) (drift hε hT β C s U z t b) t := by
-  have hH := position_deriv hε hT U z ht
-  have hIm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hH
-  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
-  dsimp only [Function.comp_def,Complex.imCLM_apply,Complex.reCLM_apply] at hIm hRe
-  have hY : HasDerivAt (LoewnerField.height hε hT U z)
-      (velocity ε (extendDriver T U t) (position hε hT U z t)).im t := by
-    apply hIm.congr_of_eventuallyEq
-    filter_upwards [hIm.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy)] with s hs
-    exact max_eq_right hs.le
-  have hR := hasDerivAt_radial_path β (hRe.sub_const (s*b)) hY (LoewnerField.height_pos hε hT U z t)
-  have hP := hY.rpow_const (p := β) (Or.inl (LoewnerField.height_pos hε hT U z t).ne')
-  apply (hR.add (hP.const_mul C)).congr_deriv
-  dsimp only [drift]
-  ring
-
-lemma regular (hTp : 0 < T) (β C s : ℝ) (hβ : β ≤ 1) (U : C(I, ℝ)) (z : ℂ)
-    (ρ δ : ℝ) (b : ℝ → ℝ) (hb : Continuous b) :
-    BrownianField.FieldRegular (field hε hT β C s U z) (drift hε hT β C s U z) b
-      (stop hε hT U z ρ δ) := by
-  constructor
-  · exact smooth hε hT β C s U z
-  · obtain ⟨K,hK,hb⟩ := coeff_uniform_bound hε hT β C s hβ z 3 (Or.inr (Or.inr rfl))
-    exact ⟨K,hK,hb U⟩
-  · intro x
-    exact ((continuous_field hε hT β C s z).comp
-      ((continuous_const.prodMk continuous_id).prodMk continuous_const)).continuousOn
-  · intro t ht x
-    have htT : t ∈ Ioo 0 T := ⟨ht.1,ht.2.trans_le (stop_bounds hε hT U z ρ δ).2⟩
-    have hh := (deficit_pos_iff hε hT U z ρ δ t).1
-      (positive_before hε hT hTp U z ρ δ ht.1.le ht.2)
-    exact time_deriv hε hT β C s U z htT hh.2.1 x
-  · exact ((continuous_drift hε hT β C s z).comp
-      ((continuous_const.prodMk continuous_fst).prodMk (hb.comp continuous_snd))).continuousOn
-  · exact ((continuous_coeff hε hT β C s z 2 (Or.inr rfl)).comp
-      ((continuous_const.prodMk continuous_id).prodMk hb)).continuousOn
-
-lemma field_prefix_eq {S : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) (U V : C(I, ℝ))
-    (z : ℂ) (β C s : ℝ) (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
-    {t : ℝ} (ht : t ∈ Icc 0 S) :
-    field hε hS β C s U z t = field hε hT β C s V z t := by
-  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
-  ext b
-  simp only [field,LoewnerField.height,hH]
-
-end SLEExactGauge.Swallowing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing GreenPhysical
-variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-
-def factor (β a : ℝ) (U : C(I, ℝ)) (x t : ℝ) : ℝ :=
-  Real.exp ((a*β/2)*logDerivative hε hT U (x:ℂ) t)
-
-def field (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) : ℝ :=
-  factor hε hT β a U x t * radial (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε
-
-def drift (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) : ℝ :=
-  let X := (position hε hT U (x:ℂ) t).re-s*b
-  factor hε hT β a U x t *
-    ((a*β/2)*variation hε hT U (x:ℂ) t * radial (-β) X ε +
-    radialX (-β) X ε * (velocity ε (extendDriver T U t) (position hε hT U (x:ℂ) t)).re)
-
-lemma factor_pos (β a : ℝ) (U : C(I, ℝ)) (x t : ℝ) :
-    0 < factor hε hT β a U x t := Real.exp_pos _
-
-lemma field_pos (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
-    0 < field hε hT β a s U x t b :=
-  mul_pos (factor_pos hε hT β a U x t) (radial_pos _ _ hε)
-
-lemma factor_bound (β a : ℝ) (U : C(I, ℝ)) (x t : ℝ) :
-    factor hε hT β a U x t ≤ Real.exp (|a*β/2| *((2/ε^2)*T)) := by
-  apply Real.exp_le_exp.mpr
-  exact (le_abs_self _).trans ((abs_mul _ _).le.trans
-    (mul_le_mul_of_nonneg_left (abs_logDerivative_le hε hT U (x:ℂ) t) (abs_nonneg _)))
-
-lemma first_eq (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
-    iteratedDeriv 1 (field hε hT β a s U x t) b =
-      factor hε hT β a U x t * (-s*radialX (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε) := by
-  simp only [iteratedDeriv_succ,iteratedDeriv_zero]
-  have hd := ((hasDerivAt_radial_x (-β) _ ε hε).comp b
-    (((hasDerivAt_id b).const_mul s).const_sub ((position hε hT U (x:ℂ) t).re))).const_mul (factor hε hT β a U x t)
-  simp only [Function.comp_def,id_eq] at hd
-  convert hd.deriv using 1 <;> first | rfl | ring
-
-lemma second_eq (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
-    iteratedDeriv 2 (field hε hT β a s U x t) b =
-      factor hε hT β a U x t * (s^2*radialXX (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε) := by
-  have he := funext (first_eq hε hT β a s U x t)
-  rw [show (2:ℕ)=1+1 by rfl,iteratedDeriv_succ,he]
-  have hd := (((hasDerivAt_radialX_x (-β) _ ε hε).comp b
-    (((hasDerivAt_id b).const_mul s).const_sub ((position hε hT U (x:ℂ) t).re))).const_mul (-s)).const_mul (factor hε hT β a U x t)
-  simp only [Function.comp_def,id_eq] at hd
-  convert hd.deriv using 1
-  first | rfl | ring
-
-lemma third_eq (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
-    iteratedDeriv 3 (field hε hT β a s U x t) b =
-      factor hε hT β a U x t * (-s^3*radialXXX (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε) := by
-  have he := funext (second_eq hε hT β a s U x t)
-  rw [show (3:ℕ)=2+1 by rfl,iteratedDeriv_succ,he]
-  have hd := (((hasDerivAt_radialXX_x (-β) _ ε hε).comp b
-    (((hasDerivAt_id b).const_mul s).const_sub ((position hε hT U (x:ℂ) t).re))).const_mul (s^2)).const_mul (factor hε hT β a U x t)
-  simp only [Function.comp_def,id_eq] at hd
-  convert hd.deriv using 1
-  first | rfl | ring
-
-lemma smooth (β a s : ℝ) (U : C(I, ℝ)) (x t : ℝ) :
-    ContDiff ℝ 3 (field hε hT β a s U x t) :=
-  contDiff_const.mul ((contDiff_radial_driver (-β) _ ε s hε).of_le (by simp))
-
-lemma coeff_uniform_bound (β a s : ℝ) (hβ : 0 ≤ β) (x : ℝ) (k : ℕ)
-    (hk : k=1 ∨ k=2 ∨ k=3) :
-    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t b : ℝ),
-      |iteratedDeriv k (field hε hT β a s U x t) b| ≤ K := by
-  let E := Real.exp (|a*β/2| *((2/ε^2)*T))
-  have hβ' : -β ≤ 1 := by linarith
-  have hf (U : C(I, ℝ)) (t : ℝ) : |factor hε hT β a U x t| ≤ E := by
-    rw [abs_of_pos (factor_pos hε hT β a U x t)]
-    exact factor_bound hε hT β a U x t
-  rcases hk with rfl|rfl|rfl
-  · refine ⟨E*(|s| *(|-β| *(ε^2)^((-β)/2-1/2))),by positivity,?_⟩
-    intro U t b
-    rw [first_eq,abs_mul,abs_mul,abs_neg]
-    exact mul_le_mul (hf U t) (mul_le_mul_of_nonneg_left
-      (radialX_bound hε hβ' _ le_rfl) (abs_nonneg _)) (by positivity) (by positivity)
-  · refine ⟨E*(|s|^2*((|-β|+|(-β)*(-β-2)|)*(ε^2)^((-β)/2-1))),by positivity,?_⟩
-    intro U t b
-    rw [second_eq,abs_mul,abs_mul,abs_pow]
-    exact mul_le_mul (hf U t) (mul_le_mul_of_nonneg_left
-      (radialXX_bound hε hβ' _ le_rfl) (by positivity)) (by positivity) (by positivity)
-  · refine ⟨E*(|s|^3*((|3*(-β)*(-β-2)|+|(-β)*(-β-2)*(-β-4)|)*(ε^2)^((-β)/2-3/2))),by positivity,?_⟩
-    intro U t b
-    rw [third_eq,abs_mul,abs_mul,abs_neg,abs_pow]
-    exact mul_le_mul (hf U t) (mul_le_mul_of_nonneg_left
-      (radialXXX_bound hε hβ' _ le_rfl) (by positivity)) (by positivity) (by positivity)
-
-lemma continuous_factor (β a x : ℝ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => factor hε hT β a q.1 x q.2) :=
-  Real.continuous_exp.comp (continuous_const.mul (continuous_logDerivative hε hT (x:ℂ)))
-
-lemma continuous_field (β a s x : ℝ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => field hε hT β a s q.1.1 x q.1.2 q.2) := by
-  have hH := (continuous_position hε hT (x:ℂ)).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
-  exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
-    (continuous_radial_comp hx continuous_const (fun _ => hε) (-β))
-
-lemma continuous_coeff (β a s x : ℝ) (k : ℕ) (hk : k=1 ∨ k=2) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv k (field hε hT β a s q.1.1 x q.1.2) q.2) := by
-  have hH := (continuous_position hε hT (x:ℂ)).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
-  rcases hk with rfl|rfl
-  · simp only [first_eq]
-    exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
-      (continuous_const.mul (continuous_radialX_comp hx continuous_const (fun _ => hε) (-β)))
-  · simp only [second_eq]
-    exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
-      (continuous_const.mul (continuous_radialXX_comp hx continuous_const (fun _ => hε) (-β)))
-
-lemma continuous_drift (β a s x : ℝ) :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => drift hε hT β a s q.1.1 x q.1.2 q.2) := by
-  have hH := (continuous_position hε hT (x:ℂ)).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hD := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
-  have hV := (continuous_velocity hε).comp (hD.prodMk hH)
-  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
-    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
-  have hR := continuous_radial_comp hx continuous_const (fun _ => hε) (-β)
-  have hRX := continuous_radialX_comp hx continuous_const (fun _ => hε) (-β)
-  exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
-    (((continuous_const.mul ((continuous_variation hε hT (x:ℂ)).comp continuous_fst)).mul hR).add
-    (hRX.mul (Complex.continuous_re.comp hV)))
-
-lemma time_deriv (β a s : ℝ) (U : C(I, ℝ)) (x : ℝ) {t : ℝ}
-    (ht : t ∈ Ioo 0 T) (b : ℝ) :
-    HasDerivAt (fun v => field hε hT β a s U x v b) (drift hε hT β a s U x t b) t := by
-  have hH := position_deriv hε hT U (x:ℂ) ht
-  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
-  dsimp only [Function.comp_def,Complex.reCLM_apply] at hRe
-  have hR := (hasDerivAt_radial_x (-β) _ ε hε).comp t (hRe.sub_const (s*b))
-  have hD := ((hasDerivAt_logDerivative hε hT U (x:ℂ) ht).const_mul (a*β/2)).exp
-  apply (hD.mul hR).congr_deriv
-  dsimp only [field,drift,factor,Function.comp_def]
-  ring
-
-lemma field_prefix_eq {S : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) (U V : C(I, ℝ))
-    (x β a s : ℝ) (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
-    {t : ℝ} (ht : t ∈ Icc 0 S) :
-    field hε hS β a s U x t = field hε hT β a s V x t := by
-  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
-  have hD := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
-  ext b
-  simp only [field,factor,hH,hD]
-
-def gap (U : C(I, ℝ)) (x t : ℝ) : ℝ :=
-  (position hε hT U (x:ℂ) t).re-extendDriver T U t
-
-def weight (U : C(I, ℝ)) (x β a t : ℝ) : ℝ :=
-  factor hε hT β a U x t * radial (-β) (gap hε hT U x t) ε
-
-def deficit (U : C(I, ℝ)) (x β a L t : ℝ) : ℝ :=
-  min (gap hε hT U x t-ε) (L-weight hε hT U x β a t)
-
-lemma continuous_gap (x : ℝ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => gap hε hT q.1 x q.2) :=
-  (Complex.continuous_re.comp (continuous_position hε hT (x:ℂ))).sub
-    (continuous_joint_extendDriver T)
-
-lemma continuous_weight (x β a : ℝ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => weight hε hT q.1 x β a q.2) :=
-  (continuous_factor hε hT β a x).mul
-    (continuous_radial_comp (continuous_gap hε hT x) continuous_const (fun _ => hε) (-β))
-
-lemma continuous_deficit (x β a L : ℝ) :
-    Continuous (fun q : C(I, ℝ) × ℝ => deficit hε hT q.1 x β a L q.2) :=
-  ((continuous_gap hε hT x).sub continuous_const).min
-    (continuous_const.sub (continuous_weight hε hT x β a))
-
-lemma deficit_pos_iff (U : C(I, ℝ)) (x β a L t : ℝ) :
-    0 < deficit hε hT U x β a L t ↔
-      ε < gap hε hT U x t ∧ weight hε hT U x β a t < L := by
-  simp only [deficit,lt_min_iff,sub_pos]
-
-lemma deficit_nonneg_iff (U : C(I, ℝ)) (x β a L t : ℝ) :
-    0 ≤ deficit hε hT U x β a L t ↔
-      ε ≤ gap hε hT U x t ∧ weight hε hT U x β a t ≤ L := by
-  simp only [deficit,le_min_iff,sub_nonneg]
-
-def exitPath (U : C(I, ℝ)) (x β a L : ℝ) : C(I, ℝ) := by
-  refine ⟨fun q => deficit hε hT U x β a L (T*q), ?_⟩
-  have hp : Continuous (fun q : I => (U,T*q)) :=
-    continuous_const.prodMk (continuous_const.mul continuous_subtype_val)
-  have hc := (continuous_deficit hε hT x β a L).comp hp
-  dsimp only [Function.comp_def] at hc
-  exact hc
-
-def stop (U : C(I, ℝ)) (x β a L : ℝ) : ℝ :=
-  T * ContinuousExit.time (exitPath hε hT U x β a L)
-
-lemma stop_bounds (U : C(I, ℝ)) (x β a L : ℝ) :
-    stop hε hT U x β a L ∈ Icc 0 T := by
-  have hh := ContinuousExit.time_bounds (exitPath hε hT U x β a L)
-  exact ⟨mul_nonneg hT hh.1, by simpa only [stop, mul_one] using mul_le_mul_of_nonneg_left hh.2 hT⟩
-
-lemma exitPath_extend (U : C(I, ℝ)) (x β a L : ℝ) {t : ℝ} (ht : t ∈ Icc 0 1) :
-    ContinuousExit.extend (exitPath hε hT U x β a L) t = deficit hε hT U x β a L (T*t) := by
-  simpa only [exitPath, ContinuousMap.coe_mk] using ContinuousExit.extend_coe
-    (exitPath hε hT U x β a L) (⟨t,ht⟩ : I)
-
-lemma exitPath_extend_div (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    ContinuousExit.extend (exitPath hε hT U x β a L) (t/T) = deficit hε hT U x β a L t := by
-  rw [exitPath_extend hε hT U x β a L
-    ⟨div_nonneg ht.1 hT, (div_le_one hTp).2 ht.2⟩, mul_div_cancel₀ _ hTp.ne']
-
-lemma positive_before (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
-    {t : ℝ} (ht : 0 ≤ t) (htτ : t < stop hε hT U x β a L) :
-    0 < deficit hε hT U x β a L t := by
-  rw [← exitPath_extend_div hε hT hTp U x β a L ⟨ht,htτ.le.trans (stop_bounds hε hT U x β a L).2⟩]
-  apply ContinuousExit.positive_before (div_nonneg ht hT)
-  exact (div_lt_iff₀ hTp).2 (by simpa only [stop, mul_comm] using htτ)
-
-lemma stop_pos (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
-    (h0 : 0 < deficit hε hT U x β a L 0) : 0 < stop hε hT U x β a L := by
-  apply mul_pos hTp (ContinuousExit.time_pos _)
-  simpa only [exitPath_extend hε hT U x β a L ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
-
-lemma nonneg_until (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
-    (h0 : 0 < deficit hε hT U x β a L 0) {t : ℝ}
-    (ht : t ∈ Icc 0 (stop hε hT U x β a L)) : 0 ≤ deficit hε hT U x β a L t := by
-  rw [← exitPath_extend_div hε hT hTp U x β a L ⟨ht.1,ht.2.trans (stop_bounds hε hT U x β a L).2⟩]
-  apply ContinuousExit.nonneg_until
-  · simpa only [exitPath_extend hε hT U x β a L ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
-  · exact ⟨div_nonneg ht.1 hT, (div_le_iff₀ hTp).2 (by simpa only [stop, mul_comm] using ht.2)⟩
-
-lemma nonpos_at_of_lt (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
-    (hτ : stop hε hT U x β a L < T) :
-    deficit hε hT U x β a L (stop hε hT U x β a L) ≤ 0 := by
-  have hs : ContinuousExit.time (exitPath hε hT U x β a L) < 1 := by
-    apply (mul_lt_mul_iff_right₀ hTp).mp
-    simpa only [mul_one, stop] using hτ
-  exact (exitPath_extend hε hT U x β a L (ContinuousExit.time_bounds _)) ▸
-    ContinuousExit.nonpos_at_of_lt_one hs
-
-lemma lt_stop_iff (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
-    {t : ℝ} (ht : t ∈ Ico 0 T) :
-    t < stop hε hT U x β a L ↔ ∀ s ∈ Icc 0 t, 0 < deficit hε hT U x β a L s := by
-  constructor
-  · intro hh s hs
-    exact positive_before hε hT hTp U x β a L hs.1 (hs.2.trans_lt hh)
-  · intro hh
-    change t < T * ContinuousExit.time (exitPath hε hT U x β a L)
-    rw [mul_comm]
-    apply (div_lt_iff₀ hTp).1
-    apply (ContinuousExit.lt_time_iff (p := exitPath hε hT U x β a L)
-      ⟨div_nonneg ht.1 hT, (div_lt_one hTp).2 ht.2⟩).2
-    intro s hs
-    rw [exitPath_extend hε hT U x β a L
-      ⟨hs.1, hs.2.trans ((div_le_one hTp).2 ht.2.le)⟩]
-    apply hh
-    exact ⟨mul_nonneg hT hs.1, (mul_le_mul_of_nonneg_left hs.2 hT).trans_eq
-      (mul_div_cancel₀ _ hTp.ne')⟩
-
-lemma isOpen_positive_on (x β a L t : ℝ) :
-    IsOpen {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U x β a L s} := by
-  let S : Set (C(I, ℝ) × Icc 0 t) := {q | deficit hε hT q.1 x β a L q.2 ≤ 0}
-  have hmap : Continuous (fun q : C(I, ℝ) × Icc 0 t => (q.1,(q.2:ℝ))) :=
-    continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)
-  have hd := (continuous_deficit hε hT x β a L).comp hmap
-  dsimp only [Function.comp_def] at hd
-  have hc : IsClosed S := isClosed_le hd continuous_const
-  have hh := isClosedMap_fst_of_compactSpace S hc
-  have he : {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U x β a L s} = (Prod.fst '' S)ᶜ := by
-    ext U
-    simp only [mem_ofPred_eq, mem_compl_iff, mem_image, S, not_exists, not_and]
-    constructor
-    · intro hp q hq hqp
-      subst U
-      exact (not_lt_of_ge hq) (hp q.2 q.2.property)
-    · intro hp s hs
-      by_contra hn
-      exact hp (U,⟨s,hs⟩) (not_lt.1 hn) rfl
-  rw [he]
-  exact hh.isOpen_compl
-
-lemma deficit_prefix_eq {S : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) (U V : C(I, ℝ))
-    (x β a L : ℝ) (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S)) :
-    EqOn (deficit hε hS U x β a L) (deficit hε hT V x β a L) (Icc 0 S) := by
-  intro t ht
-  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
-  have hD := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
-  simp only [deficit,weight,factor,gap,hH,hD,hUV ht]
-
-lemma regular (_hTp : 0 < T) (β a s : ℝ) (hβ : 0 ≤ β) (U : C(I, ℝ)) (x L : ℝ)
-    (b : ℝ → ℝ) (hb : Continuous b) :
-    BrownianField.FieldRegular (field hε hT β a s U x) (drift hε hT β a s U x) b
-      (stop hε hT U x β a L) := by
-  constructor
-  · exact smooth hε hT β a s U x
-  · obtain ⟨K,hK,hb⟩ := coeff_uniform_bound hε hT β a s hβ x 3 (Or.inr (Or.inr rfl))
-    exact ⟨K,hK,hb U⟩
-  · intro v
-    exact ((continuous_field hε hT β a s x).comp
-      ((continuous_const.prodMk continuous_id).prodMk continuous_const)).continuousOn
-  · intro t ht v
-    exact time_deriv hε hT β a s U x ⟨ht.1,ht.2.trans_le (stop_bounds hε hT U x β a L).2⟩ v
-  · exact ((continuous_drift hε hT β a s x).comp
-      ((continuous_const.prodMk continuous_fst).prodMk (hb.comp continuous_snd))).continuousOn
-  · exact ((continuous_coeff hε hT β a s x 2 (Or.inr rfl)).comp
-      ((continuous_const.prodMk continuous_id).prodMk hb)).continuousOn
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
-variable {ε : ℝ} (hε : 0 < ε)
-
-def coeffAt (k : ℕ) (s : ℝ≥0) (β a sdev : ℝ) (x : ℝ) (L : ℝ)
-    (q : C(I, ℝ) × ℝ) : ℝ := by
-  classical
-  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 x β a L t then
-    iteratedDeriv k (field hε s.coe_nonneg β a sdev q.1 x s) q.2 else 0
-
-lemma coeffAt_measurable (k : ℕ) (hk : k = 1 ∨ k = 2) (s : ℝ≥0)
-    (β a sdev : ℝ) (x : ℝ) (L : ℝ) :
-    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s β a sdev x L) := by
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
-      iteratedDeriv k (field hε s.coe_nonneg β a sdev q.1 x s) q.2) := by
-    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
-      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
-    have hh := (continuous_coeff hε s.coe_nonneg β a sdev x k hk).comp hp
-    dsimp only [Function.comp_def] at hh
-    exact hh
-  apply hc.measurable.ite _ measurable_const
-  exact ((isOpen_positive_on hε s.coe_nonneg x β a L s).preimage continuous_fst).measurableSet
-
-def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (β a sdev : ℝ) (x : ℝ) (L : ℝ) (sample : Ω) : ℝ :=
-  if s < T then coeffAt hε k s β a sdev x L (BrownianDriver.path B sdev s sample, B s sample) else 0
-
-lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (β a sdev : ℝ) (x : ℝ) (L : ℝ) :
-    Measurable[pastSigma B s] (rawCoeff hε k B T s β a sdev x L) := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  let : MeasurableSpace Ω := pastSigma B s
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s β a sdev x L).comp
-      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
-  · exact measurable_const
-
-lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
-    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
-    (β a sdev : ℝ) (x : ℝ) (L : ℝ) :
-    AEMeasurable (rawCoeff hε k B T s β a sdev x L) P := by
-  classical
-  let : MeasurableSpace C(I, ℝ) := borel _
-  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
-  unfold rawCoeff
-  split_ifs
-  · exact (coeffAt_measurable hε k hk s β a sdev x L).comp_aemeasurable
-      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
-  · exact aemeasurable_const
-
-lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (β a sdev : ℝ) (x : ℝ) (L : ℝ) (sample : Ω)
-    (hc : Continuous (fun t => B t sample)) :
-    rawCoeff hε k B T s β a sdev x L sample =
-      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L then
-        iteratedDeriv k (field hε T.coe_nonneg β a sdev (BrownianDriver.path B sdev T sample) x s) (B s sample) else 0 := by
-  classical
-  by_cases hsT : s < T
-  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
-    have hf := field_prefix_eq hε T.coe_nonneg s.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) x β a sdev he
-      ⟨s.coe_nonneg, le_rfl⟩
-    have hd := deficit_prefix_eq hε T.coe_nonneg s.coe_nonneg hsT.le
-      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) x β a L he
-    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) x β a L t) ↔
-        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L := by
-      rw [lt_stop_iff hε T.coe_nonneg hT _ x β a L ⟨s.coe_nonneg,hsT⟩]
-      exact forall₂_congr (fun t ht => by rw [hd ht])
-    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
-  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L :=
-      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ x β a L).2.trans (not_lt.1 hsT))
-    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
-
-lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
-    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
-    (hT : 0 < (T:ℝ)) (β a sdev : ℝ) (x : ℝ) (L : ℝ) (n i : ℕ) :
-    rawCoeff hε k B T (dyadicTime T n i) β a sdev x L =ᵐ[P]
-      stoppedCoeff k B T
-        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L)
-        (fun sample => field hε T.coe_nonneg β a sdev (BrownianDriver.path B sdev T sample) x) n i := by
-  filter_upwards [hB.cont] with sample hc
-  rw [rawCoeff_eq hε k B T _ hT β a sdev x L sample hc]
-  simp only [stoppedCoeff, coe_dyadicTime]
-
-theorem localized_centered {Ω : Type*} [MeasurableSpace Ω]
-    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
-    (hB : IsBrownianReal B P) (β a sdev : ℝ) (hβ : 0 ≤ β)
-    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x : ℝ) (L : ℝ) :
-    let U := BrownianDriver.path B sdev T
-    let τ := fun sample => stop hε T.coe_nonneg (U sample) x β a L
-    let F := fun sample => field hε T.coe_nonneg β a sdev (U sample) x
-    let V := fun sample => drift hε T.coe_nonneg β a sdev (U sample) x
-    let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
-      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
-      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
-    Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 := by
-  dsimp only
-  let U := BrownianDriver.path B sdev T
-  let τ := fun sample => stop hε T.coe_nonneg (U sample) x β a L
-  let F := fun sample => field hε T.coe_nonneg β a sdev (U sample) x
-  let V := fun sample => drift hε T.coe_nonneg β a sdev (U sample) x
-  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
-      (fun t => B t.toNNReal sample) (τ sample) := by
-    filter_upwards [hB.cont] with sample hc
-    exact regular hε T.coe_nonneg hT β a sdev hβ (U sample) x L _
-      (hc.comp continuous_real_toNNReal)
-  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
-    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) x β a L)
-  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg β a sdev hβ x 1 (Or.inl rfl)
-  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg β a sdev hβ x 2 (Or.inr (Or.inl rfl))
-  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
-      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
-        |iteratedDeriv k (field hε T.coe_nonneg β a sdev W x t) b| ≤ K)
-      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
-    apply Filter.Eventually.of_forall
-    intro sample
-    unfold stoppedCoeff
-    split_ifs
-    · exact hb _ _ _
-    · simpa only [abs_zero] using hK
-  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) β a sdev x L
-  let Q := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) β a sdev x L
-  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 1 T hT β a sdev x L n i
-  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
-    rawCoeff_stoppedCoeff_ae hε hB 2 T hT β a sdev x L n i
-  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
-    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ x L
-  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
-    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ x L
-  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ x L).aestronglyMeasurable K₁
-    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |A n i sample| ≤ K₁
-    rw [he]
-    exact hb
-  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
-    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ x L).aestronglyMeasurable K₂
-    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
-    rw [Real.norm_eq_abs]
-    change |Q n i sample| ≤ K₂
-    rw [he]
-    exact hb
-  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A Q hAe hCe hCm hC2
-    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
-  exact hi
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal
-namespace SLEExactGauge.BoundaryGreen
-open Swallowing GreenPhysical
-
-def generator (κ β a x ε : ℝ) : ℝ :=
-  -(a*β/x^2)*radial (-β) x ε + radialX (-β) x ε*(2/x)+κ/2*radialXX (-β) x ε
-
-lemma generator_factor (κ β a x ε : ℝ) (hx : 0 < x) (hε : 0 < ε) :
-    generator κ β a x ε = β*(x^2+ε^2)^((-β)/2-2)/x^2 *
-      ((κ*(β+1)/2-2-a)*x^4-(2+κ/2+2*a)*x^2*ε^2-a*ε^4) := by
-  have hq := (q_pos x hε).ne'
-  have hp₁ : (x^2+ε^2)^((-β)/2-1) = (x^2+ε^2)^((-β)/2-2)*(x^2+ε^2) := by
-    conv_rhs => rhs; rw [← Real.rpow_one (x^2+ε^2)]
-    rw [← Real.rpow_add (q_pos x hε)]
-    congr 1
-    ring
-  have hp₀ : (x^2+ε^2)^((-β)/2) = (x^2+ε^2)^((-β)/2-2)*(x^2+ε^2)^2 := by
-    conv_rhs => rhs; rw [← Real.rpow_natCast]
-    rw [← Real.rpow_add (q_pos x hε)]
-    congr 1
-    norm_num
-  unfold generator radial radialX radialXX
-  rw [hp₀,hp₁]
-  field_simp
-  ring
-
-lemma generator_nonpos {κ β a x ε : ℝ} (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a)
-    (hba : κ*(β+1) ≤ 4+2*a) (hx : 0 < x) (hε : 0 < ε) :
-    generator κ β a x ε ≤ 0 := by
-  rw [generator_factor κ β a x ε hx hε]
-  apply mul_nonpos_of_nonneg_of_nonpos
-  · positivity
-  · have h₁ : (κ*(β+1)/2-2-a)*x^4 ≤ 0 :=
-      mul_nonpos_of_nonpos_of_nonneg (by linarith) (by positivity)
-    have h₂ : 0 ≤ (2+κ/2+2*a)*x^2*ε^2 := by positivity
-    have h₃ : 0 ≤ a*ε^4 := by positivity
-    linarith
-
-lemma exponent_choice {κ : ℝ} (hκ : 0 < κ) (h8 : κ < 8) :
-    let β := (8/κ-1)/2
-    let a := 1+κ/8
-    0 < β ∧ 0 < a ∧ a < 2 ∧ κ*(β+1) < 4+2*a := by
-  have hd : 1 < 8/κ := (lt_div_iff₀ hκ).2 (by simpa)
-  dsimp only
-  refine ⟨by linarith,by positivity,by linarith,?_⟩
-  have he : κ*((8/κ-1)/2+1) = 4+κ/2 := by field_simp; ring
-  rw [he]
-  linarith
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology NNReal ENNReal unitInterval
-namespace SLEExactGauge.BoundaryGreen
-open BoundaryFunctional Swallowing GreenPhysical
-
-lemma field_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (β a s : ℝ) (U : C(I, ℝ)) (x : ℝ) :
-    field hε hT β a s U x 0 0 = radial (-β) x ε := by
-  simp only [field,factor,logDerivative_initial,mul_zero,Real.exp_zero,position_initial,
-    Complex.ofReal_re,sub_zero,one_mul]
-
-lemma generator_field {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
-    (κ β a : ℝ) (hκ : 0 ≤ κ) (U : C(I, ℝ)) (x t b : ℝ)
-    (hx : ε ≤ gap hε hT U x t) (hU : extendDriver T U t = Real.sqrt κ*b) :
-    drift hε hT β a (Real.sqrt κ) U x t b+
-      (1/2:ℝ)*iteratedDeriv 2 (field hε hT β a (Real.sqrt κ) U x t) b =
-    factor hε hT β a U x t * generator κ β a (gap hε hT U x t) ε := by
-  have hz : position hε hT U (x:ℂ) t = ((position hε hT U (x:ℂ) t).re:ℂ) := by
-    apply Complex.ext <;> simp [position_im_zero]
-  have hc : clampRe ε (position hε hT U (x:ℂ) t-(extendDriver T U t:ℂ)) =
-      (gap hε hT U x t:ℂ) := by
-    change BoundaryCutoff.clamp 1 ε _ = _
-    rw [BoundaryCutoff.clamp_eq (by norm_num : |(1:ℝ)|=1) (by simpa [gap] using hx)]
-    rw [hz]
-    simp only [gap,Complex.ofReal_sub]
-  have hv : velocity ε (extendDriver T U t) (position hε hT U (x:ℂ) t) =
-      (2/(gap hε hT U x t):ℝ) := by
-    change 2/clampRe ε _ = _
-    rw [hc]
-    push_cast
-    rfl
-  have hd : variation hε hT U (x:ℂ) t = -2/(gap hε hT U x t)^2 := by
-    unfold variation
-    rw [hc]
-    norm_cast
-  rw [second_eq,Real.sq_sqrt hκ]
-  dsimp only [drift]
-  rw [hv,hd,Complex.ofReal_re]
-  have he : (position hε hT U (x:ℂ) t).re-Real.sqrt κ*b = gap hε hT U x t := by
-    simp only [gap,hU]
-  rw [he]
-  unfold generator
-  ring
-
-end SLEExactGauge.BoundaryGreen
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.GreenTilts
+import OAI.Probability.SLE.IntegralLimits
+import OAI.Probability.SLE.ItoIsometry
+import OAI.Probability.SLE.StochasticIntegrals
+
+namespace OAI
+
+
+/-! Conformal localization and Swallowing estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.Swallowing
+open GreenPhysical
+
+lemma generator_eq (κ β C x y : ℝ) (hy : 0 < y) :
+    generator κ β C x y =
+      (x^2+y^2)^(β/2-1) *
+        (β*(κ-4)/2 + β*(8-2*κ+κ*β)/2*(x^2/(x^2+y^2))) -
+      (2*C*β)*y^β/(x^2+y^2) := by
+  have hq := (q_pos x hy).ne'
+  have hp : (x^2+y^2)^(β/2-2) = (x^2+y^2)^(β/2-1)/(x^2+y^2) := by
+    rw [← Real.rpow_sub_one hq]
+    congr 1
+    ring
+  have hyp : y^(β-1)*y = y^β := by
+    conv_lhs => rhs; rw [← Real.rpow_one y]
+    rw [← Real.rpow_add hy]
+    congr 1
+    ring
+  unfold generator radialXX
+  rw [hp]
+  calc
+    _ = (x^2+y^2)^(β/2-1)*
+        (β*(κ-4)/2+β*(8-2*κ+κ*β)/2*(x^2/(x^2+y^2))) -
+        2*C*β*(y^(β-1)*y)/(x^2+y^2) := by field_simp; ring
+    _ = _ := by rw [hyp]
+
+lemma radial_height_ineq {β : ℝ} (hβ : β ≤ 2) (x : ℝ) {y : ℝ} (hy : 0 < y) :
+    (x^2+y^2)^(β/2-1)*y^2 ≤ y^β := by
+  have hh := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hy)
+    (show y^2 ≤ x^2+y^2 by nlinarith [sq_nonneg x]) (show β/2-1 ≤ 0 by linarith)
+  calc
+    _ ≤ (y^2)^(β/2-1)*y^2 := mul_le_mul_of_nonneg_right hh (sq_nonneg y)
+    _ = y^β := by
+      rw [← Real.rpow_natCast_mul hy.le]
+      conv_lhs => rhs; rw [← Real.rpow_natCast]
+      rw [← Real.rpow_add hy]
+      norm_num
+      congr 1
+      ring
+
+lemma generator_negative {κ β : ℝ} (hκ : 4 < κ) (hβ : 0 < β)
+    (hβκ : β < 1-4/κ) (x : ℝ) {y : ℝ} (hy : 0 < y) :
+    let C := (2*κ-8-κ*β)/4
+    let d := β*(κ-4-κ*β)/2
+    0 < C ∧ 0 < d ∧ generator κ β C x y ≤ -d*(x^2+y^2)^(β/2-1) := by
+  have hk : 0 < κ := by linarith
+  have hbk : κ*β < κ-4 := by
+    have hh := (mul_lt_mul_of_pos_left hβκ hk)
+    field_simp at hh
+    linarith
+  have hb2 : β ≤ 2 := by
+    have h4 := div_pos (by norm_num : (0:ℝ)<4) hk
+    linarith
+  dsimp only
+  have hC : 0 < (2*κ-8-κ*β)/4 := by linarith
+  have hd : 0 < β*(κ-4-κ*β)/2 := by positivity
+  refine ⟨hC,hd,?_⟩
+  have he := radial_height_ineq hb2 x hy
+  have hq := q_pos x hy
+  have hh := mul_le_mul_of_nonneg_left he (show 0 ≤ β*(2*κ-8-κ*β)/2 from div_nonneg (mul_nonneg hβ.le (by linarith)) (by norm_num))
+  rw [generator_eq _ _ _ _ _ hy]
+  apply (mul_le_mul_iff_left₀ hq).1
+  have heq :
+      ((x^2+y^2)^(β/2-1)*(β*(κ-4)/2+β*(8-2*κ+κ*β)/2*(x^2/(x^2+y^2))) -
+        (2*((2*κ-8-κ*β)/4)*β)*y^β/(x^2+y^2))*(x^2+y^2) =
+      -(β*(κ-4-κ*β)/2)*(x^2+y^2)^(β/2-1)*(x^2+y^2)+
+        (β*(2*κ-8-κ*β)/2)*((x^2+y^2)^(β/2-1)*y^2-y^β) := by
+    field_simp
+    ring
+  rw [heq]
+  nlinarith
+
+lemma monomial_bound {ε : ℝ} (hε : 0 < ε) (x : ℝ) {y a : ℝ} (hy : ε ≤ y)
+    (n : ℕ) (ha : a+(n:ℝ)/2 ≤ 0) :
+    |x|^n*(x^2+y^2)^a ≤ (ε^2)^(a+(n:ℝ)/2) := by
+  have hq := q_pos x (hε.trans_le hy)
+  have hx : |x| ≤ Real.sqrt (x^2+y^2) := Real.abs_le_sqrt (by nlinarith [sq_nonneg y])
+  have hl : ε^2 ≤ x^2+y^2 := by nlinarith [sq_nonneg x]
+  calc
+    _ ≤ (Real.sqrt (x^2+y^2))^n*(x^2+y^2)^a := by gcongr
+    _ = (x^2+y^2)^(a+(n:ℝ)/2) := by
+      rw [Real.sqrt_eq_rpow,← Real.rpow_mul_natCast hq.le,← Real.rpow_add hq]
+      congr 1
+      ring
+    _ ≤ _ := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hε) hl ha
+
+lemma radialX_bound {ε β : ℝ} (hε : 0 < ε) (hβ : β ≤ 1) (x : ℝ) {y : ℝ} (hy : ε ≤ y) :
+    |radialX β x y| ≤ |β| *(ε^2)^(β/2-1/2) := by
+  have hh := monomial_bound (a := β/2-1) hε x hy 1 (by norm_num; linarith)
+  norm_num at hh
+  rw [radialX,abs_mul,abs_mul,abs_of_nonneg (Real.rpow_nonneg (q_pos x (hε.trans_le hy)).le _)]
+  have he : β/2-1+1/2 = β/2-1/2 := by ring
+  rw [he] at hh
+  simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh (abs_nonneg β)
+
+lemma radialXX_bound {ε β : ℝ} (hε : 0 < ε) (hβ : β ≤ 1) (x : ℝ) {y : ℝ} (hy : ε ≤ y) :
+    |radialXX β x y| ≤ (|β|+|β*(β-2)|)*(ε^2)^(β/2-1) := by
+  have h0 := monomial_bound (a := β/2-1) hε x hy 0 (by norm_num; linarith)
+  have h2 := monomial_bound (a := β/2-2) hε x hy 2 (by norm_num; linarith)
+  norm_num at h0 h2
+  rw [show β/2-2+1 = β/2-1 by ring] at h2
+  calc
+    _ ≤ |β| *(x^2+y^2)^(β/2-1)+|β*(β-2)| *(|x|^2*(x^2+y^2)^(β/2-2)) := by
+      unfold radialXX
+      exact (abs_add_le _ _).trans_eq (by simp [abs_mul,abs_pow,abs_of_nonneg (Real.rpow_nonneg (q_pos x (hε.trans_le hy)).le _)];ring)
+    _ ≤ |β| *(ε^2)^(β/2-1)+|β*(β-2)| *(ε^2)^(β/2-1) := by
+      gcongr
+      simpa only [sq_abs] using h2
+    _ = _ := by ring
+
+lemma radialXXX_bound {ε β : ℝ} (hε : 0 < ε) (hβ : β ≤ 1) (x : ℝ) {y : ℝ} (hy : ε ≤ y) :
+    |radialXXX β x y| ≤ (|3*β*(β-2)|+|β*(β-2)*(β-4)|)*(ε^2)^(β/2-3/2) := by
+  have h1 := monomial_bound (a := β/2-2) hε x hy 1 (by norm_num; linarith)
+  have h3 := monomial_bound (a := β/2-3) hε x hy 3 (by norm_num; linarith)
+  norm_num at h1 h3
+  rw [show β/2-2+1/2 = β/2-3/2 by ring] at h1
+  rw [show β/2-3+3/2 = β/2-3/2 by ring] at h3
+  calc
+    _ ≤ |3*β*(β-2)| *(|x| *(x^2+y^2)^(β/2-2))+
+        |β*(β-2)*(β-4)| *(|x|^3*(x^2+y^2)^(β/2-3)) := by
+      unfold radialXXX
+      exact (abs_add_le _ _).trans_eq (by simp [abs_mul,abs_pow,abs_of_nonneg (Real.rpow_nonneg (q_pos x (hε.trans_le hy)).le _)];ring)
+    _ ≤ |3*β*(β-2)| *(ε^2)^(β/2-3/2)+|β*(β-2)*(β-4)| *(ε^2)^(β/2-3/2) := by gcongr
+    _ = _ := by ring
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.Swallowing
+open GreenPhysical LoewnerCutoff LoewnerExit
+
+lemma deriv_lyapunov_driver (β C c y s b : ℝ) (hy : 0 < y) :
+    deriv (fun b => lyapunov β C (c-s*b) y) b = -s*radialX β (c-s*b) y := by
+  have hd := ((hasDerivAt_radial_x β (c-s*b) y hy).comp b
+    (((hasDerivAt_id b).const_mul s).const_sub c)).add_const (C*y^β)
+  convert hd.deriv using 1 <;> simp [lyapunov]; ring
+
+lemma second_lyapunov_driver (β C c y s b : ℝ) (hy : 0 < y) :
+    iteratedDeriv 2 (fun b => lyapunov β C (c-s*b) y) b = s^2*radialXX β (c-s*b) y := by
+  have he : deriv (fun b => lyapunov β C (c-s*b) y) = fun b => -s*radialX β (c-s*b) y :=
+    funext (fun b => deriv_lyapunov_driver β C c y s b hy)
+  simp only [iteratedDeriv_succ,iteratedDeriv_zero]
+  rw [he]
+  have hd := ((hasDerivAt_radialX_x β (c-s*b) y hy).comp b
+    (((hasDerivAt_id b).const_mul s).const_sub c)).const_mul (-s)
+  simp only [Function.comp_def,id_eq] at hd
+  convert hd.deriv using 1
+  ring
+
+lemma third_lyapunov_driver (β C c y s b : ℝ) (hy : 0 < y) :
+    iteratedDeriv 3 (fun b => lyapunov β C (c-s*b) y) b = -s^3*radialXXX β (c-s*b) y := by
+  have he : iteratedDeriv 2 (fun b => lyapunov β C (c-s*b) y) = fun b => s^2*radialXX β (c-s*b) y :=
+    funext (fun b => second_lyapunov_driver β C c y s b hy)
+  rw [show (3:ℕ)=2+1 by rfl,iteratedDeriv_succ,he]
+  have hd := ((hasDerivAt_radialXX_x β (c-s*b) y hy).comp b
+    (((hasDerivAt_id b).const_mul s).const_sub c)).const_mul (s^2)
+  simp only [Function.comp_def,id_eq] at hd
+  convert hd.deriv using 1
+  ring
+
+lemma continuous_radial_comp {α : Type*} [TopologicalSpace α] {X Y : α → ℝ}
+    (hX : Continuous X) (hY : Continuous Y) (hy : ∀ a, 0 < Y a) (β : ℝ) :
+    Continuous (fun a => radial β (X a) (Y a)) := by
+  exact ((hX.pow 2).add (hY.pow 2)).rpow_const (fun a => Or.inl (q_pos _ (hy a)).ne')
+
+lemma continuous_radialX_comp {α : Type*} [TopologicalSpace α] {X Y : α → ℝ}
+    (hX : Continuous X) (hY : Continuous Y) (hy : ∀ a, 0 < Y a) (β : ℝ) :
+    Continuous (fun a => radialX β (X a) (Y a)) := by
+  exact (continuous_const.mul hX).mul (((hX.pow 2).add (hY.pow 2)).rpow_const
+    (fun a => Or.inl (q_pos _ (hy a)).ne'))
+
+lemma continuous_radialXX_comp {α : Type*} [TopologicalSpace α] {X Y : α → ℝ}
+    (hX : Continuous X) (hY : Continuous Y) (hy : ∀ a, 0 < Y a) (β : ℝ) :
+    Continuous (fun a => radialXX β (X a) (Y a)) := by
+  have hp (e : ℝ) := ((hX.pow 2).add (hY.pow 2)).rpow_const (p := e)
+    (fun a => Or.inl (q_pos _ (hy a)).ne')
+  exact (continuous_const.mul (hp _)).add ((continuous_const.mul (hX.pow 2)).mul (hp _))
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.Swallowing
+open GreenPhysical LoewnerCutoff LoewnerExit
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+def field (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
+  lyapunov β C ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t)
+
+def drift (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) : ℝ :=
+  let x := (position hε hT U z t).re-s*b
+  let y := LoewnerField.height hε hT U z t
+  let v := velocity ε (extendDriver T U t) (position hε hT U z t)
+  β*(x*v.re+y*v.im)*(x^2+y^2)^(β/2-1)+C*(β*y^(β-1)*v.im)
+
+lemma first_eq (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    iteratedDeriv 1 (field hε hT β C s U z t) b =
+      -s*radialX β ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t) := by
+  simp only [iteratedDeriv_succ,iteratedDeriv_zero]
+  exact deriv_lyapunov_driver _ _ _ _ _ _ (LoewnerField.height_pos hε hT U z t)
+
+lemma second_eq (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    iteratedDeriv 2 (field hε hT β C s U z t) b =
+      s^2*radialXX β ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t) :=
+  second_lyapunov_driver _ _ _ _ _ _ (LoewnerField.height_pos hε hT U z t)
+
+lemma third_eq (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t b : ℝ) :
+    iteratedDeriv 3 (field hε hT β C s U z t) b =
+      -s^3*radialXXX β ((position hε hT U z t).re-s*b) (LoewnerField.height hε hT U z t) :=
+  third_lyapunov_driver _ _ _ _ _ _ (LoewnerField.height_pos hε hT U z t)
+
+lemma smooth (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) (t : ℝ) :
+    ContDiff ℝ 3 (field hε hT β C s U z t) := by
+  exact ((contDiff_radial_driver β _ _ s (LoewnerField.height_pos hε hT U z t)).of_le (by simp)).add contDiff_const
+
+lemma coeff_uniform_bound (β C s : ℝ) (hβ : β ≤ 1) (z : ℂ) (k : ℕ)
+    (hk : k=1 ∨ k=2 ∨ k=3) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t b : ℝ),
+      |iteratedDeriv k (field hε hT β C s U z t) b| ≤ K := by
+  rcases hk with rfl|rfl|rfl
+  · refine ⟨|s| * (|β| * (ε^2)^(β/2-1/2)),by positivity,?_⟩
+    intro U t b
+    rw [first_eq,abs_mul,abs_neg]
+    exact mul_le_mul_of_nonneg_left (radialX_bound hε hβ _ (le_max_left _ _)) (abs_nonneg _)
+  · refine ⟨|s|^2*((|β|+|β*(β-2)|)*(ε^2)^(β/2-1)),by positivity,?_⟩
+    intro U t b
+    rw [second_eq,abs_mul,abs_pow]
+    exact mul_le_mul_of_nonneg_left (radialXX_bound hε hβ _ (le_max_left _ _)) (by positivity)
+  · refine ⟨|s|^3*((|3*β*(β-2)|+|β*(β-2)*(β-4)|)*(ε^2)^(β/2-3/2)),by positivity,?_⟩
+    intro U t b
+    rw [third_eq,abs_mul,abs_neg,abs_pow]
+    exact mul_le_mul_of_nonneg_left (radialXXX_bound hε hβ _ (le_max_left _ _)) (by positivity)
+
+lemma continuous_field (β C s : ℝ) (z : ℂ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => field hε hT β C s q.1.1 z q.1.2 q.2) := by
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
+  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => LoewnerField.height hε hT q.1.1 z q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  exact (continuous_radial_comp hx hy (fun q => LoewnerField.height_pos hε hT q.1.1 z q.1.2) β).add
+    (continuous_const.mul (hy.rpow_const (fun q => Or.inl (LoewnerField.height_pos hε hT q.1.1 z q.1.2).ne')))
+
+lemma continuous_coeff (β C s : ℝ) (z : ℂ) (k : ℕ) (hk : k=1 ∨ k=2) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv k (field hε hT β C s q.1.1 z q.1.2) q.2) := by
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
+  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => LoewnerField.height hε hT q.1.1 z q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  rcases hk with rfl|rfl
+  · simp only [first_eq]
+    exact continuous_const.mul (continuous_radialX_comp hx hy (fun q => LoewnerField.height_pos hε hT q.1.1 z q.1.2) β)
+  · simp only [second_eq]
+    exact continuous_const.mul (continuous_radialXX_comp hx hy (fun q => LoewnerField.height_pos hε hT q.1.1 z q.1.2) β)
+
+lemma continuous_drift (β C s : ℝ) (z : ℂ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => drift hε hT β C s q.1.1 z q.1.2 q.2) := by
+  have hH := (continuous_position hε hT z).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hD := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hV := (continuous_velocity hε).comp (hD.prodMk hH)
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
+  have hy : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => LoewnerField.height hε hT q.1.1 z q.1.2) :=
+    continuous_const.max (Complex.continuous_im.comp hH)
+  have hp (e : ℝ) := ((hx.pow 2).add (hy.pow 2)).rpow_const (p := e)
+    (fun q => Or.inl (q_pos _ (LoewnerField.height_pos hε hT q.1.1 z q.1.2)).ne')
+  exact ((continuous_const.mul ((hx.mul (Complex.continuous_re.comp hV)).add
+    (hy.mul (Complex.continuous_im.comp hV)))).mul (hp _)).add
+    (continuous_const.mul ((continuous_const.mul (hy.rpow_const (fun q =>
+      Or.inl (LoewnerField.height_pos hε hT q.1.1 z q.1.2).ne'))).mul (Complex.continuous_im.comp hV)))
+
+lemma time_deriv (β C s : ℝ) (U : C(I, ℝ)) (z : ℂ) {t : ℝ}
+    (ht : t ∈ Ioo 0 T) (hy : ε < (position hε hT U z t).im) (b : ℝ) :
+    HasDerivAt (fun v => field hε hT β C s U z v b) (drift hε hT β C s U z t b) t := by
+  have hH := position_deriv hε hT U z ht
+  have hIm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t hH
+  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
+  dsimp only [Function.comp_def,Complex.imCLM_apply,Complex.reCLM_apply] at hIm hRe
+  have hY : HasDerivAt (LoewnerField.height hε hT U z)
+      (velocity ε (extendDriver T U t) (position hε hT U z t)).im t := by
+    apply hIm.congr_of_eventuallyEq
+    filter_upwards [hIm.continuousAt.preimage_mem_nhds (Ioi_mem_nhds hy)] with s hs
+    exact max_eq_right hs.le
+  have hR := hasDerivAt_radial_path β (hRe.sub_const (s*b)) hY (LoewnerField.height_pos hε hT U z t)
+  have hP := hY.rpow_const (p := β) (Or.inl (LoewnerField.height_pos hε hT U z t).ne')
+  apply (hR.add (hP.const_mul C)).congr_deriv
+  dsimp only [drift]
+  ring
+
+lemma regular (hTp : 0 < T) (β C s : ℝ) (hβ : β ≤ 1) (U : C(I, ℝ)) (z : ℂ)
+    (ρ δ : ℝ) (b : ℝ → ℝ) (hb : Continuous b) :
+    BrownianField.FieldRegular (field hε hT β C s U z) (drift hε hT β C s U z) b
+      (stop hε hT U z ρ δ) := by
+  constructor
+  · exact smooth hε hT β C s U z
+  · obtain ⟨K,hK,hb⟩ := coeff_uniform_bound hε hT β C s hβ z 3 (Or.inr (Or.inr rfl))
+    exact ⟨K,hK,hb U⟩
+  · intro x
+    exact ((continuous_field hε hT β C s z).comp
+      ((continuous_const.prodMk continuous_id).prodMk continuous_const)).continuousOn
+  · intro t ht x
+    have htT : t ∈ Ioo 0 T := ⟨ht.1,ht.2.trans_le (stop_bounds hε hT U z ρ δ).2⟩
+    have hh := (deficit_pos_iff hε hT U z ρ δ t).1
+      (positive_before hε hT hTp U z ρ δ ht.1.le ht.2)
+    exact time_deriv hε hT β C s U z htT hh.2.1 x
+  · exact ((continuous_drift hε hT β C s z).comp
+      ((continuous_const.prodMk continuous_fst).prodMk (hb.comp continuous_snd))).continuousOn
+  · exact ((continuous_coeff hε hT β C s z 2 (Or.inr rfl)).comp
+      ((continuous_const.prodMk continuous_id).prodMk hb)).continuousOn
+
+lemma field_prefix_eq {S : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) (U V : C(I, ℝ))
+    (z : ℂ) (β C s : ℝ) (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
+    {t : ℝ} (ht : t ∈ Icc 0 S) :
+    field hε hS β C s U z t = field hε hT β C s V z t := by
+  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V z hUV ht
+  ext b
+  simp only [field,LoewnerField.height,hH]
+
+end SLEExactGauge.Swallowing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing GreenPhysical
+variable {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+
+def factor (β a : ℝ) (U : C(I, ℝ)) (x t : ℝ) : ℝ :=
+  Real.exp ((a*β/2)*logDerivative hε hT U (x:ℂ) t)
+
+def field (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) : ℝ :=
+  factor hε hT β a U x t * radial (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε
+
+def drift (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) : ℝ :=
+  let X := (position hε hT U (x:ℂ) t).re-s*b
+  factor hε hT β a U x t *
+    ((a*β/2)*variation hε hT U (x:ℂ) t * radial (-β) X ε +
+    radialX (-β) X ε * (velocity ε (extendDriver T U t) (position hε hT U (x:ℂ) t)).re)
+
+lemma factor_pos (β a : ℝ) (U : C(I, ℝ)) (x t : ℝ) :
+    0 < factor hε hT β a U x t := Real.exp_pos _
+
+lemma field_pos (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
+    0 < field hε hT β a s U x t b :=
+  mul_pos (factor_pos hε hT β a U x t) (radial_pos _ _ hε)
+
+lemma factor_bound (β a : ℝ) (U : C(I, ℝ)) (x t : ℝ) :
+    factor hε hT β a U x t ≤ Real.exp (|a*β/2| *((2/ε^2)*T)) := by
+  apply Real.exp_le_exp.mpr
+  exact (le_abs_self _).trans ((abs_mul _ _).le.trans
+    (mul_le_mul_of_nonneg_left (abs_logDerivative_le hε hT U (x:ℂ) t) (abs_nonneg _)))
+
+lemma first_eq (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
+    iteratedDeriv 1 (field hε hT β a s U x t) b =
+      factor hε hT β a U x t * (-s*radialX (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε) := by
+  simp only [iteratedDeriv_succ,iteratedDeriv_zero]
+  have hd := ((hasDerivAt_radial_x (-β) _ ε hε).comp b
+    (((hasDerivAt_id b).const_mul s).const_sub ((position hε hT U (x:ℂ) t).re))).const_mul (factor hε hT β a U x t)
+  simp only [Function.comp_def,id_eq] at hd
+  convert hd.deriv using 1 <;> first | rfl | ring
+
+lemma second_eq (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
+    iteratedDeriv 2 (field hε hT β a s U x t) b =
+      factor hε hT β a U x t * (s^2*radialXX (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε) := by
+  have he := funext (first_eq hε hT β a s U x t)
+  rw [show (2:ℕ)=1+1 by rfl,iteratedDeriv_succ,he]
+  have hd := (((hasDerivAt_radialX_x (-β) _ ε hε).comp b
+    (((hasDerivAt_id b).const_mul s).const_sub ((position hε hT U (x:ℂ) t).re))).const_mul (-s)).const_mul (factor hε hT β a U x t)
+  simp only [Function.comp_def,id_eq] at hd
+  convert hd.deriv using 1
+  first | rfl | ring
+
+lemma third_eq (β a s : ℝ) (U : C(I, ℝ)) (x t b : ℝ) :
+    iteratedDeriv 3 (field hε hT β a s U x t) b =
+      factor hε hT β a U x t * (-s^3*radialXXX (-β) ((position hε hT U (x:ℂ) t).re-s*b) ε) := by
+  have he := funext (second_eq hε hT β a s U x t)
+  rw [show (3:ℕ)=2+1 by rfl,iteratedDeriv_succ,he]
+  have hd := (((hasDerivAt_radialXX_x (-β) _ ε hε).comp b
+    (((hasDerivAt_id b).const_mul s).const_sub ((position hε hT U (x:ℂ) t).re))).const_mul (s^2)).const_mul (factor hε hT β a U x t)
+  simp only [Function.comp_def,id_eq] at hd
+  convert hd.deriv using 1
+  first | rfl | ring
+
+lemma smooth (β a s : ℝ) (U : C(I, ℝ)) (x t : ℝ) :
+    ContDiff ℝ 3 (field hε hT β a s U x t) :=
+  contDiff_const.mul ((contDiff_radial_driver (-β) _ ε s hε).of_le (by simp))
+
+lemma coeff_uniform_bound (β a s : ℝ) (hβ : 0 ≤ β) (x : ℝ) (k : ℕ)
+    (hk : k=1 ∨ k=2 ∨ k=3) :
+    ∃ K : ℝ, 0 ≤ K ∧ ∀ (U : C(I, ℝ)) (t b : ℝ),
+      |iteratedDeriv k (field hε hT β a s U x t) b| ≤ K := by
+  let E := Real.exp (|a*β/2| *((2/ε^2)*T))
+  have hβ' : -β ≤ 1 := by linarith
+  have hf (U : C(I, ℝ)) (t : ℝ) : |factor hε hT β a U x t| ≤ E := by
+    rw [abs_of_pos (factor_pos hε hT β a U x t)]
+    exact factor_bound hε hT β a U x t
+  rcases hk with rfl|rfl|rfl
+  · refine ⟨E*(|s| *(|-β| *(ε^2)^((-β)/2-1/2))),by positivity,?_⟩
+    intro U t b
+    rw [first_eq,abs_mul,abs_mul,abs_neg]
+    exact mul_le_mul (hf U t) (mul_le_mul_of_nonneg_left
+      (radialX_bound hε hβ' _ le_rfl) (abs_nonneg _)) (by positivity) (by positivity)
+  · refine ⟨E*(|s|^2*((|-β|+|(-β)*(-β-2)|)*(ε^2)^((-β)/2-1))),by positivity,?_⟩
+    intro U t b
+    rw [second_eq,abs_mul,abs_mul,abs_pow]
+    exact mul_le_mul (hf U t) (mul_le_mul_of_nonneg_left
+      (radialXX_bound hε hβ' _ le_rfl) (by positivity)) (by positivity) (by positivity)
+  · refine ⟨E*(|s|^3*((|3*(-β)*(-β-2)|+|(-β)*(-β-2)*(-β-4)|)*(ε^2)^((-β)/2-3/2))),by positivity,?_⟩
+    intro U t b
+    rw [third_eq,abs_mul,abs_mul,abs_neg,abs_pow]
+    exact mul_le_mul (hf U t) (mul_le_mul_of_nonneg_left
+      (radialXXX_bound hε hβ' _ le_rfl) (by positivity)) (by positivity) (by positivity)
+
+lemma continuous_factor (β a x : ℝ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => factor hε hT β a q.1 x q.2) :=
+  Real.continuous_exp.comp (continuous_const.mul (continuous_logDerivative hε hT (x:ℂ)))
+
+lemma continuous_field (β a s x : ℝ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => field hε hT β a s q.1.1 x q.1.2 q.2) := by
+  have hH := (continuous_position hε hT (x:ℂ)).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
+  exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
+    (continuous_radial_comp hx continuous_const (fun _ => hε) (-β))
+
+lemma continuous_coeff (β a s x : ℝ) (k : ℕ) (hk : k=1 ∨ k=2) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => iteratedDeriv k (field hε hT β a s q.1.1 x q.1.2) q.2) := by
+  have hH := (continuous_position hε hT (x:ℂ)).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
+  rcases hk with rfl|rfl
+  · simp only [first_eq]
+    exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
+      (continuous_const.mul (continuous_radialX_comp hx continuous_const (fun _ => hε) (-β)))
+  · simp only [second_eq]
+    exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
+      (continuous_const.mul (continuous_radialXX_comp hx continuous_const (fun _ => hε) (-β)))
+
+lemma continuous_drift (β a s x : ℝ) :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => drift hε hT β a s q.1.1 x q.1.2 q.2) := by
+  have hH := (continuous_position hε hT (x:ℂ)).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hD := (continuous_joint_extendDriver T).comp (continuous_fst : Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => q.1))
+  have hV := (continuous_velocity hε).comp (hD.prodMk hH)
+  have hx := (Complex.continuous_re.comp hH).sub (continuous_const.mul continuous_snd :
+    Continuous (fun q : (C(I, ℝ) × ℝ) × ℝ => s*q.2))
+  have hR := continuous_radial_comp hx continuous_const (fun _ => hε) (-β)
+  have hRX := continuous_radialX_comp hx continuous_const (fun _ => hε) (-β)
+  exact ((continuous_factor hε hT β a x).comp continuous_fst).mul
+    (((continuous_const.mul ((continuous_variation hε hT (x:ℂ)).comp continuous_fst)).mul hR).add
+    (hRX.mul (Complex.continuous_re.comp hV)))
+
+lemma time_deriv (β a s : ℝ) (U : C(I, ℝ)) (x : ℝ) {t : ℝ}
+    (ht : t ∈ Ioo 0 T) (b : ℝ) :
+    HasDerivAt (fun v => field hε hT β a s U x v b) (drift hε hT β a s U x t b) t := by
+  have hH := position_deriv hε hT U (x:ℂ) ht
+  have hRe := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hH
+  dsimp only [Function.comp_def,Complex.reCLM_apply] at hRe
+  have hR := (hasDerivAt_radial_x (-β) _ ε hε).comp t (hRe.sub_const (s*b))
+  have hD := ((hasDerivAt_logDerivative hε hT U (x:ℂ) ht).const_mul (a*β/2)).exp
+  apply (hD.mul hR).congr_deriv
+  dsimp only [field,drift,factor,Function.comp_def]
+  ring
+
+lemma field_prefix_eq {S : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) (U V : C(I, ℝ))
+    (x β a s : ℝ) (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S))
+    {t : ℝ} (ht : t ∈ Icc 0 S) :
+    field hε hS β a s U x t = field hε hT β a s V x t := by
+  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
+  have hD := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
+  ext b
+  simp only [field,factor,hH,hD]
+
+def gap (U : C(I, ℝ)) (x t : ℝ) : ℝ :=
+  (position hε hT U (x:ℂ) t).re-extendDriver T U t
+
+def weight (U : C(I, ℝ)) (x β a t : ℝ) : ℝ :=
+  factor hε hT β a U x t * radial (-β) (gap hε hT U x t) ε
+
+def deficit (U : C(I, ℝ)) (x β a L t : ℝ) : ℝ :=
+  min (gap hε hT U x t-ε) (L-weight hε hT U x β a t)
+
+lemma continuous_gap (x : ℝ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => gap hε hT q.1 x q.2) :=
+  (Complex.continuous_re.comp (continuous_position hε hT (x:ℂ))).sub
+    (continuous_joint_extendDriver T)
+
+lemma continuous_weight (x β a : ℝ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => weight hε hT q.1 x β a q.2) :=
+  (continuous_factor hε hT β a x).mul
+    (continuous_radial_comp (continuous_gap hε hT x) continuous_const (fun _ => hε) (-β))
+
+lemma continuous_deficit (x β a L : ℝ) :
+    Continuous (fun q : C(I, ℝ) × ℝ => deficit hε hT q.1 x β a L q.2) :=
+  ((continuous_gap hε hT x).sub continuous_const).min
+    (continuous_const.sub (continuous_weight hε hT x β a))
+
+lemma deficit_pos_iff (U : C(I, ℝ)) (x β a L t : ℝ) :
+    0 < deficit hε hT U x β a L t ↔
+      ε < gap hε hT U x t ∧ weight hε hT U x β a t < L := by
+  simp only [deficit,lt_min_iff,sub_pos]
+
+lemma deficit_nonneg_iff (U : C(I, ℝ)) (x β a L t : ℝ) :
+    0 ≤ deficit hε hT U x β a L t ↔
+      ε ≤ gap hε hT U x t ∧ weight hε hT U x β a t ≤ L := by
+  simp only [deficit,le_min_iff,sub_nonneg]
+
+def exitPath (U : C(I, ℝ)) (x β a L : ℝ) : C(I, ℝ) := by
+  refine ⟨fun q => deficit hε hT U x β a L (T*q), ?_⟩
+  have hp : Continuous (fun q : I => (U,T*q)) :=
+    continuous_const.prodMk (continuous_const.mul continuous_subtype_val)
+  have hc := (continuous_deficit hε hT x β a L).comp hp
+  dsimp only [Function.comp_def] at hc
+  exact hc
+
+def stop (U : C(I, ℝ)) (x β a L : ℝ) : ℝ :=
+  T * ContinuousExit.time (exitPath hε hT U x β a L)
+
+lemma stop_bounds (U : C(I, ℝ)) (x β a L : ℝ) :
+    stop hε hT U x β a L ∈ Icc 0 T := by
+  have hh := ContinuousExit.time_bounds (exitPath hε hT U x β a L)
+  exact ⟨mul_nonneg hT hh.1, by simpa only [stop, mul_one] using mul_le_mul_of_nonneg_left hh.2 hT⟩
+
+lemma exitPath_extend (U : C(I, ℝ)) (x β a L : ℝ) {t : ℝ} (ht : t ∈ Icc 0 1) :
+    ContinuousExit.extend (exitPath hε hT U x β a L) t = deficit hε hT U x β a L (T*t) := by
+  simpa only [exitPath, ContinuousMap.coe_mk] using ContinuousExit.extend_coe
+    (exitPath hε hT U x β a L) (⟨t,ht⟩ : I)
+
+lemma exitPath_extend_div (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    ContinuousExit.extend (exitPath hε hT U x β a L) (t/T) = deficit hε hT U x β a L t := by
+  rw [exitPath_extend hε hT U x β a L
+    ⟨div_nonneg ht.1 hT, (div_le_one hTp).2 ht.2⟩, mul_div_cancel₀ _ hTp.ne']
+
+lemma positive_before (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
+    {t : ℝ} (ht : 0 ≤ t) (htτ : t < stop hε hT U x β a L) :
+    0 < deficit hε hT U x β a L t := by
+  rw [← exitPath_extend_div hε hT hTp U x β a L ⟨ht,htτ.le.trans (stop_bounds hε hT U x β a L).2⟩]
+  apply ContinuousExit.positive_before (div_nonneg ht hT)
+  exact (div_lt_iff₀ hTp).2 (by simpa only [stop, mul_comm] using htτ)
+
+lemma stop_pos (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
+    (h0 : 0 < deficit hε hT U x β a L 0) : 0 < stop hε hT U x β a L := by
+  apply mul_pos hTp (ContinuousExit.time_pos _)
+  simpa only [exitPath_extend hε hT U x β a L ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
+
+lemma nonneg_until (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
+    (h0 : 0 < deficit hε hT U x β a L 0) {t : ℝ}
+    (ht : t ∈ Icc 0 (stop hε hT U x β a L)) : 0 ≤ deficit hε hT U x β a L t := by
+  rw [← exitPath_extend_div hε hT hTp U x β a L ⟨ht.1,ht.2.trans (stop_bounds hε hT U x β a L).2⟩]
+  apply ContinuousExit.nonneg_until
+  · simpa only [exitPath_extend hε hT U x β a L ⟨le_rfl,zero_le_one⟩, mul_zero] using h0
+  · exact ⟨div_nonneg ht.1 hT, (div_le_iff₀ hTp).2 (by simpa only [stop, mul_comm] using ht.2)⟩
+
+lemma nonpos_at_of_lt (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
+    (hτ : stop hε hT U x β a L < T) :
+    deficit hε hT U x β a L (stop hε hT U x β a L) ≤ 0 := by
+  have hs : ContinuousExit.time (exitPath hε hT U x β a L) < 1 := by
+    apply (mul_lt_mul_iff_right₀ hTp).mp
+    simpa only [mul_one, stop] using hτ
+  exact (exitPath_extend hε hT U x β a L (ContinuousExit.time_bounds _)) ▸
+    ContinuousExit.nonpos_at_of_lt_one hs
+
+lemma lt_stop_iff (hTp : 0 < T) (U : C(I, ℝ)) (x β a L : ℝ)
+    {t : ℝ} (ht : t ∈ Ico 0 T) :
+    t < stop hε hT U x β a L ↔ ∀ s ∈ Icc 0 t, 0 < deficit hε hT U x β a L s := by
+  constructor
+  · intro hh s hs
+    exact positive_before hε hT hTp U x β a L hs.1 (hs.2.trans_lt hh)
+  · intro hh
+    change t < T * ContinuousExit.time (exitPath hε hT U x β a L)
+    rw [mul_comm]
+    apply (div_lt_iff₀ hTp).1
+    apply (ContinuousExit.lt_time_iff (p := exitPath hε hT U x β a L)
+      ⟨div_nonneg ht.1 hT, (div_lt_one hTp).2 ht.2⟩).2
+    intro s hs
+    rw [exitPath_extend hε hT U x β a L
+      ⟨hs.1, hs.2.trans ((div_le_one hTp).2 ht.2.le)⟩]
+    apply hh
+    exact ⟨mul_nonneg hT hs.1, (mul_le_mul_of_nonneg_left hs.2 hT).trans_eq
+      (mul_div_cancel₀ _ hTp.ne')⟩
+
+lemma isOpen_positive_on (x β a L t : ℝ) :
+    IsOpen {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U x β a L s} := by
+  let S : Set (C(I, ℝ) × Icc 0 t) := {q | deficit hε hT q.1 x β a L q.2 ≤ 0}
+  have hmap : Continuous (fun q : C(I, ℝ) × Icc 0 t => (q.1,(q.2:ℝ))) :=
+    continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)
+  have hd := (continuous_deficit hε hT x β a L).comp hmap
+  dsimp only [Function.comp_def] at hd
+  have hc : IsClosed S := isClosed_le hd continuous_const
+  have hh := isClosedMap_fst_of_compactSpace S hc
+  have he : {U : C(I, ℝ) | ∀ s ∈ Icc 0 t, 0 < deficit hε hT U x β a L s} = (Prod.fst '' S)ᶜ := by
+    ext U
+    simp only [mem_ofPred_eq, mem_compl_iff, mem_image, S, not_exists, not_and]
+    constructor
+    · intro hp q hq hqp
+      subst U
+      exact (not_lt_of_ge hq) (hp q.2 q.2.property)
+    · intro hp s hs
+      by_contra hn
+      exact hp (U,⟨s,hs⟩) (not_lt.1 hn) rfl
+  rw [he]
+  exact hh.isOpen_compl
+
+lemma deficit_prefix_eq {S : ℝ} (hS : 0 ≤ S) (hST : S ≤ T) (U V : C(I, ℝ))
+    (x β a L : ℝ) (hUV : EqOn (extendDriver S U) (extendDriver T V) (Icc 0 S)) :
+    EqOn (deficit hε hS U x β a L) (deficit hε hT V x β a L) (Icc 0 S) := by
+  intro t ht
+  have hH := position_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
+  have hD := logDerivative_prefix_eq hε hS hT hS le_rfl hST U V (x:ℂ) hUV ht
+  simp only [deficit,weight,factor,gap,hH,hD,hUV ht]
+
+lemma regular (_hTp : 0 < T) (β a s : ℝ) (hβ : 0 ≤ β) (U : C(I, ℝ)) (x L : ℝ)
+    (b : ℝ → ℝ) (hb : Continuous b) :
+    BrownianField.FieldRegular (field hε hT β a s U x) (drift hε hT β a s U x) b
+      (stop hε hT U x β a L) := by
+  constructor
+  · exact smooth hε hT β a s U x
+  · obtain ⟨K,hK,hb⟩ := coeff_uniform_bound hε hT β a s hβ x 3 (Or.inr (Or.inr rfl))
+    exact ⟨K,hK,hb U⟩
+  · intro v
+    exact ((continuous_field hε hT β a s x).comp
+      ((continuous_const.prodMk continuous_id).prodMk continuous_const)).continuousOn
+  · intro t ht v
+    exact time_deriv hε hT β a s U x ⟨ht.1,ht.2.trans_le (stop_bounds hε hT U x β a L).2⟩ v
+  · exact ((continuous_drift hε hT β a s x).comp
+      ((continuous_const.prodMk continuous_fst).prodMk (hb.comp continuous_snd))).continuousOn
+  · exact ((continuous_coeff hε hT β a s x 2 (Or.inr rfl)).comp
+      ((continuous_const.prodMk continuous_id).prodMk hb)).continuousOn
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional BrownianPredictable BrownianStopped BrownianIto TaylorMeshes
+variable {ε : ℝ} (hε : 0 < ε)
+
+def coeffAt (k : ℕ) (s : ℝ≥0) (β a sdev : ℝ) (x : ℝ) (L : ℝ)
+    (q : C(I, ℝ) × ℝ) : ℝ := by
+  classical
+  exact if ∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg q.1 x β a L t then
+    iteratedDeriv k (field hε s.coe_nonneg β a sdev q.1 x s) q.2 else 0
+
+lemma coeffAt_measurable (k : ℕ) (hk : k = 1 ∨ k = 2) (s : ℝ≥0)
+    (β a sdev : ℝ) (x : ℝ) (L : ℝ) :
+    @Measurable (C(I, ℝ) × ℝ) ℝ (@Prod.instMeasurableSpace _ _ (borel _) inferInstance) _ (coeffAt hε k s β a sdev x L) := by
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  have hc : Continuous (fun q : C(I, ℝ) × ℝ =>
+      iteratedDeriv k (field hε s.coe_nonneg β a sdev q.1 x s) q.2) := by
+    have hp : Continuous (fun q : C(I, ℝ) × ℝ => ((q.1,(s:ℝ)),q.2)) :=
+      (continuous_fst.prodMk continuous_const).prodMk continuous_snd
+    have hh := (continuous_coeff hε s.coe_nonneg β a sdev x k hk).comp hp
+    dsimp only [Function.comp_def] at hh
+    exact hh
+  apply hc.measurable.ite _ measurable_const
+  exact ((isOpen_positive_on hε s.coe_nonneg x β a L s).preimage continuous_fst).measurableSet
+
+def rawCoeff {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (β a sdev : ℝ) (x : ℝ) (L : ℝ) (sample : Ω) : ℝ :=
+  if s < T then coeffAt hε k s β a sdev x L (BrownianDriver.path B sdev s sample, B s sample) else 0
+
+lemma rawCoeff_measurable_past {Ω : Type*} [MeasurableSpace Ω]
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (β a sdev : ℝ) (x : ℝ) (L : ℝ) :
+    Measurable[pastSigma B s] (rawCoeff hε k B T s β a sdev x L) := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  let : MeasurableSpace Ω := pastSigma B s
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s β a sdev x L).comp
+      ((BrownianDriver.path_measurable_past B sdev s).prodMk (past_measurable_eval B (s := s) le_rfl))
+  · exact measurable_const
+
+lemma rawCoeff_aemeasurable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P)
+    (k : ℕ) (hk : k = 1 ∨ k = 2) (T s : ℝ≥0)
+    (β a sdev : ℝ) (x : ℝ) (L : ℝ) :
+    AEMeasurable (rawCoeff hε k B T s β a sdev x L) P := by
+  classical
+  let : MeasurableSpace C(I, ℝ) := borel _
+  let : BorelSpace C(I, ℝ) := ⟨rfl⟩
+  unfold rawCoeff
+  split_ifs
+  · exact (coeffAt_measurable hε k hk s β a sdev x L).comp_aemeasurable
+      ((BrownianDriver.path_aemeasurable hB sdev s).prodMk (hB.toIsPreBrownianReal.hasLaw_eval s).aemeasurable)
+  · exact aemeasurable_const
+
+lemma rawCoeff_eq {Ω : Type*} (k : ℕ) (B : ℝ≥0 → Ω → ℝ) (T s : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (β a sdev : ℝ) (x : ℝ) (L : ℝ) (sample : Ω)
+    (hc : Continuous (fun t => B t sample)) :
+    rawCoeff hε k B T s β a sdev x L sample =
+      if (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L then
+        iteratedDeriv k (field hε T.coe_nonneg β a sdev (BrownianDriver.path B sdev T sample) x s) (B s sample) else 0 := by
+  classical
+  by_cases hsT : s < T
+  · have he := BrownianDriver.prefix_eq B sdev s T sample hc hsT.le
+    have hf := field_prefix_eq hε T.coe_nonneg s.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) x β a sdev he
+      ⟨s.coe_nonneg, le_rfl⟩
+    have hd := deficit_prefix_eq hε T.coe_nonneg s.coe_nonneg hsT.le
+      (BrownianDriver.path B sdev s sample) (BrownianDriver.path B sdev T sample) x β a L he
+    have hi : (∀ t ∈ Icc 0 (s:ℝ), 0 < deficit hε s.coe_nonneg (BrownianDriver.path B sdev s sample) x β a L t) ↔
+        (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L := by
+      rw [lt_stop_iff hε T.coe_nonneg hT _ x β a L ⟨s.coe_nonneg,hsT⟩]
+      exact forall₂_congr (fun t ht => by rw [hd ht])
+    simp only [rawCoeff, ite_eq_left hsT, coeffAt, hi, hf]
+  · have hstop : ¬ (s:ℝ) < stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L :=
+      not_lt_of_ge ((stop_bounds hε T.coe_nonneg _ x β a L).2.trans (not_lt.1 hsT))
+    simp only [rawCoeff, ite_eq_right hsT, ite_eq_right hstop]
+
+lemma rawCoeff_stoppedCoeff_ae {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
+    {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (k : ℕ) (T : ℝ≥0)
+    (hT : 0 < (T:ℝ)) (β a sdev : ℝ) (x : ℝ) (L : ℝ) (n i : ℕ) :
+    rawCoeff hε k B T (dyadicTime T n i) β a sdev x L =ᵐ[P]
+      stoppedCoeff k B T
+        (fun sample => stop hε T.coe_nonneg (BrownianDriver.path B sdev T sample) x β a L)
+        (fun sample => field hε T.coe_nonneg β a sdev (BrownianDriver.path B sdev T sample) x) n i := by
+  filter_upwards [hB.cont] with sample hc
+  rw [rawCoeff_eq hε k B T _ hT β a sdev x L sample hc]
+  simp only [stoppedCoeff, coe_dyadicTime]
+
+theorem localized_centered {Ω : Type*} [MeasurableSpace Ω]
+    {P : Measure Ω} [IsProbabilityMeasure P] {B : ℝ≥0 → Ω → ℝ}
+    (hB : IsBrownianReal B P) (β a sdev : ℝ) (hβ : 0 ≤ β)
+    {ε : ℝ} (hε : 0 < ε) (T : ℝ≥0) (hT : 0 < (T:ℝ)) (x : ℝ) (L : ℝ) :
+    let U := BrownianDriver.path B sdev T
+    let τ := fun sample => stop hε T.coe_nonneg (U sample) x β a L
+    let F := fun sample => field hε T.coe_nonneg β a sdev (U sample) x
+    let V := fun sample => drift hε T.coe_nonneg β a sdev (U sample) x
+    let Y := fun sample => F sample (τ sample) (B (τ sample).toNNReal sample) - F sample 0 (B 0 sample) -
+      (∫ t in (0:ℝ)..(τ sample), V sample t (B t.toNNReal sample)) -
+      (1/2:ℝ)*∫ t in (0:ℝ)..(τ sample), iteratedDeriv 2 (F sample t) (B t.toNNReal sample)
+    Integrable Y P ∧ (∫ sample, Y sample ∂P) = 0 := by
+  dsimp only
+  let U := BrownianDriver.path B sdev T
+  let τ := fun sample => stop hε T.coe_nonneg (U sample) x β a L
+  let F := fun sample => field hε T.coe_nonneg β a sdev (U sample) x
+  let V := fun sample => drift hε T.coe_nonneg β a sdev (U sample) x
+  have hreg : ∀ᵐ sample ∂P, BrownianField.FieldRegular (F sample) (V sample)
+      (fun t => B t.toNNReal sample) (τ sample) := by
+    filter_upwards [hB.cont] with sample hc
+    exact regular hε T.coe_nonneg hT β a sdev hβ (U sample) x L _
+      (hc.comp continuous_real_toNNReal)
+  have htau : ∀ᵐ sample ∂P, 0 ≤ τ sample ∧ τ sample ≤ T :=
+    Filter.Eventually.of_forall (fun sample => stop_bounds hε T.coe_nonneg (U sample) x β a L)
+  obtain ⟨K₁,hK₁,hb₁⟩ := coeff_uniform_bound hε T.coe_nonneg β a sdev hβ x 1 (Or.inl rfl)
+  obtain ⟨K₂,hK₂,hb₂⟩ := coeff_uniform_bound hε T.coe_nonneg β a sdev hβ x 2 (Or.inr (Or.inl rfl))
+  have hsbound (k : ℕ) (K : ℝ) (hK : 0 ≤ K)
+      (hb : ∀ (W : C(I, ℝ)) (t b : ℝ),
+        |iteratedDeriv k (field hε T.coe_nonneg β a sdev W x t) b| ≤ K)
+      (n i : ℕ) : ∀ᵐ sample ∂P, |stoppedCoeff k B T τ F n i sample| ≤ K := by
+    apply Filter.Eventually.of_forall
+    intro sample
+    unfold stoppedCoeff
+    split_ifs
+    · exact hb _ _ _
+    · simpa only [abs_zero] using hK
+  let A := fun n i => rawCoeff hε 1 B T (dyadicTime T n i) β a sdev x L
+  let Q := fun n i => rawCoeff hε 2 B T (dyadicTime T n i) β a sdev x L
+  have hAe (n i : ℕ) : A n i =ᵐ[P] stoppedCoeff 1 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 1 T hT β a sdev x L n i
+  have hCe (n i : ℕ) : Q n i =ᵐ[P] stoppedCoeff 2 B T τ F n i :=
+    rawCoeff_stoppedCoeff_ae hε hB 2 T hT β a sdev x L n i
+  have hAm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (A n i) :=
+    rawCoeff_measurable_past hε 1 (Or.inl rfl) B T _ _ _ _ x L
+  have hCm (n i : ℕ) : Measurable[pastSigma B (dyadicTime T n i)] (Q n i) :=
+    rawCoeff_measurable_past hε 2 (Or.inr rfl) B T _ _ _ _ x L
+  have hA2 (n i : ℕ) : MemLp (A n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 1 (Or.inl rfl) T _ _ _ _ x L).aestronglyMeasurable K₁
+    filter_upwards [hAe n i, hsbound 1 K₁ hK₁ hb₁ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |A n i sample| ≤ K₁
+    rw [he]
+    exact hb
+  have hC2 (n i : ℕ) : MemLp (Q n i) 2 P := by
+    apply MemLp.of_bound (rawCoeff_aemeasurable hε hB 2 (Or.inr rfl) T _ _ _ _ x L).aestronglyMeasurable K₂
+    filter_upwards [hCe n i, hsbound 2 K₂ hK₂ hb₂ n i] with sample he hb
+    rw [Real.norm_eq_abs]
+    change |Q n i sample| ≤ K₂
+    rw [he]
+    exact hb
+  have hi := stopped_field_ito_centered_ae hB T τ F V htau hreg A Q hAe hCe hCm hC2
+    K₂ hK₂ (hsbound 2 K₂ hK₂ hb₂) hAm hA2 K₁ hK₁ (hsbound 1 K₁ hK₁ hb₁)
+  exact hi
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal
+namespace SLEExactGauge.BoundaryGreen
+open Swallowing GreenPhysical
+
+def generator (κ β a x ε : ℝ) : ℝ :=
+  -(a*β/x^2)*radial (-β) x ε + radialX (-β) x ε*(2/x)+κ/2*radialXX (-β) x ε
+
+lemma generator_factor (κ β a x ε : ℝ) (hx : 0 < x) (hε : 0 < ε) :
+    generator κ β a x ε = β*(x^2+ε^2)^((-β)/2-2)/x^2 *
+      ((κ*(β+1)/2-2-a)*x^4-(2+κ/2+2*a)*x^2*ε^2-a*ε^4) := by
+  have hq := (q_pos x hε).ne'
+  have hp₁ : (x^2+ε^2)^((-β)/2-1) = (x^2+ε^2)^((-β)/2-2)*(x^2+ε^2) := by
+    conv_rhs => rhs; rw [← Real.rpow_one (x^2+ε^2)]
+    rw [← Real.rpow_add (q_pos x hε)]
+    congr 1
+    ring
+  have hp₀ : (x^2+ε^2)^((-β)/2) = (x^2+ε^2)^((-β)/2-2)*(x^2+ε^2)^2 := by
+    conv_rhs => rhs; rw [← Real.rpow_natCast]
+    rw [← Real.rpow_add (q_pos x hε)]
+    congr 1
+    norm_num
+  unfold generator radial radialX radialXX
+  rw [hp₀,hp₁]
+  field_simp
+  ring
+
+lemma generator_nonpos {κ β a x ε : ℝ} (hκ : 0 ≤ κ) (hβ : 0 ≤ β) (ha : 0 ≤ a)
+    (hba : κ*(β+1) ≤ 4+2*a) (hx : 0 < x) (hε : 0 < ε) :
+    generator κ β a x ε ≤ 0 := by
+  rw [generator_factor κ β a x ε hx hε]
+  apply mul_nonpos_of_nonneg_of_nonpos
+  · positivity
+  · have h₁ : (κ*(β+1)/2-2-a)*x^4 ≤ 0 :=
+      mul_nonpos_of_nonpos_of_nonneg (by linarith) (by positivity)
+    have h₂ : 0 ≤ (2+κ/2+2*a)*x^2*ε^2 := by positivity
+    have h₃ : 0 ≤ a*ε^4 := by positivity
+    linarith
+
+lemma exponent_choice {κ : ℝ} (hκ : 0 < κ) (h8 : κ < 8) :
+    let β := (8/κ-1)/2
+    let a := 1+κ/8
+    0 < β ∧ 0 < a ∧ a < 2 ∧ κ*(β+1) < 4+2*a := by
+  have hd : 1 < 8/κ := (lt_div_iff₀ hκ).2 (by simpa)
+  dsimp only
+  refine ⟨by linarith,by positivity,by linarith,?_⟩
+  have he : κ*((8/κ-1)/2+1) = 4+κ/2 := by field_simp; ring
+  rw [he]
+  linarith
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology NNReal ENNReal unitInterval
+namespace SLEExactGauge.BoundaryGreen
+open BoundaryFunctional Swallowing GreenPhysical
+
+lemma field_initial {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (β a s : ℝ) (U : C(I, ℝ)) (x : ℝ) :
+    field hε hT β a s U x 0 0 = radial (-β) x ε := by
+  simp only [field,factor,logDerivative_initial,mul_zero,Real.exp_zero,position_initial,
+    Complex.ofReal_re,sub_zero,one_mul]
+
+lemma generator_field {ε T : ℝ} (hε : 0 < ε) (hT : 0 ≤ T)
+    (κ β a : ℝ) (hκ : 0 ≤ κ) (U : C(I, ℝ)) (x t b : ℝ)
+    (hx : ε ≤ gap hε hT U x t) (hU : extendDriver T U t = Real.sqrt κ*b) :
+    drift hε hT β a (Real.sqrt κ) U x t b+
+      (1/2:ℝ)*iteratedDeriv 2 (field hε hT β a (Real.sqrt κ) U x t) b =
+    factor hε hT β a U x t * generator κ β a (gap hε hT U x t) ε := by
+  have hz : position hε hT U (x:ℂ) t = ((position hε hT U (x:ℂ) t).re:ℂ) := by
+    apply Complex.ext <;> simp [position_im_zero]
+  have hc : clampRe ε (position hε hT U (x:ℂ) t-(extendDriver T U t:ℂ)) =
+      (gap hε hT U x t:ℂ) := by
+    change BoundaryCutoff.clamp 1 ε _ = _
+    rw [BoundaryCutoff.clamp_eq (by norm_num : |(1:ℝ)|=1) (by simpa [gap] using hx)]
+    rw [hz]
+    simp only [gap,Complex.ofReal_sub]
+  have hv : velocity ε (extendDriver T U t) (position hε hT U (x:ℂ) t) =
+      (2/(gap hε hT U x t):ℝ) := by
+    change 2/clampRe ε _ = _
+    rw [hc]
+    push_cast
+    rfl
+  have hd : variation hε hT U (x:ℂ) t = -2/(gap hε hT U x t)^2 := by
+    unfold variation
+    rw [hc]
+    norm_cast
+  rw [second_eq,Real.sq_sqrt hκ]
+  dsimp only [drift]
+  rw [hv,hd,Complex.ofReal_re]
+  have he : (position hε hT U (x:ℂ) t).re-Real.sqrt κ*b = gap hε hT U x t := by
+    simp only [gap,hU]
+  rw [he]
+  unfold generator
+  ring
+
+end SLEExactGauge.BoundaryGreen
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/TraceAvoidance.lean b/lean/OAI/Probability/SLELocalization/TraceAvoidance.lean
index 3f4c92795e86658a80ea33cb17dc80ff18b9ebe8..dc1974ffb6342eff38aa015fc343a16e2df6530d 100644
--- a/lean/OAI/Probability/SLELocalization/TraceAvoidance.lean
+++ b/lean/OAI/Probability/SLELocalization/TraceAvoidance.lean
@@ -1,208 +1,208 @@
-import Mathlib
-import OAI.Probability.SLELocalization.TraceModel
-
-namespace OAI
-
-
-/-! Conformal localization and Is Loewner Trace Using estimates. -/
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-
-lemma positive_solution_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {T : ℝ} (hT : 0 ≤ T)
-    {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
-      (2/(H t-(driver t:ℂ))) (Icc 0 T) t)
-    (hpos : ∀ t ∈ Icc 0 T, 0 < (H t).im) :
-    H 0 ∈ localizationDomain γ T ∧ EqOn (fun t => g t (H 0)) H (Icc 0 T) := by
-  have hc : ContinuousOn (fun t => (H t).im) (Icc 0 T) :=
-    Complex.continuous_im.comp_continuousOn (HasDerivWithinAt.continuousOn hH)
-  obtain ⟨s,hs,hmin⟩ := isCompact_Icc.exists_isMinOn ⟨0,le_rfl,hT⟩ hc
-  have hpositive := hpos s hs
-  let lip : ℝ≥0 := ⟨2 / (H s).im^2, by positivity⟩
-  have hlip (time : ℝ) : LipschitzOnWith lip
-      (fun point : ℂ => 2 / (point - (driver time : ℂ))) {point | (H s).im ≤ point.im} := by
-    have hnorm (point : ℂ) (hpoint : (H s).im ≤ point.im) :
-        (H s).im ≤ ‖point - (driver time : ℂ)‖ := by
-      simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using
-        hpoint.trans (by simpa using Complex.im_le_norm (point - (driver time : ℂ)))
-    apply LipschitzOnWith.of_dist_le_mul
-    intro left hleft right hright
-    have hleftNorm := hnorm left hleft
-    have hrightNorm := hnorm right hright
-    have hleftNe : left - (driver time : ℂ) ≠ 0 := norm_pos_iff.mp (hpositive.trans_le hleftNorm)
-    have hrightNe : right - (driver time : ℂ) ≠ 0 := norm_pos_iff.mp (hpositive.trans_le hrightNorm)
-    have hden : (H s).im^2 ≤ ‖(left - (driver time : ℂ)) * (right - (driver time : ℂ))‖ := by
-      rw [norm_mul, pow_two]
-      exact mul_le_mul hleftNorm hrightNorm hpositive.le (norm_nonneg _)
-    have hdiff : 2 / (left - (driver time : ℂ)) - 2 / (right - (driver time : ℂ)) =
-        2 * (right - left) / ((left - (driver time : ℂ)) * (right - (driver time : ℂ))) := by
-      field_simp
-      ring
-    simp only [dist_eq_norm, hdiff, norm_div, norm_mul, Complex.norm_ofNat]
-    change 2 * ‖right - left‖ / (‖left - (driver time : ℂ)‖ * ‖right - (driver time : ℂ)‖) ≤
-      (2 / (H s).im^2) * ‖left - right‖
-    calc
-      _ ≤ 2 * ‖right - left‖ / (H s).im^2 := by
-        apply div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hpositive)
-        simpa only [norm_mul] using hden
-      _ = _ := by rw [norm_sub_rev]; ring
-  let start := f T (H T)
-  have hstart : start ∈ localizationDomain γ T :=
-    h.2.2.2.2.2.2.2.2.1 T hT (hpos T ⟨hT, le_rfl⟩)
-  have hfinal : g T start = H T :=
-    h.2.2.2.2.2.2.2.2.2.2.1 T hT _ (hpos T ⟨hT, le_rfl⟩)
-  have hheight (time : ℝ) (htime : time ∈ Icc 0 T) :
-      (H s).im ≤ (g time start).im := by
-    have hanti := h.height_antitone hT hstart htime ⟨hT, le_rfl⟩ htime.2
-    dsimp only at hanti
-    rw [hfinal] at hanti
-    exact (hmin ⟨hT, le_rfl⟩).trans hanti
-  have hder (time : ℝ) (htime : time ∈ Ioc 0 T) :
-      HasDerivWithinAt (fun time => g time start)
-        (2 / (g time start - (driver time : ℂ))) (Iic time) time := by
-    exact (h.hasDerivAt_eval htime.1 (survivingDomain_mono htime.2 hstart)).hasDerivWithinAt
-  have heq : EqOn (fun time => g time start) H (Icc 0 T) :=
-    ODE_solution_unique_of_mem_Icc_left
-      (s := fun _ => {point : ℂ | (H s).im ≤ point.im}) (fun time _ => hlip time)
-      (h.continuous_eval hT hstart) hder
-      (fun time htime => hheight time ⟨htime.1.le, htime.2⟩)
-      (HasDerivWithinAt.continuousOn hH)
-      (fun time htime => (hH time ⟨htime.1.le, htime.2⟩).mono_of_mem_nhdsWithin
-        (Icc_mem_nhdsLE_of_mem htime))
-      (fun _ htime => hmin ⟨htime.1.le, htime.2⟩) hfinal
-  have hzero : start = H 0 := by
-    rw [← h.initial start hstart.1.1]
-    exact heq ⟨le_rfl, hT⟩
-  exact ⟨hzero ▸ hstart, hzero ▸ heq⟩
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.BoundaryCutoff
-
-lemma signed_difference_bound {σ : ℝ} (hσ : |σ| = 1) (z w : ℂ) (u v : ℝ) :
-    |σ*(z.re-u)-σ*(w.re-v)| ≤ dist z w+|u-v| := by
-  rw [← mul_sub,abs_mul,hσ,one_mul]
-  calc
-    |(z.re-u)-(w.re-v)| = |(z.re-w.re)-(u-v)| := by congr 1; ring
-    _ ≤ |z.re-w.re|+|u-v| := abs_sub _ _
-    _ ≤ dist z w+|u-v| := by
-      have hh := Complex.abs_re_le_norm (z-w)
-      simp only [Complex.sub_re] at hh
-      rw [dist_eq_norm]
-      linarith
-
-lemma neighborhood_live {U : ℝ → ℝ} {H : ℝ → ℂ} {T ε σ : ℝ}
-    (hT : 0 < T) (hε : 0 < ε) (hσ : |σ| = 1)
-    (_hU : ContinuousOn U (Icc 0 T))
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
-      (2/(H t-(U t:ℂ))) (Icc 0 T) t)
-    (hgap : ∀ t ∈ Icc 0 T, ε ≤ σ*((H t).re-U t)) :
-    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
-      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
-      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
-      ∀ w : ℂ, 0 < w.im → dist w (H 0) < η →
-        w ∈ localizationDomain ξ T ∧ ∀ t ∈ Icc 0 T, ε/2 ≤ σ*((G t w).re-V t) := by
-  let C := Real.exp ((4/(ε/2)^2)*T)
-  have hC : 0 < C := Real.exp_pos _
-  let η := ε/(8*(C+1))
-  have hη : 0 < η := div_pos hε (by positivity)
-  have hsmall : 2*η*C+η ≤ ε/2 := by
-    have hden : 0 < 8*(C+1) := by positivity
-    have he : η*(8*(C+1)) = ε := div_mul_cancel₀ _ hden.ne'
-    nlinarith
-  have hhalf : 0 < ε/2 := by positivity
-  have hbase : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
-      (velocity σ (ε/2) (U t) (H t)) (Icc 0 T) t := by
-    intro t ht
-    rw [velocity_eq hσ (by linarith [hgap t ht])]
-    exact hH t ht
-  refine ⟨η,hη,fun V ξ G F hξ hV hclose w hw hdist => ?_⟩
-  obtain ⟨J,hJ0,hJ⟩ := exists_solution hσ hhalf hT.le V hV w
-  have hd (t : ℝ) (ht : t ∈ Icc 0 T) : dist (J t) (H t) ≤ 2*η*C := by
-    have hh := solution_dist_le hσ hhalf hη.le hη.le hJ hbase
-      (by rw [hJ0]; exact hdist.le) (fun t ht => (hclose t ht).le) ht
-    simpa only [← two_mul] using hh
-  have hg (t : ℝ) (ht : t ∈ Icc 0 T) : ε/2 ≤ σ*((J t).re-V t) := by
-    have hh := signed_difference_bound hσ (J t) (H t) (V t) (U t)
-    have hn := neg_abs_le (σ*((J t).re-V t)-σ*((H t).re-U t))
-    have hs := hgap t ht
-    have hcl := hclose t ht
-    have hdt := hd t ht
-    linarith
-  have hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt J
-      (2/(J t-(V t:ℂ))) (Icc 0 T) t := by
-    intro t ht
-    have hh := hJ t ht
-    rw [velocity_eq hσ (hg t ht)] at hh
-    exact hh
-  have hpos := positive_height hσ hhalf hJ (by rw [hJ0]; exact hw)
-  have hl := hξ.positive_solution_live hT.le hder hpos
-  rw [hJ0] at hl
-  refine ⟨hl.1,fun t ht => ?_⟩
-  have he : G t w = J t := hl.2 ht
-  rw [he]
-  exact hg t ht
-
-end SLEExactGauge.BoundaryCutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge
-
-theorem separated_forward_path_avoids_trace
-    {U : ℝ → ℝ} {H : ℝ → ℂ} {T ε σ : ℝ}
-    (hT : 0 < T) (hε : 0 < ε) (hσ : |σ| = 1)
-    (hU : ContinuousOn U (Icc 0 T))
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
-      (2/(H t-(U t:ℂ))) (Icc 0 T) t)
-    (hgap : ∀ t ∈ Icc 0 T, ε ≤ σ*((H t).re-U t)) :
-    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
-      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
-      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
-      ∀ t ∈ Icc 0 T, η ≤ dist (ξ t) (H 0) := by
-  obtain ⟨η,hη,hlive⟩ := BoundaryCutoff.neighborhood_live hT hε hσ hU hH hgap
-  refine ⟨η,hη,fun V ξ G F hξ hV hclose t ht => ?_⟩
-  by_contra hn
-  have hlt : dist (ξ t) (H 0) < η := lt_of_not_ge hn
-  have htip := hξ.2.2.2.2.2.2.2.2.2.2.2.2.2 t ht.1
-  have he : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), dist (F t ((V t:ℂ)+(y:ℂ)*Complex.I)) (H 0) < η :=
-    htip.eventually (isOpen_ball.mem_nhds (show ξ t ∈ ball (H 0) η from hlt))
-  have hp : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), 0 < y := self_mem_nhdsWithin
-  obtain ⟨y,hy,hball⟩ := (hp.and he).exists
-  have hz : (V t:ℂ)+(y:ℂ)*Complex.I ∈ halfPlane := by
-    change 0 < ((V t:ℂ)+(y:ℂ)*Complex.I).im
-    simpa using hy
-  have hfw := hξ.2.2.2.2.2.2.2.2.1 t ht.1 hz
-  have hl := hlive V ξ G F hξ hV hclose _ hfw.1.1 hball
-  have hg := hl.2 t ht
-  rw [hξ.2.2.2.2.2.2.2.2.2.2.1 t ht.1 _ hz] at hg
-  simp only [Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.ofReal_im,
-    Complex.I_re,Complex.I_im,mul_zero,zero_mul,add_zero,sub_self] at hg
-  linarith
-
-end SLEExactGauge
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.TraceModel
+
+namespace OAI
+
+
+/-! Conformal localization and Is Loewner Trace Using estimates. -/
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+
+lemma positive_solution_live {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {T : ℝ} (hT : 0 ≤ T)
+    {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
+      (2/(H t-(driver t:ℂ))) (Icc 0 T) t)
+    (hpos : ∀ t ∈ Icc 0 T, 0 < (H t).im) :
+    H 0 ∈ localizationDomain γ T ∧ EqOn (fun t => g t (H 0)) H (Icc 0 T) := by
+  have hc : ContinuousOn (fun t => (H t).im) (Icc 0 T) :=
+    Complex.continuous_im.comp_continuousOn (HasDerivWithinAt.continuousOn hH)
+  obtain ⟨s,hs,hmin⟩ := isCompact_Icc.exists_isMinOn ⟨0,le_rfl,hT⟩ hc
+  have hpositive := hpos s hs
+  let lip : ℝ≥0 := ⟨2 / (H s).im^2, by positivity⟩
+  have hlip (time : ℝ) : LipschitzOnWith lip
+      (fun point : ℂ => 2 / (point - (driver time : ℂ))) {point | (H s).im ≤ point.im} := by
+    have hnorm (point : ℂ) (hpoint : (H s).im ≤ point.im) :
+        (H s).im ≤ ‖point - (driver time : ℂ)‖ := by
+      simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using
+        hpoint.trans (by simpa using Complex.im_le_norm (point - (driver time : ℂ)))
+    apply LipschitzOnWith.of_dist_le_mul
+    intro left hleft right hright
+    have hleftNorm := hnorm left hleft
+    have hrightNorm := hnorm right hright
+    have hleftNe : left - (driver time : ℂ) ≠ 0 := norm_pos_iff.mp (hpositive.trans_le hleftNorm)
+    have hrightNe : right - (driver time : ℂ) ≠ 0 := norm_pos_iff.mp (hpositive.trans_le hrightNorm)
+    have hden : (H s).im^2 ≤ ‖(left - (driver time : ℂ)) * (right - (driver time : ℂ))‖ := by
+      rw [norm_mul, pow_two]
+      exact mul_le_mul hleftNorm hrightNorm hpositive.le (norm_nonneg _)
+    have hdiff : 2 / (left - (driver time : ℂ)) - 2 / (right - (driver time : ℂ)) =
+        2 * (right - left) / ((left - (driver time : ℂ)) * (right - (driver time : ℂ))) := by
+      field_simp
+      ring
+    simp only [dist_eq_norm, hdiff, norm_div, norm_mul, Complex.norm_ofNat]
+    change 2 * ‖right - left‖ / (‖left - (driver time : ℂ)‖ * ‖right - (driver time : ℂ)‖) ≤
+      (2 / (H s).im^2) * ‖left - right‖
+    calc
+      _ ≤ 2 * ‖right - left‖ / (H s).im^2 := by
+        apply div_le_div_of_nonneg_left (by positivity) (sq_pos_of_pos hpositive)
+        simpa only [norm_mul] using hden
+      _ = _ := by rw [norm_sub_rev]; ring
+  let start := f T (H T)
+  have hstart : start ∈ localizationDomain γ T :=
+    h.2.2.2.2.2.2.2.2.1 T hT (hpos T ⟨hT, le_rfl⟩)
+  have hfinal : g T start = H T :=
+    h.2.2.2.2.2.2.2.2.2.2.1 T hT _ (hpos T ⟨hT, le_rfl⟩)
+  have hheight (time : ℝ) (htime : time ∈ Icc 0 T) :
+      (H s).im ≤ (g time start).im := by
+    have hanti := h.height_antitone hT hstart htime ⟨hT, le_rfl⟩ htime.2
+    dsimp only at hanti
+    rw [hfinal] at hanti
+    exact (hmin ⟨hT, le_rfl⟩).trans hanti
+  have hder (time : ℝ) (htime : time ∈ Ioc 0 T) :
+      HasDerivWithinAt (fun time => g time start)
+        (2 / (g time start - (driver time : ℂ))) (Iic time) time := by
+    exact (h.hasDerivAt_eval htime.1 (survivingDomain_mono htime.2 hstart)).hasDerivWithinAt
+  have heq : EqOn (fun time => g time start) H (Icc 0 T) :=
+    ODE_solution_unique_of_mem_Icc_left
+      (s := fun _ => {point : ℂ | (H s).im ≤ point.im}) (fun time _ => hlip time)
+      (h.continuous_eval hT hstart) hder
+      (fun time htime => hheight time ⟨htime.1.le, htime.2⟩)
+      (HasDerivWithinAt.continuousOn hH)
+      (fun time htime => (hH time ⟨htime.1.le, htime.2⟩).mono_of_mem_nhdsWithin
+        (Icc_mem_nhdsLE_of_mem htime))
+      (fun _ htime => hmin ⟨htime.1.le, htime.2⟩) hfinal
+  have hzero : start = H 0 := by
+    rw [← h.initial start hstart.1.1]
+    exact heq ⟨le_rfl, hT⟩
+  exact ⟨hzero ▸ hstart, hzero ▸ heq⟩
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.BoundaryCutoff
+
+lemma signed_difference_bound {σ : ℝ} (hσ : |σ| = 1) (z w : ℂ) (u v : ℝ) :
+    |σ*(z.re-u)-σ*(w.re-v)| ≤ dist z w+|u-v| := by
+  rw [← mul_sub,abs_mul,hσ,one_mul]
+  calc
+    |(z.re-u)-(w.re-v)| = |(z.re-w.re)-(u-v)| := by congr 1; ring
+    _ ≤ |z.re-w.re|+|u-v| := abs_sub _ _
+    _ ≤ dist z w+|u-v| := by
+      have hh := Complex.abs_re_le_norm (z-w)
+      simp only [Complex.sub_re] at hh
+      rw [dist_eq_norm]
+      linarith
+
+lemma neighborhood_live {U : ℝ → ℝ} {H : ℝ → ℂ} {T ε σ : ℝ}
+    (hT : 0 < T) (hε : 0 < ε) (hσ : |σ| = 1)
+    (_hU : ContinuousOn U (Icc 0 T))
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
+      (2/(H t-(U t:ℂ))) (Icc 0 T) t)
+    (hgap : ∀ t ∈ Icc 0 T, ε ≤ σ*((H t).re-U t)) :
+    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
+      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
+      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
+      ∀ w : ℂ, 0 < w.im → dist w (H 0) < η →
+        w ∈ localizationDomain ξ T ∧ ∀ t ∈ Icc 0 T, ε/2 ≤ σ*((G t w).re-V t) := by
+  let C := Real.exp ((4/(ε/2)^2)*T)
+  have hC : 0 < C := Real.exp_pos _
+  let η := ε/(8*(C+1))
+  have hη : 0 < η := div_pos hε (by positivity)
+  have hsmall : 2*η*C+η ≤ ε/2 := by
+    have hden : 0 < 8*(C+1) := by positivity
+    have he : η*(8*(C+1)) = ε := div_mul_cancel₀ _ hden.ne'
+    nlinarith
+  have hhalf : 0 < ε/2 := by positivity
+  have hbase : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
+      (velocity σ (ε/2) (U t) (H t)) (Icc 0 T) t := by
+    intro t ht
+    rw [velocity_eq hσ (by linarith [hgap t ht])]
+    exact hH t ht
+  refine ⟨η,hη,fun V ξ G F hξ hV hclose w hw hdist => ?_⟩
+  obtain ⟨J,hJ0,hJ⟩ := exists_solution hσ hhalf hT.le V hV w
+  have hd (t : ℝ) (ht : t ∈ Icc 0 T) : dist (J t) (H t) ≤ 2*η*C := by
+    have hh := solution_dist_le hσ hhalf hη.le hη.le hJ hbase
+      (by rw [hJ0]; exact hdist.le) (fun t ht => (hclose t ht).le) ht
+    simpa only [← two_mul] using hh
+  have hg (t : ℝ) (ht : t ∈ Icc 0 T) : ε/2 ≤ σ*((J t).re-V t) := by
+    have hh := signed_difference_bound hσ (J t) (H t) (V t) (U t)
+    have hn := neg_abs_le (σ*((J t).re-V t)-σ*((H t).re-U t))
+    have hs := hgap t ht
+    have hcl := hclose t ht
+    have hdt := hd t ht
+    linarith
+  have hder : ∀ t ∈ Icc 0 T, HasDerivWithinAt J
+      (2/(J t-(V t:ℂ))) (Icc 0 T) t := by
+    intro t ht
+    have hh := hJ t ht
+    rw [velocity_eq hσ (hg t ht)] at hh
+    exact hh
+  have hpos := positive_height hσ hhalf hJ (by rw [hJ0]; exact hw)
+  have hl := hξ.positive_solution_live hT.le hder hpos
+  rw [hJ0] at hl
+  refine ⟨hl.1,fun t ht => ?_⟩
+  have he : G t w = J t := hl.2 ht
+  rw [he]
+  exact hg t ht
+
+end SLEExactGauge.BoundaryCutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge
+
+theorem separated_forward_path_avoids_trace
+    {U : ℝ → ℝ} {H : ℝ → ℂ} {T ε σ : ℝ}
+    (hT : 0 < T) (hε : 0 < ε) (hσ : |σ| = 1)
+    (hU : ContinuousOn U (Icc 0 T))
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H
+      (2/(H t-(U t:ℂ))) (Icc 0 T) t)
+    (hgap : ∀ t ∈ Icc 0 T, ε ≤ σ*((H t).re-U t)) :
+    ∃ η : ℝ, 0 < η ∧ ∀ (V : ℝ → ℝ) (ξ : ℝ → ℂ) (G F : ℝ → ℂ → ℂ),
+      IsLoewnerTraceUsing V ξ G F → ContinuousOn V (Icc 0 T) →
+      (∀ t ∈ Icc 0 T, |V t-U t| < η) →
+      ∀ t ∈ Icc 0 T, η ≤ dist (ξ t) (H 0) := by
+  obtain ⟨η,hη,hlive⟩ := BoundaryCutoff.neighborhood_live hT hε hσ hU hH hgap
+  refine ⟨η,hη,fun V ξ G F hξ hV hclose t ht => ?_⟩
+  by_contra hn
+  have hlt : dist (ξ t) (H 0) < η := lt_of_not_ge hn
+  have htip := hξ.2.2.2.2.2.2.2.2.2.2.2.2.2 t ht.1
+  have he : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), dist (F t ((V t:ℂ)+(y:ℂ)*Complex.I)) (H 0) < η :=
+    htip.eventually (isOpen_ball.mem_nhds (show ξ t ∈ ball (H 0) η from hlt))
+  have hp : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), 0 < y := self_mem_nhdsWithin
+  obtain ⟨y,hy,hball⟩ := (hp.and he).exists
+  have hz : (V t:ℂ)+(y:ℂ)*Complex.I ∈ halfPlane := by
+    change 0 < ((V t:ℂ)+(y:ℂ)*Complex.I).im
+    simpa using hy
+  have hfw := hξ.2.2.2.2.2.2.2.2.1 t ht.1 hz
+  have hl := hlive V ξ G F hξ hV hclose _ hfw.1.1 hball
+  have hg := hl.2 t ht
+  rw [hξ.2.2.2.2.2.2.2.2.2.2.1 t ht.1 _ hz] at hg
+  simp only [Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.ofReal_im,
+    Complex.I_re,Complex.I_im,mul_zero,zero_mul,add_zero,sub_self] at hg
+  linarith
+
+end SLEExactGauge
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/TraceModel.lean b/lean/OAI/Probability/SLELocalization/TraceModel.lean
index ce8df1279bd3128a599c3ef91b16c8cd9d38b012..715ddf54a71961dc3d5e723ba1b1c1e31e91abc3 100644
--- a/lean/OAI/Probability/SLELocalization/TraceModel.lean
+++ b/lean/OAI/Probability/SLELocalization/TraceModel.lean
@@ -1,848 +1,848 @@
-import Mathlib
-import OAI.Probability.SLELocalization.SobolevMeans
-import OAI.Probability.SLE.ClippedCoordinates
-
-namespace OAI
-
-
-/-! Conformal localization and Boundary Cutoff estimates. -/
-
-section
-noncomputable section
-open Set Filter Metric
-open scoped Topology NNReal
-namespace SLEExactGauge.BoundaryCutoff
-
-lemma norm_velocity_sub_le {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
-    (u : ℝ) (z w : ℂ) : ‖velocity σ ε u z-velocity σ ε u w‖ ≤ (4/ε^2)*‖z-w‖ := by
-  let a := clamp σ ε (z-(u:ℂ))
-  let b := clamp σ ε (w-(u:ℂ))
-  have ha : a ≠ 0 := clamp_ne_zero hσ hε _
-  have hb : b ≠ 0 := clamp_ne_zero hσ hε _
-  have hd : ε^2 ≤ ‖a*b‖ := by
-    rw [norm_mul,pow_two]
-    exact mul_le_mul (clamp_norm hσ hε _) (clamp_norm hσ hε _) hε.le (norm_nonneg _)
-  have he : velocity σ ε u z-velocity σ ε u w = 2*(b-a)/(a*b) := by
-    change 2/a-2/b=2*(b-a)/(a*b)
-    field_simp
-  rw [he,norm_div,norm_mul,Complex.norm_ofNat]
-  have hn : ‖b-a‖ ≤ 2*‖z-w‖ := by
-    simpa only [a,b,norm_sub_rev,sub_sub_sub_cancel_right] using
-      clamp_norm_sub (ε:=ε) hσ (z-(u:ℂ)) (w-(u:ℂ))
-  calc
-    2*‖b-a‖/‖a*b‖ ≤ (2*(2*‖z-w‖))/ε^2 :=
-      div_le_div₀ (by positivity) (mul_le_mul_of_nonneg_left hn (by norm_num))
-        (sq_pos_of_pos hε) hd
-    _ = (4/ε^2)*‖z-w‖ := by ring
-
-lemma velocity_lipschitz {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (u : ℝ) :
-    LipschitzWith ⟨4/ε^2,by positivity⟩ (velocity σ ε u) := by
-  apply LipschitzWith.of_dist_le_mul
-  intro z w
-  simp only [dist_eq_norm]
-  exact norm_velocity_sub_le hσ hε u z w
-
-lemma continuous_velocity {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) :
-    Continuous (fun p : ℝ × ℂ => velocity σ ε p.1 p.2) := by
-  unfold velocity clamp
-  refine continuous_const.div ?_ (fun p => clamp_ne_zero hσ hε _)
-  change Continuous (fun p : ℝ × ℂ => Complex.equivRealProdCLM.symm
-    (σ*max (σ*(p.2-(p.1:ℂ)).re) ε,(p.2-(p.1:ℂ)).im))
-  apply Complex.equivRealProdCLM.symm.continuous.comp
-  apply Continuous.prodMk <;> fun_prop
-
-lemma exists_solution {σ ε T : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (hT : 0 ≤ T)
-    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
-    ∃ H : ℝ → ℂ, H 0 = z ∧ ∀ t ∈ Icc 0 T,
-      HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t := by
-  let L : ℝ≥0 := ⟨2/ε,by positivity⟩
-  let K : ℝ≥0 := ⟨4/ε^2,by positivity⟩
-  let a : ℝ≥0 := ⟨(2/ε)*T,mul_nonneg (by positivity) hT⟩
-  have hp : IsPicardLindelof (fun t => velocity σ ε (U t)) ⟨0,⟨le_rfl,hT⟩⟩ z a 0 L K := by
-    constructor
-    · exact fun t _ => (velocity_lipschitz hσ hε (U t)).lipschitzOnWith
-    · intro w _
-      have hc : ContinuousOn (fun t : ℝ => (U t,w)) (Icc 0 T) :=
-        hU.prodMk continuousOn_const
-      simpa only [Function.comp_def] using (continuous_velocity hσ hε).comp_continuousOn hc
-    · exact fun t _ w _ => norm_velocity_le hσ hε (U t) w
-    · dsimp [L,a]
-      simp only [sub_zero,sub_self,max_eq_left hT]
-      rfl
-  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
-
-lemma norm_velocity_driver_sub_le {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
-    (u v : ℝ) (z : ℂ) : ‖velocity σ ε u z-velocity σ ε v z‖ ≤ (4/ε^2)*|u-v| := by
-  have he (a : ℝ) : velocity σ ε 0 (z-(a:ℂ)) = velocity σ ε a z := by simp [velocity]
-  have hh := norm_velocity_sub_le hσ hε 0 (z-(u:ℂ)) (z-(v:ℂ))
-  rw [he,he] at hh
-  convert hh using 1
-  rw [sub_sub_sub_cancel_left,← Complex.ofReal_sub,Complex.norm_real,abs_sub_comm]
-  simp [Real.norm_eq_abs]
-
-lemma derivWithin_right {σ ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ico 0 T) :
-    HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Ici t) t :=
-  (hH t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
-
-lemma derivWithin_left {σ ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
-    {t : ℝ} (ht : t ∈ Ioc 0 T) :
-    HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Iic t) t :=
-  (hH t ⟨ht.1.le,ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
-
-lemma clamp_conj (σ ε : ℝ) (w : ℂ) :
-    clamp σ ε (starRingEnd ℂ w) = starRingEnd ℂ (clamp σ ε w) := by
-  apply Complex.ext <;> simp [clamp]
-
-lemma velocity_conj (σ ε u : ℝ) (w : ℂ) :
-    velocity σ ε u (starRingEnd ℂ w) = starRingEnd ℂ (velocity σ ε u w) := by
-  unfold velocity
-  rw [map_div₀,map_ofNat,← clamp_conj, map_sub,Complex.conj_ofReal]
-
-lemma conjugate_deriv {σ ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t) :
-    ∀ t ∈ Icc 0 T, HasDerivWithinAt (fun s => starRingEnd ℂ (H s))
-      (velocity σ ε (U t) (starRingEnd ℂ (H t))) (Icc 0 T) t := by
-  intro t ht
-  have hh := Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt t (hH t ht)
-  simpa only [Function.comp_def,Complex.conjCLE_apply,ContinuousLinearEquiv.coe_coe,
-    ← velocity_conj] using hh
-
-lemma positive_height {σ ε T : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
-    {U : ℝ → ℝ} {H : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
-    (h0 : 0 < (H 0).im) : ∀ t ∈ Icc 0 T, 0 < (H t).im := by
-  intro t ht
-  have hn (s : ℝ) (hs : s ∈ Icc 0 T) : (H s).im ≠ 0 := by
-    intro hs0
-    have hr : ∀ t ∈ Icc 0 s, HasDerivWithinAt H
-        (velocity σ ε (U t) (H t)) (Icc 0 s) t := fun t ht =>
-      (hH t ⟨ht.1,ht.2.trans hs.2⟩).mono (Icc_subset_Icc le_rfl hs.2)
-    have hc := conjugate_deriv hr
-    have hf : H s = starRingEnd ℂ (H s) := by
-      apply Complex.ext <;> simp [hs0]
-    have he := ODE_solution_unique_of_mem_Icc_left
-      (s:=fun _ => univ) (fun t _ => (velocity_lipschitz hσ hε (U t)).lipschitzOnWith)
-      (HasDerivWithinAt.continuousOn hr) (fun _ ht => derivWithin_left hr ht)
-      (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hc)
-      (fun _ ht => derivWithin_left hc ht) (fun _ _ => mem_univ _) hf
-    have hh := congrArg Complex.im (he ⟨le_rfl,hs.1⟩)
-    simp only [Complex.conj_im] at hh
-    linarith
-  by_contra hneg
-  have hi := intermediate_value_Icc' ht.1
-    ((Complex.continuous_im.comp_continuousOn (HasDerivWithinAt.continuousOn hH)).mono (Icc_subset_Icc le_rfl ht.2))
-  obtain ⟨s,hs,hs0⟩ := hi (a:=0) ⟨le_of_not_gt hneg,h0.le⟩
-  exact hn s ⟨hs.1,hs.2.trans ht.2⟩ hs0
-
-lemma solution_dist_le {σ ε T D δ : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
-    (hD : 0 ≤ D) (hδ : 0 ≤ δ) {U V : ℝ → ℝ} {H G : ℝ → ℂ}
-    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
-    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity σ ε (V t) (G t)) (Icc 0 T) t)
-    (h0 : dist (H 0) (G 0) ≤ δ) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    dist (H t) (G t) ≤ (δ+D)*Real.exp ((4/ε^2)*T) := by
-  let K : ℝ≥0 := ⟨4/ε^2,by positivity⟩
-  have hK : (0:ℝ) < K := by change 0 < 4/ε^2; positivity
-  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
-      dist (velocity σ ε (V s) (G s)) (velocity σ ε (U s) (G s)) ≤ (K:ℝ)*D := by
-    rw [dist_eq_norm]
-    calc
-      _ ≤ (4/ε^2)*|V s-U s| := norm_velocity_driver_sub_le hσ hε _ _ _
-      _ ≤ (K:ℝ)*D := by
-        rw [abs_sub_comm]
-        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1,hs.2.le⟩) hK.le
-  have hh := dist_le_of_approx_trajectories_ODE
-    (fun t => velocity_lipschitz hσ hε (U t))
-    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
-    (fun t _ => le_of_eq (dist_self (velocity σ ε (U t) (H t))))
-    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
-    hb h0 t ht
-  have he : gronwallBound δ K (0+(K:ℝ)*D) (t-0) =
-      δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) := by
-    simp only [gronwallBound,zero_add,sub_zero,ite_eq_right (ne_of_gt hK)]
-    field_simp
-  rw [he] at hh
-  have hexp : Real.exp ((K:ℝ)*t) ≤ Real.exp ((K:ℝ)*T) :=
-    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
-  apply hh.trans
-  change δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) ≤ (δ+D)*Real.exp ((K:ℝ)*T)
-  have h1 := mul_le_mul_of_nonneg_left hexp hδ
-  have h2 := mul_le_mul_of_nonneg_left hexp hD
-  nlinarith
-
-end SLEExactGauge.BoundaryCutoff
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-structure Gauge where
-  toFun : ℝ≥0 → ℝ≥0
-  continuous : Continuous toFun
-  monotone : Monotone toFun
-  zero : toFun 0 = 0
-  positive : ∀ r, 0 < r → 0 < toFun r
-
-instance : CoeFun Gauge (fun _ => ℝ≥0 → ℝ≥0) := ⟨Gauge.toFun⟩
-
-def Gauge.extended (h : Gauge) (r : ℝ≥0∞) : ℝ≥0∞ :=
-  if r = ∞ then ∞ else (h r.toNNReal : ℝ≥0∞)
-
-def Gauge.hausdorff (h : Gauge) : Measure ℂ :=
-  Measure.mkMetric h.extended
-
-def IsExactGauge (κ : ℝ) (h : Gauge) : Prop :=
-  ∃ r₀ : ℝ≥0, 0 < r₀ ∧ ∀ r : ℝ≥0, 0 < r → r < r₀ →
-    (h r : ℝ) = (r : ℝ) ^ dimension κ *
-      (Real.log (Real.log (1 / (r : ℝ)))) ^ (codimension κ / 2)
-
-def IsLargerGauge (κ : ℝ) (g : Gauge) : Prop :=
-  ∃ r₀ : ℝ≥0, 0 < r₀ ∧ ∀ r : ℝ≥0, 0 < r → r < r₀ →
-    (g r : ℝ) = (r : ℝ) ^ dimension κ * Real.log (Real.log (1 / (r : ℝ)))
-
-def halfPlane : Set ℂ := {z | 0 < z.im}
-
-def localizationSegment (γ : ℝ → ℂ) (s t : ℝ) : Set ℂ := γ '' Icc s t
-
-def localizationWholeTrace (γ : ℝ → ℂ) : Set ℂ := γ '' Ici 0
-
-def rectangle (m : ℕ) : Set ℂ :=
-  {z | -(m : ℝ) ≤ z.re ∧ z.re ≤ m ∧ 0 ≤ z.im ∧ z.im ≤ m}
-
-def localizationDomain (γ : ℝ → ℂ) (t : ℝ) : Set ℂ :=
-  {z | z ∈ halfPlane \ localizationSegment γ 0 t ∧
-    ¬ Bornology.IsBounded (connectedComponentIn (halfPlane \ localizationSegment γ 0 t) z)}
-
-def IsLoewnerTrace (driver : ℝ → ℝ) (γ : ℝ → ℂ) : Prop :=
-  ContinuousOn γ (Ici 0) ∧ γ 0 = 0 ∧
-  (∀ t, 0 ≤ t → 0 ≤ (γ t).im) ∧
-  Tendsto γ atTop (Bornology.cobounded ℂ) ∧
-  ∃ g f : ℝ → ℂ → ℂ,
-    (∀ z ∈ halfPlane, g 0 z = z) ∧
-    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (g t) (localizationDomain γ t)) ∧
-    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (f t) halfPlane) ∧
-    (∀ t, 0 ≤ t → MapsTo (g t) (localizationDomain γ t) halfPlane) ∧
-    (∀ t, 0 ≤ t → MapsTo (f t) halfPlane (localizationDomain γ t)) ∧
-    (∀ t, 0 ≤ t → ∀ z ∈ localizationDomain γ t, f t (g t z) = z) ∧
-    (∀ t, 0 ≤ t → ∀ z ∈ halfPlane, g t (f t z) = z) ∧
-    (∀ z ∈ halfPlane,
-      ContinuousOn (fun t => g t z) {t | 0 ≤ t ∧ z ∈ localizationDomain γ t}) ∧
-    (∀ t, 0 < t → ∀ z ∈ localizationDomain γ t,
-      HasDerivAt (fun u => g u z) (2 / (g t z - (driver t : ℂ))) t) ∧
-    (∀ t, 0 ≤ t → Tendsto
-      (fun y : ℝ => f t ((driver t : ℂ) + (y : ℂ) * Complex.I))
-      (𝓝[>] 0) (𝓝 (γ t)))
-
-def IsChordalSLE {Ω : Type*} [MeasurableSpace Ω]
-    (κ : ℝ) (P : Measure Ω) (γ : Ω → ℝ → ℂ) : Prop :=
-  (∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P) ∧
-  ∃ B : ℝ≥0 → Ω → ℝ, IsBrownianReal B P ∧
-    ∀ᵐ ω ∂P, IsLoewnerTrace (fun t => Real.sqrt κ * B t.toNNReal ω) (γ ω)
-
-def LocalizationMainEvent (h : Gauge) (γ : ℝ → ℂ) : Prop :=
-  (∀ s t : ℝ, 0 < s → s < t →
-    0 < h.hausdorff (localizationSegment γ s t) ∧ h.hausdorff (localizationSegment γ s t) < ∞) ∧
-  (∀ m : ℕ, 0 < m → h.hausdorff (localizationWholeTrace γ ∩ rectangle m) < ∞)
-
-def LocalizationMainConclusion {Ω : Type*} [MeasurableSpace Ω]
-    (P : Measure Ω) (γ : Ω → ℝ → ℂ) (h : Gauge) : Prop :=
-  (∀ᵐ ω ∂P, LocalizationMainEvent h (γ ω)) ∧
-  (∀ R : ℝ, 0 < R → ∃ C : ℝ≥0,
-    (∫⁻ ω, h.hausdorff (localizationWholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P) ≤ C)
-
-def LargerGaugeConclusion (g : Gauge) (γ : ℝ → ℂ) : Prop :=
-  ∀ s t : ℝ, 0 < s → s < t →
-    ¬ SigmaFinite (g.hausdorff.restrict (localizationSegment γ s t)) ∧
-      g.hausdorff (localizationSegment γ s t) = ∞
-
-def IsLoewnerTraceUsing (driver : ℝ → ℝ) (γ : ℝ → ℂ)
-    (g f : ℝ → ℂ → ℂ) : Prop :=
-  ContinuousOn γ (Ici 0) ∧ γ 0 = 0 ∧
-  (∀ t, 0 ≤ t → 0 ≤ (γ t).im) ∧
-  Tendsto γ atTop (Bornology.cobounded ℂ) ∧
-    (∀ z ∈ halfPlane, g 0 z = z) ∧
-    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (g t) (localizationDomain γ t)) ∧
-    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (f t) halfPlane) ∧
-    (∀ t, 0 ≤ t → MapsTo (g t) (localizationDomain γ t) halfPlane) ∧
-    (∀ t, 0 ≤ t → MapsTo (f t) halfPlane (localizationDomain γ t)) ∧
-    (∀ t, 0 ≤ t → ∀ z ∈ localizationDomain γ t, f t (g t z) = z) ∧
-    (∀ t, 0 ≤ t → ∀ z ∈ halfPlane, g t (f t z) = z) ∧
-    (∀ z ∈ halfPlane,
-      ContinuousOn (fun t => g t z) {t | 0 ≤ t ∧ z ∈ localizationDomain γ t}) ∧
-    (∀ t, 0 < t → ∀ z ∈ localizationDomain γ t,
-      HasDerivAt (fun u => g u z) (2 / (g t z - (driver t : ℂ))) t) ∧
-    (∀ t, 0 ≤ t → Tendsto
-      (fun y : ℝ => f t ((driver t : ℂ) + (y : ℂ) * Complex.I))
-      (𝓝[>] 0) (𝓝 (γ t)))
-
-def loewnerRadius (g : ℝ → ℂ → ℂ) (t : ℝ) (z : ℂ) : ℝ :=
-  2 * (g t z).im / ‖deriv (g t) z‖
-
-def loewnerAngleSine (κ : ℝ) (B : ℝ≥0 → ℝ)
-    (g : ℝ → ℂ → ℂ) (t : ℝ) (z : ℂ) : ℝ :=
-  (g t z).im / ‖g t z - (Real.sqrt κ * B t.toNNReal : ℂ)‖
-
-def firstLiveRadiusStop (γ : ℝ → ℂ) (g : ℝ → ℂ → ℂ)
-    (z : ℂ) (u : ℝ) : ℝ≥0∞ :=
-  sInf ((fun t : ℝ≥0 => (t : ℝ≥0∞)) ''
-    {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
-      loewnerRadius g t z = (2 * z.im) * Real.exp (-u)})
-
-def finiteStoppedDensity (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ → ℂ)
-    (g : ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) : ℝ≥0∞ :=
-  let τ := firstLiveRadiusStop γ g z u
-  if τ = ∞ then 0 else
-    ENNReal.ofReal ((loewnerRadius g τ.toReal z) ^ (-codimension κ) *
-      (loewnerAngleSine κ B g τ.toReal z) ^ (8 / κ - 1))
-
-lemma halfPlane_open : IsOpen halfPlane :=
-  isOpen_lt continuous_const Complex.continuous_im
-
-lemma halfPlane_preconnected : IsPreconnected halfPlane :=
-  (convex_halfSpace_im_gt 0).isPreconnected
-
-lemma halfPlane_not_bounded : ¬ Bornology.IsBounded halfPlane := by
-  intro hb
-  obtain ⟨R, hR, hbound⟩ := hb.exists_pos_norm_lt
-  have hm : ((R : ℂ) * Complex.I) ∈ halfPlane := by
-    simpa [halfPlane] using hR
-  have hh := hbound _ hm
-  rw [norm_mul, Complex.norm_real, Complex.norm_I, mul_one, Real.norm_eq_abs,
-    abs_of_pos hR] at hh
-  exact lt_irrefl _ hh
-
-lemma survivingDomain_mono {γ : ℝ → ℂ} {s t : ℝ} (hst : s ≤ t) :
-    localizationDomain γ t ⊆ localizationDomain γ s := by
-  have hsub : halfPlane \ localizationSegment γ 0 t ⊆ halfPlane \ localizationSegment γ 0 s := by
-    apply sdiff_subset_sdiff_right
-    exact image_mono (Icc_subset_Icc le_rfl hst)
-  intro z hz
-  exact ⟨hsub hz.1, fun hb => hz.2 (hb.subset (connectedComponentIn_mono z hsub))⟩
-
-lemma survivingDomain_zero {γ : ℝ → ℂ} (hγ : γ 0 = 0) :
-    localizationDomain γ 0 = halfPlane := by
-  have hset : halfPlane \ localizationSegment γ 0 0 = halfPlane := by
-    simp only [localizationSegment, Icc_self, image_singleton, hγ]
-    apply sdiff_singleton_eq_self
-    simp [halfPlane]
-  ext z
-  constructor
-  · exact fun h => h.1.1
-  · intro hz
-    refine ⟨by simpa only [hset] using hz, ?_⟩
-    rw [hset]
-    intro hb
-    exact halfPlane_not_bounded (hb.subset
-      (halfPlane_preconnected.subset_connectedComponentIn hz subset_rfl))
-
-lemma survivingDomain_open {γ : ℝ → ℂ} (hγ : ContinuousOn γ (Ici 0))
-    {t : ℝ} (_ht : 0 ≤ t) : IsOpen (localizationDomain γ t) := by
-  have hc : IsCompact (localizationSegment γ 0 t) :=
-    isCompact_Icc.image_of_continuousOn (hγ.mono (fun _ ha => ha.1))
-  have hF : IsOpen (halfPlane \ localizationSegment γ 0 t) := halfPlane_open.sdiff hc.isClosed
-  apply isOpen_iff_mem_nhds.2
-  intro z hz
-  have ho := hF.connectedComponentIn (x := z)
-  have hn := ho.mem_nhds (mem_connectedComponentIn hz.1)
-  apply Filter.mem_of_superset hn
-  intro w hw
-  refine ⟨connectedComponentIn_subset _ _ hw, ?_⟩
-  rw [← connectedComponentIn_eq hw]
-  exact hz.2
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-namespace IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-lemma initial (h : IsLoewnerTraceUsing driver γ g f) :
-    ∀ z ∈ halfPlane, g 0 z = z := h.2.2.2.2.1
-
-lemma continuous_eval (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (_hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun t => g t z) (Icc 0 T) := by
-  rcases h with ⟨hγ, hzero, him, htrans, hinit, hg, hf, hgmap, hfmap, hfg, hgf, hc, hd, htip⟩
-  apply (hc z hz.1.1).mono
-  intro t ht
-  exact ⟨ht.1, survivingDomain_mono ht.2 hz⟩
-
-lemma im_pos (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    0 < (g t z).im := h.2.2.2.2.2.2.2.1 t ht hz
-
-lemma analytic_eval (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    AnalyticAt ℂ (g t) z := h.2.2.2.2.2.1 t ht z hz
-
-lemma deriv_initial (h : IsLoewnerTraceUsing driver γ g f)
-    {z : ℂ} (hz : z ∈ halfPlane) : deriv (g 0) z = 1 := by
-  have heq : g 0 =ᶠ[𝓝 z] id := by
-    filter_upwards [halfPlane_open.mem_nhds hz] with w hw
-    exact h.initial w hw
-  rw [heq.deriv_eq]
-  exact deriv_id z
-
-lemma hasDerivAt_eval (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 < t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    HasDerivAt (fun u => g u z) (2 / (g t z - (driver t : ℂ))) t := by
-  rcases h with ⟨hγ, hzero, him, htrans, hinit, hg, hf, hgmap, hfmap, hfg, hgf, hc, hd, htip⟩
-  exact hd t ht z hz
-
-lemma deriv_ne_zero (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    deriv (g t) z ≠ 0 := by
-  rcases h with ⟨hγ, hzero, him, htrans, hinit, hg, hf, hgmap, hfmap, hfg, hgf, hc, hd, htip⟩
-  have hgd := (hg t ht z hz).differentiableAt.hasDerivAt
-  have hfd := (hf t ht _ (hgmap t ht hz)).differentiableAt.hasDerivAt
-  have heq : (fun w => f t (g t w)) =ᶠ[𝓝 z] id := by
-    filter_upwards [(survivingDomain_open hγ ht).mem_nhds hz] with w hw
-    exact hfg t ht w hw
-  have hh := (hfd.comp z hgd).congr_of_eventuallyEq heq.symm
-  have he := hh.unique (hasDerivAt_id z)
-  intro hn
-  simp [hn] at he
-
-lemma radius_pos (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    0 < loewnerRadius g t z := by
-  unfold loewnerRadius
-  exact div_pos (mul_pos (by norm_num) (h.im_pos ht hz))
-    (norm_pos_iff.2 (h.deriv_ne_zero ht hz))
-
-lemma radius_initial (h : IsLoewnerTraceUsing driver γ g f)
-    {z : ℂ} (hz : z ∈ halfPlane) : loewnerRadius g 0 z = 2 * z.im := by
-  simp [loewnerRadius, h.initial z hz, h.deriv_initial hz]
-
-lemma centered_ne_zero (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    g t z - (driver t : ℂ) ≠ 0 := by
-  intro he
-  have hi := congrArg Complex.im he
-  have hp := h.im_pos ht hz
-  simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hi
-  exact hp.ne' hi
-
-lemma hasDerivAt_height (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 < t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    HasDerivAt (fun s => (g s z).im)
-      (-2 * (g t z).im / Complex.normSq (g t z - (driver t : ℂ))) t := by
-  have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t (h.hasDerivAt_eval ht hz)
-  simpa [Complex.div_im, Function.comp_def, neg_div] using hh
-
-lemma height_antitone (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    AntitoneOn (fun t => (g t z).im) (Icc 0 T) := by
-  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T)
-    (Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz))
-    (f' := fun t => -2 * (g t z).im / Complex.normSq (g t z - (driver t : ℂ)))
-  · intro t ht
-    rw [interior_Icc] at ht
-    exact (h.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hz)).hasDerivWithinAt
-  · intro t ht
-    rw [interior_Icc] at ht
-    exact div_nonpos_of_nonpos_of_nonneg
-      (mul_nonpos_of_nonpos_of_nonneg (by norm_num)
-        (h.im_pos ht.1.le (survivingDomain_mono ht.2.le hz)).le)
-      (Complex.normSq_nonneg _)
-
-lemma height_le_initial (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    (g t z).im ≤ z.im := by
-  have hh := h.height_antitone ht hz ⟨le_rfl, ht⟩ ⟨ht, le_rfl⟩ ht
-  simpa only [h.initial z hz.1.1] using hh
-
-lemma continuous_velocity (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun t => 2 / (g t z - (driver t : ℂ))) (Icc 0 T) := by
-  apply continuousOn_const.div
-    ((h.continuous_eval hT hz).sub (Complex.continuous_ofReal.comp_continuousOn hU))
-  intro t ht
-  exact h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz)
-
-lemma integral_velocity (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    {a b : ℝ} (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T) (hab : a ≤ b) :
-    (∫ t in a..b, 2 / (g t z - (driver t : ℂ))) = g b z - g a z := by
-  have hsub : Icc a b ⊆ Icc 0 T := Icc_subset_Icc ha.1 hb.2
-  apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hab
-    ((h.continuous_eval hT hz).mono hsub)
-  · intro t ht
-    exact h.hasDerivAt_eval (lt_of_le_of_lt ha.1 ht.1)
-      (survivingDomain_mono (ht.2.le.trans hb.2) hz)
-  · apply ContinuousOn.intervalIntegrable
-    simpa only [uIcc_of_le hab] using (h.continuous_velocity hT hU hz).mono hsub
-
-lemma norm_velocity_le (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    {t : ℝ} (ht : t ∈ Icc 0 T) :
-    ‖2 / (g t z - (driver t : ℂ))‖ ≤ 2 / (g T z).im := by
-  have hY := h.im_pos hT hz
-  have hm : (g T z).im ≤ (g t z).im :=
-    h.height_antitone hT hz ht ⟨hT, le_rfl⟩ ht.2
-  have hn : (g t z).im ≤ ‖g t z - (driver t : ℂ)‖ := by
-    have hh := Complex.im_le_norm (g t z - (driver t : ℂ))
-    simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hh
-  rw [norm_div, Complex.norm_ofNat]
-  exact div_le_div_of_nonneg_left (by norm_num) hY (hm.trans hn)
-
-lemma norm_time_sub_le (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T)
-    {a b : ℝ} (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T) :
-    ‖g b z - g a z‖ ≤ (2 / (g T z).im) * |b - a| := by
-  wlog hab : a ≤ b generalizing a b
-  · have hh := this hb ha (le_of_not_ge hab)
-    simpa only [norm_sub_rev, abs_sub_comm] using hh
-  rw [← h.integral_velocity hT hU hz ha hb hab]
-  exact intervalIntegral.norm_integral_le_of_norm_le_const fun t ht =>
-    h.norm_velocity_le hT hz ⟨ha.1.trans (uIoc_of_le hab ▸ ht).1.le,
-      (uIoc_of_le hab ▸ ht).2.trans hb.2⟩
-
-end IsLoewnerTraceUsing
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge
-
-lemma firstLiveRadiusStop_zero {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-    (h : IsLoewnerTraceUsing driver γ g f) {z : ℂ} (hz : z ∈ halfPlane) :
-    firstLiveRadiusStop γ g z 0 = 0 := by
-  apply le_antisymm ?_ bot_le
-  apply sInf_le
-  refine ⟨0, ⟨?_, ?_⟩, rfl⟩
-  · simpa only [NNReal.coe_zero, survivingDomain_zero h.2.1] using hz
-  · simpa only [NNReal.coe_zero, neg_zero, Real.exp_zero, mul_one] using h.radius_initial hz
-
-end SLEExactGauge
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory ProbabilityTheory
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.IsLoewnerTraceUsing
-variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
-
-def differenceCoefficient (driver : ℝ → ℝ) (g : ℝ → ℂ → ℂ) (w z : ℂ) (t : ℝ) : ℂ :=
-  -2 / ((g t w - (driver t : ℂ)) * (g t z - (driver t : ℂ)))
-
-lemma continuous_differenceCoefficient (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {w z : ℂ} (hw : w ∈ localizationDomain γ T) (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (differenceCoefficient driver g w z) (Icc 0 T) := by
-  apply continuousOn_const.div
-    (((h.continuous_eval hT hw).sub (Complex.continuous_ofReal.comp_continuousOn hU)).mul
-      ((h.continuous_eval hT hz).sub (Complex.continuous_ofReal.comp_continuousOn hU)))
-  intro t ht
-  exact mul_ne_zero (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hw))
-    (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz))
-
-lemma difference_eq_exp_integral (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {w z : ℂ} (hw : w ∈ localizationDomain γ T) (hz : z ∈ localizationDomain γ T) :
-    g T w - g T z = (w-z) * Complex.exp
-      (∫ s in (0 : ℝ)..T, differenceCoefficient driver g w z s) := by
-  let a := differenceCoefficient driver g w z
-  let A : ℝ → ℂ := fun t => ∫ s in (0 : ℝ)..t, a s
-  have ha : ContinuousOn a (Icc 0 T) := h.continuous_differenceCoefficient hT hU hw hz
-  have hi : IntervalIntegrable a volume 0 T := ha.intervalIntegrable_of_Icc hT
-  have hAc : ContinuousOn A (Icc 0 T) := by
-    simpa only [uIcc_of_le hT, A] using
-      intervalIntegral.continuousOn_primitive_interval' hi (left_mem_uIcc : (0 : ℝ) ∈ uIcc 0 T)
-  have hAd {t : ℝ} (ht : t ∈ Ioo 0 T) : HasDerivAt A (a t) t := by
-    apply intervalIntegral.integral_hasDerivAt_right
-      (hi.mono_set (by rw [uIcc_of_le ht.1.le, uIcc_of_le hT]; exact Icc_subset_Icc le_rfl ht.2.le))
-    · have hh := ha.stronglyMeasurableAtFilter_nhdsWithin (μ := volume) measurableSet_Icc t
-      rwa [nhdsWithin_eq_nhds.mpr (Icc_mem_nhds ht.1 ht.2)] at hh
-    · exact ha.continuousAt (Icc_mem_nhds ht.1 ht.2)
-  let R : ℝ → ℂ := fun t => (g t w - g t z) * Complex.exp (-A t)
-  have hRc : ContinuousOn R (Icc 0 T) := by
-    exact ((h.continuous_eval hT hw).sub (h.continuous_eval hT hz)).mul
-      (Complex.continuous_exp.comp_continuousOn hAc.neg)
-  have hRd (t : ℝ) (ht : t ∈ Ioo 0 T) : HasDerivAt R 0 t := by
-    have hd := ((h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hw)).sub
-      (h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hz))).mul (hAd ht).neg.cexp
-    apply hd.congr_deriv
-    have hw' := h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hw)
-    have hz' := h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hz)
-    dsimp [a, differenceCoefficient]
-    field_simp
-    ring
-  have hr := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hT hRc hRd
-    (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => (0 : ℂ)) volume 0 T)
-  have hR : R T = w-z := by
-    apply sub_eq_zero.1
-    simpa only [intervalIntegral.integral_zero, R, A, intervalIntegral.integral_same,
-      neg_zero, Complex.exp_zero, mul_one, h.initial w hw.1.1, h.initial z hz.1.1,
-      ] using hr.symm
-  have hexp := Complex.exp_ne_zero (-A T)
-  dsimp [R] at hR
-  calc
-    g T w - g T z = (g T w - g T z) * (Complex.exp (-A T) * Complex.exp (A T)) := by
-      rw [← Complex.exp_add, neg_add_cancel, Complex.exp_zero, mul_one]
-    _ = (w-z) * Complex.exp (A T) := by rw [← mul_assoc, hR]
-
-lemma continuousAt_integral_differenceCoefficient (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousAt (fun w => ∫ s in (0 : ℝ)..T, differenceCoefficient driver g w z s) z := by
-  let ε := (g T z).im / 2
-  have hε : 0 < ε := half_pos (h.im_pos hT hz)
-  have hev : ∀ᶠ w in 𝓝 z, w ∈ localizationDomain γ T ∧ ε ≤ (g T w).im := by
-    have hc := Complex.continuous_im.continuousAt.comp (h.analytic_eval hT hz).continuousAt
-    have hlt : ε < (g T z).im := by dsimp [ε]; linarith [h.im_pos hT hz]
-    filter_upwards [(survivingDomain_open h.1 hT).mem_nhds hz,
-      hc.eventually (Ioi_mem_nhds hlt)] with w hw hh
-    exact ⟨hw, hh.le⟩
-  apply intervalIntegral.continuousAt_of_dominated_interval (bound := fun _ => 2 / ε^2)
-  · filter_upwards [hev] with w hw
-    have hc := h.continuous_differenceCoefficient hT hU hw.1 hz
-    apply ContinuousOn.aestronglyMeasurable _ measurableSet_uIoc
-    exact hc.mono (by rw [uIoc_of_le hT]; exact Ioc_subset_Icc_self)
-  · filter_upwards [hev] with w hw
-    apply ae_of_all
-    intro t ht
-    rw [uIoc_of_le hT] at ht
-    have hwheight : ε ≤ (g t w).im :=
-      hw.2.trans (h.height_antitone hT hw.1 ⟨ht.1.le, ht.2⟩ ⟨hT, le_rfl⟩ ht.2)
-    have hzheight : ε ≤ (g t z).im := by
-      have hhalf : ε ≤ (g T z).im := by dsimp [ε]; linarith [h.im_pos hT hz]
-      exact hhalf.trans (h.height_antitone hT hz ⟨ht.1.le, ht.2⟩ ⟨hT, le_rfl⟩ ht.2)
-    have hwn : ε ≤ ‖g t w - (driver t : ℂ)‖ := hwheight.trans (by
-      simpa using Complex.im_le_norm (g t w - (driver t : ℂ)))
-    have hzn : ε ≤ ‖g t z - (driver t : ℂ)‖ := hzheight.trans (by
-      simpa using Complex.im_le_norm (g t z - (driver t : ℂ)))
-    simp only [differenceCoefficient, norm_div, norm_neg, Complex.norm_ofNat, norm_mul]
-    apply div_le_div_of_nonneg_left (by norm_num) (sq_pos_of_pos hε)
-    simpa only [pow_two] using mul_le_mul hwn hzn hε.le (norm_nonneg _)
-  · exact intervalIntegrable_const
-  · apply ae_of_all
-    intro t ht
-    rw [uIoc_of_le hT] at ht
-    have hzt := survivingDomain_mono ht.2 hz
-    exact continuousAt_const.div
-      (((h.analytic_eval ht.1.le hzt).continuousAt.sub continuousAt_const).mul continuousAt_const)
-      (mul_ne_zero (h.centered_ne_zero ht.1.le hzt) (h.centered_ne_zero ht.1.le hzt))
-
-lemma spatial_deriv_eq_exp_integral (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    deriv (g T) z = Complex.exp (∫ s in (0 : ℝ)..T,
-      -2 / (g s z - (driver s : ℂ))^2) := by
-  have hd : HasDerivAt (g T) (Complex.exp (∫ s in (0 : ℝ)..T,
-      differenceCoefficient driver g z z s)) z := by
-    apply hasDerivAt_iff_tendsto_slope.2
-    have hc := Complex.continuous_exp.continuousAt.comp
-      (h.continuousAt_integral_differenceCoefficient hT hU hz)
-    apply (hc.mono_left nhdsWithin_le_nhds).congr'
-    filter_upwards [self_mem_nhdsWithin,
-      mem_nhdsWithin_of_mem_nhds ((survivingDomain_open h.1 hT).mem_nhds hz)]
-      with w hwne hw
-    have hwz : w ≠ z := hwne
-    dsimp only [slope, vsub_eq_sub, Function.comp_def]
-    rw [h.difference_eq_exp_integral hT hU hw hz]
-    simp only [smul_eq_mul, ← mul_assoc, inv_mul_cancel₀ (sub_ne_zero.2 hwz), one_mul]
-  have he : (fun s => differenceCoefficient driver g z z s) =
-      (fun s => -2 / (g s z - (driver s : ℂ))^2) := by
-    funext s
-    simp only [differenceCoefficient, pow_two]
-  simpa only [he] using hd.deriv
-
-lemma log_deriv_norm_eq (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    Real.log ‖deriv (g T) z‖ =
-      (∫ s in (0 : ℝ)..T, -2 / (g s z - (driver s : ℂ))^2).re := by
-  rw [h.spatial_deriv_eq_exp_integral hT hU hz, Complex.norm_exp, Real.log_exp]
-
-lemma continuous_spatialCoefficient (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun s => -2 / (g s z - (driver s : ℂ))^2) (Icc 0 T) := by
-  have hc := h.continuous_differenceCoefficient hT hU hz hz
-  unfold differenceCoefficient at hc
-  simpa only [pow_two] using hc
-
-lemma continuous_log_deriv_norm (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun t => Real.log ‖deriv (g t) z‖) (Icc 0 T) := by
-  have hi : IntervalIntegrable (fun s => -2 / (g s z - (driver s : ℂ))^2) volume 0 T :=
-    (h.continuous_spatialCoefficient hT hU hz).intervalIntegrable_of_Icc hT
-  have hc := Complex.continuous_re.comp_continuousOn
-    (intervalIntegral.continuousOn_primitive_interval' hi left_mem_uIcc)
-  rw [uIcc_of_le hT] at hc
-  apply hc.congr
-  intro t ht
-  exact h.log_deriv_norm_eq ht.1
-    (hU.mono (Icc_subset_Icc le_rfl ht.2)) (survivingDomain_mono ht.2 hz)
-
-lemma hasDerivAt_log_deriv_norm (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) {t : ℝ} (ht : t ∈ Ioo 0 T) :
-    HasDerivAt (fun s => Real.log ‖deriv (g s) z‖)
-      (-2 / (g t z - (driver t : ℂ))^2).re t := by
-  have hc := h.continuous_spatialCoefficient hT hU hz
-  have hi : IntervalIntegrable (fun s => -2 / (g s z - (driver s : ℂ))^2) volume 0 t :=
-    (hc.intervalIntegrable_of_Icc hT).mono_set (by
-      rw [uIcc_of_le ht.1.le, uIcc_of_le hT]
-      exact Icc_subset_Icc le_rfl ht.2.le)
-  have hm := hc.stronglyMeasurableAtFilter_nhdsWithin (μ := volume) measurableSet_Icc t
-  rw [nhdsWithin_eq_nhds.mpr (Icc_mem_nhds ht.1 ht.2)] at hm
-  have hd := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t
-    (intervalIntegral.integral_hasDerivAt_right hi hm
-      (hc.continuousAt (Icc_mem_nhds ht.1 ht.2)))
-  apply hd.congr_of_eventuallyEq
-  filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
-  exact h.log_deriv_norm_eq hs.1 (hU.mono (Icc_subset_Icc le_rfl hs.2))
-    (survivingDomain_mono hs.2 hz)
-
-lemma real_spatialCoefficient (w : ℂ) :
-    (-2 / w^2).re = -2 * (w.re^2-w.im^2) / (Complex.normSq w)^2 := by
-  simp only [Complex.div_re, Complex.neg_re, Complex.re_ofNat, Complex.neg_im,
-    Complex.im_ofNat, neg_zero, zero_mul, map_pow]
-  simp only [pow_two, Complex.mul_re]
-  ring
-
-lemma log_radius_eq (h : IsLoewnerTraceUsing driver γ g f)
-    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
-    Real.log (loewnerRadius g t z) = Real.log 2 + Real.log (g t z).im -
-      Real.log ‖deriv (g t) z‖ := by
-  rw [loewnerRadius, Real.log_div (mul_ne_zero (by norm_num) (h.im_pos ht hz).ne')
-    (norm_ne_zero_iff.2 (h.deriv_ne_zero ht hz)),
-    Real.log_mul (by norm_num) (h.im_pos ht hz).ne']
-
-lemma continuous_log_radius (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun t => Real.log (loewnerRadius g t z)) (Icc 0 T) := by
-  have hc := ((Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)).log
-    (fun t ht => (h.im_pos ht.1 (survivingDomain_mono ht.2 hz)).ne')).const_add (Real.log 2)
-  apply (hc.sub (h.continuous_log_deriv_norm hT hU hz)).congr
-  intro t ht
-  exact h.log_radius_eq ht.1 (survivingDomain_mono ht.2 hz)
-
-lemma hasDerivAt_log_radius (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) {t : ℝ} (ht : t ∈ Ioo 0 T) :
-    HasDerivAt (fun s => Real.log (loewnerRadius g s z))
-      (-4 * (g t z).im^2 / (Complex.normSq (g t z - (driver t : ℂ)))^2) t := by
-  have hzt := survivingDomain_mono ht.2.le hz
-  have hY := h.im_pos ht.1.le hzt
-  have hd := (((h.hasDerivAt_height ht.1 hzt).log hY.ne').const_add (Real.log 2)).sub
-    (h.hasDerivAt_log_deriv_norm hT hU hz ht)
-  have he : HasDerivAt (fun s => Real.log (loewnerRadius g s z))
-      ((-2 * (g t z).im / Complex.normSq (g t z - (driver t : ℂ))) / (g t z).im -
-        (-2 / (g t z - (driver t : ℂ))^2).re) t := by
-    apply hd.congr_of_eventuallyEq
-    filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
-    exact h.log_radius_eq hs.1 (survivingDomain_mono hs.2 hz)
-  apply he.congr_deriv
-  rw [real_spatialCoefficient]
-  simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
-  rw [Complex.normSq_apply]
-  simp only [Complex.sub_re, Complex.ofReal_re, Complex.sub_im, Complex.ofReal_im, sub_zero]
-  field_simp
-  ring
-
-lemma continuous_radius (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    ContinuousOn (fun t => loewnerRadius g t z) (Icc 0 T) := by
-  apply (Real.continuous_exp.comp_continuousOn (h.continuous_log_radius hT hU hz)).congr
-  intro t ht
-  exact (Real.exp_log (h.radius_pos ht.1 (survivingDomain_mono ht.2 hz))).symm
-
-lemma log_radius_strictAnti (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    StrictAntiOn (fun t => Real.log (loewnerRadius g t z)) (Icc 0 T) := by
-  apply strictAntiOn_of_hasDerivWithinAt_neg (convex_Icc 0 T)
-    (h.continuous_log_radius hT hU hz)
-    (f' := fun t => -4 * (g t z).im^2 / (Complex.normSq (g t z - (driver t : ℂ)))^2)
-  · intro t ht
-    rw [interior_Icc] at ht
-    exact (h.hasDerivAt_log_radius hT hU hz ht).hasDerivWithinAt
-  · intro t ht
-    rw [interior_Icc] at ht
-    have hzt := survivingDomain_mono ht.2.le hz
-    have hy := h.im_pos ht.1.le hzt
-    have hq := Complex.normSq_pos.2 (h.centered_ne_zero ht.1.le hzt)
-    exact div_neg_of_neg_of_pos (mul_neg_of_neg_of_pos (by norm_num) (sq_pos_of_pos hy))
-      (sq_pos_of_pos hq)
-
-lemma radius_strictAnti (h : IsLoewnerTraceUsing driver γ g f)
-    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
-    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
-    StrictAntiOn (fun t => loewnerRadius g t z) (Icc 0 T) := by
-  intro s hs t ht hst
-  exact (Real.log_lt_log_iff (h.radius_pos ht.1 (survivingDomain_mono ht.2 hz))
-    (h.radius_pos hs.1 (survivingDomain_mono hs.2 hz))).1
-      (h.log_radius_strictAnti hT hU hz hs ht hst)
-
-lemma live_radius_injective (h : IsLoewnerTraceUsing driver γ g f)
-    (hU : ContinuousOn driver (Ici 0)) {z : ℂ} {s t : ℝ}
-    (hs : 0 ≤ s) (ht : 0 ≤ t) (hzs : z ∈ localizationDomain γ s)
-    (hzt : z ∈ localizationDomain γ t) (he : loewnerRadius g s z = loewnerRadius g t z) :
-    s = t := by
-  rcases le_total s t with hst | hts
-  · exact (h.radius_strictAnti ht (hU.mono Icc_subset_Ici_self) hzt).injOn
-      ⟨hs, hst⟩ ⟨ht, le_rfl⟩ he
-  · exact (h.radius_strictAnti hs (hU.mono Icc_subset_Ici_self) hzs).injOn
-      ⟨hs, le_rfl⟩ ⟨ht, hts⟩ he
-
-lemma firstLiveRadiusStop_eq (h : IsLoewnerTraceUsing driver γ g f)
-    (hU : ContinuousOn driver (Ici 0)) {z : ℂ} {u : ℝ} {t : ℝ≥0}
-    (hzt : z ∈ localizationDomain γ t)
-    (he : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
-    firstLiveRadiusStop γ g z u = (t : ℝ≥0∞) := by
-  have heq : {s : ℝ≥0 | z ∈ localizationDomain γ s ∧
-      loewnerRadius g s z = (2*z.im)*Real.exp (-u)} = {t} := by
-    ext s
-    constructor
-    · intro hs
-      apply mem_singleton_iff.2
-      apply NNReal.coe_injective
-      exact h.live_radius_injective hU s.coe_nonneg t.coe_nonneg hs.1 hzt (hs.2.trans he.symm)
-    · rintro rfl
-      exact ⟨hzt, he⟩
-  simp only [firstLiveRadiusStop, heq, image_singleton, sInf_singleton]
-
-end SLEExactGauge.IsLoewnerTraceUsing
-
-end
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.SobolevMeans
+import OAI.Probability.SLE.ClippedCoordinates
+
+namespace OAI
+
+
+/-! Conformal localization and Boundary Cutoff estimates. -/
+
+section
+noncomputable section
+open Set Filter Metric
+open scoped Topology NNReal
+namespace SLEExactGauge.BoundaryCutoff
+
+lemma norm_velocity_sub_le {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
+    (u : ℝ) (z w : ℂ) : ‖velocity σ ε u z-velocity σ ε u w‖ ≤ (4/ε^2)*‖z-w‖ := by
+  let a := clamp σ ε (z-(u:ℂ))
+  let b := clamp σ ε (w-(u:ℂ))
+  have ha : a ≠ 0 := clamp_ne_zero hσ hε _
+  have hb : b ≠ 0 := clamp_ne_zero hσ hε _
+  have hd : ε^2 ≤ ‖a*b‖ := by
+    rw [norm_mul,pow_two]
+    exact mul_le_mul (clamp_norm hσ hε _) (clamp_norm hσ hε _) hε.le (norm_nonneg _)
+  have he : velocity σ ε u z-velocity σ ε u w = 2*(b-a)/(a*b) := by
+    change 2/a-2/b=2*(b-a)/(a*b)
+    field_simp
+  rw [he,norm_div,norm_mul,Complex.norm_ofNat]
+  have hn : ‖b-a‖ ≤ 2*‖z-w‖ := by
+    simpa only [a,b,norm_sub_rev,sub_sub_sub_cancel_right] using
+      clamp_norm_sub (ε:=ε) hσ (z-(u:ℂ)) (w-(u:ℂ))
+  calc
+    2*‖b-a‖/‖a*b‖ ≤ (2*(2*‖z-w‖))/ε^2 :=
+      div_le_div₀ (by positivity) (mul_le_mul_of_nonneg_left hn (by norm_num))
+        (sq_pos_of_pos hε) hd
+    _ = (4/ε^2)*‖z-w‖ := by ring
+
+lemma velocity_lipschitz {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (u : ℝ) :
+    LipschitzWith ⟨4/ε^2,by positivity⟩ (velocity σ ε u) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro z w
+  simp only [dist_eq_norm]
+  exact norm_velocity_sub_le hσ hε u z w
+
+lemma continuous_velocity {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) :
+    Continuous (fun p : ℝ × ℂ => velocity σ ε p.1 p.2) := by
+  unfold velocity clamp
+  refine continuous_const.div ?_ (fun p => clamp_ne_zero hσ hε _)
+  change Continuous (fun p : ℝ × ℂ => Complex.equivRealProdCLM.symm
+    (σ*max (σ*(p.2-(p.1:ℂ)).re) ε,(p.2-(p.1:ℂ)).im))
+  apply Complex.equivRealProdCLM.symm.continuous.comp
+  apply Continuous.prodMk <;> fun_prop
+
+lemma exists_solution {σ ε T : ℝ} (hσ : |σ| = 1) (hε : 0 < ε) (hT : 0 ≤ T)
+    (U : ℝ → ℝ) (hU : ContinuousOn U (Icc 0 T)) (z : ℂ) :
+    ∃ H : ℝ → ℂ, H 0 = z ∧ ∀ t ∈ Icc 0 T,
+      HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t := by
+  let L : ℝ≥0 := ⟨2/ε,by positivity⟩
+  let K : ℝ≥0 := ⟨4/ε^2,by positivity⟩
+  let a : ℝ≥0 := ⟨(2/ε)*T,mul_nonneg (by positivity) hT⟩
+  have hp : IsPicardLindelof (fun t => velocity σ ε (U t)) ⟨0,⟨le_rfl,hT⟩⟩ z a 0 L K := by
+    constructor
+    · exact fun t _ => (velocity_lipschitz hσ hε (U t)).lipschitzOnWith
+    · intro w _
+      have hc : ContinuousOn (fun t : ℝ => (U t,w)) (Icc 0 T) :=
+        hU.prodMk continuousOn_const
+      simpa only [Function.comp_def] using (continuous_velocity hσ hε).comp_continuousOn hc
+    · exact fun t _ w _ => norm_velocity_le hσ hε (U t) w
+    · dsimp [L,a]
+      simp only [sub_zero,sub_self,max_eq_left hT]
+      rfl
+  exact hp.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
+
+lemma norm_velocity_driver_sub_le {σ ε : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
+    (u v : ℝ) (z : ℂ) : ‖velocity σ ε u z-velocity σ ε v z‖ ≤ (4/ε^2)*|u-v| := by
+  have he (a : ℝ) : velocity σ ε 0 (z-(a:ℂ)) = velocity σ ε a z := by simp [velocity]
+  have hh := norm_velocity_sub_le hσ hε 0 (z-(u:ℂ)) (z-(v:ℂ))
+  rw [he,he] at hh
+  convert hh using 1
+  rw [sub_sub_sub_cancel_left,← Complex.ofReal_sub,Complex.norm_real,abs_sub_comm]
+  simp [Real.norm_eq_abs]
+
+lemma derivWithin_right {σ ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ico 0 T) :
+    HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Ici t) t :=
+  (hH t ⟨ht.1,ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
+
+lemma derivWithin_left {σ ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
+    {t : ℝ} (ht : t ∈ Ioc 0 T) :
+    HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Iic t) t :=
+  (hH t ⟨ht.1.le,ht.2⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsLE_of_mem ht)
+
+lemma clamp_conj (σ ε : ℝ) (w : ℂ) :
+    clamp σ ε (starRingEnd ℂ w) = starRingEnd ℂ (clamp σ ε w) := by
+  apply Complex.ext <;> simp [clamp]
+
+lemma velocity_conj (σ ε u : ℝ) (w : ℂ) :
+    velocity σ ε u (starRingEnd ℂ w) = starRingEnd ℂ (velocity σ ε u w) := by
+  unfold velocity
+  rw [map_div₀,map_ofNat,← clamp_conj, map_sub,Complex.conj_ofReal]
+
+lemma conjugate_deriv {σ ε T : ℝ} {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t) :
+    ∀ t ∈ Icc 0 T, HasDerivWithinAt (fun s => starRingEnd ℂ (H s))
+      (velocity σ ε (U t) (starRingEnd ℂ (H t))) (Icc 0 T) t := by
+  intro t ht
+  have hh := Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt t (hH t ht)
+  simpa only [Function.comp_def,Complex.conjCLE_apply,ContinuousLinearEquiv.coe_coe,
+    ← velocity_conj] using hh
+
+lemma positive_height {σ ε T : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
+    {U : ℝ → ℝ} {H : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
+    (h0 : 0 < (H 0).im) : ∀ t ∈ Icc 0 T, 0 < (H t).im := by
+  intro t ht
+  have hn (s : ℝ) (hs : s ∈ Icc 0 T) : (H s).im ≠ 0 := by
+    intro hs0
+    have hr : ∀ t ∈ Icc 0 s, HasDerivWithinAt H
+        (velocity σ ε (U t) (H t)) (Icc 0 s) t := fun t ht =>
+      (hH t ⟨ht.1,ht.2.trans hs.2⟩).mono (Icc_subset_Icc le_rfl hs.2)
+    have hc := conjugate_deriv hr
+    have hf : H s = starRingEnd ℂ (H s) := by
+      apply Complex.ext <;> simp [hs0]
+    have he := ODE_solution_unique_of_mem_Icc_left
+      (s:=fun _ => univ) (fun t _ => (velocity_lipschitz hσ hε (U t)).lipschitzOnWith)
+      (HasDerivWithinAt.continuousOn hr) (fun _ ht => derivWithin_left hr ht)
+      (fun _ _ => mem_univ _) (HasDerivWithinAt.continuousOn hc)
+      (fun _ ht => derivWithin_left hc ht) (fun _ _ => mem_univ _) hf
+    have hh := congrArg Complex.im (he ⟨le_rfl,hs.1⟩)
+    simp only [Complex.conj_im] at hh
+    linarith
+  by_contra hneg
+  have hi := intermediate_value_Icc' ht.1
+    ((Complex.continuous_im.comp_continuousOn (HasDerivWithinAt.continuousOn hH)).mono (Icc_subset_Icc le_rfl ht.2))
+  obtain ⟨s,hs,hs0⟩ := hi (a:=0) ⟨le_of_not_gt hneg,h0.le⟩
+  exact hn s ⟨hs.1,hs.2.trans ht.2⟩ hs0
+
+lemma solution_dist_le {σ ε T D δ : ℝ} (hσ : |σ| = 1) (hε : 0 < ε)
+    (hD : 0 ≤ D) (hδ : 0 ≤ δ) {U V : ℝ → ℝ} {H G : ℝ → ℂ}
+    (hH : ∀ t ∈ Icc 0 T, HasDerivWithinAt H (velocity σ ε (U t) (H t)) (Icc 0 T) t)
+    (hG : ∀ t ∈ Icc 0 T, HasDerivWithinAt G (velocity σ ε (V t) (G t)) (Icc 0 T) t)
+    (h0 : dist (H 0) (G 0) ≤ δ) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ D)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    dist (H t) (G t) ≤ (δ+D)*Real.exp ((4/ε^2)*T) := by
+  let K : ℝ≥0 := ⟨4/ε^2,by positivity⟩
+  have hK : (0:ℝ) < K := by change 0 < 4/ε^2; positivity
+  have hb (s : ℝ) (hs : s ∈ Ico 0 T) :
+      dist (velocity σ ε (V s) (G s)) (velocity σ ε (U s) (G s)) ≤ (K:ℝ)*D := by
+    rw [dist_eq_norm]
+    calc
+      _ ≤ (4/ε^2)*|V s-U s| := norm_velocity_driver_sub_le hσ hε _ _ _
+      _ ≤ (K:ℝ)*D := by
+        rw [abs_sub_comm]
+        exact mul_le_mul_of_nonneg_left (hUV s ⟨hs.1,hs.2.le⟩) hK.le
+  have hh := dist_le_of_approx_trajectories_ODE
+    (fun t => velocity_lipschitz hσ hε (U t))
+    (HasDerivWithinAt.continuousOn hH) (fun _ hs => derivWithin_right hH hs)
+    (fun t _ => le_of_eq (dist_self (velocity σ ε (U t) (H t))))
+    (HasDerivWithinAt.continuousOn hG) (fun _ hs => derivWithin_right hG hs)
+    hb h0 t ht
+  have he : gronwallBound δ K (0+(K:ℝ)*D) (t-0) =
+      δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) := by
+    simp only [gronwallBound,zero_add,sub_zero,ite_eq_right (ne_of_gt hK)]
+    field_simp
+  rw [he] at hh
+  have hexp : Real.exp ((K:ℝ)*t) ≤ Real.exp ((K:ℝ)*T) :=
+    Real.exp_le_exp.2 (mul_le_mul_of_nonneg_left ht.2 hK.le)
+  apply hh.trans
+  change δ*Real.exp ((K:ℝ)*t)+D*(Real.exp ((K:ℝ)*t)-1) ≤ (δ+D)*Real.exp ((K:ℝ)*T)
+  have h1 := mul_le_mul_of_nonneg_left hexp hδ
+  have h2 := mul_le_mul_of_nonneg_left hexp hD
+  nlinarith
+
+end SLEExactGauge.BoundaryCutoff
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+structure Gauge where
+  toFun : ℝ≥0 → ℝ≥0
+  continuous : Continuous toFun
+  monotone : Monotone toFun
+  zero : toFun 0 = 0
+  positive : ∀ r, 0 < r → 0 < toFun r
+
+instance : CoeFun Gauge (fun _ => ℝ≥0 → ℝ≥0) := ⟨Gauge.toFun⟩
+
+def Gauge.extended (h : Gauge) (r : ℝ≥0∞) : ℝ≥0∞ :=
+  if r = ∞ then ∞ else (h r.toNNReal : ℝ≥0∞)
+
+def Gauge.hausdorff (h : Gauge) : Measure ℂ :=
+  Measure.mkMetric h.extended
+
+def IsExactGauge (κ : ℝ) (h : Gauge) : Prop :=
+  ∃ r₀ : ℝ≥0, 0 < r₀ ∧ ∀ r : ℝ≥0, 0 < r → r < r₀ →
+    (h r : ℝ) = (r : ℝ) ^ dimension κ *
+      (Real.log (Real.log (1 / (r : ℝ)))) ^ (codimension κ / 2)
+
+def IsLargerGauge (κ : ℝ) (g : Gauge) : Prop :=
+  ∃ r₀ : ℝ≥0, 0 < r₀ ∧ ∀ r : ℝ≥0, 0 < r → r < r₀ →
+    (g r : ℝ) = (r : ℝ) ^ dimension κ * Real.log (Real.log (1 / (r : ℝ)))
+
+def halfPlane : Set ℂ := {z | 0 < z.im}
+
+def localizationSegment (γ : ℝ → ℂ) (s t : ℝ) : Set ℂ := γ '' Icc s t
+
+def localizationWholeTrace (γ : ℝ → ℂ) : Set ℂ := γ '' Ici 0
+
+def rectangle (m : ℕ) : Set ℂ :=
+  {z | -(m : ℝ) ≤ z.re ∧ z.re ≤ m ∧ 0 ≤ z.im ∧ z.im ≤ m}
+
+def localizationDomain (γ : ℝ → ℂ) (t : ℝ) : Set ℂ :=
+  {z | z ∈ halfPlane \ localizationSegment γ 0 t ∧
+    ¬ Bornology.IsBounded (connectedComponentIn (halfPlane \ localizationSegment γ 0 t) z)}
+
+def IsLoewnerTrace (driver : ℝ → ℝ) (γ : ℝ → ℂ) : Prop :=
+  ContinuousOn γ (Ici 0) ∧ γ 0 = 0 ∧
+  (∀ t, 0 ≤ t → 0 ≤ (γ t).im) ∧
+  Tendsto γ atTop (Bornology.cobounded ℂ) ∧
+  ∃ g f : ℝ → ℂ → ℂ,
+    (∀ z ∈ halfPlane, g 0 z = z) ∧
+    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (g t) (localizationDomain γ t)) ∧
+    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (f t) halfPlane) ∧
+    (∀ t, 0 ≤ t → MapsTo (g t) (localizationDomain γ t) halfPlane) ∧
+    (∀ t, 0 ≤ t → MapsTo (f t) halfPlane (localizationDomain γ t)) ∧
+    (∀ t, 0 ≤ t → ∀ z ∈ localizationDomain γ t, f t (g t z) = z) ∧
+    (∀ t, 0 ≤ t → ∀ z ∈ halfPlane, g t (f t z) = z) ∧
+    (∀ z ∈ halfPlane,
+      ContinuousOn (fun t => g t z) {t | 0 ≤ t ∧ z ∈ localizationDomain γ t}) ∧
+    (∀ t, 0 < t → ∀ z ∈ localizationDomain γ t,
+      HasDerivAt (fun u => g u z) (2 / (g t z - (driver t : ℂ))) t) ∧
+    (∀ t, 0 ≤ t → Tendsto
+      (fun y : ℝ => f t ((driver t : ℂ) + (y : ℂ) * Complex.I))
+      (𝓝[>] 0) (𝓝 (γ t)))
+
+def IsChordalSLE {Ω : Type*} [MeasurableSpace Ω]
+    (κ : ℝ) (P : Measure Ω) (γ : Ω → ℝ → ℂ) : Prop :=
+  (∀ t, 0 ≤ t → AEMeasurable (fun ω => γ ω t) P) ∧
+  ∃ B : ℝ≥0 → Ω → ℝ, IsBrownianReal B P ∧
+    ∀ᵐ ω ∂P, IsLoewnerTrace (fun t => Real.sqrt κ * B t.toNNReal ω) (γ ω)
+
+def LocalizationMainEvent (h : Gauge) (γ : ℝ → ℂ) : Prop :=
+  (∀ s t : ℝ, 0 < s → s < t →
+    0 < h.hausdorff (localizationSegment γ s t) ∧ h.hausdorff (localizationSegment γ s t) < ∞) ∧
+  (∀ m : ℕ, 0 < m → h.hausdorff (localizationWholeTrace γ ∩ rectangle m) < ∞)
+
+def LocalizationMainConclusion {Ω : Type*} [MeasurableSpace Ω]
+    (P : Measure Ω) (γ : Ω → ℝ → ℂ) (h : Gauge) : Prop :=
+  (∀ᵐ ω ∂P, LocalizationMainEvent h (γ ω)) ∧
+  (∀ R : ℝ, 0 < R → ∃ C : ℝ≥0,
+    (∫⁻ ω, h.hausdorff (localizationWholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P) ≤ C)
+
+def LargerGaugeConclusion (g : Gauge) (γ : ℝ → ℂ) : Prop :=
+  ∀ s t : ℝ, 0 < s → s < t →
+    ¬ SigmaFinite (g.hausdorff.restrict (localizationSegment γ s t)) ∧
+      g.hausdorff (localizationSegment γ s t) = ∞
+
+def IsLoewnerTraceUsing (driver : ℝ → ℝ) (γ : ℝ → ℂ)
+    (g f : ℝ → ℂ → ℂ) : Prop :=
+  ContinuousOn γ (Ici 0) ∧ γ 0 = 0 ∧
+  (∀ t, 0 ≤ t → 0 ≤ (γ t).im) ∧
+  Tendsto γ atTop (Bornology.cobounded ℂ) ∧
+    (∀ z ∈ halfPlane, g 0 z = z) ∧
+    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (g t) (localizationDomain γ t)) ∧
+    (∀ t, 0 ≤ t → AnalyticOnNhd ℂ (f t) halfPlane) ∧
+    (∀ t, 0 ≤ t → MapsTo (g t) (localizationDomain γ t) halfPlane) ∧
+    (∀ t, 0 ≤ t → MapsTo (f t) halfPlane (localizationDomain γ t)) ∧
+    (∀ t, 0 ≤ t → ∀ z ∈ localizationDomain γ t, f t (g t z) = z) ∧
+    (∀ t, 0 ≤ t → ∀ z ∈ halfPlane, g t (f t z) = z) ∧
+    (∀ z ∈ halfPlane,
+      ContinuousOn (fun t => g t z) {t | 0 ≤ t ∧ z ∈ localizationDomain γ t}) ∧
+    (∀ t, 0 < t → ∀ z ∈ localizationDomain γ t,
+      HasDerivAt (fun u => g u z) (2 / (g t z - (driver t : ℂ))) t) ∧
+    (∀ t, 0 ≤ t → Tendsto
+      (fun y : ℝ => f t ((driver t : ℂ) + (y : ℂ) * Complex.I))
+      (𝓝[>] 0) (𝓝 (γ t)))
+
+def loewnerRadius (g : ℝ → ℂ → ℂ) (t : ℝ) (z : ℂ) : ℝ :=
+  2 * (g t z).im / ‖deriv (g t) z‖
+
+def loewnerAngleSine (κ : ℝ) (B : ℝ≥0 → ℝ)
+    (g : ℝ → ℂ → ℂ) (t : ℝ) (z : ℂ) : ℝ :=
+  (g t z).im / ‖g t z - (Real.sqrt κ * B t.toNNReal : ℂ)‖
+
+def firstLiveRadiusStop (γ : ℝ → ℂ) (g : ℝ → ℂ → ℂ)
+    (z : ℂ) (u : ℝ) : ℝ≥0∞ :=
+  sInf ((fun t : ℝ≥0 => (t : ℝ≥0∞)) ''
+    {t : ℝ≥0 | z ∈ localizationDomain γ t ∧
+      loewnerRadius g t z = (2 * z.im) * Real.exp (-u)})
+
+def finiteStoppedDensity (κ : ℝ) (B : ℝ≥0 → ℝ) (γ : ℝ → ℂ)
+    (g : ℝ → ℂ → ℂ) (z : ℂ) (u : ℝ) : ℝ≥0∞ :=
+  let τ := firstLiveRadiusStop γ g z u
+  if τ = ∞ then 0 else
+    ENNReal.ofReal ((loewnerRadius g τ.toReal z) ^ (-codimension κ) *
+      (loewnerAngleSine κ B g τ.toReal z) ^ (8 / κ - 1))
+
+lemma halfPlane_open : IsOpen halfPlane :=
+  isOpen_lt continuous_const Complex.continuous_im
+
+lemma halfPlane_preconnected : IsPreconnected halfPlane :=
+  (convex_halfSpace_im_gt 0).isPreconnected
+
+lemma halfPlane_not_bounded : ¬ Bornology.IsBounded halfPlane := by
+  intro hb
+  obtain ⟨R, hR, hbound⟩ := hb.exists_pos_norm_lt
+  have hm : ((R : ℂ) * Complex.I) ∈ halfPlane := by
+    simpa [halfPlane] using hR
+  have hh := hbound _ hm
+  rw [norm_mul, Complex.norm_real, Complex.norm_I, mul_one, Real.norm_eq_abs,
+    abs_of_pos hR] at hh
+  exact lt_irrefl _ hh
+
+lemma survivingDomain_mono {γ : ℝ → ℂ} {s t : ℝ} (hst : s ≤ t) :
+    localizationDomain γ t ⊆ localizationDomain γ s := by
+  have hsub : halfPlane \ localizationSegment γ 0 t ⊆ halfPlane \ localizationSegment γ 0 s := by
+    apply sdiff_subset_sdiff_right
+    exact image_mono (Icc_subset_Icc le_rfl hst)
+  intro z hz
+  exact ⟨hsub hz.1, fun hb => hz.2 (hb.subset (connectedComponentIn_mono z hsub))⟩
+
+lemma survivingDomain_zero {γ : ℝ → ℂ} (hγ : γ 0 = 0) :
+    localizationDomain γ 0 = halfPlane := by
+  have hset : halfPlane \ localizationSegment γ 0 0 = halfPlane := by
+    simp only [localizationSegment, Icc_self, image_singleton, hγ]
+    apply sdiff_singleton_eq_self
+    simp [halfPlane]
+  ext z
+  constructor
+  · exact fun h => h.1.1
+  · intro hz
+    refine ⟨by simpa only [hset] using hz, ?_⟩
+    rw [hset]
+    intro hb
+    exact halfPlane_not_bounded (hb.subset
+      (halfPlane_preconnected.subset_connectedComponentIn hz subset_rfl))
+
+lemma survivingDomain_open {γ : ℝ → ℂ} (hγ : ContinuousOn γ (Ici 0))
+    {t : ℝ} (_ht : 0 ≤ t) : IsOpen (localizationDomain γ t) := by
+  have hc : IsCompact (localizationSegment γ 0 t) :=
+    isCompact_Icc.image_of_continuousOn (hγ.mono (fun _ ha => ha.1))
+  have hF : IsOpen (halfPlane \ localizationSegment γ 0 t) := halfPlane_open.sdiff hc.isClosed
+  apply isOpen_iff_mem_nhds.2
+  intro z hz
+  have ho := hF.connectedComponentIn (x := z)
+  have hn := ho.mem_nhds (mem_connectedComponentIn hz.1)
+  apply Filter.mem_of_superset hn
+  intro w hw
+  refine ⟨connectedComponentIn_subset _ _ hw, ?_⟩
+  rw [← connectedComponentIn_eq hw]
+  exact hz.2
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+namespace IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+lemma initial (h : IsLoewnerTraceUsing driver γ g f) :
+    ∀ z ∈ halfPlane, g 0 z = z := h.2.2.2.2.1
+
+lemma continuous_eval (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (_hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun t => g t z) (Icc 0 T) := by
+  rcases h with ⟨hγ, hzero, him, htrans, hinit, hg, hf, hgmap, hfmap, hfg, hgf, hc, hd, htip⟩
+  apply (hc z hz.1.1).mono
+  intro t ht
+  exact ⟨ht.1, survivingDomain_mono ht.2 hz⟩
+
+lemma im_pos (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    0 < (g t z).im := h.2.2.2.2.2.2.2.1 t ht hz
+
+lemma analytic_eval (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    AnalyticAt ℂ (g t) z := h.2.2.2.2.2.1 t ht z hz
+
+lemma deriv_initial (h : IsLoewnerTraceUsing driver γ g f)
+    {z : ℂ} (hz : z ∈ halfPlane) : deriv (g 0) z = 1 := by
+  have heq : g 0 =ᶠ[𝓝 z] id := by
+    filter_upwards [halfPlane_open.mem_nhds hz] with w hw
+    exact h.initial w hw
+  rw [heq.deriv_eq]
+  exact deriv_id z
+
+lemma hasDerivAt_eval (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 < t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    HasDerivAt (fun u => g u z) (2 / (g t z - (driver t : ℂ))) t := by
+  rcases h with ⟨hγ, hzero, him, htrans, hinit, hg, hf, hgmap, hfmap, hfg, hgf, hc, hd, htip⟩
+  exact hd t ht z hz
+
+lemma deriv_ne_zero (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    deriv (g t) z ≠ 0 := by
+  rcases h with ⟨hγ, hzero, him, htrans, hinit, hg, hf, hgmap, hfmap, hfg, hgf, hc, hd, htip⟩
+  have hgd := (hg t ht z hz).differentiableAt.hasDerivAt
+  have hfd := (hf t ht _ (hgmap t ht hz)).differentiableAt.hasDerivAt
+  have heq : (fun w => f t (g t w)) =ᶠ[𝓝 z] id := by
+    filter_upwards [(survivingDomain_open hγ ht).mem_nhds hz] with w hw
+    exact hfg t ht w hw
+  have hh := (hfd.comp z hgd).congr_of_eventuallyEq heq.symm
+  have he := hh.unique (hasDerivAt_id z)
+  intro hn
+  simp [hn] at he
+
+lemma radius_pos (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    0 < loewnerRadius g t z := by
+  unfold loewnerRadius
+  exact div_pos (mul_pos (by norm_num) (h.im_pos ht hz))
+    (norm_pos_iff.2 (h.deriv_ne_zero ht hz))
+
+lemma radius_initial (h : IsLoewnerTraceUsing driver γ g f)
+    {z : ℂ} (hz : z ∈ halfPlane) : loewnerRadius g 0 z = 2 * z.im := by
+  simp [loewnerRadius, h.initial z hz, h.deriv_initial hz]
+
+lemma centered_ne_zero (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    g t z - (driver t : ℂ) ≠ 0 := by
+  intro he
+  have hi := congrArg Complex.im he
+  have hp := h.im_pos ht hz
+  simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hi
+  exact hp.ne' hi
+
+lemma hasDerivAt_height (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 < t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    HasDerivAt (fun s => (g s z).im)
+      (-2 * (g t z).im / Complex.normSq (g t z - (driver t : ℂ))) t := by
+  have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivAt t (h.hasDerivAt_eval ht hz)
+  simpa [Complex.div_im, Function.comp_def, neg_div] using hh
+
+lemma height_antitone (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    AntitoneOn (fun t => (g t z).im) (Icc 0 T) := by
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T)
+    (Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz))
+    (f' := fun t => -2 * (g t z).im / Complex.normSq (g t z - (driver t : ℂ)))
+  · intro t ht
+    rw [interior_Icc] at ht
+    exact (h.hasDerivAt_height ht.1 (survivingDomain_mono ht.2.le hz)).hasDerivWithinAt
+  · intro t ht
+    rw [interior_Icc] at ht
+    exact div_nonpos_of_nonpos_of_nonneg
+      (mul_nonpos_of_nonpos_of_nonneg (by norm_num)
+        (h.im_pos ht.1.le (survivingDomain_mono ht.2.le hz)).le)
+      (Complex.normSq_nonneg _)
+
+lemma height_le_initial (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    (g t z).im ≤ z.im := by
+  have hh := h.height_antitone ht hz ⟨le_rfl, ht⟩ ⟨ht, le_rfl⟩ ht
+  simpa only [h.initial z hz.1.1] using hh
+
+lemma continuous_velocity (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun t => 2 / (g t z - (driver t : ℂ))) (Icc 0 T) := by
+  apply continuousOn_const.div
+    ((h.continuous_eval hT hz).sub (Complex.continuous_ofReal.comp_continuousOn hU))
+  intro t ht
+  exact h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz)
+
+lemma integral_velocity (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    {a b : ℝ} (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T) (hab : a ≤ b) :
+    (∫ t in a..b, 2 / (g t z - (driver t : ℂ))) = g b z - g a z := by
+  have hsub : Icc a b ⊆ Icc 0 T := Icc_subset_Icc ha.1 hb.2
+  apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hab
+    ((h.continuous_eval hT hz).mono hsub)
+  · intro t ht
+    exact h.hasDerivAt_eval (lt_of_le_of_lt ha.1 ht.1)
+      (survivingDomain_mono (ht.2.le.trans hb.2) hz)
+  · apply ContinuousOn.intervalIntegrable
+    simpa only [uIcc_of_le hab] using (h.continuous_velocity hT hU hz).mono hsub
+
+lemma norm_velocity_le (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    {t : ℝ} (ht : t ∈ Icc 0 T) :
+    ‖2 / (g t z - (driver t : ℂ))‖ ≤ 2 / (g T z).im := by
+  have hY := h.im_pos hT hz
+  have hm : (g T z).im ≤ (g t z).im :=
+    h.height_antitone hT hz ht ⟨hT, le_rfl⟩ ht.2
+  have hn : (g t z).im ≤ ‖g t z - (driver t : ℂ)‖ := by
+    have hh := Complex.im_le_norm (g t z - (driver t : ℂ))
+    simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using hh
+  rw [norm_div, Complex.norm_ofNat]
+  exact div_le_div_of_nonneg_left (by norm_num) hY (hm.trans hn)
+
+lemma norm_time_sub_le (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T)
+    {a b : ℝ} (ha : a ∈ Icc 0 T) (hb : b ∈ Icc 0 T) :
+    ‖g b z - g a z‖ ≤ (2 / (g T z).im) * |b - a| := by
+  wlog hab : a ≤ b generalizing a b
+  · have hh := this hb ha (le_of_not_ge hab)
+    simpa only [norm_sub_rev, abs_sub_comm] using hh
+  rw [← h.integral_velocity hT hU hz ha hb hab]
+  exact intervalIntegral.norm_integral_le_of_norm_le_const fun t ht =>
+    h.norm_velocity_le hT hz ⟨ha.1.trans (uIoc_of_le hab ▸ ht).1.le,
+      (uIoc_of_le hab ▸ ht).2.trans hb.2⟩
+
+end IsLoewnerTraceUsing
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge
+
+lemma firstLiveRadiusStop_zero {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+    (h : IsLoewnerTraceUsing driver γ g f) {z : ℂ} (hz : z ∈ halfPlane) :
+    firstLiveRadiusStop γ g z 0 = 0 := by
+  apply le_antisymm ?_ bot_le
+  apply sInf_le
+  refine ⟨0, ⟨?_, ?_⟩, rfl⟩
+  · simpa only [NNReal.coe_zero, survivingDomain_zero h.2.1] using hz
+  · simpa only [NNReal.coe_zero, neg_zero, Real.exp_zero, mul_one] using h.radius_initial hz
+
+end SLEExactGauge
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.IsLoewnerTraceUsing
+variable {driver : ℝ → ℝ} {γ : ℝ → ℂ} {g f : ℝ → ℂ → ℂ}
+
+def differenceCoefficient (driver : ℝ → ℝ) (g : ℝ → ℂ → ℂ) (w z : ℂ) (t : ℝ) : ℂ :=
+  -2 / ((g t w - (driver t : ℂ)) * (g t z - (driver t : ℂ)))
+
+lemma continuous_differenceCoefficient (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {w z : ℂ} (hw : w ∈ localizationDomain γ T) (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (differenceCoefficient driver g w z) (Icc 0 T) := by
+  apply continuousOn_const.div
+    (((h.continuous_eval hT hw).sub (Complex.continuous_ofReal.comp_continuousOn hU)).mul
+      ((h.continuous_eval hT hz).sub (Complex.continuous_ofReal.comp_continuousOn hU)))
+  intro t ht
+  exact mul_ne_zero (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hw))
+    (h.centered_ne_zero ht.1 (survivingDomain_mono ht.2 hz))
+
+lemma difference_eq_exp_integral (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {w z : ℂ} (hw : w ∈ localizationDomain γ T) (hz : z ∈ localizationDomain γ T) :
+    g T w - g T z = (w-z) * Complex.exp
+      (∫ s in (0 : ℝ)..T, differenceCoefficient driver g w z s) := by
+  let a := differenceCoefficient driver g w z
+  let A : ℝ → ℂ := fun t => ∫ s in (0 : ℝ)..t, a s
+  have ha : ContinuousOn a (Icc 0 T) := h.continuous_differenceCoefficient hT hU hw hz
+  have hi : IntervalIntegrable a volume 0 T := ha.intervalIntegrable_of_Icc hT
+  have hAc : ContinuousOn A (Icc 0 T) := by
+    simpa only [uIcc_of_le hT, A] using
+      intervalIntegral.continuousOn_primitive_interval' hi (left_mem_uIcc : (0 : ℝ) ∈ uIcc 0 T)
+  have hAd {t : ℝ} (ht : t ∈ Ioo 0 T) : HasDerivAt A (a t) t := by
+    apply intervalIntegral.integral_hasDerivAt_right
+      (hi.mono_set (by rw [uIcc_of_le ht.1.le, uIcc_of_le hT]; exact Icc_subset_Icc le_rfl ht.2.le))
+    · have hh := ha.stronglyMeasurableAtFilter_nhdsWithin (μ := volume) measurableSet_Icc t
+      rwa [nhdsWithin_eq_nhds.mpr (Icc_mem_nhds ht.1 ht.2)] at hh
+    · exact ha.continuousAt (Icc_mem_nhds ht.1 ht.2)
+  let R : ℝ → ℂ := fun t => (g t w - g t z) * Complex.exp (-A t)
+  have hRc : ContinuousOn R (Icc 0 T) := by
+    exact ((h.continuous_eval hT hw).sub (h.continuous_eval hT hz)).mul
+      (Complex.continuous_exp.comp_continuousOn hAc.neg)
+  have hRd (t : ℝ) (ht : t ∈ Ioo 0 T) : HasDerivAt R 0 t := by
+    have hd := ((h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hw)).sub
+      (h.hasDerivAt_eval ht.1 (survivingDomain_mono ht.2.le hz))).mul (hAd ht).neg.cexp
+    apply hd.congr_deriv
+    have hw' := h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hw)
+    have hz' := h.centered_ne_zero ht.1.le (survivingDomain_mono ht.2.le hz)
+    dsimp [a, differenceCoefficient]
+    field_simp
+    ring
+  have hr := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hT hRc hRd
+    (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => (0 : ℂ)) volume 0 T)
+  have hR : R T = w-z := by
+    apply sub_eq_zero.1
+    simpa only [intervalIntegral.integral_zero, R, A, intervalIntegral.integral_same,
+      neg_zero, Complex.exp_zero, mul_one, h.initial w hw.1.1, h.initial z hz.1.1,
+      ] using hr.symm
+  have hexp := Complex.exp_ne_zero (-A T)
+  dsimp [R] at hR
+  calc
+    g T w - g T z = (g T w - g T z) * (Complex.exp (-A T) * Complex.exp (A T)) := by
+      rw [← Complex.exp_add, neg_add_cancel, Complex.exp_zero, mul_one]
+    _ = (w-z) * Complex.exp (A T) := by rw [← mul_assoc, hR]
+
+lemma continuousAt_integral_differenceCoefficient (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousAt (fun w => ∫ s in (0 : ℝ)..T, differenceCoefficient driver g w z s) z := by
+  let ε := (g T z).im / 2
+  have hε : 0 < ε := half_pos (h.im_pos hT hz)
+  have hev : ∀ᶠ w in 𝓝 z, w ∈ localizationDomain γ T ∧ ε ≤ (g T w).im := by
+    have hc := Complex.continuous_im.continuousAt.comp (h.analytic_eval hT hz).continuousAt
+    have hlt : ε < (g T z).im := by dsimp [ε]; linarith [h.im_pos hT hz]
+    filter_upwards [(survivingDomain_open h.1 hT).mem_nhds hz,
+      hc.eventually (Ioi_mem_nhds hlt)] with w hw hh
+    exact ⟨hw, hh.le⟩
+  apply intervalIntegral.continuousAt_of_dominated_interval (bound := fun _ => 2 / ε^2)
+  · filter_upwards [hev] with w hw
+    have hc := h.continuous_differenceCoefficient hT hU hw.1 hz
+    apply ContinuousOn.aestronglyMeasurable _ measurableSet_uIoc
+    exact hc.mono (by rw [uIoc_of_le hT]; exact Ioc_subset_Icc_self)
+  · filter_upwards [hev] with w hw
+    apply ae_of_all
+    intro t ht
+    rw [uIoc_of_le hT] at ht
+    have hwheight : ε ≤ (g t w).im :=
+      hw.2.trans (h.height_antitone hT hw.1 ⟨ht.1.le, ht.2⟩ ⟨hT, le_rfl⟩ ht.2)
+    have hzheight : ε ≤ (g t z).im := by
+      have hhalf : ε ≤ (g T z).im := by dsimp [ε]; linarith [h.im_pos hT hz]
+      exact hhalf.trans (h.height_antitone hT hz ⟨ht.1.le, ht.2⟩ ⟨hT, le_rfl⟩ ht.2)
+    have hwn : ε ≤ ‖g t w - (driver t : ℂ)‖ := hwheight.trans (by
+      simpa using Complex.im_le_norm (g t w - (driver t : ℂ)))
+    have hzn : ε ≤ ‖g t z - (driver t : ℂ)‖ := hzheight.trans (by
+      simpa using Complex.im_le_norm (g t z - (driver t : ℂ)))
+    simp only [differenceCoefficient, norm_div, norm_neg, Complex.norm_ofNat, norm_mul]
+    apply div_le_div_of_nonneg_left (by norm_num) (sq_pos_of_pos hε)
+    simpa only [pow_two] using mul_le_mul hwn hzn hε.le (norm_nonneg _)
+  · exact intervalIntegrable_const
+  · apply ae_of_all
+    intro t ht
+    rw [uIoc_of_le hT] at ht
+    have hzt := survivingDomain_mono ht.2 hz
+    exact continuousAt_const.div
+      (((h.analytic_eval ht.1.le hzt).continuousAt.sub continuousAt_const).mul continuousAt_const)
+      (mul_ne_zero (h.centered_ne_zero ht.1.le hzt) (h.centered_ne_zero ht.1.le hzt))
+
+lemma spatial_deriv_eq_exp_integral (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    deriv (g T) z = Complex.exp (∫ s in (0 : ℝ)..T,
+      -2 / (g s z - (driver s : ℂ))^2) := by
+  have hd : HasDerivAt (g T) (Complex.exp (∫ s in (0 : ℝ)..T,
+      differenceCoefficient driver g z z s)) z := by
+    apply hasDerivAt_iff_tendsto_slope.2
+    have hc := Complex.continuous_exp.continuousAt.comp
+      (h.continuousAt_integral_differenceCoefficient hT hU hz)
+    apply (hc.mono_left nhdsWithin_le_nhds).congr'
+    filter_upwards [self_mem_nhdsWithin,
+      mem_nhdsWithin_of_mem_nhds ((survivingDomain_open h.1 hT).mem_nhds hz)]
+      with w hwne hw
+    have hwz : w ≠ z := hwne
+    dsimp only [slope, vsub_eq_sub, Function.comp_def]
+    rw [h.difference_eq_exp_integral hT hU hw hz]
+    simp only [smul_eq_mul, ← mul_assoc, inv_mul_cancel₀ (sub_ne_zero.2 hwz), one_mul]
+  have he : (fun s => differenceCoefficient driver g z z s) =
+      (fun s => -2 / (g s z - (driver s : ℂ))^2) := by
+    funext s
+    simp only [differenceCoefficient, pow_two]
+  simpa only [he] using hd.deriv
+
+lemma log_deriv_norm_eq (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    Real.log ‖deriv (g T) z‖ =
+      (∫ s in (0 : ℝ)..T, -2 / (g s z - (driver s : ℂ))^2).re := by
+  rw [h.spatial_deriv_eq_exp_integral hT hU hz, Complex.norm_exp, Real.log_exp]
+
+lemma continuous_spatialCoefficient (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun s => -2 / (g s z - (driver s : ℂ))^2) (Icc 0 T) := by
+  have hc := h.continuous_differenceCoefficient hT hU hz hz
+  unfold differenceCoefficient at hc
+  simpa only [pow_two] using hc
+
+lemma continuous_log_deriv_norm (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun t => Real.log ‖deriv (g t) z‖) (Icc 0 T) := by
+  have hi : IntervalIntegrable (fun s => -2 / (g s z - (driver s : ℂ))^2) volume 0 T :=
+    (h.continuous_spatialCoefficient hT hU hz).intervalIntegrable_of_Icc hT
+  have hc := Complex.continuous_re.comp_continuousOn
+    (intervalIntegral.continuousOn_primitive_interval' hi left_mem_uIcc)
+  rw [uIcc_of_le hT] at hc
+  apply hc.congr
+  intro t ht
+  exact h.log_deriv_norm_eq ht.1
+    (hU.mono (Icc_subset_Icc le_rfl ht.2)) (survivingDomain_mono ht.2 hz)
+
+lemma hasDerivAt_log_deriv_norm (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) {t : ℝ} (ht : t ∈ Ioo 0 T) :
+    HasDerivAt (fun s => Real.log ‖deriv (g s) z‖)
+      (-2 / (g t z - (driver t : ℂ))^2).re t := by
+  have hc := h.continuous_spatialCoefficient hT hU hz
+  have hi : IntervalIntegrable (fun s => -2 / (g s z - (driver s : ℂ))^2) volume 0 t :=
+    (hc.intervalIntegrable_of_Icc hT).mono_set (by
+      rw [uIcc_of_le ht.1.le, uIcc_of_le hT]
+      exact Icc_subset_Icc le_rfl ht.2.le)
+  have hm := hc.stronglyMeasurableAtFilter_nhdsWithin (μ := volume) measurableSet_Icc t
+  rw [nhdsWithin_eq_nhds.mpr (Icc_mem_nhds ht.1 ht.2)] at hm
+  have hd := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t
+    (intervalIntegral.integral_hasDerivAt_right hi hm
+      (hc.continuousAt (Icc_mem_nhds ht.1 ht.2)))
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
+  exact h.log_deriv_norm_eq hs.1 (hU.mono (Icc_subset_Icc le_rfl hs.2))
+    (survivingDomain_mono hs.2 hz)
+
+lemma real_spatialCoefficient (w : ℂ) :
+    (-2 / w^2).re = -2 * (w.re^2-w.im^2) / (Complex.normSq w)^2 := by
+  simp only [Complex.div_re, Complex.neg_re, Complex.re_ofNat, Complex.neg_im,
+    Complex.im_ofNat, neg_zero, zero_mul, map_pow]
+  simp only [pow_two, Complex.mul_re]
+  ring
+
+lemma log_radius_eq (h : IsLoewnerTraceUsing driver γ g f)
+    {t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z ∈ localizationDomain γ t) :
+    Real.log (loewnerRadius g t z) = Real.log 2 + Real.log (g t z).im -
+      Real.log ‖deriv (g t) z‖ := by
+  rw [loewnerRadius, Real.log_div (mul_ne_zero (by norm_num) (h.im_pos ht hz).ne')
+    (norm_ne_zero_iff.2 (h.deriv_ne_zero ht hz)),
+    Real.log_mul (by norm_num) (h.im_pos ht hz).ne']
+
+lemma continuous_log_radius (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun t => Real.log (loewnerRadius g t z)) (Icc 0 T) := by
+  have hc := ((Complex.continuous_im.comp_continuousOn (h.continuous_eval hT hz)).log
+    (fun t ht => (h.im_pos ht.1 (survivingDomain_mono ht.2 hz)).ne')).const_add (Real.log 2)
+  apply (hc.sub (h.continuous_log_deriv_norm hT hU hz)).congr
+  intro t ht
+  exact h.log_radius_eq ht.1 (survivingDomain_mono ht.2 hz)
+
+lemma hasDerivAt_log_radius (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) {t : ℝ} (ht : t ∈ Ioo 0 T) :
+    HasDerivAt (fun s => Real.log (loewnerRadius g s z))
+      (-4 * (g t z).im^2 / (Complex.normSq (g t z - (driver t : ℂ)))^2) t := by
+  have hzt := survivingDomain_mono ht.2.le hz
+  have hY := h.im_pos ht.1.le hzt
+  have hd := (((h.hasDerivAt_height ht.1 hzt).log hY.ne').const_add (Real.log 2)).sub
+    (h.hasDerivAt_log_deriv_norm hT hU hz ht)
+  have he : HasDerivAt (fun s => Real.log (loewnerRadius g s z))
+      ((-2 * (g t z).im / Complex.normSq (g t z - (driver t : ℂ))) / (g t z).im -
+        (-2 / (g t z - (driver t : ℂ))^2).re) t := by
+    apply hd.congr_of_eventuallyEq
+    filter_upwards [Icc_mem_nhds ht.1 ht.2] with s hs
+    exact h.log_radius_eq hs.1 (survivingDomain_mono hs.2 hz)
+  apply he.congr_deriv
+  rw [real_spatialCoefficient]
+  simp only [Complex.sub_im, Complex.ofReal_im, sub_zero]
+  rw [Complex.normSq_apply]
+  simp only [Complex.sub_re, Complex.ofReal_re, Complex.sub_im, Complex.ofReal_im, sub_zero]
+  field_simp
+  ring
+
+lemma continuous_radius (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    ContinuousOn (fun t => loewnerRadius g t z) (Icc 0 T) := by
+  apply (Real.continuous_exp.comp_continuousOn (h.continuous_log_radius hT hU hz)).congr
+  intro t ht
+  exact (Real.exp_log (h.radius_pos ht.1 (survivingDomain_mono ht.2 hz))).symm
+
+lemma log_radius_strictAnti (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    StrictAntiOn (fun t => Real.log (loewnerRadius g t z)) (Icc 0 T) := by
+  apply strictAntiOn_of_hasDerivWithinAt_neg (convex_Icc 0 T)
+    (h.continuous_log_radius hT hU hz)
+    (f' := fun t => -4 * (g t z).im^2 / (Complex.normSq (g t z - (driver t : ℂ)))^2)
+  · intro t ht
+    rw [interior_Icc] at ht
+    exact (h.hasDerivAt_log_radius hT hU hz ht).hasDerivWithinAt
+  · intro t ht
+    rw [interior_Icc] at ht
+    have hzt := survivingDomain_mono ht.2.le hz
+    have hy := h.im_pos ht.1.le hzt
+    have hq := Complex.normSq_pos.2 (h.centered_ne_zero ht.1.le hzt)
+    exact div_neg_of_neg_of_pos (mul_neg_of_neg_of_pos (by norm_num) (sq_pos_of_pos hy))
+      (sq_pos_of_pos hq)
+
+lemma radius_strictAnti (h : IsLoewnerTraceUsing driver γ g f)
+    {T : ℝ} (hT : 0 ≤ T) (hU : ContinuousOn driver (Icc 0 T))
+    {z : ℂ} (hz : z ∈ localizationDomain γ T) :
+    StrictAntiOn (fun t => loewnerRadius g t z) (Icc 0 T) := by
+  intro s hs t ht hst
+  exact (Real.log_lt_log_iff (h.radius_pos ht.1 (survivingDomain_mono ht.2 hz))
+    (h.radius_pos hs.1 (survivingDomain_mono hs.2 hz))).1
+      (h.log_radius_strictAnti hT hU hz hs ht hst)
+
+lemma live_radius_injective (h : IsLoewnerTraceUsing driver γ g f)
+    (hU : ContinuousOn driver (Ici 0)) {z : ℂ} {s t : ℝ}
+    (hs : 0 ≤ s) (ht : 0 ≤ t) (hzs : z ∈ localizationDomain γ s)
+    (hzt : z ∈ localizationDomain γ t) (he : loewnerRadius g s z = loewnerRadius g t z) :
+    s = t := by
+  rcases le_total s t with hst | hts
+  · exact (h.radius_strictAnti ht (hU.mono Icc_subset_Ici_self) hzt).injOn
+      ⟨hs, hst⟩ ⟨ht, le_rfl⟩ he
+  · exact (h.radius_strictAnti hs (hU.mono Icc_subset_Ici_self) hzs).injOn
+      ⟨hs, le_rfl⟩ ⟨ht, hts⟩ he
+
+lemma firstLiveRadiusStop_eq (h : IsLoewnerTraceUsing driver γ g f)
+    (hU : ContinuousOn driver (Ici 0)) {z : ℂ} {u : ℝ} {t : ℝ≥0}
+    (hzt : z ∈ localizationDomain γ t)
+    (he : loewnerRadius g t z = (2*z.im)*Real.exp (-u)) :
+    firstLiveRadiusStop γ g z u = (t : ℝ≥0∞) := by
+  have heq : {s : ℝ≥0 | z ∈ localizationDomain γ s ∧
+      loewnerRadius g s z = (2*z.im)*Real.exp (-u)} = {t} := by
+    ext s
+    constructor
+    · intro hs
+      apply mem_singleton_iff.2
+      apply NNReal.coe_injective
+      exact h.live_radius_injective hU s.coe_nonneg t.coe_nonneg hs.1 hzt (hs.2.trans he.symm)
+    · rintro rfl
+      exact ⟨hzt, he⟩
+  simp only [firstLiveRadiusStop, heq, image_singleton, sInf_singleton]
+
+end SLEExactGauge.IsLoewnerTraceUsing
+
+end
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/TruncatedEnergy.lean b/lean/OAI/Probability/SLELocalization/TruncatedEnergy.lean
index 49467983fd73160b4a398672526415bf0bf5b0b2..8b39e65ba736a83a182356a9375bae7cbc5ea62e 100644
--- a/lean/OAI/Probability/SLELocalization/TruncatedEnergy.lean
+++ b/lean/OAI/Probability/SLELocalization/TruncatedEnergy.lean
@@ -1,270 +1,270 @@
-import Mathlib
-import OAI.Probability.SLELocalization.CommonClocks
-
-namespace OAI
-
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma radialTruncation_fderiv_le {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r)
-    {F : ℂ → ℂ} {z q : ℂ} (hd : DifferentiableAt ℂ F z) :
-    ‖fderiv ℝ (radialTruncation H q r ∘ F) z‖ ≤ ‖deriv F z‖/r := by
-  have hb := radialTruncation_fderiv_bound hH hr (q := q) hd
-  have hs : ‖fderiv ℝ (radialTruncation H q r ∘ F) z‖^2 ≤ (‖deriv F z‖/r)^2 := by
-    by_cases hh : r < ‖F z-q‖ ∧ ‖F z-q‖ < Real.exp H*r
-    · rw [ite_eq_left hh] at hb
-      refine hb.trans ?_
-      rw [div_pow]
-      exact div_le_div_of_nonneg_left (sq_nonneg _) (sq_pos_of_pos hr)
-        (pow_le_pow_left₀ hr.le hh.1.le 2)
-    · rw [ite_eq_right hh] at hb
-      exact hb.trans (sq_nonneg _)
-  exact (sq_le_sq₀ (norm_nonneg _) (div_nonneg (norm_nonneg _) hr.le)).1 hs
-
-lemma truncated_gradient_comp_integrable
-    {α : Type*} [TopologicalSpace α] [T2Space α] [MeasurableSpace α] [BorelSpace α]
-    {μ : Measure α} [IsFiniteMeasureOnCompacts μ]
-    {K : Set α} (hK : IsCompact K) {c : α → ℂ} (hc : Continuous c)
-    {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F U)
-    (hmap : MapsTo c K U) {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
-    IntegrableOn (fun x => ‖fderiv ℝ (radialTruncation H q r ∘ F) (c x)‖^2) K μ := by
-  have hcont : ContinuousOn (fun x => deriv F (c x)) K :=
-    (hF.deriv hU).continuousOn.comp hc.continuousOn hmap
-  obtain ⟨M,hM⟩ := hK.exists_bound_of_continuousOn hcont
-  have hm : AEStronglyMeasurable
-      (fun x => ‖fderiv ℝ (radialTruncation H q r ∘ F) (c x)‖^2) (μ.restrict K) :=
-    (((measurable_fderiv ℝ _).comp hc.measurable).norm.pow_const 2).aestronglyMeasurable
-  let : IsFiniteMeasure (μ.restrict K) := isFiniteMeasure_restrict.2 hK.measure_lt_top.ne
-  apply (integrable_const ((M/r)^2)).mono' hm
-  filter_upwards [ae_restrict_mem₀ hK.nullMeasurableSet] with x hx
-  rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
-  have hb := (radialTruncation_fderiv_le hH hr (q := q) (hF.differentiableAt (hU.mem_nhds (hmap hx)))).trans
-    (div_le_div_of_nonneg_right (hM x hx) hr.le)
-  exact pow_le_pow_left₀ (norm_nonneg _) hb 2
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma truncated_line_oscillation {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F U) {z e : ℂ} (he : ‖e‖ = 1) {a b x y : ℝ}
-    (hx : x ∈ Icc a b) (hy : y ∈ Icc a b)
-    (hmap : MapsTo (fun s : ℝ => z+s • e) (Icc a b) U)
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
-    (radialTruncation H q r (F (z+y • e))-
-      radialTruncation H q r (F (z+x • e)))^2 ≤
-      (b-a)*(∫ s in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F) (z+s • e)‖^2) := by
-  let c := fun s : ℝ => z+s • e
-  let u := radialTruncation H q r ∘ F
-  have hab := hx.1.trans hx.2
-  have hc : ContDiff ℝ 1 c := contDiff_const.add (contDiff_id.smul contDiff_const)
-  have hG : ContDiffOn ℝ 1 (F ∘ c) (Icc a b) :=
-    ((hF.contDiffOn hU).restrict_scalars ℝ).comp hc.contDiffOn hmap
-  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
-  have hLip := (radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK
-  have hD := lipschitzOn_deriv_memLp hLip
-  have ht := acl_oscillation_energy
-    (radialTruncation_along_C1_AC hr H q (by simpa only [uIcc_of_le hab] using hG)) hD hx hy
-  have hgrad := truncated_gradient_comp_integrable (μ := volume) isCompact_Icc hc.continuous hU hF hmap hH hr q
-  have hderiv (s : ℝ) (hs : s ∈ Icc a b) :
-      deriv (u ∘ c) s = fderiv ℝ u (c s) e := by
-    apply radialTruncation_directional_deriv hH hr q (c := c) (t := s) (d := e) (hF.differentiableAt (hU.mem_nhds (hmap hs)))
-    simpa only [one_smul,c,id_eq] using ((hasDerivAt_id s).smul_const e).const_add z
-  have hbnd : (∫ s in a..b, (deriv (u ∘ c) s)^2) ≤
-      ∫ s in a..b, ‖fderiv ℝ u (c s)‖^2 := by
-    apply intervalIntegral.integral_mono_on hab
-      ((intervalIntegrable_iff_integrableOn_Ioc_of_le hab).2 hD.integrable_sq)
-      ((intervalIntegrable_iff_integrableOn_Icc_of_le hab).2 hgrad)
-    intro s hs
-    change (deriv (u ∘ c) s)^2 ≤ ‖fderiv ℝ u (c s)‖^2
-    rw [hderiv s hs]
-    have hh := (fderiv ℝ u (c s)).le_opNorm e
-    rw [he,mul_one] at hh
-    have hh' := pow_le_pow_left₀ (norm_nonneg _) hh 2
-    simpa only [Real.norm_eq_abs,sq_abs] using hh'
-  exact ht.trans (mul_le_mul_of_nonneg_left hbnd (sub_nonneg.2 hab))
-
-theorem truncated_rectangle_two_lines {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F U) {a b c d x₁ x₂ y₁ y₂ : ℝ}
-    (hx₁ : x₁ ∈ Icc a b) (hx₂ : x₂ ∈ Icc a b)
-    (hy₁ : y₁ ∈ Icc c d) (hy₂ : y₂ ∈ Icc c d)
-    (hrect : ∀ x ∈ Icc a b, ∀ y ∈ Icc c d, (x:ℂ)+(y:ℂ)*Complex.I ∈ U)
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
-    (radialTruncation H q r (F ((x₁:ℂ)+(y₁:ℂ)*Complex.I))-
-      radialTruncation H q r (F ((x₂:ℂ)+(y₂:ℂ)*Complex.I)))^2 ≤
-      2*(b-a)*(∫ x in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F)
-        ((x:ℂ)+(y₁:ℂ)*Complex.I)‖^2)+
-      2*(d-c)*(∫ y in c..d, ‖fderiv ℝ (radialTruncation H q r ∘ F)
-        ((x₂:ℂ)+(y:ℂ)*Complex.I)‖^2) := by
-  have hh := truncated_line_oscillation hU hF (z := (y₁:ℂ)*Complex.I) (e := 1)
-    (by simp) hx₂ hx₁ (by
-      intro x hx
-      simpa only [Complex.real_smul,Complex.ofReal_mul,mul_one,add_comm] using hrect x hx y₁ hy₁) hH hr q
-  have hv := truncated_line_oscillation hU hF (z := (x₂:ℂ)) (e := Complex.I)
-    (by simp) hy₂ hy₁ (by
-      intro y hy
-      simpa only [Complex.real_smul] using hrect x₂ hx₂ y hy) hH hr q
-  simp only [Complex.real_smul,mul_one] at hh hv
-  have hi : (∫ x in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F) ((y₁:ℂ)*Complex.I+(x:ℂ))‖^2) =
-      ∫ x in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F) ((x:ℂ)+(y₁:ℂ)*Complex.I)‖^2 := by
-    congr 1
-    funext x
-    rw [add_comm]
-  rw [hi,add_comm ((y₁:ℂ)*Complex.I) (x₁:ℂ),add_comm ((y₁:ℂ)*Complex.I) (x₂:ℂ)] at hh
-  nlinarith [sq_nonneg
-    (radialTruncation H q r (F ((x₁:ℂ)+(y₁:ℂ)*Complex.I))+
-      radialTruncation H q r (F ((x₂:ℂ)+(y₂:ℂ)*Complex.I))-
-      2*radialTruncation H q r (F ((x₂:ℂ)+(y₁:ℂ)*Complex.I)))]
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma radial_section_mem {z : ℂ} {a b s : ℝ} (ha : 0 < a) (hs : s ∈ Icc a b) (θ : ℝ) :
-    z+s • circleMap 0 1 θ ∈ closedBall z b := by
-  rw [mem_closedBall_iff_norm,add_sub_cancel_left,norm_smul,norm_circleMap_zero,abs_one,mul_one,
-    Real.norm_of_nonneg (ha.trans_le hs.1).le]
-  exact hs.2
-
-lemma truncated_radial_energy {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F U) {z : ℂ} {a b : ℝ}
-    (ha : 0 < a) (hab : a ≤ b) (hball : closedBall z b ⊆ U)
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (θ : ℝ) :
-    (radialTruncation H q r (F (z+b • circleMap 0 1 θ))-
-      radialTruncation H q r (F (z+a • circleMap 0 1 θ)))^2 ≤
-      Real.log (b/a)*(∫ s in a..b, s*‖fderiv ℝ (radialTruncation H q r ∘ F)
-        (z+s • circleMap 0 1 θ)‖^2) := by
-  let c := fun s : ℝ => z+s • circleMap 0 1 θ
-  let u := radialTruncation H q r ∘ F
-  have hc : ContDiff ℝ 1 c := contDiff_const.add (contDiff_id.smul contDiff_const)
-  have hmap : MapsTo c (Icc a b) U := fun s hs => hball (radial_section_mem ha hs θ)
-  have hG : ContDiffOn ℝ 1 (F ∘ c) (Icc a b) :=
-    ((hF.contDiffOn hU).restrict_scalars ℝ).comp hc.contDiffOn hmap
-  have ht := truncated_curve_weighted_energy hr ha hab q hG (H := H)
-  have hgrad := truncated_gradient_comp_integrable (μ := volume) isCompact_Icc hc.continuous hU hF hmap hH hr q
-  have hgradw : IntervalIntegrable
-      (fun s => s*‖fderiv ℝ u (c s)‖^2) volume a b :=
-    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).2
-      (hgrad.continuousOn_mul continuousOn_id isCompact_Icc)
-  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
-  have hLip := (radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK
-  have hl2 := lipschitzOn_weighted_deriv_memLp ha hLip
-  have hi : IntegrableOn (fun s => s*(deriv (u ∘ c) s)^2) (Ioc a b) := by
-    have hh := (memLp_two_iff_integrable_sq hl2.aestronglyMeasurable).1 hl2
-    apply hh.congr
-    filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
-    rw [mul_pow,Real.sq_sqrt (ha.trans_le hs.1.le).le]
-    rfl
-  have hderiv (s : ℝ) (hs : s ∈ Icc a b) :
-      deriv (u ∘ c) s = fderiv ℝ u (c s) (circleMap 0 1 θ) := by
-    apply radialTruncation_directional_deriv hH hr q (hF.differentiableAt (hU.mem_nhds (hmap hs)))
-    simpa only [one_smul,c,id_eq] using ((hasDerivAt_id s).smul_const (circleMap 0 1 θ)).const_add z
-  have hbnd : (∫ s in a..b, s*(deriv (u ∘ c) s)^2) ≤
-      ∫ s in a..b, s*‖fderiv ℝ u (c s)‖^2 := by
-    apply intervalIntegral.integral_mono_on hab
-      ((intervalIntegrable_iff_integrableOn_Ioc_of_le hab).2 hi) hgradw
-    intro s hs
-    rw [hderiv s hs]
-    apply mul_le_mul_of_nonneg_left _ (ha.trans_le hs.1).le
-    have hh := (fderiv ℝ u (c s)).le_opNorm (circleMap 0 1 θ)
-    rw [norm_circleMap_zero,abs_one,mul_one] at hh
-    have hh' := pow_le_pow_left₀ (norm_nonneg _) hh 2
-    simpa only [Real.norm_eq_abs,sq_abs] using hh'
-  have hlog : 0 ≤ Real.log (b/a) := Real.log_nonneg ((one_le_div ha).2 hab)
-  exact ht.trans (mul_le_mul_of_nonneg_left hbnd hlog)
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-theorem truncated_circle_energy {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
-    (hF : DifferentiableOn ℂ F U) {z : ℂ} {a b : ℝ}
-    (ha : 0 < a) (hab : a ≤ b) (hball : closedBall z b ⊆ U)
-    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
-    ((∫ θ in -Real.pi..Real.pi, radialTruncation H q r (F (z+b • circleMap 0 1 θ)))-
-      (∫ θ in -Real.pi..Real.pi, radialTruncation H q r (F (z+a • circleMap 0 1 θ))))^2 ≤
-      2*Real.pi*Real.log (b/a)*(∫ θ in -Real.pi..Real.pi, ∫ s in a..b,
-        s*‖fderiv ℝ (radialTruncation H q r ∘ F) (z+s • circleMap 0 1 θ)‖^2) := by
-  let u := radialTruncation H q r ∘ F
-  let D := fun θ => u (z+b • circleMap 0 1 θ)-u (z+a • circleMap 0 1 θ)
-  let E := fun θ => ∫ s in a..b, s*‖fderiv ℝ u (z+s • circleMap 0 1 θ)‖^2
-  have hp : -Real.pi ≤ Real.pi := by linarith [Real.pi_pos]
-  have hc (s : ℝ) (hs : s ∈ Icc a b) : Continuous (fun θ => u (z+s • circleMap 0 1 θ)) := by
-    apply (radialTruncation_continuous hr H q).comp
-    apply continuousOn_univ.1
-    exact hF.continuousOn.comp
-      (continuous_const.add ((continuous_circleMap 0 1).const_smul s)).continuousOn
-      (fun θ _ => hball (radial_section_mem ha hs θ))
-  have hca := hc a (left_mem_Icc.2 hab)
-  have hcb := hc b (right_mem_Icc.2 hab)
-  have hD : Continuous D := hcb.sub hca
-  let c := fun p : ℝ × ℝ => z+p.2 • circleMap 0 1 p.1
-  have hcc : Continuous c := continuous_const.add (continuous_snd.smul ((continuous_circleMap 0 1).comp continuous_fst))
-  have hmap : MapsTo c (Icc (-Real.pi) Real.pi ×ˢ Icc a b) U :=
-    fun p hp => hball (radial_section_mem ha hp.2 p.1)
-  have hi := truncated_gradient_comp_integrable (μ := volume) (isCompact_Icc.prod isCompact_Icc)
-    hcc hU hF hmap hH hr q
-  have hiw : IntegrableOn (fun p : ℝ × ℝ => p.2*‖fderiv ℝ u (c p)‖^2)
-      (Icc (-Real.pi) Real.pi ×ˢ Icc a b) :=
-    hi.continuousOn_mul continuous_snd.continuousOn (isCompact_Icc.prod isCompact_Icc)
-  have hiprod : Integrable (fun p : ℝ × ℝ => p.2*‖fderiv ℝ u (c p)‖^2)
-      ((volume.restrict (Ioc (-Real.pi) Real.pi)).prod (volume.restrict (Ioc a b))) := by
-    rw [Measure.prod_restrict]
-    exact hiw.mono_set (prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)
-  have hE : IntervalIntegrable E volume (-Real.pi) Real.pi := by
-    apply (intervalIntegrable_iff_integrableOn_Ioc_of_le hp).2
-    simpa only [E,intervalIntegral.integral_of_le hab,c,IntegrableOn] using hiprod.integral_prod_left
-  have hcs := integral_mul_square_le (continuousOn_memLp_two_Ioc hp hD.continuousOn)
-    (memLp_const (1:ℝ))
-  have hid : (∫ θ in Ioc (-Real.pi) Real.pi, D θ*1) =
-      (∫ θ in -Real.pi..Real.pi, u (z+b • circleMap 0 1 θ))-
-      (∫ θ in -Real.pi..Real.pi, u (z+a • circleMap 0 1 θ)) := by
-    simp only [mul_one,← intervalIntegral.integral_of_le hp,D]
-    exact intervalIntegral.integral_sub (hcb.intervalIntegrable _ _) (hca.intervalIntegrable _ _)
-  have hi1 : (∫ θ in Ioc (-Real.pi) Real.pi, (1:ℝ)^2) = 2*Real.pi := by
-    rw [← intervalIntegral.integral_of_le hp]
-    simp
-    ring
-  rw [hid,hi1,← intervalIntegral.integral_of_le hp] at hcs
-  have hbound : (∫ θ in -Real.pi..Real.pi, D θ^2) ≤
-      ∫ θ in -Real.pi..Real.pi, Real.log (b/a)*E θ := by
-    apply intervalIntegral.integral_mono_on hp ((hD.pow 2).intervalIntegrable _ _)
-      (hE.const_mul _)
-    intro θ hθ
-    exact truncated_radial_energy hU hF ha hab hball hH hr q θ
-  rw [intervalIntegral.integral_const_mul] at hbound
-  have hm := mul_le_mul_of_nonneg_left hbound (by positivity : 0 ≤ 2*Real.pi)
-  apply hcs.trans
-  convert hm using 1 <;> ring
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end OAI
+import Mathlib
+import OAI.Probability.SLELocalization.CommonClocks
+
+namespace OAI
+
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma radialTruncation_fderiv_le {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r)
+    {F : ℂ → ℂ} {z q : ℂ} (hd : DifferentiableAt ℂ F z) :
+    ‖fderiv ℝ (radialTruncation H q r ∘ F) z‖ ≤ ‖deriv F z‖/r := by
+  have hb := radialTruncation_fderiv_bound hH hr (q := q) hd
+  have hs : ‖fderiv ℝ (radialTruncation H q r ∘ F) z‖^2 ≤ (‖deriv F z‖/r)^2 := by
+    by_cases hh : r < ‖F z-q‖ ∧ ‖F z-q‖ < Real.exp H*r
+    · rw [ite_eq_left hh] at hb
+      refine hb.trans ?_
+      rw [div_pow]
+      exact div_le_div_of_nonneg_left (sq_nonneg _) (sq_pos_of_pos hr)
+        (pow_le_pow_left₀ hr.le hh.1.le 2)
+    · rw [ite_eq_right hh] at hb
+      exact hb.trans (sq_nonneg _)
+  exact (sq_le_sq₀ (norm_nonneg _) (div_nonneg (norm_nonneg _) hr.le)).1 hs
+
+lemma truncated_gradient_comp_integrable
+    {α : Type*} [TopologicalSpace α] [T2Space α] [MeasurableSpace α] [BorelSpace α]
+    {μ : Measure α} [IsFiniteMeasureOnCompacts μ]
+    {K : Set α} (hK : IsCompact K) {c : α → ℂ} (hc : Continuous c)
+    {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F U)
+    (hmap : MapsTo c K U) {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
+    IntegrableOn (fun x => ‖fderiv ℝ (radialTruncation H q r ∘ F) (c x)‖^2) K μ := by
+  have hcont : ContinuousOn (fun x => deriv F (c x)) K :=
+    (hF.deriv hU).continuousOn.comp hc.continuousOn hmap
+  obtain ⟨M,hM⟩ := hK.exists_bound_of_continuousOn hcont
+  have hm : AEStronglyMeasurable
+      (fun x => ‖fderiv ℝ (radialTruncation H q r ∘ F) (c x)‖^2) (μ.restrict K) :=
+    (((measurable_fderiv ℝ _).comp hc.measurable).norm.pow_const 2).aestronglyMeasurable
+  let : IsFiniteMeasure (μ.restrict K) := isFiniteMeasure_restrict.2 hK.measure_lt_top.ne
+  apply (integrable_const ((M/r)^2)).mono' hm
+  filter_upwards [ae_restrict_mem₀ hK.nullMeasurableSet] with x hx
+  rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+  have hb := (radialTruncation_fderiv_le hH hr (q := q) (hF.differentiableAt (hU.mem_nhds (hmap hx)))).trans
+    (div_le_div_of_nonneg_right (hM x hx) hr.le)
+  exact pow_le_pow_left₀ (norm_nonneg _) hb 2
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma truncated_line_oscillation {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F U) {z e : ℂ} (he : ‖e‖ = 1) {a b x y : ℝ}
+    (hx : x ∈ Icc a b) (hy : y ∈ Icc a b)
+    (hmap : MapsTo (fun s : ℝ => z+s • e) (Icc a b) U)
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
+    (radialTruncation H q r (F (z+y • e))-
+      radialTruncation H q r (F (z+x • e)))^2 ≤
+      (b-a)*(∫ s in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F) (z+s • e)‖^2) := by
+  let c := fun s : ℝ => z+s • e
+  let u := radialTruncation H q r ∘ F
+  have hab := hx.1.trans hx.2
+  have hc : ContDiff ℝ 1 c := contDiff_const.add (contDiff_id.smul contDiff_const)
+  have hG : ContDiffOn ℝ 1 (F ∘ c) (Icc a b) :=
+    ((hF.contDiffOn hU).restrict_scalars ℝ).comp hc.contDiffOn hmap
+  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
+  have hLip := (radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK
+  have hD := lipschitzOn_deriv_memLp hLip
+  have ht := acl_oscillation_energy
+    (radialTruncation_along_C1_AC hr H q (by simpa only [uIcc_of_le hab] using hG)) hD hx hy
+  have hgrad := truncated_gradient_comp_integrable (μ := volume) isCompact_Icc hc.continuous hU hF hmap hH hr q
+  have hderiv (s : ℝ) (hs : s ∈ Icc a b) :
+      deriv (u ∘ c) s = fderiv ℝ u (c s) e := by
+    apply radialTruncation_directional_deriv hH hr q (c := c) (t := s) (d := e) (hF.differentiableAt (hU.mem_nhds (hmap hs)))
+    simpa only [one_smul,c,id_eq] using ((hasDerivAt_id s).smul_const e).const_add z
+  have hbnd : (∫ s in a..b, (deriv (u ∘ c) s)^2) ≤
+      ∫ s in a..b, ‖fderiv ℝ u (c s)‖^2 := by
+    apply intervalIntegral.integral_mono_on hab
+      ((intervalIntegrable_iff_integrableOn_Ioc_of_le hab).2 hD.integrable_sq)
+      ((intervalIntegrable_iff_integrableOn_Icc_of_le hab).2 hgrad)
+    intro s hs
+    change (deriv (u ∘ c) s)^2 ≤ ‖fderiv ℝ u (c s)‖^2
+    rw [hderiv s hs]
+    have hh := (fderiv ℝ u (c s)).le_opNorm e
+    rw [he,mul_one] at hh
+    have hh' := pow_le_pow_left₀ (norm_nonneg _) hh 2
+    simpa only [Real.norm_eq_abs,sq_abs] using hh'
+  exact ht.trans (mul_le_mul_of_nonneg_left hbnd (sub_nonneg.2 hab))
+
+theorem truncated_rectangle_two_lines {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F U) {a b c d x₁ x₂ y₁ y₂ : ℝ}
+    (hx₁ : x₁ ∈ Icc a b) (hx₂ : x₂ ∈ Icc a b)
+    (hy₁ : y₁ ∈ Icc c d) (hy₂ : y₂ ∈ Icc c d)
+    (hrect : ∀ x ∈ Icc a b, ∀ y ∈ Icc c d, (x:ℂ)+(y:ℂ)*Complex.I ∈ U)
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
+    (radialTruncation H q r (F ((x₁:ℂ)+(y₁:ℂ)*Complex.I))-
+      radialTruncation H q r (F ((x₂:ℂ)+(y₂:ℂ)*Complex.I)))^2 ≤
+      2*(b-a)*(∫ x in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F)
+        ((x:ℂ)+(y₁:ℂ)*Complex.I)‖^2)+
+      2*(d-c)*(∫ y in c..d, ‖fderiv ℝ (radialTruncation H q r ∘ F)
+        ((x₂:ℂ)+(y:ℂ)*Complex.I)‖^2) := by
+  have hh := truncated_line_oscillation hU hF (z := (y₁:ℂ)*Complex.I) (e := 1)
+    (by simp) hx₂ hx₁ (by
+      intro x hx
+      simpa only [Complex.real_smul,Complex.ofReal_mul,mul_one,add_comm] using hrect x hx y₁ hy₁) hH hr q
+  have hv := truncated_line_oscillation hU hF (z := (x₂:ℂ)) (e := Complex.I)
+    (by simp) hy₂ hy₁ (by
+      intro y hy
+      simpa only [Complex.real_smul] using hrect x₂ hx₂ y hy) hH hr q
+  simp only [Complex.real_smul,mul_one] at hh hv
+  have hi : (∫ x in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F) ((y₁:ℂ)*Complex.I+(x:ℂ))‖^2) =
+      ∫ x in a..b, ‖fderiv ℝ (radialTruncation H q r ∘ F) ((x:ℂ)+(y₁:ℂ)*Complex.I)‖^2 := by
+    congr 1
+    funext x
+    rw [add_comm]
+  rw [hi,add_comm ((y₁:ℂ)*Complex.I) (x₁:ℂ),add_comm ((y₁:ℂ)*Complex.I) (x₂:ℂ)] at hh
+  nlinarith [sq_nonneg
+    (radialTruncation H q r (F ((x₁:ℂ)+(y₁:ℂ)*Complex.I))+
+      radialTruncation H q r (F ((x₂:ℂ)+(y₂:ℂ)*Complex.I))-
+      2*radialTruncation H q r (F ((x₂:ℂ)+(y₁:ℂ)*Complex.I)))]
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma radial_section_mem {z : ℂ} {a b s : ℝ} (ha : 0 < a) (hs : s ∈ Icc a b) (θ : ℝ) :
+    z+s • circleMap 0 1 θ ∈ closedBall z b := by
+  rw [mem_closedBall_iff_norm,add_sub_cancel_left,norm_smul,norm_circleMap_zero,abs_one,mul_one,
+    Real.norm_of_nonneg (ha.trans_le hs.1).le]
+  exact hs.2
+
+lemma truncated_radial_energy {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F U) {z : ℂ} {a b : ℝ}
+    (ha : 0 < a) (hab : a ≤ b) (hball : closedBall z b ⊆ U)
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) (θ : ℝ) :
+    (radialTruncation H q r (F (z+b • circleMap 0 1 θ))-
+      radialTruncation H q r (F (z+a • circleMap 0 1 θ)))^2 ≤
+      Real.log (b/a)*(∫ s in a..b, s*‖fderiv ℝ (radialTruncation H q r ∘ F)
+        (z+s • circleMap 0 1 θ)‖^2) := by
+  let c := fun s : ℝ => z+s • circleMap 0 1 θ
+  let u := radialTruncation H q r ∘ F
+  have hc : ContDiff ℝ 1 c := contDiff_const.add (contDiff_id.smul contDiff_const)
+  have hmap : MapsTo c (Icc a b) U := fun s hs => hball (radial_section_mem ha hs θ)
+  have hG : ContDiffOn ℝ 1 (F ∘ c) (Icc a b) :=
+    ((hF.contDiffOn hU).restrict_scalars ℝ).comp hc.contDiffOn hmap
+  have ht := truncated_curve_weighted_energy hr ha hab q hG (H := H)
+  have hgrad := truncated_gradient_comp_integrable (μ := volume) isCompact_Icc hc.continuous hU hF hmap hH hr q
+  have hgradw : IntervalIntegrable
+      (fun s => s*‖fderiv ℝ u (c s)‖^2) volume a b :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).2
+      (hgrad.continuousOn_mul continuousOn_id isCompact_Icc)
+  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
+  have hLip := (radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK
+  have hl2 := lipschitzOn_weighted_deriv_memLp ha hLip
+  have hi : IntegrableOn (fun s => s*(deriv (u ∘ c) s)^2) (Ioc a b) := by
+    have hh := (memLp_two_iff_integrable_sq hl2.aestronglyMeasurable).1 hl2
+    apply hh.congr
+    filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
+    rw [mul_pow,Real.sq_sqrt (ha.trans_le hs.1.le).le]
+    rfl
+  have hderiv (s : ℝ) (hs : s ∈ Icc a b) :
+      deriv (u ∘ c) s = fderiv ℝ u (c s) (circleMap 0 1 θ) := by
+    apply radialTruncation_directional_deriv hH hr q (hF.differentiableAt (hU.mem_nhds (hmap hs)))
+    simpa only [one_smul,c,id_eq] using ((hasDerivAt_id s).smul_const (circleMap 0 1 θ)).const_add z
+  have hbnd : (∫ s in a..b, s*(deriv (u ∘ c) s)^2) ≤
+      ∫ s in a..b, s*‖fderiv ℝ u (c s)‖^2 := by
+    apply intervalIntegral.integral_mono_on hab
+      ((intervalIntegrable_iff_integrableOn_Ioc_of_le hab).2 hi) hgradw
+    intro s hs
+    rw [hderiv s hs]
+    apply mul_le_mul_of_nonneg_left _ (ha.trans_le hs.1).le
+    have hh := (fderiv ℝ u (c s)).le_opNorm (circleMap 0 1 θ)
+    rw [norm_circleMap_zero,abs_one,mul_one] at hh
+    have hh' := pow_le_pow_left₀ (norm_nonneg _) hh 2
+    simpa only [Real.norm_eq_abs,sq_abs] using hh'
+  have hlog : 0 ≤ Real.log (b/a) := Real.log_nonneg ((one_le_div ha).2 hab)
+  exact ht.trans (mul_le_mul_of_nonneg_left hbnd hlog)
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+theorem truncated_circle_energy {U : Set ℂ} (hU : IsOpen U) {F : ℂ → ℂ}
+    (hF : DifferentiableOn ℂ F U) {z : ℂ} {a b : ℝ}
+    (ha : 0 < a) (hab : a ≤ b) (hball : closedBall z b ⊆ U)
+    {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q : ℂ) :
+    ((∫ θ in -Real.pi..Real.pi, radialTruncation H q r (F (z+b • circleMap 0 1 θ)))-
+      (∫ θ in -Real.pi..Real.pi, radialTruncation H q r (F (z+a • circleMap 0 1 θ))))^2 ≤
+      2*Real.pi*Real.log (b/a)*(∫ θ in -Real.pi..Real.pi, ∫ s in a..b,
+        s*‖fderiv ℝ (radialTruncation H q r ∘ F) (z+s • circleMap 0 1 θ)‖^2) := by
+  let u := radialTruncation H q r ∘ F
+  let D := fun θ => u (z+b • circleMap 0 1 θ)-u (z+a • circleMap 0 1 θ)
+  let E := fun θ => ∫ s in a..b, s*‖fderiv ℝ u (z+s • circleMap 0 1 θ)‖^2
+  have hp : -Real.pi ≤ Real.pi := by linarith [Real.pi_pos]
+  have hc (s : ℝ) (hs : s ∈ Icc a b) : Continuous (fun θ => u (z+s • circleMap 0 1 θ)) := by
+    apply (radialTruncation_continuous hr H q).comp
+    apply continuousOn_univ.1
+    exact hF.continuousOn.comp
+      (continuous_const.add ((continuous_circleMap 0 1).const_smul s)).continuousOn
+      (fun θ _ => hball (radial_section_mem ha hs θ))
+  have hca := hc a (left_mem_Icc.2 hab)
+  have hcb := hc b (right_mem_Icc.2 hab)
+  have hD : Continuous D := hcb.sub hca
+  let c := fun p : ℝ × ℝ => z+p.2 • circleMap 0 1 p.1
+  have hcc : Continuous c := continuous_const.add (continuous_snd.smul ((continuous_circleMap 0 1).comp continuous_fst))
+  have hmap : MapsTo c (Icc (-Real.pi) Real.pi ×ˢ Icc a b) U :=
+    fun p hp => hball (radial_section_mem ha hp.2 p.1)
+  have hi := truncated_gradient_comp_integrable (μ := volume) (isCompact_Icc.prod isCompact_Icc)
+    hcc hU hF hmap hH hr q
+  have hiw : IntegrableOn (fun p : ℝ × ℝ => p.2*‖fderiv ℝ u (c p)‖^2)
+      (Icc (-Real.pi) Real.pi ×ˢ Icc a b) :=
+    hi.continuousOn_mul continuous_snd.continuousOn (isCompact_Icc.prod isCompact_Icc)
+  have hiprod : Integrable (fun p : ℝ × ℝ => p.2*‖fderiv ℝ u (c p)‖^2)
+      ((volume.restrict (Ioc (-Real.pi) Real.pi)).prod (volume.restrict (Ioc a b))) := by
+    rw [Measure.prod_restrict]
+    exact hiw.mono_set (prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)
+  have hE : IntervalIntegrable E volume (-Real.pi) Real.pi := by
+    apply (intervalIntegrable_iff_integrableOn_Ioc_of_le hp).2
+    simpa only [E,intervalIntegral.integral_of_le hab,c,IntegrableOn] using hiprod.integral_prod_left
+  have hcs := integral_mul_square_le (continuousOn_memLp_two_Ioc hp hD.continuousOn)
+    (memLp_const (1:ℝ))
+  have hid : (∫ θ in Ioc (-Real.pi) Real.pi, D θ*1) =
+      (∫ θ in -Real.pi..Real.pi, u (z+b • circleMap 0 1 θ))-
+      (∫ θ in -Real.pi..Real.pi, u (z+a • circleMap 0 1 θ)) := by
+    simp only [mul_one,← intervalIntegral.integral_of_le hp,D]
+    exact intervalIntegral.integral_sub (hcb.intervalIntegrable _ _) (hca.intervalIntegrable _ _)
+  have hi1 : (∫ θ in Ioc (-Real.pi) Real.pi, (1:ℝ)^2) = 2*Real.pi := by
+    rw [← intervalIntegral.integral_of_le hp]
+    simp
+    ring
+  rw [hid,hi1,← intervalIntegral.integral_of_le hp] at hcs
+  have hbound : (∫ θ in -Real.pi..Real.pi, D θ^2) ≤
+      ∫ θ in -Real.pi..Real.pi, Real.log (b/a)*E θ := by
+    apply intervalIntegral.integral_mono_on hp ((hD.pow 2).intervalIntegrable _ _)
+      (hE.const_mul _)
+    intro θ hθ
+    exact truncated_radial_energy hU hF ha hab hball hH hr q θ
+  rw [intervalIntegral.integral_const_mul] at hbound
+  have hm := mul_le_mul_of_nonneg_left hbound (by positivity : 0 ≤ 2*Real.pi)
+  apply hcs.trans
+  convert hm using 1 <;> ring
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end OAI
diff --git a/lean/OAI/Probability/SLELocalization/WeightedEnergy.lean b/lean/OAI/Probability/SLELocalization/WeightedEnergy.lean
index 9b3396f1e554a71e67624692a0e370bec17cd05e..71c4f6aa09e31d6f3380442420980e1acab88796 100644
--- a/lean/OAI/Probability/SLELocalization/WeightedEnergy.lean
+++ b/lean/OAI/Probability/SLELocalization/WeightedEnergy.lean
@@ -1,859 +1,859 @@
-import Mathlib
-
-namespace OAI
-
-
-/-! Conformal localization and Rectangle Energy estimates. -/
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-
-lemma integral_mul_square_le {α : Type*} [MeasurableSpace α] {μ : Measure α}
-    {f g : α → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) :
-    (∫ x, f x*g x ∂μ)^2 ≤ (∫ x, f x^2 ∂μ)*(∫ x, g x^2 ∂μ) := by
-  have hi {a b : α → ℝ} (ha : MemLp a 2 μ) (hb : MemLp b 2 μ) :
-      inner ℝ (ha.toLp a) (hb.toLp b) = ∫ x, a x*b x ∂μ := by
-    rw [MeasureTheory.L2.inner_def]
-    apply integral_congr_ae
-    filter_upwards [ha.coeFn_toLp,hb.coeFn_toLp] with x hx hy
-    rw [hx,hy]
-    change b x*a x = a x*b x
-    ring
-  have hn := norm_inner_le_norm (𝕜 := ℝ) (hf.toLp f) (hg.toLp g)
-  have hh := pow_le_pow_left₀ (norm_nonneg _) hn 2
-  rw [mul_pow,← real_inner_self_eq_norm_sq (hf.toLp f),← real_inner_self_eq_norm_sq (hg.toLp g),
-    hi hf hg,hi hf hf,hi hg hg] at hh
-  simpa only [Real.norm_eq_abs,sq_abs,← sq] using hh
-
-lemma continuousOn_memLp_two_Ioc {a b : ℝ} (hab : a ≤ b) {f : ℝ → ℝ}
-    (hf : ContinuousOn f (Icc a b)) : MemLp f 2 (volume.restrict (Ioc a b)) := by
-  apply (memLp_two_iff_integrable_sq ((hf.mono Ioc_subset_Icc_self).aestronglyMeasurable measurableSet_Ioc)).2
-  exact ((hf.pow 2).intervalIntegrable_of_Icc hab).1
-
-lemma weighted_interval_energy {a b : ℝ} (ha : 0 < a) (hab : a ≤ b)
-    {F D : ℝ → ℝ} (hD : ContinuousOn D (Icc a b))
-    (hF : ∀ s ∈ Icc a b, HasDerivAt F (D s) s) :
-    (F b-F a)^2 ≤ Real.log (b/a)*(∫ s in a..b, s*(D s)^2) := by
-  let f := fun s : ℝ => Real.sqrt s*D s
-  let g := fun s : ℝ => (Real.sqrt s)⁻¹
-  have hs (s : ℝ) (hs : s ∈ Icc a b) : 0 < s := ha.trans_le hs.1
-  have hf : ContinuousOn f (Icc a b) := Real.continuous_sqrt.continuousOn.mul hD
-  have hg : ContinuousOn g (Icc a b) := Real.continuous_sqrt.continuousOn.inv₀
-    (fun s hs' => (Real.sqrt_pos.2 (hs s hs')).ne')
-  have hc := integral_mul_square_le (continuousOn_memLp_two_Ioc hab hf)
-    (continuousOn_memLp_two_Ioc hab hg)
-  have hfg : (∫ s in Ioc a b, f s*g s) = F b-F a := by
-    calc
-      _ = ∫ s in a..b, D s := by
-        rw [intervalIntegral.integral_of_le hab]
-        apply setIntegral_congr_fun measurableSet_Ioc
-        intro s hs'
-        dsimp [f,g]
-        have hne := (Real.sqrt_pos.2 (hs s (Ioc_subset_Icc_self hs'))).ne'
-        field_simp
-      _ = _ := intervalIntegral.integral_eq_sub_of_hasDerivAt
-        (fun s hs' => hF s (by simpa only [uIcc_of_le hab] using hs'))
-        (hD.intervalIntegrable_of_Icc hab)
-  have hff : (∫ s in Ioc a b, f s^2) = ∫ s in a..b, s*(D s)^2 := by
-    rw [intervalIntegral.integral_of_le hab]
-    apply setIntegral_congr_fun measurableSet_Ioc
-    intro s hs'
-    dsimp only [f]
-    rw [mul_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
-  have hgg : (∫ s in Ioc a b, g s^2) = Real.log (b/a) := by
-    calc
-      _ = ∫ s in a..b, s⁻¹ := by
-        rw [intervalIntegral.integral_of_le hab]
-        apply setIntegral_congr_fun measurableSet_Ioc
-        intro s hs'
-        dsimp only [g]
-        rw [inv_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
-      _ = _ := by
-        apply integral_inv
-        rw [uIcc_of_le hab]
-        intro hz
-        exact (not_le_of_gt ha) hz.1
-  rw [hfg,hff,hgg] at hc
-  simpa only [mul_comm] using hc
-
-def radialTruncation (H : ℝ) (q : ℂ) (r : ℝ) (w : ℂ) : ℝ :=
-  min H (Real.log (max r ‖w-q‖/r))
-
-lemma radialTruncation_source {r : ℝ} (hr : 0 < r) (H : ℝ) (q w : ℂ) :
-    radialTruncation H q r w = min H (max 0 (Real.log (‖w-q‖/r))) := by
-  unfold radialTruncation
-  by_cases h : ‖w-q‖ ≤ r
-  · rw [max_eq_left h,div_self hr.ne',Real.log_one,
-      max_eq_left (Real.log_nonpos (div_nonneg (norm_nonneg _) hr.le) ((div_le_one hr).2 h))]
-  · have hh : r < ‖w-q‖ := lt_of_not_ge h
-    rw [max_eq_right hh.le,max_eq_right (Real.log_nonneg ((one_le_div hr).2 hh.le))]
-
-lemma radialTruncation_bounds {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q w : ℂ) :
-    0 ≤ radialTruncation H q r w ∧ radialTruncation H q r w ≤ H := by
-  unfold radialTruncation
-  exact ⟨le_min hH (Real.log_nonneg ((one_le_div hr).2 (le_max_left _ _))),min_le_left _ _⟩
-
-lemma radialTruncation_continuous {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ) :
-    Continuous (radialTruncation H q r) := by
-  have hm : Continuous (fun w : ℂ => max r ‖w-q‖/r) :=
-    (continuous_const.max ((continuous_id.sub continuous_const).norm)).div_const r
-  exact continuous_const.min (hm.log (fun w => (div_pos (lt_of_lt_of_le hr (le_max_left _ _)) hr).ne'))
-
-lemma radialTruncation_zero {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) {q w : ℂ}
-    (hw : ‖w-q‖ ≤ r) : radialTruncation H q r w = 0 := by
-  simp only [radialTruncation,max_eq_left hw,div_self hr.ne',Real.log_one,min_eq_right hH]
-
-lemma radialTruncation_top {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) {q w : ℂ}
-    (hw : Real.exp H*r ≤ ‖w-q‖) : radialTruncation H q r w = H := by
-  have he : 1 ≤ Real.exp H := Real.one_le_exp_iff.2 hH
-  have hrw : r ≤ ‖w-q‖ := (le_mul_of_one_le_left hr.le he).trans hw
-  have hn : 0 < ‖w-q‖/r := div_pos (hr.trans_le hrw) hr
-  have hl : H ≤ Real.log (‖w-q‖/r) :=
-    (Real.le_log_iff_exp_le hn).2 ((le_div_iff₀ hr).2 hw)
-  simp only [radialTruncation,max_eq_right hrw,min_eq_left hl]
-
-lemma radialTruncation_middle {H r : ℝ} (hr : 0 < r) {q w : ℂ}
-    (hw : r < ‖w-q‖) (hw' : ‖w-q‖ < Real.exp H*r) :
-    radialTruncation H q r w = (1/2:ℝ)*Real.log (‖w-q‖^2)-Real.log r := by
-  have hn := hr.trans hw
-  have hl : Real.log (‖w-q‖/r) < H :=
-    (Real.log_lt_iff_lt_exp (div_pos hn hr)).2 ((div_lt_iff₀ hr).2 hw')
-  rw [radialTruncation,max_eq_right hw.le,min_eq_right hl.le,
-    Real.log_div hn.ne' hr.ne',Real.log_pow]
-  ring
-
-lemma log_normSq_fderiv_le {F : ℂ → ℂ} {z q : ℂ} (hd : DifferentiableAt ℂ F z)
-    (hn : 0 < ‖F z-q‖) (r : ℝ) :
-    ‖fderiv ℝ (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r) z‖ ≤
-      ‖deriv F z‖/‖F z-q‖ := by
-  let A := (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
-  have hA : ‖A‖ = ‖deriv F z‖ := by
-    dsimp [A]
-    rw [ContinuousLinearMap.norm_restrictScalars,ContinuousLinearMap.norm_toSpanSingleton]
-  have hF : HasFDerivAt F A z := hd.hasDerivAt.hasFDerivAt.restrictScalars ℝ
-  have hs := ((hF.sub_const q).norm_sq.log (pow_ne_zero 2 hn.ne')).const_mul (1/2:ℝ)
-  have hs' := hs.sub_const (Real.log r)
-  have hd' : HasFDerivAt
-      (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r)
-      ((‖F z-q‖^2)⁻¹ • (innerSL ℝ (F z-q)).comp A) z := by
-    convert! hs' using 1
-    ext v
-    simp
-    ring
-  rw [hd'.fderiv,norm_smul,Real.norm_eq_abs,abs_inv,abs_pow,abs_norm]
-  have hb := (innerSL ℝ (F z-q)).opNorm_comp_le A
-  rw [innerSL_apply_norm,hA] at hb
-  calc
-    (‖F z-q‖^2)⁻¹*‖(innerSL ℝ (F z-q)).comp A‖ ≤
-        (‖F z-q‖^2)⁻¹*(‖F z-q‖*‖deriv F z‖) :=
-      mul_le_mul_of_nonneg_left hb (by positivity)
-    _ = ‖deriv F z‖/‖F z-q‖ := by field_simp
-
-lemma radialTruncation_fderiv_bound {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r)
-    {F : ℂ → ℂ} {z q : ℂ} (hd : DifferentiableAt ℂ F z) :
-    ‖fderiv ℝ (radialTruncation H q r ∘ F) z‖^2 ≤
-      if r < ‖F z-q‖ ∧ ‖F z-q‖ < Real.exp H*r then
-        ‖deriv F z‖^2/‖F z-q‖^2 else 0 := by
-  by_cases hlow : ‖F z-q‖ ≤ r
-  · have hz : fderiv ℝ (radialTruncation H q r ∘ F) z = 0 := by
-      apply IsLocalMin.fderiv_eq_zero
-      filter_upwards with w
-      change radialTruncation H q r (F z) ≤ radialTruncation H q r (F w)
-      rw [radialTruncation_zero hH hr hlow]
-      exact (radialTruncation_bounds hH hr q _).1
-    rw [hz]
-    simp [not_lt_of_ge hlow]
-  · have hl : r < ‖F z-q‖ := lt_of_not_ge hlow
-    by_cases hhigh : Real.exp H*r ≤ ‖F z-q‖
-    · have hz : fderiv ℝ (radialTruncation H q r ∘ F) z = 0 := by
-        apply IsLocalMax.fderiv_eq_zero
-        filter_upwards with w
-        change radialTruncation H q r (F w) ≤ radialTruncation H q r (F z)
-        rw [radialTruncation_top hH hr hhigh]
-        exact (radialTruncation_bounds hH hr q _).2
-      rw [hz]
-      simp [not_lt_of_ge hhigh]
-    · have hh : ‖F z-q‖ < Real.exp H*r := lt_of_not_ge hhigh
-      have hc : ContinuousAt (fun w => ‖F w-q‖) z := (hd.continuousAt.sub continuousAt_const).norm
-      have he : (radialTruncation H q r ∘ F) =ᶠ[𝓝 z]
-          (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r) := by
-        filter_upwards [continuousAt_const.eventually_lt hc hl,
-          hc.eventually_lt continuousAt_const hh] with w hw hw'
-        exact radialTruncation_middle hr hw hw'
-      rw [he.fderiv_eq,ite_eq_left ⟨hl,hh⟩]
-      have ht := pow_le_pow_left₀ (norm_nonneg _) (log_normSq_fderiv_le hd (hr.trans hl) r) 2
-      simpa only [div_pow] using ht
-
-lemma radialTruncation_lipschitz {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ) :
-    LipschitzWith ⟨1/r,by positivity⟩ (radialTruncation H q r) := by
-  let K : ℝ≥0 := ⟨1/r,by positivity⟩
-  have hlog : LipschitzOnWith K Real.log (Ici r) := by
-    apply Convex.lipschitzOnWith_of_nnnorm_deriv_le
-      (fun x hx => Real.differentiableAt_log (hr.trans_le hx).ne') _ (convex_Ici r)
-    intro x hx
-    change ‖deriv Real.log x‖ ≤ 1/r
-    rw [Real.deriv_log,Real.norm_of_nonneg (inv_nonneg.2 (hr.trans_le hx).le),one_div]
-    exact inv_anti₀ hr hx
-  have hnorm : LipschitzWith 1 (fun w : ℂ => ‖w-q‖) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro x y
-    calc
-      dist ‖x-q‖ ‖y-q‖ ≤ ‖(x-q)-(y-q)‖ := dist_norm_norm_le _ _
-      _ = ‖x-y‖ := by congr 1; abel
-      _ = (1:ℝ≥0)*dist x y := by simp [dist_eq_norm]
-  have hmax := hnorm.const_max r
-  have hl : LipschitzWith K (fun w : ℂ => Real.log (max r ‖w-q‖)) := by
-    apply LipschitzWith.of_dist_le_mul
-    intro x y
-    exact (hlog.dist_le_mul (max r ‖x-q‖) (show r ≤ max r ‖x-q‖ from le_max_left _ _) (max r ‖y-q‖) (show r ≤ max r ‖y-q‖ from le_max_left _ _)).trans
-      (mul_le_mul_of_nonneg_left (by simpa using hmax.dist_le_mul x y) K.coe_nonneg)
-  have hsub : LipschitzWith K (fun w : ℂ => Real.log (max r ‖w-q‖)-Real.log r) := by
-    simpa using hl.sub (LipschitzWith.const (Real.log r))
-  have he : radialTruncation H q r = fun w : ℂ => min H (Real.log (max r ‖w-q‖)-Real.log r) := by
-    ext w
-    rw [radialTruncation,Real.log_div (hr.trans_le (le_max_left _ _)).ne' hr.ne']
-  rw [he]
-  exact hsub.const_min H
-
-lemma radialTruncation_along_C1_AC {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ)
-    {G : ℝ → ℂ} {a b : ℝ} (hG : ContDiffOn ℝ 1 G (uIcc a b)) :
-    AbsolutelyContinuousOnInterval (radialTruncation H q r ∘ G) a b := by
-  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_uIcc _ _) isCompact_uIcc
-  exact ((radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK).absolutelyContinuousOnInterval
-
-lemma radialTruncation_along_C1_FTC {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ)
-    {G : ℝ → ℂ} {a b : ℝ} (hG : ContDiffOn ℝ 1 G (uIcc a b)) :
-    (∫ s in a..b, deriv (radialTruncation H q r ∘ G) s) =
-      radialTruncation H q r (G b)-radialTruncation H q r (G a) :=
-  (radialTruncation_along_C1_AC hr H q hG).integral_deriv_eq_sub
-
-lemma radialTruncation_directional_deriv {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r)
-    (q : ℂ) {F : ℂ → ℂ} {c : ℝ → ℂ} {d : ℂ} {t : ℝ}
-    (hF : DifferentiableAt ℂ F (c t)) (hc : HasDerivAt c d t) :
-    deriv (radialTruncation H q r ∘ F ∘ c) t =
-      fderiv ℝ (radialTruncation H q r ∘ F) (c t) d := by
-  by_cases hl : ‖F (c t)-q‖ ≤ r
-  · have hz : fderiv ℝ (radialTruncation H q r ∘ F) (c t) = 0 := by
-      apply IsLocalMin.fderiv_eq_zero
-      filter_upwards with w
-      change radialTruncation H q r (F (c t)) ≤ radialTruncation H q r (F w)
-      rw [radialTruncation_zero hH hr hl]
-      exact (radialTruncation_bounds hH hr q _).1
-    rw [hz,zero_apply]
-    apply IsLocalMin.deriv_eq_zero
-    filter_upwards with s
-    change radialTruncation H q r (F (c t)) ≤ radialTruncation H q r (F (c s))
-    rw [radialTruncation_zero hH hr hl]
-    exact (radialTruncation_bounds hH hr q _).1
-  · have hl' := lt_of_not_ge hl
-    by_cases hh : Real.exp H*r ≤ ‖F (c t)-q‖
-    · have hz : fderiv ℝ (radialTruncation H q r ∘ F) (c t) = 0 := by
-        apply IsLocalMax.fderiv_eq_zero
-        filter_upwards with w
-        change radialTruncation H q r (F w) ≤ radialTruncation H q r (F (c t))
-        rw [radialTruncation_top hH hr hh]
-        exact (radialTruncation_bounds hH hr q _).2
-      rw [hz,zero_apply]
-      apply IsLocalMax.deriv_eq_zero
-      filter_upwards with s
-      change radialTruncation H q r (F (c s)) ≤ radialTruncation H q r (F (c t))
-      rw [radialTruncation_top hH hr hh]
-      exact (radialTruncation_bounds hH hr q _).2
-    · have hh' := lt_of_not_ge hh
-      have hcont : ContinuousAt (fun w => ‖F w-q‖) (c t) := (hF.continuousAt.sub continuousAt_const).norm
-      have he : (radialTruncation H q r ∘ F) =ᶠ[𝓝 (c t)]
-          (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r) := by
-        filter_upwards [continuousAt_const.eventually_lt hcont hl',
-          hcont.eventually_lt continuousAt_const hh'] with w hw hw'
-        exact radialTruncation_middle hr hw hw'
-      have hdiff : DifferentiableAt ℝ (radialTruncation H q r ∘ F) (c t) := by
-        apply he.differentiableAt_iff.2
-        exact (((((hF.restrictScalars ℝ).sub_const q).norm_sq (𝕜 := ℝ)).log
-          (pow_ne_zero 2 (hr.trans hl').ne')).const_mul (1/2:ℝ)).sub_const (Real.log r)
-      exact (hdiff.hasFDerivAt.comp_hasDerivAt t hc).deriv
-
-lemma acl_interval_energy {a b : ℝ} (hab : a ≤ b) {f : ℝ → ℝ}
-    (hf : AbsolutelyContinuousOnInterval f a b)
-    (hD : MemLp (deriv f) 2 (volume.restrict (Ioc a b))) :
-    (f b-f a)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) := by
-  have hc := integral_mul_square_le hD (memLp_const (1:ℝ))
-  have he : (∫ s in Ioc a b, deriv f s*1) = f b-f a := by
-    simp only [mul_one,← intervalIntegral.integral_of_le hab,hf.integral_deriv_eq_sub]
-  have he1 : (∫ s in Ioc a b, (1:ℝ)^2) = b-a := by simp [hab]
-  rw [he,he1] at hc
-  simpa only [intervalIntegral.integral_of_le hab,mul_comm] using hc
-
-lemma weighted_acl_interval_energy {a b : ℝ} (ha : 0 < a) (hab : a ≤ b)
-    {f : ℝ → ℝ} (hf : AbsolutelyContinuousOnInterval f a b)
-    (hD : MemLp (fun s => Real.sqrt s*deriv f s) 2 (volume.restrict (Ioc a b))) :
-    (f b-f a)^2 ≤ Real.log (b/a)*(∫ s in a..b, s*(deriv f s)^2) := by
-  let g := fun s : ℝ => (Real.sqrt s)⁻¹
-  have hs (s : ℝ) (hs : s ∈ Icc a b) : 0 < s := ha.trans_le hs.1
-  have hg : ContinuousOn g (Icc a b) := Real.continuous_sqrt.continuousOn.inv₀
-    (fun s hs' => (Real.sqrt_pos.2 (hs s hs')).ne')
-  have hc := integral_mul_square_le hD (continuousOn_memLp_two_Ioc hab hg)
-  have hfg : (∫ s in Ioc a b, (Real.sqrt s*deriv f s)*g s) = f b-f a := by
-    calc
-      _ = ∫ s in a..b, deriv f s := by
-        rw [intervalIntegral.integral_of_le hab]
-        apply setIntegral_congr_fun measurableSet_Ioc
-        intro s hs'
-        dsimp [g]
-        have hne := (Real.sqrt_pos.2 (hs s (Ioc_subset_Icc_self hs'))).ne'
-        field_simp
-      _ = _ := hf.integral_deriv_eq_sub
-  have hff : (∫ s in Ioc a b, (Real.sqrt s*deriv f s)^2) =
-      ∫ s in a..b, s*(deriv f s)^2 := by
-    rw [intervalIntegral.integral_of_le hab]
-    apply setIntegral_congr_fun measurableSet_Ioc
-    intro s hs'
-    dsimp only
-    rw [mul_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
-  have hgg : (∫ s in Ioc a b, g s^2) = Real.log (b/a) := by
-    calc
-      _ = ∫ s in a..b, s⁻¹ := by
-        rw [intervalIntegral.integral_of_le hab]
-        apply setIntegral_congr_fun measurableSet_Ioc
-        intro s hs'
-        dsimp only [g]
-        rw [inv_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
-      _ = _ := by
-        apply integral_inv
-        rw [uIcc_of_le hab]
-        intro hz
-        exact (not_le_of_gt ha) hz.1
-  rw [hfg,hff,hgg] at hc
-  simpa only [mul_comm] using hc
-
-lemma lipschitzOn_deriv_memLp {a b : ℝ} {K : ℝ≥0} {f : ℝ → ℝ}
-    (hf : LipschitzOnWith K f (Icc a b)) :
-    MemLp (deriv f) 2 (volume.restrict (Ioc a b)) := by
-  apply MemLp.of_bound (measurable_deriv f).aestronglyMeasurable (K:ℝ)
-  filter_upwards [ae_restrict_mem measurableSet_Ioc,
-    (ae_restrict_of_ae (μ := volume) (volume.ae_ne b))] with s hs hsb
-  have hsl : s ∈ Ioo a b := ⟨hs.1,lt_of_le_of_ne hs.2 hsb⟩
-  exact norm_deriv_le_of_lipschitzOn (mem_of_superset (isOpen_Ioo.mem_nhds hsl) Ioo_subset_Icc_self) hf
-
-lemma lipschitzOn_weighted_deriv_memLp {a b : ℝ} {K : ℝ≥0} {f : ℝ → ℝ}
-    (ha : 0 < a) (hf : LipschitzOnWith K f (Icc a b)) :
-    MemLp (fun s => Real.sqrt s*deriv f s) 2 (volume.restrict (Ioc a b)) := by
-  apply (lipschitzOn_deriv_memLp hf).of_le_mul
-    ((Real.continuous_sqrt.measurable).mul (measurable_deriv f)).aestronglyMeasurable (c := Real.sqrt b)
-  filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
-  rw [Pi.mul_apply,norm_mul,Real.norm_of_nonneg (Real.sqrt_pos.2 (ha.trans hs.1)).le]
-  exact mul_le_mul_of_nonneg_right (Real.sqrt_le_sqrt hs.2) (norm_nonneg _)
-
-lemma truncated_curve_weighted_energy {H r a b : ℝ} (hr : 0 < r)
-    (ha : 0 < a) (hab : a ≤ b) (q : ℂ) {G : ℝ → ℂ}
-    (hG : ContDiffOn ℝ 1 G (Icc a b)) :
-    (radialTruncation H q r (G b)-radialTruncation H q r (G a))^2 ≤
-      Real.log (b/a)*(∫ s in a..b, s*(deriv (radialTruncation H q r ∘ G) s)^2) := by
-  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
-  have hL := (radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK
-  exact weighted_acl_interval_energy ha hab
-    (radialTruncation_along_C1_AC hr H q (by simpa only [uIcc_of_le hab] using hG))
-    (lipschitzOn_weighted_deriv_memLp ha hL)
-
-lemma mean_difference_square_le
-    {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
-    {μ : Measure α} {ν : Measure β} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
-    {u : α → ℝ} {v : β → ℝ} (hu : MemLp u 2 μ) (hv : MemLp v 2 ν) :
-    ((∫ x, u x ∂μ)-(∫ y, v y ∂ν))^2 ≤
-      ∫ x, ∫ y, (u x-v y)^2 ∂ν ∂μ := by
-  have hD := (hu.comp_fst ν).sub (hv.comp_snd μ)
-  have hc := integral_mul_square_le hD (memLp_const (1:ℝ))
-  have hDi : Integrable (fun p : α × β => u p.1-v p.2) (μ.prod ν) :=
-    hD.integrable (by norm_num)
-  have he : (∫ p : α × β, (u p.1-v p.2)*1 ∂μ.prod ν) =
-      (∫ x, u x ∂μ)-(∫ y, v y ∂ν) := by
-    simp only [mul_one]
-    rw [integral_prod _ hDi]
-    have hiu := hu.integrable (by norm_num : (1:ℝ≥0∞) ≤ 2)
-    have hiv := hv.integrable (by norm_num : (1:ℝ≥0∞) ≤ 2)
-    simp_rw [integral_sub (integrable_const _) hiv]
-    simp only [integral_const,probReal_univ,one_smul]
-    rw [integral_sub hiu (integrable_const _)]
-    simp
-  have he1 : (∫ _ : α × β, (1:ℝ)^2 ∂μ.prod ν) = 1 := by simp
-  simp only [Pi.sub_apply] at hc
-  rw [he,he1,mul_one] at hc
-  have hsq : Integrable (fun p : α × β => (u p.1-v p.2)^2) (μ.prod ν) := hD.integrable_sq
-  rw [integral_prod _ hsq] at hc
-  exact hc
-
-lemma acl_oscillation_energy {a b : ℝ} {f : ℝ → ℝ}
-    (hf : AbsolutelyContinuousOnInterval f a b)
-    (hD : MemLp (deriv f) 2 (volume.restrict (Ioc a b)))
-    {x y : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b) :
-    (f y-f x)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) := by
-  have hab := hx.1.trans hx.2
-  suffices ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, x ≤ y →
-      (f y-f x)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) by
-    rcases le_total x y with h | h
-    · exact this x hx y hy h
-    · have hh := this y hy x hx h
-      nlinarith
-  intro x hx y hy hxy
-  have hsub : Ioc x y ⊆ Ioc a b := fun s hs => ⟨hx.1.trans_lt hs.1,hs.2.trans hy.2⟩
-  have hD' := hD.mono_measure (Measure.restrict_mono_set _ hsub)
-  have hf' : AbsolutelyContinuousOnInterval f x y := hf.mono
-    (by simpa only [uIcc_of_le hab,uIcc_of_le hxy] using (Icc_subset_Icc hx.1 hy.2))
-  have he := acl_interval_energy hxy hf' hD'
-  have hmono : (∫ s in x..y, (deriv f s)^2) ≤ ∫ s in a..b, (deriv f s)^2 := by
-    rw [intervalIntegral.integral_of_le hxy,intervalIntegral.integral_of_le hab]
-    apply setIntegral_mono_set hD.integrable_sq
-      (by filter_upwards with s; positivity) ?_
-    exact ae_of_all _ (fun s hs => hsub hs)
-  have hnon : 0 ≤ ∫ s in a..b, (deriv f s)^2 :=
-    intervalIntegral.integral_nonneg hab (fun s hs => sq_nonneg _)
-  exact he.trans ((mul_le_mul_of_nonneg_left hmono (sub_nonneg.2 hxy)).trans
-    (mul_le_mul_of_nonneg_right (by linarith [hx.1,hy.2]) hnon))
-
-lemma acl_congr {f g : ℝ → ℝ} {a b : ℝ}
-    (hf : AbsolutelyContinuousOnInterval f a b) (he : EqOn f g (uIcc a b)) :
-    AbsolutelyContinuousOnInterval g a b := by
-  rw [absolutelyContinuousOnInterval_iff] at hf ⊢
-  intro ε hε
-  obtain ⟨δ,hδ,hδ'⟩ := hf ε hε
-  refine ⟨δ,hδ,?_⟩
-  intro E hE hsum
-  have hh := hδ' E hE hsum
-  convert hh using 1
-  apply Finset.sum_congr rfl
-  intro i hi
-  rw [he ((hE.1 i hi).1),he ((hE.1 i hi).2)]
-
-lemma acl_of_interior {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
-    (hc : ContinuousOn f (Icc a b))
-    (hi : IntervalIntegrable (deriv f) volume a b)
-    (hloc : ∀ x ∈ Ioo a b, ∀ y ∈ Ioo a b, AbsolutelyContinuousOnInterval f x y) :
-    AbsolutelyContinuousOnInterval f a b := by
-  let m := (a+b)/2
-  have hm : m ∈ Ioo a b := by dsimp [m]; constructor <;> linarith
-  let g := fun x => (∫ s in m..x, deriv f s)+f m
-  have hg : AbsolutelyContinuousOnInterval g a b := by
-    have hh := hi.absolutelyContinuousOnInterval_intervalIntegral
-      (c := m) (by simpa only [uIcc_of_le hab.le] using Ioo_subset_Icc_self hm)
-    have hk : AbsolutelyContinuousOnInterval (fun _ : ℝ => f m) a b :=
-      (LipschitzWith.const (f m)).lipschitzOnWith.absolutelyContinuousOnInterval
-    exact hh.add hk
-  have he : EqOn f g (Ioo a b) := by
-    intro x hx
-    dsimp only [g]
-    rw [(hloc m hm x hx).integral_deriv_eq_sub]
-    ring
-  have he' : EqOn f g (Icc a b) := he.of_subset_closure hc
-    (by simpa only [uIcc_of_le hab.le] using hg.continuousOn)
-    Ioo_subset_Icc_self (by rw [closure_Ioo hab.ne])
-  exact acl_congr hg (by simpa only [uIcc_of_le hab.le] using he'.symm)
-
-lemma boundary_zero_energy {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
-    (hc : ContinuousOn f (Icc a b))
-    (hD : MemLp (deriv f) 2 (volume.restrict (Ioc a b)))
-    (hloc : ∀ x ∈ Ioo a b, ∀ y ∈ Ioo a b, AbsolutelyContinuousOnInterval f x y)
-    {z : ℝ} (hz : z ∈ Icc a b) (hfz : f z = 0) {y : ℝ} (hy : y ∈ Icc a b) :
-    (f y)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) := by
-  have hi : IntervalIntegrable (deriv f) volume a b :=
-    (intervalIntegrable_iff_integrableOn_Ioc_of_le hab.le).2 (hD.integrable (by norm_num))
-  have h := acl_oscillation_energy (acl_of_interior hab hc hi hloc) hD hz hy
-  simpa only [hfz,sub_zero] using h
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology
-namespace SLEExactGauge.Layer
-
-lemma compact_parameter_derivative {F F' : ℝ → ℝ → ℝ} {a b l h x : ℝ}
-    (hab : a ≤ b) (hlx : l < x) (hxh : x < h)
-    (hF : ContinuousOn (Function.uncurry F) (Icc l h ×ˢ Icc a b))
-    (hF' : ContinuousOn (Function.uncurry F') (Icc l h ×ˢ Icc a b))
-    (hd : ∀ t ∈ Icc a b, ∀ y ∈ Icc l h, HasDerivAt (fun y => F y t) (F' y t) y) :
-    HasDerivAt (fun y => ∫ t in a..b, F y t) (∫ t in a..b, F' x t) x := by
-  have hc (y : ℝ) (hy : y ∈ Icc l h) : ContinuousOn (F y) (Icc a b) :=
-    hF.comp (continuous_const.prodMk continuous_id).continuousOn (fun t ht => ⟨hy,ht⟩)
-  have hc' (y : ℝ) (hy : y ∈ Icc l h) : ContinuousOn (F' y) (Icc a b) :=
-    hF'.comp (continuous_const.prodMk continuous_id).continuousOn (fun t ht => ⟨hy,ht⟩)
-  obtain ⟨C,hC⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hF'
-  have hxx : x ∈ Icc l h := ⟨hlx.le,hxh.le⟩
-  apply (intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
-    (s := Icc l h) (F' := F') (bound := fun _ => C) (Icc_mem_nhds hlx hxh) ?_
-    (by apply ContinuousOn.intervalIntegrable; simpa only [uIcc_of_le hab] using hc x hxx)
-    ?_ ?_ intervalIntegrable_const ?_).2
-  · filter_upwards [Icc_mem_nhds hlx hxh] with y hy
-    have hh := ((hc y hy).mono Ioc_subset_Icc_self).aestronglyMeasurable (μ := volume) measurableSet_Ioc
-    simpa only [uIoc_of_le hab] using hh
-  · have hh := ((hc' x hxx).mono Ioc_subset_Icc_self).aestronglyMeasurable (μ := volume) measurableSet_Ioc
-    simpa only [uIoc_of_le hab] using hh
-  · filter_upwards with t ht y hy
-    exact hC (y,t) ⟨hy,Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)⟩
-  · filter_upwards with t ht y hy
-    exact hd t (Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)) y hy
-
-lemma rotated_trace (H : ℂ →L[ℝ] ℂ →L[ℝ] ℝ) (v : ℂ) (hv : ‖v‖ = 1) :
-    H v v + H (Complex.I*v) (Complex.I*v) = H 1 1 + H Complex.I Complex.I := by
-  have he : v = v.re • (1:ℂ) + v.im • Complex.I := by
-    simp only [Complex.real_smul, mul_one]
-    exact (Complex.re_add_im v).symm
-  have hi : Complex.I*v = (-v.im) • (1:ℂ) + v.re • Complex.I := by
-    apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im]
-  have hh : v.re^2+v.im^2 = 1 := by
-    have hsq := Complex.sq_norm v
-    rw [hv,one_pow,Complex.normSq_apply] at hsq
-    linarith
-  calc
-    _ = H (v.re • (1:ℂ) + v.im • Complex.I) (v.re • (1:ℂ) + v.im • Complex.I) +
-        H ((-v.im) • (1:ℂ) + v.re • Complex.I) ((-v.im) • (1:ℂ) + v.re • Complex.I) := by
-      rw [← he,← hi]
-    _ = (v.re^2+v.im^2)*(H 1 1+H Complex.I Complex.I) := by
-      simp only [map_add,map_smul,add_apply,smul_apply,
-        smul_eq_mul]
-      ring
-    _ = _ := by rw [hh,one_mul]
-
-lemma laplacian_eq_hessian (u : ℂ → ℝ) (z : ℂ) :
-    Laplacian.laplacian u z =
-      fderiv ℝ (fderiv ℝ u) z 1 1 + fderiv ℝ (fderiv ℝ u) z Complex.I Complex.I := by
-  simp only [InnerProductSpace.laplacian_eq_iteratedFDeriv_complexPlane,
-    iteratedFDeriv_two_apply,Matrix.cons_val_zero,Matrix.cons_val_one]
-
-lemma path_second {u : ℂ → ℝ} {c c' : ℝ → ℂ} {c'' : ℂ} {t : ℝ}
-    (hu : ContDiffAt ℝ 2 u (c t)) (hc : HasDerivAt c (c' t) t)
-    (hc' : HasDerivAt c' c'' t) :
-    HasDerivAt (fun s => fderiv ℝ u (c s) (c' s))
-      (fderiv ℝ (fderiv ℝ u) (c t) (c' t) (c' t) + fderiv ℝ u (c t) c'') t := by
-  have hd : ContDiffAt ℝ 1 (fderiv ℝ u) (c t) := hu.fderiv_right (by norm_num)
-  exact (hd.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt t hc |>.clm_apply hc'
-
-def circleMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
-  ∫ θ in -Real.pi..Real.pi, u (r • circleMap 0 1 θ)
-
-def circleDMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
-  ∫ θ in -Real.pi..Real.pi, fderiv ℝ u (r • circleMap 0 1 θ) (circleMap 0 1 θ)
-
-def circleDDMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
-  ∫ θ in -Real.pi..Real.pi,
-    fderiv ℝ (fderiv ℝ u) (r • circleMap 0 1 θ) (circleMap 0 1 θ) (circleMap 0 1 θ)
-
-lemma radial_norm (r θ : ℝ) : ‖r • circleMap 0 1 θ‖ = |r| := by
-  simp [norm_circleMap_zero,Real.norm_eq_abs]
-
-lemma radial_mem {r : ℝ} (hr : |r| < 1) (θ : ℝ) :
-    r • circleMap 0 1 θ ∈ ball (0:ℂ) 1 := by
-  rwa [mem_ball_zero_iff,radial_norm]
-
-lemma radial_continuous : Continuous (fun p : ℝ × ℝ => p.1 • circleMap 0 1 p.2) :=
-  continuous_fst.smul ((continuous_circleMap 0 1).comp continuous_snd)
-
-lemma radial_interval_maps {l h a b : ℝ} (hl : -1 < l) (hh : h < 1) :
-    MapsTo (fun p : ℝ × ℝ => p.1 • circleMap 0 1 p.2)
-      (Icc l h ×ˢ Icc a b) (ball (0:ℂ) 1) := by
-  rintro ⟨r,θ⟩ ⟨hr,hθ⟩
-  exact radial_mem (abs_lt.2 ⟨hl.trans_le hr.1,hr.2.trans_lt hh⟩) θ
-
-lemma circle_means_derivatives {u : ℂ → ℝ}
-    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : |r| < 1) :
-    HasDerivAt (circleMean u) (circleDMean u r) r ∧
-    HasDerivAt (circleDMean u) (circleDDMean u r) r := by
-  have hr' := abs_lt.1 hr
-  let l := (r-1)/2
-  let h := (r+1)/2
-  have hl : -1 < l := by dsimp [l]; linarith
-  have hh : h < 1 := by dsimp [h]; linarith
-  have hlr : l < r := by dsimp [l]; linarith
-  have hrh : r < h := by dsimp [h]; linarith
-  let K := Icc l h ×ˢ Icc (-Real.pi) Real.pi
-  have hm : MapsTo (fun p : ℝ × ℝ => p.1 • circleMap 0 1 p.2) K (ball (0:ℂ) 1) :=
-    radial_interval_maps hl hh
-  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
-    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
-  have hddu : ContDiffOn ℝ 0 (fderiv ℝ (fderiv ℝ u)) (ball 0 1) :=
-    hdu.fderiv_of_isOpen isOpen_ball (by norm_num)
-  have hE : Continuous (fun p : ℝ × ℝ => circleMap 0 1 p.2) :=
-    (continuous_circleMap 0 1).comp continuous_snd
-  have h0 : ContinuousOn (fun p : ℝ × ℝ => u (p.1 • circleMap 0 1 p.2)) K :=
-    hu.continuousOn.comp radial_continuous.continuousOn hm
-  have h1 : ContinuousOn (fun p : ℝ × ℝ => fderiv ℝ u (p.1 • circleMap 0 1 p.2) (circleMap 0 1 p.2)) K :=
-    (hdu.continuousOn.comp radial_continuous.continuousOn hm).clm_apply hE.continuousOn
-  have h2 : ContinuousOn (fun p : ℝ × ℝ =>
-      fderiv ℝ (fderiv ℝ u) (p.1 • circleMap 0 1 p.2) (circleMap 0 1 p.2) (circleMap 0 1 p.2)) K :=
-    ((hddu.continuousOn.comp radial_continuous.continuousOn hm).clm_apply hE.continuousOn).clm_apply hE.continuousOn
-  constructor
-  · apply compact_parameter_derivative
-      (F := fun s θ => u (s • circleMap 0 1 θ))
-      (F' := fun s θ => fderiv ℝ u (s • circleMap 0 1 θ) (circleMap 0 1 θ))
-      (neg_le_self Real.pi_pos.le) hlr hrh h0 h1
-    intro θ hθ s hs
-    have hmem := hm (show (s,θ) ∈ K from ⟨hs,hθ⟩)
-    exact ((hu.differentiableOn (by norm_num)).differentiableAt (isOpen_ball.mem_nhds hmem)).hasFDerivAt.comp_hasDerivAt s
-      (by simpa using (hasDerivAt_id s).smul_const (circleMap 0 1 θ))
-  · apply compact_parameter_derivative
-      (F := fun s θ => fderiv ℝ u (s • circleMap 0 1 θ) (circleMap 0 1 θ))
-      (F' := fun s θ => fderiv ℝ (fderiv ℝ u) (s • circleMap 0 1 θ) (circleMap 0 1 θ) (circleMap 0 1 θ))
-      (neg_le_self Real.pi_pos.le) hlr hrh h1 h2
-    intro θ hθ s hs
-    have hmem := hm (show (s,θ) ∈ K from ⟨hs,hθ⟩)
-    have huc := hu.contDiffAt (isOpen_ball.mem_nhds hmem)
-    have hd := path_second (c := fun t => t • circleMap 0 1 θ) (c' := fun _ => circleMap 0 1 θ)
-      huc (by simpa using (hasDerivAt_id s).smul_const (circleMap 0 1 θ))
-      (hasDerivAt_const s (circleMap 0 1 θ))
-    simpa only [map_zero,add_zero] using hd
-
-lemma circle_laplacian_continuous {u : ℂ → ℝ}
-    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : |r| < 1) :
-    Continuous (fun θ => Laplacian.laplacian u (r • circleMap 0 1 θ)) := by
-  have hc := (hu.fderiv_of_isOpen isOpen_ball (show (1:WithTop ℕ∞)+1 ≤ 2 by norm_num)).fderiv_of_isOpen
-    isOpen_ball (show (0:WithTop ℕ∞)+1 ≤ 1 by norm_num)
-  have hmaps : MapsTo (fun θ => r • circleMap 0 1 θ) univ (ball (0:ℂ) 1) := fun θ _ => radial_mem hr θ
-  have hc' : Continuous (fun θ => fderiv ℝ (fderiv ℝ u) (r • circleMap 0 1 θ)) :=
-    continuousOn_univ.1 (hc.continuousOn.comp ((continuous_circleMap 0 1).const_smul r).continuousOn hmaps)
-  simp_rw [laplacian_eq_hessian]
-  exact (hc'.clm_apply continuous_const |>.clm_apply continuous_const).add
-    (hc'.clm_apply continuous_const |>.clm_apply continuous_const)
-
-lemma circle_angular_laplacian {u : ℂ → ℝ}
-    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : 0 < r) (hr1 : r < 1) :
-    r * (∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (r • circleMap 0 1 θ)) =
-      circleDMean u r + r*circleDDMean u r := by
-  have hrabs : |r| < 1 := by rwa [abs_of_pos hr]
-  let E : ℝ → ℂ := circleMap 0 1
-  let c : ℝ → ℂ := fun θ => r • E θ
-  let H : ℂ → ℂ →L[ℝ] ℂ →L[ℝ] ℝ := fderiv ℝ (fderiv ℝ u)
-  let J : ℝ → ℝ := fun θ => H (c θ) (E θ*Complex.I) (E θ*Complex.I)
-  let D : ℝ → ℝ := fun θ => fderiv ℝ u (c θ) (E θ)
-  let Q : ℝ → ℝ := fun θ => H (c θ) (E θ) (E θ)
-  have hc : Continuous c := (continuous_circleMap 0 1).const_smul r
-  have hEm : Continuous E := continuous_circleMap 0 1
-  have hmem (θ : ℝ) : c θ ∈ ball (0:ℂ) 1 := radial_mem hrabs θ
-  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
-    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
-  have hddu : ContDiffOn ℝ 0 H (ball 0 1) :=
-    hdu.fderiv_of_isOpen isOpen_ball (by norm_num)
-  have hdc : Continuous (fun θ => fderiv ℝ u (c θ)) :=
-    continuousOn_univ.1 (hdu.continuousOn.comp hc.continuousOn (fun θ _ => hmem θ))
-  have hHc : Continuous (fun θ => H (c θ)) :=
-    continuousOn_univ.1 (hddu.continuousOn.comp hc.continuousOn (fun θ _ => hmem θ))
-  have hD : Continuous D := hdc.clm_apply hEm
-  have hJ : Continuous J := (hHc.clm_apply (hEm.mul_const Complex.I)).clm_apply (hEm.mul_const Complex.I)
-  have hQ : Continuous Q := (hHc.clm_apply hEm).clm_apply hEm
-  let V : ℝ → ℝ := fun θ => fderiv ℝ u (c θ) (r • (E θ*Complex.I))
-  have hdV (θ : ℝ) : HasDerivAt V (r^2*J θ-r*D θ) θ := by
-    have hcp : HasDerivAt c (r • (E θ*Complex.I)) θ := (hasDerivAt_circleMap 0 1 θ).const_smul r
-    have hcpp : HasDerivAt (fun t => r • (E t*Complex.I)) (-(c θ)) θ := by
-      have hh := ((hasDerivAt_circleMap 0 1 θ).mul_const Complex.I).const_smul r
-      simpa only [Pi.smul_apply,E,c,mul_assoc,Complex.I_mul_I,mul_neg_one,smul_neg] using! hh
-    have hh := path_second (hu.contDiffAt (isOpen_ball.mem_nhds (hmem θ))) hcp hcpp
-    convert hh using 1
-    dsimp only [H,J,D]
-    simp only [map_smul,smul_apply,smul_eq_mul,map_neg]
-    change r^2*((H (c θ)) (E θ*Complex.I)) (E θ*Complex.I)-r*(fderiv ℝ u (c θ)) (E θ) =
-      r*(r*((H (c θ)) (E θ*Complex.I)) (E θ*Complex.I))-(fderiv ℝ u (c θ)) (r • E θ)
-    rw [map_smul,smul_eq_mul]
-    ring
-  have hEnds : E (-Real.pi) = E Real.pi := by
-    simp [E,circleMap,Complex.exp_neg_pi_mul_I,Complex.exp_pi_mul_I]
-  have hVeq : V Real.pi = V (-Real.pi) := by
-    dsimp only [V,c]
-    rw [hEnds]
-  have hz := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun θ _ => hdV θ)
-    (((hJ.const_mul (r^2)).sub (hD.const_mul r)).intervalIntegrable (-Real.pi) Real.pi)
-  rw [hVeq,sub_self] at hz
-  rw [intervalIntegral.integral_sub ((hJ.intervalIntegrable _ _).const_mul _) ((hD.intervalIntegrable _ _).const_mul _),
-    intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul] at hz
-  have htrace (θ : ℝ) : Laplacian.laplacian u (c θ) = Q θ+J θ := by
-    rw [laplacian_eq_hessian]
-    symm
-    convert rotated_trace (H (c θ)) (E θ) (by simp [E,norm_circleMap_zero]) using 1
-    simp only [Q,J,H,mul_comm Complex.I]
-  have hli : (∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (c θ)) =
-      circleDDMean u r + ∫ θ in -Real.pi..Real.pi, J θ := by
-    calc
-      _ = ∫ θ in -Real.pi..Real.pi, Q θ+J θ := intervalIntegral.integral_congr (fun θ _ => htrace θ)
-      _ = _ := intervalIntegral.integral_add (hQ.intervalIntegrable _ _) (hJ.intervalIntegrable _ _)
-  change r*(∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (c θ)) = _
-  rw [hli]
-  change r^2*(∫ θ in -Real.pi..Real.pi, J θ)-r*circleDMean u r = 0 at hz
-  nlinarith
-
-lemma weighted_circle_derivative {u : ℂ → ℝ}
-    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : 0 < r) (hr1 : r < 1) :
-    HasDerivAt (fun s => s*circleDMean u s)
-      (r*(∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (r • circleMap 0 1 θ))) r := by
-  have hd := (hasDerivAt_id r).mul (circle_means_derivatives hu (by rwa [abs_of_pos hr])).2
-  simpa only [Pi.mul_apply,id_eq,one_mul,circle_angular_laplacian hu hr hr1] using! hd
-
-lemma compact_parameter_continuousAt {F : ℝ → ℝ → ℝ} {a b l h x : ℝ}
-    (hab : a ≤ b) (hlx : l < x) (hxh : x < h)
-    (hF : ContinuousOn (Function.uncurry F) (Icc l h ×ˢ Icc a b)) :
-    ContinuousAt (fun y => ∫ t in a..b, F y t) x := by
-  have hc (y : ℝ) (hy : y ∈ Icc l h) : ContinuousOn (F y) (Icc a b) :=
-    hF.comp (continuous_const.prodMk continuous_id).continuousOn (fun t ht => ⟨hy,ht⟩)
-  obtain ⟨C,hC⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hF
-  apply intervalIntegral.continuousAt_of_dominated_interval (bound := fun _ => C) ?_ ?_ intervalIntegrable_const ?_
-  · filter_upwards [Icc_mem_nhds hlx hxh] with y hy
-    have hh := ((hc y hy).mono Ioc_subset_Icc_self).aestronglyMeasurable (μ := volume) measurableSet_Ioc
-    simpa only [uIoc_of_le hab] using hh
-  · filter_upwards [Icc_mem_nhds hlx hxh] with y hy
-    filter_upwards with t ht
-    exact hC (y,t) ⟨hy,Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)⟩
-  · filter_upwards with t ht
-    have ht' : t ∈ Icc a b := Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)
-    have hh : ContinuousOn (fun y => F y t) (Icc l h) :=
-      hF.comp (continuous_id.prodMk continuous_const).continuousOn (fun y hy => ⟨hy,ht'⟩)
-    exact hh.continuousAt (Icc_mem_nhds hlx hxh)
-
-lemma laplacian_continuousOn {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1)) :
-    ContinuousOn (Laplacian.laplacian u) (ball (0:ℂ) 1) := by
-  have hc := ((hu.fderiv_of_isOpen isOpen_ball (show (1:WithTop ℕ∞)+1 ≤ 2 by norm_num)).fderiv_of_isOpen
-    isOpen_ball (show (0:WithTop ℕ∞)+1 ≤ 1 by norm_num)).continuousOn
-  have he : Laplacian.laplacian u = fun z =>
-      fderiv ℝ (fderiv ℝ u) z 1 1 + fderiv ℝ (fderiv ℝ u) z Complex.I Complex.I :=
-    funext (laplacian_eq_hessian u)
-  rw [he]
-  exact (hc.clm_apply continuousOn_const |>.clm_apply continuousOn_const).add
-    (hc.clm_apply continuousOn_const |>.clm_apply continuousOn_const)
-
-def circleLMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
-  ∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (r • circleMap 0 1 θ)
-
-lemma circle_lmean_continuous {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1)) :
-    ContinuousOn (circleLMean u) (Ioo (-1:ℝ) 1) := by
-  intro r hr
-  let l := (r-1)/2
-  let h := (r+1)/2
-  have hl : -1 < l := by dsimp [l]; linarith [hr.1]
-  have hh : h < 1 := by dsimp [h]; linarith [hr.2]
-  have hlr : l < r := by dsimp [l]; linarith [hr.1]
-  have hrh : r < h := by dsimp [h]; linarith [hr.2]
-  apply ContinuousAt.continuousWithinAt
-  apply compact_parameter_continuousAt (F := fun r θ => Laplacian.laplacian u (r • circleMap 0 1 θ))
-    (neg_le_self Real.pi_pos.le) hlr hrh
-  exact (laplacian_continuousOn hu).comp radial_continuous.continuousOn (radial_interval_maps hl hh)
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal
-namespace SLEExactGauge.Layer
-
-lemma polar_symm_circle (r θ : ℝ) :
-    Complex.polarCoord.symm (r,θ) = r • circleMap 0 1 θ := by
-  rw [Complex.polarCoord_symm_apply]
-  simp only [circleMap_zero,Complex.ofReal_one,one_mul,Complex.real_smul,Complex.exp_ofReal_mul_I]
-
-lemma polar_disk_lintegral_le (f : ℂ → ℝ≥0∞) :
-    (∫⁻ z in ball (0:ℂ) 1, f z) ≤
-      ∫⁻ r in Ioo (0:ℝ) 1, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-        ENNReal.ofReal r * f (r • circleMap 0 1 θ) := by
-  classical
-  let g := (ball (0:ℂ) 1).indicator f
-  have hp := Complex.lintegral_comp_polarCoord_symm g
-  rw [← lintegral_indicator measurableSet_ball f,← hp]
-  change (∫⁻ p, ENNReal.ofReal p.1 * g (Complex.polarCoord.symm p)
-    ∂(volume.prod volume).restrict (Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi)) ≤ _
-  rw [← Measure.prod_restrict]
-  apply (lintegral_prod_le _).trans
-  have he : (fun r => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-      ENNReal.ofReal r * g (Complex.polarCoord.symm (r,θ))) =ᵐ[volume.restrict (Ioi (0:ℝ))]
-      (Iio (1:ℝ)).indicator (fun r => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-        ENNReal.ofReal r * f (r • circleMap 0 1 θ)) := by
-    filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr
-    have hr0 : 0 < r := hr
-    by_cases h1 : r < 1
-    · rw [indicator_of_mem (show r ∈ Iio (1:ℝ) from h1)]
-      apply lintegral_congr
-      intro θ
-      rw [polar_symm_circle]
-      dsimp only [g]
-      rw [indicator_of_mem (radial_mem (by rwa [abs_of_pos hr0]) θ)]
-    · rw [indicator_of_notMem (show r ∉ Iio (1:ℝ) from h1)]
-      apply lintegral_eq_zero_of_ae_eq_zero
-      filter_upwards with θ
-      have hn : r • circleMap 0 1 θ ∉ ball (0:ℂ) 1 := by
-        simpa only [mem_ball_zero_iff,radial_norm,abs_of_pos hr0] using h1
-      rw [polar_symm_circle]
-      dsimp only [g]
-      simp only [indicator_of_notMem hn,mul_zero,Pi.zero_apply]
-  rw [lintegral_congr_ae he,lintegral_indicator measurableSet_Iio,
-    Measure.restrict_restrict measurableSet_Iio]
-  have hset : Iio (1:ℝ) ∩ Ioi (0:ℝ) = Ioo (0:ℝ) 1 := by ext r; simp [and_comm]
-  rw [hset]
-
-end SLEExactGauge.Layer
-
-end
-
-end
-
-section
-noncomputable section
-open Set Filter MeasureTheory Metric
-open scoped Topology ENNReal NNReal
-namespace SLEExactGauge.RectangleEnergy
-open Layer
-
-lemma polar_annulus_lintegral_le (f : ℂ → ℝ≥0∞) {r R : ℝ} (hr : 0 < r) :
-    (∫⁻ z in {z : ℂ | r < ‖z‖ ∧ ‖z‖ < R}, f z) ≤
-      ∫⁻ s in Ioo r R, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-        ENNReal.ofReal s*f (s • circleMap 0 1 θ) := by
-  classical
-  let A := {z : ℂ | r < ‖z‖ ∧ ‖z‖ < R}
-  have hA : MeasurableSet A := (measurableSet_lt measurable_const measurable_norm).inter
-    (measurableSet_lt measurable_norm measurable_const)
-  let g := A.indicator f
-  have hp := Complex.lintegral_comp_polarCoord_symm g
-  rw [← lintegral_indicator hA f,← hp]
-  change (∫⁻ p, ENNReal.ofReal p.1*g (Complex.polarCoord.symm p)
-    ∂(volume.prod volume).restrict (Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi)) ≤ _
-  rw [← Measure.prod_restrict]
-  apply (lintegral_prod_le _).trans
-  have he : (fun s => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-      ENNReal.ofReal s*g (Complex.polarCoord.symm (s,θ))) =ᵐ[volume.restrict (Ioi (0:ℝ))]
-      (Ioo r R).indicator (fun s => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
-        ENNReal.ofReal s*f (s • circleMap 0 1 θ)) := by
-    filter_upwards [ae_restrict_mem measurableSet_Ioi] with s hs
-    have hs0 : 0 < s := hs
-    by_cases hh : s ∈ Ioo r R
-    · rw [indicator_of_mem hh]
-      apply lintegral_congr
-      intro θ
-      rw [polar_symm_circle]
-      have hm : s • circleMap 0 1 θ ∈ A := by
-        simpa only [A,mem_ofPred_eq,mem_Ioo,radial_norm,abs_of_pos hs0] using hh
-      rw [show g (s • circleMap 0 1 θ) = f (s • circleMap 0 1 θ) by exact indicator_of_mem hm f]
-    · rw [indicator_of_notMem hh]
-      apply lintegral_eq_zero_of_ae_eq_zero
-      filter_upwards with θ
-      have hn : s • circleMap 0 1 θ ∉ A := by
-        simpa only [A,mem_ofPred_eq,mem_Ioo,radial_norm,abs_of_pos hs0] using hh
-      rw [polar_symm_circle]
-      dsimp only [g]
-      simp only [indicator_of_notMem hn,mul_zero,Pi.zero_apply]
-  rw [lintegral_congr_ae he,lintegral_indicator measurableSet_Ioo,
-    Measure.restrict_restrict measurableSet_Ioo]
-  have hset : Ioo r R ∩ Ioi (0:ℝ) = Ioo r R :=
-    inter_eq_left.2 (fun _ hs => hr.trans hs.1)
-  rw [hset]
-
-end SLEExactGauge.RectangleEnergy
-
-end
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+
+/-! Conformal localization and Rectangle Energy estimates. -/
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+
+lemma integral_mul_square_le {α : Type*} [MeasurableSpace α] {μ : Measure α}
+    {f g : α → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) :
+    (∫ x, f x*g x ∂μ)^2 ≤ (∫ x, f x^2 ∂μ)*(∫ x, g x^2 ∂μ) := by
+  have hi {a b : α → ℝ} (ha : MemLp a 2 μ) (hb : MemLp b 2 μ) :
+      inner ℝ (ha.toLp a) (hb.toLp b) = ∫ x, a x*b x ∂μ := by
+    rw [MeasureTheory.L2.inner_def]
+    apply integral_congr_ae
+    filter_upwards [ha.coeFn_toLp,hb.coeFn_toLp] with x hx hy
+    rw [hx,hy]
+    change b x*a x = a x*b x
+    ring
+  have hn := norm_inner_le_norm (𝕜 := ℝ) (hf.toLp f) (hg.toLp g)
+  have hh := pow_le_pow_left₀ (norm_nonneg _) hn 2
+  rw [mul_pow,← real_inner_self_eq_norm_sq (hf.toLp f),← real_inner_self_eq_norm_sq (hg.toLp g),
+    hi hf hg,hi hf hf,hi hg hg] at hh
+  simpa only [Real.norm_eq_abs,sq_abs,← sq] using hh
+
+lemma continuousOn_memLp_two_Ioc {a b : ℝ} (hab : a ≤ b) {f : ℝ → ℝ}
+    (hf : ContinuousOn f (Icc a b)) : MemLp f 2 (volume.restrict (Ioc a b)) := by
+  apply (memLp_two_iff_integrable_sq ((hf.mono Ioc_subset_Icc_self).aestronglyMeasurable measurableSet_Ioc)).2
+  exact ((hf.pow 2).intervalIntegrable_of_Icc hab).1
+
+lemma weighted_interval_energy {a b : ℝ} (ha : 0 < a) (hab : a ≤ b)
+    {F D : ℝ → ℝ} (hD : ContinuousOn D (Icc a b))
+    (hF : ∀ s ∈ Icc a b, HasDerivAt F (D s) s) :
+    (F b-F a)^2 ≤ Real.log (b/a)*(∫ s in a..b, s*(D s)^2) := by
+  let f := fun s : ℝ => Real.sqrt s*D s
+  let g := fun s : ℝ => (Real.sqrt s)⁻¹
+  have hs (s : ℝ) (hs : s ∈ Icc a b) : 0 < s := ha.trans_le hs.1
+  have hf : ContinuousOn f (Icc a b) := Real.continuous_sqrt.continuousOn.mul hD
+  have hg : ContinuousOn g (Icc a b) := Real.continuous_sqrt.continuousOn.inv₀
+    (fun s hs' => (Real.sqrt_pos.2 (hs s hs')).ne')
+  have hc := integral_mul_square_le (continuousOn_memLp_two_Ioc hab hf)
+    (continuousOn_memLp_two_Ioc hab hg)
+  have hfg : (∫ s in Ioc a b, f s*g s) = F b-F a := by
+    calc
+      _ = ∫ s in a..b, D s := by
+        rw [intervalIntegral.integral_of_le hab]
+        apply setIntegral_congr_fun measurableSet_Ioc
+        intro s hs'
+        dsimp [f,g]
+        have hne := (Real.sqrt_pos.2 (hs s (Ioc_subset_Icc_self hs'))).ne'
+        field_simp
+      _ = _ := intervalIntegral.integral_eq_sub_of_hasDerivAt
+        (fun s hs' => hF s (by simpa only [uIcc_of_le hab] using hs'))
+        (hD.intervalIntegrable_of_Icc hab)
+  have hff : (∫ s in Ioc a b, f s^2) = ∫ s in a..b, s*(D s)^2 := by
+    rw [intervalIntegral.integral_of_le hab]
+    apply setIntegral_congr_fun measurableSet_Ioc
+    intro s hs'
+    dsimp only [f]
+    rw [mul_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
+  have hgg : (∫ s in Ioc a b, g s^2) = Real.log (b/a) := by
+    calc
+      _ = ∫ s in a..b, s⁻¹ := by
+        rw [intervalIntegral.integral_of_le hab]
+        apply setIntegral_congr_fun measurableSet_Ioc
+        intro s hs'
+        dsimp only [g]
+        rw [inv_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
+      _ = _ := by
+        apply integral_inv
+        rw [uIcc_of_le hab]
+        intro hz
+        exact (not_le_of_gt ha) hz.1
+  rw [hfg,hff,hgg] at hc
+  simpa only [mul_comm] using hc
+
+def radialTruncation (H : ℝ) (q : ℂ) (r : ℝ) (w : ℂ) : ℝ :=
+  min H (Real.log (max r ‖w-q‖/r))
+
+lemma radialTruncation_source {r : ℝ} (hr : 0 < r) (H : ℝ) (q w : ℂ) :
+    radialTruncation H q r w = min H (max 0 (Real.log (‖w-q‖/r))) := by
+  unfold radialTruncation
+  by_cases h : ‖w-q‖ ≤ r
+  · rw [max_eq_left h,div_self hr.ne',Real.log_one,
+      max_eq_left (Real.log_nonpos (div_nonneg (norm_nonneg _) hr.le) ((div_le_one hr).2 h))]
+  · have hh : r < ‖w-q‖ := lt_of_not_ge h
+    rw [max_eq_right hh.le,max_eq_right (Real.log_nonneg ((one_le_div hr).2 hh.le))]
+
+lemma radialTruncation_bounds {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) (q w : ℂ) :
+    0 ≤ radialTruncation H q r w ∧ radialTruncation H q r w ≤ H := by
+  unfold radialTruncation
+  exact ⟨le_min hH (Real.log_nonneg ((one_le_div hr).2 (le_max_left _ _))),min_le_left _ _⟩
+
+lemma radialTruncation_continuous {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ) :
+    Continuous (radialTruncation H q r) := by
+  have hm : Continuous (fun w : ℂ => max r ‖w-q‖/r) :=
+    (continuous_const.max ((continuous_id.sub continuous_const).norm)).div_const r
+  exact continuous_const.min (hm.log (fun w => (div_pos (lt_of_lt_of_le hr (le_max_left _ _)) hr).ne'))
+
+lemma radialTruncation_zero {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) {q w : ℂ}
+    (hw : ‖w-q‖ ≤ r) : radialTruncation H q r w = 0 := by
+  simp only [radialTruncation,max_eq_left hw,div_self hr.ne',Real.log_one,min_eq_right hH]
+
+lemma radialTruncation_top {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r) {q w : ℂ}
+    (hw : Real.exp H*r ≤ ‖w-q‖) : radialTruncation H q r w = H := by
+  have he : 1 ≤ Real.exp H := Real.one_le_exp_iff.2 hH
+  have hrw : r ≤ ‖w-q‖ := (le_mul_of_one_le_left hr.le he).trans hw
+  have hn : 0 < ‖w-q‖/r := div_pos (hr.trans_le hrw) hr
+  have hl : H ≤ Real.log (‖w-q‖/r) :=
+    (Real.le_log_iff_exp_le hn).2 ((le_div_iff₀ hr).2 hw)
+  simp only [radialTruncation,max_eq_right hrw,min_eq_left hl]
+
+lemma radialTruncation_middle {H r : ℝ} (hr : 0 < r) {q w : ℂ}
+    (hw : r < ‖w-q‖) (hw' : ‖w-q‖ < Real.exp H*r) :
+    radialTruncation H q r w = (1/2:ℝ)*Real.log (‖w-q‖^2)-Real.log r := by
+  have hn := hr.trans hw
+  have hl : Real.log (‖w-q‖/r) < H :=
+    (Real.log_lt_iff_lt_exp (div_pos hn hr)).2 ((div_lt_iff₀ hr).2 hw')
+  rw [radialTruncation,max_eq_right hw.le,min_eq_right hl.le,
+    Real.log_div hn.ne' hr.ne',Real.log_pow]
+  ring
+
+lemma log_normSq_fderiv_le {F : ℂ → ℂ} {z q : ℂ} (hd : DifferentiableAt ℂ F z)
+    (hn : 0 < ‖F z-q‖) (r : ℝ) :
+    ‖fderiv ℝ (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r) z‖ ≤
+      ‖deriv F z‖/‖F z-q‖ := by
+  let A := (ContinuousLinearMap.toSpanSingleton ℂ (deriv F z)).restrictScalars ℝ
+  have hA : ‖A‖ = ‖deriv F z‖ := by
+    dsimp [A]
+    rw [ContinuousLinearMap.norm_restrictScalars,ContinuousLinearMap.norm_toSpanSingleton]
+  have hF : HasFDerivAt F A z := hd.hasDerivAt.hasFDerivAt.restrictScalars ℝ
+  have hs := ((hF.sub_const q).norm_sq.log (pow_ne_zero 2 hn.ne')).const_mul (1/2:ℝ)
+  have hs' := hs.sub_const (Real.log r)
+  have hd' : HasFDerivAt
+      (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r)
+      ((‖F z-q‖^2)⁻¹ • (innerSL ℝ (F z-q)).comp A) z := by
+    convert! hs' using 1
+    ext v
+    simp
+    ring
+  rw [hd'.fderiv,norm_smul,Real.norm_eq_abs,abs_inv,abs_pow,abs_norm]
+  have hb := (innerSL ℝ (F z-q)).opNorm_comp_le A
+  rw [innerSL_apply_norm,hA] at hb
+  calc
+    (‖F z-q‖^2)⁻¹*‖(innerSL ℝ (F z-q)).comp A‖ ≤
+        (‖F z-q‖^2)⁻¹*(‖F z-q‖*‖deriv F z‖) :=
+      mul_le_mul_of_nonneg_left hb (by positivity)
+    _ = ‖deriv F z‖/‖F z-q‖ := by field_simp
+
+lemma radialTruncation_fderiv_bound {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r)
+    {F : ℂ → ℂ} {z q : ℂ} (hd : DifferentiableAt ℂ F z) :
+    ‖fderiv ℝ (radialTruncation H q r ∘ F) z‖^2 ≤
+      if r < ‖F z-q‖ ∧ ‖F z-q‖ < Real.exp H*r then
+        ‖deriv F z‖^2/‖F z-q‖^2 else 0 := by
+  by_cases hlow : ‖F z-q‖ ≤ r
+  · have hz : fderiv ℝ (radialTruncation H q r ∘ F) z = 0 := by
+      apply IsLocalMin.fderiv_eq_zero
+      filter_upwards with w
+      change radialTruncation H q r (F z) ≤ radialTruncation H q r (F w)
+      rw [radialTruncation_zero hH hr hlow]
+      exact (radialTruncation_bounds hH hr q _).1
+    rw [hz]
+    simp [not_lt_of_ge hlow]
+  · have hl : r < ‖F z-q‖ := lt_of_not_ge hlow
+    by_cases hhigh : Real.exp H*r ≤ ‖F z-q‖
+    · have hz : fderiv ℝ (radialTruncation H q r ∘ F) z = 0 := by
+        apply IsLocalMax.fderiv_eq_zero
+        filter_upwards with w
+        change radialTruncation H q r (F w) ≤ radialTruncation H q r (F z)
+        rw [radialTruncation_top hH hr hhigh]
+        exact (radialTruncation_bounds hH hr q _).2
+      rw [hz]
+      simp [not_lt_of_ge hhigh]
+    · have hh : ‖F z-q‖ < Real.exp H*r := lt_of_not_ge hhigh
+      have hc : ContinuousAt (fun w => ‖F w-q‖) z := (hd.continuousAt.sub continuousAt_const).norm
+      have he : (radialTruncation H q r ∘ F) =ᶠ[𝓝 z]
+          (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r) := by
+        filter_upwards [continuousAt_const.eventually_lt hc hl,
+          hc.eventually_lt continuousAt_const hh] with w hw hw'
+        exact radialTruncation_middle hr hw hw'
+      rw [he.fderiv_eq,ite_eq_left ⟨hl,hh⟩]
+      have ht := pow_le_pow_left₀ (norm_nonneg _) (log_normSq_fderiv_le hd (hr.trans hl) r) 2
+      simpa only [div_pow] using ht
+
+lemma radialTruncation_lipschitz {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ) :
+    LipschitzWith ⟨1/r,by positivity⟩ (radialTruncation H q r) := by
+  let K : ℝ≥0 := ⟨1/r,by positivity⟩
+  have hlog : LipschitzOnWith K Real.log (Ici r) := by
+    apply Convex.lipschitzOnWith_of_nnnorm_deriv_le
+      (fun x hx => Real.differentiableAt_log (hr.trans_le hx).ne') _ (convex_Ici r)
+    intro x hx
+    change ‖deriv Real.log x‖ ≤ 1/r
+    rw [Real.deriv_log,Real.norm_of_nonneg (inv_nonneg.2 (hr.trans_le hx).le),one_div]
+    exact inv_anti₀ hr hx
+  have hnorm : LipschitzWith 1 (fun w : ℂ => ‖w-q‖) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    calc
+      dist ‖x-q‖ ‖y-q‖ ≤ ‖(x-q)-(y-q)‖ := dist_norm_norm_le _ _
+      _ = ‖x-y‖ := by congr 1; abel
+      _ = (1:ℝ≥0)*dist x y := by simp [dist_eq_norm]
+  have hmax := hnorm.const_max r
+  have hl : LipschitzWith K (fun w : ℂ => Real.log (max r ‖w-q‖)) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    exact (hlog.dist_le_mul (max r ‖x-q‖) (show r ≤ max r ‖x-q‖ from le_max_left _ _) (max r ‖y-q‖) (show r ≤ max r ‖y-q‖ from le_max_left _ _)).trans
+      (mul_le_mul_of_nonneg_left (by simpa using hmax.dist_le_mul x y) K.coe_nonneg)
+  have hsub : LipschitzWith K (fun w : ℂ => Real.log (max r ‖w-q‖)-Real.log r) := by
+    simpa using hl.sub (LipschitzWith.const (Real.log r))
+  have he : radialTruncation H q r = fun w : ℂ => min H (Real.log (max r ‖w-q‖)-Real.log r) := by
+    ext w
+    rw [radialTruncation,Real.log_div (hr.trans_le (le_max_left _ _)).ne' hr.ne']
+  rw [he]
+  exact hsub.const_min H
+
+lemma radialTruncation_along_C1_AC {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ)
+    {G : ℝ → ℂ} {a b : ℝ} (hG : ContDiffOn ℝ 1 G (uIcc a b)) :
+    AbsolutelyContinuousOnInterval (radialTruncation H q r ∘ G) a b := by
+  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_uIcc _ _) isCompact_uIcc
+  exact ((radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK).absolutelyContinuousOnInterval
+
+lemma radialTruncation_along_C1_FTC {r : ℝ} (hr : 0 < r) (H : ℝ) (q : ℂ)
+    {G : ℝ → ℂ} {a b : ℝ} (hG : ContDiffOn ℝ 1 G (uIcc a b)) :
+    (∫ s in a..b, deriv (radialTruncation H q r ∘ G) s) =
+      radialTruncation H q r (G b)-radialTruncation H q r (G a) :=
+  (radialTruncation_along_C1_AC hr H q hG).integral_deriv_eq_sub
+
+lemma radialTruncation_directional_deriv {H r : ℝ} (hH : 0 ≤ H) (hr : 0 < r)
+    (q : ℂ) {F : ℂ → ℂ} {c : ℝ → ℂ} {d : ℂ} {t : ℝ}
+    (hF : DifferentiableAt ℂ F (c t)) (hc : HasDerivAt c d t) :
+    deriv (radialTruncation H q r ∘ F ∘ c) t =
+      fderiv ℝ (radialTruncation H q r ∘ F) (c t) d := by
+  by_cases hl : ‖F (c t)-q‖ ≤ r
+  · have hz : fderiv ℝ (radialTruncation H q r ∘ F) (c t) = 0 := by
+      apply IsLocalMin.fderiv_eq_zero
+      filter_upwards with w
+      change radialTruncation H q r (F (c t)) ≤ radialTruncation H q r (F w)
+      rw [radialTruncation_zero hH hr hl]
+      exact (radialTruncation_bounds hH hr q _).1
+    rw [hz,zero_apply]
+    apply IsLocalMin.deriv_eq_zero
+    filter_upwards with s
+    change radialTruncation H q r (F (c t)) ≤ radialTruncation H q r (F (c s))
+    rw [radialTruncation_zero hH hr hl]
+    exact (radialTruncation_bounds hH hr q _).1
+  · have hl' := lt_of_not_ge hl
+    by_cases hh : Real.exp H*r ≤ ‖F (c t)-q‖
+    · have hz : fderiv ℝ (radialTruncation H q r ∘ F) (c t) = 0 := by
+        apply IsLocalMax.fderiv_eq_zero
+        filter_upwards with w
+        change radialTruncation H q r (F w) ≤ radialTruncation H q r (F (c t))
+        rw [radialTruncation_top hH hr hh]
+        exact (radialTruncation_bounds hH hr q _).2
+      rw [hz,zero_apply]
+      apply IsLocalMax.deriv_eq_zero
+      filter_upwards with s
+      change radialTruncation H q r (F (c s)) ≤ radialTruncation H q r (F (c t))
+      rw [radialTruncation_top hH hr hh]
+      exact (radialTruncation_bounds hH hr q _).2
+    · have hh' := lt_of_not_ge hh
+      have hcont : ContinuousAt (fun w => ‖F w-q‖) (c t) := (hF.continuousAt.sub continuousAt_const).norm
+      have he : (radialTruncation H q r ∘ F) =ᶠ[𝓝 (c t)]
+          (fun w => (1/2:ℝ)*Real.log (‖F w-q‖^2)-Real.log r) := by
+        filter_upwards [continuousAt_const.eventually_lt hcont hl',
+          hcont.eventually_lt continuousAt_const hh'] with w hw hw'
+        exact radialTruncation_middle hr hw hw'
+      have hdiff : DifferentiableAt ℝ (radialTruncation H q r ∘ F) (c t) := by
+        apply he.differentiableAt_iff.2
+        exact (((((hF.restrictScalars ℝ).sub_const q).norm_sq (𝕜 := ℝ)).log
+          (pow_ne_zero 2 (hr.trans hl').ne')).const_mul (1/2:ℝ)).sub_const (Real.log r)
+      exact (hdiff.hasFDerivAt.comp_hasDerivAt t hc).deriv
+
+lemma acl_interval_energy {a b : ℝ} (hab : a ≤ b) {f : ℝ → ℝ}
+    (hf : AbsolutelyContinuousOnInterval f a b)
+    (hD : MemLp (deriv f) 2 (volume.restrict (Ioc a b))) :
+    (f b-f a)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) := by
+  have hc := integral_mul_square_le hD (memLp_const (1:ℝ))
+  have he : (∫ s in Ioc a b, deriv f s*1) = f b-f a := by
+    simp only [mul_one,← intervalIntegral.integral_of_le hab,hf.integral_deriv_eq_sub]
+  have he1 : (∫ s in Ioc a b, (1:ℝ)^2) = b-a := by simp [hab]
+  rw [he,he1] at hc
+  simpa only [intervalIntegral.integral_of_le hab,mul_comm] using hc
+
+lemma weighted_acl_interval_energy {a b : ℝ} (ha : 0 < a) (hab : a ≤ b)
+    {f : ℝ → ℝ} (hf : AbsolutelyContinuousOnInterval f a b)
+    (hD : MemLp (fun s => Real.sqrt s*deriv f s) 2 (volume.restrict (Ioc a b))) :
+    (f b-f a)^2 ≤ Real.log (b/a)*(∫ s in a..b, s*(deriv f s)^2) := by
+  let g := fun s : ℝ => (Real.sqrt s)⁻¹
+  have hs (s : ℝ) (hs : s ∈ Icc a b) : 0 < s := ha.trans_le hs.1
+  have hg : ContinuousOn g (Icc a b) := Real.continuous_sqrt.continuousOn.inv₀
+    (fun s hs' => (Real.sqrt_pos.2 (hs s hs')).ne')
+  have hc := integral_mul_square_le hD (continuousOn_memLp_two_Ioc hab hg)
+  have hfg : (∫ s in Ioc a b, (Real.sqrt s*deriv f s)*g s) = f b-f a := by
+    calc
+      _ = ∫ s in a..b, deriv f s := by
+        rw [intervalIntegral.integral_of_le hab]
+        apply setIntegral_congr_fun measurableSet_Ioc
+        intro s hs'
+        dsimp [g]
+        have hne := (Real.sqrt_pos.2 (hs s (Ioc_subset_Icc_self hs'))).ne'
+        field_simp
+      _ = _ := hf.integral_deriv_eq_sub
+  have hff : (∫ s in Ioc a b, (Real.sqrt s*deriv f s)^2) =
+      ∫ s in a..b, s*(deriv f s)^2 := by
+    rw [intervalIntegral.integral_of_le hab]
+    apply setIntegral_congr_fun measurableSet_Ioc
+    intro s hs'
+    dsimp only
+    rw [mul_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
+  have hgg : (∫ s in Ioc a b, g s^2) = Real.log (b/a) := by
+    calc
+      _ = ∫ s in a..b, s⁻¹ := by
+        rw [intervalIntegral.integral_of_le hab]
+        apply setIntegral_congr_fun measurableSet_Ioc
+        intro s hs'
+        dsimp only [g]
+        rw [inv_pow,Real.sq_sqrt (hs s (Ioc_subset_Icc_self hs')).le]
+      _ = _ := by
+        apply integral_inv
+        rw [uIcc_of_le hab]
+        intro hz
+        exact (not_le_of_gt ha) hz.1
+  rw [hfg,hff,hgg] at hc
+  simpa only [mul_comm] using hc
+
+lemma lipschitzOn_deriv_memLp {a b : ℝ} {K : ℝ≥0} {f : ℝ → ℝ}
+    (hf : LipschitzOnWith K f (Icc a b)) :
+    MemLp (deriv f) 2 (volume.restrict (Ioc a b)) := by
+  apply MemLp.of_bound (measurable_deriv f).aestronglyMeasurable (K:ℝ)
+  filter_upwards [ae_restrict_mem measurableSet_Ioc,
+    (ae_restrict_of_ae (μ := volume) (volume.ae_ne b))] with s hs hsb
+  have hsl : s ∈ Ioo a b := ⟨hs.1,lt_of_le_of_ne hs.2 hsb⟩
+  exact norm_deriv_le_of_lipschitzOn (mem_of_superset (isOpen_Ioo.mem_nhds hsl) Ioo_subset_Icc_self) hf
+
+lemma lipschitzOn_weighted_deriv_memLp {a b : ℝ} {K : ℝ≥0} {f : ℝ → ℝ}
+    (ha : 0 < a) (hf : LipschitzOnWith K f (Icc a b)) :
+    MemLp (fun s => Real.sqrt s*deriv f s) 2 (volume.restrict (Ioc a b)) := by
+  apply (lipschitzOn_deriv_memLp hf).of_le_mul
+    ((Real.continuous_sqrt.measurable).mul (measurable_deriv f)).aestronglyMeasurable (c := Real.sqrt b)
+  filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs
+  rw [Pi.mul_apply,norm_mul,Real.norm_of_nonneg (Real.sqrt_pos.2 (ha.trans hs.1)).le]
+  exact mul_le_mul_of_nonneg_right (Real.sqrt_le_sqrt hs.2) (norm_nonneg _)
+
+lemma truncated_curve_weighted_energy {H r a b : ℝ} (hr : 0 < r)
+    (ha : 0 < a) (hab : a ≤ b) (q : ℂ) {G : ℝ → ℂ}
+    (hG : ContDiffOn ℝ 1 G (Icc a b)) :
+    (radialTruncation H q r (G b)-radialTruncation H q r (G a))^2 ≤
+      Real.log (b/a)*(∫ s in a..b, s*(deriv (radialTruncation H q r ∘ G) s)^2) := by
+  obtain ⟨K,hK⟩ := hG.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
+  have hL := (radialTruncation_lipschitz hr H q).comp_lipschitzOnWith hK
+  exact weighted_acl_interval_energy ha hab
+    (radialTruncation_along_C1_AC hr H q (by simpa only [uIcc_of_le hab] using hG))
+    (lipschitzOn_weighted_deriv_memLp ha hL)
+
+lemma mean_difference_square_le
+    {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
+    {μ : Measure α} {ν : Measure β} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    {u : α → ℝ} {v : β → ℝ} (hu : MemLp u 2 μ) (hv : MemLp v 2 ν) :
+    ((∫ x, u x ∂μ)-(∫ y, v y ∂ν))^2 ≤
+      ∫ x, ∫ y, (u x-v y)^2 ∂ν ∂μ := by
+  have hD := (hu.comp_fst ν).sub (hv.comp_snd μ)
+  have hc := integral_mul_square_le hD (memLp_const (1:ℝ))
+  have hDi : Integrable (fun p : α × β => u p.1-v p.2) (μ.prod ν) :=
+    hD.integrable (by norm_num)
+  have he : (∫ p : α × β, (u p.1-v p.2)*1 ∂μ.prod ν) =
+      (∫ x, u x ∂μ)-(∫ y, v y ∂ν) := by
+    simp only [mul_one]
+    rw [integral_prod _ hDi]
+    have hiu := hu.integrable (by norm_num : (1:ℝ≥0∞) ≤ 2)
+    have hiv := hv.integrable (by norm_num : (1:ℝ≥0∞) ≤ 2)
+    simp_rw [integral_sub (integrable_const _) hiv]
+    simp only [integral_const,probReal_univ,one_smul]
+    rw [integral_sub hiu (integrable_const _)]
+    simp
+  have he1 : (∫ _ : α × β, (1:ℝ)^2 ∂μ.prod ν) = 1 := by simp
+  simp only [Pi.sub_apply] at hc
+  rw [he,he1,mul_one] at hc
+  have hsq : Integrable (fun p : α × β => (u p.1-v p.2)^2) (μ.prod ν) := hD.integrable_sq
+  rw [integral_prod _ hsq] at hc
+  exact hc
+
+lemma acl_oscillation_energy {a b : ℝ} {f : ℝ → ℝ}
+    (hf : AbsolutelyContinuousOnInterval f a b)
+    (hD : MemLp (deriv f) 2 (volume.restrict (Ioc a b)))
+    {x y : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b) :
+    (f y-f x)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) := by
+  have hab := hx.1.trans hx.2
+  suffices ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, x ≤ y →
+      (f y-f x)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) by
+    rcases le_total x y with h | h
+    · exact this x hx y hy h
+    · have hh := this y hy x hx h
+      nlinarith
+  intro x hx y hy hxy
+  have hsub : Ioc x y ⊆ Ioc a b := fun s hs => ⟨hx.1.trans_lt hs.1,hs.2.trans hy.2⟩
+  have hD' := hD.mono_measure (Measure.restrict_mono_set _ hsub)
+  have hf' : AbsolutelyContinuousOnInterval f x y := hf.mono
+    (by simpa only [uIcc_of_le hab,uIcc_of_le hxy] using (Icc_subset_Icc hx.1 hy.2))
+  have he := acl_interval_energy hxy hf' hD'
+  have hmono : (∫ s in x..y, (deriv f s)^2) ≤ ∫ s in a..b, (deriv f s)^2 := by
+    rw [intervalIntegral.integral_of_le hxy,intervalIntegral.integral_of_le hab]
+    apply setIntegral_mono_set hD.integrable_sq
+      (by filter_upwards with s; positivity) ?_
+    exact ae_of_all _ (fun s hs => hsub hs)
+  have hnon : 0 ≤ ∫ s in a..b, (deriv f s)^2 :=
+    intervalIntegral.integral_nonneg hab (fun s hs => sq_nonneg _)
+  exact he.trans ((mul_le_mul_of_nonneg_left hmono (sub_nonneg.2 hxy)).trans
+    (mul_le_mul_of_nonneg_right (by linarith [hx.1,hy.2]) hnon))
+
+lemma acl_congr {f g : ℝ → ℝ} {a b : ℝ}
+    (hf : AbsolutelyContinuousOnInterval f a b) (he : EqOn f g (uIcc a b)) :
+    AbsolutelyContinuousOnInterval g a b := by
+  rw [absolutelyContinuousOnInterval_iff] at hf ⊢
+  intro ε hε
+  obtain ⟨δ,hδ,hδ'⟩ := hf ε hε
+  refine ⟨δ,hδ,?_⟩
+  intro E hE hsum
+  have hh := hδ' E hE hsum
+  convert hh using 1
+  apply Finset.sum_congr rfl
+  intro i hi
+  rw [he ((hE.1 i hi).1),he ((hE.1 i hi).2)]
+
+lemma acl_of_interior {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
+    (hc : ContinuousOn f (Icc a b))
+    (hi : IntervalIntegrable (deriv f) volume a b)
+    (hloc : ∀ x ∈ Ioo a b, ∀ y ∈ Ioo a b, AbsolutelyContinuousOnInterval f x y) :
+    AbsolutelyContinuousOnInterval f a b := by
+  let m := (a+b)/2
+  have hm : m ∈ Ioo a b := by dsimp [m]; constructor <;> linarith
+  let g := fun x => (∫ s in m..x, deriv f s)+f m
+  have hg : AbsolutelyContinuousOnInterval g a b := by
+    have hh := hi.absolutelyContinuousOnInterval_intervalIntegral
+      (c := m) (by simpa only [uIcc_of_le hab.le] using Ioo_subset_Icc_self hm)
+    have hk : AbsolutelyContinuousOnInterval (fun _ : ℝ => f m) a b :=
+      (LipschitzWith.const (f m)).lipschitzOnWith.absolutelyContinuousOnInterval
+    exact hh.add hk
+  have he : EqOn f g (Ioo a b) := by
+    intro x hx
+    dsimp only [g]
+    rw [(hloc m hm x hx).integral_deriv_eq_sub]
+    ring
+  have he' : EqOn f g (Icc a b) := he.of_subset_closure hc
+    (by simpa only [uIcc_of_le hab.le] using hg.continuousOn)
+    Ioo_subset_Icc_self (by rw [closure_Ioo hab.ne])
+  exact acl_congr hg (by simpa only [uIcc_of_le hab.le] using he'.symm)
+
+lemma boundary_zero_energy {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
+    (hc : ContinuousOn f (Icc a b))
+    (hD : MemLp (deriv f) 2 (volume.restrict (Ioc a b)))
+    (hloc : ∀ x ∈ Ioo a b, ∀ y ∈ Ioo a b, AbsolutelyContinuousOnInterval f x y)
+    {z : ℝ} (hz : z ∈ Icc a b) (hfz : f z = 0) {y : ℝ} (hy : y ∈ Icc a b) :
+    (f y)^2 ≤ (b-a)*(∫ s in a..b, (deriv f s)^2) := by
+  have hi : IntervalIntegrable (deriv f) volume a b :=
+    (intervalIntegrable_iff_integrableOn_Ioc_of_le hab.le).2 (hD.integrable (by norm_num))
+  have h := acl_oscillation_energy (acl_of_interior hab hc hi hloc) hD hz hy
+  simpa only [hfz,sub_zero] using h
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology
+namespace SLEExactGauge.Layer
+
+lemma compact_parameter_derivative {F F' : ℝ → ℝ → ℝ} {a b l h x : ℝ}
+    (hab : a ≤ b) (hlx : l < x) (hxh : x < h)
+    (hF : ContinuousOn (Function.uncurry F) (Icc l h ×ˢ Icc a b))
+    (hF' : ContinuousOn (Function.uncurry F') (Icc l h ×ˢ Icc a b))
+    (hd : ∀ t ∈ Icc a b, ∀ y ∈ Icc l h, HasDerivAt (fun y => F y t) (F' y t) y) :
+    HasDerivAt (fun y => ∫ t in a..b, F y t) (∫ t in a..b, F' x t) x := by
+  have hc (y : ℝ) (hy : y ∈ Icc l h) : ContinuousOn (F y) (Icc a b) :=
+    hF.comp (continuous_const.prodMk continuous_id).continuousOn (fun t ht => ⟨hy,ht⟩)
+  have hc' (y : ℝ) (hy : y ∈ Icc l h) : ContinuousOn (F' y) (Icc a b) :=
+    hF'.comp (continuous_const.prodMk continuous_id).continuousOn (fun t ht => ⟨hy,ht⟩)
+  obtain ⟨C,hC⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hF'
+  have hxx : x ∈ Icc l h := ⟨hlx.le,hxh.le⟩
+  apply (intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (s := Icc l h) (F' := F') (bound := fun _ => C) (Icc_mem_nhds hlx hxh) ?_
+    (by apply ContinuousOn.intervalIntegrable; simpa only [uIcc_of_le hab] using hc x hxx)
+    ?_ ?_ intervalIntegrable_const ?_).2
+  · filter_upwards [Icc_mem_nhds hlx hxh] with y hy
+    have hh := ((hc y hy).mono Ioc_subset_Icc_self).aestronglyMeasurable (μ := volume) measurableSet_Ioc
+    simpa only [uIoc_of_le hab] using hh
+  · have hh := ((hc' x hxx).mono Ioc_subset_Icc_self).aestronglyMeasurable (μ := volume) measurableSet_Ioc
+    simpa only [uIoc_of_le hab] using hh
+  · filter_upwards with t ht y hy
+    exact hC (y,t) ⟨hy,Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)⟩
+  · filter_upwards with t ht y hy
+    exact hd t (Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)) y hy
+
+lemma rotated_trace (H : ℂ →L[ℝ] ℂ →L[ℝ] ℝ) (v : ℂ) (hv : ‖v‖ = 1) :
+    H v v + H (Complex.I*v) (Complex.I*v) = H 1 1 + H Complex.I Complex.I := by
+  have he : v = v.re • (1:ℂ) + v.im • Complex.I := by
+    simp only [Complex.real_smul, mul_one]
+    exact (Complex.re_add_im v).symm
+  have hi : Complex.I*v = (-v.im) • (1:ℂ) + v.re • Complex.I := by
+    apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im]
+  have hh : v.re^2+v.im^2 = 1 := by
+    have hsq := Complex.sq_norm v
+    rw [hv,one_pow,Complex.normSq_apply] at hsq
+    linarith
+  calc
+    _ = H (v.re • (1:ℂ) + v.im • Complex.I) (v.re • (1:ℂ) + v.im • Complex.I) +
+        H ((-v.im) • (1:ℂ) + v.re • Complex.I) ((-v.im) • (1:ℂ) + v.re • Complex.I) := by
+      rw [← he,← hi]
+    _ = (v.re^2+v.im^2)*(H 1 1+H Complex.I Complex.I) := by
+      simp only [map_add,map_smul,add_apply,smul_apply,
+        smul_eq_mul]
+      ring
+    _ = _ := by rw [hh,one_mul]
+
+lemma laplacian_eq_hessian (u : ℂ → ℝ) (z : ℂ) :
+    Laplacian.laplacian u z =
+      fderiv ℝ (fderiv ℝ u) z 1 1 + fderiv ℝ (fderiv ℝ u) z Complex.I Complex.I := by
+  simp only [InnerProductSpace.laplacian_eq_iteratedFDeriv_complexPlane,
+    iteratedFDeriv_two_apply,Matrix.cons_val_zero,Matrix.cons_val_one]
+
+lemma path_second {u : ℂ → ℝ} {c c' : ℝ → ℂ} {c'' : ℂ} {t : ℝ}
+    (hu : ContDiffAt ℝ 2 u (c t)) (hc : HasDerivAt c (c' t) t)
+    (hc' : HasDerivAt c' c'' t) :
+    HasDerivAt (fun s => fderiv ℝ u (c s) (c' s))
+      (fderiv ℝ (fderiv ℝ u) (c t) (c' t) (c' t) + fderiv ℝ u (c t) c'') t := by
+  have hd : ContDiffAt ℝ 1 (fderiv ℝ u) (c t) := hu.fderiv_right (by norm_num)
+  exact (hd.differentiableAt (by norm_num)).hasFDerivAt.comp_hasDerivAt t hc |>.clm_apply hc'
+
+def circleMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
+  ∫ θ in -Real.pi..Real.pi, u (r • circleMap 0 1 θ)
+
+def circleDMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
+  ∫ θ in -Real.pi..Real.pi, fderiv ℝ u (r • circleMap 0 1 θ) (circleMap 0 1 θ)
+
+def circleDDMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
+  ∫ θ in -Real.pi..Real.pi,
+    fderiv ℝ (fderiv ℝ u) (r • circleMap 0 1 θ) (circleMap 0 1 θ) (circleMap 0 1 θ)
+
+lemma radial_norm (r θ : ℝ) : ‖r • circleMap 0 1 θ‖ = |r| := by
+  simp [norm_circleMap_zero,Real.norm_eq_abs]
+
+lemma radial_mem {r : ℝ} (hr : |r| < 1) (θ : ℝ) :
+    r • circleMap 0 1 θ ∈ ball (0:ℂ) 1 := by
+  rwa [mem_ball_zero_iff,radial_norm]
+
+lemma radial_continuous : Continuous (fun p : ℝ × ℝ => p.1 • circleMap 0 1 p.2) :=
+  continuous_fst.smul ((continuous_circleMap 0 1).comp continuous_snd)
+
+lemma radial_interval_maps {l h a b : ℝ} (hl : -1 < l) (hh : h < 1) :
+    MapsTo (fun p : ℝ × ℝ => p.1 • circleMap 0 1 p.2)
+      (Icc l h ×ˢ Icc a b) (ball (0:ℂ) 1) := by
+  rintro ⟨r,θ⟩ ⟨hr,hθ⟩
+  exact radial_mem (abs_lt.2 ⟨hl.trans_le hr.1,hr.2.trans_lt hh⟩) θ
+
+lemma circle_means_derivatives {u : ℂ → ℝ}
+    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : |r| < 1) :
+    HasDerivAt (circleMean u) (circleDMean u r) r ∧
+    HasDerivAt (circleDMean u) (circleDDMean u r) r := by
+  have hr' := abs_lt.1 hr
+  let l := (r-1)/2
+  let h := (r+1)/2
+  have hl : -1 < l := by dsimp [l]; linarith
+  have hh : h < 1 := by dsimp [h]; linarith
+  have hlr : l < r := by dsimp [l]; linarith
+  have hrh : r < h := by dsimp [h]; linarith
+  let K := Icc l h ×ˢ Icc (-Real.pi) Real.pi
+  have hm : MapsTo (fun p : ℝ × ℝ => p.1 • circleMap 0 1 p.2) K (ball (0:ℂ) 1) :=
+    radial_interval_maps hl hh
+  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
+    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
+  have hddu : ContDiffOn ℝ 0 (fderiv ℝ (fderiv ℝ u)) (ball 0 1) :=
+    hdu.fderiv_of_isOpen isOpen_ball (by norm_num)
+  have hE : Continuous (fun p : ℝ × ℝ => circleMap 0 1 p.2) :=
+    (continuous_circleMap 0 1).comp continuous_snd
+  have h0 : ContinuousOn (fun p : ℝ × ℝ => u (p.1 • circleMap 0 1 p.2)) K :=
+    hu.continuousOn.comp radial_continuous.continuousOn hm
+  have h1 : ContinuousOn (fun p : ℝ × ℝ => fderiv ℝ u (p.1 • circleMap 0 1 p.2) (circleMap 0 1 p.2)) K :=
+    (hdu.continuousOn.comp radial_continuous.continuousOn hm).clm_apply hE.continuousOn
+  have h2 : ContinuousOn (fun p : ℝ × ℝ =>
+      fderiv ℝ (fderiv ℝ u) (p.1 • circleMap 0 1 p.2) (circleMap 0 1 p.2) (circleMap 0 1 p.2)) K :=
+    ((hddu.continuousOn.comp radial_continuous.continuousOn hm).clm_apply hE.continuousOn).clm_apply hE.continuousOn
+  constructor
+  · apply compact_parameter_derivative
+      (F := fun s θ => u (s • circleMap 0 1 θ))
+      (F' := fun s θ => fderiv ℝ u (s • circleMap 0 1 θ) (circleMap 0 1 θ))
+      (neg_le_self Real.pi_pos.le) hlr hrh h0 h1
+    intro θ hθ s hs
+    have hmem := hm (show (s,θ) ∈ K from ⟨hs,hθ⟩)
+    exact ((hu.differentiableOn (by norm_num)).differentiableAt (isOpen_ball.mem_nhds hmem)).hasFDerivAt.comp_hasDerivAt s
+      (by simpa using (hasDerivAt_id s).smul_const (circleMap 0 1 θ))
+  · apply compact_parameter_derivative
+      (F := fun s θ => fderiv ℝ u (s • circleMap 0 1 θ) (circleMap 0 1 θ))
+      (F' := fun s θ => fderiv ℝ (fderiv ℝ u) (s • circleMap 0 1 θ) (circleMap 0 1 θ) (circleMap 0 1 θ))
+      (neg_le_self Real.pi_pos.le) hlr hrh h1 h2
+    intro θ hθ s hs
+    have hmem := hm (show (s,θ) ∈ K from ⟨hs,hθ⟩)
+    have huc := hu.contDiffAt (isOpen_ball.mem_nhds hmem)
+    have hd := path_second (c := fun t => t • circleMap 0 1 θ) (c' := fun _ => circleMap 0 1 θ)
+      huc (by simpa using (hasDerivAt_id s).smul_const (circleMap 0 1 θ))
+      (hasDerivAt_const s (circleMap 0 1 θ))
+    simpa only [map_zero,add_zero] using hd
+
+lemma circle_laplacian_continuous {u : ℂ → ℝ}
+    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : |r| < 1) :
+    Continuous (fun θ => Laplacian.laplacian u (r • circleMap 0 1 θ)) := by
+  have hc := (hu.fderiv_of_isOpen isOpen_ball (show (1:WithTop ℕ∞)+1 ≤ 2 by norm_num)).fderiv_of_isOpen
+    isOpen_ball (show (0:WithTop ℕ∞)+1 ≤ 1 by norm_num)
+  have hmaps : MapsTo (fun θ => r • circleMap 0 1 θ) univ (ball (0:ℂ) 1) := fun θ _ => radial_mem hr θ
+  have hc' : Continuous (fun θ => fderiv ℝ (fderiv ℝ u) (r • circleMap 0 1 θ)) :=
+    continuousOn_univ.1 (hc.continuousOn.comp ((continuous_circleMap 0 1).const_smul r).continuousOn hmaps)
+  simp_rw [laplacian_eq_hessian]
+  exact (hc'.clm_apply continuous_const |>.clm_apply continuous_const).add
+    (hc'.clm_apply continuous_const |>.clm_apply continuous_const)
+
+lemma circle_angular_laplacian {u : ℂ → ℝ}
+    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : 0 < r) (hr1 : r < 1) :
+    r * (∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (r • circleMap 0 1 θ)) =
+      circleDMean u r + r*circleDDMean u r := by
+  have hrabs : |r| < 1 := by rwa [abs_of_pos hr]
+  let E : ℝ → ℂ := circleMap 0 1
+  let c : ℝ → ℂ := fun θ => r • E θ
+  let H : ℂ → ℂ →L[ℝ] ℂ →L[ℝ] ℝ := fderiv ℝ (fderiv ℝ u)
+  let J : ℝ → ℝ := fun θ => H (c θ) (E θ*Complex.I) (E θ*Complex.I)
+  let D : ℝ → ℝ := fun θ => fderiv ℝ u (c θ) (E θ)
+  let Q : ℝ → ℝ := fun θ => H (c θ) (E θ) (E θ)
+  have hc : Continuous c := (continuous_circleMap 0 1).const_smul r
+  have hEm : Continuous E := continuous_circleMap 0 1
+  have hmem (θ : ℝ) : c θ ∈ ball (0:ℂ) 1 := radial_mem hrabs θ
+  have hdu : ContDiffOn ℝ 1 (fderiv ℝ u) (ball 0 1) :=
+    hu.fderiv_of_isOpen isOpen_ball (by norm_num)
+  have hddu : ContDiffOn ℝ 0 H (ball 0 1) :=
+    hdu.fderiv_of_isOpen isOpen_ball (by norm_num)
+  have hdc : Continuous (fun θ => fderiv ℝ u (c θ)) :=
+    continuousOn_univ.1 (hdu.continuousOn.comp hc.continuousOn (fun θ _ => hmem θ))
+  have hHc : Continuous (fun θ => H (c θ)) :=
+    continuousOn_univ.1 (hddu.continuousOn.comp hc.continuousOn (fun θ _ => hmem θ))
+  have hD : Continuous D := hdc.clm_apply hEm
+  have hJ : Continuous J := (hHc.clm_apply (hEm.mul_const Complex.I)).clm_apply (hEm.mul_const Complex.I)
+  have hQ : Continuous Q := (hHc.clm_apply hEm).clm_apply hEm
+  let V : ℝ → ℝ := fun θ => fderiv ℝ u (c θ) (r • (E θ*Complex.I))
+  have hdV (θ : ℝ) : HasDerivAt V (r^2*J θ-r*D θ) θ := by
+    have hcp : HasDerivAt c (r • (E θ*Complex.I)) θ := (hasDerivAt_circleMap 0 1 θ).const_smul r
+    have hcpp : HasDerivAt (fun t => r • (E t*Complex.I)) (-(c θ)) θ := by
+      have hh := ((hasDerivAt_circleMap 0 1 θ).mul_const Complex.I).const_smul r
+      simpa only [Pi.smul_apply,E,c,mul_assoc,Complex.I_mul_I,mul_neg_one,smul_neg] using! hh
+    have hh := path_second (hu.contDiffAt (isOpen_ball.mem_nhds (hmem θ))) hcp hcpp
+    convert hh using 1
+    dsimp only [H,J,D]
+    simp only [map_smul,smul_apply,smul_eq_mul,map_neg]
+    change r^2*((H (c θ)) (E θ*Complex.I)) (E θ*Complex.I)-r*(fderiv ℝ u (c θ)) (E θ) =
+      r*(r*((H (c θ)) (E θ*Complex.I)) (E θ*Complex.I))-(fderiv ℝ u (c θ)) (r • E θ)
+    rw [map_smul,smul_eq_mul]
+    ring
+  have hEnds : E (-Real.pi) = E Real.pi := by
+    simp [E,circleMap,Complex.exp_neg_pi_mul_I,Complex.exp_pi_mul_I]
+  have hVeq : V Real.pi = V (-Real.pi) := by
+    dsimp only [V,c]
+    rw [hEnds]
+  have hz := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun θ _ => hdV θ)
+    (((hJ.const_mul (r^2)).sub (hD.const_mul r)).intervalIntegrable (-Real.pi) Real.pi)
+  rw [hVeq,sub_self] at hz
+  rw [intervalIntegral.integral_sub ((hJ.intervalIntegrable _ _).const_mul _) ((hD.intervalIntegrable _ _).const_mul _),
+    intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul] at hz
+  have htrace (θ : ℝ) : Laplacian.laplacian u (c θ) = Q θ+J θ := by
+    rw [laplacian_eq_hessian]
+    symm
+    convert rotated_trace (H (c θ)) (E θ) (by simp [E,norm_circleMap_zero]) using 1
+    simp only [Q,J,H,mul_comm Complex.I]
+  have hli : (∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (c θ)) =
+      circleDDMean u r + ∫ θ in -Real.pi..Real.pi, J θ := by
+    calc
+      _ = ∫ θ in -Real.pi..Real.pi, Q θ+J θ := intervalIntegral.integral_congr (fun θ _ => htrace θ)
+      _ = _ := intervalIntegral.integral_add (hQ.intervalIntegrable _ _) (hJ.intervalIntegrable _ _)
+  change r*(∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (c θ)) = _
+  rw [hli]
+  change r^2*(∫ θ in -Real.pi..Real.pi, J θ)-r*circleDMean u r = 0 at hz
+  nlinarith
+
+lemma weighted_circle_derivative {u : ℂ → ℝ}
+    (hu : ContDiffOn ℝ 2 u (ball 0 1)) {r : ℝ} (hr : 0 < r) (hr1 : r < 1) :
+    HasDerivAt (fun s => s*circleDMean u s)
+      (r*(∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (r • circleMap 0 1 θ))) r := by
+  have hd := (hasDerivAt_id r).mul (circle_means_derivatives hu (by rwa [abs_of_pos hr])).2
+  simpa only [Pi.mul_apply,id_eq,one_mul,circle_angular_laplacian hu hr hr1] using! hd
+
+lemma compact_parameter_continuousAt {F : ℝ → ℝ → ℝ} {a b l h x : ℝ}
+    (hab : a ≤ b) (hlx : l < x) (hxh : x < h)
+    (hF : ContinuousOn (Function.uncurry F) (Icc l h ×ˢ Icc a b)) :
+    ContinuousAt (fun y => ∫ t in a..b, F y t) x := by
+  have hc (y : ℝ) (hy : y ∈ Icc l h) : ContinuousOn (F y) (Icc a b) :=
+    hF.comp (continuous_const.prodMk continuous_id).continuousOn (fun t ht => ⟨hy,ht⟩)
+  obtain ⟨C,hC⟩ := (isCompact_Icc.prod isCompact_Icc).exists_bound_of_continuousOn hF
+  apply intervalIntegral.continuousAt_of_dominated_interval (bound := fun _ => C) ?_ ?_ intervalIntegrable_const ?_
+  · filter_upwards [Icc_mem_nhds hlx hxh] with y hy
+    have hh := ((hc y hy).mono Ioc_subset_Icc_self).aestronglyMeasurable (μ := volume) measurableSet_Ioc
+    simpa only [uIoc_of_le hab] using hh
+  · filter_upwards [Icc_mem_nhds hlx hxh] with y hy
+    filter_upwards with t ht
+    exact hC (y,t) ⟨hy,Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)⟩
+  · filter_upwards with t ht
+    have ht' : t ∈ Icc a b := Ioc_subset_Icc_self (by simpa only [uIoc_of_le hab] using ht)
+    have hh : ContinuousOn (fun y => F y t) (Icc l h) :=
+      hF.comp (continuous_id.prodMk continuous_const).continuousOn (fun y hy => ⟨hy,ht'⟩)
+    exact hh.continuousAt (Icc_mem_nhds hlx hxh)
+
+lemma laplacian_continuousOn {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1)) :
+    ContinuousOn (Laplacian.laplacian u) (ball (0:ℂ) 1) := by
+  have hc := ((hu.fderiv_of_isOpen isOpen_ball (show (1:WithTop ℕ∞)+1 ≤ 2 by norm_num)).fderiv_of_isOpen
+    isOpen_ball (show (0:WithTop ℕ∞)+1 ≤ 1 by norm_num)).continuousOn
+  have he : Laplacian.laplacian u = fun z =>
+      fderiv ℝ (fderiv ℝ u) z 1 1 + fderiv ℝ (fderiv ℝ u) z Complex.I Complex.I :=
+    funext (laplacian_eq_hessian u)
+  rw [he]
+  exact (hc.clm_apply continuousOn_const |>.clm_apply continuousOn_const).add
+    (hc.clm_apply continuousOn_const |>.clm_apply continuousOn_const)
+
+def circleLMean (u : ℂ → ℝ) (r : ℝ) : ℝ :=
+  ∫ θ in -Real.pi..Real.pi, Laplacian.laplacian u (r • circleMap 0 1 θ)
+
+lemma circle_lmean_continuous {u : ℂ → ℝ} (hu : ContDiffOn ℝ 2 u (ball 0 1)) :
+    ContinuousOn (circleLMean u) (Ioo (-1:ℝ) 1) := by
+  intro r hr
+  let l := (r-1)/2
+  let h := (r+1)/2
+  have hl : -1 < l := by dsimp [l]; linarith [hr.1]
+  have hh : h < 1 := by dsimp [h]; linarith [hr.2]
+  have hlr : l < r := by dsimp [l]; linarith [hr.1]
+  have hrh : r < h := by dsimp [h]; linarith [hr.2]
+  apply ContinuousAt.continuousWithinAt
+  apply compact_parameter_continuousAt (F := fun r θ => Laplacian.laplacian u (r • circleMap 0 1 θ))
+    (neg_le_self Real.pi_pos.le) hlr hrh
+  exact (laplacian_continuousOn hu).comp radial_continuous.continuousOn (radial_interval_maps hl hh)
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal
+namespace SLEExactGauge.Layer
+
+lemma polar_symm_circle (r θ : ℝ) :
+    Complex.polarCoord.symm (r,θ) = r • circleMap 0 1 θ := by
+  rw [Complex.polarCoord_symm_apply]
+  simp only [circleMap_zero,Complex.ofReal_one,one_mul,Complex.real_smul,Complex.exp_ofReal_mul_I]
+
+lemma polar_disk_lintegral_le (f : ℂ → ℝ≥0∞) :
+    (∫⁻ z in ball (0:ℂ) 1, f z) ≤
+      ∫⁻ r in Ioo (0:ℝ) 1, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+        ENNReal.ofReal r * f (r • circleMap 0 1 θ) := by
+  classical
+  let g := (ball (0:ℂ) 1).indicator f
+  have hp := Complex.lintegral_comp_polarCoord_symm g
+  rw [← lintegral_indicator measurableSet_ball f,← hp]
+  change (∫⁻ p, ENNReal.ofReal p.1 * g (Complex.polarCoord.symm p)
+    ∂(volume.prod volume).restrict (Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi)) ≤ _
+  rw [← Measure.prod_restrict]
+  apply (lintegral_prod_le _).trans
+  have he : (fun r => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+      ENNReal.ofReal r * g (Complex.polarCoord.symm (r,θ))) =ᵐ[volume.restrict (Ioi (0:ℝ))]
+      (Iio (1:ℝ)).indicator (fun r => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+        ENNReal.ofReal r * f (r • circleMap 0 1 θ)) := by
+    filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr
+    have hr0 : 0 < r := hr
+    by_cases h1 : r < 1
+    · rw [indicator_of_mem (show r ∈ Iio (1:ℝ) from h1)]
+      apply lintegral_congr
+      intro θ
+      rw [polar_symm_circle]
+      dsimp only [g]
+      rw [indicator_of_mem (radial_mem (by rwa [abs_of_pos hr0]) θ)]
+    · rw [indicator_of_notMem (show r ∉ Iio (1:ℝ) from h1)]
+      apply lintegral_eq_zero_of_ae_eq_zero
+      filter_upwards with θ
+      have hn : r • circleMap 0 1 θ ∉ ball (0:ℂ) 1 := by
+        simpa only [mem_ball_zero_iff,radial_norm,abs_of_pos hr0] using h1
+      rw [polar_symm_circle]
+      dsimp only [g]
+      simp only [indicator_of_notMem hn,mul_zero,Pi.zero_apply]
+  rw [lintegral_congr_ae he,lintegral_indicator measurableSet_Iio,
+    Measure.restrict_restrict measurableSet_Iio]
+  have hset : Iio (1:ℝ) ∩ Ioi (0:ℝ) = Ioo (0:ℝ) 1 := by ext r; simp [and_comm]
+  rw [hset]
+
+end SLEExactGauge.Layer
+
+end
+
+end
+
+section
+noncomputable section
+open Set Filter MeasureTheory Metric
+open scoped Topology ENNReal NNReal
+namespace SLEExactGauge.RectangleEnergy
+open Layer
+
+lemma polar_annulus_lintegral_le (f : ℂ → ℝ≥0∞) {r R : ℝ} (hr : 0 < r) :
+    (∫⁻ z in {z : ℂ | r < ‖z‖ ∧ ‖z‖ < R}, f z) ≤
+      ∫⁻ s in Ioo r R, ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+        ENNReal.ofReal s*f (s • circleMap 0 1 θ) := by
+  classical
+  let A := {z : ℂ | r < ‖z‖ ∧ ‖z‖ < R}
+  have hA : MeasurableSet A := (measurableSet_lt measurable_const measurable_norm).inter
+    (measurableSet_lt measurable_norm measurable_const)
+  let g := A.indicator f
+  have hp := Complex.lintegral_comp_polarCoord_symm g
+  rw [← lintegral_indicator hA f,← hp]
+  change (∫⁻ p, ENNReal.ofReal p.1*g (Complex.polarCoord.symm p)
+    ∂(volume.prod volume).restrict (Ioi (0:ℝ) ×ˢ Ioo (-Real.pi) Real.pi)) ≤ _
+  rw [← Measure.prod_restrict]
+  apply (lintegral_prod_le _).trans
+  have he : (fun s => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+      ENNReal.ofReal s*g (Complex.polarCoord.symm (s,θ))) =ᵐ[volume.restrict (Ioi (0:ℝ))]
+      (Ioo r R).indicator (fun s => ∫⁻ θ in Ioo (-Real.pi) Real.pi,
+        ENNReal.ofReal s*f (s • circleMap 0 1 θ)) := by
+    filter_upwards [ae_restrict_mem measurableSet_Ioi] with s hs
+    have hs0 : 0 < s := hs
+    by_cases hh : s ∈ Ioo r R
+    · rw [indicator_of_mem hh]
+      apply lintegral_congr
+      intro θ
+      rw [polar_symm_circle]
+      have hm : s • circleMap 0 1 θ ∈ A := by
+        simpa only [A,mem_ofPred_eq,mem_Ioo,radial_norm,abs_of_pos hs0] using hh
+      rw [show g (s • circleMap 0 1 θ) = f (s • circleMap 0 1 θ) by exact indicator_of_mem hm f]
+    · rw [indicator_of_notMem hh]
+      apply lintegral_eq_zero_of_ae_eq_zero
+      filter_upwards with θ
+      have hn : s • circleMap 0 1 θ ∉ A := by
+        simpa only [A,mem_ofPred_eq,mem_Ioo,radial_norm,abs_of_pos hs0] using hh
+      rw [polar_symm_circle]
+      dsimp only [g]
+      simp only [indicator_of_notMem hn,mul_zero,Pi.zero_apply]
+  rw [lintegral_congr_ae he,lintegral_indicator measurableSet_Ioo,
+    Measure.restrict_restrict measurableSet_Ioo]
+  have hset : Ioo r R ∩ Ioi (0:ℝ) = Ioo r R :=
+    inter_eq_left.2 (fun _ hs => hr.trans hs.1)
+  rw [hset]
+
+end SLEExactGauge.RectangleEnergy
+
+end
+
+end
+
+end OAI